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+{"text":"\\section{Introduction}\n\nFast radio bursts (FRBs) are intense radio flashes occurring at high\nGalactic latitudes with anomalously high dispersion measure (DM)\n\\cite{Lorimer07,Thornton13,Masui15,Katz16b,Ravi16}. Duo to the lack\nof distance information, their physical origin is unknown. Some\npeople suggested that the high DM is dominated by the ionized\nintergalactic medium, which implies that FRBs may occur at\ncosmological distances.\n\nRecently, Keane et al. (2016) claimed to discover the first FRB host\ngalaxy, which is an elliptical galaxy at $z = 0.492\\pm0.008$\n\\cite{Keane16}. However, this conclusion was questioned by some\nsubsequent papers \\cite{Williams16,Vedantham16}. More recently,\nusing fast-dump interferometry with the Karl G. Jansky Very Large\nArray (VLA), the host galaxy of repeating FRB 121102 was discovered\n\\cite{Chatterjee17,Tendulkar17}. Optical imaging and spectroscopy\nidentify FRB 121102 a redshift of $z = 0.19273$ \\cite{Tendulkar17}.\nThe cosmological origin of FRB 121102 is confirmed. Therefore FRBs\nare promising cosmological probes. However, the physical origin of\nFRBs is mysterious until now. Many theoretical models for FRBs are\nproposed, including collapses of supra-massive neutron star into\nblack hole \\cite{Falcke14,Zhang14,Ravi14}, magnetar pulse-wind\ninteractions \\cite{Lyubarsky14}, charged black hole binary mergers\n\\cite{Zhang16}, giant pulse emissions from pulsars \\cite{Cordes16},\ngiant flares from magnetars \\cite{Popov13,Kulkarni14,Katz14,Pen15,\nKulkarni16}, unipolar inductor model \\cite{WangJ16}, and double\nneutron stars mergers \\cite{Totani13}. The FRB 121102 is repeating,\nwhich disfavors models involving cataclysmic events\n\\cite{Spitler16}. Additional six bursts \\cite{Scholz16} and nine\nbursts \\cite{Chatterjee17} for FRB 121102 are detected. So there may\nbe two populations of FRBs \\cite{Champion16,Spitler16,Li16}. Dai et\nal. (2016) proposed that the repeating bursts are produced by lots\nof asteroids encountering with highly magnetized pulsar\n\\cite{Dai16}. A neutron star-white dwarf binary model also has been\nproposed for the repeating FRB 121102 \\cite{Gu16}.\n\n\nUntil now, twenty six bursts of FRB 121102 have been observed.\nHowever, the nine bursts discovered by VLA are not observed by\nArecibo observatory. In this paper, we investigate the frequency\ndistributions of peak flux, fluence, duration and waiting time for\nFRB 121102. We also test the proposed models for FRBs using the\nderived distributions. This paper is organized as follows. The\nfrequency distributions are shown in section 2. In section 3, we\ntest theoretical models using the statistical results. Finally, the\nconclusion and discussions are given in section 4.\n\n\\section{Frequency distributions of burst parameters}\n\nFor FRB 121102, we use the parameters of eleven bursts from\n\\cite{Spitler16} and six bursts from \\cite{Scholz16}, which are\nlisted in Table 1. Because the nine bursts observed by VLA in the\n2.5-3.5 GHz \\cite{Chatterjee17}, and these bursts are not detected\nby Arecibo, only the upper limit is given. These nine bursts are not\nconsidered in our analysis. The eleven bursts in \\cite{Spitler16}\nare discovered by William E. Gordon Telescope at the Arecibo\nObservatory and the 7-beam Arecibo L-band Feed Array (ALFA). The\nALFA is a seven-beam receiver operating at 1.4 GHz with 0.3 GHz\nbandwidth \\cite{Cordes06}. The antenna gains for these beams are\ndifferent, i.e., 10.4 K Jy$^{-1}$ for the central beam at low zenith\nangles and 8.2 K Jy$^{-1}$ for the other six beams \\cite{Cordes06}.\nBecause the bursts could be detected by different beams, the\nobserved flux or fluence must be corrected. Only the last six bursts\nare pointing to the central beam \\cite{Spitler16}, so the fluxes and\nfluences of other five bursts are normalized to the central beam by\nmultiplying a factor of $1.268$. The additional six bursts are\nobserved by Green Bank Telescope and the single-pixel L-Wide\nreceiver at Arecibo observatory \\cite{Scholz16}. Therefore, the\nfluxes of these bursts are intrinsic. For each bursts, Column 2\ngives the peak time of each burst listed in Column 1. The peak flux\nis presented in Column 3 in unit of Jy. Column 4 gives the fluence\n$F$ of each burst in unit of Jy ms. The observed duration time of\nburst is given in Column 5. The waiting time is given in Column 6.\nThe waiting time $\\Delta t$ is defined as the difference of\noccurring times for two adjacent bursts, and can be calculated from\nthe time difference of Column 2. Only the continues observation is\nconsidered. When calculating the waiting time, the peak flux limit\n0.02 Jy is considered. Because the detection threshold of ALFA is\nabout 0.02 Jy \\cite{Spitler16,Scholz16}. The definition of waiting\ntime is widely used in solar physics and astrophysics.\n\nThe number of bursts $N(F)dF$ with fluence between $F$ and $F + dF$\ncan be expressed by\n\\begin{equation}\\label{fluencedis}\nN(F)dF\\propto F^{-\\alpha_F}\\,dF,\n\\end{equation}\nwhere $\\alpha_F$ is the power-law index. The number of bursts for\nFRB 121102 is small. Rather than examining the differential\ndistribution directly, it is preferable to plot the cumulative\ndistribution, which can avoid binning of the data. Because the width\nof binning can affect the fitting result. Integrating equation\n(\\ref{fluencedis}), we obtain the cumulative distribution of fluence\n\\begin{equation}\nN(>F)\\propto\\int_F^\\infty F^{-\\alpha_F}\\,dF \\propto F^{-\\alpha_F+1}.\n\\end{equation}\nFor the peak flux $S$, the differential frequency distribution is\n\\begin{equation}\\label{fluxdis}\nN(S)dS\\propto S^{-\\alpha_S}\\,dS.\n\\end{equation}\nSo the number of FRBs with peak flux larger than $S$ is\n\\begin{equation}\nN(>S)\\propto\\int_S^\\infty S^{-\\alpha_S}\\,dS \\propto S^{-\\alpha_S+1}.\n\\end{equation}\n\nWe apply the Markov Chain Monte Carlo (MCMC) method to derive the\nbest-fitting parameters. In astrophysical observations, count\nstatistics is often limited. The bursts of FRB 121102 is 17. Such\nlow count number does not fulfill the condition required for the\nGaussian approximation, a well approximation is the Poisson\ndistribution. Consider the number of observed events $N_{obs}$\nfollowing Poisson distribution, the likelihood function for MCMC\nmethod can be expressed as\n\\begin{eqnarray}\n\\mathcal{L}(\\theta) &=& \\sum_i\\ln(P_i(N_{obs,i})) \\\\ \\nonumber\n&=&\\sum_i(N_{obs,i}\\ln(N_{th}(\\theta))-\\ln(N_{obs,i}!)-N_{th}(\\theta)),\n\\end{eqnarray}\nwhere $\\theta$ is the parameter in the model to be constrained by\nthe observed data, $N_{obs,i}$ is the $i$th observed data, and\n$N_{th}$ is the theoretical number predicted by model. For the\ncumulative distribution, it has $N_{obs,i}=i$. Therefore, the\nlikelihood can be re-expressed as\n$\\mathcal{L}(\\theta)=\\sum_i^{N_{obs,tot}}(i\\ln(N_{th}(\\theta))-\\ln(i!)-N_{th}(\\theta))$,\nwhere $N_{obs,tot}$ is the total number of observed events. We use a\npython package pymc \\cite{github} to apply the MCMC method to\noptimize the parameters of theoretical distributions. In the\nfitting, we consider the priors of all the parameters $\\theta$ as\nuniform distributions in a relatively large range, because the\npriors are not important when sampling enough samples with MCMC\nmethod. We must note that the events in each bin of the differential\ndistribution are independent, but the number of events $N(>x)$ in\nthe cumulative distribution are statistically dependent.\nFortunately, we use a logarithmic binning, the fluctuations of\nevents for cumulative distribution in each bin, may follow\napproximately the same random statistics $\\sigma_{cum,i} =\n\\sqrt{N_{cum,i}}$ in each bin as for the differential distribution.\nSo the likelihood function of equation (5) may be a well\napproximation. This problem has been extensively discussed in\n\\cite{Aschwanden15}. Figure 1 shows the cumulative distributions of\nfluence (left panel) and peak flux (right panel) for seventeen\nbursts of FRB 121102, respectively. The power-law index for fluence\nis $\\alpha_F=1.80\\pm0.15$ with $1\\sigma$ confidence level. The value\nof $\\alpha_F$ is from 1.5 to 2.2 \\cite{Lu16}. While, for peak flux,\nthe power-law index is $\\alpha_S=1.07\\pm0.05$ with $1\\sigma$\nconfidence level.\n\n\nThe differential distribution of duration time $W$ can be expressed\nas\n\\begin{equation}\nN(W)dW\\propto W^{-\\alpha_W}\\,dW.\n\\end{equation}\nSo the cumulative distribution of duration time $W$ is\n\\begin{equation}\nN(>W)\\propto\\int_W^{W_{max}} W^{-\\alpha_W}\\,dW \\propto\nW^{-\\alpha_W+1}-W_{max}^{-\\alpha_W+1},\n\\end{equation}\nwhere $W_{max}$ is the maximal duration time. The Markov Chain Monte\nCarlo (MCMC) method is also used to derive the best-fitting\nparameters simultaneously. Left panel of Figure 2 presents the\ncumulative distribution of duration for FRB 121102. From this panel,\na maximal duration time is obviously shown. The best-fitting\npower-law index and maximal duration time are\n$\\alpha_W=1.95\\pm0.32$, and $W_{max}=9.80\\pm0.35$ with $1\\sigma$\nconfidence level, respectively. It should be noted that the observed\nduration time will be broadened when radio waves propagate through a\nplasma. The scatter-broadening time of a pulsed signal depends on\nthe DM and the observing frequency, and an empirical function is\ngiven \\cite{Bhat04}. However, there is no clear evidence for scatter\nbroadening of FRB 121102 \\cite{Spitler16}.\n\n\nIf the burst rate is constant, the waiting-time distribution is the\nPoisson interval distribution \\cite{Wheatland98}\n\\begin{equation}\nP(\\Delta t) = \\lambda e^{-\\lambda \\Delta t}\n\\end{equation}\nwhere $\\Delta t$ is the interval between events, and $\\lambda$ is\nthe burst rate. If the burst rate is time varying, the waiting time\ndistribution can be treated as a combination of piecewise constant\nPoisson processes. Generally, for most forms of $\\lambda(t)$, the\nwaiting time distribution can be shown as power-law form\n\\cite{Aschwanden10}\n\\begin{equation}\nP(\\Delta t) \\propto \\Delta t^{-\\alpha_{WT}}.\n\\end{equation}\nIn order to avoid binning of the data, the cumulative waiting time\ndistribution is given by\n\\begin{equation}\nN(>\\Delta t)\\propto \\int_{\\Delta t}^{\\Delta t_{max}} \\Delta\nt^{-\\alpha_{WT}}\\,dW \\propto \\Delta t^{-\\alpha_{WT}+1}-\\Delta\nt_{max}^{-\\alpha_{WT}+1}.\n\\end{equation}\nFor FRB 121102, the observation is not continues\n\\cite{Spitler16,Scholz16}. The detailed observations by different\ntelescopes are shown in figure 1 of \\cite{Scholz16}. Therefore, in\norder to obtain reliable waiting times, we select the waiting times\nduring periods of continuous observation. We use the waiting times\npresented in table 1 of \\cite{Gu16}. There are ten waiting times\nfrom tens to hundreds of seconds. Right panel of Figure 2 shows the\ncumulative waiting time distribution of FRB 121102. The best-fitting\npower-law index and maximal waiting time are\n$\\alpha_{WT}=1.09\\pm0.05$, and $\\Delta t_{max}=1020.18\\pm 10.25$ s\nwith $1\\sigma$ confidence level.\n\n\\section{Comparing with predictions of theoretical models}\nThe power-law distributions indicates the stochastic engine for FRB\n121102. There are many models proposed to explain the properties of\nFRBs. In this section, we will test theoretical models predictions\nwith statistical results.\n\nDai et al. (2016) proposed that the repeating bursts can be produced\nfrom lots of asteroids encountering with highly magnetized pulsar\n\\cite{Dai16}. In order to explain observation, the diameters of\nasteroids are small, i.e., $L<5$km \\cite{Dai16}. From their equation\n(2), the differential frequency distribution of diameter $L$ of\nasteroids is predicted to $dN\/dL=dN\/dW\\times dW\/dL$, with duration\ntime $W$. So if the index for differential frequency distribution of\nduration is $-2.0$, the value is $dN\/dL\\propto L^{-7\/3}$. From the\nobservation of Sloan Digital Sky Survey, the a broken power law was\nfound with $dN\/dL\\propto L^{-4}$ for large asteroids (5-50 km) and\n$dN\/dL\\propto L^{-2.3}$ for smaller asteroids (0.5-5 km)\n\\cite{Ivezic01}. The differential size distribution of small\nasteroids is $dN\/dL\\propto L^{-2.29}$ \\cite{Yoshida07}. These value\nare well consistent with the model prediction.\n\nCordes and Wasserman (2016) suggested that FRBs originate from\nCrab-like giant pulses of extragalactic neutron stars\n\\cite{Cordes16}. The index of peak flux cumulative distribution\n$\\alpha_S$ is from $1.3$ to $2.5$. The low limit is a little larger\nthan the best-fitting value. Lyutikov et al. (2016) argued that\nFRBs, including repeating and non-repeating FRBs, are from giant\npulses of young rapidly rotating pulsars \\cite{Lyutikov16}. In their\nmodel, the intrinsic luminosity of an FRB is proportional to the\nspin-down power of neutron star. So the predicted distribution of\nFRB flux is $N(>S)\\propto S^{-3\/2}$ \\cite{Lyutikov16}. From our\nstatistical study of 17 bursts of FRB 121102, the cumulative\ndistribution $N(>S)\\propto S^{-1.06}$ is found, which is different\nfrom their model prediction. So the repeating FRB 121102 may\ndisfavor the rotationally powered model. The distance measurement of\nthe repeating FRB 121102 also ruled out rotationally-powered radio\nemission \\cite{Lyutikov17}.\n\n\nKatz (2016) proposed that FRBs are generated by magnetic energy\nreleased in magnetar magnetospheres \\cite{Katz16}. Metzger et al.\n(2017) also argued that the repeating FRB 121102 is powered by\nmillisecond magnetar, through its rotational or magnetic energy\n\\cite{Metzger17}. From observational constraint, the magnetic energy\nis favored \\cite{Metzger17}. Generally, the soft gamma repeater\n(SGR) outbursts result from the dissipation of magnetostatic energy\nin the magnetosphere of magnetars. So we compare the statistical\nproperties of FRB 121102 and SGR 1806-20. Figure 3 shows the\ndifferential distributions of duration (left panel), and waiting\ntime (right panel) for SGR 1806-20. We use the waiting time data\nfrom \\cite{Gogus00}, and duration time data from \\cite{Gogus01}. The\nbest-fitting indices are $\\alpha_W=2.14\\pm0.22$ and\n$\\alpha_{WT}=0.95\\pm0.05$ for duration time and waiting time,\nrespectively. The distribution of SGR 1806-20 burst energies follows\na power-law $dN\\propto E^{-\\gamma} dE$ with $\\gamma \\sim 1.6$\n\\cite{Cheng96,Prieskorn12}. These indices are well consistent with\nthose of FRB 121102. This may indicate that repeating FRBs may be\nrelated to extremely magnetized neutron stars. Besides these\nstatistic distributions, there are some phenomenological similarity\nbetween FRBs and SGRs. First, they are both repeating. At least, FRB\n121102 show repeating bursts \\cite{Spitler16,Scholz16,Chatterjee17}.\nSecond, the duty factor $D=\\langle f(t) \\rangle^2\/\\langle\nf(t)^2\\rangle$ with flux $f(t)$ is similar, $D\\sim 10^{-10}$ for\nSGRs and $D<10^{-8}$ for FRBs \\cite{Law15,Katz16}. This value\ndenotes the the fraction of the time in which a source emits at\nclose to its peak flux.\n\n\n\n\\section{Discussions and Conclusion}\n\nIn this paper, we study the statistical properties of repeating FRB\n121102, including peak flux, fluence, duration time and waiting\ntime. The cumulative distributions of peak flux, fluence and\nduration show power-law forms. The waiting time distribution also\nshows power-law distribution, and is consistent with a\nnon-stationary Poisson process. Power-law size distributions have\nbeen discovered in many astrophysical phenomena, which may indicate\na stochastic central engine. We also compare the statistical results\nwith theoretical models predictions. The duration distribution from\ntheoretical model relating asteroids encountering with highly\nmagnetized pulsar is consistent with observations. Similar\ndistributions between FRB 121102 and SGR 1806-20, such as fluence,\nduration and waiting time, also support the models proposed by\n\\cite{Katz16}, in which the magnetic energy releases in magnetar\nmagnetospheres. So more observation is needed to distinguish these\ntwo models.\n\nPower-law distributions of events have been discovered in a large\nnumber of astrophysical phenomena in many wavelengths \\cite[for a\nrecent review, see][]{Aschwanden11}. The power-law frequency\ndistributions, including peak flux, fluence, duration and waiting\ntime, are predicted by self-organized criticality (SOC) systems\n\\cite{Bak87,Katz86,Aschwanden11}. These distributions also satisfy\nthe criteria that define a SOC system \\cite{Aschwanden11}, which\noccurs in many natural systems that exhibit nonlinear energy\ndissipation \\cite{Aschwanden11,Wang13, Wang15}. Therefore, FRBs may\nalso be avalanche events.\n\nIn future, some facilities, such as Chinese Five-hundred-meter\nAperture Spherical radio Telescope (FAST) \\cite{Nan11}, Canadian\nHydrogen Intensity Mapping Experiment (CHIME) \\cite{Kaspi16}, the\nSquare Kilometer Array, or other upcoming wide-field telescopes,\nwill collect a large number of FRBs. The statistics of FRBs will\ngive constraints on the nature of central engine.\n\n\n\n\\section*{Acknowledgements}\nWe thank the anonymous referee for useful comments and suggestions.\nWe also thank Z. G. Dai for helpful discussion. This work is\nsupported by the National Basic Research Program of China (973\nProgram, grant No. 2014CB845800) and the National Natural Science\nFoundation of China (grants 11422325 and 11373022), the Excellent\nYouth Foundation of Jiangsu Province (BK20140016).\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:itro}\nInteractions among genes, proteins and metabolites shed light into underlying biological mechanisms, and clarify their roles in carrying out cellular functions \\citep{Zhuetal2007, Michailidis2012}. \nThis has motivated the development of many statistical methods to incorporate existing knowledge of biological networks into data analysis \\citep[see e.g.][]{Kongetal2006, WeiPan2008, ShojaieMichailidis2009, ShojaieMichailidis2010}. \n Such methods can lead to identification of novel biological mechanisms associated with the onset and progression of complex diseases \\citep[see e.g.][]{Khatrietal2012}.\n\nExternal network information may be summarized using an undirected weighted graph $G=(V,E,W)$, whose node set $V=\\{1,...,p\\}$ corresponds to $p$ covariates. \nThe edge set $E$ of the graph encodes similarities among covariates, in the sense that two vertices $u,v\\in V$ are connected with an edge $e=(u\\sim v)\\in E$ if covariates $u$ and $v$ are ``similar'' to each other. \nThe similarity between neighboring nodes ($u\\sim v$) is captured by weights $w(u, v)$.\nSuch similarities can for instance correspond to interactions between genes or phylogenetic proximities of species.\n\nA popular approach for incorporating network information is to encourage smoothness in coefficient estimates corresponding to neighboring nodes in the network using a \\emph{network smoothing penalty} \\citep{LiLi2008, Slawskietal2010, Panetal2010, LiLi2010, Huangetal2011, Shenetal2012}. This approach can also be generalized to induce smoothness among similar covariates defined based on a distance matrix or ``kernel'' \\citep{Randolphetal2012} which, for instance, capture similarities among microbial communities according to lineages of a phylogenetic tree \\citep{Fukuyamaetal2012}. \n\nThe smoothness induced by the network smoothing penalty can result in more accurate parameter estimations, particularly when the sample size $n$ is small compared to the number of covariates $p$. \nSparsity-inducing penalties, like the $\\ell_1$ penalty \\citep{LiLi2008,LiLi2010} or the minimum convex penalty (MCP) \\citep{Huangetal2011}, can then be used to select a subset of covariates ${\\bm X}$ associated with the response ${\\bm y}$ for improved interpretability and reduced variability. \nIt has been shown that, under appropriate assumptions, the combination of network smoothing and sparsity-inducing penalties can consistently select the subset of covariates associated with the response \\citep{Huangetal2011}. \nHowever, such procedures do not account for the uncertainty of the estimator, and in particular, do not provide $p$-values.  \n\nA number of new approaches have recently been proposed for formal hypothesis testing in penalized regression, including resampling and subsampling approaches \\citep{MeinshausenBuhlmann2010}, ridge test with deterministic design matrices \\citep{Buhlmann2013}, and the low-dimensional projection estimator (LDPE) for $\\ell_1$-penalized regression \\citep{ZhangZhang2014, vandeGeeretal2014}.\nHowever, there are currently no inference procedures available for methods that incorporate external information using smoothing penalties. Inference procedures for kernel machine learning methods \\citep{Liuetal2007}, on the other hand, test the global association of covariates and are hence not appropriate for testing the association of individual covariates.\n\nAnother limitation of existing approaches that incorporate external network information, including those using network smoothing penalties, is their implicit assumption that the network is accurate and informative. However, existing networks may be incomplete or inaccurate \\citep{Hartetal2006GB}. As shown in \\citet{ShojaieMichailidis2010network}, such inaccuracies can severely impact the performance of network-based methods. Moreover, even if the network is accurate and complete, it is often unclear whether network connectivities correspond to similarities among corresponding coefficients, which is necessary for methods based on network smoothing penalties.  \n\nTo address the above shortcomings, we propose a testing framework, the \\emph{Grace test}, which incorporates external network information into high dimensional regression and corresponding inferences. The proposed framework builds upon  the graph-constrained estimation (Grace) procedure of \\citet{LiLi2008}, \\citet{Slawskietal2010} and \\citet{LiLi2010}, and utilizes recent theoretical developments for the ridge test by \\citet{Buhlmann2013}. \nAs part of our theoretical development, we generalize the ridge test with fixed design to the setting with random design matrices ${\\bm X}$. \nThis generalization was suggested in the discussion of \\citet{Buhlmann2013} as a possible extension of the ridge test, and results in improved power compared to the original proposal.\n\nOur theoretical analysis shows that the proposed testing framework controls the type-I error rate, regardless of the informativeness and accuracy of the incorporated network. We also show, both theoretically and using simulation experiments, that if the network is accurate and informative, the Grace test offers improved power over existing approaches that ignore such information. Finally, We propose an extension of the Grace test, called the Grace-ridge or \\textit{GraceR} test, for settings where the network may be inaccurate or uninformative. \n\nThe rest of the paper is organized as follows. In Section~\\ref{sec:grace}, we introduce the Grace estimation procedure and the Grace test. We also formally define the ``informativeness'' of the network. \nSection~\\ref{sec:power} investigates the power of the Grace test, in comparison to its competitors. \nIn Section~\\ref{sec:gracer}, we propose the Grace-ridge (GraceR) test for robust estimation and inference with potentially uninformative networks. We apply our methods to simulated data in Section~\\ref{sec:sim} and to data from The Cancer Genome Atlas (TCGA) in Section~\\ref{sec:rd}. We end with a discussion in Section~\\ref{sec:disc}. \nProofs of theoretical results and additional details of simulated and real-data analyses are gathered in Section~\\ref{sec:supp}.\n\nThroughout this paper, we use normal lowercase letters to denote scalars, bold lowercase letters to denote vectors and bold uppercase letters to denote matrices. We denote columns of an $n \\times p$ matrix ${\\bm X}$ by ${\\bm x}_j, j = 1,..., p$ and its rows by ${\\bm x}^i, i = 1,..., n$. For any two symmetric matrices ${\\bm A}$ and ${\\bm B}$, we denote ${\\bm A}\\preceq{\\bm B}$ if ${\\bm B}-{\\bm A}$ is positive semi-definite, or $\\lambda_0({\\bm B}-{\\bm A})\\geq0$, where $\\lambda_0$ denotes the smallest eigenvalue of a symmetric matrix. For an index set $J$, we denote by ${\\bm A}_{(J, J)}$ the $|J|\\times|J|$ sub-matrix corresponding to the rows and columns indexed by $J$. Finally, for a $p$-vector ${\\bm \\beta}$, we let $\\|{\\bm \\beta}\\|_k\\triangleq(\\sum_{i = 1}^p|\\beta_i|^k)^{1\/k}$ for $k \\in\\mathbb{Z}^+$ and $\\|{\\bm \\beta}\\|_\\infty\\triangleq\\max_i \\beta_i$. \n\n\n\\section{The Grace Estimation Procedure and the Grace Test}\n\\label{sec:grace}\n\n\\subsection{The Grace Estimation Procedure}\n\\label{sec:graceest}\n\nLet ${\\bm L}$ be the matrix encoding the external information in an undirected weighted graph $G=(V,E,W)$. \nIn general, ${\\bm L}$ can be any positive semi-definite matrix, or kernel, capturing the ``similarity'' between covariates. In this paper, however, we focus on the case where ${\\bm L}$ is the graph Laplacian matrix, \n\\[\n {\\bm L}_{(u, v)}\\triangleq \\left\\{\n     \\begin{array}{ll}\n         d_u & \\text{if } u = v \\\\\n       -w(u, v) & \\text{if } u \\text{ and } v \\text{ are connected}\\\\\n       0 &  \\text{otherwise}\n     \\end{array}\n   \\right.,\n\\]\nwith $d_u = \\sum_{v\\sim u} w(u, v)$ denoting the degree of node $u$. \nWe also assume that weights $w(u, v)$ are nonnegative. However, the definition of Laplacian and the analysis in this paper can be generalized to also accommodate negative weights \\citep{chung1997spectral}. \n\nLet ${\\bm X}=({\\bm x}_1, ...,{\\bm x}_p)\\in\\mathbb{R}^{n\\times p}$ be the $n \\times p$ design matrix and ${\\bm y}\\in\\mathbb{R}^n$ be the response vector in the linear model \n\\begin{equation}\\label{model}\n{\\bm y} = {\\bm X}{\\bm \\beta}^\\ast + {\\bm \\epsilon}, \\hspace{0.5cm} \n{\\bm \\epsilon} \\sim N_n({\\bm 0}, \\sigma_{\\bm \\epsilon}^2{\\bm I}_n), \\hspace{0.5cm} \n{\\bm x}^i \\sim^{iid} N_p({\\bm 0},{\\bm \\Sigma}) \\text{ for } i = 1,..., n.\n\\end{equation}\nMultivariate normality of covariates is commonly assumed in analysis of biological networks, particularly, when estimating interactions among genes or proteins using Gaussian graphical models \\citep[see e.g.][]{delaFuenteetal2004}. Interestingly, the underlying assumption of network smoothing penalties -- that connected covariates after scaling have similar associations with the response -- is also related to the assumption of multivariate normality \\citep{ShojaieMichailidis2010}.\nWithout loss of generality, we assume ${\\bm y}$ is centered and columns of ${\\bm X}$ are centered and scaled, i.e. \n$\\sum_{i=1}^ny_i = 0$ and $\\sum_{i=1}^n X_{(i, j)} = 0$, ${\\bm x}_j^\\top{\\bm x}_j=n$ for $j=1,...,p$.\nWe denote the scaled Gram matrix by $\\hat{\\bm \\Sigma}\\triangleq{\\bm X}^\\top{\\bm X}\/n$.\n\nFor a non-negative tuning parameter $h$, Grace solves the following optimization problem:\n\\begin{align}\n\\label{eq:grace}\n\\hat{\\bm \\beta}(h) = \\argmin_{\\bm \\beta} \\left\\{\\big\\|{\\bm y}-{\\bm X}{\\bm \\beta}\\big\\|_2^2 + h{\\bm \\beta}^\\top{\\bm L}{\\bm \\beta}\\right\\} = \\big(n\\hat{\\bm \\Sigma}+h{\\bm L}\\big)^{-1}{\\bm X}^\\top{\\bm y}.\n\\end{align}\nWhen ${\\bm L}$ is the Laplacian matrix, ${\\bm \\beta}^\\top{\\bm L}{\\bm \\beta} = \\sum_{u\\sim v} ({\\bm \\beta}_u - {\\bm \\beta}_v)^2 w(u,v)$ \\citep{Huangetal2011}. Hence, the Grace penalty ${\\bm \\beta}^\\top{\\bm L}{\\bm \\beta}$ encourages smoothness in coefficients of connected covariates, according to weights of edges. Henceforth, we call ${\\bm L}$ the penalty weight matrix. \n\nFor any tuning parameter $h>0$, Equation (\\ref{eq:grace}) will have a unique solution if $(n\\hat{\\bm \\Sigma} + h{\\bm L})$ is invertible.\nHowever, if $p > n$ and $rank({\\bm L}) < p$ this condition may not hold. \nWith a Gaussian design ${\\bm x}^i \\sim^{iid} N_p({\\bm 0},{\\bm \\Sigma})$, it follows from \\citet{Bai1999} that if $\\liminf_{n\\to\\infty}\\lambda_0({\\bm \\Sigma})>0$, and if there exists a sequence of index sets $C_n\\subset\\{1,...,p\\}$, $\\lim_{n\\to\\infty}|C_n|\/n<1$, such that $\\liminf_{n\\to\\infty}\\lambda_0({\\bm L}_{(V\\backslash C_n, V\\backslash C_n)})>0$, then $(n\\hat{\\bm \\Sigma}+h{\\bm L})$ is almost surely invertible. In this section we hence  assume that $(n\\hat{\\bm \\Sigma}+ h{\\bm L})$ is invertible. This condition is relaxed in Section~\\ref{sec:gracer}, when we propose the more general Grace-ridge (GraceR) test.\n\n\nAs mentioned in the Introduction, several methods have been proposed to select the subset of relevant covariates for Grace. For example, \\citet{LiLi2008,LiLi2010} added an $\\ell_1$ penalty to the Grace objective function,\n\\begin{align}\n\\hat{{\\bm \\beta}}_{\\ell_1}(h, h_1) = \\argmin_{\\bm \\beta} \\left\\{\\big\\|{\\bm y}-{\\bm X}{\\bm \\beta}\\big\\|_2^2  + h{\\bm \\beta}^\\top{\\bm L}{\\bm \\beta} + h_1\\big\\|{\\bm \\beta}\\big\\|_1\\right\\}.\n\\end{align}\n\\citet{Huangetal2011} instead added the MCP and proposed the sparse Laplacian shrinkage (SLS) estimator. \nWhile these methods perform automatic variable selection, they do not provide measures of uncertainty, i.e. confidence intervals or $p$-values. In this paper, we instead propose an inference procedure that  provides $p$-values for estimated coefficients from Equation~\\eqref{eq:grace}. The resulting $p$-values can then be used to assess the significance of individual covariates, and select a subset of relevant variables.\n\n\n\n\n\\subsection{The Grace Test}\n\\label{sec:gracetest}\n\nBefore introducing the Grace test, we present a lemma that characterizes the bias of the Grace estimation procedure.\n\\begin{lemma}\n\\label{lem:bias}\nFor any $h > 0$, assume $(n\\hat{\\bm \\Sigma}+h{\\bm L})$ is invertible. Then, given \n${\\bm X}$, $\\hat{{\\bm \\beta}}(h)$ as formulated in \\eqref{eq:grace} is an unbiased estimator of ${\\bm \\beta}^\\ast$ if and only if ${\\bm L}{\\bm \\beta}^\\ast = {\\bm 0}$. Moreover,\n\\begin{align}\\label{eq:gracebias}\n\\big\\|\\mathbf{Bias}(\\hat{{\\bm \\beta}}(h)|{\\bm X})\\big\\|_2&\\leq \\frac{h\\|{\\bm L}{\\bm \\beta}^\\ast\\|_2}{\\lambda_{0}(n\\hat{\\bm \\Sigma}+h{\\bm L})}. \n\\end{align}\n\\end{lemma}\nBecause the bias of the Grace estimator depends directly on the magnitude of ${\\bm L}{\\bm \\beta}^\\ast$, we consider ${\\bm L}$ to be informative if ${\\bm L}{\\bm \\beta}^\\ast$ is small. According to Lemma~\\ref{lem:bias}, the Grace estimator will be unbiased only if ${\\bm \\beta}^\\ast$ lies in the space spanned by the eigenvectors of ${\\bm L}$ with 0 eigenvalues. In reality, however, this condition cannot be checked from data. Thus, to control the type-I error rate, we must adjust for this potential estimation bias. \n\nOur testing procedure is motivated by the ridge test proposed in \\citet{Buhlmann2013}, which we briefly discuss next. \nFirst, note that ridge is also a biased estimator of ${\\bm \\beta}^\\ast$, and its \\emph{estimation bias} is negligible only if the ridge tuning parameter is close to zero. \nIn addition to the estimation bias, \\citet{Buhlmann2013} also accounted for the \\emph{projection bias} of ridge regression for a \\textit{fixed} design matrix ${\\bm X}$.  This is because for fixed design matrices with $p>n$, ${\\bm \\beta}^\\ast$ is not uniquely identifiable, as there are infinitely many ${\\bm \\beta}$'s such that $\\mathrm{E}({\\bm y})={\\bm X}{\\bm \\beta}$. Using ridge regression, ${\\bm \\beta}^\\ast$ is only estimable if it lies in the row space of ${\\bm X}$, $\\mathcal{R}({\\bm X})$, which is a proper subspace of $\\mathbb{R}^p$ when $p>n$. \nIf ${\\bm \\beta}^\\ast$ does not lie in this subspace, the ridge estimated regression coefficient is indeed the projection of ${\\bm \\beta}^\\ast$ onto $\\mathcal{R}({\\bm X})$, which is not identical to ${\\bm \\beta}^\\ast$. This gives rise to the projection bias. \n\nTo account for these two types of biases, \\citet{Buhlmann2013} proposed to shrink the ridge estimation bias to zero by shrinking the ridge tuning parameter to zero, while controlling the projection bias using a stochastic bias bound derived from a lasso initial estimator. A side effect of shrinking the ridge tuning parameter to zero is that the variance of covariates with high multi-collinearity could become large; this would hurt the statistical power of the ridge test. In addition, the stochastic bound for the projection bias is rather loose. This double-correction of bias further compromises the power of the ridge test.\n\nIn this paper, we develop a test for random design matrices, which was suggested in the discussion of \\citet{Buhlmann2013} as a potential extension. With random design matrices, we do not incur any projection bias. This is because the regression coefficients in this case are uniquely identifiable as ${\\bm \\Sigma}^{-1}\\mathrm{Cov}({\\bm X},{\\bm y})$ under the joint distribution of $({\\bm X}, {\\bm y})$. Here, ${\\bm \\Sigma}$ denotes the population covariance matrix of covariates and $\\mathrm{Cov}({\\bm X},{\\bm y})$ is the population covariance between the covariates and the response; see \\citet{ShaoDeng2012} for a more elaborate discussion of identifiability for fixed and random design matrices.\n\nTo control the type-I error rate of the Grace test, we adjust for the potential estimation bias using a stochastic bound derived from an initial estimator. By adjusting for the estimation bias using a stochastic upper bound, the Grace tuning parameter needs not be very small. Thus, the variances of Grace estimates are less likely to be unreasonably large; this results in improved power for the Grace test. \nPower properties of the Grace test are more formally investigated in Section~\\ref{sec:power}. Next, we formally introduce our testing procedure.\n\nConsider the null hypothesis $H_0: \\beta_j^\\ast =0$ for some $j\\in\\{1,...,p\\}$. Let $\\tilde{\\bm \\beta}$ be an initial estimator with asymptotic $\\ell_1$ estimation accuracy, i.e. $\\| \\tilde{\\bm \\beta} - {\\bm \\beta}^\\ast \\|_1 = \\smallO_p(1)$. \nThe Grace test statistic is defined as \n\\begin{align}\n\\hat{\\bm z}^G = \\hat{\\bm \\beta}(h) + h(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\tilde{\\bm \\beta},\n\\end{align}\nwhere $\\hat{\\bm \\beta}(h)$ is the Grace estimator from \\eqref{eq:grace} with tuning parameter $h$. Plugging in \\eqref{eq:grace} and adding and subtracting $h(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\tilde{\\bm \\beta}$, we can write\n\\begin{align}\n\\label{eq:keyrel}\n\\hat z_j^G = \\beta_j^\\ast + Z_j^G +\\gamma_j^G, \\qquad j = 1,...,p, \n\\end{align}\nwhere \n\\begin{align*}\nZ_j^G|{\\bm X}&\\sim N\\left( 0, n\\sigma_{\\bm \\epsilon}^2 \\left[ (n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1} \\hat{\\bm \\Sigma}(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1} \\right]_{(j,j)} \\right), \\\\\n{\\bm \\gamma}^G &\\triangleq h(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}(\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast).\n\\end{align*}\nNext, we derive an asymptotic stochastic bound for $\\gamma_j^G$ such that under the null hypothesis\n\\begin{equation}\\label{eq:keycon}\n|\\gamma_j^{G}|\\precsim^{asy.}\\Gamma_j^G \n\\hspace{.1cm} \\text{ or equivalently, } \\hspace{.1cm}\n\\lim_{n\\to\\infty}Pr\\left(|\\gamma_j^{G}|\\leq\\Gamma_j^G\\right)= 1. \n\\end{equation}\nThen, under the null hypothesis, $|\\hat z^{G}_j|\\precsim^{asy.}|Z_j^G|+\\Gamma_j^G$, which allows us to asymptotically control the type-I error rate. \n\nTo complete our testing framework, we use the fact under suitable conditions and with proper tuning parameter $h_{Lasso}$, described in Theorem~\\ref{thm:stbound}, the $\\ell_1$ estimation error of the lasso, \n\\begin{align}\n\\label{eq:lasso}\n\\tilde{\\bm \\beta}(h_{Lasso}) = \\argmin_{\\bm \\beta} \\left\\{\\frac{1}{n}\\big\\|{\\bm y}-{\\bm X}{\\bm \\beta}\\big\\|_2^2 + h_{Lasso}\\big\\|{\\bm \\beta}\\big\\|_1\\right\\},\n\\end{align}\nis asymptotically controlled \\citep{BuhlmannvandeGeer2011}. We thus use the lasso estimator as the initial estimator for the Grace test, i.e. $\\tilde{\\bm \\beta} \\triangleq \\tilde{\\bm \\beta}(h_{Lasso})$.\nTheorem~\\ref{thm:stbound} then constructs a $\\Gamma_j^G$ that satisfies Condition~\\eqref{eq:keycon}. \nFirst, we present required conditions.\n\\begin{itemize}\n\t\\item {\\bf{A0}}: $(n\\hat{\\bm \\Sigma}+h{\\bm L})$ is invertible.\n\t\\item {\\bf{A1}}: ${\\bm y}={\\bm X}{\\bm \\beta}^\\ast+{\\bm \\epsilon}$ where ${\\bm x}^i \\sim^{iid} N_p({\\bm 0},{\\bm \\Sigma}) \\text{ for } i = 1,..., n$ and ${\\bm \\epsilon}\\sim N_n({\\bm 0}, \\sigma_{\\bm \\epsilon}{\\bm I})$.\n\t\\item {\\bf{A2}}: Let $S_0 \\triangleq \\{j: \\beta^\\ast_j\\neq 0\\}$ be the active set of ${\\bm \\beta}^\\ast$ with cardinality $s_0\\triangleq|S_0|$. We have $s_0 = \\smallO\\left( \\big[n\/\\log{p}\\big]^\\xi \\right)$ for some $0<\\xi<1\/2$.\n\t\\item {\\bf{A3}}: The ${\\bm \\Sigma}$-compatibility condition \\citep{BuhlmannvandeGeer2011} in Definition~\\ref{def:compatibility} is met for the set $S_0$ with compatibility constant $\\liminf_{n\\to\\infty}\\phi_{{\\bm \\Sigma},n}^2= d>0$, where $d$ is a constant.\n\t\\item {\\bf{A4}}: $h$ and ${\\bm L}$ are such that \n\t\\[\n\t\\left[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}\\right]_{(j,j)} =\\mathcal{O}_p\\left(\\left[\\frac{n}{\\log{p}}\\right]^{\\frac{1}{2}-\\xi}\\right).\n\t\\]\n\\end{itemize}\n\\begin{definition}[${\\bm \\Sigma}$-Compatibility Condition]\n\\label{def:compatibility} \nFor an index set $S\\subset\\{1,...,p\\}$ with cardinality $s$, define ${\\bm \\beta}^S$ and ${\\bm \\beta}^{S^c}$ such that $\\beta_j^S\\triangleq\\beta_j 1_{\\{j\\in S\\}}$, $\\beta_j^{S^c}\\triangleq\\beta_j 1_{\\{j\\notin S\\}}$.\nWe say that the ${\\bm \\Sigma}$-compatibility condition is met for the set $S$ with compatibility constant $\\phi_{\\bm \\Sigma}>0$ if for all ${\\bm \\beta}\\in\\mathbb{R}^p$ living in the cone $\\|{\\bm \\beta}^{S^c}\\|_1\\leq 3\\|{\\bm \\beta}^{S}\\|_1$, we have\n\\begin{align}\n\\big\\|{\\bm \\beta}^S\\big\\|_1^2\\leq{\\bm \\beta}^\\top{\\bm \\Sigma}{\\bm \\beta}\\frac{s}{\\phi_{\\bm \\Sigma}^2}.\n\\end{align}\n\\end{definition}\nAs discussed in Section~\\ref{sec:graceest}, {\\bf{A0}} is required for uniqueness of the Grace estimator, and is justified by the Gaussian deign. \n{\\bf{A2}} is a standard assumption, and requires the number of relevant covariates to not grow too fast, so that the signal is not substantially diluted among those relevant covariates. Note that with $p=\\mathcal{O}\\left(\\exp(n^\\nu)\\right)$ for some $\\nu<1$,  $s_0$ can grow to infinity as $n\\to\\infty$. The ${\\bm \\Sigma}$-compatibility condition in {\\bf{A3}} is closely related to the restricted eigenvalue assumption introduced in \\citet{Bickeletal2009}. \nAssumption {\\textbf{A4}} is made for improved control of type-I error, and can be relaxed at a cost of potential loss of power with finite samples; see Remark~\\ref{rem:A4}.  \nOn the other hand, given ${\\bm X}$ and ${\\bm L}$, when $h\/n\\to\\infty$, the eigenvectors and eigenvalues of $(n\/h)\\hat{\\bm \\Sigma}+{\\bm L}$ converge to the eigenvectors and eigenvalues of ${\\bm L}$. This indicates that $(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}$ converges to a diagonal matrix with diagonal entries equal to 0 or 1, and {\\bf{A4}} is satisfied. \n\n\\begin{theorem}\\label{thm:stbound}\nSuppose Assumptions {\\bf{A0}} -- {\\bf{A4}} are satisfied, and let $\\tilde{\\bm \\beta}\\triangleq\\tilde{\\bm \\beta}(h_{Lasso})$ with the tuning parameter $h_{Lasso}\\asymp\\sqrt{\\log{p}\/n}$. \nLet\n\\begin{align}\n\\label{eq:Gamma}\n\\Gamma_j^G\\triangleq h\\left\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,-j)}\\right\\|_\\infty\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi},\n\\end{align}\nwhere $\\left\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,-j)}\\right\\|_\\infty \\triangleq \\max_{i:i\\neq j} \\big|(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big|_{(j,i)}$ is the maximum in absolute value of entries in row $j$ without the diagonal entry.\nThen $\\Gamma_j^G$ satisfies condition~\\eqref{eq:keycon}.\n\nUnder the null hypothesis $H_0: \\beta_j = 0$, for any $\\alpha>0$ we have\n\\begin{align}\n\\label{eq:alpha}\n\\limsup_{n\\to\\infty} Pr\\left(\\big|\\hat z_j^G\\big|>\\alpha\\right)\\leq\\limsup_{n\\to\\infty}\\Pr\\left(\\big|Z_j^G\\big|+\\Gamma_j^G>\\alpha\\right).\n\\end{align}\n\\end{theorem}\n\\begin{remark}\\label{rem:A4}\nIf we instead consider \n\\[\n\\Gamma_j^G=h\\left\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,.)}\\right\\|_\\infty\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi},\n\\]\nwe can relax Assumption {\\textbf{A4}} and still control the asymptotic type-I error rate. Theorem~\\ref{thm:stbound} can then be similarly proved without {\\textbf{A4}}. \nHowever, as $h\/n\\to\\infty$, $(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}$ converges to a diagonal matrix, in which case $\\left\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}\\big]_{(j,.)}\\right\\|_\\infty\\gg\\left\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}\\big]_{(j,-j)}\\right\\|_\\infty$. This looser stochastic bound may result in lower power in finite samples. \n\\end{remark}\n\n\nTheorem~\\ref{thm:stbound} shows that regardless of the choice of ${\\bm L}$, the type-I error rate of the Grace test is asymptotically controlled.  The stochastic bound $\\Gamma_j^G$ relies on the unknown sparsity parameter $\\xi$.  Following \\citet{Buhlmann2013} we suggest a small value of $\\xi$, and use $\\xi = 0.05$ in the simulation experiments in Section~\\ref{sec:sim} and real data example in Section~\\ref{sec:rd}.\n\nUsing \\eqref{eq:alpha}, we can test $H_0$ using the asymptotically valid two-sided $p$-value\n\\begin{equation}\\label{eq:p}\nP_j^G = 2\\left(1-\\Phi\\left[\\frac{(|\\hat{\\bm z}_j^G|-\\Gamma_j^G)_+}{\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})}}\\right]\\right),\n\\end{equation}\nwhere $\\Phi$ is the standard normal c.d.f., and $a_+ = \\max(a, 0)$. \nCalculating $p$-values requires estimating $\\sigma_{\\bm \\epsilon}^2$ and choosing a suitable tuning parameter $h$. We can estimate $\\sigma_{\\bm \\epsilon}^2$ using any consistent estimator, such as the scaled lasso \\citep{SunZhang2012}. In the simulation experiments and real data example, we choose $h$ using 10-fold cross-validation (CV). \n\n\n\n\n\nNote that, when simultaneously testing multiple hypotheses: $H_0: \\beta_j^\\ast = 0$ for any $j\\in J\\subseteq\\{1,...,p\\}$ versus $H_a: \\beta_j^\\ast\\neq 0 \\text{ for some } j\\in J$, we may wish to control the false discovery rate (FDR). Because covariates in the data could be correlated, test statistics on multiple covariates may show arbitrary dependency structure. We thus suggest controlling the FDR using the procedure of \\citet{BenjaminiYekutieli2001}. \nAlternatively, we can control the family-wise error rate (FWER) using, e.g. the method of \\citet{Holm1979}. \n\n\n\n\n\n\\section{Power of the Grace Test}\n\\label{sec:power}\n\nIn this section, we investigate power properties of the Grace test. \nOur first result describes sufficient conditions for detection of nonzero coefficients. \n\\begin{theorem}\\label{thm:suffdet}\nAssume Assumptions {\\bf{A0}} -- {\\bf{A4}} are met. If for some $h$, some $0<\\alpha<1$, $0<\\psi<1$, conditional on ${\\bm X}$, we have\n\\begin{align}\n\\label{eq:tuning}\n\\big|\\beta_j^\\ast\\big| >2\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})} + q_{(1-\\psi\/2)}, \n\\end{align}\nwhere $\\Phi\\left(q_{(1-\\alpha\/2)}\\right)=1-\\alpha\/2$. Then using the same tuning parameter $h$ in the Grace test, we get\n$\n\\lim_{n\\to\\infty} Pr\\left(P_j^{G}\\leq\\alpha\\big|{\\bm X}\\right) \\geq \\psi.\n$\n\\end{theorem}\n\nHaving established the sufficient conditions for detection of non-null hypotheses in Theorem~\\ref{thm:suffdet}, we next turn to comparing the power of the Grace test with its competitors: the Grace test, the ridge test with small tuning parameters $h_2=\\mathcal{O}(1)$ and no bias correction, and the GraceI test, which is the Grace test with identity penalty weight matrix ${\\bm I}$. The ridge test may be considered as a variant of the test proposed in \\citet{Buhlmann2013} without the adjustment of the projection bias -- because we assume the design matrix is random, we incur no projection bias in the estimation procedure. \n\nAs indicated in Lemma~\\ref{lem:bias}, the estimation bias of the Grace procedure depends on the informativeness of the penalty weight matrix ${\\bm L}$. When ${\\bm L}$ is informative, we are able to increase the size of the tuning parameter, which shrinks the estimation variance without inducing a large estimation bias. Thus, with an informative ${\\bm L}$, we are able to obtain a better prediction performance, as shown empirically in \\citet{LiLi2008, Slawskietal2010, LiLi2010}. In such setting, the larger value of the tuning parameter, e.g. as chosen by CV, also results in improved testing power, as discussed next.\n\nTheorem~\\ref{thm:power1} compares the power of the Grace test to its competitors in a simple setting of $p=2$ predictors, ${\\bm x}_1$ and ${\\bm x}_2$. \nIn particular, this result identifies sufficient conditions under which the Grace test has asymptotically superior power. It also gives conditions for the GraceI test to have higher power than the ridge test. \nThe setting of $p=2$ predictors is considered mainly for ease of calculations, as in this case, we can directly derive closed form expressions of the corresponding test statistics. Similar results are expected to hold for $p>2$ predictors, but require additional derivations and notations.\n\nAssume ${\\bm y}={\\bm x}_1\\beta_1^\\ast+ {\\bm x}_2\\beta_2^\\ast+{\\bm \\epsilon}$, where ${\\bm \\epsilon}\\sim N_2({\\bm 0}, \\sigma^2_{\\bm \\epsilon}{\\bm I})$, and ${\\bm x}_1$, ${\\bm x}_2$ are scaled. Denote \n\\[\n{\\bm L} \\triangleq \\left( \\begin{array}{ccc}\n1 & l \\\\\nl & 1  \\end{array} \\right), \\hspace{1cm}\n\\hat{\\bm \\Sigma}\\triangleq\\frac{1}{n}{\\bm X}^\\top{\\bm X}= \\left( \\begin{array}{ccc}\n1 & \\rho \\\\\n\\rho & 1  \\end{array} \\right).\n\\]\nTheorem~\\ref{thm:power1} considers the power for testing the null hypothesis $H_0:\\beta_1^\\ast=0$, in settings where $\\beta_1^\\ast\\neq0$, without any constraints on $\\beta_2^\\ast$. \n\n\\begin{theorem}\n\\label{thm:power1}\nSuppose Assumptions {\\bf{A0}} -- {\\bf{A4}} are met. Let $P_j^{G}(h_n^G)$, $P_j^{GI}(h_n^{GI})$ and $P_j^{R}$ be the Grace, GraceI and ridge $p$-values, respectively, with tuning parameters $h_n^G$ for Grace and $h_n^{GI}$ for GraceI. Define\n\\begin{align}\n\\Upsilon_{p,n}(h, l, \\rho, |\\beta_1|) \\triangleq \\frac{\\left[(h\/n+1)^2-(\\rho+lh\/n)^2\\right]\\cdot|\\beta_1| - [\\log{p}\/n]^{1\/2-\\xi}\\cdot|(l-\\rho)h\/n|}{\\sqrt{(1+2h\/n)(1-\\rho^2) + (h\/n)^2(1+l^2-2l\\rho)}}.\n\\end{align}\nThen, conditional on the design matrix ${\\bm X}$, under the alternative hypothesis $\\beta_1^\\ast =b\\neq 0$, the following statements hold with probability tending to 1, as $n\\to\\infty$.\n\\begin{itemize}\n\\item[a)] If $\\, \\lim\\limits_{n\\to \\infty}\\Upsilon_{p,n}(h_n^G, l,\\rho, |b|)\\geq\\lim\\limits_{n\\to \\infty}\\Upsilon_{p,n}(h_n^{GI}, 0,\\rho, |b|) \\,$, then \n\t$\\,\\lim\\limits_{n\\to\\infty} [P_1^{G}(h_n^G)\/P_1^{GI}(h_n^{GI})]\\leq 1$.\n\\item[b)] If $\\,\\lim\\limits_{n\\to \\infty}\\Upsilon_{p,n}(h_n^G, l,\\rho, |b|)\\geq \\sqrt{1-\\rho^2}\\,|b| \\,$, then $\\,\\lim\\limits_{n\\to\\infty} [P_1^{G}(h_n^G)\/P_1^{R}] \\leq 1$.\n\\item[c)] If $\\,\\lim\\limits_{n\\to \\infty}\\Upsilon_{p,n}(h_n^{GI}, 0,\\rho, |b|)\\geq \\sqrt{1-\\rho^2}\\,|b|$\\,, then $\\,\\lim\\limits_{n\\to\\infty} [P_1^{GI}(h_n^{GI})\/P_1^{R}] \\leq 1$.\n\\end{itemize}\n\\end{theorem}\n\nTheorem~\\ref{thm:power1} indicates that, as $h_n^G \/ n$ and $h_n^{GI}\/n$ diverge to infinity, both $\\Upsilon_{p,n}(h_n^G, l,\\rho, |\\beta_1^\\ast|)$ and $\\Upsilon_{p,n}(h_n^{GI}, 0,\\rho, |\\beta_1^\\ast|)$  approach infinity. This implies, on one hand, that for $h_n^G$ and $h_n^{GI}$ sufficiently large, both the Grace and GraceI tests are asymptotically more powerful than the ridge test. On the other hand, we can only compare the powers of the Grace and GraceI tests under some constraints on their tuning parameters. \nWith equal tuning parameters for Grace and GraceI, $h_n^G=h_n^{GI}$, we can show, after some algebra, that as $h_n^G\/n=h_n^{GI}\/n\\to\\infty$,  we have $\\lim_{n\\to \\infty}\\Upsilon_{p,n}(h_n^G, l,\\rho, |\\beta_1^\\ast|)\\geq\\lim_{n\\to \\infty}\\Upsilon_{p,n}(h_n^{GI}, 0,\\rho, |\\beta_1^\\ast|)$ if $(1-l^2) \\ge \\sqrt{(1+l^2-2l\\rho)}$.\nIn this case, the Grace test is more powerful than the GraceI test if $l$ is between 0 and $l^\\ast$, where $l^\\ast$ is the unique root  in $[-1,1]$ of the cubic equation $l^3-3l+2\\rho=0$. Figure~\\ref{fig:GvO}(a) compares the powers of the Grace and GraceI tests with equal tuning parameters $h_n^G \/ n=h_n^{GI} \/ n=10$ and $\\beta_1^\\ast=1$. \nIt can be seen that, the Grace test asymptotically outperforms the GraceI test when $l$ is close to $\\rho$ with equally large tuning parameters. However, when $\\l$ is far from $\\rho$, the GraceI test could be more powerful. This observation, and the empirical results in Section~\\ref{sec:sim} motivate the development of the GraceR test, introduced in Section~\\ref{sec:gracer}. \n\nA similar comparison for powers of the Grace and the ridge test, with $h_n^G \/ n=10$ and $\\beta_1^\\ast=1$, is provided in Figure~\\ref{fig:GvO}(b). These results suggest that, with large Grace tuning parameters, Grace substantially outperforms the ridge test in almost all scenarios. The result for the Grace and ridge comparison is similar with $h_n^G \/ n=1$.\n\n\\begin{figure}[h]\n\\caption{(a) The ratio of $\\Upsilon_{p,n}(h_n^G, l,\\rho, |\\beta_1^\\ast|)$ over $\\Upsilon_{p,n}(h_n^{GI}, 0,\\rho, |\\beta_1^\\ast|)$ for different $l$ and $\\rho$ with $h_n^G \/ n = h_n^{GI} \/ n = 10$, $[\\log{p}\/n]^{1\/2-\\xi}=0.25$ and $\\beta^\\ast_1=1$. A plus sign indicates the ratio is greater than 1.02, whereas a minus sign indicates the ratio is smaller than 0.98; filled circles indicate an intermediate value. (b) The log-ratio of $\\Upsilon_{p,n}(h_n^G, l,\\rho, |\\beta_1|)$ over $\\sqrt{1-\\rho^2}$ for different $l$ and $\\rho$ with $h_n^G \/ n= 10$, $[\\log{p}\/n]^{1\/2-\\xi}=0.25$ and $\\beta_1^\\ast=1$. A plus sign indicates the log-ratio is greater than 0.5 (ratio $>1.65$), whereas a minus sign indicates the log-ratio is smaller than \\textsc{-}0.5 (ratio $< 0.61$); filled circles indicate an intermediate value} \n\\label{fig:GvO}\n\\centering\n\\includegraphics[height = 20cm]{GracevsOthers.pdf}\n\\end{figure}\n\n\n\n\n\\section{The Grace-Ridge (GraceR) Test}\n\\label{sec:gracer}\n\nAs discussed in Section~\\ref{sec:grace}, an informative ${\\bm L}$ results in reduced bias of the Grace procedure, by choosing a larger tuning parameter $h$. \nThe result in Theorem~\\ref{thm:power1} goes beyond just the bias of the Grace procedure. It shows that for certain choices of ${\\bm L}$, i.e. when $l$ is close to the true correlation parameter $\\rho$, the Grace test can have asymptotically superior power. This additional insight is obtained by accounting for, not just the bias of the Grace procedure, but also its variance, when investigating the power.\n\nHowever, in practice, there is no guarantee that existing network information truly corresponds to similarities among coefficients, or is complete and accurate. \nTo address this issue, we introduce the Grace-ridge (GraceR) test. The estimator used in GraceR incorporates two Grace-type penalties induced by ${\\bm L}$ and ${\\bm I}$:\n\\begin{equation}\n\\label{eq:ridgegrace}\n\\hat{\\bm \\beta}(h_G, h_2) = \\argmin_{\\bm \\beta} \\left\\{\\big\\|{\\bm y}-{\\bm X}{\\bm \\beta}\\big\\|_2^2 + h_G{\\bm \\beta}^\\top{\\bm L}{\\bm \\beta} + h_2{\\bm \\beta}^\\top{\\bm \\beta}\\right\\} = \\big(n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I}\\big)^{-1}{\\bm X}^\\top{\\bm y}.\n\\end{equation}\nUsing data-adaptive choices of tuning parameters $h_G$ and $h_2$, we expect this test to be as powerful as the Grace test if ${\\bm L}$ is informative, and as powerful as the GraceI test, otherwise. \n\nAnother advantage of the GraceR over the Grace test is improved bias-variance tradeoff. \nIf ${\\bm L}$ is (almost) singular, the variance of the Grace test statistic, which depends on the eigenvalues of $(n\\hat{\\bm \\Sigma} + h{\\bm L})$, could be large even for reasonably large $h$. \nThus, even though our discussion in Section~\\ref{sec:graceest} shows that $(n\\hat{\\bm \\Sigma} + h{\\bm L})$ is almost surely invertible, with finite samples, its smallest eigenvalue could be very small, if not zero. \nIf ${\\bm L}$ is informative, ${\\bm L}{\\bm \\beta}$ and hence the bias in \\eqref{eq:gracebias} are small. Thus, the rank-deficiency of $(n\\hat{\\bm \\Sigma} + h{\\bm L})$ can be alleviated by choosing a large value of $h$. However, if ${\\bm L}{\\bm \\beta}$ is non-negligible, choosing a large value of $h$ may result in a large bias, even larger than the ridge estimate. \nto the extent which may offset the benefit from the  variance reduction. \nThe finite sample type-I error rate of the Grace test may thus be controlled poorly. By incorporating an additional $\\ell_2$ penalty, we can better control the eigenvalues and achieve a better bias-variance trade-off. \n\nThe GraceR optimization problem leads to the following test statistic:\n\\begin{align}\n\\hat{\\bm z}^{GR} = \\hat{\\bm \\beta}(h_G, h_2) + (n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I})^{-1}(h_G{\\bm L} + h_2{\\bm I})\\tilde{\\bm \\beta}.\n\\end{align}\nSimilar to Section~\\ref{sec:gracetest}, we can write\n\\begin{align}\n\\hat z_j^{GR} = \\beta_j ^\\ast+ Z_j^{GR} +\\gamma_j^{GR}, \\qquad j = 1,...,p, \n\\end{align}\nwhere \n\\begin{align*}\nZ_j^{GR}|{\\bm X}&\\sim N \\left( 0, n\\sigma_{\\bm \\epsilon}^2 \\left[ (n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I})^{-1} \\hat{\\bm \\Sigma}(n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I})^{-1} \\right]_{(j,j)} \\right), \\\\\n{\\bm \\gamma}^{GR} &\\triangleq (n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I})^{-1}(h_G{\\bm L} + h_2{\\bm I})(\\tilde{\\bm \\beta}-{\\bm \\beta}).\n\\end{align*}\nSimilar to the Grace test in in Section~\\ref{sec:gracetest}, we choose $\\tilde{\\bm \\beta}$ to be an initial lasso estimator, and derive an asymptotic stochastic bound for $\\gamma_j^{GR}$ such that $|\\gamma_j^{GR}|\\precsim^{asy.}\\Gamma_j^{GR}$. Equation~\\eqref{eq:p} is again used to obtain two-sided $p$-values for $H_0$. \nTheorems~\\ref{thm:stbound2} and \\ref{thm:suffdet2} parallel the previous results for the Grace test, and establish GraceR's asymptotic control of type-I error rate, and conditions for detection of non-null hypotheses. Proofs of these results are similar to Theorems~\\ref{thm:stbound} and \\ref{thm:suffdet}, and are hence omitted. \nWe first state an alternative to Assumption {\\bf{A4}}. This assumption can be justified using an argument similar to that for Assumption {\\bf{A4}}, and can also be relaxed with the cost of reduced power for the GraceR test. \n\\begin{itemize}\n\t\\item {\\bf{A4'}}: $h_G$, $h_2$ and ${\\bm L}$ are such that \n\t\\[\n\t\\left[(n\\hat{\\bm \\Sigma}+h_G{\\bm L} + h_2{\\bm I})^{-1}(h_G{\\bm L}+h_2{\\bm I})\\right]_{(j,j)} =\\mathcal{O}_p\\left(\\left[\\frac{n}{\\log{p}}\\right]^{\\frac{1}{2}-\\xi}\\right).\n\t\\]\n\\end{itemize}\n\\begin{theorem}\n\\label{thm:stbound2}\nAssume Assumptions {\\bf{A1}} -- {\\bf{A3}} and {\\bf{A4'}} are met. The following $\\Gamma_j^{GR}$ satisfies the stochastic bound for GraceR.\n\\begin{align}\n\\label{eq:gammagr}\n\\Gamma_j^{GR}\\triangleq \\left\\|\\big[(n\\hat{\\bm \\Sigma}+h_G{\\bm L}+h_2 {\\bm I})^{-1}(h_G{\\bm L} + h_2{\\bm I})\\big]_{(j,-j)}\\right\\|_\\infty\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi}.\n\\end{align}\nThen, under the null hypothesis, for any $\\alpha>0$,\n\\begin{align}\n\\limsup_{n\\to\\infty} Pr\\left(\\big|\\hat{\\bm z}_j^{GR}\\big|>\\alpha\\right)\\leq\\limsup_{n\\to\\infty}Pr\\left(\\big|Z_j^{GR}\\big|+\\Gamma_j^{GR}>\\alpha\\right).\n\\end{align}\n\\end{theorem}\n\\begin{theorem}\n\\label{thm:suffdet2}\nAssume Assumptions {\\bf{A1}} -- {\\bf{A3}} and {\\bf{A4'}} are met. If for some $h_G > 0$ and $h_2 > 0$, conditional on ${\\bm X}$, we have\n\\begin{align}\n\\label{eq:tuning2}\n\\big|\\beta_j^\\ast\\big| >2\\Gamma_j^{GR} + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^{GR}|{\\bm X})} + q_{(1-\\psi\/2)}\n\\end{align}\nfor some $0<\\alpha<1$ and $0<\\psi<1$. Then using the same $h_G$ and $h_2$ in the GraceR test, we get\n$\n\\lim_{n\\to\\infty} Pr\\left(P_j^{GR}\\leq\\alpha\\big|{\\bm X}\\right) \\geq \\psi.\n$\n\\end{theorem}\n\n\n\n\\section{Simulation Experiments}\n\\label{sec:sim}\nIn this section, we compare the Grace and GraceR tests with the ridge test \\citep{Buhlmann2013} with small tuning parameters, low-dimensional projection estimator (LDPE) for inference \\citep{ZhangZhang2014,vandeGeeretal2014} and the GraceI test. \nTo this end, we consider a graph similar to \\citet{LiLi2008}, with 50 hub covariates (genes), each connected to 9 other satellite covariates (genes). The 9 satellite covariates are not connected with each other, nor are covariates in different hub-satellite clusters. In total the graph includes  $p=500$ covariates and 450 edges; see Figure {S1} in Section~\\ref{sec:supp} for an illustration with 5 hub-satellite clusters.\nWe build the underlying true Laplacian matrix ${\\bm L}^*$ according to the graph with all edge weights equal 1. \n\nTo assess the effect of inaccurate or incomplete network information, we also consider variants of the Grace and GraceR tests with incorrectly specified graphs, where a number of randomly selected edges are added or removed. \nThe number of removed or added (perturbed) edges relative to the true graph is $\\textrm{NPE}\\in\\{$\\textsc{-165, -70, -10, 0, 15, 135, 350}$\\}$, with negative and positive numbers indicating removals and additions of edges, respectively. For example, $\\textrm{NPE}$=\\textsc{-165} indicates 165 of the 450 edges in the true graph represented by ${\\bm L}^*$ are randomly removed in the perturbed graph with corresponding perturbed Laplacian matrix ${\\bm L}$. This represents the case with incomplete network information. On the other hands, $\\textrm{NPE} = 350$ indicates that in addition to the 450 true edges in ${\\bm L}^*$, we also randomly add 350 wrong edges to ${\\bm L}$. \nThe $\\textrm{NPE}$ values considered correspond to similar normalized spectral differences for settings where edges are removed or added, i.e. $\\|{\\bm L}-{\\bm L}^*\\|_2\/\\|{\\bm L}^*\\|_2\\approx (0.75, 0.50, 0.25, 0, 0.25, 0.50, 0.75)$. Thus, the size of perturbation to the graph is roughly the same with $\\textrm{NPE}=\\textsc{-}165$ and 350. The perturbed penalty weight matrix ${\\bm L}$ is then used in the Grace and GraceR tests.  \nSince $({\\bm X}^\\top{\\bm X}+h{\\bm L})$ may not be invertible, for Grace, we add a value of 0.01 to the diagonal entries of ${\\bm L}$ to make it positive definite. No such correction is needed for GraceR and GraceI because of the $\\ell_2$ penalty.\n\n\nIn each simulation replicate, we generate $n = 100$ independent samples, where for the 50 hub covariates in each sample, $x_k^{hub}\\sim^{iid}N(0, 1)$, $k = 1,..., 50$, and for the 9 satellite covariates in the $k$-th hub-satellite cluster, $x_l^{hub_k}\\sim^{iid}N(0.9\\times x_k^{hub}, 0.9)$, $l=1,...,9$, $k = 1,..., 50$. This is equivalent to  simulating ${\\bm x}^i\\sim^{iid} N_p({\\bm 0}, {\\bm \\Sigma})$ for $i =1,...,100$ with ${\\bm \\Sigma} = ({\\bm L}^* + 0.11\\times{\\bm I})^{-1}$, where ${\\bm L}^*$ corresponds to the partial covariance structure of the covariates.\n\n\nWe consider a sparse model in which covariates in the first hub-satellite cluster are equally associated with the outcome, and those in the other 49 clusters are not. Specifically,  we let\n\\[\n{\\bm \\beta}^\\ast\\triangleq\\frac{1}{\\sqrt{10}}(\\underbrace{1,...,1}_{10}, \\underbrace{0,...,0}_{p-10})^\\top. \n\\] \nWe then simulate \n$\n{\\bm y}={\\bm X}{\\bm \\beta}^\\ast+{\\bm \\epsilon},\n$\nwith ${\\bm \\epsilon}\\sim N_n({\\bm 0}, \\sigma^2_{\\epsilon}{\\bm I}_n)$, and consider $\\sigma_{\\bm \\epsilon}\\in\\{9.5, 6.3, 4.8\\}$ to produce expected $R^2=1-\\sigma_{\\bm \\epsilon}^2\/\\mathrm{Var}({\\bm y})\\in\\{0.1, 0.2, 0.3\\}$.\n\n\nThroughout the simulation iterations, ${\\bm L}^*$ and ${\\bm \\beta}^\\ast$ are kept fixed, and ${\\bm L}$, ${\\bm X}$ and ${\\bm \\epsilon}$ are randomly generated in each repetition. We set the sparsity parameter $\\xi = 0.05$, and $h_{Lasso}=4\\hat\\sigma_{\\bm \\epsilon}\\sqrt{3\\log{p}\/n}$, where $\\hat\\sigma_{\\bm \\epsilon}$ is calculated using the scaled lasso \\citep{SunZhang2012}. \nAs suggested in \\citet{Buhlmann2013}, the tuning parameter for the ridge test is set to 1. \nTuning parameters for LDPE, Grace, GraceR and GraceI are chosen by 10-fold CV.\nWe use two-sided significance level $\\alpha=0.05$ and calculate the average and  standard error of powers from 10 non-zero coefficients and the type-I error rates of each test from 490 zero coefficients. \nFigure~\\ref{fig:power} summarizes the mean powers and type-I error rates of tests across $B=100$ simulated data sets, along with the corresponding 95\\% confidence intervals. Detail values of powers and type-I error rates, as well as an expanded simulation with a larger range of NPE, are available in Section~\\ref{sec:supp}.\n\n\\begin{figure}\n\\caption{Comparison of powers and type-I error rates of different testing methods, along with their 95\\% confidence bands. Testing methods include LDPE  \\citep{ZhangZhang2014, vandeGeeretal2014}, ridge \\citep{Buhlmann2013}, GraceI, Grace and GraceR tests. Filled circles ($\\bullet$) corresponds to powers, whereas crosses ($\\times$) are type-I error rates. Numbers on $x$-axis for Grace and GraceR tests refer to the number of perturbed edges (\\textrm{NPE}) in the network used for testing, compared to the true network used to generate the data.} \n\\label{fig:power}\n\\centering\n\\includegraphics[angle = 90, height = 20cm]{PowerMain.pdf} \n\\end{figure}\n\nComparing the power of the tests, it can be seen that the Grace test with correct choices of ${\\bm L}$ ($\\textrm{NPE} = 0$) results in highest power. The  performance of the Grace test, however, deteriorates as ${\\bm L}$ becomes less accurate. The performance of the GraceR test is, on the other hand, more stable. It is close to the Grace test when the observed ${\\bm L}$ is close to the truth, and is roughly as good as the GraceI test when ${\\bm L}$ is significantly inaccurate. As expected, our testing procedures asymptotically control the type-I error rate, in that observed type-I error rates are not significantly different from $\\alpha=0.05$. \n\n\n\n\\section{Analysis of TCGA Prostate Cancer Data}\n\\label{sec:rd}\nWe examine the Grace and GraceR tests on a prostate adenocarcinoma dataset from The Cancer Genome Atlas (TCGA) collected from prostate tumor biopsies. After removing samples with missing measurements, we obtain a dataset with $n=321$ samples. For each sample, the prostate-specific antigen (PSA) level and the RNA sequences of 4739 genes are available. Genetic network information for these genes is obtained from the Kyoto Encyclopedia of Genes and Genomes (KEGG), resulting in a dataset with $p=3450$ genes and $|E|=38541$ edges.\n\nWe center the outcome and center and scale the covariates. For the Grace and GraceR tests, we set the sparsity parameter $\\xi=0.05$ and $h_{Lasso}=4\\hat\\sigma_{\\bm \\epsilon}\\sqrt{3\\log{p}\/n}$, where $\\hat\\sigma_{\\bm \\epsilon}$ is calculated using the scaled lasso \\citep{SunZhang2012}.\nWe control the false discovery rate at $\\alpha = 0.05$ level using the method of \\citet{BenjaminiYekutieli2001}. \n\nTo increase the chance of selecting ``hub'' genes, we use the normalized Laplacian matrix ${\\bm L}^{(norm)} = \\mathbf{D}^{-1\/2} {\\bm L} \\mathbf{D}^{-1\/2}$, where $\\mathbf{D}$ is the diagonal degree matrix for the KEGG network with edge weights set to 1.\nThe Grace penalty induced by the normalized Laplacian matrix encourages smoothness of coefficient estimates based on the degrees of respective nodes, ${\\bm \\beta}^\\top{\\bm L}^{(norm)}{\\bm \\beta} = \\sum_{u\\sim v} ({\\bm \\beta}_u\/\\sqrt{d_u} - {\\bm \\beta}_v\/\\sqrt{d_v})^2 w(u,v)$ \\citep{LiLi2008}. We add 0.001 to the diagonal entries of ${\\bm L}^{(norm)}$ to induce positive definitiveness in the Grace test.\n\n\nAs shown in Figure~\\ref{fig:RD}(a), the Grace test with tuning parameter\nselected by 10-fold CV identifies 54 genes that are associated with PSA level. They consist of 42 histone genes, 11 histone deacetylase (HDAC) genes and the paired box gene 8 (PAX8). Histone and HDAC genes are densely connected in the KEGG network. With the network smoothing penalty, the Grace regression coefficients of histone and HDAC genes are all positive with a similar magnitude. Existing literature indicates that the histone and HDAC genes are associated with the occurrence, progression, clinical outcomes or recurrence of prostate cancer.\nFigure~\\ref{fig:RD}(b) shows the result for the GraceR test. \nGraceR identifies 5 histone genes, which are also identified by the Grace test. In addition, GraceR identifies 11 genes that are not identified by Grace. Prior work has identified 9 of those 11 genes to be associated with PSA level or the severity and stage of cancer. Additional details about existing evidence in support of genes identified using Grace and GraceR tests, as well as extended results on  prediction performance and stability of the Grace test are provided in Section~\\ref{sec:supp}. \n\\begin{figure}[h]\n\\centering\n\\caption{Results of analysis of TCGA prostate cancer data using the (a) \\emph{Grace} and (b) \\emph{GraceR} tests after adjusting for FDR at 0.05 level. In each case, genes found to be significantly associated with PSA level are shown, along with their interactions based on information from KEGG.} \\label{fig:RD}\n\\includegraphics[width = 15cm]{graph_both.pdf}\n\\end{figure}\n\nAs a comparison, the GraceI test with 10-fold CV identifies 16 disconnected genes, 11 of them are also identified by the GraceR test. Ridge test \\citep{Buhlmann2013} with tuning parameter $h_2 = 1$ identifies 4 disconnected genes, which are also identified by the GraceR test. The low-dimensional projection estimator (LDPE) with tuning parameters chosen by 10-fold CV identifies 10 disconnected genes. Seven of these genes are identified by GraceR and two by Grace.\n\n\n\n\n\\section{Discussion}\\label{sec:disc}\nIn this paper, we proposed the Grace and GraceR tests that incorporate external graphical information regarding the similarity between covariates. Such external information is presented in the form of a penalty weight matrix ${\\bm L}$, which is considered to be the (normalized) graph Laplacian matrix in this paper. However, any positive semi-definite matrix can be used as ${\\bm L}$. The proposed inference framework thus allows researchers in different fields to incorporate relevant external information through ${\\bm L}$. For example, we can use various distance and kernel metrics that measure the (dis)similarity between species in phylogenetic studies. We can also use the adaptive graph Laplacian matrix \\citep{LiLi2010} so that coefficients of negatively correlated covariates are penalized to have the opposite signs. Regardless of the choice of ${\\bm L}$, our proposed procedures asymptotically control the type-I error rate; the power of the Grace test, however, depends on the informativeness of ${\\bm L}$. The power of the GraceR test is on the other hand less dependent on the choice of ${\\bm L}$.\n\n\nThe Grace test introduced in this paper is not scale invariant. That is, the Grace test with the same tuning parameter could produce different $p$-values with data $({\\bm X},{\\bm y})$ and $({\\bm X}, k{\\bm y})$, where $k\\neq 1$ is a constant. This is clear as the test statistic $\\hat z_j$ depends on ${\\bm y}$ whereas the stochastic bound $\\Gamma_j^G$ does not. To make the Grace and GraceR tests scale invariant, we can simply choose the tuning parameter for our lasso initial estimator to be $h_{Lasso}=C\\sigma_{\\bm \\epsilon}\\sqrt{\\log{p}\/n}$ with a constant $C>2\\sqrt{2}$. \\citet{SunZhang2012} show that the lasso is scale invariant in this case. We would also need to use scaled invariant stochastic bounds $\\tilde\\Gamma_j^G\\triangleq \\sigma_{\\bm \\epsilon}\\Gamma_j^G$ and $\\tilde\\Gamma_j^{GR}\\triangleq \\sigma_{\\bm \\epsilon}\\Gamma_j^{GR}$ in our Grace and GraceR tests. Note that multiplying any constant in $\\Gamma_j^G$ and $\\Gamma_j^{GR}$does not change our asymptotic control of the type-I error rate. \n\nIn this paper, cross validation (CV) is used to choose tuning parameters of the Grace and GraceR tests. However, CV does not directly maximize the power of these tests. \nSelection of tuning parameters for optimal testing performance can be a fruitful direction of future research. \nAnother useful extension of the proposed framework is its adaptation to generalized linear models (GLM).\n\n\\section*{Acknowledgements}\nWe would like to thank Dr. Ruben Dezeure and Dr. Peter B\\\"{u}hlmann of the Seminar for Statistics of the Department of Mathematics at ETH Z\\\"{u}rich for providing the code for LDPE. \n\n\n\n\n\n\n\n\n\n\n\\vspace*{-8pt}\n\n\n\n\\section{Supplementary Materials}\n\\label{sec:supp}\n\n\\subsection{Proof of Lemma \\ref{lem:bias}}\n\n\\begin{proof} Given that $(n\\hat{\\bm \\Sigma}+h{\\bm L})$ is invertible and $h>0$, we have\n\\begin{align*}\n\\mathbf{Bias}\\big(\\hat{\\bm \\beta}(h)\\big|{\\bm X}\\big) &=\\mathrm{E}\\big(\\hat{{\\bm \\beta}}(h)\\big|{\\bm X}\\big) -{\\bm \\beta}^\\ast \\\\\n& = (n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}n\\hat{\\bm \\Sigma}{\\bm \\beta}^\\ast - (n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}(n\\hat{\\bm \\Sigma}+h{\\bm L}){\\bm \\beta}^\\ast\\\\\n&=-(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}{\\bm \\beta}^\\ast,\n\\end{align*}\nwhich is equal to ${\\bm 0}$ if and only if\n$\n\t{\\bm L}{\\bm \\beta}^\\ast ={\\bm 0}.\n$\nWe know that\n\\[\n(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}\\preceq \\frac{1}{{\\lambda_{0}(n\\hat{\\bm \\Sigma}+h{\\bm L})}}{\\bm I}.\n\\]\nTherefore,\n\\begin{align*}\n\\big\\|\\mathbf{Bias}(\\hat{{\\bm \\beta}}(h))\\big|{\\bm X}\\big\\|_2 &= h\\sqrt{({\\bm L}{\\bm \\beta}^\\ast)^\\top(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-2}({\\bm L}{\\bm \\beta}^\\ast)} \\\\\n&\\leq h\\sqrt{({\\bm L}{\\bm \\beta}^\\ast)^\\top\\frac{1}{{\\lambda_{0}(n\\hat{\\bm \\Sigma}+h{\\bm L})}^2}({\\bm L}{\\bm \\beta}^\\ast)} \\\\\n&= \\frac{h\\|{\\bm L}{\\bm \\beta}^\\ast\\|_2}{\\lambda_{0}(n\\hat{\\bm \\Sigma}+h{\\bm L})}.\n\\end{align*}\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:stbound}}\n\n\\begin{proof}\nUnder the null hypothesis $H_0: \\beta_j^\\ast=0$, we have\n\\begin{align*}\n\\big|\\gamma_j^{G}\\big| &=h\\big|(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}(\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast)\\big|_j \\\\\n&=h\\big|\\sum_{i=1}^p\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,i)}(\\tilde\\beta_i-\\beta^\\ast_i)\\big| \\\\\n&\\leq h\\big|\\sum_{i:i\\neq j}\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,i)}(\\tilde\\beta_i-\\beta^\\ast_i)\\big| + h\\big|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,j)}\\tilde\\beta_j\\big|\\\\\n&\\leq h\\big\\|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,-j)}\\big\\|_\\infty \\big\\|\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast\\big\\|_1 + h\\big|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,j)}\\tilde\\beta_j\\big|\n\\end{align*}\n\nBased on \\citet{BuhlmannvandeGeer2011}, Chapter 6.12, with Gaussian design, if the ${\\bm \\Sigma}$-compatibility condition is met for the set $S_0$ with compatibility constant $\\phi_{{\\bm \\Sigma}}$, with probability tending to 1, the condition is also met for $\\hat{\\bm \\Sigma}$ with compatibility constant $\\phi_{\\hat{\\bm \\Sigma}} > \\phi_{{\\bm \\Sigma}} \/ 2$. \nMoroever,\nwith $h_{Lasso}\\asymp \\sqrt{\\log{p}\/n}$ and the  $\\hat{\\bm \\Sigma}$-compatibility condition for the set $S_0$, with probability tending to 1, we have\n\\[\n\\big\\|\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast\\big\\|_1\\leq 4\\frac{h_{Lasso} s_0}{\\phi_{\\hat{\\bm \\Sigma}}^2}.\n\\]\nThen, because $s_0 = \\smallO([n\/\\log{p}]^\\xi)$ and $\\liminf\\phi_{\\hat{\\bm \\Sigma}}^2>d\/2>0$, we get\n\\[\n\\big\\|\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast\\big\\|_1=\\small\\smallO_p\\left(\\big(\\frac{\\log{p}}{n}\\big)^{\\frac{1}{2}-\\xi}\\right).\n\\]\nOn the other hand, by Assumption A4, $\\big((n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}\\big)_{(j,j)} =\\mathcal{O}_p\\big((n\/\\log{p})^{1\/2-\\xi}\\big)$. Thus\n\\[\nh\\big|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}\\big]_{(j,j)}\\tilde\\beta_j\\big| = \\big|\\big[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}h{\\bm L}\\big]_{(j,j)}\\big|\\big|\\tilde\\beta_j - \\beta^\\ast_j\\big|=\\small\\smallO_p(1),\n\\]\nand hence\n\\[\nPr\\left(\\big|\\gamma_j^{G}\\big| \\leq h\\big\\|[(n\\hat{\\bm \\Sigma}+h{\\bm L})^{-1}{\\bm L}]_{(j,-j)}\\big\\|_\\infty\\big(\\frac{\\log{p}}{n}\\big)^{\\frac{1}{2}-\\xi}\\right)\\to 1,\n\\]\nwhere the right hand side is $\\Gamma^G_j$. We can thus write\n\\begin{align*}\n\\big|\\hat{\\bm z}_j^G\\big| &= \\big|Z_j^G +\\gamma_j^{G}\\big| \\\\\n& \\leq \\big|Z_j^G\\big| +|\\gamma_j^{G}\\big|  \\\\\n&\\precsim^{asy.}\\big|Z_j^G\\big|+\\Gamma_j^G.\n\\end{align*}\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:suffdet}}\n\n\\begin{proof}\nGiven \\eqref{eq:p}, conditional on ${\\bm X}$,\nthe objective of $P_j^G \\leq \\alpha$ is satisfied if $\\big|\\hat{\\bm z}_j^G\\big| \\geq\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})}$. According to Equation (\\ref{eq:keyrel}), this is equivalent of $\\big|\\beta^\\ast_j+Z_j^G + \\gamma_j^G\\big| \\geq\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})}$, which is satisfied if\n\\[\n\\big|\\beta^\\ast_j\\big| - \\big|\\gamma_j^G\\big|-\\big|Z_j^G\\big| \\geq\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})}.\n\\]\nThis holds with probability at least $\\psi$ if \n\\[\n\\big|\\beta^\\ast_j\\big| - \\big|\\gamma_j^G\\big| \\geq\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})} + q_{(1-\\psi\/2)}.\n\\]\n\nWe know that with probability tending to 1, $\\big|\\gamma_j^G\\big|\\leq \\Gamma_j^G$.  Therefore, conditional on ${\\bm X}$, we have $P_j^G \\leq \\alpha_L$ with probability tending to at least $\\psi$,  if\n\\[\n\\big|\\beta^\\ast_j\\big| >2\\Gamma_j^G + q_{(1-\\alpha\/2)}\\sqrt{\\mathrm{Var}(Z_j^G|{\\bm X})} + q_{(1-\\psi\/2)}.\n\\]\n\n\\end{proof}\n\n\\subsection{Proof of Theorem \\ref{thm:power1}}\n\n\\begin{proof}\n\\textbf{a)} We note that $P_1^G\/P_1^{GI}\\leq 1$ is equivalent of  \n\\[\n\\frac{\\left(\\big|\\hat{\\bm z}_1^{GI}\\big|-\\Gamma_1^{GI}\\right)_+\/\\sqrt{\\mathrm{Var}(Z_1^{GI}|{\\bm X})}}{\\left(\\big|\\hat{\\bm z}_1^G\\big|-\\Gamma_1^G\\right)_+\/\\sqrt{\\mathrm{Var}(Z_1^G|{\\bm X})}}\\leq1. \n\\]\nWe first write out those components for the Grace test:\n\\begin{align*}\n\\hat{\\bm z}_1^G&=\\big(({\\bm X}^\\top{\\bm X}+h_n^G {\\bm L})^{-1}({\\bm X}^\\top{\\bm y}+h_n^G{\\bm L}\\tilde{\\bm \\beta})\\big)_1 \\\\\n&= \\frac{(n+h_n^G){\\bm x}_1^\\top{\\bm y}-(n\\rho+h_n^Gl){\\bm x}_2^\\top{\\bm y} + h_n^G\\tilde\\beta_1(n+h_n^G-n\\rho l-h_n^Gl^2)+nh_n^G\\tilde\\beta_2(l-\\rho)}{(n+h_n^G)^2-(n\\rho+h_n^G l)^2}; \\\\\n\\Gamma_1^G & = \\left|h_n^G\\big[({\\bm X}^\\top{\\bm X}+h_n^G{\\bm L})^{-1}{\\bm L}\\big]_{(1, -1)}\\right|\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi} \\\\\n&= \\left|h_n^G\\big[({\\bm X}^\\top{\\bm X}+h_n^G{\\bm L})^{-1}{\\bm L}\\big]_{(1, 2)}\\right|\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi}\\\\\n&= \\frac{|nh_n^Gl-nh_n^G\\rho|}{(n+h_n^G)^2-(n\\rho+h_n^G l)^2}\\left(\\frac{\\log{p}}{n}\\right)^{\\frac{1}{2}-\\xi}; \\\\\n\\mathrm{Var}(Z_1^G|{\\bm X})& = \\sigma^2_{\\bm \\epsilon}\\left[({\\bm X}^\\top{\\bm X}+h_n^G {\\bm L})^{-1}{\\bm X}^\\top{\\bm X}({\\bm X}^\\top{\\bm X}+h_n^G {\\bm L})^{-1}\\right]_{(1, 1)} \\\\\n& =  \\sigma^2_{\\bm \\epsilon}\\frac{(n^3+2h_n^G n^2)(1-\\rho^2)+n(h_n^G)^2(1+l^2-2l\\rho)}{[(n+h_n^G)^2-(n\\rho+h_n^G l)^2]^2}.\n\\end{align*}\nWe can also write out those components for the GraceI test likewise with $l=0$. \n\n\nIn the proof of Theorem~\\ref{thm:stbound}, we have shown that $Pr\\left(\\big\\|\\tilde{\\bm \\beta}-{\\bm \\beta}^\\ast\\big\\|_1\\leq 4h_{Lasso}s_0\/\\phi^2_{\\hat{\\bm \\Sigma}}\\right)\\to1$. With $h_{Lasso}=\\mathcal{O}(\\log{p}\/n)$, $s_0=\\smallO([n\/\\log{p}]^\\xi)$ for some $0\\leq\\xi<1\/2$, $\\liminf\\phi_{\\hat{\\bm \\Sigma}}>d\/2>0$, and $p=\\mathcal{O}(\\exp(n^\\nu))$ for some $0\\leq\\nu<1$, we have $\\|\\tilde{\\bm \\beta}-{\\bm \\beta}\\|_1=\\small\\smallO_p(1)$. Thus we get \n\\[\n\\tilde\\beta_1 = \\beta_1^\\ast+\\small\\smallO_p(1), \\hspace{0.5in} \\tilde\\beta_2 = \\beta_2^\\ast+\\small\\smallO_p(1). \n\\]\nWe also note that since our design matrix is scaled, we get\n\\begin{align*}\n{\\bm x}_1^\\top{\\bm y}&={\\bm x}_1^\\top{\\bm x}_1\\beta^\\ast_1 + {\\bm x}_1^\\top{\\bm x}_2\\beta^\\ast_2 + {\\bm x}_1^\\top{\\bm \\epsilon} = n\\beta^\\ast_1+n\\rho\\beta^\\ast_2 + nE, \\\\\n{\\bm x}_2^\\top{\\bm y}&={\\bm x}_2^\\top{\\bm x}_1\\beta^\\ast_1 + {\\bm x}_2^\\top{\\bm x}_2\\beta^\\ast_2 + {\\bm x}_2^\\top{\\bm \\epsilon} = n\\rho\\beta^\\ast_1+n\\beta^\\ast_2 + nE, \n\\end{align*}\nwhere $E\\sim N\\left({\\bm 0}, \\sigma^2_{\\bm \\epsilon}\/n\\right)=\\small\\smallO_p(1)$. \n\nDefine $k_n^G\\triangleq h_n^G\/n$ and $k_n^{GI}\\triangleq h_n^{GI}\/n$. With some algebra, We get\n\\begin{align}\n\\label{ts:G}\n\\frac{\\left(|\\hat{\\bm z}_1^G|-\\Gamma_1^G\\right)_+}{\\sqrt{\\mathrm{Var}(Z_1^G|{\\bm X})}} = \\frac{\\sqrt{n}\\left[|(k_n^G+1)^2-(\\rho+lk_n^G)^2 + \\smallO_p(1)|\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^G(l-\\rho)|\\right]_+}{\\sigma_{\\bm \\epsilon}\\sqrt{(1+2k_n^G)(1-\\rho^2) + (k_n^G)^2(1+l^2-2l\\rho)}}.\n\\end{align}\nSimilarly for the GraceI, we get\n\\begin{align}\n\\label{ts:GI}\n\\frac{\\left(|\\hat{\\bm z}_1^{GI}|-\\Gamma_1^{GI}\\right)_+}{\\sqrt{\\mathrm{Var}(Z_1^{GI}|{\\bm X})}} = \\frac{\\sqrt{n}\\left[|(k_n^{GI}+1)^2-\\rho^2 + \\smallO_p(1)|\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^{GI}\\rho|\\right]_+}{\\sigma_{\\bm \\epsilon}\\sqrt{(1+2k_n^{GI})(1-\\rho^2) + (k_n^{GI})^2}}.\n\\end{align}\n\nWe observe that $k_n^{GI}+1>1\\geq |\\rho|$ and $k_n^G+1\\geq|l|k_n^G+|\\rho|\\geq |\\rho+lk_n^G|$. We plug in those two inequalities into Equation ~\\eqref{ts:G} and ~\\eqref{ts:GI}. Hence, conditional on the design matrix ${\\bm X}$, $P_1^G\/P_1^{GI}\\leq 1$ with probability tending to 1 if\n\\begin{align*}\n&\\lim_{n\\to\\infty}\\frac{\\left\\{\\big[(k_n^G+1)^2-(\\rho+lk_n^G)^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^G(l-\\rho)|\\right\\}_+}{\\sqrt{(1+2k_n^G)(1-\\rho^2) + (k_n^G)^2(1+l^2-2l\\rho)}} \\\\\n\\geq &\\lim_{n\\to\\infty}\\frac{\\left\\{\\big[(k_n^{GI}+1)^2-\\rho^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^{GI}\\rho|\\right\\}_+}{\\sqrt{(1+2k_n^{GI})(1-\\rho^2) + (k_n^{GI})^2}}.\n\\end{align*}\n\nNote that for any two real numbers $f$ and $g$, $f\\geq g$ implies $f_+\\geq g_+$. Thus, conditional on the design matrix ${\\bm X}$, $P_1^G\/P_1^{GI}\\leq 1$ with probability tending to 1 if\n\\begin{align}\n\\label{ts:GGI}\n&\\lim_{n\\to\\infty}\\frac{\\big[(k_n^G+1)^2-(\\rho+lk_n^G)^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^G(l-\\rho)|}{\\sqrt{(1+2k_n^G)(1-\\rho^2) + (k_n^G)^2(1+l^2-2l\\rho)}} \\nonumber\\\\\n\\geq &\\lim_{n\\to\\infty}\\frac{\\big[(k_n^{GI}+1)^2-\\rho^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^{GI}\\rho|}{\\sqrt{(1+2k_n^{GI})(1-\\rho^2) + (k_n^{GI})^2}}.\n\\end{align}\n\nIf we assume $k_n^G=k_n^{GI}=k\\to\\infty$, Inequality~\\eqref{ts:GGI} is satisfied if \n\\begin{align}\n&\\lim_{n\\to\\infty}\\frac{\\big[(k+1)^2-(\\rho+lk)^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k(l-\\rho)|}{\\big[(k+1)^2-\\rho^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k\\rho|} \\nonumber \\\\\n\\times&\\frac{\\sqrt{(1+2k)(1-\\rho^2) + k^2}}{\\sqrt{(1+2k)(1-\\rho^2) + k^2(1+l^2-2l\\rho)}} \\nonumber \\\\ \n=&\\lim_{n\\to\\infty}\\frac{\\big[(1-l^2)+(2-2l\\rho)\/k+(1-\\rho^2)\/k^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|(l-\\rho)\/k|}{\\big[1+2\/k+(1-\\rho^2)\/k^2\\big]\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|\\rho\/k|} \\nonumber\\\\\n\\times& \\frac{\\sqrt{1 + (2-2\\rho^2)\/k + (1-\\rho^2)\/k^2 }}{\\sqrt{(1+l^2-2l\\rho) + (2-2\\rho^2)\/k + (1-\\rho^2)\/k^2}}  \\nonumber\\\\\n=&\\frac{(1-l^2)}{\\sqrt{(1+l^2-2l\\rho)}} \\geq1.\n\\end{align}\nThe last equality holds because $p=\\mathcal{O}(\\exp(n^\\nu))$ for some $0\\leq\\nu<1$ implies that $\\log{p}\/n\\to0$.\n\n\n\nFor the ridge test, we assume $h_n^R=\\mathcal{O}(1)$. Thus with some algebra we can similarly write out the ridge test objective:\n\\begin{align}\n\\label{ts:R}\n\\frac{|\\hat{\\bm z}_1^R|}{\\sqrt{\\mathrm{Var}(Z_1^R|{\\bm X})}} = \\frac{\\sqrt{n}|1-\\rho^2 + \\smallO_p(1)|\\cdot|\\beta_1^\\ast|}{\\sigma_{\\bm \\epsilon}\\sqrt{(1-\\rho^2)+\\smallO(1)}}.\n\\end{align}\n\n\\textbf{b)} Thus, conditional on ${\\bm X}$, we get $P_1^G\/P_1^{R}\\leq 1$ with probability tending to 1 if\n\\begin{align}\n\\label{ts:GR}\n\\lim_{n\\to\\infty}\\frac{\\big((k_n^G+1)^2-(\\rho+lk_n^G)^2\\big)\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^G(l-\\rho)|}{\\sqrt{(1+2k_n^G)(1-\\rho^2) + (k_n^G)^2(1+l^2-2l\\rho)}}\\geq \\sqrt{1-\\rho^2}\\cdot|\\beta_1^\\ast|.\n\\end{align}\n\n\\textbf{c)} We also have$P_1^{GI}\/P_1^{R}\\leq 1$ with probability tending to 1 if\n\\begin{align}\n\\label{ts:GIR}\n\\lim_{n\\to\\infty}\\frac{\\big((k_n^{GI}+1)^2-\\rho^2\\big)\\cdot|\\beta_1^\\ast| - (\\log{p}\/n)^{1\/2-\\xi}\\cdot|k_n^{GI}\\rho|}{\\sqrt{(1+2k_n^{GI})(1-\\rho^2) + (k_n^{GI})^2}}\\geq \\sqrt{1-\\rho^2}\\cdot|\\beta_1^\\ast|.\n\\end{align}\n\n\\end{proof}\n\n\n\\subsection{Illustration of the Graph Structure in the Simulation Study}\nFigure~\\ref{fig:graph} shows the graph structure used in the simulation study with 5 hub-satellite clusters. In the simulation study, we use 50 such hub-satellite clusters.\n\\begin{figure}[h]\n\\caption{An illustration of the graph structure with 5 hub-satellite clusters.} \n\\label{fig:graph}\n\\centering\n\\includegraphics[width = 12cm]{graph.pdf} \n\\end{figure}\n\n\n\\subsection{Additional Details for Analysis of TCGA Data}\n\\subsubsection{Biological Evidence}\nIn this section, we summarize some of the biological evidences in support of the association between genes identified by the Grace and GraceR tests with the onset, progression and severity of prostate cancer, as well as PSA level. \n\nAs pointed out in the main paper, the Grace and GraceR tests identify a number of histone genes and histone deacetylase (HDAC) genes. Previous research indicates that histone genes are associated with the occurrence, clinical outcomes and recurrence of prostate cancer \\citep{Seligsonetal2005, Keetal2009}. The pathological role of HDAC genes on the onset and progression of prostate cancer have also been previously studied \\citep{Halkidouetal2004, Chenetal2007, AbbasGupta2008}. \n\nIn addition to the highly connected histone and HDAC genes, the GraceR test also identifies some disconnected genes. Prior works shows that the expression of ribonucleoside-diphosphate reductase subunit M2 (RRM2) is associated with higher Gleason scores, which correlate with the severity of prostate cancer \\citep{Huangetal2014}. Protein arginine methyltransferase 1 (PRMT1) may also have an effect on the proliferation of prostate cancer cells \\citep{Yuetal2009}. Activation of olfactory receptors (OR) prevents proliferation of prostate cancer cells \\citep{Neuhausetal2009}. Interferon-$\\gamma$ (IFNG) plays a role in the differentiation of human prostate basal-epithelial cells \\citep{Untergasseretal2005}. IFNG is connected to the interleukin receptor 22 $\\alpha1$ (IL22RA1), the role of which related to prostate cancer is unknown. However, several earlier studies point out the associations between prostate cancer and several other interleukin receptors in the Janus kinase and signal transducer and activator of transcription (JAK-STAT) activating family, including IL 6, 8, 11, 13 and 17 genes\\citep{Culigetal2005, Inoueetal2000, Campbelletal2001, Mainietal1997, Zhangetal2012}. Cell-division cycle genes (CDC) may also be associated with various cancers. The association between collagen type 2 $\\alpha1$ (COL2A1) and prostate cancer is also not known, but other collagen genes, including type 1 $\\alpha2\\beta1$, type 4 $\\alpha5$ and $\\alpha6$, have been shown to be associated with prostate cancer progression \\citep{Halletal2008, Dehanetal1997}.  Although the association between phosphate cytidylyltransferase 1 choline-$\\alpha$ (PCYT1A) and prostate cancer or PSA level is not known, \\citet{Vaezietal2014} shows that PCYT1A is a prognostic factor in survival for patients with lung and head and neck squamous cell carcinomas.\n\n\n\\subsubsection{Stability of the Grace Test to the Tuning Parameter}\nFigure~\\ref{fig:ngenes} shows the number of significant genes identified by the Grace test in the TCGA data against various values of $h_G$.  The results indicate that the number of genes found by the Grace test is relatively stable for a range of tuning parameters including the CV choice. On the other hand, very few genes are identified when the tuning parameter is too small or too large. This is because, with small tuning parameters, the variance is large and thus no gene is statistically significant. On the other hand, with large tuning parameters, the stochastic bound $\\Gamma_j$ dominates $\\hat z_j$. Note that above results of power do not contradict Theorem~\\ref{thm:power1}, which shows the \\textit{asymptotic} power of the Grace test improves as we use larger $h_G$. A vital condition for Theorem~\\ref{thm:power1} to hold is $\\|\\tilde{\\bm \\beta}-{\\bm \\beta}\\|_1=\\smallO_p(1)$. \n\\begin{figure}[h]\n\\caption{Number of genes identified by the Grace test in the TCGA data against the tuning parameter of the Grace test, $h_G$. The red dashed line corresponds to the choice made by 10-fold CV ($h_G=\\exp(14.2)$).} \n\\label{fig:ngenes}\n\\centering\n\\includegraphics[width = 12cm]{nsigGrace.pdf} \n\\end{figure}\n\n\n\\subsubsection{Stability of the Grace Test to the Network}\nWe examine whether the result of the Grace test on the TCGA data is sensitive to the KEGG network structure. To this end, we randomly change the connectivity of $m$ node pairs in the KEGG network and form the new perturbed network $\\tilde G$, $|E\\Delta \\tilde E|=m$, where $\\Delta$ is the symmetric difference operator between two sets. In other words, for $m$ randomly selected node pairs $(a_i, b_i)$, $i=1,...,m$, if there is an edge $(a_i,b_i)$ in the KEGG network, we remove it in the perturbed network; otherwise, we add an edge in the perturbed network. In our examination, $m$ ranges from $10,000$ to $600,000$. Note that there are 38,541 edges in the original KEGG network. We counted the number of genes that are significant using both networks. The result shown in Figure~\\ref{fig:robust} is an average of 50 independent replications.\n\\begin{figure}[h]\n\\caption{Number of genes that are significant using both the KEGG network and the perturbed network against the number of perturbed edges. The red dashed line represents the number of genes identified by the Grace test with the KEGG network.} \n\\label{fig:robust}\n\\centering\n\\includegraphics[width = 12cm]{nperturb.pdf} \n\\end{figure}\n\n\n\\subsubsection{Prediction Performance}\nWe also compare the prediction performance by Grace, GraceR, GraceI and lasso with tuning parameters chosen by 10-fold CV, as well as ridge with $h_2=1$. The result is shown in Table~\\ref{tab:RDpred}. GraceR produced the smallest CV prediction error, followed closely by GraceI and Grace. This result may indicate the KEGG network information is in fact informative in prediction. \n\\begin{table}\n\\centering\n\\caption{Prediction performance of the Grace, GraceR, GraceI(ridge regression with tuning parameter chosen by CV), ridge ($h_2=1$) and lasso. The performance metric is the sum of 10-fold CV prediction error (CVER).} \n\\label{tab:RDpred} \n\\begin{tabular}{lccccc}\n\\hline \\hline\n\t&Grace\t&GraceR\t&GraceI\t&Ridge\t&Lasso\t\\\\\t\\hline\nCVER\t& 3473\t& 3411\t&3418\t&3917\t&3546\\\\\n\\hline\n\\end{tabular}\n\\end{table}\n \n\\subsection{Additional Simulation Studies with Extended NPE}\n\nWe performed simulation studies with extended $\\textrm{NPE}\\in\\{$\\textsc{-225, -165, -70, -10, 0, 15, 135, 350, 600, 900, 1250, 1650, 2050, 3150}$\\}$. These perturbations in the network correspond to the spectral norm of perturbations $\\|{\\bm L}-{\\bm L}^*\\|_2\/\\|{\\bm L}^*\\|_2$ equal 0.85, 0.75, 0.50, 0.25, 0, 0.25, 0.50, 0.75, 1.00, 1.25, 1.50, 1.75, 2.00 and 2.65, respectively. The power and type-I error rates are summarized in Figure~\\ref{fig:power2}, Table~\\ref{tab:power} and Table~\\ref{tab:level}. Our conclusions on the simulation study stated in the main paper do not change with this expanded version of simulation study.\n\n\\begin{figure}\n\\caption{Comparison of power and type-I error rates of different testing methods with their 95\\% confidence bands. Testing methods include LDPE, ridge, GraceI, Grace and GraceR. Filled circles ($\\bullet$) show powers, whereas crosses ($\\times$) are type-I error rates. Numbers on $x$-axis for Grace and GraceR tests refer to the number of perturbed edges (\\textrm{NPE}).} \n\\label{fig:power2}\n\\centering\n\\includegraphics[angle = 90, height = 20cm]{Power.pdf} \n\\end{figure}\n\n\\begin{table}\n\\centering\n\\caption{Mean power and the standard error for the LDPE test, ridge test, GraceI, Grace and GraceR tests with different $R^2$ values.} \n\\label{tab:power} \n\\begin{tabular}{lccc}\n\\hline \\hline\n             \t\t\t\t\t\t\t\t& $R^2=0.1$ \t\t&$R^2=0.2$ \t\t&$R^2=0.3$   \t\\\\\t\\hline\nLDPE\t\t\t\t\t\t\t\t& 0.181 (0.011)\t\t& 0.274 (0.012)\t\t& 0.343 (0.014) \\\\\nRidge        \t\t\t\t\t\t\t& 0.220 (0.016)\t\t& 0.393 (0.018) \t& 0.580 (0.019) \\\\\nGraceI\t\t\t\t\t\t\t\t& 0.493 (0.026)\t\t& 0.769 (0.021) \t& 0.868 (0.015) \\\\\nGrace $\\mathrm{NPE} = \\textsc{-225}$  \t\t& 0.623 (0.033)\t\t& 0.853 (0.018) \t& 0.918 (0.011) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{-165}$  \t\t& 0.720 (0.032)\t\t& 0.918 (0.012) \t& 0.959 (0.007) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{-70}$  \t\t& 0.780 (0.035)\t\t& 0.974 (0.005) \t& 0.985 (0.004) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{-10}$  \t\t& 0.839 (0.035)\t\t& 0.986 (0.010) \t& 0.998 (0.001) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{0}$  \t\t& 0.813 (0.039)\t\t& 1.000 (0.000) \t& 1.000 (0.000) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{15}$  \t\t& 0.760 (0.042)\t\t& 0.947 (0.022) \t& 0.989 (0.010) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{135}$  \t\t& 0.506 (0.047)\t\t& 0.791 (0.038) \t& 0.920 (0.023) \\\\\nGrace $\\mathrm{NPE}  =\\textsc{350}$  \t\t& 0.431 (0.045)\t\t& 0.732 (0.041) \t& 0.873 (0.031) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{600}$  \t\t& 0.328 (0.040)\t\t& 0.719 (0.037) \t& 0.906 (0.024) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{900}$\t  \t& 0.337 (0.037)\t\t& 0.609 (0.041) \t& 0.791 (0.032) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{1250}$  \t\t& 0.316 (0.036)\t\t& 0.672 (0.038) \t& 0.911 (0.017) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{1650}$  \t\t& 0.376 (0.040)\t\t& 0.688 (0.037) \t& 0.859 (0.025) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{2050}$  \t\t& 0.252 (0.037)\t\t& 0.558 (0.042) \t& 0.792 (0.032) \\\\\nGrace $\\mathrm{NPE}  = \\textsc{3150}$  \t\t& 0.312 (0.037)\t\t& 0.622 (0.038) \t& 0.845 (0.024) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-225}$  \t\t& 0.547 (0.033)\t\t& 0.790 (0.023) \t& 0.882 (0.015) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-165}$  \t\t& 0.606 (0.032)\t\t& 0.831 (0.018) \t& 0.923 (0.012) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{-70}$  \t\t& 0.650 (0.032)\t\t& 0.872 (0.018) \t& 0.925 (0.013) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{-10}$  \t\t& 0.722 (0.034)\t\t& 0.904 (0.019) \t& 0.959 (0.011) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{0}$  \t\t& 0.682 (0.038)\t\t& 0.901 (0.020) \t& 0.928 (0.017) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{15}$  \t\t& 0.702 (0.035)\t\t& 0.887 (0.023) \t& 0.958 (0.011) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{135}$  \t\t& 0.631 (0.037)\t\t& 0.882 (0.025) \t& 0.957 (0.013) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{350}$  \t\t& 0.628 (0.036)\t\t& 0.878 (0.018) \t& 0.940 (0.013) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{600}$  \t\t& 0.539 (0.036)\t\t& 0.785 (0.028) \t& 0.905 (0.017) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{900}$\t\t& 0.490 (0.033)\t\t& 0.781 (0.024) \t& 0.875 (0.016) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{1250}$  \t& 0.515 (0.031)\t\t& 0.822 (0.022) \t& 0.909 (0.013) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{1650}$  \t& 0.585 (0.032)\t\t& 0.821 (0.022) \t& 0.890 (0.016) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{2050}$  \t& 0.450 (0.034)\t\t& 0.748 (0.028) \t& 0.876 (0.017) \\\\\nGraceR $\\mathrm{NPE}  = \\textsc{3150}$  \t& 0.442 (0.036)\t\t& 0.767 (0.025) \t& 0.864 (0.017) \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\centering\n\\caption{Mean type-I error rate and the standard error for the LDPE test, ridge test, GraceI, Grace and GraceR tests with different $R^2$ values.} \n\\label{tab:level} \n\\begin{tabular}{lccc}\n\\hline \\hline\n             \t\t\t\t\t\t\t\t& $R^2=0.1$ \t\t&$R^2=0.2$ \t\t&$R^2=0.3$ \t\\\\\t\\hline\nLDPE\t\t\t\t\t\t\t\t& 0.048 (0.0010)\t& 0.048 (0.0010)\t& 0.047 (0.0010) \\\\\nRidge        \t\t\t\t\t\t\t& 0.046 (0.0012)\t& 0.048 (0.0013) \t& 0.050 (0.0012) \\\\\nGraceI \t\t\t\t\t\t\t\t& 0.031 (0.0010)\t& 0.027 (0.0009) \t& 0.025 (0.0008) \\\\\nGrace $\\mathrm{NPE} = \\textsc{-225}$  \t\t& 0.026 (0.0013)\t& 0.021 (0.0012) \t& 0.019 (0.0010) \\\\\nGrace $\\mathrm{NPE} = \\textsc{-165}$  \t\t& 0.025 (0.0014)\t& 0.020 (0.0013) \t& 0.017 (0.0012) \\\\\nGrace $\\mathrm{NPE} = \\textsc{-70}$  \t\t& 0.027 (0.0021)\t& 0.019 (0.0017) \t& 0.014 (0.0013) \\\\\nGrace $\\mathrm{NPE} = \\textsc{-10}$  \t\t& 0.022 (0.0021)\t& 0.015 (0.0017) \t& 0.013 (0.0015) \\\\\nGrace $\\mathrm{NPE} = \\textsc{0}$  \t\t\t& 0.024 (0.0021)\t& 0.017 (0.0017) \t& 0.011 (0.0013) \\\\\nGrace $\\mathrm{NPE} = \\textsc{15}$  \t\t& 0.032 (0.0034)\t& 0.031 (0.0031) \t& 0.028 (0.0028) \\\\\nGrace $\\mathrm{NPE} = \\textsc{135}$  \t\t& 0.040 (0.0073)\t& 0.037 (0.0059) \t& 0.029 (0.0042) \\\\\nGrace $\\mathrm{NPE} = \\textsc{350}$  \t\t& 0.059 (0.0137)\t& 0.051 (0.0102) \t& 0.036 (0.0052) \\\\\nGrace $\\mathrm{NPE} = \\textsc{600}$  \t\t& 0.060 (0.0156)\t& 0.059 (0.0155) \t& 0.040 (0.0083) \\\\\nGrace $\\mathrm{NPE} = \\textsc{900}$\t  \t& 0.041 (0.0115)\t& 0.038 (0.0101) \t& 0.027 (0.0033) \\\\\nGrace $\\mathrm{NPE} = \\textsc{1250}$  \t\t& 0.052 (0.0151)\t& 0.045 (0.0111) \t& 0.037 (0.0075) \\\\\nGrace $\\mathrm{NPE} = \\textsc{1650}$  \t\t& 0.044 (0.0141)\t& 0.045 (0.0125) \t& 0.038 (0.0104) \\\\\nGrace $\\mathrm{NPE} = \\textsc{2050}$  \t\t& 0.039 (0.0141)\t& 0.035 (0.0112) \t& 0.027 (0.0023) \\\\\nGrace $\\mathrm{NPE} = \\textsc{3150}$  \t\t& 0.039 (0.0110)\t& 0.027 (0.0024) \t& 0.026 (0.0015) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-225}$  \t\t& 0.027 (0.0012)\t& 0.023 (0.0011) \t& 0.020 (0.0009) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-165}$  \t\t& 0.028 (0.0013)\t& 0.023 (0.0011) \t& 0.019 (0.0010) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-70}$  \t\t& 0.028 (0.0014)\t& 0.022 (0.0014) \t& 0.018 (0.0012) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{-10}$  \t\t& 0.026 (0.0018)\t& 0.020 (0.0015) \t& 0.017 (0.0014) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{0}$  \t\t& 0.027 (0.0018)\t& 0.022 (0.0016) \t& 0.015 (0.0013) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{15}$  \t\t& 0.030 (0.0025)\t& 0.026 (0.0025) \t& 0.021 (0.0025) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{135}$  \t\t& 0.058 (0.0165)\t& 0.041 (0.0112) \t& 0.038 (0.0103) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{350}$  \t\t& 0.076 (0.0182)\t& 0.059 (0.0152) \t& 0.030 (0.0027) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{600}$  \t\t& 0.058 (0.0145)\t& 0.054 (0.0139) \t& 0.027 (0.0016) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{900}$\t\t& 0.044 (0.0109)\t& 0.040 (0.0099) \t& 0.025 (0.0010) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{1250}$  \t& 0.057 (0.0125)\t& 0.044 (0.0100) \t& 0.034 (0.0071) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{1650}$  \t& 0.053 (0.0138)\t& 0.047 (0.0122) \t& 0.039 (0.0104) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{2050}$  \t& 0.045 (0.0111)\t& 0.033 (0.0038) \t& 0.025 (0.0009) \\\\\nGraceR $\\mathrm{NPE} = \\textsc{3150}$  \t& 0.039 (0.0053)\t& 0.029 (0.0017) \t& 0.025 (0.0012) \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\label{lastpage}\n\\bibliographystyle{apa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\\subsection{Granular physics and wet granular media}\nGranular materials are collections of macroscopic particles,\nlike glass beads or sand,\nwhich are visible by the naked eye.\nBecause of the macroscopic size of the particles,\nthermal noise plays no role in particle motion,\nand particle-particle interactions are dissipative.\nTherefore, continuous energy input by external forces\n(gravity, vibrations, etc.) are necessary in order to keep them in motion.\nParticles may stay at rest like a solid, flow like a liquid,\nor behave like a gas,\ndepending on the rate of energy input.\nHowever, the external force is often not enough for\nthe particles to explore their phase space,\nwhich makes them \nquite different from conventional molecular systems.\n\n\nThe scientific study of granular media\nhas a long history,\nmainly in the engineering field,\nand many physicists have joined the granular research community\nover the past few decades\n(for reviews and books, see, e.g.,\n\\cite{Duran97,Jaeger96,Nedderman,deGennesGranular,\nPandG97,BagnoldBook,Jaeger92,Jaeger96-PT,Sholtz97,Nori97,\nChaos9,Edwards02}).\nMost studies on granular media, especially in the physics field,\nhave focused on {\\it dry granular materials},\nwhere the effects of \ninterstitial fluids are negligible \nfor the particle dynamics.\nFor dry granular media,\nthe dominant interactions are inelastic collisions and friction,\nwhich are short-range and {\\it non-cohesive}.\nEven in this idealised situation, dry granular media\nshow unique and striking behaviours, which have attracted\nthe attention of many scientists for centuries.\n\nIn the real world, however, \nwe often see {\\it wet granular materials},\nsuch as beach sand.\nDry and wet granular materials have many aspects in common, \nbut there is one big difference:\nWet granular materials are {\\it cohesive}\ndue to surface tension.\n\nIn the past, many research groups that studied \ngranular physics have been struggling to\nminimise humidity and avoid inter-granular\ncohesive forces (e.g., \\cite{Bretz92}).\nIndeed, some experiments were performed in \nvacuum  chambers.\nHumidity and fluids in general were\nseen as a nuisance to be avoided at all costs.\nHowever, many important real life applications\ninvolve mechanical properties of wet granular media.\nExamples include rain-induced landslides,\npharmaceuticals,\nfood processing,\nmining and construction industries.\nThus, it is important to study the mechanical \nresponse of granular matter with various degrees of \nwetness or liquid content.\n\nIn this review, we show how the cohesion \ninduced by the liquid changes the mechanical properties of\ngranular materials.\nWe mainly consider static or quasistatic situations,\nwhere the cohesion dominates over other effects of the liquid,\nsuch as lubrication and viscosity.\nSome phenomena in this field\nwhich are not well-understood are presented below as ``open problems''.\nMost references listed at the end focus on experimental results.\nTheoretical approaches can be found in the\nreviews cited below and references therein.\n\n\\subsection{What is different from dry granular media}\n\\begin{table*}\n\\begin{center}\n\\caption{\nComparison of physical properties between dry \nand wet granular matter.}\n\\label{Table:DryWet}\n\\begin{tabular}[t]{|p{3.cm}||p{3.5cm}|p{4.5cm}|}\n\\hline\nPROPERTY&DRY & WET \\\\\n\\hline\n\\hline\nCohesion&Negligible&Important\\\\\n\\hline\nSurface angle \n& Finite\\newline\nAround $35^\\circ$ for sand&\nFinite: Larger than the dry case\\newline\nCan be as large as $90^\\circ\n, or even larger\\\\\n\\hline\nTensile strength&\nNegligible&\nFinite \\\\\n\\hline\nYield shear stress & Finite \\newline \nZero at zero normal stress\n & \nFinite: Can be larger than the dry case\\newline\nNon-zero at zero normal stress\\\\\n\\hline\nHysteresis&Yes&Yes: Enhanced\\\\\n\\hline\nConfigurational phase space \nfor packing\n& Finite \n & \nFinite: Can be larger than the dry case\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\nStudies of wet granular media\nhave been made in \nmany industrial applications.\nThe mechanical properties of wet\ngranular media are also \nextremely important\nin geology and civil engineering.\nFor example, \nlet us consider the huge and expensive civil works project of\nthe construction of the Kansai international airport, \non a man-made island near Osaka.\nBecause of the weight of the \n180 million cubic meters of landfill and facilities, \nthe seabed composed of clay is compressed, and \nit is inevitable for the airport to sink by some amount; \nThe airport had sunk by 11.7 meters on average at the end of 2000,\nand the settlement is still carefully monitored \\cite{KIX}. \nThis and other examples from geology \nand civil engineering stress the need\nto better understand\nthe mechanical properties of wet granular \nassemblies, both small and large.\n\nThe biggest effect that the liquid in granular media induces\nis the cohesion between grains.\nEven humidity in the air may\nresult in a tiny liquid bridge at a contact point,\nwhich introduces cohesion.\nThe cohesion occurs in wet granular material\nunless the system becomes over-wet, i.e., \nthe granular medium is completely immersed in a liquid.\nIn this short review, we focus on the effect of this cohesion;\nthe system that we are considering is partially wet\ngranular material, which is a mixture of solid grains, liquid, and air.\n\nIn addition to cohesion, there are many effects \ninduced by the presence of the liquid. \nOne of them is the lubrication of\nsolid-solid friction \\cite{Tribology,LubricationE,FrictionBook}. \nIn addition, the liquid viscosity may induce\na velocity-dependent behaviour \nand additional dissipation.\nThese effects are often seen \nin underwater experiments \n(e.g., \\cite{Geminard99,Medved01}).\nThe time scale of liquid motion\n(how the liquid moves or flows through granular media)\nalso affects the dynamics.\nAll of these effects, of course, play\nimportant roles in the properties of wet granular media.\nHowever, these are more or less velocity-dependent\nphenomena, and in the static or quasistatic regime,\nthe cohesion often plays the most important role,\nproviding a significant qualitative difference from dry granular media.\n\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=0.4\\textwidth\n\\epsfbox{Fig1a.eps\n\\epsfxsize=0.41\\textwidth\n\\epsfbox{Fig1b.eps}\n\\end{center}\n\\caption\n(a) Dry sandpile with a well-defined surface angle. \n(b) Wet sandpile with a tunnel.\n}\n\\label{Fig:Sandpile}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\epsfxsize=0.5\\textwidth\n\\epsfbox{Fig2.eps}\n\\end{center}\n\\caption\nSandcastle made of wet sand \n(Copyright Sandscapes 2005).\n}\n\\label{Fig:Sandcastle}\n\\end{figure}\n\nThe simplest situation where we see the \neffect of cohesion in wet granular media\nwould be the sandpiles that children make\nin a sandbox.\nLet us first consider a sandpile made of dry grains \nand afterwards wet grains.\n\nWhen we make a sandpile using dry sand\n(Fig.~\\ref{Fig:Sandpile}(a)),\nthe surface of the pile is smooth, \nand makes a finite, well-defined angle, which is about $35^\\circ$.\nA slightly denser pile might be made by tapping the surface of the pile,\nbut it does not change much the shape of the pile.\nEven if we try to make a sharper pile by pouring \nmore dry sand, grains flow down along\nthe pile surface and the resulting angle is always around\nthe same value.\nIt is easy to remove part of the sand from the pile, \nbut we cannot make a tunnel through dry sandpile\nbecause the wall around the hole would collapse and the angle \nof the surface cannot be larger than the critical angle.\n\nLet us add some water to the sand,\nand make another sandpile (Fig.~\\ref{Fig:Sandpile}(b)).\nWe can try to find the optimal amount of \nwater to produce a certain shape.\nSmall amounts of water do not \nallow the creation of many shapes,\nwhile too much water results in muddy water\nthat cannot keep a shape. \nWith the proper amount of water,\nwe can make a sandpile with \nthe surface angle larger than the dry case;\nIndeed, the angle can be as large as \n$90^\\circ$, or even larger,\nallowing the construction of elaborate and stunning sandcastles\n(see, e.g., Fig.~\\ref{Fig:Sandcastle}).\nThis wet pile can be made denser and stronger \nby tapping the surface.\nNow we can make a tunnel through the pile,\nif we are sufficiently careful. \nIf we try to make a hole that is too large, \nthe pile would break down\nforming some rugged surfaces.\n\nTherefore, in a sandbox, we already learnt\nimportant properties of dry granular media:\na finite angle of repose, small hysteresis in packing,\nand small strength against loading.\nWe also learnt how drastically these properties \nare changed or enhanced by adding a liquid,\nresulting in: \na much larger angle of repose,\nstronger hysteresis in packing,\nand enhanced strength against loading.\nThese comparisons are summarised in Table \\ref{Table:DryWet}.\nIn the following sections, \nwe will see how these behaviours are studied by scientists.\n\n\\section{Wet granular media: Grains with liquid and air}\n\\subsection{Cohesion between two spheres}\n\\subsubsection{Meniscus and suction}\nCohesion in wet granular media arises from surface tension \nand capillary effects of the liquid.\nConsider a meniscus between \nair with pressure $P_a$ and liquid with pressure $P_l$.\nThe pressure difference $\\Delta P$ between liquid and air \nwith a meniscus of curvature radii $r_1$ and $r_2$ \nis given by the Young-Laplace equation as\n\\begin{equation}\n\\Delta P=P_a-P_l=\\gamma\\left[\\frac{1}{r_1}+\\frac{1}{r_2}\\right],\n\\label{eq:DeltaP}\n\\end{equation}\nwhere $\\gamma$ is the surface tension between air and the liquid,\nand the curvature is positive \nwhen the meniscus is drawn back into the liquid phase\n(e.g., \\cite{FluidMechanics}).\nWhen the curvature is positive,\n$\\Delta P$ is positive, and\nit is often called the suction.\n\nThe capillary length \n\\begin{equation}\na=\\sqrt{\\frac{2 \\gamma}{\\rho_l g}}\n\\end{equation}\ngives the length scale that compares\nthe capillary force and gravity, where $g$ is\nthe gravitational acceleration and $\\rho_l$\nis the mass density of the liquid;\n$a$ is around $3.9$ mm for water at room temperature.\nThe capillary force becomes dominant when \nthe relevant length scales are much smaller than $a$. \nHereafter we consider the situation where \nthe capillary force is dominant.\nIn general, \none needs to consider the capillary length\nwhen interpreting experimental results,\nbecause most experiments are conducted under gravity.\n\n\\subsubsection{Liquid bridge between two spheres}\n\\label{CohesionTwoParticles}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]\n{Fig3.eps}\n\\end{center}\n\\caption{\nSchematic diagram of a liquid bridge\nbetween identical spheres.\n}\n\\label{Fig:LiquidBridge}\n\\end{figure}\n\nNow let us see how the liquid induces cohesion in granular media\nby considering a liquid bridge between two identical spheres \nas shown in Fig.~\\ref{Fig:LiquidBridge}.\nThe attracting force between spheres due to the liquid menisci\nis given by the sum of the surface tension\nand the suction; \nwhen we estimate the force at the neck of the bridge,\nit is given by \n\\begin{equation}\nF_{\\rm bridge}=2\\pi r_2\\gamma+\\pi r_2^2\\Delta P,\n\\label{eq:bridge0}\n\\end{equation}\nwith\n\\begin{equation}\n\\Delta P=\\gamma\\left[\\frac{1}{r_1}-\\frac{1}{r_2}\\right].\n\\end{equation}\nThe detailed experimental analysis of the cohesion force\ndue to the liquid bridge between two spheres\nis found in \\cite{Willett00}.\nThe effect of roughness on the cohesive force\nis discussed in detail in \\cite{Herminghaus05}.\n\nIn real partially wet granular media, \nthe picture of a liquid bridge \nformed between completely spherical particles\nis often not sufficient to describe the interaction.\nHowever, the presence of a liquid,\nwhich tries to minimise its surface area, generally\nresults in suction and a cohesive force between particles.\n\n\\subsection{Wet granular media with various liquid content}\n\\subsubsection{Four states of liquid content:\npendular, funicular, capillary, and slurry state}\n\\begin{sidewaystable}\n\\caption{\nGranular media with various\namounts of liquid \\cite{Newitt58,GranulationReview}.\nIn the schematic diagrams \nin the third column,\nthe filled circles \nrepresent the grains and\nthe grey regions represent the interstitial liquid.\n}\n\\label{Table:Regimes}\n\\begin{tabular}{|p{1in}||p{.8in}|p{1.3in}|p{3.in}|}\n\\hline\n\\begin{minipage}{1in}\n{\\bf Liquid content}\n\\end{minipage}\n&\n\\begin{minipage}{.8in}\n{\\bf State}\n\\end{minipage}\n&\n\\begin{minipage}{1.3in}\n{\\bf Schematic diagram}\n\\end{minipage}\n&\n\\begin{minipage}{3.in}\n{\\bf Physical description}\n\\end{minipage}\\\\\n\\hline\n\\hline\n\\begin{minipage}{1in}\nNo\n\\end{minipage}\n&\n\\begin{minipage}{.8in}\nDry\n\\end{minipage}\n&\n\\begin{minipage}{1.3in}\n\\includegraphics[width=1in]{FigTable2-1.eps}\n\\end{minipage}\n&\n\\begin{minipage}{3in}\n\\begin{flushleft}\nCohesion between grains is negligible.\n\\end{flushleft}\n\\end{minipage}\n\\\\\n\\hline\n\\begin{minipage}{1in}\nSmall \n\\end{minipage}\n&\n\\begin{minipage}{.8in}\nPendular \n\\end{minipage}\n&\n\\begin{minipage}{1.3in}\n \\includegraphics[width=1in]{FigTable2-2.eps}\n\\end{minipage}\n&\n\\begin{minipage}{3in}\n\\begin{flushleft}\nLiquid bridges are formed at the contact points of grains.\nCohesive forces act through the liquid bridges.\n\\end{flushleft}\n\\end{minipage} \n\\\\\n\\hline\n\\begin{minipage}{1in}\nMiddle\n\\end{minipage}\n& \n\\begin{minipage}{.8in}\nFunicular\n\\end{minipage}&\n\\begin{minipage}{1.3in}\n\\includegraphics[width=1in]{FigTable2-3.eps}\n\\end{minipage}\n&\n\\begin{minipage}{3in}\n\\begin{flushleft}\nLiquid bridges around the contact points\nand liquid-filled pores coexist.\nBoth give rise to cohesion between particles.\n\\end{flushleft}\n\\end{minipage}\n\\\\\n\\hline\n\\begin{minipage}{1in}\nAlmost saturated\n\\end{minipage}\n&\n\\begin{minipage}{.8in}\nCapillary\n\\end{minipage}\n&\n\\begin{minipage}{1.3in}\n\\includegraphics[width=1in]{FigTable2-4.eps}\n\\end{minipage}\n&\n\\begin{minipage}{3in}\n\\begin{flushleft}\nAlmost all the pores are filled with the liquid,\nbut the liquid surface forms menisci and the \nliquid pressure is lower than the air pressure.\nThis suction results in a cohesive interaction between particles.\n\\end{flushleft}\n\\end{minipage}\n\\\\\n\\hline\n\\begin{minipage}{1in}\nMore \n\\end{minipage}& \n\\begin{minipage}{.8in}\nSlurry \n\\end{minipage}\n&\n\\begin{minipage}{1.3in}\n\\includegraphics[width=1in]{FigTable2-5.eps}\n\\end{minipage}\n&\n\\begin{minipage}{3in}\n\\begin{flushleft}\nThe liquid pressure is equal to, or higher than, the air\npressure. No cohesive interaction appears between particles.\n\\end{flushleft}\n\\end{minipage}\n\\\\\n\\hline\n\\end{tabular}\n\\end{sidewaystable}\n\\label{phases}\n\nIt is well known that cohesion in wet granular materials \ndepends on the amount of liquid in the system.\nThe following four regimes of liquid content\nhave been distinguished \nin wet granular media \\cite{Newitt58,GranulationReview}:\n\\begin{itemize}\n\\item Pendular state:  \nParticles are held together by liquid bridges at their contact points.\n\n\\item Funicular state: \nSome pores are fully saturated by liquid,\nbut there still remain voids filled with air.\n\n\\item Capillary state: \nAll voids between particles are filled with liquid, \nbut the surface liquid is drawn back into the pores under capillary\naction.\n\n\\item Slurry state: \nParticles are fully immersed in liquid and the surface of liquid\nis convex, i.e., no capillary action at the surface.\n\\end{itemize}\nThese four regimes are schematically shown in \nTable \\ref{Table:Regimes}.\n\nCohesion arises in the pendular, funicular,\nand capillary states. \nThus, we  consider these three states in this review.\nA priori, the mechanical properties of these three states \nwould be expected to be qualitatively different.\nIn the case of the pendular state,\nthe cohesive force between a pair of grains\nacts through a liquid bridge;\nwhile in the capillary state\nthe interface between \nthe liquid and the air\nis pressed due to the suction and \nthat pressure keeps together all the grains in the liquid phase.\nBoth the two-body cohesion due to liquid bridges \nand the suction at the liquid-air interfaces\nplay important roles in the funicular state.\n\nIt is very difficult to directly observe the liquid \ndistribution in three-dimensional granular assemblies,\nthough the liquid distribution should be readily observable\nfor two-dimensional wet granular aggregates\nconfined by Plexiglas plates (cf. \\cite{Rubio89}). \nAs we will see later, \nliquid bridges in three dimensions for rather small \nliquid content have recently been visualised by using index matching \ntechniques \\cite{Mason99,Fournier05,Herminghaus05},\nbut it seems to be difficult to extend the method for \nmuch larger liquid content.\nThe detailed liquid distribution \nin the each liquid content regimes described in Table\n\\ref{Table:Regimes} still remains as one of many ``open problems'' \nthat we list below as areas that have not been \nsufficiently well studied so far.\n\nConventionally, these liquid content regimes \nhave been distinguished by\nmeasuring the relation between the liquid content $S$\nand the suction $\\Delta P$ \\cite{Newitt58} \nas we will see below,\nwhere $S$ is \nthe ratio of the volume of liquid $V_l$ in the system \nto the volume of the voids $V_v$ in the granular media\n(when the total volume of the system is\n$V_t$ and the volume occupied by grains is $V_s$,\nthen $V_v=V_t-V_s$ and $V_v=V_l+V_a$\nwhere $V_a$ is the air volume in the system).\nSome authors define $S$ as the \npercentage of the liquid content \n(i.e., multiplying the \nratio of $S=V_l\/V_v$ by 100).\nHere, either case, ratio or percentage, is clear from the text.\n\\subsubsection{Liquid content and suction}\n\\paragraph{Measurement of suction in granular media}\n\\label{suctionmeasure}\n\\begin{table*}\n\\caption{List of methods to measure\nthe suction and practical suction range for each measurement.\nMethods based on the osmotic pressure difference (e.g., due to \ngradients of solute concentration) are not listed below.\nAdapted from \\cite{UnsaturatedBook}.\n}\n\\label{Table:UnsaturatedBookTable10-1}\n\\begin{center}\n\\begin{tabular}{|p{2.15in}|p{1.5in}|p{0.85in}|}\n\\hline\n{\\bf Technique} or {\\bf Sensor} & {\\bf Suction Range} (kPa)\n& References\\\\\n\\hline\n\\hline\nTensiometers & 0-100& \n\\cite{Cassel86,Stannard92}\\\\\n\\hline\nAxis translation techniques &\n0-1500& \n\\cite{Hilf56,Bocking80}\\\\\n\\hline\nElectrical or thermal conductivity sensors&\n0-400&\n\\cite{Phene71a,Phene71b,Fredlund89}\\\\\n\\hline\nContact filter paper method&\nEntire range\n& \n\\cite{Houston94}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThere are several ways to measure the suction\n$\\Delta P$ in granular media,\nand the appropriate method should be chosen\nregarding the measurement range of $\\Delta P$.\nTable \\ref{Table:UnsaturatedBookTable10-1} \nshows the list of methods and the range of measurements\n\\cite{UnsaturatedBook}\ncommonly used in soil mechanics\n(where the mixture of soil grains and water\nis considered as a specimen).\\footnote{\nWhen solutes are dissolved in water in soil,\nthe osmotic effects produce the chemical potential \ndifference from free water,\nwhere free water is the water that contains no dissolved solute,\nhas no interactions with other phases that produce \ncurvature to the air-water interface, and has\nno external force other than gravity \\cite{UnsaturatedBook}.\nIn the soil mechanics, this chemical potential difference\nis sometimes measured in the unit of pressure,\nand it is referred to as osmotic suction.\nThe pressure difference between the \nair and liquid (water) in soil $\\Delta P=P_a-P_l$, which is \ndue to capillary effect and \nshort-range adsorption of water to grain surfaces,\nis referred to as matric suction.\nIn this paper, we focus on the latter,\nand $\\Delta P$ is simply referred to as suction.\n}\nIt is beyond the scope of this brief review\nto describe all the experimental methods used,\nbut theses are described in the references.\nAs examples, below we describe two\noften-used methods, \ncalled the axis-translation techniques and \ntensiometers,\nwhich make use of the capillary pressure\nin a porous ceramic disk to measure and to control the suction\n\\cite{UnsaturatedBook}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=1.\\textwidth]{Fig4.eps}\n\\end{center}\n\\caption\nSchematic diagram describing the \ntensiometers and the axis-translation techniques.\nThe filled circles represent grains\nand the light blue region represents a liquid.\nThe container is separated by a High-Air-Entry (HAE) \nporous disk that has many microscopic pores.\nThe upper part is filled with air with pressure $P_a$,\nand the lower part is filled with liquid with pressure $P_l$.\nThe pores in the porous disk are filled with liquid,\nand the menisci at the top surface of the pores in the porous disk\n(a magnified view is shown in inset (a))\nmake it possible to keep $P_l<P_a$.\nA wet granular assembly is placed on the porous disk \n(a magnified view is shown in inset (b)),\nand liquid in the wet granular media\nis in good contact with the liquid in the porous disk \n(a magnified view is shown in inset (c));\nthe liquid pressure around the wet grains\nequals the pressure $P_l$\nof the bulk liquid in the bottom of the container.\nAdapted from\n\\cite{UnsaturatedBook}. \n}\n\\label{Fig:UnsaturatedBookFig5-19}\n\\end{figure}\n\nRelatively low suction can be measured or controlled\nby using tensiometers and the axis-translation technique\n\\cite{UnsaturatedBook}.\nA schematic diagram to describe these methods \nis given in Fig.~\\ref{Fig:UnsaturatedBookFig5-19};\nThese methods make use of the properties of \nwhat is called a ``high-air-entry (HAE) material'',\nwhich is a material with many microscopic pores,\nsuch as a porous ceramic. \nIn Fig.~\\ref{Fig:UnsaturatedBookFig5-19},\na container is separated in two parts by a HAE disk;\nthe bottom part of the box is filled with a liquid \nwith pressure $P_l$, while\nthe upper part is filled with air with pressure $P_a$.\nA magnified view of the HAE disk is given in the inset (a),\nwhere the pores of the HAE material are saturated with liquid.\nThere, the surface tension at the liquid-air interfaces\nformed in the pores allow a finite pressure difference between \nthe liquid and air.\nConsidering Eq.~(\\ref{eq:DeltaP})\nand the magnified view (a) in Fig.~\\ref{Fig:UnsaturatedBookFig5-19},\nthe largest possible pressure difference \nbetween the air and liquid in the HAE material is roughly given by \n\\begin{equation}\n\\Delta P_{\\max}=\\frac{2\\gamma}{R_{\\rm max}}.\n\\label{Eq:deltapmax}\n\\end{equation}\nHere, as a first approximation, \nwe assume that the pores are tube-shaped with \nmaximum radius  $R_{\\rm max}$;\nfor smaller pore size, $\\Delta P_{\\max}$ would be larger.\nThe liquid pressure $P_l$ in the lower part of the container\nand the air pressure $P_a$ in the upper part \ncan be separately controlled \nwithout allowing the air to enter the lower part of the container,\nas far as $\\ (P_a-P_l)<\\Delta P_{\\max}\\ $ is satisfied.\n\nNow, let us place a wet granular material on the HAE disk\ninside the container (Fig.~\\ref{Fig:UnsaturatedBookFig5-19}, inset (b)) \n\\cite{UnsaturatedBook}.\nWhen the equilibrium is reached after\na long enough waiting time, the \nliquid pressure in the wet granular medium should be equal to\nthe liquid pressure $P_l$ in the lower part of the container,\nif the liquid phase in the granular medium is in good contact with\nthe liquid in the pores of the HAE disk\n(Fig.~\\ref{Fig:UnsaturatedBookFig5-19}, inset (c)).\nThe air pressure $P_a$ and the liquid pressure $P_l$\nin the upper and lower parts of the container, respectively,\ncan be easily measured, or even controlled,\nand the suction $\\Delta P$ in the wet granular material \nis given by $\\Delta P=P_a-P_l$\n\\cite{UnsaturatedBook}.\n\nThe method to control the suction $\\Delta P$\nby controlling the liquid and air pressures separately \nis called the axis-translation technique \\cite{UnsaturatedBook}.\nInitially a saturated granular material is placed on \na HAE disk that separates the liquid and the air, as \nschematically shown in \nFig.~\\ref{Fig:UnsaturatedBookFig5-19},\nand the liquid in the granular assembly is drained\nuntil the system reaches the equilibrium, where \nthe liquid pressure in the granular medium becomes equal to \nthat of the bulk liquid in the lower box.\nThe tool to measure the suction by using the HAE disk\nis called a tensiometer, which is \ncomposed of a cup made of a HAE ceramic and\na sensor to measure the liquid pressure, \nconnected by a tube filled with liquid (water) \\cite{UnsaturatedBook}.\nThe HAE ceramic cup \nis placed in the wet granular assembly,\nand the pore pressure is measured by a direct exchange of liquid between \nthe sensor and the wet granular medium.\nNote that, in using either \nthe axis-translation technique or\na tensiometer,\nthe maximum possible value of the suction is limited by $\\Delta P_{\\max}$;\nother methods used to measure larger pressure differences\nare listed in Table \\ref{Table:UnsaturatedBookTable10-1} \nand Ref. \\cite{UnsaturatedBook}.\n\n\\paragraph{Relation between the liquid content and suction}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig5.eps}\n\\end{center}\n\\caption{\nA schematic plot of \nthe suction $\\Delta P$ versus the liquid content $S$. \nThe suction is shown in a log-scale.\nThe upper curve represents the drying process,\nand the lower curve represents the wetting process.\nIt changes the slope significantly near the phase boundaries between \nthe pendular, funicular, and capillary states.\n}\n\\label{Fig:SoilWater}\n\\end{figure}\n\nThe schematic description of the suction $\\Delta P$\nversus the liquid content $S$ \nis shown in Fig.~\\ref{Fig:SoilWater}\n\\cite{UnsaturatedBook,Newitt58}.\nTwo lines are shown because there is a hysteresis in the wetting\nand drying processes.\nIn both lines, the slope\nfor low and large liquid content is much larger \nthan that for intermediate liquid content.\nThis change of slope reflects the\nliquid distribution in granular media.\n\nFor simplicity, let us consider the drying process\n(the drainage of liquid)\nfrom completely saturated granular media. \nAt first the material is in the slurry state, \nand the liquid pressure \nis the same or larger than the air pressure.\nAfter an appropriate amount of liquid is drained,\nthe media is in the capillary state, and\nthe liquid pressure becomes lower than the air pressure,\ni.e., the suction $\\Delta P$ becomes positive.\nWhen we further decrease the liquid content $S$ \nin the capillary state, \n$\\Delta P$ increases very rapidly upon decreasing $S$; \n$\\Delta P$ is determined by the curvature\nof the liquid surface,\nand the change of the surface curvature \nrequires little change of liquid content $S$ because\nbulk liquid remains in the system.\nWhen $\\Delta P$ exceeds a critical value,\nwhich is mainly determined by the surface tension and\nthe geometry of granular media around the surface liquid,\nair starts to enter the bulk of the system and the funicular state\nis obtained.\nThe suction still increases as the liquid content decreases,\nbut its rate of increase becomes much smaller than that in \nthe capillary state,\nbecause the liquid in the pores in the bulk needs to be removed.\nAt some point, most of the voids will be\nfilled with air, and the rest of the liquid is\nfound around the contact points between neighbouring grains \nand in the thin liquid film that covers the grains' surfaces.\nNamely, the system is in the pendular state.\nIn this pendular state, a small change of the liquid \ncontent again results in a relatively large change of \nthe suction, because the total amount of the liquid \nis much smaller than the void volume $V_v$ and\nthe change of curvature at each liquid bridge requires \nonly a small change of the liquid content $S$.\n\nAs shown in Fig.~\\ref{Fig:SoilWater},\nfor a given value of the suction $\\Delta P$,\nthe liquid content $S$ is smaller for the wetting process\nof initially dry granular media than that of the drying process\nof initially saturated grains.\nThis is partly because of the hysteresis in\nthe wetting and drying processes at solid surfaces\n\\cite{deGennesWet,Herminghaus05,QuereBook},\nand partly because the variation of the pore size\nin the granular medium\n\\cite{UnsaturatedBook}.\nThe flow of liquid in porous media\n(including granular media)\nis an important and interesting problem;\nindeed, this problem has developed into \na separate research area.\nFor a recent review on this problem, in terms of statistical physics, \nsee \n\\cite{ImbibitionReview}.\n\n\\section{Mechanical properties}\nIn this section, we summarise what is known about the \nmechanical properties of wet granular media.\nAs mentioned above, here we focus on granular cohesion\nin the static and quasistatic regimes\nfor varying liquid content.\nTo also have some idea about the dynamics,\ndynamical experiments are briefly presented \nin the final part of this section.\n\\subsection{\nGranular cohesion in the static and quasistatic regimes}\n\\subsubsection{Compaction of wet granular media}\n\\label{compaction}\nThe configuration of grains in a \ncontainer shows history-dependence. \nIn the case of dry granular media, \nthe material just poured in a container\nis compacted when it is tapped.\nThe dynamics of this compaction process is slow,\nwhich is known to show logarithmic dependence on the number \nof taps (e.g., \n\\cite{Knight95,Nowak98,Nicodemi97}).\n\nIn the case of wet granular media, \nvery low-density configurations can be realised\neven when the grains are spherical.\nDue to the cohesion, a configuration with many large openings \nor gaps surrounded by chains of particles can be stabilised,\nwhich may not be stable for dry spherical grains \n\\cite{Xu04}.\nThe compaction via a ram\nof this low-density configuration\nby a quasistatic compression has been examined\n\\cite{Gioia02,Kong00}.\nWith increasing pressure on the system,\nthe relatively large openings between particles\ndisappear;\na grain in a chain snaps in away from the chain, \nand the chain structure changes.\nAfter the openings disappear, a high-density region is\nformed, and the density difference between \nthe initial low-density and \nnewly-formed high-density regions becomes clearly visible.\nOne can see that the domain wall \nbetween the high-density and low-density regions\ntravels through the system like a wave.\n\nThe realisability of very low-density structures\nindicates that the hysteresis of configurations\nin wet granular media can be stronger \nthan that in dry granular media.\nIn other words, the many \nhuge gaps or openings \nthat are possible in wet granular media, \nmake the configuration phase space available to the grains\nmuch larger than the dry granular case \n(see also Table \\ref{Table:DryWet}).\n\n\\subsubsection{Angle of repose for small amounts of liquid}\n\\label{AngleofRepose}\n\\paragraph{Experiments}\nThe angle of repose $\\theta_r$ and \nthe critical angle $\\theta_c$ are measurable quantities \nthat are sensitive to the cohesion.\nHere, the critical angle $\\theta_c$ is the surface angle just before an \navalanche occurs, while the angle of repose $\\theta_r$ is the surface angle \nafter the avalanche, and is typically slightly smaller than $\\theta_c$.\nWhen the amount of liquid is small,\nthe cohesion is also small, but the cohesion is still \ndetectable through $\\theta_r$ and $\\theta_c$\nof either a granular pile or\nin a drum that rotates very slowly\n\\cite{Bocquet98,Hornbaker97,Albert97,Fraysse99,Halsey98,Nase01}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig6.eps}\n\\end{center}\n\\caption{\nThe critical angle $\\theta_c$ \nof glass beads in humid air\nas a function of the waiting time $t_w$\nin a slowly rotating drum,\nshown as $\\tan\\theta_c$ versus $(\\log_{10}t_w\/\\cos\\theta_c)$.\nThe time is in units of seconds,\nand humidities are  15\\% (triangles),\n27\\% (pentagons), 36.1\\% (squares), and 45.5\\% (circles).\nAs time passes,\nmore and more liquid condenses \nat the contact points between particles,\nincreasing the cohesion between them.\nTherefore, the critical angle\nincreases with time.\nFrom\n\\cite{Bocquet98} .\n}\n\\label{Fig:Bocquet98Fig2a}\n\\end{figure}\n\nEven the humidity in the air is enough to\ninduce cohesion for initially dry granular material\n\\cite{Bocquet98,Restagno02,Fraysse99,Ovarlez01,DAnna00,\nDanna00-2,Forsyth02,Crassous99}.\nBocquet {\\it et al.} \\cite{Bocquet98}\nmeasured the critical angle $\\theta_c$\nof glass beads in humid air, and found that\n$\\theta_c$ depends on the waiting time $t_w$,\nnamely, how long they waited after preparing the sample\nuntil measurement. \nThey reported the relation \n$(\\tan\\theta_c -\\tan\\theta_0)\\propto \\log(t_w)\/\\cos\\theta_c$,\nwhere $\\theta_0$ is a constant\n(Fig.~\\ref{Fig:Bocquet98Fig2a}),\nand also found that the $t_w$-dependence \nof $\\theta_c$ varies with humidity.\n\nThis time dependence \nof the critical angle $\\theta_c$ on the waiting time $t_w$\nhas been analysed based on\nthe capillary condensation, \nwhich results from the fact that\nthe equilibrium vapour pressure\nin a narrow space where\nnon-zero suction $\\Delta P$ is allowed\nis lower than that in the bulk\n\\cite{Israelachvili}.\n\\footnote{ \n``Capillary condensation''\ncan be understood as follows.\nConsider air and liquid in a box,\nwhere the liquid pressure $p_l$,\nthe total air pressure is $p_a$,\nthe liquid vapour pressure $p_v$,\nand dry air pressure $p_{da}=p_a-p_v$.\nFrom the Gibbs-Duhem equilibrium criterion and \nideal gas approximation for the air, \nthe Kelvin's equation \\cite{Israelachvili}\nis obtained:\n$p_v=p_{v0}\\exp\\left[-(\\Delta P v_l)\/(R_gT)\\right]$,\nwhere $v_l$ is the partial molar volume of liquid,\n$p_{v0}$ is the saturated vapour pressure,\n$T$ is the absolute temperature, and $R_g$ is the gas constant.\nThus, when $\\Delta P>0$, $p_v$ becomes smaller than $p_{v0}$.\nNamely, in narrow spaces like pores in granular media,\nthe vapour can condense for lower vapour pressures.\n}\nDue to this effect,\nthe vapour can condense for lower pressures\nin a narrow space, like the point where particles are in contact,\nresulting in tiny liquid bridges at the contact points\nbetween touching grains.\nThe critical angle $\\theta_c$ increases due to \nthe cohesion from these tiny bridges,\nwhich becomes larger and increases with time\nbecause more and more vapour condenses.\n\\footnote{ \nIt should be noted, however, that\nthere are alternative explanations about the origin of \ncohesion in this experiment.\nRestagno {\\it et al.} \\cite{Restagno02}\nconducted similar experiments for glass beads with water\nand glass beads with ethanol, and found that\nthe difference in the increase of the critical angle \nis larger than the value expected from \nthe difference of the surface tension \nbetween water and ethanol.\nThis might be partly due to the chemical reaction\nbetween silica and water \n\\cite{Restagno02,Olivi-Tran02}.\n}\n\nHornbaker {\\it et al.} \\cite{Hornbaker97} \ninvestigated\nthe angle of repose $\\theta_r$ of \nspherical polystyrene beads \nmixed with either corn oil or vacuum-pump oil\nby the draining-crater method.\nThey measured the liquid content dependence of \nthe angle of repose $\\theta_r$.\nThey found that $\\theta_r$ increases\nlinearly with the average oil-layer thickness $\\delta_{\\rm liq}$\n(the volume of oil added divided by the total surface area of particles),\nindicating that the cohesion increases as the amount of liquid increases.\nHowever, as they increased the liquid content,\nparticles started to form correlated particle clusters (clumps),\nand finally $\\theta_r$ could not be determined in a well-defined manner.\nThe distribution width of the measured angle of repose \nbecame suddenly wide for a certain amount of liquid, \nsuggesting a transition to \na situation where long-range correlations dominate.\n\n\\paragraph{Cohesion and angle of repose}\n\\label{AngleofReposeThe}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig7.eps}\n\\end{center}\n\\caption{\nSchematic diagram of the force \nbalance at the failure plane,\nas determined by\nthe Mohr-Coulomb failure criterion.\n}\n\\label{Fig:Coulomb}\n\\end{figure}\nThe observed increase of $\\theta_r$ and $\\theta_c$\nwith increasing liquid content\nshould be due to the \ncohesive force between grains.\nThe relation between the surface angle and cohesion, \nhowever, has not yet\nbeen clearly understood.\nThis is an area that would greatly benefit from \nmore systematic studies.\n\nThe most well-known analysis of the critical angle $\\theta_c$\nin granular engineering is \nto use the failure condition\nfor a continuum description \\cite{Nedderman,Halsey98}.\nIn the case without cohesion, a phenomenological failure criterion\nfor granular media is given \nin terms of the shear stress $\\tau$ and the normal compressible \nstress $\\sigma$ as\n\\begin{equation}\n\\tau>\\mu \\sigma,\n\\label{Eq:CoulombCriterionNo}\n\\end{equation}\nwhere $\\mu$ is the internal friction coefficient,\nconsidered as a material parameter.\nIf there is any plane for which the ratio $\\tau\/\\sigma$\nexceeds $\\mu$, the material fails at that plane;\n$\\tau_f\/\\sigma_f=\\mu$,\nwhere the subscript $f$ for the stresses \n$\\tau_f$ and $\\sigma_f$ indicates the value at the failure plane.\nThe critical angle corresponds to the angle\nof this failure plane, where the failure starts.\nAt the same time, because the stress comes from \nthe weight above the plane, as\nschematically shown in Fig.~\\ref{Fig:Coulomb},\nwe have \n\\begin{equation}\n\\tau_f=\\rho g D \\sin\\theta_c \\qquad \\mbox{and} \\qquad\n\\sigma_f=\\rho g D \\cos\\theta_c,\n\\label{eq:taufsigmaf}\n\\end{equation}\nwith mass density $\\rho$\nand the depth $D$ of the failure plane from the \nsurface before failure.\nNamely, we have \n\\begin{equation}\n\\frac{\\tau_f}{\\sigma_f}=\\tan\\theta_c=\\mu,\n\\label{Eq:NonCohesiveTanc}\n\\end{equation}  \nwhich gives $\\theta_c$ for \ndry granular media.\n\nIn the case of wet granular media, \nit has been considered that the cohesion  \nresult in a normal cohesive \npressure $\\sigma_c$ \\cite{Nedderman,Halsey98}.\nThis additional normal stress $\\sigma_c$ \nallows to support a finite shear stress, even\nin the limit of zero applied normal stress $\\sigma$.\nThen, the criterion in Eq.~(\\ref{Eq:CoulombCriterionNo}), \nfor non-cohesive material, is modified into \\cite{Nedderman,Halsey98}\n\\begin{equation}\n\\tau > \\mu (\\sigma+\\sigma_c).\n\\label{Eq:CoulombCriterion}\n\\end{equation}\nThe criterion in Eq.~(\\ref{Eq:CoulombCriterion})\nis called the Mohr-Coulomb criterion,\nand Eq.~(\\ref{Eq:CoulombCriterionNo}) is \nthe special case of Eq.~(\\ref{Eq:CoulombCriterion}) with $\\sigma_c=0$.\nWith non-zero $\\sigma_c$, \nthe stresses at the failure plane satisfies \n$\\tau_{f} = \\mu (\\sigma_f+\\sigma_c)$,\nwhile we still have Eq.~(\\ref{eq:taufsigmaf}) for the force balance.\nThen we have\n\\begin{equation}\n\\mu=\\tan\\theta_c\\left(1+\\frac{\\sigma_c}{\\rho g D\n\\cos\\theta_c}\\right)^{-1},\n\\label{Eq:CohesiveAngle}\n\\end{equation}\nwhich shows that \n$\\theta_c$ decreases with increasing $D$.\nNamely, the criterion is \nthe strictest at the bottom of the \nsandpile, which has the largest $D$.\nTherefore, the failure occurs at the bottom\nof the sandpile, and Eq.~(\\ref{Eq:CohesiveAngle}) with\n$D$ being the total depth of the sandpile\ngives the critical angle $\\theta_c$ of that sandpile.\nNote that the relation for the non-cohesive case,\nEq.~(\\ref{Eq:NonCohesiveTanc}), \nis recovered in the limit of $D\\to \\infty$;\nthe frictional force that holds the pile increases with\nthe size of the pile because the normal pressure increases,\nbut the cohesive force remains constant, \nthus the cohesive force becomes irrelevant\nin a large enough granular pile \\cite{Nedderman,Halsey98}.\nWe have increasing critical angle \ndue to cohesion only for a finite sandpile\nwhere $\\sigma_c\/\\rho g D$ is finite,\nand within this range $\\theta_c$ \nincreases with the cohesive stress $\\sigma_c$\nand depends on the pile size.\nNote that, in this Mohr-Coulomb criterion,\nthe yield shear stress is the\nconsequence of the increased normal pressure,\nand the direct contribution of particle-particle cohesion to\nthe shear stress is not taken into account.\n\nIn the case of the pendular state,\nthe cohesive stress $\\sigma_c$\narises from the liquid bridges between particles.\nRumpf \\cite{Rumpf62} proposed a simple model \nto estimate the cohesive stress \nfrom the force per bridge\nin isotropic granular media of identical spheres with diameter $d$,\nignoring the effect of distributions in liquid bridge sizes \nand in number of bridges per particles \\cite{Fournier05}.\nHe estimated the cohesive force $\\sigma_c$ per unit area as \n\\begin{equation}\n\\sigma_c\\approx \\nu \\frac{k}{\\pi d^2}F,\n\\label{Eq:RumpfModel0}\n\\end{equation}\nwhere $\\nu$ is the packing fraction of the grains,\n$k$ is the average number of liquid bridges per particle,\nand $F$ is the average force per liquid bridge.\n\\footnote{\nIn papers on granular engineering\nincluding \\cite{Rumpf62}, the porosity $n$ \ndefined as the ratio of the void volume $V_v$ \nto the total volume $V_t$ is often\nused instead of the packing fraction $\\nu$\nas a parameter to characterise the packing.\nThese parameters are related by\n$n=1-\\nu$.\n}\n\nAccording to the Rumpf model in Eq.~(\\ref{Eq:RumpfModel0}),\n$\\sigma_c$ is proportional to\nthe cohesion per liquid bridge $F$.\nThen, the fact that $\\theta_r$ \nfor wet spherical glass beads increases with liquid content\n\\cite{Hornbaker97} may suggest increasing $F$ by\nincreasing the liquid bridge volume,\nif the number of bridge per particle $k$ stays constant.\nHowever, this cannot be understood by \nsimply considering \nthe pendular state for completely spherical beads.\nFor the case when two spheres are in contact ($h=0$) and the wetting \nangle $\\theta=0$, we get from Eq.~(\\ref{eq:bridge0}) that\n\\begin{equation}\nF_{\\rm bridge}=\\frac{2\\pi r\\gamma}{1+\\tan(\\phi\/2)},\n\\label{eq:liquidbridge}\n\\end{equation}\nwhere $r$ is the radius of the spheres\nand $\\phi$ is given in Fig.~\\ref{Fig:LiquidBridge}.\nEquation (\\ref{eq:liquidbridge})\ngives $\\partial F_{\\rm bridge}\/\\partial \\phi<0$.\nNamely, the attracting force $F_{\\rm bridge}$\nbecomes smaller as the amount of liquid in the bridge increases \nwhile keeping the particle separation equal to zero.\n\nThis discrepancy can be resolved by considering \nthe surface roughness of the grains\n\\cite{Albert97,Halsey98}.\nIt has been known that,\nin the case of the cone-plane contact, \nthe attracting force increases when the amount of liquid increases\n\\cite{Cahn70}. \nIf the liquid bridge volume is small enough\nto neglect the curvature of the \nmacroscopic spherical shape of the beads,\nthe surface roughness becomes more relevant, \nand the geometry at the contact point\ncan be considered as the cone-plane type.\nIf the liquid bridge becomes wide enough\ncompared to the macroscopic curvature of the\nsurface, then the estimation for \nspheres in Eq.~(\\ref{eq:liquidbridge}) is \nexpected to work.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig8.eps}\n\\end{center}\n\\caption\nDistribution of small amount of liquid in granular media.\nThe blue region shows the location of the liquid.\nThe average grain diameter is 240 $\\mu$m.\n(a) \nWhen the amount is liquid is small, the liquid\nis uniformly distributed at the grain surface.\n(b) \nWhen the amount of liquid is large enough,\nmost of the liquid is caught at the contact points\nbetween particles and liquid bridges are formed.\nFrom\n\\cite{Mason99}.\n}\n\\label{MasonFig}\n\\end{figure}\n\nMason {\\it et al.} \\cite{Mason99} experimentally observed\nhow the liquid distributes in spherical beads \nwith microscopic surface roughness.\nThey found that, when the amount of liquid is \nsmaller than a critical value,\nmost of the liquid is trapped on the particle surface due \nto the surface roughness \n(Fig.\\ref{MasonFig}(a)) \nand the critical angle did not depend on the surface tension of the\nliquid.\nWhen the liquid content exceeds a critical value,\nliquid bridges are formed at the contact points \n(Fig.\\ref{MasonFig}(b)), and the critical angle increases \nwith liquid volume and depends on the liquid surface tension.\nThis agrees with the behaviour expected from \nthe regime where the surface roughness \ndetermines the cohesion from a liquid bridge.\n\nHowever, one should note that,\nnot only the cohesion force $F$ but also\nthe number of liquid bridges $k$ per particles \ndepends on the liquid content in general, \nwhich should also affect the cohesive stress $\\sigma_c$.\nRecent experiments \\cite{Kohonen04,Fournier05} show that \n$k$ increase very rapidly upon liquid content \nfor small liquid content, and then saturate. \nThe effect of $k$ on the critical angle needs to be investigated\ncarefully especially for small liquid content.\n\nAlbert {\\it et al.} \n\\cite{Albert97} proposed a criterion to understand the critical\nangle different from the Mohr-Coulomb criterion. \nThey considered the criterion whether\na grain at the surface escapes from\nthe bumpy geometry that other grains form.\nIn their theory, the surface grain \nat an unstable position rolls down\nand the final angle is determined by \nthe condition that all grains \nat the surface sit at stable positions. \nThe cohesion from liquid \naffects the force balance and \nincreases the critical angle $\\theta_c$.\n\nIn either mechanism,\nthe Mohr-Coulomb criterion or the surface \nfailure criterion, \nthe angle of repose of cohesive granular media is larger than\nthat of non-cohesive granular media. \nHowever, the location of the failure plane is different.\nThe failure occurs at the bottom \nin the case of the Mohr-Coulomb criterion\nand $\\theta_c$ depends on the system size,\nwhile the surface fails for the ``grain escape'' criterion\nand $\\theta_c$ is independent of the system size.\nIn addition, the Mohr-Coulomb criterion is \na continuous description, \nwhile the grain escape criterion explicitly considers \nthe discreteness of the system. \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth\n{Fig9.eps}\n\\end{center}\n\\caption{\nAngle of repose $\\theta_r$\nversus the average oil-layer thickness $\\delta_{\\rm liq}$\nfor spherical glass beads with average diameter 0.9mm\nwith three different container diameters $d_c$\n(diamonds: 10.3cm, \ntriangles: 15.6cm, \nsquares: 20.4cm).\nThe inset shows an enlargement of small\n$\\delta_{\\rm liq}$ regime.\nThe two vertical arrows indicate the \ntransitions between the granular, \ncorrelated, and plastic regime.\nFrom \\cite{Tegzes99}. \n}\n\\label{Fig:Tegzes99Fig2}\n\\end{figure}\n\nBoth mechanisms may be responsible in determining\nthe critical angle. However, it is likely that\nthe discrete picture is dominant for\nvery small amounts of liquid content,\nwith small cohesive bulk stress,\nwhile the continuous picture is appropriate when\nthe liquid content is enough to cause the cohesive bulk stress\ncomparable to gravity.\nIn order to see the relevant mechanism,\nTegzes {\\it et al.} \\cite{Tegzes99} examined \nthe system size dependence of the angle of repose $\\theta_r$\nfor a larger range of liquid content (Fig.~\\ref{Fig:Tegzes99Fig2}). \nThey adopted the draining crater method,\nand the system size is controlled\nby changing cylinder diameters.\nThey found that $\\theta_r$ is independent of system size\nfor small enough liquid content \n(called the {\\it granular regime}, where \nsurface flow occurs at the top few layers),\nwhile it decreases with system size for\nlarger liquid content\n(called the {\\it correlated regime}, where \ngrains form clumps and the surface flow is correlated).\nAt the largest liquid content, the angle \nfirst decreases and then increases slightly with \noil-layer thickness, but still depends on the system size\n(called the {\\it plastic regime}, where\nthe medium retains a smooth crater surface \nand the motion is coherent).\nQualitative agreements have been obtained \nwith the surface failure criterion \nfor the granular regime\nand with the Mohr-Coulomb criterion for the correlated and plastic\nregime.\n\nThese results are not yet conclusive,\nand many researches on the surface angle are still going on.\nErtas {\\it et al.} \\cite{Ertas02} considered the critical \nangle in a pile made of a\nmixture of spheres and dumbbell-shape grains,\nextending the surface failure criterion by \\cite{Albert97}.\nThey found that the surface angle drastically increases\nas the fraction of dumbbell-shape grains increases,\nbecause the dumbbell-shape grains hardly roll down the slope.\nThey pointed out that, in wet granular media\nwith small liquid content,\nliquid bridges connect a few grains and make clumps;\nthese clumps may contribute to the increasing surface angle\nthrough the same mechanism that the fraction of \ndumbbell-shaped grains increased the critical angle.\nOn the other hand, Nowak {\\it et al.} \\cite{Nowak05} \napplied the geometrical consideration of force balance\nby \\cite{Albert97}\nnot only at the surface but also inside the material.\nThey claimed that the most unstable plane from \nthe geometrical consideration is inside the material.\nTheir argument does not take into account the effect of friction,\nbut it gives system-size dependent critical angle\nthat agree with the critical angle of dry grains in \nthe limit of infinite system size;\nthis feature is similar to \nthat in the Mohr-Coulomb criterion.\n\n\nIn summary, the effect of cohesion on surface angle\nin granular pile has not yet been clearly understood,\nand further systematic studies are needed.\n\n\\subsubsection{Tensile, compression, and shear tests\nfor intermediate and large liquid content}\n\\label{tests}\nFor large enough cohesion,\nthe angle of repose may exceed $90^\\circ$\nfor laboratory-scale granular piles. In such a regime,\nthe angle of repose is no longer a \ngood parameter to characterise the cohesion,\nand we need another method to characterise the cohesion \nin wet granular media.\n\nIf the failure condition is characterised by\nthe Mohr-Coulomb criterion (\\ref{Eq:CoulombCriterion}),\nthe measurement of the stresses at the \nfailure gives the cohesive stress $\\sigma_c$,\nwhich characterises the cohesion in wet granular media.\nThe analysis of granular media based on the\nMohr-Coulomb criterion has been well-established \nin the engineering field, \nand a considerable amount of data has been accumulated \nfor wet granular media.\nIn this subsection, we briefly describe several \nexperimental methods to \ndetermine the internal friction coefficient\n$\\mu$ and the cohesive stress $\\sigma_c$, \nand summarise what has been known for\nwet granular media from those measurements.\n\n\\paragraph{Test methods and Mohr circle}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Fig10.eps}\n\\end{center}\n\\caption{\nSeveral measurement techniques of strength of granular media.\nThe adhesive method (a),\nthe split-plate method (traction table method) (b),\nthe uniaxial compression test (c),\nand the diametrical compression test (d).\n}\n\\label{Fig:Schubert75-2Fig4}\n\\end{figure}\nSeveral experimental methods have been used to measure \nthe tensile strength of wet granular media\n(e.g., \\cite{Bika01,Schubert75-2}).\nThe simplest method is just to pull the sample apart until the\nmaterial fails, as shown \nin Figs.~\\ref{Fig:Schubert75-2Fig4}(a) and (b).\nHowever, it is difficult to hold the sample\nbecause wet granular media are fragile and not always uniform.\nIn the adhesive method (Fig.~\\ref{Fig:Schubert75-2Fig4}(a)), \nthe material needs to be strong enough so\nthat the bottom and top faces of the sample\ncan be bonded to the adaptors.\nIn the case of the split-plate method\n(Fig.~\\ref{Fig:Schubert75-2Fig4}(b)), \nthe sample is placed on a plate,\nand the plate is split until the material fails.\nIn either method, \nboth the stress and the strain can be measured until reaching failure.\n\nThe uniaxial and diametrical compression tests have also been \nconducted, where the sample is compressed\nin one direction until it fails.\nIn uniaxial compression tests\n(Fig.~\\ref{Fig:Schubert75-2Fig4}(c)),\na sample is pressed by a ram along one direction.\nIn the case of the diametrical compression test,\na linear load is applied to a sample of cylindrical shape\nacross one diameter of the disk \n(Fig.~\\ref{Fig:Schubert75-2Fig4}(d)),\nand the tensile stress is calculated \nbased on the Hertz theory \nfor isotropic elastic bodies \\cite{Procopio03,Hertz},\nthough the validity of the\nelastic theory for granular media \nwith the assumption of the isotropy\nis rather suspicious.\n\nIn addition to these uniaxial tests, where \nonly one component of the stress applied to the sample is controlled,\nthere are methods where two components \nof the stress are separately controlled. \nThe triaxial compression test\nhas been widely performed for soils \\cite{SoilBook,UnsaturatedBook,\nSoilJapaneseBook}.\nFigure \\ref{Fig:SoilTests}(a) shows a \nschematic description of\nthe triaxial testing system.\nThe cylindrical-shaped sample is placed in a confining cell\nfilled with fluid, and the sample is separated \nfrom the confining fluid by a flexible membrane. \nThe sample is compressed from above by a ram, \nand the pressure at side membrane can be \nseparately controlled\nby changing the pressure of the confining fluid.\nIn the case of the direct shear test\nin Fig.~\\ref{Fig:SoilTests}(b),\na sample in a shear box is sheared as the shear box is slid.\nA series of tests can be \nconducted by varying the vertical compressive stress \nfrom the top.\nNote that there are several modifications of these tests:\nIn the case of unsaturated soils, \nthe modified tests are often performed,\nwhere the suction is kept constant during the test\nby using the HAE material (see subsection \\ref{suctionmeasure})\nand allows the liquid to drain from the sample \\cite{UnsaturatedBook}. \n\n\nIt should be noted that the sample needs to be\nprepared carefully in either method\nin order to get reproducible results. \nEspecially, how the sample is compacted \nbefore the measurement is an important factor to \ndetermine its mechanical response.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=.9\\textwidth]{Fig11.eps}\n\\end{center}\n\\caption{Schematic diagrams of \na triaxial test system (a)\nand a direct shear test system (b). \n}\n\\label{Fig:SoilTests}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.9\\textwidth]{Fig12.eps}\n\\end{center}\n\\caption{\nA schematic diagram of \nMohr circles in the plane \nshear stress $\\tau$ versus normal compressive stress $\\sigma$. \nWhen a two-dimensional stress tensor \n(whose $(i,j)$ component is given by $\\sigma_{ij}$)\nis considered, the Mohr circle includes the diametrically-opposed points \n$A (\\sigma_{11}, \\sigma_{12})$\nand $B (\\sigma_{22}, -\\sigma_{12})$,\nas shown in the figure.\nThe yield locus from the Mohr-Coulomb criterion \nfor cohesive granular media is shown by the solid straight line,\nand the yield locus of the non-cohesive granular media \nis shown by the dashed line.\n}\n\\label{Fig:MohrCircle}\n\\end{figure}\nThe results of the tensile, compression, and shear tests are often analysed \nby using the Mohr circle, which\nvisualises the rotational transformation of \nthe two-dimensional stress tensor as follows\n\\cite{SoilBook,UnsaturatedBook,SoilJapaneseBook}. \nLet us consider a two-dimensional stress tensor\nwhose principal values are $\\sigma_1$ and $\\sigma_2$.\nIf the coordinate axes are rotated by an angle $\\alpha$ from the\nprincipal axes, the stress tensor $\\sigma_{ij}$ in \nthe coordinate system is given by\n\\begin{eqnarray}\n\\left(\n\\begin{array}{cc}\n\\sigma_{11} & \\sigma_{12}\\\\\n\\sigma_{21}&\\sigma_{21}\n\\end{array}\n\\right)\n&=&\n\\left(\n\\begin{array}{cc}\n\\cos\\alpha & \\sin\\alpha \\\\\n-\\sin \\alpha &\\cos \\alpha \n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\n\\sigma_{1} & 0\\\\\n0 &\\sigma_{2}\n\\end{array}\n\\right)\n\\left(\n\\begin{array}{cc}\n\\cos\\alpha & -\\sin\\alpha \\\\\n\\sin \\alpha &\\cos \\alpha \n\\end{array}\n\\right)\\nonumber \\\\\n&=&\n\\left(\n\\begin{array}{cc}\n\\frac{(\\sigma_1+\\sigma_2)}{2}+\\frac{(\\sigma_1-\\sigma_2)}{2}\\cos\n2\\alpha&\n\\frac{(\\sigma_1-\\sigma_2)}{2}\\sin2\\alpha\\\\\n\\frac{(\\sigma_1-\\sigma_2)}{2}\\sin2\\alpha&\n\\frac{(\\sigma_1+\\sigma_2)}{2}-\\frac{(\\sigma_1-\\sigma_2)}{2}\n\\cos 2\\alpha\n\\end{array}\n\\right)\n\\end{eqnarray}\nNamely, when we consider two points $A (\\sigma_{11}, \\sigma_{12})$\nand $B (\\sigma_{22}, -\\sigma_{21})$ \nin the $\\tau-\\sigma$ plane,\nthey locate at the diametrically-opposed points \non a circle\n\\begin{equation}\n\\left[\\sigma-\\frac{1}{2}(\\sigma_1+\\sigma_2)\\right]^2+\\tau^2\n=\\left[\\frac{1}{2}(\\sigma_1-\\sigma_2)\\right]^2,\n\\end{equation}\nas shown in Fig.~\\ref{Fig:MohrCircle},\nwhere the angle between the $\\sigma$ axis and\nthe line $A-B$ is $2\\alpha$.\n\nThe Mohr circle is useful when considering the\nMohr-Coulomb criterion Eq.~(\\ref{Eq:CoulombCriterion}).\nThe stress at failure, from the Mohr-Coulomb\ncriterion, gives a straight line \n\\begin{equation}\n\\tau=\\mu(\\sigma+\\sigma_c)\n\\label{tangentline} \n\\end{equation}\nin the $\\tau$-$\\sigma$ plane\nwith $\\sigma$-intercept at $-\\sigma_c$\nand slope $\\mu$.\nInstead of $\\mu$, \n$\\Phi=\\tan^{-1}\\mu$ (shown in Fig.~\\ref{Fig:MohrCircle})\nis also used as a parameter for the\nMohr-Coulomb criterion; \n$\\Phi$ is called the internal friction angle.\nIf the granular aggregate obeys the Mohr-Coulomb criterion,\nthe Mohr circle at failure should be\ntangent to this straight line (\\ref{tangentline}),\nbecause the granular aggregate fails as soon as \na stress state that satisfies the criterion appears.\nTherefore, by drawing an envelope curve \nof Mohr circles at failure\nwith various stress conditions,\nfrom the data one can construct the Mohr-Coulomb \ncriterion to determine \nthe parameters $\\mu$ and $\\sigma_c$.\n\nThis procedure can be done for the triaxial compression test \nand the shear test, where \ntwo components of the stress tensor are \nvaried separately.\nThe granular material does not always \nfollow the ideal Mohr-Coulomb criterion\nand the envelope might be curved, but\nthe response is linear\nfor small enough values of the stress,\nand this gives $\\mu$ and $\\sigma_c$.\n\nIn the case of uniaxial tensile (compression) tests, \none of the principal values is always zero.\nThus, the resulting Mohr circle crosses the origin \nof the $\\sigma$-$\\tau$ plane, \nwhere the tensile (compressive) stress\nat failure is given by $\\sigma_{\\rm tensile}$\n($\\sigma_{\\rm compress}$) as shown in Fig.~\\ref{Fig:MohrCircle}.\nFor a given internal friction coefficient \n$\\mu$, $\\sigma_{\\rm tensile}$ ($\\sigma_{\\rm compress}$) \nis proportional to $\\sigma_c$,\nand the behaviour of\n$\\sigma_{\\rm tensile}$ ($\\sigma_{\\rm compress}$) \ngives information about the cohesion in the wet granular medium.\nHowever, we cannot determine the cohesive stress $\\sigma_c$ from \na uniaxial test unless we know $\\mu$ from another experiment.\n\nBelow let us briefly review some experimental results\nabout the stresses at the failure state\nfor wet granular media with various liquid content.\n\n\\paragraph{Tests in the pendular state}\nPierrat {\\it et al.} \\cite{Pierrat98} investigated\nthe Mohr-Coulomb criterion by using direct shear tests\nfor various granular assemblies\nincluding monodisperse glass beads \nand polydisperse crushed limestone\nwith relatively small liquid content. \nThe amount of liquid was larger than that in\nthe experiments on the angle of repose presented \nin subsection \\ref{AngleofRepose}, \nbut the system was supposed to be still in the pendular state.\n\nThe yield loci in shear test for glass beads of\ndiameter 93 $\\mu$m with various liquid content\nare shown in Fig.~\\ref{Fig:Pierrat98Fig4},\nwhich shows a shift of the yield locus to the left \nfrom the dry case,\nbut the slope is unchanged.\nIt has also been found that the shift increases \nwith the liquid content.\nFor all the materials they investigated,\nthey found that the yield loci \ncan be collapsed by shifting the curve.\nNamely, in this experiment, the main effect of the\nliquid appears in the cohesive stress $\\sigma_c$,\nbut the internal friction coefficient $\\mu$ \nwas not significantly modified,\nwhich indicates that the effect of lubrication\nwas small.\n\nThey also investigated the relation \nbetween the shear test and the uniaxial tensile test.\nThey had conducted tensile tests separately\n\\cite{Pierrat97}, \nand from their empirical relation\nbetween $\\sigma_{\\rm tensile}$  and parameters that characterise the\nsample, such as packing fraction and liquid content,\nthey estimated the tensile strength $\\sigma_{\\rm tensile}$ \nfor the sample in the shear tests. \nIf the internal friction coefficient $\\mu=\\tan\\Phi$ for \ndry samples does not change for wet granular materials,\nwe can estimate $\\sigma_c$ from \n$\\sigma_{\\rm tensile}$, using the relation\n$\\sigma_c=(1+\\sin\\Phi)|\\sigma_{\\rm tensile}|\/(2\\sin\\Phi)$\n(see Fig.~\\ref{Fig:MohrCircle})\nwith $\\Phi$ being the internal friction angle in the dry case.\nThis estimation was compared \nwith observed yield loci in shear tests,\nand reasonable agreement was found.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90,width=0.6\\textwidth]{Fig13.ps}\n\\end{center}\n\\caption{\nYield locus obtained by\ndirect shear tests for glass beads (93 $\\mu$m):\nDry (circles), \nwith water (1.3\\% of liquid content is shown by squares,\nand 4\\% of liquid content is shown by triangles),\nand with n-hexadecan \n(liquid content 1\\%, shown by upside-down triangles).\nAdapted from\n\\cite{Pierrat98}.\n}\n\\label{Fig:Pierrat98Fig4}\n\\end{figure}\n\n\n\\paragraph{Tests with intermediate liquid content}\nNext, we review\nthe tensile tests and uniaxial compression tests\nof wet granular media with intermediate liquid content,\ntypically in the funicular state.\nThe mechanical response of the funicular state \nis not well understood,\nbecause in this state \neven the qualitative dependence \nof the tensile strength \n(tensile stress at the peak of stress-strain relation)\non liquid content\nvaries from one material to another,\nand the reason for this is not very clear. \nHere we just summarise a few experiments.\n\n\\label{granules}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig14.eps}\n\\end{center}\n\\caption{\nStress-strain relation of wet limestone (mean diameter 65 $\\mu$m)\nobtained by the adhesive method \\cite{Schubert75}.\nThe packing fraction $\\nu$ of the sample was $0.66$.\nThe dashed lines show the result\nfor a sample prepared by a drying process (drainage), \nwhile the solid lines show the result\nfor a sample prepared by a wetting process (imbibition).\n$S$ refers to the percentage liquid content.\nFrom\n\\cite{Schubert75}.\n}\n\\label{Fig:Schubert75Fig9}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig15.eps}\n\\end{center}\n\\caption{Stress-strain relation of\ndicalcium phosphate (diameter 21 $\\mu$m)\nwith an aqueous solution of a polymer.\n$S$ refers to the percentage liquid content.\nThe packing fraction of the sample was $0.5$.\nFrom\n\\cite{Holm85}.\n}\n\\label{Fig:Holm85Fig8.ps}\n\\end{figure}\n\nIn the field of agglomeration processing, experiments \nhave been conducted from relatively dry to\nalmost saturated conditions.\nHowever, in the funicular state,\nthe liquid content dependence of\nthe strength may either increase\nor decrease upon increasing the liquid content.\nFor example,\nincreasing tensile strength\nwith increasing liquid content has\nbeen found by \nSchubert \\cite{Schubert75} \nfor limestone with mean diameter 65$\\mu$m\nusing the adhesive method.\nFigure \\ref{Fig:Schubert75Fig9} shows\nthe stress-strain relation for various liquid content $S$,\nand the peak stress gives tensile strength.\nStress-strain curves, showing\ndecreasing strength\nas the liquid content $S$ increases from 36\\% to 70\\% \nare shown in Fig.~\\ref{Fig:Holm85Fig8.ps},\nfrom \n\\cite{Holm85}, where \ncompressive tests were performed using\ndicalcium phosphate with diameter 21 $\\mu$m.\nKristensen {\\it et al.} \\cite{Kristensen85} found increasing strength\nfor increasing liquid content using glass beads of particle size 68 $\\mu$m,\nwhile  decreasing strength\nfor dicalcium phosphate of particle size 14 $\\mu$m in \nuniaxial compression tests.\nThere have been many other experiments probing the\nmechanical strength of wet granular media \n(see \\cite{GranulationReview} for a review). \nThe effect of particle size on the competition \nbetween cohesion and lubrication was considered \\cite{GranulationReview} \nto be one of the causes of these various behaviours,\nbut the liquid content dependence in the \nintermediate liquid-content-regime \n(approximately given by $20 \\% \\lesssim S \\lesssim 90 \\%$)\nis not yet well-understood.\n\nIt has been found that the critical strain $\\epsilon_c$\n(strain at the peak in the stress-strain curve)\nalways increases with increasing liquid content,\nas shown in Figs.~\\ref{Fig:Schubert75Fig9} and \\ref{Fig:Holm85Fig8.ps}.\nThis indicates that the wet granular material\nis brittle when the amount of liquid is small,\nand tends to show visco-plastic behaviour as the liquid content is increased.\n\nIt should be also noted that,\neven for similar amounts of liquid content,\nthe strength of a sample may depend on whether it is \nprepared by either draining or adding liquid;\nSchubert \\cite{Schubert75} found different \nstress-strain curves for each processes \nas shown in Fig.~\\ref{Fig:Schubert75Fig9}.\nIn many studies, the sample is often driven by an external force\n(e.g., rotating drum, mixing, etc.) before the measurement\nin order to better distribute\nthe liquid, which would make hysteresis less obvious.\n\n\n\\paragraph{Tests in soil mechanics:\nrelatively large amounts of liquid}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90,width=0.6\\textwidth]{Fig16a.ps}\n\\includegraphics[angle=-90,width=0.6\\textwidth]{Fig16b.ps}\n\\end{center}\n\\caption{Yield loci for wet soil (Madrid Gray Clay).\n(a) Net normal stress $\\sigma$ versus shear stress $\\tau$ for\nvarious suction $\\Delta P=P_a-P_l$.\nThe slope is almost independent of suction in this regime.\n(b) Shear stress versus suction for various\nnet normal stresses $\\sigma$.\nThe slope is almost independent of the net normal stress in this regime.\nAdapted from\n\\cite{UnsaturatedBook}\n(original data from\n\\cite{Escario80}).\n}\n\\label{Fig:UnsaturatedBookFig6-6}\n\\end{figure}\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[angle=-90,width=0.6\\textwidth]%\n{Fig17.ps}\n\\end{center}\n\\caption{Shear strength $\\tau_s$ versus suction $\\Delta P$\nobtained by shear tests \nfor red clay. \nThis figure shows a\nlarger range of suction $\\Delta P$\n(compared to Fig.~\\ref{Fig:UnsaturatedBookFig6-6}(b)),\nas well as a non-linear and non-monotonic behaviour.\nAdapted from\n\\cite{Escario89}\n}\n\\label{Fig:UnsaturatedBookFig6-12}\n\\end{figure}\n\nAs pointed out in subsection \\ref{phases},\nthe suction in the capillary state largely\nvaries upon a small change of liquid content.\nIn soil mechanics, the shear tests\nof unsaturated soils have been conducted\nby decreasing the water content from initially saturated soils,\nor, in other words, \nby increasing the suction from zero.\nThe data is usually presented in terms of the suction $\\Delta P$,\nnot the liquid content $S$.\nThis is a reasonable way of arranging data in the capillary state,\nbecause the change of $\\Delta P$ upon changing $S$ is \nvery rapid for low and high liquid content \nas has been shown in subsection \\ref{phases}.\nHere we summarise these data to try to understand the \nmechanical properties with large amount of liquid content.\nIt should be noted that the grains\nthat form soils can be \nvery small (for example, clay can be the order of \n2 $\\mu$m in diameter), may have characteristic shape,\nand may be affected by, e.g., electrostatic interaction. \nThese properties affect its mechanical responses,\nbut we focus on the variation of \nthe mechanical properties upon changing the suction $\\Delta P$\nin the following, so that\nwe will be able to extract the properties \nmainly determined by the capillary effect.\n\nThe failure condition given by\nthe Mohr-Coulomb criterion\nin Eq.~(\\ref{Eq:CoulombCriterion})\nhas been experimentally investigated for soils.\nFigure \\ref{Fig:UnsaturatedBookFig6-6}(a)\nshows\\footnote{\nIn soil mechanics, the total normal stress $\\sigma_t$,\nthe air pressure $P_a$, and the water pressure $P_l$\nare often taken to be the control parameters.\nHere, the total normal stress $\\sigma_t$ \nincludes both the load on the granular sample\nand the fluid pressure on it, thus\n$\\sigma_t=P_a$ if the sample is \nplaced in air without any external load.\nWhen an external pressure is applied on the sample,\nthe net normal stress $\\sigma$ \nsustained by granular particles would be given by $\\sigma_t-P_a$.\nIn the original figures of Fig.~\\ref{Fig:UnsaturatedBookFig6-6}(a),\nthe label of the horizontal axis shows\n``Net Normal Stress $\\sigma-u_a$ (kPa)'',\nwhere $\\sigma$ in \\cite{UnsaturatedBook} represents the total normal stress,\nand $u_a$ is the air pressure ($\\sigma_t$ and $P_a$ in our notation,\nrespectively).\n}\nthe yield loci of wet clay for various \nvalues of the suction $\\Delta P$,\nwhere the slope gives the friction coefficient $\\mu$\nand the $y$-intercepts of lines give $\\mu\\sigma_c$.\nThe slopes scarcely vary with suction,\nwhich indicates that $\\mu$, \nor, the internal friction angle $\\Phi=\\tan^{-1}\\mu$,\ndoes not depend on the suction $\\Delta P$.\nThe $y$-intercept is small for $\\Delta P=0$ and \nincreases with increasing the suction $\\Delta P$.\nNamely, $\\sigma_c$ increases with increasing $\\Delta P$.\n\nFigure \\ref{Fig:UnsaturatedBookFig6-6}(b) \nshows the yield shear stress versus suction \nfor various normal compressive stresses,\nwhich shows a linear dependence on $\\Delta P$.\nThis result and the fact that the \ninternal friction angle $\\Phi$ \ndepends little on the suction $\\Delta P$\nsuggest that the cohesive stress $\\sigma_c$\nincreases linearly with $\\Delta P$. \nThe linear increase of \n$\\sigma_c$ \nupon increasing $\\Delta P$ for small $\\Delta P$\nis natural, because the compressive \nstress $\\Delta P$ at the liquid-air interface\nat the surface of the granular assembly\nis the source of the cohesion.\nThis behaviour has been found \nfor some kinds of clay \\cite{Escario80,Escario89,Bishop60},\nundisturbed decomposed granite \\cite{Ho82},\nsilt \\cite{Krahn89},\nand glacial till \\cite{Gan88}.\n\nWhen $\\Delta P$ is increased further,\na non-linear dependence of the shear strength $\\tau_s$\n(the shear stress at failure for a given suction and normal load)\non $\\Delta P$ is found \\cite{Escario89,UnsaturatedBook}, as shown in \nFig.~\\ref{Fig:UnsaturatedBookFig6-12}. \nThe increase of the shear strength $\\tau_s$ becomes \nnearly zero as $\\Delta P$ grows.\nIt is very likely that the internal friction coefficient $\\mu$ \ndoes not vary significantly upon changing $\\Delta P$ in these regimes,\nand then the obtained \nshear strength $\\tau_s$ is proportional to the cohesive stress $\\sigma_c$. \n\nSummarising these results, \n$\\sigma_c$ increases linearly \nwith $\\Delta P$ for small enough suction $\\Delta P$,\nbut for large $\\Delta P$, the increase disappears.\nThe suction $\\Delta P$ increases as the liquid content $S$ \ndecreases as shown in Fig.~\\ref{Fig:SoilWater},\ntherefore, \n$\\sigma_c$ increases upon decreasing the liquid content\nfrom $S=100\\%$ ($\\Delta P=0$)\n\\cite{Vanapalli96}.\n\n\\subsection{Dynamical behaviours}\n\\label{dynamicalbehaviours}\nIn this subsection, we briefly describe some dynamical\nbehaviours of wet granular media. \nIn the dynamical regime, not\nonly cohesion but also other effects of the liquid\n such as viscosity, lubrication, and liquid motion\nplay important roles.\nThe dynamical behaviours observed\nare far more complicated\nthan those in the quasistatic regime.\nIn addition, though there are numerous\nstudies of the dynamics of partially wet granular media \nfrom the practical point of view,\nsystematic studies \nin simple situations are rather few;\nhere we briefly summarise some of the experimental studies\non the dynamical response of wet granular media.\nThose who are interested in recent studies on \ndynamics should also refer to a recent review\n\\cite{Herminghaus05}.\n\n\\subsubsection{Dynamics in the pendular state}\n\n\\paragraph{Avalanches in rotating drums}\n\\label{Tegzes}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.8\\textwidth]{Fig18.eps}\n\\end{center}\n\\caption\nSurface shapes in a rotating drum \nduring single avalanches, for various liquid contents $S$\n(shown as percentages).\nThe coloured line for liquid content percentages $0.12\\%$ \nand 0.37\\% indicate approximate slip planes in the correlated regime, \nand the coloured lines for $5.00\\%$ \nshow the travelling quasiperiodic surface features \nin the viscoplastic regime.\nFrom\n\\cite{Tegzes03}.\n}\n\\label{Fig:Tegzes03Fig17a}\n\\end{figure}\nGranular flow and avalanches in a rotating drum\nhave been widely investigated for dry granular media\n(see, e.g., \n\\cite{Rajchenbach90,Rajchenbach02,Bonamy02,RajchenbachReview,\nBenza93,Ristow96,Khakhar97,Baumann95}).\nAt low rotation rate, the grains stay at rest\nuntil the surface angle reaches the \ncritical angle $\\theta_c$,\nand avalanche occurs;\nthe angle just after the avalanche is \nthe angle of repose $\\theta_r$.\nThis is called the intermittent regime.\nWhen the rotation rate is high enough, \nthe continuous flow occurs with keeping \na constant angle between the surface and horizontal plane;\nthis is the continuous regime.\nThe flow regimes of fine cohesive powders in a rotating drum has\nbeen also studied by \\cite{Castellanos99}.\n\nTegzes {\\it et al.} \\cite{Tegzes02,Tegzes03} investigated\nthe flow behaviours of wet granular materials\nin a thin rotating drum. In the experiments,\nthe amount of liquid is small \n(liquid content varied from 0.001\\%\nto 5\\%),\nwhere the pendular state is expected.\nThey have observed \nthe transition from intermittent to continuous flow\nwith increasing the rotation rate\nin this range of liquid content.\n\nThey observed the time evolution of the surface profile,\nwhich enable them to calculate the mean surface angle,\nstatistics of the local surface angle,\nevolution of pattern at the surface, etc.\nIn Fig.~\\ref{Fig:Tegzes03Fig17a},\nthe typical surface profiles\njust after avalanches in the intermittent flow regime are \nshown. For enhanced clarity, the gray-scale plot in the \nthird dimension of the figure shows\nthe local surface angles.\n\nBy changing the liquid content,\nthe following three regimes have been distinguished. \nFor low liquid content, the cohesion is small,\nand small avalanches are observed (the {\\it granular} regime);\nthe avalanches occur at the surface and\nthe surface profile is always smooth\n(Fig.~\\ref{Fig:Tegzes03Fig17a}, liquid content $0.04\\%$), \nsimilarly to the dry granular media\n(Fig.~\\ref{Fig:Tegzes03Fig17a}, dry case).\nWhen increasing the liquid content, \nthe cohesion becomes stronger,\nand grains move as a connected block.\nAn avalanche occurs\nthrough a succession of local slip events,\nand the surface structure fluctuates after\neach avalanche; this is the {\\it correlated} regime\n(Fig.~\\ref{Fig:Tegzes03Fig17a}, liquid content $0.12\\%$ and $0.37\\%$).\nFurther increasing the liquid content\nresults in the {\\it visco-plastic} regime.\nIn this state, the flow becomes\ncoherent over the entire sample,\nand fluctuations are suppressed \n(Fig.~\\ref{Fig:Tegzes03Fig17a}, liquid content $5\\%$). \n\n\nThey also found that \nthe critical angle $\\theta_c$ depends\nnot only on the liquid content but also on the rotation rate.\nThey varied the duration between avalanches, \nor the ``waiting time'', from about 0.1 s to 1000 s. \nIt was found that $\\theta_c$ increased logarithmically \nwith the waiting time.\nThey supposed that this was caused by the slow \nflow of liquid along the particle surface.\n\nA systematic study of liquid motion in \nwet granular media in the pendular state \nwas recently conducted by \\cite{Kohonen04}.\nThey found that, just after shaking the material to \nmix the liquid with grains,\nthe average liquid bridge volume was much less than that \nin equilibrium;\nthe volume saturated after long enough waiting times,\nmore than one hour.\nThis also suggests that the liquid-moving process\nis very slow and may induce the ageing effect\nobserved by \\cite{Tegzes02,Tegzes03}.\n\n\\paragraph{Vibrated wet granular media}\n\\label{vibratedwet}\nDry granular media placed on a horizontal plane\nexcited by vertical vibrations\nis a fundamental and widely-used setup \nto study granular gases (e.g., \n\\cite{GranularGas}),\nsegregation (e.g., \n\\cite{GranularSeg}),\nand pattern formation\n(e.g., \n\\cite{GranularPattern}).\nIn dry granular media,\nthe onset of fluidisation\nis often characterised by the dimensionless\nacceleration $\\Gamma=A\\omega^2\/g$,\nwhere $A$  is the amplitude of the vibration, $\\omega$\nis the angular frequency of the vibration,\nand $g$ is the acceleration of gravity.\nThere is a threshold $\\Gamma_{\\rm min}$ below which \nthere are no fluidisation, which was reported to be a constant\naround one, (e.g., \\cite{Duran97,Clement92,Knight93}),\nbut recently reported that\n$\\Gamma_{\\rm min}$ can be smaller than one\nand show weak $\\omega$ dependence \\cite{Renard01,Poschel00}. \nAbove the fluidisation threshold,\ndry granular media exhibits \ntransitions between localised patterns of jumping grains \n(in the case of few layers of grains)\nall the way to gas-like phases \n(see, e.g.,\n\\cite{Losert99,Melo95,DAnna03,\nUmbanhowar96,Pak94,Laroche89,DAnna01}).\n\nFor wet granular media, the cohesion force introduces \nan additional force or energy scale. \nIt is non-trivial to determine\nwhich parameters would better \ncharacterise the resulting behaviour.\nRecently, \nexperiments on this subject\nhave been conducted \\cite{Schell04,Fournier05},\nwhere fluidisation of glass beads wet by water\nunder vertical vibration has been investigated.\nThe critical dimensionless \nacceleration $\\Gamma_{\\rm crit}$ \nfor fluidisation is found to \ndepend on the frequency $f=\\omega\/2\\pi$,\nthe particle radius $R$,\nand the liquid content $W$\ndefined as the ratio of the liquid volume to the total volume,\nas shown in Fig.~\\ref{Fig:Schell04Fig1}.\nHere, $W$ is defined as the ratio of \nthe liquid volume to the total volume.\nThey \\cite{Schell04,Fournier05} \nfound that $\\Gamma_{\\rm crit}$ \ndepends weakly on $f$, but\nbecomes constant for sufficiently high frequency. \nIn this high frequency regime \\cite{Schell04,Fournier05},\n$\\Gamma_{\\rm crit}$ \nis smaller for larger $R$ for a given $W$ (Fig.~\\ref{Fig:Schell04Fig1}(a)), \nwhile it increases\nwith $W$ for a given $R$\n(Fig.~\\ref{Fig:Schell04Fig1}(b)). \n\nRef. \\cite{Schell04,Fournier05} interpreted their \nresults by considering the \ncohesion force due to liquid bridges\nbetween grains\nas well as between grains and the container wall.\nThe increase of $\\Gamma_{\\rm crit}$ \nwith $W$ was found \\cite{Schell04} to be proportional to\nthe increase of the cohesion force $F_{\\rm bridge}$ per \nliquid bridge, which they estimated from $W$.\nIt is natural that $\\Gamma_{\\rm crit}$\nincreases with increasing $F_{\\rm bridge}$,\nbut \\cite{Schell04,Fournier05} found that the proportionality \nis not trivial: Another possibility is that \n$\\Gamma_{\\rm crit}$ is ruled by the energy\nrequired to break the bridge, whose dependence on $W$\nshould be different from the dependence of the force. \nIn the case of the dependence upon $R$, \n\\cite{Schell04,Fournier05}\nexplained it by considering the shear stress \ndue to the formation of liquid \nbridges between grains and side walls;\nthe bulk material tends to move together due to the cohesion,\nbut if the shear from the side wall is large enough,\nthe shear stress makes it possible to deform \nthe bulk sample and to induce the fluidisation.\nFollowing their interpretation, the \ndecrease of $\\Gamma_{\\rm crit}$\nupon increasing $R$ is because \nthe force per bridge is proportional to $R$ but the \nnumber of liquid bridges per area is proportional to \n$1\/R^2$, which gives the shear stress \nproportional to $1\/R$.\nThey \\cite{Schell04,Fournier05} \nalso studied the fluidisation \nwith the container whose walls are covered by the \nhydrophobic material so that liquid \nbridges cannot be formed with the side walls, \nand they found that the material cannot \nbe fluidised up to $\\Gamma=20$.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.48\\textwidth]{Fig19a.eps}\n\\includegraphics[width=0.51\\textwidth]{Fig19b.eps}\n\\end{center}\n\\caption{\n(a) Critical acceleration \n$\\Gamma_{\\rm crit}$\nversus frequency $f$ for three different\nbead radii, with water content $W=0.5\\%$.\n(b) $\\Gamma_{\\rm crit}$\nversus liquid content $W$\nfor three different frequencies.\nThe particle radius R is 140 $\\mu$m.\nFrom \\cite{Fournier05}.\n}\n\\label{Fig:Schell04Fig1}\n\\end{figure}\n\n\\paragraph{Segregation}\nSegregation of grains by their size\nis one of the \nmost striking phenomena of granular materials.\nThe most well-known example is the Brazil-nuts effect, \ni.e., the large Brazil-nuts always appear on top \nof mixed-nuts containers. \nAfter two or more kinds of \ngrains are mixed together,\nsize segregation occurs very easily\nwhen they are excited by an external energy input\nin, e.g., a vibrated container or chute flows. \nSize-segregation has been studied for dry granular material \nfor various types of mixtures and excitations \n(e.g.,\n\\cite{Mobius01,Hong01,GranularSeg,Ottino00}).\nIn wet granular materials,\nthe cohesion tends to suppress the segregation\nby sticking grains together,\nbut at the same time, various factors \ncome into play in such a dynamical situation,\nas we will see below.\n\nThe size-segregation \nin wet granular materials\nthat flow down an inclined plane\nhas been investigated in \\cite{Samadani00,Samadani01}.\nThey used binary mixtures of glass beads with several \nsets of diameters. Also water, glycerol, \nand other kinds of liquid were used to observe the effect of \nthe viscosity as well as the surface tension.\nThey found that segregation is basically \nsuppressed by increasing the liquid content,\nwhich is natural because the cohesion tends to\nsuppress the particles' relative motion.\n\nSamadani and Kudrolli \\cite{Samadani00,Samadani01}\ninvestigated the phase diagrams of segregation\nin the space of particle diameter ratio and\nliquid content. \nThese found a qualitative difference of phase diagrams\nbetween water and glycerol:\nthe surface tensions are almost the same for \nthese two kinds of liquid, but the \nviscosities are different, and  \nthe segregation is lower at higher viscosity.\nThis is because \\cite{Samadani00,Samadani01}\nviscosity tends to reduce the velocity fluctuations\nrequired for segregation.\n\nSamadani and Kudrolli \\cite{Samadani00,Samadani01}\nestimated the viscous force between \nmoving particles connected by a liquid bridge \nby using the Reynolds lubrication theory, \nin which the hydrodynamic effect in \nthin space is taken into account\n\\cite{LubricationE,Tribology,Pitois00,Pitois01}.\nThere \\cite{Samadani00,Samadani01}, \nthe force is proportional to the relative velocity \nbetween moving particles, and they found that\nthe viscous force can be comparable with the cohesive force\nfor the characteristic velocity scale under gravity.\nThis also suggests that the viscosity is \nvery important for dynamical behaviour like segregation.\n\n\\begin{figure}\n\\begin{minipage}{0.3\\textwidth}\n\\includegraphics[width=\\textwidth]{Fig20a.eps}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.6\\textwidth}\n\\includegraphics[width=\\textwidth]{Fig20b.eps}\n\\end{minipage}\n\\caption{\n(a) Schematic description of the experimental setup\nin \\cite{Geromichalos03,Herminghaus05}.\nThe jar experience rapid, horizontal circular motions \n(20 revolutions per second),\nwhere the centre of the jar follows a small circle\nas shown in the figure\n(The jar itself is not spinning around its centre).\n(b) Particle-size segregation phase diagram\nfor binary mixtures of wet granular materials.\nThe radius of the bigger beads is fixed, and\nthe vertical axis shows the radius of the smaller beads.\nThe horizontal axis shows the amount of liquid \nvia the dimensionless quantity $W$, \ndefined as the volume of the liquid divided \nby the total volume of the small beads (including the space\nbetween the beads).\nThe segregation does not occur in grey region.\nThe dashed steep line separates \nthe region where the segregation is enhanced\nby adding water (left) from the region \nwhere the segregation decreases or stays the same (right).\nFrom \n\\cite{Geromichalos03}.\n}\n\\label{Fig:Geromichalos03Fig3}\n\\end{figure}\n\nOn the other hand, Geromichalos {\\it et al.} \n\\cite{Geromichalos03,Herminghaus05} \ninvestigated size segregation in granular media\nin a jar driven by a horizontal circular motion \n(Fig.~\\ref{Fig:Geromichalos03Fig3}(a)).\nThey found that segregation sometimes occurs partially,\nand they investigated the degree of segregation\nby measuring the fraction of the\nmixture zone (i.e., the region where no segregation occurs)\nto the total sample.\nFrom the experiments,\nthey distinguished three regimes:\n(Fig.~\\ref{Fig:Geromichalos03Fig3}(b));\nthe {\\it gaseous} regime,\nwhere an increase of the liquid content \nenhances size-segregation;\nthe {\\it intermediate} regime,\nwhere segregation occurs, but \nincreasing the liquid content \nsuppresses segregation;\nand the {\\it viscoplastic} regime,\nwhere no segregation occurs.\nBy increasing the liquid content\nwith a fixed ratio of radii, \nthe system state evolves\nfrom gaseous, intermediate, to the viscoplastic regimes.\n\nThe behaviour in the gaseous regime \nfound in \\cite{Geromichalos03}, where \nsegregation is enhanced \nby increasing liquid content,\nseems to contradict the results in\n\\cite{Samadani00,Samadani01}.\nGeromichalos {\\it et al.} \\cite{Geromichalos03} discussed that,\nin the gaseous regime,\nthe energy dissipation is enhanced by the breakage \nof a number of tiny liquid bridges, \nwhile the cohesion is not enough to strongly affect the \ndynamics. \nThe larger energy dissipation tends to \nenlarge the energy difference \nbetween large and small grains,\nresulting in an enhanced segregation.\nAs liquid content is increased,\nthe liquid bridge becomes larger and the\nkinetic energy becomes insufficient\nto break the bridges \\cite{Geromichalos03}. Then,\nsegregation is suppressed and\nthe system goes from the intermittent regime to \nthe viscoplastic regime.\n\n\\subsubsection{Shear experiments for various liquid content}\n\\begin{figure}\n\\begin{minipage}{0.4\\textwidth}\n\\includegraphics[width=\\textwidth]{Fig21a-1.eps}\n\\centerline{\\includegraphics[width=0.5\\textwidth]{Fig21a-2.eps}}\n\\end{minipage}\n\\hfill\n\\begin{minipage}{0.48\\textwidth}\n\\includegraphics[width=\\textwidth]{Fig21b.eps}\n\\end{minipage}\n\\caption{(a) Experimental setup to shear \nwet granular materials in a cell,\nshowing a slice of the shearing cell seen from the side\nand from the front.\n(b) The hysteresis loop \nof the pressure difference $\\Delta \\sigma$\nversus the average shear $\\Delta V\/A$\nfor densely packed glass beads.\nFrom\n\\cite{Fournier05}.\n}\n\\label{Fournier05Fig7}\n\\end{figure}\nRecently, Fournier {\\it et al.} \\cite{Fournier05} investigated the \nresponse of wet granular materials under shear \nby using a unique experimental setup, shown in Fig.~\\ref{Fournier05Fig7}.\nA cylindrical cell was filled with\nspherical glass beads wet by water, \nwhere the flat sides of the cell (area $A$)\neach consist of a thin latex membrane.\nAdjacent to each membrane was a cylindrical chamber \nfilled with liquid, connected to a syringe. \nWhen the pistons of the syringe are moved at equal speed, \nthe membranes are deformed by a volume\n$\\Delta V$ (Fig.~\\ref{Fournier05Fig7}(a)\nshows a lateral cut of the cylindrical cell (top) \nand a vertical cut (bottom)),\nwhich allows them to shear the granular material under \na constant volume at a controlled speed.\nThe pressure at each membrane is measured during the shear.\nThe pressure difference $\\Delta \\sigma$ between \nthe two membranes \nversus average shear $\\Delta V\/A$ \nis shown in Fig.~\\ref{Fournier05Fig7}(b)\nfor densely packed glass beads (packing fraction $\\nu \\approx 0.63$),\nshowing hysteresis. \nThe vertical width of the hysteresis loop reflects the \nresistance to shear of the material.\nThey defined $\\Delta \\sigma_0$ as the\nvertical width of the hysteresis loop \nat zero strain divided by two, and measured $\\Delta \\sigma_0$\nfor various liquid content and shear rate.\nThe dependence of $\\Delta \\sigma_0$ upon the amount of wetting \nliquid is shown in Fig.~\\ref{Fournier05Fig10}. \nFollowing their parametrisation,\nthe horizontal axis shows $W$, the ratio of \nthe liquid volume to the total volume.\nFor a given packing fraction $\\nu$,\nthe liquid saturation $S$ is given by $S=W\/(1-\\nu)$, thus \n$W=0.35$ is almost in a \nsaturated state ($S\\approx 0.95$) when $\\nu=0.63$.\nWe can see that $\\Delta \\sigma_0$ increases rapidly \nas $W$ is increased from zero, but it starts to drop \nat $W\\approx 0.04 \\ (S\\approx 0.1)$ and goes to zero at $W\\approx 0.35$.\nThey observed the liquid  \ndistribution for small liquid content\nby index matching techniques, and\nfound that $W\\approx0.03$ is \nthe point where liquid bridges start to \nmerge to form a liquid cluster\n(Fig.~\\ref{Fournier05Fig11});\nbeyond that point, the interfaces between \nthe air and liquid start to decrease, \nwhich may cause the decrease of the cohesive force.\n\nReference \\cite{Fournier05}  also investigated the shear rate dependence\nof $\\Delta \\sigma_0$. Within the investigated \nrange of the shear rate, $\\Delta \\sigma_0$ decreases \nfor larger shear rate,\nwhile the dry granular material \ndoes not show shear-rate dependence.\nA possible origin of the shear-rate dependence \nis the time dependence of liquid motion as described in \nsection \\ref{Tegzes},\nwhich affect the temporal evolution of liquid bridges.\n\nIn addition, they \\cite{Fournier05} \nclaim that the increase \nof the shear stress is not solely coming from \nthe frictional effect, as in the Mohr-Coulomb picture described in section \n\\ref{AngleofRepose}, \nbut it can be understood by just \nconsidering the cohesion that arises from the liquid bridges.\nOne of the tests they conducted to confirm their claim\nis to measure the absolute pressure dependence of $\\Delta \\sigma_0$;\nif the frictional shear stress proportional to the\ncohesive normal stress $\\sigma_c$ is responsible for the \nfact that $\\Delta \\sigma_0$ is greater than that of the dry \ngranular material, then applying a unidirectional pressure comparable\nwith $\\sigma_c$ for {\\it dry} granular material \nmay cause a similar effect to increase $\\Delta \\sigma_0$.\nHowever, they found that applying absolute pressure to the sample\nis not enough to make $\\Delta \\sigma_0$ increase.\nThough the fact that the applied \npressure is unidirectional while\nthe cohesive pressure is uniform\nmay be one of the causes of this difference \\cite{Herminghaus05}, \ntheir results indicates that\nthe origin of the higher yield shear stress\nin wet granular materials need to be investigated further.\n\nThe response of materials against shear is \na fundamental property\nto understand the material's rheology.\nFurther experiments of sheared wet granular material\nin various setups would be useful.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig22.eps}\n\\end{center}\n\\caption{\nDependence of $\\Delta \\sigma_0$\nupon the amount of the liquid $W$.\nFrom \\cite{Fournier05}.\n}\n\\label{Fournier05Fig10}\n\\end{figure}\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=0.5\\textwidth]{Fig23.eps}\n\\end{center}\n\\caption\nFluorescence microscope images of liquid distribution \nbetween glass beads of diameter 375 $\\mu$m for \n$W=0.03$, ``liquid clusters'' are formed.\nFrom \\cite{Fournier05}.\n}\n\\label{Fournier05Fig11}\n\\end{figure}\n\n\n\\subsubsection{\nDynamics of  \nwet granular media: practical applications}\nThere are many research fields \nthat investigate the dynamics\nof wet granular media from a practical point of view.\nBelow we summarise a few examples.\n\n\\paragraph{Agglomeration processing:\ngrains binded by liquid} \\label{Granulation}\nDuring an agglomeration or granulation process,\nparticles lump or agglomerate together into\nlarger, semi-permanent aggregates called granules\n\\cite{GranulationReview,GranulationProceedings,Cates04}.\nThere are several methods to make granules.\nOne of them is \nto spray a binder liquid onto dry powder \nand mix them in, e.g., a tumbling drum\nso that clumps of particles binded by liquid grows.\nThe granules may either grow or break down\nwhen they collide.\nAnother method is to strongly shear \na very dense mixture of binder liquid and powder (dense paste), \nand then air comes into the paste to form lumps.\nIn either method, the main source of the cohesive force\nthat binds powder particles together is the capillary force.\n\nNumerous dynamical experiments have been performed\non granulation processing:\nThe dynamics of liquid distribution during granulation, \nthe growth of granules, \nthe shear rate dependence of the growth rate, \nthe collision velocity dependence of breakage,\nand so on.\nThough most of the experimental setups seem\nrather complicated, \nthe accumulated knowledge provides\nmany insights on the dynamical properties of wet granular media.\nHowever, it is beyond the scope of this brief review\nto summarise all experiments \nin this subfield.\nInterested readers \nshould consult a rather recent review on this topic\n\\cite{GranulationReview},\na conference proceedings \\cite{GranulationProceedings},\nand references therein.\n\nIn addition, the properties of sheared dense paste \nare important not only \nin the granulation process, but also in, e.g., ceramics engineering,\nwhere the rheology of paste has been investigated.\nThe importance of cohesion due to the capillary \neffect in the rheology of paste\nhas often been pointed out \n(e.g., \\cite{CeramicProcessBook,VanDamme02}),\nand the knowledge about its rheology also provides \ninsights about wet granular media.\n\n\\paragraph{Geological events}\nGeology is one of the largest research fields \non wet granular media.\nThe failure criterion of soils\nand also their dynamical behaviours \nare very important to understand.\nSignificant, and sometimes catastrophic and tragic, \nexamples include land slides, debris flows, \nand liquefaction of ground \n(see, e.g., \\cite{Iverson97,Dikau,Coussot,\nSoilJapaneseBook,Ishihara93}).\n\nMost of the past work on \ndebris flows focused on the flow of\nsoils saturated by water.\nIn studies of debris flows,\nknowledge about dry granular media\nhas been incorporated into the analysis, and at the same time, \nthe significant effects of the liquid lubrication and\nthe liquid viscosity on debris flows have been investigated\n\\cite{Iverson97}.\nLiquefaction of wet soils triggered by earthquakes\nis also often studied in saturated situations \\cite{SoilJapaneseBook}.\nSystematic studies of wet granular media\nfrom partially-wet to the completely wet state\nshould also provide useful insights to understand \nthese important geological events.\n\n\\section{Summary and open questions}\n\\subsection{Effect of the liquid content on \nquasistatic behaviour}\n\\label{SummaryCohesion}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.6\\textwidth]{Fig24.eps}\n\\end{center}\n\\caption\nSchematic diagram summarising the variations of\nthe suction $\\Delta P$ and \nthe cohesive stress $\\sigma_c$, \nupon changing the liquid content $S$.\n}\n\\label{Fig:Schematic}\n\\end{figure}\n\nThe cohesive force due to the presence of a\nliquid arises in the pendular,\nfunicular, and capillary states.\nThis cohesion gives rise to a finite cohesive stress\nin quasistatic experiments.\nFigure \\ref{Fig:Schematic} shows a schematic \ndiagram summarising how the suction $\\Delta P$\nand the cohesive stress $\\sigma_c$\nvary upon liquid content.\n\nThe dependence \nof the suction $\\Delta P$\non the liquid content $S$\nis given by a red solid line, and\nits slope changes significantly near the phase boundaries between \nthe pendular, funicular, and capillary states,\nas discussed in subsection \\ref{phases}.\n\nThe cohesive stress $\\sigma_c$ versus $S$\nis schematically shown by the green dashed lines.\nThe overlapping dashed curves for low and high $S$ are better established,\nwhile the non-overlapping intermediate curves \nvary significantly between experiments.\nThe cohesive stress $\\sigma_c$ increases \nas we add more and more liquid to initially dry grains,\nshown by the positive \nslope, for small $S$, of the $\\sigma_c$ line \nin the pendular state.\nAt the opposite end, cohesion becomes zero \nfor completely saturated granular media ($S=100\\%$),\nschematically shown by the green dashed line with negative slope \nfor $S$ close to 100\\% in the capillary state.\n\nIn the funicular state,\nthe cohesive stress $\\sigma_c$ dependence\non the liquid content $S$\nis not clearly understood.\nThe cohesive stress $\\sigma_c$ may either increase or decrease with $S$\nin the funicular state. \nThese possible curves are shown by the green dashed lines,\nwhere both lines connect to single lines \nin the limits when $S\\to 0$ and $S\\to 100\\%$.\nWe see that there would be at least one maximum of \nthe cohesive stress at a certain liquid content, though\nit is not clear, a priori, the location of the peak.\n\n\\subsection{Open problems}\nThere are a number of open problems\non mechanical properties of wet granular media,\nsome of which have already been mentioned in the text.\nIn this subsection, we list a few examples of \nopen problems, in order to \nencourage future studies in this area.\n\\subsubsection{Jamming}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.7\\textwidth]{Fig25.eps}\n\\end{center}\n\\caption\nSchematic diagram of \na possible jamming phase diagram \nof granular media for various liquid contents.\nNumber 0 (black solid line): Dry . \nNumber 1 (green dotted line): Small amount of liquid (slightly cohesive).\nNumber 2 (blue dashed line): ``Optimal'' amount of liquid (the most\n cohesive,\nand the largest phase-space volume of the jammed phase).\nNumber 3 (red dash-dotted line): Large amount of liquid \n(less cohesive than the optimal case).\n}\n\\label{Jamming}\n\\end{figure}\n\nAs summarised in subsection \\ref{SummaryCohesion},\nin the quasistatic regime, the cohesion \nin wet granular media \nhas a maximum for a certain liquid content,\nthough whether the maximum is located\nat a rather small or large liquid content\nis not known a priori.\nThis variable cohesion will \naffect the ``jamming'' properties of the granular assembly.\nThus, an interesting problem\nworth considering \nis the jamming phase diagram in wet granular media.\n\nA jammed state occurs when\na disordered system subject to an \nexternal force is caught in a\nsmall region of phase space \nwith no possibility of escape \n\\cite{Liu98,JammingBook,DAnna01,DAnna03,OHern02,OHern03,\nTrappe01,Valverde04}.\nIn order to unjam a jammed material,\na finite yield stress or fluctuation energy is needed,\nwhich forces the elements to escape from \nthe phase space region where they are trapped.\nGranular media at rest is \na typical jammed material, where we need \na finite yield stress or external energy input\nin order to let the grains flow or move.\nThere are many kinds of materials that show\njamming, such as dense colloids, \nand pinned vortices in superconductors.\nJamming also plays a role in \nthe mechanical properties of wet granular media.\nFor example, Cates {\\it et al.} \\cite{Cates04} recently\ntried to understand the physics of granulation (section \\ref{Granulation})\nin a highly-sheared dense paste in terms of a jamming \ntransition under shear.\n\n\nLiu {\\it et al.} \\cite{Liu98} \nproposed a very schematic phase diagram for jamming,\nin the space of packing fraction $\\nu$ (density), \nexternal load, and temperature\nor fluctuation energy. The purpose of this \nschematic phase diagram is to \ntry to summarise a unified view of the jamming phenomena,\nobserved in various media.\nA schematic diagram of our jamming phase diagram\nis shown in Fig.~\\ref{Jamming};\nthe solid black curves, \nlabelled with the number 0, was proposed \nby \\cite{OHern02,OHern03} for non-cohesive (dry) granular media.\nThe axes shown are:\ninverse packing fraction $1\/\\nu$,\nthe yield stress $\\sigma_{\\rm yield}$,\nand the fluctuating energy $T$\nfrom an external energy source, such as vibration.\nOne of the features of jamming in dry granular media\nis that the behaviour in the $T=0$ plane, \ni.e., no external energy, can be easily \ninvestigated. The ``jammed'' phase-boundary\nintersects the inverse packing fraction $1\/\\nu$ axis at zero yield stress\nand temperature,\nand the packing fraction at the intersection\ncorresponds to the random closed-packing density \\cite{OHern02,OHern03}.\n\nObviously, attractive interactions modify the \njamming properties.\nTrappe {\\it et al.} \\cite{Trappe01} investigated jamming in colloidal systems\nwith varying attractive interaction potentials;\nthey found that the jammed phase boundary \nshifts to higher temperatures for stronger attractive potentials,\nbecause the attractive potential characterises\nthe energy that particles need to escape from the \nphase space where they are trapped.\nJamming in cohesive powders was investigated by\n\\cite{Valverde04}; the powder particles were large enough \nso that thermal noise was negligible.\nIt was found that the critical jamming density \nfor zero fluctuation energy and yield stress\nfor cohesive powder is smaller than the critical\ndensity in non-cohesive granular media.\nThis is because the cohesion \nmade it possible to form\nvery sparse but stable \nclusters of powder.\nIndeed, a percolating cluster appears\nin their system for relatively low densities.\n\nConsidering these results, \nthe cohesion present in wet granular media \nwill decrease the density needed to jam the system.\nTo break a cluster in \nwet granular media, a large enough \nexternal fluctuation energy is needed,\nand this energy will increase with cohesion.\nThe yield stress will increase with cohesion \nas described by the Mohr-Coulomb criterion,\nwhere the yield shear stress increases\nwith the cohesive stress $\\sigma_c$.\nThese considerations imply that, \nfor cohesive granular media,\nthe volume of phase space occupied by the jammed\nstate expands as the cohesion increases.\n\nHow is the jammed phase diagram\nfor granular media modified by adding liquid?\nA schematic phase diagram is shown in Fig.~\\ref{Jamming}.\nThe number in each curve there shows how the behaviour changes\nas we increase liquid content.\nWhen a small amount of liquid is added, \nthe phase boundaries are slightly shifted \nso that the jammed phase expands \nin all three directions of $1\/\\nu$, $\\tau$, and $T$\n(number 1, green dotted lines).\nAs liquid is further added, the cohesion {\\it increases},\nand the boundary will shift further.\nAt a certain liquid content, the\ncohesion will reach a maximum, \ncorresponding to the {\\it largest jammed region}\nin phase space (number 2, blue dashed lines).\nAs we further increase the liquid content,\nthe cohesion {\\it decreases}, and the \nphase-space volume of the jammed region decreases\n(number 3, red dash-dotted line).\nNamely, as far as cohesion is concerned, \nthere is an optimal liquid \ncontent that maximises the jammed region.\nThis argument\ndoes not consider the effects of liquid\nother than cohesion.\nFor large enough liquid content, \nthe lubrication effect may affect this conclusion,\nand could drastically shrink the size of the jammed region.\nThe liquid content dependence of the jamming behaviour\nin wet granular media\nis an important and interesting \nphenomenon that has not been systematically investigated\nand constitutes a fertile new direction of research. \n\n\\subsubsection{Statistical mechanics approach}\nRecently, many physicists are trying to \ndescribe dry granular systems using statistical mechanics approaches,\nespecially in the jammed regime (For a recent review,\nsee \\cite{Richard05}). \n\nEdwards \\cite{Edwards,Edwards02} postulated that \na dense granular assembly \nunder small external perturbations can\ntake all possible jammed configurations,\nand the density can be described by suitable ensemble averages over \nits blocked states. Later, Nowak {\\it et al.} \\cite{Nowak98} investigated\nthe compaction of slowly tapped granular assemblies,\nand found there exists a ``reversible regime'': \nWhen a loosely packed configuration is tapped,\nlarge voids are removed, which results in irreversible\nmotion of grains (the initial loosely packed configuration\ncannot be reproduced by further tapping).\nOnce the memory of this initial configuration is\nlost, after large enough tapping,\nthe density is determined by the ratio $\\Gamma$\nof the tapping amplitude to gravity,\nwhich is in the reversible regime; \nin this regime, the density is lower for larger $\\Gamma$.\nThe existence of a reversible regime is non-trivial\nin granular matter, where the frictional force depends on history.\nThe Edwards' scenario only considers non-history dependent\nsituations, and the existence of such a regime gives hope\nthat this scenario would work in the reversible regime.\nResearch to test this theory has been done\n\\cite{Brey00,Tarjus04,Nicodemi99,Coniglio01,Makse02,Fierro02,Nicodemi02}, \nand there remains many open questions.\n\nAnother example of a statistical mechanics approach\nis the recent experiment by \\cite{DAnna03}, \nwho observed the motion of \na torsion oscillator immersed in a granular medium\nperturbed by external vertical tapping,\nand measured  the susceptibility\nand the auto-correlation function of the motion \nof the oscillator.\nFor the equilibrium systems,\nthese two quantities are related by\nthe fluctuation-dissipation theorem and \nthe temperature can be consistently determined.\nThey \\cite{DAnna03} examined whether the same formalism works in \nthe granular state, and found that\nan ``effective temperature'' can be defined\nfrom the susceptibility and the auto-correlation function.\nThough this dissipative system is far from equilibrium, \nthe analogy to the fluctuation-dissipation theorem \nin the experiment is remarkable. It is an interesting question \nwhether such a relation \ncan be extended to other systems or regimes\n(e.g., wet grains).\n\nIn the case of wet granular media, the available phase space \nis larger and,\ndue to the cohesion, the history dependence is stronger than the dry case \nas mentioned in Section \\ref{compaction} \nand Table \\ref{Table:DryWet}. \nFuture experiments probing the statistical mechanics \nin wet granular assemblies will \nprovide more knowledge of its applicability and generality.\n\\subsubsection{Arches and contact-force fluctuations}\nArches (i.e., effective long-range interactions)\nand the resulting large fluctuation of contact forces\nhave been extensively studied in \nseveral types of dry granular media \n(see, e.g., \\cite{Hartley03,Albert00,Liu95,Howell99,Miller96,\nCoppersmith96,Blair01}).\nHowever, no such systematic studies exist\nfor wet granular media.\nThe clogging of cohesive powders \nin a hopper is often said to be an arching effect\n\\cite{Nedderman},\nbut it is rather rare to investigate\nthe arches directly in partially wet granular media.\nThis would be interesting \nbecause the change of the inter-particle forces\nwith liquid content\nlikely affects the strength and \ntopology of the so-called \n``force chains'' or stress-line-networks\nof the granular assembly.\n\n\\subsubsection{Simple experimental setups to \nstudy the dynamics of wet granular media}\nSeveral experimental setups have been\nwidely performed to study the fundamental \ndynamics of dry granular media.\nThe widely-used setups include\nvibrated granular media (e.g., \n\\cite{Losert99,Melo95,\nDAnna03,\nUmbanhowar96,Pak94,Laroche89,DAnna01,\nGranularSeg})),\nrotating drums (e.g., \n\\cite{Rajchenbach90,Rajchenbach02,Bonamy02,RajchenbachReview,\nBenza93,Ristow96,Khakhar97,Baumann95}),\nshear (e.g., \\cite{Xu03,Mueth00,Miller96,Nasuno97}), and\ninclined chutes (e.g.,\n\\cite{Poschel93,Mitarai02,Mitarai03,Mitarai04,\nMitarai05,\nPouliquen99,\nDaerr99,Savage89,Komatsu01,Bretz92}).\nThese setups have played important roles in \nstudying the fundamental dynamics and \nrheology of dry granular materials. \n\nSome of the setups have been used in recent experiments\nof wet granular media, as discussed in \nsection \\ref{dynamicalbehaviours},\nbut not so many systematic studies have been done\nso far. \nOne of the difficulties in fundamental studies \nof wet granular materials is their strong \ntendency to be inhomogeneous.\nIt becomes difficult to induce particle motions,\ndue to cohesive force. \nIn other words, the regions that were mobile\nin the dry case become localised when wet,\nand the bulk material becomes solid.\nSimilar behaviours are known for dry granular materials as\nwell (e.g., shear bands), but the localisation would be \nstronger for the wet case.\nIn addition, the distribution of liquid can \nalso be inhomogeneous, \nespecially for larger liquid content.\nNonetheless,\nfurther studies on the dynamics of wet granular materials\n(including their inhomogeneity)\nusing these rather simple setups\nwill certainly contribute to granular science.\n\n\\subsubsection{Numerical simulations}\n\nThe recent remarkable progress in\nthe study of dry granular material is \npartially \ndue to numerical simulations.\nMolecular dynamics simulations of soft-sphere \nor inelastic hard-sphere models help\nclarify phenomena found in experiments\nand get data in ``ideal'' situations.\nIn the case of wet granular media, however, \nits numerical simulation model \nfor the wide range of liquid content\nhas not been established yet because of its complexity.\n\nSome models \nof wet grains in the pendular state have been proposed \nfor molecular dynamics simulations \n(e.g., \\cite{Groger03,Schulz03,Lian98,Nase01,Hakuno93,\nGranulationReview,GranulationProceedings}),\nwhere most of them are based on \nsoft-particle models for dry granular materials \nwith elastic force and dissipation \\cite{Cundall79}.\nIn models for wet grains, the effect of the liquid \nis added by assuming that liquid bridges \nare formed when grains are in contact, \nand cohesion and dissipation \ndue to the liquid bridges are taken into account.\nThese types of models would be applicable to some extent\nto situations where the amount of liquid is small\nand the liquid mostly sticks to the grain surface.\nHowever, as the liquid content is increased,\nthe liquid-bridge picture becomes invalid \nand liquid motion becomes relevant.\nNo available simple models has been yet proposed \nfor this regime, as far as we know.\n\nEven for dry granular media, \nthe real granular particles are more complicated than\nthe ones used in\nmost computational models; still, the simple models have been\nfound to be very useful.\nIt is apparent that good models are also\nneeded for wet granular\n material.\nThe required level of realism of the model largely depends on \nthe phenomena that one would like to understand.\nTo help the modelling,\nmore systematic experiments in simple situations\nare necessary.\n\n\\subsubsection{Mechanical properties of snow}\nThe mechanical properties of snow have been\ninvestigated for a long time\n(e.g., \\cite{Fukue76,Nicot04,DAnna00,Bartelt01,snowski,Shapiro97}).\nIt is important to understand snow properties\nto reduce disasters caused by snow avalanches;\nthus knowledge should also be\nuseful to design equipment\nand constructions in snowbound areas \\cite{Shapiro97}.\n\nSnow is a type of granular material of grains of ice. \nSnow partially melts in many situations,\nand pores are filled with both air and water;\nit is a ``wet granular material''. \nThe knowledge obtained about the dynamics of dry granular media\nhas been found to be useful to understand snow dynamics;\nfor example, the mechanism of size segregation \nis important to understand how to save skiers caught in an avalanche\n\\cite{Bartelt01}.  In terms of cohesion among grains,\nwet snow avalanches has some aspects \nin common with avalanches in partially wet granular media\n\\cite{Tegzes03}. More research on wet granular media\nshould be useful to better understand the mechanical properties of snow.\n\nHowever, there is a big difference between \nwet sands and snow; in snow, solid ice-bonds \nexist between ice grains. \nIn new or low-density snows, ice grains \nslide over each other, while in old or dense snow,\nthe deformation of solid ice-bonds dominate the interactions\n\\cite{Bartelt01}. The formation, strength, and breakage of ice-bonds \ndepend on many factors\nsuch as the heat flux and motion of water or water vapours.\nBecause of this complex nature of interactions,\nthere remains many open questions in snow physics. \nFor example, it is known that the density of snow\nis not a good parameter to characterise snow properties,\nand one needs to classify snow by considering\nits microstructure \\cite{Shapiro97}.\nIt will be useful to find a parameter \nthat is easily measured and\ncan better characterise its mechanical properties.\n\n\n\\section{Conclusion}\nWet granular media have properties \nwhich are significantly different from\ndry grains. These include enhanced cohesion\namong grains leading to very \nsteep angles of repose.\nWet granular media are\npervasive everywhere, \nincluding numerous industrial applications\nand geological phenomena.\nIn spite of the ubiquitous \npresence of wet granular media and its importance,\nrelatively little is known about it.\nThis brief overview sketched some\nof the physical properties of wet granular media, \nand identified several open problems for future studies.\n\n\\section*{Acknowledgements}\nNM thanks H. Nakanishi for helpful discussions.\nAuthors thank G. D'Anna for his comments on an early version \nof the manuscript.\nPart of this work has been done \nwhen NM was supported by the Special Postdoctoral Researcher \nProgram by RIKEN.\nNM is supported in part by a Grant-in-Aid for\nYoung Scientists(B) 17740262 from\nThe Ministry of Education, Culture, Sports,\nScience and Technology (MEXT) and \nGrant-in-Aid for Scientific Research (C) 16540344 from \nJapan Society for the Promotion of Science (JSPS).\nFN is supported in part by the National Security Agency (NSA)\nand Advanced Research and Development Activity (ARDA) under Air\nForce Office of Research (AFOSR) contract number F49620-02-1-0334,\nand by the National Science Foundation grant No. EIA-0130383.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nAn accurate description of quantum many-electron dynamics is one of the most challenging problems in the modern theoretical physics. Because a direct application of the standard many-body techniques to interacting nonequilibrium systems becomes prohibitively complicated, the time-dependent density functional theory (TDDFT)\\cite{TDDFT2006} remains practically the only available tool for modelling dynamics of realistic systems at a reasonable computational cost. By formally avoiding a full solution of the complicated many-body problem, TDDFT gives a direct access to experimentally relevant observables, such as the density of particles or the density of current. Within the most common Kohn-Sham (KS) implementation of TDDFT the exact density is calculated by propagating mean-field-like equations of motion for a fictitious system of noninteracting KS particles, which enormously simplifies the problem. Therefore a growing popularity of TDDFT is absolutely not surprising (see, e.~g., recent special issues of PCCP \\cite{TDDFT-PCCP} and TEOCHEM \\cite{TDDFT-TEOCHEM}, as well as this volume).\n\nPractical applications of TDDFT to realistic many-electron systems unavoidably employ approximations. However, in almost every paper, where TDDFT is applied, it is introduced as a formally exact theory. Unfortunately, at the present stage, the common statements of exactness express our expectations\/believes more than the actual situation in the field of mathematical foundations of TDDFT and related theories. At the formal level TDDFT is based on two main statements: (i) A map from the external potential to the density is unique and invertable; (ii) A given density can be obtained from the Schr\\\"odinger dynamics driven by a properly chosen external potential (for example, the density coming from the interacting system can be reproduced by adjusting a potential in a fictitious noninteracting system). The first statement is known as the Runge-Gross mapping theorem \\cite{RunGro1984}. The statement (ii) is commonly referred to as a ${\\cal V}$-representability problem, which is a corner stone of the KS formalism. \n\nA disturbing fact is that the Runge-Gross proof \\cite{RunGro1984} of the mapping theorem is only valid for potentials which are analytic functions of time ($t$-analytic) and can be represented around the initial time by a convergent Taylor series. The conditions of the ${\\cal V}$-representability theorem, as formulated by van~Leeuwen, \\cite{vanLeeuwen1999} are even more restrictive. One also assumes the $t$-analyticity of the density to uniquely recover coefficients in the power series expansion of the potential supporting that density, while a uniform convergence of the recovered series is taken for granted. Therefore our present justification of TDDFT and its generalizations, such as time-dependent current density functional theory (TDCDFT) \\cite{GhoDha1988,Vignale2004}, is limited to a very narrow class of external potentials. It should be emphasized that the $t$-analyticity of the potential, and even its infinite smoothness in space, are not sufficient to guarantee the $t$-analyticity of the density \\cite{Maitra2010} required for the present formulation of the ${\\cal V}$-representability theorem \\cite{vanLeeuwen1999}. Hence, strictly speaking, the precise conditions of ${\\cal V}$-representability are still unknown. Obviously, a better understanding of the fundamental issues of TDDFT is highly desirable, especially in a view of explosively growing number of practical applications.\n\nA commonly accepted weak point of the existing proofs both in TDDFT \\cite{RunGro1984,vanLeeuwen1999} and in TDCDFT \\cite{GhoDha1988,Vignale2004} is that the mathematical statement of the problem explicitly relies on the power series representation of the densities and\/or potentials. It is absolutely clear that no further progress is possible without finding a new formulation, which does not require, at least in principle, the power series expansion of the basic quantities. In several recent works \\cite{TokatlyPRB2007,TokatlyPCCP2009,Maitra2010,Tokatly2010arxiv,RugLee2010arxiv} such a new route to the proof of existence of TDDFT-like theories has been proposed. Namely, the density-potential mapping and the ${\\mathcal V}$-representability problems can be restated in the form of the existence and uniqueness of solutions to a certain universal nonlinear Schr\\\"odinger equation (NLSE). The universal NLSE has been first formulated in the context of the time-dependent deformation functional theory (TDDefFT)\\cite{TokatlyPRB2007,TokatlyPCCP2009}, where it appears as a natural step in solving the many-body problem in the comoving reference frame. In Ref.~\\cite{Maitra2010} a conceptually similar formulation has been proposed for TDDFT, and more recently the NLSE approach has been adapted to rigorously prove the uniqueness and existence theorems for a lattice version of TDCDFT \\cite{Tokatly2010arxiv}.\n\nDespite the NLSE formulations proposed for TDDefFT in \\cite{TokatlyPRB2007,TokatlyPCCP2009} and for TDDFT in \\cite{Maitra2010} are similar conceptually, their formal realizations look very different and appear to be completely disconnected. One of the aims of the present paper is find a relation between these two theories. I will derive the NLSE formulation of TDCDFT and show that it is precisely the missing link which bridges the two abovementioned proposals. We will see that the concept of a universal many-body NLSE provides us with a unified point of view on the whole family of TDDFT-like formalisms, namely, TDCDFT, TDDefFT, and TDDFT itself. This unification is the main result of the present work.\n\nThe structure of the paper is the following. In Sec.~IIA I formulate the NLSE approach to the mapping and the ${\\mathcal V}$-representability problems in TDCDFT. An equivalence of the derived NLSE for TDCDFT and the NLSE for TDDefFT \\cite{TokatlyPRB2007,TokatlyPCCP2009} is proven in  Sec.~IIB. In Sec.~IIC I establish a connection of the NLSE formalism to the Vignale's approach to the TDCDFT mapping theorem \\cite{Vignale2004}. In Sec.~III I derive a new NLSE formulation of TDDFT, and demonstrate its connection, on one hand, to the TDCDFT-NLSE of Sec.~II, and, on the other hand to TDDFT-NLSE proposed in Ref.~\\cite{Maitra2010}. This uncovers interrelations between all members of the TDDFT-family and completes the unification.\nThe main results of the paper are summarized in Conclusion.\n\n\n\\section{Time-dependent current density functional theory}\n\n\\subsection{Nonlinear inverse many-body problem in TDCDFT}\n\nLet us consider a system of $N$ identical particles in the presence of time dependent external scalar $U({\\bf x},t)$ and vector ${\\bf A}({\\bf x},t)$ potentials. The corresponding many-body wave function \n$\\Psi({\\bf x}_{1},\\dots,{\\bf x}_{N},t)$\nis a solution to the time-dependent Schr\\\"odinger equation\n\\begin{equation}\n i\\partial_t\\Psi({\\bf x}_{1},\\dots,{\\bf x}_{N},t) = H\\Psi({\\bf x}_{1},\\dots,{\\bf x}_{N},t)\n\\label{SE}\n\\end{equation}\nwith the following Hamiltonian \n\\begin{equation} \nH = \\sum_{j = 1}^{N}\\left[\\frac{(-i\\nabla_{j} \n- {\\bf A}({\\bf x}_{j},t))^{2}}{2m} + U({\\bf x}_{j},t)\\right] +\n\\frac{1}{2}\\sum_{j\\ne k} V(|{\\bf x}_{j}-{\\bf x}_{k}|)\n\\label{H}\n\\end{equation}\nwhere $\\nabla_{j}=\\partial\/\\partial{\\bf x}_j$, and $V(|{\\bf x}-{\\bf x}'|)$ is the interaction potential. For a given initial condition, \n\\begin{equation}\n\\Psi({\\bf x}_{1},\\dots,{\\bf x}_{N},0) = \\Psi_0({\\bf x}_{1},\\dots,{\\bf x}_{N}),\n \\label{InitialPsi}\n\\end{equation}\nthe dynamics of the system is completely specified by Eq.~(\\ref{SE}). \n\nThe Schr\\\"odinger equation (\\ref{SE}) is invariant under the following gauge transformation\n\\begin{eqnarray}\n \\label{Psi-gauge}\n\\Psi(t)&\\to& e^{-i\\sum_{j}\\chi({\\bf x}_{j},t)}\\Psi(t),\\\\\n\\label{A-gauge}\n{\\bf A}({\\bf x},t) &\\to& {\\bf A}({\\bf x},t) + \\nabla\\chi({\\bf x},t), \\\\\n\\label{U-gauge}\nU({\\bf x},t) &\\to& U({\\bf x},t) - \\partial_t \\chi({\\bf x},t)\n\\end{eqnarray}\nBecause the physical results are independent of the gauge we are free to choose a gauge function $\\chi({\\bf x},t)$ which is most suitable for a particular problem. We will see that TDCDFT is most elegantly formulated in a temporal gauge that corresponds to the following choice \\cite{Vignale2004}\n\\begin{equation}\n \\label{t-gauge}\n\\chi({\\bf x},t) = \\int_{0}^{t}U({\\bf x},t')dt'.\n\\end{equation}\nThe transformation with the gauge function (\\ref{t-gauge}) eliminates the scalar potential so that the many-body Hamiltonian reduces to the form\n\\begin{equation} \nH[{\\bf A}] = \\sum_{j = 1}^{N}\\frac{(-i\\nabla_{j} - {\\bf A}({\\bf x}_{j},t))^{2}}{2m} +\n\\frac{1}{2}\\sum_{j\\ne k} V(|{\\bf x}_{j}-{\\bf x}_{k}|).\n\\label{H-t-gauge}\n\\end{equation}\n\nThe key physical observables in TDCDFT are the density of particles $n({\\bf x},t)$ and the current density ${\\bf j}({\\bf x},t)$:\n\\begin{eqnarray}\n\\label{n-def}\nn({\\bf x},t) &=& \\langle\\Psi(t)|\\hat{n}({\\bf x})|\\Psi(t)\\rangle, \\\\\n \\label{j-def}\n{\\bf j}({\\bf x},t) &=& \\langle\\Psi(t)|\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle - \\frac{n({\\bf x},t)}{m}{\\bf A}({\\bf x},t),\n\\end{eqnarray}\nwhere $\\Psi(t)$ is the solution to the Schr\\\"odinger equation (\\ref{SE}), and $\\hat{n}({\\bf x})$ and $\\hat{{\\bf j}}^{p}({\\bf x})$ are the operators of the density and of the paramagnetic current, respectively,\n\\begin{eqnarray}\n \\label{n-oper}\n\\hat{n}({\\bf x}) &=& \\sum_{j=1}^{N}\\delta({\\bf x} - {\\bf x}_j), \\\\\n\\label{jpar-oper}\n\\hat{{\\bf j}}^{p}({\\bf x}) &=& \\frac{-i}{2m}\\sum_{j=1}^{N}\\{\\nabla_{j},\\delta({\\bf x} - {\\bf x}_j)\\}.\n\\end{eqnarray}\nThe gauge invariance implies that the number of particles is locally conserved, i.~e. the density and the current are connected by the continuity equation\n\\begin{equation}\n \\label{continuity}\n\\partial_t n({\\bf x},t) + \\nabla{\\bf j}({\\bf x},t)=0.\n\\end{equation}\nThis, in particular, means that $n({\\bf x},t)$ is uniquely determined by the current ${\\bf j}({\\bf x},t)$ and the initial density distribution $n({\\bf x},0)=n_0({\\bf x})$.\n\nThe usual statement of the problem in quantum mechanics corresponds to solving the time-dependent Schr\\\"odinger equation (\\ref{SE}), (\\ref{H-t-gauge}), for a given initial condition (\\ref{InitialPsi}) and the external driving potential ${\\bf A}({\\bf x},t)$. This determines the many-body wave function $\\Psi[\\Psi_0,{\\bf A}](t)$ and thus any observable, e.~g. the current ${\\bf j}[\\Psi_0,{\\bf A}]({\\bf x},t)$, as unique functionals of the initial state $\\Psi_0$ and the vector potential ${\\bf A}$. The existence of TDCDFT assumes that the inverse problem also possess a unique solution. Namely, TDCDFT is valid if, given the initial state $\\Psi_0$ and the current ${\\bf j}({\\bf x},t)$, one can uniquely reconstruct the time-dependent wave function $\\Psi[\\Psi_0,{\\bf j}](t)$ and the potential ${\\bf A}[\\Psi_0,{\\bf j}]({\\bf x},t)$ which supports the prescribed current. \n\nThe key observation is that, by ``reinterpreting'' the definition (\\ref{j-def}) of the current, the above inverse problem can be mathematically posed in a form of the following {\\it nonlinear} system of equations\n\\begin{eqnarray}\n \\label{NLSE-TDCDFT}\ni\\partial_t\\Psi(t) &=& H[{\\bf A}]\\Psi(t), \\\\\n\\label{A(j)}\n{\\bf A}({\\bf x},t) &=& \\frac{m}{n({\\bf x},t)}\\left[\\langle\\Psi(t)|\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle - {\\bf j}({\\bf x},t)\\right],\n\\end{eqnarray}\nwhich has to be solved with the initial condition (\\ref{InitialPsi}). The Hamiltonian in (\\ref{NLSE-TDCDFT}) still has a form of Eq.~(\\ref{H-t-gauge}). However, now the vector potential ${\\bf A}$ is not fixed externally, but calculated selfconsistently from Eq.~(\\ref{A(j)}). The density $n({\\bf x},t)$ entering (\\ref{A(j)}) is determined either by intergating the continuity equation (\\ref{continuity}) or calculated directly as the expectation value of Eq.~(\\ref{n-def}). Both definitions are identical if the wave function $\\Psi(t)$ satisfies Eq.~(\\ref{NLSE-TDCDFT}). By inserting the selfconsistent vector potential (\\ref{A(j)}) into (\\ref{NLSE-TDCDFT}) we obtain NLSE with a special cubic nonlinearity. The solution of the Cauchy problem for this NLSE, provided it is well posed, returns the time-dependent many-body wave function and the vector potential as universal functionals of the initial state $\\Psi_0$ and the given current ${\\bf j}({\\bf x},t)$. Therefore the problem of existence of TDCDFT reduces to proving the existence and uniqueness of solutions to NLSE defined by Eqs.~(\\ref{NLSE-TDCDFT}) and (\\ref{A(j)}).\n\nTo illustrate how the NLSE approach works I consider the simplest case of one quantum particle. For $N=1$ the nonlinear Cauchy problem (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}), and (\\ref{InitialPsi}) simplifies as follows\n\\begin{eqnarray}\n \\label{SE-1part}\ni\\partial_t\\Psi({\\bf x},t) &=& \\frac{1}{2m}(-i\\nabla-{\\bf A}({\\bf x},t))^2\\Psi({\\bf x},t),\\\\\n\\label{A(j)-1part}\n{\\bf A}({\\bf x},t) &=& \\frac{1}{|\\Psi|^2}\\left[\\frac{-i}{2}(\\Psi^*\\nabla\\Psi - \\Psi\\nabla\\Psi^*)-m{\\bf j}({\\bf x},t)\\right],\\\\\n\\label{Init-1part}\n\\Psi({\\bf x},0) &=& \\Psi_0({\\bf x}) \\equiv \\sqrt{n_0({\\bf x})}e^{i\\varphi_0({\\bf x})}.\n\\end{eqnarray}\nOn the first sight this nonlinear problem may look complicated, but it turns out to be trivially integrable. The corresponding exact analytic solution can be represented in the form\n\\begin{eqnarray}\n \\label{Psi-1part}\n\\Psi[\\Psi_0,{\\bf j}]({\\bf x},t) &=& \\sqrt{n({\\bf x},t)}e^{i\\varphi({\\bf x},t)},\\\\\n\\label{A-1part}\n{\\bf A}[\\Psi_0,{\\bf j}]({\\bf x},t) &=& \\nabla\\varphi({\\bf x},t) - m \\frac{{\\bf j}({\\bf x},t)}{n({\\bf x},t)},\n\\end{eqnarray}\nwhere the functions $n({\\bf x},t)$ and $\\varphi({\\bf x},t)$ are defined as follows\n\\begin{eqnarray}\n \\label{n-1part}\nn({\\bf x},t) &=& n_0({\\bf x}) - \\int_0^t dt' \\nabla{\\bf j}({\\bf x},t'),\\\\\n\\label{phase-1part}\n\\varphi({\\bf x},t) &=& \\varphi_0({\\bf x}) + \\int_0^t dt'\\left[\\frac{\\nabla^2\\sqrt{n({\\bf x},t')}}{2m\\sqrt{n({\\bf x},t')}} -\n\\frac{m{\\bf j}^2({\\bf x},t')}{2n^2({\\bf x},t')}\\right].\n\\end{eqnarray}\nEquations (\\ref{Psi-1part})--(\\ref{phase-1part}) provide us with an explicit example of the universal TDCDFT functionals $\\Psi[\\Psi_0,{\\bf j}](t)$ and ${\\bf A}[\\Psi_0,{\\bf j}]({\\bf x},t)$ recovered from the NLSE (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}) for a particular case of $N=1$. \n\nApparently the general NLSE for any $N>1$ is not solvable analytically. However the mathematical structure of equations remains practically the same, which leaves a strong hope that early or later the uniqueness and existence theorems for the general nonlinear many-body problem, and thus the existence of TDCDFT, will be proved. In fact, recently I have succeeded to prove the corresponding theorems for a lattice TDCDFT \\cite{Tokatly2010arxiv} which is formulated in terms of a discrete (actually finite-difference) version of the above NLSE.\n\n\n\\subsection{Equivalence of the nonlinear universal problems in TDCDFT and TDDefFT}\nThe NLSE formulation has been first formulated in the context of TDDefFT where it appears as a natural ``universal'' step in solving the many-body in the comoving reference frame \\cite{TokatlyPRB2007,TokatlyPCCP2009}. Below I will show that there is a one-to-one correspondence between the universal problem derived in Refs.~\\cite{TokatlyPRB2007,TokatlyPCCP2009} for TDDefFT and the NLSE (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}) for TDCDFT. To prove this correspondence we proceed as follows. First, for a given current ${\\bf j}({\\bf x},t)$, we integrate the continuity equation (\\ref{continuity}) to calculate the density $n({\\bf x},t)$, and then define the velocity field:\n\\begin{equation}\n \\label{velocity}\n{\\bf v}({\\bf x},t) = \\frac{{\\bf j}({\\bf x},t)}{n({\\bf x},t)}.\n\\end{equation}\nUsing the velocity field ${\\bf v}({\\bf x},t)$ one finds a set of Lagrangian trajectories ${\\bf x}(\\bm{\\xi},t)$ by solving the following initial value problem (see, e.~g. Refs.~\\cite{TokatlyPRB2007,TokatlyPCCP2009})\n\\begin{equation}\n \\label{trajectory}\n\\dot{\\bf x}(\\bm{\\xi},t) = {\\bf v}({\\bf x}(\\bm{\\xi},t),t)\\, \\qquad {\\bf x}(\\bm{\\xi},0) = \\bm{\\xi},\n\\end{equation}\nwhere $\\dot{\\bf x}(\\bm{\\xi},t)=\\partial_t{\\bf x}(\\bm{\\xi},t)$. Physically the function ${\\bf x}(\\bm{\\xi},t)$ is a trajectory of an infinitesimal fluid element which start its motion at $t=0$ from the point $\\bm{\\xi}$. The formal significance of the trajectory function ${\\bf x}(\\bm{\\xi},t)$ is that the equation ${\\bf x}={\\bf x}(\\bm{\\xi},t)$ defines a transformation of coordinates ${\\bf x}\\to\\bm{\\xi}$ which corresponds the transformation from the laboratory frame (${\\bf x}$-space) to the comoving Lagrangian frame ($\\bm{\\xi}$-space). \n\nThe last step in proving the equivalence of NLSE in TDCDFT and TDDefFT is to rewrite Eqs.~(\\ref{NLSE-TDCDFT}) and (\\ref{A(j)}) in the comoving frame. Formally we perform the transformation of coordinates ${\\bf x}\\to\\bm{\\xi}$ : ${\\bf x}={\\bf x}(\\bm{\\xi},t)$ and define the transformed many-body wave function $\\widetilde{\\Psi}(\\bm{\\xi}_1,\\dots,\\bm{\\xi}_N,t)$ in the comoving frame as follows \\cite{TokatlyPRB2007,TokatlyPCCP2009}\n\\begin{equation}\n\\widetilde{\\Psi}({\\bm\\xi}_{1},\\dots,{\\bm\\xi}_{N},t)\n= \\prod_{j = 1}^{N}g^{\\frac{1}{4}}({\\bm\\xi}_{j},t)\ne^{-iS_{\\text{cl}}({\\bm\\xi}_{j},t)}\\Psi({\\bf x}({\\bm\\xi}_{1},t),\\dots,{\\bf x}({\\bm\\xi}_{N},t),t),\n\\label{Psi-Lagr}\n\\end{equation}\nwhere $g(\\bm{\\xi},t)={\\rm det}[g_{\\mu\\nu}(\\bm{\\xi},t)]$ is the determinant of the metric tensor induced by the transformation of coordinates ($\\sqrt{g}$ is the Jacobian of the transformation)\n\\begin{equation}\n \\label{metric}\ng_{\\mu\\nu}(\\bm{\\xi},t) = \\frac{\\partial x^{\\alpha}}{\\partial\\xi^{\\mu}}\n\\frac{\\partial x^{\\alpha}}{\\partial\\xi^{\\mu}}; \\quad\ng^{\\mu\\nu}(\\bm{\\xi},t) = [g_{\\mu\\nu}]^{-1} = \\frac{\\partial\\xi^{\\mu}}{\\partial x^{\\alpha}}\n\\frac{\\partial\\xi^{\\nu}}{\\partial x^{\\alpha}},\n\\end{equation}\nand $S_{\\text{cl}}({\\bm\\xi},t)$ is the classical action of a particle moving along the trajectory ${\\bf x}(\\bm\\xi,t)$\n\\begin{equation}\n\\label{action}\nS_{\\text{cl}}({\\bm\\xi},t)= \\int_0^t\\left[ \n\\frac{m}{2}(\\dot{\\bf x}(\\bm{\\xi},t))^{2} \n+ \\dot{\\bf x}(\\bm{\\xi},t){\\bf A}({\\bf x}(\\bm{\\xi},t),t)\\right].\n\\end{equation}\nNote that the factor $\\prod_{j = 1}^{N}g^{\\frac{1}{4}}({\\bm\\xi}_{j},t)$ in (\\ref{Psi-Lagr}) ensures the standard normalization of the wave function $\\langle\\widetilde{\\Psi}|\\widetilde{\\Psi}\\rangle=1$ after a non-volume-preserving transformation of coordinates. After straightforward algebra we find that the transformed Schr\\\"odinger equation (\\ref{NLSE-TDCDFT}) and the ``self-consistency'' equation (\\ref{A(j)}) can be reduced to the following form\n\\begin{eqnarray}\n \\label{NLSE-DFDefFT}\ni\\partial_t\\widetilde{\\Psi}({\\bm\\xi}_{1},\\dots,{\\bm\\xi}_{N},t) &=&\n\\widetilde{H}[g_{ij},\\bm{\\mathcal A}]\\widetilde{\\Psi}({\\bm\\xi}_{1},\\dots,{\\bm\\xi}_{N},t),\\\\\n\\label{A(g)}\n{\\bm{\\mathcal A}}(\\bm{\\xi},t) &=& \\frac{m}{n_0(\\bm{\\xi})}\\langle\\widetilde{\\Psi}(t)|\\hat{{\\bf j}}^{p}(\\bm{\\xi})|\\widetilde{\\Psi}(t)\\rangle.\n\\end{eqnarray}\nIn Eq.~(\\ref{NLSE-DFDefFT}) $\\widetilde{H}[g_{ij},\\bm{\\mathcal A}]$ is the Hamiltonian (\\ref{H-t-gauge}) transformed to the comoving frame:\n\\begin{equation}\n \\label{H-Lagr}\n\\widetilde{H}[g_{ij},\\bm{\\mathcal A}] = \\sum_{j = 1}^{N}g^{-\\frac{1}{4}}_{j}\n\\hat{K}_{j,\\mu}\\frac{\\sqrt{g_{j}}g^{\\mu\\nu}_{j}}{2m}\n\\hat{K}_{j,\\nu}g^{-\\frac{1}{4}}_{j}\n+\\frac{1}{2}\\sum_{k\\ne j}V(l_{\\bm\\xi_{k}\\bm\\xi_{j}})\n\\end{equation}\nwhere $\\hat{K}_{j,\\mu}=-i\\partial_{\\xi^{\\mu}_{j}}\n- {\\cal A}_{\\mu}(\\bm\\xi_{j},t)$, $g^{\\mu\\nu}_{j}=g^{\\mu\\nu}(\\bm\\xi_{j},t)$, and $l_{\\bm\\xi_{k}\\bm\\xi_{j}}$ is the distance between $j$th and $k$th particles in the moving frame (the length of geodesic connecting points $\\bm\\xi_{j}$ and $\\bm\\xi_{k}$ in the space with metric $g_{\\mu\\nu}$). The selfconsistent vector potential ${\\bm{\\mathcal A}}(\\bm{\\xi},t)$ in (\\ref{NLSE-DFDefFT}), (\\ref{A(g)}) is related to ${\\bf A}({\\bf x},t)$ entering (\\ref{NLSE-TDCDFT}) and (\\ref{A(j)}) as follows\n \\begin{equation}\n \\label{A-TDDefFT}\n{\\mathcal A}_{\\mu}(\\bm{\\xi},t) = \\frac{\\partial x^{\\nu}}{\\partial\\xi^{\\mu}}A_{\\nu}({\\bf x}(\\bm{\\xi},t),t) \n+ \\frac{\\partial x^{\\nu}}{\\partial\\xi^{\\mu}}\\dot{x}^{\\nu} - \\partial_{\\xi^{\\mu}}S_{\\text{cl}}({\\bm\\xi},t).\n\\end{equation}\nThe first term in the right hand side in this equation is the transformed left hand side in (\\ref{A(j)}). The second term originates from the term $m{\\bf j}\/n$ in (\\ref{A(j)}), while the last term in (\\ref{A-TDDefFT}) comes from the expectation value of the paramagnetic current in (\\ref{A(j)}) and is related to the extra phase factor in the transformed wave function (\\ref{Psi-Lagr}).\n\nThe nonlinear system of equations (\\ref{NLSE-DFDefFT}), (\\ref{A(g)}) is exactly the universal many-body problem formulated within TDDefFT (see Eqs.~(58), (59) in \\cite{TokatlyPRB2007} or Eqs.~(3.38), (3.39) in \\cite{TokatlyPCCP2009}). In fact, the NLSE in TDCDFT and TDDefFT correspond to the same mathematical problem formulated in the laboratory or the comoving frame respectively. Hence to prove the existence of TDCDFT we are free to choose any of the two formulations. The advantage of the comoving frame formulation is a much simpler form of the selfconsistency equation. Indeed Eq.~(\\ref{A(g)}) does not contain an inhomogeneous term and involves only the initial density distribution $n_0$. The price for that -- the appearance of the time-dependent metric tensor in the Schr\\\"odinger equation -- is probably not too high from the conceptual point of view. On the other hand, the formulation derived in this paper, Eqs.~ (\\ref{NLSE-TDCDFT}) and (\\ref{A(j)}), is much easier to digest as it is based on the usual quantum dynamics in the laboratory frame. \n\n\\subsection{Connection of the nonlinear inverse problem in TDCDFT to the Vignale proof of the mapping theorem}\n\nIn Ref.~\\cite{Vignale2004} Vignale generalized the van~Leeuwen construction \\cite{vanLeeuwen1999} to prove the mapping theorem for TDCDFT. The main idea of the proof is to relate the vector potential ${\\bf A}({\\bf x},t)$ to the current and the many-body wave function using the force balance equation. Then this relation has been interpreted as an equation of motion for ${\\bf A}({\\bf x},t)$ and solved recursively assuming $t$-analyticity of the vector potential. The present subsection is aimed at demonstrating that the statement the problem used in \\cite{Vignale2004} directly follows from the nonlinear system of Eqs.~(\\ref{NLSE-TDCDFT}) and (\\ref{A(j)}).\n\nLet us first multiply both sides of (\\ref{A(j)}) with the density $n({\\bf x},t)$ and differentiate the result with respect to time:\n\\begin{equation}\n \\label{dA(j)\/dt}\nn\\partial_t{\\bf A} + {\\bf A}\\partial_t n = m\\partial_t\\langle\\Psi(t)|\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle \n- m\\partial_t{\\bf j} .\n\\end{equation}\nThe time derivative of the paramagnetic current ${\\bf j}^{p}({\\bf x},t)\\equiv\\langle\\Psi(t)|\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle$ in the right hand side of (\\ref{dA(j)\/dt}) can be straightforwardly calculated using the Schr\\\"odinger equation (\\ref{NLSE-TDCDFT}). The result of the differentiation takes the following form\n\\begin{equation}\n \\label{djp\/dt}\n m\\partial_t j_{\\mu}^p = A_{\\mu}\\partial_t n + \\left[\\left({\\bf j}^{p} - \\frac{n}{m}{\\bf A}\\right)\\times{\\bf B}\\right]_{\\mu}\n- \\partial_{\\nu}\\Pi_{\\mu\\nu},\n\\end{equation}\nwhere ${\\bf B}=\\nabla\\times{\\bf A}$ is the magnetic field associated with the selfconsistent vector potential, and the last term is the stress force with $\\Pi_{\\mu\\nu}({\\bf x},t)=\\langle\\Psi(t)|\\hat{\\Pi}_{\\mu\\nu}({\\bf x})|\\Psi(t)\\rangle$ being the stress tensor. An explicit form of the stress force and the stress tensor can be found, for example, in \\cite{Vignale2004,TokatlyPRB2005a,TokatlyPCCP2009}. Inserting the result of (\\ref{djp\/dt}) into (\\ref{dA(j)\/dt}) we arrive at the following equation of motion for the vector potential\n\\begin{equation}\n \\label{force-balance1}\nn\\partial_t A_{\\mu} = \\left[\\left({\\bf j}^{p} - \\frac{n}{m}{\\bf A}\\right)\\times{\\bf B}\\right]_{\\mu} - \\partial_{\\nu}\\Pi_{\\mu\\nu}\n- m\\partial_t j_{\\mu},\n\\end{equation}\nwhich is exactly the force balance equation. Using the definition of (\\ref{j-def}), which is an absolutely legitimate operation at the solution point, we find that (\\ref{force-balance1}) becomes identical to Eq.~(10) in \\cite{Vignale2004}. Hence the force balance equation, which has been used in \\cite{Vignale2004} to construct a recursion for ${\\bf A}[{\\bf j}](t)$, is nothing but the time derivative of the selfconsistency equation (\\ref{A(j)}). In other words, the force balance equation, considered as the equation of motion for the vector potential ${\\bf A}({\\bf x},t)$, can be integrated explicitly and its solution is given by (\\ref{A(j)}).\n\nApparently the equation for ${\\bf A}$ (\\ref{A(j)}) is advantageous as it leads to a much simpler and cleaner form of the nonlinearity in the inverse many-body problem. Another important point is that Eq.~(\\ref{A(j)}) is local in time. Just from the structure of the nonlinear problem (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}) we immediately see that, if the solution exists, the vector potential ${\\bf A}[{\\bf j}](t)$ is a strictly retarded functional of the current ${\\bf j}({\\bf x},t)$. In particular, it can not contain even local terms which depend on the time derivative of the current. The time locality of (\\ref{A(j)}) also makes it absolutely obvious that the power series expansion of ${\\bf A}(t)$ is unique, which is the essence of the Vignale theorem \\cite{Vignale2004}. Of course, to construct such an expansion we have to assume, as usual, the $t$-analyticity of the current. I am not presenting here the explicit construction of the recursion because it is practically trivial, and basically reproduces the recursion for the selfconsistent potential ${\\bm{\\mathcal A}}(t)$ in the universal problem (\\ref{NLSE-DFDefFT}), (\\ref{A(g)}) for TDDefFT \\cite{TokatlyPRB2007,TokatlyPCCP2009}. Hence the present derivation of the nonlinear inverse problem (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}) can be also viewed as a considerable simplification of the proof for the Vignale theorem. Clearly the convergence of the series remains unproved as in all power series based approaches to the problem of TDDFT and TDCDFT \\cite{vanLeeuwen1999,Vignale2004,TokatlyPRB2007}. The hope is that a cleanly stated nonlinear initial value problem, (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}), can be treated using more advanced mathematical tools as it has been done recently for a lattice version of the theory \\cite{Tokatly2010arxiv}.\n\n\\section{Time-dependent density functional theory}\n\nIn Sec.~II we have found that the problem of existence of TDCDFT is equivalent to that of existence and uniqueness of a solution to a certain time-dependent NLSE. We have proved a one-to-one correspondence of this NLSE to the universal nonlinear problem derived earlier for TDDefFT, and demonstrated its close relation to the Vignale's power-series proof of the TDCDFT mapping theorem. Recently a conceptually similar NLSE setup has been also proposed for TDDFT \\cite{Maitra2010}. In this section I will derive an alternative formulation of the TDDFT problem, and establish a connection between the results of Sec.~II (i.~e., the NLSE formulation of TDDefFT and TDCDFT) to the NLSE proposed in Ref.~\\cite{Maitra2010}.\n\nThe main statement of TDDFT is that the density $n({\\bf x},t)$ completely determines the many-body dynamics, provided the latter is driven by a scalar potential. Hence our starting point is the many-body Schr\\\"odinger equation (\\ref{SE}) with a Hamiltonian that contains only the external scalar potential as a driving force:\n\\begin{equation} \nH = \\sum_{j = 1}^{N}\\left[-\\frac{\\nabla_{j}^2}{2m} + U({\\bf x}_{j},t)\\right] +\n\\frac{1}{2}\\sum_{j\\ne k} V(|{\\bf x}_{j}-{\\bf x}_{k}|).\n\\label{H-U}\n\\end{equation}\nAt the formal level this standard form of the Hamiltonian assumes that a given, purely longitudinal external field is described using a Coulomb gauge ($\\nabla{\\bf A} =0$). \n\nFollowing the ideas of the previous section we perform a gauge transformation (\\ref{Psi-gauge}) with the gauge function $\\chi({\\bf x},t)$ defined by Eq.~(\\ref{t-gauge}). In other words, we switch from the original Coulomb gauge to a temporal gauge, where the scalar potential is absent, but instead the dynamics is driven by a purely longitudinal vector potential ${\\bf A}({\\bf x},t)=\\nabla\\chi({\\bf x},t)$. The Hamiltonian in the temporal gauge takes the form\n\\begin{equation} \nH[\\chi] = \\sum_{j = 1}^{N}\\frac{[-i\\nabla_{j} - \\nabla_{j}\\chi({\\bf x}_{j},t)]^{2}}{2m} +\n\\frac{1}{2}\\sum_{j\\ne k} V(|{\\bf x}_{j}-{\\bf x}_{k}|).\n\\label{H-chi}\n\\end{equation}\nObviously, the driving force is still determined by a single scalar function $\\chi({\\bf x},t)$ which equals to the potential $U({\\bf x},t)$ integrated over the time, Eq.~(\\ref{t-gauge}). \n\nSolution of the Schr\\\"odinger with the Hamiltonian $H[\\chi]$ yields the many-body wave function $\\Psi[\\Psi_0,\\chi](t)$ which can be used to calculate any observable, for example the density or the current, as a functional of the initial state $\\Psi_0$ and the driving ``potential'' $\\chi({\\bf x},t)$. It is worth noting that the density $n({\\bf x},t)$ is always given by the gauge independent Eq.~(\\ref{n-def}), while the formal expression for the current ${\\bf j}({\\bf x},t)$ in the temporal gauge acquires a diamagnetic contribution:\n\\begin{equation}\n \\label{j(chi)}\n{\\bf j}({\\bf x},t) = \\langle\\Psi(t)|\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle - \\frac{n({\\bf x},t)}{m}\\nabla\\chi({\\bf x},t)\n\\end{equation}\n\nFor the discussion below the following simple observation is of primary importance. By taking the divergence of Eq.~(\\ref{j(chi)}) and using the continuity equation (\\ref{continuity}) we find that the time derivative of the density, $\\partial_t n({\\bf x},t)$, can be calculated directly as the following expectation value\n\\begin{equation}\n \\label{dn(chi)}\n\\partial_t n({\\bf x},t) = -\\langle\\Psi(t)|\\nabla\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle +\n \\frac{1}{m}\\nabla[n({\\bf x},t)\\nabla\\chi({\\bf x},t)].\n\\end{equation}\nThe relevance of this equation for TDDFT should be obvious from the familiar looking Sturm-Liouville operator \\cite{vanLeeuwen1999} in the right hand side (the second term).\n\n\nThe validity of TDDFT means that, given the initial state $\\Psi_0$ and the density $n({\\bf x},t)$, we can uniquely recover the time-dependent many-body wave function $\\Psi[\\Psi_0,n](t)$ and the corresponding potential which gives rise to the prescribed density. In the temporal gauge this inverse problem can be formulated in terms of the following nonlinear system of equations\n\\begin{eqnarray}\n \\label{NLSE-TDDFT}\ni\\partial_t\\Psi(t) &=& H[\\chi]\\Psi(t), \\\\\n\\label{chi(n)}\n\\nabla[n({\\bf x},t)\\nabla\\chi({\\bf x},t)] &=& m\\langle\\Psi(t)|\\nabla\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle,\n+m\\partial_t n({\\bf x},t).\n\\end{eqnarray}\nwhere Eq.~(\\ref{chi(n)}) corresponds to Eq.~(\\ref{dn(chi)}) reinterpreted as an equation for $\\chi({\\bf x},t)$. The system (\\ref{NLSE-TDDFT}), (\\ref{chi(n)}) has to be supplemented with the initial condition $\\Psi(0)=\\Psi_0$, and solved for a given $n({\\bf x},t)$.\n\nThe Hamiltonian in the Schr\\\"odinger equation (\\ref{NLSE-TDDFT}) is the same as in the usual linear quantum problem, i.~e. it is defined by Eq.~(\\ref{H-chi}). However, the ``potential'' function $\\chi({\\bf x},t)$ in $H[\\chi]$ is determined selfconsistently  from Eq.~(\\ref{chi(n)}) for a given density. Mathematically the system of (\\ref{NLSE-TDDFT}), (\\ref{chi(n)}) is equivalent to NLSE with a spatially nonlocal cubic nonlinearity. Provided the Cauchy problem for this NLSE is well posed, its solution returns the functionals $\\Psi[\\Psi_0,n](t)$ and $\\chi[\\Psi_0,n]({\\bf x},t)$. A functional, which corresponds to the scalar potential in the original gauge, is calculated as $U[\\Psi_0,n]({\\bf x},t) = \\partial_t\\chi[\\Psi_0,n]({\\bf x},t)$ according to the definition (\\ref{t-gauge}) of the gauge function. Hence the proof of the density-potential mapping and the ${\\mathcal V}$-representability reduces to proving the uniqueness and the existence of solutions for the nonlinear problem (\\ref{NLSE-TDDFT}), (\\ref{chi(n)}). In this setting the existence corresponds to the ${\\mathcal V}$-representability, while the uniqueness of a solution is equivalent to the Runge-Gross mapping theorem. \n\n\nA connection of the above NLSE for TDDFT to the corresponding problem in TDCDFT (see Sec.~II) is practically obvious. Indeed Eq.~(\\ref{chi(n)}) is nothing but a divergence of Eq.~(\\ref{A(j)}) when the latter is restricted to longitudinal vector potentials of the form ${\\bf A}=\\nabla\\chi$. Hence TDDFT can be considered as a particular case of TDCDFT for systems driven by a purely longitudinal vector potential. Since in this case the vector potential is uniquely determined by a single scalar function $\\chi({\\bf x},t)$, a knowledge of a single scalar collective variable, the density $n({\\bf x},t)$, is sufficient to uniquely recover both the full many-body wave function and the driving potential. \n\nLet us now demonstrate the equivalence of Eqs.~(\\ref{NLSE-TDDFT}), (\\ref{chi(n)}) to the NLSE formulation proposed in \\cite{Maitra2010}. The argumentation below is very similar to that used in Sec.~II to prove the equivalence of our NLSE formulation (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}), and the Vignale's approach to the TDCDFT mapping problem. As a first step we differentiate Eq.~(\\ref{chi(n)}) with respect to $t$:\n\\begin{equation}\n \\label{dchi\/dt}\n\\nabla[\\dot{n}\\nabla\\chi] + \\nabla[n\\nabla\\dot{\\chi}] = m\\partial_t\\langle\\Psi(t)|\\nabla\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle\n+m\\ddot{n}.\n\\end{equation}\nNext, we use the Schr\\\"odinger equation (\\ref{NLSE-TDDFT}) to compute the time derivative of the quantity $\\langle\\Psi(t)|\\nabla\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle$ entering the right hand side in (\\ref{dchi\/dt}). This is equivalent to simply setting ${\\bf A}=\\nabla\\chi$ in Eq.~(\\ref{djp\/dt}) and taking its divergence, which yields the following result\n\\begin{equation}\n \\label{divdjp\/td}\nm\\partial_t\\langle\\Psi(t)|\\nabla\\hat{{\\bf j}}^{p}({\\bf x})|\\Psi(t)\\rangle = \\nabla[\\dot{n}\\nabla\\chi]\n-\\partial_{\\mu}\\partial_{\\nu}\\Pi_{\\mu\\nu}.\n\\end{equation}\nInserting (\\ref{divdjp\/td}) into (\\ref{dchi\/dt}) we obtain the following form of the selfconsistency equation,\n\\begin{equation}\n \\label{force-balance2}\n\\nabla[n({\\bf x},t)\\nabla U({\\bf x},t)]= m\\ddot{n}({\\bf x},t)-\\partial_{\\mu}\\partial_{\\nu}\\Pi_{\\mu\\nu}({\\bf x},t),\n\\end{equation}\nwhere $U({\\bf x},t)\\equiv\\partial_t\\chi({\\bf x},t)$ [see Eq.~(\\ref{t-gauge})] is the usual scalar potential. Apparently Eq.~(\\ref{force-balance2}) is equivalent to the equation (5) in Ref.~\\cite{Maitra2010}. More precisely, after the transformation to the Coulomb gauge, equations (\\ref{force-balance2}) and (\\ref{NLSE-TDDFT}) identically reproduce equations (5) and (6) of Ref.~\\cite{Maitra2010}. This proves the equivalence of the present formulation of TDDFT problem and NLSE proposed in \\cite{Maitra2010}. \n\nNow we can answer the question raised by the authors of Ref.~\\cite{Maitra2010} -- what is the relation between the particular NLSE derived for TDDefFT in \\cite{TokatlyPRB2007,TokatlyPCCP2009}, and the one proposed for TDDFT in \\cite{Maitra2010}. Starting from the TDDefFT universal problem, (\\ref{NLSE-DFDefFT}) and (\\ref{A(g)}), we successively perform the following transformations. (i) First, we transform the TDDefFT-NLSE of Eqs.~(\\ref{NLSE-DFDefFT}), (\\ref{A(g)}) from the comoving to the laboratory frame. The result is the TDCDFT-NLSE, (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}). (ii) Then, in the TDCDFT framework we consider only longitudinal vector potentials by setting ${\\bf A}=\\nabla\\chi$ in (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}), and taking the divergence of Eq.~(\\ref{A(j)}). This brings us to the TDDFT-NLSE in the form (\\ref{NLSE-TDDFT}), (\\ref{chi(n)}). (iii) Next, we differentiate (\\ref{chi(n)}) with respect to time and, as a result, arrive at the TDDFT-NLSE of (\\ref{NLSE-TDDFT}), (\\ref{force-balance2}). (iv) Finally, we switch from the temporal to the Coulomb gauge, and transform Eqs.~(\\ref{NLSE-TDDFT}) and (\\ref{force-balance2}) to the TDDFT-NLSE proposed in Ref.~\\cite{Maitra2010}.\n\n\\section{Conclusion}\n\nThe main result of the present work is a unified approach to the density-potential mapping and the ${\\mathcal V}$-representability problems in TDDFT, TDCDFT, and TDDefFT. The existence of each theory in the above family is mapped to the uniqueness and existence of solutions to a certain nonlinear many-body problem. We have uncovered close relations between the nonlinear problems for all three theories, and established links to all previously known approaches to TDDFT and TDCDFT. The analysis performed in this work shows a special role of TDCDFT in the whole problematics of the TDDFT-like formalisms. Indeed, starting the nonlinear universal problem, (\\ref{NLSE-TDCDFT}), (\\ref{A(j)}), for TDCDFT one can easily generate the corresponding nonlinear problems for the other members of the family. The NLSE formulation of TDCDFT demonstrates the main features of the new approach in the most transparent form. It is free of the additional Sturm-Liouville-type nonlocality of TDDFT, and does not involve complications related to the time-dependent metrics as TDDefFT. This clearly indicate that first it makes a sense to concentrate on the existence and uniqueness theorems for the TDCDFT nonlinear problem of Eqs.~(\\ref{NLSE-TDCDFT}), (\\ref{A(j)}). Provided this problem is solved the corresponding results for the other members of the family should follow almost automatically.\n\n\n\\bigskip\nThis work was supported by the Spanish MICINN (FIS2010-21282-C02-01), ``Grupos Consolidados UPV\/EHU del Gobierno\nVasco'' (IT-319-07), and the European Union through e-I3 ETSF project (Contract No. 211956).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe work that is injected to, or extracted from, a microscopic system may fluctuate significantly.\nThese fluctuations are not only relevant from a practical perspective, but they also encode fundamental results concerning the second law of thermodynamics~\\cite{Jarzynski1997,Crooks1998}. Over the last two decades this has motivated a flurry of interest in understanding and characterizing work fluctuations. In the simplest scenario, the problem can be framed as that of a system driven externally by a time-dependent work protocol, which causes its Hamiltonian to be time-dependent for a certain duration $\\tau$. \nThe work distribution $P(W)$ is then obtained by measuring the system's energy before and after the drive~\\cite{Talkner2007}.\nIn particular, a problem that has received considerable attention is the work statistics of quantum critical systems. It dates back to the early days of quantum thermodynamics \\cite{prl_silva}, and is motivated by the goal of understanding how far from equilibrium the system goes when driven across its critical point \\cite{Talkner2007, WorkDistPRL, john_chapter_book}. \n\nIn this regard, the duration $\\tau$ of the drive plays a crucial role. \nExtremely rapid dynamics (the so-called ``sudden quench regime''), tend to produce a large number of excitations and have been the focus of several recent works~\\cite{prl_silva, CampbellPRB, prx_fusco, ArgentinianPRE, 2017_quan, CampbellPRL2020, LandiPRR, CampbellPRE}. Interestingly, it has been observed that in this regime $P(W)$ for a variety of closed many-body systems tends to a Gaussian \\cite{Sotiriadis-PRE, Marino_PRB_TFIM_Gaussian, Chenu2018SciRep,AndreiPRB2019, ArgentinianPRE,Abeling2016} and therefore most papers have focused only on the first and second moments.\nAt the other extreme, for slow protocols the dynamics will be effectively adiabatic. \nThis regime has also received a lot of attention because it is at the core of thermal cycles and serves as a basis to achieve protocols in which the system remains close to equilibrium \\cite{Scandi2020, Miller2021}. Furthermore, in contrast to sudden quenches, in this case it has been shown that quantum coherences lead to non-Gaussian work distributions~\\cite{Miller2019,Scandi2020}.\n\nIn between the sudden quench and slow driving regimes,  finite time protocols can give rise to highly non-trivial behaviors, such as those captured by the Kibble-Zurek mechanism~\\cite{Kibble1976, Zurek1985,Damski2005,Zurek2005,PueblaPRR,EspositoPRL2020}. Indeed, as discussed in Ref.~\\cite{KrissaPRR}, the interplay between the time-dependent driving and the characteristic energy scales implies that contributions of different excitations change dynamically. In practice, the speed of the external driving determines which states are accessible at the beginning and the end of the process. This is encoded in the conditional probabilities, $p_{m|n}$, of the system being in state $m$ given that it was initially in state $n$. \nModifying the structure of accessible states, and consequently the possible transitions $p_{m|n}$, has a strong impact on the statistics of the work distribution that extends beyond the first two moments. \nRecently, it was further shown that the statistics of the work distribution exhibit a universal scaling analogous to the Kibble-Zurek scaling of defects~\\cite{Dong2019,EspositoPRL2020}. Moreover, finite time effects have been used to refine bounds on probabilistic violations of the second law in the work distribution~\\cite{Miller2022}.\n\nIn this work, we add to this endeavour by characterising the work distribution in terms of its Gaussianity. Focusing on the transverse field Ising chain, we carry out a detailed analysis of the skewness of the distribution as a function of the chain size and the drive time. \nWe show that there is a highly non-trivial interplay between both, which precisely captures the transition from fast to slow driving. \nIn addition, we study the relative entropy of non-Gaussianity, also known as negentropy. We explain some subtleties in employing this metric for discrete distributions, but notwithstanding demonstrate that it does provide a useful quantifier of non-Gaussianity, which goes a step beyond the skewness by also incorporating information related to higher order moments of the distribution. In particular, both methods establish that there is a clear intermediate regime between the sudden quench and adiabatic limits, where the distribution tends to become increasingly non-Gaussian.\n\n\\section{Work statistics in driven quantum systems}\n\\subsection{Distribution, Moments and Cumulants}\nWe consider an isolated quantum system, prepared in an initial state $\\rho$. At time $t=0$ the system is driven by a time-dependent  Hamiltonian $H_t= \\sum_n E_n^t |n_t\\rangle\\langle n_t\\rangle$ for a total time $\\tau$. \nThe final state of the system is thus $\\rho_\\tau =  U \\rho U^\\dagger$, where\n$U=\\mathcal{T}\\exp\\left(-\\frac{i}{\\hbar}\\int_0^\\tau ds H_s\\right)$  and $\\mathcal{T}$ is the Dyson time-ordering operator.\nWe are interested in the work distribution as the system Hamiltonian is changed from $H_0$ to $H_\\tau$. This can be obtained following the standard two-point measurement~\\cite{Talkner2007}, which consists in projectively measuring the energy of the system before and after the drive. \nThe corresponding work will then be one of the possible energy differences between the final and initial Hamiltonian, $W = E_m^\\tau - E_n^0$, which occur with probability \n\\begin{equation}\\label{PW}\n    P(W) = \\sum\\limits_{n,m}^N \\langle n_0 |\\rho| n_0\\rangle~p_{m|n}~\\delta\\left[W - (E_m^\\tau - E_n^0)\\right]\n\\end{equation}\nwhere $p_{m|n}=|\\langle m_\\tau | U| n_0 \\rangle|^2$. Note that in the adiabatic limit $p_{m|n}=\\delta_{m,n}$ and for sudden quenches $p_{m|n}=|\\braket{m_\\tau}{n_0}|^2$.\n\nThe expression simplifies when the initial state is an eigenstate of $H_0$. For instance, if $\\rho$ is  the ground state, $\\rho = |{\\rm gs}_0\\rangle\\langle {\\rm gs}_0|$, the work distribution becomes \n\\begin{equation}\\label{PW_gs}\n    P(W) = \\sum\\limits_{m} |\\langle m_\\tau | U| {\\rm gs}_0 \\rangle|^2~\\delta\\left[W - \\left(E_m^\\tau - E_{\\rm gs}^0\\right)\\right].\n\\end{equation}\nThe work distribution in this case is a local density of states, which is essentially probing the spectrum $E_m^\\tau$ of the final Hamiltonian, with weights given by $|\\langle m_\\tau | U| {\\rm gs}_0 \\rangle|^2$.\n\n\nThe characteristic function (CF) of the work distribution Eq.~\\eqref{PW} is given by\n\\begin{equation}\\label{CF}\n    G(u) := \\langle e^{i u W} \\rangle = {\\mbox{tr}}\\Big(U^\\dagger e^{i u H_\\tau} U e^{-i u H_0} \\rho_d\\Big),\n\\end{equation}\nwhere $\\rho_d = \\sum_n |n_0\\rangle\\langle n_0|\\rho|n_0\\rangle\\langle n_0|$ is the initial state, dephased in the basis of $H_0$. \nFrom $G(u)$ the moments can be computed as $\\langle W^n \\rangle = i^{-n} G^{(n)}(0)$. \nCarrying out the expansion we find \n\\begin{equation}\\label{moments}\n    \\langle W^n \\rangle = \\sum\\limits_{k=0}^n \\binom{n}{k} (-1)^{n-k} {\\mbox{tr}}\\Big\\{ (U^\\dagger H_\\tau U)^k H_0^{n-k} \\rho_d\\Big\\}.\n\\end{equation}\nThis provides a convenient expression to compute all moments, without having to actually construct $P(W)$.\nFrom the CF one also builds the cumulant generating function (CGF)  $C(u)=\\ln G(u)$. A series expansion then yields the cumulants as $\\kappa_m=i^{-m} C^{(m)}(0)$. \nThe first cumulant is the mean, $\\kappa_1 \\equiv \\mu$, and the second is the variance $\\kappa_2 \\equiv \\sigma^2$. \nThe first three cumulants coincide with the corresponding central moments, $\\mu_n := \\langle (W-\\langle W \\rangle)^n \\rangle$, but this is no longer true for $n>3$. For instance, \n$\\kappa_4 = \\mu_4 -3 \\mu_2^2$, and so on. \n\nThe cumulants are particularly convenient to characterize the dependence of the work distribution with the size $L$ of the system (e.g. the number of lattice sites, or the number of particles).\nDepending on the structure of the Hamiltonian, and the type of drive employed, the dependence with $L$ may vary significantly. \nA useful reference, for comparison, is when the work is associated to $L$ independent sources. This happens, for instance, if the system is composed of $L$ non-interacting particles. This is also naturally expected in models which are exactly solveable via a mapping to free fermions~\\footnote{We remark that for truly strongly correlated systems this will necessarily break down~\\cite{ToAppearKrissia}.}.\n\nIn this case the statistics become extensive and the central limit theorem applies. \nAs a consequence, all cumulants of $W$  grow linearly with $L$:\n\\begin{equation}\\label{standard_scenario}\n\\kappa_n(W) \\sim L.    \n\\end{equation}\nIf one defines a rescaled work variable $w = W\/\\sqrt{L}$, the corresponding cumulants will then scale as\n\\begin{equation}\n    \\kappa_n(w)=(1\/\\sqrt{L})^n\\kappa_n(W) \\sim L^{1-\\frac{n}{2}}.\n\\end{equation}\nThat is\n\\begin{align*}\n    \\kappa_1(w) &\\sim \\sqrt{L}, \n    &\\kappa_2(w) &\\sim 1, \n    \\\\[0.2cm]\n    \\kappa_3(w) &\\sim 1\/\\sqrt{L}, \n    &\\kappa_4(w) &\\sim 1\/L.\n\\end{align*}\nFor large $L$ all cumulants with $n\\geqslant 3$ tend to be suppressed, and the work distribution will therefore tend to a Gaussian.\n\n\n\\subsection{Measures of Non-Gaussianity and Negentropy}\n\\label{measures}\nThe above discussion highlights the Gaussianity (or lack thereof) of $P(W)$ as an interesting feature that may provide insight in characterizing  different models and\/or dynamical regimes~\\cite{Genoni2010,Takagi2018,Albarelli2018,Zhuang2018,Santalla2017}. In this section we discuss measures to quantify the degree of non-Gaussianity. \nThe simplest approach is to analyze the cumulants $\\kappa_3(W)$, $\\kappa_4(W)$, etc. \nIt is more convenient to work with the dimensionless quantities \n\\begin{equation}\n    \\lambda_m := \\frac{\\kappa_m(W)}{\\kappa_2(W)^{m\/2}}.\n\\end{equation}\nThe skewness is \n$\\lambda_3 = \\kappa_3\/\\kappa_2^{3\/2} = \\kappa_3\/\\sigma^3$, while $\\lambda_4 = \\kappa_4\/\\kappa_2^2 = \\kappa_4\/\\sigma^4$ is related to the kurtosis as $\\lambda_4 + 3$. \nIn the standard scenario of Eq.~\\eqref{standard_scenario}, they scale according to \n\\begin{equation}\\label{standard_scenario_lambda}\n\\lambda_m \\sim L^{1-m\/2},    \n\\end{equation}\nsuch that a vanishing $\\lambda_m$ ($m\\geqslant 3$) can be used to quantify if the work distribution is tending to a Gaussian. Many normality tests used in statistics, such as the Jarque-Bera~\\cite{Jarque1980} or D'Agostino's $K$-squared tests~\\cite{D_AGOSTINO1970}, are in fact based on $\\lambda_3$ and~$\\lambda_4$. \n\nAlternatively, one may employ an information-theoretic approach, reminiscent of quantum resource theories. A monotone of non-Gaussianity can be built using the relative entropy (Kullback-Leibler divergence) between $P(W)$ and a corresponding Gaussian distribution\n$P_G(W)=\\frac{1}{\\sqrt{2 \\pi}\\sigma} e^{-(W-\\mu)^2\/2\\sigma^2}$, with the same mean $\\mu$ and variance $\\sigma^2$ as $P(W)$. \nFor a given $P(W)$, the quantifier is therefore defined as\n\\begin{eqnarray}\n\\label{eq:distance_Gaussian}\nJ(P) := D\\left(P || P_G\\right) &=& \\int_{-\\infty}^{\\infty} P(W) \\ln \\left[\\frac{P(W)}{P_G(W)}\\right] dW.\n\\end{eqnarray}\nThis quantity is usually termed negentropy.\nThe integral can actually be carried out further, leading to\n\\begin{equation}\\label{negentropy2}\n    J(P) = S(P_G)-S(P) = \\ln\\left(\\sqrt{2\\pi e} \\sigma\\right) - S(P),\n\\end{equation}\nwhere \n\\begin{equation}\\label{differential_entropy}\nS(p)=-\\int_{-\\infty}^{\\infty} p(x) \\ln p(x) \\, dx,    \n\\end{equation}\nis the differential entropy of $p(x)$. \nThe concept of negentropy can also be extended to quantum states, in the context of resource theories of non-Gaussianity~\\cite{Genoni2010,Takagi2018,Albarelli2018,Zhuang2018}. In that case, the meaningful entropic quantities are instead the von Neumann entropy, and the quantum relative entropy. \nDespite the underlying quantum system, our interest here is in the classical version of the negentropy, since our object of study is the classical probability distribution $P(W)$.\n\nFor continuous probability distributions, the negentropy is a very good quantifier of non-Gaussianity: it is non-negative and vanishes if and only if the distribution is Gaussian. \nIn our case, however, an issue arises concerning the discrete nature of Eq.~\\eqref{PW}. \nNamely, because the support (the points $W$ where $P(W) \\neq 0$) is not compact, the corresponding differential entropy~\\eqref{differential_entropy} is not well defined. In fact, this is a famous issue found by Shannon, and later resolved by Jaynes using the concept of a limiting density of discrete points~\\cite{Jaynes1968}. However, unfortunately there is no unambiguous way of addressing it. \n\nOne approach, which directly connects with the Jarque-Bera test~\\cite{Jarque1980}, is to approximate $P(W)$ by a continuous smooth distribution using the Gram-Chalier series~\\cite{Wallace1958}, \nwhich effectively provides a systematic set of corrections to a Gaussian distribution in terms of $\\lambda_3,\\lambda_4,\\ldots$. \nStopping at fourth order, the approximate distribution will have the form \n\\begin{align}\n    P_{\\rm app}(W)&= P_G(W)\\left[ 1+f(v)\\right], \\label{GCE_Series} \\\\[0.2cm]\n    f(v) &= \\frac{\\lambda_3}{3!}\\mathit{He}_3\\left(v\\right)+\\frac{\\lambda_4}{4!}\\mathit{He}_4\\left(v\\right),\n\\end{align}\nwhere $\\mathit{He}_n(x)$ are the probabilist's Hermite polynomials.\nFor $\\lambda_3 \\sim L^{-1\/2}$ and $\\lambda_4 \\sim L^{-1}$, the correction terms become vanishingly small with increasing $L$.\nThis expansion applies not only to the standard scenario of Eqs.~\\eqref{standard_scenario} and~\\eqref{standard_scenario_lambda}, but also to any scenario where cumulants of order 3 or higher are small. This includes, for instance, sub\/super extensive dependences in $L$;\nthat is, $\\kappa_m \\sim L^q$ for $q \\lessgtr1$, which then implies $\\lambda_m \\sim L^{q(1-m\/2)}$.\n\nFollowing~\\cite{Hulle2005}, when $f$ is small we can series expand \n\\begin{eqnarray}\nP_{\\rm app} \\ln P_{\\rm app} \n&\\approx &  P_G\\left[ (1+f) \\ln P_G + f+\\frac{f^2}{2}\\right].\n\\end{eqnarray}\nThe negentropy~\\eqref{negentropy2} can then be written as\n\\begin{equation*}\nJ(P_{\\rm app}) \n=  \\left[1-\\ln\\left(\\sqrt{2 \\pi}\\sigma\\right)\\right]\\left\\langle f\\right\\rangle_G -\\frac{1}{2} \\left\\langle v^2 f\\right\\rangle_G +\\frac{1}{2}\\left\\langle f^2\\right\\rangle_G,\n\\end{equation*}\nwhere $\\langle \\ldots \\rangle_G$ denotes expectation values over $P_G(W)$. \nFinally, using the orthogonality relations of the Hermite polynomials, we find that\n$\\left\\langle f\\right\\rangle_G=0$ and $\\left\\langle v^2 f\\right\\rangle_G=0$.\nThis then leads us to\n$J(P_{\\rm app}) = \\frac{1}{2} \\langle f^2\\rangle_G \\nonumber $ or, more explicitly, \n\\begin{equation}\nJ(P) = \\frac{1}{2}\\left[ \\frac{\\lambda_3^2}{3!} +\\frac{\\lambda_4^2}{4!} \\right].\n\\label{negentropy_approx}\n\\end{equation}\nThis result, which also coincides with the Jarque-Bera test~\\cite{Jarque1980} used in statistics, provides us with a clear measure of deviations from Gaussianity in terms of  $\\lambda_3$ and $\\lambda_4$. \nIn principle, one could also extend this procedure and consider higher order cumulants. \nHowever, in practice this is typically not very useful since higher order cumulants are extremely sensitive to numerical errors. \n\n\n\\begin{figure*}[t!]\n\\begin{center}\n    \\begin{tikzpicture}\n        \\node[label={[font=\\normalsize, shift={(-2.5cm,-5mm), align=left}]above:(a)}] (figa) at (0,0)\n        {\\includegraphics[ trim={3cm 0.5cm 1.75cm 1.cm}, clip, width=0.31\\textwidth]{paper_exact_distribution_L=20.png}};\n        \n        \\node[anchor=north west, \n       label={[font=\\normalsize, shift={(-2.5cm,-5mm), align=left}]above:(b)}\n        ] (figb) at ($(figa.north east)+(-0.2cm,0)$)\n        {\\includegraphics[ trim={3cm 0.5cm 1.75cm 1.cm}, clip,width=0.31\\textwidth]{paper_approximated_distribution_L=20_Kmax=4.png}};\n        \n        \n        \\node[anchor=west,\n      label={[font=\\normalsize, shift={(-3.5cm,-5mm), align=left}]above:(c)}] (figc) at ($(figb.east)+(0.1cm,0mm)$)\n {\\includegraphics[width=0.35\\textwidth,]{paper_times_distribution_L=20.png}};   \n\n    \\end{tikzpicture}\n\\end{center}\n\\caption{(a) Full work distribution, Eq.~\\eqref{PW_gs}, for an Ising spin chain with $L\\!=\\!20$ as a function of quench duration. (b) Approximation to the full distribution constructed using the first four cumulants. (c) Comparison between the exact discrete (bars) and approximate continuous (dashed lines) distributions for fixed values of ramp duration, $\\tau$. In all panels we consider a linear ramp, Eq.~\\eqref{linramp}, with the system starting in the ground state and $g_0\\!=\\!0$.}\n \\label{DistED_Approx}\n\\end{figure*}\n\\section{Numerical Results}\nWe now apply the above ideas to a paradigmatic model for critical quantum systems, namely the transverse field Ising model (TFIM) consisting of $L$ sites described by Pauli matrices $\\sigma_i^{x,y,z}$, with Hamiltonian \n\\begin{equation}\n    H_t=- \\hbar \\omega\\sum_{i=1}^{L}\\left[g(t) \\,\\sigma_i^x+\\sigma_i^z\\sigma_{i+1}^z\\right].\n\\end{equation}\nWe assume $L$ is even, and use periodic boundary conditions $\\sigma_{L+1}^{x,y,z}\\!=\\!\\sigma_1^{x,y,z}$.\nMoreover, we henceforth set $\\hbar\\omega=1$, thus fixing the scales of energy. \n\nThe model is well known to exhibit a second-order quantum phase transition at $g_c\\!\\!=\\!\\!1$ \\cite{prl_silva, Dorner-PRL-work-TFIM, Heyl-PRL-work-TFIM, EspositoPRL2020, Abeling-PRB-work-TFIM-finiteT, Russomanno2015, ParaanPRE, XuPRA2018, prx_fusco, LandiPRR, Scandi2020}. We explore the properties of the full work distribution for a system driven symmetrically through the critical point by a linearly varying drive\n\\begin{equation}\n\\label{linramp}\n    g(t)\\!=\\!g_0+2(g_c-g_0) t\/\\tau\n\\end{equation} \nwhere $\\tau$ is the duration but remark that we expect our results to be qualitatively similar for other suitable choices of ramp. We assume that the system begins in the ground state manifold for $g_0\\!=\\!0$ and we calculate the full work distribution, Eq.~\\eqref{PW_gs}, of the TFIM via Exact Diagonalization (ED). Translational and spin inversion symmetries allow us to solve systems with up to 20 spins since the ground-state only connects to states sharing the same conserved quantum numbers~\\cite{Sacco2022}.\n\n\\begin{figure*}[t]\n\\begin{center}\n\\includegraphics[width=0.95\\linewidth]{{moments_cumulants_TFIM_protocolA_symmetries_Ls_g_0=0.000_tau=}.png}\n\\end{center}\n\\caption{(a)-(d) Moments of work distribution for Ising chains of size $L=8$ (Dark red solid line), $L=10$ (Orange dotted line),  $L=12$ (Yellow dot dashed line), $L=14$ (Lime green dotted line), $L=16$ (Cyan solid line), $L=18$ (Blue solid line), $L=20$ (Indigo dotted line). (e)-(h) Cumulants of work distribution for Ising chains of size $L=8$ (Dark red solid line), $L=10$ (Orange dotted line),  $L=12$ (Yellow dot dashed line), $L=14$ (Lime green dotted line), $L=16$ (Cyan solid line), $L=18$ (Blue solid line), $L=20$ (Indigo dotted line). In all panels we consider a linear ramp, Eq.~\\eqref{linramp}, with the system starting in the ground state and $g_0\\!=\\!0$.}\n \\label{momentsED}\n\\end{figure*}\n\n\n\\begin{figure}[t]\n\\begin{center}\n\n\\begin{tikzpicture}\n       \\node[label={[yshift=-2mm](a)}] (figa) at (0,0)\n        {\\includegraphics[width=0.9\\columnwidth]{skewness_scaling_TFIM_protocolA_symmetries_Ls_g_0=0.000_tau=.png}};\n        \n       \n       \n       \n    \n\\node[anchor=north west,\n        label={[yshift=-2mm](b)}] (figc) at ($(figa.south west)+(0.cm,-0.25cm)$)\n        {\\includegraphics[width=0.9\\columnwidth]{negentropy_scaling_TFIM_protocolA_symmetries_Ls_g_0=0.000_tau=.png}};\n        \n    \n\\end{tikzpicture}\n\\end{center}\n\\caption{Measures of non-Gaussianity of the work distribution for the linear ramp protocol. (a) Skewness and (b) Negentropy for finite Ising chains. The inset in (a) shows a linear fit of each curve in the region where a power-law is observed. The inset in (b) displays the unscaled, long-time values which the negentropy reaches near $\\tau \\sim 10^2$.}\n \\label{SkewNegentropyED}\n\\end{figure}\n\nThe initial condition $g_0\\!=\\!0$ makes the ground-state twofold degenerate, with each state related by spin inversion symmetry in the $z$ direction. In an adiabatic evolution, this degeneracy implies a splitting of the transition probabilities into two peaks with equal weight $p_{m|n}\\! =\\! 1\/2$. In the limit of a sudden quench, when many excitations are created, these two ground-states can overlap with a large number of eigenstates of the final Hamiltonian. While quantitative differences will occur for other choices of initial states, particularly if one restricts to a specific symmetry sector, the qualitative behaviors discussed remain largely unaffected.\n\nIn Fig.~\\ref{DistED_Approx}(a) we show the exact work distribution for a $L\\!=\\!20$ site chain for the two-fold degenerate initial state. For $\\tau \\omega\\! \\ll \\! 1$, $P(W)$ resembles a Gaussian: it is widely spread throughout the full spectrum, with the largest contributions centered at the average $\\langle W \\rangle$. For such fast, but manifestly not instantaneous, ramps we see that the distribution is nevertheless invariant. This indicates that the sudden quench approximation remains valid even for fast finite ramps. Conversely, for very slow protocols ($\\tau \\omega \\! \\rightarrow \\! \\infty$), the work distribution splits into two peaks, each one associated with the two ground-states. An intermediate regime emerges for $1 \\!\\lesssim\\! \\tau \\omega \\! \\lesssim \\! 10$, for which the work distribution appears more involved and does not peak at the adiabatic ground-states. In this regime, the skewness of $P(W)$ is enhanced towards negative values of the work.\n\n\nFrom the full distribution we determine the first four moments $\\langle W^m \\rangle$ and cumulants $\\kappa_m$ ($m=1,2,3,4$) as a function of the duration of the ramp $\\tau$, for finite systems with sizes from $L\\!=\\!8$ to $L\\!=\\!20$. In Fig.~\\ref{momentsED}, the first row shows the scaled moments $\\langle (W\/L)^m \\rangle$, while the second  shows the scaled cumulants $\\kappa_m \/ L$. In both, we observe a crossover between $10^{-1} \\! <\\! \\tau \\omega\\! < \\! 10 $, which separates the sudden quench and the (quasi)adiabatic regimes. Note that the scaling $(W\/L)^k$ is non-linear, and the moments are sensitive to the system size. Regions where the cumulants deviate from extensive scaling (shown in the inset of Fig.~\\ref{momentsED} (e) and (f)) are due to finite size effects. While the first three scaled cumulants display a similar behavior, the fourth shows a peak centered at $\\tau \\omega\\! =\\! 1$. The fluctuations, given by the second cumulant, are amplified for a fast driving due to the spread of the spectral weight of the instantaneous wave-function through the final states.\n\nFrom the first four cumulants of the exact data in Fig.~\\ref{DistED_Approx}(a), we can build an approximated distribution for $P(W)$ using the Gram\u2013Charlier series in Eq.~\\eqref{GCE_Series}. The result is shown in Fig.~\\ref{DistED_Approx}(b). The structure of the exact distribution is preserved: close to a sudden quench, one can recover a nearly Gaussian profile; while at intermediate driving speeds, a skewed distribution emerges before then converging to a single sharp peak in the adiabatic limit. A comparison between the exact and approximated distributions for $\\tau \\omega \\! \\approx\\! 0.05,~2.5,~116 $ is shown panel Fig.~\\ref{DistED_Approx}(c). The accuracy of this approximation is expected to improve with increasing system size.\n\nTurning to the characterisitics of the distributions, we quantitatively examine deviations of the work distribution from a Gaussian and its dependence on the speed of the external drive employing the metrics outlined in Sec.~\\ref{measures}. To this aim, in Fig.~\\ref{SkewNegentropyED}(a) we inspect the skewness, $\\lambda_3$, of the full distribution calculated via ED as a function of $\\tau \\omega$ for different $L$. As remarked previously, given that all cumulants of a normal distribution vanish after second order, the skewness provides a readily accessible figure of merit to understand the effects of the finite-time dynamics in the work distribution. \nFrom Fig.~\\ref{SkewNegentropyED}(a) we observe that in the fast driving regime the skewness is constant and scales as $1\/\\sqrt{L}$. At short times, we see that there is a residual value of $\\lambda_3$, which in effect quantifies the smallest distance between the actual distribution and a perfect Gaussian for a finite system after a sudden quench. Notice that since the skewness scales inversely with the system size, it is clear that in the thermodynamic limit, one would expect $\\lambda_3 \\rightarrow 0$ for the Ising model. As we increase the ramp duration, the skewness increases for intermediate driving  times before peaking around $\\tau \\omega \\!\\sim\\! 3$, after which it decreases according to a power law in $\\tau \\omega$. We see a clear universal behavior for the sudden quench and intermediate regimes, while features due to finite size effects become evident as the ramp approaches the adiabatic limit; the onset of the power law decay emerges at progressively longer ramp durations for larger system sizes due to the fact that the ground state energy gap decreases as $L^{-1}$. Thus, the time  scales to reach adiabaticity necessarily increase, approaching infinity as $L\\!\\to\\!\\infty$.\n\n\nIn Fig.~\\ref{SkewNegentropyED}(b) we plot the scaled negentropy, $J L$, as a function of $\\omega \\tau$ for different system sizes. We immediately see some qualitative similarities with the skewness. In particular, the negentropy of the approximate distribution is constant for fast dynamics, where the sudden quench approximation is valid. However, in contrast with the skewness, the negentropy asymptotically approaches a fixed value for a given system size in the adiabatic limit, c.f. the inset of Fig.~\\ref{SkewNegentropyED}(b) shows the (unscaled) values attained at $\\tau\\sim 10^2$. Once again, this can be understood from the distribution tending towards two equally likely values of work for the given choice of initial state, and in this limit the negentropy tends to $J \\rightarrow 1\/12$. Note that there is a slower convergence towards this value for larger system sizes as the decreased energy gap results in a longer timescale required for adiabaticity, as can be inferred from the inset of Fig.~\\ref{SkewNegentropyED}(b) where $N\\!=\\!8$ is already close to the asymptotic value while larger systems clearly require longer timescales. In the intermediate regime we see that the negentropy captures the increasingly non-Gaussian characteristics of the full distribution, similarly peaking for a finite drive but now also exhibiting a sharp minimum which appears to signal the transition toward the adiabatic regime and arises due to contributions coming from the fourth cumulant. \n\n\n\\section{Conclusions}\nWe have examined how characteristics of the work distribution are dependent on the rate at which a system is driven through a quantum phase transition. In particular, we have established that there is an interesting trade-off between the (lack of) Gaussianity of a distribution and speed with which it is ramped. Higher order moments of the distribution were shown to be useful indicators of different dynamical regimes. While the first and second moments (corresponding to the mean and variance) are typically the most studied, in line with Ref.~\\cite{KrissaPRR} we have shown the the skewness provides valuable information about the response of the system to the ramp protocol. \n\nIn particular, we have developed a general framework to assess the Gaussianity of the work distribution in terms of the negentropy. This can be approximated by a series expansion of cumulants providing a simple means of calculation, however, we stress that due to the discreteness of the full distribution for finite sizes, systematic errors can emerge since the approximation attempts to resolve the fine structure of the distribution. These errors can be more critical in scenarios in which the system's spectrum and its level statistics constrain the set of accessible energy transitions to a few. For this reason, we have demonstrated that restricting to the first four cumulants is sufficient to obtain a very good approximation, while largely avoiding such pathological issues.\n\nWhile our results are demonstrated for the Ising model, we expect that qualitatively similar behaviors would be exhibited in other models that can be mapped to free-fermions. Future work could go beyond the scenario presented here, e.g. considering thermal initial states, the effect of non-integrability, or non-linear ramps. Furthermore, the characterisations of non-Gaussianity that we have proposed requires knowledge of only a few cumulants and hence could be applicable to settings where cumulants can be easily calculated, but the full distribution is difficult to access.\n\n\\acknowledgements\n A.K. and S.C acknowledge support from the Science Foundation Ireland Starting Investigator Research Grant ``SpeedDemon\" No. 18\/SIRG\/5508. K.Z. acknowledges Javier Laguna for insightful comments. The package QuSpin~\\cite{QuSpin} was employed in the ED calculations.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Methods}\n\\textbf{Device Fabrication.} Nanowires were selected by imaging with scanning electron microscopy. A double resist layer, consisting of PMMA AR-P 669.04 and PMMA AR-P 672.02, was spun onto the chip and patterned with e-beam lithography. After developing the resist with MIBK:IPA (1:2), an Ar reactive ion etch was performed to remove native oxide on the PbTe nanowires \\cite{Yang2008}. Immediately after the Ar etch, the chip was loaded into an e-beam evaporator where 5~nm Ti and 50~nm Au were deposited. Then, lift-off was carried out in acetone.\n\n\\textbf{$g$-factor fitting.} By fitting the complete set of $g$-factors extracted for all magnetic field orientations with Eq.~\\ref{eq:2}, we determined the principal $g$-factors and the principal axes of the effective $g$-factor tensor $|g^*|(\\vec{B})$. The magnetic field components $B_\\mathrm{x}$, $B_\\mathrm{y}$, $B_\\mathrm{z}$ and the measured $g$-factors formed the set of input parameters for the fit. The fit parameters were the principal $g$-factors $g_1$, $g_2$, $g_3$ and the Euler angles of rotation $\\phi$, $\\theta$, $\\psi$ \\cite{Liles2021}. With these angles, the magnetic field components were transformed from the Cartesian coordinate system to the coordinate system of the principal axes of $|g^*|(\\vec{B})$. This fitting procedure was repeated for two gate voltage regimes of Device~1 and for one regime of Device~2. Subsequently, with the Euler angles of rotation found by the fits, we transformed the principal $g$-factors to spherical coordinates to determine the orientation of each principal $g$-factor.\n\n\\section{Acknowledgments}\nWe thank W.~Riess, G.~Salis, E.~G.~Kelly, F.~J.~Schupp and the Cleanroom Operations Team of the Binnig and Rohrer Nanotechnology Center (BRNC) for their help and support. A.~Fuhrer acknowledges support from NCCR SPIN, funded by the SNSF under grant number 51NF40-180604. The work in Eindhoven is supported by the European Research Council (ERC TOCINA 834290). F.~Nichele acknowledges support from the European Research Council, grant number 804273, and the Swiss National Science Foundation, grant number 200021\\_201082.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor a function $f:X\\rightarrow\\mathbb{R}$, we say that $x$ is a\n\\emph{critical point} if $\\nabla f(x)=\\mathbf{0}$, and $y$ is a\n\\emph{critical value} if there is some critical point $x$ such that\n$f(x)=y$. A critical point $x$ is a \\emph{saddle point} if it is\nneither a local minimizer nor a local maximizer. In this paper, we\npresent algorithms based on the multidimensional mountain pass theorem\nto find saddle points numerically.\n\nThe main purpose of critical point theory is the study of variational\nproblems. These are problems (P) such that there exists a smooth functional\n$\\Phi:X\\rightarrow\\mathbb{R}$ whose critical points are solutions\nof (P). Variational problems occur frequently in the study of partial\ndifferential equations. \n\nAt this point, we make a remark about saddle points in the study of\nmin-max problems. Such saddle points occur in problems in game theory\nand in constrained optimization using the Lagrangian, and have the\nsplitting structure\\[\n\\min_{x\\in X}\\max_{y\\in Y}f(x,y).\\]\nIn min-max problems, this splitting structure is exploited in numerical\nprocedures. See \\cite{RH03} for a survey of algorithms for min-max\nproblems. In the general case, for example in finding weak solutions\nof partial differential equations, such a splitting structure may\nonly be obtained after the saddle point is located, and thus is not\nhelpful for finding the saddle point.\n\nA critical point $x$ is \\emph{nondegenerate} if its Hessian $\\nabla^{2}f(x)$\nis nonsingular and it is \\emph{degenerate} otherwise. The \\emph{Morse\nindex} of a critical point is the maximal dimension of a subspace\nof $X$ on which the Hessian $\\nabla^{2}f(x)$ is negative definite.\nIn the finite dimensional case, the Morse index is the number of negative\neigenvalues of the Hessian.\n\nLocal maximizers and minimizers of $f:X\\rightarrow\\mathbb{R}$ are\neasily found using optimization, while saddle points are harder to\nfind. To find saddle points of Morse index 1, one can use algorithms\nmotivated by the mountain pass theorem. Given points $a,b\\in X$ ,\ndefine a \\emph{mountain pass} $p^{*}\\in\\Gamma(a,b)$ to be a minimizer\nof the problem \\[\n\\inf_{p\\in\\Gamma(a,b)}\\sup_{0\\leq t\\leq1}f(p(t)),\\]\nif it exists. Here, $\\Gamma(a,b)$ is the set of continuous paths\n$p:[0,1]\\rightarrow X$ such that $p(0)=a$ and $p(1)=b$. Ambrosetti\nand Rabinowitz's \\cite{AR73} mountain pass theorem states that under\nadded conditions, there is a critical value of at least $\\max\\{f(a),f(b)\\}$.\nTo find saddle points of higher Morse index, it is instructive to\nlook at theorems establishing the existence of critical points of\nMorse index higher than 1. Rabinowitz \\cite{R77} proved the multidimensional\nmountain pass theorem which in turn motivated the study of linking\nmethods to find saddle points. We shall recall theoretical material\nrelevant for finding saddle points of higher Morse index in this paper\nas needed.\n\nWhile the study of numerical methods for the mountain pass problem\nbegan in the 70's or earlier to study problems in computational chemistry,\nChoi and McKenna \\cite{CM93} were the first to propose a numerical\nmethod for the mountain pass problem to solve variational problems.\nMost numerical methods for finding critical points of mountain pass\ntype rely on discretizing paths in $\\Gamma(a,b)$ and perturbing paths\nto lower the maximum value of $f$ on the path. There are a few other\nmethods of finding saddle points of mountain pass type that do not\ninvolve perturbing paths, for example \\cite{H04,BT07}. \n\nSaddle points of higher Morse index are obtained with modifications\nof the mountain pass algorithm. Ding, Costa and Chen \\cite{DCC99}\nproposed a numerical method for finding critical points of Morse index\n2, and Li and Zhou \\cite{LZ01} proposed a method for finding critical\npoints of higher Morse index. \n\nIn \\cite{LP08}, we suggested a numerical method for finding saddle\npoints of mountain pass type. The key observation is that the value\n\\[\n\\sup\\big\\{l\\geq\\max\\big(f(a),f(b)\\big)\\mid a,b\\mbox{ lie in different path components of }\\{x\\mid f(x)\\leq l\\}\\big\\}\\]\nis a critical value. In other words, the supremum of all levels $l$\nsuch that there is no path connecting $a$ and $b$ in the level set\n$\\{x\\mid f(x)\\leq l\\}$ is a critical value. See Figure \\ref{fig:mtn-contrast}\nfor an illustration of the difference between the two approaches.\nAn extensive theoretical analysis and some numerical results of this\napproach were provided in \\cite{LP08}. \n\nIn this paper, we extend three of the themes in the level set approach\nto find saddle points of higher Morse index, namely the convergence\nof the basic algorithm (Sections \\ref{sec:Alg-desc} and \\ref{sec:Conv-ppties}),\noptimality condition of sub-problem (Section \\ref{sec:Opti-condns}),\nand a fast locally convergent method in $\\mathbb{R}^{n}$ (Sections\n\\ref{sec:Fast-local-convergence} and \\ref{sec:Proof-of-superlinear}).\nSection \\ref{sec:Other-local-analysis} presents an alternative result\non convergence to a critical point similar to that of Section \\ref{sec:Conv-ppties}.\n\nWe refer the reader to \\cite{LP08} for examples reflecting the limitations\nof the level set approach for finding saddle points of mountain pass\ntype, which will be relevant for the design of level set methods of\nfinding saddle points of general Morse index. \n\n\\begin{figure}[h]\n\\begin{tabular}{|c|c|}\n\\hline \n\\includegraphics[scale=0.4]{perturb} & \\includegraphics[scale=0.3]{converge2}\\tabularnewline\n\\hline\n\\end{tabular}\n\n\\caption{\\label{fig:mtn-contrast}The diagram on the left shows the classical\nmethod of perturbing paths for the mountain pass problem, while the\ndiagram on the right shows convergence to the critical point by looking\nat level sets.}\n\n\\end{figure}\n\n\n\n\\section*{Notation}\n\\begin{description}\n\\item [{$\\mbox{\\rm lev}_{\\geq b}f$}] This is the level set $\\{x\\mid f(x)\\geq b\\}$,\nwhere $f:X\\rightarrow\\mathbb{R}$. The interpretations of $\\mbox{\\rm lev}_{\\leq b}f$\nand $\\mbox{\\rm lev}_{=b}f$ are similar.\n\\item [{$\\mathbb{B}$}] The ball of center $\\mathbf{0}$ and radius $1$.\n$\\mathbb{B}(x,r)$ stands for a ball of center $x$ and radius $r$.\n$\\mathbb{B}^{n}$ denotes the $n$-dimensional sphere in $\\mathbb{R}^{n}$.\n\\item [{$\\mathbb{S}^{n}$}] The $n$-dimensional sphere in $\\mathbb{R}^{n+1}$.\n\\item [{$\\partial$}] Subdifferential of a real-valued function, or the\nrelative boundary of a set. If $h:\\mathbb{B}^{n}\\to S$ is a homeomorphism\nbetween $\\mathbb{B}^{n}$ and $S$, then the relative boundary of\n$S$ is $h(\\mathbb{S}^{n-1})$.\n\\item [{$\\mbox{lin}(A)$}] For an affine space $A$, the lineality space\n$\\mbox{lin}(A)$ is the space $\\{a-a^{\\prime}\\mid a,a^{\\prime}\\in A\\}$.\n\\end{description}\n\n\\section{\\label{sec:Alg-desc}Algorithm for critical points}\n\nWe look at the critical point existence theorems to give an insight\non our algorithm for finding critical points of higher Morse index\nbelow. Here is the definition of linking sets. We take our definition\nfrom \\cite[Section II.8]{Str08}.\n\\begin{defn}\n(Linking) Let $A$ be a subset of $\\mathbb{R}^{n}$, $B$ a submanifold\nof $\\mathbb{R}^{n}$ with relative boundary $\\partial B$. Then we\nsay that $A$ and $\\partial B$ \\emph{link} if \n\n(a) $A\\cap\\partial B=\\emptyset$, and \n\n(b) for any continuous $h:\\mathbb{R}^{n}\\to\\mathbb{R}^{n}$ such that\n$h\\mid_{\\partial B}=id$ we have $h(B)\\cap A\\neq\\emptyset$.\n\\end{defn}\nFigure \\ref{fig:Linking-subsets} illustrates two examples of linking\nsubsets in $\\mathbb{R}^{3}$. In the diagram on the left, the set\n$A$ is the union of two points inside and outside the sphere $B$.\nIn the diagram on the right, the sets $A$ and $B$ are the interlocking\n'rings'. Note however that $A$ and $B$ link does not imply that\n$B$ and $A$ link, though this will be true with additional conditions.\nWe hope this does not cause confusion. \n\nWe now recall the Palais-Smale condition.\n\\begin{defn}\n(Palais-Smale condition) Let $X$ be a Banach space and $f:X\\rightarrow\\mathbb{R}$\nbe $\\mathcal{C}^{1}$. We say that a sequence $\\{x_{i}\\}_{i=1}^{\\infty}\\subset X$\nis a \\emph{Palais-Smale sequence} if $\\{f(x_{i})\\}_{i=1}^{\\infty}$\nis bounded and $\\nabla f(x_{i})\\rightarrow\\mathbf{0}$, and $f$ satisfies\nthe \\emph{Palais-Smale condition} if any Palais-Smale sequence admits\na convergent subsequence.\n\\end{defn}\nThe classical multidimensional pass theorem originally due to Rabinowitz\n\\cite{R77} states that under added conditions, if there are linking\nsets $A$ and $B$ such that $\\max_{A}f<\\min_{B}f$ and the Palais-Smale\ncondition holds, then there is a critical value of at least $\\max_{A}f$\nfor the case when $f$ is smooth. (See Theorem \\ref{thm:Rabinowitz-77}\nfor a statement of the multidimensional mountain pass theorem) Generalizations\nin the nonsmooth case are also well-known in the literature. See for\nexample \\cite{Jab03}.\n\nTo find saddle points of Morse index $m$, we consider finding a sequence\nof linking sets $\\{A_{i}\\}_{i=1}^{\\infty}$ and $\\{B_{i}\\}_{i=1}^{\\infty}$\nsuch that $\\mbox{diam}(A_{i})$, the diameter of the set $A_{i}$,\ndecreases to zero, and the set $A_{i}$ is a subset of an $m$-dimensional\naffine space. This motivates the following algorithm.\n\n\\begin{figure}\n\\begin{tabular}{|c|c|}\n\\hline \n\\includegraphics[scale=0.5]{morse1b} & \\includegraphics[scale=0.5]{morse2}\\tabularnewline\n\\hline\n\\end{tabular}\n\n\\caption{\\label{fig:Linking-subsets}Linking subsets}\n\n\\end{figure}\n \n\\begin{algorithm}\n\\label{alg:saddle-points-outer}First algorithm for finding saddle\npoints of Morse index $m\\geq1$.\n\\begin{enumerate}\n\\item Set the iteration count $i$ to $0$, and let $l_{i}$ be a lower\nbound of the critical value and $u_{i}$ be an upper bound.\n\\item Find $x_{i}$ and $y_{i}$, where $(S_{i},x_{i},y_{i})$ is an optimizing\ntriple of \\begin{equation}\n\\min_{S\\in\\mathcal{S}}\\max_{x,y\\in S\\cap(\\scriptsize\\mbox{\\rm lev}_{\\geq\\frac{1}{2}(l_{i}+u_{i})}f)\\cap U_{i}}|x-y|,\\label{eq:keyexp1}\\end{equation}\nwhere $U_{i}$ is some open set. Here, $\\mathcal{S}$ is the set of\n$m$-dimensional affine subspaces of $\\mathbb{R}^{n}$ intersecting\n$U_{i}$. In the inner maximum problem above, we take the value to\nbe $0$ if $S\\cap(\\mbox{\\rm lev}_{\\leq\\frac{1}{2}(l_{i}+u_{i})}f)\\cap U_{i}$\nis empty, making the objective function above equal to $0$. For simplicity,\nwe shall just assume that minimizers and maximizers of the above problem\nexist.\n\\item (Bisection) If the objective of \\eqref{eq:keyexp1} is zero, then\n$\\frac{1}{2}(l_{i}+u_{i})$ is a lower bound of the critical value.\nSet $l_{i+1}=\\frac{1}{2}(l_{i}+u_{i})$ and $u_{i+1}=u_{i}$. Otherwise,\nset $l_{i+1}=l_{i}$ and $u_{i+1}=\\frac{1}{2}(l_{i}+u_{i})$. \n\\item Increase $i$ and go back to step 2.\n\\end{enumerate}\n\\end{algorithm}\nThe critical step of Algorithm \\ref{alg:saddle-points-outer} lies\nin step 2. We elaborate on optimal conditions that will be a useful\napproximate for this step in Section \\ref{sec:Opti-condns}. One may\nthink of the set $A_{i}$ as the relative boundary (to the affine\nspace $S_{i}$) of $S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i}$. A\nfrequent assumption we will make is nondegenericity.\n\\begin{defn}\nWe say that a critical point is \\emph{nondegenerate} if its Hessian\nis invertible.\n\\end{defn}\nAlgorithm \\ref{alg:saddle-points-outer} requires $m>0$, but when\n$m=0$, nondegenerate critical points of Morse index zero are just\nstrict local minimizers that can be easily found by optimization.\nWe illustrate two special cases of Algorithm \\ref{alg:saddle-points-outer}.\n\\begin{example}\n(Particular cases of Algorithm \\ref{alg:saddle-points-outer}) (a)\nFor the case $m=1$, $\\mathcal{S}$ is the set of lines. The inner\nmaximization problem in \\eqref{eq:keyexp1} has its solution on the\ntwo endpoints of $S_{i}\\cap(\\mbox{\\rm lev}_{\\geq\\frac{1}{2}(l_{i}+u_{i})}f)\\cap U_{i}$.\nThis means that \\eqref{eq:keyexp1} is equivalent to finding the local\nclosest points between two components of $(\\mbox{\\rm lev}_{\\leq\\frac{1}{2}(l_{i}+u_{i})}f)\\cap U_{i}$,\nas was analyzed in \\cite{LP08}.\n\n(b) For the case $m=n$, $\\mathcal{S}$ contains the whole of $\\mathbb{R}^{n}$.\nHence the outer minimization problem in \\eqref{eq:keyexp1} is superfluous.\nThe level set $(\\mbox{\\rm lev}_{\\geq\\frac{1}{2}(l_{i}+u_{i})}f)\\cap U_{i}$\ngets smaller and smaller as $\\frac{1}{2}(l_{i}+u_{i})$ approaches\nthe maximum value, till it becomes a single point if the maximizer\nis unique.\n\\end{example}\n\n\\section{\\label{sec:Conv-ppties}Convergence properties}\n\nIn this section, we prove the convergence of $x_{i}$, $y_{i}$ in\nAlgorithm \\ref{alg:saddle-points-outer} to a critical point when\nthey converge to a common limit. We recall some facts about nonsmooth\nanalysis needed for the rest of the paper. It is more economical to\nprove our result for nonsmooth critical points because the proofs\nare not that much harder, and nonsmooth critical points are also of\ninterest in applications.\n\nLet $X$ be a Banach space, and $f:X\\rightarrow\\mathbb{R}$ be a locally\nLipschitz function at a given point $x$.\n\\begin{defn}\n(Clarke subdifferential) \\cite[Section 2.1]{Cla83} Suppose $f:X\\rightarrow\\mathbb{R}$\nis locally Lipschitz at $x$. The \\emph{Clarke generalized directional\nderivative} of $f$ at $x$ in the direction $v\\in X$ is defined\nby\\[\nf^{\\circ}(x;v)=\\limsup_{t\\searrow0,y\\rightarrow x}\\frac{f(y+tv)-f(y)}{t},\\]\nwhere $y\\in X$ and $t$ is a positive scalar. The \\emph{Clarke subdifferential}\nof $f$ at $x$, denoted by $\\partial_{C}f(x)$, is the subset of\nthe dual space $X^{*}$ given by\\[\n\\left\\{ \\zeta\\in X^{*}\\mid f^{\\circ}(x;v)\\geq\\left\\langle \\zeta,v\\right\\rangle \\mbox{ for all }v\\in X\\right\\} .\\]\nThe point $x$ is a \\emph{Clarke (nonsmooth) critical point} if $\\mathbf{0}\\in\\partial_{C}f(x)$.\nHere, $\\left\\langle \\cdot,\\cdot\\right\\rangle :X^{*}\\times X\\rightarrow\\mathbb{R}$\ndefined by $\\left\\langle \\zeta,v\\right\\rangle :=\\zeta(v)$ is the\ndual relation.\n\\end{defn}\nFor the particular case of $\\mathcal{C}^{1}$ functions, $\\partial_{C}f(x)=\\{\\nabla f(x)\\}$.\nTherefore critical points of smooth functions are also nonsmooth critical\npoints. From the definitions above, it is clear that an equivalent\ndefinition of a nonsmooth critical point is $f^{\\circ}(x;v)\\geq0$\nfor all $v\\in X$. This property allows us to prove that a point is\nnonsmooth critical without appealing to the dual space $X^{*}$.\n\nWe now prove our result of convergence to nonsmooth critical points.\n\\begin{prop}\n\\label{pro:triples-conv}(Convergence to saddle point) Let $\\bar{z}\\in X$.\nSuppose there is a ball $\\mathbb{B}(\\bar{z},r)$, a sequence of triples\n$\\{(S_{i},x_{i},y_{i})\\}_{i=1}^{\\infty}$ and a sequence $l_{i}$\nmonotonically increasing to $f(\\bar{z})$ such that $(x_{i},y_{i})\\to(\\bar{z},\\bar{z})$\nand $(S_{i},x_{i},y_{i})$ is an optimizing triple of \\eqref{eq:keyexp1}\nin Algorithm \\ref{alg:saddle-points-outer} for $l_{i}$ with $U_{i}=\\mathbb{B}(\\bar{z},r)$.\nThen $\\bar{z}$ is a Clarke critical point.\\end{prop}\n\\begin{proof}\nSeeking a contradiction, suppose there exists some direction $\\bar{v}$\nsuch that $f^{\\circ}(\\bar{z};\\bar{v})<0$. This means that there is\nsome $\\bar{\\epsilon}>0$ such that if $|z-\\bar{z}|<\\bar{\\epsilon}$\nand $\\epsilon<\\bar{\\epsilon}$, then \\begin{eqnarray*}\n\\frac{f(z+\\epsilon\\bar{v})-f(z)}{\\epsilon} & < & \\frac{1}{2}f^{\\circ}(\\bar{z};\\bar{v})\\\\\n\\Rightarrow f(z+\\epsilon\\bar{v}) & < & f(z)+\\epsilon\\frac{1}{2}f^{\\circ}(\\bar{z};\\bar{v}).\\end{eqnarray*}\nSuppose $i$ is large enough so that $x_{i},y_{i}\\in\\mathbb{B}(\\bar{z},\\frac{\\bar{\\epsilon}}{2})$,\nand that $x_{i},y_{i}\\in A_{i}:=S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap\\mathbb{B}(\\bar{z},r)$\nare such that $|x_{i}-y_{i}|=\\mbox{\\rm diam}(A_{i})$. Consider the set $\\tilde{A}:=(S_{i}+\\epsilon_{1}\\bar{v})\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap\\mathbb{B}(\\bar{z},r)$,\nwhere $\\epsilon_{1}>0$ is arbitrarily small. Let $\\tilde{x}_{i},\\tilde{y}_{i}\\in\\tilde{A}$\nbe such that $|\\tilde{x}_{i}-\\tilde{y}_{i}|=\\mbox{\\rm diam}(\\tilde{A})$. From\nthe minimality of the outer minimization, we have $|\\tilde{x}_{i}-\\tilde{y}_{i}|\\geq|x_{i}-y_{i}|$.\nNote that $f(\\tilde{x}_{i})=f(\\tilde{y}_{i})=l_{i}$. Then \\begin{eqnarray*}\nf(\\tilde{x}_{i}) & < & f(\\tilde{x}_{i}-\\epsilon_{1}\\bar{v})+\\epsilon_{1}\\frac{1}{2}f^{\\circ}(\\bar{z};\\bar{v})\\\\\n\\implies f(\\tilde{x}_{i}-\\epsilon_{1}\\bar{v}) & > & f(\\tilde{x}_{i})-\\epsilon_{1}\\frac{1}{2}f^{\\circ}(\\bar{z};\\bar{v})\\\\\n & > & l_{i}.\\end{eqnarray*}\nThe continuity of $f$ implies that we can find some $\\epsilon_{2}>0$\nsuch that $\\hat{x}_{i}:=\\tilde{x}_{i}-\\epsilon_{1}\\bar{v}+\\epsilon_{2}(\\tilde{x}_{i}-\\tilde{y}_{i})$\nlies in $A_{i}$. Similarly, $\\hat{y}_{i}:=\\tilde{y}_{i}-\\epsilon_{1}\\bar{v}$\nlie in $A_{i}$ as well. But\\begin{eqnarray*}\n|\\hat{x}_{i}-\\hat{y}_{i}| & > & |\\tilde{x}_{i}-\\tilde{y}_{i}|\\\\\n & \\geq & |x_{i}-y_{i}|.\\end{eqnarray*}\nThis contradicts the maximality of $|x_{i}-y_{i}|$ in $A_{i}$, and\nthus $\\bar{z}$ must be a critical point. \n\\end{proof}\n\n\\section{\\label{sec:Opti-condns}Optimality conditions}\n\nWe now reduce the min-max problem \\eqref{eq:keyexp1} to a condition\non the gradients $\\nabla f(x_{i})$ and $\\nabla f(y_{i})$ that is\neasy to verify numerically. This condition will help in the numerical\nsolution of \\eqref{eq:keyexp1}. We use methods in sensitivity analysis\nof optimization problems (as is done in \\cite{BS00}) to study how\nvarying the $m$-dimensional affine space $S$ in an $(m+1)$-dimensional\nsubspace affects the optimal value in the inner maximization problem\nin \\eqref{eq:keyexp1}. We conform as much as possible to the notation\nin \\cite{BS00} throughout this section. \n\nConsider the following parametric optimization problem $(P_{u})$\nin terms of $u\\in\\mathbb{R}$ as an $m+1$ dimensional model in $\\mathbb{R}^{m+1}$\nof the inner maximization problem in \\eqref{eq:keyexp1}:\\begin{eqnarray}\n(P_{u}):\\qquad v(u):= & \\min & F(x,y,u):=-|x-y|^{2}\\nonumber \\\\\n & \\mbox{s.t.} & G(x,y,u)\\in K,\\nonumber \\\\\n &  & x,y\\in\\mathbb{R}^{m+1},\\label{eq:stylized-problem}\\end{eqnarray}\nwhere $G:(\\mathbb{R}^{m+1})^{2}\\times\\mathbb{R}\\rightarrow\\mathbb{R}^{4}$\nand $K\\subset\\mathbb{R}^{4}$ are defined by \\[\nG(x,y,u):=\\left(\\begin{array}{c}\n-f(x)+b\\\\\n-f(y)+b\\\\\n(0,0,\\dots,0,u,1)x\\\\\n(0,0,\\dots,0,u,1)y\\end{array}\\right),\\qquad K:=\\mathbb{R}_{-}^{2}\\times\\{0\\}^{2}.\\]\nThe problem $(P_{u})$ reflects the inner maximization problem of\n\\eqref{eq:keyexp1}. Due to the standard practice of writing optimization\nproblems as minimization problems, \\eqref{eq:stylized-problem} is\na minimization problem instead. We hope this does not cause confusion. \n\nLet $S(u)$ be the $m$-dimensional subspace orthogonal to $(0,\\dots,0,u,1)$.\nThe first two components of $G(x,y,u)$ model the constraints $f(x)\\geq b$\nand $f(y)\\geq b$, while the last two components enforce $x,y\\in S(u)$.\nDenote an optimal solution to $(P_{u})$ to be $(\\bar{x}(u),\\bar{y}(u))$,\nand let $(\\bar{x},\\bar{y}):=(\\bar{x}(0),\\bar{y}(0))$. We make the\nfollowing assumption throughout.\n\\begin{assumption}\n\\label{ass:uniqueness}(Uniqueness of optimizers) $(P_{0})$ has a\nunique solution $\\bar{x}=\\mathbf{0}$ and $\\bar{y}=(0,\\dots,0,1,0)$\nat $u=0$. \n\\end{assumption}\nWe shall investigate how the set of minimizers of $(P_{u})$ behaves\nwith respect to $u$ at $0$.\n\nThe derivatives of $F$ and $G$ with respect to $x$ and $y$, denoted\nby $D_{x,y}F$ and $D_{x,y}G$, are\\begin{eqnarray}\nD_{x,y}F(x,y,u) & = & 2\\big(\\begin{array}{cc}\n(y-x)^{T} & (x-y)^{T}\\end{array}\\big),\\nonumber \\\\\n\\mbox{ and }D_{x,y}G(x,y,u) & = & \\left(\\begin{array}{cc}\n-\\nabla f(x)^{T}\\\\\n & -\\nabla f(y)^{T}\\\\\n(0,0,\\dots,0,u,1)\\\\\n & (0,0,\\dots,0,u,1)\\end{array}\\right),\\label{eq:D-x-y}\\end{eqnarray}\nwhere the blank terms in $D_{x,y}G(x,y,u)$ are all zero. \n\nThe \\emph{Lagrangian} is the function $L:\\mathbb{R}^{m+1}\\times\\mathbb{R}^{m+1}\\times\\mathbb{R}^{4}\\times\\mathbb{R}\\rightarrow\\mathbb{R}$\ndefined by \\[\nL(x,y,\\lambda,u):=F(x,y,u)+\\sum_{i=1}^{4}\\lambda_{i}G_{i}(x,y,u).\\]\nWe say that $\\lambda:=(\\lambda_{1},\\lambda_{2},\\lambda_{3},\\lambda_{4})$,\ndepending on $u$, is a \\emph{Lagrange multiplier }if $D_{x,y}L(x,y,\\lambda,u)=\\mathbf{0}$\nand $\\lambda\\in N_{K}(G(x,y,u))$, and the set of all Lagrange multipliers\nis denoted by $\\Lambda(x,y,u)$. Here, $N_{K}(G(x,y,u))$ stands for\nthe \\emph{normal cone} defined by \\[\nN_{K}\\big(G(x,y,u)\\big):=\\{v\\in\\mathbb{R}^{4}\\mid v^{T}[w-G(x,y,u)]\\leq0\\mbox{ for all }w\\in K\\}.\\]\nWe are interested in the set $\\Lambda(\\bar{x},\\bar{y},0)$. It is\nclear that optimal solutions must satisfy $G(\\bar{x},\\bar{y},0)=\\mathbf{0}$,\nso $\\lambda\\in N_{K}(\\mathbf{0})=\\mathbb{R}_{+}^{2}\\times\\mathbb{R}^{2}$. \n\nThe condition $D_{x,y}L(\\bar{x},\\bar{y},\\lambda,0)=\\mathbf{0}$ reduces\nto \\begin{eqnarray*}\nD_{x,y}\\left(F(x,y,0)+\\sum_{i=1}^{4}\\lambda_{i}G_{i}(x,y,0)\\right)\\mid_{x=\\bar{x},y=\\bar{y}} & = & \\mathbf{0}\\\\\n\\Rightarrow2\\left({\\bar{y}-\\bar{x}\\atop \\bar{x}-\\bar{y}}\\right)+\\lambda_{1}\\left({-\\nabla f(\\bar{x})\\atop \\mathbf{0}}\\right)+\\lambda_{2}\\left({\\mathbf{0}\\atop -\\nabla f(\\bar{y})}\\right)\\qquad\\qquad\\qquad\\\\\n+\\lambda_{3}\\left({(0,0,\\dots,0,0,1)^{T}\\atop \\mathbf{0}}\\right)+\\lambda_{4}\\left({\\mathbf{0}\\atop (0,0,\\dots,0,0,1)^{T}}\\right) & = & \\mathbf{0}.\\end{eqnarray*}\n\n\nHere, $G_{i}(\\bar{x},\\bar{y},0)$ is the $i$th row of $G(\\bar{x},\\bar{y},0)$\nfor $1\\leq i\\leq4$. This is exactly the KKT conditions, and can be\nrewritten as\\begin{eqnarray}\n2(\\bar{y}-\\bar{x})-\\lambda_{1}\\nabla f(\\bar{x})+\\lambda_{3}(0,0,\\dots,0,0,1)^{T} & = & 0,\\nonumber \\\\\n2(\\bar{x}-\\bar{y})-\\lambda_{2}\\nabla f(\\bar{y})+\\lambda_{4}(0,0,\\dots,0,0,1)^{T} & = & 0.\\label{eq:small-KKT}\\end{eqnarray}\n\n\nIt is clear that $\\lambda_{1}$ and $\\lambda_{2}$ cannot be zero,\nand so we have \\begin{eqnarray*}\n\\nabla f(\\bar{x})^{T} & = & \\left(0,0,\\dots,0,\\frac{2}{\\lambda_{1}},\\frac{\\lambda_{3}}{\\lambda_{1}}\\right),\\\\\n\\nabla f(\\bar{y})^{T} & = & \\left(0,0,\\dots,0,-\\frac{2}{\\lambda_{2}},\\frac{\\lambda_{4}}{\\lambda_{2}}\\right).\\end{eqnarray*}\nRecall that $\\lambda_{1},\\lambda_{2}\\geq0$, so this gives more information\nabout $\\nabla f(\\bar{x})$ and $\\nabla f(\\bar{y})$.\n\nWe next discuss the optimality of the outer minimization problem of\n\\eqref{eq:keyexp1}, which can be studied by perturbations in the\nparameter $u$ of \\eqref{eq:stylized-problem}, but we first recall\na result on the first order sensitivity of optimal solutions.\n\\begin{defn}\n(Robinson's constraint qualification) (from \\cite[Definition 2.86]{BS00})\nWe say that \\emph{Robinson's constraint qualification }holds at $(\\bar{x},\\bar{y})\\in\\mathbb{R}^{m+1}\\times\\mathbb{R}^{m+1}$\n if the regularity condition \\[\n\\mathbf{0}\\in\\mbox{\\rm int}\\left\\{ G(\\bar{x},\\bar{y},0)+\\mbox{\\rm Range}\\big(D_{x,y}G(\\bar{x},\\bar{y},0)\\big)-K\\right\\} \\]\nis satisfied.\\end{defn}\n\\begin{thm}\n\\label{thm:from-BS-4.26}(Parametric optimization) (from \\cite[Theorem 4.26]{BS00})\nFor problem\\eqref{eq:stylized-problem}, let $(\\bar{x}(u),\\bar{y}(u))$\nbe as defined earlier. Suppose that \n\\begin{enumerate}\n\\item [(i)] Robinson's constraint qualification holds at $(\\bar{x}(0),\\bar{y}(0))$,\nand\n\\item [(ii)] if $u_{n}\\rightarrow0$, then $(P_{u_{n}})$ possesses an\noptimal solution $(\\bar{x}(u_{n}),\\bar{y}(u_{n}))$ that has a limit\npoint $(\\bar{x},\\bar{y})$.\n\\end{enumerate}\nThen $v(\\cdot)$ is directionally differentiable at $u=0$ and \\[\nv^{\\prime}(0)=D_{u}L(x,y,\\lambda,0).\\]\n\n\\end{thm}\nWe proceed to prove our result.\n\\begin{prop}\n(Optimality condition on $\\nabla f(\\bar{y})$) \\label{pro:perturb-decrease}Consider\nthe setup so far in this section and suppose Assumption \\ref{ass:uniqueness}\nholds. If $\\nabla f(\\bar{y})$ is not a positive multiple of $(0,0,\\dots,0,1,0)^{T}$\nat $u=0$, then we can perturb $u$ so that \\eqref{eq:stylized-problem}\nhas an increase in objective. \\end{prop}\n\\begin{proof}\nWe first obtain first order sensitivity information from Theorem \\ref{thm:from-BS-4.26}.\nRecall that by definition, Robinson's constraint qualification holds\nat $(\\bar{x},\\bar{y})$ if \\[\n\\mathbf{0}\\in\\mbox{int}\\left\\{ G(\\bar{x},\\bar{y},0)+\\mbox{Range}\\big(D_{x,y}G(\\bar{x},\\bar{y},0)\\big)-K\\right\\} .\\]\nFrom \\eqref{eq:small-KKT}, it is clear that $\\nabla f(\\bar{x})$\nand $(0,\\dots,0,0,1)$ are linearly independent, and so are $\\nabla f(\\bar{y})$\nand $(0,\\dots,0,0,1)$. From the formula of $D_{x,y}G(\\bar{x},\\bar{y},0)$\nin \\eqref{eq:D-x-y}, we see immediately that $\\mbox{Range}\\left(D_{x,y}G(\\bar{x},\\bar{y},0)\\right)=\\mathbb{R}^{4}$,\nthus the Robinson's constraint qualification indeed holds.\n\nSuppose that $\\lim_{n\\to\\infty}t_{n}=0$. We prove that part (ii)\nof Theorem \\ref{thm:from-BS-4.26} holds by proving that $(\\bar{x}(t_{n}),\\bar{y}(t_{n}))$\ncannot have any other limit points. Suppose that $(x^{\\prime},y^{\\prime})$\nis a limit point of $\\{(\\bar{x}(t_{n}),\\bar{y}(t_{n}))\\}_{n=1}^{\\infty}$.\nIt is clear that $x^{\\prime},y^{\\prime}\\in S(0)$.\n\nWe can find $y_{n}\\rightarrow\\bar{y}$ such that $y_{n}\\in S(t_{n})$\nand $f(y_{n})=b$. For example, we can use the Implicit Function Theorem\nwith the constraints \\begin{eqnarray*}\nf(y) & = & b,\\\\\ng(y,u) & = & 0,\\end{eqnarray*}\nwhere $g(y,u)=(0,0,\\dots,0,u,1)^{T}y$. The derivatives with respect\nto $y_{m}$ and $y_{m+1}$ are \\[\n\\begin{array}{ccc}\n\\frac{\\partial}{\\partial y_{m}}f(\\bar{y})=-\\frac{2}{\\lambda_{2}}, &  & \\frac{\\partial}{\\partial y_{m+1}}f(\\bar{y})=\\frac{\\lambda_{4}}{\\lambda_{2}},\\\\\n\\frac{\\partial}{\\partial y_{m}}g(\\bar{y},0)=0, &  & \\frac{\\partial}{\\partial y_{m+1}}g(\\bar{y},0)=1.\\end{array}\\]\nTherefore, for $y_{1}=y_{2}=\\cdots=y_{m-2}=y_{m-1}=0$ and any choice\nof $u$ close to zero, there is some $y_{m}$ and $y_{m+1}$ such\nthat $y\\in S(u)$ and $f(y)=b$.\n\nClearly $|\\bar{x}-y_{n}|\\leq|\\bar{x}(t_{n})-\\bar{y}(t_{n})|$. Taking\nlimits as $n\\rightarrow\\infty$, we have $|\\bar{x}-\\bar{y}|\\leq|x^{\\prime}-y^{\\prime}|$.\nSince $(\\bar{x},\\bar{y})$ minimize $F$, it follows that $|\\bar{x}-\\bar{y}|=|x^{\\prime}-y^{\\prime}|$,\nand by the uniqueness of solutions to $(P_{0})$, we can assume that\n$x^{\\prime}=\\bar{x}$ and $y^{\\prime}=\\bar{y}$.\n\nTheorem \\ref{thm:from-BS-4.26} implies that $v^{\\prime}(0)=D_{u}L(x,y,\\lambda,0)$.\nWe now calculate $D_{u}L(x,y,\\lambda,0)$. It is clear that $D_{u}G(x,y,\\lambda,0)=(0,0,0,1)^{T}$,\nand so $D_{u}L(x,y,\\lambda,0)=\\lambda_{4}$. Since $\\nabla f(\\bar{y})$\nis not a multiple of $(0,0,\\dots,0,1,0)^{T}$ at $u=0$, $\\lambda_{4}\\neq0$,\nand this gives the conclusion we need.\n\\end{proof}\nA direct consequence of Proposition \\ref{pro:perturb-decrease} is\nthe following easily checkable condition.\n\\begin{thm}\n(Gradients are opposite) \\label{pro:opposite-directions}Let $(S_{i},x_{i},y_{i})$\nbe an optimizing triple to \\eqref{eq:keyexp1} for some $l_{i}$ such\nthat $S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i}$ is closed, and $(x_{i},y_{i})$\nis the unique pair of points in $S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i}$\nsatisfying $|x_{i}-y_{i}|=\\mbox{\\rm diam}(S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i})$.\nThen $\\nabla f(x_{i})$ and $\\nabla f(y_{i})$ are nonzero and point\nin opposite directions.\\end{thm}\n\\begin{proof}\nWe can look at an $m+1$ dimensional subspace which reduces to the\nsetting that we are considering so far in this section. By Proposition\n\\ref{pro:perturb-decrease}, $\\nabla f(y_{i})$ is a positive multiple\nof $x_{i}-y_{i}$ at optimality. Similarly, $\\nabla f(x_{i})$ is\na positive multiple of $y_{i}-x_{i}$ at optimality, and the result\nfollows.\n\\end{proof}\n\n\nWe remark on how to start the algorithm. We look at critical points\nof Morse index 1 first. In this case, two local minima $\\bar{x}_{1}$,\n$\\bar{x}_{2}$ are needed before the mountain pass algorithm can guarantee\nthe existence of a critical point $\\bar{x}_{3}$. For any value above\nthe critical value corresponding to the critical point of Morse index\n1, the level set contains a path connecting $\\bar{x}_{1}$ and $\\bar{x}_{2}$\npassing through $\\bar{x}_{3}$. \n\nTo find the next critical point of Morse index 2 we remark that under\nmild conditions, if $\\mbox{\\rm lev}_{\\leq a}f$ contains a closed path homeomorphic\nto $\\mathbb{S}_{1}$, the boundary of the disc of dimension 2, then\nthe linking principle guarantees the existence of a critical point\nthrough the multidimensional mountain pass theorem. Theorem \\ref{thm:Rabinowitz-77}\nwhich we quote later gives an idea how this is possible. We refer\nthe reader to \\cite{Sch99} and \\cite[Chapter 19]{Jab03} for more\ndetails on linking methods. \n\nWe now illustrate with an example that without the assumption that\n$(x_{i},y_{i})$ is the unique pair of points satisfying $|x_{i}-y_{i}|=\\mbox{\\rm diam}(S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i})$,\nthe conclusion in Theorem \\ref{pro:opposite-directions} need not\nhold.\n\n\\begin{figure}[h]\n\\begin{tabular}{|c|c|}\n\\hline \n\\includegraphics[scale=0.6]{wedge} & \\includegraphics[scale=0.4]{wedge2}\\tabularnewline\n\\hline\n\\end{tabular}\n\n\\caption{\\label{fig:The-lines}The diagram on the left illustrates the setting\nof Lemma \\ref{lem:intersecting-rays}, while the diagram on the right\nillustrates Example \\ref{exa:4-lines}.}\n\n\\end{figure}\n\n\\begin{lem}\n(Shortest line segments) \\label{lem:intersecting-rays}Suppose lines\n$l_{1}$ and $l_{2}$ intersect at the origin in $\\mathbb{R}^{2}$,\nand let $P$ be a point on the angle bisector as shown in the diagram\non the left of Figure \\ref{fig:The-lines}. The minimum distance of\nthe line segment $AB$, where $A$ is a point on $l_{1}$ and $B$\nis a point on $l_{2}$ and $AB$ passes through $P$, is attained\nwhen $OAB$ is an isosceles triangle with $AB$ as its base. \\end{lem}\n\\begin{proof}\nMuch of this is high school trigonometry and plane geometry, but we\npresent full details for completeness. Let $\\alpha$ be the angle\n$\\measuredangle AOP$, $\\beta$ be the angle $\\measuredangle PAO$,\nand $d=|OP|$. By using the sine rule, we get \\[\n|AB|=d\\left(\\frac{\\sin\\alpha}{\\sin\\theta}+\\frac{\\sin\\alpha}{\\sin(\\pi-2\\alpha-\\theta)}\\right).\\]\nThe problem is now reduced to finding the $\\theta$ that minimizes\nthe value above. Continuing the arithmetic gives:\\begin{eqnarray*}\nd\\left(\\frac{\\sin\\alpha}{\\sin\\theta}+\\frac{\\sin\\alpha}{\\sin(\\pi-2\\alpha-\\theta)}\\right) & = & d\\sin\\alpha\\left(\\frac{1}{\\sin\\theta}+\\frac{1}{\\sin(2\\alpha+\\theta)}\\right)\\\\\n & = & d\\sin\\alpha\\left(\\frac{\\sin\\theta+\\sin(2\\alpha+\\theta)}{\\sin(\\theta)\\sin(2\\alpha+\\theta)}\\right)\\\\\n & = & d\\sin\\alpha\\left(\\frac{\\sin\\theta+\\sin(2\\alpha+\\theta)}{\\sin(\\theta)\\sin(2\\alpha+\\theta)}\\right)\\\\\n & = & d\\sin\\alpha\\left(\\frac{2\\sin(\\alpha+\\theta)\\cos\\alpha}{\\frac{1}{2}[\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)]}\\right)\\\\\n & = & 2d\\sin(2\\alpha)\\left(\\frac{\\sin(\\alpha+\\theta)}{\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)}\\right)\\end{eqnarray*}\nWe now differentiate the $\\frac{\\sin(\\alpha+\\theta)}{\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)}$\nterm above, which gives\\begin{eqnarray*}\n &  & \\frac{d}{d\\theta}\\left(\\frac{\\sin(\\alpha+\\theta)}{\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)}\\right)\\\\\n & = & \\frac{1}{[\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)]^{2}}\\left[\\cos(\\alpha+\\theta)[\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)]-2\\sin(2\\alpha+2\\theta)\\sin(\\alpha+\\theta)\\right]\\end{eqnarray*}\nThe numerator is simplified to be:\\begin{eqnarray*}\n &  & \\cos(\\alpha+\\theta)[\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)]-2\\sin(2\\alpha+2\\theta)\\sin(\\alpha+\\theta)\\\\\n & = & \\cos(\\alpha+\\theta)[\\cos(2\\alpha)-2\\cos^{2}(\\alpha+\\theta)+1]-4\\sin^{2}(\\alpha+\\theta)\\cos(\\alpha+\\theta)\\\\\n & = & \\cos(\\alpha+\\theta)[\\cos(2\\alpha)-2\\cos^{2}(\\alpha+\\theta)+1-4\\sin^{2}(\\alpha+\\theta)]\\\\\n & = & \\cos(\\alpha+\\theta)[\\cos(2\\alpha)-2\\cos^{2}(\\alpha+\\theta)+4\\cos^{2}(\\alpha+\\theta)-3]\\\\\n & = & \\cos(\\alpha+\\theta)[\\cos(2\\alpha)+2\\cos^{2}(\\alpha+\\theta)-3].\\end{eqnarray*}\nWith this formula, we see that the conditions for $\\frac{d}{d\\theta}\\left(\\frac{\\sin(\\alpha+\\theta)}{\\cos(2\\alpha)-\\cos(2\\alpha+2\\theta)}\\right)=0$\nis to have $\\cos(\\alpha+\\theta)=0$ or $\\cos(2\\alpha)+2\\cos^{2}(\\alpha+\\theta)=3$.\nThe first case gives us $\\theta=\\frac{\\pi}{2}-\\alpha$, which gives\nus the required conclusion. The second case requires $\\alpha=0$ or\n$\\alpha=\\pi$ and $\\theta=0$ or $\\theta=\\pi$, which are degenerate\ncases. This gives us all optimum solutions to our problem, and concludes\nthe proof.\n\\end{proof}\nWe now create an example in $\\mathbb{R}^{3}$ that illustrates that\nthe omission of the condition of unique solutions need not give us\npoints whose gradients point in opposite directions.\n\\begin{example}\n(Gradients need not be opposite) \\label{exa:4-lines}Define the four\nlines $L_{1}$ to $L_{4}$ by \\begin{eqnarray*}\nL_{1} & := & \\{(0,0,-1)+\\lambda(1,0,1)\\mid\\lambda\\in\\mathbb{R}_{+}\\}\\\\\nL_{2} & := & \\{(0,0,-1)+\\lambda(-1,0,1)\\mid\\lambda\\in\\mathbb{R}_{+}\\}\\\\\nL_{3} & := & \\{(0,0,1)+\\lambda(0,1,-1)\\mid\\lambda\\in\\mathbb{R}_{+}\\}\\\\\nL_{4} & := & \\{(0,0,1)+\\lambda(0,-1,-1)\\mid\\lambda\\in\\mathbb{R}_{+}\\}\\end{eqnarray*}\nThe lines $L_{1}$ and $L_{2}$ lie in the $x$-$z$ plane, while\nthe lines $L_{3}$ and $L_{4}$ lie in the $y$-$z$ plane. See the\ndiagram on the right of Figure \\ref{fig:The-lines}.\n\nConsider first the problem of finding a plane $S$ that is a minimizer\nof the maximum of the distances between the points defined by the\nintersections of $S$ and the $L_{i}$'s. We now show that $S$ has\nto be the $x$-$y$ plane. The plane $S$ intersects the $z$ axis\nat some point $(0,0,p)$. When $S$ is the $x$-$y$ plane, the maximum\ndistance between the points is $2$. By Lemma \\ref{lem:intersecting-rays},\nthe distance between the points $S\\cap L_{1}$ and $S\\cap L_{2}$\nis at least $2(1-p)$, while the distance between the the points $S\\cap L_{3}$\nand $S\\cap L_{4}$ is at least $2(1+p)$. This tells us that the $x$-$y$\nplane is optimal.\n\nWith this observation, we now construct our example. Consider the\nfunction $f:\\mathbb{R}^{3}\\rightarrow\\mathbb{R}$ defined by\\[\nf(x,y,z)=-\\left(\\frac{x}{1+z}+\\frac{y}{1-z}\\right)^{4\/3}-\\left(\\frac{x}{1+z}-\\frac{y}{1-z}\\right)^{4\/3}.\\]\nThe level set $\\mbox{\\rm lev}_{\\geq-2}f$ contains the lines $L_{1}$ to $L_{4}$.\nThis means that $\\mbox{diam}(S\\cap\\mbox{\\rm lev}_{\\geq-2}f)\\geq2$. This is\nin fact an equation when $S$ is the $x$-$y$ plane, and the maximizers\nbeing the pairs $\\{\\pm(1,0,0)\\}$ and $\\{\\pm(0,1,0)\\}$.\n\nThe gradient $\\nabla f(x,y,z)$ is\\[\n\\nabla f(x,y,z)=\\left(\\begin{array}{c}\n-\\frac{4}{3}\\left(\\frac{x}{1+z}+\\frac{y}{1-z}\\right)^{1\/3}-\\frac{4}{3}\\left(\\frac{x}{1+z}-\\frac{y}{1-z}\\right)^{1\/3}\\\\\n-\\frac{4}{3}\\left(\\frac{x}{1+z}+\\frac{y}{1-z}\\right)^{1\/3}+\\frac{4}{3}\\left(\\frac{x}{1+z}-\\frac{y}{1-z}\\right)^{1\/3}\\\\\n-\\frac{4}{3}\\left(\\frac{x}{1+z}+\\frac{y}{1-z}\\right)^{1\/3}\\left(-\\frac{x}{(1+z)^{2}}+\\frac{y}{(1-z)^{2}}\\right)-\\frac{4}{3}\\left(\\frac{x}{1+z}-\\frac{y}{1-z}\\right)^{1\/3}\\left(-\\frac{x}{(1+z)^{2}}-\\frac{y}{(1-z)^{2}}\\right)\\end{array}\\right)\\]\nWith this, we can evaluate $\\nabla f$ at $\\pm(1,0,0)$ and $\\pm(0,1,0)$\nto be \\begin{eqnarray*}\n\\nabla f(1,0,0) & = & \\left(-\\frac{8}{3},0,\\frac{8}{3}\\right),\\\\\n\\nabla f(-1,0,0) & = & \\left(\\frac{8}{3},0,\\frac{8}{3}\\right),\\\\\n\\nabla f(0,1,0) & = & \\left(0,-\\frac{8}{3},-\\frac{8}{3}\\right),\\\\\n\\nabla f(0,-1,0) & = & \\left(0,\\frac{8}{3},-\\frac{8}{3}\\right).\\end{eqnarray*}\nNeither of the pairs $\\{\\pm(1,0,0)\\}$ and $\\{\\pm(0,1,0)\\}$ have\nopposite pointing gradients, which concludes our example.\n\\end{example}\n\n\\section{\\label{sec:Other-local-analysis}Another convergence property of\ncritical points}\n\nIn this section, we look at a condition on critical points similar\nto Proposition \\ref{pro:triples-conv} that can be helpful for numerical\nmethods for finding critical points. Theorem \\ref{thm:crit-from-linking}\nbelow does not seem to be easily found in the literature, and can\nbe seen as a local version of the mountain pass theorem. \n\nWe prove Theorem \\ref{thm:crit-from-linking} in the more general\nsetting of metric spaces. Such a treatment includes the case of nonsmooth\nfunctions. We recall the following definitions in metric critical\npoint theory from \\cite{DM94,IS96,Katriel94}.\n\\begin{defn}\n\\label{def:Deformation-critical}Let $(X,d)$ be a metric space. We\ncall the point $x$ \\emph{Morse regular} for the function $f:X\\rightarrow\\mathbb{R}$\nif, for some numbers $\\gamma,\\sigma>0$, there is a continuous function\n\\[\n\\phi:\\mathbb{B}(x,\\gamma)\\times[0,\\gamma]\\rightarrow X\\]\nsuch that all points $u\\in\\mathbb{B}(x,\\gamma)$ and $t\\in[0,\\gamma]$\nsatisfy the inequality \\[\nf\\big(\\phi(x,t)\\big)\\leq f(x)-\\sigma t,\\]\nand that $\\phi(\\cdot,0):\\mathbb{B}(x,\\gamma)\\to\\mathbb{B}(x,\\gamma)$\nis the identity map. The point $x$ is \\emph{Morse critical }if it\nis not Morse regular. \n\nIf for some $\\phi$, there is some $\\kappa>0$ such that $\\phi$ also\nsatisfies the inequality \\[\nd\\big(\\phi(x,t),x\\big)\\leq\\kappa t,\\]\nthen we call $x$ \\emph{deformationally regular}. The point $x$ is\n\\emph{deformationally critical }if it is not deformationally regular.\n\\end{defn}\nIt is a fact that if $X$ is a Banach space and $f$ is locally Lipschitz,\nthen deformationally critical points are Clarke critical. The following\ntheorem gives a strategy for identifying deformationally critical\npoints.\n\\begin{thm}\n\\label{thm:crit-from-linking}(Critical points from sequences of linking\nsets) Let $X$ be a metric space and $f:X\\to\\mathbb{R}$. Suppose\nthere is some open set $U$ of $\\bar{x}$ and sequences of sets $\\{\\Phi_{i}\\}_{i=1}^{\\infty}$\nand $\\{\\Gamma_{i}\\}_{i=1}^{\\infty}$ such that \n\\begin{enumerate}\n\\item $\\Phi_{i}$ and $\\partial\\Gamma_{i}$ link.\n\\item $\\Gamma_{i}$ are homeomorphic to $\\mathbb{B}^{m}$ for all $i$,\nand $\\max_{x\\in\\partial\\Gamma_{i}}f(x)<\\inf_{x\\in\\Phi_{i}\\cap U}f(x)$. \n\\item For any open set $V$ containing $\\bar{x}$, there is some $I>0$\nsuch that $\\Gamma_{i}\\subset V$ for all $i>I$. \n\\item $f$ is Lipschitz in $U$.\n\\end{enumerate}\nThen $\\bar{x}$ is deformationally critical.\\end{thm}\n\\begin{proof}\nSuppose $\\bar{x}$ is deformationally regular. Then there are $\\gamma,\\sigma,\\kappa>0$\nand $\\phi:\\mathbb{B}(\\bar{x},\\gamma)\\times[0,\\gamma]\\to X$ such that\nthe inequalities\\[\nf\\big(\\phi(x,t)\\big)\\leq f(x)-\\sigma t\\mbox{ and }d\\big(\\phi(x,t),x\\big)\\leq\\kappa t\\]\nhold for all points $x\\in\\mathbb{B}(\\bar{x},\\gamma)$ and $t\\in[0,\\gamma]$,\nand $\\phi(\\cdot,0):\\mathbb{B}(\\bar{x},\\gamma)\\to\\mathbb{B}(\\bar{x},\\gamma)$\nis the identity map. We may reduce $\\gamma$ as necessary and assume\nthat $U=\\mathbb{B}(\\bar{x},\\gamma)$. \n\nCondition (3) implies that for any $\\alpha>0$, then there is some\n$I_{1}$ such that $\\Gamma_{i}\\subset\\mathbb{B}(\\bar{x},\\alpha)$\nfor all $i>I_{1}$. Consider $\\Gamma_{i,t}:=\\phi((\\Gamma_{i}\\times\\{t\\})\\cup(\\partial\\Gamma_{i}\\times[0,t]))$.\nProvided $0<t<\\frac{\\gamma-\\alpha}{\\kappa}$, we have $\\Gamma_{i,t}\\subset U$.\n\nSince $f$ is Lipschitz in $U$, let $\\bar{\\kappa}$ be the modulus\nof Lipschitz continuity in $U$. We have $\\max_{x\\in\\Gamma_{i}}f(x)\\leq\\max_{x\\in\\partial\\Gamma_{i}}f(x)+\\bar{\\kappa}\\mbox{\\rm diam}(\\Gamma_{i})$.\nAlso,\\begin{eqnarray*}\n\\max_{x\\in\\Gamma_{i.t}}f(x) & \\leq & \\max\\left(\\max_{x\\in\\partial\\Gamma_{i}}f(x)+\\bar{\\kappa}\\mbox{\\rm diam}(\\Gamma_{i})-\\sigma t,\\max_{x\\in\\partial\\Gamma_{i}}f(x)\\right).\\end{eqnarray*}\n\n\nBy condition (3), there is some $I_{2}>0$ such that if $i>I_{2}$,\nthen $\\mbox{\\rm diam}(\\Gamma_{i})<\\frac{\\sigma t}{\\bar{\\kappa}}$. So for $i>\\max(I_{1},I_{2})$,\nwe have \\[\n\\max_{x\\in\\Gamma_{i,t}}f(x)=\\max_{x\\in\\partial\\Gamma_{i}}f(x)<\\inf_{x\\in\\Phi_{i}\\cap U}f(x).\\]\n\n\nHowever, the fact that $\\partial\\Gamma_{i}$ and $\\Phi_{i}$ link\nimplies that $\\Gamma_{i,t}$ and $\\Phi_{i}$ must intersect, and since\n$\\Gamma_{i,t}\\subset U$, $\\Gamma_{i,t}$ and $\\Phi_{i}\\cap U$ must\nintersect. This is a contradiction, so $\\bar{x}$ is deformationally\ncritical.\n\\end{proof}\nIt is reasonable to choose $\\Gamma_{i}$ to be a simplex (that is,\na convex hull of $m+1$ points) and $\\Phi_{i}$ to be an affine space.\nIf the sequence of sets $\\{\\Gamma_{i}\\}_{i=1}^{\\infty}$ converges\nto the single point $\\bar{x}$ and $f$ is $\\mathcal{C}^{2}$ there,\na quadratic approximation of $f$ using only the knowledge of the\nvalues of $f$ and $\\nabla f$ on the vertices of the simplex would\nbe good approximation of $f$ on the simplex. We outline our strategy\nbelow.\n\\begin{algorithm}\n\\label{alg:quad-approx}(Obtaining unknowns in quadratic) Let $h:\\mathbb{R}^{m}\\to\\mathbb{R}$\nbe defined by $h(x)=\\frac{1}{2}x^{T}Ax+b^{T}x+c$, and let $p_{1},\\dots,p_{m+1}$\nbe $m+1$ points in $\\mathbb{R}^{m}$. Suppose that the values of\n$h(p_{i})$ and $\\nabla h(p_{i})$ are known for all $i=1,\\dots,m+1$.\nWe seek to obtain the values of $A$, $b$ and $c$.\n\\begin{enumerate}\n\\item Let $P\\in\\mathbb{R}^{m\\times m}$ be the matrix such that the $i$th\ncolumn is $p_{i+1}-p_{1}$, and let $D\\in\\mathbb{R}^{m\\times m}$\nbe the matrix such that the $i$th column is $\\nabla h(p_{i+1})-\\nabla h(p_{1})$.\nCalculate $A$ with $A=DP^{-1}$.\n\\item Calculate $b$ with $b=\\nabla h(p_{1})-Ap_{1}$.\n\\item Calculate $c$ with $c=h(p_{1})-\\frac{1}{2}p_{1}^{T}Ap_{1}-b^{T}p_{1}$.\n\\end{enumerate}\n\\end{algorithm}\nIf $h$ is $\\mathcal{C}^{2}$ instead of being a quadratic, then the\nprocedure in Algorithm \\ref{alg:quad-approx} can be used to approximate\nthe values of $h$ on a simplex. \n\nIn Lemma \\ref{lem:quad-est} below, given $m+1$ points in $\\mathbb{R}^{n}$,\nwe need to approximate a quadratic function on $\\Delta$ as a subset\nof $\\mathbb{R}^{n}$. Even though $m<n$, the procedure to obtain\na quadratic estimate of $f$ in $\\Delta$ is a straightforward extension\nof Algorithm \\ref{alg:quad-approx}. \n\\begin{lem}\n\\label{lem:quad-est}(Quadratic estimate on simplex) Let $f:\\mathbb{R}^{n}\\to\\mathbb{R}$\nbe $\\mathcal{C}^{2}$ and $\\bar{x}\\in\\mathbb{R}^{n}$. Let $p_{1},\\dots,p_{m+1}$\nbe points close to $\\bar{x}$. Suppose the matrix $P\\in\\mathbb{R}^{n\\times m}$,\nwhose $i$th column is $p_{i+1}-p_{1}$, has full column rank. Let\n$f_{e}:\\mathbb{R}^{n}\\cap\\Delta\\to\\mathbb{R}$ be defined as the quadratic\nfunction obtained using $f(p_{i})$ and $\\nabla f(p_{i})$ for $i=1,\\dots,m+1$\nwith Algorithm \\ref{alg:quad-approx}. For any $\\epsilon>0$, there\nis some $\\delta>0$ such that if $p_{1},\\dots,p_{m+1}\\in\\mathbb{B}(\\bar{x},\\delta)$,\nthen \\[\n|f_{e}(x)-f(x)|<\\frac{1}{2}\\mbox{\\rm diam}(\\Delta)^{2}\\epsilon(1+\\kappa\\|P\\|\\|P^{\\dagger}\\|)\\mbox{ for all }x\\in\\Delta,\\]\nwhere $\\|\\cdot\\|$ stands for the matrix 2-norm, $P^{\\dagger}$ is\nthe pseudoinverse of $P$, and $\\kappa$ is some constant dependent\nonly on $n$ and $m$.\\end{lem}\n\\begin{proof}\nThe first step of this proof is to show that step 1 of Algorithm \\ref{alg:quad-approx}\ngives a matrix in $\\mathbb{R}^{n\\times n}$ which is a good approximation\nof how $A=\\nabla^{2}f(\\bar{x})$ acts on the lineality space of the\naffine hull of $\\Delta$. Since $f$ is $\\mathcal{C}^{2}$, for any\n$\\epsilon>0$, there exists $\\delta>0$ such that $|\\nabla f(x)-\\nabla f(x^{\\prime})-A(x-x^{\\prime})|<\\epsilon|x-x^{\\prime}|$\nfor all $x,x^{\\prime}\\in\\mathbb{B}(\\bar{x},\\delta)$. Thus, there\nis some $\\kappa>0$ depending only on $m$ and $n$ such that if $p_{1},\\dots,p_{m+1}\\in\\mathbb{B}(\\bar{x},\\delta)$,\nthen \\begin{equation}\n\\|D-AP\\|<\\kappa\\epsilon\\|P\\|.\\label{eq:|D-AP|}\\end{equation}\n\n\nLet $P=QR$, where $Q\\in\\mathbb{R}^{n\\times m}$ has orthonormal columns\nand $R\\in\\mathbb{R}^{m\\times m}$, be a QR decomposition of $P$.\nFor any $v\\in\\mathbb{R}^{n}$ in the range of $P$, or equivalently,\n$v=Qv^{\\prime}$ for some $v^{\\prime}\\in\\mathbb{R}^{m}$, we want\nto show that $\\|Av-DR^{-1}Q^{T}v\\|$ is small. We note that $|v|=|v^{\\prime}|$,\nand we have the following calculation. \\begin{eqnarray*}\n\\|Av-DR^{-1}Q^{T}v\\| & = & \\|AQv^{\\prime}-DR^{-1}Q^{T}Qv^{\\prime}\\|\\\\\n & = & \\|AQv^{\\prime}-DR^{-1}v^{\\prime}\\|\\\\\n & \\leq & \\|AQ-DR^{-1}\\||v^{\\prime}|\\\\\n & \\leq & \\|AQR-D\\|\\|R^{-1}\\||v|\\\\\n & = & \\|D-AP\\|\\|R^{-1}\\||v|\\\\\n & \\leq & \\kappa\\epsilon\\|P\\|\\|R^{-1}\\||v|.\\end{eqnarray*}\nNext, for $x,x^{\\prime}\\in\\mathbb{B}(\\bar{x},\\delta)$, let $d=\\mbox{\\rm unit}(x^{\\prime}-x)$.\nThen\\begin{eqnarray*}\nf(x^{\\prime})-f(x) & = & \\int_{0}^{|x^{\\prime}-x|}\\nabla f(x+sd)^{T}d\\,\\mathbf{d}s\\\\\n & = & \\int_{0}^{|x^{\\prime}-x|}\\int_{0}^{s}d^{T}\\nabla^{2}f(x+td)d\\,\\mathbf{d}t+\\nabla f(x)^{T}d\\,\\mathbf{d}s.\\end{eqnarray*}\nSince $f$ is $\\mathcal{C}^{2}$, we may reduce $\\delta$ if necessary\nso that $\\|A-\\nabla^{2}f(x+td)\\|<\\epsilon$ for all $0\\leq t\\leq|x^{\\prime}-x|$.\nThis tells us that \\begin{eqnarray*}\n|d^{T}(DR^{-1}Q^{T})d-d^{T}\\nabla^{2}f(x)d| & \\leq & |d|\\|DR^{-1}Q^{T}d-\\nabla^{2}f(x)d\\|\\\\\n & \\leq & \\|DR^{-1}Q^{T}d-Ad\\|+\\|Ad-\\nabla^{2}f(x)d\\|\\\\\n & \\leq & \\kappa\\epsilon\\|P\\|\\|R^{-1}\\||d|+\\|A-\\nabla^{2}f(x)\\|\\\\\n & \\leq & \\epsilon(1+\\kappa\\|P\\|\\|R^{-1}\\|).\\end{eqnarray*}\nWe have \\begin{eqnarray*}\n &  & \\int_{0}^{|x^{\\prime}-x|}\\int_{0}^{s}d^{T}(DR^{-1}Q^{T})d\\,\\mathbf{d}t+\\nabla f(x)^{T}d\\,\\mathbf{d}s\\\\\n & = & \\nabla f(x)^{T}(x^{\\prime}-x)+\\int_{0}^{|x^{\\prime}-x|}d^{T}(DR^{-1}Q^{T})ds\\,\\mathbf{d}s\\\\\n & = & \\nabla f(x)^{T}(x^{\\prime}-x)+\\frac{|x^{\\prime}-x|^{2}}{2}d^{T}(DR^{-1}Q^{T})d\\\\\n & = & \\nabla f(x)^{T}(x^{\\prime}-x)+\\frac{1}{2}(x^{\\prime}-x)^{T}(DR^{-1}Q^{T})(x^{\\prime}-x).\\end{eqnarray*}\nContinuing with the arithmetic earlier, we obtain\\begin{eqnarray*}\n &  & \\left|f(x^{\\prime})-f(x)-\\left(\\nabla f(x)^{T}(x^{\\prime}-x)+\\frac{1}{2}(x^{\\prime}-x)^{T}(DR^{-1}Q^{T})(x^{\\prime}-x)\\right)\\right|\\\\\n & \\leq & \\int_{0}^{|x^{\\prime}-x|}\\int_{0}^{s}\\epsilon(1+\\kappa\\|P\\|\\|R^{-1}\\|)s\\,\\mathbf{d}t\\mathbf{d}s\\\\\n & = & \\frac{1}{2}|x^{\\prime}-x|^{2}\\epsilon(1+\\kappa\\|P\\|\\|R^{-1}\\|).\\end{eqnarray*}\nLet $x=p_{1}$ and $x^{\\prime}$ be any point in $\\Delta$. Define\n$f_{e}(x^{\\prime})$ by \\[\nf_{e}(x^{\\prime})=f(x)+\\nabla f(x)^{T}(x^{\\prime}-x)+\\frac{1}{2}(x^{\\prime}-x)^{T}(DR^{-1}Q^{T})(x^{\\prime}-x),\\]\nwhich is the quadratic function obtained using Algorithm \\ref{alg:quad-approx}.\nWe have \\begin{eqnarray*}\n|f_{e}(x^{\\prime})-f(x^{\\prime})| & \\leq & \\frac{1}{2}|x^{\\prime}-x|^{2}\\epsilon(1+\\kappa\\|P\\|\\|R^{-1}\\|)\\\\\n & \\leq & \\frac{1}{2}\\mbox{\\rm diam}(\\Delta)^{2}\\epsilon(1+\\kappa\\|P\\|\\|R^{-1}\\|),\\end{eqnarray*}\nwhich gives what we need.\n\\end{proof}\nIn the statement of Lemma \\ref{lem:quad-est}, we chose the domain\nof $f$ to be $\\mathbb{R}^{n}$ so that the inequality \\eqref{eq:|D-AP|}\nfollows from the equivalence of finite dimensional norms. Next, the\naccuracy of the computed values of $\\nabla f(p_{i+1})-\\nabla f(p_{1})$\nmight be poor, which makes the quadratic approximation strategy ineffective\nonce we are too close to the critical point $\\bar{x}$. We remark\non how we can overcome this problem by exploiting concavity.\n\\begin{rem}\n(Exploiting concavity) The lineality space of the affine hull of $\\Delta$\nmay span the eigenspaces of the $m$ negative eigenvalues of $\\nabla^{2}f(\\bar{x})$\nonce we are close to the critical point $\\bar{x}$. This can be checked\nby calculating the Hessian as was done earlier. If this is the case,\n$f$ would be concave in $\\Delta$ when $p_{1},\\dots,p_{m+1}$ are\nsufficiently close to $\\bar{x}$. The estimate $f(x)\\leq f(p_{i})+\\nabla f(p_{i})^{T}(x-p_{i})$\nwould hold for all $x\\in\\Delta$ and $1\\leq i\\leq m+1$, which can\ngive a sufficiently good estimate of $\\max_{x\\in\\partial\\Delta}f(x)$\nthrough linear programming.\n\\end{rem}\n\n\\section{Fast local convergence\\label{sec:Fast-local-convergence}}\n\nIn this section, we discuss how we can find good lower bounds that\nallow us to achieve better convergence if $f$ is $\\mathcal{C}^{2}$\nand $X=\\mathbb{R}^{n}$. Our method extends the local superlinearly\nconvergent method in \\cite{LP08} for finding smooth critical points\nof mountain pass type when $X=\\mathbb{R}^{n}$.\n\nLet us recall the multidimensional mountain pass theorem due to Rabinowitz\n\\cite{R77}.\n\\begin{thm}\n(Multidimensional mountain pass theorem) \\cite{R77}\\label{thm:Rabinowitz-77}\nLet $X=Y\\oplus Z$ be a Banach space with $Z$ closed in $X$ and\n$\\dim(Y)<\\infty$. For $\\rho>0$ define\\[\n\\mathcal{M}:=\\{u\\in Y\\mid\\|u\\|\\leq\\rho\\},\\quad\\mathcal{M}_{0}:=\\{u\\in Y\\mid\\|u\\|=\\rho\\}.\\]\nLet $f:X\\to\\mathbb{R}$ be $\\mathcal{C}^{1}$, and \\[\nb:=\\inf_{u\\in Z}f(u)>a:=\\max_{u\\in\\mathcal{M}_{0}}f(u).\\]\nIf $f$ satisfies the Palais Smale condition and \\[\nc:=\\inf_{\\gamma\\in\\Gamma}\\max_{u\\in\\mathcal{M}}f\\big(\\gamma(u)\\big)\\mbox{ where }\\Gamma:=\\left\\{ \\gamma:\\mathcal{M}\\to X\\mbox{ is continuous}\\mid\\gamma|_{\\mathcal{M}_{0}}=id\\right\\} ,\\]\nthen $c$ is a critical value of $f$. \n\\end{thm}\n For the case when $X=\\mathbb{R}^{3}$, we have an illustration\nin Figure \\ref{fig:multi-MPT-fig} of the case $f:\\mathbb{R}^{3}\\rightarrow\\mathbb{R}$\ndefined by $f(x)=x_{3}^{2}-x_{1}^{2}-x_{2}^{2}$. The critical point\n$\\mathbf{0}$ has critical value $0$. Choose $Y$ to be $\\mathbb{R}^{2}\\times\\{0\\}$\nand $Z$ to be $\\{\\mathbf{0}\\}\\times\\mathbb{R}$. The union of the\ntwo blue cones is the level set $\\mbox{lev}_{=0}f:=f^{-1}(0)$, while\nthe bold red ring denotes $\\mathcal{M}_{0}$ and the red disc denotes\na possible image of $\\mathcal{M}$ under $\\gamma$. \n\n\\begin{figure}\n\n\n\\includegraphics[scale=0.5]{multimtn}\\caption{\\label{fig:multi-MPT-fig}Illustration of the multidimensional mountain\npass theorem}\n\n\\end{figure}\n\n\nIt seems intuitively clear that $\\gamma(\\mathcal{M})$ has to intersect\nthe vertical axis. This is indeed the case, since $\\mathcal{M}_{0}$\nand $Z$ link. (See for example \\cite[Example II.8.2]{Str08}.)\n\nWith this observation, we easily see that $\\max_{u\\in\\mathcal{M}}f(\\gamma(u))\\geq\\inf_{z\\in Z}f(z)$.\nThus the critical value $c=\\inf_{\\gamma\\in\\Gamma}\\max_{u\\in\\mathcal{M}}f(\\gamma(u))$\nfrom Theorem \\ref{thm:Rabinowitz-77} is bounded from below by $\\inf_{z\\in Z}f(z)$.\nThis gives a lower bound for the critical value. In the mountain pass\ncase when $m=1$, the set $\\mathcal{M}_{0}$ consists of two points,\nand the space $Z$ separates the two points in $\\mathcal{M}_{0}$\nso that any path connecting the two points in $\\mathcal{M}_{0}$ must\nintersect $Z$. \n\nA first try for a fast locally convergent algorithm is as follows:\n\\begin{algorithm}\n\\label{alg:first-local-algorithm}A first try for a fast locally convergent\nalgorithm to find saddle points of Morse index $m$ for $f:X\\to\\mathbb{R}$.\\end{algorithm}\n\\begin{enumerate}\n\\item Set the iteration count $i$ to $0$, and let $l_{i}$ be a lower\nbound of the critical value. \n\\item Find $x_{i}$ and $y_{i}$, where $(S_{i},x_{i},y_{i})$ is an optimizing\ntriple of\\begin{equation}\n\\min_{S\\in\\mathcal{S}}\\max_{x,y\\in S\\cap(\\scriptsize\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i}}|x-y|,\\label{eq:key-exp2}\\end{equation}\nwhere $U_{i}$ is an open set. Here $\\mathcal{S}$ is the set of $m$-dimensional\naffine subspaces of $\\mathbb{R}^{n}$ intersecting $U_{i}$. (The\ndifference between this formula and \\eqref{eq:keyexp1} is that we\ntake level sets of level $l_{i}$ instead of $\\frac{1}{2}(l_{i}+u_{i})$.)\n\n\\item For an optimal solution $(S_{i},x_{i},y_{i})$, let $l_{i+1}$ be\nthe lower bound of $f$ on the $(n-m)$-dimensional affine space passing\nthrough $z_{i}:=\\frac{1}{2}(x_{i}+y_{i})$ whose lineality space is\northogonal to the lineality space of $S_{i}$ .\n\\item Increase $i$ and go back to step 2.\n\\end{enumerate}\nWhile Algorithm \\ref{alg:first-local-algorithm} as stated works fine\nfor the case $m=1$ to find critical points of mountain pass type,\nthe $l_{i}$'s calculated in this manner need not increase monotonically\nto the critical value when $m>1$. We first present a lemma on the\nmin-max problem \\eqref{eq:key-exp2} for the case of a quadratic.\n\\begin{lem}\n(Analysis on exact quadratic) \\label{lem:quadratic-min-max} Consider\n$f:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$ defined by $f(x)=\\sum_{j=1}^{n}a_{j}x_{j}^{2}$,\nwhere $a_{i}$ are in decreasing order, with $a_{j}>0$ for $1\\leq j\\leq n-m$\nand $a_{j}<0$ for $n-m+1\\leq j\\leq n$. The function $f$ has one\ncritical point $\\mathbf{0}$, and $f(\\mathbf{0})=0$. Given $l<0$,\nan optimizing triple $(\\bar{S},\\bar{x},\\bar{y})$ of the problem \\begin{equation}\n\\min_{S\\in\\mathcal{S}}\\max_{x,y\\in S\\cap\\scriptsize\\mbox{\\rm lev}_{\\geq l}f}|x-y|,\\label{eq:quadratic-min-max}\\end{equation}\nwhere $\\mathcal{S}$ is the set of affine spaces of dimension $m$,\nsatisfies \\[\n\\bar{x}=\\left(0,0,\\dots,0,\\pm\\sqrt{\\frac{l}{a_{n-m+1}}},0,\\dots,0\\right),\\]\nwhere the nonzero term is in the $(n-m+1)$th position, and $\\bar{y}=-\\bar{x}$. \\end{lem}\n\\begin{proof}\nLet $S_{\\bar{z},V}:=\\{\\bar{z}+Vw\\mid w\\in\\mathbb{R}^{m}\\}$, where\n$V\\in\\mathbb{R}^{n\\times m}$ is a matrix with orthonormal columns.\nLet the matrix $A\\in\\mathbb{R}^{n\\times n}$ be the diagonal matrix\nwith entries $a_{j}$ in the $(j,j)$th position. The ellipse $S_{\\bar{z},V}\\cap\\mbox{\\rm lev}_{\\geq l}f$\ncan be written as a union of elements of the form $\\bar{z}+Vw$, where\n$w$ satisfies \\begin{eqnarray*}\n(\\bar{z}+Vw)^{T}A(\\bar{z}+Vw) & \\geq & l\\\\\n\\Leftrightarrow w^{T}V^{T}AVw+2\\bar{z}^{T}AVw+\\bar{z}^{T}A\\bar{z} & \\geq & l.\\end{eqnarray*}\nIf the matrix $V^{T}AV$ has a nonnegative eigenvalue, then $S_{\\bar{z},V}\\cap\\mbox{\\rm lev}_{\\geq l}f$\nis unbounded. Otherwise, the set \\[\n\\{\\bar{z}+Vw\\mid w^{T}V^{T}AVw+2\\bar{z}^{T}AVw+\\bar{z}^{T}A\\bar{z}\\geq l\\}\\]\nis bounded. Therefore the inner maximization problem of \\eqref{eq:quadratic-min-max}\ncorresponding to $S=S_{\\bar{z},V}$ has a (not necessarily unique)\npair of minimizers. We continue completing the square with respect\nto $w$ and let the symmetric matrix $C$ be the square root $C=[-V^{T}AV]^{\\frac{1}{2}}$.\\begin{eqnarray*}\n-w^{T}C^{2}w+2\\bar{z}^{T}AVw+\\bar{z}^{T}A\\bar{z} & \\geq & l\\\\\n\\Leftrightarrow-(Cw-C^{-1}V^{T}A\\bar{z})^{T}(Cw-C^{-1}V^{T}A\\bar{z})+\\bar{z}^{T}A\\bar{z}+\\bar{z}^{T}AVC^{-2}V^{T}A^{T}\\bar{z} & \\geq & l.\\end{eqnarray*}\nThe maximum length between two points of an ellipse is twice the distance\nbetween the center and the furthest point on the ellipse. (This fact\nis easily proved by reducing to, and examining, the two dimensional\ncase.) The distance between the center and the furthest point on\nthe ellipse $S_{\\bar{z},V}\\cap\\mbox{\\rm lev}_{\\geq l}f$ can be calculated to\nbe\\[\n\\sqrt{\\frac{1}{\\alpha}(\\bar{z}^{T}A\\bar{z}+\\bar{z}^{T}AVC^{-2}V^{T}A^{T}\\bar{z}-l)},\\]\nwhere $\\alpha$ is the square of the smallest eigenvalue in $C$,\nor equivalently the negative of the largest eigenvalue of $V^{T}AV$.\nThe term $(\\bar{z}^{T}A\\bar{z}+\\bar{z}^{T}AVC^{-2}V^{T}A^{T}\\bar{z})$\nis $\\max\\{f(x)\\mid x\\in S_{\\bar{z},V}\\}$, which we refer to as $\\max_{S_{\\bar{z},V}}f$.\nWe now proceed to minimize $\\max_{S_{\\bar{z},V}}f$ and maximize $\\alpha$\nseparately.\n\n\\textbf{Claim 1: $\\max_{S_{\\bar{z},V}}f\\geq0$.}\n\nWe first prove that the subspace \\[\nZ:=\\{z\\mid z_{n}=z_{n-1}=\\cdots=z_{n-m+1}=0\\}\\]\nmust intersect $S_{\\bar{z},V}$. Recall that $V^{T}AV$ is negative\ndefinite. Therefore for any $w\\neq\\mathbf{0}$, $w^{T}V^{T}AVw<0$.\nSince the first $n-m$ eigenvalues of $A$ are positive, $Vw$ cannot\nbe all zeros in its last $m$ components. This shows that the $m\\times m$\nmatrix $V((n-m+1):n,1:m)$ is invertible. We can find some $\\bar{w}$\nsuch that the last $m$ components of $\\bar{z}+V\\bar{w}$ are zeros.\nThis shows that $S_{\\bar{z},V}\\cap Z\\neq\\emptyset$, so $\\max_{S_{\\bar{z},V}}f\\geq\\min\\{f(x)\\mid x\\in Z\\}=0$.\n\n\\textbf{Claim 2: $\\alpha\\leq-a_{n-m+1}$.}\n\nTo find the maximum value of $\\alpha$, we recall that it is the negative\nof the largest eigenvalue of $V^{T}AV$. Since $V\\in\\mathbb{R}^{n\\times m}$,\nthe Courant-Fischer Theorem, gives $\\alpha\\leq-a_{n-m+1}$.\n\nChoose the affine space $\\bar{S}:=\\{\\mathbf{0}\\}\\times\\mathbb{R}^{m}$.\nThis minimizes $\\max_{S}f$ and maximizes $\\alpha$ as well, giving\nthe optimal solution in the statement of the lemma.\n\\end{proof}\nIt should be noted however that the minimizing subspace need not be\nunique, even if the values of $a_{j}$ are distinct. The example below\nhighlights how Algorithm \\ref{alg:first-local-algorithm} can fail.\n\\begin{example}\n(Failure of Algorithm \\ref{alg:first-local-algorithm}) \\label{exa:failure-first-try}Suppose\n$f(x)=x_{1}^{2}-x_{2}^{2}-3x_{3}^{2}$. The subspace $S=\\{x\\mid x_{1}=x_{3}\\}$\nintersects the level set $\\mbox{\\rm lev}_{\\geq-1}f$ in the disc \\[\n\\left\\{ \\lambda\\left(\\frac{1}{\\sqrt{2}}\\sin\\theta,\\cos\\theta,\\frac{1}{\\sqrt{2}}\\sin\\theta\\right)\\mid0\\leq\\theta\\leq2\\pi,0\\leq\\lambda\\leq1\\right\\} .\\]\nThe largest distance between two points on the disc is $2$, and the\nsubspace $S$ can be verified to give the optimal value to the min-max\nproblem \\eqref{eq:key-exp2} by Lemma \\ref{lem:quadratic-min-max}.\n\nOn the ray $S^{\\perp}=\\{\\lambda(1,0,-1)\\mid\\lambda\\in\\mathbb{R}\\}$,\nthe function $f$ is concave, hence there is no minimum. This example\nillustrates that Algorithm \\ref{alg:first-local-algorithm} can fail\nin general. See Figure \\ref{fig:bad-3d}.\n\\end{example}\n\\begin{figure}\n\\includegraphics[scale=0.4]{line_plane}\n\n\\caption{\\label{fig:bad-3d}An example where Algorithm \\ref{alg:first-local-algorithm}\nfails.}\n\n\n\n\\end{figure}\n\n\nExample \\ref{exa:failure-first-try} shows that even if there are\nonly 2 negative eigenvalues, it might be possible to find a two-dimensional\nsubspace $S$ on which the Hessian is negative definite on both $S$\nand $S^{\\perp}$. Therefore, we amend Algorithm \\ref{alg:first-local-algorithm}\nby determining the eigenspace corresponding to the $m$ smallest eigenvalues. \n\\begin{algorithm}\n\\label{alg:fast-local-method} Fast local method to find saddle points\nof Morse index $m$.\\end{algorithm}\n\\begin{enumerate}\n\\item Set the iteration count $i$ to $0$, and let $l_{i}$ be a lower\nbound of the critical value. \n\\item Find $x_{i}$ and $y_{i}$, where $(S_{i}^{\\prime},x_{i},y_{i})$\nis an optimizing triple of\\begin{equation}\n\\min_{S\\in\\mathcal{S}}\\max_{x,y\\in S\\cap(\\scriptsize\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap U_{i}}|x-y|,\\label{eq:key-exp}\\end{equation}\nwhere $U_{i}$ is an open set. Here $\\mathcal{S}$ is the set of $m$-dimensional\naffine subspaces of $\\mathbb{R}^{n}$ intersecting $U_{i}$. We emphasize\nthat the space where minimality is attained in the outer minimization\nproblem is $S_{i}^{\\prime}$. After solving the above problem, find\nthe subspace $S_{i}$ that approximates the eigenspace corresponding\nto the $m$ smallest eigenvalues using Algorithm \\ref{alg:find-S-prime}\nbelow.\n\\item For an optimizing triple $(S_{i},x_{i},y_{i})$ found in step 2, let\n$l_{i+1}$ be the lower bound of $f$ on the $(n-m)$-dimensional\naffine space passing through $z_{i}:=\\frac{1}{2}(x_{i}+y_{i})$ whose\nlineality space is orthogonal to the lineality space of $S_{i}$ .\n\\item Increase $i$ and go back to step 2 till convergence.\n\\end{enumerate}\nA local algorithm is needed in Algorithm \\ref{alg:fast-local-method}\nto find the subspace $S_{i}$ in step 2.\n\\begin{algorithm}\n\\label{alg:find-S-prime}Finding the subspace $S_{i}$ in step 2 of\nAlgorithm \\ref{alg:fast-local-method}:\\end{algorithm}\n\\begin{enumerate}\n\\item Let $X_{1}=\\mbox{span}\\{x_{i}-y_{i}\\}$, where $x_{i}$, $y_{i}$\nare found in step 2 of Algorithm \\ref{alg:fast-local-method}, and\nlet $j$ be $1$.\n\\item Find the closest point from $z_{i}:=\\frac{1}{2}(x_{i}+y_{i})$ to\n$\\mbox{\\rm lev}_{\\leq l_{i}}f\\cap(z_{i}+X_{j}^{\\perp})$, which we call $\\bar{p}_{j+1}$.\n\\item Let $X_{j+1}=\\mbox{span}\\{X_{j},\\bar{p}_{j+1}-z_{i}\\}$ and increase\n$j$ by $1$. If $j=m$, let $S_{i}$ be $z_{i}+X_{m}$ and the algorithm\nends. Otherwise, go back to step 2.\n\\end{enumerate}\nStep 2 of Algorithm \\ref{alg:find-S-prime} finds the negative eigenvalues\nand eigenvectors, starting from the eigenvalues furthest from zero.\nOnce all the eigenvectors are found, then $S_{i}$ is the span of\nthese eigenvectors. \n\nIn some situations, the lineality space of $S_{i}$ are known in advance,\nor do not differ too much from the previous iteration. In this case,\nwe can get around using Algorithm \\ref{alg:find-S-prime} and use\nthe estimate instead.\n\nWe are now ready to prove the convergence of Algorithm \\ref{alg:fast-local-method}. \n\n\n\\section{Proof of superlinear convergence of local algorithm\\label{sec:Proof-of-superlinear}}\n\n We prove our result on the convergence of Algorithm \\ref{alg:fast-local-method}\nin steps. The first step is to look closely at a model problem. \n\\begin{assumption}\n\\label{ass:condns-on-model}Given $\\delta>0$, suppose $h:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nis $\\mathcal{C}^{2}$, \\begin{eqnarray*}\n|\\nabla h(x)-Ax| & \\leq & \\delta|x|,\\\\\n\\mbox{{\\rm and }}|h(x)-\\frac{1}{2}x^{T}Ax| & \\leq & \\frac{1}{2}\\delta|x|^{2}\\mbox{ {\\rm for all }}x\\in\\mathbb{B},\\end{eqnarray*}\nwhere $A\\in\\mathbb{R}^{n\\times n}$ is an invertible diagonal matrix\nwith diagonal entries ordered decreasingly, of which $a_{i}=A_{ii}$\nand \\[\na_{1}>a_{2}>\\cdots>a_{n-m}>0>a_{n-m+1}>\\cdots>a_{n}.\\]\nDefine $h_{\\min}:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$ and $h_{\\max}:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nby:\\[\nh_{\\min}(x)=\\frac{1}{2}x^{T}(A-\\delta I)x\\quad\\mbox{and}\\quad h_{\\max}(x)=\\frac{1}{2}x^{T}(A+\\delta I)x.\\]\n\n\\end{assumption}\nIt is clear that $\\nabla h(\\mathbf{0})=\\mathbf{0}$, $h(\\mathbf{0})=0$,\n$\\nabla^{2}h(\\mathbf{0})=A$, and the Morse index is $m$. Here is\na simple observation that bounds the level sets of $h$:\n\\begin{prop}\n(Level set property) The level sets of $h$ satisfy\\begin{eqnarray*}\n &  & \\mathbb{B}\\cap\\mbox{\\rm lev}_{\\geq l}h_{\\min}\\subset\\mathbb{B}\\cap\\mbox{\\rm lev}_{\\geq l}h\\subset\\mathbb{B}\\cap\\mbox{\\rm lev}_{\\geq l}h_{\\max},\\\\\n & \\mbox{{\\rm and}} & \\mathbb{B}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\max}\\subset\\mathbb{B}\\cap\\mbox{\\rm lev}_{\\leq l}h\\subset\\mathbb{B}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\min}.\\end{eqnarray*}\n\\end{prop}\n\\begin{proof}\nThis follows easily from $|h(x)-\\frac{1}{2}x^{T}Ax|\\leq\\frac{1}{2}\\delta|x|^{2}$\nfor all $x\\in\\mathbb{B}$.\n\\end{proof}\nFor convenience, we highlight the standard problem below:\n\\begin{problem}\n\\label{pro:standard-prob}Suppose $g:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nis $\\mathcal{C}^{2}$, with critical point $\\mathbf{0}$ of Morse\nindex $m$, $g(\\mathbf{0})=0$ and the Hessian $\\nabla^{2}g(\\mathbf{0})$\nhas distinct eigenvalues that are all nonzero. Consider the problem\n\\[\n\\min_{S\\in\\mathcal{S}}\\max_{x,y\\in S\\cap(\\scriptsize\\mbox{\\rm lev}_{\\geq l}g)\\cap\\mathbb{B}}|x-y|,\\]\nwhere $\\mathcal{S}$ is the set of $m$ dimensional affine subspaces.\n\\end{problem}\nNote that in Problem \\ref{pro:standard-prob}, we have limited the\nregion where $x$ and $y$ lie in by $\\mathbb{B}$. Here is a result\non the optimizing pair $(\\bar{x},\\bar{y})$ of the inner maximization\nproblem in Problem \\ref{pro:standard-prob}.\n\\begin{lem}\n(Convergence to eigenvector and saddle point)\\label{lem:min-maximizer-ppties}For\nall $\\delta>0$ sufficiently small, suppose that $h:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nis such that Assumption \\ref{ass:condns-on-model} holds. Assume that\nfor the optimizing triple $(\\bar{S},\\bar{x},\\bar{y})$ of Problem\n\\ref{pro:standard-prob} for $g=h$, $(\\bar{x},\\bar{y})$ is the unique\npair of points in $\\bar{S}\\cap(\\mbox{\\rm lev}_{\\geq l}h)\\cap\\mathbb{B}$ such\nthat $|\\bar{x}-\\bar{y}|=\\mbox{\\rm diam}(\\bar{S}\\cap(\\mbox{\\rm lev}_{\\geq l}h)\\cap\\mathbb{B})$.\nThen there exists $\\epsilon>0$ such that if $l<0$ satisfies $-\\epsilon<l<0$,\nthen $(\\bar{x},\\bar{y})$ are such that $\\frac{\\bar{y}-\\bar{x}}{|\\bar{y}-\\bar{x}|}$\nconverges to the $(n-m+1)$th eigenvector as $\\delta\\to0$, and $|\\frac{1}{2}(\\bar{x}+\\bar{y})|^{2}\/|l|\\to0$\nas $\\delta\\to0$.\\end{lem}\n\\begin{proof}\nSince $\\mbox{\\rm lev}_{\\leq l}h_{\\max}\\cap\\mathbb{B}\\subset\\mbox{\\rm lev}_{\\leq l}h\\cap\\mathbb{B}$,\nwe look at the the optimal solution of the min-max problem for $h_{\\max}$\nfirst. The objective of Problem \\ref{pro:standard-prob} for $g=h_{\\max}$\nis $2\\sqrt{\\frac{l}{a_{n-m+1}+\\delta}}$ by Lemma \\ref{lem:quadratic-min-max}.\nThis gives an upper bound for the min-max problem for $h$. Similarly,\nby considering $h_{\\min}$ instead, we deduce that Problem \\ref{pro:standard-prob}\nhas optimal solution bounded from below by $2\\sqrt{\\frac{l}{a_{n-m+1}-\\delta}}$.\n\nRecall the optimality condition in Proposition \\ref{pro:opposite-directions}.\nWe now proceed to find the first pair of points with\\textbf{ }opposite\npointing gradients. Let $\\bar{x}$ and $\\bar{y}$ be optimal points\nat level $l_{i}$. Now, \\begin{eqnarray*}\n\\bar{y}-\\bar{x} & = & \\lambda_{1}\\nabla h(\\bar{x}),\\\\\n\\mbox{ and }\\bar{x}-\\bar{y} & = & \\lambda_{2}\\nabla h(\\bar{y}),\\end{eqnarray*}\n for some $\\lambda_{1},\\lambda_{2}>0$. Then \\begin{eqnarray*}\n\\lambda_{1}\\nabla h(\\bar{x})+\\lambda_{2}\\nabla h(\\bar{y}) & = & \\mathbf{0}\\\\\n|\\lambda_{1}A\\bar{x}+\\lambda_{2}A\\bar{y}| & \\leq & \\lambda_{1}|A\\bar{x}-\\nabla h(\\bar{x})|+\\lambda_{2}|A\\bar{y}-\\nabla h(\\bar{y})|\\\\\n & \\leq & \\delta(\\lambda_{1}|\\bar{x}|+\\lambda_{2}|\\bar{y}|)\\\\\n|A(\\lambda_{1}\\bar{x}+\\lambda_{2}\\bar{y})| & \\leq & \\delta(\\lambda_{1}|\\bar{x}|+\\lambda_{2}|\\bar{y}|)\\\\\n\\Rightarrow|\\lambda_{1}\\bar{x}+\\lambda_{2}\\bar{y}| & \\leq & |A^{-1}||A(\\lambda_{1}\\bar{x}+\\lambda_{2}\\bar{y})|\\\\\n & \\leq & |A^{-1}|\\delta(\\lambda_{1}|\\bar{x}|+\\lambda_{2}|\\bar{y}|).\\end{eqnarray*}\nThis means that there are points $x^{\\prime}$ and $y^{\\prime}$ such\nthat $\\lambda_{1}x^{\\prime}+\\lambda_{2}y^{\\prime}=\\mathbf{0}$, $|\\bar{x}-x^{\\prime}|\\leq|A^{-1}|\\delta|\\bar{x}|$\nand $|\\bar{y}-y^{\\prime}|\\leq|A^{-1}|\\delta|\\bar{y}|$. With this,\nwe now concentrate on pairs of points that are negative multiples\nof each other.\n\nNow,\\begin{eqnarray*}\n\\lambda_{1}\\nabla h(\\bar{x}) & = & \\bar{y}-\\bar{x}\\\\\n\\Rightarrow\\nabla h(\\bar{x}) & = & \\frac{1}{\\lambda_{1}}(\\bar{y}-\\bar{x})\\\\\n\\Rightarrow\\left|A\\bar{x}-\\frac{1}{\\lambda_{1}}(\\bar{y}-\\bar{x})\\right| & \\leq & \\delta|\\bar{x}|.\\end{eqnarray*}\nSimilarly, this gives us\\begin{eqnarray*}\n\\left|A(-\\bar{y})-\\frac{1}{\\lambda_{2}}(\\bar{y}-\\bar{x})\\right| & \\leq & \\delta|\\bar{y}|.\\end{eqnarray*}\nTherefore, \\[\n\\left|A(\\bar{y}-\\bar{x})+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)(\\bar{y}-\\bar{x})\\right|\\leq\\delta(|\\bar{x}|+|\\bar{y}|).\\]\nThis gives:\\begin{eqnarray}\n &  & \\left|A(y^{\\prime}-x^{\\prime})+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)(y^{\\prime}-x^{\\prime})\\right|\\nonumber \\\\\n & \\leq & |A(y^{\\prime}-x^{\\prime})-A(\\bar{y}-\\bar{x})|+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)|(y^{\\prime}-x^{\\prime})-(\\bar{y}-\\bar{x})|\\nonumber \\\\\n &  & \\qquad+\\left|A(\\bar{y}-\\bar{x})+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)(\\bar{y}-\\bar{x})\\right|\\nonumber \\\\\n & \\leq & |A||A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)+\\delta(|\\bar{x}|+|\\bar{y}|)\\nonumber \\\\\n & = & \\delta(|\\bar{x}|+|\\bar{y}|)\\underbrace{\\left(|A||A^{-1}|+\\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)|A^{-1}|+1\\right)}_{(1)}.\\label{eq:y_minus_x_eigenvalue}\\end{eqnarray}\nNext, we relate $|\\bar{y}-\\bar{x}|$ and $|\\bar{x}|+|\\bar{y}|$. We\nhave \\begin{eqnarray*}\n|\\bar{x}| & \\leq & |x^{\\prime}|+|A^{-1}|\\delta|\\bar{x}|\\\\\n\\Rightarrow(1-|A^{-1}|\\delta)|\\bar{x}| & \\leq & |x^{\\prime}|,\\end{eqnarray*}\nand similarly, $(1-|A^{-1}|\\delta)|\\bar{y}|\\leq|y^{\\prime}|$. It\nis clear that $|y^{\\prime}-x^{\\prime}|=|x^{\\prime}|+|y^{\\prime}|$,\nand we get\\begin{eqnarray*}\n|\\bar{x}|+|\\bar{y}| & \\leq & \\frac{1}{1-|A^{-1}|\\delta}(|x^{\\prime}|+|y^{\\prime}|)\\\\\n & = & \\frac{1}{1-|A^{-1}|\\delta}|y^{\\prime}-x^{\\prime}|\\\\\n & \\leq & \\frac{1}{1-|A^{-1}|\\delta}[|\\bar{y}-\\bar{x}|+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)]\\\\\n\\Rightarrow\\left(1-\\frac{|A^{-1}|\\delta}{1-|A^{-1}|\\delta}\\right)(|\\bar{x}|+|\\bar{y}|) & \\leq & \\frac{1}{1-|A^{-1}|\\delta}|\\bar{y}-\\bar{x}|\\\\\n\\Rightarrow(1-2|A^{-1}|\\delta)(|\\bar{x}|+|\\bar{y}|) & \\leq & |\\bar{y}-\\bar{x}|.\\end{eqnarray*}\nTo show that $(1)$ in \\eqref{eq:y_minus_x_eigenvalue} converges\nto $0$ as $\\delta\\searrow0$, we need to show that $(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}})$\nremains bounded as $\\delta\\searrow0$. Note that \\begin{eqnarray*}\n\\nabla h(\\bar{x})-\\nabla h(\\bar{y}) & = & \\left(\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right)(\\bar{y}-\\bar{x})\\\\\n\\Rightarrow\\left|\\frac{1}{\\lambda_{1}}+\\frac{1}{\\lambda_{2}}\\right| & = & \\frac{\\left|\\nabla h(\\bar{x})-\\nabla h(\\bar{y})\\right|}{|\\bar{y}-\\bar{x}|}\\\\\n & \\leq & \\frac{1}{|\\bar{y}-\\bar{x}|}[|A(\\bar{x}-\\bar{y})|+\\delta(|\\bar{x}|+|\\bar{y}|)]\\\\\n & \\leq & \\frac{1}{|\\bar{y}-\\bar{x}|}\\left(|A||\\bar{x}-\\bar{y}|+\\frac{1}{1-2|A^{-1}|\\delta}\\delta|\\bar{y}-\\bar{x}|\\right)\\\\\n & = & |A|+\\frac{\\delta}{1-2|A^{-1}|\\delta}.\\end{eqnarray*}\nSince the eigenvectors depend continuously on the entries of a matrix\nwhen the eigenvalues remain distinct, we see that $\\frac{1}{|\\bar{y}-\\bar{x}|}(y^{\\prime}-x^{\\prime})$\nconverges to an eigenvector of $A$ as $\\delta\\rightarrow0$ from\nformula \\eqref{eq:y_minus_x_eigenvalue}. \n\nNext, we show that $\\frac{1}{|\\bar{y}-\\bar{x}|}(\\bar{y}-\\bar{x})$\nconverges to an eigenvector corresponding to the eigenvalue $a_{n-m+1}$.\nRecall that $2\\sqrt{\\frac{l}{a_{n-m+1}-\\delta}}\\leq|\\bar{x}-\\bar{y}|$\nand $\\left|(\\bar{x}-\\bar{y})-(x^{\\prime}-y^{\\prime})\\right|\\leq2|A^{-1}|\\delta$.\n So $\\frac{1}{|\\bar{y}-\\bar{x}|}(\\bar{y}-\\bar{x})$ has the same\nlimit as $\\frac{1}{|\\bar{y}-\\bar{x}|}(y^{\\prime}-x^{\\prime})$. If\n$x^{\\prime}$ and $y^{\\prime}$ are such that $\\frac{1}{|\\bar{y}-\\bar{x}|}(y^{\\prime}-x^{\\prime})$\nconverges to a eigenvector corresponding to $a_{k}$, then Lemma \\ref{lem:bounds-on-primes}\nbelow gives us the following chain of inequalities: \\begin{eqnarray*}\n|\\bar{x}-\\bar{y}| & \\leq & |\\bar{x}|+|\\bar{y}|\\\\\n & \\leq & 2\\sqrt{[(1+\\theta)^{2}+(n-1)\\theta^{2}]\\frac{l}{(a_{k}+\\delta)(1-\\theta)^{2}+(n-1)(a_{1}+\\delta)\\theta^{2}}},\\end{eqnarray*}\nwhere $\\theta\\to0$ as $\\delta\\searrow0$. We note that $k\\geq n-m+1$\nbecause $a_{k}$ cannot be nonnegative. As $\\delta\\searrow0$, the\nlimit of the RHS of the above is $2\\sqrt{\\frac{l}{a_{k}}}$. This\ngives a contradiction if $k>n-m+1$, so $k=n-m+1$.\n\n\\textbf{To show $\\frac{1}{2}(\\bar{x}+\\bar{y})\\to\\mathbf{0}$ as $\\delta\\searrow0$:\n}We now work out an upper bound for $\\left|\\frac{1}{2}(\\bar{x}+\\bar{y})\\right|$\nusing Lemma \\ref{lem:bounds-on-primes}. We get\\begin{eqnarray*}\n\\left|\\frac{1}{2}(\\bar{x}+\\bar{y})\\right| & \\leq & \\left|\\frac{1}{2}(x^{\\prime}+y^{\\prime})\\right|+\\frac{1}{2}|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & = & \\frac{1}{2}\\big||x^{\\prime}|-|y^{\\prime}|\\big|+\\frac{1}{2}|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & \\leq & \\frac{1}{2}\\big||\\bar{x}|-|\\bar{y}|\\big|+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & \\leq & \\frac{1}{2}\\sqrt{[(1+\\theta)^{2}+(n-1)\\theta^{2}]\\frac{l}{(a_{n-m+1}+\\delta)(1-\\theta)^{2}+(n-1)(a_{1}+\\delta)\\theta^{2}}}\\\\\n &  & \\qquad-\\frac{1}{2}[1-\\theta]\\sqrt{\\frac{l}{(a_{n-m+1}-\\delta)(1+\\theta)^{2}+(n-1)(a_{n}-\\delta)\\theta^{2}}}+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|).\\end{eqnarray*}\nHere, $\\theta>0$ is such that $\\theta\\to0$ as $\\delta\\to0$. At\nthis point, we note that the final formula above can be written as\n$\\frac{1}{2}\\left(\\sqrt{\\frac{l}{c_{1}}}-\\sqrt{\\frac{l}{c_{2}}}\\right)+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)$,\nwhere $c_{1},c_{2}<0$, with $|c_{1}|<|c_{2}|$, and $c_{1},c_{2}\\rightarrow a_{n-m+1}$\nas $\\delta\\rightarrow0$. Therefore\\begin{eqnarray*}\n\\left|\\frac{1}{2}(\\bar{x}+\\bar{y})\\right| & \\leq & \\frac{1}{2}\\left(\\sqrt{\\frac{l}{c_{1}}}-\\sqrt{\\frac{l}{c_{2}}}\\right)+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & = & \\frac{1}{2}\\frac{\\frac{l}{c_{1}}-\\frac{l}{c_{2}}}{\\sqrt{\\frac{l}{c_{1}}}+\\sqrt{\\frac{l}{c_{2}}}}+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & \\leq & \\frac{1}{2c_{1}c_{2}}\\frac{l(c_{2}-c_{1})}{2\\sqrt{\\frac{l}{c_{2}}}}+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|)\\\\\n & \\leq & \\frac{c_{2}-c_{1}}{4c_{1}}\\sqrt{\\frac{l}{c_{2}}}+|A^{-1}|\\delta(|\\bar{x}|+|\\bar{y}|).\\end{eqnarray*}\nIt is clear that as $\\delta\\rightarrow0$, the above formula goes\nto zero, so $|\\frac{1}{2}(\\bar{x}+\\bar{y})|^{2}\/|l|\\rightarrow0$\nas $\\delta\\rightarrow0$ as needed.\n\\end{proof}\nIn Lemma \\ref{lem:bounds-on-primes} below, we say that $e_{i}$ is\nthe $i$th \\emph{elementary vector} if it is the $i$th column of\nthe identity matrix. It is also the eigenvector corresponding to the\neigenvalue $a_{k}$ of $A$. \n\\begin{lem}\n(Length estimates of vectors) \\label{lem:bounds-on-primes} Let $h:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nand $A\\in\\mathbb{R}^{n\\times n}$ satisfy Assumption \\ref{ass:condns-on-model}\nfor some $\\delta>0$. Suppose $h(\\bar{x})=h(\\bar{y})=l<0$. Suppose\n$\\theta>0$ is such that $|d_{\\bar{x}}-e_{k}|_{\\infty}<\\theta$ for\nsome $d_{\\bar{x}}$ pointing in the same direction as $\\bar{x}$,\nand that the same relation holds for $\\bar{y}$. \n\nThen $|\\bar{x}|$ and $|\\bar{y}|$ are bounded from below and above\nby\\[\n(1-\\theta)\\sqrt{\\frac{l}{(a_{k}-\\delta)(1+\\theta)^{2}+(n-1)(a_{n}-\\delta)\\theta^{2}}}\\leq|\\bar{x}|,|\\bar{y}|,\\]\n\\[\n|\\bar{x}|,|\\bar{y}|\\leq\\sqrt{[(1+\\theta)^{2}+(n-1)\\theta^{2}]\\frac{l}{(a_{k}+\\delta)(1-\\theta)^{2}+(n-1)(a_{1}+\\delta)\\theta^{2}}}.\\]\n\\end{lem}\n\\begin{proof}\nNecessarily we must have $k\\geq n-m+1$ because if $k<n-m+1$, then\nthe eigenvalue $a_{k}$ is positive, making the direction $e_{k}$\nand its nearby directions directions of ascent. Let $d:=d_{\\bar{x}}$.\nFrom $|d-e_{k}|_{\\infty}<\\theta$, we obtain $1-\\theta\\leq d_{k}\\leq1+\\theta$\nand $|d_{j}|<\\theta$ for $j\\neq k$.\n\nFor the direction $d$, we find the largest and smallest value for\nthe norm of $\\bar{x}$ if $\\mbox{unit}(\\bar{x})=\\mbox{unit}(d)$.\nThis is also the largest and smallest possible value of $t$ such\nthat $h(t\\mbox{ unit}(d))=l$. Here, $\\mbox{unit}(\\cdot)$ maps a\nnonzero vector to the unit vector of the same direction. Now\\begin{eqnarray*}\nh\\big(t\\mbox{ unit}(d)\\big) & \\leq & \\sum_{i=1}^{n}(a_{i}-\\delta)[t\\mbox{ unit}(d)_{i}]^{2}\\\\\n & = & t^{2}\\sum_{i=1}^{n}(a_{i}-\\delta)\\frac{d_{i}^{2}}{\\sum_{j=1}^{n}d_{j}^{2}}\\\\\n & \\leq & \\frac{t^{2}}{\\sum_{j=1}^{n}d_{j}^{2}}\\left((a_{k}-\\delta)d_{k}^{2}+(a_{n}-\\delta)\\sum_{j\\neq k}d_{j}^{2}\\right)\\\\\n & \\leq & \\frac{t^{2}}{(1-\\theta)^{2}}[(a_{k}-\\delta)(1+\\theta)^{2}+(n-1)(a_{n}-\\delta)\\theta^{2}].\\end{eqnarray*}\nSince $h(t\\mbox{ unit}(d))=l$, we have \\[\nt\\leq(1-\\theta)\\sqrt{\\frac{l}{(a_{k}-\\delta)(1+\\theta)^{2}+(n-1)(a_{n}-\\delta)\\theta^{2}}}.\\]\nNext, \\begin{eqnarray*}\nh(t\\mbox{ unit}(d)) & \\geq & \\sum_{i=1}^{n}(a_{i}+\\delta)[t\\mbox{ unit}(d)_{i}]^{2}\\\\\n & = & \\frac{t^{2}}{\\sum_{j=1}^{n}d_{j}^{2}}\\sum_{i=1}^{n}(a_{i}+\\delta)d_{i}^{2}\\\\\n & \\geq & \\frac{t^{2}}{(1+\\theta)^{2}+(n-1)\\theta^{2}}[(a_{k}+\\delta)(1-\\theta)^{2}+(n-1)(a_{1}+\\delta)\\theta^{2}].\\end{eqnarray*}\nAgain, since $h(t\\mbox{ unit}(d))=l$, we have \\[\nt\\geq\\sqrt{[(1+\\theta)^{2}+(n-1)\\theta^{2}]\\frac{l}{(a_{k}+\\delta)(1-\\theta)^{2}+(n-1)(a_{1}+\\delta)\\theta^{2}}}.\\]\n\n\\end{proof}\nHere are some lemmas on the completion of orthogonal matrices. In\nLemmas \\ref{lem:first-matrix-approx} and \\ref{lem:matrix-approx},\nlet $|\\cdot|$ denote a norm for matrices (which need not be a matrix\nnorm).\n\\begin{lem}\n(Completion to orthogonal matrix) \\label{lem:first-matrix-approx}Let\n$E_{k}\\in\\mathbb{R}^{n\\times k}$ be the first $k$ columns of the\n$n\\times n$ identity matrix. Then for all $\\epsilon>0$, there exists\n$\\delta>0$ such that if $V_{k}\\in\\mathbb{R}^{n\\times k}$ has orthonormal\ncolumns and $|V_{k}-E_{k}|<\\delta$ , then $V_{k}$ can be completed\nto an orthogonal matrix $V_{n}\\in\\mathbb{R}^{n\\times n}$ such that\n$|I-V_{n}|<\\epsilon$.\n\\end{lem}\nThe above lemma is an easy consequence of the following result.\n\\begin{lem}\n(Finding orthogonal vector) \\label{lem:matrix-approx}For all $\\epsilon>0$,\nthere exists a $\\delta>0$ such that if $V_{k}\\in\\mathbb{R}^{n\\times k}$\nhas orthonormal columns and $|V_{k}-E_{k}|<\\delta$ , then there is\na vector $v_{k+1}\\in\\mathbb{R}^{n}$ such that $|v_{k+1}|_{2}=1$\nand is orthogonal to all columns of $V_{k}$, and the concatenation\n$V_{k+1}:=[V_{k},v_{k+1}]$ satisfies $|V_{k+1}-E_{k+1}|<\\epsilon$.\\end{lem}\n\\begin{proof}\nSince all finite dimensional norms are equivalent, we can assume that\nthe norm $|\\cdot|$ on $\\mathbb{R}^{n\\times k}$, $\\mathbb{R}^{n\\times(k+1)}$\nis the $\\infty$-norm for vectors, that is $|M|=\\max_{i,j}|M(i,j)|$.\nSuppose $|V_{k}-E_{k}|<\\delta$. Then $|V_{k}(i,j)|<\\delta$ if $i\\neq j$\nand $|V_{k}(i,i)-1|<\\delta$. We now construct the vector $v_{k+1}$\nusing the Gram-Schmidt process. \n\nThe direction of $v_{k+1}$ obtained by the Gram-Schmidt process is:\\begin{eqnarray*}\n(I-V_{k}V_{k}^{T})e_{k+1} & = & e_{k+1}-V_{k}V_{k}^{T}e_{k+1}\\\\\n & = & e_{k+1}-\\sum_{i=1}^{k}V_{k}(i,k+1)V_{k}(:,i).\\end{eqnarray*}\nSince $|V_{k}(i,j)|<\\delta$ for all $i\\neq j$, the sum $\\alpha_{k+1}\\in\\mathbb{R}^{n}$\ndefined by $\\alpha_{k+1}=\\sum_{i=1}^{k}V_{k}(i,k+1)V_{k}(:,i)$ has\ncomponents obeying the bounds\\[\n|\\alpha_{k+1}(j)|\\leq\\left\\{ \\begin{array}{ll}\nk\\delta^{2} & \\mbox{ if }j\\geq k+1,\\\\\n(k-1)\\delta^{2}+\\delta & \\mbox{ if }j<k+1.\\end{array}\\right.\\]\nThen $v_{k+1}=\\mbox{unit}(e_{k+1}+\\alpha_{k+1})$. We first analyze\nthe maximum error in $v_{k+1}(j)$ for $j\\neq k+1$. The $2-$norm\nof $e_{k+1}+\\alpha_{k+1}$ is at least \\begin{eqnarray*}\n|e_{k+1}+\\alpha_{k+1}|_{2} & \\geq & |e_{k+1}|_{2}-|\\alpha_{k+1}|_{2}\\\\\n & \\geq & 1-\\sqrt{\\sum_{i=1}^{n}[\\alpha_{k+1}(i)]^{2}}\\\\\n & \\geq & 1-\\sqrt{(n-k)k\\delta^{2}+k\\left((k-1)\\delta^{2}+\\delta\\right)}\\\\\n & \\geq & 1-\\sqrt{nk\\delta^{2}+k\\delta}.\\end{eqnarray*}\nIf $\\delta<\\min\\{1,\\frac{1}{4(n+1)k}\\}$, then $\\sqrt{nk\\delta^{2}+k\\delta}\\leq\\frac{1}{2}$,\nand so $\\left|e_{k+1}+\\alpha_{k+1}\\right|_{2}\\geq\\frac{1}{2}$. In\nthis case, the maximum error in $v_{k+1}(j)$ is $2((k-1)\\delta^{2}+\\delta)$\nfor $j\\neq k$. For $j=k+1$, we note that $|v_{k+1}|_{2}=1$, which\ntells us that \\begin{eqnarray*}\n[v_{k+1}(k+1)]^{2} & = & 1-\\sum_{1\\leq i\\leq n,i\\neq k}|v_{k+1}(i)|^{2}\\\\\n & \\geq & 1-4(n-1)[(k-1)\\delta^{2}+\\delta]^{2}\\\\\n\\Rightarrow v_{k+1}(k+1) & \\geq & \\sqrt{1-4(n-1)[(k-1)\\delta^{2}+\\delta]^{2}}\\\\\n & = & \\sqrt{1-4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}}.\\end{eqnarray*}\nIf $\\delta<\\min\\{1,\\frac{1}{2k\\sqrt{n-1}}\\}$, then $4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}<1$,\nwhich gives \\begin{eqnarray*}\nv_{k+1}(k+1) & \\geq & \\sqrt{1-4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}}\\\\\n & \\geq & 1-4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}\\\\\n|v_{k+1}(k+1)-1| & \\leq & 4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}.\\end{eqnarray*}\nIf $\\delta<\\min\\{\\frac{\\epsilon}{2k},\\frac{1}{2k}\\sqrt{\\frac{\\epsilon}{n-1}}\\}$,\nthen \\begin{eqnarray*}\n|[V_{k},v_{k+1}]-E_{k+1}| & \\leq & \\max\\{2[(k-1)\\delta^{2}+\\delta],4(n-1)\\delta^{2}[(k-1)\\delta+1]^{2}\\}\\\\\n & < & \\epsilon,\\end{eqnarray*}\n and we are done.\n\\end{proof}\nThe next result shows that we get a closer estimate to the critical\nvalue after each iteration of step 2 in Algorithm \\ref{alg:fast-local-method}. \n\\begin{lem}\n(Lower bound on critical value) \\label{lem:bdd-on-l-(i+1)}Let $\\delta>0$\nbe sufficiently small, and suppose $h:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$\nsatisfies Assumption \\ref{ass:condns-on-model}. Let $S_{\\bar{z},V}:=\\{\\bar{z}+Vw\\mid w\\in\\mathbb{R}^{n-m}\\}$,\nand $V\\in\\mathbb{R}^{n\\times(n-m)}$ be such that $V$ has orthonormal\ncolumns, with $|\\bar{z}-\\mathbf{0}|<\\delta$ and $|V-E_{n-m}|<\\delta$,\nwhere $E_{n-m}\\in\\mathbb{R}^{n\\times(n-m)}$ is the first $n-m$ columns\nof the identity matrix. Then\\[\n-\\frac{1}{2}|A-\\delta I|\\left(1+|[V^{T}(A-\\delta I)V]^{-1}||V^{T}||A-\\delta I|\\right)^{2}|\\bar{z}|^{2}\\leq\\min_{s\\in S_{\\bar{z},V}\\cap\\mathbb{B}}h(s).\\]\n\\end{lem}\n\\begin{proof}\nWe find a lower bound for the smallest value of $h$ on $S_{\\bar{z},V}$.\nThe function $h_{\\bar{z},V}:\\mathbb{R}^{n-m}\\rightarrow\\mathbb{R}$\ndefined by $h_{\\bar{z},V}(w):=h(\\bar{z}+Vw)$ satisfies\\begin{eqnarray*}\nh_{\\bar{z},V}(w) & = & h(\\bar{z}+Vw)\\\\\n & \\geq & \\frac{1}{2}(\\bar{z}+Vw)^{T}(A-\\delta I)(\\bar{z}+Vw).\\end{eqnarray*}\nLet us denote $h_{\\bar{z},V,\\min}:\\mathbb{R}^{n-m}\\rightarrow\\mathbb{R}$\nby $h_{\\bar{z},V,\\min}(w)=\\frac{1}{2}(\\bar{z}+Vw)^{T}(A-\\delta I)(\\bar{z}+Vw)$.\nThe Hessian of $h_{\\bar{z},V,\\min}$ is $V^{T}(A-\\delta I)V$, which\ntells us that $h_{\\bar{z},V,\\min}$ is strictly convex. Therefore,\nwe seek to find the minimizer of $h_{\\bar{z},V,\\min}$.\n\nThe minimizing value of $w$, which we denote as $\\bar{w}_{\\min}$,\nsatisfies $\\nabla h_{\\bar{z},V,\\min}(\\bar{w}_{\\min})=\\mathbf{0}$.\nThis gives us \\begin{eqnarray*}\nV^{T}(A-\\delta I)\\bar{z}+V^{T}(A-\\delta I)V\\bar{w}_{\\min} & = & \\mathbf{0}\\\\\n\\Rightarrow\\bar{w}_{\\min} & = & -[V^{T}(A-\\delta I)V]^{-1}V^{T}(A-\\delta I)\\bar{z}.\\end{eqnarray*}\nAn easy bound on $\\left|\\bar{w}_{\\min}\\right|$ is $\\left|\\bar{w}_{\\min}\\right|\\leq|[V^{T}(A-\\delta I)V]^{-1}||V^{T}||A-\\delta I||\\bar{z}|$.\n So $h_{\\bar{z},V,\\min}$ is bounded from below by \\begin{eqnarray}\n\\min_{w}h_{\\bar{z},V,\\min}(w) & = & \\min_{w}\\frac{1}{2}(\\bar{z}+Vw)^{T}(A-\\delta I)(\\bar{z}+Vw)\\nonumber \\\\\n & = & \\frac{1}{2}(\\bar{z}+V\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V\\bar{w}_{\\min})\\nonumber \\\\\n & \\geq & -\\frac{1}{2}|\\bar{z}+V\\bar{w}_{\\min}||A-\\delta I||\\bar{z}+V\\bar{w}_{\\min}|\\nonumber \\\\\n & = & -\\frac{1}{2}|A-\\delta I||\\bar{z}+V\\bar{w}_{\\min}|^{2}\\nonumber \\\\\n & \\geq & -\\frac{1}{2}|A-\\delta I|[|\\bar{z}|+\\left|V\\bar{w}_{\\min}\\right|]^{2}\\nonumber \\\\\n & = & -\\frac{1}{2}|A-\\delta I|[|\\bar{z}|+|\\bar{w}_{\\min}|]^{2}\\nonumber \\\\\n & \\geq & -\\frac{1}{2}|A-\\delta I|\\left(|\\bar{z}|+\\left|[V^{T}(A-\\delta I)V]^{-1}\\right||V^{T}||A-\\delta I||\\bar{z}|\\right)^{2}\\nonumber \\\\\n & = & -\\frac{1}{2}|A-\\delta I|\\left(1+\\left|[V^{T}(A-\\delta I)V]^{-1}\\right||V^{T}||A-\\delta I|\\right)^{2}|\\bar{z}|^{2}.\\label{eq:bdd-below-h-subspace}\\end{eqnarray}\n\n\\end{proof}\nWe shall prove Lemma \\ref{lem:on-S-prime} about the approximation\nof the eigenvectors corresponding to the smallest eigenvalues. This\nlemma analyzes Algorithm \\ref{alg:find-S-prime}. We clarify our notation.\nIn the case of an exact quadratic, Algorithm \\ref{alg:fast-local-method}\nfirst finds $e_{n-m+1}$, then invokes Algorithm \\ref{alg:find-S-prime}\nto find the eigenvector $e_{n}$, followed by $e_{n-1}$, $e_{n-2}$\nand so on, all the way to $e_{n-m+2}$. \n\nWe define $I_{k}$ and $I_{k}^{\\perp}$ as subsets of $\\{1,\\dots,n\\}$\nby \\begin{eqnarray*}\nI_{k} & := & \\{n-m+1\\}\\cup\\{n-k+2,n-k+3,\\dots,n\\}\\\\\nI_{k}^{\\perp} & := & \\{1,\\dots,n\\}\\backslash I_{k}.\\end{eqnarray*}\nNext, we define $E_{k}^{\\prime}$ and $E_{k}^{\\perp}$. The matrix\n$E_{k}^{\\prime}\\in\\mathbb{R}^{n\\times k}$ has the $k$ columns $e_{n-m+1}$,\n$e_{n}$, $e_{n-1}$, ..., $e_{n-k+2}$, while the matrix $E_{k}^{\\perp}\\in\\mathbb{R}^{n\\times(n-k)}$\ncontains all the other columns in the $n\\times n$ identity matrix.\nThe columns of $E_{k}^{\\prime}$ and $E_{k}^{\\perp}$ are chosen from\nthe $n\\times n$ identity matrix from the index sets $I_{k}$ and\n$I_{k}^{\\perp}$ respectively.\n\nWe will need to analyze the eigenvalues of $(V_{k}^{\\perp})^{T}(A\\pm\\delta I)V_{k}^{\\perp}$\nin the proof of Lemma \\ref{lem:on-S-prime}, where $|V_{k}^{\\perp}-E_{k}^{\\perp}|$\nis small. Note that the matrix $(E_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}$\nis principle minor of $A\\pm\\delta I$, and its eigenvalues are the\neigenvalues of $A\\pm\\delta I$ chosen according to the index set $I_{k}^{\\perp}$.\nFurthermore, \\begin{eqnarray*}\n &  & |(V_{k}^{\\perp})^{T}(A\\pm\\delta I)V_{k}^{\\perp}-(E_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}|\\\\\n & \\leq & |(V_{k}^{\\perp})^{T}(A\\pm\\delta I)V_{k}^{\\perp}-(V_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}|\\\\\n &  & \\quad+|(V_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}-(E_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}|\\\\\n & \\leq & |(V_{k}^{\\perp})^{T}(A\\pm\\delta I)||V_{k}^{\\perp}-E_{k}^{\\perp}|\\\\\n &  & \\quad+|(V_{k}^{\\perp})^{T}-(E_{k}^{\\perp})^{T}||(A\\pm\\delta I)E_{k}^{\\perp}|\\end{eqnarray*}\nIt is clear that as $|V_{k}^{\\perp}-E_{k}^{\\perp}|\\to0$, $|(V_{k}^{\\perp})^{T}(A\\pm\\delta I)V_{k}^{\\perp}-(E_{k}^{\\perp})^{T}(A\\pm\\delta I)E_{k}^{\\perp}|\\to0$.\nThe eigenvalues of a matrix varies continuously with respect to the\nentries when the eigenvalues are distinct, so we shall let $\\hat{a}_{i}$\ndenote the eigenvalue of $(V_{k}^{\\perp})^{T}(A+\\delta I)V_{k}^{\\perp}$\nthat is closest to $a_{i}$, and $\\tilde{a}_{i}$ denote the eigenvalue\nof $(V_{k}^{\\perp})^{T}(A-\\delta I)V_{k}^{\\perp}$ that is closest\nto $a_{i}$. \n\\begin{lem}\n(Estimates of eigenvectors to negative eigenvalues) \\label{lem:on-S-prime}Let\n$h:\\mathbb{R}^{n}\\to\\mathbb{R}$. Given a fixed $l<0$ sufficiently\nclose to $0$, let $p$ be the closest point to $\\bar{z}$ in the\nset $\\mbox{\\rm lev}_{\\leq l}h\\cap S_{\\bar{z},V_{k}^{\\perp}}$, where $S_{\\bar{z},V_{k}^{\\perp}}:=\\{\\bar{z}+V_{k}^{\\perp}w\\mid w\\in\\mathbb{R}^{n-k}\\}$\nand $V_{k}^{\\perp}\\in\\mathbb{R}^{n\\times(n-k)}$. Then for all $\\epsilon>0$,\nthere exists $\\delta>0$ such that if \n\\begin{enumerate}\n\\item $h:\\mathbb{R}^{n}\\rightarrow\\mathbb{R}$ satisfies Assumption \\ref{ass:condns-on-model},\n\\item $|\\bar{z}|<\\delta$ and\n\\item $|V_{k}^{\\perp}-E_{k}^{\\perp}|<\\delta$, where $V_{k}^{\\perp}$ has\northogonal columns,\n\\end{enumerate}\nthen $|\\mbox{\\rm unit}(p-\\bar{z})-e_{n-k+1}|<\\epsilon$. As a consequence,\n$|V_{m}^{\\perp}-E_{m}^{\\perp}|\\to0$ as $\\delta\\to0$.\\end{lem}\n\\begin{proof}\nThe first step is to find an upper bound on the distance between $\\bar{z}$\nand $p$. The upper bound is obtained from looking at the closest\ndistance between $\\bar{z}$ and $\\mbox{\\rm lev}_{\\leq l}h_{\\max}\\cap S_{\\bar{z},V_{k}^{\\perp}}$.\n\nWe look at the function $h_{\\bar{z},V_{k}^{\\perp},\\max}:\\mathbb{R}^{n-k}\\rightarrow\\mathbb{R}$\ndefined by $h_{\\bar{z},V_{k}^{\\perp},\\max}(w):=h_{\\max}(\\bar{z}+V_{k}^{\\perp}w)$.\nWe have\\begin{eqnarray*}\nh_{\\bar{z},V_{k}^{\\perp},\\max}(w) & = & h_{\\bar{z},V_{k}^{\\perp},\\max}(\\bar{z}+V_{k}^{\\perp}w)\\\\\n & = & \\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}w)^{T}(A+\\delta I)(\\bar{z}+V_{k}^{\\perp}w)\\\\\n\\Rightarrow\\nabla h_{\\bar{z},V_{k}^{\\perp},\\max}(w) & = & (V_{k}^{\\perp})^{T}(A+\\delta I)\\bar{z}+(V_{k}^{\\perp})^{T}(A+\\delta I)V_{k}^{\\perp}w.\\end{eqnarray*}\nThe critical point of $h_{\\bar{z},V_{k}^{\\perp},\\max}$ is thus $\\bar{w}_{\\max}:=-[(V_{k}^{\\perp})^{T}(A+\\delta I)V_{k}^{\\perp}]^{-1}(V_{k}^{\\perp})^{T}(A+\\delta I)\\bar{z}$.\nThe critical value corresponding to this is $h_{\\bar{z},V_{k}^{\\perp},\\max}(\\bar{w}_{\\max})$.\nAn upper bound for this critical value is $\\frac{1}{2}|\\bar{z}+V_{k}^{\\perp}\\bar{w}|\\left|A+\\delta I\\right||\\bar{z}+V_{k}^{\\perp}\\bar{w}|$,\nwhich is in turn bounded by:\\begin{eqnarray*}\n &  & \\frac{1}{2}|\\bar{z}+V_{k}^{\\perp}\\bar{w}||A+\\delta I||\\bar{z}+V_{k}^{\\perp}\\bar{w}|\\\\\n & = & \\frac{1}{2}\\left|A+\\delta I\\right||\\bar{z}+V_{k}^{\\perp}\\bar{w}|^{2}\\\\\n & \\leq & \\frac{1}{2}\\left|A+\\delta I\\right|\\left(1+\\left|[(V_{k}^{\\perp})^{T}(A+\\delta I)V_{k}^{\\perp}]^{-1}\\right||(V_{k}^{\\perp})^{T}|\\left|A+\\delta I\\right|\\right)^{2}|\\bar{z}|^{2}.\\\\\n &  & \\mbox{(following calculations similar to that of \\eqref{eq:bdd-below-h-subspace}).}\\end{eqnarray*}\nThen an upper bound of the distance $d(\\bar{z},\\mbox{\\rm lev}_{\\leq l}h_{\\max}\\cap S_{\\bar{z},V_{k}^{\\perp}})$\ncan be calculated by:\n\n\\begin{eqnarray*}\n &  & d(\\bar{z},S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\max})\\\\\n & \\leq & \\left|\\bar{z}-\\left(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\max}\\right)\\right|+d(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\max},S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\max})\\\\\n & \\leq & \\underbrace{|\\bar{w}_{\\max}|+\\sqrt{\\frac{l-\\frac{1}{2}\\left|A+\\delta I\\right|\\left(1+\\left|[(V_{k}^{\\perp})^{T}(A+\\delta I)V_{k}^{\\perp}]^{-1}\\right||(V_{k}^{\\perp})^{T}|\\left|A+\\delta I\\right|\\right)^{2}|\\bar{z}|^{2}}{\\hat{a}_{n-k+1}+\\delta}}.}_{\\beta}\\end{eqnarray*}\n\n\nThe extra term $-\\frac{1}{2}\\left|A+\\delta I\\right|\\cdots|\\bar{z}|^{2}$\ncompensates for the fact that the critical value of $h_{\\bar{z},V_{k}^{\\perp},\\max}$\nis not necessarily zero. To simplify notation, let $\\beta$ be the\nright hand side of the above formula as marked.\n\nWe now figure the possible intersection between $\\mathbb{B}(\\bar{z},\\beta)$\nand $S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h$. Again, since $S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h\\subset S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\min}$,\nwe look at the intersection of $\\mathbb{B}(\\bar{z},\\beta)$ and $S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\min}$.\nWe find the critical point of $h_{\\bar{z},V_{k}^{\\perp},\\min}:\\mathbb{R}^{n-k}\\rightarrow\\mathbb{R}$\ndefined by $h_{\\bar{z},V_{k}^{\\perp},\\min}(w):=\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}w)^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}w)$.\nThe gradient of $h_{\\bar{z},V_{k}^{\\perp},\\min}$ can be found to\nbe \\[\n\\nabla h_{\\bar{z},V_{k}^{\\perp},\\min}(w)=(V_{k}^{\\perp})^{T}(A-\\delta I)\\bar{z}+(V_{k}^{\\perp})^{T}(A-\\delta I)V_{k}^{\\perp}w.\\]\nOnce again, the critical point is $\\bar{w}_{\\min}=[(V_{k}^{\\perp})^{T}(A-\\delta I)V_{k}^{\\perp}]^{-1}(V_{k}^{\\perp})^{T}(A-\\delta I)\\bar{z}$.\nSo $\\mathbb{B}(\\bar{z},\\beta)\\subset\\mathbb{B}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min},\\beta+|\\bar{w}_{\\min}|)$.\n\nConsider $p\\in\\mathbb{B}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min},\\beta+|\\bar{w}_{\\min}|)\\cap(S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\min})$.\nLet us introduce a change of coordinates such that $p=\\sum_{i\\in I_{k}^{\\perp}}\\tilde{p}_{i}\\tilde{v}_{i}+\\bar{w}_{\\min}$,\nwhere $\\tilde{v}_{i}\\in\\mathbb{R}^{n-k}$ correspond to the eigenvectors\nof $(V_{k}^{\\perp})^{T}(A-\\delta I)V_{k}^{\\perp}$ (in turn corresponding\nto the eigenvalues $\\tilde{a}_{i}$) and $\\tilde{p}_{i}\\in\\mathbb{R}$\nare the multipliers. Then the condition $\\bar{z}+V_{k}^{\\perp}p\\in\\mathbb{B}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min},\\beta+|\\bar{w}_{\\min}|)$\nand $\\bar{z}+V_{k}^{\\perp}p\\in S_{\\bar{z},V_{k}^{\\perp}}\\cap\\mbox{\\rm lev}_{\\leq l}h_{\\min}$\ncan be represented as the following constraints respectively:\\begin{eqnarray}\n\\sum_{i\\in I_{k}^{\\perp}}\\tilde{p}_{i}^{2} & \\leq & (\\beta+|\\bar{w}_{\\min}|)^{2},\\nonumber \\\\\n\\sum_{i\\in I_{k}^{\\perp}}\\tilde{a}_{i}\\tilde{p}_{i}^{2} & \\leq & l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min}).\\label{eq:linear-constraints}\\end{eqnarray}\nAs $\\delta\\rightarrow0$, the only admissible solution is $\\tilde{p}_{n-k+1}=\\sqrt{\\frac{l}{\\tilde{a}_{n-k+1}}}$\nand the rest of the $\\tilde{p}_{i}$'s are zero. The above constraints\nare linear in $\\tilde{p}_{i}^{2}$. We consider the minimum possible\nvalue of $\\frac{\\tilde{p}_{n-k+1}}{\\sqrt{\\sum\\tilde{p}_{i}^{2}}}=\\sqrt{\\frac{\\tilde{p}_{n-k+1}^{2}}{\\sum\\tilde{p}_{i}^{2}}}$,\nwhich is the dot product between the unit vectors in the direction\nof $\\tilde{v}_{n-k+1}$ and $p-(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})$.\nThis is equivalent to the linear fractional program in $\\tilde{p}_{i}^{2}$\nof minimizing $\\frac{\\tilde{p}_{n-k+1}^{2}}{\\sum\\tilde{p}_{i}^{2}}$\nsubject to the constraints in \\eqref{eq:linear-constraints}.\n\nThis linear fractional program can be transformed into a linear program\nby $q=\\frac{1}{\\sum\\tilde{p}_{i}^{2}}$ and $q_{i}=\\frac{\\tilde{p}_{i}^{2}}{\\sum\\tilde{p}_{i}^{2}}$,\nwhich gives:\\begin{eqnarray}\n\\min & q_{n-k+1}\\nonumber \\\\\n\\mbox{s.t.} & q & \\geq\\frac{1}{(\\beta+|\\bar{w}_{\\min}|)^{2}},\\label{eq:linear-constraint-2.1}\\\\\n & \\sum_{i\\in I_{k}^{\\perp}}\\tilde{a}_{i}q_{i} & \\leq\\Big[l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})\\Big]q,\\label{eq:linear-constraint-2.2}\\\\\n & \\sum_{i\\in I_{k}^{\\perp}}q_{i} & =1,\\label{eq:linear-constraint-2.3}\\\\\n & q_{i} & \\geq0\\mbox{ for all }i\\in I_{k}^{\\perp}.\\nonumber \\end{eqnarray}\nThe constraints of the linear program above gives \\begin{eqnarray*}\n\\sum_{i\\in I_{k}^{\\perp}}-\\tilde{a}_{i}q_{i}+\\tilde{a}_{n-k}\\sum_{i\\in I_{k}^{\\perp}}q_{i} & \\geq & -\\Big[l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})\\Big]q+\\tilde{a}_{n-1}\\\\\n\\Rightarrow\\sum_{i\\in I_{k}^{\\perp}}(-\\tilde{a}_{i}+\\tilde{a}_{n-k})q_{i} & \\geq & -\\Big[l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})\\Big]q+\\tilde{a}_{n-1}\\\\\n & \\geq & -\\frac{l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})}{(\\beta+|\\bar{w}_{\\min}|)^{2}}+\\tilde{a}_{n-1}.\\end{eqnarray*}\nSince only $-\\tilde{a}_{n-k+1}+\\tilde{a}_{n-k}$ is positive and the\nother $-\\tilde{a}_{i}+\\tilde{a}_{n-k}$ are nonpositive, we have\\begin{eqnarray*}\n &  & (-\\tilde{a}_{n-k+1}+\\tilde{a}_{n-k})q_{n-k+1}\\\\\n & \\geq & \\sum_{i\\in I_{k}^{\\perp}}(-\\tilde{a}_{i}+\\tilde{a}_{n-k})q_{i}\\\\\n & \\geq & -\\frac{l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})}{(\\beta+|\\bar{w}_{\\min}|)^{2}}+\\tilde{a}_{n-k}\\end{eqnarray*}\n\\[\n\\Rightarrow q_{n-k+1}\\geq\\frac{1}{-\\tilde{a}_{n-k+1}+\\tilde{a}_{n-k}}\\left(-\\frac{l+\\frac{1}{2}(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})^{T}(A-\\delta I)(\\bar{z}+V_{k}^{\\perp}\\bar{w}_{\\min})}{(\\beta+|\\bar{w}_{\\min}|)^{2}}+\\tilde{a}_{n-k}\\right).\\]\nThe limit of the right hand side goes to $1$ as $\\delta\\rightarrow0$,\nso this means that $\\bar{z}-p$ is close to the direction of the eigenvector\ncorresponding to the eigenvalue $\\tilde{a}_{n-k+1}$ in $(V_{k}^{\\perp})^{T}AV_{k}^{\\perp}$,\nwhich in turn converges to $e_{n-k+1}$. The proof of this lemma is\ncomplete.\n\nThe conclusion that $|V_{m}^{\\perp}-E_{m}^{\\perp}|\\to0$ as $\\delta\\to0$\nfollows from the first part of this lemma and Lemma \\ref{lem:first-matrix-approx}.\n\\end{proof}\nWith these lemmas set up, we are now ready to prove the fast local\nconvergence of Algorithm \\ref{alg:fast-local-method} to the critical\npoint and critical value. We recall that \\emph{Q-linear convergence}\nof a sequence of positive numbers $\\{\\alpha_{i}\\}_{i=1}^{\\infty}$\nconverging to zero is defined by $\\limsup_{i\\to\\infty}\\frac{\\alpha_{i+1}}{\\alpha_{i}}<1$,\nwhile \\emph{Q-superlinear convergence} is defined by $\\lim_{i\\to\\infty}\\frac{\\alpha_{i+1}}{\\alpha_{i}}=0$.\nNext, \\emph{R-linear convergence} and \\emph{R-superlinear convergence}\nof a sequence are defined by being bounded by a Q-linearly convergent\nsequence and a Q-superlinearly convergent sequence respectively.\n\\begin{thm}\n(Fast convergence of Algorithm \\ref{alg:fast-local-method}) \\label{thm:Wrap-up}Suppose\nthat $f:\\mathbb{R}^{n}\\to\\mathbb{R}$ is $\\mathcal{C}^{2}$ and $\\mathbf{0}$\nis a nondegenerate critical point of $f$ of Morse index $m$, and\n$h(\\mathbf{0})=0$. \n\nThere is some $R>0$ such that if $0<r<R$, then for $U_{i}=\\mathbb{B}(\\mathbf{0},r)$\nand $l_{0}<0$ (depending on $r$) sufficiently close to $0$, Algorithm\n\\ref{alg:fast-local-method} converges R-superlinearly to the critical\npoint $\\mathbf{0}$ and Q-superlinearly to the critical value $0$\nprovided that at each iteration, there exists an optimizing triple\n$(S_{i},x_{i},y_{i})$ for which $(x_{i},y_{i})$ is the unique pair\nof points $S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap\\mathbb{B}(\\mathbf{0},r)$\nsuch that $|x_{i}-y_{i}|=\\mbox{\\rm diam}(S_{i}\\cap(\\mbox{\\rm lev}_{\\geq l_{i}}f)\\cap\\mathbb{B}(\\mathbf{0},r))$. \n\nIf a sufficiently good approximate for the lineality space of each\n$S_{i}$ is available in step 2 of Algorithm \\ref{alg:fast-local-method}\ninstead, then Algorithm \\ref{alg:fast-local-method} converges R-linearly\nto the critical point and Q-linearly to the critical value.\\end{thm}\n\\begin{proof}\nAs a reminder, Q-linear convergence of the critical value is defined\nto be $\\limsup_{i\\to\\infty}\\frac{|l_{i+1}|}{|l_{i}|}<\\infty$, and\nQ-superlinear convergence of the critical value is defined to be $\\lim_{i\\to\\infty}\\frac{|l_{i+1}|}{|l_{i}|}=0$.\nFrom the Q-linear (Q-superlinear) convergence of $l_{i}$, we obtain\nthe R-linear (R-superlinear) convergence of the critical point by\nobserving that $\\limsup_{i\\to\\infty}\\frac{|z_{i}|}{\\sqrt{|l_{i}|}}<\\infty$,\nand that $\\sqrt{|l_{i}|}$ converges Q-linearly (Q-superlinearly). \n\nSince $f$ is $\\mathcal{C}^{2}$, for all $\\delta>0$, we can find\n$R>0$, such that \\[\n|f(x)-x^{T}Ax|<\\delta|x|^{2}\\mbox{ for all }x\\in\\mathbb{B}(\\mathbf{0},R).\\]\nThe function $f_{R}:\\mathbb{R}^{n}\\to\\mathbb{R}$ defined by $f_{R}(x):=\\frac{1}{R^{2}}f(Rx)$\nsatisfies Assumption \\ref{ass:condns-on-model} with $A:=\\nabla^{2}f_{R}(x)=\\nabla^{2}f(x)$. \n\nWe want to show that if $\\delta>0$ is sufficiently small, then for\nall $l<0$ sufficiently small, a step in Algorithm \\ref{alg:fast-local-method}\ngives good convergence to the critical value. Given an iterate $l_{i}$,\nthe next iterate $l_{i+1}$ is \\[\n\\min_{x\\in S_{z_{i},V_{m}^{\\perp}}\\cap\\mathbb{B}(\\mathbf{0},R)}f(x)=\\min_{x\\in S_{\\frac{1}{R}z_{i},V_{m}^{\\perp}}\\cap\\mathbb{B}}R^{2}f_{R}(x),\\]\nwhere $V_{m}^{\\perp}$, which approximates the first $n-m$ eigenvectors,\nis defined before Lemma \\ref{lem:on-S-prime}. \n\nWe seek to find $\\frac{|l_{i+1}|}{|l_{i}|}$. The value of $l_{i+1}$\ndepends on how well the last $m$ eigenvectors are approximated, and\nhow well the critical point is estimated, which in turn depends on\n$\\delta$. The ratio $\\frac{|l_{i+1}|}{|l_{i}|}$ is bounded from\nabove by\\[\n-\\frac{1}{2}|A-\\delta I|\\left(1+\\left|[(V_{m}^{\\perp})^{T}(A-\\delta I)V_{m}^{\\perp}]^{-1}\\right||(V_{m}^{\\perp})^{T}||A-\\delta I|\\right)^{2}|z_{i}|^{2}\/l_{i},\\]\nwhich converges to $0$ as $\\delta\\to0$ by Lemmas \\ref{lem:min-maximizer-ppties},\n\\ref{lem:first-matrix-approx}, \\ref{lem:bdd-on-l-(i+1)} and \\ref{lem:on-S-prime}. \n\nThe conclusion in the second part of the theorem follows a similar\nanalysis.\n\\end{proof}\n\n\n\n\\section{Conclusion and conjectures}\n\nIn this paper, we present a strategy to find saddle points of general\nMorse index, extending the algorithms for finding critical points\nof mountain pass type as was done in \\cite{LP08}. Algorithms \\ref{alg:fast-local-method}\nand \\ref{alg:find-S-prime} may not be easily implementable, especially\nwhen $m$ is large. However, Algorithm \\ref{alg:find-S-prime} can\nbe performed only as needed in a practical implementation. It is hoped\nthat this strategy can augment current methods for finding saddle\npoints, and can serve as a foundation for further research on effective\nmethods of finding saddle points.\n\nHere are some conjectures:\n\\begin{itemize}\n\\item How do the algorithms presented fare in real problems? Are there difficulties\nin the infinite dimensional case when implementing Algorithm \\ref{alg:fast-local-method}?\n\\item Are there ways to integrate Algorithms \\ref{alg:fast-local-method}\nand \\ref{alg:find-S-prime} to give a better algorithm?\n\\item Are there better algorithms than Algorithm \\ref{alg:find-S-prime}\nto approximate $S_{i}$?\n\\item Can the uniqueness assumption in Theorem \\ref{thm:Wrap-up} be lifted?\nIf not, how does it affect the design of algorithms?\\end{itemize}\n\n\\begin{acknowledgement*}\nI thank the Fields Institute in Toronto, where much of this paper\nwas written. They have provided a wonderful environment for working\non this paper.\\end{acknowledgement*}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nFractional calculus appeared in 1695, when Leibniz described the derivative \nof order $\\alpha=1\/2$ \\cite{OS,SKM,Ross}.\nDerivatives and integrals of noninteger order were studied by \nLeibniz, Liouville, Grunwald, Letnikov and Riemann.\nMany books have now been written about fractional calculus and \nfractional differential equations\n\\cite{OS,SKM,MR,Podlubny,KST,K,N}.\nDerivatives and integrals of noninteger order and \nfractional integro-differential equations \nhave found many applications in recent studies in physics\n(see, e.g., \\cite{Zaslavsky1,GM,WBG,H} and \\cite{Zaslavsky2,MS,MK1,MK2}).\n\nIn quantum mechanics, \nobservables are given by self-adjoint operators.\nThe dynamical description of a quantum system is given by superoperators.\nA superoperator is a map that assigns one operator some other operator.\n\nThe motion of a system is naturally described in terms of the \ninfinitesimal change of the system.\nThe equation for a quantum observable is called the Heisenberg equation. \nFor Hamiltonian quantum systems, the infinitesimal superoperator \nis defined by some form of derivation.\nA derivation is a linear map ${\\cal L}$ \nthat satisfies the Leibnitz rule\n${\\cal L}(AB)=({\\cal L}A)B+ A({\\cal L}B)$ for any operators $A$ and $B$.\nA fractional derivative can be defined as the fractional power\nof the derivative (see, e.g., \\cite{IJM}).\nIt is known that the infinitesimal generator ${\\cal L}=(1\/i\\hbar)[H, \\ . \\ ]$, \nwhich is used for Hamiltonian systems, is a derivation of quantum observables.\nIn \\cite{Heis}, we regarded a fractional power ${\\cal L}^{\\alpha}$ \nof the derivative operator ${\\cal L}=(1\/i\\hbar)[H, \\ . \\ ]$\nas a fractional derivative on a set of observables. \nAs a result, we obtain a fractional generalization of the Heisenberg equation,\nwhich allows generalizing the notion of Hamiltonian quantum systems. \nWe note that a fractional generalization of classical Hamiltonian systems \nwas suggested in \\cite{FracHam} (also see \\cite{JPA2006}).\nIn the general case, quantum systems are non-Hamiltonian\nand ${\\cal L}$ is not a derivation operator.\nFor a wide class of quantum systems, the infinitesimal generator ${\\cal L}$ is \ncompletely dissipative \\cite{Kossakowski,Dav,IngKos,kn3}.\nTherefore, it is interesting to consider a fractional generalization\nof the equation of motion for non-Hamiltonian quantum systems using\na fractional power of a completely dissipative superoperator.\n\nThe most general change of state of a non-Hamiltonian quantum system\nis a quantum operation \\cite{Kr1,Kr2,Kr3,Kr4,Schu,JPA}.\nA quantum operation for a quantum system can be described starting\nfrom a unitary evolution of some closed Hamiltonian system if\nthe quantum system is a part of the closed system \\cite{ALV,Weiss}.\nBut situations can arise where it is difficult or impossible \nto find a Hamiltonian system that includes the given quantum system.\nAs a result, the theory of non-Hamiltonian quantum systems \ncan be considered a fundamental generalization \nof the quantum mechanics of Hamiltonian systems \\cite{Kossakowski,Dav,IngKos,kn3}.\nThe quantum operations that describe the dynamics of non-Hamiltonian\nsystems can be regarded as real completely positive\ntrace-preserving superoperators on some operator space.\nThese superoperators form a completely positive semigroup.\nThe infinitesimal generator of this semigroup is completely dissipative. \nThe problem of non-Hamiltonian dynamics \nis to obtain an explicit form for the infinitesimal generator, \nwhich is in turn connected with the problem of determining \nthe most general explicit form of this superoperator.\nThis problem was investigated in \\cite{K1,K2,Lind1}.  \nHere, we consider superoperators that are fractional powers of \ncompletely dissipative superoperators.\nWe prove that the suggested superoperators are infinitesimal generators \nof completely positive semigroups.\nThe quantum Markovian equations with a completely dissipative superoperator\nare the most general form of the Markovian master equation \ndescribing the nonunitary evolution of a density operator\nthat is trace preserving and completely positive. \nWe consider a fractional generalization of the quantum Markovian equation,\nwhich is solved for the harmonic oscillator with friction.\nWe can assume that other solutions and properties described in\n\\cite{Lind2,SS,ISSSS,ISS,N0,N1,N2,N3,TN2} \ncan also be considered for fractional generalizations of the quantum Markovian equation\nand the Gorini-Kossakowski-Sudarshan equation \\cite{K1,K2}.\n\nA fractional power of infinitesimal generator can be considered \na parameter describing a measure of \"screening\" of the environment. \nUsing the interaction representation of the quantum Markovian equation,\nwe consider a fractional power $\\alpha$ \nof the non-Hamiltonian part of the infinitesimal generator.\nWe obtain the Heisenberg equation for Hamiltonian systems\nin the limit as $\\alpha \\rightarrow 0$. \nIn the case $\\alpha=1$, we have the usual quantum Markovian equation.\nAs a result, we can distinguish the following cases: \n(1) absence of the environmental influence ($\\alpha=0$), \n(2) complete environmental influence ($\\alpha=1$), and\n(3) powerlike screening of the environmental influence ($0<\\alpha<1$). \nThe physical interpretation of the fractional quantum Markovian equation \ncan be connected with an existence of \na powerlike \"screening\" of the environmental influence.\nQuantum computations by quantum operations with mixed states \n(see, e.g., \\cite{JPA}) can be controlled by this parameter.\nWe assume that there exist stationary states of open quantum systems \n\\cite{Dav1,Lind2,Spohn,Spohn2,AH,ISS,TN2} \nthat depend on the fractional parameter.\nWe note that it is possible to consider quantum dynamics with \na low fractionality by a generalization of the\nmethod proposed in \\cite{TZ2} (also see \\cite{TP,TG}).\n\n\nIn Section 2, we briefly review of superoperators\non an operator Hilbert space and quantum operations and introduce the notation.\nIn Section 3, we consider the fractional power of a superoperator.\nIn Section 4, we suggest a fractional generalization of the quantum Markovian equation.\nIn Section 5, we describe the properties of the fractional semigroup.\nIn Sections 6 and 7, we solve the fractional equations \nfor the quantum harmonic oscillator with and without friction.\n\n\n\n\n\n\\section{Superoperator and quantum operations}\n\n\n\n$ \\quad \\ $\nQuantum theories essentially consist of two structures: \na kinematic structure describing the initial  \nstates and observables of the system,\nand a dynamical structure describing the change of \nthese states and observables with time.\nIn quantum mechanics,  \nthe states and observables can be given by operators.\nThe dynamical description of the quantum system is given by \na superoperator, which is\na map from a set of operators into itself.\n\nLet ${\\cal M}$ be an operator space. \nWe let ${\\cal M}^{*}$ denote the space dual to ${\\cal M}$.\nHence, ${\\cal M}^{*}$ is the set of all \nlinear functionals on ${\\cal M}$. \nThe classic denotations for an element of ${\\cal M}$ \nare $|B)$ and $B$.\nThe symbols $(A|$ and $\\omega$ denote the elements of ${\\cal M}^{*}$. \nBy the Riesz-Frechet theorem, any linear continuous functional $\\omega$\non an operator Hilbert space ${\\cal M}$ has the form\n$\\omega(B)=(A|B)$ for all $B \\in {\\cal M}$,\nwhere $|A)$ is an element in ${\\cal M}$.\nTherefore, the element $A$ can be considered not \nonly an element $|A)$ of ${\\cal M}$, but also \nan element $(A|$ of the dual space ${\\cal M}^{*}$. \nThe symbol $(A|B)$ for a value of \nthe functional $(A|$ on the operator $|B)$ \nis the graphic combination of the symbols $(A|$ and $|B)$.  \\\\\n\n\n\\noindent {\\large Definition 1.} \n{\\it A linear superoperator is a map ${\\cal L}$ \nfrom an operator space ${\\cal M}$ into itself such that \nthe relation\n\\[ {\\cal L} (aA+bB) = a{\\cal L} (A) + b{\\cal L} (B) \\]\nis satisfied for all $A, B \\in D ({\\cal L}) \\subset {\\cal M} $, \nwhere $D({\\cal L})$ is the domain of ${\\cal L}$ \nand $a,b \\in \\mathbb{C}$. } \\\\\n\nA superoperator ${\\cal L}$ assigns each operator $A \\in D ({\\cal L})$ \nthe operator ${\\cal L}(A)$. \\\\\n\n\\noindent {\\large Definition 2.} \n{\\it  Let ${\\cal L}$ be a superoperator on ${\\cal M}$.\nAn adjoint superoperator of ${\\cal L}$ \nis a superoperator $\\Lambda=\\bar {\\cal L}$ on ${\\cal M}^{*}$ \nsuch that \n\\begin{equation} \\label{Lam}\n(\\Lambda (A) |B) = (A | {\\cal L} (B)) \\end{equation}\nfor all $B \\in D ({\\cal L}) \\subset {\\cal M}$ \nand $A \\in D(\\Lambda) \\subset {\\cal M}^{*}$.} \\\\\n\nLet ${\\cal M}$ be an operator Hilbert space and\n${\\cal L}$ be a superoperator on ${\\cal M}$. \nThen $(A|B)=Tr[A^{\\dagger} B]$, and equation (\\ref{Lam}) becomes\n\\[ Tr [(\\Lambda (A))^{\\dagger} B] =Tr [A^{\\dagger} {\\cal L} (B)] . \\]\n\nIf ${\\cal M}$ is an operator Hilbert space, \nthen by the Riesz-Frechet theorem,  \n${\\cal M}$ and ${\\cal M}^{*}$ are isomorphic, and\nwe can define the self-adjoint superoperators. \\\\\n\n\\noindent\n{\\large Definition 3.} \n{\\it  A self-adjoint superoperator is a superoperator ${\\cal L}$ \non a Hilbert operator space ${\\cal M}$ such that\n$({\\cal L} (A) |B) = (A | {\\cal L} (B))$ for all \n$A, B \\in D ({\\cal L}) \\subset {\\cal M}$\nand $D ({\\cal L}) =D (\\bar {\\cal L})$.} \\\\\n\n\nLet ${\\cal M}$ be a normed operator space.\nThe superoperator ${\\cal L}$ is said to be called bounded if\n$\\|{\\cal L}(A)\\|_{\\cal M} \\le c \\|A\\|_{\\cal M}$ \nfor some constant $c$ and all $A \\in {\\cal M}$.\nThe value \n\\[ \\|{\\cal L}\\|=\\sup_{A \\ne 0} \n\\frac{\\|{\\cal L}(A)\\|_{\\cal M}}{\\|A\\|_{\\cal M}}  \\]\nis called the norm of the superoperator ${\\cal L}$.\nIf ${\\cal M}$ is a normed space and \n${\\cal L}$ is a bounded superoperator, then \n$\\| \\bar {\\cal L} \\| = \\| {\\cal L} \\|$. \n\nIn quantum theory, the class of \nreal superoperators is the most important. \\\\\n\n\\noindent {\\large Definition 4.} \n{\\it  Let ${\\cal M}$ be an operator space and \n$A^{\\dagger}$ be an adjoint operator of $A\\in {\\cal M}$.\nA  real superoperator is a superoperator \n${\\cal L}$ on ${\\cal M}$ such that\n\\[ [{\\cal L} (A)]^{\\dagger} = {\\cal L} (A^{\\dagger}) \\] \nfor all $A \\in D ({\\cal L}) \\subset {\\cal M}$ \nand $A^{\\dagger}\\in D ({\\cal L})$.} \\\\\n\nIf ${\\cal L}$ is a real superoperator, then $\\Lambda=\\bar {\\cal L}$ is real.\nIf ${\\cal L}$ is a real superoperator\nand $A$ is a self-adjoint operator $A^{\\dagger}=A \\in D({\\cal L})$, \nthen the operator $B={\\cal L}(A)$ is self-adjoint.\nThen superoperators from a set of quantum observables ${\\cal M}$ \ninto itself should be real. \nAll possible dynamics of quantum systems\nmust be described by a set of real superoperators.  \\\\\n\n\\noindent {\\large Definition 5.} \n{\\it A nonnegative superoperator is a map  \n${\\cal L}$ from ${\\cal M}$ into ${\\cal M}$, such that \n${\\cal L} (A^2) \\ge 0$\nfor all $A^2=A^{\\dagger}A \\in D ({\\cal L}) \\subset {\\cal M}$.\nA positive superoperator is a map \n${\\cal L}$ from ${\\cal M}$ into itself such that\n${\\cal L}$ is nonnegative and ${\\cal L}(A)=0$ \nif and only if $A=0$.} \\\\\n\n\nLet ${\\cal M}$ denote an operator algebra.\nA left superoperator corresponding to $A \\in {\\cal M}$ \nis a superoperator $L_A$ on ${\\cal M}$ such that $L_A C=AC$ \nfor all $C \\in {\\cal M}$. \nWe can think of $L_A$ as meaning left multiplication by $A$. \nA right superoperator corresponding to $A \\in {\\cal M}$ \nis a superoperator $R_A$ on ${\\cal M}$ such that \n$R_A C=CA$ for all $C \\in {\\cal M}$. \n\nThe most general state change of a quantum system\nis a called a { \\it quantum operation} \\cite{Kr1,Kr2,Kr3,Kr4,Schu,JPA}.\nA quantum operation is described by a superoperator $\\hat{\\cal E}$\nthat is a map on a set of density operators.\nIf $\\rho$ is a density operator, then $\\hat{\\cal E}(\\rho)$\nshould also be a density operator.\nAny density operator $\\rho_t=\\rho(t)$ is\na self-adjoint ($\\rho^{\\dagger}_{t}=\\rho_{t}$),\npositive ($\\rho_{t}>0$) operator with unit trace ($ Tr\\rho_{t}=1$).\nTherefore, for a superoperator $\\hat{\\cal E}$\nto be the quantum operation,  \nthe following conditions must be satisfied:\n\\begin{enumerate}\n\\item\nThe superoperator $\\hat{\\cal E}$ is a real superoperator, i.e.\n$\\Bigl(\\hat{\\cal E}(A)\\Bigr)^{\\dagger}=\\hat{\\cal E}(A^{\\dagger})$\nfor all $A$.\nThe real superoperator $\\hat{\\cal E}$ maps the self-adjoint operator\n$\\rho$ to the self-adjoint operator $\\hat{\\cal E}(\\rho)$:\n\\ $(\\hat{\\cal E}(\\rho))^{\\dagger}=\\hat{\\cal E}(\\rho)$.\n\\item\nThe superoperator $\\hat{\\cal E}$ is a positive superoperator,\ni.e. $\\hat{\\cal E}$ maps positive operators to positive operators:\n\\ $\\hat{\\cal E}(A^{2}) >0$ for all $A\\not=0$ or\n$\\hat{\\cal E}(\\rho)\\ge 0$.\n\\item\nThe superoperator $\\hat{\\cal E}$ is a trace-preserving map, i.e.\n$(I|\\hat{\\cal E}|\\rho)=(\\hat{\\cal E}^{\\dagger}(I)|\\rho)=1$\nor $\\hat{\\cal E}^{\\dagger}(I)=I$.\n\\end{enumerate}\n\nMoreover, we assume that the superoperator $\\hat{\\cal E}$\nis not only positive but also completely positive \\cite{Arveson}.\nThe superoperator $\\hat{\\cal E}$ is a {\\it  completely positive}\nmap from an operator space ${\\cal M}$ into itself if \n\\[ \\sum^{n}_{k=1} \\sum^{n}_{l=1} B^{\\dagger}_{k}\n\\hat{\\cal E}(A^{\\dagger}_kA_l)B_l \\ge 0 \\]\nfor all operators $A_k$, $B_k \\in {\\cal M}$ and any integer $n$. \\\\\n\nLet the superoperator $\\hat{\\cal E}$ be a convex linear map\non the set of density operators, i.e.\n\\[ \\hat{\\cal E}\\Bigl(\\sum_{s} \\lambda_{s} \\rho_{s}\\Bigr)=\n\\sum_{s} \\lambda_{s} \\hat{\\cal E}(\\rho_{s}), \\]\nwhere $0<\\lambda_{s}<1$ for all $s$, and $\\sum_{s} \\lambda_{s}=1$.\nAny convex linear map of density operators\ncan be uniquely extended to a {\\it  linear} map on self-adjoint operators.\nWe note that any linear completely positive superoperator can be represented by\n\\[ \\hat{\\cal E}=\\sum^{m}_{k=1} \\hat L_{A_{k}} \\hat R_{A^{\\dagger}_{k}}: \\quad\n\\hat{\\cal E}(\\rho)=\\sum^{m}_{k=1} A_k \\rho A^{\\dagger}_k. \\]\nIf this superoperator is trace-preserving, then\n\\[ \\sum^{m}_{k=1} A^{\\dagger}_{k} A_{k}=I. \\]\n\nBecause all processes occur in time, it is natural to consider \nquantum operations $\\hat {\\cal E}(t,t_{0})$\nthat depend on time. \nLet the linear superoperators $\\hat {\\cal E}(t,t_{0})$ form a completely\npositive quantum semigroup \\cite{AL} such that \n\\begin{equation} \\label{qo1}\n\\frac{d}{dt}\\hat{\\cal E}(t,t_{0})= \\hat\\Lambda_t \\hat{\\cal E}(t,t_{0}), \\end{equation}\nwhere $\\hat \\Lambda_t$ is an infinitesimal generator of \nthe semigroup \\cite{Lind1,AL,kn3}. \nThe evolution of a density operator $\\rho$ is described by\n\\[ \\hat{\\cal E}(t,t_{0}) \\rho(t_0)= \\rho(t) . \\]\nWe consider quantum operations $\\hat{\\cal E}(t,t_0)$ \nwith an infinitesimal generator $\\hat \\Lambda$ such that \nthe adjoint superoperator $ {\\cal L}$ is completely dissipative, i.e.\n\\[ {\\cal L} (A_{k}A_{l})-\n{\\cal L}(A_{k}) A_{l}-A_{k} {\\cal L}(A_{l}) \\ge 0, \\]\nfor all $A_1,...,A_n \\in D({\\cal L})$ such that $A_kA_l \\in D({\\cal L})$.\nThe superoperator ${\\cal L}$ describes the dynamics of observables \nof a non-Hamiltonian quantum system.\nThe completely dissipative superoperators are infinitesimal generators \nof completely positive semigroups  \n$\\{\\Phi_t | \\ t>0\\}$ that are adjoint to $\\{ \\hat{\\cal E}_t | \\ t>0\\}$, \nwhere $\\hat{\\cal E}_t= \\hat{\\cal E}(t,0)$. \n\n\n\n\n\\section{Fractional power of a superoperator}\n\n\n$ \\quad \\ $ \nLet ${\\cal L}$ be a closed linear superoperator\nwith an everywhere dense domain $D({\\cal L})$\nand a resolvent $R(z,{\\cal L})$ on the negative semiaxis, and \nsatisfy the condition\n\\begin{equation} \\label{Rcond}\n\\| R(-z,{\\cal L}) \\| \\le M \/ z, \\quad (z>0, M>0) . \\end{equation}\nWe note that\n\\[ R(-z,{\\cal L})= (zL_I+{\\cal L})^{-1} . \\]\nThe superoperator\n\\begin{equation} \\label{LaA1}\n{\\cal L}^{\\alpha}=\\frac{\\sin \\pi \\alpha }{\\pi} \n\\int^{\\infty}_0 dz\\, z^{\\alpha-1} R(-z,{\\cal L}) \\, {\\cal L}\n\\end{equation}\nis defined on $D({\\cal L})$ for $0< \\alpha <1$ and \nis called a fractional power of the superoperator ${\\cal L}$ \\cite{HP,Yosida}.\nWe note that the superoperator ${\\cal L}^{\\alpha}$ allows a closure.\nIf a closed superoperator ${\\cal L}$ satisfies condition (\\ref{Rcond}), \nthen ${\\cal L}^{\\alpha} {\\cal L}^{\\beta}={\\cal L}^{\\alpha+\\beta}$ \nfor $\\alpha, \\beta>0$, and $\\alpha+\\beta<1$. \n\nLet ${\\cal L}$ be a closed generating superoperator of the semigroup \n$\\{\\Phi_{t} | \\ t \\ge 0\\}$.\nThen the fractional power ${\\cal L}^{\\alpha}$ of ${\\cal L}$ is given by\n\\[ {\\cal L}^{\\alpha}= \n\\frac{1}{\\Gamma(-\\alpha)} \\int^{\\infty}_0 dz\\, z^{-\\alpha-1} (\\Phi_z-L_I) , \\]\nwhich is called the {\\it Balakrishnan formula}.\n\nThe resolvent for the superoperator ${\\cal L}^{\\alpha}$ \ncan be found by the equation \n\\[ R(-z,{\\cal L}^{\\alpha})=\n(zL_I+ {\\cal L}^{\\alpha})^{-1}= \\]\n\\[ =\\frac{\\sin \\pi \\alpha }{\\pi} \\int^{\\infty}_0 dx\\, \n\\frac{x^{\\alpha}}{z^2+2 zx^{\\alpha} \\cos \\pi \\alpha +x^{2 \\alpha} } \n\\, R(-x,{\\cal L}) , \\]\ncalled {\\it Kato's formula}. \nIt follows from this formula that the inequality\n\\[ \\| R(-z, {\\cal L}^{\\alpha}) \\| \\le M \/ z, \\quad (z>0) ,  \\]\nis satisfied with the constant $M$ in inequality (\\ref{Rcond})\nfor the superoperator ${\\cal L}$. \nIt follows from the inequality\n\\[ \\|z R(-z,{\\cal L}) \\|=\\|z(zL_I+{\\cal L})^{-1}\\| \\le M \\]\nfor all $z>0$ that the superoperator $z(zL_I+{\\cal L})^{-1}$\nis uniformly bounded in every sector of the complex plane \ngiven by the relation $|\\arg z| \\le \\phi$ \nfor $\\phi$ not greater than some number $\\pi - \\psi$, ($0<\\psi<\\pi$).\nThen the superoperator $zR(-z, {\\cal L}^{\\alpha})$\nis uniformly bounded in every sector of the complex plane\nsuch that $|\\arg z| \\le \\phi$  for $\\phi<\\pi - \\alpha \\psi$.\n\nLet ${\\cal L}$ be a closed generating superoperator of \nthe semigroup $\\{\\Phi_{t} | \\ t \\ge 0\\}$.\nThen the superoperators\n\\begin{equation} \\label{BPf}\n\\Phi^{(\\alpha)}_t=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) \\, \\Phi_s , \n\\quad (t>0) , \\end{equation}\nform a semigroup such that\n${\\cal L}^{\\alpha}$ is an infinitesimal generator of $\\Phi^{(\\alpha)}_t$. \nEquation (\\ref{BPf}) is called the Bochner-Phillips formula. \n\nIn equation (\\ref{BPf}), we use the function\n\\begin{equation} \\label{fats}\nf_{\\alpha}(t,s)=\\frac{1}{2\\pi i} \\int^{a+\\i\\infty}_{a-i\\infty}\ndz \\, \\exp (sz-tz^{\\alpha}) ,\n\\end{equation}\nwhere $a,t>0$, $s \\ge 0 $, and $0<\\alpha <1$.\nThe branch of $z^{\\alpha}$ is chosen such that $Re(z^{\\alpha})>0$\nfor $Re(z)>0$.\nThis branch is a one-valued function in the $z$ plane \ncut along the negative real axis.\nThis integral obviously converges by virtue of the factor $\\exp(-tz^{\\alpha})$.\nThe function $f_{\\alpha}(t,s)$ has the following properties: \n\n\\begin{enumerate}\n\\item\nFor all $s>0$, the function $f_{\\alpha}(t,s)$ is nonnegative:\n$f_{\\alpha}(t,s)\\ge 0$. \n\\item\nWe have the identity\n\\[ \\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) =1 . \\]\n\\item\nFor $t>0$ and $x>0$, \n\\[ \\int^{\\infty}_0 ds \\, e^{-sx} \\, f_{\\alpha}(t,s) = e^{-tx^{\\alpha}}. \\]\n\\item\nPassing from the integration contour in (\\ref{fats})\nto the contour crossing of the two rays $r\\, \\exp(-i \\theta)$ and \n$r\\, \\exp(+i \\theta)$, where $r \\in (0,\\infty)$, \nand $\\pi\/2 \\le \\theta \\le \\pi$, we obtain\n\\[\nf_{\\alpha}(t,s)=\\frac{1}{\\pi} \\int^{\\infty}_0  dr \\,\n\\exp (sr \\cos \\theta - t r^{\\alpha} \\cos (\\alpha \\theta)) \\cdot \\]\n\\begin{equation} \\label{freal}\n \\cdot \\sin (sr \\sin \\theta - \nt r^{\\alpha} \\sin (\\alpha \\theta)+ \\theta) . \\end{equation}\n\\item\nIf $\\alpha=1\/2$, then $\\theta = \\pi$, and\n\\[ f_{1\/2}(t,s)=\\frac{1}{\\pi} \\int^{\\infty}_0 dr \\,\ne^{-sr} \\sin ( t \\sqrt{r} )=\n\\frac{t}{2 \\sqrt{\\pi} s^{3\/2}} e^{-t^2\/4s} . \\]\nwhich is a corollary of equation (\\ref{freal}).\n\\end{enumerate}\n\n\n\n\\section{Fractional quantum Markovian equation}\n\n\n$ \\quad \\ $\nThe motion of a systems is naturally described in terms of the\ninfinitesimal change.\nThis change can be described by an infinitesimal generator. \nOne problem of the non-Hamiltonian dynamics \nis to obtain an explicit form \nof the infinitesimal generator. \nFor this, it is necessary to find \nthe most general explicit form of this superoperator.\nThe problem was investigated in \\cite{K1,K2,Lind1}\nfor completely dissipative superoperators.  \nLindblad showed that there exists a one-to-one correspondence \nbetween the completely positive norm-continuous semigroups and \ncompletely dissipative generating superoperators \\cite{Lind1}.\nLindblad's structural theorem gives \nthe most general form of a completely dissipative superoperator. \\\\\n\n\n\\noindent {\\large Theorem 1.} \n{\\it  A generating superoperator ${\\cal L}_V$ of a completely positive \nunity-preserving semigroup $\\{\\Phi_t=\\exp(-t{\\cal L}_V)| \\ t \\ge 0\\}$\non an operator space ${\\cal M}$ can be represented in the form \n\\begin{equation} \\label{s153}\n-{\\cal L}_V (A) =- \\frac{1}{i\\hbar}[H, A] + \n\\frac{1}{2 \\hbar} \\sum^{\\infty}_{k=1} \n\\Bigl(V^{\\dagger}_{k}[A, V_{k}] + [V^{\\dagger}_{k}, A] V_{k} \\Bigr) ,\n\\end{equation}\nwhere $H$, $V_{k}$, $\\sum_k V^{\\dagger}_{k}, V^{\\dagger}_{k} V_{k} \\in {\\cal M}$.} \\\\\n\nWe note that the form of ${\\cal L}_V$ is not uniquely fixed by (\\ref{s153}).  \nIndeed, formula (\\ref{s153}) preserves its form under the changes\n\\[ V_k \\ \\rightarrow \\ V_k+ a_k I , \\quad\nH \\ \\rightarrow \\ H+\\frac{1}{2i\\hbar} \n\\sum^{\\infty}_{k=1} (a^{*}_kV_k-a_kV^{\\dagger}_k) , \\]\nwhere $a_k$ are arbitrary complex numbers. \n\nUsing $A_t=\\Phi_t(A)$, where $\\Phi_t=\\exp(-t{\\cal L}_V)$, we obtain the equation \n\\begin{equation} \\label{LindA1}\n\\frac{d}{dt} A_t=-\\frac{1}{i\\hbar}[H,A_t]+ \n\\frac{1}{2 \\hbar} \\sum^{\\infty}_{k=1} \\Bigl(V^{\\dagger}_k [A_t, V_k] +\n[V^{\\dagger}_k, A_t] V_k \\Bigr) , \\end{equation}\nwhere ${\\cal L}_V$ is defined by (\\ref{s153}). \nThis is called the {\\it quantum Markovian equation} for the observable $A$. \n\nThe Lindblad theorem gives an explicit form of\nthe equations of motion if the following restrictions are satisfied \n(here $\\Lambda_V$ is adjoint to ${\\cal L}_V$): \n\n\\begin{enumerate}\n\\item\n${\\cal L}_V$ and $\\Lambda_V$ are bounded superoperators and \n\\item \n${\\cal L}_V$ and $\\Lambda_V$ are completely dissipative superoperators. \n\\end{enumerate}\n\nDavies extended the Lindblad result \nto a class of quantum dynamical semigroups  \nwith unbounded generating superoperators \\cite{Davies2}. \n\n\n\nWe consider quantum Markovian equation (\\ref{LindA1})\nfor an observable $A_t$.\nWe rewrite this equation in the form\n\\begin{equation} \\label{Lind1a}\n\\frac{d}{dt} A_t= -{\\cal L}_V A_t , \\end{equation}\nwhere ${\\cal L}_V$ denotes the Markovian superoperator\n\\begin{equation} \\label{Lind1b}\n{\\cal L}_V =L^{-}_{H}+ \\frac{i}{2} \\sum^{\\infty}_{k=1} \n\\Bigl(L_{V^{\\dagger}_k} L^{-}_{V_k}-L^{-}_{V^{\\dagger}_k} R_{V_k}  \\Bigr) . \\end{equation}\nHere, we use the superoperators of left multiplication $L_V$ and\nright multiplication $R_V$ determined by the relations $L_VA=VA$ and $R_VA=AV$.\nThe superoperator $L^{-}_H$ is a left Lie multiplication by $A$ such that \n\\begin{equation} \\label{Lminus}\nL^{-}_HA=\\frac{1}{i \\hbar} [H,A] .  \\end{equation}\nIf all operators $V_k$ are equal to zero, then ${\\cal L}_V=L^{-}_H$,\nand equations (\\ref{Lind1a}) and (\\ref{Lind1b}) give \nthe Heisenberg equations for a Hamiltonian system. \nIn the general case, the quantum system is non-Hamiltonian \\cite{kn3}.\n\nWe obtain a fractional generalization\nof the quantum Markovian equation. \nFor this, we define a fractional power for \nthe Markovian superoperator ${\\cal L}_V$ in the form\n\\begin{equation} \\label{LaA3}\n-({\\cal L}_V)^{\\alpha}=\\frac{\\sin \\pi \\alpha }{\\pi} \n\\int^{\\infty}_0 dz \\, \nz^{\\alpha-1} R(-z,{\\cal L}_V) \\, {\\cal L}_V \\quad (0< \\alpha <1) .\n\\end{equation}\nThe superoperator $({\\cal L}_V)^{\\alpha}$ is called\na {\\it fractional power of the Markovian superoperator}.\nWe note that $({\\cal L}_V)^{\\alpha}({\\cal L}_V)^{\\beta}=\n({\\cal L}_V)^{\\alpha+\\beta}$ \nfor $\\alpha, \\beta>0$, and $\\alpha+\\beta<1$. \nAs a result, we obtain the equation\n\\begin{equation} \\label{Lind2} \n\\frac{d}{dt} A_t=-({\\cal L}_V)^{\\alpha} A_t , \\end{equation}\nwhere $t$, $H\/\\hbar$ and $V_k\/ \\sqrt{\\hbar}$ \nare dimensionless variables.\nWe call this is the {\\it fractional quantum Markovian equation}. \n\nIf $V_k=0$, then equation (\\ref{Lind2}) \ngives the fractional Heisenberg equation \\cite{Heis} of the form\n\\begin{equation} \\label{Heis2b} \n\\frac{d}{dt} A_t=-(L^{-}_H)^{\\alpha} A_t . \\end{equation}\nThe superoperator $(L^{-}_H)^{\\alpha}$ is \na {\\it fractional power of the left Lie superoperator} (\\ref{Lminus}). \nWe note that this equation cannot be represented in the form\n\\[ \\frac{d}{dt} A_t=-L^{-}_{H_{new}} A_t=\n\\frac{i}{\\hbar} [H_{new}, A_t] \\]\nwith some operator $H_{new}$.\nTherefore, quantum systems described by (\\ref{Heis2b}) \nare not Hamiltonian systems.\nThese systems are called the {\\it fractional Hamiltonian quantum systems} (FHQS). \nUsual Hamiltonian quantum systems can be considered a special case of FHQS. \nWe note that a fractional generalization of classical Hamiltonian systems \nwas suggested in \\cite{FracHam,JPA2006}.\n\nUsing the operators\n\\[ A_U(t) = U(t) A_t U^{\\dagger}(t) , \\quad W_k(t) = U(t) V_k U^{\\dagger}(t) , \\]\nwhere $U(t)= \\exp \\{ (1\/i\\hbar) H \\}$, \nwe can write the quantum Markovian equation in the form\n\\begin{equation} \\label{Lind-I}\n\\frac{d}{dt} A_U(t)= - \\tilde{\\cal L}_W A_U(t) . \\end{equation}\nThe superoperator\n\\begin{equation} \\label{Lind-W}\n\\tilde{\\cal L}_W =\\frac{i}{2} \\sum^{\\infty}_{k=1} \n\\Bigl(L_{W^{\\dagger}_k} L^{-}_{W_k}-L^{-}_{W^{\\dagger}_k} R_{W_k}  \\Bigr) \\end{equation}\ndescribes the non-Hamiltonian part of the evolution.\nEquation (\\ref{Lind-I}) is the quantum Markovian equation \nin the interaction representation.\nThe fractional generalization of this equation is\n\\begin{equation} \\label{Lind-I2}\n\\frac{d}{dt} A_U(t)= - (\\tilde{\\cal L}_W)^{\\alpha} A_U(t) . \\end{equation}\nEquation (\\ref{Lind-I2}) is the fractional quantum Markovian equation \nin the interaction representation.\nThe parameter $\\alpha$ can be considered \na measure of the influence of the environment.\nFor $\\alpha=1$, we have quantum Markovian equation (\\ref{Lind-I}).\nIn the limit as $\\alpha \\rightarrow 0$, we obtain the Heisenberg equation\nfor the  quantum observable $A_t$ of a Hamiltonian system.\nAs a result, we can consider the physical interpretation of equations \nwith a fractional power of the Markovian superoperator \nan influence of the environment.\nThe following cases can be considered in quantum theory: \n(1) absence of the environmental influence ($\\alpha=0$), \n(2) complete environmental influence ($\\alpha=1$), and \n(3) powerlike screening of the environmental influence ($0<\\alpha<1$). \nThe physical interpretation of fractional equation (\\ref{Lind-I2})\ncan be connected with an existence of a powerlike screening \nof the environmental influence on the system.\n\n\n\n\\section{Fractional semigroup}\n\n$ \\quad \\ $\nIf we consider the Cauchy problem for equation (\\ref{Lind1a}) \nwith the initial condition given at the time $t=0$ by $A_0$,\nthen its solution can be written in the form\n$A_t=\\Phi_t A_0$. \nThe one-parameter superoperators $\\Phi_t$, $t \\ge 0$ \nhave the properties\n\\[ \\Phi_t \\Phi_s=\\Phi_{t+s}, \\quad (t,s >0) , \\quad \\Phi_0=L_I . \\] \nAs a result, the superoperators $\\Phi_t$ form a semigroup, \nand the superoperator ${\\cal L}_V$ is a generating superoperator \nof the semigroup $\\{\\Phi_{t} | \\ t\\ge 0\\}$.\n\nWe consider the Cauchy problem for \nfractional quantum Markovian equation (\\ref{Lind2}) \nwith the initial condition given by $A_0$.\nThen its solution can be presented in the form\n\\[ A_t(\\alpha)=\\Phi^{(\\alpha)}_t A_0, \\]\nwhere the superoperators $\\Phi^{(\\alpha)}_t$, $t>0$,  \nform a semigroup, \nwhich we call the {\\it fractional semigroup}.\nThe superoperator $-({\\cal L}_V)^{\\alpha}$ is a \ngenerating superoperator of \nthe semigroup $\\{\\Phi^{(\\alpha)}_t| \\ t\\ge 0\\}$.\nWe consider some properties of the fractional semigroups\n$\\{\\Phi^{(\\alpha)}_t | \\ t>0\\}$. \n\nThe superoperators $\\Phi^{(\\alpha)}_t$ can be constructed\nin terms of $\\Phi_t$ by Bochner-Phillips formula (\\ref{BPf}), \nwhere $f_{\\alpha}(t,s)$ is defined in (\\ref{fats}). \nIf $A_t$ is a solution of quantum Markovian equation (\\ref{Lind1a}),\nthen formula (\\ref{BPf}) gives the solution\n\\[ A_t(\\alpha)=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) A_s , \\quad (t>0) \\]\nof fractional quantum Markovian equation (\\ref{Lind2}). \n\nA linear superoperator $\\Phi^{(\\alpha)}_t$ is completely positive if \n\\[ \\sum_{i,j} B_i \\Phi^{(\\alpha)}_t(A^{\\dagger}_i A_j) B_j \\ge 0  \\]\nfor any $A_i, B_i \\in {\\cal M}$. \nThe following theorem states that the fractional semigroup\n$\\{\\Phi^{(\\alpha)}_t| \\ t>0\\}$ is completely positive. \\\\\n\n\\noindent\n{\\large  Theorem 2.}\n{\\it  If $\\{\\Phi_{t} | \\ t>0\\}$ is a completely positive semigroup of \nsuperoperator $\\Phi_t$ on ${\\cal M}$, \nthen the fractional superoperators $\\Phi^{(\\alpha)}_t$ \nform a completely positive semigroup $\\{ \\Phi^{(\\alpha)}_t | \\ t>0\\}$ .} \\\\\n\n\\noindent {\\it  Proof.}\nBochner-Phillips formula (\\ref{BPf}) gives \n\\[ \\sum_{i,j} B_i \\Phi^{(\\alpha)}_t(A^{\\dagger}_i A_j) B_j =\n\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) \n\\sum_{i,j} B_i \\Phi_s(A^{\\dagger}_i A_j) B_j   \\]\nfor $t>0$.\nUsing \n\\[ \\sum_{i,j} B_i \\Phi_s(A^{\\dagger}_i A_j) B_j \\ge 0 , \\quad\nf_{\\alpha}(t,s)\\ge 0  \\quad (s>0) , \\]\nwe obtain\n\\[ \\sum_{i,j} B_i \\Phi^{(\\alpha)}_t(A^{\\dagger}_i A_j) B_j \\ge 0 . \\ \\ \\ \\Box \\]\n\n\n\\noindent\n{\\large  Corollary.}\n{\\it  If $\\Phi_t$, $t>0$, is a nonnegative one-parameter superoperator, i.e.,\n$\\Phi_t (A) \\ge 0$ for $A \\ge 0$, then\nthe superoperator $\\Phi^{(\\alpha)}_t$ is nonnegative, i.e., \n$\\Phi^{(\\alpha)}_t (A) \\ge 0 $ for $A\\ge 0$. } \\\\\n\nUsing the Bochner-Phillips formula and the property\n$f_{\\alpha}(t,s) \\ge 0$,  $s>0$, \nwe can easily prove that the superoperator $\\Phi^{(\\alpha)}_t$\nis nonnegative, \nif $\\Phi_t$, $t>0$ is a nonnegative one-parameter superoperator. \nThis corollary can also be proved by using \n$B_1=I$, $A_1=A$, and $A_i=B_i=0$ ($i=2,...$) \nin the proof of the theorem. \\\\\n\nIn quantum theory, the class of real superoperators is the most important. \nLet $A^{\\dagger} \\in {\\cal M}^{*}$ be adjoint to $A\\in {\\cal M}$.\nA {\\it  real superoperator} is a superoperator \n$\\Phi_t$ on ${\\cal M}$, such that\n$(\\Phi_t A)^{\\dagger} = \\Phi_t (A^{\\dagger})$\nfor all $A \\in D (\\Phi_t) \\subset {\\cal M}$. \nA quantum observable is a self-adjoint operator. \nIf $\\Phi_t$ is a real superoperator\nand $A$ is a self-adjoint operator, $A^{\\dagger}=A$, \nthen the operator $A_t=\\Phi_t A$ is self-adjoint, i.e., \n$(\\Phi_t A)^{\\dagger}=\\Phi_t A$. \nLet ${\\cal M}$ be a set of quantum observables. \nThen superoperators on ${\\cal M}$ into ${\\cal M}$ must be real because \nquantum dynamics, i.e., \ntemporal evolutions of quantum observables,\nmust be described by real superoperators. \\\\\n\n\\noindent\n{\\large  Theorem 3.}\n{\\it  If $\\Phi_t$ is a real superoperator, \nthen the superoperator $\\Phi^{(\\alpha)}_t$ is also real. } \\\\\n\n\\noindent {\\it Proof.}\nThe Bochner-Phillips formula gives\n\\[ (\\Phi^{(\\alpha)}_t A)^{\\dagger} =\n\\int^{\\infty}_0 ds f^{*}_{\\alpha}(t,s) \\, (\\Phi_s A)^{\\dagger} , \n\\quad (t>0) . \\]\nUsing (\\ref{freal}), we can easily see \nthat $f^{*}_{\\alpha}(t,s)=f_{\\alpha}(t,s)$ \nis a real-valued function.\nThen $(\\Phi_t A)^{\\dagger}=\\Phi_t A^{\\dagger}$ leads to \n$(\\Phi^{(\\alpha)}_t A)^{\\dagger} = \\Phi^{(\\alpha)}_t (A^{\\dagger})$\nfor all $A \\in D (\\Phi^{(\\alpha)}_t) \\subset {\\cal M}$. $\\ \\ \\ \\Box$ \\\\\n\n\n \nIf $\\Phi_t$ is a superoperator on a Hilbert operator space ${\\cal M}$, \nthen an {\\it  adjoint superoperator} of $\\Phi_t$ \nis a superoperator $\\hat{\\cal E}_t$ on ${\\cal M}^{*}$ such that \n\\begin{equation} \\label{TrPhi} (\\hat{\\cal E}_t (A) |B) = (A | \\Phi_t (B)) \\end{equation}\nfor all $B \\in D (\\Phi_t) \\subset {\\cal M}$ \nand some $A \\in {\\cal M}^{*}$. \nUsing the Bochner-Phillips formula, we obtain\nthe following theorem. \\\\\n\n\\noindent\n{\\large  Theorem 4.}\n{\\it  If $\\hat{\\cal E}_t$ is an adjoint superoperator of $\\Phi_t$, \nthen the superoperator\n\\[ \\hat{\\cal E}^{(\\alpha)}_t =\n\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) \\ \\hat{\\cal E}_s , \n\\quad (t>0) , \\]\nis an adjoint superoperator of $\\Phi^{(\\alpha)}_t$.} \\\\\n\n\\noindent {\\it Proof.}\nLet $\\hat{\\cal E}_t$ be adjoint to $\\Phi_t$, i.e.\nequation (\\ref{TrPhi}) is satisfied. \nThen\n\\[ ( \\hat{\\cal E}^{(\\alpha)}_t A| B) =\n\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) (\\hat{\\cal E}_s A|B) = \\]\n\\[ =\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s) \n(A| \\Phi_s B) = (A| \\Phi^{(\\alpha)}_t B) . \\ \\ \\ \\Box \\]\n\n\nIt is known that $\\hat{\\cal E}_t$ is a real superoperator if\n$\\Phi_t$ is real. \nAnalogously, if $\\Phi^{(\\alpha)}_t$ is a real superoperator, then\n$\\hat{\\cal E}^{(\\alpha)}_t$ is real. \n\n\nLet $\\{ \\hat{\\cal E}_t | t>0\\}$ be a completely positive semigroup  \nsuch that the density operator $\\rho_t=\\hat{\\cal E}_t \\rho_0$\nis described by\n\\begin{equation} \\label{Lindrho}\n\\frac{d}{dt} \\rho_t=- \\hat \\Lambda_V \\rho_t , \\end{equation}\nwhere $\\hat \\Lambda_V$ is adjoint to \nthe Markovian superoperator ${\\cal L}_V$.\nThe superoperator $\\hat \\Lambda_V$ can be represented in the form\n\\[ \\hat \\Lambda_V \\rho_t=-\\frac{1}{i\\hbar}[H,\\rho_t]+\n\\frac{1}{\\hbar} \\sum^{\\infty}_{k=1} \\Bigl(V_k \\rho_t V^{\\dagger}_k - \n(\\rho_t V^{\\dagger}_k V_k+V^{\\dagger}_k V_k \\rho_t) \\Bigr) . \\]\nWe note that equation (\\ref{Lindrho}) with $V_k=0$\ngives the von Neumann equation\n\\[ \\frac{d}{dt} \\rho_t=\\frac{1}{i\\hbar}[H,\\rho_t] .  \\]\nThe semigroup $\\{ \\hat{\\cal E}^{(\\alpha)}_t | \\ t>0\\}$\ndescribes the evolution of the density operator\n$\\rho_t(\\alpha)=\\hat{\\cal E}^{(\\alpha)}_t \\rho_0$\nby the fractional equation\n\\[ \\frac{d}{dt} \\rho_t(\\alpha)=- (\\hat \\Lambda_V)^{\\alpha} \\rho_t(\\alpha) . \\]\nThis is the {\\it  fractional quantum Markovian equation for the density operator}.\nFor $V_k=0$, this equation gives \n\\[ \\frac{d}{dt} \\rho_t=-(-L^{-}_H)^{\\alpha} \\rho_t . \\] \nwhich can be called the {\\it  fractional von Neumann equation}. \n\n\n\n\\section{Fractional equation for the harmonic oscillator}\n\n$ \\quad \\ $ \nWe consider a quantum harmonic oscillator such that\n\\begin{equation} \\label{oscHam}\nH=\\frac{1}{2m} P^2 +\\frac{m\\omega^2}{2} Q^2, \\quad V_k=0 , \\end{equation}\nwhere $t$ and $P$ are dimensionless variables.\nThen equation (\\ref{Lind2}) (also see (\\ref{Heis2b})) describes a harmonic oscillator.\nFor $A=Q$ and $A=P$, equation (\\ref{Lind2}) for $\\alpha=1$ gives\n\\[ \\frac{d}{dt} Q_t=\\frac{1}{m} P_t, \\quad \n\\frac{d}{dt} P_t=-m \\omega^2 Q_t . \\]\nThe well-known solutions of these equations are\n\\[ Q_t=Q_0 \\cos (\\omega t) +\\frac{1}{m \\omega} P_0 \\sin (\\omega t) , \\]\n\\begin{equation} \\label{osc1}\nP_t=P_0 \\cos (\\omega t) - m \\omega Q_0 \\sin (\\omega t) . \\end{equation}\nUsing these solutions and the Bochner-Phillips formula, \nwe can obtain solutions of the fractional equations\n\\begin{equation} \\label{ex2}\n\\frac{d}{dt} Q_t=- (L^{-}_{H})^{\\alpha} Q_t , \\quad \n\\frac{d}{dt} P_t=- (L^{-}_{H})^{\\alpha} P_t , \\end{equation}\nwhere $H$ is given by (\\ref{oscHam}). \nThe solutions of fractional equations (\\ref{ex2}) have the forms\n\\[ Q_t(\\alpha)=\\Phi^{(\\alpha)}_t Q_0=\n\\int^{\\infty}_0 ds f_{\\alpha}(t,s) Q_s , \\]\n\\begin{equation} \\label{osc2}\nP_t(\\alpha)=\\Phi^{(\\alpha)}_t P_0=\n\\int^{\\infty}_0 ds f_{\\alpha}(t,s) P_s . \\end{equation}\nSubstituting (\\ref{osc1}) in (\\ref{osc2}) gives \\cite{Heis} the equations\n\\begin{equation} \\label{Hsol2a}\nQ_t=Q_0 C_{\\alpha}(t) +\\frac{1}{m \\omega} P_0 S_{\\alpha}(t) , \\quad\nP_t=P_0 C_{\\alpha}(t) - m \\omega Q_0 S_{\\alpha}(t) , \\end{equation}\nwhere\n\\[ C_{\\alpha}(t)=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s)\\, \\cos(\\omega s) , \n\\quad \nS_{\\alpha}(t)=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s)\\, \\sin(\\omega s) . \\]\nEquations (\\ref{Hsol2a}) describe solutions of \nfractional equations (\\ref{ex2}) for the quantum harmonic oscillator. \nFor $\\alpha=1\/2$, we have \n\\[ C_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\int^{\\infty}_0 ds \\, \n\\frac{\\cos(\\omega s)}{s^{3\/2}} \\, e^{-t^2\/4s} , \\]\n\\[ S_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\int^{\\infty}_0 ds \\, \n\\frac{\\sin(\\omega s)}{s^{3\/2}} \\, e^{-t^2\/4s} . \\]\nThese functions can be represented in terms of the Macdonald function\n(see Sec. 2.5.37.1 in \\cite{Prudnikov}), \nwhich is also called the modified Bessel function of the third kind.\n\nIt is easy to obtain the expectations \n\\[ <Q_t>=x_0 C_{\\alpha}(t) +\\frac{1}{m \\omega} p_0 S_{\\alpha}(t) , \\]\n\\[ <P_t>=p_0 C_{\\alpha}(t) - m \\omega x_0 S_{\\alpha}(t) , \\]\nand the dispersions \n\\[ D_t(Q)=\\frac{a^2}{2} C^2_{\\alpha}(t) + \n\\frac{\\hbar^2}{2a^2 m^2 \\omega^2} S^2_{\\alpha}(t) , \\]\n\\[ D_t(P)=\\frac{\\hbar^2}{2a^2}\nC^2_{\\alpha}(t) +\\frac{a^2 m^2 \\omega^2}{2} S^2_{\\alpha}(t) . \\]\nHere, we use the coordinate representation and the pure state\n\\begin{equation} \\label{Psi0}\n\\Psi(x)=<x|\\Psi>= \\frac{1}{\\sqrt{a\\sqrt{\\pi}}} \\exp\\Bigl\\{-\\frac{(x-x_0)^2}{2a} +\n\\frac{i}{\\hbar} p_0x \\Bigr\\} . \\end{equation}\nThe expectation and dispersion are defined by usual.\n\n\n\\section{Fractional quantum Markovian equation for the oscillator with friction}\n\n$ \\quad \\ $ \nWe consider the fractional quantum Markovian equation with $V_k \\ne 0$.\nThe basic assumption is that the general form of a \nbounded completely dissipative superoperator \ngiven by the quantum Markovian equation\nalso holds for an unbounded completely \ndissipative superoperator ${\\cal L}_V$.\nAnother condition imposed on the operators $H$ and $V_k$\nis that they are functions of the operators $Q$ and $P$ \nsuch that the obtained model is exactly solvable \\cite{Lind2,SS}\n(also see \\cite{ISSSS,ISS}).\nWe assume that $V_k=V_k(Q,P)$ are \nthe first-degree polynomials in $Q$ and $P$,\nand that $H=H(Q,P)$ is a second-degree polynomial in $Q$ and $P$.\nThese assumptions are analogous to those used in classical\ndynamics when friction forces proportional to the velocity are considered.\nThen $V_k$ and $H$ are given in the forms:\n\\begin{equation} \\label{v-h} \nH = \\frac{1}{2m} P^2 + \\frac{m \\omega^2}{2} Q^2 + \\frac{\\mu}{2} (PQ+QP)  ,\n\\quad V_k=a_kP+b_kQ , \n\\end{equation} \nwhere $a_k$ and $b_k $, $k=1,2$, are complex numbers.\nIt is easy to obtain\n\\[ {\\cal L}_V Q = \\frac{1}{m} P + \\mu Q - \\lambda Q , \\]\n\\[ {\\cal L}_V P = -m \\omega^2 Q - \\mu P - \\lambda P , \\] \nwhere\n\\[ \\lambda =Im\\Bigl (\\sum^{n=2}_{k=1} a_kb^*_k\\Bigr) =\n-Im \\Bigl(\\sum^{n=2}_{k=1} a^*_k b_k\\Bigr) . \\]\nUsing the matrices \n\\[ A = \n\\left (\\begin{array}{c} Q \\\\ P \\\\\n\\end{array}\n\\right)  , \\quad \nM = \\left (\\begin{array}{cc} \n\\mu - \\lambda & \\frac{1}{m} \\\\\n-m \\omega^2 & - \\mu - \\lambda\n\\end{array} \\right) , \\]\nwe write the quantum Markovian equation for $A_t$ as\n\\begin{equation} \\label{Lineq} \n\\frac{d}{dt} A_{t} =MA_{t} , \\end{equation}\nwhere ${\\cal L}_V A_t =MA_t$. \nThe solution of (\\ref{Lineq}) is \n\\[ A_{t} = \\Phi_t A_0 = \n\\sum^{\\infty}_{n=0} \\frac{t^n}{n!}{\\cal L}^n_V A_0 =\n\\sum^{\\infty}_{n=0} \\frac{t^n}{n!} M^n A_0 . \\] \nThe matrix $M$ can be represented in the form $M=N^{-1} FN$, \nwhere $F$ is a diagonal matrix. \nLet $\\nu$ be a complex parameter such that\n$\\nu^2 = \\mu^2 - \\omega^2$. Then we have\n\\[ N =\n\\left (\\begin{array}{cc} \nm \\omega^2 & \\mu + \\nu \\\\ \nm \\omega^2 & \\mu - \\nu\n\\end{array} \\right), \\quad\nN^{-1} = \\frac{1}{2m \\omega^2 \\nu}\n\\left (\\begin{array}{cc} \n- (\\mu - \\nu) & \\mu + \\nu \\\\ \nm \\omega^2 & -m \\omega^2\n\\end{array} \\right) , \\]\n\\[ F = \\left (\n\\begin{array}{cc}\n- (\\lambda + \\nu) & 0 \\\\ \n0 & - (\\lambda - \\nu)\n\\end{array} \\right) . \\]\nTaking\n\\[ \\Phi_t = \\sum^{\\infty}_{n=0} \\frac{t^n}{n!} M^n =\nN^{-1} \\left(\\sum^{\\infty}_{n=0} \\frac{t^n}{n!} F^n\\right) N , \\]\ninto account, we obtain the superoperator $\\Phi_t$ in the form\n\\[ \\Phi_t=e^{tM} =N^{-1} e^{tF} N = \\]\n\\[ =e^{-\\lambda t}\n\\left (\\begin{array}{cc} \n\\cosh (\\nu t) + (\\mu\/\\nu) \\sinh (\\nu t) &\n(1\/m \\nu) \\sinh (\\nu t) \\\\ \n- (m \\omega^2\/\\nu) \\sinh (\\nu t) & \n\\cosh(\\nu t) - (\\mu\/\\nu) \\sinh (\\nu t)\n\\end{array} \\right) . \\]\nAs a result, we obtain\n\\[\nQ_t=e^{-\\lambda t}[\\cosh (\\nu t) + \\frac{\\mu}{\\nu} \\sinh (\\nu t)] Q_0 +\n\\frac{1}{m \\nu} e^{-\\lambda t} \\sinh (\\nu t) P_0  , \\]\n\\begin{equation} \\label{LindP1}\nP_t = - \\frac{m \\omega^2}{\\nu} e^{-\\lambda t} \\sinh (\\nu t) Q_0 +\ne^{-\\lambda t}[\\cosh (\\nu t) - \\frac{\\mu}{\\nu} \\sinh (\\nu t)] P_0 . \\end{equation}\n\n\n\nThe fractional quantum Markovian equations for $Q_t$ and $P_t$ are\n\\begin{equation} \\label{Lind3} \n\\frac{d}{dt} Q_t=-({\\cal L}_V)^{\\alpha} Q_t , \\quad\n\\frac{d}{dt} Q_t=-({\\cal L}_V)^{\\alpha} Q_t , \\end{equation}\nwhere $t$ and $V_k\/ \\sqrt{\\hbar}$ are dimensionless variables.\nThe solutions of these fractional equations\nare given by the Bochner-Phillips formula,  \n\\[\nQ_t(\\alpha)=\\Phi^{(\\alpha)}_t Q_0=\n\\int^{\\infty}_0 ds f_{\\alpha}(t,s) Q_s , \n\\quad (t>0) , \\]\n\\begin{equation} \\label{LindP2}\nP_t(\\alpha)=\\Phi^{(\\alpha)}_t P_0=\n\\int^{\\infty}_0 ds f_{\\alpha}(t,s) P_s , \n\\quad (t>0) , \\end{equation}\nwhere $Q_s$ and $P_s$ are given by (\\ref{LindP1}) and \nthe function $f_{\\alpha}(t,s)$ is defined in (\\ref{fats}). \nSubstituting (\\ref{LindP1}) in (\\ref{LindP2}) gives\n\\[\nQ_t(\\alpha)=[Ch_{\\alpha}(t) + \\frac{\\mu}{\\nu} Sh_{\\alpha}(t)] Q_0 +\n\\frac{1}{m \\nu} Ch_{\\alpha}(t) P_0  , \\]\n\\begin{equation} \\label{LindP3}\nP_t(\\alpha) = - \\frac{m \\omega^2}{\\nu} Sh_{\\alpha}(t) Q_0 +\n[Ch_{\\alpha}(t) - \\frac{\\mu}{\\nu} Sh_{\\alpha}(t)] P_0 , \\end{equation}\nwhere\n\\[ Ch_{\\alpha}(t)=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s)\\, \ne^{-\\lambda s} \\cosh (\\nu s) , \\]\n\\[ Sh_{\\alpha}(t)=\\int^{\\infty}_0 ds \\, f_{\\alpha}(t,s)\\, \ne^{-\\lambda s} \\sinh (\\nu s) . \\]\nFor $\\alpha=1\/2$, we have\n\\[ Ch_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\int^{\\infty}_0 ds \\, \n\\frac{\\cosh(\\nu s)}{s^{3\/2}} \\, e^{-t^2\/4s - \\lambda s} , \\]\n\\[ Sh_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\int^{\\infty}_0 ds \\, \n\\frac{\\sinh (\\nu s)}{s^{3\/2}} \\, e^{-t^2\/4s-\\lambda s} . \\]\nThese functions can be represented in terms of the Macdonald function\n(see Sec. 2.4.17.2 in \\cite{Prudnikov}) such that\n\\[ Ch_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\Bigl[\nV(t,\\lambda,-\\nu)+V(t,\\lambda,\\nu) \\Bigr] , \\]\n\\[ Sh_{1\/2}(t)=\\frac{t}{2 \\sqrt{\\pi}} \\Bigl[\nV(t,\\lambda,-\\nu)-V(t,\\lambda,\\nu) \\Bigr] , \\]\nwhere we use the notation\n\\[ V(t,\\lambda,\\nu)=\\Bigl(\\frac{t^2+4\\nu}{4 \\lambda}\\Bigr)^{1\/4} \nK_{-1\/2} \\Bigl( 2 \\sqrt{\\frac{\\lambda(t^2+4\\nu)}{4}} \\Bigr) , \\]\nwhere $Re(t^2)> Re(\\nu)$, $Re(\\lambda)>0$,\nand $K_{\\alpha}(z)$ is the Macdonald function \\cite{OS,SKM}. \n\nAs a result, equations (\\ref{LindP3}) define a solution\nof the fractional quantum Markovian equation for \nthe harmonic oscillator with friction. \n\n\n\\section{Conclusion}\n\nQuantum dynamics can be described by superoperators.\nA map assigning each operator exactly one operator is called a superoperator.\nIt is natural to describe motion in terms of the \ninfinitesimal change of a system.\nThe equation of motion for a quantum observable\nis called the Heisenberg equation. \nFor Hamiltonian quantum systems, the infinitesimal superoperator \nis some form of derivation. \nA linear map ${\\cal L}$ satisfying the Leibnitz rule\n${\\cal L}(AB)=({\\cal L}A)B+ A({\\cal L}B)$ for all operators $A$ and $B$\nis called derivation.\nIt is known that the infinitesimal generator ${\\cal L}=(1\/i\\hbar)[H, \\ . \\ ]$, \nwhich is used for Hamiltonian systems, is a derivation of quantum observables.\nWe can regard a fractional power ${\\cal L}^{\\alpha}$ of the derivative \n${\\cal L}=(1\/i\\hbar)[H, \\ . \\ ]$ as  a fractional derivative on\na set of quantum observables \\cite{Heis}. \nAs a result, we obtain a fractional generalization of \nthe Heisenberg equation \\cite{Heis}, which allows generalizing \nthe notion of Hamiltonian quantum systems. \nIn the general case, quantum systems are non-Hamiltonian\nand ${\\cal L}$ is not a derivation.\nFor a wide class of quantum systems, the infinitesimal generator ${\\cal L}$ is \ncompletely dissipative \\cite{Kossakowski,Dav,IngKos,kn3}.\n\nHere, we consider a fractional generalization of \nthe equation of motion for non-Hamiltonian quantum systems using\na fractional power of a completely dissipative superoperator.\nWe suggested a generalization of the quantum Markovian equation \nfor quantum observables.\nIn this equation, we used a superoperator that is a fractional power of \na completely dissipative superoperator.\nWe proved that the suggested superoperator \nis an infinitesimal generator of a completely positive semigroup and \ndescribed properties of this semigroup.\nWe solved the proposed fractional quantum Markovian equation exactly  \nfor the harmonic oscillator with linear friction.\nA fractional power $\\alpha$ of the quantum Markovian superoperator \ncan be considered a parameter \ndescribing a measure of \"screening\" of the environment.\nWe can separate the cases where $\\alpha=0$,  \nabsence of the environmental influence;\nwhere $\\alpha=1$, complete environmental influence; \nand where $0<\\alpha<1$, a powerlike environmental influence. \nA one-parameter description of a screening of the coupling between\nthe quantum system and the environment is thus \na physical interpretation of a fractional power of \nthe quantum Markovian superoperator.\n\nWe note that the quantum Markovian equation describes\na coupling between a quantum system and an environment (see \\cite{ALV}).\nAnother physical interpretation of a fractional power of \nthe infinitesimal generator is connected with \nBochner-Phillips formula (\\ref{BPf}) as follows.\nUsing the properties\n\\[ \\int^{\\infty}_0 f_{\\alpha}(t,s)=1 , \\quad \\quad f_{\\alpha}(t,s) \\ge 0 \\quad \n(for \\ all \\ \\ s>0) , \\]\nwe can assume that $f_{\\alpha}(t,s)$ is the density of a probability distribution.\nThen Bochner-Phillips formula (\\ref{BPf}) can be considered \na smoothing of the evolution $\\Phi_t$ with respect to the time $s>0$.\nThis smoothing can be considered a screening of \nthe environment of the quantum system. \n\nThe function $f_{\\alpha}(t,s)$\ncan be represented as the Levy distribution using a reparametrization. \nWe note that Levy distributions are solutions of fractional \nequations (see, e.g., \\cite{Zaslavsky2,SZ,CNSNS2008-1,Y})\nthat describe anomalous diffusion.\nIt is known that quantum Markovian equations are used to describe\nthe Brownian motion of quantum systems \\cite{Lind2}.\nPerhaps, the fractional generalization of \nquantum Markovian equations can be used to describe \nanomalous processes and random walks \n\\cite{Zaslavsky2,MS,MK1,MK2} in quantum systems.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nMany knowledge-driven applications are built on knowledge graphs (KGs), which store massive amounts of structured facts about the real world~\\cite{KG_survey}.\nThroughout the life-cycle of KG construction and application, \nnew facts, unseen entities, and unseen relations continually emerge into the KG on a regular basis. \nAs a result, the real-world KG is rarely a static graph, but rather evolves and grows alongside the development.\nFigure~\\ref{fig:illustration} illustrates an example excerpted from Wikidata~\\cite{Wikidata}, which shows the growth of the KG along with the continuous knowledge extraction. \nHowever, KG embedding, a critical task for downstream applications, has primarily focused on static KGs over the years~\\cite{KGE_survey}. \nLearning from scratch every time is inefficient and wastes previously acquired knowledge, and simply fine-tuning new facts would quickly disrupt previously acquired knowledge. \nHence, this paper proposes to investigate lifelong embedding learning and transfer for growing KGs, with the goal of learning new facts while retaining old knowledge without re-training from scratch.\n\nThe key idea of this paper comes from the human learning process.\nHumans are typical lifelong learners, with knowledge transfer and retention being the most important aspects of lifelong learning. \nHumans, in particular, can continually learn new knowledge given new facts and use previously learned knowledge to help new knowledge learning (\\textit{knowledge learning and transfer}),\nas well as update old knowledge while retaining useful knowledge (\\textit{knowledge update and retention}). \nMotivated by this, we seek to build a lifelong KG embedding model, namely \\textbf{LKGE\\xspace}, which is capable of learning, transferring and retaining knowledge for growing KGs efficiently.\nExisting related work, such as inductive KG embedding~\\cite{MEAN,LAN}, mainly focuses on knowledge transfer, ignoring new knowledge learning and old knowledge update.\n\n\\begin{figure}[!t]\n\\includegraphics[width=\\columnwidth]{figs_new\/illustration2.pdf}\n\\caption{An example of a growing KG. In each snapshot $i$, new facts are added into the KG, and previously unseen entities and relations emerge with the new facts.}\n\\label{fig:illustration}\n\\vspace{-10pt}\n\\end{figure}\n\nThe proposed lifelong KG embedding task faces two major challenges.\nFirst, how to strike a balance between new knowledge learning and old knowledge transfer? \nLearning embeddings for new entities and relations from scratch cannot leverage previously learned knowledge, and inductively generating embeddings for them ignores new knowledge in the new snapshot.\nSecond, how to update old knowledge while retaining useful knowledge?\nLearning new facts about an old entity usually requires updating the previously learned embeddings, which can be harmful to the old model.\nThis is because updating an old entity embedding would affect many other old embeddings of related entities.\nThis would cause the catastrophic forgetting issue, and therefore affect the applications built on the old KG snapshot.\n\nTo resolve the above challenges, we propose three solutions in our LKGE\\xspace model.\nFirst, as the base embedding model for new knowledge learning and old knowledge update, we design a masked KG autoencoder that masks and reconstructs the entities or relations in new facts. \nIt builds connections between locally-related old and new entities, acting as a bridge for knowledge transfer.\nSecond, to aid in the learning of new knowledge, we propose a knowledge embedding transfer strategy that uses previously learned knowledge to initialize the embeddings of new entities and relations.\nThese embeddings are used by the KG autoencoder to learn new facts.\nThird, to avoid catastrophic forgetting in the old knowledge update, \nwe propose embedding regularization to balance the learning of new facts and the update of old embeddings.\n\nWe build four datasets to assess the lifelong KG embedding performance, including the\nentity-centric, relation-centric, fact-centric, and hybrid growth.\nEach dataset examines a different aspect of KG growth.\nBy contrast, existing datasets \\cite{MEAN,DiCGRL,CKGE} all assume that a KG grows in an ideal way, with balanced new entities or facts in each new snapshot. \nIn our experiments, we compare the link prediction accuracy, knowledge transfer ability, and learning efficiency of the proposed model against baselines on the four datasets. \nThe results show that the proposed LKGE\\xspace not only achieves the best performance on the four datasets, but also has the best forward knowledge transfer ability and learning efficiency.\nThe main contributions of this paper are summarized as follows:\n\\begin{itemize}\n\\item We study \\textit{lifelong KG embedding learning and transfer}. It is a practical task since the real-world KGs continually evolve and grow, which requires the embedding model to be capable of handling the knowledge growth.\n\n\\item We propose \\textit{a novel lifelong learning model}, LKGE\\xspace. It includes a masked KG autoencoder as the basis of embedding learning and update, an embedding transfer strategy for knowledge transfer, and an embedding regularization to prevent catastrophic forgetting in knowledge update.\n\n\\item We conduct \\textit{extensive experiments on four new datasets}. The results demonstrate the effectiveness and efficiency of LKGE\\xspace against a variety of state-of-the-art models.\n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:related_work}\n\nIn this section, we review two lines of related work, i.e., KG embedding and lifelong learning.\n\n\\subsection{Knowledge Graph Embedding}\n\\label{subsec:knowledge_graph_embedding}\n\nKG embedding seeks to encode the symbolic representations of KGs into vector space to foster the application of KGs in downstream tasks. \nMost existing KG embedding models \\cite{TransE,TransH,ConvE,R-GCN,RSN,InteractE,CompGCN} focus on static graphs and cannot continually learn new knowledge on the growing KGs.\n\nTo embed unseen entities, inductive KG embedding models learn to represent an entity by aggregating its existing neighbors in the previous KG snapshot. \nMEAN \\cite{MEAN} uses a graph convolutional network (GCN) \\cite{GCN} for neighborhood aggregation. \nWhen an unseen entity emerges, the GCN would aggregate its previously seen neighboring entities to generate an embedding. \nLAN \\cite{LAN} adopts an attention mechanism to attentively aggregate different neighbors. \nAs MEAN and LAN rely on the entity neighborhood for embedding learning, they cannot handle the new entities that have no neighbors in the previous snapshot.\nFurthermore, inductive KG embedding disregards learning the facts about new entities.\n\nOur work is also relevant to dynamic KG embedding. \npuTransE \\cite{puTransE} trains several new models when facts are added. \nDKGE \\cite{DKGE} learns contextual embeddings for entities and relations, which can be automatically updated as the KG grows.\nThey both need partial re-training on old facts, but our model does not.\nIn addition, some subgraph-based models, such as GraIL \\cite{GraIL}, INDIGO \\cite{INDIGO}, and TACT \\cite{TACT}, can also represent unseen entities using the entity-independent features and subgraph aggregation.\nTheir subgraph-building process is time-consuming, making them only applicable to small KGs.\nIn order to run on large-scale KGs, NBFNet \\cite{NBFNet} proposes a fast node pair embedding model based on the Bellman-Ford algorithm, and NodePiece \\cite{NodePiece} uses tokenized anchor nodes and relational paths to represent new entities. However, they do not consider learning new knowledge and cannot support new relations.\n\n\\subsection{Lifelong Learning}\n\\label{subsec:lifelong_learning}\n\nLifelong learning seeks to solve new problems quickly without catastrophically forgetting previously acquired knowledge. \nLifelong learning models are broadly classified into three categories.\n(\\romannumeral1) Dynamic architecture models \\cite{PNN,CWR} extend the network to learn new tasks and avoid forgetting acquired knowledge. \n(\\romannumeral2) Regularization-based models \\cite{EWC,SI} capture the importance of model parameters for old tasks and limit the update of important parameters. \n(\\romannumeral3) Rehearsal-based models \\cite{GEM,EMR} memorize some data from old tasks and replay them when learning new knowledge. \n\nFew lifelong learning models focus on KG embedding. \nDiCGRL \\cite{DiCGRL} is a disentangle-based lifelong graph embedding model. \nIt splits node embeddings into different components and replays related historical facts to avoid catastrophic forgetting. \nThe work \\cite{CKGE} combines class-incremental learning models with TransE~\\cite{TransE} for continual KG embedding. \nHowever, it does not propose a specific lifelong KG embedding model. \n\n\n\n\\section{Lifelong Knowledge Graph Embedding}\n\\label{sec:method}\n\nIn this section, we first introduce our problem setting. \nThen, we present our model, LKGE\\xspace, in detail. \n\n\\begin{figure*}[!t]\n\\centering\n\\includegraphics[width=0.9\\textwidth]{figs_new\/overview_new.pdf}\n\\caption{Overview of the proposed model for lifelong KG embedding.}\n\\label{fig:overview}\n\\centering\n\\vspace{-5pt}\n\\end{figure*}\n\n\\subsection{Preliminaries}\n\n\\textbf{Growing KG.} \nThe growth process of a KG yields a snapshot sequence, i.e., $\\mathcal{G}=\\{\\mathcal{S}_1, \\mathcal{S}_2, \\ldots, \\mathcal{S}_t\\}$. \nEach snapshot $\\mathcal{S}_i$ is defined as a triplet $(\\mathcal{T}_i,\\mathcal{E}_i, \\mathcal{R}_i)$, \nwhere $\\mathcal{T}_i, \\mathcal{E}_i$ and $\\mathcal{R}_i$ denote the fact, entity and relation sets, respectively.\nWe have $\\mathcal{T}_i \\subseteq \\mathcal{T}_{i+1}$, $\\mathcal{E}_i \\subseteq \\mathcal{E}_{i+1}$ and $\\mathcal{R}_i \\subseteq \\mathcal{R}_{i+1}$.\nWe use $\\mathcal{T}_{\\Delta i}=\\mathcal{T}_{i}-\\mathcal{T}_{i-1}$, $\\mathcal{E}_{\\Delta i}=\\mathcal{E}_{i}-\\mathcal{E}_{i-1}$, and $\\mathcal{R}_{\\Delta i}=\\mathcal{R}_{i}-\\mathcal{R}_{i-1}$ to denote the new facts, entities and relations, respectively.\nEach fact is in the form of $(s, r, o)\\in\\mathcal{T}_i$, where $s,o\\in\\mathcal{E}_i$ are the subject and object entities, respectively, and $r\\in\\mathcal{R}_i$ is their relation. \n\n\\smallskip\n\\noindent\\textbf{Lifelong KG embedding.} \nKG embedding seeks to encode the symbolic representations of entities and relations into vector space and capture KG semantics using vector operations. \nFor a growing KG, a lifelong KG embedding model learns to represent the snapshot sequence $\\mathcal{G}=\\{\\mathcal{S}_1, \\mathcal{S}_2, \\ldots, \\mathcal{S}_t\\}$ continually. \nWhen a new fact set $\\mathcal{T}_{\\Delta i}$ emerges, the current KG embedding model $\\mathcal{M}_{ i-1}$ needs update to fit the new facts and learn embeddings for the new entities $\\mathcal{E}_{\\Delta i}$ and new relations $\\mathcal{R}_{\\Delta i}$. \nThe resulting model is denoted by $\\mathcal{M}_i$.\n\n\\smallskip\n\\noindent\\textbf{Lifelong link prediction.} \nThe link prediction task asks the KG embedding model to predict the missing subject or object entity in an incomplete fact like $(s, r, ?)$ or $(?, r, o)$.\nFor each snapshot $\\mathcal{S}_i$, the new fact set $\\mathcal{T}_{\\Delta i}$ is divided into a training set $\\mathcal{D}_i$, a validation set $\\mathcal{V}_i$ and a test set $\\mathcal{Q}_i$. \nIn the lifelong setting, the model is required to learn the training data sets, $\\mathcal{D}_1,\\mathcal{D}_2,\\ldots,\\mathcal{D}_t$, in turn. \nAfter finishing the learning on $\\mathcal{D}_i$, the model is evaluated on the accumulated test data, which is $\\cup_{j=1}^i\\mathcal{Q}_j$, to assess the overall learning performance.\nAfter learning $\\mathcal{D}_i$, then $\\mathcal{D}_i,\\mathcal{V}_i$ would no longer be available for the following learning.\nNote that the goal of lifelong KG embedding is to improve the overall performance on all snapshots, which requires the KG embedding model to continually learn new knowledge and retain the learned knowledge.\n\n\\subsection{Model Overview}\nThe overview of our model, LKGE\\xspace, is shown in Figure~\\ref{fig:overview}.\nIt continually learns knowledge over a sequence of KG snapshots without re-training on previously seen data. \nThe foundation is a masked KG autoencoder that can reconstruct the entity and relation embeddings from the masked related subgraph.\nTo enable knowledge transfer from old snapshots to the new one, we propose embedding transfer to inject learned knowledge into unseen entities and relations and then iteratively optimize the model to fit the new facts.\nWe also propose a lightweight regularization method to retain old knowledge. \n\n\\subsection{Masked Knowledge Graph Autoencoder}\nIn lifelong KG embedding learning, the new facts, e.g., $\\mathcal{T}_{\\Delta i}$, bring new entities $\\mathcal{E}_{\\Delta i}$, and new relations $\\mathcal{R}_{\\Delta i}$. \nIt would also involve some old entities from $\\mathcal{E}_{i-1}$, and old relations from $\\mathcal{R}_{i-1}$.\nThus, a new entity may be connected with both new and old entities (referred to as new knowledge).\nThe involved old entities also receive more facts, and therefore their previously learned embeddings (referred to as old knowledge) need to be updated to fit the new facts.\nHence, the base KG embedding model for lifelong learning should be capable of capturing new knowledge as well as updating old knowledge.\nTo this end, we propose a masked KG autoencoder motivated by the recent success of self-supervised learning~\\cite{GraphMAE}.\nThe key idea is to reconstruct the embedding for an entity or relation based on its masked subgraph, which may include both other new entities and some old entities.\nSpecifically, we use the first-order subgraph of an entity or relation to reconstruct its embedding $\\bar{\\mathbf{x}}_i$:\n\\begin{equation}\n    \\bar{\\mathbf{x}}_i = \\text{MAE}\\big(\\cup_{j=1}^i\\mathcal{N}_{j}(x)\\big),\n\\end{equation}\nwhere $x$ denotes either an entity or a relation, and $\\mathcal{N}_{j}\\subseteq\\mathcal{D}_j$ denotes the involved facts of $x$ in the $j$-th snapshot. $\\text{MAE}()$ is an encoder to represent the input subgraph.\nThe objective of our KG encoder is to align the entity or relation embedding with the reconstructed representation as follows:\n\\begin{equation}\n    \\mathcal{L}_\\text{MAE} = \\sum_{e\\in\\mathcal{E}_i}\\Vert\\mathbf{e}_i-\\bar{\\mathbf{e}}_i\\Vert_2^2 + \\sum_{r\\in\\mathcal{R}_i}\\Vert\\mathbf{r}_i-\\bar{\\mathbf{r}}_i\\Vert_2^2.\n\\end{equation}\n\nThe key then becomes how to design an effective and efficient encoder for lifelong learning.\nGCN~\\cite{Kipf_GCN} and Transformer~\\cite{Transformer} are two common choices for the encoder. \nThe two encoders both introduce additional model parameters (e.g., the weight matrices).\nIn our lifelong learning setting, the encoder needs to be updated to fit new facts or subgraphs.\nIn this case, once the GCN or Transformer is updated, the changed model parameters would affect the embedding generation of \\emph{all} old entities (not just the involved old entities in new facts), increasing the risk of catastrophically forgetting previous snapshots. \nTo avoid this issue, we use the entity and relation embedding transition functions as encoders, which do not introduce additional parameters.\nWe borrow the idea of TransE \\cite{TransE} and interpret a relation embedding as the translation vector between the subject and object entity embeddings, i.e., $\\mathbf{s} + \\mathbf{r} \\approx \\mathbf{o}$,\nwhere $\\mathbf{s}, \\mathbf{r}, \\mathbf{o}$ denote the embeddings of subject entity, relation and object entity, respectively. \nBased on this, we can deduce two transition functions for entity and relation embeddings.\nThe subject entity of $(s,r,o)$ can be represented by $f_{\\rm{sub}}(\\mathbf{r}, \\mathbf{o})=  \\mathbf{o} - \\mathbf{r}$,\nand the relation embedding is $f_{\\rm{rel}}(\\mathbf{s}, \\mathbf{o})=  \\mathbf{o} - \\mathbf{s}$.\nWe can define the encoders as\n\\begin{align}\n\\label{eq:auto_ent_lifelong_1}\n    \\bar{\\mathbf{e}}_i &= \\frac{\\sum_{j=1}^i\\sum_{(s, r, o) \\in \\mathcal{N}_j(e)} f_{\\rm{sub}}(\\mathbf{r}_i, \\mathbf{o}_i)}{\\sum_{j=1}^{i} |\\mathcal{N}_j(e)|},\\\\\n\\label{eq:auto_rel_lifelong_1}\n\\bar{\\mathbf{r}}_i &= \\frac{\\sum_{j=1}^i\\sum_{(s, r, o) \\in \\mathcal{N}_j(r)} f_{\\rm{rel}}(\\mathbf{s}_i, \\mathbf{o}_i)}{\\sum_{j=1}^{i} |\\mathcal{N}_j(r)|},\n\\end{align}\nwhere $\\mathbf{s}_i, \\mathbf{r}_i, \\mathbf{o}_i$ are the embeddings of $s,r,o$ during the training on $\\mathcal{D}_i$. \n$\\mathcal{N}_j(x)\\subseteq\\mathcal{D}_j$ is the set of facts containing $x$.\n\nIn lifelong learning, the model learns from a snapshot sequence. \nEqs.~(\\ref{eq:auto_ent_lifelong_1}) and (\\ref{eq:auto_rel_lifelong_1}) require training samples from the first $i$ snapshots, which are not in line with lifelong learning. \nTo reduce the reliance on learned data, we use $\\mathbf{e}_{i-1}$ and $\\mathbf{r}_{i-1}$ as the approximate average embeddings of $e$ and $r$ in the first $i-1$ snapshots, respectively, and rewrite the encoders as\n\\begin{equation}\n\\resizebox{.9\\columnwidth}{!}{$\n\\label{eq:auto_ent_lifelong_2}\n\\bar{\\mathbf{e}}_i \\approx \\frac{\\sum_{j=1}^{i-1}|\\mathcal{N}_j(e)| \\mathbf{e}_{i-1}+\\sum_{(s, r, o) \\in \\mathcal{N}_i(e)} f_{\\rm{sub}}(\\mathbf{r}_i, \\mathbf{o}_i)}\n    {\\sum_{j=1}^{i-1}|\\mathcal{N}_j(e)|+ |\\mathcal{N}_i(e)|},\n$}\n\\end{equation}\n\\begin{equation}\n\\resizebox{.9\\columnwidth}{!}{$\n\\label{eq:auto_rel_lifelong_2}\n\\bar{\\mathbf{r}}_i \\approx \\frac{\\sum_{j=1}^{i-1}|\\mathcal{N}_j(r)| \\mathbf{r}_{i-1}+\\sum_{(s, r, o) \\in \\mathcal{N}_i(r)} f_{\\rm{rel}}(\\mathbf{s}_i, \\mathbf{o}_i)}\n    {\\sum_{j=1}^{i-1}|\\mathcal{N}_j(r)| + |\\mathcal{N}_i(r)|}.\n$}\n\\end{equation}\n\nThe encoders use the facts involving both old and new entities and relations for embedding reconstruction and they build a bridge for knowledge transfer.\n\nFor each snapshot, to learn the knowledge from the new data and update the learned parameters, we leverage TransE \\cite{TransE} to train the embedding model:\n\\begin{equation}\n\\resizebox{.9\\columnwidth}{!}{$\n    \\mathcal{L}_\\text{new} = \\sum\\limits_{(s, r, o)\\in\\mathcal{D}_i} \\max\\big(0,\\gamma+f(\\mathbf{s}, \\mathbf{r}, \\mathbf{o})-f(\\mathbf{s}', \\mathbf{r}, \\mathbf{o}')\\big),\n$}\n\\end{equation}\nwhere $\\gamma$ is the margin. $(\\mathbf{s}', \\mathbf{r}, \\mathbf{o}')$ is the embedding of a negative fact. \nFor each positive fact, we randomly replace the subject or object entity with a random entity $e'\\in\\mathcal{E}_i$.\n\n\n\\subsection{Embedding Transfer}\nDuring the lifecycle of a growing KG, there are abundant unseen entities and some unseen relations emerge with the new facts. \nLearning effective embeddings for them is an essential aspect of lifelong KG embedding learning.\nHowever, these unseen ones are not included in any learned snapshots, so only inheriting the learned parameters cannot transfer the acquired knowledge to their embeddings.\nTo avoid learning from scratch, we propose embedding transfer that seeks to leverage the learned embeddings to help represent unseen entities and relations.\nSpecifically, we initialize the embeddings of each unseen entity by aggregating its facts:\n\\begin{equation}\n    \\mathbf{e}_{i} = \\frac{1}{|\\mathcal{N}_i(e)|}\\sum_{(e, r, o) \\in \\mathcal{N}_i(e)} f_{\\rm{sub}}(\\mathbf{r}_{i-1}, \\mathbf{o}_{i-1}) ,\n\\end{equation}\nwhere $\\mathcal{N}_i(e)\\subseteq\\mathcal{D}_i$ is the set of facts containing $e$. \nFor the new entities that do not have common facts involving existing entities, we randomly initialize their embeddings.\nWe also use this strategy to initialize the embeddings of unseen relations:\n\\begin{equation}\n    \\mathbf{r}_i = \\frac{1}{|\\mathcal{N}_i(r)|}\\sum_{(s, r, o) \\in \\mathcal{N}_i(r)} f_{\\rm{rel}}(\\mathbf{s}_{i-1}, \\mathbf{o}_{i-1}),\n\\end{equation}\nwhere $\\mathcal{N}_i(r)\\subseteq\\mathcal{D}_i$ is the set of facts containing $r$.\n\n\\subsection{Embedding Regularization}\nLearning new snapshots is likely to overwrite the learned knowledge from old snapshots. \nTo avoid catastrophic forgetting, some regularization methods \\cite{EWC,GEM} constrain the updates of parameters that are important to old tasks.\nThe loss function of regularization methods is\n\\begin{equation}\n\\resizebox{.88\\columnwidth}{!}{$\n    \\mathcal{L}_\\text{old} = \\sum\\limits_{e\\in\\mathcal{E}_{i-1}}\\omega(e)\\Vert\\mathbf{e}_i-\\mathbf{e}_{i-1}\\Vert_2^2 + \\sum\\limits_{r\\in\\mathcal{R}_{i-1}}\\omega(r)\\Vert\\mathbf{r}_i-\\mathbf{r}_{i-1}\\Vert_2^2,\n    $}\n\\end{equation}\nwhere $\\omega(x)$ is the regularization weight for $x$.\n\nConventional regularization-based methods for classification tasks such as \\cite{EWC} model the importance of each parameter at a high cost based on the gradient or parameter change during training. \nThis problem is even more severe for KG embedding models that have a large number of embedding parameters (i.e. entity and relation embeddings).\nTo resolve this problem, we propose a lightweight embedding regularization method, which calculates the regularization weight of each entity or relation by the number of new and old facts containing it:\n\\begin{equation}\n    \\omega(x) = 1 - \\frac{|\\mathcal{N}_{i}(x)|}{\\sum_{j=1}^i|\\mathcal{N}_{j}(x)|}.\n\\end{equation}\n\nAs a lightweight technique, it only keeps the total number of involved trained facts for each entity or relation and only updates regularization weights once per snapshot.\n\n\\subsection{Overall Learning Objective}\nTo learn new knowledge while retaining acquired knowledge, the overall lifelong learning objective $\\mathcal{L}$ is defined as follows:\n\\begin{equation}\n    \\mathcal{L} = \\mathcal{L}_\\text{new} + \\alpha\\,\\mathcal{L}_\\text{old} + \\beta\\,\\mathcal{L}_\\text{MAE},\n\\end{equation}\nwhere $\\alpha,\\beta$ are hyperparameters for balancing the objectives.\n\n\\subsection{Complexity Analysis}\nCompared with fine-tuning, the proposed model requires few extra resources. \nIt does not increase the size of training samples like the rehearsal models \\cite{EMR, GEM, DiCGRL}. \nIn addition to fine-tuning, the proposed model calculates the loss of masked autoencoder and embedding regularization. \nThe additional time complexity in each iteration is $O(|\\mathcal{E}|+|\\mathcal{R}|+|\\mathcal{D}|)$. \nIn practice, we find that the loss of autoencoder can accelerate learning, and its time consumption is close to that of fine-tuning. \nThe space complexity of fine-tuning is $O((|\\mathcal{E}|+|\\mathcal{R}|)\\times d)$, and the space complexity of the proposed model is $O((|\\mathcal{E}|+|\\mathcal{R}|)\\times(d+1))$, where $d$ is the dimension of embeddings.\n\n\\begin{table*}\n\\centering\n\\resizebox{\\linewidth}{!}{\n    \\begin{tabular}{lcccccccccccccccccccc}\n    \\toprule\n    \\multirow{2}{*}{Datasets} & \\multicolumn{3}{c}{Snapshot 1} & \\multicolumn{3}{c}{Snapshot 2} &    \\multicolumn{3}{c}{Snapshot 3} & \\multicolumn{3}{c}{Snapshot 4} & \\multicolumn{3}{c}{Snapshot 5} \\\\ \n    \\cmidrule(lr){2-4} \\cmidrule(lr){5-7} \\cmidrule(lr){8-10} \\cmidrule(lr){11-13} \\cmidrule(lr){14-16} & $|\\mathcal{T}_{\\Delta 1}|$ & $|\\mathcal{E}_1|$ & $|\\mathcal{R}_1|$ & $|\\mathcal{T}_{\\Delta 2}|$ & $|\\mathcal{E}_2|$ & $|\\mathcal{R}_2|$ & $|\\mathcal{T}_{\\Delta 3}|$ & $|\\mathcal{E}_3|$ & $|\\mathcal{R}_3|$ & $|\\mathcal{T}_{\\Delta 4}|$ & $|\\mathcal{E}_4|$ & $|\\mathcal{R}_4|$ & $|\\mathcal{T}_{\\Delta 5}|$ & $|\\mathcal{E}_5|$ & $|\\mathcal{R}_5|$ \\\\\n    \\midrule\n    \\textsc{Entity} & 46,388 & \\ \\ 2,909 & 233 & 72,111 & \\ \\ 5,817 & 236 & 73,785 & \\ \\ 8,275 & 236 & \\ \\ 70,506 & 11,633 & 237 & 47,326 & 14,541 & 237  \\\\\n    \\textsc{Relation} & 98,819 & 11,560 & \\ \\ 48 & 93,535 & 13,343 & \\ \\ 96 & 66,136 & 13,754 & 143 & \\ \\ 30,032 & 14,387 & 190 & 21,594 & 14,541 & 237  \\\\\n    \\textsc{Fact} & 62,024 & 10,513 & 237 & 62,023 & 12,779 & 237 & 62,023 & 13,586 & 237 & \\ \\ 62,023 & 13,894 & 237 & 62,023 & 14,541 & 237  \\\\\n    \\textsc{Hybrid} & 57,561 & \\ \\ 8,628 & \\ \\ 86 & 20,873 & 10,040 & 102 & 88,017 & 12,779 & 151 & 103,339 & 14,393 & 209 & 40,326 & 14,541 & 237 \\\\\n    \\bottomrule\n\\end{tabular}}\n\\caption{Statistical data of the four constructed growing KG datasets. \nFor the $i$-th snapshot, $\\mathcal{T}_{\\Delta i}$ denotes the set of new facts in this snapshot, and $\\mathcal{E}_i, \\mathcal{R}_{i}$ denote the sets of cumulative entities and relations in the first $i$ snapshots, respectively.}\n\\label{tab:datasets}\n\\end{table*} \n\n\\section{Dataset Construction}\n\\label{sec:datasets}\n\nTo simulate a variety of aspects of KG growth, we create four datasets based on FB15K-237 \\cite{FB237}, which are entity-centric, relation-centric, fact-centric, and hybrid.\nWe denote them by \\textsc{Entity}, \\textsc{Relation}, \\textsc{Fact} and \\textsc{Hybrid}, respectively. \nGiven a KG $\\mathcal{G}=\\{\\mathcal{E}, \\mathcal{R}, \\mathcal{T}\\}$, we construct five snapshots with the following steps:\n\\begin{enumerate}\n\\item \\textbf{Seeding.} \nWe randomly sample 10 facts from $\\mathcal{T}$ and add them into $\\mathcal{T}_1$ for initialization. \nThe entities and relations in the 10 facts form the initial $\\mathcal{E}_1$ and $\\mathcal{R}_1$, respectively.\n\n\\item \\textbf{Expanding.} \nTo build \\textsc{Entity}, \\textsc{Relation} and \\textsc{Fact}, we iteratively sample a fact containing at least one seen entity in $\\mathcal{E}_i$, add it into $\\mathcal{T}_i$, and extract the unseen entity and relation from it to expand $\\mathcal{E}_i$ and $\\mathcal{R}_i$. \nFor \\textsc{Entity}, once $|\\mathcal{E}_i| \\geq \\frac{i+1}{5}|\\mathcal{E}|$, we add all new facts $\\big\\{(s, r, o)\\,|\\,s\\in\\mathcal{E}_i\\wedge o\\in\\mathcal{E}_i \\big\\}$ into $\\mathcal{T}_i$ and start building the next snapshot. \nIn the same way, we construct \\textsc{Relation} and \\textsc{Fact}. \nAs for \\textsc{Hybrid}, we uniformly sample an entity, relation or fact without replacement from $\\mathcal{U}=\\mathcal{E}\\cup\\mathcal{R}\\cup\\mathcal{T}$ to join $\\mathcal{E}_i$, $\\mathcal{R}_i$ and $\\mathcal{T}_i$. \nNote that when the sampled fact contains an unseen entity or relation, we re-sample a fact that only contains seen entities and relations to replace it.\nAfter each iteration, we terminate the expansion of this snapshot with a probability $\\frac{5}{|\\mathcal{U}|}$. \nConsequently, the expansion of \\textsc{Hybrid} is uneven, making it more realistic and challenging.\nFor all datasets, we take the whole KG as the last snapshot, i.e., $\\mathcal{T}_5=\\mathcal{T}$, and $\\mathcal{E}_5=\\mathcal{E}, \\mathcal{R}_5=\\mathcal{R}$.\n\n\\item \\textbf{Dividing.} \nFor each snapshot, we randomly divide the new fact set $\\mathcal{T}_{\\Delta i}$ into a training set $\\mathcal{D}_i$, a validation set $\\mathcal{V}_i$ and a test set $\\mathcal{Q}_i$ by a split ratio of 3:1:1. \n\\end{enumerate}\n\nThe statistics of the four datasets are presented in Table~\\ref{tab:datasets}. \n\n\\section{Experimental Results}\n\\label{sec:experiments}\n\nWe conduct experiments regarding link prediction accuracy, knowledge transfer capability, and learning efficiency to validate the proposed model, LKGE.\nThe datasets and source code are available at \\url{https:\/\/github.com\/nju-websoft\/LKGE}.\n\n\\subsection{Experiment Settings}\n\\noindent\\textbf{Competitors.} We compare our model with 12 competitors, including \n(\\romannumeral1) three baseline models: snapshot only, re-training, and fine-tuning; \n(\\romannumeral2) two inductive models: MEAN \\cite{MEAN}, and LAN \\cite{LAN}; \n(\\romannumeral3) two dynamic architecture models: PNN \\cite{PNN}, and CWR \\cite{CWR};\n(\\romannumeral4) two regularization-based models: SI \\cite{SI}, and EWC \\cite{EWC}; \nand (\\romannumeral5) three rehearsal-based models: GEM \\cite{GEM}, EMR \\cite{EMR}, and DiCGRL \\cite{DiCGRL}.\n\n\\smallskip\n\\noindent\\textbf{Evaluation metrics.} Following the convention, we conduct the experiments on link prediction. \nGiven a snapshot $\\mathcal{S}_i$, for each test fact $(s, r, o)\\in\\mathcal{Q}_i$, we construct two queries $(s, r, ?)$ and $(?, r, t)$. \nWhen evaluating on $\\mathcal{Q}_i$, we set all seen entities in $\\mathcal{E}_i$ as candidate entities. \nWe select seven metrics to evaluate all models, including \n(\\romannumeral1) Four metrics on link prediction accuracy: mean reciprocal rank (MRR) and Hits@$k$ ($k=1,3,10$, and H@$k$ for shot). \nWe conduct the model $\\mathcal{M}_5$ trained on the last snapshot to evaluate on the union of the test sets in all snapshots.\n(\\romannumeral2) Two metrics on knowledge transfer capability: forward transfer (FWT) and backward transfer (BWT) \\cite{GEM}.\nFWT is the influence of learning a task to the performance on the future tasks, while BWT is the influence of learning to the previous tasks: \n\\begin{align}\n\\resizebox{.88\\columnwidth}{!}{$\n\\text{FWT} = \\frac{1}{n-1}\\sum\\limits_{i=2}^{n} h_{i-1,i},\\ \\text{BWT} = \\frac{1}{n-1}\\sum\\limits_{i=1}^{n-1} (h_{n,i} - h_{i,i}),\n$}\n\\end{align}\nwhere $n$ is the number of snapshots, $h_{i,j}$ is the MRR scores on $\\mathcal{Q}_j$ after training the model $\\mathcal{M}_i$ on the $i$-th snapshot.\nHigher scores indicate better performance.\n(\\romannumeral3) Time cost: The cumulative time cost of the learning on each snapshot.\n\n\n\\begin{table*}[!t]\n\\centering\n\\resizebox{1.0\\textwidth}{!}{\\Large\n\\begin{tabular}{lcccccccccccccccc}\n\\toprule\n\\multirow{2}{*}{Models} & \\multicolumn{4}{c}{\\textsc{Entity}} & \\multicolumn{4}{c}{\\textsc{Relation}} & \\multicolumn{4}{c}{\\textsc{Fact}} & \\multicolumn{4}{c}{\\textsc{Hybrid}} \\\\\n\\cmidrule(lr){2-5} \\cmidrule(lr){6-9} \\cmidrule(lr){10-13} \\cmidrule(lr){14-17} & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 \\\\\n\\midrule\nSnapshot\t& $.084_{\\pm .001}$ \t& $.028_{\\pm .000}$ \t& $.107_{\\pm .002}$ \t& $.193_{\\pm .003}$ \t& $.021_{\\pm .000}$ \t& $.010_{\\pm .000}$ \t& $.023_{\\pm .000}$ \t& $.043_{\\pm .001}$ \t& $.082_{\\pm .001}$ \t& $.030_{\\pm .001}$ \t& $.095_{\\pm .002}$ \t& $.191_{\\pm .006}$ \t& $.036_{\\pm .001}$ \t& $.015_{\\pm .001}$ \t& $.043_{\\pm .001}$ \t& $.077_{\\pm .003}$ \t\\\\\nRe-train\t& $.236_{\\pm .001}$ \t& $.137_{\\pm .001}$ \t& $.274_{\\pm .001}$ \t& $.433_{\\pm .001}$ \t& $.219_{\\pm .001}$ \t& $.128_{\\pm .001}$ \t& $.250_{\\pm .001}$ \t& $.403_{\\pm .002}$ \t& $.206_{\\pm .001}$ \t& $.118_{\\pm .001}$ \t& $.232_{\\pm .001}$ \t& $.385_{\\pm .001}$ \t& $.227_{\\pm .001}$ \t& $.134_{\\pm .001}$ \t& $.260_{\\pm .002}$ \t& $.413_{\\pm .001}$ \t\\\\\nFine-tune\t& $.165_{\\pm .002}$ \t& $.085_{\\pm .002}$ \t& $.188_{\\pm .003}$ \t& $.321_{\\pm .003}$ \t& $.093_{\\pm .003}$ \t& $.039_{\\pm .002}$ \t& $.106_{\\pm .003}$ \t& $.195_{\\pm .007}$ \t& $.172_{\\pm .003}$ \t& $.090_{\\pm .002}$ \t& $.193_{\\pm .004}$ \t& $.339_{\\pm .005}$ \t& $.135_{\\pm .002}$ \t& $.069_{\\pm .001}$ \t& $.151_{\\pm .003}$ \t& $.262_{\\pm .005}$ \t\\\\\n\\midrule\nMEAN\t& $.117_{\\pm .005}$ \t& $.068_{\\pm .003}$ \t& $.123_{\\pm .006}$ \t& $.212_{\\pm .007}$ \t& $.039_{\\pm .004}$ \t& $.024_{\\pm .005}$ \t& $.040_{\\pm .005}$ \t& $.067_{\\pm .008}$ \t& $.084_{\\pm .008}$ \t& $.051_{\\pm .005}$ \t& $.088_{\\pm .008}$ \t& $.146_{\\pm .015}$ \t& $.046_{\\pm .004}$ \t& $.029_{\\pm .003}$ \t& $.049_{\\pm .003}$ \t& $.080_{\\pm .004}$ \t\\\\\nLAN\t& $.141_{\\pm .004}$ \t& $.082_{\\pm .004}$ \t& $.149_{\\pm .003}$ \t& $.256_{\\pm .005}$ \t& $.052_{\\pm .003}$ \t& $.033_{\\pm .003}$ \t& $.052_{\\pm .004}$ \t& $.092_{\\pm .008}$ \t& $.106_{\\pm .007}$ \t& $.056_{\\pm .006}$ \t& $.113_{\\pm .007}$ \t& $.200_{\\pm .011}$ \t& $.059_{\\pm .005}$ \t& $.032_{\\pm .005}$ \t& $.062_{\\pm .005}$ \t& $.113_{\\pm .007}$ \t\\\\\n\\midrule\nPNN\t& $.229_{\\pm .001}$ \t& $.130_{\\pm .001}$ \t& $.265_{\\pm .001}$ \t& $\\textbf{.425}_{\\pm .001}$ \t& $.167_{\\pm .002}$ \t& $.096_{\\pm .001}$ \t& $.191_{\\pm .001}$ \t& $.305_{\\pm .001}$ \t& $.157_{\\pm .000}$ \t& $.084_{\\pm .002}$ \t& $.188_{\\pm .001}$ \t& $.290_{\\pm .001}$ \t& $.185_{\\pm .001}$ \t& $.101_{\\pm .001}$ \t& $.216_{\\pm .001}$ \t& $.349_{\\pm .001}$ \t\\\\\nCWR\t& $.088_{\\pm .002}$ \t& $.028_{\\pm .001}$ \t& $.114_{\\pm .004}$ \t& $.202_{\\pm .007}$ \t& $.021_{\\pm .000}$ \t& $.010_{\\pm .000}$ \t& $.024_{\\pm .000}$ \t& $.043_{\\pm .000}$ \t& $.083_{\\pm .001}$ \t& $.030_{\\pm .002}$ \t& $.095_{\\pm .002}$ \t& $.192_{\\pm .005}$ \t& $.037_{\\pm .001}$ \t& $.015_{\\pm .001}$ \t& $.044_{\\pm .002}$ \t& $.077_{\\pm .002}$ \t\\\\\n\\midrule\nSI\t& $.154_{\\pm .003}$ \t& $.072_{\\pm .003}$ \t& $.179_{\\pm .003}$ \t& $.311_{\\pm .004}$ \t& $.113_{\\pm .002}$ \t& $.055_{\\pm .002}$ \t& $.131_{\\pm .002}$ \t& $.224_{\\pm .002}$ \t& $.172_{\\pm .004}$ \t& $.088_{\\pm .003}$ \t& $.194_{\\pm .004}$ \t& $.343_{\\pm .005}$ \t& $.111_{\\pm .004}$ \t& $.049_{\\pm .003}$ \t& $.126_{\\pm .006}$ \t& $.229_{\\pm .006}$ \t\\\\\nEWC\t& $.229_{\\pm .001}$ \t& $.130_{\\pm .001}$ \t& $.264_{\\pm .002}$ \t& $.423_{\\pm .001}$ \t& $.165_{\\pm .005}$ \t& $.093_{\\pm .005}$ \t& $.190_{\\pm .005}$ \t& $.306_{\\pm .006}$ \t& $.201_{\\pm .001}$ \t& $.113_{\\pm .001}$ \t& $.229_{\\pm .001}$ \t& $.382_{\\pm .001}$ \t& $.186_{\\pm .004}$ \t& $.102_{\\pm .003}$ \t& $.214_{\\pm .004}$ \t& $.350_{\\pm .004}$ \t\\\\\n\\midrule\nGEM\t& $.165_{\\pm .002}$ \t& $.085_{\\pm .002}$ \t& $.188_{\\pm .002}$ \t& $.321_{\\pm .002}$ \t& $.093_{\\pm .001}$ \t& $.040_{\\pm .002}$ \t& $.106_{\\pm .002}$ \t& $.196_{\\pm .002}$ \t& $.175_{\\pm .004}$ \t& $.092_{\\pm .003}$ \t& $.196_{\\pm .005}$ \t& $.345_{\\pm .007}$ \t& $.136_{\\pm .003}$ \t& $.070_{\\pm .001}$ \t& $.152_{\\pm .004}$ \t& $.263_{\\pm .005}$ \t\\\\\nEMR\t& $.171_{\\pm .002}$ \t& $.090_{\\pm .001}$ \t& $.195_{\\pm .002}$ \t& $.330_{\\pm .003}$ \t& $.111_{\\pm .002}$ \t& $.052_{\\pm .002}$ \t& $.126_{\\pm .003}$ \t& $.225_{\\pm .004}$ \t& $.171_{\\pm .004}$ \t& $.090_{\\pm .003}$ \t& $.191_{\\pm .004}$ \t& $.337_{\\pm .006}$ \t& $.141_{\\pm .002}$ \t& $.073_{\\pm .001}$ \t& $.157_{\\pm .002}$ \t& $.267_{\\pm .003}$ \t\\\\\nDiCGRL\t& $.107_{\\pm .009}$ \t& $.057_{\\pm .009}$ \t& $.110_{\\pm .008}$ \t& $.211_{\\pm .009}$ \t& $.133_{\\pm .007}$ \t& $.079_{\\pm .005}$ \t& $.147_{\\pm .009}$ \t& $.241_{\\pm .012}$ \t& $.162_{\\pm .007}$ \t& $.084_{\\pm .007}$ \t& $.189_{\\pm .008}$ \t& $.320_{\\pm .007}$ \t& $.149_{\\pm .005}$ \t& $.083_{\\pm .004}$ \t& $.168_{\\pm .005}$ \t& $.277_{\\pm .008}$ \t\\\\\n\\midrule\nLKGE\t& $\\textbf{.234}_{\\pm .001}$ \t& $\\textbf{.136}_{\\pm .001}$ \t& $\\textbf{.269}_{\\pm .002}$ \t& $\\textbf{.425}_{\\pm .003}$ \t& $\\textbf{.192}_{\\pm .000}$ \t& $\\textbf{.106}_{\\pm .001}$ \t& $\\textbf{.219}_{\\pm .001}$ \t& $\\textbf{.366}_{\\pm .002}$ \t& $\\textbf{.210}_{\\pm .002}$ \t& $\\textbf{.122}_{\\pm .001}$ \t& $\\textbf{.238}_{\\pm .002}$ \t& $\\textbf{.387}_{\\pm .002}$ \t& $\\textbf{.207}_{\\pm .002}$ \t& $\\textbf{.121}_{\\pm .002}$ \t& $\\textbf{.235}_{\\pm .002}$ \t& $\\textbf{.379}_{\\pm .003}$ \t\\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Result comparison of link prediction on the union of the test sets in all snapshots.}\n\\label{tab:main_results}\n\\vspace{-5pt}\n\\end{table*}\n\n\\smallskip\n\\noindent\\textbf{Implementation details.}\nWe use TransE \\cite{TransE} as the base model and modify the competitors to do our task:\n\\begin{itemize}\n    \\item \\textit{Snapshot only}. For the $i$-th snapshot, we reinitialize and train a model only on the training set $\\mathcal{D}_i$.\n    \n    \\item \\textit{Re-training}. For the $i$-th snapshot, we reinitialize and train a model on the accumulated training data $\\cup_{j=1}^i \\mathcal{D}_j$.\n    \n    \\item \\textit{Fine-tuning}. For the $i$-th snapshot, the model inherits the learned parameters of the model trained on the previous snapshots, and we incrementally train it on $\\mathcal{D}_i$.\n    \n    \\item \\textit{Inductive} models. We train each model on the first snapshot and obtain the embeddings of unseen entities in the following snapshots by neighborhood aggregation.\n    \n    \\item \\textit{Dynamic architecture} models. For PNN, \n    following the implementation of \\cite{CKGE}, \n   \n    we freeze the parameters learned on previous snapshots and update new parameters. For CWR, after training on $\\mathcal{D}_1$, we replicate a model as the consolidated model. For the following $i$-th snapshot, we reinitialize and train a temporal model on $\\mathcal{D}_i$, and merge the temporal model into the consolidated model by copying new parameters or averaging old ones.\n    \n    \\item \\textit{Regularization} models. Since the base model parameters increase with the emergence of unseen entities and relations, we only use the parameters learned from the previous snapshot to calculate the regularization loss.\n    \n    \\item \\textit{Rehearsal} models. We store 5,000 training facts from previous snapshots and add them to the current training set of the $i$-th snapshot. After the learning, we randomly replace half of these facts with those in $\\mathcal{D}_i$.\n\\end{itemize}\n\nFor a fair comparison, we first tune the hyperparameters of the base model using grid-search: learning rate in \\{0.0005, 0.0001, 0.001\\}, batch size in \\{1024, 2048\\}, embedding dimension in \\{100, 200\\}.\nThen, we use the same base model for LKGE\\xspace and all competitors, and tune other hyperparameters.\nFor the regularization models, the $\\alpha$ of regularization loss is in \\{0.01, 0.1, 1.0\\}. \nFor our model, the $\\beta$ of MAE loss is in \\{0.01, 0.1, 1.0\\}.\nFor all competitors, we use Adam optimizer and set the patience of early stopping to 3.\nAll experiments are conducted with two NVIDIA RTX 3090 GPUs, two Intel Xeon Gold 5122 CPUs, and 384GB RAM.\n\n\n\\subsection{Link Prediction Accuracy}\n\nIn Table~\\ref{tab:main_results}, we run 5-seeds experiments for all models on our four link prediction datasets and report the means and standard deviations.  \nThe results show that:\n(\\romannumeral1) our model consistently achieves the best performance across all datasets. Some results of our model in \\textsc{Fact} even outperforms the re-training model. \nThis is because our masked KG autoencoder effectively improves information propagation based on both old and new embeddings, and the embedding regularization avoids catastrophic forgetting. \nFurthermore, most competitors only work well on \\textsc{Entity}, while our model shows stable and promising results on all these datasets.\n(\\romannumeral2) Re-training is far superior to most of other baseline models on \\textsc{Relation} and \\textsc{Hybrid}, while the gaps on \\textsc{Entity} and \\textsc{Fact} are small. \nWe believe this is because the KG embedding model learns two aspects of knowledge: relational patterns and entity embeddings. \nIn \\textsc{Entity} and \\textsc{Fact}, the relational patterns are stable, while in \\textsc{Relation} and \\textsc{Hybrid}, their relational patterns are constantly changing due to unseen relations. \nThese phenomena illustrate that the variation of relational patterns is more challenging for lifelong KG embedding.\n(\\romannumeral3) The inductive models are only trained on the first snapshot, and cannot transfer knowledge to unseen relations. \nSo their results are lower than other models, especially on \\textsc{Relation} and \\textsc{Hybrid} which contain many unseen relations. \n(\\romannumeral4) Since the learned parameters are not updated, PNN preserves the learned knowledge well. \nBut on \\textsc{Fact}, due to a few unseen entities, it lacks new learnable parameters, and the performance is not well. \nConversely, CWR averages the old and new model parameters, which does not work well on the embedding learning task. \n(\\romannumeral5) EWC performs well because it can model the importance of each parameter using Fisher information matrices to the learned snapshots to avoid catastrophic forgetting. \n(\\romannumeral6) Unlike the classification tasks, most parameters of KG embedding models correspond to specific entities or relations, so the training data cannot be divided into a few types, and we cannot use 5,000 old samples to replay the learned facts for all entities. \nTherefore, the performance of GEM, EMR and DiCGRL is limited.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=.82\\columnwidth]{figs_new\/mrr.pdf}\n\\caption{MRR changes. $\\mathcal{M}_i$ is trained for the $i$-th snapshot and evaluated using the test data of previous snapshots 1 to $i$.}\n\\label{fig:mrr}\n\\vspace{-5pt}\n\\end{figure}\n\n\\begin{table*}[!t]\n\\centering\n\\resizebox{1.0\\textwidth}{!}{\n\\begin{tabular}{lcccccccccccccccc}\n\\toprule\n\\multirow{2}{*}{Variants} & \\multicolumn{4}{c}{\\textsc{Entity}} & \\multicolumn{4}{c}{\\textsc{Relation}} & \\multicolumn{4}{c}{\\textsc{Fact}} & \\multicolumn{4}{c}{\\textsc{Hybrid}} \\\\\n\\cmidrule(lr){2-5} \\cmidrule(lr){6-9} \\cmidrule(lr){10-13} \\cmidrule(lr){14-17} & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 & MRR & H@1 & H@3 & H@10 \\\\\n\\midrule\nLKGE (full) & 0.234 \t& 0.136 \t& 0.269 \t& 0.425 \t& 0.192 \t& 0.106 \t& 0.219 \t& 0.366 \t& 0.210 \t& 0.122 \t& 0.238 \t& 0.387 \t& 0.207 \t& 0.121 \t& 0.235 \t& 0.379 \t\\\\\n$-$ fine-tuning\t& 0.123\t& 0.068\t& 0.136\t& 0.225\t& 0.126\t& 0.073\t& 0.146\t& 0.231\t& 0.154\t& 0.091\t& 0.176\t& 0.269\t& 0.109\t& 0.060\t& 0.126\t& 0.201\t\\\\\n$-$ masked KG autoencoder\t& 0.222\t& 0.124\t& 0.255\t& 0.415\t& 0.185\t& 0.100\t& 0.212\t& 0.355\t& 0.191\t& 0.105\t& 0.215\t& 0.369\t& 0.198\t& 0.111\t& 0.227\t& 0.367\t\\\\\n$-$ embedding transfer\t& 0.240\t& 0.141\t& 0.275\t& 0.433\t& 0.174\t& 0.091\t& 0.201\t& 0.339\t& 0.210\t& 0.123\t& 0.237\t& 0.390\t& 0.200\t& 0.112\t& 0.229\t& 0.372\t\\\\\n$-$ regularization\t& 0.166\t& 0.089\t& 0.184\t& 0.316\t& 0.040\t& 0.014\t& 0.049\t& 0.089\t& 0.175\t& 0.095\t& 0.195\t& 0.338\t& 0.154\t& 0.079\t& 0.171\t& 0.300\t\\\\\n\\bottomrule\n\\end{tabular}}\n\\caption{Ablation results of link prediction on the union of the test sets in all snapshots.}\n\\label{tab:ablation}\n\\vspace{-5pt}\n\\end{table*}\n\nTo show the performance evolution of our model during the learning process, we evaluate the model $\\mathcal{M}_i$ trained for the $i$-th snapshot using the test data from previous snapshots.\nWe report the MRR results in Figure~\\ref{fig:mrr}. \nWe can see that LKGE\\xspace can stably maintain the learned knowledge during lifelong learning. \nMoreover, on some snapshots like \\textsc{Entity} Snapshot 3, the knowledge update improves the performance on old test data,\nwhich shows that our old knowledge update has the potential for backward knowledge transfer.\n\n\\subsection{Knowledge Transfer Capability}\n\nTo evaluate the knowledge transfer and retention capability of all models, we report the FWT and BWT of MRR results in Figure~\\ref{fig:knowledge_transfer}. \nBecause of the embedding transfer, the FWT of LKGE\\xspace is higher than all lifelong learning competitors. \nEven on \\textsc{Relation} and \\textsc{Hybrid} where the KG schema changes, LKGE\\xspace still keeps the FWT capability well. \nMEAN and LAN are designed to transfer knowledge forward to embed new entities.  \nSo, they work well on \\textsc{Entity}.\nHowever, their FWT capability is limited on other datasets since they cannot update the old embeddings to adapt to new snapshots.\n\nOn the other hand, BWT is usually negative due to the overwriting of learned knowledge. \nPNN, MEAN, and LAN do not update the learned parameters, so their BWT scores are ``NA''. \nThe poor BWT scores of CWR show the harmful effects of the average operation.\nThe BWT capability of rehearsal models is also not good because they cannot store enough facts. \nLKGE\\xspace achieves good BWT scores as the embedding regularization can well balance the learning of new and the update of old embeddings.\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{figs_new\/FWT_BWT.pdf}\n\\caption{Forward transfer and backward transfer of MRR.}\n\\label{fig:knowledge_transfer}\n\\vspace{-5pt}\n\\end{figure}\n\n\\subsection{Learning Efficiency}\nHere, we compare the training time. \nWe report the time cost on \\textsc{Fact} as all snapshots of \\textsc{Fact} have the same training set size, which is easier for comparison.\nFigure~\\ref{fig:time_cost} shows the results.\nUnsurprisingly, re-training is most time-consuming. \nSnapshot is also costly because it cannot inherit knowledge from previous snapshots. \nBy contrast, our model is most efficient, and its advantage is more significant in the final snapshot. \nThis is because the embedding transfer can use the learned knowledge to accelerate the learning of new facts.\n\n\n\\begin{figure}[t]\n\\includegraphics[width=\\columnwidth]{figs_new\/time.pdf}\n\\caption{Cumulative time cost on \\textsc{Fact}.}\n\\label{fig:time_cost}\n\\vspace{-5pt}\n\\end{figure}\n\n\n\\subsection{Ablation Study}\n\\label{subsec:ablation}\nWe conduct an ablation study to validate the effectiveness of each model component. \nWe design four variants of LKGE\\xspace: ``w\/o fine-tuning'', ``w\/o masked KG autoencoder'', ``w\/o embedding transfer'' and ``w\/o regularization''. \nThe ``w\/o fine-tuning'' variant is trained on $\\mathcal{D}_1$ and performs the embedding transfer on other $\\mathcal{D}_i$. \nThe latter three variants disable the specific components from our model. \nThe results are shown in Table~\\ref{tab:ablation}. \nWe observe that \n(\\romannumeral1) although fine-tuning is disabled, ``w\/o fine-tuning'' can still perform well with only the knowledge from the first snapshot, showing that embedding transfer can effectively transfer learned knowledge to unseen entities and relations. \n(\\romannumeral2) Compared with the full model, both ``w\/o masked KG autoencoder'' and ``w\/o regularization'' significantly drop, demonstrating the effects of masked KG autoencoder and knowledge retention.\n(\\romannumeral3) Embedding transfer enables the model to be trained at a starting point closer to the optimal parameters and stabilizes the embedding space. There are declines when using the embedding transfer on \\textsc{Entity}. This is because the \\textsc{Entity} contains massive new entities and needs more plasticity rather than stability. However, \non \\textsc{Relation} and \\textsc{Hybrid}, the results of ``w\/o embedding transfer''  are lower than the full model, showing that embedding transfer can reduce the interference caused by the KG schema changes.\nOn \\textsc{Fact}, the results of ``w\/o embedding transfer'' are similar to the full model. \nThis shows that, even without embedding transfer, the model can still capture the knowledge.\n  \n\n\\section{Conclusion and Future Work}\nThis paper proposes and studies lifelong embedding learning for growing KGs. \nAiming at better knowledge transfer and retention, we propose a lifelong KG embedding model consisting of a masked KG autoencoder, embedding transfer, and embedding regularization. \nThe experimental results on four newly-constructed datasets show better link prediction accuracy, knowledge transfer capability, and learning efficiency of our model. \nIn future work, we plan to investigate lifelong embedding learning in long-tail and low-resource settings.\n\n\\section*{Acknowledgments} \nThis work was supported by National Natural Science Foundation of China (No. 62272219).\n \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConsider the system\n\\begin{equation}\\label{eq:1}\n\\ddot{u}=-\\dfrac{u}{|u|^{3}}+ \\varepsilon \\nabla_{u} U(t, u, \\varepsilon), u \\in \\mathbb{R}^{3}\n\\end{equation}\nwhere $U$ is a $C^{\\infty}$-function defined on $\\mathbb{R} \\times \\mathcal{U} \\times [0, \\varepsilon_{*}]$ for an open set $\\mathcal{U} \\subset \\mathbb{R}^{3}$ containing the origin $u=0$ and some $\\varepsilon_{*}>0$.  The function $U(t, u, \\varepsilon)$ is assumed to be $T$-periodic in $t$.  This system models a Kepler problem with an external periodic force. \\par\n\nIt is a classical problem in the theory of perturbations to look for periodic solutions for small $\\varepsilon$. {Traditionally, due to the proper-degeneracy of the Kepler problem, all the collisionless bounded orbits are closed and some non-degeneracy condition has to be imposed on the function $U(t,u,\\varepsilon )$. This condition has to be verified for concrete $U(t,u,\\varepsilon )$}. In typical situations the periodic orbits are found in the neighbourhood of a prescribed\nclosed orbit of the unperturbed Kepler problem and in particular they have no collisions.\\par \n\nA result of different nature was obtained in \\cite{BOZ}, in which it has been shown that for any smooth function $U$, the equation \\eqref{eq:1} has an arbitrarily large number of $T$-periodic solutions if $\\varepsilon$\nhas been accordingly chosen small enough. {As it is usually the case in the study of periodic orbits with singular potentials,} in the framework of \\cite{BOZ} the function $U$ was supposed to be globally defined; that is,\n\\begin{equation}\\label{global}\n\\mathcal{U}=\\mathbb{R}^{3}.\n\\end{equation} Also, the periodic solutions are understood in a generalized sense. They can have collisions but the energy and the direction of motion must be preserved after each bouncing at a time $t_*$ with $u(t_* )=0$.\n\\par There are good topological reasons for the introduction of generalized solutions. Let $\\mathcal{M} $ be the set of $T$-periodic solutions of the Kepler problem ($\\varepsilon =0$). After including solutions with collisions it becomes a manifold $$\\tilde{\\mathcal{M}}=\\bigcup_{n=1}^{\\infty} \\mathcal{M}_n$$ with infinitely many\nconnected components $\\mathcal{M}_n$, where each of them is compact. The periodic solutions of \\eqref{eq:1} for $\\varepsilon \\neq 0$ are obtained as bifurcations from these manifolds. The compactness is essential if we want to guarantee that\nthis bifurcation is always produced.\\par On the other hand, as we willexplain in this note, since our method is perturbative, the condition \\eqref{global} is not\nso essential. Since the components $\\mathcal{M}_n$ converge to the origin as $n\\to \\infty$, the projection of $\\mathcal{M}_n$ on the configuration space will lie inside any neighbourhood $\\mathcal{U}$ of the origin for $n$ large enough. In consequence, when \\eqref{global}\ndoes not hold,  it is still possible\nto obtain bifurcations from $\\mathcal{M}_n$ when $n$ is large and the projection of $\\mathcal{M}_{n}$ to the configuration space has a neighborhood in which the function $U$ is well-defined. This observation may seem minor but it is of importance if we want to apply this technique to some classical problems in Celestial Mechanics. A typical situation arises in the so-called circular or elliptic restricted three-body problem, where the perturbation $U$ on the asteroid is due to a periodic gravitational force from another massive body. {The function $U$} will then be singular at the position of this massive body which is at some distance of $u=0$. Note that for the circular problem we shall consider the problem in a fixed inertial frame, and shall not go to some rotating frame to reduce the system to an autonomous one.\n\\par\nWe shall then apply this to show that an arbitrarily large number of generalized $T$-periodic solutions of a general class of circular or elliptic restricted three body problems with any eccentricity $e \\in [0, 1)$  can be found  when the mass ratio between the primaries is taken as an external parameter that we assume to be sufficiently small. These solutions may collide with the big primary.  In fact we will find periodic motions of the small body in a neighbourhood of this big primary. The common year of the primaries will also be a period for the third body and each year this small body will make a large number of revolutions around the big primary. There are many other results on the existence of periodic solutions of the circular or elliptic restricted problem and many families of periodic solutions have been identified. See for instance \\cite{CPS, AL, {PYFN}}. A possible novelty of our result is mainly in the absence of additional non-degeneracy conditions and the result is more global in the sense that the continuation is made from manifolds made of periodic orbits of the Kepler problem instead of particular periodic orbits of an approximating system.\nIt seems reasonable to expect other applications in similar models. For instance, in an elliptic restricted N-body problem where the primaries are assumed to move on an elliptic homographic motion and the infinitesimal body is assumed to stay close to a primary.  \n\\par\nThe rest of the paper is organized in three sections. In Section \\ref{22} we work with the equation \\eqref{eq:1} and go back to the main theorem in \\cite{BOZ}. We explain the modifications in the proof allowing to eliminate the condition (\\ref{global}). The result in \\cite{BOZ} was obtained via the use of Levi-Civita regularization in dimension 2 and Kustaanheimo-Stiefel regularization in dimension 3, together with the application of a theorem of Weinstein \\cite{Weinstein}. Generalized periodic solutions can be characterized equivalently by periodic orbits of the regularized system. This was proved in \\cite{BOZ} in dimension 2 and we will prove that it is also the case in dimension 3. This proof will require a delicate topological result on the lifting of piecewise smooth paths via the Hopf fibration. For smooth paths this is a direct consequence of general results in the theory of Ehresmann connections, and the modifications required for the piecewise smooth case are explained in Section \\ref{33}. Finally, the application to the circular or elliptic restricted three body problem is presented in Section \\ref{44}.\\par\nIn this note we will work in the space $u \\in \\mathbb{R}^{3}$ but the results are easily adapted to the simpler case of the plane $u \\in \\mathbb{R}^{2}$ or the line $u \\in \\mathbb{R}$. \n\n\\section{Periodic Solutions via Regularization}\\label{22}\nFollowing \\cite{BOZ} we say that a continuous and $T$-periodic function $u: \\mathbb{R} \\to \\mathbb{R}^{3}$ is a generalized $T$-periodic solution of \\eqref{eq:1} if it satisfies the following conditions:\n\\begin{enumerate}\n\\item $\\mathcal{Z}=\\{t \\in \\mathbb{R}: u(t)=0\\}$ is a discrete set;\n\\item for any open interval $I \\subset \\mathbb{R} \\setminus \\mathcal{Z}$ the function u is in $C^{\\infty}(I, \\mathbb{R}^{3})$ and satisfies \\eqref{eq:1} on this interval;\n\\item For any $t_{0} \\in \\mathcal{Z}$ the limits below exist\n$$\\lim_{t \\to t_{0}} \\dfrac{u(t)}{|u(t)|}, \\qquad \\qquad \\lim_{t \\to t_{0}} \\Bigl( \\dfrac{1}{2} |\\dot{u}(t)|^{2}-\\dfrac{1}{|u(t)|} \\Bigr).$$\n\\end{enumerate}\n\nNote that for any classical solution of \\eqref{eq:1} tending to a collision at $t_{0}$, the left and right limits $t \\to t_{0}^{-}$ and $t \\to t_{0}^{+}$ always exist (see \\cite{Sperling}). The crucial point in the above definition is that they coincide. \n\nThe above definition extends in an obvious way for the case of sub-harmonic solutions having period $\\eta T$ with $\\eta \\in \\mathbb{Z}, \\eta \\ge 2$.\n\\par We will prove the existence of generalized $T$-periodic solutions using a regularization technique. We refer to \\cite{BDP} and \\cite{BOV} for an alternative use of variational techniques. \nFollowing \\cite{ZhaoKS, BOZ} we consider the Kustaanheimo-Stiefel regularization. The skew-field of quaternions is denoted by $\\mathbb{H}$. The space of purely imaginary quaternions $\\mathbb{IH}=\\{z \\in \\mathbb{H}: \\Re(z)=0\\}$ is naturally a three-dimensional vector field over the reals. The map \n$$ \\Pi:  \\mathbb{H} \\to \\mathbb{IH}, \\qquad z \\mapsto \\bar{z} i z$$ \nplays an important role. \n\nWe set $\\mathbb{T}=\\mathbb{R}\/T \\mathbb{Z}$ for the quotient space of the real line of time by the lattice $T \\mathbb{Z}$, so for $t \\in \\mathbb{R}$ we denote by $\\bar{t}$ its corresponding quotient. The manifold \n$$\\mathcal{M}=\\mathbb{H} \\times \\mathbb{H} \\times \\mathbb{T} \\times \\mathbb{R}$$\nwith points $(z=z_{0}+z_{1} i + z_{2} j + z_{3} k, w=w_{0}+w_{1} i + w_{2} j + w_{3} k, \\bar{t}, \\tau)$ is endowed with the symplectic two-form\n$$\\omega=\\sum_{l=0}^{3} d z_{l} \\wedge d w_{l} + d t \\wedge d \\tau$$\nOn the symplectic manifold $(\\mathcal{M}, \\omega)$ we consider the Hamiltonian function\n$$K_{\\varepsilon}: \\mathcal{M} \\to \\mathbb{R}, K_{\\varepsilon}(z, w, t,\\tau)=\\dfrac{|w|^{2}}{8}+\\tau |z|^{2} -1 + \\varepsilon P(t, z, \\varepsilon)$$\nwith $P(t,z,\\varepsilon)=|z|^{2} U(t, \\bar{z} i z, \\varepsilon)$. This is the regularized system of the spatial forced Kepler problem in extended phase space \\cite{BOZ}. \n\nThis function is invariant under the Hamiltonian $S^{1}$-action\n$$g \\ast (z, w, \\bar{t}, \\tau)=(g z, g w, \\bar{t}, \\tau),$$\nwhere we realize $S^{1}$ as $\\{g \\in \\mathbb{C}: |g|=1\\}$. By standard theory, the corresponding moment map\n$$BL(z, w, \\bar{t}, \\tau)=\\bar{z} i w$$\nis a first integral of the system. \n\nAssume now that $X(s)=(z(s), w(s), \\bar{t}(s), \\tau(s))$ is a solution of the Hamiltonian system $(\\mathcal{M}, \\omega, K_{\\varepsilon})$ which lies in $K_{\\varepsilon}^{-1}(0) \\cap BL^{-1}(0)$ and $X(s+S)=g \\ast X(s), s \\in \\mathbb{R}$ for some $S >0$ and $g \\in S^{1}$. Then $X(s)$ gives a generalized periodic solution of \\eqref{eq:1}, as a consequence of Lem 5.1, Rem 5.2 in \\cite{BOZ}, with period $\\eta T$ where $\\eta$ is the degree of the map $\\bar{t}(s)$, given via a lift of this mapping as \n$$t(s+S)=t(s)+\\eta T.$$\n\nThese solutions were produced by a perturbation argument in the symplectically reduced manifold\n$$\\mathcal{M}_{0}=BL^{-1}(0)\/S^{1}.$$\nIndeed by applying a result of Weinstein \\cite{Weinstein}, we obtained a continuation from the sets $\\bar{\\Lambda}_{k}$ of periodic orbits of $K_{0}$ which are obtained as quotients from the sets\n$$\\Lambda_{k}=\\{X=(z, w, \\bar{t}, \\tau) \\in K_{0}^{-1}(0) \\cap BL^{-1}(0): \\tau=\\tau_{k}\\}$$\nwith $\\tau_{k}=\\Bigl(\\dfrac{\\sqrt{2} k \\pi}{T}\\Bigr)^{\\frac{2}{3}}, \\quad k=1,2,\\cdots$\n\nIn principle the set \n$$E_{k}=\\{\\bar{z} i z: X=(z, w, \\bar{t}, \\tau) \\in \\Lambda_{k}\\}$$\ncan occupy any region in $\\mathbb{IH}$. For this reason, it was assumed in \\cite{BOZ} that (\\ref{global}) holds. Nevertheless, by the result by Weinstein, the only requirement for continuation of periodic orbits from the periodic manifold  $\\overline{\\Lambda}_{k}$  is that the perturbation be defined in a neighborhood of  $\\overline{\\Lambda}_{k}$ and is sufficiently small. The computations in \\cite{BOZ} therefore remain valid when $\\mathcal{U} \\subsetneq  \\mathbb{R}^{3}$ as long as $E_{k} \\subset \\mathcal{U}$ holds.\n\nThe set $\\mathcal{U}$ can be otherwise chosen arbitrarily. For instance we may choose $\\mathcal{U}$ small. The above-mentioned inclusion might not hold for all $k \\ge 1$. Nevertheless we now remark that this inclusion holds for $k$ sufficiently large.\n\nFrom $X \\in K_{0}^{-1} (0)$ we deduce that \n$$\\tau_{k} |z|^{2}+\\dfrac{|w|^{2}}{8}=1,$$\nwhich implies that $|z| \\le \\tau_{k}^{-1\/2} \\to 0$ as $k \\to + \\infty$. \n\nThe Hamiltonian function\n$$\\bar{K}_{\\varepsilon}(\\bar{X})=K_{\\varepsilon}(X)$$\nis not well-defined on the whole manifold $\\mathcal{M}_{0}$ but it is well-defined on the open set\n$$\\mathcal{D}_{\\delta}=\\{ \\bar{X}\\in \\mathcal{M}_0 :\\; X=(z,w,\\bar{t},\\tau),\\; |z| < \\delta \\},$$\nwhere $\\delta >0$ is such that\n$$\\{u \\in \\mathbb{R}^{3}: |u| < \\delta^{2}\\} \\subset \\mathcal{U}.$$\n\nThe periodic manifold $\\bar{\\Lambda}_{k}$ is thus contained in $\\mathcal{D}_{\\delta}$ for $k$ large enough. For these values of $k$, the argument in \\cite{BOZ} holds without change.\n\nWe sum up the previous discussions in the following theorem:\n\\begin{theo} \\label{theo: local} Assume that $U: \\mathbb{R} \\times \\mathcal{U} \\times [0, \\varepsilon_{*}] \\to \\mathbb{R}$ is a $C^{\\infty}$-function satisfying\n$$U(t+T, u, \\varepsilon)=U(t, u, \\varepsilon)$$\nfor some fixed $T >0$ and an open set $\\mathcal{U} \\subset \\mathbb{R}^{3}$ containing $u=0$. Given an integer $l \\ge 1$ there exists $\\varepsilon_{l}>0$ such that \\eqref{eq:1} has at least $l$ generalized $T$-periodic solutions (lying in $\\mathcal{U}$) for each $\\varepsilon \\in ]0, \\varepsilon_{l}[$.\n\\end{theo}\n\n\\begin{rem} The same result holds in the plane and in a line as well, which can be obtained by following the same line of argument and using Levi-Civita regularization instead. \n\\end{rem}\n\nIn the previous proof we have used that the solutions $X(s)=(z(s), w(s), t(s), \\tau(s))$ of the Hamiltonian system $(M, \\omega, K_{\\varepsilon})$ lying in \n$$\\{K_{\\varepsilon}(X)=0, BL(X)=0\\}$$ \nand satisfying \n$$X(s+S)=g \\ast X(s), t(s+S)=t(s)+\\eta T, s \\in \\mathbb{R}$$\nfor some $S>0, g \\in S^{1}$ and $\\eta \\in \\mathbb{Z}$ lead to generalized periodic solutions of \\eqref{eq:1} with period $\\eta T$. These are given by \n$$u(t)=\\overline{z(s(t))} i z(s(t)),$$\nwhere $s=s(t)$ is the inverse of the homeomorphism $t=t(s)$. Note that in the previous proof $\\eta=1$ for $\\varepsilon=0$ and so $\\eta=1$ also for small $\\varepsilon$, since $\\eta=\\eta_{\\varepsilon}$ is a continuous function taking values in $\\mathbb{Z}$.\n\nThe rest of the Section will be devoted to show that it is possible to go from  generalized solutions of \\eqref{eq:1} to solutions $X(s)$ of $(M,\\omega, K_{\\varepsilon})$ satisfying the above conditions. \n\nA similar discussion for planar solutions and Levi-Civita regularization can be found in Section 4 of \\cite{BOZ}. We now explain that the same holds for the spatial case as well. \n\nNow assume that $u(t)$ is a generalized $T$-periodic solution of  \\eqref{eq:1} and consider the function $\\sigma(t)=\\dfrac{u(t)}{|u(t)|}.$ In principle $\\sigma(t)$ is only defined for $t \\in \\mathbb{R} \\setminus \\mathcal{Z}$ but the notion of generalized solution  implies that it has a continuous extension to the whole real line. At collisions, the function $\\sigma$ is not necessarily smooth but there is some control on its velocity. More precisely for each $t_{0} \\in \\mathcal{Z}$, there holds\n$$\\dot{\\sigma}(t)=O((t-t_{0})^{-1\/3})\\,\\, \\quad \\hbox{ as } t \\to t_{0}.$$\nThis asymptotic expansion follows from the formula\n$$\\dot{\\sigma}(t)=\\dfrac{|u(t)|^{2} \\dot{u}(t)  - \\langle u(t), \\dot{u}(t) \\rangle u(t)}{|u(t)|^{3}}, \\qquad t \\in \\mathbb{R} \\setminus \\mathcal{Z}$$\ntogether with the classical estimates at collisions (see \\cite{Sperling})\n$$u(t)=a (t-t_{0})^{2\/3} + b(t) (t-t_{0})^{4\/3},$$\n$$\\dot{u}(t)=\\dfrac{2}{3}a (t-t_{0})^{-1\/3} + c(t) (t-t_{0})^{1\/3},$$\nwhere $a \\in \\mathbb{R}^{3} \\setminus \\{0\\}$ and $b(t), c(t)$ are bounded functions defined in a neighborhood of $t-t_{0}$.\nWe now identify $\\mathbb{R}^{3}$ and $\\mathbb{IH}$, $u=u(t), u=u_{1} i + u_{2} j + u_{3} k$ and now view $S^{2}$ and $S^{3}$ as unit spheres in $\\mathbb{IH}$ and $\\mathbb{H}$ respectively. The properties of $\\sigma: \\mathbb{R} \\to S^{2}$ suggests the following definition.\n\n\\begin{defi} Given an interval $[0, T]$ and a partition of it \n$$\\mathcal{P}:b_{0}=0 < t_{1 } < \\cdots < t_{N}=T,$$\n{the path $\\alpha:[0,  T] \\to S^{d}$, $d=2$ or $3$,} is a $\\mathcal{P}$-path, if it is continuous, the restriction $\\alpha |_{]t_{i}, t_{i+1}[}$ is $C^{\\infty}$ and the integral below is finite:\n$$\\int_{t_{i}}^{t_{i+1}} |\\dot{\\alpha} (t) | d t < \\infty, i=0, \\cdots, N-1.$$\n\\end{defi}\nThe map $\\Pi :z \\mapsto \\bar{z} i z$ sends $S^{3}$ onto $S^{2}$ and we can define the Hopf map as the restriction \n$$\\pi: S^{3} \\to S^{2}, z \\mapsto \\bar{z} i z.$$\nNote that, in the notation of \\cite[pp. 376]{Cushman}, $x_{1}=-z_{0}, x_{2}=z_{1}, x_{3}=z_{2}, x_{4}=z_{3}$.\n\nWe will need the following result on the lifting of $\\mathcal{P}$-paths.\n\\begin{lem} \\label{lem: ehresmann} Assume that $\\gamma: [0, T] \\to S^{2}$ is a $\\mathcal{P}$-path. Then there exists a $\\mathcal{P}$-path $\\Gamma: [0, T] \\to S^{3}$ satisfying $\\pi \\circ \\Gamma=\\gamma$ and\n$$\\Re\\{\\bar{\\Gamma}(t) i \\dot{\\Gamma}(t)\\}=0, \\hbox{  when } t \\neq t_{i}.$$\nMoreover, if $\\gamma(0)=\\gamma(T)$ then $\\Gamma(T)=g \\Gamma(0)$ for some $g \\in S^{1}$.\n\\end{lem}\n\nThe proof of this result is postponed to the next Section. It is worth to observe that the last conclusion on closed paths follows easily from the  structure of the fibers of the Hopf map\n$$\\pi^{-1}(\\gamma(0))=\\{g \\Gamma(0): g \\in S^{1}\\}.$$\n\nLet us take our generalized $T$-periodic solution $u(t)$. The set $[0, T] \\cap \\mathcal{Z}$ defines a partition $\\mathcal{P}$. Then $\\sigma$ is a $\\mathcal{P}$-path and Lemma \\ref{lem: ehresmann} can be applied to $\\gamma=\\sigma$. In consequence there exists a $\\mathcal{P}$-path $\\Sigma: [0, T] \\to S^{3}$ with $\\pi \\circ \\Sigma=\\sigma$ and $\\Re\\{\\bar{\\Sigma} i \\Sigma\\}=\\Re\\{\\bar{\n\\Sigma} i \\dot{\\Sigma}\\}=0$. Note that $\\bar{\\Sigma} i \\Sigma \\in S^{2} \\subset \\mathbb{IH}$.\n\nSince $\\sigma$ is $T$-periodic we extend $\\Sigma$ to the whole real line via the formula\n$\\Sigma(t+T)=g \\Sigma(t), t \\in \\mathbb{R}.$\nThen $\\Sigma: \\mathbb{R} \\to S^{3}$ is continuous.\n\nWe will define $X(s)=(z(s), w(s), t(s), \\tau(s))$ in terms of $\\Sigma$. The coordinate $t(s)$  is defined from Sundman integral as in \\cite{BOZ}. The set $\\mathcal{Z}^{*}=t^{-1} (\\mathcal{Z})$ is discrete. Then we have\n$$z(s)=r(s) \\Sigma(t(s)), s \\in \\mathbb{R},$$\nwhere \n$$r(s)=|u(t(s))|^{1\/2}, w(s)=4 z'(s), \\tau(s)=-E(t(s))+\\varepsilon  U(t(s), \\bar{z}(s) i z(s), \\varepsilon), s \\in \\mathbb{R} \\setminus \\mathcal{Z}^{*}.$$ \n\nAfter this definition the proof follows along the same line of \\cite{BOZ}. The only essential difference is the verification of the additional condition $BL(X(s))=0$ which is equivalent to \n$$\\Re\\{\\bar{z}(s) i z'(s)\\}=0.$$\nTo check this condition, it is sufficient to observe that\n$$z'(s)=r'(s) \\Sigma(t(s))+r(s)^{3} \\dot{\\Sigma} (t(s)) $$\nand thus the condition follows directly.\n\n\\section{Lifting of paths via the Hopf map}\\label{33}\nFor each $z \\in S^{3}$ we consider the tangent space\n$$T_{z}(S^{3})=\\{w \\in \\mathbb{H}: w \\perp z\\}$$\nwith vertical and horizontal subspaces\n$$\\hbox{Vert}_{z}=\\ker (d \\Pi )_{z} \\cap T_{z} (S^{3})$$\n$$\\hbox{Hor}_{z}=[\\ker (d   \\Pi  )_{z}]^{\\perp} \\cap T_{z} (S^{3})$$\nThe real vector space $\\hbox{Vert}_{z}$ has dimension one and is spanned  by $i z$. The space $\\hbox{Hor}_{z}$ has dimension two and is spanned by  the vectors $jz, kz $. Moreover\n$$(d \\pi)_{z} (\\hbox{Hor}_{z})=T_{\\pi (z)}(S^{2}).$$\nThe splitting\n$$T_{z}(S^{3})=\\hbox{Vert}_{z} \\oplus \\hbox{Hor}_{z}$$\nis clearly smooth with respect to the base points and thus defines an Ehresmann connection associated to the submersion $\\pi$.  We refer to \\cite[Chapter VIII]{Cushman} for more details.\n\nThe results in \\cite{Cushman} imply that this connection is ``good''. This means that, given a $C^{\\infty}$ curve $\\alpha: [t_{0}, t_{1}]  \\to S^{2}$ and a point $\\xi \\in \\pi^{-1}  (\\alpha (t_{0}))$, there exists a horizontal lift starting at $\\xi$. This lift is a $C^{\\infty}$ curve $A:[t_{0}, t_{1}] \\to S^{3}$ satisfying $A(t_{0})=\\xi, \\pi \\circ A = \\alpha, \\dot{A}(t) \\in \\hbox{Hor}_{A(t)}$ for each $t \\in [t_{0}, t_{1}]$. \n\nFrom our point of view, the key point is the following characterization of the horizontal component\n$$\\hbox{Hor}_{z}=\\{w \\in \\mathbb{H}: w \\perp z, \\Re\\{\\bar{z} i w\\}=0\\}.$$\nFrom this observation we could derive a version of Lemma \\ref{lem: ehresmann} in the class of $C^{\\infty}$-paths, or even in the class of piecewise $C^{\\infty}$-paths. This second case will follow by an iterative application of the above-mentioned lifting principle for good Ehresmann connections. However we must work in the larger class of $\\mathcal{P}$-paths and the proof uses specific properties of the Hopf map. To prove Lemma \\ref{lem: ehresmann} we will restrict to the case $\\mathcal{P}: t_{0}=0<t_{1}=T$ so that $[t_{0}, t_{1}]=[0, T].$ The iterative argument for a general partition is left to the reader. We also assume that $\\gamma (t)\\neq i$ for each $t\\in [t_0,t_1]$ and, at the end of the proof, we will show that there is no loss of generality. \n\n\nFollowing \\cite{Cushman}, we shall obtain $\\Gamma(t)$ from the solution of a system of differential equations in the plane. \nWe sketch some computations which are similar to those in \\cite{Cushman}, pp. 376-377. Note that our map $\\pi$ does not coincide with the map $\\pi$ in \\cite{Cushman}, although they are conjugate. \n\\par The condition $\\dot{\\Gamma}(t) \\in \\hbox{Hor}_{\\Gamma(t)}$ implies that\n$$\\dot{\\Gamma}(t)=(\\lambda_{1}(t)j + \\lambda_{2}(t)k) \\Gamma(t) ,$$\nwhere $\\lambda_{1}, \\lambda_{2}: [t_{0}, t_{1}] \\to \\mathbb{R}$ are functions to be determined. With the notation $\\Gamma=\\Gamma_{0} + \\Gamma_{1} i + \\Gamma_{2} j + \\Gamma_{3} k$ we are led to the equations\n\n\\begin{equation}\\label{eq: one}\n\\dot{\\Gamma}_{0}=-\\lambda_{1} \\Gamma_{2} -\\lambda_{2} \\Gamma_{3}, \\qquad \\dot{\\Gamma}_{1}=\\lambda_{1} \\Gamma_{3} -\\lambda_{2} \\Gamma_{2}\n\\end{equation}\n\\begin{equation}\\label{eq: two}\n\\dot{\\Gamma}_{2}=\\lambda_{1} \\Gamma_{0} +\\lambda_{2} \\Gamma_{1}, \\qquad \\dot{\\Gamma}_{3}=-\\lambda_{1} \\Gamma_{1} +\\lambda_{2} \\Gamma_{0}.\n\\end{equation}\nFrom the notation $\\gamma=\\gamma_{1} i +\\gamma_{2} j + \\gamma_{3} k$ and the definition of $\\pi$, \n\\begin{equation}\\label{eq: arbol}\n\\gamma_{1}=\\Gamma_{0}^{2}+\\Gamma_{1}^{2}-\\Gamma_{2}^{2}-\\Gamma_{3}^{2}, \\qquad \\begin{pmatrix} \\Gamma_{2} & - \\Gamma_{3} \\\\ \\Gamma_{3} & \\Gamma_{2} \\end{pmatrix} \\begin{pmatrix} \\Gamma_{1}  \\\\ \\Gamma_{0}\\end{pmatrix}=\\dfrac{1}{2} \\begin{pmatrix} \\gamma_{2}  \\\\ \\gamma_{3}\\end{pmatrix}.\n\\end{equation}\nIn particular, $\\Gamma_{2}^{2} + \\Gamma_{3}^{2}=\\dfrac{1}{2} (1-\\gamma_{1})$. \n\nWe recall that the curve $\\gamma$ does not pass through the point $i$ and therefore $\\gamma_1 <1$ and  $\\Gamma_{2}^{2}+\\Gamma_{3}^{2}>0$ everywhere. This allows us to use the equations in \\eqref{eq: one} to solve $\\lambda_{1}$ and $\\lambda_{2}$. \n\nPlugging the corresponding formulas in \\eqref{eq: two} and taking into account the equations in \\eqref{eq: arbol} we obtain\n\\begin{equation}\\label{eq: liason}\n(1-\\gamma_1 )\\begin{pmatrix} \\dot{\\Gamma}_{2}  \\\\ \\dot{\\Gamma}_{3}\\end{pmatrix}=-M\\begin{pmatrix} \\dot{\\Gamma}_{1}  \\\\ \\dot{\\Gamma}_{0}\\end{pmatrix},\\; \\; M=\\begin{pmatrix} \\gamma_{2}&\\gamma_3 \\\\ \\gamma_3 &-\\gamma_2 \\end{pmatrix}.\n\\end{equation}\nAlso, from the second identity in \\eqref{eq: arbol},\n\\begin{equation}\\label{eq:deuxl}\n(1-\\gamma_1 )\\begin{pmatrix} \\Gamma_{1}  \\\\ \\Gamma_{0}\\end{pmatrix}=  \\begin{pmatrix} \\Gamma_{2}&\\Gamma_3 \\\\ -\\Gamma_3 &\\Gamma_2 \\end{pmatrix}\n\\begin{pmatrix}\\gamma_2  \\\\ \\gamma_3 \\end{pmatrix} =M\\begin{pmatrix} \\Gamma_{2} \\\\ \\Gamma_3  \\end{pmatrix}.\n\\end{equation}\nDifferentiating with respect to $t$ and substituting the result in \\eqref{eq: liason} we find\n$$\\begin{pmatrix} \\dot{\\Gamma}_{2}  \\\\ \\dot{\\Gamma}_{3}\\end{pmatrix}=-\\frac{1}{2(1-\\gamma_1)} [\\dot{\\gamma}_1 M\\begin{pmatrix} \\Gamma_1  \\\\ \\Gamma_0 \\end{pmatrix}+M\\dot{M} \\begin{pmatrix} \\Gamma_{2}  \\\\ \\Gamma_{3}\\end{pmatrix}].$$\nIn these computation we have used that $M^2 =(1-\\gamma_1^2)I$. Combining this identity with \\eqref{eq:deuxl}, we are led to a planar linear system of the type\n\\begin{equation}\\label{eq: ps}\n\\begin{pmatrix} \\dot{\\Gamma}_{2}  \\\\ \\dot{\\Gamma}_{3}\\end{pmatrix}=B(t) \\begin{pmatrix} \\Gamma_{2}  \\\\ \\Gamma_{3}\\end{pmatrix}.\n\\end{equation}\nwhere the coefficients of the matrix $B(t)$ are linear combinations of functions of the type $\\dfrac{\\gamma_{i} \\dot{\\gamma}_{j}}{1-\\gamma_{1}}$ and $ \\dfrac{\\dot{\\gamma}_{i}}{1-\\gamma_{1}}$. \n\nMost probably this matrix is not continuous at $t=t_0=0$ and $t=t_1=T$. \nNevertheless, since we know that $\\gamma $ is $C^{\\infty}$ in $]0, T[$ and $\\int_{0}^{T} |\\dot{\\gamma}(t)| d t < \\infty$, we deduce that the coefficients of $B(t)$ belong to the Lebesgue space $L^{1}(]0,T[)$ .  We are assuming that $\\gamma_{1}(t) \\neq 1$ if $t \\in [0, T].$ In consequence, the matrix $B$ is integrable and the system \\eqref{eq: ps} satisfies the conditions of Carath\\'eodory's theorem (see for instance \\cite[Chapter 2]{CL}). Given $\\xi \\in \\pi^{-1}(\\gamma(0))$, we impose the initial condition \n$$\\Gamma_{2}(0)=\\xi_{2}, \\Gamma_{3}(0)=\\xi_{3}$$\nto obtain a unique solution of the Cauchy problem for \\eqref{eq: ps} defined on the whole interval $[0,T]$. These functions $\\Gamma_{2}$ and $\\Gamma_{3}$ are absolutely continuous in $[0, T]$. The functions $\\Gamma_0$ and $\\Gamma_1$ are\ndefined from the identity \\eqref{eq:deuxl} and they are also absolutely continuous. In particular there holds \n$$\\int_{0}^{T} |\\dot{\\Gamma}(t)| d t < \\infty$$\nand $\\Gamma$ is a $\\mathcal{P}$-path. Going back to the previous construction we observe that $\\Gamma$ is the desired lift. \n\\par\nTo complete the proof we must remove the extra assumption $\\gamma_1 \\neq 1$. Assume now that $\\gamma (t)$ is an arbitrary $\\mathcal{P}$-path. Since $\\gamma$ is smooth on $]t_0,t_1[$, the set $\\gamma ([t_0 ,t_1])$ has zero measure in $S^2$.\nLet us take a point $\\xi \\in S^2$ such that $\\gamma (t)\\neq \\xi$ for each $t\\in [t_0 ,t_1]$. We select a rotation of $S^2$ sending $\\xi$ into $i$. This rotation can be expressed in the form $z'=qz\\overline{q}$ for some $q\\in S^3$. Then $\\gamma_* (t)\n=q\\gamma (t)\\overline{q}$ is a $\\mathcal{P}$-path with $\\gamma_* (t)\\neq i$ for each $t\\in [t_0 ,t_1]$. The possible lifts of $\\gamma$ and $\\gamma_*$ are linked, for if $\\Gamma =\\Gamma (t)$ satisfies $\\pi \\circ \\Gamma =\\gamma$ then\n$\\pi \\circ \\Gamma_* =\\gamma_*$, where $\\Gamma_* (t)=\\Gamma (t)\\overline{q}$. Moreover, it is easy to check that $\\Re\\{\\bar{\\Gamma}(t) i \\dot{\\Gamma}(t)\\}=0$ is equivalent to $\\Re\\{\\bar{\\Gamma}_*(t) i \\dot{\\Gamma}_*(t)\\}=0.$ \n\n\\section{A restricted three-body problem}\\label{44}\nLet us assume that the $C^{\\infty}$ functions $X, x: \\mathbb{R} \\to \\mathbb{R}^{3},  X=X(t), x=x(t),$ are $T$-periodic and satisfy \n$$X(t) \\neq x(t)\\qquad \\forall t \\in \\mathbb{R}.$$\nIn addition we assign positive masses $M$ and $m$ to them, {by normalization}\n$$M+m=1.$$\nWe consider the system\n\\begin{equation}\\label{eq: 3R}\n\\ddot{\\xi}=\\dfrac{M(X(t)-\\xi)}{|X(t)-\\xi|^{3}}  + \\dfrac{m(x(t)-\\xi)}{|x(t)-\\xi|^{3}}. \n\\end{equation}\nThis system describes the motion of an infinitesimal body attracted by two moving centers $X(t)$ and $x(t)$. When $(X(t), x(t))$ solves the corresponding two-body problem, then the system describes a restricted spatial three-body problem. By assuming that $(X(t), x(t))$ move on a circular or elliptic Keplerian orbit we obtain a periodic system. Note that for {the} restricted circular three-body problem many studies have been made in a proper rotating coordinate system and for the elliptic problem in a proper {rotating-pulsating coordinate system}. \nWe shall just consider the problems in the inertial system. \n\nIn principle we could have collisions of the infinitesimal particle $\\xi$  with any of the two primaries: $X=\\xi$ or $x=\\xi$. Nevertheless we shall only consider collisions with the first primary $X$.  \n\n A continuous and $T$-periodic function $\\xi : \\mathbb{R} \\to \\mathbb{R}^{3},\\; \\xi=\\xi(t)$ is called a generalized periodic solution of the first kind if the following conditions hold:\n \\begin{itemize}\n \\item $\\mathcal{Z}_{M}=\\{t \\in \\mathbb{R}: \\xi(t)=X(t)\\}$ is discrete;\n \\item $\\mathcal{Z}_{m}=\\{t \\in \\mathbb{R}: \\xi(t)=x(t)\\}$ is empty;\n \\item In each interval $I \\subset \\mathbb{R} \\setminus \\mathcal{Z}_{M}$, the function $\\xi(t)$ is $C^{\\infty}$ and satisfies \\eqref{eq: 3R};\n \\item For each $t_{0} \\in \\mathcal{Z}_{M}$ the limits below exist\n $$\\lim_{t  \\to t_{0}} \\dfrac{\\xi(t)-X(t)}{|\\xi(t)-X(t)|}, \\quad \\lim_{t  \\to t_{0}} \\{\\dfrac{1}{2} |\\dot{\\xi}(t)-\\dot{X}(t)|^{2}-\\dfrac{M}{|\\xi(t)-X(t)|}\\}.$$\n \\end{itemize}\n\nLet us now assume that the primaries depend upon a parameter $\\varepsilon \\in [0, 1]$, $X_{\\varepsilon}=X_{\\varepsilon}(t), x_{\\varepsilon}=x_{\\varepsilon}(t)$. We assume that $(X_{\\varepsilon}(t), x_{\\varepsilon}(t))$ is a circular or elliptic solution of the two body problem with masses $M_{\\varepsilon}, m_{\\varepsilon}$ and with center of mass placed at the origin\n\n\\begin{equation}\\label{eq:cm}\nM_{\\varepsilon} X_{\\varepsilon} + m_{\\varepsilon} x_{\\varepsilon}=0. \n\\end{equation}\n\nFrom the general theory of Kepler problem we know that $X_{\\varepsilon}(t)$ satisfies\n\\begin{eqnarray*}\nX_{\\varepsilon}(t)= R_{\\varepsilon} \\left(a_{\\varepsilon} (\\cos u(t)-e_{\\varepsilon}), a_{\\varepsilon} \\sqrt{1-e_{\\varepsilon}^{2}} \\sin u(t), 0 \\right)^* \\\\\nu(t)-e_{\\varepsilon} \\sin u(t)=\\dfrac{m_{\\varepsilon}^{3\/2}}{a_{\\varepsilon}^{3\/2}} t.\n\\end{eqnarray*}\nWe are assuming that $t=0$ is the time of passage through the pericenter and the matrix $R_{\\varepsilon}$ is in the group of rotations $SO(3)$. From $X_{\\varepsilon}(t)$ one determines $x_{\\varepsilon}(t)$ from the center of mass condition Eq. \\eqref{eq:cm}. \n\nFrom now on it will be assumed that the functions\n$$\\varepsilon \\in [0, 1] \\mapsto R_{\\varepsilon} \\in SO(3)$$\nand\n$$\\varepsilon \\in [0, 1] \\mapsto m_{\\varepsilon}, M_{\\varepsilon}, a_{\\varepsilon}, e_{\\varepsilon}$$\nare all $C^{\\infty}$ and, for each $\\varepsilon>0$ there holds\n$$m_{\\varepsilon}, M_{\\varepsilon}>0, m_{\\varepsilon}+M_{\\varepsilon}=1, a_{\\varepsilon}>0, \\,\\, 0 \\le e_{\\varepsilon} <1.$$\n\nAccording to the third Kepler law, the system \\eqref{eq: 3R} is periodic in time with period\n\n\\begin{equation}\\label{eq:per}\nT_{\\varepsilon}=\\dfrac{2 \\pi}{m_{\\varepsilon}^{3\/2}} a_{\\varepsilon}^{3\/2}.\n\\end{equation}\n\nIn the following result we will assume that the primary $X_{0}$ is fixed at the origin while $x_{0}$ describes a circular or elliptic Keplerian orbit with mass $m_{0}=0$. \n\n\\begin{theo}\\label{theo: g3R}\nAssume in addition that \n$$m_{0} = 0 ,  e_0<1 , T_{\\varepsilon} \\to T_{0} >0 \\hbox{ as } \\varepsilon \\to 0^{+}. $$\nThen , for any given integer $l \\ge 1$, there exists $\\varepsilon_{l}>0$ such that the equation \\eqref{eq: 3R} has at least $l$ generalized $T_{\\varepsilon}$-periodic solutions of the first kind for $\\varepsilon \\in ]0, \\varepsilon_{l}[$.\n\\end{theo}\n\nIn contrast to many other results \\cite{CPS, AL, {PYFN}}, we do not impose any further resonance or non-degeneracy conditions and the dependence with respect to parameters is very general. On the other hand, our solutions are understood in a generalized sense.\n\n\\begin{proof} \nTo fix a uniform period  we change time $s \\to t$ according to the relation\n$$T_{\\varepsilon} s=t$$\nand set $\\eta(s)=\\xi(t)$,\nso that \\eqref{eq: 3R} is transformed into\n$$\\eta^{''}=\\dfrac{T_{\\varepsilon}^{2} M_{\\varepsilon} (\\phi_{\\varepsilon}(s)-\\eta)}{|\\phi_{\\varepsilon}(s)-\\eta|^{3}}+ \\dfrac{T_{\\varepsilon}^{2} m_{\\varepsilon} (\\psi_{\\varepsilon}(s)-\\eta)}{|\\psi_{\\varepsilon}(s)-\\eta|^{3}},$$\nwhere $\\phi_{\\varepsilon}(s)=X_{\\varepsilon} (T_{\\varepsilon} s), \\psi_{\\varepsilon}(s)=x_{\\varepsilon} (T_{\\varepsilon} s)$.\n\nWe then introduce \n$$u={\\lambda^{-1}} (\\eta-\\phi_{\\varepsilon}(s))$$\nwhere $\\lambda>0$ is a normalization parameter to be adjusted. In this way we obtain an equation of the form of \\eqref{eq:1} if $\\lambda=\\lambda_{\\varepsilon}$ is given by\n\\begin{equation}\\label{eq:para}\n\\lambda_{\\varepsilon}^{3} =T_{\\varepsilon}^{2} M_{\\varepsilon}.\n\\end{equation}\nNamely,\n$$u''=-\\dfrac{u}{|u|^{3}}+\\varepsilon \\nabla_{u} U(s, u, \\varepsilon)$$\nwith\n$$U(s, u, \\varepsilon)=\\dfrac{1}{\\varepsilon} \\lambda_{\\varepsilon}^{-3} T_{\\varepsilon}^{2} m_{\\varepsilon} \\dfrac{1}{|\\lambda_{\\varepsilon}^{-1} (\\psi_{\\varepsilon} (s)-\\phi_{\\varepsilon}(s)) - u|} - \\dfrac{1}{\\varepsilon} \\lambda_{\\varepsilon}^{-1} \\langle \\phi''_{\\varepsilon}(s), u \\rangle.$$\nThis function belongs to $C^{\\infty} (\\mathbb{R} \\times \\mathcal{U} \\times [0, \\varepsilon_{*}])$ where $\\mathcal{U}$ is a neighborhood of $u=0$ and $\\varepsilon_{*}>0$ is small enough. \n\nTo prove this, we first observe that\n$$a_{\\varepsilon} (1-e_{\\varepsilon}) \\le |\\phi_{\\varepsilon} (s)| \\le a_{\\varepsilon} (1+e_{\\varepsilon}), s \\in \\mathbb{R}.$$\nThen, in view of \\eqref{eq:cm}, for small enough $\\varepsilon$ we have\n$$|\\lambda_{\\varepsilon}^{-1} (\\psi_{\\varepsilon}(s)-\\phi_{\\varepsilon}(s))| \\ge |\\lambda_{\\varepsilon}^{-1} \\phi_{\\varepsilon}(s)| \\ge  \\lambda_{\\varepsilon}^{-1} \\dfrac{1}{m_{\\varepsilon}} a_{\\varepsilon} (1-e_{\\varepsilon}).$$ \nUsing \\eqref{eq:per} and \\eqref{eq:para} we see that the lower bound converges to $\\dfrac{1-e_{0}}{(2 \\pi)^{2\/3}}$ as $\\varepsilon \\to 0$. Consequently there exist $\\varepsilon_{*}>0$ and $\\delta>0$ such that\n$$|\\lambda_{\\varepsilon}^{-1} (\\psi_{\\varepsilon} (s) -\\phi_{\\varepsilon}(s)) | \\ge \\delta \\hbox{ for } \\varepsilon \\in [0, \\varepsilon_{*}], s \\in \\mathbb{R}.$$ \n\nWe define\n$$\\mathcal{U}=\\{u \\in \\mathbb{R}^{3}: |u| < \\delta\\}.$$\nFrom the explicit definition of $\\phi_{\\varepsilon}$ and $\\psi_{\\varepsilon}$ it is clear that $U$ is smooth on $\\mathbb{R} \\times \\mathcal{U} \\times ]0, \\varepsilon_{*}]$. It remains to analyze $\\varepsilon=0$.\n\nFirst we observe that \n$$\\phi_{\\varepsilon} (s)=a_{\\varepsilon} R_{\\varepsilon} (\\cos u-e_{\\varepsilon}, \\sqrt{1-e_{\\varepsilon}^{2}} \\sin u, 0)^*, u-e_{\\varepsilon}\\sin u=2 \\pi s$$\nand therefore $$\\lambda_{\\varepsilon}^{-1} (\\psi_{\\varepsilon} (s) -\\phi_{\\varepsilon}(s)) =-\\lambda_{\\varepsilon}^{-1} (\\frac{T_{\\varepsilon}}{2\\pi})^{2\/3} R_{\\varepsilon} (\\cos u-e_{\\varepsilon}, \n\\sqrt{1-e_{\\varepsilon}^{2}} \\sin u, 0)^*$$  is $C^{\\infty}$ in $\\mathbb{R}\\times [0,1]$. \nAlso, the function $f(\\varepsilon)=\\dfrac{1}{\\varepsilon} \\lambda_{\\varepsilon}^{-3} T_{\\varepsilon}^{2} m_{\\varepsilon}=\\dfrac{m_{\\varepsilon}}{ \\varepsilon M_{\\varepsilon}}$ belongs to $C^{\\infty}[0, \\varepsilon_{*}]$ and therefore the first summand in the definition of $U$ is smooth. \n\nTo analyze the second summand we differentiate twice the function $\\phi_{\\varepsilon}$. \nThen we have\n$$\\phi''_{\\varepsilon}(s)=a_{\\varepsilon} \\chi(s, \\varepsilon)$$ \nwith\n$$\\chi(s, \\varepsilon)=u'' R_{\\varepsilon} (-\\sin u, \\sqrt{1-e_{\\varepsilon}^{2}} \\cos u,0)^*-(u')^{2} R_{\\varepsilon}  (\\cos u, \\sqrt{1-e_{\\varepsilon}^{2}} \\sin u, 0)^*   $$\nand\n$$u'=\\dfrac{\\partial u}{\\partial s}=\\dfrac{2 \\pi}{1-e_{\\varepsilon} \\cos u}, u''=\\dfrac{\\partial^{2} u}{\\partial s^{2}}=-\\dfrac{2 \\pi e_{\\varepsilon} \\sin u}{(1-e_{\\varepsilon} \\cos u)^{2}} u'. $$\nThe function $\\chi(s, \\varepsilon)$ thus belongs to $C^{\\infty} (\\mathbb{R} \\times [0, \\varepsilon_{*}], \\mathbb{R}^{3})$. In addition, \n$$g(\\varepsilon)=\\dfrac{1}{\\varepsilon} \\lambda_{\\varepsilon}^{-1} a_{\\varepsilon}=\\dfrac{m_{\\varepsilon}}{\\varepsilon M_{\\varepsilon}^{1\/3} (2 \\pi)^{2\/3}}$$ is in $C^{\\infty}[0, \\varepsilon_{*}]$.\n\nTherefore $U$ is smooth in $\\mathbb{R} \\times \\mathcal{U} \\times [0, \\varepsilon_{*}]$ and Theorem \\ref{theo: local} is applicable. Undoing the change of variables we obtain generalized $T_{\\varepsilon}-$periodic solutions. To check the continuity of the energy\nat collisions it is convenient to use the identity $$\\frac{1}{2} |u'(s)|^2 -\\frac{1}{|u(s)|} =\\frac{T_{\\varepsilon}^2}{\\lambda_{\\varepsilon}^2}\\Bigl(\\frac{1}{2} |\\dot{\\xi}(t)-\\dot{X}_{\\varepsilon} (t)|^2 -\\frac{M_{\\varepsilon}}{|\\xi (t)-X_{\\varepsilon}(t)|}\\Bigr).$$ \n\\end{proof}\n\n\\begin{rem} After the change of variables $\\xi =R_{\\varepsilon} \\xi_1$ we can assume that the primaries lie on the fixed plane $x_3=0$. Then we can apply the planar version of  Theorem \\ref{theo: local} to conclude that, for each $\\varepsilon$, the three bodies move in a common plane. \n\\end{rem}\n\nWe end this note with a comparison of our result with classical results concerning periodic orbits of the planar circular restricted three-body problem of the first and second kind in a rotating frame, where first and second kind refer to continuations of periodic orbits from circular and elliptic periodic orbits of the limiting Kepler problem in a rotating frame respectively. Indeed the period of rotations of the reference frame is derived from the corresponding period $T$ of the Keplerian elliptic motions of the primaries. When a periodic orbit has minimal period $T\/n$ in the rotating frame, then { it will also have }period $T$ and therefore {this period is preserved} in the initial fixed reference frame. The existence of such orbits of the first kind has been obtained by Poincar\\'e \\cite{P} and has been explained in \\cite{MZ}. \n \n \\begin{ack} R. O. is supported by  MTM2017-82348-C2-1-P, L. Z. is supported by DFG ZH 605\/1-1. \n \\end{ack}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\nMillisecond pulsars (MSPs) are rapidly spinning and highly magnetized neutron stars (NSs), formed through mass and angular momentum accretion from an evolving companion star to a slowly rotating NS \\citep{alpar82,stairs04,tauris11}. At the end of the accretion stages, which are commonly observed in low-mass X-ray binaries \\citep[e.g.][]{tauris06,papitto13,ferraro15}, the re-accelerated NS is reactivated as a MSP in the radio bands and the companion star is expected to be an exhausted and deeply peeled star, typically a white dwarf (WD) with a He core \\citep[e.g.][]{stairs04}, although deviations from this scenario exist \\citep[e.g.][]{ferraro01a,ransom05,lynch12,pallanca12,pallanca13b,cadelano15a}.\nGlobular clusters (GCs) are the ideal factory for the formation of MSPs. In fact, the large stellar densities in the cores of GCs favor dynamical interactions, such as exchange interactions and tidal captures, which can lead to the formation of a large variety of binary systems whose evolution generates stellar exotica like blue stragglers \\citep[e.g.][]{ferraro09,ferraro12,ferraro18a}, low-mass X-ray binaries \\citep[e.g.][]{pooley03}, cataclysmic variables \\citep[e.g.][]{paresce92,ivanova06} and MSPs as well \\citep[e.g.][]{ransom05,hessels07,cadelano18}. As a consequence, the number of MSPs per unit mass in the global GC system of the Milky Way turns out to be $\\sim 10^3$ times larger than in the Galactic field. \n\n{To date, 150 MSPs are known in 28 different GCs. Among these clusters, the case of Terzan 5 and 47 Tucanae are particularly worthy of attention: the former is a stellar system known to host $\\sim25\\%$ of the entire MSP population in GCs \\citep{ransom05,cadelano18,andersen18} and proposed to be the remnant of a pristine fragment of the Galactic bulge \\citep{ferraro09,lanzoni10,massari14,ferraro16,prager17}; the latter is the second cluster in terms of MSP abundance, hosting 25 systems that allowed studies ranging from binary evolution to the cluster structure and gas content \\citep[see e.g.][]{freire01,cadelano15b, riverasandoval15,ridolfi16,freire17,abbate18}}. Therefore the study of MSPs deserves special attention since it allows to study the physical properties of the host stellar system \\citep[e.g.][]{freire17,prager17,abbate18} and also to probe binary  and stellar evolution under extreme conditions. This can be done, for example, through the identification of the optical counterparts, whose emission is dominated by the companion stars \\citep{ferraro01a,ferraro03a,mucciarelli13,pallanca14,cadelano15b,cadelano17}. In the case of degenerate companions, like WDs or NSs, precise mass measurements of both the components of the binary can be performed directly through timing analysis if relativistic effects are observed. Such values, not only are highly valuable to study in detail the formation and evolution of these systems, but are also of extreme importance to put constraints on the equation of state of ultra-dense matter and to test general relativity \\citep[e.g.][]{freire08,demorest10,antoniadis13}. If relativistic effects are not observed, mass measurements can be obtained through the optical identification of the WD companion. In fact, by comparing the observed magnitudes with appropriate WD cooling sequences, the mass of the companion star can be evaluated and such a value, combined with the mass function obtained through radio timing, provides the determination of the NS mass. { Furthermore, if the companion star is bright enough, the radial velocity curve, and thus the mass ratio,  can be determined through spectroscopic observations, together with more precise measurements of the companion mass} \\citep[see e.g.][]{ferraro03b,antoniadis12,antoniadis13}. \n\nM3 (NGC 5272) is a bright and well studied GC located at about 10 kpc from the Sun \\citep[see][]{ferraro93,buonanno94,ferraro97a,ferraro97b,ferraro97c,carretta98,rood99,ferraro18b}. Its stellar population is characterized by an intermediate metallicity ($[Fe\/H]=-1.5$), a very small reddening $E(B-V)=0.01$ and a large number of variable stars \\citep[see][for more information about the cluster properties]{harris10}. \\citet{hessels07} reported on the secure identification of 3 MSPs in this cluster plus a fourth candidate. Among these, only two MSPs, namely J1342$+$2822B and J1342$+$2822D (hereafter M3B and M3D) have been clearly detected  in a number of observations large enough to provide the astrometric and orbital properties of the binaries, although the position of M3D is still affected by a large uncertainty in both right ascension and declination. The properties of these two systems, as reported by \\citet{hessels07}, are summarized in Table~\\ref{tab:MSP}. M3B is a 2.39 ms MSP, located in binary with a $\\sim 1.4$ days circular orbit. Its general radio properties suggest that it is a canonical system and therefore it should be in a binary system with a WD companion. On the other hand, M3D has a spin period of 5.45 ms and it is in a very long and slightly eccentric orbit of about 129 days. The vast majority of binary MSPs in GCs have orbital periods shorter than few days, while just a dozen have orbital periods longer than 10 days\\footnote{see \\url{http:\/\/www.naic.edu\/~pfreire\/GCpsr.html}}. Therefore M3D likely underwent an anomalous evolution; indeed, binaries with orbital periods longer then 10 days are believed to be formed through an exchange interaction between an isolated NS and a primordial cluster binary \\citep{hut92,sigurdsson93}. Its eccentricity, which is likely the result of fly-by encounters with other stars \\citep{phinney92,phinney93,bagchi09}, is unexpected for systems formed through the canonical low-mass X-ray binary channel and confirms a non-standard evolution for this system.\n\nHere we report on the secure identification of the companion star to M3B and on a candidate companion to M3D. These companions have been discovered through high resolution near-ultraviolet (near-UV) and optical observations. In Section~\\ref{sec:data} we present the data-set we used in this work and the data reduction procedures. In Section~\\ref{sec:comM3B} we present the identification and characterization of the companion star to M3B, while in Section~\\ref{sec:comM3D} we discuss a possible candidate companion to M3D. Finally, in Section~\\ref{sec:conc} we draw our conclusions.\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablecaption{Main radio timing\nparameters for M3B and M3D\\tablenotemark{a}, from \\citet{hessels07}.\\label{tab:MSP}}\n\\tablehead{\\colhead{Parameter} & \\colhead{M3B} & \\colhead{M3D}}\n\\startdata\nRight ascension, $\\alpha$ (J2000)\\dotfill  & 13$^{\\rm h}\\,42^{\\rm m}\\,11\\fs0871(1)$  & 13$^{\\rm h}\\,42^{\\rm m}\\,10\\fs2(6)$    \\\\\nDeclination, $\\delta$ (J2000)  \\dotfill  & $28^\\circ\\,22'\\,40\\farcs141(2)$   & $28^\\circ\\,22'\\,36(14)\\arcsec$  \\\\\nAngular offset from cluster center, $\\theta_{\\perp}$ (\\arcsec) \\dotfill & 8.4 & 14(7) \\\\\nSpin period, $P$ (ms)  \\dotfill  & 2.389420757786(1) &   5.44297516(6) \\\\\nOrbital period, $P_b$ (days) \\dotfill & 1.417352298(2) & 128.752(5) \\\\\nTime of ascending node passage, $T_{asc}$ (MJD) \\dotfill & 52485.9679712(6) & 52655.38(4) \\\\\nProjected semi-major axis, $x$ (s) \\dotfill & 1.875655(2) & 38.524(4)\\\\\nEccentricity, $e$ \\dotfill & ... & 0.0753(5) \\\\\nMSP mass function, $f \\ (M_{\\odot})$ \\dotfill  & 0.003526842(6) & 0.0037031(8) \\\\\n\\hline\n\\enddata\n\\tablenotetext{a}{Numbers in parentheses are uncertainties in the last digits quoted.}\n\\end{deluxetable*} \n\n\n\\section{Observations and DAta analysis} \\label{sec:data}\n\nThis work is based on a archival data-set of deep, high resolution near-UV and optical observations obtained through the UVIS channel of the Wide Field Camera 3 mounted on the Hubble Space Telescope (GO 12605, PI: Piotto). The data-set is composed of 6 images in the F275W {(near-ultraviolet)} filter with exposure time of 415 s, 4 images in the F336W (U) filter with exposure time of 350 s and 4 images in the F438W (B) filter with exposure time of 42 s. All the images {have been obtained on May 15th 2012 during $\\sim5$ hours of continual observations and thus they only cover $\\sim15\\%$ and $\\sim0.16\\%$ of the orbits of M3B and M3D, respectively}. The images are dithered by few arcseconds each, in order to avoid spurious effects due to bad pixels and to fill the inter-chip gap. The photometric analysis has been performed with {\\rm DAOPHOT IV} \\citep{stetson87} adopting the so-called ``UV-route'' as described in \\citealt{raso17} (see also \\citealt{ferraro97b,ferraro01b,ferraro03c,dalessandro18a,dalessandro18b}). As a first step, about 200 stars have been selected in each image in order to model the point spread function, which has been then applied to all the sources detected at more than $5\\sigma$ from the background level. Then, we built a master catalog with stars detected in at least half the near-UV (F275W) images. At the corresponding position of these stars, the photometric fit was forced in all the other frames  by using {\\rm DAOPHOT\/ALLFRAME} \\citep{stetson94}. Using such a near-UV master list, the crowding effect due to the presence of giants and turn-off stars is mitigated and several blue stars like blue stragglers and white dwarfs are recovered. Finally, for each star we homogenized the magnitudes estimated in different images, and their weighted mean and standard deviation have been adopted as the star magnitude and its related photometric error. The instrumental magnitudes have been reported to the VEGAMAG system by cross-correlation with the catalog of the ``{\\it The Hubble Space Telescope UV Legacy Survey of Galactic globular clusters}'' \\citep{piotto15}, obtained from the same data-set used in this work and publicly available.\n\nThe star detector positions have been corrected from geometric distortion following the procedure described by \\citet{bellini11}. Then, these positions have been reported to the absolute coordinate system ($\\alpha$,$\\delta$) using the stars in common with the Gaia DR2 catalog \\citep{gaia18}. The coordinate system of this catalog is based on the International Celestial Reference System, which allows an appropriate comparison with the MSP positions derived from timing using solar system ephemerids, being the latter referenced to the same celestial system. The resulting $1\\sigma$ astrometric uncertainty is of $\\sim 0.1 \\arcsec$ in both $\\alpha$ and $\\delta$.\n\n\n\\section{IDENTIFICATION OF THE COMPANION STAR to M3B} \\label{sec:comM3B}\n\n\\begin{figure}[t] \n\\centering\n\\includegraphics[scale=0.72]{chartB.png}\n\\caption{$3\\arcsec \\times 3 \\arcsec$ finding chart of the region around the radio position of M3B. The left, middle and right panels are extracted from a F275W, F336W and F438W image, respectively. The red cross is centered on the MSP position and the red circle has a radius $r=0.15\\arcsec$. The only star located within this circle is the identified companion to M3B.}\n\\label{fig:chartB}\n\\end{figure}\n\n\\begin{figure}[h!] \n\\centering\n\\includegraphics[scale=0.49]{cmd_M3B.pdf}\n\\caption{($m_{F275W}$, $m_{F275W}-m_{F336W}$) CMD of M3 obtained from the data-set used in this work. The positions of the companion star to M3B is marked with a red square.}\n\\label{fig:cmd}\n\\end{figure}\n\nIn order to search for the companion star to M3B, we investigated all the stellar sources detected in a $3\\arcsec \\times 3 \\arcsec$ region surrounding the MSP timing position reported in Table~\\ref{tab:MSP}. The stellar source closest to the radio position turned out to be a very blue object, located at $\\alpha=13^{\\rm h}\\,42^{\\rm m}\\,11\\fs0881$ and $\\delta=28^\\circ\\,22'\\,40\\farcs141$, at only $0.01 \\arcsec$ from the radio MSP position. The finding chart of this object in the three different filters is reported in Figure~\\ref{fig:chartB}. In the color-magnitude diagram (CMD) this star is located along the red side of the WD cooling sequence, in a region where WDs with a He core are expected to be found (see Figure~\\ref{fig:cmd}). The combination of the excellent agreement between the radio and the optical position and the peculiar location in the CMD allows us to safely conclude that the identified WD is the companion star to M3B. Its magnitudes in the three filters are the following: $m_{F275W}=22.45\\pm0.05$, $m_{F336W}=22.75\\pm0.05$ and $m_{F438W}=23.4\\pm0.2$. \nAlthough multiple epoch images are available for each filter, no evidence of photometric variability has been observed for the companion star. While this could be due to the poor and sparse orbital period coverage provided by the available data-set, {we stress that the photometric variability is only rarely observed for degenerate companion stars and is not due to the re-heating of the star by the MSP emitted energy, which is generally negligible, but most likely to pulsations (global stellar oscillations) of the WD itself \\citep[e.g.][]{maxted13,kilic15}.} \n\n\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=0.4]{cmdwdB.pdf}\n\\caption{Same as in Figure~\\ref{fig:cmd} but zoomed into the WD region. As reported in the top legend, the blue curve is a cooling track for $0.55 \\, M_{\\odot}$ C-O WDs from the {\\it BaSTI} database, while the cyan curves are He WD cooling tracks with masses, from left to right, of $0.24 \\, M_{\\odot}$, $0.23 \\, M_{\\odot}$, $0.20 \\, M_{\\odot}$, $0.19 \\, M_{\\odot}$ and $0.18 \\, M_{\\odot}$, calculated at the cluster metallicity $(Z\\sim0.0005)$ following \\citet{istrate14,istrate16}. Points at different cooling ages are highlighted with gray symbols as reported in the secondary legend.}\n\\label{fig:cmdwd}\n\\end{figure}\n\nWe can now compare the measured magnitudes with those predicted by He WD cooling sequences in order to derive the main physical properties of the companion star. As a first step, we checked the validity of the photometric calibration by comparing the observed sequence of standard WDs with a theoretical cooling model for C-O WDs with masses $M=0.55 \\ M_{\\odot}$ taken from the {\\it BaSTI} database \\citep{salaris10}. Absolute magnitudes have been reported to the observed frame by adopting a color-excess $E(B-V)=0.01$, a distance modulus $(m-M)_V=15.07$ \\citep{harris10}, and appropriate extinction coefficients for each filter: $A_{F275W}\/A_V =1.94$, $A_{F336W}\/A_V =1.66$ and $A_{F438W}\/A_V =1.33$ \\citep{cardelli89,odonnell94}. The corresponding curve is shown in Figure~\\ref{fig:cmdwd} and its agreement with the observed WD sequence confirms the accuracy of the adopted photometric calibration and cluster parameters.\n\nIn the case of the He WDs, we built a set of theoretical tracks using the new models for extremely low-mass He WD by \\citet{istrate14,istrate16}. These models are particularly appropriate for the present case study, since they simulate the entire evolution of systems within the low-mass X-ray binary channel, also including the effects of rotational mixing and element diffusion {and adopting the appropriate cluster metallicity for the secondary stars}. A selection of the obtained cooling tracks is shown in Figure~\\ref{fig:cmdwd} (cyan lines).\n\nIt is now possible to use the theoretical tracks to constrain the main physical properties of the detected WD such as its mass, surface temperature and cooling age\\footnote{The WD cooling age is defined, according to \\citet{istrate16}, as the time passed since the proto-WD reached the maximum surface temperature along the evolutionary track.}. To do this, we used an approach similar to that implemented by \\citealt{dai17} (see also \\citealt{pallanca13a}, \\citealt{testa15}). We interpolated all the available models to create a fine grid in the mass range $0.16 \\, M_{\\odot}$ - $0.35 \\, M_{\\odot}$, temperature range 9000~K~-~21000~K and cooling age range 0.1~Gyr~-~3~Gyr. Assuming Gaussian photometric errors,  we thus estimated the likelihood of each point of the grid (i.e. of each possible combination of the three parameters) as: \n$$\nL = \\prod_{j} \\frac{exp\\left[\\frac{-\\left(m_j - m_j^{mod}\\right)^2}{2\\delta_j^2}\\right]}{\\sqrt{2\\pi \\delta_j^2}} \n$$\nwhere the index $j$ runs through the three photometric filters F275W, F336W and F438W, $m_j$ and $\\delta_j$ are the observed magnitudes and related errors and $m_j^{mod}$ is the magnitude predicted by the models at that point of the grid (see equation 5 in \\citealt{dai17}).\n\nThe marginalized 1D and 2D likelihood distributions are presented in the corner plot\\footnote{see \\url{https:\/\/corner.readthedocs.io\/en\/latest\/} \\citep{foreman16}} of Figure~\\ref{fig:cornerM3B}. For each parameter, the best-fit value, the lower and the upper uncertainties have been estimated as the $50^{th}$, $16^{th}$ and $84^{th}$ percentile of its likelihood distribution, respectively. We therefore found that the He WD has a mass $M_{COM}~=~0.19~\\pm~0.02~ \\ M_{\\odot}$, a cooling age of $1.0^{+0.2}_{-0.3}$ Gyr and an effective temperature of $T=12\\pm 1\\cdot10^3$ K. The derived cooling age is consistent with the MSP spin-down age ($>1.1$~Gyr) estimated by \\citet{hessels07}.\n\\begin{figure}[b] \n\\centering\n\\includegraphics[scale=0.45]{cornerM3B.pdf}\n\\caption{Constraints on the mass, cooling age and surface temperature of the companion star to M3B. The 1D histograms show the likelihood marginalized distributions for each of the three parameters and the blue solid and black dashed lines are, respectively, the $50^{th}$, $16^{th}$ and $84^{th}$ percentiles of each distribution, that have been used as estimates of the best-fit value of each parameter and their related uncertainty. The contours in the 2D histograms correspond to $1\\sigma$, $2\\sigma$ and $3\\sigma$ levels and the best values for each parameter are marked with the blue point and lines. The text at the top  reports the derived mass, cooling age and temperature values.}\n\\label{fig:cornerM3B}\n\\end{figure}\nFrom the best-fit model we have also inferred that the proto-WD phase\\footnote{{ It is the phase following the mass-transfer stage, when the He core contracts at an almost constant luminosity before starting its cooling phase \\citep[see][]{istrate14,istrate16}}.} lasted for $1.0^{+0.1}_{-0.5}$ Gyr and that this star is composed of a core with a mass of about $0.18 \\ M_{\\odot}$ and a thin envelope with mass of about $0.01 \\ M_{\\odot}$. The total He mass is around $0.187 \\ M_{\\odot}$, while the H mass is only around $0.003 \\ M_{\\odot}$. Finally, its central density and  temperature are around $1.3 \\cdot 10^{5}$~g\/cm$^{-3}$ and $2\\cdot 10^{7}$~K, respectively.\n\n{{The companion star parameters just derived can be used to investigate the physical properties of its progenitor star.} We used the {\\rm PARSEC} evolutionary tracks \\citep{bressan12,bressan13} to study the evolution of the He core of isolated stars with different masses and ages at the cluster metallicity. Assuming a cluster stellar population age of $12.5\\pm0.5$ Gyr \\citep{dotter10} and using the derived cooling age and proto-WD phase duration, we can estimate that the mass-transfer stopped (i.e. the Roche-Lobe detachment occurred) when the cluster had an age around $10.5\\pm0.8$ Gyr. {Within this age range, only stars in the mass range $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ have grown a He core with a mass comparable with that derived for the companion to M3B (see left panel of Figure~\\ref{fig:massacore}). On the other hand, at an age $t=10.5\\pm0.8$ Gyr and at the cluster metallicity, the mass of a star at the main sequence turn-off is $0.81^{+0.01}_{-0.02} \\ M_{\\odot}$ and stars as massive as $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ are already evolved toward the red giant branch. Indeed, as shown in the evolutionary tracks plotted in right panel of Figure~\\ref{fig:massacore}, stars with mass in the range $0.82 \\ M_{\\odot} - 0.85 \\ M_{\\odot}$ have grown a $0.19\\pm0.02 \\ M_{\\odot}$ He core at the base of the red giant branch. By assuming that the mass-transfer phase lasted approximately $\\sim1$ Gyr, we can infer that the it started just after the star left the main sequence stage and therefore the bulk of the mass-transfer essentially occurred during the sub-giant branch phase, where the star is expected to expand as a consequence of its canonical evolution and so is able to fill its Roche-Lobe. Therefore the observational properties of the M3B companion measured here appear to be fully consistent with a suitable scenario for the formation of the MSP.}}\n\n\\begin{figure}[b!]\n\\centering\n\\includegraphics[scale=0.32]{masscore2.png}\n\\includegraphics[scale=0.32]{hr.png}\n\\caption{{{\\it Left Panel:} He core mass as a function of the stellar mass  as predicted by \\citet{bressan12,bressan13} evolutionary tracks. The black solid curve and the light gray shaded area represent the values for an age of $10.5\\pm0.8$ Gyr. The horizontal line (and dark gray band) marks the mass of the He WD companion to M3B (and its related uncertainty) as derived in Section~\\ref{sec:comM3B}. {\\it Right Panel:} evolutionary tracks for different stellar masses as reported in the top-left legend. The red region of the tracks highlights the phase where the stars have grown a He core with a mass comparable with the one measured for the companion to M3B at an age of $10.5\\pm0.8$ Gyr (corresponding to the mass-transfer end, i.e. the Roche-Lobe detachment phase). The blue shaded area marks the region of the tracks 1 Gyr before the Roche-Lobe detachment, when the mass-transfer probably started.}}\n\\label{fig:massacore}\n\\end{figure}\n\n\nThe He WD mass here obtained ($M_{COM}$), combined with the MSP mass function ($f$) derived from radio timing (see Table~\\ref{tab:MSP}) can be use to constrain the orbital inclination angle $i$ and, most importantly, the mass of the NS ($M_{NS}$). Indeed these quantities are related by the following equation:\n\\begin{equation}\nf(M_{NS}, M_{COM}, i) = \\frac{(M_{COM}\\sin{i})^3}{(M_{NS}+M_{COM})^2}\n\\end{equation}\nWe used the affine-invariant Markov Chain Monte Carlo (MCMC) ensemble sampler {\\rm emcee\\footnote{\\url{https:\/\/emcee.readthedocs.io\/en\/stable\/}}} \\citep{foreman13} to constrain $M_{NS}$ and $i$. We set a Gaussian prior on $M_{NS}$, centered at $1.4 \\ M_{\\odot}$, which is the mass typically measured for NSs in binary MSPs \\citep{ozel16}, and with a standard deviation of $0.5 \\ M_{\\odot}$, {large enough to include the observed NS mass distribution \\citep{antoniadis16}}. $M_{NS}$ is sampled in the range $0.5 \\ M_{\\odot}$ - $2.5 \\ M_{\\odot}$. On the other hand, we set an uniform prior on the distribution of $\\cos{i}$ in the range 0 - 1. The results are presented in Figure~\\ref{fig:mpsrB}. The best value for the NS mass is $M_{NS}=1.1\\pm 0.3 \\ M_{\\odot}$, {thus suggesting that this system hosts a low-mass NS, although the value is still compatible with the typical NS masses measured for these kind of systems. Results do not change if wider NS mass distributions are used}. The probability distribution of the inclination angle is shaped as a truncated Gaussian and clearly shows that this system is observed almost edge-on. We therefore assumed that the best value for the inclination angle corresponds to the maximum in the probability distribution and its lower uncertainty to the $16^{th}$ percentile of an identical but symmetric probability distribution. Doing so we have estimated the inclination angle to be $i=89^{+1}_{-26}$ degrees. { All the physical properties of the binary here derived are summarized in Table~\\ref{tab:comMSP}.}\n\\begin{figure}[h]\n\\centering\n\\includegraphics[scale=0.6]{corner_mpsrB.pdf}\n\\caption{Constraints on the mass of the NS and on the orbital inclination angle of M3B. The 1D histograms are the marginalized probability distributions of the two parameters, where the solid blue and black dashed lines are the best values and their related uncertainties (see text). The bottom left panel is the joint 2D posterior probability distribution and the contours corresponds to $1\\sigma$, $2\\sigma$ and $3\\sigma$ confidence levels.}\n\\label{fig:mpsrB}\n\\end{figure}\n\n\n\n\n\\section{A He WD ORBITING M3D TOO?}\\label{sec:comM3D}\n\n\\begin{figure}[h] \n\\centering\n\\includegraphics[scale=0.16]{chartD.png}\n\\caption{Left panel: $15\\arcsec \\times 18\\arcsec$ chart of the region surrounding M3D, obtained from a F275W image. The black box is centered on the MSP position and corresponds to the error box quoted in \\citet{hessels07}. The red circle indicates the position of the candidate companion. Right panel: same as in the left panel, but zoomed into the position of the candidate companion.}\n\\label{fig:chartD}\n\\end{figure}\n\n\\begin{figure}[h!] \n\\centering\n\\includegraphics[scale=0.35]{cmd_M3D.pdf}\n\\includegraphics[scale=0.35]{cmdwdD.pdf}\n\\caption{{{\\it Left panel:} same as in Figure~\\ref{fig:cmd}. The position of the candidate companion star to M3D is marked with a large red square, while the positions of all the other stars within the investigated area is marked with smaller red squares. {\\it Right panel:} same as in Figure~\\ref{fig:cmdwd} but for the candidate companion to M3D.}}\n\\label{fig:cmdD}\n\\end{figure}\n\nThe timing solution of M3D reported by \\citet{hessels07} provides the celestial position of this binary affected by a modest uncertainty ($0.6\\arcsec$) in $\\alpha$ and a large uncertainty ($14\\arcsec$) in $\\delta$. Therefore the search for the optical counterpart to this system is challenging. We carefully investigated all the stellar sources within and nearby the position error box: {this region of interest is centered on the best MSP position (see Table~\\ref{tab:MSP}) and is as large as the $3\\sigma$ uncertainty in $\\alpha$ and $2\\sigma$ uncertainty in $\\delta$. We only found a possible candidate in a position marginally compatible with the radio one (see Figure~\\ref{fig:chartD} and Figure~\\ref{fig:cmdD}). In fact, all the stars detected within the investigated area likely belong to the canonical evolutionary sequences but one source located again on the red side of the WD cooling sequence, in a position compatible with that of the He WDs}. This candidate counterpart is located at $\\alpha=13^{\\rm h}\\,42^{\\rm m}\\,10\\fs3041$ and $\\delta=28^\\circ\\,22'\\,44\\farcs786$. The right ascension is shifted by $1.5\\arcsec$ from the radio value, which is larger than the combined radio uncertainty and optical astrometric precision.  Its observed magnitudes are: $m_{F275W}=23.06\\pm0.08$, $m_{F336W}=23.35\\pm0.08$ and $m_{F438W}=24.1\\pm0.3$.\nFollowing the same method used for M3B and presented in Section~\\ref{sec:comM3B}, we found that this star has a mass $M_{COM}=0.22 \\pm 0.02 \\ M_{\\odot}$, a cooling age of $1.1^{+0.7}_{-0.6}$ Gyr and an effective temperature of $13\\pm 2 \\cdot 10^{3}$ K. Using the MSP mass function (Table~\\ref{tab:MSP}) we estimated the NS mass to be $M_{NS}=1.3 \\pm 0.3 \\ M_{\\odot}$ and the orbital inclination angle to be $i=72\\pm16$ degrees (see Table~\\ref{tab:comMSP}). Again, the $M_{NS}$ value would be in agreement with the typical values observed for binary MSPs.\nWe tried to establish phase-connection of the MSP timing solution using the available radio data and fixing the MSP position to that of the candidate counterpart. Unfortunately, this did not lead to an improvement of the timing solution obtained by \\citet{hessels07}. \nGiven all this, we cannot solidly confirm that the detected He WD is the optical counterpart to M3D, {which could be still under the detection threshold. Furthermore, the peculiar evolution that this system likely underwent implies that the companion star might not be a classical He WD. However it is worth noting that a He WD has been identified orbiting B1620$-$26 in M4, the only similar MSP with an optical counterpart to date \\citep{sigurdsson03}.} Future radio observations and timing analysis are needed to improve the position measurement of this object.\n\n\n\\begin{deluxetable*}{lcc}\n\\tablecolumns{3}\n\\tablewidth{0pt}\n\\tablecaption{Physical properties of the binaries M3B and M3D as derived in Section~\\ref{sec:comM3B} and ~\\ref{sec:comM3D}. \\label{tab:comMSP}}\n\\tablehead{\\colhead{Parameter} & \\colhead{M3B} & \\colhead{M3D?}}\n\\startdata\nCompanion mass, $M_{COM} \\ (M_{\\odot})$\\dotfill  & $0.19\\pm0.02$ & $0.22\\pm0.02$    \\\\\nEffective temperature, $T$ ($10^3$ K) \\dotfill & $12\\pm1$ & $13\\pm2$ \\\\\nCooling age, Age (Gyr) \\dotfill & $1.0^{+0.2}_{-0.3}$ & $1.1^{+0.7}_{-0.6}$ \\\\\nNeutron star mass, $M_{NS} \\ (M_{\\odot})$\\dotfill  & $1.1\\pm0.3$ & $1.3\\pm0.3$    \\\\\nInclination angle, $i$ (deg) \\dotfill & $89^{+1}_{-26}$ & $72\\pm16$ \\\\\n\\hline\n\\enddata\n\\end{deluxetable*} \n\n\\section{CONCLUSIONS}\\label{sec:conc}\n\nWe used deep and high resolution images obtained at near-UV and optical wavelengths to search for the companion stars to the binary MSPs in the GC M3. By exploiting the ``UV route'' to dilute the crowding issues in the cluster center and increase the sensitivity to blue\/hot stars, we have been able to firmly identify the companion star to the canonical MSP M3B and find a candidate counterpart to the anomalous (long period and mild eccentricity) system M3D.\nThe companion star to M3B turned out to be a WD with a He core, as expected from the canonical formation scenario. Interestingly, despite the fact that this cluster hosts a stellar population with an intermediate metallicity, the companion is an extremely low-mass object, with a mass of about $0.19 \\ M_{\\odot}$. Indeed, the lower is the metal content of the secondary stars, the larger is expected to be the minimum mass of the WD remnants. This is due to the fact that low metallicity stars have shorter evolutionary timescales and smaller radii and therefore their Roche Lobe is filled (i.e. the mass-transfer starts) in a more advanced stage of their evolution, when the He core is grown more massive than it would have for a more metal-rich system \\citep{istrate16}. Our mass measurement is indeed close to the minimum possible mass produced by the adopted models at the cluster metallicity. This stresses the importance of the study of extremely low-mass WD systems, with special care to the physical processes occurring during the evolution. In fact, the models we used take into account effects like rotational mixing and element diffusion. Models not including these effects are characterized by bluer cooling tracks that would have not been able to reproduce the observed magnitudes of the companion or, in the case they do, the resulting companion mass would have been so small that the corresponding NS mass becomes unreasonable (i.e. $<1 \\ M_{\\odot}$). {We have also shown that the progenitor of this WD was likely a $\\sim0.83 \\ M_{\\odot}$ star which filled its Roche-Lobe after leaving the main sequence, thus implying that the bulk of the mass-transfer activity occurred during the sub-giant branch phase. All the derived physical properties of the companion star, combined with the information obtained through radio timing, allowed to infer that this binary is observed almost edge-on and probably hosts a low-mass NS with a mass around $\\sim 1.1 \\ M_{\\odot}$.}\n\nIn the case of the candidate companion star to M3D, we have identified a He WD at a position marginally compatible with the highly uncertain radio one. Therefore the association between this degenerate object and the MSP cannot be confirmed yet. This WD is again a low-mass object with a mass around $0.22 \\ M_{\\odot}$ and a cooling age of $\\sim1$~Gyr\n\n{\\citet{tauris99} discussed the correlation between the masses of the WD companion stars and the orbital periods of the binary MSPs (the so-called ``TS99'' relation; see also \\citealt{istrate14} for a more recent investigation on this). According to this relation, the $\\sim1.4$~days orbit of M3B implies a companion star with a mass of $\\sim0.21 \\ M_{\\odot}$, fully compatible, within the uncertainties, with the mass value derived in this work. On the other hand, the long $\\sim 129$ days orbit of M3D implies a companion star with a mass around $\\sim 0.35 \\ M_{\\odot}$, a value significantly larger than the one here derived for its candidate companion star. This could suggests that the proposed optical counterpart is not the real companion star, even though its mass is compatible with the measured MSP mass function if a typical NS mass is assumed. If this WD is truly the companion star to M3D, the discrepancy between the measured mass and the one predicted by the TS99 relation could be explained by the formation of this system through an exchange interaction occurred after the NS recycling phase. Such an exchange, however, is expected to produce highly eccentric binaries \\citep[see][]{prince91,freire04,lynch12}, while only a mild eccentricity is measured for this system.}\n\n\\acknowledgments\n{We thank the anonymous referee for the careful reading of the manuscript}. This research is part of the scientific project Cosmic-Lab at the Department of Physics and Astronomy at Bologna University. \n\n\\vspace{5mm}\n\\facilities{HST(WFC3)}\n\\software{DAOPHOT \\citep{stetson87}, {\\rm emcee} \\citep{foreman13}, {\\rm corner.py} \\citep{foreman16}}\n\n\n\n\n\\section{} \n\n\\textit{Research Notes of the \\href{https:\/\/aas.org}{American Astronomical Society}}\n(\\href{http:\/\/rnaas.aas.org}{RNAAS}) is a publication in the AAS portfolio\n(alongside ApJ, AJ, ApJ Supplements, and ApJ Letters) through which authors can \npromptly and briefly share materials of interest with the astronomical community\nin a form that will be searchable via ADS and permanently archived.\n\nThe astronomical community has long faced a challenge in disseminating\ninformation that may not meet the criteria for a traditional journal article.\nThere have generally been few options available for sharing works in progress,\ncomments and clarifications, null results, and timely reports of observations\n(such as the spectrum of a supernova), as well as results that wouldn't\ntraditionally merit a full paper (such as the discovery of a single exoplanet\nor contributions to the monitoring of variable sources). \n\nLaunched in 2017, RNAAS was developed as a supported and long-term\ncommunication channel for results such as these that would otherwise be\ndifficult to broadly disseminate to the professional community and persistently\narchive for future reference.\n\nSubmissions to RNAAS should be brief communications - 1,000 words or fewer\n\\footnote{An easy way to count the number of words in a Research Note is to use\nthe \\texttt{texcount} utility installed with most \\latex\\ installations. 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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nThe state of the inhomogeneous cosmic matter distribution (and the time-dependent\ndark energy density) can be measured by\nthe gravitational lensing shearing and magnification of background luminous\nsources. Using galaxies as the back lights for gravitational lensing (``cosmic shear'')\nallows tomographic reconstruction of the matter distribution along the line of sight.\nBut, inference of galaxy lensing shears is clouded by the unknown distribution\n(and possibly correlation) of galaxy intrinsic morphologies as well as image systematics\nthat induce spurious ellipticity correlations.\n\nCommon inference algorithms for gravitational lensing shear involve cross-correlating\nthe ellipticities of galaxies in a two-point function estimator under the assumption\nof calibrated intrinsic ellipticity distributions and alignments~\\citep[e.g.,][]{jee2015}.\nThis is a lossy procedure because the angular phase information in the lensing\nshear and magnification fields on the sky is discarded. Traditional algorithms\nare also necessarily biased because of the need to calibrate the unknown galaxy\nellipticity distributions. Said another way, the two-point function summary statistics of\ncosmological large-scale structure do not capture the full statistical information in the multivariate\ndistribution of the lensing observable\n\\citep[e.g.,][]{pan2005,hamilton2006,takada2013,carron2014,2016arXiv160501100P}\n\nIn this paper we revisit the problem of inferring the lensing convergence posterior distribution\nfrom a catalog of galaxy ellipticities to potentially capture more cosmological information from\ncosmic shear than is available in two-point function estimators of the shear. We pursue a\nprobabilistic approach to lensing convergence inference as a means to propagate the large\nuncertainties that can come in the presence of galaxy shape noise, finite survey areas, masking,\nand sample selection. Moreover, we aim to build an inference framework that can fit within our previous\nwork on probabilistic cosmic shear~\\citep{mbi-theory}.\n\n\\citet{1990ApJ...349L...1T} first presented a method to reconstruct the projected mass distribution\nfrom weak gravitational lensing shears of source galaxy images.\n\\citet{1993ApJ...404..441K} derived the lensing convergence in terms\nof the shear (a nonlocal relation) and applied this theory to estimators of the convergence given a\nmeasured galaxy ellipticity catalog. The method of \\citet{1993ApJ...404..441K}, while theoretically sound,\nrequires unbroken sky coverage and low-pass filtering to yield finite noise in the lensing\nconvergence estimator.\nThe \\citet{1993ApJ...404..441K} method remains useful, however, for visualization\npurposes~\\citep{2015PhRvL.115e1301C,2015PhRvD..92b2006V}.\nSubsequent papers have extended, applied, and explored the limitations of filtering algorithms for\ncluster mass mapping~\\citep{1995A&A...303..643B,1995A&A...294..411S,1998A&A...335....1L,2002ApJ...568..141G,2005A&A...440..453D,2007Natur.445..286M,2012A&A...540A..34D,2012MNRAS.424..553A,2013MNRAS.433.3373V}. Remaining data analysis challenges\nusing such methods include finite survey boundaries and masks,\nseparation of E and B modes in the shear\nfield, noise or significance characterization for shear or convergence extrema, and the requirement\nin many algorithms to smooth or average the ellipticities of galaxies before the convergence inference\nprocess.\n\n\nMaximum likelihood (ML) estimators for the lensing\nconvergence~\\citep[e.g.,][]{1996ApJ...464L.115B,1998A&A...337..325S,2008MNRAS.385.1431H,2009ApJ...702..980K}\ncan help mitigate biases arising from survey masks and admit mathematically consistent noise\ncharacterization~\\citep{2000MNRAS.313..524V}. However such estimators are biased. And many\napproaches often still require a preliminary smoothing of the observed galaxy ellipticity field.\n\n\\citet{bridle1998} and \\citet{bridle2000} introduced a `maximum entropy'\nBayesian prior for the lensing convergence from information theoretic and Bayesian analysis perspectives to\nderive an estimator for the projected mass distribution of galaxy clusters with desirable noise\nproperties in a finite field.\n\\citet{Marshall:531310} and \\citet{2001A&A...368..730S} refined the maximum entropy method to\nspecify the optimal smoothing\nlength scale for galaxy cluster mass inference via the Bayesian evidence of the observed\nellipticities. \\citet{2013ascl.soft08004M} released a code\\footnote{{\\sc LensEnt2}\\xspace, \\url{http:\/\/www.slac.stanford.edu\/~pjm\/lensent\/version2\/index.html}}\nimplementing the algorithm from \\citet{Marshall:531310}.\nThe algorithm of \\citet{Marshall:531310} is close to meeting all the requirements for the current\nanalysis, except the choice of smoothing scale and application to field rather than cluster lensing\nis not demonstrated in the literature. We will show further benefits of the algorithm developed\nin this paper below.\n\\citet{2006A&A...451.1139S} applied a modification of the maximum entropy method to a cosmic\nshear analysis using $N$-body cosmological simulations to create mock observations.\nThe work of \\citet{2006A&A...451.1139S} and also \\citet{jiao2011} include maximum entropy algorithm\nmodifications to better handle masking, shape noise, and degrees of smoothing.\n\nFor constraining cosmological parameters, several groups have considered the abundance of peaks\nin lensing convergence maps~\\citep{2010PhRvD..81d3519K,2011ApJ...735..119S,2014MNRAS.442.2534S,2015PhRvD..91f3507L,2016MNRAS.456..641R,2016PhRvD..94d3533L,PhysRevLett.117.051101,2016arXiv160501100P}. These studies have a common\napproach in direct calculation or estimation of lensing peaks without an attempt to infer\nassociated peaks in the 3D cosmological mass density. The measurement process is thus direct as\nlensing by multiple structures or voids along any given line of sight can easily confuse the\n2D to 3D mass inference. Just as the abundance of galaxy clusters provides tight constraints on the\ncosmological model, so too can the abundance of lensing peaks.\nHowever, the estimators for lensing peaks often involve averages or line integrals over contiguous\nsky areas, which are confounded by survey masks~\\citep{0004-637X-784-1-31}.\nThe bias introduced by masking may be overcome with forward simulations of the nonlinear lensing\nconvergence field, which has been achieved with $N$-body cosmological simulations of large-scale\nstructure combined with ray-tracing predictions of the lensing statistics~\\citep[e.g.,][]{2016ApJ...819..158B,2016arXiv160501100P}.\n\nIn a complimentary approach to cosmological parameter inference, other groups have considered a \nlinear theory approximation to (suitably smoothed) lensing mass maps. \nSpecifically, if (i) the shear field can be approximated as Gaussian distributed, (ii) the noise \ndistribution in the galaxy ellipticity measurements (and intrinsic shape distribution) can also \nbe approximated as Gaussian distributed, and (iii) the weak shear approximation is valid, then \nmass maps can be obtained using the Wiener Filter (WF)~\\citep{wiener1949extrapolation}. \nThe WF is a well-studied signal-inference algorithm that has been applied to cosmic microwave \nbackground (CMB) analysis with great success in a Bayesian context~\\citep{2004PhRvD..70h3511W}.\n\\citet{Alsing2015} and \\citet{2016arXiv160700008A} have recently extended the WF technology from\nthe CMB literature to the problem of weak lensing shear field and power spectrum inference. However,\nthese WF approaches are limited not only by the WF assumptions above but also by requirements\nto smooth the measured galaxy ellipticity field to a uniform grid on the sky and to work solely\nwithin a linear theory approximation for the distribution of cosmological mass density perturbations.\nWorking within the stated assumptions, the primary challenge for WF approaches is the computation\nof large matrix solve operations. Novel and effective algorithms for sampling from the WF\ndistribution have been demonstrated for the CMB~\\citep{2013A&A...549A.111E,2016ApJ...820...31R}\nthat are also effective for smoothed, weak-shear, and linear theory cosmic shear\ninferences~\\citep{2016arXiv160700008A}.\n\n\nThis paper is structured as follows. In \\autoref{sec:method} we describe the statistical\nframework for lensing convergence and shear inference given galaxy images or a galaxy\nellipticity catalog. In \\autoref{sec:framework} we describe the Gaussian Process (GP)\nprior for the lensing potential and how this informs the correlated inferences of shear\nand convergence. We give a prescription for GP parameter optimization in\n\\autoref{sub:optimizing_interpolation_parameters}.\nWe apply our method to infer lensing convergence maps of simulated ellipticity catalogs\nin \\autoref{sub:simulation_study} and of an observed galaxy cluster in \\autoref{sub:abell}.\nWe describe our main conclusions in \\autoref{sec:conclusions}.\nWe provide details of the GP covariance derivation for lensing shear and convergence\nfields in \\autoref{sec:Gaussian process covariances} and provide\nlensing map inferences under an analogous covariance\nfor a cosmological model with a linear theory approximation in\n\\autoref{sec:cosmology_dependent_covariance_model}.\n\n\\section{Method}\n\\label{sec:method}\n\nWe describe a joint probabilistic model for the lensing shear and convergence given galaxy imaging\ndata. We previously presented the complete statistical framework for cosmic shear\ninference~\\citep{mbi-theory} and here focus on the specifics of lensing convergence inference\nas a function of sky coordinates in both linear and nonlinear regimes of the cosmological\nmass density perturbations. In \\autoref{fig:pgm} (left panel) we show the relationships in\nthe probabilistic model for CCD galaxy imaging data for multiple observation epochs, multiple\ngalaxies, and multiple galaxy samples (e.g., samples selected in different photometric redshift bins).\n\\begin{figure*}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.48\\textwidth]{shear_general_pgm.png}\n    \\includegraphics[width=0.4\\textwidth]{shear_gp_pgm.png}\n  }\n  \\caption{Two successive levels of approximation for our statistical model for sampling\n  probabilistic lensing shear and convergence $\\Upsilon$ fields.\n  The unshaded circles indicate sampling parameters.\n  Shaded circles indicate observed parameters while dots indicate parameters with fixed values\n  rather than being sampled.\n  %\n  Left panel: The model for galaxy image pixel data ${\\mathbf{d}}_{n,i}$ for each galaxy $n$ and observation\n  epoch $i$ requires specification of the pixel noise $\\sigma_{n,i}$, the intrinsic (i.e., unlensed)\n  galaxy ellipticity $e_n$, the lensing fields $\\Upsilon$, and the PSF at each galaxy location in\n  each epoch ${\\Pi}_{n,i}$. The distribution of galaxy intrinsic ellipticities is described by the\n  parameters $\\alpha_n$, which can specify distinct distributions for each galaxy $n$.\n  In \\citet{mbi-theory} we infer marginal constraints on $\\alpha_n$ under a Dirichlet Process (DP)\n  prior. For this initial study, we assert a Gaussian Process (GP) prior on the lens fields\n  $\\Upsilon$ in this paper, with the assumption that the posterior inferences of $\\Upsilon$\n  will be related to a cosmological model in a separate analysis pipeline as we describe in the\n  text. Here we assert a known PSF ${\\Pi}_{n,i}$ at every galaxy location, again deferring the\n  inference and marginalization of PSFs to a separate paper. The inference of the lens fields also\n  depends on our assertion of the survey window function $W$.\n  %\n  Right panel: The approximate statistical model when the galaxy imaging pixel data is summarized as\n  a galaxy ellipticity catalog ($e_{n}, \\sigma_{e,n}$ for $n=1,\\dots,n_{\\rm gal}$) requires specification of\n  the intrinsic ellipticity distribution parameters $\\alpha$ (now assumed the same for all galaxies),\n  the ellipticity measurement uncertainties $\\sigma_{\\rm ms,n}$ per galaxy, and the lens fields\n  $\\Upsilon$.\n  We now also assert the variance, $\\alpha$ of the assumed zero-mean\n  intrinsic ellipticity distribution.\n  }\n  \\label{fig:pgm}\n\\end{figure*}\n\n\n\\begin{table*\n\\begin{center}\n\\caption{Parameters for the statistical model. }\n\\label{tab:sampling_parameters}\n\\begin{tabular}{cl}\n\\hline\nParameter & Description \\\\\n\\hline\n$i$ & index over epochs, or different exposures of a galaxy\\\\\n$n$ & index over galaxies\\\\\n${\\Pi}_{n,i}$ & Point Spread Function (PSF) for galaxy $n_s$ in epoch $i$ \\\\\n$e_{n}$ & Intrinsic (pre-lensing) ellipticity of a galaxy\\\\\n$\\Upsilon_{s}$ & lensing shear and convergence that modify a galaxy image\\\\\n$\\psi_{s}$ & lensing potential \\\\\n$\\theta$ & cosmological parameters\\\\\n$W$ & survey window function \\\\\n$\\sigma_{n,i}$ & noise properties in a galaxy image\\\\\n$\\alpha_{n}$ & parameters specifying the intrinsic ellipticity distribution for a galaxy\\\\\n$a_{\\rm DP}$ & parameters specifying the distribution over $\\alpha_{n_s}$\\\\\n${\\bf d_{n}}$ & data vector (measured $e_{1,2}$ for each galaxy $n$)\\\\\n$\\sigma_{{\\rm ms}; n}$ & ellipticity measurement error for galaxy $n$ \\\\\n$\\mathbf{x}$ & vector of 2D spatial locations of galaxies \\\\\n$\\epsilon^\\mathrm{int}_{n}$ & intrinsic galaxy ellipticity for galaxy $n$ \\\\\n$\\alpha\\equiv\\sigma_{e}^2$ & parameters of the distribution of galaxy parameters \\\\\nW & window function for the survey footprint \\\\\n$a_{\\rm GP}$ & parameters of the GP kernel\\\\\n$\\lambda_{\\rm GP}$ & precision of the GP kernel (element of $a_{\\rm GP}$) \\\\\n$\\ell^{2}_{\\rm GP}$ & squared GP correlation length (element of $a_{\\rm GP}$) \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table*}\n\nThe model for observed galaxy images requires (at least) specification of the point-spread\nfunction (PSF), ${\\Pi}_{n_s,i}$, at all $n_s$ galaxy locations in all epochs $i$, the intrinsic\nor pre-lensing shape or morphology of the galaxy,\n$e_{n_s}$, the applied lensing shear and magnification, $\\Upsilon_{s}$, and the noise properties\nof the pixelated image, $\\sigma_{n_s,i}$ (e.g., Gaussian with an asserted r.m.s.).\n\nTo simplify the discussion for this paper focused on shear inference, we assume the PSF ${\\Pi}_{n_s,i}$ is\nknown at the locations of all galaxies in all epochs. We do not further consider errors in the\nPSF inference in this work. Following \\citet{mbi-theory} we allow for the intrinsic ellipticity\n(and potentially size, fluxes, etc.) of each galaxy to be drawn from galaxy-specific distributions\nwith parameters $\\alpha_{n_s}$. The parameters $\\alpha_{n_s}$ in turn are hierarchically \ndistributed under a distribution with with parameters $a_{\\rm DP}$, allowing \ninference of the effective `shape noise' in the \nshear inference. We use the label $\\Upsilon$ for \nthe model for the lensing shear $\\gamma$ and convergence $\\kappa$, which uniquely specify the \ncomponents of the trace-free linear distortion matrix  under a weak lensing ($\\kappa\\ll1$) \napproximation of the lens equation~\\citep[e.g.,][]{bartelmann01}.\n\nAs illustrated by the `plates' in the left panel of \\autoref{fig:pgm}, the PSF is unique to each\nobservation epoch\nbut the intrinsic properties of a galaxy image are common across epochs (for each unique galaxy).\nA key feature in the graphical model of the left panel in \\autoref{fig:pgm} is that the lens fields\n$\\Upsilon_{s}$ are common across all galaxy images in all epochs, indicating that the we require\na spatially correlated model for the coherent lensing shear and magnification patterns on the sky.\nSpecifying an appropriate correlated model for the lens fields is the main focus of this paper.\nWe allow the lens field models to be distinct for different galaxy samples $s$ in\n\\autoref{fig:pgm} because, e.g., galaxies in different photometric redshift bins will be lensed\nby partially different foreground mass distributions. Only at the top level of the graphical\nmodel do the lens field inferences from different galaxy samples become connected under a\ncosmological model with parameters $\\theta$.\n\nWe therefore explore an interim probabilistic model for the lens fields inferred from distinct\ngalaxy samples $s$ such that inferences of $\\Upsilon_s$ are statistically independent for\ndifferent $s$. This will allow us to separate computationally expensive components of a cosmic\nshear inference pipeline, explore multiple cosmological and systematics models for the data,\nand eventually perform rigorous uncertainty propagation and marginalization of image and intrinsic\ngalaxy nuisance parameters as outlined in \\citet{mbi-theory}.\n\nIn measuring lensing shear of galaxy images, we must marginalize over the\nintrinsic ellipticities, $\\epsilon^\\mathrm{int}$, of the images. This is often done by averaging the ellipticities\nof galaxies in neighboring regions of the sky and redshift, where the weights can include the\nmeasurement and shape noise models. But, here we perform a\nmarginalization over an explicit intrinsic ellipticity distribution,\n\\begin{multline}\\label{eq:marg_like}\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha} = \\prod_{n=1}^{n_{\\rm gal}}\n  \\int d^{m}\\epsilon^\\mathrm{int}_n\\,\n  \\prf{{\\mathbf{d}}_n | \\epsilon^\\mathrm{int}_n, \\Upsilon(\\mathbf{x}_n)}\n  \\\\ \\times\n  \\prf{\\epsilon^\\mathrm{int}_n|\\alpha},\n\\end{multline}\nwhere we assume the likelihood functions for each galaxy image $n=1,\\dots,n_{\\rm gal}$ are\nstatistically independent~\\citep[see][for more discussion of this assumption]{mbi-theory}.\nThe data vector ${\\mathbf{d}}_{n}$ is composed of either the pixel values contributing to the image of\ngalaxy $n$ or a summary statistic of those pixel values.\n\nAs a pedagogical step in the development of our probabilistic lens field model,\nwe will consider an approximate likelihood function for summary statistics of the pixel data;\nnamely estimators for the\nellipticity of each galaxy image $e\\equiv e_1 + i e_2$ along with an associated measurement\nerror per ellipticity component $\\sigma_{\\rm ms}$. See the right panel of \\autoref{fig:pgm}.\nWe will develop the approximate model for ellipticity measurements as a data vector\nin this paper but advocate for the more complete algorithm of \\citet{mbi-theory}\nfor any data analysis because of the known large biases in using ellipticity estimators for\ncosmic shear~\\citep[e.g.,][]{Refregier++2012,Kacprzak++2012}.\n\nAssuming Gaussian distributed ellipticity measurement errors the likelihood\nfunction is then,\n\\begin{equation}\\label{eq:likelihood}\n  \\prf{{\\mathbf{d}}_n|e_n, \\Upsilon(\\mathbf{x}_n)} = \\mathcal{N}_{\\hat{e}_n}\n  \\left(e_n, \\sigma^2_{{\\rm ms}; n}\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right),\n\\end{equation}\nwhere we explicitly label the data ${\\mathbf{d}}_n\\equiv\\hat{e}_{n}$ as ellipticity estimators,\nand the distribution is bivariate given the two ellipticity components.\n\nA general likelihood function depends on the observable galaxy properties\nsuch as ellipticity, size, and flux, which are modified by lensing from the intrinsic\nproperties described by $\\epsilon^\\mathrm{int}$.\nSo, to evaluate \\autoref{eq:likelihood}, we define the lensed galaxy parameters,\n\\begin{equation}\\label{eq:lens_transform}\n  e_n(\\Upsilon(\\mathbf{x}_n)) \\equiv f(\\epsilon^\\mathrm{int}_n, \\Upsilon(\\mathbf{x}_n)),\n\\end{equation}\nwhere $f(\\cdot)$ denotes the function that transforms the intrinsic, unlensed, galaxy\nellipticities $\\epsilon^\\mathrm{int}_n$ to those in the lensed model $e_n$\nunder the action of the lensing convergence and shear specified by $\\Upsilon(\\mathbf{x}_n)$\nat the galaxy sky location $\\mathbf{x}_n$.\nIn the weak shear limit defined by $\\kappa \\ll 1$, \\autoref{eq:lens_transform} reduces to\n\\begin{equation}\\label{eq:weak_shear}\n  \\tilde{e}_{n}^{\\rm weak-shear} = \\epsilon^\\mathrm{int}_{n} + g(\\mathbf{x}_n),\n\\end{equation}\nwhere $g\\equiv\\gamma\/(1-\\kappa)\\approx\\gamma$ is the reduced shear.\n\nFor our pedagogy, we specify a Gaussian distribution for the \\emph{unlensed}\ngalaxy properties $\\epsilon^\\mathrm{int}$ to use in evaluating \\autoref{eq:marg_like},\n\\begin{equation}\n  \\prf{\\epsilon^\\mathrm{int}_{n}|\\alpha,\\Upsilon(\\mathbf{x}_n)} =\n  \\mathcal{N}_{\\epsilon^\\mathrm{int}_n} \\left(0, \\sigma_{e}^2\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right).\n\\end{equation}\n\nUsing the weak shear approximation in \\autoref{eq:weak_shear}, we can perform\nthe marginalization integral in \\autoref{eq:marg_like} analytically,\n\\begin{align}\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha} &= \\prod_{n=1}^{n_{\\rm gal}}\n    \\int d^{2}\\epsilon^\\mathrm{int}_n\\,\n    \\mathcal{N}_{\\epsilon^\\mathrm{int}}\\left(\\hat{e}_n - g(\\mathbf{x}_n), \\sigma_{{\\rm ms};n}^2\\mathbb{1}} % identity matrix: \\usepackage{bbold_2\\right)\n    \\notag\\\\\n    &\\quad\\times\n    \\mathcal{N}_{\\epsilon^\\mathrm{int}}\\left(0, \\sigma_{e}^{2}\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right)\n    \\\\\n    &= \\prod_{n=1}^{n_{\\rm gal}}\n    \\mathcal{N}_{g(\\mathbf{x}_n)}\\left(\\hat{e}_n, \\left(\\sigma_{{\\rm ms};n}^2 + \\sigma_{e}^2\\right)\\mathbb{1}} % identity matrix: \\usepackage{bbold_{2}\\right)\n    \\\\\n    &= \\mathcal{N}_{\\Matrix{g}}\\left(\\hat{\\Matrix{e}}, \\mathsf{N}\\right),\n    \\label{eq:marg_like_analytic}\n\\end{align}\nwhere in the final line we defined the $2n_{\\rm gal}$-length vector of observed ellipticities $\\hat{\\Matrix{e}}$,\nthe same-length vector of reduced shear components at each galaxy locations $\\Matrix{g}$, and\nthe diagonal $2n_{\\rm gal}\\times 2n_{\\rm gal}$ dimensional covariance matrix $\\mathsf{N}$ with the $n$th diagonal\nentry equal to $\\sigma^2_{{\\rm ms};n} + \\sigma^2_{e}$.\nNote the weak shear likelihood does not depend on the lensing convergence $\\kappa$ (because the\nreduced shear is approximated as equal to the non-reduced shear and we ignore lensing magnification\neffects on the galaxy sizes and fluxes).\n\n\n\\subsection{A maximum entropy prior for lensing fields}\n\\label{sec:framework}\n\nWe want an interim prior on the lensing convergence and shear that is not only\nindependent of cosmology but also broadly encompassing of the possible\ncosmological interpretations and spatially varying systematics contributions.\nThat is, by choosing a functional form of a prior\nfor interpolating shear over the sky and marginalizing statistical uncertainties,\nwe should not restrict the class of physical models that might explain the data.\nA mathematical version of this sentiment is that we want a\nmaximum entropy prior~\\citep{PhysRev.106.620,shore1980} on the lensing fields.\n\nFor an assumed mean and (co-)variance, the Gaussian distribution is the maximum entropy\ndistribution~\\citep{10.2307\/2984828}. Therefore, because it is principally the second moment of the\nlensing convergence that we use to constrain cosmological models, we choose a\nGaussian Process (GP) interim prior for sampling convergence (and shear) fields on\nthe sky given measurements of galaxy image moments.\n\nGravitational lensing shear is a spin-2 field and is non-locally\nrelated to the lensing convergence, which both present modeling challenges.\nHowever, both the lensing convergence and shear can be derived as second\nderivatives of a scalar valued lensing potential $\\psi$. So we impose the\nGP prior on the potential $\\psi$ and derive the related priors on the\nconvergence $\\kappa$ and shear $\\gamma$ from this starting assertion.\n\nEach of the lensed observables, $\\Upsilon(\\mathbf{x}) \\equiv \\left[\\kappa, \\gamma_1, \\gamma_2\\right]$\nis related to the lensing potential $\\psi$ via derivatives in the form of:\n\\begin{align}\n  \\kappa &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1^2} +\n    \\frac{\\partial^2 \\psi}{\\partial x_2^2 }\\right) = \\frac{1}{2}(\\psi_{,11} +\n    \\psi_{,22})\n  \\label{eq:kappa_from_psi}\n  \\\\\n  \\gamma_1 &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1^2} -\n    \\frac{\\partial^2 \\psi}{\\partial x_2^2}\\right) = \\frac{1}{2}(\\psi_{,11} -\n    \\psi_{,22})\n  \\label{eq:gamma1_from_psi}\n  \\\\\n  \\gamma_2 &=\\frac{1}{2}\\left(\\frac{\\partial^2 \\psi}{\\partial x_1 \\partial\n    x_2} +\\frac{\\partial^2 \\psi}{\\partial x_2 \\partial x_1}\\right) =\n    \\frac{1}{2}(\\psi_{,12} + \\psi_{,21}).\n  \\label{eq:gamma2_from_psi}\n\\end{align}\nIt is straightforward to derive via integration by parts on the moments of the field \nthat if $\\psi(\\mathbf{x})$ is Gaussian\ndistributed for given $\\mathbf{x}$, then so too is $\\Upsilon(\\mathbf{x})$, but with a\nmodified covariance.\nBy specifying a GP prior on the lens potential, we therefore can derive\na GP prior on the combination of lensing convergence and shear fields that preserves the\nphysical correlations between these fields. We will show that we can infer the lensing \nconvergence and shear via correlated draws from a GP distribution given a galaxy \nellipticity catalog. We do not then compute lensing convergence and shear from \nspatial derivatives of a GP-distributed lens potential. The latter operation is only \nprecisely defined when we have (in principle) knowledge of the lens potential at all \nsky locations. However, galaxies provide a non-contiguous background for measuring the lens \npotential. The relations in \\autoref{eq:kappa_from_psi}, \\autoref{eq:gamma1_from_psi}, and \n\\autoref{eq:gamma2_from_psi} are then imposed only by the GP covariance structure during \nsampling and marginalization.\n\nThe derivations of using a GP to represent the lensing convergence and shear fields were\nfirst presented in \\citet{ng2016}. We refer readers to \\citet{ng2016} for an introduction to\nthe basics of a GP. We show the derivations from \\citet{ng2016} in\n\\autoref{sec:Gaussian process covariances} for the convenience of the reader.\nIn particular, the GP prior for $\\Upsilon$\nderived from that for $\\psi$ should not mix E and B modes in the two-point function\nbecause we only allow for GP realizations that preserve the combinations of fields that satisfy\nEquations~\\ref{eq:kappa_from_psi}--\\ref{eq:gamma2_from_psi} in the two-point correlations.\nIt is also straightforward\nto extend the scalar valued potential $\\psi$ to a complex-valued potential\n$\\psi\\rightarrow\\psi_{E} + i\\psi_{B}$\nto model or infer both E and B mode contributions to a measured shear signal.\n\nWe choose a squared exponential kernel for the GP model of the lensing potential $\\psi$,\n\\begin{align}\\label{eq:gp_cov}\n  \\mathsf{S}_{\\psi}(\\mathbf{x}, \\mathbf{y}; \\lambda_{\\rm GP}, \\ell^{2}_{\\rm GP}) =\n \n  \\lambda_{\\rm GP}^{-1} \\exp\\left(-\\frac{1}{2} \\frac{s^2(\\mathbf{x},\\mathbf{y})}{\\ell^{2}_{\\rm GP}}\\right),\n\\end{align}\nwhere $\\lambda_{\\rm GP}$ is a precision parameter that sets the amplitude of fluctuations of the GP,\n$\\ell^{2}_{\\rm GP}$ is a squared distance defining the correlation length of the GP kernel,\nand the squared distance $s^2$ between pairs of galaxy locations is,\n\\begin{equation}\\label{eq:euclid_dist}\n  s^2 \\equiv (\\mathbf{x} - \\mathbf{y})^T (\\mathbf{x} - \\mathbf{y}).\n\\end{equation}\nThe squared exponential kernel is useful in interpolating smooth response functions between \nthe observed galaxy locations, as we expect for low resolution or low signal-to-noise ratio (SNR)\nweak lensing mass reconstructions. Exploration of other kernel choices for different mass \nreconstruction resolutions and SNRs is an interesting question that we leave for later work.\nA practical motivation for our kernel choice is that the two parameters of the squared exponential \nkernel can be optimized for different data sets without particular numerical or computational \nchallenges. A further justification for our smooth kernel choice comes from our intention to use the \nGP as merely an interim prior for sampling lens fields. We will later describe how these interim lens \nfield realizations may be re-weighted under cosmologically informed priors.\nWe list the derivatives of \\autoref{eq:gp_cov} in Appendix \\ref{sec:Gaussian process covariances} \nthat are required to build the joint covariance for the lensing shear and convergence.\nWe will denote the covariance constructed from second derivatives of \\autoref{eq:gp_cov} as\n$\\mathsf{S}_{\\rm GP}$.\n\nWe now return to the pedagogical derivation of the likelihood function for galaxy ellipticity\nestimators adding the GP distribution for the shear.\nBecause both the marginal likelihood in \\autoref{eq:marg_like_analytic} and the shear prior are\nGaussian distributions in the shear, we can specify the shear (and convergence) joint posterior\nfor all galaxy and grid locations as a multivariate Gaussian distribution (otherwise known\nas the Wiener filter),\n\\begin{multline}\\label{eq:shear_posterior}\n  \\prf{\\Upsilon(\\mathbf{x},\\mathbf{x}') | {\\mathbf{d}}, \\alpha, a_{\\rm GP}}\n  =\n  \\mathcal{N}_{\\Upsilon(\\mathbf{x},\\mathbf{x}')}\n  \\left({\\bm \\mu}_{\\Upsilon}, \\mathsf{S}_{\\Upsilon} \\right)\n  \\\\ \\times\n  \\mathcal{N}_{\\hat{\\Matrix{e}}}\\left(0, \\mathsf{N} + \\mathsf{S}_{\\rm GP}\\right),\n\\end{multline}\nwhere,\n\\begin{align}\n  \\mathsf{S}_{\\Upsilon} &\\equiv \\mathsf{S}_{\\rm GP}\\left(\\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right)^{-1}\\mathsf{N}\n  \\label{eq:posterior_cov}\n  \\\\\n  {\\bm \\mu}_{\\Upsilon} &\\equiv \\mathsf{S}_{\\rm GP} \\left(\\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right)^{-1} \\hat{\\Matrix{e}}.\n  \\label{eq:posterior_mean}\n\\end{align}\nCalculation of ${\\bm \\mu}_{\\Upsilon}$ in \\autoref{eq:posterior_mean} yields a posterior mean\nestimate of the lensing shear and convergence at every galaxy location. However, this estimator\nrequires inversion of an $2n_{\\rm gal}\\times2n_{\\rm gal}$ covariance matrix, which can be computationally expensive.\n\nOther works have attempted to reduce the dimensionality of the covariances\nin \\autoref{eq:shear_posterior} by interpolating and averaging\nthe measured galaxy ellipticities onto a grid of coarser resolution than that\nsampled by the galaxy angular distribution~\\citep[e.g.,][]{Alsing2015}.\nHowever, such interpolations not only restrict the measured dynamic range,\nbut also can be expected to introduce artefacts in the inferred shear field on the grid based on\nthe shape of the smoothing kernel, to propagate\nshape measurement systematics to a broad range of angular scales,\nand to ignore error propagation from individual galaxy shape measurements to the shear inference.\nOur method defines an explicit interpolation from galaxy to other sky locations, with error\npropagation included.\n\nWe can marginalize the lensing fields at the galaxy locations to obtain the marginal\nposterior for the lensing fields at just the smaller number of grid locations $\\mathbf{x}'$,\n\\begin{multline}\\label{eq:lens_posterior}\n  \\prf{\\Upsilon(\\mathbf{x}') | {\\mathbf{d}}, \\alpha, a_{\\rm GP}} \\propto\n  \\int d\\Upsilon(\\mathbf{x})\\,\n  \\prf{{\\mathbf{d}} | \\Upsilon(\\mathbf{x}), \\alpha}\n  \\\\ \\times\n  \\prf{\\Upsilon(\\mathbf{x}') | \\Upsilon(\\mathbf{x}), a_{\\rm GP}}\n  \\prf{\\Upsilon(\\mathbf{x}) | a_{\\rm GP}}.\n\\end{multline}\nWhen all distributions are Gaussians, \\autoref{eq:lens_posterior} reduces to evaluating\n\\autoref{eq:shear_posterior} at only the locations of the grid $\\mathbf{x}'$ given the input\nellipticities $\\hat{\\Matrix{e}}$. Note however that we still require in \\autoref{eq:posterior_mean}\nthe evaluation of the shear model at every galaxy location, which is then interpolated by the\nWF to arbitrary sky locations. Our approach is therefore different from algorithms that require\nan initial averaging of galaxy ellipticity components over local sky regions.\n\\autoref{eq:lens_posterior} thus specifies the interim distribution of lensing shear and convergence\nthat can be later propagated to cosmological model analyses.\n\n\n\n\\subsubsection{Optimizing interpolation parameters}\n\\label{sub:optimizing_interpolation_parameters}\n\nWe can further marginalize the lens fields $\\Upsilon$ at all locations $\\mathbf{x}$ and $\\mathbf{x}'$ to obtain\nthe marginal likelihood for the GP parameters,\n\\begin{equation}\\label{eq:gp_marg_like}\n  \\prf{{\\mathbf{d}} | \\alpha, a_{\\rm GP}} = \\mathcal{N}_{{\\mathbf{d}}}\n  \\left(0, \\mathsf{S}_{\\rm GP} + \\mathsf{N}\\right).\n\\end{equation}\nWe maximize the density in \\autoref{eq:gp_marg_like} to determine suitable values of the\nGP parameters for interpolation of the lens fields for a given ellipticity\ncatalog~\\citep[see a similar approach in][]{Marshall:531310}.\n\n\\autoref{eq:gp_marg_like} is informative on the GP parameters if the data vector ${\\mathbf{d}}$ includes\nsufficient numbers of galaxies to beat down the shape noise. In cases with smaller\nsignal to noise ratios we can also add a cosmologically informed prior to help constrain\nthe GP parameters. In this case we can replace the data vector in \\autoref{eq:gp_marg_like}\nwith a simulation of the data vector derived from a cosmological model. Then, by marginalizing\nover realizations of the simulated data vector we get a marginal prior for the GP parameters,\n\\begin{align}\\label{eq:marg_dist_cosmo}\n  \\prf{a_{\\rm GP}| \\theta, \\alpha} &=\n  \\int d{\\mathbf{d}}^{\\rm sim}\\,\n  \\prf{a_{\\rm GP} | {\\mathbf{d}}^{\\rm sim}, \\alpha} \\prf{{\\mathbf{d}}^{\\rm sim} | \\theta},\n  \\notag\\\\\n  %\n  &=\n  \\prf{a_{\\rm GP}}\n  \\\\\n  &\\times\n  \\int d{\\mathbf{d}}^{\\rm sim}\\, \\frac{\\prf{{\\mathbf{d}}^{\\rm sim} | a_{\\rm GP}, \\alpha}}{\\prf{{\\mathbf{d}}^{\\rm sim} | \\alpha}}\n  \\prf{{\\mathbf{d}}^{\\rm sim} | \\theta}.\n  \\notag\n\\end{align}\nIn \\autoref{eq:marg_dist_cosmo} we specify a model for drawing simulated data realizations\n${\\mathbf{d}}^{\\rm sim}$ given cosmological parameters $\\theta$. A standard approach to such a model\nis to first draw a realization of the 3D mass density perturbations from a Gaussian distribution\nwith a cosmologically determined initial power spectrum and then to evolve the initial\nmass density perturbations under a numerical (e.g., $N$-body) simulation of gravitational\nevolution.\n\nModeling only linear density perturbation evolution, the late-time model for the\nprojected lensing mass density, and hence the shear, is also a Gaussian distribution,\n\\begin{equation}\\label{eq:linear_cosmo_model}\n  \\prf{{\\mathbf{d}}^{\\rm sim}|\\theta}=\\mathcal{N}_{{\\mathbf{d}}^{\\rm sim}}(0, \\mathsf{\\Sigma}(\\theta)),\n\\end{equation}\nfor some cosmological covariance $\\mathsf{\\Sigma}(\\theta)$.\nTo evaluate \\autoref{eq:marg_dist_cosmo}  we also must determine the evidence\n\\begin{equation}\\label{eq:evidence}\n  \\prf{{\\mathbf{d}}^{\\rm sim}|\\alpha, I} \\equiv\n  \\int da_{\\rm GP}\\, \\prf{{\\mathbf{d}}^{\\rm sim} | \\alpha, a_{\\rm GP}}\n  \\prf{a_{\\rm GP} | I},\n\\end{equation}\nwhere $I$ denotes prior information on the GP parameters.\nEven under a linear density perturbation approximation the evidence\ncannot be calculated analytically. \n\nFor illustration of the GP parameter prior informed by cosmology, we use \\autoref{eq:linear_cosmo_model} \nto evaluate \\autoref{eq:marg_dist_cosmo} via Monte Carlo integration. We draw $N$ samples \nof ${\\mathbf{d}}^{\\rm sim}_{i}$ from $\\prf{{\\mathbf{d}}^{\\rm sim}|\\theta}$ and evaluate \n$\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|a_{\\rm GP}, \\alpha}$ for each $i=1,\\dots,N$. We approximate \nthe evidence in \\autoref{eq:evidence} via 2D numerical integration and then calculate \n\\begin{equation}\n  \\prf{a_{\\rm GP}| \\theta, \\alpha} \\approx \\prf{a_{\\rm GP}}\n  \\frac{1}{N} \\sum_{i=1}^{N} \\frac{\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|a_{\\rm GP}, \\alpha}}\n  {\\prf{{\\mathbf{d}}^{\\rm sim}_{i}|\\alpha, I}}.\n\\end{equation}\nWe show the resulting posterior constraints on the GP parameters in \\autoref{fig:gp_cosmo_prior}.\n\\begin{figure}[!htb]\n    \\centerline{\n        \\includegraphics[width=0.4\\textwidth]{gp_cosmo_prior.png}\n    }\n    \\caption{Posterior constraints on the GP parameters after marginalizing realizations of \n    simulated lens fields under a cosmological model. The correlations and variance in the simulated \n    data serve to communicate the cosmological covariance structure into constraints on the \n    GP parameters. The tight degeneracy shows the tradeoff between low correlations and large \n    GP precision (equivalent to white noise as the GP precision becomes large) and larger correlations \n    and smaller precision as the simulated maps are fit with smoother models. The lines show the 3-$\\sigma$\n    confidence interval.}\n    \\label{fig:gp_cosmo_prior}\n\\end{figure}\nThe cosmological simulation realizations impose a tight constraint in a linear combination \nof the logarithm of the two GP parameters. The constraints in \\autoref{fig:gp_cosmo_prior} \nindicate the cosmological lens field simulations can be fit either with a large GP precision \nand small correlation length (i.e., essentially as white noise), or with a smaller precision \nand larger correlation lengths (i.e., as a smooth correlated map). \nWe thus obtain tight constraints on the GP parameters if we can include some prior knowledge \nof the correlation length scale via $\\prf{a_{\\rm GP}}$.\n\nSee \\autoref{sec:cosmology_dependent_covariance_model} for\na description of our linear theory cosmological model for the lensing convergence.\n\n\n\n\n\\subsection{Cosmological parameter inference}\n\\label{sub:cosmological_parameter_inference}\n\nUnder a standard cosmological model the lensing potential $\\psi$ is related to the\n3D cosmological late-time gravitational potential $\\Psi^{\\rm LT}$ by a projection along\nthe line of sight weighted by the lensing efficiency $K(a;A_s)$,\n\\begin{equation}\\label{eq:lenspot_proj}\n  \\bar{\\psi}_{s}(\\mathbf{x}) \\equiv\n \n  W(\\mathbf{x})\n  \\int dz\\,\n  \\Psi^{\\rm LT}(\\mathbf{x},z)\n  K(z ; Z_s),\n\\end{equation}\nwhere $W(\\mathbf{x})$ is the survey window function, $z$ is the cosmological redshift,\nand $Z_s$ defines the redshift distribution of source galaxies for a galaxy sample $s$.\nThe late-time gravitational potential is in turn related to the potential of the\ncosmological initial conditions $\\Psi^{\\rm IC}$ via a deterministic function $G$\ndefining gravitational evolution~\\citep{2013MNRAS.432..894J},\n\\begin{equation}\\label{eq:gravpot_ic}\n  \\Psi^{\\rm LT}(\\mathbf{x}, z) = G\\left(\\Psi^{\\rm IC}, \\theta, z\\right).\n\\end{equation}\nWe assume the initial conditions $\\Psi^{\\rm IC}$ are Gaussian distributed with mean\nzero and covariance $\\mathsf{\\Sigma}^{\\rm IC}(\\theta)$, which is entirely determined by the\npotential power spectrum from inflation.\n\nA common method to simulate $\\Psi^{\\rm LT}$ according to \\autoref{eq:gravpot_ic} is\nto run a cosmological $N$-body simulation with initial conditions $\\Psi^{\\rm IC}$\ndrawn from a Gaussian distribution with a specified power spectrum.\nThe numerical $N$-body solver allows evaluation of the function $G$, which does not\nhave a known analytic form beyond low-order perturbation theory.\nThe standard approach to evaluate \\autoref{eq:lenspot_proj} for the lens potential\nin a numerical simulation is to trace light rays through the observer's light cone\ngiven the simulated $\\Psi^{\\rm LT}(\\mathbf{x}, a)$. The bundle of light rays is\nevaluated at a discrete set of sky locations to predict the lensing shear and\nconvergence. The predicted lens fields must then be interpolated over the sky to\ngalaxy locations to complete the numerical cosmological model prediction.\n\nWe have already defined a probabilistic interpolation of lens fields over the sky\nvia the GP. We therefore define the conditional distribution of the lens potential\nat galaxy locations $\\mathbf{x}$ given the simulations evaluated at positions $\\mathbf{x}'$ via the\nmodel of \\autoref{eq:gravpot_ic} and \\autoref{eq:lenspot_proj}\nfollowed by GP interpolation,\n\\begin{equation}\\label{eq:cond_model_comparison}\n  \\prf{\\psi(\\mathbf{x}) | \\bar{\\psi}(\\mathbf{x}', \\theta),a_{\\rm GP}}\n  = \\mathcal{N}_{\\psi} \\left(\n    \\mu_{\\psi}, \\mathsf{\\Sigma}_{\\psi}\n  \\right),\n\\end{equation}\nwhere,\n\\begin{align}\n  \\mu_{\\psi} &\\equiv \\mathsf{S}(\\mathbf{x},\\mathbf{x}')\\mathsf{S}^{-1}(\\mathbf{x}',\\mathbf{x}')\n  \\bar{\\psi}_{s}(\\mathbf{x}'; \\Psi^{\\rm IC}, \\theta, A_s, W)\n  \\label{eq:mu_psi}\n  \\\\\n  \\mathsf{\\Sigma}_{\\psi} &\\equiv\n  \\mathsf{S}(\\mathbf{x}, \\mathbf{x}) - \\mathsf{S}(\\mathbf{x},\\mathbf{x}')\\mathsf{S}^{-1}(\\mathbf{x}',\\mathbf{x}')\\mathsf{S}(\\mathbf{x}',\\mathbf{x}),\n  \\label{eq:sigma_psi}\n\\end{align}\ndefine the mean and covariance for the conditional multivariate Gaussian.\nWith \\autoref{eq:cond_model_comparison} we have thus derived a conditional probability\ndistribution to compare theoretical predictions of the lens potential with interim\nsamples of the potential drawn under the GP prior.\nSaid another way, our comparison of conditional posterior\nsamples of the lens fields with the cosmological models is mediated by the interpolation\nover the sky using the GP.\n\nBy marginalizing the initial conditions realizations and the lens fields realizations\nwe can now derive the posterior distribution for the cosmological parameters,\n\\begin{multline}\\label{eq:marg_post_cosmo}\n  \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\propto\n  \\prf{\\theta}\n  \\\\ \\times\n  \\int d\\Upsilon\\, \\int d\\Upsilon'\\, \\int d\\Psi^{\\rm IC}\\,\n  \\delta_{D}\\left(\\Upsilon' - h(\\Psi^{\\rm IC})(\\mathbf{x}')\\right)\n  \\\\ \\times\n  \\frac{\\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}}\n    {\\prf{\\Upsilon' | a_{\\rm GP}}}\n  \\\\ \\times\n  \\prf{{\\mathbf{d}} | \\Upsilon, \\alpha}\n  \\prf{\\Upsilon | \\Upsilon', a_{\\rm GP}} \\prf{\\Upsilon' | a_{\\rm GP}},\n\\end{multline}\nwhere $h(\\Psi^{\\rm IC})$ indicates the deterministic gravitational evolution of the\ninitial conditions potential to late times where the lensing is observed.\n\nThe final line of \\autoref{eq:marg_post_cosmo} is the interim sampling distribution\nwe defined in the previous section. We can thus perform the integrals in\n\\autoref{eq:marg_post_cosmo} via Monte Carlo with the same interim lens field\nsamples that are generated in the map making algorithm.\n\\begin{figure}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.5\\textwidth]{shear_gp_pgm_cosmo_split.png}\n  }\n  \\caption{\n   \n   \n   \n   \n   \n    In our approximate cosmological parameter inference pipeline we assume\n    the lens fields at the galaxy locations $\\Upsilon$ are entirely determined\n    by GP interpolation from the lens fields at a regular set of sky locations\n    $\\Upsilon'$ at which we compute theory predictions.\n    The lens fields on the regular set are informed by the cosmological\n    model with parameters $\\theta$.\n    \n  }\n  \\label{fig:pgm_cosmo}\n\\end{figure}\n\n\nWe propose two sampling algorithms depending on whether or not we can directly evaluate the\nprobability density for the lens fields under the cosmological model.\n\\begin{description}\n  \\item[Sampling algorithm 1] This is an algorithm to use when it is possible to analytically\n  evaluate the cosmological probability density function.\n  With $K$ samples of $\\Upsilon(\\mathbf{x}')$ from the interim sampling distribution,\n  we use importance sampling to approximate the marginalizations in\n  \\autoref{eq:marg_post_cosmo},\n  \\begin{multline}\\label{eq:marglikecosmo}\n    \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\approx\n    \\prf{\\theta}\n   \n    \\frac{1}{K}\\sum_{k=1}^{K}\n    \\frac{\\prf{\\Upsilon_{k}(\\mathbf{x}') | \\theta}}\n      {\\prf{\\Upsilon_{k}(\\mathbf{x}') | a_{\\rm GP}}},\n  \\end{multline}\n  where from \\autoref{eq:marg_post_cosmo},\n  \\begin{multline}\n    \\prf{\\Upsilon(\\mathbf{x}') | \\theta} \\equiv\n    \\int d\\Psi^{\\rm IC}\\,\n    \\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}\n    \\\\ \\times\n    \\delta_{D}\\left(\\Upsilon' - h(\\Psi^{\\rm IC}(\\mathbf{x}'))\\right).\n  \\end{multline}\n\n  \\item[Sampling algorithm 2] This is an algorithm to use when cosmological modeling of the lens\n  fields is only possible via forward simulation.\n  For most cosmological models of interest, we can only predict the model for the\n  lens fields via forward simulation of the cosmological mass density perturbation evolution.\n  In this case, we have no direct mechanism to evaluate the density\n  $\\prf{\\Upsilon(\\mathbf{x}')|\\theta}$.\n  Instead, we follow the steps,\n  \\begin{enumerate}\n    \\item Draw $\\Psi^{\\rm IC}(\\mathbf{x}')$ from $\\prf{\\Psi^{\\rm IC}(\\mathbf{x}') | \\mathsf{\\Sigma}^{\\rm IC}(\\theta)}$.\n    \\item Compute the predicted lens fields given the drawn initial conditions\n    $h(\\Psi^{\\rm IC})$ (e.g., via $N$-body simulation and ray-tracing prediction of the lensing\n    shear and convergence).\n    \\item Select $K$ samples of $\\Upsilon(\\mathbf{x})$ from the set of interim samples under the\n    GP prior.\n    \\item Evaluate the density, $\\prf{\\Upsilon|\\Upsilon', a_{\\rm GP}}$, as in\n    \\autoref{eq:cond_model_comparison} for each sample $K$ and compute,\n    \\begin{equation}\n      \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\approx \\prf{\\theta}\n      \\frac{1}{K} \\sum_{k=1}^{K}\n      \\prf{\\Upsilon_{k}(\\mathbf{x})|\\Upsilon(\\mathbf{x}',\\theta), a_{\\rm GP}}.\n    \\end{equation}\n    Note that $\\Upsilon'$ here is that from the cosmological forward model, not that from the\n    interim sampling.\n  \\end{enumerate}\n\\end{description}\n\n\nFor Sampling Algorithm 1, if we assume linear cosmological perturbations we can approximate,\n  \\begin{equation}\n    \\prf{\\Upsilon(\\mathbf{x}')| \\theta} \\approx\n    \\mathcal{N}_{\\Upsilon(\\mathbf{x}')}\n    \\left(\\mathbf{0}, \\mathsf{\\Sigma}(P_{\\Upsilon}(\\theta))\\right).\n  \\end{equation}\n  Under this approximation we can combine the terms from \\autoref{eq:marg_post_cosmo},\n  \\begin{equation}\\label{eq:lens_field_cond_factor}\n    \\prf{\\Upsilon | \\Upsilon',a_{\\rm GP}} \\prf{\\Upsilon'|\\theta} =\n    \\mathcal{N}_{\\Upsilon(\\mathbf{x},\\mathbf{x}')} \\left(\n      0, \\mathsf{M}(\\mathbf{x},\\mathbf{x}'; a_{\\rm GP}, \\theta)\n    \\right),\n  \\end{equation}\n  where $\\mathsf{M}(\\mathbf{x},\\mathbf{x}'; a_{\\rm GP}, \\theta)$ is defined in \\autoref{eq:signal_cov_cosmo}.\n \n \n \n \n \n \n \n \n \n \n \n \n \n \n \n\n  If we further assume, as above, that the likelihood function is a Gaussian distribution in the\n  galaxy ellipticities with a linear weak shear applied, then \\autoref{eq:marg_post_cosmo} becomes,\n  \\begin{equation}\\label{eq:marg_post_cosmo_linear}\n    \\prf{\\theta | {\\mathbf{d}}, a_{\\rm GP}, \\alpha} \\propto \\prf{\\theta}\n    \\mathcal{N}_{{\\mathbf{d}}}\\left(\n      0, \\mathsf{M}(a_{\\rm GP}, \\theta) + \\mathsf{N}\n    \\right),\n  \\end{equation}\n  with the signal covariance as defined in \\autoref{eq:signal_cov_cosmo}.\n\n  To summarize, the approximations required for \\autoref{eq:marg_post_cosmo_linear} are,\n  \\begin{enumerate}\n    \\item a likelihood function that is Gaussian in the galaxy ellipticities,\n    \\item galaxy ellipticity measurements that are unbiased estimators of the reduced shear,\n      \\item weak shear ($\\kappa \\ll 1$),\n    \\item linear cosmological perturbations.\n  \\end{enumerate}\n  We can drop assumptions 1, 2, and 3 and still use Sampling Algorithm 1. But to avoid assumption\n  4 we must resort to Sampling Algorithm 2.\n\n\n\n\\section{Results}\n\\label{sec:results}\n\nWe evaluate the mean and covariance of the marginal posterior for the lensing shear and convergence\nfrom \\autoref{eq:shear_posterior} for a simulated data set, where we can compare with a known\ntruth (\\autoref{sub:simulation_study}). We also analyze\nthe ellipticity catalog~\\citep{jee2013} in the\nDeep Lens Survey (DLS)\\footnote{\\url{http:\/\/dls.physics.ucdavis.edu}}~\\citep{2002SPIE.4836...73W}\nin the vicinity of a massive galaxy cluster where the\namplitudes of the lensing fields are large (\\autoref{sub:abell}).\n\n\\subsection{Simulation study}\n\\label{sub:simulation_study}\n\nTo validate that \\autoref{eq:posterior_mean} can recover the correct shear and convergence fields,\nwe create simulated galaxy ellipticity catalogs with artificially small shape noise so we can\nmeasure the lensing shear and convergence to a tuneable precision.\n\nOur procedure for simulating galaxy ellipticity catalogs is,\n\\begin{enumerate}\n  \\item Calculate a lensing convergence angular power spectrum using the cosmology theory\n  code {\\sc CHOMP}\\xspace\\footnote{\\url{https:\/\/github.com\/karenyyng\/chomp}}~\\citep{2013JCAP...11..009M}.\n    We assume a standard\n  $\\Lambda$CDM cosmology with $\\Omega_m=0.3$, $\\sigma_8=0.8$ and a source redshift distribution\n  with a narrow peak at $z=1$.\n  \\item Simulate Gaussian-distributed lensing shear and convergence maps on a grid using the\n  code {\\sc GalSim}\\xspace\\footnote{\\url{https:\/\/github.com\/GalSim-developers\/GalSim}}~\\citep{galsim}.\n  \\item Place one galaxy in each grid cell of the simulated lensing shear maps. These galaxies are simply\n  sources of illumination for measuring the lensing fields, not cosmologically clustered galaxies.\n  \\item For each galaxy, draw intrinsic ellipticity components from a 2D Gaussian distribution\n  with mean 0 and a specified variance, $\\sigma_e^2$.\n  \\item Calculate lensed ellipticities for each galaxy by adding the lensing shear to the intrinsic\n  ellipticities, assuming a weak shear approximation.\n  \\item Save the galaxy angular sky locations and ellipticity components to a catalog file.\n\\end{enumerate}\nGiven a simulated ellipticity catalog, we find the GP parameters\n$\\lambda_{\\rm GP}, \\ell^{2}_{\\rm GP}$ that maximize \\autoref{eq:gp_marg_like}.\nWe then evaluate \\autoref{eq:posterior_mean} and \\autoref{eq:posterior_cov} using the optimized\nGP parameters to obtain the\nmarginal posterior distribution of the lensing fields $\\kappa, \\gamma$ at all galaxy locations\nas well as on a regular grid of locations.\nWe expect that in practical applications to large data sets the shear will only need to be\nevaluated at the galaxy locations and the converngence (or lens potential) will only need to be\nevaluated at a smaller number of sky grid locations, thus reducing the overall dimensionality of\nthe linear system to be solved.\n\n\nWe show an example of the output of this procedure in\n\\autoref{fig:mass_map_comparison_galsim}\ncompared to the input shear fields used to generate the mock ellipticity catalog. In the\nleft column in \\autoref{fig:mass_map_comparison_galsim} we show the posterior mean fields from\n\\autoref{eq:posterior_mean}.\nIn the adjacent panel we show the input shear fields.\nWe show the `true' convergence that we calculated at the same time as the input shear, but we\ndo not use the convergence at any point in our calculation. The ``estimated convergence'' comes\nfrom interpolating the measured galaxy ellipticities with the GP kernel.\nThe right panel in \\autoref{fig:mass_map_comparison_galsim} shows the\nsignal-to-noise ratio (SNR) of the lens field maps, defined as,\n\\begin{equation}\\label{eq:snr_def}\n  {\\rm SNR} \\equiv \\Upsilon \/ \\sqrt{{\\rm diag}\\left(\\mathsf{S}_{\\Upsilon}\\right)},\n\\end{equation}\nwith $\\mathsf{S}_{\\Upsilon}$ defined in \\autoref{eq:posterior_cov}.\nIn the limit that the variance of the lens fields dominates the intrinsic shape\nand ellipticity measurement variances, \\autoref{eq:posterior_cov} reduces to\n$\\mathsf{N}$. Then \\autoref{eq:snr_def} becomes\n${\\rm SNR} \\rightarrow \\Upsilon \/ \\sqrt{\\sigma_e^2 + \\sigma_{\\rm ms}^{2}}$,\nwhich is similar to SNR definitions in other weak lensing mass mapping\nanalyses~\\citep{2000MNRAS.313..524V,2014MNRAS.442.2534S}.\n\nThe simulation in\n\\autoref{fig:mass_map_comparison_galsim} has an artificially low intrinsic ellipticity r.m.s.\\\nof $\\sigma_e=0.0026$ compared to $\\sigma_e=0.26$ in the Deep Lens Survey that\nwe analyze in \\autoref{sub:abell}.\nWe choose a small shape noise r.m.s.\\ to\nallow us to validate our convergence inference with a small number of only 1600 galaxies.\nWe place the simulated galaxies on a $40\\times40$ grid, so that we do not have to interpolate\nthe simulated shear fields to build the mock ellipticity catalog.\n\nBecause the Gaussian model for the shape noise r.m.s.\\ scales with the number of galaxies\n$n_{\\rm gal}$ as $n_{\\rm gal}^{-1\/2}$, the shape noise r.m.s.\\ in\nour simulation is equivalent to a DLS-like galaxy sample with 16 million galaxies.\nThis is about 64 times the number of galaxies we have in a single four square degree field\nof the DLS. We show the effect of increasing the shape noise r.m.s.\\ by a factor\nof $\\sqrt{64}$ later in this section.\n\\begin{figure*}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.54\\textwidth]{mass_map_comparison_galsim.png}\n    \\includegraphics[width=0.35\\textwidth]{mass_map_snr_galsim.png}\n  }\n  \\caption{Left: Comparison of the convergence and shear maps in our simulation study between that\n  used to generate our mock galaxy ellipticity catalog (right column) and the output of our\n  GP interpolation (left column). The rows show the maps for the two shear components $\\gamma_{1,2}$\n  and the convergence $\\kappa$. These maps cover a $2\\times2$ square degree field. The simulated\n  intrinsic ellipticity r.m.s.\\ is set to an artificially small value of $\\sigma_e=0.0026$,\n  which is 100 times smaller than that observed for the complete Deep Lens Survey catalog.\n  Right: signal-to-noise ratio (SNR) maps for the same simulations. We calculate SNR as the\n  ratio of the map to the square root of the diagonal of the covariance\n  in \\autoref{eq:posterior_cov}.\n  These simulations use 1600 galaxies and grid sizes for the GP interpolated shear and convergence\n  with 24 grid cells per dimension.\n  }\n  \\label{fig:mass_map_comparison_galsim}\n\\end{figure*}\n\nThe reconstructed shear and convergence maps in \\autoref{fig:mass_map_comparison_galsim} are a\ngood match to the simulation inputs, but are noticeably smoothed. This is for two reasons;\n(1) we evaluate the interpolated lens fields on a smaller $24\\times24$ grid than the\n$40\\times40$ grid on which the inputs are evaluated,\n(2) we use a value of $\\ell^{2}_{\\rm GP}=0.0123$ that is larger than the Nyquist scale in the maps, even\nfor the $24\\times24$ interpolated grid because of the way the GP parameters are optimized.\nWe will discuss the GP parameter optimization below.\n\nIn \\autoref{fig:power_spectra_galsim} we show the E and B mode power spectrum estimators obtained\nfrom the posterior mean shear fields shown in \\autoref{fig:mass_map_comparison_galsim}.\nWe compare the E-mode power spectrum estimator from the mean posterior shear maps with that\nusing the higher-resolution shear maps that were used to generate the mock ellipticity\ncatalog (which we label `simulation truth'). We also show in \\autoref{fig:power_spectra_galsim}\nthe `theory' power spectrum that we used to generate the `simulation truth' shear maps.\nThe `theory' and `simulation truth' spectra agree on scales below the Nyquist frequency,\nshown in \\autoref{fig:power_spectra_galsim} as the vertical blue line. The mean posterior\npower spectrum agrees with the `simulation truth' spectrum on scales below the effective\nsmoothing frequency derived from the value of $\\ell^{2}_{\\rm GP}$ and shown by the vertical\ndot-dashed line in \\autoref{fig:power_spectra_galsim}.\n\\begin{figure}[!htb]\n  \\centerline{\n    \\includegraphics[width=0.5\\textwidth]{power_galsim_lownoise.png}\n  }\n  \\caption{Comparison of power spectrum estimators in our simulation study to the `truth' input\n  spectrum.\n  The vertical solid lines show the Nyquist frequencies of the grids.\n  The vertical dot-dashed line shows the multipole corresponding to the length scale\n  $\\sqrt{\\ell_{\\rm GP}^{2}}$ set by the GP kernel parameter $\\ell_{\\rm GP}^{2}=0.0123$ for\n  field coordinates are normalized to the unit square.\n  }\n  \\label{fig:power_spectra_galsim}\n\\end{figure}\n\nThe B-mode power spectrum estimated\nfrom the mean posterior shear maps is shown by the dashed red line in \\autoref{fig:power_spectra_galsim}.\nWe expect the B-mode power to be consistent with zero because we simulated only E-mode power.\nThe nonzero B-mode power spectrum in \\autoref{fig:power_spectra_galsim} is explained by\nexamining \\autoref{fig:EB_maps}, which shows the E and B mode mean posterior maps from which the\nE and B mode mean posterior power spectra were derived. The B-mode map in \\autoref{fig:EB_maps}\nis near zero throughout the field except near the boundaries. These edge effects in the B-mode map\ncan be explained by mathematical ambiguities in the definition of E and B mode separation in a\nfinite field~\\citep{bunn2003}. The small value of the B-mode map in \\autoref{fig:EB_maps} away from\nthe field boundaries indicates our GP interpolation method does not create spurious B-modes.\n\\begin{figure*}\n  \\centerline{\n    \\includegraphics[width=0.6\\textwidth]{EB_maps_galsim.png}\n  }\n  \\caption{E and B mode maps derived from the posterior mean shear maps shown in the left column\n  of \\autoref{fig:mass_map_comparison_galsim}.\n  The E-mode map closely matches the interpolated convergence map in\n  \\autoref{fig:mass_map_comparison_galsim} as expected.\n  The B-mode map is near zero throughout the center of the field, but shows non-zero\n  values around the field edge because of mathematical ambiguities in the E\/B mode separation\n  at the field boundaries~\\citep{bunn2003}. Note the value of the plotted B-mode map\n  has been multiplied by 10 for easier visualization.\n  }\n  \\label{fig:EB_maps}\n\\end{figure*}\n\nTo illustrate the GP parameter optimization procedure we show in \\autoref{fig:sigma_e_maps} the\nmean posterior convergence, the associated SNR maps, and the marginal likelihood surface for\nthe GP parameters (left to right columns) for increasing intrinsic ellipticity variance\n$\\sigma_e^2$ (top to bottom rows). The top row of panels in \\autoref{fig:sigma_e_maps} show a repeat\nof the $\\sigma_e^2$ and $a_{\\rm GP}$ values used in \\autoref{fig:mass_map_comparison_galsim}.\nFor the very small shape noise in this case, there is a narrow peak in the log-likelihood surface\nfor the two GP kernel parameters. We thus select the maximum likelihood (ML) values for $a_{\\rm GP}$\nand obtain the convergence and SNR maps that closely resemble the `true' convergence\nas shown in \\autoref{fig:mass_map_comparison_galsim}. However, the ML value of $\\ell^{2}_{\\rm GP}$ is somewhat\nlarger than the Nyquist frequency of the grid to which we interpolate as shown in\n\\autoref{fig:power_spectra_galsim} (compare dot-dashed to red solid vertical lines). Because the\nnoise is sub-dominant in this example, we would expect a value of $\\ell^{2}_{\\rm GP}$ matching the grid\nNyquist frequency to yield a more accurate convergence map reconstruction. We are likely to\nobtain more accurate results, therefore, if we impose a prior on $a_{\\rm GP}$ that encodes this\nexpectation.\n\\begin{figure*}\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0026$}}{kappa_galsim_sigmae1.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae1.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.012$}}{lnp_galsim_sigmae1.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0077$}}{kappa_galsim_sigmae2.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae2.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.061$}}{lnp_galsim_sigmae2.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.013$}}{kappa_galsim_sigmae3.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae3.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.061$}}{lnp_galsim_sigmae3.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.023$}}{kappa_galsim_sigmae5.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_sigmae5.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{6}, \\ell^2=0.30$}}{lnp_galsim_sigmae5.png}\n  }\n  \\caption{Posterior mean convergence maps (left), signal-to-noise ratio (SNR) maps (middle),\n  and marginal likelihood contours (right) for simulated galaxy catalogs with varying intrinsic\n  ellipticity r.m.s.\\ $\\sigma_e$.\n  From top to bottom, $\\sigma_e = \\left(0.00258, 0.00774, 0.0129, 0.0232\\right)$ (these values are\n  also annotated in the bottom left corner of each convergence map).\n  The top row matches the simulation shown in \\autoref{fig:mass_map_comparison_galsim}.\n  Following the right column from top to bottom shows how the maximum-likelihood estimates for the\n  GP parameters shifts to longer correlation lengths and smaller precision parameters with increasing\n  shape noise. That is, as the data becomes more noise dominated, the marginal GP parameter likelihood\n  changes shape to prefer smoothing, or effectively averaging, more galaxies to retain a more\n  significant shear and convergence signal. The ML values for the GP parameters are listed\n  in each panel showing the log-likelihood contours.\n  }\n  \\label{fig:sigma_e_maps}\n\\end{figure*}\n\nAs the shape noise increases (for the same input signal and measurement uncertainties), the peak\nin the marginal log-likelihood surface for $a_{\\rm GP}$ becomes broader and eventually disappears\nas shown by the rows of panels from top to bottom in \\autoref{fig:sigma_e_maps}. The ML value\nfor $\\ell^{2}_{\\rm GP}$, while less well defined, continues to yield maps that are more smoothed as\n$\\sigma_e^2$ increases. This helps to preserve large amplitudes of the peaks in the SNR maps (see\nthe color bar scales in the SNR maps of \\autoref{fig:sigma_e_maps}), but\nin compensation erases structures at all but the lowest spatial frequencies in the maps. This\nprocedure, with flat priors in the log of $a_{\\rm GP}$, appears useful for visualizing the posterior\nconvergence maps, but is undesirable for subsequent cosmological analyses. We see again that\nwe would prefer a prior favoring smaller $\\ell^{2}_{\\rm GP}$ even as the shape noise becomes large.\n\nWe assert such a prior in \\autoref{fig:sigma_e_maps_with_prior}, by imposing Gaussian priors\nseparately in $\\ln(\\lambda_{\\rm GP})$ and $\\ln(\\ell^{2}_{\\rm GP})$ with parameters given in\n\\autoref{tab:prior_params_galsim}. Our Gaussian prior is informed by the cosmological simulation study \nshown in \\autoref{fig:gp_cosmo_prior} combined with our prior that the GP correlation length be large enough \nso that white noise does not dominate the fits to the lens fields. \nFor each value of $\\sigma_e$ in \\autoref{fig:sigma_e_maps_with_prior}\nwe see that the signal-to-noise ratio (SNR) is comparable to that in \\autoref{fig:sigma_e_maps}\nbut the convergence maps include higher spatial frequency structures.\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:prior_params_galsim}Parameters for the Gaussian prior on the\n      GP parameters for our simulation study.}\n    \\begin{tabular}{lcc}\n      \\hline\\hline\n      Parameter & mean & std. dev. \\\\\n      \\hline\n      $\\ln\\left(\\lambda_{\\rm GP}\\right)$ & $18$ & 0.43 \\\\\n      $\\ln\\left(\\ell^{2}_{\\rm GP}\\right)$    & $-4.0$ & 0.1\\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\n\n\\begin{figure*}\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0026$}}{kappa_galsim_prior_sigmae1.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae1.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.012$}}{lnp_galsim_prior_sigmae1.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.0077$}}{kappa_galsim_prior_sigmae2.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae2.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.3\\times 10^{7}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae2.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.013$}}{kappa_galsim_prior_sigmae3.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae3.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae3.png}\n  }\n  \\centerline{\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\sigma_{e}=0.023$}}{kappa_galsim_prior_sigmae5.png}\n    \\includegraphics[width=0.3\\textwidth]{snr_galsim_prior_sigmae5.png}\n    \\subfigimg[width=0.3\\textwidth,pos='LOWER LEFT',hsep=15pt,font=\\footnotesize]{\\textcolor{white}{$\\lambda_{\\rm GP}=2.2\\times 10^{8}, \\ell^2=0.027$}}{lnp_galsim_prior_sigmae5.png}\n  }\n  \\caption{Same as \\autoref{fig:sigma_e_maps} except with a prior asserted for the GP\n  parameters $a_{\\rm GP}$ that limits the degree of smoothing in the posterior mean maps.\n  }\n  \\label{fig:sigma_e_maps_with_prior}\n\\end{figure*}\n\n\n\\subsubsection{Cosmological parameter constraints}\n\\label{sub:cosmo_params}\n\nAlthough we assert a non-cosmological GP prior on the lens fields to infer regular convergence\nand shear maps, we now demonstrate how we can recover cosmological information in a manner\nsimilar to common algorithms in the literature. That is, we compute an angular power spectrum\nestimator for the lensing convergence from the lens field posterior distribution.\n\nThe posterior distribution for the lens fields given the GP parameters is a multivariate\nGaussian distribution characterized by a mean field and covariance as given in\n\\autoref{eq:posterior_mean} and \\autoref{eq:posterior_cov}. The posterior mean is thus a\nconvenient and useful summary statistic (as we have shown above).\nAlso, because an isotropic Gaussian random field is fully described by the angular power spectrum,\nit is common in the literature to reduce cosmological large-scale structure statistics\nto two-point function estimators for cosmological parameter estimation.\n\nWe showed the angle-averaged (E-mode) convergence power spectrum estimator for our low-noise\nGaussian simulated maps in \\autoref{fig:power_spectra_galsim}. We use this power spectrum\nestimator as a summary statistic derived from the observed ellipticity catalog. We further\nassert a multivariate Gaussian likelihood function for the power spectrum estimator\nwith covariance~\\citep[e.g.,][]{2001ApJ...554...56C},\n\\begin{equation}\n  \\mathrm{Cov}\\left(P_{\\kappa}(\\ell)\\right) = \\frac{2}{N_{\\ell}}P_{\\kappa}^{2}(\\ell),\n\\end{equation}\nwhere $N_{\\ell}$ is the number of modes contributing to the band power estimator for a multipole\nbin centered at $\\ell$.\n\nIn \\autoref{fig:cosmo_param_contours} we show 68\\% and 95\\% contours of the 2D posterior\ndistribution for the cosmological mass density $\\Omega_m$ and density fluctuations r.m.s.\\\n$\\sigma_8$ given the angular power spectrum likelihood just described and flat priors.\nThe values used to generate the mock data are $\\Omega_m=0.3, \\sigma_8=0.8$.\nWe limit the multipole range of the power spectrum in the likelihood to $100 < \\ell < 1300$,\nwhere the upper bound is set by the effective smoothing length imposed by the asserted GP correlation\nparameter $\\ell^{2}_{\\rm GP}$.\n\\begin{figure}\n  \\centerline{\n    \\includegraphics[width=0.47\\textwidth]{cosmo_param_contours.png}\n  }\n  \\caption{Marginal constraints on the cosmological parameters using the E-mode power spectrum\n  estimator derived from the mean posterior convergence field.}\n  \\label{fig:cosmo_param_contours}\n\\end{figure}\n\nWe see from \\autoref{fig:cosmo_param_contours} that the GP interim prior used to derive the\nconvergence map from the galaxy ellipticity catalog has not biased the cosmological\nparameter constraints (within the uncertainties) obtained from a reduced summary statistic\nof the lens field posterior. We defer to later work more complete demonstrations of the\ncosmological parameter inference algorithms in \\autoref{sub:cosmological_parameter_inference}\nbased on marginalizing the lens field realizations.\n\n\n\\subsection{Abell cluster in the Deep Lens Survey}\n\\label{sub:abell}\n\nWe apply our lens field inference algorithm to the galaxy ellipticity catalog derived from the\nDLS\\footnote{\\url{http:\/\/dls.physics.ucdavis.edu}}~\\citep{2002SPIE.4836...73W}.\nThe DLS is a 20 square degree optical imaging survey optimized for cosmic shear measurements.\nWe analyze a $\\sim1$~square~degree field centered on the Abell cluster 781, which was\npreviously analyzed using DLS lensing shear measurements in\n\\citet{2006ApJ...643..128W,2009ApJ...702..980K,wittman2014}.\nAbell 781 consists of four massive galaxy clusters, which is a useful case study for our\nalgorithm because it provides a large lensing signal with a modest number of galaxies\nand because the distribution of mass density perturbations is decidedly not Gaussian\ndistributed.\n\n\\citet{jee2013} presented galaxy ellipticity measurements with the DLS $R$-band imaging calibrated\nto produce shear estimates with biases well below the statistical uncertainties for two-point\ncosmic shear correlation function estimators. The DLS shear pipeline in \\citet{jee2013} includes\ncorrelated PSF size and ellipticity corrections in each DLS exposure, calibration of additive\nand multiplicative shear biases with image simulations, and a set of null tests validating the PSF\nshear calibration corrections. The shear estimation pipeline used in \\citet{jee2013}, \\textsf{sFIT},\nwas further validated as the winning algorithm in the blinded community shear measurement\nchallenge \\textsf{GREAT3}~\\citep{great3-paper1}.\nThe DLS was performed with four optical pass-bands ($BVRz$) that allow photometric\nredshift (photo-$z$) estimates for all lensing source galaxies~\\citep{schmidt2013}.\n\\citet{jee2015} extended the DLS shear analysis to a tomographic cosmic shear measurement using\nthe photo-$z$ estimates.\n\nThe DLS galaxy ellipticity catalog produced for \\citet{jee2013} includes a catalog-level selection\nbased on measured galaxy magnitudes, sizes, ellipticity measurement error, photo-$z$ estimates, and\nproximity to masks as listed in \\citet{jee2013} Table 2.\nWe perform a further set of selections on this catalog as listed in\n\\autoref{tab:dls_selection}.\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:dls_selection}Selection criteria applied to the DLS galaxy catalog.}\n    \\begin{tabular}{ccc}\n    \\hline\\hline\n    Parameter        & min         & max \\\\\n    \\hline\n    $z_b$            & 0.45        & --  \\\\\n    $de$             & --          & 0.1 \\\\\n    $\\sqrt{a^2+b^2}$ & $0.8^{''}$  & --  \\\\\n    $R$              & 22          & 23  \\\\\n    \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\nWe choose the lower bound on (maximum posterior) photo-$z$, $z_b$, to select source galaxies\nthat are likely to be at redshifts larger than that of the highest redshift sub-cluster in the field\nat $z\\approx0.43$~\\citep{wittman2014}. We further select only those galaxies in the ellipticity\ncatalog with ellipticity measurement errors, $de$, less than 0.1 and sizes greater than 0.8~arcseconds to\nobtain galaxies likely to have more precisely measured shapes for informing the lensing shear.\nWe exclude galaxies with sizes, as determined from the geometric mean of the semi-major and semi-minor axes $a,b$,\nless than 0.8 arcseconds because the ellipticities tend to be less\nwell measured when the galaxy size is similar to that of the PSF.\nWe select the brighter galaxies based on $R$-band magnitude that are still likely to be faint\nenough to avoid significant contamination from cluster members. After all the selections listed\nin \\autoref{tab:dls_selection} we measure an ellipticity r.m.s.\\ of $\\sigma_e=0.21$. This measurement\nincludes the ellipticities with lensing effects included, but because lensing is sub-dominant\nto the intrinsic ellipticity dispersion we assert $\\sqrt{\\alpha} = \\sigma_e=0.21$ for the posterior\ninference for A781.\n\nWe show our mean posterior inference of the lensing convergence of A781 in\n\\autoref{fig:mass_map_Abell} using 6000 galaxies randomly selected from the cut sample described\nin \\autoref{tab:dls_selection}. We select only 6000 galaxies to limit the size of the lens field\njoint covariance that we must invert.\nThe left column of panels shows the convergence while the right\ncolumn shows the signal-to-noise ratio for the mean posterior. Our calculation is in the\ntop row of panels in \\autoref{fig:mass_map_Abell}, which we compare with the aperture mass\nalgorithm of \\citet{2012ApJ...747L..42D} in the bottom row of panels.\nNote we use the same galaxy sample as input to each mass mapping algorithm.\nThe algorithm of \\citet{2012ApJ...747L..42D},\ncalled `aperture densomitry', provides a mass estimator that is more localized on the sky. The shear\nin apertures is averaged with weighting functions that account for both angular selections and\nthe expected line-of-sight lensing kernel with the aide of the photometric redshift information\nin the DLS catalog. However, to make a more direct comparison with the algorithm in this paper,\nwe recomputed the aperture densomitry weights without using any photometric redshift information\nfor the source galaxies.\n\n\n\\autoref{fig:mass_map_Abell} shows that we obtain consistent results for the two main A781\nsub-clusters using our mean posterior map and the method used in a previous analysis. The white\ncrosses in \\autoref{fig:mass_map_Abell} indicate the locations of all sub-clusters detected\nin \\citet{wittman2014}. We do not detect all the same sub-clusters, which is likely\nbecause we use a significantly\nsmaller number of galaxies (6000 versus $\\sim 50000$) while we test the performance and scaling\nof our codes. However one sub-cluster denotedin \\autoref{fig:mass_map_Abell} is only detected\nin the literature in x-ray emission~\\citep{sehgal2008}.\n\\citet{wittman2014} also weight the source galaxies according to the expected\nlensing kernel and the photo-$z$ estimates. We make no use of photo-$z$ information other than\nin the sample selection.\n\\begin{figure*}[!htb]\n\t\\centerline{\n\t\t\\includegraphics[width=0.47\\textwidth]{kappa_Abell781.png}\n    \\includegraphics[width=0.47\\textwidth]{snr_Abell781.png}\n\t}\n  \\centerline{\n    \\includegraphics[width=0.47\\textwidth]{kappa_Abell781_WD.png}\n    \\includegraphics[width=0.47\\textwidth]{snr_Abell781_WD.png}\n  }\n\n\t\\caption{Posterior mean lens field maps (left) and SNR maps (right) for a\n  field centered on the galaxy cluster Abell 781. We use 6000 galaxies selected\n  to have photometric redshifts larger than the known redshifts of the two primary\n  clusters in this field of view ($z=0.296$ for Abell 781 and $z=0.43$ for a chance\n  alignment of a second cluster)\n  The top row of panels show the posterior mean map and SNR using our GP\n  prior. The bottom row of panels show the algorithm of \\citet{2012ApJ...747L..42D}\n  applied to the same galaxies with the same uniform per-galaxy weighting\n  as in the top row, but evaluated on a finer grid. The normalization of the\n  convergence in the lower left panel is arbitrary~\\citep[see][]{2012ApJ...747L..42D}.\n  The white crosses indicate sub-clusters identified by~\\citet{sehgal2008} with\n  x-ray detections. There is no associated mass for the white cross second\n  from the right in any published analyses of this system.\n\t}\n\t\\label{fig:mass_map_Abell}\n\\end{figure*}\n\n\\begin{table}[!htb]\n  \\begin{center}\n    \\caption{\\label{tab:prior_params_abell}Parameters for the Gaussian prior on the\n      GP parameters for A781.}\n    \\begin{tabular}{lcc}\n      \\hline\\hline\n      Parameter & mean & std. dev. \\\\\n      \\hline\n      $\\ln\\left(\\lambda_{\\rm GP}\\right)$ & $\\ln\\left(10^{6}\\right)$ & 0.1 \\\\\n      $\\ln\\left(\\ell^2\\right)$           & $\\ln\\left(10^{-4}\\right)$ & 0.5\\\\\n      \\hline\\hline\n    \\end{tabular}\n  \\end{center}\n\\end{table}\nIn \\autoref{fig:abell_lnp} we show the marginal log-likelihood for the A781 ellipticities in\nthe plane of the two GP parameters. Unlike in \\autoref{fig:sigma_e_maps} the noise covariance\nis now significant in defining the contours in \\autoref{fig:abell_lnp} such that a large\nGP precision and small GP correlation length is favored (indicating a sub-dominant\nsignal covariance). We therefore impose a Gaussian prior in the logarithm of the GP parameters\nwith parameters listed in \\autoref{tab:prior_params_abell}. We infer maximum posterior values\nof $\\lambda_{\\rm GP}=2.7\\times10^{6}$ and $\\ell^2=0.012$, which we use in the convergence inference\nin \\autoref{fig:mass_map_Abell}. However, as shown in \\autoref{fig:abell_lnp}, the marginal\nposterior for the GP parameters is only weakly peaked for this data set, and a range of GP\nparameters would be acceptable for the mass map inference.\n\\begin{figure}[!htb]\n    \\centerline{\n        \\includegraphics[width=0.45\\textwidth]{lnp_Abell781.png}\n    }\n    \\caption{The marginal posterior distribution for the GP parameters with\n    priors imposed to favor correlation lengths smaller than the expected cluster sizes.}\n    \\label{fig:abell_lnp}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have demonstrated a probabilistic model inference of lensing shear and convergence\nusing a Gaussian Process (GP) prior. Our method is an extension of previous Bayesian maximum entropy\nmass mapping methods that is applicable in both cosmic shear (i.e., `field') and cluster\nlensing regimes. We validated our algorithm using simulated Gaussian distributed\nshear maps with comparisons to the input `truth' and by reconstructing the mass distribution\nin a cluster field in the Deep Lens Survey where we compare with the results of a previously\npublished `aperture densometry' method.\n\nBecause our GP kernel for the lensing shear and convergence is derived from a consistent lens\npotential, we find excellent separation of E and B modes in the posterior lens field maps. \nRecently \\citet{2016arXiv161003345B} presented an algorithm to project Wiener Filter maps into \n`pure' E and B modes, removing ambiguity in the E\/B decomposition from survey masks. Because we compute \nWiener Filter solutions for the lensing convergence from galaxy ellipticities, the method of \n\\citet{2016arXiv161003345B} is a simple extension of the algorithm we present in this paper. \n\nWe have also described an algorithm for optimization of the GP kernel parameters given a\nmeasured galaxy ellipticity catalog or a cosmologically-informed prior based on simulation\nof lensing fields. An interesting future extension of this work could include both galaxy \npositions and ellipticities in the reconstruction of the gravitational potential. \n\nOur algorithm is computationally challenging in the solution of linear systems of dimension equal\nto the number of galaxies. Next generation surveys will have $10^{7}$--$10^{10}$ galaxies, requiring\nboth parallelization of the linear solver routines and optimizations such as sparse matrix\napproximations via tapering~\\citep{Kaufman2008} or Fast Fourier Transform (FFT)-based matrix\nmultiplications~\\citep[exploiting the special structure of the isotropic cosmological covariances][]{padmanabhan++2003}.\nAs a next step in the validation and scaling of\nour code we plan to apply the algorithms in this paper to the variable shear `branches' of the\n{\\sc GREAT3}\\xspace challenge\\footnote{\\url{http:\/\/great3challenge.info\/}}~\\citep{Mandelbaum++2013}, which\ncontain $2\\times 10^{5}$ galaxies per simulated field. Coincidentally, each of the five\nDLS four square degree fields contains a comparable number of $\\sim2\\times10^5$ lensing source\ngalaxies.\n\nFor the algorithm we validated in this paper we assumed an approximate data vector of galaxy\nellipticity measurements rather than the more informative imaging pixel data for each galaxy, as we\ndescribed previously in \\citet{mbi-theory}.\nIf we use galaxy samples rather than an ellipticity catalog then we have to go back to\n\\autoref{eq:marg_like} and replace the integral over the intrinsic ellipticities with a numerical\nimportance sampling formula from \\citet{mbi-theory}. In this case we do not need to assume\nconjugacy of the intrinisc ellipticity prior and the likelihood function, which is good because\nthe pixel-level likelihood will not be conjugate. The algorithms we presented in\n\\autoref{sub:cosmo_params} for cosmological parameter estimation are applicable, however, when using\neither an approximate ellipticity data vector or the interim sampling of galaxy image model\nparameters. \n\nWe can still derive a Wiener Filter from the product of the Gaussian intrinsic ellipticity\ndistribution (or DP base distribution) and the interim variable shear GP prior assuming a weak\nshear approximation. But then we need to evaluate \\autoref{eq:posterior_cov} and\n\\autoref{eq:posterior_mean} for every interim sample.\nThe parameters of $\\mathsf{S}+\\mathsf{N}$ include the GP parameters in $\\mathsf{S}$ and the $\\alpha$ intrinsic\nellipticity distribution parameter in $\\mathsf{N}$. If we update either parameter we need to perform a\nnew matrix factorization and solve operation at every step (but these operations can be done once\nfor all interim samples of galaxy image model parameters in a given step). Exploring optimized\nlinear algebra approaches and effective sampling strategies for this more complete framework will\nbe a focus of future work.\n\nWe have not included any redshift information about the source galaxies in our model. However, the \nuse of such information (via photometric redshifts) is a critical component of the weak lensing \nanalyses for cosmic shear surveys as well as cluster mass reconstructions with optimized \nsignal-to-noise ratios. A simple extension of our work to include redshifts could be to impose \nseparable GP priors on the lens fields for galaxy samples binned in redshift (or redshift estimator).\nThe physical correlations between the lens fields inferred in each source bin can be modeled in \nthe hierarchical inference stage when marginalizing lens fields to infer cosmological parameters\n\\citep[as outlined in][]{mbi-theory}. A more thorough approach would be to include probabilistic \nredshifts for each source galaxy and marginalize over each source redshift distribution as \npart of the lens field inference. \n\n\\section*{Acknowledgments}\nWe thank Chris Paciorek for discussion about the\nstatistical framework for performing shear inference and the use of Gaussian Processes.\nWe thank Alex Malz for critical reviews of draft versions of this work.\nThanks to M. James Jee for providing the Deep Lens Survey shear catalog.\nPart of this work performed under the auspices of the U.S. Department of Energy\nby Lawrence Livermore National Laboratory under Contract DE-AC52-07NA27344.\nFunding for this work was provided by LLNL Laboratory Directed Research and Development\ngrant 16-ERD-013.\nThis work was also supported by the Director, Office of Science, Office of Advanced\nScientific Computing Research, Applied Mathematics program of the U.S. Department of Energy under\nContract No.DE-AC02-05CH11231.\nThis research used resources of the National Energy Research Scientific\nComputing Center (NERSC), a DOE Office of Science User Facility supported by\nthe Office of Science of the U.S. Department of Energy under Contract No.\nDE-AC02-05CH11231.\nThis work uses a modified version of the public code \\texttt{George} available at\n\\url{https:\/\/github.com\/karenyyng\/george}, which was forked from\n\\url{https:\/\/github.com\/dfm\/george}. We also made use of the \\texttt{daft} package \n(developed by Dan Foreman-Mackey, David W. Hogg, and contributors, and available at \n\\url{http:\/\/daft-pgm.org\/}) for plotting probabilistic graphical models.\n\n\n\\bibliographystyle{apj3auth}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nThe probably presence of an unknown form of energy in the universe, called dark energy, and confirmed by a large number of observations, starting by the data of Supernovae IA in 1998 \\cite{riess}, have leaded to explore the possible theoretical origin for this fluid. Since it is the direct responsible of the present accelerating expansion of the universe, a negative pressure is required which leads to a negative equation of state parameter. The most popular candidate, the cosmological constant, which posses a constant equation of state (EoS), $p_{\\Lambda}=-\\rho_{\\Lambda}$, can explain quite well the cosmological evolution. However,  the open possibility that the EoS is not completely constant but evolutes dynamically (even crossing the phantom barrier more), and the quite large difference between the observed dark energy density and the vacuum energy density predicted by quantum field theories, have leaded to explore other possibilities, as the existence of scalar fields, vector fields or modifications of General Relativity (GR), among others (for a review on dark energy candidates, see \\cite{copeland}). \n\\par\nIn the context of modified gravities, a wide range of possibilities have been explored, being $f(R)$ gravity probably the most popular one due to its simplicity since it generalizes the Hilbert-Einstein action to a more complex function of the Ricci scalar (for a review on $f(R)$ gravity, see \\cite{reviews,Bamba:2012cp}). Nevertheless, other kind of theories have been suggested, where other curvature invariants are included as the Gauss-Bonnet gravity. In this paper, we study the so-called $f(T)$ gravity, which in analogy to f(R) gravity, consists in a generalization of the action of Teleparallel gravity, a theory that assumes Weitzenbock connection instead of the Levi-Civita connection, which yields to a null curvature but a non-vanishing torsion (for a review see \\cite{pereira1}). In this gravitational theory, the main field is represented by the so-called tetrads instead of the metric as in GR. This kind of theories has become very popular recently as can also explain the accelerated expansion of the universe with no need of dark energy, and even the inflationary epoch (see \\cite{ferraro1}-\\cite{ratbay}). Then, a wide number of aspects have been studied in the context of $f(T)$ gravity, as  its local Lorentz invariance \\cite{barrow}, static solutions \\cite{daouda1}, non-diagonal tetrads \\cite{daouda3}, or the presence of wormholes \\cite{boehmer}, as well as other aspects \\cite{x,cemsinan}. Also a large effort has been done to study cosmological solutions for this class of theories, as well as possible cosmological predictions (see Refs.~\\cite{houndjo}-\\cite{li}).\n\\par\nAt the present work, we are interested to study some particular cosmological solutions in $f(T)$ gravity, where the appropriate action is reconstructed for each case. Specifically, the Bianchi type-I, Kantowski-Sachs (KS) and Bianchi type-III models are considered, and particularly some important solutions, such as power law and de Sitter (dS) expansion, or more complex ones as exponential functions for the scale factor in each direction of the space. Since power law and dS solutions can provide a good description for some specific phases of the universe evolution, their reconstruction in $f(T)$ gravity becomes a crucial point in order to consider this class of theories as serious candidates for explaining the whole cosmological history. In addition, here we assume more general cosmological metrics than Friedmann-Lema\\^itre-Robertson-Walker (FLRW) metrics,  in particular anisotropic universes described by the Bianchi type-I, Kantowski-Sachs (KS) and Bianchi type-III metrics, in order to provide the most general description of the cosmological evolution in the context of $f(T)$ gravity.  Moreover, exponential solutions are also considered, this kind of expansions has become very popular recently as they may conduct the universe to a non-singular state, where some bounded systems may be broken. Such state suggested in Ref.~\\cite{LittleRip}, and called {\\it Little Rip},  has already been studied in $f(R)$ gravity (see Ref.~\\cite{Nojiri:2011kd}), as well as in $f(T)$ theories \\cite{bambamyr}. Even more,  the possible occurrence of a   {\\it Little Rip} has been also explored in the context of the so-called viable modified gravities (see Ref.~\\cite{SaezGomez:2012ek}). Note that anisotropic\ncosmological metrics have been already studied in the context of GR with the presence of isotropic and anisotropic fluids, as well as the stability of the solutions \\cite{barrow3,stability}.\n\\par\nFurthermore, the use of an auxiliary scalar field, in analogy to the equivalence of Brans-Dicke theories for $f(R)$ gravity (see for instance Ref.~\\cite{STFR}), is also implemented, from which may result a useful tool to reconstruct the appropriate action as well as for studying the properties of $f(T)$ gravity. \n\\par\nThe main motivations of assuming  the assumption of a model with anisotropic geometry are based on several physical aspects as: the famous problem of the CMB quadrupole can be solved by considering a universe with planar symmetry \\cite{campanelli1} where eccentricity in decoupling, generated by a uniform cosmic magnetic field whose current strength,  $ B(t_0) \\sim 10^{-9}$ Gauss, should be close to $e_{dec}\\sim 10^{-2}$; the Bianchi type models in Loop Quantum Cosmology \\cite{lqc}, $^4He$ abundance \\cite{campanelli2}, cosmic parallax \\cite{campanelli3,fontanini}, small anisotropic pressures \\cite{barrow2}, cosmological solutions of the low energy string effective action \\cite{massimo}, anisotropic inflationary universe \\cite{inflation} and some other \\cite{x-1}. In the $f(R)$ theory, we already have some good results \\cite{can,sharif2}, therefore, we propose to establish the equations here and get the first results in $f(T)$ gravity, for the Bianchi type-I, type-III and KS models.\n\\par\nThen, the paper is organized as follows: in section \\ref{sec2}, the basic concepts of $f(T)$ gravity are introduced. In section \\ref{BK} , the equations for general  Bianchi type-I, type-III and Kantowski-Sachs (KS) models are deduced in a particular coordinate system and diagonal tetrads. Section \\ref{sec2a} deals with the reconstruction of the $f(T)$ action for some relevant solutions, and where several techniques are considered, including a kind of scalar-tensor theory for torsion gravity. Finally, section \\ref{conclusions} is devoted to the conclusions and discussions on the results found in the paper.\n\\section{\\large Preliminary definitions and equations of motion}\\label{sec2}\n\n\nAs previously mentioned, the $f(T)$ theory of gravity is defined in the Weitzenbock's space time in which the line element is described by \n\\begin{equation}\\label{el}\ndS^{2}=g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}\\; ,\n\\end{equation} \nwhere $g_{\\mu\\nu}$ are the components of the metric which is symmetric and possesses $10$ degrees of freedom. One can describe the theory in the spacetime or in the tangent space, which allows us to rewrite the line element (\\ref{el}) as follows \n\\begin{eqnarray}\ndS^{2} &=&g_{\\mu\\nu}dx^{\\mu}dx^{\\nu}=\\eta_{ij}\\theta^{i}\\theta^{j}\\label{1}\\; ,\\\\\ndx^{\\mu}& =&e_{i}^{\\;\\;\\mu}\\theta^{i}\\; , \\; \\theta^{i}=e^{i}_{\\;\\;\\mu}dx^{\\mu}\\label{2}\\; ,\n\\end{eqnarray} \nwhere $\\eta_{ij}=diag[1,-1,-1,-1]$ and $e_{i}^{\\;\\;\\mu}e^{i}_{\\;\\;\\nu}=\\delta^{\\mu}_{\\nu}$ or  $e_{i}^{\\;\\;\\mu}e^{j}_{\\;\\;\\mu}=\\delta^{j}_{i}$. The square root of the metric determinant is given by  $\\sqrt{-g}=\\det{\\left[e^{i}_{\\;\\;\\mu}\\right]}=e$ and the matrix $e^{a}_{\\;\\;\\mu}$ are called tetrads and represent the dynamic fields of the theory.\n\\par\nBy using theses fields, one can define the Weitzenbock's connection as \n\\begin{eqnarray}\n\\Gamma^{\\alpha}_{\\mu\\nu}=e_{i}^{\\;\\;\\alpha}\\partial_{\\nu}e^{i}_{\\;\\;\\mu}=-e^{i}_{\\;\\;\\mu}\\partial_{\\nu}e_{i}^{\\;\\;\\alpha}\\label{co}\\; .\n\\end{eqnarray}\nThe main geometrical objects of the spacetime are constructed from this connection. The components of the tensor torsion are defined by the antisymmetric part of this connection\n\\begin{eqnarray}\nT^{\\alpha}_{\\;\\;\\mu\\nu}&=&\\Gamma^{\\alpha}_{\\nu\\mu}-\\Gamma^{\\alpha}_{\\mu\\nu}=e_{i}^{\\;\\;\\alpha}\\left(\\partial_{\\mu} e^{i}_{\\;\\;\\nu}-\\partial_{\\nu} e^{i}_{\\;\\;\\mu}\\right)\\label{tor}\\;.\n\\end{eqnarray}\nThe components of the contorsion are defined as \n\\begin{eqnarray}\nK^{\\mu\\nu}_{\\;\\;\\;\\;\\alpha}&=&-\\frac{1}{2}\\left(T^{\\mu\\nu}_{\\;\\;\\;\\;\\alpha}-T^{\\nu\\mu}_{\\;\\;\\;\\;\\alpha}-T_{\\alpha}^{\\;\\;\\mu\\nu}\\right)\\label{contor}\\; .\n\\end{eqnarray}\nIn order to make more clear the definition of the scalar equivalent to the curvature scalar of RG, we first define a new tensor $S_{\\alpha}^{\\;\\;\\mu\\nu}$, constructed from the components of the tensors torsion and contorsion as\n\\begin{eqnarray}\nS_{\\alpha}^{\\;\\;\\mu\\nu}&=&\\frac{1}{2}\\left( K_{\\;\\;\\;\\;\\alpha}^{\\mu\\nu}+\\delta^{\\mu}_{\\alpha}T^{\\beta\\nu}_{\\;\\;\\;\\;\\beta}-\\delta^{\\nu}_{\\alpha}T^{\\beta\\mu}_{\\;\\;\\;\\;\\beta}\\right)\\label{s}\\;.\n\\end{eqnarray}\nWe can now define the torsion scalar by the following contraction\n\\begin{eqnarray}\nT=T^{\\alpha}_{\\;\\;\\mu\\nu}S^{\\;\\;\\mu\\nu}_{\\alpha}\\label{te}\\; .\n\\end{eqnarray}\n\nThe action of the theory is defined by generalizing  the Teleparallel theory, as \n\\begin{eqnarray}\\label{action}\nS=\\int e\\left[f(T)+\\mathcal{L}_{Matter}\\right]d^4x\\;,\n\\end{eqnarray}\nwhere $f(T)$ is an algebraic function of the torsion scalar $T$. Making the functional variation of the action  (\\ref{action}) with respect to the tetrads, we get the following equations of motion \\cite{barrow,daouda1,daouda2}\n\\begin{eqnarray}\nS^{\\;\\;\\nu\\rho}_{\\mu}\\partial_{\\rho}Tf_{TT}+\\left[e^{-1}e^{i}_{\\mu}\\partial_{\\rho}\\left(ee^{\\;\\;\\alpha}_{i}S^{\\;\\;\\nu\\rho}_{\\alpha}\\right)+T^{\\alpha}_{\\;\\;\\lambda\\mu}S^{\\;\\;\\nu\\lambda}_{\\alpha}\\right]f_{T}+\\frac{1}{4}\\delta^{\\nu}_{\\mu}f=4\\pi\\mathcal{T}^{\\nu}_{\\mu}\\label{em}\\; ,\n\\end{eqnarray}\nwhere $\\mathcal{T}^{\\nu}_{\\mu}$ is the energy momentum tensor, $f_{T}=d f(T)\/d T$ and $f_{TT}=d^{2} f(T)\/dT^{2}$. By setting $f(T)=a_1T+a_0$,  the equations of motion (\\ref{em}) are the same as that of the Teleparallel theory with a cosmological constant, and this is dynamically equivalent to the GR.  These equations clearly depend on the choice made for the set of tetrads \\cite{cemsinan}. \n\\par\nThe contribution of the interaction with the matter fields is given by the energy momentum tensor which, is this case, is defined as \n\\begin{eqnarray}\n\\mathcal{T}^{\\,\\nu}_{\\mu}=  diag\\left(1,-\\omega_x,-\\omega_y,-\\omega_z\\right)\\rho                         \\label{tem}\\; ,\n\\end{eqnarray}\nwhere the $\\omega_i$ ($i=x,y,z$) are the parameters of equations of state related to the pressures $p_x$, $p_y$ and $p_z$.\n\n\\section{Field equations for Bianchi type-I, type-III and Kantowski-Sachs models} \\label{BK}\n\n\nLet us first establish the equations of motion of a set of diagonal tetrads using the Cartesian coordinate metric, for  describing models of Bianchi type-I, type-III and Kantowski-Sachs (KS). We propose to start with the Bianchi type-III case, from which Bianchi type-I and KS can be recovered. For the Bianchi type-III case, the metric reads \n\\begin{equation}\ndS^2=dt^2-A^2(t)dx^2-e^{-2\\alpha x}B^2(t)dy^2-C^2(t)dz^2\\,\\,\\,,\\label{metrictype3}\n\\end{equation}\nwhere $\\alpha$ is a constant parameter. Note that the Bianchi type-I is recovered by setting $\\alpha=0$ from the Bianchi type-III, while KS is recovered when one takes $\\alpha=0$ and $B(t)=C(t)$. Let us choose the following set of diagonal tetrads related to the metric (\\ref{metrictype3})\n\\begin{eqnarray}\n\\left[e^{a}_{\\;\\;\\mu}\\right]=diag\\left[1,A,e^{-\\alpha x}B,C\\right]\\;. \\label{matrixtype3}\n\\end{eqnarray}\nThe determinant of the matrix (\\ref{matrixtype3}) is $e=e^{-\\alpha x}ABC$. The components of the tensor torsion (\\ref{tor}) for the tetrads (\\ref{matrixtype3}) are given by\n\\begin{eqnarray}\nT^{1}_{\\;\\;01}=\\frac{\\dot{A}}{A}\\,,\\,T^{2}_{\\;\\;02}=\\frac{\\dot{B}}{B}\\,,\\, T^{2}_{\\;\\;21}=\\alpha\\,,\\,T^{3}_{\\;\\;03}=\\frac{\\dot{C}}{C}\\;,\\label{torsiontype3}\n\\end{eqnarray}\nand the components of the corresponding tensor contorsion are \n\\begin{eqnarray}\nK^{01}_{\\;\\;\\;\\;1}=\\frac{\\dot{A}}{A}\\,,\\,K^{02}_{\\;\\;\\;\\;2}=\\frac{\\dot{B}}{B}\\,,\\,K^{12}_{\\;\\;\\;\\;2}=\\frac{\\alpha}{A^2}\\,,\\,K^{03}_{\\;\\;\\;\\;3}=\\frac{\\dot{C}}{C}\\;.\\label{contorsiontype3}\n\\end{eqnarray} \nThe components of the tensor $S_{\\alpha}^{\\;\\;\\mu\\nu}$, in (\\ref{s}),  are given by\n\\begin{eqnarray}\nS_{0}^{\\;\\;01}=S_{3}^{\\;\\;31}=\\frac{\\alpha}{2A^2}\\,,\\,S_{1}^{\\;\\;10}=\\frac{1}{2}\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\,,\\,S_{2}^{\\;\\;20}=\\frac{1}{2}\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\,,\\,S_{3}^{\\;\\;30}=\\frac{1}{2}\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\;.\\label{tensortype3}\n\\end{eqnarray}\nBy using the components (\\ref{torsiontype3}) and (\\ref{tensortype3}),  the torsion scalar (\\ref{te}) is given by\n\\begin{eqnarray}\nT=-2\\left(\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right)\\; \\label{torsionScalar1}.\n\\end{eqnarray}\nThe equations of motion are given by \n\\begin{eqnarray}\n16\\pi \\rho &=& f+4f_T\\Big[\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}-\\frac{\\alpha^2}{2A^2}\\Big]\\,\\,,\\label{densitytype3}\\\\\n-16\\pi p_x &=& f+2f_T\\left[\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+2\\frac{\\dot{B}\\dot{C}}{BC}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{radialpressuretype3}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+2\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure1type3}\\\\\n-16\\pi p_z=f&+&2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+2\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}-\\frac{\\alpha^2}{A^2}\\right]\\nonumber\\\\&+&2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure2type3}\\\\\n\\frac{\\alpha}{2A^2}\\left[\\left(\\frac{\\dot{A}}{A}-\\frac{\\dot{B}}{B}\\right)f_T-\\dot{T}f_{TT}\\right]&=&0 \\,\\,\\,, \\label{constraint1} \\\\\n\\alpha\\left(\\frac{\\dot{A}}{A}-\\frac{\\dot{B}}{B}\\right)f_T&=&0\\;.\\label{constraint2}\n\\end{eqnarray}\nIn the particular case where $f(T)=T-2\\Lambda$, the equations (\\ref{densitytype3})-(\\ref{constraint2}) are identical to that of the GR \\cite{akarsu}. The equation of constraint (\\ref{constraint2}) appears in both the GR as in $f(R)$ gravity \\cite{sharif2}. But here we have a second equation of constraint (\\ref{constraint1}), which appears as a generalization of the previous one, because here we have a contribution of a term of second derivative of the function $f(T)$ with respect to $T$.\n\\par\n\nBy setting $\\alpha=0$, the Bianchi type-I case is recovered and the equations of motions read\n\\begin{eqnarray}\n16\\pi \\rho &=& f+4f_T\\Big[\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\Big]\\,\\,\\,,\\label{densitytype1}\\\\\n-16\\pi p_x &=& f+2f_T\\left[\\frac{\\ddot{B}}{B}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+2\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{B}}{B}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{radialpressuretype1}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{C}}{C}+\\frac{\\dot{A}\\dot{B}}{AB}+2\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{C}}{C}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressure1type1}\\\\\n-16\\pi p_z&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+2\\frac{\\dot{A}\\dot{B}}{AB}+\\frac{\\dot{A}\\dot{C}}{AC}+\\frac{\\dot{B}\\dot{C}}{BC}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;\\label{tangentialpressure2type1}\\;.\n\\end{eqnarray}\nThe equations of motion corresponding to KS model are obtained by setting $\\alpha=0$ and $B=C$, yielding\n\\begin{eqnarray}\n16\\pi\\rho&=&f+4f_T\\left[\\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\right]\\;,\\label{densityks}\\\\\n-16\\pi p_x&=&f+4f_T\\left[\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+\\frac{\\dot{A}\\dot{B}}{AB}\\right]+4\\frac{\\dot{B}}{B}\\dot{T}f_{TT}\\;,\\label{radialpressureks}\n\\end{eqnarray}\n\\begin{eqnarray}\n-16\\pi p_y&=&f+2f_T\\left[\\frac{\\ddot{A}}{A}+\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+3\\frac{\\dot{A}\\dot{B}}{AB}\\right]+2\\left(\\frac{\\dot{A}}{A}+\\frac{\\dot{B}}{B}\\right)\\dot{T}f_{TT}\\;,\\label{tangentialpressureks}\\\\\np_y&=&p_z\\nonumber\\,\\,\\,.\n\\end{eqnarray}\nIn the next section we will perform the reconstruction scheme of the action of the system for some particular cases.\n\n\\section{ Reconstructing $f(T)$ gravity in inhomogeneous universes}\\label{sec2a}\n\n\nLet us now consider the reconstruction of the $f(T)$ action for some particular solutions of the class of metrics explored in the previous section. Specifically, we consider solutions of the type of de Sitter, power law evolutions and exponential solutions. Note that de Sitter and power law solutions have been widely explored in other contexts of modified gravity, as $f(R)$ and Gauss-Bonnet gravities (see Ref.~\\cite{Nojiri:2009kx}), since they can provide a well description of  the cosmological evolution along its particular phases.\\\\\nLet's start by considering for simplicity Bianchi type-I and Kantowski-Sachs $(\\alpha=0)$ metrics. Then, the conservation equation for the energy momentum tensor (\\ref{tem}) can be easily obtained,\n\\begin{equation}\n\\dot{\\rho}+\\left(H_x+H_y+H_z\\right)\\rho+H_x p_x+H_y p_y +H_z p_z=0\\ ,\n\\label{D1}\n\\end{equation}\nwhere we have defined $H_x=\\frac{\\dot{A}}{A}\\ \\ H_y=\\frac{\\dot{B}}{B}\\ \\ H_z=\\frac{\\dot{C}}{C}$. We can now analyse de Sitter, power law solutions and exponential expansion in  Bianchi type-I metric by one side, and Kantowski-Sachs metric by the other, where $B=C$ that implies $p_y=p_z$.\n\n\\subsection{De Sitter solutions}\n\nDe Sitter solutions are well known in the context of cosmology since the current epoch, where the universe expansion is being accelerated, can be described approximately with a de Sitter solution. This kind of solutions consists on an exponential expansion of the scale factor, which yields a constant Hubble parameter. In the case of Bianchi type-I and Kantowski-Sachs metrics $(\\alpha=0)$ in (\\ref{metrictype3}), we may assume an exponential expansion for each spatial direction,\n\\begin{equation}\nA=A_0 {\\rm e}^{a t}\\ \\ B=B_0 {\\rm e}^{b t}\\ \\ C=C_0 {\\rm e}^{c t}\\ ,\n\\label{D2}  \n\\end{equation}\nand the rates of the expansion for each direction can be defined as,\n\\begin{equation}\nH_x=\\frac{\\dot{A}}{A}=H_{x0}\\ \\ H_{y}=\\frac{\\dot{B}}{B}=H_{y0}\\ \\ H_c=\\frac{\\dot{C}}{C}=H_{z0}\\ ,\n\\label{D3}\n\\end{equation}\nwhere $\\{H_{x0}=a,H_{y0}=b,H_{z0}=c\\}$ are constants. The torsion scalar defined in (\\ref{torsionScalar1}) is given by,\n\\begin{equation}\nT_0=-2\\left(H_{x0}H_{y0}+H_{x0}H_{z0}+H_{y0}H_{z0}\\right)\\ .\n\\label{D4}\n\\end{equation}\nThen, by assuming $p_x=p_y=p_z=p$ and an equation of state $p=w\\rho$, the conservation equation (\\ref{D1}) can be easily solved for the ansatz  (\\ref{D2}),\n\\begin{equation}\n\\rho=\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}\\ .\n\\label{D5}\n\\end{equation}\nHence, the field equations (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}) become,\n\\begin{eqnarray}\n16\\pi \\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+4\\left[H_{x0}H_{y0}+H_{z0}(H_{x0}+H_{y0})\\right]f_T(T_0)\\ , \\label{D6} \\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{y0}+H_{z0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ , \\label{D7}\\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{x0}+H_{z0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ , \\label{D8}\\\\\n-16\\pi w\\rho_0{\\rm e}^{-(H_{x0}+H_{y0}+H_{z0})(1+w)t}=f(T_0)+2(H_{x0}+H_{y0})(H_{x0}+H_{y0}+H_{z0})f_T(T_0)\\ . \\label{D9}\n\\end{eqnarray}\nNote that the only possible solution in the presence of a perfect fluid is one with $w=-1$ as the r.h.s. of equations (\\ref{D6})-(\\ref{D9}) is independent of time, according to the expression of the scalar torsion for a pure de Sitter solution (\\ref{D4}), unless $H_{x0}+H_{y0}+H_{z0}=0$, which would imply a decelerating expansion in a particular direction, being $H_{i0}<0$. Moreover, for a particular $f(T)$ action, the system of equations (\\ref{D4})-(\\ref{D9}) reduces to an algebraic system of equations for the variables $\\{H_{x0},H_{y0},H_{z0}\\}$. Since  the system of equations (\\ref{D4})-(\\ref{D9}) are composed by four equations, while there are only three variables,\nthe above 4-equations system has to be reduced. However, even in the case of  Kantowski-Sachs metric, where $B(t)=C(t)\\rightarrow H_{y0}=H_{z0}$, the system (\\ref{D4})-(\\ref{D9}) still posses three independent equations with two variables. Hence, the only possible solution imposes,\n\\begin{equation}\n A(t)=B(t)=C(t)\\rightarrow H_{x0}=H_{y0}=H_{z0}=H_0\\ ,\n \\label{D10}\n\\end{equation}\nAnd the metric (\\ref{metrictype3}) reduces to the well known Friedmann-Lema\\^itre-Robertson-Walker metric with an exponential expansion, $A(t)=A_0{\\rm e}^{H_0\\ t}$. Hence, the only solution for a pure de Sitter expansion in  Bianchi type-I and Kantowski-Sachs metrics gives a FLRW universe\\footnote{Recall that we have assumed here that the pressures are equal, $p_x=p_y=p_z$.}, and the system of equations (\\ref{D4})-(\\ref{D9}) reduces now to a unique independent equation,\n \\begin{equation}\n 16 \\pi \\rho_0=f(T_0)+12H_{0}^2f_T(T_0)\\ .\n \\label{D11}\n \\end{equation}\n Then, the roots of the algebraic equation (\\ref{D11}) give the de Sitter points of a particular $f(T)$ action. In order to illustrate such possibility, let us consider the action,\n \\begin{equation}\n f(T)=\\left(-T\\right)^n\\ ,\n \\label{D12}\n \\end{equation}\n where $n$ is a real constant. Then, the equation (\\ref{D11}) is rewritten as,\n \\begin{equation}\n 16\\pi \\rho_0=(1-2n)(6H_{0}^2)^n\\ ,\n \\label{D13}\n \\end{equation}\n whose solution is given by,\n \\begin{equation}\n H_{0}^2=\\frac{1}{6}\\left(\\frac{16\\pi\\rho_0}{1-2n}\\right)^{1\/n}\\ .\n \\label{D14}\n\\end{equation}\n Hence, the only physical solution ($\\rho_0,H_0^2\\geq0$) imposes $n\\leq1\/2$. Then, the de Sitter solution is a direct consequence of the energy density $\\rho_0$, which can be interpreted as a cosmological constant according to the condition imposed above for its equation of state, $w=-1$. Nevertheless, in vacuum the equation (\\ref{D13}) reduces to $0=(1-2n)(6H_{x0}^2)^n$, whose only solution is given by $n=1\/2$, rising to $f(T)=\\sqrt{-T}$ that posses an infinite number of de Sitter points. Moreover, we may consider in vacuum the action,\n \\begin{equation}\n f(T)=C_1 T+C_2 \\left(-T\\right)^n\\ ,\n \\label{D15}\n \\end{equation}\nwhere $\\{C_1,C_2\\}$ are the coupling constants and $n$ is a real constant. The field equation (\\ref{D11}) in vacuum yields,\n \\begin{equation}\n0= C_1 6H_{0}^2+C_2(1-2n)(6H_{0}^2)^n\\ .\n\\label{D16}\n\\end{equation}\nSo the roots of this equation give the dS points allowed by the class of theories expressed in (\\ref{D15}). Note that now, the exponential expansion is a direct consequence of the action instead of the contribution of a kind of cosmological constant as in the case shown above. For instance, $n=2$, it yields the solution,\n\\begin{equation}\nH_{0}=\\sqrt{\\frac{C_1}{18C_2}}\\ . \n\\label{D17}\n\\end{equation}\nWhile for higher powers of $n$, more de Sitter points can be obtained for the action (\\ref{D15}). Note that in $f(R)$ theories, dS points constitutes the critical points of the dynamical system, which may be (un)stable, and could explain both the inflationary and dark energy epochs (see \\cite{Cognola:2008zp}), which may be the case also in $f(T)$ gravity.\n\n\\subsection{Power law solutions}\n\nLet us now explore a cosmological evolution described by a power law in each direction of the space expansion. In such case, the scale parameters for the Bianchi type-I and Kantowski-Sachs  metric (\\ref{metrictype3}), where we set $(\\alpha=0)$, can be expressed as,\n\\begin{equation}\nA(t)=A_0t^a\\ , \\ B(t)=B_0t^b\\ , \\ C(t)=C_0t^c\\ ,\n\\label{D18}\n\\end{equation}\nwhere $\\{a,b,c\\}$ and $\\{A_0,B_0,C_0\\}$ are constants to be determined by the field equations, and initial conditions respectively. The expansion rates are given by,\n\\begin{equation}\nH_x=\\frac{a}{t}\\ , \\ H_y=\\frac{b}{t}\\ , \\ H_z=\\frac{c}{t}\\ .\n\\label{D19}\n\\end{equation}\nWhile the expression for the torsion scalar (\\ref{torsionScalar1}) yields,\n\\begin{equation}\nT=-2\\left(\\frac{ab}{t^2}+\\frac{ac}{t^2}+\\frac{bc}{t^2}\\right)\\ .\n\\label{D20}\n\\end{equation}\nThen, introducing the above quantities in the field equations (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}), we get the following system of differential equations in $f(T)$,\n\\begin{eqnarray}\n16\\pi \\rho(T)=f(T)-2Tf_T(T)\\ , \\label{D21} \\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(b+c)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(b+c)}{bc+a(b+c)}T^2f_{TT}(T)\\ , \\label{D22}\\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(a+c)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(a+c)}{bc+a(b+c)}T^2f_{TT}(T)\\ ,\\label{D23} \\\\\n-16\\pi w\\rho(T)=f(T)+\\frac{(a+b)(1-a-b-c)}{bc+a(b+c)}Tf_T(T)+2\\frac{(a+b)}{bc+a(b+c)}T^2f_{TT}(T)\\ , \\label{D24}\n\\end{eqnarray}\nwhere we have assumed for simplicity that $p_x=p_y=p_z=p$ and an EoS $p=w\\rho$. Hence, the system (\\ref{D21})-(\\ref{D24}) is a set of differential equations in f(T) with the torsion scalar $T$ as the independent variable. \\\\ \n\nFirstly let us consider the vacuum case, or in other words, the homogeneous part of the first equation (\\ref{D21}), which becomes $f(T)-2Tf_T(T)=0$ and whose solution is given by,\n\\begin{equation}\nf(T)=C_1 \\sqrt{-T}\\ ,\n\\label{D26}\n\\end{equation}\nwhere $C_1$ is an integration constant. In order to satisfy the rest of the equations (\\ref{D22})-(\\ref{D24}), the condition $a=b=c$ must be imposed, so that the Hubble parameters (\\ref{D19}) reduce to the usual FLRW cosmology reproducing  power law solution. \\\\\n\nIn the presence of a kind of isotropic perfect fluid $p=w\\rho$, we can first solve the continuity equation (\\ref{D1}) in order to obtain $\\rho=\\rho(T)$, which yields,\n\\begin{equation}\n\\rho=\\rho_0 t^{-(a+b+c)(1+w)}=\\rho_0 \\left(-\\frac{T}{2(ab+ac+bc)}\\right)^{\\frac{(a+b+c)(1+w)}{2}}\\ .\n\\label{D25}\n\\end{equation}\nHence, the solution for the set of equations (\\ref{D21})-(\\ref{D24}) is given by $f(T)=f_h(T)+f_p(T)$, where $f_h(T)$ is the solution of the homogeneous equation, which coincides with the vacuum solution (\\ref{D26}), and $f_p(T)$ is the particular solution. Then, by using (\\ref{D25}) in the equation (\\ref{D21}), the particular solution can be easily found,\n\\begin{equation}\nf_{p}(T)=\\chi\\ T^{\\frac{(1+w)(a+b+c)}{2}}\\ , \n\\label{D27} \n\\end{equation}\n where $\\chi$ is a constant given by\n\\begin{equation}\n \\chi=\\frac{2^{4-(1+w)(a+b+c)\/2}\\pi \\rho_0}{\\left[-1+(1+w)(a+b+c)\\right] \\left[-bc-a(b+c)\\right]^{\\frac{(1+w)(a+b+c)}{2}}}\\ .\n \\label{D28}\n\\end{equation}\nNote that the condition $(1+w)(a+b+c)=2\\ n$ with $n$ being any natural number,  has to be imposed in order to avoid a complex gravitational action that would lacks any physical sense, recall  that $T<0$ for an expanding universe according to (\\ref{D20}).   In order to satisfy the complete set of equations (\\ref{D21})-(\\ref{D24}), we introduce the solution (\\ref{D27}) into the field equations (\\ref{D22})-(\\ref{D24}), and the following solutions for the parameters  $\\{a,b,c\\}$ are found,\n\\begin{enumerate}[ i.]\n\\item $\\ c=\\frac{1-w(a+b)}{w}\\ $, where $w\\neq 0$. This provides an anisotropic solution in $f(T)$ gravity with $A(t)$, $B(t)$ and $C(t)$ being different functions in (\\ref{D18}), and recalling that the perfect fluid assumed is an isotropic fluid. This does not hold in GR or Teleparallel Theory (TT) but is possible here due to the presence of second derivatives of the function $f(T)$ with respect to the torsion scalar $T$ in (\\ref{D21})-(\\ref{D24}). Note that field equations may be rewritten as the usual equations in TT, \n\\[\nH_xH_y+H_xH_z+H_yH_z=16\\pi(\\rho +\\rho_{f(T)})\\ ,\\; \\;\n-\\dot{H}_y-H_{y}^2-\\dot{H}_z-H_{z}^2-H_y\\ H_z=8\\pi (w\\rho +p_{f(T)}^x)\\ ,\n\\]\n\\begin{equation}\n-\\dot{H}_x-H_{x}^2-\\dot{H}_z-H_{z}^2-H_x\\ H_z=8\\pi (w\\rho +p_{f(T)}^y)\\ ,\\; \\;\n-\\dot{H}_x-H_{x}^2-\\dot{H}_y-H_{y}^2-H_x\\ H_y=8\\pi (w\\rho +p_{f(T)}^z)\\ .\n\\end{equation}\nHere, the extra terms coming from $f(T)$ are defined as an energy density $\\rho_{f(T)}$ and pressures $\\{p_{f(T)}^x, \\ p_{f(T)}^y, \\ p_{f(T)}^z\\}$, which are the origin of the anisotropic evolution. In this case, we have to fix $C_1=0$  in (\\ref{D26}) in order to satisfy the whole system.\n\\item $a=b=c$. The cosmological evolution expressed by \\ref{D18} reduces to a FLRW metric as in the homogeneous part of the equations, so that $C_1\\neq0$. \n\\item $c=-ab\/(a+b)$. In spite of this satisfies equations (\\ref{D21})-(\\ref{D24}) once (\\ref{D27}) is substituted in the equations, it gives $T=0$, and the r.h.s of equations (\\ref{D21})-(\\ref{D24}) become null while the l.h.s. are not, since $\\rho=\\rho(t)$ as given in (\\ref{D25}), so this is not a real solution.\n\n\\end{enumerate}\ufffd\n\\par\nTherefore,  we have obtained a complete set of power law solutions for Bianchi type-I  universe and Kantowski-Sachs metrics in the context of $f(T)$ gravity. Nevertheless, the action is clearly dependent on the EoS parameter $w$. Note also that in vacuum, the only possible solution reduces to a FLRW metric.\n\n\\subsection{General exponential solutions}\n\nIn this subsection we consider a more general exponential expansion for each spatial direction by \n\\begin{eqnarray}\nA=A_0e^{g_x(t)}\\,\\,\\,, \\quad B=B_0e^{g_y(t)}\\,\\,\\,,\\quad C=C_0e^{g_z(t)}\\,\\,\\,,\n\\end{eqnarray}\nwhere the function $g_i(t)$ is assumed as \n\\begin{eqnarray}\ng_i(t)=h_i(t)\\ln{\\left(t\\right)}\\,\\,,\\quad i=x,y,z\\;\\;,\n\\end{eqnarray}\n and $A_0$, $B_0$ and $C_0$ are positive constants. Note that the previous cases, the de Sitter solutions and power law solutions can be recovered from this one by setting $h_i(t)=a_it\/(\\ln{(t)})$ and $h_i(t)=a_i$, respectively, where $\\{a_i\\}=\\{a,b,c\\}$. In what follows, we will use an adiabatic approximation for the expansion in each spatial direction and neglect the derivatives of $h_i(t)$, i.e., setting $(\\dot{h}_i\\sim \\ddot{h}_i\\sim 0)$. The expansion rates in this case are given by\n\\begin{eqnarray}\nH_x=\\frac{h_x(t)}{t}\\,\\,,\\quad H_y=\\frac{h_y(t)}{t}\\,\\,,\\quad H_z=\\frac{h_z(t)}{t}\\,\\,.\n\\label{steph62}\n\\end{eqnarray}\nThus, the torsion scalar (\\ref{torsionScalar1}) becomes\n\\begin{eqnarray}\nT=-2\\left[\\frac{h_x(t)h_y(t)}{t^2}+\\frac{h_x(t)h_z(t)}{t^2}+\\frac{h_y(t)h_z(t)}{t^2}\\right]\\,\\,.\\label{steph63}\n\\end{eqnarray} \nThe acceleration in each direction is given by\n\\begin{eqnarray}\n\\ddot{A}=\\frac{h_x(h_x-1)}{t^2}A\\,\\,,\\quad \\ddot{B}=\\frac{h_y(h_y-1)}{t^2}B\\,\\,,\\quad \\ddot{C}=\\frac{h_z(h_z-1)}{t^2}C\\,\\,.\\label{steph64}\n\\end{eqnarray}\nSince $A$, $B$ and $C$ are positives, the acceleration is guaranteed in each direction when $h_i>1$, while for $0<h_i<1$, the universe  is in deceleration.\\par\nThe simplest example of $h_i(t)$ is\n\\begin{eqnarray}\nh_i(t)=\\frac{h_{i\\,in}+h_{i\\,out}\\,qt^2}{1+qt^2}\\,\\,\\,,\\label{steph65}\n\\end{eqnarray}\nwhere $h_{i\\,in}$, $h_{i,out}$ and $q$ are positive constants, and the $q$ is assumed to be enough small in order to make $h_i(t)$ varying slowly. Thus, the torsion scalar is always negative. From (\\ref{steph65}), we see that at early time $t=0$, $h_i\\rightarrow h_{i\\,in}$ and for late universe, $h_i\\rightarrow h_{i\\,out}$. By using \n(\\ref{steph65}) and (\\ref{steph63}), one gets the following equation\n\\begin{eqnarray}\nq^2Tt^6+\\left(2qT+2q^2X\\right)t^4+\\left(T+2qY\\right)t^2+2Z=0\\,\\,\\,,\\\\\nX=h_{x\\,out}h_{y\\,out}+h_{x\\,out}h_{z\\,out}+h_{y\\,out}h_{z\\,out}\\,\\,,\\quad\nY=h_{x\\,in}h_{y\\,out}+h_{x\\,out}h_{y\\,in}+h_{x\\,in}h_{z\\,out}+\\nonumber\\\\\nh_{x\\,out}h_{z\\,in}+h_{y\\,in}h_{z\\,out}+h_{y\\,out}h_{z\\,in}\\,\\,,\\quad \nZ=h_{x\\,in}h_{y\\,in}+h_{x\\,in}h_{z\\,in}+h_{y\\,in}h_{z\\,in}\\,\\,,\\nonumber\n\\end{eqnarray}\nwhose solutions read\n\\begin{eqnarray}\nt^2&=&\\left\\{\\Psi_0(T),\\;\\;\\Psi_{\\pm}(T)\\right\\}\\,,\\label{steph67}\\\\\n\\Psi_0(T)&\\equiv& \\alpha^{1\/3}+\\beta_0\\alpha^{-1\/3}+\\beta_1\\,\\;\\;\\Psi_{\\pm}(T)\\equiv e^{\\mp 2i\\pi\/3}\\alpha^{1\/3}+e^{\\pm 2i\\pi\/3}\\alpha^{-1\/3}+\\beta_1\\,\\,,\\;\\;\\,\\nonumber\\\\ \\alpha&=&\\frac{\\sqrt{\\alpha_1}}{q^2T^2}-\\alpha_2\\,\\,,\\;\n\\beta_0=\\frac{qT(8X-6Y)+4q^2X^2+T^2}{9q^2T^2}\\,\\,,\\quad \\beta_1=-\\frac{2qX+2T}{3qT}\\,\\,,\\nonumber\\\\\n\\alpha_1&=&\\frac{1}{27qT}\\Big[27qT^2Z^2+[(-36q^2TX-36qT^2)Y+16q^3X^3+48q^2TX^2+30qT^2X-2T^3]Z\\nonumber\n\\\\&+&8q^2TY^3+(-4q^3X^2-8q^2TX+8qT^2)Y^2+(-4q^2TX^2-8qT^2X+2T^3)Y-qT^2X^2-2T^3X\\Big]\\,\\,,\\nonumber\\\\\n\\alpha_2&=&\\frac{qT^2(27Z-18Y+15X)+q^2T(24X^2-18XY)+8q^3X^3-T^3}{27q^3T^3}\\,\\,.\\nonumber\n\\end{eqnarray}\nWe see from (\\ref{steph67}) that there are one real positive solution, $\\Psi_0(T)$, and two complex solutions $\\Psi_{\\pm}(T)$. By using (\\ref{steph62})-(\\ref{steph64}) the system of equations of motion (\\ref{densitytype1})-(\\ref{tangentialpressure2type1}) becomes\n\\begin{eqnarray}\n16\\pi\\rho&=&f-2Tf_T\\,\\,\\,,\\label{steph68}\\\\\n-16\\pi p_x&=&f+2f_T\\left[\\frac{(h_y+h_z)^2-(h_y+h_z)-h_yh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_y+h_z}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph69}\\\\\n-16\\pi p_y&=&f+2f_T\\left[\\frac{(h_x+h_z)^2-(h_x+h_z)-h_xh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_x+h_z}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph70}\\\\\n-16\\pi p_z&=&f+2f_T\\left[\\frac{(h_x+h_y)^2-(h_x+h_y)-h_xh_y}{\\Psi_0(T)}-\\frac{T}{2}\\right]+4\\left(\\frac{h_x+h_y}{\\Psi_0(T)}\\right){T}f_{TT}\\,\\,,\\label{steph71}\\\\\nh_i&=&\\frac{h_{i in}+h_{i out}q\\Psi_0(T)}{1+q\\Psi_0(T)}\\,\\,\\,.\\label{steph72}\n\\end{eqnarray} \n\\subsubsection{A special case}\\label{numero1}\n\nHere, we assume that the expansion rates are equal in the three spatial directions ($h_x=h_y=h_z$), and the system (\\ref{steph68})-(\\ref{steph71}) reduces to\n\\begin{eqnarray}\n16\\pi\\rho&=&f-2Tf_{T}\\,\\,,\\label{steph73}\\\\\n-16\\pi p_x&=&f-2f_T\\left(T+\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}\\right)+4T\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}f_{TT}\\,\\,,\\label{steph74}\\\\\np_x&=&p_y=p_z\\,\\,\\,,\\nonumber\n\\end{eqnarray}\nwhich means that the assumption of having the same rate in the three spatial direction leads to an isotropic matter content. By combining (\\ref{steph73}) and (\\ref{steph74}), one gets \n\\begin{eqnarray}\n4T\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}f_{TT}-2f_T\\left[(1+\\omega)T+\\sqrt{\\frac{-2T}{3\\Psi_0(T)}}\\,\\right]+(1+\\omega)f=0\\,\\,\\,,\\label{steph75}\n\\end{eqnarray}\nwhere we used the barotropic equation $p_x=\\omega\\rho$. Let us consider an asymptotic analysis, looking for the early universe (small time) and the late time one (large time), for which the function $h_i(t)$ yields $h_{i\\,in}$ and $h_{i\\,out}$, respectively. Thus, for the early universe Eq. (\\ref{steph75}) reduces to \n\\begin{eqnarray}\n4T^2f_{TT}+6Tf_{T}\\left[3h_{i\\,in}(\\omega+1)-1\\right]-3h_{i\\,in}(\\omega+1)f=0\\,\\,\\,,\\\\\nh_{i\\,in}=h_{x\\,in}=h_{y\\,in}=h_{z\\,in}\\,\\,\\,,\\nonumber\n\\label{steph76}\n\\end{eqnarray}\nwhose general solution reads \n\\begin{eqnarray}\nf(T)=C_3T^{\\gamma_+}+C_4T^{\\gamma_-}\\,\\,,\\quad \\gamma_{\\pm}=\\frac{5-9h_{x\\,in}(1+\\omega)\\pm\\sqrt{25-78h_{x\\,in}(1+\\omega)+81h^2_{x\\,in}(1+\\omega)^2}}{4}\\,\\,,\\label{steph77}\n\\end{eqnarray}\nwhere $C_3$ and $C_4$ are integration constants. From (\\ref{steph77}), by writing the radicand as $\\left[5-9h_{x\\,in}(1+\\omega)\\right]^2+12h_{x\\,in}(1+\\omega)$, one observes that, for any $\\omega>-1$, $\\gamma_{+}>0$ and $\\gamma_{-}<0$. Moreover, in this context of asymptotic analysis, we observe from (\\ref{steph63}) that for small $t$, the torsion scalar $T$ is large, while for large $t$, the torsion is small. Thus, for small $t$ with $h_{x\\,in}>1$, corresponding to the inflation, the algebraic expression of $f(T)$ is given by\n\\begin{eqnarray}\nf(T)=C_3T^{\\gamma_{+}}\\,\\,\\,.\n\\end{eqnarray}\nSince $h_x(t)$ reduces to $h_{x\\,out}$ in the late universe, the model corresponding to the late accelerated universe can be obtained by replacing $h_{x\\,in}$ by $h_{x\\,out}$. Precisely, for large $t$, the torsion scalar is small, and for $h_{x\\,out}>1$, the dominate term in (\\ref{steph77}), corresponding to the model of late time universe, is\n\\begin{eqnarray}\nf(T)=C_4T^{\\gamma_{-}'}\\,\\,\\,,\\;\\;\\gamma_{\\pm}'=\\frac{5-9h_{x\\,out}(1+\\omega)\\pm\\sqrt{25-78h_{x\\,out}(1+\\omega)+81h^2_{x\\,out}(1+\\omega)^2}}{4}\\,\\,.\n\\end{eqnarray}\nThis model is equivalent to the teleparallel gravity for $C_4=1$ and $h_{x\\,out}=2\/(5+5\\omega)$. It is easy to see from this that, for any ordinary matter, i.e., $\\omega>0$, one gets $h_{x\\,out}<1$, meaning that the teleparallel gravity without cosmological constant cannot provide the late acceleration of the universe (remembering that the acceleration is guaranteed for $h_{x\\,out}>1$, and $0<h_{x\\,out}<1$ characterising a decelerated expanding universe). Thus, the contribution of the $f(T)$ terms plays the role of the dark energy.\\par\nLooking for the expression of $f(T)$ for large cosmic time $t$, i.e., the expression (\\ref{steph77}) (replacing $h_{x\\,in}$ by $h_{x\\,out}$), a similarity can be observed with a result of Bamba et al in \\cite{bambamyr}. In this work, they undertook $f(T)$ theory in the FLRW metric, first assuming a power-law expression for $f(T)$ in the Eq.~(4.6) for investigating what type of finite future time singularities may appear. Later, they introduced a correction term, still in the form of power-law, their Eq.~(4.22), in order to analyse the possible avoidance of the singularities. Then, they obtained the global expression (4.23) of \\cite{bambamyr}. Note that this expression is equivalent to our Eq.~(\\ref{steph77}). Moreover, they shown in their ``TABLE II\" that the Big Rip and the Big Freeze can be removed if the product of the exponents is negative. This is exactly our result, since $\\gamma_{+}'$ and $\\gamma_{-}'$ are always positive and negative, respectively ($\\gamma_{\\pm}'$ are obtained from $\\gamma_{\\pm}$ by replacing $h_{x\\,in}$ by $h_{x\\,out}$). This means that for both Bianchi type-I and KS, if the expansion of the universe occurs with the same rate in all direction, models that can realize the late time accelerated expansion of the universe, and able to prevent the Big Rip and the Big Freeze can be reconstructed.\n\n\\subsubsection{Using an auxiliary scalar field}\\label{numero2}\nIn this subsection, we would like to use the reconstruction scheme, for which an auxiliary scalar field is introduced in the action of the theory. By this way, the functional form of $f(T)$ can be found through two other scalar functions $P$ and $Q$\n(for more clarification on the method, see the case of $f(R)$ gravity \\cite{odintsov-asf} and other of $f(T)$ gravity \\cite{daouda3}).\nHere, we return to the general system (\\ref{steph68})-(\\ref{steph72}), where we introduce the auxiliary scalar field $\\phi$, by defining the algebraic function $f(T)$ by\n\\begin{eqnarray}\nf(T)=P(\\phi)T+Q(\\phi)\\,\\,\\,. \\label{steph78}\n\\end{eqnarray}\nBy using (\\ref{steph78}) and varying the action (\\ref{action}) with respect to the scalar field $\\phi$, one gets\n\\begin{eqnarray}\nP'(\\phi)T+Q'(\\phi)=0\\,\\,\\,,\n\\end{eqnarray}\nwhich may be solved with respect to $\\phi$, yielding  $\\phi=\\phi(T)$. Here, the prime (') denotes the derivative with respect to $\\phi$. By using (\\ref{steph78}), one obtains\n\\begin{eqnarray}\nf_T(T)=P(\\phi(T)) \\,\\,\\,,\\quad f_{TT}(T)=P_T(\\phi(T))\\,\\,\\,. \\label{steph80}\n\\end{eqnarray}\nMaking use of (\\ref{steph80}), one can rewrite the system (\\ref{steph68})-(\\ref{steph71}) as \n\\begin{eqnarray}\n16\\pi\\rho&=&P(\\phi)T+Q(\\phi)-2TP(\\phi)\\,\\,\\,,\\label{steph81}\\\\\n-16\\pi p_x&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_y+h_z)^2-(h_y+h_z)-h_yh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\&+&4\\left(\\frac{h_y+h_z}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,,\\label{steph82}\n\\end{eqnarray}\n\\begin{eqnarray}\n-16\\pi p_y&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_x+h_z)^2-(h_x+h_z)-h_xh_z}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\\n&+&4\\left(\\frac{h_x+h_z}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,,\\label{steph83}\\\\\n-16\\pi p_z&=&P(\\phi)T+Q(\\phi)+2P(\\phi)\\left[\\frac{(h_x+h_y)^2-(h_x+h_y)-h_xh_y}{\\Psi_0(T)}-\\frac{T}{2}\\right]\\nonumber\\\\\n&+&4\\left(\\frac{h_x+h_y}{\\Psi_0(T)}\\right){T}P_T(\\phi)\\,\\,\\label{steph84} \\,\\,.\n\\end{eqnarray}\nLet us consider that $p_y=p_z$. Then, by equating (\\ref{steph83}) with (\\ref{steph84}), one gets\n\\begin{eqnarray}\nTP_T(\\phi)=-\\frac{1}{2}P(\\phi)\\left(h_x+h_y+h_z-1\\right)\\,\\,\\,.\\label{steph85}\n\\end{eqnarray}\nUsing (\\ref{steph85}), the system (\\ref{steph81})-(\\ref{steph84}) reduces to\n\\begin{eqnarray}\n16\\pi\\rho&=&Q(\\phi)-TP(\\phi)\\,\\,\\,,\\label{steph86}\\\\\n-16\\pi p_x&=&Q(\\phi)+TP(\\phi)\\,\\,\\,,\\label{steph87}\\\\\np_x&=& p_y=p_z \\,,\\,\\omega_x=\\omega_y=\\omega_z\\label{steph88}\\,.\n\\end{eqnarray} \nBy using (\\ref{steph86}), one can determine $Q(\\phi)$ as\n\\begin{eqnarray}\nQ(\\phi)=16\\pi~\\rho+TP(\\phi)\\,\\,\\,,\\label{steph90}\n\\end{eqnarray}\nwhich, substituted in (\\ref{steph87}) and (\\ref{steph88}), yields\n\\begin{eqnarray}\n-16\\pi(\\omega_x+1)\\rho=2TP(\\phi)\\label{steph91}\\,.\n\\end{eqnarray}\nAs one can redefine the scalar field properly, we may choose $\\phi=t$, then $P(\\phi)=P(t)=\\bar{P}(T)$. Note that in the case where $\\omega_x=-1$, one gets $\\bar{P}(T)=0$, and the algebraic function $f(T)=Q(t)=16\\pi\\rho=\\bar{Q}(T)$. But this requires to have the complete expression of $\\rho$ depending on $T$, i.e., solving the equation of continuity. Let us consider in general the case where $\\omega_x\\neq-1$ and try to solve the equation of continuity, which will help us to determine  $\\bar{P}(T)$ and $\\bar{Q}(T)$, leading to the reconstruction of the algebraic function $f(T)$. We choose the case of KS geometry ($h_y=h_z$), where the equation of continuity is written as\n\\begin{eqnarray}\n\\dot{\\rho}+\\rho(1+\\omega_x)(H_x+2H_y)=0\\,\\,\\,,\n\\end{eqnarray}\nwhich can be solved, giving\n\\begin{eqnarray}\n\\rho(t)&=&C_5\\exp{\\left[G_1(t)+G_2(t)\\right]}\\label{steph93}\\,\\,\\,,\\\\\nG_1(t)&=&\\frac{(1+\\omega_x)(h_{x\\,in}+2h_{y\\,in})}{t}\\,\\,\\,,\\\\\nG_2(t)&=&\\sqrt{q}\\left[(1+\\omega_x)(h_{x\\,in}-h_{x\\,out}+2h_{y\\,in}-2h_{y\\,out})\\right]\n\\arctan{\\left(\\sqrt{q}t\\right)}\\,\\,\\,,\n\\end{eqnarray}\nwhere $C_5$ is an integration constant.\nMaking use of (\\ref{steph67}), one can cast $\\rho(t)$ in terms of $T$. Also, from Eq.~(\\ref{steph91}), one obtains $P(t)$ in functions of $T$, and from (\\ref{steph90}), one gets $Q(t)$ in terms of $T$. Therefore, we can reconstruct the algebraic function  $f(T)$, given in  (\\ref{steph78}), as\n\\begin{eqnarray}\nf(T)&=&-16\\pi C_3 \\omega_x \\exp\\left[\\frac{g_1}{\\sqrt{\\Psi_0 (T)}}+g_2 \\arctan\\left(\\sqrt{q\\Psi_0 (T)}\\right)\\right]\\label{steph103}\\,,\n\\end{eqnarray}\nwhere $g_1=(1+\\omega_x)(h_{x\\,in}+2h_{y\\,in})$ and $g_2=\\sqrt{q}(1+\\omega_x)(h_{x\\,in}-h_{x\\,out}+2h_{y\\,in}-2h_{y\\,out})$.\n\\par\nIn principle, with (\\ref{steph103}), some cosmological models can now be reproduced. Let us focus our attention to the  early universe (which may be characterized by the inflation), and the late time universe (characterized by its accelerated expansion) in the three spatial direction. \\par\nAt early time, i.e., for small $t$  (or  large $T$), $\\Psi_0(T)$ is very small, and the corresponding algebraic function is\n\\begin{eqnarray}\nf(T)=-16\\pi C_3\\omega_x\\exp{\\left[\\frac{g_1}{\\sqrt{\\Psi_0 (T)}}\\right]}\\,\\,\\,.\n\\end{eqnarray}\nAt late time, the time $t$ is large (corresponding to small torsion scalar and large $\\Psi_0(T)$), and the algebraic function reads\n\\begin{eqnarray}\nf(T)=-16\\pi C_3\\omega_x\\exp{\\left[g_2 \\arctan\\left(\\sqrt{q\\Psi_0 (T)}\\right)\\right]}\\,\\,\\,.\n\\end{eqnarray}\nThus, we see that models corresponding to the inflation and the late time accelerated universe can be reconstructed within KS metrics where the matter content is partially isotropic ($p_y=p_z$). Since in this work we fall in the situation where $\\omega_x=\\omega_y=\\omega_y$ one could use the WMAP data and try to check if they fit with this anisotropic model. Because of having ultimately $p_x=p_y=p_z$ and $h_y=h_z$, one can just use the first two equations of motion of KS case, i.e., (\\ref{densityks})-(\\ref{radialpressureks}). In order to put out the contribution of the modified part of the algebraic function $f(T)$, we cast it into the form $f(T)=T+j(T)$, as the teleraparallel term plus the modified additive algebraic function $j(T)$. Thus, the equations (\\ref{densityks}) and (\\ref{tangentialpressureks}) become\n\\begin{eqnarray}\n8\\pi \\rho_{eff}&=& \\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\,\\,,\\label{effectivedensity}\\\\\n-8\\pi p_{x\\,eff}&=& 2 \\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2\\,\\,,\\label{effectivepressure}\n\\end{eqnarray}\nwhere $\\rho_{eff}$ and $p_{x\\,eff}$ are the effective energy density and effective pressure, respectively, and defined by\n\\begin{eqnarray}\n\\rho_{eff}&=&\\rho-\\frac{1}{16\\pi}\\left\\{j+4j_T\\left[ \\left(\\frac{\\dot{B}}{B}\\right)^2+2\\frac{\\dot{A}\\dot{B}}{AB}\\right]\\right\\}\\,\\,\\,,\\\\\np_{x\\,eff}&=&p_{x}+\\frac{1}{16\\pi}\\left\\{j+4j_T\\left[\\frac{\\ddot{B}}{B}+\\left(\\frac{\\dot{B}}{B}\\right)^2+\\frac{\\dot{A}\\dot{B}}{AB}\\right]+4\\frac{\\dot{B}}{B}\\dot{T}j_{TT}\\right\\}\\,\\,\\,.\n\\end{eqnarray}\nBy dividing (\\ref{effectivepressure}) by (\\ref{effectivedensity}), using $\\omega_{eff}=p_{x\\,eff}\/\\rho_{eff}$, one gets\n\\begin{eqnarray}\n\\omega_{eff}=\\frac{2-3h_y}{h_y+2h_x}\\label{steph104}\\,.\n\\end{eqnarray}\nSince the observational data from the WMAP project do not depend on the spatial direction, i.e., are not based on an anisotropic geometry, we have to impose $h_x=h_y$ in (\\ref{steph104}), for getting a suitable cosmological feature. Thus, assuming that the accelerated expansion of the universe is realized with the same rate in all directions, $h_x=h_y=h_z>1$, and that the universe is essentially filled  by the dark energy, where we can neglect the usual matter content such that $\\omega_{eff}\\sim \\omega_{DE}$. Therefore, using (\\ref{steph104}), we have the possibility to regain the well known range of values  allowed by the current 7-year WMAP data for the parameter of the equation of state of the  dark energy, $\\omega_{DE}= -1.1 \\pm 0.14$  WMAP \\cite{wmap}.\n\n\\subsection{On Bianchi type-III solutions}\n\nIn this section, we propose to present some comments on Bianchi type-III solutions. This case is quite exceptional since we do not have the freedom of making cosmological reconstruction as in the case of Bianchi type-I and KS, due to the constraints equations \n(\\ref{constraint1}) and (\\ref{constraint2}). \\par\nFrom (\\ref{constraint2}), since the parameter is different from zero and the algebraic function cannot be a constant, one gets\n\\begin{eqnarray}\n\\frac{\\dot{A}}{A}=\\frac{\\dot{B}}{B}\\,\\,\\,,\n\\end{eqnarray}\nwhich, injected in (\\ref{constraint1}) leads to\n\\begin{eqnarray}\n\\dot{T}f_{TT}=0\\,\\,\\,,\\label{constraint3}\n\\end{eqnarray}\nmeaning that one has $\\dot{T}=0$ or $f_{TT}=0$. The first case, $\\dot{T}=0$ implies that one has a constant torsion scalar, i.e.,\n\\begin{eqnarray}\n\\frac{\\dot{A}^2}{A^2}+2\\frac{\\dot{A}\\dot{C}}{AC}=K\\,\\,,\\label{steph111}\n\\end{eqnarray}\nwhere $K$ is positive constant. Let us consider $A=C^n$, with $n$ bigger than zero or less  than $-2$. Thus, Eq.~(\\ref{steph111}) can be solved yielding\n\\begin{eqnarray}\nC(t)=C_0'\\exp{\\left(\\pm\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\\label{soluC}\n\\end{eqnarray}\nleading to \n\\begin{eqnarray}\nA(t)=B(t)=(C_0')^n\\exp{\\left(\\pm n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\n\\end{eqnarray}\nwhere $C_0'$ is a positive constant. It is important to note that in order to guarantee the expansion of the universe, one need to have\n\\begin{eqnarray}\\label{vincent18}\nA(t)=B(t)=\\left\\{\\begin{array}{ll}\n(C_0')^n\\exp{\\left(- n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,,\\quad \\mbox{for}\\,\\,\\, n<-2 \\\\\n(C_0')^n\\exp{\\left( n\\sqrt{\\frac{K}{n(n+2)}}\\;t\\;\\right)}\\,\\,\\,, \\;\\;\\;\\quad \\mbox{for}\\,\\,\\, n>0\\,\\,\\,.\\end{array}\\right.\n\\end{eqnarray}\n\nIn this case, we see that the rate of expansion is constant for the three spatial direction: this is the de Sitter universe.\\par\nNow we can perform the reconstruction of the algebraic function $f(T)$.  One can cast Eqs.~(\\ref{densitytype3})-(\\ref{tangentialpressure2type3}) in the following system \n\\begin{eqnarray}\n16\\pi\\rho&=&f+4f_T\\left(K-\\frac{\\alpha^2}{2A^2}\\right)\\,\\,\\,,\\label{jesuis}\\\\\n-16\\pi\\omega_x\\rho&=&f+2Kf_T\\left[\\frac{2n^2+3n+1}{n(n+2)}\\right]\\,\\,\\,,\\label{tues}\\\\\n-16\\pi\\omega_z\\rho&=&f+4Kf_T\\left(\\frac{2n+1}{n+2}\\right)-2\\frac{\\alpha^2}{A^2}f_T\\;\\;,\\label{ilest}\\\\\np_x&=&p_y\\,\\,\\,.\n\\end{eqnarray}\nBy combining (\\ref{jesuis}) with (\\ref{ilest}), one can eliminate the term containing $\\alpha$, obtaining\n\\begin{eqnarray}\n-16\\pi(\\omega_z+1)\\rho=\\frac{4K(n-1)}{n+2}f_T \\,\\,.\\label{noussommes}\n\\end{eqnarray}\nThe energy density $\\rho$ can be eliminated by combining (\\ref{tues}) with (\\ref{noussommes}) yielding the following differential equation\n\\begin{eqnarray}\n2K\\left[2n(n-1)\\omega_x-(\\omega_z+1)(2n^2+3n+1)\\right]f_T-n(n+2)(\\omega_z+1)f=0\\,\\,\\,,\n\\end{eqnarray}\nwhose general solution is \n\\begin{eqnarray}\nf(T)&=&C_6\\exp{\\left[R(n)T\\right]}\\,\\,\\,,\\\\\nR(n)&=&\\frac{n(n+2)(\\omega_z+1)}{2K\\left[2n(n-1)\\omega_x-(\\omega_z+1)(2n^2+3n+1)\\right]}\\,\\,,\\nonumber\n\\end{eqnarray}\nwhere $C_6$ is an integration constant. Note that for $n=1$ and $\\omega_x=\\omega_z$, Eq.~(\\ref{D10}) is recovered.\n\\par\nThe second case from (\\ref{constraint3}), $f_{TT}=0$, implies that $f_{T}$ is constant, that we choose to be minus two times the cosmological constant $\\Lambda$, then, $f(T)$ is written as\n\\begin{eqnarray}\nf(T)=T-2\\Lambda\\,\\,\\,,\n\\end{eqnarray}\nwhich is the teleparallel gravity with cosmological constant.\n\n\n\\section{Conclusion} \\label{conclusions}\n\nAlong the paper, the Bianchi type-I, Kantowski-Sachs, and Bianchi type-III metrics have been studied in the context of $f(T)$ gravities. Particularly, we have shown the reconstruction of some important cosmological solutions, obtaining the corresponding $f(T)$ action. We have assumed initially a particular choice of coordinates and tetrads, specifically cartesian coordinates and a diagonal set of tetrads have been imposed in order to avoid the well known constraint $f_{TT}=0$, which reduces trivially to the action of teleparallel gravity (see Ref.~\\cite{Tamanini:2012hg}).\\par\nThen, several important cosmological solutions have been considered. In particular, dS solutions, where the scale factor is an exponential function of the cosmic time, has been considered for Bianchi type-I and Kantowski-Sachs metrics by imposing a particular exponential expansion in each direction of the space.  We have shown that the only possible solution turns out to the FLRW metric, such that no possible dS anisotropic evolution can be found in $f(T)$, unless one considers an anisotropic fluid. Nevertheless, in the case of power law solutions, we have found that in the presence of a perfect isotropic fluid, an anisotropic cosmological evolution can be found for a particular choice of the action $f(T)$, while in vacuum the action reduces to FLRW metric.\\par\nMoreover, we have extended the cosmological reconstruction scheme to a general exponential solutions, from which the above de Sitter law and power law solutions are particular cases. We have assumed an adiabatic approximation for the expansion in each spatial direction. We undertook two cases, a special case and a second where an auxiliary field is used. In the both cases, we shown that the models can realize the early accelerated universe, characterized by the inflation, and the late time acceleration of our current universe. In the special case, the model presents an interesting aspect because it ensures the avoidance of the Big Rip and the Big Freeze. In the case where the auxiliary field is used, the model corresponding to the late time accelerated universe fits with the 7-year WMAP data, confirming the consistency of the result.\n\\par\nThe Bianchi type-III case  presents some constraints from which only two forms of the algebraic function $f(T)$ can be obtained. The first is the well known teleparallel gravity with cosmological constant, and the second is a de Sitter type solution.\n\n\\vspace{0,25cm}\n{\\bf Acknowlegments} The authors thank Prof. J. D. Barrow for useful discussions. MER thanks  UFES for the hospitality during the development of this work. MJSH thanks CNPq\/FAPES for financial support. DSG acknowledges support from a postdoctoral contract from the University of the Basque Country (UPV\/EHU) under the program ``Specialization of research staff'', and support from the research project FIS2010-15640, and also by the Basque Government through the special research action KATEA and UPV\/EHU under program UFI 11\/55.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nMultiferroics\\textemdash materials exhibiting a coexistence of both magnetic and ferroelectric orders \\cite{Kim-2003a,Che-2007}\\textemdash have attracted substantial technological and scientific interest recently. The technological interest stems from the multifunctional properties exhibited by multiferroics, which make them potentially useful in device applications such as magnetoelectric memories and switches. Multiferroics are scientifically interesting, in part, because they exhibit a variety of microscopic mechanisms that can result in an interesting interplay between ferroelectric and magnetic orders; \\cite{Che-2007} among other consequences, this interplay can spawn interesting dynamical properties in multiferroic materials, including electromagnons, i.e., hybrid excitations involving a coupling between optical phonons and spin waves via the magnetoelectric interaction, \\cite{Bar-1970, Pim-2006, Kid-2008a, Kid-2008b, Sus-2008, Val-2009, Ste-2009, Moc-2010, Tiw-2010, Har-2011, Jon-2014, Cao-2015} and magnetodielectric effects. \\cite{Kim-2003b, Law-2003, Yan-2012} \n\nMaterials in which geometric frustration leads to non-collinear spin order and strong spin-lattice coupling are particularly rich material environments to find novel magnetoelectric behavior. \\cite{Kim-2003a, Got-2004} Transition-metal-oxide spinel materials (\\textit{AB}$_2$O$_4$), for example, exhibit both non-collinear spin orders and strong spin-lattice coupling that can lead to magnetoelectric coupling, because the presence of magnetic ions on the \\textit{B}-site pyrochlore lattice of the spinel structure often leads to strong geometric frustration and consequent non-collinear orders that can generate multiferroic phenomena. \\cite{Che-2007} Magnetoelectric effects are indeed realized in some \\textit{A}Cr$_2$O$_4$\\, spinels (\\textit{e.g.}, \\textit{A}=Co$^{2+}$ and Fe$^{2+}$), in which the competition among the various exchange interactions, J$_\\text{A-A}$, J$_\\text{A-Cr}$, and J$_\\text{Cr-Cr}$, involving the \\textit{A}$^{2+}$ ions and the Cr$^{3+}$ $S=3\/2$ spins lead to complex magnetic orders. \\cite{Bor-2009, Sin-2011} \n\n$\\text{CoCr}_2\\text{O}_4$, in particular, exhibits a succession of magnetic orders, including ferrimagnetic order below $T_C \\sim 94$~K, incommensurate conical spiral order below $T_S\\sim 26$ K, commensurate order below $T_L \\sim 14$ K, \\cite{Tom-2004, Tsu-2013} as well as spin-driven multiferroic behavior and dielectric anomalies below $T_S$. \\cite{Yam-2006, Law-2006, Cho-2009} Yet, the nature and origin of magnetoelectric behavior in $\\text{CoCr}_2\\text{O}_4$\\,remains uncertain. Multiferroicity in $\\text{CoCr}_2\\text{O}_4$\\,has been associated with the spin-current mechanism \\cite{Kat-2005} involving cycloidal spin order, \\cite{Yam-2006} in which the induced electric polarization is generated by the non-collinear spins \\cite{Mos-2006} via the inverse Dzyaloshinskii-Moriya interaction, $\\boldsymbol P \\sim \\boldsymbol e_{ij}\\times(\\boldsymbol S_i\\times \\boldsymbol S_j)$. However, evidence for multiferroicity, \\cite{Sin-2011, Yan-2012} structural distortion, \\cite{Yan-2012} and magnetodielectric behavior \\cite{Yan-2012} have also been reported above $T_S$ in the ferrimagnetic state of $\\text{CoCr}_2\\text{O}_4$, raising questions about the origin of multiferroic behavior in this material. Yang \\textit{et al.}, for example, have suggested that magnetodielectric behavior in $\\text{CoCr}_2\\text{O}_4$\\,results from the presence of multiferroic domains that are reoriented in the presence of a magnetic field. \\cite{Yan-2012} But magnetodielectric behavior in magnetic materials can also arise from magnetic fluctuations that induce shifts in optical phonon frequencies via strong spin-lattice coupling. \\cite{Law-2003}\n\nUnfortunately, a lack of microscopic information regarding spin-lattice coupling has prevented a clear identification of the mechanism for magnetodielectric behavior in $\\text{CoCr}_2\\text{O}_4$. The intersublattice exchange magnon has been observed in $\\text{CoCr}_2\\text{O}_4$\\,using infrared and terahertz spectroscopies \\cite{Tor-2012, Kam-2013} and optical phonons in $\\text{CoCr}_2\\text{O}_4$\\,have been identified using Raman scattering \\cite{Kus-2009, Pta-2014, Eft-2015} and optical absorption \\cite{Tor-2012} measurements. However, to our knowledge, there have been no microscopic studies of spin-lattice coupling in $\\text{CoCr}_2\\text{O}_4$\\,that could clarify the origin of magnetodielectric behavior in this material. The application of pressure \\cite{Kan-1988, Tam-1993, Che-2013} would be a useful means of studying spin-lattice coupling and its role in magnetoelectric behavior in spinels such as $\\text{CoCr}_2\\text{O}_4$; indeed, \\textit{ab initio} calculations predict that pressure should enhance the macroscopic polarization in the multiferroic regime of $\\text{CoCr}_2\\text{O}_4$. \\cite{Eft-2015} However, the effects of pressure on the magnetoelectric behavior and spin-lattice coupling in $\\text{CoCr}_2\\text{O}_4$\\,have not yet been experimentally investigated.\n\nRaman scattering is a powerful tool for studying magnons, \\cite{Gle-2014, Gim-2016} strong spin-lattice coupling \\cite{Gle-2014, Byr-2016} and electromagnons \\cite{Caz-2008, Sin-2008, Rov-2010, Rov-2011} in complex oxide materials. When used in conjunction with pressure and magnetic-field tuning, Raman scattering can provide pressure- and magnetic-field-dependent information about the energy and lifetime of phonons, magnons, and spin-phonon coupling effects. In this paper, we report an inelastic light (Raman) scattering study of magnon and phonon excitations in $\\text{CoCr}_2\\text{O}_4$\\,as simultaneous functions of temperature, pressure, and magnetic field. Below $T_C=94$ K, we report the development in $\\text{CoCr}_2\\text{O}_4$\\,of  a\\, $\\sim$ $16 \\,\\text{cm}^{-1}$ (2 meV) $\\boldsymbol q=0$ magnon excitation with T$_{1g}$\\,symmetry. The anomalously large Raman scattering susceptibility associated with the T$_{1g}$\\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is indicative of a large magneto-optical response arising from large magnetic fluctuations that couple strongly to the dielectric response; this coupling is likely associated with the dielectric anomalies \\cite{Sin-2011} observed in the ferrimagnetic phase of $\\text{CoCr}_2\\text{O}_4$. We also show that the Raman intensity of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,exhibits a strong suppression with increasing magnetic field, suggesting that the dramatic magneto-dielectric behavior \\cite{Muf-2010, Yan-2012} observed in $\\text{CoCr}_2\\text{O}_4$\\,results from the magnetic-field-induced suppression of magnetic fluctuations that are strongly coupled to phonons. \\cite{Law-2003} Using applied pressure to increase the magnetic anisotropy in $\\text{CoCr}_2\\text{O}_4$\\,results in a decreased magnetic field-dependence of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$, suggesting that pressure or epitaxial strain can be used to control magnetodielectric behavior and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$\\,by suppressing magnetic fluctuations. \n\n\\section{Experimental Methods}\n\n\\subsection{Crystal Growth and Characterization}\n$\\text{CoCr}_2\\text{O}_4$\\,crystals were grown by chemical vapor transport (CVT) following a procedure described by Ohgushi et al.\\cite{Ohg-2008} Polycrystalline powder samples of $\\text{CoCr}_2\\text{O}_4$\\,were first synthesized using cobalt nitrate hexahydrate (Strem Chemicals 99\\%) and chromium nitrate nonahydrate (Acros 99\\%). The nitrates were combined in stoichiometric amounts and dissolved in water. The solution was heated to $350^\\circ$ C and stirred using a magnetic stir bar at $300$ rpm until all of the liquid evaporated. The remaining powder was heated in an alumina crucible at $900^\\circ$ C for $16$ hours and then air quenched. Crystal samples of $\\text{CoCr}_2\\text{O}_4$\\,were grown by CVT using CrCl$_3$ as a transport agent. $2.0$ g of polycrystalline samples and $0.04$ g of CrCl$_3$ were sealed in an evacuated quartz ampoule, which was placed inside a three-zone furnace having $950 ^\\circ $C at the center with a temperature gradient of $10 ^\\circ $C\/cm for one month. Crystals with typical dimensions of $2 \\times 2 \\times 2$\\,mm$^3$ were obtained.\n\n\\begin{figure}\n\\includegraphics[height=7.4cm,width=8.5cm]{xray.pdf}\n\\caption{\\label{fig1}X-ray diffraction pattern and Rietveld fit of $\\text{CoCr}_2\\text{O}_4$\\,at 298 K. The Miller indices for a cubic unit cell with cell parameter $a=8.334(1) \\text{\\AA}$ are also shown.}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[scale=0.4]{MPMS.pdf}\n\\caption{\\label{fig2}Molar susceptibility of $\\text{CoCr}_2\\text{O}_4$\\,as a function of temperature measured in an applied field of 100 Oe.}\n\\end{figure}\n\nThe $\\text{CoCr}_2\\text{O}_4$\\,crystals were characterised using x-ray diffraction and magnetization measurements. Crystals of $\\text{CoCr}_2\\text{O}_4$\\,were ground to a powder to obtain the x-ray diffraction pattern using a Siemens-Bruker D5000 diffractometer using Cu-K$\\alpha$ radiation shown in Fig.~\\ref{fig1}. Rietveld refinement of the $\\text{CoCr}_2\\text{O}_4$\\,cell to the XRD data was performed using XND Rietveld, \\cite{xnd} and indicates a pure sample with $Fd \\bar{3} m$ symmetry and a lattice constant of $8.334(1)$\\AA , which agrees with the established structure. \\cite{Tor-2012} The $<110>$ reflections from a single crystal of $\\text{CoCr}_2\\text{O}_4$\\,were measured, and no evidence of twinning imperfections was found. The field-cooled dc magnetization data on the $\\text{CoCr}_2\\text{O}_4$\\,powder from which our crystal sample was obtained was collected using a Quantum Design MPMS-3 and is shown as a function of temperature in Fig.~\\ref{fig2}. Our results are similar to existing data. \\cite{Law-2006} In particular, the sudden increase in the molar susceptibility, $\\chi_{\\text m}$ at $T\\sim 94$ K marks the onset of ferrimagnetic ordering. The change in slope of the graph at $T\\sim26$ K and an additional small anomaly at $T\\sim14$ K correspond to the incommensurate and commensurate spiral ordering, respectively, in $\\text{CoCr}_2\\text{O}_4$. \n\n\\subsection{Raman Scattering Measurements}\nRaman scattering measurements were performed using the $647.1\\,\\text{nm}$ excitation line from a Kr$^+$ laser. The incident laser power was limited to $5-10$ mW, and was focused to a $\\sim \\text{\\SI{50}{\\micro\\meter}}$-diameter spot to minimize laser heating of the sample. Sample heating by the laser was estimated to be in the range $5-7$ K, and this estimated laser heating is included in the temperatures given in the results section. The scattered light from the samples was collected in a backscattering geometry, dispersed through a triple stage spectrometer, and then detected with a liquid-nitrogen-cooled CCD detector. The samples were inserted into a continuous He-flow cryostat, which was horizontally mounted in the open bore of a superconducting magnet. \\cite{Kim-2011} This experimental arrangement allows Raman scattering measurements under the simultaneous conditions of low temperature ($3-300$ K), high magnetic fields ($0-9$ T), and high pressures ($0-100$ kbar). To determine the symmetries of the measured Raman excitations in zero magnetic field, linearly polarized incident and scattered light were used for various crystallographic orientations of the sample. In the magnetic field measurements, circularly polarized light was used to avoid Faraday rotation of the light polarization.\n\n\n\\begin{figure}\n\\subfloat{\\includegraphics[height=5.2cm,width=8.5cm]{exa.pdf} \\label{exa}}\\\\\n\\subfloat{\\includegraphics[height=3.8cm,width=4.1cm]{exb.pdf} \\label{exb}} \n\\subfloat{\\includegraphics[height=3.8cm,width=4.1cm]{excc.pdf} \\label{exc}}\n\\caption{\\label{fig3}Illustrations showing the experimental arrangements used for different high-magnetic-field and high temperature Raman scattering experiments at low temperatures in this study. \\cite{Kim-2011} (a) Configuration for high-magnetic field measurements in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol H$) geometry, where $\\boldsymbol k$ is the wavevector of the incident light and $\\boldsymbol H$ is the applied magnetic field direction. (b) Configuration for high-magnetic-field measurements in the Voigt ($\\boldsymbol k \\perp \\boldsymbol H$) geometry. (c) Configuration for high pressure measurements using a diamond anvil cell.}\n\\end{figure}\n\nMagnetic field measurements were performed in both Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) and Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometries, where $\\boldsymbol k$ is the wavevector of the incident light and $\\boldsymbol M$ is the magnetization direction. \\cite{Kim-2011} Because of the very small anisotropy field in $\\text{CoCr}_2\\text{O}_4$\\,($H_A\\leq0.1$ T), \\cite{Kam-2013} the net magnetization $\\boldsymbol M$ was assumed to follow the applied field $\\boldsymbol H$ in all experiments performed. To verify this, we confirmed that the field-dependence of the Raman spectrum was independent of the crystallographic orientation of the applied field. The field measurements in the Faraday geometry were performed by mounting the sample at the end of the insert, as illustrated in Fig.~\\subref*{exa}, so that the wavevector of the incident light is parallel to the applied field. The Voigt geometry was achieved by mounting the sample on an octagon plate, which was mounted sideways on the sample rod, as illustrated in Fig.~\\subref*{exb}. The incident light was guided to the sample surface with a 45$^\\circ$ mirror mounted on the sample rod. This sample mounting arrangement allows the magnetic field to be applied perpendicular to the wavevector of the incident light,  $\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$. \n\n High pressure measurements were performed using a miniature cryogenic diamond anvil cell (MCDAC) to exert pressure on the sample via an argon liquid medium. The high-pressure cell was inserted into the cryostat as illustrated in Fig.~\\subref*{exc}, allowing the pressure to be changed \\textit{in situ} at low temperatures without any extra warming\/cooling procedure. This arrangement also allows simultaneous high-pressure and high-magnetic field measurements in the Faraday  ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry, as illustrated in Fig.~\\subref*{exc}. \\cite{Kim-2011} The pressure was determined from the shift in the fluorescence line of a ruby chip loaded in the cell along with the sample piece.\n\n\\section{Temperature dependence of the magnetic excitation at \\textit{P}=0 and \\textit{B}=0}\n\\subsection{Results}\n\\begin{figure}\n\\includegraphics[scale=0.4]{T1.pdf}\n\\caption{\\label{fig4}Temperature-dependence of the Raman scattering intensity, $S(\\omega)$, for $\\text{CoCr}_2\\text{O}_4$\\,at 10 K and 130 K, showing the phonon modes above 150 cm$^{-1}$ and the T$_{1g}$\\,symmetry magnon near 16 cm$^{-1}$ that evolves for $T<90$ K. Inset shows the polarization dependence of the magnon in $\\text{CoCr}_2\\text{O}_4$; the presence of this mode only in the depolarized geometry for all crystallographic orientations is indicative of the T$_{1g}$\\,symmetry, which transforms like an axial vector.}\n\\end{figure}\n\nFig.~\\ref{fig4} shows the $T=10$ K and $T=130$ K Raman spectra of $\\text{CoCr}_2\\text{O}_4$\\,between 0-700 cm$^{-1}$ in a scattering geometry with circularly polarized incident light and unanalyzed scattered light. The $T=10$ K spectrum exhibits the five Raman-active phonon modes expected and previously observed \\cite{Kus-2009, Pta-2014, Eft-2015} for $\\text{CoCr}_2\\text{O}_4$, including phonon modes at 199 cm$^{-1}$, 454 cm$^{-1}$, 518 cm$^{-1}$, 609 cm$^{-1}$, and 692 cm$^{-1}$ (at $T=10$ K). In addition to the phonon modes, the $T=10$ K spectrum in Fig.~\\ref{fig4} has an additional mode that develops near 16 cm$^{-1}$ ($\\sim$2 meV) below $T=90$ K. The inset of Fig.~\\ref{fig4} shows that the 16 cm$^{-1}$ mode intensity is present only in the ``depolarized\" scattering geometry, \\textit{i.e.}, only when the incident and scattered light polarizations are perpendicular to one another, independent of the crystallographic orientation. This polarization dependence indicates that the 16 cm$^{-1}$ mode symmetry transforms like the fully antisymmetric representation, T$_{1g}$, which has the symmetry properties of an axial vector, characteristic of a magnetic excitation. \\cite{Cot-1986, Ram-1991} Consequently, we identify the 16 cm$^{-1}$ excitation as a $\\boldsymbol q=0$ T$_{1g}$ \\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$. This interpretation is supported by the temperature-dependence of the 16 cm$^{-1}$ T$_{1g}$ -symmetry mode Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$ (see Fig.~\\subref*{T2a}), where $\\text{Im}\\,\\chi(\\omega)=S(\\boldsymbol q=0,\\omega)\/[1+n(\\omega,T)]$, $S(\\boldsymbol q=0,\\omega)$ is the measured Raman scattering response, and $[1+n(\\omega,T)]$ is the Bose thermal factor with $n(\\omega,T)=[e^{\\hbar\\omega\/k_BT})-1]^{-1}$. Fig.~\\subref*{T2b} shows that the $\\sim16\\,\\text{cm}^{-1}$ T$_{1g}$\\,symmetry mode energy (solid squares) decreases in energy (``softens\") with increasing temperature toward $T_C$\\textemdash consistent with the temperature-dependence of the Co$^{2+}$ sublattice magnetization \\cite{Kam-2013}\\textemdash indicative of a single-magnon excitation. \\cite{Cot-1986} Fig.~\\subref*{T2b} also shows that the amplitude of the Raman susceptibility, $\\text{Im}\\,\\chi(\\omega)$, associated with the 16 cm$^{-1}$ T$_{1g}$ -symmetry magnon mode (solid circles) is relatively insensitive to temperature and is  comparable to that of the 199 cm$^{-1}$ T$_{2g}$ phonon. Notably, the 16 cm$^{-1}$ T$_{1g}$\\,symmetry magnon we observe in $\\text{CoCr}_2\\text{O}_4$\\,has a similar energy and temperature dependence to that of the exchange magnon observed previously in terahertz \\cite{Tor-2012} and infrared spectroscopy \\cite{Kam-2013} measurements of $\\text{CoCr}_2\\text{O}_4$. Nevertheless, it is unlikely that the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon we observe in $\\text{CoCr}_2\\text{O}_4$\\,is the same as the intersublattice exchange mode reported in infrared measurements, because T$_{1g}$\\,is not an infrared-active symmetry. Note in this regard that the spinel structure of $\\text{CoCr}_2\\text{O}_4$\\,is expected to exhibit six $\\boldsymbol q=0$ magnon modes with 5 closely spaced optical branches, \\cite{Kap-1953, Bri-1966, Sah-1974, Tor-2012} so we are likely observing a different optical magnon that is close in energy to that observed in infrared measurements. \\cite{Tor-2012, Kam-2013}\n\n\\subsection{Discussion and Analysis}\nThe finite $\\boldsymbol q=0$ energy of the $\\omega \\sim$ 16 \\text{cm}$^{-1}$ (2 meV) T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,primarily reflects the finite exchange, $H_E$, and anisotropy, $H_A$, fields in $\\text{CoCr}_2\\text{O}_4$, according to $\\omega=\\gamma(2H_AH_E + {H_A}^2)^{1\/2}$, where $\\gamma$ is the gyromagnetic ratio $g\\mu_B\/\\hbar$. \\cite{Ram-1991} Fig. ~\\ref{fig5} also shows that the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is apparent to temperatures as high as $T\\sim60$ K, indicating that the T$_{1g}$\\,symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,is dominated by the Co$^{2+}$ sublattice spins, which order at a significantly higher temperature (94 K) than the Cr$^{3+}$ sublattice (49 K). \\cite{Kam-2013}\n\n\\begin{figure}\n\\subfloat{\\includegraphics[height=3.6cm,width=3.6cm]{T2a.pdf}\n\\label{T2a}}\n\\quad\n\\subfloat{\\includegraphics[height=3.6cm,width=4cm]{T2b.jpg}\n\\label{T2b}}\n\\caption{\\label{fig5}(a) Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$, of the T$_{1g}$-symmetry magnon of $\\text{CoCr}_2\\text{O}_4$\\,as a function of temperature. (b) Summary of the temperature dependence of the T$_{1g}$\\,symmetry magnon energy (filled squares). Also shown in filled circles is a summary of the temperature dependence of the T$_{1g}$\\,symmetry magnon Raman susceptibility amplitude normalized to the susceptibility amplitude of the 199 cm$^{-1}$ T$_{2g}$ optical phonon, $\\text{Im}\\,\\chi_{mag}(\\omega)\/\\text{Im}\\,\\chi_{ph}(\\omega)$.}\n\\end{figure}\n\nImportantly, the Raman susceptibility of the 16 cm$^{-1}$ T$_{1g}$\\, symmetry magnon at T=10 K (for $H=0$ T and $P=0$ kbar) (see Fig.~\\ref{fig5}) reflects the degree to which this magnon modulates the dielectric response, $\\epsilon=4\\pi\\chi_E$ (where $\\chi_E$ is the electric susceptibility). \\cite{Dem-1987, Kum-2011} Consequently, while Raman scattering from magnons is generally much weaker than Raman scattering from phonons, \\cite{Cot-1986} Figs.~\\ref{fig4} and \\ref{fig5} show that Raman intensity of the T$_{1g}$-symmetry magnon is comparable to that of the Raman-active phonons in $\\text{CoCr}_2\\text{O}_4$, indicative of a strong influence of this magnon on the dielectric response of $\\text{CoCr}_2\\text{O}_4$.\n\nThe large Raman susceptibility of the T$_{1g}$\\, symmetry magnon reflects a large magneto-optical response in $\\text{CoCr}_2\\text{O}_4$, and is likely associated with strong magnetic fluctuations that modulate the dielectric response via strong spin-lattice coupling. \\cite{Bar-1983, Dem-1987, Bor-1988} Such large magnetic fluctuations are attributable to the weak anisotropy field in $\\text{CoCr}_2\\text{O}_4$, $H_A\\leq$0.1 T, \\cite{Kam-2013} and can contribute in several ways to fluctuations in the dielectric response:\\cite{Bar-1983, Dem-1987, Bor-1988}\n\n\\begin{equation}\n\\delta \\epsilon(m, l) = i \\, f \\delta m + g (\\delta l )^2 + a (\\delta m)^2 \n\\label{eq1}\n\\end{equation}\nwhere $\\delta\\epsilon$ is the dielectric response fluctuation, $\\delta m$ represents longitudinal fluctuations in the magnetization, $\\delta l$ represents fluctuations of the antiferromagnetic vector, and $a$, $f$, and $g$ are constants. The first term in Eq.~\\eqref{eq1} is associated with the linear magneto-optical Faraday effect, the second term is associated with linear magnetic birefringence, and the final term is an isotropic ``exchange\" mechanism for magnon scattering that is present in non-collinear antiferromagnets. \\cite{Dem-1987, Vit-1991} In non-collinear antiferromagnetic and ferrimagnetic materials with weak anisotropy\\textemdash such as $\\text{CoCr}_2\\text{O}_4$\\textemdash strong single-magnon scattering can result from large fluctuations of both $l$ and $m$. In particular, the one-magnon Raman scattering intensity, $S$, associated with large magnetic fluctuations of the antiferromagnetic vector at $H=0$ is limited only by the anisotropy field, $H_A$ (i.e., $S \\propto 1\/H_A$), \\cite{Dem-1987} which is very small in $\\text{CoCr}_2\\text{O}_4$, $H_A\\leq$0.1 T. \\cite{Kam-2013}\n\n\\section{Magnetic-field-dependence of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$}\n\\subsection{Results}\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.25]{F1aa.pdf}\n\\label{F1a}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.25]{F1b.jpg}\n\\label{F1b}}\n\\\\\n\\subfloat{\\includegraphics[scale=0.24]{F1c.jpg}\n\\label{F1c}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.24]{F1d.jpg}\n\\label{F1d}}\n\\caption{\\label{fig6} Magnetic-field-dependence of the Raman scattering susceptibility, $\\text{Im}\\,\\chi(\\omega)$, of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K in the (a) Faraday geometry ($\\boldsymbol k\\parallel\\boldsymbol M\\parallel\\boldsymbol H$) and the (b) Voigt geometry ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$). (c) Summary of the field dependences of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,at (filled squares) $T=10$ K and (filled circles) $T=55$ K in the Faraday geometry and at (filled triangles) $T=10$ K in the Voigt geometry. (d) Summary of the field dependences of the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility normalized to the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon Raman susceptibility at (filled squares) $T=10$ K and (filled circles) $T=55$ K in the Faraday geometry and at (filled triangles) $T=10$ K in the Voigt geometry.}\n\\end{figure}\n\nFig.~\\ref{fig6} shows the magnetic-field-dependence of the Raman susceptibility, $\\text{Im}\\,\\chi(\\omega)$, for the T$_{1g}$-symmetry magnon of $\\text{CoCr}_2\\text{O}_4$\\,at $P=0$ kbar and $T=10$ K with an applied magnetic field in both the (Fig.~\\subref*{F1a}) Faraday ($\\boldsymbol k\\parallel\\boldsymbol M \\parallel \\boldsymbol H$) and (Fig.~\\subref*{F1b}) Voigt ($\\boldsymbol k \\perp\\boldsymbol M \\parallel \\boldsymbol H$) geometries. Fig.~\\subref*{F1c} summarizes the field-dependences of the T$_{1g}$-symmetry magnon energy at both $T=10$ K and $T=55$ K, showing that the T$_{1g}$-symmetry magnon energy exhibits a linear increase with increasing field. The shift in the T$_{1g}$-symmetry magnon energy with field, $d\\omega \/dH \\sim$ 1.1 cm$^{-1}\/$T corresponds to a dimensionless ratio $\\hbar\\omega\/\\mu_BH=2.4$. This ratio is close to the $T=4$ K value of $\\hbar\\omega\/\\mu_BH=2.5$  measured for the exchange magnon in $\\text{CoCr}_2\\text{O}_4$\\,\\cite{Kam-2013} and is consistent with the gyromagnetic ratio of 2.2 for Co$^{2+}$. \\cite{Alt-1974, Tor-2012} Fig.~\\subref*{F1d} compares the field-dependence of the normalized T$_{1g}$-symmetry magnon intensity, $\\text{Im}\\,\\chi_{mag}(\\omega)\/\\text{Im}\\,\\chi_{ph}(\\omega)$, in both the (filled circle and square) Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) and (filled triangle) Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometries, where $\\text{Im}\\,\\chi_{mag}(\\omega)$ and $\\text{Im}\\,\\chi_{ph}(\\omega)$ are the Raman susceptibilities of the T$_{1g}$-symmetry magnon and 199 cm$^{-1}$ T$_{2g}$ phonon, respectively. Fig.~\\subref*{F1d} shows that there is a substantial decrease in the normalized T$_{1g}$-symmetry magnon intensity of $\\text{CoCr}_2\\text{O}_4$\\,with increasing field in both the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) and Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometries at $T=10$ K and $T=55$ K. Note that the field-dependent decrease we observe in the T$_{1g}$-symmetry magnon intensity \\textemdash which is particularly dramatic in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) \\textemdash cannot be attributed to field-dependent changes in polarization or crystallographic orientation: the use of circularly polarized incident light in these experiments precludes field-dependent rotation of the incident polarization; and T$_{1g}$-symmetry modes appear in the depolarized scattering geometry independent of the crystallographic orientation of the sample.\n\n\\subsection{Discussion and Analysis}\nThe anomalously large decrease in the 16 cm$^{-1}$ T$_{1g}$-magnon Raman intensity with increasing field in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) of $\\text{CoCr}_2\\text{O}_4$\\,(see Fig.~\\ref{fig6}) is quite different than the field-independent magnon Raman intensities observed in other spinel materials, such as Mn$_3$O$_4$ and MnV$_2$O$_4$. \\cite{Gle-2014} To clarify the anomalously strong field-dependence of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$, note that the magnon Raman intensity in the Faraday geometry is expected to be dominated by the linear magnetic birefringence contribution to dielectric fluctuations, $\\delta\\epsilon=g (\\delta l )^2$. \\cite{Dem-1987, Bor-1988, Vit-1991} Thus, the strong decrease in the T$_{1g}$-symmetry magnon Raman intensity in the Faraday geometry likely reflects a field-induced decrease in fluctuations of the antiferromagnetic vector, $\\delta l$. A similar field-dependent decrease in the single-magnon inelastic light scattering response associated with fluctuations of the antiferromagnetic vector was also observed in the canted antiferromagnet EuTe. \\cite{Dem-1987}\n\nFig.~\\subref*{F1b}, \\subref*{F1d} shows that there is a similar, albeit less dramatic, field-dependent decrease in the T$_{1g}$-symmetry magnon Raman intensity measured in the Voigt ($\\boldsymbol k \\perp \\boldsymbol M \\parallel \\boldsymbol H$) geometry. This geometry is primarily sensitive to the Faraday ($\\delta\\epsilon=i \\, f \\delta m$) and isotropic exchange ($\\delta\\epsilon=a (\\delta m)^2$) contributions to dielectric fluctuations, which are dominated by longitudinal fluctuations in the magnetization. \\cite{Dem-1987, Bor-1988, Vit-1991} Altogether, the suppression of the T$_{1g}$-symmetry magnon Raman scattering intensities in both Faraday and Voigt geometries is indicative of a field-induced suppression of both transverse and longitudinal magnetic fluctuations in $\\text{CoCr}_2\\text{O}_4$.\n\nThe field-dependent suppression of the T$_{1g}$-symmetry magnon Raman intensity in $\\text{CoCr}_2\\text{O}_4$\\,points to a specific microscopic mechanism for the magnetodielectric response observed in $\\text{CoCr}_2\\text{O}_4$. \\cite{Sin-2011, Muf-2010, Yan-2012} Lawes \\textit{et al.} have pointed out that the field-induced suppression of magnetic fluctuations can contribute to the magnetodielectric response of a material via the coupling of magnetic fluctuations to optical phonons. \\cite{Law-2003} This spin-phonon coupling contributes to the magnetodielectric response of a material through field-induced changes to the net magnetization. \\cite{Smo-1982, Kim-2003b, Law-2003, Yan-2012} A simple phenomenological description for how the magnetization of a magnetoelectric material influences the dielectric response of the material is obtained by considering the free energy, $F$, in a magnetoelectric material with a coupling between the magnetization $\\boldsymbol M$ and polarization $\\boldsymbol P$: \\cite{Smo-1982, Kim-2003b, Yan-2012}\n\n\\begin{eqnarray}\nF(M,P) = F_0 + aP^2 &+& bP^4 - PE + cM^2 + \\nonumber \n\\\\  &&dM^4 - MH + eM^2P^2 ,\n\\label{eq2}\n\\end{eqnarray}\nwhere $F_0$, $a$, $b$, $c$, $d$, and $e$ are temperature-dependent constants, and $M$, $P$, $E$, and $H$ are the magnitudes of the magnetization, polarization, applied electric field, and applied magnetic field, respectively. The dependence of the dielectric response on magnetization in a magnetoelectric material, $\\epsilon(M)$ , can be obtained from the second derivative of the free energy with respect to polarization $P$: \\cite{Smo-1982, Kim-2003b, Yan-2012}\n\n\\begin{equation}\n[\\epsilon(M)]^{-1} \\sim (\\partial^2F\/\\partial P^2) = 2a + 12bP^2 + 2eM^2,\n\\end{equation}\nwhich, for a negligible macroscopic polarization $P$ in the material, can be written: \\cite{Kim-2003b, Yan-2012}\n\\begin{equation}\n\\epsilon(M) = 1\/[2a + 2eM^2].\n\\end{equation}\nThus, the dielectric response, $\\epsilon = 4\\pi\\chi_E$, decreases with increasing squared magnetization, $M^2$ and decreasing magnetic fluctuations. \\cite{Law-2003}\n\nThe above results suggest that both the magnetic-field-dependent decrease in the intensity of the 16 cm$^{-1}$ T$_{1g}$-symmetry magnon (see Fig.~\\ref{fig6}) and the magnetodielectric response, $\\Delta\\epsilon(H)\/\\epsilon(0)=[\\epsilon(H)-\\epsilon(0)]\/\\epsilon(0)$, in $\\text{CoCr}_2\\text{O}_4$\\,\\cite{Muf-2010, Yan-2012} reflect magnetic-field-induced changes to magnetic fluctuations\\textemdash particularly fluctuations associated with the antiferromagnetic vector\\textemdash that are strongly coupled to phonons \\cite{Law-2003} via the biquadratic contribution to the free energy, $M^2P^2$ (see Eq.~\\eqref{eq2}).\n\n\\section{Pressure dependence of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$}\n\n\\subsection{Results}\n\nAs discussed above, the strong T$_{1g}$-symmetry magnon Raman intensity of $\\text{CoCr}_2\\text{O}_4$\\,is believed to reflect strong magnetic fluctuations that are coupled to long-wavelength phonons, which should also be associated with significant magneto-optical responses (both linear Faraday and linear magnetic birefringence) in $\\text{CoCr}_2\\text{O}_4$. Our results show that the application of a magnetic field suppresses these fluctuations, leading to the substantial magnetodielectric response observed in $\\text{CoCr}_2\\text{O}_4$. An alternative approach to suppressing magnetic fluctuations is to use applied pressure or strain to increase the crystalline anisotropy of $\\text{CoCr}_2\\text{O}_4$. To investigate this possibility, magnetic-field-dependent measurements of the T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,were performed for different applied pressures.\n\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.23]{P1aa.pdf}\n\\label{P1a}}\n\\,\\,\n\\subfloat{\\includegraphics[scale=0.23]{P1bb.pdf}\n\\label{P1b}}\n\\,\\,\n\\subfloat{\\includegraphics[scale=0.23]{P1cc.pdf}\n\\label{P1c}}\n\\caption{\\label{fig7}Field-dependence in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry of the T$_{1g}$-symmetry magnon Raman susceptibility of $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K and at various applied pressures, including (a) $P=0$ kbar, (b) $P=15$ kbar, and (c) $P=21$ kbar.}\n\\end{figure}\n\nFig.~\\ref{fig7} shows the field-dependence of the T$_{1g}$-symmetry magnon spectrum of $\\text{CoCr}_2\\text{O}_4$\\,in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry for different applied pressures at $T=10$ K. Fig.~\\subref*{P2a} summarizes the field dependence of the T$_{1g}$-symmetry magnon energy at $T=10$ K for different applied pressures, and Fig.~\\subref*{P2b} shows the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility (normalized by the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon susceptibility) at $T=10$ K for different applied pressures. The inset of Fig.~\\subref*{P2b} summarizes the pressure-dependence of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,for $H=0$ T and $T=10$ K.\n\n\\subsection{Discussion and Analysis}\nThe inset of Fig.~\\subref*{P2b} shows that the T$_{1g}$-symmetry magnon energy increases linearly with applied pressure at a rate of $d\\omega\/dP=$0.46 cm$^{-1}$\/kbar. This increase likely reflects a systematic increase in the anisotropy field, $H_A$, with increasing pressure, according to the relationship $\\omega\\sim(2H_AH_E)^{1\/2}$. Additionally, the magnetic field dependence of the Raman spectrum of $\\text{CoCr}_2\\text{O}_4$\\,at different fixed pressures summarized in Fig.~\\subref*{P2a} shows that the field-dependent slope associated with the T$_{1g}$-symmetry magnon frequency, $d\\omega\/dH$, is insensitive to applied pressure up to roughly 21 kbar, indicating that the gyromagnetic ratio associated with Co$^{2+}$ is not strongly affected by these pressures in $\\text{CoCr}_2\\text{O}_4$.\n\n\\begin{figure}\n\\subfloat{\\includegraphics[scale=0.25]{P2aa.pdf}\n\\label{P2a}}\n\\quad\n\\subfloat{\\includegraphics[scale=0.25]{P2b.pdf}\n\\label{P2b}}\n\\caption{\\label{fig8}(a) Summary of the field dependences in the Faraday ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$) geometry of the T$_{1g}$-symmetry magnon energy of $\\text{CoCr}_2\\text{O}_4$\\,at (filled squares) $T=10$ K and at various pressures, including (filled squares) $P=0$ kbar, (filled circles) $P=4.5$ kbar, (filled triangles) $P=15$ kbar, and (filled diamonds) $P=21$ kbar. (b) Summary of the field dependences of the amplitude of the T$_{1g}$-symmetry magnon Raman susceptibility normalized to the amplitude of the 199 cm$^{-1}$ T$_{2g}$ phonon Raman susceptibility at $T=10$ K and various pressures, including (filled squares) $P=0$ kbar, (filled circles) $P=4.5$ kbar, (filled triangles) $P=15$ kbar, and (filled diamonds) $P=21$ kbar. ((b) inset) Summary of the pressure-dependence of the T$_{1g}$-symmetry magnon energy in $\\text{CoCr}_2\\text{O}_4$\\,at $T=10$ K and $H=0$ T.}\n\\end{figure}\n\nOn the other hand, Figs.~\\ref{fig7} and \\ref{fig8} also show that $H=0$ T$_{1g}$-symmetry magnon Raman intensity, $\\text{Im}\\,\\chi(\\omega)$, systematically decreases relative to the T$_{2g}$ phonon intensity, illustrating that increasing pressure suppresses the magnetic fluctuations and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$\\,by increasing the anisotropy field. Additionally, Fig.~\\subref*{P2b} shows that increasing pressure reduces the strong suppression of the T$_{1g}$-symmetry magnon intensity with increasing magnetic field in the Faraday geometry ($\\boldsymbol k \\parallel \\boldsymbol M \\parallel \\boldsymbol H$), providing evidence that the magnetodielectric response of $\\text{CoCr}_2\\text{O}_4$\\,decreases with increasing pressure. Altogether, these results show that, by tuning magnetic anisotropy and reducing magnetic fluctuations of the Co$^{2+}$ spins, pressure and epitaxial strain can be used as effective tuning parameters for controlling the magnetodielectric response of $\\text{CoCr}_2\\text{O}_4$.\n\n\\section{Summary and Conclusions}\n\nIn this paper, we showed that the $\\boldsymbol q=0$ T$_{1g}$-symmetry magnon in $\\text{CoCr}_2\\text{O}_4$\\,exhibits an anomalously large Raman scattering intensity, which reflects a large magneto-optical response that likely results from large magnetic fluctuations that couple strongly to the dielectric response. The strong suppression of the T$_{1g}$-symmetry magnon Raman intensity in an applied field is consistent with the magnetodielectric response observed previously in this material, \\cite{Muf-2010, Yan-2012} and suggests that the strong magnetodielectric response in $\\text{CoCr}_2\\text{O}_4$\\,results from the magnetic-field-induced suppression of magnetic fluctuations that are strongly coupled to phonons. \\cite{Law-2003} Using pressure to increase the magnetic anisotropy in $\\text{CoCr}_2\\text{O}_4$, we found that we can suppress the magnetic field-dependence of the T$_{1g}$-symmetry magnon Raman intensity by suppressing magnetic fluctuations, demonstrating that pressure or epitaxial strain should be an effective means of controlling magnetodielectric behavior and the magneto-optical response in $\\text{CoCr}_2\\text{O}_4$. This Raman study also reveals conditions that are conducive for the substantial magneto-optical responses and magneto-dielectric behaviors in materials, including the presence of strong spin-orbit coupling and weak magnetic anisotropy, both of which create favorable conditions for large magnetic fluctuations that strongly modulate the dielectric response.\n\n\n\\acknowledgements{\nResearch was supported by the National Science Foundation under Grant NSF DMR 1464090. RDM and DPS thank the Illinois Department of Materials Science and Engineering for support. X-ray diffraction and magnetic susceptibility measurements were performed in the Frederick Seitz Materials Research Laboratory.}\n\n\\bibliographystyle{apsrev}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAdequate semigroups form a class of semigroups in which the cancellation properties\nof elements are reflected in the cancellation properties of idempotents.\nThey form a natural common generalisation of inverse semigroups and cancellative\nmonoids. Their importance was first recognised by Fountain \nin the 1970's \\cite{Fountain79}, but for many years their study was restricted by\na lack of applicable methods. In the last few years, interest has been\nreawakened by the development of several new techniques and results \n(see for example \\cite{Araujo11,Branco09a,Gomes09,\nK_freeadequate,K_onesidedadequate}).\n\n\\textit{Free algebras} form a natural focus of attention when studying any class of algebras\nin which they exist; indeed, an understanding of the free objects in a class of algebras usually\nyields considerable information about the class as a whole. In the case of adequate semigroups,\nthe fact that free adequate semigroups of every rank exist follows from\nelementary principles of universal algebra (see for example \\cite[Proposition~VI.4.5]{Cohn81}),\nbut an explicit description proved elusive until recently. In \\cite{K_freeadequate}, the first\nauthor gave a concrete geometric realisation of the free adequate semigroups (and monoids),\ninspired by Munn's celebrated representation of the free inverse semigroups, in terms of directed,\nlabelled, birooted trees under a natural combinatorial multiplication operation. In \\cite{K_onesidedadequate} he showed further\nthat the certain natural subsemigroups are the free objects in the\nrelated categories of \\textit{left adequate} and \\textit{right adequate}\nsemigroups (and monoids). The free left, right and two-sided adequate semigroups also\nturn out to be free objects in the larger classes of left, right and two-sided\n\\textit{Ehresmann semigroups} \\cite{Branco09a,Gomes09,Lawson91}.\n\nThis representation immediately gave rise to a non-deterministic\npolynomial-time\nalgorithm for the\nword problem in finite rank free adequate semigroups and\nmonoids (and hence also in finite rank free left adequate and right adequate\nsemigroups and monoids). Since a relation holds in a free algebra in a category\nexactly if the corresponding identity holds in \\textit{all} algebras in the\ncategory, this also yields an algorithm to check whether a given identity\nholds in all adequate (or left adequate or right adequate) semigroups or\nmonoids.\nThis algorithm has proved surprisingly practical for human application to \nshort words, with the intuitive geometric nature of the representation \noften allowing an effective use of guesswork to circumvent the issue\nof non-determinism. However, a non-deterministic algorithm is clearly\nnot well-suited to computer implementation for larger words, and it would\nalso be more satisfactory for theoretical reasons to know the precise\nasymptotic complexity of the problem.\n\nIn this paper, we apply some ideas from constraint satisfaction theory\nto refine the algorithm into a deterministic form, thus showing that the\nword problems for free adequate, free left adequate and free adequate\nsemigroups and monoids, and hence also the problem of checking whether identities hold\nfor all adequate semigroups or monoids, are decidable in quadratic (in the\nRAM model of computation) time. Moreover, we show how to efficiently (again, in quadratic\ntime in the RAM model) compute normal forms (either trees\nor words) for elements of the free adequate semigroup or monoid.\n\n\\section{Preliminaries}\\label{sec_prelim}\n\nIn this section we very briefly recall the definitions of (left, right and two-sided) adequate\nsemigroups, and the first author's characterisation of the free (left, right\nand two-sided) adequate semigroups monoids. The reader seeking a more complete\nintroduction with examples is referred to \\cite{Fountain79} for adequate semigroups in general and\n\\cite{K_freeadequate} for free adequate semigroups and monoids.\n\nLet $S$ be a semigroup whose idempotent elements commute. Denote by $S^1$ the\nmonoid consisting of $S$ with a new identity element $1$ adjoined. Then $S$ is\ncalled \\textit{left adequate} if for every element $x \\in S$ there\nis an idempotent element $x^+ \\in S$ such that\n$a x = bx \\iff a x^+ = b x^+$ for all $a, b \\in S^1$. If $S$ is left adequate then\nthe choice of $x^+$ is uniquely determined by $x$, and it is usual to\nconsider $S$ as a $(2,1)$-algebra with the binary operation of multiplication\nand the unary operation $x \\mapsto x^+$. In particular, we restrict attention\nto morphisms respecting both operations.\nDually, $S$ is \\textit{right adequate} if for every $x \\in S$ there is an\nidempotent $x^*$ with $xa = xb \\iff x^* a = x^* b$ for all $a, b \\in S^1$;\nright adequate semigroups are also $(2,1)$-algebras. The semigroup $S$ is\ncalled \\textit{(two-sided) adequate} if it is both left adequate and right\nadequate; the two maps $x \\to x^+$ and $x \\to x^*$, which in general will\nbe different, make $S$ into a $(2,1,1)$-algebra.\n\nNow let $\\Sigma$ be an alphabet. A \\textit{$\\Sigma$-tree} (or just\na \\textit{tree} if the alphabet $\\Sigma$ is clear) is a finite\ndirected graph with edges labelled by letters from $\\Sigma$, whose\nunderlying undirected graph is a tree, together\nwith two distinguished\nvertices (the \\textit{start} vertex and the \\textit{end} vertex) such that\nthere is a (possibly empty) directed path\nfrom the start vertex to the end\nvertex. The (unique) simple path from the start vertex to the end vertex\nis termed the \\textit{trunk} of the tree; vertices and edges lying on it are\ncalled \\textit{trunk vertices} and \\textit{trunk edges} respectively. If\n$e$ is an edge in such a tree, we denote by $\\alpha(e)$,\n$\\omega(e)$ and $\\lambda(e)$ respectively the source vertex, target vertex\nand label of $e$. We say that a vertex $v$ is a \\textit{descendant} of a vertex\n$u$ if the unique simple undirected path between $v$ and the start vertex passes\nthrough $u$.\n\nAs a notational convenience, we let $\\Sigma' = \\lbrace j' \\mid j \\in \\Sigma \\rbrace$\nbe an alphabet disjoint from and in bijective correspondence with $\\Sigma$, and say\nthat a $\\Sigma$-tree $X$ has \\textit{an edge from $u$ to $v$ labelled $j'$} to mean that\nit has an edge from $v$ to $u$ labelled $j$. (Intuitively, the elements of $\\Sigma'$ can\nbe thought of as labelling directed edges when read ``in the wrong direction''. This\nnotation will allow a unified consideration of labels and directions of edges;\nsince label and direction play similar roles as obstructions to a morphism mapping one\nedge to another, considering them together simplifies our arguments in several places.)\n\nA \\textit{morphism} $\\rho : X \\to Y$ of $\\Sigma$-trees $X$ and $Y$ is a\nmap taking edges to edges and vertices to vertices which commutes with\n$\\alpha$, $\\lambda$ and $\\omega$ and maps the start and end vertex of $X$\nto the start and end vertex of $Y$ respectively. An \\textit{isomorphism} is a\nmorphism which is bijective on both edges and vertices. A \\textit{retraction} is\nan idempotent morphism from a $\\Sigma$-tree to itself; its image is called a \\textit{retract}.\nA tree is called \\textit{pruned} if it does not admit a\nnon-identity retraction. (Structures without retractions are often called\n\\textit{cores} in graph theory.)\n\nThe $\\Sigma$-tree with a single vertex and no edges is called \\textit{trivial}.\nThe set of all isomorphism types of $\\Sigma$-trees (including the trivial\n$\\Sigma$-tree) is denoted $UT^1(\\Sigma)$\nwhile the set of isomorphism types of non-trivial $\\Sigma$-trees is\ndenoted $UT(\\Sigma)$. The set of all isomorphism types of pruned trees\n[respectively, non-trivial pruned trees] is denoted $T^1(\\Sigma)$\n[respectively, $T(\\Sigma)$]. For any $X \\in UT^1(\\Sigma)$ there is a unique\n$Y \\in T^1(\\Sigma)$ which is isomorphic to a retract of $X$ \\cite[Proposition~3.5]{K_freeadequate};\nwe denote this pruned tree $\\ol{X}$ and call it the \\textit{pruning} of $X$.\n\nIf $X, Y \\in UT^1(\\Sigma)$ then the \\textit{unpruned product} $X \\times Y$ is (the isomorphism type of)\nthe tree obtained by glueing together $X$ and $Y$, identifying the end vertex\nof $X$ with the start vertex of $Y$ and keeping all other vertices and\nall edges distinct; this is a well-defined, associative binary operation\n\\cite[Proposition~4.2]{K_freeadequate}. If\n$X \\in UT^1(\\Sigma)$ then $X^{(+)}$ is (the isomorphism type of) the tree with\nthe same labelled graph and start vertex of $X$, but with end vertex of $X^{(+)}$ the\nstart vertex of $X$.\nDually, $X^{(*)}$ is the isomorphism type of the idempotent tree with the\nsame underlying graph and end vertex as $X$, but with start vertex the end vertex of $X$.\nWe define corresponding \\textit{pruned operations} on $T^1(\\Sigma)$ by\n$XY = \\ol{X \\times Y}$, $X^* = \\ol{X^{(*)}}$ and $X^+ = \\ol{X^{(+)}}$.\n\nA tree with a single edge and distinct start and end vertices is called a\n\\textit{base tree}; we identify each base tree with the label of its\nedge, thus viewing $\\Sigma$ itself as a set of $\\Sigma$-trees.\nThe main\nresult of \\cite{K_freeadequate} is that $T^1(\\Sigma)$ is the free adequate monoid\non $\\Sigma$, being freely generated under pruned multiplication, $*$ and $+$ by the base $\\Sigma$-trees\n\\cite[Theorem~5.16]{K_freeadequate}. The map\n$X \\to \\ol{X}$ is a $(2,1,1,0)$-morphism from $UT^1(\\Sigma)$ onto $T^1(\\Sigma)$\n\\cite[Theorem~4.5]{K_freeadequate}.\nMoreover, the submonoid of $T^1(\\Sigma)$ generated by the base trees under pruned\nmultiplication and $*$ [respectively, $+$] is the free left adequate [respectively,\nright adequate] monoid on $\\Sigma$ \\cite[Theorem~3.18]{K_onesidedadequate}. Free\nadequate, left adequate or right adequate semigroups can all be obtained by discarding\nthe trivial tree (which is the identity element) in the corresponding monoids\n(see \\cite[Proposition~2.2]{K_freeadequate} and \\cite[Proposition~2.6]{K_onesidedadequate}).\n\n\\section{Computing with Formulas and Trees}\\label{sec_exptree}\n\nIn this section we study the computational complexity of converting between\nwell-formed formulas, over a generating set $\\Sigma$ and the binary and unary\noperations in an adequate semigroup, and $\\Sigma$-trees. This will allow\nus, in later sections, to use algorithms operating on $\\Sigma$-trees to\nsolve computational problems involving formulas.\n\nFor our complexity\nanalysis throughout this paper, we shall work in the RAM\nmodel of computation, in which integer operations and indirection\n(finding a value stored at a known position in an array) take unit time.\nFor simplicity we will analyse \nthe complexity of problems for a fixed rank semigroup, say on an alphabet\n$\\Sigma$, rather than the uniform complexity as the rank grows.\nIn places where it is necessary to be formal, we shall regard formulas as\nwords over the alphabet $\\Omega$ consisting of generators from $\\Sigma$ plus\nthe symbols\n$($, $)$, $*$ and $+$ with the obvious meaning. We denote by $\\Omega^*$\nthe set of all words over the alphabet $\\Omega$, including the empty word\nwhich we denote $\\epsilon$. Our measure of the size of an expression will\nbe its length as a word over $\\Omega$.\n\nWe assume\n$\\Sigma$-trees are by default stored as a natural number representing the\nstart vertex, a natural number representing the end vertex\nand a list of edges (in no particular order), each\nbeing a triple consisting of a label from $\\Sigma$ and two natural numbers\nencoding its start vertex and its end vertex. Sometimes it\nwill be expedient to convert trees to an alternative\nrepresentation. Note that the same abstract $\\Sigma$-tree can admit\nmultiple representations, by numbering the vertices and ordering the edges\ndifferently. Our measure of the size of a $\\Sigma$-tree will be the\nnumber of edges.\n\n\\begin{proposition}\\label{prop_exptotree}\nGiven a well-formed formula $\\omega$, one can compute in quadratic time\nthe (unpruned) $\\Sigma$-tree which is its evaluation in $UT^1(\\Sigma)$.\nMoreover, this tree has size linear in the length of $\\omega$.\n\\end{proposition}\n\\begin{proof}\nA formula of length $n$ can be evaluated by a depth-first traversal\nof a parse tree; this will clearly involve performing at most $n$ unpruned\noperations with trees whose size is $O(n)$.\nClearly the unpruned $(+)$ and $(*)$ operations on trees can be performed\nin constant time. Unpruned multiplication of trees can be performed in\ntime linear in the number of edges in the trees, by first relabelling\nthe vertices in the second tree (so that all references to its start\nvertex become the end vertex of the first tree, and all its other vertices\nare distinct from those in the first tree) and then concatenating the edge\nlists and setting start and end vertices appropriately.\n\nThus, the $O(n)$ unpruned operations can each be performed in $O(n)$ time,\nand the evaluation of the expression takes time $O(n^2)$.\nMoreover, the resulting tree clearly has exactly one edge for each\noccurrence of a generator in the expression, and hence has size linear\nin the size of the expression.\n\\end{proof}\n\nFor our present purpose, the computations we wish to perform with\ntrees will all take quadratic time, so there is no particular benefit in\nbeing able to compute the trees in faster than quadratic time.\nHowever, we remark that the complexity of the algorithm given above can be\nimproved by a more sophisticated approach, using what is known in the computer science literature as a\n``Union Find'' algorithm. Under this approach, when\nperforming multiplication, instead of merging the end vertex of one\ntree with the start vertex of another, we keep them separate (allowing\nthe data structure to become a forest, rather than a tree) and\nmaintain another data structure recording which vertices are to be\nmerged at the end. An efficient implemention of this algorithm is extremely\nclose to being linear time; see \\cite[Section~21.3]{Cormen01}\nfor more details.\n\nNext, we shall show how a (not necessarily pruned) $\\Sigma$-tree can be efficiently\nconverted into an well-formed formula.\nWe will define a function $\\sigma : UT^1(\\Sigma) \\to \\Omega^*$ such\nthat for each tree $X$, $\\sigma(X)$ is a well-formed formula which\nevaluates to $X$\nin $UT^1(X)$, and then show that this function can be computed in quadratic time.\nNote that since $\\sigma$ is a function defined on abstract trees, the\nalgorithm produces a formula depending only on the abstract tree,\nand not on its representation. We shall exploit this in\nSection~\\ref{sec_normalforms}\nbelow\nto compute normal forms (as formulas) for elements of the free adequate\nsemigroup.\nTo do this, we shall need a linear order on the set of all\nformulas; for now, we will assume that we have such an order fixed. We\nwill discuss the choice and implementation of this order when we come to\nanalyse the complexity of the algorithm.\n\nLet $X$ be a tree. We begin by defining a function $\\rho$\nfrom the vertex set of $X$ to $\\Omega^*$; this is done inductively\nby downwards induction on the distance of the vertex from the trunk.\nLet $v$ be a vertex, and suppose $\\rho$ is already defined on all vertices\nstrictly further from the trunk than $v$. Let\n$v_1, \\dots, v_p$ be the vertices adjacent to $v$ and strictly further from\nthe trunk, noting that $\\rho(v_i)$ is already defined for each $i$.\nFor each $i$, let $e_i$ be the edge connecting $v$ to $v_i$,\nand let $a_i \\in \\Sigma$ be its label. Define a formula $\\tau_i \\in \\Omega^*$\nby:\n$$\\tau_i = \\begin{cases}\n (a_i \\rho(v_i))+ &\\textrm{ if $e_i$ is orientated away from $v$} \\\\\n (\\rho(v_i) a_i)* &\\textrm{ if $e_i$ is orientated towards $v$}\n\\end{cases}$$\nNow we define $\\rho(v) \\in \\Omega^*$ to be the word obtained by sorting the words\n$\\tau_i$ according to our ordering of formulas, and then concatenating.\n(If $p = 0$, that is, if $v$ is a ``leaf'', this means $\\rho(v) = \\epsilon$.)\n\nNow let $t_0, \\dots t_q$ be\nthe trunk vertices of $X$ and $b_1, \\dots, b_q$ the labels of the edges between\nthem, both in the obvious order. We define\n$$\\sigma(X) = \\rho(t_0) b_1 \\rho(t_1) b_2 \\dots \\rho(t_{q-1}) b_q \\rho(t_q).$$\n\nA simple but tedious inductive argument, akin to those in \\cite{K_freeadequate},\nshows that $\\sigma(X)$ evaluates to the tree $X$ in $UT^1(\\Sigma)$, and that\nthe number of characters in $\\sigma(X)$ is at most four times the number\nof edges in $X$.\n\nTo compute $\\sigma(X)$, we start by precomputing adjacency matrices for $X$\ncorresponding to each possible edge label and direction; it is easily seen\nthat this can be done in $O(n^2)$ time where $n$ is the number of edges in $X$.\nIt is immediate from the inductive method of definition how to compute\n$\\sigma(X)$ by a simple depth first traversal (following non-trunk edges)\nfrom each of the trunk vertices; this involves\nconsidering each of $O(n)$ vertices once.\n\nAt each vertex, the only non-trivial operation is to sort the words $\\tau_i$ into order\nand then concatenate; the complexity of this of course depends on the choice of order. The\nsum length of all the words $\\tau_i$ is clearly $O(n)$. If\nwe choose the order to be lexicographic order (with respect to some arbitrary linear\norder on $\\Omega$), then a careful implementation of radix sort gives us a lexicographically\nsorted list of formulas in $O(n)$ time, and concatenation is\nclearly also $O(n)$.\n\nThus, the total time required for the algorithm is $O(n^2)$, and we have established:\n\n\\begin{proposition}\\label{prop_treetoexp}\nGiven an unpruned $\\Sigma$-tree $X$, we can in quadratic time compute\na well-formed formula which evaluates to $X$ in $UT^1(\\Sigma)$. Moreover,\nthe formula has size linear in the size of $X$, and depends only\non the isomorphism type of $X$ and not on its representation.\n\\end{proposition}\n\n\\section{The Word Problem}\\label{sec_wp}\n\nRecall that the \\textit{word problem} for an algebra $A$ with a given generating set\nis the algorithmic problem of determining, given as input two well-formed formulas\nover the generating set and the operations of the algebra, whether the formulas\nrepresent the same element of the algebra. The word problem for free objects\nin a variety of algebras is of particular importance, since it is trivially\nequivalent\nto the problem of testing whether a given identity holds in all algebras of\nthe variety.\n\nIn this section, we shall exhibit a quadratic time algorithm to solve\nthe word problem in a free adequate monoid $T^1(\\Sigma)$. In fact in\nSection~\\ref{sec_normalforms} below, we shall see that it is also possible\nto compute normal forms of elements of $T^1(\\Sigma)$ in quadratic time;\nthis automatically yields another algorithm for the word\nproblem (by computing normal forms and comparing), of the same asymptotic\ncomplexity. However, we present an explicit word problem algorithm first\nsince this is simpler, potentially easier to implement, and illustrates in\na simple context some of the ideas we will need in Section~\\ref{sec_normalforms}.\n\nBy Proposition~\\ref{prop_exptotree} we can efficiently convert\nwell-formed formulas in the free adequate monoid into unpruned\n$\\Sigma$-trees of comparable\nsize. It follows that to test\n(efficiently) whether two given expression $x$ and $y$ represent the same\nelement of the free adequate monoid, that is, to solve the word problem,\nit suffices to compute corresponding $\\Sigma$-trees\n$X, Y \\in UT^1(\\Sigma)$,\nand then check (efficiently) if $\\ol{X} = \\ol{Y}$ in $T^1(\\Sigma)$.\n\nTo solve\nthis latter problem, we begin with an elementary proposition, which\nreduces it a constraint satisfaction problem (formulated in terms of\nmorphisms between structures, in the manner usual in the literature\nof areas such as graph theory and universal algebra --- see for example\n\\cite{Hell04}).\n\n\\begin{proposition}\\label{prop_equiv}\nLet $X$ and $Y$ be $\\Sigma$-trees. Then the following are equivalent:\n\\begin{itemize}\n\\item[(i)] $\\ol{X} = \\ol{Y}$;\n\\item[(ii)] $X$ and $Y$ admit isomorphic retracts;\n\\item[(iii)] there is a morphism from $X$ to $Y$ and a morphism from $Y$ to $X$.\n\\end{itemize}\n\\end{proposition}\n\\begin{proof}\nThe equivalence of (i) and (ii) follows from \\cite[Proposition~3.5]{K_freeadequate}, so\nit suffices to establish the equivalence of (ii) and (iii).\n\nIf (ii) holds then, in particular, some retract of $X$ is isomorphic to a\nsubstructure of $Y$; composing the retraction of $X$ with the isomorphism\nyields a morphism from $X$ to $Y$. By symmetry of assumption there is also\na morphism from $Y$ to $X$, so (iii) holds.\n\nNow suppose (iii) holds, say $\\sigma : X \\to Y$ and $\\tau : Y \\to X$ are\nmorphisms. Then the compositions $\\tau \\circ \\sigma : X \\to X$ and\n$\\sigma \\circ \\tau : Y \\to Y$ are maps on finite sets, and it follows that\nwe may choose $n$ such that both\n$(\\tau \\circ \\sigma)^n : X \\to X$ and $(\\sigma \\circ \\tau)^n : Y \\to Y$\nare idempotent, that is, are retractions of $X$ and $Y$ respectively.\nLet $X'$ and $Y'$ be the retracts which are the respective images of\nthese retractions. Now it is easily verified that \n$\\sigma$ and $\\tau \\circ (\\sigma \\circ \\tau)^{n-1}$ restrict to mutually\ninverse isomorphisms between the retracts $X'$ and $Y'$, showing that\n(ii) holds.\n\\end{proof}\n\nProposition~\\ref{prop_equiv} implies that to check if two $\\Sigma$-trees\nare equivalent, and hence by the preceding arguments to solve the word problem\nfor the free adequate semigroup on $\\Sigma$, it suffices to check whether each\n$\\Sigma$-tree admits a morphism to the other. Our main goal in the rest of\nthis section, then, is an efficient algorithm to test, given an ordered pair of\n$\\Sigma$-trees, whether there is a morphism from the first to the second.\nOur approach is essentially a constraint propagation\nalgorithm, with the correctness of the result being shown by an \narc consistency argument utilising the tree-like nature of\nour geometric representatives for elements. The ideas behind the proof\nare well known in the fields of constraint satisfaction and artificial\nintelligence (see for example\n\\cite{Dechter87}), but for the benefit of semigroup theorists who may not be\nfamiliar with these fields we present the algorithm in an elementary form:\n\n\\begin{algorithm}\\label{algH} \\ \\\\\n\n\\noindent\\textbf{Input:} Two $\\Sigma$-trees $T_1$ and $T_2$ on $n$ and $m$ vertices respectively.\n\n\\noindent\\textbf{Output:} ``Yes'' if there exists a homomorphism from $T_1$ to $T_2$. ``No'' otherwise.\n\n\\begin{enumerate}\n\\item Consider the start vertex of $T_1$, label this vertex $1$, and then use\na depth-first traversal (ignoring direction of edges) to label the remaining vertices\nfrom $2$ to $n$ in ascending order.\n\n\\item For each $i$ in $\\{1,\\dots,n\\}$, let $B_i$ be the set of vertices\nin $T_2$.\n\n\\item For the start [end] vertex $i$ set $B_i$ to be the singleton set containing the\nstart [end] vertex of $T_2$\n\\item For $i$ descending from $n$ to $1$, and each vertex $j > i$ adjacent to $i$, do the following:\n\\begin{itemize}\n\\item[(i)] Let $a \\in \\Sigma \\cup \\Sigma'$ be the label of the edge from $i$ to $j$ in $T_1$;\n\\item[(ii)] Let $B_i:= B_i \\cap B_j^\\star$ where\n\\begin{align*}\nB_j^\\star = \\lbrace x \\mid & \\textrm{ $T_2$ has an edge labelled $a$ } \\\\\n&\\textrm{ from $x$ to some $y \\in B_j$ } \\rbrace.\n\\end{align*}\n\\end{itemize}\n\\item If $B_1=\\emptyset$, output ``No''; otherwise output ``Yes.''\n\\end{enumerate}\n\\end{algorithm}\n\n\\begin{proposition}\\label{prop_wpcorrect}\nAlgorithm~\\ref{algH} is correct, that is, $B_1$ is non-empty on completion of the\nalgorithm if and only if there is a morphism from $T_1$ to $T_2$.\n\\end{proposition}\n\\begin{proof}\nFor brevity, we identify the vertices with the labels from $1$ to $n$\nassigned in the algorithm.\nSuppose first that there is a morphism $\\sigma : T_1 \\to T_2$. We claim\nthat $B_i$ contains $\\sigma(i)$ for all $i$, from which it follows in\nparticular that $B_1$ contains $\\sigma(1)$ so that $B_1$ is\nnon-empty as required. Indeed, if not, choose $i$ maximal such that\n$\\sigma(i) \\notin B_i$. Clearly $\\sigma(i)$ was in $B_i$ after Step 2 of the\nalgorithm and, because $\\sigma$ preserves start and end vertices, also after\nStep 3; therefore, it must have been removed during Step 4.\nFor this to have happened, there must have been a $j > i$ and an edge from $i$\nto $j$ (labelled $a \\in \\Sigma \\cup \\Sigma'$, say) such that\n$\\sigma(i) \\notin B_j^\\star$.\nBy the definition of $B_j^\\star$, this means there was (at the time of removal)\nno edge labelled $a$ from $\\sigma(i)$ to any $y \\in B_j$.\n But because $\\sigma$ is a morphism, $\\sigma(j) \\in B_j$\nis connected to $\\sigma(i)$ by such an edge, so it must be that\n$\\sigma(j)$ was not in $B_j$ at the time $\\sigma(i)$ was removed from\n$B_i$. Now since $B_j$ only gets smaller, $\\sigma(j)$ is not in $B_j$\nat the end of the algorithm. But $j > i$, so this contradicts the\nmaximality of $i$.\n\nConversely, suppose $B_1$ is non-empty at the end of the algorithm. We\ndefine a morphism $\\sigma : T_1 \\to T_2$ inductively as follows. First,\nchoose $\\sigma(1) \\in B_1$ arbitrarily. Now assume $1 < i < n$ and\nwe have defined\n$\\sigma$ on the vertices $1$, \\dots $i-1$ and all edges between them, in\nsuch a way as to preserve adjacency, labels and directions of edges, and\nthe start and end vertices if appropriate, and such that $\\sigma(p) \\in B_p$\nfor $1 \\leq p \\leq i-1$.\n\nSince $T_1$ is a tree and the edges were numbered by a depth-first \ntraversal, it follows that vertex $i$ is connected to vertex $k$ for some \nunique $k < i$; suppose $T_1$ has an edge from $k$ to $i$ labelled\n$a \\in \\Sigma \\cup \\Sigma'$. \n\nConsidering the way $B_k$ is constructed, we see that every vertex in $B_k$,\nincluding $\\sigma(k)$, is connected to some $v \\in B_i$. \nMoreover, if $i$ happens to be the start [respectively, end] vertex of $T_1$,\nthen $B_i$ was originally set to contain only the start [end] vertex of $T_2$,\nso it must be that $v \\in B_i$ is the start [end] vertex of $T_2$.\nThus, by defining $\\sigma(i) = v$ and $\\sigma(e)$ to be the \nappropriate edge, we extend $\\sigma$ to be defined on the vertices \n$1, \\dots, i$ and all edges between, with the appropriate properties.\n\\end{proof}\n\nWe now analyse the complexity of Algorithm~\\ref{algH}.\nAt the start of the algorithm, we can precompute for\neach vertex in $T_1$ a list of edges adjacent to that vertex; this can be done\nin $O(n)$ time.\n\nHaving done this, Step 1 of the algorithm (a simple depth first traversal\nof the tree $T_1$) has complexity $O(n)$. If we store the lists $B_i$ as\narrays of $m$ boolean flags then Step 2 has complexity $O(mn)$\nsince we need to initialise $mn$ values. Step 3 has complexity $O(m)$,\nsince we must reset $m-1$ values for each of the start and end vertices.\n\nThe most interesting part is the complexity of Step 4. The number of\niterations of the outer loop is clearly bounded by the number of edges\nin $T_1$, so it is $O(n)$ and the precomputed lists of edges mean there is\nno extra overhead in finding the edges in the correct order.\nIn each iteration, the fact that the corresponding edge has been found\nmeans Step 4(i) takes constant time. In Step 4(ii), computing $B_j^\\star$\ninvolves\npassing through the list of all $O(m)$ edges of $T_2$ and for each edge\nchecking (in constant time) if one of the ends lies in $B_j$ and if the\nlabel is correct; this takes\n$O(m)$ time. Computing the intersection is simply\na boolean ``and'' operation on two arrays of length $m$, and so also takes $O(m)$\ntime. Thus, Step 4 takes time $O(mn)$, and the total complexity of the\nalgorithm is $O(mn)$.\n\nCombining the above arguments with Proposition~\\ref{prop_exptotree},\nwe have established the following main result:\n\\begin{theorem}\nThe word problem for any finite rank free left adequate, free right\nadequate or free adequate semigroup is decidable in time\npolynomial (quadratic, in the RAM model of computation) in the combined\nlength of the two formulas.\n\\end{theorem}\n\n\\section{Pruned Trees and Normal Forms}\\label{sec_normalforms}\n\nIn this section, we show how to efficiently compute the minimal retract of a \n$\\Sigma$-tree. Combined with the results of Section~\\ref{sec_exptree}, this\nwill allow us to compute normal forms (as formulas) for elements of\nfree adequate monoids. Our main algorithm is the following, the first four\nsteps of which are essentially the same as in Algorithm~\\ref{algH}:\n\n\\begin{algorithm}\\label{algN} \\ \\\\\n\n\\noindent\\textbf{Input:} A $\\Sigma$-tree $T$ on $n$ vertices.\n\n\\noindent\\textbf{Output:} The vertex set of a pruned subtree of $T$,\nisomorphic to the $\\ol{T}$.\n\n\\begin{itemize}\n\\item[(1)] Consider the start vertex of $T$, label this vertex $1$, and then use\na depth-first traversal (ignoring direction of edges) to label the remaining vertices\nfrom $2$ to $n$ in ascending order.\n\n\\item[(2)] For each $i$ in $\\{1,\\dots,n\\}$, set $B_i = \\lbrace 1, \\dots, n \\rbrace$.\n\n\\item[(3)] For the start [end] vertex $i$ set $B_i = \\lbrace i \\rbrace$.\n\n\\item[(4)] For $i$ descending from $n$ to $1$ and each $j$ with $j > i$ and $i$ connected to $j$, do the following:\n\\begin{itemize}\n\\item[(i)] Let $a \\in \\Sigma \\cup \\Sigma'$ be the label of the edge in $T$\nfrom $i$ to $j$.\n\\item[(ii)] Let $B_i := B_i \\cap B_j^\\star$ where\n\\begin{align*}\nB_j^\\star = \\lbrace x \\mid & \\textrm{ $T$ has an edge labelled $a$ } \\\\\n&\\textrm{ from $x$ to some $y \\in B_j$ } \\rbrace.\n\\end{align*}\n\\end{itemize}\n\n\\item[(5)] Set $X = \\lbrace 1, \\dots, n \\rbrace$.\n\n\\item[(6)] For $w$ ascending from $1$ to $n$ and $a \\in \\Sigma \\cup \\Sigma'$,\ndo the following:\n\\begin{itemize}\n\\item[(i)] If $w \\notin X$ then go to the next $w$.\n\\item[(ii)] Otherwise, find all vertices $u$ such that $a$ labels\nan edge from $w$ to $u$ and put them in a list $K$.\n\\item[(iii)] For each $u \\in K$ such that $u > w$:\n\\begin{itemize}\n\\item[(a)] Check if $K \\cap B_u = \\lbrace u \\rbrace$.\n\\item[(b)] If \\textbf{not}, then remove $u$ from $K$, and traverse the\ntree below $u$, removing $u$ and all its descendant vertices from $X$.\n\\end{itemize}\n\\end{itemize}\n\\item[(7)] Output $X$.\n\\end{itemize}\n\\end{algorithm}\n\nOur next aim is to prove the correctness of this algorithm.\n\n\\begin{lemma}\\label{lemma_retract}\nThe subtree $X$, as computed at the end of Algorithm~\\ref{algN}, is a retract of $T$.\n\\end{lemma}\n\\begin{proof}\nWe shall show that each time a vertex and its descendants are removed from\n$X$ at Step 6(iii)(b), there is a retraction from the tree $X$ prior\nto the removal, onto the tree $X$ after the removal. Since the successive\nsubtrees $X$ form a chain under inclusion, it is clear that composing these\nretractions in the appropriate order yields a retraction from $T$ onto\nthe final tree $X$, as required.\n\nIndeed, suppose $u$ and its descendants are removed from $X$ at some point.\nLet $w$, $a$ and $K$ be as in the algorithm at that point, and let $X_1$ and $X_2$\nbe the values of $X$ immediately before and after\nthe deletion, respectively.\n\nNote that, since the identity map is a morphism, it is easily verified\nthat $i \\in B_i$ for all vertices $i$ of $T$. The\nfact that $u$ was removed means that $K \\cap B_u \\neq \\lbrace u \\rbrace$, and\nwe know $u \\in K \\cap B_u$, so we may choose some vertex $v \\in K \\cap B_u$\nwith $v \\neq u$.\n\nFirst, we follow the procedure from the proof of Proposition~\\ref{prop_wpcorrect}\nto inductively define a morphism $\\sigma : T \\to T$, but\nbeing  more careful about our choices in order to ensure that\n$\\sigma(u) = \\sigma(v) = v$.\nWe start by setting $\\sigma(i) = i$ for all $i < u$; since $i \\in B_i$ for all\n$i$ it is easily verified that this is consistent with the procedure\nin Proposition~\\ref{prop_wpcorrect}. Note in particular\nthat $w < u$, so this means $\\sigma(w) = w$. Now since $u, v \\in K$, there\nare edges from $w = \\sigma(w)$ to $u$ and $v$ both labelled $a$, so in\nfollowing the procedure of Proposition~\\ref{prop_wpcorrect}\nwe may choose to set $\\sigma(u) = v$.\nWe now continue the process from the proof of Proposition~\\ref{prop_wpcorrect}. For each vertex\n$r$ in turn, if $r$ is a descendant of $u$, then we define $\\sigma(r)$\nas in the proof of Proposition~\\ref{prop_wpcorrect}, making any choices arbitrarily. If $r$ is\nnot a descendant of $u$ then the unique vertex $k < r$ adjacent to $r$\nis also not a descendant of $u$; thus, we have already defined\n$\\sigma(k) = k$ and we may set $\\sigma(r) = r$.\n\nNow $\\sigma$ is a map on a finite set, and so has an idempotent power,\nsay $\\sigma^i$. Since $v$ is not a descendant of $u$, we have\n$\\sigma(v) = v$, and hence $\\sigma^i(u) = \\sigma^{i-1}(v) = v$, so $u$\nis not in the image of $\\sigma^i$. Since the image of $\\sigma^i$ is a\n$\\Sigma$-tree, it must contain the start\nvertex and be connected, so we deduce that no descendants of $u$ are\nin the image of $\\sigma^i$. It follows that $\\sigma^i$ maps $X_1$ to\n$X_2$. Moreover, $\\sigma$ fixes $X_2$, so restricting $\\sigma^i$ to\n$X_1$ gives the required retraction of $X_1$ onto $X_2$.\n\\end{proof}\n\n\\begin{lemma}\nThe retract $X$, as computed at the end of Algorithm~\\ref{algN}, is pruned.\n\\end{lemma}\n\\begin{proof}\nSuppose not, say $X$ admits a proper retraction $\\sigma : X \\to X$. Let\n$u$ be a vertex in $X$ but not in the image of $\\sigma$, and suppose $u$\nis minimal with respect to this condition. Then $u \\neq 1$, since $1$ labels\nthe start vertex which is fixed by every retraction. Thus, we may let $w$ be\nthe unique vertex with $w < u$ and $w$ adjacent to $u$.\n\nSince $w < u$, by the minimality\nof the choice of $u$, we have $\\sigma(w) = w$. It follows that $\\sigma(u)$\nis connected to $w$ by an edge of the same label and orientation as that\nconnecting $u$ to $w$. This means that, when considering $w$ at step\n6(iii), we would initially have had $\\sigma(u) \\in K$. Since\n$\\sigma(u)$ is in the final tree $X$, it was never removed from $K$.\nMoreover, composing the retraction of $T$ onto $X$ (given by\nLemma~\\ref{lemma_retract}) with\n$\\sigma$ gives a morphism of $T$ mapping $u$ to $\\sigma(u)$; it follows\nfrom the argument in the proof of Proposition~\\ref{prop_wpcorrect} that\n$\\sigma(u) \\in B_u$.\n\nThis means that at the time\n$u$ was considered in Step 6(iii)(a) we had\n$\\sigma(u) \\in K \\cap B_u$. But then $K \\cap B_u \\neq \\lbrace u \\rbrace$,\nso $u$ would have been removed from $X$, giving a contradiction.\n\\end{proof}\n\nTurning to the complexity of the algorithm, Steps (1)-(4) are exactly\nas in Algorithm~\\ref{algH} (except that the source and target trees for\nthe morphism are the same, so $m=n$), and by the same analysis as in\nSection~\\ref{sec_wp} take time $O(n^2)$.\n\nFor efficiency, we store the set $X$ as an array of boolean flags. The time\nrequirement for Step\n(5) is clearly $O(n)$. The loop in Step 6 is iterated at most $O(n)$ times.\nIn each such iteration, step (i) takes constant time. Step (ii) cannot\ninvolve checking more than $O(n)$ vertices, so the total contribution\nto the time required will be $O(n^2)$. In step (iii), note that each\nelement of $L$ is uniquely determined (across the entire algorithm) by\nthe ordered pair $(w,u)$ where there is always an edge between $w$ and $u$;\nthus, the number of iterations of this step across the whole algorithm is\nat most twice the number of edges in the tree, which is $O(n)$.\nWithin each iteration, each step takes $O(n)$ time, so the total contribution is\n$O(n^2)$.\n\nThus, we have established:\n\\begin{theorem}\\label{thm_pruning}\nGiven a $\\Sigma$-tree $T$, one can compute in polynomial time (quadratic\ntime in the RAM model of computation) the pruned $\\Sigma$-tree $\\ol{T}$.\n\\end{theorem}\n\nCombining with the results of Section~\\ref{sec_exptree}, Theorem~\\ref{thm_pruning}\nallows us to compute normal forms (as formulas) in the free adequate monoid.\nIndeed, given a formula $w$, by Proposition~\\ref{prop_exptotree} we may\nconvert it in quadratic time to a corresponding unpruned $\\Sigma$-tree $T$ of comparable size. By\nTheorem~\\ref{thm_pruning} we may then compute the pruned tree $\\ol{T}$ in\ntime quadratic in the size of $T$ and hence in the size of $w$. Finally, by\nProposition~\\ref{prop_treetoexp} we can convert $\\ol{T}$ into the uniquely\ndefined formula $\\sigma(\\ol{T})$ in time quadratic in the size of $\\ol{T}$; since $\\ol{T}$\nis no larger than $T$, this is also quadratic in the size of $T$, and hence\nin the size of $w$.\n\n\\begin{theorem}\nGiven a formula in the free adequate, left adequate or right adequate\nsemigroup or monoid, one can compute\na normal form in polynomial (quadratic in the RAM model of computation) time.\n\\end{theorem}\n\nWe note that the resulting language of normal forms for elements, which by\ndefinition is the set\n$$\\lbrace \\sigma(\\ol{T}) \\mid T \\textrm{ is a pruned $\\Sigma$-tree} \\rbrace,$$\ndoes not appear to have a completely elementary description without reference to trees. Of\ncourse one may check (in quadratic time) whether a given formula $w$ is a\nnormal form by following the above procedure to convert $w$ to a normal form\nand then comparing with $w$; we do not know of a fundamentally easier method.\n\nWe also note that in the case of free \\textit{inverse} monoids (and semigroups), it is known\n\\cite[Theorem 11]{Lohrey07} that the word problem is decidable in (RAM) linear \ntime. In the inverse case computations appear to be inherently simpler, as\nthe operation corresponding to computing a minimal retract (namely, computing a minimal\nmorphic image) can be performed by an iterative process of identifying vertices, where\nthe fact a pair of vertices can be identified is determined ``locally'', by looking only\nin the immediate neighbourhood of the vertices. It seems unlikely that quite such a\nfast algorithm can be obtained in the adequate case, but one might still ask whether our\nalgorithms can be significantly improved upon. Also shown in \\cite[Theorem 11]{Lohrey07}\nis that the word problem for a free inverse monoid is decidable (using a different\nalgorithm to the linear time one) in logarithmic space: the space complexity of the word\nproblem for free adequate monoids and semigroups is a natural topic for future research.\n\n\\section*{Acknowledgements}\n\nThe second author was supported by the Czech Government Grant Agency GA\\v CR\nproject 13-01832S.\nThe authors thank the organisers of the 4th Novi Sad Algebraic Conference\n(NSAC2013), which by bringing together researchers in semigroup theory\nand universal algebra catalysed this research. They also thank Victoria Gould\nfor some helpful comments on the draft, and Stuart Margolis for pointing them\nto the work of Lohrey and Ondrusch \\cite{Lohrey07}.\n\n\\bibliographystyle{plain}\n\n\\def$'$} \\def\\cprime{$'${$'$} \\def$'$} \\def\\cprime{$'${$'$}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{SUPPLEMENTARY MATERIAL}\nMore computational details (with the definition of $N_c$ and $L_\\mathrm{p}$);\nevolution of $N_c$ for cMD at various temperatures;\npotential energy distribution for each replica in T-REMD and REST;\nadditional statistical data and MD snapshots\nfor assembly structures; replica traversal in temperature space for T-REMD;\nan electronic archive for the configuration\nand topology files of the present BTA system.\n\n\n\n\\vspace{-0.5cm}\n\\begin{acknowledgments}\nThe authors acknowledge support from JSPS Grant-in-Aid for Scientific Research\non Innovative Areas ``Dynamical ordering of biomolecular systems\nfor creation of integrated functions'' (Grant No. 25102002).\n\\end{acknowledgments}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\label{sec:Intro}\n\nDue to their large dipole moments \\cite{G94}, Rydberg atoms experience\nstrong long-range dipole-dipole interactions. Within the last decade,\nthis feature has been put forward as the key ingredient of different\npromising atomic quantum-processing scenarios \\cite{SWM10}. For instance,\nRydberg-Rydberg interactions can be used to perform two-qubit logic\noperations in individual-atom systems by shifting a transition off-resonance\nin an atom, depending on the internal state of another atom in its\nimmediate neighborhood \\cite{JCZ00,PRS02,SW05}. In a mesoscopic ensemble,\ndipole-dipole interactions are able to inhibit transitions into collective\nstates that contain more than one Rydberg excitation, thus leading\nto the so-called Rydberg blockade. First predicted in \\cite{LFC01},\nthis phenomenon was locally observed in laser-cooled atomic systems\n\\cite{TFS04,SRA04,CRB05,AVG98,VVZ06} and could in principle be used\nin the future to manipulate and entangle collective excitation states\nof mesoscopic ensembles of cold atoms which could therefore be run\nas quantum processors \\cite{LFC01,BMM07,BMS07,BPS08} or repeaters\n\\cite{ZMH10,HHH10,BCA12}. Rydberg atomic ensembles are also investigated\non quantum non-linear optical purposes \\cite{WA13}: converting photons\ninto so-called Rydberg polaritons which strongly interact through\ndipole-dipole interaction, it seems indeed possible to generate giant\nnon-linearities in the quantum regime, \\emph{i.e.\\@} to effectively\nimplement photon-photon interactions \\cite{PFL12,PBS12,MSP13}.\n\nThe exact calculation of the time-dependent state of an ensemble of\natoms resonantly laser-driven towards a Rydberg level constitutes\na highly non-trivial coupled many-body problem. Such a complex system,\nhowever, often shows an effective thermalization behavior \n\\cite{LOG10,AL12,BMFALA13}: \nobservables, such as the number of Rydberg excitations, indeed tend\nto quasistationary values which can be computed assuming the system\nis in a thermal equilibrium state, either in the canonical \\cite{LOG10}\nor microcanonical \\cite{AL12,BMFALA13} ensembles. Considering the\nsame system as in \\cite{AL12,BMFALA13}, we compare the predictions\nof the microcanonical ensemble assumption to a numerical simulation\nof the unitary evolution of a $100$-particle system, confirmed by\na simplified analytical treatment. The discrepancies we observe allow\nus to show the limitations of the equilibrium hypothesis and to precise\nits applicability conditions.\n\nThe paper is organized as follows. In Sec.~\\ref{sec:PhysicalModel},\nwe present the physical system and simplified model. In Sec.~\\ref{sec:Thermal},\nwe give an analytical description of the distribution of excitations\naccording to the microcanonical ensemble. In Sec.~\\ref{Calculations},\nwe numerically compute the distribution of excitations and apply a\nsimplified analytical treatment which allows us to satisfactorily\nreproduce the results of the full simulation, in the regime of at\nmost 2 excitations. In Sec.~\\ref{sec:Comparison}, we discuss and\ncompare the results obtained according to the different approaches,\nbefore concluding in Sec.~\\ref{sec:Conclusion}.\n\n\n\\section{Model and approximations}\n\n\\label{sec:PhysicalModel}\n\nWe consider a system of $N$ identical atoms located along a line\nof length $L$. The Hilbert space of each atom is assumed to be restricted\nto the ground state $\\left|g\\right\\rangle $ and a highly excited\n(so-called) Rydberg state $\\left|r\\right\\rangle $. In the ensemble\n``vacuum state'' $\\left|\\varnothing\\right\\rangle $, all atoms are\nin the ground state: $\\left|\\varnothing\\right\\rangle \\equiv\\left|g\\dots g\\right\\rangle $.\nDenoting by $\\sigma_{+}\\equiv\\left|e\\right\\rangle \\left\\langle g\\right|$\nand $\\sigma_{-}\\equiv\\sigma_{+}^{\\dagger}$ the usual raising and\nlowering operators for a two-level atom, one defines the ensemble\nstate $\\left|i\\right\\rangle \\equiv\\sigma_{+}^{i}\\left|\\varnothing\\right\\rangle =\\left|g_{1},\\dots,g_{i-1},r_{i},g_{i+1},\\dots,g_{N}\\right\\rangle $\nin which the $i^{\\text{th}}$ atom is Rydberg-excited while the others are\nin the ground state. In the same way, one can define the doubly excited\nstate $\\left|i,j\\right\\rangle \\equiv\\sigma_{+}^{\\left(i\\right)}\\sigma_{+}^{\\left(j\\right)}\\left|\\varnothing\\right\\rangle $\nwhich contains only two Rydberg excitations at positions $i$ and\n$j$, and more generally any arbitrary multiply excited state $\\left|i,j,k,\\dots\\right\\rangle \\equiv\\sigma_{+}^{\\left(i\\right)}\\sigma_{+}^{\\left(j\\right)}\\sigma_{+}^{\\left(k\\right)}\\dots\\left|\\varnothing\\right\\rangle $.\n\nThe atomic ensemble is subject to a laser beam which resonantly drives\nthe transition $\\left|g\\right\\rangle \\leftrightarrow\\left|r\\right\\rangle $:\nin the rotating wave approximation, this process is simply described\nby the Hamiltonian $H_{L}=\\hbar\\Omega\\sum_{k=1}^{n}\\left(\\sigma_{+}^{k}+\\sigma_{-}^{k}\\right)$,\nwhere $\\Omega$ denotes the laser Rabi frequency. Moreover, when lying\nin their Rydberg state, two atoms interact through the (strong) dipole-dipole\ninteraction (this interaction is negligible when at least one atom\nin the pair is in the ground state): the corresponding Hamiltonian\nis \n\\begin{equation}\nV_{dd}=\\hbar C_{6}\\sum_{k\\neq m}\\frac{n_{m}n_{k}}{d(m,k)^{6}}\\label{V_dd}\n\\end{equation}\nwhere $C_{6}$ is the van der Waals interaction coefficient, $n_{k}\\equiv\\sigma_{+}^{k}\\sigma_{-}^{k}$\nthe projector onto the Rydberg state for the $k^{\\text{th}}$ atom and $d(m,k)$\nis the distance between the $m^{\\text{th}}$ and the $k^{\\text{th}}$ atoms.\nFinally,\nthe full Hamiltonian governing the dynamics of the system is \n\\begin{align}\nH & =H_{L}+V_{dd}\\label{Ham}\n\\end{align}\n\n\nStarting in the ensemble vacuum state $\\left|\\varnothing\\right\\rangle $,\nin the absence of interatomic interactions, each atom in the sample\nwould independently undergo Rabi oscillations. Because of dipole-dipole\ninteractions, atoms actually get entangled during their evolution,\naccording to the so-called Rydberg blockade phenomenon \\cite{LFC01}.\nTo understand this mechanism, let us first consider the simple case\nof two atoms. If they are ``close enough'' so that their dipole-dipole\ninteraction overwhelms the laser Rabi frequency, their simultaneous\nexcitation into the Rydberg state becomes impossible since the doubly\nexcited state is strongly shifted out of resonance. As a rule of thumb,\none can define the typical distance $R_{b}$, called the blockade\nradius, at which the blockade starts being effective as the distance\nfor which the van der Waals interaction becomes comparable with the\nlaser excitation, \\emph{i.e} $R_{b}\\approx\\left(\\frac{C_{6}}{\\Omega}\\right)^{\\frac{1}{6}}$.\nNow turning to the full sample, it is clear that dipole-dipole interactions\nforbid the system to explore its full Hilbert space: too off-resonant\nconfigurations, \\emph{i.e.\\@} ensemble states in which two Rydberg excited\natoms are closer than the radius $R_{b}$, will indeed never be substantially\npopulated. In other words, due to the Rydberg blockade the system\nis bound to essentially evolve in the subspace of ``allowed states''\nin which excited atoms are separated at least by $R_{b}$ (Note that\nin a 3D geometric arrangement, each Rydberg excited atom creates an\n``exclusion'' sphere of radius $R_{b}$ often called a ``Rydberg\nbubble'').\n\nThough simple in its form, the Hamiltonian Eq.~\\eqref{Ham} leads\nto complex many-body dynamics. In particular, besides the Rydberg\nblockade phenomenon qualitatively described above, it was shown to\nyield thermalization effects \\cite{LOG10}. The full computation of\nthe dynamics is intractable for large numbers of atoms and one must\nresort to approximations. Following \\cite{LOG10}, we shall make the\nhardcore Rydberg sphere assumption, that is we shall merely discard\nall atomic configurations in which two Rydberg excitations are closer\nthan $R_{b}$, while keeping the others; moreover, we shall make the\nsimplistic approximation that in the allowed subspace the dipole-dipole\nHamiltonian is zero. In this approximation the full Hamiltonian therefore\nbecomes \n\\begin{equation}\nH\\approx\\hbar\\Omega\\sum_{k=1}^{n}(\\tilde{\\sigma}_{+}^{k}+\\tilde{\\sigma}_{-}^{k})\\label{Happrox}\n\\end{equation}\nwhere $\\tilde{\\sigma}_{+}^{k}$ is the raising operator of the $k^{\\text{th}}$\natom restricted to the allowed configuration subspace, \\emph{i.e.\\@}\nthe operator which excites the $k^{\\text{th}}$ atom into the Rydberg\nstate provided that no other Rydberg atom is in the range $R_{b}$.\n\n\n\\section{Thermalized state: Analytical results from the microcanonical ensemble\nassumption }\n\n\\label{sec:Thermal}\n\\\nAnalytical \\cite{AGL12,JAG13} and numerical \\cite{LOG10} investigations\nof the approximate Hamiltonian Eq.~\\eqref{Happrox} both predict\nthermalization to occur in the system. Intuitively, this phenomenon\nresults from the destructive interferences between different frequency\ncomponents of the evolved vector state: for large times, due to the\ncomplexity of the Hilbert space and the high connectivity of the basis\nstates, observables, such as the number of Rydberg-excited atoms,\nare expected to stop oscillating and tend to quasistationary values.\nAccording to the microcanonical ensemble assumption \\cite{AL12,BMFALA13},\nthese values can be accounted for by assuming an effective thermal\nequilibrium-like state for the system which consists of an equiprobable\nstatistical mixture of all allowed states.\n\nThe common probability of all the components in this mixture is therefore\nsimply given by the inverse of the total number $\\mathcal{N}$ of\nallowed states. This number can be determined by summing all numbers\n$\\mathcal{N}(\\nu)$ of allowed configurations with exactly\n$\\nu$ excitations, which, as we show below, can be calculated through\na straightforward combinatorial argument. From $\\mathcal{N}(\\nu)$,\none easily computes the average number of excitations $\\left\\langle \\nu\\right\\rangle $\nand, in the limit of a continuous distribution, one can even deduce\na simple expression of the spatial density of Rydberg excitations.\n\nIn this section, we present our analytical calculations in detail\nand compare our results to numerical Monte Carlo simulations presented\nin \\cite{BMFALA13}.\n\n\n\\subsection{Number of allowed states and average excitation number: combinatorial\nanalysis}\n\nThe goal of this subsection is to compute the average number of Rydberg\nexcitations observed in the thermalized state according to the microcanonical\nensemble assumption. For sake of simplicity, we assume that the atoms\nare located at the nodes of a regular 1D lattice of step $a$. The\ndistance between the $i^{\\text{th}}$ and $j^{\\text{th}}$ atoms is therefore $d(i,j)=a\\left|i-j\\right|$\nwhile the total length of the line is given by $L=\\left(N-1\\right)a$.\nThe quantity $n_{b}\\equiv\\left\\lfloor \\frac{R_{b}}{a}\\right\\rfloor $,\nwhere $\\left\\lfloor \\cdot\\right\\rfloor $ denotes the lower integer\npart, represents the minimal number of ground-state atoms which must\nlie between two Rydberg excitations in an allowed atomic configuration\naccording to the hardcore Rydberg sphere assumption. Finally, we introduce\nthe real parameter $\\Lambda\\equiv\\frac{L}{R_{b}}$. Adding one to its integer part\ngives the maximum number of Rydberg excitations the sample can accommodate for: \n$\\nu_{\\max}=\\left\\lfloor \\Lambda\\right\\rfloor + 1$.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{Fig1_2}%\n\n{\\Large{}{}$\\overbrace{\\bigstar\\bigstar\\bigstar}^{n'_{0}}|\\overbrace{\\phantom{|}}^{n'_{1}}|\\overbrace{\\bigstar\\bigstar\\bigstar\\bigstar}^{n'_{2}}|\\overbrace{\\bigstar\\bigstar\\bigstar}^{n'_{3}}$}\n\\protect\\protect\\caption{\\label{fig:StarsBars}: Description of a configuration of the excitations\nand mapping to the ``Stars and bars'' problem. An excited atom with\nall the atoms on its right closer than $R_b$ correspond to a bar, except\nfor the last bar constituted only by the last excited atom. Each remaining\natom represent a star. The depicted configuration corresponds to $L=16$,\n$N=17$, $2a\\le R_{b}<3a$, $n_{b}=2$, $\\{n_{k}\\}_{k}=\\{3,2,6,3\\}$,\n$\\{n'_{k}\\}_{k}=\\{3,0,4,3\\}$.}\n\\end{figure}\n\n\nTo begin with, we compute the number of allowed states which comprise\na given number of excitations $\\nu$. In such a state, the $\\nu$\nRydberg excitations split the sample into $\\left(\\nu+1\\right)$ groups\nof $n_{k=0,\\dots,\\nu}$ ground-state atoms (see Fig.~\\ref{fig:StarsBars}),\nwith the convention that the zeroth and $\\nu^{\\text{th}}$ groups are on the\nleft and the right of the leftmost and rightmost excited atoms, respectively,\nand allowing $n_{0}$ and $n_{\\nu}$ to be zero. The state indeed\ncorresponds to an allowed configuration if it satisfies the hardcore\nRydberg sphere condition, \\emph{i.e.\\@} $n_{k}\\geq n_{b}$ for $1\\leq k\\leq\\left(\\nu-1\\right)$,\nunder the prescription $\\sum_{k=0}^{\\nu}n_{k}=N-\\nu$: finding the\nnumber of allowed states with $\\nu$ excitations is therefore equivalent\nto computing the number of sets of integers $\\left\\{ n_{k=0,\\dots,\\nu}\\right\\} $\nwhich satisfy the two previous conditions. A slight modification in\nthe formulation of this problem turns it into a standard combinatorial\ncalculation as we shall now show. We first note that an allowed atomic\nconfiguration can be uniquely determined by the alternative set of\nnumbers $\\left\\{ n{}_{k}'\\right\\} $ defined by \n\\begin{align*}\nn_{0}' & \\equiv n_{0}\\\\\nn_{k}' & \\equiv n_{k}-n_{b}\\;\\text{for }1\\leq k\\leq\\nu-1\\\\\nn_{\\nu}' & \\equiv n_{\\nu}\n\\end{align*}\nwhich satisfy the conditions $n_{k}'\\geq0$ and $\\sum_{k=0}^{\\nu}n'_{k}=N-1-(\\nu-1)(n_{b}+1)$.\nThis change of variables suggests to associate the original atomic\narrangement with an abstract linear distribution of $\\left[N-1-(\\nu-1)(n_{b}+1)\\right]$\n``stars'' split by $\\nu$ ``bars'' into $\\left(\\nu+1\\right)$\ngroups labelled by $k=0,\\dots,\\nu$ and respectively comprising $n_{k}'$\nelements. As shown in Fig.~\\ref{fig:StarsBars}, the first $\\left(\\nu-1\\right)$\nbars symbolize the first $\\left(\\nu-1\\right)$ Rydberg excited atoms\nwith their first $n_{b}$ (ground-state) right neighbors, while the\nlast bar represents the last Rydberg excited atom only; stars then\nsimply stand for the remaining ground state atoms. Calculating the\nnumber $\\mathcal{N}(\\nu)$ of such configurations is a\nstandard combinatorial problem whose solution is given by the binomial\ncoefficient \n\\begin{align}\n\\mathcal{N}(\\nu) & =\\binom{N-(\\nu-1)n_{b}}{\\nu}\\nonumber \\\\\n & =\\frac{N^{\\nu}}{\\nu!}\\prod_{i=0}^{\\nu-1}\\left(1-\\frac{(\\nu-1)n_{b}+i}{N}\\right)\\label{eq:Nnudiscrete}\n\\end{align}\nNote that $\\mathcal{N}(\\nu)=0$ when $\\nu-1\\geq\\frac{N}{\\left\\lfloor R_{b}\\right\\rfloor +1}=\\frac{L+1}{\\left\\lceil R_{b}\\right\\rceil }$.\nIn the limit of large $N$ and $R_{b}$, this essentially means that\nwe only have to consider configurations with a number of excitations\nsmaller than $\\nu\\lesssim\\Lambda$. In this limit, when $\\Lambda\\ll R_b,N$,\nwe can approximate equation \\eqref{eq:Nnudiscrete} by \n\\begin{equation}\n\\mathcal{N}(\\nu)=\\frac{N^{\\nu}}{\\nu!}\\left[1-\\frac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}+O(N^{\\nu-1}),\\label{eq:apNnudiscrete}\n\\end{equation}\nwhere $[x]_{+}^{\\nu}=0$ if $x\\le0$ and $[x]_{+}^{\\nu}=x^{\\nu}$\nif $x\\ge0$.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{ExcProba}\\protect\\caption{\\label{fig:ExcProba} Probability to have $\\nu$ excitation considering\nthe microcanonical ensemble as a function of $\\Lambda$ for $N=10^{4}$.\nThe successive peaks correspond to increasing value of $\\nu$. For\nexample, $P(\\nu=2)$ is close to 1 when $\\Lambda$ is between 1 and\n2. }\n\\end{figure}\n\n\nFrom $\\mathcal{N}(\\nu)$, one easily computes the total number of\nallowed configurations $\\mathcal{N}=\\sum_{\\nu}\\mathcal{N}(\\nu)$,\nthe probability to have $\\nu$ excitations in the sample $\\mathcal{P}(\\nu)=\\mathcal{N}(\\nu)\/\\mathcal{N}$\nand the average excitation number $\\langle\\nu\\rangle=\\sum_{\\nu}\\nu\\mathcal{P}(\\nu)$\nas well as its standard deviation $\\left\\langle \\sigma_{\\nu}\\right\\rangle $\nas a function of $\\Lambda$. The family of curves $\\left\\{ \\mathcal{P}(\\nu),\\nu=0,1,\\dots\\right\\} $\nis plotted on Fig.~\\ref{fig:ExcProba} as a function of $\\Lambda$\nfor $N=10^{4}$ and on Fig.~\\ref{fig:Probabilities} for $N=10^{2}$;\n$\\left\\langle \\nu\\right\\rangle $ and $\\left\\langle \\sigma_{\\nu}\\right\\rangle $\nare represented on Fig.~\\ref{fig:AvrgNu} as functions of $\\Lambda$\nfor $N=10^{4}$. Our results show perfect quantitative agreement with\n\\cite{BMFALA13}, as detailed in Appendix~\\ref{App:Comparison}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{AvrgNu}\\protect\\caption{\\label{fig:AvrgNu} Average $\\nu$ and standard deviation in the microcanonical\npredictions as a function of $\\Lambda$ for $N=10^{4}$. The crosses\ncorrespond to the Monte-Carlo results from \\cite[fig 2c]{BMFALA13}.}\n\\end{figure}\n\n\n\n\\subsection{Spatial density of Rydberg excitations}\n\nWe can go further in our analysis and compute how Rydberg excitations\nare distributed along the line in average. Calculations turn to be\nmuch easier in the limit of a homogeneous and continuous atomic distribution,\nof constant linear density $\\delta\\equiv\\frac{1}{a}$ which is a good\napproximation of our model when $R_{b},L\\gg a$.\n\nLet us denote by $\\mathcal{N}(\\nu,l)$ the number of configurations\nwith $\\nu$ excitations on a line of length $l$ with the density\n$\\delta$. We have \n\\begin{align*}\n\\forall l\\ge0,\\;\\mathcal{N}(0,l) & =1\\\\\n\\forall l<0,\\forall\\nu,\\;\\mathcal{N}(\\nu,l) & =0\n\\end{align*}\nWith these notations, $\\mathcal{N}(\\nu)=\\mathcal{N}(\\nu,L)$\nand if $\\nu>0$, the number of configurations with the leftmost excited\natom at position $x$ is given by $\\mathcal{N}(\\nu-1,L-R_{b}-x)$.\nIntegrating over $x$, we get the recurrence relation \n\\begin{equation}\n\\mathcal{N}(\\nu+1,L)=\\int_{0}^{L}dx\\:\\delta\\:\\mathcal{N}(\\nu,L-R_{b}-x),\n\\end{equation}\nand \n\\begin{align}\n\\mathcal{N}(\\nu,L) & =\\frac{\\delta^{\\nu}}{\\nu!}\\left[l-(\\nu-1)R_b\\right]_{+}^{\\nu}\\nonumber \\\\\n & =\\frac{N^{\\nu}}{\\nu!}\\left[1-\\frac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}\\label{eq:Nnu2}\n\\end{align}\nwhich is consistent with Eq.~\\eqref{eq:apNnudiscrete}.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{ExcDistribution}\\protect\\caption{\\label{fig:ExcDistribution} Probability distribution of Rydberg excitations\nalong the chain as a function of $\\Lambda$ and the position for $N=10^{4}$\natoms. This figure quantitatively reproduces the Monte-Carlo simulation \nof \\cite[fig 2a]{BMFALA13}, as detailed in Appendix~\\ref{App:Comparison}.}\n\\end{figure}\n\n\nThe probability density to have the $n^{\\text{th}}$ excited atom out of $\\nu$\nat the position $x$ is: \n\\begin{align}\n  p(\\nu,n,x)= \n    & \\delta\\:\\frac{\\mathcal{N}(n-1,x-R_b)\\times\\mathcal{N}(\\nu-n,L-R_b-x)}{\\mathcal{N}(\\nu)} \n    \\notag\\\\\n  = & \\frac{\\nu!\\left[\\xi-\\tfrac{n-1}{\\Lambda}\\right]_{+}^{n-1}%\n          \\left[1-\\xi-\\tfrac{\\nu-n}{\\Lambda}\\right]_{+}^{\\nu-n}}%\n    {\\left(n-1\\right)!\\left(\\nu-n\\right)!\\left[1-\\tfrac{\\nu-1}{\\Lambda}\\right]_{+}^{\\nu}}\n    \\label{eq:excdensity}\n\\end{align}\nwhere we introduced the normalized dimensionless position $\\xi\\equiv\\frac{x}{L}$.\nNote that it does not depend on $N$; as seen above, however, $N$\nplays a role in the global probability for having $\\nu$ excitations.\nThis allows to plot the spatial distribution of excitation $\\mathcal{P}\\left(x\\right)=\\sum_{\\nu}\\sum_{n\\leq\\nu}p\\left(\\nu,n,x\\right)$,\nas in Fig.~\\ref{fig:ExcDistribution} for $N=10^{4}$, which quantitatively\nagrees with the Monte-Carlo simulations provided in \\cite[fig 2a]{BMFALA13,BMFALA13data}\n(see Appendix~\\ref{App:Comparison}).\nThe spatial distribution of excitations is also plotted in Fig.~\\ref{fig:exc_proba_1,5_}\n(red curve) for $N=100$ and $\\Lambda=1.5$.\n\n\n\\section{Numerical and simplified analytical calculations of the thermalized\nstate \\label{Calculations}}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig5_new}\\protect\\caption{\\label{fig:localisationExc2}: Numerically computed probability distribution\n$\\mathcal{P}_{k}$ of Rydberg excitations along the chain, as a function\nof $\\Lambda$ and the position (see section \\ref{sec:Numerics}) for\n$N=100$ atoms. The blue curve is the predicted position of the excitation\npeak by our simplified analytical treatment.}\n\\end{figure}\n\n\nIn this section, we present the results we obtained through a direct\nnumerical calculation of the thermalized state averages and recover\nsome of their interesting features through approximately diagonalizing\nthe Hamiltonian in a conveniently truncated basis. \n\n\n\\subsection{Numerical calculation of the thermalized state \\label{sec:Numerics}}\n\nOur numerical method consists in time-averaging the observables of\ninterest: if the average is performed on a very long (ideally infinite)\ntime, the obtained average must indeed coincide with the thermalized\nvalue. Due to computational complexity, we restricted our study to\na modest system of $N=100$ atoms equally spaced along the chain.\nEven with this relatively small value of $N$, the dimension $2^{N}$\nof the complete Hilbert space makes the full dynamical treatment intractable.\nWe therefore restricted ourselves to the regime $\\Lambda<2$, \\emph{i.e.\\@}\nthe chain is shorter than two Rydberg radii ($L\\leq2R_{b}$) and the\nmaximum number of excitations distributed along the chain is $2$.\nWe only need to take into account the states allowed by the Rydberg\nblockade whose number is given by Eq.~\\eqref{eq:Nnu2}:\n$\\sum_{\\nu\\leq2}\\mathcal{N}(\\nu)\\simeq\\mathcal{N}(\\nu=2)=\\frac{N^{2}}{2}\\left[1-\\frac{1}{\\Lambda}\\right]_{+}^{2}$.\nWe generate this set of allowed states through an arborescent\nsearch starting from $\\left|\\varnothing\\right\\rangle $ and adding\nallowed excitations.\n\nIn this subspace, we numerically diagonalize the Hamiltonian of\nEq.~\\eqref{Happrox}, yielding the (possibly degenerate) eigenenergies\n$E_{n}$ and the associated eigenvectors \n$\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}} $\nwhere $\\alpha_{n}=1\\dots d_{n}$, $d_{n}$ are the degeneracy index\nof the eigenenergy $E_{n}$. Fig.~\\ref{fig:DiffEnergy} presents the\nnumerical results of the diagonalization of $H$: more explicitly,\nthe red curve shows the absolute value $\\left|E_{n}\\right|$ versus\nthe rank of the corresponding eigenvectors \n$\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}}$,\narranged in increasing order of their eigenenergy; the blue curve\nrepresents the energy difference between two successive eigenvectors\nand therefore allows to check degeneracy. We take as a numerical criterion\nthat two energies coincide when their difference is less than $10^{-13}\\Omega$,\nconsistent with the precision of IEEE 754 floating-point arithmetics.\nOne first observes a wide central area corresponding to the highly\ndegenerate eigenenergy $E\\approx0$; in addition, on both\nsides of the spectrum, there exist two pairs of eigenstates with degenerate\nenergies.\n\nIf the system is initially prepared in \n$\\ket{\\Psi\\left(0\\right)} \\equiv \n \\sum_{n,\\alpha_{n}}c_{n}^{\\alpha_{n}}\\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}}$,\nits state at time $t$ is given by \n$\\ket{\\Psi\\left(t\\right)} =\n  \\sum_{n,\\alpha_{n}}c_{n}^{\\alpha_{n}}e^{-\\mathrm{i}\\frac{E_{n}}{\\hbar}t}\n  \\ket{\\psi_{n}^{\\left(\\alpha_{n}\\right)}} $.\nThe time-averaged probability $\\mathcal{P}_{k}$ to have a Rydberg\nexcitation in site $k$ is therefore given by $\\mathcal{P}_{k}=\\mathrm{Tr}\\left[\\bar{\\rho}\\sigma_{rr}^{\\left(k\\right)}\\right]$\nwhere the average state $\\bar{\\rho}$ is \n\\begin{align}\n\\bar{\\rho} & =\\overline{\\left|\\Psi\\left(t\\right)\\right\\rangle \\left\\langle \\Psi\\left(t\\right)\\right|}\\nonumber \\\\\n & =\\sum_{m,\\alpha_{m}}\\sum_{n,\\beta_{n}}c_{m}^{\\alpha_{m}}\\left(c_{n}^{\\beta_{n}}\\right)^{*}\\left|\\psi_{m}^{\\left(\\alpha_{m}\\right)}\\right\\rangle \\left\\langle \\psi_{n}^{\\left(\\beta_{n}\\right)}\\right|\\times\\overline{e^{-\\mathrm{i}\\frac{E_{m}-E_{n}}{\\hbar}t}}\\nonumber \\\\\n & =\\sum_{n,\\alpha_{n},\\beta_{n}}c_{n}^{\\alpha_{n}}\\left(c_{n}^{\\beta_{n}}\\right)^{*}\\left|\\psi_{n}^{\\left(\\alpha_{n}\\right)}\\right\\rangle \\left\\langle \\psi_{n}^{\\left(\\beta_{n}\\right)}\\right|\\label{EqState}.\n\\end{align}\nWe have used the time average $\\overline{e^{-\\mathrm{i}\\frac{E_{m}-E_{n}}{\\hbar}t}}=\\delta_{mn}$\nto simplify the double sum. \n\nThe probability distribution $\\mathcal{P}_{k}$ is represented on\nFig.~\\ref{fig:localisationExc2} as a function of $\\Lambda$. For\n$\\Lambda\\gtrsim1.2$, two one-atom-wide black lines appear, revealing\na strong localization of Rydberg excitations. In the next subsection,\nwe account for this phenomenon through the approximate diagonalization\nof the Hamiltonian in a conveniently truncated basis.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig6_new}\\protect\\caption{\\label{fig:DiffEnergy} In red: $|E_{n}|$ versus $\\left(n,\\alpha_{n}\\right)$ with $\\Lambda = 1.5$\n(eigenstates are arranged in increasing order of their eigenenergy).\nIn blue: difference between two successive eigenvalues, \\emph{i.e.\\@}\n$\\left|E_{n+1}-E_{n}\\right|$. The dashed line shows the degeneracy\nlimit: below this line, any values can be assumed to be zero, up to numerical artifacts (see text). \n}\n\\end{figure}\n\n\n\n\\subsection{Simplified analytical treatment \\label{SimpleAnalyticalTreatment}}\n\nWe are now looking for a simple description of the system which would\nretain the basic physical features of the model, in particular the\nlocalization effect observed in the previous subsection. To this end,\nwe shall try and restrict the basis of the whole Hilbert space to\nonly the relevant states, \\emph{i.e.\\@} those which get significantly\npopulated during the evolution.\n\nAs a first try, we consider the four-dimensional basis $\\left\\{ \\left|\\varnothing\\right\\rangle ,\\left|\\phi_{1}\\right\\rangle ,\\left|\\phi_{2}\\right\\rangle ,\\left|\\phi_{3}\\right\\rangle \\right\\} $\ndefined by\n\\begin{align}\n  \\left|\\phi_{1}\\right\\rangle  \n  & \\equiv\\frac{H\\left|\\varnothing\\right\\rangle }%\n\t\t{\\left\\Vert H\\left|\\varnothing\\right\\rangle \\right\\Vert } \\label{state1}\\\\\n\t\\left|\\phi_{2}\\right\\rangle  \n\t& \\equiv\\frac{\\Pi_{2}H\\left|\\phi_1\\right>}%\n\t\t\t{\\left\\Vert\\Pi_{2}H\\left|\\phi_1\\right>\\right\\Vert}\n\t\t=\\frac{\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle \n\t\t\t{\\left\\Vert \\Pi_{2}H^{2}\\left|\t\\varnothing\\right\\rangle \\right\\Vert }%\n\t \\label{state2}\\\\\n\t\\left|\\phi_{3}\\right\\rangle  &\n\t \\equiv\\frac{H\\left|\\phi_2\\right>}%\n\t\t\t{\\left\\Vert H\\left|\\phi_2\\right>\\right\\Vert}\n\t\t=\\frac{H\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle }%\n\t\t\t{\\left\\Vert H\\Pi_{2}H^{2}\\left|\\varnothing\\right\\rangle \\right\\Vert  }%\n\t  \\label{state3}\n\\end{align}\nwhere $\\Pi_{2}$ denotes the projector onto the subspace of states\nwith exactly two Rydberg excitations. The diagonalization of $H$\nin this subspace yields four eigenstates $\\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle $\nand eigenenergies $\\pm E_{i=1,2}$, such that $H\\left|\\psi_{i}^{s}\\right\\rangle =s\\times E_{i}\\left|\\psi_{i}^{s}\\right\\rangle $,\nwhose explicit expressions can be found in Appendix~\\ref{App:DiagH2}. We conventionally\nchoose $E_{2}\\geq E_{1}\\geq0$. The eigenenergies $E_{i}$ are plotted\nas functions of $\\Lambda$ on Fig.~\\ref{fig:EvsLambda}. Note that\nfor $\\Lambda>1$, all four eigenenergies $\\pm E_{i=1,2}$ take different\nvalues, there is hence no degeneracy. \n\nSince the eigenstates $\\left|\\psi_{i}^{s}\\right\\rangle $ describe\nconfigurations where excitations are delocalized (see Appendix~\\ref{App:Localization}),\nthe probability $\\mathcal{P}_{k}$ computed from the time-averaged\nstate Eq.~(\\ref{EqState})\n\\begin{align*}\n\\bar{\\rho} & =\\sum_{i=1,2}\\sum_{s=\\pm}\\left|c_{i}^{s}\\right|^{2}\\left|\\psi_{i}^{s}\\right\\rangle \\left\\langle \\psi_{i}^{s}\\right|\n\\end{align*}\nwill not exhibit the observed strong localization effect. \nNote that the four eigenstates\n$\\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle $ contribute to the statistical\nmixture $\\bar{\\rho}$ with the respective weights \n$\\left|c_{i}^{s}\\right|^{2}\\equiv\\left|\\Braket{\\psi_{i}^{s}|\\Psi(0)}\\right|^{2}$\ndetermined by the initial state vector \n$\\ket{\\Psi(0)}=\\ket{\\varnothing}$. \n\nTo correctly account for the observed localization phenomenon, we\nmust therefore slightly extend the basis. To this end, we consider\nthe family of states $\\left\\{ \\ket{\\varphi_{k=1,\\dots,N-n_b-1}^{s=\\pm}}\\right\\} $\ndefined by\n\\begin{equation}\n  \\Ket{\\varphi_{k=1,\\dots,N-R_b-1}^{\\pm}} \\equiv\n    \\frac{\\ket{\\Phi_{k}^{\\left(1\\right)}} \\pm \\ket{\\Phi_{k}^{\\left(2\\right)}}}%\n         {\\sqrt{2}} \\label{state}\n\\end{equation}\nwith\n\\begin{align*}\n\\left|\\Phi_{k}^{\\left(1\\right)}\\right\\rangle  & \\equiv\\frac{\\left|k\\right\\rangle +\\left|N-k\\right\\rangle }{\\sqrt{2}}\\\\\n\\left|\\Phi_{k}^{\\left(2\\right)}\\right\\rangle  & \\equiv\\sum_{l=0}^{N-n_b-k-1}\\frac{\\left|k,N-l\\right\\rangle +\\left|N-k,l\\right\\rangle }{\\sqrt{2\\left(N-n_b-k\\right)}}\n\\end{align*}\nNote that $\\ket{\\Phi_{k}^{\\left(1\\right)}}$ describes\na configuration with exactly one Rydberg excited atom, localized either\nat position $k$ or $\\left(N-k\\right)$; \n$\\ket{\\Phi_{k}^{\\left(2\\right)}} $\ndescribes a configuration with two Rydberg excitations, one being\nlocalized in $k$ or $\\left(N-k\\right)$ while the other is fully\ndelocalized along the chain. The states $\\left|\\varphi_{k}^{s}\\right\\rangle $\nare therefore coherent superpositions of states with either one or\ntwo excitations, one being localized with certainty either at position\n$k$ or $\\left(N-k\\right)$.\n\nThe states $\\left|\\varphi_{k}^{s}\\right\\rangle $ are found to be\napproximately orthogonal to $\\left|\\psi_{k}^{s}\\right\\rangle $, \\emph{i.e.\\@}\n\\[\n\\left\\langle \\varphi_{k}^{s}\\middle|\\psi_{k'}^{s'}\\right\\rangle =O\\left(\\frac{1}{\\sqrt{N}}\\right).\n\\]\nThey are, moreover, only very weakly coupled to $\\left|\\psi_{k}^{s}\\right\\rangle $\nby the Hamiltonian, \\emph{i.e.\\@}\n\\begin{equation}\n\\left\\langle \\varphi_{k}^{s}\\middle| H\\middle|\\psi_{k'}^{s'}\\right\\rangle =O\\left(\\frac{1}{\\sqrt{N}}\\right).\\label{HamDiag}\n\\end{equation}\nFinally, for any $k=1,\\dots,\\left(N-R_b-1\\right)$ and $s=\\pm$, one\nhas \n\\[\n\\left\\langle \\varphi_{k}^{s}\\left|H\\right|\\varphi_{k}^{s}\\right\\rangle =s\\times\\varepsilon_{k}.\n\\]\nThe expression of $\\varepsilon_{k}$ is given in Appendix~\\ref{App:Localization}.\nFig.~\\ref{fig:EvsLambda} shows the quasi-continuum formed by the\ndifferent $\\varepsilon_{k}$'s plotted as functions of $\\Lambda$.\n\nIf the system starts in a superposition of $\\ket{\\Psi_{k}^{s}}$,\n\\emph{i.e.\\@} $\\ket{\\psi\\left(0\\right)} =\\sum_{k,s}c_{k}^{s}\\ket{\\psi_{k}^{s}}$,\none could be tempted, due to Eq.~(\\ref{HamDiag}), to assume that\nnone of the states $\\ket{\\varphi_{k}^{s}}$ ever gets\nsubstantially populated and to discard the whole family \n$\\left\\{\\ket{\\varphi_{k}^{s}} \\right\\} $\nfrom our description. This would actually be incorrect: it may indeed\nhappen that, for a given $k=K$, $\\left|\\varphi_{K}^{s}\\right\\rangle $\nbecomes resonant with $\\left|\\psi_{1}^{s}\\right\\rangle $, \\emph{i.e.\\@}\n$\\varepsilon_{K}=E_{1}$ (as can be checked on Fig.~\\ref{fig:EvsLambda},\nsuch a resonance exists only for $\\Lambda\\geq\\frac{7}{6}$; \nin Appendix~\\ref{App:Localization}, this\nresult is also analytically deduced from the expressions of $E_{1}$\nand $\\varepsilon_{k}$). In that case, though very weak,\nthe coupling term $\\left\\langle \\varphi_{K}^{s}\\left|H\\right|\\psi_{1}^{s}\\right\\rangle $\nstrongly mixes the states $\\left|\\varphi_{K}^{s}\\right\\rangle $ and\n$\\left|\\psi_{1}^{s}\\right\\rangle $ and the two vectors $\\left|\\varphi_{K}^{s=\\pm}\\right\\rangle $\nmust be adjoined to the previous set $\\left\\{ \\left|\\psi_{i=1,2}^{s=\\pm}\\right\\rangle \\right\\} $.\nIn this subspace, the six eigenvectors of the Hamiltonian now read\n\\[\n\\left\\{ \\left|\\chi_{\\pm}^{s=\\pm}\\right\\rangle \\equiv\\frac{\\left|\\psi_{1}^{s}\\right\\rangle \\pm\\left|\\varphi_{K}^{s}\\right\\rangle }{\\sqrt{2}},\\left|\\chi_{0}^{s=\\pm}\\right\\rangle \\equiv\\left|\\psi_{2}^{s}\\right\\rangle \\right\\} \n\\]\nand the energy degeneracy is lifted. An initial state of the form\n$\\ket{\\Psi(0)} =\\sum_{k,s}c_{k}^{s}\\ket{\\psi_{k}^{s}}$\nnow has components on the six new eigenvectors, \\emph{i.e.\\@} \n$\\ket{\\Psi(0)} =\\sum_{r=\\pm,0}\\sum_{s=\\pm}d_{r}^{s}\\ket{\\chi_{r}^{s}}$\nand therefore the time-averaged state \n\\begin{align}\n\\label{rho}\n\\bar{\\rho} & =\\sum_{r=\\pm,0}\\sum_{s=\\pm}\\left|d_{r}^{s}\\right|^{2}\\left|\\chi_{r}^{s}\\right\\rangle \\left\\langle \\chi_{r}^{s}\\right|\n\\end{align}\nnow contains a highly localized component, on the atom at position\n$K$ or $\\left(N-K\\right)$. Accordingly, the probability distribution\n$\\mathcal{P}_{k}$ exhibits a strongly peaked behavior at $k=K,\\left(N-K\\right)$.\nThis localization phenomenon is in good qualitative agreement with\nwhat we observe with the full simulation: in particular, the appearance\nof the localization lines indeed happens when $\\Lambda\\approx\\frac{7}{6}$\n(see Fig.~\\ref{fig:localisationExc2}). This validates the simplified\nanalytical treatment we have just carried out which indeed seems to\nretain the main physical ingredients of the system and its evolution.\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig7}\\protect\\caption{\\label{fig:EvsLambda} $E_{1},E_{2}$ and $\\varepsilon_{k}$ as functions\nof $\\Lambda$, computed by our simplified analytical treatment for\n$N=100$. The values of $\\varepsilon_{k}$ form a quasi-continuum.\nAs discussed in the text, localization peaks arise when a resonance\ntakes place,\\emph{i.e.\\@} when there exists a value $k=K$ such that\n$\\varepsilon_{K}=E_{1}$. This happens for $\\Lambda\\geq\\frac{7}{6}$\nas can be shown analytically (see Appendix~\\ref{App:Localization}) and graphically checked\non the present Figure. }\n\\end{figure}\n\n\n\\section{Discussion}\n\n\\label{sec:Comparison}\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig8-eps-converted-to}\\protect\\caption{\\label{fig:Probabilities} Probability to have $\\nu$ excitations\nas a function of $\\Lambda$, with $N=100$, according to the microcanonical\npredictions (red), our numerical simulation (green) and the analytical\ntreatment (blue). }\n\\end{figure}\n\n\n\\begin{figure}\n\\centering{}\\includegraphics[width=8cm]{Fig9-eps-converted-to} \\protect\\caption{\\label{fig:exc_proba_1,5_} Spatial distribution of excitations at\n$\\Lambda=1.5$, with $N=100$, according to the microcanonical predictions\n(red), our numerical simulation (green) and the analytical model (blue).}\n\\end{figure}\n\nThis section is devoted to the comparison of the results obtained\nabove following the different approaches \nand to discussions on their differences.\n\nFig.~\\ref{fig:Probabilities} displays plots of the probability $\\mathcal{P}(\\nu)$\nto have $\\nu$ Rydberg excitations in the sample, as a function\nof $\\Lambda$ (for $0.9\\leq\\Lambda\\leq2$), calculated according to:\ni) the microcanonical hypothesis (Sec.~\\ref{sec:Thermal}); ii)\nthe full simulation of the system (Sec.~\\ref{sec:Numerics}); iii)\nthe approximate diagonalization of $H$ in a reduced $6$-dimensional\nHilbert space (Sec.~\\ref{SimpleAnalyticalTreatment}). While the schemes\nii) and iii) yield very similar results (as expected), assumption\ni) induces quite different behaviors. The same comparison can be performed\non the spatial probability distribution $\\mathcal{P}_{k}$ which is\ndisplayed on Fig.~\\ref{fig:exc_proba_1,5_}. Again, the shapes obtained\nvia schemes ii) and iii) are in very good qualitative agreement: in\nboth cases, one observes two localization peaks on a ``background\ncurve'', which coincide satisfactorily. (Note that, according to\nour calculations, excitations are more likely to be localized at the\nborders). The spatial probability distribution obtained according\nto assumption i) differs strongly: no excitation localization effect\nis observed and the background curve is far from what is observed\nin the full simulation.\n\nThe discrepancies observed above can be partly explained by the following\n``parity balance property'' established in Appendix~\\ref{App:Parity}: \nfor any eigenstate $\\ket\\psi$ of the Hamiltonian $H$ of nonzero energy, \nthe\nprojections $\\left|\\psi_{\\text{odd}}\\right\\rangle $ and \n$\\left|\\psi_{\\text{even}}\\right\\rangle $\nonto the orthogonal and plementary subspaces $\\mathcal{H}_{\\text{odd}}$\nand $\\mathcal{H}_{\\text{even}}$, respectively spanned by the states with\nan odd and even number of Rydberg excitations, have the same norm,\n\\emph{i.e.\\@} $\\left|\\psi\\right\\rangle =\\left|\\psi_{\\text{odd}}\\right\\rangle +\\left|\\psi_{\\text{even}}\\right\\rangle $\nwith $\\left\\Vert \\left|\\psi_{\\text{odd}}\\right\\rangle \\right\\Vert =\\left\\Vert \\left|\\psi_{\\text{even}}\\right\\rangle \\right\\Vert =\\frac{1}{\\sqrt2}$.\nThis property conflicts directly with the microcanonical predictions\naccording to which the probability to have $\\nu<\\nu_{\\max}$ excitations\nis negligible compared to the probability to have the maximum number\nof excitations. For example, suppose $\\nu_{\\max}=1$, the microcanonical\nensemble implies that $P(\\nu=0)=\\frac{1}{1+N}$ and $P(\\nu=1)=\\frac{N}{1+N}$.\nBy contrast, the parity balance property implies $P(\\nu=0)=P(\\nu=1)=0.5$.\nFurthermore, one can see that in Fig.~\\ref{fig:ExcProba}, each time\none of the probability curve is above $\\frac{1}{2}$, the parity balance\ncondition is therefore impossible to fulfill. In almost all cases,\nthe even\/odd parity balance property and the simple microcanonical\napproach presented in Sec.~\\ref{sec:Thermal} disagree.\n\nThe inaccuracy of the predictions deduced from the microcanonical\nassumption can also be explained by the choice of \n$\\left|\\varnothing\\right\\rangle $\nas initial state: the low connectivity of this state to the rest\nof the Hilbert space constitutes indeed a strongly limiting factor\nto the thermalization process \\cite{OML10}. In particular, the vaccum\nstate being symmetric as well as the Hamiltonian, the system remains\nin a symmetric state during its evolution. The direct application\nof the microcanonical assumption, taking into account all the states\nwhich are allowed by the Rydberg blockade, is therefore incorrect\n: for a proper use of the microcanonical hypothesis, one should actually\ntake this extra symmetry selection rule into consideration and count\nonly the accessible, \\emph{i.e.\\@} symmetric, states. Note that the\nvacuum state is the natural starting point from an experimental perspective\nto study the build-up of excitations and is therefore widely used\n\\cite{AL12,BMFALA13}.\n\nAnother choice of initial state can actually be considered. Starting\nwith a random initial state, Ates et al. \\cite{AGL12} showed that\nin the regime of strong nearest neighbor interaction ($\\Lambda>\\frac{N}{2}$),\nthe dynamics of the system is well described by the microcanonical\nensemble. In the regime studied in the present article, $\\Lambda\\ll N$,\na similar random choice of initial state leads to an essentially ``frozen\nevolution'' as seen by the following dimensionality arguments. From\nEq.~\\eqref{eq:Nnu2}, the number of states containing at most $\\nu_{\\max}-1$\nexcitations is $\\propto N^{\\nu_{\\max}-1}$ and the dimension of the\ngenerated subspace $\\mathcal{H}_{\\nu\\leq\\nu_{\\max}-1}$ is a small\nfraction $O(\\frac{1}{N})$ of the dimension of the total Hilbert space\n$\\mathcal{H}$. As $N$ increases, the Hilbert space is therefore\nessentially composed by states containing $\\nu_{\\max}$ excitations.\nFurthermore, since all eigenvectors of $H$ with non-zero eigenvalue\nfollow the parity balance property, \n\\begin{align*}\n\\text{dim}(\\mathcal{H}\\setminus\\text{ker}(H)) & \\leq2\\min\\left(\\text{dim}(\\mathcal{H}_{\\text{even}}),\\text{dim}(\\mathcal{H}_{\\text{odd}})\\right)\\\\\n & \\sim2\\text{dim}\\left(\\mathcal{H}_{\\nu=\\nu_{\\max}-1}\\right)\\\\\n & \\sim O\\left(N^{\\nu_{\\max}-1}\\right)\n\\end{align*}\nAs a consequence, the Hilbert space is mainly spanned by the states\nin $\\mathrm{ker}\\left(H\\right)$ with $\\nu_{\\max}$ excitations. \nThis can be seen, for example, on Fig.~\\ref{fig:DiffEnergy}. \nTherefore,\nthe projector on $\\text{ker}(H)$ is a ``gentle'' operator \\cite{W99} for the\nensemble of states picked uniformly at random: with high probability,\na state from this ensemble will have a large component on $\\text{ker}(H)$\nand its evolution will essentially be ``frozen'', which contradicts\nthe microcanonical predictions. \n\nConversely, if one chooses the initial\nstate in the $\\mathcal{H}_{\\nu\\leq\\nu_{\\max}-1}$ subspace, the system\nwill not explore $\\text{ker}(H)$: the dimensionality of the actual\nmicrocanonical ensemble is therefore again much less than the number\nof states allowed by the Rydberg blockade.\nNote that this initial state choice is a natural generalization of \n$\\ket{\\varnothing}$ to study the buildup of excitations.\n\n\\section{Conclusion}\n\n\\label{sec:Conclusion}\n\nIn the present article, we studied the dynamics of a 1D-Rydberg ensemble\nin the regime of at most 2 excitations. In the same conditions as\nin \\cite{AL12,BMFALA13}, we tested the validity of the microcanonical\npredictions and found it cannot be used straightforwardly to account\nfor the thermalization process which occurs in this particular regime.\nThough the observed discrepancies can be related to our specific choice\nof initial state and its particular symmetry properties, we also proved,\nby an argument involving the dimension of the kernel of the Hamiltonian,\nthat the same restriction holds for a randomly chosen initial state.\n\nFurther investigations are needed to better understand when and how\nto apply the (micro)canonical predictions. In particular, the results\npresented here all rely on the hardcore sphere assumption. Refining\nthe model and considering the full Rydberg-Rydberg interaction Hamiltonian\nEq.~\\eqref{V_dd} might actually change our conclusions and make\nthe microcanonical assumption more adapted, as shown in \\cite{LOG10}.\nIndeed, in that case, all states become, strictly speaking, allowed,\nthough more or less accessible, and the connectivity accordingly increases\nbetween states of the Hilbert space. Moreover, as suggested by our\ndiscussion, the systematic study of symmetry properties of the system\nat stake, as well as the selection rules they impose, appears to be\na crucial point in the proper application of microcanonical assumption.\n\n\n\\section*{Acknowledgments}\nWe thank Stefano Bettelli for digging up the raw data \\cite{BMFALA13data}.\nMaurice Raoult's highlighting of the simplicity of \n$2\\times2$ matrix diagonalization inspired us for\nthe simplified analytical treatment of section \\ref{SimpleAnalyticalTreatment}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe motivation of this paper is the classical family of \\defn{stack-sortable} permutations introduced by D.~Knuth in his textbook~\\cite[Sect.~2.2.1]{Knuth-TAOCP1} and characterized by the following equivalent conditions for a permutation~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i)]\n\\item \\label{condition:stackSorting}\n$\\pi$ is sent to the identity by the stack sorting~$S$ defined inductively by~$S(\\tau n \\rho) \\eqdef S(\\tau) S(\\rho) n$.\n\\item \\label{condition:patternStack}\n$\\pi$ avoids the pattern $231$ (\\ie there is no~$p < q < r$ such that~$\\pi_r < \\pi_p < \\pi_q$).\n\\item \\label{condition:minimalLinearExtentionBinaryTree}\n$\\pi$ is minimal among all linear extensions of a binary tree on $n$ nodes (seen as a poset, where the nodes are labeled in inorder and the edges are oriented towards the leaves).\n\\item \\label{condition:alignedStack}\nFor $i < j < k$, the inversion set~$\\inv(\\pi) \\eqdef \\set{(\\pi_p, \\pi_q)}{p < q \\text{ and } \\pi_p > \\pi_q}$ of~$\\pi$ contains the inversion $(k,j)$ as soon as it contains the inversion~$(k,i)$.\n\\item \\label{condition:sortableStack}\n$\\pi$ admits a reduced expression of the form~$\\pi = c_{I_1} \\cdots c_{I_p}$ with nested subsets~$I_1 \\supseteq \\dots \\supseteq I_p$, where~$c_{\\{i_1 < \\dots < i_j\\}} \\eqdef s_{i_j} \\cdots s_{i_1}$ is a product of the simple transpositions~$s_i \\eqdef (i \\;\\; i+1)$.\n\\end{enumerate}\nIt follows from~\\eqref{condition:minimalLinearExtentionBinaryTree} that these permutations are counted by the Catalan number~$C_n \\eqdef \\frac{1}{n+1} \\binom{2n}{n}$.\n\nIn his seminal work on lattice congruences~\\cite{Reading-latticeCongruences, Reading-CambrianLattices, Reading-CoxeterSortable}, N.~Reading defined natural counterparts to conditions~\\eqref{condition:minimalLinearExtentionBinaryTree}, \\eqref{condition:alignedStack}, and~\\eqref{condition:sortableStack} above, parametrized by the choice of a Coxeter element~$c$ in a finite Coxeter group~$W$: the minimality in $c$-Cambrian classes, the $c$-alignment, and the $c$-sortability.\n(We skip the general definitions of these conditions here as we stick with the combinatorics of the symmetric group.)\nIn the situation of the symmetric group~$\\fS_n$, we can think of a Coxeter element on~$\\fS_n$ as an orientation of an $(n-1)$-path, or equivalently as a partition of~$\\{2, \\dots, n-1\\}$ into two subsets~$U$ and~$D$.\nThe Cambrian analogues of the conditions~\\eqref{condition:patternStack}, \\eqref{condition:minimalLinearExtentionBinaryTree}, \\eqref{condition:alignedStack} and~\\eqref{condition:sortableStack} above are the following equivalent conditions for a permutation~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i')]\n\\addtocounter{enumi}{1}\n\\item \\label{condition:patternCambrian}\nFor~$i < j < k$, the permutation~$\\pi$ does not contain the subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$.\n\\item \\label{condition:minimalLinearExtentionCambrian}\n$\\pi$ is minimal among all linear extensions of a $c$-Cambrian tree on $n$ nodes. A $c$-Cambrian tree is an oriented tree on~$[n]$ where node~$j$ has one parent if~$j \\notin U$ and two parents if~$j \\in U$, and one child if~$j \\notin D$ and two children if~$j \\in D$, with an additional local condition at each node similar to the binary search tree condition~\\cite{ChatelPilaud}.\n\\item \\label{condition:alignedCambrian}\nFor~$i < j < k$, if~$\\inv(\\pi)$ contains~$(k,i)$, then it also contains $(k,j)$ if~$j \\in U$ and $(j,i)$ if $j \\in D$.\n\\item \\label{condition:sortableCambrian}\n$\\pi$ admits a reduced expression of the form~$\\pi = c_{I_1} \\cdots c_{I_p}$ with nested subsets~$I_1 \\supseteq I_2 \\supseteq I_p$, where~$c_I \\eqdef c_{i_1} \\cdots c_{i_{|I|}}$ denotes the subword of~$c \\eqdef c_1 \\cdots c_{n-1}$ indexed by~$I \\eqdef \\{i_1 < \\dots < i_j\\}$.\n\\end{enumerate}\nIt turns out that for any Coxeter element~$c$, the permutations satisfying these conditions are still counted by the Catalan number~$C_n$.\n\nThese Cambrian combinatorics motivated the introduction of permutree combinatorics~\\cite{PilaudPons-permutrees}.\nPermutrees generalize and interpolate between permutations, binary trees, and binary sequences, and explain the combinatorial, geometric, and algebraic similarities between them.\nThe data is now given by two subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$ that are not anymore required to form a partition of~$\\{2, \\dots, n-1\\}$ (they may intersect and may not cover all the set).\nIt was proved in~\\cite{PilaudPons-permutrees, ChatelPilaudPons} that the conditions~(\\ref{condition:patternCambrian}'), (\\ref{condition:minimalLinearExtentionCambrian}'), and~(\\ref{condition:alignedCambrian}') are still equivalent for a permutation~$\\pi \\in \\fS_n$.\nThe number of permutations satisfying these conditions is called $(U,D)$-factorial-Catalan number and admits recursive formulae interpolating between the formulae for the factorial and for the Catalan number.\n\nThe objective of this paper is to discuss characterizations of permutree minimal permutations in terms of their reduced expressions.\nIn other words, we aim at a condition playing the role of condition~(\\ref{condition:sortableCambrian}') and equivalent to conditions~(\\ref{condition:patternCambrian}'), (\\ref{condition:minimalLinearExtentionCambrian}'), and~(\\ref{condition:alignedCambrian}') for arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nWe first focus on the case where~$U = \\varnothing$ and~$D = \\{j\\}$ for some~${j \\in \\{2, \\dots, n-1\\}}$, or the opposite.\nTo characterize the permutree minimal permutations in terms of their reduced expressions in that situation, we use two automata~$\\automatonU(j)$ and $\\automatonD(j)$ defined inductively as shown in \\cref{fig:automataRecursive}.\nThe induction stops at~$\\automatonU(n)$ and~$\\automatonD(1)$, which are defined by deleting the transitions~$s_n$ and~$s_0$ respectively in \\cref{fig:automataRecursive}.\n\\cref{fig:automataComplete} presents the complete automaton $\\automatonU(j)$ after all recursion is done, and \\cref{fig:TreePartialOrientations} shows the automata~$\\automatonU(2)$, $\\automatonD(2)$, $\\automatonU(3)$, and~$\\automatonD(3)$.\nIn all these pictures the initial state is marked with ``start'', the accepting states are doubly circled, all transitions are labeled with simple transpositions~$s_i$ for~$i \\in [n-1]$, and all missing transitions are loops (we assume the reader familiar with basic automata theory, see for instance~\\cite{HopcroftUllman}).\nOur main tool is the following statement, proved in \\cref{sec:proofPatternAvoidance}.\n\n\\begin{theorem}\\label{thm:patternAvoidance}\nFix $j \\in \\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$.\n\\end{itemize}\n\\end{theorem}\n\nLet us warn the reader on the fact that~$j$ is fixed in \\cref{thm:patternAvoidance}, while $i$ and~$k$ are arbitrary such that~$1 \\le i < j < k \\le n$.\nA priori, we should try all possible reduced expressions of~$\\pi$ to decide if one is accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nHowever, we can show that if~$\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$ and has a descent~$s_\\ell$ distinct from~$s_{j-1}$ (resp.~$s_j$), then it has a reduced expression starting with~$s_\\ell$ and accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nIn other words, there is no loss of generality in starting constructing a reduced expression for~$\\pi$ as long as we stay in the states of the top row of~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nThis yields a simple algorithm to construct a reduced expression accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\nIt also yields natural tree structures on the permutations characterized by \\cref{thm:patternAvoidance}, which can be glanced upon in \\cref{fig:TreePartialOrientations}.\nThese algorithmic and combinatorial consequences of \\cref{thm:patternAvoidance} are explored in \\cref{sec:algorithmicCombinatorialConsequences}.\nMost results of \\cref{sec:proofPatternAvoidance,sec:algorithmicCombinatorialConsequences} are stated with respect to both automata~$\\automatonU(j)$ and $\\automatonD(j)$ but proved only for $\\automatonU(j)$ as all proofs for~$\\automatonD(j)$ are symmetric.\n\n\\begin{figure}\n\t\\centerline{\n\t\t$\\automatonU(j) \\eqdef$\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (q_0) {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (q_1) [below= 1.5cm of q_0] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (q_2) [below= 1.5cm of q_1] {}; \n\t\t\t\\node (q_3) [right= 2.5cm of q_0] {$\\automatonU(j+1)$};\n\t\t\t\\path[->] \n\t\t\t\t(q_0) edge node [swap] {$s_{j-1}$} (q_1)\n\t\t\t\t\t  edge node {$s_j$} (q_3)\n\t\t\t\t(q_1) edge node [swap] {$s_j$} (q_2);\n\t\t\\end{tikzpicture}\n\t\t\\qquad\\qquad\n\t\t$\\automatonD(j) \\eqdef$\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm] \n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (q_0) {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (q_1) [below= 1.5cm of q_0] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (q_2) [below= 1.5cm of q_1] {}; \n\t\t\t\\node (q_3) [right= 2.5 cm of q_0] {$\\automatonD(j-1)$};\n\t\t\t\\path[->] \n\t\t\t\t(q_0) edge node [swap] {$s_{j}$} (q_1)\n\t\t\t\t\t  edge node {$s_{j-1}$} (q_3)\n\t\t\t\t(q_1) edge node [swap] {$s_{j-1}$} (q_2);\n\t\t\\end{tikzpicture}\n\t}\n\t\\caption{The automata $\\automatonU(j)$ (left) and $\\automatonD(j)$ (right) defined recursively.}\n\t\\label{fig:automataRecursive}\n\\end{figure}\n\n\\begin{figure}\n\t\\centerline{\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj-1)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj) [right= 2.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 2.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {};\t  \n\t\t\t%\n\t\t\t\\node (void) [right= 2.5cm of hj+1] {\\dots};\n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hn-2) [right= 2.5cm of void] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (in-2) [below= 1.5cm of hn-2] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dn-2) [below= 1.5cm of in-2] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hn-1) [right= 2.5cm of hn-2] {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (in-1) [below= 1.5cm of hn-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj-1) edge node [swap] {$s_{j-1}$} (ij-1)\n\t\t\t\t\t   edge node {$s_j$} (hj)\n\t\t\t\t(ij-1) edge node [swap] {$s_j$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj) edge node [swap] {$s_{j}$} (ij)\n\t\t\t\t\t edge node {$s_{j+1}$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_{j+1}$} (dj)\t\t\t\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_{j+1}$} (ij+1)\n\t\t\t\t\t   edge  node {$s_{j+2}$} (void)\n\t\t\t\t(ij+1) edge  node [swap] {$s_{j+2}$} (dj+1)\t\t\t  \n\t\t\t\t%\n\t\t\t\t(void) edge node {$s_{n-1}$} (hn-2)\n\t\t\t\t%\n\t\t\t\t(hn-2) edge node [swap] {$s_{n-2}$} (in-2)\n\t\t\t\t\t   edge node {$s_{n-1}$} (hn-1)\n\t\t\t\t(in-2) edge node [swap] {$s_{n-1}$} (dn-2)\n\t\t\t\t%\n\t\t\t\t(hn-1) edge node [swap] {$s_{n-1}$} (in-1);\n\t\t\\end{tikzpicture}\n\t\t}\n\t\\caption{The complete automaton~$\\automatonU(j)$.}\n\t\\label{fig:automataComplete}\n\\end{figure}\n\n\\enlargethispage{-.5cm}\nConsider now arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nIt follows from \\cref{thm:patternAvoidance} that a permutation is minimal in its $(U,D)$-permutree class if and only if it admits a reduced expression accepted by~$\\automatonU(j)$ for each~$j \\in U$ and by~$\\automatonD(j)$ for each~$j \\in D$.\nIn general, the reduced expressions accepted by the automata~$\\automatonU(j)$ for each~$j \\in U$ and by~$\\automatonD(j)$ for each~$j \\in D$ are distinct.\nWe prove however in \\cref{sec:intersectionsAutomata} that there is a reduced expression simultaneously accepted by all these automata when~$U$ and~$D$ are disjoint.\n\n\\begin{theorem}\\label{thm:permutreeMinimal}\nConsider two disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by all automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~${j \\in D}$,\n\t\\item $\\pi$ contains no subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ for any~$i < j < k$.\n\\end{itemize}\n\\end{theorem}\n\n\\cref{thm:permutreeMinimal} implies that given any permutation~$\\pi$ avoiding $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$, we can sort~$\\pi$ while preserving these avoiding conditions.\nThe resulting sorting procedures, that we call \\defn{$(U,D)$-permutree sorting}, are discussed in~\\cref{subsec:permutreeSorting}.\nFor instance, stack sorting is a $(\\{2, \\dots, n-1\\}, \\varnothing)$-permutree sorting.\n\nFinally, in the particular situation when the subsets~$U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$, we actually show that the reduced expression simultaneously accepted by the automata $\\automatonU(j)$ for~${j \\in U}$ and~$\\automatonD(j)$ for~$j \\in D$ is the $c$-sorting word of~$\\pi$ as defined in~\\cite{Reading-CoxeterSortable}.\nThis yields in particular an alternative proof that condition~(\\ref{condition:sortableCambrian}') characterizes the Cambrian minimal permutations.\nThis new perspective on $c$-sortability is explored in \\cref{sec:coxeterSortable}.\n\n\n\\section{Automata for reduced expressions}\\label{sec:proofPatternAvoidance}\n\n\\subsection{Reduced expressions, automata, and subword avoiding}\n\nWe start with properly fixing the few notations needed in this paper.\nWe consider the \\defn{symmetric group}~$\\fS_n$ of permutations of the set~$[n] \\eqdef \\{1,\\dots,n\\}$.\nIt is generated by the \\defn{transpositions}~$s_i \\eqdef (i \\;\\; i+1)$ for $i \\in [n-1]$ which are involutions~$s_i^2 = id$ and satisfy the commutation relations $s_i \\cdot s_j =s_j \\cdot s_i$ if $|i-j|>1$ and the braid relations~$s_i \\cdot s_{i+1} \\cdot s_i = s_{i+1} \\cdot s_i \\cdot s_{i+1}$.\nNote that we multiply permutations as usual, so that the left multiplication by~$s_i$ exchanges the entries with values~$i$ and~$i+1$, while the right multiplication by~$s_i$ exchanges the entries at positions~$i$ and~$i+1$.\nEach permutation~$\\pi$ decomposes into products of transpositions of the form~$\\pi = s_{i_1} \\cdots s_{i_k}$ with~$i_1, \\dots, i_k \\in [n-1]$.\nThe minimal number of transpositions in such a decomposition is the \\defn{length}~$\\ell(\\pi)$ of~$\\pi$ and the decompositions of length~$\\ell(\\pi)$ are the \\defn{reduced expressions} for~$\\pi$.\n\nConsider now the automata $\\automatonU(j)$ and $\\automatonD(j)$ described in the introduction, see \\cref{fig:automataRecursive,fig:automataComplete,fig:TreePartialOrientations}.\nWe call a state \\defn{healthy}, \\defn{ill}, or \\defn{dead} depending on whether it belongs to the top, middle, or bottom row of the automata.\nEach state has $n-1$ possible transitions, one for each $s_i$ for~$i \\in [n-1]$, but we only explicitly indicate the ones between different states.\nThe automata $\\automatonU(j)$ and $\\automatonD(j)$ take as entry a reduced expression~$s_{i_1} \\cdots s_{i_\\ell}$ for a permutation of~$\\fS_n$ and read it from left to right.\nWe start at the initial state (marked with ``start''), and at step~$t$ we follow the transition marked by the letter~$s_{i_t}$ if any, or stay in the current state otherwise.\nAfter~$\\ell$ steps, the reduced expression~$s_{i_1} \\cdots s_{i_\\ell}$ is declared accepted if the current state is accepting (doubly circled, healthy or ill states), and rejected otherwise (dead states).\n\nFor a fixed~$j \\in \\{2, \\dots, n-1\\}$, we say that a permutation~$\\pi$ \\defn{avoids} $jki$ (resp.~$kij$) if for any~$i < j < k$, the word~$jki$ (resp.~$kij$) does not appear as a subword of the one-line notation of~$\\pi$, or said differently if there are no positions~$p < q < r$ such that~$\\pi(r) < \\pi(p) = j < \\pi(q)$ (resp.~$\\pi(q) < \\pi(r) = j < \\pi(p)$).\nWe insist on the fact that while the value~$j$ is fixed, $i$ and~$k$ take all possible values such that~$1 \\le i < j < k \\le n$.\nThis convenient notion here should not be mixed up with the notion of pattern avoidance where~$j$ is not fixed.\nFor instance, a permutation avoids the pattern~$231$ if and only if it avoids $jki$ for all~$j \\in \\{2, \\dots, n-1\\}$.\n\n\\begin{example}\\label{exm:patternAvoidance}\n\tThe permutation~$42135$ avoids~$2ki$, $3ki$, and~$4ki$ (and therefore the pattern~$231$), but contains~$ki3$ (and therefore the pattern~$312$) because its one-line notation contains~$423$. \n\\end{example}\n\n\\subsection{Behavior under left multiplication}\n\nIn the perspective of proving \\cref{thm:patternAvoidance}, we study the two properties ``$\\pi$ admits a reduced expression accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$)'' and ``$\\pi$ avoids $jki$ (resp.~$kij$)''.\nIn this section, we study the behavior of these properties under left multiplication.\nWe treat separately the cases when we multiply by a permutation commuting with both~$s_{j-1}$ and~$s_j$ (\\cref{lem:vincent2}), by $s_{j-1}$ (\\cref{lem:vincent3}), and by~$s_j$ (\\cref{lem:vincent4}).\n\n\\begin{lemma}\\label{lem:vincent2}\nIf two permutations~$\\sigma, \\tau \\in \\fS_n$ are such that $\\sigma([j-1]) = [j-1]$, $\\sigma(j) = j$, and $\\sigma([n]\\ssm [j]) = [n]\\ssm [j]$ and $\\length(\\sigma \\cdot \\tau) = \\length(\\sigma) + \\length(\\tau)$, then:\n\\begin{enumerate}\n\t\\item $\\tau$ admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) if and only if $\\sigma \\cdot \\tau$ admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\tau$ avoids $jki$ (resp.~$kij$) if and only if $\\sigma \\cdot \\tau$ avoids $jki$ (resp.~$kij$).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item The conditions on~$\\sigma$ imply that none of its reduced expressions contain the transpositions $s_{j-1}$ or $s_j$. Therefore, while reading any reduced expression for~$\\sigma$, the automaton~$\\automatonU(j)$ stays in the initial state. The result immediately follows.\n\t\\item Since~$\\sigma$ permutes only values smaller than $j$ between themselves and values greater than $j$ between themselves, we see a subword~$jki$ with~$i < j < k$ in~$\\tau$ if and only if we see a subword~$jk'i'$ with~$i' < j < k'$ in~$\\sigma \\cdot \\tau$, where~$i' = \\sigma(i)$ and~$k' = \\sigma(k)$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent2}\nConsider~$j \\eqdef 4$ and the permutations~$\\sigma \\eqdef 312465 = s_2 \\cdot s_1 \\cdot s_5$, $\\tau_1 \\eqdef 143256 = s_3 \\cdot s_2 \\cdot s_3$, and~$\\tau_2 \\eqdef 124536 = s_3 \\cdot s_4$.\nMultiplying we obtain~$\\sigma\\cdot\\tau_1=342165$ and~$\\sigma\\cdot\\tau_2 = 314625$. Observe that\n\\begin{enumerate}\n\t\\item $\\automatonU(4)$ accepts all reduced expressions of both~$\\tau_1$ and~$\\sigma\\cdot \\tau_1$ on its first ill state, and rejects all reduced expressions of both~$\\tau_2$ and~$\\sigma\\cdot \\tau_2$,\n\t\\item both~$\\tau_1$ and~$\\sigma\\cdot\\tau_1$ avoid~$4ki$, while both~$\\tau_2$ and~$\\sigma\\cdot\\tau_2$ contain~$4ki$.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{lem:vincent3}\nIf a permutation $\\tau \\in \\fS_n$ has a reduced expression starting with $s_{j-1}$ (resp.~$s_j$) and accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$), then\n\\begin{enumerate} \n\t\\item $\\tau$ does not permute $j$ and $j+1$ (resp.~$j-1$ and $j$),\n\t\\item $\\tau$ avoids $jki$ (resp.~$kij$).\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nConsider a reduced expression $w$ starting with $s_{j-1}$ and accepted by~$\\automatonU(j)$. \nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item Since $w$ starts with~$s_{j-1}$, the values~$j-1$ and~$j$ are reversed in~$\\tau$. If~$j$ and~$j+1$ were also reversed in~$\\tau$, we would obtain that~$j-1$ and~$j+1$ are reversed. It follows that $w$ must contain a~$s_j$ at some point after the~$s_{j-1}$. But this would lead to a dead state, contradicting the assumption that~$w$ is accepted.\n\t\\item Let $\\tau = s_{j-1}\\cdot \\rho$. Since any reduced expression of $\\rho$ cannot contain~$s_j$ (as $w$ is be accepted by $\\automatonU(j)$), we have that $\\rho([j]) = [j]$ and $\\rho([n+1] \\ssm [j]) = [n+1] \\ssm [j]$ and find that $\\rho$ contains no subword~$ki$ with $i < j < k$. Therefore, $\\tau$ avoids $jki$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent3}\nConsider~$j \\eqdef 4$ and the permutation~$\\tau \\eqdef 413265$, whose reduced expression~$s_3 \\cdot s_5 \\cdot s_2 \\cdot s_1 \\cdot s_3$ is accepted by~$\\automatonU(4)$.\nObserve that\n\\begin{enumerate}\n\t\\item $\\tau$ indeed does not permute the values~$4$ and~$5$,\n\t\\item $\\tau$ avoids~$4ki$.\n\\end{enumerate}\n\\end{example}\n\n\\begin{lemma}\\label{lem:vincent4}\nIf a permutation~$\\tau \\in \\fS_n$ does not permute $j$ and $j+1$ (resp.~$j-1$ and~$j$), then\n\\begin{enumerate} \n\t\\item $s_j \\cdot \\tau$ (resp.~$s_{j-1} \\cdot \\tau$) admits a reduced expression accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) if and only if $\\tau$ admits a reduced expression accepted by $\\automatonU(j+1)$ (resp.~$\\automatonD(j-1)$),\n\t\\item $s_j \\cdot \\tau$ (resp.~$s_{j-1} \\cdot \\tau$) avoids $jki$ (resp.~$kij$) if and only if $\\tau$ avoids $(j+1)ki$ (resp.~$ki(j-1)$).\n\\end{enumerate}\n\\end{lemma}\n\n\\pagebreak\n\\begin{proof}\nWe deal with the two statements separately:\n\\begin{enumerate} \n\t\\item Suppose that $w$ is a reduced expression for $\\tau$ accepted by $\\automatonU(j+1)$. Since $\\tau$ does not permute $j$ and $j+1$, we know that $s_j \\cdot w$ is a reduced expression for~$s_j \\cdot \\tau$, and it is accepted by~$\\automatonU(j)$ by construction. Conversely assume that $s_j \\cdot \\tau$ admits a reduced expression $w$ accepted by $\\automatonU(j)$. Since $s_j \\cdot \\tau$ permutes $j$ and $j+1$, $w$ must contain a $s_j$ and cannot start by $s_{j-1}$ by \\cref{lem:vincent3}. Due to \\cref{lem:vincent2} we can also assume that $w$ starts with $s_j$. Thus the suffix is a reduced expression for $\\tau$ that is accepted by $\\automatonU(j+1)$.\n\t\\item Observe that since~$j$ and~$j+1$ are reversed in~$s_j \\cdot \\tau$ and not in~$\\tau$, the value $j+1$ cannot serve as~$k$ in a subword~$jki$ of~$s_j \\cdot \\tau$ and the value~$j$ cannot serve as~$i$ in a subword $(j+1)ki$ in~$\\tau$. The result thus immediately follow from the fact that the left multiplication by~$s_j$ only exchanges the values $j$ and $j+1$.\n\t\\qedhere\n\\end{enumerate}\n\\end{proof}\n\n\\begin{example}\\label{exm:lemVincent4}\nConsider~$j \\eqdef 4$ and the permutations~$\\tau_1 \\eqdef 142536$ and~$\\tau_2 \\eqdef 142563$ that do not permute~$4$ and~$5$. \nMultiplying we obtain~$s_4 \\cdot \\tau_1 = 152436$ and~$s_4 \\cdot \\tau_2 = 152463$. Observe that\n\\begin{enumerate}\n\t\\item the reduced expression~$s_4 \\cdot s_3 \\cdot s_4 \\cdot s_2$ of~$s_4 \\cdot \\tau_1$ is accepted by~$\\automatonU(4)$ and the reduced expression~$s_3 \\cdot s_4 \\cdot s_2$ of~$\\tau_1$ is accepted by~$\\automatonU(5)$, while all reduced expressions of~$s_4 \\cdot \\tau_2$ are rejected by~$\\automatonU(4)$ and all reduced expressions of~$\\tau_2$ are rejected by~$\\automatonU(5)$,\n\t\\item $s_4 \\cdot \\tau_1$ avoids~$4ki$ and $\\tau_1$ avoids~$5ki$, while $s_4 \\cdot \\tau_2$ contains~$463$ and $\\tau_2$ contains~$463$.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Proof of \\cref{thm:patternAvoidance}}\n\nWith these lemmas in hand, we are now ready to show \\cref{thm:patternAvoidance} that we repeat here for convenience.\n\n\\newtheorem*{thm:patternAvoidance}{\\cref{thm:patternAvoidance}}\n\\begin{thm:patternAvoidance}\nFix $j \\in \\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonU(j)$ (resp.~$\\automatonD(j)$),\n\t\\item $\\pi$ contains no subword $jki$ (resp.~$kij$) with~$i < j < k$.\n\\end{itemize}\n\\end{thm:patternAvoidance}\n\n\\begin{proof}[Proof of \\cref{thm:patternAvoidance}]\nWe work by induction on the length of the permutations.\nAssume that a permutation $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$. Let $s_i$ be the first letter of this reduced expression and let $\\tau$ be such that $\\pi = s_i \\cdot \\tau$. We distinguish three cases:\\begin{itemize}\n\t\\item if $i = j-1$, then $\\pi$ avoids $jki$ by \\cref{lem:vincent3}\\,(2).\n\t\\item if $i = j$, then $\\tau$ admits a reduced expression accepted by $\\automatonU(j+1)$ by \\cref{lem:vincent4}\\,(1). We obtain by induction that $\\tau$ avoids $(j+1)ki$. Thus $\\pi = s_j \\cdot \\tau$ avoids $jki$ by \\cref{lem:vincent4}\\,(2).\n\t\\item otherwise, $\\tau$ admits a reduced expression accepted by $\\automatonU(j)$ by \\cref{lem:vincent2}\\,(1), so that $\\tau$ avoids $jki$ by induction. Thus $\\pi = s_i \\cdot \\tau$ avoids $jki$ by \\cref{lem:vincent2}\\,(2).\n\\end{itemize}\nIn all three cases, we proved that $\\pi$ avoids $jki$.\n\nAssume now that a permutation $\\pi$ avoids $jki$. Here, we have to be careful because not all reduced expressions for $\\pi$ will be accepted by $\\automatonU(j)$ \\apriori. So we have to construct a good reduced expression for~$\\pi$. We distinguish two cases:\n\\begin{itemize}\n\t\\item Assume first that there is $m > j$ such that $\\pi$ reverses $j$ and $m$, and pick $m$ minimal for this property. It follows that $\\pi$ reverses $\\ell$ and $m$ for all $\\ell$ in $\\{j,\\dots,m-1\\}$. In other words, $\\pi$ admits a reduced expression starting by the cyclic permutation $(j, j+1, ..., m) = s_{m-1} \\cdot s_{m-2} \\cdots s_{j+1} \\cdot s_j$. Define $\\sigma = s_{m-1} \\cdot s_{m-2}\\cdots s_{j+1}$ and $\\tau$ such that $\\pi = \\sigma \\cdot s_j \\cdot \\tau$ and so that this expression is reduced. By Lemmas \\ref{lem:vincent2}\\,(2) and \\ref{lem:vincent4}\\,(2), $\\tau$ avoids $(j+1)ki$. By induction, we obtain that it admits a reduced expression accepted by $\\automatonU(j+1)$. By Lemmas \\ref{lem:vincent2}\\,(1) and \\ref{lem:vincent4}\\,(1), we conclude that $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$.\n\t\\item Assume now that $j$ appears before all $m > j$ in $\\pi$. Consider any reduced expression for $\\pi$. If this expression is accepted by $\\automatonU(j)$, we are done. Otherwise, it first uses $s_{j-1}$ (otherwise, $j$ and some $m > j$ would be exchanged) and then $s_j$. Call $i$ and $k$ the two elements that are exchanged when the reduced expression first uses $s_j$. We have $i < j < k$ and $jki$ in $\\pi$ (because $j$ and $k$ are not exchanged in $\\pi$, and $i$ and $k$ are already exchanged so they will remain exchanged in $\\pi$), a contradiction.\n\\end{itemize}\nIn both cases, we proved that $\\pi$ admits a reduced expression accepted by $\\automatonU(j)$.\n\\end{proof}\n\n\n\\pagebreak\n\\section{Structure of accepted reduced expressions}\\label{sec:algorithmicCombinatorialConsequences}\n\nIn this section, we explore some additional properties of the set of reduced expressions accepted by the automata~$\\automatonU(j)$ and~$\\automatonD(j)$ and derive relevant algorithmic and combinatorial consequences.\n\n\\subsection{The set of accepted reduced expressions}\\label{subsec:principles}\n\nObserve that a given permutation~$\\pi$ may admit both accepted and rejected reduced expressions.\nFor instance, the (non-simple) transposition~${(j-1 \\;\\; j+1)}$ has reduced expressions~$s_j \\cdot s_{j-1} \\cdot s_j$ accepted by~$\\automatonU(j)$ and~$s_{j-1} \\cdot s_j \\cdot s_{j-1}$ rejected by~$\\automatonU(j)$.\nHowever, \\cref{prop:prefixesReducedExpressions,prop:algorithm,prop:sameStateAcceptedReducedExpressions} below show that the set of accepted reduced expressions satisfies the following three principles:\n\\begin{itemize}\n\t\\item \\textbf{Who can do more can do less!} --- The set of accepted reduced words is closed by~prefix.\n\t\\item \\textbf{When health goes, everything goes!} --- If $\\pi$ admits an accepted reduced expression, then $\\pi$ admits an accepted reduced expression starting with any descent that remains in the healthy states.\n\t\\item \\textbf{All roads lead to Rome!} --- All accepted reduced expressions for~$\\pi$ end at the same~state.\n\\end{itemize}\n\n\\begin{proposition}\\label{prop:prefixesReducedExpressions}\nThe set of reduced words accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$) is closed by prefix.\n\\end{proposition}\n\n\\begin{proof}\nThis immediately follows from the fact that the set of reduced words is closed by prefix, and that the set of accepting states of~$\\automatonU(j)$ is connected and contains the initial state.\n\\end{proof}\n\n\\begin{proposition}\\label{prop:algorithm}\nLet $\\ell \\in [n-1]$ distinct from $j-1$ (resp.~$j$).\nA permutation $\\pi \\in \\fS_n$ that avoids $jki$ (resp.~$kij$) and reverses $\\ell$ and $\\ell+1$ admits a reduced expression starting with $s_\\ell$ and accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$).\n\\end{proposition}\n\n\\begin{proof}\nSince $\\pi$ reverses $\\ell$ and $\\ell+1$, it admits a reduced expression of the form~$\\pi = s_\\ell \\cdot \\tau$. Now consider two cases depending on the value of $\\ell$:\n\\begin{itemize}\n\t\\item If $\\ell = j$, then $\\tau$ avoids $(j+1)ki$ by \\cref{lem:vincent4}\\,(2), thus $\\tau$ has a reduced expression accepted by $\\automatonU(j+1)$ by \\cref{thm:patternAvoidance}, and we conclude by \\cref{lem:vincent4}\\,(1).\n\t\\item Otherwise, $\\ell$ is neither $j-1$ nor $j$, so that $\\tau$ avoids $jki$ by \\cref{lem:vincent2}\\,(2), thus $\\tau$ has a reduced expression accepted by $\\automatonU(j)$ by \\cref{thm:patternAvoidance}, and we conclude by \\cref{lem:vincent2}\\,(1).\n\t\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressions}\nGiven a permutation $\\pi \\in \\fS_n$, all the reduced expressions for~$\\pi$ accepted by $\\automatonU(j)$ (resp.~$\\automatonD(j)$) end at the same state.\n\\end{proposition}\n\nTo prove \\cref{prop:sameStateAcceptedReducedExpressions}, it would be enough to check that any two reduced words accepted by~$\\automatonU(j)$ that differ by a single commutation or a single braid relation indeed end at the same state.\nHowever, we prefer to prove instead the following stronger but more technical version of \\cref{prop:sameStateAcceptedReducedExpressions}.\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressionsRefined}\nFor a permutation~$\\pi \\in \\fS_n$, let~$\\ninv^j(\\pi) = |\\set{(j,i)}{i < j \\text{ and } \\pi^{-1}(i) > \\pi^{-1}(j)}|$ and~$\\ninv_j(\\pi) = |\\set{(k,j)}{j < k \\text{ and } \\pi^{-1}(j) > \\pi^{-1}(k)}|$.\n\\begin{enumerate}[(i)]\n\t\\item if $\\ninv^j(\\pi) = 0$, then all reduced expressions for~$\\pi$ end at the same healthy state of~$\\automatonU(j)$,\n\t\\item if~$\\ninv_j(\\pi) = 0$, then all reduced expressions for~$\\pi$ end at the same state of~$\\automatonU(j)$, which might be healthy if $\\pi$ avoids~$ji$, ill if~$\\pi$ contains~$ji$ but avoids~$jki$, or dead if~$\\pi$ contains~$jki$,\n\t\\item if $\\ninv^j(\\pi) \\ne 0 \\ne \\ninv_j(\\pi)$, all accepted reduced expressions for~$\\pi$ end at the same ill state of~$\\automatonU(j)$ while the rejected reduced expressions may end at distinct dead states of~$\\automatonU(j)$.\n\\end{enumerate}\nMoreover, all reduced expressions for~$\\pi$ accepted by~$\\automatonU(j)$ end in the $(\\ninv_j(\\pi)+1)$st column of~$\\automatonU(j)$.\nA similar statement holds for~$\\automatonD(j)$ by exchanging $\\ninv_j(\\pi)$ and~$\\ninv^j(\\pi)$.\n\\end{proposition}\n\n\\begin{proof}\nThe proof works by induction on the length of~$\\pi$.\nConsider an arbitrary reduced expression~$w$ for~$\\pi$, starting with a transposition~$s_\\ell$, and write~$w = s_\\ell \\cdot w'$ and~$\\pi = s_\\ell \\cdot \\tau$.\nObserve~that:\n\\begin{itemize}\n\t\\item if $\\ell \\notin \\{j-1, j\\}$, then~$s_\\ell$ loops in~$\\automatonU(j)$, $\\ninv^j(\\pi) = \\ninv^j(\\tau)$ and $\\ninv_j(\\pi) = \\ninv_j(\\tau)$,\n\t\\item if~$\\ell = j$, then~$s_j$ goes to~$\\automatonU(j+1)$, $\\ninv^j(\\pi) = \\ninv^{j+1}(\\tau)$ and~${\\ninv_j(\\pi) = \\ninv_{j+1}(\\tau) + 1}$,\n\t\\item if~$\\ell = j-1$, then~$s_{j-1}$ goes to the first ill state of~$\\automatonU(j)$, $\\ninv^j(\\pi) = \\ninv^{j+1}(\\tau) + 1$ and~$\\ninv_j(\\pi) = \\ninv_{j+1}(\\tau)$.\n\\end{itemize}\nBy induction, we obtain that the reduced expression~$w'$ for~$\\tau$ ends as predicted in the statement.\nThe previous observations ensure that the reduced expression~$w$ for~$\\pi$ also does.\n\\end{proof}\n\n\\begin{example}\\label{exm:sameStateAcceptedReducedExpressionsRefined}\nWe present an example of each case:\n\\begin{enumerate}[(i)]\n\t\\item For~$\\pi \\eqdef 4312$, we have that~$\\ninv^2(\\pi)=0$ and all of its~$5$ reduced expressions end at the third healthy state of $\\automatonU(2)$.\n\t\\item For~$\\pi \\eqdef 32145$ (resp.~$\\pi \\eqdef 43215$, resp.~$\\pi \\eqdef 43251$), we have~$\\ninv_4(\\pi) = 0$ and all its~$2$ (resp.~$16$, resp.~$35$) reduced expressions end at the first healthy (resp.~ill, resp.~dead) state~of~$\\automatonU(4)$.\n\t\\item For~$\\pi \\eqdef 4321$, we have~$\\ninv^2(\\pi) = |\\{(2,1)\\}| = 1$ and $\\ninv_2 = |\\{(3,2),(4,2)\\}| = 2$. Among the~$16$ reduced expressions of~$\\pi$, the automaton~$\\automatonU(2)$ accepts~$7$ at its third ill state, rejects~$7$ at its first dead state, and rejects the other~$2$ at its second dead state.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Finding accepted reduced expressions}\n\n\\cref{prop:algorithm} has a strong algorithmic consequence.\nImagine we want to test whether a permutation~$\\pi \\in \\fS_n$ is minimal in its permutree class for~$U = \\{j\\}$ and~$D = \\varnothing$.\nOf course, the quickest way is to check for all~$i < j < k$ whether~$\\pi$ contains the subword~$jki$.\nBut since this interpretation will be lost beyond type~$A$, let us impose the use of reduced expressions for~$\\pi$ to make this test.\nWhile it would be \\apriori{} necessary to check all reduced expressions on the automaton~$\\automatonU(j)$, \\cref{prop:algorithm} enables us to construct without loss of generality a candidate reduced expression for~$\\pi$ and we will just neeed to check that this one is accepted by~$\\automatonU(j)$.\nSomewhat dually, one can also construct a reduced word accepted by~$\\automatonU(j)$ that is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$.\nThis is done in the following algorithm, that we call \\defn{$(\\{j\\}, \\varnothing)$-permutree sorting}.\nThe reader is invited to write down the symmetric $(\\varnothing, \\{j\\})$-permutree sorting.\nWe will discuss further permutree sorting in \\cref{subsec:permutreeSorting}.\n\n\\bigskip\n\\IncMargin{1em}\n\\SetKwFor{Repeat}{repeat}{}{}\n\\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{}\n\\DontPrintSemicolon\n\\begin{algorithm}[H]\n\t\\KwData{a permutation $\\pi \\in \\fS_n$ and an integer~$j \\in [n]$}\n\t\\KwResult{a reduced word accepted by~$\\automatonU(j)$, candidate reduced expression for~$\\pi$}\n\t$w \\eqdef \\varepsilon$ \\;\n\t\\Repeat{}{\n\t\t\\If{$\\exists \\; \\ell \\ne j-1$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$}{\n\t\t\t$\\pi \\eqdef s_\\ell \\cdot \\pi$, \\quad $w \\eqdef w \\cdot s_\\ell$ \\;\n\t\t\t\\lIf{$\\ell = j$}{\n\t\t\t\t$j \\eqdef j+1$\n\t\t\t}\n\t\t}\n\t}\n\t\\If{$j-1$ and $j$ are reversed in~$\\pi$}{\n\t\t$\\pi \\eqdef s_{j-1} \\cdot \\pi$, \\quad $w \\eqdef w \\cdot s_{j-1}$ \\;\n\t\t$w \\eqdef w \\cdot w' \\cdot w''$ where~$w'$ sorts~$\\pi_{[j]}$ and $w''$ sorts $\\pi_{[n] \\ssm [j]}$ \\;\n\t}\n\t\\Return $w$\n\t\\caption{$(\\{j\\}, \\varnothing)$-permutree sorting}\n\t\\label{algo:shortcutsUj}\n\\end{algorithm}\n\\bigskip\n\n\\begin{example}\\label{exm:algo1}\n\tLet us present the $({2},\\varnothing)$-permutree sorting algorithm in action for the permutations~$\\pi_1 \\eqdef 3421$ and~$\\pi_2 \\eqdef 4231$. The steps of the algorithm are presented in \\cref{tab:algo1}. Each row contains the states of the permutation~$\\pi$ and of the word~$w$ and the current values of~$j$ and~$\\ell$ in use at each step. Notice that for~$\\pi_1 \\eqdef 3421$ the algorithm ends with the identity, which coincides with the fact that~$\\pi_1 \\eqdef 3421$ avoids~$2ki$. In contrast, for~$\\pi_2 \\eqdef 4231$ the algorithm ends with the permutation~$1243$, meaning that~$\\pi_2$ is not~$(\\{2\\},\\varnothing)$-sortable, which coincides with the fact that~$\\pi_2 \\eqdef 4231$ contains~$2ki$. \n\n\t\\begin{table}[h!]\n\t\t\\begin{tabular}[t]{l|l|c|c}\n\t\t\t$\\pi_1$ & $w_1$ & $j_1$ & $\\ell_1$ \\\\\n\t\t\t\\hline\n\t\t\t$3421$ & $\\varepsilon$ & $2$ & $2$ \\\\\n\t\t\t$2431$ & $s_2$ & $3$ & $1$ \\\\\n\t\t\t$1432$ & $s_2 \\cdot s_1$ & $3$ & $3$ \\\\\n\t\t\t$1342$ & $s_2 \\cdot s_1 \\cdot s_3$ & $4$ & $2$ \\\\\n\t\t\t$1243$ & $s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2$ & $4$ & $3$ \\\\\n\t\t\t$1234$ & $s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2 \\cdot s_3$ & $4$ &\n\t\t\\end{tabular}\n\t\t\\qquad\\qquad\n\t\t\\begin{tabular}[t]{l|l|c|c}\n\t\t\t$\\pi_2$ & $w_2$ & $j_2$ & $\\ell_2$ \\\\\n\t\t\t\\hline\n\t\t\t$4231$ & $\\varepsilon$ & $2$ & $3$ \\\\\n\t\t\t$3241$ & $s_3$ & $2$ & $2$ \\\\\n\t\t\t$2341$ & $s_3 \\cdot s_2$ & $3$ & $1$ \\\\\n\t\t\t$1342$ & $s_3 \\cdot s_2 \\cdot s_1$ & $3$ & $2$ \\\\\n\t\t\t$1243$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2$ & $3$ &\n\t\t\\end{tabular}\n\t\\caption{The~$(\\{2\\},\\varnothing)$-permutree sorting of~$\\pi_1 \\eqdef 3421$ and~$\\pi_2 \\eqdef 4231$.}\n\t\\label{tab:algo1}\n\t\\end{table}\n\\end{example}               \n\n\\begin{corollary}\\label{coro:algorithm}\nFor any permutation~$\\pi$ and~$j \\in \\{2, \\dots, n-1\\}$, \\cref{algo:shortcutsUj} returns a reduced~word~$w$ accepted by $\\automatonU(j)$ with the property that~$w$ is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$.\n\\end{corollary}\n\n\\begin{proof}\nThis algorithm constructs a candidate reduced word for $\\pi$ while following the automaton~$\\automatonU(j)$ and prioritizing healthy states over ill states.\nLines 2 to 5 start by all possible transitions~$s_\\ell$ that remain in healthy states, updating~$j$ to~$j+1$ when~$\\ell = j$ according to \\cref{lem:vincent4} (if condition at line 5).\nWhen we have exhausted all these transitions, if we need to go to an ill state (if condition at line 6) applying~$s_{j-1}$, then we are not anymore allowed to use~$s_j$ and we obtain a candidate reduced word by sorting independently the first $j$ positions of~$\\pi$ with a reduced expression in~$\\{s_1, \\dots, s_{j-1}\\}^*$ and the last $[n] \\ssm [j]$ positions of~$\\pi$ with a reduced expression in~$\\{s_{j+1}, \\dots, s_{n-1}\\}^*$.\nThe resulting reduced word~$w$ is clearly accepted by~$\\automatonU(j)$ because we never allow the transition from an ill state to the corresponding dead state.\nIf~$w$ is a reduced expression for~$\\pi$, then $\\pi$ avoids~$jki$ by \\cref{thm:patternAvoidance}.\nConversely, if~$\\pi$ avoids~$jki$, then $w$ must be a reduced expression for~$\\pi$ since the choice to start with~$s_\\ell$ is valid in lines 2 to 5 by \\cref{prop:algorithm} and forced in lines 6 to 8 (since all reduced expressions of~$\\pi$ then start by~$s_\\ell$).\n\\end{proof}\n\n\\begin{remark}\nWe really wrote \\cref{algo:shortcutsUj} as a sorting algorithm.\nIt first tries to sort the permutation~$\\pi \\in \\fS_n$ while avoiding to swap~$j-1$ and~$j$ for a certain token~$j$ (and changing the token when swapping~$j$ and~$j+1$).\nOnce it is forced to swap~$j-1$ and~$j$, it tries to sort the permutation~$\\pi$ while avoiding to swap any value of~$[j]$ with a value of~$[n] \\ssm [j]$.\nIf we were only interested in deciding whether the permutation~$\\pi$ is $(\\{j\\}, \\varnothing)$-sortable, then we could stop and accept the permutation as soon as we reach $j = n$, and we could just check at line 8 of the algorithm whether~$\\pi([j]) = [j]$ and~$\\pi([n] \\ssm [j]) = [n] \\ssm [j]$.\n\\end{remark}\n\n\\subsection{Generating trees on accepted reduced expressions}\\label{subsec:trees}\n\n\\cref{prop:prefixesReducedExpressions,prop:sameStateAcceptedReducedExpressions} also have a relevant consequence, more combinatorial this time.\nNamely, they naturally define generating trees for the $(\\{j\\}, \\varnothing)$-permutree minimal permutations, following certain special reduced expressions for them.\nTo construct these trees, pick an arbitrary priority order~$\\prec$ on~$\\{s_1, \\dots, s_{n-1}\\}$.\nFor a $(\\{j\\}, \\varnothing)$-permutree minimal permutation~$\\pi \\in \\fS_n$, denote by~$\\pi(\\{j\\}, \\varnothing, \\prec)$ the $\\prec$-lexicographic minimal reduced expression for~$\\pi$ that is accepted by~$\\automatonU(j)$.\nDenote by~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ the set of reduced words of the form~$\\pi(\\{j\\}, \\varnothing, \\prec)$ for all $(\\{j\\}, \\varnothing)$-permutree minimal permutations~$\\pi \\in \\fS_n$.\nThe following statement is an analogue of \\cref{prop:prefixesReducedExpressions}.\n\n\\begin{proposition}\\label{prop:generatingTrees}\nThe set $\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ is closed by prefix.\n\\end{proposition}\n\n\\begin{proof}\nConsider a reduced expression~$w = u \\cdot v$ where~$u$ is not in~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$.\nIf~$u$ is not accepted by~$\\automatonU(j)$, neither is~$w$ by \\cref{prop:prefixesReducedExpressions}.\nOtherwise, there exists a reduced expression~$u'$ representing the same permutation as~$u$, accepted by~$\\automatonU(j)$ and $\\prec$-lexicographic smaller than~$u$.\nBy \\cref{prop:sameStateAcceptedReducedExpressions}, the reduced expressions~$u$ and~$u'$ end at the same state of~$\\automatonU(j)$.\nTherefore, if~$w = u \\cdot v$ is accepted by~$\\automatonU(j)$, so is~$u' \\cdot v$.\nSince~$u' \\cdot v$ is $\\prec$-lexicographically smaller than~$u \\cdot v$ and represents the same permutation, this ensures that~$w$ is not in~$\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$.\n\\end{proof}\n\n\\cref{prop:generatingTrees} yields a natural generating tree for $\\lexmin(n, \\{j\\}, \\varnothing, \\prec)$ where the parent of a reduced word~$w$ is obtained by deleting its last letter.\nReplacing each reduced expression by the corresponding permutation, this provides a generating tree for the $(\\{j\\}, \\varnothing)$-permutree minimal permutations of~$\\fS_n$.\nOf course there is a similar generating tree for the $(\\varnothing, \\{j\\})$-permutree minimal permutations of~$\\fS_n$.\n\\cref{fig:TreePartialOrientations} presents these generating trees for~$n = 4$ and~$j = 2, 3$, with the priority order~$s_1 \\prec s_2 \\prec s_3$.\nIt is natural to draw these trees on top of the Hasse diagram of the right weak order on permutations, defined by inclusion of inversion sets.\nIn other words, the cover relations in weak order correspond to the swap of the values at two consecutive positions in a permutation, \\ie to a right multiplication by a simple transposition.\nThe edges of the trees corresponding to the right multiplications by $s_1$, $s_2$ and $s_3$ are colored by blue, red, and green respectively.\n\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\t\\begin{tabular}{cccc}\n\t\t$\\automatonU(2)$\n\t\t&\n\t\t$\\automatonD(2)$\n\t\t&\n\t\t$\\automatonU(3)$\n\t\t&\n\t\t$\\automatonD(3)$\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n    \t\t\\node[blue](P1234) at (0,0){1234};\n    \t\t%\n    \t\t\\node[blue](P2134) at (-1,1.5){2134};\n    \t\t\\node[blue](P1324) at ( 0,1.5){1324};\n    \t\t\\node[blue](P1243) at ( 1,1.5){1243};\n    \t\t%\n    \t\t\\node(P2314) at (-2,3){2314};\n    \t\t\\node[blue](P3124) at (-1,3){3124};\n    \t\t\\node[blue](P2143) at ( 0,3){2143};\n    \t\t\\node[blue](P1342) at ( 1,3){1342};\n    \t\t\\node[blue](P1423) at ( 2,3){1423};\n    \t\t%\n    \t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n    \t\t\\node(P2341) at (-1.5,4.5){2341};\n    \t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n    \t\t\\node(P2413) at ( 0.5,4.5){2413};\n    \t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n    \t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n    \t\t%\n    \t\t\\node(P3241) at (-2,6){3241};\n    \t\t\\node(P2431) at (-1,6){2431};\n    \t\t\\node[blue](P3412) at ( 0,6){3412};\n    \t\t\\node[blue](P4213) at ( 1,6){4213};\n    \t\t\\node[blue](P4132) at ( 2,6){4132};\n    \t\t%\n    \t\t\\node[blue](P3421) at (-1,7.5){3421};\n    \t\t\\node(P4231) at ( 0,7.5){4231};\n    \t\t\\node[blue](P4312) at ( 1,7.5){4312};\n    \t\t%\n    \t\t\\node[blue](P4321) at (0,9){4321};\n    \t\t%\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n    \t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n    \t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n    \t\t%\n    \t\t\\draw(P2134) -- (P2314);\n    \t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n    \t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n    \t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n    \t\t\\draw(P1243) -- (P2143);\n    \t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n    \t\t%\n    \t\t\\draw(P2314) -- (P3214);\n    \t\t\\draw(P2314) -- (P2341);\n    \t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n    \t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n    \t\t\\draw(P2143) -- (P2413);\n    \t\t\\draw(P1342) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n    \t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n    \t\t\\draw(P1423) -- (P1432);\n    \t\t%\n    \t\t\\draw(P3214) -- (P3241);\n    \t\t\\draw(P2341) -- (P3241);\n    \t\t\\draw(P2341) -- (P2431);\n    \t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n    \t\t\\draw(P2413) -- (P4213);\n    \t\t\\draw(P2413) -- (P2431);\n    \t\t\\draw[line width=0.5mm,red](P4123) -- (P4213);\n    \t\t\\draw(P4123) -- (P4132);\n    \t\t\\draw[line width=0.5mm,blue](P1432) -- (P4132);\n    \t\t%\n    \t\t\\draw(P3241) -- (P3421);\n    \t\t\\draw(P2431) -- (P4231);\n    \t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n    \t\t\\draw[line width=0.5mm,green](P3412) -- (P3421);\n    \t\t\\draw(P4213) -- (P4231);\n    \t\t\\draw(P4132) -- (P4312);\n    \t\t%\n    \t\t\\draw(P3421) -- (P4321);\n    \t\t\\draw(P4231) -- (P4321);\n    \t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\t\n    \t\\end{tikzpicture}\n    \t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n    \t\t\\node[blue](P1234) at (0,0){1234};\n    \t\t%\n    \t\t\\node[blue](P2134) at (-1,1.5){2134};\n    \t\t\\node[blue](P1324) at ( 0,1.5){1324};\n    \t\t\\node[blue](P1243) at ( 1,1.5){1243};\n    \t\t%\n    \t\t\\node[blue](P2314) at (-2,3){2314};\n    \t\t\\node(P3124) at (-1,3){3124};\n    \t\t\\node[blue](P2143) at ( 0,3){2143};\n    \t\t\\node[blue](P1342) at ( 1,3){1342};\n    \t\t\\node[blue](P1423) at ( 2,3){1423};\n    \t\t%\n    \t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n    \t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n    \t\t\\node(P3142) at (-0.5,4.5){3142};\n    \t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n    \t\t\\node(P4123) at ( 1.5,4.5){4123};\n    \t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n    \t\t%\n    \t\t\\node[blue](P3241) at (-2,6){3241};\n    \t\t\\node[blue](P2431) at (-1,6){2431};\n    \t\t\\node(P3412) at ( 0,6){3412};\n    \t\t\\node[blue](P4213) at ( 1,6){4213};\n    \t\t\\node(P4132) at ( 2,6){4132};\n    \t\t%\n    \t\t\\node[blue](P3421) at (-1,7.5){3421};\n    \t\t\\node[blue](P4231) at ( 0,7.5){4231};\n    \t\t\\node(P4312) at ( 1,7.5){4312};\n    \t\t%\n    \t\t\\node[blue](P4321) at (0,9){4321};\n    \t\t%\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n    \t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n    \t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n    \t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n    \t\t\\draw(P1324) -- (P3124);\n    \t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n    \t\t\\draw(P1243) -- (P2143);\n    \t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n    \t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n    \t\t\\draw(P3124) -- (P3214);\n    \t\t\\draw(P3124) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n    \t\t\\draw(P1342) -- (P3142);\n    \t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n    \t\t\\draw(P1423) -- (P4123);\n    \t\t\\draw(P1423) -- (P1432);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n    \t\t\\draw(P2341) -- (P3241);\n    \t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n    \t\t\\draw(P3142) -- (P3412);\n    \t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n    \t\t\\draw(P2413) -- (P2431);\n    \t\t\\draw(P4123) -- (P4213);\n    \t\t\\draw(P4123) -- (P4132);\n    \t\t\\draw(P1432) -- (P4132);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n    \t\t\\draw[line width=0.5mm,blue](P2431) -- (P4231);\n    \t\t\\draw(P3412) -- (P4312);\n    \t\t\\draw(P3412) -- (P3421);\n    \t\t\\draw(P4213) -- (P4231);\n    \t\t\\draw(P4132) -- (P4312);\n    \t\t%\n    \t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n    \t\t\\draw(P4231) -- (P4321);\n    \t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node[blue](P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node[blue](P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw[line width=0.5mm,green](P4213) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,red](P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,red](P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n    \t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n\t\t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node[blue](P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node[blue](P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\\end{tabular}\n}\n{Generating trees for the $(\\{j\\}, \\varnothing)$- and $(\\varnothing, \\{j\\})$-permutree minimal permutations of~$\\fS_4$, with priority order $s_1 \\prec s_2 \\prec s_3$.}\n{fig:TreePartialOrientations}\n\n\n\\newpage\n\\section{Intersection of automata}\\label{sec:intersectionsAutomata}\n\nWe now consider arbitrary subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nWe already know from~\\cite{PilaudPons-permutrees} and \\cref{thm:patternAvoidance} that the following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{enumerate}[(i)]\n\t\\item the permutation $\\pi$ is minimal in its~$(U,D)$-permutree class, \n\t\\item for~$i < j < k$, the permutation $\\pi$ does not contain the subword~$jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$,\n\t\\item for each~$j \\in U$ (each~$j \\in D$), there is a reduced expression for~$\\pi$ accepted by~$\\automatonU(j)$ (resp.~by~$\\automatonD(j)$).\n\\end{enumerate}\nA natural question is whether there is a reduced expression simultaneously accepted by all these automata.\nWe start with an example showing that this is not always the case.\n\n\\begin{example}\\label{exm:problemUDintersect}\nFor~$j \\in \\{2, \\dots, n-1\\}$, consider $U = \\{j\\} = D$, and~$\\pi = s_{j-1} \\cdot s_j \\cdot s_{j-1} = s_j \\cdot s_{j-1} \\cdot s_j$.\nThen, the expression~$s_{j-1} \\cdot s_j \\cdot s_{j-1}$ is accepted by $\\automatonD(j)$ but not by~$\\automatonU(j)$, while the expression $s_j \\cdot s_{j-1} \\cdot s_j$ is accepted by $\\automatonU(j)$ but not by~$\\automatonD(j)$.\n\\end{example}\n\nThis example clearly extends to all subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$ with a non-empty intersection.\nIn contrast, we will now show that this situation cannot occur when~$U$ and~$D$ are disjoint.\n\n\\subsection{Proof of \\cref{thm:permutreeMinimal}}\n\n\\cref{exm:problemUDintersect} motivates \\cref{thm:permutreeMinimal} that we repeat here for convenience.\n\n\\newtheorem*{thm:permutreeMinimal}{\\cref{thm:permutreeMinimal}}\n\\begin{thm:permutreeMinimal}\nConsider two disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$.\nThe following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\\item $\\pi$ admits a reduced expression accepted by all automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~${j \\in D}$,\n\\item $\\pi$ contains no subword $jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ for any~$i < j < k$.\n\\end{itemize}\n\\end{thm:permutreeMinimal}\n\n\\begin{proof}\nThe direct implication is immediate from \\cref{thm:patternAvoidance}.\nFor the converse implication, consider a permutation~$\\pi$ that avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$.\nConsider $\\bar U \\eqdef \\set{j \\in U}{\\ninv_j(\\pi) \\ne 0}$ and~$\\bar D \\eqdef \\set{j \\in D}{\\ninv^j(\\pi) \\ne 0}$.\nBy \\cref{prop:sameStateAcceptedReducedExpressionsRefined}\\,(ii), any reduced expression for~$\\pi$ is accepted by~$\\automatonU(j)$ for~$j \\in U \\ssm \\bar U$ and by~$\\automatonD(j)$ for~$j \\in D \\ssm \\bar D$.\nWe can therefore assume that~$\\bar U = U$ and~$\\bar D = D$ and one of them is non-empty, say~$\\bar U = U \\ne \\varnothing$.\nLet~$j_\\circ \\eqdef \\max(U)$ and~$m$ be minimal such that~$j_\\circ < m$ and~$\\pi^{-1}(j_\\circ) > \\pi^{-1}(m)$.\nBy minimality of~$m$, we obtain that~$\\pi$ contains the subword~$mj_\\circ\\ell$ for any~$j_\\circ < \\ell < m$.\nIt implies that\n\\begin{itemize}\n\\item $\\ell$ is neither in~$U$ by maximality of~$j_\\circ$, nor in~$D$ by assumption on~$\\pi$, for all~$j_\\circ < \\ell < m$,\n\\item $\\pi$ reverses~$\\ell$ and $m$ for all $\\ell$ in $\\{j_\\circ,\\dots,m-1\\}$, so that $\\pi$ admits a reduced expression of the form $\\pi = s_{m-1} \\cdots s_{j_\\circ} \\cdot \\tau$.\n\\end{itemize}\nLemmas \\ref{lem:vincent2}\\,(2) and \\ref{lem:vincent4}\\,(2) ensure that\n\\begin{itemize}\n\\item $\\tau$ avoids~$jki$ for all~$j \\in U \\ssm \\{j_\\circ\\}$ and~$kij$ for all~$j \\in D \\ssm \\{m\\}$ (because~$j_\\circ, \\dots, m-1$ are all distinct from~$j-1$ and~$j$ in these cases by the second observation above),\n\\item $\\tau$ avoids $(j_\\circ+1)ki$,\n\\item if~$m \\in D$, then~$\\tau$ avoids~$kij_\\circ$.\n\\end{itemize}\nBy induction, it implies that~$\\tau$ admits a reduced expression~$w$ simultaneously accepted by all automata~$\\automatonU(j)$ for~$j \\in U \\ssm \\{j_\\circ\\}$ and~$j = j_\\circ + 1$, and all automata~$\\automatonD(j)$ for~$j \\in D \\ssm \\{m\\}$ and~$j = j_\\circ$ if~$m \\in D$.\nBy Lemmas \\ref{lem:vincent2}\\,(1) and \\ref{lem:vincent4}\\,(1), we conclude that $s_{m-1} \\cdots s_{j_\\circ} \\cdot w$ is a reduced expression for~$\\pi$ simultaneously accepted by all $\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$.\n\\end{proof}\n\n\\subsection{Intersection of automata}\n\n\\cref{thm:permutreeMinimal} can be rephrased in terms of intersection of automata.\nRecall that the intersection of some automata~$\\automatonA_1, \\dots, \\automatonA_p$ is the automaton~$\\automatonA = \\bigcap_{i \\in [p]} \\automatonA_i$ such that a word is accepted by~$\\automatonA$ if and only if it is accepted by all~$\\automatonA_1, \\dots, \\automatonA_p$.\nA state of the automaton~$\\automatonA$ is $p$-tuple formed by states of the automata~$\\automatonA_1, \\dots, \\automatonA_p$, and a transition~$t$ simultaneously changes all entries of the $p$-tuple corresponding to states modified by~$t$.\nSee~\\cite[p.\\,59--60]{HopcroftUllman} for details.\nWe denote by~$\\automatonP(U,D)$ the intersection of the automata~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$.\nWe thus obtain the following statement.\n\n\\begin{corollary}\n\\label{coro:permutreeMinimal}\nWhen~$U$ and~$D$ are disjoint, the following conditions are equivalent for~$\\pi \\in \\fS_n$:\n\\begin{itemize}\n\t\\item $\\pi$ admits a reduced expression accepted by the automaton~$\\automatonP(U,D)$,\n\t\\item $\\pi$ contains no subword~$jki$ if~$j \\in U$ and~$kij$ if~$j \\in D$ with~$i < j < k$.\n\\end{itemize}\n\\end{corollary}\n\nWe say that a state of~$\\automatonP(U,D)$ is \\defn{healthy} (resp.~\\defn{ill}, resp.~\\defn{dead}) when the corresponding states in~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$ are all healthy (resp.~contain at least one ill state, but no dead one, resp.~contains at least one dead state).\n\\cref{fig:automataProduct} illustrates the automata~$\\automatonP(\\{4\\},\\{2\\})$ when~$n = 5$ (left), $\\automatonP(\\{3\\},\\{2\\})$ for $n=4$ (middle), and~$\\automatonP(\\{2\\},\\{4\\})$ for $n=5$ (right).\nFor the first two automata, we have drawn the complete automata on top, and their skeleta on the bottom.\nHere, we call skeleton a simplification of the automaton that recognizes the same reduced words.\nIt is obtained using the fact that the word is rejected as soon as we reach a dead state, and that the automata~$\\automatonU(n)$ and~$\\automatonD(1)$ accept all reduced expressions.\nFor the last automaton, the complete intersection is too big, so we only draw the reachable healthy states.\nIn the picture, we color the transitions in red, blue, or purple depending on whether only~$\\automatonU$, only $\\automatonD$, or both~$\\automatonU$ and $\\automatonD$ change state.\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\\begin{tabular}{l@{\\hspace{-.8cm}}l@{\\hspace{-.8cm}}l}\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (11) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (12) [right= 1.5cm of 11] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (21) [right= 2.5cm of 12] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (22) [right= 1.5cm of 21] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (13) [below= 1cm of 11] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (14) [right= 1.5cm of 13] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (23) [right= 2.5cm of 14] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (24) [right= 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[color=red] node [swap] {$s_3$} (25);\n\t\t\\path[->] (24) edge [color=blue] node [left=-1mm] {$s_1$} (44);\n\t\t\\path[->] (25) edge [color=blue] node [below=1mm] {$s_1$} (45);\n\t\t%\n\t\t\\path[->] (32) edge [color=blue] node [below=1mm] {$s_1$} (52) edge [color=red] node {$s_3$} (34);\n\t\t\\path[->] (33) edge [color=red] node [swap] {$s_3$} (35) edge [color=blue] node [below=1mm] {$s_1$} (53);\n\t\t\\path[->] (34) edge [color=blue] node [below=1mm] {$s_1$} (54);\n\t\t\\path[->] (35) edge [color=blue] node [below=1mm] {$s_1$} (55);\n\t\t%\n\t\t\\path[->] (41) edge [color=red] node {$s_2$} (43) edge [color=red] node [below left=-1mm] {$s_3$} (42);\n\t\t\\path[->] (42) edge [color=red] node {$s_3$} (44);\n\t\t\\path[->] (43) edge [color=red] node {$s_3$} (45);\n\t\t%\n\t\t\\path[->] (52) edge [color=red] node {$s_3$} (54);\n\t\t\\path[->] (53) edge [color=red] node {$s_3$} (55);\n\t\\end{tikzpicture}\n\t&\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (11) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (12) [right= 1.5cm of 11] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (23) [right= 1.5cm of 12] {};\n\t\t\\node[state,accepting,,minimum size=0.5cm] (24) [right= 1.5cm of 23] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (21) [below= 1.5cm of 11] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (32) [below= 1.5cm of 12] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (33) [below= 1.5cm of 23] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (34) [below= 1.5cm of 24] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (42) [below= 1.5cm of 32] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (43) [below= 1.5cm of 33] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (44) [below= 1.5cm of 34] {};\n\t\t\\path[->] (11) edge [color=red] node [swap] {$s_2$} (21) edge [color=blue] node {$s_3$} (12);\n\t\t\\path[->] (12) edge [color=violet] node {$s_2$} (23);\n\t\t\\path[->] (23) edge [color=red] node [swap] {$s_3$} (33) edge [color=blue] node {$s_1$} (24);\n\t\t\\path[->] (24) edge [color=red] node [swap] {$s_3$} (34);\n\t\t\\path[->] (21) edge [color=violet] node {$s_3$} (32);\n\t\t\\path[->] (32) edge [color=blue] node {$s_2$} (33) edge [color=red] node [swap] {$s_4$} (42);\n\t\t\\path[->] (33) edge [color=blue] node {$s_1$} (34) edge [color=red] node [swap] {$s_4$} (43);\n\t\t\\path[->] (34) edge [color=red] node [swap] {$s_4$} (44);\n\t\t\\path[->] (42) edge [color=blue] node {$s_2$} (43);\n\t\t\\path[->] (43) edge [color=blue] node {$s_1$} (44);\n\t\\end{tikzpicture}\n\t\\\\\n\t\\\\[.3cm]\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (h1) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (h2) [right= 1.5cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h3) [right= 2.5cm of h2] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h4) [right= 1.5cm of h3] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i1) [below= 1cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (i11) [right= 1cm of i1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (i12) [below= 1cm of i11] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i2) [right= 1.5cm of i11] {};\n\t\t\\node[state,minimum size=0.5cm] (d) [below= 2.5cm of i2] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i3) [below= 1.1cm of h3] {};\n\t\t%\n\t\t\\path[->] (h1) edge [bend left,color=blue] node {$s_1$} (h3) edge [swap,color=red] node {$s_3$} (i1) edge [color=red] node [swap] {$s_4$} (h2) edge [color=blue] node [below left=-2mm] {$s_2$} (i11) ;\n\t\t\\path[->] (i1) edge [bend right=50,swap,color=red] node {$s_4$} (d) edge [bend left, color=blue] node {$s_1$} (i3) edge [color=blue] node [below left=-2mm] {$s_2$} (i12);\n\t\t\\path[->] (i11) edge [swap,color=red] node {$s_3$} (i12) edge [color=blue] node {$s_1$} (d) edge [color=red] node [swap] {$s_4$} (i2);\n\t\t\\path[->] (i12) edge [color=violet] node [below left=-2mm] {${\\color{blue}s_1},{\\color{red}s_4}$} (d);\n\t\t%\n\t\t\\path[->] (h2) edge [bend left,color=blue] node {$s_1$} (h4) edge [color=blue] node [below left=-2mm] {$s_2$} (i2);\n\t\t\\path[->] (i2) edge [color=blue] node {$s_1$} (d);\n\t\t%\n\t\t\\path[->] (h3) edge [color=red] node [swap] {$s_4$} (h4) edge [swap,color=red] node {$s_3$} (i3);\n\t\t\\path[->] (i3) edge [color=red] node {$s_4$} (d);\n\t\\end{tikzpicture}\n\t&\n\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto, baseline=-1.5cm]\n\t\t\\node[state,initial,accepting,minimum size=0.5cm] (h1) {}; \n\t\t\\node[state,accepting,,minimum size=0.5cm] (h2) [right= 1.5cm of h1] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h3) [right= 2.5cm of h2] {};\n\t\t\\node[state,accepting,minimum size=0.5cm] (h4) [right= 1.5cm of h3] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i1) [below= 1cm of h1] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i2) [below= 1cm of h2] {};\n\t\t\\node[state,minimum size=0.5cm] (d) [below= 1cm of i2] {};\n\t\t%\n\t\t\\node[state,accepting,minimum size=0.5cm] (i3) [below= 1cm of h3] {};\n\t\t%\n\t\t\\path[->] (h1) edge [bend left,color=blue] node {$s_1$} (h3) edge [color=violet] node [swap] {$s_2$} (i1) edge [color=red] node [swap] {$s_3$} (h2);\n\t\t\\path[->] (i1) edge [color=violet] node [below left=-2mm] {${\\color{blue}s_1},{\\color{red}s_3}$} (d);\n\t\t%\n\t\t\\path[->] (h2) edge [bend left,color=blue] node {$s_1$} (h4) edge [color=blue] node {$s_2$} (i2);\n\t\t\\path[->] (i2) edge [color=blue] node {$s_1$} (d);\n\t\t%\n\t\t\\path[->] (h3) edge [color=red] node [swap] {$s_3$} (h4) edge [swap,color=red] node {$s_2$} (i3);\n\t\t\\path[->] (i3) edge [color=red] node {$s_3$} (d);\n\t\\end{tikzpicture}\n\\end{tabular}\n}\n{The automaton~$\\automatonP(\\{4\\},\\{2\\})$ for $n = 5$ and its skeleton (left), the automaton~$\\automatonP(\\{3\\},\\{2\\})$ for $n=4$ and its skeleton (middle), and the healthy states of the automaton~$\\automatonP(\\{2\\},\\{4\\})$ for $n=5$ (right).}\n{fig:automataProduct}\n\n\\subsection{The set of accepted reduced expressions of~$\\automatonP(U,D)$}\n\nApplying the principles of \\cref{subsec:principles} to each automaton~$\\automatonU(j)$ for~$j \\in U$ and~$\\automatonD(j)$ for~$j \\in D$, we derive similar principles for the automaton~$\\automatonP(U,D)$.\nThe following statements are direct consequences of \\cref{prop:prefixesReducedExpressions,prop:algorithm,prop:sameStateAcceptedReducedExpressions}.\n\n\\begin{proposition}\\label{prop:prefixesReducedExpressionsIntersection}\nThe set of reduced words accepted by~$\\automatonP(U,D)$ is closed by prefix.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:algorithmIntersection}\nIf a permutation~$\\pi$ avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$, and admits a reduced expression starting with~$s_\\ell$ such that the transition~$s_\\ell$ leads to an healthy state of~$\\automatonP(U,D)$, then it admits a reduced expression starting with~$s_\\ell$ and accepted by~$\\automatonP(U,D)$.\n\\end{proposition}\n\n\\begin{proposition}\\label{prop:sameStateAcceptedReducedExpressionsIntersection}\nGiven a permutation $\\pi \\in \\fS_n$, all the reduced expressions for~$\\pi$ accepted by $\\automatonP(U,D)$ end at the same state.\n\\end{proposition}\n\n\\subsection{Permutree sorting}\\label{subsec:permutreeSorting}\n\nWe have seen that for any~$(U,D)$-permutree minimal permutation, the set of reduced expressions for~$\\pi$ that are accepted by~$\\automatonP(U,D)$ is non-empty (by \\cref{coro:permutreeMinimal}) and closed by prefix (by \\cref{prop:prefixesReducedExpressionsIntersection}).\nTherefore, it is possible to sort~$\\pi$ passing only through $(U,D)$-permutree minimal permutations along the way.\nThis motivates the following definition.\n\n\\begin{definition}\nAn \\defn{$(U,D)$-permutree sorting algorithm} is a sorting procedure such that\n\\begin{itemize}\n\\item applied to an $(U,D)$-permutree minimal permutation~$\\pi$, it only passes through $(U,D)$-permutree minimal permutations and arrives to the identity permutation, \n\\item it fails to sort a non $(U,D)$-permutree minimal permutation~$\\pi$.\n\\end{itemize}\n\\end{definition}\n\n\\begin{example}\nThe stack sorting algorithm is a $(\\{2, \\dots, n-1\\}, \\varnothing)$-permutree sorting algorithm.\n\\end{example}\n\n\\enlargethispage{.1cm}\nSaid differently, any procedure that looks for a reduced expression accepted by~$\\automatonP(U,D)$ gives a $(U,D)$-permutree sorting algorithm.\nFor instance, \\cref{algo:shortcutsUj} is a $(\\{j\\}, \\varnothing)$-permutree sorting algorithm.\nWe generalize it in the following algorithm, where we opted for a recursive style.\nAs in \\cref{algo:shortcutsUj}, the algorithm will read the automaton~$\\automatonP(U,D)$ without actually constructing it.\nTo virtually follow the edges of the automaton~$\\automatonP(U,D)$, we use the following two operations on our sets $U$ and $D$:\n\\[\n\\moveU(U,\\ell) = \n\\begin{cases}\nU & \\text{ if } \\ell \\notin U, \\\\\n(U \\ssm \\{ \\ell \\}) \\cup \\{ \\ell + 1 \\} & \\text{ if } \\ell \\in U,\n\\end{cases}\n\\]\n\n\\[\n\\moveD(D,\\ell) = \n\\begin{cases}\nD & \\text{ if } \\ell + 1 \\notin D, \\\\\n(D \\ssm \\{ \\ell + 1 \\}) \\cup \\{ \\ell \\} & \\text{ if } \\ell + 1 \\in D.\n\\end{cases}\n\\]\n\n\\bigskip\n\\IncMargin{1em}\n\\SetKwFor{Repeat}{repeat}{}{}\n\\SetKwIF{If}{ElseIf}{Else}{if}{then}{else if}{else}{}\n\\DontPrintSemicolon\n\\begin{algorithm}[H]\n\t\\KwData{a permutation $\\pi \\in \\fS_n$ and two disjoint subsets~$U$ and~$D$ of~$[n]$}\n\t\\KwResult{a reduced word accepted by~$\\automatonP(U,D)$, candidate reduced expression for~$\\pi$}\n\t\\If{$\\exists \\; \\ell \\in [n-1]$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$, and $\\ell+1 \\notin U$ and~$\\ell \\notin D$}{\n\t\t\\Return $s_\\ell \\cdot \\text{permutreeSort}(s_\\ell \\cdot \\pi, \\; \\moveU(U, \\ell), \\; \\moveD(D,\\ell))$ \\;\n\t}\n\t\\If{$\\exists \\; \\ell \\in [n-1]$ such that $\\ell$ and $\\ell+1$ are reversed in~$\\pi$,\n\t\\\\ and ($\\ell + 1 \\notin U$ or $\\pi([\\ell+1]) = [\\ell +1]$) and ($\\ell \\notin D$ or $\\pi([\\ell -1]) = [\\ell - 1]$)\n\t}{\n\t\t\\Return $s_\\ell \\cdot \\text{permutreeSort}(s_\\ell \\cdot \\pi, \\; \\moveU(U \\ssm \\{ \\ell + 1 \\}, \\ell), \\; \\moveD(D \\ssm \\{ \\ell \\},\\ell))$ \\;\n\t}\n\t\\Return $\\varepsilon$ \\;\n\t\\caption{$(U,D)$-permutree sorting}\n\t\\label{algo:permutreeSorting}\n\\end{algorithm}\n\\bigskip\n\nNote that in \\cref{algo:permutreeSorting}, we could ignore~$n$ in the list~$U$ (resp.~$1$ in the list~$D$) since~$\\automatonU(n)$ (resp.~$\\automatonD(1)$) accepts all reduced words.\nWe have decided not to do it to be coherent with our recursive definition of~$\\automatonU(j)$ and~$\\automatonD(j)$.\n\n\\begin{example}\\label{exm:algo2}\n\tWe present in \\cref{tab:algo2-3-2} the $(\\{3\\},\\{2\\})$-permutree sorting algorithm in action for the permutations~$\\pi_1 \\eqdef 3214$, $\\pi_2 \\eqdef 1324$ and~$\\pi_3 \\eqdef 1342$, and in \\cref{tab:algo2-2-4} the $(\\{2\\},\\{4 \\})$-permutree sorting algorithm in action for the permutations~$\\pi_4 \\eqdef 54213$ and~$\\pi_5 \\eqdef 15342$. \n\tThe corresponding automata~$\\automatonP(\\{3\\},\\{2\\})$ and~$\\automatonP(\\{2\\},\\{4\\})$ are represented in \\cref{fig:automataProduct}.\n\tEach row in these tables contains the states of the permutation~$\\pi$ and of the word~$w$, the current values of~$j$ and~$\\ell$ in use at each step, and the values of~$k$ for which we have to check that~$\\pi([k]) = [k]$, crossed in red when it fails. These tables show that~$\\pi_1$ and~$\\pi_2$ are $(\\{3\\},\\{2\\})$-permutree sortable while $\\pi_3$ is not, and that~$\\pi_4$ is $(\\{2\\},\\{4 \\})$-permutree sortable while $\\pi_5$ is not.\n\n\\begin{table}[h!]\n\t\\centerline{\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_1$ & $w_1$ & $U_1$ & $D_1$ & $\\ell_1$ & $k_1$ \\\\\n\t\t\\hline\n\t\t$3214$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $1$ & . \\\\\n\t\t$3124$ & $s_1$ & $\\{3\\}$ & $\\{1\\}$ & $2$ & $3$ \\\\\n\t\t$2134$ & $s_1 \\cdot s_2$ & $\\varnothing$ & $\\{1\\}$ & $1$ & $0$ \\\\\n\t\t$1234$ & $s_1 \\cdot s_2 \\cdot s_1$ & & &\n\t\\end{tabular}\n\t\\quad\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_2$ & $w_2$ & $U_2$ & $D_2$ & $\\ell_2$ & $k_2$ \\\\\n\t\t\\hline\n\t\t$1324$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $2$ & $1$, $3$ \\\\\n\t\t$1234$ & $s_2$ & & &\n\t\\end{tabular}\n\t\\quad\n\t\\begin{tabular}[t]{l|l|c|c|c|c} \n\t\t$\\pi_3$ & $w_3$ & $U_3$ & $D_3$ & $\\ell_3$ & $k_3$ \\\\\n\t\t\\hline\n\t\t$1342$ & $\\varepsilon$ & $\\{3\\}$ & $\\{2\\}$ & $2$ & $1$, \\textcolor{red}{$\\xcancel{\\textcolor{black}{3}}$}\n\t\\end{tabular}\n\t}\n\t\\caption{The~$(\\{3\\},\\{2 \\})$-permutree sorting of~$\\pi_1 \\eqdef 3214$, $\\pi_2 \\eqdef 1324$ and~$\\pi_3 \\eqdef 1342$.}\n\t\\label{tab:algo2-3-2}\n\\end{table}\n\n\\begin{table}[h!]\n\t\\centerline{\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_4$ & $w_4$ & $U_4$ & $D_4$ & $\\ell_4$ & $k_4$ \\\\\n\t\t\\hline\n\t\t$54213$ & $\\varepsilon$ & $\\{2\\}$ & $\\{4\\}$ & $3$ & . \\\\\n\t\t$53214$ & $s_3$ & $\\{2\\}$ & $\\{3\\}$ & $2$ & . \\\\\n\t\t$52314$ & $s_3 \\cdot s_2$ & $\\{3\\}$ & $\\{2\\}$ & $1$ & . \\\\\n\t\t$51324$ & $s_3 \\cdot s_2 \\cdot s_1$ & $\\{3\\}$ & $\\{1\\}$ & $4$ & . \\\\\n\t\t$41325$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4$ & $\\{3\\}$ & $\\{1\\}$ & $3$ & . \\\\\n\t\t$31425$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3$ & $\\{4\\}$ & $\\{1\\}$ & $2$ & . \\\\\n\t\t$21435$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2$ & $\\{4\\}$ & $\\{1\\}$ & $1$ & . \\\\\n\t\t$12435$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1$ & $\\{4\\}$ & $\\{1\\}$ & $3$ & $4$ \\\\\n\t\t$12345$ & $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3$ & $\\{4\\}$ & $\\{1\\}$ &\n\t\\end{tabular}\n\t\\qquad\n\t\\begin{tabular}[t]{l|l|c|c|c|c}\n\t\t$\\pi_5$ & $w_5$ & $U_5$ & $D_5$ & $\\ell_5$ & $k_5$ \\\\\n\t\t\\hline\n\t\t$15342$ & $\\varepsilon$ & $\\{2\\}$ & $\\{4\\}$ & $2$ & . \\\\\n\t\t$15243$ & $s_2$ & $\\{3\\}$ & $\\{4\\}$ & $3$ & . \\\\\n\t\t$15234$ & $s_2 \\cdot s_3$ & $\\{4\\}$ & $\\{3\\}$ & $4$ & . \\\\\n\t\t$14235$ & $s_2 \\cdot s_3 \\cdot s_5$ & $\\{5\\}$ & $\\{3\\}$ & $3$ & \\textcolor{red}{$\\xcancel{\\textcolor{black}{2}}$}\n\t\\end{tabular}\n\t}\n\t\\caption{The~$(\\{2\\},\\{4 \\})$-permutree sorting of~$\\pi_4 \\eqdef 54213$ and~$\\pi_5 \\eqdef 15342$.}\n\t\\label{tab:algo2-2-4}\n\\end{table}\n\\end{example}\n\n\\begin{corollary}\\label{coro:algorithmIntersection}\nFor any permutation~$\\pi$ and any disjoint subsets~$U$ and~$D$ of~$\\{2, \\dots, n-1\\}$, \\cref{algo:permutreeSorting} returns a reduced word~$w$ accepted by $\\automatonP(U,D)$ with the property that~$w$ is a reduced expression for~$\\pi$ if and only if~$\\pi$ avoids~$jki$ for~$j \\in U$ and~$kij$ for~$j \\in D$.\n\\end{corollary}\n\n\\begin{proof}\nThis algorithm constructs a candidate reduced word for $\\pi$ following the automaton~$\\automatonP(U,D)$ and prioritizing healthy states over ill states. \nIt begins by checking all possible transitions~$s_\\ell$ that keep $\\automatonP(U,D)$ in healthy states following \\cref{lem:vincent4} (if condition in line~1).\nDoing this in the intersection of automata translates to updating~$\\ell$ to~$\\ell+1$ when~$\\ell \\in U$ and $\\ell+1$ to~$\\ell$ when~$\\ell+1 \\in D$ (line~2).\nWhen we have exhausted all these transitions, we need to go to an ill state of~$\\automatonP(U,D)$, \\ie to apply a transposition that sends at least one automaton of the intersection to an ill state.\nIf there is~$\\ell+1 \\in U$ (resp.~$\\ell \\in D$) such that~$s_\\ell$ is a descent of~$\\pi$ and~$\\pi([\\ell+1]) = [\\ell+1]$ (resp.~$\\pi([\\ell-1]) = [\\ell-1]$), then any reduced expression for~$\\pi$ is accepted by the automaton~$\\automatonU(\\ell)$ (resp.~$\\automatonD(\\ell)$) by \\cref{prop:sameStateAcceptedReducedExpressionsRefined}\\,(i). \nWe can thus start with~$s_\\ell$ and forget about the automaton~$\\automatonU(\\ell)$ (resp.~$\\automatonD(\\ell)$) (lines 3, 4 and 5).\nFinally, if none of these options are possible, any reduced expression for~$\\pi$ will lead to a dead state in at least one of the automata, so that $\\pi$ is not $(U,D)$-sortable.\nWe thus return the empty reduced expression (line 6).\n\\end{proof}\n\n\\subsection{Generating trees}\n\nAs in \\cref{subsec:trees}, we can define natural generating trees for the $(U,D)$-permutree minimal permutations.\nNamely, fix an arbitrary priority order~$\\prec$ on~$\\{s_1, \\dots, s_{n-1}\\}$.\nFor an $(U,D)$-permutree minimal permutation~$\\pi$, we denote by~$\\pi(U, D, \\prec)$ the $\\prec$-lexicographic minimal reduced expression for~$\\pi$ that is accepted by~$\\automatonP(U,D)$.\nWe denote by~$\\lexmin(n, U, D, \\prec)$ the set of reduced words of the form~$\\pi(U, D, \\prec)$ for all $(U,D)$-permutree minimal permutations~$\\pi \\in \\fS_n$.\nThe same proof as that of \\cref{prop:generatingTrees} shows that $\\lexmin(n, U, D, \\prec)$ is closed by prefix.\nThis yields a natural generating tree on~$\\lexmin(n, U, D, \\prec)$ where the parent of a reduced word~$w$ is obtained by deleting its last letter.\nReplacing each reduced expression by the corresponding permutation, this provides a generating tree for the $(U,D)$-permutree minimal permutations of~$\\fS_n$.\n\\cref{fig:TreeCompleteOrientations} presents these generating trees for different values of~$U$ and~$D$.\n\n\\hvFloat[floatPos=p, capWidth=h, capPos=r, capAngle=90, objectAngle=90, capVPos=c, objectPos=c]{figure}\n{\n\t\\begin{tabular}{cccc}\n\t\t$\\automatonU(2) \\cap \\automatonU(3)$\n\t\t&\n\t\t$\\automatonU(2) \\cap \\automatonD(3)$\n\t\t&\n\t\t$\\automatonU(3) \\cap \\automatonD(2)$\n\t\t&\n\t\t$\\automatonD(2) \\cap \\automatonD(3)$\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj+1) [below= 1.5cm of ij+1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+2) [right= 1.5cm of hj+1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+2) [below= 1.5cm of hj+2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_1$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_2$} (ij+1)\n\t\t\t\t\t edge node {$s_3$} (hj+2)\n\t\t\t\t(ij+1) edge node [swap] {$s_3$} (dj+1)\n\t\t\t\t%\n\t\t\t\t(hj+2) edge node [swap] {$s_3$} (ij+2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj+1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij+1) [below= 1.5cm of hj+1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_3$} (hj+1)\n\t\t\t\t(ij) edge node [swap] {$s_3$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj+1) edge node [swap] {$s_3$} (ij+1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_2$} (ij)\n\t\t\t\t\t   edge node {$s_1$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_1$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_1$} (ij-1);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[shorten >=1pt, node distance=2cm, on grid, auto]\n\t\t\t\\node[state,initial,accepting,minimum size=0.5cm] (hj)   {}; \n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij) [below= 1.5cm of hj] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj) [below= 1.5cm of ij] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-1) [right= 1.5cm of hj] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-1) [below= 1.5cm of hj-1] {}; \n\t\t\t\\node[state,minimum size=0.5cm] (dj-1) [below= 1.5cm of ij-1] {}; \n\t\t\t%\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (hj-2) [right= 1.5cm of hj-1] {};\n\t\t\t\\node[state,accepting,minimum size=0.5cm] (ij-2) [below= 1.5cm of hj-2] {}; \n\t\t\t\\path[->] \n\t\t\t\t(hj) edge node [swap] {$s_3$} (ij)\n\t\t\t\t\t   edge node {$s_2$} (hj-1)\n\t\t\t\t(ij) edge node [swap] {$s_2$} (dj)\n\t\t\t\t%\n\t\t\t\t(hj-1) edge node [swap] {$s_2$} (ij-1)\n\t\t\t\t\t edge node {$s_1$} (hj-2)\n\t\t\t\t(ij-1) edge node [swap] {$s_1$} (dj-1)\n\t\t\t\t%\n\t\t\t\t(hj-2) edge node [swap] {$s_1$} (ij-2);\n\t\t\\end{tikzpicture}\n\t\t\\\\[.5cm]\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node(P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node[blue](P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node(P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node[blue](P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw(P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw(P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw[line width=0.5mm,blue](P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P4123) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,red](P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node(P2314) at (-2,3){2314};\n\t\t\t\\node[blue](P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node[blue](P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node(P2431) at (-1,6){2431};\n\t\t\t\\node[blue](P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node[blue](P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw(P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,blue](P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw(P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw[line width=0.5mm,red](P3124) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw[line width=0.5mm,red](P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw[line width=0.5mm,blue](P3412) -- (P4312);\n\t\t\t\\draw[line width=0.5mm,green](P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,green](P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n\t\t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node(P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node(P1342) at ( 1,3){1342};\n\t\t\t\\node[blue](P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node(P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node[blue](P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node(P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node[blue](P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node(P3421) at (-1,7.5){3421};\n\t\t\t\\node[blue](P4231) at ( 0,7.5){4231};\n\t\t\t\\node(P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw(P1324) -- (P3124);\n\t\t\t\\draw(P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw[line width=0.5mm,red](P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw(P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw(P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw[line width=0.5mm,green](P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw(P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw(P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw[line width=0.5mm,blue](P2413) -- (P4213);\n\t\t\t\\draw[line width=0.5mm,green](P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw(P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw[line width=0.5mm,green](P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw(P3421) -- (P4321);\n\t\t\t\\draw[line width=0.5mm,red](P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n\t\t\\end{tikzpicture}\n\t\t&\n    \t\\begin{tikzpicture}[xscale=.9, yscale=0.7, color=lightgray]\n\t\t\t\\node[blue](P1234) at (0,0){1234};\n\t\t\t%\n\t\t\t\\node[blue](P2134) at (-1,1.5){2134};\n\t\t\t\\node[blue](P1324) at ( 0,1.5){1324};\n\t\t\t\\node[blue](P1243) at ( 1,1.5){1243};\n\t\t\t%\n\t\t\t\\node[blue](P2314) at (-2,3){2314};\n\t\t\t\\node(P3124) at (-1,3){3124};\n\t\t\t\\node[blue](P2143) at ( 0,3){2143};\n\t\t\t\\node[blue](P1342) at ( 1,3){1342};\n\t\t\t\\node(P1423) at ( 2,3){1423};\n\t\t\t%\n\t\t\t\\node[blue](P3214) at (-2.5,4.5){3214};\n\t\t\t\\node[blue](P2341) at (-1.5,4.5){2341};\n\t\t\t\\node(P3142) at (-0.5,4.5){3142};\n\t\t\t\\node(P2413) at ( 0.5,4.5){2413};\n\t\t\t\\node(P4123) at ( 1.5,4.5){4123};\n\t\t\t\\node[blue](P1432) at ( 2.5,4.5){1432};\n\t\t\t%\n\t\t\t\\node[blue](P3241) at (-2,6){3241};\n\t\t\t\\node[blue](P2431) at (-1,6){2431};\n\t\t\t\\node(P3412) at ( 0,6){3412};\n\t\t\t\\node(P4213) at ( 1,6){4213};\n\t\t\t\\node(P4132) at ( 2,6){4132};\n\t\t\t%\n\t\t\t\\node[blue](P3421) at (-1,7.5){3421};\n\t\t\t\\node(P4231) at ( 0,7.5){4231};\n\t\t\t\\node(P4312) at ( 1,7.5){4312};\n\t\t\t%\n\t\t\t\\node[blue](P4321) at (0,9){4321};\n\t\t\t%\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P1234) -- (P2134);\n\t\t\t\\draw[line width=0.5mm,red](P1234) -- (P1324);\n\t\t\t\\draw[line width=0.5mm,green](P1234) -- (P1243);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P2134) -- (P2314);\n\t\t\t\\draw[line width=0.5mm,green](P2134) -- (P2143);\n\t\t\t\\draw(P1324) -- (P3124);\n\t\t\t\\draw[line width=0.5mm,green](P1324) -- (P1342);\n\t\t\t\\draw(P1243) -- (P2143);\n\t\t\t\\draw(P1243) -- (P1423);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P2314) -- (P3214);\n\t\t\t\\draw[line width=0.5mm,green](P2314) -- (P2341);\n\t\t\t\\draw(P3124) -- (P3214);\n\t\t\t\\draw(P3124) -- (P3142);\n\t\t\t\\draw(P2143) -- (P2413);\n\t\t\t\\draw(P1342) -- (P3142);\n\t\t\t\\draw[line width=0.5mm,red](P1342) -- (P1432);\n\t\t\t\\draw(P1423) -- (P4123);\n\t\t\t\\draw(P1423) -- (P1432);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,green](P3214) -- (P3241);\n\t\t\t\\draw(P2341) -- (P3241);\n\t\t\t\\draw[line width=0.5mm,red](P2341) -- (P2431);\n\t\t\t\\draw(P3142) -- (P3412);\n\t\t\t\\draw(P2413) -- (P4213);\n\t\t\t\\draw(P2413) -- (P2431);\n\t\t\t\\draw(P4123) -- (P4213);\n\t\t\t\\draw(P4123) -- (P4132);\n\t\t\t\\draw(P1432) -- (P4132);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,red](P3241) -- (P3421);\n\t\t\t\\draw(P2431) -- (P4231);\n\t\t\t\\draw(P3412) -- (P4312);\n\t\t\t\\draw(P3412) -- (P3421);\n\t\t\t\\draw(P4213) -- (P4231);\n\t\t\t\\draw(P4132) -- (P4312);\n\t\t\t%\n\t\t\t\\draw[line width=0.5mm,blue](P3421) -- (P4321);\n\t\t\t\\draw(P4231) -- (P4321);\n\t\t\t\\draw(P4312) -- (P4321);\n    \t\\end{tikzpicture}\n\t\\end{tabular}\n}\n{Generating trees for the $(U,D)$-permutree minimal permutations of~$\\fS_4$, with priority order $s_1 \\prec s_2 \\prec s_3$.}\n{fig:TreeCompleteOrientations}\n\n\n\\section{Permutree sorting versus Coxeter sorting}\\label{sec:coxeterSortable}\n\nIn this section, we discuss the particular case when~$U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$.\nIn that situation, we connect the $(U,D)$-permutree sorting with the $c$-sorting of N.~Reading~\\cite{Reading-sortableElements}.\n\n\\subsection{Coxeter sorting word and Coxeter sortable permutations}\\label{subsec:csorting}\n\nWe first recall the theory of $c$-sorting developed by N.~Reading in~\\cite{Reading-sortableElements}.\nWhile it was defined in arbitrary finite Coxeter groups, we focus on the symmetric group in this presentation.\n\nWe consider a \\defn{Coxeter element}~$c$ of~$\\fS_n$, \\ie the product of all simple transpositions $\\{s_1, \\dots, s_{n-1}\\}$ in an arbitrary order.\nFor a permutation $\\pi \\in \\fS_n$, the \\defn{$c$-sorting word} $\\pi(c)$ is the lexicographically smallest reduced expression for $\\pi$ in the infinite word $c^\\infty = c \\cdot c \\cdot c \\cdot c \\cdots$.\nNote that strictly speaking, $\\pi(c)$ depends on a reduced expression for~$c$, not only on the Coxeter element~$c$.\nHere, we assume that we have chosen a reduced expression and hide this dependence.\nWe let~$I_1, \\dots, I_p$ denote the subsets of~$[n-1]$ such that~$\\pi(c) = c_{I_1} \\cdot c_{I_2} \\cdots c_{I_p}$ where~$c_I$ is the subword of~$c$ obtained by keeping only the letters~$s_i$ for~$i \\in I$.\nThe permutation $\\pi$ is \\defn{$c$-sortable} if~$I_1 \\supseteq I_2 \\supseteq \\dots \\supseteq I_p$.\nNote that this does not depend on the choice of the reduced expression~$c$, only on the Coxeter element~$c$.\n\nFor our proofs we will need some simple yet useful facts from \\cite{Reading-sortableElements} on how prefixes of words influence sortability.\n\n\\begin{lemma}\\label{lem:coxeterElementFacts}\nConsider a Coxeter element of the form~$c = s_\\ell \\cdot d$ and let $\\pi \\in \\fS_n$. Then\n\\begin{itemize}\n\t\\item if $\\pi = s_\\ell \\cdot \\tau$ with $\\ell(\\pi) = \\ell(\\tau) + 1$, then $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$,\n\t\\item otherwise, $\\pi(c) = \\pi(d \\cdot s_\\ell)$.\n\\end{itemize}\n\\end{lemma}\n\n\\begin{lemma}\\label{lem:vincent8}\nLet~$s_i \\ne s_j$ be two letters that appear in the $c$-sorting word $\\pi(c)$ of a $c$-sortable permutation~$\\pi$. Then\n\\begin{enumerate} \n\t\\item if $s_i$ appears before $s_j$ in $c$, then $s_i$ appears before $s_j$ in $\\pi(c)$,\n\t\\item if $s_j$ does not appear in between two occurrences of $s_i$ in $\\pi(c)$, then it does not appear after these occurrences either.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{proof}\nWe deal with the two statements separately:\n\t\\begin{enumerate} \n\t\t\\item Immediate from the definition since both $s_i$ and $s_j$ appear in $\\pi(c)$.\n\t\t\\item Since $\\pi$ is $c$-sortable, $\\pi(c)$ is formed by a succession of subwords that are nested. Thus if $s_j$ were to appear after the occurrences of $s_i$ it would have to appear between them as well.\n\t\t\\qedhere\n\t\\end{enumerate}\n\\end{proof}\n\n\\subsection{Coxeter sorting via permutree automata}\n\nA Coxeter element~$c$ of~$\\fS_n$ defines a partition ${\\{2, \\dots, n-1\\} = U_c \\sqcup D_c}$, where~$U_c$ (resp.~$D_c$) consists of the elements~$j \\in \\{2, \\dots, n-1\\}$ such that~$s_j$ appears before (resp.~after) $s_{j-1}$ in~$c$.\nFor instance, when~$c = s_2 \\cdot s_5 \\cdot s_4 \\cdot s_3 \\cdot s_1 \\cdot s_6$, we obtain~$U_c = \\{2,4,5\\}$ and~$D_c = \\{3,6\\}$.\nSaid differently, $j \\in U$ (resp.~$j \\in D$) if~$c$ is accepted by~$\\automatonU(j)$ but not by~$\\automatonD(j)$ (resp.~by~$\\automatonD(j)$ but not by~$\\automatonU(j)$).\nThe goal of this section is the following connection between the $c$-sorting of \\cref{subsec:csorting} and the $(U_c, D_c)$-permutree sorting of \\cref{sec:intersectionsAutomata}.\n\n\\pagebreak\n\\begin{theorem}\n\\label{thm:csorting}\nFor any Coxeter element~$c$ and any permutation~$\\pi$, the following assertions are equivalent:\n\\begin{enumerate}[(i)]\n\t\\item $\\pi$ is $c$-sortable,\n\t\\item the $c$-sorting word~$\\pi(c)$ is accepted by the automaton~$\\automatonP(U_c, D_c)$,\n\t\\item there exists a reduced expression for~$\\pi$ accepted by the automaton~$\\automatonP(U_c, D_c)$,\n\t\\item for each~$j \\in \\{2, \\dots, n-1\\}$, there exists a reduced expression for $\\pi$ that is accepted by the automaton~$\\automatonU(j)$ if~$j \\in U_c$ and~$\\automatonD(j)$ if~$j \\in D_c$,\n\t\\item $\\pi$ avoids $jki$ for~$j \\in U_c$ and $kij$ for~$j \\in D_c$.\n\\end{enumerate}\n\\end{theorem}\n\nThe equivalences (iii) $\\iff$ (iv) $\\iff$ (v) were already established earlier.\nHere, we aim at identifying the $c$-sorting word as a reduced expression for~$\\pi$ accepted by~$\\automatonP(U_c, D_c)$.\nWe split the proof of the equivalences (i) $\\iff$ (ii) $\\iff$ (iii) into the following few lemmas.\n\n\\begin{lemma}\\label{lem:vincent9}\nThe $c$-sorting word of a $c$-sortable permutation is recognized by~$\\automatonU(j)$ for~$j \\in U_c$ and by~$\\automatonD(j)$ for~$j \\in D_c$.\n\\end{lemma}\n\n\\begin{proof}\nConsider~$j \\in U_c$ (the proof for~$j \\in D_c$ is symmetric).\nWe distinguish two possible cases:\n\\begin{itemize}\n\t\\item If $\\pi(c)$ contains no $s_{j-1}$, then $\\pi(c)$ either remains in the first healthy state or ends in the first ill state of~$\\automatonU(j)$.\n\t\\item If $\\pi(c)$ contains $s_{j-1}$, then by \\cref{lem:vincent8}\\,(1) $s_{j-1}$ appears before $s_j$ in $\\pi(c)$ and $\\pi(c)$ leads to the second healthy state of~$\\automatonU(j)$. From here on out, notice that $\\pi(c)$ cannot end at a dead state because of \\cref{lem:vincent8}\\,(2). \n\\end{itemize}\nIn both cases, $\\pi(c)$ is accepted by~$\\automatonU(j)$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Vincent10}\nA permutation $\\pi \\in \\fS_n$ whose $c$-sorting word $\\pi(c)$ is accepted by~$\\automatonP(U_c, D_c)$ is $c$-sortable.\n\\end{lemma}\n\n\\begin{proof}\nSuppose that $\\pi$ is not $c$-sortable. We will find an automaton that rejects $\\pi(c)$ among the automata~$\\automatonU(j)$ for~$j \\in U_c$ and~$\\automatonD(j)$ for~$j \\in D_c$.\nOnce again, we work by induction on the length of $\\pi$ and the size of~$c$.\nLet~$s_\\ell$ be the first letter of~$c$ and write~$c = s_\\ell \\cdot d$.\nSince~$s_\\ell$ appears before~$s_{\\ell-1}$ and~$s_{\\ell+1}$ in~$c$, we have~$\\ell \\in U_c$ and~$\\ell+1 \\in D_c$.\nMoreover, the letter~$s_\\ell$ yields to the next healthy state in both automata~$\\automatonU(\\ell)$ and~$\\automatonD(\\ell+1)$, and remains in the initial state for all other automata~$\\automatonU(j)$ for~$j \\in U_c \\ssm \\{\\ell\\}$ and~$\\automatonD(j)$ for~$j \\in D_c \\ssm \\{\\ell+1\\}$.\nWe now distinguish two cases, depending on whether~$\\ell$ and~$\\ell+1$ are reversed in~$\\pi$.\n\nAssume first that $\\ell$ and $\\ell+1$ are reversed in $\\pi$ and write $\\pi = s_\\ell \\cdot \\tau$.\nWe then have $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ by \\cref{lem:coxeterElementFacts}, so that~$\\tau$ is not $d \\cdot s_\\ell$-sortable.\nBy induction hypothesis, $\\tau(d \\cdot s_\\ell)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_{d \\cdot s_\\ell}$ and~$\\automatonD(j)$ for~$j \\in D_{d \\cdot s_\\ell}$.\nSince~$U_{d \\cdot s_\\ell} = U_c \\symdif \\{\\ell, \\ell+1\\}$ and~$D_{d \\cdot s_\\ell} = D_c \\symdif \\{\\ell, \\ell+1\\}$, \\cref{lem:vincent2,lem:vincent4} ensure that $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_c$ and~$\\automatonD(j)$ for~$j \\in D_c$.\n\nAssume now that $\\ell$ and $\\ell+1$ are not reversed in $\\pi$.\nThen~$\\pi(c)$ does not use~$s_\\ell$ and~$\\pi$ is not $d$-sortable in~$W_{\\langle s_\\ell \\rangle}$.\nBy induction hypothesis, $\\pi(c)$ is rejected by one of the automata~$\\automatonU(j)$ for~$j \\in U_d$ and~$\\automatonD(j)$ for~$j \\in D_d$.\nThis concludes the proof since~$U_d \\subseteq U_c$ and~$D_d \\subseteq D_c$.\n\\end{proof}\n\n\\begin{lemma}\\label{lem:Vincent11}\nIf a permutation~$\\pi \\in \\fS_n$ admits a reduced expression accepted by~$\\automatonP(U_c, D_c)$, then its $c$-sorting word~$\\pi(c)$ is accepted by~$\\automatonP(U_c, D_c)$.\n\\end{lemma}\n\n\\begin{proof}\nOnce agin, we work by induction on the length of $\\pi$.\nLet~$s_\\ell$ be the first letter of~$c$ and write~$c = s_\\ell \\cdot d$.\nSince~$s_\\ell$ appears before~$s_{\\ell-1}$ and~$s_{\\ell+1}$ in~$c$, we have~$\\ell \\in U_c$ and~$\\ell+1 \\in D_c$.\nMoreover, the letter~$s_\\ell$ yields to the next healthy state in both automata~$\\automatonU(\\ell)$ and~$\\automatonD(\\ell+1)$, and remains in the initial state for all other automata~$\\automatonU(j)$ for~$j \\in U_c \\ssm \\{\\ell\\}$ and~$\\automatonD(j)$ for~$j \\in D_c \\ssm \\{\\ell+1\\}$.\nWe now distinguish two cases, depending on whether~$\\ell$ and~$\\ell+1$ are reversed in~$\\pi$.\n\nAssume first that $\\ell$ and $\\ell+1$ are reversed in $\\pi$ and write $\\pi = s_\\ell \\cdot \\tau$. Using \\cref{lem:coxeterElementFacts} it suffices now to show that after $s_\\ell$, there is a reduced expression for $\\tau$ accepted by the automata. For each $j$, since the automaton $\\automatonD(j)$ or $\\automatonU(j)$ that we see accepts at least one reduced expression for $\\pi$ and $s_\\ell$ does not lead to a ill state, it also accepts a reduced expression for $\\pi$ starting with $s_\\ell$ by \\cref{prop:algorithm}. Observe moreover that:\n\\begin{itemize}\n\\item $\\tau$ admits a reduced expression accepted by~$\\automatonU(j)$ for each~$j \\notin \\{\\ell, \\ell+1\\}$ by \\cref{lem:vincent2}. This lines up with the fact that the order of $s_{j-1}$ and $s_j$ has not changed from $c = s_\\ell \\cdot d$ to~$d \\cdot s_\\ell$.\n\\item $\\tau$ admits a reduced expression accepted by~$\\automatonU(\\ell+1)$ and a reduced expression accepted by~$\\automatonD(\\ell)$ by \\cref{lem:vincent4}. This fits the fact that $\\ell$ now appears after $\\ell-1$ and $\\ell+1$ in $d \\cdot s_\\ell$.\n\\end{itemize}\nBy induction, we obtain that $\\tau(d \\cdot s_\\ell)$ is accepted by~$\\automatonP(U_{d \\cdot s_\\ell}, D_{d \\cdot s_\\ell})$, so that $\\pi(c) = s_\\ell \\cdot \\tau(d \\cdot s_\\ell)$ is accepted~$\\automatonP(U_c, D_c)$.\n\t\nAssume now that $\\ell$ and $\\ell+1$ are not reversed in $\\pi$. We want to show that $\\ell$ never appears in the reduced expressions for $\\pi$, ie. that $\\pi([\\ell]) = [\\ell]$ and $\\pi([n] \\ssm [\\ell]) = [n] \\ssm [\\ell]$. Otherwise, the reduced expression $w_\\ell$ accepted by $\\automatonU(\\ell)$ would see first a $s_{\\ell+1}$, and then a $s_\\ell$ before it sees any $s_{\\ell-1}$, so that we would have an inversion $k \\ell$ in $\\pi$ for some $\\ell < k$. Similarly, the reduced expression $w_{\\ell+1}$ accepted by $\\automatonD(\\ell+1)$ should see first a $s_{\\ell-1}$ and then a $s_\\ell$ before it sees any $s_{\\ell+1}$, so that we would have an inversion $(\\ell+1) i$ in $\\pi$ for some $i < \\ell+1$. Since $\\ell$ and $\\ell+1$ are not reversed, we see $k \\ell (\\ell+1) i$ which contradicts twice \\cref{thm:patternAvoidance}. We conclude that this case never happens, so that we can work in the parabolic subgroup of permutations that never use $s_\\ell$ in their reduced expressions.\n\\end{proof}\n\n\\subsection{Some negative observations}\n\nWe conclude this paper with some negative observations and warnings about the connection between $c$-sorting and $(U_c,D_c)$-permutree sorting.\nFirst, we want to underline that using $c$-sorting words to test whether a permutation avoids~$jki$ or $kij$ for a fixed~$j$ is dangerous for the following two reasons.\n\n\\begin{remark}\nEven if a permutation~$\\pi$ avoids~$jki$ (resp.~$kij$) for a given~$j$, there might be no Coxeter element~$c$ for which $\\pi$ is $c$-sortable and~$j \\in U_c$ (resp.~$j \\in D_c$).\nFor instance, the permutation $41325 \\in \\fS_5$ avoids $2ki$ and~$ki4$, but contains~$352$ and~$413$, so it is not $c$-sortable for any Coxeter element~$c$.\n\\end{remark}\n\n\\begin{remark}\nWhen a permutation~$\\pi$ is not $c$-sortable, there might exist~$j \\in U_c$ (resp.~$j \\in D_c$) for which the $c$-sorting word~$\\pi(c)$ is not accepted by~$\\automatonU(j)$ (resp.~$\\automatonD(j)$) even if~$\\pi$ avoids~$jki$ (resp.~$kij$).\nFor instance, consider~$c = s_2 \\cdot s_1 \\cdot s_3$ and~$\\pi = 4213 = s_3 \\cdot s_1 \\cdot s_2 \\cdot s_1 =  s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2 =  s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$.\nThen~$2 \\in U_c$, and the $c$-sorting word~$\\pi(c) = s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$ is rejected by~$\\automatonU(2)$ while~$\\pi$ contains no~$2ki$ (and indeed~$s_3 \\cdot s_2 \\cdot s_1 \\cdot s_2$ is accepted by~$\\automatonU(2)$).\n\\end{remark}\n\nWe conclude the paper with an observation about sorting networks and permutree sorting.\n\n\\begin{remark}\nGiven a Coxeter element $c$, the word $c^\\infty$ which is used to compute $\\pi(c)$ is a \\defn{sorting network}. This means that we decide \\emph{beforehand} a list of transpositions to apply if appropriate. On the other hand, the permutree sorting given in Algorithm~\\ref{algo:permutreeSorting} is \\emph{not} a sorting network. Indeed, the order on transpositions depends on the permutation and more specifically on the state of the automaton we are at. A natural question then occurs: can we replace the permutree sorting algorithm by a sorting network? Or said differently, when~$U$ and~$D$ are disjoint but do not cover~$\\{2, \\dots, n-1\\}$, can we find a word $\\tilde{c}$ which plays the role of $c^\\infty$ in the sense that looking at $\\pi(\\tilde{c})$ would be enough to check whether $\\pi$ is accepted by $\\automatonP(U,D)$?\n\nThe answer is negative in general. A counter-example is found for $n = 5$, $U = \\{ 2\\}$, and $D=\\{ 4 \\}$. In this case one can check through computer exploration that no reduced word $\\tilde{c}$ of the maximal permutation $54321$ can be used as a sorting network. Namely, for all choices of $\\tilde{c}$, there exist a permutation $\\pi$ which is accepted by $\\automatonP(U,D)$ whereas the reduced expression $\\pi(\\tilde{c})$ is rejected. The healthy states of $\\automatonP(\\{ 2\\},\\{ 4 \\})$ are shown in Figure~\\ref{fig:automataProduct}. We see that accepted reduced expressions can start with either $s_2$ or $s_3$. For some permutations, such as $54213$ shown in Example~\\ref{exm:algo2}, all accepted reduced expressions start with $s_3$ whereas for some other permutations such as $35421$, all accepted reduced expressions start with $s_2$. This eventually leads to an empty intersection for the choice of $\\tilde{c}$.\n\nNevertheless, it seems interesting to study in which case the answer is positive. The Cambrien case with the $c$-sorting word when $U$ and~$D$ form a partition of~$\\{2, \\dots, n-1\\}$ is one example. The case where $|U| + |D| = 1$ is another one. This is the case corresponding to Theorem~\\ref{thm:patternAvoidance} where we have only one automaton. In this case, we can construct a word $\\tilde{c}$ by reading the healthy states of the automaton linearly, adding at each state the word $(s_{i_1} \\cdots s_{i_k})^k s_j$ where $s_{i_1}, \\dots, s_{i_k}$ are the looping transitions and $s_j$ is the transition going to the next healthy state. This process gives a prefix that can be extended in any way to obtain a proper sorting word $\\tilde{c}$. For example, if $U = \\{2\\}$, we obtain the prefix $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3$ and indeed $s_3 \\cdot s_2 \\cdot s_1 \\cdot s_3 \\cdot s_2 \\cdot s_1$ acts as a sorting network equivalent to the $(\\{2\\},\\varnothing)$-permutree sorting. This process actually seems to extend to all cases where, at each healthy state of the intersection automaton, the choices for the healthy transitions commute. For example, in the case where $n=5$, $U = \\{ 4 \\}$ and $D = \\{ 2 \\}$ as illustrated in Figure~\\ref{fig:automataProduct}, the word $s_1 \\cdot s_2 \\cdot s_4 \\cdot s_3 \\cdot s_2 \\cdot s_1 \\cdot s_4 \\cdot s_3 \\cdot s_2$ gives a proper sorting network. \n\\end{remark}\n\n\n\\bibliographystyle{alpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe fine topology on an open set $\\Omega\\subset{\\mathbb R}^n$ was introduced\nby H.\\ Cartan in classical potential theory. It is defined as the\nsmallest topology on $\\Omega$ in which every super\\-harmonic function on\n$\\Omega$ is continuous.\nPotential theory on a finely open set, for example in ${\\mathbb R}^n$, was introduced\nand studied in the 1970's by the second named author\n\\cite{F1}. The harmonic and super\\-harmonic functions and the potentials in this\ntheory are termed finely [super]harmonic functions and fine\npotentials. Generally one distinguishes by the prefix `fine(ly)'\nnotions in fine potential theory from those in classical potential\ntheory on a usual (Euclidean) open set. Large parts of classical\npotential theory have been extended to fine potential theory.\n\nThe integral representation of positive ($=$ nonnegative) finely super\\-harmonic\nfunctions by using Choquet's method of extreme points was studied by the first\nnamed author in \\cite{El1}, where it was shown that the cone of positive\nsuper\\-harmonic functions equipped with the natural topology has a compact base.\nThis allowed the present authors\nin \\cite{EF1} to define the Martin compactification and the Martin boundary of\na fine domain $U$ in ${\\mathbb R}^n$. The Martin compactification $\\overline U$ of $U$\nwas defined by injection of $U$ in a compact base of the cone $\\cal S(U)$ of\npositive finely super\\-harmonic functions on $U$. While the Martin boundary of\na usual domain is closed and hence compact, all\nwe can say in the present setup is that the Martin boundary $\\Delta(U)$ of $U$\nis a $G_\\delta$ subset of the compact Riesz-Martin space\n$\\overline U=U\\cup\\Delta(U)$ endowed with the natural topology. Nevertheless\nwe have defined in \\cite{EF1} a suitably measurable Riesz-Martin kernel\n$K:U\\times\\overline U\\longrightarrow[0,+\\infty]$.\nEvery function $u\\in\\cal S(U)$ has an integral representation\n$u(x)=\\int_{\\overline U}K(x,Y)d\\mu(Y)$ in terms of a Radon measure $\\mu$ on\n$\\overline U$. This representation is unique if it is required that $\\mu$ be\ncarried by $U\\cup\\Delta_1(U)$, where $\\Delta_1(U)$ denotes the minimal Martin\nboundary of $U$, which likewise is a $G_\\delta$ in $\\overline U$. In this case\nof uniqueness we write $\\mu=\\mu_u$. It was shown that $u$ is a\nfine potential, resp.\\ an invariant function, if and only if $\\mu_u$ is\ncarried by $U$, resp.\\ by $\\Delta(U)$. The invariant functions, likewise\nstudied in \\cite{EF1}, generalize the positive harmonic functions in the\nclassical Riesz decomposition theorem. Finite valued invariant functions are\nthe same as positive finely harmonic functions.\n\nThere is a notion of minimal thinness of a set $E\\subset U$ at a point\n$Y\\in\\Delta_1(U)$, and an associated minimal-fine filter $\\cal F(Y)$,\nwhich allowed the authors\nin \\cite{EF1} to obtain a generalization of the classical Fatou-Na{\\\"\\i}m-Doob\ntheorem.\n\nIn a continuation \\cite{EF2} of \\cite{EF1} we studied sweeping on a subset\nof the Riesz-Martin space,\nboth relative to the natural topology and to the minimal-fine topology on\n$\\overline U$, and we showed that the two notions of sweeping are identical.\nIn the present further continuation of \\cite{EF1} and \\cite{EF2} we investigate\nthe Dirichlet problem at the Martin boundary of our given fine domain $U$\nby adapting the Perron-Wiener-Brelot (PWB) method to the present setup.\nIt is a complication that there is no Harnack convergence theorem for finely\nharmonic functions, and hence the infimum of a sequence of  upper\nPWB-functionss on $U$ may equal $-\\infty$ precisely\non some nonvoid proper finely closed subset of $U$.\nWe define resolutivity of a numerical function on $\\Delta(U)$ in a standard\nway and show that it is equivalent to a weaker, but technically supple\nconcept called quasi\\-resolutivity, which possibly has not been considered\nbefore in the literature (for the classical case where $U$ is Euclidean open).\nOur main result implies the corresponding known result for the classical case,\ncf.\\ \\cite[Theorem 1.VIII.8]{Do}.\nAt the end of Section 3 we obtain analogous results for the case where the\nupper and lower PWB-classes are defined in terms of the minimal-fine topology\non $\\overline U$ instead of the natural topology. It follows that the two\ncorresponding concepts of resolutivity are compatible. This result is\npossibly new even in the classical case.\nA further alternative, but actually equivalent, concept of resolutivity is\ndiscussed in the closing Section 4.\n\n{\\bf Notations}: If $U$ is a fine domain in $\\Omega$ we\ndenote by ${\\cal S}(U)$ the convex cone of positive finely super\\-harmonic\nfunctions on $U$ in the sense of \\cite{F1}. The convex cone of fine potentials\non $U$ (that is, the functions in ${\\cal S}(U)$ for which every finely\nsubharmonic minorant is $\\le 0$) is denoted by ${\\cal P}(U)$. The cone of\ninvariant functions on $U$ is the\northogonal band to ${\\cal P}(U)$ relative to ${\\cal S}(U)$.\nBy $G_U$ we denote the (fine) Green kernel for $U$, cf.\\ \\cite{F2}, \\cite{F4}.\nIf $A\\subset U$ and $f:A\\longrightarrow[0,+\\infty]$ one denotes by $R{}_f^A$,\nresp.\\ ${\\widehat R}{}_f^A$, the reduced function, resp.\\ the swept function,\nof $f$ on $A$ relative to $U$, cf.\\ \\cite[Section 11]{F1}. If $u\\in\\cal S(U)$\nand $A\\subset U$ we may write ${\\widehat R}{}_u^A$ for\n${\\widehat R}{}_f$ with $f:=1_Au$.\n\n\n\\section{The upper and lower PWB$^h$-classes of a function on $\\Delta(U)$}\n\\label{sec2}\n\nWe shall study the Dirichlet problem at $\\Delta(U)$ relative to a fixed\nfinely harmonic function $h>0$ on $U$. We denote by $\\mu_h$ the\nmeasure on $\\Delta(U)$ carried by $\\Delta_1(U)$ and representing $h$,\nthat is $h=\\int K(.,Y)d\\mu_h(Y)=K\\mu_h$. A function $u$ on $U$ (or on\nsome finely open subset of $U$) is said to be finely\n$h$-hyper\\-harmonic, finely $h$-super\\-harmonic, $h$-invariant, or a fine\n$h$-potential, respectively, if it has the form $u=v\/h$, where\n$v$ is finely hyper\\-harmonic, finely super\\-harmonic, invariant, or a\nfine potential, respectively.\n\nLet $f$ be a function on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$.\nA finely $h$-hyper\\-harmonic function $u=v\/h$ on $U$ is said\nto belong to the upper PWB$^h$-class, denoted by\n${\\overline {\\cal U}}{}_f^h$, if $u$ is lower bounded and if\n$$\\underset{x\\to Y,\\,x\\in U}{\\liminf}u(x)\\ge f(Y)\n\\quad\\text{ for every }\\;Y\\in \\Delta(U).$$\nWe define\n $$\\dot H{}_f^h=\\inf\\,{\\overline{\\cal U}}{}_f^h,\n\\quad{\\overline H}{}_f^h=\\widehat{\\dot H{}_f^h}\n={\\widehat\\inf}\\,{\\overline {\\cal U}}{}_f^h\\;(\\le\\dot H{}_f^h).$$\nBoth functions $\\overline H{}_f^h$ and $\\dot H{}_f^h$ are needed here,\nunlike the classical\ncase where we have the Harnack convergence theorem and hence\n$\\overline H{}_f^h=\\dot H{}_f^h$.\nIn our setup, $\\dot H{}_f^h$\nmay be neither finely $h$-hyperharmonic nor identically $-\\infty$, but only\nnearly finely $h$-hyperharmonic on the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}$ which can be a nonvoid proper subset of $U$,\nsee Example \\ref{example2.0} below, which also shows that $\\Delta(U)$ can be\nnon-compact.\nAccording to the fundamental convergence theorem \\cite[Theorem 11.8]{F1}\n${\\overline H}{}_f^h$ is finely $h$-hyper\\-harmonic on\n$\\{\\overline H{}_f^h>-\\infty\\}$\nand ${\\overline H}{}_f^h={\\dot H}{}_f^h$ quasieverywhere (q.e.)\\! there;\nfurthermore, since ${\\overline{\\cal U}}{}_f^h$ is lower directed, there is a\ndecreasing sequence\n$(u_j)\\subset\\overline{\\cal U}{}_f^h$ such that $\\inf_ju_j=\\dot H{}_f^h$.\nClearly, $\\dot H{}_f^h$ is finely u.s.c.\\ on all of  $U$, and\n${\\overline H}{}_f^h$ is finely l.s.c.\\ there.\n\nThe lower PWB$^h$ class ${\\underline{\\cal U}}{}_f^h$\nis defined by\n${\\underline {\\cal U}}{}_f^h=-{\\overline {\\cal U}}_{-f}^h$, and we have\n$\\dot H{}_0^h=0$, hence also\n$\\overline H{}_0^h={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_0^h\n=\\underline H_0^h=0$.\nIt follows that if $f\\ge0$ then\n$\\dot H_f^h\\ge\\overline H_f^h\\ge0$ and therefore only positive functions\nof class $\\overline{\\cal U}{}_f^h$ need to be considered in the\ndefinition of $\\dot H{}_f^h$ and hence of $\\overline H_f^h$.\nMoreover, $\\dot H{}_{\\alpha f+\\beta}^h=\\alpha\\dot H{}_f^h+\\beta$ and hence\n$\\overline H_{\\alpha f+\\beta}^h=\\alpha\\overline H_f^h+\\beta$ for real\nconstants $\\alpha\\ge0$ and $\\beta$ (when $0$ times $\\pm\\infty$ is defined\nto be $0$).\n\n\\begin{example}\\label{example2.0} In $\\Omega={\\mathbb R}^n$ with the Green kernel\n$G(x,y)=|x-y|^{2-n}$, $n\\ge4$, let $\\omega\\subset\\Omega$ be a bounded H\\\"older\ndomain such that $\\omega$ is irregular with a single\nirregular boundary point $z$, cf.\\ e.g.\\ \\cite[Remark 6.6.17]{AG}. Take\n$U=\\omega\\cup\\{z\\}$. According to \\cite[Theorems 1 and 3.1]{Ai} the\nEuclidean boundary $\\partial\\omega$ of $\\omega$ is topologically contained\nin the Martin boundary $\\Delta(\\omega)$.\nIn particular, $z$ is non-isolated as a point of $\\Delta(\\omega)$.\nBut $\\Delta(U)=\\Delta(\\omega)\\setminus\\{z\\}$, where $z$ is identified\nwith $P_z$ (see \\cite[Section 3]{EF1}), and since $\\Delta(\\omega)$ is\ncompact we infer that $\\Delta(U)$ is noncompact.\nIn ${\\mathbb R}^n$ choose a sequence $(z_j)$ of points of\n$\\complement{\\overline\\omega}$ such that $|z_j-z|\\le2^{-j}$.\nThen $u:=\\sum_j2^{-j}G(.,z_j)$ is infinite at $z$, but finite and\nharmonic on $\\omega$. Furthermore, $u=\\sup_ku_k$,\nwhere $u_k:=\\sum_{j\\le k}2^{-j}G(.,z_j)$ is harmonic and bounded on\n$\\overline\\omega$ ($\\subset{\\mathbb R}^n$). It follows that $(u_k)_{|U}$\nis of class $\\underline{\\cal U}{}_f^h$, where $f:=u_{|\\Delta(U)}$.\nIn fact,\n$$\n\\underset{x\\to Y,\\,x\\in U}{\\lim}\\,u_k(x)=u_k(Y)\\le u(Y)=f(Y)\n$$\nfor $Y\\in\\Delta(U)$ (natural limit on $U\\cup\\Delta(U)$, or equivalently\nEuclidean limit on $\\omega\\cup((\\partial\\omega)\\setminus\\{z\\})$. Thus\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge(u_k)_{|U}$, and hence\n$$\n\\underline H{}_f^h(z)\\ge\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(z)\n\\ge\\sup_ku_k(z)=u(z)=+\\infty.\n$$\nTo show that $\\underline H{}_f^h<+\\infty$ on $U\\setminus\\{z\\}$ ($\\cong\\omega$),\nlet $v\\in\\underline{\\cal U}{}_f^h$. Being upper bounded on the bounded open\nset $\\omega$, $v$ is subharmonic on $\\omega$ by \\cite[Theorem 9.8]{F1},\nand so is therefore $v-u$.\nFor any $Y\\in\\Delta(U)$ ($\\cong(\\partial\\omega)\\setminus\\{z\\}$) we have\n$$\n\\underset{x\\to Y,\\,x\\in\\omega}{\\limsup}\\,v(x)\\le f(Y)<+\\infty\n$$\n(also with Euclidean limit), or equivalently\n$$\n\\underset{x\\to Y,\\,x\\in\\omega}{\\limsup}\\,(v(x)-u(x))\\le0.\n$$\nSince $\\{z\\}$ is polar and $v-u\\le v$ is upper bounded, it follows by a\nboundary minimum principle that $v-u\\le0$, that is, $v\\le u$ on $\\omega$.\nBy varying $v\\in\\underline{\\cal U}{}_f^h$ we conclude that\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\\le u$\non $\\omega\\cong U\\setminus\\{z\\}$ and hence by regularization that\n$\\underline H{}_f^h\\le u<+\\infty$ on $U\\setminus\\{z\\}$.\nAltogether, $\\overline H{}_{-f}^h=-\\underline H_f^h$ equals\n$-\\infty$ at $z$, but is finite on $U\\setminus\\{z\\}$.\n\\end{example}\n\nHenceforth we fix the finely harmonic function $h>0$ on $U$, relative to\nwhich we shall study the Dirichlet problem at $\\Delta(U)$. Similarly to\nthe classical case, cf.\\ \\cite[p.\\ 108]{Do}, we pose the following definition,\ndenoting by $1_A$ the indicator function of a set $A\\subset\\Delta(U)$:\n\n\\begin{definition}\\label{def4.1}\n A subset $A$ of $\\Delta(U)$ is said to be $h$-harmonic measure null if\n  $\\overline H{}_{1_A}^h=0$.\n\\end{definition}\n\nIt will be shown in Corollary \\ref{cor6.10c} that $A$ is $h$-harmonic measure\nnull if and only if $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$.\n\n\\begin{prop}\\label{prop4.2}\n{\\rm{(a)}} Every countable union of $h$-harmonic measure null sets is $h$-harmonic\nmeasure null.\n\n{\\rm{(b)}} A set $A\\subset\\Delta(U)$ is $h$-harmonic measure null if and only\nif there is a finely $h$-super\\-harmonic function $u$ (positive if we like)\non $U$ such that $\\lim_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every $Y\\in A$.\n\n{\\rm{(c)}} If $f:\\Delta(U)\\to[0,+\\infty]$ has $\\overline H{}_f^h=0$ then\n$\\{f>0\\}$ is $h$-harmonic measure null.\n\n{\\rm{(d)}} If $f:\\Delta(U)\\to[0,+\\infty]$ has $\\overline H{}_f^h<+\\infty$ then\n$\\{f=+\\infty\\}$ is $h$-harmonic measure null.\n\n{\\rm{(e)}} If $f,g:\\Delta(U)\\to[0,+\\infty]$ and if $f\\le g$ off some\n$h$-harmonic measure null set then $\\overline H{}_f^h\\le\\overline H{}_g^h$.\n\\end{prop}\n\n\\begin{proof} We adapt the proof in \\cite[p.\\ 108, 111]{Do} for the\nclassical case.\n\n(a) Fix a point $x_0$ of the co-polar subset\n$\\bigcap_j\\{\\dot H{}_{1_{A_j}}^h=0\\}$ of $U$. For given $\\varepsilon>0$ and\nintegers $j>0$ there are functions $u_j\\in\\overline{\\cal U}{}_{1_{A_j}}^h$ with\n$u_j(x_0)<2^{-j}\\varepsilon$. It follows that the function\n$u:=\\sum_ju_j$ is of class $\\overline{\\cal U}{}_{1_A}^h$ because\n$\\sum_j1_{A_j}\\ge1_A$ on $\\Delta(U)$. Consequently,\n$\\overline H{}_{1_A}^h(x_0)\\le\\dot H{}_{1_A}^h(x_0)\\le u(x_0)<\\varepsilon$,\nand the positive finely $h$-hyperharmonic function $\\overline H{}_{1_A}^h$\ntherefore equals $0$ at $x_0$ and so indeed everywhere on $U$.\n\n(b) If $\\overline H{}_{1_A}=0$ then $\\dot H{}_{1_A}^h=0$ q.e., so\nwe may choose $x_0\\in U$ with $\\dot H{}_{1_A}^h(x_0)=0$. For integers\n$j>0$ there exist positive finely $h$-superharmonic functions\n$u_j\\in\\overline{\\cal U}{}_{1_A}^h$ on $U$ such that $u_j(x_0)<2^{-j}\\varepsilon$.\nThe function $u:=\\sum_ju_j$ is positive and finely $h$-superharmonic on $U$\nbecause $u(x_0)<+\\infty$. Furthermore,\n $\\liminf_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every  $Y\\in A$.\nConversely, if there exists a function $u$ as described in (b), we may arrange\nthat $u\\ge0$ after adding a constant. Then\n$\\varepsilon u\\in\\overline{\\cal U}{}_{1_A}^h$ for every $\\varepsilon>0$. It follows that\n$\\dot H{}_{1_A}^h\\le\\varepsilon u$ and by varying $\\varepsilon$ that\n$\\dot H{}_{1_A}^h=0$ off the polar set of infinities of $u$, and hence\nq.e.\\ on $U$. It follows that indeed $\\overline H{}_f^h=0$.\n\n(c) For integers $j\\ge1$ let $f_j$ denote the indicator function on $U$\nfor the set $\\{f>1\/j\\}$. Then $0=\\overline H{}_f^h\\ge\\overline H{}_{f_j}^h\/j$,\nso the sets $\\{f>1\/j\\}$ are $h$-harmonic measure null, and so is by (a) the\nunion $\\{f>0\\}$ of these sets.\n\n(d) Choose $x_0\\in U$ so that $\\dot H{}_f^h(x_0)=\\underline H{}_f^h(x_0)$\n($<+\\infty$) and $u\\in\\overline{\\cal U}{}_f^h$. Then\n$\\lim_{x\\to Y,\\,x\\in U}u(x)=+\\infty$ for every\n$Y\\in A:=\\{f=+\\infty\\}$. After adding a constant we arrange that the finely\n$h$-hyperharmonic function $u$ is positive. According to (b) it follows that\nindeed $\\overline H{}_{1_A}^h=0$.\n\n(e) Let $v\\in\\overline{\\cal U}{}_g^h$ and let $u$ be a positive\n$h$-superharmonic function on $U$ with limit $+\\infty$ at every point of\nthe $h$-harmonic measure null subset $\\{f>g\\}$ of $\\Delta(U)$. Then\n$\\dot H{}_f^h\\le v+\\varepsilon u\\in\\overline{\\cal U}{}_f^h$\nfor every $\\varepsilon>0$. Hence $\\dot H{}_f^h\\le v$ q.e., and so\n$\\overline H{}_f^h\\le v$ everywhere on $U$. By varying $v$ it follows that\n$\\overline H{}_f^h\\le\\dot H{}_g^h$ and so indeed by finely l.s.c.\\\nregularization $\\overline H{}_f^h\\le\\overline H{}_g^h$.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.1}\nLet $f$ be a function on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$.\n\n{\\rm{(a)}} $\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nand hence $\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ and $\\dot H{}_f^h\n\\ge\\underline H{}_f^h$.\n\n{\\rm{(b)}} $\\overline H{}_f^h\n\\ge\\underline H{}_f^h$ on\n$\\{{\\overline H}{}_f^h>-\\infty\\}\\cup\n\\{{\\underline H}{}_f^h<+\\infty\\}$.\n\n{\\rm{(c)}} If $f$ is lower bounded then\n$\\overline H{}_f^h(x)=\\dot H{}_f^h(x)$ at any point $x\\in U$\nat which $\\dot H{}_f^h(x)<+\\infty$. If \\,$f\\ge0$ on $\\Delta(U)$ then\n${\\overline H}{}_f^h$ $(\\ge0)$ is either identically $+\\infty$ or\n$h$-invariant on $U$.\n\n\\end{prop}\n\n\\begin{proof} Clearly, ${\\overline{\\cal U}}{}_f^h$ is lower directed and\n$\\underline{\\cal U}{}_f^h$ is upper directed. The constant function $+\\infty$\nbelongs to ${\\overline{\\cal U}}{}_f^h$. If \\,$+\\infty$ is the only function\nof class ${\\overline{\\cal U}}{}_f^h$ then obviously $\\dot H{}_f^h=+\\infty$\nand hence ${\\overline H}{}_f^h=+\\infty$. In the remaining case it suffices\nto consider finely $h$-super\\-harmonic functions in the definition of\n$\\dot H{}_f^h$ and hence of \\,${\\overline H}{}_f^h$.\n\n(a) Let $u\\in\\overline{\\cal U}{}_f^h$ and $v\\in\\underline{\\cal U}{}_f^h$.\nThen $u-v$ is well defined, finely $h$-hyper\\-harmonic, and lower bounded\non $U$, and\n\\begin{eqnarray*}\\underset{x\\to Y,\\,x\\in U}{\\liminf}(u(x)-v(x))\\!\\!\\!\n&\\ge&\\!\\!\\!\\underset{x\\to Y,\\,x\\in U} {\\liminf}u(x)\n-\\underset{x\\to Y,\\,x\\in U} {\\limsup}v(x)\\\\\n&\\ge&\\!\\!\\! f(Y)-f(Y)=0\n\\end{eqnarray*}\nif $f(Y)$ is finite; otherwise $\\liminf u(x)-\\limsup v(x)=+\\infty\\ge0$,\nfor if for example $f(Y)=+\\infty$ then $\\liminf u=+\\infty$ whereas\n$\\limsup v<+\\infty$ since $v$ is upper bounded.\nBy the minimal-fine boundary minimum property given in\n\\cite[Corollary 3.13]{EF2} together with \\cite[Proposition 3.5]{EF2}\napplied to the finely super\\-harmonic function $hu-hv$\n(if $\\ne+\\infty$) it then follows that $u-v\\ge0$, and hence\n$u\\ge v$. By varying $u$ and $v$ in either order we obtain\n$\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$.\nSince\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\sup\\underline{\\cal U}{}_f^h$\nis finely l.s.c.\\ it follows that\n${\\overline H}{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\nand similarly $\\dot H{}_f^h\n\\ge{\\underline H}{}_f^h$.\n\n(b) Consider any point $x_0$ of the finely open set\n$V=\\{\\overline H{}_f^h>-\\infty\\}$.\nSince ${\\overline H}{}_f^h$ is finely $h$-hyper\\-harmonic\nand hence finely continuous on $V$\nwe obtain by (a)\n$$\n{\\overline H}{}_f^h(x_0)\n=\\underset{x\\to x_0,\\,x\\in V}{\\fine\\lim}\\,{\\overline H}{}_f^h(x)\n\\ge\\underset{x\\to x_0,\\,x\\in V\\setminus E}{\\fine\\lim\\sup}\\,\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n={\\underline H}{}_f^h(x_0).\n$$\nThe case $x_0\\in\\{\\underline H{}_f^h<+\\infty\\}$ is treated similarly,\nor by replacing $f$ with $-f$.\n\n(c) The former assertion reduces easily to the case $f\\ge0$, whereby\n$\\overline{\\cal U}{}_f^h$ consists of positive functions. We may assume that\n$\\overline H{}_f^h\\not\\equiv+\\infty$, and hence\n$h\\overline{\\cal U}{}_f^h\\subset\\cal S(U)$.\nConsider the cover of $U$ by the finely open sets $V_k$ from\n\\cite[Lemma 2.1 (c)]{EF2}. Then $h\\overline{\\cal U}{}_f^h$ is a Perron family\nin the sense of \\cite[Definition 2.2]{EF2}. It therefore follows by\n\\cite[Theorem 2.3]{EF2} that indeed\n${\\overline H}{}_f^h=\\widehat{\\inf}\\,\\overline{\\cal U}{}_f^h$ is\n$h$-invariant, and that $\\overline H{}_f^h(x)=\\dot H{}_f^h(x)$ at any point\n$x\\in U$ at which $\\dot H{}_f^h(x)<+\\infty$.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.2} Let $f,g$ be two functions on $\\Delta(U)$ with\nvalues in $\\overline{\\mathbb R}$.\n\n{\\rm{1.}} If $f+g$ is well defined everywhere on $\\Delta(U)$ then\nthe inequality $\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$\nholds at each point of $U$ where $\\dot H{}_f^h+\\dot H{}_g^h$ is\nwell defined.\n\n{\\rm{2.}} If $(f+g)(Y)$ is defined arbitrarily at points $Y$ of\nundetermination then the inequality\n$\\overline H{}_{f+g}^h\\le\\overline H{}_f^h+\\overline H{}_g^h$\nholds everywhere on $\\{\\overline H{}_f^h,\\overline H{}_g^h>-\\infty\\}$.\n\n{\\rm{3.}} For any point $x\\in U$ we have $\\dot H{}_f^h(x)<+\\infty$\nif and only if $\\dot H{}_{f\\vee0}^h(x)<+\\infty$.\n\n{\\rm{4.}} Let $(f_j)$ be an increasing sequence of lower bounded functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$. Writing $f=\\sup_jf_j$ we have\n$\\overline H{}_f^h=\\sup_j\\overline H{}_{f_j}^h$ and\n$\\dot H{}_f^h=\\sup_j\\dot H{}_{f_j}^h$.\n\\end{prop}\n\n\\begin{proof} For 1., 2., and 4.\\ we proceed much as in\n\\cite[1.VIII.7, Proof of (c), (b), and (e)]{Do}.\nFor Assertion 1., consider any two functions $u\\in\\overline{\\cal U}{}_f^h$\nand $v\\in\\overline{\\cal U}{}_g^h$.\nThen $u+v\\in\\overline{\\cal U}{}_{f+g}^h$ and hence $\\dot H{}_{f+g}^h\\le u+v$.\nBy varying $v$ it follows that $\\dot H{}_{f+g}^h\\le u+\\dot H{}_g^h$ on\n$\\{\\dot H{}_g^h>-\\infty\\}$.\nBy varying $u$ this leads to\n$\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$ whereever the sum is well defined\non $\\{\\dot H{}_g^h>-\\infty\\}$.\nBy interchanging $u$ and $v$ we infer that\n$\\dot H{}_{f+g}^h\\le\\dot H{}_f^h+\\dot H{}_g^h$ altogether holds whereever the\nsum is well defined on $\\{\\dot H{}_f^h,\\dot H{}_g^h>-\\infty\\}$.\nOn the residual set $\\{\\dot H{}_f^h=-\\infty\\}\\cup\\{\\dot H{}_g^h=-\\infty\\}$\nit is easily seen that\n$\\dot H{}_{f+g}^h=-\\infty=\\dot H{}_f^h+\\dot H{}_g^h$ whereever the sum is well\ndefined.\n\nFor Assertion 2., suppose first that $f,g<+\\infty$ (and so $f+g$ is well\ndefined). In the proof of Assertion 1.\\ we had\n$\\dot H{}_{f+g}^h\\le u+\\dot H{}_g^h$ for $\\dot H{}_g^h>-\\infty$, which is\nsatisfied on $\\{\\overline H{}_g^h>-\\infty\\}$. It follows that\n$\\overline H{}_{f+g}^h\\le u+\\overline H{}_g^h$ there, and hence that\n$\\overline H{}_{f+g}^h\\le\\overline H{}_f^h+\\overline H{}_g^h$ there (whereever\nwell defined). In the general case\ndefine functions $f_0<+\\infty$, resp.\\ $g_0<+\\infty$, which equal $f$,\nresp.\\ $g$, except on the set $\\{f=+\\infty\\}$, resp.\\ $\\{g=+\\infty\\}$.\nWe may assume that these exceptional sets are $h$-harmonic measure null,\nfor if e.g.\\ $\\{f=+\\infty\\}$ is not $h$-harmonic measure null then\n$\\overline H{}_f^h\\equiv=+\\infty$ by Proposition \\ref{prop4.2} (d),\nin which case 1.\\ becomes trivial.\nIt therefore follows in view of Proposition\n\\ref{prop4.2} (a), (e) that $f+g=f_0+g_0$ off the $h$-harmonic measure\nnull set $\\{f=+\\infty\\}\\cup\\{g=+\\infty\\}$ and hence by Proposition\n\\ref{prop4.2} (e) that\n$$\n\\overline H{}_{f+g}^h=\\overline H{}_{f_0+g_0}^h\n\\le\\overline H{}_{f_0}^h+\\overline H{}_{g_0}^h\n\\le\\overline H{}_f^h+\\overline H{}_g^h$$\non the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}\\cap\\{\\overline H{}_g^h>-\\infty\\}$,\nthe second inequality because $f_0\\le f$ and $g_0\\le g$.\n\nFor Assertion 3., let $x\\in U$ be given, and suppose that\n$\\dot H{}_f^h(x)<+\\infty$.\nThere is then $u\\in\\overline{\\cal U}{}_f^h$ with $u(x)<+\\infty$, $u$ being\nfinely $h$-superharmonic $\\ge-c$ for some constant $c\\ge0$. It follows that\n$u+c\\in\\overline{\\cal U}{}_{f\\vee0}^h$ and hence\n$\\dot H{}_{f\\vee0}^h(x)\\le u(x)+c<+\\infty$. The converse follows from\n$\\dot H{}_f^h\\le\\dot H{}_{f\\vee0}^h$.\n\nAssertion 4.\\ reduces easily to the case of positive functions $f_j$.\nConsider first the case of $\\overline H$. Then\n${\\overline H}{}_f^h$ and each ${\\overline H}{}_{f_j}^h$ are\npositive and hence finely $h$-hyper\\-harmonic by Proposition \\ref{prop6.1} (c).\nThe inequality ${\\overline H}{}_f^h\\ge\\sup_j{\\overline H}{}_{f_j}^h$ is obvious,\nand we may therefore assume that the positive finely $h$-hyper\\-harmonic\nfunction $\\sup_j{\\overline H}{}_{f_j}^h$ is not identically $+\\infty$, and\ntherefore is $h$-invariant, again according to\nProposition \\ref{prop6.1} (c). Denote $E_j$ the polar subset\n$\\{\\overline H{}_{f_j}^h<\\dot H{}_{f_j}^h\\}$\nof $U$ and write $E:=\\bigcup_jE_j$ (polar). For a fixed $x\\in U\\setminus E$\nand for given $\\varepsilon>0$ choose functions $u_j\\in{\\overline{\\cal U}}{}_{f_j}^h$\nso that\n\\begin{eqnarray}u_j(x)\\!\\!\\!\n&<&\\!\\!\\!\\dot H{}_{f_j}^h(x)+2^{-j}\\varepsilon=\\overline H{}_{f_j}^h(x)+2^{-j}\\varepsilon.\n\\end{eqnarray}\nIn particular, $u_j$ is finely $h$-super\\-harmonic.\n Define a finely $h$-hyper\\-harmonic function $u$ by\n\\begin{eqnarray}u\\!\\!\\!\n&=&\\!\\!\\!\\sup_j{\\overline H}{}_{f_j}^h\n+\\sum_j(u_j-{\\overline H}{}_{f_j}^h)\n\\ge{\\overline H}{}_{f_k}^h+(u_k-{\\overline H}{}_{f_k}^h)=u_k\n\\end{eqnarray}\nfor any index $k$. Then\n$$\\liminf_{x\\to Y,\\,x\\in U}u(x)\n\\ge\\liminf_{x\\to Y,\\,x\\in U}u_k(x)\n\\ge f_k(Y)$$\nfor every $Y\\in\\Delta(U)$ and every index $k$.\nThus $u\\in{\\overline{\\cal U}}{}_f^h$ and\n${\\overline H}{}_f^h\\le{\\dot H}{}_f^h\\le u$.\nIn particular, by the former equality (2.2) and by (2.1),\n\\begin{eqnarray}{\\overline H}{}_f^h(x)\\!\\!\\!\n&\\le&\\!\\!\\!u(x)\n\\le\\sup_j{\\overline H}{}_{f_j}^h(x)+\\varepsilon,\n\\end{eqnarray}\nand hence the finely $h$-hyper\\-harmonic function ${\\overline H}{}_f^h$ is\nfinely $h$-super\\-harmonic.\nBecause $\\sup_j{\\overline H}{}_f^h$ is $h$-invariant and majorized by\n${\\overline H}{}_f^h$, the function\n${\\overline H}{}_f^h-\\sup_j {\\overline H}{}_{f_j}^h$ is finely\n$h$-super\\-harmonic $\\ge0$ on $U$ by \\cite [Lemma 2.2]{EF1},\nand $\\le\\varepsilon$ at $x$. For $\\varepsilon\\to0$ we obtain the remaining inequality\n${\\overline H}{}_f^h\\le\\sup_j {\\overline H}{}_{f_j}^h$.\n\nIn the remaining case of $\\dot H$ we have\n$\\sup_j\\dot H{}_{f_j}^h(x)\\le\\dot H{}_f^h$. For any point $x\\in U$ at which\n$\\sup_j\\dot H{}_{f_j}^h(x)<\\dot H{}_f^h$ and for any $\\varepsilon>0$ choose functions\n$u_j\\in{\\overline{\\cal U}}{}_{f_j}^h$ so that\n\\begin{eqnarray}u_j(x)\\!\\!\\!\n&<&\\!\\!\\!\\dot H{}_{f_j}^h(x)+2^{-j}\\varepsilon.\n\\end{eqnarray}\nProceed as in the above case of $\\overline H$ by defining the finely\n$h$-hyperharmonic function by (2.2), replacing throughout $\\bar H$\nby $\\dot H$. Corresponding to (2.3) we now end by\n$$\n\\dot H{}_f^h(x)\\le u(x)\\le\\sup_j{\\dot H}{}_{f_j}^h(x)+\\varepsilon,\n$$\nfrom which the remaining inequality\n$\\dot H{}_f^h(x)\\le\\sup_j{\\dot H}{}_{f_j}^h(x)$ follows for $\\varepsilon\\to0$.\n\\end{proof}\n\n\n\\section{$h$-resolutive and $h$-quasiresolutive functions}\\label{sec3}\n\n\\begin{definition}\\label{def6.9}\nA function $f$ on $\\Delta(U)$ with values in ${\\overline {\\mathbb R}}$ is said\nto be $h$-resolutive if $\\overline H{}_f^h=\\underline H{}_f^h$ on $U$\nand if this function, also denoted by $H_f^h$,\nis neither identically $+\\infty$ nor identically $-\\infty$.\n\\end{definition}\n\nIt follows that $H_f^h$ is finely $h$-harmonic on the finely open set\n$\\{\\overline H{}_f^h>-\\infty\\}\\cap\\{\\underline H{}_f^h>+\\infty\\}\n=\\{-\\infty<H_f^h<+\\infty\\}$.\n\nFor any function $f:\\Delta(U)\\longrightarrow{\\mathbb R}$ we consider the following\ntwo subsets of $U$:\n$$\nE_f^h=\\{\\overline H{}_f^h=-\\infty\\}\\cup\n\\{\\underline H{}_f^h=+\\infty\\}\\cup\n\\{\\overline H{}_f^h\\ne\\underline H{}_f^h\\},\n$$\n$$\nP_f^h=\\{\\dot H{}_f^h>\\overline H{}_f^h>-\\infty\\}\n\\cup\\{{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n<\\underline H{}_f^h<+\\infty\\},\n$$\nof which $P_f^h$ is polar.\n\n\\begin{definition}\\label{def2.3} A function $f$ on $\\Delta(U)$\nwith values in ${\\overline {\\mathbb R}}$ is said to be\n$h$-quasi\\-resolutive if $E_f^h$ is polar, or equivalently if the relations\n$\\overline H{}_f^h>-\\infty$,\n$\\underline H{}_f^h<+\\infty$, and\n$\\overline H{}_f^h=\\underline H{}_f^h$ hold quasieverywhere on $U$.\n\\end{definition}\n\nWhen $f$ is $h$-quasi\\-resolutive on $U$ the functions $\\overline H{}_f^h$\nand $\\underline H{}_f^h$ are finely $h$-hyperharmonic and finely\n$h$-hypoharmonic, respectively, off the polar set\n$\\{\\overline H{}_f^h=-\\infty\\}\\cup\\{\\underline H{}_f^h=+\\infty\\}$, and they\nare actually equal and hence finely $h$-harmonic off the smaller\npolar set $E_f^h$. We then denote\nby $H_f^h$ the common restriction of $\\overline H{}_f^h$ and\n$\\underline H{}_f^h$ to $U\\setminus E_f^h$.\nSince $\\overline H{}_f^h$ is finely $h$-hyperharmonic on $U\\setminus E_f$, and\n$\\underline H{}_f^h$ is finely $h$-hypoharmonic there, it\nfollows by Proposition \\ref{prop6.1} (a) that the equalities\n$$\n\\overline H{}_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h=\n\\underline H{}_f^h\n$$\nhold q.e.\\ on $U\\setminus E_f^h$ and hence q.e.\\ on $U$.\n\n\\begin{lemma}\\label{lemma3.3} Every $h$-resolutive function $f$ is\n$h$-quasi\\-resolutive.\n\\end{lemma}\n\n\\begin{proof} The sets $E_+:=\\{H_f^h=+\\infty\\}$ and $E_-:=\\{H_f^h=-\\infty\\}$\nare finely closed and disjoint. For any fine component $V$ of\n$U\\setminus E_-$ such that $V\\cap E_+$ is nonpolar we have $V\\cap E_+=V$,\nthat is, $V\\subset E_+$, because $\\overline H{}_f^h$ is finely $h$-hyperharmonic\non $V$. Denote by $W$ the union of these fine components $V$, and by $W'$ the\n(countable) union of the remaining fine components $V'$ of $U\\setminus E_-$.\nThen $W\\subset E_+$ whereas the set $P:=W'\\cap E_+$ is polar along with each\n$V'\\cap E_+$. Since $E_+\\cap E_-=\\varnothing$ we obtain\n$$\nE_+=(U\\setminus E_-)\\cap E_+=(W\\Cup W')\\cap E_+\n=(W\\cap E_+)\\Cup(W'\\cap E_+)=W\\Cup P,\n$$\n$\\Cup$ denoting disjoint union.\nNow, $(U\\setminus P)\\cap E_+=E_+\\setminus P$ is finely closed relatively to\nthe nonvoid fine domain $U\\setminus P$ (cf.\\ \\cite[Theorem 12.2]{F1}), but also\nfinely open, being equal to $W$ as seen from the above display. Thus either\n$W=U\\setminus P$ or $W=\\varnothing$. But $W=U\\setminus P$ would imply $E_+=U$,\ncontradicting $H_f^h\\not\\equiv+\\infty$, and so actually $E_+=P$ (polar).\nSimilarly (or by replacing $f$ with $-f$) it is shown that $E_-$ is polar,\nand so $f$ is $h$-quasiresolutive because\n$\\overline H{}_f^h=\\underline H{}_f^h$ even holds everywhere on $U$.\n\\end{proof}\n\nIn view of Lemma \\ref{lemma3.3} an $h$-quasi\\-resolutive function\n$f$ is $h$-resolutive if and only if $E_f^h=\\varnothing$ (any polar subset\nof $\\Delta(U)$ being a proper subset). This implies that 1.\\ and 2.\\ in\nthe following proposition remain valid with `$h$-quasi\\-resolutive' replaced\nthroughout by `$h$-resolutive'.\nIt will be shown in Corollary \\ref{cor6.3b} that $h$-resolutivity and\n$h$-quasi\\-resolutivity are actually identical concepts.\n\n\\begin{prop}\\label{prop6.3} Let $f,g:\\Delta(U)\\longrightarrow\\overline{{\\mathbb R}}$\nbe $h$-quasi\\-resolutive. Then\n\n{\\rm{1.}} For $\\alpha\\in{\\mathbb R}$ we have $E_{\\alpha f}^h\\subset E_f^h$ and hence\n$\\alpha f$ is $h$-quasi\\-resolutive.\nFurthermore, $H{}_{\\alpha f}^h=\\alpha H{}_f^h$ on $U\\setminus E_f^h$.\n\n{\\rm{2.}} If $f+g$ is defined arbitrarily at points of undetermination then\n$E_{f+g}^h\\subset E_f^h\\cup E_g^h$ and hence $f+g$ is $h$-quasi\\-resolutive.\nFurthermore, $H{}_{f+g}^h=H{}_f^h+H{}_g^h$ on $U\\setminus(E_f^h\\cup E_g^h)$.\n\n{\\rm{3.}} $E_{f\\vee g}^h,E_{f\\wedge g}^h$ $\\subset E_f^h\\cup E_g^h\\cup P_f^h$ and\nhence $f\\vee g$ and $f\\wedge g$ are $h$-quasi\\-resolutive.\nIf for example $H_f^h\\vee H_g^h\\ge0$\nthen $H{}_{f\\vee g}^h=(1\/h){\\widehat R}{}_{(H_f^h\\vee H{}_g^h)h}$ on\n$U\\setminus(E_f^h\\cup E_g^h\\cup P_f^h)$.\n\\end{prop}\n\n\\begin{proof} For Assertion 1., consider separately the cases\n$\\alpha>0$, $\\alpha<0$, and $\\alpha=0$.\nFor 2.\\ and 3. we proceed as in \\cite[1.VIII.7 (d)]{Do}.\nFor 2.\\ we have\n\\begin{eqnarray}\nH_f^h+H_g^h\\ge\\overline H{}_{f+g}^h\\ge\\underline H{}_{f+g}^h\\ge H_f^h+H_g^h\n\\end{eqnarray}\non $U\\setminus(E_f^h\\cup E_g^h)$, the first inequality by 2.\\ in\nProposition \\ref{prop6.2},\nthe third inequality by replacing $f$ with $-f$ in the\nfirst inequality, and the second inequality holds by Proposition\n\\ref{prop6.1} (b) on\n$$\n\\{\\overline H_{f+g}^h>-\\infty\\}\n\\supset\\{\\overline H_f^h>-\\infty\\}\\cap\\{\\overline H_g^h>-\\infty\\}\n\\supset U\\setminus(E_f\\cup E_g).\n$$\nThus equality prevails on $U\\setminus(E_f\\cup E_g)$ (and hence q.e.\\ on $U$)\nin both of these inclusion relations. It follows that\n$$\n\\{\\overline H{}_{f+g}^h=-\\infty\\}\n\\subset\\{\\overline H{}_f^h=-\\infty\\}\\cup\\{\\overline H{}_g^h=-\\infty\\}\n\\subset E_f^h\\cup E_g^h.\n$$\nand similarly\n$\\{\\underline H_{f+g}^h=-\\infty\\}\\subset E_f^h\\cup E_g^h$. Finally, by (3.1)\nwith equality throughout,\n$$\\{\\overline H{}_{f+g}^h\\ne\\underline H{}_{f+g}^h\\}\n\\subset\\{\\overline H{}_f^h\\ne\\underline H{}_f^h\\}\n\\cup\\{\\overline H{}_g^h\\ne\\underline H{}_g^h\\}\n\\subset E_f^h\\cup E_g^h.\n$$\nAltogether, $E_{f+g}^h\\subset E_f^h\\cup E_g^h$,\nand so $f+g$ is indeed $h$-quasi\\-resolutive along with $f$ and $g$.\n\nFor the notation in the stated equation in 3., see \\cite[Definition 11.4]{F1}.\nSince $f\\wedge g=-[(-f)\\vee(-g)]$ and $f\\vee g=[(f-g)\\vee0]+g$  it follows by\n1.\\ and 2.\\ that 3.\\ reduces to $E_{f\\vee0}^h\\subset E_f\\cup P_f^h$,\nwhich implies the $h$-quasi\\-resolutivity of $f^+=f\\vee0$\nand the stated expression for $H_{f\\vee g}^h$ with $g=0$.\nFor given $x\\in U\\setminus(E_f^h\\cup P_f^h)$ and integers $j>0$\nchoose $u_j\\in\\overline{\\cal U}{}_f^h$ with\n$u_j(x)\\le\\dot H{}_f^h(x)+2^{-j}=\\overline H{}_f^h(x)+2^{-j}$.\nThe series $\\sum_{j=k}^\\infty(u_j-H_f^h)$ of\npositive finely $h$-super\\-harmonic functions on $U\\setminus(E_f^h\\cup P_f^h)$\n($u_j$ being likewise restricted to $U\\setminus(E_f^h\\cup P_f^h)$) has a positive\nfinely $h$-super\\-harmonic sum, finite at $x$.\nRecall that $H_f^h$ is  defined and finely $h$-harmonic on $U\\setminus E_f^h$\nand in particular on $U\\setminus(E_f^h\\cup P_f^h)$. Consequently,\n$H_f^h\\vee0$ is finely $h$-subharmonic (and positive)\non $U\\setminus(E_f^h\\cup P_f^h)$ and majorized there by $\\overline H{}_{f\\vee0}^h$,\nwhich is finite valued on $U\\setminus(E_f^h\\cup P_f^h)$ by 3.\\ in Proposition\n\\ref{prop6.2} because $\\overline H{}_f^h<+\\infty$ on\n$U\\setminus(E_f^h\\cup P_f^h)$ and because $\\overline H{}_f^h=\\dot H{}_f^h$\nthere. It follows by \\cite[Theorem 11.13]{F1}, applied with $f$ replaced by\n$h\\overline H{}_f^h\\vee0$ on $U$,\nwhich is finely subharmonic on $U\\setminus(E_f^h\\cup P_f^h)$,\nthat $\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h$ (sweeping relative to $U$)\nis finely $h$-harmonic on $U\\setminus(E_f^h\\cup P_f^h)$,\nbeing majorized there by\n$\\overline H{}_f^h\\vee0\\le\\overline H{}_{f\\vee0}^h<+\\infty$.\nThe positive function\n\\begin{eqnarray}\n\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h\n+\\sum_{j=k}^\\infty(u_j-H_f^h)\n\\end{eqnarray}\nrestricted to $U\\setminus(E_f^h\\cup P_f^h)$ is therefore finely\n$h$-super\\-harmonic.\nMoreover, this positive finely  $h$-super\\-harmonic function on\n$U\\setminus(E_f^h\\cup P_f^h)$ majorizes $u_k\\in{\\overline{\\cal U}}{}_f^h$ there\n(being $\\ge\\frac1h{\\widehat R}{}_{h\\overline H{}_f\\vee0}^h+(u_k-H_f^h)\\ge u_k$ there),\nand this majorization remains in force after extension by fine continuity\nto $U$, cf.\\ \\cite[Theorem 9.14]{F1}. Thus the extended positive function (3.2)\nbelongs to $\\overline{\\cal U}{}_{f\\vee 0}^h$. For $k\\to\\infty$ it follows that\n${\\overline H}{}_{f\\vee 0}^h\n\\le\\frac1h{\\widehat R}{}_{h\\overline H{}_{f\\vee 0}^h}$\non $U$. On the other hand, $\\underline H{}_{f\\vee0}^h$ majorizes both\n $\\underline H{}_f^h$ and $0$,\nso $\\underline H{}_{f\\vee0}^h\\ge\\frac1h{\\widehat R}{}_{h\\underline H{}_f^h\\vee0}\n=\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}$ on $U$, the equality because\n$\\underline H{}_f^h=\\overline H{}_f^h$ on $U\\setminus(E_f^h\\cup P_f^h)$ and hence\nq.e.\\ on $U$.\nIt follows that\n$${\\overline H}{}_{f\\vee0}^h\\le\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}\n=\\frac1h{\\widehat R}{}_{h\\underline H{}_f^h\\vee0}\n\\le\\underline H{}_{f\\vee0}^h\\le{\\overline H}{}_{f\\vee0}^h<+\\infty\n$$ because\n$h\\overline H{}_f^h\\vee0=h\\underline H{}_f^h\\vee0$\non $U\\setminus(E_f^h\\cup P_f^h)$ and hence q.e.\\ on $U$. (The last inequality\nin the above display follows by Proposition \\ref{prop6.1} (b) because\n$f\\vee0>-\\infty$.)\nSince $\\overline H{}_{f\\vee0}^h\\ge0\\ge-\\infty$ we conclude that $f\\vee0$\nindeed is $h$-resolutive, resp.\\ $h$-quasi\\-resolutive, and that\n$E_{f\\vee0}^h\\subset E_f^h\\cup P_f^h$ and\n$H_{f\\vee0}^h=\\frac1h{\\widehat R}{}_{h\\overline H{}_f^h\\vee0}$ on $E_f^h\\cup P_f^h$.\n\\end{proof}\n\nA version of Proposition \\ref{prop6.3} for $h$-resolutive\nfunctions instead of $h$-quasi\\-resolutive functions will\nof course follow when the identity of $h$-resolutivity and\n$h$-quasi\\-resolutivity has been established in Corollary \\ref{cor6.3b}.\nBefore that, we do however need the following step in that direction,\nbased on Proposition \\ref{prop6.2}.\n\n\\begin{lemma}\\label{lemma6.3c} Let $f$ be an $h$-quasi\\-resolutive function\non $\\Delta(U)$. If $f^+$ and $f^-$ are $h$-resolutive then so is $f$,\nand the function\n$H_f^h=\\overline H{}_f^h=\\underline H{}_f^h$ on $U$ is finite valued.\n\\end{lemma}\n\n\\begin{proof} According to 3.\\ in Proposition \\ref{prop6.3}, $f^+$ and $f^-$\nare $h$-quasi\\-resolutive (besides being $h$-resolutive), and the functions\n$H_{f+}^h:=\\overline H{}_{f^+}^h=\\underline H{}_{f^+}^h$\n(defined on $\\{\\overline H{}_{f_+}^h>-\\infty\\}=U$ since $f^+\\ge0$)\nand similarly $H_{f-}^h:=\\overline H{}_{f^-}^h=\\underline H{}_{f^-}^h$\nare therefore finite valued. Since $-f^-\\le f\\le f^+$ it follows that\n$-\\infty<-H_{f^-}^h\\le\\underline H{}_f^h,\\overline H{}_f^h\\le H_{f^+}^h<+\\infty$.\nApplying 2.\\ in Proposition \\ref{prop6.2} to the sums\n$f=f^++(-f)^+=f^+-f^-$ and $-f=f^--f_+$, which are well defined on $\\Delta(U)$,\nwe obtain\n$$\n\\overline H{}_f^h\\le H_{f^+}^h-H_{f^-}^h\\le\\underline H{}_f^h\n$$\non all of $U$, and hence $\\overline H{}_f^h=\\underline H{}_f^h$ there because\n$\\overline H{}_f^h\\ge\\underline H{}_f^h$ on all of $U$, again by\nProposition \\ref{prop6.1} (b) since we have seen that for example\n$\\overline H{}_f^h>-\\infty$.\n\\end{proof}\n\n\\begin{cor}\\label{cor4.8} Let $(f_j)$ be an increasing sequence of lower\nbounded $h$-resolutive, resp.\\  $h$-quasi\\-resolutive functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$, and let $f=\\sup_jf_j$.\nIf $\\overline H{}_f^h\\not\\equiv+\\infty$ then $f$ is\n$h$-resolutive, resp.\\ $h$-quasi\\-resolutive.\n \\end{cor}\n\n\\begin{proof} By adding a constant to $f$ we reduce the claim to the case\n$f_j\\ge0$. For every $j$ we have\n$\\underline H{}_f^h\\ge\\underline H{}_{f_j}^h=\\overline H{}_{f_j}^h$,\nby Proposition \\ref{prop6.1} (b) because $\\overline H{}_{f_j}^h>-\\infty$. Hence\n$\\underline H{}_f^h\\ge\\sup_j\\overline H{}_{f_j}^h=\\overline H{}_f^h$\naccording to 4.\\ in Proposition \\ref{prop6.2}.\nBy definition of $\\underline H{}_f^h$ we have at any point $x_0\\in U$\n$$\n\\underline H{}_f^h(x_0)=\\underset{x\\to x_0,\\,x\\in U}{\\fine\\lim\\sup}\\,\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(x)\n\\le\\underset{x\\to x_0,\\,x\\in U}{\\fine\\lim\\sup}\\,\n\\overline H{}_f^h(x)\n=\\overline H{}_f^h(x_0)\n$$\naccording to Proposition \\ref{prop6.1} (a), $\\overline H{}_f^h$ being finely\n$h$-hyper\\-harmonic by Proposition \\ref{prop6.1} (c). By Proposition\n\\ref{prop6.1} (b) we have\n$\\underline H{}_f^h\\le\\overline H{}_f^h$ on $\\{\\overline H{}_f^h>-\\infty\\}=U$,\nand we conclude that $\\underline H{}_f^h=\\overline H{}_f^h$. By hypothesis\nthis finely hyperharmonic function on $U$ is finite q.e.\\ on $U$, and in\nparticular this positive function is not identically $+\\infty$.\nConsequently, $f$ is indeed $h$-resolutive, resp.\\ $h$-quasi\\-resolutive.\n\\end{proof}\n\nRecall that $\\mu_h$ denotes the unique measure on $\\overline U$\ncarried by $\\Delta_1(U)$ and representing $h$,\nthat is, $h=K\\mu_h=\\int K(.,Y)d\\mu_h(Y)$.\n\n\\begin{prop}\\label{prop6.4} For any $\\mu_h$-measurable subset $A$\nof $\\Delta(U)$ the indicator function $1_A$ is $h$-resolutive, and\n\\begin{eqnarray}H{}_{1_A}^h\\!\\!\\!\n&=&\\!\\!\\!\\frac{1}{h}\\int_AK(.,Y)d\\mu_h(Y)=\\frac1h\\widehat R{}_h^A\n\\end{eqnarray}\non $U$.\nIn particular, the constant function $1$ on $\\Delta(U)$ is\n$h$-resolutive and $H_1^h=1$.\n\\end{prop}\n\n\\begin{proof} Because $h=K\\mu_h$ and because $\\mu_h$ is carried by\n$\\Delta_1(U)$ we have by \\cite[Theorem 3.10 and Proposition 3.9]{EF2}\n$$\n\\widehat R{}_h^A=\\widehat R{}_{K\\mu_h}^A\n=\\int_{\\Delta_1(U)}\\widehat R{}_{K(.,Y)}^Ad\\mu_h(Y)\n=\\int K(.,Y)1_A(Y)d\\mu_h(Y).\n$$\nConsider any finely $h$-hyper\\-harmonic function $u=v\/h\\ge0$ on $U$ such that\n$u\\ge1$ on some open set $W\\subset\\overline U$ with $W\\supset A$. Then\n$u\\in\\overline{\\cal U}{}_{1_A}^h$ and hence\n$u\\ge\\dot H{}_{1_A}^h\\ge\\overline H{}_{1_A}^h$.\nBy varying $W$ it follows by \\cite[Definition 2.4]{EF2} that\n$\\frac1h\\widehat R{}_h^A\\ge\\overline H{}_{1_A}^h$.\nWe have\n\\begin{eqnarray}\\frac1h\\int K(.,Y)1_A(Y)d\\mu_h(Y)\n=\\frac1h \\widehat R{}_h^A\\ge{\\overline H}{}_{1_A}^h.\n\\end{eqnarray}\nApplying this inequality to the $\\mu_h$-measurable set $\\Delta(U)\\setminus A$\nin place of $A$ we obtain\n\\begin{eqnarray}\\frac{1}{h}\\int K(.,Y)1_{\\Delta(U)\\setminus A}(Y)d\\mu_h(Y)\n\\ge {\\overline H}{}_{1_{\\Delta(U)\\setminus A}}^h.\n\\end{eqnarray}\nBy adding the left hand, resp.\\ right hand, members of (3.4) and (3.5)\nthis leads by 2.\\ in Proposition \\ref{prop6.2} to\n\\begin{eqnarray}1=\\frac1h\\int K(.,Y)d\\mu_h(Y)\n\\ge{\\overline H}{}_{1_A}^h+{\\overline H}{}_{1_{\\Delta(U)\\setminus A}}^h\n\\ge{\\overline H}{}_1^h=1 .\n\\end{eqnarray}\nThus equalities prevail throughout in (3.4), (3.5), and (3.6). It follows\naltogether that\n\\begin{eqnarray*}\\underline H{}_{1_A}^h\\!\\!\\!\n&=&-\\overline H{}_{-1_A}^h=1-\\overline H{}_{1-1_A}^h\n=1-\\overline H{}_{1_{\\Delta(U)\\setminus A}}^h\\\\\n&=&\\!\\!\\!\\overline H{}_{1_A}^h\n=\\frac1h\\widehat R{}_h^A=\\frac{1}{h}\\int_AK(.,Y)d\\mu_h(Y),\n\\end{eqnarray*}\nso that indeed $1_A$ is $h$-resolutive and (3.3) holds.\n\\end{proof}\n\nFor any function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ we define\n$f(Y)K(x,Y)=0$ at points $(x,Y)$ where $f(Y)=0$ and $K(x,Y)=+\\infty$.\nIf $f$ is $\\mu_h$-measurable then so is $Y\\longmapsto f(Y)K(x,Y)$ for each\n$x\\in U$ because $K(x,Y)>0$ is $\\mu_h$-measurable (even l.s.c.) as a function\nof $Y\\in\\Delta(U)$ according to \\cite[Proposition 2.2 (i)]{EF1}.\n\n\\begin{prop}\\label{prop6.5} Let $f$ be a $\\mu_h$-measurable\nlower bounded function on $\\Delta(U)$. Then\n$$\n{\\overline H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y) d\\mu_h(Y)>-\\infty,\n$$\nand ${\\overline H}{}_f^h$ is either identically $+\\infty$ or the sum of\nan $h$-invariant function and a constant $\\le0$.\nFurthermore $f$ is $h$-quasi\\-resolutive if and only if $f$ is $h$-resolutive,\nand that holds if and only if\n$\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$\nq.e.\\ on $U$, or equivalently: everywhere on $U$.\nIn particular, every bounded $\\mu_h$-measurable function\n$f:\\Delta(U)\\longrightarrow{\\mathbb R}$ is $h$-resolutive.\n\\end{prop}\n\n\\begin{proof} Consider first the case of a positive $\\mu_h$-measurable\nfunction $f$. Then $f$ is the pointwise supremum of an\nincreasing sequence of positive $\\mu_h$-measurable step\nfunctions $f_j$ (that is, finite valued functions $f_j$ taking only finitely\nmany values, each finite and each on some $\\mu_h$-measurable set; in other\nwords: affine combinations of indicator functions of $\\mu_h$-measurable sets).\nFor any index $j$ it follows by Proposition \\ref{prop6.4} and by 1.\\ and 2.\\\nin Proposition \\ref{prop6.3} (the latter extended to finite sums\nand with `$h$-resolutive' throughout in place of\n`$h$-quasi\\-resolutive', cf.\\ the paragraph preceding Proposition\n\\ref{prop6.3}) that each $f_j$ is $h$-resolutive and that\n$$\nH_{f_j}^h=\\frac{1}{h}\\int f_j(Y)K(.,Y)d\\mu_h(Y)\n$$\non $U$, whence\n\\begin{eqnarray*}0\\!\\!\\!\n&\\le&\\!\\!\\!\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)\\\\\n&=&\\!\\!\\!\\frac1{h}\\sup_j\\int f_j(Y)K(.,Y)d\\mu_h(Y)\n=\\sup_j H_{f_j}^h={\\overline H}{}_f^h\n\\end{eqnarray*}\nby 4.\\ in Proposition \\ref{prop6.2}.\nFor a general lower bounded $\\mu_h$-measurable function $f$ on $\\Delta(U)$\nthere is a constant $c\\ge0$ such that\n$g:=f+c\\ge0$ and hence $\\overline {}H_g^h=\\overline {}H_f^h+c\\ge0$.\nIt follows that\n$$\n{\\overline H}{}_f^h=\\frac1h\\int g(Y)K(.,Y)d\\mu_h(Y)-c\n=\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)>-\\infty\n$$\nand hence by Proposition \\ref{prop6.1} (c) applied to $g$ that\n${\\overline H}{}_f^h=\\overline H{}_g^h-c$ has the asserted form.\n\nNext, consider a bounded $\\mu_h$-measurable function $f$ on $\\Delta(U)$.\nAs just shown, we have\n$${\\overline H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)$$\nand the same with $f$ replaced by $-f$, whence\n${\\overline H}{}_f^h={\\underline H}{}_f^h$,\nfinite valued because $f$ is bounded. Thus $f$ is $h$-resolutive.\nLet $c\\ge0$ be a constant such that $|f|\\le c$.\nThen ${\\overline H}{}_f^h=c-{\\underline H}{}_{c-f}^h$\nwhich is finely $h$-harmonic because $c-f\\ge0$ and so $H_{c-f}^h$ is\n$h$-invariant by Proposition \\ref{prop6.1} (c) and\nhence finely $h$-harmonic, being finite valued.\n\nReturning to a general lower bounded $\\mu_h$-measurable function $f$,\nsuppose first that $f$ is $h$-quasi\\-resolutive. Then, as shown in the first\nparagraph of the proof,\n$\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)={\\overline H}_f^h$ is finite q.e.\\ on $U$.\nConversely, if $\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$ q.e.,\nthat is $\\overline H{}_f^h\\not\\equiv+\\infty$,\nthen Corollary \\ref{cor4.8} applies to the increasing\nsequence of bounded $\\mu_h$-measurable and hence $h$-resolutive functions\n$f\\wedge j$ converging to $f$, and we conclude that $f$ is $h$-resolutive\n(in particular $h$-quasi\\-resolutive) and hence that\n$\\frac1h\\int f(Y)K(.,Y)d\\mu_h(Y)={\\overline H}_f^h$ is finite everywhere on $U$.\n\\end{proof}\n\n\\begin{cor}\\label{cor4.10a} Let $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$\nbe $\\mu_h$-measurable. Then $f$ is $h$-resolutive\nif and only if $|f|$ is $h$-resolutive.\n\\end{cor}\n\n\\begin{proof} If $f$ is $h$-resolutive, and therefore $h$-quasiresolutive by\nLemma \\ref{lemma3.3}, then $|f|=f\\vee(-f)$ is\n$h$-quasi\\-resolutive according to 3.\\ and 1.\\ in Proposition \\ref{prop6.3}.\nSince $|f|$ is lower bounded (and $\\mu_h$-measurable) then by Proposition\n\\ref{prop6.5} $|f|$ is even $h$-resolutive and $|f|K(x,.)$ is\n$\\mu_h$-integrable for every $x\\in U$. So are therefore $f^+K(x,.)$ and\n$f^-K(x,.)$, and it follows, again by Proposition \\ref{prop6.5},\nthat $f^+$ and $f^-$ are $h$-resolutive.\nSo is therefore $f=f^+-f^-$ by Lemma \\ref{lemma6.3c}.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.10b} Every $h$-quasi\\-resolutive\nfunction $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is $\\mu_h$-measurable.\n\\end{prop}\n\n\\begin{proof}\nWe begin by proving this for $f=1_A$, the indicator function of a subset $A$ of\n$\\Delta(U)$, cf.\\ \\cite[p.\\ 113]{Do}. Clearly, $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h(x)$ have their\nvalues in $[0,1]$, and hence $\\overline H{}_f^h=\\dot H{}_f^h$ and\n$\\underline H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\naccording to Proposition \\ref{prop6.1} (c).\nSince $\\overline{\\cal U}{}_f^h$ is lower directed there is a decreasing\nsequence of functions $u_j\\in\\overline{\\cal U}{}_f^h$ such that\n$\\overline H{}_f^h(x_0)=\\inf_ju_j(x_0)$. Replacing $u_j$ by\n$u_j\\wedge1\\in\\overline{\\cal U}{}_f^h$ we arrange that $u_j\\le1$.\nDenote by $g_j$ the function defined on ${\\overline U}$ by\n$$g_j(Y)=\\underset{z\\to Y,\\,z\\in U}{\\liminf}u_j(z)$$ for any\n$Y\\in {\\overline U}.$\nClearly, $g_j$ is  l.s.c.\\ on ${\\overline U}$ and $1_A\\le g_j\\le1$\non $\\Delta(U)$. Write $f_2=\\inf_jg_j$ (restricted to $\\Delta(U)$).\nThen $f_2$ is Borel measurable and\n$1\\ge f_2\\ge f=1_A$, whence $\\overline H{}_{f_2}^h\\ge\\overline H{}_f^h$. For the\nopposite inequality note that $u_j\\in\\overline{\\cal U}{}_{f_2}^h$ because\n$g_j\\ge f_2$. Hence\n$\\overline H{}_f^h=\\inf_ju_j\\ge\\overline H{}_{f_2}^h$ with equality at $x_0$.\nFurthermore, $\\overline H{}_f^h$ is invariant according to\nProposition \\ref{prop6.1} (c), and hence\n$\\overline H{}_{f_2}^h-\\overline H{}_f^h$ is positive and finely\n$h$-superharmonic on $U$. Being $0$ at $x_0$ it is identically $0$, and so\n$\\overline H{}_{f_2}^h=\\overline H{}_f^h$. Similarly there is\na positive Borel measurable function $f_1\\le f$ such that\n$\\underline H{}_{f_1}^h=\\underline H{}_f^h$. Clearly, $0\\le f_1\\le f_2\\le1$.\nSince $f$ is $h$-quasi\\-resolutive\nwe obtain from Proposition \\ref{prop6.1} (a) q.e.\\ on $U$\n$$\nH_f^h=\\underline H{}_{f_1}^h\\le\\overline H{}_{f_1}^h\\le H_f^h\n\\le\\underline H{}_{f_2}^h\\le\\overline H{}_{f_2}^h=H_f^h,\n$$\nthus with equality q.e.\\ all through. Hence $f_1$ and $f_2$ are\n$h$-quasi\\-resolutive, and so is therefore $f_2-f_1$ by 1.\\ and 2.\\ in\nProposition \\ref{prop6.3}, which also shows that $H_{f_2-f_1}^h=H_{f_2}-H_{f_1}=0$\nq.e. Because $f_2-f_1$ is positive and Borel measurable on $\\Delta(U)$ it\nfollows by Proposition \\ref{prop6.5} that\n$\\frac1h\\int(f_2(Y)-f_1(Y))K(.,Y)d\\mu_h(Y)=0$,\nand hence $f_1=f_2$ $\\mu_h$-a.e.\nIt follows that $f=1_A$ is $\\mu_h$-measurable, and so is therefore $A$.\n\nNext we treat the case of a finite valued $h$-quasi\\-resolutive function $f$ on\n$\\Delta(U)$. Adapting the proof given in \\cite[p.\\ 115]{Do} in the classical\nsetting we consider the space $\\cal C(\\overline{\\mathbb R},{\\mathbb R})$ of continuous (hence\nbounded) functions $\\overline{\\mathbb R}\\to{\\mathbb R}$, and denote by $\\Phi$ the space of\nfunctions $\\phi\\in\\cal C(\\overline{\\mathbb R},{\\mathbb R})$ such that $\\phi\\circ f$ is\n$h$-quasi\\-resolutive.\nIn view of Proposition \\ref{prop6.3}, $\\Phi$ is a vector lattice,\nclosed under uniform convergence because $|\\phi_j-\\phi|<\\varepsilon$ implies\n$|H_{\\phi_j\\circ f}^h-\\overline H{}_{\\phi\\circ f}^h|\\le\\varepsilon$ and\n$|H_{\\phi_j\\circ f}^h-\\underline H_{\\phi\\circ f}^h|\\le\\varepsilon$ on $U\\setminus E^h_f$,\nand so $|\\overline H{}_{\\phi\\circ f}^h-\\underline H{}_{\\phi\\circ f}^h|\\le2\\varepsilon$\non $U\\setminus E^h_f$. We infer that\n$\\overline H{}_{\\phi\\circ f}^h=\\underline H{}_{\\phi\\circ f}^h$ (finite values) q.e.\\\non $U$, and so $\\phi\\circ f$ is indeed resolutive.\nFurthermore, $\\Phi$ includes the fuctions $\\phi_n:t\\longmapsto(1-|t-n|)\\vee0$ on\n${\\mathbb R}$ for integers $n\\ge1$, again by Proposition \\ref{prop6.3}. These functions\nseparate points of ${\\mathbb R}$. In fact, for distinct $s,t\\in{\\mathbb R}$, say $s<t$, take\n$n=[s]$ (that is, $n\\le s<n+1$). If also $n\\le t<n+1$ then clearly\n$\\phi_n(t)<\\phi_n(s)\\le1$, and in the remaining case $t\\ge n+1$ we have\n$\\phi_n(t)=0<\\phi_n(s)$. It therefore follows by the lattice version of the\nStone-Weierstrass theorem that $\\Phi=\\cal C(\\overline{\\mathbb R},{\\mathbb R})$.\nNext, the class $\\Psi$ of (not necessarily continuous) functions\n$\\psi:{\\mathbb R}\\longrightarrow{\\mathbb R}$ for which $\\psi\\circ f$ is $h$-quasi\\-resolutive\nis closed under bounded monotone convergence, by Corollary \\ref{cor4.8}\n(adapted to a bounded monotone convergent sequence of functions $f_j$).\nAlong with the continuous functions ${\\mathbb R}\\longmapsto{\\mathbb R})$, $\\Psi$ therefore\nincludes every bounded Borel measurable function ${\\mathbb R}\\longmapsto{\\mathbb R}$.\nIn particular, the indicator function $1_J$ of an interval $J\\subset{\\mathbb R}$\nbelongs to $\\Psi$, and hence $1_J\\circ f$ is $h$-quasi\\-resolutive.\nWe conclude by the first part of the proof that the $h$-quasi\\-resolutive\nindicator function $1_J\\circ f=1_{f^{-1}(J)}$ is $\\mu_h$-measurable.\n\nFinally, for an arbitrary $h$-quasi\\-resolutive function\n$f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$, write $A_+:=\\{f=+\\infty\\}$ and\n$A_-:=\\{f=-\\infty\\} $. By 3.\\ in Proposition \\ref{prop6.3}, $f\\vee0$ is\n$h$-quasi\\-resolutive and\n$$\n\\overline H{}_{(+\\infty)1_{A_+}}^h=\\overline H{}_{f1_{A_+}}^h\n\\le\\overline H{}_{f\\vee0}^h=\\dot H{}_{f\\vee0}^h<+\\infty\n$$\non $U\\setminus(E^h_f\\cup P_f^h)$, cf.\\ the opening of the present section.\nIt follows that $\\overline H{}_{1_{A_+}}=0$ q.e.\\ on $U$\nand hence $\\overline H{}_{(+\\infty)1_{A_+}}^h=0$ q.e.\\ according to 3.\\ in\nProposition \\ref{prop6.2}. Since $-f$ likewise is $h$-quasi\\-resolutive\nwe have $\\underline H{}_{(+\\infty)1_{A_-}}^h=0$ q.e. Furthermore,\n$0\\le\\underline H{}_{(+\\infty)1_{A_+}}\\le\\overline H{}_{(+\\infty)1_{A_+}}=0$ q.e.\\ by\nProposition \\ref{prop6.1} (a), and similarly $\\overline H{}_{(+\\infty)1_{A_-}}=0$\nq.e. Writing $A=A_+\\cup A_-=\\{|f|=+\\infty\\}$ we infer from 2.\\ in\nProposition \\ref{prop6.3} that $1_A=1_{A_+}+1_{A_-}$ is $h$-quasi\\-resolutive,\nand hence so is\n$(+\\infty)1_A$ in view of Corollary \\ref{cor4.8}. As shown in the first\nparagraph of the proof it follows that $A$ is $\\mu_h$-measurable.\nDefine $g:\\Delta(U)\\longrightarrow{\\mathbb R}$ by $g=f$ except that $g=0$ on\n$A=\\{|f|=+\\infty\\}$. Then $g\\le f+(+\\infty)1_A$ and $f\\le g+(+\\infty)1_A$, and\nhence by Proposition \\ref{prop6.2} (recall that $\\overline H{}_{1_{A_+}}=0$\nq.e.\\ on $U$)\n$$\n\\overline H{}_g^h\\le H_f^h+\\overline H{}_{(+\\infty)1_A}^h=H_f^h\n\\le\\underline H{}_g^h+\\underline H{}_{(+\\infty)1_A}^h=\\underline H{}_g^h\n$$\nquasieverywhere, actually with equalities q.e., and hence $g$ is\n$h$-quasi\\-resolutive along with $f$.\nAs established in the beginning of the proof, $g$ is $\\mu_h$-measurable and so\nis $f$. In fact, $\\{f\\ne g\\}=A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$ by the\nfollowing corollary.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.10c} {\\rm{(Cf.\\ \\cite[p.\\ 114]{Do}.)}}\nA set $A\\subset\\Delta(U)$ is $h$-harmonic null\nif and only if $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$.\n\\end{cor}\n\n\\begin{proof} If $A$ is $\\mu_h$-measurable with $\\mu_h(A)=0$ then\n$\\overline H{}_{1_A}^h=\\frac1h\\int1_AK(.,Y)d\\mu_h(Y)=0$ according to Proposition\n\\ref{prop6.5}. Conversely, if $\\overline H{}_{1_A}^h=0$ then\n$0\\le\\underline H{}_{1_A}^h\\le\\overline H{}_{1_A}^h=0$ and so $1_A$ is\n$h$-resolutive. It follows by Proposition \\ref{prop6.10b} that $1_A$\nis $\\mu_h$-measurable, and again by Proposition \\ref{prop6.5} that\n$\\frac1h\\int1_AK(.,Y)d\\mu_h(Y)=\\overline H{}_{1_A}^h=0$. Since $K(x,.)>0$ is\nl.s.c.\\ by \\cite[Proposition 3.2]{EF1} we conclude that $\\mu_h(A)=0$.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.3b} A function $f$ on $\\Delta(U)$ with values in\n$\\overline{\\mathbb R}$ is $h$-resolutive if and only if $f$ is $h$-quasi\\-resolutive.\n\\end{cor}\n\n\\begin{proof} The `only if' part is contained in Lemma \\ref{lemma3.3}.\nFor the `if' part, suppose that $f$ is $h$-quasi\\-resolutive and hence\n$\\mu_h$-measurable, by Proposition \\ref{prop6.10b}. If $f\\ge0$ then\n$f$ is $h$-resolutive according to Proposition \\ref{prop6.5}.\nFor arbitrary $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ this applies to\n$f^+$ and $f^-$, which are $h$-quasi\\-resolutive\naccording to 3.\\ in Proposition \\ref{prop6.3}. Consequently, $f=f^+-f^-$\nis likewise finely $h$-resolutive by 1.\\ and 2.\\ in Proposition \\ref{prop6.3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm6.8}\nA function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is\n$h$-resolutive if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta(U)$ is $\\mu_h$-integrable\nfor quasievery $x\\in U$. In the affirmative  case $Y\\longmapsto f(Y)K(x,Y)$\nis $\\mu_h$-integrable for every $x\\in U$, and we have everywhere on $U$\n$$\nH{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{proof} Suppose first that $f$ is $h$-resolutive.\nBy Proposition \\ref{prop6.10b} $f$ is then $\\mu_h$-measurable.\nAccording to 3.\\ in Proposition \\ref{prop6.3} the function $|f|$ is also\n$h$-resolutive, and it follows by Proposition \\ref{prop6.5} that\n$\\overline H{}_{|f|}^h(x)=\\frac1h\\int |f(Y)|K(x,Y)d\\mu_h(Y)<+\\infty$\nfor every $x\\in U$. Conversely, suppose that $fK(x,.)$ is $\\mu_h$-integrable\nfor quasi\\-every\n$x\\in U\\setminus E$. For any $x\\in U$, $K(x,.)>0$ is l.s.c.\\ and hence\n$\\mu_h$-measurable on $\\overline U$ by \\cite[Proposition 3.2]{EF1}, and so\n$f^+$ and $f^-$ must be $\\mu_h$-measurable. By Proposition \\ref{prop6.5},\n$f^+$ and $f^-$ are therefore $h$-quasi\\-resolutive, that is $h$-resolutive\nby Corollary \\ref{cor6.3b}, and so is therefore\n$f=f^+-f^-$ by 1.\\ and 2.\\ in Proposition \\ref{prop6.3}.\n\\end{proof}\n\n\\begin{remark}\\label{remark6.12}\nIn the case where $U$ is Euclidean open it\nfollows by the Harnack convergence theorem for harmonic functions (not\nextendable to finely harmonic functions) for any numerical function\n$f$ on $\\Delta(U)$ that $\\overline H{}_f^h$ is\n$h$-hyper\\-harmonic on $U$ (in particular $>-\\infty$) and hence equal to\n$\\dot H{}_f^h$ (except if $\\overline H{}_f^h\\equiv-\\infty$). It follows\nby Proposition \\ref{prop6.1} (a) that\n$\\overline H{}_f^h=\\dot H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\underline H{}_f^h$. If $f$ is resolutive then\nby Definition \\ref{def6.9} equality prevails on all of $U$.\nSumming up, Theorem \\ref{thm6.8} is a (proper) extension of\nthe corresponding classical result, cf.\\ e.g.\\ \\cite[Theorem 1.VIII.8]{Do}.\n\\end{remark}\n\nWe close this section with a brief discussion of an alternative, but\ncompatible concept of $h$-resolutivity based on the minimal-fine\n(mf) topology, cf.\\ \\cite[Definition 3.4]{EF2}. The mf-closure of $U$\nis $U\\cup\\Delta_1(U)$, and the relevant boundary functions $f$ are therefore\nnow defined only on the mf-boundary $\\Delta_1(U)$.\n\nGiven a function $f$ on $\\Delta_1(U)$ with values in ${\\overline {\\mathbb R}}$,\na finely $h$-hyper\\-harmonic function $u$ on $U$ is now said\nto belong to the upper PWB$^h$ class, denoted again by\n${\\overline {\\cal U}}{}_f^h$, if $u$ is lower bounded and if\n$$\\underset{x\\to Y,\\,x\\in U}{\\mfliminf}\\,u(x)\\ge f(Y)\n\\quad\\text{ for every }\\;Y\\in \\Delta_1(U).$$\nThis leads to new, but similarly denoted concepts\n $$\\dot H{}_f^h=\\inf{\\overline{\\cal U}}{}_f^h,\n\\quad{\\overline H}{}_f^h=\\widehat{\\dot H{}_f^h}\n={\\widehat\\inf}\\,{\\overline {\\cal U}}{}_f^h\\;(\\le\\dot H{}_f^h),$$\nand hence new concepts of $h$-quasi\\-resolutivity and $h$-resolutivity.\n\nWhen considering reduction $R_u^A$ and sweeping $\\widehat R{}_u^A$ of a\nfinely $h$-hyperharmonic function $u$ on $U$ onto a set\n$A\\subset\\overline U$ we similarly use the alternative, though actually\nequivalent mf-versions\n\\cite[Definition 3.14]{EF2}, cf.\\ \\cite[Theorem 3.16]{EF2}.\nThis occurs in the proof of Proposition \\ref{prop6.4} (after the first\ndisplay), where now $W\\subset\\overline U$ is mf-open (and contains $A$).\n\nThe changes as compared with the case of $h$-resolutivity relative to the\nnatural topology are chiefly as follows. A set $A\\subset U\\cup\\Delta_1(U)$\nis of course now said to be $h$-harmonic null if $\\overline H{}_{1_A}^h=0$\n(with the present mf-version of $\\overline H$). In Proposition\n\\ref{prop6.1} (b) we apply \\cite[Proposition 3.12]{EF2} in place of its\ncorollary. In the beginning of the proof of Proposition \\ref{prop6.10b}\nthe function $g_j$ shall now be defined at $Y\\in\\Delta_1(U)$ by\n$$\ng_j(Y)=\\underset{z\\to Y,\\,z\\in U}{\\mfliminf}\\,u_j(z).\n$$\nAnd $g_j$ is $\\mu_h$-measurable on $\\Delta_1(U)$ because $g_j$ equals\n$\\mu_h$-a.e.\\ the $\\mu_h$-measurable function defined $\\mu_h$-a.e.\\ on\n$\\Delta_1(U)$ by\n$Y\\longmapsto\\mflim_{z\\to Y,\\,z\\in U}\\,u_j(z)=\\frac{d\\mu_{u_j}}{d\\mu_h}(Y)$\naccording to the version of the\nFatou-Na{\\\"i}m-Doob theorem established in \\cite[Theorem 4.5]{EF1}. Here\n$\\mu_{u_j}$ denotes the representing measure for $u_j$, that is $K\\mu_{u_j}=u_j$,\nand $d\\mu_{u_j}\/d\\mu_h$ denotes the Radon-Nikod\\'ym derivative of the\n$\\mu_h$-continuous part of $\\mu_{u_j}$ (carried by $\\Delta_1(U)$) with respect\nto $\\mu_h$. The $\\mu_h$-measurability of $g_j$ on $\\Delta_1(U)$ thus established\nis all that is needed for the proof of the mf-version of Proposition\n\\ref{prop6.10b}, replacing mostly $\\Delta(U)$ with $\\Delta_1(U)$.\n\nThe following result is established like Theorem \\ref{thm6.8}\n\n\\begin{theorem}\\label{thm6.8a} A function\n$f:\\Delta_1(U)\\longrightarrow\\overline{\\mathbb R}$ is $h$-resolutive relative to the\nmf-topology if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta_1(U)$ is $\\mu_h$-integrable\nfor quasievery $x\\in U$. In the affirmative  case $Y\\longmapsto f(Y)K(x,Y)$\nis $\\mu_h$-integrable for every $x\\in U$, and we have everywhere on $U$\n$$\nH{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{cor}\\label{cor6.8b} For any $h$-resolutive function\n$f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ relative to the natural topology,\nthe restriction of $f$ to $\\Delta_1(U)$ is resolutive relative to the\nmf-topology. Conversely, for any $h$-resolutive function\n$f:\\Delta_1(U)\\longrightarrow\\overline{\\mathbb R}$ relative to the mf-topology,\nthe extension of $f$ by $0$ on $\\Delta(U)\\setminus\\Delta_1(U)$ is\n$h$-resolutive relative to the natural topology.\n\\end{cor}\n\n\n\\section{Further equivalent concepts of $h$-resolutivity}\\label{sec4}\n\nWe again consider functions $f:\\Delta(U)\\longrightarrow\\overline{{\\mathbb R}}$.\nWe show that the equivalent concepts of $h$-resolutivity and\n$h$-quasiresolutivity do not alter when $\\overline H{}_f^h,\\underline H{}_f^h$\nin Definitions \\ref{def6.9}, \\ref{def2.3} are\nreplaced by $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\nrespectively. Recall from Proposition \\ref{prop6.1} (c) that\n$\\dot H{}_f^h=\\overline H{}_f^h$ if $\\dot H{}_f^h<+\\infty$. This applies,\nin particular, to the indicator function $1_A$ of a set $A\\subset\\Delta(U)$.\nTherefore the ``dot''-version of the concept of an $h$-harmonic null set\n$A$ is identical with the version considered in Definition \\ref{def4.1}.\n\n\\begin{definition}\\label{def4.0} A function $f$ on $\\Delta(U)$ with values in\n${\\overline {\\mathbb R}}$ is said to be $h$-dot-resolutive if $\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ on $U$\nand if this function, also denoted by $H_f^h$,\nis neither identically $+\\infty$ nor identically $-\\infty$.\n\\end{definition}\n\nFor any function $f:\\Delta(U)\\longrightarrow{\\mathbb R}$ we consider the following\nsubset of $U$:\n$$\n\\dot E{}_f^h=\\{\\dot H{}_f^h=-\\infty\\}\\cup\n\\{{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h=+\\infty\\}\n\\cup\\{\\dot H{}_f^h\\ne\n{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\\}.\n$$\n\n\\begin{lemma}\\label{lemma4.3} A function $f$ on $\\Delta(U)$ with values in\n${\\overline {\\mathbb R}}$ is $h$-quasi\\-resolutive if and only if $f$ is\n$h$-dot-quasi\\-resolutive in the sense that $\\dot E{}_f^h$ is polar, or\nequi\\-valently that the relations\n$\\dot H{}_f^h>-\\infty$,\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h<+\\infty$, and\n$\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nall hold quasi\\-everywhere on $U$.\n\\end{lemma}\n\n\\begin{proof} If these three relations hold q.e.\\ on $U$ then analogously\n$\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\dot H{}_f^h>-\\infty$ q.e.\\ and similarly\n$\\underline H{}_f^h<+\\infty$ q.e. But $\\overline H{}_f^h=\\dot H{}_f^h>-\\infty$ q.e.\\\non $\\{\\overline H{}_f^h>-\\infty\\}$, hence also q.e.\\ on $U$. Similarly,\n$\\underline H{}_f^h<+\\infty$ q.e.\\ on $U$, and altogether\n$\\overline H{}_f^h=\\underline H{}_f^h$ q.e.\\ on $U$. Thus $f$ is\nquasi\\-resolutive. The converse is obvious.\n\\end{proof}\n\n\\begin{lemma}\\label{lemma4.4} Every $h$-dot-resolutive function $f$ is\n$h$-resolutive, and hence $h$-quasi\\-resolutive (now also termed\n$h$-dot-quasi\\-resolutive).\n\\end{lemma}\n\n\\begin{proof} Suppose that $f$ is $h$-dot-resolutive then $f$. Then $f$ is\n$h$-resolutive, for $\\dot H{}_f^h,\\overline H{}_f^h,\\underline H{}_f^h$, and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\nare all equal because there is equality in the general inequalities\n$\\dot H{}_f^h\\ge\\overline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$ and\n$\\dot H{}_f^h\\ge\\underline H{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$,\ncf.\\ Proposition \\ref{prop6.1} (a). The rest follows from Lemma \\ref{lemma3.3}.\n\\end{proof}\n\nIn view of Lemma \\ref{lemma4.4}, an $h$-(dot-)quasi\\-resolutive function\nis $h$-dot-resolutive if and only if $\\dot E_f^h=\\varnothing$.\nAssertions 1.\\ and 2.\\ of Proposition \\ref{prop6.3} therefore remain valid\nwhen $E$ is replaced throughout by $\\dot E$.\nThe proof of the dot-version of eq.\\ (3.1) uses 1.\\ of Proposition\n\\ref{prop6.2} in place of 2.\\ there.\nThe following lemma is analogous to Lemma \\ref{lemma6.3c}:\n\n\\begin{lemma}\\label{lemma4.5}  Let $f$ be an $h$-quasi\\-resolutive function\non $\\Delta(U)$. If $f^+$ and $f^-$ are $h$-dot-resolutive then so is $f$,\nand the function\n$H_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\non $U$ is finite valued.\n\\end{lemma}\n\n\\begin{proof} Since $f$ is $h$-quasi\\-resolutive, so are $f^+,f^-$\n(besides being $h$-dot-resolutive) by 3.\\ in Proposition \\ref{prop6.3}.\nHence the functions $H_{f^{\\pm}}^h=\\dot H{}_{f^{\\pm}}^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f^{\\pm}}^h$\nare finite valued (co-polar subets of $U$ being non-void). From\n$-f^-\\le f\\le f^+$ it therefore follows by Proposition \\ref{prop6.1} (a) that\n$$-\\infty\n<-{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f^-}^h\n\\le{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\le\\dot H{}_f^h\\le\\dot H{}_{f^+}^h<+\\infty.$$\nApplying 1.\\ in Proposition \\ref{prop6.2} to the sums\n$f=f^++(-f)^+=f^+-f^-$ and $-f=f^--f_+$, which are both well defined on\n$\\Delta(U)$, we obtain\n$$\n\\dot H{}_f^h\\le H_{f^+}^h-H_{f^-}^h\n\\le{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n$$\non all of $U$. It follows that $\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=H_{f^+}^h-H_{f^-}^h$ holds there, again by Proposition \\ref{prop6.1} (a).\n\\end{proof}\n\n\\begin{cor}\\label{cor4.8a} Let $(f_j)$ be an increasing sequence of lower\nbounded $h$-dot-resolutive functions\n$\\Delta(U)\\longrightarrow\\,]-\\infty,+\\infty]$, and let $f=\\sup_jf_j$.\nIf $\\dot H{}_f^h\\not\\equiv+\\infty$ then $f$ is $h$-dot-resolutive.\n \\end{cor}\n\n\\begin{proof} For every $j$ we have\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge{H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_{f_j}^h\n=\\dot H{}_{f_j}^h$, and hence\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n\\ge\\sup_j\\dot H{}_{f_j}^h=\\dot H{}_f^h$\naccording to 4.\\ in Proposition \\ref{prop6.2}.\nHere equality prevails on account of Proposition \\ref{prop6.1} (a).\nBy hypothesis, $H_f^h\\not\\equiv+\\infty$, and clearly $H_f^h>-\\infty$, so we\nconclude that $f$ indeed is $h$-dot-resolutive.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.4a} For any $\\mu_h$-measurable subset $A$\nof $\\Delta(U)$ the indicator function $1_A$ is $h$-dot-resolutive and\n{\\rm{(3.3)}} holds.\nIn particular, the constant function $1$ on $\\Delta(U)$ is\n$h$-dot-resolutive and $H_1^h=1$.\n\\end{prop}\n\n\\begin{proof} Since $\\dot H{}_{1_A}^h\\le\\dot H_1^h=1<+\\infty$ it follows from\nProposition \\ref{prop6.1} (c) that $\\overline H{}_{1_A}^h=\\dot H{}_{1_A}^h$,\nand the assertions reduce to the analogous Proposition \\ref{prop6.4}.\n\\end{proof}\n\n\\begin{prop}\\label{prop6.5a} Let $f$ be a $\\mu_h$-measurable\nlower bounded function on $\\Delta(U)$. Then\n$$\n{\\dot H}{}_f^h=\\frac{1}{h}\\int f(Y)K(.,Y) d\\mu_h(Y)>-\\infty,$$\nand ${\\dot H}{}_f^h$ is either identically $+\\infty$ or the sum of\nan $h$-invariant function and a constant $\\le0$.\nFurthermore $f$ is $h$-quasi\\-resolutive if and only if $f$ is\n$h$-dot-resolutive,\nand that holds if and only if\n$\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y)<+\\infty$\nq.e.\\ on $U$, or equivalently: everywhere on $U$.\nIn particular, every bounded $\\mu_h$-measurable function\n$f:\\Delta(U)\\longrightarrow{\\mathbb R}$ is $h$-dot-resolutive.\n\\end{prop}\n\n\\begin{proof} In view of the case of $\\dot H$ in 4.\\ of Proposition\n\\ref{prop6.2} the proof of the analogous Proposition \\ref{prop6.5} carries over\n{\\it{mutatis mutandis}}.\n\\end{proof}\n\n\\begin{cor}\\label{cor6.10a} Let $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$\nbe $\\mu_h$-measurable. Then $f$ is $h$-dot-resolutive\nif and only if $|f|$ is $h$-dot-resolutive.\n\\end{cor}\n\n\\begin{proof} If $f$ is $h$-dot-resolutive, and therefore\n$h$-quasiresolutive by Lemma \\ref{lemma4.4}, then $|f|=f\\vee(-f)$ is\n$h$-quasi\\-resolutive according to 1.\\ and 3.\\ in Proposition \\ref{prop6.3}.\nNow, $|f|$ is lower bounded (and $\\mu_h$-measurable), and $|f|$ is therefore\neven $h$-dot-resolutive, by Proposition \\ref{prop6.5a}.\nConsequently, $|f|K(x,.)$ is $\\mu_h$-integrable for every\n$x\\in U$. So are therefore $f^+K(x,.)$ and $f^-K(x,.)$.\nAgain by Proposition \\ref{prop6.5a} it follows that $f^+$ and $f^-$ are\n$h$-dot-resolutive along with $f^+$ and $f^-$ (positive) by Lemma\n\\ref{prop6.5a}. So is therefore $f=f^+-f^-$ by Lemma \\ref{lemma4.5}.\n\\end{proof}\n\n\n\\begin{cor}\\label{cor4.6} A function $f$ on $\\Delta(U)$ with values in\n$\\overline{\\mathbb R}$ is $h$-dot-resolutive if and only if $f$ is\n$h$-quasi\\-resolutive.\n\\end{cor}\n\n\\begin{proof} The `only if' part is contained in Lemma \\ref{lemma4.4}.\nFor the `if' part, suppose that $f$ is $h$-quasi\\-resolutive and hence\n$\\mu_h$-measurable according to Proposition \\ref{prop6.10b}. If $f\\ge0$ then\n$f$ is $h$-dot-resolutive according to Proposition \\ref{prop6.5a}.\nFor arbitrary $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ this applies to\n$f^+$ and $f^-$, which are $h$-quasi\\-resolutive\naccording to 3.\\ in Proposition \\ref{prop6.3}. Consequently, $f=f^+-f^-$\nis likewise finely $h$-dot-resolutive by 1.\\ and 2.\\ in Proposition\n\\ref{prop6.3}.\n\\end{proof}\n\n\\begin{theorem}\\label{thm6.8c}\nA function $f:\\Delta(U)\\longrightarrow\\overline{\\mathbb R}$ is\n$h$-dot-resolutive if and only if the\nfunction $Y\\longmapsto f(Y)K(x,Y)$ on $\\Delta(U)$ is $\\mu_h$-integrable  for\nevery $x\\in U$, or equivaletly for quasievery $x\\in U$.\nIn the affirmative case the solution of the PWB-problem\non $U$ with boundary function $f$ is\n$$H{}_f^h:=\\overline H{}_f^h=\\underline H{}_f^h=\\dot H{}_f^h\n={H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h\n=\\frac{1}{h}\\int f(Y)K(.,Y)d\\mu_h(Y).\n$$\n\\end{theorem}\n\n\\begin{proof} The proof of the analogous Theorem \\ref{thm6.8a} carries over\n{\\it{mutatis mutandis}}.\n\\end{proof}\n\nThe alternative concept of $h$-resolutivity relative to the mf-topology\ndiscussed at the end of the preceding section likewise has a similarly\nestablished compatible version based on $\\dot H{}_f^h$ and\n${H}{\\vrule width 0pt depth 0.2em }_{\\kern-0.7em\\hbox{.}}\\,\\,{}_f^h$\ninstead of $\\overline H{}_f^h$ and $\\underline H{}_f^h$.\n\n\n\\thebibliography{9}\n\n\\bibitem{Ai} Aikawa, H.:\\textit{Potential Analysis on non-smooth domains --\nMartin boundary and boundary Harnack principle}, Complex Analysis and\nPotential Theory, 235--253, CRM Proc. Lecture Notes 55, Amer. Math. Soc.,\nProvidence, RI, 2012.\n\n\\bibitem{Al} Alfsen, E.M.: \\textit{Compact Convex Sets and Boundary\nIntegrals}, Ergebnisse der Math., Vol. 57, Springer, Berlin, 2001.\n\n\\bibitem{AG} Armitage, D.H., Gardiner, S.J.: \\textit{Classical\nPotential Theory}, Springer, London, 2001.\n\n\\bibitem{Do} Doob, J.L.: \\textit{Classical Potential Theory and Its\nProbabilistic Counterpart}, Grundlehren Vol. 262, Springer, New York, 1984.\n\n\\bibitem{El1} El Kadiri, M.: \\textit{Sur la d\\'ecomposition de\nRiesz et la repr\\'esentation int\\'egrale des fonctions finement\nsurharmoniques}, Positivity {\\bf 4} (2000), no. 2, 105--114.\n\n\\bibitem{EF1} El Kadiri, M., Fuglede, B.: \\textit{Martin boundary of a\nfine domain and a Fatou-Naim-Doob theorem for finely super\\-harmonic\nfunctions}, Manuscript (2013).\n\n\\bibitem{EF2} El Kadiri, M., Fuglede, B.: \\textit{Sweeping at the Martin\nboundary of a fine domain}, Manuscript (2013).\n\n\\bibitem{F1} Fuglede, B.: \\textit{Finely Harmonic Functions}, Lecture Notes\nin Math. 289, Springer, Berlin, 1972.\n\n\\bibitem{F2} Fuglede, B.: \\textit{Sur la fonction de Green pour un\ndomaine fin}, Ann. Inst. Fourier \\textbf{25}, 3--4 (1975), 201--206.\n\n\\bibitem{F4} Fuglede, B.: \\textit{Integral representation of fine\n potentials}, Math. Ann. \\textbf{262} (1983), 191--214.\n\n\\end{document}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nCertain nonstandard Lagrangian and Hamiltonian dynamical systems \\cite{mukunda,arnold,Parra} encompass very interesting classes of nonlinear oscillators and admit fascinating dynamical properties \\cite{musie,cari,jose} such as   isochronous oscillations,  linearization, nonlocal transformations and so on \\cite{CSL:02,Nucci,cmee,ChML,Ch:02,EAAR,Chandru;jPA}.   In particular, the Li\\'enard class of oscillators appear in the study of a wide range of fields such as seismology \\cite{cartwright}, biological regulatory systems \\cite{nicolas}, in the study of a self graviting stellar gas cloud \\cite{shapiro}, optoelectronics, fluid mechanics \\cite{kalashnik}.  Many of these Li\\'enard class of equations admit limit cycle\/periodic oscillations which are used to model many physical phenomenon.  Identifying such classes of coupled Li\\'enard-type equations admitting isochronous oscillations is an interesting area of research.  Several procedures have been developed to construct and identify classes of isochronous oscillators.  In particular, Calogero \\cite{Calogero:08} and Calogero and Leyvraz \\cite{Calogero:08c,Calogero:08c1,Calogero:08c2,Calogero:08d,Partha:0a} have developed many techniques to generate isochronous oscillators.  In a recent paper a procedure to generate scalar isochronous systems recursively from a given Hamiltonian \\cite{CDL:01} and a method to construct higher dimensional isochronous nonsingular Hamiltonian systems have  been discussed \\cite{DGCL:01}.  In this paper we obtain a class of coupled Li\\'enard type oscillator equations admitting isochronous oscillations by generalizing the nonstandard Lagrangian of the scalar system to coupled systems.  In order to do so let us consider the scalar Li\\'enard-type linear\/ quadratic and mixed type (in velocities) systems.  For example the Li\\'enard equation with linear velocity term,\n\\begin{eqnarray}\n\\ddot x+F(x)\\dot x+G(x)=0,\\label{meee}\n\\end{eqnarray}\nadmits a class of interesting nonstandard type conserved Hamiltonian \\cite{CSL:02} and isochronous solutions for the choice \\cite{ChML},\n\\begin{eqnarray}\nF(x)=3kx, \\quad G(x)=k^2x^3+\\omega^{2}x.\\label{mee}\n\\end{eqnarray}\nFor the choice $\\omega=0$ the above equation reduces to the modified Emden equation which is well studied in the literature and it occurs in the study of equilibrium configurations of a spherical cloud acting under the mutual attraction of its molecules and subject to the laws of thermodynamics \\cite{Ch:02} and in the modelling of the fusion of pellets \\cite{EAAR}. For the choice $\\omega\\ne0$ the above equation exhibits isochronous behaviour, that is it admits periodic oscillations with frequency of oscillation independent of the amplitude \\cite{CSL:02}.\nThe general solution of this equation is given as \n\\begin{eqnarray}\nx(t)=\\frac{A\\sin(\\omega t+\\delta)}{1-\\frac{kA}{\\omega}\\cos(\\omega t+\\delta)}.\n\\end{eqnarray}\nThe corresponding system admits a nonstandard Hamiltonian\n\\begin{eqnarray}\nH=\\frac{1}{2}\\hat{F}(p)x^2+U(p), \\label{Hammee}\n\\end{eqnarray}\nwith \n\\begin{eqnarray}\n\\hat{F}(p)=\\omega^2\\bigg(1-\\frac{2 k}{\\omega^2}p  \\bigg), \\quad U(p)=\\frac{\\omega^{4}}{2k^2}\\bigg(\\sqrt{1-\\frac{2k}{\\omega^2}p}   -1\\bigg)^2,\\label{Hammee2}\n\\end{eqnarray}\nwhere the canonically conjugate momentum\n\\begin{eqnarray}\np= \\frac{\\omega^2}{2k}\\bigg(1-\\frac{\\omega^{4}}{(k\\dot x+k^2x^2+\\omega^2)^2}  \\bigg) .\n\\end{eqnarray}\n\nSystem (\\ref{Hammee}) is \\emph{PT} symmetric $(x\\rightarrow-x$, $t\\rightarrow-t$, $\\dot {x}\\rightarrow \\dot {x})$ and is also exactly quantizable in momentum space \\cite{chithi}.\nThe class of nonlinear oscillators containing both quadratic and linear terms in $\\dot x$ is called a mixed Li\\'enard-type equation.  It can be written in the form \n\\begin{eqnarray}\n\\ddot x+J(x)\\dot x^2+F(x)\\dot x+G(x)=0,\\label{lienard2}\n\\end{eqnarray}\nwhere $F(x)$, $G(x)$ and $J(x)$ are functions of $x$.  For example with the choice\n \\begin{eqnarray}\nJ(x)=\\frac{f_x}{f},\\quad F(x)=\\frac{(r+2)h_x}{(r+1)f}, \\quad G(x)=\\frac{hh_x}{(r+1)f^2}, \\,r\\ne-1\\label{func}\n\\end{eqnarray}\nthe system admits a nonstandard Lagrangian and Hamiltonian functions of the form\n\\begin{eqnarray}\nL=\\frac{1}{(f(x)\\dot x+h(x))^r},\\label{scalag}\n\\end{eqnarray}\nwhere $f(x)$ and $h(x)$ are arbitrary functions of $x$, $r$ is a suitably chosen real positive parameter and the Hamiltonian\n\\begin{eqnarray}\nH=\\frac{p}{f}\\bigg(\\bigg(\\frac{-r f}{p}\\bigg)^{\\frac{1}{r+1}}-h   \\bigg)-\\bigg(\\frac{-r f}{p}  \\bigg)^{\\frac{-r}{r+1}},\n\\end{eqnarray}\nwith the conjugate momentum\n\\begin{eqnarray}\np=-rf(f\\dot x+h)^{-(r+1)}.\\label{scalagp}\n\\end{eqnarray}\nIn the above, $r$ is a real positive number so chosen such that $H$ is real and in (\\ref{scalag})-(\\ref{scalagp}) only the principal branch is taken when fractional powers appear.  Note that the MEE is a special case of the above system with  $f=1$, $r=1$ and \n $h=kx^2$.  We also note here that this class of Lagrangian and Hamiltonian systems are also drawing considerable interest in connection with supersymmetry related partner systems \\cite{curtright,bijan} and $\\emph PT$-symmetric systems \\cite{chithi}.  \n Now, it is of considerable importance to extend the study of above type nonstandard Lagrangian and Hamiltonian systems to higher degrees of freedom.  Particularly we wish to identify two dimensional nonstandard Hamiltonian systems which are isochronous and in future to study their exact quantization as in the case of MEE \\cite{chithi}.  \n\nA natural way of generalising the above one dimensional nonstandard Lagrangian is to modify \n(\\ref {scalag}) to the form, \n \\begin{eqnarray}\nL=\\frac{1}{(f\\dot x+g\\dot y+h)^r}\\label{arblag}\n\\end{eqnarray}\nwhere $f=f(x,y), g=g(x,y), h=h(x,y)$.\nIn this case, however the Lagrangian turns out to be singular or degenerate \\cite{Parra} \nwhich can be verified from the vanishing the Hessian,\n\\begin{eqnarray}\n\\Delta\\equiv\\left|\n\\begin{array}{ccc}\n\\frac{\\partial^2 L}{\\partial \\dot{x}^2}&\\frac{\\partial^2 L}{\\partial \\dot{x}\\partial \\dot{y}}\\\\\n\\frac{\\partial^2 L}{\\partial \\dot{y}\\partial \\dot{x}}&\\frac{\\partial^2 L}{\\partial \\dot{y}^2}\n\\end{array}\n\\right|\n=0.\n\\end{eqnarray}\nTo obtain a nonsingular Lagrangian, a modification of the above Lagrangian with suitable terms is essential.  For this purpose, we make a simple minded extension of the above form (\\ref{arblag}) judiciously and succeed to identify non-trivial coupled nonstandard and nonsingular Lagrangian type nonlinear evolution equations.  We then show how from the associated equations of motion one can obtain coupled mixed Li\\'enard-type equations, which may be considered as the natural extension of the above single degree of freedom mixed Li\\'enard-type equation (\\ref{lienard2}).  Then the procedure is specifically illustrated for the MEE equation to obtain a two dimensional isochronous extension of the MEE equation.  \n\n\nThe plan of the paper is as follows.  In section 2, we  introduce an  arbitrary form of nonsingular Lagrangian and use it to obtain the corresponding Newton's equation of motion.  However the resultant equation of motion is not in the  form of mixed Li\\'enard-type class of oscillators.  By demanding the resultant dynamical equations are of generalized coupled mixed Li\\'enard-type equations, we deduce the functional form of the allowed nonstandard Lagrangian.  In section 3, we show that the conditions obtained from the coupled mixed Li\\'enard-type oscillators allow one to deduce another class of Li\\'enard-type oscillators which exhibits periodic and quasiperiodic motions.  Also we have shown that the corresponding Hamiltonian obtained from the coupled nonsingular Lagrangian can be transformed into a two dimensional harmonic oscillator Hamiltonian through appropriate canonical transformations.  In section 4, we have shown that a special case of Li\\'enard-type of oscillators leads to a two dimensional version of  the modified Emden equation which exhibits isochronous property.  \nThe general solution is shown to admit periodic as well as quasiperiodic solutions for suitable choices of parameters. Finally, in section 5, we present our conclusions.  \n\n\\section{Two dimensional coupled mixed Li\\'enard-type equations}\n\nWe wish to identify a coupled mixed Li\\'enard-type system which possesses a nonstandard Hamiltonian and admits isochronous solution by a suitable generalization of the scalar Lagrangian (\\ref{scalag}) of MEE.  As noted earlier, the Lagrangian (\\ref{arblag}) which is a natural generalization in two dimensions is singular.  One can overcome this problem by suitably redefining the Lagrangian in the form,  \n\n\n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(f_{i}\\dot x+g_{i}\\dot y+h_{i})^{r_{i}}}.\\label{eq7}\n\\end{eqnarray}\nwhere $f_{i}=f_{i}(x,y)$, $g_{i}=g_{i}(x,y)$, $h_{i}=h_{i}(x,y)$, $i=1,2$.  With this choice of the Lagrangian we can show that Hessian in the present case is non-zero,\n\\begin{eqnarray}\n\\Delta\\equiv\\left|\n\\begin{array}{ccc}\n\\frac{\\partial^2 L}{\\partial \\dot{x}^2}&\\frac{\\partial^2 L}{\\partial \\dot{x}\\partial \\dot{y}}\\\\\n\\frac{\\partial^2 L}{\\partial \\dot{y}\\partial \\dot{x}}&\\frac{\\partial^2 L}{\\partial \\dot{y}^2}\n\\end{array}\n\\right|\n\\ne0.\n\\end{eqnarray}\nConsequently, the Lagrangian is nonsingular.\nFrom the above Lagrangian, the equation of motion can be obtained as  \n\\begin{eqnarray}\n&&\\ddot x= \\bigg(g_{2}(x,y)q_{1}(\\dot x,\\dot y,x,y)+g_{1}(x,y)q_{2}(\\dot x,\\dot y,x,y)  +  d_{1}(x,y)\\dot x^2\n\\nonumber\\\\\n&&\\hspace{0.5cm}+d_{2}(x,y)\\dot y^2+d_{3}(x,y)\\dot x \\dot y+ s_{1}(x,y)\\dot x+s_{2}(x,y)\\dot y+u_{1}(x,y)    \\bigg),\\label{seceq1}\\\\\n&&\\ddot y= -\\bigg(f_{2}(x,y)q_{1}(\\dot x,\\dot y,x,y)+f_{1}(x,y)q_{2}(\\dot x,\\dot y,x,y)+d_{4}(x,y)\\dot x^2 \n\\nonumber\\\\\n&&\\hspace{0.5cm}+d_{5}(x,y)\\dot y^2+d_{6}(x,y)\\dot x \\dot y+ s_{3}(x,y)\\dot x+ s_{4}(x,y)\\dot y+u_{2}(x,y)     \\bigg),\\label{seceq2}\\\\\n&&\\hspace{-1cm}\\mbox{where}\\nonumber\\\\ \n&&q_{1}=\\frac{(r_{2}+1)}{\\Delta^{2}r_{1}}\\bigg(r_{2}(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{(r_{1}+2)} (f_{2}\\dot x+g_{2}\\dot y+h_{2})^{-(r_{2}+1)}\\nonumber\\\\\n&&\\hspace{0.5cm}\\times\\bigg(f_{2}h_{2y}-g_{2}h_{2x}+(f_{2y}-g_{2x})(\\dot x f_{2}+\\dot y g_{2})\\bigg)\\bigg),\\label{q1}\\\\\n&&q_{2}=\\frac{(r_{1}+1)}{\\Delta^{2}r_{2}}\\bigg(r_{1}(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{-(r_{1}+1)} (f_{2}\\dot x+g_{2}\\dot y+h_{2})^{(r_{2}+2)}\\nonumber\\\\\n&&\\hspace{0.5cm}\\times\\bigg(f_{1}h_{1y}-g_{1}h_{1x}+(f_{1y}-g_{1x})(\\dot x f_{1}+\\dot y g_{1})\\bigg)\\bigg),\\label{q2}\n\\end{eqnarray}\n\\begin{eqnarray}\n&&d_{1}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}f_{2}(g_{1}(f_{2y}-g_{2x})(r_{1}+1)+g_{2}(f_{1y}-g_{1x})(r_{2}+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+(r_{1}+1)(r_{2}+1)(g_{1}g_{2}(f_{2}f_{1x}+f_{1}f_{2x})-f_{2}f_{2x}g_{1}^2-f_{1}f_{1x}g_{2}^2) \\bigg),\\\\\n&&d_{2}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}^2(r_{1}+1)(g_{2}(f_{2y}-g_{2x})-f_{2}g_{2y}(r_{2}+1))+g_{2}^2(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(g_{1}(f_{1y}-g_{1x})-f_{1}g_{1y}(r_{1}+1))+g_{1}g_{2}(r_1+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_{2}+1)(f_{1}g_{2y}+f_{2}g_{1y})   \\bigg),\\\\\n&&d_{3}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}g_{2}(f_{2}(r_{2}+1)( f_{1y}(r_{1}+2)+g_{1x}r_{1})+f_{1}(r_{1}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{2y}(r_{2}+2)+g_{2x}r_{2}))-g_{1}^2f_{2}(r_{1}+1)(f_{2y}r_{2}+ g_{2x}(r_{2}+2))\\nonumber\\\\\n&&\\hspace{0.5cm}-g_{2}^2f_{1}(r_{2}+1)(f_{1y}r_{1}+ g_{1x}(r_{1}+2))  \\bigg),\\\\\n&&d_{4}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2(r_{1}+1)(f_{2}(f_{2y}-g_{2x})+ f_{2x}g_{2}(r_{2}+1))+f_{2}^{2}(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{1}(f_{1y}-g_{1x})+ f_{1x}g_{1}(r_1+1))- f_{1}f_{2}(r_1+1)(r_2+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{2x}g_{1}+f_{1x}g_{2}) \\bigg),\n\\end{eqnarray}\n\\begin{eqnarray}\n&&d_{5}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}g_{2}(f_{1}(f_{2y}-g_{2x})(r_1+1)+f_{2}(f_{1y}-g_{1x})(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+(f_{2}g_{1}-f_{1}g_{2})(f_{2}g_{1y}-f_{1}g_{2y})(r_{1}+1)(r_{2}+1) \\bigg), \\\\\n&&d_{6}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2g_{2}(r_1+1)(g_{2x}r_{2}+f_{2y}(r_2+2))+f_{2}^2g_{1}(r_2+1)(g_{1x}r_{1}\\nonumber\\\\\n&&\\hspace{0.5cm}+ f_{1y}(r_1+2))-f_{1}f_{2}(g_{1}(r_1+1) (f_{2y}r_{2}+g_{2x}(r_2+2))+g_{2}(r_2+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(f_{1y}r_{1}+ g_{1x}(r_1+2)))\\bigg),\\\\\n&&s_{1}=\\frac{1}{\\Delta^{2}}\\bigg(g_{1}(r_1+1)(f_{1}(h_{2}(f_{2y}-g_{2x})+f_{2}h_{2y})-f_{2}g_{1}h_{2x}(r_2+2))\\nonumber\\\\\n&&\\hspace{0.5cm}+g_{2}(r_2+1)(f_{2}(h_{1}(f_{1y}-g_{1x})+f_{1}h_{1y})-f_{1}g_{2}h_{1x}(r_1+2))\\nonumber\\\\\n&&\\hspace{0.5cm}+ g_{1}g_{2}(r_1+1)(r_2+1) (f_{2}h_{1x}+f_{1}h_{2x})   \\bigg), \\\\\n&&s_{2}=\\frac{1}{\\Delta^{2} }\\bigg(g_{1}^2(r_1+1)(h_{2}(f_{2y}-g_{2x})-g_{2}h_{2x}-f_{2}h_{2y}(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}\n+g_{2}^2(r_2+1) (h_{1}(f_{1y}-g_{1x})-g_{1}h_{1x}-f_{1}h_{1y}(r_1+1))+g_{1}g_{2}\n\\nonumber\\\\\n&&\\hspace{0.5cm}\n\\times(f_{2}h_{1y}(r_1+2)(r_2+1)+f_{1}h_{2y}(r_1+1)(r_2+2))  \\bigg),\\\\\n&&s_{3}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}^2(r_1+1)(h_{2}(f_{2y}-g_{2x})+f_{2}h_{2y}+g_{2}h_{2x}(r_2+1))\\nonumber\\\\\n&&\\hspace{0.5cm}+f_{2}^2(r_2+1)(h_{1}(f_{1y}-g_{1x})+f_{1}h_{1y}+g_{1}h_{1x}(r_1+1))\n\\nonumber\\\\\n&&\\hspace{0.5cm}-f_{1}f_{2}(g_{2}h_{1x}(r_1+2)(r_2+1)\n+g_{1}h_{2x}(r_1+1)(r_2+2))  \\bigg),\\\\ \n&&s_{4}=\\frac{1}{\\Delta^{2}}\\bigg(f_{1}g_{1}(r_{1}+1)(h_{2}(f_{2y}-g_{2x})-g_{2}h_{2x})+f_{2}g_{2}(r_{2}+1)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(h_{1}(f_{1y}-g_{1x})-g_{1}h_{1x})-f_{1}f_{2}(r_{1}+1)(r_{2}+1)(g_{2}h_{1y}+g_{1}h_{2y})\\nonumber\\\\\n&&\\hspace{0.5cm}+f_{1}^2g_{2}h_{2y}(r_{1}+1)(r_{2}+2)+f_{2}^2g_{1}h_{1y}(r_{1}+2)(r_{2}+1) \\bigg),\\\\\n&&u_{1}=\\frac{1}{\\Delta^{2}}\\bigg[g_{1}h_{2}(f_{1}h_{2y}-g_{1}h_{2x})(r_1+1)\n+g_{2}h_{1}(f_{2}h_{1y}-g_{2}h_{1x})\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_2+1)\\bigg], \\\\\n&&\nu_{2}=\\frac{1}{\\Delta^{2}}\\bigg[f_{1}h_{2}(f_{1}h_{2y}-g_{1}h_{2x})(r_1+1)+f_{2}h_{1}(f_{2}h_{1y}-g_{2}h_{1x})\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(r_2+1)\\bigg], \\\\\n&&\\Delta=(f_{2}g_{1}-f_{1}g_{2})\\sqrt{(r_{1}+1)(r_{2}+1)}. \n\\end{eqnarray}\n\n\n\n\n\n\n\nThe mixed scalar Li\\'enard-type equation  (\\ref{lienard2}) has only linear and quadratic terms in $\\dot x$.  On the other hand  the coupled system of second order equations of motion (\\ref{seceq1}) and (\\ref{seceq2}), obtained from the nonsingular Lagrangian (\\ref{eq7}) is not in the class of mixed Li\\'enard-type oscillators because it contains higher\/different powers of $\\dot x$ and  $\\dot y$ than quadratic and linear powers.  By equating  these higher\/different degree coefficients to zero, analyzing them and making use of the results in the original coupled equations (\\ref{seceq1}) and (\\ref{seceq2}), we can obtain the relevant evolution equations.  For this purpose we equate the terms $q_{1}$ and $q_{2}$ in equations (\\ref{seceq1}) and (\\ref{seceq2}) to zero as they contain higher-degree terms in $\\dot x$, $\\dot y$ and their products.  Therefore we take\n\\begin{eqnarray}\nq_{1}(\\dot x,\\dot y, x ,y)=0, \\quad q_{2}(\\dot x,\\dot y, x ,y)=0. \\label{q1q2}\n\\end{eqnarray}\nSolving equation (\\ref{q1}) and (\\ref{q2}),  we get a set of partial differential equations for the variables $f_{i}$ and $g_{i}$, $(i=1,2)$. \nWe can easily see that from $q_{1}(\\dot x,\\dot y, x ,y)=0$, we get\n\\begin{eqnarray}\nf_{2y}=g_{2x},\\quad  f_{2}=\\frac{g_{2}h_{2x}}{h_{2y}},\\label{condition1}\n\\end{eqnarray}\n\nSimilarly from $q_{2}(\\dot x,\\dot y, x ,y)=0$, we get\n\\begin{eqnarray}\nf_{1y}=g_{1x},\\quad f_{1}=\\frac{g_{1}h_{1x}}{h_{1y}} , \\label{condition}\n\\end{eqnarray}\n\nOn substituting the above forms in equations (\\ref{seceq1}) and (\\ref{seceq2}), we obtain the coupled system of mixed  Li\\'enard-type class of oscillators of the form\n \n\\begin{eqnarray}\n&&\\ddot x=\\frac{-1}{\\hat{\\Delta}g_{1}g_{2}r_{12}}\\bigg[g_{1}g_{2}\\bigg((r_{1}+1)(r_{2}+1)\\bigg((f_{2x}g_{1}-f_{1x}g_{2})\\dot x^2-(g_{1y}g_{2}-g_{1}g_{2y})\\dot y^2\\nonumber\\\\\n&&\\hspace{0.7cm}-2(g_{1x}g_{2}-g_{1}g_{2x})\\dot x\\dot y\\bigg)   +(r_{1}+1)(r_{2}+2)(h_{2x}\\dot x+h_{2y}\\dot y)g_{1}\\nonumber\\\\\n&&\\hspace{0.7cm}-(r_{1}+2)(r_{2}+1)(h_{1x}\\dot x+h_{1y}\\dot y)g_{2}\\bigg)-g_{2}^2h_{1}h_{1y}(r_2+1)\\nonumber\\\\\n&&\\hspace{0.7cm}+g_{1}^2h_{2}h_{2y}(r_1+1)    \\bigg] ,  \\label{eq91}     \n\\end{eqnarray}\n\\begin{eqnarray}\n&&\\ddot y=\\frac{1}{\\hat{\\Delta}g_{1}g_{2}r_{12}h_{1y}h_{2y}}\\bigg[g_{1}g_{2}\n\\bigg((r_1+1)(r_2+1)  \\bigg((f_{2x}g_{1}h_{1x}h_{2y}-f_{1x}g_{2}h_{1y}h_{2x})\\dot x^2\\nonumber\\\\\n&&\\hspace{0.7cm}+(g_{2y}g_{1}h_{1x}h_{2y}-g_{1y}g_{2}h_{1y}h_{2x})\\dot y^2+2(g_{1}g_{2x}h_{1x}h_{2y}-g_{1x}g_{2}h_{1y}h_{2x})\\dot x\\dot y\\bigg)\n\\nonumber\\\\\n&&\\hspace{0.7cm}-(r_{1}+2)(r_{2}+1)g_{2}h_{1y}h_{2x}(h_{1x}\\dot x+h_{1y}\\dot y)\\nonumber\\\\\n&&\\hspace{0.7cm}+(r_{1}+1)(r_{2}+2)g_{1}h_{2y}h_{1x}(h_{2x}\\dot x+h_{2y}\\dot y)\\bigg)\n+g_{1}^2h_{2}h_{1x}h_{2y}^2(r_1+1)\\nonumber\\\\\n&&\\hspace{0.7cm}-g_{2}^2h_{1}h_{1y}^2h_{2x}(r_2+1)    \\bigg]. \\label{eq92}  \n\\end{eqnarray}\nThe Hamiltonian associated with (\\ref{eq91}) and (\\ref{eq92}) corresponding to the Lagrangian (\\ref{eq7}) can now be written down as \n\n\\begin{eqnarray}\n&&H=\\frac{r_{12}}{\\hat{\\Delta}}\\bigg[\\frac{g_{1}}{h_{1y}}(p_{2}h_{1x}-p_{1}h_{1y})\\bigg(h_{2}-\\bigg(\\frac{g_{1}(h_{1x}p_{2}-h_{1y}p_{1})r_{12}}{h_{1y}r_{2}\\hat{\\Delta}}  \\bigg)^\\frac{-1}{r_2+1}\\bigg)\\nonumber\\\\\n&&\\hspace{0.7cm}  +\\frac{g_{2}}{h_{2y}}(p_{1}h_{2y}-p_{2}h_{2x})\\bigg(h_{1}-\\bigg(\\frac{g_{2}(h_{2y}p_{1}-h_{2x}p_{2})r_{12}}{h_{2y}r_{1}\\hat{\\Delta}}\\bigg) ^\\frac{-1}{r_{1}+1}\\bigg)\\bigg]\\nonumber\\\\\n&&\\hspace{0.7cm}-\\bigg(\\frac{g_{1}(h_{1x}p_{2}-h_{1y}p_{1})r_{12}}{h_{1y}r_{2}\\hat{\\Delta}}\\bigg)^\\frac{r_{2}}{r_2+1}-\\bigg(\\frac{g_{2}(h_{2y}p_{1}-h_{2x}p_{2})r_{12}}{h_{2y}r_{1}\\hat{\\Delta}}\\bigg)^\\frac{r_{1}}{r_1+1},\n\\end{eqnarray}\nwhere $\\hat{\\Delta}=g_{1}g_{2}(h_{1y}h_{2x}-h_{1x}h_{2y})(h_{1y}h_{2y})^{-1}r_{12},\\,\\,r_{12}=[(r_{1}+1)(r_{2}+1)]^{\\frac{1}{2}}$.\nHere the conjugate momenta $p_{1}$ and $p_{2}$ are defined as \n\\begin{eqnarray}\n&&\\hspace{-2cm}p_{1}=L_{\\dot x}=-\\frac{f_{1}r_{1}}{(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{r_{1}+1}}-\\frac{f_{2}r_{2}}{(f_{2}\\dot x+g_{2}\\dot y+h_{2})^{r_{2}+1}},\\\\\n&&\\hspace{-2cm}p_{2}=L_{\\dot y}=-\\frac{g_{1}r_{1}}{(f_{1}\\dot x+g_{1}\\dot y+h_{1})^{r_{1}+1}}-\\frac{g_{2}r_{2}}{(f_{2}\\dot x+g_{2}\\dot y+h_{2})^{r_{2}+1}}.\n\\end{eqnarray}\n\n\\section{\\bf{Reduction to a subclass exhibiting quasiperiodic motion}}\n~~~~~~~~\tIn our further analysis, for simplicity, we assume the parameters $r_{1}$ = $r_{2}$ = $1$ in the Lagrangian given by (\\ref{eq7}).  Now, let the quantities $(f_{i}\\dot x+g_{i}\\dot y)$, $i=1,2,$ be the total derivatives (when $r_{1}$ = $r_{2}$ = $1$) of certain functions $\\rho_{i}(x,y)$.  \n\\begin{eqnarray}\nf_{i}\\dot x+g_{i}\\dot y=\\frac{d}{dt}[\\rho_{i}(x,y)]=\\rho_{ix}\\dot x+\\rho_{iy}\\dot y.\\label{totder}\n\\end{eqnarray}\nFrom the above equation, we find $f_{i}=\\rho_{ix}$, $g_{i}=\\rho_{iy}$, $i=1,2$.  Substituting Eq. (\\ref{totder}) in Eq. (\\ref{eq7}) we get\n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(\\rho_{ix}\\dot x+\\rho_{iy}\\dot y+h_{i})}\\label{modlag}.\n\\end{eqnarray}\n\n\n\nSimilarly, the condition (\\ref{condition}) reduces to \n\\begin{eqnarray}\n\\rho_{ix}=\\frac{\\rho_{iy}h_{ix}}{h_{iy}}. \\label{frac}\n\\end{eqnarray}\nEquation (\\ref{frac}) can also be written as\n\\begin{eqnarray}\n\\left|\n\\begin{array}{ccc}\n\\rho_{ix}&\\rho_{iy}\\\\\nh_{ix}&h_{iy}\n\\end{array}\n\\right|=0.\n\\end{eqnarray}\nConsequently the term $h_{i}$ is functionally dependent on $\\rho_{i}$ that is $h_{i}=Q_{i}(\\rho_{i})$.\\\\\nThen the Lagrangian (\\ref{modlag}) can also be written as \n\\begin{eqnarray}\nL=\\sum_{i=1}^{2}\\frac{1}{(\\rho_{ix}\\dot x+\\rho_{iy}\\dot y+Q_{i}(\\rho_{i}))}.\\label{rearr1}\n\\end{eqnarray}\n~~~~~~~~Now the modified Emden equation is a special case of  Li\\'enard type of oscillators \\cite{CSL:02} which is obtained for specific forms of $Q_{i}(\\rho_{i})=\\rho_{i}^2+\\lambda_{i}$, where $\\lambda_{i}'s$ are constants. \\\\\nFrom the Lagrangian (\\ref{rearr1}), the corresponding equation of motion is \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\label{equation12}\n\\addtocounter{equation}{1}\n&&\\ddot x=\\bigg((\\rho_{1xx}\\rho_{2y}-\\rho_{1y}\\rho_{2xx})\\dot x^2+(\\rho_{1yy}\\rho_{2y}-\\rho_{1y}\\rho_{2yy})\\dot y^2+3(\\rho_{1}\\rho_{1x}\\rho_{2y}\\nonumber\\\\\n&&\\hspace{0.5cm}-\\rho_{2}\\rho_{1y}\\rho_{2x})\\dot x+3 \\rho_{1y}\\rho_{2y}(\\rho_{1}-\\rho_{2})\\dot y+2(\\rho_{1xy}\\rho_{2y}-\\rho_{1y}\\rho_{2xy})\\dot x\\dot y\\nonumber\\\\\n&&\\hspace{0.5cm}+\\rho_{1}\\rho_{2y}(\\rho_{1}^2+\\lambda_{1})-\\rho_{2}\\rho_{1y}(\\rho_{2}^2+\\lambda_{2})\\bigg)(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1},\\label{equation1}\\\\\n&&\\ddot y=\\bigg( (\\rho_{1x}\\rho_{2xx}-\\rho_{1xx}\\rho_{2x})\\dot x^{2}+(\\rho_{1x}\\rho_{2yy}-\\rho_{1yy}\\rho_{2x})\\dot y^{2}+3\\rho_{1x}\\rho_{2x}(\\rho_{2}\\nonumber\\\\\n&&\\hspace{0.5cm}-\\rho_{1})\\dot x+3(\\rho_{2}\\rho_{1x}\\rho_{2y}-\\rho_{1}\\rho_{1y}\\rho_{2x})\\dot y+2(\\rho_{1x}\\rho_{2xy}-\\rho_{2x}\\rho_{1xy})\\dot x\\dot y\\nonumber\\\\\n&&\\hspace{0.5cm}+\\rho_{2}\\rho_{1x}(\\rho_{2}^2+\\lambda_{2})-\\rho_{1}\\rho_{2x}(\\rho_{1}^2+\\lambda_{1})\\bigg)(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1}.\\label{equation2}\n\\end{eqnarray}\nThe associated Hamiltonian becomes\n\\begin{eqnarray}\n&&H=\\bigg((\\rho_{2y}p_{1}-\\rho_{2x}p_{2})(\\rho_{1}^2+\\lambda_{1}) +(\\rho_{1x}p_{2}-\\rho_{1y}p_{1})(\\rho_{2}^2+\\lambda_{2})\\nonumber\\\\\n&&\\hspace{0.5cm} -2\\sqrt{(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})}(\\sqrt{(\\rho_{1x}p_{2}-\\rho_{1y}p_{1})}+\\sqrt{(\\rho_{2y}p_{1}-\\rho_{2x}p_{2})}) \\bigg)\\nonumber\\\\\n&&\\hspace{0.5cm}\\times(\\rho_{1y}\\rho_{2x}-\\rho_{1x}\\rho_{2y})^{-1}.\\label{arbham}\n\\end{eqnarray}\nHere the conjugate momenta $p_{1}$ and $p_{2}$ are defined as\n\\begin{eqnarray}\n&&\\hspace{-1cm}p_{1}=L_{\\dot x}=-\\frac{\\rho_{1x}}{(\\rho_{1x}\\dot x+\\rho_{1y}\\dot y+\\rho_{1}^2+\\lambda_{1})^2}-\\frac{\\rho_{2x}}{(\\rho_{2x}\\dot x+\\rho_{2y}\\dot y+\\rho_{2}^2+\\lambda_{2})^2},\\\\\n&&\\hspace{-1cm}p_{2}=L_{\\dot y}=-\\frac{\\rho_{1y}}{(\\rho_{1x}\\dot x+\\rho_{1y}\\dot y+\\rho_{1}^2+\\lambda_{1})^2}-\\frac{\\rho_{2y}}{(\\rho_{2x}\\dot x+\\rho_{2y}\\dot y+\\rho_{2}^2+\\lambda_{2})^2}.\n\\end{eqnarray}\nThe Hamiltonian (\\ref{arbham}) is connected to the Hamiltonian of  a system of uncoupled  linear harmonic oscillators \n $\\tilde{H}=\\frac{1}{2}(P_{1}^2+P_{2}^2+\\lambda_{1}U_{1}^2+\\lambda_{2}U_{2}^2)$ through the following canonical transformation,\n\n\n\\begin{eqnarray}\n&&P_1=\\left[\\lambda_1+\\left(\\lambda_1^2-2\\lambda_1\\left[\\frac{p_1\\rho_{2y}-p_2\\rho_{2x}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right]\\right)^{\\frac{1}{2}}\\right],\\label{xx}\\\\\n&&P_2=\\left[\\lambda_2+\\left(\\lambda_2^2-2\\lambda_2\\left[\\frac{p_2\\rho_{1x}-p_1\\rho_{1y}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right]\\right)^{\\frac{1}{2}}\\right],\\\\\n&&U_1=-\\frac{\\rho_1}{\\lambda_1}\\left[\\lambda_1^2-2\\lambda_1\\left(\\frac{p_1\\rho_{2y}-p_2\\rho_{2x}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right)\\right]^{\\frac{1}{2}},\\\\\n&&U_2=-\\frac{\\rho_2}{\\lambda_2}\\left[\\lambda_2^2-2\\lambda_2\\left(\\frac{p_2\\rho_{1x}-p_1\\rho_{1y}}{\\rho_{2y}\\rho_{1x}-\\rho_{1y}\\rho_{2x}}\\right)\\right]^{\\frac{1}{2}}.\\label{yy}\n\\end{eqnarray}\n\n\nThe general solution of (\\ref{equation1}) and (\\ref{equation2}) can be found after choosing the forms of $f_{1}$ and $f_{2}$ by using the relations (\\ref{xx}) -(\\ref{yy}) and the harmonic oscillator solution\n\\begin{eqnarray}\nU_{i}=A_{i}\\sin(\\omega_{i}t+\\delta_{i})\\nonumber\\\\\nP_{i}=A_{i}\\omega_{i}\\cos(\\omega_{i}t+\\delta_{i}),\\quad \\omega_i=\\sqrt{\\lambda_i}\n\\end{eqnarray}\nwhere $A_{i}$ and $\\delta_{i}$, ($i=1,2,$) are arbitrary constants. The above canonical transformations are identified by generalizing the knowledge of the canonical transformation for the scalar equation which is discussed in the appendix.\n\n\\subsection{ Nonlocal transformation}\nThe system of coupled mixed MEE equations (\\ref{equation1}) and (\\ref{equation2}) can also be related to a system of uncoupled simple harmonic oscillators,\n\\begin{eqnarray}\n\\ddot{u}+\\lambda_1u=0,\\qquad \\ddot{v}+\\lambda_2v=0,\\label{sho}\n\\end{eqnarray}\nthrough the nonlocal transformations,\n\\begin{eqnarray}\nu=\\rho_1(x,y)e^{\\int \\rho_1(x,y)dt},\\qquad v=\\rho_2(x,y)e^{\\int \\rho_2(x,y)dt}.\\label{nonlocal}\n\\end{eqnarray}\nEquations having such type of nonlocal transformations and their solutions have been studied in Ref. \\cite{nonlocal-connection}.  In addition to using the canonical transformation obtained in the previous subsection to find the solution of equation (\\ref{equation12}), one can also obtain the solution of it by solving the following set of coupled first order equations arising from (\\ref{sho}) and (\\ref{nonlocal}), \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\label{coupled-eq1}\n\\addtocounter{equation}{1}\n\\dot{x}=\\frac{uv(\\rho_2^2\\rho_{1y}-\\rho_1^2\\rho_{2y})+\\dot{u}v\\rho_1\\rho_{2y}-u\\dot{v}\\rho_2\\rho_{1y}}{uv(\\rho_{1x}\\rho_{2y}-\\rho_{1y}\\rho_{2x})},\\\\\n\\dot{y}=\\frac{uv(\\rho_1^2\\rho_{2x}-\\rho_2^2\\rho_{1x})+u\\dot{v}\\rho_2\\rho_{1x}-\\dot{u}v\\rho_1\\rho_{2x}}{uv(\\rho_{1x}\\rho_{2y}-\\rho_{1y}\\rho_{2x})}\\label{coupled-eq2}.\n\\end{eqnarray}\nHere $u$ and $v$ are the solutions of the simple harmonic oscillator equations (\\ref{sho}). The difficulty in solving the equations (\\ref{coupled-eq1}) depends on the form of $\\rho_1$ and $\\rho_2$ (see \\cite{nonlocal-connection}).\n\n\\section{\\bf{A system of coupled Li\\'enard-type equations}}\n Next we can obtain a special case of Li\\'enard-type of oscillators (\\ref{rearr1}) by choosing \n\\begin{eqnarray}\n\\rho_{1}=k_{1}(\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}}), \\quad \\rho_{2}=k_{2}(\\alpha_{3}y^{m_{3}}+\\alpha_{4}x^{m_{4}}). \n\\end{eqnarray}\nwhere $k_{i}$'s and $\\alpha_{j}$'s are arbitrary real parameters and $m_{j}$'s are positive integers and $i=1,2,$ and $j=1,2,3,4.$\n  In this case the specific form of the nonstandard and nonsingular Lagrangian with two degrees of freedom takes the form\n\\begin{eqnarray}\n&&\\hspace{-1.3cm}L=\\frac{1}{\\lambda_{1}+k_{1}^2(\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}})^2+k_{1}(m_{1}\\alpha_{1}x^{m_{1}-1}\\dot x+m_{2}\\alpha_{2}y^{m_{2}-1}\\dot y)}\\nonumber\\\\\n&&\\hspace{-.5cm}+\\frac{1}{\\lambda_{2}+k_{2}^2(\\alpha_{3}y^{m_{3}}+\\alpha_{4}x^{m_{4}})^2+k_{2}(m_{3}\\alpha_{3}y^{m_{3}-1}\\dot y+m_{4}\\alpha_{4}x^{m_{4}-1}\\dot x)}.\\label{lagpowerm}\n\\end{eqnarray}\nThe equation of motion of the mixed Li\\'enard-type class of oscillators can be obtained from the above Lagrangian.  It is of the form\n\\label{secpowerm1&m2}\n\\begin{eqnarray}\n&&\\ddot x=\\frac{-1}{xy^2\\delta_{m}}\\bigg(\\dot x^2 y^2\\bigg[m_{1}m_{3}\\alpha_{1}\\alpha_{3}(m_{1}-1)x^{m_{1}}y^{m_{3}}-m_{2}m_{4}\\alpha_{2}\\alpha_{4}(m_{4}-1)x^{m_{4}}y^{m_{2}}  \\bigg]\\nonumber\\\\\n&&+\n\\dot y^2 x^2\\bigg[    m_{2} m_{3} \\alpha_{2}\\alpha_{3}(m_{2}-m_{3})y^{m_{2}+m_{3}}\\bigg]+ \\bigg[3k_{1}y^{m_{3}} (m_{1}\\alpha_{1}x^{m_{1}}y\\dot x+m_{2}\\alpha_{2}y^{m_{2}}x\\dot y)\\nonumber\\\\\n&& +k_{1}^2xy^{m_{3}+1}(\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})^2+\\lambda_{1}x y^{m_{3}+1} \\bigg]m_{3} \\alpha_{3} x y (\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})\\nonumber\\\\\n&&-\\bigg[3k_{2}y^{m_{2}} (m_{4}\\alpha_{4}x^{m_{4}}y\\dot x+m_{3}\\alpha_{3}y^{m_{3}}x\\dot y) +k_{2}^2xy^{m_{2}+1}(\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}})^2\\nonumber\\\\\n&&+\\lambda_{2}x y^{m_{2}+1} \\bigg]m_{2} \\alpha_{2} x y (\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}}) \\bigg),\\label{secpowerm1}\n\\end{eqnarray}\n\n\\begin{eqnarray}\n&&\\ddot y=\\frac{1}{yx^2\\delta_{m}}\\bigg(\\dot y^2x^2\\bigg[m_{2}m_{4}\\alpha_{2}\\alpha_{4}(m_{2}-1)x^{m_{4}}y^{m_{2}} -m_{1}m_{3}\\alpha_{1}\\alpha_{3}(m_{3}-1)x^{m_{1}}y^{m_{3}}\\bigg]\\nonumber\\\\\n&&-\\dot x^2y^{2}\\bigg[m_{1}m_{4}\\alpha_{1}\\alpha_{4}(m_{4}-m_{1})x^{m_{1}+m_{4}} \\bigg] \n+ \\bigg[3k_{1} x^{m_{4}}(m_{1}\\alpha_{1}x^{m_{1}}y\\dot x+m_{2}\\alpha_{2}y^{m_{2}}x\\dot y)\\nonumber\\\\\n&& +k_{1}^2x^{m_{4}+1}y(\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})^2+\\lambda_{1}yx^{m_{4}+1}  \\bigg]m_{4} \\alpha_{4} x y (\\alpha_{1} x^{m_{1}}+\\alpha_{2} y^{m_{2}})\n\\nonumber\\\\\n&&-\\bigg[3k_{2}x^{m_{1}} (m_{4}\\alpha_{4}x^{m_{4}}y\\dot x+m_{3}\\alpha_{3}y^{m_{3}}x\\dot y) +k_{2}^2x^{m_{1}+1}y(\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}})^2\\nonumber\\\\\n&&+\\lambda_{2}x^{m_{1}+1} y \\bigg]m_{1} \\alpha_{1} x y (\\alpha_{4} x^{m_{4}}+\\alpha_{3} y^{m_{3}}) \\bigg),\\label{secpowerm2}\n\\end{eqnarray}\n\nwhere $\\delta_{m}=m_{1}m_{3}\\alpha_{1}\\alpha_{3}x^{m_{1}}y^{m_{3}}-m_{2}m_{4}\\alpha_{2}\\alpha_{4}x^{m_{4}}y^{m_{2}}$. \\\\\n\n\\noindent The associated Hamiltonian becomes\n\\begin{eqnarray}\n&&\\hspace{-0.3cm}H=\\frac{1}{k_{1}k_{2}\\delta_{m}}\\bigg[k_{2}(m_{4}\\alpha_{4}p_{2}x^{m_{4}}y-m_{3}\\alpha_{3}p_{1}xy^{m_{3}})\\bigg(k_{1}^2 X_{m}^2+\\lambda_{1}  \\bigg)\\nonumber\\\\\n&&\\hspace{0.3cm}+k_{1}(m_{2}\\alpha_{2}p_{1}xy^{m_{2}}-m_{1}\\alpha_{1}p_{2}x^{m_{1}}y)\\bigg(k_{2}^2 Y_{m}^2+\\lambda_{2}  \\bigg)-2\\delta_{m}^{\\frac{1}{2}}\\bigg(k_{1}k_{2}^{\\frac{1}{2}}(m_{2}\\alpha_{2}p_{1}xy^{m_{2}}\\nonumber\\\\\n&&\\hspace{0.5cm}-m_{1}\\alpha_{1}p_{2}x^{m_{1}}y)^{\\frac{1}{2}}+k_{2}k_{1}^{\\frac{1}{2}}(m_{4}\\alpha_{4}p_{2}x^{m_{4}}y-m_{3}\\alpha_{3}p_{1}xy^{m_{3}})^{\\frac{1}{2}}  \\bigg)\\bigg].\n\\end{eqnarray}\nwhere $X_{m}=\\alpha_{1}x^{m_{1}}+\\alpha_{2}y^{m_{2}}$, $Y_{m}=\\alpha_{4}x^{m_{4}}+\\alpha_{3}y^{m_{3}}$.\\\\\n\\subsection{\\bf{Coupled Modified Emden Equation: A special case}}\n\tThe coupled system of equations (\\ref{secpowerm1}) and (\\ref{secpowerm2}) reduces to a coupled generalization of the modified Emden equation (\\ref{meee})-(\\ref{mee}) for the choice $m_{1}$ = $m_{2}$ = $m_{3}$ = $m_{4}$ = $1$ and is of the form \n\\begin{eqnarray}\n\\addtocounter{equation}{-1}\n\\addtocounter{equation}{1}\n&\\hspace{-1cm}\\ddot x=\\frac{-1}{\\hat{\\delta_{1}}}\\bigg[\\bigg(3 k_{1}(\\alpha_{1}\\dot x+\\alpha_{2}\\dot y)+k_{1}^2(\\alpha_{1}x+\\alpha_{2}y)^2 +\\lambda_{1}\\bigg)(\\alpha_{3}(\\alpha_{1}x+\\alpha_{2}y))\\nonumber\\\\\n&\\hspace{-1cm}-\\bigg(3 k_{2}(\\alpha_{3}\\dot y+\\alpha_{4}\\dot x)+k_{2}^2(\\alpha_{3}y+\\alpha_{4}x)^2 +\\lambda_{2}\\bigg)(\\alpha_{2}(\\alpha_{3}y+\\alpha_{4}x)) \\bigg],\\label{eq15}\n\\end{eqnarray}\n\\begin{eqnarray}\n&\\hspace{-1cm}\\ddot y=\\frac{1}{\\hat{\\delta_{1}}}\\bigg[\\bigg(3 k_{1}(\\alpha_{1}\\dot x+\\alpha_{2}\\dot y)+k_{1}^2(\\alpha_{1}x+\\alpha_{2}y)^2 +\\lambda_{1}\\bigg)(\\alpha_{4}(\\alpha_{1}x+\\alpha_{2}y))\\nonumber\\\\&\\hspace{-1cm}-\\bigg(3 k_{2}(\\alpha_{3}\\dot y+\\alpha_{4}\\dot x)+k_{2}^2(\\alpha_{3}y+\\alpha_{4}x)^2 +\\lambda_{2}\\bigg)(\\alpha_{1}(\\alpha_{3}y+\\alpha_{4}x)) \\bigg],\\label{eq15a}\n\\end{eqnarray}\nwhere $\\hat{\\delta_{1}}=(\\alpha_{1}\\alpha_{3}-\\alpha_{2}\\alpha_{4})$.  Note that the above system of equations is \\emph{PT} symmetric under the combined transformations, $(t\\rightarrow-t$, $x\\rightarrow-x$, $ y\\rightarrow- y)$.  There are other ways of generalizing the modified Emden equation to two dimensions, for example see Ref. \\cite{cmee}.  Even though the generalized version identified as the coupled modified Emden equation in Ref. \\cite{cmee} is isochronous, the system lacks a Hamiltonian description in order to quantize it.  However, we find that the system (\\ref{eq15}) and (\\ref{eq15a}) has the well defined Hamiltonian \n\\begin{eqnarray}\n&&\\hspace{-0.7cm}H=\\frac{1}{k_{1}k_{2}\\hat{\\delta_{1}}}\\bigg[k_{2}(p_{2}\\alpha_{4}-p_{1}\\alpha_{3})(k_{1}^2 X^2+\\lambda_{1})+k_{1}(p_{1}\\alpha_{2}-p_{2}\\alpha_{1})(k_{2}^2 Y^2+\\lambda_{2})\\nonumber\\\\\n&&\\hspace{-0.1cm}  -2\\sqrt{\\hat{\\delta_{1}}}(k_{1}k_{2}^{\\frac{1}{2}}(p_{1}\\alpha_{2}-p_{2}\\alpha_{1})^{\\frac{1}{2}}+   k_{2}k_{1}^{\\frac{1}{2}}(p_{2}\\alpha_{4}-p_{1}\\alpha_{3})^{\\frac{1}{2}}) \\bigg],\\label{eq16}\n\\end{eqnarray}\nwhere $X=(\\alpha_{1}x+\\alpha_{2}y)$ and $Y=(\\alpha_{3}y+\\alpha_{4}x)$.   Here we note that one can use a variable transformation $X=(\\alpha_{1}x+\\alpha_{2}y)$ and $Y=(\\alpha_{3}y+\\alpha_{4}x)$ in equation (\\ref{eq15}) to obtain two uncoupled modified Emden equations. \n\nIn order to find the general solution of the equations (\\ref{eq15}) and (\\ref{eq15a}) we use suitable canonical transformation to the above Hamiltonian to reduce it to a simpler form.  We find that the above Hamiltonian is connected to the Hamiltonian of a two dimensional linear harmonic oscillator through the following canonical transformation, see (\\ref{xx})-(\\ref{yy}),\t\n\n\\begin{eqnarray}\n&&P_1=\\lambda_1+\\left[\\lambda_1^2-\\frac{2\\lambda_1(\\alpha_3p_1-\\alpha_4p_2)}{k_1(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\label{eq119}\\\\\n&&P_2=\\lambda_2+\\left[\\lambda_2^2-\\frac{2\\lambda_2(\\alpha_1p_2-\\alpha_2p_1)}{k_2(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\\\\n&&U_1=-\\frac{k_1(\\alpha_1x+\\alpha_2y)}{\\lambda_1}\\left[\\lambda_1^2-\\frac{2\\lambda_1(\\alpha_3p_1-\\alpha_4p_2)}{k_1(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}},\\\\\n&&U_2=-\\frac{k_2(\\alpha_4x+\\alpha_3y)}{\\lambda_2}\\left[\\lambda_2^2-\\frac{2\\lambda_2(\\alpha_1p_2-\\alpha_2p_1)}{k_2(\\alpha_1\\alpha_3-\\alpha_2\\alpha_4)}\\right]^{\\frac{1}{2}}\\label{eq120},\n\\end{eqnarray}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.1\\columnwidth]{figure1}\n\\caption{(color online) Periodic oscillations with $\\omega_{1}:\\omega_{2}$ = $4:4$, $k_{1}$ = $k_{2}$ = 1,  $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$ (a) Time series plot (b) Projected phase portrait.  Similar plots can be given for the $y$ variable.}\n\\end{figure}\n\\begin{figure}\n\\centering \n\\includegraphics[width=1.0\\columnwidth]{figure2}\n\\caption{(color online) Quasiperiodic oscillations with $\\omega_{1}:\\omega_{2}$ = $4$ : $\\sqrt{3}$, $k_{1}$ = $k_{2}$ = $1$,  $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$. (a) Time series plot (b) Projected phase portrait (c) Poincar\\'e section.  Similar plots can be given for the $y$ variable.}\n\\end{figure}\n\n\nThe general solution of equations (\\ref{eq15}) and (\\ref{eq15a}) can then be obtained by substituting the general solution of the two dimensional harmonic oscillator given by the expressions\n\\begin{eqnarray}\nU_{1}=A \\sin(\\omega_{1} t+ \\delta_{1}),\\quad U_{2}=B \\sin(\\omega_{2} t+ \\delta_{2}),\\nonumber \\\\\nP_{1}=A \\omega_{1}\\cos(\\omega_{1}t+\\delta_{1}),\\quad P_{2}=B \\omega_{2}\\cos(\\omega_{2}t+\\delta_{2}),\\label{harmq}\n\\end{eqnarray}\nwhere $\\omega_{j}=\\sqrt{\\lambda_j}$, j=1,2 into Eqs. (\\ref{eq119})-(\\ref{eq120}) and solving the resultant equations.  We obtain\n\\begin{eqnarray}\nx=\\frac{A\\alpha_{3}\\omega_{1}\\sin(\\omega_{1}t+\\delta_{1})}{k_{1}\\hat{\\delta_{1}}\\left(\\omega_{1}-A\\cos(\\omega_{1}t+\\delta_{1})\\right)}-\\frac{B\\alpha_{2}\\omega_{2}\\sin(\\omega_{2}t+\\delta_{2})}{k_{2}\\hat{\\delta_{1}}\\left(\\omega_{2}-B\\cos(\\omega_{2}t+\\delta_{2})\\right)},\n\\end{eqnarray}\n\\begin{eqnarray}\ny=-\\frac{A\\alpha_{4}\\omega_{1}\\sin(\\omega_{1}t+\\delta_{1})}{k_{1}\\hat{\\delta_{1}}\\left(\\omega_{1}-A\\cos(\\omega_{1}t+\\delta_{1})\\right)}+\\frac{B\\alpha_{1}\\omega_{2}\\sin(\\omega_{2}t+\\delta_{2})}{k_{2}\\hat{\\delta_{1}}\\left(\\omega_{2}-B\\cos(\\omega_{2}t+\\delta_{2})\\right)},\n\\end{eqnarray}\\\\\nwhere $A$,\\,$B$,\\,$\\delta_{1}$, $\\delta_{2}$ are arbitrary constants. \nThe above solution is oscillatory and is periodic and bounded for suitable choice of parameters namely $0<A<\\omega_{1}$ and $0<B<\\omega_{2}$.\nHere one can note that the frequency of oscillations is again independent of the amplitude for the two dimensional coupled modified Emden equation.  Figures 1 and 2 show two types of oscillatory behaviour, namely periodic and quasiperiodic oscillations, depending upon the ratio of frequencies $\\omega_{1}$ and $\\omega_{2}$.  Fig.1 shows  periodic oscillations for $\\omega_1$ : $\\omega_2$ = $4:4$, $k_{1}$ = $k_{2}$ = $1$, $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$ and Fig.2 shows quasiperiodic oscillations for $\\omega_1$ : $\\omega_2$ = $4$ : $\\sqrt{3}$, $k_{1}$ = $k_{2}$ = $1$, $\\alpha_{1}$ = $\\alpha_{3}$ = $5.5$, $\\alpha_{2}$ = $\\alpha_{4}$ = $3$.\n\n\n  \n\n\n\\section{Conclusion}\nIn this paper we have identified a class of coupled mixed Li\\'enard type nonlinear evolution equations.  This class of equation is identified by generalizing the nonstandard Lagrangian of the scalar MEE to a suitable nonsingular two dimensional Lagrangian.  Imposing the condition that the resultant Euler-Lagrange equation should be of mixed Li\\'enard type form, we have identified a specific class of equations which admits isochoronous solutions. The procedure is illustrated for the special case of  two dimensional modified Emden equation and is found to be isochronous as well as $\\emph PT$ symmetric for suitable choice of parameters, exhibiting quasiperiodic and periodic oscillations.  The above procedure can in principle be extended to higher degrees of freedom also.  The problem of quantization of system  (\\ref{eq16}) is also worth further consideration.\n\n\\section{Acknowledgements}\nThe work of ADD and ML is supported by a DST-IRHPA research project and the work of ML is also supported by a DAE Raja Ramanna Fellowship.   The work of V.K.C. is supported by the SERB-DST Fast Track scheme for young scientists under Grant No. YSS\/2014\/000175.\n\n\n\\section*{Appendix}\n\n\nLet us consider the following mixed Li\\'enard type equation,\n\\begin{eqnarray}\n\\ddot{x}+\\dot{x}^2\\frac{u_{xx}}{u_x}+3ku\\dot{x}+\nk^2\\frac{u^3}{u_x}+\\lambda\\frac{u}{u_x}=0,\n\\end{eqnarray}\nwhere $u(x)$ is an arbitrary function of $x$.  This equation is obtained by applying the transformation $y=u(x)$ in the linearizable MEE\n\\begin{eqnarray}\n\\ddot{y}+3ky\\dot{y}+k^2y^3+\\lambda y=0.\n\\end{eqnarray}\nThe resultant equation\npossess the Hamiltonian structure\n\\begin{eqnarray}\nH=-kp\\frac{u^2}{u_x}-\\frac{\\lambda p}{k u_{x}}+2\\sqrt{-\\frac{p}{k u_x}},\n\\end{eqnarray}\nwhich can be deduced from the Lagrangian (\\ref{arblag}).  This Hamiltonian can be transformed to the Hamiltonian of the simple\nharmonic oscillator equation through the canonical transformation\n\\begin{eqnarray}\nU=\\frac{u(x)\\sqrt{\\lambda(\\lambda-\\frac{2kp}{u_x})}}{\\lambda},\\qquad P=\\frac{\\lambda-\\sqrt{\\lambda(\\lambda-\\frac{2kp}{u_x})}}{k}.\n\\end{eqnarray}\nGeneralizing the above canonical transformation to two dimensions we obtain the canonical transformations (vide Eqs. (\\ref{xx})-(\\ref{yy})) for the coupled MEE.\\\\\n\\section*{References}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nDiffusion was invented by Fourier to describe the dynamics of heat \\cite{fourier1822theorie}. Heat or energy transport is ubiquitous and of relevance to essentially any physical system at nonzero temperature. In modern parlance, diffusion is understood more generally as the universal late-time hydrodynamic description of systems governed by a single conservation law, barring spontaneously broken symmetries. The diffusion constant may itself depend on the background value of the quantity that is being transported. Classically, this results in the diffusion equation being augmented to a non-linear differential equation and can lead to rich phenomena (see e.g.~\\cite{crank1979mathematics}), in analogy with turbulence in the Navier-Stokes equation. At the quantum or statistical physics level, the consequence of these non-linearities is that the effective theory of hydrodynamic fluctuations is interacting.\n\nThe traditional approach to address this class of problems is to couple the degrees of freedom of interest to stochastic noise fields and solve perturbatively a non-linear Langevin equation \\cite{PhysRevA.8.423,POMEAU197563}. This approach is also familiar in the context of the KPZ equation \\cite{PhysRevLett.56.889}. These methods have revealed striking effects in hydrodynamics such as long-time tails \\cite{DESCHEPPER19741,POMEAU197563,Kovtun:2003vj,PhysRevB.73.035113} and potentially large renormalizations of transport parameters \\cite{PhysRevA.16.732,Kovtun:2011np,Kovtun:2014nsa}. However, where standard classical hydrodynamic stood on firm symmetry principles \\cite{chaikin1995principles}, the physical principles governing stochastic hydrodynamics -- in particular how `noise' fields interact with conserved densities -- were less transparent. This was remedied recently by a decade-long effort culminating in a first principles construction of the general effective field theory of hydrodynamic fluctuations about a thermal equilibrium state \\cite{Dubovsky:2011sj,Grozdanov:2013dba,Haehl:2015pja,Crossley:2015evo,Jensen:2017kzi}, see \\cite{Glorioso:2018wxw} for a recent review.\n\nAnother motivation for a systematic study of hydrodynamic fluctuations is thermalization. The local thermalization (or equilibration) time $\\tau_{\\rm th}$ is loosely defined as the time it takes a system to reach local thermodynamic equilibrium. At times $t>\\tau_{\\rm th}$, hydrodynamics governs the slower relaxation to global thermodynamic equilibrium.\nIt is tempting to identify the thermalization time with the exponential decay of non-hydrodynamic correlators $\\langle \\mathcal{O}(t) \\mathcal{O}\\rangle\\sim e^{-t\/\\tau_{\\rm th}}$. Such correlators are however sensitive in general to hydrodynamic long-time tails and therefore strictly do not decay exponentially \\cite{POMEAU197563,Kovtun:2003vj}. A better understanding of long-time tails may therefore help provide a sharp definition of $\\tau_{\\rm th}$. See e.g.~\\cite{Han:2018hlj,Lucas:2018wsc} for recent alternative approaches to $\\tau_{\\rm th}$.\n\n\n\nIn the following we use the general formalism of Ref.~\\cite{Crossley:2015evo} to uncover the universal structure of late-time  response functions for interacting systems with a single continuous symmetry, focusing on time translation invariance (and therefore heat transport) for concreteness. We find that the thermal dc conductivity and diffusion constant both receive independent non-vanishing radiative corrections, even in the case of a single conserved density, and that the correction is not sign definite. Both of these statements are different to the results obtained from a traditional approach \\cite{Kovtun:2014nsa}, for reasons we shall explain. Moreover, we compute the one-loop retarded Green's function $G^R_{\\varepsilon \\varepsilon}(\\omega,k)$ at finite frequency and wavevector, revealing its analytic structure. We conclude by discussing experimental signatures of hydrodynamic fluctuations with applications to insulators, bad metals and cold atoms.\n\n\n\n\\section{Formalism}\n\nOur objective is to understand the structure of energy density correlation functions in non-integrable quantum systems at nonzero temperature\n\\begin{equation}\\label{eq_corr}\n\\langle \\varepsilon (t, x) \\varepsilon (t', x') \\cdots\\rangle_\\beta\n\t\\equiv \\Tr \\Bigl(\\rho_{\\beta}\\,  \\varepsilon (t,x) \\varepsilon(t',x')\\cdots \\Bigr)\\, ,\n\\end{equation}\nwhere the thermal density matrix $\\rho_\\beta = e^{-\\beta H}\/\\Tr e^{-\\beta H}$. Here we will be interested in the case where energy is the only conserved quantity. The systematic study of a single diffusive charge was initiated in Ref.~\\cite{Crossley:2015evo}. In that formalism, furthermore, the contribution of ghosts (or lack thereof) has been well understood \\cite{Gao:2018bxz}. A self-contained review of this special case of the formalism is given in appendix \\ref{app_Hong}. The output of this method is an effective field theory that provides a perturbative expansion for computing the correlators \\eqref{eq_corr}:\n\\begin{equation}\nZ = \\int D \\varepsilon D \\varphi_a \\, e^{i \\int  \\mathcal L[\\varepsilon,\\, \\varphi_a]} \\,.\n\\end{equation}\nHere $\\varepsilon$ is the energy density and $\\varphi_a$ is an auxiliary field (the $a$ subscript is not an index). The most general Lagrangian to cubic order in fields was constructed in Ref.~\\cite{Crossley:2015evo}. In appendix \\ref{app_Hong} we extend their construction to quartic interactions, which will play a role below. The resulting Lagrangian to leading order in derivatives that is at most quartic in fields is given by\n\\begin{equation}\\label{eq_action_main}\n\\mathcal L\n\t= iT^2\\kappa (\\nabla \\varphi_a)^2 - \\varphi_a \\left(\\dot \\varepsilon -\n\t D \\nabla^2 \\varepsilon\\right) \n\t+ \\nabla^2\\varphi_a\\left[\\frac12 \\lambda\\varepsilon^2 + \\frac13\\lambda'\\varepsilon^3\\right] +\n\ticT^2(\\nabla\\varphi_a)^2 \\left[\\widetilde\\lambda\\varepsilon + \\widetilde\\lambda'\\varepsilon^2\\right]\n\t+ \\cdots \\, ,\n\\end{equation}\nwhere $T$ is the temperature, $D$ the diffusivity, $c$ the specific heat and $\\kappa = c D$ the thermal conductivity. These are all `bare' values that will be renormalized by the interactions in (\\ref{eq_action_main}). The couplings $\\lambda,\\lambda',\\widetilde \\lambda, \\widetilde \\lambda'$ themselves can be written as linear combinations of the following derivatives of the transport parameters\n\\begin{equation}\\label{eq_couplings_raw}\nT\\partial_T \\kappa\\, , \\qquad\\quad\nT^2\\partial_T^2 \\kappa\\, , \\qquad\\quad\nT\\partial_T D\\, , \\qquad\\quad\nT^2\\partial_T^2 D\\, .\n\\end{equation}\nTheir explicit expressions are given in \\eqref{eq_couplings}.\n\nThe traditional stochastic approach to hydrodynamic fluctuations with Gaussian noise \\cite{PhysRevA.8.423,POMEAU197563,PhysRevLett.56.889,Kovtun:2003vj} can be recovered from the general effective action \\eqref{eq_action_main} when the interactions that are quadratic in auxiliary fields (i.e. the $\\widetilde\\lambda$ and $\\widetilde \\lambda'$ terms) are absent, by performing a Legendre transform and introducing the noise field $\\xi = \\partial \\mathcal L \/\\partial \\varphi_a$ \\cite{Crossley:2015evo}. However, when $\\widetilde \\lambda$ or $\\widetilde \\lambda'$ is non-vanishing, the resulting theory will contain  interactions of the form $\\xi^2 \\varepsilon$ or $\\xi^2 \\varepsilon^2$. The noise correlations will therefore not be strictly Gaussian because they now depend on energy fluctuations.%\n\t\\footnote{That these interactions should arise is already clear from the stochatistic approach, where the fluctuation dissipation theorem imposes $\\langle \\xi(x)\\xi\\rangle = -2T^2\\kappa(\\varepsilon) \\nabla^2 \\delta(x)$. These interactions only vanish if $\\kappa(\\varepsilon) = \\rm const$, which is also apparent in \\eqref{eq_couplings}.} \n\nIn the remainder we will show precisely how\nthe interactions in (\\ref{eq_action_main}) lead to non-analyticities in response functions and renormalize the transport parameters themselves \\cite{DESCHEPPER19741,PhysRevB.73.035113,PhysRevA.16.732,Kovtun:2011np,Kovtun:2012rj}. Concretely, we are interested in the one-loop correction to the retarded Green's function, which is simply diffusive in the absence of interactions\n\\begin{equation}\\label{eq:bare}\nG_{\\varepsilon \\varepsilon}^{R,0}(\\omega,k)\n\t= \\frac{i\\kappa T k^2}{\\omega + iDk^2}\\, .\n\\end{equation}\nThe diagrams contributing at one loop are shown in Fig. \\ref{fig_loops}, and computed in appendix \\ref{app_loops}. These loops are all UV divergent and should be truncated at the hydrodynamic cutoff $k_{\\rm max} = 2\\pi\/\\ell_{\\rm th}$, which defines the thermalization length $\\ell_{\\rm th}$. Perturbation theory is controlled because all couplings in the Lagrangian (\\ref{eq_action_main}) are power counting irrelevant, and therefore have small effects at low, hydrodynamic energy scales. Indeed, the appropriate dimensional analysis is set by the diffusive pole in (\\ref{eq:bare}), so that $[\\omega] = 2 [k]$ and $[D]=0$. It follows that $[\\varphi_a] = [\\varepsilon]= \\frac{d}{2}[k]$ and hence the cubic couplings $[\\lambda] = [\\widetilde\\lambda] = -\\frac{d}{2}[k]$ are irrelevant. These dimensions suggest that the one loop corrections to tree-level diffusion, which are quadratic in coupling, will be of the schematic form $Dk^2(1+\\lambda^2 k^d)$, as we verify below.\n\n\n\n\\begin{figure}[h]\n\\centerline{\n\\subfigure{\\label{sfig_label1}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d1.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d4.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/tadpole2.png}}\n}\n\\centerline{\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d2.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/d3.png}}\n\\subfigure{\\label{sfig_label2}\n\\includegraphics[width=0.22\\linewidth, angle=0]{fig\/tadpole1.png}}\n}\\caption{\\label{fig_loops}The one-loop diagrams contributing to $G_{\\varepsilon \\varepsilon}$. Solid lines denote the energy density field $\\varepsilon$ and squiggly lines denote the auxiliary field $\\varphi_a$.}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\\section{Results}\n\nThe diagrams shown in Fig. \\ref{fig_loops} sum up to give the one loop retarded Green's function\n\\begin{equation}\\label{eq_GR}\nG^R_{\\varepsilon \\varepsilon}(\\omega,k)\n\t= \\frac{i \\left(\\kappa + \\delta\\kappa(\\omega,k)\\right) Tk^2}{\\omega+ i D k^2 + \\Sigma(\\omega,k)}\\, ,\n\\end{equation}\nwhere both $\\delta\\kappa(\\omega,k)$ and $\\Sigma(\\omega,k)$ receive analytic and non-analytic contributions. Separating these contributions as\n\\begin{equation}\\label{eq_ana_nonana}\n\\delta \\kappa(\\omega,k)\n\t= \\delta \\kappa + \\kappa_\\star(\\omega,k) \\, , \\qquad\\qquad\n\\Sigma(\\omega,k)\n\t= i\\delta D k^2 + \\Sigma_\\star(\\omega,k) \\, ,\n\\end{equation}\none finds that the analytic pieces have the form\n\\begin{equation}\\label{eq_res_ana}\n\\frac{\\delta \\kappa}{\\kappa} = \\frac{f_d}{c\\, \\ell_{\\rm th}^d} \\lambda_\\kappa\\, , \\qquad\\qquad\n\\frac{\\delta D}{D} = \\frac{f_d}{c\\, \\ell_{\\rm th}^d} \\lambda_D\\, ,\n\\end{equation}\nwith $f_d = {\\rm Vol} (B_{d})  =  2,\\, {\\pi},\\, \\frac{4\\pi}{3}$ for spatial dimensions $d=1,2,3$, and where $\\lambda_\\kappa$, $\\lambda_D$ are dimensionless effective couplings. Their explicit form will be given below. The non-analytic parts of \\eqref{eq_ana_nonana} have the form\n\\begin{equation}\\label{eq_res_nonana}\n\\kappa_\\star(\\omega,k)\n\t= f_\\kappa(\\omega,k) \\alpha_d (\\omega,k) \\, , \\qquad\\qquad\n\\Sigma_\\star(\\omega,k)\n\t= k^2 f_\\Sigma(\\omega,k) \\alpha_d (\\omega,k) \\, , \n\\end{equation}\nwhere $f_\\kappa$, $f_\\Sigma$ are analytic functions, shown below, that do not depend on dimension, and the non-analyticity is \n\\begin{subequations}\\label{eq_alpha}\n\\begin{align}\n&&\\alpha_1(\\omega,k) &= \\frac{1}{4} \\left[k^2 - \\frac{2i\\omega}{D}\\right]^{-1\/2}\\, , \n& (d=1)\\\\\n&&\\alpha_2(\\omega,k) &= -\\frac{1}{16\\pi}\\log \\left[k^2 - \\frac{2i\\omega}{D}\\right]\\, , \n& (d=2)\\\\\n&&\\alpha_3(\\omega,k) &= -\\frac{1}{32\\pi} \\left[k^2 - \\frac{2i\\omega}{D}\\right]^{1\/2}\\, .\n& (d=3)\n\\end{align}\n\\end{subequations}\nThe effect of these non-analyticities is suppressed by powers of momenta and frequency appearing in $f_\\kappa$, $f_\\Sigma$, as we will see below.\n\nThe retarded Green's function is analytic in the upper-half frequency plane, as required by causality. \nThe interactions have induced a branch point at $\\omega = -\\frac{i}{2} D k^2$. Moreover, the diffusive pole is split into two poles with small real parts $\\omega = -i(D+\\delta D)k^2 \\pm O(k^2 |k|^{d}).$\\footnote{Additional poles in \\eqref{eq_GR} are outside the validity of the resummation. The non-analytic corrections to diffusion are seen to be more important than the $O(k^4)$ higher derivative corrections to the diffusion equation in $d=1$ and also in $d=2$, where the dispersion receives an additional imaginary part $O(k^4 \\log k)$. \n}  The location of the branch point can be understood from simple kinematics, by putting both internal legs on-shell (in either the retarded or advanced Green's functions) as in Fig.~\\ref{fig_non_ana}. The frequencies $\\omega$ for which the on-shell condition is satisfied form a half-line in the complex plane parametrized by the loop momentum $k'$, where the Green's function has a branch cut. The branch point is located at the smallest frequency $\\omega$ (in magnitude) that can satisfy the on-shell conditions:\n\\begin{equation}\n\\omega_\\star\n\t= -i D \\min_{k'} \\left[ k^2 + 2k\\cdot k' + 2 k'^2\\right]\n\t= -\\frac{i}{2}Dk^2\\, .\n\\end{equation}\n\n\n\\begin{figure}\n\\centerline{\n\\subfigure{\\label{sfig_label1}\n\\begin{overpic}[width=0.40\\textwidth,tics=10]{fig\/cut}\n\t \\put (55,47) {$\\omega'=iD k'^2$}\n\t \\put (55,16) {$\\omega+\\omega'=-iD (k+k')^2$}\n\t \\put (13,36) {$\\omega,k$} \n\t \\put (29,49) {$\\omega',k'$} \n\\end{overpic} \n}\n\\hspace{50pt}\n\\subfigure{\\label{sfig_label2}\n\\begin{overpic}[width=0.35\\textwidth,tics=10]{fig\/non_ana_new}\n\t \\put (5,66) {$\\omega$}\n\t \\put (18,43) {$ - \\frac{i}{2}D k^2$} \n\t \\put (-4,26) {$ - i(D+\\delta D)k^2$} \n\\end{overpic}\n}\n}\n\\caption{\\label{fig_non_ana} On-shell condition for the two internal legs (left), and analytic structure of the retarded Green's function $G^R_{\\varepsilon\\varepsilon}(\\omega,k)$ at one loop (right). In imposing the on-shell condition, it is important to consider two poles in opposite halves of the complex $\\omega'$ plane, otherwise the loop contribution vanishes. The pole in the upper half plane arises from an advanced Green's function in the loop: $G^A = \\left(G^R\\right)^*$.}\n\\end{figure}\n\nIn previous treatments, similar physics to what we have just described was found in the coupled diffusion of two modes \\cite{PhysRevB.73.035113,Kovtun:2014nsa}. Due to the absence of a systematic formalism for hydrodynamic fluctuations at that time, those works did not account --- among other things --- for interactions that are quadratic in the auxiliary field (in particular $\\widetilde\\lambda$), nor the quartic terms $\\lambda'$ and $\\widetilde\\lambda'$. While the systematic approach modifies the results for two coupled modes, see appendix \\ref{app_2n}, the most qualitative difference is seen for diffusion of a single conserved density. We have found that renormalization of the diffusion constant and conductivity occurs even in this case, \ncontrolled by the effective couplings in \\eqref{eq_res_ana}\n\\begin{equation}\\label{eq:ll}\n\\lambda_\\kappa = \\frac{c^2 T^2 }{D}\\tilde\\lambda'\\, , \\qquad\\qquad\n\t\\lambda_D = - \\frac{c^2 T^2}{2D^2} \\left[\\lambda(\\lambda+\\tilde \\lambda) + 2\\lambda' D\\right]\\,,\n\\end{equation}\nwhich are not sign-definite in general.\n\nFurthermore, we have also found non-analytic corrections to the Green's function even with a single diffusing mode. These are not as strong as those arising with two modes, as we now explain.\nThe functions $f_\\kappa,\\,f_\\Sigma$ appearing in the non-analytic contributions \\eqref{eq_res_nonana} are\n\\begin{equation}\nf_\\kappa(\\omega,k)\n\t= \\frac{cT^2}{D^2} k^2 \\lambda\\tilde \\lambda\\, , \\qquad\\qquad\nf_\\Sigma(\\omega,k)\n\t=\\frac{cT^2}{D^2}\\left[\\omega \\lambda(\\lambda+\\tilde \\lambda) + iDk^2 \\lambda\\tilde \\lambda\\right]\\, .\n\\end{equation}\nWhile $f_\\Sigma$ has both the $O(\\omega)$ and $O(k^2)$ terms expected at this order in the derivative expansion, $f_{\\kappa}$ only has an $O(k^2)$ term.\nThe subsequent suppression of $f_{\\kappa}$ as $k \\to 0$ implies that the optical conductivity does not receive non-analytic corrections, and is instead constant in the hydrodynamic regime\n\\begin{equation}\nT\\kappa(\\omega)\n\t\\equiv \\lim_{k\\to 0} \\frac{\\omega}{k^2}\\Im  G^R_{\\varepsilon \\varepsilon}(\\omega,k)\n\t= T(\\kappa + \\delta \\kappa)\\, .\n\\end{equation}\nThis result can be contrasted with the case of two interacting diffusive densities, wherein the optical conductivity receives a non-analytic fluctuation correction \\cite{DESCHEPPER19741,PhysRevB.73.035113}. We revisit this case in appendix \\ref{app_2n}, where the analytic structure is discussed in the light of a systematic inclusion of fluctuation effects. We find a non-analytic correction to the optical conductivity of the form (we use $\\sigma$ to denote a generic conductivity)\n\\begin{equation}\n\\delta\\sigma(\\omega,k)\n\t\\sim \\omega\\, \\alpha_d(\\omega,k) + \\cdots\\, ,\n\\end{equation}\nwhere $\\alpha_d$ is as in (\\ref{eq_alpha}) and $\\cdots$ denote terms that are further $k^2$ suppressed. In particular, the correction in $d=2$ is $\\delta\\sigma(\\omega) \\sim \\omega\\log\\omega$.\n\n\n\\section{Discussion and Applications}\n\nStrong renormalization of the transport parameters due to hydrodynamic fluctuations occurs if the ratios in (\\ref{eq_res_ana}) are large. When the dimensionless couplings are order unity, $\\lambda_\\kappa\\sim \\lambda_D \\sim 1$,\\footnote{Using equations (\\ref{eq:ll}) and (\\ref{eq_couplings}), this holds in any scaling regime where $\\kappa \\sim T^a$ and $c \\sim T^b$.} the strength of fluctuations is controlled by the specific heat per `thermal volume' $c\\ell_\\text{th}^d$. This quantity can be thought of as the number of degrees of freedom in the smallest volume that can reach local thermodynamic equilibrium. When there are many degrees of freedom in a thermal volume, fluctuation effects are small. One might further expect that a sufficient number of degrees of freedom are necessary in order for a region to locally thermalize, and hence $c\\ell_\\text{th}^d \\gtrsim 1$. Indeed, such a bound has been established in the presence of operators with microscopic positivity properties, using the eigenstate thermalization hypothesis \\cite{Delacretaz:2018cfk,ETHnew}. Thus hydrodynamic fluctuation corrections to thermal transport parameters are expected to be at most comparable to the bare values.\n\nIf microscopic interactions are weak then $\\ell_\\text{th} \\sim \\ell_\\text{mfp}$, the (inelastic) quasiparticle mean free path, will be large. Fluctuation corrections to transport are therefore small in weakly interacting systems. In contrast, in strongly correlated systems $\\ell_\\text{th}$ can become very short and fluctuations may be important. For example, at high temperatures in a lattice model, with order one degrees of freedom per unit cell,\nthe bound mentioned above is only saturated when \n$\\ell_\\text{th} \\sim a$, the lattice spacing. This roughly coincides with the `minimal' mean free path for thermal transport by well-defined phonons in insulators \\cite{SLACK19791, PhysRevB.49.9073}. In strongly correlated regimes, however, the notion of a mean free path is likely not a useful concept. Recent measurements of thermal diffusivity in cuprates \\cite{Zhang:2016ofh,2018arXiv180807564Z} and perovskites \\cite{doi:10.1063\/1.3371815,PhysRevLett.120.125901,AhBh} suggest (assuming that the microscopic sound speed is the relevant velocity) that a transport lengthscale reaches and possibly surpasses the lattice spacing at high temperatures. The specific heat in these materials is roughly $c a^2 \\sim 40$ and $c a^3 \\sim 15$, respectively. It may be interesting to look for signatures of diffusive fluctuations in the thermal transport of these systems.\n\nFluctuation effects can also become important for transport close to a thermal phase transition. The thermalization length diverges as $\\ell_\\text{th} \\sim \\tau^{-\\nu}$ as the reduced temperature $\\tau \\to 0$, while the specific heat scales as $c \\sim \\tau^{2-\\alpha}$. It follows that $\\delta D\/D \\sim \\tau^{\\alpha + d\\nu-2} \\sim 1$ if hyperscaling is obeyed, so fluctuations are important in that case. Above the upper critical dimension hyperscaling is violated and fluctuations are small.  A more sophisticated discussion must include fluctuations of the order parameter in the analysis \\cite{RevModPhys.49.435}.\n\nTransport lengthscales approaching or exceeding the lattice spacing are also seen in `bad metals' \\cite{PhysRevLett.74.3253,RevModPhys.75.1085,hussey}. All of our expressions above are easily adapted to describe the diffusion of a single conserved $U(1)$ charge, instead of heat.\\footnote{If thermoelectric effects are strong, one should instead work with coupled heat and charge diffusion, as in appendix \\ref{app_2n}.} Particle-hole symmetry should be broken, typically by a background charge density, otherwise many terms we have considered are forced to be zero.\nThe correction to the dc electrical conductivity $\\sigma$, for example, is found to be\n\\begin{equation}\\label{eq:sigma}\n\\frac{\\delta \\sigma}{\\sigma}\n\t= \\frac{f_d}{\\ell_{\\rm th}^d} \\frac{T}{\\chi \\mu^2} \\lambda_\\sigma\\, .\n\\end{equation}\nHere $\\chi$ is the charge susceptibility and $\\mu$ the chemical potential.\nIn the definition of the couplings in (\\ref{eq_action_main}) in terms of the thermodynamic derivatives (\\ref{eq_couplings_raw}), one replaces $T \\to \\mu$.\n\nCondensed matter systems --- including most bad metals --- are typically at degenerate temperatures $T < E_F$, below the Fermi energy. At these temperatures $\\chi \\sim k_F^d\/E_F$ and $\\mu \\sim E_F$. Here $k_F$ is the Fermi momentum. The contribution (\\ref{eq:sigma}) of fluctuations to the conductivity is therefore small, even when the thermalization length becomes of order $\\ell_\\text{th} \\sim a \\sim 1\/k_F$. This is the shortest length consistent with local thermalization \\cite{ETHnew}. In contrast, at high temperatures where fermions are non-degenerate, $\\chi \\sim 1\/(T a^d)$. If the total charge is held fixed, then $\\mu \\sim T$. It follows that as the thermalization length becomes short, of order $\\ell_\\text{th} \\sim a$, fluctuation corrections to the conductivity are order one. Diffusive transport by strongly correlated but non-degenerate fermions has recently been probed in an ultracold atom realization of the Hubbard model \\cite{2018arXiv180209456B}, and earlier in e.g. \\cite{Schneider2012}, as well as in numerics \\cite{2018arXiv180308054M, 2018arXiv180608346H}. Indeed, $\\ell_\\text{th}$ is found to saturate around the lattice scale at high temperatures, and so fluctuation effects may be important.\n\n\nFinally, diffusion with a short thermalization length has also been seen in spin transport in strongly interacting ultracold atoms in a trap \\cite{2018arXiv180505354E}. The formulae we have developed can be applied directly to longitudinal spin diffusion in a magnetic field (to break spin reversal symmetry) or to transverse spin diffusion without a magnetic field or spontaneous magnetization (so that isotropy prevents mixing of the two transverse modes).\nAt temperatures $T \\lesssim E_F$, with electrons on the verge of becoming non-degenerate, the thermalization length is found to be $\\ell_\\text{th} \\sim 1\/k_F$ (there is no lattice scale in these experiments). Diffusive fluctuations may therefore again be important for transport.\n\nIn summary, long wavelength fluctuations about diffusive dynamics may be relevant in condensed matter and cold atom systems of widespread interest. We have seen that a systematic derivation of these effects leads to different results than previous, more phenomenological, approaches. For this reason, it will be important to revisit the computation of fluctuations in relativistic hydrodynamics \\cite{Kovtun:2011np, Stephanov:2017ghc}, which includes a sound mode in addition to transverse momentum diffusion. Fluctuations in relativistic hydrodynamics may have direct consequences for the quark-gluon plasma.\n\n\n\n\n\\section*{Acknowledgments} \n \nWe would like to thank Erez Berg, Debanjan Chowdhury, Paolo Glorioso and Andrew Lucas for illuminating discussions. SAH and LVD are partially supported by the US Department of Energy Office of Science under Award Number DE-SC0018134. XCL is supported by the Knut and Alice Wallenberg Foundation.\n\n\\pagebreak\n\n\\bibliographystyle{ourbst}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\nThe Unites States (U.S.) is one of the countries with the highest firearms--related death rate, exceptionally high compared to other industrialized countries \\citep{pallin19}. Furthermore, it is the country with the highest firearms possession rate by citizens, thanks to loose legislation in several of its member states \\citep{hemenway17}. The relation between firearms--related death rate and firearms possession rate has been largely investigated and a causal relation going from the latter to the former seems very plausible, e.g. \\citet{krug98, bangalore13} and \\citet{siegel13}. If this fact has prompted part of the public opinion to ask for stricter guns laws, the constitutional right to posses and bear firearms has been strenuously defended by the opposite faction, rendering this one of the most popular and controversial topic in the country.\\par\nAlmost every aspect of the relation between firearms legislation and gun violence has been extensively researched. Besides the mentioned studies that have tried to establish a causal link between firearms--related death rate and firearms possession rate \\citep{krug98, bangalore13, siegel13}, others have investigated the effect of firearms legislation\\textbackslash possession rate on suicide rates \\citep{kellermann92, anestis18}, on pediatric firearms--related mortality \\cite{goyal19}, on homicides rate \\citep{duggan01, kovandzic13, siegel13, siegel14}, and on the death rate of police officers on duty \\citep{lester87, mustard01, swedler15}.\\par\nWith approximately 1000 deaths per year, United States (U.S.) holds another inglorious record among industrialized countries: the highest rate of homicides committed by police forces \\citep{hemenway19}. This raises the question of whether the high rate of police fatal shootings results from relaxed firearms legislation\\textbackslash high possession rate. From a speculative point of view, one can argue that the diffusion of firearms increases the probability of police officers to face armed people while on duty, thus increasing the probability of being involved in potentially dangerous situations that require the use of guns from their part. Furthermore, the increased probability to face dangerous situations is a factor of stress that may lead law enforcement officers to overreact or, more generally, to commit mistakes. A positive relation between firearms diffusion and deadly assaults to police officers \\citep{swedler15} corroborates the theory of an increase danger to officers for higher levels of firearms ownership.\\par\nThanks to the availability of independent datasets that remedy the underreporting of police fatal use of force in official statistics \\citep{conner19}, this topic has recently been investigated. \\citet{hemenway19} find a positive association between firearms prevalence and fatal police shooting rates. \\citet{kivisto17} report that U.S. states with stricter firearms legislation have lower incidence rates of fatal police shootings. Both studies are, however, cross-sectional and thus, despite the use of several controls common in the dedicated literature, may suffer from the omitted variable problem. \\citet{kivisto17}, for example, mention, among the limitations of their paper, exactly this problem, stating that ``it is possible that states with stricter gun legislation also have better training for police officers and more stringent hiring practices, or that states that are already safe are more likely to implement stricter gun laws''. \\par \nThe novelty of the present study is to use a panel dataset to investigate the relation between firearms legislation\\textbackslash possession rate and fatal police shootings. This allows to control for unobserved fixed characteristics at state level that may have biased previous analysis, providing, therefore, more robust results.\n\\section{Literature Review}\nAs discussed before, firearms legislation and ownership is a strongly investigated topic, particularly in the United States. A number of studies document negative impacts of firearms legislation and prevalence, such as increased suicide and homicide rate, although evidence is not unambiguous. From an extensive literature review, \\citet{kleck15} concludes that guns diffusion is a positive determinant of crime rate, but this relation looses statistical significance in the most methodologically rigorous papers. \\citet{branas09} find that possessing a gun increases the probability of being shot during an assault, thus dismantling the opinion of weapons having a protective role. On the other side, \\citet{kleck93}, by comparing 170 U.S. cities, find scarce evidence that guns restrictions have some positive role in reducing the rate of violence. Similarly, \\citet{altheimer08} evidences that gun availability has no effect in determining the number of total individual assaults and robbery, but only increases the number of the ones committed with a fire--weapon.\\par \nRegarding one of the most serious violent crimes, homicide, the results are mixed. \\citet{duggan01}, \\citet{siegel13} and \\citet{siegel14} find a significant and positive relation between guns diffusion and the number of homicides. However, this view is strongly opposed by \\citet{kovandzic13}, which, on the contrary, document a negative relation. They cite a potential deterrent effect of guns as an explanation for their results. A similar debate has surrounded the permission of carrying concealed weapons, with \\citet{lott97} showing a positive role of such law in reducing violent crimes whereas \\citet{dezhbakhsh98} rejecting this finding and claiming the opposite. Findings regarding firearms diffusion and legislation are therefore very discordant even when considering aspects such as crime and homicide rate that are among the most studied.\\par \nShifting the attention on police forces, there is a paucity of research regarding the association between firearms diffusion and legislation and the number of killings of, and by, police officers. When considering the former, the killing of law enforcement officers, we have again contrasting findings. \\citet{lester87} and \\citet{swedler15} find that increasing levels of households gun ownership are a clear factor of risk for police officers, whereas \\citet{mustard01}, limiting the attention to the possibility of carrying concealed weapons, puts in evidence a potential protective role of this law. \\par\n\\citet{kivisto17} and \\citet{hemenway19} investigate the relation between firearms and killings committed by police officers. The two studies both rely on independent and open source databases to retrieve the number of fatal police shootings: \\citet{kivisto17} on \\href{https:\/\/www.theguardian.com\/us-news\/series\/counted-us-police-killings}{The Counted}, maintained by The Guardian, and \\citet{hemenway19} on \\href{https:\/\/www.washingtonpost.com\/graphics\/investigations\/police-shootings-database\/}{Fatal Force}, created by The Washington Post. Furthermore, they both rely on a cross sectional analysis using the fraction of suicides with a fire--weapon on the total number of suicides as a proxy for guns diffusion, as previously done in other papers, e.g. \\citet{kleck04} and \\citet{azrael04}. Compared to \\citet{hemenway19}, whose focus is exclusively on guns diffusion as a cause of fatal police shootings, \\citet{kivisto17} further consider firearms legislation, using the \\href{https:\/\/www.bradyunited.org\/}{Brady Campaign} scorecards as an indicator of law strength. In both papers, firearms ownership is found to positively affect the number of fatal police shootings. Even after controlling for firearms prevalence, firearms regulations on gun trafficking and on child and consumer safety significantly reduces fatal police shootings \\citep{kivisto17}.\\par \nGiven the contrasting evidence emerged in other topics related to firearms diffusion and legislation and since both the last mentioned papers rely on a cross sectional analysis that may be plagued by the omitted variable bias, the extension to a panel data setting seems a necessary further step. This may help to strengthen the findings reached so far or to contest their validity as the result of a biased analysis.   \n\\section{Data and Methods}\nFollowing \\citet{kivisto17} and \\citet{hemenway19}, our units of observation are the 50 U.S. states, with District of Columbia having being excluded for lack of data in several covariates. The covered time period is from January 1, 2012, to December 31, 2018, and all variables are expressed as yearly values, forming a dataset with seven time periods. Different databases have been consulted and merged in order to have all the variables of interest and the necessary controls: \\hyperlink{https:\/\/fatalencounters.org\/}{Fatal Encounters}, \\href{https:\/\/giffords.org\/lawcenter\/resources\/scorecard\/}{Giffords scorecards}, the \\href{https:\/\/www.census.gov\/}{U.S. Census Bureau} data portal, the \\href{https:\/\/crime-data-explorer.fr.cloud.gov\/explorer\/national\/united-states\/crime}{Federal Bureau of Investigation's (FBI's) Crime Data Explorer} and the Centers for Disease Control\nand Prevention's (CDC's) \\href{https:\/\/www.cdc.gov\/injury\/wisqars\/index.html}{Web-based Injury Statistics Query and Reporting System} (WISQARS). Following is a description of all variables.\n\\subsection{Description of Variables}\nOur dependent variable, the number of deaths caused by police shooting per million inhabitants (\\textit{Pol\\_Shoot}), is retrieved from the \\hyperlink{https:\/\/fatalencounters.org\/}{Fatal Encounters} database. We choose an independent database to alleviate the likely problem of underreporting of such episodes in the FBI's official statistics \\citep{williams19}. Compared to other open source repositories, e.g. \\href{https:\/\/www.theguardian.com\/us-news\/series\/counted-us-police-killings}{The Counted}, \\href{https:\/\/www.washingtonpost.com\/graphics\/investigations\/police-shootings-database\/}{Fatal Force}, \\href{https:\/\/mappingpoliceviolence.org\/}{Mapping Police Violence} and \\href{https:\/\/www.gunviolencearchive.org\/}{Gun Violence Archive}, the Fatal Encounters database covers the longest time span -- from 2000 to present. Specific cases of police shooting can be retrieved by selecting the category ``Deadly use of force'' and the subcategory ``Gunshot''.\\par\nOur independent variable of interest is the strength of firearms regulations at state level (\\textit{Giff\\_Score}). In order to obtain a synthetic measure of the strength of a state legislation, we rely on the \\href{https:\/\/giffords.org\/lawcenter\/resources\/scorecard\/}{Giffords scorecards}, available for the period 2010-2018, with the exclusion of the year 2011, hence the need to drop the year 2010. The overall score is an aggregation of seven component scores, namely: background checks and access to firearms (\\textit{BCAF}), other regulations of sales and transfers (\\textit{ORST}), classes of weapons and magazines\\textbackslash ammunitions (\\textit{CWAM}), consumers and child safety (\\textit{CCS}), gun owner accountability (\\textit{GOA}), firearms in public places (\\textit{FPP}) and a residual class (\\textit{OTH}). Disaggregation of the overall score allows us to test the role of each component in explaining fatal police shootings, following \\citet{kivisto17}. This is helpful in identifying the areas where intervention should be prioritized to reduce police shooting episodes. Since the scoring system has been slightly modified several times during the study period, we have implemented a harmonization procedure, retaining only the sub-indicators that remained unaltered over time. \\citet{kivisto17}, using data from the same source, eliminate the weighting system in favor of a ``1 law = 1 point'' scale. They argue that a weighting system necessarily entails a degree of arbitrariness. However, we think that the equal weighting implied by the ``1 law = 1 point'' scale is analogously arbitrary. We, therefore, prefer to rely on the weights assigned by professional lawyers, thus leaving the Giffords scorecard scale unaltered.\\par\nStricter legislation on firearms may reduce the quantity of fire--weapons owned by citizens, but may also promote safer use, e.g. by denying dangerous subjects access to guns or by increasing the safety of circulating weapons. Besides examining if laws to promote safe gun use are effective in reducing fatal police shootings, we can test if the effect of firearms legislations operate via the former channel by looking at the relation between fatal police shootings and gun diffusion. Lacking state--level data on gun ownership for our study period, we rely on a commonly adopted proxy (\\textit{Suicide}) -- the percentage of suicides committed with a fire--weapon over the total of suicides \\citep{kleck04, azrael04, kivisto17, hemenway19}. These data are retrieved from the Web-based Injury Statistics Query and Reporting System (WISQARS).\\par \n\\begin{table}[H]\n\t\\centering\n\t\\caption{Descriptive Statistics}\\label{tab1}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lccccc}\n\t\t\t\\hline\\hline\n\t\t\t\\textbf{Variable} & \\textbf{N. Obs.} & \\textbf{Mean} & \\textbf{Std. Dev.} & \\textbf{Min.} & \\textbf{Max.} \\\\\n\t\t\t\\hline\\hline\n\t\t\t\\textit{Pol\\_Shoot} & 350 & 3.51 & 2.11 & 0 & 10.85 \\\\\n\t\t\t\\textit{Giff\\_Score} & 350 & 31.98 & 24.42 & 4 & 105.50 \\\\\n\t\t\t\\textit{Crime} & 350 & 3627.32 & 1388.63 & 1026.24 & 8849.56 \\\\\n\t\t\t\\textit{Suicide} & 350 & 51.54 & 12.35 & 13.20 & 74.30 \\\\\n\t\t\t\\textit{PC\\_Income} & 350 & 29712.58 & 4828.93 & 20119 & 52500.00 \\\\\n\t\t\t\\textit{Urban} & 350 & 73.59 & 14.44 & 38.70 & 95 \\\\\n\t\t\t\\textit{Poverty} & 350 & 9.99 & 2.80 & 4 & 19.20 \\\\\n\t\t\t\\textit{White} & 350 & 76.95 & 12.67 & 24.30 & 95.10 \\\\\n\t\t\t\\textit{Low\\_Edu} & 350 & 11.19 & 2.97 & 6.10 & 18.60 \\\\\n\t\t\t\\textit{Unemp.} & 350 & 3.96 & 1.18 & 1.80 & 7.90 \\\\\n\t\t\t\\textit{Young} & 350 & 23.15 & 1.31 & 19.80 & 27.50 \\\\ \\hline\n\t\t\t\\multicolumn{6}{c}{\\textit{\\textbf{Giff\\_Score disentangled}}} \\\\\n\t\t\t\\textit{BCAF} & 350 & 7.19 & 5.51 & 0 & 22 \\\\\n\t\t\t\\textit{ORST} & 350 & 4.12 & 5.89 & 0 & 24 \\\\\n\t\t\t\\textit{CWAM} & 350 & 1.82 & 3.86 & 0 & 14 \\\\\n\t\t\t\\textit{CCS} & 350 & 2.25 & 2.02 & 0 & 9 \\\\\n\t\t\t\\textit{GOA} & 350 & 2.76 & 5.00 & 0 & 17.50 \\\\\n\t\t\t\\textit{FPP} & 350 & 9.16 & 4.36 & 0 & 19\\\\ \n\t\t\t\\textit{OTH} & 350 & 4.67 & 2.18 & 0 & 10\\\\ \\hline\\hline\n\t\\end{tabular}}\n\\end{table}\n\\indent \nRegarding control variables, we retrieved data on the number of violent crimes (per million inhabitants, \\textit{Crime}) from the FBI's Crime Data Explorer, where a crime is defined as any of the four offenses -- murder and non-negligent manslaughter, rape, robbery, and aggravated assault. All the other controls are retrieved from the U.S. Census Bureau. Note that all values are projections on the 2010 census data. These controls include per--capita income in 2010 inflation--adjusted dollars (\\textit{PC\\_Income}) and the percentage of people living in urban areas (\\textit{Urban}). It must be noted that, for this last variable, only figures for the year 2010 were available, thus it is treated as a time--invariant covariate. Other socio--economic characteristics, such as poverty rate (\\textit{Poverty}),  unemployment rate (\\textit{Unemp.}), and the percentage of adults with an education lower than high school diploma (\\textit{Low\\_Edu}), are also included. The percentage of young population, aged 18--34, (\\textit{Young}) and the percentage of white Caucasians (\\textit{White}) over the whole population are controlled for in the analysis. The last variable is added since several studies find a racial bias in police shootings \\citep{ross15, nix17, mesic18}. Table \\ref{tab1} reports some key statistics of all the mentioned variables.     \n\\subsection{Statistical Analysis}\nThe statistical analysis is divided into two main parts. In the first part, we focus on the role of the legislative strength as a whole, thus considering the overall score provided by Giffords for each U.S. state. In the second part, we analyze the component Giffords scores separately. This analysis should provide more specific policy indications with regard to the legislative field where intervention may be more productive in reducing fatal police shootings.\\par \nThe effects of any changer in legislation may take time to be observed. In all our analysis, therefore, both the overall and the component Giffords scores enter in their first lags. The inclusion of lags more distant in time (two or three years) is precluded by the limited number of time periods at our disposal. We have actually run regressions with the contemporaneous level of the Giffords scores, but none of them has resulted in significant coefficients (results are available from the authors upon request). The possibility to include both the lag and the current level is prevented by their high correlation ($\\rho = 0.99$) that most likely causes a problem of collinearity. \\par \nFigure \\ref{fig1} shows the Spearman's rank correlation coefficients of all variables, except the component Giffords score. Note \\textit{Giff\\_Score\\_L}  denotes the lagged Giffords score. From Figure \\ref{fig1} it is possible to observe that several covariates have a relatively low correlation coefficient. The exceptions are the violent crime rate ($\\rho = 0.49$) and the per--capita income ($\\rho = -0.29$). \\par \n\\begin{figure}[htbp!]\n\t\\centering\n\t\\caption{Correlation Matrix}\\label{fig1}\n\t\\includegraphics[scale=0.75]{Fig\/Corr_fig.eps}\n\\end{figure} \n\\indent  \nDespite the low correlation of these covariates with the dependent variable, our full specifications (specifications (1) and (2) in Table \\ref{tab2}) include all of them. Their coefficients are not significant in these models. Failing to reject the null hypothesis that the joint significance of the socio--economic covariates (\\textit{Poverty}, \\textit{White}, \\textit{Low\\_Edu}, \\textit{Unemp.} and \\textit{Young}) is equal to zero (p--value = 0.6098 for model (1) and p--value = 0.8595 for model (2)), we omit these variables in any subsequent analysis. Despite its lack of significance, \\textit{Urban} is kept in all models because of its high correlation with the lagged Giffords score ($\\rho = 0.62$) and because it is a significant control in previous studies \\citep{hemenway19}. Models (3) and (4) in Table  \\ref{tab2} report the results with the parsimonious set of explanatory variables. From Table \\ref{tabs1} in the Appendix, it is possible to observe non significant p--values for the RESET and for the Mundlak test. Compared to models (1) and (2), the coefficients of models (3) and (4) are negligibly different.\\par\nA conditional fixed effect (FE) and a random effect (RE) Poisson regressions (models (1) and (2) in Table \\ref{tab2}) are our main models of interest. The functional form specification is checked through a RESET test, by adding the squared residuals of the Poisson regressions (FE and RE) and checking their significance \\citep{ramsey74}. Results are reported in Table \\ref{tabs1} in the  \\hyperref[app]{Appendix} together with the coefficients of the year dummies (year 2013 as base) included in all models. The p--value of the squared residuals is far above the 10\\% significance level, thus dismissing possible concerns about misspecification. The choice to report both the FE and RE estimates is due to the fact that the Mundlak test -- also reported in Table \\ref{tabs1} -- has a significance level very close to the 5\\% level \\citep{mundlak78}. Although this test, chosen compared to the more common Hausman test given the presence of year dummies and a time invariant covariate (\\textit{Urban}), suggests the use of the random effect model, the closeness of the p--value to the threshold level and the concerns about the distributional assumptions of the Poisson RE model lead us to report the fixed effect estimates as well. However, it can be observed that the estimated coefficients are very similar in both specifications.\\par\n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Total Giffords Score (Lagged)}\\label{tab2}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t\\multirow{3}{*}{} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(1) \\\\ Poisson FE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(2) \\\\ Poisson RE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(3) \\\\ Poisson FE\\end{tabular}}} & \\multirow{2}{*}{\\textbf{\\begin{tabular}[c]{@{}c@{}}(4) \\\\ Poisson RE\\end{tabular}}} \\\\\n\t\t\t&  &  &  &  \\\\\\hline\\hline\n\t\t\t& \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} & \\multicolumn{1}{l}{\\textit{Pol\\_Shoot}} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Giff\\_Score\\_L}} & 0.990* & 0.992+ & 0.990* & 0.993+ \\\\\n\t\t\t& (-2.05) & (-1.65) & (-2.10) & (-1.82) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.000 & 1.000*** & 1.000 & 1.000*** \\\\\n\t\t\t& (1.61) & (4.04) & (1.46) & (4.69) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & 0.993 & 1.007 & 0.993 & 1.008 \\\\\n\t\t\t& (-0.77) & (1.03) & (-0.76) & (1.21) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & 1.000* & 1.000* & 1.000 & 1.000+ \\\\\n\t\t\t& (-2.15) & (-2.05) & (-1.49) & (-1.94) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 1.004 &  & 1.005 \\\\\n\t\t\t&  & (0.90) &  & (1.32)\\\\\n\t\t\t\\multirow{2}{*}{\\textit{Poverty}} & 0.958 & 0.965 &  &  \\\\\n\t\t\t& (-1.51) & (-1.04) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{White}} & 0.958 & 0.996 &  &  \\\\\n\t\t\t& (-1.21) & (-0.66) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Low\\_Edu}} & 1.025 & 1.012 &  &  \\\\\n\t\t\t& (0.90) & (0.38) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Unemp.}} & 1.064 & 1.032 &  &  \\\\\n\t\t\t& (1.20) & (0.80) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Young}} & 0.946 & 1.008 &  &  \\\\\n\t\t\t& (-1.01) & (0.23) &  &  \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}\n\\indent\nAs a robustness check, we also report the results of linear models, both FE and RE, after that the dependent variable has been transformed with a Yeo--Johnson power transform in order to render its distribution more normal--like \\citep{yeo00}. The results of the linear models are shown in Table \\ref{tabs2}. Although the transformation of the dependent variable prevents the computation of meaningful marginal effects, the sign and significance of the coefficients serve to confirm the results of, or to signal a possible problem in, the Poisson regressions. The distribution of the dependent variable before and after the Yeo--Johnson transformation is shown in Figure \\ref{figs1} in the \\hyperref[app]{Appendix}.\\par \nFor our second purpose, we regress the rate of fatal police shootings on the component Giffords scores rather than the overall score. Here we present only the parsimonious models. Four models have been run, two Poisson -- FE and RE -- and two analogous linear regressions. Results are reported in Table \\ref{tab3}, models (5), (6), (7) and (8), and auxiliary information can be found in Table \\ref{tabs1} in the \\hyperref[app]{Appendix}. From this last table, it is possible to see that the p--values of the RESET and of the Mundlak test are all above conventional significance levels. As for the previous models, the difference in the significance of the coefficients of the Poisson and of the linear regressions are very modest, so as the difference in the coefficients between the FE and RE models. \\par \n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Linear Models}\\label{tabs2}\n\t\\resizebox{!}{9cm}{\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t\\multirow{2}{*}{} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(1b) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(2b) \\\\ Linear RE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(3b) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(4b) \\\\ Linear RE\\end{tabular}} \\\\\n\t\t\t\\hline\\hline\n\t\t\t& \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Giff\\_Score\\_L}} & -0.00879* & -0.00924** & -0.00925* & -0.00803* \\\\\n\t\t\t& (-2.32) & (-2.86) & (-2.34) & (-2.52) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.47-E004+ & 1.36-E004*** & 1.60-E004+ & 149-E004*** \\\\\n\t\t\t& (1.76) & (3.75) & (1.95) & (4.43) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & -0.00515 & 0.00193 & -0.00562 & 0.00237 \\\\\n\t\t\t& (-0.55) & (0.36) & (-0.59) & (0.45) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & -1.83-E005* & -1.95-E005* & -1.44-E005+ & -155-E005* \\\\\n\t\t\t& (-2.07) & (-2.22) & (-1.75) & (-2.12) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 0.00255 &  & 0.00357 \\\\\n\t\t\t&  & (0.66) &  & (0.92) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Poverty}} & -0.0248 & -0.0259 &  &  \\\\\n\t\t\t& (-0.80) & (-0.87) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{White}} & -0.0159 & -0.00411 &  &  \\\\\n\t\t\t& (-0.41) & (-1.03) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Low\\_Edu}} & 0.0142 & 0.0124 &  &  \\\\\n\t\t\t& (0.51) & (0.52) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Unemp.}} & 0.0393 & 0.0327 &  &  \\\\\n\t\t\t& (0.64) & (0.69) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Young}} & -0.0476 & -0.0138 &  &  \\\\\n\t\t\t& (-0.92) & (-0.47) &  &  \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Const.}} & 4.304 & 2.195* & 1.938** & 1.291* \\\\\n\t\t\t& (1.34) & (2.06) & (2.83) & (2.42) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2014}} & 0.0538 & 0.0639 & 0.0313 & 0.0418 \\\\\n\t\t\t& (0.76) & (0.92) & (0.46) & (0.60) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2015}} & 0.0661 & 0.0829 & 0.0359 & 0.0522 \\\\\n\t\t\t& (0.84) & (1.15) & (0.52) & (0.76) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2016}} & 0.137 & 0.151+ & 0.113* & 0.120* \\\\\n\t\t\t& (1.55) & (1.95) & (2.24) & (2.48) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2017}} & 0.163 & 0.189+ & 0.129+ & 0.146* \\\\\n\t\t\t& (1.42) & (1.75) & (1.99) & (2.31) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{2018}} & 0.255* & 0.282* & 0.220** & 0.236*** \\\\\n\t\t\t& (2.11) & (2.40) & (3.10) & (3.36) \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}\n\\indent\n\n\\begin{table}[H]\n\t\\centering\n\t\\caption{Regressions Results: Giffords Score Disentangled (Lagged)}\\label{tab3}\n\t{\\renewcommand{\\arraystretch}{1.3}\n\t\t\\begin{tabular}{lcccc}\n\t\t\t\\hline\\hline\n\t\t\t& \\textbf{\\begin{tabular}[c]{@{}c@{}}(5) \\\\ Poisson FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(6) \\\\ Poisson RE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(7) \\\\ Linear FE\\end{tabular}} & \\textbf{\\begin{tabular}[c]{@{}c@{}}(8) \\\\ Linear RE\\end{tabular}} \\\\ \\hline\\hline\n\t\t\t& \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} & \\textit{Pol\\_Shoot} \\\\\n\t\t\t\\multirow{2}{*}{\\textit{BCAF\\_L}} & 1.001 & 1.000 & 0.00131 & -6.63-E004 \\\\\n\t\t\t& (0.15) & (0.03) & (0.22) & (-0.12) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{ORST\\_L}} & 1.032 & 1.008 & 0.0231 & 0.0110 \\\\\n\t\t\t& (1.44) & (0.47) & (1.17) & (0.93) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{CWAM\\_L}} & 0.981 & 1.014 & -0.0168 & 0.00372 \\\\\n\t\t\t& (-0.93) & (0.44) & (-1.01) & (0.22) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{CCS\\_L}} & 0.926 & 0.958 & -0.0732 & -0.0409 \\\\\n\t\t\t& (-1.09) & (-1.24) & (-1.19) & (-1.47) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{GOA\\_L}} & 0.963* & 0.962* & -0.0351* & -0.0352** \\\\\n\t\t\t& (-2.57) & (-2.32) & (-2.58) & (-2.93) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{FPP\\_L}} & 0.984 & 0.983 & -0.0191 & -0.0193 \\\\\n\t\t\t& (-1.25) & (-1.38) & (-1.54) & (-1.63) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Crime}} & 1.000 & 1.000*** & 1.58-E004+ & 1.57-E004*** \\\\\n\t\t\t& (1.45) & (4.96) & (1.90) & (4.64) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Suicide}} & 0.992 & 1.002 & -0.00597 & -0.00118 \\\\\n\t\t\t& (-0.83) & (0.37) & (-0.63) & (-0.21) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{PC\\_Income}} & 1.000 & 1.000* & -1.39-E005+ & -1.62E-005* \\\\\n\t\t\t& (-1.48) & (-2.18) & (-1.70) & (-2.14) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Urban}} &  & 1.004 &  & 0.00351 \\\\\n\t\t\t&  & (1.19) &  & (0.94) \\\\\n\t\t\t\\multirow{2}{*}{\\textit{Const.}} &  &  & 1.955** & 1.503** \\\\\n\t\t\t&  &  & (2.72) & (2.66) \\\\ \\hline\\hline\n\t\t\t\\multicolumn{5}{l}{\\footnotesize{t-statistics in parentheses. P-value: + p \\textless{}0.1, * p \\textless{}0.05, ** p \\textless{}0.01, *** p \\textless{}0.001.}}\n\t\\end{tabular}}\n\\end{table}   \n\\section{Results and Discussion}    \nIn the presentation of results, we will focus solely on the Poisson regressions, given the difficult interpretation of the coefficients of the linear models discussed earlier. Note that all the reported coefficients relative to the Poisson models are incidence rate ratios (IRR). From Table \\ref{tab2}, it is possible to observe that the Giffords score (lagged) is always statistically significant, although only at 10\\% level in the RE models, while at 5\\% in the FE models. An increase of one point in the overall Giffords score causes an approximately 1\\% reduction in the number of fatal police shootings per million of inhabitants if considering the FE model. The percentage reduction is slightly lower for the RE model: 0.8\\% when including all controls -- model (3) -- and 0.7\\% when excluding the subset of jointly non-significant covariates -- model (4).\\par \nAn important point to notice is the lack of significance, in all models of Table \\ref{tab2}, of the proxy for firearms diffusion: \\textit{Suicide}. The strong correlation with the lag of the Giffords score ($\\rho = -0.83$) may suggest that the effect of this last masks the one of the former. However, if running the same regressions without the inclusion of the Giffords score, the p--value of \\textit{Suicide} remains above 0.1 in all models: 0.152, 0.106, 0.140 and 0.118 for, respectively, models (1), (2), (3) and (4) without \\textit{Giff\\_Score\\_L} (full results available from the authors upon request). \\par \nWhen considering Table \\ref{tab3} and the disentangled categories composing the Giffords score, only the coefficient related to the lag of one category, gun owner accountability, is significant (at 5\\% level). This happens to be true both in the FE and in the RE Poisson regressions, so as in the linear models. In particular, an increase of one point in the strength of the gun owner accountability category is associated with a decrease in per--million inhabitants fatal police shootings of 3.7\\% (FE model) or of 3.8\\% (RE model).\\par \nWe can further notice that in all models, both in Table \\ref{tab2} and \\ref{tab3}, the sign of the coefficients of the main control variables is as expected. In particular, the number of violent crimes positively impacts the number of fatal police shooting episodes whereas per--capita income has the opposite effect. A last word is dedicated to the significance of the violent crime rate that is very high (0.1\\%) in all RE Poisson models but absent in the FE models. This is possibly due to the persistent nature in time of this phenomenon that, in the FE model, gets captured by the fixed effect component.           \n\\subsection{Discussion}\nThe present study shows that increasing levels of firearms regulation are significantly associated with a lower number of fatal police shooting cases. In particular, a point increase in the overall Giffords score leads to a decrease of in fatal police shootings of 0.7\\%--1\\%, depending from the model. When considering separately the various categories composing the Giffords score, one point increase in the strength of gun owner accountability leads to a decrease of approximately 3.7\\%--3.8\\% in the number of people killed by police officers. The diffusion of fire--weapons, instead, has no statistically significant role in determining the considered outcome. This finding has been achieved through the use of a panel dataset, thus controlling for unobserved heterogeneity through the use of FE Poisson models.\\par\nIt is interesting to compare our results with previous findings. Regarding the effect of firearms diffusion, our results clearly contradicts the previous findings of \\citet{kivisto17} and of \\citet{hemenway19}. In fact, we do not find a statistically significant effect of guns diffusion in determining the number of fatal police shootings. \\par \nConsidering the strength of firearms regulations, our analysis basically confirms the findings of \\citet{kivisto17}. However, this is true only for the overall score. When evaluating each category separately, significant differences emerge. First of all, it must be noted that the results are not easily comparable, given the different scoring system used in \\citet{kivisto17}, namely the Brady scorecards, and in the present analysis. However, a comparison is not impossible. In \\citet{kivisto17}, two categories remained significant after all controls were added, namely promoting safe storage via child and consumer safety laws and curbing gun trafficking. The former corresponds to the category consumers and child safety (\\textit{CSS}) in the Giffords scorecards and the latter is included in other regulation of sales and transfers (\\textit{ORST}), both not significant in our models. The gun owner accountability, the category found significant in the present analysis, is instead composed by three elements: licensing of gun owners and purchasers, having the highest weights, followed by registration of firearms and reporting lost or stolen firearms. This is an important difference, with potentially relevant implications for policy--makers. Furthermore, it is reasonable to think that the first sub-category, namely the need of gun owners to have a license, has a great discriminant power in determining the final identity of gun owners. This further suggests that the qualitative side of gun diffusion (who owns a gun) is more important in limiting the number of police shootings than the quantitative side (how many guns are owned).     \n\\section*{Conclusions}\nThe present analysis has shown that police shooting episodes are significantly reduced by stricter levels of firearms law. While this finding partially confirms what emerged in previous studies, we also find that the diffusion of fire--weapons is inconsequential in determining the number of police shootings, thus contradicting the precedent evidence.\\par \nThe policy recommendations that can be derived from the present paper are quite straightforward. Improving the strength of firearms regulations seems an effective way for reducing the number of people killed by law enforcement officers. Actually, the policy prescriptions can be even more specific. In fact, from the analysis it emerges that the most effective intervention for reducing fatal police shooting episodes would be to strengthen the rules of gun owner accountability, namely licensing of gun owners and purchasers, registration of firearms and reporting lost or stolen firearms. These policy prescriptions are different from the ones provided by previous studies.\\par\nAnother important lesson, and a departure from previous findings, is the lack of statistical significance of the diffusion of guns in causing fatal police shootings. This suggests that the cause may be more qualitative (who owns the guns) rather than quantitative (how many guns). The fact that the only significant legislative category emerged from this study is the gun owner accountability further strengthens this hypothesis.\\par  \nThere are several possible ways in which the analysis could be expanded in order to have more precise and specific prescriptions. One possibility would be to further disaggregate each category of the Giffords score into its subcategories. We have not pursued this road due to the limited number of observations at our disposal. Another interesting extension would be to consider the episodes of police shootings directed towards unarmed citizens \\citep{hemenway19}. The lack of this information in the Fatal Encounters database has prevented us to conduct this analysis.    \n\\newpage \n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\bf NOTATION AND DEFINITIONS}\\label{nota} \n\n\\noindent $\\bullet$ Throughout this article units are used in which \n$\\hbar = c = 1$.  \n\n\\noindent $\\bullet$ Everywhere the {\\it repeated} indexes imply the\nsummation.\n\n\\noindent $\\bullet$ $\\delta^{ik} = \\delta_{ik} = \\delta^i_k = \\delta^k_i$\n$(i,k = 1, \\ldots , n)$ is Kronecker symbol,\n\\[\n\\delta^{ik} = 0 \\quad \n i \\ne k, \\quad \\delta^{11} = \\delta^{22} = \\ldots \\delta^{nn} = 1.\n\\]\n\\noindent Metric tensor in Minkowski space $g^{\\mu \\nu} = g_{\\mu \\nu}$ \n$(\\mu, \\nu = 0, 1, \\ldots , n)$ equals:\n\\[\ng^{\\mu \\nu} = 0 \\quad  \\mu \\ne \\nu, \\quad \ng^{00} = 1, \\quad g^{11} = g^{22} = \\ldots = g^{nn} = -1.\n\\]\n\\noindent The tensor $g^{\\mu \\nu}$ is used for raising and lowering of the \nLorentz subscripts and superscripts. \n\n\\noindent $\\bullet$ The scalar products of any two $p$ and $q$ vectors (both \nin Euclidean and in Minkowski spaces) is denoted as follows: \n\\[\n p \\cdot q \\quad {\\rm or} \\quad (pq), \\quad {\\rm i.e.} \\quad p \\cdot q = \n(pq).\n\\]\nThe scalar products of any two $p$ and $q$ Euclidean vectors would be also\ndenoted as:\n\\[\n\\vec p \\vec q, \\quad {\\rm i.e.} \\quad \\vec p \\vec q = p \\cdot q = (pq).\n\\]\n\n\\noindent $\\bullet$ 4--vector $p^{\\mu}$ in Minkowski space is given by\n\\[\np^{\\mu} \\equiv (E, \\, \\vec p) \\; = \\; (p_0, \\, p_1, \\, p_2, \\, p_3), \\; \\;\np_{\\mu} = g_{\\mu \\nu} p^{\\nu} \\, = \\, (E, \\, -\\vec p).\n\\]\nThe scalar product of any two vectors $p$ and $q$ in Minkowski space is given\nby\n\\[\np^{\\mu} g_{\\mu \\nu} q^{\\nu} \\, = \\, p^{\\mu} q_{\\mu} \\, = \\, p_{\\nu} q^{\\nu} \\,\n = \\, p_0 q_0 - p_1 q_1 - p_2 q_2 - p_3 q_3. \n\\]\nThe products of the 4--vector $p^{\\mu}$ with Dirac $\\gamma^{\\mu}$ matrix \ndenotes as usual\n\\[\n \\hat p \\; \\equiv \\; p^{\\mu} g_{\\mu \\nu}  \\gamma^{\\nu} \\, = \\,\n p^{\\mu} \\gamma_{\\mu} \\, = \\, p_{\\nu} \\gamma^{\\nu}. \n\\]\n\n\\noindent $\\bullet$ {\\bf Totally antisymmetric tensor\n $\\varepsilon^{A B \\ldots N}$}.\n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in two dimensions}:\n$\\varepsilon^{AB} \\quad (A,B=1, 2)$:\n\\begin{eqnarray*} \n && \\varepsilon_{12}=\\varepsilon^{12} = 1; \\; \\varepsilon_{21}=\n \\varepsilon^{21} = -1; \\; \n \\varepsilon_{AB}\\ =\\ \n \\left( \\begin{array}{c c} 0\\ &\\ 1 \\\\ -1 \\ & \\ 0 \\end{array} \\right); \\\\\n && \\varepsilon^{AB}=\\varepsilon_{AB}, \\; \\varepsilon_{BA}=-\\varepsilon_{AB}, \n \\;   \\varepsilon_{AB} \\varepsilon^{AB}\\ =\\ 2; \\;  \n\\varepsilon_{AB} \\varepsilon^{BC}\\ =\\ - \\delta_A^C; \\\\\n&& \\varepsilon_{AB} \\varepsilon^{CD}\\ =\\ \n\\delta_A^C \\delta_B^D \\ -\\ \\delta_A^D \\delta_B^C; \\\\ \n &&\\varepsilon_{AB} \\varepsilon_{CD} + \\varepsilon_{AC} \\varepsilon_{DB} +\n\\varepsilon_{AD} \\varepsilon_{BC} =\\ 0.\n\\end{eqnarray*}\n $\\varepsilon^{AB}$--symbol is used for rising and lowering of the spinor \nindexes (see Subsection~\\ref{sigpaul}). \n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in three dimensions}: \n$\\varepsilon^{ijk} \\quad (i,j,k = 1,2,3)$: \n\\begin{displaymath}\n \\varepsilon^{123}=\\varepsilon_{123}=1, \\quad \n \\varepsilon^{ijk}=\\varepsilon_{ijk}, \\quad \n\\varepsilon_{ijk}\\varepsilon^{lmn}= \n\\left| \\begin{array}{ccc}\n\\delta_i^l & \\delta_j^l & \\delta_k^l \\\\\n\\delta_i^m & \\delta_j^m & \\delta_k^m \\\\\n\\delta_i^n & \\delta_j^n & \\delta_k^n \n\\end{array} \\right|\n\\end{displaymath}\n\n\\[\\varepsilon_{ijk}\\varepsilon^{lmk}= \n\\delta_i^l\\delta_j^m-\\delta_i^m\\delta_j^l, \\ \\ \\ \n\\varepsilon_{ijk}\\varepsilon^{ljk}=2\\>\\delta^{l}_{i},\\ \\ \\\n\\varepsilon_{ijk}\\varepsilon^{ijk}=6. \\]\n{\\it Schouten identity}. For any 3--vector $p^i$ one has:\n\\[ p_{i_1} \\varepsilon_{i_2 i_3 i_4}\n- p_{i_2} \\varepsilon_{i_1 i_3 i_4}\n+ p_{i_3} \\varepsilon_{i_1 i_2 i_4}\n- p_{i_4} \\varepsilon_{i_1 i_2 i_3}\\ =\\ 0. \\]\n\n\\noindent $\\bullet$ {\\it $\\varepsilon$-symbol in four--dimensional Minkowski\n space}: \n$\\varepsilon^{\\alpha \\beta \\mu \\nu}$ $(\\alpha, \\ldots \\nu = 0,1,2,3)$:\n\\[\\varepsilon^{0123}\\ =\\ -\\ \\varepsilon_{0123}\\ =\\ 1. \\]\n\\begin{displaymath}\n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\sigma\\tau} =\n-\\left| \\begin{array}{cccc}\n\\delta_\\mu^\\lambda & \\delta_\\nu^\\lambda & \\delta_\\alpha^\\lambda &\n\\delta_\\beta^\\lambda \\\\\n\\delta_\\mu^\\rho & \\delta_\\nu^\\rho & \\delta_\\alpha^\\rho & \\delta_\\beta^\\rho \\\\\n\\delta_\\mu^\\sigma & \\delta_\\nu^\\sigma & \\delta_\\alpha^\\sigma &\n\\delta_\\beta^\\sigma \\\\\n\\delta_\\mu^\\tau & \\delta_\\nu^\\tau & \\delta_\\alpha^\\tau & \\delta_\\beta^\\tau\n\\end{array} \\right|, \\; \\; \n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\sigma\\beta} = -\n\\left| \\begin{array}{ccc}\n\\delta_\\mu^\\lambda & \\delta_\\nu^\\lambda & \\delta_\\alpha^\\lambda \\\\\n\\delta_\\mu^\\rho & \\delta_\\nu^\\rho & \\delta_\\alpha^\\rho \\\\\n\\delta_\\mu^\\sigma & \\delta_\\nu^\\sigma & \\delta_\\alpha^\\sigma \n\\end{array} \\right|,\n\\end{displaymath}\n\\[\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\rho\\alpha\\beta}\n=\\ -2 (\\delta^\\lambda_\\mu \\delta^\\rho_\\nu -\n\\delta^\\rho_\\mu \\delta^\\lambda_\\nu),\\ \\ \\ \\\n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\lambda\\nu\\alpha\\beta}\n =\\ -6 \\delta_\\mu^\\lambda, \\ \\ \\ \\ \n\\varepsilon_{\\mu\\nu\\alpha\\beta}\\varepsilon^{\\mu\\nu\\alpha\\beta} =\\ -24. \\]\n\\noindent {\\it Schouten identity}. For any 4--vector $p_{\\mu}$ one has:\n\\[p_{\\mu_1} \\varepsilon_{\\mu_2 \\mu_3 \\mu_4 \\mu_5}\n+ p_{\\mu_2} \\varepsilon_{\\mu_3 \\mu_4 \\mu_5 \\mu_1}\n+ p_{\\mu_3} \\varepsilon_{\\mu_4 \\mu_5 \\mu_1 \\mu_2}\n+ p_{\\mu_4} \\varepsilon_{\\mu_5 \\mu_1 \\mu_2 \\mu_3} \n+ p_{\\mu_5} \\varepsilon_{\\mu_1 \\mu_2 \\mu_3 \\mu_4} = 0. \\]\n\n\\noindent $\\bullet$ {\\it Generalized Kronecker deltas} \\\\\nSometimes one can make no difference between a vector and\nindex. For example, one can write:\n\\[ \n\\varepsilon^{p_1 p_2 p_3 p_4} \\; {\\rm or} \\; \n\\varepsilon (p_1, p_2, p_3, p_4) \\quad {\\rm instead} \\; {\\rm of} \\; \n\\varepsilon_{\\mu\\nu\\rho\\sigma} p_1^\\mu p_2^\\nu p_3^\\rho p_4^\\sigma.\n\\] \nThese notation can be used in operations with generalized Kronecker deltas:\n\\begin{displaymath}\n\\delta_{i_1 ... i_n}^{j_1 ... j_n}\\equiv\n\\left| \\begin{array}{ccc}\n\\delta_{i_1}^{j_1} & ... & \\delta_{i_n}^{j_1} \\\\\n... & ... & ... \\\\\n\\delta_{i_n}^{j_1} & ... & \\delta_{i_n}^{j_n} \n\\end{array} \\right| , \\ \\mbox{or} \\ \n\\delta_{p_1 ... p_n}^{q_1 ... q_n}\\equiv\n\\left| \\begin{array}{ccc}\np_1\\cdot q_1 & ... & p_n\\cdot q_1 \\\\\n... & ... & ... \\\\\np_1\\cdot q_n & ... & p_n\\cdot q_n \n\\end{array} \\right|.\n\\end{displaymath}\nIn $n$-dimensional Euclidean space one has:\n\\[\n\\delta_{p_1 ... p_m}^{q_1 ... q_m}=\\frac{1}{(n-m)!}\n\\varepsilon^{q_1 ... q_m \\alpha_{m+1} ... \\alpha_n}\n\\varepsilon_{p_1 ... p_m \\alpha_{m+1} ... \\alpha_n}.\n\\]\nIn Minkowski space the minus sign appears:\n\\[ \\delta_{p_1 p_2 p_3}^{q_1 q_2 q_3}=\\ -\n\\varepsilon_{p_1 p_2 p_3 \\mu} \\varepsilon^{q_1 q_2 q_3 \\mu}, \\ \\ \n\\delta_{p_1 p_2}^{q_1 q_2} =\\ -\\frac{1}{2}\n\\varepsilon_{p_1 p_2 \\mu \\nu}\\varepsilon^{q_1 q_2 \\mu \\nu}. \\]\n\n\\noindent $\\bullet$ {\\it Matrices}\n\n\\noindent For any matrix $A = (a_{ik}) \\quad (i,k = 1, \\ldots n)$ we use the \nfollowing notation: \\\\\n\\noindent $I$ is the {\\it unit} matrix, i.e. $I = \\delta_{ik}$ (sometimes, the\nunit matrix will be denote just $1$); \\\\\n\\noindent $A^{-1}$ is the {\\it inverse} matrix, i.e. \n$A^{-1} A = A A^{-1} = I$;  \\\\\n\\noindent $A^{\\top}$ is the {\\it transposed} matrix, i.e. \n$a^{\\top}_{ik} = a_{ki}$; \\\\\n\\noindent $A^{\\ast}$ is the {\\it complex conjugated} matrix, i.e. \n$(a^{\\ast})_{ik} = (a_{ik})^{\\ast}$; \\\\\n\\noindent $A^{\\dagger}$ is the {\\it Hermitian conjugated} matrix, i.e. \n$a^{\\dagger}_{ik} = a^{\\ast}_{ki}$; \\\\\n$H$ -- {\\it Hermitian} and  $U$ -- {\\it unitary} matrices\nshould satisfy the following conditions:\n\\begin{eqnarray*}\n &&  H^{\\dagger} = H, \\\\\n &&U = (U^{\\dagger})^{-1}, \\quad {\\rm hence} \\quad \n U^{\\dagger} = U^{-1}, \\quad  U U^{\\dagger} = U^{\\dagger} U = I.\n\\end{eqnarray*}\n\\noindent $\\det A $ is the {\\it determinant} of matrix $A$\n\\begin{eqnarray*}\n\\det A &=& \\varepsilon^{i_1 i_2 \\ldots i_n} a_{i_1 1} a_{i_2 2} \\cdots \na_{i_n n} \\\\\n &=& \\frac{1}{n !} \\varepsilon^{i_1 i_2 \\ldots i_n} \n                   \\varepsilon^{k_1 k_2 \\ldots k_n} \n  a_{i_1 k_1} a_{i_2 k_2} \\cdots a_{i_n k_n}.\n\\end{eqnarray*}\n\n\n\\noindent ${\\rm Tr} A $ is the {\\it trace} of matrix $A$ : \n ${\\rm Tr} A = a_{ii} \\, (= \\sum_{i=1}^n a_{ii})$. The chief properties of the \ntrace are  as follows (below $\\lambda$ and $\\mu$ are parameters):\n\\begin{eqnarray*}\n&& {\\rm Tr} (\\lambda A + \\mu B) = \\lambda {\\rm Tr} A + \\mu {\\rm Tr} B, \\\\\n&& {\\rm Tr}A^{\\top} = {\\rm Tr}A, \\quad {\\rm Tr} A^{\\ast} = \n {\\rm Tr} A^{\\dagger} = ({\\rm Tr}A)^{\\ast},\n\\quad {\\rm Tr} I = n, \\\\\n&&{\\rm Tr} (AB) = {\\rm Tr} (BA), \\qquad \\det(e^A) = e^{{\\rm Tr}A}. \n\\end{eqnarray*}\n\n\\noindent For any two matrices $A$ and $B$ the {\\it commutator} \n$[A,B]$ and {\\it anticommutator}  $\\lbrace A,B \\rbrace$ are denoted as usual:\n\\[\n [A,B] \\equiv AB - BA, \\quad \\lbrace A,B \\rbrace \\equiv AB + BA.\n\\]\n\n\n\n\n\\section{\\bf PAULI MATRICES}\\label{pauli} \n\n\\subsection{\\it Main Properties}\nThe Pauli matrices $\\sigma_i$ $(i=1,2,3)$ are generators of the group\n$SU(2)$. The $\\sigma_i$ are equal \\cite{bogol,land,itzu,okun}:\n\\begin{displaymath}\n \\sigma_1 =\\sigma^1 = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right),\n\\quad\n \\sigma_2 = \\sigma^2 = \\left( \\begin{array}{ccc} 0 & -i \\\\ i & 0 \\end{array} \\right),\n\\quad  \n \\sigma_3 =  \\sigma^3 =  \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right). \n \\end{displaymath}\nThe main properties of $\\sigma_i$ are as follows:\n\\begin{eqnarray}\n  \\sigma^{\\dagger}_i \\, = \\, \\sigma_i, \\quad {\\rm Tr}\\sigma_i \\,=\\,0,\\quad \n \\det \\sigma_i \\, = \\, -1, \\quad \n\\sigma_i \\sigma_k \\, = \\, i\\varepsilon_{ikj} \\sigma_j \\, + \\, \\delta_{ik}. \n\\label{p1} \n\\end{eqnarray}\nUsing relation (\\ref{p1}), one gets:\n\\begin{eqnarray*}\n&& \\sigma^{2}_i \\, = \\, I, \\quad [\\sigma_i, \\sigma_k] \\, = \\, \n 2 i\\varepsilon_{ikj} \\sigma_j, \\quad \n \\lbrace \\sigma_i, \\sigma_k \\rbrace  \\, = \\, 2\\delta_{ik}, \\\\\n&& \\sigma_i \\sigma_k \\sigma_l \\, = \\, i\\varepsilon_{ikl}I + \\delta_{ik} \n\\sigma_l  - \\delta_{il} \\sigma_k + \\delta_{kl} \\sigma_i, \\\\\n&& {\\rm Tr} (\\sigma_i \\sigma_k) \\, = \\, 2\\delta_{ik}, \\quad \n {\\rm Tr} (\\sigma_i \\sigma_k \\sigma_l) \\, = \\, 2 i\\varepsilon_{ikl}, \\\\\n&&{\\rm Tr}(\\sigma_i \\sigma_k \\sigma_l \\sigma_m) \\, \n = 2(\\delta_{ik} \\delta_{lm} \\, \n+  \\, \\delta_{im} \\delta_{kl} \\, - \\, \\delta_{il} \\delta_{km}).\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities}\nThe Fiertz identities for the Pauli matrices have the form: \n\\begin{eqnarray} \n \\sigma^i_{AB} \\sigma^i_{CD} \\, &=& \\, 2 \\delta_{AD}\\delta_{CB} \n - \\delta_{AB} \\delta_{CD}, \\label{p2} \\\\\n \\sigma^i_{AB} \\sigma^i_{CD} \\, &=& \\, \\frac{3}{2} \\delta_{AD}\\delta_{CB} \\,\n - \\, \\frac{1}{2} \\sigma^i_{AD} \\sigma^i_{CB}. \\label{p3}\n\\end{eqnarray}\n\nUsing (\\ref{p2}), one can obtain the following relations:\n\\begin{eqnarray*} \n \\delta_{AB} \\sigma^i_{CD} \\, &=& \\, \\frac{1}{2}[\\delta_{AD}\\sigma^i_{CB} +\n\\sigma^i_{AD} \\delta_{CB} + i\\varepsilon^{ikl} \\sigma^k_{AD} \\sigma^l_{CB}], \\\\\n \\sigma^i_{AB} \\delta_{CD} \\, &=& \\, \\frac{1}{2}[\\delta_{AD}\\sigma^i_{CB} +\n\\sigma^i_{AD} \\delta_{CB} - i\\varepsilon^{ikl} \\sigma^k_{AD} \\sigma^l_{CB}], \\\\\n \\delta_{AB} \\sigma^i_{CD} &+& \\sigma^i_{AB} \\delta_{CD} \\, = \\, \n \\sigma^i_{AD} \\delta_{CB} + \\delta_{AD} \\sigma^i_{CB}, \\\\\n  \\sigma^i_{AB} \\sigma^k_{CD} \\, &=& \\, \\frac{1}{2} \n [\\sigma^i_{AD} \\sigma^k_{CB} + \\delta^{ik} \\delta_{AD} \\delta_{CB}\n - \\delta^{ik} \\sigma^l_{AD} \\sigma^l_{CB} + \\\\\n &&  + i\\varepsilon^{ikl} \\sigma^l_{AD} \\delta_{CB}\n  - i\\varepsilon^{ikl} \\delta_{AD} \\sigma^l_{CB}].\n\\end{eqnarray*}\n\n\\subsection{$\\sigma_+$ {\\it and} $\\sigma_-$ {\\it Matrices}}\nThe $\\sigma_+$ and $\\sigma_-$ matrices are defined as follows:\n\\begin{displaymath}\n \\sigma_+ \\equiv \\frac{1}{2}(\\sigma_1 + i \\sigma_2)  \n = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 0 & 0 \\end{array} \\right), \n\\quad\n \\sigma_- \\equiv  \\frac{1}{2}(\\sigma_1 - i \\sigma_2)  \n  = \\left( \\begin{array}{ccc} 0 & 0 \\\\ 1 & 0 \\end{array} \\right).\n \\end{displaymath}\nThe relations for these matrices are given by \n\\begin{eqnarray*}\n && (\\sigma_{\\pm})^{\\dagger} = \\sigma_{\\mp}, \\quad {\\rm Tr}\\sigma_{\\pm}=0,\\quad\n \\det \\sigma_{\\pm} = 0, \\\\\n && [\\sigma_{\\pm}, \\sigma_1] = \\pm \\sigma_3, \\quad\n [\\sigma_{\\pm}, \\sigma_2] = i \\sigma_3, \\quad\n [\\sigma_{\\pm}, \\sigma_3] = \\mp 2 \\sigma_{\\pm}, \\quad\n  [\\sigma_+, \\sigma_-] = \\sigma_3, \\\\\n && \\lbrace \\sigma_{\\pm}, \\sigma_1 \\rbrace  = I, \\quad\n \\lbrace \\sigma_{\\pm}, \\sigma_2 \\rbrace = \\pm iI, \\quad\n \\lbrace \\sigma_{\\pm}, \\sigma_3 \\rbrace  = 0, \\quad\n \\lbrace \\sigma_+, \\sigma_- \\rbrace = I, \\\\\n &&  \\sigma_+^2 = \\sigma_-^2 = 0, \\quad \n \\sigma_+ \\sigma_3 = - \\sigma_+, \\quad\n \\sigma_3 \\sigma_+ =  \\sigma_+, \\\\ \n && \\sigma_+ \\sigma_- = \\frac{1}{2} ( I + \\sigma_3), \\quad\n  \\sigma_- \\sigma_+ = \\frac{1}{2} ( I - \\sigma_3), \\\\\n && (\\sigma_+ \\sigma_-)^n = \\sigma_+ \\sigma_-, \\quad \n  (\\sigma_- \\sigma_+)^n = \\sigma_- \\sigma_+.\n\\end{eqnarray*}\nFor any parameter $\\xi$ one gets:\n\\[\n exp(\\xi \\frac{\\sigma_3}{2}) \\sigma_{\\pm} exp(-\\xi \\frac{\\sigma_3}{2}) =\n  \\sigma_{\\pm} exp(\\pm \\xi).\n\\] \nIf $f(\\sigma_+ \\sigma_-)$ (or $f(\\sigma_- \\sigma_+))$ is an arbitrary function\nof $\\sigma_+ \\sigma_-$ (or of $\\sigma_- \\sigma_+)$, and this function can \nbe expanded into power series with respect to $\\sigma_+ \\sigma_-$ (or with\nrespect to $\\sigma_- \\sigma_+)$,\nthen\n\\begin{eqnarray*}\n f(\\sigma_+ \\sigma_-) = f(0) + [f(1) - f(0)] \\sigma_+ \\sigma_-, \\\\\n f(\\sigma_- \\sigma_+) = f(0) + [f(1) - f(0)] \\sigma_- \\sigma_+.\n\\end{eqnarray*}\n\n\\subsection{\\it Various Relations} \nAny $2 \\times 2$ matrix $A$ can be expanded over the set $\\{ I, \\sigma_i \\}$:\n\\[\n A = a_0 I + a_i \\sigma_i,\n\\]\nwhere $a_0 = \\frac{1}{2}{\\rm Tr} A$, and $a_i = \\frac{1}{2}\n {\\rm Tr}(\\sigma_i A)$.\n\n\\noindent Let $\\alpha_i$ be the 3--vector. Then\n\\begin{eqnarray}\ne^{\\alpha_i \\sigma_i} = \\cosh \\sqrt{\\vec \\alpha^2} + \n \\frac{\\sinh \\sqrt{\\vec \\alpha^2}}{\\sqrt{\\vec \\alpha^2}} (\\alpha_i\\sigma_i)\n = p_0 + p_i \\sigma_i.  \\label{p4}\n\\end{eqnarray}\nThe components of the 4--vector $p^{\\mu}$ equal:\n\\begin{eqnarray}\n p_0 = \\cosh \\sqrt{\\vec \\alpha^2}, \\quad \n p_i = \\frac{\\sinh \\sqrt{\\vec \\alpha^2}}{\\sqrt{\\vec \\alpha^2}} \\alpha_i, \\quad\n p_0^2 - \\vec p^{\\,2} = p^2 = 1, \\label{p5}\n\\end{eqnarray}\nand we have \n\\begin{eqnarray}\n \\alpha_i = \\frac{p_i}{\\sqrt{\\vec p^{\\, 2}}} \\ln (p_0 + \\sqrt{\\vec p^{\\, 2}}). \n\\label{p6}\n\\end{eqnarray}\nLet $p$ and $q$ be two 4--vectors, and $p^2 = q^2 = 1$, then \n\\begin{eqnarray}\n (p_0 + p_i \\sigma_i) (q_0 + q_k \\sigma_k) = a_0 + a_l \\sigma_l = \n e^{\\beta_i \\sigma_i}, \\label{p7}\n\\end{eqnarray}\nwhere $a_0 = p_0q_0 + (\\vec p \\vec q), \\quad a_j = p_0 q_j + p_j q_0 +\ni\\varepsilon^{jkl} p_k q_l$, and the 3--vector $\\beta_i$ in the relation \n(\\ref{p7}) is expressed through $a_0$ and $\\vec a$ as in (\\ref{p6}). \n\n\\subsection{\\it 4--dimensional $\\sigma^{\\mu}$ Matrices} \\label{sigpaul}\nHere we present the various properties of $2 \\times 2$ matrices $\\sigma^{\\mu}$ \nand  $\\bar \\sigma^{\\mu}$ ($\\mu \\: = \\: 0,1,2,3$): \n\\begin{eqnarray}\n \\sigma^{\\mu}_{A \\dot B} \\equiv (I, -\\sigma_i); \\quad\n \\bar \\sigma^{\\mu \\dot A B} \\equiv (I, \\sigma_i), \\quad \n\\mu = 0, \\, 1, \\, 2, \\, 3, \\label{p10}\n\\end{eqnarray}\nwhere $\\sigma_i$ are Pauli matrices. \\\\\nWith the help of $\\sigma^{\\mu}$--matrices any tensor in Minkowski space can be\nunambiguously rewritten in spinorial form. In order to deal only with \nLorentz--covariant expressions one should clearly distinguish between dot and\nundot, lower and upper Weyl indices. The $\\varepsilon$--symbol (see\nSection~\\ref{nota}) used here for rising and lowering indices. \n\n\\noindent The main properties of the $\\sigma^{\\mu}$ matrices are as follows: \n\\begin{eqnarray*}\n &&  \\bar \\sigma^{\\mu \\dot A A} = \\varepsilon^{\\dot A \\dot B} \\varepsilon^{AB} \n \\sigma^{\\mu}_{B \\dot B}, \\quad\n \\sigma^{\\mu}_{\\dot A A} = \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B} \n \\bar \\sigma^{\\mu \\dot B B}, \\\\ \n &&  (\\sigma^{\\mu})^{\\dagger} = \\sigma^{\\mu}, \\quad \n  (\\bar \\sigma^{\\mu})^{\\dagger} = \\bar \\sigma^{\\mu}, \\quad \n \\det \\sigma^{\\mu} = \\det \\bar \\sigma^{\\mu} = 1(-1), \\: {\\rm for}\n\\: \\mu = 0(1,2,3).\n\\end{eqnarray*}\nFor any 4--vector $p^{\\mu}$ one has: \n\\[ \n\\det p_{\\mu} \\sigma^{\\mu} = \\det p_{\\mu} \\bar \\sigma^{\\mu} = p^2.\n\\]\n\n\n\\noindent Various products of $\\sigma^{\\mu}$ matrices have the form: \n\\begin{eqnarray*}\n && \\sigma^{\\mu}_{A \\dot C} \\bar \\sigma^{\\nu \\dot C B}\n+ \\sigma^{\\nu}_{A \\dot C} \\bar \\sigma^{\\mu \\dot C B} = \n 2 g^{\\mu \\nu} \\delta_A{}^B, \\quad \n \\bar \\sigma^{\\mu \\dot A C} \\sigma^{\\nu}_{C \\dot B}\n + \\bar \\sigma^{\\nu \\dot A C} \\sigma^{\\mu}_{C \\dot B} =\n 2 g^{\\mu \\nu} \\delta^{\\dot A}{}_{\\dot B}, \\\\\n && \\sigma^{\\mu}_{A \\dot C} \\bar \\sigma_{\\mu}^{\\dot C B}\n =  4 \\delta_A{}^B, \\quad \\bar \\sigma^{\\mu \\dot A C} \\sigma_{\\mu C \\dot B}\n = 4 \\delta^{\\dot A}{}_{\\dot B}, \\\\\n && \\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B} \\varepsilon^{\\dot A \\dot B} \n= 4 \\varepsilon_{AB},  \\quad \n \\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B}  \\varepsilon^{A B} \n= 4 \\varepsilon_{\\dot A \\dot B}, \\\\\n && \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B}\n  \\varepsilon_{\\dot A \\dot B} = 4 \\varepsilon^{AB},  \\quad \n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B}  \\varepsilon_{A B} \n= 4 \\varepsilon^{\\dot A \\dot B}, \\\\\n&& \\sigma^{\\mu}_{A \\dot A} \\sigma^{\\nu}_{B \\dot B} \\varepsilon^{A B} \n \\varepsilon^{\\dot A \\dot B} \\, = \\, \n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma^{\\nu \\dot B B}  \\varepsilon_{A B} \n \\varepsilon_{\\dot A \\dot B} = \\, 2 g^{\\mu \\nu}, \\\\\n&&\\sigma^{\\mu} \\bar \\sigma^{\\lambda} \\sigma^{\\nu} = g^{\\mu \\lambda}\\sigma^{\\nu}\n + g^{\\nu \\lambda}\\sigma^{\\mu} - g^{\\mu \\nu}\\sigma^{\\lambda}\n   -i\\varepsilon^{\\mu \\lambda \\nu \\rho}\\sigma^{\\rho}, \\\\\n&& \\bar \\sigma^{\\mu} \\sigma^{\\lambda} \\bar \\sigma^{\\nu} = \ng^{\\mu \\lambda}\\sigma^{\\nu}+g^{\\nu\\lambda}\\sigma^{\\mu}\n -g^{\\mu \\nu}\\sigma^{\\lambda}\n   +i\\varepsilon^{\\mu \\lambda \\nu \\rho}\\sigma^{\\rho}, \\\\\n&& \\varepsilon^{\\mu\\nu\\rho\\lambda}=i\\sigma^{\\mu\\dot A A}\n\\sigma^{\\nu\\dot B B}\\sigma^{\\rho\\dot C C}\\sigma^{\\lambda \\dot D D} \n(\\varepsilon_{AC}\\varepsilon_{BD}\\varepsilon_{\\dot A\\dot D}\n \\varepsilon_{\\dot B\\dot C} -\n\\varepsilon_{AD}\\varepsilon_{BC}\\varepsilon_{\\dot A\\dot C}\n\\varepsilon_{\\dot B\\dot D}). \n\\end{eqnarray*}\nThe commutators of $\\sigma^{\\mu}$ and $\\bar \\sigma^{\\mu}$ matrices have the\nspecial notation:\n\\[ \\sigma^{\\mu\\nu B}_A \\equiv \\frac{1}{4}\n  (\\sigma^{\\mu}_{A \\dot C} \\bar \\sigma^{\\nu \\dot C B} -\n  \\sigma^{\\nu}_{A \\dot C} \\bar \\sigma^{\\mu \\dot C B}), \\; \\;  \n \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot B} \\equiv \\frac{1}{4}\n  (\\bar \\sigma^{\\mu \\dot A C} \\sigma^{\\nu}_{C \\dot B} -\n  \\bar \\sigma^{\\nu \\dot A C} \\sigma^{\\mu}_{C \\dot B}).\n\\]\nThe main properties of $\\sigma^{\\mu \\nu}$ are as follows:\n\\begin{eqnarray*}\n &&  \\sigma^{0i}=\\frac{1}{2}\\sigma^i, \\, \\, \n \\sigma^{ik}=-\\frac{i}{2} \\varepsilon^{ikl} \\sigma^l, \\quad\n \\bar \\sigma^{0i}=-\\frac{1}{2}\\sigma^i, \\, \\, \n \\bar \\sigma^{ik} = \\sigma^{ik}, \\\\\n&& \\sigma^{\\mu \\nu} = - \\sigma^{\\nu \\mu}, \\quad\n \\bar \\sigma^{\\mu \\nu} = - \\bar \\sigma^{\\nu \\mu}, \\\\ \n&& (\\sigma^{\\mu} \\bar \\sigma^{\\nu})_A{}^B =\n  g^{\\mu \\nu} \\delta_A{}^B + 2 \\sigma^{\\mu\\nu B}_A, \\quad\n (\\bar \\sigma^{\\mu} \\sigma^{\\nu})^{\\dot A}{}_{\\dot B} = \ng^{\\mu\\nu} \\delta^{\\dot A}{}_{\\dot B}+2 \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot B}, \n\\\\\n&& \\sigma^{\\mu\\nu K}_A \\varepsilon_{KB} = \n   \\sigma^{\\mu\\nu K}_B \\varepsilon_{KA}, \\quad\n \\bar \\sigma^{\\mu\\nu \\dot A}{}_{\\dot K} \\varepsilon^{\\dot K \\dot B} = \n  \\bar \\sigma^{\\mu\\nu \\dot B}{}_{\\dot K} \\varepsilon^{\\dot K \\dot A}, \\\\\n&& \\varepsilon^{\\mu \\nu \\lambda \\rho} \\sigma_{\\lambda \\rho} = \n     -2i \\sigma^{\\mu \\nu}, \\quad \n \\varepsilon^{\\mu \\nu \\lambda \\rho} \\bar \\sigma_{\\lambda \\rho} =\n     2i \\bar \\sigma^{\\mu \\nu}. \n\\end{eqnarray*}\n\n\\subsection{\\it Traces of $\\sigma^{\\mu}$ Matrices} \n\\begin{eqnarray*}\n&&{\\rm Tr}\\sigma^{\\mu}={\\rm Tr}\\bar \\sigma^{\\mu}\\,=\\,2(0)\\quad {\\rm for} \\quad \n \\mu = 0 \\, (1,2,3), \\\\\n&&{\\rm Tr}\\sigma^{\\mu \\nu}={\\rm Tr}\\bar \\sigma^{\\mu \\nu} = 0, \\quad\n{\\rm Tr}(\\sigma^{\\mu} \\bar \\sigma^{\\nu})\n  = {\\rm Tr}(\\bar \\sigma^{\\mu} \\sigma^{\\nu}) = 2 g^{\\mu \\nu}, \\\\ \n&&{\\rm Tr}(\\sigma^{\\mu} \\bar \\sigma^{\\nu} \\sigma^{\\lambda} \\bar \\sigma^{\\rho})\n =  2(g^{\\mu \\nu}g^{\\lambda \\rho} + g^{\\mu \\rho}g^{\\nu \\lambda} \n - g^{\\mu \\lambda}g^{\\nu \\rho} - i \\varepsilon^{\\mu \\nu \\lambda \\rho}), \\\\\n&&{\\rm Tr}(\\sigma^{\\mu \\nu} \\sigma^{\\lambda \\rho}) =\n {\\rm Tr}(\\bar \\sigma^{\\mu \\nu}\\bar \\sigma^{\\lambda \\rho}) = \n \\frac{1}{2} (g^{\\mu \\rho}g^{\\nu \\lambda} - g^{\\mu \\lambda}g^{\\nu \\rho}\n   - i \\varepsilon^{\\mu \\nu \\lambda \\rho}).\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities for $\\sigma^{\\mu}$ Matrices} \n\nThe Fiertz identities for $\\sigma^{\\mu}$ equal:\n\\begin{eqnarray}\n\\sigma^{\\mu}_{A \\dot A} \\bar \\sigma_{\\mu}^{\\dot B B} = \n 2 \\delta_A{}^B \\delta_{\\dot A}{}^{\\dot B}, \\; \\;\n\\sigma^{\\mu}_{A \\dot A} \\sigma_{\\mu B \\dot B} = \n 2 \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B}, \\; \\;\n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma_{\\mu}^{\\dot B B} = \n 2 \\varepsilon^{\\dot A \\dot B} \\varepsilon_{AB}.\n\\label{p11} \n\\end{eqnarray}\nFrom the relations (\\ref{p11}) one gets:\n\\begin{eqnarray*}\n\\sigma^{\\mu}_{A \\dot A} \\bar \\sigma^{\\nu \\dot B B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\delta_A{}^B \\delta_{\\dot A}{}^{\\dot B} \n - \\delta_A{}^B \\bar \\sigma^{\\mu \\nu \\dot B}{}_{\\dot A}  \n + \\sigma^{\\mu \\nu B}_A \\delta_{\\dot A}{}^{\\dot B} \n + 2 \\sigma^{\\nu \\lambda B}_A \\bar \\sigma^{\\mu \\lambda \\dot B}{}_{\\dot A}, \\\\\n \\sigma^{\\mu}_{A \\dot A} \\sigma^{\\nu}_{B \\dot B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\varepsilon_{AB} \\varepsilon_{\\dot A \\dot B} \n + \\varepsilon_{AB} (\\varepsilon_{\\dot A \\dot C} \n     \\bar \\sigma^{\\mu \\nu \\dot C}{}_{\\dot B}) + \n  (\\sigma^{\\mu \\nu C}_A \\varepsilon_{CB}) \\varepsilon_{\\dot A \\dot B} \\\\\n && - 2 (\\sigma^{\\mu \\lambda C}_A \\varepsilon_{CB}) \n  (\\varepsilon_{\\dot A \\dot C} \\bar \\sigma^{\\nu \\lambda \\dot C}{}_{\\dot B}), \\\\\n \\bar \\sigma^{\\mu \\dot A A} \\bar \\sigma^{\\nu \\dot B B} &=& \n \\frac{1}{2} g^{\\mu \\nu} \\varepsilon^{AB} \\varepsilon^{\\dot A \\dot B} \n + (\\varepsilon^{AC} \\sigma^{\\mu \\nu B}_C) \\varepsilon^{\\dot A \\dot B} \n +  \\varepsilon^{AB}\n  (\\bar \\sigma^{\\mu \\nu \\dot A}{}_{\\dot C} \\varepsilon^{\\dot C \\dot B}) \\\\  \n && - 2 (\\varepsilon^{AC} \\sigma^{\\mu \\lambda C}_B) \n  (\\bar \\sigma^{\\nu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}), \n\\end{eqnarray*}\n\\[ \n (\\varepsilon^{AC} \\sigma^{\\mu \\lambda C}_B)  \n  (\\bar \\sigma^{\\nu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}) = \n (\\varepsilon^{AC} \\sigma^{\\nu \\lambda C}_B) \n  (\\bar \\sigma^{\\mu \\lambda \\dot A}{}_{\\dot C}\\varepsilon^{\\dot C \\dot B}).\n\\] \n\n\n\n\\section{\\bf DIRAC MATRICES}\\label{dirac}\n\n\\subsection{\\it Main Properties}\n\n\\noindent The main properties of the Dirac $\\gamma$-matrices are as follows \n\\cite{bogol,land,itzu,okun,velt}:\n\\begin{eqnarray}\n&& \\gamma^{\\mu} \\gamma^{\\nu} + \\gamma^{\\nu} \\gamma^{\\mu} = 2 g^{\\mu \\nu}, \n  \\label{d1} \\\\\n && (\\gamma^0)^2 = I, \\quad (\\gamma^i)^2 = -I, \\quad, (\\gamma^0)^{\\dagger} = \n \\gamma^0, \\quad (\\gamma^i)^{\\dagger} = - \\gamma^i. \\label{d2} \\\\\n &&  \\sigma^{\\mu \\nu} \\; \\equiv \\; \\frac{1}{2} \n(\\gamma^{\\mu} \\gamma^{\\nu} - \\gamma^{\\nu} \\gamma^{\\mu}), \\quad\n\\sigma^{\\mu \\nu} \\; = \\; - \\sigma^{\\nu \\mu} \\label{d13}\n\\end{eqnarray}\nThe definition of the $\\gamma^5$ matrix and its properties are as follows: \n\\begin{eqnarray}\n&& \\gamma^5 \\equiv i\\gamma^0 \\gamma^1 \\gamma^2 \\gamma^3 \\, = \n\\, -\\frac{i}{4!} \\varepsilon_{\\alpha\\beta\\mu\\nu} \\gamma^{\\alpha}\n\\gamma^{\\beta}\\gamma^{\\mu}\\gamma^{\\nu}. \\label{d3} \\\\\n&&  (\\gamma^5)^2 \\, = \\, I, \\quad  (\\gamma^5)^{\\dagger} \\, = \\, \\gamma^5, \\quad\n\\gamma^5 \\gamma^{\\mu} + \\gamma^{\\mu} \\gamma^5 = \\{\\gamma^{\\mu}, \\, \\gamma^5 \\}\n= 0 \\nonumber\n\\end{eqnarray}\nNote, that\n\\begin{eqnarray*}\n  \\gamma^{0} = \\gamma_{0}, \\; \\gamma^{i} = -\\gamma_{i}, \\;\n  (\\gamma^{\\mu})^{\\dagger} =  \\gamma^0 \\gamma^{\\mu} \\gamma^{0} = \\gamma_{\\mu} \n\\end{eqnarray*}\n  \n\\noindent The {\\bf \\it Dirac conjugation} of any $4\\times4$--matrix $A$ is\ndefined as follows: \n\\begin{eqnarray}\n \\bar A \\equiv \\gamma^0 A^{\\dagger} \\gamma^0. \\label{d4}\n\\end{eqnarray}\nFrom (\\ref{d4}) one gets:\n\\begin{eqnarray*} \n && \\overline{\\gamma^{\\mu}} = \\gamma^{\\mu}, \\quad \n\\overline{ \\gamma^5} = -\\gamma^5, \\quad\n \\overline{\\gamma^{\\alpha} \\gamma^{\\beta} \\cdots \\gamma^{\\lambda}} =\n \\gamma^{\\lambda} \\cdots \\gamma^{\\beta} \\gamma^{\\alpha}, \\\\\n && \\overline{\\gamma^{\\alpha} \\gamma^{\\beta} \\cdots \\gamma^{\\sf m } \\gamma^5\n  \\cdots \\gamma^{\\lambda}} =\n \\gamma^{\\lambda}\\cdots (-\\gamma^5) \\gamma^{\\sf m}\\cdots \\gamma^{\\beta}\n \\gamma^{\\alpha} = \\gamma^{\\lambda} \\cdots \\gamma^{\\sf m} \\gamma^5 \\cdots\n \\gamma^{\\beta} \\gamma^{\\alpha}.\n\\end{eqnarray*}\nIn this Section for the string of the $\\gamma$--matrices we shall use the\nspecial notation:\n\\begin{eqnarray}\n S = S^n \\equiv \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\cdots \n       \\gamma^{\\alpha_n}, \\quad \n S_R = S^n_R \\equiv \\gamma^{\\alpha_n} \\cdots \\gamma^{\\alpha_2}  \n       \\gamma^{\\alpha_1}.  \\label{dd1}\n\\end{eqnarray}\nOdd-- and even--numbered string of $\\gamma$--matrices will be denoted as \nfollows:\n\\begin{eqnarray}\n S^{odd} \\equiv \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\cdots \n       \\gamma^{\\alpha_{2k+1}}, \\quad \n S^{even}  \\equiv \\gamma^{\\alpha_1}  \\gamma^{\\alpha_2}  \\cdots\n       \\gamma^{\\alpha_{2k}}.  \\label{dd2}\n\\end{eqnarray}\n\n\\subsection{\\it Representations of the Dirac Matrices} \\label{ref_gamma}\n\nThe non--singular transformation $\\gamma \\to U \\gamma U^{\\dagger}$ connects the\ndifferent representations of the $\\gamma$--matrices (Pauli lemma). Here we  \npresent three representations of the Dirac matrices.\n\n\\noindent $\\bullet$ {\\bf Dirac (standard)} representation\n\\begin{displaymath}\n \\gamma^0_D = \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right) ,\n\\quad\n \\gamma^i_D = \\left( \\begin{array}{ccc} 0 & \\sigma^i \\\\ -\\sigma^i & 0 \n\\end{array} \\right) , \\quad\n \\gamma^5_D = \\left( \\begin{array}{ccc} 0 & 1 \\\\ 1 & 0 \\end{array} \\right).\n\\end{displaymath}\n\n\\noindent $\\bullet$ {\\bf Chiral (spinorial)} representation \n\\begin{eqnarray*}\n  &&  \\gamma^{\\mu}_C \\, = \\,  U_C \\, \\gamma^{\\mu}_D U_C^{\\dagger}, \\qquad \n U_C \\, = \\, \\frac{1}{\\sqrt{2}}(1 - \\gamma^5_D \\gamma^0_D) \\, = \\,\n  \\frac{1}{\\sqrt{2}}\n \\left( \\begin{array}{ccc} 1 & 1 \\\\ -1 & 1 \\end{array} \\right), \\\\\n &&  \\gamma^0 = \\left( \\begin{array}{ccc} 0 & -1 \\\\ -1 & 0 \\end{array} \\right) ,\n\\quad\n \\gamma^i = \\left( \\begin{array}{ccc} 0 & \\sigma^i \\\\ -\\sigma^i & 0 \n\\end{array} \\right) , \\quad\n \\gamma^5 = \\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & -1 \\end{array} \\right), \\\\\n&& \\gamma_{R}=\\frac{1}{2} \\left(1 \\, + \\, \\gamma^5 \\right)  = \\, \n\\left( \\begin{array}{ccc} 1 & 0 \\\\ 0 & 0 \\end{array} \\right), \\quad \n\\gamma_{L}=\\frac{1}{2} \\left( 1 \\, - \\, \\gamma^5 \\right) \\, = \\, \n\\left( \\begin{array}{ccc} 0 & 0 \\\\ 0 & 1 \\end{array} \\right).\n\\end{eqnarray*}\n\n\n\\noindent $\\bullet$ {\\bf Majorana} representation \n\\begin{eqnarray*}\n&& \\gamma^{\\mu}_M \\, = \\,  U_M \\, \\gamma^{\\mu}_D U_M^{\\dagger}, \\qquad \n U_M \\, = \\frac{1}{\\sqrt{2}}\n\\left( \\begin{array}{ccc} 1 & \\sigma_2 \\\\ \\sigma_2 & -1 \\end{array} \\right), \n \\\\\n&& \\gamma^0 = \\left( \\begin{array}{ccc} 0 & \\sigma_2 \\\\ \\sigma_2 & 0 \n\\end{array} \\right), \\quad\n \\gamma^1 = \\left( \\begin{array}{ccc} i\\sigma_3 & 0 \\\\ 0 & i\\sigma_3 \n\\end{array} \\right), \\quad\n \\gamma^2 = \\left( \\begin{array}{ccc} 0 & -\\sigma_2 \\\\ \\sigma_2 & 0 \n\\end{array} \\right), \\quad \\\\\n&& \\gamma^3 = \\left( \\begin{array}{ccc} -i\\sigma_1 & 0 \\\\ 0 & -i\\sigma_1 \n\\end{array} \\right), \\qquad\n\\gamma^5 = \\left( \\begin{array}{ccc} \\sigma_2 & 0 \\\\ 0 & -\\sigma_2 \n \\end{array} \\right).\n\\end{eqnarray*}\n\n\\subsection{\\it Expansion of $4 \\times 4$ Matrices}\n\nThe following 16 matrices $\\Gamma_A \\quad (A=1, \\ldots , 16)$\n\\begin{eqnarray}\nI, \\quad \\gamma^5, \\quad \\gamma^{\\mu}, \\quad \\gamma^5 \\gamma^{\\mu}, \\quad\n\\sigma^{\\mu \\nu}  \\label{d5} \n\\end{eqnarray}\nare the full set of $4 \\times 4$--matrices. \\\\\nThe main properties of $\\Gamma_A$ are as follows:\n\\begin{eqnarray}\n {\\rm Tr} I = 4, \\quad {\\rm Tr} \\gamma^5 = {\\rm Tr} \\gamma^{\\mu} = \n {\\rm Tr} \\gamma^5 \\gamma^{\\mu} = {\\rm Tr} \\sigma^{\\mu \\nu} = 0 \\label{d7}\n\\end{eqnarray}\nAny $4 \\times4$--matrix $A$ can be expanded over set of the\n$\\Gamma_A$-matrices:\n\\begin{eqnarray}\n A = a_0 I + a_5 \\gamma^5 + v_{\\mu} \\gamma^{\\mu}+ a_{\\mu} \\gamma^5 \\gamma^{\\mu}\n  + T_{\\mu \\nu} \\sigma^{\\mu \\nu}, \\label{d8}\n\\end{eqnarray}\nwhere the coefficients could be found from the following relations:\n\\begin{eqnarray*}\n a_0 = \\frac{1}{4} {\\rm Tr} A, \\quad \n a_5 = \\frac{1}{4} {\\rm Tr} (\\gamma^5 A), \\quad \n v^{\\mu} = \\frac{1}{4} {\\rm Tr}(\\gamma^{\\mu} A), \\\\\n a^{\\mu} = -\\frac{1}{4} {\\rm Tr}( \\gamma^5 \\gamma^{\\mu} A), \\quad\n T^{\\mu \\nu} = -T^{\\nu \\mu} = -\\frac{1}{8} {\\rm Tr}(\\sigma^{\\mu \\nu} A).\n\\end{eqnarray*}\nFor the expansion of a matrix $A$ one can use another set of $\\Gamma'_A$ \n($\\Gamma'_A = X, \\; Y, \\; U^{\\mu}, \\; V^{\\mu}, \\; \\sigma^{\\mu \\nu}$):\n\\begin{eqnarray*}\n && X = I + \\gamma^5, \\quad Y = I - \\gamma^5, \\quad \n U^{\\mu} = (I + \\gamma^5)\\gamma^{\\mu},  \\quad \n V^{\\mu} = (I - \\gamma^5)\\gamma^{\\mu}, \\\\\n &&X^2 = 2X, \\quad Y^2 = 2Y.\n\\end{eqnarray*}\nThese matrices have the following properties: \n\\begin{eqnarray*} \n && U^2 = V^2 = XY = YX = X U^{\\mu} =  Y V^{\\mu} = 0, \\\\\n && {\\rm Tr} X = {\\rm Tr} Y = 4, \\; {\\rm Tr} U^{\\mu} = {\\rm Tr} U^{\\mu} =\n {\\rm Tr} \\sigma^{\\mu \\nu} = 0.\n\\end{eqnarray*}\nThe expansion of any $4 \\times4$--matrix $A$ over set of $\\Gamma'$-matrices has\nthe form:\n\\begin{eqnarray*}\n A = a_x X + a_y Y + b_{\\mu} U^{\\mu} + c_{\\mu} V^{\\mu}\n  + T_{\\mu \\nu} \\sigma^{\\mu \\nu}, \n\\end{eqnarray*}\nwhere \n\\begin{eqnarray*}\n && a_x = \\frac{1}{8} {\\rm Tr} (XA), \\quad \n a_y = \\frac{1}{8} {\\rm Tr} (YA), \\quad \n b^{\\mu} = \\frac{1}{8} {\\rm Tr}(V^{\\mu} A), \\quad \n c^{\\mu} = \\frac{1}{8} {\\rm Tr}(U^{\\mu} A), \\\\\\\n && T^{\\mu \\nu} = -\\frac{1}{8} {\\rm Tr}(\\sigma^{\\mu \\nu} A).\n\\end{eqnarray*}\n\n\\subsection{\\it Products of the Dirac Matrices}\n\\begin{eqnarray*}\n  \\gamma^{\\mu} \\gamma^{\\nu} &=& g^{\\mu \\nu} + \\sigma^{\\mu \\nu}, \\quad\n \\sigma^{\\mu \\nu} = \\gamma^{\\mu} \\gamma^{\\nu} - g^{\\mu \\nu} = \n - \\gamma^{\\nu} \\gamma^{\\mu} + g^{\\mu \\nu}, \\\\\n  \\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu} &=& g^{\\mu \\nu} \\gamma^5 \n + \\frac{i}{2} \\varepsilon^{\\mu \\nu \\alpha \\beta} \\sigma_{\\alpha \\beta}, \\quad \n  \\gamma^5 \\sigma^{\\mu \\nu} = \n  + \\frac{i}{2} \\varepsilon^{\\mu \\nu \\alpha \\beta} \\sigma_{\\alpha \\beta}, \\\\ \n \\gamma^{\\lambda} \\sigma^{\\mu \\nu} &=& (g^{\\mu \\lambda} \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^{\\mu}) - i\\varepsilon^{\\lambda \\mu \\nu \\alpha}\n   \\gamma^5 \\gamma_{\\alpha}, \\\\\n \\sigma^{\\mu \\nu} \\gamma^{\\lambda} &=& -(g^{\\mu \\lambda} \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^{\\mu}) - i\\varepsilon^{\\lambda \\mu \\nu \\alpha}\n   \\gamma^5 \\gamma_{\\alpha}, \\\\\n \\gamma^5 \\gamma^{\\lambda} \\sigma^{\\mu \\nu} &=& \n  (g^{\\mu \\lambda} \\gamma^5 \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^5 \\gamma^{\\mu}) \n   - i\\varepsilon^{\\lambda \\mu \\nu \\alpha} \\gamma_{\\alpha}, \\\\\n  \\sigma^{\\mu \\nu} \\gamma^5 \\gamma^{\\lambda} &=&\n  -(g^{\\mu \\lambda} \\gamma^5 \\gamma^{\\nu} \n  -g^{\\nu \\lambda} \\gamma^5 \\gamma^{\\mu}) \n   - i\\varepsilon^{\\lambda \\mu \\nu \\alpha} \\gamma_{\\alpha}, \\\\\n  \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} &=& \n  g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu} \n  - i\\varepsilon^{\\alpha \\beta \\mu \\nu} \\gamma^5 \\\\\n &&+ ( g^{\\alpha \\nu} g^{\\beta \\lambda} g^{\\mu \\sigma} \n   - g^{\\alpha \\mu} g^{\\beta \\lambda} g^{\\nu \\sigma}\n   - g^{\\beta \\nu} g^{\\alpha \\lambda} g^{\\mu \\sigma}\n + g^{\\beta \\mu} g^{\\alpha \\lambda} g^{\\nu \\sigma}) \\sigma_{\\lambda \\sigma}, \\\\\n \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} & + &\\sigma^{\\mu \\nu} \n\\sigma^{\\alpha \\beta}= \n 2(g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu} \n  - i\\varepsilon^{\\alpha \\beta \\mu \\nu} \\gamma^5).\n\\end{eqnarray*}\n\nThe totally antisymmetric tensor $\\gamma^{\\mu \\nu \\lambda}$ is defined as \nfollows:\n\\begin{eqnarray*}\n&&  \\gamma^{\\mu \\nu \\lambda} \\equiv \\frac{1}{6} \n (\\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\lambda} \n +\\gamma^{\\nu} \\gamma^{\\lambda} \\gamma^{\\mu} \n +\\gamma^{\\lambda} \\gamma^{\\mu} \\gamma^{\\nu} \n -\\gamma^{\\nu} \\gamma^{\\mu} \\gamma^{\\lambda} \n -\\gamma^{\\lambda} \\gamma^{\\nu} \\gamma^{\\mu} \n -\\gamma^{\\mu} \\gamma^{\\lambda} \\gamma^{\\nu}), \\\\\n&& \\gamma^{\\mu} \\gamma^{\\nu} \\gamma^{\\lambda} = \\gamma^{\\mu \\nu \\lambda}\n + g^{\\mu \\nu} \\gamma^{\\lambda} - g^{\\mu \\lambda} \\gamma^{\\nu} \n + g^{\\nu \\lambda} \\gamma^{\\mu}, \\\\\n&& \\gamma^{\\mu \\nu \\lambda} = -i \\varepsilon^{\\mu \\nu \\lambda \\alpha} \n  \\gamma^5 \\gamma_{\\alpha}, \\quad \n \\gamma^5 \\gamma^{\\alpha}=\\frac{i}{6} \\varepsilon^{\\alpha \\mu \\nu \\lambda} \n \\gamma_{\\mu \\nu \\lambda}.\n\\end{eqnarray*} \n\nThe products of the type $\\sum_{A=1}^{16} \\Gamma^A \\Gamma^B \\Gamma^A$ are\npresented in the Table~\\ref{dirac}.1. \n\n\\noindent \n\\underline{ {\\bf Table~\\ref{dirac}.1.}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\\hline\n $\\Gamma^B$ & $\\gamma^5 \\Gamma^B \\gamma^5$ &\n $\\gamma^{\\nu} \\Gamma^B \\gamma^{\\nu}$ & \n $\\gamma^5 \\gamma^{\\nu} \\Gamma^B \\gamma^5 \\gamma^{\\nu}$ & \n $\\sigma^{\\mu \\nu} \\Gamma^B \\sigma^{\\mu \\nu}$ \\\\ \\hline\n $I$ & $I$ & $4$ & $-4$ & $-1$ \\\\ \\hline\n $\\gamma^5$ & $\\gamma^5$ & $-4\\gamma^5$ & \n $-4\\gamma^5$ & $-12\\gamma^5$ \\\\ \\hline\n $\\gamma^{\\alpha}$ & $-\\gamma^{\\alpha}$ & \n  $-2\\gamma^{\\alpha}$ & $-2\\gamma^{\\alpha}$ & $0$ \\\\ \\hline\n $\\gamma^5 \\gamma^{\\alpha}$ &\n  $-\\gamma^5 \\gamma^{\\alpha}$ & $2\\gamma^5 \\gamma^{\\alpha}$ &\n  $2\\gamma^5 \\gamma^{\\alpha}$ & $0$ \\\\ \\hline\n $\\sigma^{\\alpha \\beta}$ & $\\sigma^{\\alpha \\beta}$ &\n $0$ & $0$& $4\\sigma^{\\alpha \\beta}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\noindent So--called Chisholm identities are given by \\cite{velt}:\n\\begin{eqnarray}\n &&\\gamma^{\\mu} S^{odd} \\gamma_{\\mu} = -2 S^{odd}_R,  \\label{dd3} \\\\\n &&\\gamma^{\\mu} S^{even} \\gamma_{\\mu} = \n \\gamma^{\\mu} \\gamma^{\\lambda} S'^{odd} \\gamma_{\\mu} = \n 2 \\gamma^{\\lambda} S'^{odd}_R + 2 S'^{odd}\\gamma^{\\lambda}, \\label{dd4}\n\\end{eqnarray}\nwhere in the last relation $ S^{even} = \\gamma^{\\lambda} S'^{odd}$. \\\\\nUsing the relations (\\ref{dd3}) and (\\ref{dd4}), one gets:\n\\begin{eqnarray*}\n \\gamma^{\\mu} S^{even} \\gamma_{\\mu} &=& \n {\\rm Tr} (S^{even}) I - {\\rm Tr} (\\gamma^5S^{even}) \\gamma^5,  \\\\\n \\hat p S^{even} \\hat p &=& - p^2 S^{even}_R + \\frac{1}{2}\n {\\rm Tr}(\\hat p \\gamma^{\\alpha} S^{even}_R) \\gamma_{\\alpha} \\hat p, \\\\\n \\hat p S^{even} \\hat p &=& - \\hat p S^{even}_R \\hat p, \n \\quad {\\rm for} \\quad p^2=0, \\\\ \n \\gamma^{\\mu} S^{odd} \\gamma_{\\mu} &=& -\\frac{1}{2}\n {\\rm Tr} (\\gamma^{\\alpha}S^{odd}) \\gamma_{\\alpha} + \\frac{1}{2}\n  {\\rm Tr} (\\gamma^5\\gamma^{\\alpha}S^{odd}) \\gamma_{\\alpha} \\gamma^5,  \\\\\n \\hat p S^{odd} \\hat p &=& - p^2 S^{odd}_R + \\frac{1}{2}\n {\\rm Tr}(\\hat p S^{odd}_R) \\hat p + \\frac{1}{2} \n  {\\rm Tr}(\\gamma^5 \\hat p S^{odd}_R) \\hat p \\gamma^5, \\\\\n S^{odd} &=& \\frac{1}{4} {\\rm Tr}(\\gamma^{\\alpha} S^{odd}) \\gamma_{\\alpha}\n + \\frac{1}{4} {\\rm Tr}(\\gamma^5 \\gamma^{\\alpha} S^{odd}) \n    \\gamma_{\\alpha}\\gamma^5,\\\\\n S^{odd} &+& S^{odd}_R = \\frac{1}{2} {\\rm Tr}(\\gamma^{\\alpha} S^{odd})\n    \\gamma_{\\alpha}.\n\\end{eqnarray*}\nUsing (\\ref{dd3}) and (\\ref{dd4}), one can write also the well known relations \nfor $S^1$, $S^2$, $S^3$, $S^4$:\n\\begin{eqnarray*}\n\\gamma^{\\mu} \\gamma^{\\alpha} \\gamma_{\\mu} &=& -2 \\gamma^{\\alpha}, \\quad\n\\gamma^{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta} \\gamma^{\\delta} \\gamma_{\\mu} =\n -2 \\gamma^{\\delta} \\gamma^{\\beta} \\gamma^{\\alpha}, \\\\\n \\gamma^{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta} \\gamma_{\\mu} \n   &=& 4 g^{\\alpha \\beta}, \\\\\n \\gamma^{\\mu} \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3} \n  \\gamma^{\\alpha_4} \\gamma_{\\mu} &=& \n 2( \\gamma^{\\alpha_4} \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3}\n +\\gamma^{\\alpha_3} \\gamma^{\\alpha_2} \\gamma^{\\alpha_1} \\gamma^{\\alpha_4})  \\\\\n &=& 2( \\gamma^{\\alpha_1} \\gamma^{\\alpha_4} \\gamma^{\\alpha_3} \\gamma^{\\alpha_2}\n +\\gamma^{\\alpha_2} \\gamma^{\\alpha_3} \\gamma^{\\alpha_4} \\gamma^{\\alpha_1}), \\\\\n \\gamma^{\\mu} \\sigma^{\\alpha \\beta} \\gamma_{\\mu} &=& 0, \\quad\n \\gamma^{\\mu} \\sigma^{\\alpha \\beta} \\gamma^{\\delta} \\gamma_{\\mu} = \n 2 \\gamma^{\\delta} \\sigma^{\\alpha \\beta}, \\quad\n \\gamma^{\\mu} \\gamma^{\\delta} \\sigma^{\\alpha \\beta} \\gamma_{\\mu} = \n 2 \\sigma^{\\alpha \\beta} \\gamma^{\\delta}.\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identities}\n\nFiertz identities for $\\gamma$--matrices could be obtained from\nthe basic formula: \n\\begin{eqnarray}\n\\delta_{ij} \\delta_{kl} = \n\\frac{1}{4} [ \\delta_{il} \\delta_{kj}\n + \\gamma^5_{il} \\gamma^5_{kj} + \\gamma^{\\mu}_{il} \\gamma_{\\mu\\;kj}  \n - (\\gamma^5 \\gamma^{\\mu})_{il} (\\gamma^5 \\gamma_{\\mu})_{kj}  \n - \\frac{1}{2} \\sigma^{\\mu \\nu}_{il} (\\sigma_{\\mu \\nu})_{kj}]. \\label{d11}\n\\end{eqnarray}\nUsing (\\ref{d11}) one can obtain the well known relations:\n\\begin{eqnarray}\n \\Gamma^M_{ij} \\Gamma^M_{kl} = \n \\sum^{16}_{N=1} C_{MN} \\Gamma^N_{il} \\Gamma^N_{kj}, \\label{d12}\n\\end{eqnarray}\nThe coefficients $C_{MN}$ are presented in Table~\\ref{dirac}.2, where we  use \nthe traditional notations:\n\\[\nS=I, \\quad P=\\gamma^5, \\quad V = \\gamma^{\\mu}, \\quad \nA = \\gamma^5 \\gamma^{\\mu}, \\quad T = \\sigma^{\\mu \\nu}.\n\\]\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{dirac}.2.}}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n   & $N = S$ & $V$ & $T$ & $A$ & $P$ \\\\ \\hline\n $M=I$ & $\\frac{1}{4}$ & $\\frac{1}{4}$ & $-\\frac{1}{8}$ & $-\\frac{1}{4}$ & \n $\\frac{1}{4}$ \\\\ \\hline\n $V$ &  $1$ & $-\\frac{1}{2}$ & $0$ & $-\\frac{1}{2}$ & $-1$ \\\\ \\hline\n $T$ & $-3$ & $0$  & $-\\frac{1}{2}$ & $0$ & $-3$ \\\\ \\hline\n $A$ & $-1$ & $-\\frac{1}{2}$ & $0$ & $-\\frac{1}{2}$ & $1$ \\\\ \\hline\n $P$ & $\\frac{1}{4}$ & $-\\frac{1}{4}$ & $-\\frac{1}{8}$ & $\\frac{1}{4}$ & \n $\\frac{1}{4}$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\nUsing relation (\\ref{d11}) one gets:\n\\begin{eqnarray*}\n&& (1 \\pm \\gamma^5)_{ij} \\delta_{kl} = \\\\\n&&\\frac{1}{8} [ 2(1 \\pm \\gamma^5)_{il} (1 \\pm \\gamma^5)_{kj}\n + 2((1 \\pm \\gamma^5)\\gamma^{\\mu})_{il} ((1\\mp \\gamma^5)\\gamma_{\\mu})_{kj} \n  - ((1 \\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& \\delta_{ij} (1 \\pm \\gamma^5)_{kl} = \\\\\n&& \\frac{1}{8} [ 2(1 \\pm \\gamma^5)_{il} (1 \\pm \\gamma^5)_{kj}\n + 2((1\\mp \\gamma^5)\\gamma^{\\mu})_{il} ((1 \\pm \\gamma^5)\\gamma_{\\mu})_{kj} \n  - ((1 \\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& (1 \\pm \\gamma^5)_{ij} (1 \\pm \\gamma^5)_{kl} = \n \\frac{1}{4} [ 2(1\\pm \\gamma^5)_{il} (1\\pm \\gamma^5)_{kj}\n  - ((1\\pm \\gamma^5)\\sigma_{\\mu \\nu})_{il} \\sigma^{\\mu \\nu}_{kj}], \\\\\n&& (1\\pm \\gamma^5)_{ij} (1\\mp \\gamma^5)_{kl} =  \\frac{1}{2} \n [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{il}   [(1\\mp \\gamma^5)\\gamma_{\\mu}]_{kj}, \\\\\n&& [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{ij} [(1\\pm \\gamma^5)\\gamma_{\\mu}]_{kl} = \n -[(1\\pm \\gamma^5)\\gamma^{\\mu}]_{il} [(1\\pm \\gamma^5)\\gamma_{\\mu}]_{kj}, \\\\ \n&& [(1\\pm \\gamma^5)\\gamma^{\\mu}]_{ij} [(1\\mp \\gamma^5)\\gamma_{\\mu}]_{kl} = \n 2 (1\\pm \\gamma^5)_{il} (1\\mp \\gamma^5)_{kj}, \\\\ \n && (\\gamma^{\\mu})_{ij} (\\gamma_{\\mu})_{kl} + \n  (\\gamma^5 \\gamma^{\\mu})_{ij} (\\gamma^5 \\gamma_{\\mu})_{kl} = \n - [(\\gamma^{\\mu})_{il} (\\gamma_{\\mu})_{kj} + \n  (\\gamma^5 \\gamma^{\\mu})_{il} (\\gamma^5 \\gamma_{\\mu})_{kj}].\n\\end{eqnarray*}\n \n\\subsection{\\it Traces of the $\\gamma$-matrices} \n\nThe trace of any odd--numbered string of $\\gamma$--matrices\n(including any number of $\\gamma^5$ matrices) and trace of the \n$\\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu}$ product are equal to zero:\n\\[\n {\\rm Tr} S^{odd} = {\\rm Tr} \\bigl( S^{odd}(\\cdots \\gamma^5 \\cdots) \\bigr) = \n {\\rm Tr} (\\gamma^5 \\gamma^{\\mu} \\gamma^{\\nu})  = 0.\n\\]\nIn this Subsection we use the following notation:\n\\[\nT^{\\mu_1 \\mu_2 ... \\mu_n} \\equiv \\frac{1}{4} {\\rm Tr}(\\gamma^{\\mu_1} \n\\gamma^{\\mu_2} \\, ... \\, \\gamma^{\\mu_n}). \n\\]\nThen\n\\begin{eqnarray*}\n&&T^{\\mu \\nu} = g^{\\mu \\nu}, \\qquad\nT^{\\alpha \\beta \\delta \\sigma} = g^{\\alpha \\beta} g^{\\delta \\sigma} + \ng^{\\alpha \\sigma} g^{\\beta \\delta} - g^{\\alpha \\delta} g^{\\beta \\sigma}, \\\\\n &&T^{\\alpha \\beta \\delta \\lambda \\rho \\sigma} = \n  g^{\\alpha \\beta} T^{\\delta \\lambda \\rho \\sigma} \n - g^{\\alpha \\delta} T^{\\beta \\lambda \\rho \\sigma} \n + g^{\\alpha \\lambda} T^{\\beta \\delta \\rho \\sigma}\n - g^{\\alpha \\rho} T^{\\beta \\delta \\lambda \\sigma} \n + g^{\\alpha \\sigma} T^{\\beta \\delta \\lambda \\rho}, \\\\\n&&{\\rm Tr}(\\gamma^5) = 0, \\quad \n  {\\rm Tr}(\\gamma^5\\gamma^{\\mu}\\gamma^{\\nu}) = 0, \\quad\n{\\rm Tr}(\\gamma^5\\gamma^{\\alpha}\\gamma^{\\beta}\\gamma^{\\delta}\\gamma^{\\lambda})\n= -4i\\varepsilon^{\\alpha\\beta\\delta\\lambda}, \\\\\n&&{\\rm Tr}(\\gamma^5 \\gamma^{\\alpha_1} \\gamma^{\\alpha_2} \\gamma^{\\alpha_3}\n \\gamma^{\\alpha_4} \\gamma^{\\alpha_5} \\gamma^{\\alpha_6}) = \n4i(g^{\\alpha_1 \\alpha_2}\\varepsilon^{\\alpha_3\\alpha_4\\alpha_5\\alpha_6} \n - g^{\\alpha_1 \\alpha_3}\\varepsilon^{\\alpha_2\\alpha_4\\alpha_5\\alpha_6} \\\\\n && + g^{\\alpha_2 \\alpha_3}\\varepsilon^{\\alpha_1\\alpha_4\\alpha_5\\alpha_6} \n  + g^{\\alpha_4 \\alpha_5}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_6} \n - g^{\\alpha_4 \\alpha_6}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_5} \n + g^{\\alpha_5 \\alpha_6}\\varepsilon^{\\alpha_1\\alpha_2\\alpha_3\\alpha_4}), \\\\\n&&{\\rm Tr} \\sigma^{\\alpha \\beta} \\sigma^{\\mu \\nu} = \n 4 (g^{\\alpha \\nu} g^{\\beta \\mu} - g^{\\alpha \\mu} g^{\\beta \\nu}).\n\\end{eqnarray*}\nUsing the relation (\\ref{d11}), one can rewrite the trace of the \nproduct of two $4 \\times 4$ matrices $A$ and $B$ as follows:\n\\begin{eqnarray*}\n 4 {\\rm Tr}(AB) &=& {\\rm Tr}(A) {\\rm Tr}(B) + {\\rm Tr}(\\gamma^5 A) \n {\\rm Tr}(\\gamma^5 B) + {\\rm Tr}( \\gamma^{\\mu} A) {\\rm Tr}(\\gamma_{\\mu} B) \\\\ \n &-&{\\rm Tr}( \\gamma^5 \\gamma^{\\mu} A) {\\rm Tr}(\\gamma^5 \\gamma_{\\mu} B)\n - \\frac{1}{2}{\\rm Tr}(\\sigma^{\\mu \\nu} A){\\rm Tr}(\\sigma_{\\mu \\nu} B).\n\\end{eqnarray*}\nThe additional equation can be obtained using the Chisholm identities \n(\\ref{dd3}) and (\\ref{dd4}):\n\\[\n {\\rm Tr}(A \\gamma^{\\mu} B) {\\rm Tr}(\\gamma_{\\mu} S^{odd}) = \n2\\Bigl [{\\rm Tr}(A S^{odd}B) + {\\rm Tr}(A S^{odd}_R B) \\Bigr ]. \n\\]\n\n\\subsection{\\it Dirac Matrices Algebra in $n$--dimensions}\n\nIn the framework of dimensional regularization one gets:\n\\begin{eqnarray*}\n && {\\rm Tr} \\; I \\; = f(n), \\;\\; f(4) = 4, \\quad g^{\\mu\\nu}g_{\\mu\\nu} = n, \\\\\n && \\gamma^{\\mu} \\gamma^{\\nu} \\, + \\,  \\gamma^{\\nu} \\gamma^{\\mu} \\, = \\,\n 2g^{\\mu \\nu}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha}\\gamma^{\\mu} \\, = \\, (2-n)\\gamma^{\\alpha}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha}\\gamma^{\\beta} \\gamma^{\\mu} \\, = \\, \n 4g^{\\alpha\\beta} + (n-4)\\gamma^{\\alpha}\\gamma^{\\beta}, \\\\\n&& \\gamma_{\\mu} \\gamma^{\\alpha} \\gamma^{\\beta}\\gamma^{\\delta}\\gamma^{\\mu} \\, =\n \\, -2\\gamma^{\\delta}\\gamma^{\\beta}\\gamma^{\\alpha} + \n (4-n) \\gamma^{\\alpha} \\gamma^{\\beta}\\gamma^{\\delta}.\n\\end{eqnarray*}\n\n\n\\section{\\bf GELL--MANN MATRICES}\\label{gellmann}\n\n\\subsection{\\it The main properties}\n\nThe Gell-Mann $3 \\times 3$ matrices $\\lambda_i (i=1, \\ldots ,8)$ are\ngenerators\nof the group $SU(3)$. Their properties were presented elsewhere \n \\cite{itzu,okun,gw,ditn,cvit}. \\\\\nUsually in QCD instead of $\\lambda_i$ one deals with matrices $t_i$:\n\\[\nt_i \\equiv \\frac{1}{2} \\lambda_i.\n\\]\nEight $\\lambda_i$ matrices equal: \n\\begin{displaymath}\n \\lambda_1 = \\left( \\begin{array}{ccc} 0 & 1 & 0 \\\\ 1 & 0 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_2 = \\left( \\begin{array}{ccc} 0 & -i & 0 \\\\ i & 0 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_3 = \\left( \\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & -1 & 0 \\\\ 0 & 0 & 0\n\\end{array} \\right) ,\n\\end{displaymath}\n\\begin{displaymath}\n \\lambda_4 = \\left( \\begin{array}{ccc} 0 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_5 = \\left( \\begin{array}{ccc} 0 & 0 & -i \\\\ 0 & 0 & 0 \\\\ i & 0 & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_6 = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & 1 \\\\ 0 & 1 & 0\n\\end{array} \\right) ,\n\\end{displaymath}\n\\begin{displaymath}\n \\lambda_7 = \\left( \\begin{array}{ccc} 0 & 0 & 0 \\\\ 0 & 0 & -i \\\\ 0 & i & 0\n\\end{array} \\right) ,\n\\quad\n \\lambda_8 = \\frac{1}{\\sqrt{3}}\n\\left(\\begin{array}{ccc} 1 & 0 & 0 \\\\ 0 & 1 & 0 \\\\ 0 & 0 & -2\n\\end{array} \\right).\n\\end{displaymath}\nThe main properties of $t_i$ (or $\\lambda_i$) are as follows:\n\\begin{eqnarray}\n &&t^{\\dagger}_i = t_i, \\quad  \\det t_i = 0, \\quad (i=1, \\ldots ,7), \\quad \n \\det t_8 = -\\frac{2}{\\sqrt{3}}, \\nonumber \\\\\n&& [ t^a , t^b] = i f^{abc} t^c, \\qquad \\{ t^a , t^b \\} = \n\\frac{1}{3}\\delta^{ab}+d^{abc}t^c, \\label{t1}\n\\end{eqnarray}\nwhere $d^{abc} (f^{abc})$ is totally symmetric (anti-symmetric) tensor. The\nnon-zero elements of $f^{abc}$ and $d^{abc}$ are equal to: \n\\begin{eqnarray*}\n&& f_{123} = 1, \\, f_{147}=-f_{156}=f_{246}=f_{257}=f_{345}=-f_{367}=\n\\frac{1}{2}, \\, f_{458}=f_{678}=\\frac{\\sqrt{3}}{2}, \\\\\n&& d_{146} = d_{157}=-d_{247}=d_{256}=d_{344}=d_{355}=-d_{366}=\n-d_{377} = \\frac{1}{2}, \\\\\n&& d_{118} = d_{228}=d_{338}=-d_{888} = \\frac{1}{\\sqrt{3}}, \\, \nd_{448} =d_{558}=d_{668}=d_{778}=-\\frac{1}{2\\sqrt{3}}.\n\\end{eqnarray*}\nThroughout this Section we use two additional notations: \n\\begin{eqnarray*}\n && h^{abc} = d^{abc} + i f^{abc}, \\quad \n h^{abc} = h^{bca} = h^{cab}, \\quad h^{aab} = 0, \\\\\n && S(a_1 a_2 \\ldots a_n) \\equiv t^{a_1} t^{a_2} \\ldots t^{a_n}, \\; \n S_R(a_1 a_2 \\ldots a_n) \\equiv t^{a_n} \\ldots t^{a_2} t^{a_1}.\n\\end{eqnarray*}\nThus, from (\\ref{t1}) one has:\n\\begin{eqnarray}\n t^a t^b = \\frac{1}{6}\\delta^{ab} + \\frac{1}{2}(d^{abk}+if^{abk})t^k \\; \n = \\frac{1}{6}\\delta^{ab} +  \\displaystyle {\\frac{1}{2}} h^{abk} t^k. \\label{t2}\n\\end{eqnarray}\n\n\\subsection{\\it Traces of the $t^a$--matrices} \n\nTrace of any string of $t^a$ matrices can be evaluated recursively using the \nrelation (\\ref{t2}):\n\\begin{eqnarray}\n{\\rm Tr} S(a_1 a_2 \\ldots a_n) = \\frac{1}{6} \\delta^{a_{n-1} a_n} \n {\\rm Tr}S(a_1 \\ldots a_{n-2}) +  \\displaystyle \\frac{1}{2}  h^{a_{n-1} a_n k} \\, \n {\\rm Tr} S(a_1 \\ldots a_{n-2} k) \\label{t3} \n\\end{eqnarray}\nUsing (\\ref{t1}) and (\\ref{t3}) one gets: \n\\begin{eqnarray*}\n&&{\\rm Tr}(t^a) = 0, \\quad {\\rm Tr}(t^at^b) = \\frac{1}{2}\\delta^{ab}, \\quad \n{\\rm Tr}(t^at^bt^c) = \\frac{1}{4}(d^{abc}+if^{abc}) = \\frac{1}{4}h^{abc}, \\\\\n&&{\\rm Tr}(t^at^bt^ct^d) = \\frac{1}{12}\\delta^{ab}\\delta^{cd} + \n\\frac{1}{8} h^{abn} h^{ncd}, \\\\ \n&&{\\rm Tr}(t^at^bt^ct^d t^e) = \\frac{1}{24} h^{abc}\\delta^{de} + \n\\frac{1}{24} \\delta^{ab} h^{cde} + \\frac{1}{16} h^{abn} h^{nck} h^{kde}.\n\\end{eqnarray*}\n\n\\subsection{\\it Fiertz Identity}\n\nThe Fiertz identity for $t^a$ has the form:\n\\begin{eqnarray}\n t^a_{ik} t^a_{jl} = \\frac{1}{2}(\\delta_{il}\\delta_{kj}  \n - \\frac{1}{3} \\delta_{ik}\\delta_{jl}). \\label{t5}\n\\end{eqnarray}\nAny $3 \\times 3$ matrix $A$ can be expanded over set $\\{I, t^a\\}$:\n\\[\nA=a_0I + a^i t^i, \\quad {\\rm where} \\quad a_0 = \\frac{1}{3} {\\rm Tr} A, \\quad\n a^i = 2 \\, {\\rm Tr} (t^i A).\n\\]\nDecomposition of the two $u_i$ and $\\bar u_i$ color spinors products into\ncolor--singlet and color--octet parts has the form:\n\\[u_i \\bar u_j = \\frac{\\delta_{ij}}{\\sqrt{3}} + \\sqrt{2}\\varepsilon^kt^k_{ij},\n\\quad \\varepsilon^k \\varepsilon^l = \\delta^{kl}.\n\\]\n\n\\subsection{\\it Products of the $t^a$--matrices}\n\nThe product of $n$ matrices $t^a$ could be written in the form\n$a_0 + a_i t^i$\nusing the following relations (see ({\\ref{t2})):\n\\begin{eqnarray}\nS(a_1 a_2 \\ldots a_n) = \n \\frac{1}{6} \\delta^{a_{n-1} a_n} S(a_1 a_2 \\ldots a_{n-2})\n+  \\displaystyle \\frac{1}{2} h^{a_{n-1} a_n k} S(a_1 a_2 \\ldots a_{n-2} k) \\label{t4}\n\\end{eqnarray}\nThus, the products of two, three, and four matrices equal: \n\\begin{eqnarray*}\n t^a t^b &=& \\frac{1}{6}\\delta^{ab} + \\frac{1}{2}(d^{abk}+if^{abk})t^k \\; \n = \\frac{1}{6}\\delta^{ab} + \\frac{1}{2} h^{abk} t^k, \\\\ \nt^at^bt^c &=& \\frac{1}{6}\\delta^{ab}t^c + \\frac{1}{12} h^{abc} \n +  \\frac{1}{4} h^{abk} h^{kcn} t^n, \\\\\nt^at^bt^ct^d &=& \\frac{1}{36}\\delta^{ab}\\delta^{cd} \n  + \\frac{1}{24} h^{abk} h^{kcd} + \\frac{1}{12} [ h^{abk} \\delta^{cd} \n+ \\delta^{ab} h^{cdk}] t^k\n \n  +  \\frac{1}{8} h^{abn} h^{cdk} h^{nkp} t^p.\n\\end{eqnarray*}\nThe products of the type $t^k S t^k$ have the form:\n\\begin{eqnarray*}\n && {\\sf t^k} S {\\sf t^k} = \\frac{1}{2} {\\rm Tr}(S) - \\frac{1}{6}S, \\\\\n&& t^k t^k = \\frac{4}{3}I, \\quad {\\sf t^k}t^a{\\sf t^k} = -\\frac{1}{6}t^a, \\quad \n{\\sf t^k}t^a t^b {\\sf t^k} = \\frac{1}{4}\\delta^{ab} -\n \\frac{1}{6}t^at^b, \\quad  \n{\\sf t^k}t^at^bt^c{\\sf t^k}  = \\frac{1}{8} h^{abc} -\\frac{1}{6}t^at^bt^c, \\\\\n&&{\\sf t^k}t^at^bt^ct^d{\\sf t^k} = -\\frac{1}{6}t^at^bt^ct^d +\n \\frac{1}{24}\\delta^{ab}\\delta^{cd} + \n \\frac{1}{16} h^{abn} h^{ncd}.\n\\end{eqnarray*}\nThe products of the type $S S^{\\Pi}$ (here $ S^{\\Pi}$ is denoted any\npermutation of the $t^{a_i}$--matrices) are given by\n\\begin{eqnarray*}\n&&t^{a_1}t^{a_2} \\ldots t^{a_n} \\; \\; t^{a_n} \\ldots t^{a_2}t^{a_1} = \n \\Bigl ( \\frac{4}{3} \\Bigr )^n, \\\\\n && t^a t^b t^a t^b = -\\frac{2}{9}I, \\; \\; t^a t^b t^b t^a = \\frac{16}{9}I.\n\\end{eqnarray*}\nThe products of the $S(abc) S^{\\Pi}(abc)$ and $S(abcd) S^{\\Pi}(abcd)$ are\npresented on the following tables (in these tables symbol $(abc)$ stands for\n$t^a t^b t^c$, etc). \n\n\\noindent \n\\underline{ {\\bf Table~\\ref{gellmann}.1.}} \n  The products of the $(abc)$ on the \n  $(abc)^{\\Pi}$. All products are contain the common factor\n $ \\displaystyle \\frac{1}{27}I$.\n\\begin{center} \n\\begin{tabular}{|c|c|c|c|c|c|}\\hline\n   $(abc)$ & $10$ & $(bac)$ &  $1$ & $(cab)$ & $-8$ \\\\ \\hline\n   $(acb)$ & $1$  & $(bca)$ & $-8$ & $(cba)$ & $64$ \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{gellmann}.2.}} \n The products of the $(abcd)$ on the $(abcd)^{\\Pi}$. All products are contain \nthe common factor $\\displaystyle \\frac{1}{81}I$.\n\\begin{center}\n\\begin{tabular}{|c|c||c|c|c|c|c|c|}\\hline\n$(abcd)$ & $-14$ & $(bacd)$ & $+31$ & $(cabd)$ & $-5$ & $(dabc)$ & $+40$ \n  \\\\ \\hline\n$(abdc)$ & $+31$ & $(badc)$ & $+\\frac{71}{2}$ & $(cadb)$ & $-\\frac{1}{2}$ & \n   $(dacb)$ & $+4$    \\\\ \\hline\n$(acbd)$ & $+31$ & $(bcad)$ & $-5$ & $(cbad)$ & $-\\frac{1}{2}$ & \n   $(dbac)$ & $+4$   \\\\ \\hline\n$(acdb)$ & $-5$ & $(bcda)$ & $+40$ & $(cbda)$ & $+4$ & \n   $(dbca)$ & $-32$   \\\\ \\hline\n$(adbc)$ & $-5$ & $(bdac)$ & $-\\frac{1}{2}$ & $(cdab)$ & $+4$ & \n   $(dcab)$ & $-32$   \\\\ \\hline\n$(adcb)$ & $-\\frac{1}{2}$ & $(bdca)$ & $+4$ & $(cdba)$ & $-32$ & \n   $(dcba)$ & $+256$   \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\n\\subsection{\\it Convolutions of $d^{abc}$ and $f^{abc}$ with $t^a$}\nThe convolutions of the coefficients $d^{abc}$ and $f^{abc}$ with the \n$t^a$--matrices equal:\n\\begin{eqnarray*}\n&&d^{abc} t^c = t^a t^b + t^b t^a -\\frac{1}{3}\\delta^{ab}, \\quad \n f^{abc} t^c = i(t^b t^a - t^a t^b), \\\\\n &&h^{abc} t^c = 2 t^a t^b - \\frac{1}{2} \\delta^{ab}\n \\end{eqnarray*}\n \\begin{eqnarray*}\n&&d^{abk} d^{kcl} t^l = \n(t^a t^b t^c + t^b t^a t^c + t^c t^a t^b + t^c t^b t^a) -\\frac{1}{3} d^{abc}I \n - \\frac{2}{3} \\delta^{ab} t^c, \\\\\n&&d^{abk} f^{kcl} t^l = \n i(-t^a t^b t^c - t^b t^a t^c + t^c t^a t^b + t^c t^b t^a), \\\\\n&&f^{abk} d^{kcl} t^l = \n i(-t^a t^b t^c + t^b t^a t^c - t^c t^a t^b + t^c t^b t^a) \n      -\\frac{1}{3} f^{abc}I, \\\\ \n&&f^{abk} f^{kcl} t^l = \n (-t^a t^b t^c + t^b t^a t^c + t^c t^a t^b - t^c t^b t^a), \\\\\n&&d^{abc}t^at^bt^c = \\frac{10}{9}I, \\; \nf^{abc}t^at^bt^c = 2iI, \\; h^{abc}t^at^bt^c = -\\frac{8}{9}I, \\\\ \n && d^{abc}t^at^b = \\frac{5}{6}t^c, \\; \nf^{abc}t^at^b = \\frac{3}{2} i t^c, \\; \nh^{abc}t^at^b = -\\frac{2}{3} t^c.\n\\end{eqnarray*}\nThe Jacobi identities for the coefficients $f^{abc}$ and $d^{abc}$ equal: \n\\begin{eqnarray*}\nf_{abk}f_{kcl} \\, &+& \\,f_{bck}f_{kal}\\, +\\,f_{cak}f_{kbl}\\, = 0, \\\\\nd_{abk}f_{kcl} \\, &+& \\,d_{bck}f_{kal}\\, +\\,d_{cak}f_{kbl}\\, = 0.\n\\end{eqnarray*}\nThe various relations of a such type were presented in \\cite{ditn}:\n\\begin{eqnarray*}\n&&d_{abk}d_{kcl} \\, + \\,d_{bck}d_{kal}\\, +\\,d_{cak}d_{kbl}\\, = \\,\\frac{1}{3} \n(\\delta_{ab}\\delta_{cl}\\,+\\,\\delta_{ac}\\delta_{bl}\\,+\\,\n\\delta_{al}\\delta_{bc}), \\\\\n&& f_{abk}f_{kcl} \\, = \\,\\frac{2}{3}(\\delta_{ac}\\delta_{bl}\\,-\\,\n\\delta_{al}\\delta_{bc})\\,+\\,d_{ack}d_{blk}\\, -\\,d_{alk}d_{bck}, \\\\ \n&&3d_{abk}d_{kcl} \\, = \\,\\delta_{ac}\\delta_{bl}\\,+\\,\\delta_{al}\n\\delta_{bc}\\,-\\,\n\\delta_{ab}\\delta_{cl}\\,+\\,f_{ack}f_{blk}\\, +\\,f_{alk}f_{bck}, \\\\ \n&& d_{aac} \\; = \\; f_{aac} \\; = \\; d_{abc} f_{abm} \\; = \\; 0.\n\\end{eqnarray*}\n\\begin{eqnarray*}\nf_{akl}f_{bkl} &=& 3 \\delta_{ab}, \\qquad d_{akl}d_{bkl} = \\frac{5}{3}\n\\delta_{ab}, \\\\\nf_{pak}f_{kbl}f_{lcp} &=& -\\frac{3}{2}f_{abc}, \\quad \nd_{pak}f_{kbl}f_{lcp} = -\\frac{3}{2}d_{abc}, \\\\\nd_{pak}d_{kbl}f_{lcp} &=& \\frac{5}{6}f_{abc}, \\quad \nd_{pak}d_{kbl}d_{lcp} = -\\frac{1}{2}d_{abc}, \\\\  \nd_{piq}d_{qjm}d_{mkt}d_{tlp} &=& \\frac{1}{36}\n(13 \\delta_{ij} \\delta_{kl} - 7 \\delta_{ik} \\delta_{jl} \n + 13 \\delta_{il} \\delta_{jk} - d_{ikm} d_{mjl}), \\\\ \nd_{piq}d_{qjm}d_{mkt}f_{tlp} &=& \\frac{1}{12}\n(-7 d_{ijm} f_{mkl} + d_{ikm} f_{mjl} + 9 d_{ilm} f_{mjk}), \\\\\nd_{piq}d_{qjm}d_{mkt}d_{tlp} &=& \\frac{1}{36}\n(-21 \\delta_{ij} \\delta_{kl} + 19 \\delta_{ik} \\delta_{jl} \n - \\delta_{il} \\delta_{jk} ) \\\\\n& +& \\frac{1}{6} (d_{ikm} d_{mjl}  - 4 d_{ilm} d_{mjk}), \\\\\nd_{piq}f_{qjm}f_{mkt}d_{tlp} &=& \\frac{3}{4}\n(d_{ikm} f_{mil} + d_{ilm} f_{mkj}), \\\\ \nf_{piq}f_{qjm}f_{mkt}f_{tlp} &=& \\frac{1}{4}\n(5 \\delta_{ij} \\delta_{kl} + \\delta_{ik} \\delta_{jl} \n + 5 \\delta_{il} \\delta_{jk}  - 6 d_{ikm} d_{mjl}).\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n  \\begin{array}{lcrlcr}   \nd^{abc} d^{abc} & = & 40\/3,  &  f^{abc} f^{abc} &=& 24, \\\\ \nh^{abc} h^{abc} & = & -32\/3, & h^{abc} h^{bac}  &=& 112\/3, \\\\ \n \\displaystyle d^{abk}d^{klc}d^{cbn}d^{nla} &  = & - 20\/3, &\nd^{abk}d^{klc}d^{cbn}f^{nla} &=& 0, \\\\\nd^{abk}d^{klc}f^{cbn}f^{nla} & = & 20, &\nd^{abk}f^{klc}d^{cbn}f^{nla} &=& -20, \\\\\nd^{abk}f^{klc}f^{cbn}f^{nla} & =&  0, & \nf^{abk}f^{klc}f^{cbn}f^{nla}  &=& 36, \\\\\n\\displaystyle h^{abk}h^{klc}h^{cbn}h^{nla} & = & - 32\/3 & {}\n\\end{array} \n \\end{eqnarray*} \n\n\\subsection{\\it Gell-Mann Matrices  in $n$--dimensions}\nThe generators for an arbitrary $SU(n)$ group are:\n\\begin{eqnarray}\n  t^a, \\quad a = 1,2,..,N, \\quad N=n^2-1  \\label{sun-1}\n\\end{eqnarray}\nwith the properties as follows:\n\\begin{eqnarray}\n &&  (t^a)^{\\dagger} = t^a, \\;\\; Tr t^a = 0,  \\label{sun-2} \\\\\n  && [t^a, t^b] = i \\, f^{abc} \\, t^c   \\label{sun-3} \\\\\n  && \\{t^a, t^b\\} =  d^{abc} \\, t^c \\, + \\frac{1}{n} \\delta^{ab}  \\label{sun-4} \\\\\n  &&   [t^a, t^b] \\equiv t^a t^b - t^b t^a, \\quad\n  \\{t^a, t^b\\} \\equiv t^a t^b + t^b t^a, \\nonumber \n\\end{eqnarray}\nThe constants $f^{abc} (d^{abc})$ is totally antisymmetric (symmetric) tensor.\n\\\\\n$SU(n)$ group has the invariants as follows:\n\\begin{eqnarray}\n   f^{a \\, kl} f^{b \\, kl} = C_A \\delta^{ab}, \\;\\;\n  d^{\\, a \\, kl} d^{\\, b \\, kl} = C_D \\delta^{ab}, \\;\\;\n  t^a t^a = C_F I, \\;\\; Tr \\, t^at^b = T_F \\delta^{ab}\n  \\label{app-gr-5}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n  C_A = n, \\;\\; C_F = \\frac{n^2-1}{2n}, \\;\\; C_D = \\frac{n^2-4}{n}, \\;\\;\n  T_F = \\frac{1}{2} \\label{app-gr-6}\n\\end{eqnarray}\nThe Jacobi identities have the form: \n\\begin{eqnarray*}\n && f^{ab \\, K} f^{K\\, cl} + f^{bc \\, K} f^{K\\, al} + f^{ca \\, K} f^{K\\, bl}  = 0 \\\\ \n && d^{ab \\, K} f^{K\\, cl} + d^{bc \\, K} f^{K\\, al} + d^{ca \\, K} f^{K\\, bl}  = 0  \n\\end{eqnarray*}\n\n\n\\section{\\bf VECTOR ALGEBRA}\\label{vecal}\n\nLet $\\{ p_1, \\ldots ,p_n\\}$ be some basis and scalar products $p_i \\cdot p_j$ \ndefine a matrix $M$: $M_{ij}=p_i\\cdot p_j$. The dual basis is the set of \nvectors $\\{\\xi_1,...,\\xi_n\\}$, which satisfy the conditions:\n\\[ \\xi_i\\cdot p_j = \\delta_{ij},\\ \\ \\ \n\\xi_i \\cdot \\xi_j=(M^{-1})_{ij}. \\]\nThen\n\\[\\xi_i^\\alpha=\\delta^{p_1,...,p_{i-1},\\alpha,p_{i+1},...,p_n}\n_{p_1, .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ .\\ ,p_n}\/\\Delta_n, \\]\nwhere $\\Delta_n=\\delta_{p_1,...,p_n}^{p_1,...,p_n}$.\nSometimes one needs to represent some vector $Q$ in the form \\cite{vermaseren}:\n\\[ Q_\\alpha = \\cal{P}_\\alpha+V_\\alpha, \\]\nwhere $\\p_\\alpha$ is a linear combination of $p_1,...,p_m\\ \\ (m<n)$,\nand $V\\cdot p_i=0$ for $i=1,...,m$.\n\\begin{eqnarray*}\nm=1 &  {\\cal P}_{\\alpha}=\\frac{p_1 \\cdot Q}{p_1 \\cdot p_1} p_{1\\alpha}, \\; \nV_\\alpha=\\frac{1}{p_1\\cdot p_1}\\delta^{p_1\\alpha}_{p_1 Q} & \\\\\nm=2 &  {\\cal P}_\\alpha= \\frac{1}{\\Delta_2} \n\\delta^{Q\\mu}_{p_1 p_2}\\delta^{\\mu \\alpha}_{p_1 p_2}=\\frac{1}{\\Delta_2}\n\\left(\\delta^{Q p_2}_{p_1 p_2}p_{1\\alpha}\\right.\n\\left.+\\delta^{p_1 Q }_{p_1 p_2}p_{2\\alpha}\\right) & \\\\\n & V_\\alpha=\\frac{1}{\\Delta_2}\\delta^{p_1 p_2 \\alpha}_{p_1 p_2 Q} \\\\\nm=3 & {\\cal P}_\\alpha= \\frac{1}{2\\Delta_3} \n\\delta^{Q\\mu\\nu}_{p_1 p_2 p_3}\\delta^{\\alpha \\mu\\nu}_{p_1 p_2 p_3}\n=\\frac{1}{\\Delta_3} \\left(\\delta^{Q p_2 p_3}_{p_1 p_2 p_3}p_{1\\alpha}\\right.\n+\\delta^{p_1 Q p_3}_{p_1 p_2 p_3}p_{2\\alpha}\n\\left.+\\delta^{p_1 p_2 Q }_{p_1 p_2 p_3}p_{3\\alpha}\\right) & \\\\\n & V_\\alpha=\\frac{1}{\\Delta_3}\n\\delta^{p_1 p_2 p_3 \\alpha}_{p_1 p_2 p_3 Q}. &\n\\end{eqnarray*}\n\n\\subsection{\\it Representation of 3--dimensional Vectors, Reflections and \nRotations Using the Pauli Matrices}\n\\noindent $\\bullet$ {\\bf Vectors} \\\\\nAny vector $\\vec x$ in 3--dimensional Euclidean space can be represented in \nthe matrix form:\n\\begin{eqnarray*}\n\\hat x\\ &\\equiv &  \\vec x \\cdot \\vec \\sigma\\ = \n x^i\\sigma^i \\ =\\ \\left( \\begin{array}{cc}\nx^3 & x^1-ix^2 \\\\\nx^1+ix^2 & -x^3 \n\\end{array} \\right), \\\\\n\\det\\> \\hat x\\ &=&\\ -\\vec x\\cdot \\vec x \\ =\\ -(x_1^2+x_2^2+x_3^2),\n\\end{eqnarray*}\nwhere $\\sigma^i$ is the Pauli matrices (see Section~\\ref{pauli}). \\\\\nThe fundamental property of this representation is\n\\begin{equation}\n(\\hat x)^2= \\vec x\\cdot \\vec x\\> I,\\ \\ \\mbox{hence}\\ \\ \\\n\\hat x\\hat y+\\hat y\\hat x =2\\> (\\vec x\\cdot \\vec y)\\> I.\n\\end{equation}\nOne should also note that \n\\[\\hat x\\hat y-\\hat y\\hat x\\ =\\ 2i\\> \\widehat{\\vec x\\times \\vec y}.\\]\nIf components of $\\vec x$ are real, then $\\hat x^\\dagger =\\hat x$.\nHowever, in some practically important cases $(\\hat x)^2=0$ and,\nhence, $\\vec x\\cdot \\vec x=0$, components of $\\vec x$ are complex, say,\n$x^2=ix^0$. Then the matrix\n\\begin{displaymath}\n\\hat x=\\left( \\begin{array}{cc}\nx^3 & x^1+x^0 \\\\\nx^1-x^0 & -x^3\n\\end{array}\\right)\n\\end{displaymath}\nrepresents a vector from 3--dimensional space--time, in that\ncase $\\hat x^\\dagger \\neq \\hat x$.\n\n\\noindent $\\bullet$ {\\bf Reflections} \\\\ \nLet $\\vec x$ be an arbitrary vector and $S$ be the plane\northogonal to some unit vector $\\vec s$. Then, vector $\\vec x'$\nwhich results from $\\vec x$ after the reflection in the plane $S$\nis equal to: \n\\[\\vec x' = \\vec x - 2 (\\vec x \\vec s) \\vec s, \\]\nor, in the considered matrix representation\n\\[ \\hat x'  = - \\hat s\\hat x\\hat s. \\]\n\n\\noindent $\\bullet$ {\\bf Rotations} \\\\ \nLet $\\vec p$ and $\\vec q$ be the two unit vectors with the\nangle $\\theta\/2$ between them:\n\\[ \\vec p^{\\; 2} = \\vec q^{\\; 2} = 1, \\ \\ (\\vec p\\vec q) = \\cos(\\theta\/2).\\]\nSince any spatial rotation is a composition of two reflections,\nthe rotation by the angle $\\theta$ in the direction from $\\vec p$\nto $\\vec q$ is given by the matrix \n\\[ M\\ =\\ \\hat q\\hat p, \\]\ni.e. an arbitrary vector $\\vec x$ transforms as follows:\n\\[ \\hat x' = M\\hat x M^{-1} = \\hat q\\hat p\\hat x\\hat p\\hat q. \\]\nThe matrix $M$ can be rewritten in widely used form:\n\\begin{eqnarray} \nM = \\hat q\\hat p = \\vec q \\cdot \\vec p\\> I + i\\varepsilon_{qpr}\n\\hat r = \\cos(\\theta\/2) I\\>-\\>i\\>\\vec n\\vec \\sigma \\> \\sin(\\theta\/2), \n\\label{ve2}\n\\end{eqnarray} \nwhere $\\vec n\\ \\sin(\\theta\/2)\\ =\\ \\vec p \\times \\vec q,\\ \\ \n-\\pi<\\theta<\\pi$, {\\bf positive values of $\\theta$ correspond to \ncounterclockwise rotations if one sees from the head of\nvector $\\vec n$}.\n\n\\noindent So, we get the two forms of representation of a spatial rotation:\n\\begin{itemize}\n\\item{} The rotation by angle $\\theta$ about a unit vector $\\vec n$\nis given by \n\\[\nM = \\cos(\\theta\/2)I \\>-\\>i\\>\\vec n\\vec \\sigma \\> \\sin(\\theta\/2).\n\\] \n\\item{}The rotation in the plane of unit vectors $\\vec p$ and\n$\\vec q$ which transforms $\\vec p$ into $\\vec q$ is represented by\n\\[M\\ =\\ \\frac{I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1+\\vec q\\cdot \\vec p)}}\\]\n\\end{itemize}\n\n\\subsection{\\it Representation of 4--dimensional Vectors, Reflections and \nRotations Using the Dirac Matrices}\n\n\\noindent $\\bullet$ {\\bf Vectors} \\\\\n$4\\times 4$ matrix $\\hat x$, which represents the 4--vector $x^{\\mu}$ in\nMinkowski space looks as follows:\n\\begin{displaymath}\n\\hat x \\equiv x^\\mu\\gamma_\\mu=\\left(\\begin{array}{cccc}\n0 & 0 & -x^0-x^3 & -x^1+ix^2 \\\\\n0 & 0 & -x^1-ix^2 & -x^0+x^3 \\\\\n-x^0+x^3 & x^1-ix^2 & 0 & 0 \\\\\nx^1+ix^2 & -x^0-x^3 & 0 & 0 \n\\end{array} \\right).\n\\end{displaymath}\nThis matrix satisfies the fundamental property\n\\begin{eqnarray} \n(\\hat x)^2 =x\\cdot x I \\ \\ \\Rightarrow \\ \\ \n\\hat x\\hat y +\\hat y\\hat x=2\\> x\\cdot y. \\label{ve3}\n\\end{eqnarray} \n\n\\noindent $\\bullet$ {\\bf Reflections} \\\\ \nUsing the relation (\\ref{ve3}) one can easily derive formulas for the\nreflections in 3--hyperplanes. Let $x^{\\mu}$ be an arbitrary vector and \n$S$ be the 3-hyperplane orthogonal to some unit vector $s$. Then, vector $x'$\nwhich results from $x$ after the reflection in the hyperplane $S$\nis equal to \n\\[ x' =  x - 2 \\ x\\cdot s \\ s, \\]\nor, in the considered matrix representation\n\\[ \\hat x'  = - \\hat s\\hat x\\hat s. \\]\n\n\\noindent $\\bullet$ {\\bf Lorentz transformations} \\\\ \nLet $p$ and $q$ be the two unit time-like vectors:\n\\[ p\\cdot p = q\\cdot q = 1.\\]\nLorentz transformation, which is a composition of the reflections\nin 3-hyper\\-pla\\-nes determined by the vectors $p$ and $q$ is given by\nthe matrix:\n\\[ M\\ =\\ \\hat q\\hat p, \\]\ni.e. an arbitrary vector $x$ transforms as follows:\n\\[ \\hat x' = M\\hat x M^{-1} = \\hat q\\hat p\\hat x\\hat p\\hat q. \\]\nThe Lorentz transformation in the 2-plane (defined by the vectors $p$ and $q$), \nwhich transforms $p$ into $q$, is represented by\n\\[M\\ =\\ \\frac{I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1+ q\\cdot p)}}. \\]\nFor space-like unit vectors $p$ and $q$ we arrive at \n\\[M\\ =\\ \\frac{- I\\ +\\ \\hat q\\hat p}{\\sqrt{2(1 - q\\cdot p)}}. \\]\n\nLet $\\Lambda^{\\mu}_{\\nu}$ be the matrix of Lorentz boost\nthat moves the vector\\\\\n$m\\mathbf{e_0}^\\nu=(m,0,0,0)\\ $ to $\\ p^\\mu=(p^0, p^1, p^2, p^3)$, \n$$\np^\\mu = \\Lambda^{\\mu}_{\\nu} \\ m\\mathbf{e_0}^\\nu\\ ,\n$$\nand leaves all vectors orthogonal to $p$ and $\\mathbf{e_0}$\ninvariant. Then \n$$\n\\hat p = S(\\Lambda)\\; m \\gamma^0\\; S^{-1}(\\Lambda)\\;,\n$$\nwhere\n$$\nS(\\Lambda) = {\\displaystyle {\\hat p \\gamma^0 + m \\over \\sqrt{2m(m+p_0)}}  }\\;.\n$$\nThis being so,\n\\begin{eqnarray*}\n\\Lambda^{\\mu}_{\\nu} = {1\\over 4} \\;T\\!r\\; \\gamma^\\mu S(\\Lambda) \n\\gamma_\\nu S^{-1}(\\Lambda)\\;=\\;\\left(\n\\begin{array}{cc}\n\\displaystyle{{p^0\\over m}} & \\displaystyle{{p^i\\over m}}\\\\[4mm]\n\\displaystyle{{p^i\\over m}} & \\displaystyle{\\delta^{ij}+{p^ip^j\\over m(p^0+m)}}\n\\end{array}\\right)\\ .\n\\end{eqnarray*}\n\n\n\n\\noindent \nThe reference frame where\nthe momentum of the reference body is equal to $p$\nis usually referred to as the laboratory frame {\\it Lab}-frame;\nthe frame where the it is equal to $m\\mathbf{e_0}$\nis the rest frame {\\it R}-frame;\n\nLet $k^{\\mu}$ be some 4-vector defined in the {\\it Lab}-frame  and\n$k^{* \\mu}$ be the same 4-vector in the {\\it R}-frame;\n$k^{\\mu}=\\Lambda^{\\mu}_{\\nu} k^{* \\nu}$ and\n$k^{* \\mu}=\\big(\\Lambda^{-1}\\big)^{\\mu}_{\\nu} k^{\\nu}\\;.$\n\nThe Lorentz transormations form {\\it R}(est)-frame to {\\it Lab}-frame and vice versa \ncan also be represented in the form\n\\begin{eqnarray}\n \\begin{array}{l} { k^{*} \\to k} \\\\ {R \\to Lab} \\end{array}: \\; { \\left \\{ \\begin{array}{ccl}\n k_0 & = & \\displaystyle \\frac{k_{0}^{*}p_{0} + (\\vec{k}^*\\vec{p})} {m} \\\\[2mm]\n \\vec{k} & =  & \\vec{k}^{*} + \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n \\begin{array}{r} {k \\to k^{*}} \\\\ {Lab \\to R} \\end{array}: \\;\n              { \\left \\{ \\begin{array}{ccl}\n k_0^{*} & = & \\;\\; \\displaystyle \\frac{(pk)} {m} \\\\[2mm]\n \\vec{k}^* & =  & \\vec{k} - \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n \\label{lorentz-10}\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n \\alpha = \\frac {k^{*}_0 + k_0} {p_0 + m}\n\\end{eqnarray*}\n\n\n\n\n\\section{\\bf DIRAC SPINORS}\\label{spin2}\n\n\\subsection{\\it General Properties}\n\nDirac spinors $u(p, s)$ and $v(p, s)$ describe the solutions of the Dirac \nequation with positive and negative energy:\n\\begin{eqnarray}\n  && (\\hat p -m)\\> u(p,s)=0,\\ \\  \\ (\\hat p+m)\\> v(p,s)=0 \\\\\n  && \\bar u(p,n)(\\hat p-m)=0, \\ \\ \\ \\bar v(p,n)(\\hat p+m)=0 \n\\end{eqnarray}\nwhere  The {\\it conjugated} spinors are defined as \nfollows:\n\\[ \n\\bar u=u^\\dagger \\gamma^0,\\ \\ \\ \\bar v=v^\\dagger \\gamma^0,\n\\] \nThese spinors are the functions of 4-momentum $p^{\\mu}$ on the mass shell \n$p^0=\\sqrt{\\vec p^{\\; 2}+m^2}$. The normalization conditions are\nas follows:\n\\begin{eqnarray*}\n\\bar u(p, s) u(p,s) &=&+2m, \\\\\n\\bar v(p, s) v(p,s) &=& -2m.\n\\end{eqnarray*}\nSymbol $s$ stands for the polarization of the fermion. The axial--vector \n$s^\\mu$ of the fermion spin is defined by the relations: \n\\begin{eqnarray*}\n && \\bar u(p,s)\\gamma^\\mu \\gamma^5 u(p,s)\\ =\\ m\\> s^\\mu, \\quad  \n  (s \\, s) = -1,\\ \\ \\ (s \\, p) = 0. \n\\end{eqnarray*}\n\n\\noindent It is helpful to clear up the relation between $n$\nand the Pauli--Lubanski pseudovector\n$$\nW_\\mu=\\varepsilon_{\\mu\\alpha\\beta\\nu}M^{\\alpha\\beta}P^{\\nu}\\;,\n$$\nwhere $M^{\\alpha\\beta}$ and $P^{\\nu}$ are the generators of \nLorentz transformations and translations; it acts on the\nfermion fields as follows:\n\\begin{eqnarray*}\n&& \n  W_\\mu \\psi(x) = {i\\over 2}\\;\\sigma_{\\mu\\nu}\\gamma^5 \\partial^{\\nu}\\psi(x),\n  \\quad \n\\gamma^5 \\hat s = -\\;{2\\over m}\\; s\\cdot W \n\\end{eqnarray*}\nTo make a bridge between the above formulas and nonrelativistic\nconcept of spin it is well to recollect that the operator of spin\nof the fermion at rest is given by\n\\begin{displaymath}\n\\Sigma^k = {1\\over 2} \n\\left( \\begin{array}{ccc} \\sigma^k & 0 \\\\ 0 & \\sigma^k \\end{array} \\right) \n\\ =\\;-\\;{1\\over 2}\\; \\gamma^5 \\gamma^k \\gamma^0 \n\\end{displaymath}\nFor the fermion of momentum $p$, the spin has the form:\n$$\n\\Sigma^\\mu = {1\\over 2} \\gamma^5 \\gamma^\\mu {\\hat p\\over m} \n$$\nIts projection on the direction defined by \nan arbitrary 4-vector $s$ such that  $(s\\, p)=0$ and $(s \\, s)=-1$\nis given by the formula:\n$$\n\\Sigma\\cdot s = {1\\over 2} \\gamma^5 \\hat s\\; {\\hat p\\over m}\\;.\n$$\nThus, the spin is the Pauli--Lubanski pseudovector times the sign of energy:\n$$\n\\Sigma = W\\; {\\hat p\\over m}\\;,\n$$\nThat is why the spin of the fermions $u(p,s)$ and $v(p,-s)$ is \n$\\displaystyle \\Sigma= {s\\over 2}$ and the spin of the fermions\n$u(p,-s)$ and $v(p,s)$ is \n$\\displaystyle \\Sigma = -\\;{s\\over 2}$.\n\nNote that the axial vector $s$ describing  \nthe spin of the fermion has only spatial non-zero components in its rest\nframe  \nand transforms together with the vector $p$ under Lorentz transformations.\n\nThe spinor $u(p,s)$ describes a fermion with momentum $p$ and the vector of \nspin $n$. The spinor $v(p,s)$ describes an antifermion with momentum $p$ and \nthe vector of spin $-s$. (One should note, that axial vector $s$ describing  \nspin of a fermion has only spatial non-zero components in the rest frame of \nthis fermion. However, it transforms together with the vector $p$ under \nLorentz transformations.)\n\n\\noindent Spinors $u(p,s)$ and $v(p,s)$ satisfy the following relations:\n\\begin{eqnarray} \nu(p,s)\\> \\bar u(p,s)\\ &=&\\ \\frac{(\\hat p+m)\\>(1+\\gamma^5\\hat s)}{2}, \\\\\nv(p,s)\\> \\bar v(p,s)\\ &=&\\ \\frac{(\\hat p-m)\\>(1+\\gamma^5\\hat s)}{2}, \\\\\n\\hat s \\gamma^5 u(p,s) = u(p,s),\\ && \\ \\hat s \\gamma^5 v(p,s) = v(p,s),\n\\end{eqnarray} \nas well as the Gordon identities:\n\\begin{eqnarray*}\n \\bar u(p_1,s_1)&\\gamma^\\mu& u(p_2,s_2) \\\\ &=& \n\\frac{1}{2m}\\bar u(p_1,s_1)\n\\left[ (p_1+p_2)^\\mu+\\sigma^{\\mu\\nu}(p_1-p_2)_\\nu\\right] u(p_2,s_2),\\\\\n \\bar u(p_1,s_1)&\\gamma^\\mu &\\gamma^5 u(p_2,s_2) \\\\ &=& \n\\frac{1}{2m}\\bar u(p_1,s_1)\n\\left[ (p_1-p_2)^\\mu\\gamma^5+\\sigma^{\\mu\\nu}(p_1+p_2)_\\nu\\gamma^5 \\right]\nu(p_2,s_2),\n\\end{eqnarray*}\nBoth $(p^{\\mu}+mn^{\\mu})$ and $(p^{\\mu}-mn^{\\mu})$ are light-like vectors.\n\n\\subsection{\\it Bispinors in the Dirac represenetaion of the\n  $\\gamma$ matrices}\nFor the massive fermion with the mass $m$ and 4-momentum $p_{\\mu}$\n\\[\np^{\\mu} = (p^0, p^1, p^2, p^3) = (p^0, \\vec{p}\\,), \\;;\np^0 = \\sqrt{m^2 + \\vec{p}^{\\,\\, 2}}\n  \\]\nWe define two reference frames, namely, {\\it R}-frame (the fermion rest-frame)\n  and {\\it L}-frame (where the fermion momentum $p^{\\mu}$ defined above):\n \\begin{eqnarray*}\n  && \\hbox{{\\it R}-frame}: \\quad p^{\\mu} = (m, 0) \\\\\n   && \\hbox{{\\it L}-frame}: \\quad p^{\\mu} = (p^0, p^1, p^2, p^3)\n \\end{eqnarray*}\nIn the fermion rest-frame we introduce four orts are as follows:\n\\begin{eqnarray}\n&& {\\mathbf e_0}^\\mu = (1,0,0,0), \\;\\; {\\mathbf e_1}^\\mu=(0,1,0,1), \\;\\;\n {\\mathbf e_2}^\\mu = (0,0,1,0), \\;\\; {\\mathbf e_3}^\\mu=(0,0,0,1)\n \\label{orts} \\\\\n &&  {\\mathbf e_0}^2 = 1, \\;\\; {\\mathbf e_i}^2 = -1, \\; i=1,2,3, \\;\\;\n  ({\\mathbf e_i} \\, {\\mathbf e_j}) = 0, i,j=0,1,2,3, \\;  i \\ne j \\nonumber\n\\end{eqnarray} \nIn the {\\it L}-frame the spatial orts have the form: \n\\begin{eqnarray*}\n && e_i^{\\mu} = {\\mathbf e_i}^{\\mu} + \\frac{p^i}{m} \\, V^{\\mu}; \\; \\; i =1,2,3, \\quad \nV^{\\mu} = \\left( 1;  \\frac{\\vec{p}}{m+p_0} \\right); \\;\nV^2 = \\frac{2m}{m+p_0} \\\\  \n&&   (e_i \\,  e_j) = -\\delta_{ij}, \\; (e_i \\, p) = 0, \\;\\; i=1,2,3  \n\\end{eqnarray*}\nThe standard spinors $u$ and $v$\nare most conveniently defined  in the fermion {\\it R}est-frame,\n\\begin{eqnarray*}\n & u_1^{(0)}  \n  =   \\left( \\begin{array}{c} \\sqrt{2m} \\\\ 0 \\\\ 0 \\\\ 0 \\end{array} \\right),\n  \\;\\; \n& \\bar{u}_1^{(0)}  = \n \\left( \\begin{array}{rrrr} \\sqrt{2m}, & 0, & 0, & 0 \\end{array} \\right), \\\\\n& u_2^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ \\sqrt{2m} \\\\ 0 \\\\ 0 \\end{array} \\right),  \\;\\; \n &\\bar{u}_2^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & \\sqrt{2m}, & 0, & 0 \\end{array} \\right), \\\\\n & v_1^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ \\sqrt{2m} \\\\ 0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_1^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & 0, & -\\;\\sqrt{2m}, & 0 \\end{array} \\right),\n \\\\\n & v_2^{(0)} = \n \\left( \\begin{array}{c} 0 \\\\ 0 \\\\ 0 \\\\ \\sqrt{2m} \\end{array} \\right),  \\;\\; \n & \\bar{v}_2^{(0)} = \n \\left( \\begin{array}{rrrr} 0, & 0, & 0, & -\\; \\sqrt{2m}\n \\end{array} \\right)  \n\\end{eqnarray*}\n These spinors $u_1$ and $v_2$ describe the positive polarization\n  along the $Z$ axis, while $u_2$ and $v_1$ correspond to the\n  negative polarization.\n\nThen, the standard spinors in the {\\it L}-frame of the fermion  \n  can be obtained by making the Lorentz boost $S_1$:\n  \\begin{eqnarray*}\n&&\\hat p = S_1\\ m\\hat{\\mathbf e_0} \\ S_1^{-1} = S_1\\ m\\gamma^0 \\ S_1^{-1} \\\\\n && S_1 = {\\displaystyle {\\hat p \\gamma^0 + m \\over \\sqrt{2m(m+p_0)}}  } \\;=\\;\n {\\displaystyle {1\\over \\sqrt{2m(m+p_0)}}  }\n  \\left( \n  \\begin{array}{cc} (p^0 + m) I & \\sigma^i p^i \\\\ \\sigma^i p^i & (p^0+m) I\n  \\end{array} \\right)\\; ; \\\\ \n  && S_1^{-1}=S_1^\\dagger = {\\displaystyle {\\gamma^0 \\hat p  + m \\over\n      \\sqrt{2m(m+p_0)}}  } \\;=\\;\n {\\displaystyle {1\\over \\sqrt{2m(m+p_0)}}  }\n  \\left( \n  \\begin{array}{cc} (p^0 + m) I & -\\;\\sigma^i p^i \\\\ -\\;\\sigma^i p^i & (p^0+m)\n    I \\end{array} \\right)\\; ; \\\\ \n&&\nu_r(p,z)\\;=\\;S_1 u_r(m {\\mathbf e_0}, {\\mathbf e_3} ), \\qquad\nv_r(p,z)\\;=\\;S_1 v_r(m {\\mathbf e_0}, {\\mathbf e_3} ), \\qquad r=1,2\n\\end{eqnarray*}\nThus, int the {\\it L}-frame one has:\n\\begin{eqnarray*}\n & u_1(p) = {\\displaystyle  \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} m+p^0 \\\\ 0 \\\\ p^3 \\\\ p_+ \\end{array} \\right),  \\;\\; \n& \\bar{u}_1(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} m+p^0\\;, & 0\\;, & -p^3\\;, & -p_- \\end{array} \\right), \\\\\n& u_2(p) = {\\displaystyle \\frac{1}{\\sqrt{m+p_0}} }\n \\left( \\begin{array}{c} 0 \\\\ m+p^0 \\\\ p_- \\\\ -p^3 \\end{array} \\right),\n \\;\\; \n &\\bar{u}_2(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} 0\\;, & m+p^0\\;, & -p_+\\;, & p^3 \\end{array} \\right), \\\\\n & v_1(p) = {\\displaystyle \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} p^3 \\\\ p_+ \\\\ m+p^0 \\\\ 0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_2(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} p^3\\;, & p_-\\;, & -(m+p^0)\\;, & 0  \\end{array} \\right),\n \\\\ \n & v_2(p) = {\\displaystyle  \\frac{1}{\\sqrt{m+p^0}} }\n \\left( \\begin{array}{c} p_- \\\\ -p^3 \\\\ 0 \\\\ m+p^0 \\end{array} \\right),  \\;\\; \n & \\bar{v}_1(p) = \\frac{1}{\\sqrt{m+p^0}}  \n \\left( \\begin{array}{rrrr} p_+\\;, & -p^3\\;, & 0\\;, & -(m+p^0) \\end{array}\n \\right)\n\\end{eqnarray*}\nwhere\n\\[\n p_{\\pm} = p^1 \\pm i p^2 \n\\] \nThese spinors have the properties as follows:\n\\begin{eqnarray*} \n&& (\\hat{p} -m)u_i=0; \\; i=1,2; \\quad \n \\bar{u}_1 u_1 = \\bar{u}_2 u_2 = 2m; \\; \\;\n  \\bar{u}_1 u_2 = \\bar{u}_1 u_2 = 0 \\\\\n&& (\\hat{p} + m)v_i=0; \\; i=1,2; \\quad \n  \\bar{v}_1 v_1 = \\bar{v}_2 v_2 = -2m; \\; \\;\n  \\bar{v}_1 v_2 = \\bar{v}_1 v_2 = 0  \\\\\n&& \nu_1\\bar{u}_1 + u_2 \\bar{u}_2 = \\hat{p} + m; \\;\\; \nv_1\\bar{v}_1 + v_2 \\bar{v}_2 = \\hat{p} - m  \\label{spinor-11}\n\\end{eqnarray*}\n\nIn the both frames there are the following relations:\n\\begin{eqnarray*}\n  && \\hbox{{\\it R}-frame}: \\;\\; \\leftrightarrow \\;\\;\n  {\\mathbf e_i}^\\mu, \\;\\; u^{(0)}, \\; v^{(0)}  \\\\\n   && \\hbox{{\\it L}-frame}: \\;\\; \\leftrightarrow \\;\\;\n    e_i^\\mu, \\;\\; u(p), \\; v(p) \\\\ \n &&U-spinors \\\\\n &&\n{\\begin{array}{llcll}  \n \\gamma^5 \\hat{e}_1 u_1 = u_2 &  \\gamma^5 \\hat{e}_1 u_2 = u_1 & &\n \\bar{u}_1 \\gamma^5 \\hat{e}_1 = \\bar{u}_2 & \\bar{u}_2 \\gamma^5 \\hat{e}_1 =\n \\bar{u}_1 \\\\\n\\gamma^5 \\hat{e}_2 u_1 = i u_2 & \\gamma^5 \\hat{e}_2 u_2 = -i u_1 & &\n\\bar{u}_1 \\gamma^5 \\hat{e}_2 = -i \\bar{u}_2 & \n \\bar{u}_2 \\gamma^5 \\hat{e}_2 = i \\bar{u}_1 \\\\ \n\\gamma^5 \\hat{e}_3 u_1 = u_1 & \\gamma^5 \\hat{e}_3 u_2 = -u_2 & &\n \\bar{u}_1 \\gamma^5 \\hat{e}_3 = \\bar{u}_1 &\n \\bar{u}_2 \\gamma^5 \\hat{e}_3 = -\\bar{u}_2 \n\\end{array} }\n\\\\\n&& V-spinors \\\\\n&& { \\begin{array}{llcll}  \n \\gamma^5 \\hat{e}_1 v_1 = v_2 &  \\gamma^5 \\hat{e}_1 v_2 = v_1 & &\n \\bar{v}_1 \\gamma^5 \\hat{e}_1 = \\bar{v}_2 & \\bar{v}_2 \\gamma^5 \\hat{e}_1 =\n \\bar{v}_1 \\\\\n\\gamma^5 \\hat{e}_2 v_1 = i v_2 & \\gamma^5 \\hat{e}_2 v_2 = -i v_1 & &\n\\bar{v}_1 \\gamma^5 \\hat{e}_2 = -i \\bar{v}_2 & \n \\bar{v}_2 \\gamma^5 \\hat{e}_2 = i \\bar{v}_1 \\\\ \n\\gamma^5 \\hat{e}_3 v_1 = v_1 & \\gamma^5 \\hat{e}_3 v_2 = -v_2 & &\n \\bar{v}_1 \\gamma^5 \\hat{e}_3 = \\bar{v}_1 &\n \\bar{v}_2 \\gamma^5 \\hat{e}_3 = -\\bar{v}_2 \n\\end{array} }\n\\end{eqnarray*}\n\n\\noindent \nThe polarization vector \n$z$ is defined by the relation \n$$\n\\hat{z} = S_1\\; {\\hat{\\mathbf e_3}}\\; S_1^{-1} = S\\; (-\\;\\gamma^3)\\; S_1^{-1}\\;\n$$\nIt has the form\n$$\nz^\\mu={1\\over m(p^0+m)} \\Big( p^3(p^0+m)\\,, \\ \\ p^1p^3, \\ \\ p^2p^3,\n\\ \\ (p^3)^2+m(p^0+m) \\Big)\\ .\n$$\nSince $z^2=-1$ and $z\\cdot p=0$, any polarization vector $n$ \nof the fermion of momentum $p$ can be obtained \nby the Lorentz transformation $S_2$ from the Wigner little group of the\nmomentum $p$,\n$$\n\\hat s = S_2\\hat z S_2^{-1}, \\quad\nS_2={\\hat s \\hat z +1\\over \\sqrt{2(1 + z\\cdot s)}}.\n$$\nThus we arrive at the explicit expressions\nfor the spinors $u(p,s)$ and $v(p,s)$:\n$$\nu_r(p,s)=S_2 u_r(p,z), \\qquad v_r(p,s)=S_2 v_r(p,z)\\;;\n$$\nto put it differently, the spinors $u_1(p,s)$,\n$u_2(p,s)$, $v_1(p,s)$, and $v_2(p,s)$ are the columns \nof the matrix \n$$\n{\\sqrt{2m}} S_2 S_1 = {\\hat s\\hat z +1\\over \\sqrt{2(1 + z\\cdot s)}}\n{\\hat p \\gamma^0 + m \\over \\sqrt{(m+p_0)}}\\; ,\n$$\nwhere\n$$\n\\hat z = -\\; {(\\hat p \\gamma^0 + m)\\gamma^3(\\gamma^0\\hat p  + m)\n  \\over {2m(m+p_0)}}\\; .\n$$\n\n\n\\noindent\nThe case when the spatial component $\\vec s$ of the polarization vector $s^{\\mu}$\nis directed along the 3-momentum $\\vec p$ is of particular interest.\nIn this case, $u_r$ and $v_r$ describe states with definite helicity.\n\nGiven $p=(p^0\\;, p^1\\;, p^2\\;, p^3\\;)=\n(E\\;,\\ |\\vec p|\\cos\\, \\theta\\, \\cos\\varphi\\;, \\ |\\vec p|\\cos\\, \\theta\\,\n\\sin\\varphi\\;,\n\\ |\\vec p|\\sin\\, \\theta\\,)$,\nwe obtain in the both ({\\it R} and {\\it L}) frames\n\\begin{eqnarray} \nR:  s^{(0)\\;\\; \\mu} = \\left( 0, \\, \\frac{\\vec{p}}{|{\\vec{p}}\\,| } \\right), \\;\\;\nLf:  s^{\\mu} = \\frac{1}{ m |\\vec p\\,|} ( |\\vec{p}\\,|^2, \\,\n  p^0 \\vec{p} \\,) \\label{vspin}\n\\end{eqnarray}\nAs a result  we can obtain the so-called ``helicity'' spinors with\nthe properties as follows:\n\\begin{eqnarray} \n  \\left\\{ \\begin{array}{l|l}\n    \\gamma^5 \\hat{s} \\; u_R = u_R  &  \\gamma^5 \\hat{s} \\; u_L = -u_L  \\\\\n    \\gamma^5 \\hat{s} \\; v_R = -v_R &  \\gamma^5 \\hat{s} \\; v_L = v_L \n        \\end{array}\\right.\n   \\label{spir-1}\n\\end{eqnarray}\nFor the massless fermions, these spinors have the well-known projectors\n\\begin{eqnarray} \n  \\left\\{ \\begin{array}{ll|ll}\n     P_R u_R = u_R & P_L u_R = 0 &  P_R u_L = 0 & P_L u_L = u_L  \\\\\n     P_R v_R = v_R & P_L v_R = 0  & P_R v_L = 0 & P_L v_L = v_L\n  \\end{array}\\right.; \\quad\n  P_{R\/L} = \\frac{1}{2} \\left(1 \\pm \\gamma^5 \\right) \\quad \n   \\label{spir-2}\n\\end{eqnarray}\nThe explicit expressions for these spinors (in the Dirac representation of the\n$\\gamma$-matrices) are as follows:\n\\begin{eqnarray*} \\displaystyle\n  &&u_R =  \\left( \\begin{array}{r}\n     \\chi_1 \\\\  \\chi_2 \\end{array}\\right), \\;\\; \n  \\bar{u}_R =  \\left( \\chi^{\\dagger}_1, \\, - \\chi^{\\dagger}_2 \\right) \n\\\\\n  &&u_L =  \\left( \\begin{array}{r}\n     \\chi_3 \\\\   \\chi_4 \\end{array}\\right), \\;\\; \n  \\bar{u}_L = \\left( \\chi^{\\dagger}_3, \\, - \\chi^{\\dagger}_4 \\right) \n  \\\\\n && v_R =  \\left( \\begin{array}{r}\n      \\chi_2 \\\\   \\chi_1 \\end{array}\\right), \\;\\; \n  \\bar{v}_R = \\left( \\chi^{\\dagger}_2, \\, -\\chi^{\\dagger}_1 \\right) \n  \\\\\n && v_L =   \\left( \\begin{array}{r}\n     \\chi_4 \\\\   \\chi_3 \\end{array}\\right), \\;\\; \n   \\bar{v}_L = \\left( \\chi^{\\dagger}_4, \\, -\\chi^{\\dagger}_3 \\right)\n\\end{eqnarray*}\nwhere\n\\begin{eqnarray*}\n  \\chi_1 = \\left( \\begin{array}{c}\n    \\sqrt{E+m}\\;\\cos {\\theta\\over 2} \\\\[2mm] \n    \\sqrt{E+m}\\;\\sin {\\theta\\over 2}\\ e^{i\\varphi}\n\\end{array}     \\right)\n    =   \\sqrt{\\frac{p_0 + m}{2 |\\vec{p}\\, | (|\\vec{p}\\, | + p_3)}}\n    \\left( \\begin{array}{c}\n      |\\vec{p}\\, | + p^3 \\\\[2mm] \n      p_{+}\n    \\end{array} \\right)\n \\\\\n \\chi_2= \\left( \\begin{array}{c}\n    \\sqrt{E-m}\\;\\cos {\\theta\\over 2} \\\\[2mm] \n    \\sqrt{E-m}\\;\\sin {\\theta\\over 2}\\ e^{i\\varphi}\n\\end{array}     \\right)\n  =  \\sqrt{\\frac{p_0 - m}{2 |\\vec{p}\\,| (|\\vec{p}\\,| + p_3)}}\n  \\left( \\begin{array}{c}\n    |\\vec{p}\\,| + p^3 \\\\[2mm] \n      p_{+}   \\end{array} \\right) \\\\\n\n\n  \\chi_3= \\left( \\begin{array}{c}\n    -\\sqrt{E-m}\\;\\sin {\\theta\\over 2} e^{-i\\varphi} \\\\[2mm] \n     \\sqrt{E-m}\\;\\cos {\\theta\\over 2}\n \\end{array}     \\right)\n  =  \\sqrt{\\frac{p_0 - m}{2 |\\vec{p}\\,|  (|\\vec{p}\\,| + p_3)}}\n  \\left( \\begin{array}{c}\n    -p_{-} \\\\[2mm]\n    |\\vec{p}\\,| + p^3\n  \\end{array} \\right)\n  \\\\\n  \\chi_4 = \\left( \\begin{array}{c}\n    \\sqrt{E+m}\\;\\sin {\\theta\\over 2}\\ e^{-i\\varphi}\n    \\\\[2mm] \n    -\\sqrt{E+m}\\;\\cos {\\theta\\over 2}\n\\end{array}     \\right)\n    =   \\sqrt{\\frac{p_0 + m}{2 |\\vec{p}\\,| (|\\vec{p}\\,| + p_3)}}\n \\left( \\begin{array}{c} p_{-} \\\\[2mm] -|\\vec{p}\\,| - p^3  \\end{array} \\right)\n\\end{eqnarray*}\nNote, that\n\\[\n v_R = \\gamma^5 u_R, \\quad v_L =  \\gamma^5 u_L \\nonumber \n\\]\n\n\n\\section{\\bf STANDARD MODEL LAGRANGIAN}\\label{sml}\n\nIn this Section we present the basic Lagrangian of the Standard Model(SM),\ncorresponding to the $SU(3)\\times SU(2)\\times U(1)$ local gauge symmetry\n(see, \nfor example, \\cite{itzu,okun,aoki}). The algebra of the semisimple group \n$SU(3)\\times SU(2)\\times U(1)$ is generated by Gell-Mann matrices \n$t^a = \\frac{1}{2}\\lambda^a$ (a =1,...8) (Section~\\ref{gellmann}), \nPauli matrices $\\tau^i = \\sigma^i\/2$ (Section~\\ref{pauli}) \nand hypercharge $Y$ with the following commutation relations\n\\begin{eqnarray*}\n\\left[ t^a , t^b \\right] &=& i\\;f^{abc} t^c, \\\\ \n\\left[ \\tau^i , \\tau^j \\right] &=& i\\;\\epsilon^{ijk} \\tau^k, \\\\\n\\left[ \\tau^i , Y \\right] &=& \\left[ t^a, \\tau^j \\right] = \n\\left[ t^a, Y \\right] = 0.\n\\end{eqnarray*}\nThe full SM Lagrangian has the form \\cite{itzu,okun}:\n\\begin{equation}\n{\\cal L = L_G + L_F + L_H + L_M + L_{GF} + L_{FP}.} \\label{sm91}\n\\end{equation}\nHere ${\\cal L_G}$ is the Yang-Mills Lagrangian without matter fields\n\\begin{eqnarray}\n{\\cal L_G} = -\\frac{1}{4} F^i_{\\mu\\nu}(W) F^{\\mu\\nu}_i(W) \n -\\frac{1}{4} F^{\\mu\\nu}(W^0) F^{\\mu\\nu}(W^0) \n-\\frac{1}{4} F^a_{\\mu\\nu}(G) F^{\\mu\\nu}_a(G), \\label{sm92}\n\\end{eqnarray} \nwhere $ F^i_{\\mu\\nu}(W), F^a_{\\mu\\nu}(G),  F_{\\mu\\nu}(W^0)$ are given by\n\\begin{eqnarray*}\n F^i_{\\mu\\nu}(W) &=& \\partial_{\\mu}W^i_{\\nu} - \\partial_{\\nu}W^i_{\\mu}\n+ g\\;\\epsilon^{ijk} W^j_{\\mu} W^k_{\\nu}, \\\\\nF_{\\mu\\nu}(W^0) &=& \\partial_{\\mu}W^0_{\\nu} - \\partial_{\\nu}W^0_{\\mu}, \\\\\nF^a_{\\mu\\nu}(G) &=& \\partial_{\\mu}G^a_{\\nu} - \\partial_{\\nu}G^a_{\\mu}\n+ g_s\\;f^{abc} G^b_{\\mu} G^c_{\\nu},\n\\end{eqnarray*}\nwith $W^i_{\\mu}, W^0_{\\mu}$ the $SU(2)\\times U(1)$ original gauge fields \nand $G^a_{\\mu}$ the gluon fields. The infinitesimal gauge transformations \nof these fields are given by\n\\begin{eqnarray*}\n\\delta W^0_{\\mu} &=& \\partial_{\\mu}\\theta(x), \\\\\n\\delta W^i_{\\mu} &=& \\partial_{\\mu}\\theta^i -\ng\\epsilon^{ijk}\\theta^j W^k_{\\mu} = {\\cal D}^{ij}_{\\mu}(W) \\theta^j \\\\\n\\delta G^a_{\\mu} &=& \\partial_{\\mu}\\epsilon^a -\ng_s f^{abc}\\epsilon^b G^c_{\\mu} = {\\cal D}^{ab}_{\\mu}(G) \\epsilon^b\n\\end{eqnarray*}\nHere ${\\cal D}^{ij}_{\\mu}(W)$ and $  {\\cal D}^{ab}_{\\mu}(G)$ stand for the \ncovariant derivatives, $g_s$ and $g$ are the $ SU(3)$ and $SU(2)$ gauge \ncoupling constants, respectively, $\\epsilon$ and $\\theta^{a(i)}$ are an \narbitrary functions depending on the space-time coordinates. It can be easily \nchecked that Lagrangian (\\ref{sm92}) is invariant under \nthese gauge transformations. \n\nLagrangian ${\\cal L_F}$ describes coupling of fermions with gauge fields. \nFor simplicity we shall consider one lepton generation, say $e^-$ and\n$\\nu_e$, and three quark generations. Fermions constitute only doublets and \nsinglets in $SU(2)\\times U(1)$\n\\begin{eqnarray*}\nR &=& e^-_R, \\ \\ \\ L = \n\\left( \\begin{array}{c} \\nu_L \\\\ e^-_L \\end{array} \\right)   \\\\\nR_I &=& \\left( q_I \\right)_R, \\ \\ R_i = \\left( q_i \\right)_R, \\ \\ \nL_I = \\left( \\begin{array}{c} q_I \\\\ \\ \\\\ V^{-1}_{Ii}q_i \\end{array} \\right)\n\\end{eqnarray*}\nwhere $L$ and $R$ denote left- and right-handed components of the spinors, \nrespectively: \n\\begin{eqnarray*}\ne_{R,L} = \\frac{ 1\\pm \\gamma_5}{2} e.\n\\end{eqnarray*}\nThe neutrino is assumed to be left-handed, while right-handed components of \nboth up- and down-quarks enter in the ${\\cal L_F}$. Indices $I$ and $i$ \nnumerate three quark generations: $I,i = 1,2,3$, and $I(i)$ refers to the \nup (down) quarks. A possible mixing of quark generations was taken into \naccount by introduction of Kobayashi-Maskava matrix $V_{iI}$ (see, for\nexample, \n\\cite{okun,pdg} for details).\nThe infinitesimal gauge transformations of fermion fields looks as follows:\n\\begin{eqnarray*}\n\\delta \\psi_{lep} &=& \\left(\nig'\\frac{Y}{2}\\theta(x) + ig\\frac{\\sigma^i}{2}\\theta^i(x) \\right)\n\\psi_{lep},  \\\\\n\\delta \\psi_{quark} &=& \\left(\nig'\\frac{Y}{2}\\theta(x) + ig\\frac{\\sigma^i}{2}\\theta^i(x)\n +ig_s t^a \\theta^a(x) \\right)\n\\psi_{quark},\n\\end{eqnarray*}\nwhere $g'$ is $U(1)$ gauge coupling constant.\nObviously, lepton and quark fields belong to the fundamental \nrepresentation of the $ SU(3)\\times SU(2) \\times U(1)$.\nUnder the requirements of the $ SU(3)\\times SU(2) \\times U(1)$\nlocal gauge symmetry and renormalizability of the theory, the Lagrangian\n${\\cal L_F}$ acquires the following expression\n\\begin{eqnarray}\n{\\cal L_F} = i \\bar {L} \\hat {D}_L L +  i \\bar {R} \\hat {D}_R R \n+ i \\sum_{I}\\left( \\bar {L}_I \\hat {D}_L^q L_I + \n \\bar {R}_I \\hat {D}^q_R R_I \\right)\n+i\\sum_{i} \\bar {R}_i \\hat {D}^q_R R_i, \\label{sm93}\n\\end{eqnarray}\nwhere covariant derivatives are given by\n\\begin{eqnarray*}\nD_{L\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig\\frac{\\sigma^i}{2}W^i_{\\mu}, \\\\\nD_{R\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu},  \\\\\nD^q_{L\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig\\frac{\\sigma^i}{2}W^i_{\\mu} - ig_s t^a G^a_{\\mu}, \\\\\nD^q_{R\\;\\mu} &=& \\partial_{\\mu} - ig'\\frac{Y}{2}W^0_{\\mu} -\n ig_s t^a G^a_{\\mu}.\n\\end{eqnarray*}\nWe remind that the value of hypercharge $Y$ is determined by the following\nrelation $Q~ =~ \\tau_3~ +~ Y\/2$ with $Q$ being the charge operator.\n\nBoth the gauge fields and fermion ones described above have zero mass, while \nin the reality all charged fermions are massive and intermediate bosons are \nknown to be very heavy. To make the weak bosons massive one can use Higgs \nmechanism of spontaneous breakdown of the $SU(2)\\times U(1)$ symmetry to the \n$U(1)$ symmetry. The widely accepted way to do that consists in the \nintroduction of the Higgs $SU(2)$ doublet $\\Phi$ (with $Y=1$). This doublet \nacquires the nonzero vacuum expectation value:\n\\begin{eqnarray*}\n <\\Phi> = \\left( \\begin{array}{c} 0 \\\\ \\ \\\\ \n \\frac{v}{\\sqrt2} \\end{array} \\right). \n\\end{eqnarray*}\nThe potential term $V(\\Phi)$, which can give rise to the symmetry violation,\n reads\n\\begin{eqnarray*}\nV(\\Phi) = -\\mu^2 \\Phi^+ \\Phi + \\lambda \\left(\\Phi^+\\Phi\\right)^2.\n\\end{eqnarray*}\nOne can easily verify that the vacuum expectation value satisfies to the \nconditions:\n\\begin{eqnarray*}\n\\tau^i <\\Phi> &=& \\frac{1}{2}\\sigma^i <\\Phi> \\neq 0, \\\\\nQ<\\Phi> &=& \\frac{1}{2}\\left(\\sigma_3+ Y\\right) = 0.\n\\end{eqnarray*}\nIt means, that only the symmetry generated by $Q$ is not broken on\nthis vacuum. \nLet us choose the Lagrangian for the Higgs field interaction with\ngauge fields in the form:\n\\begin{equation}\n{\\cal L_H} = \\left( D_{L\\;\\mu}\\Phi\\right)^+\\left( D_L^{\\mu}\\Phi\\right)\n-V(\\Phi).\n\\end{equation}\nThen one finds that only gauge boson coupling to $Q$ ( i.e. photon)\nremains massless, while other bosons acquire masses. Diagonalization of the\nmass matrix gives\n\\begin{eqnarray}\nW^{\\pm}_{\\mu} &=& \\frac{1}{\\sqrt2}( W^1_{\\mu} \\mp iW^2_{\\mu} ), \\;\\; \n M_W = \\frac{1}{2} g v, \\\\ \nZ_{\\mu} &=& \\frac{1}{\\sqrt{g^2+g'^2}} (g W^3_{\\mu} - g'W^0_{\\mu} ), \\;\\; \n M_Z = \\frac{1}{2}\\sqrt{g^2 +g'^2}v, \\\\\nA_{\\mu} &=& \\frac{1}{\\sqrt{g^2+g'^2}} (g' W^3_{\\mu} + g W^0_{\\mu} ), \\;\\;\n M_A = 0,\n\\end{eqnarray}\nwhere $W^{\\pm}_{\\mu}, Z_{\\mu}$ are charged and neutral weak bosons, $A_{\\mu}$ \nis the photon. It is suitable to introduce rotation angle $\\vartheta_W$\nbetween $(W^3,W^0)$ and $(Z,A)$, which is called the {\\it Weinberg angle} \n\\begin{equation}\n\\sin\\vartheta_W \\equiv g'\/\\sqrt{g^2+g'^2}.\n\\end{equation}\nThe relation of constants $g,g'$ with electromagnetic coupling constants $e$ \nfollows from (\\ref{sm93}). Since  the photon coupling with charged particles\nis proportional to $g g' \/\\sqrt{g^2 +g'^2}$, we should identify this \nquantity with the electric charge $e$:\n\\begin{equation}\n e = \\frac{g g'}{\\sqrt{g^2+g'^2}}.\n\\end{equation}\nIn order to find mass spectrum in the Higgs sector, let us \nexpress doublet $\\Phi$ in the form\n\\begin{eqnarray*}\n\\Phi = \\left( \\begin{array}{c} i\\omega^+ \\\\\n\\frac{1}{\\sqrt2}\\left( v +H -i z \\right)  \\end{array}\\right).\n\\end{eqnarray*}\nOne can verify that Nambu-Goldstone bosons $\\omega^{\\pm}, z$ have zero masses \nand may be cancelled away by suitable choice of the $SU(2)\\times U(1)$ \nrotation. The only {\\bf physical component of the Higgs doublet}\nis $H$, which acquires mass \n\\[ \nm_H = \\sqrt2 \\mu.\n\\] \n\nThe Lagrangian ${\\cal L_M}$ generates fermion mass terms. Supposing the \nneutrinos to be massless, we write the Yukawa interaction of the \nfermions with Higgs doublet in the form\n\\begin{eqnarray}\n{\\cal L_M} = - f_e \\bar {L}\\Phi R  - \\sum_{i} f_i \\bar {L}_i\\Phi R_i\n-\\sum_{I} f_I \\bar {L}_I \\left(i\\sigma^2\\Phi^*\\right)R_I + h.c.\n\\end{eqnarray}\nHere we introduced doublet $L_i $ related with $L_I$ by\n\\begin{eqnarray*}\nL_i = V_{i\\;I}\\;L_I,\n\\end{eqnarray*}\nand $f_{I,\\;i}$ are the Yukawa coupling constants. Then the masses of \nfermions in the tree approximation are given by\n\\begin{equation}\nm_{I,\\;i} = \\frac{f_{I,\\;i}\\;v}{\\sqrt2}.\n\\end{equation}\n\nIt is well known that quantization of dynamical systems is governed by \nLagrangians having local gauge symmetry requires an additional care. Freedom \nof redefining gauge and matter fields without changing the Lagrangians leads \nto the vanishing of some components of the momenta, canonically conjugate to \nthe gauge fields, say\n\\begin{eqnarray*}\n\\frac{\\delta L}{\\delta \\partial_0 A_{\\mu}} = - F^{0\\;\\mu} = 0 \n \\quad ({\\rm for} \\;\\; \\mu = 0).\n\\end{eqnarray*}\nTo perform the quantization procedure, one should add to the Lagrangian \na gauge fixing terms, breaking explicitly the local symmetry. In the \nfunctional integral formulation it leads, in the case of non-Abelian gauge \nsymmetry, to modification of the path integral measure \\cite{faddeev}. As a \nresult, the measure of the path integral will be multiplied by functional \ndeterminant $\\Delta (W^a_{\\mu})$. In order to apply the well known methods of \nperturbation theory, one may \nexponentiate $\\Delta(W^a_{\\mu})$ and redefine the initial Lagrangian. It can \nbe made by introducing auxiliary fields $c^a$ and $\\bar c^a$ which are scalar \nfields anticommuting with themselves and belonging  to the adjoint \nrepresentation of the Lie algebra.\nThe fields $c^a$ and $\\bar {c}^a$ are called Faddeev-Popov ghosts (FP ghosts).\n\nThe gauge fixing terms are usually chosen in the form \n\\begin{eqnarray*}\nL_{GF} = B^a F^a(W) + \\frac{\\xi}{2}\\left( B^a\\right)^2,\n\\end{eqnarray*}\nwhere $B^a$ are auxiliary fields introduced to linearize this expression,\n$\\xi$ is the gauge parameter, $ F^a = \\partial^{\\mu}W^a_{\\mu}$.\nThen FP ghosts enter in the Lagrangian in the following way\n\\begin{eqnarray}\nL_{FP} = -\\bar {c}^a \\frac{\\partial F^a}{\\partial W^c_{\\mu}}\nD^{cb}_{\\mu}(W) c^b. \\label{sm912}\n\\end{eqnarray}\nAs it was pointed above, these additional terms violate local gauge invariance, \nbut the final Lagrangian becomes invariant under the global transformations  \nmixing the gauge fields and FP ghosts. This symmetry, found by by Becchi, \nRouet, and Stora, was called BRS symmetry. The BRS infinitesimal \ntransformations are defined by the following relations\n\\begin{eqnarray*}\n\\delta^{BRS}\\psi(x) &=& i\\beta g c^a(x) t^a \\psi(x),  \\;\\;  \n\\delta^{BRS}W^a_{\\mu}(x) = \\beta D^{ab}_{\\mu}c^b(x),  \\\\\n\\delta^{BRS}\\bar {c}^a (x) &=& \\beta B^a(x), \\;\\;\n\\delta^{BRS} c^a(x) = -\\frac{\\beta}{2} g f^{abc} c^b(x) c^c(x),  \\\\\n\\delta^{BRS} B^a(x) &=& 0.\n\\end{eqnarray*}\nHere $\\psi$ denotes any matter field, the parameter $\\beta$ does not depend on \n$x$ and anticommutes with $c^a$ and $\\bar {c}^a$, as well as with all fermion \nfields. Using these relations, the formula (\\ref{sm912}) can be written in the \nbrief form:\n\\begin{eqnarray}\nL_{GF} = \\frac{\\delta}{\\delta \\beta}\\left(\n\\bar {c}^a \\delta^{BRS}\\left( \\partial^{\\mu}W^a_{\\mu}\\right)\\right),\n\\end{eqnarray}\nwhere $ \\delta\/\\delta \\beta$ means left differentiation.\n\nIn our case we choose the gauge fixing part of the Lagrangian in the form\n\\begin{eqnarray}\n{\\cal L_{GF}} &=& B^+ (\\partial^{\\mu} W^-_{\\mu} + \\xi_W M_W\n\\omega^- ) + B^- (\\partial^{\\mu} W^+_{\\mu} + \\xi_W M_W\n\\omega^+ ) \\\\\n&+& B^Z (\\partial^{\\mu} Z_{\\mu} + \\xi_Z M_Z z) \n+ B^A (\\partial^{\\mu} A_{\\mu})\n+ B^a (\\partial^{\\mu} G^a_{\\mu})  \\nonumber \\\\\n&+& \\xi_W B^+ B^- + \\frac{\\xi_Z}{2} B^Z B^Z\n + \\frac{\\xi_A}{2} B^A B^A + \\frac{\\xi_G}{2} B^a_G B^a_G,\n\\nonumber \n\\end{eqnarray}\nthen FP--ghost Lagrangian looks as follows:\n\\begin{eqnarray}\n&&{\\cal L_{FP}} =   \\\\\n&&\\frac{\\delta}{\\delta \\beta} \\Bigl \\{\n\\bar {c}^{\\;+} \\delta^{BRS}\\left( \\partial^{\\mu}W^-_{\\mu} + \\xi_W \nM_W \\omega^- \\right) +\n\\bar {c}^{\\;-} \\delta^{BRS}\\left( \\partial^{\\mu}W^+_{\\mu} + \\xi_W \nM_W \\omega^+ \\right)  \\nonumber \\\\\n&&+ \\bar {c}^{\\;Z} \\delta^{BRS}\\left( \\partial^{\\mu}Z_{\\mu} + \\xi_Z\nM_Z \\;z \\right)  \n+ \\bar {c}^{\\;A} \\delta^{BRS}\\left( \\partial^{\\mu}A_{\\mu} \\right)\n+ \\bar {c}^{\\;a} \\delta^{BRS}\\left( \\partial^{\\mu}G^a_{\\mu} \\right)\\Bigr\\},\n \\nonumber\n\\end{eqnarray}\nwhere the fields $c^A,c^Z$ are constructed from original ghosts\n$c^0,c^3$ just like the bosons $Z_{\\mu}, A_{\\mu}$ from initial fields\n$W^0_{\\mu}, W^3_{\\mu}$.\n\n{\\bf The total Lagrangian of the Standard Model} \\\\\nNow, we are ready to present the total Lagrangian of the \n{\\bf Standard Model} rewritten in the terms of physical fields \\cite{aoki}.\n\\begin{eqnarray}\n{\\cal L_G} &=& -\\frac{1}{2}F^+_{\\mu\\nu}F^{-\\;\\mu\\nu}\n -\\frac{1}{4}( F^Z_{\\mu\\nu})^2  -\\frac{1}{4}( F^A_{\\mu\\nu})^2\n -\\frac{1}{4}( G^a_{\\mu\\nu})^2     \\label{sm917} \\\\\n&+& i e \\cot\\vartheta_W \\left(  \ng^{\\alpha\\gamma}g^{\\beta\\delta} - g^{\\alpha\\delta}g^{\\beta\\gamma} \\right)\n\\Bigl( W^-_{\\gamma}Z_{\\delta}\\partial_{\\alpha}W^+_{\\beta}    \n+Z_{\\gamma}W^+_{\\delta}\\partial_{\\alpha}W^-_{\\beta} \\nonumber \\\\\n&+& W^+_{\\gamma}W^-_{\\delta}\\partial_{\\alpha}Z_{\\beta} \\Bigr)        \n + i e  \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\delta}g^{\\beta\\gamma} \\right)\n\\Bigl(  W^-_{\\gamma}A_{\\delta}\\partial_{\\alpha}W^+_{\\beta}  \\nonumber \\\\\n& + & A_{\\gamma}W^+_{\\delta}\\partial_{\\alpha}W^-_{\\beta} \n+W^+_{\\gamma}W^-_{\\delta}\\partial_{\\alpha}A_{\\beta} \\Bigr)          \n- \\frac{1}{2}\ng_s f^{abc} G^a_{\\mu} G^b_{\\nu} \\partial^{\\mu} G^{c\\;\\nu}  \\nonumber \\\\\n&+& e^2 \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta} A_{\\gamma}A_{\\delta}   \\nonumber \\\\\n& + & e^2  \\cot^2\\vartheta_W \\left( g^{\\alpha\\gamma}g^{\\beta\\delta} -\ng^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta} Z_{\\gamma}Z_{\\delta})  \\nonumber \\\\\n &+& e^2 \\cot\\vartheta_W \\left(\n g^{\\alpha\\delta}g^{\\beta\\gamma} + g^{\\alpha\\gamma}g^{\\beta\\delta} \n- 2 g^{\\alpha\\beta}g^{\\gamma\\delta} \\right)\nW^+_{\\alpha}W^-_{\\beta}A_{\\gamma}Z_{\\delta} \\nonumber \\\\\n &+& \\frac{e^2}{2\\sin^2\\vartheta_W} \\left(\ng^{\\alpha\\beta}g^{\\gamma\\delta} - g^{\\alpha\\gamma}g^{\\beta\\delta} \\right)\nW^+_{\\alpha}W^+_{\\beta}W^-_{\\gamma}W^-_{\\delta} \\nonumber \\\\\n &-&\\frac{1}{4} g^2_s f^{rab} f^{rcd} G^a_{\\mu} G^b_{\\nu} G^{c\\;\\mu} \nG^{d\\;\\nu},    \\nonumber \n\\end{eqnarray}\nwhere the field sthrenghtes $G^a_{\\mu\\nu}, F^+_{\\mu\\nu}, \\ldots$\nare given by\n\\begin{eqnarray*}\n&&F^+_{\\mu\\nu} = \\partial_{\\mu}W^+_{\\nu} - \\partial_{\\nu}W^+_{\\mu}, \\\\\n&&G^a_{\\mu\\nu} = \\partial_{\\mu}G^a_{\\nu} - \\partial_{\\nu}G^a_{\\mu}, \\\\\n&& \\cdots\n\\end{eqnarray*}\n\n\\begin{eqnarray}\n{\\cal L_F} &=&  i \\bar {e} \\hat {\\partial} e + \ni \\bar{\\nu}_L \\hat {\\partial} \\nu_L  + \ni \\sum_{n} \\bar {q}_n \\hat {\\partial} q_n  \\label{sm918} \\\\\n&+& \\frac{e}{\\sqrt2 \\sin\\vartheta_W} \\left(\n\\bar {\\nu}_L \\hat {W}^+ e_L +\\bar {e}_L \\hat {W}^- \\nu_L \\right)\n+ \\frac{e}{\\sin 2\\vartheta_W} \n\\bar {\\nu}_L \\hat {Z} \\nu_L   \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W } \\left(\n\\bar {e}\\hat {Z} ( 2\\sin^2\\vartheta_W - \\frac{1 - \\gamma_5}{2} ) e \\right)\n- e  \\bar {e} \\hat {A} e  \\nonumber \\\\\n&+& \\frac{e}{\\sqrt2 \\sin\\vartheta_W} \\sum_{I,i} \\left(\n\\bar {q}_I \\hat {W}^+ q_{i\\;L} (V^+)_{Ii} \n+ \\bar {q}_i \\hat {W}^- q_{I\\;L} V_{iI} \\right)  \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W}\\sum_{I} \\left(\n\\bar {q}_I \\hat {Z} ( \\frac{1 - \\gamma_5}{2} - 2 Q_I\\sin^2\\vartheta_W ) q_I\n\\right) \\nonumber \\\\\n&+& \\frac{e}{\\sin 2\\vartheta_W}\\sum_{i} \\left(\n\\bar {q}_i \\hat {Z} ( \\frac{-1 + \\gamma_5}{2} - 2 Q_i\\sin^2\\vartheta_W ) q_i\n\\right) \\nonumber \\\\\n&+& e \\sum_{n} Q_n  \\bar {q}_n \\hat {A} q_n\n+ g_s \\sum_{n} \\bar {q}_n G^a_{\\mu} \\gamma^{\\mu} t^a q \\nonumber\n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\cal L_H} &=& \\frac{1}{2} (\\partial_{\\mu}H)^2 - \\frac{m^2_H}{2}H^2\n+ \\frac{1}{2} (\\partial_{\\mu} z)^2 + \n\\partial_{\\mu}\\omega^+ \\partial^{\\mu}\\omega^-   \\label{sm919}  \\\\\n&+& M^2_W W^+_{\\mu}W^{-\\;\\mu} +  \\frac{1}{2}M_Z^2 Z^2_{\\mu} -M_W \\left(\nW^-_{\\mu}\\partial^{\\mu}\\omega^+ + W^+_{\\mu}\\partial^{\\mu}\\omega^-\n\\right)                                       \\nonumber \\\\\n&-& M_Z Z_{\\mu} \\partial^{\\mu} z                   \n+ \\frac{e M_W}{\\sin\\vartheta_W} H W^+_{\\mu} W^{-\\;\\mu}\n+ \\frac{e M_Z}{\\sin2\\vartheta_W} H\\,  Z^2_{\\mu}  \\nonumber \\\\\n&+& \\frac{e}{2\\sin\\vartheta_W} W^{+\\;\\mu} \\left(\\omega^-  \n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} ( H - i z)\\right) \n+ \\frac{e}{2\\sin\\vartheta_W} W^{-\\;\\mu} \\left(\\omega^+ \n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} ( H + i z)\\right) \\nonumber  \\\\\n&+& i e ( A^{\\mu}+ \\cot 2\\vartheta_W Z^{\\mu}) \\left(\\omega^-\n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu}\\omega^+\\right) \n+ \\frac{e}{\\sin 2\\vartheta_W} Z^{\\mu} \\left( z\n\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} H \\right)   \\nonumber \\\\\n&+& i e M_Z \\sin\\vartheta_W Z^{\\mu} ( W^+_{\\mu}\\omega^- \n- W^-_{\\mu}\\omega^+ )  +\ni e M_W A^{\\mu} (  W^-_{\\mu}\\omega^+ -  W^+_{\\mu}\\omega^-) \\nonumber \\\\  \n&+& \\frac{e^2}{4\\sin^2\\vartheta_W} H^2 ( W^+_{\\mu}W^{-\\;\\mu} +\n2\\;Z^2_{\\mu} )\n+ \\frac{i e^2}{2\\cos\\vartheta_W} H Z^{\\mu} ( W^+_{\\mu}\\omega^-\n- W^-_{\\mu}\\omega^+ )                           \\nonumber \\\\\n&+& \\frac{i e^2}{2\\sin\\vartheta_W} H A^{\\mu} ( W^-_{\\mu}\\omega^+\n- W^+_{\\mu}\\omega^- )\n+ \\frac{e^2}{4\\sin^2\\vartheta_W} z^2 ( W^+_{\\mu}W^{-\\;\\mu} +\n 2\\;Z^2_{\\mu })        \\nonumber    \\\\\n&+& \\frac{e^2}{2\\cos\\vartheta_W} z Z^{\\mu} ( W^+_{\\mu}\\omega^-\n+ W^-_{\\mu}\\omega^+ ) \n- \\frac{e^2}{2\\sin\\vartheta_W} z A^{\\mu} ( W^+_{\\mu}\\omega^-\n+ W^-_{\\mu}\\omega^+ )                                \\nonumber \\\\\n&+& \\frac{e^2}{2\\sin^2\\vartheta_W}\\omega^+\\omega^- W^+_{\\mu}W^{-\\;\\mu}\n+ e^2 \\cot^2 2\\vartheta_W \\omega^+\\omega^- Z_{\\mu}^2\n+ e^2 \\omega^+\\omega^- A_{\\mu}^2    \\nonumber \\\\\n&+& 2 e^2 \\cot(2\\vartheta) \\omega^+\\omega^- A^{\\mu}Z_{\\mu}  \n-\\frac{e m^2_H}{4 M_W \\sin\\vartheta_W} H^3\n- \\frac{e m^2_H}{2 M_W \\sin\\vartheta_W} \\omega^+\\omega_- H \\nonumber \\\\\n&-& \\frac{e m^2_H}{4 M_W \\sin\\vartheta_W} z^2 H\n-\\frac{e^2 m^2_H}{32 M^2_W \\sin^2\\vartheta_W} H^4\n-\\frac{e^2 m^2_H}{32 M^2_W \\sin^2\\vartheta_W} z^4   \\nonumber \\\\\n&-& \\frac{e^2 m^2_H}{8 M^2_W \\sin^2\\vartheta_W}\\omega^+ \\omega^- \n( H^2 + z^2)\n-\\frac{e^2 m^2_H}{16 M^2_W \\sin^2\\vartheta_W} z^2 H^2\n\\nonumber \\\\\n&-& \\frac{e^2 m^2_H}{8 M^2_W \\sin^2\\vartheta_W} \n(\\omega^+\\omega^-)^2 \\nonumber\n\\end{eqnarray}\nHere symbol $f\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} g$ is\nused as usual: $f\\stackrel{\\leftrightarrow}{\\partial}_{\\mu} g \\equiv \nf \\partial_{\\mu} g - ({\\partial}_{\\mu} f) g$.\n\n\\begin{eqnarray}\n{\\cal L_M} &=& -\\frac{e m_e}{M_Z \\sin 2\\vartheta_W}\nH \\bar {e} e  - \\frac{e}{M_Z \\sin 2\\vartheta_W}\n\\sum_{n} m_n H \\bar {q}_n q_n        \\label{sm920} \\\\\n&+& \\frac{i e \\sqrt2 m_e}{M_Z \\sin 2\\vartheta_W} \\left(\n\\omega^- \\bar {e} \\nu_L - \\omega^+ \\bar {\\nu}_L e \\right)\n+ \\frac{i e m_e}{M_Z \\sin 2\\vartheta_W} z\n\\bar {e} \\gamma_5 e                    \\nonumber \\\\\n&+& \\frac{i e  }{\\sqrt2 M_Z \\sin 2\\vartheta_W}\n\\omega^+ \\sum_{I,i}( V^+)_{Ii} \\bar {q}_I \\left(\nm_I - m_i - ( m_I + m_i) \\gamma_5 \\right) q_i  \\nonumber \\\\\n&+& \\frac{i e  }{\\sqrt2 M_Z \\sin 2\\vartheta_W}\n\\omega^- \\sum_{I,i}( V)_{iI} \\bar {q}_i \\left(\nm_i - m_I - ( m_I + m_i) \\gamma_5 \\right) q_I  \\nonumber \\\\\n&-& \\frac{i e  }{ M_Z \\sin 2\\vartheta_W} \\sum_{I}\nm_I\\bar {q}_I \\gamma_5 q_I\n + \\frac{i e  }{ M_Z \\sin 2\\vartheta_W} \\sum_{i}\nm_i\\bar {q}_i \\gamma_5 q_i \\nonumber \n\\end{eqnarray}\n\n\\begin{eqnarray}\n{\\cal L_{FP}} &=& -\\bar {c}^{\\;+} (\\partial^2 + \\xi_W M_W^2 )c^-\n-\\bar {c}^{\\;-} (\\partial^2 + \\xi_W M_W^2 )c^+ \n -\\bar {c}^{\\;A} \\partial^2 c^A      \\label{sm921} \\\\\n&-&\\bar {c}^{\\;Z} (\\partial^2 + \\xi_Z\\; M_Z^2 )c^Z           \n-\\bar {c}^{\\;a} \\partial^2 c^a\n+ i e \\cot\\vartheta_W W^{+\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;-} c^Z - \\partial_{\\mu}\\bar {c}^{\\;Z} c^-\n\\right)                                    \\nonumber \\\\\n&+& i e W^{+\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;-} c^A - \\partial_{\\mu}\\bar {c}^{\\;A} c^-\n\\right)\n- i e \\cot\\vartheta_W W^{-\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^Z - \\partial_{\\mu}\\bar {c}^{\\;Z} c^+\n\\right)                                    \\nonumber \\\\\n&-& i e W^{-\\;\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^A - \\partial_{\\mu}\\bar {c}^{\\;A} c^+\n\\right)\n+ i e \\cot\\vartheta_W Z^{\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^-  - \\partial_{\\mu}\\bar {c}^{\\;-} c^+\n\\right)                                  \\nonumber \\\\\n&+& i e A^{\\mu}\\left(\n\\partial_{\\mu}\\bar {c}^{\\;+} c^-  - \\partial_{\\mu}\\bar {c}^{\\;-} c^+\n\\right)       \\nonumber \\\\\n&+& i \\omega^+ \\left(\n-\\xi_W e M_W \\cot 2\\vartheta_W \\bar {c}^{\\;-} c^Z\n-\\xi_W e M_W \\bar {c}^{\\;-} c^A\n+\\frac{\\xi_Z e}{2\\sin\\vartheta_W} M_Z \\bar {c}^{\\;Z} c^-\n\\right)                                    \\nonumber \\\\\n&+& i \\omega^- \\left(\n\\xi_W e M_W \\cot 2\\vartheta_W \\bar {c}^{\\;+} c^Z\n+ \\xi_W e M_W \\bar {c}^{\\;+} c^A\n-\\frac{\\xi_Z e}{2\\sin\\vartheta_W} M_Z \\bar {c}^{\\;Z} c^+\n\\right)                                     \\nonumber \\\\\n&+& \\frac{i \\xi_W e}{2} M_Z \\cot\\vartheta_W z \\left(\n\\bar {c}^- c^+ - \\bar {c}^{\\;+} c^- \\right)  \n- \\frac{ \\xi_W e}{2\\sin\\vartheta_W} M_W H \\left(\n\\bar {c}^{\\;-} c^+ + \\bar {c}^{\\;+} c^-\n\\right)                                \\nonumber \\\\\n&-& \\frac{ \\xi_Z e}{\\sin 2\\vartheta_W} M_Z H \n\\bar {c}^{\\;Z} c^Z   \\nonumber\n\\end{eqnarray}\n\n\n\n\\section{\\bf FEYNMAN RULES}\\label{fr}\n\n\\subsection{\\it General Remarks}\nThis Section presents the complete list of Feynman rules\ncorresponding to the Lagrangian of SM (see (\\ref{sm917} -- \\ref{sm921})).\n\nFirst of all we define the propagators by the relation\n\\begin{equation}\n\\Delta_{ij}(k) = i \\int d^4 x\\; e^{-ikx} <0|T(\\phi_i(x)\\phi_j(0)|0>,\n\\end{equation}\nwhere $\\phi_i$ presents any field. Curly, wavy and zigzag lines denote \ngluons, photons and weak bosons respectively, while full, dashed and dot\nlines \nstand for fermions (leptons and quarks), Higgs particles and ghosts fields,\nrespectively. Arrows on the propagator lines show : for the $W^+$ and \n$\\omega^+$ fields the flow of the positive charge, for the fermion that of\nthe \nfermion number, and for the ghost that of the ghost number.\n \nThe vertices are derived using ${\\it L_I}$, instead of usual usage of\n${\\it i\\; L_I}$. All the momenta of the particles are supposed to flow in.\nThe only exception was made for the ghost fields, where\ndirection of momentum  coincides with the direction of ghost number flow.\nThis convention permits to minimize the number of times when the \nimaginary unit $i$ appears. \n\nIt should be noted ones more, that all fields can be ''divided'' into two\nparts:  \n\\begin{verse}\n$\\bullet$ {\\it physical fields}: \n$A$ (photon), $W^{\\pm}$, $Z$, $G$ (gluon), $\\psi$, $H$ (Hiigs). \\\\\n$\\bullet$ {\\it non--physical fields}: \n$\\omega^{\\pm}$, $z$ (pseudogoldstones), $c^{\\pm}$, $c^z$, $c^A$, $c_a$ (\nghosts).\n\\end{verse}\n\n\\noindent Charged fermions have the electric charges (in the positron charge\n$e$ units):\n\\begin{eqnarray*}\n &&Q(e^-) = Q(\\mu^-) = Q(\\tau^-) = -e, \\\\ \n &&Q(e^+) = Q(\\mu^+) = Q(\\tau^+) = +e, \\\\\n &&Q(u) = Q(c) = Q(t) = +\\frac{2}{3}e, \\\\ \n &&Q(d) = Q(s) = Q(b) = -\\frac{1}{3}e. \n\\end{eqnarray*}\nThe electric charge $e$ (or strong coupling constant $g_s$ in QCD) is related\nto the fine structure constant $\\alpha$ (or $\\alpha_s$ in QCD) as follows:\n\\[\n\\alpha_{QED} \\equiv \\alpha = \\frac{e^2}{4 \\pi}, \\quad\n\\alpha_{QCD} \\equiv \\alpha_s = \\frac{g_s^2}{4 \\pi}.\n\\]\nThe electric charge, the $\\sin \\vartheta_W$, and Fermi constant\n$G_F$ are related as follows:\n\\begin{eqnarray}\n \\frac{e}{2 \\sqrt{2} \\sin \\vartheta_W} = M_W \\sqrt{\\frac{G_F}{\\sqrt{2}}}.\n \\label{vr3}\n\\end{eqnarray}\nFinally, every loop integration is performed by the rule\n\\begin{equation}\n\\int \\frac{d^d \\;k}{i \\, (2\\pi)^d},\n\\end{equation}\nand with  every fermion or ghost loop we associate extra factor $(-1)$.  \n\n\\subsection{\\it Propagators}\\label{prop}\n\\vspace{5mm}\n\n\\includegraphics[scale=0.95]{figs\/fig_prop.pdf} \n\n\n\\vspace{5mm}\n\n\\subsection{\\it Some Popular Gauges}\\label{gauges} \n\nHere we discuss the explicit forms of the propagators for some popular\ngauges.\nLet us consider a theory with free  boson Lagrangian:\n\\[\n{\\cal L} = -\\frac{1}{4} F^2_{\\mu \\nu}, \\quad F_{\\mu \\nu} = \n\\partial_{\\mu} A_{\\nu} - \\partial_{\\nu} A_{\\mu}.\n\\]\nOne can fix a gauge in one of three ways \\cite{itzu,lieb}: \n\\begin{itemize}\n\\item{\\bf i} to impose a gauge condition,\n\\item{\\bf ii} to add a {\\it Gauge Fixing Term} (GFT) to the Lagrangian\n\\item{\\bf iii} to fix a form of the Hamiltonian.\n\\end{itemize}\n\nIn a rigorous theory one should impose two gauge conditions. However,\nas it is \nusually accepted, we write only one condition. It should be considered\nrather \nas a symbol which denotes acceptable for a given gauge procedure of \nquantization, described somewhere in literature.\n\nIn practical calculations one needs an explicit form of a propagator with \nsatisfactory prescription for poles (which plays a key role in the loop \ncalculations). For this purposes it is sufficient to fix a gauge as\nmentioned \nin {\\bf ii} and {\\bf iii}. Polarization vectors of physical bosons and\nghosts \nshould be chosen in accordance with a detailed quantization procedure\napplicable for a given gauge. \n\n{\\bf Covariant gauges.}\n\n1. {\\it Generalized Lorentz gauge.}\n  \n\\begin{itemize}\n\\item{Notation}  $\\partial ^\\mu A_\\mu (x) = B(x)$\n\\item{GFT} $L_{GF}=-\\displaystyle \\frac{1}{2\\xi}(\\partial ^\\mu A_\\mu)^2$\n\\item{Propagator}\n\\[\nD^{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g^{\\mu\\nu} \n- (1-\\xi)\\frac{k^{\\mu}k^{\\nu}}{k^2+i\\varepsilon}].\n\\]\n\\item{Comments} \n\n$\\xi =1$ is Feynman gauge, while $\\xi =0$ is Landau gauge. For the photon \n(gluon) propagator one should write $\\xi_G (\\xi_A)$ (see\nSubsection~\\ref{prop}). \n\\end{itemize}\n\n2. {\\it 't Hooft gauges ($R_\\xi$-gauges).}\n\n\\begin{itemize}\n\\item{Notation}  $\\partial ^\\mu A^a_\\mu (x) -i\\xi (v, \\tau^a \\phi) = B^a(x)$\n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{2\\xi}(\\partial ^\\mu A^a_\\mu)^2$\n\\item{Propagator}\n\\[\nD^{ab}_{\\mu\\nu} = \\frac{\\delta^{ab}}{k^2-M^2+i\\varepsilon}[g_{\\mu\\nu} \n- (1-\\xi)\\frac{k_{\\mu}k_{\\nu}}{k^2-\\xi M^2+i\\varepsilon}].\n\\]\n\\item{Comments} \n\nThe gauge parameter $\\xi = \\xi_W (\\xi_Z)$ for the case of $W(Z)$ boson (see \nSubsection~\\ref{prop}), $v\/\\sqrt{2}$ is the vacuum expectation\nvalue of the gauge \nfield, $\\tau^a$ are generators, $M$ is the vector boson mass.  $\\xi =1$ is \n't Hooft--Feynman gauge, $\\xi = 0$ is Landau gauge, $\\xi\\rightarrow\\infty$\ncorresponds to {\\it unitary} gauge. Non--physical gauge bosons should\nalso be \ntaken into account in loop calculations. They also have gauge--dependent \npropagator, see Subsection~\\ref{prop}. \n\\end{itemize}\n\n{\\bf Non-covariant gauges}\n\n3. {\\it Coulomb gauge.}\n\\begin{itemize}\n\\item{Notation}  $\\vec \\partial \\vec A =0,\\ k=1,2,3.$ \n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{2\\xi}(\\partial_kA_k)^2.$\n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{k_{\\mu}k_{\\nu}-k_0 k_\\mu g_{\\nu 0} - k_0 k_\\nu g_{\\mu 0}}\n{|\\vec k |^2} -\\frac{\\xi k^2k_\\mu q_\\nu}{|\\vec k|^4}].\n\\]\n\\item{Comments} \n\nThe proper {\\it Coulomb gauge} corresponds to the case $\\xi = 0$.\n\\end{itemize}\n\n\n4. {\\it The general axial gauge.} \n\n\\begin{itemize}\n\\item{Notation} $n^\\mu A_\\mu(x)=B(x).$\n\\item{GFT} $L_{GF}= \\displaystyle -\\frac{1}{4\\xi}[n^*\\cdot \\partial\\ \nn\\cdot A]^2.$ \n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu} + k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} \n- \\frac{(\\xi k^2 - n^2)\\ (n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k\n +i\\varepsilon)^2} k_{\\mu}k_{\\nu}].\n\\]\n\\item{Comments}\n\nFeynman rules in this gauge usually do not contain ghosts. As it has been shown\nin \\cite{lieb} one has to consider an additional gauge vector $n^{\\ast \\mu}$ in\norder to have a correct prescription for poles. The quantization\nin this gauge was considered, for example, in \\cite{burnel,bassetto}.\n\nThe gauge vector $n^\\mu$ has the form: \n\\[ n^\\mu\\ =\\ (n_0;\\vec n)\\ =\\ (n_0; \\vec n_\\bot,n_3)\\ =\\ (n_0;n_1,n_2,n_3).\\]\nThe explicit form of the component structure of $n^\\mu$ and $n^{\\ast \\mu}$ \nshould be considered separately in the cases $n^2>0,\\ n^2=0$ and $n^2<0$. \nThe following widely used gauges are obtained in the limit $\\xi = 0$:\n\\end{itemize}\n\n4a. {\\it Temporal gauge:} $n^2>0$.\n\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} + \\frac{n^2\\ \n(n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k+i\\varepsilon)^2} k_{\\mu} k_{\\nu}],\n\\]\n\\[\nn^\\mu\\ =\\ (n_0; \\vec n_\\bot,-i|\\vec n_\\bot|);\\ \\ \nn^{\\ast \\, \\mu}\\ =\\ (n_0; \\vec n_\\bot, i| \\vec n_\\bot|).\n\\]\n\n4b. {\\it Light--cone gauge:} $n^2=0$.\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} ],\n\\]\n\\[\nn^\\mu\\ =\\ (|\\vec n|;\\vec n);\\ \\ n^{* \\, \\mu}\\ =\\ (|\\vec n|;-\\vec n).\n\\]\n\n4c. {\\it Proper axial gauge:} $n^2<0$.\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon}[g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} + \\frac{ n^2\\ \n(n^*\\cdot k)^2}{(n\\cdot k\\ n^*\\cdot k+i\\varepsilon)^2} k_{\\mu} k_{\\nu}],\n\\]\n\\[\nn^\\mu\\ =\\ (|\\vec n_\\bot|;\\vec n);\\ \\ n^{* \\, \\mu}\\ =\\ (|\\vec n_\\bot|;-\\vec n).\n\\]\n\n5. {\\it Planar gauge.}\n  \n\\begin{itemize}\n\\item{Notation} $n^\\mu A_\\mu(x)=B(x),\\ n^2\\neq 0.$\n\\item{GFT} $L_{GF}=\\displaystyle \\frac{1}{2n^2}[\\partial_\\mu (n\\cdot A)]^2.$ \n\\item{Propagator}\n\\[\nD_{\\mu\\nu} = \\frac{1}{k^2+i\\varepsilon} \\left [ g_{\\mu\\nu} \n- \\frac{(n_{\\mu}k_{\\nu}+k_{\\mu}n_{\\nu})\\ n^*\\cdot k}\n{n\\cdot k\\ n^*\\cdot k+i\\varepsilon} \\right ],\n\\]\n\\[\nn^\\mu\\ =\\ (n_0; \\vec n_\\bot,-i| \\vec n_\\bot|),\\ \\ \nn^{* \\, \\mu}\\ =\\ (n_0; \\vec n_\\bot,i|\\vec n_\\bot|),\\ \\ \\mbox{if}\\ \\ n^2>0;\n\\]\n\\[\nn_\\mu\\ =\\ (| \\vec n_\\bot|;\\vec n),\\ \\ n^*_\\mu\\ =\\ (|\\vec n_\\bot|;-\\vec n)\n\\ \\ \\mbox{if} \\ \\ n^2<0.\n\\]\n\n\\item{Comments}\n\nYang-Mills theory is not multiplicatively renormalizable in this gauge. \nQuantization in this gauge is also poorly understood. This gauge has the same \ndenotation as the axial gauge, that is not suitable. However, that should not \nlead to confusion (see the beginning of this Subsection).\n\n\\end{itemize}\n\n\n\n\\section{\\bf INTEGRATION IN $N$-DIMENSIONS}\\label{dimreg}\n\n\\subsection{\\it Dimensional Regularization }\n\nA powerful method of the evaluation of the loop integrals (which very often\nare\ndivergent) is {\\it Dimensional Regularization } (DR) \\cite{hooft}. The idea\nof\nDR is to consider the loop integral as an {\\it analytic function} of $n$ --\nnumber of dimensions. Then one can calculate this integral in that region of\nthe complex $n$ plane, where this function is convergent.\n\nA typical loop integral looks as follows:\n\\[\n\\int_{}^{} \\frac{d^4 p} {(2 \\pi)^4} \\frac{P(q_i^{\\nu}, m_i, p^{\\nu})}\n{ \\prod_{i=1}^{l} ( m_i^2 - (p - k_i)^2)},\n\\]\nwhere $q_i$ ($m_i$) are 4--momenta (masses) of external particles; \n$P(q_i^{\\nu}, m_i, p^{\\nu})$ is a function of masses $m_i$ and momenta \n$q_i$ and $p$.\n\nTo use the DR method one needs to transform the product of denominators into\nexpression such as : $p^2 + (pk) + M^2$, where $k^{\\nu}$ is the linear\ncombination of $q_i$ momenta and $M$ is a combination of $q_i^2$, $(q_i q_j)$,\nand $m_i^2$. That can be done by using of {\\it Feynman parameterization}:\n\\begin{eqnarray*}\n\\frac{1}{a^{\\alpha}b^{\\beta}} &=& \n\\frac{\\Gamma(\\alpha+\\beta)}{\\Gamma(\\alpha)\\Gamma(\\beta)}\n\\int_{0}^{1} dx \\frac{x^{\\alpha-1}(1-x)^{\\beta-1} }\n { [ax+b(1-x)]^{\\alpha+\\beta}}, \\\\ \n\\frac{1}{a^n} - \\frac{1}{b^n} &=& \\int_{0}^{1}  \n \\frac{n(a-b) dx}{ [(a-b)x+b]^{n+1} }, \\\\\n\\frac{1}{b_1^{\\alpha_1}b_2^{\\alpha_2} \\ldots b_m^{\\alpha_m}} &=& \n\\frac{\\Gamma(\\alpha_1+ \\ldots +\\alpha_m)}\n{\\Gamma(\\alpha_1)\\Gamma(\\alpha_2) \\ldots \\Gamma(\\alpha_m)}\n\\int_{0}^{1} dx_1 \\int_{0}^{x_1} dx_2 \\ldots \\int_{0}^{x_{m-2}} dx_{m-1} \\\\\n&& \\frac{\nx_{m-1}^{\\alpha_1-1}(x_{m-2}-x_{m-1})^{\\alpha_2-1} \\ldots \n(1-x_1)^{\\alpha_m-1}}\n{ [b_1x_{m-1}+b_2(x_{m-2}-x_{m-1})+\\ldots+b_m(1-x_1)]^\n{\\alpha_1+\\ldots+\\alpha_m}},\n\\end{eqnarray*}\nwhere $\\Gamma(z)$ is the Euler {\\it Gamma function}.\n\nUsing the Wick rotation $p_0 \\; \\to \\; i p_0$ and replacement $4 \\to n$, one\ncan obtain a typical integral in $n$-dimensional Euclidean space:\n\\[\n J = \\int d^np \\frac{P(q_i^{\\nu}, m_i, p^{\\nu})}\n{(p^2+2(pk)+M^2)^{\\alpha}}, \\qquad {\\rm Re} \\; \\alpha > 0.\n\\]\nThe differential $d^n p$ has the form:\n\\begin{eqnarray}\n d^n p &=& p^{n-1} dp \\; d\\Omega_n, \\quad \n \\int d\\Omega_n = \\Omega_n = 2\\pi^\\frac{n}{2} \/ \\Gamma(\\frac{n}{2}), \n \\label{int1} \\\\\n d\\Omega_n &=& (\\sin^{n-2}\\vartheta_{n-1} d\\vartheta_{n-1}) d\\Omega_{n-1}  \n \\label{int2} \\\\\n & = &  (\\sin^{n-2}\\vartheta_{n-1} d\\vartheta_{n-1})\n(\\sin^{n-3}\\vartheta_{n-2} d\\vartheta_{n-2}) \\ldots  d\\vartheta_1, \\nonumber\n\\end{eqnarray}\nwhere $0 \\leq \\vartheta_i \\leq \\pi, \\quad  0 \\leq \\vartheta_1 \\leq 2\\pi.$\n(The\nlast equality in (\\ref{int2}) obeys for the integer $n$.)\n\n\\subsection{\\it Integrals}\n\nLet us introduce the following notation:\n\\begin{eqnarray}\n \\hat J \\; f(p) \\; &\\equiv& \\; \\int d^np  \\; \\frac{1} \n{(p^2+2(pk)+M^2)^{\\alpha}} \\; f(p), \\label{int3} \\\\\nJ_0 \\; &\\equiv& \\; \\frac{i\\pi^{n\/2}\\;\\; i^n}{(M^2-k^2)^{\\alpha-\\frac{n}{2}}\n\\Gamma(\\alpha)}.   \\label{int4}\n\\end{eqnarray} \nThen :\n\\begin{eqnarray*}\nI_0 &=& \\hat J \\; 1 =  \\frac{i\\pi^{n\/2}}{(M^2-k^2)^{\\alpha-n\/2}} \n\\frac{\\Gamma(\\alpha-\\frac{n}{2})}{\\Gamma(\\alpha)} =\n\\Gamma(\\alpha-\\frac{n}{2}) \n J_0. \\\\\nI^{\\mu} &=& \\hat J \\; p^{\\mu} = (-k^{\\mu}) I_0, \\\\\nI_2  &=& \\hat J \\; p^2 = J_0 \\{ k^2\\Gamma(\\alpha-\\frac{n}{2}) + \n\\frac{n}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2)\\}, \\\\\nI^{\\mu\\nu} &=& \\hat J \\; p^{\\mu}p^{\\nu} = J_0 \\{k^{\\mu}k^{\\nu} \n\\Gamma(\\alpha-\\frac{n}{2}) + \\frac{1}{2}\\Gamma(\\alpha-1-\\frac{n}{2})\n g^{\\mu\\nu}(M^2-k^2) \\}, \\\\\nI^{\\mu\\nu\\lambda} &=& \\hat J \\; p^{\\mu}p^{\\nu}p^{\\lambda} = \n J_0 \\{-k^{\\mu}k^{\\nu}k^{\\lambda}\\Gamma(\\alpha-\\frac{n}{2}) \\\\\n &&- \\frac{1}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2)(g^{\\mu\\nu}k^{\\lambda}\n + g^{\\mu\\lambda}k^{\\nu} + g^{\\nu\\lambda}k^{\\mu}) \\}, \\\\\nI_2^{\\mu} &=&  \\hat J \\; p^2p^{\\mu}= \n -J_0 \\, k^{\\mu} \\, \\{k^2\\Gamma(\\alpha-\\frac{n}{2}) +\n \\frac{n+2}{2}\\Gamma(\\alpha-1-\\frac{n}{2})(M^2-k^2) \\}.\n\\end{eqnarray*}\nFor calculation of the basic integral $I_0$ one can use the well--known \nrelation \\cite{be}:\n\\[ \\displaystyle\n\\int_{0}^{\\infty} \\frac{x^{\\beta}}{(x^2+M^2)^{\\alpha}} dx =\n\\frac{ \\Gamma \\left(\\displaystyle \\frac{\\beta+1}{2}\n  \\right) \\Gamma \\left( \\displaystyle \\frac{2\\alpha-\\beta-1}{2} \\right) }\n{2\\Gamma(\\alpha)\\displaystyle (M^2)^{\\alpha - \\frac{\\beta+1}{2}} }.\n\\]\n\n\\subsection{\\it Spence Integral (Dilogarithm)}\n\nAs a rule the final expressions for the loop integrals include so--called\n{\\it Spence integral} or {\\it Euler dilogarithm} \\cite{be,pbm,lewin}:\n\\begin{eqnarray}\n\\li_2(z) = \\li(z) \\equiv - \\int_{0}^{z}\\frac{\\ln(1 - t)}{t}dt =\n  \\int_{0}^{1}\\frac{\\ln t}{t - z^{-1}}dt \\;\\; [arg(1-z) < \\pi].  \\label{int5} \n\\end{eqnarray}\nDilogarithm is a special case of the polylogarithm \\cite{be,pbm,lewin}:\n\\begin{eqnarray}\n\\li_{\\nu}(z) \\equiv \\sum_{k=1}^{\\infty} \\frac{z^k}{k^{\\nu}} \\quad \n[|z| < 1, \\;\\;\\; {\\rm or} \\, |z| = 1, \\, {\\rm Re} \\nu > 1].\n\\end{eqnarray}\nThe main properties of $\\li(z)$ are as follows:\n\\begin{eqnarray*}\n&&\\li_n(z) + \\li_n(-z) = 2^{1-n} \\li_n(z^2), \\\\\n&&\\li_n(iz) + \\li_n(-iz) = 4^{1-n} \\li_n(z^4) - 2^{1-n} \\li_n(z^2), \\\\\n&&\\li_n(iz) - \\li_n(-iz) = 2i \\sum_{k=0}^{\\infty} \\frac{(-1)^k z^{2k+1}}\n{(2k+1)^n}  \\quad  [|z| < 1], \\\\\n&&\\li_n(z) = \\int_0^z \\frac{\\li_{n-1}(t)}{t} dt \\quad (n = 1, 2, \\ldots), \\\\\n&&\\li_0(z) = \\frac{z}{1-z}, \\quad \\li_1(z) = -\\ln(1-z).\n\\end{eqnarray*}\nThe Riemann sheet of the $\\li_2(z)$ has a cut along the real axes when $z > 1$,\nand\n\\[\n{\\rm Im} \\; \\li_2(z \\pm i\\varepsilon) = \\pm \\pi \\Theta(z-1) \\ln (z),\n\\]\nwhere the $\\Theta(x)$ is the step function (see Subsection~\\ref{miscel}). \\\\\nThe equation ${\\rm Re} \\li_2(z) = 0$ has two solutions on the real axes\n\\[\nz_1 = 0, \\;\\;\\; {\\rm and} \\;\\; z_2 \\approx 12.6.\n\\]\n${\\rm Re} \\li_2(z)$ achieves its maximum at $z = 2$:\n\\[\n\\li_2(2) = \\frac{\\pi^2}{4},\n\\]\nand at this point the $\\li_2(z)$ has the expansion as follows \\cite{vermaseren}:\n\\[\n\\li_2 (2 - \\delta) = \\frac{\\pi^2}{4} - \\frac{\\delta^2}{4} \n- \\frac{\\delta^3}{6} - \\frac{5 \\delta^4}{48} - \\frac{\\delta^5}{15} - \\ldots\n\\]\nOne easily gets:\n\\begin{eqnarray*}\n&&\\li_2(0) = 0, \\quad \\li_2(1) = \\frac{\\pi^2}{6}, \\quad \n\\li_2(-1) = -\\frac{\\pi^2}{12}, \\\\\n&&\\li_2(\\frac{1}{2}) = \\frac{\\pi^2}{12} - \\frac{1}{2} \\ln^2 2, \\\\\n&&\\li_2(\\pm i) = -\\frac{\\pi^2}{48} \\pm i {\\bf G}, \\;\\; \n{\\bf G} = \\sum_{k=0}^{\\infty} \\frac{(-1)^k}{(2k+1)^2} = 0.915965594\\ldots \n\\end{eqnarray*}\nThe various relations with $\\li_2$ are as follows \\cite{be,pbm,lewin}: \n\\begin{eqnarray*}\n&&\\li_2(z) = -\\li_2(1-z) + \\frac{\\pi^2}{6} - \\ln z \\, \\ln(1-z) \\;\\; \n[|argz|, \\, |arg(1-z)| < \\pi], \\\\\n&&\\li_2(z) = -\\li_2(\\frac{1}{z}) - \\frac{1}{2} \\ln^2 z + i \\pi \\ln z\n+ \\frac{\\pi^2}{3}  \\;\\;\\; [|arg(-z)| < \\pi], \\\\\n&&\\li_2(z) = \\li_2(\\frac{1}{1-z}) + \\frac{1}{2} \\ln^2 (1-z) - \\ln (-z) \\ln (1-z)\n- \\frac{\\pi^2}{6} \\\\ \n && \\hspace{10mm} [|arg(-z)| < \\pi].\n\\end{eqnarray*}\nThe Hill identity has the form \\cite{vermaseren,pbm}:\n\\begin{eqnarray*}\n\\li_2(\\omega z)& =& \\li_2(\\omega) + \\li_2(z) \n- \\li_2 \\left ( \\frac{\\omega - \\omega z}{1-\\omega z} \\right ) \n- \\li_2 \\left ( \\frac{z - \\omega z}{1-\\omega z} \\right ) \\\\\n &-& \\ln \\left (\\frac{1-\\omega}{1-\\omega z} \\right )\n  \\ln \\left (\\frac{1-z}{1-\\omega z} \\right ) \\\\\n&-& \\eta \\left [ 1- \\omega, \\; \\frac{1}{1- \\omega z} \\right ] \\; \\ln \\omega \\; \n - \\eta \\left [ 1- z, \\; \\frac{1}{1- \\omega z} \\right ] \\; \\ln z,\n\\end{eqnarray*} \nwhere the function $\\eta$ compensates for the cut in the Riemann\nsheet of the logarithm \\cite{vermaseren}:\n\\[\n\\ln x y = \\ln x + \\ln y + \\eta (x,y).\n\\]\nA typical integral, which can be expressed via the dilogarithm, is, for \nexample: \n\\begin{eqnarray*}\n\\int_a^b \\frac{\\ln(p+qt)}{t} dt = \\ln p \\ln \\frac{b}{a} \n- \\li_2(-b \\frac{q}{p}) + \\li_2(-a \\frac{q}{p}). \n\\end{eqnarray*}\nThe Euler {\\it Gamma function} $\\Gamma(z)$ is given by the integral \nrepresentation \\cite{be}:\n\\[ \n \\Gamma(z) \\equiv \\int_{0}^{\\infty} dt \\, t^{z-1} \\, e^{-t}, \n\\qquad {\\rm Re} \\; z > 0.\n\\]\nThe main properties of the $\\Gamma(z)$ are as follows \\cite{be}:\n\\begin{eqnarray*}\n&& \\Gamma(1+z) = z \\Gamma(z), \\quad \\Gamma(n+1) = n !, \\\\\n&& \\Gamma(z) \\Gamma(-z) = - \\frac{\\pi}{z \\; \\sin(\\pi z)}, \\;\\;\\;\n   \\Gamma(z) \\Gamma(1-z) =  \\frac{\\pi}{\\sin(\\pi z)}, \\\\\n&& \\Gamma(\\frac{1}{2}+z) \\Gamma(\\frac{1}{2}-z) = \n     \\frac{\\pi}{ \\cos(\\pi z)}, \\;\\;\\;\n \\Gamma(2z) =  \\frac{2^{(2z-1)}}{\\sqrt{\\pi}} \\Gamma(z) \\Gamma(\\frac{1}{2}+z),\\\\\n&&\\Gamma(1) = \\Gamma(2) = 1, \\;\\;\\; \\Gamma \\left ( \\frac{1}{2} \\right ) = \n    \\sqrt{\\pi}, \\\\\n&&\\Gamma(z)|_{z \\to 0} \\simeq \\frac{1}{z} + \\Gamma'(1);  \\quad \n    \\Gamma'(1) = \\Gamma(1) \\Psi(1) = \\Psi(1) = -\\gamma = -0.57721 \\, 56649\n    \\ldots, \n\\end{eqnarray*}\nwhere $\\gamma$ is Euler constant.\n\n\n\\section{\\bf KINEMATICS}\\label{kinem} \n\nThe nice book by E.~Byckling and K.~Kajantie \\cite{bk} contains a lot of \ninformation about relativistic kinematics. Here we present a brief description \nof\nrelativistic kinematics following the Review of Particle Properties \\cite{pdg}.\n\n\\subsection{\\it Lorentz transformation}\nLet  $p^{\\mu}$ be some four-momentum of the massive particle with the mass M:\n\\begin{eqnarray} \n  p^{\\mu} = (p_0, \\vec{p}); \\;\\; \\vec{p} = (p_x, p_y, p_z),  \\;\n  p_0=\\sqrt{M^2 + \\vec{p}^2} \\label{kin_eq-1}\n\\end{eqnarray} \n\n\\noindent The reference frame where this $p^{\\mu}$-vector is defined is usually\nreferred as the {\\it laboratory} frame ({\\it L}-frame). This momentum in its\nrest frame ({\\it R}-frame) is as follows:\n\\begin{eqnarray} \n  Rf: \\;\\; p^{\\mu} = (M, 0)  \\label{kin_eq-2}\n\\end{eqnarray}\n\n\\noindent \nLet $k^{\\mu}$ be some 4-vector (4-momentum) defined in the {\\it Lab}-frame\nand $k^{* \\mu}$\nbe the same 4-vector in the {\\it R}-frame. \nThe Lorentz transormations form {\\it R}(est)-frame to {\\it Lab}-frame and\nvice versa can be written in the form as follows:\n\\begin{eqnarray}\n  \\begin{array}{l} { k^{*} \\to k} \\\\ {R \\to Lab} \\end{array}: \\;\n     { \\left \\{ \\begin{array}{ccl}\n k_0 & = & \\displaystyle \\frac{k_{0}^{*}p_{0} + (\\vec{k}^*\\vec{p})} {M} \\\\\n \\vec{k} & =  & \\vec{k}^{*} + \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad\n     \\begin{array}{r} {k \\to k^{*}} \\\\ {Lab \\to R} \\end{array}: \\;\n          { \\left \\{ \\begin{array}{ccl}\n k_0^{*} & = & \\;\\; \\displaystyle \\frac{(pk)} {M} \\\\\n \\vec{k}^* & =  & \\vec{k} - \\alpha \\vec{p} \n   \\end{array} \\right. } \\quad  \\quad \\quad  (\\ref{lorentz-10}) \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n \\alpha = \\frac {k^{*}_0 + k_0} {p_0 + M}\n\\end{eqnarray*}\n\n\n\\subsection{\\it Variables} \n\nInitial (final) particles total momentum (energy) squared will be denoted by:\n\\begin{eqnarray}\n s \\equiv \\Big( \\sum_{initial} p_i \\Big)^2 = \\Big( \\sum_{final} p_j \\Big)^2. \n \\label{kin33}\n\\end{eqnarray}\nLet $E$ and $\\vec p$ be energy and momentum of a particle. The energy and \nmomentum of this particle ($E', \\vec p'$) in the frame moving with the velocity\n$\\vec \\beta$ are given by the Lorentz transformation:\n\\begin{eqnarray}\n E' = \\gamma (E + \\beta p _{||}), \\quad  p'_{||} = \\gamma (p_{||} + \\beta E), \n\\quad \\vec p{\\;}'_{\\top} = \\vec p_{\\top}, \\label {kin1}\n\\end{eqnarray}\nwhere $\\gamma = 1\/\\sqrt{1 - \\beta^2}$ and $\\vec p_{\\top} (p_{||})$ are the \ncomponents of $\\vec p$ perpendicular (parallel) to $\\vec \\beta$. \\\\\nThe beam direction choose along the $z$--axes.  4--momentum of a particle \n$p^{\\mu} = (E, \\vec p)$ can be written as:\n\\begin{eqnarray}\n &&E = p_0, \\; \\vec p_{\\top} = (p_x, p_y), \\; p_z, \\nonumber \\\\\n && p_x = |\\vec p| \\cos \\phi \\sin \\vartheta, \\;\n p_y = |\\vec p| \\sin \\phi \\sin \\vartheta, \\;\n p_z = |\\vec p| \\cos \\vartheta, \\label{kin2}\n\\end{eqnarray}\nwhere $\\phi$ is the azimuthal angle $(0\\leq~\\phi~\\leq~2\\pi)$;\n $\\vartheta$ is the polar angle $(0\\leq \\vartheta \\leq \\pi )$.\\\\\nAnother parameterization of $p^{\\mu}$ looks as follows:\n\\begin{eqnarray}\n E = m_{\\top} \\cosh y, \\; p_x, \\; p_y, \\; p_z = m_{\\top} \\sinh y, \\label{kin3}\n\\end{eqnarray}\nwhere $ m_{\\top}^2 = m^2 + p_{\\top}^2$ is the transverse mass (''old''\ndefinition), $y$ is the rapidity. \\\\\n{\\it Rapidity} $y$ is defined by\n\\begin{eqnarray}\n y \\equiv \\frac{1}{2} \\ln \\Big (\\frac{E+p_z}{E-p_z}\\Big) = \\ln \\Big(\n\\frac{E + p_z} {m_{\\top}} \\Big) =\n \\tanh^{-1} \\Big(\\frac{p_z}{E}\\Big). \\label{kin4}\n\\end{eqnarray}\nUnder a boost along $z$--direction to a frame with velocity $\\beta$,\n\\[\ny \\; \\to \\; y + \\tanh^{-1} \\beta.\n\\] \n{\\it Pseudorapidity} $\\eta$ is defined by:\n\\begin{eqnarray}\n &&\\eta \\equiv -\\ln (\\tan(\\vartheta \/2)), \\\\\n&& \\sinh \\eta = \\cot \\vartheta, \\; \\; \\cosh \\eta = \\frac{1}{\\sin \\vartheta}, \n \\; \\;  \\tanh \\eta = \\cos  \\vartheta. \\nonumber\n\\end{eqnarray}\nFor $p \\gg m$ and $\\vartheta \\gg  1 \/\\gamma$ one has : $\\quad \\eta \\approx y$.\n\n\\noindent Feynman's $x_F = x$ variable is given by\n\\begin{eqnarray}\n x = \\frac{p_z}{p_{z\\; {\\hbox to 0pt {max\\hss}}}} \\hspace{5mm}\n\\approx \\frac{(E + p_z)}{\n (E + p_z)_{\\hbox to 0pt {max\\hss}}}, \\quad \n  {\\rm in} \\; {\\rm cms} \\;\\; x = \\frac{2 p_z}{\\sqrt{s}}.\n\\end{eqnarray}\nThe last equation is valid for two particles collisions, and here $s$ is\ntotal energy squared (see (\\ref{kin33})).  \\\\\nIn the collider's experiments the following additional variables are used:\n\\begin{eqnarray*}\n\\begin{array}{lcrl}\nE_{\\bot} &=& E \\sin\\vartheta & \\; {\\rm -} \\; {\\rm transverse} \\; {\\rm energy},\n  \\\\\n \\vec{p}_{\\bot mis} &=& -(\\Sigma \\vec{p}_{\\bot}) & \\; {\\rm - } \\; { \\rm missing } \n \\; {\\rm transverse} \\; {\\rm momentum},\n \\\\\n \\vec{ E}_{\\bot mis} &=& -(\\Sigma \\vec E_{\\bot}) & \\; {\\rm -} \\; {\\rm  missing} \n \\; {\\rm transverse} \\; {\\rm energy}\n\\end{array}\n\\end{eqnarray*}\nwhere sum is performed over all detected particles. \\\\\nThe ''distance'' in $(\\eta, \\phi)$--plane between two particles (clusters)\n$1$ and $2$ is given by\n\\[\n\\Delta R \\equiv \\sqrt{(\\Delta \\phi)^2 + (\\Delta \\eta)^2 }, \\;\n \\Delta \\phi = \\phi_1 - \\phi_2, \\; \\Delta \\eta = \\eta_1 - \\eta_2.\n\\]\nThe \"transverse\" mass of the particle (cluster) $c$ with momentum $\\vec p_c$ \nand the \"missing\" transverse momentum (energy) $\\vec p_{\\bot \\; mis}$\n($\\vec E_{\\bot \\; mis}$) is given by:\n\\[\nM_{\\bot}^2(c, \\vec p_{\\bot \\; mis}) \\equiv \n(\\sqrt{m_c^2 + p^2_{\\bot c}} + p_{\\bot \\; mis})^2 -\n (\\vec p_{\\bot c} + \\vec p_{\\bot \\; mis})^2.\n\\]\n\n\\subsection{\\it Event Shape Variables} \n\nIn this Subsection we describe in brief event shape variables for $n$--particle\nfinal state (for details, see, for example \\cite {sj}). None of the\nvariables presented in this Subsection are Lorentz invariant. \n\n\\noindent $\\bullet$ {\\bf Sphericity} \\\\\nThe {\\it sphericity} tensor is defined as \\cite{sj,bjbr}:\n\\begin{eqnarray}\n S^{ab} \\equiv \\frac{ \\sum_i p^a_i p^b_i}{\\sum_i |\\vec p_i|^2},\n\\end{eqnarray}\nwhere $a,b = 1,2,3$ corresponds to the $x, y$ and $z$ components. By\nstandard diagonalization of $S^{ab}$ one can find three eigenvalues\n\\[ \n \\lambda_1 \\geq \\lambda_2 \\geq \\lambda_3, \\quad {\\rm with} \\quad\n \\lambda_1 + \\lambda_2 + \\lambda_3 = 1.\n\\]\nThen, the {\\it sphericity} is defined as:\n\\begin{eqnarray} \n S  \\equiv \\frac{3}{2} ( \\lambda_2 + \\lambda_3), \\quad 0 \\leq S \\leq 1. \n\\end{eqnarray}\nEigenvectors $\\vec s_i$ can be found that correspond to the three eigenvalues\n$\\lambda_i$. The $\\vec s_1$ eigenvector is called the {\\it sphericity\naxes}, while the {\\it sphericity event plane} is spanned by $\\vec s_1$ and\n $\\vec s_2$. \\\\\nSphericity is essentially a measure of the summed $\\vec p_{\\top}$ with\nrespect to sphericity axes. So, one can use another definition of the\n sphericity:\n\\begin{eqnarray} \n S = \\frac{3}{2} \\min_{\\vec n} \\frac{\\sum_i \\vec p_{\\top i}^{\\; 2}}\n  {\\sum_i |\\vec p_i|^2}, \\label{kin5} \n\\end{eqnarray}\nwhere $\\vec p_{\\top i}$ is a component of $\\vec p_i$ perpendicular to\n$\\vec n$. So, the sphericity axes $\\vec s_i$ given (\\ref{kin5}) by the\n$\\vec n$ vector for which minimum is attained. A 2--jet event corresponds\nto $S \\approx 0$ and isotropic event to $S \\approx 1$. \n\nSphericity is not an infrared safe quality in QCD perturbation theory.\nSometimes one can use the generalization of the sphericity tensor, given by\n\\begin{eqnarray} \n S^{(r)ab} \\equiv \\frac{\\sum_i |\\vec p_i|^{r-2} p^a_i p^b_i} \n  {\\sum_i |\\vec p_i|^r}, \\label{kin6} \n\\end{eqnarray}\n\n\\noindent $\\bullet$ {\\bf Aplanarity} \\\\\nThe {\\it aplanarity} $A$ is define as \\cite{sj,mrkj}:\n\\begin{eqnarray} \n A \\equiv  \\frac{3}{2} \\lambda_2, \\quad 0 \\leq A \\leq \\frac{1}{2}.\n\\end{eqnarray}\nThe aplanarity measures the transverse momentum component out of the event\nplane. A planar event has $A \\approx 0$ and isotropic one $A \\approx \n\\frac{1}{2}$. \\\\\n\n\\noindent $\\bullet$ {\\bf Thrust} \\\\\nThe {\\it thrust} $T$ is given by \\cite{sj,bran} \n\\begin{eqnarray} \n T \\equiv  \\max_{|\\vec n|=1} \\frac{\\sum_i |(\\vec n \\vec p_i)|}\n   {\\sum_i |\\vec p_i|}, \\quad  \\frac{1}{2} \\leq T \\leq 1.\n\\end{eqnarray}\nand the {\\it thrust axes} $\\vec t_i$ is given by the $\\vec n$ vector for\nwhich maximum is attained. 2--jet event corresponds to $T \\approx 1$ and\nisotropic event to $T \\approx \\frac{1}{2}$. \n\n\\noindent $\\bullet$ {\\bf Major and \\bf minor values} \\\\\nIn the plane perpendicular to the thrust axes, a {\\it major axes} $\\vec m_a$ \nand {\\it major value} $M_a$ may be defined in just the same fashion as thrust\n\\cite{sj}, i.e. \n\\begin{eqnarray} \n M_a \\equiv \\max_{|\\vec n|=1, \\; (\\vec n \\vec t_1)=0} \n \\frac{\\sum_i |(\\vec n \\vec p_i)|} {\\sum_i |\\vec p_i|}. \n\\end{eqnarray}\nFinally, a third axes, the {\\it minor axes}, is defined perpendicular to\nthe thrust ($\\vec t_1$) and major ($\\vec m_a$) axes. The {\\it minor value}\n$M_i$ is calculated just as thrust and major values.\n\n\\noindent $\\bullet$ {\\bf Oblatness} \\\\\nThe {\\it oblatness} $O$ is given by \\cite{sj}\n\\[\nO \\equiv M_a - M_i.\n\\]\nIn general, $O \\approx 0$, corresponds to an event symmetrical around the\nthrust axes $\\vec t_1$ and high $O$ to aplanar event.\n\n\\noindent $\\bullet$ {\\bf Fox--Wolfram moments} \\\\\nThe {\\it Fox--Wolfram moments} $H_l$, $l = 0, 1, 2, \\ldots,$ are defined by\n\\cite{sj,fox}:\n\\begin{eqnarray} \n H_l \\equiv \\sum_{i,j = 1} \\frac{|\\vec p_i| |\\vec p_j|}{E^2_{vis}} \n P_l(\\cos \\vartheta _{ij}),\n\\end{eqnarray}\nwhere $\\vartheta _{ij}$ is the opening angle between hadron $i$ and $j$,\nand $E_{vis}$ is the total visible energy of the event. $P_l(z)$ are the\nLegendre polynomials \\cite{be}:\n\\begin{eqnarray*} \n && P_0(z) = 1, \\; P_1(z) = z, \\; P_2(z) = \\frac{1}{2}(3 z^2 - 1), \\ldots \\\\\n && P_k(z) = \\frac{1}{k} \\big [ (2k-1)z P_{k-1}(z) - (k-1) P_{k-2}(z) \\big ].\n\\end{eqnarray*}\nNeglecting the masses of all the particles, one gets $H_0 = 1$. If\nmomentum is balanced, then $H_1=0$. 2--jet events tend to give $H_l \\approx 1$\nfor $l$ even and $H_l \\approx 0$ for $l$ odd. \n\nThe summary of the discussed quantities are presented in Table~\\ref{kinem}.1. \n\n\\vspace{0.5cm}\n\\noindent \n\\underline{ {\\bf Table~\\ref{kinem}.1.}} \\\\\n Summary of event shape variables. \n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|}\\hline\n & $S$ & $A$ & $T$ & $O$ &\n  $\\begin{array}{c} H_0 \\\\ {\\rm all} \\; m_i=0 \\end{array}$ & $H_l$ \\\\ \\hline\n  isotropic & $1$ & $\\frac{1}{2}$ & $\\frac{1}{2}$ & - & $1$ & - \\\\ \\hline\n  2--jet  &  $0$  & - & $1$ & $0$ & $1$ & \n  $\\begin{array}{c} H_1 = 0 \\\\ H_{2k} \\approx 1, \\; H_{2k+1} \\approx 0 \n   \\end{array} $ \\\\  \\hline\n   planar & - & $0$ & - & $\\gg 0$ & $1$ & - \\\\ \\hline \n\\end{tabular}\n\\end{center}\n\n\\subsection {\\it Two--body Final State}\n\nIn the collision of two particles of mass $m_1$ and $m_2$ and momenta $p_1$\nand $p_2$ \n\\[\n s = (p_1 + p_2)^2 = m_1^2 + m_2^2 + 2 E_{1\\;Lab}m_2,\n\\]\nwhere the last equation is valid in the frame, where one particle (second one)\nis at rest (Lab frame).\n\nThe energies and momenta of the particles $1$ and $2$ in their center--of--mass\nsystem (cms) are equal to:\n\\begin{eqnarray}\n&& E^{\\ast}_1 = \\frac{s + m_1^2 - m_2^2}{2 \\sqrt{s}}, \\quad\n  E^{\\ast}_2 = \\frac{s - m_1^2 + m_2^2}{2 \\sqrt{s}}, \\label{kin7} \\\\\n&& \\vec p_1^{\\; \\ast} = - \\vec p_2^{\\; \\ast}, \\quad\n|\\vec p_1^{\\; \\ast}| = \\frac{\n \\sqrt{[s - (m_1 + m_2)^2][s - (m_1-m_2)^2]}}{2 \\sqrt{s}}, \\label{kin8}\n\\end{eqnarray}\nor\n\\[\n|\\vec p_1^{\\; \\ast}| = \\frac{1}{2\\sqrt{s}} \\lambda^{1\/2}(s, m^2_1, m^2_2),\n\\]\nwhere $\\lambda (x,y,z)$ is the so--called {\\it kinematical function} \\cite{bk}:\n\\begin{eqnarray}\n \\lambda (x,y,z) & \\equiv & (x - y - z)^2 - 4yz \\label{kint1} \\\\\n & = & x^2 + y^2 + z^2 - 2xy - 2yz - 2 zx   \\nonumber \\\\\n & = & \\bigl \\{ x - (\\sqrt{y} + \\sqrt{z})^2 \\bigr \\}\n  \\bigl \\{ x - (\\sqrt{y} - \\sqrt{z})^2 \\bigr \\}.\n\\end{eqnarray}\nLet us now consider the two--body reaction (4--momenta of the particles are \npresented in the parentheses):\n\\begin{eqnarray*}\na(p_a) + b(p_b) &\\to& 1(p_1) + 2(p_2) \\\\\n p_a  + p_b &=& p_1 + p_2\n\\end{eqnarray*}\nThe Lorentz--invariant Mandelstam variables for reaction $2 \\to 2$ are defined \nby:\n\\begin{eqnarray}\ns &=& (p_a  + p_b)^2 = (p_1 + p_2)^2, \\quad  \nt = (p_a  - p_1)^2 = (p_b - p_2)^2, \\label{kin11} \\\\ \nu &=& (p_a  - p_2)^2 = (p_b - p_1)^2, \\nonumber\n\\end{eqnarray}\nand they satisfy\n\\[\ns + t + u = m_a^2 + m_b^2 + m_1^2 + m_2^2.\n\\]\nTwo limits of t (corresponding to $\\vartheta_{cm} = 0$ and $\\pi$) equal:\n\\begin{eqnarray}\n t_{\\pm} &=& m^2_a + m^2_1 - 2E^{\\ast}_a E^{\\ast}_1 \\pm 2|\\vec p^{\\; \\ast}_a|\n|\\vec p^{\\; \\ast}_1| = \\\\\n   &=& m^2_a + m_1^2 - \\frac{1}{2s}(s + m_a^2 - m_b^2)(s+m_1^2-m_2^2) \n\\nonumber \\\\\n &&\\pm \\frac{1}{2s} \\lambda^{1\/2}(s, m^2_a, m^2_b) \n                    \\lambda^{1\/2}(s, m^2_1, m^2_2).  \\nonumber\n\\end{eqnarray}\n\n\\subsection {\\it Three--body Final State}\n\nLet us consider three--body decay of particle $a$ with mass $M$ \n\\[ \n a(P) \\to 1(p_1) + 2(p_2) + 3(p_3).\n\\]\nDefining\n\\begin{equation}\np_{ij} \\equiv p_i + p_j, \\;\\; m_{ij}^2 \\equiv p_{ij}^2, \\label{kin10}\n\\end{equation}\nthen\n\\begin{eqnarray*}\n && m_{12}^2 + m_{23}^2 + m_{13}^2 =  M^2 + m_1^2 + m_2^2 + m_3^2, \\\\\n{\\rm and} \\;\\; &&m_{ij}^2 = (P - p_k)^2 = M^2 + m_k^2 - 2ME_k.\n\\end{eqnarray*}\nThe $1 \\to 3$ decay is described by two variables (for example, $m_{12}$\nand $m_{13}$). If $m_{12}$ is fixed, then limits of $m_{13}^2$ variation are\nequal to:\n\\begin{eqnarray*}\n \\Big(m_{13}^2 \\Big)_{\\pm} &=& m_1^2 + m_3^2 - \n\\frac{1}{2m^2_{12}}(m^2_{12}-M^2+m^2_3)(m^2_{12}+m^2_1- m^2_2) \\\\\n &&\\pm \\frac{1}{2m^2_{12}} \\lambda^{1\/2}(m^2_{12}, M^2, m^2_3) \n                    \\lambda^{1\/2}(m^2_{12}, m^2_1, m^2_2) = \\\\  \n &=& (E^{\\ast}_1 + E^{\\ast}_3 )^2 - (\\sqrt{E^{\\ast\\, 2}_1-m^2_1} \\mp\n   (\\sqrt{E^{\\ast\\, 2}_3-m^2_3})^2,\n\\end{eqnarray*}\nwhere $E^{\\ast}_1 = \\frac{m^2_{12}+m^2_1-m^2_2}{2m_{12}}$ and\n $E^{\\ast}_3 = \\frac{M^2 - m^2_{12} - m^2_3}{2m_{12}}$. \n\n\\noindent $2 \\to 3$ scattering is described by five independent variables.\nFor example, \n\\[\ns = (p_a + p_b)^2, \\; m^2_{12}, \\; m^2_{23}, \\; t_1 = (p_q - p_1)^2, \n\\; t_2 = (p_b - p_3)^2.\n\\]\n\n\\subsection{\\it Lorentz Invariant Phase Space}\n\nLorentz invariant phase space (LIPS) of $n$ particles with 4--momenta $p_j$\n($j = 1, 2, \\ldots n$) and the total momentum \n$ P = \\sum^{n}_{j=1} p_j$ is given by:\n\\begin{eqnarray} \nd R_n(P; \\, p_1, p_2, \\ldots p_n)  \\equiv \\delta^{(4)}(P - \\sum_{j=1}^n p_j) \n \\prod_{j=1}^{n} \\frac{ d^3p_j}{(2\\pi)^3 2E_j}.\n\\end{eqnarray}\nThrough of this Subsection we use the following notation:\n\\[\n s \\equiv P^2.\n\\]\nThis LIPS can be generated recursively as follows \\cite{pdg,bk}:\n\\begin{eqnarray} \nd R_n = d R_2(P; \\, p_n, q) (2\\pi)^3 dq^2 \\, d R_{n-1}(q; \\, p_1, \n\\ldots p_{n-1}), \\label{lips2}\n\\end{eqnarray}\nwhere $q = \\sum^{n-1}_{i=1} p_i$ and $(m_1+m_2+ \\ldots + m_{n-1})^2 \\leq q^2\n\\leq (\\sqrt{P^2} - m_n)^2$, or:\n\\begin{eqnarray} \nd R_n = d R_{n-j+1}(P; \\, q,  p_{j+1}, \\ldots p_n) (2\\pi)^3 dq^2 \\, \nd R_{j}(q; \\, p_1, \\ldots p_{j}), \\label{lips3}\n\\end{eqnarray}\nhere $q = \\sum^j_{l=1} p_l$ and\n\\[\n(m_1 + \\ldots + m_j)^2 \\leq q^2 \\leq (\\sqrt{P^2} - \\sum^n_{l=j+1}m_l)^2.\n\\]\n\n\\noindent The integrated LIPS for $m_1 = m_2= \\ldots = m_n = 0$ equals:\n\\[\nR_n(0) = \\frac{1}{(2\\pi)^{3n}} \\frac{(\\pi \/ 2)^{n-1}}{(n-1)!(n-2)!} \n(P^2)^{n-2}.\n\\]\n\n\\noindent Two--particle LIPS equals:\n\\[\nR_2 = \\frac{1}{(2\\pi)^6} \\frac{p^{\\ast}_1}{4\\sqrt{P^2}} \\int d \\Omega^{\\ast}_1\n =  \\frac{1}{(2\\pi)^6} \\frac{\\pi p^{\\ast}_1}{\\sqrt{P^2}} \n =  \\frac{1}{(2\\pi)^6} \\frac{\\pi p^{\\ast}_1}{\\sqrt{s}} ,\n\\]\nwhere $p^{\\ast}_1$ is momentum of first (second) particle in cms \n(see (\\ref{kin8})). \\\\\n\n\\noindent Different choice of $m_1$ and $m_2$ leads to:\n\\begin{eqnarray*} \n R_2 &=& \\frac{1}{(2\\pi)^6} \n \\frac{\\pi \\sqrt{[s - (m_1+m_2)^2][s - (m_1-m_2)^2]}}{2 s}, \n\\quad (m_1 + m_2) \\leq \\sqrt{s}, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2} \\sqrt{1 - \\frac{4 m^2}{s}}, \n    \\quad m_1 = m_2 = m, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2} ( 1 - \\frac{m^2_1}{s}), \n    \\quad m_2=0, \\\\\n R_2 &=& \\frac{1}{(2\\pi)^6} \\frac{\\pi}{2}, \\quad m_1=m_2=0. \n\\end{eqnarray*}\n\n\\noindent Three body decay final state LIPS equals:\n\\begin{eqnarray*} \n d R_3 = \\frac{1}{(2\\pi)^9} \n \\frac{\\pi^2}{4s} d m^2_{12} d m^2_{13}  \n  = \\frac{1}{(2\\pi)^9} \\pi^2 d E_1 \\, d E_2,\n\\end{eqnarray*}\nwhere $m_{12}$ and $m_{13}$ are defined in (\\ref{kin10}), $E_1(E_2)$ is the\n energy of\nthe first (second) particle in $P$ rest frame. This is the standard form of the\nDalitz plot.\n\n\\subsection{\\it Width and Cross Section}\n\nThe partial decay rate ({\\it partial width}) of a particle of mass $M$ into \n$n$ bodies in its rest\nframe is given in terms of the Lorentz--invariant matrix element $M_{fi}$ by:\n\\begin{eqnarray}\nd\\Gamma = \\frac{(2\\pi)^4}{2M} |M_{fi}|^2 d R_n(P; \\; p_1, p_2, \\ldots, p_n).\n \\label{kin12}\n\\end{eqnarray}\nThe differential cross section of the reaction $a \\; + b \\; \\to \\; 1 \\; + 2 \\;\n+ \\ldots + \\; n$ ($p_a+p_b \\equiv P)$  is given by:\n\\begin{eqnarray}\n&&d\\sigma = \\frac{(2\\pi)^4}{2 I} |M_{fi}|^2 d R_n(P; \\; p_1, p_2, \\ldots, p_n),\n \\label{kin13}  \\\\ \n&& I^2 = [s-(m_a+m_b)^2][s-(m_a-m_b)^2] = 4[(p_a p_b)^2 - m^2_a m^2_b].\n\\nonumber\n\\end{eqnarray}\n\n\n\\section{\\bf DECAYS}\\label{decays}\n\n\\subsection{\\it Standard Model Higgs Decays Rates} \n\nStandard Model Higgs is expected to have a mass between 45 Gev and 1 TeV, and,\nsince it couples directly to the masses of other particles, to decay into\nheaviest available particles. The SM Higgs decay rates, calculated without \nradiative corrections  are as follows (see \\cite{pl} and references therein):\n\\begin{eqnarray*}\n &H& \\to f \\bar f, \\quad \\Gamma = \\frac{N_c G_F m^2_f}{4 \\sqrt{2}\\pi} m_H \n\\beta^3, \n\\end{eqnarray*}\nwhere $\\beta = \\sqrt{1 - 4m^2_f \/ m^2_H}$ and $N_c = 1(3)$ for $f=$~lepton \n(quark).\n\\begin{eqnarray*}\n &H& \\to W^+ W^- (ZZ), \\quad \\Gamma = \\frac{G_F^2 M^2 m_H}{8 \\sqrt{2}\\pi}\n \\frac{\\sqrt{1-x}}{x} (3x^2-4x+4), \n\\end{eqnarray*}\nwhere $x = 4M^2 \/ m^2_H$, $M$ is $W^{\\pm}(Z)$--boson mass. \nHiggs decay into two photons or two gluons proceeds via loops. Its decay rates\nare equal \\cite{hf}: \n\\begin{eqnarray*}\n &H& \\to \\gamma \\gamma, \\quad \\Gamma = \\frac{\\alpha^2 G_F}{8 \\sqrt{2}\\pi^3}\n m^3_H |I|^2, \n\\end{eqnarray*}\nwhere $I = I_{lepton} + I_{hadron} + I_W + \\ldots$, and $|I| \\approx O(1)$.\n\\begin{eqnarray*}\n &H& \\to g g, \\quad \\Gamma = \\frac{\\alpha^2_s G_F}{4 \\sqrt{2}\\pi^3}\n \\frac{ m^3_H}{9}  |N|^2, \n\\end{eqnarray*}\nwhere $N \\equiv  3 \\sum_j N_j$ is the sum of the quark's contributions \n$j = 1,2,\\ldots,$ given by \\cite{ggmn}:\n\\[ \nN_j  = \\int^1_0 dx \\int^{1-x}_0 dy \n\\frac{1 - 4xy}{1 - xy \\frac{m^2_H}{m^2_j} - i\\varepsilon} = \n  2 \\lambda_j + \\lambda_j (4 \\lambda_j -1) G( \\lambda_j),\n\\] \nwhere $\\lambda_j \\equiv  m^2_j \/ m^2_H$, and\n\\begin{eqnarray*}\n G(z) &=& -2 [\\arcsin(\\frac{1}{2\\sqrt{z}})]^2, \\quad z \\geq \\frac{1}{4}, \\\\\n G(z) &=& \\frac{1}{2} \\ln^2 \\left [ \\frac{1+\\sqrt{1-4z}}{1-\\sqrt{1-4z}} \n \\right ]\n - \\frac{\\pi^2}{2} + i \\pi \\ln \\left [ \\frac{1+\\sqrt{1-4z}}{1-\\sqrt{1-4z}} \n\\right ],  \\quad z \\le \\frac{1}{4}.\n\\end{eqnarray*}\n$N_q$ vanishes for $m_q \\ll m_H$ and $N_q \\to 1\/3$ for $m_q \\gg m_H$.\n\n\\begin{eqnarray*}\n H \\to W^{\\pm} f \\bar f', \\quad &\\Gamma& = \\frac{g^4 m_H}{307 \\pi^3} \n F(\\epsilon), \\quad \\epsilon = \\frac{m_W}{m_H},\n\\end{eqnarray*}\n\\begin{eqnarray*} \n H \\to W^{\\pm} \\sum f \\bar f', \\quad &&({\\rm except} \\; W^+ \\to t \\bar b) \\; \\\\\n &  \\Gamma& = \\frac{3 g^4 m_H}{512 \\pi^3} F(\\epsilon),\n\\quad \\epsilon = \\frac{m_W}{m_H}, \\\\\n H \\to Z \\sum f \\bar f, \\quad &\\Gamma& = \\frac{g^4 m_H}\n {2048 \\pi^3 \\cos^4 \\vartheta_W} \\\\\n &\\times& (7 - \\frac{40}{3} \\sin^2 \\vartheta_W\n  + \\frac{160}{9} \\sin^4 \\vartheta_W) F(\\epsilon'),\n\\quad \\epsilon' = \\frac{m_Z}{m_H}, \\\\\n F(z) &=& \\frac{3(1-8z^2 + 20 z^4)}{\\sqrt{4 z^2 - 1}} \\arccos\n (\\frac{3z^2-1}{2z^3}) \\\\\n & - &(1-z^2)(\\frac{27}{2}z^2-\\frac{13}{2}+\\frac{1}{z^2}) \n - 3(1-6z^2+4z^4) \\ln z. \n\\end{eqnarray*}\n\n\\subsection{\\it $W$ and $Z$ Decays} \n\nThe partial decay widthes for gauge bosons to decay into massless fermions \n$f_1 \\bar f_2$ are equal to \\cite{okun,pdg}: \n\\begin{eqnarray*}\n W^+ \\to e^+ \\nu_e, \\quad \\Gamma &=& \\frac{G_F M_W^3}{6\\sqrt{2}\\pi} \\approx \n 227 \\pm 1 \\; {\\rm MeV}, \\\\ \n W^+ \\to u_i \\bar d_i, \\quad \\Gamma &=& C\\frac{G_F M_W^3}{6\\sqrt{2}\\pi} \n  |V_{ij}|^2 \\approx (707 \\pm 3) |V_{ij}|^2 \\; {\\rm MeV}, \\\\ \n Z \\to \\psi_i \\bar \\psi_i, \\quad \\Gamma &=& C\\frac{G_F M_Z^3}{6\\sqrt{2}\\pi} \n  [g_{iV}^2 + g_{iA}^2] \\approx \\\\\n &=& \\left \\{\n \\begin{array}{cc} 167.1 \\pm 0.3 \\; {\\rm MeV} \\; (\\nu \\bar \\nu), & \n  83.9 \\pm 0.2 \\; {\\rm MeV} \\; (e^+ e^-), \\\\ \n 298.0 \\pm 0.6 \\; {\\rm MeV} \\; (u \\bar u), & \n 384.5 \\pm 0.8 \\; {\\rm MeV} \\; ( d \\bar d), \\\\ \n 375.2 \\pm 0.4 \\; {\\rm MeV} \\; (b \\bar b), & {} \\end{array} \\right.\n\\end{eqnarray*}\nFor lepton $C=1$, while for quarks $C=3(1+ \\frac{\\alpha_s(M_V)}{\\pi} \n + 1.409 \\frac{\\alpha_s^2}{\\pi^2} - 12.77 \\frac{\\alpha_s^3}{\\pi^3})$,\n where $3$ is due to color and the factor in parentheses is a QCD\n correction~\\cite{Chetyrkin:1979bj}.\n\n\\subsection{\\it Muon Decay}\n\nIn the SM the total muon decay width is equal (up to $100 \\%$ accuracy) to the\nwidth of the decay\n\\[\n\\mu^- \\to e^- \\bar \\nu_e \\nu_{\\mu}.\n\\]\nThe matrix element squared for this decay equals \\cite{okun}:\n\\[\n|M|^2 = 128 G_f^2 (p_{\\mu} p_{\\nu_e}) \\, (p_e p_{\\nu_{\\mu}}).\n\\]\nThen the total muon width is given by \\cite{mudec}:\n\\begin{eqnarray}\n \\Gamma^{tot}_{\\mu} =  \\frac{G_F^2 m^5_{\\mu}} {192 \\pi^3} \nF(\\frac{m_e^2}{m^2_{\\mu}}) (1 + \\frac{3}{5}\\frac{m^2_{\\mu}}{M^2_W})\n [ 1 +\\frac{\\alpha(m_{\\mu})}{2 \\pi} (\\frac{25}{4} - \\pi^2)],\n\\end{eqnarray}\nwhere $F(x) = 1 - 8x + 8x^3 - x^4 - 12 x^2 \\ln x$, and\n\\[\n\\alpha(m_{\\mu})^{-1} = \\alpha^{-1} - \\frac{2}{3 \\pi} \\ln (\\frac{m_{\\mu}}{m_e})\n + \\frac{1}{6 \\pi} \\approx 136.\n\\]\nFor pure $V-A$ coupling (and neglecting of the electron mass) in the rest \nframe of the polarized muon ($\\mu^{\\mp}$) the differential decay rate is:\n\\[\n d \\Gamma (\\mu^{\\mp}) = \\frac{G_F^2 m^5_{\\mu}}{192 \\pi ^3} [3-2x \\pm (1-2x) \n  \\cos \\vartheta] x^2 dx d(\\cos \\vartheta),\n\\]\nwhere $\\vartheta$ is the angle between the $e^{\\pm}$  momentum and the $\\mu$ spin,\nand $x = 2 E_{\\mu} \/ m_{\\mu}$. \n\n\\subsection{\\it Charged Meson Decay}\n\n\\noindent The decay constant $f_P$ for pseudoscalar meson $P$ is defined by\n\\cite{pdg}\n\\[\n <0|A_{\\mu}(0)|P(k)> = i f_P k_{\\mu}.\n\\]\nThe state vector is normalized by $<P(k)|P(k')> = (2\\pi)^3 2 E_q \\delta^3\n(\\vec k - \\vec k')$. The annihilation rate of the \n$P(q_1 \\bar q'_2) \\to f \\bar f'$ decay is given by\n\\begin{eqnarray}\n\\Gamma(P \\to f \\bar f') = C \\frac{G_F^2 |V_{q_1 q'_2}|^2}{8 \\pi}\n f^2_P m^2_f M_P (1 - \\frac{m^2_f}{M^2_P}),\n\\end{eqnarray}\nwhere $C = 1$ for $P \\to l \\nu_l$ decay and $C = (3 |V_{q_3 q'_4}|^2)$ for\n$ P \\to q_3 \\bar q'_4$ one, and $m_f$ is the heaviest final fermion mass.\n\n\\subsection{\\it Quark Decay} \n\nIn the region $m_q \\ll M_W$ the total quark width is given by \\cite{okun}:\n\\begin{eqnarray}\n\\Gamma(Q \\to q_2 q_3 \\bar q'_4)  = \\frac{G_F^2 m^5_Q} {64 \\pi^3}\n |V_{Q q_2}|^2 |V_{q_3 q'_4}|^2.\n\\end{eqnarray}\nFor the case of $m_Q \\gg m_W + m_q$ the width of the heavy quark decay \n$Q \\to W + q$ equals \\cite{bigi}: \n\\begin{eqnarray}\n\\Gamma (Q \\rightarrow W + q) &=& \\frac{G_F m_Q^3}{8\\pi \\sqrt{2}} \\,\n |V_{Qq}|^2 \\,  \\frac{2k}{m_Q} f_Q(\\frac{m^2_q}{m^2_Q}, \\frac{M^2_W}{m^2_Q}) \\, \n \\approx  180 \\; ({\\rm MeV}) \\, \\quad |V_{Qq}|^2 \n \\left ( \\frac{m_Q}{m_W} \\right )^3, \\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*} \nf_Q(x,y) &=& (1-x)^2 + (1+x)y - 2y^2, \\;\\;\\; \nk = \\frac{1}{2 m_Q}\\sqrt{ [ m^2_Q - (m_W+m_q)^2][ m^2_Q - (m_W-m_q)^2]},\n\\end{eqnarray*}\nhere $k$ is $W$ (or $q$) momentum in the $Q$--quark rest frame. \\\\\nThe width of the heavy $Q$ decay\n\\[\n  Q \\to q + W (\\to l \\nu)\n\\]\nis given by \\cite{bigi}: \n\\[\n\\Gamma (Q \\rightarrow q + W(\\rightarrow l \\nu) = \n \\frac{G_F^2 m_Q^5}{192 \\pi^3} |V_{Qq}|\n F(\\frac{m_Q^2}{m_W^2};\\frac{m_q^2}{m_Q^2};\\frac{\\Gamma_W^2}{m_W^2}),\n\\] \nwhere\n\\begin{eqnarray*}\n && F(a,b,c) = 2\\int_{0}^{(1-\\sqrt{b})^2} d t \n\\frac{ f_Q(b,t) \\sqrt{1+b^2+t^2 - 2(b+bt+t)} }\n{[(1-at)^2+c]}, \\\\\n &&F(a,0,c) = \\\\\n &&\\frac{2}{a^4} {[c-3(1-a)]A+2a(1-a)-a[3(2-a)c-(2+a)(1-a)^2]B}, \\\\\n&& A = \\ln \\frac{c+1}{c+(1-a)^2}, \\quad\n B = \\frac{1}{a\\sqrt{c}} [ \\arctan (\\frac{1}{\\sqrt{c}}) - \n\\arctan (\\frac{1-a}{\\sqrt{c}})].\n\\end{eqnarray*}\n\n\\subsection{\\it Heavy Quarkonia $(Q \\bar Q)$ Decays} \n\nSuppose that the matrix element of the vector state $V$ decay $V \\to l^+ l^-$\nis given by\n\\[ \n M = g_V e^{\\nu}_V \\bar u(l^+) \\gamma^{\\nu} u(-l^-).\n\\] \nThen\n\\begin{eqnarray*}\n \\Gamma(V \\to l^+ l^-) = \\frac{g^2_V}{12\\pi}M_V, \n\\quad  g_V = \\sqrt{ \\frac{12\\pi \\Gamma(V \\to l^+ l^-)}{M_V}}.\n\\end{eqnarray*}\nDenote $R^2_0 \\equiv 4 \\pi |\\psi(0)|^2$, where $\\psi(0)$ is bound state wave\nfunction in the origin.\n\n\\noindent The width of the decay of the quark antiquark vector state $1^{--}$\nequals: \n\\[\n\\Gamma (1^{--} \\rightarrow l^+ l^-) = N_c\\frac{4 }{3} \\frac{\\alpha^2Q^2_q}\n{M^2} R^2_0. \n\\]\nwhere $N_c = 1(3)$ for colorless (color) quarks, $Q_q$ is the effective \ncharge: \n\\begin{displaymath}\n\\begin{array}{ccccccccc}\n\\rho & = & \\frac{1}{\\sqrt{2}} (u \\bar u - d \\bar d) &\\Rightarrow&\nQ^2_q& =& |\\frac{1}{\\sqrt{2}}(\\frac{2}{3} + \\frac{1}{3})|^2 &=& \\frac{1}{2}, \\\\\n\\omega &=& \\frac{1}{\\sqrt{2}} (u \\bar u + d \\bar d) &\\Rightarrow&\n Q^2_q &=& |\\frac{1}{\\sqrt{2}}(\\frac{2}{3} - \\frac{1}{3})|^2& =&\n \\frac{1}{18}, \\\\\n \\phi &=& s \\bar s  &\\Rightarrow& Q^2_q & = & \\frac{1}{9},& & \\\\\n J \/ \\psi &=& c \\bar c  &\\Rightarrow& Q^2_q & = & \\frac{4}{9},& & \\\\\n \\Upsilon  &=& b \\bar b  &\\Rightarrow& Q^2_q & =& \\frac{1}{9}.& &\n\\end{array}\n\\end{displaymath}\nFor positron annihilation (with $Q_e = 1$) one has: \n\\begin{eqnarray*}\n&&\\Gamma (0^- \\rightarrow \\gamma \\gamma ) = \\frac{4\\alpha^2}{M^2} R_0^2, \\\\\n&&\n\\Gamma (1^{--} \\rightarrow \\gamma\\gamma\\gamma ) = \\frac{16}{9\\pi}\n(\\pi^2-9)\\frac{\\alpha^3}{M^2} R_0^2.\n\\end{eqnarray*}\nFor quarkonia annihilation one gets:\n\\[\n\\Gamma (0^- \\rightarrow \\gamma \\gamma ) = \\frac{12\\alpha^2Q_q^4}{M^2} R^2_0.\n\\]\nFor the two (three) gluon annihilation one need to change :\n$\\alpha^2 Q^4_q \\rightarrow 2\\alpha^2_s \/9$ \n($\\alpha^3 \\rightarrow 5\\alpha^3_s \/18$): \n\\begin{eqnarray*}\n&&\\Gamma (0^- \\rightarrow gg) = \\frac{8\\alpha^2_s}{3M^2} R_0^2, \\\\\n&&\\Gamma (1^{--} \\rightarrow ggg) = \\frac{40}{81\\pi}\n(\\pi^2-9)\\frac{\\alpha^3_s}{M^2} R_0^2.\n\\end{eqnarray*}\n\n\n\\section{\\bf CROSS SECTIONS }\\label{cros}\n\n\\subsection {\\it $e^+ e^-$ Annihilation}\\label{annihil}\nFor pointlike spin--$\\frac{1}{2}$ fermions the differential cross section\nin the cms for $e^+ e^- \\to f \\bar f$ via single photon and $Z$--boson (with\nmass $M_Z$ and total width $\\Gamma_Z$) is given by \\cite{pdg}:\n\\begin{eqnarray}\n\\frac{d \\sigma}{d \\Omega} &=& \\frac{\\alpha^2}{4s}\\beta Q^2_f \n\\left \\{1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta \\right \\} \n\\label {sig1} \\\\\n &+& \\frac{\\alpha^2}{4s}\\beta \\chi_2 \\Bigl \\{ V^2_f(1+V^2)\n [1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta] \\label{sig2} \\\\\n &&+ \\beta^2 a^2_f(1+V^2)[1+\\cos^2\\vartheta] - 8\\beta V V_f a_f \n \\cos \\vartheta \\Bigr \\} \\nonumber  \\\\\n &-& \\frac{\\alpha^2}{4s}\\beta 2Q_f \\chi_1 \\Bigl \\{ V V_f\n[1 + \\cos^2\\vartheta + (1-\\beta^2)\\sin^2\\vartheta] \\label{sig3} \\\\\n && - 2a_f\\beta \\cos \\vartheta \\Bigr \\},  \\nonumber \n\\end{eqnarray}\nwhere $\\beta = \\sqrt{1 - 4m^2_f\/s}$ is the velocity of the final\nstate fermion in the center of mass, $Q_f$ is the charge of the\nfermion in units of the proton charge,\n\\begin{eqnarray*}\n \\chi_1 &=& \\frac{1}{16 \\sin^2\\vartheta_W \\cos^2\\vartheta_W}\n \\frac{s(s-M^2_Z)}{(s-M^2_Z)^2 + \\Gamma^2_Z M^2_Z}, \\\\\n \\chi_2 &=& \\frac{1}{256 \\sin^4\\vartheta_W \\cos^4\\vartheta_W}\n \\frac{s^2}{(s-M^2_Z)^2 + \\Gamma^2_Z M^2_Z}, \\\\\n V &=& -1 + 4 \\sin^2\\vartheta_W, \\quad \n V_f = 2T_{3f} - 4 Q_f \\sin^2\\vartheta_W, \\quad \n a_f = 2 T_{3f}, \n\\end{eqnarray*}\nhere the subscript $f$ refers to the particular fermion and\n\\begin{eqnarray*}\n T_3 &=& + \\frac{1}{2} \\quad {\\rm for} \\quad \\nu, u, c, t, \\\\\n T_3 &=& - \\frac{1}{2} \\quad {\\rm for} \\quad l^-, d, s, b.\n\\end{eqnarray*}\nThe first (\\ref{sig1}), second (\\ref{sig2}), and third\n(\\ref{sig3}) terms correspond to the $e^+ e^- \\to f \\bar f$\nprocess via single photon annihilation, via $Z$--boson exchange,\nand photon~--~$Z$--boson interference, respectively. \\\\\nFor $s \\gg m^2_f$ (i.e. $\\beta \\to 1$) the annihilation via\nsingle photon exchange (\\ref{sig1}) tends to:\n\\begin{eqnarray}\n\\sigma = \\frac{4 \\pi \\alpha^2}{3s} Q^2_f \\approx \\frac{86.3 Q^2_f}\n{s \\; ({\\rm GeV}^2)} \\; {\\rm nb}.\n\\end{eqnarray}\n\n\\subsection {\\it Two--photon Process at $e^+ e^-$ Collisions}\nWhen an $e^+$ and $e^-$ collide with energies $E_1$ and $E_2$,  they emit\n$d n_1$ and $d n_2$ virtual photons with energies $\\omega_1$ and\n$\\omega_2$ and 4--momenta $q_1$ and $q_2$. In the equivalent\nphoton approximation (EPA) \\cite{eqf}, the cross section for the reaction\n\\begin{eqnarray}\n e^+ e^- \\to e^+ e^- X \\label {sig4}\n\\end{eqnarray}\nis related to the cross section for $\\gamma \\gamma \\to X$ by:\n\\begin{eqnarray}\nd \\sigma_{EPA} (s) \\equiv d \\sigma_{e^+ e^- \\to e^+ e^- X}(s) = d n_1 \\, d n_2 \n d \\sigma_{\\gamma \\gamma \\to X} (W^2), \\label{sig5}\n\\end{eqnarray}\nwhere $s = 4 E_1 E_2$, $\\;\\;$ $W^2 = 4\\omega_1 \\omega_2$ and\n\\[\n d n_i = \\frac{\\alpha}{\\pi} \\Bigl [ 1 - \\frac{\\omega_i}{E_i} \n + \\frac{\\omega_i^2}{2E_i^2} \n - \\frac{m^2_e \\omega_i^2}{(-q^2_i)E_i^2} \\Bigr ] \n \\frac{d \\omega_i}{\\omega_i} \\frac{d q^2_i}{q^2_i}.\n\\]\nAfter integration (including that over $q^2_i$ in the region \n$m^2_e \\omega^2_i \/E_i (E_i \\omega_i) \\leq -q^2_i \\leq (-q^2)_{max}$), the \ncross section is\n\\begin{eqnarray}\n \\sigma_{EPA}(s) = \\frac{\\alpha^2}{\\pi^2} \n\\int^1_{z_{th}} \\frac{dz}{z}\\Biggl [f(z) \\left ( \\ln \\frac{(-q^2)_{max}}\n{m^2_ez} - 1 \\right )^2 - \\frac{\\ln^3 z}{3}\\Biggr] \n \\sigma_{\\gamma \\gamma \\to X} (zs), \\label{sig6}\n\\end{eqnarray}\nwhere $z = W^2\/s$, and\n\\[\nf(z) = (1+ \\frac{z}{2})^2 \\ln\\frac{1}{z} - \\frac{1}{2} (1-z)(3+z).\n\\]\nThe value $(-q^2)_{max}$ depends on properties of the produced\nsystem $X$. For example, $(-q^2)_{max} \\sim m^2_{\\rho}$ for  hadron \nproduction $(X = h)$, and $(-q^2)_{max} \\sim M^2_{ll}$ for\nthe lepton pair production $(X = l^+ l^-)$. \\\\\nFor the production of a resonance of mass $M_R$ and spin $J \\neq 1$  one has:\n\\begin{eqnarray}\n \\sigma_{EPA}(s) &=& (2J+1) \n \\frac{8\\alpha^2 \\Gamma(R \\to \\gamma \\gamma)}{M^3_R} \\label{sig7} \\\\\n &\\times& \\Biggl [ f(\\frac{M^2_R}{s})(\\ln \\frac{s M^2_0}{m^2_eM^2_R}-1)^2 \n - \\frac{1}{3}(\\ln\\frac{s}{M^2_R})^3 \\Biggr], \\nonumber \n\\end{eqnarray}\nwhere $M_0$ is the mass that enters into the from factor of the\n$\\gamma \\gamma \\to R$ transition: $M_0 \\sim m_{\\rho}$ for $R =\n\\pi^0, \\rho^0, \\omega, \\phi, \\ldots$ and $M_0 \\sim M_R$ for $R\n= c \\bar c$ or $b \\bar b$ resonances.\n\n\\subsection {\\it $l \\; h $ Reactions}\nThe reaction of the lepton hadron deep inelastic scattering (DIS)\n\\begin{eqnarray}\nl(k, m_l) \\quad h(P, M) \\; \\to \\; l'(k', m_{l'}) \\quad X, \\label{sig8}\n\\end{eqnarray}\nis described by the following invariant kinematic variables (the\n4--momenta and masses of the particles are denoted in the parentheses)\n\\cite{pdg}: \n\\begin{description}\n\\item[$q = k - k'$] is four--momentum transferred by exchanged particle\n($\\gamma$, $Z$, or $W^{\\pm}$) to the target, \n\\item[$\\nu = \\frac{q \\cdot P}{M} = E - E'$] is the lepton's energy\nloss in the lab frame, $E$ and $E'$ are the initial and final\nlepton energies in the lab,\n\\item[$Q^2 = -q^2 = 2(E E' - \\vec k \\cdot \\vec k') - m^2_l - m^2_{l'},$] if \n $E E'\\sin^2(\\vartheta \/ 2) \\gg m^2_l, \\; m^2_{l'},$ then $Q^2  \\approx \n 4E E'\\sin^2(\\vartheta \/ 2)$, where $\\vartheta$ is the lepton's\nscattering angle in the lab,\n\\item[$x = \\frac{Q^2}{2 M \\nu} = \\frac{Q^2}{2 q \\cdot P},$] in\nthe parton model, $x$ is the fraction of the target hadron's\nmomentum carried by the struck quark,  \n\\item[$y = \\frac{ q \\cdot P}{ k \\cdot P} = \\frac{\\nu}{E},$] is\nthe fraction of the lepton's energy lost in the lab,\n\\item[$W^2 = (P + q)^2 = M^2 + 2 M \\nu - Q^2,$] is the mass\nsquared of the system recoiling against the lepton,\n\\item[$s = (P + k)^2 = M^2 + \\frac{Q^2}{xy}.$] \n\\end{description}\nThe differential cross section of the reaction (\\ref{sig8}) as a\nfunction of the different variables is given by\n\\[\n\\frac{d^2 \\sigma}{ dx d y} = \\nu(s-M^2) \\frac{d^2 \\sigma}{ d\\nu d Q^2} =\n\\frac{2\\pi M\\nu}{E'}\\frac{d^2 \\sigma}{ d\\Omega_{lab}d E'} = \n x (s-M^2) \\frac{d^2 \\sigma}{ d x d Q^2}.\n\\] \nParity conserving neutral current process, $l^{\\pm} h \\to l^{\\pm} X$, \ncan be written in terms of two structure functions $F^{NC}_1(x, Q^2)$\nand $F^{NC}_2(x, Q^2)$:\n\\begin{eqnarray}\n\\frac{d^2 \\sigma}{d x d y} &=& \\frac{4 \\pi \\alpha^2 (s-M^2)}{Q^4} \n \\label{sig9} \\\\\n &\\times& \\Bigl [ (1-y)F^{NC}_2 +y^2 x F^{NC}_1 \n-\\frac{M^2}{(s-M^2)} x y F^{NC}_2 \\Bigr ]. \\nonumber \n\\end{eqnarray}\nParity violating charged current processes, $l h \\to \\nu X$ and\n$\\nu h \\to l X$, can be written in terms of three structure functions \n$F^{CC}_1(x, Q^2)$, $F^{CC}_2(x, Q^2)$, and $F^{CC}_3(x, Q^2)$:\n\\begin{eqnarray}\n\\frac{d^2 \\sigma}{d x d y} &=& \\frac{G^2_F (s-M^2)}{2 \\pi} \n \\frac{M^4_W}{(Q^2  + M^2_W)^2}   \\label{sig10} \\\\\n &\\times& \\Bigl \\{ [(1-y - \\frac{M^2 xy}{(s-M^2)}]F^{CC}_2 +y^2 x F^{CC}_1 \n \\pm (y - \\frac{y^2}{2})x F^{CC}_3 \\Bigr \\}, \\nonumber\n\\end{eqnarray}\nwhere the last term is positive for $l^-$ and $\\nu$ reactions\nand negative for $l^+$ and $\\bar \\nu$ reaction. \n\n\\subsection {\\it Cross Sections in the Parton Model}\nIn the {\\it parton model} framework the reaction\n\\begin{eqnarray}\n h_1 \\quad h_2 \\; \\; \\to \\; \\; C \\quad X, \\label{sig11}\n\\end{eqnarray}\nwhere $C$ is a particle (or group of the particles) with large mass \n(invariant mass) or with high $p_{\\top}$  can be considered  as a result \nof the hard interaction of the one $i$--parton from  $h_1$ hadron with\n$j$--parton from $h_2$ hadron. Then the cross section of the  reaction \n(\\ref{sig11}) can be written as follows:\n\\begin{eqnarray}\n\\sigma(h_1 h_2 \\to C X) = \\sum_{ij} \\int f^{h_1}_i(x_1, Q^2)\n f^{h_2}_j(x_2, Q^2) \\hat \\sigma(i j \\to C) d x_1 d x_2, \\label{sig12}\n\\end{eqnarray}\nwhere sum is performed over all partons, participating in the  subprocess \n$i j \\to C$; $f^{h}_i(x, Q^2)$ is {\\it parton distribution} in $h$--hadron;\n$Q$ is a typical momentum transfer in partonic process $ij \\to C$ and \n$\\hat \\sigma$ is partonic cross section.\n\n\\newpage\n\\subsection {\\it Vector Boson Polarization Vectors}\\label{vecpol}\n\nLet us consider a vector boson with mass $m$ and 4--momentum $k^{\\mu} \\; \n(k^2=m^2)$. Three polarization vectors of this boson can expressed in terms of\n$k^{\\mu}$,\n\\[ \n k^{\\mu} = (E, k_x, k_y, k_z), \\;  k_{\\top} = \\sqrt{ k^2_x + k^2_y} \n\\]\nas folows \\cite{hz}:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\varepsilon^{\\mu}(k, \\lambda = 2) = \\frac{1}{k_{\\top}}\n (0, \\, k_y, \\, -k_x, 0), \\\\\n  \\varepsilon^{\\mu}(k, \\lambda=1) = \\frac{1}{|\\vec k| k_{\\top}}\n (0, \\, k_x k_z, \\, k_y k_z, \\, -k^2_{\\top}), \\\\\n \\varepsilon^{\\mu}(k, \\lambda = 3) = \\frac{E}{ m |\\vec k|}\n (\\frac{\\vec k^{\\; 2}}{E}, \\, k_x, \\, k_y, \\, k_z). \n\\end{array} \\right \\} \\label{vp1}\n\\end{eqnarray} \nIt is easy to verify that\n\\begin{eqnarray}\n p^{\\mu} \\varepsilon_{\\mu}(k, \\lambda) = 0, \\quad \n \\varepsilon^{\\mu}(k, \\lambda) \\varepsilon_{\\mu}(k, \\lambda') = \n -\\delta^{\\lambda \\lambda'}. \\label{vp2}\n\\end{eqnarray} \nFor $k_{\\top} = 0$ (i.e. $k^{\\mu} = (E, 0, 0, k)$) these polarization\nvectors can be chosen as follows:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\varepsilon^{\\mu}(k, \\lambda=1) = {}(0, \\, 0, \\, 1, \\, 0), \\\\ \n \\varepsilon^{\\mu}(k, \\lambda = 2) = {}(0, \\, 1, \\, 0, \\, 0), \\\\\n \\varepsilon^{\\mu}(k, \\lambda = 3) = \\frac{1}{m}\n (k, \\, 0, \\, 0, \\, E). \n\\end{array} \\right \\} \\label{vp3}\n\\end{eqnarray}\n\n\\noindent\nGluon is the massless vector boson. Any massless vector boson has only two\npolarization states, $\\lambda=1$ and $2$, on its mass-shell. \\\\\nThe gluon density matrix (in the axial gauge) has the form\n(see Subsection~\\ref{gauges}):\n\\begin{eqnarray} \n\\displaystyle\n\\rho^{\\mu \\nu} = -g^{\\mu \\nu} + \\frac{k^{\\mu} n^{\\nu} + k^{\\nu} n^{\\mu}}{k\\cdot n } -\n\\frac{n^2 k^{mu} k^{\\nu}} {(k\\cdot n)^2} \\; = \\;\n\\epsilon^{\\mu}_{1} \\epsilon^{\\nu}_{1} + \\epsilon^{\\mu}_{2} \\epsilon^{\\nu}_{2},\n\\end{eqnarray}\nwhere $n$ is axial gauge fixing vector and $\\epsilon^{\\mu}_{i}$ are the gluon\npolarization vectors.\nIn the axial gauge there appears an additional condition (see Subsection~\\ref{gauges}): \n\\[ \n\\epsilon^{\\mu}_i  n^{\\mu} = 0, \\;\\; i=1,2 \n\\]\nFor this case polarization vectors $\\epsilon^{\\mu}_g(p, \\lambda=1,2)$ can\nbe chosen as follows:\n\\begin{equation}\n\\epsilon^{\\mu}_g(p, \\lambda) = \\varepsilon^{\\mu}(p, \\lambda) \n - \\frac{ \\varepsilon(p, \\lambda) \\cdot n}{p \\cdot n} p^{\\mu}, \\label{vp5}\n\\end{equation}\nwhere $\\varepsilon^{\\mu}(p, \\lambda)$ are given in (\\ref{vp1}) or (\\ref{vp3}). \\\\\nFor numerical calculations it is convenient to set\n\\begin{equation}\n  n^{\\mu} = (1, \\vec{0})\n\\end{equation}\nAs a result the first two vectors from (\\ref{vp1}) can be used:\n\\begin{eqnarray} \n\\displaystyle\n\\left. \\begin{array}{l}\n \\epsilon^{\\mu}_1 = \\frac{1}{k_{\\top}}\n (0, \\, k_y, \\, -k_x, 0) \\\\\n \\epsilon^{\\mu}_2 = \\frac{1}{|\\vec k| k_{\\top}}\n (0, \\, k_x k_z, \\, k_y k_z, \\, -k^2_{\\top}) \n\\end{array} \\right \\}, \\; k_{\\top} >0; \\;\\;\\;\n\\left. \\begin{array}{l}\n \\epsilon^{\\mu}_1 = (0, \\, 1, \\, 0, \\, 0) \\\\\n \\epsilon^{\\mu}_2 = (0, \\, 0, \\, 1, \\, 0) \n\\end{array} \\right \\}, k_{\\top} =0\n\\label{vp10}\n\\end{eqnarray}\nFor gluon being {\\bf off}-shell we should introduce third ``polarization'' vector:\n\\begin{eqnarray} \n\\displaystyle\n\\rho^{\\mu \\nu} = \\epsilon^{\\mu}_{1} \\epsilon^{\\nu}_{1} + \\epsilon^{\\mu}_{2} \\epsilon^{\\nu}_{2}\n+ \\epsilon^{\\mu}_{3} \\epsilon^{\\nu}_{3}, \\;\\;\n\\epsilon^{\\mu}_{3} \\cdot \\epsilon^{\\mu}_{3} = - \\frac{k^2}{k^2_0}\n\\end{eqnarray}\nwhere one has:\n\\begin{eqnarray} \n\\displaystyle\n \\begin{array}{ll}\nk^2 > 0: \\epsilon^{\\mu}_3 = \\frac{\\sqrt{k^2}}{k_0 |\\vec{k}|} (0, \\, \\vec{k}),  \\\\\nk^2 < 0: \\epsilon^{\\mu}_3 = i \\frac{\\sqrt{|k^2|}}{|k_0| |\\vec{k}|} (0, \\, \\vec{k})\n\\end{array}\n\\label{vp101}\n\\end{eqnarray}\nNote, that for space-like momentum third vector becomes a complex one.  \n\n\\noindent $\\bullet$ {\\bf Two Photons (Gluons) System}\n \nFor the system of two photons (gluons) with momenta $p_1$ and $p_2$ the \npolarization vectors $\\varepsilon^{\\mu}_{1(2)}$ can be written in the\nexplicitly covariant form: \n\\begin{equation}\n\\varepsilon^{\\mu}_i(\\pm) \\, = \\, \\frac{1}{\\sqrt{2\\Delta_3}}\n  \\bigl [ (p_1 p_2) q^\\mu - (q p_2) p_1^{\\mu} - (q p_1) p_2^{\\mu} \n  \\pm i \\varepsilon^{\\mu \\nu \\alpha \\beta} q^{\\nu} \n  p_{1 \\, \\alpha} p_{2 \\, \\beta}  \\bigr]. \\label{vp6}\n\\end{equation}\nwhere sign $+(-)$ corresponds to positive (negative) helicity, $q$ is any \narbitrary vector, which is independent on $p_1$ and $p_2$ (it may be a \nmomentum of some  particle), and \n\\[ \\Delta_3=\\delta_{qp_1p_2}^{qp_1p_2} = (p_1 p_2) \n(2\\> (q p_1) (q p_2) - q^2 (p_1 p_2)). \\]\nThese vectors were considered also in Subsection~\\ref{sbc65}. \\\\\nProjectors on various combinations of the helicity states look as follows:\n\\begin{eqnarray*}\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(-)  \n    + \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(+) \\right)\n = \\frac{1}{2(p_1 p_2)} (p_1^\\nu \\> p_2^\\mu - (p_1 p_2)\\>g^{\\mu\\nu}), \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(-)  \n    - \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(+) \\right)\n=-\\frac{i}{2\\> (p_1 p_2)}\\varepsilon^{p_1p_2\\mu\\nu}, \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(+)  \n    + \\varepsilon^\\mu_1(-) \\varepsilon^\\nu_2(-) \\right)\n=\\frac{1}{2\\Delta_3}\\{2 [(p_1p_2)(q p_1)(q p_2) g^{\\mu\\nu} \\\\\n&& + q^\\mu q^\\nu (p_1 p_2)^2  \n -(p_1 p_2) ((q p_1) p_2^\\mu q^\\nu + (q p_2) p_1^\\nu q^\\mu)] \\\\\n&&+q^2 (p_1 p_2) (p_1^\\nu p_2^\\mu - (p_1 p_2) g^{\\mu\\nu})\\}, \\\\\n&&\\frac{1}{2}\n\\left(\\varepsilon^\\mu_1(+) \\varepsilon^\\nu_2(+)  \n    - \\varepsilon^\\mu_1(-) \\varepsilon^\\nu u_2(-) \\right) = \\\\\n &&\\frac{i}{2\\Delta_3}\\{ ((p_1 p_2) q^\\mu - (q p _1) p_2^\\mu )\n\\varepsilon^{\\nu  q p_1p_2}  \n+((p_1 p_2) q^\\nu - (q p_2) p_1^\\nu)\\varepsilon^{\\mu q p_1p_2} \\},  \\\\ \n&&= \\frac{i\\ (p_1 p_2)}{2\\Delta_3}\n\\left( q^\\mu \\varepsilon^{\\nu q p_1p_2}+q^\\nu \\varepsilon^{\\mu q p_1p_2}\\right.\n(qp_1) \\varepsilon^{p_2 q\\mu\\nu}+ (qp_2) \\left. \\varepsilon^{p_1q\\mu\\nu}\\right).\n\\end{eqnarray*}\n\n\n\n\\section{\\bf MATRIX ELEMENTS }\\label{matrel}\n\n\\subsection {\\it General Remarks}\nIn this Section we present the matrix elements squared $|M|^2$ for various \nprocesses in the Standard Model. Almost all of these $|M|^2$ were presented \nin the book by R.~Gastmans and Tai~Tsun~Wu \\cite{gw}. The symbol $|M|^2$ is \nused to denote the square of the absolute value of the matrix element $M$\nsummed over the {\\bf initial} and {\\bf final} degrees of freedom (polarization \nand color), but {\\bf without} averaging over the {\\bf initial} state \ndegrees of freedom.\n\nSo, one can use  the  well--known crossing  relations to obtain\n$\\overline{|M|^2}$ for processes differing from each other by repositioning the\nfinal and\/or initial particles. The averaged over the initial state degrees of \nfreedom matrix element squared $\\overline{|M|^2}$ can be obtained from\n$|M|^2$ by trivial procedure:\n\\begin{eqnarray*}\n e^+ e^-, \\; e^{\\pm} \\gamma, \\; \\gamma \\gamma &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \\hspace{22mm} \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{4} |M|^2, \\\\\n q \\bar q, \\; q q, \\; \\bar q \\bar q &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{3} \\cdot \\frac{1}{3} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{36} |M|^2, \\\\\n g q, \\; g \\bar q &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{8} \\cdot \\frac{1}{3} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{96} |M|^2, \\\\\n g g &:& \n\\frac{1}{2} \\cdot \\frac{1}{2} \\;({\\rm spin}) \n \\frac{1}{8} \\cdot \\frac{1}{8} \\;({\\rm color}) \\; \\Rightarrow \n   \\overline{|M|^2} = \\frac{1}{256} |M|^2.\n\\end{eqnarray*}\nFor the $2 \\to 2$ processes the differential cross section is\nrelated to the \n$\\overline{|M|^2}$ as follows: \n\\begin{eqnarray}\n \\frac{d \\sigma(2 \\to 2)}{dt} = \\frac{\\overline{|M|^2}}{16 \\pi I^2}, \\quad\n I^2 \\approx s^2,\n\\end{eqnarray}\nwhere $t$ and $I$ are defined in (\\ref{kin11}) and (\\ref{kin13}).\n\nThe notations, used through of this Section, are the same as in \nSection~\\ref{fr}:\n\\begin{description}\n\\item[$e$] is the electric charge of the positron,\n  $\\alpha_{QED} \\equiv \\alpha \n = \\frac{e^2}{4 \\pi} \\approx \\frac{1}{137}$,\n\\item[$Q_f$] is the charge of the quark in units of the positron charge,\n\\item[$g_s$] is the QCD coupling constant, $\\alpha_{QCD} \\equiv \\alpha_s = \n\\frac{g_s^2}{4 \\pi} $,\n\\item[$G_F$] is the Fermi constant. \n\\end{description}\nAs in Section~\\ref{kinem} for the reaction $2 \\to 2$ \n\\begin{eqnarray*}\na(p_1) + b(p_2) &\\to& 1(q_1) + 2(q_2) \\\\\n p_1  + p_2 &=& q_1 + q_2\n\\end{eqnarray*}\nthe Lorentz--invariant Mandelstam variables for reaction are given by \n\\begin{eqnarray*}\n&& s = (p_1  + p_2)^2 = (q_1 + q_2)^2, \\quad  \nt = (p_1  - q_1)^2 = (p_2 - q_2)^2, \\\\ \n&&u = (p_1  - q_2)^2 = (p_2 - q_1)^2, \\\\\n&&s + t + u = m_a^2 + m_b^2 + m_1^2 + m_2^2.\n\\end{eqnarray*}\n\n\\subsection{\\it Matrix Elements} \n\n\\subsubsection{\\it $e^+ e^- \\to f \\bar f$ (no $Z$--boson exchange)}\n\\noindent $\\bullet$ $e^+ e^- \\to l^+ l^-$ ($l \\ne e$, $l = \\mu, \\tau$). \n\\begin{eqnarray}\n|M_e|^2 &=& 8 e^4 \\frac{1}{s^2} \\bigl [ t^2 + u^2 + (m_e^2 + m_f^2)\n(2s - m_e^2 - m_f^2) \\bigr ], \\label{ms2} \\\\\n &=& 8 e^4 \\frac{t^2 + u^2}{s^2}, \\quad {\\rm for} \\; \\; m_e = m_f=0. \\nonumber\n\\end{eqnarray}\n\\noindent $\\bullet$ $e^+ e^- \\to q \\bar q$ \n\\[ \n|M_q|^2  = 3 Q^2_f |M_e|^2.\n\\]\nThe detailed description of the process\n$e^+ e^- \\to f \\bar f$ with $Z$--boson\nexchange is presented in Subsection~\\ref{annihil}.\n\n\\subsubsection{\\it $e^+ e^- \\to e^+ e^-$ (no $Z$--boson exchange)}\n\\begin{eqnarray}\n|M|^2 &=& 8 e^4 \\Bigl \\{ \\frac{1}{s^2} \\bigl [ t^2 + u^2 + 8m^2(s - m^2) \n \\bigr]   \n + \\frac{2}{st} (u - 2 m^2) (u - 6 m^2) \\Bigr \\},  \\label{ms3}  \\\\\n &=& 8 e^4 \\frac{s^4 + t^4 + u^4}{s^2 t^2}, \\quad {\\rm for} \\; \\; m = 0. \n\\nonumber\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to \\gamma \\gamma \\gamma$} \n\\noindent $\\bullet$ $e^+(p_1) + e^-(p_2) \\to \\gamma(k_1) + \\gamma(k_2) \n  + \\gamma(k_3), \\;\\;m_e = 0.$\n\\begin{eqnarray}\n|M|^2 = 8 e^6 \\;\\; \\frac{\\sum\\limits_{i=1}^{3} (p_1 k_i) (p_2 k_i) \n \\bigl [ (p_1 k_i)^2 + (p_2 k_i)^2 \\bigr]}\n {\\prod\\limits_{i=1}^{3} (p_1 k_i) (p_2 k_i) }. \\label{ms4}\n\\nonumber\n\\end{eqnarray}\n\n\\noindent\n$\\bullet$ $e^+ e^- \\to \\gamma \\gamma \\gamma, \\;\\; m_e = m \\ne 0.$ \\\\\nFor the case of $s = (p_{e^+} + p_{e^-})^2 \\to 4m^2$, i.e. in the limit \n$p_{e^+} = p_{e^-} = (m, 0)$, \nthe $|M|^2$ is given by \\cite{land}: \n\\begin{eqnarray}\n|M|^2 = 64 e^6 \n \\left [ \\Bigl (\\frac{m -\\omega_1}{\\omega_2 \\omega_3} \\Bigr )^2 \n  + \\Bigl (\\frac{m -\\omega_2}{\\omega_1 \\omega_3} \\Bigr )^2\n  + \\Bigl (\\frac{m -\\omega_3}{\\omega_1 \\omega_2} \\Bigr )^2 \\right ],\n \\label{ms5}\n\\end{eqnarray}\nwhere $\\omega_i$ is $i$--photon energy in cms.\n\n\\subsubsection{\\it $e^+ e^- \\to l^+ l^- \\gamma$} \n\\[\n e^+(p_1) + e^-(p_2) \\to l^+(q_1) + l^-(q_2) + \\gamma(k),\n  \\;\\; m_e = m_{l} = 0.\n\\]\nInvariants:\n\\begin{eqnarray}\n  s &=& 2(p_1 p_2), \\quad t = -2(p_1 q_1), \\quad u = -2(p_1 q_2),\n  \\label{ms6} \\\\\n s' &=& 2(q_1 q_2), \\quad t' = -2(p_2 q_2), \\quad u' = -2(p_2 q_1). \\nonumber\n\\end{eqnarray}\n\n\\noindent $\\bullet$ $l \\ne e$, for example, $e^+ e^- \\to \\mu^+ \\mu^- \\gamma$\n\\begin{eqnarray}\n|M|^2 = -4 e^6 (v_p - v_q)^2 \\frac{t^2 + t'^2 + u^2 + u'^2}{s s'}. \\label{ms7}\n\\end{eqnarray}\n\n\\noindent $\\bullet$ $l = e$, i.e. $e^+ e^- \\to e^+ e^- \\gamma$\n\\begin{eqnarray}\n|M|^2 = -4 e^6 (v_p - v_q)^2 \n \\frac{s s'(s^2 + s'^2) + t t'(t^2 + t'^2) + u u'(u^2 + u'^2)}{s s' t t'}. \n \\label{ms8}\n\\end{eqnarray}\nwhere in (\\ref{ms7}) and (\\ref{ms8}) we use:\n\\begin{eqnarray} \n v_p^{\\mu} \\equiv \\frac{p_1^{\\mu}}{(p_1 k)} - \\frac{p_2^{\\mu}}{(p_2 k)}, \\quad\n v_q^{\\mu} \\equiv \\frac{q_1^{\\mu}}{(q_1 k)} - \\frac{q_2^{\\mu}}{(q_2 k)}.\n\\label{ms9}\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to q \\bar q g$}\nFor this reaction the invariants are the same as in (\\ref{ms6}).\n\\begin{eqnarray}\n|M|^2 = 16 e^4 Q^2_f g_s^2 \\frac{t^2 + t'^2 + u^2 + u'^2}{s (q_1k) (q_2 k)}. \n\\label{ms10}\n\\end{eqnarray}\n\n\\subsubsection{\\it $e^+ e^- \\to q \\bar q \\gamma$}\nFor this reaction the invariants are the same as in (\\ref{ms6}).\n\\begin{eqnarray}\n|M|^2 = - 12 e^6 (v_p + Q_f v_q)^2  \\frac{t^2 + t'^2 + u^2 + u'^2}{s s'},\n \\label{ms11}\n\\end{eqnarray}\nwhere $v_p$ and $v_q$ are defined in (\\ref{ms9}). \n\n\\subsubsection{\\it $g g \\to q \\bar q$, $m_q = m \\ne 0$} \nThe final $q \\bar q$--pair can be in color {\\it singlet} or color {\\it octet} \nfinal states.\n\\begin{eqnarray}\n|M_{singl}|^2 &=& 16 g_s^4 \\chi_0 \\left [ \\frac{1}{3} \\right ], \\quad\n|M_{oct}|^2 = 16 g_s^4 \\chi_0 \\left [ \\frac{7}{3} - 6 \\chi_1 \\right ], \n \\nonumber \\\\\n|M_{tot}|^2 &=& |M_{singl}|^2 + |M_{oct}|^2 = 16 g_s^4 \\chi_0 \n  \\left [ \\frac{8}{3} - 6 \\chi_1 \\right ], \\label{ms12}\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\n \\chi_0 & = & \\frac{m^2 - t}{m^2 - u} + \\frac{m^2 - u}{m^2 - t} \n + 4 \\left ( \\frac{m^2}{m^2 - t} + \\frac{m^2}{m^2 - u} \\right )\n -4 \\left ( \\frac{m^2}{m^2 - t} + \\frac{m^2}{m^2 - u} \\right )^2  \\label{ms13}  \\\\\n \\chi_1 & = & \\frac{(m^2 - t)(m^2 - u)}{s^2} \\label{ms14}\n\\end{eqnarray}\nFor $m_q = 0$,\n\\[\n\\chi_0 = \\frac{t^2 + u^2}{ut}, \\quad \\chi_1 = \\frac{ut}{s^2}.\n\\]\n\n\\subsubsection{\\it $\\gamma g (\\gamma \\gamma) \\; \\to \\; f \\bar f$}\n\n\\noindent $\\bullet$ $g \\gamma \\to q \\bar q: \\quad$ \n$\\quad  \n|M|^2 = 32 g_s^2 e^2 Q^2_f \\chi_0.\n$\n\n\\noindent $\\bullet$ $\\gamma \\gamma \\to q \\bar q: \\quad$ \n$\\quad  \n|M|^2  = 24 e^4 Q^4_f \\chi_0.\n$\n\n\\noindent $\\bullet$ $\\gamma \\gamma \\to e^+ e^-:  \\quad$ \n$ \\quad  \n|M|^2  = 8 e^4 \\chi_0.\n$\n\n\\subsubsection{\\it $q \\bar q \\; \\to \\; Q \\bar Q$, $m_q = 0$, $m_Q = m \\ne 0$}\n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\frac{t^2 + u^2 + 2m^2(2s - m^2)}{s^2}. \\label{ms15}\n\\end{eqnarray}\n\n\\subsubsection{\\it $q q \\; \\to \\; q q$, $m_q = 0$} \n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\left [\\frac{s^4 + t^4 + u^4}{t^2u^2} - \\frac{8}{3}\n\\frac{s^2}{tu} \\right ]. \\label{ms16}\n\\end{eqnarray}\n\n\\subsubsection{\\it $q \\bar q \\; \\to \\; q \\bar q$, $m_q = 0$} \n\\begin{eqnarray}\n |M|^2  = 16 g^4_s \\frac{1}{s^2 t^2} [s^4 + t^4 + u^4 - \\frac{8}{3}stu^2].\n \\label{ms17}\n\\end{eqnarray}\n\n\\subsubsection{\\it $g g \\; \\to \\; g g$}\n\\begin{eqnarray}\n |M|^2  = 288 g^4_s \\frac{(s^4 + t^4 + u^4)(s^2+t^2+u^2)}{s^2t^2u^2}.\n \\label{ms18}\n\\end{eqnarray}\n\n\\subsubsection{\\it $ f_1 \\bar f_2 \\to W^{\\ast} \\to f_3 \\bar f_4$}\n\\[\n f_1(p_1) + \\bar f_2(p_2) \\to f_3(p_3) + \\bar f_4(p_4), \\quad\n m_{1,2,3,4} \\ne 0.\n\\]\n\\begin{eqnarray}\n|M|^2 = C 128 G^2_F M^4_W \\frac{(p_1 p_4) (p_2 p_3)}\n {(s - M^2_W)^2 + \\Gamma^2_W M^2_W}, \\label{ms19}\n\\end{eqnarray}\nwhere $C=1$ for $l^- \\bar \\nu \\to l'^- \\bar \\nu'$, \n$C=3$ for $l^- \\bar \\nu \\to q \\bar q' ( q \\bar q' \\to l^- \\bar \\nu)$, and \n$C=9$ for $q_1 \\bar q_2 \\to q_3 \\bar q_4$, $M_W$ and $\\Gamma_W$ are the mass \nand total width of the $W$--boson.\n\n\\subsubsection{\\it $ l^- \\bar \\nu \\to d \\bar u g $}\n\n\\[\n l^-(p_1) + \\bar \\nu(p_2) \\to d(p_3) + \\bar u(p_4) + g(k), \\quad\n m_{d, u} \\ne 0.\n\\]\n\\begin{eqnarray}\n|M|^2 &=& 256 G^2_F M^4_W g_s^2 \\frac{ A_1 - A_2 - A_3}\n {(s - M^2_W)^2 + \\Gamma^2_W M^2_W}, \\label{ms20} \\\\\n A_1 &=& \\frac{1}{(k p_3) (k p_4)} \\Bigl \\{ s \\bigl [(p_1 p_4)^2 + (p_2 p_3)^2\n \\bigr ] \\nonumber \\\\\n &-& (m^2_u+m^2_d) \\bigl [\\frac{s}{2} ((p_1 p_4)+(p_2p_3)) \n  - (p_1p_3)(p_2p_3) - (p_1p_4)(p_2p_4)\\bigr] \\Bigr \\}, \\nonumber \\\\\n A_2 &=& \\frac{2m_u^2}{(k p_4)^2} (p_2 p_3) \\bigl[(p_1 k) + (p_1 p_4) \\bigr],\n  \\quad \n A_3 = \\frac{2m_d^2}{(k p_3)^2} (p_1 p_4) \\bigl[(p_2 k) + (p_2 p_3) \\bigr].\n  \\nonumber \n\\end{eqnarray}\n\n\n\\section{\\bf MISCELLANEA}\\label{misc} \n\n\\subsection{\\it Miscellanea}\\label{miscel} \n\n\n\\noindent $\\bullet$ Let us consider the recursion $A_n = aA_{n-1}+bA_{n-2}$ \nfor given $A_0$ and $A_1$. Then \n\\begin{eqnarray*}\n&& A_n = \\alpha z_1^n + \\beta z_2^n,\n \n  z_{1,2} = \\frac{a}{2}[1 \\pm \\sqrt{1+4b\/a^2}], \\quad\n\\alpha = \\frac{A_1-z_2A_0}{z_1-z_2}, \\quad\n\\beta = \\frac{z_1A_0-A_1}{z_1-z_2}.\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Various representations of the {\\it Dirac} \n$\\delta$--function:\n\\begin{eqnarray*}\n&& \\delta(x) \\equiv \\frac{1}{(2 \\pi)} \\int^{\\infty}_{-\\infty}\n e^{ixt} dt, \\\\\n&& \\delta(x, \\alpha) = \\frac{\\alpha}{\\pi(\\alpha^2 x^2 + 1)}, \\; \n \\alpha \\to \\infty; \\quad \n \\delta(x, \\beta) = \\frac{\\beta}{\\pi(x^2 + \\beta^2)}, \\; \n \\beta \\to 0, \\\\ \n&& \\delta(x, \\alpha) = \\frac{\\alpha}{\\sqrt{\\pi}}\n e^{-\\alpha^2 x^2}, \\;\\;\\;  \\alpha \\to \\infty, \\quad \n \\delta(x, \\alpha) = \\frac{\\alpha}{\\pi}\n \\frac{ \\sin(\\alpha x)}{(\\alpha x)}, \\;  \\alpha \\to \\infty, \\\\ \n&& \\frac{1}{x \\pm i\\varepsilon} = \\p \\frac{1}{x} \\mp i \\pi \\delta(x).\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ {\\it Step}--functions $\\Theta (x)$ and $\\varepsilon (x)$\n\\begin{eqnarray*}\n&& \\Theta (x) \\equiv \\frac{1}{(2 \\pi i)} \\int^{\\infty}_{-\\infty}\n \\frac{e^{ixt}}{t - i \\varepsilon} dt \\; = \\; \n\\left \\{ \\begin{array}{rl} 1, & x>0 \\\\ 0, & x<0 \\end{array} \\right. \n \\\\\n&& \\varepsilon (x) \\equiv \\frac{1}{(i \\pi )} \\p \\int^{\\infty}_{-\\infty}\n \\frac{e^{ixt}}{t} dt \\; \\hspace{5mm} = \\;\n\\left \\{ \\begin{array}{rl} 1, & x>0 \\\\ -1, & x<0 \\end{array} \\right. \n\\end{eqnarray*}\n\n\\noindent $\\bullet$ \n\\begin{eqnarray*}\n&& \\frac{1}{(a - i \\varepsilon)^k} = \\frac{i^k}{\\Gamma(k)} \n\\int^{\\infty}_0 e^{i \\alpha (-a + i \\varepsilon)} \\alpha^{k-1} d \\alpha, \n \\; k \\ge 0, \\\\\n&& \\int^{\\infty}_0 (e^{i t a} - e^{i t b}) \\frac{d t}{t} \\; = \\; \n \\ln \\biggl ( \\frac{b + i \\varepsilon}{a + i \\varepsilon} \\biggr ).\n\\end{eqnarray*}\n\n\\subsection{\\it Properties of Operators} \n\nThe various properties of the operators can be found, for example, in\n\\cite{velt,louis,wilcox}. Let $f(A)$ be any function from the operator \n(matrix) $A$, \nwhich can expanded into series with respect to operators (matrices) $A^n$:\n\\[ f(A) = \\sum_{n=0}^\\infty c_n A^n.\n\\]\n\n\\noindent $\\bullet$ Let $\\xi$ be a parameter, then: \n\\begin{eqnarray*}\n&& e^{\\xi A} e^{-\\xi A} \\, =\\, 1, \\quad e^{\\xi A} A e^{-\\xi A}\\,=\\, A,\n  \\quad e^{\\xi A} A^n e^{-\\xi A} \\, =\\, A^n, \\quad\n   e^{\\xi A} f(A) e^{-\\xi A}\\,=\\, f(A).\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Let $A$ and $B$ be noncommuting operators, $\\xi$ and\n$n$ be parameters ($n$ integer). Then: \n\\begin{eqnarray*}\n&& e^{\\xi A} B^n e^{-\\xi A} = (e^{\\xi A} B e^{-\\xi A})^n, \\\\\n&& e^{\\xi A} F(B) e^{-\\xi A} = F(e^{\\xi A} B e^{-\\xi A}), \\\\\n&& e^{\\xi A} B e^{-\\xi A} = B + \\xi[A,B] + \\frac{\\xi^2}{2!} [A,[A,B]] \n+ \\frac{\\xi^3}{3!} [A,[A,[A,B]]] + \\cdots\n\\end{eqnarray*}\n\n\\noindent $\\bullet$ Let $A$ be an operator and there exists the inverse \noperator $A^{-1}$. Then for any integer $n$ :\n\\[ \n A B^n A^{-1} = (A B A^{-1})^n, \\qquad   A f(B) A^{-1} = f(A B A^{-1}).\n\\]\n\n\\noindent $\\bullet$ Let $A(x)$ be an operator, depending on the scalar\nvariable\n$x$, then \n\\begin{eqnarray*}\n  \\frac{d A^{-1}(x)}{d x} = -A^{-1}(x) \\frac{d A(x)}{d x} A^{-1}(x),\n  \\quad \\frac{d e^{A(x)}}{d x} = \\int_0^1 e^{(1-t)A(t)} \n \\frac{d A(t)}{d t} e^{tA(t)} dt.\n\\end{eqnarray*}\n\n\\subsection {\\it The Baker-Campbell-Hausdorff Formula}\n\n Let $A$ and $B$ be non--commuting operators, then :\n\\begin{equation} \ne^A \\; e^B = e^{ \\sum^{\\infty}_{i = 1} Z_i}, \\label{misc10}\n\\end{equation} \nwhere\n\\begin{eqnarray}\n Z_1 & = & A+B; \\label{misc1} \\\\ \n Z_2 &=& \\frac{1}{2}[A,B] ; \\label{misc2} \\\\ \n Z_3 & = & \\frac{1}{12} \\bigl [A,[A,B]\\bigr] + \\frac{1}{12}\\bigl [[A,B],B\n\\bigr]; \\label{misc3} \\\\ \n Z_4 & = & \\frac{1}{48}\\Bigl[A,\\bigl[[A,B],B\\bigr]\\Bigr] + \n  \\frac{1}{48}\\Bigl[\\bigl[A,[A,B]\\bigr],B\\Bigr]; \\label{misc4} \\\\ \n Z_5 & = & \\frac{1}{120}\\biggl[\\Bigl[A,\\bigl[[A,B],B\\bigr]\\Bigr],B\\biggr] + \n  \\frac{1}{120} \\biggl[A,\\Bigl[\\bigl[A,[A,B]\\bigr],B\\Bigr]\\biggr] \n \\label{misc5} \\\\ \n   & - & \\frac{1}{360} \\biggl[A,\\Bigl[\\bigl[[A,B],B\\bigr],B\\Bigr]\\biggr] - \n \\frac{1}{360} \\biggl[\\Bigl[A,\\bigl[A,[A,B]\\bigr]\\Bigr],B\\biggr] \\nonumber \\\\ \n   & - & \\frac{1}{720}\\biggl[A,\\Bigl[A,\\bigl[A,[A,B]\\bigr]\\Bigr]\\biggr] - \n \\frac{1}{720} \\biggl[\\Bigl[\\bigl[[A,B],B\\bigr],B\\Bigr],B\\biggr], \\ldots \n\\nonumber \n\\end{eqnarray}\nThe other terms can be evaluated from the relation (see \\cite{velt,wilcox}): \n\\begin{eqnarray} \n \\sum_{k=0}^{\\infty} \\frac{1}{(k+1)!} [\\![ Z^k,Z' ]\\!] = A + \n \\sum_{j=0}^{\\infty} \\xi^j \\frac{ [\\![ A^j,B ]\\!] }{j!}, \\label{o1} \n\\end{eqnarray}\nwhere $e^Z = e^{\\xi A} e^{\\xi B}$; $Z = \\sum_{n=1}^{\\infty} \\xi^n Z_n$; \n$Z' = \\sum_{n=1}^{\\infty} n \\xi^{n-1} Z_n$.\nThe repeated commutator bracket is defined as follows\n\\begin{eqnarray*}\n [\\![A^0,B ]\\!] = B, \\quad [\\![ A^{n+1},B ]\\!] = \\biggl [ A, [\\![\nA^n,B ]\\!] \\biggr ].\n\\end{eqnarray*}\nSince relation (\\ref{o1}) must be satisfied identically in $\\xi$, one can \nequate the coefficients of $\\xi^j$ on the two sides of this relation. In\nparticular,  $j=0,1,2,3,4$ gives (\\ref{misc1}, \\ref{misc2}, \\ref{misc3}, \n\\ref{misc4}, \\ref{misc5}), respectively.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:intro}Introduction}\n\nThe propagation of mechanical vibrations in particulate matter is a nontrivial phenomenon that has implications in several fields, both in basic and applied research~\\cite{deGennes1999,Porter2015}. The complexity of the contact network~\\cite{Radjai1996,Lherminier2014} and the nonlinearity of the interaction force between particles~\\cite{Popov2010} determines how the waves propagate in granular media~\\cite{Norris1997}. In dry packings, the repulsive nonlinear interaction between two spheres with radius $R$ and elastic modulus $E$ relies on the Hertz contact force, $F_H \\propto ER^{1\/2}\\delta^{3\/2}$~\\cite{Landau1986}. The latter implies that the contact stiffness increases rapidly with the deformation $\\delta$ and vanishes in absence of mechanical contact. In the dry case, there is thus no tensile force, and the particles can eventually loose contact~\\cite{Tournat2004}. In one-dimensional lattices of particles~\\cite{Nesterenko2001}, the nonlinear elastic interaction given by the Hertzian contact dictates the dynamics; a mechanical impact can propagate weakly to strongly nonlinear waves~\\cite{Daraio2005,Job2005} depending on the amplitude of the pulse. Similarly, in random granular packing, a low-amplitude mechanical excitation generates a linear ballistic waves, both in the longitudinal (namely a {\\em P-wave}) and in the transverse (namely a {\\em S-wave}) directions~\\cite{Norris1997,Makse2004,Johnson2005}. The coherent perturbation, resulting from an ensemble average~\\cite{Page1995}, travels straight from the source to the receiver, as in an effective medium~\\cite{Norris1997,Makse2004}. Owing to the randomness, the coherent pulse is followed by an incoherent and long-lasting {\\em coda wave}, which corresponds to the multiple scattering~\\cite{Snieder2006} of the initial excitation, across the contact network. Interestingly, the ballistic waves possess the reminiscent features of the microscopic scale~\\cite{Makse2004}. Recently, a continuous description bridging the linear and the nonlinear regimes in random packing of particles has been proposed~\\cite{Wildenberg2013}, depending on the confinement pressure applied to the packing and the amplitude of the propagating impulse~\\cite{Santibanez2016}.\\\\\n\nWeakening of granular materials is another nonlinear mechanism that is reported to occur in particulate systems. Its origin is a softening induced by an acoustic fluidization mechanism~\\cite{Jia2011, Espindola2012, Giacco2015, Olson2015}. Such nonlinear process has been demonstrated to be responsible for triggering secondary earthquakes after the occurrence of a main seismic event~\\cite{Johnson2005}. The propagation of the mechanical impulse interacts with the grains in the material and mobilizes the particles with weaker contacts~\\cite{Johnson2008, Johnson2016}. This leads to a modification of the contact network that can trigger major faults and lead to the emission of new events~\\cite{Johnson2005,Johnson2008}. Since the material weakening involves weakly consolidated contacts, it can be modified by the presence of an interstitial fluid, owing to an enhanced cohesion due to capillary effects~\\cite{Dorostkar2018} or to the viscous lubrication between the grains~\\cite{Galaz2018}.\\\\\n\nWet granular media are ubiquitous in nature, as for instance the sediments, the mud, or the sand of a beach. In this paper, we look at understanding how the presence of an interstitial fluid can affect the nonlinear dynamics of particulate matter. The behavior of wet particles has been investigated in recent years due to the great number of industrial applications, from mining to food and pharmaceutical industries. A wet granular medium has a fluid phase that partially occupies the interstitial volume available between grains and their motions are likely determined by interactions mediated by the fluid. In the literature, it has been shown that viscous forces, surface tension, and capillary bridges among others, have a great importance in systems where the energy dissipation occurs due to liquid films trapped either in the asperities~\\cite{Brunet2008} or nearby the contact region~\\cite{Marshall2011}. On one hand, capillarity bridges between grains~\\cite{Semprebon2016} have been studied owing to significant effects on the static cohesion of granular packings~\\cite{Pacheco2012,Saingier2017}. On the other hand, wave propagation in wet granular media relies on the dynamics of the interstitial fluid~\\cite{Herminghaus2005,Moller2007}, which generates a viscous repulsion~\\cite{Donahue2010,Arutkin2017} that generally cannot be neglected at acoustic frequencies; the fluid modifies the acoustic features due to a velocity-dependent viscous contribution~\\cite{Herbold2006,Job2008,Griffiths2010}, in addition to increasing the overall dissipation~\\cite{Brunet2008}. When the fluid is highly confined, in the lubrication regime~\\cite{Davis1986}, and under high-frequency or high-amplitude vibrations, the viscous forces can ultimately induce elastic deformations of the particles, via an elastohydrodynamic interaction~\\cite{Leroy2011,Leroy2012,Villey2013,Wang2015}. Understanding all these behaviors is fundamental for unraveling all the phenomena ranging from the stability of a pile of wet grains~\\cite{Scheel2008} to the dynamics of dense suspensions~\\cite{Waitukaitis2012,Han2016,Peters2016,Buttinoni2017}.\\\\\n\nIn this paper, we aim (i) at ruling out the mechanisms involved in the propagation of mechanical impulses in a wet model granular medium and (ii) at unraveling how the fluid can affect the dynamic weakening of such a material. In Sec.~\\ref{sec:setup_observations}, we present the experimental setup, the protocols of analysis, and a description of the experimental observations. In Sec.~\\ref{sec:model}, we analyze and interpret the features of the high-frequency spectrum of the transmitted pulses, in terms of the propagation velocity. These features allow probing the elasticity of the medium, and how it is affected by the presence of an interstitial fluid. In Sec.~\\ref{sec:weakening}, we analyze the low-frequency spectrum of the mechanical response. Our measurements reveal the elastic weakening of dry granular media at low confinement and high amplitude, which disappears either at large confinement pressure or when the contacts are lubricated by a sufficiently viscous fluid. Finally, Sec.~\\ref{sec:conclusion} summarizes the results and the observations presented in this paper.\\\\\n\n\\section{\\label{sec:setup_observations} Experimental Setup and observations}\n\n\\begin{figure}[b]\n\\begin{center}\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_1.pdf}\n\\caption{\\label{fig:setup} Sketch of the experimental setup depicting a cylindrical granular medium enclosed in a soft elastic sheet, the shock initiator pendulum, and the positions of the dynamic sensors.}\n\\end{center}\n\\end{figure}\n\nThe experimental sample under study (see Fig.~\\ref{fig:setup} and Ref.~\\cite{Santibanez2016}) consists in a packing of approximately $3000$ spherical glass particles (density $\\rho_g=2400$~kg\/m$^3$, radius $R_g=2.5$~mm, Young modulus $E_g=69$~GPa, Poisson's ratio $\\nu_g=0.2$, and surface roughness $Ra\\simeq10$~nm~\\cite{Brunet2008,Buttinoni2017}) confined inside a thin deformable latex sheet. The sheet is hermetically sealed and clamped between two square plates made of a rigid plastic. The two plates are $1$~cm thick and have a $5$~cm in diameter circular aperture. Thin lateral holes on the side of the plates allow the emergence of sensors cables and a vacuum hose. The soft sheet allows maintaining a controlled isotropic stress on the granular medium, by evacuating the interstitial air from the hose with a vacuum pump. While the sample is set under pressure, it can be molded by hand in the form of a cylinder of length $L_s=15$~cm and radius $R_s=2.5$~cm that fits in the holes of the two supporting plastic plates. This finally leads to a solidlike granular medium with compactness approximately equal to the random close packing fraction, $\\phi_s\\simeq0.63$. The pump allows reaching a hydrostatic pressure of as high as $P_0\\approx83$~kPa, which is probed by a static pressure sensor ({\\em Honeywell 19C015PV5K} with its {\\em INA114} low-noise amplifier). At one extremity of the sample, a dynamic force sensor ({\\em PCB Piezotronics 208C01}) is placed in direct contact with the grains through a central and hermetic hole cut in the sheet. At the opposite side of the sample, a miniature accelerometer ({\\em PCB Piezotronics 352A24}) is located on the axis of the cylindrical sample. A single short mechanical impulse is initiated in the sample by impacting the back of the dynamic force sensor with a $21$~cm long pendulum. The impacting head of the pendulum consists in a piece of brass in front of which a spherical glass particles has been glued. The contact duration depends on the mass ($57$~g) of the head~\\cite{Landau1986}, which has been chosen so that the initial excitation is about a fraction of a millisecond~\\cite{Santibanez2016}. A collision thus generates a broadband excitation from dc to few kilohertz. The strength of the impact is controlled by adjusting the initial release angle of the pendulum. At the opposite end, the accelerometer records the outgoing pulse that travels through the sample. The signals of the dynamic force sensor and the accelerometer are routed through a signal conditioner ({\\em PCB Piezotronics model 482C}) and acquired simultaneously, together with the signal of the static pressure, with an analog-to-digital converter ({\\em National Instruments USB-6356}) at $500$~kS\/s sample rate.\\\\\n\n\\begin{figure*}\n\\includegraphics[width=0.900\\textwidth]{1808_03150_v2_Fig_2.pdf}\n\\caption{\\label{fig:signals_low_P} Examples of experimental data at low confinement pressure ($P_0=3.2$~kPa). The left column (0) shows the input force for each experiment: the solid line corresponds to the dry configuration, the dashed line is $\\mu=1$~Pa.s and the dotted line is $\\mu=30$~Pa.s, whose acceleration outputs are shown in the next three columns (1)--(3), respectively. Rows (a)--(c) correspond to different strength of the initial excitation, as shown in the right-side text labels.}\n\\end{figure*}\n\n\\begin{figure*}\n\\includegraphics[width=0.900\\textwidth]{1808_03150_v2_Fig_3.pdf}\n\\caption{\\label{fig:signals_high_P} Examples of experimental data at high confinement pressure ($P_0=83$~kPa). The left column (0) shows the input force for each experiment: the solid line corresponds to the dry configuration, the dashed line is $\\mu=1$~Pa.s and the dotted line is $\\mu=30$~Pa.s, whose acceleration outputs are shown in the next three columns (1)--(3), respectively. Rows (a)--(c) correspond to different strength of the initial excitation, as shown in the right-side text labels.}\n\\end{figure*}\n\nThe experimental setup allows probing the acoustic response of dry and wet granular media by measuring the speed of the waves that propagate through the samples. The wet samples are prepared by homogeneously filling a fraction of the interparticle space with a viscous fluid. The amount of added liquid corresponds to a third of the available volume; given the estimated packing fraction $\\phi_s$, the liquid thus approximately occupies $12\\%$ of the total volume of a sample. If all the spherical particles were homogeneously lubricated~\\cite{Galaz2018}, the liquid coating would have a thickness $D_{coat}=R_g([1+(1-\\phi_s)\/3\\phi_s]^{1\/3}-1)\\simeq 150$~$\\mu$m. Before pouring the wet particles into the soft elastic sheet, we make sure that all the grains are uniformly coated with fluid by carefully stirring the mixture in a container. After having used a fluid, the grains are washed several times with alcohol and then gently dried in an oven. The fluids are silicone oils from {\\em Sigma Aldrich}. Four different fluids' viscosity were considered in this study, ranging over more than two decades, from $\\mu=0.1$~Pa.s to $\\mu=30$~Pa.s at room temperature. The sound speed and the mass density of these fluids are close to the features of water, $c_f\\simeq1500$~m\/s and $\\rho_f\\simeq1000$~kg\/m$^3$. Their surface tension is $\\gamma_f\\simeq21.5$~mN\/m~\\cite{Langley2017} and their storage modulus is $G\\simeq5$~kPa~\\cite{Oswald2014}, both independently of the viscosity. In particular, these fluids remain Newtonian below a critical flow's shear rate $\\dot{\\gamma}_f=G\/\\mu$~\\cite{Oswald2014,Guyon2001}. This condition is always satisfied in all our experiments, as demonstrated in Sec.~\\ref{sec:model}. Above the critical shear rate, the behavior of these fluids is known to be relatively independent of the viscosity~\\cite{Oswald2014,Barlow1964}.\\\\\n\nA set of typical waveforms propagated through different samples under $P_0=3.2$~kPa and $P_0=83.0$~kPa confinement pressures is shown in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, respectively. In both figures, the rows stand for three different excitation amplitude. The leftmost columns show the initial force perturbation as a function of time, resulting from the impact of the pendulum. The short initial impact is about $0.2$~ms wide: its spectrum (not shown; see Ref.~\\cite{Santibanez2016}) extends from dc to $8$~kHz approximately. The three rightmost columns present the acceleration of the transmitted waves, first in a dry medium and then with two different interstitial viscosities. The long-lasting transmitted {\\em coda wave} corresponds to the multiple scattering of the incident pulse~\\cite{Snieder2006}, along different paths through the random network of particles. The very first transmitted event [see the magnified plots in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)] corresponds to the fastest ballistic contribution~\\cite{Page1995}, that is a longitudinal pressure wave traveling straight from the source to the receiver~\\cite{Norris1997, Makse2004, Johnson2005}. In all the transmitted signals presented in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, one clearly distinguishes two distinct frequency components. On one hand, the ballistic wave carries a high-frequency content, lying in the range of $1$~kHz [see the rise time in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)]. This spectrum matches the bandwidth of the excitation attenuated by the frequency-dependent scattering~\\cite{Page1995} and the viscoelastic dissipation at the contact between grains~\\cite{Job2005, Popov2010}. In the presence of a fluid, the amplitude of the ballistic pulse decreases slightly more: the viscous dissipation enhances the attenuation~\\cite{Herbold2006,Job2008,Brunet2008}. The analysis of the speed of the ballistic wave is presented in more details in Sec.~\\ref{sec:model}. On the other hand, the transmitted waves shown in Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P} also exhibit a long-lasting, superimposed, low-frequency oscillation of the order of $100$~Hz. It has been demonstrated to rely on a longitudinal resonance of the sample~\\cite{Santibanez2016} and to reveal the nonlinear weakening~\\cite{Jia2011, Espindola2012, Olson2015, Giacco2015, Johnson2008, Johnson2016} of the granular matter. The detailed analysis of the low-frequency oscillation is presented in Sec.~\\ref{sec:weakening}.\\\\\n\n\\begin{figure}[b]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_4}\n\\caption{\\label{fig:wave_speed} Examples of the time of flight (tof) measurement for (a) dry and (b) wet ($\\mu=30$~Pa.s) cases, at low confinement pressure. Wave speed as a function of the magnitude of the initial collision force in dry and wet granular media, for both low (c) and high (d) confinement pressures. Markers correspond to different interstitial fluid viscosity: $\\circ:$ dry, $\\triangle:0.1$~Pa.s, $\\square:1$~Pa.s, $\\triangledown:10$~Pa.s, and $\\diamond:30$~Pa.s. The dashed lines are guides for the eyes; the vertical arrows point toward increasing viscosities.}\n\\end{figure}\n\nThe measurements of the speed of the ballistic pulses are presented in Fig.~\\ref{fig:wave_speed}. The wave speed $c$ is estimated as the distance $L_s$ between the input (force sensor) and the output (accelerometer) divided by the time of flight. The latter is measured as the time difference between the earliest in and out events [see for instance the dashed line in Figs.~\\ref{fig:wave_speed}(a) and~\\ref{fig:wave_speed}(b)], which are detected systematically as the instants at which the amplitude of each waveform emerges above a threshold (three times the average noise level). In Figs.~\\ref{fig:wave_speed}(c) and~\\ref{fig:wave_speed}(d), we show the propagation velocity for both low and high confinement pressures, respectively. In both cases, the wave speed is measured first in a dry medium as a reference, then with the four fluids with different viscosities. The dry measurements at low confinement pressure, $P_0=3.2$~kPa, are shown as circles in Fig.~\\ref{fig:wave_speed}(c): the log-log representation reveals that the wave speed $c$ approximately increases as a power law of the magnitude of the excitation, as it is predicted from a Hertzian interaction~\\cite{Norris1997, Makse2004, Santibanez2016, Wildenberg2013}, see Sec.~\\ref{sec:model}. In contrast, with interstitial fluids, the wave speed increases more slowly with the strength of the impact and quickly tends to a constant value, independent of the input force, at large viscosities. Remarkably, the propagation velocity noticeably increases with the viscosity of the fluid; this nontrivial feature has been observed yet in one-dimensional granular media~\\cite{Herbold2006,Job2008} and is analyzed in Sec.~\\ref{sec:model}. On the opposite, at the highest confinement pressure, $P_0=83$~kPa, the propagation velocity shown in Fig.~\\ref{fig:wave_speed}(d) appears independent on the amplitude of the perturbation in the dry configuration: the dynamical stress being negligible compared to the static pressure, the response is linear. With fluid, the wave speed does not depend either on the perturbation strength and a weaker dependence of the wave speed with the viscosity of the fluid is observed, compared to the low confinement case.\\\\\n\nThese observations suggest a competition between the elastic deformations of the particles resulting from the confinement or the dynamic pressure, and an effect of the viscous fluid, which likely resides in between the particles and at the periphery of their contacts~\\cite{Marshall2011}. Here, it is possible to rule out the contribution of capillary effects since all our fluids have the same surface tension, $\\gamma_f\\simeq21.5$~mN\/m, independently of their viscosity~\\cite{Langley2017}: it thus cannot explain the viscosity-dependent wave speed observed in the wet media. Moreover, equating the Laplace pressure~\\cite{Guyon2001} to the confinement pressure, $P_0\\simeq \\gamma_f\/R_c$, indicates excessively small capillary bridges curvatures, compared to the size of the particles and the liquid's filling fraction, to produce a significant contribution: $R_c\\simeq P_0\/\\gamma_f\\sim 7$~$\\mu$m at $P_0=3.2$~kPa and $R_c\\sim 0.25$~$\\mu$m at $P_0=83$~kPa. The next section aims at deriving an alternative framework, in order to relate the increase of the wave speed, i.e., the enhancement of the effective elasticity of the medium, as an effect of the viscous flow in the interstices of the particles.\\\\\n\n\\section{\\label{sec:model} Wave speed analysis and interpretation}\n\n\\subsection{\\label{subsec:model_dry} Wave speed in dry granular media}\n\nMeasuring the speed of mechanical waves gives access to the elastic features of the media, at an effective macroscopic scale. Considering that the fastest event corresponds to a ballistic pressure wave in the longitudinal direction~\\cite{Norris1997, Makse2004}, one can extract the effective longitudinal modulus $M$ from the experimental measurements of the wave speed $c$ shown in Fig.~\\ref{fig:wave_speed},\n\\begin{equation}\nM = K+(4\/3)G = \\rho_g\\phi_s c^2.\\label{eq:model_Mexp}\n\\end{equation}\n\nAccording to the effective medium theory (EMT)~\\cite{Norris1997, Makse2004}, the effective bulk and shear moduli, $K$ and $G$, of a random packing of frictional spheres can be related to the interactions at the interparticles scale; these moduli are given by\n\\begin{eqnarray}\nK &\\propto& (Z\\phi_s\/R_{\\ast})\\kappa_n,\\label{eq:model_Kemt}\\\\\nG &\\propto& (Z\\phi_s\/R_{\\ast})[\\kappa_n+(3\/2)\\kappa_t],\\label{eq:model_Gemt}\n\\end{eqnarray}\nwhere $\\kappa_{n,t}=\\partial F_{n,t}\/\\partial \\delta_{n,t}$, $F_{n,t}$, and $\\delta_{n,t}$ are the normal and tangential stiffnesses, forces, and deformations at a single contact between two particles, respectively. $Z$ is the coordination number and $R_\\ast=R_g\/2$ is the reduced radius of curvature at the contacts. The interparticles stiffness depends on the size of the mechanical contact; in the case of dry spheres, the Hertz-Mindlin interaction potential~\\cite{Landau1986, Popov2010} provides\n\\begin{eqnarray}\na_{dry} &=& (R_\\ast\\delta_n)^{1\/2}\\propto R_\\ast(p\/E_\\ast)^{1\/3},\\label{eq:model_Adry}\\\\\n\\kappa_{n,t} &\\propto& F_{n,t}\/\\delta_{n,t} \\propto E_\\ast a_{dry},\\label{eq:model_Kdry}\n\\end{eqnarray}\nwhere $a_{dry}$ is the radius of the flat contact disk between two spheres, $E_{\\ast}=E_g\/2(1-\\nu_g^2)$ is the reduced elastic modulus and $p\\propto Z\\phi_sF_n\/R_{\\ast}^2$~\\cite{Norris1997} is the confining pressure. Note that the scaling given in Eq.~\\ref{eq:model_Kdry} relies on the estimation of the normal force, $F_n\\propto\\pi a_{dry}^2p_{max}\\propto E_\\ast a_{dry}\\delta_n$, from the maximal pressure inside the contact region given by the Hooke's law, $p_{max}\\propto E_\\ast(\\delta_n\/a_{dry})$. Note also that in Eq.~\\ref{eq:model_Gemt}, the tangential contribution $\\kappa_t$ stands for a nonsliding (i.e., sticking) contact resulting from an infinite Coulomb's friction coefficient between particles. In the case of frictionless particles, the tangential stiffness is zero, $\\kappa_t=0$. Hence, it turns out that the effective longitudinal modulus non-linearly depends on the confinement pressure $p$,\n\\begin{equation}\n(M_{dry}\/E_{\\ast}) = \\alpha_{dry}(p\/E_{\\ast})^{1\/3},\\label{eq:model_Mdry}\n\\end{equation}\nwhere $\\alpha_{dry}$ is a numerical prefactor of the order of unity, which only depends on the topology of the granular packing (via $Z$ and $\\phi_s$) and wether the contacts of the particles stick or slip. Quantitatively (see Eqs. (1), (12), (13) and (14) in Ref.~\\cite{Makse2004}), a lower and an upper bounds of the prefactor are $\\alpha_{dry}\\simeq 0.37$ for frictionless particles with $Z=2$ and $\\alpha_{dry}\\simeq 1.70$ for frictional particles with $Z=6$, at $\\phi_s=0.63$.\\\\\n\n\\begin{figure}[b]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_5.pdf}\n\\caption{\\label{fig:elastic_moduli} (a) Effective longitudinal modulus in the dry case as a function of the pressure $p$, where $p$ stands either for the static pressure $P_0$ at high confinement or for the magnitude of the perturbation in the low confinement limit. (b) Ratio of the wet to dry longitudinal modulus, $M_{wet}\/M_{dry}$, as a function of $\\mu\\omega\/p$. (c) Ratio of the elastohydrodynamic contribution to the dry longitudinal modulus as a function of $\\mu\\omega\/p$, with $M_{ehd}=M_{wet}-M_{dry}$, see Eq.~\\ref{eq:model_Mwet}. (d) Linear plot of the elastohydrodynamic contribution to the elastic modulus as a function of $(\\mu\\omega\/E_{\\ast})^{1\/3}$. In (a)--(d), the markers refer to the definition given in Fig.~\\ref{fig:wave_speed} and the shaded region show the $50\\%$ mean deviation error. In (a) and (c), the straight lines have a slope of $1\/3$, according to Eqs.~\\ref{eq:model_Mdry},~\\ref{eq:model_Mehd} and~\\ref{eq:model_Mwet}.}\n\\end{figure}\n\nThe estimations of the dry longitudinal modulus, $M_{dry}$, obtained from the measurements of the wave speeds $c$ shown in Fig.~\\ref{fig:wave_speed}, are presented in Fig.~\\ref{fig:elastic_moduli}(a) in a nondimensional form. Experimentally, $p$ stands for the static pressure at the highest confinement, $p=P_0+P_m\\simeq P_0\\gg P_m$, where $P_m=F_m\/\\pi R_s^2$ is an estimation of the magnitude of the dynamic stress and $F_m$ is the magnitude of the measured excitation force. At the lowest confinement, $p$ stands for the magnitude of the perturbation, $p\\simeq P_m\\gg P_0$. Matching the data shown in Fig.~\\ref{fig:elastic_moduli}(a) to the Eq.~\\ref{eq:model_Mdry} provides two estimations for the prefactor, both being of the order of unity: $\\alpha_{dry}=0.25\\pm38\\%$ at low confinement and $\\alpha_{dry}=0.88\\pm3\\%$ at high confinement. The order of magnitude of these coefficients are in fair agreement with the EMT prediction given in Eq.~\\ref{eq:model_Mdry}. In particular, the experimental data at low confinement pressure reveal the nonlinear nature of the amplitude-dependent elastic modulus. In this regime, the slightly lower exponent, in comparison to the $1\/3$ expectation, and the slightly low experimental prefactor $\\alpha_{dry}$ are presumably a trace of the nonlinear softening of the material~\\cite{Jia2011, Espindola2012, Giacco2015, Olson2015, Johnson2016} described in Sec.~\\ref{sec:weakening}.\n\n\\subsection{\\label{subsec:model_wet}elastohydrodynamic interactions mediated by the fluid}\n\n\\begin{figure}[t]\n\\begin{center}\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_6.pdf}\n\\caption{\\label{fig:sketch_contact} Sketch of the normal deformation at the contact between two spheres in the dry case (dashed line) and with an interstitial fluid (solid line), depicting the spatial extent of the deformations, $a_{dry}$ and $a_{ehd}$, and a fluid layer with thickness $D_c$.}\n\\end{center}\n\\end{figure}\n\nGetting a deeper insight on the fluid-dependent elastic features shown in Fig.~\\ref{fig:elastic_moduli} can be achieved by analyzing the interaction between two elementary elastic spheres separated by a thin layer of viscous fluid, as sketched in Fig.~\\ref{fig:sketch_contact}. This case has been extensively addressed in the literature untill recently, via the elastohydrodynamic rebound of an elastic sphere on a liquid layer~\\cite{Davis1986} or via the oscillatory excitations of a spherical tip immersed in a fluid near an elastic plane~\\cite{Leroy2011, Leroy2012, Villey2013}. The details presented in these four studies are recalled in the following, in order to derive a useful description of our observations. In the following, $D>0$ denotes the initial separation between two spheres separated by an interstitial layer of viscous fluid in the lubrication limit, $D\\ll R_\\ast$. The relative displacement of the particles and their elastic deformation, mediated by the fluid, are denoted by $d(t)$ and $\\delta_n(r,t)$, respectively. They result from a normal collision, along the direction $z$ in Fig.~\\ref{fig:sketch_contact}, at relative velocity $v_n=\\dot{d}\\propto\\omega d$. Here, $\\omega=2\\pi f$ is the angular frequency relying on the duration of the shock, $\\tau\\propto\\omega^{-1}$. The thickness of the fluid as a function of the radial coordinate $r$ and the time $t$ thus reads $\\Delta=D+\\delta_n(r,t)-d(t)+r^2\/2R_\\ast$.\\\\\n\n{\\em Rigid particles ($\\delta_n=0$)}. The case of rigid particles is considered first. The weak collision ($d\\ll D$) between the two spheres squeezes the fluid out from the interstitial region and induces a radial flow, which is assumed laminar and incompressible~\\cite{Davis1986, Guyon2001}. Within the incompressible assumption, the mean radial velocity of the fluid $\\langle v_r \\rangle=(1\/\\Delta)\\int_0^\\Delta{v_rdz}$ can be estimated from the flow rate conservation, $\\pi r^2v_n=2\\pi r\\Delta\\langle v_r \\rangle$, as $\\langle v_r \\rangle\\propto r\\dot{d}\/(D+r^2\/2R_\\ast)$. The latter expression reveals the radial extent of the hydrodynamic field $a_{h}=(2R_\\ast D)^{1\/2}$~\\cite{Davis1986}: the radial velocity vanishes at $r=0$ and $r\\gg a_h$, and is maximal and of the order of $\\langle v_r \\rangle \\propto a_h\\dot{d}\/D$ at $r\\propto a_h$. The laminar assumption implies that $v_r$ is parabolic along the normal $z$ axis~\\cite{Leroy2011} ($v_r=0$ at the solid\/fluid interfaces due to the non-slip condition and is maximal at $z=0$ for symmetry reason). The local shear rate in the fluid is thus $\\dot{\\gamma}=\\partial v_r\/\\partial z\\propto a_h\\dot{d}\/D^2$ and its mean value over the whole interstitial region is $\\langle\\dot{\\gamma}\\rangle=(1\/\\pi R^2)\\int_0^R{\\dot{\\gamma}2\\pi rdr}\\propto(a_h\/R)^2\\dot{\\gamma}\\propto\\dot{d}\/a_h$. The flow generates a hydrodynamic pressure given by the Stokes equation, $(\\partial p_h\/\\partial r\\propto p_h\/a_h)\\simeq(\\mu\\partial\\dot{\\gamma}\/\\partial z\\propto \\mu a_h\\dot{d}\/D^3)$~\\cite{Davis1986,Leroy2011}, such that $p_h\\propto\\mu a_h^2\\dot{d}\/D^3\\propto\\mu\\omega R_\\ast d\/D^2$. In the case of rigid particles, the interstitial flow consequently induces the well-known Reynolds force~\\cite{Guyon2001}, which counteracts the relative approach of the particles, $F_h\\simeq\\pi a_h^2p_h\\propto \\mu\\omega R_\\ast^2 d\/D$.\\\\ \n\n{\\em Interstitial fluid flow}. The details of the interparticles flow regime can be inferred from the long-wavelength experimental data shown in Figs.~\\ref{fig:signals_low_P},~\\ref{fig:signals_high_P}, and~\\ref{fig:wave_speed}. The largest acceleration is typically $\\Gamma\\propto\\omega V\\sim10$~m\/s$^2$, see Figs.~\\ref{fig:signals_low_P} and~\\ref{fig:signals_high_P}, with $V$ denoting the velocity field. The rise duration revealing a frequency content at around $f\\sim1$~kHz, see Fig.~\\ref{fig:wave_speed}, then $V\\sim1.6$~mm\/s at the most. The normal relative velocity between two particles, $v_n=\\dot{d}$, can be deduced from an estimation of the gradient of the velocity field, $V\/\\lambda$ where $\\lambda=c\/f$ is the wavelength, as $v_n\\propto RV\/\\lambda\\propto R\\Gamma\/c$. The wave speed being at least $c\\sim200$~m\/s with fluids, see Fig.~\\ref{fig:wave_speed}, then $\\dot{d}\\sim0.25$~mm\/s at the most. This allows estimating then the typical fluid's shear rate in the interstitial region, $\\langle\\dot{\\gamma}\\rangle\\propto\\dot{d}\/a_h\\propto (R\/D)^{1\/2}(\\Gamma\/c)$. The latter requires an estimation of the fluid's thickness $D$. As a worst situation, the surface roughness of the particles, $Ra\\sim10$~nm~\\cite{Brunet2008,Buttinoni2017}, can be considered as the minimal achievable separation, $D\\sim Ra$, below which the fluid may be trapped in between asperities~\\cite{Marshall2011,Buttinoni2017}. At worst, the typical shear rate is thus $\\langle\\dot{\\gamma}\\rangle\\sim25$~s$^{-1}$. This value remains an order of magnitude, or smaller, below the critical shear rate, $\\langle\\dot{\\gamma}\\rangle\\ll\\dot{\\gamma}_f=G\/\\mu$, above which the fluid becomes non-Newtonian: with $G\\simeq5$~kPa~\\cite{Oswald2014}, the critical shear rate is $\\dot{\\gamma}_f\\sim166$~s$^{-1}$ at the highest viscosity $\\mu=30$~Pa.s and $\\dot{\\gamma}_f\\sim50.000$~s$^{-1}$ at the lowest viscosity $\\mu=0.1$~Pa.s. The fluid thus remains Newtonian in all our experiments. Finally, the largest radial velocity of the fluid, $\\langle v_r \\rangle \\propto (R\/D)^{1\/2}(\\Gamma R\/c)\\sim6.25$~cm\/s, indicates a Mach number $M_f=\\langle v_r \\rangle\/c_f\\ll1$ and a Reynolds number, $Re=\\rho_f \\langle v_r \\rangle D\/\\mu\\ll1$ well below unity. The interstitial flow thus remains incompressible and laminar~\\cite{Guyon2001}, in agreement with previous assumptions. As a consequence, the increase of the effective elasticity observed in wet samples cannot be attributed either to a non-Newtonian viscoelastic behavior of the fluid, or to an effect of its compressibility.\\\\\n\n\\begin{figure*}\n\\includegraphics[width=0.882\\textwidth]{1808_03150_v2_Fig_7.pdf}\n\\caption{\\label{fig:weakening_spectra} Frequency spectra at $P_0=3.2$~kPa (solid black line) and $P_0=83$~kPa (dashed red line) for increasing amplitudes of excitation. (a) dry contacts, (b) $\\mu=1$~Pa.s, (c) $\\mu=10$~Pa.s, and (d) $\\mu=30$~Pa.s show the weakening behavior of the lowest mode. The vertical arrows are guides for the eye, pointing toward increasing amplitudes of excitation.}\n\\end{figure*}\n\n{\\em Elastic particles ($\\delta_n>0$)}. Nevertheless, the hydrodynamic pressure field $p_h\\propto\\mu\\omega R_\\ast d\/D^2$ can reach sufficiently high values, for instance when the fluid's thickness vanishes, to involve the elasticity of the particles~\\cite{Davis1986,Leroy2011}. Given the Hooke's law, $p_h\\propto E_\\ast(\\delta_n\/a_h)$, the elastic deformation of the particles can be rewritten as a fraction of their relative displacement, $(\\delta_n\/d)\\propto (D_c\/D)^{3\/2}$, where~\\cite{Leroy2011}\n\\begin{equation}\nD_c=8R_\\ast(\\mu\\omega\/E_\\ast)^{2\/3}\n\\label{eq:model_D_cutoff}\n\\end{equation}\nis a cutoff thickness at which the deformation accommodates the displacement, $\\delta_n=d$. This situation was referred to as an {\\em elastic confinement} of the fluid~\\cite{Villey2013}: the fluid being clamped by its viscosity, it does not flow but instead mediates the elastic deformations of the bodies as a rigid layer. The cutoff thickness thus stands as a minimal achievable fluid thickness. Quantitatively, it is approximately $D_c\\simeq70$~nm with the lowest viscosity and $D_c\\simeq3$~$\\mu$m with the highest viscosity, under our experimental conditions. These values are larger than the typical surface roughness, $D_c\\gg Ra\\sim10$~nm, and smaller than the typical thickness of the liquid coating, $D_c\\ll D_{coat}\\sim150$~$\\mu$m, this regime is thus accessible in our experiments. In such an elastohydrodynamic regime, the typical extent of the field and the normal stiffness thus become, respectively\n\\begin{eqnarray}\na_{ehd} &=& (2R_\\ast D_c)^{1\/2}\\propto R_\\ast(\\mu\\omega\/E_\\ast)^{1\/3}, \\label{eq:model_Aehd}\\\\\n\\kappa_n &\\propto& F_{ehd}\/\\delta_n\\propto E_{\\ast}a_{ehd}, \\label{eq:model_Kehd}\n\\end{eqnarray}\nwhere the elastohydrodynamic force $F_{ehd}\\propto\\pi a_{ehd}^2p_{ehd}$ is proportional to the pressure given by the Hooke's law, $p_{ehd}\\propto E_\\ast(\\delta_n\/a_{ehd})$, with $p_{ehd}\\propto \\mu\\omega R_\\ast d\/D_c^2$. In addition, the tangential interaction presumably becomes frictionless in presence of the lubricating layer of fluid, $\\kappa_t=0$. Consequently, the effective longitudinal modulus given by Eqs.~\\ref{eq:model_Kemt},~\\ref{eq:model_Gemt},~\\ref{eq:model_Aehd}, and~\\ref{eq:model_Kehd} becomes\n\\begin{equation}\n(M_{ehd}\/E_{\\ast}) = \\alpha_{ehd} (\\mu\\omega\/E_{\\ast})^{1\/3},\\label{eq:model_Mehd}\n\\end{equation}\nwhere $\\alpha_{ehd}$ is a numerical prefactor depending on $Z$ and $\\phi_s$ only. Interestingly, the comparison of the expressions of the dry and the wet elastic moduli, given by Eqs.~\\ref{eq:model_Mdry} and~\\ref{eq:model_Mehd}, respectively, shows that their ratio is a function of a single nondimensional parameter, $(M_{ehd}\/M_{dry})\\propto(\\mu\\omega\/p)^{1\/3}$. This result substantiates the observation of a nontrivial competition between a viscous contribution of the fluid and the elastic deformation of the particles under the action of the confinement pressure. This feature is confirmed by representing the experimental wet-to-dry elastic moduli as a function of $\\mu\\omega\/p$, see Fig.~\\ref{fig:elastic_moduli}(b). The data set, aggregating two different confinement pressures and four different viscosities, indeed fairly collapses along a master curve when represented in such a nondimensional form. However, the ratio of the elastic moduli asymptotically saturates at one for small values of $\\mu\\omega\/p$ in experiments. This is consistent with the fact that a wet sample under a high pressure and with a small interstitial viscosity should tend to behave as a dry sample, $M_{wet}\\simeq M_{dry}$ at $\\mu\\omega\/p\\ll1$. Moreover, within a constant confinement pressure only, the fluid quasistatically flows out from the contact between particles, leaving a central region in mechanical contact surrounded by a peripheral region filled with fluid~\\cite{Marshall2011}, see Fig.~\\ref{fig:sketch_contact}. A more convenient ansatz would thus correspond to a Hertzian elastic response, coming from the central and flat dry region, acting in parallel to a peripheral elastohydrodynamic response, due to the fluid pinched at the edge between the elastic solids:\n\\begin{equation}\nM_{wet} = M_{dry}+M_{ehd}.\\label{eq:model_Mwet}\n\\end{equation}\n\nThe ansatz given by Eq.~\\ref{eq:model_Mwet} is probed in Fig.~\\ref{fig:elastic_moduli}(c), which represents the ratio $(M_{ehd}\/M_{dry})=(M_{wet}\/M_{dry}-1)$ as a function of $(\\mu\\omega\/p)$. As in Fig.~\\ref{fig:elastic_moduli}(b), the data set still demonstrates fair correlations, but now shows a clear power law with an exponent close to $1\/3$, in agreement with our analysis, see Eqs.~\\ref{eq:model_Mdry} and~\\ref{eq:model_Mehd}. Quantitatively, the fit of the data shown in Fig.~\\ref{fig:elastic_moduli}(c) provides $(\\alpha_{ehd}\/\\alpha_{dry})=1.01\\pm43\\%$ at low confinement pressure and $(\\alpha_{ehd}\/\\alpha_{dry})=0.46\\pm20\\%$ at high confinement pressure; this corresponds to prefactors of the order of unity, $\\alpha_{ehd}\\simeq0.25$ and $\\alpha_{ehd}\\simeq0.40$, respectively. Finally, the Fig.~\\ref{fig:elastic_moduli}(d) shows $(M_{ehd}\/E_{\\ast})$ as a function of $(\\mu\\omega\/E_{\\ast})^{1\/3}$ for the high confinement pressure data set. The plot confirms that $M_{ehd}$ increases monotonically according to Eq.~\\ref{eq:model_Mehd}, i.e., the effective elastic modulus increases with the viscosity of the fluid; matching the curve to Eq.~\\ref{eq:model_Mehd} provides a consistent value of the prefactor, $\\alpha_{ehd}=0.37\\pm37\\%$.\\\\\n\nThe analysis of the experimental wave speed presented in Fig.~\\ref{fig:wave_speed}, based on an effective medium theory, thus demonstrates that the propagation of mechanical waves in wet granular samples induces an elastohydrodynamic mechanism at the interparticle level. The mechanical response of a wet sample results from the competition between (i) an elastic contribution related to the static confinement pressure within the contact region between grains and (ii) an elastohydrodynamic interplay between the particles and the fluid, which resides at the periphery of the contacts. The crossover between these two contributions is fairly described by a unique nondimensional number, $(\\mu\\omega\/p)$.\n\n\n\\section{\\label{sec:weakening} Material weakening}\n\nMaterial weakening has been proposed as a triggering mechanism for the emission of secondary pulses after the passage of a principal mechanical event. Several authors~\\cite{Johnson2005,Jia2011,Johnson2016} have linked the breaking of unconsolidated and weak contacts with the loss of material strength and the dynamical change of the shape of the {\\em coda wave}. Here, the material softening is probed by tracking the frequency shift of the lowest prominent mode in the spectrum of the long-lasting outgoing acceleration. The dry configuration is first considered, see Fig.~\\ref{fig:weakening_spectra}(a), at both low and high confinement pressure. It is observed that for the high confinement pressure, the low-frequency component, at around $100$~Hz, remains unaffected by the strength of the dynamical perturbation. This results is coherent with the fact that the dynamics of the sample is linear at high confinement, owing to a negligible dynamical perturbation: the sample is highly consolidated and no material weakening is observable. On the other hand, at the lowest confinement pressure, the low-frequency component decreases with increasing impulse amplitude: a $20\\%$ variation in respect to the initial frequency is observed, consistently with the observations available in the literature~\\cite{Johnson2005}. The decrease of the prominent frequency reveals the nonlinear nature of the contact dynamics and the weakening of the material as the dynamical strength is progressively increased.\\\\\n\n\\begin{figure}[t]\n\\includegraphics[width=0.450\\textwidth]{1808_03150_v2_Fig_8.pdf}\n\\caption{\\label{fig:weakening_frequency} Frequency of the lowest mode as a function of the input force, for different fluid viscosities at (a) low and (b) high confinement pressures. The markers refer to the definitions given in Fig.~\\ref{fig:wave_speed}.}\n\\end{figure}\n\nNext, the same procedure is repeated in samples containing interstitial fluids with different viscosities, see Figs.~\\ref{fig:weakening_spectra}(b)--\\ref{fig:weakening_spectra}(d). In Fig.~\\ref{fig:weakening_frequency}, the evolution of the lowest mode for every fluid is shown as a function of the impulse amplitude. The weakening of the sample remains observable for the lowest viscosity only, and progressively disappears when the viscosity is increased. This means that an interstitial fluid with a sufficiently high viscosity consolidates the sample and tends to linearize its mechanical response. Indeed, using the nondimensional parameter $(\\mu\\omega\/p)$ derived in Sec.~\\ref{sec:model} as an indicator of the regime, one find a crossover viscosity $\\mu_{\\ast}=p\/\\omega$. Below this value, the dynamics is essentially Hertzian and nonlinear, see Eqs.~\\ref{eq:model_Mdry}. Above the crossover, the dynamics tends to becomes linear owing to a dominant elastohydrodynamic interplay, see Eqs.~\\ref{eq:model_Mehd} and~\\ref{eq:model_Mwet}. Such a crossover is $\\mu_{\\ast}\\simeq3.6$~Pa.s at low confinement pressure, where $f\\sim140$~Hz, in agreement with our observations. Similarly, the crossover is $\\mu_{\\ast}\\simeq82$~Pa.s at high confinement pressure, where $f\\sim160$~Hz. In this case, the behavior of all the wet samples relies on the Hertzian elastic interaction: it is thus reminiscent to the dry configuration, which proved to be consolidated and where no weakening is expected.\\\\\n\n\\section{\\label{sec:conclusion} Conclusions}\n\nThe experimental evidences presented in this paper showed that the presence of an interstitial fluid in a granular media significantly modifies the contact dynamics between particles and, consequently, the features of mechanical waves propagation in the long wavelength approximation. In both low and high confinement cases, it was observed that the fluid induces an elastohydrodynamic mechanism that enhances the rigidity of the contacts, rendering a higher wave speed as compared with the dry configuration. All our results were discussed in terms of an effective mean-field theory, coupled to a description of the elastohydrodynamic interaction between spherical elastic particles mediated by an interstitial Newtonian viscous fluid. Our analysis suggested a nontrivial competition between the elastohydrodynamic interaction and the elastic deformation due to the confining pressure. The interplay was shown to be fairly described by a unique nondimensional parameter $\\mu\\omega\/p$, which allowed defining a threshold viscosity above which the effect of the fluid dominates. It was also observed that the dry granular material weakens when submitted to strong dynamical perturbations, due to the breaking of unconsolidated contacts. This nonlinear softening can be impeded by either increasing the confinement pressure, or by adding an interstitial fluid with a viscosity above the threshold. Our analysis and description qualitatively match the experimental observations; all these results might prove to be useful in practical situations and pave the way to the need of a more quantitative and precise description.\\\\\n\n\\begin{acknowledgments}\nThe authors thank Francisco Melo for many fruitful discussions. R.Z. acknowledges CONICYT National Doctoral Program Grant No. 21161404 and the financial support of {\\em Supm{\\'e}ca} during his stay in France. S.J. acknowledges the {\\em Pontificia Universidad Cat{\\'o}lica de Valpara{\\'i}so} and the Franco-Chilean {\\em Laboratoire International Associ{\\'e} LIA-MSD} for the financial support during his stay in Chile. F.S. acknowledges the financial support from FONDECYT Project No. 11140556.\n\\end{acknowledgments}\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nIn the present paper, we study spectral estimates for the logarithmic Laplacian \n$L_{\\text{\\tiny $\\Delta \\,$}}\\!= \\log (-\\Delta)$, which is a (weakly) singular integral operator with Fourier symbol $2\\log |\\eta|$ and arises as formal derivative $\\partial_s \\Big|_{s=0} (-\\Delta)^s$ of fractional Laplacians at $s= 0$. The study of $L_{\\text{\\tiny $\\Delta \\,$}}\\!$ has been initiated recently in \\cite{HW}, where its relevance for the study of asymptotic spectral properties of the family of fractional Laplacians in the limit $s \\to 0^+$ has been discussed. A further motivation for the study of $L_{\\text{\\tiny $\\Delta \\,$}}\\!$ is given in \\cite{jarohs-saldana-weth}, where it has been shown that this operator allows to characterize the $s$-dependence of solution to fractional Poisson problems for the full range of exponents $s \\in (0,1)$.  The logarithmic Laplacian also arises in the geometric context of the $0$-fractional perimeter, which has been studied recently in \\cite{DNP}. \n\nFor matters of convenience, we state our results for the operator $\\mathcal H= \\frac{1}{2}L_{\\text{\\tiny $\\Delta \\,$}}\\!$ \nwhich corresponds to the quadratic form \n\\begin{equation}\n\\label{log-quadratic}\n\\varphi \\mapsto (\\varphi,\\varphi)_{log} := \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi.\n\\end{equation}\nHere and in the following, we let $\\widehat{\\varphi}$ denote the Fourier transform \n$$\n\\xi \\mapsto \\widehat{\\varphi}(\\xi)= \\int_{{\\mathbb R}^d} e^{-ix \\xi} \\varphi(x)\\,dx\n$$\nof a function $\\varphi\\in L^2({\\mathbb R}^d)$. Let $\\Omega\\subset \\Bbb R^d$ be an open set of finite measure, and let \n${\\mathbb H}(\\Omega)$ denote the closure of $C^\\infty_c(\\Omega)$ with respect to the norm\n\\begin{equation}\n  \\label{eq:def-norm--star}\n\\varphi \\mapsto \\|\\varphi\\|_{*}^2:= \\int_{\\Bbb R^d} \\log(e + |\\xi|)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi.\n\\end{equation}\nThen $(\\cdot,\\cdot)_{log}$  defines a closed, symmetric and semibounded quadratic form with domain ${\\mathbb H}(\\Omega) \\subset L^2(\\Omega)$, see Section~\\ref{sec:prel-basic-prop} below. Here and in the following, we identify $L^2(\\Omega)$ with the space of functions $u \\in L^2({\\mathbb R}^d)$ with $u \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$. Let \n$$\n\\mathcal H : {\\mathcal D}(\\mathcal H) \\subset L^2(\\Omega) \\to L^2(\\Omega) \n$$\nbe the unique self-adjoint operator associated with the quadratic form. The eigenvalue problem for $\\mathcal H$ then writes as \n\\begin{equation}\\label{D}\n\\left\\{\n  \\begin{aligned}\n\\mathcal H \\varphi &= \\lambda \\varphi, &&\\qquad \\text{in $\\Omega$,}\\\\\n\\varphi &= 0, &&\\qquad \\text{on ${\\mathbb R}^d \\setminus \\Omega$.}\n  \\end{aligned}\n\\right.\n\\end{equation}\nWe understand (\\ref{D}) in weak sense, i.e. \n$$\n\\varphi \\in {\\mathbb H}(\\Omega) \\quad \\text{and}\\quad (\\varphi,\\psi)_{log}= \\lambda \\int_{\\Omega}\\varphi(x)\\psi(x)\\,dx \\quad \\text{for all $\\psi \\in {\\mathbb H}(\\Omega)$.}\n$$\nAs noted in \\cite[Theorem 1.4]{HW}, there exists a sequence of eigenvalues \n$$\n\\lambda_1(\\Omega)< \\lambda_2(\\Omega) \\le \\dots, \\qquad \\lim_{k \\to \\infty} \\lambda_k(\\Omega) = \\infty \n$$\nand a corresponding complete orthonormal system of eigenfunctions. We note that the discreteness of the spectrum is a consequence of the fact that the embedding ${\\mathbb H}(\\Omega) \\hookrightarrow L^2(\\Omega)$ is compact. In the case of bounded open sets, the compactness of this embedding follows easily by Pego's criterion~\\cite{Pego}. In the case of unbounded open sets of finite measure, the compactness can be deduced from \\cite[Theorem 1.2]{jarohs-weth} and estimates for $\\|\\cdot\\|_*$, see Corollary~\\ref{cor-compact-embedding} below. \n\nIn Section \\ref{sec:prel-basic-prop}, using the results from \\cite{HW} and \\cite{FKV}, we discuss properties  of functions from  $\\mathcal D(\\mathcal H)$. In particular, we show that $e^{ix\\xi}\\big|_{x\\in\\Omega} \\in \\mathcal D(\\mathcal H)$, $\\xi\\in\\Bbb R^d$, provided $\\Omega$ is an open bounded sets with Lipschitz boundary.\n\nIn Section \\ref{sec:deriving-an-upper-1} we obtain a sharp upper bound for the Riesz means and for the number of eigenvalues  $N(\\lambda)$ of the operator $\\mathcal H$ below $\\lambda$. Here we use technique developed in papers \\cite{Bz1}, \\cite{Bz2}, \\cite{LY} and \\cite{L}. In \\cite{Lap} it was noticed that such technique could be applied for a class of pseudo-differential operators with Dirichlet boundary conditions in domains of finite measure without any requirements on the smoothness of the boundary. \n\n We discuss  lower bounds for $\\lambda_1(\\Omega)$ in Section \\ref{sec:lower-bound-lambd}. In Theorem \\ref{lower-bound-lambda_1-first} we present an estimate that is valid for arbitrary open sets of finite measure. For sets with Lipschitz boundaries, H.Chen and T.Weth \\cite{HW} have proved a Faber-Krahn inequality for the operator $\\mathcal H$ that reduces the problem to the estimate of \n$\\lambda_1(B)$, where $B$ is a ball satisfying $|B| = |\\Omega|$, see Corollary \\ref{cor-faber-krahn}. In Theorem \\ref{lower-bound-lambda-1-second} we find an estimate for $\\lambda_1(B_d)$, where $B_d$ is the unit ball, that is better in lower dimensions than the one obtained in Theorem \\ref{lower-bound-lambda_1-first}. We also compare our results with bounds resulting from previously known spectral inequalities obtained in  \\cite{BK} and \\cite{B}.\n\nIn Section \\ref{LowB1} we obtain asymptotic lower bounds using the coherent states transformation approach given in \\cite{G}. It allows us to derive, in Section \\ref{Weyl}, asymptotics for the Riesz means of eigenvalues in  Theorem \\ref{3.1} and for $N(\\lambda)$ in Corollary \\ref{3.2}. Here $\\Omega \\subset {\\mathbb R}^N$ is an arbitrary open set of finite measure without any additional restrictions on the boundary. \n\nFinally in Section \\ref{LowB2} we obtain uniform bounds on the Riesz means of the eigenvalues using the fact that for bounded open sets with Lipschitz boundaries we have $e^{ix\\xi}\\big|_{x\\in\\Omega} \\in \\mathcal D(\\mathcal H)$.\n\n\n\n \n\n\n\\section{Preliminaries and basic properties of eigenvalues}\n\\label{sec:prel-basic-prop}\n\nAs before, let $(\\cdot,\\cdot)_{log}$ denote the quadratic form defined in (\\ref{log-quadratic}), and let, for an open set $\\Omega \\subset {\\mathbb R}^d$, \n ${\\mathbb H}(\\Omega)$ denote the closure of $C^\\infty_c(\\Omega)$ with respect to the norm $\\|\\cdot\\|_*$ defined in (\\ref{eq:def-norm--star}). \n\n\\begin{lem}\n\\label{closed-semibounded}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then $(\\cdot,\\cdot)_{log}$  defines a closed, symmetric and semibounded quadratic form with domain ${\\mathbb H}(\\Omega) \\subset L^2(\\Omega)$.\n\\end{lem}\n\n\\begin{proof}\nObviously, the form $(\\cdot,\\cdot)_{log}$ is  symmetric. For functions $\\varphi \\in C^\\infty_c(\\Omega)$, we have \n\\begin{equation}\n  \\label{eq:basic-fourier-ineq}\n(2\\pi)^{d} \\|\\varphi\\|_2^2 =\\|\\widehat \\varphi\\|_2^2 \\le \\|\\varphi\\|_*^2. \n\\end{equation}\nMoreover, with $c_1:= \\log (e+2)+ \\sup \\limits_{t \\ge 2}\\frac{\\log (e+t)}{\\log t}$\nwe have\n\\begin{align}\n\\frac{\\|\\varphi\\|_{*}^2}{c_1} &\\le \\| \\widehat \\varphi\\|_{2}^2+  \\int_{|\\xi| \\ge 2}\\ln |\\xi| |\\widehat{\\varphi}(\\xi)|^2  \\, d\\xi \\nonumber\\\\\n&\\le (2\\pi)^d \\bigl( \\|\\varphi\\|_{2}^2 +  (\\varphi,\\varphi)_{log}\\bigr) \n- \\int_{|\\xi| \\le 2}\\ln |\\xi| |\\widehat{\\varphi}(\\xi)|^2  \\, d\\xi \\nonumber \\\\\n&\\le (2\\pi)^d \\bigl( \\|\\varphi\\|_{2}^2 +  (\\varphi,\\varphi)_{log}\\bigr) \n+ \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} \\|\\widehat{\\varphi}\\|_\\infty^2 \\label{closed-semibounded-est-1}\n\\end{align}\nwhile \n\\begin{equation}\n  \\label{eq:closed-semibounded-est-2}\n\\|\\widehat{\\varphi}\\|_\\infty^2 \\le \\|\\varphi\\|_1^2 \\le |\\Omega|\\, \\|\\varphi\\|_2^2.\n\\end{equation}\nConsequently, \n\\begin{align}\n(\\varphi,\\varphi)_{log} &\\ge   \\frac{\\|\\varphi\\|_*^2}{(2\\pi)^d c_1}-\n\\left(1+ \\frac{|\\Omega|\\, \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}}{(2\\pi)^{d}}\\right)\\|\\varphi\\|_2^2 \\label{intermediate-est}\\\\\n&\\ge  \\left(\\frac{1}{c_1}\\,-\\,1\\,-\\,\\frac{|\\Omega|\\, \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}}{(2\\pi)^{d}}\\right)\\|\\varphi\\|_2^2. \\nonumber\n\\end{align}\nIn particular, $(\\varphi,\\varphi)_{log}$ is semibounded. Moreover, it follows from (\\ref{intermediate-est}) and the completeness of $({\\mathbb H}(\\Omega),\\|\\cdot\\|_*)$ that the form $(\\varphi,\\varphi)_{log}$ is closed on ${\\mathbb H}(\\Omega)$.  \n\\end{proof}\n\n\\begin{lem}\n\\label{equivalent-norms}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then \n\\begin{equation}\n  \\label{eq:def-norm-double-star}\n\\varphi \\mapsto \\|\\varphi\\|_{**}^2:= \n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy\n\\end{equation}\ndefines an equivalent norm to the norm $\\|\\cdot\\|_*$ defined in (\\ref{eq:def-norm--star}) on $C^\\infty_c(\\Omega)$. \n\\end{lem}\n\n\\begin{proof}\nLet $\\varphi \\in C^\\infty_c(\\Omega)$. By \\cite[Lemma 2.7]{FKV}, we have \n\\begin{equation}\n  \\label{eq:FKV-lemma}\n\\|\\varphi\\|_2 \\le c_2 \\|\\varphi\\|_{**}  \\qquad \\text{with a constant $c_2>0$ independent of $\\varphi$.}  \n\\end{equation}\nIn particular, $\\|\\cdot\\|_{**}$ defines a norm on $C^\\infty_c(\\Omega)$.\nNext we note that, by \\cite[Theorem 1.1(ii) and Eq. (3.1)]{HW},   \n$$\n(\\varphi,\\varphi)_{log} = \\frac{1}{2}\\int_{{\\mathbb R}^d}[L_{\\text{\\tiny $\\Delta \\,$}}\\! \\varphi(x)]\\varphi(x)\\,dx = \\kappa_d \\|\\varphi\\|_{**}^2 - \\int_{\\Bbb R^d} [j * \\varphi] \\varphi \\,dx + \\zeta_d  \\|\\varphi\\|_2^2\n$$\nwith \n\\begin{equation}\n  \\label{eq:def-zeta_d}\n\\kappa_d:= \\frac{\\pi^{- \\frac{d}{2}}  \\Gamma(d\/2)}{4}, \\qquad \\zeta_d:= \n\\log 2 + \\frac{1}{2}\\left(\\psi\\left(d\/2\\right) -\\gamma\\right)\n\\end{equation}\nand \n$$\nj: {\\mathbb R}^d \\setminus \\{0\\} \\to {\\mathbb R}, \\qquad j(z)= 2 \\kappa_d 1_{{\\mathbb R}^d \\setminus B_d}(z)|z|^{-d}.\n$$\nHere $\\psi:= \\frac{\\Gamma'}{\\Gamma}$ is the Digamma function and $\\gamma= -\\Gamma'(1)$ is the Euler-Mascheroni constant. \nConsequently, we have \n\\begin{align}\n\\Bigl|(\\varphi,\\varphi)_{log}- \\kappa_d \\|\\varphi\\|_{**}^2\\Bigr| &\\le \n\\|j\\|_\\infty \\|\\varphi\\|_1^2 + \\zeta_d  \\|\\varphi\\|_2^2 \\nonumber\\\\\n&\\le \\Bigl(\\|j\\|_\\infty |\\Omega| +\\zeta_d\\Bigr)\\|\\varphi\\|_2^2.  \\label{modulus-ineq-quad-form}\n\\end{align}\nAs a consequence of (\\ref{eq:basic-fourier-ineq}) and (\\ref{modulus-ineq-quad-form}), we find that \n\\begin{align*}\n\\|\\varphi\\|_{**}^2 &\\le \\frac{1}{\\kappa_d}\\Bigl[ (\\varphi,\\varphi)_{log} + \n\\bigl(\\|j\\|_\\infty |\\Omega|+\\zeta_d\\bigr) \\|\\varphi\\|_2^2 \\Bigr]\\\\\n&\\le \\frac{1}{(2\\pi)^d \\kappa_d}\\Bigl(1  + \\|j\\|_\\infty |\\Omega|+\\zeta_d\\Bigr)\\|\\varphi\\|_*^2.\n\\end{align*}\nMoreover, by (\\ref{closed-semibounded-est-1}), (\\ref{eq:closed-semibounded-est-2}), (\\ref{eq:FKV-lemma}) and (\\ref{modulus-ineq-quad-form}) we have \n\\begin{align*}\n&\\frac{\\|\\varphi\\|_{*}^2}{c_1}  \n\\le (2\\pi)^d \\bigl(\\|\\varphi\\|_{2}^2 + (\\varphi,\\varphi)_{log}\\bigr) \n+   \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} |\\Omega| \\|\\varphi\\|_2^2\\\\\n&\\le (2\\pi)^d \\Bigl( \\kappa_d \\|\\varphi\\|_{**}^2\n+ \\bigl(1+ \\|j\\|_\\infty |\\Omega| +\\zeta_d\\bigr)\\|\\varphi\\|_2^2 \\Bigr) \n+   \\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))} |\\Omega| \\|\\varphi\\|_2^2\\\\\n&\\le c_3 \\|\\varphi\\|_{**}^2\n\\end{align*}\nwith $c_3 = (2\\pi)^d \\kappa_d + c_2\\bigl[(2\\pi)^d \\bigl(1+ \\|j\\|_\\infty |\\Omega|+\\zeta_d\\bigr) +\\bigl\\| \\ln |\\cdot| \\bigr\\|_{L^1(B_2(0))}|\\Omega| \\bigr]$. Hence the norms $\\|\\cdot\\|_{*}$ and $\\|\\cdot\\|_{**}$ are equivalent on $C^\\infty_c(\\Omega)$.\n\\end{proof}\n\n\\begin{cor}\n\\label{cor-compact-embedding}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then the embedding ${\\mathbb H}(\\Omega) \\hookrightarrow L^2(\\Omega)$ is compact. \n\\end{cor}\n\n\\begin{proof}\nLet $\\tilde {\\mathbb H}(\\Omega)$ be defined as the space of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and \n$$\n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy <\\infty.\n$$  \nBy \\cite[Theorem 1.2]{jarohs-weth}, the Hilbert space $(\\tilde {\\mathbb H}(\\Omega),\\|\\cdot\\|_{**})$ is compactly embedded in $L^2(\\Omega)$. \nSince, by Lemma~\\ref{equivalent-norms}, the norms $\\|\\cdot\\|_*$ and $\\|\\cdot\\|_{**}$ are equivalent on $C^\\infty_c(\\Omega)$, the space ${\\mathbb H}(\\Omega)$ is embedded in $\\tilde {\\mathbb H}(\\Omega)$. Hence the claim follows.    \n\\end{proof}\n\n\n\\begin{cor}\n\\label{space-equivalence}\nLet $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary.\n\\begin{enumerate}\n\\item[(i)] The space ${\\mathbb H}(\\Omega)$ is equivalently given as the set of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and \n  \\begin{equation}\n    \\label{eq:kernel-finiteness-cond}\n\\int \\!\\!\\! \\int_{|x-y|\\le 1} \\frac{(\\varphi(x)-\\varphi(y))^2}{|x-y|^d}\\,dxdy <\\infty.\n  \\end{equation}\n\\item[(ii)] ${\\mathbb H}(\\Omega)$ contains the characteristic function $1_\\Omega$ of $\\Omega$ and also the restrictions of exponentials $x \\mapsto 1_\\Omega(x) \\, e^{ix \\xi}$, $\\xi \\in {\\mathbb R}^d$.\n\\end{enumerate}\n\\end{cor}\n\n\\begin{proof}\n(i) Let, as in the proof of Corollary~\\ref{cor-compact-embedding}, $\\tilde {\\mathbb H}(\\Omega)$ be the space of functions $\\varphi \\in L^2({\\mathbb R}^d)$ with $\\varphi \\equiv 0$ on ${\\mathbb R}^d \\setminus \\Omega$ and with (\\ref{eq:kernel-finiteness-cond}), endowed with the norm $\\|\\cdot\\|_{**}$. Since $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary, it follows from \\cite[Theorem 3.1]{HW} that $C_0^\\infty(\\Omega) \\subset \\tilde {\\mathbb H}(\\Omega)$ is dense. Hence the claim follows from Lemma~\\ref{equivalent-norms}.\n\n(ii) follows from (i) and a straightforward computation.   \n\\end{proof}\n\nNext we note an observation regarding the scaling properties of the eigenvalues $\\lambda_k(\\Omega)$. \n\\begin{lem}\n\\label{lemma-scaling-properties}  \nLet $\\Omega \\subset {\\mathbb R}^d$ be a bounded open set with Lipschitz boundary, and let \n$$\nR\\Omega:= \\{R x\\::\\: x \\in \\Omega\\}. \n$$\nThen we have\n$$\n\\lambda_k(R \\Omega) = \\lambda_k(\\Omega) -  \\log R \\qquad \\text{for all $k \\in {\\mathbb N}$.}\n$$\n\\end{lem}\n\n\\begin{proof}\nSince $C_0^\\infty(\\Omega) \\subset {\\mathbb H}(\\Omega)$ is dense, it suffices to note that\n\\begin{equation}\n  \\label{eq:scaling-test-functions}\n(\\varphi_R,\\varphi_R)_{log} = (\\varphi,\\varphi)_{log} - \\log R \\|\\varphi\\|_{L^2({\\mathbb R}^d)}^2 \\qquad \\text{for $\\varphi \\in C^\\infty_c({\\mathbb R}^d)$}\n\\end{equation}\nwith $\\varphi_R \\in C^\\infty_c({\\mathbb R}^d)$ defined by $\\varphi_R(x)= R^{-\\frac{d}{2}}\\varphi(\\frac{x}{R})$, whereas $\\|\\varphi_R\\|_{L^2({\\mathbb R}^d)}= \\|\\varphi\\|_{L^2({\\mathbb R}^d)}$. Since \n$$\n\\widehat{\\varphi_R}= R^{\\frac{d}{2}} \\widehat{\\varphi}(R \\,\\cdot \\,)\n$$\nwe have \n\\begin{align*}\n&(\\varphi_R,\\varphi_R)_{log}\\\\\n&= \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi_R}(\\xi)|^2\\, d\\xi = \\frac{R^{d}}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\log(|\\xi|)\\,  |\\widehat{\\varphi}(R \\xi)|^2\\, d\\xi\\\\\n&= \\frac{1}{(2\\pi)^{d}} \\, \\int_{\\Bbb R^d} \\bigl(\\log(|\\xi|)-\\log R\\bigr)\\,  |\\widehat{\\varphi}(\\xi)|^2\\, d\\xi =(\\varphi,\\varphi)_{log} - \\log R \\|\\varphi\\|_{L^2({\\mathbb R}^d)}^2,\n\\end{align*}\nas stated in (\\ref{eq:scaling-test-functions}).\\\\ \n\\end{proof}\n \n\\section{An upper trace bound}\n\\label{sec:deriving-an-upper-1}\n\n\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. Let $\\{\\varphi_k\\}$ and $\\{\\lambda_k\\}$ be the orthonormal in $L^2(\\Omega)$ system of eigenfunctions and the eigenvalues of the operator $\\mathcal H$  respectively. In what follows we denote\n$$\n(\\lambda - t)_+ = \n\\begin{cases}\n\\lambda - t, & {\\rm if} \\quad t <\\lambda, \\\\\n0, \\quad & {\\rm if} \\quad  t \\ge \\lambda.\n\\end{cases}\n$$\nThen we have\n\\begin {thm}\\label{1.1}\nFor the eigenvalues of the problem \\eqref{D} and any $\\lambda\\in \\Bbb R$ we have\n\\begin{equation}\\label{BU}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\le \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\lambda} \\, |B_d|\\, d^{-1},\n\\end{equation}\nwhere $|B_d|$ is the measure of the unit ball in $\\Bbb R^d$.\n\\end{thm}\n\n\\begin{proof}\nExtending the eigenfunction $\\varphi_k$ by zero outside $\\Omega$ and using the Fourier transform we find\n\\begin{multline*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ = \\sum_{k}\\left(\\lambda (\\varphi_k, \\varphi_k) - (\\mathcal H\\varphi_k, \\varphi_k) \\right)_+ \\\\\n= \\frac{1}{(2\\pi)^{d}}\\, \\left(\\sum_k \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right) \\, |\\widehat{\\varphi_k}(\\xi)|^2 \\, d\\xi \\right)_+\\\\\n\\le\n \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right)_+ \\,  \\sum_k |\\widehat{\\varphi_k}(\\xi)|^2 \\, d\\xi. \n\\end{multline*} \nUsing that $\\{\\varphi_k\\}$ is an orthonormal basis in $L^2(\\Omega)$ and denoting \n$e_\\xi = e^{-i (\\cdot,\\xi)}$we have \n$$\n\\sum_k |\\widehat{\\varphi_k}(\\xi)|^2 = \\sum_k |(e_\\xi, \\varphi_k)|^2 = \\|e_\\xi\\|^2_{L^2(\\Omega)} = |\\Omega|,\n$$\nand finally obtain\n\\begin{align*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ & \\le  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, \\int_{\\Bbb R^d} \\left(\\lambda - \\log(|\\xi|) \\right)_+ \\\\\n& =  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\lambda} \\,  \\int_{|\\xi|\\le 1} \\log(|\\xi|^{-1}) \\, d\\xi.\n\\end{align*}\nWe complete the proof by computing the last integral. \n\\end{proof}\n\n\\noindent\nLet $\\eta >\\lambda$ and let us consider the function\n$$\n\\psi_\\lambda(t) = \\frac{1}{\\eta - \\lambda} (\\eta - t)_+.\n$$\nDenote by $\\chi$ the step function \n$$\n\\chi_\\lambda (t) = \n\\begin{cases} \n1,  \\quad & {\\rm if} \\quad t<\\lambda,\\\\\n0,\\quad & {\\rm if} \\quad t \\ge \\lambda,\n\\end{cases}\n$$\nand let  \n$$\nN(\\lambda) = \\# \\{k:\\, \\lambda_k<\\lambda\\},\n$$\nbe the number of the eigenvalues below $\\lambda$ of the operator $\\mathcal H$.\n\nThen by using the previous statement we have \n$$\nN(\\lambda) \\le \\frac{1}{\\eta - \\lambda} \\, \\sum_k (\\eta - \\lambda_k)_+ \\le \n\\frac{1}{\\eta - \\lambda} \\,  \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, e^{d\\eta} \\,  |B_d|\\, d^{-1}.\n$$\nMinimising the right hand side w.r.t. $\\eta$ we find $\\eta = \\lambda + \\frac1d\n$ and thus obtain the following\n\n\\begin{cor}\n\\label{cor-N-lambda}\nFor the number $N(\\lambda)$ of the eigenvalues of the operator $\\mathcal H$ below $\\lambda$ we have\n\\begin{equation}\\label{Numb}\nN(\\lambda) \\le \n e^{\\lambda d +1} \n\\frac{1}{(2\\pi)^{d}}\\, |\\Omega| \\, |B_d|.\n\\end{equation}\n\\end{cor}\n\n\n\n\\section{A lower bound for $\\lambda_1(\\Omega)$}\n\\label{sec:lower-bound-lambd}\n\nIn this section, we focus on lower bounds for the first eigenvalue $\\lambda_1= \\lambda_1(\\Omega)$. From Corollary~\\ref{cor-N-lambda}, we readily deduce the following bound. \n\n\\begin{thm} \n\\label{lower-bound-lambda_1-first}\nLet $\\Omega \\subset {\\mathbb R}^d$ be an open set of finite measure. Then we have \n\\begin{equation}\n  \\label{eq:est-lambda_1-first}\n\\lambda_1(\\Omega) \\ge \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}.\n\\end{equation}\nIn particular, if $|\\Omega| \\le \\frac{(2\\pi)^{d}}{e\\, |B_d|}$, then the operator $\\mathcal H$ does not have negative eigenvalues.\n\\end{thm}\n\n\\begin{proof}\nIf $\\lambda < \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}$, then \n$N(\\lambda)<1$ by (\\ref{Numb}), and therefore $N(\\lambda)=0$. Consequently, $\\mathcal H$ does not have eigenvalues below $\\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |\\Omega| \\, |B_d|}$.\n\\end{proof}\n\n\n\\begin{rem}\nNote that the inequalities \\eqref{BU}, \\eqref{Numb} and \\eqref{eq:est-lambda_1-first} hold for any open set $\\Omega$ of finite measure without any additional conditions on its boundary.\n\\end{rem}\n\n\nIn the following, we wish to improve the bound given in Theorem~\\ref{lower-bound-lambda_1-first} in low dimensions $d$ for open boundary sets with Lipschitz boundary. We shall use the following Faber-Krahn type inequality.\n\n\\begin{thm} (\\cite[Corollary 1.6]{HW})\\\\\n\\label{sec:faber-Krahn-main}\nLet $\\rho>0$. Among all bounded open sets $\\Omega$ with Lipschitz boundary and $|\\Omega| = \\rho$, the ball $B=B_r(0)$ with $|B|=\\rho$ minimizes $\\lambda_1(\\Omega)$.\n\\end{thm}\n\n \n\\begin{cor}\n\\label{cor-faber-krahn}\nFor every open bounded sets $\\Omega$ with Lipschitz boundary we have \n\\begin{equation}\n  \\label{eq:sharp-lower-bound}\n\\lambda_1(\\Omega) \\ge \\lambda_1(B_d) + \\frac{1}{d}\\log \\frac{|B_d|}{|\\Omega|},\n\\end{equation}\nand equality holds if $\\Omega$ is a ball.   \n\\end{cor}\n\n\\begin{proof}\nThe result follows by combining Theorem~\\ref{sec:faber-Krahn-main} with the identity \n$$\n\\lambda_1(B_r(0)) = \\lambda_1(B_d) + \\log \\frac{1}{r}\\qquad \\text{for $r>0$,}\n$$\nwhich follows from the scaling property of $\\lambda_1$ noted in Lemma~\\ref{lemma-scaling-properties}.\n\\end{proof}\n\nCorollary~\\ref{cor-faber-krahn} gives a sharp lower bound, but it contains the  \nunknown quantity $\\lambda_1(B_d)$. By Theorem~\\ref{lower-bound-lambda_1-first}, we have \n\\begin{align}\n\\lambda_1(B_d) &\\ge \\frac{1}{d} \\log \\frac{(2\\pi)^{d}}{e |B_d|^2} = \n\\log (2\\pi) -\\frac{1}{d}\\bigl(1+ 2 \\log |B_d|\\bigr) \\nonumber \\\\\n&= \\frac{2}{d} \\log \\Gamma\\left(d\/2\\right)  + \\log 2 + \\frac{2}{d} \\log \\frac{d}{2} -\\frac{1}{d}.\n\\label{lower-bound-lambda-1-first}  \n\\end{align}\nThe following theorem improves this lower bound in low dimensions $d \\ge 2$. \n\n\\begin{thm} \n\\label{lower-bound-lambda-1-second}\nFor $d \\ge 2$, we have \n\\begin{equation}\n  \\label{eq:est-lambda-1-first}\n\\lambda_1(B_d) \\ge \\log \\bigl(2 \\sqrt{d +2}\\bigr) -  \n\\frac{2^{d+1} |B_d|^2 (d +2)^{\\frac{d}2} }{d (2\\pi)^{2d}}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\n Let $u \\in L^2(B_d)$ be radial with $\\|u\\|_{L^2}=1$. Then $\\widehat u$ is also radial, and \n\\begin{align*}\n|\\widehat u(\\xi)|&=|\\widehat u(s)|= s^{1-\\frac{d}{2}}\\left|\\int_{0}^{1} u(r)J_{\\frac{d}{2}-1}(rs)r^{\\frac{d}{2}}dr\\right| \\\\\n  &\\le s^{1-\\frac{d}{2}} \\left( \\int_0^{1}r^{d-1} u^2(r)\\,dr\\right)^{1\/2} \n\\left(\\int_0^{1}rJ_{\\frac{d}{2}-1}^2(sr) \\,dr\\right)^{1\/2}\\\\ \n  &=\\frac{s^{1-\\frac{d}{2}}}{\\sqrt{|S^{d-1}|}} \n\\left(s^{-2} \\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau\\right)^{1\/2} \\\\\n&=\\frac{s^{-\\frac{d}{2}}}{\\sqrt{|S^{d-1}|}} \n\\left(\\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau\\right)^{1\/2}\\qquad \\text{for $\\xi \\in {\\mathbb R}^d$ with $s = |\\xi|$.}\n\\end{align*}\nConsequently, \n$$\n|S^{d-1}| |\\widehat u(s)|^2 \\le s^{-d} \\int_0^{s} \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau.\n$$\nIn the case where, in addition, $u$ is a radial eigenfunction of (\\ref{D}) corresponding to $\\lambda_1$ in $\\Omega= B_d$, it follows that, for every $\\lambda \\in {\\mathbb R}$,  \n\\begin{align*}\n&(2\\pi)^{d}[\\lambda-\\lambda_1] =  \\int_{{\\mathbb R}^d} (\\lambda -\\ln |\\xi|)|\\widehat u(\\xi)|^2\\,d\\xi \\le  \\int_{{\\mathbb R}^d} (\\lambda -\\ln |\\xi|)_+|\\widehat u(\\xi)|^2\\,d\\xi\\\\\n&=|S^{d-1}| \n\\int_0^\\infty s^{d-1} (\\lambda -\\ln s)_+|\\widehat u(s)|^2\\,ds \\le \\int_0^\\infty \\frac{(\\lambda -\\ln s)_+}{s}  \\int_0^{s} \\!\\!\\!\\tau J_{\\frac{d}{2}-1}^2(\\tau) \\,d\\tau \\,ds\\\\\n&= \\int_0^\\infty  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\tau}^\\infty \\frac{(\\lambda -\\ln s)_+}{s}  \\,ds d\\tau= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\tau}^{e^\\lambda} \\frac{\\lambda -\\ln s}{s} \\,ds d\\tau\\\\\n&= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{\\ln \\tau}^{\\lambda}(\\lambda - s) \\,ds d\\tau= \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau) \\int_{0}^{\\lambda- \\ln \\tau} s \\,ds d\\tau\\\\\n&= \\frac{1}{2}  \\int_0^{e^\\lambda}  \\tau J_{\\frac{d}{2}-1}^2(\\tau)\n\\bigl(\\lambda- \\ln \\tau \\bigr)^2 \\,d\\tau = \\frac{e^{2\\lambda}}{2}  \\int_0^{1}  \\tau J_{\\frac{d}{2}-1}^2(e^\\lambda \\tau) \\ln^2 \\tau \\,d\\tau. \n\\end{align*}\nWe now use the following estimate for Bessel functions of the first kind:\n\\begin{equation}\n  \\label{eq:bessel-est-proof}\nJ_\\nu(x) \\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\quad \\text{for $\\:\\nu > \\sqrt{3}-2$, $\\:0 \\le x < 2 \\sqrt{2(\\nu+2)}$.}\n\\end{equation}\nA proof of this elementary estimate is given in the Appendix. We wish to apply (\\ref{eq:bessel-est-proof}) with $\\nu = \\frac{d}{2}-1$. This gives \n$$\ne^{2\\lambda} J_{\\frac{d}{2}-1}^2(r_0 e^\\lambda \\tau) \\le e^{d\\lambda} \\frac{\\tau^{d-2}}{2^{d-2}\\Gamma^2 (\\frac{d}{2})}=\n \\frac{d^2 |B_d|^2e^{d\\lambda}}{(2\\pi)^{d}} \\tau^{d-2}\n \\qquad \\text{for $\\tau \\in [0,1]$} \n$$ \nif $d \\ge 2$ and $e^\\lambda \\le 2 \\sqrt{d +2}$, i.e., if   \n\\begin{equation}\n\\label{condition-proof}\nd \\ge 2 \\quad \\text{and}\\quad \\lambda \\le \\log \\bigl(2 \\sqrt{d +2}\\bigr).\n\\end{equation}\nHere we used that $|B_d|= \\frac{2}{d} \\frac{\\pi^{\\frac{d}{2}}}{\\Gamma(d\/2)}$. Consequently, if (\\ref{condition-proof}) holds, we find that  \n$$\n(2\\pi)^{d}[\\lambda-\\lambda_1] \\le  \\frac{d^2 |B_d|^2e^{d\\lambda}}{(2\\pi)^{d}}\\int_0^{1} \n \\tau^{d-1} \\ln^2 \\tau \\,d\\tau,\n$$\nwhere \n$$\n\\int_0^{1} \\tau^{d-1} \\ln^2 \\tau d\\tau = - \\frac{2}{d}\n\\int_0^1 \\tau^{d-1} \\ln \\tau d\\tau = \\frac{2}{d^2} \\int_{0}^1 \\tau^{d-1}d\\tau\n= \\frac{2}{d^3}.\n$$\nHence \n$$\n(2\\pi)^{d}[\\lambda -\\lambda_1] \\le \\frac{2|B_d|^2}{d (2\\pi)^{d}}e^{d\\lambda}, \\quad \\text{i.e.,}\\quad \\lambda_1 \\ge \\lambda- \\frac{2|B_d|^2 }{d (2\\pi)^{2d}} e^{d\\lambda}.\n$$\nInserting the value $\\lambda = \\log \\bigl(2 \\sqrt{d +2}\\bigr)$ from (\\ref{condition-proof}), we deduce that \n$$\n\\lambda_1= \\lambda_1(B_d) \\ge  \\log \\bigl(2 \\sqrt{d +2}\\bigr) - \\frac{2^{d+1} |B_d|^2 (d +2)^{\\frac{d}2} }{d(2\\pi)^{2d}},  \n$$\nas claimed. \n\\end{proof}\n\n\n\\begin{rem}{\\rm \n\\label{rem-comparison-of-other-bounds}\nIt seems instructive to compare the lower bounds given in (\\ref{lower-bound-lambda-1-first}) and (\\ref{eq:est-lambda-1-first}) with other bounds obtained from spectral estimates which are already available in the literature. We first mention Beckner's logarithmic estimate of uncertainty \\cite[Theorem 1]{B}, which implies that\\footnote{We note here that a different definition of Fourier transform is used in \\cite{B} and therefore the inequality looks slightly different}\n\\begin{equation*}\n(\\varphi,\\varphi)_{log} \\ge \\int_{{\\mathbb R}^d} \\left[\\psi\\left(d\/4\\right)+ \\log \\frac{2}{|x|}\\right]\\varphi^2(x) dx \\ge \\left[\\psi\\left(d\/4\\right)+ \\log 2\\right]\\|\\varphi\\|_2^2 \n\\end{equation*}\nfor functions $\\varphi \\in C^\\infty_c(B_d)$ and therefore \n\\begin{equation}\n  \\label{eq:beckner-lambda-1-est}\n\\lambda_1(B_d) \\ge \\psi\\left(d\/4\\right)+ \\log 2 .\n\\end{equation}\nHere, as before, $\\psi = \\frac{\\Gamma'}{\\Gamma}$ denotes the Digamma function. Next we state a further lower bound for $(\\varphi,\\varphi)_{log}$ which follows from \\cite[Proposition 3.2 and Lemma 4.11]{HW}. We have\n\\begin{equation} \n\\label{cw-inequality}\n(\\varphi,\\varphi)_{log} \\ge \\zeta_d \\|\\varphi\\|_2^2 \\qquad \\text{for $\\varphi \\in C^\\infty_c(B_d)$,}\n\\end{equation}\nwhere $\\zeta_d$ is given in (\\ref{eq:def-zeta_d}), i.e., \n$$\n\\zeta_d =  \\log 2 + \\frac{1}{2}\\left( \\psi(d\/2)-\\gamma\\right) = \\left\\{\n  \\begin{aligned}\n  &- \\gamma + \\sum_{k=1}^{\\frac{d-1}{2}} \\frac{1}{2k-1},&&\\qquad \\text{$d$ odd,}\\\\\n  &\\log 2 - \\gamma + \\sum_{k=1}^{\\frac{d-2}{2}} \\frac{1}{k},&&\\qquad \\text{$d$ even.}   \n  \\end{aligned}\n\\right.\n$$\nInequality (\\ref{cw-inequality}) implies that\n\\begin{equation} \n\\label{cw-lambda-1-bound}\n\\lambda_1(B_d) \\ge \\zeta_d.\n\\end{equation}\nThe latter inequality can also be derived from a lower bound of Ba$\\rm{\\tilde{n}}$uelos and Kulczycki for the \nfirst Dirichlet eigenvalue $\\lambda_1^\\alpha(B_d)$ of the fractional Laplacian $(-\\Delta)^{\\alpha\/2}$ in $B_d$. In \n\\cite[Corollary 2.2]{BK}, it is proved that \n$$\n\\lambda_1^\\alpha(B_d) \\ge 2^\\alpha \\frac{\\Gamma(1+\\frac{\\alpha}{2}) \\Gamma(\\frac{d+\\alpha}{2})}{\\Gamma(\\frac{d}{2})}\\qquad \\text{for $\\alpha \\in (0,2)$.}\n$$\nCombining this inequality with the characterization of $\\lambda_1(B_d)$ given in \\cite[Theorem 1.5]{HW}, we deduce that   \n$$\n\\lambda_1(B_d)= \\lim_{\\alpha \\to 0^+}\\frac{\\lambda_1^\\alpha(B_d)-1}{\\alpha}\\ge \\frac{d}{d\\alpha}\\Big|_{\\alpha=0}\\, 2^\\alpha \\frac{\\Gamma(1+\\frac{\\alpha}{2}) \\Gamma(\\frac{d+\\alpha}{2})}{\\Gamma(\\frac{d}{2})} = \\zeta_d, \n$$\nas stated in (\\ref{cw-lambda-1-bound}). \n\nWe briefly comment on the quality of the lower bounds obtained here in low and high dimensions. In low dimensions $d \\ge 2$, (\\ref{eq:est-lambda-1-first}) is better than the bounds (\\ref{lower-bound-lambda-1-first}), (\\ref{eq:beckner-lambda-1-est}) and (\\ref{cw-lambda-1-bound}). In dimension $d=1$ where the bound (\\ref{eq:est-lambda-1-first}) is not available, the bound (\\ref{lower-bound-lambda-1-first}) yields the best value. The following table shows numerical values of the bounds $b_1(d)$, $b_2(d)$, $b_3(d)$ resp. $b_4(d)$ given by (\\ref{lower-bound-lambda-1-first}), (\\ref{eq:est-lambda-1-first}), (\\ref{eq:beckner-lambda-1-est}), (\\ref{cw-lambda-1-bound}), respectively. \n\\medskip\n\n\\begin{center}\n{\\tiny\n\\renewcommand{\\arraystretch}{1.6}\n    \\begin{tabular}{ l | l | l | l | l |l | l | l | l | l | l |}\n    $d$ & 1 & 2 & 3 & 4&5&6&7&8&9&10\\\\ \\hline\n    $b_1(d)$ & $-0,55$ & $0,19$ & $0,55$ & $0,79$ &$0,97$&$1,12$ & $1,25$ & $1,36$ & $1,46$ & $1,55$\n\\\\ \\hline\n$b_2(d)$ & $\\quad\/$& $1,28 $& $1,48 $ & $1,59 $& $1,67$&$1,73$ &$1,79$ & $1,84$& $1,89$ & $1,94$\\\\ \\hline\n$b_3(d)$ &$-3.53$ & $-1,27$ & $-0,39$ &$0,12$&$0,47$ & $0,73$&$0,94$& $1,12$ &$1,27$ & $1,40$ \\\\ \\hline\n$b_4(d)$ &$-0,58$& $0,12$ &$0,42$ &$0,62$& $0,76$ &$0,87$ &$0,96$ & $1,03$ & $1,10$ & $1,16$\n\\end{tabular}\n\\renewcommand{\\arraystretch}{1}\n}\n\\end{center}\n\\medskip\n\nTo compare the bounds in high dimensions, we consider the asymptotics as $d \\to \\infty$. \nSince $\\frac{\\log \\Gamma(t)}{t} = \\log t - 1 + o(t)$ as $t \\to \\infty$, the bound (\\ref{lower-bound-lambda-1-first}) yields \n\\begin{equation}\n\\label{lower-bound-lambda-1-first-asymptotics}\n\\lambda_1(B_d) \\ge \\log d - 1 + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nwhereas (\\ref{eq:est-lambda-1-first}) obviously gives \n\\begin{equation}\n\\label{lower-bound-est-lambda-1-first-asymptotics}\n\\lambda_1(B_d) \\ge \\log \\sqrt{d+2} + \\log 2  + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nMoreover, from (\\ref{eq:beckner-lambda-1-est}) and the fact that \n\\begin{equation}\n  \\label{eq:Digamma-asymptotics}\n\\psi(t) = \\log t + o(1)\\qquad\\text{as $t \\to \\infty$,}   \n\\end{equation}\nwe deduce that \n\\begin{equation}\n  \\label{eq:beckner-lambda-1-est-asymptotics}\n\\lambda_1(B_d) \\ge \\log d - \\log 2 + o(1) \\qquad \\text{as $d \\to \\infty$,} \n\\end{equation}\nFinally, (\\ref{cw-inequality}) and (\\ref{eq:Digamma-asymptotics}) yield\n\\begin{equation} \n\\label{cw-lambda-1-bound-asymptotics}\n\\lambda_1(B_d) \\ge \\log \\sqrt{d} +  \\log 2  -\\frac{\\gamma}{2} + o(1) \\qquad \\text{as $d \\to \\infty$.} \n\\end{equation}\nSo (\\ref{eq:beckner-lambda-1-est-asymptotics}) provides the best asymptotic bound as $d \\to \\infty$.\n\nNumerical computations indicate that the bound (\\ref{eq:est-lambda-1-first}) is better than the other bounds for $2 \\le d \\le 21$, and (\\ref{eq:beckner-lambda-1-est}) is the best among these bounds for $d \\ge 22$.\n}\n\n\\end{rem}\n\n\n\n\n\n\\section{An asymptotic lower trace bound}\\label{LowB1}\n\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. In this section we prove the following asymptotic lower bound. A similar statement was obtained in \\cite{G} for the Dirichlet boundary problem for a fractional Laplacian.\n\n\\begin {thm}\\label{2.1}\nFor the eigenvalues of the problem \\eqref{D} and any $\\lambda\\in \\Bbb R$ we have\n\\begin{equation}\\label{BLow}\n\\liminf_{\\lambda\\to\\infty} e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\, \\, |B_d|\\, d^{-1}.\n\\end{equation}\n\\end{thm}\n\n\\begin{proof}\nLet us fix $\\delta>0$ and consider \n$$\n\\Omega_\\delta = \\{ x\\in \\Omega: \\, {\\rm dist}(x, \\Bbb R^d \\setminus \\Omega) >\\delta\\}.\n$$\nSince $\\delta$ is arbitrary it  suffices to show the lower bound \\eqref{BLow}, where $\\Omega$ is replaced by $\\Omega_\\delta$.\nLet $g\\in C_0^\\infty(\\Bbb R^d)$ be a real-valued even function, $\\|g\\|_{L^2(\\Bbb R^d)} = 1$ with support in $\\{x\\in \\Bbb R^d: \\, |x| \\le \\delta\/2\\}$. For $\\xi\\in \\Bbb R^d$ and $x\\in \\Omega_\\delta$ we introduce the \\lq\\lq coherent state\" \n$$\ne_{\\xi,y}(x) =  e^{-i\\xi x} g(x-y).\n$$\nNote that $\\|e_{\\xi,y}\\|_{L^2(\\Bbb R^d)} = 1$. Using the properties of coherent states \\cite[Theorem 12.8]{LL} we obtain\n$$\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \n \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\int_{\\Omega_\\delta} (e_{\\xi,y}, (\\lambda - \\mathcal H)_+ e_{\\xi,y})_{L^2(\\Omega)} \\, dy d\\xi.\n$$\nSince $t \\mapsto (\\lambda-t)_+$ is convex then applying Jensen's inequality to the spectral measure of $\\mathcal H$ we obtain\n\\begin{equation}\\label{jensen}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge   \\frac{1}{(2\\pi)^{d}}\\, \\int_{\\Bbb R^d} \\int_{\\Omega_\\delta}\n\\left(\\lambda - (\\mathcal H e_{\\xi,y}, e_{\\xi,y})_{L^2(\\Omega)} \\right)_+ \\, dy d\\xi.\n\\end{equation}\nNext we consider the quadratic form \n\\begin{multline*}\n\\left(\\mathcal H e_{\\xi,y},  e_{\\xi,y} \\right)_{L^2(\\Omega)} = \\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^d} \\int_{\\Omega} \\int_{\\Omega} e^{i(x-z)(\\eta-\\xi)} g(x-y)g(z-y) \\log(|\\eta|)  \\, dz dx d\\eta \\\\\n=\n\\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-z)\\rho} g(x-y)g(z-y) \\log(|\\xi-\\rho|) \\, dz dx d\\rho\\\\\n= \\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-z)\\rho} g(x-y)g(z-y) \\left( \\log|\\xi| + \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)\\right)\n\\, dz dx d\\rho\\\\\n=  \\log|\\xi| + R(y,\\xi).\n\\end{multline*}\nSince $g\\in C_0^\\infty(\\Bbb R^d)$ we have for any $M>0$\n\\begin{multline*}\nR(y,\\xi) =  \\\\\n\\frac{1}{(2\\pi)^d} \\, \\int_{\\Bbb R^{d}} \\int_{\\Omega} \\int_{\\Omega} \ne^{i(x-y)\\rho} g(x-y) e^{i(y-z)\\rho} g(z-y)  \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)\n\\, dz dx d\\rho\\\\\n= \\int_{\\Bbb R^{d}} |\\widehat{g}|^2\\, \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right) \\, d\\rho\n \\le C_M\\,\n\\int_{\\Bbb R^{d}} (1+ |\\rho|)^{-M}   \\log \\left(\\left|\\xi -\\rho\\right|\/|\\xi| \\right)  \\,  d\\rho\\\\\n\\le C\\, |\\xi|^{-1}.\n\\end{multline*}\nTherefore from \\eqref{jensen} we find \n\\begin{equation}\\label{below1}\n\\sum_{k}(\\lambda - \\lambda_k)_+  \\ge (2\\pi)^{-d}\\, |\\Omega_\\delta| \\, \\int_{\\Bbb R^d}  (\\lambda -  \\log|\\xi| -  C|\\xi|^{-1})_+ \\, d\\xi.\n\\end{equation}\nLet us redefine the spectral parameter $\\lambda =  \\ln \\mu$.\nThen introducing polar coordinates we find\n\\begin{multline}\\label{below22} \n\\int_{\\Bbb R^d}  (\\lambda -  \\log|\\xi| -  C|\\xi|^{-1})_+ \\, d\\xi = \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\ln \\frac{\\mu}{r} - \\frac{C}{r}\\right)_+ \\, r^{d-1}dr\\\\\n= \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)_+ \\, r^{d-1}dr\n\\end{multline} \nThe expression in the latter integral is positive if $ - r\\ln r > C\\mu^{-1}$. The function $ -r\\ln r $ is concave. \n\n\\smallskip\n\n\\qquad\\qquad\\qquad \\qquad \\qquad{\\centering \n{\\includegraphics[scale=.4]{concave.png}} }\n \n\\smallskip\n \\noindent\nIts maximum is achieved at $r=1\/e$ at the value $1\/e$. The equation\n$ - r\\ln r = C\\mu^{-1}$ has two solutions $r_1(\\mu)$ and $r_2(\\mu)$ such that $r_1(\\mu) \\to 0$ and $r_2(\\mu)\\to 1$ as $\\mu \\to\\infty$\nTherefore \n\\begin{multline}\\label{below3}\n\\int_0^\\infty \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)_+ \\, r^{d-1}dr \\ge \n\\int_{r_1(\\mu)}^{r_2(\\mu)}     \\left(\\ln \\frac{1}{r} - \\frac{C}{\\mu r}\\right)  \\, r^{d-1}dr \\\\\n=-\\frac1d\\, r^d \\ln r  \\Big|_{r_1(\\mu)}^{r_2(\\mu)}+ \\frac{C}{\\mu(d+1)} r^{d+1}  \\Big|_{r_1(\\mu)}^{r_2(\\mu)}\n+ \\frac{1}{d^2}r^d   \\Big|_{r_1(\\mu)}^{r_2(\\mu)} \\to \\frac{1}{d^2} \\quad {\\rm as} \\quad \\mu\\to\\infty.\n\\end{multline} \nPutting together \\eqref{below1}, \\eqref{below22} and \\eqref{below3} and using $\\mu = e^\\lambda$ we obtain\n$$\n\\liminf_{\\lambda\\to\\infty} e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \\frac{1}{(2\\pi)^{d}}\\, |\\Omega_\\delta|\\, \\, |B_d|\\, d^{-1}.\n$$\nSince $\\delta>0$ is arbitrary we complete the proof of  Theorem \\ref{2.1}.\n\n\\end{proof}\n\n\\section{Weyl asymptotics}\\label{Weyl}\n\n\\noindent\nThroughout this section, we let $\\Omega \\subset {\\mathbb R}^d$ denote an open set of finite measure. Combining Theorems \\ref{1.1} and \\ref{2.1} we have\n\n\n\\begin {thm}\\label{3.1}\nThe Riesz means of the eigenvalues of the Dirichlet boundary value problem  \\eqref{D} satisfy the following asymptotic formula \n\\begin{equation}\\label{Weyl1}\n \\lim_{\\lambda\\to\\infty} e^{-d\\lambda}\\, \\sum_{k}(\\lambda - \\lambda_k)_+ = \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\,  |B_d|\\, d^{-1}.\n\\end{equation}\n\\end{thm}\nAs a corollary we can obtain asymptotics of the number of  the eigenvalues of the operator $\\mathcal H$.\n\n\n\\begin{cor} \\label{3.2}\nThe number of the eigenvalues $N(\\lambda)$ of the Dirichlet boundary value problem  \\eqref{D} below $\\lambda$ satisfies the following asymptotic formula \n\\begin{equation}\\label{Weyl2}\n\\lim_{\\lambda\\to\\infty} e^{-d\\lambda} \\, N(\\lambda) =   \\frac{1}{(2\\pi)^{d}}\\, |\\Omega|\\,  |B_d|.\n\\end{equation}\n\\end{cor} \n\n\\begin{proof}\nIn order to prove \\eqref{Weyl2} we use two simple inequalities. If $h>0$, then\n\\begin{equation}\\label{Nabove}\n\\frac{(\\lambda + h - \\lambda_k)_+   - (\\lambda - \\lambda_k)_+}{h} \\ge 1_{\\text{\\tiny $(-\\infty,\\lambda)$}}(\\lambda_k)\n\\end{equation}\nand\n\\begin{equation}\\label{Nbelow}\n\\frac{(\\lambda - \\lambda_k)_+   - (\\lambda - h- \\lambda_k)_+}{h}\n \\le 1_{\\text{\\tiny $(-\\infty,\\lambda)$}}(\\lambda_k)\n\\end{equation}\n\nThe inequality \\eqref{Nabove} implies, together with Theorems \\ref{1.1} and~\\ref{2.1}, that \n\\begin{align*}\n&\\limsup_{\\lambda \\to \\infty}e^{-d\\lambda}N(\\lambda) \\le \n\\limsup_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}\\frac{(\\lambda + h - \\lambda_k)_+   - (\\lambda - \\lambda_k)_+}{h}\\\\\n&\\le \\frac{1}{h}\\Bigl[e^{dh} \\limsup_{\\lambda \\to \\infty}e^{-d(\\lambda+h)} \\sum_{k}(\\lambda + h - \\lambda_k)_+   -\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+\\Bigr]\\\\\n&\\le \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\:\\frac{e^{dh}-1}{h} \\qquad \\text{for every $h>0$}\n\\end{align*}\nand thus \n\\begin{equation}\n  \\label{eq:limsup-N-lambda-ineq}\n\\limsup_{\\lambda\\to\\infty}e^{-d\\lambda}N(\\lambda) \\le \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\lim_{h \\to 0^+}\\frac{e^{dh}-1}{h}= \\frac{|\\Omega|\\,  |B_d|}{(2\\pi)^{d}}.\n\\end{equation}\nMoreover, \\eqref{Nabove} implies, together with Theorems \\ref{1.1} and~\\ref{2.1}, that \n\\begin{align*}\n&\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda}N(\\lambda) \\ge \n\\liminf_{\\lambda \\to \\infty}e^{-d\\lambda} \\sum_{k}\\frac{(\\lambda - \\lambda_k)_+   - (\\lambda -h - \\lambda_k)_+}{h}\\\\\n&\\ge \\frac{1}{h}\\Bigl[e^{dh} \\liminf_{\\lambda \\to \\infty}e^{-d \\lambda} \\sum_{k}(\\lambda - \\lambda_k)_+   -e^{-dh}\\limsup_{\\lambda \\to \\infty} e^{-d(\\lambda-h)} \\sum_{k}(\\lambda -h - \\lambda_k)_+\\Bigr]\\\\\n&\\ge \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\:\\frac{1-e^{-dh}}{h} \\qquad \\text{for every $h>0$}\n\\end{align*}\nand therefore \n\\begin{equation}\n  \\label{eq:liminf-N-lambda-ineq}\n\\liminf_{\\lambda\\to\\infty}e^{-d\\lambda}N(\\lambda) \\ge \\frac{|\\Omega| |B_d|}{d(2\\pi)^d}\\lim_{h \\to 0^+}\\frac{1-e^{-dh}}{h}= \\frac{|\\Omega|\\,  |B_d|}{(2\\pi)^{d}}.\n\\end{equation}\nThe claim follows by combining (\\ref{eq:limsup-N-lambda-ineq}) and (\\ref{eq:liminf-N-lambda-ineq}).  \n\\end{proof}\n\n\n\\section{An exact lower trace bound}\\label{LowB2}\n\nIn this section we prove the following exact lower bound in the case of bounded open sets with Lipschitz boundary.\n\n\\begin {thm}\\label{2.1-new-lower-bound}\nLet $\\Omega \\subset {\\mathbb R}^d$, $N \\ge 2$ be an open bounded set with Lipschitz boundary, let $\\tau \\in (0,1)$, and let \n\\begin{equation}\n\\label{def-C-Omega}\nC_{\\Omega,\\tau} := \\frac{1}{|\\Omega|(2\\pi)^d} \\int_{\\Bbb R^d}(1+|\\rho|)^\\tau \\log(1+|\\rho|) |\\widehat{1_\\Omega}(\\rho)|^2\\,d\\rho, \n\\end{equation}\nwhere $1_\\Omega$ denotes the indicator function of $\\Omega$. \n\nFor any $\\lambda \\ge 2 C_{\\Omega,\\tau}$,  we have\n\\begin{equation}\\label{BL}\n\\sum_{k}(\\lambda - \\lambda_k)_+\n\\ge \\frac{|\\Omega|\\, |B_d|}{(2\\pi)^{d}\\,d} \\Bigl[e^{d\\lambda} \\,- \\,a_\\tau\\, C_{\\Omega,\\tau}\\,e^{(d-\\tau)\\lambda} \\,- \\,b_\\tau\\, C_{\\Omega,\\tau}^2\\, \ne^{(d-2\\tau)\\lambda} \\,-\\, (d \\lambda + 1)  \\Bigr] \\nonumber\n\\end{equation}\nwith $a_\\tau:= \\frac{d(d-\\tau)-1}{d-\\tau}$ and $b_\\tau := 4d \\tau$.\n\\end{thm}\n\n\\begin{rem}\nIn the definition of $C_{\\Omega,\\tau}$, we need $\\tau<1$, otherwise the integral might not converge. In particular, if $\\Omega=B_d$ is the unit ball in ${\\mathbb R}^d$, we have \n$$\n\\widehat{1_\\Omega}(\\rho)= (2\\pi)^{\\frac{d}{2}} |\\rho|^{-\\frac{d}{2}}J_{\\frac{d}{2}}(|\\rho|)\n$$\nwhere $J_{\\frac{d}{2}}(r)= O(\\frac{1}{\\sqrt{r}})$ as $r \\to \\infty$. Hence the integral defining $C_{\\Omega,\\tau}$ converges if $\\tau <1$. A similar conclusion arises for cubes or rectangles, where \n$$\n\\widehat{1_\\Omega}(\\rho) = f_1(\\rho_1) \\cdot \\dots \\cdot f_d(\\rho_d)\n$$\nand $f_j(s) = O(\\frac{1}{s})$ as $|s| \\to \\infty$, $j=1,\\dots,d$.\n\nOn the other hand, if $\\Omega \\subset {\\mathbb R}^d$ is an open bounded set with Lipschitz boundary, we have \n\\begin{equation}\n  \\label{eq:C-Omega-tau-finiteness}\nC_{\\Omega,\\tau}<\\infty \\qquad \\text{for $\\tau \\in (0,1)$.}  \n\\end{equation}\nIndeed, in this case, $\\Omega$ has finite perimeter, i.e., $1_\\Omega \\in BV({\\mathbb R}^d)$. Therefore, as noted e.g. in \\cite[Theorem 2.14]{Lombardini}, $\\Omega$ also has finite fractional perimeter\n$$\nP_\\tau(\\Omega)= \\int_{\\Omega}\\int_{{\\mathbb R}^d \\setminus \\Omega} |x-y|^{-d-\\tau}\\,dxdy = \\frac{1}{2} \\int \\!\\! \\int_{{\\mathbb R}^{2d}}\\frac{(1_\\Omega(x)-1_\\Omega(y))^2}{|x-y|^{d+\\tau}}\\,dxdy\n$$\nfor every $\\tau \\in (0,1)$. Moreover, $P_\\tau(\\Omega)$ coincides, up to a constant, with the integral\n$$\n\\int_{\\Bbb R^d}|\\rho|^\\tau |\\widehat{1_\\Omega}(\\rho)|^2\\,d\\rho\n$$\nwhich therefore is also finite for every $\\tau \\in (0,1)$. Since moreover $1_\\Omega$ and therefore also $\\widehat{1_\\Omega}$ are functions in $L^2(\\Bbb R^d)$ and for every $\\varepsilon>0$ there exists $C_\\varepsilon>0$ with \n$$\n(1+|\\rho|)^\\tau \\log(1+|\\rho|) \\le C_\\varepsilon \\bigl(1 + |\\rho|^{\\tau+\\varepsilon}\\bigr) \\qquad \\text{for $\\rho \\in {\\mathbb R}^d$,}\n$$\nit follows that (\\ref{eq:C-Omega-tau-finiteness}) holds. \n\\end{rem}\n\nIn the proof of Theorem~\\ref{2.1-new-lower-bound}, we will use the following elementary estimate. \n\n\\begin{lem}\n\\label{elem-lemma}\nFor $r \\ge 0$, $s>0$ and $\\tau \\in (0,1)$, we have \n\\begin{equation}\n  \\label{eq:first-elem-ineq}\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\frac{1}{s} \\log(1+r) \\qquad \\text{if $s \\in (0,1)$}\n\\end{equation}\nand \n\\begin{equation}\n  \\label{eq:second-elem-ineq}\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r) \\qquad \\text{if $s \\ge 1$.}\n\\end{equation}\nIn particular, \n$$\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\max \\left\\{\\frac{1}{s}, \\frac{1}{s^\\tau} \\right\\} \n(1+r)^\\tau\\log(1+r) \\qquad \\text{for $r,s>0$.}\n$$\n\\end{lem}\n\n\\begin{rem}\nThe obvious bound $\\log(1 + \\frac{r}{s}) \\le \\frac{r}{s}$ will not be enough for our purposes. We need an upper bound of the form $g(s)h(r)$ where $h$ grows less than linearly in $r$.    \n\\end{rem}\n\n\\begin{proof}[Proof of Lemma~\\ref{elem-lemma}]\nLet first $s \\in (0,1)$. Since \n$$\n\\log\\left(1 + \\frac{r}{s}\\right)\\Big|_{r=0} = 0 = \\frac{1}{s} \\log(1+r)\\Big|_{r=0}\n$$\nand, for every $r>0$, \n$$\n\\frac{d}{dr} \\log\\left(1 + \\frac{r}{s}\\right) = \n\\frac{1}{s+r} \\le \\frac{1}{s+ sr} =  \\frac{d}{dr} \\frac{1}{s} \\log(1+r),\n$$\ninequality (\\ref{eq:first-elem-ineq}) follows. To see (\\ref{eq:second-elem-ineq}), we fix $s>1$, and we note that\n$$\n\\log\\left(1 + \\frac{r}{s}\\right)\\Big|_{r=0} = 0 = \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r)\\Big|_{r=0}.\n$$\nMoreover, for $0 < r \\le s-1$, we have  \n\\begin{align*}\n&\\frac{d}{dr} \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r)= \n\\frac{(1+r)^{\\tau-1}}{s^\\tau}(1+ \\tau \\log(1+r))\\\\\n&\\ge \\frac{(1+r)^{\\tau-1}}{s^\\tau} \n\\ge \\frac{1}{s}\\ge \\frac{1}{s+r}= \\frac{d}{dr} \\log\\left(1 + \\frac{r}{s}\\right),\n\\end{align*}\nso the inequality holds for $r \\le s-1$. If, on the other hand, $r \\ge s-1$, we have obviously\n$$\n\\log\\left(1 + \\frac{r}{s}\\right) \\le \\log(1 + r) \\le \\frac{(1+r)^\\tau}{s^\\tau} \\log(1+r).\n$$\n\\end{proof}\n\nWe may now complete the \n\n\\begin{proof}[Proof of Theorem~\\ref{2.1-new-lower-bound}]\nFor $\\xi \\in {\\mathbb R}^d$, we define $f_\\xi \\in L^2(\\Bbb R^d)$ by $f_\\xi(x)= \\frac{1}{\\sqrt{|\\Omega|}}1_{\\Omega} e^{-i x \\xi}$. Note that  $\\|f_{\\xi}\\|_{L^2(\\Bbb R^d)} = 1$ for any $\\xi \\in \\Bbb R^d$. We write\n\\begin{align*}\n\\sum_{k}(\\lambda - \\lambda_k)_+ &= \\sum_{k}(\\lambda - \\lambda_k)_+ \\|\\varphi_k\\|_{L^2(\\Omega)}^2 \n= \\frac{1}{(2\\pi)^{d}} \\sum_{k}(\\lambda - \\lambda_k)_+ \n\\|\\widehat{\\varphi_k}\\|_{L^2(\\Bbb R^{d})}^2 \n\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\sum_{k}(\\lambda - \\lambda_k)_+ \\int_{\\Bbb R^{d}} |\\langle f_\\xi,\\varphi_k \\rangle|^2\\,d\\xi\n\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\sum_{k}(\\lambda - \\lambda_k)_+  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\,d\\xi.\n\\end{align*}\nSince $\\sum \\limits_{k} |\\langle f_\\xi,\\varphi_k \\rangle|^2 = \\|f_{\\xi}\\|_{L^2(\\Bbb R^d)}^2 = 1$ for $\\xi \\in \\Bbb R^d$, Jensen's inequality gives \n\\begin{align}\n\\sum_{k}(\\lambda - \\lambda_k)_+ &\\ge \n\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda \\sum_{k}|\\langle f_\\xi,\\varphi_k \\rangle|^2    - \\sum_k \\lambda_k  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\Bigr)_+\\,d\\xi \\nonumber\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda   - \\sum_k \\lambda_k  |\\langle f_\\xi,\\varphi_k \\rangle|^2\\Bigr)_+\\,d\\xi \\nonumber\\\\\n&=\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl( \\lambda   - ( \\mathcal H  f_\\xi, f_\\xi ) \\Bigr)_+\\,d\\xi. \\label{jensen-new}\n\\end{align}\nHere, since \n$$\n\\sqrt{|\\Omega|}\\, \\widehat {f_\\xi}(\\eta-\\xi)= \n\\int_{\\Omega}e^{-i(\\eta-\\xi) x}e^{-ix \\xi}\\,dx = \n\\int_{\\Omega}e^{-i \\eta x} \\,dx = \\widehat{1_\\Omega}(\\eta)\n$$\nfor $\\eta,  \\xi \\in {\\mathbb R}^d$, we have \n\\begin{align}\n&|\\Omega|(2\\pi)^d \\left(\\mathcal H  f_{\\xi}, f_{\\xi}    \\right) = |\\Omega| \\int_{\\Bbb R^d}\n\\log |\\eta| |\\widehat {f_\\xi}(\\eta)|^2d \\eta = |\\Omega| \\int_{\\Bbb R^d}\n\\log |\\eta-\\xi| |\\widehat {f_\\xi}(\\eta-\\xi)|^2d \\eta \\nonumber\\\\\n&=\\int_{\\Bbb R^d} \\log|\\eta-\\xi| |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta \\le \\int_{\\Bbb R^d}\n\\left[ \\log |\\xi| +\\log (1+|\\eta|\/|\\xi|)\\right] |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta \\nonumber\\\\\n&\\le \\log |\\xi| \\int_{\\Bbb R^d} |\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta+ \\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\}\n\\int_{\\Bbb R^d}(1+|\\eta|)^{\\tau} (\\log(1+|\\eta|)|\\widehat{1_\\Omega}(\\eta)|^2\\,d\\eta\n\\nonumber\\\\\n&= |\\Omega|(2\\pi)^d \\Bigl(\\log |\\xi| +  \\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\} C_{\\Omega,\\tau} \\Bigr)\\qquad \\text{for $\\xi \\in {\\mathbb R}^d$,}\\label{jensen-new-compl} \n\\end{align}\nwhere $C_{\\Omega,\\tau}$ is defined in (\\ref{def-C-Omega}). Here we used Lemma~\\ref{elem-lemma}. Combining (\\ref{jensen-new}) and (\\ref{jensen-new-compl}), we get\n\\begin{equation}\n\\sum_{k}(\\lambda - \\lambda_k)_+ \\ge \n\\frac{|\\Omega|}{(2\\pi)^{d}}  \\int_{\\Bbb R^{d}} \\Bigl(\\lambda - \\log |\\xi| - \n\\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau}\\right\\} C_{\\Omega,\\tau}\\Bigr)_+d\\xi. \\label{intermediate-new} \n\\end{equation}\nLet us redefine the spectral parameter $\\lambda =  \\log \\mu$ again.\nThen we find\n\\begin{align}\n&\\int_{\\Bbb R^{d}} \\Bigl(\\lambda - \\log |\\xi| - \n\\max \\left\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\right\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\nonumber\n\\\\\n&= \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(\\log \\frac{\\mu}{r} - \n\\max \\left\\{ \\frac{1}{r}, \\frac{1}{r^\\tau}\\right\\} C_{\\Omega,\\tau}\n\\right)_+ \\, r^{d-1}dr  \\nonumber \\\\\n&= \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_0^\\infty \\left(-\\log r  - \n\\max \\left\\{ \\frac{1}{\\mu^{1-\\tau} r}, \\frac{1}{r^\\tau}\\right\\} \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr\n \\nonumber \\\\\n&\\ge \n\\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^\\infty \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr.  \\label{below2}\n\\end{align}\nFor the last inequality, we used the fact that $\\frac{1}{\\mu^{1-\\tau} r} \\le \n\\frac{1}{r^\\tau}$ for $r \\ge \\frac{1}{\\mu}$.\n\nNext we note that the function $r \\mapsto f_\\mu(r) = \n-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}$ satisfies \n\\begin{equation}\n  \\label{eq:boundary-conditions}\nf_\\mu(r)<0 \\quad \\text{for $r \\ge 1$}\\qquad \\text{and}\\qquad \n\\lim_{r \\to 0^+}f_\\mu(r)= -\\infty.\n\\end{equation}\nMoreover, this function has two zeros $r_1(\\mu), r_2(\\mu)$ with $0<r_1(\\mu)< \\frac{1}{\\mu} < r_2(\\mu)<1$ and \n$$\nf_\\mu(r)\\ge 0 \\qquad \\text{if and only if}\\quad r_1(\\mu) \\le r \\le r_2(\\mu).\n$$\nTo see this, we write \n$$\nf_\\mu(r)= \\frac{1}{r^\\tau}g(r^\\tau) \\qquad \\text{with}\\qquad g(s)=-\\frac{s}{\\tau} \\log s - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n$$\nand note that $g$ is strictly concave since $s \\mapsto g'(s)= -\\frac{1}{\\tau} - \\log s$ is strictly decreasing. Consequently, $g$ has at most two zeros, and the same is true for $f$. Combining this with (\\ref{eq:boundary-conditions}) and the fact that \n$$\nf(1\/\\mu) = \\log \\mu  - C_{\\Omega,\\tau} > 0 \n$$\nsince $\\lambda \\ge 2 C_{\\Omega,\\tau} >C_{\\Omega,\\tau}$ by assumption, the claim above follows. From (\\ref{below2}), we thus obtain the lower bound\n\\begin{align}\n  \\label{eq:r-2-mu-est}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^{r_2(\\mu)} \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr. \\nonumber  \n\\end{align}\nNext, we claim that\n\\begin{equation}\n\\label{r-2-mu-lower-bound}  \nr_2(\\mu) \\ge r_3(\\mu):= e^{\\frac{1}{2\\tau}\\bigl(\\sqrt{1- \\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}}\\;-1\\bigr)}.\n\\end{equation}\nHere we note that $\\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}=\\frac{4\\tau C_{\\Omega,\\tau}}{e^{\\tau \\lambda}} <1$ since $\\lambda \\ge 2 C_{\\Omega,\\tau}$ by assumption. \nTo see (\\ref{r-2-mu-lower-bound}), we write \n$$\nr_3(\\mu)= e^{- c_\\mu \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}}\\qquad \\text{with}\\qquad  \nc_\\mu = \\frac{\\mu^\\tau}{2 \\tau  C_{\\Omega,\\tau}}\\Bigl(1 - \\sqrt{1- \\frac{4\\tau C_{\\Omega,\\tau}}{\\mu^\\tau}}\\Bigr),\n$$\nnoting that \n$$\n\\frac{\\tau C_{\\Omega,\\tau}}{\\mu^\\tau} c_\\mu^2 -c_\\mu +1= 0\n$$\nand therefore \n\\begin{align*}\n&f(r_3(\\mu)) = f(e^{-\\frac{c_\\mu  C_{\\Omega,\\tau}}{\\mu^\\tau}})=\\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau} - \n\\frac{1}{e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}}\n\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\\\\\n&=\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}} \\Bigl( c_\\mu e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}} - 1\\Bigr) h\\ge \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau e^{-\\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}}}\\Bigl( c_\\mu \\bigl(1 - \\tau \\frac{c_\\mu C_{\\Omega,\\tau}}{\\mu^\\tau}\\bigr)-1\\Bigr) = 0.\n\\end{align*}\nThis proves (\\ref{r-2-mu-lower-bound}). As a consequence of the inequality $\\sqrt{1-a} \\ge 1-\\frac{a}{2} -\\frac{a^2}{2}$ for $0 \\le a \\le 1$, we also have \n$$\nr_3(\\mu) \\ge e^{- \\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} = : r_4(\\mu).\n$$\nConsequently, \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\int_{\\frac{1}{\\mu}}^{r_4(\\mu)} \\left(-\\log r  - \n\\frac{1}{r^\\tau}\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}\n\\right)_+ \\, r^{d-1}dr\\\\  \n&= \n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\, \\Bigl[-\\frac{r^d}{d} \\log r + \\frac{1}{d^2}r^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r^{d-\\tau}  \\Bigr]_{\\frac{1}{\\mu}}^{r_4(\\mu)}\\\\\n&= \n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl( \\Bigl[-\\frac{r_4(\\mu)^d}{d} \\log r_4(\\mu) + \\frac{1}{d^2}r_4(\\mu)^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r_4(\\mu)^{d-\\tau}  \\Bigr]\\\\\n&- \\Bigl[\\frac{\\mu^{-d}}{d} \\log \\mu + \\frac{1}{d^2}\\mu^{-d} - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}\\mu^{\\tau-d}  \\Bigr]\\Bigr),\n\\end{align*}\nwhich implies that \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge\n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl(\\frac{1}{d^2}r_4(\\mu)^d - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}r_4(\\mu)^{d-\\tau}- \\frac{\\mu^{-d}}{d} \\log \\mu - \\frac{1}{d^2}\\mu^{-d}  \\Bigr)\\\\\n&=\n  \\mu^d\\, \\left|\\Bbb S^{d-1} \\right| \\,\\Bigl(\\frac{1}{d^2}e^{- d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)}\n - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}e^{- (d-\\tau)\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)}\\\\\n&- \\frac{\\mu^{-d}}{d} \\log \\mu - \\frac{1}{d^2}\\mu^{-d}  \\Bigr).\n\\end{align*}\nSince \n$$\ne^{- d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} \\ge 1-  d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)\n$$\nand \n$$\ne^{- (d-\\tau)\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)} \\le 1,\n$$\nwe conclude that \n\\begin{align*}\n\\int_{\\Bbb R^{d}} &\\Bigl(\\lambda - \\log |\\xi| - \n\\max \\bigl\\{ \\frac{1}{|\\xi|}, \\frac{1}{|\\xi|^\\tau} \\bigl\\} C_{\\Omega,\\tau} \\Bigr)_+d\\xi \\\\\n&\\ge \n  \\mu^d\\, \\frac{\\left|\\Bbb S^{d-1} \\right|}{d^2} \\,\\Bigl(1-  d\\bigl(\\frac{C_{\\Omega,\\tau}}{\\mu^\\tau}+\\frac{4\\tau C_{\\Omega,\\tau}^2}{\\mu^{2\\tau}}\\bigr)\n - \\frac{C_{\\Omega,\\tau}}{\\mu^\\tau(d-\\tau)}- \\mu^{-d}(d \\log \\mu + 1) \\Bigr)\\\\\n&= \n\\frac{\\left|B_d \\right|}{d} \\,\\Bigl(\\mu^d \n- C_{\\Omega,\\tau}(d-\\frac{1}{d-\\tau})\\mu^{d-\\tau} - 4d\\tau C_{\\Omega,\\tau}^2 \\mu^{d-2\\tau} - (d \\log \\mu + 1) \\Bigr)\\\\\n&= \n\\frac{\\left|B_d \\right|}{d} \\,\\Bigl(e^{d \\lambda} \n- \\frac{d(d-\\tau)-1}{d-\\tau} C_{\\Omega,\\tau}e^{(d-\\tau)\\lambda} - 4d\\tau C_{\\Omega,\\tau}^2 \ne^{(d-2\\tau)\\lambda} - (d \\lambda + 1) \\Bigr).\n\\end{align*}\nCombining the last estimate with (\\ref{intermediate-new}), we get the asserted lower bound.\n\\end{proof}\n\n\n\\section{Appendix: Note on a bound for Bessel functions}\n\\label{sec:appendix:-note-bound}\n\nThe following elementary bound might be known but seems hard to find in this form. \n\n\\begin{lem}\nFor $\\nu \\ge \\sqrt{3}-2$ and $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$ we have \n$$\n|J_\\nu(x)| \\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n$$\n\\end{lem} \n\n\\begin{proof}\nWe use the representation \n$$\nJ_\\nu(x)= \\Bigl(\\frac{x}{2}\\Bigr)^{\\nu} \\sum_{m=0}^\\infty \\frac{(-1)^m}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}.\n$$\nFor $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$ and $m \\ge 1$, we have \n$$\n\\Bigl(\\frac{x}{2}\\Bigr)^{2} \\le (m+1)(m+\\nu+1) = \\frac{(m+1) \\Gamma(m+\\nu + 2)}{\\Gamma(m+\\nu + 1)}\n$$\nand therefore \n\\begin{equation}\n  \\label{eq:bessel-proof-1}\n\\frac{\\Gamma(\\nu+1)}{(m+1)! \\Gamma(m+\\nu + 2)} \\Bigl(\\frac{x}{2}\\Bigr)^{2(m+1)} \n\\le \\frac{\\Gamma(\\nu+1)}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}.\n\\end{equation}\nConsequently, \n\\begin{align*}\nJ_\\nu(x) &= \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}\\Bigl[1 + \\sum_{m=1}^\\infty \\frac{(-1)^m \\Gamma(\\nu+1)}{m! \\Gamma(m+\\nu + 1)} \\Bigl(\\frac{x}{2}\\Bigr)^{2m}\\Bigr]\\le \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n\\end{align*}\nFrom (\\ref{eq:bessel-proof-1}) we also deduce that \n\\begin{align*}\nJ_\\nu(x) &\\ge \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1 - \\frac{\\Gamma(\\nu+1)}{\\Gamma(\\nu + 2)} \\Bigl(\\frac{x}{2}\\Bigr)^{2}+ \\frac{\\Gamma(\\nu+1)}{2\\Gamma(\\nu + 3)} \\Bigl(\\frac{x}{2}\\Bigr)^{4}- \\frac{\\Gamma(\\nu+1)}{6\\Gamma(\\nu + 4)} \\Bigl(\\frac{x}{2}\\Bigr)^{6}\\Bigr]\\\\\n&= \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1-\\frac{1}{\\nu + 1}f\\bigl(\\bigl(\\frac{x}{2}\\bigr)^{2} \\bigr)\\Bigr]\n\\end{align*}\nwith $f: {\\mathbb R} \\to {\\mathbb R}$ given by $f(t)= t - \\frac{t^2}{2(\\nu+2)}+ \\frac{t^3}{6(\\nu+2)(\\nu+3)}$. Since \n$$\nf'(t)= 1- \\frac{t}{\\nu+2} + \\frac{t^2}{2(\\nu+2)(\\nu+3)}, \\qquad \\text{and}\\qquad f''(t)= \\frac{1}{\\nu+2}\\bigl(\\frac{t}{\\nu+3}- 1\\bigr)\n$$\nwe have \n$$\nf'(t) \\ge f'(\\nu+3) = 1-  \\frac{\\nu+3 }{\\nu+2} + \\frac{\\nu+3}{2(\\nu+2)}\n= 1 - \\frac{1}{2}\\frac{\\nu+3 }{\\nu+2} \\ge 0 \n\\quad \\text{for $t \\in {\\mathbb R}$ if $\\nu \\ge -1$}\n$$\nand therefore \n$$\nf(t) \\le f(2(\\nu +2))= 2(\\nu+2) - \\frac{[2(\\nu+2)]^2}{2(\\nu+2)}+ \\frac{[2(\\nu+2)]^3}{6(\\nu+2)(\\nu+3)}= \\frac{4(\\nu+2)^2}{3(\\nu+3)} \n$$\nfor $t \\le 2(\\nu+2)$ if $\\nu \\ge -1$. Since $\\frac{4(\\nu+2)^2}{3(\\nu+3)} \\le \\frac{2}{\\nu+1}$ for $\\nu \\ge \\sqrt{3}-2$, we conclude that\n$$\nJ_\\nu(x) \\ge \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)} \\Bigl[1-\\frac{1}{\\nu + 1}f\\bigl(\\bigl(\\frac{x}{2}\\bigr)^{2} \\bigr)\\Bigr]\\ge - \\frac{x^\\nu}{2^\\nu \\Gamma(\\nu+1)}.\n$$\nfor $\\nu \\ge \\sqrt{3}-2$ and $0 \\le x \\le 2 \\sqrt{2(\\nu+2)}$. The claim thus follows. \n\\end{proof}\n\n\\noindent\n{\\it Acknowledgements}.\nAL was supported by the RSF grant  19-71-30002.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \\label{sec:intro}\n\n\n\n\n\\par\n\nThe W Ursa Majoris (W~UMa)-type binary, also known as an EW-type binary, is a contact system where the two components fill their Roche lobes and share a common envelope. \nThe spectral types of EW binaries usually range from F to K \\citep{2017RAA....17...87Q}.\nTheir variability amplitudes are generally less than 1 magnitude, while the typical orbital period ranges from 0.2 to 1.0 day, but strongly\npeaked between 0.2-0.5 days \\citep[see][Figure~1]{2017RAA....17...87Q}.\nThe iconic feature of their optical light curves is the continuous variation of luminosity with nearly equal depths of the\nprimary and secondary minima, which indicates that the two components are in thermal contact, characterized by nearly identical temperatures. Furthermore, EWs can be subsequently divided into two subtypes based on mass and temperature, namely: the W subtype and A subtype. The primary star of the former is a more-massive and hotter component, while that of the latter is a more-massive and cooler one.\nA fraction of W~UMa systems are X-ray sources, which we refer to as EW-type binaries with X-ray emission (hereafter EWXs).\nAlthough the X-ray emission mechanism remains puzzling, the stellar dynamo magnetic activity generated by rapid rotation and envelope convection is usually in consideration \\citep{2004A&A...415.1113G}.\nThe studies of BH~Cas \\citep{2019PASP..131h4202L} and 2MASS~J11201034$-$2201340 \\citep{2016AJ....151..170H} indicate that the X-ray light curves do not show any obvious occultation or modulation as optical light curves would. The X-ray spectra of the former can be described by thermal models, while those of the latter can be both fitted by a thermal or a power-law model. The X-ray grating spectra of VW~Cep \\citep{2006ApJ...650.1119H} reveal that the compact corona is mainly located in the polar region of the primary star.\n\n\\par\n\n\\citet{2001A&A...370..157S} examined a sample of 102 W~UMa systems and found 57 of them were X-ray sources detected by the ROSAT All Sky Survey (RASS), which indicates a high fraction of W~UMa binaries having X-ray emission. \n\\citet{2006AJ....131..990C} also obtained X-ray fluxes for 34 W~UMa  systems from the RASS database. In these studies, they calculated the hardness ratio between the X-ray counts in the hard (H; 0.5-2.0 keV) and soft (S; 0.1-0.4 keV) bands, and then the X-ray flux and X-ray luminosity based on the energy conversion factor derived from hardness ratio \\citep{1996A&A...310..801H}. The X-ray luminosities of W~UMa type binaries within 400 pc range from $4.4 \\times 10^{29}$ to $2.3 \\times 10^{31}$ erg s$^{-1}$ \\citep{2006AJ....131..990C}. Combining the samples of contact binaries observed by ROTSE-1 and the RASS catalog, \\citet{2006AJ....131..633G}  found that 140 contact binaries have X-ray emission with typical luminosities of $\\sim 1.0 \\times 10^{30}$erg s$^{-1}$. \\citet{2008AcA....58..405S} compiled a catalog containing 379 X-ray emitting contact eclipsing binaries for which they applied somewhat different selection criteria from the widely-used EW classification. They found evidence of an X-ray saturation effect, while their sample exhibits large scatter in the X-ray activity and period relation. \n\n\n\\par\n\nFor late-type main sequence stars (G- to F-types), their X-ray luminosities, $L_{\\textrm{X}}$, tend to reach a maximum value at 10$^{-3}$ of the star's bolometric luminosity $L_{\\textrm{bol}}$, namely, log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) $\\sim -3$, which is known as the \"saturation limit,\" while the $\\log(L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) can be used to describe the X-ray activity level \\citep{1984A&A...133..117V, 1987ApJ...321..958V, 1993ApJ...410..387F}. For single stars with $P<$~0.4~days, the phenomenon of log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) decreasing with the shortening period is referred to as \"supersaturation\"\\ \\citep{1984ApJ...277..263C, 2001A&A...370..157S}. The $L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ ratio also represents the X-ray activity level of binary systems; this value increases as the orbital period becomes shorter \\citep{1996AJ....112.1570P, 2001A&A...370..157S}. Previous studies of the relationship between period and X-ray emission of EWs often suffer from a limited sample size and large scattering of the data, making it difficult to describe them quantitatively, thus impeding further studies of the physical mechanism. Therefore, enlarging the sample size and improving the X-ray coverage and data quality are very important for elucidating the origins and properties of X-ray emission and further investigating the evolution of binary systems. \n\n\\par\n\nIn this paper, we identify a large number of EW-type binaries with X-ray emission by cross-matching the All-Sky Automated Survey for Supernovae Variable Stars Database (AVSD) with the 4XMM-Newton Data Release 9 (4XMM-DR9) and the RASS catalogs. Combining with the spectral parameters from the seventh data release of the Large Sky Area Multi object Fiber Spectroscopic Telescope (LAMOST DR7) and binary absolute parameters collected from lectures, we investigate the possible mechanisms of X-ray radiation. In Section~\\ref{sec:dataselection}, we describe the selection of our sample. In Section~\\ref{sec:data_analysis}, we mainly provide the correlation analyses for X-ray luminosity and activity level versus stellar rotation, along with the spectral and component parameters. We discuss the physical implications of our results in Section~\\ref{sec:Discu} and provide a summary of this work in Section~\\ref{sec:Summary}.\n\n\\par\n\n\\section{Sample selection} \\label{sec:dataselection}\n\n\\subsection{EWXs in 4XMM-DR9 }\\label{sec:Sample_selection_XMM}\n\n\\par\n\nThe All-Sky Automated Survey for Supernovae (ASAS-SN) is a ground-based optical survey regularly scanning the full visible sky with a cadence of between two and three days, with a sensitivity limit down to $V$ $\\lesssim$ 17 mag \\citep{2018MNRAS.477.3145J}. ASAS-SN discovered new EW binaries by using V-band light curves with a random forest classifier based on 16 Fourier features and 10 other features describing the statistical and mathematical characteristics that the EW binaries are expected to exhibit \\citep{2019MNRAS.486.1907J}. Through a cross-matching the variable stars with Gaia DR2 \\citep{2018yCat.1345....0G}, 2MASS \\citep{2006AJ....131.1163S}, and ALLWISE \\citep{2010AJ....140.1868W, 2014yCat.2328....0C}, the AVSD \\footnote{https:\/\/asas-sn.osu.edu\/variables} provides V-bands light curves, the parallaxes, proper motions, photometry, and color or reddening information for most variable sources. Up until September 2021, 76378 objects have been classified as EW-type binaries from the analysis of $\\sim$ 660000 variable stars listed in AVSD. We chose these EW-type binaries as our primary catalog (hereafter, ASAS-SN-EW). The 4XMM-DR9 released 550124 unique X-ray sources detected over the 11204 pointed XMM-Newton EPIC observations \\citep{2020A&A...641A.136W}. The long period of data accumulation, high sensitivity, and deep exposures make XMM-Newton very suitable for searches of the X-ray counterparts of EW-type binaries, which are often weak X-ray sources. Firstly, we cross-match the ASAS-SN-EW catalog with the 4XMM-DR9 full catalog with a matching radius of $6^{\\prime\\prime}$. This process leads to 723 unique X-ray sources (1205 observations in total) with XMM-Newton detections, defined as the Parent Group. Secondly, in order to further purify the Parent Group, we cross-matched it with the ATLAS \\citep{2018AJ....156..241H} and WISE \\citep{2018ApJS..237...28C} catalogs that provide classifications for binary systems. We use a matching radius of $3^{\\prime\\prime}$, and only the closest object from multiple matches is selected as the counterpart (less than 1.7\\%). There are 433 and 261 unique counterparts from these two catalogs, respectively. We further divided these counterparts into Group A and Group B. \nGroup A includes 407 objects labeled as close binaries in one or both of the two catalogs.\\footnote{CBF or CBH types in ATLAS, EW or EW\/EA types in WISE.} The 123 objects in Group B are those labeled as types other than close binaries in both catalogs. The remaining 193 objects from the Parent Group that have counterparts in neither catalogs are named as Group C.  \n\n\n\\par\n \nWe consider the Group A objects as having reliable classifications, since they are consistent in at least two catalogs (one is the ASAS-SN-EW, while the other is either ATLAS or WISE). The objects in Groups B and C were screened again via a visual inspection of their light curves, which is crucial for distinguishing genuine W~UMa binaries from other types of variables that may be misclassified by the mathematical screening criteria used in the production of different catalogs. We calculated the distance of each system in the three groups using Gaia DR2 parallax data and eliminated those with a period $>$ 1~day or distance $>$ 1~kpc (see below), or an uncertainty of distance $>$ 20\\% (parallax error\/parallax $>$ 20\\%). There are 255, 39, and 82 objects retained in Groups A, B, and C, respectively. All these objects combined constitute the main sample of this work (STW hereafter). The full process of this sample compilation is illustrated in the flowchart in Figure~\\ref{fig:Flowchart}. The STW contains 376 objects in total, listed in Table~\\ref{table:Properties_EWXs}. The columns are organized as ASAS-SN name, right ascension (R.A.; J2000), declination (DEC.; J2000), period, parallax, distance, 4XMM-DR9 name, log$L_{\\textrm{X}}$, ATLAS classification, and WISE classification. All the X-ray fluxes of our objects are taken from the 4XMM-DR9, which provides an average unabsorbed flux value in cases of multiple observations for a given source. The X-ray luminosity (log$L_{\\textrm{X}}$) is calculated from the full-band X-ray flux in 0.2-12~keV and the Gaia DR2 distance. Almost all of the sources in the STW have $L_{\\textrm{X}} > 10^{29}$~erg~s$^{-1}$. \n\n\\par\n\n\\begin{sidewaystable*}[!htpb]\n  \\centering\n  \\caption{Properties of EWXs in STW (Groups A, B, and C)}\n  \\label{table:Properties_EWXs}\n  \\small\n  \\begin{tabular}{rrrcccrccccrr}\n  \\hline\\noalign{\\smallskip}\n  ASAS-SN Name & R.A. (J2000) & DEC. (J2000) & Period& Parallax & Distance & 4XMM-DR9 Name & log$L_{\\textrm{X}}$ & ATLAS Type & WISE Type & $A_{\\rm G}$ & log$L_{\\textrm{bol}}$ & log($L_{\\textrm{X}}\/L_{\\textrm{bol}}$) \\\\\n(ASASSN-V) & ($^{\\circ}$) & ($^{\\circ}$) & (days) &  & (pc) & (4XMM) & (erg s$^{-1}$) &  & & mag & (erg s$^{-1}$) & \\\\\n  \\hline\\noalign{\\smallskip}\n                & & & & &  &  Group A & & & & & & \\\\\n        J001322.67+054009.6 & 3.34447 & 5.66933 & 0.28948 & 1.9764 & 498.85 & J001322.7+054008 & 29.738  & EW\/EA    & CBF & 0.525 & 33.538 & -3.800\\\\ \n                J001445.74-391435.4 & 3.69058 & -39.24317 & 0.36436 & 5.3361 & 186.40 & J001445.7-391435 & 30.255  & EW     &   & 0.997 & 33.812 & -3.557\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&...\\\\\n                J184128.52+622409.4 & 280.36883  & 62.40270 & 0.28545 & 1.0850 & 898.01 &J184128.5+622408 & 30.222 & EW  &dubious  & 0.600* & 33.655 & -3.433\\\\\n                J195923.73+225703.6   & 299.84887 & 22.95099 & 0.38876 & 1.4855 & 660.76 & J195923.6+225702 & 29.800  & EW               & MSINE& 0.467  & 34.275 & -4.475 \\\\ \n                J232739.41-004346.6   & 351.91420 & -0.72962 & 0.41252 & 1.8069 & 544.93 & J232739.2-004345 & 30.204  & EW               & SINE  & 0.393 & 34.202 & -3.998\\\\ \n                 & & & & & & Group B & & & & &    &  \\\\\n                J000525.84-084035.2   & 1.35768 & -8.67644 & 0.26309 & 1.8841 & 523.14 & J000525.8-084035 & 29.879  &      & MPULSE & 0.566 & 33.856 & -3.977\\\\ \n                J013143.20+302327.9   & 22.93000 & 30.39107 & 0.26660 & 2.2050 & 447.81 & J013143.1+302329 & 29.773  &      & MSINE &  0.702  & 33.464 & -3.691\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&... \\\\\n                J232008.39+241055.1   & 350.03494 & 24.18197 & 0.32222 & 1.7312 & 568.21 & J232008.3+241055 & 30.036  &      & MSINE  &0.637  & 33.856 & -3.820\\\\ \n                J232907.52+145731.2   & 352.28132 & 14.95866 & 0.26678 & 1.8248 & 547.99 & J232907.4+145731 & 29.714  &      & MSINE  &0.720  & 33.516 & -3.802\\\\\n                & & & & & & Group C & & & &  &    & \\\\\n                J002150.83-704642.5   & 5.46179 & -70.77846 & 0.27176 & 1.9298 & 510.58 & J002150.5-704640 & 29.963 &  &  &0.639  & 34.025 & -4.062\\\\ \n                J002234.47+614417.2   & 5.64364 & 61.73811 & 0.35770 & 1.3647 & 717.89 &J002234.3+614417 & 30.032 &  &  &1.208 &  34.107 & -4.075\\\\ \n                ... & ...&...&...&...&...&...&...&...&...&...&...&...\\\\\n                J234658.78-545310.2   & 356.74490 & -54.88617 & 0.28150 & 2.6239 & 376.96 & J234658.8-545308 & 29.719 &  &   &0.531  & 33.475 & -3.756\\\\ \n                J235142.03-395949.8   & 357.92512 & -39.99716 & 0.28395 & 1.2965 & 754.65 & J235142.1-395946 & 29.892 &  &   &0.475  & 33.777 & -3.885\\\\ \n\n  \\hline\\noalign{\\smallskip}\n  \\end{tabular}  \n  \\flushleft\n      \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n \\end{sidewaystable*}\n\n\n\\par\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 7cm]{Fig1.png}\n\\caption{Flowchart of compiling STW sample.}\n\\label{fig:Flowchart}\n\\end{figure}\n\n\\par\n\nOne factor which could potentially affect the accuracy of X-ray luminosity is the presence of X-ray flares. Since there has been sparse mention in the literature on this aspect of EWXs, we refer to the occurrence frequency of X-ray flares for main sequence stars. \\citet{2019A&A...628A..41P} selected 102 X-ray emitting main sequence stars with spectral types ranging from A to M to investigate their magnetic activities and X-ray flares. They detected six flares on five stars, indicating a low X-ray flare rate. Therefore, we did not consider the effects of X-ray flares in our analysis.\n\n\\par\n\n\\subsection{EWXs in RASS} \\label{sec:sec:Sample_selection_RASS}\n\n\n\\par\n\nBy reprocessing the RASS data, \\citet{2016A&A...588A.103B} released an updated catalog with 135118 X-ray sources (the 2RXS catalog). We compiled a sample of 190 EWXs detected by RASS following the same procedure as that in Section~\\ref{sec:Sample_selection_XMM}, but with the X-ray-to-optical matching radius set to $20^{\\prime\\prime}$. \nWe follow \\citet{2001A&A...370..157S} and \\citet{2006AJ....131..990C} to convert the count rate to X-ray flux using the energy conversion factor.  The X-ray luminosity is calculated with the Gaia DR2 distance.\n\n\\par\n\n\nIn addition, we collected the EWXs identified with RASS from the literature and updated their distances and X-ray luminosities using the Gaia DR2 distances. \\citet{2001A&A...370..157S} found that log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) declines with the decreasing color index $B-V < 0.6$. \\citet{2006AJ....131..990C} investigated the relationship between the X-ray luminosity and the stellar parameters, such as the rotation period and color index. In combining their sample with those of \\citet{2001A&A...370..157S} and \\citet{1996MNRAS.280..627M}, these authors reinforced a relation in which the shorter the rotation period, the weaker the X-ray emission when the rotational period is shorter than $\\sim0.5$ days. \\citet{2006AJ....131..633G} also used the RASS database to estimate the X-ray emissivity of contact binaries in our Galaxy. These previous studies utilized a variety of approaches to calculate the distance of each binary. \\citet{2006AJ....131..633G} used a period-color-luminosity relation with $J-H$ colors. The distances of the \\citet{2006AJ....131..990C} sample were obtained via the absolute magnitude calibrated with period and $B-V$ colors, while those of \\citet{2001A&A...370..157S} were derived from the parallaxes released by the Hipparcos satellite. In total, they identified 231 EWXs (including duplicates; for more details,  see below). \nWe cross-matched these 231 objects with Gaia DR2 using the $3^{\\prime\\prime}$ radius to obtain the updated parallaxes. For cases with multiple matches (about 18 \\%), the closest object is selected as the counterpart. In order to ensure the consistency across different samples, we only kept objects with parallax values in Gaia DR2. In total, there are 54\/57 objects from \\citet{2001A&A...370..157S}, 117\/140 from \\citet{2006AJ....131..633G}, and 9\/34 from \\citet{2006AJ....131..990C} remaineing. The X-ray luminosity of each object in these three groups was updated with the new distance from Gaia parallax. Duplicated objects among these three samples were removed. We finally obtained 165 EWXs from the literature (54 from \\citealt{2001A&A...370..157S}, 106 from \\citealt{2006AJ....131..633G} and 8 from \\citealt{2006AJ....131..990C}).\n\n\\par\n\n\nFinally, after combining the EWXs we obtained from 2RXS catalog and those from the literature and then removing 36 duplicated objects, we obtained a sample of 319 objects consistently selected from RASS, named as the SRASS sample (listed in Table~\\ref{table:SRASS_EWXs}). All the EWXs in SRASS are within 1~kpc from us. The STW objects are also limited within 1~kpc for consistency. Because of the substantially different X-ray detection sensitivity between 4XMM and RASS, we could not merge these two samples. Instead, we performed the same analyses for the two samples separately and compared their results, as we present in the following sections. Eight sources are duplicated in STW and SRASS, and each of them has similar X-ray luminosities in both catalogs. In order to maintain the completeness of the samples, we kept them in their respective datasets in the analysis. \n\n\n\\par\n \\begin{table*}[!htpb]\n   \\caption{Properties of objects in SRASS}\n     \\label{table:SRASS_EWXs}\n   \\begin{center}\n   \\begin{tabular}{rrccclccc}\\hline \\hline\n\nR.A. (J2000)  &DEC. (J2000) & Period & Distance & log$L_{\\textrm{X}}$ & $A_{\\rm G}$ & log$L_{\\textrm{bol}}$  & log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) & Reference   \\\\\n($^{\\circ}$) & ($^{\\circ}$)       & (days) & (pc)         & (erg s$^{-1}$)            & mag          & (erg s$^{-1}$)       & &    \\\\ \\hline\n324.70584 &  26.69278 & 0.28040 & 152.54 & 29.880 & 0.612 & 33.689 & -3.809 & (1)   \\\\\n150.41876 &  17.40888 & 0.28410  & 62.87 & 30.188  & 1.211  & 33.883 & -3.695 & (1)   \\\\\n...&...&...&...&...&...&...&...&... \\\\\n330.69898 & -12.31137 & 0.30678 & 182.27 & 30.261 & 0.454 & 33.732 & -3.471 & (5)    \\\\\n347.74778 & 21.71198 & 0.25817 & 105.88 & 30.168 & 1.058 & 33.505 & -3.337 & (5)  \\\\\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] Ref.(1): \\citet{2001A&A...370..157S}, (2):\\citet{2006AJ....131..633G}, (3):\\citet{2006AJ....131..990C}, (4): Duplications in STW, (5): This Work.\n    \\item[2] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\n\\subsection{Spectra from LAMOST}\\label{sec:Spectra_of_LAMOST}\n\n\\par\n\nWe utilized the LAMOST stellar spectral database to investigate the relationships between the optical spectral parameters and X-ray properties of EW-type binaries. \nThe LAMOST is a spectroscopic survey telescope located at the Xinglong station, National Astronomical Observation of China, with a $\\sim4$-meter effective aperture and a field of view of $5^\\circ$ \\citep{2012RAA....12.1197C}. LAMOST can obtain 4,000 spectra covering wavelength from 3700 to 9000 {\\AA} with a resolving power of 1800 in a single exposure. We input the coordinates of STW and SRASS samples into the LAMOST DR7 database\\footnote{http:\/\/dr7.lamost.org} with a matching radius of $3^{\\prime\\prime}$ to obtain stellar spectral parameters, including the effect temperature, $T_{\\rm eff}$, surface gravity, $\\log g$, metallicity [Fe\/H], and radial velocity, $V_{\\rm r}$; these values were automatically derived by the LAMOST stellar parameters pipeline \\citep{2011RAA....11..924W, 2014IAUS..306..340W}.  A total of 139 unique sources are matched in LAMOST DR7 (hereafter, the Spec-EWX sample), which are listed in Table~\\ref{table:Spectra_EWXs}. The designation of each column is defined as follows: right ascension (J2000), declination (J2000), log$L_{\\textrm{X}}$, period, effective temperature ($T_{\\rm eff}$), error of $T_{\\rm eff}$, metallicity ([Fe\/H]), error of [Fe\/H], surface gravity ($\\log g$), and its error.\n\n\\par\n\\begin{table*}[!htpb]\n  \\centering\n  \\caption{Stellar spectral parameters of EWXs in the Spec-EWX sample}\n  \\label{table:Spectra_EWXs}\n  \\small\n  \\begin{tabular}{rrcccccccc}\n  \\hline\\noalign{\\smallskip}\nR.A. (J2000) & DEC. (J2000) & log$L_{\\textrm{X}}$ & Period  &$T_{\\rm eff}$ & $T_{\\rm eff}$ error & [Fe\/H] &  [Fe\/H] error & $\\log g$  &$\\log g$ error \\\\\n($^{\\circ}$) & ($^{\\circ}$) & (erg s$^{-1}$) & (days) & (K) & (K) & (dex) &  (dex) & (dex) & (dex) \\\\\n  \\hline\\noalign{\\smallskip}\n283.39054 &47.43649 &30.213 &0.31503 &5589.50&21.82&-0.10&0.02&4.29& 0.04 \\\\ \n8.52526 &39.69754 &29.078 &0.28669 &5220.21&63.34&0.25&0.06&4.43& 0.10 \\\\ \n...&...&...&...&...&...&...\\\\\n349.62904 &42.92814 &30.072 &0.34180 &5818.81&25.68&0.09&0.02&4.20& 0.04 \\\\ \n350.03494 &24.18197 &30.036 &0.32222 &5416.84&32.06&0.19&0.03&4.291& 0.05 \\\\\n  \\hline\\noalign{\\smallskip}\n  \\end{tabular}  \n  \\flushleft\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\n\n\\par\n\n\n\n\\subsection{Upper limits for X-ray nondetections} \\label{sec:sec:upper limits}\n\nFor the completeness of the sample selection, we determined that EW binaries with X-ray nondetections should also be considered. We screened the ASAS-SN-EW samples following the same procedures as in Section~\\ref{sec:Sample_selection_XMM} (corresponding to the ``Group A\" objects after applying the cuts on period, distance, and parallax precision) without matching X-ray source catalogs. A total of 14096 reliably-classified EW binaries are obtained. After removing the X-ray detected objects we already obtained in Section~\\ref{sec:Sample_selection_XMM}, we calculated the X-ray flux upper limits using the web client {\\it HIgh-energy LIght curve GeneraTor} (\\citealt{2022A&C....3800531S})\\footnote{http:\/\/xmmuls.esac.esa.int\/hiligt\/scripts\/\nhiligt.py} which can poll individual servers for the chosen X-ray missions, and return the X-ray flux and\/or upper limits for given targets or coordinates.\nThe default parameter settings were adopted, namely: a 2$\\sigma$ upper limit significance, an absorbed power law spectral model for flux conversion with a photon index  of $\\Gamma=2,$ and the hydrogen column density of $N_{\\rm H} =3 \\times 10^{20}\\ \\rm cm^{-2}$. The XMM-Newton observations in both slew mode \\citep{2012A&A...548A..99W} and pointed mode were searched, while the entire RASS was utilized. When multiple upper limits were returned for the same coordinate, we chose the lower value (to achieve a tighter constraint). In particular, we used the upper limits from pointed observations for XMM-Newton rather than those from the slew mode when both exist. Finally, we get 12279 (39 from pointed mode and 12240 from slew mode) and 13769 X-ray flux upper limits from the XMM-Newton and ROSAT servers, respectively. Combining the distance data of Gaia DR2, we further calculated the X-ray luminosity upper limits, listed in Tables~\\ref{table:XMMnewtonupperlimits} and \\ref{table:ROSATupperlimits} for XMM-Newton and ROSAT, respectively. These X-ray upper limits are also considered in the subsequent analyses.\n\n\\par\n\n \\begin{table*}[!htpb]\n   \\caption{Properties of EW-type binaries with X-ray luminosity upper limits from XMM-Newton pointed and slew surveys}\n   \\begin{center}\n   \\begin{tabular}{rrcccc}\\hline \\hline\n\nR.A. (J2000) & DEC. (J2000) & Modes & Period & Distance & log$L_{\\textrm{X}}$\\\\\n($^{\\circ}$)   & ($^{\\circ}$)     &             & (days)    & (pc)       & (erg s$^{-1}$)  \\\\ \\hline\n  0.01355 & 69.37062 & slew & 0.37280 & 902.25 & $<$32.072 \\\\\n  0.14513 & 53.81984 & slew  & 0.23166  & 583.38 & $<$31.536 \\\\\n... & ... & ... &... &... & ...  \\\\\n359.97490 & 66.15284 & slew  & 0.32750 & 835.48 & $<$31.985 \\\\\n359.98450 & 51.50363  & slew & 0.29762 & 579.29 & $<$31.561 \\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:XMMnewtonupperlimits}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n \\begin{table*}[!htpb]\n   \\caption{Properties of EW-type binaries with X-ray luminosity upper limits from the ROSAT survey}\n   \\begin{center}\n   \\begin{tabular}{rrcccc}\\hline \\hline\n\nR.A. (J2000) & DEC. (J2000) & Modes & Period & Distance & log$L_{\\textrm{X}}$\\\\\n($^{\\circ}$)   & ($^{\\circ}$)     &             & (days)    & (pc)       & (erg s$^{-1}$)  \\\\ \\hline\n   0.01355 & 69.37062& RosatSurvey& 0.37280 &902.25 & $<$31.385 \\\\\n    0.14513  &53.81984 &RosatSurvey &0.23166 &583.38 & $<$30.956 \\\\\n... & ... & ... &... &... & ...  \\\\\n359.97490 & 66.15284& RosatSurvey &0.32750 &835.48 & $<$31.492 \\\\\n359.98450  &51.50363 &RosatSurvey &0.29762 &579.29 & $<$31.047 \\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:ROSATupperlimits}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] (This table is available in its entirety in the online machine-readable form.)\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\n\n\\section{Data analysis} \\label{sec:data_analysis}\n\n\\subsection{X-ray emission versus the period} \\label{sec:X-ray_P}\n\n\\par\n\nIn this section, we investigate the relationship between the X-ray emission and rotation for EW-type binaries. Contact binaries have circular orbits and synchronous co-rotation, which mean that the orbital period of the binary equals to the spin period of each component. In Figure~\\ref{fig:Lx_P} (1) and (2), we plot the period against log$L_{\\textrm{X}}$ of the EWXs in STW and SRASS, respectively. Objects with X-ray detections and upper limits are both plotted. Most of the upper limits do not provide physically meaningful constraints because of the low sensitivity. We discuss those X-ray nondetections in Section~\\ref{sec:completeness}. For the X-ray detected EW binaries, the X-ray luminosity has a significant positive correlation with the orbital period of the binary stars between $\\sim$0.2 and $\\sim$0.45 days. Meanwhile, in the period interval [0.4,0.5], there appears to be a break point (designated ``P1\") after which the correlations between the period and log$L_{\\textrm{X}}$ no longer hold, and the data points become more scarce and scattered. \n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig2.png}\n\\caption{Orbital period vs. log$L_{\\textrm{X}}$: for STW (1) shown as black points. The magenta and gray `$tri\\_down$' symbols represent the X-ray upper limits for some EW-type binaries from XMM-Newton pointed and slew survey, respectively;  for SRASS (2)  shown in black points. The gray `$tri\\_right$' symbols represent the X-ray upper limits for some EW-type binaries from ROSAT survey. The period vs. slopes with their corresponding error bars of STW (black) and SRASS (red) with different limiting period ranges (3). The black dashed lines are at $P=0.44$ days. The solid blue lines stand for best-fit linear correlations, and the light blue dashed lines represent 95\\% uncertainty ranges.}\n\\label{fig:Lx_P}\n\\end{figure*}\n\n\\par\n\nAs shown in panel (3) of Figure~\\ref{fig:Lx_P}, the value of each point on the black and red lines represents the slope $a$ of the least-squares quadratic fitting (in the form of $\\log L_{\\rm X} = a \\times P + b $) to all objects with a period less than its abscissa value  \\textit{P}. The black broken line with error bars is for STW, while the red one line is for SRASS. The fitting range is from 0.3 to 0.8 days with the step of 0.01 days. We performed the Kendall$^{\\prime}$s $\\tau$ test \\citep{1990BJpc....25..86} between \\textit{P} and log$L_{\\textrm{X}}$ for objects in these two samples. The null-hypothesis (i.e., no correlation exists) probability is always less than 10$^{-5}$ for the period range of [0.36, 0.80], which suggests that the periods of EWXs show a strong linear correlation ($>5\\sigma$) with the log$L_{\\textrm{X}}$ for both samples.\nWith the increasing period (and thus the number of objects), the slopes of the two samples gradually converge. The fitting slopes of STW and SRASS are both within the 1$\\sigma$ uncertainty range of each other, showing that these two samples are highly consistent with each other in the relationship between \\textit{P} and $\\log L_{\\rm X}$. Especially in the period range of [0.38, 0.44], the difference between the slopes is less than 1$\\sigma$. Beyond $P=0.44$ days, the slope difference starts to increase as the period gets longer, which means that P1 is likely to be at 0.44 days. Meanwhile, only a few data points have a period greater than 0.44 days. Therefore, we took 0.44~days as the upper bound for the period of STW ($\\sim$96\\% of its objects) and SRASS ($\\sim$90\\%) samples for the correlation analysis; that is, data points with period values less than 0.44 days were selected for the least-squares fitting. The blue lines in panels (1) and (2) of Figure~\\ref{fig:Lx_P} represent the best-fit relation and the light blue dashed lines represent the 95\\% uncertainty range. We formulate, for the first time, a clear linear correlation between the period $P$ (log$P$) and X-ray luminosity log$L_{\\textrm{X}}$ for EWXs, which is described by the following Equations~(\\ref{equ:STW_LX_P})--(\\ref{equ:SRASS_LX_logP}) for the STW and SRASS: \n\n\\par\n\n\\begin{equation}\\label{equ:STW_LX_P}\n    \\log L_{X_{STW}} = 2.81(27) \\times P + 29.10(9),\n\\end{equation}\n\n\\begin{equation}\\label{equ:SRASS_LX_P}\n    \\log L_{X_{SRASS}} = 3.17(35) \\times P + 29.17(12),\n\\end{equation}\n\n\\begin{equation}\\label{equ:STW_LX_logP}\n    \\log L_{X_{STW}} = 2.14(24) \\times \\log P + 31.08(10),\n\\end{equation}\n\n\\begin{equation}\\label{equ:SRASS_LX_logP}\n   \\log L_{X_{SRASS}} = 2.45(27) \\times \\log P + 31.41(13).\n\\end{equation}\n\n\\par \n\nWhile the dependency of X-ray luminosity upon period has been pointed out by previous studies (e.g., \\citealt{2001A&A...370..157S}, \\citealt{2006AJ....131..990C}, and \\citealt{2006AJ....131..633G}), quantitative descriptions had not been provided. The linear correlations between orbital period and X-ray luminosity for the STW and SRASS samples we present here are consistent with each other ($\\simlt1\\sigma$). The use of Gaia's more accurate parallax information, which results in more accurate log$L_{\\textrm{X}}$ values in our work help reduce the scatter of the data, making the correlations more clear and better defined. The linear correlations shown in both the STW and SRASS have high statistical significance: the probabilities of null-hypothesis (i.e., no correlation exists) in both cases are $<10^{-5}$, corresponding to a $>5\\sigma$ confidence level. For EWs (regardless of their X-ray emission), \\citet{2018ApJ...859..140C} has established the positive correlation between period and luminosity in the optical and mid-infrared bands. The period of their sample ranges from 0.25 to 0.56 days. As a comparison, the positive correlation between period and X-ray luminosity is well maintained at $P=0.2$ to 0.44 days. \n\n\\par\n\nBased on the method provided by \\citet{2018A&A...616A...8A}, we can calculate the bolometric luminosity $L_{\\textrm{bol}}$ of the EWXs with the following equations:\n\\begin{equation}\\label{equ:bolometricL}\n    -2.5~\\textrm{log}_{10}~(L_{\\textrm{bol}}\/L_{\\odot}) = M_{\\textrm{G}} + BC - M_{\\rm sun},\n\\end{equation}\n \\begin{equation}\\label{equ:Mg}\n    M_{\\textrm{G}} = G + 5 - 5\\log_{10} D - A_{\\rm G},\n\\end{equation}\nwhere $M_{\\textrm{G}}$ is the absolute $G$-band magnitude, $BC$ is the temperature-dependent bolometric correction (based on the effective temperatures in Gaia DR2 estimated from the $BP$-band, $RP$-band, and $G$-band magnitudes; see \\citealt{2018A&A...616A...8A} for details), $M_{\\rm sun}$ is the solar bolometric magnitude 4.74~mag, $G$ is the apparent magnitude in $G$-band, $D$ is the distance, and $A_{\\rm G}$ is the $StarHorse$ extinction \\citep{2019A&A...628A..94A} in the $G$-band.\\footnote{For sources without this value, we set it to 0.600 mag according to the statistical distributions of extinction and distance of our samples.} The values of $A_{\\rm G}$, log$L_{\\rm bol}$, and log$(L_{\\rm X}\/L_{\\rm bol})$ of 370 EWXs from STW are listed in the last three columns of Table~\\ref{table:Properties_EWXs}, while those of 318 EWXs from SRASS are listed in the Columns (6)--(8) of Table~\\ref{table:SRASS_EWXs}; for the remaining 6 sources in STW and 1 source in SRASS, these three parameters are not calculated because of the lack of $BP$\/$RP$-band coverage. In Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} (1) and (2), we plot the period against log$L_{\\textrm{bol}}$ and log$(L_{\\rm X}\/L_{\\rm bol})$ of the EWXs in STW and SRASS with blue and red symbols, respectively. The blue and red solid lines represent the best-fit linear relations ($P<0.44$~days), while the light blue and light red dashed lines represent the 95\\% uncertainty ranges for these two samples, respectively.  The Kendall$^{\\prime}$s $\\tau$ tests suggest that the confidence levels of $P$ correlated with both the log$L_{\\textrm{X}}$ and log$(L_{\\rm X}\/L_{\\rm bol})$ are all greater than 5~$\\sigma$. The fitting parameters are listed in Table~\\ref{table:ste_sub_srassLX\/Lbol}.\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig3.png}\n\\caption{Orbital period vs. log$L_{\\textrm{bol}}$: for STW in blue points and SRASS in red points (1).  Orbital period vs. log$(L_{\\rm X}\/L_{\\rm bol})$ for STW in blue points and SRASS in red points (2). Black dashed lines are at $P=0.44$ days. Solid blue and red lines stand for best-fit linear correlations ($P<0.44$~days) for STW and SRASS respectively, while the light blue and red dashed lines are their corresponding 95\\% uncertainty ranges.}\n\\label{fig:Lx_Lbol_STW_sub_SRASS}\n\\end{figure*}\n\n\\par\n\n\\begin{table}[!htpb]\n   \\caption{Best-fit parameters for $P$-log$L_{\\textrm{bol}}$ and $P$-\\-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of STW and SRASS}\n   \\begin{center}\n   \\begin{tabular}{llrrrr}\\hline \\hline\nRelations &  Samples & Slope & Intercept  \\\\\n\\hline \\\\\n  $P$-log$L_{\\textrm{bol}}$               & STW              &    4.19 $\\pm$ 0.21 &  32.47 $\\pm$ 0.07  \\\\\n                          & SRASS &  4.57 $\\pm$ 0.23 &  32.34 $\\pm$ 0.08  \\\\ \n$P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$)               &   STW     &  -1.36 $\\pm$ 0.29 &  -3.37 $\\pm$ 0.10  \\\\\n                          & SRASS&   -1.39 $\\pm$ 0.37 &  -3.17 $\\pm$ 0.13  \\\\      \n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:ste_sub_srassLX\/Lbol}\n\\end{table}\n\n\n\n\n\\par\n\n\n\\subsection{Spectral parameters analysis}\\label{sec:Spectral_parameters}\n\n\\par\n\nThe Spec-EWX sample has stellar spectral parameters measured from LAMOST spectra. First, we sought to address whether they indeed constitute a representative sample of the STW and SRASS.  \nWe performed K-S tests on the distributions of period between Spec-EWX and these two samples; the probability $P_{\\rm K-S}$ are 93.2\\% and 22.1\\%, respectively, which suggest they follow the same distributions. As shown in Figure \\ref{fig:LAMOST_P_Lx}, we also provide the linear fitting for the period versus log$L_{\\textrm{X}}$ relation (black points) for Spec-EWX sample with $P<0.44$~days. The best-fit slope and intercept values are 2.53(46) and 29.31(16), respectively, which is consistent within $\\sim 1.5\\sigma$ error of those values in Equations (\\ref{equ:STW_LX_P}) and (\\ref{equ:SRASS_LX_P}).\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 15cm]{Fig4.png}\n\\caption{Orbital period vs. log$L_{\\textrm{X}}$ of Spec-EWX sample (black points) with LAMOST stellar parameters. Solid black line shows the least-squares linear fitting for the data with a period less than 0.44 days. Light black dashed lines represents 95\\% uncertainty range. The size of the light blue circles represents the binary temperature ($\\sim4500$--6600~K). }\n\\label{fig:LAMOST_P_Lx}\n\\end{figure*}\n\n\\par\n\nThe statistical distributions of binary temperature $\\log T_{\\rm eff}$, metallicity  ([Fe\/H]), and the surface gravity $\\log g$ are shown in the upper panels of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-1, 2-1, 3-1). For all of the Spec-EWX sample, the effective temperature, $T_{\\rm eff}$, ranges from $\\sim4450$~K (K5) to $\\sim6600$~K (F2), while the majority are distributed between 5150~K and 6000~K (corresponding to $\\sim$ K0V to G0V spectral type), indicating that they are solar-like main sequence stars. The metallicity [Fe\/H] ranges from $\\sim-1.00$ to $\\sim0.6$ dex, with a mean value of -0.05 dex. The distribution peaks at $-0.05$ dex which is slightly lower than that of Sun, that is, [Fe\/H] = 0. The surface gravity $\\log g$ ranges from $\\sim3.4$ to $\\sim4.7$, with the mean value of 4.22 and the distribution peak at $4.15$.  We also plot the normalized distributions of $\\log T_{\\rm eff}$, [Fe\/H], and $\\log g$ of EW-type binaries collected by \\citet{2017RAA....17...87Q} in panel (1-1), (2-1) and (3-1) of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, respectively. K-S tests between our Spec-EWX  and their sample were performed for these three parameters. The values of $P_{\\textrm{K-S}}$ are 2.72 $\\times 10^{-4}$, 7.25$\\times 10^{-10}$, and 4.63$\\times 10^{-5}$, respectively, suggesting that for each of the three parameters, the statistical distributions of these two samples differ at the $> 3\\sigma$ level. \n\n\\par\n\n\\begin{table*}[!htpb]\n   \\caption{Best-fit parameters for log$T$, [Fe\/H] and $\\log g$ with log$L_{\\textrm{X}}$ and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, and corresponding correlation probabilities from the Kendall$^{\\prime}$s $\\tau$ tests}\n   \\begin{center}\n   \\begin{tabular}{llrrrrr}\\hline \\hline\nRelations &Parameters&  Slope & Intercept &  $\\tau$ & $1-P_{\\tau} $ \\\\\n\\hline \\\\\nlog$L_{\\textrm{X}}$& log$T$ &     3.78 $\\pm$ 0.65 &  16.01 $\\pm$ 2.44  &   0.320  &   $> 99.99\\%$  \\\\\n                            & $[\\textrm{Fe}\/\\textrm{H}]$&    0.26 $\\pm$ 0.09 &  30.19 $\\pm$ 0.02  &      0.188   & 99.89\\%    \\\\ \n                            & $\\log g$&    $-$0.49 $\\pm$ 0.17 &  32.26 $\\pm$ 0.72     &    $-$0.198 & 99.94\\%      \\\\\n                            \nlog~($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$)& log$T$ &     $-$2.70 $\\pm$ 0.67 &  6.41 $\\pm$ 2.50  &  $-$0.232  &   $> 99.99\\%$  \\\\\n                            & $[\\textrm{Fe}\/\\textrm{H}]$&    0.06 $\\pm$ 0.09 &  $-$3.73 $\\pm$ 0.02  &      0.047  & 59.09\\%    \\\\ \n                            & $\\log g$&    0.23 $\\pm$ 0.17 &  $-$4.69 $\\pm$ 0.72  &     0.083 & 84.79\\%      \\\\                           \n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:leastsquare_parameters}\n\\end{table*}\n\n\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 18cm]{Fig5.png}\n\\caption{Normalized distributions of three spectral parameters ($\\log T_{\\rm eff}$, [Fe\/H] , and $\\log g$) of EWXs, and their relationships with log$L_{\\textrm{X}}$ and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$). Upper panels: Normalized distributions of the binary temperature (1-1), metallicity [Fe\/H] (2-1), and surface gravity $\\log g$ (3-1). The red and black broken lines represent the EW-type binaries collected by \\citet{2017RAA....17...87Q} and our Spec-EWX sample, respectively. \nMiddle panels: Relationships between  log$L_{\\textrm{X}}$ and the binary temperature (1-2), metallicity [Fe\/H] (2-2), and surface gravity $\\log g$ (3-2). \nLower panels: Relationships between log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) and the binary temperature (1-3), metallicity [Fe\/H] (2-3), and surface gravity $\\log g$ (3-3).\nSolid black lines are the least-squares linear fits and the light black dashed lines represent the 95\\% uncertainty ranges.}\n\\label{fig:LAMOST_logT_logg_FeH_Lx_N}\n\\end{figure*}\n\n\\par\n\nThe component stars of an EW binary, especially the more massive primary, still maintain the mass-temperature relation of single main sequence stars \\citep{2005ApJ...629.1055Y, 2018MNRAS.473.5043E}. Therefore, it is reasonable to use spectral temperature to infer the mass of the primary star of an EW. In terms of temperature, as shown in Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-1), both the EWXs and EWs are mainly distributed ($\\sim$67\\% vs. 48\\%) between $\\sim5150$~K (corresponding to $\\sim$0.79 M$_{\\odot}$ for a main sequence star) and $\\sim6000$~K ($\\sim$1.1 M$_{\\odot}$). In particular, the ratio of EWXs with spectral temperatures below 6000~K is 82.7\\%, which may indicate that a large fraction of EWXs have primary stars with masses lower than 1.1~M$_{\\odot}$. However, EWXs have a higher percentage of objects in this range ($T<6000$~K) than the full EW-binary population has, while the opposite is true for the case of $T>6000$~K. Meanwhile, only a few EWXs have temperatures higher than 6500~K, which indicates that few EWXs have primary stars heavier than $\\sim1.4$ M$_{\\odot}$.\n\n\\par\n\nIn Figure~\\ref{fig:LAMOST_P_Lx}, the size of the light blue circles around each data point represents the effective temperature of Spec-EWX sample from LAMOST spectroscopy. This plot suggests the positive correlations between X-ray luminosity and both period and effective temperature. However, some sources with a period from $\\sim0.40$ to 0.44 days have high temperature but low X-ray luminosity. This can be explained by the general trend of X-ray luminosity exhibited by single main sequence stars. With the ascending temperatures from K to G type stars, the typical X-ray luminosity increases gradually from $\\sim$10$^{29}$ erg s$^{-1}$ to $\\sim$10$^{30}$ erg s$^{-1}$. However, this trend does not continue to F-type stars as their typical X-ray luminosity ($\\sim$10$^{30}$ erg s$^{-1}$) is comparable to that of G-type stars \\citep[see][the left panel of Figure 6]{2020ApJ...902..114W}.\n\n\\par\n\nAs shown in Figure \\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (2-1), the peak value of Spec-EWX metallicity distribution is at $\\sim-0.05$. About 77\\% of the Spec-EWX objects have [Fe\/H] $> -0.25$. In contrast, for general EWs (represented by the red line), the distribution peak is at  $\\sim-0.25$; the population is distributed almost evenly around this peak value (53\\% vs. 47\\%). This indicates that EWs with higher metallicity are more likely to produce X-ray emission. \n\n\\par\n\nIn Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (3-1), we can see that the surface gravity of EWXs and of EWs shows nearly the same distribution peaks ($\\sim$4.15). At the interval of $4.2\\leq \\log g \\leq 4.5$, the percentage of EWXs is slightly higher than that of EWs. \n\n\\par\n\n\nThe middle and bottom rows of Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} demonstrates the correlations between the above three stellar parameters and X-ray luminosity log$L_{\\textrm{X}}$, and X-ray activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of EWXs, respectively. Linear regressions and Kendall$^{\\prime}$s $\\tau$ tests were performed and the results are listed in Table~\\ref{table:leastsquare_parameters}. The binary temperature $\\log T_{\\rm eff}$ has a strong positive correlation with log$L_{\\textrm{X}}$ at $>5\\sigma$. The metallicity is also positively correlated with log$L_{\\textrm{X}}$, while the surface gravity $\\log g$ is negatively correlated with log$L_{\\textrm{X}}$. The latter two correlations are somewhat weaker, but still at the $ >3\\sigma$ level. Regarding the X-ray activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$), it has a negative correlation with the temperature $\\log T_{\\rm eff}$ ($>5\\sigma$), while being almost independent of [Fe\/H] and $\\log g$. \n\n\\par\n\n\n\\subsection{Binary components parameters analysis}\\label{sec:binaryparameters}\n\n\\par\n\nFor the SRASS sample, we collected 23 sources (hereafter, sub-SRASS) that feature measurements of their absolute stellar parameters for each of the two individual components, that is: mass ($M_{1}$ and $M_{2}$), radius ($R_{1}$ and $R_{2}$), and temperature ($T_{1}$ and $T_{2}$), as well as the orbital inclination ($i$) from the literature. Here, the more massive component of the binary is defined as Star$_{1}$, and its physical parameters have the subscript of 1, while the parameters with the subscript of 2 represent the less massive component Star$_{2}$. These parameters are listed in Table~\\ref{table:SRASSsamples}. The K-S tests between the sub-SRASS and the whole SRASS samples are performed upon the orbital period and log$L_{\\textrm{X}}$. The null-hypothesis probabilities, $P_{\\rm K-S}$,  are 79.5\\% and 37.9\\%, respectively, suggesting that they follow the same distributions. For the STW sample, there are no such measurements for the absolute stellar parameters for each of the binary components. \n\n\\par\n\n \\begin{table*}[!htpb]\n   \\caption{Absolute stellar parameters for each component of the EWXs in sub-SRASS}\n   \\begin{center}\n   \\begin{tabular}{lcccccccccccc}\\hline \\hline\n\nName &  log$L_{\\textrm{X}}$ & Period & $M_{1}$ & $M_{2}$  &  $R_{1}$ & $R_{2}$  & $T_{1}$  & $T_{2}$ & $i$ &  Subtype &  Reference \\\\\n & (erg s$^{-1}$) & (days) & (M$_{\\odot}$) & (M$_{\\odot}$) &  (R$_{\\odot}$) & (R$_{\\odot}$) & (K) & (K) & ($^{\\circ}$) & &  \\\\ \\hline\nV523 Cas & 29.707 & 0.2337 & 0.740 & 0.380 & 0.770 & 0.590 & 5104 & 5076 & 84.36  &  A & (1) \\\\\nBX Peg   & 29.880 & 0.2804 & 1.020 & 0.380 & 0.966 & 0.623 & 5872 & 5300 & 87.00  &  A & (2) \\\\\nSX Crv   & 30.376 & 0.3166 & 1.246 & 0.098 & 1.347 & 0.409 & 6340 & 6160 & 61.21  &  A & (3) \\\\\nV508 Oph & 29.983 & 0.3448 & 1.010 & 0.520 & 1.060 & 0.800 & 5980 & 5893 & 83.78  &  A & (4) \\\\\nGR Vir   & 30.081 & 0.3470 & 1.370 & 0.170 & 1.420 & 0.610 & 6300 & 6163 & 83.36  &  A & (5) \\\\\nAH Cnc   & 30.488 & 0.3604 & 1.188 & 0.185 & 1.332 & 0.592 & 6300 & 6151 & 83.11  &  A & (6) \\\\\nU Peg    & 30.161 & 0.3748 & 1.149 & 0.379 & 1.224 & 0.744 & 5860 & 5785 & 77.51  &  A & (7) \\\\\nV566 Oph & 29.859 & 0.4096 & 1.500 & 0.380 & 1.490 & 0.810 & 6456 & 6247 & 80.40  &  A & (8) \\\\\nAQ Psc   & 30.470 & 0.4756 & 1.260 & 0.280 & 1.220 & 1.180 & 6445 & 5946 & 68.90  &  A & (9) \\\\\nSW Lac   & 30.453 & 0.3207 & 1.207 & 0.991 & 1.090 & 1.000 & 5371 & 5529 & 80.95  &  W & (10)\\\\\nRW Com   & 29.638 & 0.2373 & 0.800 & 0.380 & 0.770 & 0.540 & 4720 & 4900 & 74.90  &  W & (11)\\\\\nRW Dor   & 29.919 & 0.2854 & 0.820 & 0.520 & 0.881 & 0.703 & 5238 & 5560 & 76.90  &  W & (12)\\\\\nBW Dra   & 30.198 & 0.2923 & 0.920 & 0.260 & 0.980 & 0.550 & 5980 & 6164 & 74.42  &  W & (13)\\\\\nTW Cet   & 30.241 & 0.3117 & 1.060 & 0.610 & 0.990 & 0.760 & 5450 & 5600 & 83.70  &  W & (14)\\\\\nFG Hya   & 30.139 & 0.3278 & 1.444 & 0.161 & 1.405 & 0.591 & 5900 & 6012 & 82.25  &  W & (15)\\\\\nV781 Tau & 29.993 & 0.3449 & 1.060 & 0.430 & 1.130 & 0.760 & 5536 & 6000 & 65.89  &  W & (16)\\\\\nAC Boo   & 30.060 & 0.3524 & 1.200 & 0.360 & 1.190 & 0.690 & 6241 & 6250 & 86.03  &  W & (17)\\\\\nV752 Cen & 30.285 & 0.3702 & 1.310 & 0.390 & 1.300 & 0.770 & 6014 & 6138 & 82.07  &  W & (18)\\\\\nRT LMi   & 29.662 & 0.3749 & 1.290 & 0.490 & 1.280 & 0.840 & 6400 & 6513 & 84.10  &  W & (19)\\\\\nTX Cnc   & 30.182 & 0.3829 & 1.350 & 0.610 & 1.270 & 0.890 & 6250 & 6537 & 62.10  &  W & (20)\\\\\nUV Lyn   & 30.083 & 0.4150 & 1.430 & 0.550 & 1.400 & 0.920 & 5736 & 5960 & 66.13  &  W & (16)\\\\\nUX Eri   & 30.993 & 0.4453 & 1.450 & 0.540 & 1.450 & 0.910 & 6046 & 6100 & 76.89  &  W & (21)\\\\\nV502 Oph & 30.263 & 0.4534 & 1.370 & 0.460 & 1.510 & 0.940 & 5900 & 6140 & 76.40  &  W & (22)\\\\\n\n\\hline\\noalign{\\smallskip}\n  \\end{tabular}\n  \\end{center}\n  \\label{table:SRASSsamples}\n    \\begin{tablenotes}\n     \\footnotesize\n    \\item[1] Ref.(1):\\citet{2016NewA...44...78M}, (2):\\citet {2015NewA...41...17L}, (3): \\citet {2004AcA....54..299Z}, (4): \\citet{2015AJ....149...62X} , (5): \\citet{2004AJ....128.2430Q} , (6): \\citet{2016RAA....16..157P} , (7): \\citet{2002CoSka..32...79P} , (8): \\citet{2018Ap&SS.363...34S} , (9): \\citet{2020MNRAS.491.6065Z} ,  (10): \\citet{2014arXiv1402.2929E} , (11): \\citet{2011A&A...525A..66D} , (12): \\citet{2019PASJ...71...34S} , (13): \\citet{1986AJ.....92..666K} , (14): \\citet{1982A&AS...47..211R} , (15): \\citet{2005MNRAS.356..765Q} , (16): \\citet{2020ApJ...901..169L} , (17): \\citet{2010IBVS.5951....1N}, (18): \\citet{2019MNRAS.489.4760Z} , (19): \\citet{2008PASJ...60...77Q} , (20): \\citet{1987PNAS...84..610C},  (21): \\citet{2007AJ....134.1769Q} , (22): \\citet{2016PASJ...68..102X}\n    \\end{tablenotes}\n\\end{table*}\n\n\\par\n\nWe investigate the possible relationships between log$L_{\\textrm{X}}$ and the temperature and mass of the different components in Figures~\\ref{fig:DIS_T_Lx} and \\ref{fig:DIS_M_Lx}, respectively. The temperatures of the primary (more massive), secondary, and the binary (average value of the two components) are plotted in Figure~\\ref{fig:DIS_T_Lx} (1), (2) and (3), respectively. The primary and secondary components, as well as the W\/A-subtypes, are indicated in the subscripts of the labels in each panel. The black dashed lines are located at $P=0.44$ days, the break point determined in Section~\\ref{sec:X-ray_P}. For all three panels, the temperature increases along with the period when $P < 0.44$ days. The similarity in this tendency is expected since the temperature difference between the two components of an EW binary is small, about several hundred kelvins. The relation between the average temperature of two components and X-ray luminosity is also plotted in panel (4). We also plot Spec-EWX sample and its best-fit line (see Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}, panel 1-2) in light blue points and gray line in the background, respectively. In this panel, the sub-SRASS objects are distributed in the region generally similar to where the Spec-EWX sample occupies. \n\n\\par\n\nSimilarly, we plot the masses of the primary and secondary from sub-SRASS in Figure~\\ref{fig:DIS_M_Lx} (1) and (2), respectively. The relation between log$L_{\\textrm{X}}$ and the masses of primary and secondary are plotted in panels (3) and (4). In the last two panels, the black dots with error bars represent the typical saturated X-ray luminosity of the main sequence stars within the mass range (0.22 to 1.29 M$_{\\odot}$). Each horizontal error bar indicates that stars with that mass range reach saturation when $P < 1$ day at the corresponding X-ray luminosity  \\citep[see][Table~3]{2003A&A...397..147P}. \nIn addition, for those EWXs with the mass of the primary star between $\\sim$1.29 and $\\sim$ 1.7 M$_{\\odot}$ (corresponding to F-type), we adopted a saturation X-ray luminosity value of $\\sim$3.2 $\\times$ 10$^{29}$ to $\\sim$3.2 $\\times$ 10$^{30}$ erg s$^{-1}$ \\citep{2019A&A...628A..41P, 2020ApJ...902..114W}. \n\n\n\\par\n\nThe mass ranges of the primary and secondary stars are  $\\sim$0.7 to $\\sim$1.5 M$_{\\odot}$ and $\\sim$ 0.1 to $\\sim$1 M$_{\\odot}$, respectively. It is clearly shown in panels (1) that the mass of the primary star is positively correlated with the period below $P=0.44$ days, while the mass of the secondary shows a much weaker correlation (panel 2). For all the relationships discussed in this subsection, it is clear that there is no significant difference between the behaviors of W-subtype and A-subtype systems.\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 14cm]{Fig6.png}\n\\caption{Plots of the period vs. log$T$ for primary stars (1), secondary stars (2), and the binary systems (3) of the W-subtype (`+' symbols) and A-subtype (`x' symbols) from sub-SRASS. \n (4): Plot of log T vs. log$L_{\\textrm{X}}$. The dashed lines are at $P=0.44$ days. The light blue points and gray line in the background of panel (4) are the Spec-EWX sample points and its fitting line from Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N} (1-2).}\n\\label{fig:DIS_T_Lx}\n\\end{figure*}\n\n\\par\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width = 14cm]{Fig7.png}\n\\caption{Plots of the period vs. stellar mass for primary (1) and secondary (2) from sub-SRASS, and the mass of primary (3) and secondary (4) vs. log$L_{\\textrm{X}}$. The black dashed line in panels (1) \\& (2) is at 0.44 days. The small black points with dashed line in panels (3) \\& (4) indicate the X-ray luminosity of the main sequence star at saturation in the mass range (0.22 to 1.7 M$_{\\odot}$). In panel (1), the blue contours contain the data points from the 10,000 simulated samples (see text) with masses corresponding to the spectral temperatures from Spec-EWX (extra adding $0$--200~K random errors of uniform distribution for each object) against the period, while the gray line and  light blue dashed lines are the best-fit relation of these points with periods of $\\leq$~0.44~days and its 95\\% uncertainty range. The blue and red histograms (with 0.02-day steps) with axis on the right represent the period distributions of the primary stars with masses less than 1.1~M$_{\\odot}$ and greater than 1.1~M$_{\\odot}$, respectively (the average value of 10,000 simulations). }\n\\label{fig:DIS_M_Lx}\n\\end{figure*}\n\n\\par\n\n\\section{Discussion}\\label{sec:Discu}\n\n\\subsection{Sample completeness}\\label{sec:completeness}\n\nIn this section, we discuss the completeness of our EWX sample, especially with regard to how the X-ray nondetections would affect our results. For our reliably classified EW binaries without X-ray detections, most of the X-ray luminosity upper limits have been obtained from the XMM Slew Survey or the RASS (see the gray symbols in the upper two panels of Figure~\\ref{fig:Lx_P}). These upper limits are generally substantially higher than the X-ray luminosity range of detected EWXs because of the low sensitivity of the two surveys. For the correlation analyses we  carried out (presented in Section~\\ref{sec:data_analysis}), adding these upper limits will not provide further constraints.\n\n\n\nFrom the pointed XMM-Newton observations that were utilized in the 4XMM-DR9 catalog, we obtained X-ray luminosity upper limits for 39 EWXs. These objects and the 255 EWXs detected by XMM-Newton in Group A are all from the 14096 EW binaries that we selected using the same criteria as those of the Group A (after applying the cuts on period, distance, and parallax precision; see Section~\\ref{sec:sec:upper limits}). Therefore, the covering fraction (the number of EWXs over the number of EWs) of pointed XMM-Newton observations for EWXs in the whole sky region is estimated to be $2.09\\%$, which is similar to the $2.85\\%$ sky coverage of XMM-Newton pointed observations (1152 deg$^{2}$).\n\n\nWe repeated the correlation analysis between the period and X-ray luminosity by taking those 39 upper limits into account. We employed the software package ASURV Rev 1.2 \\citep{1990BAAS...22..917I, 1992ASPC...25..245L}, which implements the methodology proposed by \\citet{1986ApJ...306..490I} and contains the Expectation-Maximization regression algorithm for astronomical data with detection limits, that is, censored data. The best-fit correlations of $P$-$\\rm log L_{\\rm X}$, $P$-$\\rm log L_{\\rm bol}$, and $P$-$L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ for the objects with periods of less than 0.44~days are $\\log L_{\\rm X} = 2.84(27) \\times P + 29.07(9)$, $\\log L_{\\rm bol} = 4.34(31) \\times P + 32.38(10),$ and $\\log L_{\\textrm{X}}\/L_{\\textrm{bol}}= -1.42(29) \\times P -3.37(10)$, which are values that are highly consistent with previous results on X-ray detections only shown in Equation~(\\ref{equ:STW_LX_P}) and Table~\\ref{table:ste_sub_srassLX\/Lbol}. This indicates that whether or not including the upper limits from pointed XMM-Newton observations would not affect the correlation studies of EWXs. We do not include these upper limits in the discussions in Sections~\\ref{sec:magnetic}--\\ref{sec:primary_Xemission}.\n\nThere are four objects having tight X-ray upper limits from the pointed XMM-Newton observations that are well below the best-fit period-luminosity relation of EWXs. Labeled as bigger magenta `$tri\\_down$' symbols in Figure~\\ref{fig:Lx_P}~(1) from left to right, these objects are ASASSN-V J152949.56-445113.1, J023156.36+605519.0,  J074108.82+251600.7, and J071826.31-243845.6. \nThe exposure times of their corresponding observations are 14ks, 7ks, 9ks, and 8ks, respectively, suggesting that they do not suffer from insufficient exposure depth. They do indeed have lower X-ray luminosity than typical EWXs. The estimated temperatures from Gaia DR2 for these four objects (4198K, 4864K, 6655K, and 6948K) may provide clues to the reason. Based on the temperature and period information from Spec-EWX sample in Table~\\ref{table:Spectra_EWXs}, we calculate the average temperature of the six objects with closest periods to each of the four sources. The mean temperature values are 5500 $\\pm$ 77K, 5831 $\\pm$ 84K, 6113 $\\pm$ 102K, and 6098 $\\pm$ 61K, respectively. The temperatures of the four sources are $> 3\\sigma$ below or above their respective mean temperature. This may indicate that the temperature deviation of EW-type binaries may affect their X-ray emission. As discussed in Section~\\ref{subsubsec:Lx_Lbol_P} (below), EW-type binaries with lower temperature may generate less overall radiation across the full spectrum, while stars hotter than typical would also produce weaker X-ray radiation due to their thinner convection zone. Deeper X-ray observations on these sources are needed to replace the current upper limits with definite X-ray luminosity level and, thus, to facilitate further investigations on their lower X-ray emission level compared to typical EW binaries.\n\n\n\n\\subsection{$L_{\\rm X}$ and $L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$ of whole EWX system}\\label{sec:magnetic}\n\n\\subsubsection{Linear relationships with $P$}\\label{subsubsec:Lx_Lbol_P}\n\n\\par\n\nAs shown in Figures~\\ref{fig:Lx_P} and \\ref{fig:Lx_Lbol_STW_sub_SRASS}, we obtained clear correlations ($>5\\sigma$) of $P$-log$L_{\\textrm{X}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) for EWXs with $P<0.44$~days using the high-quality X-ray and optical data of 4XMM-DR9 and Gaia DR2, and we provide the first linear parametrization of these relationships in this work. The same correlations for the SRASS sample are generally consistent ($\\lower 2pt \\hbox{$\\, \\buildrel {\\scriptstyle <}\\over {\\scriptstyle \\sim}\\,$} 1\\sigma$) with those of the STW sample (see Equations~\\ref{equ:STW_LX_P} \\& \\ref{equ:SRASS_LX_P} and Table~\\ref{table:ste_sub_srassLX\/Lbol}). Since the STW has a larger sample size and higher-quality X-ray data from XMM-Newton, the  analyses presented in the following are mainly based on the correlation analysis results of the STW objects with periods of less than 0.44~days. As the number of EWXs with $P >0.44$~days is very small, only a qualitative description can be given, which is that the above relationships may appear to remain flat or weakly negative. \n\n\\par\n\nBased on the best-fit linear relationships we have obtained, the period can be treated as a good predictor of the X-ray luminosity (Section \\ref{sec:X-ray_P}) and activity level for EWXs. The slope of the $P$-$\\log(L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation is $-1.36\\pm0.29$, which is fully consistent with the difference between the slopes of the $P$-$\\log L_{\\textrm{X}}$ and $P$-$\\log L_{\\textrm{bol}}$ relationships ($2.81\\pm0.27$ and $4.19\\pm0.21,$ respectively). When the period of an EWX is shorter than 0.44~days, both its X-ray luminosity and bolometric luminosity rise with the increasing period, but the growth rate of former is slower than the latter, making the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation display a downward trend. The linear relationships of $P$-log$L_{\\textrm{X}}$, $P$-log$L_{\\textrm{bol}}$, and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$), as listed in Equations~(\\ref{equ:STW_LX_P})--(\\ref{equ:SRASS_LX_logP}), and Table~\\ref{table:ste_sub_srassLX\/Lbol}, provide a convenient check of whether a given EWX has X-ray luminosity, bolometric luminosity, and X-ray activity level that are consistent with the typical population.\n\n\n\\par\n\nWith the period increasing from 0.2 to 0.44 days, the average X-ray luminosity, $L_{\\rm X}$, of EWXs increases from 4.60$\\times 10^{29}$ erg s$^{-1}$ to 2.17$\\times 10^{30}$ erg s$^{-1}$, while the average bolometric luminosity, $L_{\\rm bol}$, rises from 2.03$\\times 10^{33}$ erg s$^{-1}$ to 2.04$\\times 10^{34}$ erg s$^{-1}$. This makes that the average ratio of the X-ray radiation flux to the total radiation flux $L_{\\rm X}\/L_{\\rm bol}$ decreases from $2.2 \\times 10^{-4}$ to $1.0 \\times 10^{-4}$. If we assume that the X-ray emission region is proportional to the total surface region of the EWXs, one reason for this phenomenon may be that the EWXs with short periods have higher X-ray activity level due to their thicker convective zone on its surface. Given the parallel correlation of increasing effective temperature with increasing period (i.e., $P$-$\\log L_{\\textrm{bol}}$), as the temperature rises, the convection zone on the surface will be thinner to generate less X-ray per unit area, while convection and rotation are the essential ingredients to power the magnetic dynamos \\citep{1966ApJ...144..695W, 1967ApJ...150..551K, 2003A&A...397..147P}. Therefore, the X-ray emitting area of EWXs with shorter periods is smaller than that of EWXs with longer period, which makes the total X-ray luminosity increase along with the period. However, the convection zone of EWXs with short periods is thicker, providing higher X-ray activity level than that of longer-period EWXs. \n\n\n\n\\par\n\n\n\\subsubsection{Plateau of saturation}\\label{subsubsec:Lx_Lbol_platreau}\n\n\\par\n\nAs shown in Figure~\\ref{fig:LAMOST_logT_logg_FeH_Lx_N}~(1-3), for our Spec-EWX sample, the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) shows a monotonically decreasing trend with the increasing temperature, that is, the higher the temperature, the lower the X-ray activity level. This finding supports the magnetic dynamo theory we discuss in Section~\\ref{subsubsec:Lx_Lbol_P}. However, based on the $B-V$ color index, \\citet{2001A&A...370..157S} divided EWXs into cool ($>$ 0.6) and hotter ($<$ 0.6) stars, and concluded that the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) (also called normalized X-ray flux) of cool variables reaches a plateau while that of hotter objects decreases with the decreasing color index.\nThis means that at $B-V=0.6$ ($T_{\\textrm{eff}}~\\approx~5880$~K and log$T_{\\textrm{eff}}\\approx~3.77$), there is a \"break\" in the log$T_{\\textrm{eff}}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation.\\footnote{One should note that the $B-V$ color index corresponds almost linearly with temperatures between 0.30 and 1.15 ($T=$ 7300~K and 4410~K, respectively; \\citealt{2000asqu.book.....C}).} We suggest that the difference between our results and that of \\citet{2001A&A...370..157S} is due to the data completeness on the cool end. \n\\citet{2001A&A...370..157S} stressed that it was implausible to determine the positive correlation between the X-ray activity level and color index with their available data  \\citep[see][Section 3.4]{2001A&A...370..157S}. \n\n\n\\par\n\nSimilarly, the STW objects also show a negative $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) correlation, namely,  objects with shorter periods generally have higher X-ray activity level, as shown in Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} (2). The nature of this similarity is that EWXs has a strong period-temperature (or bolometric luminosity) correlation. \\citet{2001A&A...370..157S} invoked a surface horizontal flow to explain the X-ray emission level of EWXs.\nAs the period gradually reaches $\\sim$0.2~days, the activity level of EWXs gradually approaches the state of saturation limit (log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}})=-3$), but there is no saturation plateau similar to that of ultra fast rotating stars (UFRs). We further discuss the interpretation of this relation in the last paragraph of Section~\\ref{sec:primary_Xemission}.\n\n\\par\n\nAt $P<$0.44~days, the monotonously decreasing trend of the X-ray activity level (i.e., no plateau) may indicate that the EWXs are all in the same state (i.e., regarding whether they are in X-ray saturation or supersaturation). In the case of a mixture of multiple states, it is likely that the correlation will exhibit a changing slope or no correlation exists at all. For late-type main-sequence stars, the X-ray radiation reaches saturation when the rotation period is less than one day \\citep{1984A&A...133..117V, 1987ApJ...321..958V, 1993ApJ...410..387F, 2003A&A...397..147P}. For EWs, the orbital period is synchronized with the rotational period of their individual components. These EWX objects have a period far less than one day, which suggests that their X-ray emission is likely to reach saturation as well. \n\n\\par\n\n\n\\subsection{Binary components versus X-ray emission}\\label{sec:primary-dominating}\n\n\\par\n\nIn this section, we compare the X-ray luminosity of the EWXs with that of the main sequence stars for given mass ranges when they reach saturation, as shown by the black points with error bars in Figure~\\ref{fig:DIS_M_Lx} (3) and (4). For the main sequence stars, the saturated X-ray luminosity gradually increases from $\\sim$ $1.2\\times10^{29}$ erg s$^{-1}$ to $\\sim$4.0 $\\times$ 10$^{30}$ erg s$^{-1}$ with increasing stellar mass from 0.6 to 1.1 M$_{\\odot}$.  However, as the mass of the stars continues to grow (in the range of 1.1 to 1.7 M$_{\\odot}$), the saturated X-ray luminosity remains at $\\sim10^{30}$ erg s$^{-1}$. The panel (3) demonstrates that in the range of $\\sim$0.6 to $\\sim$1.5 M$_{\\odot}$, the X-ray luminosity of an EWX, whether for A-subtype or W-subtype, is generally consistent with the saturated X-ray luminosity of a single main sequence star with the same mass value as that of the primary component of this EWX, but not the mass of the secondary (panel 4). More generally, we further infer that EWXs may inherit the changing trends of the X-ray luminosities of the single stars with different masses. These may be the results of a combination of the primary star dominating the X-ray radiation of the binary system \\citep{2004A&A...426.1035G, 2006ApJ...650.1119H} and the X-ray emission being saturated for the EWXs with $P<$0.44~days. \n\n\n\n\\par\n\nGiven that the X-ray emissions of single stars and EWXs are all determined by the coverage area of the X-ray emitting regions on their surface \\citep{2001A&A...370..157S}, the consistent X-ray luminosities of EWXs and single stars we mention above may imply that the coverage area of the EWXs is close to that of the single star with the mass of the primary component. In this scenario, it can be understood that the different log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) distributions exhibited by UFRs and EWXs with the same period \\citep{2001A&A...370..157S} is mainly related to the differences in the bolometric flux of UFRs and EWXs. The bolometric luminosity of an EWX binary is likely greater than that of an UFR (e.g., those from \\citealt{2003A&A...397..147P}) with similar mass to the primary star of that EWX, which will result in lower log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) of EWXs. Meanwhile, the difference between the bolometric luminosity of UFRs and EWXs also likely varies with changing periods (for $P<$~0.44 days). For EWXs, the $P$-$L_{\\textrm{bol}}$ relationship (Figure~\\ref{fig:Lx_Lbol_STW_sub_SRASS} panel 1) is essentially the same relationship between the period and mass (Section~\\ref{sec:primary_Xemission}). Based on this degeneracy, we can find that the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) trend of our EWXs (all with primary mass greater than 0.5M$_\\odot$) is basically similar to the mass-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation of single stars under saturation  in the middle panel of Figure 7 (we note the direction of the mass axis in that figure) in \\citet{2003A&A...397..147P}, which supports our above analyses.\n\n\\par\n\n\n\n\\subsection{Period and mass versus X-ray emission}\\label{sec:primary_Xemission}\n\n\\par\n\nSince the LAMOST DR7 survey used single-star models to fit unresolved binaries, \\citet{2018MNRAS.473.5043E} derived that the LAMOST temperatures derivation is systematically lower than the temperature from the spectrum of the primary star by  $\\lower 2pt \\hbox{$\\, \\buildrel {\\scriptstyle <}\\over {\\scriptstyle \\sim}\\,$} 200$~K. Therefore, we generated 10,000 simulated samples by adding random errors uniformly distributed in [0, 200]~K to the spectral temperature of each Spec-EWX object. Then we derived their corresponding masses \n(linear interpolated using Tables 15.7 and 15.8 of \\citealt{2000asqu.book.....C}) and performed linear fittings between the mass and period of the simulated samples with $P<$~0.44~days. The blue-colored contours in Figure \\ref{fig:DIS_M_Lx} (1) contain the 10,000 simulated samples, while the gray line and light blue dashed lines represent the best-fitting relations between period and mass, and its 95\\% uncertainty range, respectively.\\footnote{This blue bubble structure in the lower right corner is due to the simulation of one binary system ASASSN-V J141756.07+210554.7, which has a period of 0.61 days but a spectral temperature of only 5321K.}  As shown in this figure, the primary stars' masses of sub-SRASS are mostly distributed in the range of that of Spec-EWX. We find that $83.7\\pm$2.7~\\% (from statistical distribution and error of the 10,000 simulated samples) objects of EWXs have masses lower than 1.1~M$_{\\odot}$ in this sample. The best-fit gray line (slope is 1.82~$\\pm$~0.02, intercept is 0.35~$\\pm$~0.04) is intercepted by the black dashed line corresponding to $P=0.44$~days at mass value of $1.15\\pm$0.04~M$_{\\odot}$. Combined with the fact that the X-ray luminosity increases with mass until reaching $\\sim$~1.1~M$_{\\odot}$ (see Figure~\\ref{fig:DIS_M_Lx}, panel 3) and the X-ray emission of EWXs is dominated by the primary, this naturally explains the positive correlation between period and $\\log L_{\\rm X}$ until $P=0.44$~days (corresponding to $\\sim1.1$M$_{\\odot}$) that we found in Section~\\ref{sec:X-ray_P}. Considering each finer period interval (see the blue and red histograms with axis on the right in Figure~\\ref{fig:DIS_M_Lx} panel 1), the fraction of objects with a primary mass lower than 1.1~M$_{\\odot}$ generally decreases with increasing period. It is apparent that when the period approaching $0.44$~days, the objects with primary more massive than 1.1~M$_{\\odot}$ start to dominate. Meanwhile, the X-ray luminosity of the primary stars in this mass range ($>1.1$~M$_{\\odot}$) remains constant ($\\sim$ 10$^{30}$ erg s$^{-1}$). Therefore, the $P-\\log L_{\\rm X}$ correlation breaks at $P=0.44$~days.\n\n\\par\n\nCertainly, we cannot rule out the contribution of secondary stars to the binary X-ray luminosity. Long-period binaries also usually have larger secondary stars. This would reinforce the positive correlation between X-ray luminosity and period. On the other hand, this would not affect the correlation break. Based on the mass range of the secondary, their saturated X-ray luminosity is $\\geqslant 1.5$ dex lower than that of the primary. \nThe X-ray emission from the primary remains dominating. It is notable that the sub-SRASS sample has a higher fraction of objects with more massive primary ($>1.1$~M$_{\\odot}$) than that of the Spec-EWX sample, as shown in Figure~\\ref{fig:DIS_M_Lx} (1). This could be a selection effect since obtaining the spectral parameters for each of the binary component would require higher quality optical spectra from more luminous (thus more massive) stars.\n\n\\par\n\n\nAs shown in Figure~\\ref{fig:DIS_M_Lx} (1), the increase of the main mass is strongly correlated with the increase of the period of EWXs. Given the positive correlation between effective temperature and mass, at a certain mass, the temperature of primary star is so high that the convective zone on its surface would not have adequate material to support a normal magnetic dynamo, which is essentially driven by convection and rotation \\citep{1966ApJ...144..695W, 1967ApJ...150..551K, 2003A&A...397..147P}. This critical mass\/temperature relation occurs at the mass of a primary star is $\\geqslant 1.1$~M$_{\\odot}$ (P$\\sim$0.44 days), which may result in decreased X-ray luminosity for binaries with longer rotation periods. Most of EWXs in this work have short periods, which provide a unique sample to test the magnetic dynamo theories under the extreme physical conditions from fast-rotating stars.\n\n\\par\n\nIn combination with sections 4.1, 4.2, and 4.3, we can conclude that the physical nature of the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation for single stars and for EWXs is quite different. For single stars, the log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) increases monotonically as the period decreases, meaning that they are not saturated with X-ray radiation. In a fixed mass range, their rotation periods are distributed over a wide range (e.g., 0.1 to 100 days). Only when the period is below a certain value will there be a plateau of X-ray luminosity and activity level \\citep{2003A&A...397..147P, 2019A&A...628A..41P}, and this plateau represents the saturated X-ray activity level. Single stars with different masses have different saturated $P$-log$L_{\\textrm{X}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) plateaus. For EWXs, as we discuss in Section \\ref{subsubsec:Lx_Lbol_platreau}, although their $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relationship has a slope, they are still likely to be all saturated with X-ray emission. Given the primary star dominating the X-ray radiation of an EWX, the X-ray saturation luminosities of the EWXs are likely to behave as the collection of the X-ray saturation luminosities from single stars with a range of masses and, thus, $Mass$ and log$L_{\\textrm{X}}$ are correlated with each other (Figure~\\ref{fig:DIS_M_Lx} panel 3). Since the period and mass of EWXs are highly degenerated, the $P$-log$L_{\\textrm{X}}$ relation in Figure~\\ref{fig:Lx_P} (1) is similar to the trend of $Mass$-log$L_{\\textrm{X}}$ in Figure~\\ref{fig:DIS_M_Lx} (3), which explains why EWXs have no X-ray luminosity plateau although they are all in saturation. At $P<$0.44~days, $P$-log$L_{\\textrm{X}}$ relation is the part that maintains a nearly monotonous increase. This, combined with the linear relationship of $P$-log$L_{\\textrm{bol}}$, naturally leads to our finding that the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation maintains a monotony with no plateau. Essentially, for EWXs, the (primary) mass is probably the most fundamental physical parameter that determines the X-ray emission level, and the period and $L_{\\textrm{bol}}$ are the observed manifestations of the mass. More EWXs with measured stellar parameters (such as mass, temperature of each component) and more single star X-ray data with wider mass ranges will strongly facilitate the study of X-ray radiation mechanisms of binary and single stars.\n\n\\par\n\n\\section{Summary}\\label{sec:Summary}\n\n\\par\n\nIn this work, by cross-matching the AVSD with 4XMM-Newton DR9 and the 2RXS catalogs, we compile the largest sample to date of X-ray emitting EW-type binaries with periods of less than 1 day and distance less than 1~kpc. We also added in the RASS-selected EWXs from literature \\citep{2001A&A...370..157S,2006AJ....131..633G,2006AJ....131..990C} and updated their distance and X-ray luminosity with the Gaia DR2 parallax data. The full EWX sample in this work contains 376 objects detected by XMM-Newton and 319 objects detected by ROSAT. For EW binaries with X-ray coverage but without any detections, most of the upper limits are from the low-sensitivity RASS and XMM slew survey, which could not provide useful constraints on the X-ray emission. There are 39 objects having tight X-ray upper limits from pointed XMM-Newton observations. Adding these upper limits to the sample does not affect the results of the correlation analysis. Our sample of EWXs is sufficiently complete for studying the X-ray properties of EW binaries (see Section~\\ref{sec:completeness}).\n\nThe statistical distributions of parameters (period, temperature, mass, etc.) of EWXs and their relationships with the X-ray luminosity and X-ray activity level are investigated in Section~\\ref{sec:data_analysis}. We discuss the properties of $\\log L_{\\textrm{X}}$, $\\log L_{\\textrm{bol}}$, and log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) for the whole EWX systems in Section~\\ref{sec:magnetic}. Then, at the level of component stars, we further investigate the possible physical mechanisms of the observed characteristics of EWXs in Sections~\\ref{sec:primary-dominating} and \\ref{sec:primary_Xemission}. Our main results are detailed as follows.\n\n\\begin{enumerate}\n\n\n\\par\n\n\\item  We provide the quantitative formulation of the strong positive correlation ($>5\\sigma$) between the period, $P,$ and X-ray luminosity, log$L_{\\textrm{X}}$, for EWXs (Equations~\\ref{equ:STW_LX_P}--\\ref{equ:SRASS_LX_logP}) at $P<0.44$~days. We also present the best-fit $P$-log$L_{\\textrm{bol}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relations for EWXs. These linear relationships effectively constrain the model of X-ray emission mechanisms for contact binary stars. Based on the fitting results of $P$-$\\log L_{\\textrm{X}}$ and $P$-$\\log L_{\\textrm{bol}}$, we provide the quantitative description of EWX activity levels with different periods, which may relate to the thickness of the convection zone in the magnetic dynamo theories. An EWX with a short period has a smaller but thicker X-ray emitting region than a relative longer period EWX. The relationship between log$T_{\\rm eff}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) shows the higher the temperature, the lower the X-ray activity level, which is the evidence to support magnetic dynamo theories adopted to interpret the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation of EWXs.\n\n\\par\n\n\\item At $P<0.44$~days, neither the $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) nor the log$T_{\\rm eff}$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relationship has a saturation plateau similar to that of single ultra fast rotating stars. The X-ray activity level decreases monotonically with the period, which, combined with the short periods of EWXs, may indicate that EWXs are all in X-ray emission saturated state. \n\n\\par\n\n\\item  We compiled the spectral parameters, including effective temperature, $T_{\\rm eff}$,  metallicity [Fe\/H], and surface gravity $\\log g$, for the Spec-EWX sample using LAMOST spectroscopy. We find that EWXs and the general EW population have different statistical distributions ($>3\\sigma$) of all the above three spectral parameters. In particular, the proportions of EWXs on the low mass (temperature) end is higher than that of EWs. The values of $T_{\\rm eff}$ and [Fe\/H] are positively correlated with the X-ray luminosity while $\\log g$ is anti-correlated with log$L_{\\textrm{X}}$. There is a high confidence ($>5\\sigma$) negative correlation between temperature and activity level log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$).\n\n\\par\n\n\\item Based on the relation between saturated X-ray luminosity and the mass of main sequence stars, we find that the X-ray luminosity of an EWX is generally consistent with the saturated X-ray luminosity of a main sequence star with the same mass as the primary component of the EWX. The X-ray saturation luminosity values of the EWXs are similar to the collection of single stars with different masses when they all reach saturation. \n\n\n\\par\n\n\\item Most of our EWXs with $P < 0.44$~days have primary stars that are less massive than 1.1~M$_{\\odot}$. In this mass range, the X-ray luminosity increases with mass while it remains constant when the primary mass exceeds 1.1~M$_{\\odot}$. As the period increases, the temperature, primary mass and the X-ray luminosity of the binary systems change in concordance. The mass distribution of the primary stars may be the direct reason for the positive $P$-$\\log L_{\\textrm{X}}$ correlation and also contribute to break for this relation at $P\\sim$ 0.44~days. Because there is no plateau in $P$-$\\log L_{\\textrm{X}}$ and the $P$-$\\log L_{\\textrm{bol}}$ remains monotonically increasing for periods less than 0.44 days, we  strictly confirm that there is a decreasing tendency in $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) with no plateau. The degeneracy between the mass and the period of EWXs results in the monotonous relationships of $P$-log$L_{\\textrm{X}}$, $P$-log$L_{\\textrm{bol}}$ and $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$). \n\n\\end{enumerate}\n\nIn conclusion, most, if not all, of the EWXs are in X-ray saturation state. The $P$-log($L_{\\textrm{X}}$\/$L_{\\textrm{bol}}$) relation for EWXs is the manifestation of the X-ray saturation and the degeneracy between mass, period, and temperature. The mass of the primary star is the most fundamental physical parameter determining the X-ray emission properties of EWXs.\n\n\n\n\\begin{acknowledgements}\n\nWe thank the anonymous referee very much for the constructive suggestions, which helped to improve the paper. This work is supported by the National Natural Science Foundation of China (NSFC) under the grant numbers U1938105 (J.L. and J.Wu), U2031209 (A.E.), 11925301 (W.-M.G), 11973002 (M.Y.S), and U1831205 (J.Wang). We acknowledge the data support from Guoshoujing Telescope. Guoshoujing Telescope (the Large Sky Area Multi-Object Fiber Spectroscopic Telescope LAMOST) is a National Major Scientific Project built by the Chinese Academy of Sciences. Funding for the project has been provided by the National Development and Reform Commission. LAMOST is operated and managed by the National Astronomical Observatories, Chinese Academy of Sciences.\nWe also acknowledge the support of X-ray data based on observations obtained with \\textit{XMM-Newton}, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA. This work has made use of data from the European Space Agency (ESA) mission \\textit{Gaia} (https:\/\/www.cosmos.esa.int\/gaia), processed by the \\textit{Gaia} Data Processing and Analysis Consortium. We thank the Las Cumbres Observatory and its staff for its continuing support of the ASAS-SN project. ASAS-SN is supported by the Gordon and Betty Moore Foundation through grant GBMF5490 to the Ohio State University, and NSF grants AST-1515927 and AST-1908570. Development of ASAS-SN has been supported by NSF grant AST-0908816, the Mt. Cuba Astronomical Foundation, the Center for Cosmology and AstroParticle Physics at the Ohio State University, the Chinese Academy of Sciences South America Center for Astronomy (CAS-SACA), the Villum Foundation, and George Skestos.\n\n\n\nSoftware: Matplotlib \\citep{2007CSE.....9...90H}, Numpy (https:\/\/numpy.org\/), Scipy (http:\/\/www.scipy.org), Seaborn \\citep{2021JOSS....6.3021W}.\n\n\\end{acknowledgements}\n\n\\par\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nReconstruction of non-linear dynamic processes based on sparse observations is an important and difficult problem. The problem traditionally requires knowledge of the governing equations or processes to be able to generalize from the the sparse observations to a wider area around, in-between and beyond the measurements. Alternatively it is possible to learn the underlying processes or equations based on data itself, so called data driven methods. In geophysics and environmental monitoring measurements is often only available at sparse locations. For instance, within the field of meteorology, atmospheric pressures, temperatures and wind are only measured at limited number of stations. To produce accurate and general weather predictions, requires methods that both forecast in the future, but also reconstruct where no data is available. Within oceanography one faces the same problem, that in-situ information about the ocean dynamics is only available at sparse locations such as buoys or sub-sea sensors. \\\\\n\nBoth the weather and ocean currents can be approximated with models that are governed by physical laws, e.g. the Navier-Stokes Equation. However, to get accurate reliable reconstructions and forecasts it is of crucial importance to incorporate observations. \\\\ \n\nReconstruction and inference based on sparse observations is important in many applications both in engineering and physical science \\cite{brunton2015closed, kong2018application, bolton2019applications, venturi2004gappy,callaham2018robust, manohar2018data}. Bolton et. al. \\cite{bolton2019applications}  used convolutional neural networks to hindcast ocean models, and in \\cite{yeo2019data} K. Yeo reconstructs time series of nonlinear dynamics from sparse observation.  Oikonomo et. al. \\cite{oikonomou2018novel} proposed a method for filling data gaps in groundwater level observations and Kong. et. al \\cite{kong2018application} used reconstruction techniques to modeling the characteristics of cartridge valves.  \\\\\n\nThe above mentioned applications are just some of the many examples of reconstruction of a dynamic process based on limited information. Here we focus on reconstruction of flow. This problem can be formulated as follows. Let $\\bm w \\in {\\mathbb R}^d,$ $d \\in {\\mathbb N},$ represent a state of the flow, for example velocity, pressure, temperature, etc. Here, we will focus on incompressible unsteady flows and ${\\bm w}=(u,v)\\in {\\mathbb R}^2$ where $u$ and $v$ are the horizontal and vertical velocities, respectively. The velocities ${\\bm w}$ are typically obtained from computational fluid dynamic simulations on a meshed spatial domain $\\mathcal{P}$ at discrete times $\\mathcal{T} = \\{t_1,...,t_K\\} \\subset {\\mathbb R}$. \\\\\n\nLet $ \\mathcal{P}=\\{p_1,...,p_N\\}$ consist of $N$ grid points $p_n,$ $n=1,...,N.$  \nThen the state of the flow $\\bm w$ evaluated on ${\\mathcal P}$ at a time $t_i \\in {\\mathcal T}$ can be represented as a vector \n$\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N},$ \n\\begin{equation}\n\\label{eq:x_i}\n\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}=(u(p_1,t_i),...,u(p_N,t_i), v(p_1,t_i),...,v(p_N,t_i))^T.\n\\end{equation}\nThe collection of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,K,$ constitutes the data set $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi.$ In order to account for incompressibility, we introduce a discrete divergence operator $L_{div}$, which is given by a $N \\times 2N$ matrix associated with a finite difference scheme, and\n\\begin{equation}\n\\label{eq:L_div}\n(L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)_{k} \\approx (\\nabla \\cdot w)(p_k)=0.\n\\end{equation}\n\nFurther, we assume that the state can be measured only at specific points in ${\\mathcal P},$ that is, at ${\\mathcal Q}=\\{q_1,...,q_M\\} \\subset {\\mathcal P}$ where $M$ is typically much less than $N.$  Hence, there is $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in {\\mathbb R}^{2M}: \\, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C\\,\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\, \\forall \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi \\},$ where ${\\bm C} \\in {\\mathbb R}^{2M \\times 2N}$ is a sampling matrix. More specifically, $\\bm C$ is a two block matrix \n\n$$\n{\\bm C}=\\begin{pmatrix}\n{\\bm C}_{1\/2}& O\\\\\nO&{\\bm C}_{1\/2}\\\\\n\\end{pmatrix}, \\quad \n({\\bm C}_{1\/2})_{ij}= \\left\\{\n\\begin{array}{ll}\n1,& \\mbox{if } \\, q_i =p_j\\\\\n0,& \\mbox{otherwise}\n\\end{array} \\right. , \\quad i=1,...,N \\quad j=1,...,M,\n$$\nand ${\\bm O}\\in {\\mathbb R}^{M\\times N}$ is a zero matrix. The problem of reconstructing fluid flow $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}\\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ is presented as a schematic plot in \\cref{Sketch_map}.\n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.70\\linewidth]{Subset_M_of_D_v2_new_notation_i.png}\n\t\t\\captionof{figure}{Sketch of reconstruction of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$. The dots on the right side represent the grid ${\\mathcal P}$, and those on  the left side represent the measurement locations ${\\mathcal Q}.$}\\label{Sketch_map}\n\n\\end{figure}\nThere have been a wide range of methods for solving the problem, e.g. \\cite{sirovich1987turbulence, everson1995karhunen, donoho2006compressed, schmid2010dynamic, ELMS2018, raissi2019physics}. In particular, use of proper orthogonal decomposition (POD) \\cite{sirovich1987turbulence} techniques has been popular. \\\\ \n\nPOD \\cite{sirovich1987turbulence} is a traditional dimensional reduction technique where based on a data set, a number of basis functions are constructed. The key idea is that a linear combination of the basis functions can reconstruct the original data within some error margin, efficiently reducing the dimension of the problem. In a modified version of the POD, the Gappy POD (GPOD) \\citep{everson1995karhunen}, the aim is to fill the gap in-between sparse measurements. Given a POD basis one can minimize the $L_2$-error of the measurements and find a linear combination of the POD-basis that complements between the measurements. If the basis is not know, a iterative scheme can be formulated to optimize the basis based on the measurements. The original application of GPOD \\citep{everson1995karhunen} was related to reconstruction of human faces, and it has later been applied to fluid flow reconstruction \\cite{venturi2004gappy}. We will use the GPOD approach for comparison later in this study. \\\\ \n\nA similar approach is the technique of Compressed Sensing (CS) \\cite{donoho2006compressed}. As for the GPOD method, we want to solve a  linear system. However, in the CS-case this will be a under-determined linear system. That is we need some additional information about the system to be able to solve it, typically this can be a condition\/constraint related to the smoothness of the solution. The core difference between CS and GPOD is however the sparsity constraint. That is, instead of minimizing the L2-norm, we minimize the L1-norm. Minimizing the L1-norm favours sparse solutions, i.e. solutions with a small number of nonzero coefficients. \\\\ \n\nAnother reconstruction approach is Dynamical Mode Decomposition (DMD) \\cite{schmid2010dynamic}. Instead of using principal components in the spatial domain, DMD seek to find modes or representations that are associated with a specific frequency in the data, i.e. modes in the temporal domain. Again, the goal is to find a solution to an undetermined linear system and reconstruct based on the measurements, by minimizing the error of the observed values. \\\\\n\nDuring the last decade, data driven methods have become tremendously popular, partly because of the growth and availability of data, but also driven by new technology and improved hardware. To model a non-linear relationships with linear approximations is one of the fundamental limitation of the DMD, CS and GPOD. Recently we have seen development in methods where the artificial neural networks is informed with a physical law, the so called physic-informed neural networks (PINN) \\cite{raissi2019physics}. In PINNs the reconstruction is informed by a Partial Differential Equation (PDE) (e.g. Navier Stokes), thus the neural network can learn to fill the gap between measurements that are in compliance with the equation. This is what Rassi et. al. \\cite{raissi2018hidden} have shown for the benchmark examples such as flow around a 2D and 3D cylinder. Although PINNs are showing promising results, we have yet to see applications to complex systems such as atmospheric or oceanographic systems, where other aspect have to be accounted for, e.g. in large scale oceanic circulation models that are driven by forcing such as tides, bathymetry and river-influx. That being said, these problems may be resolved through PINNs in the future. Despite the promise of PINNs, they will not be a part of this study, as our approach is without any constraint related to the physical properties of the data. \\\\\n\nAnother non-linear data driven approaches for reconstruction of fluid flow are different variations of auto-encoders \\cite{ELMS2018, grover2019uncertainty}. An auto-encoder \\cite{Rumelhart86_autoencoder} is a special configuration of an artificial neural network that first encodes the data by gradually decreasing the size of the hidden layers. With this process, the data is represented in a lower dimensional space. A second neural network then takes the output of the encoder as input, and decodes the representation back to its original shape. These two neural networks together constitute an auto-encoder. Principal Component Analysis (PCA) \\cite{pearson1901_PCA} also represent the data in a different and more compact space. However, PCA reduce the dimension of the data by finding orthogonal basis functions or principal components through singular value decomposition. In fact, it has been showed with linear activation function, PCA and auto-encoders produces the same basis function \\cite{bourlard1988auto}. Probabilistic version of the auto-encoder are called Variational Auto-Encoders (VAEs) \\cite{kingma2013auto}. CVAEs \\cite{sohn2015learning} are conditional probabilistic auto-encoders, that is, the model is dependent on some additional information such that it is possible to create representations that are depend on this information. \\\\\n\nHere, we address the mentioned problem from a probabilistic point of view. Let $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi: {\\mathcal P} \\to {\\mathbb R}^{2N}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi: {\\mathcal Q} \\to {\\mathbb R}^{2M}$ be two multivariate random variables associated with the flow on ${\\mathcal P}$ and on ${\\mathcal Q}$, respectively. Then the data sets $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ consist of the realizations of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi$, respectively. Using $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi,$ we intend to approximate the probability distribution $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ This would not only allow to predict $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},$ but also to estimate an associated uncertainty. In this paper, we use a variational auto-encoder to approximate $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi| \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi)$. The method we use is a Bayesian Neural Network \\cite{MacKay92} approximated through variational inference \\cite{hoffman2013stochastic,blei2017variational}, that we have called \\textit{Semi-Conditional Variational Auto-encoder}, SCVAE. A detailed description of the SCVAE method for reconstruction and associated uncertainty quantification is given in \\cref{SCVAE_section}. \\\\\n\nHere we focus on fluid flow, being the main driving mechanism behind transport and dilution of tracers in marine waters. The world's oceans are  under tremendous stress \\citep{Halpern:2012hs}, UN has declared 2021-2030 as the ocean decade\\footnote{\\url{https:\/\/en.unesco.org\/ocean-decade}}, and an ecosystem based Marine Spatial Planning initiative has been launched by IOC \\citep{DominguezTejo:2016dt}. \\\\\n\nLocal and regional current conditions determines transport of tracers in the ocean \\cite{drange2001ocean,BARSTOW1983211}. Examples are accidental release of radioactive, biological or chemical substances from industrial complexes, e.g. organic waste from fish farms in Norwegian fjords \\citep{Ali:2011hd}, plastic \\cite{Law:2017}, or other contaminants that might have adverse effects on marine ecosystems \\citep{Hylland:2015gt}. \\\\ \n\nTo be able to predict the environmental impact of a release, i.e. concentrations as function of distance and direction from the source, requires reliable current conditions \\citep{Ali:2016go,Blackford:2020}. Subsequently, these transport predictions support design of marine environmental monitoring programs \\citep{Hvidevold:2015,Hvidevold:2016cx,Alendal:2017b, oleynik2020optimal}. The aim here is to model current conditions in a probabilistic manner using SCVAEs. This allows for predicting footprints in a Monte Carlo framework, providing simulated data for training networks used for, e.g., analysing environmental time series \\cite{gundersen2020binary}.\\\\ \n\nIn this study we will compare results  with the GPOD method \\cite{willcox2006unsteady}. We are aware that there recent methods (e.g. PINNS and traditional Auto-encoder) that may perform better on the specific data sets than the GPOD, however, the GPODs simplicity, versatility and not least its popularity \\cite{jo2019effective, mifsud2019fusing, callaham2019robust}, makes it a great method for comparison. \\\\\n\nThe reminder of this manuscript is outlined in the following: \\cref{A_Motivating_Example} presents a motivating example for the SCVAE-method in comparison with the GPOD-method. In \\cref{methods} we review both the VAE and CVAE method and present the SCVAE. Results of experiments on two different data sets are presented in \\cref{Experiment}. \\cref{discussion} summarize and discuss the method, experiments, drawbacks and benefits and potential extensions and further work.  \n\n\\section{A Motivating Example}\\label{A_Motivating_Example}\nHere we illustrate the performance of the proposed method vs the GPOD method in order to give a motivation for this study. We use simulations of a two dimensional viscous flow around a cylinder at the Raynolds number of $160,$ obtained from \\url{https:\/\/www.csc.kth.se\/~weinkauf\/notes\/cylinder2d.html}. The simulations were performed  by Weinkauf et. al. \\cite{weinkauf2010streak} with the Gerris Flow Solver software \\cite{gerrisflowsolver}. The data set consists of a horizontal $u$ and a vertical $v$ velocities on an uniform $400 \\times 50 \\times 1001$ grid of $[-0.5, 7.5] \\times [-0.5, 0.5] \\times [15, 23]$ spatial-temporal domain.\\footnote{The simulations are run from $t=0$ to $t=23$, but velocities are only extracted from $t=15$ to $t=23$} In particular, we have $400$ points in the horizontal, and $50$ points in the vertical  direction, and $1001$ points in time.  \n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_original_data_u.png}\n    \t\\includegraphics[width=0.99\\linewidth]{CW_original_data_v.png}\n\t\t\\captionof{figure}{Typical data instance from the original 2D flow around a cylinder data set with $u$ and $v$ component presented at the upper and lower panel, respectively} \\label{Cw_data_motivating}\n\n\\end{figure}\nThe cylinder has the diameter of $0.125$ and is centered at the origin, see \\cref{Cw_data_motivating}. The left vertical boundary (inlet) has Dirichlet boundary condition $u=1$ and $v=0$. The homogeneous Neumann boundary condition is given at the right boundary (outlet), and the homogeneous Dirichlet conditions on the remaining boundaries. At the start of simulations, $t=0$, both velocities were equal to zero.  We plot the velocities at the time $t \\approx 19$ (time step $500$) in \\cref{Cw_data_motivating}. \\\\\n\nFor simplicity, in the experiment below we extract the data downstream from the cylinder, that is, from grid point $40$ to $200$ in the horizontal direction, and keep all grid points in vertical direction. Hence, $\\mathcal{P}$ contains $N = 8000$ points, $160$ points in the horizontal and $50$ in the vertical direction. The temporal resolution is kept as before, that is, the number of time steps in $\\mathcal{T}$ is $K=1001$. For validation purposes, the data set was split into a train, validation and test data set. The train and validation data sets were used for optimization of the model parameters. For both the SCVAE and the GPOD, the goal was to minimize the $L2$ error between the true and the modeled flow state.  \nThe restriction of the GPOD is that the number of components $r$ could be at most $2M.$ \nTo deal with this problem, and to account for the flow incompressibility, we added the  regularization term  $\\lambda \\|L_{div} x^{(i)}\\|,$ $\\lambda>0$, to the objective function, see \\cref{Appendix_C}. For the GPOD method, the parameters $r$ and\/or $\\lambda$ where optimized on the validation data set in order to have the smallest mean error.\nWe give more details about objective functions for the SCVAE in \\cref{SCVAE_section}. For now we mention that there are two versions, where one version uses an additional divergence regularization term similar to GPOD.\\\\\n\nIn \\cref{Error_boxplot_CW} we plot the mean  of the relative $L_2$ error calculated on the test data for both methods with and without the div-regularization. The results are presented for $3,$ $4,$ and $5$ measurement locations, that is, $M=3,4,5.$ For each of these three cases, we selected $20$ different configurations of $M.$. In particular, we created $20$ subgrids ${\\mathcal Q}$, each containing $5$ randomly sampled spatial grid points. Next we removed one and then two points from each of the $20$ subgrids ${\\mathcal Q},$ to create new subgrids of $4$ and $3$ measurements, respectively.\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.75\\linewidth]{Motivating_example_boxplot_3M_v1.png}\n\t\t\\captionof{figure}{The mean relative error for two reconstruction methods. \n\t\tThe orange and blue label correspond to the SCVAE with (div-on) and without (div-off) additional divergence regularization. The green and red labels correspond to the GPOD method. }\\label{Error_boxplot_CW}\n\\end{figure}\nAs it can be seen in \\cref{Error_boxplot_CW}, both methods perform well for the $5$ measurements case. The resulting relative errors have comparable mean and variance. When reducing the number of observations, the SCVAE method maintains low errors, while the GPOD error increases. The SCVAE seems to benefit from the additional regularization of minimizing the divergence, in terms of lower error and less variation in the error estimates. The effect is more profound with fewer measurements. \\\\\n\nThe key benefit of the SCVAE is that its predictions are optimal for the given measurement locations.  In a contrast, the POD based approaches, and in particular the GPOD, create a set of basis functions (principal components) based on the training data independently of the measurements. While this has an obvious computational advantage, the number of principle components for complex flows can be high and, as a result, many more measurements are needed, \\cite{willcox2006unsteady,manohar2018data,Proctor2014}. There are number of algorithms that aim to optimize to measurement locations to achieve the best performance of the POD based methods, see e.g., \\cite{jo2019effective,willcox2006unsteady,YILDIRIM2009160}. In practice, however, the locations are often fixed and another approaches are needed. The results in \\cref{Error_boxplot_CW} suggest that the SCVAE could be one of these approaches.\n\n\\section{Methods}\\label{methods}\nBefore we introduce the model used for reconstruction of flows, we give a brief introduction to VAEs and CVAEs. For a detailed introduction, see \\cite{vae_intro}. VAEs are neural network models that has been used for learning structured representations in a  wide variety of applications, e.g., image generation \\cite{gregor2015draw}, interpolation between sentences \\cite{bowman2015generating} and compressed sensing \\cite{grover2019uncertainty}. \n\n\\subsection{Preliminary}\nLet us assume that the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is generated by a random process that involves an unobserved continuous random variable $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.$ The process consists of two steps: (i) a value $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ is sampled from a prior $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi);$ and (ii)  $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ is generated from a conditional distribution  $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ In the case of flow reconstruction, $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ could be thought of as unknown boundary or initial conditions, tidal and wind forcing, etc. However, generally $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ is just a convenient construct to represent $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ rather than a physically explained phenomena. Therefore it is for convenience assumed that $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ come from parametric families of distributions $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ and their density functions are differentiable almost everywhere w.r.t. both $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ and $\\theta$. A probabilistic auto-encoder is neural network that is trained to represent its input $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ as $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ via \\textit{latent representation} $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ that is,\n\n\\begin{equation} \\label{eq:p_theta(x)}\np_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi) = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi,\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{equation}\nAs $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ is unknown and observations $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ are not accessible, we must use $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ in order to generate $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ That is, the network can be viewed as consisting of two parts: an \\textit{encoder} $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ and a \\textit{decoder} $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ Typically the true posterior distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ is intractable but could be approximated with variational inference \\cite{hoffman2013stochastic,blei2017variational}.  That is, we define a so called recognition model $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ with variational parameters $\\phi$, which aims to approximate $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ The recognition model is often parameterized as a Gaussian. Thus, the problem of estimating $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$, is reduced to finding the best possible estimate for $\\phi$, effectively turning the problem into an optimization problem. \\\\\n\nAn auto-encoder that uses a recognition model is called Variational Auto-Encoder (VAE). In order to get good prediction we need to estimate the parameters $\\phi$ and $\\theta.$ The marginal likelihood  is equal to the sum over the marginal likelihoods of the individual samples, that is, $\\sum_{i=1}^K \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Therefore, we further on present estimates for an individual sample. The Kullback -Leibler divergence between two probability distributions  $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$, defined as $$D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] = \\int q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\log\\left(\\frac{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}{p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}\\right) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,$$ can be interpreted as a measure of distinctiveness between these two distributions \\cite{kullback1951information}. It can be shown, see \\cite{vae_intro}, that\n\\begin{equation}\\label{eq:DKL_via_L}\n\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})=D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] +\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}),\n\\end{equation}\nwhere\n$$\n\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = \n{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ -\\log q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})+\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right].\n$$\nSince KL-divergence is non-negative, we have $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\geq\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and\n$\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$\nis called Evidence Lower Bound (ELBO) for the marginal likelihood $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$\nThus, instead of maximizing the marginal probability, one can instead maximize its variational lower bound to which we also refer as an objective function. It can be further shown that the ELBO can be written as\n\\begin{equation}\n\t\\label{eq:VLB:2_no_beta}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)].\n\\end{equation}\nReformulating the traditional VAE framework as a constraint optimization problem, it is possible to obtain the $\\beta$-VAE \\cite{higgins2016beta} objective function if $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) =\\mathcal{N}({\\bf 0},{\\bm I}),$\n\\begin{equation}\n\t\\label{eq:VLB:2}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - \\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)],\n\\end{equation}\nwhere $\\beta>0.$ Here $\\beta$ is a regularisation coefficient that constrains the capacity of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right]$ can be interpreted as the reconstruction term, while the KL-term, $\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]$ as regularization term. \n\n\nConditional Variational Auto-encoders \\cite{sohn2015learning} (CVAE) are similar to VAEs, but differ by conditioning on an additional property of the data (e.g. a label or class), here denoted $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$. Conditioning both the recognition model and the true posteriori on both $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ and $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ results in the CVAE ELBO \n\\begin{align}\n    \\begin{split}\n        \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)={\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi) \\right] - D_{KL}[q(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)]. \\label{CVAE_objective}\n    \\end{split}\n\\end{align}\nIn the decoding phase, CVAE allows for conditional probabilistic reconstruction and permits sampling from the conditional distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)$, which has been useful for generative modeling of data with known labels, see \\cite{sohn2015learning}. Here we investigate a special case of the CVAE when $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ is a partial observation of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi.$ We call this Semi Conditional Variational Auto-encoder (SCVAE).\n\n\\subsection{Semi Conditional Variational Auto-encoder}\\label{SCVAE_section}\nThe SCVAE takes the input data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$, conditioned on $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ and approximates the probability distribution $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ Then we can generate $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$, based on the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. As $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ where $C$ is a non-stochastic sampling matrix, we have \n$$p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\, \\mbox{ and } \\, q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$$ Therefore, from \\cref{CVAE_objective} the ELBO for SCVAE is \n\\begin{equation}\n    \\begin{split}\n        \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})\\geq \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=\n        &{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\\\ & - D_{KL}[q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})]\\label{eq:ELBO:SCVAE_no_beta}\n    \\end{split}\n\\end{equation}\nwhere $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ Similarly as for the $\\beta$-VAE \\cite{higgins2016beta} we can obtain a relaxed version of \\cref{eq:ELBO:SCVAE_no_beta} by maximizing the parameters $\\{\\phi, \\theta\\}$ of the expected log-likelihood ${\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])$ and treat it as an constrained optimization problem. That is, \n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\leq \\epsilon  \n    \\end{split}\\label{beta_SCVAE_opt_prob}\n\\end{align}\nwhere $\\epsilon>0$ is small. The subscript $q_\\phi(\\cdot)$ is short for $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Since $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is dependent on $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ we have that $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ \\cref{beta_SCVAE_opt_prob}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions \\cite{kuhn2014nonlinear, karush1939minima}. Hence,\n\\begin{align}\n    \\begin{split}\n    \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & = \n    {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]) \\\\ & + \n    \\beta(D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})-\\epsilon)   \n    \\end{split}\\label{beta_SCVAE_constrained}\n\\end{align}\nAccording to the complementary slackness KKT condition $\\beta \\geq 0,$ we can rewrite \\cref{beta_SCVAE_constrained} as\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}).   \n    \\end{split}\\label{beta_SCVAE_constrained_2}\n\\end{align}\nObjective functions in \\cref{eq:ELBO:SCVAE_no_beta} and \\cref{beta_SCVAE_constrained_2}, and later \\cref{eq:ELBO:SCVAE:J}, show that if conditioning on a feature which is a known function of the original data, such as measurements, we do not need to account for them in the encoding phase.The measurements are then coupled with the encoded data in the decoder. We sketch the main components of the SCVAE in \\cref{DAE_neural_network_sketch}.\n\\begin{figure}[ht]\n    \\centering\n\t\t\\includegraphics[width=0.80\\linewidth]{Model_Sparse_autoencoder_new_notation_i.png}\n\t\t\\captionof{figure}{The figure shows a sketch of the model used to estimate \n\t\t$p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})$. During training both the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and the data $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ will be used. After the model is trained, we can predict using only the decoder part of the neural network. The input to the decoder will then only be the observations and random samples from the latent space.}\\label{DAE_neural_network_sketch}\n\\end{figure}\nIn order to preserve some physical properties of the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ we can condition yet on another feature. Here we utilize the incompressibility property of the fluid, i.e., $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}=L_{div} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\approx 0,$ see \\cref{eq:L_div}. \\\\ \n\nWe intend to maximize a log-likelihood under an additional constrain $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}$, compared to \\cref{beta_SCVAE_opt_prob}. That is\n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\leq \\epsilon \\quad \\text{and} \\quad \\\\ &\n    -{\\mathbb E}_{q_\\phi(\\cdot)}[\\log p_\\theta(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)] \\leq \\delta \n    \\end{split}\\label{Constraint_optimization_prob_v3}\n\\end{align}\nwhere $\\epsilon, \\delta>0$ are small. \\cref{Constraint_optimization_prob_v3}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions as before and as a consequence of the complementary slackness condition $\\lambda,\\beta \\geq 0,$ we can obtain the objective function \n\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\lambda\\,{\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & -\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}),   \n    \\end{split}\\label{eq:ELBO:SCVAE:J}\n\\end{align}\nwhere $p(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ For convenience of notation we refer to the objective function \\cref{beta_SCVAE_constrained_2} as the case with $\\lambda=0$, and the objective function \\cref{eq:ELBO:SCVAE:J} as the case with $\\lambda > 0.$ Observe that under the Gaussian assumptions on the priors, \\cref{eq:ELBO:SCVAE:J} is equivalent to \\cref{beta_SCVAE_constrained_2} if $\\lambda=0.$ Thus, from now one we will refer to it as a special case of \\cref{eq:ELBO:SCVAE:J} and denote as $\\mathcal{L}_{0}.$ \\\\\n\nSimilarly to \\cite{kingma2013auto} we obtain $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= \\mathcal{N}({\\mu}^{(i)} \\mathbf{1},(\\sigma^{(i)})^2\\,\\mathbf{I})\n$, that is, $\\phi=\\{ \\mu, \\sigma\\}.$  This allows to express the KL-divergence terms in a closed form and avoid issues related to differentiability of the ELBOs.  Under these assumptions, the KL-divergence terms can be integrated analytically while the term\n${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $ and ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $\nrequires estimation by sampling\n\\begin{equation}\\label{eq:E-estimate}\n    \\begin{array}{l}\n         {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        \\mbox{where } \\, \\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)} = g_{\\phi}(\\bm{\\epsilon}^{(i,l)}, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\quad \n        \\bm{\\epsilon}^{l} \\sim p(\\bm{\\epsilon}).\n    \\end{array}\n\\end{equation}\nHere $\\bm{\\epsilon}^{l}$ is an auxiliary (noise) variable with independent marginal $p(\\bm{\\epsilon})$, and $g_{\\phi}(\\cdot)$ is a differentiable transformation of $\\bm{\\epsilon},$ parametrized by $\\phi,$  see for details \\cite{kingma2013auto}. We denote $\\mathcal{L}_\\lambda,$ $\\lambda \\geq 0$  \\cref{eq:ELBO:SCVAE:J} with the approximation above as $\\mathcal{\\widehat{L}_\\lambda},$ that is,\n\\begin{align}\n    \\begin{split}\n        & \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})  =   \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\\\\n         &+ \\lambda \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) -\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})].\n        \\label{Loss_function}\n    \\end{split}\n\\end{align}\nThe objective function $\\widehat{\\mathcal{L}}_\\lambda$ can be maximized by gradient descent. Since the gradient $\\nabla_{\\theta,\\phi}\\,\\widehat{\\mathcal{L}}_\\lambda$ cannot be calculated for large data sets, Stochastic Gradient Descent methods, see \\cite{kiefer1952stochastic, robbins1951stochastic} are typically used where \n\\begin{equation}\n    \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi, \\bm{D}) \\approx \\widehat{\\mathcal{L}}^{R}(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^R, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^R, \\bm{D}^R) = \n    \\frac{K}{R}\\sum\\limits_{r=1}^{R} \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}), \\quad \\lambda \\geq 0.\\label{obj_function_SCVAE}\n\\end{equation}\nHere  $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^{R}=\\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}\\right\\}_{r=1}^{R},$  $R<K$ is a minibatch consisting of randomly sampled datapoints, $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^{R}=\\left\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}$ and $\\bm{D}^{R}=\\left\\{\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}.$ After the network is optimized, a posterior predictive distribution $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi)$ can be approximated with a Monte Carlo estimator. \\\\\n\n\n\\subsubsection{Uncertainty Quantification}\\label{sec:posterior}\nLet  $\\hat{\\theta}$ and $\\hat{\\phi}$ be an estimation of generative and variational parameters, as described in \\cref{SCVAE_section}. Then the decoder can be used to predict the posterior as\n\\begin{align}\\label{eq:p_pred}\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\approx \\frac{1}{N_{MC}} \\sum_{j=1}^{N_{MC}} p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(j)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\xrightarrow[N_{MC} \\rightarrow \\infty]{} \\int\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)p_{\\hat{\\theta}}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{align}\nWhile sampling from the latent space has been viewed typically as an approach for generating new samples with similar properties, here we use it to estimate the prediction uncertainty of the trained model. From \\cref{eq:p_pred} we are able to estimate the mean prediction  $\\hat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and empirical covarience matrix $\\widehat{{\\bm \\Sigma}}$ using  a Monte Carlo estimator. We get\n\\begin{align} \n      \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*=\n      \\frac{1}{N_{MC}} \\sum\\limits_{j=1}^{N_{MC}} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}\\quad \\mbox{and} \\quad\n      \\widehat{{\\bm \\Sigma}} =  \\frac{1}{N_{MC}-1} \\sum\\limits_{j=1}^{N_{MC}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)^T,\n        \\label{eq:mean_and_cov} \n               \n \\end{align}\nwhere $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}  \\sim p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*).$ The empirical standard deviation is then simply $\\widehat{\\bm \\sigma} = \\sqrt{diag(\\widehat{{\\bm \\Sigma}})}.$ To estimate the confidence region we assume that the predicted $p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{*},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)$ is well approximated by a normal distribution $N({\\bm \\mu}, \\bm{\\Sigma}).$ Given that $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and $\\widehat{{\\bm \\Sigma}}$ are approximations of $\\bm{\\mu}$ and $\\bm{\\Sigma},$ obtained from $N_{MC}$ samples as above, a confidence region estimate for a prediction $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{*}$ can be given as\n\\begin{equation}\\label{eq:ConfRegion:1}\n    \\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N}: \\,(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}-{\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})^T \\hat{\\bm \\Sigma}^{+} (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}- \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)\\leq \\chi^2_k(p)\\right\\}\n\\end{equation}\nwhere $\\chi^2_k(p)$ is the quantile function for probability $p$ of the chi-squared distribution with $k=\\min\\{N_{MC},2N\\}$ degrees of freedom, and $\\widehat{\\bm \\Sigma}^{+}$ is the pseudoinverse of $\\widehat{\\bm \\Sigma}.$ Using the singular value decomposition, ${\\widehat{\\bm \\Sigma}}=\\bm{U S U}^T,$ the corresponding interval for $(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^*)_n,$ $n=1,\\dots,2N,$ is\n\\begin{equation}\\label{eq:ConfRegion:2}\n   \\left[\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n-\\sqrt{\\chi^2_k(p)}\\,\\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2,\\quad\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n+\\sqrt{\\chi^2_k(p)}\\,\n \\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2\n  \\right]   \n\\end{equation}\nwhere ${\\bm u}_n^T$ is $n$th row of the matrix ${\\bm U}.$\n\n\n\\section{Experiments}\\label{Experiment}\nWe will present the SCVAE method on two different data sets. The first one is the 2D flow around a cylinder described in \\cref{A_Motivating_Example}, and the second is ocean currents on the seafloor created by the Bergen Ocean Model \\cite{berntsen2000users}. The data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ consists of the two dimensional velocities ${\\bm w} =(u, v).$ To illustrate the results we will plot $u$ and $v$ components of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ see \\cref{eq:x_i}. For validation of the models, the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is split into train, test and validation subsets, which are subscripted accordingly, if necessary. The data sets, spitting and preprocessing for each case are described in \\cref{Cylinder_wake_experiment,BOM_experiment}. \\\\\n\nWe use a schematically simple architecture to explore the SCVAE. The main ingredient of the encoder is convolutional neural network (CNN)  \\cite{lecun1998gradient, krizhevsky2012imagenet} and for the decoder we use transposed CNN-layers \\cite{noh2015learning}. The SCVAE has a slightly different architecture in each case, which we present in \\cref{Appendix_A}. \\\\\n\nThe SCVAE is trained to maximize the objective functions in \\cref{obj_function_SCVAE} with the backpropagation algorithm \\cite{lecun1989backpropagation} and the Adam algorithm \\cite{kingma2014adam}. We use an adaptive approach of weighing the reconstruction term with KL-divergence and\/or divergence terms \\cite{heydari2019softadapt}, that is, finding the regularization parameters $\\beta$ and $\\lambda$. Specifically, we calculate the proportion of each term contribution of the total value of the objective function, and scale the terms accordingly. This approach prevents posterior collapse. Posteriori collapse occurs if the KL-divergence term becomes too close to zero, resulting in a non probabilistic reconstruction. The approach of weighing the terms proportionally iteratively adjusts the weight of the KL-divergence term, $\\beta$, such that posterior collapse is avoided. For the result shown here we trained the SCVAEs  with an early stopping criteria of $50$ epochs, i.e., the optimization is stopped if we do not see any improvement after $50$ epochs, and returns the best model. We use a two-dimensional Gaussian  distribution for $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})$ in all the experiments.\\\\\n\nLet the test data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi_{test}$ consist of $n$ instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,n,$ and $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ denote a prediction of the true $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ In the case of the SCVAE, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is the mean prediction obtained as in \\cref{eq:mean_and_cov}. For the GPOD method, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is a deterministic output of the optimization problem, see \\cref{Appendix_C}. In order to compare the SCVAE results with the results of the GPOD method, we introduce the following measures; the mean of the relative error for the prediction\n\\begin{equation}\\label{L2_error}\n  \\mathcal{E} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\frac{||\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)} - \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}{||\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}\n\\end{equation}\nand the mean of the absolute error for the divergence \n\\begin{equation}\n        \\mathcal{E}_{div} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\| L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}\\|_2.\\label{divergence_error_1}\n\\end{equation}\n\n\\subsection{2D Flow Around a Cylinder}\\label{Cylinder_wake_experiment}\nHere we return to the example in \\cref{A_Motivating_Example}. Below we give some additional details of the data preprocessing and model implementation.\n\\subsubsection{Preprocessing}\nThe data is reduced as described in \\cref{A_Motivating_Example}. \nWe assess the SCVAE with a sequential split for train, test and validation. The sequantial split is defined as follows. The last $15 \\%$ of the data is used for test, the last $30\\%$ of the remaining data is used for validation, and the first $70\\%$ for training. To improve the conditioning of the optimization problem we scale the data as decribed in  \\cref{Appendix_D}. The errors $\\mathcal{E}$ (\\cref{L2_error}) and $\\mathcal{E}_{div} $ (\\cref{divergence_error_1}) are calculated after scaling the data back. The input to the SVAE $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ was shaped as $(160 \\times 50 \\times 2)$ in order to apply convolutional layers. Here we use  $5,4,3$ and $2$ fixed spatial measurements, that is, four different subgrids ${\\mathcal Q}$ \n\\begin{align}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_5 = & \\{(12,76),(47,8),(30,40),(153,34),(16,10)\\}, \\\\\n    {\\mathcal Q}_4 = & \\{(12,76),(47,8),(30,40),(153,34)\\},  \\\\\n    {\\mathcal Q}_3 = & \\{(12,76),(47,8),(30,40)\\}, \\\\\n    {\\mathcal Q}_2 = & \\{(12,76),(47,8)\\}.\n    \\end{split}\\label{CW_measurements}\n\\end{align}\nThe flow state at these specific locations constitute $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model}\nA schematic description of the model is given in \\cref{Appendix_A}. \nThe first layer of the encoder is a zero-padding layer that expands the horizontal and vertical dimension by adding zeros on the boundaries. Here we add zero-padding of four in the horizontal and three in the vertical direction. The subsequent layers consists of two convolutional layers, where the first and second layer have $160$ and $200$ filters, respectively. We use a kernel size and strides of $2$ in both convolutional layers and ReLu activation functions. This design compresses the data into a $(42 \\times 14 \\times 200)$ shape. The compressed representation from the convolutional layers are flattened and are further compressed into a $64$ dimensional vector through a traditional dense layer. Two outputs layers are defined to represent the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The reparametrization trick is realized in a third layer, a so called  lambda layer, which takes the mean and log-variance as an input and generates $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The output of the encoder are the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. \\\\\n\nThe decoder takes the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ as input. The input $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened and then concatenated with $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$. The next layer is a dense layer with shape $(42 \\times 14 \\times 200)$. Afterwards there are two transposed convolutional layers with filters of $200$ and $160$. The strides and the kernel size is the same as for the encoder. The final layer is a transposed convolutional layer, with same dimension as the input to the encoder, that is the dimension of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. A linear activation function is used for this output layer. The last layer of the model is a lambda layer that removes the zero-padding. In the next section we show statistics of the probabilistic reconstruction and compare with the GPOD method. \n\n\\subsubsection{Results}\nIn \\cref{Cylinder_wake_pred} we have plotted the true velocity field, the reconstructed velocities, the standard deviation of the velocities and the absolute error between the true and reconstructed velocity fields. The observations placement is shown as stars (black and white). The SCVAE with the objective function  $\\widehat{\\mathcal{L}}_0$, see \\cref{CVAE_objective}, was used for this prediction. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $100$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty estimates, see \\cref{eq:mean_and_cov}. \\\\\n\nWe emphasise here again that the SCVAE with $\\widehat{\\mathcal{L}}_0$ and with $\\widehat{\\mathcal{L}}_\\lambda,$ $\\lambda>0,$ are two different models. For the notation sake we here refer to $\\lambda=0$ when we mean the model with the objective function in \\cref{CVAE_objective}, and to $\\lambda>0$ when in \\cref{eq:ELBO:SCVAE:J}. The same holds for the GPOD method, see \\cref{Appendix_D}. When $\\lambda=0,$ the number of the principle components $r$ is less $2M.$ The number $r$ is chosen such that the prediction on the validation data has the smallest possible error on average. If $\\lambda>0,$ no restrictions on $r$ are imposed. In this case both  $\\lambda$ and $r$ are estimated from the validation data. \\\\\n\nThe general observation is that the SCVAE reconstruction fits the data well, with associated low uncertainty. \nThis can be explained by the periodicity in the data. In particular, the training and validation data sets represent the  test data well enough.\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_recon_3M_v1_CR.png}\n\t\\captionof{figure}{Left panels shows the u-velocities, and the right panel v-velocities. The results are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{First panels:} The true solutions \\textbf{Second panels:} Reconstructed solution based on the SCVAE model \\textbf{Third panels:} Standard deviation of the predicted solution \\textbf{Fourth panels:} Absolute error between the true and predicted solution.}\\label{Cylinder_wake_pred}\n\n\\end{figure}\n\nIn \\cref{Cylinder_wake_time_series} we  have plotted four time series of the reconstructed test data at two specific grid points, together with the confidence regions constructed as in \\cref{eq:ConfRegion:2} with $p=0.95.$ The two upper panels represents the reconstruction at the grid point $(6, 31)$, and the lower at $(101,25)$ for $u$ and $v$ on the left and right side, respectively. The SCVAE reconstruction is significantly better than the GPOD, and close to the true solution for all time steps. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_time_series_with_GPOD_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$ The results are obtained from the model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(6, 31).$ \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(101, 25).$ }\\label{Cylinder_wake_time_series}\n\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_error_with_GPOD_CR.png}\n\t\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange). The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. The estimates are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(6,31)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(101,25)$ for $u$ (left) and $v$ (right).}\\label{Cylinder_wake_error}\n\n\\end{figure}\n\\cref{Cylinder_wake_error} shows the difference between the true values and the model prediction in time for the same two locations. This figure has to be seen in context with \\cref{Cylinder_wake_pred}. In \\cref{tab:comparison_CW}  we display the relative errors, \\cref{L2_error}, for the SCVAE and the GPOD method, both with and without divergence regularization, for $5, 4, 3,$ and $2$ measurement locations given in \\cref{CW_measurements}. \\\\\n\nThe results of the SCVAE depend on two stochastic inputs which are (i) randomness in the initialization of the prior weights and (ii) random mini batch sampling. We have trained the model with a each measurement configuration $10$ times, and chose the model that performs the best on the validation data set. Ideally we would run test cases where we used all the values as measurements,i.e., $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ and test how well the model would reconstruct in this case. This would then give us the lower bound of the best reconstruction that is possible for this specific architecture and hyper parameter settings. However, this scenario was not possible to test, due to limitations in memory in the GPU. Therefore we have used a large enough $M$ which still allowed us to run the model. In particular, we used every fifth and second pixel in the horizontal and vertical direction, which resulted in a total of $(32 \\times 25)$ measurement locations, or $M=800$. We believe that training the model with these settings, gave us a good indication of the lower bound of the reconstruction error. The error observed was of the magnitude of $10^{-3}$. \\\\\n\nThis lower bound has been reached for all measurement configurations \\cref{CW_measurements}. \nHowever, larger computational cost was needed to reach the lower bound for fewer measurement locations. \\cref{Epochs_per_measurement_CW} shows the number of epochs as a boxplot diagram. In comparison with GPOD, the SCVAE error is 10 times lower than the GPOD error, and this difference becomes larger with fewer measurements. Note that adding regularization did not have much effect on  the relative error. From the motivating example we observed that regularizing with $\\lambda>0$ is better in terms of a more consistent and low variable error estimation. Here we selected from the 10 trained models the one that performed best on the validation data set. This model selection approach shows that there are no significant differences between the two regularization techniques. The associated error in the divergence of the velocity fields is reported in  \\cref{tab:comparison_CW_div}. \n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n         & 0.30e-02 & 0.33e-02 & 0.26e-02 & 0.28e-02    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.31e-02 & 0.32e-02 & 0.30e-02 & 0.28e-02 \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$  & 2.35e-02& 2.49e-02  \n           & 3.38e-02& 17.38e-02\\\\ \\cline{2-6}\n         & $\\lambda > 0$ &   2.12e-02 & 2.33e-02& 3.15e-02& 16.38e-02    \\\\ \\hline\n    \\end{tabular}\n    \\caption{The mean relative error $\\mathcal{E}$ (\\cref{L2_error}) for the  SCVAE prediction  and  the GPOD prediction with or without div-regularization, and different number of measurements.}\n    \\label{tab:comparison_CW}\n\\end{table}\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n          & 0.1439 & 0.1580 & 0.1383 & 0.1432    \\\\ \\cline{2-6}\n         & $\\lambda > 0$ & 0.1533 & 0.1408 & 0.1468 & 0.1410    \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$ & 0.1052 & 0.1047 & 0.0943 & 0.08866    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.1039 & 0.1051 & 0.0966 & 0.0669   \\\\ \\hline\n    \\end{tabular}\n    \\caption{Comparison of the divergence error $\\mathcal{E}_{div}$ as calculated in \\cref{divergence_error_1} for the different methods and regularization techniques. The true divergence error on the entire test data set is $0.1058$}\n    \\label{tab:comparison_CW_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{Epochs_per_Measurement.png}\n \t    \\captionof{figure}{Number of epochs trained depending on the number of measurements. For each measurement configuration and regularization technique the model is run $10$ times. The variation of number of epochs for for each measurement locations is due to different priors of the weights and random mini-batch sampling.}\\label{Epochs_per_measurement_CW}\n\\end{figure}\n\n\\subsection{Current data from Bergen ocean model}\\label{BOM_experiment}\nWe tested the SCVAE on simulations from the Bergen Ocean Model (BOM) \\cite{berntsen2000users}. BOM is a three-dimensional terrain-following nonhydrostatic ocean model with capabilities of resolving mesoscale to large-scale processes. Here we use velocities simulated by Ali. et. al \\cite{Ali:2016go}. The simulations where conducted on the entire North Sea with 800 meter horizontal and vertical grid resolution and 41 layers for the period from 1st to 15th of January 2012. Forcing of the model consist of wind, atmospheric pressure, harmonic tides, rivers, and initial fields for salinity and temperature. For details of the setup of the model, forcing and the simulations we refer to \\cite{Ali:2016go}. \\\\\n\nHere, the horizontal and vertical velocities of an excerpt of 25.6 $\\times$ 25.6 km$^2$ at the bottom layer centered at the Sleipner CO2 injection site ($58.36^\\circ N, \\, 1.91^\\circ E$) is used as data set for reconstruction. In \\cref{Data_set_time_series_BOM} we have plotted the mean and extreme values of $u$ and $v$ for each time $t$ in ${\\mathcal T}$. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{Data_CW_U.png}\n        \\includegraphics[width=0.99\\linewidth]{Data_CW_V.png}\n\t\t\\captionof{figure}{The light-blue line represent the maximum, the orange the minimum and the green mean value of $u$ and $v$ for each time $t$ in ${\\mathcal T}.$ The horizontal lines indicate the sequential data split. }\\label{Data_set_time_series_BOM}\n\n\\end{figure}\n\\subsubsection{Preprocessing}\nWe extract $32 \\times 32$ central grid from the bottom layer velocity data. Hence, $\\mathcal{P}$ contains $N = 1024$ points, $32$ points in the horizontal and $32$ in the vertical direction. The temporal resolution is originally $105000$ and the time between each time step is $1$ minute. We downsample the temporal dimension of the original data uniformly such that the number of time steps in $\\mathcal{T}$ is $K=8500.$ We train and validate the SCVAE with two different data splits: randomized and sequential in time. For the sequential split we have used the last $15 \\%$ for the test, the last $30\\%$ of the remaining data is used for validation, and the fist $70\\%$ for training. In \\cref{Data_set_time_series_BOM}, the red and blue vertical lines indicate the data split for this case. For the random split, the instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ are drawn randomly from $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ with the same percentage. The data was scaled as described in \\cref{Appendix_D}.  The input $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ to the SCVAE was shaped as $(32 \\times 32 \\times 2)$ in order to apply convolutional layers. We use  $9,5$ and $3$ fixed spatial measurement locations. In particular, the subgrid ${\\mathcal Q}$ is given as \n\\begin{align}\\label{BOM_measurements}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_9 = & \\{(6,6),(6,17),(6,27),(17,17),(17,27),(17,6),(27,6), (27,17),(27,27) \\}, \\\\\n    {\\mathcal Q}_5 = & \\{(6,6),(17,17),(27,27),(6,27),(27,6) \\}, \\\\\n    {\\mathcal Q}_3 = & \\{ (6,27),(17,17),(27,6) \\}. \n    \\end{split}\n\\end{align}\nAs before, the values of $u$ and $v$ at these specific locations constitute the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model} \nA schematic description of the model is given in \\cref{Appendix_A.3,Appendix_A.4}. \nThe first two layers of the encoder are convolutional layers with $64$ and $128$ filters with strides and kernel size of $2$ and ReLu activation functions. This compresses the data into a shape of $(8 \\times 8 \\times 128)$. The next layers are flattening and dense layers, where the latter have $16$ filters and ReLu activation. The subsequent layers defines the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$, which is input to a lambda layer for realization of the reparametrization trick. The encoder outputs the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$.\\\\\n\nInput to the decoder is the output $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ of the encoder and the measurement $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ To concatenate the inputs, $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened. After concatenation of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$, the next layer is a dense layer with shape $(8 \\times 8 \\times 128)$ and ReLu activation. This allows for use of transposed convolutional layers to obtain the original shape of the data. Hence, the following layers are two transposed convolutional layers with $64$ and $128$ filters, strides and kernel size of $2$ and ReLu activation's. The final layer is a transposed convolutional with linear activation functions and filter size  of shape $(32 \\times 32 \\times 2),$ i.e., the same shape as $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. \n\n\\subsubsection{Results}\\label{BOM_results}\nWe illustrate the obtained posterior predictive distribution in terms the predictive mean and standard deviation for the prediction at a specific time. The SCVAE is compared with the GPOD method, both with $\\lambda >0$ and $\\lambda = 0$ for measurement locations given in \\cref{BOM_measurements} for random and sequential split cases. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $200$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty\nestimates, see \\cref{eq:mean_and_cov}. \\cref{UV_prediction} shows the results of the prediction at time step $1185$ for both the $u$ and $v$ component and associated uncertainty estimates for a trained model with $\\lambda=0$ and $Q_3$ measurement locations (see \\cref{BOM_measurements}).  \n\n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Sleipner_prediction_std_abs_v1.png}\n\t\t\\captionof{figure}{Presentation of statistics for\n\t\tthe reconstruction of the $u$ and $v$ component of the velocity for sample $1185$ in the test data set based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Left panels:} From top to bottom; True velocity in $u$, predicted mean velocity field of $u$, the standard deviation of the prediction for $u$ and the absolute error of $u.$ \\textbf{Lower panels:} Similar as describe for the upper panels, but for $v$ }\\label{UV_prediction}\n\\end{figure} \nIn \\cref{BOM_time_series} we plot the true solution and the predicted mean velocity \\cref{eq:mean_and_cov} with the associated uncertainty, see \\cref{eq:ConfRegion:2}, for two grid points. We plot only the first $600$ time steps for readability. The first grid point is $(26,6)$ and $(4,1).$ One location is approximately $5.1$ km from the nearest observation, and another one is about $16.1$ km away.\n\\begin{figure}[H]\n        \\centering\n\t\t\\includegraphics[width=0.85\\linewidth]{BOM_plots_time_series_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$.  \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(26,6)$, approximately $5.1$ km from nearest observation. \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(4,1)$ approximately $16.1$ km from nearest observation.}\\label{BOM_time_series}\n\\end{figure} \n\\cref{BOM_error} has to be viewed in context with \\cref{BOM_time_series} and show the difference between the true and the predicted solutions with associated difference marginal in time for the two locations as in \\cref{BOM_time_series}. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{BOM_error_plot_CR.png}\n\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange) based on the $\\lambda=0$ model and $Q_3$ measurement locations. The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(26,6)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(4,1)$ for $u$ (left) and $v$ (right).}\\label{BOM_error}\n\\end{figure}\nIntegrating over the latent space generates a posterior distribution of the reconstruction, as described in \\cref{sec:posterior}. It is also possible to use the latent space to generate new statistically sound versions of $u$ and $v$. This is presented in \\cref{z_space_sampling_v} where it is sampled uniformly over the 2 dimensional latent space $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ and the result shows how different variations can be created with the SCVAE model, given only the sparse measurements.    \n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Sleipner_generating_data_v1.png}\n\t\t\\captionof{figure}{The left panels shows 9 different generated velocity-field-snapshots for the $\\ifmmode\\bm{u}\\else\\textbf{\\textit{u}}\\fi$ and $\\ifmmode\\bm{v}\\else\\textbf{\\textit{v}}\\fi$ component for test sample number $1185$. The predictions are generated from the model with $\\lambda=0$ and $Q_3$ measurement locations. We sample uniformly over the latent space and predicts with the decoder, given the measurements.}\\label{z_space_sampling_v}\n\\end{figure}\nThese sampled velocities could be used for ensemble simulations when estimating uncertainty in a passive tracer transport, see e.g., \\cite{oleynik2020optimal}. \n\nThe SCVAE results are are compared with results of the GPOD method, see \\cref{tab:comparison_BOM} and \\cref{tab:comparison_BOM_div}. The tables show the errors as calculated in \\cref{L2_error} and \\cref{divergence_error_1} of the test data set for both sequential and random split.\nFor the sequential splitting, the SCVAE is better for $3$ measurement locations, while the GPOD method performs better for $9$ and $5$ locations.  From  \\cref{Data_set_time_series_BOM}, we observe that test data set seems to arise from a different process than the train and validation data (especially for $v$). Thus, the SCVAE generalize worse than a simpler model such as the GPOD, \\cite{model_selection}. For the $3$ location case, the number of components in the GPOD is not enough to compete with the SCVAE. \\\\\n\nWith random split on the train, test and validation data, we see that the SCVAE is significantly better than the GPOD. The training data and measurements represent the test data and test measurements better with random splitting. This highlights the importance of large data sets that cover as many outcomes as possible. Demanding that $\\lambda > 0$ in \\cref{obj_function_SCVAE} do not improve the result. The SCVAE-models with $\\lambda = 0$ learns that the reconstructed representations should have low divergence without explicitly demanding it during optimization. However, as discussed in the 2D flow around cylinder experiment, demanding $\\lambda>0$ seems to improve the conditioning of the optimization problem and give more consistent results.  \n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3 \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.1379 & 0.2097 & 0.2928   \\\\ \\cline{3-6}\n         & & GPOD & 0.3300 & 0.3822 & 0.4349  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.1403 & 0.2025 & 0.3016   \\\\ \\cline{3-6}\n         & & GPOD & 0.2971 & 0.3579 & 0.4039  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.3493 & 0.3913 & 0.4155  \\\\ \\cline{3-6}\n         & & GPOD & 0.3767 & 0.4031 & 0.4678  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.3527 & 0.3889 & 0.4141  \\\\ \\cline{3-6}\n         & & GPOD &0.3362 & 0.3695 & 0.4462  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Errors as calculated in \\cref{L2_error} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements}\n    \\label{tab:comparison_BOM}\n\\end{table}\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3  \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 3.75e-05 & 3.62e-05 & 3.42e-05   \\\\ \\cline{3-6}\n         & & GPOD & 6.51e-05 & 5.88e-05 & 5.02e-05   \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE &  3.60e-05 & 3.60e-05 & 3.13e-05  \\\\ \\cline{3-6}\n         & & GPOD & 6.23e-05 & 4.77e-05 & 4.14e-05  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 2.02e-05 & 1.80e-05 & 1.69e-05   \\\\ \\cline{3-6}\n         & & GPOD & 5.09e-05 & 4.03e-05 & 4.15e-05  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 2.05e-05 & 1.99e-05 & 1.85e-05  \\\\ \\cline{3-6}\n         & & GPOD & 4.39e-05 & 3.65e-05 & 2.92e-05  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Divergence errors as calculated in \\cref{divergence_error_1} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements. The true divergence of the test data is of order  $10^{-4}.$ }\n    \\label{tab:comparison_BOM_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{BOM_epochs_random_split.png}\n \t    \\captionof{figure}{The figure shows number of epochs and number of measurement locations. For each measurement configuration and regularization technique the model is optimized 10 times. The variation in the number of epochs for each measurement and regularization technique is due to different priors of the weights and mini-batch sampling.}\\label{Epochs_per_measurement}\n\\end{figure}\n\n\\section{Discussion}\\label{discussion}\nWe have presented the SCVAE method for efficient data reconstruction based on sparse observations. The derived objective functions for the network optimization  show that the encoding is independent of measurements. This allows for a simpler model structure with fewer model parameters than a CVAE and results in an optimization procedure that requires less computations.  \\\\\n\nWe have shown that the SCVAE is suitable for reconstruction of fluid flow. The method is showcased on two different data sets, velocity data from simulations of 2D flow around a cylinder, and bottom currents from the BOM. The fact that the fluids studied in the experiments are incompressible served as a motivation for adding an extra term to the objective function, see \\cref{obj_function_SCVAE} with $\\lambda>0$. \\\\\n\nOur investigation of additional regularization showed that the mean reconstruction error over all models was lower with $\\lambda>0$ compared to the model where $\\lambda=0$, but the best reconstruction error was similar for $\\lambda=0$ and $\\lambda>0$. In \\cref{Experiment} we optimized 10 models for every experiment, and chose the model that performed best on the validation data sets. With this approach we did not observe significant differences between optimizing with  $\\lambda=0$ and $\\lambda>0$. However, the reconstruction became less sensitive to the stochasticity involved in optimization (minibatch selection, network weights priors) when the regularization was used, see \\cref{A_Motivating_Example}. \\\\ \n\nThe SCVAE is a probabilistic model, which allows to make predictions, estimate their uncertainty, see \\cref{sec:posterior},  and draw multiple samples from the predictive distribution, see \\cref{z_space_sampling_v}. The last two properties make the SCVAE a useful method especially when the predictions are used in another application, i.e., ensemble simulation of tracer transport. Motivated by \\cite{YILDIRIM2009160}, we compared the SCVAE predictions with the predictions of a modified GPOD method, see \\cref{Appendix_C}. \\\\\n\nUnlike the GPOD-method, a benefit with the SCVAE-method is that it scales well to larger data sets. Another aspect and as the experiments in \\cref{Experiment} suggest, the GPOD seems more sensitive to the number of measurement locations than the SCVAE. On the other hand, the experiments suggested that GPOD is better than SCVAE with a larger number of measurement locations if the training data and the test data are too different, see BOM experiment with sequential splitting \\cref{BOM_results}. Essentially the SCVAE overfit to the training data, and as a result performing poorly on the test data set. This fact shows the importance of training the the SCVAE on large data sets, which covers as many potential flow patterns as possible. Further, the results show that the GPOD is more sensitive to the measurement location choice than the SCVAE, see \\cref{A_Motivating_Example}, and the GPOD-method is not expected to preform well on a complex flow with very few fixed measurement locations. \\\\\n\nVAEs has been used for generating data in e.g. computer vision \\cite{kingma2013auto}, and auto-encoders is a natural to use in reconstruction tasks \\cite{ELMS2018}. Many reconstruction approaches, including the GPOD approach, first create a basis, then use the basis and minimize the error of the observations \\cite{willcox2006unsteady, bui2004aerodynamic}. This makes the GPOD suitable for fast optimization of measurement locations that minimize the reconstruction error. On the other hand,  the SCVAE optimizes the basis function given the measurements, i.e. they are known and fixed. This makes it challenging to use the framework for optimizing sensor layout. But if the measurement locations are fixed and large amounts of training data are available, the SCVAE outperforms the GPOD for reconstruction. SCVAE optimize the latent representation and the neural network model parameters, variational and generative parameters, given the measurements. This ensures that the reconstruction is adapted to the specific configuration of measurements. \\\\\n\nA limitation of our experiments is that we used only $100$ and $200$ samples and constructed the confidence region under further simplifying assumptions. The uncertainty estimate could be improved by increasing the sample size and better model for the confidence region. \\\\\n\nNatural applications for the SCVAE are related to environmental data, where we often have sparse measurements. It is for example possible to optimize sensor layout to best possible detect unintentional discharges in the marine environment by using a simple transport model \\cite{oleynik2020optimal}. Oleynik. et al. used deterministic flow fields to transport the contaminant and thus obtain a footprint of the leakage. SCVAE can be used to improve that method and efficiently generate probabilistic footprints of a discharges. This may be important as input to design, environmental risk assessments, and emergency preparedness plans. \n\nWe have highlighted the SCVAE through the reconstruction of currents and flow field reconstruction, however, the SCVAE method is not limited to fluid flow problems. For instance, the same principles could be used in computer vision to generate new picture based on sparse pixel representations or in time series reconstruction. \\\\\n\nA natural extension of the SCVAE is to set it up as a partially hidden Markov model. That is to predict the current state $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_t|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi_t, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_{t-1}),$ given the measurements and the reconstruction from the previous time step. This could potentially improve the reconstruction further.\n\n\\section*{Acknowledgements}\nThis work is part of the project ACTOM, funded through the ACT programme (Accelerating CCS Technologies, Horizon2020 Project No 294766). Financial contributions made from; The Research Council of Norway, (RCN), Norway, Netherlands Enterprise Agency (RVO), Netherlands, Department for Business, Energy \\& Industrial Strategy (BEIS) together with extra funding from NERC and EPSRC research councils, United Kingdom, US-Department of Energy (US-DOE), USA. Kristian Gundersen has been supported by the Research Council of Norway, through the CLIMIT program (project 254711, BayMode) and the European Union Horizon 2020 research and innovation program under grant agreement 654462, STEMM-CCS. The authors would like to acknowledge NVIDIA Corporation for providing their GPUs in the academic GPU Grant Program.\n\n\\clearpage\n\n\\bibliographystyle{unsrt}\n\\clearpage\n\n\\section{Introduction}\nReconstruction of non-linear dynamic processes based on sparse observations is an important and difficult problem. The problem traditionally requires knowledge of the governing equations or processes to be able to generalize from the the sparse observations to a wider area around, in-between and beyond the measurements. Alternatively it is possible to learn the underlying processes or equations based on data itself, so called data driven methods. In geophysics and environmental monitoring measurements is often only available at sparse locations. For instance, within the field of meteorology, atmospheric pressures, temperatures and wind are only measured at limited number of stations. To produce accurate and general weather predictions, requires methods that both forecast in the future, but also reconstruct where no data is available. Within oceanography one faces the same problem, that in-situ information about the ocean dynamics is only available at sparse locations such as buoys or sub-sea sensors. \\\\\n\nBoth the weather and ocean currents can be approximated with models that are governed by physical laws, e.g. the Navier-Stokes Equation. However, to get accurate reliable reconstructions and forecasts it is of crucial importance to incorporate observations. \\\\ \n\nReconstruction and inference based on sparse observations is important in many applications both in engineering and physical science \\cite{brunton2015closed, kong2018application, bolton2019applications, venturi2004gappy,callaham2018robust, manohar2018data}. Bolton et. al. \\cite{bolton2019applications}  used convolutional neural networks to hindcast ocean models, and in \\cite{yeo2019data} K. Yeo reconstructs time series of nonlinear dynamics from sparse observation.  Oikonomo et. al. \\cite{oikonomou2018novel} proposed a method for filling data gaps in groundwater level observations and Kong. et. al \\cite{kong2018application} used reconstruction techniques to modeling the characteristics of cartridge valves.  \\\\\n\nThe above mentioned applications are just some of the many examples of reconstruction of a dynamic process based on limited information. Here we focus on reconstruction of flow. This problem can be formulated as follows. Let $\\bm w \\in {\\mathbb R}^d,$ $d \\in {\\mathbb N},$ represent a state of the flow, for example velocity, pressure, temperature, etc. Here, we will focus on incompressible unsteady flows and ${\\bm w}=(u,v)\\in {\\mathbb R}^2$ where $u$ and $v$ are the horizontal and vertical velocities, respectively. The velocities ${\\bm w}$ are typically obtained from computational fluid dynamic simulations on a meshed spatial domain $\\mathcal{P}$ at discrete times $\\mathcal{T} = \\{t_1,...,t_K\\} \\subset {\\mathbb R}$. \\\\\n\nLet $ \\mathcal{P}=\\{p_1,...,p_N\\}$ consist of $N$ grid points $p_n,$ $n=1,...,N.$  \nThen the state of the flow $\\bm w$ evaluated on ${\\mathcal P}$ at a time $t_i \\in {\\mathcal T}$ can be represented as a vector \n$\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N},$ \n\\begin{equation}\n\\label{eq:x_i}\n\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}=(u(p_1,t_i),...,u(p_N,t_i), v(p_1,t_i),...,v(p_N,t_i))^T.\n\\end{equation}\nThe collection of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,K,$ constitutes the data set $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi.$ In order to account for incompressibility, we introduce a discrete divergence operator $L_{div}$, which is given by a $N \\times 2N$ matrix associated with a finite difference scheme, and\n\\begin{equation}\n\\label{eq:L_div}\n(L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)_{k} \\approx (\\nabla \\cdot w)(p_k)=0.\n\\end{equation}\n\nFurther, we assume that the state can be measured only at specific points in ${\\mathcal P},$ that is, at ${\\mathcal Q}=\\{q_1,...,q_M\\} \\subset {\\mathcal P}$ where $M$ is typically much less than $N.$  Hence, there is $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in {\\mathbb R}^{2M}: \\, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C\\,\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\, \\forall \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi \\},$ where ${\\bm C} \\in {\\mathbb R}^{2M \\times 2N}$ is a sampling matrix. More specifically, $\\bm C$ is a two block matrix \n\n$$\n{\\bm C}=\\begin{pmatrix}\n{\\bm C}_{1\/2}& O\\\\\nO&{\\bm C}_{1\/2}\\\\\n\\end{pmatrix}, \\quad \n({\\bm C}_{1\/2})_{ij}= \\left\\{\n\\begin{array}{ll}\n1,& \\mbox{if } \\, q_i =p_j\\\\\n0,& \\mbox{otherwise}\n\\end{array} \\right. , \\quad i=1,...,N \\quad j=1,...,M,\n$$\nand ${\\bm O}\\in {\\mathbb R}^{M\\times N}$ is a zero matrix. The problem of reconstructing fluid flow $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}\\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ is presented as a schematic plot in \\cref{Sketch_map}.\n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.70\\linewidth]{Subset_M_of_D_v2_new_notation_i.png}\n\t\t\\captionof{figure}{Sketch of reconstruction of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ from $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$. The dots on the right side represent the grid ${\\mathcal P}$, and those on  the left side represent the measurement locations ${\\mathcal Q}.$}\\label{Sketch_map}\n\n\\end{figure}\nThere have been a wide range of methods for solving the problem, e.g. \\cite{sirovich1987turbulence, everson1995karhunen, donoho2006compressed, schmid2010dynamic, ELMS2018, raissi2019physics}. In particular, use of proper orthogonal decomposition (POD) \\cite{sirovich1987turbulence} techniques has been popular. \\\\ \n\nPOD \\cite{sirovich1987turbulence} is a traditional dimensional reduction technique where based on a data set, a number of basis functions are constructed. The key idea is that a linear combination of the basis functions can reconstruct the original data within some error margin, efficiently reducing the dimension of the problem. In a modified version of the POD, the Gappy POD (GPOD) \\citep{everson1995karhunen}, the aim is to fill the gap in-between sparse measurements. Given a POD basis one can minimize the $L_2$-error of the measurements and find a linear combination of the POD-basis that complements between the measurements. If the basis is not know, a iterative scheme can be formulated to optimize the basis based on the measurements. The original application of GPOD \\citep{everson1995karhunen} was related to reconstruction of human faces, and it has later been applied to fluid flow reconstruction \\cite{venturi2004gappy}. We will use the GPOD approach for comparison later in this study. \\\\ \n\nA similar approach is the technique of Compressed Sensing (CS) \\cite{donoho2006compressed}. As for the GPOD method, we want to solve a  linear system. However, in the CS-case this will be a under-determined linear system. That is we need some additional information about the system to be able to solve it, typically this can be a condition\/constraint related to the smoothness of the solution. The core difference between CS and GPOD is however the sparsity constraint. That is, instead of minimizing the L2-norm, we minimize the L1-norm. Minimizing the L1-norm favours sparse solutions, i.e. solutions with a small number of nonzero coefficients. \\\\ \n\nAnother reconstruction approach is Dynamical Mode Decomposition (DMD) \\cite{schmid2010dynamic}. Instead of using principal components in the spatial domain, DMD seek to find modes or representations that are associated with a specific frequency in the data, i.e. modes in the temporal domain. Again, the goal is to find a solution to an undetermined linear system and reconstruct based on the measurements, by minimizing the error of the observed values. \\\\\n\nDuring the last decade, data driven methods have become tremendously popular, partly because of the growth and availability of data, but also driven by new technology and improved hardware. To model a non-linear relationships with linear approximations is one of the fundamental limitation of the DMD, CS and GPOD. Recently we have seen development in methods where the artificial neural networks is informed with a physical law, the so called physic-informed neural networks (PINN) \\cite{raissi2019physics}. In PINNs the reconstruction is informed by a Partial Differential Equation (PDE) (e.g. Navier Stokes), thus the neural network can learn to fill the gap between measurements that are in compliance with the equation. This is what Rassi et. al. \\cite{raissi2018hidden} have shown for the benchmark examples such as flow around a 2D and 3D cylinder. Although PINNs are showing promising results, we have yet to see applications to complex systems such as atmospheric or oceanographic systems, where other aspect have to be accounted for, e.g. in large scale oceanic circulation models that are driven by forcing such as tides, bathymetry and river-influx. That being said, these problems may be resolved through PINNs in the future. Despite the promise of PINNs, they will not be a part of this study, as our approach is without any constraint related to the physical properties of the data. \\\\\n\nAnother non-linear data driven approaches for reconstruction of fluid flow are different variations of auto-encoders \\cite{ELMS2018, grover2019uncertainty}. An auto-encoder \\cite{Rumelhart86_autoencoder} is a special configuration of an artificial neural network that first encodes the data by gradually decreasing the size of the hidden layers. With this process, the data is represented in a lower dimensional space. A second neural network then takes the output of the encoder as input, and decodes the representation back to its original shape. These two neural networks together constitute an auto-encoder. Principal Component Analysis (PCA) \\cite{pearson1901_PCA} also represent the data in a different and more compact space. However, PCA reduce the dimension of the data by finding orthogonal basis functions or principal components through singular value decomposition. In fact, it has been showed with linear activation function, PCA and auto-encoders produces the same basis function \\cite{bourlard1988auto}. Probabilistic version of the auto-encoder are called Variational Auto-Encoders (VAEs) \\cite{kingma2013auto}. CVAEs \\cite{sohn2015learning} are conditional probabilistic auto-encoders, that is, the model is dependent on some additional information such that it is possible to create representations that are depend on this information. \\\\\n\nHere, we address the mentioned problem from a probabilistic point of view. Let $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi: {\\mathcal P} \\to {\\mathbb R}^{2N}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi: {\\mathcal Q} \\to {\\mathbb R}^{2M}$ be two multivariate random variables associated with the flow on ${\\mathcal P}$ and on ${\\mathcal Q}$, respectively. Then the data sets $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ consist of the realizations of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi$, respectively. Using $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ and $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi,$ we intend to approximate the probability distribution $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ This would not only allow to predict $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},$ but also to estimate an associated uncertainty. In this paper, we use a variational auto-encoder to approximate $p(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi| \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi)$. The method we use is a Bayesian Neural Network \\cite{MacKay92} approximated through variational inference \\cite{hoffman2013stochastic,blei2017variational}, that we have called \\textit{Semi-Conditional Variational Auto-encoder}, SCVAE. A detailed description of the SCVAE method for reconstruction and associated uncertainty quantification is given in \\cref{SCVAE_section}. \\\\\n\nHere we focus on fluid flow, being the main driving mechanism behind transport and dilution of tracers in marine waters. The world's oceans are  under tremendous stress \\citep{Halpern:2012hs}, UN has declared 2021-2030 as the ocean decade\\footnote{\\url{https:\/\/en.unesco.org\/ocean-decade}}, and an ecosystem based Marine Spatial Planning initiative has been launched by IOC \\citep{DominguezTejo:2016dt}. \\\\\n\nLocal and regional current conditions determines transport of tracers in the ocean \\cite{drange2001ocean,BARSTOW1983211}. Examples are accidental release of radioactive, biological or chemical substances from industrial complexes, e.g. organic waste from fish farms in Norwegian fjords \\citep{Ali:2011hd}, plastic \\cite{Law:2017}, or other contaminants that might have adverse effects on marine ecosystems \\citep{Hylland:2015gt}. \\\\ \n\nTo be able to predict the environmental impact of a release, i.e. concentrations as function of distance and direction from the source, requires reliable current conditions \\citep{Ali:2016go,Blackford:2020}. Subsequently, these transport predictions support design of marine environmental monitoring programs \\citep{Hvidevold:2015,Hvidevold:2016cx,Alendal:2017b, oleynik2020optimal}. The aim here is to model current conditions in a probabilistic manner using SCVAEs. This allows for predicting footprints in a Monte Carlo framework, providing simulated data for training networks used for, e.g., analysing environmental time series \\cite{gundersen2020binary}.\\\\ \n\nIn this study we will compare results  with the GPOD method \\cite{willcox2006unsteady}. We are aware that there recent methods (e.g. PINNS and traditional Auto-encoder) that may perform better on the specific data sets than the GPOD, however, the GPODs simplicity, versatility and not least its popularity \\cite{jo2019effective, mifsud2019fusing, callaham2019robust}, makes it a great method for comparison. \\\\\n\nThe reminder of this manuscript is outlined in the following: \\cref{A_Motivating_Example} presents a motivating example for the SCVAE-method in comparison with the GPOD-method. In \\cref{methods} we review both the VAE and CVAE method and present the SCVAE. Results of experiments on two different data sets are presented in \\cref{Experiment}. \\cref{discussion} summarize and discuss the method, experiments, drawbacks and benefits and potential extensions and further work.  \n\n\\section{A Motivating Example}\\label{A_Motivating_Example}\nHere we illustrate the performance of the proposed method vs the GPOD method in order to give a motivation for this study. We use simulations of a two dimensional viscous flow around a cylinder at the Raynolds number of $160,$ obtained from \\url{https:\/\/www.csc.kth.se\/~weinkauf\/notes\/cylinder2d.html}. The simulations were performed  by Weinkauf et. al. \\cite{weinkauf2010streak} with the Gerris Flow Solver software \\cite{gerrisflowsolver}. The data set consists of a horizontal $u$ and a vertical $v$ velocities on an uniform $400 \\times 50 \\times 1001$ grid of $[-0.5, 7.5] \\times [-0.5, 0.5] \\times [15, 23]$ spatial-temporal domain.\\footnote{The simulations are run from $t=0$ to $t=23$, but velocities are only extracted from $t=15$ to $t=23$} In particular, we have $400$ points in the horizontal, and $50$ points in the vertical  direction, and $1001$ points in time.  \n\\begin{figure}[h]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_original_data_u.png}\n    \t\\includegraphics[width=0.99\\linewidth]{CW_original_data_v.png}\n\t\t\\captionof{figure}{Typical data instance from the original 2D flow around a cylinder data set with $u$ and $v$ component presented at the upper and lower panel, respectively} \\label{Cw_data_motivating}\n\n\\end{figure}\nThe cylinder has the diameter of $0.125$ and is centered at the origin, see \\cref{Cw_data_motivating}. The left vertical boundary (inlet) has Dirichlet boundary condition $u=1$ and $v=0$. The homogeneous Neumann boundary condition is given at the right boundary (outlet), and the homogeneous Dirichlet conditions on the remaining boundaries. At the start of simulations, $t=0$, both velocities were equal to zero.  We plot the velocities at the time $t \\approx 19$ (time step $500$) in \\cref{Cw_data_motivating}. \\\\\n\nFor simplicity, in the experiment below we extract the data downstream from the cylinder, that is, from grid point $40$ to $200$ in the horizontal direction, and keep all grid points in vertical direction. Hence, $\\mathcal{P}$ contains $N = 8000$ points, $160$ points in the horizontal and $50$ in the vertical direction. The temporal resolution is kept as before, that is, the number of time steps in $\\mathcal{T}$ is $K=1001$. For validation purposes, the data set was split into a train, validation and test data set. The train and validation data sets were used for optimization of the model parameters. For both the SCVAE and the GPOD, the goal was to minimize the $L2$ error between the true and the modeled flow state.  \nThe restriction of the GPOD is that the number of components $r$ could be at most $2M.$ \nTo deal with this problem, and to account for the flow incompressibility, we added the  regularization term  $\\lambda \\|L_{div} x^{(i)}\\|,$ $\\lambda>0$, to the objective function, see \\cref{Appendix_C}. For the GPOD method, the parameters $r$ and\/or $\\lambda$ where optimized on the validation data set in order to have the smallest mean error.\nWe give more details about objective functions for the SCVAE in \\cref{SCVAE_section}. For now we mention that there are two versions, where one version uses an additional divergence regularization term similar to GPOD.\\\\\n\nIn \\cref{Error_boxplot_CW} we plot the mean  of the relative $L_2$ error calculated on the test data for both methods with and without the div-regularization. The results are presented for $3,$ $4,$ and $5$ measurement locations, that is, $M=3,4,5.$ For each of these three cases, we selected $20$ different configurations of $M.$. In particular, we created $20$ subgrids ${\\mathcal Q}$, each containing $5$ randomly sampled spatial grid points. Next we removed one and then two points from each of the $20$ subgrids ${\\mathcal Q},$ to create new subgrids of $4$ and $3$ measurements, respectively.\n\\begin{figure}[h!]\n    \\centering\n\t\\includegraphics[width=0.75\\linewidth]{Motivating_example_boxplot_3M_v1.png}\n\t\t\\captionof{figure}{The mean relative error for two reconstruction methods. \n\t\tThe orange and blue label correspond to the SCVAE with (div-on) and without (div-off) additional divergence regularization. The green and red labels correspond to the GPOD method. }\\label{Error_boxplot_CW}\n\\end{figure}\nAs it can be seen in \\cref{Error_boxplot_CW}, both methods perform well for the $5$ measurements case. The resulting relative errors have comparable mean and variance. When reducing the number of observations, the SCVAE method maintains low errors, while the GPOD error increases. The SCVAE seems to benefit from the additional regularization of minimizing the divergence, in terms of lower error and less variation in the error estimates. The effect is more profound with fewer measurements. \\\\\n\nThe key benefit of the SCVAE is that its predictions are optimal for the given measurement locations.  In a contrast, the POD based approaches, and in particular the GPOD, create a set of basis functions (principal components) based on the training data independently of the measurements. While this has an obvious computational advantage, the number of principle components for complex flows can be high and, as a result, many more measurements are needed, \\cite{willcox2006unsteady,manohar2018data,Proctor2014}. There are number of algorithms that aim to optimize to measurement locations to achieve the best performance of the POD based methods, see e.g., \\cite{jo2019effective,willcox2006unsteady,YILDIRIM2009160}. In practice, however, the locations are often fixed and another approaches are needed. The results in \\cref{Error_boxplot_CW} suggest that the SCVAE could be one of these approaches.\n\n\\section{Methods}\\label{methods}\nBefore we introduce the model used for reconstruction of flows, we give a brief introduction to VAEs and CVAEs. For a detailed introduction, see \\cite{vae_intro}. VAEs are neural network models that has been used for learning structured representations in a  wide variety of applications, e.g., image generation \\cite{gregor2015draw}, interpolation between sentences \\cite{bowman2015generating} and compressed sensing \\cite{grover2019uncertainty}. \n\n\\subsection{Preliminary}\nLet us assume that the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is generated by a random process that involves an unobserved continuous random variable $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.$ The process consists of two steps: (i) a value $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ is sampled from a prior $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi);$ and (ii)  $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ is generated from a conditional distribution  $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ In the case of flow reconstruction, $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ could be thought of as unknown boundary or initial conditions, tidal and wind forcing, etc. However, generally $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ is just a convenient construct to represent $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ rather than a physically explained phenomena. Therefore it is for convenience assumed that $p_{\\theta^*}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta^*}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ come from parametric families of distributions $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ and $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ and their density functions are differentiable almost everywhere w.r.t. both $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$ and $\\theta$. A probabilistic auto-encoder is neural network that is trained to represent its input $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ as $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ via \\textit{latent representation} $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi),$ that is,\n\n\\begin{equation} \\label{eq:p_theta(x)}\np_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi) = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi,\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi = \\int p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{equation}\nAs $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)$ is unknown and observations $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ are not accessible, we must use $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ in order to generate $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ That is, the network can be viewed as consisting of two parts: an \\textit{encoder} $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ and a \\textit{decoder} $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi).$ Typically the true posterior distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ is intractable but could be approximated with variational inference \\cite{hoffman2013stochastic,blei2017variational}.  That is, we define a so called recognition model $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$ with variational parameters $\\phi$, which aims to approximate $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi).$ The recognition model is often parameterized as a Gaussian. Thus, the problem of estimating $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi)$, is reduced to finding the best possible estimate for $\\phi$, effectively turning the problem into an optimization problem. \\\\\n\nAn auto-encoder that uses a recognition model is called Variational Auto-Encoder (VAE). In order to get good prediction we need to estimate the parameters $\\phi$ and $\\theta.$ The marginal likelihood  is equal to the sum over the marginal likelihoods of the individual samples, that is, $\\sum_{i=1}^K \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Therefore, we further on present estimates for an individual sample. The Kullback -Leibler divergence between two probability distributions  $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$, defined as $$D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] = \\int q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\log\\left(\\frac{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}{p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})}\\right) d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,$$ can be interpreted as a measure of distinctiveness between these two distributions \\cite{kullback1951information}. It can be shown, see \\cite{vae_intro}, that\n\\begin{equation}\\label{eq:DKL_via_L}\n\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})=D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})] +\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}),\n\\end{equation}\nwhere\n$$\n\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = \n{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ -\\log q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})+\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right].\n$$\nSince KL-divergence is non-negative, we have $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) \\geq\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$ and\n$\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})$\nis called Evidence Lower Bound (ELBO) for the marginal likelihood $\\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$\nThus, instead of maximizing the marginal probability, one can instead maximize its variational lower bound to which we also refer as an objective function. It can be further shown that the ELBO can be written as\n\\begin{equation}\n\t\\label{eq:VLB:2_no_beta}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)].\n\\end{equation}\nReformulating the traditional VAE framework as a constraint optimization problem, it is possible to obtain the $\\beta$-VAE \\cite{higgins2016beta} objective function if $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) =\\mathcal{N}({\\bf 0},{\\bm I}),$\n\\begin{equation}\n\t\\label{eq:VLB:2}\n\t\\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right] - \\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)],\n\\end{equation}\nwhere $\\beta>0.$ Here $\\beta$ is a regularisation coefficient that constrains the capacity of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi) \\right]$ can be interpreted as the reconstruction term, while the KL-term, $\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]$ as regularization term. \n\n\nConditional Variational Auto-encoders \\cite{sohn2015learning} (CVAE) are similar to VAEs, but differ by conditioning on an additional property of the data (e.g. a label or class), here denoted $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$. Conditioning both the recognition model and the true posteriori on both $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ and $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ results in the CVAE ELBO \n\\begin{align}\n    \\begin{split}\n        \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)={\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi) \\right] - D_{KL}[q(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)]. \\label{CVAE_objective}\n    \\end{split}\n\\end{align}\nIn the decoding phase, CVAE allows for conditional probabilistic reconstruction and permits sampling from the conditional distribution $p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi)$, which has been useful for generative modeling of data with known labels, see \\cite{sohn2015learning}. Here we investigate a special case of the CVAE when $\\ifmmode\\bm{c}\\else\\textbf{\\textit{c}}\\fi$ is a partial observation of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi.$ We call this Semi Conditional Variational Auto-encoder (SCVAE).\n\n\\subsection{Semi Conditional Variational Auto-encoder}\\label{SCVAE_section}\nThe SCVAE takes the input data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$, conditioned on $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$ and approximates the probability distribution $p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi,\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi).$ Then we can generate $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$, based on the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. As $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}=C \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ where $C$ is a non-stochastic sampling matrix, we have \n$$p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\, \\mbox{ and } \\, q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$$ Therefore, from \\cref{CVAE_objective} the ELBO for SCVAE is \n\\begin{equation}\n    \\begin{split}\n        \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})\\geq \\mathcal{L}(\\theta,\\phi;\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})=\n        &{\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\\\ & - D_{KL}[q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})]\\label{eq:ELBO:SCVAE_no_beta}\n    \\end{split}\n\\end{equation}\nwhere $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ Similarly as for the $\\beta$-VAE \\cite{higgins2016beta} we can obtain a relaxed version of \\cref{eq:ELBO:SCVAE_no_beta} by maximizing the parameters $\\{\\phi, \\theta\\}$ of the expected log-likelihood ${\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])$ and treat it as an constrained optimization problem. That is, \n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\leq \\epsilon  \n    \\end{split}\\label{beta_SCVAE_opt_prob}\n\\end{align}\nwhere $\\epsilon>0$ is small. The subscript $q_\\phi(\\cdot)$ is short for $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ Since $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is dependent on $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ we have that $q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}) = q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}).$ \\cref{beta_SCVAE_opt_prob}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions \\cite{kuhn2014nonlinear, karush1939minima}. Hence,\n\\begin{align}\n    \\begin{split}\n    \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & = \n    {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)]) \\\\ & + \n    \\beta(D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})-\\epsilon)   \n    \\end{split}\\label{beta_SCVAE_constrained}\n\\end{align}\nAccording to the complementary slackness KKT condition $\\beta \\geq 0,$ we can rewrite \\cref{beta_SCVAE_constrained} as\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}).   \n    \\end{split}\\label{beta_SCVAE_constrained_2}\n\\end{align}\nObjective functions in \\cref{eq:ELBO:SCVAE_no_beta} and \\cref{beta_SCVAE_constrained_2}, and later \\cref{eq:ELBO:SCVAE:J}, show that if conditioning on a feature which is a known function of the original data, such as measurements, we do not need to account for them in the encoding phase.The measurements are then coupled with the encoded data in the decoder. We sketch the main components of the SCVAE in \\cref{DAE_neural_network_sketch}.\n\\begin{figure}[ht]\n    \\centering\n\t\t\\includegraphics[width=0.80\\linewidth]{Model_Sparse_autoencoder_new_notation_i.png}\n\t\t\\captionof{figure}{The figure shows a sketch of the model used to estimate \n\t\t$p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)})$. During training both the observations $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ and the data $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ will be used. After the model is trained, we can predict using only the decoder part of the neural network. The input to the decoder will then only be the observations and random samples from the latent space.}\\label{DAE_neural_network_sketch}\n\\end{figure}\nIn order to preserve some physical properties of the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ we can condition yet on another feature. Here we utilize the incompressibility property of the fluid, i.e., $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}=L_{div} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\approx 0,$ see \\cref{eq:L_div}. \\\\ \n\nWe intend to maximize a log-likelihood under an additional constrain $\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}$, compared to \\cref{beta_SCVAE_opt_prob}. That is\n\\begin{align}\n    \\begin{split}\n    & \\max\\limits_{\\phi,\\theta} {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\text{ subject to} \\\\ &\n    D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\leq \\epsilon \\quad \\text{and} \\quad \\\\ &\n    -{\\mathbb E}_{q_\\phi(\\cdot)}[\\log p_\\theta(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)] \\leq \\delta \n    \\end{split}\\label{Constraint_optimization_prob_v3}\n\\end{align}\nwhere $\\epsilon, \\delta>0$ are small. \\cref{Constraint_optimization_prob_v3}  can expressed as a Lagrangian under the Karush\u2013Kuhn\u2013Tucker (KKT) conditions as before and as a consequence of the complementary slackness condition $\\lambda,\\beta \\geq 0,$ we can obtain the objective function \n\n\\begin{align}\n    \\centering\n    \\begin{split}\n      \\mathcal{F}(\\theta, \\phi, \\beta, \\alpha, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) \\geq  \\mathcal{L}(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) & =  {\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & +\n     \\lambda\\,{\\mathbb E}_{q_\\phi(\\cdot)} [\\log p_\\theta (\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi)])  \\\\ & -\n     \\beta D_{KL}(q_\\phi(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_\\theta (\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}),   \n    \\end{split}\\label{eq:ELBO:SCVAE:J}\n\\end{align}\nwhere $p(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}) = \\mathcal{N}({\\bf 0},{\\bf I}).$ For convenience of notation we refer to the objective function \\cref{beta_SCVAE_constrained_2} as the case with $\\lambda=0$, and the objective function \\cref{eq:ELBO:SCVAE:J} as the case with $\\lambda > 0.$ Observe that under the Gaussian assumptions on the priors, \\cref{eq:ELBO:SCVAE:J} is equivalent to \\cref{beta_SCVAE_constrained_2} if $\\lambda=0.$ Thus, from now one we will refer to it as a special case of \\cref{eq:ELBO:SCVAE:J} and denote as $\\mathcal{L}_{0}.$ \\\\\n\nSimilarly to \\cite{kingma2013auto} we obtain $q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})= \\mathcal{N}({\\mu}^{(i)} \\mathbf{1},(\\sigma^{(i)})^2\\,\\mathbf{I})\n$, that is, $\\phi=\\{ \\mu, \\sigma\\}.$  This allows to express the KL-divergence terms in a closed form and avoid issues related to differentiability of the ELBOs.  Under these assumptions, the KL-divergence terms can be integrated analytically while the term\n${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $ and ${\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] $\nrequires estimation by sampling\n\\begin{equation}\\label{eq:E-estimate}\n    \\begin{array}{l}\n         {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        {\\mathbb E}_{q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})} \\left[ \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\right] \\approx \n        \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}),\\\\\n        \\mbox{where } \\, \\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)} = g_{\\phi}(\\bm{\\epsilon}^{(i,l)}, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}), \\quad \n        \\bm{\\epsilon}^{l} \\sim p(\\bm{\\epsilon}).\n    \\end{array}\n\\end{equation}\nHere $\\bm{\\epsilon}^{l}$ is an auxiliary (noise) variable with independent marginal $p(\\bm{\\epsilon})$, and $g_{\\phi}(\\cdot)$ is a differentiable transformation of $\\bm{\\epsilon},$ parametrized by $\\phi,$  see for details \\cite{kingma2013auto}. We denote $\\mathcal{L}_\\lambda,$ $\\lambda \\geq 0$  \\cref{eq:ELBO:SCVAE:J} with the approximation above as $\\mathcal{\\widehat{L}_\\lambda},$ that is,\n\\begin{align}\n    \\begin{split}\n        & \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})  =   \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log  p_{\\theta}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) \\\\\n         &+ \\lambda \\frac{1}{L}\\sum\\limits_{l=1}^{L} \\log p_{\\theta}(\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)}|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i,l)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}) -\\beta D_{KL}[q_{\\phi}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)})||p_{\\theta}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})].\n        \\label{Loss_function}\n    \\end{split}\n\\end{align}\nThe objective function $\\widehat{\\mathcal{L}}_\\lambda$ can be maximized by gradient descent. Since the gradient $\\nabla_{\\theta,\\phi}\\,\\widehat{\\mathcal{L}}_\\lambda$ cannot be calculated for large data sets, Stochastic Gradient Descent methods, see \\cite{kiefer1952stochastic, robbins1951stochastic} are typically used where \n\\begin{equation}\n    \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi, \\bm{D}) \\approx \\widehat{\\mathcal{L}}^{R}(\\theta, \\phi; \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^R, \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^R, \\bm{D}^R) = \n    \\frac{K}{R}\\sum\\limits_{r=1}^{R} \\widehat{\\mathcal{L}}_\\lambda(\\theta, \\phi; \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}), \\quad \\lambda \\geq 0.\\label{obj_function_SCVAE}\n\\end{equation}\nHere  $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi^{R}=\\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i_r)}\\right\\}_{r=1}^{R},$  $R<K$ is a minibatch consisting of randomly sampled datapoints, $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi^{R}=\\left\\{\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}$ and $\\bm{D}^{R}=\\left\\{\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i_r)}\\right\\}_{r=1}^{R}.$ After the network is optimized, a posterior predictive distribution $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi)$ can be approximated with a Monte Carlo estimator. \\\\\n\n\n\\subsubsection{Uncertainty Quantification}\\label{sec:posterior}\nLet  $\\hat{\\theta}$ and $\\hat{\\phi}$ be an estimation of generative and variational parameters, as described in \\cref{SCVAE_section}. Then the decoder can be used to predict the posterior as\n\\begin{align}\\label{eq:p_pred}\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\approx \\frac{1}{N_{MC}} \\sum_{j=1}^{N_{MC}} p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(j)},\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*) \\xrightarrow[N_{MC} \\rightarrow \\infty]{} \\int\n    p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi, \\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)p_{\\hat{\\theta}}(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)d\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi.\n\\end{align}\nWhile sampling from the latent space has been viewed typically as an approach for generating new samples with similar properties, here we use it to estimate the prediction uncertainty of the trained model. From \\cref{eq:p_pred} we are able to estimate the mean prediction  $\\hat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and empirical covarience matrix $\\widehat{{\\bm \\Sigma}}$ using  a Monte Carlo estimator. We get\n\\begin{align} \n      \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*=\n      \\frac{1}{N_{MC}} \\sum\\limits_{j=1}^{N_{MC}} \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}\\quad \\mbox{and} \\quad\n      \\widehat{{\\bm \\Sigma}} =  \\frac{1}{N_{MC}-1} \\sum\\limits_{j=1}^{N_{MC}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)} -  \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)^T,\n        \\label{eq:mean_and_cov} \n               \n \\end{align}\nwhere $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(j)}  \\sim p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^*,\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*).$ The empirical standard deviation is then simply $\\widehat{\\bm \\sigma} = \\sqrt{diag(\\widehat{{\\bm \\Sigma}})}.$ To estimate the confidence region we assume that the predicted $p_{\\hat{\\theta}}(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{*},\\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^*)$ is well approximated by a normal distribution $N({\\bm \\mu}, \\bm{\\Sigma}).$ Given that $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*$ and $\\widehat{{\\bm \\Sigma}}$ are approximations of $\\bm{\\mu}$ and $\\bm{\\Sigma},$ obtained from $N_{MC}$ samples as above, a confidence region estimate for a prediction $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{*}$ can be given as\n\\begin{equation}\\label{eq:ConfRegion:1}\n    \\left\\{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in {\\mathbb R}^{2N}: \\,(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}-{\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})^T \\hat{\\bm \\Sigma}^{+} (\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}- \\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*)\\leq \\chi^2_k(p)\\right\\}\n\\end{equation}\nwhere $\\chi^2_k(p)$ is the quantile function for probability $p$ of the chi-squared distribution with $k=\\min\\{N_{MC},2N\\}$ degrees of freedom, and $\\widehat{\\bm \\Sigma}^{+}$ is the pseudoinverse of $\\widehat{\\bm \\Sigma}.$ Using the singular value decomposition, ${\\widehat{\\bm \\Sigma}}=\\bm{U S U}^T,$ the corresponding interval for $(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^*)_n,$ $n=1,\\dots,2N,$ is\n\\begin{equation}\\label{eq:ConfRegion:2}\n   \\left[\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n-\\sqrt{\\chi^2_k(p)}\\,\\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2,\\quad\n ({\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^*})_n+\\sqrt{\\chi^2_k(p)}\\,\n \\|{\\bm u}_n^T\\,{\\bm S}^{1\/2}\\|_2\n  \\right]   \n\\end{equation}\nwhere ${\\bm u}_n^T$ is $n$th row of the matrix ${\\bm U}.$\n\n\n\\section{Experiments}\\label{Experiment}\nWe will present the SCVAE method on two different data sets. The first one is the 2D flow around a cylinder described in \\cref{A_Motivating_Example}, and the second is ocean currents on the seafloor created by the Bergen Ocean Model \\cite{berntsen2000users}. The data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ consists of the two dimensional velocities ${\\bm w} =(u, v).$ To illustrate the results we will plot $u$ and $v$ components of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)} \\in \\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ see \\cref{eq:x_i}. For validation of the models, the data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ is split into train, test and validation subsets, which are subscripted accordingly, if necessary. The data sets, spitting and preprocessing for each case are described in \\cref{Cylinder_wake_experiment,BOM_experiment}. \\\\\n\nWe use a schematically simple architecture to explore the SCVAE. The main ingredient of the encoder is convolutional neural network (CNN)  \\cite{lecun1998gradient, krizhevsky2012imagenet} and for the decoder we use transposed CNN-layers \\cite{noh2015learning}. The SCVAE has a slightly different architecture in each case, which we present in \\cref{Appendix_A}. \\\\\n\nThe SCVAE is trained to maximize the objective functions in \\cref{obj_function_SCVAE} with the backpropagation algorithm \\cite{lecun1989backpropagation} and the Adam algorithm \\cite{kingma2014adam}. We use an adaptive approach of weighing the reconstruction term with KL-divergence and\/or divergence terms \\cite{heydari2019softadapt}, that is, finding the regularization parameters $\\beta$ and $\\lambda$. Specifically, we calculate the proportion of each term contribution of the total value of the objective function, and scale the terms accordingly. This approach prevents posterior collapse. Posteriori collapse occurs if the KL-divergence term becomes too close to zero, resulting in a non probabilistic reconstruction. The approach of weighing the terms proportionally iteratively adjusts the weight of the KL-divergence term, $\\beta$, such that posterior collapse is avoided. For the result shown here we trained the SCVAEs  with an early stopping criteria of $50$ epochs, i.e., the optimization is stopped if we do not see any improvement after $50$ epochs, and returns the best model. We use a two-dimensional Gaussian  distribution for $p_\\theta(\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}, \\ifmmode\\bm{d}\\else\\textbf{\\textit{d}}\\fi^{(i)})$ in all the experiments.\\\\\n\nLet the test data $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi_{test}$ consist of $n$ instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)},$ $i=1,\\dots,n,$ and $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ denote a prediction of the true $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ given $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ In the case of the SCVAE, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is the mean prediction obtained as in \\cref{eq:mean_and_cov}. For the GPOD method, $\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)}$ is a deterministic output of the optimization problem, see \\cref{Appendix_C}. In order to compare the SCVAE results with the results of the GPOD method, we introduce the following measures; the mean of the relative error for the prediction\n\\begin{equation}\\label{L2_error}\n  \\mathcal{E} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\frac{||\\widehat{\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi}^{(i)} - \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}{||\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}||_2}\n\\end{equation}\nand the mean of the absolute error for the divergence \n\\begin{equation}\n        \\mathcal{E}_{div} = \\frac{1}{n}\\sum\\limits_{i=1}^{n}\\| L_{div} \\, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}\\|_2.\\label{divergence_error_1}\n\\end{equation}\n\n\\subsection{2D Flow Around a Cylinder}\\label{Cylinder_wake_experiment}\nHere we return to the example in \\cref{A_Motivating_Example}. Below we give some additional details of the data preprocessing and model implementation.\n\\subsubsection{Preprocessing}\nThe data is reduced as described in \\cref{A_Motivating_Example}. \nWe assess the SCVAE with a sequential split for train, test and validation. The sequantial split is defined as follows. The last $15 \\%$ of the data is used for test, the last $30\\%$ of the remaining data is used for validation, and the first $70\\%$ for training. To improve the conditioning of the optimization problem we scale the data as decribed in  \\cref{Appendix_D}. The errors $\\mathcal{E}$ (\\cref{L2_error}) and $\\mathcal{E}_{div} $ (\\cref{divergence_error_1}) are calculated after scaling the data back. The input to the SVAE $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ was shaped as $(160 \\times 50 \\times 2)$ in order to apply convolutional layers. Here we use  $5,4,3$ and $2$ fixed spatial measurements, that is, four different subgrids ${\\mathcal Q}$ \n\\begin{align}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_5 = & \\{(12,76),(47,8),(30,40),(153,34),(16,10)\\}, \\\\\n    {\\mathcal Q}_4 = & \\{(12,76),(47,8),(30,40),(153,34)\\},  \\\\\n    {\\mathcal Q}_3 = & \\{(12,76),(47,8),(30,40)\\}, \\\\\n    {\\mathcal Q}_2 = & \\{(12,76),(47,8)\\}.\n    \\end{split}\\label{CW_measurements}\n\\end{align}\nThe flow state at these specific locations constitute $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model}\nA schematic description of the model is given in \\cref{Appendix_A}. \nThe first layer of the encoder is a zero-padding layer that expands the horizontal and vertical dimension by adding zeros on the boundaries. Here we add zero-padding of four in the horizontal and three in the vertical direction. The subsequent layers consists of two convolutional layers, where the first and second layer have $160$ and $200$ filters, respectively. We use a kernel size and strides of $2$ in both convolutional layers and ReLu activation functions. This design compresses the data into a $(42 \\times 14 \\times 200)$ shape. The compressed representation from the convolutional layers are flattened and are further compressed into a $64$ dimensional vector through a traditional dense layer. Two outputs layers are defined to represent the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The reparametrization trick is realized in a third layer, a so called  lambda layer, which takes the mean and log-variance as an input and generates $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. The output of the encoder are the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$. \\\\\n\nThe decoder takes the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ as input. The input $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened and then concatenated with $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$. The next layer is a dense layer with shape $(42 \\times 14 \\times 200)$. Afterwards there are two transposed convolutional layers with filters of $200$ and $160$. The strides and the kernel size is the same as for the encoder. The final layer is a transposed convolutional layer, with same dimension as the input to the encoder, that is the dimension of $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. A linear activation function is used for this output layer. The last layer of the model is a lambda layer that removes the zero-padding. In the next section we show statistics of the probabilistic reconstruction and compare with the GPOD method. \n\n\\subsubsection{Results}\nIn \\cref{Cylinder_wake_pred} we have plotted the true velocity field, the reconstructed velocities, the standard deviation of the velocities and the absolute error between the true and reconstructed velocity fields. The observations placement is shown as stars (black and white). The SCVAE with the objective function  $\\widehat{\\mathcal{L}}_0$, see \\cref{CVAE_objective}, was used for this prediction. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $100$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty estimates, see \\cref{eq:mean_and_cov}. \\\\\n\nWe emphasise here again that the SCVAE with $\\widehat{\\mathcal{L}}_0$ and with $\\widehat{\\mathcal{L}}_\\lambda,$ $\\lambda>0,$ are two different models. For the notation sake we here refer to $\\lambda=0$ when we mean the model with the objective function in \\cref{CVAE_objective}, and to $\\lambda>0$ when in \\cref{eq:ELBO:SCVAE:J}. The same holds for the GPOD method, see \\cref{Appendix_D}. When $\\lambda=0,$ the number of the principle components $r$ is less $2M.$ The number $r$ is chosen such that the prediction on the validation data has the smallest possible error on average. If $\\lambda>0,$ no restrictions on $r$ are imposed. In this case both  $\\lambda$ and $r$ are estimated from the validation data. \\\\\n\nThe general observation is that the SCVAE reconstruction fits the data well, with associated low uncertainty. \nThis can be explained by the periodicity in the data. In particular, the training and validation data sets represent the  test data well enough.\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{CW_recon_3M_v1_CR.png}\n\t\\captionof{figure}{Left panels shows the u-velocities, and the right panel v-velocities. The results are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{First panels:} The true solutions \\textbf{Second panels:} Reconstructed solution based on the SCVAE model \\textbf{Third panels:} Standard deviation of the predicted solution \\textbf{Fourth panels:} Absolute error between the true and predicted solution.}\\label{Cylinder_wake_pred}\n\n\\end{figure}\n\nIn \\cref{Cylinder_wake_time_series} we  have plotted four time series of the reconstructed test data at two specific grid points, together with the confidence regions constructed as in \\cref{eq:ConfRegion:2} with $p=0.95.$ The two upper panels represents the reconstruction at the grid point $(6, 31)$, and the lower at $(101,25)$ for $u$ and $v$ on the left and right side, respectively. The SCVAE reconstruction is significantly better than the GPOD, and close to the true solution for all time steps. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_time_series_with_GPOD_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$ The results are obtained from the model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(6, 31).$ \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(101, 25).$ }\\label{Cylinder_wake_time_series}\n\n\\end{figure}\n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.85\\linewidth]{CW_error_with_GPOD_CR.png}\n\t\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange). The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. The estimates are based on a model trained with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(6,31)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(101,25)$ for $u$ (left) and $v$ (right).}\\label{Cylinder_wake_error}\n\n\\end{figure}\n\\cref{Cylinder_wake_error} shows the difference between the true values and the model prediction in time for the same two locations. This figure has to be seen in context with \\cref{Cylinder_wake_pred}. In \\cref{tab:comparison_CW}  we display the relative errors, \\cref{L2_error}, for the SCVAE and the GPOD method, both with and without divergence regularization, for $5, 4, 3,$ and $2$ measurement locations given in \\cref{CW_measurements}. \\\\\n\nThe results of the SCVAE depend on two stochastic inputs which are (i) randomness in the initialization of the prior weights and (ii) random mini batch sampling. We have trained the model with a each measurement configuration $10$ times, and chose the model that performs the best on the validation data set. Ideally we would run test cases where we used all the values as measurements,i.e., $\\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi=\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi,$ and test how well the model would reconstruct in this case. This would then give us the lower bound of the best reconstruction that is possible for this specific architecture and hyper parameter settings. However, this scenario was not possible to test, due to limitations in memory in the GPU. Therefore we have used a large enough $M$ which still allowed us to run the model. In particular, we used every fifth and second pixel in the horizontal and vertical direction, which resulted in a total of $(32 \\times 25)$ measurement locations, or $M=800$. We believe that training the model with these settings, gave us a good indication of the lower bound of the reconstruction error. The error observed was of the magnitude of $10^{-3}$. \\\\\n\nThis lower bound has been reached for all measurement configurations \\cref{CW_measurements}. \nHowever, larger computational cost was needed to reach the lower bound for fewer measurement locations. \\cref{Epochs_per_measurement_CW} shows the number of epochs as a boxplot diagram. In comparison with GPOD, the SCVAE error is 10 times lower than the GPOD error, and this difference becomes larger with fewer measurements. Note that adding regularization did not have much effect on  the relative error. From the motivating example we observed that regularizing with $\\lambda>0$ is better in terms of a more consistent and low variable error estimation. Here we selected from the 10 trained models the one that performed best on the validation data set. This model selection approach shows that there are no significant differences between the two regularization techniques. The associated error in the divergence of the velocity fields is reported in  \\cref{tab:comparison_CW_div}. \n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n         & 0.30e-02 & 0.33e-02 & 0.26e-02 & 0.28e-02    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.31e-02 & 0.32e-02 & 0.30e-02 & 0.28e-02 \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$  & 2.35e-02& 2.49e-02  \n           & 3.38e-02& 17.38e-02\\\\ \\cline{2-6}\n         & $\\lambda > 0$ &   2.12e-02 & 2.33e-02& 3.15e-02& 16.38e-02    \\\\ \\hline\n    \\end{tabular}\n    \\caption{The mean relative error $\\mathcal{E}$ (\\cref{L2_error}) for the  SCVAE prediction  and  the GPOD prediction with or without div-regularization, and different number of measurements.}\n    \\label{tab:comparison_CW}\n\\end{table}\n\\begin{table}[H]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Method} & \\multirow{2}{*}{Regularization} & \\multicolumn{4}{|c|}{Measurement Locations} \\\\  \\cline{3-6} \n          & & 5 & 4 & 3 & 2  \\\\  \\hline\n         \n         \\multirow{2}{*}{SCVAE} & $\\lambda = 0$\n          & 0.1439 & 0.1580 & 0.1383 & 0.1432    \\\\ \\cline{2-6}\n         & $\\lambda > 0$ & 0.1533 & 0.1408 & 0.1468 & 0.1410    \\\\ \\cline{3-6} \\hline\n         \\multirow{2}{*}{GPOD} & $\\lambda = 0$ & 0.1052 & 0.1047 & 0.0943 & 0.08866    \\\\ \\cline{2-6}\n         & $\\lambda > 0$  & 0.1039 & 0.1051 & 0.0966 & 0.0669   \\\\ \\hline\n    \\end{tabular}\n    \\caption{Comparison of the divergence error $\\mathcal{E}_{div}$ as calculated in \\cref{divergence_error_1} for the different methods and regularization techniques. The true divergence error on the entire test data set is $0.1058$}\n    \\label{tab:comparison_CW_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{Epochs_per_Measurement.png}\n \t    \\captionof{figure}{Number of epochs trained depending on the number of measurements. For each measurement configuration and regularization technique the model is run $10$ times. The variation of number of epochs for for each measurement locations is due to different priors of the weights and random mini-batch sampling.}\\label{Epochs_per_measurement_CW}\n\\end{figure}\n\n\\subsection{Current data from Bergen ocean model}\\label{BOM_experiment}\nWe tested the SCVAE on simulations from the Bergen Ocean Model (BOM) \\cite{berntsen2000users}. BOM is a three-dimensional terrain-following nonhydrostatic ocean model with capabilities of resolving mesoscale to large-scale processes. Here we use velocities simulated by Ali. et. al \\cite{Ali:2016go}. The simulations where conducted on the entire North Sea with 800 meter horizontal and vertical grid resolution and 41 layers for the period from 1st to 15th of January 2012. Forcing of the model consist of wind, atmospheric pressure, harmonic tides, rivers, and initial fields for salinity and temperature. For details of the setup of the model, forcing and the simulations we refer to \\cite{Ali:2016go}. \\\\\n\nHere, the horizontal and vertical velocities of an excerpt of 25.6 $\\times$ 25.6 km$^2$ at the bottom layer centered at the Sleipner CO2 injection site ($58.36^\\circ N, \\, 1.91^\\circ E$) is used as data set for reconstruction. In \\cref{Data_set_time_series_BOM} we have plotted the mean and extreme values of $u$ and $v$ for each time $t$ in ${\\mathcal T}$. \n\\begin{figure}[H]\n\t\\centering\n\n\t\t\\includegraphics[width=0.99\\linewidth]{Data_CW_U.png}\n        \\includegraphics[width=0.99\\linewidth]{Data_CW_V.png}\n\t\t\\captionof{figure}{The light-blue line represent the maximum, the orange the minimum and the green mean value of $u$ and $v$ for each time $t$ in ${\\mathcal T}.$ The horizontal lines indicate the sequential data split. }\\label{Data_set_time_series_BOM}\n\n\\end{figure}\n\\subsubsection{Preprocessing}\nWe extract $32 \\times 32$ central grid from the bottom layer velocity data. Hence, $\\mathcal{P}$ contains $N = 1024$ points, $32$ points in the horizontal and $32$ in the vertical direction. The temporal resolution is originally $105000$ and the time between each time step is $1$ minute. We downsample the temporal dimension of the original data uniformly such that the number of time steps in $\\mathcal{T}$ is $K=8500.$ We train and validate the SCVAE with two different data splits: randomized and sequential in time. For the sequential split we have used the last $15 \\%$ for the test, the last $30\\%$ of the remaining data is used for validation, and the fist $70\\%$ for training. In \\cref{Data_set_time_series_BOM}, the red and blue vertical lines indicate the data split for this case. For the random split, the instances $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ are drawn randomly from $\\ifmmode\\bm{X}\\else\\textbf{\\textit{X}}\\fi$ with the same percentage. The data was scaled as described in \\cref{Appendix_D}.  The input $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$ to the SCVAE was shaped as $(32 \\times 32 \\times 2)$ in order to apply convolutional layers. We use  $9,5$ and $3$ fixed spatial measurement locations. In particular, the subgrid ${\\mathcal Q}$ is given as \n\\begin{align}\\label{BOM_measurements}\n    \\centering\n    \\begin{split}\n    {\\mathcal Q}_9 = & \\{(6,6),(6,17),(6,27),(17,17),(17,27),(17,6),(27,6), (27,17),(27,27) \\}, \\\\\n    {\\mathcal Q}_5 = & \\{(6,6),(17,17),(27,27),(6,27),(27,6) \\}, \\\\\n    {\\mathcal Q}_3 = & \\{ (6,27),(17,17),(27,6) \\}. \n    \\end{split}\n\\end{align}\nAs before, the values of $u$ and $v$ at these specific locations constitute the measurements $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)} \\in \\ifmmode\\bm{M}\\else\\textbf{\\textit{M}}\\fi$.\n\n\\subsubsection{Model} \nA schematic description of the model is given in \\cref{Appendix_A.3,Appendix_A.4}. \nThe first two layers of the encoder are convolutional layers with $64$ and $128$ filters with strides and kernel size of $2$ and ReLu activation functions. This compresses the data into a shape of $(8 \\times 8 \\times 128)$. The next layers are flattening and dense layers, where the latter have $16$ filters and ReLu activation. The subsequent layers defines the mean and log-variance of the latent representation $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$, which is input to a lambda layer for realization of the reparametrization trick. The encoder outputs the samples $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and the mean and the log-variance of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi$.\\\\\n\nInput to the decoder is the output $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ of the encoder and the measurement $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}.$ To concatenate the inputs, $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$ is flattened. After concatenation of $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi^{(i)}$ and $\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi^{(i)}$, the next layer is a dense layer with shape $(8 \\times 8 \\times 128)$ and ReLu activation. This allows for use of transposed convolutional layers to obtain the original shape of the data. Hence, the following layers are two transposed convolutional layers with $64$ and $128$ filters, strides and kernel size of $2$ and ReLu activation's. The final layer is a transposed convolutional with linear activation functions and filter size  of shape $(32 \\times 32 \\times 2),$ i.e., the same shape as $\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi^{(i)}$. \n\n\\subsubsection{Results}\\label{BOM_results}\nWe illustrate the obtained posterior predictive distribution in terms the predictive mean and standard deviation for the prediction at a specific time. The SCVAE is compared with the GPOD method, both with $\\lambda >0$ and $\\lambda = 0$ for measurement locations given in \\cref{BOM_measurements} for random and sequential split cases. To generate the posterior predictive distributions, \\cref{eq:p_pred},  we sample $200$ realizations from $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ , which allows for calculation mean prediction and uncertainty\nestimates, see \\cref{eq:mean_and_cov}. \\cref{UV_prediction} shows the results of the prediction at time step $1185$ for both the $u$ and $v$ component and associated uncertainty estimates for a trained model with $\\lambda=0$ and $Q_3$ measurement locations (see \\cref{BOM_measurements}).  \n\n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.75\\linewidth]{Sleipner_prediction_std_abs_v1.png}\n\t\t\\captionof{figure}{Presentation of statistics for\n\t\tthe reconstruction of the $u$ and $v$ component of the velocity for sample $1185$ in the test data set based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. \\textbf{Left panels:} From top to bottom; True velocity in $u$, predicted mean velocity field of $u$, the standard deviation of the prediction for $u$ and the absolute error of $u.$ \\textbf{Lower panels:} Similar as describe for the upper panels, but for $v$ }\\label{UV_prediction}\n\\end{figure} \nIn \\cref{BOM_time_series} we plot the true solution and the predicted mean velocity \\cref{eq:mean_and_cov} with the associated uncertainty, see \\cref{eq:ConfRegion:2}, for two grid points. We plot only the first $600$ time steps for readability. The first grid point is $(26,6)$ and $(4,1).$ One location is approximately $5.1$ km from the nearest observation, and another one is about $16.1$ km away.\n\\begin{figure}[H]\n        \\centering\n\t\t\\includegraphics[width=0.85\\linewidth]{BOM_plots_time_series_CR.png}\n\t\t\\captionof{figure}{Velocities $u$ and $v$ at specific locations based on the trained model with $\\lambda=0$ and $Q_3$ measurement locations. The red line corresponds to the true values, blue to the SCVAE mean prediction, and orange to the GPOD reconstruction. Light blue shaded area represents the confidence region obtained in \\cref{eq:ConfRegion:2} with $p=0.95.$.  \\textbf{Upper panels:} The time series and confidence region for $u$ (left) and $v$ (right) at  grid point $(26,6)$, approximately $5.1$ km from nearest observation. \\textbf{Lower panels:} The time series and confidence region for $u$ (left) and $v$ (right) at grid point $(4,1)$ approximately $16.1$ km from nearest observation.}\\label{BOM_time_series}\n\\end{figure} \n\\cref{BOM_error} has to be viewed in context with \\cref{BOM_time_series} and show the difference between the true and the predicted solutions with associated difference marginal in time for the two locations as in \\cref{BOM_time_series}. \n\\begin{figure}[H]\n\t\\centering\n\t\\includegraphics[width=0.85\\linewidth]{BOM_error_plot_CR.png}\n\t\\captionof{figure}{The difference between the true and predicted estimate for the SCVAE (blue) and for the GPOD (orange) based on the $\\lambda=0$ model and $Q_3$ measurement locations. The light blue shaded region represents the difference marginals, obtained from the confidence region in \\cref{BOM_time_series}. \\textbf{Upper panels:} The difference between the true and predicted estimate at grid point $(26,6)$ for $u$ (left) and $v$ (right), \\textbf{Lower panels:} The difference between the true and predicted estimate at point $(4,1)$ for $u$ (left) and $v$ (right).}\\label{BOM_error}\n\\end{figure}\nIntegrating over the latent space generates a posterior distribution of the reconstruction, as described in \\cref{sec:posterior}. It is also possible to use the latent space to generate new statistically sound versions of $u$ and $v$. This is presented in \\cref{z_space_sampling_v} where it is sampled uniformly over the 2 dimensional latent space $\\ifmmode\\bm{z}\\else\\textbf{\\textit{z}}\\fi \\sim \\mathcal{N}(\\bm{0},\\bm{I})$ and the result shows how different variations can be created with the SCVAE model, given only the sparse measurements.    \n\\begin{figure}[H]\n    \\centering\n\t\t\\includegraphics[width=0.70\\linewidth]{Sleipner_generating_data_v1.png}\n\t\t\\captionof{figure}{The left panels shows 9 different generated velocity-field-snapshots for the $\\ifmmode\\bm{u}\\else\\textbf{\\textit{u}}\\fi$ and $\\ifmmode\\bm{v}\\else\\textbf{\\textit{v}}\\fi$ component for test sample number $1185$. The predictions are generated from the model with $\\lambda=0$ and $Q_3$ measurement locations. We sample uniformly over the latent space and predicts with the decoder, given the measurements.}\\label{z_space_sampling_v}\n\\end{figure}\nThese sampled velocities could be used for ensemble simulations when estimating uncertainty in a passive tracer transport, see e.g., \\cite{oleynik2020optimal}. \n\nThe SCVAE results are are compared with results of the GPOD method, see \\cref{tab:comparison_BOM} and \\cref{tab:comparison_BOM_div}. The tables show the errors as calculated in \\cref{L2_error} and \\cref{divergence_error_1} of the test data set for both sequential and random split.\nFor the sequential splitting, the SCVAE is better for $3$ measurement locations, while the GPOD method performs better for $9$ and $5$ locations.  From  \\cref{Data_set_time_series_BOM}, we observe that test data set seems to arise from a different process than the train and validation data (especially for $v$). Thus, the SCVAE generalize worse than a simpler model such as the GPOD, \\cite{model_selection}. For the $3$ location case, the number of components in the GPOD is not enough to compete with the SCVAE. \\\\\n\nWith random split on the train, test and validation data, we see that the SCVAE is significantly better than the GPOD. The training data and measurements represent the test data and test measurements better with random splitting. This highlights the importance of large data sets that cover as many outcomes as possible. Demanding that $\\lambda > 0$ in \\cref{obj_function_SCVAE} do not improve the result. The SCVAE-models with $\\lambda = 0$ learns that the reconstructed representations should have low divergence without explicitly demanding it during optimization. However, as discussed in the 2D flow around cylinder experiment, demanding $\\lambda>0$ seems to improve the conditioning of the optimization problem and give more consistent results.  \n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3 \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.1379 & 0.2097 & 0.2928   \\\\ \\cline{3-6}\n         & & GPOD & 0.3300 & 0.3822 & 0.4349  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.1403 & 0.2025 & 0.3016   \\\\ \\cline{3-6}\n         & & GPOD & 0.2971 & 0.3579 & 0.4039  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 0.3493 & 0.3913 & 0.4155  \\\\ \\cline{3-6}\n         & & GPOD & 0.3767 & 0.4031 & 0.4678  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 0.3527 & 0.3889 & 0.4141  \\\\ \\cline{3-6}\n         & & GPOD &0.3362 & 0.3695 & 0.4462  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Errors as calculated in \\cref{L2_error} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements}\n    \\label{tab:comparison_BOM}\n\\end{table}\n\\begin{table}[ht]\n    \\centering\n    \\begin{tabular}{|c|c|c|c|c|c|} \n        \\hline\n         \\multirow{2}{*}{Split} & \\multirow{2}{*}{Regularization} & \\multirow{2}{*}{Method} & \\multicolumn{3}{|c|}{Measurement Locations} \\\\  \\cline{4-6} \n          & & & 9 & 5 & 3  \\\\  \\hline\n         \n         \\multirow{4}{*}{Random} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 3.75e-05 & 3.62e-05 & 3.42e-05   \\\\ \\cline{3-6}\n         & & GPOD & 6.51e-05 & 5.88e-05 & 5.02e-05   \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE &  3.60e-05 & 3.60e-05 & 3.13e-05  \\\\ \\cline{3-6}\n         & & GPOD & 6.23e-05 & 4.77e-05 & 4.14e-05  \\\\ \\cline{3-6} \\hline\n         \n         \\multirow{4}{*}{\\makecell{Time \\\\ Dependent}} & \\multirow{2}{*}{$\\lambda=0$}\n         & SCVAE & 2.02e-05 & 1.80e-05 & 1.69e-05   \\\\ \\cline{3-6}\n         & & GPOD & 5.09e-05 & 4.03e-05 & 4.15e-05  \\\\ \\cline{2-6}\n         & \\multirow{2}{*}{$\\lambda>0$}\n         & SCVAE & 2.05e-05 & 1.99e-05 & 1.85e-05  \\\\ \\cline{3-6}\n         & & GPOD & 4.39e-05 & 3.65e-05 & 2.92e-05  \\\\ \\hline\n    \\end{tabular}\n    \\caption{Divergence errors as calculated in \\cref{divergence_error_1} for the different methods, regularization techniques ($\\lambda=0$ or $\\lambda > 0$), split regimes and measurements. The true divergence of the test data is of order  $10^{-4}.$ }\n    \\label{tab:comparison_BOM_div}\n\\end{table}\n\\begin{figure}[H]\n    \\centering\n \t    \\includegraphics[width=0.75\\linewidth]{BOM_epochs_random_split.png}\n \t    \\captionof{figure}{The figure shows number of epochs and number of measurement locations. For each measurement configuration and regularization technique the model is optimized 10 times. The variation in the number of epochs for each measurement and regularization technique is due to different priors of the weights and mini-batch sampling.}\\label{Epochs_per_measurement}\n\\end{figure}\n\n\\section{Discussion}\\label{discussion}\nWe have presented the SCVAE method for efficient data reconstruction based on sparse observations. The derived objective functions for the network optimization  show that the encoding is independent of measurements. This allows for a simpler model structure with fewer model parameters than a CVAE and results in an optimization procedure that requires less computations.  \\\\\n\nWe have shown that the SCVAE is suitable for reconstruction of fluid flow. The method is showcased on two different data sets, velocity data from simulations of 2D flow around a cylinder, and bottom currents from the BOM. The fact that the fluids studied in the experiments are incompressible served as a motivation for adding an extra term to the objective function, see \\cref{obj_function_SCVAE} with $\\lambda>0$. \\\\\n\nOur investigation of additional regularization showed that the mean reconstruction error over all models was lower with $\\lambda>0$ compared to the model where $\\lambda=0$, but the best reconstruction error was similar for $\\lambda=0$ and $\\lambda>0$. In \\cref{Experiment} we optimized 10 models for every experiment, and chose the model that performed best on the validation data sets. With this approach we did not observe significant differences between optimizing with  $\\lambda=0$ and $\\lambda>0$. However, the reconstruction became less sensitive to the stochasticity involved in optimization (minibatch selection, network weights priors) when the regularization was used, see \\cref{A_Motivating_Example}. \\\\ \n\nThe SCVAE is a probabilistic model, which allows to make predictions, estimate their uncertainty, see \\cref{sec:posterior},  and draw multiple samples from the predictive distribution, see \\cref{z_space_sampling_v}. The last two properties make the SCVAE a useful method especially when the predictions are used in another application, i.e., ensemble simulation of tracer transport. Motivated by \\cite{YILDIRIM2009160}, we compared the SCVAE predictions with the predictions of a modified GPOD method, see \\cref{Appendix_C}. \\\\\n\nUnlike the GPOD-method, a benefit with the SCVAE-method is that it scales well to larger data sets. Another aspect and as the experiments in \\cref{Experiment} suggest, the GPOD seems more sensitive to the number of measurement locations than the SCVAE. On the other hand, the experiments suggested that GPOD is better than SCVAE with a larger number of measurement locations if the training data and the test data are too different, see BOM experiment with sequential splitting \\cref{BOM_results}. Essentially the SCVAE overfit to the training data, and as a result performing poorly on the test data set. This fact shows the importance of training the the SCVAE on large data sets, which covers as many potential flow patterns as possible. Further, the results show that the GPOD is more sensitive to the measurement location choice than the SCVAE, see \\cref{A_Motivating_Example}, and the GPOD-method is not expected to preform well on a complex flow with very few fixed measurement locations. \\\\\n\nVAEs has been used for generating data in e.g. computer vision \\cite{kingma2013auto}, and auto-encoders is a natural to use in reconstruction tasks \\cite{ELMS2018}. Many reconstruction approaches, including the GPOD approach, first create a basis, then use the basis and minimize the error of the observations \\cite{willcox2006unsteady, bui2004aerodynamic}. This makes the GPOD suitable for fast optimization of measurement locations that minimize the reconstruction error. On the other hand,  the SCVAE optimizes the basis function given the measurements, i.e. they are known and fixed. This makes it challenging to use the framework for optimizing sensor layout. But if the measurement locations are fixed and large amounts of training data are available, the SCVAE outperforms the GPOD for reconstruction. SCVAE optimize the latent representation and the neural network model parameters, variational and generative parameters, given the measurements. This ensures that the reconstruction is adapted to the specific configuration of measurements. \\\\\n\nA limitation of our experiments is that we used only $100$ and $200$ samples and constructed the confidence region under further simplifying assumptions. The uncertainty estimate could be improved by increasing the sample size and better model for the confidence region. \\\\\n\nNatural applications for the SCVAE are related to environmental data, where we often have sparse measurements. It is for example possible to optimize sensor layout to best possible detect unintentional discharges in the marine environment by using a simple transport model \\cite{oleynik2020optimal}. Oleynik. et al. used deterministic flow fields to transport the contaminant and thus obtain a footprint of the leakage. SCVAE can be used to improve that method and efficiently generate probabilistic footprints of a discharges. This may be important as input to design, environmental risk assessments, and emergency preparedness plans. \n\nWe have highlighted the SCVAE through the reconstruction of currents and flow field reconstruction, however, the SCVAE method is not limited to fluid flow problems. For instance, the same principles could be used in computer vision to generate new picture based on sparse pixel representations or in time series reconstruction. \\\\\n\nA natural extension of the SCVAE is to set it up as a partially hidden Markov model. That is to predict the current state $p_\\theta(\\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_t|\\ifmmode\\bm{m}\\else\\textbf{\\textit{m}}\\fi_t, \\ifmmode\\bm{x}\\else\\textbf{\\textit{x}}\\fi_{t-1}),$ given the measurements and the reconstruction from the previous time step. This could potentially improve the reconstruction further.\n\n\\section*{Acknowledgements}\nThis work is part of the project ACTOM, funded through the ACT programme (Accelerating CCS Technologies, Horizon2020 Project No 294766). Financial contributions made from; The Research Council of Norway, (RCN), Norway, Netherlands Enterprise Agency (RVO), Netherlands, Department for Business, Energy \\& Industrial Strategy (BEIS) together with extra funding from NERC and EPSRC research councils, United Kingdom, US-Department of Energy (US-DOE), USA. Kristian Gundersen has been supported by the Research Council of Norway, through the CLIMIT program (project 254711, BayMode) and the European Union Horizon 2020 research and innovation program under grant agreement 654462, STEMM-CCS. The authors would like to acknowledge NVIDIA Corporation for providing their GPUs in the academic GPU Grant Program.\n\n\\clearpage\n\n\\bibliographystyle{unsrt}\n\\clearpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\n\\subsection{Training SNNs}\n\n\n\\begin{figure}[t!]\n  \\centering\n  \\includegraphics[width=0.8\\linewidth]{figures\/overview.pdf}\n  \\caption{Overview of the knowledge distillation method for SNNs.}\n  \\label{fig:overview}\n\\end{figure}\n\n\\subsection{Analyzing conventional knowledge distillation applied to SNNs}\n\\label{conventional_KD_analysis}\nAs already mentioned in the previous section, knowledge distillation has been conventionally applied to SNN models based on the method proposed in \\cite{hinton2015distilling}.\nThe overall process of knowledge distillation in SNNs is visualized in Fig.~\\ref{fig:overview}.\nIn this method, the loss term is formulated as\n\\begin{equation}\n\\label{eq:total_loss}\n    L = \\alpha L_{\\mathrm{CE}} + (1-\\alpha) L_{\\mathrm{KD}} \\textrm{,}\n\\end{equation}\nwhere $L_{\\mathrm{CE}}$ is the original cross entropy loss, $L_{\\mathrm{KD}}$ is additional distillation loss, and the two loss terms are balanced by a weighting hyperparameter $\\alpha$.\nMoreover, $L_{\\mathrm{KD}}$ can be computed as\n\\begin{equation}\n\\label{eq:KD_loss_equal_T}\n    L_{\\mathrm{KD}} = \\tau^{2}D_{\\mathrm{KL}}(\\sigma(z^{\\mathrm{s}};T=\\tau)||\\sigma(z^{\\mathrm{t}};T=\\tau)) \\textrm{,}\n\\end{equation}\nwhere Kullback Leibler divergence is adopted, and temperature $T$ with $\\tau > 1$ is used to soften the output logits of the student and the teacher, notated $z^s$ and $z^t$ respectively.\nThe softened output logits are normalized by a softmax function $\\sigma(\\cdot)$, which can be described by\n\\begin{equation}\n\\label{eq:softmax}\n    \\sigma(z_k;T=\\tau)=\\frac{e^{z_k\/\\tau}}{\\sum_{i}e^{z_i\/\\tau}} \\textrm{.}\n\\end{equation}\n\nThe performance of knowledge distillation depends on the choices of the two hyperparameters, $T$ and $\\alpha$.\nIn past studies, $T$ and $\\alpha$ have been commonly decided in a heuristic manner, usually following the popular choices \\cite{cho2019efficacy}, to derive only the best inference accuracy.~\\cite{kushawaha2020distilling,takuya2021training}\nHowever, when applying knowledge distillation to SNNs, both accuracy and energy efficiency should be considered together as important metrics when evaluating the performance of the models.\nAccordingly, in this paper we first investigate the performance of knowledge distillation in SNN models in terms of inference accuracy and energy efficiency by changing the values of $T$ and $\\alpha$.\nFor energy efficiency, we measure the number of spikes occurring inside the network, which serves as an effective indicator for estimating energy efficiency.\nA baseline SNN model trained from scratch is also evaluated in the same way for fair comparison.\n\nFrom the aforementioned analysis, we find that the hyperparameter sets that lead to student SNN models with high accuracy also result in the increase in the number of spikes, which will be described in detail in Section~\\ref{result:conventional_KD_analysis}.\nThis can act as a major drawback when trying to apply knowledge distillation to SNNs, as the number of spikes is strongly related to the energy efficiency of the model.\nTo address this issue, we conduct a second analysis on the weight and output distributions to determine the cause for this increase.\nThe related results can be found in Section~\\ref{result:why_spikes_increase}.\n\n\\subsection{Knowledge distillation with heterogeneous temperature}\n\\label{KD_proposed}\nWe come to the conclusion that the temperature is the key factor, which has been conventionally applied to both the teacher and the student outputs with the same value without doubt, as in Eq.\\ref{eq:KD_loss_equal_T}.\nHowever, we find that in SNNs, $T$ applied to the teacher output and $T$ applied to the student output play different roles in the distilling process.\nWith this in mind, we propose a novel knowledge distillation method that significantly reduces the number of spikes while boosting the inference accuracy of the student SNN models when compared with the conventional method.\n\nWe construct a new loss term by introducing heterogeneous temperatures for the outputs of student and teacher models.\nThe reformed KD loss $L_{\\mathrm{KD}}$ is as follows:\n\\begin{equation}\n\\label{eq:KD_loss_hetero_T}\n    L_{\\mathrm{KD}} = T_{\\mathrm{s}}T_{\\mathrm{t}}D_{\\mathrm{KL}}(\\sigma(z^{\\mathrm{s}};T=T_{\\mathrm{s}})||\\sigma(z^{\\mathrm{t}};T=T_{\\mathrm{t}})) \\textrm{,}\n\\end{equation}\nwhere $T_{\\mathrm{s}}$ and $T_{\\mathrm{t}}$ are the temperatures applied to the student SNN model and the teacher ANN model, respectively.\nWith this simple change in the loss term, by only tweaking the existing temperature parameter, we can effectively regulate the number of spikes in student SNN models while maintaining their accuracy.\nDetailed results are covered in the next section.\n\n\n\n\n\n\n\n\n\\section{Introduction}\n\\label{intro}\n\\input{1-introduction}\n\n\n\\section{Background}\n\\label{background}\n\n\\subsection{Spiking neural networks}\n\\label{bg:SNN}\n\\input{2.1-spiking_neural_networks}\n\n\\subsection{Knowledge distillation}\n\\label{bg:KD}\n\\input{2.2-knowledge_distillation}\n\n\n\\section{Methodology}\n\\label{method}\n\\input{3-methodology}\n\n\n\\section{Results}\n\\label{result}\n\n\\subsection{Experimental settings}\n\\label{result:settings}\n\\input{4.1-experimental_settings}\n\n\\subsection{Analyzing the performance of conventional knowledge distillation}\n\\label{result:conventional_KD_analysis}\n\\input{4.2-conventional_KD_analysis}\n\n\\subsection{Why does the number of spikes increase?}\n\\label{result:why_spikes_increase}\n\\input{4.3-why_spikes_increase}\n\n\\subsection{Analyzing the performance of proposed knowledge distillation}\n\\label{result:proposed_KD_analysis}\n\\input{4.4-proposed_KD_analysis}\n\n\\subsection{How does the number of spikes decrease?}\n\\label{result:how_spikes_decrease}\n\\input{4.5-how_spikes_decrease}\n\n\\subsection{Comparison with other methods}\n\\label{result:comparison}\n\\input{4.6-comparison_with_other_methods}\n\n\n\\section{Conclusion}\n\\label{conclusion}\n\\input{5-conclusion}\n\n\n\\bibliographystyle{abbrvnat}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \\label{sec:intro} Recent research on route planning in transportation networks~\\cite{bdgmpsww-rptn-14} has produced several speedup techniques varying in preprocessing time, space, query performance, and simplicity. Overall, queries on road networks are several orders of magnitude faster than on public transit~\\cite{bdgmpsww-rptn-14}. Our aim is to reduce this gap.\n\nThere are many natural query types in public transit. An \\emph{earliest arrival} query seeks a journey that arrives at a target stop \\targetStop as early as possible, given a source stop \\sourceStop and a departure time~(\\eg, ``now''). A \\emph{multicriteria} query also considers the number of transfers when traveling from \\sourceStop\\ to \\targetStop. A \\emph{profile} query reports all quickest journeys between two stops within a time range.\n\nThese problems can be approached by variants of Dijkstra's algorithm~\\cite{d-ntpcg-59} applied to a graph modeling the public transit network, with various techniques to handle time-dependency~\\cite{pswz-emtip-08}. In particular, the \\emph{time-expanded}~(TE) graph encodes time in the vertices, creating a vertex for every \\emph{event} (\\eg, a train departure or arrival at a stop at a specific time). Newer approaches, like CSA~\\cite{dpsw-isftr-13} and RAPTOR~\\cite{dpw-rbptr-14}, work directly on the timetable. Speedup techniques~\\cite{bdgmpsww-rptn-14} such as Transfer Patterns~\\cite{bceghrv-frvlp-10,bs-fbspt-14}, Timetable Contraction Hierarchies~\\cite{g-ctnrt-10}, and ACSA~\\cite{sw-csa-13} use preprocessing to create auxiliary data that is then used to accelerate queries.\n\nFor aperiodic timetables, the TE model yields a \\emph{directed acyclic graph}~(DAG), and several public transit query problems translate to reachability problems. Although these can be solved by simple graph searches, this is too slow for our application. Different methodologies exist to enable faster reachability computation~\\cite{chwf-tflat-13,jw-sfsro-13,ms-prcha-14,sabw-ferra-13,yaiy-fsrqg-13,ycz-grail-10,zlwx-rqldg-14}. In particular, the \\emph{2-hop labeling}~\\cite{chkz-rdqhl-03} scheme associates with each vertex two labels (forward and backward); reachability (or shortest-path distance) can be determined by intersecting the source's forward label and the target's backward label. On continental road networks, 2-hop labeling distance queries take less than a microsecond~\\cite{adgw-hhlsp-12}.\n\nIn this work, we adapt 2-hop labeling to public transit networks, improving query performance by orders of magnitude over previous methods, while keeping preprocessing time practical. Starting from the time-expanded graph model~(Section~\\ref{sec:basic}), we extend the labeling scheme by carefully exploiting properties of public transit networks (Section~\\ref{sec:leverage}). Besides earliest arrival and profile queries, we address multicriteria and location-to-location queries, as well as reporting the full journey description quickly~(Section~\\ref{sec:practical}). We validate our Public Transit Labeling~(PTL) algorithm by careful experimental evaluation on large metropolitan and national transit networks (Section~\\ref{sec:exp}), achieving queries within microseconds.\n\n\\section{Preliminaries} \\label{sec:prelim}\n\nLet~$G = (V,A)$ be a (weighted) \\emph{directed graph}, where~$V$ is the set of vertices and~$A$ the set of arcs. An arc between two vertices $\\avertex,\\bvertex \\in V$ is denoted by~$(\\avertex,\\bvertex)$. A \\emph{path} is a sequence of adjacent vertices. A vertex~\\bvertex is \\emph{reachable} from a vertex~\\avertex if there is a path from~\\avertex to \\bvertex. A \\emph{DAG} is a graph that is both directed and acyclic.\n\nWe consider \\emph{aperiodic} timetables, consisting of sets of stops~\\stopSet, events~\\eventSet, trips~\\tripSet, and footpaths~\\footSet. \\emph{Stops} are distinct locations where one can board a transit vehicle~(such as bus stops or subway platforms). \\emph{Events} are the scheduled departures and arrivals of vehicles. Each event~$\\event \\in \\eventSet$ has an associated stop~$\\eventStop(\\event)$ and time~$\\eventTime(\\event)$. Let $\\stopEventList{\\astop} = \\{\\event_0(\\astop),\\ldots,\\event_{k_\\astop}(\\astop)\\}$ be the list (ordered by time) of events at a stop~\\astop. We set~$\\eventTime(\\event_i(\\astop)) = -\\infty$ for $i < 0$, and~$\\eventTime(\\event_i(\\astop)) = \\infty$ for $i > k_\\astop$. For simplicity, we may drop the index of an event (as in~$\\event(\\astop) \\in \\stopEventList{\\astop}$) or its stop (as in $\\event \\in \\eventSet$). A \\emph{trip} is a sequence of events served by the same vehicle. A pair of a consecutive departure and arrival events of a trip is a \\emph{connection}. \\emph{Footpaths} model transfers between nearby stops, each with a predetermined walking duration.\n\nA journey planning algorithm outputs a set of \\emph{journeys}. A journey is a sequence of trips (each with a pair of pick-up and drop-off stops) and footpaths in the order of travel. Journeys can be measured according to several criteria, such as arrival time or number of transfers. A journey~\\ajourney \\emph{dominates} a journey~\\bjourney if and only if~\\ajourney is no worse in any criterion than~\\bjourney. If~$\\ajourney$ and~$\\bjourney$ are equal in all criteria, we break ties arbitrarily. A set of non-dominated journeys is called a \\emph{Pareto set}. Multicriteria Pareto optimization is NP-hard in general, but practical for natural criteria in public transit networks~\\cite{dpw-rbptr-14,dpsw-isftr-13,mw-op-06,pswz-emtip-08}. A journey is \\emph{tight} if there is no other journey between the same source and target that dominates it in terms of departure and arrival time, \\eg, that departs later and arrives earlier.\n\nGiven a timetable, stops~$\\sourceStop$ and~$\\targetStop$, and a departure time~$\\depTime$, the~\\emph{$(\\sourceStop,\\targetStop,\\depTime)$-earliest arrival}~(EA) problem asks for an~$\\sourceStop$--$\\targetStop$ journey that arrives at~\\targetStop as early as possible and departs at~\\sourceStop no earlier than~$\\depTime$. The~\\emph{$(\\sourceStop,\\targetStop)$-profile} problem asks for a Pareto set of all tight journeys between~$\\sourceStop$ and~$\\targetStop$ over the entire timetable period. Finally, the~\\emph{$(\\sourceStop,\\targetStop,\\depTime)$-multicriteria}~(MC) problem asks for a Pareto set of journeys departing at~$\\sourceStop$ no earlier than~$\\depTime$ and minimizing the criteria arrival time and number of transfers. We focus on computing the \\emph{values} of the associated optimization criteria of the journeys~(\\ie, departure time, arrival times, number of transfers), which is enough for many applications. Section~\\ref{sec:practical} discusses how the full journey description can be obtained with little overhead.\n\nOur algorithms are based on the 2-hop~labeling scheme for directed graphs~\\cite{chkz-rdqhl-03}. It associates with every vertex~\\bvertex a \\emph{forward label}~$\\flabel(\\bvertex)$ and a \\emph{backward label}~$\\rlabel(\\bvertex)$. In a \\emph{reachability labeling}, labels are subsets of~$V$, and vertices $\\avertex \\in \\flabel(\\bvertex) \\cup \\rlabel(\\bvertex)$ are \\emph{hubs} of \\bvertex. Every hub in $\\flabel(\\bvertex)$ must be reachable from $\\bvertex$, which in turn must be reachable by every hub in $\\rlabel(\\bvertex)$. In addition, labels must obey the \\emph{cover property}: for any pair of vertices~\\avertex and~\\bvertex, the intersection~$\\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ must contain at least one hub on a~\\avertex--\\bvertex~path~(if it exists). It follows from this definition that~$\\flabel(\\avertex) \\cap \\rlabel(\\bvertex) \\neq \\emptyset$ if and only if~\\bvertex is reachable from~\\avertex.\n\nIn a \\emph{shortest path labeling}, each hub~$\\avertex \\in \\flabel(\\bvertex)$ also keeps the associated distance~$\\dist(\\avertex, \\bvertex)$, or~$\\dist(\\bvertex,\\avertex)$ for backward labels, and the cover property requires $\\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ to contain at least one hub on a \\emph{shortest} \\mbox{\\avertex--\\bvertex~path}. If labels are kept sorted by hub ID, a \\emph{distance label query} efficiently computes $\\dist(\\avertex,\\bvertex)$ by a coordinated linear sweep over~$\\flabel(\\avertex)$ and $\\rlabel(\\bvertex)$, finding the hub~$\\cvertex \\in \\flabel(\\avertex) \\cap \\rlabel(\\bvertex)$ that minimizes $\\dist(\\avertex,\\cvertex) + \\dist(\\cvertex,\\bvertex)$. In contrast, a \\emph{reachability label query} can stop as soon as any matching hub is found.\n\nIn general, smaller labels lead to less space and faster queries. Many algorithms to compute labelings have been proposed~\\cite{adgw-hhlsp-12,aiy-f-13,chwf-tflat-13,jw-sfsro-13,yaiy-fsrqg-13,zlwx-rqldg-14}, often for restricted graph classes. We leverage (as a black box) the recent RXL algorithm~\\cite{dgpw-rdqmn-14}, which efficiently computes small shortest path labelings for a variety of graph classes at scale. It is a sampling-based greedy algorithm that builds labels one hub at a time, with priority to vertices that cover as many relevant paths as possible.\n\nDifferent approaches for transforming a timetable into a graph exist~(see~\\cite{pswz-emtip-08} for an overview). In this work, we focus on the \\emph{time-expanded model}. Since it uses scalar arc costs, it is a natural choice for adapting the labeling approach. In contrast, the \\emph{time-dependent model}~(another popular approach) associates functions with the arcs, which makes adaption more difficult.\n\n\\section{Basic Approach} \\label{sec:basic}\n\nWe build the time-expanded graph from the timetable as follows. We group all departure and arrival events by the stop where they occur. We sort all events at a stop by time, merging events that happen at the same stop and time. We then add a vertex for each unique event, a \\emph{waiting arc} between two consecutive events of the same stop, and a \\emph{connection arc} for each connection (between the corresponding departure and arrival event). The cost of arc $(\\avertex,\\bvertex)$ is $\\eventTime(\\bvertex) - \\eventTime(\\avertex)$, \\ie, the time difference of the corresponding events. To account for footpaths between two stops $a$ and $b$, we add, from each vertex at stop $a$, a \\emph{foot arc} to the first reachable vertex at~$b$ (based on walking time), and vice versa. As events and vertices are tightly coupled in this model, we use the terms interchangeably.\n\nAny label generation scheme~(we use RXL~\\cite{dgpw-rdqmn-14}) on the time-expanded graph creates two~(forward and backward) \\emph{event labels} for every vertex~(event), enabling \\emph{event-to-event queries}. For our application \\emph{reachability} labels~\\cite{yaiy-fsrqg-13}, which only store hubs~(without distances), suffice. First, since all arcs point to the future, time-expanded graphs are DAGs. Second, if an event \\event\\ is reachable from another event $\\event'$ (\\ie, $\\flabel(\\event') \\cap \\rlabel(\\event) \\neq \\emptyset$), we can compute the time to get from $\\event'$\\ to \\event\\ as $\\eventTime(\\event) - \\eventTime(\\event')$. In fact, \\emph{all} paths between two events have equal cost.\n\nIn practice, however, event-to-event queries are of limited use, as they require users to specify both departure \\emph{and} arrival times, one of which is usually unknown. Therefore, we discuss earliest arrival and profile queries, which \\emph{optimize} arrival time and are thus more meaningful. See Section~\\ref{sec:practical} for multicriteria queries.\n\n\\paragraph{Earliest Arrival Queries.} Given event labels, we answer an ($\\sourceStop,\\targetStop,\\depTime$)-EA query as follows. We first find the earliest event $\\event_i(\\sourceStop) \\in \\stopEventList{\\sourceStop}$ at the source stop~\\sourceStop that suits the departure time, \\ie, with $\\eventTime(\\event_i(\\sourceStop)) \\geq \\depTime$ and \\mbox{$\\eventTime(\\event_{i-1}(\\sourceStop)) < \\depTime$}. Next, we search at the target stop~\\targetStop for the earliest event $\\event_j(\\targetStop) \\in \\stopEventList{\\targetStop}$ that is reachable from~$\\event_i(\\sourceStop)$ by testing whether~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) \\neq \\emptyset$ and \\mbox{$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_{j-1}(\\targetStop)) = \\emptyset$}. Then, $\\eventTime(\\event_j(\\targetStop))$ is the earliest arrival time. One could find $\\event_j(\\targetStop)$ using linear search (which is simple and cache-friendly), but binary search is faster in theory and in practice. To accelerate queries, we \\emph{prune} (skip) all events~$\\event(\\targetStop)$ with~$\\eventTime(\\event(\\targetStop)) < \\depTime$, since~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event(\\targetStop)) = \\emptyset$ always holds in such cases. Moreover, to avoid evaluating $\\flabel(\\event_i(\\sourceStop))$ multiple times, we use \\emph{hash-based queries}~\\cite{dgpw-rdqmn-14}: we first build a hash set of the hubs in $\\flabel(\\event_i(\\sourceStop))$, then check the reachability for an event $\\event(\\targetStop)$ by probing the hash with hubs $h \\in \\rlabel(\\event(\\targetStop))$.\n\n\\paragraph{Profile Queries.} To answer an ($\\sourceStop,\\targetStop$)-profile query, we perform a coordinated sweep over the events at $\\sourceStop$ and $\\targetStop$. For the current event~$\\event_i(\\sourceStop) \\in \\stopEventList{\\sourceStop}$ at the source stop~(initialized to the earliest event~$\\event_0(\\sourceStop) \\in \\stopEventList{\\sourceStop}$), we find the first event~\\mbox{$\\event_j(\\targetStop) \\in \\stopEventList{\\targetStop}$} at the target stop that is reachable, \\ie, such that \\mbox{$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) \\neq \\emptyset$} and~$\\flabel(\\event_i(\\sourceStop)) \\cap \\rlabel(\\event_{j-1}(\\targetStop)) = \\emptyset$. This gives us the earliest arrival time~$\\eventTime(\\event_j(\\targetStop))$. To identify the latest departure time from \\sourceStop\\ for that earliest arrival event (and thus have a tight journey), we increase $i$ until~$\\flabel(\\event_{i}(\\sourceStop)) \\cap \\rlabel(\\event_j(\\targetStop)) = \\emptyset$, then add~$(\\eventTime(\\event_{i-1}(\\sourceStop)),\\eventTime(\\event_j(\\targetStop)))$ to the profile. We repeat the process starting from the events~$\\event_{i}(\\sourceStop)$ and~$\\event_{j+1}(\\targetStop)$. Since we increase either~$i$ or~$j$ after each intersection test, the worst-case time to find all tight journeys is linear in the number of events~(at~$\\sourceStop$ and~$\\targetStop$) multiplied by the size of their largest label.\n\n\\section{Leveraging Public Transit} \\label{sec:leverage} Our approach can be refined to exploit features specific to public transit networks. As described so far, our labeling scheme maintains reachability information for \\emph{all pairs} of events (by covering all paths of the time-expanded graph, breaking ties arbitrarily). However, in public transit networks we actually are only interested in \\emph{certain paths}. In particular, the labeling does \\emph{not} need to cover any path ending at a departure event (or beginning at an arrival event). We can thus discard forward labels from arrival events and backward labels from departure events.\n\n\\paragraph{Trimmed Event Labels.} Moreover, we can disregard paths representing dominated journeys that depart earlier and arrive later than others (\\ie, journeys that are not tight, \\cf~Section~\\ref{sec:prelim}). Consider all departure events of a stop. If a certain hub is reachable from event~$\\event_i(\\sourceStop)$, then it is also reachable from $\\event_0(\\sourceStop), \\ldots, \\event_{i-1}(\\sourceStop)$, and is thus potentially added to the forward labels of all these earlier events. In fact, experiments show that on average the same hub is added to 1.8--5.0 events per stop~(depending on the network). We therefore compute \\emph{trimmed event labels} by discarding all but the latest occurrence of each hub from the forward labels. Similarly, we only keep the earliest occurrence of each hub in the backward labels. (Preliminary experiments have shown that we obtain very similar label sizes with a much slower algorithm that greedily covers tight journeys explicitly~\\cite{adgw-hhlsp-12,dgpw-rdqmn-14}.)\n\nUnfortunately, we can no longer just apply the query algorithms from Section~\\ref{sec:basic} with trimmed event labels: if the selected departure event at~$\\sourceStop$ does not correspond to a tight journey toward~$\\targetStop$, the algorithm will not find a solution~(though one might exist). One could circumvent this issue by also running the algorithm from subsequent departure events at~$\\sourceStop$, which however may lead to quadratic query complexity in the worst case~(for both EA and profile queries).\n\n\\paragraph{Stop Labels.} We solve this problem by working with \\emph{stop labels}: For each stop~\\astop, we merge all forward event labels $\\flabel(\\event_0(\\astop)), \\ldots, \\flabel(\\event_k(\\astop))$ into a forward stop label \\fslabel(\\astop), and all backward event labels into a backward stop label \\rslabel(\\astop). Similar to distance labels, each stop label~$\\stoplabel(\\astop)$ is a list of pairs~$(\\ahub, \\atime_\\astop(\\ahub))$, each containing a hub and a time, sorted by hub. For a forward label, $\\atime_\\astop(\\ahub)$ encodes the latest departure time from~\\astop to reach hub~\\ahub. More precisely, let $h$ be a hub in an event label~$\\flabel(\\event_i(\\astop))$: we add the pair~$(\\ahub, \\eventTime(\\event_i(\\astop)))$ to the stop label \\fslabel(\\astop) only if $\\ahub \\notin \\flabel(\\event_j(\\astop)), j>i$, \\ie, only if \\ahub does not appear in the label of another event with a later departure time at the stop. Analogously, for backward stop labels, $\\atime_\\astop(\\ahub)$ encodes the earliest arrival time at $p$ from~$h$.\n\nBy restricting ourselves to these entries, we effectively discard dominated~(non-tight) journeys to these hubs. It is easy to see that these stop labels obey a \\emph{tight journey cover property}: for each pair of stops \\sourceStop and \\targetStop, $\\fslabel(\\sourceStop) \\cap \\rslabel(\\targetStop)$ contains at least one hub on each tight journey between them (or any equivalent journey that departs and arrives at the same time; recall from Section~\\ref{sec:prelim} that we allow arbitrary tie-breaking). This property does \\emph{not}, however, imply that the label intersection \\emph{only} contains tight journeys: for example, $\\fslabel(\\sourceStop)$ and $\\rslabel(\\targetStop)$ could share a hub that is important for long distance travel, but not to get from~\\sourceStop~to~\\targetStop. The remainder of this section discusses how we handle this fact during queries.\n\n\\paragraph{Stop Label Profile Queries.} To run an (\\sourceStop,\\targetStop)-profile query on stop labels, we perform a coordinated sweep over both labels~\\fslabel(\\sourceStop) and \\rslabel(\\targetStop). For every matching hub~$\\ahub$, \\ie, $(\\ahub, \\atime_\\sourceStop(\\ahub)) \\in \\fslabel(\\sourceStop)$ and $(\\ahub, \\atime_\\targetStop(\\ahub)) \\in \\rslabel(\\targetStop)$, we consider the journey induced by~$(\\atime_\\sourceStop(\\ahub), \\atime_\\targetStop(\\ahub))$ for output. However, since we are only interested in reporting tight journeys, we maintain~(during the algorithm) a tentative set of tight journeys, removing dominated journeys from it on-the-fly. (We found this to be faster than adding all journeys during the sweep and only discarding dominated journeys at the end.) We can further improve the efficiency of this approach in practice by (globally) reassigning hub IDs by the time of day. Note that every hub~$\\ahub$ of a stop label is still also an event and carries an event time~$\\eventTime(\\ahub)$. (Not to be confused with $\\atime_\\sourceStop(\\ahub)$ and $\\atime_\\targetStop(\\ahub)$.) We assign sequential IDs to all hubs~$\\ahub$ in order of increasing $\\eventTime(\\ahub)$, thus ensuring that hubs in the label intersection are enumerated chronologically. Note that this does not imply that journeys are enumerated in order of departure or arrival time, since each hub $\\ahub$ may appear anywhere along its associated journey. However, preliminary experiments have shown that this approach leads to fewer insertions into the tentative set of tight journeys, reducing query time. Moreover, as in shortest path labels~\\cite{dgpw-rdqmn-14}, we improve cache efficiency by storing the values for hubs and times separately in a stop label, accessing times only for matching hubs.\n\nOverall, stop and event labels have different trade-offs: maintaining the profile requires less effort with event labels~(any discovered journey is already tight), but fewer hubs are scanned with stop labels~(there are no duplicate hubs).\n\n\\paragraph{Stop Label Earliest Arrival Queries.} Reassigned hub IDs also enable fast $(\\sourceStop,\\targetStop,\\depTime)$-EA queries. We use binary search in $\\fslabel(\\sourceStop)$ and $\\rslabel(\\targetStop)$ to find the earliest relevant hub~\\ahub, \\ie, with $\\eventTime(\\ahub) \\geq \\depTime$. From there, we perform a linear coordinated sweep as in the profile query, finding $(\\ahub, \\atime_\\sourceStop(\\ahub)) \\in \\fslabel(\\sourceStop)$ and $(\\ahub, \\atime_\\targetStop(\\ahub)) \\in \\rslabel(\\targetStop)$. However, instead of maintaining tentative profile entries~$(\\atime_\\sourceStop(\\ahub), \\atime_\\targetStop(\\ahub))$, we ignore solutions that depart too early~(\\ie, $\\atime_\\sourceStop(\\ahub) < \\depTime$), while picking the hub~$\\ahub^*$ that minimizes the tentative best arrival time~$\\atime_\\targetStop(\\ahub^*)$. (Note that $\\eventTime(\\ahub) \\geq \\depTime$ does not imply $\\atime_\\sourceStop(\\ahub) \\geq \\depTime$.) Once we scan a hub~$h$ with~\\mbox{$\\eventTime(\\ahub) \\geq \\atime_\\targetStop(\\ahub^*)$}, the tentative best arrival time cannot be improved anymore, and we stop the query. For practical performance, \\emph{pruning} the scan, so that we only sweep hubs~\\ahub between $\\depTime \\leq \\atime(\\ahub) \\leq \\atime_\\targetStop(\\ahub^*)$, is very important.\n\n\\section{Practical Extensions} \\label{sec:practical}\n\nSo far, we presented stop-to-stop queries, which report the departure and arrival times of the quickest journey(s). In this section, we address multicriteria queries, general location-to-location requests, and obtaining detailed journey descriptions.\n\n\\paragraph{Multicriteria Optimization and Minimum Transfer Time.} Besides optimizing arrival time, many users also prefer journeys with fewer transfers. To solve the underlying multicriteria optimization problem, we adapt our labeling approach by (1)~encoding transfers as arc costs in the graph, (2)~computing shortest path labels based on these costs (instead of reachability labels on an unweighted graph), and (3)~adjusting the query algorithm to find the Pareto set of solutions.\n\nReconsider the earliest arrival graph from Section~\\ref{sec:basic}. As before, we add a vertex for each unique event, linking consecutive events at the same stop with waiting arcs of cost~0. However, each connection arc~$(u,w)$ in the graph is subdivided by an intermediate \\emph{connection vertex}~$v$, setting the cost of arc~$(u,v)$ to 0 and the cost of arc~$(v,w)$ to 1. By interpreting costs of 1 as leaving a vehicle, we can count the number of trips taken along any path. To model staying in the vehicle, consecutive connection vertices of the same trip are linked by zero-cost arcs.\n\nA shortest path labeling on this graph now encodes the number of transfers as the shortest path distance between two events, while the duration of the journey can still be deduced from the time difference of the events. Consider a fixed source event~$\\event(\\sourceStop)$ and the arrival events of a target stop~$\\event_0(\\targetStop), \\event_1(\\targetStop), \\ldots$ in order of increasing time. The minimum number of transfers required to reach the target stop~$\\targetStop$ never increases with arrival times. (Hence, the whole Pareto set~$P$ of multicriteria solutions can be computed with a single Dijkstra run~\\cite{pswz-emtip-08}.)\n\nWe exploit this property to compute~$(\\sourceStop,\\targetStop,\\depTime)$-EA multicriteria~(MC) queries from the labels as follows. We initialize~$P$ as the empty set. We then perform an~$(\\sourceStop,\\targetStop,\\depTime)$-EA query~(with all optimizations described in Section~\\ref{sec:basic}) to compute the \\emph{fastest} journey in the solution, \\ie, the one with most transfers. We add this journey to~$P$. We then check~(by performing distance label queries) for each subsequent event at~$\\targetStop$ whether there is a journey with fewer transfers~(than the most recently added entry of~$P$), in which case we add the journey to~$P$ and repeat. The MC~query ends once the last event at the target stop has been processed. We can stop earlier with the following optimization: we first run a distance label query on the \\emph{last} event at~\\targetStop\\ to obtain the \\emph{smallest} possible number of transfers to travel from~$\\sourceStop$ to~$\\targetStop$. We may then already stop the MC~query once we add a journey to~$P$ with this many transfers. Note that, since we do not need to check for domination in~$P$ explicitly, our algorithm maintains~$P$ in constant time per added journey.\n\n\\paragraph{Minimum Transfer Times.} Transit agencies often model an entire station with multiple platforms as a single stop and account for the time required to change trips inside the station by associating a \\emph{minimum transfer time}~$\\mtt(\\astop)$ with each stop~$\\astop$. To incorporate them into the EA graph, we first locally replace each affected stop~$\\astop$ by a \\emph{set} of new stops~$\\superstop$, distributing \\emph{conflicting} trips~(between which transferring is impossible due to~$\\mtt(\\astop)$) to different stops of~$\\superstop$. We then add footpaths between all pairs of stops in~$\\superstop$ with length~$\\mtt(\\astop)$. A small set~$\\superstop$ can be computed by solving an appropriate coloring problem~\\cite{dkp-pcbcp-12}. For the MC graph, we need not change the input. Instead, it is sufficient to \\emph{shift} each arrival event~$\\event \\in \\stopEventList{\\astop}$ by adding~$\\mtt(\\astop)$ to~$\\eventTime(\\event)$ before creating the vertices.\n\n\\paragraph{Location-to-Location Queries.}\n\nA query between arbitrary locations~$\\sourceLocation$ and~$\\targetLocation$, which may employ walking or driving as the first and last legs of the journey, can be handled by a two-stage approach. It first computes sets~$\\sourceSet$ and~$\\targetSet$ of relevant stops near the origin~$\\sourceLocation$ and destination~$\\targetLocation$ that can be reached by car or on foot. With that information, a \\emph{forward superlabel}~\\cite{adfgw-hldbl-12} is built from all forward stop labels associated with \\sourceSet. For each entry $(\\ahub, \\atime_\\astop(\\ahub)) \\in \\fslabel(\\astop)$ in the label of stop~$\\astop \\in \\sourceSet$, we adjust the departure time~$\\atime_\\sourceLocation(\\ahub) = \\atime_\\astop(\\ahub) - \\dist(\\sourceLocation, \\astop)$ so that the journey starts at \\sourceLocation and add $(\\ahub, \\atime_\\sourceLocation(\\ahub))$ to the superlabel. For duplicate hubs that occur in multiple stop labels, we keep only the latest departure time from \\sourceLocation. This can be achieved with a coordinated sweep, always adding the next hub of minimum ID. A \\emph{backward superlabel} (for $\\targetSet$) is built analogously. For location-to-location queries, we then simply run our stop-label-based EA and profile query algorithms using the superlabels. In practice, we need not build superlabels explicitly but can simulate the building sweep during the query~(which in itself is a coordinated sweep over two labels). A similar approach is possible for event labels. Moreover, point-of-interest queries~(such as finding the closest restaurants to a given location) can be computed by applying known techniques~\\cite{adfgw-hldbl-12} to these superlabels.\n\n\\paragraph{Journey Descriptions.} While for many applications it suffices to report departure and arrival times (and possibly the number of transfers) per journey, sometimes a more detailed description is needed. We could apply known path unpacking techniques~\\cite{adfgw-hldbl-12} to retrieve the full sequence of connections~(and transfers), but in public transit it is usually enough to report the list of trips with associated transfer stops. We can accomplish that by storing with each hub the sequences of trips~(and transfer stops) for travel between the hub and its label vertex.\n\n\\section{Experiments} \\label{sec:exp}\n\n\\paragraph{Setup.}\n\nWe implemented all algorithms in C++ using Visual Studio 2013 with full optimization. All experiments were conducted on a machine with two 8-core Intel Xeon E5-2690 CPUs and 384\\,GiB of DDR3-1066 RAM, running Windows 2008R2 Server. All runs are \\emph{sequential}. We use at most 32 bits for distances.\n\n\\begin{table} \\centering \\caption{\\label{tab:sizes}Size of timetables and the earliest arrival~(EA) and multicriteria~(MC) graphs.} \\setlength{\\tabcolsep}{0.9ex} \\begin{tabular}{@{}lrrrrrrrrr@{}} \\toprule &&&&&& \\multicolumn{2}{c}{EA Graph} & \\multicolumn{2}{c}{MC Graph} \\\\ \\cmidrule(lr){7-8}\\cmidrule(l){9-10} Instance & Stops & Conns & Trips & Footp. & Dy. & $|V|$ & $|A|$ & $|V|$ & $|A|$ \\\\ \\midrule London&20.8\\,k&5,133\\,k&133\\,k&45.7\\,k&1&4,719\\,k&51,043\\,k&9,852\\,k&72,162\\,k\\\\ Madrid&4.7\\,k&4,527\\,k&165\\,k&1.3\\,k&1&3,003\\,k&13,730\\,k&7,530\\,k&34,505\\,k\\\\ Sweden&51.1\\,k&12,657\\,k&548\\,k&1.1\\,k&2&8,151\\,k&34,806\\,k&20,808\\,k&93,194\\,k\\\\ Switzerland&27.1\\,k&23,706\\,k&2,198\\,k&29.8\\,k&2&7,979\\,k&49,656\\,k&31,685\\,k&170,503\\,k\\\\ \\bottomrule \\end{tabular} \\end{table} We consider four realistic inputs: the metropolitan networks of London (\\url{data.london.gov.uk}) and Madrid (\\url{emtmadrid.es}), and the national networks of Sweden (\\url{trafiklab.se}) and Switzerland (\\url{gtfs.geops.ch}). London includes all modes of transport, Madrid contains only buses, and the national networks contain both long-distance and local transit. We consider 24-hour timetables for the metropolitan networks, and two days for national ones (to enable overnight journeys). Footpaths were generated using a known heuristic~\\cite{dkp-pcbcp-12} for Madrid; they are part of the input for the other networks. See~\\tablename~\\ref{tab:sizes} for size figures of the timetables and resulting graphs. The average number of unique events per stop ranges from 160 for Sweden to 644 for Madrid. (Recall from Section~\\ref{sec:basic} that we merge all coincident events at a stop.) Note that no two instances dominate each other~(\\wrt number of stops, connections, trips, events per stop, and footpaths).\n\n\\paragraph{Preprocessing.}\n\n\\begin{table}[b] \\setlength{\\tabcolsep}{0.9ex} \\centering \\caption{Preprocessing figures. Label sizes are averages of forward and backward labels.} \\label{tab:prepro} \\begin{tabular}{@{}lrrrrrrrrrr@{}} \\toprule & \\multicolumn{6}{c}{\\textbf{Earliest Arrival}} & \\multicolumn{4}{c}{\\textbf{Multicriteria}}\\\\ \\cmidrule(lr){2-7}\\cmidrule(l){8-11} & & \\multicolumn{3}{c}{Event Labels} & \\multicolumn{2}{c}{Stop Labels} & & \\multicolumn{3}{c}{Event Labels} \\\\ \\cmidrule(lr){3-5}\\cmidrule(lr){6-7}\\cmidrule(l){9-11} & RXL & Hubs & Hubs & Space & Hubs & Space & RXL & Hubs & Hubs & Space \\\\ Instance & [h:m] & p.\\,lbl & p.\\,stop & [MiB] & p.\\,stop & [MiB] & [h:m] & p.\\,lbl & p.\\,stop & [MiB]\\\\ \\midrule London&0:54&70&15,480&1,334&7,075&1,257&49:19&734&162,565&26,871\\\\ Madrid&0:25&77&49,247&963&9,830&403&10:55&404&258,008&10,155\\\\ Sweden&0:32&37&5,630&1,226&1,536&700&36:14&190&29,046&12,637\\\\ Switzerland&0:42&42&11,189&1,282&2,970&708&61:36&216&58,022&12,983\\\\ \\bottomrule \\end{tabular} \\end{table}\n\n\\tablename~\\ref{tab:prepro} reports preprocessing figures for the unweighted earliest arrival graph~(which also enables profile queries) and the multicriteria graph. For earliest arrival~(EA), preprocessing takes well below an hour and generates about one gigabyte, which is quite practical. Although there are only 37--70 hubs per label, the total number of hubs per stop (\\ie, the combined size of all labels) is quite large~(5,630--49,247). By eliminating redundancy (\\cf~Section~\\ref{sec:leverage}), stop labels have only a fifth as many hubs (for Madrid). Even though they need to store an additional distance value per hub, total space usage is still smaller. In general, \\emph{average} labels sizes (though not total space) are higher for metropolitan instances. This correlates with the higher number of daily journeys in these networks.\n\nPreprocessing the multicriteria~(MC) graph is much more expensive: times increase by a factor of~26.2--54.8 for the metropolitan and~67.9--88 for the national networks. On Madrid, Sweden, and Switzerland labels are five times larger compared to EA, and on London the factor is even more than ten. This is immediately reflected in the space consumption, which is up to 26\\,GiB~(London).\n\n\\paragraph{Queries.}\n\nWe now evaluate query performance. For each algorithm, we ran~100,000 queries between random source and target stops, at random departure times between~0:00 and~23:59~(of the first day). \\tablename~\\ref{tab:ea-queries} reports detailed figures, organized in three blocks: event label EA queries, stop label EA queries, and profile queries (with both event and stop labels). We discuss MC queries later.\n\nWe observe that event labels result in extremely fast EA queries (6.9--14.7\\,\\musec), even without optimizations. As expected, pruning and hashing reduce the number of accesses to labels and hubs~(see columns ``Lbls.'' and ``Hubs''). Although binary search cannot stop as soon as a matching hub is found~(see the ``='' column), it accesses fewer labels and hubs, achieving query times below~3\\,\\musec\\ on all instances.\n\nUsing stop labels~(\\cf~Section~\\ref{sec:leverage}) in their basic form is significantly slower than using event labels. With pruning enabled, however, query times (3.6--6.2\\,\\musec) are within a factor of two of the event labels, while saving a factor of 1.1--2.4 in space. For profile queries, stop labels are clearly the best approach. It scans up to a factor of 5.1 fewer hubs and is up to 3.3 times faster, computing the profile of the full timetable period in under 80\\,\\musec~on all instances. The difference in factors is due to the overhead of maintaining the Pareto set during the stop label query.\n\n\\begin{table} \\setlength{\\tabcolsep}{0.7ex} \\centering \\caption{Evaluating earliest arrival queries. 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Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document{\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document$) indicate different features: profile query~(Prof.), stop labels~(St.\\,lbs.), pruning~(Prn.), hashing~(Hash), and binary search~(Bin.). The column ``='' indicates the average number of matched hubs.} \\label{tab:ea-queries} \\begin{tabular}{@{}cccccrrrrrrrrrrrr@{}} \\toprule &&&&&\\multicolumn{4}{c}{\\textbf{London}}&\\multicolumn{4}{c}{\\textbf{Sweden}}&\\multicolumn{4}{c}{\\textbf{Switzerland}}\\\\ \\cmidrule(lr){6-9}\\cmidrule(lr){10-13}\\cmidrule(lr){14-17} \\feature{Prof.} & \\feature{St.\\,lbs.} & \\feature{Prn.} & \\feature{Hash} & \\feature{Bin.} & Lbls. & Hubs & = & [\\textmu{}s] & Lbls. & Hubs & = & [\\textmu{}s] & Lbls. & Hubs & = & [\\textmu{}s] \\\\ \\midrule \\disabled&\\disabled&\\disabled&\\disabled&\\disabled&108.4&6,936&1&14.7&68.0&2,415&1&6.9&89.0&3,485&1&8.7\\\\ \\disabled&\\disabled&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). 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The first block of techniques considers the EA problem, the second the MC problem and the third the profile problem.} \\label{tab:comparison} \\begin{tabular}{@{}llrrrcccrrr@{}} \\toprule &\\multicolumn{4}{c}{\\textbf{Instance}}&\\multicolumn{3}{c}{\\textbf{Criteria}}\\\\ \\cmidrule(lr){2-5}\\cmidrule{6-8} &&Stops&Conns&&&&&Prep.&&Query\\\\ Algorithm&Name&[$\\cdot 10^3$]&[$\\cdot 10^6$]&Dy.&\\feature{Arr.}&\\feature{Tran.}&\\feature{Prof.}&[h]&Jn.&[ms]\\\\ \\midrule CSA~\\cite{dpsw-isftr-13}&London&20.8&4.9&1&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). 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Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&0.5&12.7&0.0121\\\\ PTL&Switzerland&27.1&23.7&2&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document{\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&\\disabled&\\ensuremath{\\bullet}}\\newcommand{\\disabled}{\\ensuremath{\\circ}}\\newcommand{\\feature}[1]{\\begin{rotate}{60}\\hspace{-.33ex}#1\\end{rotate}}\\newcommand{\\Xcomment}[1]{}\\newcommand{\\tabhead}{\\sc}\\newcommand{\\musec}{\\textmu{}s\\xspace}\\newcommand{\\wrt}{w.\\,r.\\,t.\\xspace}\\newcommand{\\eg}{e.\\,g.\\xspace}\\newcommand{\\ie}{i.\\,e.\\xspace}\\newcommand{\\cf}{cf.\\xspace}\\newcommand{\\depTime}{\\ensuremath{\\tau}\\xspace}\\newcommand{\\astop}{\\ensuremath{p}\\xspace}\\newcommand{\\superstop}{\\ensuremath{p^*}\\xspace}\\newcommand{\\sourceStop}{\\ensuremath{s}\\xspace}\\newcommand{\\targetStop}{\\ensuremath{t}\\xspace}\\newcommand{\\sourceSet}{\\ensuremath{\\cal S}\\xspace}\\newcommand{\\targetSet}{\\ensuremath{\\cal T}\\xspace}\\newcommand{\\sourceLocation}{\\ensuremath{s^*}\\xspace}\\newcommand{\\targetLocation}{\\ensuremath{t^*}\\xspace}\\newcommand{\\event}{\\ensuremath{e}}\\newcommand{\\lab}{\\ensuremath{L}}\\newcommand{\\flabel}{\\ensuremath{\\lab_f}}\\newcommand{\\rlabel}{\\ensuremath{\\lab_b}}\\newcommand{\\stoplabel}{\\ensuremath{SL}}\\newcommand{\\fslabel}{\\ensuremath{\\stoplabel_f}}\\newcommand{\\rslabel}{\\ensuremath{\\stoplabel_b}}\\newcommand{\\reachLabelTrue}{\\ensuremath{\\texttt{reach}_\\lab}}\\newcommand{\\reachLabelFalse}{\\ensuremath{\\texttt{!reach}_\\lab}}\\newcommand{\\eventTime}{\\ensuremath{\\texttt{time}}}\\newcommand{\\eventStop}{\\ensuremath{\\texttt{stop}}}\\newcommand{\\stopSet}{\\ensuremath{S}}\\newcommand{\\routeSet}{\\ensuremath{R}}\\newcommand{\\tripSet}{\\ensuremath{T}}\\newcommand{\\footSet}{\\ensuremath{F}}\\newcommand{\\eventSet}{\\ensuremath{E}}\\newcommand{\\conn}{\\ensuremath{c}}\\newcommand{\\journey}{\\ensuremath{j}}\\newcommand{\\ajourney}{\\ensuremath{\\journey_1}\\xspace}\\newcommand{\\bjourney}{\\ensuremath{\\journey_2}\\xspace}\\newcommand{\\stopEventList}[1]{\\ensuremath{E(#1)}}\\newcommand{\\avertex}{\\ensuremath{u}\\xspace}\\newcommand{\\bvertex}{\\ensuremath{v}\\xspace}\\newcommand{\\cvertex}{\\ensuremath{w}\\xspace}\\newcommand{\\ahub}{\\ensuremath{h}\\xspace}\\newcommand{\\atime}{\\eventTime\\xspace}\\DeclareMathOperator{\\dist}{dist}\\DeclareMathOperator{\\rank}{rank}\\DeclareMathOperator{\\mtt}{mtt}\\def\\comment#1{}\\def\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document{\\comment}\\def\\withcomments{\\newcounter{mycommentcounter}\\def\\comment##1{\\refstepcounter{mycommentcounter}\\ifhmode\\unskip{\\dimen1=\\baselineskip \\divide\\dimen1 by 2 \\raise\\dimen1\\llap{\\tiny\\bfseries \\textcolor{red}{-\\themycommentcounter-}}}\\fi\\marginpar[{\\renewcommand{\\baselinestretch}{0.8}\\hspace*{-1em}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter] \\raggedright ##1\\end{minipage}}]{\\renewcommand{\\baselinestretch}{0.8}\\begin{minipage}{10em}\\footnotesize [\\themycommentcounter]: \\raggedright ##1\\end{minipage}}}}\\renewcommand\\Affilfont{\\small}\\renewcommand\\Authands{ and }\\newcommand{\\email}[1]{\\texttt{#1}}\\title{Public Transit Labeling\\thanks{An extended abstract of this paper has been accepted at the 14th International Symposium on Experimental Algorithms~(SEA'15). Work done mostly while all authors were at Microsoft Research Silicon Valley.}}\\author[1]{Daniel Delling}\\author[2]{Julian Dibbelt}\\author[3]{Thomas Pajor}\\author[4]{Renato~F.~Werneck}\\affil[1]{Sunnyvale, USA, \\email{daniel.delling@gmail.com}}\\affil[2]{Karlsruhe Institute of Technology, Germany, \\email{dibbelt@kit.edu}}\\affil[3]{Microsoft Research, USA, \\email{tpajor@microsoft.com}}\\affil[4]{San Francisco, USA, \\email{rwerneck@acm.org}}\\date{May 5, 2015}\\begin{document&0.7&31.5&0.0245\\\\ \\bottomrule \\end{tabular} \\end{table}\n\n\\tablename~\\ref{tab:comparison} compares our new algorithm (indicated as \\emph{PTL}, for Public Transit Labeling) to the state of the art and also evaluates multicriteria queries. In this experiment, PTL uses event labels with pruning, hashing and binary search for earliest arrival~(and multicriteria) queries, and stop labels for profile queries. We compare PTL to~CSA~\\cite{dpsw-isftr-13} and RAPTOR~\\cite{dpw-rbptr-14}~(currently the fastest algorithms without preprocessing), as well as Accelerated CSA~(ACSA)~\\cite{sw-csa-13}, Timetable Contraction Hierarchies~(CH)~\\cite{g-ctnrt-10}, and Transfer Patterns~(TP)~\\cite{bceghrv-frvlp-10,bs-fbspt-14}~(which make use of preprocessing). Since RAPTOR always optimizes transfers~(by design), we only include it for the MC problem. Note that the following evaluation should be taken with a grain of salt, as no standardized benchmark instances exist, and many data sets used in the literature are proprietary. Although precise numbers are not available for several competing methods, it is safe to say they use less space than PTL, particularly for the MC problem.\n\n\\tablename~\\ref{tab:comparison} shows that PTL queries are very efficient. Remarkably, they are faster on the national networks than on the metropolitan ones: the latter are smaller in most aspects, but have more frequent journeys (that must be covered). Compared to other methods, PTL is~\\mbox{2--3}~orders of magnitude faster on London than CSA and RAPTOR for EA~(factor~643), profile~(factor~2,167), and MC~(factor~203) queries. We note, however, that PTL is a point-to-point algorithm~(as are ACSA, TP, and CH); for one-to-all queries, CSA and RAPTOR would be faster.\n\nPTL has 1--2~orders of magnitude faster preprocessing and queries than TP for the EA and profile problems. On Madrid, EA queries are 233~times faster while preprocessing is faster by a factor of~48. Note that Sweden~(PTL) and Germany~(TP) have a similar number of connections, but PTL queries are~95~times faster. (Germany does have more stops, but recall that PTL query performance depends more on the frequency of trips.) For the MC problem, the difference is smaller, but both preprocessing and queries of PTL are still an order of magnitude faster than TP~(up to 48~times for MC queries on Madrid).\n\nCompared to ACSA and CH~(for which figures are only available for the EA and profile problems), PTL has slower preprocessing but significantly faster queries~(even when accounting for different network sizes).\n\n\\section{Conclusion} \\label{sec:conclusion}\n\nWe introduced PTL, a new preprocessing-based algorithm for journey planning in public transit networks, by revisiting the time-expanded model and adapting the Hub Labeling approach to it. By further exploiting structural properties specific to timetables, we obtained simple and efficient algorithms that outperform the current state of the art on large metropolitan and country-sized networks by orders of magnitude for various realistic query types. Future work includes developing tailored algorithms for hub computation~(instead of using RXL as a black box), compressing the labels~(\\eg, using techniques from~\\cite{bs-fbspt-14} and~\\cite{dgpw-rdqmn-14}), exploring other hub representations~(\\eg, using trips instead of events, as in 3-hop labeling~\\cite{yaiy-fsrqg-13}), using multicore- and instruction-based parallelism for preprocessing and queries, and handling dynamic scenarios~(\\eg, temporary station closures and train delays or cancellations~\\cite{bdgmpsww-rptn-14}).\n\n\\bibliographystyle{plain} \\begin{small} ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nMany conjectures on the moduli of Higgs bundles require a deeper understanding of the ends of the moduli space, among which the most well-known are the GMN (Gaiotto-Moore-Neitzke) conjecture \\cite{gaiotto2010four} on the asymptotic metric and Hausel's conjecture \\cite{hausel2001geometry} on the vanishing of the $L^2$-cohomology. A consequence of the GMN conjecture is that $g_{L^2}-g_{\\mathrm{sf}}=O(\\mathrm{e}^{-c t})$ along a generic ray parametrized by $t\\in \\mathbb{R}^+$, where $g_{L^2}$ is the natural complete hyperk\u00e4hler metric on the moduli space, and $g_{\\mathrm{sf}}$ is the semiflat metric \\cite{freed1999special} arising from the algebraic integrability. With analytic techniques, many progresses have been made towards proving this result. The pioneering work \\cite{mazzeo2019asymptotic} used fiducial solutions and limiting configurations in \\cite{mazzeo_swoboda_weiss_witt_2016} to establish the polynomial decay of the above difference. In \\cite{dumas2019asymptotics}, Dumas and Neitzke used the local biholomorphic flow to show that the decay is exponential on the Hitchin section. These techniques were combined in \\cite{Fredrickson:2018fun, fredrickson_2019} to prove the exponential decay for $\\mathrm{SL}(n,\\mathbb{C})$-Higgs bundles. Recently in \\cite{fredrickson2022asymptotic}, the authors extended this result to the moduli space of parabolic $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundles, where the Higgs fields admit simple poles, and they showed that the moduli space of $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundles with four strongly parabolic points over $\\mathbb{C}P^1$ is an ALG gravitational instanton.\n\nThe goal of this paper is to extend the result of \\cite{fredrickson2022asymptotic} to the moduli space of irregular (wild) Higgs bundles, which means the Higgs fields can have higher order poles. Irregular Higgs bundles have received an increasing interest in recent years, because of their appearances in isomonodromic deformations \\cite{boalch2001symplectic}, wild character varieties \\cite{boalch2014geometry}, and the geometric Langlands program \\cite{witten2008gauge}. In \\cite{biquard_boalch_2004}, Biquard and Boalch established the wild non-abelian Hodge correspondence, which exhibits the moduli space of irregular Higgs bundles as a complete hyperk\u00e4hler manifold. When the moduli space is four-dimensional, it is expected to be a gravitational instanton of ALG type \\cite{boalch2012hyperkahler}. In this paper we consider the moduli space $\\mathcal{M}$ of rank two irregular Higgs bundles over $\\mathbb{C}P^1$. Using the methods of \\cite{fredrickson2022asymptotic} and the analytic results of \\cite{biquard_boalch_2004}, we obtain the following main result.\n\n\\begin{theorem}\\label{Main_thm}\n  Fix a generic curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$, and an infinitesimal deformation $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$. As $t\\to \\infty$, there exist positive constants $c,\\sigma$ such that\n  \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{sf}}}^2=O(\\mathrm{e}^{-ct^{\\sigma}}).\\]\n\\end{theorem}\nThe precise definition of the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ will be given in Section \\ref{approxsol_sec}. Later in Section 6.3, we will specialize to the case when the moduli space is (real) four dimensional, with this result, we are able to show that $g_{L^2}$ is polynomially close to an ALG\/ALG$^\\ast$ model metric $g_{\\mathrm{model}}$. Moreover, the constants in the difference $g_{L^2}-g_{\\mathrm{model}}$ can be shown to be independent of the choice of the curve $[(\\bar{\\partial}_E,\\Phi_t)]$. The correspondences between the singularity configurations and the types of the model metrics are listed below (see also \\cite[p.~448]{boalch2018wild}). The tilde $(\\tilde{~})$ on the pole indicates a twisted irregular type (defined in Section 2).\n\\renewcommand{\\arraystretch}{1.3}\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{|c|c|c|c|}\n    \\hline\n Kodaira type&$\\uppercase\\expandafter{\\romannumeral2}^\\ast$ & $\\uppercase\\expandafter{\\romannumeral3}^\\ast$ & $\\uppercase\\expandafter{\\romannumeral4}^\\ast$\\\\\nDynkin diagram &$A_0$&$A_1$&$A_2$\\\\\n $\\beta$&$\\frac{5}{6}$&$\\frac{3}{4}$&$\\frac{2}{3}$\\\\\n$\\tau$&$\\mathrm{e}^{2\\pi \\mathrm{i}\/3}$&$\\mathrm{i}$&$\\mathrm{e}^{2\\pi \\mathrm{i}\/3}$\\\\\n $D$&$4\\cdot\\{\\tilde{0}\\}$&$4\\cdot\\{0\\}$ or $3\\cdot\\{\\tilde{0}\\}+\\{\\infty\\}$&$3\\cdot\\{0\\}+\\{\\infty\\}$\\\\\n \\hline\n  \\end{tabular}\n  \\vspace{0.5em}\n  \\caption{ALG}\n\\end{table}\n\\vspace{-1em}\n\nIn the previous table, $(\\beta, \\tau)$ means that $\\mathcal{M}$ is asymptotic to the standard ALG model $(X,G_{\\beta,\\tau})$ of type $(\\beta,\\tau)$. Here $X$ is obtained by identifying two boundary components of \\[\\{u\\in \\mathbb{C}\\,|\\,\\mathrm{Arg}(u)\\in [0,2\\pi\\beta]\\text{ and }|u|\\geq R\\}\\times \\mathbb{C}_v\/(\\mathbb{Z}\\oplus \\mathbb{Z}\\tau)\\] via the gluing map $(u,v)\\sim (\\mathrm{e}^{2\\pi \\mathrm{i}\\beta}u,\\mathrm{e}^{-2\\pi \\mathrm{i}\\beta}v)$. $G_{\\beta,\\tau}$ is a flat hyperk\u00e4hler metric on $X$ such that $\\omega^1=\\mathrm{i}\/2 (\\mathrm{d}u\\wedge \\mathrm{d}\\bar{u}+\\mathrm{d}v\\wedge \\mathrm{d}\\bar{v})$ and $\\omega^2+\\mathrm{i}\\omega^3=\\mathrm{d}u\\wedge \\mathrm{d}v$. Kodaira type means that $\\mathcal{M}$ is biholomorphic to a rational elliptic surface minus a fiber with the given Kodaira type \\cite{chen_chen_2020}. Dynkin diagram means that $H^2(\\mathcal{M})$ is generated by the given extended Dynkin diagram. This makes sense because ALG gravitational instantons with the same $\\beta$ are diffeomorphic to each other \\cite{chen2021gravitational}.\n\n\\begin{table}[h]\n  \\centering\n  \\begin{tabular}{|c|c|c|c|c|}\n    \\hline\n Kodaira type&$\\uppercase\\expandafter{\\romannumeral1}_4^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_3^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_2^\\ast$&$\\uppercase\\expandafter{\\romannumeral1}_1^\\ast$\\\\\nDynkin diagram &$D_0$&$D_1$&$D_2$&$D_3=A_3$\\\\\n $D$&$2\\cdot\\{\\tilde{0}\\}+2\\cdot\\{\\tilde{\\infty}\\}$&$2\\cdot\\{0\\}+2\\cdot\\{\\tilde{\\infty}\\}$&\\makecell{$2\\cdot\\{0\\}+2\\cdot\\{\\infty\\}$ \\\\or $\\{0\\}+\\{1\\}+2\\cdot\\{\\tilde{\\infty}\\}$}&$\\makecell{\\{0\\}+\\{1\\}\\\\+2\\cdot\\{\\infty\\}}$\\\\\n \\hline\n  \\end{tabular}\n  \\vspace{0.5em}\n  \\caption{ALG$^\\ast$}\n\\end{table}\n\\vspace{-1em}\n\nIn the previous table,\nKodaira type means that $\\mathcal{M}$ is biholomorphic to a rational elliptic surface minus a fiber with the given Kodaira type \\cite{chen2021gravitational}. Dynkin diagram means that $H^2(\\mathcal{M})$ is generated by the given extended Dynkin diagram. This makes sense because ALG$^*$ gravitational instantons with the same Kodaira type at infinity are diffeomorphic to each other \\cite{chen2021gravitational}.\n\nTo prove Theorem \\ref{Main_thm}, we wish to use a similar strategy as that in \\cite{fredrickson2022asymptotic}:\n\\begin{itemize}\n\\item\tConstruct the harmonic metric $h_t$ for $(\\bar{\\partial}_E,t\\Phi)$ by a gluing method, with the following building blocks.\n\\begin{itemize}\n  \\item Limiting metric $h_\\infty$, which is singular at the points where the spectral cover is ramified.\n  \\item Fiducial solutions $h_{t}^{\\mathrm{model}}$ in some fixed disks around the ramification points.\n  \\item Desingularize $h_\\infty$ by $h_{t}^{\\mathrm{model}}$ to obtain $h_t^{\\mathrm{app}}$, and perturb $h_t^{\\mathrm{app}}$ to $h_t$.\n\\end{itemize}\n\\item Interpolate $g_{\\mathrm{app}}$ between $g_{L^2}$ and $g_{\\mathrm{sf}}$, compare tangent vectors in Coulomb gauges using $L^2$ descriptions of these metrics.\n\\end{itemize}\nHowever, there are two major new challenges in our case. The first is the lack of a natural $\\mathbb{C}^\\ast$ action on $\\mathcal{M}$ since the irregular types are fixed. Fortunately, irregular Higgs bundles over $\\mathbb{C}P^1$ can be explicitly described, and we can find some curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$ tending to infinity in place of the ray $[(\\bar{\\partial}_E,t\\Phi)]$ considered in \\cite{fredrickson2022asymptotic}. The second is that the analysis of the linearized operator requires modifications because of the presence of irregular singularities. We use the analytic tools from \\cite{biquard_boalch_2004} to handle these singularities.\n\nThis paper is organized as follows. In Section 2, we give some definitions relating to irregular Higgs bundles, and describe the Hitchin base and generic Hitchin fibers explicitly. In Section 3, we introduce the building blocks, the fiducial solutions and the limiting metric, and construct an approximate solution by gluing these together. Note that in our case the zeros of $\\mathrm{det}\\,\\Phi_t$ are moving with $t$, so the gluing regions also depend on $t$. The decaying rates of the error terms in different regions depend on some local parameters, which are called \\emph{local masses} by analogy with those appearing in \\cite{foscolo2017gluing}. We analyze the linearized Hitchin equation in Section 4, and nonlinear terms in Section 5, then a genuine solution to the Hitchin equation is obtained. Finally in Section 6, we prove Theorem \\ref{Main_thm}, and analyze the semiflat metrics in detail for the four-dimensional moduli spaces.\n\n\n\\section{Preliminaries}\\label{Prelim_sec}\nLet $E$ be a complex rank two vector bundle over $C=\\mathbb{C}P^1$, $S$ be a set of points on $C$. Cover $C$ by the usual coordinate charts $U=\\mathbb{C}_z$, $V=\\mathbb{C}_w$ with $w=1\/z$, and such that $\\{w=0\\}\\notin S$. For $x\\in S$, we also denote its $z$-coordinate by $x\\in \\mathbb{C}$. Furthermore, we fix the \\emph{parabolic structure} at each $x\\in S$, which consists of a one-dimensional subspace $L_x\\subset E_x$ (a full flag) and the associated parabolic weights $1\/2<\\alpha_{x,2}<1$, $\\alpha_{x,1}=1-\\alpha_{x,2}$.\n\nRecall that an $\\mathrm{SL}(2,\\mathbb{C})$-irregular Higgs bundle \\cite{biquard_boalch_2004} is a pair $(\\bar{\\partial}_E,\\Phi)$, where $\\bar{\\partial}_E$ is a holomorphic structure on $E$, and the Higgs field $\\Phi\\in H^0(C,\\mathfrak{sl}(E)\\otimes K(D))$, for the divisor $D=\\sum_{x\\in S} m_x\\cdot \\{x\\}$ with $m_x\\in \\mathbb{Z}^+$. Write $S=I\\cup T$, where $m_x>1$ for $x\\in I$ and $m_x=1$ for $x\\in T$. Then $\\Phi$ is a meromorphic section of $\\Omega^{1,0}(\\mathfrak{sl}(E))$ with singularities at points in $S$, which are called \\emph{irregular singularities} when they belong to $I$, otherwise called \\emph{tame singularities}. We require that $\\Phi$ has a fixed \\emph{irregular type} at each $x\\in I$, meaning that there exists a holomorphic trivialization of $(E,\\bar{\\partial}_E)$ over a neighborhood $U_x$ of $x$ such that\n\\[\\Phi=\\left(\\frac{\\phi_{x,m_x}}{z_x^{m_x}}+\\cdots+\\frac{\\phi_{x,1}}{z_x}+\\text{holomorphic terms}\\right)\\,\\mathrm{d}z_x,\\]\nwhere $z_x=z-x$, and $\\phi_{x,m_x},\\ldots,\\phi_{x,1}\\in \\mathfrak{sl}(2,\\mathbb{C})$ are given. Moreover, $\\Phi$ is adapted to the parabolic structure, i.e., $\\phi_{x,m_x}(L_x)\\subset L_x$. We assume that $\\Phi$ is \\emph{strongly parabolic} at each $x\\in T$ (see Remark \\ref{WeakParabolic_rmk}), meaning that $\\phi_{x,1}$, the residue of $\\Phi$ at $x$, is nilpotent and not fixed.\n\\begin{definition}\nAn irregular Higgs bundle $(\\bar{\\partial}_E,\\Phi)$ is \\emph{stable} if for every rank one $\\Phi$-invariant holomorphic subbundle $F$ of $(E,\\bar{\\partial}_E)$,\n\\[\\mathrm{pdeg}\\,F<\\frac{1}{2}\\mathrm{pdeg}\\,E, \\]\nwhere $\\mathrm{pdeg}\\,E=\\mathrm{deg}\\,E+|S|$, and $\\mathrm{pdeg}\\,F=\\mathrm{deg}\\,F+\\sum_{x\\in S}\\alpha_{x,F}$, $\\alpha_{x,F}=\\alpha_{x,2}$ if $F_x=L_x$, otherwise $\\alpha_{x,F}=\\alpha_{x,1}$.\n\\end{definition}\n\n\\begin{definition}\n  The \\emph{moduli space} of irregular Higgs bundles is\n\\begin{align*}\n  \\mathcal{M}=\\{(\\bar{\\partial}_E,\\Phi) \\text{ stable and compatible with the parabolic }&\\\\\\text{structure and the irregular type at each }x\\in S\\}&\\,\/\\,\\mathscr{G}_{\\mathbb{C}},\n\\end{align*}\nwhere $\\mathscr{G}_{\\mathbb{C}}=\\Gamma(\\mathrm{ParEnd}(E)\\cap SL(E))$, which consists of sections of $\\mathrm{SL}(E)$ preserving $L_x$ for each $x\\in S$. Alternatively,\n $\\mathcal{M}=\\{(\\bar{\\partial}_E,\\Phi,L)\\}\\,\/\\,\\Gamma(SL(E))$, where $L=\\{L_x\\}_{x\\in S}$, $L_x\\in \\mathbb{CP}^1_x$ which parametrizes one-dimensional subspaces of $E_x$.\n  The gauge group acts as\n\\[g\\cdot (\\bar{\\partial}_E,\\Phi,L)=(g^{-1}\\comp \\bar{\\partial}_E\\comp g,g^{-1}\\Phi g,g^{-1}\\cdot L).\\]\nBy \\cite[Th.~3.1]{inaba_michi_2016}, $\\mathrm{dim}_{\\mathbb{C}}\\,\\mathcal{M}=2(N-3)$, where $N=\\mathrm{deg}\\,D$. We assume that $N\\geqslant 4$.\n\\end{definition}\n\nWe further decompose $I=I_u\\cup I_t$. For $x\\in I_u$, the leading term $\\phi_{x,m_x}$ is diagonalizable with opposite nonzero eigenvalues $\\pm \\rho_{x,m_x}$ ($x$ is called an \\emph{untwisted singularity}), while for $x\\in I_t$, $\\phi_{x,m_x}$ is nilpotent ($x$ is called a \\emph{twisted singularity}). In the triple $(\\bar{\\partial}_E,\\Phi,L)$, as $\\Phi$ is compatible with $L$, we can determine $L_x$ form $\\Phi$ as the kernel of $\\phi_{x,m_x}$ for $x\\in I_t\\cup T$, and as one of the eigenspace of $\\phi_{x,m_x}$ for $x\\in I_u$. There are $2^{|I_u|}$ choices, depending on whether $L_x$ corresponds to $\\rho_{x,m_x}$ or $-\\rho_{x,m_x}$, where $0\\leqslant \\mathrm{Arg}(\\rho_{x,m_x})<\\pi$. These $2^{|I_u|}$ components of $\\mathcal{M}$ are equivalent, so we only need to consider the one where $L_x$ corresponds to $\\rho_{x,m_x}$, and still denote this component by $\\mathcal{M}$. We also suppress $[(\\bar{\\partial}_E,\\Phi,L_{\\Phi})]\\in\\mathcal{M}$ as $[(\\bar{\\partial}_E,\\Phi)]$.\n\n\\begin{definition}\n  The \\emph{Hitchin fibration (map)} is defined as \\[H:\\mathcal{M}\\to\\mathcal{B},~[(\\bar{\\partial}_E,\\Phi)]\\mapsto \\det\\Phi.\\]\n  Here $\\mathcal{B}\\subset H^0(C,K(D)^2)$ consists of quadratic differentials $\\nu$ which has the expansion\n  \\[\\nu=\\left(\\frac{\\mu_{x,2m_x}}{z_x^{2m_x}}+\\cdots+\\frac{\\mu_{x,m_x+1}}{z_x^{m_x+1}}+\\text{higher order terms}\\right)\\,\\mathrm{d}z_x^2,\\]\n  near each $x\\in I$. $\\mu_{x,j}$'s are complex numbers fixed by the irregular type at $x$, and clearly $\\mu_{x,2m_{x}}\\neq 0$ for $x\\in I_u$, while $\\mu_{x,2m_{x}}= 0$ for $x\\in I_t$. Moreover, we assume that $\\mu_{x,2m_{x}-1}\\neq 0$ for $x\\in I_t$ (when it is zero, by choosing a different extension of the holomorphic structure over $x$, this case can be reduced to the nonzero case or the case where the leading coefficient is diagonalizable \\cite[p.~53]{witten2008gauge}).\n\\end{definition}\n\\begin{lemma}\n  The \\emph{Hitchin base} $\\mathcal{B}$ is a dimension $N-3$ affine subspace of $H^0(C,K(D)^2)$, which can be explicitly written as\n  \\begin{align}\n    \\mathcal{B}=\\left\\{\\,\\left(\\sum_{x\\in I_{\\geqslant 3}}\\sum_{a=m_x+1}^{2m_x}\\frac{\\mu_{x,a}}{(z-x)^a}+\\sum_{x\\in I_{=2}}\\left( \\frac{\\mu_{x,4}}{(z-x)^4}+\\frac{\\mu_{x,3}(x-y_x)}{(z-x)^3(z-y_x)}\\right)\\hspace{1.15cm}\\right.\\right.\\notag\\\\+\\left.\\left.\\left.\\frac{\\sum_{b=0}^{N-4}\\nu_{b}z^b}{\\prod_{x\\in S}(z-x)^{m_x}}\\right)\\,\\mathrm{d}z^2\\,\\right|\\, \\nu_0,\\ldots,\\nu_{N-4}\\in \\mathbb{C} \\right\\},\\label{HitBase_eq}\n  \\end{align}\n    where $I_{\\geqslant 3}$ consists of poles with order $\\geqslant 3$, and $I_{=2}$ consists of order two poles. If $I_{=2}\\neq \\varnothing$, then we can choose $y_x\\in S\\backslash\\{x\\}\\neq \\varnothing$ since $N\\geqslant 4$. Choosing a different $y_x$ for $x\\in I_{=2}$ amounts to shifting $(\\nu_0,\\ldots,\\nu_{N-4})$ by an element of $\\mathbb{C}^{N-3}$.\n\\end{lemma}\n\\begin{proof}\nBy considering the difference of two elements in $\\mathcal{B}$, one can see that $\\mathcal{B}$ is an affine space modeled on $H^0(C,K^2\\otimes \\mathcal{O}(D))$, which has dimension $N-3$. Let the right hand side of \\eqref{HitBase_eq} be $\\mathcal{B}_1$, then $\\mathcal{B}_1\\subset \\mathcal{B}$. Since $\\prod_{x\\in S}(z-x)^{-m_x}z^i$ ($i=0,\\ldots,N-4$) are linearly independent, we have $\\mathrm{dim}\\,\\mathcal{B}_1=N-3$ and $\\mathcal{B}_1=\\mathcal{B}$. For $y_{x,1},y_{x,2}\\in S\\backslash \\{x\\}$, \\[\\frac{\\mu_{x,3}(x-y_{x,1})}{(z-x)^3(z-y_{x,1})}-\\frac{\\mu_{x,3}(x-y_{x,2})}{(z-x)^3(z-y_{x,2})}=\\frac{\\mu_{x,3}(y_{x,2}-y_{x,1})}{(z-x)^2(z-y_{x,1})(z-y_{x,2})},\\]\nthe last statement follows.\n\\end{proof}\n\\begin{remark}\n  Let $\\widetilde{\\mathcal{M}}$ be a moduli space of $\\mathrm{GL}(2,\\mathbb{C})$-irregular Higgs bundles, defined in the same way as for $\\mathcal{M}$ except that $\\Phi$ belongs to $H^0(C,\\mathfrak{gl}(E)\\otimes K(D))$. Then for two different Higgs fields $\\Phi_1,\\Phi_2$ in $\\widetilde{\\mathcal{M}}$, since the irregular types are fixed, we have $\\mathrm{tr}(\\Phi_1)-\\mathrm{tr}(\\Phi_2)\\in H^0(C,K)$ which is trivial. This means that Higgs fields in $\\widetilde{\\mathcal{M}}$ have constant traces, and there is no loss in considering $\\mathrm{SL}(2,\\mathbb{C})$-irregular Higgs bundles over $C$.\n\\end{remark}\nLet $\\pi:K(D)\\to C$ be the projection, and $\\lambda$ be the tautological section of $\\pi^\\ast K(D)$. Each point $\\nu$ in $\\mathcal{B}$ defines a section of $(\\pi^\\ast K(D))^2$ of the form $\\lambda^2+\\pi^\\ast \\nu$ whose zero locus $S_\\nu$ is called the \\emph{spectral curve} determined by $\\nu$. From now on we assume that $S_\\nu$ is smooth, which is true for generic $\\nu$ by Bertini's theorem. $\\pi: S_\\nu\\to C$ is a two-fold ramified cover with ramification divisor $R_\\nu$. By the Riemann-Hurwitz formula, the genus of $S_\\nu$ is $N-3$. For $\\Phi$ with $\\mathrm{det}\\,\\Phi=\\nu$, one can associate the \\emph{spectral line bundle} $\\mathcal{L}_\\Phi$ over $S_\\nu$ such that $\\mathcal{L}_\\Phi(-R_\\nu)\\subset \\pi^{\\ast} E$ and $\\pi_\\ast \\mathcal{L}_\\Phi=E$. Away from the support of $R_\\nu$, the fiber $\\mathcal{L}_{\\Phi,q}$ is the eigenspace in $E_{\\pi(q)}$ of $\\Phi$ with the eigenvalue $q$. By the Grothendieck\u2013Riemann\u2013Roch theorem \\cite[p.~96]{logares_martens_2010},\n\\begin{equation}\\label{SpecLinedeg_eq}\n  \\mathrm{deg}\\,\\mathcal{L}_\\Phi=\\mathrm{deg}\\,E+N-2.\n\\end{equation}\n\nAssume for simplicity that $\\mathrm{pdeg}\\,E=0$ (if $\\mathrm{pdeg}\\,E=2d$ for some $d\\in \\mathbb{Z}$, one can replace $E$ by $E\\otimes \\mathcal{O}(-d)$ \\cite[p.~624]{fredrickson2021moduli}). Then $\\mathrm{deg}\\,E=-|S|$ and $\\mathcal{E}=(E,\\bar{\\partial}_E)\\cong \\mathcal{O}(m)\\oplus \\mathcal{O}(-|S|-m)$, $m\\geqslant -|S|\/2$, by the Birkhoff\u2013Grothendieck theorem. Now \\[\\mathrm{End}\\,\\mathcal{E}\\cong \\begin{pmatrix}\n  \\mathcal{O}&\\mathcal{O}(2m+|S|)\\\\ \\mathcal{O}(-|S|-2m)&\\mathcal{O}\n\\end{pmatrix}.\\]\nIn a trivialization of $\\mathrm{End}\\,\\mathcal{E}$ over $U$, we have\n\\begin{equation}\\label{GlobalHiggs_eq}\n  \\Phi=\\frac{\\mathrm{d}z}{\\prod_{x\\in S}(z-x)^{m_x}}\\begin{pmatrix}\n    a(z)&b(z)\\\\ c(z)&-a(z)\n  \\end{pmatrix},\\text{ where }a(z),b(z),c(z)\\in \\mathbb{C}[z].\n\\end{equation}\n\n\\begin{lemma}\nEvery $\\Phi$ in \\eqref{GlobalHiggs_eq} with $\\det\\Phi\\in\\mathcal{B}$ of \\eqref{HitBase_eq} is compatible with some fixed irregular data at points in $I$, i.e.,\n\\begin{enumerate}[label=(\\roman*)]\n  \\item there exists a local holomorphic frame around $x\\in I_u$ in which\n    \\begin{equation}\\label{DiagLocHiggs_eq}\n      \\Phi=\\sum_{j=1}^{m_{x}}\\rho_{x,j}z_{x}^{-j}\\sigma_3\\,\\mathrm{d}z_{x}+\\text{holomorphic terms},\n    \\end{equation}\n  where $\\sigma_3$ is the Pauli matrix $\\mathrm{diag}(1,-1)$ and $\\rho_{x,j}$'s are determined by $\\mu_{x,a}$'s in \\eqref{HitBase_eq};\n  \\item there exists a local holomorphic frame around $x\\in I_t$ in which\n    \\begin{equation}\\label{NilLocHiggs_eq}\n      \\Phi=\\begin{pmatrix}\n        0&-\\sum_{j=1}^{m_{x}-1}\\mu_{x,m_{x}+j}z_{x}^{-j}\\\\ z_{x}^{-m_{x}}&0\n      \\end{pmatrix}\\,\\mathrm{d}z_{x}+\\text{holomorphic terms}.\n    \\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) This is essentially \\cite[Lem.~1.1]{biquard_boalch_2004}. By \\eqref{HitBase_eq} and \\eqref{GlobalHiggs_eq}, in a neighborhood of $x$, we have the expansion\n  \\begin{align}\n    \\det\\Phi(z_{x})&=\\bigg(\\sum_{j=m_{x}+1}^{2m_{x}}\\mu_{x,j}z_{x}^{-j}+\\text{higher order terms}\\bigg)\\,\\mathrm{d}z_{x}^2,\\label{DiagLocDetExp_eq}\\\\\n    \\Phi(z_{x})&=\\bigg(\\sum_{j=1}^{m_{x}}\\phi_{x,j}z^{-j}+\\text{holomorphic terms}\\bigg)\\,\\mathrm{d}z_{x},\\label{DiagLocHiggsExp_eq}\n  \\end{align}\n $\\mu_{x,2m_{x}}=\\det\\phi_{x,m_{x}}\\neq 0$, so we can find a frame where $\\phi_{x,m_{x}}=\\rho_{x,m_{x}}\\sigma_3$ with $\\rho_{x,m_{x}}=\\mu_{x,2m_{x}}^{1\/2}$. Then we use gauge transformations of the form $g=1-g_jz_{x}^j$ ($j=1,\\ldots,m_{x}-1$) to successively cancel the off-diagonal terms of $\\phi_{x,m_{x}-j}$. In each stage, if necessary, we shrink the neighborhood of $0$ so that $g$ is invertible. For example, for $g=1-g_1z$, \\[g^{-1}\\Phi g=\\left(\\phi_{x,m_{x}}z^{-m_{x}}+(\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1])z^{-m_{x}+1}+O\\left(z^{-m_{x}+2}\\right)\\right)\\,\\mathrm{d}z_{x},\\]\n  and $\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1]$ becomes diagonal if we choose\n  \\[g_1=\\begin{pmatrix}\n  0&\\phi_{x,m_{x}-1,12}\/2\\\\-\\phi_{x,m_{x}-1,21}\/2&0\n\\end{pmatrix},\\text{ where }\\phi_{x,m_{x}-1,ij}\\text{ is the }(i,j)-\\text{entry of }\\phi_{x,m_{x}-1}.\\]\nTherefore, in some local holomorphic frame around $x$, $\\Phi$ has the form in \\eqref{DiagLocHiggs_eq}. Now $\\rho_{x,m_{x}}=\\mu_{x,2m_{x}}^{1\/2}\\neq 0$, and $\\rho_{x,j}$ ($j=m_{x}-1,\\ldots,1$) can be recursively determined as\n\\[\\rho_{x,j}=(-2\\rho_{x,m_{x}})^{-1}\\bigg(\\mu_{x,j+m_{x}}+\\sum_{k=1}^{m_{x}-j-1}\\rho_{x,j+k}\\rho_{x,m_{x}-k}\\bigg).\\]\n(\\romannumeral2) The proof is similar to (\\romannumeral1). Around $x$, we can expand $\\det\\Phi$ and $\\Phi$ as in \\eqref{DiagLocDetExp_eq}, \\eqref{DiagLocHiggsExp_eq}. Since $\\mu_{x,2m_{x}}=0$ and $\\mu_{x,2m_{x}-1}\\neq 0$, then $\\phi_{x,m_{x}}\\neq 0$ and is nilpotent, so we can find a local frame in which $\\phi_{x,m_{x}}=\\left(\\begin{smallmatrix}\n    0&0\\\\1&0\n  \\end{smallmatrix}\\right)$. Then we use gauge transformations of the form $g=1-g_jz_{x}^j$ ($j=1,\\ldots,m_{x}-1$) to successively cancel the lower-triangular part of $\\phi_{x,m_{x}-j}$. For example if we choose \\[g_1=\\begin{pmatrix}\n    \\phi_{x,m_{x}-1,21}&\\phi_{x,m_{x}-1,22}\\\\0&0\n  \\end{pmatrix}\\]\n  in $g=1-g_1z$, then $\\phi_{x,m_{x}-1}-[\\phi_{x,m_{x}},g_1]$ becomes strictly upper-triangular. Therefore, in some local holomorphic frame around $x$, $\\Phi$ has the form in \\eqref{NilLocHiggs_eq}.\n\\end{proof}\nIn $V$, we can write $\\Phi$ as\n\\[\\Phi=\\frac{-w^{N-2}\\,\\mathrm{d}w}{\\prod_{x\\in S}(1-xw)^{m_x}}\\begin{pmatrix}\n  a(w^{-1})&w^{2m+|S|}b(w^{-1})\\\\ w^{-2m-|S|}c(w^{-1})&-a(w^{-1})\n\\end{pmatrix}.\\]\n$\\Phi$ is regular at $\\infty$, so $\\mathrm{deg}\\,a\\leqslant N-2$, $\\mathrm{deg}\\,b\\leqslant 2m+|S|+N-2$, $\\mathrm{deg}\\,c\\leqslant -2m-|S|+N-2$. Since $ \\mathrm{deg}\\,c\\geqslant 0$, and recall that $m\\geqslant -|S|\/2$, we have\n\\[-|S|\\leqslant 2m\\leqslant -|S|+N-2,\\]\nwhich yields a stratification of $\\mathcal{M}$:\n\\begin{equation}\\label{Strata_eq}\n  \\mathcal{M}=\\bigsqcup_{m=\\lceil -|S|\/2\\rceil}^{\\lfloor (-|S|+N-2)\/2 \\rfloor}\\mathcal{M}_m,\n\\end{equation}\nwhere $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}_m$ if $(E,\\bar{\\partial}_E)\\cong \\mathcal{O}(m)\\oplus \\mathcal{O}(-|S|-m)$. Let $H_m$ be the restriction of $H$ to $\\mathcal{M}_m$. Note that\n\\begin{equation}\\label{detHiggs_eq}\n  -a(z)^2-b(z)c(z)=\\prod_{x\\in S}(z-x)^{2m_x}\\nu(z):=\\tilde{\\nu}(z) \\text{ where } \\det\\Phi=\\nu=\\nu(z)\\,\\mathrm{d}z^2,\n\\end{equation}\n then $b(z)=(-a(z)^2-\\tilde{\\nu}(z))\/c(z)$. The spectral cover $S_\\nu\\to C$ ramifies exactly at the $2N-4$ zeros (counted with multiplicity) of $\\tilde{\\nu}(z)$. We say that $\\nu\\in\\mathcal{B}'$, the \\emph{regular locus}, if $\\tilde{\\nu}(z)$ only has simple zeros. When $\\nu=\\det\\Phi\\in \\mathcal{B}'$, the Higgs bundle is stable \\cite[p.~7]{mazzeo_swoboda_weiss_witt_2016} and $S_\\nu$ is smooth \\cite[p.~10]{fredrickson2022asymptotic}. Fix $\\nu\\in\\mathcal{B}'$, and next we study the fiber $H_m^{-1}(\\nu)$. Parabolic Higgs bundles have been studied in \\cite{fredrickson2022asymptotic}, so from now on we suppose that $I\\neq\\varnothing$. If $I_u\\neq \\varnothing$, by a coordinate change when necessary, we may assume that $0\\in I_u$, or else we assume that $0\\in I_t$.\n\nIf $m=-|S|\/2$, then $|S|$ is even, the holomorphic gauge transformations are of the form\n\\[g=\\begin{pmatrix}\n  g_{11}&g_{12}\\\\g_{21}&g_{22}\n\\end{pmatrix},\\quad g_{11},g_{12},g_{21},g_{22}\\in \\mathbb{C},\\det g=1.\\]\nLet $a_i,b_i,c_i,\\tilde{\\nu}_i$ be the coefficients of $z^i$ in $a(z),b(z),c(z),\\tilde{\\nu}(z)$, and make the generic assumption that $\\tilde{\\nu}_{2N-4},\\tilde{\\nu}_{2N-5}\\neq 0$. By some gauge transformation $g$ above, we can make $b_0=c_0=0$ when $0\\in I_u$, or $a_0=c_0=0$ when $0\\in I_t$. Now $c(z)\\neq 0$, otherwise $\\tilde{v}(z)=-a(z)^2$, contradicting that $\\tilde{\\nu}(z)$ only has simple zeros. If $\\mathrm{deg}\\,c=N-2$, using the diagonal gauge transformation $g=\\mathrm{diag}(c_{N-2}^{-1\/2},c_{N-2}^{1\/2})$ we can make $c(z)$ monic. Then we apply \\[g=\\begin{pmatrix}\n  1& a_{N-2}\\\\ 0&1\n\\end{pmatrix}\\]\nto make $a_{N-2}=0$. There is no gauge freedom left. The remaining $N-2$ (if $0\\in I_u$, or $N-3$ if $0\\in I_t$) coefficients of $a(z)$ can be determined (up to finitely many choices) by $c_1,c_2,\\ldots,c_{N-3}$ and the relation \\eqref{detHiggs_eq}. For example, if $c(z)$ has distinct roots $x_1=0,\\ldots,x_{N-2}$, then $a_0,a_1,\\ldots,a_{N-3}$ satisfy $N-2$ linear equations \\[a(x_i)=y_i,~ i=1,\\ldots, N-2,\\text{ where }y_i=\\pm (-\\tilde{\\nu}(x_i))^{1\/2}.\\] If $\\mathrm{deg}\\,c<N-2$, then $a_{N-2}^2= -\\tilde{\\nu}_{2N-4}$ and $\\mathrm{deg}\\,c=N-3$. In fact, if $a_{N-2}^2\\neq -\\tilde{\\nu}_{2N-4}$, then $\\mathrm{deg}(-a(z)^2-\\tilde{\\nu}(z))=2N-4>2N-5\\geqslant\\mathrm{deg}(b(z)c(z))$, a contradiction. Then we have $ \\mathrm{deg}(-a(z)^2-\\tilde{\\nu}(z))=2N-5$, and $\\mathrm{deg}\\,c=2N-5-\\mathrm{deg}\\,b\\geqslant N-3$, and $\\mathrm{deg}\\,c=N-3$. Applying gauge transformations as above we can make $c(z)$ monic and $a_{N-3}=0$. As before, the remaining coefficients of $a(z)$ are determined by $c_1,\\ldots, c_{N-4}$. Therefore \\[\\mathrm{dim}(\\mathcal{M}_{-|S|\/2})=N-3.\\]\n\nIf $m>-|S|\/2$, the holomorphic gauge transformations are of the form\n\\[g=\\begin{pmatrix}\n  g_{11}&g_{12}(z)\\\\\n  0&g_{11}^{-1}\n\\end{pmatrix},\\quad g_{11}\\in \\mathbb{C},g_{12}(z)\\in \\mathbb{C}[z], \\mathrm{deg}\\,g_{12}(z)\\leqslant 2m+|S|.\\]\nIf $\\mathrm{deg}\\,c=-2m-|S|+N-2$, we may assume that $c(z)$ is monic after applying a diagonal gauge transformation as above. Then we successively apply\n\\[g=\\begin{pmatrix}\n  1& a_{N-2-i}z^{2m+|S|-i}\\\\0&1\n\\end{pmatrix},\\quad i=0,1,\\ldots, 2m+|S|,\\]\nto make $a_{N-2}=a_{N-3}=\\cdots=a_{N-2-2m-|S|}=0$. There is no gauge freedom left. The remaining $N-2-2m-|S|$ coefficients of $a(z)$ are determined by $c(z)$ and \\eqref{detHiggs_eq}. If $\\mathrm{deg}\\,c<-2m-|S|+N-2$, then $a_{N-2}^2=-\\tilde{\\nu}_{2N-4}$ and $\\mathrm{deg}\\,c=-2m-|S|+N-3$. Again we can find a representative in each gauge orbit with $c(z)$ monic and $a_{N-3}=\\cdots=a_{N-3-2m-|S|}=0$. The remaining coefficients of $a(z)$ are determined by $c(z)$. Therefore \\[\\mathrm{dim}(\\mathcal{M}_{m})=-2m-|S|+N-2.\\]\n\n\\section{Approximate Solutions}\\label{approxsol_sec}\nWe consider the case $I_u\\neq \\varnothing$, then $0\\in I_u$ by assumption. Fix $\\nu\\in\\mathcal{B}'$ satisfying the generic conditions $\\tilde{\\nu}_{2N-4}, \\tilde{\\nu}_{2N-5}\\neq 0$, and $\\nu_{N-4}\\neq 0$. By \\eqref{HitBase_eq}, we can represent $\\nu$ by $(\\nu_0,\\ldots,\\nu_{N-4})\\in \\mathbb{C}^{N-3}$. Consider a curve $\\nu_t=(f_0(t)\\nu_0,\\ldots,f_{N-4}(t)\\nu_{N-4})$ in $\\mathcal{B}$ with $\\nu_{t=1}=\\nu$ and its norm tending to $\\infty$, where $f_i(t)$ is positive and nondecreasing in $t$. For simplicity, let $f_{N-4}(t)=t$ and $f_i(t)=o(t^{(m_0+i)\/(N-4+m_0)})$ for $i=0,\\ldots,N-5$, so that the following holds.\n\\begin{lemma}\\label{HiggsDetRoot_lem}\n  For $t$ sufficiently large, $\\tilde{\\nu}_t(z)$ has $2N-4$ distinct roots, with leading terms given by\n\\begin{align*}\n  z_{0,j}(t)&\\sim \\Bigg(-\\frac{\\mu_{0,2m_0}}{\\nu_{N-4}}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{m_y}\\Bigg)^{\\frac{1}{m_0+N-4}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_0+N-4}}t^{-\\frac{1}{m_0+N-4}},~ j=0,\\ldots, m_0+N-5;\\\\\n  z_{x,j}(t)&\\sim x+ \\Bigg(-\\frac{\\mu_{x,2m_x}}{\\nu_{N-4}}x^{4-N}\\prod_{y\\in S\\backslash \\{x\\}}(x-y)^{m_y}\\Bigg)^{\\frac{1}{m_x}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_x}}t^{-\\frac{1}{m_x}},~j=0,\\ldots,m_x-1,\\\\\\text{ for }x&\\in I_u\\backslash\\{0\\};\\\\\n  z_{x,j}(t)&\\sim x+ \\Bigg(-\\frac{\\mu_{x,2m_x-1}}{\\nu_{N-4}}x^{4-N}\\prod_{y\\in S\\backslash \\{x\\}}(x-y)^{m_y}\\Bigg)^{\\frac{1}{m_x-1}}\\mathrm{e}^{\\frac{2j\\pi \\mathrm{i}}{m_x-1}}t^{-\\frac{1}{m_x-1}},~j=0,\\ldots,m_x-2,\\\\\\text{ for }x&\\in I_t;\\text{ and }\n  z_{x,m_x-1}(t)=x,\\text{ for }x\\in I_t\\cup T.\n\\end{align*}\nHere the notation $A\\sim B$ means $A=B+o(B)$. Therefore $\\nu_t\\in \\mathcal{B}'$ for $t$ large enough.\n\\end{lemma}\n\\begin{proof}\n  By \\eqref{HitBase_eq}, we can write\n  \\[\\tilde{\\nu}_t(z)=\\bigg(\\sum_{b=0}^{N-4}f_b(t)\\nu_b z^b\\bigg)z^{m_0}\\prod_{y\\in S\\backslash \\{0\\}}(z-y)^{m_y}+\\mu_{0,2m_0}\\prod_{y\\in S\\backslash\\{0\\}}(z-y)^{2m_y}+zg_0(z),\\]\n  where $g_0(z)$ is some polynomial independent of $t$. By Rouch\u00e9's theorem, there is a constant $\\kappa$ such that in the disk $B_{\\kappa t^{-1\/(m_0+N-4)}}(0)$, $\\tilde{\\nu}_t(z)$ has the same number of zeros as that for\n  \\[\\tilde{\\nu}_t^{(0)}(z):=\\nu_{N-4}tz^{m_0+N-4}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{m_y}+\\mu_{0,2m_0}\\prod_{y\\in S\\backslash\\{0\\}}(-y)^{2m_y},\\]\n  for $t$ large enough. Choose $\\kappa$, so that $\\tilde{\\nu}_t^{(0)}$ has $m_0+N-4$ distinct roots in the disk. Using the scaling $z\\mapsto t^{1\/(m_0+N-4)}z$, we see that the roots $z_{0,j}(t)$ ($j=0,\\ldots,m_0+N-5$) of $\\tilde{\\nu}_t(z)$ in the disk are asymptotic to those of $\\tilde{\\nu}_t^{(0)}(z)$, which are given in the statement of the lemma. The analysis near $x\\in I\\backslash\\{0\\}$ is similar, using the coordinate $z_x$ in pace of $z$.\n\\end{proof}\n\nThe hyperk\u00e4hler metric on $\\mathcal{M}$ is defined through the nonabelian Hodge correspondence, which realizes $\\mathcal{M}$ as a hyperk\u00e4hler quotient, i.e., unitary gauge equivalence classes of solutions to Hitchin's equations. Thus, we need to find the solution of the following \\emph{Hitchin's equation} for $[(\\bar{\\partial}_E,\\Phi)]\\in\\mathcal{M}$:\n\\[F_h+[\\Phi,\\Phi^{\\ast_h}]=0,\\]\nwhere $F_h$ is the curvature of the Chern connection $D(\\bar{\\partial}_E,h)$. $h$ is called a \\emph{harmonic metric} if it satisfies the above equation and adapted to the parabolic structure, which means for each $x$ the filtration of the fiber $E_x$ given by $E_{x,\\beta}:=\\{s(0):s\\text{ holomorphic},|s(z)|_{h}=O(|z|^\\beta)\\}$ coincides with that of the parabolic structure. When $I_t=\\varnothing$, such a metric $h$ always exists on a stable Higgs bundle by \\cite{biquard_boalch_2004}. However, if $I_t\\neq \\varnothing$, the existence of $h$ will impose further constraints on the parabolic weights $\\alpha_{x,1}$, $\\alpha_{x,2}$ for $x\\in I_t$.\n\\begin{lemma}\\label{TIrregWeight_lem}\n  Let $(E,\\Phi)$ be an $\\mathrm{SL}(2,\\mathbb{C})$-Higgs bundle over a Riemann surface $C$, where $\\Phi$ has an order $n\\geqslant 2$ pole at $x$, with nilpotent leading coefficient $\\phi_{-n}$. The kernel $L$ of $\\phi_{-n}$ determines a filtration $0\\subset L\\subset E_x$ and let the associated parabolic weights be $1>\\alpha_2>\\alpha_1>0$, $\\alpha_1+\\alpha_2=1$. Suppose $\\mathrm{pdeg}\\,E=0$. If $h$ is a harmonic metric adapted to the parabolic structure, then $\\alpha_1=1\/4,\\alpha_2=3\/4$.\n\\end{lemma}\n\\begin{proof}\nAs in \\cite[Lem.~3.6]{mazzeo_swoboda_weiss_witt_2016}, we first find a standard form of $\\Phi$ near $x$. Since $\\phi_{-n}$ is nilpotent, $\\det\\Phi$ has an order $2n-1$ pole at $p_0$. By \\cite[p.~28]{strebel_1984}, one can find a local coordinate $z$ centered at $p_0$, such that $\\det\\Phi=-z^{1-2n}\\,\\mathrm{d}z^2$. Again by the nilpotency of the leading term, in some local holomorphic frame\n  \\[z^n\\phi(z)=\\begin{pmatrix}\na(z)&b(z)\\\\c(z)&-a(z)\n\\end{pmatrix},\\quad z^n\\phi(z)|_{z=0}=\\begin{pmatrix}\n    0&0\\\\1&0\n  \\end{pmatrix}, \\text{ where }\\Phi(z)=\\phi(z)\\,\\mathrm{d}z.\\]\n  Now $\\sqrt{c(z)}$ is well-defined near $0$ since $c(0)=1$, and \\[g^{-1}\\Phi g=\\begin{pmatrix}\n    0&\\frac{1}{z^{n-1}}\\\\\\frac{1}{z^n}&0\n  \\end{pmatrix}\\,\\mathrm{d}z,\\quad\\text{where }g=\\frac{1}{\\sqrt{c(z)}}\\begin{pmatrix}\n    1&a(z)\\\\0&c(z)\n  \\end{pmatrix}.\\]\n  In the local holomorphic frame determined by $g$, we have \\[h=\\begin{pmatrix}\n    h_{11}(z)&h_{12}(z)\\\\\\widebar{{h}_{12}(z)}&h_{22}(z)\n  \\end{pmatrix},\\quad h_{11}(z)\\sim c_1r^{2\\alpha_1},h_{22}(z)\\sim c_2r^{2\\alpha_2},\\det h=Q(z)r^{2},h_{12}(z)=O(r),\\]\n  where $c_1,c_2$ are positive constants, $r=|z|$, $Q(z)$ is determined by $h_{\\det E}$, the metric on $\\det E$ induced by $h$, $Q(0)>0$. The $(1,1)$-entry of $F_h$ is $O(r^{-2})$ while that of $[\\Phi,\\Phi^{\\ast_h}]$ is $O(r^{4\\alpha_1-2n})+O(r^{4\\alpha_2-2-2n})$ (the leading terms in these $O(\\cdot)$'s are nonzero), which grows faster than $r^{-2}$ near $0$ if the sum of the leading terms is nonzero. Therefore $F_h+[\\Phi,\\Phi^{\\ast_h}]=0$ implies that $4\\alpha_1-2n=4\\alpha_2-2-2n$, and $\\alpha_1=1\/4,\\alpha_2=3\/4$.\n\\end{proof}\n\nHenceforth, we assume that $\\alpha_{x,1}=1\/4, \\alpha_{x,2}=3\/4$, for $x\\in I_t$. To get a more explicit description of the hyperk\u00e4hler metric near the ends of $\\mathcal{M}$, we need more information on $h$, which can be obtained via gluing constructions as in \\cite{fredrickson2022asymptotic}, and we also follow this strategy. Fix $[(\\bar{\\partial}_E,\\Phi)]\\in H^{-1}(\\nu)$, and choose $[(\\bar{\\partial}_E,\\Phi_t)]\\in H^{-1}(\\nu_t)$, such that $\\Phi_t(z)$ and $\\Phi(z)$ have the same $c(z)$ when written as in \\eqref{GlobalHiggs_eq} (by the previous section, there are only finitely many gauge inequivalent choices of $\\Phi_t$). Note that if $[(\\bar{\\partial}_E,\\Phi)]$ belongs to $\\mathcal{M}_m$ of \\eqref{Strata_eq}, then the whole curve $[(\\bar{\\partial}_E,\\Phi_t)]$ remains in $\\mathcal{M}_m$ as $t$ varies.\n\nNow we aim to find a solution $h_t$ of $F_{h_t}+[\\Phi_t,\\Phi_t^{\\ast_{h_t}}]=0$ for large $t$. First we construct an approximate solution $h_t^{\\mathrm{app}}$, and later we will deform it into a genuine solution. The construction of $h_t^{\\mathrm{app}}$ is similar to that in \\cite[Sec.~3]{fredrickson2022asymptotic}. Since we are considering $\\mathrm{SL}(2,\\mathbb{C})$ Higgs bundles, the data on $\\mathrm{det}\\,E\\simeq \\mathcal{O}(-|S|)$ is fixed. In other words, we fix the holomorphic structure induced from $\\bar{\\partial}_E$ and the harmonic metric $h_{\\mathrm{det}\\,E}=\\prod_{x\\in S}|z-x|^2 $ adapted to the induced parabolic weight $\\alpha_{x,1}+\\alpha_{x,2}=1$ at $x$ (in $V$, $h_{\\mathrm{det}\\,E}=\\prod_{x\\in S}|1-xw|^2$). Let $(S_t,\\mathcal{L}_t)$ be the spectral data associated with $(\\bar{\\partial}_E,\\Phi_t)$. The spectral cover $\\pi:S_t\\to C$ ramifies at $Z_t$, the zero locus of $\\tilde{\\nu}_t(z)$. By \\eqref{SpecLinedeg_eq}, $\\mathrm{deg}\\,\\mathcal{L}_t=-|S|+N-2$. Now $\\mathcal{L}_t$ has rank one, a parabolic structure at a point is equivalent to a parabolic weight since there is only a trivial filtration. For each of the two points in $\\pi^{-1}(x)$, $x\\in I_u$, let the weight be $\\alpha_{x,1}$ if it corresponds to the eigenvalue $\\rho_{x,m_{x}}$ of $\\phi_{x,m_{x}}$ and otherwise be $\\alpha_{x,2}$. For each point in $\\pi^{-1}(Z_t\\backslash(I_t\\cup T))$, let the weight be $-1\/2$. For each point $\\pi^{-1}(x)$, $x\\in I_t\\cup T$, let the weight be $\\alpha_{x,1}+\\alpha_{x,2}-1\/2=1\/2$. Endowed with this parabolic structure we have\n\\[\\mathrm{pdeg}\\,\\mathcal{L}_t=-|S|+N-2+|I_u|+|I_t|\/2+|T|\/2-(2N-4-|I_t|-|T|)\/2 =0=\\mathrm{pdeg}\\, E.\\] By \\cite{simpson_1990}, there is a unique (up to scaling) Hermitian-Einstein metric $h_{\\mathcal{L}_t}$ on $\\mathcal{L}_t$ adapted to the parabolic structure, which is flat in our setting. Finally, let $h_{t,\\infty}$ be the metric on $E|_{C\\backslash (Z_t\\cup S)}$ defined as the orthogonal pushforward of $h_{\\mathcal{L}_t}$. Then $h_{t,\\infty}$ solves the decoupled Hitchin's equations:\n\\[F_{h_{t,\\infty}}=0,\\quad [\\Phi_t,\\Phi_t^{\\ast_{h_{t,\\infty}}}]=0.\\]\n The metric on $\\mathrm{det}\\,E|_{C\\backslash (Z_t\\cup S)}$ induced by $h_{t,\\infty}$ can be extended to $C\\backslash\\{S\\}$ and is a harmonic metric. By uniqueness, we may assume that this metric is $h_{\\mathrm{det}\\,E}$ defined above, by rescaling $h_{\\mathcal{L}_t}$ if necessary. Near $x\\in Z_t\\cup S$, $\\Phi_t$ and $h_{t,\\infty}$ can be explicitly described using the following normal forms.\n\\begin{proposition}\\label{NormalForm_prop}\n  There exist constants $1>\\kappa,\\kappa_1,\\kappa_2>0$, such that for $t$ large enough, the followings hold.\n  \\begin{enumerate}[label=(\\roman*)]\n    \\item There is a holomorphic coordinate $\\zeta_{x,j,t}$ centered at $z_{x,j}(t)$ for $x\\in I$, $j=0,\\ldots,j(x)-1$ where\n    \\[j(x)=\\begin{cases}\n      m_0+N-4,&\\text{ if }x=0,\\\\\n      m_x,&\\text{ if }x\\in I_u\\backslash\\{0\\},\\\\\n      m_x-1,&\\text{ if }x\\in I_t,\n    \\end{cases}\\]\n    and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,j,t}:=\\{\\,|\\zeta_{x,j,t}|<\\kappa t^{-1\/j(x)} \\,\\}$ in which\n    \\begin{equation}\\label{NormalFormZ_eq}\n      \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n        0&1\\\\\\zeta_{x,j,t}&0\n      \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |\\zeta_{x,j,t}|^{1\/2}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\n      \\end{pmatrix}.\n    \\end{equation}\n    Here the \\emph{local mass} $\\lambda_{x,j}(t)=|\\nu_{x,j,t}(z_{x,j}(t))|^{1\/2}$, where $\\nu_{x,j,t}(z)=(z-z_{x,j}(t))^{-1}\\nu_t(z)$.\n    \\item There is a holomorphic coordinate $\\zeta_{x,t}$ centered at $x$ for $x\\in T$, and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\kappa\\,\\}$ in which\n  \\begin{equation}\\label{NormalFormT_eq}\n    \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n        0&1\\\\ \\frac{1}{\\zeta_{x,t}}&0\n      \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |\\zeta_{x,t}|^{1\/2}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\n      \\end{pmatrix}.\n  \\end{equation}\n    Here $\\lambda_{x}(t)=|\\nu_{x,t}(x)|^{1\/2}$, and $\\nu_{x,t}(z)=(z-x)\\nu_t(z)$.\n    \\item There is a holomorphic coordinate $\\zeta_{x,t}$ centered at $x$ for $x\\in I_t$, and a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\kappa t^{-1\/(m_x-1)}\\,\\}$ in which\n\\begin{equation}\\label{NormalFormIt_eq}\n  \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n    0&\\frac{1}{\\zeta_{x,t}^{m_x-1}}\\\\ \\frac{1}{\\zeta_{x,t}^{m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n    |\\zeta_{x,t}|^{1\/2}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\n  \\end{pmatrix}.\n\\end{equation}\n    Here $\\lambda_{x}(t)=|\\nu_{x,t}(x)|^{1\/2}$, and $\\nu_{x,t}(z)=(z-x)^{2m_x-1}\\nu_t(z)$.\n    \\item For $x\\in I_u$, there is a local holomorphic frame of $(E,\\bar{\\partial}_E)$ over $\\widetilde{B}_{x,t}:=\\{\\,|z-x|<\\kappa t^{-1\/j(x)}\\,\\}$ in which\n    \\begin{equation}\\label{NormalFormIu_eq}\n      \\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\frac{1}{z_x^{m_x}}(z_x^{2m_x}\\nu_t(z))^{1\/2}\\sigma_3\\,\\mathrm{d}z,\\quad h_{t,\\infty}=\\begin{pmatrix}\n        |z_x|^{2\\alpha_{x,1}}&0\\\\0&|z_x|^{2\\alpha_{x,2}}\n      \\end{pmatrix}\n    \\end{equation}\n  \\end{enumerate}\nThe disks $\\widetilde{B}_{x(,j),t}$ are disjoint, and $B_{x(,j),t,\\kappa_1} \\subset \\widetilde{B}_{x(,j),t}\\subset B_{x(,j),t,\\kappa_2}$, where $B_\\bullet$ are disks defined using the coordinate $z$, for example, $B_{x,j,t,\\kappa_1}=\\{\\,|z-z_{x,j}(t)|<\\kappa_1 t^{-1\/j(x)}\\,\\}$, for $x\\in I, j=0,\\ldots,j(x)-1$.\n\\end{proposition}\n\\begin{proof}\n  By \\cite[Th.~6.1]{strebel_1984}, any quadratic differential having a critical point of odd order can be written in a standard form using some local holomorphic coordinate around that point. This can be applied to (\\romannumeral1)-(\\romannumeral3), where the quadratic differential $\\det\\Phi_t(z)$ has a critical point of order $1$, $-1$, $-(2m_x-1)$ respectively.\n\n  (\\romannumeral1) By a straightforward adaptation of the proof in \\cite{strebel_1984}, one can find a biholomorphic map $\\sigma_{x,j,t}(z-z_{x,j}(t))=\\zeta_{x,j,t}$ from some disk $B_{x,j,t,\\kappa_2}$ to its image, such that $\\det \\Phi_t=-\\lambda_{x,j}(t)^2 \\zeta_{x,j,t}\\,\\mathrm{d}\\zeta_{x,j,t}^2$. $\\sigma_{x,j,t}(0)=0$, and moreover for $t$ large enough, $|\\sigma_{x,j,t}'|$ is bounded above and below by some positive constants. Therefore one can find constants $\\kappa,\\kappa_1$ so that $\\widetilde{B}_{x,j,t}$ lies inside the image $\\sigma_{x,j,t}(B_{x,j,t,\\kappa_2})$ and its preimage contains a smaller disk $B_{x,j,t,\\kappa_1}$, for all sufficiently large $t$. As in \\cite[Lem.~3.6]{mazzeo_swoboda_weiss_witt_2016}, we will write $\\Phi_t$ in a standard form using local holomorphic gauge transformations, well defined over $\\widetilde{B}_{x,j,t}$. Recall that $c(z)\\neq 0$ in $\\Phi_t$ is fixed, for $t$ large, $c(z)$ is nonzero on $\\widetilde{B}_{x,j,t}$. So we can write $\\Phi_t$ in the coordinate $\\zeta_{x,j,t}$ as $\\Phi_t=\\phi_{x,j,t}(\\zeta_{x,j,t})\\,\\mathrm{d}\\zeta_{x,j,t}$,   with $\\phi_{x,j,t,21}(\\zeta_{x,j,t})$ nonvanishing on $\\widetilde{B}_{x,j,t}$, $\\phi_{x,j,t,ij}(\\zeta_{x,j,t})$ being the $(i,j)$-entry of $\\phi_{x,j,t}(\\zeta_{x,j,t})$. By a constant gauge transformation\n  \\[\\begin{pmatrix}\n    \\phi_{x,j,t,11}(0)&1\\\\\\phi_{x,j,t,21}(0)&0\n  \\end{pmatrix},\\]\n  we can make $\\Phi_t=\\phi^{(1)}_{x,j,t}(\\zeta_{x,j,t})\\,\\mathrm{d}\\zeta_{x,j,t}$, with\n\\[\\phi^{(1)}_{x,j,t}(0)=\\begin{pmatrix}\n  0&1\\\\0&0\n\\end{pmatrix},\\text{ and }\\phi^{(1)}_{x,j,t,12}=\\phi_{x,j,t,21}(\\zeta_{x,j,t})\/\\phi_{x,j,t,21}(0),\\]\nNow for $\\kappa$ suitably small, $|\\phi^{(1)}_{x,j,t,12}-1|<1\/2$ and $\\sqrt{\\phi^{(1)}_{x,j,t,12}}$ is well defined on $\\widetilde{B}_{x,j,t}$. Take the gauge transformation\n\\[\\frac{1}{\\sqrt{\\phi^{(1)}_{x,j,t,12}}}\\begin{pmatrix}\n  \\phi^{(1)}_{x,j,t,12}&0\\\\-\\phi^{(1)}_{x,j,t,11}&1\n\\end{pmatrix}, \\text{ such that }\\Phi_t=\\begin{pmatrix}\n  0&1\\\\ \\lambda_{x,j}(t)^2 \\zeta_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t}.\\]\nUse the constant gauge transformation\n\\[\\begin{pmatrix}\n  1&0\\\\0&\\lambda_{x,j}(t)\n\\end{pmatrix}, \\text{ and then }\\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n  0&1\\\\\\zeta_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t}.\\]\nFinally, as in \\cite[Prop.~3.5]{Fredrickson:2018fun}, $h_{t,\\infty}$ has the form given above, after applying a holomorphic gauge transformation on $\\widetilde{B}_{x,j,t}$ preserving $\\Phi_t$.\n\n(\\romannumeral2) is similar to \\cite[Prop.~3.5]{fredrickson2022asymptotic}. (\\romannumeral3) is similar to Lemma \\ref{TIrregWeight_lem} and (\\romannumeral2). (\\romannumeral4) is similar to \\cite[Prop.~3.7]{fredrickson2022asymptotic}, noting that $(z_x^{2m_x}\\nu_t(z))^{1\/2}$ is well defined on $\\widetilde{B}_{x,t}$ for $\\kappa$ suitably small. We can also choose $\\kappa$ so that all the disks above are disjoint.\n\\end{proof}\n\nNear the points in (\\romannumeral3) and (\\romannumeral4) above, $h_{t,\\infty}$ is already adapted to the parabolic structure. Near the points in (\\romannumeral1), $h_{t,\\infty}$ is singular, while near the points in (\\romannumeral2), $h_{t,\\infty}$ is not adapted to the parabolic weights. So we need to desingularize $h_{t,\\infty}$ using the following \\emph{fiducial solutions} $h_{t}^{\\mathrm{model}}$ \\cite[Prop.s~3.8~\\&~3.9]{fredrickson2022asymptotic}, and obtain the approximate metric $h_{t}^{\\mathrm{app}}$.\n\\begin{definition}\\label{ApproxMet_def}\n  \\begin{enumerate}[label=(\\roman*)]\n    \\item For $x\\in I$, in the above holomorphic frame over $\\widetilde{B}_{x,j,t}$, define\n    \\[h_{t}^{\\mathrm{model}}=\\begin{pmatrix}\n      |\\zeta_{x,j,t}|^{1\/2}\\mathrm{e}^{l_{x,j,t}(|\\zeta_{x,j,t}|)}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\\mathrm{e}^{-l_{x,j,t}(|\\zeta_{x,j,t}|)}\n    \\end{pmatrix},\\]\n    where $l_{x,j,t}(r)=\\psi_1(8\\lambda_{x,j}(t)r^{3\/2}\/3)$, and $\\psi_1$ satisfies the ODE\n    \\begin{equation}\\label{PainleveODE_eq}\n      \\left(\\frac{\\mathrm{d}^2}{\\mathrm{d}\\rho^2}+\\frac{1}{\\rho}\\frac{\\mathrm{d}}{\\mathrm{d}\\rho}\\right)\\psi_1(\\rho)=\\frac{1}{2}\\sinh(2\\psi_1(\\rho)),\n    \\end{equation}\n    with asymptotics\n    \\[\\psi_1(\\rho)\\sim \\frac{1}{\\pi}K_0(\\rho)\\text{ as }\\rho\\to\\infty,\\quad \\psi_1(\\rho)\\sim -\\frac{1}{3}\\log\\rho\\text{ as }\\rho\\to 0.\\]\n  Here  $K_0$ is the modified Bessel function of the second kind, $K_0(\\rho)\\sim \\rho^{-1\/2}\\mathrm{e}^{-\\rho}$ as $\\rho\\to\\infty$. Then we define\n\\begin{equation}\\label{approxMetricZ_eq}\n  h_t^{\\mathrm{app}}=\\begin{pmatrix}\n      |\\zeta_{x,j,t}|^{1\/2}\\mathrm{e}^{\\chi(|\\zeta_{x,j,t}|\\kappa^{-1}t^{1\/j(x)})l_{x,j,t}(|\\zeta_{x,j,t}|)}&0\\\\0&|\\zeta_{x,j,t}|^{-1\/2}\\mathrm{e}^{-\\chi(|\\zeta_{x,j,t}|\\kappa^{-1}t^{1\/j(x)})l_{x,j,t}(|\\zeta_{x,j,t}|)}\n    \\end{pmatrix},\n\\end{equation}\n  where $\\chi$ is a nonincreasing smooth cutoff function with $\\chi(r)=1$ for $r\\leqslant 1\/2$ and $\\chi(r)=0$ for $r\\geqslant 1$.\n  \\item For $x\\in T$, in the above holomorphic frame over $\\widetilde{B}_{x,t}$, define\n  \\[h_{t}^{\\mathrm{model}}=\\begin{pmatrix}\n    |\\zeta_{x,t}|^{1\/2}\\mathrm{e}^{m_{x,t}(|\\zeta_{x,t}|)}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\\mathrm{e}^{-m_{x,t}(|\\zeta_{x,t}|)}\n  \\end{pmatrix},\\]\n  where $m_{x,t}(r)=\\psi_{2,x}(8\\lambda_x(t)r^{1\/2})$, and $\\psi_{2,x}$ satisfies \\eqref{PainleveODE_eq} with asymptotics\n  \\[\\psi_{2,x}(\\rho)\\sim \\frac{1}{\\pi}K_0(\\rho)\\text{ as }\\rho\\to\\infty,\\quad \\psi_{2,x}(\\rho) \\sim (1+2\\alpha_{x,1}-2\\alpha_{x,2})\\log\\rho\\text{ as }\\rho\\to 0.\\]\n  Then we define\n\\begin{equation}\\label{approxMetricT_eq}\n  h_t^{\\mathrm{app}}=\\begin{pmatrix}\n      |\\zeta_{x,t}|^{1\/2}\\mathrm{e}^{\\chi(|\\zeta_{x,t}|\\kappa^{-1})m_{x,t}(|\\zeta_{x,t}|)}&0\\\\0&|\\zeta_{x,t}|^{3\/2}\\mathrm{e}^{-\\chi(|\\zeta_{x,t}|\\kappa^{-1})m_{x,t}(|\\zeta_{x,t}|)}\n    \\end{pmatrix}.\n\\end{equation}\n    \\end{enumerate}\n  It is proved in \\cite{mccoy1977painleve} that there exist unique functions $\\psi_1$, $\\psi_{2,x}$ solving \\eqref{PainleveODE_eq} with the prescribed asymptotics. Finally, set $h_{t}^{\\mathrm{app}}=h_{t,\\infty}$ on the complement of the above disks.\n\\end{definition}\nIt is straightforward to check that $h_{t}^{\\mathrm{model}}$ satisfies Hitchin's equation over $\\widetilde{B}_{x(,j),t}$, smooth near $z_{x,j}(t)\\in Z_t\\backslash(I_t\\cup T)$ and adapted to the parabolic structure at $x\\in T$. $h_t^{\\mathrm{app}}$ approximately solves Hitchin's equation in the following sense.\n\\begin{proposition}\n  There exists positive constants $t_0$, $c,c'$, such that for any $t\\geqslant t_0$,\n  \\begin{equation}\\label{ErrorEst_eq}\n  \\left\\lVert F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right] \\right\\rVert_{L^2}\\leqslant c\\mathrm{e}^{-c' t^{\\sigma}},\n\\end{equation}\nwhere the $L^2$-norm is taken with respect to $h_t^{\\mathrm{app}}$ and a fixed Riemannian metric on $C$, and\n\\[\\sigma=\\min\\left(\\frac{m_0-1}{m_0+N-4},\\min_{x\\in I_u\\backslash\\{0\\}}\\frac{m_x-1}{m_x},\\min_{x\\in I_t}\\frac{2m_x-3}{2(m_x-1)},\\frac{1}{2}\\delta_{T\\neq \\varnothing}\\right),\\]\n$\\delta_{T\\neq \\varnothing}=1$ if $T\\neq \\varnothing$, otherwise it equals to $2$.\n\\end{proposition}\n\\begin{proof}\n  Since $h_t^{\\mathrm{app}}$ coincides with $h_{t}^{\\mathrm{model}}$ and $h_{t,\\infty}$ away from some annuli, solving Hitchin's equation, we only need to estimate the error in these regions. First, we consider $\\mathrm{Ann}_{x,j}=\\{\\,\\kappa t^{-1\/j(x)}\/2<|\\zeta_{x,j,t}|<\\kappa t^{-1\/j(x)}\\,\\}$. Denote $|\\zeta_{x,j,t}|$ by $r$ for simplicity.\n  \\begin{align*}\n  F_{h_t^{\\mathrm{app}}}&=-\\frac{1}{4}\\left(\\Delta l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\, \\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar{\\zeta}_{x,j,t}\\\\\n  &=-\\frac{1}{4}\\left(\\vphantom{\\frac{1}{r}}l_{x,j,t}''(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)+2\\kappa^{-1}t^{1\/j(x)}l_{x,j,t}'(r)\\chi'\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right.\\\\&\\quad+\\kappa^{-2}t^{2\/j(x)}l_{x,j,t}(r)\\chi''\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\\\&\\quad+\\left.\\frac{1}{r}l_{x,j,t}'(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)+\\frac{1}{r}\\kappa^{-1}t^{1\/j(x)}l_{x,j,t}(r)\\chi'\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}.\n\\end{align*}\nOn the other hand,\n\\[  \\left[\\Phi,\\Phi^{\\ast_{h_t^{\\mathrm{app}}}}\\right]=2\\lambda_{x,j}(t)^2r\\sinh\\left(2l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}.\\]\nBy \\eqref{PainleveODE_eq} and the definition of $l_{x,j,t}$, we have\n\\[l''_{x,j,t}(r)+r^{-1}l_{x,j,t}'(r)=8\\lambda_{x,j}(t)^2 r\\sinh(2l_{x,j,t}),\\quad l_{x,j,t}=O(\\mathrm{e}^{-c'\\lambda_{x,j}(t)t^{-3\/2j(x)}}).\\]\nWe can write $F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right]=E_{x,j,t}\\sigma_3\\,\\mathrm{d}\\zeta_{x,j,t}\\mathrm{d}\\bar\\zeta_{x,j,t}$, the $L^2$ norm of which is bounded by $ct^{-1\/j(x)}\\sup|E_{x,j,t}|$. Here and below, $c,c'>0$ are generic constants which may vary, but does not depend on $t$. By Lemma \\ref{HiggsDetRoot_lem}, as $t\\to\\infty$,\n\\[\\lambda_{x,j}(t)\\sim t^{1\/2}\\prod_{y\\in S}|z_{x,j}(t)-y|^{-m_x}\\prod_{(y,k)\\neq(x,j)}|z_{x,j}(t)-z_{y,k}(t)|^{1\/2}\\sim \\begin{cases}\n  c' t^{\\frac{2m_x+1}{2j(x)}}&\\text{ if }x\\in I_u,\\\\\n  c' t^{\\frac{m_x}{j(x)}}&\\text{ if }x\\in I_t.\n\\end{cases}\\]\nThen for $x\\in I_u$,\n\\begin{align*}\n  |E_{x,j,t}|&\\leqslant c\\left(\\lambda_{x,j}(t)^2r(\\left|\\sinh(2l_{x,j,t})-\\sinh(2l_{x,j,t}\\chi)\\right|+|\\chi-1||\\sinh(2l_{x,j,t})|)\\right.\\\\&\\left.\\quad+t^{1\/j(x)}|l'_{x,j,t}|+t^{2\/j(x)}|l_{x,j,t}|\\right)\\\\\n  &\\leqslant c\\left(t^{2m_x\/j(x)}\\cosh(2l_{x,j,t})|l_{x,j,t}|+t^{1\/j(x)}|l'_{x,j,t}|+t^{2\/j(x)}|l_{x,j,t}|\\right)\\\\&\\leqslant c\\left(t^{2m_x\/j(x)}|l_{x,j,t}|+t^{2\/j(x)}|l'_{x,j,t}|\\right)\\leqslant c \\mathrm{e}^{-c' t^{(m_x-1)\/j(x)}}.\n\\end{align*}\nFor $x\\in I_t$, $|E_{x,j,t}|\\leqslant c\\mathrm{e}^{-c' t^{(2m_x-3)\/2j(x)}}$. Let $\\mathrm{Ann}_x=\\{\\,\\kappa \/2<|\\zeta_{x,t}|<\\kappa \\,\\}$ for $x\\in T$, and denote $r=|\\zeta_{x,t}|$. Similar as before,\n\\begin{align*}\n  F_{h_t^{\\mathrm{app}}}  &=-\\frac{1}{4}\\left(\\vphantom{\\frac{1}{r}}m_{x,t}''(r)\\chi\\left(r\\kappa^{-1}\\right)+2\\kappa^{-1}m_{x,t}'(r)\\chi'\\left(r\\kappa^{-1}\\right)+\\kappa^{-2}m_{x,t}(r)\\chi''\\left(r\\kappa^{-1}\\right)\\right.\\\\&\\quad+\\left.\\frac{1}{r}m_{x,t}'(r)\\chi\\left(r\\kappa^{-1}\\right)+\\frac{1}{r}\\kappa^{-1}m_{x,t}(r)\\chi'\\left(r\\kappa^{-1}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,t}\\mathrm{d}\\bar{\\zeta}_{x,t},\\\\\n  \\left[\\Phi,\\Phi^{\\ast_{h_t^{\\mathrm{app}}}}\\right]\n &=2\\lambda_{x}(t)^2r^{-1}\\sinh\\left(2m_{x,t}(r)\\chi\\left(r\\kappa^{-1}\\right)\\right)\\sigma_3\\,\\mathrm{d}\\zeta_{x,t}\\mathrm{d}\\bar{\\zeta}_{x,t}.\n\\end{align*}\nNow we have\n\\[m_{x,t}''(r)+r^{-1}m_{x,t}'(r)=8\\lambda_{x}(t)^2 r^{-1}\\sinh(2m_{x,t}),\\quad m_{x,j,t}=O(\\mathrm{e}^{-c'\\lambda_{x}(t)}),\\]\nwhere $\\lambda_{x}(t)\\sim c' t^{1\/2}$. For $E_{x,t}$ defined similarly as above, we have the estimate \\[|E_{x,t}|\\leqslant c\\mathrm{e}^{-c' t^{1\/2}}.\\]\nCombining all the estimates above, the result follows.\n\\end{proof}\n\\begin{remark}\\label{WeakParabolic_rmk}\n  If we allow tame untwisted singularities (also called weakly parabolic), then $\\lambda_{x,j}(t)\\sim c' t^{3\/2}$, and $l_{x,j,t}=O(\\mathrm{e}^{-c' t^{3\/2}t^{-3\/2}})=O(1)$ which does not decay to $0$ as $t\\to\\infty$. Thus, the error cannot be arbitrarily small when tame untwisted poles exist.\n\\end{remark}\n\n\\section{Linear Analysis}\nWe wish to obtain the solution $h_t$ to Hitchin's equation by perturbing $h_t^{\\mathrm{app}}$. This is equivalent to finding an $\\mathrm{SL}(E)$-valued $h_t^{\\mathrm{app}}$-Hermitian section $H_t$ such that $h_t(v,w)=h_t^{\\mathrm{app}}(H_tv,w)$. Let $\\mathrm{e}^{\\gamma_t}=H_t^{-1\/2}$, then $\\mathrm{e}^{\\gamma_t}\\cdot h_t(v,w)=h_t(H_t^{-1\/2}v,H_t^{-1\/2}w)=h_t^{\\mathrm{app}}(v,w)$. By the gauge invariance of Hitchin's equations, $(\\bar{\\partial}_E,\\Phi_t,h_t)$ is a solution is equivalent to that \\[\\mathrm{e}^{\\gamma_t}\\cdot (\\bar{\\partial}_E,\\Phi_t,h_t)=(\\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E\\mathrm{e}^{\\gamma_t},\\mathrm{e}^{-\\gamma_t}\\Phi_t \\mathrm{e}^{\\gamma_t},h_t^{\\mathrm{app}})\\]\nis a solution. Therefore, we need to find $\\gamma_t\\in \\Omega^0(i\\mathfrak{su}(E))$ (the bundle is defined over $C\\backslash S$ using $E$ and $h_t^{\\mathrm{app}}$), such that $F_t(\\gamma_t)=0$, where\n\\[F_t(\\gamma):=F_{A_t^{\\exp(\\gamma)}}+\\big[\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma},\\mathrm{e}^{\\gamma}\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}\\mathrm{e}^{-\\gamma}\\big],\\]\n$A_t$ is the Chern connection $D(\\bar{\\partial}_E,h_t^{\\mathrm{app}})$ and $A_t^{\\exp(\\gamma)}=D(\\mathrm{e}^{-\\gamma}\\bar{\\partial}_E\\mathrm{e}^{\\gamma},h_t^{\\mathrm{app}})$. Let $\\mathrm{d}_{A_t}=\\partial_{A_t}+\\bar{\\partial}_E$ be the exterior derivative associated with $A_t$, $\\nabla_{A_t}$ be the covariant derivative. We have\n\\[ F_{A_t^{\\mathrm{exp}(\\gamma)}}=\\mathrm{e}^{-\\gamma}\\left(F_{A_t}+\\bar{\\partial}_{A_t}\\left(\\mathrm{e}^{2\\gamma} \\partial_{A_t}\\left(\\mathrm{e}^{-2\\gamma}\\right)\\right)\\right)\\mathrm{e}^\\gamma.\\]\nLinearize $F_t$ at $\\gamma=0$ we get\n\\begin{align*}\n  DF_t(\\gamma)&=[F_{A_t},\\gamma]-2\\bar{\\partial}_{A_t}\\partial_{A_t}\\gamma+[[\\Phi_t,\\gamma],\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}]+[\\Phi_t,[\\gamma,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}]]\\\\&=\\mathrm{i}\\star\\Delta_{A_t}\\gamma+M_{\\Phi_t}\\gamma,\n\\end{align*}\nwhere $\\Delta_{A_t}=\\mathrm{d}_{A_t}^\\ast \\mathrm{d}_{A_t}$ on $\\Omega^0(i\\mathfrak{su}(E))$, $M_{\\Phi_t}\\gamma=[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},[\\Phi_t,\\gamma]]-[\\Phi_t,[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\gamma]]$, and we used the identity $2\\bar{\\partial}_A\\partial_A=F_A-\\mathrm{i}\\star\\Delta_A$. By composing $-\\mathrm{i}\\star: \\Omega^2(\\mathfrak{su}(E))\\to\\Omega^0(i\\mathfrak{su}(E))$ we get the operator\n\\begin{equation}\\label{LinOp_eq}\n  L_t(\\gamma)=\\Delta_{A_t}\\gamma-\\mathrm{i}\\star M_{\\Phi_t}\\gamma.\n\\end{equation}\nConsider the irregular connection defined by $D_t^\\gamma=\\mathrm{d}_{A_t^{\\mathrm{exp}(\\gamma)}}+\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma}+\\mathrm{e}^{\\gamma}\\Phi_t^{\\ast}\\mathrm{e}^{-\\gamma}$, here and below we drop the subscript of $\\ast_{h_t^\\mathrm{app}}$. Then its curvature $\\left(D_t^\\gamma\\right)^2=F_t(\\gamma)$, since $\\bar{\\partial}_E \\Phi_t=\\partial_{A_t}\\Phi_t^\\ast=0$. So we can also regard the problem as finding $\\gamma$ so that this connection is flat. When $\\gamma=0$, write the operator as $D_t$, which is the sum of the unitary part $\\mathrm{d}_{A_t}$ and the self-adjoint part $\\Psi_t:=\\Phi_t+\\Phi_t^\\ast$.\n\\begin{lemma}(Weitzenb\u00f6ck formulas)\\label{weitz_lemma}\n  Suppose $\\gamma\\in\\Omega^0(i\\mathfrak{su}(E))$, $w\\in \\Omega^p(\\mathrm{End}\\,E)$, then \\begin{enumerate}[label=(\\roman*)]\n    \\item $(D_t^\\ast D_t+D_tD_t^\\ast)w=\\nabla_{A_t}^\\ast\\nabla_{A_t} w+(\\Psi_t\\otimes )^\\ast\\Psi_t\\otimes w+\\mathscr{F}(w)+\\mathscr{R}(w)$, where $\\mathscr{F},\\mathscr{R}$ are curvature operators, $\\Psi_t\\otimes\\cdot$ is a combination of tensor product on the manifold part and Lie bracket on the bundle part. More explicitly, let $e_1,e_2$ be a local orthonormal frame of $TC$ and $e^1,e^2$ be the dual frame, write the curvature $D_t^2$ as $F_{t,12}e^1\\wedge e^2$, then $\\mathscr{F}(w)=[F_{t,12},(e^1\\wedge i_{e_2}-e^2\\wedge  i_{e_1})w]$, and\n  \\begin{align*}\n    \\mathscr{R}(w)(X_1,\\ldots,X_p)&=\\sum_{j=1}^pw(X_1,\\ldots,\\mathrm{Ric}(X_j),\\ldots,X_p)\\\\&+\\sum_{a=1}^2\\sum_{j<k}(-1)^{j+k+1}w(e_a,R(X_j,X_k)e_a,X_1,\\ldots,\\hat{X}_j,\\ldots,\\hat{X}_k,\\ldots,X_p).\\hspace{-1.5cm}\n  \\end{align*}\n    \\item $D_t^\\ast D_t\\gamma=\\mathrm{d}_{A_t}^\\ast \\mathrm{d}_{A_t}\\gamma+([\\Psi_t,\\cdot])^\\ast[\\Psi_t,\\gamma]=L_t\\gamma$.\n    \\item $\\mathrm{Re}\\left(\\langle \\mathrm{d}_{A_t}\\gamma,[\\Psi_t,\\gamma]\\rangle\\right)=0$, so the following pointwise equality holds\n   \\[\\langle D_t\\gamma,D_t\\gamma\\rangle= |D_t\\gamma|^2=|\\mathrm{d}_{A_t}\\gamma|^2+|[\\Psi_t,\\gamma]|^2.\\]\n  \\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) This follows from \\cite[Theorem 5.4]{biquard_1997}, since $\\mathrm{d}_{A_t}^\\ast \\Psi_t=0$.\\\\\n  (\\romannumeral2) The first equality is a special case of (\\romannumeral1). The adjoint of $[\\Phi_t,\\cdot]$ is $-\\bar{\\ast}[\\Phi_t,\\bar{\\ast}\\cdot]$, where $\\bar{\\ast}$ is the conjugate linear operator combining $\\bar{\\star}$ and the Hermitian adjoint. Actually, for a fixed holomorphic coordinate $z$ where $\\Phi=\\varphi\\,\\mathrm{d}z$ (subscript $t$ omitted),\n  \\begin{align*}\n    \\langle [\\Phi,a],b\\,\\mathrm{d}z\\rangle&=\\langle [\\varphi,a]\\,\\mathrm{d}z,b\\,\\mathrm{d}z\\rangle=2\\langle [\\varphi,a],b\\rangle,\\\\\n    \\langle a,-\\bar{\\ast}[\\Phi,\\bar{\\ast} b\\,\\mathrm{d}z]\\rangle&=\\langle a,-\\bar{\\ast} [\\Phi,ib^\\ast \\mathrm{d}\\bar{z}]\\rangle=\\langle a,-[b,\\varphi^\\ast]\\bar{\\star} (\\mathrm{i}\\,\\mathrm{d}z \\mathrm{d}\\bar{z})\\rangle=2\\langle a,[\\varphi^\\ast,b]\\rangle=2\\langle [\\varphi,a],b\\rangle.\n  \\end{align*}\nSimilarly, the adjoint of $[\\Phi_t^\\ast,\\cdot]$ is $-\\bar{\\ast}[\\Phi_t^\\ast,\\bar{\\ast}\\cdot]$. Then (recall that $\\bar{\\ast}\\mathrm{d}z=\\mathrm{i}\\,\\mathrm{d}\\bar{z}\n$, $\\bar{\\ast}\\mathrm{d}\\bar{z}=-\\mathrm{i}\\,\\mathrm{d}z$)\n\\begin{align*}\n  ([\\Psi_t,\\cdot])^\\ast[\\Psi_t,\\gamma]&=-\\bar{\\ast}[\\Phi_t,\\bar{\\ast}[\\Phi_t,\\gamma]]-\\bar{\\ast}[\\Phi_t^\\ast,\\bar{\\ast}[\\Phi_t^\\ast,\\gamma]]\\\\\n  &=-\\bar{\\ast}[\\Phi_t,\\mathrm{i}[\\Phi_t^\\ast,\\gamma]]+\\bar{\\ast}[\\Phi_t^\\ast,\\mathrm{i}[\\Phi_t,\\gamma]]\\\\\n  &=-\\mathrm{i}\\star [\\Phi_t^\\ast,[\\Phi_t,\\gamma]]+\\mathrm{i}\\star [\\Phi_t,[\\Phi_t^\\ast,\\gamma]]=-\\mathrm{i}\\star M_{\\Phi_t}\\gamma.\n\\end{align*}\n(\\romannumeral3) Write $\\mathrm{d}_{A_t}\\gamma=\\gamma_z\\,\\mathrm{d}z+\\gamma_{\\bar{z}}\\,\\mathrm{d}\\bar{z}$. $\\gamma_{\\bar{z}}=\\gamma_z^\\ast$, since $\\gamma=\\gamma^\\ast$ and $\\mathrm{d}_{A_t}$ is a Hermitian connection. As above, write $\\Phi=\\varphi\\, \\mathrm{d}z$, then \\begin{align*}\n  \\mathrm{Re}\\left(\\langle \\mathrm{d}_{A_t}\\gamma,[\\Phi+\\Phi^\\ast,\\gamma]\\rangle\\right)&=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)+\\mathrm{tr}(\\gamma_{\\bar{z}}[\\varphi^\\ast,\\gamma]^\\ast)\\right)\\\\\n  &=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)+\\mathrm{tr}(\\gamma_z^\\ast[\\gamma,\\varphi])\\right)\\\\\n  &=2\\mathrm{Re}\\left(\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)-\\overline{\\mathrm{tr}(\\gamma_z[\\varphi,\\gamma]^\\ast)}\\right)=0.\n  \\end{align*}\n\\end{proof}\nChoose a local holomorphic trivialization as in Proposition \\ref{NormalForm_prop} (\\romannumeral3). Consider the map $\\sigma_{x,t}:D_{x,t}\\to \\widetilde{B}_{x,t},~\\xi_{x,t}\\to \\xi_{x,t}^2=\\zeta_{x,t}$, where $D_{x,t}=\\{\\,|\\xi_{x,t}|<\\kappa^{1\/2}t^{-1\/2(m_x-1)}\\,\\}$. Then for the pullbacks $\\widetilde{E}=\\sigma_{x,t}^\\ast E$, $\\widetilde{\\Phi}_t=\\sigma_{x,t}^\\ast \\Phi_t$, $\\tilde{h}_t^{\\mathrm{app}}=\\sigma_{x,t}^\\ast h_t^{\\mathrm{app}}$, we have\n\\[ \\tilde{\\bar{\\partial}}_E=\\bar{\\partial},\\quad \\widetilde{\\Phi}_t=2\\lambda_x(t)\\begin{pmatrix}\n    0&\\frac{1}{\\xi_{x,t}^{2m_x-3}}\\\\\\frac{1}{\\xi_{x,t}^{2m_x-1}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\xi_{x,t},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\begin{pmatrix}\n  |\\xi_{x,t}|&0\\\\0&|\\xi_{x,t}|^3\n\\end{pmatrix}. \\]\nAfter applying the gauge transformation $\\left(\\begin{smallmatrix}\n  \\xi_{x,t}&-\\xi_{x,t}\\\\1&1\n\\end{smallmatrix}\\right)$ (noninvertible at $0$),\n\\begin{equation}\\label{DiagHiggs2Cover_eq}\n  \\widetilde{\\Phi}_t=2\\lambda_x(t)\\begin{pmatrix}\n      \\frac{1}{\\xi_{x,t}^{2(m_x-1)}}&0\\\\0&\\frac{1}{\\xi_{x,t}^{2(m_x-1)}}\n  \\end{pmatrix}\\,\\mathrm{d}\\xi_{x,t},\\quad \\tilde{h}_t^{\\mathrm{app}}=2\\begin{pmatrix}\n    |\\xi_{x,t}|^3&0\\\\0&|\\xi_{x,t}|^3\n  \\end{pmatrix}.\n\\end{equation}\nFor $x\\in T$, denote $\\hat{B}_{x,t}:=\\{\\,|\\zeta_{x,t}|<\\lambda_x(t)^{-2}\\,\\}$. Choose $t_0\\geqslant 1$ so that when $t\\geqslant t_0$, we have $\\lambda_x(t)^{-2}<\\kappa\/2$, and then $h_t^{\\mathrm{app}}=h_t^{\\mathrm{model}}$ over $\\hat{B}_{x,t}$. From now on, we assume that $t\\geqslant t_0$.\n\\begin{definition}\n  Let $r_t$ be a smooth weight function on $C$ such that\n\\begin{enumerate}[label=(\\roman*)]\n  \\item  $r_t=1$ in $C\\backslash (\\bigcup_{x\\in I}\\widetilde{B}_{x,t}\\cup\\bigcup_{x\\in T}\\hat{B}_{x,t})$.\n  \\item  In $\\widetilde{B}_{x,t}$, $x\\in S$, $r_t$ is nondecreasing in $|\\zeta_{x,t}|$ (or $|z_x|$ for $x\\in I_u$).\n  \\item \\[r_t=\\begin{cases}\n      |t^{1\/j(x)}z_x|\\quad &\\text{if }x\\in I_u,~|z_x|\\leqslant \\kappa t^{-1\/j(x)}\/2,  \\\\\n      |t^{1\/(m_x-1)}\\zeta_{x,t}|^{1\/2}\\quad &\\text{if }x\\in I_t,~|\\zeta_{x,t}|\\leqslant \\kappa t^{-1\/(m_x-1)}\/2,\\\\\n      |\\lambda_x(t)^2\\zeta_{x,t}|\\quad &\\text{if }x\\in T,~|\\zeta_{x,t}|\\leqslant \\lambda_x(t)^{-2}\/2.\n    \\end{cases}\\]\n\\end{enumerate}\n\\end{definition}\n  Denote $B_x=\\{\\,|z_x|<\\kappa_2\\,\\}$, where $\\kappa_2$ is from Proposition \\ref{NormalForm_prop}, and we can assume that $\\kappa_2$ is small enough so that the disks $B_{x}$ are disjoint.\n\\begin{definition}\\label{Metricgt_def}\n  Let $g_t=f_t g_C$ be the metric on $C$ conformal to the round metric $g_C$ (Hitchin's equations are conformally invariant, we consider $L_t$ above with $\\star$ and $\\Delta_{A_t}$ defined using $g_t$), which satisfies\n\\begin{enumerate}[label=(\\roman*)]\n  \\item $f_t=1$ in $C\\backslash(\\bigcup_{x\\in S}B_x)$, independent of $t$ in $C\\backslash(\\bigcup_{x\\in I_t\\cup T}B_x)$.\n  \\item $g_t$ induces the volume form $(i\/8|\\zeta_{x,t}|)\\,\\mathrm{d}\\zeta_{x,t} \\mathrm{d}\\bar{\\zeta}_{x,t}$ when restricted to $\\widetilde{B}_{x,t}$ for $x\\in I_t$, its pullback under $\\sigma_{x,t}$ is $(i\/2)\\,\\mathrm{d}\\xi_{x,t} \\mathrm{d}\\bar{\\xi}_{x,t}$, the Euclidean volume form on $D_{x,t}$.\n  \\item $g_t$ is Euclidean in $B_x$ for $x\\in I_u$.\n  \\item $g_t$ is Euclidean in $\\widetilde{B}_{x,t}$ for $x\\in T$.\n  \\item $|\\nabla_{g_C}^j \\log\\,f_t|_{g_C}\\leqslant c$ in $B_x$ for $x\\in I_u\\cup T$, $j=0,1,2$, and $c^{-1}|z_x|^{-1}\\leqslant f_t\\leqslant c|z_x|^{-1}$, $|\\nabla_{g_C}^j \\log\\,f_t|_{g_C}\\leqslant ct^{j\/(m_x-1)}$ in $B_x\\backslash \\widetilde{B}_{x,t}$ for $x\\in I_t$, $j=1,2$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\n  For smooth sections $w\\in\\Omega^p(\\mathrm{End} E)$, define\n\\begin{equation}\\label{SobolevSpace_eq}\n\\lVert w\\rVert_{L_\\delta^p}=\\lVert r_t^{-\\delta-2\/p} w\\rVert_{L^p},~  \\lVert w\\rVert_{L_\\delta^{k,p}}^p=\\lVert w\\rVert_{L_\\delta^{k-1,p}}^p+\\lVert \\nabla_{A_t}w\\rVert_{L_\\delta^{k-1,p}}^p+\\lVert \\Psi_t\\otimes w\\rVert_{L_\\delta^{k-1,p}}^p\n\\end{equation}\ninductively for $k=1,2\\ldots$ Here the $L^p$ norms are taken with respect to $g_t$ and $h_t^{\\mathrm{app}}$. Let the spaces $L_\\delta^p$ and $L_\\delta^{k,p}$ be the completions of smooth sections over $C\\backslash S$ with respect to these norms. We also introduce spaces with additional weighted conditions,\n\\[\\hat{L}_{\\delta}^{k,p}=\\left\\{\\,f\\in L_{\\delta}^{k,p},r_t^{j-k}\\nabla_{A_t}^j f\\in L_\\delta^p\\text{ for }0\\leqslant j\\leqslant k\\right\\}.\\]\nBy using cutoff functions of the form $1-\\chi(nr_t)$, one can see that $C_0^\\infty(C\\backslash S)$ is dense in $\\hat{L}_\\delta^{k,p}$.\n\\end{definition}\n\nIn $\\widetilde{B}_{x,t}$ for $x\\in I_u$, we can decompose $\\mathrm{End}\\, E$ as $\\mathrm{End}_D\\, E\\oplus \\mathrm{End}_T\\, E$ in the holomorphic frame above, where $\\mathrm{End}_D$ is the diagonal part. Then $\\mathrm{End}_D\\, E=\\ker \\mathrm{ad}(\\Phi_t), \\mathrm{End}_T\\, E=\\left(\\ker \\mathrm{ad}(\\Phi_t)\\right)^\\perp$. $\\Phi_t$ acts on $u\\in\\mathrm{End}_T\\, E$ as\n\\[[\\Phi_t,u]= 2 \\frac{\\sqrt{-z_x^{2m_x}\\nu_t(z)}}{z_x^{m_x}}\\begin{pmatrix}\n0&u_{12}\\\\-u_{21}&0\n\\end{pmatrix}\\mathrm{d}z\\Rightarrow |[\\Phi_t,u]|=2\\frac{|-z_x^{2m_x}\\nu_t(z)|^{1\/2}}{|z_x|^{m_x}}|u|,\\]\nand then\n\\begin{equation}\\label{HiggsActBd_eq}\nc^{-1}t^{m_x\/j(x)} |u|r_t^{-m_x}  \\leqslant |[\\Phi_t,u]|\\leqslant ct^{m_x\/j(x)} |u|r_t^{-m_x}\\text{ in }\\widetilde{B}_{x,t}.\n\\end{equation}\n Therefore, over $\\widetilde{B}_{x,t}$, the spaces defined above are equivalent (may not be uniformly in $t$) to the spaces defined in \\cite{biquard_boalch_2004}, using the irregular Hermitian connection $A_t+\\Phi_t-\\Phi_t^\\ast$ and the decomposition of $\\mathrm{End}\\, E$ into components $\\mathrm{End}(E)_0=\\mathrm{End}_D\\, E$, $\\mathrm{End}(E)_{m_x}=\\mathrm{End}_T\\, E$, from which we collect the following facts over $\\widetilde{B}_{x,t}$.\n\\begin{lemma}\\label{SobEmbed_lem}\n\\begin{enumerate}[label=(\\roman*)]\n\\item For functions, we have $\\hat{L}_{\\delta-1}^{1,p}\\hookrightarrow L_\\delta^q$ if $1\/2\\geqslant 1\/p-1\/q$ (with strict inequality for $q=\\infty$). In particular, for $p>2$ we have $\\hat{L}_{\\delta-1}^{1,p}\\hookrightarrow C_\\delta^0$.\n\\item If $\\delta<0$ and the function $f$ vanishes on $\\partial \\widetilde{B}_{x,t}$, or if $\\delta>0$ and $f$ vanishes near $x$, then\n\\[\\lVert \\partial f\/\\partial r\\rVert_{L_{\\delta-1}^p}\\geqslant c^{-1}t^{1\/j(x)}\\lVert f\\rVert_{L_{\\delta}^p},\\]\nwhere $r=|z_x|$. This implies that $\\hat{L}_{\\delta-1}^{1,p}=L_{\\delta-1}^{1,p}$ for $\\delta<0$ and $p\\in[1,\\infty)$.\n\\item If $u\\in L_{-2+\\delta}^{2,2}$ is a section of $\\mathrm{End}_D\\, E$, then $u$ is continuous and $u-u(x)\\in\\hat{L}_{-2+\\delta}^{2,2}\\subset C_\\delta^0\\cap L_\\delta^p$.\n\\item For $\\delta'<\\delta$, $p>2$, there are compact Sobolev embeddings for sections of $\\mathrm{End}_T\\, E$: \\[\nL_{\\delta}^{1,2}\\hookrightarrow L_{\\delta'+1+(2m_x-2)\/p}^p,\\quad L_{\\delta}^{1,p}\\hookrightarrow C_{\\delta'+m_x-2(m_x-1)\/p}^0,\\quad\nL_{\\delta}^{2,2}\\hookrightarrow C_{\\delta'+m_x+1}^0.\n\\]\n\\end{enumerate}\n\\end{lemma}\n\nNow we consider the Dirichlet problem on $\\widetilde{B}_{x,t}$ and derive some a priori estimates. The proofs of the following two results are adapted from \\cite[Cor.~4.2,~Lem.~4.4]{biquard_boalch_2004}, with the $t$-dependence of the operator $L_t$ taken into account.\n\n\\begin{lemma}\\label{aprioriEst_lem}\n  Suppose $w\\in L_{-2+\\delta}^{1,2}(\\Omega^1(\\mathrm{End}_T\\, E))$, which vanishes on $\\partial \\widetilde{B}_{x,t}$, then\n  \\[\\lVert D_t w\\rVert_{L_{-2+\\delta}^2}+\\lVert D_t^\\ast w\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\big(\\lVert\\nabla_{A_t} w\\rVert_{L_{-2+\\delta}^2}+\\lVert \\Psi_t\\otimes w\\rVert_{L_{-2+\\delta}^2}\\big),\\]\n  for all weights $\\delta$, when $t$ is sufficiently large.\n\\end{lemma}\n\\begin{proof}\n   By Lemma \\ref{weitz_lemma} (\\romannumeral1), noting that the curvature terms vanish,\n   \\[\\lVert (D_t+D_t^\\ast)(r_t^{1-\\delta}w)\\rVert_{L^2}^2=\\lVert \\nabla_{A_t}(r_t^{1-\\delta}w)\\rVert_{L^2}^2+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}^2.\\]\n   The commutator can be controlled as\n \\[|[D_t+D_t^\\ast,r_t^{1-\\delta}]w|+|[\\nabla_{A_t},r_t^{1-\\delta}]w|\\leqslant ct^{1\/j(x)}|r_t^{-\\delta}w|.\\]\n Then we have \\[\\lVert r_t^{1-\\delta}(D_t+D_t^\\ast)w\\rVert_{L^2}\\geqslant c^{-1}\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right)-ct^{1\/j(x)}\\lVert r_t^{-\\delta}w\\rVert_{L^2}.\\]\nOn the other hand, \\[\\lVert r_t^{-\\delta}w\\rVert_{L^2}\\leqslant \\lVert r_t^{1-\\delta}r_t^{-m_x}w\\rVert_{L^2}\\leqslant ct^{-m_x\/j(x)} \\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}.\\]\n Thus \\begin{align*}\n   \\lVert r_t^{1-\\delta}(D_t+D_t^\\ast)w\\rVert_{L^2}&\\geqslant (c^{-1}-ct^{-(m_x-1)\/j(x)})\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right)\\\\\n   &\\geqslant (c^{-1}\/2)\\left(\\lVert r_t^{1-\\delta}\\nabla_{A_t} w\\rVert_{L^2}+\\lVert r_t^{1-\\delta}\\Psi_t\\otimes w\\rVert_{L^2}\\right),\n \\end{align*}\n when $t$ is large enough.\n\\end{proof}\n\\begin{proposition}\\label{analysisIu_prop}\n  On $\\widetilde{B}_{x,t}$ for $x\\in I_u$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet boundary condition for small weights $\\delta>0$. When restricted to the off-diagonal part, the same holds for any $\\delta$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{proposition}\n\\begin{proof}\nObserve that the decomposition $\\mathrm{End}\\, E=\\mathrm{End}_D\\, E\\oplus \\mathrm{End}_T\\, E$ is preserved by $L_t$, we can consider the two components separately. For the off-diagonal part, we find a solution of $L_tu=v$ by minimizing the functional (recall Lemma \\ref{weitz_lemma} (\\romannumeral3))\n\\[S(u)=\\frac{1}{2}\\int_{\\widetilde{B}_{x,t}} |D_t u|^2\\,\\mathrm{dvol}-\\langle u,v\\rangle_{L^2}=\\frac{1}{2}\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+\\lVert[\\Phi_t,u]\\rVert_{L^2}^2-\\langle u,v\\rangle_{L^2}\\]\namong $u\\in L_{-1}^{1,2}(i\\mathfrak{su}_T(E))$ vanishing on the boundary, where $v\\in L_{-m_x-1}^2(i\\mathfrak{su}_T(E))$. $S(u)$ is continuous: \\[S(u)\\leqslant \\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert \\Psi_t\\otimes u\\rVert_{L^2}^2+\\lVert r_t^{-m_x}u \\rVert_{L^2}\\lVert r_t^{m_x} v\\rVert_{L^2}\\leqslant \\lVert u\\rVert_{L_{-1}^{1,2}}^2+c\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2},\\]\nwhere we used \\eqref{HiggsActBd_eq} in the last inequality. On the other hand,\n\\begin{align}\n  S(u)&=\\frac{1}{2}\\left(\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+2\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right)-\\langle u,v\\rangle_{L^2}\\notag\\\\&\\geqslant c^{-1}\\left(\\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert \\Psi_t\\otimes u\\rVert_{L^2}^2+t^{2m_x\/j(x)} \\left\\lVert r_t^{-m_x}u\\right\\rVert_{L^2}^2\\right)- \\lVert r_t^{-m_x}u\\rVert_{L^2}\\lVert v\\rVert_{L_{-m_x-1}^2}\\notag\\\\\n  &\\geqslant c^{-1}\\lVert u\\rVert_{L_{-1}^{1,2}}^2-c\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2}.\\label{lowerbdS_eq}\n\\end{align}\nTherefore $S(u)$ is coercive. If $\\mathrm{d}_{A_t}u=0$, then $|u|^2$ is constant and $u=0$ since it vanishes on the boundary. This implies that $S$ is strictly convex, and a unique minimum of $S$ can be found, being a weak solution of the equation of the form \\[\\Delta_{A_t}\\left( r_t^{m_x} u\\right)=r_t^{m_x}v+P_0\\left(r_t^{-m_x}u\\right)+P_1\\left(\\nabla_{A_t} u\\right),\\]\nwhere $P_0,P_1:L^2\\to L^2$ are bounded. By elliptic regularity, $r_t^{m_x} \\nabla_{A_t}^2 u\\in L^2$, and then $u\\in L_{-m_x-1}^{2,2}$. With the Dirichlet boundary condition, we have obtained an isomorphism $L_t: L_{-m_x-1}^{2,2}\\to L_{-m_x-1}^2$. The solution satisfies $\\langle D_tu,D_tf\\rangle_{L^2}=\\langle v,f\\rangle_{L^2}$ for any $f\\in L_{-1}^{1,2}$ vanishing on the boundary. Choose $f=u$ (the first inequality follows by setting $v=0$ in \\eqref{lowerbdS_eq}),\n\\begin{align}\n  \\lVert u\\rVert_{L_{-1}^{1,2}}^2\\leqslant c\\lVert D_tu\\rVert_{L^2}^2= c\\langle v,u\\rangle_{L^2}&\\leqslant ct^{-m_x\/j(x)}\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\lVert r_t^{m_x}v\\rVert_{L^2}\\notag\\\\&\\leqslant ct^{-m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}\\lVert v\\rVert_{L_{-m_x-1}^2}\\notag\\\\\\Rightarrow~\\lVert u\\rVert_{L_{-1}^{1,2}}&\\leqslant ct^{-m_x\/j(x)}\\lVert v\\rVert_{L_{-m_x-1}^2}.\\label{estimate1_eq}\n\\end{align}\nFor a one form $w$ with coefficient in $i\\mathfrak{su}(E)$, by Lemma \\ref{aprioriEst_lem} we have \\[\\lVert\\nabla_{A_t} w\\rVert_{L_{-m_x-1}^2}\\leqslant c\\left(\\lVert D_t w\\rVert_{L_{-m_x-1}^2}+\\lVert D_t^\\ast w\\rVert_{L_{-m_x-1}^2}\\right).\\]\nThen for $w=\\nabla_{A_t}u$,\n\\begin{align*}\n  D_tw&=(\\mathrm{d}_{A_t}+[\\Psi_t,\\cdot])(\\mathrm{d}_{A_t}u)=[\\Psi_t,\\mathrm{d}_{A_t}u],\\\\\n  D_t^\\ast w&=D_t^\\ast D_tu-D_t^\\ast[\\Psi_t,u]=D_t^\\ast D_tu+\\bar{\\ast}[\\bar{\\ast}\\Psi_t,\\mathrm{d}_{A_t}u]-[\\Psi_t,\\cdot]^\\ast[\\Psi_t,u],\\\\\n  \\lVert r_t^{m_x} (\\Psi_t\\otimes\\cdot)^2 u\\rVert_{L^2}&\\leqslant t^{m_x\/j(x)}\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}, \\\\\n  \\lVert r_t^{m_x}\\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant t^{m_x\/j(x)}\\lVert \\nabla_{A_t}u\\rVert_{L^2}\\leqslant t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}},\\\\\n  \\lVert r_t^{m_x}\\nabla_{A_t}^2 u\\rVert_{L^2}&\\leqslant c\\left(t^{m_x\/j(x)}\\lVert u\\rVert_{L_{-1}^{1,2}}+\\lVert D_t^\\ast D_tu\\rVert_{L_{-m_x-1}^2}\\right)\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2},\n\\end{align*}\nby \\eqref{estimate1_eq}. Similarly, with $w=[\\Psi_t,u]$ one deduces that $\\lVert r_t^{m_x} \\nabla_{A_t}[\\Psi_t,u]\\rVert_{L^2}\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2}$. Hence\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-m_x-1}^{2,2}}&\\leqslant c\\left(\\lVert u\\rVert_{L_{-1}^{1,2}}+\\lVert r_t^{m_x}\\nabla_{A_t}^2u\\rVert_{L^2}+\\lVert r_t^{m_x}\\nabla_{A_t}[\\Psi_t,u]\\rVert_{L^2}\\right.\\\\&+\\left.\\lVert r_t^{m_x}(\\Psi_t\\otimes\\cdot)^2u\\rVert_{L^2}+\\lVert r_t^{m_x}\\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}\\right)\\leqslant c\\lVert v\\rVert_{L_{-m_x-1}^2}.\n\\end{align*}\n\nNext we prove that the isomorphism extends to all weights, by showing that the inverse is continuous in other weighted spaces. For any weight $\\delta$, it suffices to prove\n\\[\\lVert r_t^\\delta D_t^\\ast D_t u\\rVert_{L_{-m_x-1}^2}=\\lVert D_t^\\ast D_t u\\rVert_{L_{-m_x-1-\\delta}^2}\\geqslant c^{-1}\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}},\\]\nusing the established a priori estimates\n\\begin{equation}\n  \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}\\leqslant c\\lVert D_t^\\ast D_t r_t^\\delta u\\rVert_{L_{-m_x-1}^2}.\\label{wtApriori_eq}\n\\end{equation}\nWe have\n \\begin{align*}\n  \\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}&\\leqslant ct^{2\/j(x)}\\lVert r_t^{\\delta-2} u\\rVert_{L_{-m_x-1}^2}\\\\&\\quad+ct^{1\/j(x)}\\left(\\lVert r_t^{\\delta-1} \\nabla_{A_t} u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^{\\delta-1} [\\Phi_t,u]\\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant ct^{2\/j(x)} \\lVert r_t^\\delta r_t^{-2m_x} u\\rVert_{L_{-m_x-1}^2}\\\\&\\quad+ct^{1\/j(x)}\\left(\\lVert r_t^\\delta r_t^{-m_x}\\nabla_{A_t} u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^\\delta r_t^{-m_x}[\\Phi_t, u] \\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant c t^{(1-m_x)\/j(x)} \\left(\\lVert r_t^\\delta \\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L_{-m_x-1}^2}+\\lVert r_t^\\delta (\\Psi_t\\otimes \\cdot)^2u\\rVert_{L_{-m_x-1}^2}\\right)\\\\\n  &\\leqslant ct^{(1-m_x)\/j(x)} \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}.\n\\end{align*}\n Similarly, noting that $|[\\nabla_{A_t},r_t^\\delta]w|\\leqslant c t^{(1-m_x)\/j(x)}|r_t^\\delta \\Psi_t\\otimes w|$,\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}&\\leqslant \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}+\\lVert  [\\nabla_{A_t}^2,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}+\\lVert [\\nabla_{A_t},r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\&\\hspace{2.75cm}+\\lVert \\Psi_t\\otimes [\\nabla_{A_t},r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}+\\lVert [\\nabla_{A_t},r_t^\\delta]\\Psi_t\\otimes u\\rVert_{L_{-m_x-1}^2} \\\\\n  &\\leqslant \\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}+ct^{(1-m_x)\/j(x)} \\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}.\n\\end{align*}\nFinally by \\eqref{wtApriori_eq},\n\\begin{align*}\n  \\lVert r_t^\\delta D_t^\\ast D_t u\\rVert_{L_{-m_x-1}^2}&\\geqslant \\lVert D_t^\\ast D_tr_t^\\delta u \\rVert_{L_{-m_x-1}^2}-\\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\&\\geqslant c^{-1}\\lVert r_t^\\delta u\\rVert_{L_{-m_x-1}^{2,2}}-\\lVert [D_t^\\ast D_t,r_t^\\delta]u\\rVert_{L_{-m_x-1}^2}\\\\\n  &\\geqslant (c^{-1}-ct^{(1-m_x)\/j(x)})\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}}\\geqslant (c^{-1}\/2)\\lVert u\\rVert_{L_{-m_x-1-\\delta}^{2,2}},\n\\end{align*}\nfor $t$ large. We have proved that for the off-diagonal part, $L_t: L_{-2+\\delta}^{2,2}\\to L_{-2+\\delta}^2$ is an isomorphism for all weights $\\delta$ with the norm of the inverse uniformly bounded in $t$.\n\nFor the diagonal part, the equation becomes the Laplace equation $\\Delta u=v$ with the Dirichlet boundary condition. Let $\\tau: \\widetilde{B}_{x,t}\\to B_{0}(\\kappa):=\\{\\,|\\tilde{z}|<\\kappa\\,\\},~ z_x\\mapsto t^{1\/j(x)}z_x:=\\tilde{z}$ be the rescaling map, and $\\tilde{u}={(\\tau^{-1})}^\\ast u$, $\\tilde{v}= {(\\tau^{-1})}^\\ast v$. Then $\\Delta u=v$ is equivalent to $t^{2\/j(x)}\\Delta \\tilde{u}=\\tilde{v}$ on $B_{0}(\\kappa)$, where $\\Delta: L_{-2+\\delta}^{2,2}\\to L_{-2+\\delta}^2$ is an isomorphism by classical elliptic theory on cylinders (see \\cite{biquard_boalch_2004}), and the weighted spaces on $B_{0}(\\kappa)$ are defined similarly as in \\eqref{SobolevSpace_eq}, using the Euclidean distance from $0$ as the weight function, which is equivalent to ${(\\tau^{-1})}^\\ast r_t$ uniformly in $t$. Then $\\lVert  \\tilde{v}\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\lVert t^{2\/j(x)} \\tilde{u}\\rVert_{L_{-2+\\delta}^{2,2}}$, which implies that $\\lVert v\\rVert_{L_{-2+\\delta}^2}\\geqslant c^{-1}\\lVert u\\rVert_{L_{-2+\\delta}^{2,2}}$.\n\\end{proof}\n\nNext we study the behavior of $L_t$ around $x\\in I_t$. Define $L_\\delta^{k,p}(D_{x,t}),\\hat{L}_\\delta^{k,p}(D_{x,t})$ as in \\eqref{SobolevSpace_eq} using $\\tilde{r}_t=\\sigma_{x,t}^\\ast r_t$, $\\widetilde{\\Phi}_t$, $\\tilde{h}_t^{\\mathrm{app}}$, the Euclidean metric on $D_{x,t}$ (which is $\\sigma_{x,t}^{\\ast}g_t$), and $\\widetilde{A}_t=D(\\tilde{\\bar{\\partial}}_E,\\tilde{h}_t^{\\mathrm{app}})$. For $w\\in \\Omega^q(\\mathrm{End}\\,E)$ we have \\begin{equation}\\label{NormRelation_eq}\n     \\lVert \\sigma_{x,t}^\\ast w\\rVert_{L_\\delta^{k,p}(D_{x,t})}=2\\lVert w\\rVert_{L_\\delta^{k,p}(\\widetilde{B}_{x,t})}.\n   \\end{equation}\nLet $\\widetilde{L}_t(\\gamma)=\\Delta_{\\widetilde{A}_t}\\gamma-\\mathrm{i}\\star M_{\\widetilde{\\Phi}_t}\\gamma$, then $\\widetilde{L}_t(\\sigma^\\ast\\gamma)=\\sigma^\\ast (L_t\\gamma)$. The previous analysis near $x\\in I_u$ yields the following.\n\n\\begin{corollary}\\label{analysisIt_cor}\n  On $\\widetilde{B}_{x,t}$ for $x\\in I_t$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet condition on the boundary for small weights $\\delta>0$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{corollary}\n\\begin{proof}\n Let $v\\in L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))(\\widetilde{B}_{x,t})$ then $\\tilde{v}=\\sigma_{x,t}^\\ast v\\in L_{-2+\\delta}^2(i\\mathfrak{su}(E))(D_{x,t})$, and by Proposition \\ref{analysisIu_prop} with $m_x$ and $j(x)$ both replaced by $2(m_x-1)$, there exists $\\tilde{u}\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))(D_{x,t})$ such that $\\widetilde{L}_t \\tilde{u}=\\tilde{v}$ and $\\tilde{u}=0$ on $\\partial D_{x,t}$. Note that $\\widetilde{L}_t$ is $\\mathbb{Z}_2$-equivariant, meaning that $\\iota^\\ast(\\widetilde{L}_t (\\tilde{u}))=\\widetilde{L}_t (\\iota^\\ast \\tilde{u})$ for the involution $\\iota:\\xi_{x,t}\\mapsto -\\xi_{x,t}$. Then $\\widetilde{L}_t(\\iota^\\ast \\tilde{u}-\\tilde{u})=\\iota^\\ast \\tilde{v}-\\tilde{v}=0$, and $\\iota^\\ast \\tilde{u}=\\tilde{u}$ since $\\widetilde{L}_t$ is injective. $\\tilde{u}$ descends to $\\widetilde{B}_{x,t}$ to give $u\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))(\\widetilde{B}_{x,t})$ with $L_t u=v$ and $u=0$ on $\\partial \\widetilde{B}_{x,t}$. If $L_t u=0$, then $\\widetilde{L}_t(\\sigma_{x,t}^\\ast u)=0$ and $\\sigma_{x,t}^\\ast u=0$ which implies that $u=0$. Therefore $L_t$ is an isomorphism. The uniform boundedness of $L_t^{-1}$ follows from that of $\\widetilde{L}_t^{-1}$ and \\eqref{NormRelation_eq}.\n\\end{proof}\n The local behavior of $L_t$ around $x\\in T$ has been studied in \\cite{fredrickson2022asymptotic}, and we have the following result.\n\\begin{proposition}\\label{analysisT_prop}\n  On $\\hat{B}_{x,t}$, for $x\\in T$, $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ is an isomorphism with zero Dirichlet condition on the boundary for small weights $\\delta>0$. Moreover, the norm of $L_t^{-1}$ is uniformly bounded in $t$.\n\\end{proposition}\n\\begin{proof}\nConsider $\\tau:\\hat{B}_{x,t}\\to B_0(1):=\\{\\,|\\tilde{z}|<1\\,\\},~\\zeta_{x,t}\\to \\lambda_x(t)^2\\zeta_{x,t}:=\\tilde{z}$. Let $\\widetilde{\\Phi}_t=(\\tau^{-1})^\\ast\\Phi_t$, and $\\tilde{h}_t^{\\mathrm{app}}=(\\tau^{-1})^\\ast \\tilde{h}_t^{\\mathrm{app}}$. Then by \\eqref{NormalFormT_eq} and \\eqref{approxMetricT_eq},\n\\[\\widetilde{\\Phi}_t=\\begin{pmatrix}\n  0&\\frac{1}{\\lambda_x(t)}\\\\ \\frac{\\lambda_x(t)}{\\tilde{z}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\tilde{z},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\begin{pmatrix}\n  \\frac{1}{\\lambda_x(t)}\\tilde{r}^{1\/2}\\mathrm{e}^{\\psi_{2,x} (8\\tilde{r}^{1\/2})}&0\\\\0&\\frac{1}{\\lambda_x(t)^3}\\tilde{r}^{3\/2}\\mathrm{e}^{-\\psi_{2,x}(8\\tilde{r}^{1\/2})}\n\\end{pmatrix},\\]\nwhere $\\tilde{r}=|\\tilde{z}|$. By the gauge transformation $g=\\left(\\begin{smallmatrix}\n  \\lambda_x(t)^{-1\/2}&0\\\\0&\\lambda_x(t)^{1\/2}\n\\end{smallmatrix}\\right)$, we have\n\\[\\widetilde{\\Phi}_t=\\begin{pmatrix}\n  0&1\\\\ \\frac{1}{\\tilde{z}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\tilde{z},\\quad \\tilde{h}_t^{\\mathrm{app}}=\\frac{1}{\\lambda_x(t)^2}\\begin{pmatrix}\n  \\tilde{r}^{1\/2}\\mathrm{e}^{\\psi_{2,x} (8\\tilde{r}^{1\/2})}&0\\\\0&\\tilde{r}^{3\/2}\\mathrm{e}^{-\\psi_{2,x}(8\\tilde{r}^{1\/2})}\n\\end{pmatrix}.\\]\nThen in this trivialization, $\\widetilde{\\Phi}_t$ and $\\lambda_x(t)^2 \\tilde{h}_t^{\\mathrm{app}}$ are independent of $t$, which together with the Euclidean metric $g_e$ on $B_0(1)$ define an operator $\\tilde{L}$ as in \\eqref{LinOp_eq}. The indicial root analysis as in \\cite[Sec.~4.2,~Lem. 5.8]{fredrickson2022asymptotic} implies that $\\tilde{L}: L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2}(i\\mathfrak{su}(E))$ (these spaces are defined on $B_0(1)$ as in \\eqref{SobolevSpace_eq} using $g_e$, $\\widetilde{\\Phi}_t$, $\\lambda_x(t)^2 \\tilde{h}_t^{\\mathrm{app}}$) is an isomorphism for $\\delta>0$ small. Therefore $L_t$ is also an isomorphism, and the uniform boundedness of $L_t^{-1}$ follows in the same way as in Proposition \\ref{analysisIu_prop} for the diagonal part.\n\\end{proof}\n\nLet $U_{\\mathrm{ext}}:=C\\backslash (\\bigcup_{x\\in I} \\frac{1}{2}\\widetilde{B}_{x,t}\\cup\\bigcup_{x\\in T}\\frac{1}{2}\\hat{B}_{x,t})$, where $\\frac{1}{2}\\widetilde{B}_{x,t}=\\{\\,|\\zeta_{x,t}|< \\kappa t^{-1\/(m_x-1)}\/2\\,\\}$ for $x\\in I_t$, $\\frac{1}{2}\\widetilde{B}_{x,t}=\\{\\,|z_x|< \\kappa t^{-1\/j(x)}\/2\\,\\}$ for $x\\in I_u$, and $\\frac{1}{2}\\hat{B}_{x,t}=\\{\\,|\\zeta_{x,t}|< \\lambda_x(t)^{-2}\/2\\,\\}$ for $x\\in T$. In $U_{\\mathrm{ext}}$, $c^{-1}\\leqslant r_t\\leqslant 1$, which is immaterial in the analysis, so we omit the subscript of $L^{k,2}_{-2+\\delta}$.\n\n\\begin{proposition}\\label{analysisUext_prop}\n  On $U_{\\mathrm{ext}}$, $L_t:L^{2,2}(i\\mathfrak{su}(E))\\to L^2(i\\mathfrak{su}(E))$ is an isomorphism when imposing the zero Dirchlet boundary condition. The norm of $L_t^{-1}$ is bounded by $ct^4$.\n\\end{proposition}\n\\begin{proof}\n  Invertibility is standard (see \\cite[Sec.~5.2]{mazzeo_swoboda_weiss_witt_2016}). Let $u\\in L^{2,2}(i\\mathfrak{su}(E))$ be vanishing on $\\partial U_{\\mathrm{ext}}$, then \\begin{align*}\n    \\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert [\\Psi_t,u]\\rVert_{L^2}^2 =\\langle D_t^\\ast D_t u,u\\rangle_{L^2}\\leqslant \\lVert L_t u\\rVert_{L^2}\\lVert u\\rVert_{L^2}.\n  \\end{align*}\nLet $\\check{U}_{\\mathrm{ext}}=\\{|w|<2\\kappa^{-1}t^{1\/(m_0+N-4)}\\}$, where $w=1\/z$. Then $\\check{U}_{\\mathrm{ext}}$ is a disk centered at $\\infty$ and contains $U_{\\mathrm{ext}}$. Let $L_\\mathrm{e}^2$ be the $L^2$ norm on $\\check{U}_{\\mathrm{ext}}$ defined using the Euclidean metric $g_e$ on this disk. By Kato's inequality and Poincar\u00e9 inequality, $\\lVert u\\rVert_{L_\\mathrm{e}^2}\\leqslant ct^{1\/(m_0+N-4)}\\lVert \\nabla_{A_t} u\\rVert_{L_\\mathrm{e}^2}\\leqslant ct^{1\/2}\\lVert \\nabla_{A_t} u\\rVert_{L_\\mathrm{e}^2}$. By the Definition \\ref{Metricgt_def} of $g_t$, we have $c^{-1}t^{-1}g_t\\leqslant g_e\\leqslant ct^{4\/(m_0+N-4)} g_t\\leqslant ct^2g_t$ on $U_{\\mathrm{ext}}$. Therefore,\n\\[\\lVert u\\rVert_{L^2}^2\\leqslant ct\\lVert u\\rVert_{L_\\mathrm{e}^2}^2\\leqslant ct^2\\lVert \\nabla_{A_t}u\\rVert_{L_\\mathrm{e}^2}^2= ct^2\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2,\\]\nwhere the last equality follows from the conformal invariance of the $L^2$ norm of a $1$-form. Then we have\n  \\begin{align*}\n    \\lVert u\\rVert_{L^2}^2&\\leqslant ct^2\\lVert L_t u\\rVert_{L^2}\\lVert u\\rVert_{L^2}\\,\\Rightarrow\\, \\lVert u\\rVert_{L^2}\\leqslant ct^2\\lVert L_t u\\rVert_{L^2},\\\\\n    \\lVert \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant ct\\lVert L_tu\\rVert_{L^2},~\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant ct\\lvert L_t u\\rVert_{L^2}.\n  \\end{align*}\nWe deduce that \\begin{align*}\n    \\lVert (\\Psi_t\\otimes\\cdot)^2 u\\rVert_{L^2}&\\leqslant c\\sup |\\Phi_t|\\lVert \\Psi_t\\otimes u\\rVert_{L^2}\\leqslant ct\\sup |\\Phi_t|\\lVert L_tu\\rVert_{L^2}\\\\\n    \\lVert \\Psi_t\\otimes \\nabla_{A_t}u\\rVert_{L^2}&\\leqslant c\\sup |\\Phi_t|\\lVert  \\nabla_{A_t}u\\rVert_{L^2}\\leqslant ct\\sup |\\Phi_t|\\lVert L_tu\\rVert_{L^2}.\n    \\end{align*}\n  By Lemma \\ref{weitz_lemma} (\\romannumeral1), and the conformal change formula for $\\mathscr{R}$,\n  \\begin{align*}\n    \\lVert \\nabla_{A_t}^2 u\\rVert_{L^2}&\\leqslant c\\left(\\lVert D_t\\nabla_{A_t} u\\rVert_{L^2}+\\lVert D_t^\\ast \\nabla_{A_t} u\\rVert_{L^2}+t^3\\lVert\\nabla_{A_t} u\\rVert_{L^2}+\\lVert \\mathscr{F}(\\nabla_{A_t} u)\\rVert_{L^2}\\right)\\\\\n    &\\leqslant c\\left(t\\sup|\\Phi_t|+t^4+t\\max\\sup |E_{x(,j),t}|\\right)\\lVert L_t u\\rVert_{L^2},\n  \\end{align*}\n  where the error terms $E_{x(,j),t}$ decay exponentially in $t$. In $U_{\\mathrm{ext}}\\backslash \\big(\\bigcup_{x(,j)} \\widetilde{B}_{x(,j),t}\\big)$, $\\Phi_t$ can be diagonalized as $\\sqrt{-\\nu_t(z)}\\sigma_3\\,\\mathrm{d}z$, and $h_t^{\\mathrm{app}}$ is also diagonal so that (recall that the norm $|\\mathrm{d}z|$ is taken with respect to $g_t$) \\[|\\Phi_t|^2=2|\\nu_t(z)||\\mathrm{d}z|^2\\leqslant ct^2.\\]\n  In $\\widetilde{B}_{x,j,t}$, $\\Phi_t$ and $h_t^\\mathrm{app}$ are given by \\eqref{NormalFormZ_eq} and \\eqref{approxMetricZ_eq} in some holomorphic frame, so we have\n  \\begin{align*}\n    |\\Phi_t|^2&=2\\lambda_{x,j}(t)^2r\\cosh\\left(2l_{x,j,t}(r)\\chi\\left(r\\kappa^{-1}t^{1\/j(x)}\\right)\\right)|\\mathrm{d}\\zeta_{x,j,t}|^2\\\\\n    &\\leqslant c\\lambda_{x,j}(t)^{4\/3} \\rho^{2\/3}\\cosh(2\\psi_1(\\rho)\\chi)|\\mathrm{d}\\zeta_{x,j,t}|^2\\leqslant ct^2,\n  \\end{align*}\nwhere $r=|\\zeta_{x,j,t}|, \\rho=8\\lambda_{x,j}(t)r^{3\/2}\/3$. Similarly, in $\\widetilde{B}_{x,t}\\cap U_{\\mathrm{ext}}$,  $\\Phi_t$ and $h_t^\\mathrm{app}$ are given by \\eqref{NormalFormT_eq} and \\eqref{approxMetricT_eq} in some holomorphic frame, then\n\\begin{align*}\n  |\\Phi_t|^2&=2\\lambda_{x}(t)^2r\\cosh\\left(2m_{x,t}(r)\\chi\\left(r\\kappa^{-1}\\right)\\right)|\\mathrm{d}\\zeta_{x,t}|^2\\\\\n  &\\leqslant c \\rho^2\\cosh(2\\psi_{2,x}(\\rho)\\chi)|\\mathrm{d}\\zeta_{x,t}|^2\\leqslant ct,\n\\end{align*}\nwhere $r=|\\zeta_{x,t}|, \\rho=8\\lambda_{x}(t)r^{1\/2}$. Therefore, $\\lVert \\nabla_{A_t}^2 u\\rVert_{L^2}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$. Similarly, applying the Weitzenb\u00f6ck formula for $w=[\\Psi_t,u]$ gives $\\lVert \\nabla_{A_t} [\\Psi_t,u] \\rVert_{L^2}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$. Combining all the estimates above, $\\lVert u\\rVert_{L^{2,2}(U_{\\mathrm{ext}})}\\leqslant ct^4\\lVert L_t u\\rVert_{L^2}$.\n\\end{proof}\nFinally, we consider the global behavior of $L_t$.\n\\begin{lemma}\\label{GlobIso_lem}\n  $L_t:L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^2(i\\mathfrak{su}(E))$ is an isomorphism for small $\\delta>0$.\n\\end{lemma}\n\\begin{proof}\n  By \\cite[Lemma 5.1]{biquard_boalch_2004}, $L_t$ is Fredholm of index zero (the proof still work for $x\\in I_t$ by considering the local lifted problem as in Corollary \\ref{analysisIt_cor} and for $x\\in T$ by \\cite[Lem. 5.8]{fredrickson2022asymptotic}). For $u\\in\\ker L_t$, near $x\\in I_u\\cup T$ we have $|D_t u|=O(r^{\\delta-1})$ ($r=|\\zeta_{x,t}|$ for $x\\in T$, and $r=|z_x|$ for $x\\in I_u$) by the proof of \\cite[Lem.~4.6]{biquard_boalch_2004}. The boundary term in the integration by parts is $\\lim_{\\epsilon\\to 0^+}\\int_{r=\\epsilon} \\langle \\partial_r u,u\\rangle r\\,\\mathrm{d}\\theta$ \\cite[Lem.~5.5]{fredrickson2022asymptotic} which vanishes for $x\\in I_u\\cup T$ since $\\langle[\\Phi_t,u],u\\rangle=0$ and $\\langle \\partial_r u,u\\rangle=O(r^{\\delta-1})$, and similarly vanishes for $x\\in I_t$ by considering the double lifting. Therefore \\begin{equation}\\label{intbyParts_eq}\n  \\langle L_t u,u\\rangle_{L^2}=\\lVert \\mathrm{d}_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Psi_t,u]\\rVert_{L^2}^2.\n\\end{equation}\nNow we have $\\mathrm{d}_{A_t}u=[\\Phi_t,u]=0$, which implies that $u\\equiv 0$ as in the proof of \\cite[Lem.~5.6]{fredrickson2022asymptotic} since $Z_t\\backslash (I_t\\cup T)\\neq \\varnothing$. $L_t$ has trivial kernel, and then it is an isomorphism.\n\\end{proof}\n\n\\begin{lemma}\\label{PoincareIneq_lem}Let $u\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$, then for $t$ large enough,\n  \\begin{equation}\\label{PoincareIneq_eq}\n    \\lVert u\\rVert_{L_{-\\delta}^2}^2\\leqslant ct^2\\left(\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right).\n  \\end{equation}\n\\end{lemma}\n\\begin{proof}\nThe idea is to use $[\\Phi_t,u]$ to control $u$ around a zero of $\\tilde{\\nu}_t(z)$ near each $x\\in I$ (we choose $z_{x,0}(t)$, which has asymptotics given in Lemma \\ref{HiggsDetRoot_lem}), and use $\\nabla_{A_t}u$ to control $u$ elsewhere. The inequality is obtained by gluing these estimates together.\n\nFor $t$ large, $x\\in I_u$, define $W_x:=\\{\\,|\\zeta_{x,0,t}|<t^{-(2m_x+1)\/3j(x)}\\}\\subset \\frac{1}{2}\\widetilde{B}_{x,0,t}$, a disk centered at $z_{x,0}(t)$. Then $\\chi(r\\kappa^{-1}t^{1\/j(x)})=1$ on $W_x$, where $r=|\\zeta_{x,0,t}|$, and \\[\\rho=8\\lambda_{x,0}(t)r^{3\/2}\/3\\leqslant c\\text{ on }W_x.\\]  Recall that $\\Phi_t$ is given by \\eqref{NormalFormZ_eq} in some holomorphic frame over $W_x$. Since $u$ is a section of $i\\mathfrak{su}(E)$, it has form\n\\[u=\\begin{pmatrix}\n  p& \\bar{s} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\\\ s&-p\n\\end{pmatrix},\\]\nwhere $p$ is real valued. $\\Phi_t$ acts on $u$ as\n\\[[\\Phi_t,u]=\\lambda_{x,0}(t)\\begin{pmatrix}\n  s-\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}&-2p\\\\\n  2\\zeta_{x,0,t} p&-s+\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,0,t}.\\]\nThen on $W_x$ we have \\begin{align}\n  |u|^2&=2p^2+2|s|^2 r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\leqslant 2p^2+c|s|^2t^{(2m_x+1)\/3j(x)}\\rho^{-2\/3}\\mathrm{e}^{-2\\psi_1(\\rho)}\\notag\\\\&\\leqslant 2p^2+ct^{(2m_x+1)\/3j(x)}|s|^2,\\notag  \\\\\n  |[\\Phi_t,u]|^2&= \\lambda_{x,0}(t)^2\\left(8p^2 r\\cosh\\left(2l_{x,0,t}(r)\\right)+2\\left|s-\\bar{s}\\zeta_{x,0,t} r^{-1}\\mathrm{e}^{-2l_{x,0,t}(r)}\\right|^2\\right)\\notag\\\\\n  &\\geqslant c^{-1}t^{(2m_x+1)\/j(x)}\\left(p^2t^{-(2m_x+1)\/3j(x)}\\rho^{2\/3}\\cosh\\left(2\\psi_1(\\rho)\\right)+|s|^2\\left(1-\\mathrm{e}^{-2\\psi_1(\\rho)}\\right) \\right)\\notag\\\\&\\geqslant c^{-1}t^{(4m_x+2)\/3j(x)}|u|^2.\\label{HiggsIuEst_eq}\n\\end{align}\n\nConsider $\\chi_x(r):=\\chi(t^{(2m_x+1)\/3j(x)}r)$, then $\\chi_x=0$ in $C\\backslash W_x$ and $\\chi_x=1$ in $\\hat{W}_x:=\\frac{1}{2}W_x$. Let $r_{t,x}$ be the function obtained by extending $r_t|_{\\widetilde{B}_{x,t}}$ by $1$, and $\\hat{g}_t=\\hat{f}_tg_C$ where $\\hat{f}_t$ is the extension of $f_t|_{C\\backslash \\bigcup_{y\\in I_t}B_y}$ by $1$. We claim that for all $v\\in L_{-2+\\delta}^{2,2}$ with $v$ vanishing in $\\hat{W}_x$,\n\\begin{equation}\\label{hardyineqIu_eq}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{(6+(4m_x-4)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}v\\rVert_{L_{-\\delta-1,x}^2}^2,\n\\end{equation}\nwhere the subscripts indicate the norms are defined using $r_{t,x},\\hat{g}_t$ instead of $r_t,g_t$. By Kato's inequality, we may assume that $v$ is a function, and prove the inequality with $\\nabla_{A_t}v$ replaced by $\\mathrm{d}v$. First consider the biholomorphism $\\tau_x:C\\to C_1=\\mathbb{CP}^1$, defined by $z_x\\mapsto c_xt^{1\/j(x)}z_x$, where $c_x$ is the constant satisfying $\\lim_{t\\to\\infty}\\tau_x(z_{x,0}(t))=x+1$. Define weighted spaces on $C_1$ using the round metric on $C_1$ and the weight function $r_x$ given by the geodesic distance from $x$, then $r_x$ is uniformly equivalent to $(\\tau_x^{-1})^\\ast r_{t,x}$, and $c^{-1}t^{-2\/j(x)}(\\tau_x^{-1})^\\ast \\hat{g}_t\\leqslant g_{C_1}$. Then for $v_1=(\\tau_x^{-1})^\\ast v$ we have\n\\begin{align*}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2&=\\int_{C} r_{t,x}^{2\\delta-2}|v|^2\\,\\mathrm{dvol}_{\\hat{g}_t}\\leqslant c \\int_{C_1} r_x^{2\\delta-2}|v_1|^2\\,(\\tau_x^{-1})^\\ast(\\mathrm{dvol}_{\\hat{g}_t})\\\\&\\leqslant ct^{2\/j(x)}\\int_{C_1} r_x^{2\\delta-2}|v_1|^2\\,\\mathrm{dvol}_{C_1}=ct^{2\/j(x)}\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\\\\n  \\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2&=\\int_{C_1}r_x^{2\\delta}|\\mathrm{d}v_1|_{g_{C_1}}^2\\,\\mathrm{dvol}_{C_1}\\leqslant c\\int_C r_{t,x}^{2\\delta}|\\mathrm{d}v|_{\\hat{g}_t}^2\\,\\mathrm{dvol}_{\\hat{g}_t}=c\\lVert \\mathrm{d}v\\rVert_{L_{-\\delta-1,x}^2}^2.\n\\end{align*}\n\\eqref{hardyineqIu_eq} will follow from\n\\begin{equation}\\label{hardyineqIu_eq1}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{equation}\nSuppose $C_1$ is embedded in $\\mathbb{R}^3$ so that $x=(0,1,0)$ and $x+1=(0,0,1)$. Consider the stereographic projection $\\hat{\\tau}_x:C_1\\backslash \\{x+1\\}\\to \\mathbb{R}^2$, which sends $x+1$ to $\\infty$ and $x$ to $(0,1)$. Note that as $t\\to\\infty$, $\\tau_x(z_{x,0}(t))\\to x+1$ and the diameter of $\\tau_x(\\hat{W}_x)$ is $O(t^{-(2m_x-2)\/3j(x)})$, then for some constant $\\kappa_x$, we have $\\hat{\\tau}_x^{-1}\\left(\\mathbb{R}^2\\backslash C_2\\right)\\subset \\tau_x(\\hat{W}_x)$, where $C_2:=B_{(0,1)}(\\kappa_xt^{(2m_x-2)\/3j(x)})$. Define weighted spaces on $C_2$ using the Euclidean metric and the weight function $\\hat{r}$ given by the Euclidean distance from $(0,1)$. Then $v_2=(\\hat{\\tau}_x^{-1})^\\ast v_1$ vanishes on $\\partial C_2$, $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast r_x\\leqslant \\hat{r}\\leqslant ct^{(2m_x-2)\/3j(x)}(\\hat{\\tau}_x^{-1})^\\ast r_x$, and $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast g_{C_1}\\leqslant g_{C_2}$ on $C_2$. We deduce that\n\\begin{align*}\n\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2&=\\int_{C_1}r_x^{2\\delta-2}|v_1|^2\\,\\mathrm{dvol}_{C_1}\\leqslant c\\int_{C_2}\\hat{r}^{2\\delta-2}|v_2|^2\\,\\mathrm{dvol}_{C_2}=c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\\\\n\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2&=\\int_{C_2}\\hat{r}^{2\\delta}|\\mathrm{d}v_2|_{g_{C_2}}^2\\,\\mathrm{dvol}_{C_2}\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{align*}\nAn analogue of Lemma \\ref{SobEmbed_lem} (\\romannumeral2) implies that \\[\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2,\\]\nsince $v_2$ vanishes on $\\partial C_2$. Therefore\n\\begin{align*}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2\\leqslant ct^{(4m_x-4)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2,\n\\end{align*}\nverifying \\eqref{hardyineqIu_eq1}, and \\eqref{hardyineqIu_eq} holds.\n\nFor $t$ large, $x\\in I_t$, $W_x:=\\{\\,|\\zeta_{x,0,t}|<t^{-2m_x\/3j(x)}\\,\\}\\subset \\frac{1}{2}\\widetilde{B}_{x,0,t}$, so that $\\chi(r\\kappa^{-1}t^{1\/j(x)})=1$ and $\\rho\\leqslant c$ on $W_x$, where $r=|\\zeta_{x,0,t}|$ and $\\rho=8\\lambda_{x,0}(t)r^{3\/2}\/3$. Similar to \\eqref{HiggsIuEst_eq},\n\\begin{equation}\n  |[\\Phi_t,u]|^2\\geqslant c^{-1}t^{4m_x\/3j(x)} |u|^2.\\label{HiggsItEst_eq}\n\\end{equation}\nConsider $\\chi_x(r):=\\chi(t^{2m_x\/3j(x)}r)$, then $\\chi_x=0$ in $C\\backslash W_x$ and $\\hat{\\chi}=1$ in $\\hat{W}_x:=\\frac{1}{2}W_x$. Let $r_{t,x}$ be the function on $C$ which is the extension of $r_t|_{\\widetilde{B}_{x,t}}$ by $1$, and $g_{t,x}=f_{t,x}g_C$ where $f_{t,x}$ is the extension of $f_t|_{C\\backslash \\bigcup_{y\\in I_t\\backslash\\{x\\}}B_y}$ by $1$. We claim that for all $v\\in L_{-2+\\delta}^{2,2}$ with $v$ vanishing in $\\hat{W}_x$,\n\\begin{equation}\\label{hardyineqIt_eq}\n  \\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{(6+(2m_x-3)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}v\\rVert_{L_{-\\delta-1,x}^2}^2,\n\\end{equation}\nwhere the subscripts indicate the norms are defined using $r_{t,x},g_{t,x}$ instead of $r_t$, $g_t$. As above we prove the inequality for $v$ being a function. Consider the biholomorphism $\\tau_x: C\\to C_1$, defined by $z_x\\mapsto c_xt^{1\/j(x)}z_x$, where $c_x$ is the constant chosen so that $\\lim_{t\\to\\infty}\\tau_x(z_{x,0}(t))=x+1$.  Define weighted spaces on $C_1$ using the singular metric $g_{C_1}=r_x^{-1}g_{\\mathrm{FS}(C_1)}$ where $g_{\\mathrm{FS}(C_1)}$ is the Fubini-Study metric on $C_1$, and the weight function $r_x^{1\/2}$ where $r_x$ is the geodesic distance from $x$. Note that $r_x^{1\/2}$ is uniformly equivalent to $(\\tau_x^{-1})^\\ast r_{t,x}$, and $c^{-1}t^{-2\/j(x)}(\\tau_x^{-1})^\\ast g_{t,x}\\leqslant g_{C_1}$. Then for $v_1=(\\tau^{-1})^\\ast v$ we have\n\\[\\lVert v\\rVert_{L_{-\\delta,x}^2}^2\\leqslant ct^{2\/j(x)}\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2,\\quad \\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2\\leqslant c\\lVert \\mathrm{d}v\\rVert_{L_{-\\delta-1,x}^2}^2.\\]\n\\eqref{hardyineqIt_eq} follows once we prove\n\\begin{equation}\\label{hardyineqIt_eq1}\n  \\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant ct^{(2m_x-3)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\n\\end{equation}\nSuppose $C_1$ is embedded in $\\mathbb{R}^3$ as above and consider the stereographic projection $\\hat{\\tau}_x:C_1\\backslash \\{x+1\\}\\to \\mathbb{R}^2$, which sends $x+1$ to $\\infty$ and $x$ to $(0,1)$. Note that the diameter of $\\tau_x(\\hat{W}_x)$ is $O(t^{-(2m_x-3)\/3j(x)})$, then for some constant $\\kappa_x$, we have $\\hat{\\tau}_x^{-1}\\left(\\mathbb{R}^2\\backslash C_2\\right)\\subset \\tau_x(W_2)$, where $C_2:=B_{(0,1)}(\\kappa_xt^{(2m_x-3)\/3j(x)})$. Define weighted spaces on $C_2$ using the singular metric $g_{C_2}=\\hat{r}^{-1}g_{e(C_2)}$ where $g_{e(C_2)}$ is the Euclidean metric, and the weight function $\\hat{r}^{1\/2}$ where $\\hat{r}$ is the Euclidean distance from $(0,1)$. Then $v_2=(\\hat{\\tau}_x^{-1})^\\ast v_1$ vanishes on $\\partial C_2$, $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast r_x\\leqslant \\hat{r}\\leqslant ct^{(2m_x-3)\/3j(x)}(\\hat{\\tau}_x^{-1})^\\ast r_x$, and $c^{-1}(\\hat{\\tau}_x^{-1})^\\ast g_{C_1}\\leqslant g_{C_2}$ on $C_2$. We have\n\\[\\lVert v_1\\rVert_{L_{-\\delta}^2(C_1)}^2\\leqslant c\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2,\\quad \\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2\\leqslant ct^{(2m_x-3)\\delta\/3j(x)}\\lVert \\mathrm{d}v_1\\rVert_{L_{-\\delta-1}^2(C_1)}^2.\\]\nLet $\\check{L}_{\\delta}^{k,p}(C_2)$ be defined using the Euclidean metric $g_{e(C_2)}$ and the weight function $\\hat{r}$. An analogue of Lemma \\ref{SobEmbed_lem} (\\romannumeral2) implies that \\[\\lVert v_2\\rVert_{\\check{L}_{-\\delta\/2}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{\\check{L}_{-\\delta\/2-1}^2(C_2)}^2,\\]\nsince $v_2$ vanishes on $\\partial C_2$. This is equivalent to $\\lVert v_2\\rVert_{L_{-\\delta}^2(C_2)}^2\\leqslant c\\lVert \\mathrm{d}v_2\\rVert_{L_{-\\delta-1}^2(C_2)}^2$. Therefore \\eqref{hardyineqIt_eq1} and \\eqref{hardyineqIt_eq} are verified.\n\nFor $x\\in T$, let $\\hat{\\chi}_x(r):=\\chi(\\lambda_x(t)^2 r)$, where $r=|\\zeta_{x,t}|$. In $\\hat{B}_{x,t}$, similar to \\eqref{HiggsIuEst_eq}, we have \\begin{equation}\\label{HiggsTEst_eq}\n|[\\Phi_t,u]|^2\\geqslant c^{-1}t^2|u|^2.\n\\end{equation}\nLet $v$ be vanishing on $\\partial\\hat{B}_{x,t}$, then as before,\n\\begin{equation}\\label{hardyineqT_eq}\n  \\lVert v\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\leqslant ct^{-2}\\lVert \\nabla_{A_t}u\\rVert_{L_{-\\delta-1}^2(\\hat{B}_{x,t})}^2.\n\\end{equation}\n\nApplying \\eqref{hardyineqIu_eq}, \\eqref{hardyineqIt_eq} to $v=(1-\\chi_x)u$ and \\eqref{hardyineqT_eq} to $v=\\hat{\\chi}_x u$, and using \\eqref{HiggsIuEst_eq}, \\eqref{HiggsItEst_eq}, \\eqref{HiggsTEst_eq},\n\\begin{align*}\n  \\lVert u\\rVert_{L_{-\\delta}^2}^2&\\leqslant c\\sum_{x\\in I}\\lVert u\\rVert_{L_{-\\delta,x}^2}^2+ c\\sum_{x\\in T} \\lVert\\hat{\\chi}_x u\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\\\& \\leqslant c\\sum_{x\\in I}\\lVert \\chi_xu\\rVert_{L^2(W_x)}^2+c\\sum_{x\\in I}\\lVert (1-\\chi_x)u\\rVert_{L_{-\\delta,x}^2}^2+ c\\sum_{x\\in T} \\lVert\\hat{\\chi}_x u\\rVert_{L_{-\\delta}^2(\\hat{B}_{x,t})}^2\\\\&\\leqslant c\\left(\\sum_{x\\in I_u}t^{-(4m_x+2)\/3j(x)}+\\sum_{x\\in I_t}t^{-4m_x\/3j(x)}\\right)\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\\\&+c\\sum_{x\\in I_u} t^{(6+(4m_x-4)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}(1-\\chi_x)u\\rVert_{L_{-\\delta-1,x}^2}^2\\\\&+c\\sum_{x\\in I_t} t^{(6+(2m_x-3)\\delta)\/3j(x)}\\lVert \\nabla_{A_t}(1-\\chi_x)u\\rVert_{L_{-\\delta-1,x}^2}^2+ct^{-2}\\sum_{x\\in T}\\lVert \\nabla_{A_t}\\hat{\\chi}_x u\\rVert_{L_{-\\delta-1,x}^2}^2\\\\\n  &\\leqslant ct^2\\left(\\lVert \\nabla_{A_t}u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right),\n\\end{align*}\nwhere in the last inequality we used $\\delta<1$, $(4m_x+2)\/3j(x)<4\/3+2\/3=2$, and the $L^2$ norm is uniformly stronger than various $L_{-\\delta-1}^2$ norms.\n\\end{proof}\n\n\\begin{proposition}\\label{InvNormBd_prop}\n  The norm of $L_t^{-1}:L_{-2+\\delta}^2(i\\mathfrak{su}(E))\\to L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$ is bounded by $ct^8$.\n\\end{proposition}\n\\begin{proof}\nLet $v\\in C_0^\\infty(C\\backslash S)$, then $u=L_t^{-1}(v)\\in L_{-2+\\delta}^{2,2}$ and the boundary terms leading to the integration by parts vanish as above, so \\eqref{intbyParts_eq} holds. By the density of $C_0^\\infty(C\\backslash S)$ in $L_{-2+\\delta}^{2}$ and the continuity of $L_t^{-1}, \\mathrm{d}_{A_t}, [\\Psi_t,\\cdot]$, \\eqref{intbyParts_eq} holds for all $u\\in L_{-2+\\delta}^{2,2}$. Using \\eqref{intbyParts_eq} and \\eqref{PoincareIneq_eq},\n  \\begin{align*}\n   \\lVert u\\rVert_{L_{-\\delta}^2}^2&\\leqslant ct^2\\left(\\lVert \\nabla_{A_t} u\\rVert_{L^2}^2+\\lVert [\\Phi_t,u]\\rVert_{L^2}^2\\right)\\leqslant ct^2\\langle L_t u,u\\rangle_{L^2}\\\\&\\leqslant ct^2\\lVert u\\rVert_{L_{-\\delta}^2}\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\n   \\lVert u\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert u\\rVert_{L_{-\\delta}^2}\\leqslant ct^2\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\n  \\lVert \\nabla_{A_t}u\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert\\nabla_{A_t}u\\rVert_{L^2}\\leqslant ct\\lVert L_t u\\rVert_{L_{-2+\\delta}^2},\\\\\\lVert [\\Phi_t,u]\\rVert_{L^2}&\\leqslant ct\\lVert L_t u\\rVert_{L_{-2+\\delta}^2}.\n \\end{align*}\nFor $x\\in I$, let $\\hat{\\chi}_x$ be the function supported in $\\widetilde{B}_{x,t}$ where it satisfies $\\hat{\\chi}_x(r)=\\chi(\\kappa^{-1} t^{1\/j(x)} r)$ ($r=|z_x|$ for $x\\in I_u$ and $r=|\\zeta_{x,t}|$ for $x\\in I_t$). Recall that we have defined $\\hat{\\chi}_x$ for $x\\in T$ in a similar way. Using Propositions \\ref{analysisIu_prop}, \\ref{analysisIt_cor}, \\ref{analysisT_prop}, \\ref{analysisUext_prop} we have\n \\begin{align}\n   \\lVert u\\rVert_{L_{-2+\\delta}^{2,2}}&\\leqslant \\sum_{x\\in S}\\lVert \\hat{\\chi}_x u\\rVert_{L_{-2+\\delta}^{2,2}}+\\lVert (1-\\sum_{x\\in S} \\hat{\\chi}_x)u \\rVert_{L^{2,2}}\\\\&\\leqslant c\\sum_{x\\in S}\\lVert L_t\\hat{\\chi}_x u\\rVert_{L_{-2+\\delta}^2}+ct^4\\lVert L_t(1-\\sum_{x\\in S}\\hat{\\chi}_x) u\\rVert_{L^2}\\notag\\\\\n   &\\leqslant ct^4\\left(\\lVert  L_t u\\rVert_{L_{-2+\\delta}^2}+t^2\\lVert u\\rVert_{L_{-\\delta}^2}+t\\lVert  \\nabla_{A_t} u\\rVert_{L^2}+t\\lVert [\\Phi_t,u]\\rVert_{L^2}\\right)\\notag\\\\\n   &\\leqslant ct^8\\lVert L_t u\\rVert_{L_{-2+\\delta}^2}.\\label{InverseBound_eq}\n \\end{align}\n\\end{proof}\n\\section{Deforming to a Solution}\nNow we find a solution to Hitchin's equation by deforming the approximate solution $(A_t,\\Phi_t)$ via a complex gauge transformation $\\exp(\\gamma)$ for $\\gamma\\in\\Omega^0(i\\mathfrak{su}(E))$, which satisfies\n\\[F_t(\\gamma)=F_{A_t^{\\exp(\\gamma)}}+\\left[\\mathrm{e}^{-\\gamma}\\Phi_t\\mathrm{e}^{\\gamma},\\mathrm{e}^{\\gamma}\\Phi_t^\\ast \\mathrm{e}^{-\\gamma}\\right]=:F_{A_t}+[\\Phi_t,\\Phi_t^\\ast]+L_t\\gamma +Q_t\\gamma=0,\\]\nwhere by abuse of notation we still denote by $L_t$ the linear operator in the previous section composed with the isomorphism $\\mathrm{i}\\star$. We will analyze the nonlinear term $Q_t$ in a similar way as that in \\cite{mazzeo_swoboda_weiss_witt_2016}. Denote\n\\begin{align*}\n  A_t^{\\exp \\gamma}&=A_t+(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma),\\\\ \\Phi_t^{\\exp \\gamma}&=\\Phi_t+[\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma),\\\\\n  R_{A_t}(\\gamma)&=\\mathrm{e}^{-\\gamma}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma})-(\\partial_{A_t}\\mathrm{e}^{\\gamma})\\mathrm{e}^{-\\gamma}-(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma,\\\\\n  R_{\\Phi_t}(\\gamma)&=\\mathrm{e}^{-\\gamma}\\Phi_t \\mathrm{e}^{\\gamma}-[\\Phi_t,\\gamma]-\\Phi_t.\n\\end{align*}\nThen we have \\begin{align}\n  Q_t(\\gamma)&=\\mathrm{d}_{A_t}(R_{A_t}(\\gamma))+[R_{\\Phi_t}(\\gamma),\\Phi_t^\\ast]+[\\Phi_t,R_{\\Phi_t}(\\gamma)^\\ast]\\notag\\\\&\\quad+\\frac{1}{2}[(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma),(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma+R_{A_t}(\\gamma)]\\notag\\\\\n  &\\quad+[[\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma),([\\Phi_t,\\gamma]+R_{\\Phi_t}(\\gamma))^\\ast].\\label{NonlinTerm_eq}\n\\end{align}\nFor $u\\in \\Omega^p(\\mathrm{End}\\, E)$, define \\[\\lVert u\\rVert_{\\widetilde{L}_{\\delta}^{k,q}}:=\\left(\\sum_{j=0}^k\\lVert \\nabla_{A_t}^j u\\rVert_{L_{\\delta}^q}^q\\right)^{1\/q}.\\]\nClearly $\\lVert u\\rVert_{\\widetilde{L}_{\\delta}^{k,q}}\\leqslant \\lVert u\\rVert_{L_{\\delta}^{k,q}}$.\n\n\\begin{lemma}\n  $\\widetilde{L}_{-2+\\delta}^{2,2}(\\mathrm{End}\\,E)$ is an algebra, and $\\widetilde{L}_{-2+\\delta}^{1,2}(\\mathrm{End}\\,E)$ is a module over this algebra. Moreover,\n  \\begin{align}\n    \\lVert uv\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}&\\leqslant ct^5 \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}},\\\\\n    \\lVert uv\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant ct^{9\/2} \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}},\\\\\n    \\lVert u\\rVert_{L^\\infty}&\\leqslant ct^3 \\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align}\n\\end{lemma}\n\\begin{proof}\n  Let $\\tau_x$ and $\\hat{\\chi}_x$ be defined as in the previous section, and denote $u_x:=\\hat{\\chi}_xu$, $u_{\\mathrm{ext}}:=(1-\\sum_{x\\in S}\\hat{\\chi}_x)u$. For $x\\in I_u$, $\\tau_x$ maps $\\widetilde{B}_{x,t}$ to $B_0(\\kappa |c_x|):=\\{\\,|z|<\\kappa |c_x|\\,\\}$. Denote the Euclidean metric on this disk by $g_e$, then $c^{-1}t^{2\/j(x)}(\\tau_x^{-1})^\\ast g_t\\leqslant g_e\\leqslant ct^{2\/j(x)}(\\tau_x^{-1})^\\ast g_t$. Define the Sobolev spaces on $B_0(\\kappa |c_x|)$ using the weight function $r=|z|$ and the metric $g_e$. For $w$ a $p$-form with values in $\\mathrm{End} E$, using the Sobolev embedding $L_{-2+\\delta}^{1,2}\\hookrightarrow L_{-1+\\delta}^q$ on $B_0(\\kappa |c_x|)$ for any fixed $q\\in[2,\\infty)$,\n  \\begin{align*}\n      \\lVert w_x\\rVert_{L_{-1+\\delta}^q}&\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa |c_x|))}\\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{L_{-2+\\delta}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{L_{-2+\\delta}^2(B_0(\\kappa |c_x|))}\\right)\\\\\n      &\\leqslant ct^{-2\/qj(x)}\\left(t^{1\/j(x)}\\lVert w_x\\rVert_{L_{-2+\\delta}^2}+\\lVert \\nabla_{A_t}w_x\\rVert_{L_{-2+\\delta}^2}\\right)\\leqslant ct^{(q-2)\/qj(x)}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n  \\end{align*}\n  Similarly for $x\\in T$, we have $\\lVert w_x\\rVert_{L_{-1+\\delta}^q}\\leqslant ct^{(q-2)\/q}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}$. For $x\\in I_t$, let $\\check{L}_{\\delta}^{k,m}(B_0(\\kappa |c_x|))$ be defined using $r^{-1}g_e$ and the weight function $r^{1\/2}$, then $\\check{L}_{\\delta}^{q}$ is equivalent to $L_{\\delta\/2}$ for functions on $B_0(\\kappa |c_x|)$, and we have\n  \\begin{align*}\n    \\lVert w_x\\rVert_{L_{-1+\\delta}^q}&\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{\\check{L}_{-1+\\delta}^q(B_0(\\kappa |c_x|))}\\leqslant ct^{-2\/qj(x)}\\lVert (\\tau_x^{-1})^{\\ast}|w_x|\\rVert_{L_{(-1+\\delta)\/2}^q(B_0(\\kappa |c_x|))}\n    \\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{L_{(-1+\\delta)\/2-1}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{L_{(-1+\\delta)\/2-1}^2(B_0(\\kappa |c_x|))}\\right)\n    \\\\&\\leqslant ct^{-2\/qj(x)}\\left(\\lVert (\\tau_x^{-1})^\\ast |w_x|\\rVert_{\\check{L}_{-2+\\delta}^2(B_0(\\kappa |c_x|))}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|w_x|\\rVert_{\\check{L}_{-2+\\delta}^2(B_0(\\kappa |c_x|))}\\right)\\\\\n    &\\leqslant ct^{-2\/qj(x)}\\left(t^{1\/j(x)}\\lVert w_x\\rVert_{L_{-2+\\delta}^2}+\\lVert \\nabla_{A_t}w_x\\rVert_{L_{-2+\\delta}^2}\\right)\\leqslant ct^{(q-2)\/qj(x)}\\lVert w_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n\\end{align*}\nOn $U_{\\mathrm{ext}}$, $c^{-1}g_C\\leqslant g_t\\leqslant ctg_C$, and $r_t$ is uniformly bounded, then\n\\begin{align}\n  \\lVert w_{\\mathrm{ext}}\\rVert_{L^q}&\\leqslant ct^{1\/q}\\lVert w_{\\mathrm{ext}}\\rVert_{L^q(g_C)}\\leqslant ct^{1\/q}\\left(\\lVert w_{\\mathrm{ext}}\\rVert_{L^2(g_C)}+\\lVert \\nabla_{A_t}w_{\\mathrm{ext}}\\rVert_{L^2(g_C)}\\right)\\notag\\\\\n  &\\leqslant ct^{1\/q}\\left(t^{\\max(p-1,0)\/2}\\lVert w_{\\mathrm{ext}}\\rVert_{L^2}+t^{p\/2}\\lVert \\nabla_{A_t}w_{\\mathrm{ext}}\\rVert_{L^2}\\right).\\label{SobineqExt_eq}\n\\end{align}\nCombining the above inequalities,\n  \\[\\lVert w\\rVert_{L_{-1+\\delta}^q}\\leqslant \\sum_{x\\in S}\\lVert w_x\\rVert_{L_{-1+\\delta}^q}+\\lVert w_{\\mathrm{ext}}\\rVert_{L_{-1+\\delta}^q}\\leqslant ct^{2+p\/2}\\lVert w\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\]\nFor $w_i\\in \\Omega^{p_i}(\\mathrm{End} E)$ ($i=1,2$),\n  \\begin{align}\n    \\lVert w_1w_2\\rVert_{L_{-2+\\delta}^2}&=\\lVert r_t^{1-\\delta}w_1w_2\\rVert_{L^2}\\leqslant \\lVert r_t^{1-2\\delta}w_1w_2\\rVert_{L^2}\\leqslant \\lVert r_t^{1\/2-\\delta}w_1\\rVert_{L^4}\\lVert r_t^{1\/2-\\delta}w_2\\rVert_{L^4}\\notag\\\\&=\\lVert w_1\\rVert_{L_{-1+\\delta}^4}\\lVert w_2\\rVert_{L_{-1+\\delta}^4}\\leqslant ct^{4+(p_1+p_2)\/2}\\lVert w_1\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert w_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\label{SobMultIneq_eq}\n  \\end{align}\nFor $2<q<2\/(1-\\delta)$, $x\\in I_u$, using $L_{-1+\\delta}^q\\subset L^q$, $ L^{1,q}\\subset L^\\infty$ on $B_0(\\kappa|c_x|)$,\n  \\begin{align*}\n    \\lVert u_x\\rVert_{L^\\infty}&=\\lVert (\\tau_x^{-1})^\\ast |u_x|\\rVert_{L^\\infty(B_0(\\kappa|c_x|))}\\\\&\\leqslant c\\left(\\lVert (\\tau_x^{-1})^{\\ast}|u_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa|c_x|)}+\\lVert (\\tau_x^{-1})^\\ast \\mathrm{d}|u_x|\\rVert_{L_{-1+\\delta}^q(B_0(\\kappa|c_x|)}\\right)\\\\\n    &\\leqslant c\\left(t^{2\/qj(x)}\\lVert u_x\\rVert_{L_{-1+\\delta}^q}+t^{(2-q)\/qj(x)}\\lVert\\nabla_{A_t}u_x\\rVert_{L_{-1+\\delta}^q}\\right)\\\\\n    &\\leqslant c\\left(t^{2\/qj(x)}t^{(q-2)\/qj(x)}\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert \\nabla_{A_t}u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\right)\\leqslant ct^{1\/j(x)}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align*}\n  Similarly, we have $\\lVert u_x\\rVert_{L^\\infty}\\leqslant ct\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}$ for $x\\in T$, and $\\lVert u_x\\rVert_{L^\\infty}\\leqslant ct^{1\/j(x)}\\lVert u_x\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}$ for $x\\in I_t$. On $U_{\\mathrm{ext}}$, by \\eqref{SobineqExt_eq},\n  \\[  \\lVert u_{\\mathrm{ext}}\\rVert_{L^\\infty}\\leqslant c\\left(\\lVert u_{\\mathrm{ext}}\\rVert_{L^q(g_C)}+\\lVert \\nabla_{A_t}u_{\\mathrm{ext}}\\rVert_{L^q(g_C)}\\right)\\leqslant ct^{1\/2}\\lVert u_{\\mathrm{ext}}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\\]\nTherefore,\n  \\[\\lVert u\\rVert_{L^{\\infty}}\\leqslant \\sum_{x\\in S}\\lVert u_x\\rVert_{L^{\\infty}}+\\lVert u_{\\mathrm{ext}}\\rVert_{L^{\\infty}}\\leqslant ct^3\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\\]\n  Then by \\eqref{SobMultIneq_eq},\n  \\begin{align*}\n    \\lVert \\nabla_{A_t}(uv)\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert (\\nabla_{A_t}u)v\\rVert_{L_{-2+\\delta}^2}+\\lVert u\\nabla_{A_t}v\\rVert_{L_{-2+\\delta}^2}\\\\&\\leqslant ct^{9\/2}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert u\\rVert_{L^\\infty}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\n  \\leqslant ct^{9\/2}\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\n  \\end{align*}\n  Finally we estimate\n  \\begin{align*}\n    \\lVert\\nabla_{A_t}^2 uv\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\lVert u\\rVert_{L^\\infty}\\lVert \\nabla_{A_t}^2 v\\rVert_{L_{-2+\\delta}^2}+ct^5\\lVert \\nabla_{A_t}u\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert \\nabla_{A_t}v\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert v\\rVert_{L^\\infty}\\lVert \\nabla_{A_t}^2 u\\rVert_{L_{-2+\\delta}^2}\\\\\n    &\\leqslant ct^5\\lVert u\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert v\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n  \\end{align*}\n\\end{proof}\n\\begin{theorem}\\label{DefSol_thm}\n There exists $t_1,c,c'$ such that for every $t\\geqslant t_1$, there exists a unique $h_t^{\\mathrm{app}}$-Hermitian $\\gamma_t\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E))$ with $\\lVert \\gamma_t\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant c\\exp(-c't^\\sigma)$, such that $F_t(\\gamma_t)=0$. Hence $h_t$ defined by $h_t(v,w)=h_t^{\\mathrm{app}}(\\mathrm{e}^{-\\gamma_t}v,\\mathrm{e}^{-\\gamma_t}w)$ is a harmonic metric associated to $(\\bar{\\partial}_E,\\Phi_t)$ and is compatible with the parabolic structure at each $x\\in S$.\n\\end{theorem}\n\\begin{proof}\nLet $0<\\rho\\leqslant 1$ and $\\gamma_1,\\gamma_2\\in B_\\rho:=\\{\\gamma\\in L_{-2+\\delta}^{2,2}(i\\mathfrak{su}(E)):\\lVert \\gamma\\rVert_{L_{-2+\\delta}^{2,2}}<\\rho\\}$. We estimate each of the five terms in the difference $Q_t(\\gamma_1)-Q_t(\\gamma_2)$, with $Q_t$ given by \\eqref{NonlinTerm_eq}. First, \\begin{align}\n  \\lVert R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant\\lVert \\left(\\mathrm{e}^{-\\gamma_1}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_1})-\\mathrm{e}^{-\\gamma_2}(\\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_2})-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2)\\right)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\notag\\\\\n  &\\quad +\\lVert \\left((\\partial_{A_t}\\mathrm{e}^{\\gamma_1})\\mathrm{e}^{-\\gamma_1}-(\\partial_{A_t}\\mathrm{e}^{\\gamma_2})\\mathrm{e}^{-\\gamma_2}+\\partial_{A_t}(\\gamma_1-\\gamma_2)\\right)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}.\\label{NonlinearAaDiff_eq}\n\\end{align}\nThe first term is bounded by\n\\begin{align*}\n  \\lVert (\\mathrm{e}^{-\\gamma_1}-\\mathrm{e}^{-\\gamma_2})\\bar{\\partial}_{A_t} \\mathrm{e}^{\\gamma_1}\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert \\mathrm{e}^{-\\gamma_2}(\\bar{\\partial}_{A_t}(\\mathrm{e}^{\\gamma_1}-\\mathrm{e}^{\\gamma_2}))-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}:=\\uppercase\\expandafter{\\romannumeral1}+\\uppercase\\expandafter{\\romannumeral2}.\n\\end{align*}\nWrite $\\mathrm{e}^\\gamma=1+\\gamma+S(\\gamma)$, then\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}&\\leqslant c t^{9\/2}\\lVert \\mathrm{e}^{-\\gamma_1}-\\mathrm{e}^{-\\gamma_2}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\bar{\\partial}_{A_t}\\mathrm{e}^{\\gamma_1}\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\leqslant c t^{9\/2}\\lVert \\mathrm{e}^{-\\gamma_1}- \\mathrm{e}^{-\\gamma_2}\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\gamma_1+S(\\gamma_1)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\n  \\\\&\\leqslant c t^{9\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\left(\\sum_{j=0}^\\infty \\frac{(\\rho ct^5)^j}{j!}\\right)\\rho\\left(\\sum_{j=0}^\\infty \\frac{(\\rho ct^5)^j}{(j+1)!}\\right)\\\\\n  &\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}},\n\\end{align*}\nfor $\\rho<t^{-5}$. Similarly,\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral2}&=\\lVert (1-\\gamma_2+S(-\\gamma_2))\\left(\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2))\\right)-\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant \\lVert \\bar{\\partial}_{A_t}(S(\\gamma_1)-S(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}+\\lVert (-\\gamma_2+S(-\\gamma_2))\\bar{\\partial}_{A_t}(\\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant \\lVert S(\\gamma_1)-S(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}+  ct^{9\/2}\\lVert -\\gamma_2+S(-\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\lVert \\gamma_1-\\gamma_2+S(\\gamma_1)-S(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\\\\n  &\\leqslant ct^{9\/2}\\rho \\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThe estimate for the other term in \\eqref{NonlinearAaDiff_eq} is analogous, and we obtain \\begin{align}\n    \\lVert R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}}\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}},\\label{RAtIneq_eq}\\\\\n    \\lVert \\mathrm{d}_{A_t}(R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2))\\rVert_{L_{-2+\\delta}^2}&\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\\notag\n\\end{align}\nLet $\\beta_1,\\ldots,\\beta_n\\in L_{-2+\\delta}^{2,2}$, where $n\\geqslant 2$. By the Jacobi identity, $\\left|[\\Phi_t^\\ast,[\\beta_1,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\right|$ is bounded by\n\\begin{align*}\n  & \\left|[[\\Phi_t^\\ast,\\beta_1],[\\beta_2,\\cdots,[\\Phi_t,\\beta_n]\\cdots]]\\right|+\\left|[\\beta_1,[[\\Phi_t^\\ast,\\beta_2],[\\beta_3,\\cdots[\\Phi_t,\\beta_n]\\cdots]]]\\right|\\\\\n  +\\cdots&+\\left|[\\beta_1,\\cdots[[\\Phi_t^\\ast,\\beta_{n-1}],[\\Phi_t,\\beta_n]]\\cdots]\\right|+\\left|[\\beta_1,\\cdots[\\beta_{n-1},[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]]\\cdots]\\right|\\\\\n  &\\leqslant 2^n\\left(\\sum_{j=1}^{n-1}\\left(|[\\Phi_t^\\ast,\\beta_j]||[\\Phi_t,\\beta_n]|\\prod_{m\\neq j,n}\\lVert \\beta_m\\rVert_{L^\\infty}\\right)+|[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]|\\prod_{m=1}^{n-1}\\lVert \\beta_m\\rVert_{L^\\infty}\\right).\n\\end{align*}\nIntegrate both sides, then $\\left\\lVert[\\Phi_t^\\ast,[\\beta_1,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\right\\rVert_{L_{-2+\\delta}^2}$ is bounded by\n\\begin{align*}\n    & 2^n\\left(\\sum_{j=1}^{n-1}ct^5\\lVert [\\Phi_t^\\ast,\\beta_j]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert [\\Phi_t,\\beta_n]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}(ct^3)^{n-2}\\prod_{m\\neq j,n}\\lVert \\beta_m\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}} \\right.\\\\\n    &\\qquad+\\left. \\lVert[\\Phi_t^\\ast,[\\Phi_t,\\beta_n]]\\rVert_{L_{-2+\\delta}^2}(ct^3)^{n-1}\\prod_{m=1}^{n-1}\\lVert \\beta_m\\rVert_{\\widetilde{L}_{-2+\\delta}^{2,2}} \\right)\\\\\n    &\\leqslant (ct^{9\/2})^{n-1}t^{1\/2}\\prod_{m=1}^n\\lVert \\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThen for $\\gamma_1,\\gamma_2\\in B_\\rho$,\n\\begin{align*}\n\\left\\lVert [\\Phi_t^\\ast,(\\mathrm{ad}_{\\gamma_1})^n\\Phi_t-(\\mathrm{ad}_{\\gamma_2})^n\\Phi_t]\\right\\rVert_{L_{-2+\\delta}^2}&\\leqslant \\left\\lVert[\\Phi_t^\\ast,[\\gamma_1-\\gamma_2,[\\gamma_1,\\cdots,[\\gamma_1,\\Phi_t]\\cdots]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\&\\quad+\\left\\lVert[\\Phi_t^\\ast,[\\gamma_2,[\\gamma_1-\\gamma_2,[\\gamma_1,\\cdots,[\\gamma_1,\\Phi_t]\\cdots]]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\&\\quad+\\cdots+\\left\\lVert[\\Phi_t^\\ast,[\\gamma_2,[\\gamma_2,\\cdots,[\\gamma_1-\\gamma_2,\\Phi_t]\\cdots]]]\\right\\rVert_{L_{-2+\\delta}^2}\\\\\n  &\\leqslant  n(ct^{9\/2}\\rho)^{n-1}t^{1\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThen we compute that\n\\begin{align*}\n  \\lVert [R_{\\Phi_t}(\\gamma_1),\\Phi_t^\\ast]-[R_{\\Phi_t}(\\gamma_2),\\Phi_t^\\ast]\\rVert_{L_{-2+\\delta}^2}&=\\lVert[\\Phi_t^\\ast,\\sum_{j=2}^\\infty \\frac{1}{j!}(\\mathrm{ad}_{-\\gamma_1})^j\\Phi_t-(\\mathrm{ad}_{-\\gamma_2})^j\\Phi_t]\\rVert_{L_{-2+\\delta}^2}\\\\\n  &\\leqslant \\sum_{j=2}^\\infty \\frac{1}{j!}j(c t^{9\/2}\\rho)^{j-1}t^{1\/2}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}\\\\\n  &\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nSimilarly,\n\\[\\lVert [\\Phi_t,R_{\\Phi_t}(\\gamma_1)^\\ast]-[\\Phi_t,R_{\\Phi_t}(\\gamma_2)^\\ast]\\rVert_{L_{-2+\\delta}^2}\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\\]\nThe $L_{-2+\\delta}^2$ norm of the fourth term in $Q_t(\\gamma_1)-Q_t(\\gamma_2)$ is bounded by\n\\begin{align*}\n&\\,\\quad \\lVert[(\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2),(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_1+R_{A_t}(\\gamma_1)]\\rVert_{L_{-2+\\delta}^2}\\\\\n  &+\\lVert[(\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_2+R_{A_t}(\\gamma_2),(\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2)]\\rVert_{L_{-2+\\delta}^2}:=\\uppercase\\expandafter{\\romannumeral1}'+\\uppercase\\expandafter{\\romannumeral2}'.\n\\end{align*}\nBy \\eqref{SobMultIneq_eq}, \\eqref{RAtIneq_eq},\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}'&\\leqslant ct^5\\lVert (\\bar{\\partial}_{A_t}-\\partial_{A_t})(\\gamma_1-\\gamma_2)+R_{A_t}(\\gamma_1)-R_{A_t}(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert (\\bar{\\partial}_{A_t}-\\partial_{A_t})\\gamma_1+R_{A_t}(\\gamma_1)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant ct^5\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}(1+ct^{9\/2}\\rho)^2\\rho\\leqslant ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nSimilarly, $\\uppercase\\expandafter{\\romannumeral2}'$ is also bounded by $ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}$. Consider $\\beta_1,\\ldots,\\beta_n\\in L_{-2+\\delta}^{2,2}$, then\n\\begin{align*}\n  \\lVert [\\beta_1,[\\beta_2,\\cdots,[\\beta_n,\\Phi_t]\\cdots]]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}&\\leqslant (  ct^{9\/2})^{n-1}\\prod_{m=1}^{n-1}\\lVert\\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}\\lVert [\\Phi_t,\\beta_n]\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant (  ct^{9\/2})^{n-1}\\prod_{m=1}^{n}\\lVert\\beta_m\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{align*}\nThis implies that\n\\[\n  \\lVert R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2)\\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\leqslant \\sum_{j=2}^\\infty \\frac{1}{j!}j(  ct^{9\/2}\\rho)^{j-1}\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{9\/2}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\]\nThe $L_{-2+\\delta}^2$ norm of the last term in $Q_t(\\gamma_1)-Q_t(\\gamma_2)$ is bounded by\n\\begin{align*}\n&\\,\\quad\\lVert [[\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2),([\\Phi_t,\\gamma_1]+R_{\\Phi_t}(\\gamma_1))^\\ast]\\rVert_{L_{-2+\\delta}^2}\\\\&+ \\lVert [[\\Phi_t,\\gamma_2]+R_{\\Phi_t}(\\gamma_2),([\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2))^\\ast]\\rVert_{L_{-2+\\delta}^2}:=\\uppercase\\expandafter{\\romannumeral1}''+\\uppercase\\expandafter{\\romannumeral2}'',\n\\end{align*}\nwhere $\\uppercase\\expandafter{\\romannumeral1}''$ can be estimated as\n\\begin{align*}\n  \\uppercase\\expandafter{\\romannumeral1}''&\\leqslant ct^5\\lVert[\\Phi_t,\\gamma_1-\\gamma_2]+R_{\\Phi_t}(\\gamma_1)-R_{\\Phi_t}(\\gamma_2) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\lVert [\\Phi_t,\\gamma_1]+R_{\\Phi_t}(\\gamma_1) \\rVert_{\\widetilde{L}_{-2+\\delta}^{1,2}}\\\\\n  &\\leqslant ct^5\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}(1+ct^{9\/2}\\rho)^2\\rho\\leqslant ct^5\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}},\n\\end{align*}\nsimilar for $\\uppercase\\expandafter{\\romannumeral2}''$. Therefore\n\\begin{equation}\\label{NonlinearEst_eq}\n  \\lVert Q_t(\\gamma_1)-Q_t(\\gamma_2)\\rVert_{L_{-2+\\delta}^2}\\leqslant ct^{5}\\rho\\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}.\n\\end{equation}\nNow $F_t(\\gamma)=0$ is equivalent to \\[\\gamma=-L_t^{-1}(F_{A_t}+[\\Phi_t,\\Phi_t^\\ast]+Q_t(\\gamma)):=T_t(\\gamma).\\]\nBy \\eqref{InverseBound_eq}, \\eqref{NonlinearEst_eq}, $\\lVert T_t(\\gamma_1)-T_t(\\gamma_2)\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{13}\\rho \\lVert \\gamma_1-\\gamma_2\\rVert_{L_{-2+\\delta}^{2,2}}$. Furthermore, by \\eqref{ErrorEst_eq} (which is still valid when the norm is taken with respect to $g_t$), \\[\\lVert T_t(\\gamma)\\rVert_{L_{-2+\\delta}^{2,2}}\\leqslant ct^{8}\\mathrm{e}^{-c' t^\\sigma}+ct^{13}\\rho^2,\\]\nfor $\\gamma\\in B_\\rho$. Let $\\rho_t=\\mathrm{e}^{(-c'\/2+\\epsilon)t^\\sigma}$, where $\\epsilon<c'\/2$ is small. Then for $t$ large, $\\rho_t<t^{-5}$, $T_t$ maps $B_{\\rho_t}$ to $B_{\\rho_t}$, and is a contraction mapping, hence there is a unique fixed point $\\gamma_t$ in $B_{\\rho_t}$. In other words, the metric $h_t$ is harmonic, which is also compatible with the parabolic structure as $h_t^{\\mathrm{app}}$ is and $\\gamma_t\\in L_{-2+\\delta}^{2,2}$.\n\\end{proof}\n\n\\section{Asymptotic Metric}\nIn this section, we study the asymptotic geometry of $\\mathcal{M}$ using the framework provided by \\cite{fredrickson_2019, fredrickson2022asymptotic}.\n\nElements in $\\mathcal{M}$ are gauge equivalence classes of pairs $(\\bar{\\partial}_E,\\Phi)$, with infinitesimal deformations denoted by $(\\dot{\\eta},\\dot{\\Phi})$. Linearize $\\bar{\\partial}_E\\Phi=0$ and we get \\begin{equation}\\label{LinHol_eq}\n  \\bar{\\partial}_E\\dot{\\Phi}+[\\dot{\\eta},\\Phi]=0.\n\\end{equation}\nFor $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}$, let $h$ be the unique harmonic metric (if exists) which induces $h_{\\det E}$, and denote its infinitesimal deformation by $\\dot{h}$. The infinitesimal gauge transformation $\\dot{\\gamma}$ acts as\n\\[(\\dot{\\eta},\\dot{\\Phi},\\dot{h})\\mapsto (\\dot{\\eta}+\\bar{\\partial}_E\\dot{\\gamma},\\dot{\\Phi}+[\\Phi,\\dot{\\gamma}],\\dot{h}+\\dot{\\gamma}).\\]\n The linearized Hitchin equation and the Coulomb gauge condition can be combined to form a single equation \\cite[Prop.~2.2]{fredrickson_2019}\n\\begin{equation}\\label{LinHitCou_eq}\n  \\partial_E^h\\bar{\\partial}_E\\dot{h}-\\partial_E^h\\dot{\\eta}-[\\Phi^{\\ast_h},\\dot{\\Phi}+[\\dot{h},\\Phi]]=0.\n\\end{equation}\nRecall that $2\\partial_E^h\\bar{\\partial}_E=F_{A_h}+\\mathrm{i}\\star \\Delta_{A_h}=-[\\Phi,\\Phi^{\\ast_h}]+\\mathrm{i}\\star \\Delta_{A_h}$, the equation can be written as \\[-\\frac{1}{2}[[\\Phi,\\Phi^{\\ast_h}],\\dot{h}]+\\frac{1}{2}\\mathrm{i}\\star \\Delta_{A_h}\\dot{h}-[\\Phi^{\\ast_h},[\\dot{h},\\Phi]]=\\partial_{E}^h\\dot{\\eta}+[\\Phi^{\\ast_h},\\dot{\\Phi}].\\]\nComposing with $-2\\mathrm{i}\\star$ and simplifying, we obtain\n\\[L_{\\Phi}\\dot{h}:=\\Delta_{A_h}\\dot{h}-\\mathrm{i}\\star([\\Phi^{\\ast_h},[\\Phi,\\dot{h}]] -[\\Phi,[\\Phi^{\\ast_h},\\dot{h}]])=-2\\mathrm{i}\\star(\\partial_E^h\\dot{\\eta}+[\\Phi^{\\ast_h},\\dot{\\Phi}]).\\]\nNote that $L_{\\Phi}$ is exactly $L_t$ in \\eqref{LinOp_eq} with $(\\bar{\\partial}_E,\\Phi_t,h_{t}^{\\mathrm{app}})$ replaced by $(\\bar{\\partial}_E,\\Phi,h)$. Now we consider the solvability of \\eqref{LinHitCou_eq}. Let $(\\bar{\\partial}_E,\\Phi_t)$ and $L_{\\delta}^{k,p}$ (depending on $g_t$, $\\Phi_t$, and $h_t^{\\mathrm{app}}$) be defined as before, $\\nabla_t$ be the Chern connection determined by $\\bar{\\partial}_E$ and $h_t$.\n\\begin{lemma}\\label{InvLin_lem}\nFix a pair $[(\\bar{\\partial}_E,\\Phi_t)]\\in\\mathcal{M}$, $t\\geqslant t_1$ so that the harmonic metric $h_t$ exists. Then $L_{\\Phi_t}: L_{-2+\\delta}^{2,2}(\\mathfrak{sl}(E))\\to L_{-2+\\delta}^2(\\mathfrak{sl}(E))$ is an isomorphism.\n\\end{lemma}\n\\begin{proof}\nThat $L_t$ is Fredholm of index zero follows from Lemma \\ref{GlobIso_lem}, and the fact that $L_t$ preserves the decomposition $\\mathfrak{sl}(E)=\\mathfrak{su}(E)\\oplus i\\mathfrak{su}(E)$. Since $\\gamma_t\\in L_{-2+\\delta}^{2,2}$, $L_{\\Phi_t}$ differs from $L_t$ by a compact perturbation, it is also Fredholm of index zero. If $u\\in \\ker L_{\\Phi_t}$, then as before the integration by parts yields $\\nabla_t u=[\\Phi_t,u]=0$. By the proof of \\cite[Prop. 7.4]{fredrickson2022asymptotic}, we have $u=0$. Therefore $L_{\\Phi_t}$ is an isomorphism.\n\\end{proof}\n\\begin{lemma}\\label{InfsmRep_lem}\n  Let $[(\\bar{\\partial}_E,\\Phi)]\\in \\mathcal{M}$ and $(\\dot{\\eta}_1,\\dot{\\Phi}_1)$ be an infinitesimal variation. There exists $(\\dot{\\eta}_2,\\dot{\\Phi}_2)$ infinitesimally gauge equivalent to $(\\dot{\\eta}_1,\\dot{\\Phi}_1)$ such that the followings hold.\n\\begin{enumerate}[label=(\\roman*)]\n  \\item In a neighborhood of $x\\in T$ and a holomorphic frame where \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi=\\begin{pmatrix}\n    0&1\\\\\\zeta_x^{-1}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\]\n  we have\n  \\[\\dot{\\eta}_2=0,\\quad\\dot{\\Phi}_2=\\begin{pmatrix}\n    0&0\\\\\\frac{\\dot{P}}{\\zeta_x}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\quad\\dot{P}\\text{ holomorphic}.\\]\n  \\item In a neighborhood of $x\\in I_u$ and a holomorphic frame where $\\bar{\\partial}_E=\\bar{\\partial}$, $\\Phi=fz_x^{-2m_x} \\sigma_3\\,\\mathrm{d}z_x$ with $f$ being holomorphic and nowhere vanishing, we have $\\dot{\\eta}_2=0, \\dot{\\Phi}_2=\\dot{P}\\sigma_3\\,\\mathrm{d}z$, $\\dot{P}$ is holomorphic.\n  \\item In a neighborhood of $x\\in I_t$ and a holomorphic frame where \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi=\\begin{pmatrix}\n    0&\\frac{1}{\\zeta_{x}^{m_x-1}}\\\\ \\frac{1}{\\zeta_{x}^{m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x},\\] we have \\[\\dot{\\eta}_2=0,\\quad \\dot{\\Phi}_2=\\begin{pmatrix}\n    0&\\dot{P}\\\\\\frac{\\dot{P}}{\\zeta_x}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x,\\quad \\dot{P}\\text{ is holomorphic}.\\]\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n  (\\romannumeral1) is just \\cite[Lem.~7.3]{fredrickson2022asymptotic}.\\\\\n  (\\romannumeral2) By the Poincar\u00e9 lemma we may assume that $\\dot{\\eta}_1=0$ in the disk. The leading term of $\\dot{\\Phi}_1$ preserves the flag at $x$, then we can write $\\dot{\\Phi}_1$ as \\[\\dot{\\Phi}_1=\\frac{1}{z_x^{2m_x}}\\begin{pmatrix}\n    \\dot{P}_1&\\dot{P_2}z_x\\\\ \\dot{P}_3&-\\dot{P}_1\n  \\end{pmatrix}.\\]\n  where $\\dot{P}_1,\\dot{P}_2,\\dot{P}_3$ are holomorphic by \\eqref{LinHol_eq}. Then $\\dot{\\Phi}_2=\\dot{\\Phi}_1+[\\Phi,\\dot{\\gamma}]=\\dot{P}_1z_x^{-2m_x}\\sigma_3\\,\\mathrm{d}z$ if we choose \\[\\dot{\\gamma}=\\begin{pmatrix}\n    0&-\\frac{\\dot{P}_2 z_x}{2f}\\\\\\frac{\\dot{P}_3}{2f}&0\n  \\end{pmatrix},\\]\n  which satisfies $\\bar{\\partial}\\dot{\\gamma}=0$ and preserves the flag. Higgs fields in $\\mathcal{M}$ have fixed polar parts, implying that $\\dot{P}_1z_x^{-2m_x}=\\dot{P}$ for some holomorphic $\\dot{P}$.\\\\\n  (\\romannumeral3) As above, assume that $\\dot{\\eta}_1=0$. The leading term of $\\dot{\\Phi}_1$ preserves the flag at $x$ and the leading term of the Higgs field is nilpotent, then we can write $\\dot{\\Phi}_1$ as \\[\\dot{\\Phi}_1=\\frac{1}{\\zeta_x^{2m_x-1}}\\begin{pmatrix}\n    \\dot{P}_1&\\dot{P_2}\\\\ \\frac{\\dot{P}_3}{\\zeta_x}&-\\dot{P}_1\n  \\end{pmatrix}.\\]\n  where $\\dot{P}_1,\\dot{P}_2,\\dot{P}_3$ are holomorphic by \\eqref{LinHol_eq}. We choose \\[\\dot{\\gamma}=\\begin{pmatrix}\n    \\frac{\\dot{P}_2-\\dot{P}_3}{4}&\\zeta_x\\dot{P}_1\\\\0&-\\frac{\\dot{P}_2-\\dot{P}_3}{4}\n  \\end{pmatrix},\\]\n  which satisfies $\\bar{\\partial}\\dot{\\gamma}=0$ and preserves the flag. Then \\[\\dot{\\Phi}_2=\\dot{\\Phi}_1+[\\Phi,\\dot{\\gamma}]=\\begin{pmatrix}\n    0&\\frac{\\dot{P}_2+\\dot{P}_3}{2\\zeta_x^{2m_x-1}}\\\\ \\frac{\\dot{P}_2+\\dot{P}_3}{2\\zeta_x^{2m_x}}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_x.\\] Higgs fields in $\\mathcal{M}$ have fixed polar parts, implying that $(\\dot{P}_2+\\dot{P}_3)\/(2\\zeta_x^{2m_x-1})=\\dot{P}$ for $\\dot{P}$ holomorphic.\n\\end{proof}\n\nSuppose $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, by the above lemma we may assume that $\\dot{\\eta}=0$ and $\\dot{\\Phi}$ has the above forms near every $x\\in S$. Then $\\partial_E^{h_t}\\dot{\\eta}+[\\Phi_t^{\\ast_{h_t}},\\dot{\\Phi}]\\in L_{-2+\\delta}^2$, as $\\partial_E^{h_t}\\dot{\\eta}=0$ near $x\\in S$, $[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\dot{\\Phi}]\\in L_{-2+\\delta}^2$ for $\\delta$ small and $\\gamma_t\\in L_{-2+\\delta}^{2,2}$. By Lemma \\ref{InvLin_lem}, a unique $\\dot{h}\\in L_{-2+\\delta}^{2,2}$ solving \\eqref{LinHitCou_eq} can be found. Now the hyperk\u00e4hler metric is given by \\cite[(7.5)]{fredrickson2022asymptotic}\n\\begin{equation}\\label{HKahlerMetric_eq}\n\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2=2\\int_C |\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}|_{h_t}^2+|\\dot{\\Phi}+[\\dot{h},\\Phi_t]|_{h_t}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h},\\dot{\\eta}\\rangle_{h_t}+\\langle \\dot{\\Phi}+[\\dot{h},\\Phi_t],\\dot{\\Phi}\\rangle_{h_t},\n\\end{equation}\nwhere the second equality follows from \\eqref{LinHitCou_eq} and the integration by parts which is justified as in the proof of Lemma \\ref{GlobIso_lem} and \\cite[p.~43]{fredrickson2022asymptotic}.\n\nFor the pair $[(\\bar{\\partial}_E,\\Phi_t)]\\in\\mathcal{M}$, one can associate metrics $h_t^{\\mathrm{app}}$ and $h_{t,\\infty}$, with infinitesimal deformations denoted by $\\dot{h}_{\\mathrm{app}}$ and $\\dot{h}_{\\infty}$. The metric $g_{\\mathrm{app}}$ on $\\mathcal{M}$ is defined by\n\\begin{equation}\\label{ApproxMetric_eq}\n\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}},\\dot{\\eta}\\rangle_{h_t^{\\mathrm{app}}}+\\langle \\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t],\\dot{\\Phi}\\rangle_{h_t^{\\mathrm{app}}},\n\\end{equation}\nwhere $\\dot{h}_{\\mathrm{app}}$ satisfies\n\\begin{equation}\\label{AppLinHitCol_eq}\n  \\partial_E^{h_t^{\\mathrm{app}}}\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}-\\partial_E^{h_t^{\\mathrm{app}}}\\dot{\\eta}-[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]]=0.\n\\end{equation}\nSimilarly, the metric $g_\\infty$ is\n\\begin{equation}\\label{InftyMetric_eq}\n  \\lVert[(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_\\infty}^2=2\\int_C \\langle \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{\\infty}},\\dot{\\eta}\\rangle_{h_{t,\\infty}}+\\langle \\dot{\\Phi}+[\\dot{h}_\\infty,\\Phi_t],\\dot{\\Phi}\\rangle_{h_{t,\\infty}},\n\\end{equation}\nwhere $\\dot{h}_{\\infty}$ satisfies the decoupled equations\n\\begin{equation}\\label{DecoupInftyDef_eq}\n  \\partial_E^{h_{t,\\infty}}\\bar{\\partial}_E\\dot{h}_\\infty-\\partial_{E}^{h_{t,\\infty}}\\dot{\\eta}=0,\\quad [\\Phi_t^{\\ast_{h_{t,\\infty}}},\\dot{\\Phi}+[\\dot{h}_\\infty,\\Phi_t]]=0.\n\\end{equation}\nRecall that for each $[(\\bar{\\partial}_E,\\Phi_t)]\\in \\mathcal{M}$, we can view the associated spectral data as a pair $(\\bar{\\partial}_{L,t},\\tau_t)$, where $L$ is a complex line bundle of degree $-|S|+N-2$ over a surface $S_t$ with genus $N-3$, $\\tau_t$ is the tautological eigenvalue of $\\pi^\\ast \\Phi_t$ which embeds $S_t$ in $\\pi^\\ast K(D)$. The tangent map induces an isomorphism\n\\[T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}\\to H^0(S_t,K_{S_t})^\\ast_{\\mathrm{odd}}\\oplus H^0(S_t,K_{S_t})_{\\mathrm{odd}},\\quad [(\\dot{\\eta},\\dot{\\Phi})]\\mapsto (\\dot{\\xi},\\dot{\\tau}),\\] where $H^0(S_t,K_{S_t})_{\\mathrm{odd}}$ consists of elements of $H^0(S_t,K_{S_t})$ which are odd under the exchange of two sheets of $\\pi:S_t\\to C$, and $\\dot{\\xi},\\dot{\\tau}$ are the infinitesimal deformations of $\\bar{\\partial}_{L,t}$ and $\\tau$. The metric $h_{t,\\infty}$ is the orthogonal pushforward of the harmonic metric $h_{\\mathcal{L}_t}$ on the spectral line bundle $\\mathcal{L}_t=(L,\\bar{\\partial}_{L,t})\\to S_t$, and the infinitesimal deformation $\\dot{h}_L$ of $h_{\\mathcal{L}_t}$ is determined by\n\\[\\partial_L^{h_{\\mathcal{L}_t}}\\bar{\\partial}_{L,t}\\dot{h}_L-\\partial_L^{h_{\\mathcal{L}_t}}\\dot{\\xi}=0.\\]\n\\begin{lemma}[{\\cite[Prop.~2.14]{fredrickson_2019}}]The semiflat metric is characterized by three properties:\n\\begin{enumerate}[label=(\\arabic*)]\n  \\item On horizontal deformations, the semiflat metric is $\\int_{S_t}2|\\dot{\\tau}|^2$.\n  \\item On vertical deformations, the semiflat metric is $\\int_{S_t} 2|\\dot{\\xi}-\\bar{\\partial}_L\\dot{h}_L|^2$.\n  \\item Horizontal and vertical deformations are orthogonal.\n\\end{enumerate}\n\\end{lemma}\n\\begin{lemma}[{\\cite[Cor.~7.9]{fredrickson2022asymptotic}}]\n  The semiflat metric $g_{\\mathrm{sf}}$ on $\\mathcal{M}$ coincides with $g_{\\infty}$.\n\\end{lemma}\n\\subsection{Comparing $g_{L^2}$ and $g_{\\mathrm{app}}$}\nThe difference $\\frac{1}{2}|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2|$ is bounded by\n\\[  \\left|\\int_C | \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}|_{h_t^{\\mathrm{app}}}^2-| \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}|_{h_t}^2\\right|+\\left|\\int_C |\\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]|_{h_t^{\\mathrm{app}}}^2-| \\dot{\\Phi}+[\\dot{h},\\Phi_t]|_{h_t}^2\\right|:=\\uppercase\\expandafter{\\romannumeral1}+\\uppercase\\expandafter{\\romannumeral2}.\\]\nWe first estimate $\\uppercase\\expandafter{\\romannumeral1}$, \\begin{align}\n\\uppercase\\expandafter{\\romannumeral1}&\\leqslant \\int_C |\\langle (\\mathrm{Id}-\\mathrm{e}^{-\\gamma_t})(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}),\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rangle_{h_t^{\\mathrm{app}}}|\\notag\\\\&\\quad+|\\langle \\mathrm{e}^{-\\gamma_t}(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}),(\\mathrm{Id}-\\mathrm{e}^{-\\gamma_t})(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}})\\rangle_{h_t^{\\mathrm{app}}}|\\notag\\\\\n&\\quad+2|\\langle \\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E(\\dot{h}_{\\mathrm{app}}-\\dot{h}),\\mathrm{e}^{-\\gamma_t}(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rangle_{h_t^{\\mathrm{app}}}|+|\\mathrm{e}^{-\\gamma_t}\\bar{\\partial}_E(\\dot{h}_{\\mathrm{app}}-\\dot{h})|_{h_t^{\\mathrm{app}}}^2.\\label{L2AppCompTerm1_eq}\n\\end{align}\nSince the $L_{-2+\\delta}^{2,2}$ norm of $\\gamma_t$ is exponentially decaying, the first two terms are bounded by \\[c\\mathrm{e}^{-c't^\\sigma}\\lVert \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rVert_{L^2}^2.\\]\nHere and following, unless specified, the norms are taken with respect to $h_t^{\\mathrm{app}}$ as in \\eqref{SobolevSpace_eq}. It remains to estimate $\\lVert\\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}$, where $\\dot{\\mathbf{v}}=\\dot{h}_{\\mathrm{app}}-\\dot{h}$. By \\eqref{LinHitCou_eq}, \\eqref{AppLinHitCol_eq}, we have\n\\begin{align*}\n  &\\partial_{E}^{h_t^{\\mathrm{app}}}\\bar{\\partial}_E \\dot{\\mathbf{v}}-[\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},[\\dot{\\mathbf{v}},\\Phi_t]]+T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})+T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])=0,\\text{ where }\\\\\n  &T_t^1=\\partial_E^{h_t}-\\partial_E^{h_t^{\\mathrm{app}}}=-2[\\partial_E^{h_t^{\\mathrm{app}}}\\gamma_t,\\cdot]+R_t^1,\\quad T_t^2=[\\Phi_t^{\\ast_{h_t}}-\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}},\\cdot]=-2[[\\gamma_t,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}],\\cdot]+R_t^2,\n\\end{align*}\n$R_t^1,R_t^2$ are nonlinear in $\\gamma_t$. As before, by composing with $-2\\mathrm{i}\\star$, the equation becomes\n\\[L_t\\dot{\\mathbf{v}}=2\\mathrm{i}\\star(T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})+T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])+E_t\\dot{\\mathbf{v}}):=\\dot{\\mathbf{w}},\\]\nwhere $E_t=[F_{h_t^{\\mathrm{app}}}+[\\Phi_t,\\Phi_t^{\\ast_{h_t^{\\mathrm{app}}}}],\\cdot]$. Note that $\\nabla_{A_t}\\gamma_t, [\\Psi_t,\\gamma_t]\\in L_{-2+\\delta}^{1,2}$ and $L_{-2+\\delta}^{1,2}\\hookrightarrow L_{-1+\\delta}^p$ for any fixed $p\\in [2,\\infty)$, $L_{-1+\\delta}^p\\cdot L^2\\subset L_{-2+\\delta}^{2p\/(p+2)}$, similar to the proof of Theorem \\ref{DefSol_thm}, we have\n \\[\\lVert\\star T_t^1(\\dot{\\eta}-\\bar{\\partial}_E\\dot{h})\\rVert_{L_{-2+\\delta}^{p_1}}+\\lVert\\star T_t^2(\\dot{\\Phi}+[\\dot{h},\\Phi_t])\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}},\\]\n where $p_1=2p\/(p+2)<2$. By the construction of $h_t^{\\mathrm{app}}$, $E_t$ is supported on a union of annuli and the $C^0$ norm is exponentially decaying, so that\n \\[\\lVert E_t\\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}.\\] Therefore,\n\\begin{equation}\\label{p1NormEst_eq1}\n  \\lVert \\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}}).\n\\end{equation} By \\eqref{intbyParts_eq},\n\\begin{equation}\\label{p1NormEst_eq2}\n  \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}^2+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}^2=\\langle \\dot{\\mathbf{w}},\\dot{\\mathbf{v}}\\rangle_{L^2}\\leqslant \\lVert \\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}},\n\\end{equation}\n where $1\/p_1+1\/p_1^\\ast=1$. Let $u$ be a function on the unit disk vanishing on the boundary, then we have the Sobolev inequality $\\lVert u\\rVert_{L^{p_1^\\ast}}\\leqslant c\\lVert \\nabla u\\rVert_{L^2}$. On the other hand, by Lemma \\ref{SobEmbed_lem}, $L_{-\\delta-1}^{1,2}=\\hat{L}_{-\\delta-1}^{1,2}\\hookrightarrow L_{-\\delta}^{p_1^\\ast}$, then \\[\\lVert u\\rVert_{L_{-\\delta}^{p_1^\\ast}}\\leqslant c(\\lVert\\nabla u\\rVert_{L_{-\\delta-1}^2}+\\lVert u\\rVert_{L_{-\\delta-1}^2})\\leqslant c(\\lVert\\nabla u\\rVert_{L_{-\\delta-1}^2}+\\lVert\\nabla u\\rVert_{L_{-\\delta-2}^2})\\leqslant c\\lVert \\nabla u\\rVert_{L^2}.\\] These estimates can be translated to $C$ as in the proof of \\eqref{PoincareIneq_eq}, and we obtain\n\\[  \\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}}\\leqslant ct^M(\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}),\\]\nfor some constant $M$. Combining with \\eqref{p1NormEst_eq1}, \\eqref{p1NormEst_eq2}, we deduce that \\begin{align*}\n  \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}&\\leqslant ct^M \\lVert\\dot{\\mathbf{w}}\\rVert_{L_{-2+\\delta}^{p_1}}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\dot{\\mathbf{v}}\\rVert_{L_{-2+\\delta}^{p_1}})\\\\\n  &\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert\\dot{\\mathbf{v}}\\rVert_{L_{-\\delta}^{p_1^\\ast}})\\\\\n  &\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}+\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}),\\\\\n   \\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}+\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}&\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}},\n\\end{align*}\nfor $t$ large. By \\eqref{L2AppCompTerm1_eq}, we conclude that\n\\[\\uppercase\\expandafter{\\romannumeral1}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert \\dot{\\eta}-\\bar{\\partial}_E\\dot{h}_{\\mathrm{app}}\\rVert_{L^2}^2+\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2).\\]\nSimilarly, with the roll of $\\lVert \\nabla_{A_t}\\dot{\\mathbf{v}}\\rVert_{L^2}$ replaced by $\\lVert [\\Psi_t,\\dot{\\mathbf{v}}]\\rVert_{L^2}$, we have\n\\[\\uppercase\\expandafter{\\romannumeral2}\\leqslant c\\mathrm{e}^{-c't^\\sigma}(\\lVert \\dot{\\Phi}+[\\dot{h}_{\\mathrm{app}},\\Phi_t]\\rVert_{L^2}^2+\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2).\\]\nTherefore $|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2|\\leqslant c\\mathrm{e}^{-c't^\\sigma}\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}^2$, meaning that along the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$, $g_{L^2}$ and $g_{\\mathrm{app}}$ are exponentially close in $t$.\n\n\\subsection{Comparing $g_{\\mathrm{app}}$ and $g_{\\mathrm{sf}}$}\nBy the construction, $h_t^{\\mathrm{app}}$ and $h_{t,\\infty}$ coincide on the complement of the disks $\\widetilde{B}_{x,j,t}$ ($x\\in I$) and $\\widetilde{B}_{x,t}$ ($x\\in T$), and so do the infinitesimal deformations $\\dot{h}_{\\mathrm{app}}$ and $\\dot{h}_\\infty$. Accordingly, the difference $g_{\\mathrm{app}}-g_{\\mathrm{sf}}$ localizes on these disks.\n\n  On $\\widetilde{B}_{x,j,t}$, in a local holomorphic coordinate $\\zeta_{x,j,t}$ and a holomorphic frame we have \\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x,j}(t)\\begin{pmatrix}\n    0&1\\\\\\zeta_{x,j,t}&0\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,j,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n    |\\zeta_{x,j,t}|^{1\/2}&0\\\\0& |\\zeta_{x,j,t}|^{-1\/2}\n  \\end{pmatrix},\\]\nand $h_t^{\\mathrm{app}}$ is given by \\eqref{approxMetricZ_eq}. Let $\\hat{\\zeta}_{x,j,t}=\\kappa^{-1}t^{1\/j(x)}\\zeta_{x,j,t}$ and $\\hat{r}_{x,j,t}=|\\hat{\\zeta}_{x,j,t}|$, then the disk $\\widetilde{B}_{x,j,t}=\\{\\,\\hat{r}_{x,j,t}<1\\,\\}$. Applying the local holomorphic gauge transformation\n\\[  g=\\begin{pmatrix}\n    \\kappa^{-1\/4}t^{1\/4j(x)}&0\\\\0&\\kappa^{1\/4}t^{-1\/4j(x)}\n  \\end{pmatrix},\\text{ then }\n  g^{-1}\\Phi_tg =t_{x,j}\\begin{pmatrix}\n    0&1\\\\\\hat{\\zeta}_{x,j,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,j,t},\\]\nwhere $t_{x,j}=\\lambda_{x,j}(t)\\kappa^{3\/2}t^{-3\/2j(x)}$.\n\\[ g^\\ast h_{t,\\infty}g=\\begin{pmatrix}\n  \\hat{r}_{x,j,t}^{1\/2}&0\\\\0&\\hat{r}_{x,j,t}^{-1\/2}\n\\end{pmatrix},\\quad g^\\ast h_t^{\\mathrm{app}}g=\\begin{pmatrix}\n  \\hat{r}_{x,j,t}^{1\/2}\\mathrm{e}^{l_{t_{x,j}(\\hat{r}_{x,j,t})}\\chi(\\hat{r}_{x,j,t})}&0\\\\0&\\hat{r}_{x,j,t}^{-1\/2}\\mathrm{e}^{-l_{t_{x,j}}(\\hat{r}_{x,j,t})\\chi(\\hat{r}_{x,j,t})}\n\\end{pmatrix},\\]\nwhere $l_{t_{x,j}}(\\hat{r}_{x,j,t})=\\psi_1(8t_{x,j}\\hat{r}_{x,j,t}^{3\/2}\/3)$. In this local holomorphic frame, the following result holds.\n\\begin{proposition}[{\\cite[Th.~3.1]{fredrickson_2019}}]\nLet $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, then on the disk $\\{\\,\\hat{r}_{x,j,t}<1\\,\\}$ there is a unique representative in the equivalence class where\n\\[\\dot{\\eta}=0,\\quad \\dot{\\Phi}=t_{x,j}\\begin{pmatrix}\n  0&0\\\\\\dot{P}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,j,t},\\quad \\dot{h}_{\\infty}=\\frac{\\dot{P}}{4\\hat{\\zeta}_{x,j,t}}\\begin{pmatrix}\n  1&0\\\\0&-1\n\\end{pmatrix}\\quad\\text{for }\\bar{\\partial}\\dot{P}=0.\\]\nThese infinitesimal deformations satisfy \\eqref{DecoupInftyDef_eq}, and $\\dot{h}_{\\mathrm{app}}$ solving \\eqref{AppLinHitCol_eq} is diagonal. For any $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, there exists a constant $c>0$ such that \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}(\\widetilde{B}_{x,j,t})}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\infty}(\\widetilde{B}_{x,j,t})}^2=O(\\mathrm{e}^{-ct_{x,j}}).\\]\n\\end{proposition}\nOn $\\widetilde{B}_{x,t}$ ($x\\in T$), in a local holomorphic coordinate $\\zeta_{x,t}$ and holomorphic frame we have\n\\[\\bar{\\partial}_E=\\bar{\\partial},\\quad \\Phi_t=\\lambda_{x}(t)\\begin{pmatrix}\n  0&1\\\\\\frac{1}{\\zeta_{x,t}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{x,t},\\quad h_{t,\\infty}=\\begin{pmatrix}\n  |\\zeta_{x,t}|^{1\/2}&0\\\\0& |\\zeta_{x,t}|^{3\/2}\n\\end{pmatrix},\\]\nand $h_t^{\\mathrm{app}}$ is given by \\eqref{approxMetricT_eq}. Let $\\hat{\\zeta}_{x,t}=\\kappa^{-1}\\zeta_{x,t}$ and $\\hat{r}_{x,t}=|\\hat{\\zeta}_{x,t}|$, then the disk $\\widetilde{B}_{x,t}=\\{\\,\\hat{r}_{x,t}<1\\,\\}$. Applying the local holomorphic gauge transformation\n\\[  g=\\kappa^{-1\/2}\\begin{pmatrix}\n    \\kappa^{1\/4}&0\\\\0&\\kappa^{-1\/4}\n  \\end{pmatrix},\\text{ then }\n  g^{-1}\\Phi_tg =t_x\\begin{pmatrix}\n    0&1\\\\\\hat{\\zeta}_{x,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,t},\\]\nwhere $t_x=\\lambda_x(t)\\kappa^{1\/2}$.\n\\[ g^\\ast h_{t,\\infty}g=\\begin{pmatrix}\n  \\hat{r}_{x,t}^{1\/2}&0\\\\0&\\hat{r}_{x,t}^{3\/2}\n\\end{pmatrix},\\quad g^\\ast h_t^{\\mathrm{app}}g=\\begin{pmatrix}\n  \\hat{r}_{x,t}^{1\/2}\\mathrm{e}^{m_{t_{x}(\\hat{r}_{x,t})}\\chi(\\hat{r}_{x,t})}&0\\\\0&\\hat{r}_{x,t}^{3\/2}\\mathrm{e}^{-m_{t_{x}}(\\hat{r}_{x,t})\\chi(\\hat{r}_{x,t})}\n\\end{pmatrix},\\]\nwhere $m_{t_x}(\\hat{r}_{x,t})=\\psi_{2,x}(8t_x\\hat{r}_{x,t}^{1\/2})$. In this local holomorphic frame, the following result holds.\n\\begin{proposition}[{\\cite[Lem.~7.10,~Prop.~7.11]{fredrickson2022asymptotic}}]\nLet $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, then on the disk $\\{\\,\\hat{r}_{x,t}<1\\,\\}$ there is a unique representative in the equivalence class where\n\\[\\dot{\\eta}=0,\\quad \\dot{\\Phi}=t_x\\begin{pmatrix}\n  0&0\\\\\\frac{\\dot{P}}{\\hat{\\zeta}_{x,t}}&0\n\\end{pmatrix}\\,\\mathrm{d}\\hat{\\zeta}_{x,t},\\quad \\dot{h}_{\\infty}=\\frac{\\dot{P}}{4}\\begin{pmatrix}\n  1&0\\\\0&-1\n\\end{pmatrix}\\quad\\text{for }\\bar{\\partial}\\dot{P}=0.\\]\nThese infinitesimal deformations satisfy \\eqref{DecoupInftyDef_eq}. For any $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$, there exists a constant $c>0$ such that \\[\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{app}}(\\widetilde{B}_{x,t})}^2-\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\infty}(\\widetilde{B}_{x,t})}^2=O(\\mathrm{e}^{-ct_{x}}).\\]\n\\end{proposition}\nThe above two propositions immediately imply $g_{\\mathrm{app}}-g_{\\mathrm{sf}}=O(\\mathrm{e}^{-ct^{\\sigma}})$ along the curve $[(\\bar{\\partial}_E,\\Phi_t)]$ in $\\mathcal{M}$. Hence Theorem \\ref{Main_thm} is established.\n\\begin{remark}\n $g_{\\mathrm{L^2}}-g_{\\mathrm{app}}$ and $g_{\\mathrm{app}}-g_{\\mathrm{sf}}$ are still exponentially decaying when $I_u=\\varnothing$, only slight modifications of the previous proofs are needed.\n\\end{remark}\n\n\\subsection{Comparing $g_{\\mathrm{sf}}$ and $g_{\\mathrm{model}}$}\nIn this subsection we assume that $\\mathrm{dim}_{\\mathbb{C}}\\,\\mathcal{M}=2$, or equivalently $N=\\mathrm{deg}\\,D=4$. The Hitchin base now has complex dimension one, parametrized by $t\\in \\mathbb{C}$. The constants in this subsection will not depend on $\\mathrm{Arg}(t)$ and the choice of the Higgs bundle in the Hitchin fiber $H^{-1}(t)$.\n\\subsubsection{An untwisted order four pole}\\label{U4_subsubsec}\nThe Hitchin base is given by\n\\begin{equation}\\label{HitBaseU4_eq}\n  \\mathcal{B}=\\left\\{\\,\\left(\\sum_{k=5}^8\\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in \\mathbb{C}\\,\\right\\},\n\\end{equation}\nwhere $\\mu_5,\\ldots,\\mu_8$ are constants determined by the irregular type at $0$, and $\\mu_8\\neq 0$. We may assume that $\\mu_8=-1$, using a rescaling of $z$ if necessary.\nFix \\[\\nu=\\nu(z)\\,\\mathrm{d}z^2=\\left(\\sum_{k=5}^8 \\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^4}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen the special K\u00e4hler metric $g_{\\mathrm{sK}}$, which is the restriction of $g_{\\mathrm{sf}}$ on $\\mathcal{B}'$, is given by \\cite[Prop.~8.3]{fredrickson2022asymptotic} \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|t|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z},\n\\end{align*}\nwhere $z_k(t)=t^{-1\/4}\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}+t^{-1\/2}(-1)^k \\mu_7 \/ 4 + t^{-3\/4} \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2} (-\\mu_6\/4 - \\mu_7^2\/32)+O(|t|^{-1})$ ($k=1,\\ldots,4$) are the four roots of $z^8\\nu(z):=\\tilde{\\nu}(z)$.  Let $z=t^{-1\/4}\\xi$ and $z_k=t^{-1\/4}\\xi_k$, then the integral becomes\n\\[\n|t|^{1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar\\xi.\n\\]\nWe define a funtion \\[F(\\xi_1,...\\xi_4,\\bar\\xi_1,...\\bar\\xi_4) := \\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar\\xi.\\]\nThen we can prove that\n\\[\\frac{\\partial F}{\\partial \\xi_k}= \\lim_{\\epsilon \\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3} \\frac{1}{\\left|\\prod_{l\\not =k}(\\xi-\\xi_l)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\,\\quad\\frac{\\partial F}{\\partial \\bar\\xi_k}=\\overline{\\frac{\\partial F}{\\partial \\xi_k}}.\\]\nThen, for $l\\not= k$,\n\\begin{align*}\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\bar\\xi-\\bar\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\\\\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\bar\\xi-\\bar\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\xi-\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi},\\\\\n\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_l}= &\\lim_{\\epsilon\\to 0}\\int_{|\\xi-\\xi_k|>\\epsilon, |\\xi-\\xi_l|>\\epsilon}\\frac{\\xi-\\xi_k}{2\\left|\\xi-\\xi_k\\right|^3}\\frac{\\xi-\\xi_l}{2\\left|\\xi-\\xi_l\\right|^3} \\frac{1}{\\left|\\prod_{m\\not =k,l}(\\xi-\\xi_m)\\right|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d} \\bar{\\xi}.\n\\end{align*}\nFor any $k=1,2,3,4$,\n\\[\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l},\\,\n\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l},\\,\n\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_k}=-\\sum_{l\\not=k}\\frac{\\partial^2 F}{\\partial \\bar\\xi_k\\partial \\bar\\xi_l}\\]\nbecause we can shift $\\xi$ by any variation of $\\xi_k$ in $\\frac{\\partial F}{\\partial \\xi_k}$ or $\\frac{\\partial F}{\\partial \\bar{\\xi}_k}$.\n\nNow we use these to calculate\n\\begingroup\n\\allowdisplaybreaks\n\\begin{align*}\n  &g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu}) =|t|^{1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\xi-\\xi_k)\\right|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi} = |t|^{1\/2}F(\\xi_k, \\bar\\xi_k)\\\\\n =&|t|^{1\/2}\\left( F(\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}, \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2}) + \\sum_{k=1}^{4}2\\mathrm{Re}\\left(\\frac{\\partial F}{\\partial \\xi_k}\\frac{(-1)^k\\mu_7t^{-1\/4}}{4}\\right)\\right.\\\\\n &+\\sum_{k=1}^{4}2\\mathrm{Re}\\left(\\frac{\\partial F}{\\partial \\xi_k}(t^{-1\/2} \\mathrm{e}^{-\\mathrm{i}(k-1)\\pi\/2})(-\\mu_6\/4 - \\mu_7^2\/32)\\right)\\\\\n&+\\mathrm{Re}\\left(\\sum_{k=1}^{4}\\sum_{l=1}^{4}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\xi_l}(-1)^{k+l}\\frac{\\mu_7^2 t^{-1\/2}}{16}\\right)\\\\& +\\left. \\mathrm{Re}\\left(\\sum_{k=1}^{4}\\sum_{l=1}^{4}\\frac{\\partial^2 F}{\\partial \\xi_k\\partial \\bar\\xi_l}\\right)(-1)^{k+l}\\frac{|\\mu_7|^2 |t|^{-1\/2}}{16}+O(|t|^{-3\/4})\\right).\n\\end{align*}\n\\endgroup\nThe terms involving $\\frac{\\partial F}{\\partial \\xi_k}$ cancel with each other. So in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re} (C_1 \\mu_7^2 t^{-1\/2}) + C_2 |\\mu_7|^2 |t|^{-1\/2} + O(r^{-3\/4})))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{\\sqrt{r}},\\]\nwhich is asymptotic to a conic metric with cone angle $3\\pi\/2$.\n\nNext we consider $g_{\\mathrm{sf}}$ along the fiber direction. By the discussion in Section \\ref{Prelim_sec}, the Hitchin fiber $H^{-1}(t)$ can be written as\n\\begin{align}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    \\pm (-t)^{1\/2}z^2&1-\\mu_7z-\\mu_6z^2-\\mu_5z^3\\\\1&\\mp(-t)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\notag\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z - {} \\\\\n      \\mu_7+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\\label{HitFibU4_eq}\n\\end{align}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}\\oplus\\mathcal{O}(-1)$. Therefore $H^{-1}(t)$ is the closure of the second set, which is the elliptic curve \\[a_0^2=-\\tilde{\\nu}(-c_0)=-t\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$. Here $\\lambda$ is the modular lambda function, and\n\\begin{align*}\n  l(t)&=\\frac{(z_4(t)-z_1(t))(z_3(t)-z_2(t))}{(z_3(t)-z_1(t))(z_4(t)-z_2(t))}=\\frac{1}{2}+\\frac{3\\mathrm{i}\\mu_7^2}{32}t^{-1\/2}+O(|t|^{-3\/4}), \\\\\n  \\tau(t)&=\\mathrm{i}+\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}+O(|t|^{-3\/4}),\n\\end{align*}\nwhere $K(x)=\\int_0^{\\pi\/2} (1-x^2\\sin^2\\theta)^{-1\/2}\\,\\mathrm{d}\\theta$ is the complete elliptic integral of the first kind. The semiflat metric induces a constant multiple of the Euclidean metric on $H^{-1}(t)$:\n\\begin{proposition}[{\\cite[Proposition 8.4]{fredrickson2022asymptotic}}]\n  The area of $H^{-1}(t)$ in the semiflat metric is $4\\pi^2$, so $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$ with Euclidean metric $\\mathrm{d}x^2+\\mathrm{d}y^2$ and $c_t=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\tau(t)}$.\n\\end{proposition}\nLet $T_{\\hat{\\tau}}^2=\\mathbb{C}\\,\/\\,\\hat{c}(\\mathbb{Z}\\oplus \\hat{\\tau}\\mathbb{Z})$, where $\\hat{\\tau}=i, \\hat{c}=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\hat{\\tau}}$. Define $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as \\[x+\\mathrm{i}y\\mapsto \\frac{c_t}{\\hat{c}}\\left(x+\\frac{\\mathrm{Re}\\,(\\tau(t)-\\hat{\\tau})}{\\mathrm{Im}\\,\\hat{\\tau}}y+\\mathrm{i}\\frac{\\mathrm{Im}\\,\\tau(t)}{\\mathrm{Im}\\,\\hat{\\tau}}y\\right)=\\frac{c_t}{\\hat{c}}(x+y\\mathrm{Re}\\,\\tau(t)+\\mathrm{i}y \\mathrm{Im}\\,\\tau(t)).\\]\nThen we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\frac{c_t^2}{\\hat{c}^2}\\left((\\mathrm{d}x+\\mathrm{Re}\\,\\tau(t)\\,\\mathrm{d}y)^2+(\\mathrm{Im}\\,\\tau(t))^2\\,\\mathrm{d}y^2\\right)\n\\\\&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\mathrm{Im}\\left(\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+2\\mathrm{Re}\\left(\\frac{3\\pi \\mu_7^2}{32K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-3\/4}).\n\\end{align*}\n\n\\subsubsection{A twisted order four pole}\nIn this case,\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\sum_{k=5}^7\\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_5,\\mu_6,\\mu_7$ are fixed and $\\mu_7\\neq 0$, we may assume that $\\mu_7=-1$. Fix \\[\\nu=\\left(\\sum_{k=5}^7 \\frac{\\mu_k}{z^k}+\\frac{t}{z^4}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^4}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|t|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 z(z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z},\n\\end{align*}\nwhere $z_k(t)=t^{-1\/3}\\mathrm{e}^{2(k-1)\\pi \\mathrm{i}\/3}-\\mu_6\\mathrm{e}^{-2(k-1)\\pi \\mathrm{i}\/3}t^{-2\/3}\/3-\\mu_5t^{-1}\/3+O(|t|^{-4\/3})$ ($k=1,2,3$) are the three roots of $z^7\\nu(z):=\\tilde{\\nu}(z)$. Let $z=t^{-1\/3}\\xi$, the integral becomes\n\\[\n  |t|^{2\/3}\\int_{\\mathbb{C}}\\frac{1}{|\\xi\\prod_{k=1}^3 (\\xi-t^{1\/3}z_k(t))|}~\\mathrm{i}\\,\\mathrm{d} \\xi \\mathrm{d}\\bar\\xi.\n\\]\nAs before, the linear terms cancel with each other.\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1 \\mu_5 t^{-2\/3})+\\mathrm{Re}(C_2 \\mu_6^2t^{-2\/3})+C_3|\\mu_6|^2|t|^{-2\/3}+O(r^{-1}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{1\/3}},\\]\nwhich is asymptotic to a conic metric with cone angle $5\\pi\/3$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    \\pm (-t)^{1\/2}z^2&z-\\mu_6z^2-\\mu_5z^3\\\\1&\\mp(-t)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z + {} \\\\\n      1+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix},a_0^2=c_0\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}\\oplus\\mathcal{O}(-1)$. Therefore $H^{-1}(t)$ is the closure of the second set, which is the elliptic curve \\[a_0^2=c_0\\tilde{\\nu}(-c_0)=tc_0\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_2(t)(z_3(t)-z_1(t))}{z_3(t)(z_2(t)-z_1(t))}=\\mathrm{e}^{-\\pi \\mathrm{i}\/3}+\\frac{\\mathrm{e}^{\\pi \\mathrm{i}\/6}}{3\\sqrt{3}}(3\\mu_5+\\mu_6^2)t^{-2\/3}+O(|t|^{-4\/3}), \\\\\n  \\tau(t)&=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}+\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}+O(|t|^{-4\/3}).\n\\end{align*}\nNow $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$. Let $T_{\\hat{\\tau}}^2=\\mathbb{C}\\,\/\\,\\hat{c}(\\mathbb{Z}\\oplus \\hat{\\tau}\\mathbb{Z})$, where $\\hat{\\tau}=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}, \\hat{c}=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\hat{\\tau}}$. Define $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as \\begin{align*}\nx+\\mathrm{i}y&\\mapsto \\frac{c_t}{\\hat{c}}\\left(x+\\frac{\\mathrm{Re}\\,(\\tau(t)-\\hat{\\tau})}{\\mathrm{Im}\\,\\hat{\\tau}}y+\\mathrm{i}\\frac{\\mathrm{Im}\\,\\tau(t)}{\\mathrm{Im}\\,\\hat{\\tau}}y\\right)\\\\&=\\frac{c_t}{\\hat{c}}(x+2y(\\mathrm{Re}\\,\\tau(t)+1\/2)\/\\sqrt{3}+2\\mathrm{i}y \\mathrm{Im}\\,\\tau(t)\/\\sqrt{3}).\n\\end{align*}\nThen we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\frac{c_t^2}{\\hat{c}^2}\\left((\\mathrm{d}x+2(\\mathrm{Re}\\,\\tau(t)+1\/2)\/\\sqrt{3}\\,\\mathrm{d}y)^2+(2 \\mathrm{Im}\\,\\tau(t)\/\\sqrt{3})^2\\,\\mathrm{d}y^2\\right)\n\\\\&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\frac{2}{\\sqrt{3}}\\mathrm{Im}\\left(\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+\\frac{4}{\\sqrt{3}}\\mathrm{Re}\\left(\\frac{\\mathrm{e}^{-\\pi \\mathrm{i}\/3}\\pi (3\\mu_5+\\mu_6^2)}{12\\sqrt{3}K(\\mathrm{e}^{-\\pi \\mathrm{i}\/6})^2}t^{-2\/3}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-4\/3}).\n\\end{align*}\n\n\n\\subsubsection{An untwisted order three pole and a simple pole} Suppose these poles are located at $0$ and $\\infty$ respectively, then\n\\begin{align*}\n  \\mathcal{B}&=\\left\\{\\,\\left(\\sum_{k=4}^6\\frac{\\mu_k}{z^k}+\\frac{t}{z^3}+\\frac{\\mu_2}{z^2}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\\\\\n  &=\\left\\{\\,\\frac{1}{w^2}\\left(\\mu_6w^4+\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\n\\end{align*}\nwhere $w=1\/z$, $\\mu_2,\\mu_4,\\mu_5,\\mu_6$ are fixed and $\\mu_6\\neq 0$, we may assume that $\\mu_6=-1$. Here we do not require that the simple pole is twisted, if it is, then $\\mu_2=0$. Fix \\[\\nu=\\frac{1}{w^2}\\left(-w^4+\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{w}\\,\\mathrm{d}w^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4 (w-w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $w_k(t)=t^{1\/3}\\mathrm{e}^{2(k-1)\\pi \\mathrm{i}\/3}+1+O(|t|^{-1\/3})$ ($k=1,2,3$), $w_4(t)=\\mu_2(-t^{-1}-\\mu_2\\mu_4t^{-3}+O(|t|^{-5}))$ are the four roots of $w^2\\nu(w):=\\tilde{\\nu}(w)$. Let $w=t^{1\/3}\\eta $, the integral becomes\n\\begin{align*}\n  &|t|^{-2\/3}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^4(\\eta -t^{-1\/3}w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }\\\\\n  =&|t|^{-2\/3}\\left(\\int_{\\mathbb{C}}\\frac{1}{|\\eta |\\left|\\prod_{k=1}^3(\\eta -\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2})\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }+O(|t|^{-1\/3})\\right)\\\\\n  =&|t|^{-2\/3}(C_0+\\mathrm{Re}(C_1t^{-1\/3})+O(|t|^{-2\/3}))\n\\end{align*}\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1t^{-1\/3})+O(r^{-2\/3}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{2\/3}},\\]\nwhich is asymptotic to a conic metric with cone angle $4\\pi\/3$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    0&w^4-\\mu_5w^3-\\mu_4w^2-tw-\\mu_2\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    -\\mu_5w\/2+w^2&-(\\mu_5^2\/4+\\mu_4)w^2-tw-\\mu_2\\\\1&\\mu_5w\/2-w^2\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n  a_0+w^2&\\begin{smallmatrix}\n    -\\mu_5w^2+(c_0\\mu_5-\\mu_4-2a_0)w-{}\\\\c_0^2\\mu_5+c_0\\mu_4+2a_0c_0-t\n  \\end{smallmatrix}\\\\c_0+w & -a_0-w^2\n\\end{pmatrix},(a_0+c_0^2)^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[(a_0+c_0^2)^2=-\\tilde{\\nu}(-c_0)=\\prod_{k=1}^4(-c_0-w_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{(w_1(t)-w_2(t))(w_3(t)-w_4(t))}{(w_3(t)-w_2(t))(w_1(t)-w_4(t))}=\\mathrm{e}^{-\\pi \\mathrm{i}\/3}+\\sqrt{3}\\mathrm{i} t^{-1\/3} +O(|t|^{-2\/3}), \\\\\n  \\tau(t)&=\\mathrm{e}^{2\\pi \\mathrm{i}\/3}+\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}+O(|t|^{-2\/3}).\n\\end{align*}\nDefine $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as above, then we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\frac{2}{\\sqrt{3}}\\mathrm{Im}\\left(\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+\\frac{4}{\\sqrt{3}}\\mathrm{Re}\\left(\\frac{\\sqrt{3}\\pi}{4K(\\mathrm{e}^{-\\pi\/6})^2}t^{-1\/3}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-2\/3}).\n\\end{align*}\n\n\\subsubsection{A twisted order three pole and a simple pole} Suppose these poles are located at $0$ and $\\infty$ respectively, then\n\\begin{align*}\n  \\mathcal{B}&=\\left\\{\\,\\left(\\frac{\\mu_5}{z^5}+\\frac{\\mu_4}{z^4}+\\frac{t}{z^3}+\\frac{\\mu_2}{z^2}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\\\\\n  &=\\left\\{\\,\\frac{1}{w^2}\\left(\\mu_5w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\}\n\\end{align*}\nwhere $w=1\/z$, $\\mu_2,\\mu_4,\\mu_5$ are fixed and $\\mu_5\\neq 0$, we may assume that $\\mu_5=-1$. Fix \\[\\nu=\\frac{1}{w^2}\\left(-w^3+\\mu_4w^2+tw+\\mu_2\\right)\\,\\mathrm{d}w^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{w}\\,\\mathrm{d}w^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 (w-w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $w_k(t)=(-1)^{k-1}(t^{1\/2}+\\mu_4^2t^{-1\/2}\/8)+\\mu_4\/2+O(|t|^{-1})$ ($k=1,2$), $w_3(t)=\\mu_2(-t^{-1}-\\mu_2\\mu_4t^{-3}+O(|t|^{-4}))$ are the three roots of $w^2\\nu(w):=\\tilde{\\nu}(w)$. Let $w=t^{1\/2}\\eta $, the integral becomes\n\\begin{align*}\n  &|t|^{-1\/2}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3(\\eta -t^{-1\/2}w_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }\\\\\n  =&|t|^{-1\/2}\\left(\\int_{\\mathbb{C}}\\frac{1}{|\\eta (\\eta -1)(\\eta +1)|}~\\mathrm{i}\\,\\mathrm{d}\\eta \\mathrm{d}\\bar{\\eta }+O(|t|^{-1\/2})\\right)\\\\\n  =&|t|^{-1\/2}(C_0+\\mathrm{Re}(C_1\\mu_4t^{-1\/2})+O(|t|^{-1}))\n\\end{align*}\nTherefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(C_0+\\mathrm{Re}(C_1\\mu_4t^{-1\/2})+O(r^{-1}))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r^{1\/2}},\\]\nwhich is asymptotic to a conic metric with cone angle $3\\pi\/2$.\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n    0&w^3-\\mu_4w^2-tw-\\mu_2\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}w}{w}\\begin{pmatrix}\n  a_0&\\begin{smallmatrix}\n    w^2-(\\mu_4+c_0)w+{}\\\\\\mu_4c_0+c_0^2-t\n  \\end{smallmatrix}\\\\c_0+w & -a_0\n\\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=\\prod_{k=1}^3(-c_0-w_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{w_3(t)-w_2(t)}{w_1(t)-w_2(t)}=\\frac{1}{2}-\\mu_4t^{-1\/2}\/4 +O(|t|^{-1}), \\\\\n  \\tau(t)&=\\mathrm{i}+\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}+O(|t|^{-1}).\n\\end{align*}\nDefine $\\mu_t: T_{\\hat{\\tau}}^2\\to H^{-1}(t)$ as above, then we have\n\\begin{align*}\n\\mu_t^\\ast (g_{\\mathrm{sf}}|_{H^{-1}(t)})&=\\mathrm{d}x^2+\\mathrm{d}y^2+\\mathrm{Im}\\left(\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)(-\\mathrm{d}x^2+\\mathrm{d}y^2)\\\\\n&+2\\mathrm{Re}\\left(\\frac{\\mu_4\\pi \\mathrm{i} }{4K(\\sqrt{1\/2})^2}t^{-1\/2}\\right)\\,\\mathrm{d}x\\otimes \\mathrm{d}y+O(|t|^{-1}).\n\\end{align*}\n\n\\subsubsection{Two untwisted order two poles} Suppose these poles are located at $0$ and $\\infty$ respectively, then \\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}+\\mu_0\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3,\\mu_4$ are fixed and $\\mu_0,\\mu_4\\neq 0$, we may assume that $\\mu_4=-1$. Fix \\[\\nu=\\left(\\frac{-1}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}+\\mu_0\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_0\\prod_{k=1}^4 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|z|\\leqslant 1}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{\\left|\\prod_{k=1}^4(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $z_k(t)=(-1)^{k-1}t^{-1\/2}-\\mu_3t^{-1}\/2+O(|t|^{-3\/2})$ ($k=1,2$), $z_k(t)=(-1)^{k-1}(-\\mu_0)^{-1\/2}t^{1\/2}-\\mu_1\/2\\mu_0+O(|t|^{-1\/2})$ ($k=3,4$) are the four roots of $z^4\\nu(z)=:\\tilde{\\nu}(z)$. In $\\{|z|\\leqslant 1\\}$, $|z-z_k(t)|=|\\mu_0^{-1\/2}t^{1\/2}|(1+O(|t|^{-1\/2}))$ for $k=3,4$, then the first integral is\n\\begin{align*}\n  &\\int_{|z|\\leqslant 1}\\frac{|\\mu_0t^{-1}|}{|z-z_1(t)||z-z_2(t)|}(1+O(|t|^{-1\/2}))~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\quad (\\text{let }\\xi=t^{1\/2}z)\\\\=&\\int_{|\\xi|\\leqslant |t^{1\/2}|}\\frac{1}{|\\xi-t^{1\/2}z_1(t)||\\xi-t^{1\/2}z_2(t)|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi}\\cdot|\\mu_0t^{-1}|(1+O(|t|^{-1\/2}))\\\\\n  =&\\int_{|\\xi|\\leqslant |t^{1\/2}|}\\frac{1}{|\\xi-1||\\xi+1|}~\\mathrm{i}\\,\\mathrm{d}\\xi \\mathrm{d}\\bar{\\xi}\\cdot|\\mu_0t^{-1}|(1+O(|t|^{-1\/2}))\\\\\n  =&2\\pi |\\mu_0||t|^{-1}\\log|t|+O(|t|^{-1}).\n\\end{align*}\nSimilarly, the second integral is\n\\[2\\pi |t|^{-1}\\log|t|+O(|t|^{-1}).\\]\nTherefore in polar coordinates, as $|a|=r\\to\\infty$, the special K\u00e4hler metric is \\[(4\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&1-\\mu_3z-tz^2-\\mu_1z^3-\\mu_0z^4\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n  -\\frac{\\mu_1z}{2(-\\mu_0)^{1\/2}}+(-\\mu_0)^{1\/2}z^2&-(t+\\mu_1^2\/4\\mu_0)z^2-\\mu_3z+1\\\\1&  \\frac{\\mu_1z}{2(-\\mu_0)^{1\/2}}-(-\\mu_0)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0+(-\\mu_0)^{1\/2}z^2&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-2a_0(-\\mu_0)^{1\/2}-t)z-{}\\\\\n     \\mu_1c_0^2+2a_0c_0(-\\mu_0)^{1\/2}+tc_0-\\mu_3\n   \\end{smallmatrix}\\\\c_0+z&-a_0-(-\\mu_0)^{1\/2}z^2\n \\end{pmatrix},\\right.\\notag\\\\&\\left.\\hspace{8cm}\\vphantom{\\begin{pmatrix}\n     1+a_1z&\\begin{smallmatrix}\n       \\mu_1c_0-2a_0(-\\mu_0)^{1\/2}\\\\a_0c_0(-\\mu_0)^{1\/2}\n     \\end{smallmatrix}\\\\1+c_1z&-(-\\mu_0)^{1\/2}z^2\n  \\end{pmatrix}}  (a_0+(-\\mu_0)^{1\/2}c_0^2)^2=-\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-1)$, $(E,\\bar{\\partial}_E^1)\\cong \\mathcal{O}\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[(a_0+(-\\mu_0)^{1\/2}c_0^2)^2=-\\tilde{\\nu}(-c_0)=-\\mu_0\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{(z_1(t)-z_2(t))(z_3(t)-z_4(t))}{(z_3(t)-z_2(t))(z_1(t)-z_4(t))}\\\\&=4(-\\mu_0)^{1\/2}t^{-1}+(8\\mu_0+(-\\mu_0)^{-1\/2}(\\mu_1^2-\\mu_0\\mu_3^2)\/2)t^{-2}+O(|t|^{-3}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\log(4(-\\mu_0)^{-1\/2}t)+\\frac{\\mathrm{i}}{\\pi}(\\mu_0^{-1}(\\mu_1^2-\\mu_0\\mu_3^2)\/8)t^{-1}+O(|t|^{-2}).\n\\end{align*}\n $H^{-1}(t)\\cong \\mathbb{C}\\,\/\\,c_t(\\mathbb{Z}\\oplus\\tau(t)\\mathbb{Z})$ with Euclidean metric $\\mathrm{d}x^2+\\mathrm{d}y^2$ and $c_t=2\\pi\/\\sqrt{\\mathrm{Im}\\,\\tau(t)}$.\n\n\\subsubsection{Two order two poles with one twisted} Suppose the pole at $0$ is untwisted and the pole at $\\infty$ is twisted, then\n\\[\\mathcal{B}=\\left\\{\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_1,\\mu_3,\\mu_4$ are fixed and $\\mu_1,\\mu_4\\neq 0$, we may assume that $\\mu_4=-1$. Fix \\[\\nu=\\left(\\frac{\\mu_4}{z^4}+\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{\\mathbb{C}}\\frac{1}{\\left|\\prod_{k=1}^3 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{|z|\\leqslant 1}\\frac{1}{|\\prod_{k=1}^3(z-z_k(t))|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{|w||\\prod_{k=1}^3(w-z_k(t)^{-1})|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w},\n\\end{align*}\nwhere $z_k(t)=(-1)^{k-1}t^{-1\/2}-\\mu_3t^{-1}\/2+O(|t|^{-3\/2})$ ($k=1,2$), $z_3(t)=-t\/\\mu_1+\\mu_3t^{-1}+O(|t|^{-2})$ are the three roots of $z^4\\nu(z)=:\\tilde{\\nu}(z)$. In $\\{|z|\\leqslant 1\\}$, $|z-z_3(t)|=|t\/\\mu_1|(1+O(|t|^{-1}))$, then the first integral is\n\\begin{align*}\n &\\int_{|z|\\leqslant 1}\\frac{|\\mu_1t^{-1}|}{|z-z_1(t)||z-z_2(t)|}(1+O(|t|^{-1}))~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\=&2\\pi|\\mu_1||t|^{-1}\\log\\,|t|+O(|t|^{-1}),\n\\end{align*}\nas before. Similarly, the second integral is $4\\pi |t|^{-1}\\log\\,|t|+O(|t|^{-1})$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(6\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&1-\\mu_3z-tz^2-\\mu_1z^3\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\notag\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-t)z-{}\\\\\n     \\mu_3+tc_0-\\mu_1c_0^2\n   \\end{smallmatrix}\\\\c_0+z&-a_0\n \\end{pmatrix},a_0^2=-\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $\\bar{\\partial}_E^0,\\bar{\\partial}_E^1$ are holomorphic structures as before. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_1\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_1(t)-z_2(t)}{z_1(t)-z_3(t)}=2\\mu_1t^{-3\/2}+\\mu_1\\mu_3^2t^{-5\/2}\/4+O(|t|^{-3}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\left(\\log(8\\mu_1^{-1}t^{3\/2})-\\mu_3^2t^{-1}\/8-\\mu_1t^{-3\/2}\\right)+O(|t|^{-2}).\n\\end{align*}\n\n\\subsubsection{Two twisted order two poles}\nSuppose the poles are $0$ and $\\infty$, then\n\\[\\mathcal{B}=\\left\\{\\left(\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_1,\\mu_3\\neq 0$ are fixed, we may assume that $\\mu_3=-1$. Fix\n\\[\\nu=\\left(\\frac{\\mu_3}{z^3}+\\frac{t}{z^2}+\\frac{\\mu_1}{z}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z^2}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nBy a similar computation as before, $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is \\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{\\mathbb{C}}\\frac{1}{\\left|z(z-z_1(t))(z-z_2(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_1|}\\int_{|z|\\leqslant 1}\\frac{1}{|z||z-z_1(t)||z-z_2(t)|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&\\quad+|\\dot{t}|^2\\int_{|w|\\leqslant 1}\\frac{1}{|w||w-z_1(t)^{-1}||w-z_2(t)^{-1}|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\&=|\\dot{t}|^2( 8\\pi |t|^{-1}\\log\\,|t|+O(|t|^{-1})).\n\\end{align*}\nHere $z_1(t)=t^{-1}-\\mu_1t^{-3}+O(|t|^{-5})$, $z_2(t)=-t\/\\mu_1-t^{-1}+O(|t|^{-3})$ are the two roots of $z^3\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(8\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^1,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n    0&z(1-tz+\\mu_1z^2)\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n  &\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z^2}\\begin{pmatrix}\n   a_0&\\begin{smallmatrix}\n     -\\mu_1z^2+(\\mu_1c_0-a)z+{}\\\\1+tc_0-\\mu_1c_0^2\n   \\end{smallmatrix} \\\\c_0+z&-a_0\n \\end{pmatrix},a_0^2=c_0\\tilde{\\nu}(-c_0)\\,\\right\\},\n\\end{align*}\nwhere $\\bar{\\partial}_E^0,\\bar{\\partial}_E^1$ are as above. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=c_0\\tilde{\\nu}(-c_0)=\\mu_1c_0(c_0+z_1(t))(c_0+z_2(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\n  l(t)&=\\frac{z_1(t)-0}{z_1(t)-z_2(t)}=\\mu_1t^{-2}-3\\mu_1^2t^{-4}+O(|t|^{-6}),\\\\\n  \\tau(t)&=\\frac{\\mathrm{i}}{\\pi}(\\log(16\\mu_1^{-1}t^2)+5\\mu_1t^{-2}\/2)+O(|t|^{-4}).\n\\end{align*}\n\n\\subsubsection{An untwisted order two pole and two simple poles}\nSuppose the two simple poles are $0$ and $1$, the order two pole is $\\infty$, then\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\mu_4+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3,\\mu_4$ are fixed, $\\mu_4\\neq 0$. Fix\n\\[\\nu=\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\mu_4+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z(z-1)}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_4\\prod_{k=1}^4 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_4|}\\int_{|z|\\leqslant 2}+\\int_{2\\leqslant |z|\\leqslant |\\mu_4|^{-1\/2}|t|^{1\/2}\/2}\\frac{1}{\\left|\\prod_{k=1}^4(z-z_k(t))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&+\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|w|\\leqslant 2|\\mu_4|^{1\/2}|a|^{-1\/2}}\\frac{1}{\\left|\\prod_{k=1}^4(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\\n  &=|\\dot{t}|^2(O(1\/|t|)+2\\pi |t|^{-1}\\log\\,|t|+O(1\/|t|))=2\\pi |\\dot{t}|^2|t|^{-1}(\\log\\,|t|+O(1)).\n\\end{align*}\nHere $z_1(t)=\\mu_0t^{-1}+\\mu_0(\\mu_3-\\mu_0)t^{-2}+O(|t|^{-3})$, $z_2(t)=1-\\mu_1t^{-1}+\\mu_1^2t^{-2}+O(|t|^{-3})$, $z_k(t)=(-1)^{k-1}(-\\mu_4)^{-1\/2}t^{1\/2}+(1-\\mu_3\/\\mu_4)\/2+O(|t|^{-1\/2})$ ($k=3,4$) are the four roots of $z^2(z-1)^2\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(2\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\n\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n    \\pm (-\\mu_4)^{1\/2}z^2&\\mu_4z^4-\\tilde{\\nu}(z)\\\\1&\\mp(-\\mu_4)^{1\/2}z^2\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\hspace{-2em}\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n a_0&\\begin{smallmatrix}\n   -\\mu_4z^3+(\\mu_4c_0+2\\mu_4-\\mu_3)z^2+{}\\\\\n   (2\\mu_3-\\mu_4-\\mu_1-t-\\mu_4c_0^2-2\\mu_4c_0+\\mu_3c_0)z+{}\\\\2\\mu_0+t-\n  \\mu_3+(\\mu_4-2\\mu_3+\\mu_1+t)c_0+{}\\\\(2\\mu_4-\\mu_3)c_0^2+\\mu_4c_0^3\n \\end{smallmatrix}\\\\c_0+z&-a_0\n\\end{pmatrix}, a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_4\\prod_{k=1}^4(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\nl(t)&=\\frac{(z_1(t)-z_2(t))(z_3(t)-z_4(t))}{(z_3(t)-z_2(t))(z_1(t)-z_4(t))}=-2(-\\mu_4)^{1\/2}t^{-1\/2}+2\\mu_4t^{-1}+O(|t|^{-3\/2}),\\\\\n\\tau(t)&=\\frac{\\mathrm{i}}{\\pi}\\log(-8(-\\mu_4)^{-1\/2}t^{1\/2})+O(|t|^{-1}).\n\\end{align*}\n\n\\subsubsection{A twisted order two pole and two simple poles} Let $0,1$ be the two simple poles and $\\infty$ be the order two pole, then\n\\[\\mathcal{B}=\\left\\{\\,\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\,\\middle|\\,t\\in\\mathbb{C}\\,\\right\\},\\]\nwhere $\\mu_0,\\mu_1,\\mu_3$ are fixed, $\\mu_3\\neq 0$. Fix\n\\[\\nu=\\left(\\frac{\\mu_0}{z^2}+\\frac{\\mu_1}{(z-1)^2}+\\frac{\\mu_3}{z}+\\frac{t}{z(z-1)}\\right)\\,\\mathrm{d}z^2\\in\\mathcal{B}',~\\dot{\\nu}=\\frac{\\dot{t}}{z(z-1)}\\,\\mathrm{d}z^2\\in T_\\nu\\mathcal{B}'.\\]\nThen $g_{\\mathrm{sK}}$ on $\\mathcal{B}'$ is\n\\begin{align*}\n  g_{\\mathrm{sK}}(\\dot{\\nu},\\dot{\\nu})&=\\int_{C}\\frac{|\\dot{\\nu}|^2}{|\\nu|}\\,\\mathrm{dvol}_C=|\\dot{t}|^2\\int_{\\mathbb{C}}\\frac{1}{\\left|\\mu_3\\prod_{k=1}^3 (z-z_k(t))\\right|}~\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\\n  &=\\frac{|\\dot{t}|^2}{|\\mu_3|}\\int_{|z|\\leqslant 2}+\\int_{2\\leqslant |z|\\leqslant |t|\/(2|\\mu_3|)}\\frac{1}{\\left|\\prod_{k=1}^3(z-z_k(a))\\right|}\\,\\mathrm{i}\\,\\mathrm{d}z\\mathrm{d}\\bar{z}\\\\&+\\frac{|\\dot{t}|^2}{|\\mu_0|}\\int_{|w|\\leqslant 2|\\mu_3|\/|t|}\\frac{1}{|w|\\left|\\prod_{k=1}^3(w-z_k(t)^{-1})\\right|}~\\mathrm{i}\\,\\mathrm{d}w\\mathrm{d}\\bar{w}\\\\\n  &=|\\dot{t}|^2(O(1\/|t|)+4\\pi |t|^{-1}\\log\\,|t|+O(1\/|t|))=4\\pi |\\dot{t}|^2 |t|^{-1}(\\log\\,|t|+O(1)).\n\\end{align*}\nHere $z_1(t)=\\mu_0t^{-1}+\\mu_0(\\mu_3-\\mu_0)t^{-2}+O(|t|^{-3})$, $z_2(t)=1-\\mu_1t^{-1}+\\mu_1^2t^{-2}+O(|t|^{-3})$, $z_3(t)=-t\/\\mu_3+(1-(\\mu_0+\\mu_1)\/\\mu_3)+O(|t|^{-1})$ are the three roots of $z^2(z-1)^2\\nu(z):=\\tilde{\\nu}(z)$. Therefore in polar coordinates, as $|t|=r\\to\\infty$, the special K\u00e4hler metric is \\[(4\\pi\\log\\,r+O(1))\\frac{\\mathrm{d}r^2+r^2\\,\\mathrm{d}\\theta^2}{r}.\\]\nThe fiber $H^{-1}(t)$ can be written as\n\\begin{align*}\n  H^{-1}(t)&=\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n    0&-\\tilde{\\nu}(z)\\\\1&0\n\\end{pmatrix}\\,\\right\\}\\\\\n&\\cup\\left\\{\\,\\left[\\left(\\bar{\\partial}_{E}^0,\\Phi\\right)\\right]\\,\\middle|\\,\\Phi=\\frac{\\mathrm{d}z}{z(z-1)}\\begin{pmatrix}\n a_0&\\begin{smallmatrix}\n  -\\mu_3z^2+(\\mu_3c_0+2\\mu_3-\\mu_0-\\mu_1-t)z+{}\\\\\n  t+2\\mu_0-\\mu_3+{}\\\\(t+\\mu_0+\\mu_1-2\\mu_3)c_0-\\mu_3c_0^2\n \\end{smallmatrix}\\\\c_0+z&-a_0\n\\end{pmatrix}, a_0^2=-\\tilde{\\nu}(-c_0)\\right\\},\n\\end{align*}\nwhere $(E,\\bar{\\partial}_E^0)\\cong \\mathcal{O}(-1)\\oplus \\mathcal{O}(-2)$. $H^{-1}(t)$ is the closure of the last set, which is the elliptic curve\n\\[a_0^2=-\\tilde{\\nu}(-c_0)=-\\mu_3\\prod_{k=1}^3(-c_0-z_k(t)),~(a_0,c_0)\\in \\mathbb{C}^2,\\]\nwith modulus $\\tau(t)=\\lambda^{-1}(l(t))$, where\n\\begin{align*}\nl(t)&=\\frac{(z_1(t)-z_2(t))}{(z_1(t)-z_3(t))}=-\\mu_3t^{-1}-\\mu_3t^{-2}+O(|t|^{-3}),\\\\\n\\tau(t)&=\\frac{\\mathrm{i}}{\\pi}(\\log(-16\\mu_3^{-1}t)+(\\mu_3\/2-1)t^{-1})+O(|t|^{-2}).\n\\end{align*}\n\n\\subsection{The Uniformness}\nIn Theorem \\ref{Main_thm}, the constants in the decay rate of $g_{L^2}-g_{\\mathrm{sf}}$ depend on the choice of the curve. Now we aim to find constants uniform in these choices for a four-dimensional moduli space $\\mathcal{M}$. For simplicity, here we only consider the moduli space in Section \\ref{U4_subsubsec}.\n\nAs in \\eqref{HitBaseU4_eq}, $t\\in \\mathbb{C}$ parametrizes the quadratic differential $\\nu_t$ in the Hitchin base, and is no longer a positive real number as in the previous sections. Let $(\\bar{\\partial}_E,\\Phi_t)$ be any Higgs bundle in the Hitchin fiber $H^{-1}(t)$. By Lemma \\ref{HiggsDetRoot_lem}, the four roots of $\\tilde{\\nu}_t(z)$ have asymptotics\n\\[z_k(t)=t^{-1\/4}\\mathrm{e}^{\\mathrm{i}(k-1)\\pi\/2}+O(|t|^{-1\/2}),\\]\nas $|t|\\to\\infty$. The local mass at $z_k(t)$ $(k=1,2,3,4)$ is\n\\[\\lambda_k(t)=\\Big|t\\prod_{j\\neq k}(z_k(t)-z_j(t))z_k(t)^{-8}\\Big|^{1\/2}=4t^{9\/4}(1+O(|t|^{-1\/4})).\\]\nHere and below, the constants are independent of $\\mathrm{Arg}(t)$ and the choice of $(\\bar{\\partial}_E,\\Phi_t)$ in $H^{-1}(t)$. Proposition \\ref{NormalForm_prop} still hold on disks $\\widetilde{B}_{k,t}:=\\{\\,|\\zeta_{k,t}|<\\kappa |t|^{-1\/4}\\,\\}$, where $\\zeta_{k,t}$ is the local holomorphic coordinate centered at $z_k(t)$ such that $q_t=-\\lambda_k(t)^2\\zeta_{k,t}\\,\\mathrm{d}\\zeta_{k,t}^2$, and $\\kappa$ is independent of the above choices. One only need to check that the gauge transformations in the proof is well-defined on the whole disk $\\widetilde{B}_{k,t}$ for suitable $\\kappa$. This is clear for $\\Phi_t$ in the first subset of \\eqref{HitFibU4_eq}. For $\\Phi_t$ in the second subset, now $a_0,c_0$ may depend on $t$. Suppose $|c_0+z_k(t)|\\geqslant |t|^{-1\/4}\/2$, the first gauge transformation is $\\left(\\begin{smallmatrix}\n  a_0&1\\\\ z_k(t)+c_0&0\n\\end{smallmatrix}\\right)$, which is constant and invertible over $\\widetilde{B}_{k,t}$. The second gauge transformation is well-defined since \\[\\left|\\frac{z+c_0}{z_k(t)+c_0}-1\\right|= \\left|\\frac{z-z_k(t)}{z_k(t)}\\right|\\left|\\frac{z_k(t)}{c_0+z_k(t)}\\right|\\leqslant c\\kappa\\leqslant 1\/2,\\]\nfor $\\kappa$ small. The rest of the proof remains the same. If $|c_0+z_k(t)|\\leqslant |t|^{-1\/4}\/2$ for some $k$, then $|c_0+z_j(t)|\\geqslant |t|^{-1\/4}\/2$ for $j\\neq k$ and $|t|$ large, and the previous discussion applies for the disks $\\widetilde{B}_{j,t}$. On $\\widetilde{B}_{k,t}$,\n\\[\\Phi_t=\\frac{\\mathrm{d}z}{z^4}\\begin{pmatrix}\n    a_0&\\begin{smallmatrix}\n      -tz^3+(-\\mu_5+tc_0)z^2+(-\\mu_6+\\mu_5c_0-tc_0^2)z - {} \\\\\n      \\mu_7+\\mu_6c_0-\\mu_5c_0^2+tc_0^3\n    \\end{smallmatrix}\\\\ c_0+z&-a_0\n  \\end{pmatrix}:=\\begin{pmatrix}\n    a_t(\\zeta_{k,t})&b_t(\\zeta_{k,t})\\\\c_t((\\zeta_{k,t}))&-a_t(\\zeta_{k,t})\n  \\end{pmatrix}\\,\\mathrm{d}\\zeta_{k,t},\\]\n  where \\begin{align*}\n    b_t(\\zeta_{k,t})&=\\frac{\\mathrm{d}z}{\\mathrm{d}\\zeta_{k,t}}(\\zeta_{k,t})\\frac{-t}{z^4}(z^3-c_0z^2+c_0^2z-c_0^3)(1+O(|t|^{-1\/4}))\\\\\n    &=\\frac{\\mathrm{d}z}{\\mathrm{d}\\zeta_{k,t}}(\\zeta_{k,t})\\frac{-t}{z^4}(z-c_0)(z^2+c_0^2)(1+O(|t|^{-1\/4})).\n\\end{align*}\nThen $\\sqrt{b_t(\\zeta_{k,t})}$ is well-defined on $\\widetilde{B}_{k,t}$ by the assumption that $|c_0+z_k(t)|\\leqslant |t|^{-1\/4}\/2$, and we have \\[g^{-1}\\Phi_t g=\\begin{pmatrix}\n  0&1\\\\\\lambda_k(t)^2\\zeta_{k,t}&0\n\\end{pmatrix}\\,\\mathrm{d}\\zeta_{k,t}, \\text{ for }g=\\frac{1}{\\sqrt{b_t(\\zeta_{k,t})}}\\begin{pmatrix}\n  b_t(\\zeta_{k,t})&0\\\\-a_t(\\zeta_{k,t})&1\n\\end{pmatrix}.\\]\nThe remaining proof is the same. The approximate metric $h_t^{\\mathrm{app}}$ is constructed as in Definition \\ref{ApproxMet_def}, and the proof of the error estimate in \\eqref{ErrorEst_eq} works uniformly. Then there are uniform constants $t_0,c,c'$, such that for any $|t|\\geqslant t_0$,\n\\[\n\\left\\lVert F_{h_t^\\mathrm{app}}+\\left[\\Phi_t,\\Phi_t^{\\ast_{h_t^\\mathrm{app}}}\\right] \\right\\rVert_{L^2}\\leqslant c\\mathrm{e}^{-c' |t|^{3\/4}}.\n\\]\nThe computations in Sections 4, 5, 6 only use the expression of $q_t=\\det\\,\\Phi_t$ and the local forms of $\\Phi_t$ provided by Proposition \\ref{NormalForm_prop}, so the estimates remain valid uniformly for the choice of $\\Phi_t$ in $H^{-1}(t)$. Moreover, the constants can also be chosen to be independent of $\\mathrm{Arg}(t)$. Therefore, we find uniform constants $t_0,c,c'$ so that for $[(\\dot{\\eta},\\dot{\\Phi})]\\in T_{[(\\bar{\\partial}_E,\\Phi_t)]}\\mathcal{M}$ and $|t|\\geqslant t_0$,\n  \\[|\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{L^2}}\/\\lVert [(\\dot{\\eta},\\dot{\\Phi})]\\rVert_{g_{\\mathrm{sf}}}-1|\\leqslant c\\mathrm{e}^{-c'|t|^{3\/4}}.\\]\n\n\\bibliographystyle{amsalpha}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{introduction}\n\\label{introduction}\nBy far the most acute and clear-cut tests of MOND \\cite{milgrom83a} come from the dynamical analysis of rotation curves (RCs) of disc galaxies. For reviews of MOND see, e.g. Refs. \\cite{fm12,milgrom14}.\n\\par\nIn particular, the MOND acceleration constant, $\\az$, appears in such analyses in different roles: as the boundary constant marking the transition from Newtonian behavior to deep-MOND, as setting behavior in the deep MOND regime, for example, fixing the normalization of the MOND mass-asymptotic-speed relation (MASR) -- underlying the baryonic Tully-Fisher relation (BTFR), and in dictating the RCs of low-acceleration (or low-surface-brightness) galaxies. It appears in several roles in the much discussed MOND prediction \\cite{milgrom83a} of the mass-discrepancy-acceleration relation (for tests of this prediction see, e.g., \\cite{sanders90,mcgaugh04,tiret09,wu15,mcgaugh16a}). It also appears in the no-less-striking central-surface-densities relation, which is different and independent of the other MOND relations \\cite{milgrom09a,lelli16,milgrom16a}.\n\\par\nThe value that has emerged for $\\az\\approx 1.2\\times 10^{-8}\\cmss$, as been recognized early on (Milgrom 1983a) to have cosmological connotations. In particular we have:\n\\beq \\bar\\az\\equiv 2\\pi \\az\\approx cH_0\\approx c^2(\\Lambda\/3)^{1\/2}, \\eeqno{coinc}\nwhere $H_0$ is the Hubble constant, and $\\Lambda$ the observed equivalent of a cosmological constant.\n\\par\nThe Former of these near equalities, and the realization that MOND may well be an effective theory rooted somehow in cosmology, have pointed to the possibility that $\\az$, or some aspects of MOND, may be varying with cosmological time  so as to retain the first equality at all times.\n\nThe obvious way to test this possibility, given that $\\az$ is sharply determined by rotation-curve analysis, is to analyze RCs of high-$z$ galaxies to see if their dynamics can be accounted for by MOND, and whether this requires $\\az$ to be $z$ dependent (an early attempt at this is described in Ref. \\cite{milgrom08}).\n\\par\nIn recent years, there have been several studies of the internal kinematics of high-$z$ galaxies (e.g., \\cite{price16,wuyts16,lang17}). These are, by and large, statistical in nature.\n\\par\nGenzel et al. \\cite{genzel17} have recently published the individual RCs of six high-redshift galaxies\n($z\\sim 0.9-2.4$), and have presented a thorough dynamical analysis of them.\nThese are selected from a large sample of several hundred, according to criteria that are conducive to cleaner analysis.\nThis sample now also affords a closer, if preliminary, examination of the dynamics of high-$z$ in light of MOND.\n\\par\nThe main general conclusions of Ref. \\cite{genzel17} are that these galaxies are `baryon dominated' within the studies radii, and that they show marked decline in the RC still within the optical image.\nIn both regards, this is very reminiscent of the findings of Ref. \\cite{romanowsky03} of `dearth of dark matter in ordinary elliptical galaxies' (at low redshift) -- based on planetary-nebulae velocities.\n\\par\nI shall show that both of these characteristics of the high-$z$ disc galaxies of Ref. \\cite{genzel17} follow from MOND because these galaxies have accelerations within the studied regions that are higher than $\\az$. The MOND analysis by Ref. \\cite{ms03} and Ref. \\cite{tian16} showed this for low-$z$, elliptical galaxies.\n\\par\nIt is important to keep in mind that for natural reasons the data of Ref. \\cite{genzel17} are not up to the standard afforded by RCs and baryon distributions available for dynamical analysis in the nearby Universe. In comparison with the latter they are limited in scope, and they are subject to large uncertainties (partly reflected in their large quoted errors).\nSome concerns that come to mind are: a. The inclinations of the six galaxies are $i({\\rm deg})=75\\pm 5,~30\\pm 5,~62\\pm 5,~25\\pm 12,~45\\pm 10,~34\\pm 5$. Three of them have low inclinations $i< 35$ degrees. Such low inclinations are generally considered problematic because it is difficult to measure such low inclinations accurately, and because the actual rotational speeds, and the acceleration deduced from them are sensitive to the exact value (the accelerations scaling as $1\/sin^2 i$).\nb. This is further compounded by the fact that\nthese RCs are not based on 2-D tilted-ring derivation as the standard has come to require, and hence do not account for possible variable position angle and inclination, especially problematic for low-inclination galaxies.\nc. Large random motions are present in these galaxies; so large (and uncertain) asymmetric-drift corrections have to be applied. d. The luminosity distribution is measured in the rest-frame optical-band, not as good for converting light to mass compared with far IR now used routinely for local galaxies. e. Some of the galaxies have a substantial bulge, and the necessary separation to components, with possibly different M\/L values, is problematic.\nf. Kinematics are measured from $H_\\a$ velocities, so are confined to the optical image with no analog of the extended HI RCs.\\footnote{In many of these regards the quality of these RCs may be likened to that of RCs available in the late 1970s for low-$z$ galaxies, before extended HI RCs became available.}\n\\par\nStill, these are the best RC data we have at present for such high redshift, and thus are valuable in constraining cosmological evolution of either DM scenarios, or, as here, $z$-dependence of MOND.\n\\par\nIn Sec. \\ref{MOND}, I give some MOND formulae needed here. Section \\ref{results}\ncompare the MOND predictions with the results of Ref. \\cite{genzel17},\nand Sec. \\ref{discussion} is a discussion.\n\n\\section{Relevant MOND formulae}\n\\label{MOND}\nIf at some radius, $R$, in the midplane of a disc galaxy $\\gN(R)$ is the Newtonian acceleration calculated from the baryon distribution, and $g$ is the dynamically determined acceleration, then MOND predicts the relation \\cite{milgrom83a}:\n\\beq g\\m(g\/\\az)=\\gN, \\eeqno{mdarmu}\nwhere $\\az$ is the MOND acceleration constant, and $\\m(x)$ the `interpolating function' for rotation curves.\nThis can be written equivalently in terms of the $\\n(y)$ interpolating function\nas\n\\beq g=\\gN\\n(\\gN\/\\az), \\eeqno{mdarnu}\nwhere $\\n(y)$ is related to $\\m(x)$  by $\\m(x)=1\/\\n(y)$, where $x=y\\n(y)$ [and so $y=x\\m(x)$].\nThese equivalent forms are known as the mass-discrepancy-acceleration relation (MDAR) since\n\\beq \\eta\\equiv g\/\\gN=1\/\\m(g\/\\az)=\\n(\\gN\/\\az) \\eeqno{mdar}\ncan be identified as the mass discrepancy.\n\\par\nAs convention goes, Ref. \\cite{genzel17} define the dark-matter fraction at $R$ -- better referred to in the present context as the `phantom-matter' fraction --as\n$\\zeta(R)=[V\\_{DM}(R)\/V(R)]^2$, or in terms of the accelerations\n\\beq \\zeta(R)\\equiv (g-\\gN)\/g. \\eeqno{frac}\nThus, MOND predicts\n\\beq \\zeta=1-\\eta^{-1}=1-\\m=1-1\/\\n.  \\eeqno{etamu}\n\nI will show the MOND predictions for $\\zeta$ for two forms of the MOND interpolating function used routinely for RC fits and fits to the MDAR.\nThe first is\n\\beq \\m(x)=\\frac{x}{1+x}, \\eeqno{i}\nwhich gives a `phantom-matter' fraction of $\\zeta=(1+x)^{-1}$.\nThe other takes a simple form in the $\\n(y)$ language \\cite{ms08,mcgaugh08,mls16,milgrom16}:\n\\beq \\n(y)=(1-e^{-\\sqrt{y}})^{-1},   \\eeqno{ii}\nwhich gives $\\zeta=e^{-\\sqrt{y}}$.\nThese two interpolating functions differ by at most $\\sim 5\\%$ over the full range of arguments and so predict almost indistinguishable rotation curves.\nHowever, in the region of high accelerations, where both functions are nearly 1, they differ substantially in their exact departure from 1. So, when the predicted\nfractions of `phantom matter' are small, the two functions can give different predictions for this small quantity.\n\\par\nAnother MOND prediction I will need: Given the total baryonic mass, $M_b$, MOND predicts \\cite{milgrom83a,milgrom83b} for an isolated galaxy, an asymptotically flat RC, with the constant rotation speed\n\\beq \\vinf^4=M_bG\\az.  \\eeqno{iii}\nThis is the MASR mentioned in Sec. \\ref{introduction}\n\\section{Results}\n\\begin{table*}[t]\n \\begin{tabular}{lccccccccl}\n \\hline\\hline\n\n\nGalaxy& $z$ & $\\raf$ & $V_c(\\raf)$ & $M_b$& $\\zef$& $\\vinf$ & $x\\_{1\/2}$& $\\zeta\\^M\\_{1\/2,a}$ & $\\zeta\\^M\\_{1\/2,b}$  \\\\\n& & $\\kpc$ & $\\kms$ & $10^{11}\\msun$ & & $\\kms$&&&\\\\\n\\hline\nCOS4 01351   &  0.854 & 7.3  & 276 & 1.7 & $0.21 (\\pm0.1$) & 228 & 2.8 &0.26& 0.22 \\\\\nD3a 6397 & 1.500&  7.4      & 310 & 2.3  & 0.17 ($<$0.38)\t& 246 & 3.5 &0.22& 0.18 \\\\\nGS4 43501  & 1.613 &  4.9  & 257 & 1.0 & $0.19 (\\pm0.09)$ & 200 & 3.6 & 0.22 & 0.17 \\\\\nzC 406690 &2.196\t&  5.5   & 301  &  1.7  & $0 (<0.08)$ & 228 & 4.4 & 0.18 & 0.14 \\\\\nzC 400569 & 2.242  &  3.3      & 364    \t & 1.7 & $0 (<0.07)$ & 228 & 10.8 & 0.08 & 0.04 \\\\\nD3a 15504 & 2.383  &  6      & 299    \t & 2.1 & $0.12 (<0.26)$ & 240 & 4.0 & 0.20& 0.16\\\\\n\\hline\n\\end{tabular}\n\\caption{Galaxy name \\{column 1\\}, its redshift \\{2\\}. Columns 3-6 are best-fit attributes deduced by Ref. \\cite{genzel17}: the half-light radius, $\\raf$, (in the rest-frame optical band) \\{3\\}; the rotational speed there (corrected for inclination and asymmetric drift) \\{4\\}; the total baryonic mass \\{5\\}; and the dark-matter fraction, $\\zef$ at $\\raf$, with errors or upper limits \\{6\\}. Column 7-10 show calculated MOND quantities: the predicted asymptotic rotational speed, $\\vinf$, based on $M_b$, from eq.(\\ref{iii}) \\{7\\}, The acceleration at $\\raf$ in units of $\\az$ \\{8\\}, the expected MOND value of $\\zef$ based on the interpolating function of eq. (\\ref{i}), $\\zeta\\^M\\_{1\/2,a}$ \\{9\\}, and that based on eq. (\\ref{ii}), $\\zeta\\^M\\_{1\/2,b}$ \\{10\\}, all calculated for the nearby-Universe value of $\\az=1.2\\times 10^{-8}\\cmss$.}\n\\label{table}\n\\end{table*}\n\\label{results}\nTable \\ref{table} shows the values of the relevant parameters as they appear in Table 1 of Ref. \\cite{genzel17}. I show in the table, and use, the relevant quantities given in Ref. \\cite{genzel17} as their best fit model parameters (resulting from fitting the rotation curves to mass models that include baryons and dark matter): the half light radius, $\\raf$, the dynamical rotational speed at $\\raf$, and the total baryonic mass, $M_b$. For $M_b$ and $\\raf$ they also give their pre-fit, directly estimated values. In most cases, the former values, which I use, lie within the error range of the latter. I also show their deduced values of the `phantom-matter' fractions, $\\zef$, at $\\raf$, and the values of $\\zef$ predicted by MOND for the two commonly used interpolating functions, all as detailed in Sec. \\ref{MOND}.\n\\par\nWe see that as found by Ref. \\cite{genzel17} the MOND $\\zef$ values are small -- a few tens of percents at most. Furthermore, except for the rogue zC 406690, where the upper limit is lower than my estimates, the MOND predictions are, case by case, in good agreement with what Ref. \\cite{genzel17} give.\\footnote{But beware that the $\\zef$ values of Ref. \\cite{genzel17} are based on model best-fits with NFW dark-matter distributions. Given their large uncertainties on $M_b$, their RCs are probably also consistent with sub-maximal discs, and rather larger values of $\\zef$. The distinction between maximal and sub-maximal discs is moot even with much better data.\\label{fnq}} And note that zC 406690 has a quoted inclination of $i=25\\pm 12$ degrees; so it's kinematic analysis is practically useless.\n\n\n\n\n\\subsection{Falling rotation curves}\nThe RCs shown by Ref. \\cite{genzel17} exhibit some decline beyond their maximum. Such decline is also typical of high-surface-brightness galaxies in the local Universe (see, e.g. some early-type galaxies in the sample of Ref. \\cite{sn07}, in particular, their RC for UGC 4458, which drops from $\\sim 500\\kms$ to $\\sim 300\\kms$ within $10 \\kpc$ and then becomes flat at $\\sim 250\\kms$ to $55\\kpc$).\n\\par\nSuch a decline seems to be more prevalent in the high-$z$ samples at hand (see also Ref. \\cite{lang17}). Part of the reason, as extensively discussed by Refs. \\cite{genzel17,lang17}, is that rather more than in low-$z$ galaxies, velocity dispersions in the disc contribute substantially to the balance against gravity, hence diminishing the role of rotational support. It is notoriously difficult and uncertain to correct for this important effect. Indeed, in the stacked RCs of Ref. \\cite{lang17} (see their Fig. 8), galaxies with\nhigh rotation-to-dispersion ratio show much less marked decline than those with small values.\n\\par\nOne should also consider the effects of selection: High-surface-brightness galaxies -- where such declines are also observed at low-$z$ -- are naturally more amenable to measurements at high redshift, and are easier to follow to larger radii. Indeed, Fig. 5 of Ref. \\cite{lang17} shows that the number of galaxies contributing at the outer radii, where the decline is evident, is much smaller than the total in the sample: $\\sim 12$ galaxies that contribute down to the outer stacked-data point, compared with $\\sim 90$ that contribute at low radii. These may well be selecting preferentially higher-surface-brightness galaxies.\n\\par\nIn MOND, we do expect marked decline beyond the maximum in galaxies with mean accelerations that are so high compared with $\\az$. For example, MOND-predicted rotation curves of such model galaxies are shown in Figs. 1 and 2\nof Ref. \\cite{milgrom83b} (the models with high $\\xi\\sim 5$ there). And see also Fig. 2 of Ref. \\cite{ms03} for the predicted MOND RC of the elliptical NGC 3379, which drops from $\\sim 300\\kms$ at maximum to $\\sim 200 \\kms$.\n\\par\nWe can estimate the room for a drop in the velocity allowed by MOND for the six galaxies under study, by comparing the observed maximum speed with the predicted asymptotic rotational speed, $\\vinf$, which can be deduced from the estimates of the baryonic masses, using eq. (\\ref{iii}) -- assuming that the galaxy is isolated. These estimates are given in Table \\ref{table} based on the best-fit values that Ref. \\cite{genzel17} give for $M_b$. Note that the direct estimates of $M_b$ given by Ref. \\cite{genzel17} have large quoted errors given in all cases as $\\pm 50\\%$ (i.e., a factor of $\\sim 3$ in range), corresponding to a relative error of $+0.1~-0.15$ in $\\vinf$. From Fig. 2 of Ref. \\cite{genzel17} one sees that the maximum speed for the galaxies is about\n$1.1V(\\raf)$, and occurs at $\\sim 1.5 \\raf$. We see then that the estimated ratio $\\vinf\/V_{max}$ is as low as $\\sim 0.55~ (\\pm 0.1)$ (for one of the 6 galaxies, zC 400569), and is $\\sim 0.7 (\\pm 0.1)$ for most others. This would allow the drops Ref. \\cite{genzel17} estimate (these are subject to substantial uncertainties due to the uncertain asymmetric-drift correction, and possible unaccounted for warps). For one galaxy, zC 406690, Ref. \\cite{genzel17} estimate a very large drop. But, as I pointed out above, this is quite unreliable as the stated inclination for this galaxy is $i=25\\pm 12$ degrees.\n\\par\nIn MOND, the presence of neighboring bodies can also contribute to the decline of the RCs through the external-field effect (e.g., Refs. \\cite{milgrom83a,wu15,haghi16,mcgaugh16}. According to Ref. \\cite{genzel17} their 6 galaxies are relatively isolated, so this should not be a factor, but it is hard to asses the importance of the effect in statistical studies such as that of Ref. \\cite{lang17}.\n\n\n\\section{Discussion}\n\\label{discussion}\nThe results of Ref. \\cite{genzel17} are well accounted for by MOND in the very form that has been applied successfully to low-$z$ galaxies, with the canonical value of $\\az$.\n\\par\nAlthough these RCs do not probe the deep MOND regime -- where MOND enters in full glory -- they do vindicate an important prediction of MOND that does not arise naturally in the dark-matter paradigm. Namely, that mass anomalies should be small (sub-dominance of `phantom matter') at accelerations above $\\az$.\nThat this is now seen to be the case also at high $z$ even sharpens the case for MOND: It shows this prediction to be independent of the evolutionary status of the galaxies, strengthening the case for a law of nature as the origin, rather than some complicated and contrived evolutionary processes.\n\\par\nIt appears that these results cannot accommodate much higher values of $\\az$ at high redshift. Looking at Table \\ref{table}, we see that, for example, a value of the MOND acceleration of $4\\az$ would have resulted in $x\\_{1\/2}$ values for the higher-$z$ galaxies of order 1. This would have predicted $\\zef$ values of order 0.5, which would be uncomfortably in tension with the values estimated by Ref. \\cite{genzel17}\\footnote{Values of the MOND constant smaller than $\\az$ cannot be excluded, but they are anyhow less motivated.} (But, remember footnote \\ref{fnq}.) This constraint makes use, essentially of the role of $\\az$ in MOND as `boundary acceleration'. Another, independent constraint is based on the role of $\\az$ as setting the MASR normalization: With a value of the MOND constant as high as $4\\az$ the predicted values of $\\vinf$ in  Table \\ref{table} should be increased by a factor of $4^{1\/4}\\sim 1.4$, making $\\vinf\/V_{max}\\sim 1$, not leaving room for decline beyond the maximum, unless the baryonic masses are substantially lower.\n\\par\nIdeally, we could test for variations of $\\az$ by searching for evolution in the proportionality constant of the MASR, eq. (\\ref{iii}). But this is not possible with the present data, as clearly they do not reach the asymptotic speeds, as required by the MOND MASR.  `Evolution' of the zero point of some versions of the BTFR, using available velocity measures such as the maximum speed have been studied. But these are not what the MOND MASR dictates, and cannot be used to constrain cosmological variations of the MOND constant.\nIt is an opportunity to stress again the distinction between various versions of the BTFR, and the specific version MOND predicts as the MASR, which employs the asymptotic speed.\n\\par\nThis result may help constrain ideas that rest on the MOND constant varying with cosmic time, such as the suggestion that the first of the near equalities in eq. (\\ref{coinc}) held at all times, or other possible variations (see discussion in Ref. \\cite{milgrom09} and references therein). Ref. \\cite{milgrom99} offers a possible causal connection between $\\az$ and $\\Lambda$.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction} \nBecause of the dominance of lines of atomic iron in the spectra of cool stars, the iron abundance is often used as a proxy for total metal content, or metallicity. Neutral and singly ionised iron with different properties are also frequently used for determination of spectroscopic stellar parameters. This makes [Fe\/H] arguably the most important abundance indicator when studying the evolution of stars and galaxies. The iron abundance of the Sun itself is important not only as an anchor for the cosmic [Fe\/H]-scale, but it also influences the structure and evolution of stars because it is a large opacity contributor \\citep[][and references therein]{Bailey15}. \n\nIt has long been known that the ionisation balance of FeI--FeII departs from local thermodynamic equilibrium (LTE) in the photospheres of late-type stars \\citep[e.g.][]{Athay72,Rutten84,Thevenin99, Korn03}. However, non-LTE (hereafter NLTE) calculations of neutral iron have suffered from large systematic uncertainties due to poorly constrained atomic data, in particular the efficiency of collisions with electrons and neutral hydrogen \\citep[e.g.][]{Mashonkina11b}. Further, the large complexity of the atomic structure of iron has prevented consistent NLTE calculations to be performed with realistic 3D radiation-hydrodynamical simulations of solar and stellar photospheres. \n\nIn Paper I and II of this publication series \\citep{Bergemann12,Lind12a}, we presented a model atom for iron with the efficiency of H collisions calibrated using high-quality spectroscopic observations of well-studied benchmark stars, including the Sun. We employed both standard 1D atmospheric models and so called average-3D models (hereafter $\\rm\\langle3D\\rangle$), which are spatial and temporal averages of full 3D radiation-hydrodynamical simulations of stellar atmospheres. In Paper III \\citep{Amarsi16b} we presented a new model atom including quantum mechanical calculations of hydrogen collisions (Barklem, in prep.) and demonstrated its performance for metal-poor benchmark stars using full 3D NLTE calculations. Here we further improve the atom and confront our 3D NLTE predictions with the observed centre-to-limb variation of iron lines in Sun. \n\n\\citet{Nordlund84,Nordlund85} pioneered the investigation of NLTE line formation of iron in 3D hydrodynamical model atmospheres more than three decades ago. The first paper studied the departure of FeI--FeII from Saha ionisation balance and reported significant $\\rm(0.2\\,dex)$ over-ionisation of the neutral species. The second paper used a two-level FeI atom, coupled to a FeII continuum, and predicted significant line weakening of the example FeI line at 5225\\AA\\  due to a superthermal source function. \n\n\\citet{Shchukina01} later studied NLTE line formation in a hydrodynamical model of the Sun in the so called 1.5D approximation, neglecting horizontal radiative transfer. They used a 248 level FeI+FeII atom and concluded that NLTE effects vary strongly with the granulation pattern and the FeI line properties, with a net NLTE correction to FeI line abundances of up to $+0.12$\\,dex for the lowest-excitation lines. The only previous work investigating NLTE line formation of iron in the Sun using a multi-level atom and full 3D radiative transfer is the series by \\citet{Holzreuter12,Holzreuter13,Holzreuter15}, in which the authors rigorously compare synthetic line profiles generated under different assumptions. However, they were limited to using a strongly simplified 23-level atom and made no quantitative comparison to observations. These earlier studies have in common that they included only experimentally known energy levels of iron and neglected the influence of hydrogen collisions on the statistical equilibrium, both of which exaggerate the NLTE effects.  \n\nWe present full 3D NLTE calculations using a comprehensive 463 level atom with realistic atomic data to enable a direct comparison to the most constraining observations possible, i.e. high-spectral resolution and high signal-to-noise (S\/N) observations of the Sun at different viewing angles. The paper is divided in the following sections: Sect.\\,\\ref{sect:method} outlines the observations, the assembly and reduction of the model atom, and the method used for spectral synthesis.  Sect.\\,\\ref{sect:results} presents the results for the solar centre-to-limb variation of iron lines and the solar iron abundance. Sect.\\,\\ref{sect:conc} summarises our conclusions.  \n\n\\section{Method}\n\\label{sect:method}\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.43,viewport=3cm 1cm 21cm 21cm]{Positions}\n\\caption[]{Overview of the SST pointings on the solar disk, inclined by the heliographic latitude of the observer.  The blue circles mark the targeted $\\mu$-angles and $\\mu=0.999$ for reference.} \n\\label{fig:pos} \n\\end{center} \n\\end{figure}\n\n\n\\subsection{Observations}  \nWe acquired spectroscopic data with high spatial and spectral resolution using the TRIPPEL \\citep{Kiselman11} instrument at the Swedish 1-m Solar Telescope \\citep[SST,][]{Scharmer03} on La Palma. The observing campaign lasted from 23 June to 8 July, 2011. \n\nThree spectrographic cameras and three imaging cameras were operated simultaneously.  Three  different setups were used, resulting in a total of nine spectral windows with wavelength bands specified in Table \\ref{tab:bands}. Two slit-jaw cameras recorded simultaneous images at approximately 5320\\AA\\, and 6940\\AA\\,, respectively. The third camera was used to monitor the magnetic activity of the region with a 1.1\\AA\\ filter centered on the Ca\\,II H line. Five different heliocentric angles on the solar disk were targeted, corresponding to $\\mu\\equiv\\cos{\\theta}=0.2, 0.4, 0.6, 0.8$ and $1.0$, where $\\theta$ is the angle between the ray direction and the surface normal. The number of observations at each pointing is listed in Table \\ref{tab:bands}, discarding exposures that failed due to suspected tracking problems (usually due to very bad seeing), or regions with obvious activity as deemed from the Ca\\,II~H core emission.\n\nThe intensity contrast peaks at disk centre and exposures were made while scanning the spectroscopic slit over a small region in order to reduce the imprint of the local granulation pattern. At other pointings, the slit position was held fixed and aligned parallel to the closest part of the solar limb. The telescope field rotation caused the actual position selected in this way to depend on the time of day, as is evident in Fig.~\\ref{fig:pos}. Of the two possible choices for a specific $\\mu$ value and time of day, the one showing the least activity was preferred. \n\nFor the $\\mu=0.2$ pointings, the position of the slit could be measured accurately using the slit-jaw images, which include the solar limb. For the other pointings, $\\mu$ was determined from the output of the telescope tracking system. In order to get the readings as accurate as possible, frequent calibrations by pointing at the limb at four position angles to find the solar centre were made. Somewhat conservatively, we estimate the accuracy in the the nominal values to be $\\pm 15\\hbox{$^{\\prime\\prime}$}$.\n\nAs evident from Fig. \\ref{fig:pos} and Table \\ref{tab:bands}, the pointing accuracy as expressed in $\\mu$ degrades with decreasing $\\mu$ value, i.e. from the centre towards the limb. The $\\mu=0.2$ pointing deviates from this trend since it was determined from the slit-jaw images and the largest uncertainties thus affect the $\\mu=0.4$ observations. In total we kept 147 pointings, 4-20 for each $\\mu$-value and configuration, as detailed in Table \\ref{tab:bands}. For each $\\mu$ pointing and wavelength, Table~\\ref{tab:bands} gives the mean value of $\\mu$ and its error estimate calculated from the standard deviation of the nominal position readings combined with a systematic error. For $\\mu\\ge0.4$ the systematic component was computed using the $15\\hbox{$^{\\prime\\prime}$}$ calibration error and for $\\mu=0.2$, we used the approximate spatial extent of the slit in the same way as \\citet{Pereira09a}. \n\nThe data were reduced using the same method and software as described in \\citet{Pereira09a}. First, the data were corrected for dark current and flat fielded using calibration exposures taken in close connection to the observations. \nGeometrical distortions in the spectrograms were then removed using polynomial fits to the location of selected spectral lines (for smile) and with the help of a grid place across the slit (for keystone). Wavelength calibration was made by cross-correlation of spectral lines with the disk-centre FTS atlas of Brault \\& Noyes (1987)\\footnote{\\url{ftp:\/\/ftp.hs.uni- hamburg.de\/pub\/outgoing\/FTS- Atlas}}. The same atlas was used to model the internal straylight of the spectrograph (assumed constant over each spectrogram) as well as systematic spectral artefacts that the flat-fielding cannot correct for. The result of the reduction procedure is to force mean spectra from the quiet disk centre to be as close as possible to the reference atlas spectrum. The same corrections are then applied to all spectra.\n\nIn this paper, we analyse the centre-to-limb behaviour of iron lines and create a single 1D spectrum for each pointing by coadding the individual spectrograms and then forming an average along the slit direction. This increases the S\/N and, following \\citet{Pereira09b}, removes the need for Fourier filtering of photon noise that was applied by \\citet{Pereira09a}. The S\/N per pixel of the average spectra ranges between 1000--4000 at a spectral resolution of $\\lambda\/\\delta\\lambda\\approx150,000$. The S\/N ratio was estimated from the median standard deviation of all measurements at a given wavelength.\n\n\\begin{table*}\n      \\caption{Summary of the observational configuration. Columns A-C give the wavelength band of each of the three spectrographic cameras. $\\#$ represents the number of pointings.}\n         \\label{tab:bands}\n         \\centering\n         \\begin{tabular}{llllllllllllll}\n                \\hline\\hline\n                 Set & A & B & C &  $\\#$ & $\\mu$ &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$  &  $\\#$ & $\\mu$ \\\\\n                 & [\\AA\\ ]& [\\AA\\ ]& [\\AA\\ ]   \\\\\n                                \\hline\n\t      1\t& 5366-5377 &  6147-6159 & 8710-8728 & 6 \t& 0.201 \t\t& 4 \t\t& 0.380\t      \t& 7 \t& 0.600     \t& 7 \t&  0.802 \t\t& 4 \t\t& 1.0000 \t\t\t\\\\%\t      1 &  537 &  615 & 873 \\\\\n\t         &     \t\t      &                       &                      &\t& $\\pm0.007$ \t&   \t\t& $\\pm0.032$\t&  \t& $\\pm0.014$\t&  \t& $\\pm0.005$ \t&  \t\t& $\\pm0.0005$\t\t\\\\\n\t      2 & 5378-5390 &  6159-6172 & 8727-8744 & 7 \t& 0.205\t\t& 7 \t\t& 0.393\t  \t& 7 \t& 0.604\t   \t& 6 \t&  0.803\t \t& 12 \t\t& 1.0000\t \t\t\\\\%\t      2 &  538 &  616 & 874 \\\\\n\t         &     \t\t      &                       &                      &\t& $\\pm0.005$ \t&   \t\t& $\\pm0.019$\t&  \t& $\\pm0.009$\t&  \t& $\\pm0.005$ \t&  \t\t& $\\pm0.0003$\t\t\\\\\n\t      3 & 8656-8668 &  7825-7842 & 8691-8708 & 16\t& 0.203\t \t& 16\t\t& 0.397\t\t& 19 & 0.603\t  \t& 20 \t&  0.801\t \t& 9 \t\t& 1.0000\t \t\t\\\\\n\t      \t &     \t\t      &                       &                      &\t& $\\pm0.006$ \t&   \t\t& $\\pm0.027$\t&  \t& $\\pm0.006$\t&  \t& $\\pm0.004$ \t&  \t\t& $\\pm0.0005$\t\t\\\\\t      \t      \n                   \\hline \n         \\end{tabular}\n\\end{table*} \n\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.33,viewport=2cm 1cm 25cm 21cm]{LabK14}\n\\includegraphics[scale=0.33,viewport=2cm 0cm 25cm 21cm]{LabK13}\n\\caption[]{Comparison between theoretically predicted \\citep{K13,K14} and experimentally measured (see text for references) oscillator strengths of FeI and FeII. The experimental uncertainties are plotted as error bars. For FeII, a representative experimental uncertainly was set to $0.05$\\,dex.} \n\\label{fig:fvalues} \n\\end{center} \n\\end{figure}\n\n\n\\subsection{Atomic data} \n\\label{sect:atomdata}\n\nThe non-LTE calculations are performed by iterative solutions of the radiative transfer and statistical equilibrium equations, until the level populations have converged at all points in the atmosphere. The statistical equilibrium solution requires knowledge of the relevant radiative and collisional transition probabilities, which are collected in a model atom. The literature sources and databases used to assemble the model atom were listed in \\cite{Amarsi16b}. In this section we reiterate the main points and present more detail. \n\n\\begin{table*}\n      \\caption{Atomic data for the iron lines used for the centre-to-limb analysis. The $W_\\lambda$ columns list the equivalent widths measured for the five different $\\mu$-angles using direct integration over the wavelength range specified by $\\lambda_{\\rm int}$.}\n         \\label{tab:lines}\n         \\centering\n         \\begin{tabular}{lllllllllllllc}\n                \\hline\\hline\n\t\tIon & $\\lambda_{\\rm air}$ & $E_{\\rm low}$ & $\\log(gf)$ & $\\log(\\gamma)$ & $\\sigma^{(b)}$& $\\alpha^{(b)}$& $C_4^{(c)}$ & \\multicolumn{5}{c}{$W_{\\lambda}$ [m\\AA ]}   &$\\lambda_{\\rm int.}$\\\\\n                          & [\\AA\\ ]          & [eV]                   &                   &   Rad.\\,$^{(a)}$   &                   &                          &                        &    $\\mu=1.0$  &    $\\mu=0.8$  &    $\\mu=0.6$  &    $\\mu=0.4$  &    $\\mu=0.2$  &  [\\AA ] \\\\\n                                \\hline\nFeI & 5367.4659  \t& 4.415\t&   0.443$^{(d)}$   \t& 8.32 \t&    972 & 0.280\t&  -13.11 &   168.5  &   166.3   &  164.4  &   158.7   &  143.4   & $5367.10-5368.10$ \\\\\n&      &            \t\t&     \t\t&                \t\t\t& \t\t&    \t&         &  $\\pm1.3$  & $\\pm1.3$  & $\\pm1.4$  & $\\pm1.2$  & $\\pm1.5$ \\\\\nFeI & 5373.7086  & 4.473&  -0.710$^{(e)}$   & 8.13 &   1044 & 0.282  &  -13.76 &    59.4  &    58.4   &   58.5  &    58.5   &   54.6   & $5373.62-5373.82$ \\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.4$  & $\\pm0.6$  & $\\pm0.6$  & $\\pm0.7$ \\\\\nFeI & 5379.5736  & 3.695&  -1.514$^{(d)}$   & 7.85 &    363 & 0.249  &  -15.51 &    62.8  &    63.1   &   64.2  &    66.4   &   66.1   & $5379.20-5379.72$\\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.7$  & $\\pm0.6$  & $\\pm0.9$  & $\\pm0.8$ \\\\\nFeI & 5383.3685  & 4.313&   0.645$^{(d)}$   & 8.30 &    836& 0.278  &  -13.83 &   219.3  &   217.7   &  214.9  &   208.3   &  188.4   & $5382.70-5384.00$\\\\\n &   &           &     &                & &            &         &  $\\pm1.8$  & $\\pm1.9$  & $\\pm1.6$  & $\\pm1.4$  & $\\pm1.6$ \\\\\nFeI & 5386.3331  & 4.154&  -1.670$^{(f)}$   & 8.45 &    930& 0.278  &  -13.02 &    31.0  &    31.9   &   33.1  &    34.7   &   34.8   & $5386.10-5386.45$\\\\\n &   &           &     &                & &            &         &  $\\pm0.3$  & $\\pm0.4$  & $\\pm0.3$  & $\\pm0.5$  & $\\pm0.4$ \\\\\nFeI & 5389.4788  & 4.415&  -0.418$^{(g)}$   & 8.32 &    959& 0.280  &  -13.53 &    87.9  &    86.8   &   86.2  &    85.4   &   80.4   & $5389.30-5389.65$\\\\\n &   &           &     &                & &            &         &  $\\pm0.6$  & $\\pm0.6$  & $\\pm0.7$  & $\\pm0.9$  & $\\pm0.7$ \\\\\nFeII& 6149.2459  & 3.889&  -2.840$^{(h)}$   & 8.50 &    186& 0.269  &  -16.11 &    38.3  &    38.4   &   37.2  &    36.7   &   31.4   & $6149.05-6149.40$\\\\\n &   &           &     &                & &            &         &  $\\pm0.3$  & $\\pm0.5$  & $\\pm0.6$  & $\\pm0.6$  & $\\pm0.5$ \\\\\nFeI & 6151.6173  & 2.176&  -3.299$^{(i)}$   & 8.29 &    277& 0.263  &  -15.55 &    49.0  &    49.6   &   52.2  &    55.4   &   55.8   &$6151.30-6151.85$ \\\\\n &   &           &     &                & &            &         &  $\\pm0.8$  & $\\pm0.7$  & $\\pm0.5$  & $\\pm0.7$  & $\\pm1.6$ \\\\\nFeI & 6157.7279  & 4.076&  -1.160$^{(f)}$   & 7.89 &    375& 0.255  &  -15.36 &    62.6  &    61.6   &   62.4  &    63.2   &   60.2   & $6157.50-6157.85$\\\\\n &   &           &     &                & &            &         &  $\\pm0.7$  & $\\pm0.6$  & $\\pm0.7$  & $\\pm0.9$  & $\\pm1.2$ \\\\\nFeI & 6165.3598  & 4.143&  -1.473$^{(d)}$   & 8.00 &    380& 0.250  &  -15.34 &    44.5  &    44.8   &   45.2  &    45.9   &   44.6   & $6165.25-6165.55$\\\\\n &   &           &     &                & &            &         &  $\\pm0.4$  & $\\pm0.5$  & $\\pm0.4$  & $\\pm0.6$  & $\\pm0.6$ \\\\\nFeI & 8699.4540  & 4.956&  -0.370$^{(e)}$   & 8.74 &    817& 0.272  &  -14.59 &    73.7  &    72.4   &   70.7  &    68.0   &   61.9   & $8699.20-8699.85$\\\\\n &   &           &     &                & &            &         &  $\\pm0.8$  & $\\pm0.9$  & $\\pm1.0$  & $\\pm1.0$  & $\\pm0.9$ \\\\\n              \\hline\n              \\multicolumn{14}{l}{$^{(a)}$ Radiative broadening is given by the logarithm (base 10) of the FWHM given in $\\rm rad\\,s^{-1}$.} \\\\\n              \\multicolumn{14}{l}{$^{(b)}$ \\citet{Anstee95} notation for the broadening cross-section ($\\sigma$) for collisions by H\\,I at 10\\,$\\rm km\\,s^{-1}$ and its velocity dependence ($\\alpha$).} \\\\\n             \\multicolumn{14}{l}{$^{(c)}$ Stark broadening constant.} \\\\\n               \\multicolumn{14}{l}{$^{(d)}$ \\citet{BWL}, $^{(e)}$ \\citet{2014MNRAS.441.3127R}, $^{(f)}$ \\citet{MRW}, $^{(g)}$ \\citet{FMW}, $^{(h)}$ \\citet{RU}, $^{(i)}$ \\citet{GESB82c}.} \\\\\n         \\end{tabular}\n\\end{table*} \n\n\\subsubsection{Energy levels}\n\nEnergy levels were downloaded from Robert Kurucz's online database, updated in 2013 for FeII \\footnote{\\url{http:\/\/kurucz.harvard.edu\/atoms\/2601}} and 2014 for FeI \\footnote{\\url{http:\/\/kurucz.harvard.edu\/atoms\/2600}}. These data are referenced in the VALD3 \\citep{Ryabchikova15} data base as \\citet[][\"K13\"]{K13} and \\citet[][\"K14\"]{K14}, respectively, and include both observed and theoretically predicted energy levels. The importance of the inclusion of predicted energy levels was demonstrated by \\citet{Mashonkina11b}. There are 2,980 energy levels of FeI below the first ionisation potential ($63,737\\,\\rm cm^{-1}=7.902\\,eV$), approximately two thirds of which have not been observed. For FeII we consider the 116 energy levels below 60,000\\,$\\rm cm^{-1}$, all of them observed, as more highly excited levels  are not relevant in late-type stellar atmospheres (the second ionisation potential of Fe is $130,655\\,\\rm cm^{-1}=16.199\\,eV$). \n\nWe have homogenised the nomenclature of electron configurations and terms to enable energy levels to be merged. Energies with terms given in jj-coupling notation, e.g. \"2+[1+]\" have instead been designated by the leading eigenvector's term in LS-coupling notation, e.g. \"7F\".  This convention is used in the creation of the term diagrams shown in Fig.\\,\\ref{fig:Fe804} and Fig.\\,\\ref{fig:Fe463}.\n\n\\subsubsection{Transition probabilities}\n\nThe \\citet{K13,K14} database contains 533,772 radiative transitions between bound levels of FeI and 1174 between the bound levels we consider for FeII. We have cross-referenced these data with laboratory measurements of transition probabilities carried out since the late 1970's and identified 2080 matches  (0.4\\% of all lines) for FeI and 115 matches for FeII (10\\% of all lines). The references used for FeI are \\citet{BIPS,GESB79b,GESB82c,GESB82d,GESB86,BKK,BK,BWL,2014ApJS..215...23D,2014MNRAS.441.3127R} and for FeII we use the re-normalised compilation by \\citet{Melendez10}. The source with smallest quoted uncertainty was adopted for lines with multiple sources. \n\nFig.\\,\\ref{fig:fvalues} compares theoretical and experimental values of $\\log(gf)$ for both ionisation states. We find that the agreement is typically better for strong transitions; for $\\log(gf)_{\\rm K14}>-2$, theoretical values for FeI show a bias and scatter with respect to experiment of $0.08\\pm0.29$\\,dex, which increases in magnitude and changes sign to $-0.40\\pm0.79$\\,dex at $\\log(gf)_{\\rm K14}<-2$. For weak lines, there appears to be a correlation with the energy of the upper level involved in the transitions, such that the disagreement is very strong for lines with highly excited upper energy levels, while the least excited are in as good agreement with theory as stronger lines. For FeII lines, we find a bias of $-0.11\\pm0.24$\\,dex. The comparison suggests that the use of theoretical data for diagnostic lines should be avoided for precision spectroscopy. However, sensitivity tests that we carried out indicate that Fe NLTE level populations in the Sun are not sensitive to uncertainties in oscillator strengths of this magnitude. All lines selected for abundance analysis in Sect. \\ref{sect:feabund} have laboratory measurements of $\\log(gf)$. \n\n\n\\subsubsection{Photo-ionisation cross-sections}\n\nWe computed total and partial (state-to-state) photoionisation cross-sections for Fe I with the R-matrix method for atomic scattering as implemented in the RMATRX package \\citep{Berrington95}.  These calculations employed close coupling expansion of 157 states of the Fe II target ion from 35 configurations made by atomic orbitals up to principal quantum number $n=6$. The atomic dataset includes cross-sections for 936 LS terms of Fe I with $n\\le 10$ and $l\\le 7$. Details of this calculation will be presented elsewhere (Bautista \\& Lind 2016, in preparation). This calculation is considerably larger and more accurate than our previous computations of atomic data in \\citet{Bautista97}. We use the total, not partial, photoionisation cross-sections in our model atom to limit the number of bound-free transitions.  Each FeI level is thus bound to a single FeII level, as shown in Fig.\\,\\ref{fig:Fe463}.\n\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.67,viewport=2cm 10.5cm 26cm 20.0cm]{Fe804}\n\\caption[]{The complete Fe model atom without fine structure. FeI levels are shown below the dashed line, which indicates the first ionisation potential, and the associated terms are listed at the bottom x-axis. The FeII levels considered in this work are shown above the dashed line and the associated terms are listed at the top axis. Even parity terms are displayed in red and odd parity terms in blue. The left-hand panel shows all radiative bound-bound transitions and the right-hand panel shows all bound-free transitions.} \n\\label{fig:Fe804} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.67,viewport=2cm 10.5cm 26cm 20.0cm]{Fe463}\n\\caption[]{Same as in Fig.\\,\\ref{fig:Fe804}, but for the reduced model atom used for 3D NLTE calculations. Merged levels are indicated with longer horisontal lines.} \n\\label{fig:Fe463} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.60]{clvprof}\n\\caption[]{Normalised observed (bullets) and synthetic (red lines) centre-to limb profiles for two iron lines, where the numbers below each spectrum correspond to the approximate $\\mu$-angle. The synthetic line profiles have been computed in 3D NLTE and the iron abundance has been calibrated for each line to match the disk centre intensity. The calibrated abundance used for synthesis is indicated at the bottom of each panel. Both observed and synthetic spectra have been radial-velocity corrected so that the line centres coincide with the rest wavelength. Spectra for $\\mu\\ge0.4$ have been incrementally offset vertically by $+0.2$. } \n\\label{fig:clvprof} \n\\end{center} \n\\end{figure*}\n\n\\subsubsection{Electron collisions}\nWe adopt the results of \\citet{Zhang95}, who used the R-matrix method to compute collision rates between electrons and 18 low-excitation states of singly ionised Fe. When not available for bound-bound and bound-free electron impact collisions, we follow the semi-empirical recipes given by \\citet{Allen00} for FeI and FeII, which are originally from \\citet{vanRegemorter62} and \\citet{Bely70}. The same formula was used for optically allowed and forbidden transitions, assuming $f=0.005$ for the latter, which gives the two types of transitions similar efficiencies. A comparison between rate coefficients computed by van Regmorter and \\citet{Zhang95} for bound-bound FeII transitions gives a root mean square deviation of $0.6$\\,dex in the temperature interval $3,000-10,000$\\,K. For bound-free transitions, \\citet{Allen00} mentions a probable uncertainty of 0.3\\,dex. We note that more recent collisional data for FeII now exist and should be used for NLTE calculations \\citep{Bautista15}. For the Sun, NLTE effects on FeII lines are insignificant, so the new data would not influence our results. \n \n\\subsubsection{Hydrogen collisions} \nCollision rates for excitation processes, Fe($\\alpha ^{2S+1}L$) + H($1s$) $\\rightarrow$ Fe($\\alpha'^{2S'+1}L'$) + H($1s$), and charge transfer processes, Fe($\\alpha^{2S+1}L$) + H($1s$) $\\rightarrow$ Fe$^+$($\\alpha'^{2S'+1}L'$) + H$^-$, due to low-energy hydrogen atom collisions on neutral iron have been calculated with the asymptotic two-electron method presented by \\citet{Barklem16}. The calculation used here includes 138 states of FeI, and 11 cores of FeII, leading to the consideration of 17 symmetries of the FeH molecule.  These data will be the subject of a future publication (Barklem, in prep.).\n\nFor transitions with no data available, we approximated values using robust fits to the behaviour of the (logarithmic) quantum mechanical rate coefficients with transition energy at a given temperature. Linear fits were used for de-excitation rates and second order polynomials were used for charge exchange rates. The dispersion around the fits are approximately 1.2\\,dex in the temperature interval $3,000-10,000$\\,K. A more elaborate discussion about the appropriate functional forms of such fits is given by Ezzeddine et al. (submitted).\n\n\\subsection{Atom reduction}\n\\label{sect:atomred}\n\nIn its complete form, our Fe model atom contains more than 3,000 fine-structure energy levels, coupled by half a million radiative transitions. To establish the statistical equilibrium using this atom would mean having to solve the radiative transfer equation for at least hundreds of thousands of frequency points, which is not feasible in 3D. The atom must therefore be simplified, while preserving the overall NLTE behaviour.\n\nThe traditional method used to reduce the size of complex model atoms is to merge close energy levels, implicitly assuming that the levels have the same departure coefficients. The degeneracies of the levels that are merged are used as weights in the calculation of the mean energy and the radiative transition probabilities corresponding to the merged level, in such a way that the sum of $gf$ is preserved. We start by following this approach for the collapse of the fine-structure levels, resulting in 762 bound levels of FeI, 41 levels of FeII, and the FeIII ground state. These levels are coupled by 92,567 transitions between bound states of FeI and 226 transitions between bound states of FeII. All FeI levels are coupled to a core FeII state and the photoionisation cross-sections are tabulated over 1,000-2,000 frequency points each. This model will be used as reference model atom and its term diagram is illustrated in Fig.\\,\\ref{fig:Fe804}. The simplification process has so far preserved level configuration, term, and parity for all levels.\n\nWe thereafter proceed to test how much further the atomic level structure can be simplified without causing a significant change in the departure coefficients. We use the $\\rm\\langle3D\\rangle$ structure of the Sun as the default test model in this section, adopting a depth-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$. Above a certain energy limit, FeI energy levels are now merged that share the same multiplicity, parity, and configuration. We gradually decrease this energy limit, while monitoring the difference in equivalent width with respect to the reference model atom, for lines between $200$\\,nm and $2\\rm\\mu m$. The number of levels were thereby reduced from 804 to 463. \n\nIf all radiative transitions were kept, the 463 level atom would still contain approximately 37,000 transitions. However, many transitions do not contribute significantly to make the level populations depart from LTE. To reduce the number of transitions and enable full 3D calculations, we first computed the net radiative imbalance for each transitions in the $\\rm\\langle3D\\rangle$ model of the Sun, assuming LTE populations, i.e., $\\Delta_{ij}=|n_{i}R_{ij}-n_jR_{ji}|$. We then selected a point in the atmosphere ($\\tau_{\\rm500\\,nm}\\approx0.01$), where the NLTE effects are noticeable and relevant for line formation, and removed radiative transitions with a relatively small value of $\\Delta_{ij}$. Thereby, only 3,000 bound-bound transitions and 100 bound-free transitions were kept. We note that the choice of reference depth does not strongly influence which transitions are discarded. Finally, the wavelength grids of the photo-ionisation cross-sections were down-sampled heavily, to a factor 30 fewer points. The final reduced atom contains approximately 17,000 frequency points and preserves $\\rm\\langle3D\\rangle$ equivalent widths for the Sun within 0.01\\,dex, compared to the reference atom. We note that, within these small uncertainties, the smaller atom gives slightly less efficient over-ionisation of FeI than the larger atom, but that further merging of energy levels would have the opposite effect because the collisional coupling between FeI to the FeII reservoir is reduced.      \n\n\\subsection{Spectral synthesis}\n\\label{sect:spec}\nThe restricted NLTE problem, which neglects feedback effects on the atmospheric temperature and density structure, is solved using the 3D radiative transfer code \\textsc{Multi3D}, developed by \\citet{Botnen97} and \\citet{Leenaarts09}. \\citet{Amarsi16a} and Paper III describe a range of improvements recently made to the code, most importantly a new equation-of-state and background opacity package, frequency parallelisation, and improved numerical precision.  We use the same version of the code and same settings here as described in Paper III, except that we use a finer angle quadrature for the radiative transfer solution while the system converges. The Carlson A4 quadrature has 24 angles in total, four azimuthal and six inclined to the normal direction \\citep{Alder63}. After the level populations have converged, the final spectrum is computed at $\\mu=0.2, 0.4, 0.6, 0.8$ and 1.0, in four azimuthal directions. \n\n\\textsc{Multi3D} calculations were performed on three atmospheric snapshots drawn from the most recent 3D radiation-hydrodynamical simulation with the \\textsc{Stagger} code \\citep[e.g.][]{Stein98,Collet11b,Magic13}. A detailed description of the updated simulation run will be given in a future paper (Amarsi et al. in prep.). The snapshots were resized from their original $240\\times240\\times230$ resolution to $60\\times60\\times101$, as described and tested for an earlier Solar simulation by e.g. \\citet{Amarsi17}. The physical sizes of the snapshots are $6\\times6\\times1.5$Mm.\n\nIn addition to LTE and non-LTE line profiles computed with \\textsc{Multi3D}, we computed line profiles from a larger number of 15 snapshots in LTE using \\textsc{Scate} \\citep{Hayek11}. Subtle differences, of the order of $2-3\\%$, were noticed in the centre-to-limb behaviour of equivalent widths between the two codes, with the latter more closely resembling observations. We therefore computed our final NLTE profiles by multiplying the NLTE\/LTE profile ratio found by \\textsc{Multi3D} with the LTE profiles computed by \\textsc{Scate}. The average effective temperature of the 15 snapshots is $5776\\pm16$\\,K, close enough for our purposes to the nominal $T_{\\rm eff}=5772$\\,K \\citep{Prsa16}.\n\n\\begin{figure} \n\\begin{center} \n\\includegraphics[scale=0.28]{1panel6151}\n\\caption[]{The coloured image and bar on the right-hand side represent the NLTE\/LTE equivalent width ratio of FeI 6151\\AA\\ at disk centre for a single snapshot from the solar convection simulation. In the up-flowing granules, over-ionisation causes the line to weaken in NLTE, while the inter-granular lanes display the opposite effect. The y-axis on the left-hand side indicates the spatial scale.} \n\\label{fig:ratio} \n\\end{center} \n\\end{figure}\n\n\\section{Results and discussion}\n\\label{sect:results}\n\nIt is well-known that level populations of Fe do not strongly depart from LTE in the line-forming regions of the Sun and NLTE effects on line strengths are therefore small \\citep[e.g.][]{Mashonkina11a,Bergemann12}.  The $\\rm\\langle3D\\rangle$ solar model predicts significant over-ionisation of FeI to be important only at very optically thin layers ($\\log(\\tau_{500\\rm nm})<-3.5$), while over-recombination barely dominates in deeper layers ($-2<\\log(\\tau_{500\\rm nm})<-3$), and even deeper layers are fully thermalised. Line strengths are typically affected by less than 0.01\\,dex. In full 3D, the NLTE effects vary with the convection pattern and all but the highest excited levels experience under-population in the up-flowing granules and over-population in the inter-granular lanes (Fig.\\,\\ref{fig:ratio}). This variation is expected given the much steeper temperature gradients of the granules and the behaviour is qualitatively similar to that found by \\citet{Shchukina01}, although they predict stronger over-ionisation overall. This difference is likely due to the model atoms; our atom contains many more highly excited levels and collisions with neutral hydrogen, which strengthen the collisional coupling between FeI and the FeII reservoir and reduces NLTE effects. The surface variation can also be compared to the NLTE effects of Li\\,I, Na\\,I, Mg\\,I and Ca\\,I in metal-poor stars \\citep{Asplund03,Lind13,Nordlander16}. The net effect from our 3D NLTE modelling is more over-ionisation of FeI compared to the $\\rm\\langle3D\\rangle$ model and low-excitation lines in particular are substantially weakened. In Sect.\\,\\ref{sect:feabund}, we describe how these effects propagate into iron abundance corrections.    \n\nFor three FeI lines, we can compare our predicted NLTE effects with those of \\citet{Holzreuter13}. Their Fig.\\,8 shows histograms of the equivalent-width ratios between LTE and NLTE at each pixel in the $xy$-plane for FeI 5250\\AA , 6301\\AA , and 6302\\AA .  For the bluer line, which has low excitation potential, we find a mean ratio of $+4$\\%; significantly less than their $+15$\\%. For the redder lines, we find $-1$\\%, compared to their $+1$\\%. Again, differences in model atom structure and adopted collisional cross-sections are most likely responsible for their stronger over-ionisation. \n\n\\subsection{Centre-to-limb variation}\n\\label{sect:clv}\nAfter performing an assessment of blends, we selected eleven iron lines, including one FeII line, within the SST wavelength ranges (See Table \\ref{tab:lines}). The ten FeI lines span a wide range in wavelength and strength, but unfortunately a narrow range in lower level excitation potential. As mentioned above, low-excitation lines are most sensitive to NLTE effects, but the only observed line, 5371\\AA , connected to a level below $2\\,\\rm eV$ in our wavelength regions is too blended to have diagnostic value and we therefore excluded it. \n\nThe centre-to-limb behaviour is depicted in Fig.\\,\\ref{fig:CLVew}. The observed data points correspond to average equivalent widths measured  at each $\\mu$-angle and the vertical error bars to the standard deviation of the individual pointings added to an estimated 0.5\\% error due to continuum placement. Equivalent widths were measured by direct integration within wavelength ranges that were considered blend-free (see Table \\ref{tab:lines}), after applying a radial velocity correction that aligns the deepest point of the line profile with the rest wavelength. The curves correspond to the predicted equivalent widths at a given abundance for each model and line, optimised to match disk-centre line strengths. The model spectra were similarly corrected to rest wavelength and integrated over the same wavelength interval as the observations. We chose this approach to enable a comparison between the models that is as fair as possible, because the 3D velocity field gives rise to a differential radial velocity effect with $\\mu$ that is not captured in 1D or $\\rm\\langle3D\\rangle$.\n\nFull 3D modelling matches the observed centre-to-limb behaviour well; the equivalent widths are reproduced to within $\\sim$5\\% in both LTE and NLTE.  Comparing the two, the latter performs better for strong lines, 5367\\AA\\ and 5383\\AA , and for the only line that becomes weaker in NLTE, 6151\\AA , which has the lowest excitation potential of our lines. LTE is slightly better for 5373\\AA\\ and 5389\\AA\\ , but the differences are small and restricted to $\\mu=0.2$. There is a general tendency for the 3D equivalent widths of weak lines, $W_\\lambda<100\\rm\\,m\\AA$, to be over-predicted by a few percent at $\\mu=0.2$.  \n\nWe have investigated if a better match to the limb observations could be achieved by modifying the model atom. A single 3D snapshot was run with model atoms for which all hydrogen collision and electron collision rates, respectively, were reduced by an order of magnitude. The results for the atom with modified hydrogen collisions is labeled $\\rm H\\times0.1$ in Fig.\\,\\ref{fig:CLVew} and the atom with modified electron collisions is labeled $\\rm e\\times0.1$.  We find that reduced hydrogen collisions systematically strengthen the limb equivalent widths compared to the disk centre, such that the discrepancy with the observations increases for most lines. The effect on the level populations is such that the departures from LTE are simply shifted to deeper layers. Reducing the electron collisions also makes NLTE effects set in at deeper layers, but it also gradually enhances the over-ionisation of FeI with decreasing atmospheric depth. This has a small differential effect on the centre-to-limb variation that improves the agreement with observations in most cases. \n\nThe change in line strength as a function of viewing angle is not well predicted by the $\\rm\\langle3D\\rangle$ model, which gives systematically too small equivalent widths at the limb compared to the line centre. NLTE line formation alleviates the problem slightly for FeI lines, but the line strength at $\\mu=0.2$ is still $5-20$\\% too small. The different behaviour to full 3D modelling can be largely attributed to the treatment of velocity fields; Fig.\\,\\ref{fig:CLVew2} shows the results of 3D LTE modelling with the velocity field at all points and in all directions set to zero, but with a constant microturbulence of $1\\rm\\,km\\,s^{-1}$. Evidently, this method reproduces the $\\rm\\langle3D\\rangle$ centre-to-limb behaviour very closely, in particular for the FeII line and the high-excitation FeI lines. Fe\\,I 6151\\AA\\ shows a slightly larger difference, which is probably caused by its lower excitation potential and thus greater sensitivity to temperature inhomogeneities. Fig.\\,\\ref{fig:CLVew2} also shows the results of using a 1D \\textsc{MARCS} model atmosphere \\citep{Gustafsson08} with $1\\rm\\,km\\,s^{-1}.$ microturbulence, which even more strongly underestimates the the line strengths at the limb, in agreement with the Fe line analysis of \\citet{Pereira09b}. The difference with respect to $\\rm\\langle3D\\rangle$ may be attributed to the slightly steeper temperature gradient around continuum optical depth unity.\n\nThis failure of 1D models is well-known and was reported already by \\citet{Holweger78}, who demonstrated that a $\\mu$-dependent microturbulence may solve the problem. Their Fe line analysis found empirically that a value of $1.6\\rm\\,km\\,s^{-1}$ is suitable at $\\mu=0.3$, compared to $1.0\\rm\\,km\\,s^{-1}$ at the disk centre, thus strengthening lines at the limb compared to centre. The same qualitative behaviour of 1D models has also been demonstrated for the centre-to-limb behaviour of the O\\,I 777\\,nm triplet \\citep{Steffen15}. We refrain from deriving an empirical $\\mu$-dependent microturbulence for 1D and $\\rm\\langle3D\\rangle$ modelling to match our observations, but emphasize that models can now predict the 3D velocity field and thus the line broadening without invoking free parameters. We note that strengthening of lines toward the limb can occur also in 1D models as a consequence of the change in temperature gradient, without considering the velocity field, as shown e.g. for very weak ($<15\\rm\\,m\\AA$) O\\,I, Sc\\,II, and Fe\\,I lines by \\citep{Pereira09b}.\n\n\\citet{Mashonkina13} modelled the centre-to-limb behaviour of two FeI lines, 6151\\AA\\ and 7780\\AA , using the SST observations of \\citet{Pereira09b}. They report 1D LTE modelling based on \\textsc{MAFAGS-OS} model atmosphere \\citep{Grupp09}, 3D LTE modelling based on a \\textsc{CO$^5$BOLDT} model atmosphere \\citep{Freytag12}, and $\\rm\\langle3D\\rangle$ LTE and NLTE modelling. For the bluer line, also studied in this paper, their 1D and $\\rm\\langle3D\\rangle$ LTE results are in good agreement with ours, but their 3D LTE modelling predicts more line strengthening toward the limb, implying that the 3D velocity field is characteristically different from our \\textsc{Stagger} model. For the redder line, not studied here, they find $\\rm\\langle3D\\rangle$ NLTE to well reproduce the centre-to-limb behaviour.\n\n{The importance of velocity fields aside, little can be found in the literature to explain the model centre-to-limb behaviour of different lines from basic principles. In general, we find that the $\\rm\\langle3D\\rangle$ model predicts line strengthening toward the limb for blue lines  ($<4000\\AA$) and line weakening for red lines. Strong lines tend to be more weakened than weak lines at a given wavelength, as can be seen in Fig.\\,\\ref{fig:CLVew}. We find that this behaviour can be partly explained by equation 17.183 in \\citet{Hubeny14}:\n\n\\begin{equation}\n\tr_\\nu(\\mu)\\equiv I_\\nu(0,\\mu)\/I_c(0,\\mu)=[a_\\nu+b_\\nu\\mu\/(1+\\beta_\\nu)]\/(a_\\nu+b_\\nu\\mu)\n\\end{equation}\n\nTo derive this expression, the authors assume a line formed in true absorption and a linear dependence of the Planck function with continuum optical depth at a given frequency, such that $B_\\nu=a_\\nu+b_\\nu\\tau_c$, where $a_\\nu$ and $b_\\nu$ are positive constants. {Scattering and velocity fields are neglected. $I_\\nu(0,\\mu)$ is the emergent intensity at a given $\\mu$-angle, $I_c(0,\\mu)$ is the corresponding continuum intensity, and $\\beta_\\nu$ is the ratio between line and continuous opacity. Since $1+\\beta_\\nu>1$, the residual intensity at the limb is always higher than at disk centre for a given wavelength. Lines are therefore always predicted to weaken with decreasing $\\mu$, which we have seen is true at least for red lines according to $\\rm\\langle3D\\rangle$ modelling. It may further explain why strong lines typically decrease more in line strength than weak lines, because the inverse dependence on $\\beta_\\nu$ has higher influence on the residual intensity at higher $\\mu$. When $\\beta_\\nu\\gg1$, Eq. 1 approaches $a_\\nu\/(a_\\nu+b_\\nu\\mu)$, which implies that the behaviour for strong lines at a given frequency is similar. This is true for the two strongest lines in our sample. Further, we estimated values for the coefficients $a_\\nu$ and $b_\\nu$ in the region around continuum optical depth unity for our lines and found that they change in such a way that it can explain why redder lines are more weakened than bluer (see Fig.\\,\\ref{fig:CLVew} and \\ref{fig:CLVew2}) . However, the dependence is weaker than what the detailed modelling predicts. The validity of Eq.\\,1 thus appears limited by the assumptions made.  \n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.8]{CLVew}\n\\caption[]{Centre-to-limb variation of solar iron lines. The black bullets are observed equivalent widths and the lines represent predictions in LTE and NLTE for different model atmospheres and atomic data. A depth- and $\\mu$-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$ was adopted for the $\\rm\\langle3D\\rangle$ models. In the $\\rm H\\times0.1$ and $\\rm e\\times0.1$ models, Hydrogen and electron collisional rates were reduced by a factor ten, respectively.} \n\\label{fig:CLVew} \n\\end{center} \n\\end{figure*}\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.8]{CLVew2}\n\\caption[]{The black bullets and red dashed lines are the same as in Fig.\\,\\ref{fig:CLVew}. The red solid lines represent 3D LTE modelling without velocity fields. For all models, we assume a depth- and $\\mu$-independent microturbulence value of $1\\rm\\,km\\,s^{-1}$. }\n\\label{fig:CLVew2} \n\\end{center} \n\\end{figure*}\n\n\\subsection{Iron abundance}\n\\label{sect:feabund}\n\\citet{Scott15} revised the solar iron abundance of \\citet{Asplund09} using disk-centre intensities of 31 FeI and FeII lines, carefully selected based on blending properties, line strength, and atomic data.  They employed an earlier version of a 3D hydrodynamical \\textsc{Stagger} simulation of the solar photosphere and the same Fe model atom as in Papers I and II in this series.  Abundances were first computed in 3D LTE and then corrected using NLTE calculations based on a $\\rm\\langle3D\\rangle$ model. \\citeauthor{Scott15} recommended a weighted mean abundance $\\log(\\epsilon_{\\rm Fe})=7.47\\pm0.04\\rm\\,dex$. They find that the excitation balance of FeI is well established in 3D LTE, whilst the NLTE abundances show a slight negative trend with excitation potential. FeI and FeII lines give a difference in mean abundance of 0.07\\,dex in LTE, which decreases to 0.06\\,dex after NLTE corrections have been applied. \n\nIn this study, we re-determined iron abundances for the lines selected by \\citeauthor{Scott15}, but using consistent 3D NLTE modelling. As described in the beginning of Sect.\\,\\ref{sect:results}, full 3D calculations predict a higher degree of over-ionisation than $\\rm\\langle3D\\rangle$ calculations. Our new analysis technique and new model atomic data for Fe result in more positive abundance corrections for low-excitation FeI lines; between $+0.03$ and $+0.06\\rm\\,dex$ for $E_{\\rm low}<1\\rm\\,eV$), while high-excitation lines ($E_{\\rm low}>4\\rm\\,eV$) are at most affected by $-0.01$\\,dex. This can be compared to $+0.11$\\,dex and $+0.06$\\,dex predicted for low and high-excitation lines, respectively, by \\citet{Shchukina01}. Our FeI line abundances move slightly further away from fulfilling excitation balance, while the offset in ionisation balance is reduced to 0.04\\,dex. The weighted mean abundance of all lines becomes slightly larger; $7.48\\pm0.04$\\,dex.\n\nWe repeated the model atom modifications described in Sect.\\,\\ref{sect:clv}, in order to evaluate if better agreement between different iron lines can be achieved. The $\\rm H\\times0.1$ model with altered hydrogen collisions improves neither excitation nor ionisation balance, while the $\\rm e\\times0.1$ model with altered electron collisions reduces the ionisation imbalance to 0.01\\,dex, but at the expense of further strengthening the excitation imbalance. Turning to other potential sources of error, we note our use of electron densities computed in LTE, although important electron donors (including hydrogen) have been shown to have significant NLTE effects. We also remind the reader that the atom reduction itself may have a small impact (see Sect.\\,\\ref{sect:atomred}).   \n\nFinally, \\citet{Scott15} discussed the influence on FeI line abundances by magnetic fields, referencing the work of \\citet{Fabbian12}, and concluded that an ionisation imbalance of order 0.02\\,dex may be amended by using realistic magneto-hydrodynamic simulations with an average field strength of 100\\,G \\citep{TrujilloBueno04}. However, the simulations by \\citet{Moore15} showed that the magnetic field must be concentrated and coherent to have an impact; a small-scale, randomly oriented field of 80\\,G would not affect the iron abundance determination significantly. \\citet{Shchukina15} concluded, based on the magneto-convection simulation by \\citet{Rempel12}, that a small-scale dynamo with no net magnetic flux would have a typical influence on FeI line abundances of the order $+0.014$\\,dex. \n\n\\section{Conclusions}\n\\label{sect:conc}\n\nWe have demonstrated that full 3D, NLTE modelling of iron line formation of the Sun, using a comprehensive model atom with 463 levels, is now feasible and can successfully reproduce observed data without invoking free parameters  \\citep[see also Paper III and][]{Nordlander16}. In particular we conclude here:\n\n\\begin{itemize}\n\n\\item 3D NLTE effects on low-excitation FeI lines ($<1$\\,eV) are stronger than predicted by $\\rm\\langle3D\\rangle$ modelling, resulting in 0.03-0.06\\,dex higher abundances for these lines.\n\n\\item When normalised to disk-centre line strength, full 3D NLTE modelling typically over-predicts limb (here $\\mu=0.2$) line strengths by approximately 5\\,\\%. 1D and $\\rm\\langle3D\\rangle$ modelling in LTE and NLTE perform significantly worse, assuming a constant microturbulence of $1\\rm\\,km\\,s^{-1}$, independent of depth and viewing angle. We stress the importance of proper treatment of the 3D velocity field for centre-to-limb modelling. \n\n\\item The iron abundance of the Sun is found to be $\\log(\\epsilon_{\\rm Fe})=7.48\\pm0.04$\\,dex, using consistent 3D NLTE modelling of the lines selected by \\citet{Scott15}.\n\n\\item The ionisation imbalance between FeI and FeII line abundances in the Sun is reduced to 0.04\\,dex compared to 0.06\\,dex found by \\citet{Scott15}. FeI line abundances show a negative slope with respect to excitation potential, similarly to metal-poor standard stars (see Paper III).\n\n\\item Rates of collisional excitation and ionisation of FeI by electrons still rely on simple semi-empirical recipes. Our tests show that less efficient electron collisions than employed in this work can improve agreement with solar observations in certain respects. This highlights the urgent need of improved data for such transitions, e.g.\\ using the R-matrix method. \n\n\\item High-quality solar observations at different viewing angles pose excellent challenges for spectral line formation models, testing the accuracy of atomic data as well as physical assumptions. Low-excitation FeI lines are of particular diagnostic importance and more data should be obtained.    \n\n\\end{itemize}\n\n\n\\section*{Acknowledgments} \nKL acknowledges funds from the Alexander von Humboldt Foundation in the framework of the Sofja Kovalevskaja Award endowed by the Federal Ministry of Education and Research as well as funds from the Swedish Research Council (Grant nr. 2015-00415 3) and Marie Sk\\l odowska Curie Actions (Cofund Project INCA 600398). The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at UPPMAX under project p2013234. AMA and MA are supported by the Australian Research Council (grant FL110100012). PSB acknowledges support from the Royal Swedish Academy of Sciences, the Wenner-Gren Foundation, G\\\"oran Gustafssons Stiftelse and the Swedish Research Council. For much of this work PSB was a Royal Swedish Academy of Sciences Research Fellow supported by a grant from the Knut and Alice Wallenberg Foundation. PSB is presently partially supported by the project grant The New Milky Way from the Knut and Alice Wallenberg Foundation. Funding for the Stellar Astrophysics Centre is provided by The Danish National Research Foundation (Grant agreement no.: DNRF106). TMDP was supported by the European Research Council under the European Union's Seventh Framework Programme (FP7\/2007-2013) \/ ERC Grant agreement No. 291058. The Swedish 1-m Solar Telescope was at the time of our observations operated on the island of La Palma by the Royal Swedish Academy of Sciences in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrof\\'{i}sica de Canarias. Finally, we thank the Max Planck Institute for Astrophysics in Garching for our SST observing time. \n\n\\bibliographystyle{mn2emod} \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe discovery of quasars beyond $z\\sim7$ (eg: \\cite{banados2018800,mortlock2011luminous}) poses a crucial question: which cosmic era marks the birth of radio-loud \\ac{AGN}? \nRadio loud \\ac{AGN}s are amongst the most radio-luminous sources at all cosmic epochs. Their large radio luminosity is attributed to a radio-jet launched by an accreting \\ac{SMBH} at their centre. So far, radio-\\ac{AGN}s are only known upto $z\\sim6$, and the most distant radio quasar is at $z=6.82$ \\citep{banados2021discovery}, while the  highest redshift radio galaxy lies at $z=5.72$ \\citep{saxena2018discovery} and the highest-redshift blazar is at $z=6.1$ \\citep{belladitta2020first}. Of $\\sim$200 quasars discovered beyond $z\\sim6$, only five are found to be radio sources.\n\n\\ac{HzRG}s have been targeted either by looking for ultra steep ($\\alpha < -1.3$) radio spectra \\citep[USS, ][]{de2000sample}, or by selecting sources with a very faint K-band (2.2\\,$\\mu$m) counterpart \\citep{jarvis2009discovery}. The former technique is the most widely used and is based on the observed steepening of the radio spectrum with redshift. This method has been used to discover almost all known \\ac{HzRG}s, including the most distant known source at $z = 5.72$ \\citep{saxena2018discovery}. However, some studies \\citep{yamashita2020wide,jarvis2009discovery,miley2008distant} have reported the discovery of HzRGs with non-steep spectral index at $z \\ge 4$, showing that the USS selection method does not give a complete view of high-redshift radio-\\ac{AGN}s. \n\n\nSimulations of forthcoming radio surveys estimated the source count of radio emitters as a function of redshift \\citep{bonaldi2019tiered,wilman2008semi} and it predicts hundreds of thousands of radio sources beyond $z \\sim 4$. Although radio emitters such as radio-loud \\ac{AGN}s, Starbursts, \\ac{SFG}s and radio-quiet \\ac{AGN}s contribute to this total radio-source count, simulations demonstrate that $z\\ge4$ sky is dominated by radio-AGNs (see \\cite[Figure~2]{raccanelli2012cosmological}).\nCurrently, $\\sim$112 published radio-\\acp{AGN} are known at redshift $z \\gtrsim 4$ (listed in Appendix, Table~\\ref{tab:known_radio_AGN}). This number thus indicates that the known radio source population at $z\\gtrsim4$ represents a small fraction of the total radio source population.\n\nIn some cases, it is unclear whether a  detected high-redshift radio source is a radio galaxy, blazar, quasar, etc., and so we adopt the neutral term ``\\ac{HzRS}'' to describe any radio source detected at high redshift ($z\\ge4$) .\n\nThe mismatch between models and data indicates that known \\acp{HzRS} are only the tip of the iceberg. The dearth of radio sources at high redshift can be attributed to the following factors: \n(i) many \\acp{HzRS} are probably in existing radio catalogues but their redshifts have not been measured  due to \ntheir faintness at optical\/IR wavelengths, \nand (ii) previous radio surveys were not sensitive enough to detect faint \\acp{HzRS}\n\n\n Since the ultimate goal of this series of papers is to establish the \\ac{HzRS} count and thus %\ntest the simulations\n\\citep{bonaldi2019tiered,wilman2008semi}, we expect that some missing \\acp{HzRS} are already in the literature, but are not classified as high-redshift radio sources.  \n We demonstrate this by visually cross-matching \\ac{SDSS} spectroscopy \\citep[DR12,][]{sdss} with the \\ac{FIRST} \\citep{becker95} and \\ac{NVSS} \\citep{condon98} catalogues. This search resulted in a further 33 sources at $z \\gtrsim 4$, listed in Appendix, Table \\ref{tab:new_radio_AGN}. In each case the \\ac{SDSS} spectrum has been checked for supporting evidence of the redshift, such as a Lyman break or other spectral features. We note that a further list of candidates is available in the MILLIQUAS \\citep{milliquas} catalogue\\footnote{\\url{https:\/\/heasarc.gsfc.nasa.gov\/W3Browse\/all\/milliquas.html}}, but to the best of our knowledge the spectroscopy has not been checked and so that list may include some spurious candidates.\n\n\\ac{EMU} is one of the deepest and the largest forthcoming radio continuum surveys \\citep{norris2011emu}, to be delivered by the \\ac{ASKAP} telescope \\citep{johnston2007science}. The \\ac{EMU} project started with a series of Early Science observations, followed by the \\ac{EMU} Pilot Survey \\citep[PS;][]{norris21}. In this paper, we use the \\ac{EMU} Early Science Observations of the GAMA23 field (hereafter referred to as ``G23''), which is one of the Galaxy And Mass Assembly (GAMA) survey fields \\citep{driver2008galaxy}. \n\nMotivated by the challenge of finding missing \\acp{HzRS}, we make use of a search technique different from conventional radio based techniques, the Lyman Dropout technique  ( a.k.a. Lyman Break Galaxy technique), to identify potential \\acp{HzRS} at $ z \\gtrsim 4-7$ in the G23 field. The Lyman dropout technique has been a popular technique in optical astronomy over the past two decades for discovering high-redshift galaxies up to  $z\\sim11$. However, only one radio galaxy at $z=4.72$ has been identified using the Lyman dropout technique to date \\citep{yamashita2020wide}.\nTherefore, the primary goal of this study is to test the efficiency of Lyman dropout technique in finding \\acp{HzRS}. \nA second goal is to determine the properties of our sample of \\acp{HzRS}, \n a detailed study of which will be discussed in a future paper. \n\nThe Lyman dropout technique looks for the redshifted spectral signature of the \\emph{Lyman limit} at 91.2\\,nm (Far-UV regime). This is the longest wavelength of light  that can ionise a ground-state hydrogen atom. Light at wavelengths shorter than 91.2\\,nm (ie. at higher energies) will be absorbed by sufficiently optically-thick atomic hydrogen present in the galaxy or its circumgalactic medium. This missing radiation creates a break in the observed spectrum. For high-$z$ galaxies, the Lyman break gets redshifted into the optical region, and can be identified using images taken in multiple filters.\n\nThe structure of this paper is as follows. In Section~\\ref{sec:data} we describe how we select \\acp{HzRS} in GAMA23 field from the \\ac{ASKAP} 887.5\\,MHz radio catalogue using 8-band \\textit{ugriZYJK\\textsubscript{s}} KiDS\/VIKING photometry. In Section~\\ref{sec:result}, we present our sample of \\ac{HzRS} candidates selected at $z \\sim 4$, 5, 6, and 7. In Section~\\ref{sec:discuss} we present the analysis of radio and IR properties of our sample. Finally, we summarise our results in Section~\\ref{sec:concl}.\n\nThis study adopts a $\\Lambda$CDM cosmology with $\\Omega\\textsubscript{m} = 0.3$, $\\Omega\\textsubscript{$\\Lambda$} = 0.7$ and $H_0 = 70~$km~s$^{-1}$Mpc$^{-1}$. \n\n\n\n\\section{Data and Methods}\n\\label{sec:data}\n\nTo find \\acp{HzRS}, we cross-match the G23 radio observations with the Kilo Degree Survey optical catalogue \\citep[KiDS;][]{kuijken2019fourth} and the VIKING DR5 \\& CATWISE2020 \\citep{marocco2021catwise2020} infrared catalogues. This results in a sample of G23 radio sources with optical and infrared photometry. We then apply the redshift specific Lyman dropout colour cuts \\citep{ono2018great,venemans2013discovery} to select the radio source candidates at $z \\gtrsim 4$ in the G23 field. This paper is the first in a series describing our search for \\acp{HzRS} using the Lyman dropout technique as part of the \\ac{EMU} survey.\n\nWe use the 887.5\\,MHz radio continuum data of the G23 field, produced by \\ac{ASKAP} as part of the \\ac{EMU} Early Science program in early 2019. \\ac{ASKAP} consists of 36 antennas, each of which is equipped with a Phased Array Feed (PAF). It operates in a frequency range from 700 to 1800\\,MHz. \\ac{ASKAP} data products have been created using the ASKAPsoft pipeline, aided by Selavy software in source extraction. This study examined the following \\ac{ASKAP} catalogues from project AS034: (i) selavy-image.i.SB8132.cont.taylor.0.restored.components and (ii) selavy-image.i.SB8137.cont.taylor.0.restored.components, retrieved from the CSIRO Data Access portal\n\\footnote{\\url{https:\/\/data.csiro.au\/domain\/casdaObservation}}.  The observational parameters of G23-ASKAP data are given in Table~\\ref{tab:survey_info}.\n\n A total of 38\\,080 radio sources are present in these 2 catalogues, of which 2107 are complex or multi-component (number of components $\\ge 2$). In this paper, we focus on simple (or single component) radio sources only, which are fitted by a single Gaussian. \n\n  For the selection of $z\\ge4-6$ radio sources, we used optical data from the complementary Kilo Degree Survey \\citep[KiDS,][]{kuijken2019fourth}, in particular we exploited the KiDS DR4.1 multiband source catalogue, featuring Gaussian Aperture and PSF photometry ( GAaP ; see \\cite{kuijken2015gravitational} for details) measurements of KiDS-$ugri$ and VIKING-ZYJHK bands for $r$-band detected sources.\n\nTo select $z\\sim7$ radio sources, we utilized VIKING photometry in the DR5 catalog obtained from the VISTA archive. \\footnote{\\url{http:\/\/horus.roe.ac.uk\/vsa\/index.html}} \nWe converted the Vega magnitudes in the VIKING DR5 catalog to AB magnitudes using the Cambridge Astronomical Survey Unit (CASU) recommendations \\footnote{\\url{http:\/\/casu.ast.cam.ac.uk\/surveys-projects\/vista\/technical\/filter-set}}.\n \n\nThe mid-IR (MIR) data used in this paper comes from the Wide-field Infrared Survey Explorer \\cite[WISE,][]{wright2010wide}, which is an all-sky survey centred at 3.4, 4.6, 12, and 22\\,$\\mu$m (referred to as bands W1, W2, W3 and W4), with an angular resolution of 6.1, 6.4, 6.5, and 12.0~arcsec respectively, and typical 5$\\sigma$ sensitivity levels of 0.08, 0.11, 1, and 6~mJy\/beam. Here, we use data from the CATWISE2020 \\citep{marocco2021catwise2020} catalogue.\n\nFar-IR (FIR) observations in the G23 field come from the {\\it Herschel} space observatory.  {\\it Herschel} carried out observations using two photometric instruments on board, (i) Photodetecting Array Camera and Spectrometer (PACS, \\cite{poglitsch2010photodetector} ) and (ii) Spectral and Photometric Imaging Receiver (SPIRE, \\cite{griffin2010herschel} ). PACS observations centred at 70\\,$\\mu$m, 100\\,$\\mu$m, and 160\\,$\\mu$m mainly trace the rest-frame mid-IR emission of the high-$z$ ($z>2$) \\ac{AGN}. SPIRE observed simultaneously in three wavebands centred at 250\\,$\\mu$m, 350\\,$\\mu$m, and 500\\,$\\mu$m, picking up the starburst emission in high-$z$ \\ac{AGN}s \\citep{hatziminaoglou2010hermes}. {\\it Herschel} ceased operation on 29$^{\\rm th}$~April~2013 when the telescope ran out of liquid helium, which is essential for cooling the instruments. This study utilized the SPIRE \\citep{schulz2017spire} and PACS point source catalogues \\citep{marton2017herschel} available in \\ac{IRSA}\\footnote{\\url{https:\/\/irsa.ipac.caltech.edu\/}}.\n\n\\begin{table}\n    \\centering\n      \\caption{ Summary of G23-ASKAP survey.}\n    \\begin{tabular}{l|c}\n      Parameters   & G23-ASKAP \\\\\n         \\hline\n       Frequency  (MHz)         & 887.5     \\\\\n       Bandwidth  (MHz)         & 288       \\\\\n       Synth. beam size ($\\,arcsec$) &  10         \\\\\n       RMS ($\\mu$Jy\/beam)   &    38           \\\\\n       Survey area (deg\\textsuperscript{2})  & 50      \\\\\n       Astrometric accuracy ($\\,arcsec$)&      $\\sim$ 1    \\\\\n      \\hline\n    \\end{tabular}\n    \\label{tab:survey_info}\n\\end{table}\n\n\\subsection{Finding optical and infrared counterparts}\n\\label{sec:optical_crossmatch}\n\nTo find the optical counterparts of radio sources, we use a simple nearest-neighbour technique. We need to choose a search radius that maximises the number of cross-matches while minimising the number of false identifications (hereafter called false-IDs). We achieve this by cross-correlating the \\ac{ASKAP} radio catalogue with the KiDS DR4.1 multiband optical (\\textit{ugri}) $+$ NIR (ZYJHK\\textsubscript{s}) photometry at a range of search radii, measuring the number of cross-matches at each radius. \nWe then estimate the false-ID rate by shifting the radio position by 1~arcmin (so that all matches are spurious) and repeating the cross-match at the same set of radii. The false-ID rate is calculated by dividing the number of shifted cross-matches by the number of unshifted cross-matches. The result is shown in Table~\\ref{tab:false_id}. Based on this, we have chosen 2~arcsec as the optimal search radius for this study, corresponding to a false-ID rate of 14.64\\% and a total cross-match rate of 63.9\\%. We reject sources that had multiple matches within 2~arcsec. This reduces the final number of radio-optical cross-matches to 17\\,447.\n\nWe followed the same procedure to select infrared counterparts to our radio sources. We cross-matched the optical (KiDS) positions of our sample with the CATWISE2020 catalog at a search radius of 2$\\,arcsec$. This gives a false id rate of 11.9\\%.\n\n\\begin{table}\n \\centering\n \\caption{False-ID rate estimated for \\ac{ASKAP}-KiDS cross-match as a function of separation radius, using the single-component source list.}\n \\label{tab:false_id}\n \\begin{tabular}{c c c c}\n \\hline\n  Match Radius & No. of Matches & No. of Matches & False-ID Rate \\\\\n  (arcsec) & (unshifted) & (1~arcmin offset) & (\\%) \\\\\\hline\n  1 & 14\\,067 & 833 & 5.92 \\\\\n  2 & 23\\,002 & 3\\,368 & 14.64 \\\\\n  3 & 29\\,350 & 7\\,592 & 25.87 \\\\\n  4 & 36\\,417 & 13\\,410 & 36.82\\\\\n  5 & 44\\,884 & 20\\,821 & 46.39 \\\\\n  \\hline\n \\end{tabular}\n\\end{table}\n\n\n\\subsection{Selection of radio sources at $z\\gtrsim$ 4 -- 6}\n\n The Lyman dropout technique relies on finding the wavelength or passband at which the Lyman break is detected, which in turn tells us the redshift  of the source, given that rest-frame wavelength of Lyman limit is 91.2\\,nm. For example, the Lyman break of a galaxy at $z=4$ will be observed at wavelength 4560\\,\\AA\\,  and hence can be imaged in g-band. Similarly, for higher redshift objects, the Lyman break moves into the $r$ or $i$ or $Z$ bands.\n\\subsubsection{Applying  \\textit{g, r, \\& i}-band  dropout technique}\n\\label{sec:initial_sample}\n\n\n\\begin{table}\n\\caption{KiDS \\& VIKING filters, their central wavelength, and their mean 5$\\sigma$ limiting magnitude. }\n \\centering\n \\begin{tabular}{c c c}\n \\hline\n  Filters & $\\lambda$ & 5$\\sigma$ Mag. Lim. \\\\\n    & (\\AA)    & (AB) \\\\\n    \\hline\n  $u$ & 3\\,550  & 24.23 \\\\\n  $g$ & 4\\,775 & 25.12\\\\\n  $r$ & 6\\,230  & 25.02 \\\\\n  $i$ & 7\\,630  & 23.68 \\\\\n  Z & 8\\,770  &23.1 \\\\\n  Y & 10\\,200 & 22.3 \\\\\n  J & 12\\,520  & 22.1 \\\\\n  H & 16\\,450  & 21.5\\\\\nK\\textsubscript{s} & 21\\,470  & 21.2 \\\\\n\\hline\n \\end{tabular}\n \\label{tab:kiDS_filters}\n\\end{table}\n\n\n\\begin{table*}\n \\begin{threeparttable}\n \\caption{\n The criteria used to identify Lyman dropouts at $z\\sim$ 4, 5, 6, adopted from \\protect \\citet{ono2018great} and at $z\\sim$ 7, taken from \\protect \\cite{venemans2013discovery}. The last row is based on our COSMOS tests, to remove low-z interlopers in $z\\sim$ 4, 5, 6 sample, as described in subsection \\protect \\ref{sec:interloper1}.} \n \n \n \\begin{tabular}{p{2.9cm} | p{2.9cm} | p{2.9cm} | p{2.9cm} |p{3.9cm}}\n \\hline\n  \\multirow{2}{*}{$z\\sim4$ (\\textit{g} dropouts)} & \\multicolumn{2}{c}{$z\\sim5$ (\\textit{r} dropouts)} & \\multirow{2}{*}{$z\\sim6$ (\\textit{i} dropouts)} &  \\multirow{2}{*}{$z\\sim7$ (Z dropouts)} \\\\\n  \\cline{2-3}\n  &  criteria I \\tnote{a} &  criteria II \\tnote{b} & & \\\\\n  \\hline\\hline\n   S\/N(i) $>$5 & S\/N(z) $>$5 & S\/N(z) $>$5 & S\/N(z) $>$5  &  S\/N(Y) $>$7\\\\\n   & S\/N(g) $<$2  & S\/N(g) $<$2 &S\/N(g) $<$2 ; S\/N(r) $<$2 & Z-Y $\\ge$ 1.1  \\\\\n   \\textit{g-r $>$ 1.0} &\\textit{r-i $>$ 1.2}&\\textit{r-i $>$ 1.0} &\\textit{i-z $>$ 1.5} & -\u2212 0.5 $<$ Y \u2212 J $\\le$ 0.5 \\\\\n   \\textit{r-i $<$ 1.0} &\\textit{i-z $<$ 0.7} &\\textit{i-z $<$ 0.5} & \\textit{z-Y $<$ 0.5} & Z \u2212 Y $>$ Y \u2212 J + 0.7 \\\\\n   \\textit{g-r $>$1.5 (r-i) + 0.8} & \\textit{r-i $>$1.5(i-Z) + 0.8} & \\textit{r-i $>$1.5(i-Z) + 0.8} & \\textit{i-z $>$2.0 (z-Y) + 1.1} & \u22120.5 $<$ Y \u2212 K $<$ 1.0 \\\\\n  \\cline{1-4}\n  \\multirow{3}{*}{\\textit{$i_{AB} > 22.2$}}  & \\multirow{3}{*}{\\textit{$z_{AB} > 23$}} & \\multirow{3}{*}{\\textit{$z_{AB} > 23$}}& \\multirow{3}{*}{\\textit{$z_{AB} > 22$}} & J \u2212 K $<$ 0.8 \\\\\n    &  &  & & undetected in \\textit{ugri} bands if available\\\\\n   \\hline\n \\end{tabular}\n \\begin{tablenotes}\n    \\item[a] \\cite{ono2018great}\n    \\item[b]  Our relaxed {\\it r} dropout criteria (see text for details.)\n    \\end{tablenotes}\n \\label{tab:dropouts_colour_criteria}\n  \\end{threeparttable}\n\\end{table*}\n\n\n\n\nTo find \\acp{HzRS}, we adopt the dropout criteria of \\cite{ono2018great}, shown in Table~\\ref{tab:dropouts_colour_criteria}.   The spectroscopically confirmed redshift ranges covered by each dropout, adopted from \\citet[Figure~6]{ono2018great}, are as follows: (i) g-dropout: $3\\le z \\le 4.5$ (ii) r-dropout: $4.3\\le z \\le 5.4$ (iii) i-dropout: $5.6\\le z \\le 6.2$.  To keep it simple, we use redshifts, $z\\sim4$, 5 and 6 to represent $g$, $r$, and $i$-band dropouts respectively. We use photometry from the KiDS DR4.1 multi-band catalogue (shown in Table~\\ref{tab:kiDS_filters}), based on the GAaP magnitudes corrected for both zero-point and Galactic extinction. The 5$\\sigma$ limiting magnitude for the $g$-band was used in the \\textit{$g - r$} colour if objects were undetected ( i.e. no entry in the catalogue) in $g$-band. Similarly, a 5$\\sigma$ limiting magnitude for the $r$ and $i$-bands were used to estimate \\textit{$r - i$} and \\textit{$i - z$} colours for $r$ and $i$ dropouts if objects were undetected in $r$ and $i$ band respectively.  In Table~\\ref{tab:total_rg-dropout}, we describe the photometric selection and number of sources remaining after applying $z \\sim 4$, 5, and 6 colour cuts. We present examples for all three dropouts, taken from our final sample, in Figure~\\ref{fig:example_dropouts}, showing their cutouts at each of the \\textit{ugri}ZYJHKs bands.\n\n Ideally, the signal-to-noise-ratio (SNR) of the sources should be used to \nmeasure\ntheir detection or non-detection in a given band. Since such information is not present in the KiDS catalogue, we utilized the mean 5$\\sigma$ limiting magnitude of each passband to define detection and undetection. Given that the 5$\\sigma$ limiting magnitude follows a continuous distribution \\citep{kuijken2019fourth}, this could  result in some dropouts being missed.\n\n\n\n\\begin{table*}\n    \\centering\n    \\begin{threeparttable}\n     \\caption{ Our initial sample: the number of radio sources remaining after each criterion. Sources lacking detection in a given dropout (no entry in the catalog) band is replaced with their respective mean 5$\\sigma$ limiting magnitude.}\n    \\begin{tabular}{c|c|c}\n    \\hline\n       Sample & Criteria & Radio source \\\\\n              & &  Count        \\\\\n       \\hline\n       G23-ASKAP &-- & 35,973 \\\\\n                                & cross-match with KiDS at 2\\,arcsec & 23\\,002 \\\\\n                                & After removing multiple matches & 17\\,447 \\\\\n       & imaflags\\_iso = 0 \\& &  \\multirow{2}{*}{17\\,396}  \\\\    \n       & nimaflags\\_iso = 0  & \\\\\n       & flag\\_gaap\\_$gri$ = 0  & 17\\,396 \\\\\n       & flag\\_gaap\\_$griZ$ = 0  & 17\\,376 \\\\\n                 \\hline\n        & col~1,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{4}{*}{229}\\\\\n        $z\\sim4$ sample  & Mag\\_gaap\\_i < 23.8 \\& & \\\\\n        ($g$ detected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                 \\hline\n         & col~1,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{4}{*}{6}\\\\\n                          & Mag\\_gaap\\_g = 25.12 \\& &\\\\\n         $z\\sim4$ sample  & Mag\\_gaap\\_i < 23.8 \\tnote{a} \\,\\& & \\\\\n         ($g$ undetected) &  Mag\\_gaap\\_u > 24.23 or undetected & \\\\\n                         \n                 \\hline\n         &   col~3,Table~\\ref{tab:dropouts_colour_criteria} colour cuts\\& & \\multirow{4}{*}{58}\\\\\n        $z\\sim5$ sample  & Mag\\_gaap\\_Z < 23.6 \\tnote{b} \\,\\& & \\\\\n        ($r$ detected)& Mag\\_gaap\\_u > 24.23 or undetected \\& & \\\\\n                      & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                 \\hline\n      &  col~4,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{5}{*}{6}\\\\\n        $z\\sim6$ sample  & Mag\\_gaap\\_Z < 23.6\\tnote{b} \\,\\& & \\\\\n        ($i$ detected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                      & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                      & Mag\\_gaap\\_r > 23.68 or undetected \\\\\n                 \\hline\n        & \n        col~4,Table~\\ref{tab:dropouts_colour_criteria} colour cuts \\& & \\multirow{5}{*}{9}\\\\\n        $z\\sim6$ sample  & Mag\\_gaap\\_Z < 23.6\\tnote{b} \\,\\& & \\\\\n        ($i$ undetected)& Mag\\_gaap\\_u > 24.23 or undetected\\\\\n                        & Mag\\_gaap\\_g > 25.12 or undetected \\\\\n                        & Mag\\_gaap\\_r > 23.68 or undetected \\\\\n                 \\hline         \n\n    \\end{tabular}\n    \\begin{tablenotes}\n    \\item[a]  $i$-band $5\\sigma$ limiting magnitude (AB) correspond to the end of the peak of the distribution \\citep[Figure~3]{kuijken2019fourth}.\n    \\item[b]  $Z$ band $5\\sigma$ limiting magnitude (AB) distribution correspond to the end of the peak of the distribution, assuming a spread of $\\pm0.5\\sigma$, given that the nominal $5\\sigma$ $Z_{lim} = 23.1$. \n    \\end{tablenotes}\n    \\label{tab:total_rg-dropout}\n    \\end{threeparttable}\n\\end{table*}\n\n\n\n\n\\begin{figure*}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=1.01\\textwidth]{g_dropout3.pdf}\n \\caption{An example of g dropout at $z\\sim4$ taken from our final sample:   J223541-311145. The size of the cutouts is $\\sim20\\,arcsec\\times15\\,arcsec$ and the circle is of radius $\\sim7.4\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=\\textwidth]{r_dropout2.pdf}\n \\caption{An example of r dropout at $z\\sim5$ taken from our final sample:   J230551-343338. The size of the cutouts is $\\sim24\\,arcsec\\times18\\,arcsec$ and the circle is of radius $\\sim8\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\\begin{subfigure}{0.9\\textwidth}\n \\centering\n\n \\includegraphics[trim={100 290 90 0},width=\\textwidth]{i_dropout4.pdf}\n \\caption{An example of i dropout at $z\\sim6$ taken from our final sample:     J230246-293923. The size of the cutouts is $\\sim20\\,arcsec\\times15\\,arcsec$ and the circle is of radius $\\sim6\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source.}\n\\end{subfigure}\n\n\\caption{KiDS-\\textit{ugri} and ViKING-ZYJHKs images of Lyman dropouts at each redshift are illustrated above in the order of increasing wavelength. The Lyman break can be identified by combining photometry in three consecutive bands. At $z \\sim 4$, the Lyman limit ($\\lambda\\textsubscript{rest}=912\\textup{\\AA}$) falls in the g-band and shorter wavelengths are significantly absorbed, as indicated by faint $u$ and $g$ bands. Similarly for $z \\sim 5$ and 6 sources, the Lyman limit falls in the $r$ and $i$-bands respectively.}\n\\label{fig:example_dropouts}\n\\end{figure*}\n\\begin{figure*}\n    \\centering\n    \\includegraphics[trim={0 370 0 25 clip=true},width=\\textwidth]{z_dropout3.pdf}\n    \\caption{VIKING bands of a Z dropout at $z\\sim7$ taken from our final sample.  The size of the cutouts is $\\sim25\\,arcsec\\times20\\,arcsec$ and the circle is of radius $\\sim8\\,arcsec$ which represents the apparent size of the \\ac{ASKAP} detected radio source. }\n    \\label{fig:my_label}\n\\end{figure*}\n\n\n\n\n\n\n\n\\subsubsection{Removing low-$z$ interlopers from our $z \\gtrsim 4-6$ sample}\n\\label{sec:interloper1}\n\nLow-$z$ objects such as dwarf stars and passive galaxies can enter the selection region defined by Lyman dropout colour cuts due to photometric errors, even though intrinsically they do not enter the colour selection window. We evaluate the contamination rate in the $z \\sim 4$, 5 and 6 selection window by testing the Lyman dropout technique on a set of objects with known spectroscopic redshifts. We chose the COSMOS field \\citep{scoville2007cosmic}, a well-studied region of the sky where both broadband optical photometry and spectroscopic redshifts are available.  We used the following catalogues of the COSMOS field, available in the public \\ac{IRSA} domain, to test $g$, and $r \\&$ $i$-band dropout techniques respectively ; (i) COSMOS Photometry Catalogue January 2006 (hereafter, COSMOS2006 catalog; \\cite{capak2007first})  and (ii) COSMOS2015 Catalog \\citep{laigle2016cosmos2015}. The spectroscopy for COSMOS sources was obtained from COSMOS DEIMOS Catalogue \\citep{hasinger2018deimos}.  We utilized the  Subaru-\\textit{griz} photometry in the COSMOS catalogues to test the colour cuts.  Furthermore, in this paper, we follow lowercase and uppercase notation in the literature for the Subaru {\\it z} filter and the VIKING Z filter respectively.\n\n We crossmatched the COSMOS 2006 catalogue and the COSMOS DEIMOS Catalogue at 1\\,arcsec, resulting in 8906 matches. We further excluded multiple matches and apply the following criteria as recommended in \\cite{capak2007first} to obtain a cleanest sample: blend$\\_$mask = 0, i$\\_$mask = 0, b$\\_$mask = 0, and v$\\_$mask = 0. This results in 6\\,787 unique sources that have been deblended and are without any photometric flags to test the $g$-band dropout colour cuts. We further corrected the $g$, $r$, and $i$ magnitudes for Galactic extinction following the recommendations in \\cite{capak2007first}.\n\n Similarly, we crossmatched the COSMOS2015 catalogue and the COSMOS DEIMOS Catalogue at 1\\,arcsec, resulting 7640 matches. The sources with multiple matches were excluded and the following criteria were applied as per \\cite{laigle2016cosmos2015}: $flag\\_HJMCC=0 \\& flag\\_cosmos=1 \\& flag\\_peter=0$, giving a source count of 7\\,502. We finally applied respective Galactic extinction corrections to $r$, $i$, $z$, and $Y$ bands as per \\cite{laigle2016cosmos2015}. \n\nTable~\\ref{tab:dropouts_colour_criteria} shows that $z \\sim 4$, 5, and 6 galaxy candidates can be selected based on their \\textit{gri}, \\textit{riz}, \\textit{izy} colours respectively. We demonstrate this in Figure~\\ref{fig:cc_plot_cosmos} by plotting the spectroscopic redshift distribution of COSMOS sources in \\textit{g$-$r} vs. \\textit{r$-$i}, \\textit{r$-$i} vs. \\textit{i$-$z} and \\textit{i$-$z} vs. \\textit{z$-$Y} colour-colour space. \n To test the $z \\sim 7$ or Z-band dropout colour cuts, deep spectroscopic data ($z_{spec}>6.4$) is needed, which is not available in the COSMOS DEIMOS catalog. It is evident that the photometric selection window of g-dropouts (the black box) encompasses almost all $z \\sim 4$ sources, with a small contribution from low-$z$ ``interloper'' sources. By contrast, the r-dropout selection window misses a significant fraction of $z \\sim 5$ sources.   Therefore, we relaxed the $r$ dropout colour cuts as follows:\n\n\\begin{equation} \\label{eq:1}\n r-i > 1.0; \\\\\ni-z < 0.75 ; \\\\\nr-i > 1.5*(i-z)+0.8\n\\end{equation}\n\nThe resulting colour locus is indicated by black dashed lines, showing \n that more $z \\sim 5$ sources get included than low-$z$ ones.  \n\n\nIn Figure~\\ref{fig:mag_redshift_cosmos}, we demonstrate that $g$ and $r$ dropouts suffer significant contamination from low-$z$ sources at the bright end. Based on this, we introduce further selection criteria in the form of a magnitude cut-off in $i$-band (i\\textsubscript{AB} $>$ 22.2) for $g$-dropouts and in $z$-band  (z\\textsubscript{AB} $>$ 23) for $r$-dropouts, which helps to reduce the contribution from low-z sources (see bottom row of Table~\\ref{tab:dropouts_colour_criteria}). \n\n We further note that our sample size of $i$-band dropouts is \ntoo small\nto draw any conclusion. The $Z$ band magnitude cut-off ($Z_{AB}>24$) implied by the figure~\\ref{fig:mag_redshift_cosmos} is too high and the VIKING data is also not deep enough to apply this cut-off.  Therefore, we performed a literature search to to find the $Z$-band magnitude of known $z\\sim6$ quasars. Based on the spectroscopically confirmed $i$ - band dropouts in \\citet[Table~3]{venemans2015first}, we applied a magnitude cut-off, Z\\textsubscript{AB} $>$ 22 for $i$-dropouts.  \n\n\n\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{gir_cc_cosmos2.png} \n\\end{subfigure}\\hspace{0.5cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{riz_cc_cosmos2.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{izy_cc_cosmos2.png} \n\\end{subfigure}\n\\caption{\\textbf{(Top Left)} \nSelection of $z\\sim4$ sources by means of $g$-band dropout technique using $gri$ broadband filters: \\textit{(g-r)} vs. \\textit{(r-i)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box represents the $g$ dropout selection criteria in \\protect\\cite{ono2018great}. It encompasses almost all $z\\sim4$ sources, but with a small contamination from low-$z$ sources too. \\textbf{(Top Right)} Selection of $z\\sim5$ sources by means of $r$-band dropout technique using $riz$ broadband filters: \\textit{(r-i)} vs. \\textit{(i-z)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box (solid lines) represents the $r$ dropout selection criteria in \\protect\\cite{ono2018great}.  The black box bordered by dashed black lines represent the colour locus from relaxed $r$ band colour-cuts. \n \\textbf{(Bottom)} Selection of $z\\sim6$ sources by means of $i$-band dropout technique using $izY$ broadband filters: \\textit{(i-z)} vs. \\textit{(z-Y)} colour-colour diagram of COSMOS sources, colour coded according to their spectroscopic redshift. The black box represents the $i$ dropout selection criteria in \\protect\\cite{ono2018great}. }\n\n\\label{fig:cc_plot_cosmos}\n\\end{figure*}\n\n\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{imag_zspec_final.png} \n\\end{subfigure}\\hspace{0.5cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{zmag_zspec_z5_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{zmag_zspec_z6_final.png} \n\\end{subfigure}\n\\caption{Selecting low-$z$ interlopers in $g$ and $r$ dropout samples in COSMOS field by utilising the respective detection bands of each dropout: \\textbf{(Top Left)} $i$ magnitude of $g$ dropouts together with their spectroscopic redshifts. \\textbf{(Top Right)} $z$ magnitude of $r$ dropouts together with their spectroscopic redshifts.   Blue solid circles represent the $r$ dropouts selected using \\protect\\cite{ono2018great} criteria and red crosses, the $r$ dropouts selected using our relaxed colour-cuts (Equation~\\ref{eq:1}).    (Bottom:) $z$ magnitude of $i$ dropouts together with their spectroscopic redshifts. } \n\\label{fig:mag_redshift_cosmos}\n\\end{figure*}\n\\begin{figure*}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{g_dropout_final.png}\n\\end{subfigure} \\hspace{1 cm}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{r_dropout_final.png}\n\\end{subfigure}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{i_dropout_final.png}\n\\end{subfigure}\\hspace{1 cm}\n\\begin{subfigure}{0.35\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{z_dropout_final.png}\n\\end{subfigure}\n\\caption{ Colour-colour diagram of our final sample of \\acp{HzRS} at $z\\gtrsim$4-7 selected via the Lyman dropout technique. The black box shows the selection criteria of $g$, $r$, $i$, and $Z$ dropouts in $gir$, $riZ$, $iZY$, and $ZYJ$ colour-colour plane respectively.}\n\\label{fig:cc_dropouts}\n\\end{figure*}\n\n\n\n\n\n\\subsection{Selection of radio sources at $z\\sim7$}\n\n\nWe utilised VIKING photometry to search for radio sources in the redshift range, $6.44\\le z \\le 7.44$. \\cite{venemans2013discovery} demonstrated that Lyman dropouts at $z\\sim7$ (a.k.a Z dropouts) can be selected using  ZYJ near-IR filters. We cross-matched G23-ASKAP radio catalogue and VIKING DR5 catalogue at a search radius of 2$\\,arcsec$,  which  gives a false ID rate of 7.9\\% . Using the selection method in \\cite{venemans2013discovery} (see Table~\\ref{tab:dropouts_colour_criteria}), we select a sample of 3 radio source candidates at $z\\sim7$, of which one is a known radio-quasar, VIK J2318\u22123113, at $z=6.44$ \\citep{ighina2021radio}. Scattering of foreground galaxies into the Z-dropout selection region in the ZYJ color-color space is minimised by selecting point sources only by applying the criterion: pGalaxy$<0.95$ ( see \\cite{venemans2013discovery} and reference therein for details), where pGalaxy is the probability that the source is a galaxy.   \n\n\\subsection{Estimate of Reliability}\n\\label{reliability}\nFinally, we estimate the success rate of our magnitude cutoff by counting the fraction of low-$z$ sources remaining after applying the $i$-band and $z$-band cutoff to $g$ and $r$ \\& $i$ dropouts respectively. \n A sample of  294 sources were identified from the COSMOS catalog as satisfying the $z \\sim 4$ colour cuts (Table~\\ref{tab:dropouts_colour_criteria}), of which  75 sources have $z_\\textrm{spec}$ below 3.0, resulting in a contamination rate of 25.5\\%. The further application of the i\\textsubscript{AB} $>$ 22.2 criterion reduces the low-z source count to 29 and the total sample size to 248. This gives a final contamination rate of  11.7\\%.\n\n A total of 78 sources were identified as $r$ dropouts using \\cite{ono2018great} criteria, \nof which 19\nhave $z_\\textrm{spec} < 4.3$ giving a contamination rate of $\\sim24.4\\%$. On the other hand, relaxed $r$ band colour cuts (Equation~\\ref{eq:1}) selected 127 sources in total, of which 29 have $z_\\textrm{spec} < 4.3$ resulting a contamination rate of $\\sim22.8\\%$. This shows that relaxed $r$ band colour cuts select $\\sim1.7$ times more $z_\\textrm{spec} > 4.3$ sources than the \\cite{ono2018great} criteria while the contamination rate remains almost constant. We further reduce the contamination rate to 19\\% by applying a  z\\textsubscript{AB} $>$ 23 cutoff. \n\n Due to the dearth of deep spectroscopic data, only 14 sources were selected as $i$ dropouts, 6 of which have a $z_\\textrm{spec}<5.6$ classifying as interlopers. This gives an initial contaminant rate of $\\sim 43 \\%$ and the further application of magnitude cut-off, $Z_{AB} > 22$, results a final contaminants rate of $\\sim 38.4\\%$.\n\n This shows that the $z \\sim 4$ sample is  $\\sim88\\%$ reliable while the $z \\sim 5$ sample is around 81\\% reliable. \nThe $z \\sim 6$ sample is $\\sim62\\%$ reliable, which is likely to be a lower limit given that the sample size was smaller.\nThere is no robust way of estimating the reliability for the $z\\sim7$ sample, as there is a lack of deep spectroscopic data in COSMOS. \n\n We applied these magnitude cut-offs to the sources in Table~\\ref{tab:total_rg-dropout} to select our final sample of radio source candidates at $z\\gtrsim4-6$ in the G23-\\ac{ASKAP} field. The resulting source count of each dropout is shown in Table~\\ref{tab:final}. As a further check on reliability, we cross-matched our sample with the GAMA spectroscopic redshift \\citep{baldry18} catalogue and the Gaia Early Data Release 3 \\citep[EDR3,][]{brown2020gaia}. The Gaia satellite measures parallaxes and proper motions for nearby stars in our Milky Way, and allow us to identify cool dwarf stars, if any, in our final sample. Gaia EDR3 is the latest release, providing information for about 1.8 billion objects.\n\nNone of our sources were found to have either GAMA spectroscopic or GAIA counterparts out to a search radius of 4~arcsec. The lack of GAMA spectroscopic counterparts is consistent with a high redshift, as the GAMA spectroscopic survey is complete to $z\\sim0.4$. Similarly, the lack of GAIA counterparts confirms that (i) no Milky Way stars are present in our sample, and (ii) no low-z quasars that are bright enough to be detected by GAIA are in our final sample.\n\n\n\n\\section{Results}\n\\label{sec:result}\n\n\n \n  Using the Lyman Dropout photometric technique and our additional magnitude cut-offs (Table~\\ref{tab:dropouts_colour_criteria}) to reduce low-$z$ interlopers, we select our final sample of 148 radio source candidates at $z \\gtrsim 4-7$, where (i) {\\it g} \\& {\\it i} dropouts criteria come from \\cite{ono2018great} (ii) {\\it r } dropout criteria from this study (Equation~\\ref{eq:1}) and (iii) Z dropout criteria from \\cite{venemans2013discovery}. \n \n  The colour-colour plot of our final sample is shown in Figure~\\ref{fig:cc_dropouts}. Furthermore, we identify a known radio quasar at $z=6.44$ and 2 radio source candidates at $z\\sim7$ using the Z-dropout selection technique from \\cite{venemans2013discovery}.  Thus our final sample of 149 radio sources in 50\\,deg\\textsuperscript{2} implies a sky density of $\\sim3$ per $deg^{2}$ for $S_{888}\\gtrsim0.1$\\,mJy radio sources at $z\\gtrsim4$. For comparison, \\cite{norris21} suggested a sky density of $\\sim5$ per $deg^{2}$ beyond $z\\sim4$ at EMU flux limit,\n based on simulations.  We present the catalogues of $z \\gtrsim 4-6$ and $z\\sim7$ sources, including KiDS\/VIKING and WISE W1 photometry, in Table~\\ref{tab:catalog} and Table~\\ref{tab:z7_sample} respectively.\n \n\n\n\n\n\\begin{table}\n \\caption{Our final sample of  $ z\\gtrsim4-7$ candidates in G23 field selected using Lyman dropout colour cuts (Table~\\ref{tab:dropouts_colour_criteria}) together with our magnitude cutoffs from Figure~\\ref{fig:mag_redshift_cosmos}}.\n \\centering\n \\begin{tabular}{c|c}\n \\hline\n  Redshift ($z$) & Sample Size \\\\\n  \\hline\n  $z\\sim4$ &  117\\\\\n  $z\\sim5$ &  14 \\\\\n  $z\\sim6$ &  15\\\\\n  $z\\sim7$ & 3\\\\\n  \\hline\n \\end{tabular}\n \\label{tab:final}\n\\end{table}\n\n\n\n\\clearpage\n\\onecolumn\n\\begin{longtable}[c]{ccccccccccc}\n \\caption{Our final $z\\gtrsim4-6$ sample, showing radio properties from \\ac{ASKAP} and optical\/ IR photometry from KiDS and CATWISE catalogues respectively. Missing optical magnitudes indicate non-detection in that respective filter. The missing WISE magnitude (W1) indicate no counterpart within 2~arcsec of the KiDS position.   KiDS\/VIKING magnitudes are given in the AB system and the WISE magnitudes in the Vega unit. }\\\\\n \\hline\n Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5}&Mag\\_u &Mag\\_g& Mag\\_r & Mag\\_i & Mag\\_Z & W1 \\\\\n  & name & (deg) & (deg) & (mJy) & & & & & & (mag)  \\\\\n  \\hline\n \\endfirsthead\n \\hline\n  Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5}&Mag\\_u &Mag\\_g& Mag\\_r & Mag\\_i & Mag\\_Z & W1 \\\\\n  & name & (deg) & (deg) & (mJy) & & & & & & (mag)  \\\\\n  \\hline\n  \\endhead\n  \\multicolumn{11}{c}{$z\\sim4$ Sample ($g$ dropouts) } \\\\\n  \\hline\n\n1&  J225745-311209 & 344.437  & -31.203   & 0.32     & 25.067 & 25.779  & 24.108  & 23.549  & 22.928  &        \\\\\n2&  J225802-345205 & 344.511  & -34.868    & 0.948    &            & 25.954  & 23.293  & 22.808  & 21.720  & 17.588 \\\\\n3&  J225835-325509 & 344.649  & -32.919   & 0.237    &            & 25.220  & 23.268  & 22.689  & 21.573  & 15.633 \\\\\n4&  J225915-314116 & 344.815   & -31.688   & 0.57     &            & 24.348  & 22.954  & 22.649  & 21.796  & 16.055 \\\\\n5&  J230041-322125 & 345.172  & -32.357   & 0.247    &            & 25.854  & 23.887  & 23.248  & 21.810  & 15.95  \\\\\n6&  J230122-294717 & 345.342  & -29.788   & 0.254    & 24.800 & 25.084   & 23.678  & 23.685  & 23.143  &        \\\\\n7&  J230156-331615 & 345.484  & -33.271   & 0.261    & 24.993  & 25.791  & 23.483  & 22.506  & 21.295   & 15.556 \\\\\n8&  J230223-294300 & 345.599  & -29.716   & 0.584    & 24.999  & 25.341   & 23.511  & 22.884   & 22.187 & 17.686 \\\\\n9&  J230234-335010 & 345.645  & -33.836   & 0.246    &            & 25.306  & 23.750 & 23.314  & 23.198 &        \\\\\n10&  J230249-322437 & 345.708  & -32.411   & 0.233    &            & 26.273  & 24.163   & 23.632   & 22.896  & 16.641 \\\\\n11&  J230306-333335 & 345.778  & -33.559   & 0.282    &            &            & 23.845  & 23.724   &            &        \\\\\n12&  J230317-345556 & 345.824  & -34.932   & 2.428    &            & 25.179  & 23.003 & 22.217 & 21.372 & 16.771 \\\\\n13&  J230432-341722 & 346.133   & -34.289   & 6.517    & 25.132 & 24.487  & 23.360   & 23.481  & 23.491  &        \\\\\n14&  J230504-292342 & 346.267  & -29.395   & 0.646    &            & 25.831  & 23.852  & 23.297  & 22.074  & 17.746 \\\\\n15&  J230516-331612 & 346.317  & -33.270   & 0.421    &            & 25.096  & 23.024   & 22.381   & 22.067   &        \\\\\n16&  J230544-351610 & 346.434  & -35.269   & 0.302    &            & 24.140  & 23.057  & 23.002  & 22.886  &        \\\\\n17&  J230547-293327 & 346.449  & -29.557   & 2.886    & 24.393   &            & 23.511  & 23.343  & 22.727   &        \\\\\n18&  J230718-310125 & 346.826   & -31.024   & 0.364    &            & 24.013  & 22.692  & 22.784  & 22.728  &        \\\\\n19&  J230831-350015 & 347.133  & -35.004   & 0.203    &            & 24.927  & 23.111   & 22.451  & 21.681  & 16.86  \\\\\n20&  J230832-340644 & 347.137   & -34.112   & 8.009    & 25.594  & 24.677  & 23.069  & 22.766  & 22.104  & 16.219 \\\\\n21&  J230905-334318 & 347.272  & -33.722   & 0.204    &            & 25.661   & 23.539  & 22.862  & 21.849  &        \\\\\n22&  J230937-323421 & 347.405  & -32.573   & 0.272    &            & 25.599   & 23.829  & 23.760   & 22.196  & 16.971 \\\\\n23&  J230940-335143 & 347.418  & -33.862     & 0.298    &            & 24.765  & 23.522  & 23.691  & 24.487  &        \\\\\n24&  J230942-335049 & 347.427  & -33.847   & 0.304    &            &            & 23.385  & 22.792  & 21.804  & 16.54  \\\\\n25&  J231057-294135 & 347.740   & -29.693   & 0.431    &            & 25.159  & 23.852  & 23.621  & 22.309  & 16.91  \\\\\n26&  J231117-323952 & 347.822   & -32.664   & 0.247    &            & 25.543  & 24.052 & 23.674  & 23.917  &        \\\\\n27&  J231148-311231 & 347.950  & -31.209   & 0.323    & 25.134  & 24.726  & 22.381  & 22.753  & 22.904  & 18.202 \\\\\n28&  J231149-304758 & 347.958  & -30.799   & 0.292    & 24.414   & 24.550   & 23.228   & 23.087  & 22.582   & 16.862 \\\\\n29&  J231210-332436 & 348.045  & -33.410   & 0.26     & 24.598  & 25.670  & 24.111  & 23.718  & 23.112  & 17.172 \\\\\n30&  J231421-344141 & 348.587   & -34.695   & 0.749    & 24.784   & 24.396  & 22.718  & 22.233  & 22.000  & 16.423 \\\\\n31&  J231444-291949 & 348.684  & -29.330   & 0.276    &            & 26.165   & 23.711  & 22.739  & 21.711  & 16.301 \\\\\n32&  J231449-293938 & 348.706  & -29.661   & 1.205    & 24.960  & 25.635  & 23.636  & 23.097  & 21.851  & 16.812 \\\\\n33&  J231508-312105 & 348.784  & -31.351   & 0.244    &            & 25.858  & 24.062  & 23.455  & 22.462   & 17.192 \\\\\n34&  J231508-341955 & 348.787  & -34.332    & 98.088   &            & 25.778   & 23.454   & 22.511   & 21.470  & 15.897 \\\\\n35&  J231555-311458 & 348.979  & -31.249   & 0.472    &            & 26.288  & 23.857  & 23.423   & 22.709   & 17.889 \\\\\n36&  J231604-324740 & 349.020  & -32.795   & 0.151    &            & 26.329  & 24.171  & 23.403  & 22.477  & 16.847 \\\\\n37&  J231617-303200 & 349.073  & -30.533   & 3.779    &            & 25.763  & 23.302   & 22.391  & 21.255  & 15.878 \\\\\n38&  J231632-331953 & 349.134  & -33.332   & 1.342    & 24.946   & 24.309  & 22.669  & 22.263  & 21.888  & 17.111 \\\\\n39&  J231645-301948 & 349.189  & -30.330  & 0.763    &            & 24.803  & 23.166  & 22.959  & 21.994   & 18.09  \\\\\n40&  J231648-303629 & 349.201  & -30.608   & 0.356    &            & 25.818  & 23.713  & 23.062  & 22.658  & 16.939 \\\\\n41&  J231719-315344 & 349.332  & -31.896   & 0.205    & 25.291  & 24.586  & 23.584  & 23.451    & 23.077   &        \\\\\n42&  J231723-301556 & 349.348  & -30.266   & 0.464    &            & 24.145   & 22.748  & 22.487  & 22.705  & 18.284 \\\\\n43&  J231737-313228 & 349.407  & -31.541   & 0.259    & 24.763  & 26.235  & 23.565  & 22.605   & 21.880   & 16.663 \\\\\n44&  J231752-311151 & 349.468  & -31.197   & 0.324    &            & 24.784  & 23.772  & 23.705  & 22.739  & 16.736 \\\\\n45&  J231826-323746 & 349.610  & -32.629    & 0.374    & 24.867  & 25.198  & 23.653  & 23.676  & 22.280  & 16.01  \\\\\n46&  J231828-310408 & 349.618  & -31.069   & 0.197    &            & 24.853  & 23.426   & 23.403  & 22.139  & 17.125 \\\\\n47&  J231850-303818 & 349.708  & -30.638   & 0.2      &            & 25.686  & 22.949  & 22.236  & 21.583  & 16.557 \\\\\n48&  J231853-293420 & 349.722  & -29.572   & 0.361    & 24.365   & 26.231  & 24.235  & 23.583  & 22.366  & 16.635 \\\\\n49&  J232019-294205 & 350.081  & -29.702  & 0.572    &            & 25.674  & 24.177  & 23.777  & 22.864  & 15.97  \\\\\n50&  J232020-343818 & 350.084  & -34.638   & 0.257    &            & 25.217  & 23.124  & 22.267  & 21.1660  & 16.308 \\\\\n51&  J232034-305242 & 350.143  & -30.879   & 0.506    &            & 26.007  & 23.508  & 22.591  & 21.225   & 16.13  \\\\\n52&  J232043-320457 & 350.182  & -32.083   & 3.006    & 25.217  & 25.887  & 23.793   & 23.129  & 22.256  & 16.822 \\\\\n53&  J232130-320208 & 350.377  & -32.036   & 0.294    & 25.261  & 24.562  & 23.421  & 23.215  & 21.940  & 17.03  \\\\\n54&  J232135-320117 & 350.399  & -32.022   & 0.266    &            & 24.778   & 22.919   & 22.309  & 21.839  & 16.359 \\\\\n55&  J232140-311522 & 350.417  & -31.256   & 0.47     &            & 25.638  & 23.374  & 22.705   & 21.997  & 17.304 \\\\\n56&  J232246-331448 & 350.691  & -33.247     & 0.277    & 24.752  & 26.199  & 23.584  & 22.589  & 21.546  & 16.676 \\\\\n57&  J232331-345632 & 350.882  & -34.942   & 0.639    &            & 26.101  & 23.673  & 22.688  & 21.728  & 15.694 \\\\\n58&  J232338-350353 & 350.910  & -35.065   & 0.988    &            & 24.471   & 23.451  & 23.767  & 22.534  & 18.065 \\\\\n59&  J232347-344625 & 350.948   & -34.774   & 0.726    &            & 25.682  & 23.363   & 22.491  & 21.452   & 15.994 \\\\\n60&  J232413-303039 & 351.057  & -30.511   & 0.457    &            & 26.281  & 23.783   & 22.797  & 21.864  & 16.669 \\\\\n61&  J232515-295957 & 351.314  & -29.999   & 1.317    &            & 24.565  & 22.944  & 22.509   & 22.097  & 17.487 \\\\\n62&  J232515-314448 & 351.314  & -31.747   & 1.431    & 24.848  & 26.250  & 23.447  & 22.489  & 21.277  & 16.122 \\\\\n63& J222915-334311  & 337.314  & -33.719   & 1.992    &            & 25.057  & 23.829   & 23.792   & 23.329    &        \\\\\n64& J223018-304213  & 337.576  & -30.704   & 2.954    &            & 25.314  & 23.629  & 23.089   & 22.573  & 17.405 \\\\\n65& J223054-322552  & 337.727 & -32.431    & 75.829   &            & 25.564  & 23.444  & 22.627  & 21.861  & 17.538 \\\\\n66& J223215-301935  & 338.066   & -30.327   & 10.809   &            & 26.243  & 23.488   & 23.135   & 22.872  & 18.29  \\\\\n67& J223508-295809  & 338.785  & -29.969   & 1.081    &            & 24.139  & 22.897   & 22.607  & 22.438  &        \\\\\n68& J223531-291912  & 338.879  & -29.320   & 0.744    & 24.549  & 25.323   & 23.572  & 23.050   & 22.325   & 16.968 \\\\\n69& J223533-343305  & 338.889  & -34.552   & 1.942    & 24.406  & 24.286    & 23.031  & 22.894   & 21.702  & 17.279 \\\\\n70& J223541-311145  & 338.922  & -31.196   & 0.689    &            & 26.473  & 24.012  & 23.065  & 21.906  & 16.757 \\\\\n71& J223710-305549  & 339.294  & -30.930   & 2.54     &            & 25.052  & 23.322  & 23.006  & 22.722  & 16.801 \\\\\n72& J223729-325838  & 339.374  & -32.977  & 0.337    & 24.749  & 25.078   & 23.796  & 23.485  & 22.158   & 16.236 \\\\\n73& J223743-333305  & 339.432  & -33.551   & 1.05     &            & 24.747  & 23.514   & 23.384   & 22.600  & 16.724 \\\\\n74& J223831-330728  & 339.629  & -33.124   & 0.249    &            & 25.445  & 23.981  & 23.727  & 24.024   &        \\\\\n75& J223936-295303  & 339.900  & -29.884    & 2.703    &            & 25.995  & 23.209 & 22.222  & 21.617   & 16.493 \\\\\n76& J224017-325128  & 340.073  & -32.858   & 0.398    & 24.662  & 25.188  & 23.179  & 22.451   & 22.441 & 16.653 \\\\\n77& J224040-295600  & 340.168  & -29.933    & 2.673    & 24.283  & 24.770  & 22.952   & 22.466  & 20.989  & 16.221 \\\\\n78& J224047-331331  & 340.197  & -33.225   & 0.646    &            & 25.308  & 23.958  & 23.689   & 23.193  &        \\\\\n79& J224138-331346  & 340.411  & -33.229    & 0.242    &            & 25.162  & 23.246  & 22.570  & 21.842  & 17.368 \\\\\n80& J224145-340622  & 340.439   & -34.106    & 66.186   &            & 26.2801  & 23.891   & 22.992  & 21.688  & 16.254 \\\\\n81& J224203-333606  & 340.513  & -33.602   & 0.769    & 25.436   & 25.747  & 23.598  & 23.154  & 24.637   & 16.188 \\\\\n82& J224246-335928  & 340.692  & -33.991   & 1.751    &            & 25.566  & 23.892  & 23.647  & 23.481  &        \\\\\n83& J224249-333814  & 340.706  & -33.637   & 0.55     &            & 25.751  & 23.515   & 22.745  & 21.976  & 17.565 \\\\\n84& J224308-313622  & 340.784  & -31.606   & 0.662    & 24.432 & 25.657  & 24.005  & 23.593  & 22.933  & 17.425 \\\\\n85& J224311-332250  & 340.796  & -33.381   & 4.432    &            & 25.643  & 23.491  & 22.924  & 21.404   & 16.056 \\\\\n86& J224354-333900  & 340.976  & -33.650   & 0.272    & 25.384  & 25.028  & 23.303   & 23.028  & 21.557  & 16.502 \\\\\n87& J224401-315431  & 341.0056   & -31.909   & 11.74    &            & 26.160  & 23.101   & 22.309  & 21.1800  & 15.864 \\\\\n88& J224417-350540  & 341.072  & -35.094   & 0.622    & 24.869  & 25.819  & 23.907  & 23.741  & 22.267   & 17.523 \\\\\n89& J224445-314547  & 341.188 & -31.763   & 4.325    &            & 25.334  & 23.252     & 22.967  & 21.969  &        \\\\\n90& J224519-342942  & 341.329  & -34.495   & 0.535    &            & 25.644  & 23.446  & 22.529  & 21.990  & 16.562 \\\\\n91& J224540-313747  & 341.418  & -31.629   & 0.221    & 25.059  & 24.642  & 22.851   & 22.319  & 21.532  & 16.265 \\\\\n92& J224540-330700  & 341.419  & -33.117   & 1.679    & 24.517   & 25.802  & 23.269  & 22.568  & 21.588   & 16.067 \\\\\n93& J224552-312733  & 341.468  & -31.4592   & 0.409    &            & 26.263  & 23.615  & 22.636  & 22.315   & 16.989 \\\\\n94& J224601-310331  & 341.508 & -31.059   & 0.685    & 25.250  & 26.278  & 24.116  & 23.278  & 22.031  & 16.936 \\\\\n95& J224615-285257  & 341.565  & -28.883   & 1.177    &            & 24.328  & 22.526  & 22.237  & 22.035  &        \\\\\n96& J224623-311641  & 341.597  & -31.278     & 0.206    & 24.845   & 25.129  & 23.845  & 23.735  & 22.802  & 18.402 \\\\\n97& J224629-323825  & 341.622 & -32.640  & 0.824    &            & 25.533  & 23.647   & 23.141  & 22.331  & 16.925 \\\\\n98& J224630-343853  & 341.628  & -34.648   & 0.599    & 24.742  & 26.381  & 24.066   & 23.071   & 21.513  & 16.402 \\\\\n99& J224642-311527  & 341.677 & -31.258   & 0.189    & 24.730  & 25.230   & 23.807   & 23.410  & 22.712  &        \\\\\n100& J224710-321939  & 341.792   & -32.328   & 0.476    & 24.249  &            & 23.011   & 22.249   & 21.336  & 16.425 \\\\\n101& J224716-310058  & 341.817  & -31.016   & 0.32     &            & 26.009  & 23.856  & 23.021  & 21.859  & 17.402 \\\\\n102& J224812-335035  & 342.051  & -33.843   & 0.36     & 25.272  & 26.052  & 24.037  & 23.753  & 21.790  & 16.874 \\\\\n103& J224918-314457  & 342.327  & -31.749   & 4.265    &            & 25.877  & 23.815  & 23.540  & 24.035  &        \\\\\n104& J224955-294536  & 342.479  & -29.760     & 0.294    & 24.707  & 26.023  & 23.892  & 23.036  & 22.479  & 16.346 \\\\\n105& J225029-350452  & 342.622  & -35.0811   & 0.663    &            & 25.287  & 23.936  & 23.575  & 22.385  & 16.907 \\\\\n106& J225052-331301  & 342.719  & -33.217   & 0.49     & 24.649  & 25.228  & 23.448  & 22.800  & 21.656  & 16.539 \\\\\n107& J225214-310846  & 343.061  & -31.146   & 2.607    &            & 26.549  & 24.020   & 23.083  & 22.204  & 16.635 \\\\\n108& J225240-302302  & 343.166  & -30.384   & 0.384    &            & 26.053  & 23.786   & 23.082  & 21.805  & 16.802 \\\\\n109& J225257-351113  & 343.238  & -35.187   & 0.275    & 25.510  & 25.626  & 23.263  & 22.320  & 21.177  & 16.124 \\\\\n110& J225300-302322  & 343.253  & -30.389   & 0.49     &            & 24.503  & 23.148  & 22.869  & 22.856  &        \\\\\n111& J225312-305344  & 343.303  & -30.896   & 0.188    & 24.612  & 25.492  & 23.359  & 22.624 & 22.381  & 16.758 \\\\\n112& J225314-295139  & 343.311  & -29.861   & 0.365    &            & 25.334  & 23.804  & 23.712  & 24.634  &        \\\\\n113& J225332-313319  & 343.386  & -31.555   & 0.242    & 24.613  & 25.953  & 23.599   & 23.023   & 21.794  & 16.304 \\\\\n114& J225343-313305  & 343.429  & -31.552  & 0.367    &            & 25.312  & 23.191  & 22.481  & 21.788   & 16.938 \\\\\n115& J225647-285027  & 344.196  & -28.841   & 2.049    &            & 24.798  & 23.615   & 23.382   & 22.680  & 17.25  \\\\\n116& J225733-313857  & 344.388  & -31.649   & 0.249    &            & 25.377  & 23.456  & 22.837  & 21.364  & 16.001 \\\\\n117& J225827-342715  & 344.614  & -34.454   & 0.286    & 25.139  & 26.231  & 24.271  & 23.650  & 22.228   & 16.669 \\\\\n\n\n\\hline\n\\multicolumn{11}{c}{$z\\sim5$ Sample  ($r$ dropouts)} \\\\\n\\hline\n\n1&  J231423-331509 & 348.596  & -33.253    & 0.232    &            & 26.248  & 24.738 & 23.642  & 23.575  & 16.47  \\\\\n2& J224105-345956  & 340.271  & -34.999   & 0.261    &            & 25.664  & 24.804  & 23.343  & 23.205  & 16.581 \\\\\n3&  J232503-340057 & 351.264  & -34.016    & 0.293    & 25.392  &            & 24.578 & 23.471  & 23.267  & 17.989 \\\\\n4&  J231919-320058 & 349.831  & -32.016  & 0.294    &            &            & 25.372  & 23.793  & 23.308  & 17.586 \\\\\n5& J224820-301317  & 342.086  & -30.221    & 0.295    & 24.808  & 25.333  & 24.632   & 23.402    & 23.171 & 17.793 \\\\\n6&  J230855-335352 & 347.229  & -33.898   & 0.323    &            &            & 24.749  & 23.372  & 23.063 & 17.508 \\\\\n7& J224858-343550  & 342.242  & -34.597   & 0.356    & 25.101   & 26.083  & 24.681  & 23.457  & 23.569     &        \\\\\n8&  J232021-315838 & 350.090  & -31.977   & 0.382    &            & 25.374  & 24.706  & 23.483   & 23.327  & 17.392 \\\\\n9&  J230235-292456 & 345.648  & -29.415    & 0.443    &            & 25.676  & 24.440  & 23.003 & 23.181   & 16.789 \\\\\n10&  J232035-314851 & 350.149  & -31.814   & 0.453    & 25.137  & 25.209  & 24.388  & 23.235    & 23.167  &        \\\\\n11& J225340-300626  & 343.417  & -30.107   & 0.604    & 24.703  & 26.143  & 24.253   & 23.164  & 23.186  & 16.329 \\\\\n12&  J230551-343338 & 346.463  & -34.561   & 1.248    &            & 26.491  & 25.365  & 23.768  & 23.568   & 15.901 \\\\\n13&  J230946-314050 & 347.446  & -31.681   & 1.925    & 25.160  & 25.341  & 24.632   & 23.556  & 23.590   & 16.995 \\\\\n14&  J230301-305405 & 345.755  & -30.902   & 48.28    & 24.559  & 25.334  & 24.356  & 22.638  & 23.045 & 17.474 \\\\\n\n\\hline\n\\multicolumn{11}{c}{$z\\sim6$ Sample  ($i$ dropouts)} \\\\\n\\hline\n\n1& J223445-332701  & 338.687  & -33.450   & 0.418    & 25.773  &            & 24.113  & 24.442   & 22.824  & 18.457 \\\\\n2& J223719-333857  & 339.330  & -33.649   & 0.253    & 24.446   & 25.272  & 24.209 & 24.738  & 22.934  & 16.201 \\\\\n3& J224115-303408  & 340.313  & -30.569   & 1.534    &            & 25.440   & 24.481  & 24.176  & 22.531  & 17.176 \\\\\n4& J224130-302918  & 340.377 & -30.488   & 0.414    & 24.261 & 25.51  & 24.133  & 23.502  & 22.001  & 16.495 \\\\\n5& J224652-340238  & 341.717  & -34.044   & 0.364    &            & 25.277  & 24.919  & 24.649  & 22.740  & 17.127 \\\\\n6& J224957-332626  & 342.487  & -33.441   & 0.348    &            &            & 24.518  & 24.346  & 22.288  & 17.592 \\\\\n7&  J230246-293923 & 345.693  & -29.656   & 0.255    & 24.531  & 25.785  & 24.542  & 24.67  & 23.152  & 17.14  \\\\\n8&  J230423-322732 & 346.098  & -32.459    & 0.323    &            & 25.833  & 24.599  & 24.326  & 22.393  & 17.412 \\\\\n9&  J230617-335108 & 346.574  & -33.852   & 1.338    & 24.927  & 26.045    & 24.540 & 24.729  & 22.707   & 17.383 \\\\\n10&  J231051-334628 & 347.713  & -33.775   & 2.471    &            & 25.472  & 24.659  & 25.338 & 23.557 & 18.277 \\\\\n11&  J231125-342657 & 347.857  & -34.449   & 0.212    & 25.349   & 25.270   & 24.544  & 24.442  & 22.541  & 16.731 \\\\\n12&  J231245-291817 & 348.188  & -29.305   & 0.447    &            & 25.961  & 24.485  & 24.084 & 22.301  & 15.885 \\\\\n13&  J231004-334153 & 347.519  & -33.698   & 0.287    &            & 25.541  & 25.099  & 24.757 & 23.125  &        \\\\\n14& J223430-311836  & 338.627  & -31.310   & 1.23     & 25.057  & 25.216  & 24.858 & 25.361  & 23.528 &        \\\\\n15&  J232457-322915 & 351.241  & -32.488   & 0.353    & 25.209    & 25.133  & 24.645   & 24.775   & 23.102  &        \\\\\n\n\n \\hline\n \\label{tab:catalog}\n\\end{longtable}\n\\twocolumn\n\n\\begin{table*}\n    \\centering\n    \\caption{  Our final $z\\sim7$ sample  ($Z$ dropouts), showing radio properties from \\ac{ASKAP} and IR photometry from VIKING and CATWISE catalogues respectively. Missing WISE magnitudes (W1\/W2) indicate no counterpart within 2~arcsec of the VIKING position. pGalaxy indicates the probability that the source is a galaxy, obtained from the VIKING DR5 catalogue.}\n    \\begin{tabular}{cccccccccccc}\n    \\hline\n  Index & \\ac{ASKAP} &RA & DEC & S\\textsubscript{887.5} & Mag\\_Z & Mag\\_Y & Mag\\_J & Mag\\_Ks & W1\\_mag & W2\\_mag &pGalaxy \\\\\n  & name & (deg) & (deg) & (mJy) & (AB) &(AB) & (AB)&(AB) &(Vega) & (Vega) &  \\\\\n         \\hline\n1 &  J230535--341213 & 346.399453 & -34.203674 & 0.251 & 22.53 &21.16  &21.65  &23.08 & -- &-- & $9.5\\times10^{-6}$  \\\\\n2 &  J223833--320822 & 339.63998 & -32.139507 & 0.335 & 21.41 &20.19 &19.86 &21.99 &17.09 &16.10 &0.00017  \\\\ \n3 &  J231818--311345 & 349.576502 & -31.229435 & 0.662 &21.91  &20.78 &20.79 &22.35 &18.02 &18.11 &0.000171 \\\\ \n\\hline\n    \\end{tabular}\n    \\label{tab:z7_sample}\n\\end{table*}\n\n\\section{Discussion}\n\\label{sec:discuss}\n\n\\subsection{Radio flux density}\n\\label{sec:radio_f_de}\n\n\\begin{figure}\n \\centering\n\n \\includegraphics[width=0.45\\textwidth]{flux_density_dist.png} \n \\caption{887.5\\,MHz Total radio flux density distribution of our final sample of   148 \\ac{HzRS} candidates and a known radio-quasar (selected as a Z-dropout).}\n \\label{fig:fluxD}\n\\end{figure}\n\\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_1p3_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p9_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p7_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p4_final.png} \n\\end{subfigure}\\\\\n\\vspace{0.3cm}\n\\begin{subfigure}{.55\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{L_1p4_0p2_final.jpg} \n\\end{subfigure}\\vspace{-0.2cm}\n\\caption{ The 1.4\\,GHz radio luminosity distribution of our sample estimated for a set of spectral indices and known radio-\\ac{AGN}s (Appendix~\\ref{A}, yellow stars) as a function of redshift. Irrespective of spectral index, our sample is 1-2 orders of magnitude less luminous than known \\ac{AGN}s at the similar redshift. }\n\\label{fig:FD}\n\\end{figure*}\n\nThe observed 887.5\\,MHz flux density distribution of our final sample is shown in Figure~\\ref{fig:fluxD}. Our sample is selected from the G23 \\ac{ASKAP} radio catalogue and no additional radio properties are used in the selection.\nThe distribution of flux densities of such a flux-limited sample is expected to peak at low flux densities, and this is seen in our \\acp{HzRS} sample, which peaks at 0.2--0.4\\,mJy.\n\n Although the $z \\sim 5$ and $z \\sim 6$ sample  are smaller, they also peak at low flux densities and follow a distribution similar to that of $z\\sim4$. We  confirmed the statistical significance of this similarity by performing a Kolmogorov-Smirnov (KS) test, comparing the $z \\sim 5$ and $z \\sim 6$ samples with the $z \\sim 4$ sample. Using the KS test, the probability that the $z \\sim 5$, $z \\sim 6$, and $z\\sim7$ flux density distributions are drawn from the same population as the $z \\sim 4$ sample is 0.44, 0.31, and 0.65. Thus, the test suggests that the $z \\sim 4$, 5, 6, and 7 sources are most likely to represent a similar type of radio source population.\n\n\n \n\nWe calculate the rest-frame 1.4\\,GHz luminosity of our sources from the total flux density observed at 887.5\\,MHz, using Equation~\\ref{eq:lum_1.4GHz}, by assuming a power-law ($S_\\nu \\propto \\nu^\\alpha$) radio spectral energy distribution (SED) and adopting a  set of radio spectral indices $\\alpha =\\{-1.3, -0.9, -0.7, -0.4, 0.2 \\}$, as indicative of our complete \\acp{HzRS} sample. We assumed redshift, $z=$ 4, 5, 6, and 7 for \\textit{g, r, i,} and Z dropouts respectively in the Equation~\\ref{eq:lum_1.4GHz}. The luminosity distance (D\\textsubscript{L}) is calculated using the online cosmology calculator \\citep{wright2006cosmology}. In other words:\n\n\\begin{align}\n  L_{\\nu_{1}} &= \\frac{4\\pi D_\\textrm{L}^{2}}{(1+z)^{(1+\\alpha)}} \\left ( \\frac{\\nu_{1}}{\\nu_{2}}\\right)^{\\alpha}S_{\\nu_{2}} (\\textrm {W\/Hz}),\n  \\label{eq:lum_1.4GHz}\n\\end{align}\nwhere L\\textsubscript{$\\nu$\\textsubscript{1}} is the radio luminosity at rest-frame $\\nu\\textsubscript{1}$ derived from the observed flux density S\\textsubscript{$\\nu$\\textsubscript{2}} at $\\nu\\textsubscript{2}$ and  (1+z)\\textsuperscript{-($1+\\alpha$)} denotes the standard radio K-correction.\n\n The resulting 1.4\\,GHz luminosity of our sources for a given spectral index, shown in Figure~\\ref{fig:FD},  as a function of redshift indicates that our sample probes less powerful \\acp{HzRS} compared to known radio sources (listed in the Appendix) at the same redshift. This is expected because of the greater sensitivity of the \\ac{ASKAP} survey. Very little is known about the properties of \\acp{HzRS} with $L_{1.4} < 10^{26}$\\,W\/Hz as the known \\acp{HzRS} are mostly brighter. \n\nWe therefore used the following diagnostics to investigate the physical origin of radio emission in our sample: (i) radio luminosity at 1.4\\,GHz, (ii) 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio, (iii) WISE colour, (iv) FIR detection at 250\\,$\\mu$m, and (v) SED modelling. \n\n\n\n\n\n\n\\subsection{1.4~GHz luminosity diagnostic }\n\nIf we assume that all the radio emission is caused by star formation processes, we can calculate the \\ac{SFR} following \\citet{bell03},\n \\begin{align}\n SFR(L\\textsubscript{1.4})& = L\\textsubscript{1.4} \\times 5.52 \\times 10^{-22} M\\textsubscript{$\\odot$}\/yr.\n \\label{eq:sfr}\n\\end{align}\nWe show the results in Table~\\ref{tab:sfr}.\n\nAmong extragalactic objects, \\acp{SMG} have the highest reported \\ac{SFR}: $\\sim 6000~$M\\textsubscript{$\\odot$}\/yr \\citep{barger2014there}. By comparison, most Lyman dropouts do not have as high \\ac{SFR} as SMGs, with a maximum of $\\sim 300~$M\\textsubscript{$\\odot$}\/yr \\citep{barger2014there}. Furthermore,  \\citet[Figure~23]{barger2014there} demonstrated a turn-down in the observed \\ac{SFR} distribution function beyond 2000\\,M\\textsubscript{$\\odot$}\/yr. In Table~\\ref{tab:sfr}, we present the \\acp{SFR} estimated  for each dropout at the following two flux densities at a given spectral index: (i) the minimum of 887.5\\,MHz flux density distribution for each dropout (ii) flux density which marks the  exceeding of \\ac{SFR} beyond the highest known limit (6000\\,M\\textsubscript{$\\odot$}\/yr); \\cite{barger2014there}. In some cases, the minimum of 887.5\\,MHz flux density distribution itself results in an unphysical SFR. \nThus, most of the \\ac{SFR} shown in Table~\\ref{tab:sfr} exceed by far the highest-known \\ac{SFR} of a galaxy \\citep{barger2014there}, and we conclude that most of the radio emission from these galaxies is unlikely to be primarily generated by star formation,  implying the presence of a radio-\\ac{AGN} as well.\n\nThis finding is also consistent with the simulation \\citep{bonaldi2019tiered,wilman2008semi} of the redshift evolution of radio sources, which predicts that star forming galaxies (SFGs) are a negligible fraction of the observed radio sources beyond redshift $z > 2$ at the flux limit of 0.1\\,mJy. \n  We acknowledge that \\textit{radio}-\\ac{SFR} of the faintest end ($S_{887}\\le0.2$\\,mJy; 5 sources) of our $z\\sim4$ sample lie in the range $\\sim$4000-8000\\,M\\textsubscript{$\\odot$}\/yr,  which is very high but in priciple possible given that estimated space density of \\acp{SMG} with SFRs above 2000\\,M\\textsubscript{$\\odot$}\/yr is  $1.4\\times10^{-5}$\\,M\\textsubscript{$\\odot$}\/yr\/Mpc\\textsuperscript{3} \\citep{fu2013rapid}. Therefore, it is likely that faintest end of $z\\sim4$ sample  have a \\ac{SB} component or be pure \\acp{SB}.   \n\n \n \n\n\\begin{table}\n \\centering\n \\caption{ \n \\ac{SFR} estimated for different flux densities at $z \\sim 4$, 5, 6 and 7, assuming all radio emission is powered by star formation. \\ac{SFR} is calculated using Equations \\ref {eq:lum_1.4GHz} at spectral index, $\\alpha = -0.8$, -0.7, -0.6 given that mean spectral index of \\acp{SFG} is $\\sim0.75$ }\n \\begin{tabular}{c|c|c|c|c|c}\n \\hline\n Redshift & D\\textsubscript{L} & S\\textsubscript{887.5}& S\\textsubscript{1.4}  &L\\textsubscript{1.4} & SFR\\textsubscript{1.4}\\\\\n  $z$ & (10\\textsuperscript{3}Mpc) & (mJy) & (mJy) &  (W~Hz$^{-1}$) & (10\\textsuperscript{3}M\\textsubscript{$\\odot$}\/yr)\\\\\n  \\hline \n \n \n \n\n   \\multicolumn{6}{c}{$\\alpha=-0.8$} \\\\\n \\hline\n 4 & 36.6 & 0.15 & 0.10 &1.211$\\times 10^{25}$ & 6.67 \\\\\n\n 5 & 47.6   &0.2 &0.14 & 2.63$\\times 10^{25}$ &14.55 \\\\\n\n 6 & 58.98  & 0.2 &0.14 &3.92$\\times 10^{25}$  &21.62 \\\\\n \n7 & 70.54   & 0.2 &0.14 &5.45$\\times 10^{25}$  &30.11 \\\\\n\n  \\hline\n  \n   \\multicolumn{6}{c}{$\\alpha=-0.7$} \\\\\n \\hline\n  \\multirow{2}{*}{4} & \\multirow{2}{*}{36.6} & 0.15 & 0.11 &1.07$\\times 10^{25}$ & 5.95 \\\\\n       &       & 0.2 & 0.15  &1.44$\\times 10^{25}$ & 7.93\\\\\n\\hline\n  5 & 47.6  &0.2 &0.15 & 2.31$\\times 10^{25}$ &12.73 \\\\\n\n  6 & 58.98  & 0.2 &0.15 &3.37$\\times 10^{25}$  &18.63 \\\\\n \n7 & 70.54  & 0.2 &0.15 &4.64$\\times 10^{25}$  &25.53 \\\\\n  \\hline\n  \n   \\multicolumn{6}{c}{$\\alpha=-0.6$} \\\\\n \\hline\n  \\multirow{2}{*}{4} & \\multirow{2}{*}{36.6} & 0.15 & 0.11 &6.9$\\times 10^{24}$ & 3.84 \\\\\n       &       & 0.24 & 0.18  &1.11$\\times 10^{25}$ & 6.15\\\\\n\\hline\n\n 5 & 47.6   &0.2 &0.15 & 1.41$\\times 10^{25}$ &7.78 \\\\\n\n 6 & 58.98  & 0.2 &0.15 &1.97$\\times 10^{25}$  &10.87 \\\\\n \n7 & 70.54   & 0.2 &0.15 &2.6$\\times 10^{25}$  &14.36 \\\\\n\n  \\hline\n \\end{tabular}\n \\label{tab:sfr}\n\\end{table}\n\n\n\n\\subsection{Radio-to-3.4\\,$\\mu$m flux density ratio}\n\\label{sec:NIR}\n\n\n\\citet[][Figure 4]{norris2011deep} presented a new measure for radio-loudness of a source, 1.4\\,GHz-to-3.6\\,$\\mu$m flux density ratios. They compared the 1.4\\,GHz-to-3.6\\,$\\mu$m flux density ratios for different galaxy populations  namely, \\acp{HzRG}, \\acp{SMG}, \\acp{SB}, RL \\& RQ quasars and \\acp{ULIRG}, as a function of redshift.  They showed that galaxies powered primarily by star forming activity have a lower ratio in contrast to radio-AGNs at all redshifts. \n We use 3.4\\,$\\mu$m photometry from CATWISE2020 catalogue \\citep{marocco2021catwise2020} since there is no substantial difference between 3.4 \\& 3.6\\,$\\mu$m passbands, and 3.4\\,$\\mu$m magnitude (W1) is converted to flux density \\citep{cutri2012explanatory} using,\n\\begin{align}\n  S\\textsubscript{3.4\\,$\\mu$m} = 306.682 \\times 10^{-W1\/2.5} ~ \\textrm{Jy.}\n\\end{align}.\n\n\nWe compare \\cite{norris2011deep} model with the 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio of our sample at redshifts, $ z \\sim 4$, 5, and 6 in Figure~\\ref{fig:radio_ir_ratio}.  The measured flux density at 887.5\\,MHz was used to estimate the 1.4\\,GHz flux density, assuming spectral indices,  $\\alpha$ of -1.3, -0.7, and 0.2 as indicative of our entire \\ac{HzRS} sample.\nAccording to that model, radio-\\acp{AGN} have a radio-to-3.6\\,$\\mu$m flux density ratio above 100 at all redshifts. However, \\cite{maini2016infrared} showed that the ratio for Type 1 \\& 2 \\ac{AGN}\/QSO and high power radio galaxies start to decrease (between 10 and 100) beyond $z\\sim4$ and $z\\sim5$ . We contend that reason for this difference is that the \\cite{norris2011deep} model refers to \\emph{powerful} radio-\\ac{AGN}s only, at all redshifts.  About $55\\%$ of our sample have a 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio between 1 and 10 and $\\sim41\\%$ between 10 and 100. Only 5 sources in our sample have a radio-to-3.4\\,$\\mu$m ratio greater than 100. \n\n\nIt therefore seems that high-$z$ radio-\\ac{AGN}s can extend to even lower values ($<10$) of radio\/IR ratios than previously thought.  The following are some possible explanations for their low ratio: (i) radio-\\ac{AGN} is accompanied by SB activity, and the optical emission being redshifted to W1 band causing an increase in the 3.4\\,$\\mu$m flux density (ii) the inherently low radio powers of high-$z$ radio-\\acp{AGN}, as predicted by simulation in \\cite{saxena2017modelling} (iii) optical emission from un-obscured quasars at $z\\ge4$ being redshifted to W1 band. We discuss this further in Section \\ref{sec:FIR}.\n\n\n \\begin{figure*}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_1p3.png} \n\\end{subfigure}\n\\hspace{0.1cm}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_0p7.png} \n\\end{subfigure}\n\\hspace{0.1cm}\n\\begin{subfigure}{.32\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{ratio_0p2.png} \n\\end{subfigure}\n\n \\caption{ 887.5\\,MHz flux density vs. 3.4\\,$\\mu$m flux density for $z\\gtrsim4$ sample, colour coded according to their 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio. 1.4\\,GHz flux density is estimated from observed flux density at 887.5\\,MHz assuming spectral indices, $\\alpha = -1.3$, -- 0.7, 0.2.  }\n \\label{fig:radio_ir_ratio}\n\\end{figure*}\n\n\n\n\n\n\n\n\n\\subsection{FIR detection}\nDust heated by stars or \\ac{AGN} generally re-radiates the absorbed energy at FIR wavelengths. Each galaxy will have a distribution of dust temperatures which is determined by the size and distribution of dust with respect to the heating source (\\ac{SF} or \\ac{AGN}). Cool ($T\\sim20\\,K$) dust arises from the diffuse interstellar medium (ISM), warm ($T\\sim40\\,K$) dust from SF regions, and even warmer $T\\sim60-100\\,K$ dust results from \\ac{AGN} activity. \n\nWe queried the {\\it Herschel} database to check whether any of our sources have been detected at FIR wavelengths. We searched the SPIRE point source catalogues at 250\\,$\\mu$m, 350\\,$\\mu$m, and 500\\,$\\mu$m \\citep{schulz2017spire} separately in a 5~arcsec cone around the WISE position of each of our HzRS sample,  which yielded a false-id rate of $\\sim16\\%$. Several of our sources have a SPIRE counterpart, but none was found in the PACS point source catalogues.  A total of 14 and 3 sources from our $z\\sim4$ and $z\\sim6$ sample respectively were found to have a SPIRE counterpart, representing only 10\\% of our entire \\ac{HzRS} sample.\n \nWe present the results in Table~\\ref{tab:spire_sources}.  The SPIRE 250\\,$\\mu$m band probes a rest wavelength $\\sim$45 - 63\\,$\\mu$m  at $z=3-4.3$ ($g$ dropouts) and $\\sim$34 - 38\\,$\\mu$m  at $z=5.6-6.2$ ($i$ dropouts). At wavelength $<50$\\,$\\mu$m in the rest-frame, an \\ac{AGN} heated dust torus \nmay contribute significantly to the FIR emission.\nTherefore the SPIRE detection of our radio sources suggests that the observed 250\\,$\\mu$m emission may arise from either (i) a dust torus heated by an \\ac{AGN} or (ii) \\ac{SB} activity or (iii) a combination of both \\ac{SF} and \\ac{AGN}.\n\n\\begin{table}\n    \\centering\n     \\caption{Our \\acp{HzRS} with a FIR detection in either of {\\it Herschel}-SPIRE 250\\,$\\mu$m, 350\\,$\\mu$m $\\&$ 500\\,$\\mu$m bands, selected from a 5~arcsec search around their WISE position at which false-ID rate is $\\sim16\\%$.  The probability of a source's measured SPIRE flux being contaminated by a nearby neighbour is about 1\/14.}\n    \\begin{tabular}{c|c|c|c}\n    \\hline\n        Source ID & S\\textsubscript{250} & S\\textsubscript{350} & S\\textsubscript{500}  \\\\\n         &  (mJy) & (mJy) & (mJy)   \\\\\n        \n        \\hline\n\\multicolumn{4}{c}{$z\\sim4$ candidates}        \\\\\n\\hline\n\nJ224308-313622 & 59.3 &\t &  \\\\\nJ223533-343305 & 67.4 & 40.7 &  --    \\\\\nJ231210-332436 & 49.9 &\t32.3 &   --   \\\\\nJ224203-333606 & 59.5 &\t68.5 &  --    \\\\\nJ231826-323746 & 39.9 & -- & -- \\\\\nJ223729-325838 & 34.8 & -- & -- \\\\\nJ224017-325128 & 66.2 & -- & -- \\\\\nJ224552-312733 & 51.2 &--   &--   \\\\\nJ225343-313305 & 65.4     & --  & --  \\\\\nJ231648-303629 & 65.6     & --  &--   \\\\\nJ232019-294205 & 93.6     &77.2   &--   \\\\\nJ232135-320117 & 44.5     & --   &   \\\\\nJ225915-314116 & --& 78.1 & 62.9 \\\\\nJ224955-294536 & -- & -- & 49 \\\\\n\n\\hline\n\\multicolumn{4}{c}{$z\\sim6$ candidates}  \\\\\n\\hline \nJ224652-340238 & 92.9 & 62.4 & 47.9 \\\\\nJ231245-291817 & 98.5 & 86.8 & 48.9 \\\\\nJ223719-333857 & 65.6 & 52.5 & -- \\\\\n\\hline\n    \\end{tabular}\n    \\label{tab:spire_sources}\n\\end{table}\n\n\\subsubsection{Radio-FIR relation}\n\n\nIn this section, we explore the 250\\,$\\mu$m-to-1.4\\,GHz luminosity ratio of our sources, as defined by Equation~\\ref{eq:q250},  which is a tracer of star formation activity. Here we are only interested in order-of-magnitude estimates and hence we neglect evolution. Another paper in which we model evolutionary effects is in preparation. \n$q\\textsubscript{250}$ is defined as:\n\\begin{align}\n  q_{250} = \\log_{10}\\left (\\frac{L_{250}}{L_{1.4}}\\right ) ~,\n  \\label{eq:q250}\n\\end{align}\nwhere $L\\textsubscript{250}$ and $L\\textsubscript{1.4}$ are the rest-frame luminosities at 250\\,$\\mu$n and 1.4\\,GHz respectively.\n\nTo estimate rest-frame $L\\textsubscript{250}$, we must first estimate the \nK-correction (k\\textsubscript{corr}) defined as \n\\begin{align}\n  k_{corr} = \\frac{S_{rest}}{S_{obs}} ~,\n  \\label{eq:k_corr}\n\\end{align}\nwhere $S_{\\rm obs}$ and $S_{\\rm rest}$ are the observed and the rest-frame flux densities respectively. \n\nAssuming a greybody thermal emission, the spectral flux density at a given frequency and temperature, $S_{\\nu}(T)$, would be a modified Planck's radiation law $B(\\nu,T)$,\n\\begin{align}\n S_{\\nu}(T) &\\propto \\nu^{\\beta}B(\\nu,T) ~, \\label{eq:S_nu}\\\\  \\intertext{where}\n B(\\nu,T) &= \\frac{2h}{c^{2}}\\frac{\\nu^{3}}{e^{\\frac{h\\nu}{k_{B}T}}-1}\n \\label{eq:planck}\n\\end{align}\n and $\\beta$ represents the dust emissivity index, for which we assume a value of  1.5 \\citep{kirkpatrick2015role}.\n\n\nWe calculate the K-correction using Equations~\\ref{eq:k_corr}, \\ref{eq:S_nu} and \\ref{eq:planck}, assuming 2 dust temperatures, 45\\, K, $\\&$ 80\\,K, corresponding to warm and warmer dust components  based on \\cite{kirkpatrick2015role} study, and present the results in Table~\\ref{tab:k_corr}.\n\\begin{table}\n \\centering\n \\caption{K-correction estimated for the observed frame at 250\\,$\\mu$m,  assuming the dust emission either dominated by a warmer dust ($T_{w} \\sim$ 80\\,K) or a warm dust ($T_{w} \\sim$ 45\\,K).}\n\n \\begin{tabular}{c|c|c|c|c}\n \\hline\n  redshift & $\\nu$\\textsubscript{obs} & $\\lambda$\\textsubscript{rest} & $\\nu$\\textsubscript{rest}  & k\\textsubscript{corr} \\\\\n  ($z$) & (Hz) & ($\\mu$m) & (Hz)  & \\\\\n\\hline\n  \\multicolumn{5}{c}{$T_{dust} \\sim 45\\,K$} \\\\\n  \\hline\n   4 &\\multirow{3}{*}{$1.2\\times10^{12}$} &50 & $5.9\\times10^{12}$ &  6.22 \\\\\n  5 & & 42 & $7.2\\times10^{12}$  & 3.8 \\\\\n  6 & &36 & $8.4\\times10^{12}$  &  2.11\\\\\n  \\hline\n  \\multicolumn{5}{c}{$T_{dust} \\sim 80\\,K$} \\\\\n  \\hline\n  4 &\\multirow{3}{*}{$1.2\\times10^{12}$} &50 & $5.9\\times10^{12}$ &  40.77 \\\\\n  5 & & 42 & $7.2\\times10^{12}$  & 44.93 \\\\\n  6 & &36 & $8.4\\times10^{12}$  & 43.46 \\\\\n  \\hline\n \\end{tabular}\n \\label{tab:k_corr}\n\\end{table}\n\nWe then calculate the luminosity $L_{250}$,\n\\begin{align}\n  L_{250} = \\frac {4\\pi S_{250}K_{cor} D_\\textrm{L}^{2}}{1+z} ~, \n  \\label{eq:L250}\n\\end{align}\nwhere $S_{250}$ is the observed flux densities of our sources in the SPIRE database and D\\textsubscript{L} is the luminosity distance. \n\n\n\n The resulting $q\\textsubscript{250}$ estimated for two extremes of assumed radio spectral indices (see Section~\\ref{sec:radio_f_de}) is shown in Figure~\\ref{fig:q250}. All sources with a reliable detection as indicated by SNR>5 and 3>SNR>5 have q$_{250}$ exceeding the threshold of 1.3 \\citep{jarvis2010herschel, virdee2013herschel} indicating luminous SFG.   The high values of $q\\textsubscript{250}$,  irrespective of radio spectral index, suggest the presence of an active \\ac{SB} component in SPIRE detected sources but on the other hand our K-correction is very uncertain because of our lack of knowledge of the FIR SED, and so we do not consider this a definitive indicator of SF. Furthermore, SPIRE detected sources have $S_{887.5} > 0.2$\\,mJy implying the presence of a radio-\\ac{AGN} component according to Table~\\ref{tab:sfr}.   Therefore, we attempt to estimate \\ac{SFR} (see Section~\\ref{sec:FIR}), from the observed 250\\,$\\mu$m emission to verify whether the resulting SFR is sufficient enough to generate the observed radio luminosity. We also note  that q\\textsubscript{250} criterion is based on the study of powerful radio-\\acp{AGN} in the literature, whereas this study probe low power radio-\\acp{AGN}, suggesting q$_{250}>1.3$ may not be applicable in this case.  \n\n\n\n \\begin{figure*}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_45k_1p3_final.png} \n\\end{subfigure}\n\\hspace{0.3cm}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_45k_0p2_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_80k_1p3_final.png} \n\\end{subfigure}\n\\begin{subfigure}{.45\\textwidth}\n \\centering\n\n \\includegraphics[width=\\textwidth]{q250_80k_0p2_final.png} \n\\end{subfigure}\n \\caption{ 887.5\\,MHz flux density vs. 250\\,$\\mu$m flux density for $z\\gtrsim4$ sample, colour coded according to their q\\textsubscript{250} measurements.  }\n \\label{fig:q250}\n\\end{figure*}\n\n\\subsubsection{\\ac{SFR} from S\\textsubscript{250}}\n\\label{sec:FIR}\n\nWe estimate the SFR from S\\textsubscript{250} via the rest-frame 24\\,$\\mu$m luminosity, $L_{24\\,\\mu m}$ (in erg\/s), using the following equations from \\citet{brown2017calibration},\n\\begin{align}\n \\log L_{24_{\\mu m}} &= 40.93+1.3(\\log L_{H\\alpha} - 40)\n \\label{eq:Halpha}\\\\\n SFR &= L_{H\\alpha}\\times5.5\\times10^{-42}~\\rm{M_\\odot \/yr}~.\n  \\label{eq:sfr_ha}\n\\end{align}\n\nFor the calculation of $\\text{L}_{24\\,\\mu m}$, we consider 4 different spectral energy distribution (SED) models corresponding to: (i) an ultra luminous infrared galaxy (Arp~220), (ii) a starburst (M\\,82),  (iii) Mrk\\,231 (luminous infrared \\ac{AGN}), and (iv) a composite system (IRAS F00183--7111), in which the optical\/IR SED is dominated by the starburst surrounding the \\ac{AGN}, with the \\ac{AGN} emerging only at radio wavelengths \\citep{norris12}. In each case, we interpolate the SED using flux density measurements obtained via NASA\/IPAC Extragalactic Database (NED). \n\n\nWe perform the following calculations to estimate $\\text{L}_{24\\,\\mu m}$ of our sources:\n\n\\begin{enumerate}\n \\item We calculate the luminosity of our sources corresponding to their emitted wavelengths from their observed flux density at 250\\,$\\mu$m using the equation,\n \n \\( L_{\\lambda} = \\frac{4\\pi D_\\textrm{L}^{2}S_{250}}{1+z}\\), \\\\\n where $\\lambda$ represents the respective emitted (or rest-frame) wavelength at $z\\sim4,5,6$ as shown in column 3 of  Table~\\ref{tab:k_corr}.\n \\item For each model source (Arp~220, M\\,82, Mrk\\,231, IRAS F00183--7111) we assume that the ratio $L_{24\\mu m}\/L_{\\lambda}$ for our sources is the same as that of the model source as measured from their SED: \n \n \n  \\(\\left(\\frac{L_{24\\mu m}}{L_{\\lambda}}\\right)_{our source}\\) =\n  \\(\\left(\\frac{S_{24\\mu m}}{S_{\\lambda}}\\right)_{model}\\).\n \n\\end{enumerate}\n\n\n \n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[trim={15 210 15 0},width=\\textwidth]{SFR_z4z6.pdf}\n    \\caption{ \\ac{SFR} estimated from S\\textsubscript{250} by assuming various SED models.}\n    \\label{fig:sfr}\n\\end{figure*}\nWe present the results of this calculation in Figure~\\ref{fig:sfr}. The resulting \\ac{SFR}s  for $z\\sim4$ \\& $z\\sim6$ samples lie in the range $\\sim 400-3500$\\, M\\textsubscript{$\\odot$}\/yr \\& $\\sim 2000-8200$\\, M\\textsubscript{$\\odot$}\/yr respectively, which is high but not unphysical, and similar to the \\ac{SFR}s reported for \\ac{SB}s and SMGs at similar redshifts \\citep{barger2014there}. These SFRs would generate a radio power in the order of  $\\sim 10^{24}-10^{25}$~W~Hz$^{-1}$, which is typically a small fraction ($\\le 10\\%$) of the observed radio luminosities.\n\n We conclude that these galaxies detected by \\textit{Herschel}-SPIRE are likely to be composite galaxies containing both a radio-\\ac{AGN} and a starburst. We further note that the radio-to-3.4\\,$\\mu$m flux density ratio of SPIRE detected sources is $<100$. 7 out of 15 250\\,$\\mu$m detected sources found to have $W1-W2 > 0.8$ WISE colour, suggesting an \\ac{AGN} component. At $z\\gtrsim4$, W1 WISE band observations probe the optical rest-frame including H$\\alpha$ emission, an indicator of SF activity or  quasar light or both. This suggests that a fraction of  high-$z$ radio-\\ac{AGN}s ($\\log L\\textsubscript{1.4}=25$) at $z\\gtrsim4$ may have a \\ac{SB} host, contrary to low-z radio-\\ac{AGN}s which are usually hosted by quiescent galaxies. Similarly, \\cite{rees2016radio} \nfound that $z > 1.5$ radio-\\ac{AGN}s tend to be hosted by SFGs. Our study suggests that this may be true at even higher redshifts.\n\n\n \n\\subsection{WISE colours}\n\\label{sec:wise_colours}\n\n\n\nMid-IR colour selection criteria are an efficient tool to identify a hot accretion disk, which is an indicator of \\ac{AGN} in galaxies. The dust reprocesses the emission from the accretion disk into the IR, which dominates the \\ac{AGN} SED at wavelengths from $\\sim1$\\,$\\mu$m to a few tens of microns, the wavelength regime covered by the WISE survey. \n\n\\cite{stern2012mid} demonstrated that a simple WISE colour cut, $W1-W2>0.8$, selects \\ac{AGN} with a reliability of 95\\% and with a completeness of 78\\%. However, this colour cut works efficiently for low-z sources ($z\\sim1$) only, given that is defined using observed fluxes not rest-frame ones. \\cite{stern2012mid} therefore discussed a number of possible scenarios to interpret W1-W2 colour at higher redshifts.\n\nFollowing are the possibilities, taken from \\cite{stern2012mid}, applicable to our $z\\gtrsim4$ sample, \n\\begin{enumerate}\n \\item At $z\\gtrsim3.5$, W1\/W2 bands probe optical \\& near-IR rest-frame emission\n  causing the $\\sim1$\\,$\\mu$m minimum  seen in some starburst galaxies\n shifting to W2 band and H$\\alpha$ emission shifting to W1. This results in a blue W1-W2 (<0.8) colour \n\n \\item A highly obscured \\ac{AGN} will have $W1-W2>0.8$ at all redshifts.\n \\item For a composite system (\\ac{AGN} $+$ \\ac{SB}), dilution by host-galaxy light can limit the identification of low luminosity AGNs.\n\\end{enumerate}\n \n\nWe looked at the W1-W2 colour of our sources, utilizing CATWISE magnitudes. We considered only those sources without any flags and with SNR$\\ge$5 in the W1 and W2 bands. This identifies 106 sources, of which only 28 sources satisfy the $W1-W2>0.8$ \\ac{AGN} criterion. We further found that all but 2 of the sources satisfying the $W1-W2>0.8$ \\ac{AGN} criterion have S\\textsubscript{887}$\\gtrsim$0.2-2.5\\,mJy and  1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio between 1 and 100. This supports our argument \nthat (i) low power radio sources in our sample are powered by a radio-\\ac{AGN} (ii) the radio-to-IR flux density ratio of low power radio-\\acp{AGN} can extend to lower values (<100). On the other hand, the sources with $W1-W2<0.8$ could be either unobscured quasars, as they have a blue $W1-W2$ colour at high-$z$ as demonstrated in \\cite{ross2020near} or the composite galaxies where the SB component results in a blue $W1-W2$ colour.\nIn addition to this, we verified the WISE magnitudes of a known starbust at $z=4.1$ \\citep{ciesla2020hyper} and found that its $W1-W2 = 0.63$ matches our $z\\gtrsim4$ sample with S\\textsubscript{1.4}$<$1\\,mJy.\n\n\n\n\n\n\n\n\n\n\\subsection{SED modelling}\n\\label{sec:sed_modelling}\n\n It is beyond the scope of this paper to perform full SED modelling of these galaxies, but here we make an illustrative comparison of these galaxies with two representative low-redshift galaxies. To do so, we shift two local radio sources to $z\\gtrsim4$ for comparison: (i) Cygnus~A, a radio galaxy at $z = 0.05607$, and (ii) IRAS F00183--7111, an ultraluminous infrared galaxy (ULIRG) at $z=0.327$ showing composite emission from a radio loud-\\ac{AGN} and a \\ac{SB}. Each of these sources is distinct: their radio emission either comes from an \\ac{AGN} or a combination of SF and \\ac{AGN}. Their broadband observed photometry is obtained from NED.  We added more radio data from  \\citet[Table~1]{norris2012radio} to the IRAS F00183--7111 template since the radio coverage is poor and this wavelength regime is critical for this analysis. \n\nWe do not consider galaxies exclusively powered by star formation activity, like M\\,82 and Arp~220, as they produce radio flux densities three or four orders of magnitude below the current detection limit when shifted to $z\\gtrsim4$, indicating that none of our sources represent typical SFGs. \n \nWe create the SED at $z\\gtrsim4$ by shifting the observed SED in the frequency space by a factor of \\textit{(1+$z$)\\textsuperscript{-1}}, and in flux density space by a factor of (\\textit{D\\textsubscript{obs}}\/\\textit{D\\textsubscript{z})\\textsuperscript{2}}$\\times {\\it (1+z_{\\rm {\\it shift}}}\/{\\it 1+z_{\\rm {\\it obs}}})$.\\textit{ D\\textsubscript{obs}} and \\textit{D\\textsubscript{z}} are the luminosity distance to the observed redshift and shifted redshift ($z_{\\rm {\\it shift}}\\sim4$, 5, 6 or 7) respectively, estimated using an online cosmology calculator \\citep{wright2006cosmology}.\n\n\nFigure~\\ref{fig:shifted_SED} shows the resulting shifted SEDs. The extracted 1.4\\,GHz flux density from the shifted SEDs, shown in Table~\\ref{tab:shifted_sed}, demonstrates that (i) active galaxies powered by a radio-\\ac{AGN}, such as Cygnus~A, can represent the brighter sources in our sample and (ii) a composite system like IRAS F00183-7111 consisting of a radio-\\ac{AGN} and a significant \\ac{SB} component, can represent fainter sources in our sample. Thus, our SED modelling shows that a radio-\\ac{AGN} and a composite system can reproduce the galaxies in our sample implying that properties of our \\acp{HzRS}  are not that extraordinary compared to typical local galaxies. At the same time, we note that IRAS F00183-7111 does not reproduce our sample's observed SPIRE flux densities, as its SFR ($\\sim 220$\\,$M_{\\odot}$\/yr; \\cite{mao2014star}) is not sufficient.\n\n\n\n\\begin{figure*}\n \\centering\n  \\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{CygnusA_4_final.png} \n\\end{subfigure}\n\\hspace{0.5cm}\n \\begin{subfigure}{.45\\textwidth}\n \\centering\n \\includegraphics[width=\\textwidth]{f00183_3_final.png} \n\\end{subfigure}\n \\caption{Template SEDs shifted to redshifts, $z=4, 5, 6, 7$ in grey, black, red and blue colours respectively. SED templates are built by simply connecting the datapoints and performed 1D interpolation using python-scipy (interp1d) package to obtain 1.4\\,GHz flux density information from the shifted SEDs. 1.4\\,GHz is marked in black solid line. Pink points represent the observed SED, obtained from NED.\n }\n \\label{fig:shifted_SED}\n\\end{figure*}\n\\begin{table}\n \\centering\n  \\caption{1.4\\,GHz flux density extracted from the template SEDs shifted to $z=4,5,6$ by performing 1d interpolation using interp1d function in python-scipy package.}\n \\begin{tabular}{c|c|c|c|c}\n \\hline\n  \\multirow{2}{*}{Source} & \\multicolumn{3}{c}{S\\textsubscript{1.4} (mJy)}\\\\\n  \\cline{2-5}\n   & $z\\sim4$ & $z\\sim5$ & $z\\sim6$ & $z\\sim7$ \\\\\n   \\hline\n  Cygnus A & 60 & 29 & 20 & 14 \\\\ \n  IRAS F00183 & 0.6 & 0.3 & 0.2 &  0.1 \\\\\n   \\hline\n \\end{tabular}\n \\label{tab:shifted_sed}\n\\end{table}\n\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have used the Lyman dropout technique to identify a sample of 149 radio sources at $z\\gtrsim4-7$, of which one is  a known radio-quasar (VIK J2318\u22123113; selected as a Z-dropout) at $z=6.44$, and 148 are newly-identified. Our study reveals a new population of high-redshift ($z\\sim$ 4--7) radio-\\ac{AGN}s, $\\sim1-2$ orders of magnitudes fainter than currently known radio-\\ac{AGN}s at similar redshifts, but with 1.4\\,GHz luminosities typical of lower-redshift radio-loud \\ac{AGN}. \n\nComparison with spectroscopic redshifts in the COSMOS field indicate that our $z\\sim4$  sample is about 88\\% reliable, the  $z\\sim5$  sample is about 81\\% reliable, and the  $z\\sim6$  sample is  at least about 62\\% reliable. We do not have a reliability estimate for the  $z\\sim7$  sample. \n\nWe have explored radio and IR observations to understand  the origin of the radio emission of our  \\ac{HzRS} sample.  Our conclusions are as follows.\n\n\\begin{enumerate}\n \\item The radio (1.4\\,GHz) estimated \\ac{SFR} of our sample (Table~\\ref{tab:sfr}) gives unphysical values, indicating that radio emission in our sources is not solely powered by star formation. \n \n \\item The faint ($\\log L\\textsubscript{1.4}=25$) and bright ($\\log L\\textsubscript{1.4}\\ge26$)\n end of our sample spans the low and high power radio galaxy luminosity classes respectively, suggesting the presence of a radio-\\ac{AGN} component in our sample. This study presents a new population of less powerful radio-\\ac{AGN}s candidates at $z\\sim4$, 5 and 6 that have been missed by previous surveys.\n \n\n\n \\item  $\\sim10\\%$ of our \\ac{HzRS} sample are detected in the {\\it Herschel}-SPIRE bands which probe \\ac{SB} heated dust emission and \\ac{AGN} heated dust torus emission at $z>2$. This suggests that SPIRE detected sources are likely to represent composite systems. \n \n \\item We demonstrate that some high-$z$ radio-\\ac{AGN}s tend to have hosts that are \\ac{SB} galaxies, in contrast to low-$z$ radio-\\ac{AGN}s, which are usually hosted by quiescent elliptical galaxies.\n \n  \\item  Using the $W1-W2>0.8$ AGN indicator, we identified 28 radio-\\acp{AGN},  26 of which are found to be on the faint end of the observed 887.5\\,MHz flux density distribution.  We further demonstrate that the 1.4\\,GHz-to-3.4\\,$\\mu$m flux density ratio of these weak radio-\\ac{AGN}s extends to lower values (1-100) than previously thought.\n \n \\item SED modelling confirms that a composite system (radio-\\ac{AGN} $+$ \\ac{SB}) and a radio galaxy at $z\\gtrsim4$ can produce the radio flux densities similar to ones observed at the faint and bright end respectively.\n\n\\end{enumerate}\n\\section{Future work}\n\n Spectroscopic follow-up of our sample is essential (i) to confirm the redshifts of these candidate sources identified by the Lyman dropout technique (ii) to verify the reliability of magnitude cut-off introduced in $Z$-band for $i$ dropouts (iii) to establish the criteria to identify interlopers, if present. The above three goals are critical in producing reliable criteria to select \\acp{HzRS} in the full \\ac{EMU} survey and thus verifying the model \\citep{raccanelli2012cosmological} for the redshift distribution of radio sources.\n\n\n\n\\section*{Acknowledgements}\n\nThe Australian SKA Pathfinder is part of the Australia Telescope National Facility which is managed by \\ac{CSIRO}. Operation of \\ac{ASKAP} is funded by the Australian Government with support from the National Collaborative Research Infrastructure Strategy. \\ac{ASKAP} uses the resources of the Pawsey Supercomputing Centre. \nEstablishment of \\ac{ASKAP}, the Murchison Radio-astronomy Observatory and the Pawsey Supercomputing Centre are initiatives of the Australian Government, with support from the Government of Western Australia and the Science and Industry Endowment Fund. \nWe acknowledge the Wajarri Yamatji people as the traditional owners of the Observatory site. \n\nBased on observations made with ESO Telescopes at the La Silla Paranal Observatory under programme IDs 177.A-3016, 177.A-3017, 177.A-3018 and 179.A-2004, and on data products produced by the KiDS consortium. The KiDS production team acknowledges support from: Deutsche Forschungsgemeinschaft, ERC, NOVA and NWO-M grants; Target; the University of Padova, and the University Federico II (Naples).\n\nIsabella Prandoni acknowledges support from INAF under the PRIN MAIN stream \"SAuROS\" project, and from CSIRO under its Distinguished Research Visitor Programme. \n\n We thank an anonymous referee for helpful comments and some excellent suggestions.\n\\section*{Data Availability}\n\n \n\nThis study utilised the data available in the following public domains: \n\\begin{enumerate}\n    \\item https:\/\/data.csiro.au\/domain\/casdaObservation\n    \\item https:\/\/kids.strw.leidenuniv.nl\/DR4\/access.php\n    \\item http:\/\/horus.roe.ac.uk\/vsa\/index.html\n    \\item https:\/\/irsa.ipac.caltech.edu\/\n\\end{enumerate}\n\nThe derived final datasets are available in the article.\n\n\n\n\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\n\nIf $X_1,X_2,\\ldots,X_n \\stackrel{\\mathrm{ind}}{\\sim}\nf$ are a sample from a nonincreasing density $f$ on $[0,\\infty)$,\nthen the Grenander estimator, the nonparametric maximum likelihood\nestimator (NPMLE) $\\tilde f_n$ of $f$ [obtained by maximizing the\nlikelihood $\\prod_{i=1}^n f(X_i)$ over all nonincreasing densities],\nmay be described as follows: let $\\mathbb{F}_n$ denote the empirical\ndistribution function (EDF) of the data, and $\\tilde{F}_n$ its least\nconcave majorant. Then the NPMLE $\\tilde{f}_n$ is the left-hand\nderivative of $\\tilde{F}_n$. This result is due to Grenander (\\citeyear\n{grenander56}) and\nis described in detail by Robertson, Wright and Dykstra (\\citeyear\n{RWD88}), pages\n326--328. Prakasa Rao (\\citeyear{prakasa69}) obtained the asymptotic\ndistribution of\n$\\tilde{f}_n$, properly normalized: let $\\mathbb{W}$ be a two-sided\nstandard Brownian motion on $\\mathbb{R}$ with $\\mathbb{W}(0) = 0$ and\n\\[\n{\\mathbb C} = \\mathop{\\arg\\max}_{s\\in{\\mathbb R}} [{\\mathbb W}(s) - s^2].\n\\]\nIf $0 < t_0 < \\infty$ and $f'(t_0) \\ne0$, then\n\\begin{equation}\n\\label{eq:chrnff}\nn^{1\/3} \\{ \\tilde f_n(t_0) - f(t_0) \\} \\Rightarrow\n2 \\bigl|\\tfrac{1}{2} f(t_0) f'(t_0) \\bigr|^{1\/3} {\\mathbb C},\n\\end{equation}\nwhere $\\Rightarrow$ denotes convergence in distribution. There are\nother estimators that exhibit similar asymptotic\nproperties; for example, Chernoff's (\\citeyear{chernoff64}) estimator\nof the mode, the\nmonotone regression estimator [Brunk (\\citeyear{brunk70})],\nRousseeuw's (\\citeyear{rousseeuw84}) least median of squares estimator,\nand the estimator\nof the shorth [Andrews et al. (\\citeyear{andrewsetal72}) and Shorack\nand Wellner (\\citeyear{shorackWe86})].\nThe seminal paper by Kim and Pollard (\\citeyear{kimPo90}) unifies\n$n^{1\/3}$-rate\nof convergence problems in the more general \\mbox{$M$-estimation} framework.\nTables and a survey of statistical problems in which the distribution\nof ${\\mathbb C}$ arises are provided by Groeneboom and Wellner\n(\\citeyear{GW01}).\n\nThe presence of nuisance parameters in the limiting distribution (\\ref\n{eq:chrnff}) complicates the construction of\nconfidence intervals. Bootstrap intervals avoid the problem of\nestimating nuisance parameters and are generally reliable in problems\nwith $\\sqrt{n}$ convergence rates. See Bickel and Freedman (\\citeyear{BF81}),\nSingh (\\citeyear{singh81}), Shao and Tu (\\citeyear{shaoTu95}) and its\nreferences. Our aim in this\npaper is to study the consistency of bootstrap methods for the\nGrenander estimator with the hope that the monotone density estimation\nproblem will shed light on the behavior of bootstrap methods in similar\ncube-root convergence problems.\n\nThere has been considerable recent interest in this question. Kosorok\n(\\citeyear{kosorok07}) show that bootstrapping from the EDF ${\\mathbb\nF}_n$ does not\nlead to a consistent estimator of the distribution of $n^{1\/3} \\{\n\\tilde f_n(t_0) - f(t_0)\\}$. Lee and Pun (\\citeyear{leePun06}) explore\n$m$ out of $n$\nbootstrapping from the empirical distribution function in similar\nnonstandard problems and prove the consistency of the method. L\\'eger\nand MacGibbon (\\citeyear{legerMa06}) describe conditions for a\nresampling procedure to\nbe consistent under cube root asymptotics and assert that these\nconditions are generally not met while bootstrapping from the EDF. They\nalso propose a smoothed version of the bootstrap and show its\nconsistency for Chernoff's estimator of the mode. Abrevaya and Huang\n(\\citeyear{AH05}) show that bootstrapping from the EDF leads to inconsistent\nestimators in the setup of Kim and Pollard (\\citeyear{kimPo90}) and propose\ncorrections. Politis, Romano and Wolf (\\citeyear{PRW99}) show that subsampling\nbased confidence intervals are consistent in this scenario.\n\nOur work goes beyond that cited above as follows: we show that\nbootstrapping from the NPMLE $\\tilde{F}_n$ also leads to inconsistent\nestimators, a result that we found more surprising, since $\\tilde{F}_n$\nhas a density. Moreover, we find that \\textit{the bootstrap estimator,\nconstructed from either the EDF or NPMLE, has no limit in probability}.\nThe finding is less than a mathematical proof, because one step in the\nargument relies on simulation; but the simulations make our point\nclearly. As described in Section~\\ref{discussion}, our findings are\ninconsistent with some claims of Abrevaya and Huang (\\citeyear{AH05}).\nAlso, our\nway of tackling the main issues differs from that of the existing\nliterature: we consider conditional distributions in more detail than\nKosorok (\\citeyear{kosorok07}), who deduced inconsistency from\nproperties of\nunconditional distributions; we directly appeal to the characterization\nof the estimators and use a continuous mapping principle to deduce the\nlimiting distributions instead of using the ``switching'' argument [see\nGroeneboom (\\citeyear{groene85})] employed by Kosorok (\\citeyear\n{kosorok07}) and Abrevaya and Huang\n(\\citeyear{AH05}); and at a more technical level, we use the Hungarian\nRepresentation theorem whereas most of the other authors use empirical\nprocess techniques similar to those described by van der Vaart and\nWellner (\\citeyear{VW00}).\n\nSection \\ref{prelim} contains a uniform version of (\\ref{eq:chrnff})\nthat is used later on to study the consistency of different bootstrap\nmethods and may be of independent interest. The main results on\ninconsistency are presented in Section \\ref{boots_prob}. Sufficient\nconditions for the consistency of a bootstrap method are presented in\nSection \\ref{smooth_boots} and applied to show that bootstrapping from\nsmoothed versions of $\\tilde{F}_n$ does produce consistent estimators.\nThe $m$ out of $n$ bootstrapping procedure is investigated, when\ngenerating bootstrap samples\nfrom $\\mathbb{F}_n$ and $\\tilde{F}_n$. It is shown that both the\nmethods lead to consistent estimators under mild conditions on $m$. In\nSection \\ref{discussion}, we discuss our findings, especially the\nnonconvergence and its implications. The\n\\hyperref[app]{Appendix}, provides the details of some arguments used\nin proving the\nmain results.\n\n\\section{Uniform convergence}\\label{prelim}\n\nFor the rest of the paper, $F$ denotes a distribution function with\n$F(0) = 0$ and a density $f$ that is nonincreasing on $[0,\\infty)$ and\ncontinuously differentiable near $t_0 \\in(0,\\infty)$ with nonzero\nderivative $f'(t_0) < 0$. If $g\\dvtx I \\to{\\mathbb R}$ is a bounded\nfunction, write $\\Vert g\\Vert:= {\\sup_{x\\in I}} |g(x)|$. Next, let $F_n$\nbe distribution functions with $F_n(0) = 0$, that converge weakly to\n$F$ and, therefore,\n\\begin{equation}\n\\label{eq:cndtn0}\n{\\lim_{n\\to\\infty}} \\| F_n-F \\| = 0.\n\\end{equation}\nLet $X_{n,1},X_{n,2},\\ldots,X_{n,m_n} \\stackrel{\\mathrm{ind}}{\\sim}\nF_n$, where $m_n\n\\le n$ is a nondecreasing sequence of integers for which $m_n \\to\\infty\n$; let ${\\mathbb F}_{n,m_n}$ denote the EDF of $X_{n,1},X_{n,2},\n\\ldots,\\break X_{n,m_n}$; and let\n\\[\n\\Delta_n := m_n^{1\/3} \\{\\tilde f_{n,m_n}(t_0) - f_n(t_0)\\},\n\\]\nwhere $\\tilde f_{n,m_n}(t_0)$ is the Grenander estimator computed from\n$X_{n,1},X_{n,2},\\ldots,\\break X_{n,m_n}$ and $f_n(t_0)$ is the density of\n$F_n$ at $t_0$ or a surrogate. Next, let $I_m = [-t_0m^{1\/3},\\infty\n)$ and\n\\begin{equation}\\label{eq:Z_process}\n\\quad \\mathbb{Z}_n(h) := m_n^{2\/3} \\{ \\mathbb{F}_{n,m_n}(t_0 +\nm_n^{-{1\/3}}h) - \\mathbb{F}_{n,m_n}(t_0) - f_n(t_0) m_n^{-{1\/3}}h \\}\n\\end{equation}\nfor $h \\in I_{m_n}$ and observe that $\\Delta_n$ is the left-hand\nderivative at $0$ of the least concave majorant of ${\\mathbb Z}_n$. It\nis fairly easy to obtain the asymptotic distribution of $\\mathbb{Z}_n$.\nThe asymptotic distribution of $\\Delta_n$ may then be obtained from the\nContinuous Mapping theorem. Stochastic processes are regarded as random\nelements in $D(\\mathbb{R})$, the space of right continuous functions on\n$\\mathbb{R}$ with left limits, equipped with the projection $\\sigma\n$-field and the topology of uniform convergence on compacta. See\nPollard (\\citeyear{polllard84}), Chapters IV and V for background.\n\n\\subsection{Convergence of ${\\mathbb Z}_n$}\n\nIt is convenient to\ndecompose $\\mathbb{Z}_n$ into the sum of $\\mathbb{Z}_{n,1}$ and $\\mathbb\n{Z}_{n,2}$ where\n\\begin{eqnarray*}\n\\mathbb{Z}_{n,1}(h) &:=& m_n^{2\/3} \\{ (\\mathbb\n{F}_{n,m_n} - F_{n})(t_0 + m_n^{-{1\/3}}h) - (\\mathbb{F}_{n,m_n}-\nF_{n})(t_0) \\},\\\\\n\\mathbb{Z}_{n,2}(h) &:=& m_n^{2\/3} \\{\nF_{n}(t_0 + m_n^{-{1\/3}} h)- F_{n}(t_0) - f_n(t_0) m_n^{-{1\/3}}h \\}.\n\\end{eqnarray*}\nObserve that ${\\mathbb Z}_{n,2}$ depends only on $F_n$ and $f_n$; only\n${\\mathbb Z}_{n,1}$ depends on $X_{n,1},\\ldots,\\break X_{n,m_n}$. Let\n$\\mathbb{W}_1$ be a standard two-sided Brownian motion on $\\mathbb{R}$\nwith $\\mathbb{W}_1(0) = 0$, and $\\mathbb{Z}_{1}(h) = \\mathbb{W}_1 [f(t_0)h]$.\n\\begin{prop}\\label{prop:Z_conv}\nIf\n\\begin{equation}\n\\label{eq:cndtn1}\n\\lim_{n\\to\\infty} m_n^{1\/3} | F_n(t_0+m_n^{-{1\/3}}h) -\nF_n(t_0) - f(t_0) m_n^{-{1\/3}}h | = 0\n\\end{equation}\nuniformly on compacts (in $h$), then ${\\mathbb Z}_{n,1} \\Rightarrow\n{\\mathbb Z}_1$; and if there is a continuous\nfunction ${\\mathbb Z}_2$ for which\n\\begin{equation}\n\\label{eq:cndtn2}\n\\lim_{n\\to\\infty} {\\mathbb Z}_{n,2}(h) = {\\mathbb Z}_2(h)\n\\end{equation}\nuniformly on compact intervals, then ${\\mathbb Z}_n \\Rightarrow\n{\\mathbb Z} := {\\mathbb Z}_1 + {\\mathbb Z}_2$.\n\\end{prop}\n\\begin{pf}\nThe Hungarian Embedding theorem of K\\'omlos, Major and\\break  Tusn\\'ady\n(\\citeyear{kmt75})\nis used. We may suppose that\n$X_{n,i} = F_n^{\\#}(U_{i})$, where $F_n^{\\#}(u) = \\inf\\{x\\dvtx F_n(x)\n\\ge\nu\\}$ and $U_{1},U_2,\\ldots$ are i.i.d. Uniform$(0,1)$ random variables.\nLet $\\mathbb{U}_n$ denote the EDF of $U_{1},\\ldots, U_{n}$, $\\mathbb\n{E}_n(t) = \\sqrt{n} [\\mathbb{U}_n(t) - t]$, and $\\mathbb{V}_n = \\sqrt\n{m_n} (\\mathbb{F}_{n,m_n} - F_n)$. Then $\\mathbb{V}_n = \\mathbb\n{E}_{m_n} \\circ F_n$. By Hungarian Embedding, we may also suppose that\nthe probability space supports a sequence of Brownian Bridges $\\{\\mathbb\n{B}_n^0\\}_{n \\ge1}$ for which\n\\begin{equation}\n\\label{eq:kmt}\n{\\sup_{0 \\le t \\le1}} | \\mathbb{E}_{n}(t) - \\mathbb{B}_n^0(t)| = O\n\\biggl[{\\log(n)\\over\\sqrt{n}} \\biggr] \\qquad\\mbox{a.s.},\n\\end{equation}\nand a standard normal random variable $\\eta$ that is independent of $\\{\n\\mathbb{B}_n^0\\}_{n \\ge1}$. Define a version $\\mathbb{B}_n$ of\nBrownian motion by $\\mathbb{B}_n(t) = \\mathbb{B}_n^0(t) + \\eta t$, for\n$t \\in[0,1]$. Then\n\\begin{eqnarray}\\label{eq:boots_proc1}\n\\mathbb{Z}_{n,1} (h) & = & m_n^{1\/6} \\{\\mathbb\n{E}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{E}_{m_n}[F_{n}(t_0)] \\} \\nonumber\\\\[-8pt]\\\\[-8pt]\n& = & m_n^{1\/6} \\{\\mathbb{B}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{B}_{m_n}[F_n(t_0)] \\} + \\mathbb{R}_{n}(h),\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray*}\n|\\mathbb{R}_n(h)| & \\le& 2 m_n^{1\/6} {\\sup_{0\n\\le t \\le1}} |\\mathbb{E}_{m_n}(t) - \\mathbb{B}_{m_n}^0(t)| \\\\\n&&{} + m_n^{1\/6}|\\eta| |F_n(t_0 + m_n^{-{1\/3}}h) - F_n(t_0)|\n\\rightarrow0\n\\end{eqnarray*}\nuniformly on compacta w.p. 1 using (\\ref{eq:cndtn1}) and (\\ref\n{eq:kmt}). Let\n\\[\n\\mathbb{X}_n(h) := m_n^{1\/6}\\{\\mathbb\n{B}_{m_n}[F_n(t_0 + m_n^{-{1\/3}}h)] -\n\\mathbb{B}_{m_n}[F_n(t_0)] \\}\n\\]\nand observe that $\\mathbb{X}_n$ is a mean zero Gaussian process defined\non $I_{m_n}$ with independent increments and\ncovariance kernel\n\\[\nK_n(h_1,h_2) = m_n^{1\/3} | F_n[t_0 + \\operatorname{sign} \\{h_1\\}\nm_n^{-{1\/3}}(|h_1| \\wedge|h_2|)] - F_n(t_0) | \\mathbf{1}\\{\nh_1 h_2 > 0\\}.\n\\]\nIt now follows from Theorem V.19 in Pollard (\\citeyear{polllard84}) and\n(\\ref\n{eq:cndtn1}) that $\\mathbb{X}_n(h)$ converges in\ndistribution to $\\mathbb{W}_1[f(t_0)h]$ in $D([-c,c])$ for every $c >\n0$, establishing the first assertion of the\nproposition. The second then follows from Slutsky's theorem.\n\\end{pf}\n\n\\subsection{Convergence of $\\Delta_n$}\n\nUnfortunately, $\\Delta_n$ is not\nquite a continuous functional of ${\\mathbb\nZ}_n$. If $f\\dvtx I \\to{\\mathbb R}$, write $f|J$ to denote the restriction\nof $f$ to $J \\subseteq I$; and if $I$ and $J$ are intervals and $f$ is\nbounded, write $L_Jf$ for the least concave majorant of the\nrestriction. Thus, $\\tilde{F}_n = L_{[0,\\infty)}{\\mathbb F}_n$ in the\n\\hyperref[intro]{Introduction}.\n\\begin{lemma}\\label{lem:ww} Let $I$ be a closed interval; let $f\\dvtx I\n\\to\n{\\mathbb R}$ be a bounded upper semi-continuous function on $I$; and\nlet $a_1,a_2,b_1,b_2 \\in I$ with $b_1 < a_1 < a_2 < b_2$. If\n$2f[{1\\over2}(a_i+b_i)] > L_{I}\nf(a_i) + L_{I} f(b_i), i = 1,2$, then $L_If(x) = L_{[b_1,b_2]}f(x)$ for\n$a_1 \\le x \\le a_2$.\n\\end{lemma}\n\\begin{pf} This follows from the proof of Lemmas 5.1 and 5.2 of\nWang and Woodroofe (\\citeyear{wangWoo07}). In that lemma\ncontinuity was assumed, but only upper semi-continuity was used in the\n(short) proof.\n\\end{pf}\n\nRecall Marshall's lemma: if $I$ is an interval, $f: I \\to{\\mathbb R}$\nis bounded, and $g\\dvtx I \\to{\\mathbb R}$ is concave, then $\\Vert\nL_If-g\\Vert\\le\\Vert f-g\\Vert$. See, for example, Robertson, Wright\nand Dykstra [(\\citeyear{RWD88}), page 329] for a proof. Write $\\tilde\n{F}_{n,m_n} =\nL_{[0,\\infty)}{\\mathbb F}_{n,m_n}$.\n\\begin{lemma}\\label{lem:loc}\nIf $\\delta> 0$ is so small that $F$ is strictly concave on\n$[t_0-2\\delta,t_0+2\\delta]$ and (\\ref{eq:cndtn0}) holds then ${\\tilde\nF}_{n,m_n} = L_{[t_0-2\\delta,t_0+2\\delta]}{\\mathbb F}_{n,m_n}$ on\n$[t_0-\\delta,t_0+\\delta]$ for all large $n$ w.p. 1.\n\\end{lemma}\n\\begin{pf} Since $F$ is strictly concave on $[t_0-2\\delta,t_0+2\\delta\n], 2F(t_0\\pm{3\\over2}\\delta) >\nF(t_0\\pm\\delta) + F(t_0\\pm2\\delta)$. Then\n\\begin{eqnarray*}\n\\| \\tilde{F}_{n,m_n} - F \\| & \\le& \\|{\\mathbb F}_{n,m_n} - F \\|\n\\\\\n& \\le& \\|{\\mathbb F}_{n,m_n} - F_n \\| + \\|F_n - F\\| \\\\\n& \\le& \\frac{1}{\\sqrt{m_n}} \\|\\mathbb{E}_{m_n}\\| + \\|F_n - F \\| \\to0\n\\qquad\\mbox{w.p. } 1\n\\end{eqnarray*}\nby Marshall's lemma, (\\ref{eq:cndtn0}) and the Glivenko--Cantelli\ntheorem. Thus,\\break $2{\\mathbb F}_{n,m_n}(t_0\\pm\n{3\\over2}\\delta) > \\tilde{F}_{n,m_n}(t_0\\pm\\delta) + \\tilde\n{F}_{n,m_n}(t_0\\pm2\\delta)$, for all large $n$ w.p. 1, and Lemma \\ref\n{lem:loc} follows from Lemma \\ref{lem:ww}.\n\\end{pf}\n\\begin{prop}\\label{prop:loc} \\textup{(i)} Suppose that (\\ref{eq:cndtn0}) and\n(\\ref{eq:cndtn1}) hold and given $\\gamma> 0$, there are $0 < \\delta<\n1$ and $C > 0$ for which\n\\begin{equation}\n\\label{eq:cndtn3}\n\\bigl| F_n(t_0+h) - F_n(t_0) - f_n(t_0)h - \\tfrac{1}{2} f'(t_0) h^2\n\\bigr| \\le\\gamma h^2 + C m_n^{-{2\/3}}\n\\end{equation}\nand\n\\begin{equation}\n\\label{eq:cndtn4}\n| F_n(t_0+h) - F_n(t_0) | \\le C (|h| + m_n^{-{1\/3}})\n\\end{equation}\nfor $|h| \\le\\delta$ and for all large $n$. If $J$ is a compact\ninterval and $\\varepsilon>0$, then there is a compact $K \\supseteq J$,\ndepending only on $\\varepsilon, J, C, \\gamma$, and $\\delta$, for which\n\\begin{equation}\n\\label{eq:loc}\nP [L_{I_{m_n}}{\\mathbb Z}_n = L_K{\\mathbb Z}_n \\mbox{ on } J ]\n\\ge1 - \\varepsilon\n\\end{equation}\nfor all large $n$.\n\n\\textup{(ii)} Let $\\mathbb{Y}$ be an a.s. continuous stochastic process on\n$\\mathbb{R}$ that is a.s. bounded above. If $\\lim_{|h| \\rightarrow\n\\infty} \\mathbb{Y}(h)\/|h| = -\\infty$ a.e., then the compact $K\n\\supseteq J$ can be chosen so that\n\\begin{equation}\n\\label{eq:loc2}\nP [L_{\\mathbb R}{\\mathbb Y} = L_K{\\mathbb Y} \\mbox{ on } J ]\n\\ge1 - \\varepsilon.\n\\end{equation}\n\\end{prop}\n\\begin{pf} For a fixed sequence ($F_n \\equiv F$) (\\ref{eq:loc}) would\nfollow from the assertion in Example 6.5 of Kim and Pollard (\\citeyear\n{kimPo90}), and\nit is possible to adapt their argument to a triangular array using (\\ref\n{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) in place of Taylor series expansion. A different\nproof is presented in the \\hyperref[app]{Appendix}.\n\\end{pf}\n\nWe will use the following easily verified fact. In its statement, the\nmetric space $\\mathcal{X}$ is to be endowed with\nthe projection $\\sigma$-field. See Pollard (\\citeyear{polllard84}),\npage 70.\n\\begin{lemma}\\label{lemma:con_derv} Let $\\{X_{n,c}\\}, \\{Y_n\\}, \\{W_c\\}$\nand $Y$ be sets of random elements taking values in a metric space\n$(\\mathcal{X}$,$d)$, $n=0,1,\\ldots,$ and $c \\in\\mathbb{R}$. If for any\n$\\delta> 0$,\n\\begin{longlist}\n\\item $\\lim_{c \\rightarrow\\infty} \\limsup_{n \\rightarrow\\infty}\nP\\{d(X_{n,c},Y_n) > \\delta\\} = 0$,\n\n\\item $\\lim_{c \\rightarrow\\infty} P\\{d(W_{c},Y) > \\delta\\} = 0$,\n\n\\item $X_{n,c} \\Rightarrow W_c$ as $n \\rightarrow\\infty$ for\nevery $c \\in\\mathbb{R}$,\n\\end{longlist}\nthen $Y_n \\Rightarrow Y$ as $n \\rightarrow\\infty$.\n\\end{lemma}\n\\begin{cor}\\label{cor:convdelta} If (\\ref{eq:loc}) and (\\ref{eq:loc2})\nhold, and $\\mathbb{Z}_n \\Rightarrow\\mathbb{Y}$, then\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Y}$ in\n$D({\\mathbb R})$ and $ \\Delta_n \\Rightarrow(L_{\\mathbb R}{\\mathbb Y})'(0)$.\n\\end{cor}\n\\begin{pf} It suffices to show that $L_{I_{m_n}}{\\mathbb Z}_n|J\n\\Rightarrow L_{\\mathbb R}{\\mathbb Y}|J$ in $D(J)$, for every compact\ninterval $J \\subseteq\\mathbb{R}$. Given $J$ and $\\varepsilon> 0$, there\nexists $K_\\varepsilon$, a compact, $K_\\varepsilon\\supseteq J$, such that\n(\\ref{eq:loc}) and (\\ref{eq:loc2}) hold. This verifies (i) and (ii)\nof Lemma \\ref{lemma:con_derv} with $c = 1\/\\varepsilon$, $X_{n,c}=\nL_{K_\\varepsilon}{\\mathbb Z}_n$, $Y_n = L_{I_{m_n}}{\\mathbb Z}_n$, $W_c =\nL_{K_\\varepsilon}{\\mathbb Y}$, $Y = L_{\\mathbb R}{\\mathbb Y}$ and $d(x,y)\n= \\sup_{t \\in J} |x(t) - y(t)|$. Clearly, $L_{K_\\varepsilon}{\\mathbb\nZ}_n|J \\Rightarrow L_{K_\\varepsilon}{\\mathbb Y}|J$ in $D(J)$, by the\nContinuous Mapping theorem, verifying condition (iii). Thus,\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Y}$ in\n$D(\\mathbb R)$. Another application of the Continuous Mapping theorem\n[via the lemma on page 330 of Robertson, Wright and Dykstra (\\citeyear\n{RWD88})] in\nconjunction with (\\ref{eq:loc}), (\\ref{eq:loc2}) and Lemma \\ref\n{lemma:con_derv} then shows that $\\Delta_n = (L_{I_{m_n}}{\\mathbb\nZ}_n)'(0) \\Rightarrow(L_{\\mathbb R}{\\mathbb Y})'(0)$.\n\\end{pf}\n\\begin{cor}\\label{cor:convdelta2} If (\\ref{eq:cndtn0}), (\\ref\n{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) hold and\n\\[\n\\lim_{|h| \\rightarrow\\infty}\n\\mathbb{Z}(h)\/|h| = -\\infty,\n\\]\nthen\n$L_{I_{m_n}}{\\mathbb Z}_n \\Rightarrow L_{\\mathbb R}{\\mathbb Z}$ in\n$D({\\mathbb R})$ and $ \\Delta_n \\Rightarrow\n(L_{\\mathbb R}{\\mathbb Z})'(0)$; and if $\\mathbb{Z}_{2}(h) = f'(t_0)\nh^2\/2$, then $ \\Delta_n \\Rightarrow2 |\\frac{1}{2} f(t_0) f'(t_0)|^{1\/3}\n\\mathbb C$, where $\\mathbb C$ has Chernoff's distribution.\n\\end{cor}\n\\begin{pf} The convergence follows directly from Proposition \\ref\n{prop:loc} and\\break Corollary~\\ref{cor:convdelta}. Note that if $\\mathbb\n{Z}_{2}(h) = f'(t_0) h^2\/2$, then (\\ref{eq:loc}) and (\\ref{eq:loc2})\nhold\\break and~Corollary~\\ref{cor:convdelta} can be applied. That\n$(L_{\\mathbb R}{\\mathbb Z})'(0)$ is distributed as\\break $2 |\\frac{1}{2}\nf(t_0) f'(t_0)|^{1\/3} \\mathbb C$ when $\\mathbb{Z}_{2}(h) =\nf'(t_0) h^2\/2$ follows from elementary properties of Brownian motion\nvia the ``switching'' argument of Groeneboom (\\citeyear{groene85}).\n\\end{pf}\n\n\\subsection{Remarks on the conditions}\\label{remarks}\n\nIf $F_n \\equiv F$\nand $f_n \\equiv f$, then clearly (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}),\n(\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and (\\ref{eq:cndtn4}) all hold\nwith ${\\mathbb Z}_2(h) = f'(t_0)h^2\/2$ for some $0 < \\delta< 1$ and $C\n\\ge f(t_0 - \\delta)$ by a Taylor expansion of $F$ and the continuity of\n$f$ and $f'$ around $t_0$.\n\\begin{cor}\\label{cor:smoothF} If there is a $\\delta> 0$ for which\n$F_n$ has a continuously differentiable density\n$f_n$ on $[t_0 - \\delta, t_0 + \\delta]$, and\n\\begin{equation}\n\\label{eq:cndsmoothF}\\quad\n\\lim_{n \\rightarrow\\infty} \\Bigl[ \\|\nF_n - F\\| + \\sup_{|t - t_0| < \\delta} \\bigl(\n|f_n(t) - f(t)| + |f_n'(t) - f'(t)| \\bigr) \\Bigr] = 0,\n\\end{equation}\nthen (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref\n{eq:cndtn3}) and (\\ref{eq:cndtn4}) hold with\n${\\mathbb Z}_2(h) = f'(t_0)h^2\/2$, and $\\Delta_n \\Rightarrow2 |\\frac\n{1}{2} f(t_0) f'(t_0)|^{1\/3} \\mathbb C$.\n\\end{cor}\n\\begin{pf} The result can be immediately derived from Taylor expansion\nof $F_n$ and the continuity of $f$ and $f'$\naround $t_0$. To illustrate the idea, we show that (\\ref{eq:cndtn3})\nholds. Let $\\gamma> 0$ be given. Clearly,\n\\begin{eqnarray}\\label{sm_boot_t6}\n&&\\biggl| F_n(t_0 + h) - F_n(t_0) - f_n(t_0) h - \\frac{1}{2} h^2 f'(t_0)\n\\biggr| \\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\le{\\frac{1}{2} h^2 \\sup_{|s| \\le|h|}} |f_n'(t_0 + s) -\nf'(t_0)|.\\nonumber\n\\end{eqnarray}\nLet $\\delta> 0$ be so small that $|f'(t) - f'(t_0)| \\le\\gamma$ for\n$|t - t_0| < \\delta$, and let $n_0$ be so large that ${\\sup_{|t - t_0|\n\\le\\delta} }|f_n'(t) - f'(t)| \\le\\gamma$ for $n \\ge n_0$. Then the\nlast line in\n(\\ref{sm_boot_t6}) is at most $\\gamma h^2$ for $|h| \\le\\delta$ and $n\n\\ge n_0$.\n\\end{pf}\n\nAnother useful remark, used below, is that if $\\lim_{n \\rightarrow\n\\infty} m_n^{1\/3} \\|F_{m_n} - F\\| = 0$, then (\\ref{eq:cndtn0}),\n(\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4}) hold.\n\nIn the next three sections, we apply Proposition \\ref{prop:Z_conv} and\nCorollary \\ref{cor:convdelta} to bootstrap samples drawn from the EDF,\nits LCM, and smoothed versions thereof. Thus, let $X_1,X_2,\\ldots\n\\stackrel{\\mathrm{ind}}{\\sim} F$; let ${\\mathbb F}_n$ be the EDF of\n$X_1,\\ldots,X_n$; and\nlet $\\tilde{F}_n$ be its LCM. If $F_n = {\\mathbb F}_n$, then\n(\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4}) hold almost\nsurely by the above remark, since\n\\begin{equation}\n\\label{eq:lil}\n\\Vert{\\mathbb F}_n - F\\Vert= O \\Biggl[\\sqrt{\\log\\log(n)\\over n}\n\\Biggr]\\qquad\\mbox{a.s.}\n\\end{equation}\nby the Law of the Iterated Logarithm for the EDF, which may be deduced\nfrom Hungarian Embedding; and the same is true if $F_n = \\tilde{F}_n$\nsince $\\Vert\\tilde{F}_n-F\\Vert\\le\\Vert{\\mathbb F}_n-F\\Vert$, by\nMarshall's lemma.\n\nIf $m_n = n$ and $f_n = \\tilde{f}_n$, then (\\ref{eq:cndtn2}) is not\nsatisfied almost surely or in probability by either ${\\mathbb F}_n$ or\n$\\tilde{F}_n$. For either choice, (\\ref{eq:cndtn3}) is satisfied in\nprobability if $f_n = f$.\n\\begin{prop}\\label{prop:edflcm}\nSuppose that $m_n = n$ and that $f_n = f$. If $F_n$ is either the EDF\n${\\mathbb F}_n$ or its LCM $\\tilde{F}_n$, then\nfor any $\\gamma, \\varepsilon> 0$, there are $C > 0$ and $0 < \\delta< 1$\nfor which (\\ref{eq:cndtn3}) holds with\nprobability at least $1- \\varepsilon$ for all large $n$.\n\\end{prop}\n\nThe proof is included in the \\hyperref[app]{Appendix}.\n\n\\section{Inconsistency and nonconvergence of the bootstrap}\\label{boots_prob}\n\nWe begin with a brief discussion of the bootstrap.\n\n\\subsection{Generalities}\\label{generalities}\n\nNow, suppose that $X_1,X_2,\\ldots\\stackrel{\\mathrm{ind}}{\\sim} F$ are\ndefined on\na probability space $(\\Omega,\\mathcal{A}, P)$. Write ${\\mathbf X}_n =\n(X_1,\\ldots,X_n)$ and suppose that the distribution function, $H_n$\nsay, of the random variable $R_n(\\mathbf{X}_n,F)$ is of interest. The\nbootstrap methodology can be broken into three simple steps:\n\n\\begin{longlist}\n\\item Construct an estimator $\\hat{F}_n$ of $F$ from ${\\mathbf X}_n$;\n\n\\item let $X_1^{*},\\ldots,X_{m_n}^{*} \\stackrel{\\mathrm{ind}}{\\sim\n}\\hat{F}_n$ be\nconditionally i.i.d. given ${\\mathbf X}_n$;\n\n\\item then let ${\\mathbf X}_n^{*} = (X_1^{*},\\ldots,X_{m_n}^{*})$\nand estimate $H_n$ by the conditional\ndistribution function of $R_n^{*} = R({\\mathbf X}_n^{*},\\hat{F}_n)$\ngiven ${\\mathbf X}_n$; that is\n\\[\nH_{n}^*(x) = P^*\\{R^*_n \\le x\\},\n\\]\nwhere $P^*\\{\\cdot\\}$ is the conditional probability given the data\n$\\mathbf{X}_n$, or equivalently, the entire sequence $\\mathbf{X} =\n(X_1,X_2,\\ldots)$.\n\\end{longlist}\nChoices of $\\hat{F}_n$ considered below are the EDF ${\\mathbb F}_n$,\nits least concave majorant\n$\\tilde{F}_n$, and smoothed versions thereof.\n\nLet $d$ denote the Levy metric or any other metric metrizing weak\nconvergence of distribution functions. We say that\n$H_{n}^*$ is \\textit{weakly}, \\textit{respectively}, \\textit{strongly},\n\\textit{consistent} if\n$d(H_n,H_n^*)\\stackrel{P}{\\rightarrow} 0$, respectively, $d(H_n,H_n^{*})\n\\to0$ a.s. If $H_{n}$ has a weak limit $H$, then consistency requires\n$H_{n}^*$ to converge weakly to $H$, in probability; and if $H$ is\ncontinuous, consistency requires\n\\[\n{\\sup_{x \\in\\mathbb{R}}} |H_{n}^*(x) - H(x)| \\stackrel{P}{\\rightarrow} 0\n\\qquad\\mbox{as } n \\rightarrow\\infty.\n\\]\nThere is also the apparent possibility that $H_n^{*}$ could converge to\na random limit; that is, that there is a\n$G\\dvtx\\Omega\\times\\mathbb{R} \\to[0,1]$ for which $G(\\omega,\\cdot)$ is a\ndistribution function for each $\\omega\\in\n\\Omega$, $G(\\cdot,x)$ is measurable for each $x \\in\\mathbb{R}$, and\n$d(G,H_n^{*}) \\stackrel{P}{\\rightarrow} 0$. This\npossibility is only apparent, however, if $\\hat{F}_n$ depends only on\nthe order statistics. For if $h$ is a bounded\ncontinuous function on $\\mathbb{R}$, then any limit in probability of\n$\\int_{\\mathbb{R}} h(x) H_n^*(\\omega;dx)$ must be invariant under\nfinite permutations of $X_1,X_2,\\ldots$ up to equivalence, and thus,\nmust be almost surely constant by the Hewitt--Savage zero--one law\n[Breiman (\\citeyear{breiman68})]. Let $\\bar G(x) = \\int_{\\Omega}\nG(\\omega;x) P(d\\omega\n)$. Then $\\bar G$ is a distribution function and $\\int_{\\mathbb{R}}\nh(x) G(\\omega;dx) = \\int_{\\mathbb{R}} h(x) \\bar G(dx)$ a.s. for each\nbounded continuous~$h$, and therefore for any countable collection of\nbounded continuous $h$. It follows that $G(\\omega;x) =\\bar G(x)$ a.e.\n$\\omega$ for all $x$ by letting $h$ approach indicator functions.\n\nNow let\n\\[\n\\Delta_n = n^{1\/3} \\{\\tilde f_n(t_0) - f(t_0) \\}\n\\quad\\mbox{and}\\quad \\Delta_n^* = m_n^{1\/3}\n\\{\\tilde f_{n,m_n}^*(t_0) - \\hat f_n(t_0) \\},\n\\]\nwhere $\\hat f_n(t_0)$ is an estimate of $f(t_0)$, for example, $\\tilde\nf_n(t_0)$, and $\\tilde f_{n,m_n}^*(t_0)$ is the\nGrenander estimator computed from the bootstrap sample $X_1^{*},\\ldots\n,X_{m_n}^{*}$. Then weak (strong) consistency of the bootstrap means\n\\begin{equation}\n\\label{eq:boot_limit}\n{\\sup_{x \\in\\mathbb R}} |P^*[\\Delta_n^* \\le x] - P[\\Delta_n \\le x]|\n\\rightarrow0\n\\end{equation}\nin probability (almost surely), since the limiting distribution (\\ref\n{eq:chrnff}) of $\\Delta_n$ is continuous.\n\n\\subsection{Bootstrapping from the NPMLE $\\tilde{F}_n$}\\label{bootsNPMLE}\n\nConsider now the case in which $m_n = n$, $\\hat F_n = \\tilde{F}_n$, and\n$\\hat{f}_n(t_0) = \\tilde{f}_n(t_0)$. Let\n\\[\n\\mathbb{Z}_n^*(h) := n^{2\/3} \\{ \\mathbb{F}_n^*(t_0 +\nn^{-{1\/3}}h) -\n\\mathbb{F}_n^*(t_0) - \\tilde f_n(t_0)n^{-{1\/3}}h \\}\n\\]\nfor $h \\in I_n = [-n^{1\/3}t_0, \\infty)$, where $\\mathbb{F}_{n}^*$\nis the EDF of the bootstrap sample\n$X_1^*,\\ldots,X_n^* \\sim\\tilde F_n$. Then $\\mathbb{Z}_n^* = \\mathbb\n{Z}_{n,1}^* + \\mathbb{Z}_{n,2}$, where\n\\begin{eqnarray}\n\\mathbb{Z}_{n,1}^* (h) &=& n^{2\/3} \\{ (\\mathbb{F}_n^*-\\tilde\n{F}_n)(t_0 + n^{-{1\/3}}h) -\n(\\mathbb{F}_n^*-\\tilde{F}_n)(t_0) \\},\n\\\\\n\\mathbb{Z}_{n,2}(h) &=& n^{2\/3} \\{\\tilde{F}_n(t_0 + h\nn^{-{1\/3}}) - \\tilde{F}_n(t_0) - \\tilde f_n(t_0) n^{-{1\/3}}h \\}.\n\\end{eqnarray}\nFurther, let $\\mathbb{W}_1$ and $\\mathbb{W}_2$ be two independent\ntwo-sided standard Brownian motions on $\\mathbb{R}$\nwith $\\mathbb{W}_1(0) = \\mathbb{W}_2(0) = 0$,\n\\begin{eqnarray*}\n\\mathbb{Z}_{1}(h) &=& \\mathbb{W}_1[f(t_0)h],\\\\\n\\mathbb{Z}_{2}^0(h) &=& \\mathbb{W}_2[f(t_0)h] +\n\\tfrac{1}{2}f'(t_0)h^2,\\\\\n\\mathbb{Z}_{2}(h) &=& L_{\\mathbb R} \\mathbb{Z}_{2}^0(h) - L_{\\mathbb R}\n\\mathbb{Z}_{2}^0(0) - (L_{\\mathbb{R}}\n\\mathbb{Z}_2^0)'(0)h,\\\\\n\\mathbb{Z} &=& \\mathbb{Z}_1 + \\mathbb{Z}_{2}.\n\\end{eqnarray*}\nThen $\\Delta_n^*$ equals the left derivative at $h = 0$ of the LCM of\n$\\mathbb{Z}_n^*$. It is first shown that\n$\\mathbb{Z}_n^*$ converges in distribution to $\\mathbb{Z}$ but the\nconditional distributions of $\\mathbb{Z}_n^*$ do not have a limit. The\nfollowing two lemmas are needed.\n\\begin{lemma}\\label{lemma:independence} Let $W_n$ and $W_n^*$ be random\nvectors in $\\mathbb{R}^l$ and $\\mathbb{R}^k$,\nrespectively; let $Q$ and $Q^*$ denote distributions on the Borel sets\nof $\\mathbb{R}^l$ and $\\mathbb{R}^k$; and let\n$\\mathcal{F}_n$ be sigma-fields for which $W_n$ is $\\mathcal\n{F}_n$-measurable. If the distribution of $W_n$ converges to $Q$ and\nthe conditional distribution of $W_n^*$ given $\\mathcal{F}_n$ converges\nin probability to $Q^*$, then the joint distribution of $(W_n,W_n^*)$\nconverges to the product measure $Q \\times Q^*$.\n\\end{lemma}\n\\begin{pf} The above lemma can be proved easily using characteristic\nfunctions. Kosorok (\\citeyear{kosorok07}) includes a detailed proof.\n\\end{pf}\n\nThe next lemma uses a special case of the Convergence of Types theorem\n[Lo\\`{e}ve (\\citeyear{loeve63}), page 203]: let $V, W, V_n$ be random\nvariables and\n$b_n$ be constants; if $V$ has a nondegenerate distribution, $V_n\n\\Rightarrow V$ as $n \\to\n\\infty$, and $V_n + b_n \\Rightarrow W$, then $b = \\lim_{n \\to\\infty}\nb_n$ exists and $W$ has the same distribution as $V+b$.\n\\begin{lemma}\\label{lemma:emp_boot1} Let $\\mathbf{X}_n^*$ be a\nbootstrap sample generated from the data $\\mathbf{X}_n$. Let $Y_n :=\n\\psi_n(\\mathbf{X}_n)$ and $Z_n := \\phi_n(\\mathbf{X}_n,\\mathbf{X}_n^*)$\nwhere $\\psi_n\\dvtx\\mathbb{R}^n \\to\\mathbb{R}$ and $\\phi_n\\dvtx\\mathbb{R}^{2\nn} \\to\\mathbb{R}$ are measurable functions; and let $K_n$ and $L_n$ be\nthe conditional distribution functions of $Y_n + Z_n$ and $Z_n$ given\n${\\mathbf X}_n$, respectively. If there are distribution functions $K$\nand $L$ for which $L$ is nondegenerate, $d(K_n,K) \\stackrel\n{P}{\\rightarrow} 0$ and $d(L_n,L) \\stackrel{P}{\\rightarrow} 0$ then\nthere is a random variable $Y$ for which $Y_n \\stackrel{P}{\\rightarrow} Y$.\n\\end{lemma}\n\\begin{pf} If $\\{n_k\\}$ is any subsequence, then there exists a further\nsubsequence $\\{n_{k_l}\\}$ for which $d(K_{n_{k_l}},K) \\rightarrow 0$\na.s. and $d(L_{n_{k_l}},L) \\rightarrow 0$ a.s. Then $Y := \\lim_{l\n\\rightarrow\\infty} Y_{n_{k_l}}$ exists a.s. by the Convergence of\nTypes theorem,\napplied conditionally given ${\\mathbf X} := (X_1,X_2,\\ldots)$ with\n$b_l = Y_{n_{k_l}}$. Note that $Y$ does not depend on the subsequence\n$n_{k_l}$, since two such subsequences can be joined to form another\nsubsequence using which we can argue the uniqueness.\n\\end{pf}\n\\begin{theorem}\\label{thm:bootmle}\n\\textup{(i)} The conditional distribution of $\\mathbb{Z}_{n,1}^*$ given\n$\\mathbf{X} = (X_1,\\break X_2,\\ldots)$ converges a.s. to the distribution\nof $\\mathbb{Z}_1$.\n\n{\\smallskipamount=0pt\n\\begin{longlist}[(iii)]\n\\item[(ii)] The unconditional distribution of $\\mathbb{Z}_{n,2}$\nconverges to that of $\\mathbb{Z}_2$ and the\nunconditional distributions of $(\\mathbb{Z}_{n,1}^*,\\mathbb{Z}_{n,2})$,\nand $\\mathbb{Z}_n^*$ converge to those of\n$(\\mathbb{Z}_{1}, \\mathbb{Z}_{2})$ and~$\\mathbb{Z}$.\n\n\\item[(iii)] The unconditional distribution of $\\Delta_n^*$ converges\nto that of $(L_{\\mathbb{R}}\\mathbb{Z})'(0)$, and (\\ref{eq:boot_limit}) fails.\n\n\\item[(iv)] Conditional on $\\mathbf{X}$, the distribution of $\\mathbb\n{Z}_n^*$ does not have a weak limit in\nprobability.\n\n\\item[(v)] If the conditional distribution function of $\\Delta_n^*$\nconverges in probability, then $(L_{\\mathbb{R}} {\\mathbb{Z}})'(0)$ and\n${\\mathbb Z}_2$ must be independent.\n\\end{longlist}}\n\\end{theorem}\n\\begin{pf} (i) The conditional convergence of $\\mathbb\n{Z}_{n,1}^*$ follows from Proposition \\ref{prop:Z_conv} with $m_n = n$,\n$F_n = \\tilde{F}_n$, $\\mathbb{F}_{n,m_n} = \\mathbb{F}_n^*$, applied\nconditionally given ${\\mathbf X}$. It is only necessary to show that\n(\\ref{eq:cndtn1}) holds a.s., and this follows from the Law of the\nIterated Logarithm for ${\\mathbb F}_n$ and Marshall's lemma, as\nexplained in Section \\ref{remarks}. The unconditional limiting\ndistribution of\n$\\mathbb{Z}_{n,1}^*$ must also be that of $\\mathbb{Z}_1$.\n\n(ii) Let\n\\[\n{\\mathbb Z}_{n,2}^0(h) = n^{2\/3}[{\\mathbb F}_n(t_0+n^{-{1\/3}}h) -\n{\\mathbb F}_n(t_0) - f(t_0)n^{-{1\/3}}h]\n\\]\nand observe that\n\\[\n{\\mathbb Z}_{n,2}(h) = L_{I_n} {\\mathbb Z}_{n,2}^0(h) -\n[L_{I_n}{\\mathbb Z}_{n,2}^0(0) + (L_{I_n}{\\mathbb Z}_{n,2}^0)'(0)h\n].\n\\]\nThe unconditional convergence of ${\\mathbb Z}_{n,2}^0$ and\n$L_{I_n}{\\mathbb Z}_{n,2}^0$ follow from Corollary \\ref{cor:convdelta2}\napplied with $F_n \\equiv F$, as explained in Section \\ref{remarks}. The\nconvergence in distribution of ${\\mathbb Z}_{n,2}$ now follows from the\nContinuous Mapping theorem, using Lemma \\ref{lemma:con_derv} and\narguments similar to those in the proof of Corollary \\ref{cor:convdelta}.\n\nIt remains to show that $\\mathbb{Z}_{n,1}^*$ and $\\mathbb{Z}_{n,2}^0$\nare asymptotically independent, for example, the joint\nlimit distribution of $\\mathbb{Z}_{n,1}^*$ and $\\mathbb{Z}_{n,2}^0$ is\nthe product of their marginal limit\ndistributions. For this, it suffices to show that $(\\mathbb\n{Z}_{n,1}^*(t_1),\\ldots,\\break\\mathbb{Z}_{n,1}^*(t_k))$ and\n$(\\mathbb{Z}_{n,2}^0(s_1),\\ldots, \\mathbb{Z}_{n,2}^0(s_l))$ are\nasymptotically independent, for all choices $-\\infty< t_1 < \\cdots<\nt_k <\\infty$ and $-\\infty< s_1 < \\cdots< s_l <\\infty$. This is an\neasy consequence of\nLemma \\ref{lemma:independence} applied with $W_n^* = (\\mathbb\n{Z}_{n,1}^*(t_1), \\ldots,\\mathbb{Z}_{n,1}^*(t_k))$ and $W_n = (\\mathbb\n{Z}_{n,2}^0(s_1),\\ldots, \\mathbb{Z}_{n,2}^0(s_l))$, and $\\mathcal{F}_n\n= \\sigma(X_1,X_2,\\ldots,X_n)$.\\vspace*{1pt}\n\n(iii) We will appeal to Corollary \\ref{cor:convdelta} to find the\nunconditional distribution of $\\Delta_n^*$. We already know that\n$\\mathbb{Z}_n^*$ converges in distribution to $\\mathbb{Z}$. That (\\ref\n{eq:loc2}) holds for the limit $\\mathbb{Z}$ can be directly verified\nfrom the definition of the process. We only have to show that (\\ref\n{eq:loc}) holds unconditionally with $\\mathbb{Z}_n = \\mathbb{Z}_n^*$.\n\nLet $\\varepsilon> 0$ and $\\gamma> 0$ be given. By Proposition \\ref\n{prop:edflcm}, there exists $\\delta> 0$ and $C>0$ such that $P(A_n)\n\\ge1- \\varepsilon$ for all $n > N_0$, where\n\\[\nA_n:= \\bigl\\{ \\bigl|\\tilde F_n(t_0 + h) + \\tilde F_n(t_0) - f(t_0)h -\n\\tfrac{1}{2} f'(t_0) h^2 \\bigr| \\le\\gamma h^2 + C n^{-{2\/3}},\n\\forall|h| \\le\\delta\\bigr\\}.\n\\]\nWe can also assume that $|F(t_0 + h) + F(t_0) - f(t_0)h - (1\/2) f'(t_0)\nh^2| \\le\\gamma h^2$ for $|h| \\le\\delta$. Let ${\\mathbb Y}_n^*(h) =\nn^{2\/3}[{\\mathbb F}_n^{*}(t_0+n^{-{1\/3}}h) - {\\mathbb\nF}_n^{*}(t_0) -\nf(t_0)n^{-{1\/3}}h]$, so that ${\\mathbb Z}_n^*(h) = {\\mathbb\nY}_n^*(h) - \\Delta_n h$ for all $h \\in I_n$, and\n\\[\nL_K{\\mathbb Z}^*_n = L_K{\\mathbb Y}^*_n - \\Delta_n h\n\\]\nfor all $h \\in K$ for any interval $K \\subseteq I_n$.\n\nLet $G_n = \\tilde F_n \\mathbf{1}_{A_n} + F \\mathbf{1}_{A_n^c}$ and let\n$P_{G_n}^\\infty$ denote the\nprobability when generating the bootstrap samples from $G_n$. Then\n$G_n$ satisfies (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}), (\\ref{eq:cndtn3})\nand (\\ref{eq:cndtn4}) a.s. with $m_n = n$, $F_n = G_n$, $\\mathbb\n{F}_{n,m_n} = \\mathbb{F}^*_n \\mathbf{1}_{A_n} + \\mathbb{F}_n \\mathbf\n{1}_{A_n^c}$ and $f_n = f$. Let $J$ be a compact interval. By\nProposition \\ref{prop:loc}, applied conditionally, there exists a\ncompact interval $K$ (not depending on $\\omega$, by the \\textit{remark}\nnear the end of the proof of Proposition~\\ref{prop:loc}) such that $K\n\\supseteq J$ and\n\\[\nP_{G_n}^\\infty[L_{I_n}{\\mathbb Y}^*_n = L_{K} {\\mathbb Y}^*_n \\mbox{\non } J] (\\omega) \\ge1-\\varepsilon\n\\]\nfor $n \\ge N(\\omega) $ for a.e. $\\omega$. As $N(\\cdot)$ is bounded in\nprobability, there exists $N_1 > 0$ such that $P(B) \\ge1 - \\varepsilon$,\nwhere $B := \\{\\omega\\dvtx N(\\omega) \\le N_1\\}$. By increasing $N_1$ if\nnecessary, let us also suppose that $N_1 \\ge N_0$. Then\n\\begin{eqnarray*}\nP[L_{I_{m_n}} \\mathbb{Z}_n^* = L_K \\mathbb{Z}_n^* \\mbox{ on } J] & = &\nP[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb{Y}_n^* \\mbox{ on } J]\n\\\\\n& \\ge& \\int_{A_n} P^*[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb{Y}_n^*\n\\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& = & \\int_{A_n} P_{G_n}^\\infty[L_{I_{m_n}} \\mathbb{Y}_n^* = L_K \\mathbb\n{Y}_n^* \\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& \\ge& \\int_{A_n \\cap B} P_{G_n}^\\infty[L_{I_{m_n}} \\mathbb{Y}_n^* =\nL_K \\mathbb{Y}_n^* \\mbox{ on } J](\\omega) \\,dP(\\omega) \\\\\n& \\ge& \\int_{A_n \\cap B} (1 - \\varepsilon) \\,dP(\\omega) \\ge1 - 3\n\\varepsilon\\qquad \\mbox{for all } n \\ge N_1\n\\end{eqnarray*}\nas $P(A_n \\cap B) \\ge1 - 2 \\varepsilon$ for $n \\ge N_1$. Thus, (\\ref\n{eq:loc}) holds and Corollary \\ref{cor:convdelta} gives $\\Delta_n^*\n\\Rightarrow(L_{\\mathbb{R}} \\mathbb{Z})'(0)$.\n\nIf (\\ref{eq:boot_limit}) holds in probability, then the unconditional\nlimit distribution of $\\Delta_n^{*}$ would be that of $2 |\\frac{1}{2}\nf(t_0) f'(t_0)|^{1\/3} \\mathbb C$, which is different from the\ndistribution of $(L_{\\mathbb R}{\\mathbb Z})'(0)$, giving rise to a\ncontradiction.\n\n(iv) We use the method of contradiction. Let $Z_n := \\mathbb\n{Z}_{n,1}^*(h_0)$ and $Y_n := \\mathbb{Z}_{n,2}(h_0)$ for some fixed\n$h_0 > 0$ (say $h_0 = 1$) and suppose that the conditional distribution\nfunction of $Z_n + Y_n = \\mathbb{Z}_n^*(h_0)$ converges in probability\nto the distribution function~$G$. By Proposition \\ref{prop:Z_conv}, the\nconditional distribution of $Z_n$ converges in probability to a normal\ndistribution, which is obviously nondegenerate. Thus, the assumptions\nof Lemma \\ref{lemma:emp_boot1} are satisfied and we conclude that $Y_n\n\\stackrel{P}{\\rightarrow} Y$, for some random variable $Y$. It then\nfollows from the Hewitt--Savage zero--one law that $Y$ is a constant, say\n$Y = c_0$ w.p. 1. The contradiction arises since $Y_n$ converges in\ndistribution to $\\mathbb{Z}_2(h_0)$ which is not a constant a.s.\n\n\\begin{figure}\n\n\\includegraphics{777f01.eps}\n\n\\caption{Scatter plot of $10\\mbox{,}000$ random draws of $((L_{\\mathbb\nR}{\\mathbb Z})'(0),(L_{\\mathbb R}{\\mathbb\nZ}_2^{0})'(0))$ when $f(t_0) = 1$ and $f'(t_0) = -2$.}\\label{fig:simul}\n\\end{figure}\n\n(v) We can show that the (unconditional) joint distribution of\n$(\\Delta_n^*, \\mathbb{Z}_{n,2}^0)$ converges to that of\n$((L_{\\mathbb{R}} {\\mathbb{Z}})'(0), {\\mathbb Z}_2^0)$. But\n$\\Delta_n^*$ and $\\mathbb{Z}_{n,2}^0$ are asymptotically independent by\nLemma \\ref{lemma:independence} applied to $W_n =\n(\\mathbb{Z}_{n,2}^0(t_1),\\mathbb{Z}_{n,2}^0(t_2),\n\\ldots,\\mathbb{Z}_{n,2}^0(t_l))$, where $t_i \\in\\mathbb{R}$, $W_n^* =\n\\Delta_n^*$ and $\\mathcal{F}_n = \\sigma(X_1,X_2,\\ldots,X_n)$. Therefore,\n$(L_{\\mathbb{R}} {\\mathbb{Z}})'(0)$ and ${\\mathbb Z}_2^0$ are\nindependent. The proposition follows directly since $\\mathbb{Z}_2$ is a\nmeasurable function of $\\mathbb{Z}_{2}^0$.\n\\end{pf}\n\nIf the conditional distribution of $\\Delta^*_n$ converges in\nprobability, as a consequence of (v) of Theorem\n\\ref{thm:bootmle}, $(L_{\\mathbb R}{\\mathbb Z})'(0)$ and $(L_{\\mathbb\nR}{\\mathbb Z}_2^{0})'(0)$ must also be independent. Figure \\ref\n{fig:simul} shows the scatter plot of $(L_{\\mathbb R}{\\mathbb Z})'(0)$\nand $(L_{\\mathbb R}{\\mathbb Z}_2^{0})'(0)$ obtained from a simulation\nstudy with $10\\mbox{,}000$ samples, $f(t_0) = 1$ and $f'(t_0) = -2$. The\ncorrelation coefficient obtained $-0.2999$ is highly significant\n($p$-value $< 0.0001$). Thus, when combined with simulations, (v) of\nTheorem \\ref{thm:bootmle} strongly suggests that the conditional\ndistribution of $\\Delta_n^*$ does not converge in probability.\n\n\\subsection{Bootstrapping from the EDF}\n\nA similar, slightly simpler\npattern arises if the bootstrap sample is drawn\nfrom $\\hat{F}_n = {\\mathbb F}_n$. Define ${\\mathbb Z}_n^*$ as before,\nand let $\\mathbb{Z}_{n,1}^* (h) = n^{2\/3} \\{\n(\\mathbb{F}_n^*-{\\mathbb F}_n)(t_0 + n^{-{1\/3}}h) - (\\mathbb\n{F}_n^*- {\\mathbb F}_n)(t_0)\\}$ and\n$\\mathbb{Z}_{n,2}(h) = n^{2\/3} \\{{\\mathbb F}_n(t_0 + h n^{-{1\/3}}) -\n{\\mathbb F}_n(t_0) - \\tilde f_n(t_0)\nn^{-{1\/3}}h\\}$. Then ${\\mathbb Z}_n^* = {\\mathbb Z}_{n,1}^* +\n{\\mathbb Z}_{n,2}$. Recall the definition of the\nprocesses $\\mathbb{W}_1$, $\\mathbb{W}_2$, $\\mathbb{Z}_1$, $\\mathbb\n{Z}^0_2$ in Section \\ref{bootsNPMLE}. Define\n\\[\n\\mathbb{Z}_2(h) = \\mathbb{Z}_2^0(h) -(L_{\\mathbb{R}} \\mathbb\n{Z}_2^0)'(0) h.\n\\]\n\\begin{theorem}\\label{thm:bootmle2}\n\\textup{(i)} The conditional distribution of $\\mathbb{Z}_{n,1}^*$ given\n$\\mathbf{X} = (X_1,X_2,\\break\\ldots)$ converges a.s. to the distribution\nof $\\mathbb{Z}_1$.\n\n{\\smallskipamount=0pt\n\\begin{longlist}[(iii)]\n\\item[(ii)] The unconditional distribution of $\\mathbb{Z}_{n,2}$\nconverges to that of $\\mathbb{Z}_2$ and the\nunconditional distributions of $(\\mathbb{Z}_{n,1}^{*},\\mathbb\n{Z}_{n,2})$, and $\\mathbb{Z}_n^*$ converge to those of $(\\mathbb\n{Z}_{1},\\mathbb{Z}_{2})$ and~$\\mathbb{Z}$.\n\n\\item[(iii)] The unconditional distribution of $\\Delta_n^*$ converges\nto that of $(L_{\\mathbb{R}}\\mathbb{Z})'(0)$, and (\\ref{eq:boot_limit}) fails.\n\n\\item[(iv)] Conditional on $\\mathbf{X}$, the distribution of $\\mathbb\n{Z}_n^*$ does not have a weak limit in\nprobability.\n\n\\item[(v)] If the conditional distribution function of $\\Delta_n^*$\nconverges in probability, then\n$(L_{\\mathbb{R}}{\\mathbb{Z}})'(0)$ and ${\\mathbb Z}_2$ must be independent.\n\\end{longlist}}\n\\end{theorem}\n\\begin{Remark*} The proof of this theorem runs along similar lines to\nthat of Theorem \\ref{thm:bootmle}. We briefly\nhighlight the differences.\n\\end{Remark*}\n\n\\begin{longlist}\n\\item The conditional convergence of $\\mathbb{Z}_{n,1}^*$ follows\nfrom Proposition \\ref{prop:Z_conv} with $m_n = n$, $F_n = \\mathbb\n{F}_n$, $\\mathbb{F}_{n,m_n} = \\mathbb{F}_n^*$, applied conditionally\ngiven ${\\mathbf X}$. It is only necessary to show that (\\ref\n{eq:cndtn1}) is satisfied almost surely, and this follows from the Law\nof the Iterated Logarithm for ${\\mathbb F}_n$, as explained in\nSection \\ref{remarks}. Then the unconditional limiting distribution of\n$\\mathbb\n{Z}_{n,1}^*$ must also be that of $\\mathbb{Z}_1$.\n\n\\item The proof is similar to that of (ii) of Theorem \\ref\n{thm:bootmle}, except that now $\\mathbb{Z}_{n,2}(h) = \\mathbb\n{Z}^0_{n,2}(h) - (L_{I_n} \\mathbb{Z}^0_{n,2})'(0) h$.\n\\end{longlist}\n\nThe proofs of (iii)--(v) are very similar to that of (iii)--(v) of\nTheorem \\ref{thm:bootmle}.\n\n\\subsection{Performance of the bootstrap methods in finite\nsamples}\\label{simul}\n\nIn this subsection, we illustrate the poor finite sample performance\nof the two inconsistent bootstrap schemes, namely, bootstrapping from\nthe EDF $\\mathbb{F}_n$ and the NPMLE $\\tilde F_n$. Table~\\ref\n{PerfBootsMeth} shows the estimated coverage probabilities of nominal\n95\\% confidence intervals for $f(1)$ using the two bootstrap methods\n\\begin{table}[b]\n\\caption{Estimated coverage probabilities of nominal 95\\% confidence\nintervals for $f(1)$ while bootstrapping from the EDF $\\mathbb{F}_n$\nand NPMLE $\\tilde F_n$, with varying sample size $n$ for the two\nmodels: $\\operatorname{Exponential}(1)$ (left) and $|Z|$ where $Z \\sim\n\\operatorname{Normal}(0,1)$ (right)}\\label{PerfBootsMeth}\n\\begin{tabular*}{\\tablewidth}{@{\\extracolsep{\\fill}}lccccc@{}}\n\\hline\n$\\bolds n$ & \\textbf{EDF} & \\textbf{NPMLE} & $\\bolds n$ & \\textbf{EDF} & \\textbf{NPMLE} \\\\\n\\hline\n\\phantom{0}$50$ & 0.747 & 0.720 & \\phantom{0}$50$ & 0.761 & 0.739 \\\\\n$100$ & 0.776 & 0.755  &  $100$ & 0.778 & 0.757 \\\\\n$200$ & 0.802 & 0.780  &  $200$ & 0.780 & 0.762 \\\\\n$500$ & 0.832 & 0.797  &  $500$ & 0.788 & 0.755 \\\\\n\\hline\n\\end{tabular*}\n\\end{table}\nfor different sample sizes, when the true distribution is assumed to be\nExponential(1) and $|\\mathrm{Normal}(0,1)|$, respectively. We used 1000\nbootstrap samples to compute each confidence interval and then\nconstructed 1000 such confidence intervals to estimate the actual\ncoverage probabilities. As is clear from the table the coverage\nprobabilities fall well short of the nominal 0.95 value. Leger and\nMacGibbon (\\citeyear{legerMa06}) also illustrate such a discrepancy in\nthe nominal and\nactual coverage probabilities while bootstrapping from the EDF for the\nChernoff's estimator of the mode.\n\n\nFigure \\ref{fig:InConsEDF_Hist} shows the histograms (computed from\n10,000 bootstrap samples) of the two inconsistent bootstrap\ndistributions obtained from a single sample of 500 Exponential(1)\nrandom variables along with the histogram of the exact distribution of\n$\\Delta_n$ (obtained from simulation). The bootstrap distributions are\nskewed and have very different shapes and supports compared to that on\nthe left panel of Figure \\ref{fig:InConsEDF_Hist}. The histograms\nillustrate the inconsistency of the bootstrap procedures.\n\n\\begin{figure}\n\n\\includegraphics{777f02.eps}\n\n\\caption{Histograms of the exact distribution of $\\Delta_n$ (left\npanel) and the two bootstrap distributions while drawing bootstrap\nsamples from $\\mathbb{F}_n$ (middle panel) and $\\tilde F_n$ (right\npanel) for $n = 500$.}\\label{fig:InConsEDF_Hist}\n\\end{figure}\n\n\\begin{figure}[b]\n\n\\includegraphics{777f03.eps}\n\n\\caption{Estimated 0.95 quantile of the bootstrap distribution while\ngenerating the bootstrap samples from $\\mathbb{F}_n$ (dashed lines) and\n$\\tilde F_n$ (solid-dotted lines) for two independent data sequences\nalong with the 0.95 quantile of the limit distribution of $\\Delta_n$\n(solid line) for the two models: $\\operatorname{Exponential}(1)$ (left panel) and $|Z|$\nwhere $Z \\sim\\operatorname{Normal}(0,1)$ (right panel).}\\label{fig:InConsQuantile}\n\\end{figure}\n\nThe estimated coverage probabilities in Table \\ref{PerfBootsMeth} are\nunconditional [see (iii) of Theorems \\ref{thm:bootmle} and \\ref\n{thm:bootmle2}] and do not provide direct evidence to suggest that the\nconditional distribution of $\\Delta_n^*$ does not converge in\nprobability. Figure \\ref{fig:InConsQuantile} shows the estimated 0.95\nquantile of the bootstrap distribution for two independent data\nsequences as the sample size increases from 500 to 10,000, for the two\nbootstrap procedures, and for both the models (exponential and normal).\nThe bootstrap quantile fluctuates enormously even at very large sample\nsizes and shows signs of nonconvergence. If the bootstrap were\nconsistent, the estimated quantiles should converge to 0.6887 (0.8269),\nthe 0.95 quantile of the limit distribution of $\\Delta_n$, indicated by\nthe solid line in Figure \\ref{fig:InConsQuantile}. From the left panel\nof Figure \\ref{fig:InConsQuantile}, we see that the estimated bootstrap\n0.95 quantiles (obtained from the two procedures) for one data sequence\nstays below 0.6887, while for the other, the 0.95 quantiles stay above\n0.6887, indicating the strong dependence on the sample path. Note that\nif the bootstrap distributions had a limit, then Figure \\ref\n{fig:InConsQuantile} suggests that the limit varies with the sample\npath, and that is impossible as explained in Section \\ref\n{generalities}. This provides evidence for the nonconvergence of the\nbootstrap estimator.\n\n\n\\section{Consistent bootstrap methods}\\label{smooth_boots}\n\nThe main reason for the inconsistency of bootstrap methods discussed in\nthe previous section is the lack of smoothness of the distribution\nfunction from which the bootstrap samples are generated. The EDF\n$\\mathbb{F}_n$ does not have a density, and $\\tilde F_n$ does not have\na differentiable density, whereas $F$ is assumed to have a nonzero\ndifferentiable density at $t_0$. At a more technical level, the lack of\nsmoothness manifests itself through the failure of (\\ref{eq:cndtn2}).\n\nThe results from Section \\ref{prelim} may be directly applied to derive\nsufficient conditions on the smoothness of the distribution from which\nthe bootstrap samples are generated. Let\n$X_1,X_2,\\ldots\\stackrel{\\mathrm{ind}}{\\sim}\nF$; let $\\hat F_n$ be an estimate of $F$ computed from $X_1,\\ldots\n,X_n$; and let $\\hat f_n$ be the density of $\\hat F_n$ or a surrogate,\nas in Section \\ref{boots_prob}.\n\\begin{theorem}\\label{thm:cons_sm_boot} If (\\ref{eq:cndtn0}), (\\ref\n{eq:cndtn1}), (\\ref{eq:cndtn2}), (\\ref{eq:cndtn3}) and\n(\\ref{eq:cndtn4}) hold a.s. with $F_n = \\hat F_n$ and $f_n = \\hat f_n$,\nthen the bootstrap estimate is strongly\nconsistent, for example, (\\ref{eq:boot_limit}) holds w.p. 1. In\nparticular, the bootstrap estimate is strongly consistent if\nthere is a $\\delta> 0$ for which $\\hat F_n$ has a continuously\ndifferentiable density $\\hat f_n$ on $[t_0 - \\delta,t_0 + \\delta]$, and\n(\\ref{eq:cndsmoothF}) holds a.s. with $F_n = \\hat F_n$ and $f_n = \\hat f_n$.\n\\end{theorem}\n\\begin{pf} That $\\Delta_n^*$ converges weakly to the distribution on\nthe right-hand side of (\\ref{eq:chrnff}) a.s. follows from Corollary\n\\ref\n{cor:convdelta2} applied conditionally given $\\mathbf{X}$ with $F_n =\n\\hat F_n$ and $f_n = \\hat f_n$. The second assertion follows similarly\nfrom Corollary \\ref{cor:smoothF}.\n\\end{pf}\n\n\\subsection{Smoothing $\\tilde F_n$}\n\nWe show that generating bootstrap\nsamples from a suitably smoothed version of\n$\\tilde{F}_n$ leads to a consistent bootstrap procedure. To avoid\nboundary effects and ensure that the smoothed version has a decreasing\ndensity on $(0,\\infty)$, we use a logarithmic transformation. Let $K$\nbe a twice continuously differentiable symmetric density for which\n\\begin{equation}\n\\label{eq:kay}\n\\int_{-\\infty}^{\\infty} [K(z)+|K'(z)|+|K''(z)|]e^{\\eta|z|}\\,dz < \\infty\n\\end{equation}\nfor some $\\eta> 0$. Let\n\\begin{eqnarray}\\label{eq:fchck1}\nK_h(x,u) & = & {1\\over hx}K \\biggl[{1\\over h}\\log\\biggl({u\\over x}\\biggr) \\biggr]\\quad\n\\mbox{and} \\nonumber\\\\[-8pt]\\\\[-8pt]\n\\check{f}_n(x) & = & \\int_{0}^{\\infty} K_h(x,u)\\tilde{f}_n(u)\\,du = \\int\n_{0}^{\\infty} K_h(1,u)\\tilde{f}_n(xu)\\,du.\\nonumber\n\\end{eqnarray}\nThus, $e^y\\check{f}_n(e^y) = \\int_{-\\infty}^{\\infty}\nh^{-1}K[h^{-1}(y-z)] \\tilde{f}_n(e^z)e^z \\,dz$. Integrating and using\ncapital letters to denote distribution functions,\n\\begin{eqnarray*}\n\\label{eq:fchck2}\n\\check{F}_n(e^y) & = & \\int_{-\\infty}^y \\check f_n(e^s) e^s \\,ds\n\\\\\n& = & \\int_{-\\infty}^y \\int_{-\\infty}^\\infty\\frac{1}{h} K \\biggl( \\frac\n{s - v}{h} \\biggr) \\tilde f_n(e^v) e^v \\,dv \\,ds\n\\\\\n& = & \\int_{-\\infty}^{\\infty} K(z)\\tilde{F}_n(e^{y-hz})\\,dz.\n\\end{eqnarray*}\nAlternatively, integrating (\\ref{eq:fchck1}) by parts yields\n\\[\n\\check{f}_n(x) = -\\int_{0}^{\\infty} {\n\\partial\\over\n\\partial u}K_h(x,u)\\tilde{F}_n(u)\\,du.\n\\]\nThe proof of (\\ref{eq:boot_limit}) requires showing that $\\check{F}_n$\nand its derivatives are sufficiently close to\nthose of $F$, and it is convenient to separate the estimation error\n$\\check{F}_n-F$ into sampling and approximation\nerror. Thus, let\n\\begin{equation}\n\\label{eq:fbar}\n\\bar{F}_h(e^y) = \\int_{-\\infty}^{\\infty} K(z) F(e^{y-hz})\\,dz.\n\\end{equation}\nWe denote the first and second derivatives of $\\bar{F}_h$ by $\\bar\n{f}_h$ and $\\bar{f}_h'$, respectively. Recall that $F$ is assumed to\nhave a nonincreasing density on $(0,\\infty)$ that is continuously\ndifferentiable near $t_0$.\n\\begin{lemma}\\label{lem:fbar} ${\\lim_{h \\rightarrow0}} \\| \\bar F_h - F\\|\n= 0$, and there is a $\\delta> 0$ for which\n\\begin{equation}\n\\label{eq:fbar2}\n\\lim_{h\\to0} \\sup_{|x-t_0|\\le\\delta} [ |\\bar{f}_h(x)-f(x)| +\n|\\bar{f}_h'(x)-f'(x)| ] = 0.\n\\end{equation}\n\\end{lemma}\n\\begin{pf} First, observe that\n\\[\n\\bar{F}_h(e^y)-F(e^y) = \\int_{-\\infty}^{\\infty}\nK(z)[F(e^{y-hz})-F(e^y)]\\,dz\n\\]\nby (\\ref{eq:fbar}). That $\\lim_{h \\rightarrow0} \\bar F_h(x) = F(x)$\nfor all $x \\ge0$ follows easily from the Dominated Convergence\ntheorem, and uniform convergence then follows from Polya's theorem.\nThis establishes the first assertion of the lemma. Next, consider (\\ref\n{eq:fbar2}). Given $t_0 > 0$, let $y_0 = \\log(t_0)$ and let $\\delta>\n0$ be so small that $e^yf(e^y)$ is continuously differentiable (in $y$)\non $[y_0-2\\delta,y_0+2\\delta]$. Then\n\\begin{eqnarray*}\n\\bar{f}_h(x) - f(x) & = & \\int_{-\\infty}^{\\infty} K(z)[f(xe^{hz})-f(x)]\ne^{h z} \\,dz \\\\\n&&{} + f(x) \\int_{-\\infty}^{\\infty} (e^{hz} - 1) K(z) \\,dz\n\\end{eqnarray*}\nand thus\n\\[\n{\\sup_{|x-t_0|\\le\\delta}} |\\bar{f}_h(x)-f(x)| \\le{\\int_{-\\infty}^{\\infty\n} \\sup_{|x-t_0|\\le\\delta}} |f(xe^{hz}) - f(x)|\ne^{h z} K(z)\\,dz + O(h^2)\n\\]\nfor any $0 < \\delta< t_0$. For sufficiently small $\\delta$, the\nintegrand approach zero as $h \\to0$; and it is bounded by $\\sup\n_{|x-t_0|\\le\\delta} (e^{-hz}\/x + f(x)) e^{hz} K(z)$, since $f(x) \\le\n1\/x$ for all $x > 0$. So the right-hand side approaches zero as $h \\to\n0$ by\nthe Dominated Convergence theorem. That ${\\sup_{|x-t_0|\\le\\delta}} |\\bar\n{f}_h'(x)-f'(x)| \\to0$ may be established similarly.\n\\end{pf}\n\\begin{theorem} Let $K$ be a twice continuously differentiable,\nsymmetric density for which (\\ref{eq:kay}) holds. If\n\\[\nh = h_n \\to0 \\quad\\mbox{and}\\quad h_n^{2}\\sqrt{n\\over\\log\\log(n)}\n\\to\\infty,\n\\]\nthen the bootstrap estimator is strongly consistent; that is, (\\ref\n{eq:boot_limit}) holds a.s.\n\\end{theorem}\n\\begin{pf} By Theorem\\vspace*{1pt} \\ref{thm:cons_sm_boot}, it suffices to show that\n(\\ref{eq:cndsmoothF}) holds a.s. with $\\hat\nF_n = \\check F_n$ and $\\hat f_n = \\check f_n$; and this would follow\nfrom\n\\[\n\\|\\check F_n - \\bar F_h\\| + \\sup_{|x - t_0| \\le\\delta} [|\\check f_n(x)\n- \\bar f_h(x)| + |\\check f_n'(x) - \\bar\nf_h'(x)|] \\rightarrow0 \\qquad\\mbox{a.s.}\n\\]\nfor some $\\delta> 0$ and Lemma \\ref{lem:fbar}. Clearly, using (\\ref\n{eq:fchck2}),\n\\begin{equation}\\label{eq:Fn-Fh}\n\\check{F}_n(e^y)-\\bar{F}_h(e^y) = \\frac{1}{h} \\int_{-\\infty}^{\\infty}\n[\\tilde{F}_n(e^{t}) - F(e^{t})] K \\biggl( \\frac{y\n- t}{h} \\biggr) \\,dt\n\\end{equation}\nfor all $y$, so that\n\\[\n\\| \\check{F}_n - \\bar F_h \\| \\le\\|{\\tilde F}_n - F \\| \\le\\|{\\mathbb\nF}_n - F \\| = O \\bigl[ \\sqrt{\\log\\log(n)\/n}\n\\bigr] \\qquad\\mbox{a.s.}\n\\]\nby Marshall's lemma and the Law of the Iterated Logarithm.\nDifferentiating (\\ref{eq:Fn-Fh}) gives\n\\[\n\\check f_n(e^{y}) - \\bar f_h(e^{y}) = \\frac{e^{-y}}{h^2} \\int_{-\\infty\n}^{\\infty} [\\tilde{F}_n(e^{t}) - F(e^{t})]\nK' \\biggl( \\frac{y - t}{h} \\biggr) \\,dt.\n\\]\nDifferentiating (\\ref{eq:Fn-Fh}) again and then taking absolute values\nand considering $0 < h \\le1$, we get\n\\begin{eqnarray*}\n&&\\hspace*{-6pt} \\sup_{|x - t_0| \\le\\delta} \\{|\\check f_n(x) - \\bar f_h(x)| +\n|\\check f_n'(x) - \\bar f_h'(x)| \\} \\\\\n&&\\hspace*{-6pt}\\qquad \\le \\frac{M}{h^3} \\sup_{|x - t_0| \\le\\delta} \\int_{-\\infty\n}^{\\infty} |\\tilde{F}_n(e^{t}) - F(e^{t})| \\biggl[ \\biggl| K' \\biggl( \\frac\n{\\log x -\nt}{h} \\biggr) \\biggr| + \\biggl| K'' \\biggl( \\frac{\\log x - t}{h} \\biggr)\n\\biggr| \\biggr] \\,dt \\\\\n&&\\hspace*{-6pt}\\qquad \\le \\frac{M}{h^2} \\|{\\mathbb F}_n - F \\| \\int_{-\\infty}^{\\infty}\n[|K'(z)| + |K''(z)|] \\,dz\n\\rightarrow 0 \\qquad\\mbox{a.s.}\n\\end{eqnarray*}\nfor a constant $M > 0$, as $h_n^2 \\sqrt{n\/ \\log\\log(n)} \\rightarrow\n\\infty$, where Marshall's lemma and the Law of Iterated Logarithm have\nbeen used again.\n\\end{pf}\n\n\\subsection{$m$ out of $n$ bootstrap}\\label{m_n_boots}\n\nIn Section \\ref\n{boots_prob}, we showed that the two most intuitive methods of\nbootstrapping are inconsistent. In this section, we show that the\ncorresponding $m$ out of $n$ bootstrap procedures are weakly consistent.\n\\begin{theorem}\\label{thm:m_n_boot1} If $\\hat F_n = \\mathbb{F}_n$,\n$\\hat f_n = \\tilde f_n$, and $m_n = o(n)$ then the\nbootstrap procedure is weakly consistent, for example, (\\ref\n{eq:boot_limit}) holds in probability.\n\\end{theorem}\n\\begin{pf}\nConditions (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref{eq:cndtn4})\nhold a.s. from (\\ref{eq:lil}), as\nexplained in Section \\ref{remarks}. To verify (\\ref{eq:cndtn3}), let\n$\\gamma> 0$ be given. From the proof of Proposition \\ref{prop:loc}\n[also see Kim and Pollard (\\citeyear{kimPo90}), page 218], there exists\n$\\delta>0$\nsuch that $|\\mathbb{F}_{n}(t_0 + h) - \\mathbb{F}_{n}(t_0) - F(t_0 + h)\n- F(t_0)| \\le\\gamma h^2 + \\mathcal{C}_n n^{-2\/3}$,\nfor $|h| \\le\\delta$, where $\\mathcal{C}_n$'s are random variables of\norder $O_P(1)$. We can also assume that $|F(t_0 + h) + F(t_0) - f(t_0)h\n- (1\/2) f'(t_0) h^2| \\le(1\/2) \\gamma h^2$ for $|h| \\le\\delta$. Then,\nusing the inequality $2|a b| \\le\\gamma a^2 + b^2\/\\gamma$,\n\\begin{eqnarray}\\label{eq:A_nB_nC_n}\\qquad\n&& \\biggl| \\mathbb{F}_{n}(t_0 + h) - \\mathbb{F}_{n}(t_0)\n- h \\tilde f_n(t_0) - \\frac{1}{2}\nh^2 f'(t_0) \\biggr| \\nonumber\\\\\n&&\\qquad \\le \\biggl| \\mathbb{F}_{n}(t_0 + h) - \\mathbb\n{F}_{n}(t_0) - h f(t_0) - \\frac{1}{2}\nh^2 f'(t_0) \\biggr| + |h| |\\tilde f_n(t_0) - f(t_0) |\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad \\le \\biggl\\{\\gamma h^2 + \\mathcal{C}_n n^{-{2 \/\n3}} + \\frac{1}{2} \\gamma h^2 \\biggr\\} + \\biggl\\{ \\frac{1}{2} \\gamma h^2\n+ \\frac{1}{2 \\gamma} |\\tilde\nf_n(t_0) - f(t_0) |^2 \\biggr\\}\\nonumber\\\\\n&&\\qquad \\le 2 \\gamma h^2 + \\mathcal{C}_n n^{-{2 \/3}} +\nO_P(n^{-2\/3}) \\le2 \\gamma h^2 + o_P(m_n^{-{2 \/3}}).\\nonumber\n\\end{eqnarray}\nFor (\\ref{eq:cndtn2}), write\n\\begin{eqnarray}\n\\label{eq:m_n_boot_t2}\n&& m_n^{{2\/3}} \\{\\mathbb{F}_n(t_0 + m_n^{-{1\/3}}h) - \\mathbb\n{F}_n(t_0) - m_n^{-{1\/3}}\\tilde\nf_n(t_0)h\\} \\nonumber\\\\\n&&\\qquad = m_n^{{2\/3}} \\{ (\\mathbb{F}_n - F)(t_0 + m_n^{-{1\/3}}h) -\n(\\mathbb{F}_n - F)(t_0) \\}\n\\nonumber\\\\[-8pt]\\\\[-8pt]\n&&\\qquad\\quad{} + m_n^{{1\/3}} [f(t_0)-\\tilde{f}_n(t_0)]h + \\tfrac{1}{\n2}f'(t_0)h^2 + o(1) \\nonumber\\\\\n&&\\qquad \\stackrel{P}{\\rightarrow} \\tfrac{1}{2}f'(t_0)h^2\\nonumber\n\\end{eqnarray}\nuniformly on compacts using Hungarian Embedding to bound the second\nline and (\\ref{eq:chrnff}) (and a two-term Taylor\nexpansion) in the third.\n\nGiven any subsequence $\\{n_k\\} \\subset\\mathbb{N}$, there exists a\nfurther subsequence $\\{n_{k_l}\\}$ such that\n(\\ref{eq:A_nB_nC_n}) and (\\ref{eq:m_n_boot_t2}) hold a.s. and\nTheorem \\ref{thm:cons_sm_boot} is applicable. Thus,\n(\\ref{eq:boot_limit}) holds for the subsequence $\\{n_{k_l}\\}$, thereby\nshowing that (\\ref{eq:boot_limit}) holds in\nprobability.\n\\end{pf}\n\nNext consider bootstrapping from $\\tilde{F}_n$. We will assume slightly\nstronger conditions on $F$, namely, conditions (a)--(d) mentioned in\nTheorem 7.2.3 of Robertson, Wright and Dykstra (\\citeyear{RWD88}):\n\\begin{itemize}\n\\item[(a)] $\\alpha_1(F) = \\inf\\{x\\dvtx F(x) = 1\\} < \\infty$,\n\n\\item[(b)] $F$ is twice continuously differentiable on $(0,\\alpha_1(F))$,\n\n\\item[(c)] $\\gamma(F) = \\frac{{\\sup_{0 < x < \\alpha_1(F)}} |f'(x)|}{\\inf\n_{0 < x < \\alpha_1(F)} f^2(x)} < \\infty$,\n\n\\item[(d)] $\\beta(F) = {\\inf_{0 < x < \\alpha_1(F)}} |\\frac\n{-f'(x)}{f^2(x)}| > 0$.\n\\end{itemize}\n\\begin{theorem}\\label{thm:m_n_boot2} Suppose that \\textup{(a)--(d)} hold. If\n$\\hat{F}_n = \\tilde{F}_n$, $\\hat f_n = \\tilde f_n$, and $m_n = o[n\n(\\log n)^{-{3\/2}}]$ then (\\ref{eq:boot_limit}) holds in probability.\n\\end{theorem}\n\\begin{pf} Conditions (\\ref{eq:cndtn0}), (\\ref{eq:cndtn1}) and (\\ref\n{eq:cndtn4}) again follow from (\\ref{eq:lil}), as explained in\nSection \\ref{remarks}. The verification of (\\ref{eq:cndtn3}) is\nsimilar to the argument in the proof of Theorem \\ref{thm:m_n_boot1}. We\nshow that (\\ref{eq:cndtn2}) holds. Adding and subtracting $m_n^{{2\/\n3}} [ \\mathbb{F}_n(t_0 + m_n^{-{1\/3}} h) - \\mathbb{F}_n(t_0)]$\nfrom $\\mathbb{Z}_{n,2}(h)$ and using\n(\\ref{eq:m_n_boot_t2}) and the result of Kiefer and Wolfowitz (\\citeyear{KW76})\n\\begin{eqnarray*}\n\\sup_{|h| \\le c} \\biggl| {\\mathbb Z}_{n,2}(h) - {1\\over2}f'(t_0)h^2\n\\biggr| & \\le& 2 m_n^{2\/3} \\| \\tilde{F}_n - {\\mathbb F}_n\\|\n+ o_P(1) \\\\\n& \\le& 2 m_n^{2 \/3} \\| \\tilde{F}_n - {\\mathbb F}_n\\| + o_P(1)\n\\\\\n& = & O_P[m_n^{2\/3} n^{-{2\/3}}\\log(n)] + o_P(1)\n\\end{eqnarray*}\nfor any $c > 0$ from which (\\ref{eq:cndtn2}) follows easily.\n\\end{pf}\n\n\\section{Discussion}\\label{discussion}\n\nWe have shown that bootstrap estimators are inconsistent when bootstrap\nsamples are drawn from either the EDF ${\\mathbb F}_n$ or its least\nconcave majorant ${\\tilde F}_n$ but consistent when the bootstrap\nsamples are drawn from a smoothed version of $\\tilde{F}_n$ or an $m$\nout of $n$ bootstrap is used. We have also derived necessary conditions\nfor the bootstrap estimator to have a conditional weak limit, when\nbootstrapping from either ${\\mathbb F}_n$ or ${\\tilde F}_n$ and\npresented compelling numerical evidence that these conditions are not\nsatisfied. While these results have been obtained for the Grenander\nestimator, our results and findings have broader implications for the\n(in)-consistency of the bootstrap methods in problems with an $n^{1\/3}$\nconvergence rate.\n\nTo illustrate the broader implications, we contrast our finding with\nthose of Abrevaya and Huang (\\citeyear{AH05}), who considered a more\ngeneral framework, as in Kim and Pollard (\\citeyear{kimPo90}). For\nsimplicity, we use the same notation as in Abrevaya and Huang\n(\\citeyear{AH05}). Let $W_n := r_n (\\theta _n - \\theta_0)$ and $\\hat\nW_n := r_n (\\hat\\theta_n - \\theta_n)$ be the sample and bootstrap\nstatistics of interest. In our case $r_n = n^{1\/3}$, $\\theta_0 =\nf(t_0)$, $\\theta_n = \\tilde f_n(t_0)$ and $\\hat \\theta_n = \\tilde\nf_n^*(t_0)$. When specialized to the Grenander estimator, Theorem $2$\nof Abrevaya and Huang (\\citeyear{AH05}) would imply [by calculations\nsimilar to those in their Theorem~5 for the NPMLE in a binary choice\nmodel] that\n\\[\n\\hat W_n \\Rightarrow\\arg\\max\\hat Z(t) - \\arg\\max Z(t)\n\\]\nconditional on the original sample, in $P^\\infty$-probability, where\n$Z(t) = W(t) - ct^2$ and $\\hat Z(t) = W(t) + \\hat W(t) - ct^2$, $W$ and\n$\\hat W$ are two independent two sided Brownian motions on $\\mathbb{R}$\nwith $W(0) = \\hat W(0) = 0$ and $c$ is a positive constant depending on\n$F$. We also know that $W_n \\Rightarrow\\arg\\max Z(t)$\nunconditionally. By (v) of Theorem \\ref{thm:bootmle}, this would\nforce the independence of $\\arg\\max Z(t)$ and $\\arg\\max\\hat Z(t) -\n\\arg\\max Z(t)$; but, there is overwhelming numerical evidence that\nthese random variables are correlated.\n\n\\begin{appendix}\\label{app}\n\\section*{Appendix}\n\n\\begin{lemma}\\label{lemma:slope_lcm} Let $\\Psi\\dvtx\\mathbb{R}\n\\rightarrow\n\\mathbb{R}$ be a function such that $\\Psi(h) \\le M$ for all $h \\in\n\\mathbb{R}$, for some $M > 0$, and\n\\begin{equation}\n\\label{eq:Psi_prop} \\lim_{|h| \\rightarrow\n\\infty} \\frac{\\Psi(h)}{|h|} = -\\infty.\n\\end{equation}\nThen for any $b > 0$, there exists $c_0 > b$ such that for any $c \\ge\nc_0$, $L_{\\mathbb{R}} \\Psi(h) = L_{[-c,c]} \\Psi(h)$ for all $|h| \\le b$.\n\\end{lemma}\n\\begin{pf} Note that for any $c > 0$, $L_{\\mathbb{R}} \\Psi(h)\n\\ge L_{[-c,c]} \\Psi(h)$ for all $h \\in[-c,c]$. Given $b > 0$,\nconsider $c > b$ and $\\Phi_c(h) = L_{[-c,c]} \\Psi(h)$ for $h \\in\n[-b,b]$, and let $\\Phi_c$ be the linear extension of $L_{[-c,c]} \\Psi\n|_{[-b,b]}$ outside $[-b,b]$. We will show that there\nexists $c_0 > b + 1$ such that $\\Phi_{c_0} \\ge\\Psi$. Then $\\Phi_{c_0}$\nwill be a concave function everywhere greater than $\\Psi$, and thus\n$\\Phi_{c_0} \\ge L_{\\mathbb{R}} \\Psi$. Hence, $L_{\\mathbb{R}} \\Psi(h)\n\\le\\Phi_{c_0}(h) = L_{[-c_0,c_0]} \\Psi(h)$ for $h \\in[-b,b]$,\nyielding the desired result.\n\nFor any $c > b + 1$, $\\Phi_c(h) = \\Phi_c(b) - \\Phi_c'(b) + \\Phi_c'(b)\n(h - b + 1)$ for $h \\ge b$. Using the min--max formula,\n\\begin{eqnarray*}\n\\Phi_c'(b) & = & \\min_{-c \\le s \\le b} \\max_{b \\le t \\le c}\n\\frac{\\Psi(t) - \\Psi(s)}{t - s} \\\\\n& \\ge& \\min_{-c \\le s \\le b} \\frac{\\Psi( b + 1) - \\Psi(s)}{ (b + 1) -\ns} \\\\\n&\\ge& \\Psi(b + 1) - M =: B_0 \\le0.\n\\end{eqnarray*}\nThus,\n\\begin{eqnarray*}\n\\Phi_c(h) & = & \\Phi_c(b) - \\Phi_c'(b) + \\Phi_c'(b) (h - b + 1)\n\\\\\n& \\ge& \\{\\Psi(b) - \\Phi_c'(b)\\} +\n\\Phi_c'(b) (h - b + 1) \\\\\n& \\ge& \\Psi(b) + (h - b) B_0\n\\end{eqnarray*}\nfor $h \\ge b + 1$. Observe that $B_0$ does not depend on $c$. Combining\nthis with a similar calculation for $h < -(b + 1)$, there are $K_0 \\ge\n0$ and $K_1 \\ge0$, depending only on $b$, for which $\\Phi_c(h) \\ge K_0\n- K_1 |h|$ for $|h| \\ge b + 1$. From (\\ref{eq:Psi_prop}), there is $c_0\n> b + 1$ for which $\\Psi(h) \\le K_0 - K_1 |h|$ for all $|h| \\ge c_0$ in\nwhich case $\\Psi(h) \\le\\Phi_{c_0} (h)$ for all $h$. It follows that\n$L_{\\mathbb R} \\Psi\\le\\Phi_{c_0}(h)$ for $|h| \\le b$.\n\\end{pf}\n\\begin{lemma}\\label{lem:abc}\nLet ${\\mathbb B}$ be a standard Brownian motion. If $a, b, c > 0,\na^3b = 1$, then\n\\begin{equation}\\label{eq:abc1}\nP \\biggl[\\sup_{t \\in\\mathbb{R}} {|{\\mathbb B}(t)|\\over a+bt^2} > c\n\\biggr] = P\\biggl[ \\sup_{s \\in\\mathbb{R}} {|{\\mathbb B}(s)|\\over1+s^2}\n> c \\biggr].\n\\end{equation}\n\\end{lemma}\n\\begin{pf} This follows directly from rescaling properties of Brownian\nmotion by letting $t = a^2s$.\n\\end{pf}\n\\begin{pf*}{Proof of Proposition \\ref{prop:loc}}\nLet $J = [a_1,a_2]$ and $\\varepsilon> 0$ be as in the statement of the\nproposition; let $\\gamma= |f'(t_0)|\/16$; and recall (\\ref\n{eq:kmt}) and (\\ref{eq:boots_proc1}) from the proof of Proposition \\ref\n{prop:Z_conv}. Then there exists $0 < \\delta< 1$, $C \\ge1$, and $n_0\n\\ge1$ for which (\\ref{eq:cndtn3}) and (\\ref{eq:cndtn4}) hold for all\n$n \\ge n_0$. Let $I_{m_n}^* := [-\\delta m_n^{1\/3}, \\delta\nm_n^{1\/3}]$. By making $\\delta$ smaller,\\vspace*{-1pt} if necessary, and using\nLemma \\ref{lem:loc}, $L_{I_{m_n}}{\\mathbb Z}_n(h) =\nL_{I_{m_n}^*}{\\mathbb Z}_n(h)$ for $|h| \\le\\delta m_n^{1\/3}\/2 $\nfor all but a finite number of $n$ w.p. 1. By increasing the values of\n$C$ and $n_0$, if necessary, we may suppose that the right-hand side of\n(\\ref\n{eq:abc1}) (with $c=C$) is less than $\\varepsilon\/3$, that $P[|\\eta| > C]\n+ P[\\sup_{0 \\le t \\le1} m_n^{1\/6}|{\\mathbb E}_{m_n}(t)-{\\mathbb\nB}_{m_n}^0(t)| > C] \\le\\varepsilon\/3$, and that $L_{I_{m_n}}{\\mathbb Z}_n\n= L_{I_{m_n}^*}{\\mathbb Z}_n$ on $[-{1\\over2}\\delta m_n^{1\/3},\n{1\\over2}\\delta m_n^{1\/3}]$ with probability at least $1 -\n\\varepsilon\/3$ for all $n \\ge n_0$. We can also assume that $\\alpha:=\n8C^3\/\\gamma> 1$. Then, using Lemma \\ref{lem:abc} with $a = \\alpha\nm_n^{-{1\/6}}$ and $b = a^{-3}$, the following relations hold\nsimultaneously with probability at least $1-\\varepsilon$ for $n \\ge n_0$:\n\\begin{eqnarray*}\n|{\\mathbb B}_{m_n}[F_n(t_0) + s] - {\\mathbb B}_{m_n}[F_n(t_0)]|\n& \\le &\nC\\bigl(\\alpha m_n^{-{1\/6}} + \\alpha^{-3}\\sqrt{m_n}s^2\\bigr)\\qquad \\mbox{for\nall } s, \\\\\nL_{I_{m_n}}{\\mathbb Z}_n & = & L_{I_{m_n}^*}{\\mathbb Z}_n \\qquad\\mbox{on }\n\\biggl[-\\frac{\\delta}{2} m_n^{1\/3},\\frac{\\delta}{2} m_n^{1\/3}\\biggr],\n|\\eta| \\le C,\n\\end{eqnarray*}\nand\n\\[\n\\sup_{0 \\le t \\le1} m_n^{1\/6}|\n{\\mathbb E}_{m_n}(t) - {\\mathbb B}_{m_n}^0(t)| \\le C.\n\\]\nLet $B_n$ be the event that these four conditions hold. Then $P(B_n)\n\\ge1-\\varepsilon$ for $n \\ge n_0$, and from (\\ref{eq:boots_proc1}),\n$B_n$ implies\n\\begin{eqnarray}\\label{eq:boundZn1Crude}\n|{\\mathbb Z}_{n,1}(h)| & \\le & C \\{\\alpha+ \\alpha^{-3}m_n^{2\/3}\n[F_n(t_0+m_n^{-{1\/3}}h) - F_n(t_0)]^2 \\} + 2C \\nonumber\\\\\n&&{} + Cm_n^{1\/6}|F_n(t_0 + m_n^{-{1\/3}}h) -\nF_n(t_0)| \\\\\n&\\le& 4 C \\{\\alpha+ \\alpha^{-1}m_n^{2\/3}[F_n(t_0+m_n^{-{1\/3}}h) -\nF_n(t_0)]^2 \\}\\nonumber\n\\end{eqnarray}\nusing the inequalities $|F_n(t_0 + m_n^{-{1\/3}} h) - F_n(t_0)| \\le\n\\alpha m_n^{-{1\/6}} + \\alpha^{-1}m_n^{1\/6}[F_n(t_0 +\nm_n^{-{1\/3}} h) - F_n(t_0)]^2$ and $\\alpha> 1$. For sufficiently\nlarge $n$, using (\\ref{eq:cndtn4}), we have\n\\begin{eqnarray}\\label{eq:boundZn1}\n|\\mathbb{Z}_{n,1} (h)| & \\le& 4 C [\\alpha+ \\alpha^{-1}C^2\nm_n^{2\/3} (m_n^{-{1\/3}}|h| + m_n^{-{1\/3}})^2]\n\\nonumber\\\\\n& \\le& 4 C [\\alpha+ 2\\alpha^{-1} C^2 (h^2 + 1) ] \\\\\n& = & \\gamma h^2 + \\mathcal{C}\\nonumber\n\\end{eqnarray}\nfor $|h| \\le\\delta m_n^{1\/3}$ with $\\mathcal{C} = 4C\\alpha+\n8C^3\\alpha^{-1}$. Also, we can show that $|{\\mathbb Z}_{n,2}(h) -\nf'(t_0)h^2 \/2| \\le\\gamma h^2 + \\mathcal{C}$\nfor all $|h| \\le\\delta m_n^{1\/3}$ by (\\ref{eq:cndtn3}). Let $b_2\n> a_2$ be such that $- 5 \\gamma(a_2 + b_2)^2 + 6 \\gamma(a_2^{2} +\nb_2^2) - 8\\mathcal{C} > 0$.\n\nRecalling that $\\gamma= -f'(t_0)\/16$, $B_n$ implies\n\\[\n-10 \\gamma h^2 - 2\\mathcal{C} \\le{\\mathbb Z}_n(h) = {\\mathbb\nZ}_{n,1}(h) + {\\mathbb Z}_{n,2}(h) \\le- 6 \\gamma h^2 + 2\\mathcal{C}\n\\]\nfor $|h| \\le\\delta m_n^{1\/3}$ and sufficiently large $n$. Since\nthe right-hand side is concave, $B_n$ also implies\n$L_{I_{m_n}^*}{\\mathbb\nZ}_n(h) \\le- 6\\gamma h^2 + 2\\mathcal{C}$ for $|h| \\le\\delta m_n^{1\/3}$. Therefore, for\nsufficiently large $n$, using the upper bound on $L_{I_{m_n}^*}{\\mathbb\nZ}_n$, the lower bound on $\\mathbb{Z}_n$ obtained above, and\n$L_{I_{m_n}}{\\mathbb Z}_n(h) = L_{I_{m_n}^*}{\\mathbb Z}_n(h)$ for $|h|\n\\le\\delta m_n^{1\/3}\/2$ on $B_n$, and $[a_2,b_2] \\subset I_{m_n}^*$, we have\n\\begin{eqnarray*}\n&&2{\\mathbb Z}_n \\biggl( {a_2 + b_2 \\over2} \\biggr) -\n[L_{I_{m_n}}{\\mathbb Z}_n(a_2) + L_{I_{m_n}}{\\mathbb\nZ}_n(b_2) ] \\\\\n&&\\qquad\\ge- 5 \\gamma(a_2 + b_2)^2 + 6 \\gamma(a_2^{2} + b_2^2) - 8\\mathcal\n{C} > 0\n\\end{eqnarray*}\nwith probability at least $1-\\varepsilon$. Thus, $B_n$ implies $2\n{\\mathbb\nZ}_n[{1\\over2}(a_2 + b_2)] > L_{I_{m_n}}{\\mathbb Z}_n(a_2) +\nL_{I_{m_n}}{\\mathbb Z}_n(b_2)$ with probability at least $1-\\varepsilon$.\nSimilarly, $B_n$ implies that there is a $b_1 < a_1$ for which $2\n{\\mathbb Z}_n[{1\\over2}(a_1 + b_1)] > L_{I_{m_n}}{\\mathbb Z}_n(a_1) +\nL_{I_{m_n}}{\\mathbb Z}_n(b_1)$ with probability at least $1-\\varepsilon$.\nRelation (\\ref{eq:loc}) then follows from Lemma \\ref{lem:ww}. It is\nworth noting as a \\textit{remark} that $b_1, b_2$ do not depend on the\nsequence $F_n$.\n\nNext, consider (\\ref{eq:loc2}). Given a compact $J = [-b,b]$, let\n$c_{0}(\\omega)$ be the smallest positive integer such that for any $c\n\\ge c_{0}$, $L_{\\mathbb{R}} \\mathbb{Z}(h) = L_{[-c,c]} \\mathbb{Z} (h) $\nfor $h \\in J$. That $c_{0}$ exists and is finite w.p. 1 follows from\nLemma \\ref{lemma:slope_lcm}. Defining $W_c := L_{[-c,c]} \\mathbb{Z}$\nand $Y = L_{\\mathbb{R}} \\mathbb{Z}$, the event $\\{W_c \\ne Y \\mbox{ on }\nJ\\} \\subset\\{c_{o} > c\\}$. Now given any $\\varepsilon> 0$, there exist\n$c$ such that $P[c_{o} \\le c] > 1 - \\varepsilon$. Therefore,\n\\[\nP\\bigl[L_{\\mathbb{R}} \\mathbb{Z} = L_{[-c,c]} \\mathbb{Z} \\mbox{ on } J \\bigr] \\ge\nP[c_{o} \\le c] > 1 - \\varepsilon.\n\\]\n\\upqed\\end{pf*}\n\\begin{pf*}{Proof of Proposition \\ref{prop:edflcm}} First, consider\n${\\mathbb F}_n$. Let $0 < \\gamma< |f'(t_0)|\/2$ be given. There is a $0\n< \\delta< {1\\over2}t_0$ such that\n\\begin{eqnarray}\n\\label{eq:2diffF}\n\\bigl| F(t_0 + h) - F(t_0) - f(t_0)h - \\tfrac{1}{2} f'(t_0) h^2 \\bigr|\n\\le\\tfrac{1}{2} \\gamma h^2\n\\end{eqnarray}\nfor $|h| \\le2\\delta$. From the proof of Proposition \\ref{prop:loc},\nusing arguments similar to deriving (\\ref{eq:boundZn1Crude}) and (\\ref\n{eq:boundZn1}), we can show that\n\\[\n|({\\mathbb F}_n-F)(t_0+h)-({\\mathbb F}_n-F)(t_0)| < \\tfrac{1}{2}\\gamma\nh^2 + C n^{-{2 \/3}}\n\\]\nfor $|h| \\le2 \\delta$ with probability at least $1 - \\varepsilon$ for\nsufficiently large $n$. Therefore, by adding and subtracting $F(t_0+h)\n- F(t_0)$ and using (\\ref{eq:2diffF}),\n\\begin{equation}\n\\label{eq:edf}\n\\bigl| {\\mathbb F}_n(t_0 + h) - {\\mathbb F}_n(t_0) - f(t_0)h - \\tfrac\n{1}{2} f'(t_0) h^2 \\bigr|\n\\le\\gamma h^2 + C n^{-{2\/3}}\n\\end{equation}\nfor $|h| \\le2 \\delta$ with probability at least $1-\\varepsilon$ for\nlarge $n$.\n\nNext, consider $\\tilde{F}_n$. Let $B_n$ denote the event that (\\ref\n{eq:edf}) holds. Then $P(B_n)$ is eventually\nlarger than $1 - \\varepsilon$ and on $B_n$, we have\n\\[\n{\\mathbb F}_n(t_0+h) - \\mathbb{F}_n(t_0) - f(t_0) h \\le\\bigl\\{ \\gamma\n- \\tfrac{1}{2} |f'(t_0)| \\bigr\\} h^2 + Cn^{-{2\/3}}\n\\]\nfor $|h| \\le2 \\delta$. Let $E_n$ be the event that $\\tilde{F}_n(h) =\nL_{[t_0-2\\delta,t_0+2\\delta]}{\\mathbb F}_n(h)$ for $h \\in[t_0-\n\\delta,t_0+ \\delta]$. Then by Lemma \\ref {lem:loc}, $P(E_n) \\ge1 -\n\\varepsilon$, for all sufficiently large $n$. Taking concave majorants\non either side of the above display for $|h| \\le2 \\delta$ and noting\nthat the right-hand side of the display is already concave, we have:\n${\\tilde F}_n(t_0+h) - \\mathbb{F}_n(t_0) - f(t_0) h \\le\\{ \\gamma-\n\\frac{1}{2} |f'(t_0)| \\} h^2 + C n^{-{2\/3}}$, for $|h| \\le\\delta$ on\n$B_n \\cap E_n$. Setting $h = 0$ shows that on $E_n \\cap B_n$, $\\tilde\nF_n(t_0) - \\mathbb{F}_n(t_0) \\le C n^{-{2\/3}}$. Now, as ${\\mathbb\nF}_n(t_0) \\le\\tilde{F}_n(t_0)$, it is also the case that on $E_n \\cap\nB_n$, for $|h| \\le\\delta$,\n\\begin{equation}\n\\label{eq:cncvbnd2}\n{\\tilde F}_n(t_0+h) - \\tilde F_n(t_0) - f(t_0) h \\le\\bigl\\{ \\gamma-\n\\tfrac{1}{2} |f'(t_0)| \\bigr\\} h^2 + C n^{-{2\/3}}.\n\\end{equation}\nFurthermore on $E_n \\cap B_n$,\n\\begin{eqnarray}\n\\label{eq:cncvbnd3}\n&& \\tilde F_n(t_0 + h) - \\tilde F_n(t_0) - f(t_0) h - \\tfrac{1}{2}\nf'(t_0) h^2 \\nonumber\\\\\n&&\\qquad \\ge \\mathbb F_n(t_0 + h) - \\{\\mathbb F_n(t_0) + C n^{-{2\/3}}\\}\n- f(t_0) h - \\tfrac{1}{2} f'(t_0) h^2\n\\\\\n&&\\qquad \\ge -\\gamma h^2 - 2 C n^{-{2\/3}}.\\nonumber\n\\end{eqnarray}\nTherefore, combining (\\ref{eq:cncvbnd2}) and (\\ref{eq:cncvbnd3}),\n\\[\n\\bigl| {\\tilde F}_n(t_0 + h) - {\\tilde F}_n(t_0) - f(t_0)h - \\tfrac\n{1}{2} f'(t_0) h^2 \\bigr| \\le\\gamma h^2 + 2C n^{-{2\/3}}\n\\]\nfor $|h| \\le\\delta$ with probability at least $1 - 2\\varepsilon$ for\nlarge $n$.\n\\end{pf*}\n\\end{appendix}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe main purpose of this work is to expose the role of gravity in the dynamics of particles in a rotating trap. In addition, we present a complete analysis of the stability regions for a rotating trap in 3D. We prove that in the generic case there are three separate regions of stability with different characteristics. Gravity induced resonances are relevant only if they occur in the regions of stability, otherwise, they are swamped by the exponential behavior of trajectories.\n\nHarmonic traps are often used in optics and atomic physics (especially in the form of TOP traps \\cite{petrich} in the study of Bose-Einstein condensates) and yet a complete theory of these devices has not been developed. The solution to the problem of a trap rotating around one of the trap axes is effectively two-dimensional and its solution has been known for at least hundred years. In the classic textbook on analytical dynamics by Whittaker \\cite{whitt} we find a solution of a mathematically equivalent problem of small oscillations of ``a heavy particle about its position of equilibrium at the lowest point of a surface which is rotating with constant angular velocity about a vertical axis through the point''. The quantum-mechanical counterpart of the Whittaker problem has also been completely solved \\cite{linn,oktel} and the statistical mechanics of a classical gas was studied in \\cite{gd}.\n\nIn this work we present a complete solution to the problem of the motion of a particle moving in a most general anisotropic rotating harmonic trap in 3D and in the presence of gravity. This is an exactly soluble problem but technical difficulties apparently served so far as a deterrent in developing a full description. A full description of the particle dynamics in a rotating anisotropic trap in three dimensions so far has not been given despite a new significance of this problem brought about by experimental and theoretical studies of Bose-Einstein condensates and the accompanying thermal clouds in rotating traps \\cite{rzs,mad,sc,gg,ros,cozz,abo,gd}. Explicit formulas describing the complete mode structure in the three-dimensional case are indeed quite cumbersome \\cite{cmm} because we deal here with third-order polynomials and on top of that they have rather complicated coefficients. However, many important features may be exhibited without straining the reader's patience. In particular, we can identify various stability regions for an arbitrary orientation of the angular velocity and we can give conditions for a resonance.\n\nThe standard arrangement \\cite{rzs,mad,sc,gg,ros,cozz,abo,gd} is to choose a vertical axis of rotation of the trap. Slight tiltings of this axis were introduced to excite the scissors modes \\cite{gds,marago,smith}. However, for such very small tilting angles the effects described in the present paper would not be noticeable. In the case of a vertical axis of rotation, there are no resonances. The only effect of gravity is a displacement of the equilibrium position. The situation completely changes and new phenomena will occur when the axis of rotation is tilted away from the vertical position. In this generic case, for every anisotropic three-dimensional trap there exist two (not three as one might expect) characteristic frequencies at which resonances occur. The motion in a trap that is rotating at a {\\em resonant} frequency will become unbounded and all particles will be expelled from the trap. The position of the resonance does not depend on the mass but only on the characteristic frequencies of the trap and on the direction of the angular velocity.\n\nAll our results are valid not only for a single particle but also for the center of mass motion in many-body (classical or quantum) theory since for all quadratic Hamiltonians the center of mass motion completely separates from the internal motion \\cite{kohn,dob,cmm}. Therefore, a trap rotating at the resonant frequency will not hold the Bose-Einstein condensate. Owing to the linearity of the equations of motion for a harmonic trap, all conclusions hold both in classical and in quantum theory. A resonant behavior caused by an application of a static force may seem counterintuitive, but it is explained by the fact that in a rotating frame the force of gravity acts as a {\\em periodically changing} external force.\n\n\\section{Equations of motion}\n\nThe best way to analyze the behavior of particles in a uniformly rotating trap is to first perform the transformation to the rotating frame. In this frame the harmonic trap potential is frozen but the force of gravity is rotating with the angular velocity of the trap rotation. In the rotating frame the Hamiltonian has the form\n\\begin{equation}\\label{ham}  {\\cal H} = \\frac{\\bm{p}^{2}}{2m}  + \\bm{r}\\!\\cdot\\!\\hat{\\Omega}\\!\\cdot\\!\\bm{p}  + \\frac{m}{2}\\bm{r}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{r} - m\\bm{r}\\!\\cdot\\!\\bm{g}(t).\n\\end{equation}\nThe potential matrix $\\hat{V}$ is symmetric and positive definite. The eigenvalues of this matrix are the squared frequencies of the oscillations in the non-rotating trap. The angular velocity matrix $\\hat{\\Omega}$ is related to the components of the angular velocity vector through the formula ${\\Omega}_{ik}=\\epsilon_{ijk}\\Omega_j$. The vector of the gravitational acceleration $\\bm{g}(t)$, as seen in the rotating frame, can be expressed in the form\n\\begin{eqnarray}\\label{rotg}\n{\\bm g}(t) = {\\bm g}_{\\parallel} + {\\bm g}_{\\perp}\\cos(\\Omega t) - ({\\bm n}\\times{\\bm g}_{\\perp})\\sin(\\Omega t),\n\\end{eqnarray}\nwhere ${\\bm n}$ denotes the direction and $\\Omega$ denotes the length of the angular velocity vector ${\\bm\\Omega}$. The parallel and the transverse components of the gravitational acceleration vector ${\\bm g}={\\bm g}(0)$ are defined as ${\\bm g}_{\\parallel} = {\\bm n}({\\bm n}\\!\\cdot\\!{\\bm g})$ and ${\\bm g}_{\\perp} = {\\bm g}-{\\bm n}({\\bm n}\\!\\cdot\\!{\\bm g})$, respectively. Note that the time-dependent part vanishes when the rotation axis is vertical.\n\nThe equations of motion determined by the Hamiltonian (\\ref{ham}) have the following form\n\\begin{subequations}\n\\begin{eqnarray} \\label{3dimeq}\n\\frac{d{\\bm r}(t)}{dt} &=& \\frac{{\\bm p}(t)}{m} - \\hat{\\Omega}\\!\\cdot\\!{{\\bm r}(t)},\\\\\n\\frac{d{\\bm p}(t)}{dt} &=& -m\\hat{V}\\!\\cdot\\!{{\\bm r}(t)} - \\hat{\\Omega}\\!\\cdot\\!{{\\bm p}(t)} + m {\\bm g}(t).\n\\end{eqnarray}\n\\end{subequations}\nThese equations describe an oscillator in a rotating frame displaced by a constant force (the longitudinal part of ${\\bm g}$) and driven by a periodic force (the transverse part of ${\\bm g}$). It is convenient to rewrite the expression (\\ref{rotg}) as a real part of a complex function\n\\begin{eqnarray}\\label{rotg1}\n {\\bm g}(t) = \\Re\\left({\\bm g}_{\\parallel}+({\\bm g}_{\\perp} +  i({\\bm n}\\times{\\bm g}_{\\perp}))e^{i\\Omega t}\\right).\n\\end{eqnarray}\nIn compact notation Eqs.~(\\ref{3dimeq}) have the form\n\\begin{equation} \\label{eqnmot}\n\\frac{d{\\cal R}(t)}{dt} = \\hat{\\cal M}(\\Omega)\\!\\cdot\\!{\\cal R}(t) + \\Re({\\cal G}_{\\parallel} + {\\cal G}_{\\perp}{e}^{i\\Omega t}),\n\\end{equation}\nwhere\n\\begin{eqnarray}\n{\\cal R}(t) &=& \\left(\\begin{array}{c} \\bm{r}(t) \\\\ \\bm{p}(t) \n\\end{array}\\right),\\;\\;\n\\hat{\\cal M}(\\Omega) = \\left(\\begin{array}{cc} - \\hat{\\Omega} & m^{-1}\\hat{I} \\\\ -m\\hat{V} & - \\hat{\\Omega}\\end{array}\\right), \\\\\n{\\cal G}_{\\parallel} &=&  m\\left(\\begin{array}{c} 0 \\\\ \\bm{g}_{\\parallel} \\end{array}\\right),\\;\\;\n{\\cal G}_{\\perp} = m\\left(\\begin{array}{c} 0 \\\\ \\bm{g}_{\\perp} + i(\\bm{n}\\times\\bm{g}_{\\perp}) \\end{array}\\right). \n\\end{eqnarray}\nWe shall now replace the equations of motion by their complex counterpart\n\\begin{equation} \\label{eqnmotc}\n\\frac{d{\\cal W}(t)}{dt} = \\hat{\\cal M}(\\Omega)\\!\\cdot\\!{\\cal W}(t) + {\\cal G}_{\\parallel} + {\\cal G}_{\\perp}{e}^{i\\Omega t}.\n\\end{equation}\nThe physical trajectory in phase space is described by the real part of the complex vector ${\\cal W}(t)$. Let us introduce a basis of six eigenvectors of $\\hat{\\cal M}(\\Omega)$\n\\begin{equation}\n\\hat{\\cal M}(\\Omega){\\cal X}_{k} \n= i \\omega_{k}(\\Omega){\\cal X}_{k},\\qquad k = 1,\\ldots, 6\n\\end{equation}\nand expand ${\\cal W}(t)$ and ${\\cal G}(t)$ in this base as follows\n\\begin{eqnarray}\\label{expan}\n{\\cal W}(t) &=&\\sum_{k=1}^{6} \\alpha^k(t) \\, {e}^{i\\omega_k(\\Omega)t} \\, {\\cal X}_k, \\\\\n{\\cal G}_\\parallel &=& \\sum_{k=1}^{6} \\gamma^k_{\\parallel}\\, {\\cal X}_{k},\\;\\;\\;\n{\\cal G}_\\perp = \\sum_{k=1}^{6} \\gamma^k_{\\perp}\\, {\\cal X}_{k}. \n\\end{eqnarray}\nOwing to a simple block structure of $\\hat{\\cal M}(\\Omega)$, the basis vectors ${\\cal X}_{k}$ can be determined by reducing effectively the problem to three dimensions. We use this method in the Appendix B to determine the resonant solution. \n\nThe equation of motion (\\ref{eqnmotc}) can be rewritten now as a set of equations for the coefficient functions $\\alpha^k(t)$\n\\begin{equation} \\label{alphaeq}\n\\frac{d{\\alpha}^k(t)}{dt} = \\gamma^{k}_{\\parallel}{e}^{-i \\omega_k(\\Omega)t} + \\gamma^{k}_{\\perp}{e}^{i(\\Omega-\\omega_k(\\Omega))t},\\;\\; k = 1,\\ldots,6.\n\\end{equation}\nIt is clear now that the mode amplitude $\\alpha_k(t)$ will grow linearly in time --- the signature of a resonance --- whenever either one of the two terms on the right hand side becomes time independent. This happens to the first term if one of the frequencies $\\omega_k(\\Omega)$ vanishes but the corresponding coefficient $\\gamma^{k}_{\\parallel}$ does not vanish. This case is not interesting, since it means that we are just at the border of the lower instability region and the trap is not holding particles, as discussed in the next section. The second term becomes time independent when the angular velocity of trap rotation $\\Omega$ satisfies the resonance condition $\\Omega=\\omega_k(\\Omega)$ and, of course, $\\gamma^{k}_{\\perp} \\neq 0$. This resonance {\\em is different} from a resonance in a standard periodically driven oscillator. In the present case the characteristic frequencies of the trap depend on the frequency $\\Omega$ of the driving force. Therefore, the position of the resonance has to be determined selfconsistently. A full description of these gravity induced resonances requires the knowledge of the behavior of $\\omega_k(\\Omega)$'s as functions of $\\Omega$. In particular, it is important to know whether a resonance occurs in a region where the system undergoes stable oscillations. This will be discussed in the next Section.\n\n\\section{Regions of stability}\n\nThe stability of motion for a harmonic oscillator is determined by the values of its characteristic frequencies $\\omega$ --- the roots of the characteristic polynomial. In the present case, these frequencies are determined by the characteristic equation for the matrix $\\hat{\\cal M}(\\Omega)$\n\\begin{equation}\n\\mathrm{Det}\\left\\{ \\hat{\\cal M}(\\Omega) - i\\omega \\right\\} = 0.\n\\end{equation}\nThe characteristic polynomial is tri-quadratic\n\\begin{equation} \\label{charpoly}\nQ(\\chi)=\\chi^3 + A\\,\\chi^2 + B\\,\\chi + C,\\;\\;\\;\\chi=\\omega^2,\n\\end{equation}\nwhere the coefficients $A$, $B$ i $C$ can be expressed in a rotationally invariant form \\cite{cmm}\n\\begin{eqnarray}\nA &\\!=\\!&-2\\Omega^2 - \\mathrm{Tr}\\{\\hat{V}\\},\\nonumber\\\\\nB &\\!=\\!&\\Omega^4\\!+\\!\\Omega^2(3\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\mathrm{Tr}\\{\\hat{V}\\})\\!\n+\\!\\frac{\\mathrm{Tr}\\{\\hat{V}\\}^2\\!-\\!\\mathrm{Tr}\\{\\hat{V}^2\\}}{2},\\nonumber\\\\\nC &\\!=\\!&{\\Omega}^2(\\mathrm{Tr}\\{\\hat{V}\\}\\!\n-\\!{\\Omega}^2)\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\Omega^2\\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n}\\!\n-\\!\\mathrm{Det}\\{\\hat{V}\\}.\\qquad\n\\end{eqnarray}\nStable oscillations take place when all characteristic $\\omega$'s are real. This means that all three roots of the polynomial $Q(\\chi)$ must be real and positive. Without rotation, when $\\Omega=0$, the three roots of $Q(\\chi)$ are equal to the eigenvalues of the potential matrix $\\hat{V}$. We have then a simple system of three harmonic oscillators vibrating independently along the principal directions of the trap. As $\\Omega$ increases, our system will, in general, go through two regions of instability: the lower region when one of the roots of $Q(\\chi)$ is negative and the upper region when two roots are complex. We shall exhibit this behavior by plotting the zero contour lines of $Q(\\chi)$ in the $\\Omega\\chi$-plane. We assume that the trap potential and the direction of rotation are fixed and we treat the characteristic polynomial $Q(\\chi)$ as a function of $\\Omega$ and $\\chi$ only. Contour lines representing the zeroes of $Q(\\chi)$ in the generic case are shown in Fig.~\\ref{fig1}. There is a region of $\\Omega$, where only one real root of $Q(\\chi)$ exists. However, this region is bounded, so for sufficiently large $\\Omega$ the system is always stable.\n\nIt has been argued in Ref.~\\cite{cmm} that there is always a region of instability when one of the roots of $Q(\\chi)$ is negative. The corresponding modes grow exponentially with time. As seen in Fig.~\\ref{fig1}, this region of instability is bounded by the two values $\\Omega_{1,2}$ at which the curve crosses the vertical axis. These values are given by the zeroes of $C$, treated as a biquadratic expressions in $\\Omega$\n\\begin{equation}\\label{zeroes}\n\\Omega_{1,2} = \\sqrt{\\frac{b\\pm\\sqrt{b^2-4ac}}{2a}},\n\\end{equation}\nwhere $a=\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n},\\;\nb=\\mathrm{Tr}\\{\\hat{V}\\}\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\n-\\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n}$, and $c=\\mathrm{Det}\\{\\hat{V}\\}$. Since $a, b$, and $c$ are positive and $b^2\\geq 4ac$, both values $\\Omega_{1}$ and $\\Omega_{2}$ are real. A degenerate case is possible, when $\\Omega_{1}=\\Omega_{2}$ then the region of instability shrinks to zero. In order to determine, when this can happen, we may use the (explicitly non-negative) representation of the discriminant $b^2-4ac$ given in Ref.~\\cite{cmm}. Assuming for definitness that $V_x<V_y<V_z$, we find that this happens in two cases: when the trap is not fully anisotropic ($V_x=V_y$ or $V_y=V_z$) or the axis of rotation lies in the $xz$-plane and its azimuthal angle satisfies the condition $\\sin^2\\theta=(1-V_x\/V_y)\/(1-V_x\/V_z)$. The second possibility has not been noticed in Ref.~\\cite{cmm}.\n\nGraphical representation of the solutions for $n_x=1$, $n_y=n_z=0$ is shown in Fig.~\\ref{fig2}. In this plot only one instability region is present, where one of the roots of (\\ref{charpoly}) is negative. Owing to the stabilizing effect of the Coriolis force, for fast rotations the system becomes again stable. We can also see that one of the characteristic frequencies remains constant, it does not vary with $\\Omega$. It is so, because the rotation does not influence the motion in the direction of the rotation axis. This is the degenerate case described by Whittaker and thoroughly studied in connection with the BEC traps \\cite{linn,oktel,gd,ros}. In this case there is only one instability region, where the square of the frequency is negative. When the direction of angular velocity is not parallel to one of the axes of the trap, a second kind of instability appears. In addition to the region of (purely exponential) instability, described before, when one root of $Q(\\chi)$ was negative there is also, in general, an additional region of (oscillatory) instability where two roots are complex. Since the coefficients of the characteristic polynomial are real, the square of the second frequency is complex conjugate to the first one. In this case the instability has the form of expanding oscillations. In Fig.~\\ref{fig3} we show how the second kind of instability develops when the rotation axis is tilted away a little bit from the $z$ direction.\n\nWe will show now, that the second kind of instability does not occur in the most common case, when the rotation vector is along one of the axes of the trap. In particular, it can never happen in a two dimensional trap. Without any loss of generality, we may assume that the direction of rotation is along the $z$ axis. In this degenerate case the characteristic polynomial (\\ref{charpoly}) factorizes because the motion in the $z$ direction is not influenced by rotation,\n\\begin{eqnarray}\n\\label{factor}\nQ(\\chi) = (\\chi - V_z)\\hspace{6cm}\\nonumber\\\\\n\\times(\\chi^2-\\chi(2\\Omega^2+V_x+V_y)+\\Omega^4\n-\\Omega^2(V_x+V_y)+V_x V_y).\\nonumber\\\\\n\\end{eqnarray}\nThe factor quadratic in $\\chi$ has real zeros if its discriminant $\\Delta$ is positive and this is, indeed, the case since\n\\begin{eqnarray} \\label{delta}\n\\Delta = 8\\Omega^2(V_x+V_y)+(V_x-V_y)^2 \\geq 0.\n\\end{eqnarray}\nIt follows, however, from the topology of the curves representing the zeros that the second kind of instability always exists if the direction of rotation does not coincide with one of the axes of the harmonic trap. \n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig1.eps}\n\\caption{The contour lines in this figure represent the zeroes of the characteristic polynomial (\\ref{charpoly}) for $V_x=1$, $V_y=2$, $V_z=3$ and $n_x=1\/\\sqrt{3}$, $n_y=1\/\\sqrt{3}$, $n_z=1\/\\sqrt{3}$ plotted as functions of the magnitude of angular velocity $\\Omega$ and the square of the characteristic frequency $\\chi=\\omega^2$. In addition to the lower region of instability (between dashed lines) where one of the roots of (\\ref{charpoly}) is negative, there is also an upper region (between dotted lines) where there exists only one real root (determined by the continuation of the line that begins at $\\chi=3$, not seen in this figure because it corresponds to a very large value of $\\chi$). Since there is no absolute scale of frequencies involved in the analysis of stability regions, we have chosen the lowest trap frequency as a unit.}\\label{fig1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig2.eps}\n\\caption{This plot is for the same trap as in Fig.~\\ref{fig1} but for the rotation around the trap axis $n_x=0$, $n_y=0$, $n_z=1$. In this degenerate case there exists only one region of instability, when one of the roots of (\\ref{charpoly}) is negative.} \\label{fig2}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig3.eps}\n\\caption{This plot shows how the degenerate case (Fig.~\\ref{fig2}) merges with the general case (Fig.~\\ref{fig1}). Here we have chosen the following values of the parameters: $V_x=1$, $V_y=2$, $V_z=3$ and $n_x=\\sin(1\/10)$, $n_y=0$, $n_z=\\cos(1\/10)$.} \\label{fig3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig4.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{2\\pi}{5})$, $n_y=0$, $n_z=\\cos(\\frac{2\\pi}{5})$. Two horizontal lines enclose the lower region of instability. Both resonant frequencies lie in the lower region of stability.} \\label{fig4}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig5.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{\\pi}{4})$, $n_y=0$, $n_z=\\cos(\\frac{\\pi}{4})$. Higher resonant frequency lies in the lower region of instability.} \\label{fig5}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig6.eps}\n\\caption{$V_x=1$, $V_y=2$, $V_z=3$ $n_x=\\sin(\\frac{\\pi}{60})$, $n_y=0$, $n_z=\\cos(\\frac{\\pi}{60})$. Resonant frequencies lie in two different regions of stability.} \\label{fig6}\n\\end{figure}\n\n\\section{Gravity induced resonances}\n\nThere are, in general, two resonant values of $\\Omega$. They occur when $\\chi=\\Omega^2$. In this case, from Eq.~(\\ref{charpoly}) we get a biquadratic equation for the resonant values of $\\Omega$:\n\\begin{equation} \\label{reseq}\n D \\Omega^4 + E \\Omega^2 +F = 0,\n\\end{equation}\nwhere\n\\begin{eqnarray}\nD &=& -2(\\,\\mathrm{Tr}\\{\\hat{V}\\}-\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\,), \\nonumber \\\\\nE &=& \\frac{\\mathrm{Tr}\\{\\hat{V}\\}^2\\!-\\!\\mathrm{Tr}\\{\\hat{V}^2\\}}{2}\n+\\mathrm{Tr}\\{\\hat{V}\\}\\bm{n}\\!\\cdot\\!\\hat{V}\\!\\cdot\\!\\bm{n}\\! - \\bm{n}\\!\\cdot\\!\\hat{V^2}\\!\\cdot\\!\\bm{n},\\nonumber \\\\\nF &=& -\\mathrm{Det}\\{\\hat{V}\\}.\n\\end{eqnarray}\nThe properties of the two solutions $\\Omega_{\\pm}^2$ of Eq.~(\\ref{reseq}) are described in the Appendix A. We may also analyze the resonance condition graphically by superposing the parabola $\\chi=\\Omega^2$ (shown as a thick line in Figs.~\\ref{fig4}--\\ref{fig7}) on the plots of characteristic frequencies. The lower value is more interesting because, as shown in the Appendix A, it always falls into the range of the first region of stable oscillations where we would expect the confinement of particles in the trap. The upper value may fall in the lower region of stability (Fig.~\\ref{fig4}), in the lower region of instability (Fig.~\\ref{fig5}), or in the higher region of stability (Fig.~\\ref{fig6}). A degenerate case is also possible (cf. Fig.~\\ref{fig7}) when the two resonance frequencies coincide but this happens only under very special circumstances (see Appendix A). The existence of only two resonant frequencies (we would expect three resonant frequencies for a three dimensional oscillator) clearly shows that we are dealing here with a more complex dynamical system than a simple driven harmonic oscillator.\n\\begin{figure}\n\\centering\n\\includegraphics[angle=-90,width=0.45\\textwidth]{fig7.eps}\n\\caption{In the degenerate case, when two resonant frequencies coincide, their common value is also equal to the lowest frequency of the trap. In this case the trap parameters are: $V_x = 1, V_y = 10\/3, V_z = 5$, and $n_x = 1$.}\\label{fig7}\n\\end{figure}\n\nThe essential difference between the resonant and the nonresonant behavior is illustrated in Figs.~\\ref{fig8}-\\ref{fig10}. The role of gravity is best seen by comparing Fig.~\\ref{fig8} (with gravity) and Fig.~\\ref{fig11} (gravity switched off). The calculations were performed in the coordinate system in which the potential matrix $\\hat V$ is diagonal and it was assumed that the force of gravity at $t=0$ is directed along the $z$ axis of the trap. All four figures were generated for the same trap and under the same initial conditions: the particle is initially placed in the center of the trap and it is given the initial velocity of 1 cm\/s in the $x$ direction. These simple initial conditions result in the excitation of all the modes of the oscillator. However, at resonance the mode growing linearly with $t$ dominates the time evolution. This can be seen by comparing, for the same parameters of the trap and at the resonant frequency $\\Omega\/2\\pi = 6.49421$ Hz, the motion depicted in Fig.~\\ref{fig8}, that was calculated numerically, with the motion depicted in Fig.~\\ref{fig12}. The second plot was obtained from an analytic solution of the equations of motion (\\ref{eqnmotc}), that contains only the resonant part. This analytic solution is derived in the Appendix B. Both plots are essentially the same except for very small differences that are due to the fact that the initial conditions assumed in Fig.~\\ref{fig8} require some admixture of the nonresonant solutions while the analytic solution does not contain any nonresonant pieces.\n\nGravity induced resonances are significant only if they occur in the region of stability. In all regions of instability, where the trajectories exhibit exponential growth, the resonances cannot be detected.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig8.eps}\n\\caption{The trajectory of a particle in the coordinate frame rotating with the trap. The characteristic frequencies of the trap in the $x$, $y$, and $z$ directions are 10 Hz, 15 Hz, and 20 Hz, respectively. The angular velocity vector has the direction $(1,1,1)$ and its length $\\Omega\/2\\pi = 6.49421$ Hz satisfies the resonance condition, as calculated in the Appendix A. The distances in this plot are measured in centimeters. The direction of the gravitational force ($g=9.81 {\\rm m s}^{-2}$) is assumed to coincide at $t=0$ with the $z$-axis of the trap. We can see that the amplitude of oscillations increases linearly with time.}\\label{fig8}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig9.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the value of the angular velocity  $\\Omega\/2\\pi = 4$ Hz is below the resonance. The scale in this plot is reduced by a factor of 10 as compared to Fig.~\\ref{fig8}.}\\label{fig9}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig10.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the value of the angular velocity $\\Omega\/2\\pi = 9$ Hz is above the resonance. The scale in this plot is reduced by a factor of 10 as compared to Fig.~\\ref{fig8}.}\\label{fig10}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig11.eps}\n\\caption{The trajectory of a particle under the same conditions as in Fig.~\\ref{fig8}, except that the gravitational field has been turned off. The angular velocity $\\Omega\/2\\pi = 6.49421$ Hz has the resonant value. The scale in this plot is reduced by a factor of 100 as compared to Fig.~\\ref{fig8}. Clearly, there is no sign of any resonant behavior.}\\label{fig11}\n\\end{figure}\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.45\\textwidth]{fig12.eps}\n\\caption{The trajectory of a particle described by the analytic formulas given in the Appendix B. The trap parameters and the angular velocity are the same as in Fig.~\\ref{fig8}.}\\label{fig12}\n\\end{figure}\n\n\\section{Conclusions}\n\nWe have analyzed the stability of motion in an anisotropic, rotating harmonic trap in 3D. We have found that, in general, there are three regions of stability. The second region and the third region merge only when the rotation axis coincides with the one of the trap axes. We have demonstrated the presence of resonances in a rotating harmonic trap subjected to the force of gravity. The nature of these resonances is different from the standard forced harmonic oscillator since the resonant frequencies must be determined selfconsistently. The resonances occur at two rotation rates but they exist only when the rotation axis is tilted away from the vertical direction. However, the rotation axis can still be directed along one of the trap axes. The lower resonance always falls in the region of stable oscillations where as the higher resonance may fall in the lower region of stability, in the first region of instability, or in the higher region of stability. Resonant rotation rates depend solely on the properties of the trap and not on the masses of particles. The resonances cause the escape of a particle, or the center of mass for a collection of interacting particles, from the trap. In view of recent experiments in which the axis of rotation was tilted away from the trap axis \\cite{marago,smith}, it should be possible to confirm experimentally the existence of the gravity-induced resonances for Bose-Einstein condensates in rotating traps. In particular, the presence of resonances makes it possible to quickly expel all particles from the trap.\n\nThis research was supported by a grant from the Polish Ministry of Scientific Research and Information Technology under the Grant No. PBZ-MIN-008\/P03\/2003.\n\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction} \n\n\nUltracold quantum gases provide an excellent and highly controllable testbed for realizing a multitude of systems without the inherent complexity of their \ncondensed matter counterparts \\cite{Lewe}. \nKey features of ultracold atoms include the ability to manipulate their interparticle interactions by employing Feshbach resonances \\cite{Fesh1,Fesh2}, \nto tune the dimensionality of the system \\cite{Petrov,lower-D}, as well as to trap few-body ensembles possessing unique properties \\cite{Brouzos,Greene,Blume,Sowinski,Lompe}. \nTwo-dimensional (2D) systems are of particular interest due to their peculiar scattering properties, the emergent phase transitions, such as the \nBerezinskii-Kosterlitz-Thouless transition \\cite{Dalibard,Bloch,Dalibard2,Cirone,Shlyapnikov,Pricoup} and \nthe existence of long-range thermal fluctuations in the homogeneous case. \nThese thermal fluctuations in turn prohibit the development of a condensed phase, but can allow the occurence of a residual \nquasi-ordered state \\cite{Thouless}.  \n\nOne among the few solvable quantum problems, is the system of two ultracold atoms confined in an isotropic harmonic oscillator. \nHere the two atoms interact via a contact pseudo-potential where only $s$-wave scattering is taken into account \\cite{Huang}, an \napproximation which is valid at ultralow temperatures where two-body interactions dominate \\cite{Tannoudji}. \nThe stationary properties of this system have been extensively studied for various dimensionalities and for arbitrary values of the coupling \nstrength \\cite{Busch,Zyl,Shea,Richard}. \nGeneralizations have also been reported including, for instance, the involvement of anisotropic traps \\cite{Calarco}, higher partial waves \\cite{Stock,Zinner} \nand very recently long-range interactions \\cite{Koscik} and hard-core interaction potentials \\cite{Diakonos}. \nRemarkably enough, exact solutions of few-body setups have also been obtained regarding the stationary properties of three harmonically trapped \nidentical atoms in all dimensions \\cite{Pethick,Drummond,Portegies,Harshman,Polls,Deck}. \n\nA quench of one of the intrinsic system's parameters is the most simple way to drive it out-of-equilibrium \\cite{Langen}. \nQuenches of $^{87}$Rb condensates confined in a 2D pancake geometry have been employed, for instance, by changing abruptly the trapping frequency \nto excite collective breathing modes \\cite{Chevy,Perrin} in line with the theoretical predictions \\cite{Pitaevskii,Pitaevskii2}. \nOn the contrary, the breathing frequency of two-dimensional Fermi gases has been recently measured experimentally \\cite{Anomaly1,Anomaly2} \nand found to deviate from theoretical predictions at strong interactions, a behavior called quantum anomaly. \nAlso, oscillations of the density fluctuations \nbeing reminiscent of the Sakharov oscillations \\cite{Sakharov} have been \nobserved by quenching the interparticle repulsion. \nFurthermore, it has been shown that the dynamics of an expanding Bose gas when switching off the external trap leads to the fast and slow equilibration of \nthe atomic sample in one- and two-spatial dimensions respectively \\cite{Demler}. \nMoreover, the collisional dynamics of two $^6$Li atoms has been experimentally monitored after quenching the frequencies of a three-dimensional harmonic trap \\cite{Jochim}.\nTurning to two harmonically trapped bosons, the existing analytical solutions have been employed in order to track the interaction quench dynamics mainly in \none- \\cite{Simos,Bolsinger,Garcia,Corson2}, but also in three-dimensional systems \\cite{Sykes}. \nFocusing on a single dimension, an analytical expression regarding the eigenstate transition amplitudes after the quench has \nbeen derived \\cite{Simos}. \nMoreover, by utilizing the Bose-Fermi mapping theorem \\cite{Tonks,Girardeau} a closed form of the time-evolved two-body wavefunction for quenches \ntowards the infinite interaction strength has been obtained \\cite{Bolsinger}, observing also a dynamical crossover from bosonic \nto fermionic properties. \n\nBesides these investigations the interaction quench dynamics of the two-boson system in two spatial dimensions employing an \nanalytical treatment has not been addressed. \nHere, the existence of a bound state for all interaction strengths might be crucial giving rise to a very different dynamics compared to its one-dimensional analogue. \nAlso, regarding the strongly interacting regime the Bose-Fermi theorem does not hold. \nTherefore it is not clear whether signatures of fermionic \nproperties can be unveiled although there are some suggestions for their existence \\cite{Mujal,Yannouleas,Mujal2}. \nAnother interesting feature is the inherent analogy between three bosons interacting via a three-body force in one-dimension and two bosons \ninteracting via a two-body force in two spatial dimensions \\cite{Valiente2,Trimers,Guijarro,Nishida,Sekino}. \nTherefore, our work can provide additional hints on the largely unexplored three-body dynamics of three bosons in one spatial dimension \\cite{Pastukhov}. \nThe present investigation will enable us to unravel the role of the different eigenstates for the dynamical response of the system and might inspire \nfuture studies examining state transfer processes \\cite{Fogarty,Reshodko} which are currently mainly restricted to one-dimensional setups. \n\nIn this work we study the interaction quench dynamics of two harmonically confined bosons \nin two spatial dimensions for arbitrary interaction strengths. \nTo set the stage, we briefly review the analytical solution of the system for an arbitrary stationary eigenstate and discuss the \ncorresponding two-body energy eigenspectrum \\cite{Busch}. \nSubsequently, the time-evolving two-body wavefunction is derived as an expansion over the postquench eigenstates of the system with \nthe expansion coefficients acquiring a closed form. \nThe quench-induced dynamical response of the system is showcased via inspecting the fidelity evolution. \nThe underlying eigenstate transitions that predominantly participate in the dynamics are identified in the fidelity spectrum \\cite{Mistakidis1,Mistakidis2,Thies}. \nIt is found that initializing the system in its ground state, characterized by finite interactions of either sign, it is driven more efficiently out-of-equilibrium \nwhen employing an interaction quench in the vicinity of the non-interacting limit.  \nDue to the interaction quench the two bosons perform a breathing motion, visualized in the temporal evolution of the single-particle density and \nthe radial probability density in both real and momentum space. \nThese observables develop characteristic structures which signal the participation of the bound and energetically higher-lying excited states of the postquench system.\nThe dynamics of the short-range correlations is captured by the two-body contact, which is found to perform an oscillatory motion possessing a multitude of frequencies. \nIn all cases the predominantly involved frequency corresponds to the energy difference between the bound and the ground state. \nAdditionally, the amplitude of these oscillations is enhanced when quenching the system from weak to infinite interactions. \nMoreover, it is shown that the system's dynamical response crucially depends on the initial state and in particular starting from an energetically higher excited state, \nthe system is perturbed to a lesser extent, and a fewer amount of postquench eigenstates contribute in the dynamics \\cite{Sowinski_ent,Katsimiga_diss_flow,Pia,Katsimiga_quantum_DBs,Katsimiga_bent}. \nHowever, if the quench is performed from the bound state \nthe system is perturbed in the most efficient manner compared to \nany other initial state configuration. \nFinally, we observe that quenching the system from its ground state at zero interactions towards the infinitely strong ones the time-evolved wavefunction becomes almost \northogonal to the initial one at certain time intervals. \n\nThis work is structured as follows. \nIn Sec. \\ref{theory} we introduce our setup, provide a brief summary of its energy spectrum and most importantly derive a closed \nform of the time-evolved wavefunction discussing also basic observables. \nSubsequently, we investigate the interaction quench dynamics from attractive to repulsive interactions in Sec. \\ref{quench_att} and vice versa in Sec. \\ref{quench_rep} as well as from zero to infinitely large coupling strengths in Sec. \\ref{inf_quench}. \nWe summarize our results and provide an outlook in Sec. \\ref{conclusions}.   \n\t\n\t\n\n\n\\section{Theoretical framework} \\label{theory}\n\n\\subsection{Setup and its stationary solutions} \\label{stationary_sol}\n         \n         \nWe consider two ultracold bosons trapped in a 2D isotropic harmonic trap. \nThe interparticle interaction is modeled by a contact $s$-wave pseudo-potential, which is an adequate approximation \nwithin the ultracold regime. \nThe Hamiltonian of the system, employing harmonic oscillator units ($\\hbar=m=\\omega=1$), reads\n\\begin{equation} \\label{hamilt}\n\\mathcal{H}= \\frac{1}{2} \\sum_{i=1}^{2} \\left[-\\nabla_i^2 +\\boldsymbol{r}_i ^{2} \\right] + 2V_{\\textrm{pp}}(\\boldsymbol{r}_1-\\boldsymbol{r}_2),\n\\end{equation}\nwhere $\\boldsymbol{r}_1$ and $\\boldsymbol{r}_2$ denote the spatial coordinates of each boson. \nNote that the prefactor 2 is used for later convenience in the calculations. \nThe contact regularized pseudo-potential can be expressed as \\cite{Olshanii}  \n\\begin{equation}\nV_{\\textrm{pp}}(\\boldsymbol{r})= -\\frac{\\pi \\delta(\\boldsymbol{r})}{\\ln(Aa\\Lambda)}\\left(1-\\ln(A\\Lambda r)r\\frac{\\partial}{\\partial r}  \\right),\n\\end{equation}\nwith $\\Lambda$ being an arbitrary dimensionful parameter possessing the dimension of a wavevector and $A= e^{\\gamma}\/2$ where $ \\gamma= 0.577\\ldots $ is the \nEuler-Mascheroni constant.  \nWe remark that the parameter $\\Lambda$ does not affect the value of any observable or the energies and eigenstates of the system as it has been shown \nin \\cite{Pricoup,Olshanii}. \nFurthermore, the 2D $s$-wave scattering length is given by $a$. \n\nTo proceed, we perform a separation of variables in terms of the center-of-mass, $\\boldsymbol{R}=\\frac{1}{\\sqrt{2}}(\\boldsymbol{r}_1+\\boldsymbol{r}_2)$, and the relative \ncoordinates $ \\boldsymbol{\\rho}= \\frac{1}{\\sqrt{2}}(\\boldsymbol{r}_1-\\boldsymbol{r}_2)$. \nEmploying this separation, the Hamiltonian \\eqref{hamilt} acquires the form $\\mathcal{H}= \\mathcal{H}_{\\textrm{CM}}+\\mathcal{H}_{\\textrm{rel}}$ with\n\\begin{equation} \\label{separation_hamilt}\n\\mathcal{H}_{\\textrm{CM}}= -\\frac{1}{2} \\nabla_{\\boldsymbol{R}}^2+\\frac{1}{2}R^2,  \n\\end{equation}\nbeing the Hamiltonian of the center-of-mass and\n\\begin{equation}\n  \\mathcal{H}_{\\textrm{rel}} = -\\frac{1}{2} \n  \\nabla_{\\boldsymbol{\\rho}}^2 +\\frac{1}{2}\\rho^2 +V_{\\textrm{pp}}(\\boldsymbol{\\rho}),\n\\end{equation}\nis the Hamiltonian corresponding to the motion in the relative coordinate frame.\n \nAs a result, the Schr\\\"odinger equation can be casted into the form $\\mathcal{H}\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=E\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)$. \nHere the total energy of the system has two contributions, namely $E=E_{\\textrm{CM}}+E_{\\textrm{rel}}$, and the system's wavefunction is a product of a \ncenter-of-mass and a relative coordinate part i.e. $\\Psi(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=\\Psi_{\\textrm{CM}}(\\boldsymbol{R})\\Psi_{\\textrm{rel}}(\\boldsymbol{\\rho})$. \nSince the center-of-mass hamiltonian $\\mathcal{H}_{\\textrm{CM}}$ is interaction independent [see Eq. \\eqref{separation_hamilt}] its eigenstates \ncorrespond to the well-known non-interacting 2D harmonic oscillator states \\cite{Sakurai}. \nWe assume that the center-of-mass wavefunction takes the form $\\Psi_{\\textrm{CM}}(\\boldsymbol{R})=\\frac{e^{-\\boldsymbol{R}^2\/2}}{\\sqrt{\\pi}}$, namely the non-interacting \nground state of the 2D harmonic oscillator. \nSince we are interested in the interaction quench dynamics of the two interacting bosons we omit the center-of-mass wavefunction in what follows \nfor simplicity. \nFollowing the above-mentioned separation of coordinates, the problem boils down to solving the relative part of the Hamiltonian, $\\mathcal{H}_{\\textrm{rel}}$, \nwhich is interaction dependent. \nFor this purpose, we assume an ansatz for the relative wavefunction, which involves an expansion over the non-interacting energy eigenstates \nof the 2D harmonic oscillator \n\\begin{equation} \\label{states}\n\\begin{split}\n\\varphi_{n,m}(\\rho,\\theta&)=\\\\&\\sqrt{\\frac{n!}{\\pi\\Gamma(n+|m|+1)}}e^{-\\rho^2\/2}\\rho^{|m|}L_n^{(m)}(\\rho^2)e^{{i\\mkern1mu} m\\theta}.\n\\end{split}\n\\end{equation} \nIn this expression, $\\Gamma(n)$ is the gamma function while $L_n^{(m)}$ refer to the generalized Laguerre polynomials of degree $n$ and value of angular momentum $m$. \nAlso, $\\boldsymbol{\\rho}=(\\rho,\\theta)$ where \n$\\rho$ is the relative polar coordinate and $\\theta$ is the relative angle. \nThe energy of the non-interacting 2D harmonic oscillator eigenstates in harmonic oscillator units is $E_{\\textrm{rel},n,m}=2n+|m|+1$ \\cite{Sakurai}.  \nWithin our relative coordinate wavefunction ansatz [see Eq. \\eqref{ansatz} below] we will employ, however, only those states that are affected by the \npseudo-potential and thus have a non-vanishing value at $\\rho=0$. \nThese are the states with bosonic symmetry $m=0$, i.e. zero angular momentum. \nThe states with odd m are fermionic, since under the exchange $\\theta \\rightarrow \\theta -\\pi$, they acquire an extra minus sign due to \nthe term $e^{{i\\mkern1mu} m\\theta}$. \nTherefore, the ansatz for the relative wavefunction reads \n\\begin{equation} \\label{ansatz}\n\\Psi_{\\textrm{rel}}(\\rho)= \\sum_{n=0}^{\\infty} c_n \\varphi_n(\\rho), \n\\end{equation} \nwhere the summation is performed over the principal quantum number $n$ and we omit the angle $\\theta$ since only the states with $m=0$ \nare taken into account. \nNote that this ansatz has already been reported previously e.g. in Refs. \\cite{Busch,Simos}.\nIn order to determine the expansion coefficients $c_n$, we plug Eq. \\eqref{ansatz} into the Schr\\\"odinger equation that $\\mathcal{H}_{\\textrm{rel}}$ satisfies \nand project the resulting equation onto the state $\\varphi_{n'}^*(\\rho)$. \nFollowing this procedure we arrive at\n\\begin{equation} \\label{exp_coeff}\n\\begin{split}\nc_{n'}&(E_{\\textrm{rel},n'}-E_{\\textrm{rel}})=\\frac{\\pi\\varphi_{n'}^*(0) }{\\ln(Aa\\Lambda)}\\\\& \\times\\left[ \\left(1-\\ln(\\sqrt{2}A\\Lambda\\rho)\\rho\\frac{\\partial}{\\partial \\rho}  \n\\right)\\sum_{n=0}^{\\infty} c_n \\varphi_n(\\rho) \\right]_{\\rho\\rightarrow 0}.\n\\end{split}\n\\end{equation} \nThe right hand side of Eq. \\eqref{exp_coeff} is related to a normalization factor of the relative wavefunction $\\ket{\\Psi_{\\textrm{rel}}}$. \nIndeed it has been shown \\cite{Busch,Simos} that the coefficients take the form\n\\begin{equation}\n  c_n=A_1\\frac{\\varphi_n^*(0)}{E_{\\textrm{rel},n}-E_{\\textrm{rel}}},\n\\end{equation}\n with \n$A_1=\\frac{2\\sqrt{\\pi}}{\\sqrt{\\psi^{(1)}\\left(\\frac{1-E_{\\textrm{rel}}}{2}\\right)}}$ being a normalization constant and $\\psi^{(1)}(z)$ the trigamma function. \n\nBy inserting this expression of $c_n$ into Eq. \\eqref{ansatz}, we can determine the relative wavefunction. \nThis can be achieved by making use of the generating function of the Laguerre polynomials i.e. $\\sum_{n=0}^{\\infty} t^n L_n(x)=\\frac{1}{1-t}e^{-\\frac{tx}{1-t}}$. \nThus, the relative wavefunction takes the form \\cite{Drummond}\n\\begin{equation} \\label{station}\n\\begin{split}\n\\Psi_{\\textrm{rel},\\nu_i}(\\rho)=\\frac{\\Gamma(-\\nu_i)}{\\sqrt{\\pi\\psi^{(1)}(-\\nu_i)}} e^{-\\rho^2\/2}U(-\\nu_i,1,\\rho^2),\n\\end{split}\n\\end{equation}\nwhere $U(a,b,z)$ refers to the confluent hypergeometric function of the second type (also known as Tricomi's function) and $2\\nu_i+1$ is the energy of the $i=0,1,\\dots$ interacting eigenstate \\cite{Stegun}.\nIn what follows we will drop the subscript rel and denote these relative coordinate states by $\\ket{\\Psi_{\\nu_i}}$. \nIt is important to note at this point that this relative wavefunction ansatz solves also the problem of three one-dimensional harmonically trapped bosons interacting \nvia three-body forces, see e.g. Ref. \\cite{Pastukhov} for more details.\n\nTo find the energy spectrum of $\\mathcal{H}_{\\textrm{rel}}$, we employ Eq. \\eqref{exp_coeff} along with the form of $c_{n,i}=\\frac{\\sqrt{\\pi}\\varphi_n^*(0)}{(n-\\nu_i)\\sqrt{\\psi^{(1)}(-\\nu_i)}}$. \nNote that in order to determine the right hand side of Eq. \\eqref{exp_coeff}, we make use of the behavior of the relative wavefunction \\eqref{station} \nclose to $\\rho=0$. \nIn this way, we obtain the following algebraic equation regarding the energy of the relative coordinates \\cite{Busch,Zyl}, \n$2\\nu_i+1$, \n\\begin{equation} \\label{spectrum}\n\\psi(-\\nu_i)= \\ln\\left( \\frac{1}{2a^2}\\right) +2\\ln2 -2\\gamma,\n\\end{equation}\nwhere $\\psi(x)$ is the digamma function. \nNote here that a different form of the algebraic Eq. \\eqref{spectrum} can be found in \\cite{Busch} and stems from a different definition of the \nscattering length $a$ \\cite{Zyl}. \nIt is also important to emphasize that the energy spectrum given by Eq. \\eqref{spectrum} is independent of the form of the pseudo-potential, $V_{\\textrm{pp}}(\\boldsymbol{r})$, i.e. independent of $\\Lambda$, A, or any short range potential, \nas long as its range is much smaller than the harmonic oscillator length \\cite{Zyl}. \nDenoting $a_0\\equiv \\frac{a}{2}e^{\\gamma}$, the algebraic Eq. \\eqref{spectrum} can be casted into the simpler form \n$\\psi(-\\nu_i)= \\ln\\left( \\frac{1}{2a_0^2}\\right)$. \nAlso, we define the interparticle interaction strength \\cite{Doganov,Busch,Zyl,Petrov,Shlyapnikov,comment} to be\n\\begin{equation}\ng=\\frac{1}{\\ln \\left( \\frac{1}{2a_0^2}\\right)}.\n\\end{equation} \n         \n\\begin{figure*}[t!] \n\\centering\n\\includegraphics[width=\\textwidth,height=8cm]{figure1-eps-converted-to.pdf}\n\\caption{Energy spectrum of two bosons trapped in a 2D harmonic trap for varying interaction strength $g$. \nIn the spectrum for $g>-0.51$ we display the bound state, $\\nu_0$, and higher-lying eigenstates up to the fourth excited state, $\\nu_5$. \nOn the other hand, for $g<-0.51$ the spectrum contains the bound state, $\\nu_0$, as well as higher excited states up to the third excited, $\\nu_4$.\nThe black solid horizontal lines indicate the asymptotic values of the energy determined by $\\psi(-\\nu_i)=0$, in the limit of strong interactions. \nThe black solid vertical line at $g=-0.51$ marks the boundary at which the bound state for negative interaction strengths becomes the ground state for $g>-0.51$.\nThe insets show the radial probability density of the bound states $\\nu_0$ for different attractive (left panel) and repulsive (right panel) interactions, \nas well as the radial probability density of the ground state, $\\nu_1$, at $g=0.3$ (left panel).} \n\\label{fig:en_spectrum}\n\\end{figure*} \n\nThe energy $E_{\\textrm{rel}}$ of the two bosons as a function of the interparticle interaction strength is presented in Fig. \\ref{fig:en_spectrum}. \nAs it can be seen, for $g=0$ $E_{\\textrm{rel}}$ has the simple form $E_{\\textrm{rel},n}=2n+1$, and thus we recover the non-interacting energy spectrum of a 2D harmonic oscillator \nwith zero angular momentum \\cite{Sakurai,Tannoudji}. \nIn this case the energy spacing between two consecutive eigenenergies is independent of n, i.e. $\\Delta E=E_{\\textrm{rel},n+1}-E_{\\textrm{rel},n}=2$. \nFor repulsive (attractive) interactions, the energy is increased (lowered) with respect to its value at $g=0$. \nAlso and in contrast to the one-dimensional case, there are bound states $\\ket{\\Psi_{\\nu_0}}$, namely eigenstates characterized by negative energy, in both interaction regimes. \nNote that herein we shall refer to these eigenstates with negative energy as bound states ($\\nu_0$) whilst the corresponding eigenstates with positive energy in increasing energetic order \nwill be denoted e.g. as the first ($\\nu_1$), second ($\\nu_2$) etc eigenstates and called ground, first excited state etc.\nThe presence of these bound states can be attributed to the existence of the centripetal term $-\\frac{1}{4r^2}$, in the 2D radial Schr\\\"odinger equation \\cite{Sakurai}, which \nsupports a bound state even for weakly attractive potentials, in contrast to the 3D case \\cite{Cirone,Gezerlis}. \nThese energy states, $\\nu_0$, correspond to the molecular branch of two cold atoms in two dimensions. \nThis is clearly captured by the lowest energy branch of Fig. \\ref{fig:en_spectrum}, as has been demonstrated in Ref. \\cite{Drummond}. \nNote that due to a different definition of the coupling constant compared to Ref. \\cite{Drummond}, which possesses a bijective mapping to our definition of \nthe coupling strength \\cite{comment}, the molecular branch \nmaps to the bound states ($\\nu_0$) herein in both the repulsive and the attractive interaction regime. \nTo further appreciate the influence of these bound states we also provide in the insets of Fig. 1 their radial probability densities $2\\pi\\rho|\\Psi|^2$ \\cite{Cirone}\nfor various interaction strengths as well as the radial probability density of the ground state $\\ket{\\Psi_{\\nu_1}}$ at $g=0.3$. \nIn the repulsive regime of interactions (right panel) the full-width-at-half-maximum of $2\\pi\\rho|\\Psi|^2$ is smaller \nthan the one of the attractive regime (left panel). \nThis behavior is caused by the much stronger energy of the bound state at $g>0$ compared to the $g<0$ case. \nFor large interaction strengths, $|g|>8$, the widths of $2\\pi\\rho|\\Psi|^2$ tend to be the same. \nAnother interesting feature of the 2D energy spectrum is the occurrence of a boundary signifying a crossover from the bound to the ground state ($\\nu_0\\rightarrow\\nu_1$) at \n$g=-0.51$, see the corresponding vertical line in Fig. \\ref{fig:en_spectrum}. \nThis means that the negative eigenenergy of $\\ket{\\Psi_{\\nu_0}}$ crosses the zero energy axis and \nbecomes the positive eigenenergy of $\\ket{\\Psi_{\\nu_1}}$ at $g=-0.51$. \nThis crossover is captured, for instance, by $2\\pi\\rho|\\Psi|^2$ which changes from a delocalized [e.g. at $g=0.3$] to a localized [e.g. at $g=-1$] distribution. \nThe existence of this boundary affects the labeling of all the states and therefore $\\nu_i$ becomes $\\nu_{i+1}$ as it is crossed from the \nrepulsive side of interactions. \nWe note here that with $\\ket{\\Psi_{\\nu_1}}$ [$\\ket{\\Psi_{\\nu_0}}$] we label the ground [bound] state and with $\\ket{\\Psi_{\\nu_i}}, \\, i>1$, the \ncorresponding excited states. \nFor repulsive interactions the energy of the bound state diverges at $g=0$ as $-1\/a_0^2$ \\cite{Gezerlis,Zinner} or as $-2e^{1\/g}$ in terms of the \ninterparticle strength, while it approaches its asymptotic value for very strong interactions [see Fig. \\ref{fig:en_spectrum}]. \nThe two bound states share the same asymptotic value $E_{\\textrm{rel}}=-1.923264$ at $g\\to \\pm \\infty$. \nWe remark that this behavior of the bound state in the vicinity of $g=0$ is the same as the one of the so-called universal bound state of \ntwo cold atoms in two dimensions in the absence of a trap \\cite{Zinner}.  \nWe also note that the states $\\ket{\\Psi_{\\nu_i}}$ with $i \\neq 0$, approach their asymptotic values faster (being close to their asymptotic value already for $g=2$) than the \nbound states. \nThe asymptotic values are determined via the algebraic equation $\\psi(-\\nu_i)=0$. \nMoreover, it can be shown that approximately the positive energy in the infinite interaction limit is given by the formula \n$E_{\\textrm{rel}} \\approx2n+1-\\frac{2}{\\ln (n) }+\\mathcal{O}\\left((\\ln n)^{-2}\\right)$ when $n \\gg 1$ \\cite{Stegun}.  \n\n\n\n\\subsection{Time-evolution of basic observables} \n\n\nTo study the dynamics of the two harmonically trapped bosons, we perform an interaction quench starting from a stationary state of the system, \n$\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$, at $g^{\\textrm{in}}$ to the value $g^f$. \nLet us also remark in passing that the dynamics of two bosons in a 2D harmonic trap employing an analytical treatment has not yet been reported. \nThe time-evolution of the system's initial wavefunction reads \n\\begin{equation} \\label{quench_wave}\n\\begin{split}\n\\ket{\\Psi_{\\nu_i}(t)}&= e^{-{i\\mkern1mu} \\hat{H}t}\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}\\\\&= \\sum_{j} e^{-{i\\mkern1mu} (2\\nu_j^f+1) t}\\ket{\\Psi_{\\nu_j}^f}\\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}(0)},\n\\end{split}\n\\end{equation} \nwhere $\\ket{\\Psi_{\\nu_j}^f}$ denotes the $j$-th eigenstate of the postquench Hamiltonian $\\hat{H}$ with energy $(2\\nu_j^f+1)$. \nNote that the indices in and $f$ indicate that the corresponding quantities of interest refer to the initial (prequench) and final (postquench) state of the system respectively. \nMoreover, the overlap coefficients, $\\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$, between the initial wavefunction and a final eigenstate \n$\\ket{\\Psi_{\\nu_j}^f}$ determine the degree of participation of this postquench eigenstate in the dynamics. \nRecall also here that the center-of-mass wavefunction, $\\Psi_{\\textrm{CM}}(\\boldsymbol{R})$, is not included in Eq. \\eqref{quench_wave} since the latter \nis not affected by the quench [see also Sec. \\ref{stationary_sol}] and therefore does not play any role in the description of the dynamics. \n\nIt can be shown that initializing the system in the eigenstate $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ at $g^{\\textrm{in}}$, the probability to \noccupy the eigenstate $\\ket{\\Psi_{\\nu_j}^f}$ after the quench is given by \n\\begin{eqnarray} \nd_{\\nu_j^f,\\nu_i^{\\textrm{in}}}&\\equiv& \\braket{\\Psi_{\\nu_j}^f|\\Psi_{\\nu_i}^{\\textrm{in}}}= \\frac{\\Gamma(-\\nu_i^{\\textrm{in}})\\Gamma(-\\nu_j^f)}{\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}} \\nonumber \\times \\\\\n& & \\times \\int_0^{\\infty}dr e^{-r}U(-\\nu_i^{\\textrm{in}},1,r)U(-\\nu_j^f,1,r) \\nonumber \\\\ & =& \\frac{\\Gamma(-\\nu_j^f)G^{32}_{33}\\left( \\begin{array}{l|lll} \n1& 0&0 &-\\nu_j^f \\\\ & 0& 0 & -1-\\nu_i^{\\textrm{in}} \n\\end{array}\\right)} {\\Gamma(-\\nu_i^{\\textrm{in}})\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}},\n\\end{eqnarray} \nwith $G^{p,q}_{m,n}\\left( \\begin{array}{l|l} \nz& a_1, \\ldots a_p \\\\ & b_1, \\ldots b_q \n\\end{array}\\right)$ being the Meijer G-function \\cite{Gradshteyn}. \nRemarkably enough, the coefficients $d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}$ can also be expressed in a much simpler form if we make use of the ansatz of Eq. \\eqref{ansatz}. \nIndeed, by employing the orthonormality properties of the non-interacting eigenstates $\\varphi_n(\\rho)$ and the explicit expression of the expansion coefficients appearing in the \nansatz \\eqref{ansatz}, the overlap coefficients between a final and the initial eigenstate reads \n\\begin{equation}\\label{overlap} \n\\begin{split}\nd_{\\nu_j^f,\\nu_i^{\\textrm{in}}} = \\frac{\\left[\\frac{1}{g^f}-\\frac{1}{g^{\\textrm{in}}}  \\right]}{(\\nu_i^{\\textrm{in}}-\\nu_j^f)\\sqrt{\\psi^{(1)}(-\\nu_i^{\\textrm{in}})\\psi^{(1)}(-\\nu_j^f)}}.  \n\\end{split}\n\\end{equation} \nIt should be emphasized here that this is a closed form of the overlap coefficients and the only parameters that need to be determined are the energies, \nwhich are determined from the algebraic equation \\eqref{spectrum}. \nAs a result in order to obtain the time-evolution of $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}(0)}$ we need to numerically evaluate Eq. (\\ref{quench_wave}) which is an infinite \nsummation over the postquench eigenstates denoted by $\\ket{\\Psi_{\\nu_j}^f}$. \nIn practice this infinite summation is truncated to a finite one with an upper limit which ensures that the values of all observables have been converged \nwith respect to a further adding of eigenstates. \n\nHaving determined the time-evolution of the system's wavefunction [Eq. (\\ref{quench_wave})] enables to determine any observable of interest in the course of the dynamics. \nTo inspect the dynamics of the system from a single-particle perspective we monitor its one-body density \n\\begin{widetext} \n\\begin{gather} \n\\rho^{(1)}(\\boldsymbol{r}_1,t)=\\int d\\boldsymbol{r}_2 \\tilde{\\Psi}(\\boldsymbol{r}_1,\\boldsymbol{r}_2;t)\\tilde{\\Psi}^*(\\boldsymbol{r}_1,\\boldsymbol{r_2};t) \\nonumber \\\\ \n= \\frac{e^{-(x^2+y^2)}}{\\pi^2}  \\sum_{j,k} \\frac{e^{2{i\\mkern1mu}(\\nu_j^f-\\nu_k^f)t}\\Gamma(-\\nu_k^f)\\Gamma^*(-\\nu_j^f)d_{\\nu_k^f,\\nu_i^{\\textrm{in}}}d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}^*}{\\sqrt{\\psi^{(1)}(-\\nu_k^f)\\psi^{(1)*}(-\\nu_j^f)}} \\times \\nonumber \\\\ \n\\times\\int_{-\\infty}^{\\infty} dz dw e^{-z^2-w^2}U^*\\left(-\\nu_j^f,1,(x-z)^2\/2+(y-w)^2\/2 \\right) \nU\\left(-\\nu_k^f,1,(x-z)^2\/2+(y-w)^2\/2 \\right). \n\\label{density_matrix} \n\\end{gather} \n\\end{widetext}\nIn this expression, the total wavefunction of the system is denoted by \n$\\tilde{\\Psi}(\\boldsymbol{r}_1,\\boldsymbol{r}_2)=\\Psi_{\\textrm{CM}}(R(\\boldsymbol{r}_1,\\boldsymbol{r}_2),t)\\Psi_{\\textrm{rel},\\nu_i}(\\rho(\\boldsymbol{r}_1,\\boldsymbol{r}_2),t)$ \\cite{Sakmann}. \nTo arrive at the second line of Eq. \\eqref{density_matrix} we have expressed the relative, $\\rho^2= \\frac{1}{2} (r_1^2+r_2^2-2\\boldsymbol{r}_1\\cdot\\boldsymbol{r}_2)$, and the \ncenter-of-mass coordinates, $R^2= \\frac{1}{2} (r_1^2+r_2^2+2\\boldsymbol{r}_1\\cdot\\boldsymbol{r}_2)$, in terms of the Cartesian coordinates ($\\boldsymbol{r_1}$, $\\boldsymbol{r_2}$) and integrated \nout the ones pertaining to the other particle. \nIn particular, we adopted the notation $\\boldsymbol{r}_1=(x,y)$ and $\\boldsymbol{r}_2=(z,w)$ for the coordinates that are being integrated out. \nMoreover, the integral $I_{\\nu_j^f,\\nu_k^f}$ appearing in the last line of Eq. \\eqref{density_matrix} can be further simplified by employing the replacements \n$z'=x-z$ ,$w'=y-w$ and then express the new variables in terms of polar coordinates. \nThe emergent angle integration can be readily performed and the integral with respect to the radial coordinate becomes \n\\begin{eqnarray} \n&I_{\\nu_j^f,\\nu_k^f}= 2\\pi e^{-(x^2+y^2)} \\int_0^{\\infty} dr\\,re^{-r^2}  I_0\\left(2r\\sqrt{x^2+y^2} \\right)\\times \\nonumber \\\\& \\times \nU^*\\left(-\\nu_j^f,1,\\frac{r^2}{2}\\right) U\\left(-\\nu_k^f,1,\\frac{r^2}{2}\\right).\n\\end{eqnarray} \nHere, $I_0(x)$ is the zeroth order modified Bessel function of the first kind \\cite{Stegun,Gradshteyn}. \n\nAnother interesting quantity which provides information about the state of the system on the two-body level is the radial probability density of the relative \nwavefunction \n\\begin{equation} \\label{prob_dens}\n\\mathcal{B}(\\rho,t)=2\\pi\\rho|\\Psi(\\rho,t)|^2.\n\\end{equation} \nIt provides the probability density to detect two bosons for a fixed time instant $t$ at a relative distance $\\rho$. \nIt can be directly determined by employing the overlap coefficients of Eq. \\eqref{overlap}.\nMoreover, the corresponding radial probability density in momentum space reads \n\\begin{equation}\n\\mathcal{C}(k,t)=2\\pi k |\\tilde{\\Psi}(k,t)|^2. \n\\end{equation} \nHere, the relative wavefunction in momentum space is obtained from the two dimensional Fourier transform \n\\begin{equation}\n\\tilde{\\Psi}(k,t)= 2\\pi \\int_{0}^{\\infty} d\\rho\\, \\rho \\Psi(\\rho,t) J_0(2\\pi\\rho k) \\quad,\n\\label{Fourier_wave}\n\\end{equation}\nwhere $J_0(x)$ is the zeroth order Bessel function. \n       \nTo estimate the system's dynamical response after the quench we resort to the fidelity evolution $F(t)$. \nIt is defined as the overlap between the time-evolved wavefunction at time $t$ and the initial one \\cite{Gorin}, namely\n\\begin{equation} \\label{f(t)}\n\\begin{split}\nF(t)=&\\bra{\\Psi(0)}e^{-{i\\mkern1mu} \\hat{H}t}\\ket{\\Psi(0)} \\\\&= \\sum_{j} e^{-{i\\mkern1mu} (2 \\nu_j^f+1 ) t}|d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}|^2.\n\\end{split}\n\\end{equation} \nEvidently, $F(t)$ is a measure of the deviation of the system from its initial state \\cite{Simos}.   \nIn what follows, we will make use of the modulus of the fidelity, $\\abs{F(t)}$. \nMost importantly, the frequency spectrum of the modulus of the fidelity $F(\\omega)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} dt\\, |F(t)|e^{{i\\mkern1mu} \\omega t}$ grants \naccess to the quench-induced dynamical modes \\cite{Mistakidis1,Mistakidis2,Mistakidis3,Mistakidis4,Jannis}. \nIndeed, the emergent frequencies appearing in the spectrum correspond to the energy differences of particular postquench eigenstates of the system and therefore  \nenable us to identify the states that participate in the dynamics (see also the discussion below).\n\nAnother observable of interest is the two-body contact $\\mathcal{D}$. \nThe latter is defined from the momentum distribution in the limit of very large momenta i.e. $\\mathcal{C}(k,t) \\xrightarrow{k\\rightarrow \\infty} \\frac{2\\pi\\mathcal{D}(t)}{k^3}$ \nand captures the ocurrence of short-range two-body correlations \\cite{Bellotti,Valiente,momentum_2}. \nMoreover, this quantity can be experimentally monitored \\cite{Contact_1,Contact_2} and satisfies a variety of universal relations independently \nof the quantum statistics, the number of particles or the system's dimensionality \\cite{Tan1,Tan2,Tan3,momentum_2}. \nHaving at hand the eigenstates of the system, we can expand the time evolved contact after a quench from $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ at \n$g^{\\textrm{in}}$ to an arbitrary $g^f$ in terms of the contacts of the postquench eigenstates \\cite{Colussi}. \nNamely\n\\begin{equation}\n   \\mathcal{D}(t)=\\left| \\sum_j e^{-{i\\mkern1mu} (2\\nu_j^f+1)t}d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}\\sqrt{|\\mathcal{D}_j|} \\right|^2.\n\\end{equation}\nThe contacts $\\mathcal{D}_j$ of the postquench eigenstates $\\ket{\\Psi_{\\nu_j}^f}$ can be inferred by employing the behavior of the \neigenstates [Eq. \\eqref{station}] close to zero distance, $\\rho \\rightarrow 0$, between the atoms\n\\begin{equation}\n   \\Psi_{\\nu_j}(\\rho) \\xrightarrow[\\rho\\rightarrow 0]{} -\\frac{2 \\ln \\rho}{\\sqrt{\\pi \\psi^{(1)}(-\\nu_j)}}.\n   \\label{waves_small} \n\\end{equation}\nBy plugging Eq. \\eqref{waves_small} into Eq. \\eqref{Fourier_wave}  and restricting ourselves to small $\\rho$ values we obtain the contact from \nthe leading order term ($\\sim 1\/k^2$) of the resulting expression. \nThe contact for the postquench eigenstates $\\ket{\\Psi_{\\nu_j}^f}$ reads\n\\begin{equation}\n   \\mathcal{D}_j=\\frac{1}{\\pi^3\\psi^{(1)}(-\\nu_j)}.\n\\end{equation}\nNote that in order to capture the quench-induced dynamical modes that participate in the dynamics of the contact, \nwe employ its corresponding frequency spectrum i.e. $\\mathcal{D}(\\omega)=\\frac{1}{\\sqrt{2\\pi}}\\int_{-\\infty}^{\\infty} \\mathcal{D}(t)e^{{i\\mkern1mu} \\omega t}$.\t \n\t \nHaving analyzed the exact solution of the two bosons trapped in a 2D harmonic trap both for the stationary and the time-dependent cases, \nwe subsequently explore the corresponding interaction quench dynamics. \nIn particular, we initialize the system into its ground state $ \\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ for attractive interactions and perform interaction quenches towards the repulsive \nregime (Sec. \\ref{quench_att}) and vice versa (Sec. \\ref{quench_rep}). \n\n\\section{Quench dynamics of two attractive bosons to repulsive interactions} \\label{quench_att} \n\nWe first study the interaction quench dynamics of two attractively interacting bosons confined in a 2D isotropic harmonic trap. \nMore specifically, the system is initially prepared in its corresponding ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{\\textrm{in}}}=-1$. At $t=0$ we perform an interaction quench towards the \nrepulsive interactions letting the system evolve. \nOur main objective is to analyze the dynamical response of the system and identify the underlying dominant microscopic mechanisms.   \n      \n     \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure2-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of the two bosons following an interaction quench from $g^{\\textrm{in}}=-1$ and \n$\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ to various postquench interaction strengths. \n(b) Fidelity evolution at representative postquench interaction strengths (see legend).}\n\\label{fig:attractive_ground} \n\\end{figure} \n\n\n\\subsection{Dynamical response} \\label{response_attract_repul}\n       \nTo examine the dynamical response of the system after the quench we employ the corresponding fidelity evolution $\\abs{F(t)}$ [see Eq. \\eqref{f(t)}] \\cite{Fogarty2}. \nFigure \\ref{fig:attractive_ground} (a) shows $\\abs{F(t)}$ for various postquench interaction strengths $g^f$. \nWe observe the emergence of four distinct dynamical regions where the fidelity exhibits a different behavior. \nIn region I, $-1<g^f<-0.27$, $\\abs{F(t)}$ performs small amplitude oscillations in time [see also $\\abs{F(t)}$ for $g^f=-0.5$ in Fig. \\ref{fig:attractive_ground} (b)] \nand therefore the system remains essentially unperturbed. \nNote that the oscillation period is slightly smaller than $\\pi$ [see also the discussion below], e.g. see Fig. \\ref{fig:attractive_ground} (b) for $g^f=-0.5$. \nEntering region II, $-0.27<g^f<0.8$, the system departs significantly from its initial state since $\\abs{F(t)}$ exhibits large amplitude oscillations in time \n(see the blue lobes in Fig. \\ref{fig:attractive_ground} (a) within region II) deviating \nappreciably from unity [see also Fig. \\ref{fig:attractive_ground} (b) at $g^f=0.5$]. \nA more careful inspection of $\\abs{F(t)}$ reveals that it oscillates with at least two frequencies, namely a faster and a slower one. \nIndeed, $\\abs{F(t)}$ oscillates rapidly (fast frequency) within a large amplitude envelope of period $\\simeq \\pi$ (slow frequency). \nWithin region III, $0.8<g^f<2.7$, the oscillation amplitude of $\\abs{F(t)}$ becomes smaller when compared to region II. \nMost importantly, we observe the appearance of irregular minima and maxima in $\\abs{F(t)}$ being shifted with time \n[Fig. \\ref{fig:attractive_ground} (b) at $g^f=1$]. \nFor strong interactions, $2.7<g^f<10$, we encounter region IV in which $|F(t)|>0.9$ performs small amplitude oscillations that resemble the \nones already observed within region I [Fig. \\ref{fig:attractive_ground} (b) at $g^f=7$]. \nAn important difference with respect to region I is that the oscillations of $\\abs{F(t)}$ are faster and there is more than one frequency involved, compare $\\abs{F(t)}$ at $g^f=-0.5$ and $g^f=7$ in \nFig. \\ref{fig:attractive_ground} (b).\n       \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure3-eps-converted-to.pdf} \n\\caption{(a) The fidelity spectrum $F(\\omega)$ after an interaction quench from $g^{\\textrm{in}}=-1$ to different final interaction strengths $g^f$. \n(b) The corresponding largest overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ (see legend). \nThe black dashed vertical line at $g^f=-0.51$ marks the boundary at which the bound state for negative interaction strengths becomes the \nground state for $g^f>-0.51$, see also Fig. \\ref{fig:en_spectrum}. The inset presents a magnification of $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ for $-1\\leq g^f \\leq-0.4$ .} \n\\label{fig:attractive_spec} \n\\end{figure}\n       \nTo gain more insights onto the dynamics, we next resort to the frequency spectrum of the fidelity $F(\\omega)$, shown in Fig. \\ref{fig:attractive_spec} (a) for a varying \npostquench interaction strength. \nThis spectrum provides information about the contribution of the different postquench states that participate in the dynamics. \nIndeed, the square of the fidelity [see Eq. \\eqref{f(t)}] can be expressed as \n\\begin{equation} \\label{fidelity}\n\\begin{split}\n|F(t)|^2=& \\sum_{j} |d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^4\\\\&+2\\sum_{j \\neq k}|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2|d_{\\nu_k^f,\\nu_1^{\\textrm{in}}}|^2\\cos(\\omega_{\\nu_j^f,\\nu_k^f}t),\n\\end{split}\n\\end{equation}\nwhere $d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}$ are the overlap coefficients between the initial (prequench) $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and the final (postquench) $\\ket{\\Psi_{\\nu_j}^f}$ eigenstates. \nThe corresponding overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ for an increasing postquench interaction strength are presented in Fig. \\ref{fig:attractive_spec} (b). \nMoreover, the frequencies $\\omega_{\\nu_j^f,\\nu_k^f}$ are determined by the energy differences between two distinct eigenstates of the postquench Hamiltonian, \nnamely $\\omega_{\\nu_j^f,\\nu_k^f}=2( \\nu_j^f- \\nu_k^f)\\equiv \\omega_{\\nu_j,\\nu_k}$ with $\\quad j\\neq k$. \nNote also that the amplitudes of the frequencies [encoded in the colorbar of Fig. \\ref{fig:attractive_spec} (a)] mainly depend on the product of their respective overlap coefficients, \ni.e. $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2|d_{\\nu_k^f,\\nu_1^{\\textrm{in}}}|^2$. \nFinally, the values of the frequencies $\\omega_{\\nu_j,\\nu_k}$ along with the coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:attractive_spec} (b)] determine the \ndominantly participating postquench eigenstates \\cite{Simos,Mistakidis1,Mistakidis2,Mistakidis3}. \n\nFocusing on region I we observe that in $F(\\omega)$ there are two frequencies, hardly visible in Fig. \\ref{fig:attractive_spec} (a). The most dominant one corresponds to $\\omega_{\\nu_1,\\nu_0}$ for \n$-1<g^f<-0.51$ and to $\\omega_{\\nu_2,\\nu_1}$ for $-0.51<g^f<-0.27$. It is larger than 2 giving thus rise to a period of $|F(t)|$ smaller than $\\pi$. The fainter one corresponds to $\\omega_{\\nu_2,\\nu_1}$ for $-1<g^f<-0.51$ and to $\\omega_{\\nu_3,\\nu_2}$ for $-0.51<g^f<-0.27$. \nFor reasons of clarity let us mention that each of these frequencies, of course, coincide with the corresponding energy difference between the respective eigenstates \nof the system's eigenspectrum [Fig. \\ref{fig:en_spectrum}]. \nRecall that at $g^f=-0.51$ indicated by the vertical line in Fig. \\ref{fig:attractive_spec} [see also Fig. \\ref{fig:en_spectrum}], the labeling of the eigenstates \nchanges and e.g. the frequency $\\omega_{\\nu_1,\\nu_0}$ becomes $\\omega_{\\nu_2,\\nu_1}$. \nAs it can be seen from Fig. \\ref{fig:attractive_spec} (a) $\\omega_{\\nu_1,\\nu_0}$ decreases for increasing $g^f$ which is in accordance \nwith the behavior of the energy gap $\\omega_{\\nu_1,\\nu_0}=2( \\nu_1^f- \\nu_0^f)$ in the system's eigenspectrum [Fig. \\ref{fig:en_spectrum}]. \nTurning to region II, a multitude of almost equidistant frequencies appears. \nThis behavior is clearly captured in the vicinity of $g^f=0$, where the energy difference between consecutive eigenenergies exhibits an almost \nequal spacing of the order of $\\Delta E\\simeq2$ [see also Fig. \\ref{fig:en_spectrum}]. \nTo characterize the observed frequency branches in terms of transitions between the system's eigenstates we determine the corresponding overlap coefficients \n$d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}$ shown in Fig. \\ref{fig:attractive_spec} (b) and also the respective eigenstate energy differences known from the eigenspectrum of the \nsystem [Fig. \\ref{fig:en_spectrum}]. \nIn this way, we identify the most prominent frequency $\\omega_{\\nu_2,\\nu_1}$ appearing in $F(\\omega)$ which is near $\\omega \\approx2$. \nAdditionally, a careful inspection of Fig. \\ref{fig:attractive_spec} (b) reveals that there is a significant decrease of $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2$ for a larger $g^f$ and subsequently \nenergetically higher excited states come into play, e.g. $\\ket{\\Psi_{\\nu_3}^f}$. \nThese latter contributions give rise to the appearance of energetically higher frequencies in $F(\\omega)$. \nIndeed the bound state, $\\ket{\\Psi_{\\nu_0}^f}$, possesses a non-negligible population already for $g^f>0.27$ [Fig. \\ref{fig:attractive_spec} (b)] \ngiving rise to the frequency branch $\\omega_{\\nu_1,\\nu_0}$ that at $g^f\\approx 0.54$ has a quite large value of approximately 14.9 and decreases rapidly as $g^f$ increases. \nOf course, this behavior stems directly from the energy gap between the bound, $\\ket{\\Psi_{\\nu_0}^f}$, and the ground, $\\ket{\\Psi_{\\nu_1}^f}$, states as it can be easily confirmed \nby inspecting the eigenspectrum [Fig. \\ref{fig:en_spectrum}]. \nIn the intersection between regions II and III, $\\omega_{\\nu_1,\\nu_0}$ becomes degenerate with the other frequency branches [see the black circles in \nFig. \\ref{fig:attractive_spec} (a)], e.g. $\\omega_{\\nu_4,\\nu_1}$ in the vicinity of $g^f=1$ and $\\omega_{\\nu_3,\\nu_1}$ close to $g^f=3$ [Fig. \\ref{fig:attractive_spec} (a)]. \nThe aforementioned frequency branches are much fainter when compared to $\\omega_{\\nu_1,\\nu_0}$, since the overlap coefficients between the relevant eigenstates are small, \ne.g. $|d_{\\nu_3^f,\\nu_1^{\\textrm{in}}}|^2<|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:attractive_spec} (b)]. \nFinally in region IV, there are mainly two dominant frequencies, namely $\\omega_{\\nu_1,\\nu_0}$ and $\\omega_{\\nu_2,\\nu_1}$, that acquire constant values as $g^f$ increases. \nIndeed, in this region $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$, $|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2$ are the most significantly populated coefficients \n[Fig. \\ref{fig:attractive_spec} (b)], which in turn yield these two frequencies. \n      \n       \n\\subsection{Role of the initial state}\\label{role_intial_state_attract_repul}\n      \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure4-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution when performing a quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from energetically \nhigher excited states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, as well as the bound state $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$ (see legend). \nThe corresponding fidelity spectrum when initializing the system in (b) $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and (c) $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$.}\n\\label{excited_att}\n\\end{figure} \n\nTo investigate the role of the initial eigenstate in the dynamical response of the two bosons, we consider an interaction quench from \n$g^{\\textrm{in}}=-1$ to $g^f=1$ but initializing the system at energetically different excited states i.e. $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, and the bound \nstate $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$. \nIn particular, Fig. \\ref{excited_att} (a) illustrates $\\abs{F(t)}$ with a prequench eigenstate being the bound state, the first, the third, the fifth and the \nseventh excited state. \nIn all cases, $\\abs{F(t)}$ exhibits an irregular oscillatory motion as in the case of $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, see also Fig. \\ref{fig:attractive_ground} (b). \nEvidently, for an energetically higher initial eigenstate (but not the bound state) $\\abs{F(t)}$ takes larger values and therefore the system is less perturbed. \nHowever, when the two bosons are prepared in the bound state, $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$, of the system then $\\abs{F(t)}$ drops to smaller \nvalues as compared to the case of energetically higher initial states and the system becomes more perturbed. \n\nThe impact of the initial state on the oscillation amplitude of $\\abs{F(t)}$ is reflected on the values of the corresponding overlap coefficients that appear in \nthe expansion of the fidelity in Eq. (\\ref{fidelity}). \nMore precisely, when an overlap coefficient possesses a dominant population with respect to the others then $\\abs{F(t)}$ exhibits a smaller oscillation \namplitude than in the case where at least two overlap coefficients possess a non negligible population. \nFor convenience and in order to identify the states that take part in the dynamics, we provide the relevant overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, \nfor the quench $g^{\\textrm{in}}=-1\\rightarrow g^f=1$ in Table \\ref{table1} for various initial eigenstates $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nIndeed, an initial energetically higher-lying excited state results in the dominant population of one postquench state while the other states \nexhibit a very small contribution, e.g. see the last column of Table \\ref{table1}. \nFor this reason an initially energetically higher excited state leads to a smaller oscillation amplitude of $\\abs{F(t)}$.  \nMoreover, the large frequency oscillations appearing in $\\abs{F(t)}$ are caused by the presence of several higher than first order eigenstate transitions as e.g. \n$\\omega_{\\nu_6,\\nu_4}$, $\\omega_{\\nu_7,\\nu_4}$, $\\omega_{\\nu_4,\\nu_0}$ in the case of starting from $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ [Fig. \\ref{excited_att} (b)]. \nThe transition mainly responsible for these large frequency oscillations of $\\abs{F(t)}$ involves the bound state $\\ket{\\Psi_{\\nu_0}^f}$.\nIndeed, by inspecting $\\abs{F(t)}$ of different initial configurations shown in Fig. \\ref{excited_att} (a) we observe that starting from energetically higher excited states \nsuch that $\\nu_j>\\nu_4$ the respective contribution of $\\ket{\\Psi_{\\nu_0}^f}$ diminishes [see also Table \\ref{table1}] leading to a decay of the amplitude of these large frequency oscillations of $\\abs{F(t)}$. \nThe aforementioned behavior becomes evident e.g. by comparing $\\abs{F(t)}$ for $\\nu_2^{\\textrm{in}}$ and $\\nu_8^{\\textrm{in}}$ in Fig \\ref{excited_att} (a). \n\nOn the other hand, in order to unveil the participating frequencies in the dynamics of $\\abs{F(t)}$ we calculate its spectrum $|F(\\omega)|$, shown in Figs. \\ref{excited_att}(b), (c). \nWe observe that starting from an energetically higher excited state several frequencies, referring to different eigenstate transitions, are triggered. \nMost of these frequencies which refer to different initial states almost coincide e.g. $\\omega_{\\nu_5,\\nu_4}$ with $\\omega_{\\nu_9,\\nu_8}$, \nsince the energy gap of the underlying eigenstates is approximately the same [see also Fig. \\ref{fig:en_spectrum}]. They possess however a distinct amplitude.  \nAdditionally, there are also distinct contributing frequencies e.g. compare $\\omega_{\\nu_4,\\nu_0}$ with $\\omega_{\\nu_8,\\nu_0}$. \nThe latter are in turn responsible for the dependence of the oscillation period of $\\abs{F(t)}$ on the initial eigenstate of the system. \nFinally, let us note that if the system is quenched to other final interaction strengths (not shown here for brevity reasons), \nacross the four dynamical regions identified in Fig. \\ref{fig:attractive_ground}(a), then $\\abs{F(t)}$ follows a similar pattern as discussed \nin Fig. \\ref{excited_att} (a). \n\n\n      \n\\begin{table}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|}\\hline\n&  {$|d_{\\nu_j^f,\\nu_0^{\\textrm{in}}}|^2$} &  {$|d_{\\nu_j^f,\\nu_2^{\\textrm{in}}}|^2$}  & {$|d_{\\nu_j^f,\\nu_4^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_6^{\\textrm{in}}}|^2$} &  {$|d_{\\nu_j^f,\\nu_8^{\\textrm{in}}}|^2$} \\\\\n\\hline\n{$\\nu_j^f=\\nu_0$} & 0.7896 & 0.0367 & 0.0147 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_1$} & 0.1214 & 0.0198 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_2$} & 0.0351 & 0.8765 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_3$} & 0.0163 & 0.0464 & 0.0187 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_4$} & 0.0092 & 0.0092 & 0.9078 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_5$} & - & - & 0.0351 & 0.0164 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_6$} & - & - & - & 0.9249 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_7$} & - & - & - & 0.0286 & 0.0145 \\\\ \\hline\n{$\\nu_j^f=\\nu_8$} & - & -& -& - & 0.9358 \\\\ \\hline\n{$\\nu_j^f=\\nu_9$} & - & -& -& - & 0.0243 \\\\ \\hline\n\\end{tabular}\n\\caption{Overlap coefficients $|d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}|^2$ for the quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from various excited states, \nnamely $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_2}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_6}^{\\textrm{in}}}$, and $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$.  \nOnly the coefficients with a value larger than 0.9\\% are presented. }\\label{table1}\n\\end{table} \n\n\n\\subsection{One-body density evolution}\n       \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5 \\textwidth]{figure5-eps-converted-to.pdf}\n\\caption{(a)-(f) Time-evolution of the one-body density following an interaction quench from $g^{\\textrm{in}}=-1$ to $g^f=1$. \nThe system of two bosons is initialized in its ground state, $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, trapped in a 2D harmonic oscillator. \n(g)-(j) The corresponding one-body densities for the pre- and postquench eigenstates (see legends) whose overlap coefficients are the dominant ones \nfor the specific quench.} \n\\label{fig:1RD_-1_1} \n\\end{figure} \n       \nTo monitor the dynamical spatial redistribution of the two atoms after the quench at the single-particle level, we next examine the evolution of the \none-body density $\\rho^{(1)}(x,y,t)$ [Eq. \\eqref{density_matrix}]. \nFigures \\ref{fig:1RD_-1_1} (a)-(f) depict $\\rho^{(1)}(x,y,t)$ following an interaction quench from $g^{\\textrm{in}}=-1$ to $g^f=1$ when the system is initialized \nin its ground state configuration $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nNote that the shown time-instants of the evolution lie in the vicinity of the local minima and maxima of the fidelity [see also Fig. \\ref{fig:attractive_ground} (b)], where the \nsystem deviates strongly and weakly from its initial state respectively. \nOverall, we observe that the atoms undergo a breathing motion manifested as a contraction and expansion dynamics of $\\rho^{(1)}(x,y,t)$, see for instance the \nincrease of the density close to $x=y=0$ [Figs. \\ref{fig:1RD_-1_1} (b), (c)] and its subsequent spread [Figs. \\ref{fig:1RD_-1_1} (d), (e)]. \nTo provide further hints on the dynamical superposition \\cite{Sowinski_ent,Katsimiga_diss_flow,Katsimiga_bent} of states we show in Figs. \\ref{fig:1RD_-1_1} (g)-(j) \nthe corresponding $\\rho^{(1)}(x,y,t=0)$ of the initial state, i.e. $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, and the densities of the three most significant, in terms of the overlap coefficients, final states namely \n$\\ket{\\Psi_{\\nu_1}^f},\\ket{\\Psi_{\\nu_0}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$. \nComparing these $\\rho^{(1)}(x,y,t=0)$ with the $\\rho^{(1)}(x,y,t)$ we can deduce that during evolution the one-body density of the system is mainly in a superposition of the \n$\\ket{\\Psi_{\\nu_1}^f}$ and the $\\ket{\\Psi_{\\nu_0}^f}$. \nThe excited state $\\ket{\\Psi_{\\nu_2}^f}$ has a smaller contribution to the dynamics of $\\rho^{(1)}(x,y,t)$ [e.g. see Fig. \\ref{fig:1RD_-1_1} (e)] compared to the other \nstates.\n\n\n\n\\subsection{Evolution of the radial probability density} \n\n\nIn order to gain a better understanding of the nonequilibrium dynamics of the two bosons, we also employ the time-evolution of the radial probability density of \nthe relative wavefunction $\\mathcal{B}(\\rho,t)$ [Eq. \\eqref{prob_dens}]. \nRecall that this quantity provides the probability density of finding the two bosons at a distance $\\rho$ apart for a fixed time-instant. \nThe dynamics of $\\mathcal{B}(\\rho,t)$ after a quench from $g^{\\textrm{in}}=-1$ to $g^f=1$, starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, is \nillustrated at selected time-instants in Fig. \\ref{fig:2RD_attractive} (a). \nWe can infer that the emergent breathing motion of the two bosons is identified via the succession in time of a single [e.g. at $t=0.46 ,1.31$] and a double peak [e.g. at $t=0.84,2.63$] \nstructure in the dynamics of $\\mathcal{B}(\\rho,t)$. \nHere, the one peak is located close to $\\rho=0$ and the other close to the harmonic oscillator length (unity in our choice of units). \nMoreover, by comparing $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_attractive} (a)] with $\\rho^{(1)}(x,y,t)$ [Fig. \\ref{fig:1RD_-1_1}] suggests that a double peak \nstructure in $\\mathcal{B}(\\rho,t)$ refers to an expansion of $\\rho^{(1)}(x,y,t)$ [e.g. at $t=6.09$], while a single peaked $\\mathcal{B}(\\rho,t)$ corresponds to a contraction of $\\rho^{(1)}(x,y,t)$ [e.g. at $t=1.31$].  Indeed, for a double peak structure of $\\mathcal{B}(\\rho,t)$, its secondary maximum always occurs at slightly larger radii than the maximum of a single peak distribution of $\\mathcal{B}(\\rho,t)$, possessing also a more extended tail. This further testifies the expanding (contracting) tendency of the cloud in the former (latter) case.\nTo reveal the microscopic origin of the structures building upon $\\mathcal{B}(\\rho,t)$ we also calculate this quantity [see the inset of Fig. \\ref{fig:2RD_attractive} (a)] \nfor the states $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_0}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$ that primarily contribute to the dynamics in terms \nof the overlap coefficients [see also Fig. \\ref{fig:attractive_spec} (b)]. \nIndeed, comparing $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_attractive} (a)] with $\\mathcal{B}(\\rho)$ of the stationary eigenstates \n[inset of Fig. \\ref{fig:2RD_attractive} (a)], enables us to deduce that $\\mathcal{B}(\\rho,t)$ resides mainly in a superposition of the \nground ($\\ket{\\Psi_{\\nu_1}^f}$), the bound ($\\ket{\\Psi_{\\nu_0}^f}$) and the first excited ($\\ket{\\Psi_{\\nu_2}^f}$) eigenstates. \nAlso, it can be clearly seen that the main contribution stems from the ground state, while the other two states possess a smaller contribution. \nIn particular, the participation of the bound state can be inferred due to the existence of the peak close to $\\rho=0$, which e.g. for $t=0.84$ becomes \nprominent, whereas the presence of the excited state $\\ket{\\Psi_{\\nu_2}^f}$ is discernible from the spatial extent of the $\\mathcal{B}(\\rho,t)$ e.g. \nat $t=2.63$ [Fig. \\ref{fig:2RD_attractive} (a)]. \n\n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure6-eps-converted-to.pdf} \n\\caption{(a) Time-evolution of the radial probability density, $\\mathcal{B}(\\rho,t)$, of the two atoms at selected time-instants (see legend) for an interaction quench \nfrom $g^{\\textrm{in}}=-1$ to $g^f=1$ starting from the ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset illustrates $\\mathcal{B}(\\rho)$ of the initial state and different postquench eigenstates (see legend). \n(b) Temporal evolution of the corresponding radial probability density in momentum space, $\\mathcal{C}(k,t)$ at specific time-instants (see legend). \nThe inset depicts $\\mathcal{C}(k)$ of the initial state and various postquench eigenstates (see legend).}\n\\label{fig:2RD_attractive}\n\\end{figure} \n\nTo showcase the motion of the two atoms in momentum space we invoke the evolution of the radial probability density in momentum space $\\mathcal{C}(k,t)$ \\cite{Selim_momentum} illustrated in Fig. \\ref{fig:2RD_attractive} (b) \nfor the quench $g^{\\textrm{in}}=-1\\rightarrow g^f=1$ starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nWe observe that in the course of the dynamics a pronounced peak close to $k=0$ and a secondary one located at values of larger $k$ appear in $\\mathcal{C}(k,t)$. \nMoreover, the breathing motion in momentum space is manifested by the lowering and raising of the zero momentum peak accompanied by a subsequent \nenhancement or reduction of the tail of $\\mathcal{C}(k,t)$, as shown e.g. at $t=0.84, 6.09$. \nNote also that the tail of $\\mathcal{C}(k,t)$ decays in a much slower manner compared to the tail of $\\mathcal{B}(\\rho,t)$. \nIndeed, the latter decays asymptotically as $\\sim e^{-\\rho^2}$ [see also Eq. (\\ref{station})] while by fitting the tail of $\\mathcal{C}(k,t)$ \nwe observe a decay law $\\sim 1\/k^3$ (not shown here for brevity reasons) \\cite{Bellotti,Valiente,momentum_1,momentum_2}. \nAdditionally, in order to unveil the corresponding superposition of states that contribute to the momentum distribution, the inset of Fig. \\ref{fig:2RD_attractive} (b) \npresents $\\mathcal{C}(k)$ of the postquench eigenstates that possess the most significantly populated overlap coefficients [see also Fig. \\ref{fig:attractive_spec} (b)]. \nAs it can be seen, the bound state ($\\ket{\\Psi_{\\nu_0}^f}$) exhibits a broad momentum distribution with a tail that extends to large values of $k$, \nwhile $\\mathcal{C}(k)$ of the ground state ($\\ket{\\Psi_{\\nu_1}^f}$) contributes the most and has a main peak around $k=0$. \nOn the other hand, the excited state ($\\ket{\\Psi_{\\nu_2}^f}$) contributes to a lesser extent, and its presence is mainly identified when the momentum \ndistribution exhibits two nodes, e.g. at $t=2.63$.\n\n\\subsection{Evolution of the contact}\n\n\\begin{figure}[t!]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure7-eps-converted-to.pdf}\n\t\\caption{(a) Temporal evolution of the normalized contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ upon considering an interaction quench from $g^{\\textrm{in}}=-1$ \n\tto $g^f=1$. \n\t(b) The corresponding frequency spectrum.}\n\t\\label{fig:Contact_attractive}\n\\end{figure} \n\nSubsequently we examine the contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ in the course of the evolution after a quench from $g^{\\textrm{in}}=-1$ to \n$g^f=1$, see Fig. \\ref{fig:Contact_attractive} (a). \nRecall that the contact reveals the existence of short-range two-body correlations. \nEvidently $\\mathcal{D}(t)\/\\mathcal{D}(0)$ exhibits an irrregular oscillatory behavior containing a variety of different frequencies. \nIndeed, by inspecting the corresponding frequency spectrum depicted in Fig. \\ref{fig:Contact_attractive} (b), a multitude of frequencies appear. \nThe most predominant frequencies possessing the largest amplitude originate from the energy difference between the bound state, $\\ket{\\Psi_{\\nu_0}}$ and \nenergetically higher-lying states, such as  $\\omega_{\\nu_1,\\nu_0}, \\omega_{\\nu_2,\\nu_0}$ and $\\omega_{\\nu_3,\\nu_0}$. \nAlso here $\\omega_{\\nu_2,\\nu_1}$ has a comparable value to $\\omega_{\\nu_3,\\nu_0}$ and thus contributes non-negligibly to the dynamics of \n$\\mathcal{D}(t)\/\\mathcal{D}(0)$. \nMoreover, there is a multitude of other contributing frequencies e.g. $\\omega_{\\nu_8,\\nu_0}$ having an amplitude smaller than $\\omega_{\\nu_3,\\nu_0}$. \nThese frequencies indicate the presence of higher-lying states in the dynamics of the contact. \nThe above-described behavior of $\\mathcal{D}(t)\/\\mathcal{D}(0)$ is expected to occur since the contact is related to short-range \ntwo-body correlations, and as such its dynamics \ninvolves a large number of postquench eigenstates, giving rise to the frequencies observed in Fig. \\ref{fig:Contact_attractive} (b). \n\n\n\\section{Quench dynamics of two repulsive bosons to attractive interactions} \\label{quench_rep} \n\nAs a next step, we shall investigate the interaction quench dynamics of two initially repulsive bosons towards the attractive \nside of interactions. \nIn particular, throughout this section we initialize the system in its ground state configuration $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ \nat $g^{\\textrm{in}}=1$ (unless it is stated otherwise) and perform an interaction quench to the attractive side of the spectrum. \n       \n\\subsection{Dynamical response}\n      \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure8-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of two bosons after an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to different \nfinal interaction strengths $g^f$. \n(b) Time-evolution of the fidelity for selected postquench interaction strengths (see legend).}\n\\label{fig:repulsive_ground} \n\\end{figure} \n      \nIn order to study the dynamical response of the system, we invoke the fidelity evolution [Eq. \\eqref{f(t)}] \\cite{Fogarty2} shown in Fig. \\ref{fig:repulsive_ground} (a) \nwith respect to $g^f$. \nWe observe the appearance of three different dynamical regions, in a similar fashion with the response of the reverse quench scenario \ndiscussed in Section \\ref{response_attract_repul}. \nWithin region I, $0.35<g^f<1$, $\\abs{F(t)}$ undergoes small amplitude oscillations [see also Fig. \\ref{fig:repulsive_ground} (b)] and the system remains close to \nits initial state. \nHowever, in region II characterized by $-2.36<g^f<0.35$ the system becomes significantly perturbed since overall $\\abs{F(t)}$ oscillates between unity and zero. \nFor instance, see $\\abs{F(t)}$ in Fig. \\ref{fig:repulsive_ground} (b) at $g^f=-0.2$ where e.g. at $t\\simeq\\pi\/2,3\\pi\/2$ $|F(t)|\\simeq 0.07$. \nRegion III where $-10<g^f<-2.36$ incorporates the intermediate and strongly attractive regime of interactions. \nHere, $\\abs{F(t)}$ oscillates with a small amplitude, while its main difference  compared to region I is that the oscillation period is larger. \nAnother interesting feature of $\\abs{F(t)}$ is that as we enter deeper into region III the oscillation amplitude decreases and the corresponding period \nbecomes smaller (see also the discussion below). \n      \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure9-eps-converted-to.pdf} \n\\caption{(a) Frequency spectrum of the fidelity, $F(\\omega)$, when performing an interaction quench from $g^{\\textrm{in}}=1$ to various final \ninteraction strengths. \n(b) The corresponding most significantly contributing overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$. \nThe black dashed vertical line at $g^f=-0.51$ indicates the boundary at which the bound state for negative interactions becomes the \nground state for $g^f>-0.51$, see also Fig. \\ref{fig:en_spectrum}.}\n\\label{fig:repulsive_spec}\n\\end{figure} \n\nTo identify the postquench eigenstates that participate in the nonequilibrium dynamics of the two bosons, we next calculate the fidelity spectrum $F(\\omega)$ \n[Fig. \\ref{fig:repulsive_spec} (a)] as well as the most notably populated overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:repulsive_spec} (b)] for a varying \npostquench interaction strength. \nIn region I we observe the occurrence of a predominant frequency, namely $\\omega_{\\nu_2,\\nu_1}$, in $F(\\omega)$. \nThis frequency is associated with the notable population of the coefficients $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2$ [Fig. \\ref{fig:repulsive_spec} (b)]. \nRecall that the amplitude of the frequency peaks appearing in $F(\\omega)$ depends on the participating overlap coefficients, as it \nis explicitly displayed in Eq. \\eqref{fidelity}. \nEntering region II there is a multitude of contributing frequencies, the most prominent of them being $\\omega_{\\nu_2,\\nu_1}$. \nThe appearance of the different frequencies is related to the fact that in this regime $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2$ drops significantly for more attractive interactions \naccompanied by the population of other states such as $\\ket{\\Psi_{\\nu_2}^f}$ and $\\ket{\\Psi_{\\nu_3}^f}$ [see Fig. \\ref{fig:repulsive_spec} (b)]. \nIt is important to remember here that at the vertical line $g^f=-0.51$ [see also Fig. \\ref{fig:en_spectrum}] there is a change in the labeling of the eigenstates, \nresulting in the alteration of the frequencies from $\\omega_{\\nu_j,\\nu_k}$ to $\\omega_{\\nu_{j-1},\\nu_{k-1}}$ when crossing this line towards the attractive regime. \nIn region III there are essentially two excited frequencies, namely $\\omega_{\\nu_1,\\nu_0}$ and $\\omega_{\\nu_2,\\nu_1}$. \nThe former is the most dominant since here the mainly contributing states are $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_0}^f}$ as it can be seen \nfrom Fig. \\ref{fig:repulsive_spec} (b). \nNote also that $\\omega_{\\nu_1,\\nu_0}$ increases for decreasing $g^f$, a behavior that reflects the increasing energy gap in the system's energy \nspectrum [Fig. \\ref{fig:en_spectrum}]. \nOn the other hand, the amplitude of $\\omega_{\\nu_2,\\nu_1}$ is weaker and essentially fades away for strong attractive interactions. \nThis latter behavior can be attributed to the fact that the contribution of the  $\\ket{\\Psi_{\\nu_2}^f}$ state in this region decreases substantially. \n\n\n\n\n\\subsection{Role of the initial state}\\label{role_intial_state_repul_attract}\n      \n      \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure10-eps-converted-to.pdf}\n\\caption{(a) Fidelity evolution of the two bosons when performing a quench from $g^{\\textrm{in}}=1$ to $g^f=-1$ starting from various \nexcited states (see legend). \nThe fidelity spectrum when the system is initially prepared in (b) $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and (c) $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$.}\n\\label{excited_rep}\n\\end{figure} \n\nIn order to expose the role of the initial state for the two-boson dynamics, we explore interaction quenches from $g^{\\textrm{in}}=1$ towards $g^f=-1$ but \ninitializing the system in various excited states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$, $k>1$, or the bound state $\\ket{\\Psi_{\\nu_0}^{\\textrm{in}}}$. \nThe emergent dynamical response of the system as captured via $\\abs{F(t)}$ is depicted in Fig. \\ref{excited_rep} (a) starting from the bound, the first, the third, the fifth \nand the seventh excited state. \nInspecting the behavior of $\\abs{F(t)}$ we can infer that the system becomes more perturbed when it is prepared in an energetically lower excited state since the oscillation \namplitude of $\\abs{F(t)}$ increases accordingly, compare for instance $\\abs{F(t)}$ for $\\nu_2^{\\textrm{in}}$ and $\\nu_6^{\\textrm{in}}$. \nMoreover, starting from the bound state the system is significantly perturbed compared to the previous cases and $\\abs{F(t)}$ showcases an irregular oscillatory \nbehavior. \nThis pattern is maintained if the quench is performed to other values of $g^f$ which belong to the attractive regime (not shown here for brevity reasons). \nRecall that a similar behavior of $\\abs{F(t)}$ occurs for the reverse quench process, see Sec. \\ref{role_intial_state_attract_repul} and also Fig. \\ref{excited_att} (a).  \n\nThe above-mentioned behavior of the fidelity evolution can be understood via employing the corresponding overlap coefficients $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, see also Eq. (\\ref{fidelity}). \nAs already discussed in Sec. \\ref{role_intial_state_attract_repul}, the fidelity remains close to its initial value in the case that one overlap coefficient dominates with respect \nto the others and deviates significantly from unity when at least two overlap coefficients possess a notable population. \nThe predominantly populated overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, are listed in Table \\ref{Table2} when starting from different initial eigenstates $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nA close inspection of this Table reveals that starting from an energetically higher excited state leads to a lesser amount of contributing overlap coefficients \nwith one among them becoming the dominant one. \nThis behavior explains the decreasing tendency of the oscillation amplitude of $\\abs{F(t)}$ for an initially energetically higher excited state, e.g. compare $\\abs{F(t)}$ of \n$\\ket{\\Psi_{\\nu_2}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_6}^{\\textrm{in}}}$ in Fig. \\ref{excited_rep} (a). \nAccordingly, an initially lower (higher) lying excited state results in a larger (smaller) amount of excitations and thus to more (less) contributing frequencies. The latter can be readily seen by resorting to the fidelity spectrum $|F(\\omega)|$ show in Figs. \\ref{excited_rep} (b) and (c) when starting from $\\ket{\\Psi_{\\nu_4}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_8}^{\\textrm{in}}}$ respectively.\n\n\n    \n\\begin{table}\n\\centering\n\\begin{tabular}{|l|c|c|c|c|c|}\\hline\n&  {$|d_{\\nu_j^f,\\nu_0^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_2^{\\textrm{in}}}|^2$}   & {$|d_{\\nu_j^f,\\nu_4^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_6^{\\textrm{in}}}|^2$} & {$|d_{\\nu_j^f,\\nu_8^{\\textrm{in}}}|^2$} \\\\\n\\hline\n{$\\nu_j^f=\\nu_0$} & 0.7896 & 0.0351 & 0.0092 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_1$} & 0.0729 & 0.0556 & - & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_2$} & 0.0367 & 0.8765  & 0.0092 & - &- \\\\ \\hline\n{$\\nu_j^f=\\nu_3$} & 0.0221 & 0.0198 & 0.0399 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_4$} & 0.0147 & - & 0.9078 & - & - \\\\ \\hline\n{$\\nu_j^f=\\nu_5$} & - & -& 0.0175 & 0.0315 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_6$} & - &- & - & 0.9248 & - \\\\ \\hline\n{$\\nu_j^f=\\nu_7$} & - &- &- & 0.0154 & 0.0262 \\\\ \\hline\n{$\\nu_j^f=\\nu_8$} & -& -& -& - & 0.9357 \\\\ \\hline\n{$\\nu_j^f=\\nu_9$} & - & -& -&- &  0.0138 \\\\ \\hline\n\\end{tabular}\n\\caption{The most significantly populated overlap coefficients, $|d_{\\nu_j^f,\\nu_k^{\\textrm{in}}}|^2$, for the quench from $g^{\\textrm{in}}=1$ to $g^f=-1$ initializing the system at various \ninitial states. \nOnly the coefficients with a value larger than 0.9\\% are shown.}\\label{Table2}\n\\end{table}\n      \t      \n\\begin{figure}[t!]\n\\centering\n\\includegraphics[width=0.5 \\textwidth]{figure11-eps-converted-to.pdf}\n\\caption{(a)-(f) Snapshots of the one-body density evolution following an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \n(g)-(j) The corresponding one-body densities for different stationary eigenstates (see legend), that possess the largest  overlap coefficients.}\n\\label{fig:1RD_1_-0.2}\n\\end{figure}\n      \n      \n\\subsection{One-body density evolution} \n\n\nTo visualize the nonequilibrium dynamics of the two-bosons, we next monitor the time-evolution of the one-body density [Eq. \\eqref{density_matrix}] depicted \nin Figs. \\ref{fig:1RD_1_-0.2} (a)-(f) for a quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \nNote that the time-instants portrayed in Fig. \\ref{fig:1RD_1_-0.2} refer to roughly the minima and maxima of the respective fidelity evolution [see Fig. \\ref{fig:repulsive_ground} (b)]. \nOverall, the atomic cloud performs a breathing motion during evolution, namely it expands and contracts in a periodic manner. \nMoreover, we deduce that when the fidelity is minimized [e.g. at $t=1.5,4.53,7.54$], the one-body density expands [Figs. \\ref{fig:1RD_1_-0.2} (a), (c) and (e)], while for the case of a maximum fidelity \n$\\rho^{(1)}(x,y,t)$ contracts [Figs. \\ref{fig:1RD_1_-0.2} (b), (f)]. \nTo understand which states are imprinted in $\\rho^{(1)}(x,y,t)$ we further show in Figs. \\ref{fig:1RD_1_-0.2} (g)-(j) $\\rho^{(1)}(x,y,t=0)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ \nand the three most significantly populated, according to the overlap coefficients $|d_{\\nu_j^f,\\nu_1^{\\textrm{in}}}|^2$, final states i.e. $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$ and \n$\\ket{\\Psi_{\\nu_3}^f}$ \\cite{Katsimiga_bent,Katsimiga_quantum_DBs}. \nComparing the $\\rho^{(1)}(x,y,t=0)$ of these stationary states with $\\rho^{(1)}(x,y,t)$ it becomes evident that during evolution $\\rho^{(1)}(x,y;t)$ is mainly in a superposition of the ground \nstate [Fig. \\ref{fig:1RD_1_-0.2} (i)] and the first excited state [Fig. \\ref{fig:1RD_1_-0.2} (h)]. \n      \n      \n      \n\\subsection{Evolution of the radial probability density} \n\nAs a next step, we examine the evolution of the radial probability density $\\mathcal{B}(\\rho,t)$ [Eq. \\eqref{prob_dens}] presented \nin Fig. \\ref{fig:2RD_repulsive} (a) for a quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $g^{\\textrm{in}}=1$ to $g^f=-0.2$. \nNote that the snapshots of $\\mathcal{B}(\\rho,t)$ depicted in Fig. \\ref{fig:2RD_repulsive} (a) correspond again to time-instants at which the \nfidelity evolution exhibits local minima and maxima [see also Fig. \\ref{fig:repulsive_ground} (b)]. \nWe observe that when $\\abs{F(t)}$ is minimized, e.g. at $t=1.50,4.00,7.74$, $\\mathcal{B}(\\rho,t)$ shows a double peak structure around \n$\\rho \\approx0.5$ and $\\rho \\approx 2$ respectively. \nHowever, for times that correspond to a maximum of the fidelity, e.g. at $t=3.1,6.17$, $\\mathcal{B}(\\rho,t)$ deforms to a single peak \ndistribution around $\\rho \\approx1.2$. \nTo relate this alternating behavior of $\\mathcal{B}(\\rho,t)$ with the breathing motion of the two bosons we can infer that when \n$\\mathcal{B}(\\rho,t)$ possesses a double peak distribution the cloud expands while in the case of a single peak \nstructure it contracts, see also Fig. \\ref{fig:1RD_1_-0.2}. \nIt is also worth mentioning here that for the times at which $\\mathcal{B}(\\rho,t)$ exhibits a double peak structure there \nis a quite significant probability density tail for $\\rho>1.5$. \nThis latter behavior is a signature of the participation of energetically higher-lying excited states as we shall discuss below. \n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure12-eps-converted-to.pdf}\n\\caption{(a) Temporal evolution of the radial probability density, $\\mathcal{B}(\\rho,t)$, upon considering a quench from $g^{\\textrm{in}}=1$ to $g^f=-0.2$ \nstarting from the ground state, $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset shows $\\mathcal{B}(\\rho)$ of the prequench state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of the postquench eigenstates $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$ \nwith the most relevant overlap coefficients. \n(b) The corresponding $\\mathcal{C}(k,t)$ of (a). \nThe inset presents $\\mathcal{C}(k)$ of the $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of the $\\ket{\\Psi_{\\nu_1}^f}$, $\\ket{\\Psi_{\\nu_2}^f}$. }\n\\label{fig:2RD_repulsive}\n\\end{figure} \n\nIndeed, the inset of Fig. \\ref{fig:2RD_repulsive} (a) depicts $\\mathcal{B}(\\rho)$ of the initial ($\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$) and the postquench \n($\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$) states that have the major contribution for this specific quench in terms of the overlap coefficients [see also Fig. \\ref{fig:repulsive_spec} (b)]. \nComparing $\\mathcal{B}(\\rho,t)$ with $\\mathcal{B}(\\rho)$ we can deduce that mainly the ground, $\\ket{\\Psi_{\\nu_1}^f}$, and the first excited, $\\ket{\\Psi_{\\nu_2}^f}$, states \nof the postquench system are imprinted in the dynamics of the relative density. \nMore specifically, $\\ket{\\Psi_{\\nu_2}^f}$ gives rise to the enhanced tail of $\\mathcal{B}(\\rho,t)$ [Fig. \\ref{fig:2RD_repulsive} (a)], while the participation of $\\ket{\\Psi_{\\nu_1}^f}$ \n(possessing also the major contribution) leads to the central peak of $\\mathcal{B}(\\rho,t)$ close to $\\rho=0$. \n\nThe radial probability density in momentum space \\cite{Selim_momentum}, $\\mathcal{C}(k,t)$, is shown in Fig. \\ref{fig:2RD_repulsive} (b) for selected time instants \nof the evolution following the quench $g^{\\textrm{in}}=1\\rightarrow g^f=-0.2$. \nWe observe that $\\mathcal{C}(k,t)$ exhibits always a two peak structure with the location and amplitude of the emergent peaks being changed \nin the course of the evolution. \nIn particular, when the atomic cloud contracts e.g. at $t=3.10,9.19$, see also Figs. \\ref{fig:1RD_1_-0.2} (b), (f), $\\mathcal{C}(k,t)$ has a large amplitude peak around $k\\approx0.1$ \nand a secondary one of small amplitude close to $k\\approx0.4$. \nHowever, for an expansion of the two bosons e.g. at $t=1.50$ [Figs. \\ref{fig:1RD_1_-0.2} (a)] the radial probability density in momentum space shows a small \nand a large amplitude peak around $k\\approx0.05$ and $k\\approx0.3$ respectively.  \nMoreover, the momentum distribution during evolution is mainly in a superposition of the ground $\\ket{\\Psi_{\\nu_1}^f}$ and the first excited state $\\ket{\\Psi_{\\nu_2}^f}$, see \nin particular the inset of Fig. \\ref{fig:2RD_repulsive} (b) which illustrates $\\mathcal{C}(k)$ of these stationary states. \nAs it can be readily seen, $\\ket{\\Psi_{\\nu_2}^f}$ is responsible for the secondary peak of $\\mathcal{C}(k,t)$ at higher momenta, while the ground state contributes \nmainly to the peak close to $k=0$.\n\n\\subsection{Dynamics of the contact}\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure13-eps-converted-to.pdf}\n\t\\caption{(a) Time-evolution of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ following a quench from $g^{\\textrm{in}}=1$ to $g^f=-1$. \n\t(b) The corresponding frequency spectrum.}\n\t\\label{fig:Contact_repulsive}\n\\end{figure}\n\nTo unravel the emergence of short-range two-body correlations we next track the time-evolution of the rescaled contact \n$\\mathcal{D}(t)\/\\mathcal{D}(0)$ after an interaction quench from $g^{\\textrm{in}}=1$ to $g^f=-1$, see Fig. \\ref{fig:Contact_repulsive} (a). \nAs it can be seen, the rescaled contact exhibits an irregular multifrequency oscillatory pattern in time. \nIt is also worth mentioning that here the involved frequencies in the dynamics of $\\mathcal{D}(t)\/\\mathcal{D}(0)$ are smaller \nwhen compared to the ones excited in the reverse quench scenario, see in particular Fig. \\ref{fig:Contact_repulsive} (b) and \nFig. \\ref{fig:Contact_attractive} (b). \nBy inspecting the corresponding frequency spectrum presented in Fig. \\ref{fig:Contact_repulsive} (b), we can deduce that the most prominent \nfrequency $\\omega_{\\nu_1,\\nu_0}\\approx2.5$ corresponds to the energy difference between the bound and the ground state. \nMoreover this predominant frequency is smaller than the corresponding dominant frequency $\\omega_{\\nu_1,\\nu_0}\\approx7.5$ occuring at the reverse \nquench process [Fig. \\ref{fig:Contact_attractive} (b)]. \nThere is also a variety of other contributing frequencies which signal the participation of higher-lying states in the evolution of the contact, \nsuch as $\\omega_{\\nu_7,\\nu_0}$, $\\omega_{\\nu_2,\\nu_1}$, $\\omega_{\\nu_3,\\nu_1}$ and $\\omega_{\\nu_2,\\nu_0}$, \nexhibiting however a much smaller amplitude as compared to $\\omega_{\\nu_1,\\nu_0}$. \nThese frequencies are essentially responsible for the observed irregular motion of $\\mathcal{D}(t)\/\\mathcal{D}(0)$. \n\n\n\n\\section{Quench from zero to Infinite interactions} \\label{inf_quench} \n\nUp to now we have discussed in detail the interaction quench dynamics of two bosons trapped in a 2D harmonic trap for weak, intermediate and strong coupling in both the \nattractive and the repulsive regime. \nNext, we aim at briefly analyzing the corresponding interaction quench dynamics from $g^{\\textrm{in}}=0$ to $g^f=\\infty$. \nWe remark here that when the system is initialized at $g^{\\textrm{in}}=0$ the formula of Eq. (\\ref{overlap}) is no longer valid and the overlap coefficients between the \neigenstates $\\ket{\\Psi_{\\nu_i}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_j}^{f}}$ are given by \n\\begin{eqnarray}\n  d_{\\nu_j^f,\\nu_i^{\\textrm{in}}}&=&\\frac{2\\Gamma(-\\nu_j^f)}{\\sqrt{\\psi^{(1)}(-\\nu_j^f)}} \\int_0^{\\infty} dr \\,r e^{-r^2} U(-\\nu_j^f,1,r^2) L_{\\nu_i^{\\textrm{in}}} (r^2) \\nonumber  \\\\ &= &\\frac{1}{(\\nu_i^{\\textrm{in}}-\\nu_j^f)\\sqrt{\\psi^{(1)}(-\\nu_j^f)}}.\n\\end{eqnarray}\n\n      \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure14-eps-converted-to.pdf}\n\\caption{Fidelity evolution when applying an interaction quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$. \nThe system is initialized in different eigenstates (see legend).}\n\\label{inf_fid}\n\\end{figure} \n\nThe dynamical response of the system after such a quench [$g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$] as captured by the fidelity evolution [Eq. \\eqref{f(t)}] is illustrated \nin Fig. \\ref{inf_fid} when considering different initial states $\\ket{\\Psi_{\\nu_k}^{\\textrm{in}}}$. \nEvidently, when the system is initialized in its ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$, $\\abs{F(t)}$ performs large amplitude oscillations. \nThe latter implies that the time-evolved wavefunction becomes almost orthogonal to the initial one at certain time intervals and as a consequence the system \nis significantly perturbed. \nAlso, it can directly be deduced by the fidelity evolution that when the system is prepared in an energetically higher excited state it is less perturbed since the \noscillation amplitude of $\\abs{F(t)}$ is smaller, e.g. compare $\\abs{F(t)}$ for $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_5}^{\\textrm{in}}}$. \nThis tendency which has already been discussed in Secs. \\ref{role_intial_state_attract_repul} and  \\ref{role_intial_state_repul_attract} can be explained \nin terms of the distribution of the amplitude of the overlap coefficients, see also Eq. (\\ref{fidelity}). \nIndeed, if there is a single dominant overlap coefficient then $|F(t)|\\approx1$, while if more than one overlap coefficients possess large values $\\abs{F(t)}$ deviates \nappreciably from unity. \nHere, for instance, the first two most dominant overlap coefficients when starting from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and $\\ket{\\Psi_{\\nu_5}^{\\textrm{in}}}$ \nare $|d_{\\nu_0^f,\\nu_1^{\\textrm{in}}}|^2=0.4837$, $|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2=0.4402$ and $|d_{\\nu_4^f,\\nu_5^{\\textrm{in}}}|^2=0.6453$, $|d_{\\nu_5^f,\\nu_5^{\\textrm{in}}}|^2=0.1894$ respectively. \n      \n\\begin{figure}\n\\centering\n\\includegraphics[width=0.47 \\textwidth]{figure15-eps-converted-to.pdf}\n\\caption{(a) Radial probability, $\\mathcal{B}(\\rho,t)$, at specific time-instants of the evolution following an interaction quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$. \nThe system is prepared in its ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nThe inset illustrates $\\mathcal{B}(\\rho)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and some of the postquench eigenstates $\\ket{\\Psi_{\\nu_0}^f}$, $\\ket{\\Psi_{\\nu_1}^f}$ \nand $\\ket{\\Psi_{\\nu_2}^f}$. \n(b) Time-evolution of the corresponding radial probability density in momentum space, $\\mathcal{C}(k,t)$. \nThe inset shows $\\mathcal{C}(k)$ of the initial state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ and of certain postquench eigenstates, namely $\\ket{\\Psi_{\\nu_0}^f}$, \n$\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$.}\n\\label{inf_relwave}\n\\end{figure} \n\nTo further unravel the motion of the two bosons we next employ the time-evolution of their radial probability density, $\\mathcal{B}(\\rho,t)$, in real space [see also Eq. \\eqref{prob_dens}]. \nFigure \\ref{inf_relwave} (a) shows snapshots of $\\mathcal{B}(\\rho,t)$ after an interaction quench from $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$ at $g^{\\textrm{in}}=0$ to\n$g^f=\\infty$. \nAs it can be seen for the time intervals that $\\abs{F(t)}$ is minimized [Fig. \\ref{inf_fid}], e.g. at $t=0.78,2.42,5.61$, $\\mathcal{B}(\\rho,t)$ exhibits a pronounced peak close to $\\rho=0$ \nand a secondary one at a larger radii $\\rho\\approx 1.5$. \nHowever, when $|F(t)|\\approx1$ ($t=1.62, 3.13, 8.04$) $\\mathcal{B}(\\rho,t)$ shows a more delocalized distribution. \nTo explain this behavior of $\\mathcal{B}(\\rho,t)$ we next calculate $\\mathcal{B}(\\rho)$ of the initial state (i.e. $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$) and of the postquench eigenstates \nthat possess the most dominant overlap coefficients, namely $\\ket{\\Psi_{\\nu_0}^f}$, $\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$, following the above-described \nquench scenario [see the inset of Fig. \\ref{inf_relwave} (a)]. \nComparing $\\mathcal{B}(\\rho,t)$ with $\\mathcal{B}(\\rho)$ we observe that the bound state, $\\ket{\\Psi_{\\nu_0}^f}$, gives rise to the prominent peak close \nto $\\rho=0$ [see Fig. \\ref{inf_relwave} (a)]. \nMoreover, the states $\\ket{\\Psi_{\\nu_1}^f}$ and $\\ket{\\Psi_{\\nu_2}^f}$ are responsible for the emergent spatial delocalization of $\\mathcal{B}(\\rho,t)$. \nOf course, the ground state ($\\ket{\\Psi_{\\nu_1}^f}$) plays a more important role here than the first excited state ($\\ket{\\Psi_{\\nu_2}^f}$), since \n$|d_{\\nu_1^f,\\nu_1^{\\textrm{in}}}|^2=0.4402$ and $|d_{\\nu_2^f,\\nu_1^{\\textrm{in}}}|^2=0.0406$ respectively [see the inset of Fig. \\ref{inf_relwave} (a)]. \n\nTurning to the dynamics in momentum space, Fig. \\ref{inf_relwave} (b) presents $\\mathcal{C}(k,t)$ at specific time-instants for the quench $g^{\\textrm{in}}=0\\rightarrow g^f=\\infty$ \nstarting from the ground state $\\ket{\\Psi_{\\nu_1}^{\\textrm{in}}}$. \nWe observe that when the system deviates notably from its initial state (i.e. $t=0.78,2.42,5.61$) meaning also that $|F(t)|\\ll1$, then $\\mathcal{C}(k,t)$ shows a two peak structure \nwith the first peak located close to $k=0$ and the second one at $k\\approx0.4$. \nNotice also here that the tail of $\\mathcal{C}(k,t)$ has an oscillatory behavior. \nOn the other hand, if $\\abs{F(t)}$ is close to unity (e.g. at $t=1.62, 3.13, 8.04$) where also $\\mathcal{B}(\\rho,t)$ is spread out [Fig. \\ref{inf_relwave} (a)], \nthe corresponding $\\mathcal{C}(k,t)$ has a narrow momentum peak close to zero and a fastly decaying tail at large $k$. \n\nThe inset of Fig. \\ref{inf_relwave} (b) illustrates $\\mathcal{C}(k)$ of the initial eigenstate and some specific postquench ones which possess the \nlargest contributions for the considered quench according to the overlap coefficients. \nIt becomes evident that both the bound state, $\\ket{\\Psi_{\\nu_0}^f}$, and the ground state, $\\ket{\\Psi_{\\nu_1}^f}$, of the postquench system are mainly \nimprinted in $\\mathcal{C}(k,t)$. \nIndeed, the bound state has a broad momentum distribution whereas the ground state possesses a main peak close to $k=0$. \nOn the other hand, the first excited state ($\\ket{\\Psi_{\\nu_2}^f}$) has a smaller contribution compared to the previous ones and its presence can be discerned in Fig. \\ref{inf_relwave} (b) \nfrom the oscillatory tails of $\\mathcal{C}(k,t)$ at large momenta. \n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=0.47 \\textwidth]{figure16-eps-converted-to.pdf}\n\t\\caption{(a) Time-evolution of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ for the interaction quench from $g^{\\textrm{in}}=0.2$ to $g^f=\\infty$. \n\t(b) The respective frequency spectrum $D(\\omega)$.}\n\t\\label{fig:Contact_inf}\n\\end{figure}\n\nFinally, we examine the dynamics of the rescaled contact $\\mathcal{D}(t)\/\\mathcal{D}(0)$ illustrated in Fig. \\ref{fig:Contact_inf} (a) \nfollowing a quench from $g^{\\textrm{in}}=0.2$ to $g^f=\\infty$. \nNote here that we choose $g^{\\textrm{in}}=0.2$, and not exactly $g^{\\textrm{in}}=0$, since the contact is well-defined only for interacting eigenstates \\cite{Tan1}. \nEvidently $\\mathcal{D}(t)\/\\mathcal{D}(0)$ undergoes a large amplitude multifrequency oscillatory motion. \nThe large amplitude of these oscillations stems from the fact that the system is quenched to unitarity and therefore the built up of short-range two-body \ncorrelations is substantial especially when compared to the correlations occuring for finite interactions as e.g. the ones displayed in \nFig. \\ref{fig:Contact_attractive} (a) and Fig. \\ref{fig:Contact_repulsive} (a). \nWe remark that similar large amplitude oscillations of the contact, at the frequency of the two-body bound state, have already been observed in Ref. \\cite{Corson} \nduring the interaction quench dynamics of a three dimensional homogeneous BEC from zero to very large interactions.\nRegarding the participating frequencies identified in the spectrum of the contact shown in Fig. \\ref{fig:Contact_inf} (b), we can clearly infer \nthat the dominant frequencies refer to the energy differences between the bound state, $\\ket{\\Psi_{\\nu_0}}$ and higher-lying states \ne.g. $\\omega_{\\nu_1,\\nu_0}$, $\\omega_{\\nu_2,\\nu_0}$. \nThe existence of other contributing frequencies in the spectrum, such as $\\omega_{\\nu_2,\\nu_1}$ and $\\omega_{\\nu_3,\\nu_0}$, has also an impact on the \ndynamics of the contact and signal the involvement of higher-lying states.\n\n\n\n\\section{Conclusions}\\label{conclusions} \n\n\nWe have explored the quantum dynamics of two bosons trapped in an isotropic two-dimensional harmonic trap, and interacting via a contact \n$s$-wave pseudo-potential. \nAs a first step, we have presented the analytical solution of the interacting two-body wavefunction for an arbitrary stationary eigenstate. \nWe also briefly discuss the corresponding two-body energy eigenspectrum covering both the attractive and \nrepulsive interaction regimes, showcasing the importance of the existing bound state. \n\nTo trigger the dynamics we consider an interaction quench from repulsive to attractive interactions and vice versa as well as a quench from zero to \ninfinite interactions. \nHaving the knowledge of the stationary properties of the system the form of the time-evolving two-body wavefunction is provided. \nMost importantly, we showcase that the expansion coefficients can be derived in a closed form and therefore the dynamics of the two-body wavefunction can be \nobtained by numerically determining its expansion with respect to the eigenstates of the postquench system. \nIn all cases, the dynamical response of the system has been analyzed in detail and the underlying eigenstate transitions that mainly contribute to the dynamics have \nbeen identified in the fidelity spectrum together with the system's eigenspectrum. \n\nWe have shown that initializing the system in its ground state, characterized by either repulsive or attractive interactions, it is driven more efficiently out-of-equilibrium, \nas captured by the fidelity evolution, when performing an interaction quench towards the vicinity of zero interactions. \nHowever, if we follow a quench towards the intermediate or strong coupling regimes of either sign, then the system remains close to its initial state. \nAs a consequence of the interaction quench the two bosons undergo a breathing motion which has been visualized by monitoring the temporal evolution \nof the single-particle density and the radial probability density, in both real and momentum space. \nThe characteristic structures building upon the above-mentioned quantities enable us also to infer about the participation of energetically higher-lying \nexcited states of the postquench system. \n\nTo inspect the dependence of the system's dynamical response we have examined also quenches for a variety of different initial states such as the bound state \nor an energetically higher excited state in both the repulsive and attractive interaction regimes. \nIt has been found that starting from energetically higher excited states, the system is perturbed to a lesser extent, and a fewer amount of postquench eigenstates \ncontribute in the emergent dynamics. \nA crucial role here is played by the bound state of the postquench system, both in the attractive and the repulsive regime, whose contribution is essentially \ndiminished as we initialize the two bosons at higher excited states. \nOn the other hand, when the quench is performed from the bound state, independently of the interaction strength, the system is driven out-of-equilibrium \nin the most efficient manner than any other initial state configuration.  \n\nAdditionally, upon quenching the system from zero to infinite interactions starting from its ground state the time-evolved wavefunction becomes even orthogonal to the initial \none at certain time intervals. \nAgain here, if the two bosons are prepared in an energetically higher excited state then the system becomes more unperturbed. \nInspecting the evolution of the radial probability density we have identified that it mainly resides in a superposition of the bound and the ground state alternating from a \ntwo peaked structure to a more spread distribution.\n\nTo unveil the emergence of short-range two-body correlations we have examined the dynamics of the Tan's contact in all of the above-mentioned quench scenaria. \nIn particular, we have found that the contact performs a multifrequency oscillatory motion in time. \nThe predominant frequency of these oscillations refers to the energy difference between the bound and the ground states.  \nThe participation of other frequencies possessing a comparable smaller amplitude signals the contribution of higher-lying states in the dynamics of the contact. \nMoreover, upon quenching the system from weak to infinite interactions, the oscillation amplitude of the contact is substantially enhanced indicating the significant development \nof short-range two-body correlations as compared to the correlations occuring at finite postquench interactions. \n\nThere is a variety of fruitful directions to follow in future works. \nAn interesting one would be to consider two bosons confined in an anisotropic two-dimensional harmonic trap and examine the stationary properties of this system \nin the dimensional crossover from two- to one-dimensions. \nHaving at hand such an analytical solution would allow us to study the corresponding dynamics of the system upon changing its dimensionality e.g. by considering \na quench of the trap frequency in one of the spatial directions which enable us to excite higher than the monopole mode. \nAlso one could utilize the spectra with respect to the different anisotropy in order to achieve controllable state transfer processes \\cite{Fogarty,Reshodko}. \nBesides the dimensionality crossover, it would be interesting to study the effect of the presence of the temperature in the interaction quench dynamics examined herein. \nFinally, the dynamics of three two-dimensional trapped bosons requires further investigation. \nEven though the Efimov effect is absent in that case \\cite{Nielsen}, the energy spectrum is rich possessing dimer and trimer states \\cite{Drummond} and the corresponding \ndynamics might reveal intriguing dynamical features when quenching from one to another configuration.  \n\n\n\\begin{acknowledgements} \n\nG. B. kindly acknowledges financial support by the State Graduate Funding Program Scholarships (HmbNFG). S. I. M and P. S gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) in the framework of the SFB \n925 ``Light induced dynamics and control of correlated quantum systems\". The authors thank G.M. Koutentakis for fruitful discussions. \n\n\\end{acknowledgements}\n      \n      \n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Conformal generators on the circle}\n\n\\subsection{Virasoro generators}\n\nA basis of generators of conformal symmetry on the circle $x \\in [0,L)$ is given by the Virasoro generators $L_n$ and $\\bar{L}_{n}$ \\cite{CFT2}, \n\\begin{eqnarray}\nL_n &\\equiv& \\frac{L}{(2\\pi)^2} \\int_{0}^{L} dx~e^{i\\frac{2\\pi}{L}nx } T(x) + \\frac{c}{24}\\delta_{n,0},\\\\\n\\bar{L}_n &\\equiv& \\frac{L}{(2\\pi)^2} \\int_{0}^{L} dx~e^{-i\\frac{2\\pi}{L}nx } \\bar{T}(x) + \\frac{c}{24}\\delta_{n,0},\n\\end{eqnarray}\nwhich are the Fourier modes of the holomorphic and antiholomorphic components of the stress tensor\n\\begin{eqnarray}\nT(x) &\\equiv&  2\\pi\\frac{h(x) + p(x)}{2},\\\\\n\\bar{T}(x) &\\equiv& 2\\pi\\frac{h(x) - p(x)}{2}.\n\\end{eqnarray}\nHere $h(x)$ and $p(x)$ are the CFT hamiltonian and momentum densities, which we can also use to write the CFT hamiltonian and momentum operators on the circle,\n\\begin{eqnarray}\nH^{\\mbox{\\tiny CFT}} &\\equiv& \\int_0^{L} dx~h(x) = \\frac{2\\pi}{L} \\left(L_0 +\\bar{L}_0 - \\frac{c}{12}\\right),\\\\\nP^{\\mbox{\\tiny CFT}} &\\equiv& \\int_0^{L} dx~p(x) =  \\frac{2\\pi}{L} \\left(L_0 -\\bar{L}_0\\right).\n\\end{eqnarray}  \n\n\n\nThe Virasoro generators $L_n, \\bar{L}_n$ can be seen to close two copies of the Virasoro algebra,\n\\begin{eqnarray} \\label{eq:Virasoro}\n\\big[L_n, L_m\\big] &=& (n-m)L_{n+m} + \\frac{c}{12}n(n^2-1)\\delta_{n+m,0},~~~~~ \\\\\n\\big[ L_n, \\bar{L}_m \\big] &=& 0, ~~~\\\\\n\\big[\\bar{L}_n, \\bar{L}_m\\big] &=& (n-m)\\bar{L}_{n+m} + \\frac{c}{12}n(n^2-1)\\delta_{n+m,0},~~~~~\n\\end{eqnarray} \nwhere $c$ is the central charge of the CFT.\n\n\n\n\\subsection{Generators of non-uniform euclidean time evolution}\n\nLet us now consider the Fourier mode expansion of the Hamiltonian density $h(x)$,\n\\begin{eqnarray}\nh(x) &\\equiv&  \\left(\\frac{2\\pi}{L}\\right) \\frac{1}{L} \\sum_n H_n ~e^{-i\\frac{2\\pi}{L}nx},\\\\\nH_n &\\equiv& \\left(\\frac{L}{2\\pi}\\right) \\int_{0}^{L}dx~ e^{i\\frac{2\\pi}{L}nx} ~h(x),\n\\end{eqnarray}\nwhere we use the unusual factor $L\/2\\pi$ so that \n\\begin{equation}\nH_n \\equiv L_{n} + \\bar{L}_{-n} -\\frac{c}{12}\\delta_{n,0}.\n\\end{equation} \nIn particular the Hamiltonian reads $H^{\\mbox{\\tiny CFT}} = (2\\pi\/L) H_0$.\nSimilarly, let $a(x)$ be a real function on the circle $x \\in [0,L)$, with Fourier expansion\n\\begin{eqnarray}\na(x) &\\equiv&   \\sum_n a_n ~e^{i\\frac{2\\pi}{L}nx},\\\\\na_n &\\equiv& \\frac{1}{L} \\int_{0}^{L} dx~ e^{-i\\frac{2\\pi}{L}nx}~a(x).\n\\end{eqnarray}\nThen we have that the generator $Q_0$ of a non-uniform euclidean time evolution with profile function $a(x)$ reads\n\\begin{eqnarray}\nQ_0 &\\equiv& \\int_0^{L} dx~ a(x) h(x) \\\\\n&=& \\frac{2\\pi}{L}\\frac{1}{L}\\int_0^L dx~  \\sum_n a_n ~e^{i\\frac{2\\pi}{L}nx} \\sum_m H_m ~e^{-i\\frac{2\\pi}{L}mx}~~~~\\\\\n&=&  \\frac{2\\pi}{L}~  \\sum_n a_n ~ \\sum_m H_m \\left(\\frac{1}{L}\\int_0^L dx~e^{i\\frac{2\\pi}{L}(n-m)x}\\right)~~~~\\\\\n&=&  \\frac{2\\pi}{L} \\sum_{n} a_nH_n,\n\\end{eqnarray}\nwhere we used that $(1\/L)\\int_0^L dx ~e^{i\\frac{2\\pi}{L}(n-m)x} = \\delta_{m,n}$.\n\n\\subsection{Generators of non-uniform rescaling}\n\nLet us now consider the Fourier mode expansion of the momentum density $p(x)$,\n\\begin{eqnarray}\np(x) &\\equiv&  \\left(\\frac{2\\pi}{L}\\right) \\frac{1}{L} \\sum_n P_n ~e^{-i\\frac{2\\pi}{L}nx},\\\\\nP_n &\\equiv& \\left(\\frac{L}{2\\pi}\\right) \\int_{0}^{L}dx~ e^{i\\frac{2\\pi}{L}nx} ~p(x),\n\\end{eqnarray}\nwhere we again use the unusual factor $L\/2\\pi$ so that \n\\begin{equation}\nP_n \\equiv L_{n} - \\bar{L}_{-n},\n\\end{equation}\nwith the momentum operator reading $P^{\\mbox{\\tiny CFT}} = (2\\pi\/L)P_0$. Let $b(x)$ be a real function on the circle $x \\in [0,L)$, with Fourier expansion\n\\begin{eqnarray}\nb(x) &\\equiv&   \\sum_n b_n ~e^{i\\frac{2\\pi}{L}nx},\\\\\nb_n &\\equiv& \\frac{1}{L} \\int_{0}^{L} dx~e^{-i\\frac{2\\pi}{L}nx}~b(x).\n\\end{eqnarray}\nThen the generator $Q_1$ of a non-uniform rescaling with profile function $b(x)$ reads\n\\begin{eqnarray}\nQ_1 \\equiv \\int_0^{L} dx~ b(x) p(x) =  \\frac{2\\pi}{L} \\sum_{n} b_nP_n.\n\\end{eqnarray}\n\n\\section{Low energy matrix elements}\n \nConsider a CFT on a circle of perimeter $L$ and a conformal transformation $V^{\\mbox{\\tiny CFT}} \\equiv e^{- Q}$, with generator $Q$, acting on its Hilbert space. Our goal is to compute the matrix elements\n\\begin{eqnarray}\nV^{\\mbox{\\tiny CFT}}_{\\alpha \\beta} &\\equiv& \\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} V^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}\\\\\n&=& \\sum_{n=0}^{\\infty} \\frac{(-1)^{n}}{n!}\\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} Q^n \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}},\n\\end{eqnarray}\nbetween simultaneous eigenstates of the hamiltonian and momentum operators $H^{\\mbox{\\tiny CFT}}$ and $P^{\\mbox{\\tiny CFT}}$,\n\\begin{eqnarray}\nH^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} &=& E^{\\mbox{\\tiny CFT}}_{\\alpha} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}, \\\\\nP^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} &=& P^{\\mbox{\\tiny CFT}}_{\\alpha} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}},\n\\end{eqnarray}\nwhere, by the CFT operator-state correspondence \\cite{CFT2}, \n\\begin{eqnarray}\nE^{\\mbox{\\tiny CFT}}_{\\alpha} &=& \\frac{2\\pi}{L}\\left( \\Delta_{\\alpha} -\\frac{c}{12}\\right),\\\\\nP^{\\mbox{\\tiny CFT}}_{\\alpha} &=& \\frac{2\\pi}{L}S_{\\alpha}.\n\\end{eqnarray}\nHere $\\Delta_{\\alpha} \\equiv h_{\\alpha}+\\bar{h}_{\\alpha}$ and $S_{\\alpha}\\equiv h_{\\alpha} -\\bar{h}_{\\alpha}$ are the scaling dimension and conformal spin of the scaling operator $\\phi_{\\alpha}$ corresponding to state $\\ket{\\phi_{\\alpha}}$, with $h_{\\alpha}$ and $\\bar{h}_{\\alpha}$ its holomorphic and antiholomorphic conformal dimensions \\cite{CFT2}. \n\nThe resulting $\\bra{\\phi_\\beta^{\\mbox{\\tiny CFT}}} V^{\\mbox{\\tiny CFT}} \\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}$ for specific choices of profile functions $a(x)$ and $b(x)$ (described below), were compared in Figs. \\ref{fig:time}-\\ref{fig:scale} of the main text with matrix elements $\\bra{\\phi_\\beta} V \\ket{\\phi_{\\alpha}}$ numerically obtained on the spin chain, where $\\ket{\\phi_{\\alpha}}$ and $V$ denote an energy\/momentum eigenstate of the spin chain and a linear map implemented by a tensor network, respectively.\n\nThe generator $Q = Q_0 + iQ_1$ can be expanded in terms of the Fourier modes $H_n = L_n +\\bar{L}_{-n}$ and $P_n=L_n -\\bar{L}_{-n}$ of the hamiltonian and momentum densities $h(x)$ and $p(x)$, namely\n\\begin{equation}\nQ = \\frac{2\\pi}{L}\\sum_{n\\in \\mathbb{Z}} \\left(a_n H_n + ib_n P_n\\right)\n\\end{equation} \nThus, we would like to compute the matrix elements $\\bra{\\phi_\\beta} Q^n \\ket{\\phi_{\\alpha}}$, were $Q^n$ is a sum of products of powers of $H_n$ and $P_n$. To do this computation, we express $Q^n$ as a sum of products of Virasoro generators such as $X \\equiv \\left(L_{n_1} \\dots L_{n_r}\\right) \\left(\\bar{L}_{\\bar{n}_1} \\dots \\bar{L}_{\\bar{n}_s} \\right)$ for some integers $r$ and $s$ and apply the Virasoro algebra. As a first step, we express the states $|\\phi_\\alpha^{\\mbox{\\tiny CFT}}\\rangle$ in terms of Virasoro generators acting on primary states.\n\n\\subsection{Descendant states in terms of primary states}\n\nTo keep the notation relatively simple, given a primary state $\\ket{h,\\bar{h}}$ we consider a descendant state $\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}$ that is obtained by applying a sequence $L_{n_1}L_{n_2} \\cdots L_{n_q}$ of holomorphic Virasoro generators $L_n$,\n\\begin{equation} \n\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} = \\frac{1}{N_{\\alpha}} L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}},\n\\end{equation}\nwhere $n_i \\leq -1$ and $N_{\\alpha}$ is a normalization factor such that $\\braket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}}=1$. $N_{\\alpha}$ can be expanded as\n\\begin{eqnarray}\nN_{\\alpha}^2 &=& \\bra{h,\\bar{h}}\\left(L_{n_1}L_{n_2} \\cdots L_{n_q}\\right)^{\\dagger} L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}} ~~~\\\\\n&=& \\bra{h,\\bar{h}} \\left(L_{n_q}^{\\dagger} \\cdots L_{n_2}^{\\dagger} L_{n_1}^{\\dagger} \\right) L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}}~~~\\\\\n&=& \\bra{h,\\bar{h}} \\left(L_{-n_q} \\cdots L_{-n_2} L_{-n_1} \\right) L_{n_1}L_{n_2} \\cdots L_{n_q} \\ket{h,\\bar{h}},~~~~~\n\\end{eqnarray}\nwhere we used that $L_n^{\\dagger} = L_{-n}$. A more general descendant state is descended from a primary state via both $L_n$ and $\\overline{L}_{\\bar{n}}$ generators:\n\\begin{equation}\\label{eq:desc_state}\n\\ket{\\phi_{\\alpha}^{\\mbox{\\tiny CFT}}} = \\frac{1}{N_{\\alpha}} \\left(L_{n_1}L_{n_2} \\cdots L_{n_q}\\right)  \\left(\\bar{L}_{\\bar{n}_1}\\bar{L}_{\\bar{n}_2} \\cdots \\bar{L}_{\\bar{n}_q} \\right)\\ket{h,\\bar{h}},\n\\end{equation}\nwith $n_i, \\bar{n}_i \\leq -1$, and its norm $N_{\\alpha}$ can be expanded similarly.\n\n\\subsection{Matrix elements of virasoro generators}\n\nGiven a product of Virasoro generators $X$, we wish to compute\n\\begin{equation}\n  X_{\\alpha\\beta} \\equiv \\langle \\phi_\\beta^{\\mbox{\\tiny CFT}} | X | \\phi_\\alpha^{\\mbox{\\tiny CFT}} \\rangle.\n\\end{equation}\nWe first expand the states $|\\phi_\\alpha^{\\mbox{\\tiny CFT}}\\rangle$ and  $|\\phi_\\beta^{\\mbox{\\tiny CFT}}\\rangle$ in terms of their primary states and products of Virasoro generators as in the previous section. If the two primary states are different, thus belonging to different representations of the Virasoro algebra, $X_{\\alpha\\beta}=0$. If the primary states are the same, we have\n\\begin{equation}\n  X_{\\alpha\\beta} \\equiv \\frac{1}{N_\\alpha N_\\beta}\\langle h,\\bar{h} | \\left(L_{n_1} \\dots L_{n_r}\\right) \\left(\\bar{L}_{\\bar{n}_1} \\dots \\bar{L}_{\\bar{n}_s} \\right) | h,\\bar{h} \\rangle,\n\\end{equation}\nwhere we have also expanded $X$, resulting in an expectation value on the state $|h,\\bar{h}\\rangle$ of a long sequence of generators. We have also used the commutativity of $L_n$ with $\\bar{L}_m$. To evaluate the expectation value, we apply the Virasoro commutators of Eq.~(\\ref{eq:Virasoro}) to produce ``normal ordered'' terms, moving all generators with $n>0$ ($\\bar{n}>0$) to the right and all generators with $n<0$ ($\\bar{n}<0$) to the left. Since $L_n\\bar{L}_{m}|h,\\bar{h}\\rangle = 0$ for $n>0$ or $m>0$, and $L_n^\\dagger = L_{-n}$, $\\bar{L}_n^\\dagger = \\bar{L}_{-n}$, these terms are zero and we are left with terms proportional to\n\\begin{equation}\n  \\langle h,\\bar{h}|(L_0)^a(\\bar{L}_0)^b|h,\\bar{h}\\rangle = (h)^a (\\bar{h})^b,\n\\end{equation}\nfor integers $a,b \\ge 0$, where we have used $L_0|h,\\bar{h}\\rangle=h|h,\\bar{h}\\rangle$ and $\\bar{L}_0|h,\\bar{h}\\rangle=\\bar{h}|h,\\bar{h}\\rangle$. We sum these terms to compute the final result.\n\nThe normalization factors $N_\\alpha$ can be computed in the same way. For example, the norm-squared of the state $|\\phi^{\\mbox{\\tiny CFT}}\\rangle = L_{-2}|h,\\bar{h}\\rangle$ is\n\\begin{align}\n  N_\\phi^2 &= \\langle h,\\bar{h} | L_2 L_{-2}  | h,\\bar{h} \\rangle \\\\\n           &= \\langle h,\\bar{h} | 4L_0 + \\frac{c}{2}  | h,\\bar{h} \\rangle\\\\\n  &= 4h+\\frac{c}{2},\n\\end{align}\nwhere we applied the Virasoro commutator once. In particular, the norm of the state $|T^{\\mbox{\\tiny CFT}}\\rangle = L_{-2}|\\mathbb{I}^{\\mbox{\\tiny CFT}}\\rangle$ is $N_T = \\sqrt{c\/2}$, since for the identity primary $h=\\bar{h}=0$. An exemplary off-diagonal matrix element involving this state is\n\\begin{align}\n&  \\langle T^{\\mbox{\\tiny CFT}} | \\bar{L}_3 L_{-2}\\bar{L}_{-3} |\\mathbb{I}^{\\mbox{\\tiny CFT}}\\rangle \\\\\n=~ &\\frac{1}{N_T}\\langle \\mathbb{I}^{\\mbox{\\tiny CFT}} | \\bar{L}_3 \\bar{L}_{-3} L_2 L_{-2} | \\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n=~ & N_T \\langle \\mathbb{I}^{\\mbox{\\tiny CFT}} | \\bar{L}_3 \\bar{L}_{-3} | \\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n= ~& N_T \\langle \\mathbb{I}^{\\mbox{\\tiny CFT}}| 6L_0 + 2c |\\mathbb{I}^{\\mbox{\\tiny CFT}} \\rangle \\\\\n= ~& \\sqrt{2}c^{3\/2}.\n\\end{align}\n\n\n\\subsection{Finite non-uniform euclidean evolution}\n\nWe wish to compute matrix elements of a conformal transformation equivalent to evolving non-uniformly in euclidean time. In other words, the generator should have the form from Eq.~(\\ref{eq:Q0})\n\\begin{equation} \nQ_0 \\equiv \\int_0^{L} dx~a(x)~h(x), \n\\end{equation}\nwith $a(x)$ a real-valued periodic function of $x$ so that $a(0) = a(L)$. Since we want to compare the action of the finite transformation $V^{\\mbox{\\tiny CFT}}_0 \\equiv e^{-Q_0}$ on eigenstates of the CFT Hamiltonian to the action of the tensor network of Fig.~\\ref{fig:time}(b) on corresponding low-energy eigenstates of a critical spin chain, we guess a form for $a(x)$ based on the structure of the tensor network. We first take each lattice site to represent an interval of the continuum one lattice-spacing in length and centered on that site. Then we interpret each euclideon as evolving its interval by one unit of euclidean time, which is consistent with the action of the full transfer matrix $\\mathcal{T}$. We then guess that a smoother evolves its interval by, on average, \\emph{half} a unit of time, with the amount of evolution changing linearly from zero at one end of the interval associated with the smoother, to one unit at the other end. This leads to the profile function shown in Fig.~\\ref{fig:time}(a), where the way the smoother is ``cut'' from the euclideon determines the direction of its sloping profile: See section \\ref{sec:time_smoothers}.\n\nThe profile function, defined as a continuous function of $x \\in [0,L)$, is\n\\begin{equation}\n  a(x) = \\begin{cases}\n    x & 0 < x \\le 1 \\\\\n    1 & 1 < x \\le L\/4 \\\\\n    -x + L\/4 & L\/4 < x \\le L\/4+1 \\\\\n    0 & L\/4+1 < x \\le L\/2 \\\\\n    x - L\/2 & L\/2 < x \\le L\/2+1 \\\\\n    1 & L\/2+1 < x \\le 3L\/4 \\\\\n    -x + 3L\/4 & 3L\/4 < x \\le 3L\/4+1 \\\\\n    0 & 3L\/4+1 < x \\le L\n  \\end{cases},\n\\end{equation}\nand has the Fourier coefficients\n\\begin{equation} \\label{eq:Fourier_boxy}\n  a_n = e^{in S \\frac{2\\pi}{N}} \\frac{1}{4} \\;\\mathrm{sinc}\\left(\\frac{n}{4}\\right) \\;\\mathrm{sinc}\\left(\\frac{n}{N}\\right) \\left(1 + e^{in \\pi}\\right),\n\\end{equation}\nwhere the shift $S$ reflects the freedom of choosing the origin along the x-axis. The corresponding conformal generator is\n\\begin{equation} \\label{eq:Q_boxy}\n  Q_0 = \\frac{2\\pi}{N} \\sum_{n=-N\/2}^{N\/2} a_n H_n,\n\\end{equation}\nwhere we have summed over only those modes that can be distinguished on a lattice of $N$ sites ($N$ is assumed to be even). For the plot in Fig.~\\ref{fig:time}(b), where $N=24$, we expand the exponential $V^{\\mbox{\\tiny CFT}}_0 \\equiv e^{-Q_0}$ to 4th order in $Q_0$ so that all the matrix elements shown are reasonably well converged.\n\nNote that there are some reasons to expect mismatches between $V^{\\mbox{\\tiny CFT}}_0$ and the euclideon tensor network of Fig.~\\ref{fig:time}(a), even with an appropriate choice of $a(x)$. Perhaps most significantly, due to the smoothers, the tensor network is not Hermitian, unlike $V^{\\mbox{\\tiny CFT}}_0$. This is the cause of the slight differences in magnitude between matrix elements of the tensor network and their Hermitian conjugate counterparts in the plot of Fig.~\\ref{fig:time}(b). Furthermore, we neglect the effect of finite-size corrections to the eigenstates of the critical spin chain due to irrelevant terms present in the lattice model, as well as the effects of any relevant or irrelevant operator contributions to the smoothers, which are indeed present, as demonstrated by the observed matrix elements connecting different conformal towers.\n\n\\subsection{Finite non-uniform scale transformation}\n\nAn appropriate conformal transformation for comparison with the local scaling tensor network of Fig.~\\ref{fig:scale}(b) is generated by a position-dependent translation\n\\begin{equation} \nQ_1 \\equiv \\int_0^{L} dx~b(x)~p(x), \n\\end{equation}\nas in Eq.~(\\ref{eq:Q1}), where $b(x)$ is a real-valued, periodic function of $x \\in [0,L)$. Unlike in the euclidean time case, however, the function $b(x)$ cannot simply be scaled to produce the position-dependent translation function associated with a \\emph{finite} transformation. Indeed, $b(x)$ represents a position-dependent \\emph{velocity} and a point $x(s_0)$ on the x-axis is translated by an infinitesimal transformation $e^{i\\epsilon Q_1}$ as\n\\begin{equation}\n  x(s_0+\\epsilon) = x(s_0) + \\epsilon b(x(s_0)).\n\\end{equation}\nTo find the final location of a point $x(s_0)$ under a finite transformation $e^{isQ_1}$, this equation must be integrated, for example numerically. By doing this for every starting point of interest $x_j$, the translation\n\\begin{equation}\n  B(x_j) \\equiv x_j(s-s_0) - x_j(s_0)\n\\end{equation}\nthese points experience under the finite transformation can be computed.\n\nBy inspecting the tensor network of Fig.~\\ref{fig:scale}(b) we can guess at the appropriate form for $B(x)$. We consider the action of the network on a state at the top, with each lattice site representing an unit interval, centered on the site, of continuous $x$-axis. The network first performs fine-graining on sites $1$ and $8$. We thus assign a constant scale-factor of $2$ to the interval $(8,1]$, \\emph{between} sites $1$ and $8$. Since we do not know exactly how the smoothers contribute, we leave the behavior in the intervals $(7.5,8]$ and $(1,1.5]$ undetermined. The network then performs a coarse-graining of sites $4$ and $5$, hence we assign the scale factor $1\/2$ to the interval $(3.5,5.5]$. We further assume there is no scaling in the neighborhood of points $2$ and $7$, since these each mark the midpoint of a pair of smoothers and, by symmetry, the scale factor should pass through zero here.\n\nTo derive shifts, we must further pay attention to how the outgoing legs at the bottom of the network are arranged relative to the ingoing legs at the bottom. Although the unscaled sites $2$ and $7$ pass through the smoothers without being translated, translations are needed at the bottom to match up the outgoing legs with lattice sites. Hence site $2$ is translated by $1$ unit (to the right) and site $7$ is translated by $-1$ (to the left). By assuming, based on the symmetry of the network, that the points $4.5$ and $8.5$ are not translated, we can then derive the translation of points in the intervals $(8,1]$ and $(3.5,5.5]$ from the above scale factors (a constant scale factor corresponds to a constant derivative of the translation function). We use only the values in the stated regions and at the stated points in arriving at a possible generator, reflecting our lack of knowledge concerning the precise action of the smoothers. The resulting profile\n\\begin{align}\n  B(x) = \\begin{cases}\n    1 & x = 2 \\\\\n    -x\/2 + 7\/4 & 3.5 < x \\le 5.5 \\\\\n    -1 & x = 7 \\\\\n    2x - 16 & 8 < x \\le 9 \\;(\\equiv 1)\n  \\end{cases}\n\\end{align}\nis shown in Fig.~\\ref{fig:scale}(a).\n\nTo constrain the set of generating functions $b(x)$, we restrict to those described by just the first two odd Fourier modes, which is enough to reproduce the regions of constant scaling quite accurately. We may restrict to odd functions since we know there must be nodes at $x=4.5$ and $x=8.5$. While it is possible to fit these parameters to $B(x)$, we choose Fourier coefficients that are very close to the optimal ones, but which better match the numerical matrix elements of Fig.~\\ref{fig:scale}(b):\n\\begin{align} \\label{eq:Fourier_cosy}\n  b_1 = e^{i S \\frac{2\\pi}{N}} \\; 0.5, \\quad b_3 = e^{i 2 S \\frac{2\\pi}{N}} \\; 0.0275,\n\\end{align}\nwhere $S$ is a shift determined by the location of the origin along the x-axis. The generator of the corresponding scale transformation is\n\\begin{equation}\n  Q_1 = \\frac{2\\pi}{N}\\left( b_1 P_1 + b_1^* P_{-1} + b_3 P_3 + b_3^* P_{-3} \\right).\n\\end{equation}\nFor the plot in Fig.~\\ref{fig:scale}(b), where $N=8$, we expand the exponential $V^{\\mbox{\\tiny CFT}}_1 \\equiv e^{-iQ_1}$ to 7th order so that the plotted matrix elements are reasonably well converged.\n \n\\subsection{Final remark on $V_0^{\\mbox{\\tiny CFT}}$ and $V_1^{\\mbox{\\tiny CFT}}$ } \n \nAbove we have carefully specified two profiles $a(x)$ and $b(x)$ that led to generators $Q_{0}$ and $Q_1$ for the specific conformal transformations $V_0^{\\mbox{\\tiny CFT}}$ and $V_1^{\\mbox{\\tiny CFT}}$ acting on the the CFT that we used in Figs.~\\ref{fig:time} and \\ref{fig:scale} for comparison with the linear maps implemented with tensor networks. The match, both qualitative and quantitative, between matrix elements of the linear maps $V$ and $V^{\\mbox{\\tiny CFT}}$ obtained on the lattice with tensor networks and on the CFT with conformal transformations is remarkable, confirming that the proposed tensor networks indeed implement lattice versions of non-uniform euclidean time evolution and rescaling.\n\nWe remark that although the profiles $a(x)$ and $b(x)$ were obtained above through a somewhat convoluted derivation that required making ad hoc decisions, slightly different choices of profile $a(x)$ and $b(x)$ were also tested and seen to lead to very similar $V^{\\mbox{\\tiny CFT}}$ whose matrix elements continued to accurately match the matrix elements of the proposed lattice linear map $V$. \n \n\\section{Euclideons, disentanglers, isometries, and smoothers} \n\nIn this section we briefly review the well-established construction of the tensors called eucliodeons $e$, disentanglers $u$, and isometries $w$, and sketch how to build smoothers $e_{L}, e_{R}, u_{L}, u_{R}$. \n\n\\subsection{Euclideons}\n\nAn \\textit{euclideon}, denoted $e$, is a tensor that implements euclidean time evolution. In Refs.~\\cite{TNR,TNRMERA,TNRscale} the tensor $e$ was instead denoted as $A$. Two possible ways of building an euclideon are (i) from a quantum spin chain Hamiltonian $H$ (see e.g.\\ Supplemental Material in \\cite{TNRMERA}) and (ii) from the Boltzmann weights of the statistical partition function of a classical two-dimensional lattice system (see e.g.\\ Ref.~\\cite{TNR}). We briefly review those constructions for completeness. \n\nGiven a local quantum spin chain Hamiltonian $H$, the defining property of an euclideon $e$ for $H$ is that a periodic row $\\mathcal{T}$ of $N$ euclideons should implement the euclidean time evolution $\\exp(-\\tau H)$ for one unit of euclidean time $\\tau=1$. If the spin chain is described at low energies by a relativistic quantum field theory (QFT), then we normalize $H$ such that the ground state energy is zero in the thermodynamic limit ($N\\rightarrow \\infty$) and the speed of light is $1$ \\cite{Ash}. When the relativistic QFT is in addition a conformal field theory (CFT), the above normalization of $H$ implies that the euclideons are (up to lattice effects) isotropic in the two-dimensional euclidean space-time. \nAfter normalizing the Hamiltonian $H$ one first builds a matrix product operator (MPO) for $\\exp(-\\delta \\tau H)$ for small $\\delta \\tau \\ll 1$, e.g.\\ using a Suzuki-Trotter decomposition, and then multiplies together $1\/\\delta \\tau$ copies of the resulting MPO while appropriately truncating the bond indices of the resulting product $\\left(\\exp (-\\delta \\tau H) \\right)^{1\/\\delta \\tau} = \\exp(- \\tau H)$. This results in a new MPO made of (roughly) isotropic euclideons $e$, see e.g.\\ part A of the Supplemental Material of Ref.~\\cite{TNRMERA} for further details.\n\nThe construction of euclideons is even simpler if the quantum spin chain relates to a two-dimensional statistical model, since in this case we can express an euclideon $e$ directly in terms of Boltzmann weights. For instance, for the critical quantum Ising spin chain $H = -\\sum_l \\sigma^{x}_l\\sigma^{x}_{l+1} - \\sigma_l^{z}$, we can build euclideons using the Boltzmann weights $e^{\\sigma_i\\sigma_j\/T}$ of the statistical partition function of the (isotropic) critical 2d classical Ising model, namely\n\\begin{equation} \\label{eq:A}\ne_{ijkl} \\equiv e^{\\left(\\sigma_i\\sigma_j + \\sigma_j\\sigma_k + \\sigma_k\\sigma_l + \\sigma_l\\sigma_i \\right)\/T},\n\\end{equation}\nwhere $\\sigma_i = \\pm 1$ labels the two possible values of a classical Ising spin on site $i$ of a two-dimensional lattice, see e.g.\\ Ref.~\\cite{TNR} for further details.\n\n\n\\subsection{Euclidean time smoothers}\n\\label{sec:time_smoothers}\n\nWe can use a truncated row of euclideons $e$ on a region of a quantum spin chain in order to apply an euclidean time evolution that only evolves the spins on that region. However, at both ends of the truncated row of euclideons we must place special tensors that we call \\textit{smoothers} $e_L$ and $e_R$, see Fig.~\\ref{fig:timesmoother0}. The purpose of each smoother is to smooth out lattice effects that occur at the ends of the truncated row of euclideons. Without smoothers there would be an open bond index at each end. Thus a first role of the smoothers, which only have one bond index (as opposed to the two bond indices of regular euclideons) is to eliminate open bond indices. However, not any tensor with a single bond index will do. Indeed, a generic choice of coefficients inside the smoother will result in a transformation that is not the intended conformal transformation (non-uniform euclidean time evolution). As a matter of fact, the resulting transformation is generically not even approximately diagonal in the conformal towers and therefore does not correspond to any conformal transformation.\n\n\n\\begin{figure}\n\\includegraphics[width=7cm]{timesmoother0_v2.pdf}\n\\caption{\n(a) A truncated row of euclideons $e$ has a dangling bond index at each end. As a result, it does not define a linear map in the Hilbert space of the spin chain.\n(b) Left and right smoother $e_L$ and $e_R$ are special tensors placed at the two ends of the truncated row of euclideons in order to eliminate the dangling bond indexes, producing a linear map in the spin chain. \n\\label{fig:timesmoother0} \n}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=7cm]{timesmoother1_v2.pdf}\n\\caption{\nComputation of smoothers $e_L$ and $e_R$ starting from an euclideon $e$.\n(a) In order to create a smoother $e_L$, first we temporarily regard an euclideon $e$ as a matrix $M$ (by joining its indexes into two pairs) and then compute its singular value decomposition $M= USV^{\\dagger}$, where $U$ and $V$ are unitary matrices and $S$ is a diagonal matrix with the singular values of $M$ in its diagonal. Then we create tensors $a_L = \\sqrt{S}V^{\\dagger}$ and $\\tilde{a}_L \\equiv U\\sqrt{S}$, where $\\sqrt{S}$ is a diagonal matrix with the square root of the singular values in the diagonal. Tensor $a_L$, which is a precursor of the smoother $e_L$, has two indices (the ones which originally belonged to the euclideon $e$, depicted in black) of dimension $d$ and one index (the one coming from the singular value decomposition, depicted in red) of dimension $d^2$. \n(b) The smoother $e_L$ is then obtained by multiplying $a_L$ by an isometry $v$ of size $d\\times d^2$ that maps the $d^2$-dimensional (red) index into a $d$-dimensional (black) index, thus implementing a truncation of the former. The variational parameters in $v$ correspond to a choice of truncation basis and are determined as indicated in Fig.~\\ref{fig:timesmoother2}. \n(c) Notice that the product of $\\tilde{e}_L$ (obtained analogously to $e_L$) and $e_L$ amounts to the original euclideon $e$, up to effects due to the truncation implemented through the isometry $v$.\n(d)-(f) The smoother $e_R$ is created analogously. \n\\label{fig:timesmoother1} \n}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{timesmoother2_v2.pdf}\n\\caption{\nIn order to determine the variational parameters in the smoother $e_L$, which are contained in the isometry $v$ of Fig.~\\ref{fig:timesmoother1}(b), we require that the dominant eigenvector of the transfer matrix $\\mathcal{T}$ (a), corresponding to the ground state on the circle, has maximal overlap with the dominant eigenvector of other transfer matrices, such as those in (b) and (c), which can be seen to become close to the ground state on the circle. The variational parameters in the smoother $e_R$ are determined analogously using the dominant eigenvectors of transfer matrices $(d)$ and $(e)$.\n\\label{fig:timesmoother2} \n}\n\\end{figure}\n\nIn practice we build the smoothers through a two-step procedure, see Fig.~\\ref{fig:timesmoother1}. First we split an euclideon diagonally using a singular value decomposition (SVD). Then we multiply it by an isometry, that we determine by demanding that it optimally connects to the physical indices of the original euclideon $e$, in the following sense. We build a \\textit{diagonal transfer matrix} $M'$ made of euclideons and smoothers and demand that its dominant eigenvector has maximal overlap with the dominant eigenvector of the horizontal transfer matrix $M$ made of euclideons. The resulting smoothers, when placed at the ends of a truncated row of euclideons, are then seen to indeed produce linear maps that not only act (approximately) diagonally in the conformal towers (as any conformal transformation does) but whose matrix elements between low energy states accurately correspond to the intended non-uniform euclidean time evolution, as seen in the example of Fig.~\\ref{fig:time} in the main text.\n\n\n\\subsection{Optimized disentanglers and isometries} \n \nDisentanglers $u$ and isometries $w$, the tensors in the multi-scale entanglement renormalization ansatz (MERA), are in general full of variational parameters constrained in such a way that the tensors are unitary\/isometric. In the context of this work, however, by a disentangler $u$ and an isometry $w$ we refer exclusively to such tensors after the variational parameters have been chosen so that the MERA represents the ground state of the quantum spin Hamiltonian $H$. \n\nThis optimization can be carried out variationally using iterative energy minimization algorithms \\cite{siMERA}, as we did here. Alternatively, one can extract optimized disentanglers and isometries from the tensor network renormalization (TNR) algorithm \\cite{TNRMERA}, which manipulates a network of euclideons.\n\n\\subsection{Scale smoothers}\n\nWe can use a truncated double layer of optimized disentanglers and isometries on a region of a quantum spin chain in order to apply a non-uniform scale transformation that only rescales the spins in that region. Again, at both ends of the truncated layer of disentanglers and isometries we must place special tensors called smoothers $u_L$ and $u_R$. One first reason for including these smoothers is that otherwise there is a mismatch in index connectivity, since the lower indices of an isometry $w$ must connect with the upper indices of a disentangler $u$ (by design of the MERA). However, a generic choice of coefficients inside the scale smoothers $u_L$ and $u_R$ (even when compatible with the unitary\/isometric character of these tensors) will produce a linear map that does not correspond to a conformal transformation. \n\nIn practice, we build the smoothers $u_L$ and $u_R$ by demanding that the equality in Fig.~\\ref{fig:scalesmoother1} be (approximately) fulfilled. These equalities ensure that the scale smoothers properly connect upper and lower indices. The resulting smoothers $u_L$ and $u_R$, when placed at the ends of a truncated double layer of optimized disentanglers and isometries, indeed produce a linear map that not only acts (approximately) diagonally in the conformal towers but whose matrix elements between low energy states accurately correspond to the intended non-uniform scale transformation.\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{scalesmoother1_v2.pdf}\n\\caption{\n(a) Isometry $w$, disentangler $u$, and smoothers $u_L$ and $u_R$, with two types of indices: in red, the top index of an isometry $w$, which connects with the bottom indices of a disentangler $u$; in black, the bottom indices of an isometry $w$, which connect with the top indices of a disentangler $u$. Smoothers map a pair of red and black top indices into a pair of bottom red indices.\n(b) At each end of a truncated double layer of MERA tensors $\\mathcal{W}$ there is a dangling black index that cannot be directly connected to a red index. Scale smoothers $u_L$ and $u_R$ can map these dangling black indexes, together with an adjacent red index, into a pair of red indexes. \n(c) The left scale smoother $u_L$ is determined variationally by demanding that this tensor network equality be fulfilled (approximately, but as accurately as possible).\n(d) Similarly, the right scale smoother $u_R$ is determined variationally by demanding that this other tensor network equality be approximately fulfilled.\n\\label{fig:scalesmoother1} \n}\n\\end{figure}\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\n\nReasoning about actions and planning concern the representation of a dynamic\nsystem. This representation consists of a description of the interaction\nbetween an agent and its environment and aims at enabling reasoning and\ndeliberation on the possible course of action for the agent \\cite{Reit01}.\nPlanning in fully observable nondeterministic domains (FOND), say in Planning Domain Definition Language (PDDL),\n\\cite{GhNT04,GeBo13} exemplifies the standard methodology for expressing\ndynamic systems: it represents the world using finitely many \\emph{fluents}\nunder the control of the \\emph{environment} and a finitely many \\emph{actions}\nunder the control of the \\emph{agent}. Using these two elements a model of the\ndynamics of world is given. Agent goals, e.g., reachability objectives, or,\nsay, temporally extended objectives written in \\text{{LTL}}\\xspace\n\\cite{BacchusK00,CTMBM17,DR-IJCAI18}, are expressed over such models in terms of\nsuch fluents and actions.\n\nAn important observation is that, in devising plans, the agent takes\nadvantage of such a representation of the world. Such a representation\ncorresponds to knowledge that the agent has of the world. In other\nwords, the agent \\emph{assumes} that the world works in a certain\nway, and \\emph{exploits such an assumption in devising its plans}.\nA question immediately comes to mind: \n \\begin{quote}\n \\emph{Which kinds of environment assumptions can the agent make?}\n \\end{quote}\nObviously the planning domain itself (including the initial state) with its\npreconditions and effects is such an assumption. That is, as long as the agent \nsticks to its preconditions, the environment acts as described by the domain. \nSo, the agent can exploit the effect of its actions in order to reach a certain goal \n{(state of affairs)}.\n{Another common assumption is to assume the domain is \\emph{fair}, i.e.,\nso-called \\emph{fair FOND}~\\cite{DaTV99,PistoreT01,Cimatti03,CTMBM17,DIppolitoRS18}.  In this case\nthe agent can exploit not only the effects, but also the guarantee that by\ncontinuing to execute an action {from a given state} the environment will\neventually respond with all its possible nondeterministic effects.}\\footnote{There are two notions of fairness in planning. One stems from the fact that nondeterminism is resolved stochastically. The other is a logical notion analogous to that used in the formal-methods literature. These two notions coincide in the context reachability goals~\\cite{DIppolitoRS18}, but diverge with more general LTL goals~\\cite{Pnueli:STOC83,Pnueli:IC93}. In this paper, we focus on the logical notion.}\nMore recently \\cite{DBLP:conf\/ijcai\/BonetG15,DBLP:conf\/ijcai\/BonetGGR17} \ntrajectory constraints over the domain, expressed in \\text{{LTL}}\\xspace, have been proposed\nto model general restrictions on the possible environment behavior.\nBut is any kind of \\text{{LTL}}\\xspace formula on the fluents and actions of the domain a\npossible trajectory constraint for the environment?\nThe answer is obviously not! To see this, consider a formula expressing that\neventually a certain possible action must actually be performed (the agent may\ndecide not to do it). But then \n\\begin{quote}\n\\emph{Which trajectory constraints are suitable as assumptions in a given\ndomain?} \n\\end{quote}\nFocusing on \\text{{LTL}}\\xspace, the question can be rephrased as: \n\\begin{quote}\n\\emph{Can any {linear-time} specification be used as an assumption for the environment?}\n\\end{quote}\nWe can summarize these questions, ultimately, by asking:\n\\begin{quote}\n\\emph{What is an environment assumption?}\n\\end{quote}\n\n\nThis is what we investigate in this paper. We take the view that environment\nassumptions are ways to talk about \\emph{the set of strategies the environment can enact}. Moreover, \nthe plan for the goal, i.e., the agent strategy for fulfilling\nthe goal, need only fulfill the goal against the strategies of the\nenvironment {from the given set of environment strategies}.\nWe formalize this insight and {\\emph{define}} synthesis\/planning under assumptions and\nthe relationship between the two in a general linear-time setting. {In particular, our definitions\nonly allow linear-time properties to be assumptions if the environment can enforce them.}\nIn doing this \\emph{we answer the above questions}.\n\nWe also concretize the study and express goals and assumptions in \\text{{LTL}}\\xspace, automata over infinite  words (deterministic parity word automata)~\\cite{ALG02}, as well as formalisms over finite traces, i.e., \\text{{LTLf}}\\xspace\/\\text{{LDLf}}\\xspace~\\cite{DegVa13,DR-IJCAI18} and finite word automata. \nThis allows us to study \\emph{how to solve} synthesis\/planning under assumptions problems. One may\nthink that the natural way to solve such synthesis problems is to have \nthe agent synthesize a strategy for the implication \n\\[ Assumption \\supset Goal\\] \nwhere both $Assumption$ and $Goals$ are expressed, say, in \\text{{LTL}}\\xspace. \nA first problem with such an implication is that the agent should not devise\nstrategies that make $Assumption$ false, because in this case the agent would\nlose its model of the world without \\emph{necessarily} fulfilling its $Goal$.  \n{This undesirable situation is avoided by our very notion of environment assumption.}\n{A second issue is this:}\n\\begin{quote}\n\\emph{Does synthesis\/planning under assumptions amount to\nsynthesizing for the above implication?} \n\\end{quote}\n{We show that this is not the case. \nNote that an agent that synthesizes for the implication is too pessimistic: the agent, having chosen a candidate agent strategy, considers as possible all environment strategies that satisfy $Assumption$ against the specific candidate strategy it is analyzing. But, in this way the agent gives too much power to the environment, since, in fact, the environment does not know the agent's chosen strategy. On the other hand, surprisingly, we show that if there is an agent strategy fulfilling $Goal$ under $Assumption$, then also there exists one that indeed enforces the implication.\n}\nThus, even if the implication \n\\emph{cannot be used for characterizing the problem of\nsynthesis\/planning under assumptions}, it \\emph{can be used to solve it}.\nExploiting this result, we give techniques to solve synthesis\/planning under\nassumptions, and study the worst case complexity of the problems when goals and assumptions are\nexpressed {in the logics and automata mentioned above.}\n\n\n\n\n\\section{Synthesis and Linear-time specifications} \\label{sec:prelims}\n\n\\emph{Synthesis} is the problem of producing a module that satisfies a given property no matter how the environment behaves ~\\cite{PnueliR89}.\nSynthesis can be thought of in the terminology of games. Let ${\\sf{Var}}$ be a finite set of Boolean variables (also called atoms), and assume it is partitioned into two sets: $A$, those controllable by the agent, and $E$, those controllable by the environment. \nLet $\\modecal{A}} \\newcommand{\\B}{\\modecal{B} = 2^A$ be the set of \\emph{actions} and $\\modecal{E}} \\newcommand{\\F}{\\modecal{F} = 2^E$ the set of \\emph{environment states} (note the symmetry: we \nthink of $\\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ as a set of actions that are compactly\nrepresented as assignments of the variables in $A$). The game consists of infinitely many phases. \nIn each phase of the game, both players assign values to their variables, with the environment going first. These assignments \nare given by \\emph{strategies}: an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace:\\modecal{E}} \\newcommand{\\F}{\\modecal{F}^+ \\to \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ and an environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$. The resulting infinite sequence of assignments is denoted $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$.\\footnote{Formally, \nwe say that $\\pi = \\pi_0 \\pi_1 \\cdots$ complies with $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ if $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace((\\pi_0 \\cap E) \\cdots (\\pi_k \\cap E)) = \\pi_k \\cap A$ for all $k$; and we say that $\\pi$ complies with $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ if $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace((\\pi_0 \\cap A) \\cdots (\\pi_k \\cap A)) = \\pi_{k+1} \\cap E$ for all $k$. Then $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is defined to be the unique infinite trace that complies with both strategies.} \n\n{In classic synthesis the agent is trying to ensure that the produced sequence satisfies a given linear-time property. In what follows \\emph{we write \\SF to denote a generic formalism for defining linear-time properties.} Thus, the reader may substitute their favorite formalism for \\SF, e.g., one can take \\SF to be linear temporal logic, or deterministic parity automata. We use logical notation throughout. For instance, when $\\phi$ refers to a logical formula, then $\\phi_1 \\wedge \\phi_2$ refers to conjunction of formulas, but when $\\phi$ refers to an automaton then $\\phi_1 \\wedge \\phi_2$ refers to intersection of automata. If $\\phi \\in \\SF$ write $[[\\phi]] \\in (2^{\\sf{Var}})^\\omega$ for the set it defines. For instance, if $\\phi \\in \\text{{LTL}}\\xspace$ then $[[\\phi]]$ is the set of infinite sequences that satisfy $\\phi$, but when $\\phi$ is an automaton operating on infinite sequences, then $[[\\phi]]$ is the set of infinite sequences accepted by the automaton. Moreover, in both cases we say that the sequence \\emph{satisfies} $\\phi$.\n}\n\nWe say that \\emph{$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]$, i.e., if no matter which strategy the environment uses, the resulting sequence satisfies $\\phi$. Similarly, we say that \\emph{$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]$. We write $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\phi)$ (resp. $Str_\\ensuremath{\\mathsf {ag}}\\xspace(\\phi)$) for the set of environment (resp. agent) strategies that realize $\\phi$, and in case this set is non-empty we say that \\emph{$\\phi$ is environment (resp. agent) realizable}. We write $Str_\\ensuremath{\\mathsf {env}}\\xspace$ (resp. $Str_\\ensuremath{\\mathsf {ag}}\\xspace$) for the set of all environment (resp. agent) strategies.\n\n\\emph{Solving $\\SF$ environment- (resp. agent-) synthesis} asks, given $\\phi \\in \\SF$ to decide if $\\phi$ is environment- (resp. agent-) \nrealizable, and to return such a finite-state strategy (if one exists). \n{In other words, realizability is the {recognition problem} associated to synthesis.}\nWe now recall two concrete specification formalisms $\\SF$, namely \\text{{LTL}}\\xspace (linear temporal logic) and \\text{{DPW}}\\xspace (deterministic parity word automata), and then state that results about solving \\text{{LTL}}\\xspace\/\\text{{DPW}}\\xspace synthesis.\n\n\\subsubsection{Linear temporal Logic (LTL)} \\label{sec:prelims:LTL}\n\n\t \\emph{Formulas of $\\text{{LTL}}\\xspace({\\sf{Var}})$}, or simply \\emph{$\\text{{LTL}}\\xspace$}, are generated by the following grammar:\n\t\\[\\varphi \\!::=\\! p \\!\\mid\\! \\varphi \\vee \\varphi \\!\\mid\\! \\neg \\varphi \\!\\mid\\!  \\nextX \\! \\varphi \\!\\mid \\! \\varphi \\until \\varphi\\]\n\twhere $p \\in {\\sf{Var}}$. The \\emph{size $|\\varphi|$} of a formula $\\varphi$ is the number of symbols in it.\n\t\\text{{LTL}}\\xspace formulas are interpreted over infinite sequences $\\pi \\in (2^{{\\sf{Var}}})^\\omega$. Define the satisfaction relation\n\t$\\models$ as follows:\n\t\\begin{enumerate*}\n\t\\item[] $(\\pi,n) \\models p$ iff $p \\in \\pi_n$;\n\t\\item[]  $(\\pi,n) \\models \\varphi_1 \\vee \\varphi_2$ iff $(\\pi,n) \\models \\varphi_i$ for some $i \\in \\{1,2\\}$;\n\t\\item[] \t$(\\pi,n) \\models \\neg \\varphi$ iff it is not the case that $(\\pi,n) \\models \\varphi$;\n\t\\item[]   $(\\pi,n) \\models \\nextX \\varphi$ iff $(\\pi,n+1) \\models \\varphi$;\n\t\\item[]  $(\\pi,n) \\models \\varphi_1 \\until \\varphi_2$ iff there exists $i \\geq n$ such that $(\\pi,i) \\models \\varphi_2$ and for all $i \\leq j < n$, $(\\pi,j) \\models \\varphi_1$.\n\t\\end{enumerate*}\n\tWrite $\\pi \\models \\varphi$ if $(\\pi,0) \\models \\varphi$ and say that $\\pi$ \\emph{satisfies} $\\varphi$ and $\\pi$ is a \\emph{model} of $\\varphi$. \n\tAn \\text{{LTL}}\\xspace formula $\\varphi$ \\emph{defines} the set $[[\\pi]] \\doteq \\{\\pi \\in (2^{\\sf{Var}})^\\omega: \\pi \\models \\varphi\\}$.\n\tWe use the usual abbreviations, $\\varphi \\supset \\varphi' \\doteq \\neg \\varphi \\vee \\varphi'$, $\\mathsf{true} := p \\vee \\neg p$, $\\mathsf{false} \\doteq \\neg \\mathsf{true}$, \n\t$\\eventually \\varphi \\doteq \\mathsf{true} \\until \\varphi$,\n\t$\\always \\varphi \\doteq \\neg \\eventually \\neg \\varphi$.  \n\tWrite $Bool({\\sf{Var}})$ for the set of Boolean formulas over ${\\sf{Var}}$. We remark that every result in this paper that mentions \\text{{LTL}}\\xspace also holds for \\text{{LDL}}\\xspace (linear dynamic logic)~\\cite{Var11,Eisner:2006tu}. \n\t\n\\subsubsection{Deterministic Parity Word Automata (DPW)}\n\nA \\emph{\\text{{DPW}}\\xspace} over ${\\sf{Var}}$ is a tuple $M = (Q,q_{in},T,col)$ where $Q$ is a finite set of \\emph{states}, \n$q_{in} \\in Q$ is an \\emph{initial state}, $T:Q \\times 2^{{\\sf{Var}}} \\to Q$ is the \\emph{transition function}, and $col:Q \\to \\mathbb{Z}$ \nis the \\emph{coloring}. The \\emph{run} $\\rho$  of $M$ on the \\emph{input} word $x_0 x_1 x_2 \\cdots \\in (2^{{\\sf{Var}}})^\\omega$ \nis the infinite sequence of transitions $(q_0,x_0,q_1) (q_1,x_1,q_2) (q_2,x_2,q_3) \\cdots$ such that $q_0 = q_{in}$. \nA run is \\emph{successful} if the largest color occurring infinitely often is even. \nIn this case, we say that the input word is \\emph{accepted}. The \\text{{DPW}}\\xspace $M$ \n\\emph{defines} the set $[[M]]$ consisting of all input words it accepts. The \\emph{size of $M$}, written $|M|$, is the cardinality of $Q$.\nThe \\emph{number of colors} of $M$ is the cardinality of $col(Q)$. \n\n{\\text{{DPW}}\\xspace{s} are effectively closed under Boolean operations, see e.g., \\cite{ALG02}:\n\\begin{lemma} \\label{lem:DPW Boolean combinations} \nLet $M_i$ be \\text{{DPW}}\\xspace with $n_i$ states and $c_i$ colors, respectively.\n\\begin{enumerate} \n \\item One can effectively form a \\text{{DPW}}\\xspace with $n_1$ states and $c_1$ colors for the complement of $M_1$.\n \\item One can effectively form a \\text{{DPW}}\\xspace with with $O(n_1 n_2 d^2d!)$ many states and $O(d)$ many colors, where $d = c_1 + c_2$, for the disjunction $M_1 \\lor M_2$.\n\\end{enumerate}\n\\end{lemma}\nThus, e.g., from \\text{{DPW}}\\xspace $M_1,M_2$ one can build a DPW for $M_1 \\supset M_2$ whose number of states is $O(n_1 n_2 d^2d!)$ and whose number of colors is $O(d)$.\n \n\nEvery \\text{{LTL}}\\xspace formula $\\varphi$ can be translated into an equivalent \\text{{DPW}}\\xspace $M$, i.e., $[[\\varphi]] = [[M]]$, see e.g. \\cite{DBLP:conf\/banff\/Vardi95,DBLP:journals\/lmcs\/Piterman07}.\nMoreover, the cost of this translation and the size of $M$ are at most doubly exponential in the size of $\\varphi$, \nand the number of colors of $M$ is at most singly exponential in the size of $\\varphi$.\n\nHere is a summary of the complexity of solving synthesis:\n\\begin{theorem}[Solving Synthesis] \\label{fact:synthesis} \\hspace{0cm}\n\\begin{enumerate} \n\\item Solving \\text{{LTL}}\\xspace environment (resp. agent) synthesis is $2$\\textsc{exptime}\\xspace-complete~\\cite{PnueliR89}.\n\\item Solving \\text{{DPW}}\\xspace environment (resp. agent) synthesis is \\textsc{ptime}\\xspace \nin the size of the automaton and $\\textsc{exptime}\\xspace$ in the number of its colors~\\cite{PnueliR89,finkbeiner2016synthesis}. \n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Synthesis under Assumptions}\n\nIn this section we give core definitions of environment assumptions and synthesis under such assumptions.\nIntuitively, the assumptions are used to select the environment strategies that the agent considers possible, i.e., although the agent does not know \nthe particular environment strategy it will encounter, it knows that it comes from such a set.\nWe begin in the abstract, and then move to declarative specifications. Unless explicitly specified, we assume fixed sets $E$ and $A$ of environment and agent atoms.\n\n{Here are the main definitions of this paper:}\n\\begin{definition}[Environment Assumptions -- abstract] \\label{def:abstract:assumptions}\nWe call any {non-empty} set $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ of environment strategies an \\emph{environment assumption}.\n\\end{definition}\n\nInformally, the set $\\Omega$ represents the set of environment strategies that the agent considers possible.\n\\begin{definition}[Agent Goals -- abstract] \nWe call any set $\\Gamma$ of traces an \\emph{agent goal}. \n\\end{definition}\n\n\\begin{definition}[Synthesis under assumptions -- abstract] \\label{def:abstract:sua}\nLet $\\Omega$ be an environment assumption and $\\Gamma$ an agent goal. \nWe say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\Gamma$ assuming $\\Omega$} if \n\\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\in \\Omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in \\Gamma\\]  \n\\end{definition}\n\n\\begin{remark}[On the non-emptiness of $\\Omega$]\nNote that the requirement that $\\Omega$ be non-empty is a consistency requirement; if it were empty then there would be no $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ to test for membership in $\\Gamma$ and so synthesis under assumptions would trivialize and all agent strategies would realize all goals. \n\\end{remark}\n\nFor the rest of this paper we will specify agent goals and environment assumptions as linear-time properties. \\textbf{In particular, we assume that $\\SF$ is a formalism for specifying linear-time properties over ${\\sf{Var}}$, e.g., $\\SF = \\text{{LTL}}\\xspace$ or $\\SF = \\text{{DPW}}\\xspace$.  }\n\nHow should $\\omega \\in \\SF$ determine an assumption $\\Omega$? In general, $\\omega$ talks about the interaction between the agent and the environment.\nHowever, we want that the agent can be guaranteed that whatever it does the resulting play satisfies $\\omega$. Thus, a given $\\omega$ induces the set $\\Omega$ consisting of {all} environment strategies $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ such that for all agent strategies $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ the resulting trace satisfies $\\omega$. In particular, for $\\Omega$ to be non-empty (as required for it to be an environment assumption) we must have that $\\omega$ is environment realizable. This justifies the following definitions.\n\n\n\\begin{definition}[Synthesis under Assumptions -- linear-time] \\label{dfn:LT:assumptions} \\hspace{0cm}\n\\begin{enumerate}\n \\item  We call $\\omega \\in \\SF$ an \\emph{environment assumption} if it is environment realizable.\n \\item We call any $\\gamma \\in \\SF$ an \\emph{agent goal}.\n \\item An \\emph{$\\SF$ synthesis under assumptions problem} is a tuple $P = (E,A,\\omega,\\gamma)$ where $\\omega \\in \\SF$ is an environment assumption and $\\gamma \\in \\SF$ is an agent goal.\n \\item We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\gamma$ assuming $\\omega$}, or that it \\emph{solves} $P$, if \n $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma$. \n \\item The corresponding decision problem is to decide, given $P$, if there is an agent strategy solving $P$.\n\\end{enumerate}\n\\end{definition}\nFor instance, \\emph{solving \\text{{LTL}}\\xspace synthesis under assumptions} means, given $P = (E,A,\\omega,\\gamma)$ with environment assumption \n$\\omega \\in \\text{{LTL}}\\xspace(E \\cup A)$ and agent goal $\\gamma \\in \\text{{LTL}}\\xspace(E \\cup A)$, to decide if there is an agent strategy solving $P$, and to return such a finite-state strategy (if one exists).\nWe remark that solving \\text{{LTL}}\\xspace synthesis under assumptions is not immediate; we will provide algorithms in the next section. For now, we point out that \n\\emph{deciding whether $\\omega$ is an environment assumption amounts to checking if $\\omega$ is environment realizable}, itself a problem that can be solved by {known results (i.e., Theorem~\\ref{fact:synthesis}).}\n\\begin{theorem}\n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTL}}\\xspace$ formula is an environment assumption is $2$\\textsc{exptime}\\xspace-complete. \n \\item Deciding if a $\\text{{DPW}}\\xspace$ is an environment assumption is in \\textsc{ptime}\\xspace in the size of the \\text{{DPW}}\\xspace and exponential in its number of colors.\n\\end{enumerate}\n\\end{theorem}\n\nWe illustrate such notions with some examples.\n\\begin{example} \\label{ex:asmp}\n\\begin{enumerate}\n   \\item The set $\\Omega = Str_\\ensuremath{\\mathsf {env}}\\xspace$, definable in \\text{{LTL}}\\xspace by the formula $\\omega \\doteq \\mathsf{true}$, is an environment assumption. It captures the situation that the agent assumes that the environment will use any of the strategies in $Str_\\ensuremath{\\mathsf {env}}\\xspace$.\n   \n   \\item In robot-action planning problems, typical environment assumptions encode the physical space, e.g., ``if robot is in Room 1 and does action $Move$ then in the next step it can only be in Rooms 1 or 4''. \n   The set $\\Omega$ of environment strategies that realize these properties is an environment assumption, \n   definable in \\text{{LTL}}\\xspace by a conjunction of formulas of the form $\\always((R_1 \\wedge Move) \\supset \\nextX (R_1 \\vee R_4))$. \n   We will generalize this example by showing that the set of environment strategies in a planning domain $D$ can be viewed as an environment assumption definable in \\text{{LTL}}\\xspace. \n   \\end{enumerate}\n\\end{example}\n\n\n\n\n\\section{Solving Synthesis under Assumptions}\n\nIn this section we show how to solve synthesis under assumptions when the environment assumptions and agent goals are given in \\text{{LTL}}\\xspace or by \\text{{DPW}}\\xspace.\n{The general idea is to reduce synthesis under assumptions to ordinary synthesis, i.e., synthesis of the implication $\\omega \\supset \\gamma$. \nAlthough correct, understanding why it is correct is not immediate.}\n\n\\begin{lemma}\nLet $\\omega \\in \\SF$ be an environment assumption and $\\gamma \\in \\SF$ an agent goal. Then, \nevery agent strategy that realizes $\\omega \\supset \\gamma$ also realizes $\\gamma$ assuming $\\omega$. \n\\end{lemma}\n\\begin{proof}\nLet $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ be an agent strategy realizing $\\omega \\supset \\gamma$ (a). To show that $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega$ let \n$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ be an environment strategy realizing $\\omega$ (b). Now consider the trace $\\pi = \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$. We must show that $\\pi$ satisfies $\\gamma$. By (a) $\\pi$ satisfies $\\omega \\supset \\gamma$ and by (b) $\\pi$ satisfies $\\omega$. \n\\end{proof}\n\nWe now observe that the converse is not true. Consider $A \\doteq \\{x\\}$ and $E \\doteq  \\{y\\}$, and let $\\omega \\doteq y \\supset x$ and $\\gamma \\doteq y \\supset \\neg x$. First note that $\\omega$ is an environment assumption formula (indeed, the environment can realize $\\omega$ by playing $\\neg y$ at the first step). Moreover, every environment strategy realizing $\\omega$ begins by playing $\\neg y$ (since otherwise the agent could play $\\neg x$ on its first turn and falsify $\\omega$). Thus, every agent strategy realizes $\\gamma$ assuming $\\omega$ (since the environment's first move is to play $\\neg y$ which makes $\\gamma$ true no matter what the agent does). On the other hand, not every agent strategy realizes $\\omega \\supset \\gamma$ \n(indeed, the strategy which plays $x$ on its first turn fails to satisfy the implication on the trace in which the environment plays $y$ on its first turn).\nIn spite of the failure of the converse, the realizability problems are inter-reducible:{\\footnote{For all reasonable expressions $\\omega$, e.g., that define Borel sets~\\cite{Mar75}.}}\n\\begin{theorem} \\label{thm:det}\nSuppose $\\omega \\in \\SF$ is an environment assumption.\n The following are equivalent:\n \\begin{enumerate}\n  \\item There is an agent strategy realizing $\\omega \\supset \\gamma$.\n  \\item There is an agent strategy realizing $\\gamma$ assuming $\\omega$.\n \\end{enumerate}\n\\end{theorem}\n\n\\begin{proof}\nThe previous lemma gives us $1 \\rightarrow 2$. For the converse, suppose $1$ does not hold, i.e., $\\omega \\supset \\gamma$ is not agent-realizable. \nNow, an immediate consequence of Martin's Borel Determinacy Theorem~\\cite{Mar75} is that for every $\\phi$ in any reasonable specification formalism (including all the ones mentioned in this paper), $\\phi$ is not agent realizable iff $\\neg \\phi$ is environment realizable. \nThus, $\\neg (\\omega \\supset \\gamma)$ is environment-realizable, i.e., \n$\\exists \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\omega \\wedge \\neg \\gamma$. Note in particular that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\omega$, i.e., $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega$. Now, suppose for a contradiction that $2$ holds, and take $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizing $\\gamma$ assuming $\\omega$. Then by definition of realizability under assumptions and using the fact that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega$ we have that $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma$. On the other hand, we have already seen that $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\neg \\gamma$, a contradiction.\n\\end{proof}\n\nMoreover, we see that one can actually extract a strategy solving \nsynthesis by assumptions simply by extracting a strategy for solving\nthe implication $\\omega \\supset \\gamma$, which itself can be done by\nknown results, i.e., for \\text{{LTL}}\\xspace use Theorem~\\ref{fact:synthesis} (part 1), and for \\text{{DPW}}\\xspace use Lemma~\\ref{lem:DPW Boolean combinations} and Theorem~\\ref{fact:synthesis} (part 2).\n\\begin{theorem}\\label{thm:solving:SUA}\n\\begin{enumerate} \n\\item Solving \\text{{LTL}}\\xspace synthesis under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\item Solving \\text{{DPW}}\\xspace synthesis under assumptions is in \\textsc{ptime}\\xspace in the size of the automata and in \\textsc{exptime}\\xspace in the number of colors of the automata.\n\\end{enumerate}\n\\end{theorem}\n\n\n\n\n\\section{Planning under Assumptions} \\label{sec:Planning Under Assumptions}\nIn this section we define planning under assumptions,\n{that is synthesis wrt a domain\\footnote{Domains can be thought of as compact representations of the arenas in games on graphs~\\cite{ALG02}. The player chooses actions, also represented compactly, and the environment resolves the nondeterminism. In addition, not every action needs to be available in every vertex of the arena.}.}\nWe begin with a representation of fully-observable non-deterministic (FOND) domains~\\cite{GhNT04,GeBo13}. Our representation considers actions symmetrically to fluents, i.e., as assignments to certain variables.\n\nA \\emph{domain} $D = (E,A,I,Pre,\\Delta)$ consists of:\n\\begin{itemize} \n \\item a non-empty set $E$ of \\emph{environment} Boolean variables, also called \\emph{fluents}; \n the elements of $\\modecal{E}} \\newcommand{\\F}{\\modecal{F} = 2^E$ are called \\emph{environment states},\n \\item a non-empty set $A$ (disjoint from $E$) of \\emph{action} Boolean variables; the elements of  $\\modecal{A}} \\newcommand{\\B}{\\modecal{B} = 2^A$ are called \\emph{actions}, \n \\item a non-empty set $I \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ of \\emph{initial environment states},\n \\item a relation $Pre \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ of \\emph{available actions} such that for every $s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ there is an $a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ with $(s,a) \\in Pre$ (we say that $a$ is \\emph{available} in $s$), and \n \\item a relation $\\Delta \\subseteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B} \\times \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ such that $(s,a,t) \\in \\Delta$ implies that $(s,a) \\in Pre$.\n\\end{itemize}\n\nAs is customary in planning and reasoning about actions, we assume domains are represented compactly by tuples $(E,A,init,pre,\\delta)$ where \n$init \\in Bool(E)$, \n$pre \\in Bool(E \\cup A)$, \nand $\\delta \\in Bool(E \\cup A \\cup E')$ (here $E' \\doteq \\{e' : e \\in E\\}$). This data induces the domain $(E,A,I,Pre,\\Delta)$ where \n\\begin{enumerate}\n \\item $s \\in I$ iff $s \\models init$,\n \\item $(s,a) \\in Pre$ iff $s \\cup a \\models pre$, \n \\item $(s,a,t) \\in \\Delta$ iff $s \\cup a \\cup \\{e' : e \\in t\\} \\models \\delta$.\n\\end{enumerate}\n\nWe emphasize that when measuring the size of $D$ we use this compact representation:\n\\begin{definition} \nThe \\emph{size of $D$}, written $|D|$, is $|E| + |A| + |init| + |pre| + |\\delta|$. \n\\end{definition}\n\nWe remark that in PDDL action preconditions are declared using $\\texttt{:precondition}$, conditional effects using the $\\texttt{when}$ operator, and nondeterministic outcomes using the $\\texttt{oneof}$ operator (note that we code actions with action variables).\n\n\\begin{example}[Universal Domain]\nGiven $E$ and $A$ define the \\emph{universal} domain $U = (E,A,I,Pre,\\Delta)$ where $I \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$, $Pre \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ and $\\Delta \\doteq \\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B} \\times \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$. \n\\end{example}\n\nWe now define the set of environment strategies induced by a domain. We do this by describing a property $\\omega_D$, that itself can be \nrepresented in \\text{{LTL}}\\xspace and \\text{{DPW}}\\xspace, as shown below.\n\n\\begin{definition} \\label{dfn:domain-env} \nFix a domain $D$.\nDefine a property $\\omega_D$ (over atoms $E \\cup A$) as consisting of all traces $\\pi = \\pi_0 \\pi_1 \\ldots$ such that \n\\begin{enumerate} \n\\item $\\pi_0 \\in I$ and \n\\item for all $n \\geq 1$, if $\\pi_i \\cap A$ is available in $\\pi_i \\cap E$ for every $i \\in [0,n-1]$ then $(\\pi_{n-1} \\cap E, \\pi_{n-1} \\cap A, \\pi_{n} \\cap E) \\in \\Delta$.\n\\end{enumerate}\n\n\\end{definition}\nObserve that $\\omega_D$ is an environment assumption since, by the definition of domain, whenever an action is available in a state there is at least one possible successor state. \nIntuitively, an environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ is in $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D)$ if i) its first move is to pick an initial environment state, and ii) thereafter, if the current action $a$ is available in the current environment state $x$ (and the same holds in all earlier steps) then the next environment state $y \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ is constrained so that $(x,a,y) \\in \\Delta$. Notice that $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ is unconstrained the moment $a$ is not available in $x$, e.g., in PDDL these would be actions for which the preconditions are not satisfied. Intuitively, this means that it is in the interest of the agent to play available actions because otherwise the agent can't rely {on the fact that the trace comes from the domain.}\n\n\\begin{remark}\nThe reader may be wondering why the above definition does not say i') $\\pi_0 \\in I$ and ii') for all $n \\geq 1$, $(\\pi_{n-1} \\cap E, \\pi_{n-1} \\cap A, \\pi_{n} \\cap E) \\in \\Delta$. Consider the linear-time property $\\omega'_D$ consisting of traces $\\pi$ satisfying i' and ii'. Observe that, in general, $\\omega'_D$ is not environment realizable. Indeed, condition ii' implies that $\\pi_n \\cap A$ is available in $\\pi_n \\cap E$. However, no environment strategy can force the agent to play an available action.\n\\end{remark}\n\n\nWe now observe that one can express $\\omega_D$ in \\text{{LTL}}\\xspace.\n\n\\begin{lemma} \\label{lem:omegaD:LTL}\nFor every domain $D$ there is an \\text{{LTL}}\\xspace formula equivalent to $\\omega_D$. \nFurthermore, the size of the \\text{{LTL}}\\xspace formula is linear in the size of $D$. \n\\end{lemma}\n\nTo see this, say domain $D = (E,A,I,Pre,\\Delta)$ is represented compactly by $(E,A,init,pre,\\delta)$. For the \\text{{LTL}}\\xspace formula, let $\\delta'$ be the $\\text{{LTL}}\\xspace(E \\cup A)$ formula formed from the formula $\\delta \\in Bool(E \\cup A \\cup E')$ by replacing every term of the form $e'$ by $\\nextX e$. Note that $(\\pi,n) \\models \\delta'$ iff $(\\pi_n \\cap E, \\pi_n \\cap A, \\pi_{n+1} \\cap E) \\in \\Delta$. \nThe promised $\\text{{LTL}}\\xspace(E \\cup A)$ formula is \n\\[ init \\wedge (\\always \\delta' \\vee \\delta' \\until \\neg pre).\\]\n\n{One can also express $\\omega_D$ directly by a \\text{{DPW}}\\xspace.}\n\\begin{lemma} \\label{lem:omegaD:DPW}\nFor every domain $D$ there is a \\text{{DPW}}\\xspace $M_D$ equivalent to $\\omega_D$. \nFurthermore, the size of the \\text{{DPW}}\\xspace is at most exponential in the size of $D$ and has two colors.\n\\end{lemma}\n\nTo do this we define the \\text{{DPW}}\\xspace directly rather than translate the \\text{{LTL}}\\xspace formula (which would give a double exponential bound).  \nDefine the \\text{{DPW}}\\xspace $M_D \\doteq (Q,q_{in},T,col)$ over $E \\cup A$ as follows. \nIntroduce fresh symbols $q_{in}, q_{+},q_{-}$. Let $q_{in}$ be the initial state. Define $Q \\doteq \\{q_{in},q_{+},q_{-}\\} \\cup (\\modecal{E}} \\newcommand{\\F}{\\modecal{F} \\times \\modecal{A}} \\newcommand{\\B}{\\modecal{B})$.\nDefine $col(q_{-}) = 1$, and $col(q) = 0$ for all $q \\neq q_{-}$.\nFor all $e,e' \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}, a,a' \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ the transitions are given in Table~\\ref{tab:trans}. Intuitively, on reading the input $e' \\cup a'$ the \\text{{DPW}}\\xspace goes to the rejecting sink $q_{-}$ if $\\Delta$ (resp. $I$) is not respected, \nit goes to the accepting sink $q_{+}$ if $\\Delta$ (resp. $I$) is respected but $Pre$ is not, \nand otherwise it continues (and accepts).\n  \\begin{table}[h!]\n \\begin{tabular}{llll}\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $q_{-}$   & if $e' \\not \\in I$\\\\\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $(e',a')$ & if $e' \\in I$ and $(e',a') \\in Pre$ \\\\\n$q_{in}$ & $\\xrightarrow{e' \\cup a'}$& $q_{+}$   & if $e' \\in I$ and $(e',a') \\not \\in Pre$\\\\\n$(e,a)$  & $\\xrightarrow{e' \\cup a'}$& $q_{-}$   & if $(e,a,e') \\not \\in \\Delta$\\\\\n$(e,a)$  & $\\xrightarrow{e' \\cup a'}$& $(e',a')$ & if $(e,a,e') \\in \\Delta$ and $(e',a') \\in Pre$\\\\\n$(e,a)$  & $\\xrightarrow{e' \\cup a'}$& $q_{+}$   & if $(e,a,e') \\in \\Delta$ and $(e',a') \\not \\in Pre$\\\\\n$q_{-}$  & $\\xrightarrow{e' \\cup a'}$& $ q_{-}$  &\\\\\n$q_{+}$  & $\\xrightarrow{e' \\cup a'}$& $ q_{+}$  &\\\\\n\\end{tabular}\n  \\caption{Transitions for \\text{{DPW}}\\xspace for $\\omega_D$}\n  \\label{tab:trans}\n\\end{table}\n\n\n\\begin{definition}\\label{dfn:planning:assumption}\nLet $D$ be a domain. \n\\begin{itemize} \\item A set $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ is an \\emph{environment assumption for the domain $D$} if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap \\Omega$ is non-empty. \n \\item $\\omega \\in \\SF$ is an \\emph{environment assumption for the domain $D$} if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega)$ is non-empty, i.e.,  if $Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega)$ is an environment assumption for the domain $D$.\n\\end{itemize}\n\n\\end{definition}\n\n\n{We illustrate the notion with some examples.}\n\\begin{example} \\label{ex} \\hspace{0cm}\n\\begin{enumerate} \n \n \\item $\\omega \\doteq \\mathsf{true}$ is an environment assumption for $D$ since $\\omega_D \\wedge \\omega \\equiv \\omega_D$ is environment realizable.\n \n \\item Let $\\omega_{D,fair}$ denote the following property: $\\pi \\in \\omega_{D,fair}$ iff for all $(s,a) \\in Pre$, if there are infinitely many $n$ such that $s = \\pi_n \\cap E$ and $a = \\pi_n \\cap A$, then for every $t \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ with $(s,a,t) \\in \\Delta$ there are infinitely many $n$ such that $s = \\pi_n \\cap E, a = \\pi_n \\cap A$ and $t = \\pi_{n+1} \\cap E$. In words, this says that if a state-action pair occurs infinitely often, then infinitely often this is followed by every possible effect.\n \n Note that $\\omega_{D,fair}$ is an environment assumption for domain $D$ since, e.g., the strategy that\n resolves the effects in a round-robin way realizes $\\omega_D \\wedge \\omega_{D,fair}$. Note that $\\omega_{D,fair}$ is definable in \\text{{LTL}}\\xspace by a formula of size exponential in $D$: \n \\[\n\\bigwedge_{s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}} \\bigwedge_{a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}} (\\always \\eventually (s \\wedge a)  \\supset \\bigwedge_{s': (s,a,s') \\in \\Delta} \\always \\eventually (s \\wedge a \\wedge \\nextX s')). \n\\]\n\n \n\n\\item In planning, trajectory constraints, e.g., expressed in LTL,  have been introduced for expressing temporally extended goals ~\\cite{BacchusK00,DBLP:journals\/ai\/GereviniHLSD09}.\nMore recently, especially in the context of generalized planning, they have been used to describe restrictions on the environment as well \\cite{DBLP:conf\/ijcai\/BonetG15,DeGiacomoMRS16,DBLP:conf\/ijcai\/BonetGGR17}. \nHowever, not all trajectory constraints $\\omega$ can be used as assumptions.  In fact, Definition~\\ref{dfn:planning:assumption}, which says that a formula $\\omega$ is an environment assumption for the domain $D$ if $\\omega_D \\wedge \\omega$ is environment realizable, characterizes those formulas that can serve as trajectory constraints.\n\\end{enumerate}\n\\end{example}\n\n\n\nWe can check if $\\omega \\in \\text{{LTL}}\\xspace$ is an environment assumption for $D$ by converting it to a DPW $M_\\omega$, converting $D$ into the \\text{{DPW}}\\xspace $M_D$ (as above), and then checking if the \\text{{DPW}}\\xspace $M_D \\wedge M_\\omega$ is environment realizable. Hence we have:\n\\begin{theorem} \n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTL}}\\xspace$ formula $\\omega$ is an environment assumption for the domain $D$ is $2$\\textsc{exptime}\\xspace-complete. Moreover, it can be solved in \n \\textsc{exptime}\\xspace in the size of $D$ and $2$\\textsc{exptime}\\xspace in the size of $\\omega$. \n \\item Deciding if a \\text{{DPW}}\\xspace $\\omega$ is an environment assumption for the domain $D$ is in \\textsc{exptime}\\xspace. Moreover, it can be solved in \\textsc{exptime}\\xspace in the size \n of $D$ and \\textsc{ptime}\\xspace in the size of $\\omega$ and \\textsc{exptime}\\xspace in the number of colors of $\\omega$.\n\\end{enumerate}\n\\end{theorem}\nFor the lower bound take $D \\doteq U$ to be the universal domain and apply the lower bound from Theorem~\\ref{fact:synthesis}.\n\n\n{Now we turn to planning under assumptions.}\n\\begin{definition}[Planning under Assumptions -- abstract] \\label{def:PUA}\\hspace{0cm}\n\\begin{enumerate}\n\\item A \\emph{planning under assumptions problem $P$} is a tuple $((D,\\Omega),\\Gamma)$ where \n \\begin{itemize} \n  \\item $D$ is a domain,\n  \\item $\\Omega \\subseteq Str_\\ensuremath{\\mathsf {env}}\\xspace$ is an environment assumption for $D$, and \n  \\item $\\Gamma$ is an agent goal.\n \\end{itemize}\n \\item  We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{solves} $P$ if \n \\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\in Str_\\ensuremath{\\mathsf {env}}\\xspace(\\omega_D) \\cap \\Omega. \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in \\Gamma\\]  \n\\end{enumerate}\n\\end{definition}\n\n\n\nWe can instantiate this definition to environment assumptions and agent goals definable in $\\SF$.\n\\begin{definition}[Planning under Assumptions -- linear-time] \\hspace{0em}\n\\begin{enumerate}\n \\item An \\emph{$\\SF$ planning under assumptions problem} is a tuple $P = ((D,\\omega),\\gamma)$ where $\\omega \\in \\SF$ is an environment assumption for $D$ and $\\gamma \\in \\SF$ is an agent goal.\n \\item We say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{realizes $\\gamma$ assuming $\\omega$}, or that it \\emph{solves} $P$, if \n \\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright (\\omega_D \\wedge \\omega). \\, \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma\\] \n\\end{enumerate}\n\\end{definition}\n\nThe corresponding decision problem asks, given an \\SF planning under assumptions problem $P$ to decide whether there is an agent strategy that solves $P$. For instance, \\emph{\\text{{LTL}}\\xspace planning under assumptions} asks, given $P = ((D,\\omega),\\gamma)$ with $\\omega,\\gamma \\in \\text{{LTL}}\\xspace$, \nto decide if there is an agent strategy that solves $P$, and to return such a finite-state strategy (if one exists). Similar definitions apply to \\text{{DPW}}\\xspace planning under assumptions, etc.\n\n \n\n \n{It turns out that {virtually all  forms of planning {(with linear-time temporally extended goals)}} in the literature are special cases of planning under \\text{{LTL}}\\xspace assumptions, i.e., {the set of strategies that solve a given planning problem are exactly the set of strategies that solve the corresponding planning under assumptions problem.} \nIn the following, $\\mathit{Goal} \\in Bool(E \\cup A)$, and \n$\\mathit{Exec}$ is the \\text{{LTL}}\\xspace formula $\\always \\bigwedge_{a \\in A} (a \\supset \\mathit{pre_a})$ expressing that if an action is done then its precondition holds.}\n\n\n\\begin{example}\n\n \\begin{enumerate}\n\\item FOND planning with reachability goals~\\cite{Rintanen:ICAPS04} corresponds to \\text{{LTL}}\\xspace planning under assumptions with $\\omega \\doteq \\mathsf{true}$ and {$\\gamma \\doteq \\mathit{Exec} \\land \\eventually Goal$.}\n\n\\item FOND planning with \\text{{LTL}}\\xspace (temporally extended) goals $\\gamma$ \n\\cite{BacchusK00,PistoreT01,CTMBM17}.\ncorresponds to \\text{{LTL}}\\xspace planning under assumptions with $\\omega \\doteq \\mathsf{true}$ {and goal $\\mathit{Exec} \\land \\gamma$.}\n\n\\item FOND planning with \\text{{LTL}}\\xspace trajectory constraints $\\omega$ and \\text{{LTL}}\\xspace (temporally extended) goals $\\gamma$ \\cite{DBLP:conf\/ijcai\/BonetG15,DeGiacomoMRS16,DBLP:conf\/ijcai\/BonetGGR17} corresponds to \\text{{LTL}}\\xspace planning under assumptions {with assumptions $\\omega$ and goal $\\mathit{Exec} \\land \\gamma$.}\n\n\\item {Fair FOND planning} with reachability goals~\\cite{DaTV99,GeBo13,DIppolitoRS18} corresponds to planning under assumptions with \n$\\omega \\doteq \\omega_{D,fair}$ and $\\gamma \\doteq \\textit{Exec} \\land \\eventually Goal$.\n\n\\item {Fair FOND planning} with (temporally extended) goals $\\gamma$ as defined in~\\cite{DBLP:conf\/ijcai\/PatriziLG13,CTMBM17} corresponds to planning under assumptions with $\\omega \\doteq \\omega_{D,fair}$ and goal $\\mathit{Exec} \\land \\gamma$.\n\n\\item Obviously adding \\text{{LTL}}\\xspace trajectory constraints $\\omega_{tc}$ to {fair FOND planning} with (temporally extended) goals  corresponds to planning under assumptions with $\\omega \\doteq \\omega_{D,fair} \\wedge \\omega_{tc}$ {and goal $\\mathit{Exec} \\land \\gamma$.} \n\\end{enumerate}\n\n\\end{example}\n\n\n{We also observe that the Fair FOND planning problems just mentioned can be captured by \\text{{LTL}}\\xspace planning under assumptions since $\\omega_{D,fair}$ can be written in \\text{{LTL}}\\xspace (see Example~\\ref{ex}).\n\n\n\n\n\n\\section{Translating between planning and synthesis}\nIn this section we ask the question if there is a {fundamental} difference between synthesis and planning in our setting (i.e., assumptions and goals given as linear-time properties). We answer by observing that there are translations between them. The next two results follow immediately from the definitions:\n\n\\begin{theorem}[Synthesis to Planning] \\label{prop:synthesis to planning}\nLet $(E,A,\\omega,\\gamma)$ be a synthesis under Assumptions problem, and let \n$P = ((U,\\omega),\\gamma)$ be the corresponding \nPlanning under Assumptions problem where $U$ is the universal domain. Then, for every agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ we have that \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ solves $P$ iff $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega$.\n\\end{theorem}\n\n\\begin{theorem}[Planning to Synthesis] \\label{prop:planning to synthesis}\nLet $D = (E,A,I,Pre,\\Delta)$ be a domain and let $P = ((D,\\omega),\\gamma)$ be a Planning under Assumptions problem. \nLet $(E,A,\\omega_D \\wedge \\omega,\\gamma)$ be the corresponding Synthesis under Assumptions problem. Then, for every agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ we have that \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ solves $P$ iff $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\gamma$ assuming $\\omega_D \\wedge \\omega$.\n\\end{theorem}\n\nThus, we can solve \\text{{LTL}}\\xspace planning under assumptions by reducing to \\text{{LTL}}\\xspace synthesis under assumptions, which itself can be solved by known results (i.e., Theorem~\\ref{fact:synthesis}):\n\\begin{corollary}\\label{thm:solving:PUA:LTL:combined}\n Solving \\text{{LTL}}\\xspace planning under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\end{corollary}\nHowever, this does not distinguish the complexity measured in the size of the domain from that in the size of the assumption and goal formulas. \n{We take this up next.}\n\n\n\n\\section{Solving Planning under Assumptions}\n\nIn this section we show how to solve Planning under Assumptions for concrete specification languages \\SF, i.e., \\SF = \\text{{LTL}}\\xspace and \\SF = \\text{{DPW}}\\xspace. We measure the complexity in two different ways: we fix the domain $D$ and measure the complexity with respect to the size of the formulas or automata for the environment assumption and the agent goal, this is called \\emph{goal\/assumption complexity}; \nand  we fix the formulas\/automata and measure the complexity with respect to the size of the domain, this is called the \\emph{domain complexity}. \\footnote{Formally, if $C$ is a complexity class, we say that \\emph{goal\/assumption complexity is in $C$} if for every domain $D_0$ the complexity of deciding if there is an agent strategy solving $P = ((D_0,\\omega),\\gamma)$, is in $C$. A similar definition holds for domain complexity. Also, we say that the \\emph{goal\/assumption complexity is $C$-hard} if there exists a domain $D_0$ such that the problem of deciding if there is an agent strategy solving $P = ((D_0,\\omega),\\gamma)$, is $C$-hard.} \n\n\nWe begin with \\SF = \\text{{DPW}}\\xspace and consider the following algorithm:\nGiven $P = ((D,\\omega),\\gamma)$ in which $\\omega$ is represented by a \\text{{DPW}}\\xspace $M_\\omega$ and $\\gamma$ is represented by a \\text{{DPW}}\\xspace $M_\\gamma$, perform the following steps:\n\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 1. Solving \\text{{DPW}}\\xspace planning under assumptions}\\\\\nGiven domain $D$, assumption $M_\\omega$, goal $M_\\gamma$.\\\\\n1:\\>Form \\text{{DPW}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form \\text{{DPW}}\\xspace $M$ for $(M_D \\wedge M_\\omega) \\supset M_\\gamma$.\\\\\n3:\\>Solve the parity game on $M$.\n\\end{tabbing}\n\nThe first step results in a \\text{{DPW}}\\xspace whose size is exponential in the size of $D$ and with a constant number of colors (Lemma~\\ref{lem:omegaD:DPW}). \nThe second step results in a \\text{{DPW}}\\xspace whose size is polynomial in the number of states of the \\text{{DPW}}\\xspace{s} involved (i.e., $M_D,M_\\omega$ and $M_\\gamma$), and exponential in the number of their colors (Lemma~\\ref{lem:DPW Boolean combinations}).\nFor the third step, the think of the \\text{{DPW}}\\xspace $M$ as a parity game: play starts in the initial state, and at each step, if $q$ is the current state of $M$, first the environment picks $s \\in \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ and then the agent picks an action $a \\in \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$, i.e., an evaluation of the action variables. The subsequent step starts in the state of $M$ resulting from taking the unique transition from $q$ labeled $s \\cup a$. This produces a run of the \\text{{DPW}}\\xspace which the agent is trying to ensure is successful (i.e., the largest color occurring infinitely often is even). \n\nFormally, we say that an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ is \\emph{winning} if \nfor every environment strategy $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$, the unique run of the \\text{{DPW}}\\xspace on input word $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is successful. \nDeciding if the a player has a winning strategy, and returning a finite-state strategy (it one exists), is called \\emph{solving} the game. \nParity games can be solved in \ntime polynomial in the size of $M$ and exponential in the number of colors of $M$~\\cite{ALG02}.\\footnote{Better algorithms are known, e.g.~\\cite{CaludeJKL017}, but are not helpful for this paper.}  \n\n{The analysis of the above algorithm shows the following.}\n\\begin{theorem} \\label{thm:solving:PUA:DPW} \\hspace{0em}\n\\begin{enumerate} \\item The domain complexity of solving \\text{{DPW}}\\xspace planning under assumptions is in \\textsc{exptime}\\xspace. \n\\item The goal\/assumption complexity of solving \\text{{DPW}}\\xspace planning under assumptions is in \\textsc{ptime}\\xspace in their sizes and \\textsc{exptime}\\xspace in the number of their colors.\n\\end{enumerate}\n\\end{theorem}\n\n{Moreover,} by converting \\text{{LTL}}\\xspace formulas to \\text{{DPW}}\\xspace with exponentially many colors and double-exponential many states \\cite{DBLP:conf\/banff\/Vardi95,DBLP:journals\/lmcs\/Piterman07}, we get the upper bounds in the following:\n\\begin{theorem} \\label{thm:solving:PUA:LTL}\n\\begin{enumerate}\n\\item The domain complexity of solving \\text{{LTL}}\\xspace planning under assumptions is \\textsc{exptime}\\xspace-complete.\n\\item The goal\/assumption complexity of solving \\text{{LTL}}\\xspace planning under assumptions is $2$\\textsc{exptime}\\xspace-complete. \n\\end{enumerate}\n\\end{theorem}\n{For the matching lower-bounds, we have that}\nthe domain complexity is \\textsc{exptime}\\xspace-hard follows from the fact that planning with reachability goals and no assumptions is \\textsc{exptime}\\xspace-hard~\\cite{Rintanen:ICAPS04}; to see that the goal\/assumption complexity is $2$\\textsc{exptime}\\xspace-hard note that \\text{{LTL}}\\xspace synthesis, known to be $2$\\textsc{exptime}\\xspace-hard~\\cite{PnueliR89,rosner1992modular}, is a special case (take $\\omega \\doteq \\mathsf{true}$ and $D$ to be the universal domain). \n\n{Similarly, one can apply this technique to solving Fair \\text{{LTL}}\\xspace planning under assumptions. The exact complexity, however, is open. See the conclusion for a discussion.}\n\n\n\n\n\n\\section{Focusing on finite traces}\nIn this section we revisit the definitions and results in case that assumptions and goals are expressed as linear-time properties over \\emph{finite} traces. There are two reasons to do this. \nFirst,  in AI and CS applications executions of interest are often finite~\\cite{DegVa13}.\nSecond, the algorithms presented for the infinite-sequence case involve complex constructions on automata\/games that are notoriously hard to optimize~\\cite{DFogartyKVW13}. Thus, we will not simply reduce the finite-trace case to the infinite-trace case~\\cite{GMM14}. We begin by carefully defining the setting.\n\n\\subsubsection{Synthesis and linear-time specifications over finite traces}\nWe define synthesis over finite traces in a similar way to the infinite-trace case, {cf.~\\cite{DegVa15,Camacho:KR18}.}\nThe main difference is that agent strategies \n$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace:\\modecal{E}} \\newcommand{\\F}{\\modecal{F}^+ \\to \\modecal{A}} \\newcommand{\\B}{\\modecal{B}$ can be partial. This represents the situation that the agent stops the play. Environment strategies $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace:\\modecal{A}} \\newcommand{\\B}{\\modecal{B}^* \\to \\modecal{E}} \\newcommand{\\F}{\\modecal{F}$ are total (as before). Thus, the resulting play $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ may be finite, if the agent chooses to stop, as well as infinite.\\footnote{Formally,  $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is redefined to be the longest trace (it may be finite or infinite) that complies with both strategies.} Objectives may be expressed in general specification formalisms \\SFf for finite traces, e.g., \\SFf = \\text{{LTLf}}\\xspace (\\text{{LTL}}\\xspace over finite traces\\footnote{All our results for \\text{{LTLf}}\\xspace also hold for linear-dynamic logic over finite traces (\\text{{LDLf}}\\xspace)~\\cite{DegVa13}. \n}), \\SFf = \\text{{DFA}}\\xspace (deterministic finite word automata). For $\\phi \\in \\SFf$, we overload notation and write $[[\\phi]]$ for the set of finite traces $\\phi$ defines.\n\nWe now define realizability in the finite-trace case:\n\\begin{definition}\nLet $\\phi \\in \\SFf$.\n\\begin{enumerate} \n\\item We say that \\emph{$\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace. \\left(\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite and } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\phi]]\\right)$.  \n\\item We say that \\emph{$\\sigma_\\ensuremath{\\mathsf {env}}\\xspace$ realizes $\\phi$ (written $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\phi$)} if $\\forall \\sigma_\\ensuremath{\\mathsf {ag}}\\xspace. \\left(\\text{if } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite, then } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}\\in [[\\phi]]\\right)$. \n\\end{enumerate}\n\\end{definition}\nThe asymmetry in the definition results from the fact that stopping is controlled by the agent.\n\n\nDuality still holds, and is easier to prove since it amounts to determinacy of reachability games~\\cite{ALG02}:\n\\begin{lemma}[Duality] \\label{lem:duality:finite}\nFor every $\\phi \\in \\SFf$ we have that $\\phi$ is not agent realizable iff $\\neg \\phi$ is environment realizable.\n\\end{lemma}\n\n\\subsubsection{Linear temporal logic on finite traces (\\text{{LTLf}}\\xspace)} The logic $\\text{{LTLf}}\\xspace$ has the same syntax as $\\text{{LTL}}\\xspace$ but is interpreted on finite traces $\\pi \\in (2^{\\sf{Var}})^+$. Formally, for $n \\leq len(\\pi)$ (the length of $\\pi$) we only reinterpret the temporal operators:\n\\begin{itemize}\n\t\\item  $(\\pi,n) \\models \\nextX \\varphi$ iff $n < len(\\pi)$ and $(\\pi,n+1) \\models \\varphi$;\n\t\\item  $(\\pi,n) \\models \\varphi_1 \\until \\varphi_2$ iff \n\tthere exists $i$ with $n \\leq i \\leq len(\\pi)$ such that $(\\pi,i) \\models \\varphi_2$ and for all $i \\leq j < n$, $(\\pi,j) \\models \\varphi_1$.\n\\end{itemize}\nLet $\\tilde{\\nextX}$ denote the dual of $\\nextX$, i.e., $\\tilde{\\nextX} \\doteq \\neg \\nextX \\neg \\varphi$. Semantically we have that\n\\begin{itemize}\n\t\\item  $(\\pi,n) \\models \\tilde{\\nextX} \\varphi$ iff $n < len(\\pi)$ implies $(\\pi,n+1) \\models \\varphi$.\n\\end{itemize}\n\n\n\\subsubsection{Deterministic finite automata (\\text{{DFA}}\\xspace)} \n\nA \\text{{DFA}}\\xspace over ${\\sf{Var}}$ is a tuple $M = (Q,q_{in},T,F)$ which is like a \\text{{DPW}}\\xspace except that $col$ is replaced by a set $F \\subseteq Q$ of final states. The run on a finite input trace $\\pi \\in (2^{\\sf{Var}})^*$ is successful if it ends in a final state. We recall that \\text{{DFA}}\\xspace are closed under Boolean operations using classic algorithms (e.g., see \\cite{DBLP:conf\/banff\/Vardi95}).\nAlso, \\text{{LTLf}}\\xspace formulas $\\varphi$ (and also \\text{{LDLf}}\\xspace formulas) can be effectively translated into \\text{{DFA}}\\xspace. This is done in three classic simple steps that highlight the power of the automata-theoretic approach: convert $\\varphi$ to an alternating automaton (poly), then into a nondeterministic finite automaton (exp), and then into a \\text{{DFA}}\\xspace (exp). These steps are outlined in detail in, e.g., \\cite{DegVa13}. \n\n\n\\subsubsection{Solving Synthesis over finite traces} \\SFf agent synthesis is the problem, given $\\phi \\in \\SFf$, of deciding if the agent can realize $\\phi$. Now, solving \\text{{DFA}}\\xspace agent synthesis is \\textsc{ptime}\\xspace-complete: it amounts to solving a reachability game on the given \\text{{DFA}}\\xspace $M$, which can be done with an algorithm that captures how close the agent is to a final state, i.e., a least-fixpoint of the operation. Finally, to solve \\text{{LTLf}}\\xspace agent synthesis first translate the \\text{{LTLf}}\\xspace formula to a \\text{{DFA}}\\xspace and then run the fixpoint algorithm  (also, \\text{{LTLf}}\\xspace agent synthesis is $2$\\textsc{exptime}\\xspace-complete)~\\cite{DegVa15}.\n\n\n\nNote that, by Duality, solving \\text{{LTLf}}\\xspace environment realizability and solving \\text{{LTLf}}\\xspace agent realizability are inter-reducible (and thus the former is also $2$\\textsc{exptime}\\xspace-complete). Thus, to decide if $\\phi$ is environment realizable we simply negate the answer to whether $\\neg \\phi$ is agent realizable. However, to extract an environment strategy, one solves the dual safety game.\n\n\n\n\\subsubsection{Synthesis under assumptions}\nWe say that $\\omega \\in \\SFf$ is an \\emph{environment assumption} if $\\omega$ is environment realizable. Solving \n\\SFf synthesis under assumptions means to decide if there is an agent strategy $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ such that\n\\[\\forall \\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega. \\left(\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\text{ is finite and } \\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\models \\gamma\\right).\\]\n\nWe now consider the case that $\\SFf = \\text{{LTLf}}\\xspace$. Checking if $\\omega \\in \\text{{LTLf}}\\xspace$ is an environment assumption is, by definition, \nthe problem of deciding if $\\omega$ is environment realizable, as just discussed. Hence we can state the following:\n\\begin{theorem} \\hspace{0em}\n\\begin{enumerate}\n \\item Deciding if an $\\text{{LTLf}}\\xspace$ formula $\\omega$ is an environment assumption is $2$\\textsc{exptime}\\xspace-complete. \n \\item Deciding if a $\\text{{DFA}}\\xspace$ $\\omega$ is an environment assumption is \\textsc{ptime}\\xspace-compete (cf.~\\cite{ALG02}).\n\\end{enumerate}\n\\end{theorem}\n\nTurning to \\text{{LTLf}}\\xspace synthesis under assumptions  we have \nthat synthesis under assumptions and synthesis of the implication are equivalent. Indeed, as before, the key point is the duality which we have in \nLemma~\\ref{lem:duality:finite}:\n\n\n\n\n\\begin{theorem} \\label{thm:det:finite}\nSuppose $\\omega \\in \\SFf$ is an environment assumption.\n The following are equivalent:\n \\begin{enumerate}\n  \\item There is an agent strategy realizing $\\omega \\supset \\gamma$.\n  \\item There is an agent strategy realizing $\\gamma$ assuming $\\omega$.\n \\end{enumerate}\n\\end{theorem}\n\nHence to solve synthesis under assumptions we simply solve agent synthesis for the implication. Hence we have:\n\\begin{theorem}\\label{thm:solving:SUA:finite} \\hspace{0em}\n\\begin{enumerate} \n\\item Solving \\text{{LTLf}}\\xspace synthesis under assumptions is $2$\\textsc{exptime}\\xspace-complete.\n\\item Solving \\text{{DFA}}\\xspace synthesis under assumptions is \\textsc{ptime}\\xspace-complete.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsubsection{Planning under assumptions}\nPlanning and fair planning have recently been studied for \\text{{LTLf}}\\xspace goals~\\cite{DR-IJCAI18,Camacho:KR18,Camacho:ICAPS18}. Here we define and study how to add environment assumptions.\n\nRecall that we represent a planning domain $D$ by the linear-time property $\\omega_D$ (Definition~\\ref{dfn:domain-env}) which itself was defined as those infinite traces satisfying two conditions. The exact same conditions determine a set of finite traces, also denoted $\\omega_D$. Moreover, \nthis $\\omega_D$ is equivalent to an \\text{{LTLf}}\\xspace formula of size linear in $D$ and a \\text{{DFA}}\\xspace of size at most exponential in $D$. \nTo see this, replace $\\nextX$ by $\\tilde{\\nextX}$ in the \\text{{LTL}}\\xspace formula from Lemma~\\ref{lem:omegaD:LTL}. That is, \nlet $\\delta''$ be the $\\text{{LTLf}}\\xspace$ formula formed from $\\delta$ by replacing every term of the form $e'$ by $\\tilde{\\nextX} e$. Note that if \n$n < len(\\pi)$ then $(\\pi,n) \\models \\delta''$ iff $(\\pi_n \\cap E, \\pi_n \\cap A, \\pi_{n+1} \\cap E) \\in \\Delta$, and if $n = len(\\pi)$ then $(\\pi,n) \\models \\delta''$ iff $(\\pi_n \\cap E, \\pi_n \\cap A) \\in Pre$. \nThe promised $\\text{{LTLf}}\\xspace(E \\cup A)$ formula is \n$init \\wedge (\\always \\delta'' \\vee \\delta'' \\until \\neg pre)$. \nAlso, similar to the \\text{{DPW}}\\xspace before there is a \\text{{DFA}}\\xspace of size at most exponential in the size of $D$ equivalent to $\\omega_D$. \nTo see this, take the \\text{{DPW}}\\xspace $M_D \\doteq (Q,q_{in},T,col)$ from Lemma~\\ref{lem:omegaD:DPW} and instead of $col$ define the set \nof final states to be the set $col^{-1}(0)$.\n\nAs before, say that $\\omega \\in \\SFf$ is an \\emph{environment assumption for the domain $D$} if $\\omega_D \\wedge \\omega$ is environment realizable. \nDefine an \\emph{\\SFf planning under assumptions problem} to be a tuple $P = ((D,\\omega),\\gamma)$ with $\\omega,\\gamma \\in \\SFf$ \nsuch that $\\omega$ is an environment assumption for $D$. To decide if $\\omega \\in \\text{{LTLf}}\\xspace\/\\text{{DFA}}\\xspace$ is an environment assumption for $D$ \nwe use the next algorithm:\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 2. Deciding if $\\omega$ is an environment assumption for $D$}\\\\\nGiven domain $D$, and \\text{{DFA}}\\xspace $M_\\omega$.\\\\\n1:\\>Convert $D$ into a \\text{{DFA}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form the \\text{{DFA}}\\xspace $M$ for $(M_D \\wedge M_\\omega)$.\\\\\n3:\\>Decide if $M$ is environment realizable.\n\\end{tabbing}\nFurther, if $\\omega$ is given as an \\text{{LTLf}}\\xspace formula, first convert it to a \\text{{DFA}}\\xspace $M_\\omega$ and then run the algorithm. We then have:\n\\begin{theorem}  \\hspace{0em}\n\\begin{enumerate}\n \\item Deciding if $\\text{{LTLf}}\\xspace$ formula $\\omega$ is an environment assumption for the domain $D$ is $2$\\textsc{exptime}\\xspace-complete. Moreover, it can be solved in \n \\textsc{exptime}\\xspace in the size of $D$ and $2$\\textsc{exptime}\\xspace in the size of $\\omega$. \n \\item Deciding if $\\text{{DFA}}\\xspace$ $\\omega$ is an environment assumption for the domain $D$ is in \\textsc{exptime}\\xspace. Moreover, it can be solved in \\textsc{exptime}\\xspace in the size \n of $D$ and \\textsc{ptime}\\xspace in the size of $\\omega$.\n\\end{enumerate}\n\\end{theorem}\n\n\n\\subsubsection{Solving Planning under Assumptions}\nAs before, there are simple translations between \\SFf planning under assumptions and \\SFf synthesis under assumptions. And again, solving \\text{{LTLf}}\\xspace planning under assumptions via such a translation is not fine enough to analyze the complexity in the domain vs the goal\/assumption. \nTo solve \\text{{DFA}}\\xspace\/\\text{{LTLf}}\\xspace planning under assumptions use the following simple algorithm:\n\n \\begin{tabbing}\n===\\===\\===\\===\\=\\+\\kill\n\\textbf{Alg 3. Solving \\text{{DFA}}\\xspace planning under assumptions}\\\\\nGiven domain $D$, assumption $M_\\omega$, goal $M_\\gamma$.\\\\\n1:\\>Convert $D$ into a \\text{{DFA}}\\xspace $M_D$ equivalent to $\\omega_D$.\\\\\n2:\\>Form the \\text{{DFA}}\\xspace $M$ for $(M_D \\wedge M_\\omega) \\supset M_\\gamma$.\\\\\n3:\\>Solve the reachability game on \\text{{DFA}}\\xspace $M$.\n\\end{tabbing}\n\nFurther, if $\\omega$ is given as an \\text{{LTLf}}\\xspace formula, first convert it to a \\text{{DFA}}\\xspace $M_\\omega$ and then run the algorithm. This gives the upper bounds in the following:\n\\begin{theorem} \\label{thm:solving:PUA:DFW} \\hspace{0em}\n\\begin{enumerate} \n\\item The domain complexity of solving \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) planning under assumptions is \\textsc{exptime}\\xspace-complete.\n\\item The goal\/assumption complexity of solving \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) planning under assumptions is \\textsc{ptime}\\xspace-complete (resp. $2$\\textsc{exptime}\\xspace-complete). \n\\end{enumerate}\n\\end{theorem}\nFor the lower bounds, setting $\\omega \\doteq \\mathsf{true}$ results in FOND with reachability goals, known to be \\textsc{exptime}\\xspace-hard~\\cite{Rintanen:ICAPS04}; and additionally taking the domain $D$ to be the universal domain results in \\text{{DFA}}\\xspace (resp. \\text{{LTLf}}\\xspace) synthesis, known to be \\textsc{ptime}\\xspace-hard~\\cite{ALG02} (resp. $2$\\textsc{exptime}\\xspace-hard~\\cite{DegVa15}).\n\nFinally, if $P = ((D,\\omega),\\gamma)$ is an \\text{{LTLf}}\\xspace planning under assumptions problem, say that $\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace$ \\emph{fairly solves} $P$ if for every $\\sigma_\\ensuremath{\\mathsf {env}}\\xspace \\vartriangleright \\omega_D \\wedge \\omega$ we have that if $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace} \\in [[\\omega_{D,fair}]]$ then $\\pi_{\\sigma_\\ensuremath{\\mathsf {ag}}\\xspace,\\sigma_\\ensuremath{\\mathsf {env}}\\xspace}$ is finite and satisfies $\\gamma$ (here $\\omega_{D,fair}$ from Example~\\ref{ex} is defined so that it now also includes all finite traces). We remark that Alg $2$ applies unchanged. \n{However, to solve the fair \\text{{LTLf}}\\xspace planning problem, we do not know a better way, in general, than translating the problem into one over infinite traces and applying the techniques from the previous section.}\n\n\n\n\n\\section{Conclusion and Outlook}\n \n\nWhile we illustrate synthesis and planning under assumptions expressed in\nlinear-time specifications, our definitions immediately apply to assumptions\nexpressed in branching-time specifications, e.g., $\\ensuremath{\\mathsf{CTL}^*}\\xspace$, $\\mu$-calculus, and\ntree automata.  As future work, it is of great interest to study synthesis\nunder assumptions in the branching time setting so as to devise restrictions on\n\\emph{possible agent behaviors} with certain guarantees, e.g., remain in an\narea from where the agent can enforce the ability to reach the recharging doc,\nwhenever it needs to, in the spirit of~\\cite{LagoPT02}.\n\nAlthough our work is in the context of reasoning about actions and planning, we\nexpect it can also provide insights to verification and to multi-agent systems.\nIn particular, the undesirable drawback of the agent being able to falsify an\nassumption when synthesizing $Assumption \\supset Goal$ is well known, and it has\nbeen observed that it can be overcome when the $Assumption$ is environment\nrealizable~\\cite{DBLP:journals\/tosem\/DIppolitoBPU13,DBLP:journals\/acta\/BrenguierRS17}.\nOur Theorem~\\ref{thm:det} provides the principle for such a solution.\nInterestingly, various degrees of cooperation to fulfill assumptions among\nadversarial agents has been considered, e.g.,\n\\cite{CH07,BEK15,DBLP:journals\/acta\/BrenguierRS17} and we believe that a work\nlike present one is needed to establish similar principled foundations.\n\nTurning to the multi-agent setting, there, agents in a common environment\ninteract with each other and may have their own objectives. Thus, it makes\nsense to model agents not as hostile to each other, but as rational, i.e.,\nagents that act to achieve their own objectives. \\emph{Rational\nsynthesis}~\\cite{KPV14} (as compared to classic synthesis) further requires\nthat the strategy profile chosen by the agents is in equilibrium (various\nnotions of equilibrium may be used). It would be interesting to investigate\nrational synthesis under environment assumptions, in the sense that all agents\nalso make use of their own assumptions about their common environment. We\nbelieve that considering assumptions as sets of strategies rather than sets of\ntraces will serve as a clarifying framework also for the multi-agent setting.\n\n{Finally, there are a number of open questions regarding the computational complexity \nof solving synthesis\/planning under assumptions, i.e., what is the exact complexity of {Fair \\text{{LTL}}\\xspace\/\\text{{LTLf}}\\xspace planning under assumptions}? what is the assumption complexity of \\text{{LTL}}\\xspace\/\\text{{LTLf}}\\xspace synthesis under assumptions? Here, the \\emph{assumption} complexity is the complexity of the problem assuming the domain and goal are fixed, and the only input to the problem is the assumption formula\/automaton.}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sectionintroduction}\n\nThe use of techniques and concepts from effective field theories\n(EFT's) has lead to an encouraging record of achievements for the\ndescription of nonrelativistic fermion-antifermion systems. The\nformulation of a nonrelativistic EFT for such systems has gone through\na number of important conceptual states~\\cite{Reviews}. Bodwin,\nBraaten and \nLepage~\\cite{Caswell,BBL} initially formulated a method for separating \nfluctuations at short distances of order the heavy particle mass $m$\nfrom those at long distances related to the nonrelativistic momentum\nand energy, $mv$ and $mv^2$ ($v$ being the relative velocity) and the\nhadronic scale $\\Lambda_{\\rm QCD}$. Subsequent work helped to clarify\nthe power counting in $v$, and the relevant degrees of freedom needed\nto construct an EFT. \n\nFor very heavy quarkonium systems, characterized by the scale\nhierarchy $m\\gg m v\\gg mv^2 > \\Lambda_{\\rm QCD}$ the formulation has\nreached a mature state. In Ref.~\\cite{Labelle} it was realized that it\nis necessary to distinguish soft ($\\sim mv$) and ultrasoft ($\\sim\nmv^2$) fluctuations and in Ref.~\\cite{LM} that the original NRQCD\naction~\\cite{BBL} is problematic concerning the simultaneous power\ncounting of soft and ultrasoft terms. In particular, the Lagrangian\nneeds to be multipole-expanded in the presence of ultrasoft\ngluons~\\cite{Labelle,GR}. Technically this can be formulated by\nintroducing soft as well as ultrasoft gluons in the theory~\\cite{LS}.\n\nIn Refs.~\\cite{PS,PS2,PS3}, based on the hierarchy  $m\\gg m v\\gg\nmv^2$, it was suggested to employ a series of EFT's, one for the\nscales $m > \\mu \\mathrel{\\raise.3ex\\hbox{$>$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}} mv$ and one for $mv > \\mu \\mathrel{\\raise.3ex\\hbox{$>$\\kern-.75em\\lower1ex\\hbox{$\\sim$}}} mv^2$, called pNRQCD\n(\"potential\" NRQCD). The pNRQCD is derived in a two-step matching\nprocedure where the intermediate soft matching scale and the pNRQCD\nrenormalization scale are independent at the field theoretic level. \nThis is appropriate e.g. for static quarks where the momentum is fixed\nby the quark separation $r\\sim 1\/mv$ and not correlated with the\ndynamical energy fluctuations $E\\sim mv^2$. The potential interactions\nare generated from integrating out soft fluctuations at the soft\nmatching scale.  \n\nIn Ref.~\\cite{LMR} is was pointed out that for dynamical heavy\nquarkonium systems the dispersive correlation between ultrasoft energy\nand soft momentum scales, $E\\sim \\mathbf p^2\/m$, can be implemented\nsystematically at the field theoretic level by matching to the proper\nEFT directly at the hard scale $\\mu\\sim m$ (one-step matching). The EFT,\ncalled vNRQCD \n(\"velocity\" NRQCD) has a strict power counting in $v$.\nIt contains soft and ultrasoft degrees of freedom as well as soft and\nultrasoft renormalization scales, $\\mu_S$ and $\\mu_U$. Through the\ndifferent velocity counting of soft and ultrasoft fields they are\ncorrelated as $\\mu_U\\propto \\mu_S^2\/m$, where usually the constant of\nproportionality is set to $1$, i.e. $\\mu_U=\\mu_S^2\/m=m\\nu^2$. The\nrenormalization group running of the theory is then conveniently\nexpressed in terms of the dimensionless parameter $\\nu$. The matching\nis carried out at the hard scale $\\nu=1$ and the theory is evolved to\n$\\nu\\sim v\\sim \\alpha_s$ of order of the relative velocity of the\ntwo-body system for computations of matrix elements. The running\nproperly sums logarithms of the momentum and the energy scale at the\nsame time~\\cite{LMR,amis4,mss1,HoangStewartultra} \nand is referred to as the velocity renormalization group\n(VRG)~\\cite{LMR}. Within dimensional regularization the powers of\n$\\mu_S^\\epsilon$ and $\\mu_U^\\epsilon$ multiplying the \noperators of the Lagrangian are uniquely determined by the $v$\ncounting and the dimension of the operators in $d=4-2\\epsilon$\ndimensions. \n\nIn Ref.~\\cite{Hoang3loop}, where a three-loop renormalization of\nquark-antiquark vertex diagrams was carried out for fermion-antifermion\nS-wave states, is was demonstrated that the one-step matching principle, upon\nwhich vNRQCD is based, is consistent under renormalization at the subleading\norder level, when subdivergences need to be subtracted and when UV\ndivergences from heavy quark-antiquark loops and from soft or\nultrasoft gluons appear simultaneously in single diagrams. Up to now there is \nno analogous demonstration for the two-step matching principle. \n\nWith the VRG, the running of potentials relevant for the\nnext-to-next-to-leading logarithmic (NNLL) description of the\nnonrelativistic dynamics of pairs of heavy quarks and colored scalars\nhas been determined in Refs.~\\cite{amis,amis2,amis3,HoangStewartultra}\nand \\cite{HoangRuizSvNRQCD}. The NLL running of leading (in the $v$\nexpansion) S-wave currents for heavy quark-antiquark production and of\nthe leading S- and P-wave currents for pairs of colored scalars were\ndetermined in Refs.~\\cite{amis3,HoangStewartultra} and\n\\cite{HoangRuizSvNRQCD}. Corresponding work in the pNRQCD formalism\nfor heavy quarks was carried out in\nRefs.~\\cite{PSstat,Pineda1,Pineda2,Pineda3}. In these computations,\ndiagrams which simultaneously contain both UV divergences from heavy\nquark-antiquark loops and from soft or ultrasoft gluon loops do not arise. The\nfinal results based on the VRG and on pNRQCD agree, after the\nscale correlations from the one-step matching procedure of vNRQCD are\nimposed in pNRQCD. Concerning the NNLL running of \nthe production currents, only the contributions from three-loop vertex\ndiagrams for the leading currents describing quark-antiquark pair\nproduction in an S-wave configuration~\\cite{Hoang3loop} are \nknown at this time in vNRQCD. For the NNLL contributions that arise from the \nsubleading order evolution of the coefficients that appears in the NLL\nanomalous dimension of the current at present only the results for the\nCoulomb potential~\\cite{PSstat,HoangStewartultra}, the spin-dependent\n$1\/m^2$ potentials~\\cite{Peninspin1,Peninspin2} and from\nultrasoft quark loops~\\cite{Stahlhofen1} have been determined.  \n\nThe main phenomenological application of the nonrelativistic EFT for\nthe situation $m\\gg mv\\gg mv^2>\\Lambda_{\\rm QCD}$ is top quark pair\nproduction close to threshold at a future Linear Collider~\\cite{LC}. \nHere the summation of logarithms of $v$ is crucial to control the\nlarge normalization uncertainties~\\cite{hmst,hmst1} that arise in \nfixed-order nonrelativistic computations that only account for the\nsummation of Coulomb potential insertions~\\cite{HT1,Hoangsynopsis}. One\ncan expect that the summation of logarithms of the velocity is also\nimportant for the description of squark pair production at\nthreshold~\\cite{HoangRuizSvNRQCD}. \nRecently, it was also found that it is crucial to apply\nthe EFT for predictions of the $e^+e^-\\to t\\bar t H$ cross section at \n$\\sqrt{s}=500$~GeV~\\cite{Farrell1,Farrell2} since here the $t\\bar t$\nfinal state interactions are dominated by nonrelativistic dynamics.\n\nIn this work we address the form and construction of non-relativistic\ninterpolating currents for production and annihilation of a color\nsinglet heavy particle-antiparticle pair with general quantum\nnumbers ${}^{2S+1}L_J$ in $n=d-1=3-2\\epsilon$ dimensions for quarks\nand colored scalars within NRQCD. In general, the interpolating\ncurrents for a specific ${}^{2S+1}L_J$ state are associated to\nirreducible tensor representations of the SO(n) rotation group that\ncan be built from  \nthe particle-antiparticle relative momentum and the bilinear\ncovariants of the particle-antiparticle field operators. \nWhile in three dimensions the basis of interpolating currents for the \ndifferent ${}^{2S+1}L_J$ states is, by construction, unique, in\ngeneral $n\\neq 3$ dimensions for fermions there exist evanescent\noperator structures that make the choice of basis for the\ninterpolating currents ambiguous. This is because the structure of\nirreducible SO(n) representations for $n\\neq 3$ is inherently\ndifferent and in general richer than in three dimensions. Technically\nthe ambiguity is related to the number of sigma \nmatrices that generate the SO(n) rotations for the spinor wave\nfunctions. For explicit computations a specific choice of basis has\nto be made, which is associated to a specific choice of a\nrenormalization scheme. A very similar problem has been treated\nsystematically already some time ago in the framework of relativistic\nQCD and the effective weak\nHamiltonian~\\cite{DuganGrin,HerrlichNierste,ChetyrkinMisiak}, and much\nof the statements that apply for the relativistic case can be directly\ntransferred to the nonrelativistic theory. However, due to the \nnonrelativistic power counting some even more specific statements,\ncan be made. In particular, \nthe NLL order matching conditions and anomalous dimensions of the\n${}^{2S+1}L_J$ interpolating currents that are leading in the\nnonrelativistic expansion are scheme independent.\n\nUsing the explicit form of the interpolating currents obtained in this\nwork we also determine the NLL anomalous dimensions of the leading (in\nthe $v$ expansion) currents describing color singlet heavy \nquark-antiquark and squark-antisquark pair production for arbitrary\nspin and angular momentum ${}^{2S+1}L_J$ configurations. The results\nfor low angular \nmomentum states (S,P,D) are e.g. relevant for angular distributions at\nthe threshold or for production and decay rates of squark-antisquark\nresonances in certain supersymmetric scenarios where squarks can\nhave a very long lifetime. The results also shed some light on the\nimportance of summing logarithms of \n$v$ for the production and annihilation rates of high angular momentum\nstates. In this respect our results help to complete analogous higher\norder considerations in previous literature for the energy\nlevels~\\cite{hms1} and the\nwave-functions~\\cite{Pineda2,Melnikov:1998ug}.\n\nFor the presentations in this work we employ the\nnotations from vNRQCD based on a label formalism for soft fields and\nthe heavy quarks (or scalars)~\\cite{LMR}. We note, however, that  \nmost of the results obtained in this work are applicable to NRQCD in\ngeneral. In particular, the NLL order anomalous dimensions obtained\nhere are also valid in pNRQCD once the vNRQCD scale correlations from\nthe one-step matching are imposed. \n\nThe outline of the paper is as follows: In Sec.~\\ref{sectionsinglet}\nwe discuss the form of the spherical harmonics in $n$ dimensions and\npresent the form of currents describing states with arbitrary angular\nmomentum $L$ and total spin zero. We also present the $n$-dimensional\nform of the Legendre polynomials and demonstrate in an example why the\n$n$-dimensional form of the spherical harmonics is needed in\ndimensional regularization. \nIn Sec.~\\ref{sectiontriplet} we discuss the properties of\n$\\sigma$-matrices for $n\\neq 3$ and the importance of evanescent\noperator structures that can be built from the $\\sigma$-matrices.\nWe present a simple basis of interpolating currents describing\nfermion-antifermion pairs in a spin triplet state and having arbitrary\n$L$ and total angular momentum $J$. \nIn Sec.~\\ref{sectionNLLsinglet} we determine the anomalous dimension\nof the current for spin singlets, and in \nSec.~\\ref{sectionNLLtriplet} those of  the current for spin triplets.\nSection~\\ref{sectionNLLtriplet} also contains a discussion on the\nscheme-dependence of the spin-dependent potentials.\nIn Sec.~\\ref{sectiondiscussion} we determine and solve the resulting\nanomalous dimensions for heavy quarks and colored scalars. The results\nare analysed numerically for a few cases.\nIn Sec.~\\ref{sectioncomments} we comment on the scheme-dependence of\nresults that can be found in previous literature. \nThe conclusions are given in Sec.~\\ref{sectionconclusion}.\nWe have added a few appendices: Appendices~\\ref{appendix2} and\n\\ref{appendix3} contain the derivation of the tree level NRQCD\nmatching conditions for the well known processes $q\\bar q\\to 2\\gamma$\nand $3\\gamma$ accounting properly for the existence of evenescent \noperator structures. In App.~\\ref{appendix4} some\nmore details are given on the determination of the spin triplet\ncurrents shown in Sec.~\\ref{sectionNLLtriplet}. We also derive an\nalternative set of currents that is equivalent for $n=3$ but \ninequivalent for $n\\neq3$. Finally in App.~\\ref{appendix5} we give \nresults for the UV-divergences for the elementary integrals\nthat are needed for the determination of the anomalous dimensions\nof the currents.\n\n\n\\section{Spin Singlet Currents} \n\\label{sectionsinglet}\n\nIn this section we discuss the form of the interpolating currents\ndescribing the production of a particle-antiparticle pair with total\nspin zero for arbitrary relative angular momentum $L$\n(${}^{2S+1}L_J={}^1L_L$). They are relevant for pairs of scalars or for\nquark-antiquark pairs in a spin singlet state. The currents naturally\nhave their simplest form in the c.m.\\,frame where relevant angular\ndependence can only arise from the c.m.\\,spatial momenta of the\npair. Thus the generic structure of the production currents is\n\\begin{eqnarray}\n  \\psi_{\\mathbf p}^\\dagger(x)\\,\\Gamma(\\mathbf p)\\,\\tilde \\chi_{-\\mathbf p}^*(x)\n\\,,\n\\end{eqnarray}\nwhere $\\Gamma(\\mathbf p)$ represents an arbitrary tensor depending on the\nc.m.\\,momentum label $\\mathbf p$. For the case of scalars  $\\psi_{\\mathbf p}$ and\n$\\tilde\\chi_{\\mathbf p}$ are scalar fields with\n$\\psi_{\\mathbf p}^\\dagger=\\psi_{\\mathbf p}^*$, while for fermions $\\psi_{\\mathbf p}$ and\n$\\chi_{\\mathbf p}$ are Pauli spinor fields with $\\tilde\\chi_{-\\mathbf p}^*\\equiv\n(i\\sigma_2)\\chi_{-\\mathbf p}^*$. (Note that we adopt the \nstandard nonrelativistic spinor convention of antiparticles with {\\it\npositive} energies. In this convention fermions and antifermions have\nthe same spin operators.) The corresponding\nannihilation current can be obtained by hermitian conjugation. \nThe interpolating currents associated to a definite angular momentum\nstate $L$ are related to irreducible representations of the tensor\n$\\Gamma$ with respect to the rotation group SO(n). Since the transformation\nproperties of the fields are not relevant in this respect for the spin\nzero case, it is sufficient to identify the irreducible tensors that\ncan be built from the spatial momentum vector $p^i$, $i=1,\\ldots,n$,\nwhere $n=d-1$. The irreducible tensors associated to the\nangular momentum $L$ are up to normalization just the spherical\nharmonics of degree $L$. In the following we give a general discussion\nin $n$ dimensions. All results reduce to the well known results for $n=3$.\n\n\\subsection*{Spherical Harmonics}\n\nThe spherical harmonics of degree $L$ are the\npolynomials $u(\\mathbf x)$ in $\\mathbb{R}^n$ which are homogeneous of degree $L$\n(i.e. $u(r \\mathbf x)=r^L u(\\mathbf x)$), harmonic (i.e. they satisfy the Laplace equation \n$\\Delta_{\\mathbb{R}^n}u(\\mathbf x)\\equiv\\nabla^2 u(\\mathbf x)=0$) and restricted to the unit\nsphere  $S^{n-1}$. The \nspherical harmonics ${Y}_{LM}(n,\\bar\\mathbf x)$ of degree $L$, with\n$M=1,\\dots,n_L$ and $\\bar\\mathbf x=\\mathbf x\/|\\mathbf x|$, form an orthogonal basis of a\n$n_L$-dimensional vector space with\n\\begin{equation}\nn_L\\,=\\,\n\\left(\\begin{array}{c}n+L-1\\\\ L\\end{array}\\right)\n-\\left(\\begin{array}{c}n+L-3\\\\ L-2\\end{array}\\right)\n\\, = \\,\n(2L+n-2)\\,\\frac{\\Gamma(n+L-2)}{\\Gamma(n-1)\\Gamma(L+1)}\\,.\n\\label{nL}\n\\end{equation}\nAny differentiable function on the unit sphere $S^{n-1}$ in\n$\\mathbb{R}^n$ can be expanded in terms of a convergent series of\nspherical harmonics.  \nA representation in terms of cartesian coordinates~\\footnote{\nAlthough it is possible to write down results with spherical\ncoordinates for integer $n$ (see e.g. Ref.~\\cite{Normand:1981dz}), it\nis more convenient in practice and for actual \ncomputations to use cartesian coordinates. \n} \nof the spherical\nharmonics of degree $L$ is given by the totally symmetric and traceless\ntensors with $L$ indices \n$T^{i_1\\dots i_L}(\\mathbf x)$, where the indices $i_1,\\ldots,i_L$ are cartesian\ncoordinates~\\cite{grouptheory}. The restriction to the unit sphere is not\nessential regarding the transformation properties under SO(n) rotations and\ncan be dropped for our purposes. An explicit expression for \n$T^{i_1\\dots i_L}(x)$ reads: \n\\begin{eqnarray}\nT^{i_1\\dots i_L}(\\mathbf x) \n &=& x^{i_1}\\ldots x^{i_L}\n   -\\frac{\\mathbf x^2 \\, \\Phi_1^{i_1\\dots i_L}(\\mathbf x)}{(2L+n-4)}\n   +\\frac{(\\mathbf x^2)^2 \\, \\Phi_2^{i_1\\dots i_L}(\\mathbf x)}{(2L+n-4)(2L+n-6)}\n   - \\dots\n\\nonumber\\\\ & & \n\\dots + (-1)^{[L\/2]}\n   \\frac{(\\mathbf x^2)^{[L\/2]} \\, \\Phi_{[L\/2]}^{i_1\\dots i_L}(\\mathbf x)}\n         {(2L+n-4)\\dots(2L+n-2-2[L\/2])}\n\\nonumber\\\\[3mm] \n &=& \n \\sum_{k=0}^{[L\/2]}\\, C_k^L \\, \\mathbf x^{2k}\\, \\Phi_k^{i_1\\dots i_L}(x) \\;,\n\\nonumber\\\\[3mm]  \nC_k^L &=& (-1)^k \\, 2^{-k} \\frac{\\Gamma(\\frac{n}{2}+L-k-1)}{\\Gamma(\\frac{n}{2}+L-1)} \\;,\n\\nonumber\\\\[3mm]  \n\\Phi_k^{i_1\\dots i_L}(\\mathbf x) & = & \n  \\sum_{\\mathrm{inequ. \\,\\,permut.}\\atop(p_1\\dots p_L)\\,\\mathrm{of}\\, (1\\dots L) }\n  \\underbrace{ \\,\\delta^{i_{p_1}i_{p_2}}\\dots\n \\delta^{i_{p_{(2k-1)}}i_{p_{2k}}} }_{k \\;\\delta'\\mathrm{s}} \n \\, x^{i_{p_{(2k+1)}}}\\dots x^{i_{p_L}} \n\\nonumber\\\\ & = &\n \\delta^{i_1i_2}\\dots\\delta^{i_{2k-1}i_{2k}}\\,x^{i_{2k+1}}\\dots x^{i_L}\n \\, + \\, \\mbox{(inequ. permut.'s)}\n\\,,\n\\label{Tdef}\n\\end{eqnarray}\nwhere the symbol $[\\ldots]$ denotes the Gauss bracket. \nThe sum in the $\\Phi_k$ tensors extends over all sets of $k$ pairs of\nindices that can be constructed  \nfrom $L$ indices ($L\\ge 2k$); the number of terms in the sum is thus \n${L\\choose 2k}(2k-1)!!$. Note that we call two\npermutations equivalent if they lead to the same term in the sum.\nAlthough we believe that the expression for $T^{i_1\\dots i_L}(x)$ has been shown somewhere in the\nliterature before, we were not able to locate a suitable reference.  \nA few simple examples are\n\\begin{eqnarray} &&\nT^i(\\mathbf x) \\, = \\, x^i\\,,\\qquad\nT^{ij}(\\mathbf x) \\, = \\, x^ix^j-\\frac{\\mathbf x^2}{n}\\delta^{ij}\\,,\n\\nonumber\\\\ & &\nT^{ijk}(\\mathbf x) \\, = \\,  x^ix^jx^k-\\frac{\\mathbf x^2}{n+2}\n   \\left(x^i\\delta^{jk}+x^j\\delta^{ik}+x^k\\delta^{ij}\\right)\\,.\n\\end{eqnarray}\nFrom the Laplacian \n\\begin{eqnarray}\n\\Delta_{\\mathbb{R}^n} & \\equiv & \n\\frac{\\partial^2}{\\partial  x^{i\\,2}} \\, = \\,\n\\left(\\frac{\\partial^2}{\\partial r^2} +\n  \\frac{n-1}{r}\\frac{\\partial}{\\partial r}\\right) -\n\\frac{1}{r^2}\\,\\mathbf L^2\n\\,,\n\\nonumber\\\\[2mm]\n-\\mathbf L^2 & = &\n\\frac{\\partial}{\\partial x^j}x^k\\frac{\\partial}{\\partial x^j} x^k-\n\\frac{\\partial}{\\partial x^j} x^k \\frac{\\partial}{\\partial x^k} x^j\n\\, = \\,\n\\mathbf x^2\\nabla^2 - \nx^k x^j \\frac{\\partial}{\\partial x^k}\\frac{\\partial}{\\partial x^j} -\n(n-1) x^k \\frac{\\partial}{\\partial x^k}\n\\,,\n\\end{eqnarray}\none can find an explicit form for the squared angular momentum\noperator and, using the homogeneity relation \n$\\Delta_{\\mathbb{R}^n}T^{i_1\\dots i_L}=0$, the eigenvalue equation\n\\begin{eqnarray}\n\\mathbf L^2\\,\\, T^{i_1\\dots i_L}(\\mathbf x) & = & L(L+n-2)\\,\\, T^{i_1\\dots i_L}(\\mathbf x)\n\\,.\n\\end{eqnarray}\nWe define the interpolating currents that describe production of a\nspin singlet and angular momentum $L$ state (${}^{2S+1}L_J={}^1L_L$)\nas\n\\begin{eqnarray}\n(j_L^{S=0})^{i_1\\ldots i_L} & \\equiv & \n\\psi_{\\mathbf p}^\\dagger(x)\\, T^{i_1\\dots i_L}(\\mathbf p)\\,\\tilde\n\\chi_{-\\mathbf p}^*(x)\n\\,.\n\\label{singletcurrent1}\n\\end{eqnarray}\nThe current in Eq.~(\\ref{singletcurrent1}) is the dominant ${}^1L_L$\ncurrent in the $v$-expansion. Higher order currents are obtained by\nincluding additional powers of the scalar term $\\mathbf p^2$.  \n\nAs an example, let us consider the currents with angular momenta $S,\\,P$ and $D$.\nThey are relevant in the electromagnetic production of\nheavy charged scalars from $e^+e^-$ and $\\gamma\\gamma$ collisions. \nThe $e^+e^-$ annihilation at lowest \norder in the expansion of the electromagnetic coupling produces $P$-wave states:\n\\begin{eqnarray}\ni{\\cal M}_{e^+e^-} =\n-i\\,\\frac{8\\pi\\alpha}{s}\\,Q_q \\,\\bar{v}(k^\\prime)\\,\\gamma^i\\, u(k)\n\\,T^i(\\mathbf p)\n\\,,\n\\label{pwave}\n\\end{eqnarray}\nwith $k,\\,k^\\prime$ the momenta of the incoming $e^+e^-$ in the c.m.\\,frame \nand $Q_q$ denoting the electromagnetic\ncharge of the scalars. Note that potential color indices are always implied.\nPairs of heavy scalars in $S$- and $D$-waves are first\nproduced in  $\\gamma\\gamma$ collisions. The first terms of the amplitude in the\nexpansion in the c.m.\\,three-momentum of the outgoing particles read:\n\\begin{eqnarray}\ni{\\cal M}_{\\gamma\\gamma} &=&\n32 i\\,\\pi\\alpha\\,Q_q^2 \\,\\epsilon_1^i\\epsilon_2^j\\,\n\\left[\\,-\\frac{1}{4}\\,\\delta^{ij}+\\frac{p^i p^j}{2m^2} \n+ \\dots \\,\\right]\n\\nonumber\\\\[3mm]\n&=& -8i\\,\\pi\\alpha\\,Q_q^2 \\,\\epsilon_1^i\\epsilon_2^j\\left(\n1-\\frac{2\\,\\mathbf p^2}{n \\,m^2}\n\\right) +\n\\frac{16i\\,\\pi\\alpha}{m^2}\\,Q_q^2 \\,\n\\epsilon_1^i\\epsilon_2^j \\, T^{ij}(\\mathbf p)\n+ \\dots\\,.\n\\label{s-d-waves}\n\\end{eqnarray}\nThe first term in the last equality is the $S$-wave\ncontribution ($T=1$) whereas the second is the $D$-wave contribution described\nby the spherical harmonic of degree 2, $T^{i_1i_2}(\\mathbf p)$.\n\nSome useful relations for the $T$ tensors read:\n\\begin{eqnarray}\nT^{\\,i_1\\ldots i_L}(\\mathbf x)\\, x^{i_L}\n &=& \\frac{L+n-3}{2L+n-4}\\,\\mathbf x^2\\,T^{\\,i_1\\ldots i_{L-1}}(\\mathbf x) \\;,\n \\nonumber\\\\[3mm] \nT^{\\,i_1\\ldots i_L}(\\mathbf x)\\,  x^{i_{L+1}} \n &=&  T^{\\,i_1\\ldots i_{L+1}}(\\mathbf x) \n     - \\frac{2 \\,\\mathbf x^2}{(2L+n-2)(2L+n-4)}\\, \\sum_{j< k \\atop j=1}^L \\, \\delta^{i_j i_k}   \\,  \n     T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots i_{L+1} } (\\mathbf x)\n \\nonumber\\\\[1mm]    \n && +\\;\\frac{\\mathbf x^2}{2L+n-2} \\, \\sum_{j=1}^L \\, \\delta^{i_j i_{L+1}} \\, \n     T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_{L} } (\\mathbf x) \\;,\n \\nonumber\\\\[3mm]   \n\\frac{\\partial}{\\partial x^\\ell}\\,\\Phi_k^{i_1\\dots i_L}(\\mathbf x) &=&\n \\sum_{j=1}^L \\, \\delta^{i_j\\ell}\\,\\Phi_{k}^{i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_L}(\\mathbf x)\n\\, = \\,\n \\Phi_{k+1}^{i_1\\dots i_L \\ell}(\\mathbf x) - x^\\ell\\,\\Phi_{k+1}^{i_1\\dots i_L}(\\mathbf x) \\;,\n \\nonumber\\\\[3mm]\nx^\\ell\\,\\Phi_{k}^{i_1\\dots i_L}(\\mathbf x) &= &\n   \\Phi_{k}^{i_1\\dots i_L \\ell}(\\mathbf x) - \n   \\sum_{j=1}^L\\,\\delta^{i_j\\ell}\\,\\Phi_{k-1}^{i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_L}(\\mathbf x) \\;,\n\\nonumber\\\\[3mm]\nx^{[\\ell}\\,\\partial^{\\,k]}_x\\,T^{\\,i_1\\ldots i_L}(\\mathbf x) &=& \n  \\frac{1}{2}\\sum_{j=1}^{L} \\,  \\delta^{i_j k}  \\,\n     T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\dots i_{L} \\ell} (\\mathbf x) - \\{ \\ell \\leftrightarrow   k \\}\\;,   \n\\label{Trelations}\n\\end{eqnarray}\nwhere we use the notation that hatted indices are dropped, and \n$A^{[ij]}\\equiv\\frac{1}{2}(A^{ij}-A^{ji})$.\n\n\\subsection*{Legendre Polynomials}\n\nThe spherical harmonics of degree $L$ satisfy the addition theorem\n\\begin{equation}\n\\sum_{M=1}^{n_L} {Y}_{LM}(n,\\bar \\mathbf x) \\, {Y}_{LM}^*(n,\\bar\n\\mathbf y)=\\frac{n_L}{\\sigma_n}\\,P_L(n,\\bar \\mathbf x.\\bar\\mathbf y) \n\\,,\n\\label{addtheorem1}\n\\end{equation}\nwhere $P_L(n,t)$, $t\\in[-1,1]$, is the Legendre polynomial\nof degree  $L$ generalized to $n$ dimensions, with $P_L(n,1)=1$ and\n$\\sigma_n=\\frac{2\\pi^{n\/2} }{\\Gamma(n\/2)}$ is the area of the unit sphere $S^{n-1}$. The\nLegendre polynomial is related to the Gegenbauer polynomial\n$C_L^{\\lambda}(t)$.\\footnote{\n  The index $\\lambda$ is\n  chosen such that the $C_L^{\\lambda}$ are orthogonal on the\n  $n$-dimensional sphere\n} \nOne has\n$P_L(n,t)=\\frac{\\Gamma(n-2)\\Gamma(L+1)}{\\Gamma(L+n-2)}C_L^{\\lambda}(t)$,\nwhere $\\lambda=\\frac{n-2}{2}$.\nThe explicit form of the generalized Legendre polynomial\nreads~\\cite{Gradstheyn},  \n\\begin{equation}\nP_L(n,t)=\\frac{\\Gamma(n-2)\\,\\Gamma(L+1)}{\\Gamma(\\frac{n-2}{2})\\, \n\\Gamma(n+L-2)}\\,\\sum_{k=0}^{\\left[\\frac{L}{2}\\right]}\n\\,\\frac{(-1)^k\\,\\Gamma(\\frac{n-2}{2}+L-k)}{\\Gamma(k+1)\\,\\Gamma(L-2k+1)}\n\\;(2t)^{L-2k} \n\\;.\n\\label{Legendredef}\n\\end{equation}\nFrom Eq.~(\\ref{addtheorem1}) follows an addition theorem for the $T$ tensors\nupon total contraction of their indices:\n\\begin{equation}\nT_{L,n}(\\mathbf x,\\mathbf y)\n\\, \\equiv \\,\nT^{\\,i_1\\dots i_{L}}(\\mathbf x)\\, T^{\\,i_1\\dots i_{L}}(\\mathbf y) \n\\, = \\, \n|\\mathbf x|^L\\,|\\mathbf y|^L\\,N_L \\, P_L(n,t) \\;,\n\\qquad t=\\frac{\\mathbf x.\\mathbf y}{|\\mathbf x||\\mathbf y|}\\;,\n\\label{Tcontract}\n\\end{equation}\nwhere\n\\begin{equation}\nN_L \\, \\equiv \\,\n\\frac{\\Gamma(L+n-2)\\,\\Gamma(\\frac{n-2}{2})}{2^{L}\\,\\Gamma(L+\\frac{n}{2}-1)\\,\\Gamma(n-2)}\\,.\n\\end{equation}\nAnother very useful contraction is:\n\\begin{eqnarray}\n\\lefteqn{\nT^{\\,i_1\\dots i_{L-1}j}(\\mathbf x)\\,T^{\\,i_1\\dots\n  i_{L-1}\\ell}(\\mathbf y) \\,\\left(y^j x^{\\ell}-x^j y^\\ell \\right) \n\\,=\\,\n}\n\\nonumber\\\\ & &\n|\\mathbf x|^{L+1}\\,|\\mathbf y|^{L+1}\\,\nN_L \\,\n\\frac{L+n-2}{2L+n-2}\\,\\left[ P_{L+1}(n,t) -P_{L-1}(n,t)  \\right]\n\\;.\n\\label{TyTx}\n\\end{eqnarray}\n\n\n\\subsection*{Consistency Consideration}\n\nThe use of the generalized currents in\nEqs.~(\\ref{Tdef}) and~(\\ref{singletcurrent1}) is mandatory to obtain\nconsistent results in dimensional regularization in accordance with\nSO(n) rotational invariance. As an example let us consider the\nnonrelativistic three-loop vacuum polarization diagram shown in\nFig.~\\ref{fig:3loopCoul} with two insertions of the Coulomb potential.\nInserting the currents that produce and annihilate the\nparticle-antiparticle pair with angular momentum $L$ and fully\ncontracting the indices, the quantity we want to compute is, up to a\nglobal factor \n($D^n\\mathbf p\\equiv \\tilde{\\mu}_s^{2\\epsilon}  d^n\\mathbf p\/(2\\pi)^{n},\\;\n\\tilde{\\mu}_s^{2\\epsilon}=\\mu_s^{2\\epsilon} (4\\pi)^{-\\epsilon} e^{\\epsilon\\gamma}$):\n\\begin{figure}[t] \n\\begin{center}\n \\leavevmode\n \\epsfxsize=6cm\n \\leavevmode\n \\epsffile{figs\/3loopCoul.eps}\n \\vskip  0.0cm\n \\caption{Three-loop diagram with two insertions of the Coulomb\n potential (black dots). \n \\label{fig:3loopCoul} }\n\\end{center}\n\\end{figure}\n\\begin{eqnarray}\n\\frac{1}{N_L}\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n\\frac{T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)\\, T^{\\,i_1\\dots i_{L}}(\\mathbf q_3) }\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,,  \n\\end{eqnarray}\nwith $\\delta=-mE-i\\epsilon$. We can now proceed in two different ways\nto do the computation. From Eq.~(\\ref{Tcontract}), the contraction of\nthe $T$'s at the ends generates a Legendre polynomial  \ndepending on the angle between the loop momenta on the sides:\n\\begin{eqnarray}\n{\\cal A}_1^{(L)} = \\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\\frac{|\\mathbf q_1|^L |\\mathbf q_3|^L \\,\nP_L(n,\\bar \\mathbf q_1 . \\bar \\mathbf q_3) }\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,.\n\\label{A1}\n\\end{eqnarray}\nOn the other hand, we can first shift the loop momenta dependence of\nthe current $T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)$ by using the \nrelation\n\\begin{eqnarray}\n &&\\int d^n \\mathbf q_1 \\, T^{\\,i_1\\dots i_{L}}(\\mathbf q_1)\\,f( \\mathbf q_1, \\mathbf q_2)= A\\,\n  T^{\\,i_1\\dots i_{L}}(\\mathbf q_2)\\,,\n\\nonumber\\\\[2mm]\n&&\nA= \\int d^n \\mathbf q_1 \\,|\\mathbf q_1|^L |\\mathbf q_2|^{-L} \\,\nP_L(n,\\bar \\mathbf q_1 . \\bar \\mathbf q_2) \\,f( \\mathbf q_1, \\mathbf q_2) \\,,\n\\label{shift}\n\\end{eqnarray}\nwhich is a consequence of rotational invariance if \n$f( \\mathbf q_1, \\mathbf q_2)$ is a scalar function depending on\n$\\mathbf q_1^2,\\,\\mathbf q_2^2,\\,\\mathbf q_1\\cdot\\mathbf q_2$. \nThis gives an alternative expression\nwith two Legendre polynomials:\n\\begin{eqnarray}\n{\\cal A}_2 ^{(L)} = \n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n\\frac{|\\mathbf q_1|^L |\\mathbf q_3|^L \\,\nP_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_2)\\,P_L(n,\\bar\\mathbf q_2.\\bar\\mathbf q_3)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,.  \n\\label{A2}\n\\end{eqnarray}\nHad we worked in $n=3$ from the beginning, we would have written down\nEqs.~(\\ref{A1}) and~(\\ref{A2}) with \nthe generalized Legendre polynomials replaced by their $n=3$\nexpressions. Let us call the corresponding \nexpressions as $\\tilde{{\\cal A}}_1^{(L)},\\;\\tilde{{\\cal A}}_2^{(L)}$. \nSince the integrals are \npower divergent, we can compute them using dimensional regularization in $n=3-2\\epsilon$\ndimensions and check whether they give the same result. \nThe first non-trivial case is $L=2$, because one has $P_{0,1}(3,x)=P_{0,1}(n,x)$. \nThe results for $\\tilde{{\\cal A}}_1^{(2)},\\;\\tilde{{\\cal A}}_2^{(2)}$ then read\n\\begin{eqnarray}\n\\tilde{{\\cal A}}_1^{(2)} &=& \\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{15+4\\pi^2+\\epsilon\\left[32\\pi^2-120\\,\\zeta(3)\n +231\\right]+{\\cal O}(\\epsilon^2) \\Big\\}\\,,\n\\nonumber\\\\[3mm]\n\\tilde{{\\cal A}}_2^{(2)} &=& \\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{\n18+4\\pi^2+\\epsilon\\left[ 32\\pi^2-120\\,\\zeta(3)\n +264-4\\log 8 \\right]+{\\cal O}(\\epsilon^2)\\Big\\}\\,.\n\\nonumber\\\\\n\\end{eqnarray}\nA difference arises even in the first term in the $\\epsilon$\nexpansion, which reflects the fact that \n$\\tilde{{\\cal A}}_1^{(2)},\\;\\tilde{{\\cal A}}_2^{(2)}$ are not the same\nquantity. The reason is that the shift (\\ref{shift}) performed in\n$n=3$ dimensions and the following evaluation using dimensional\nregularization do not commute if the integrals are divergent. \nRotational invariance in $n$ dimensions\nmust be satisfied at every step if dimensional regularization is\nused. This is automatically achieved by using the SO(n) tensors $T$ as\nshown in Eqs.~(\\ref{A1}) and~(\\ref{A2}). The computation \nwith generalized Legendre polynomials corresponding to ${\\cal\n  A}_1^{(L)}$ and ${\\cal A}_2^{(L)}$ in Eqs.~(\\ref{A1},\\ref{A2}) can\nbe shown to produce the correct outcome. For $L=2$ the correct result\nreads \n\\begin{eqnarray}\n{\\cal A}_1^{(2)} = {\\cal A}_2^{(2)} =\\frac{2^{-6\\epsilon}}{3072\\pi^3}\\,\n(\\delta)^{\\frac{3n}{2}-3}\\mu_s^{6\\epsilon}\\,\n\\Big\\{\n15+4\\pi^2+\\epsilon\\left[ 32\\pi^2-120\\,\\zeta(3)\n +236\\right]+{\\cal O}(\\epsilon^2) \\Big\\}\\,,\n \\nonumber\n\\end{eqnarray}\nwhich disagrees with $\\tilde{{\\cal A}}_1^{(2)}$ in terms proportional\nto $\\epsilon^m$ for $m>0$, and with $\\tilde{{\\cal A}}_2^{(2)}$ for all\nterms in the $\\epsilon$ expansion. \n\n\\section{Spin Triplet Currents} \n\\label{sectiontriplet}\n\nIn this section we discuss the form of the interpolating currents describing\nthe production of a fermion-antifermion pair in a spin triplet $S=1$ state for\narbitrary relative angular momentum $L$ (${}^{2S+1}L_J={}^3L_J$). As in\nSec.~\\ref{sectionsinglet} the currents are defined in the c.m.\\,frame.\n\n\\subsection*{Pauli Matrices}\n\nSince the treatment of Pauli $\\sigma$-matrices in $n$ dimensions involves a\nnumber of subtleties, we briefly review some of their properties relevant for\nthe formulation of the currents. Many of the properties can be directly\nobtained from the corresponding properties of the $\\gamma$-matrices in $d=n+1$\ndimensions. The \n$\\sigma$-matrices $\\sigma^i$ ($i=1,\\ldots,n$) are the generators of SO(n)\nrotations for spin $1\/2$. They are traceless, hermitian, satisfy the Euclidean\nClifford algebra $\\{\\sigma^i,\\sigma^j\\}=2\\delta^{ij}$, and can be defined\nsuch that $\\mathbf \\mbox{\\boldmath $\\sigma$}^T=\\mathbf \\mbox{\\boldmath $\\sigma$}^*=-\\sigma_2\\mathbf \\mbox{\\boldmath $\\sigma$}\\sigma_2$. While traces of\na product of an even number of $\\sigma$-matrices in $n$ dimensions can be\nexpressed as a \nmultiple of $\\mbox{Tr}[\\mathbf 1]$ using the anticommutator, traces of a product\nof an odd number are identically zero and for the case of three\n$\\sigma$-matrices can require additional\nrules to yield results that are consistent with the relations known from $n=3$. \nIn this respect the product of three different $\\sigma$-matrices can be\nsomewhat considered the three-dimensional\nanalog of $\\gamma_5$ in four dimensions.\\footnote{ \nThe vanishing of traces of a product of an odd number of $\\sigma$-matrices\nin $n$ dimensions\nis related to the simultaneous use of the cyclicity property of the trace\noperators and the Euclidean Clifford algebra relations for all  \n$\\sigma$-matrices. Products of an odd number of $\\sigma$-matrices,\nhowever, do not arise from insertions of\npotentials which we consider in this work. Such traces can, however, arise\ne.g.\\,when the spin-dependent radiation of ultrasoft gluons or annihilation\npotentials need to be considered. Thus, in general, the cyclicity\nproperty of the trace operation should only be used after\nrenormalization is completed - if the Euclidean Clifford algebra is applied\nfor all  $\\sigma$-matrices. \n}\nAs for the case of the $\\gamma$-matrices~\\cite{DuganGrin} products of\n$\\sigma$-matrices in arbitrary number of dimensions cannot be reduced to a\nfinite basis, but represent an infinite set of independent structures. In\nanalogy to Ref.\\,\\cite{DuganGrin} it is convenient to define \n\\begin{eqnarray}\n\\sigma^{i_1\\cdots i_m} & \\equiv &\n\\sigma^{[i_1}\\sigma^{i_2}\\cdots \\sigma^{i_m]}\n\\,,\\quad (m=1,2,\\ldots)\n\\end{eqnarray}\nfor the averaged antisymmetrized product of $\\sigma$-matrices, so e.g.\\,\n$\\sigma^{ij}=1\/2(\\sigma^i\\sigma^j-\\sigma^j\\sigma^i)$. It is straightforward to\nderive the following useful relations:\n\\begin{eqnarray}\n\\sigma^{i_1\\cdots i_m}\\sigma^\\ell & = &\n(-1)^m\\,\\Big(\\, \\sigma^{\\ell i_1\\cdots i_m} \\, + \\,\n\\sum_{j=1}^{m}(-1)^j\\,\\delta^{\\ell i_j}\\,\\sigma^{i_1\\cdots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\cdots i_m}\\Big)\n\\,,\n\\label{sigma1}\n\\\\\n\\sigma^\\ell\\sigma^{i_1\\cdots i_m} & = &\n\\sigma^{\\ell i_1\\cdots i_m} \\, + \\,\n\\sum_{j=1}^{m}(-1)^{j+1}\\,\\delta^{\\ell i_j}\\,\\sigma^{i_1\\cdots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\cdots i_m}\n\\,,\n\\label{sigma2}\n\\\\\n\\left[ [\\sigma^j,\\sigma^\\ell],\\sigma^{i_1\\ldots i_m} \\right]\n  & = & 4(-1)^m\\, \\sum_{k=1}^{m}\\, (-1)^{k+1}\n   \\left( \\delta^{j i_k}\\sigma^{i_1\\ldots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots  i_m \\ell}   \n  -  \\delta^{\\ell i_k}\\sigma^{i_1\\ldots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots  i_m j} \\right)\n\\,,\n\\label{sigma3}\n\\\\\n\\frac{1}{2} \\left( \\sigma^{j}\\sigma^{i_1\\ldots i_m}\\sigma^{\\ell}\n+ \\sigma^{\\ell}\\sigma^{i_1\\ldots i_m}\\sigma^{j} \\right)\n& = &\n(-1)^m\\,\\delta^{j\\ell}\\,\\sigma^{i_1\\dots i_m} \n\\nonumber \\\\ \n&& + \\sum_{k=1}^{m}\\, (-1)^{k+1}\\left( \\delta^{j i_k}\\sigma^{i_1\\ldots\n  \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots  i_m \\ell} \n  +  \\delta^{\\ell i_k}\\sigma^{i_1\\dots \\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\ldots  i_m j} \\right)\n\\,,\n\\label{sigma4}\n\\\\\n\\sum_k\\,\\sigma^k\\sigma^{i_1\\dots i_m}\\sigma^k & = &\n(-1)^m\\,(n-2m)\\,\\sigma^{i_1\\dots i_m}\n\\,,\n\\label{sigma5}\n\\\\\n\\sum_{i_1\\dots i_k}\\,\\sigma^{i_1\\dots i_k}\\,\\sigma^{i_1\\dots i_k}\n& = &\n(-1)^{\\frac{k(k-1)}{2}}\\,n\\dots (n-k+1)\n\\, = \\,\n(-1)^{\\frac{k(k-1)}{2}}\\,\\frac{\\Gamma(n+1)}{\\Gamma(n-k+1)}\\,,\\quad\n\\nonumber\\\\\n\\label{sigma6}\n\\\\\n\\mbox{Tr}(\\sigma^{i_1}\\dots \\sigma^{i_{2m}}) &=& \\mbox{Tr}\\,[\\mathbf 1]\n \\!\\!\\! \\!\\!\\!\\sum_{\\mathrm{inequ. \\,\\,permut.}\\atop (p_1\\dots p_{2m})\\,\\mathrm{of}\\, (1\\dots 2m) }\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\delta^{i_{p_1}i_{p_2}}\\dots\\delta^{i_{p_{2m-1}}i_{p_{2m}}}\\,\\delta_P\\,.\n\\label{sigma7}\n\\end{eqnarray}\nIn the last relation, $\\delta_P$ is the signature (sign) of the respective\npermutation  \n${1\\;\\,2 \\,\\ldots\\, \\;2m \\choose p_1\\,p_2\\ldots \\,p_{2m}}$, and for \neach $\\delta^{i_m i_n}$ only $m<n$ is allowed. Note that we call two\npermutations equivalent if they lead to the same term in the sum.\nFor $n=3$ we have $\\sigma^{ij}=i \\epsilon^{ijk}\\sigma^k$, \n$\\sigma^{ijk}=i \\epsilon^{ijk}\\mathbf 1$, and $\\sigma^{i_1\\cdots i_m}=0$ for\n$m>3$. Thus the latter are evanescent for $n\\neq 3$. For $m\\leq 3$ the\n$\\sigma^{i_1\\cdots i_m}$ are related to physical spin operators.\n\n\n\\subsection*{S-Wave Currents}\n\nThe general interpolating current describing the production of a\nfermion-antifermion pair in an S-wave state and in an arbitrary spin state in\n$n$ dimensions has the form\n\\begin{eqnarray}\n(j_{L=0})^{[i_1\\cdots i_m]} & = &\n\\psi_{\\mathbf p}^\\dagger(x)\\, \\sigma^{i_1\\cdots i_m}\\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*(x)\n\\,.\n\\label{Swavecurrent1}\n\\end{eqnarray}\nA SO(n) rotation by an angle $\\theta$ around the axis $\\mathbf n$ leads to the\ntransformed current \n$D_\\theta(\\psi_{\\mathbf p}^\\dagger\\, \\sigma^{i_1\\cdots i_m}\\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*)=\\psi_{\\mathbf p}^\\dagger S(\\theta)^\\dagger \\sigma^{i_1\\cdots i_m}\\,\nS(\\theta)(i\\sigma_2)\\chi_{-\\mathbf p}^*$, where \n$S(\\theta)=\\exp(-i \\theta \\mathbf n.\\mathbf \\mbox{\\boldmath $\\sigma$}\/2)$. Since \n$S(\\theta)^\\dagger \\sigma^i S(\\theta)=D^{ij}(\\theta)\\sigma^j$, where \n$D^{ij}(\\theta)$ is the rotation matrix for $n$-vectors, the currents in\nEq.\\,(\\ref{Swavecurrent1}) are tensors with respect to SO(n) rotations. The\ntensors are irreducible due to the antisymmetry of the \n$\\sigma^{i_1\\cdots i_m}$~\\cite{grouptheory} and each have \n$(^n_m)$ independent free components. Their eigenvalues with respect to the\nsquare of the total spin operator \n$\\mathbf S^2=[\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\otimes\\mathbf 1_{ap}+\\mathbf 1_{p}\\otimes\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}]^2\/4=\n(n[\\mathbf 1_a\\otimes\\mathbf 1_{ap}]+[\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\otimes\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}])\/2$, where\nthe indices \nof the $\\sigma$-matrices are summed over, read\n\\begin{eqnarray}\n\\mathbf S^2\\,(\nj_{L=0})^{[i_1\\cdots i_m]}\n & = &\n\\frac{1}{2}\n\\psi_{\\mathbf p}^\\dagger\\,\\Big( \nn\\,\\sigma^{i_1\\cdots i_m}(i\\sigma_2) + \n\\mathbf \\mbox{\\boldmath $\\sigma$}\\sigma^{i_1\\cdots i_m}(i\\sigma_2)\\mathbf \\mbox{\\boldmath $\\sigma$}^T \\Big)\\,\n\\chi_{-\\mathbf p}^*\n\\nonumber\\\\\n& = &\n\\frac{1}{2}\n\\left(n+(-1)^m\\,(2m-n)\n\\right)\\,\n(j_{L=0})^{[i_1\\cdots i_m]}\n\\,.\\qquad\n\\label{S2op}\n\\end{eqnarray}\nFor the physical currents with $m=(0,1,2,3)$ one thus finds the spin\neigenvalues $(0,n-1,2,n-3)$. Note that the spin eigenvalue for the unphysical\nS-wave currents $\\sigma^{i_1\\cdots i_m}$ for $m>3$ are non-zero.\nWhile for $n=3$ the spin singlet currents for\n$m=0,3$ are equivalent, and likewise the triplet currents for $m=1,2$, each of\nthe currents represents a different irreducible representation of SO(n) for $n\\neq\n3$. It is a very instructive fact that the action of $\\mathbf S^2$ onto the\nsinglet current $j_{L=0}^{[i_1i_2i_3]}$ does not give zero for $n\\neq\n3$. Thus to achieve that spin-dependent potentials do not contribute for\nphysical predictions involving spin singlet currents, in general additional finite\nrenormalizations are required in analogy to Ref.~\\cite{DuganGrin}, unless a\nscheme for potentials or currents is chosen such that they vanish\nautomatically. However, even if such a scheme is adopted, matrix elements,\nmatching conditions and anomalous dimensions can depend on the spin-dependent\npotentials at nontrivial subleading order\\footnote{\nTo be more specific,\nwe refer to orders of perturbation theory where subdivergences need to be\nrenormalized.\n}.\n\nAll of the physical currents can arise in important processes. In\nTab.~\\ref{tab1} the leading order matching for a number of\ndifferent currents is displayed. In general there are several\nrelativistic currents that can lead to the same nonrelativistic\ncurrent~\\cite{Fadin:1991zw}. Note that the respective production\nand annihilation currents are related either by hermitian or\nantihermitian conjugation. Except for $\\gamma_5$, which is\ntreated as fully anticommuting ($\\gamma_5=(^{0\\,\\mathbf I}_{\\mathbf I\\,0})$),\nthree dimensional relations have not been used.\nThe full theory \ncurrent with the $\\gamma$-structure $\\gamma^{ijk}$ for example arises for\nfermion pair production in $\\gamma\\gamma$ collisions, as shown in \nAppendix~\\ref{appendix2}.\nHowever, also evanescent currents naturally occur in standard processes, such as\nthe current $j_{L=0}^{[i_1\\cdots i_5]}$ that arises in fermion-antifermion\npair annihilation into three photons. The explicit computation can be found in\nAppendix~\\ref{appendix3}. Also note that the differences between\nthe two different singlet and triplet currents correspond to evanescent\noperators as well. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||cc|}\\hline  Full theory $\\Gamma$ & &\n$\\Gamma(\\mathbf p,\\bm{\\sigma})$  \\\\ \\hline \\hline \n $\\gamma^{i_1\\ldots i_{2k}}$ && $(-1)^{k+1} \\frac{p^\\ell}{m}\\,\\sigma^{\\ell\n i_1\\ldots i_{2k}}$ \\\\ \\hline \n $\\gamma^{i_1\\ldots i_{2k+1}}$ && $(-1)^{k} \\,\\sigma^{ i_1\\ldots i_{2k+1}}$\n \\\\ \\hline \n $\\gamma^0\\gamma^{i_1\\ldots i_{2k}}$ &  $(\\star)$ & $(-1)^{k} \\sum_{j=1}^{2k}\n (-1)^{j+1}\\frac{p^{i_j}}{m} \\sigma^{ i_1\\ldots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\ldots i_{2k}}$ \\\\\n \\hline \n $\\gamma^0\\gamma^{i_1\\ldots i_{2k+1}}$ & $(\\star)$ & $(-1)^{k} \\,\\sigma^{ i_1\\ldots\n i_{2k+1}}$  \\\\ \\hline \n $\\gamma^{i_1\\ldots i_{2k}}\\gamma^5$ & $(\\star)$ & $(-1)^{k} \\,\\sigma^{ i_1\\ldots i_{2k}}$\n \\\\ \\hline \n$\\gamma^{i_1\\ldots i_{2k+1}}\\gamma^5$ & $(\\star)$ & $(-1)^{k}\\frac{p^{\\ell}}{m}\n \\sigma^{\\ell i_1\\dots i_{2k+1}}$ \\\\\n \\hline \n$\\gamma^0\\gamma^{i_1\\ldots i_{2k}}\\gamma^5$ && $(-1)^{k} \\,\\sigma^{ i_1\\ldots\n   i_{2k}}$    \\\\ \\hline \n$\\gamma^0\\gamma^{i_1\\ldots i_{2k+1}}\\gamma^5$ && $(-1)^{k} \\sum_{j=1}^{2k+1}\n (-1)^{j}\\frac{p^{i_j}}{m} \\sigma^{ i_1\\ldots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j\\ldots i_{2k+1}}$    \\\\\n \\hline \n \\end{tabular} \n\\end{center}\n\\caption{Nonrelativistic production currents \n$\\psi_{\\mathbf p}^\\dagger\\,\\Gamma(\\mathbf p,\\bm{\\sigma}) \\,\n(i\\sigma_2)\\chi_{-\\mathbf p}^*$ arising from the leading order matching with the full\ntheory currents $\\bar{\\psi}\\,\\Gamma \\, \\psi$. The nonrelativistic\nnormalization of the full theory spinors has been used (see\nEq.~(\\ref{spinors})). The notation $\\gamma^{i_1\\ldots i_{m}}$ stands for the\naveraged antisymmetrized product of $\\gamma$-matrices for indices other than\nzero. For the corresponding annihilation currents \n$\\chi_{-\\mathbf p}^T(-i\\sigma_2)\\,\\Gamma^\\prime(\\mathbf p,\\bm{\\sigma})\\,\\psi_{\\mathbf p}$ \none has $\\Gamma^\\prime=-\\Gamma$ for all entries \nwith the $(\\star)$ symbol and $\\Gamma^\\prime=\\Gamma$ for the others.\n}  \n\\label{tab1} \n\\end{table}\nIt is well known from subleading order computations based on the effective\nweak Hamiltonian that one needs to consistently account for the evanescent\noperator structures that arise in matrix elements of physical operators when\nbeing dressed with gluons. In Ref.\\,\\cite{DuganGrin} it was shown that a\nrenormalization scheme can be adopted such that a mixing of evanescent\noperators into physical ones does not arise. Moreover it is also\nknown~\\cite{HerrlichNierste,ChetyrkinMisiak} that modifications of the\nevanescent operator basis (e.g.\\,by adding physical operators multiplied by\nfunctions of $\\epsilon$ that vanish for $\\epsilon\\to 0$) and similar\nmodifications to the physical operator basis correspond to a change of the\nrenormalization scheme. While this does not affect physical predictions, it\ndoes affect matrix elements, matching conditions and anomalous dimensions at\nnontrivial subleading order.\nThus precise definitions of the schemes being employed have to be given to\nrender such intermediate results useful. \n\nIn the framework of the nonrelativistic EFT these properties still apply.\nHowever, the velocity power counting in the EFT allows for even\nmore specific statements. Concerning interactions through potentials,\ntransitions between S-wave currents in Eq.\\,(\\ref{Swavecurrent1}) that are\ninequivalent cannot occur because the potentials are \nSO(n) scalars and the currents are (due to the different symmetry\npatterns of their indices~\\cite{grouptheory}) inequivalent irreducible\nrepresentations of SO(n). Even for currents with $L\\neq 0$ and for the\nspin-dependent spin-orbit and tensor potentials (which is all we need\nto consider at NNLL order) one can show with\nEqs.\\,(\\ref{sigma3}) and (\\ref{sigma4}) that transitions between currents\ncontaining $\\sigma^{i_1\\cdots i_m}$ with a different number of indices\ncannot occur (see Sec.\\,\\ref{sectionNLLtriplet}.1). Transitions can,\nhowever, arise between currents with \ndifferent angular momentum, such as for the tensor potential that can\ngenerate transitions between $L$ and $L^\\prime=L\\pm 2$ (see\nSec.\\,\\ref{sectionNLLtriplet}). The same arguments apply to the\nexchange of soft gluons (in the framework of vNRQCD) since they cannot appear\nas external particles and furthermore need to be exchanged in pairs due to\nenergy conservation. In this respect the effects from soft gluon exchange\neffectively represent modifications to the potential interactions (see\ne.g. Ref.\\,\\cite{LMR}). Concerning the \nexchange of ultrasoft gluons, transitions between currents\ncontaining $\\sigma^{i_1\\cdots i_m}$ with a different number of indices\ncan arise, but only if the interaction is spin-dependent. The dominant\namong these interactions corresponds to the operator \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ij} k^j \\psi_{\\mathbf p} A^i$, where $A^i$ is\nan ultrasoft gluon. This operator can only\ncontribute at N${}^4$LL order, which is beyond the present need and\ntechnical capabilities. Thus in practice, transitions between\ncurrents containing $\\sigma^{i_1\\cdots i_m}$ with a different number\nof indices do not need to be considered. \n\nSo for the S-wave currents in $n$ dimensions one can employ either one \nof the two spin singlet ($k=0,3$) or triplet currents ($k=1,2$) in the\nEFT and the difference corresponds to a change in the renormalization\nscheme. This means in particular that as long as the renormalization\nprocess is restricted e.g.\\,to time-ordered products of the currents, one\ncan (but does not have to) freely use three-dimensional relations to reduce\nthe basis of the physical currents. However, once the basis of the\nphysical currents is fixed (where each current is irreducible with\nrespect to SO(n)), one has to consistently apply the computational\nrules in $n$  dimensions discussed in the previous sections. As an\nexample, instead of using the current \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ijk}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ that arises\nin $\\gamma\\gamma$ collision, one can employ the current \n$\\psi_{\\mathbf p}^\\dagger(i\\sigma_2)\\chi_{-\\mathbf p}^*$, defined in $n$ dimensions,\ntimes the $\\epsilon$-tensor $i\\epsilon^{ijk}$ defined in three dimensions,\nwhich means that the $\\epsilon$-tensor is zero if any of its indices takes a\nvalue different from $1,2,3$. Concerning the\n$\\epsilon$-tensor, this works because the $\\epsilon$-tensor does not\nplay any role during \nthe renormalization procedure as long as we only consider time-ordered\nproducts of the currents.\\footnote{This procedure cannot be applied in\nthis simple form, if the initial state that is involved in the\nquark-antiquark production process is also involved in the\nrenormalization procedure, as can be the case for QED corrections.} \nThis justifies the approach in Sec.\\,\\ref{sectionsinglet} where we\nhave only considered spin singlet currents for fermions involving the\ncurrent $\\psi_{\\mathbf p}^\\dagger(i\\sigma_2)\\chi_{-\\mathbf p}^*$. This scheme is\nalso advantageous practically because for the current\n$\\psi_{\\mathbf p}^\\dagger\\sigma^{ijk}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ the\nmatrix elements of the spin-dependent potentials that vanish for $n=3$\ncan, as  discussed above, give evanescent contributions for $n\\ne 3$. \nMoreover, from these considerations we can also conclude that currents\ncontaining evanescent $\\sigma^{i_1\\cdots i_m}$ matrices ($m>3$) can\n(but do not have to) be dropped from the very beginning in the EFT as\nlong as one does not need to account for spin-dependent ultrasoft gluon\ninteractions.  \n\nAn important lesson to learn from this discussion is that partial results at\nnontrivial subleading order obtained from the threshold \nexpansion~\\cite{Beneke} of full theory diagrams such as e.g. the \ncontributions from the hard regions obtained in\nRefs.~\\cite{Czarnecki1,Beneke4,firstc0,2loop_hard}, \nare equal to the EFT matching conditions only in schemes for the effective \ntheory currents and potentials that are compatible with the nonrelativistic\nreduction of the $\\gamma$ matrices that has been used during the threshold \nexpansion. In general, there are additional contributions to EFT matching\nconditions to account for the scheme choices made in the EFT.\nNote that in the threshold expansion different scheme choices can also \nbe possible, in particular for the treatment of $\\gamma_5$.\nIn Sec.~\\ref{sectioncomments} we comment on scheme dependences\nof a number of partial results that can be found in the literature.\n\nAt this point one might also ask whether $\\gamma_5$ needs special\ntreatment in the nonrelativistic EFT in $n$ dimensions. Chirality\nand the flavor symmetries are not relevant in the EFT for a single heavy\nparticle-antiparticle system and their potential\neffects are contained in the matching contributions of the EFT to \nthe full theory. For the\nmatching relations shown in Tab.~\\ref{tab1} we have used a totally\nanticommuting $\\gamma_5$. Since the resulting effective theory\ncurrents have well defined SO(n) transformation properties and reduce\nto the proper nonrelativistic currents for $n\\to 3$, this represents a\nconsistent scheme choice from the point of view of the effective theory\ncomputations. A different ansatz for $\\gamma_5$ such as\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ (which is the fully consistent\none in the full theory) just corresponds to a different choice of scheme for  \nthe currents in the nonrelativistic EFT, and both schemes can be used\nconsistently. Note that from the results in Tab.~\\ref{tab1} without an\nexplicit $\\gamma_5$ one can also \nderive the form of the EFT currents for the ansatz\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ (see also\nSec.~\\ref{sectioncomments}). \n\n \n\\subsection*{Arbitrary Angular Momentum}\n\nBased on the results obtained in the previous sections it is\nstraightforward to construct spin-triplet currents with arbitrary\nangular momentum $L$ (${}^{2S+1}L_J={}^3L_J$)). They can be obtained\nby determining irreducible SO(n) representations from products of the\ntensors $T^{i_1\\dots i_L}(\\mathbf p)$ describing angular momentum $L$ and\nthe spin-triplet $S=1$ currents discussed in the previous\nparagraphs. Due to the different symmetry patterns of the symmetric\n$T^{i_1\\dots i_L}(\\mathbf p)$ tensors and the antisymmetric \n$\\psi_{\\mathbf p}^\\dagger\\sigma^{i_1\\cdots i_k}(i\\sigma_2)\\chi_{-\\mathbf p}^*$\ncurrents one needs to apply the construction principles for general\ntensors~\\cite{grouptheory} which state that tensors are irreducible\nwith respect to SO(n) if and only if they are traceless and if their\nindices have a symmetry pattern according to a standard Young tableau. \nAs for the case of the S-wave currents the physical basis for\narbitrary spatial angular momentum is not unique due to the existence\nof evanescent operator structures. \n\nHere we construct currents with fully symmetric indices because of\ntheir comparatively simple form and because their number of indices is\nequal to the total angular momentum $J$ quantum number. For the\ncase $J=L\\pm 1$, these currents are contained in the reduction of the \nreducible tensor $A^{i_1\\cdots i_{L+1}} = \\psi_{\\mathbf p}^\\dagger \n\\,T^{i_1\\dots i_L}(\\mathbf p)\\sigma^{i_{L+1}}(i\\sigma_2)\\chi_{-\\mathbf p}^*$. Upon\nsymmetrization and removal of all traces one obtains the \n${}^{2S+1}L_J={}^3L_{L+1}$ current\n\\begin{eqnarray}\n \\left(j_{J=L+1}^{S=1}\\right)^{i_1\\dots i_{L+1}} \n & \\equiv &  \n\\psi_{\\mathbf p}^\\dagger\n\\,\\Big[\\Gamma^{S=1}_{L+1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L+1}} \\Big]\n(i\\sigma_2)\\chi_{-\\mathbf p}^* \n\\,, \\nonumber\\\\\n \\Gamma^{S=1}_{L+1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L+1}}  \n&\\equiv& \n\\sum_{k=1}^{L+1}\\,T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots i_{L+1}}(\\mathbf p)\\,\\sigma^{i_k}\n  - \\frac{2}{2L+n-2}\\,\\sum_{k<j \\atop k=1}^{L+1}\\,\\delta^{i_k\n  i_j}\\,T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_j \\dots i_{L+1}\n  \\ell}(\\mathbf p)\\,\\sigma^{\\ell}\n\\,.\\nonumber\\\\\n\\label{tripletcurrent1}\n\\end{eqnarray}\nThe subtractions in the second term of Eq.~(\\ref{tripletcurrent1})\nneeded to achieve zero traces are for themselves\n${}^{2S+1}L_J={}^3L_{L-1}$ currents, so we can define\n\\begin{eqnarray}\n \\left(j_{J=L-1}^{S=1}\\right)^{i_1\\dots i_{L-1}} \n&\\equiv &\n \\psi_{\\mathbf p}^\\dagger\n \\Big[\\,\\Gamma^{S=1}_{L-1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L-1}}\\Big]\n(i\\sigma_2)\\chi_{-\\mathbf p}^* \n\\,,\\nonumber\\\\[2mm]\n\\Gamma^{S=1}_{L-1}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L-1}}\n  & \\equiv & \nT^{\\,i_1\\dots i_{L-1}\\ell}(\\mathbf p)\\,\\sigma^{\\ell}\n\\,.\n\\label{tripletcurrent2}\n\\end{eqnarray}\nThe tensor  $A^{i_1\\cdots i_{L+1}}$ also contains ${}^3L_{L}$\ncurrents. However, their indices are not fully symmetric. Their form\nis discussed in the Appendix~\\ref{appendix3} together with a more\ndetailed derivation of the two currents displayed above. A fully\nsymmetric ${}^3L_{L}$ current can be obtained from the reduction of\nthe tensor  $B^{i_1\\dots i_{L+2}} = \\psi_{\\mathbf p}^\\dagger \n\\,T^{i_1\\dots i_L}(\\mathbf p)\\sigma^{i_{L+1}\n  i_{L+2}}(i\\sigma_2)\\chi_{-\\mathbf p}^*$. Upon \nremoval of traces of  $B^{i_1\\dots i_{L+2}}$ one\ncan identify the  ${}^3L_{L}$ current\n\\begin{eqnarray}\n \\left(j_{J=L}^{S=1}\\right)^{i_1\\dots i_{L}}  \n &\\equiv &\n \\psi_{\\mathbf p}^\\dagger\n \\Big[\\,\\Gamma^{S=1}_{L}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L}}\\Big]\n(i\\sigma_2)\\chi_{-\\mathbf p}^* \n\\,,\\nonumber\\\\\n\\Gamma^{S=1}_{L}(\\mathbf p,\\bm{\\sigma})^{i_1\\dots i_{L}}\n & \\equiv & \n\\sum_{k=1}^{L}\\,T^{\\,i_1\\dots\\,\\,\\!\\widehat{\\!\\textit{\\i}}_k\\dots i_{L} \\ell}(\\mathbf p) \\,\\sigma^{\\ell\\,\n i_k}  \n\\,.\n\\label{tripletcurrent3}\n\\end{eqnarray}\nThe tensor $B^{i_1\\dots i_{L+2}}$ also contains evanescent currents and also\ncurrents describing  ${}^3L_{L\\pm 1}$ states that differ from the currents\ngiven above for $n\\neq 3$. Their form is also discussed in the\nAppendix~\\ref{appendix3}. The number of independent components of the\nspin-triplet currents defined above is that of a fully symmetric tensor with\n$J$ indices, {\\it i.e.} equal to $n_J$ given in Eq.~(\\ref{nL}). All three\ncurrents have parity and charge conjugation $P=(-1)^{L+1}$ and $C=(-1)^{L+S}$,\nrespectively. Also note that that \n$\\Gamma^{S=1}_{L\\pm 1}(\\mathbf p,\\bm{\\sigma})$ and \n$i\\Gamma^{S=1}_{L}(\\mathbf p,\\bm{\\sigma})$ are hermitian matrices.\n\nThe eigenvalue equation of the currents with respect to the spin-orbit\noperator $\\mathbf S.\\mathbf L= - [S^j,S^k] \\,p^j \\partial\/ \\partial p^k \n= - 1\/4( \\, [\\bm{\\sigma}_p^j,\\bm{\\sigma}_p^k]\\otimes\\mathbf 1_{ap} +\n\\mathbf 1_p\\otimes[\\bm{\\sigma}_{ap}^j,\\bm{\\sigma}_{ap}^k] \\, )\\,p^j \\partial\/\n\\partial p^k $ has the form  \n\\begin{eqnarray}\n\\lefteqn{\n(\\mathbf S.\\mathbf L)\\,\\Big(\n\\psi_{\\mathbf p}^\\dagger \\, \\Gamma(\\mathbf p,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_J}(i\\sigma_2)\\,\\chi_{-\\mathbf p}^*\\Big) \n = } \\nonumber\\\\\n&& =\n-\\frac{1}{4}\\, \\psi_{\\mathbf p}^\\dagger\\Big( [\\bm{\\sigma}^j,\\bm{\\sigma}^k]\n\\,p^j \\frac{\\partial}{\\partial p^k} \\,\\Gamma(\\mathbf p,\\bm{\\sigma})^{i_1\\cdots\n  i_J}(i\\sigma_2)+ \n\\,p^j \\frac{\\partial}{\\partial p^k} \\,\\Gamma(\\mathbf p,\\bm{\\sigma})^{i_1\\cdots\n  i_J}(i\\sigma_2)[ \n\\bm{\\sigma}^j,\\bm{\\sigma}^k]^T \\Big) \\chi_{-\\mathbf p}^*\n\\,,\\mbox{\\quad}\n\\label{SLop}\n\\end{eqnarray}\nwhere $\\Gamma(\\mathbf p,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_J}$ represents any of the\ntensors between the fermion fields.\nThe eigenvalues of the different currents defined in \nEqs.~(\\ref{singletcurrent1},\\ref{tripletcurrent1},\\ref{tripletcurrent2},\\ref{tripletcurrent3})   \nwith respect to the operators $\\mathbf L^2$,\n$\\mathbf S^2$, $\\mathbf L.\\mathbf S$ and $\\mathbf J^2$ are summarized in Tab.~\\ref{tab2}. \n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c||c|c|c|c|}\\hline     \n& \\raisebox{-1mm}[1mm]{$j_{J=L}^{S=0}$} \n& \\raisebox{-1mm}[1mm]{$j_{J=L-1}^{S=1}$} \n& \\raisebox{-1mm}[1mm]{$j_{J=L}^{S=1}$} \n& \\raisebox{-1mm}[1mm]{$j_{J=L+1}^{S=1}$}\n  \\\\[2mm] \\hline \\hline  \n$\\mathbf L^2$  & $L(L+n-2)$ & $L(L+n-2)$ & $L(L+n-2)$ & $L(L+n-2)$ \\\\[2mm] \\hline \n$\\mathbf S^2$ & 0 & $n-1$ & 2 & $n-1$   \\\\[2mm] \\hline \n$\\mathbf S.\\mathbf L$ & 0 & $-(L+n-2)$ & $-(n-2)$ & $L$ \\\\ \\hline \n$\\mathbf J^2$ &  $L(L+n-2)$ & $J(J+n-2)$ & $J(J+n-2)+6-2n$ & $J(J+n-2)$\\\\ \\hline  \n \\end{tabular} \n\\end{center}\n\\caption{\nSpin and orbital angular momentum quantum numbers of the currents defined in\nEqs.~(\\ref{singletcurrent1},\\ref{tripletcurrent1},\\ref{tripletcurrent2},\\ref{tripletcurrent3})\nin $n$ dimensions.}\n\\label{tab2} \n\\end{table}\nFor the computation of the NLL order anomalous dimensions of the\ncurrents from time-ordered products of two currents\n(Secs.~\\ref{sectionNLLsinglet} and \\ref{sectionNLLtriplet}), the total \ncontractions of spatial and spin indices (i.e. taking the trace of\nproducts of $\\sigma$-matrices) are useful:\n\\begin{eqnarray}\n\\mbox{Tr}\\Big[\\Gamma^{S=1\\,\\dagger}_{L+1}(\\mathbf p,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L+1}}\\,\n\\Gamma^{S=1}_{L+1}(\\mathbf q,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L+1}}\\Big] \n& = & \n(L+1)(2L+n)\\,\\frac{L+n-2}{2L+n-2}\\,T_{L,n}(\\mathbf p,\\mathbf q)\\,\n\\mbox{Tr}\\,[\\mathbf 1]\n\\,,\n\\nonumber\\\\[2mm]\n\\mbox{Tr}\\Big[\\Gamma^{S=1\\,\\dagger}_{L}(\\mathbf p,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L}}\\,\n\\Gamma^{S=1}_{L}(\\mathbf q,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L}}\\Big] \n& = & \nL(L+n-2)\\,T_{L,n}(\\mathbf p,\\mathbf q)  \\, \\mbox{Tr}\\,[\\mathbf 1]\n\\,,\n\\nonumber\\\\[2mm]\n\\mbox{Tr}\\Big[\\Gamma^{S=1\\,\\dagger}_{L-1}(\\mathbf p,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L-1}}\\,\n\\Gamma^{S=1}_{L-1}(\\mathbf q,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\cdots i_{L-1}}\\Big] \n& = & \nT_{L,n}(\\mathbf p,\\mathbf q) \\, \\mbox{Tr}\\,[\\mathbf 1]\n\\,,\n\\end{eqnarray}\nwhere the function $T_{L,n}$ is defined in Eq.~(\\ref{Tcontract}).\nNote that the expressions only involve traces of products of an even\nnumber of $\\sigma$-matrices which can be computed unambiguously from\nthe anticommutator of two $\\sigma$-matrices. \n\n \n\\section{NLL Order anomalous dimensions: Singlet Case} \n\\label{sectionNLLsinglet}\n\nFor the computation of the renormalization constants of the physical\ncurrents defined in the previous sections one has to determine the\noverall UV-divergence of particle-antiparticle-to-vacuum on-shell\nmatrix elements of the currents from vertex diagrams. However, this\nmethod is complicated by the fact that for on-shell external particles\nwith $E=\\mathbf p^2\/m$ the vertex diagrams also contain IR-divergent\nCoulomb phases that have to be distinguished from the\nUV-divergences. To avoid this issue we use the method from\nRef.~\\cite{Hoang3loop} which considers current-current correlator\ngraphs obtained from closing the external particles-antiparticle line\nof the vertex diagrams with an additional insertion of the (hermitian\nconjugated) current. This means that one has to determine diagrams with\none more loop, but in this way all IR-divergences associated to the\nCoulomb phase cancel and one needs to compute a fewer number of\ndiagrams. Technically, the UV-divergences associated to the currents\nappear as subdivergences of the correlator diagrams, which can,\nhowever, easily identified from the absorptive parts of the\ncorrelators. \n\nSince there are no UV divergences in the EFT that can lead to a LL\nanomalous dimension of the currents that are leading in the\nnonrelativistic expansion~\\cite{LMR}, the NLL order level represents\nthe lowest order in which these currents are renormalized.\nThe corresponding three-loop current correlator diagrams\nare depicted in Fig.\\,\\ref{fignllcorrelators}.\n\\begin{figure}[t]\n\\begin{center}\n \\leavevmode\n \\epsfxsize=14cm\n \\leavevmode\n \\epsffile[45 460 545 575]{figs\/3loopvac.eps}\n \\vskip  0.0cm\n \\caption{\nThree-loop current correlator diagrams and the counterterm diagram for\nthe computation of the NLL anomalous dimension of the\ncurrents. Combinatorial factors are suppressed.\n \\label{fignllcorrelators} }\n\\end{center}\n\\end{figure}\nThe terms\n${\\cal V}_c^{(s)}$ and ${\\cal V}_2^{(s)}$, ${\\cal V}_r^{(s)}$ are the\ncolor singlet Wilson coefficients of the spin-independent potentials of \norder $\\alpha_s v^{-1}$ ($1\/\\mathbf k^2$) and $\\alpha_s v$ ($1\/m^2$,\n$(\\mathbf p^2+\\mathbf p^{\\prime 2})\/(2m^2\\mathbf k^2)$),\nrespectively~\\cite{amis,amis2}, while ${\\cal V}_{k,k1,k2}^{(s)}$ are \nthe Color singlet Wilson coefficients of the sum operators \n${\\cal O}_{k}^{(s)}$, ${\\cal O}_{k1}^{(1)}$, ${\\cal O}_{k2}^{(T)}$ \nintroduced in Refs.\\,\\cite{HoangStewartultra,Hoang3loop}. The latter operators\nare the analogue of the $1\/m|\\mathbf k|$ non-Abelian-potential that was used in\nearly computations where no systematic renormalization group\nimprovement was carried out (see e.g.~\\cite{HT1}). We note that the\nWilson coefficient  ${\\cal V}_2^{(s)}$ differs for quarks and for\nscalars~\\cite{HoangRuizSvNRQCD}. This is, however, irrelevant for the result\nof the anomalous dimension and only affects the solution of the resulting\nrenormalization group equations (RGE's). \nThe diagram (e) contains the renormalization constant of the current\nwhich we determine in the ${\\overline{\\rm MS}}$ scheme. Generically we define the\nrenormalization constant as\n\\begin{eqnarray}\nZ_c & = & 1 + \\frac{\\delta z_c^{\\rm NLL}}{\\epsilon}\\,+\\,\\ldots\n\\,.\n\\end{eqnarray}\nThe resulting NLL order RGE for the current\nWilson coefficients $c(\\nu)$ reads \n\\begin{eqnarray}\n\\nu\\frac{\\partial}{\\partial\\nu}\\ln[c(\\nu)] & = &\n4\\,\\delta z_c^{\\rm NLL}\n\\,.\n\\label{rgeCurrent}\n\\end{eqnarray}\n\nDue to rotational invariance and the fact that the currents are\nirreducible, all components of any given current have the same\nanomalous dimension. Therefore we can compute the correlators for\nall spatial and spin indices being contracted. As an example, for the\ncorrelators in Fig.~\\ref{fignllcorrelators}a with the two different\ninsertions of the \npotential $-i{\\cal V}_2\/m^2$, and after applying the contraction formula\n(\\ref{Tcontract}), one has to compute the three-loop diagram \n\\begin{eqnarray}\n-2i m N_c\\,{\\cal V}_c^{(s)}\\,{\\cal V}_2^{(s)}\\,N_L\n\\int\\!\\! D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n\\frac{|\\mathbf q_1|^L|\\mathbf q_3|^L P_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_3)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_2^2+\\delta)\\,(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\n\\,.\n\\label{V2correlatorexample}\n\\end{eqnarray}\nThe expression in Eq.~(\\ref{V2correlatorexample}) is equivalent to the\ncorresponding S-wave result up to the weight factor \n$N_L|\\mathbf q_1|^L |\\mathbf q_3|^L P_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_3)$, where\n$\\mathbf q_1$ and $\\mathbf q_3$ are the loop-momenta on the sides of the\ndiagram. The weight factor is a homogeneous polynomial of degree $L$\nof the terms $\\mathbf q_1^2$, $\\mathbf q_3^2$ and $\\mathbf q_1.\\mathbf q_3$, see\nEq.~(\\ref{Legendredef}). The \ncontributions to the renormalization constant $Z_c$ of each of the\ndiagrams in Fig.~\\ref{fignllcorrelators} with the weight factor\n$(\\mathbf q_1.\\mathbf q_3)^m(\\mathbf q_1^2)^{\\frac{L-m}{2}}(\\mathbf q_3^2)^{\\frac{L-m}{2}}$\nand proper combinatorial \nfactors are as follows:\n\\begin{eqnarray}\n\\delta z^{\\rm NLL}_{c,(a)} & = & \n-\\frac{{\\cal V}_c^{(s)}(\\nu){\\cal V}_r^{(s)}(\\nu)}\n   {64\\pi^2}\\,\n   {}_2F_1({\\textstyle\\frac{1}{2},-m,\\frac{3}{2},2}) \n\\, + \\,\n\\left\\{\n\\begin{array}{c}\n0\\,,\\qquad m\\,\\,\\mbox{odd}\\\\[2mm]\n{\\displaystyle -\\frac{{\\cal V}_c^{(s)}(\\nu){\\cal V}_2^{(s)}(\\nu)}{64(1+n)\\pi^2}}\n\\,,m\\,\\,\\mbox{even}\n\\end{array}\n\\right.\n\\,,\n\\nonumber\\\\[2mm]\n\\delta z^{\\rm NLL}_{c,(b)} & = &\n\\Big(\n\\frac{\\alpha_s(m\\nu)^2}{4} \\big(3{\\cal V}_{k1}^{(s)}(\\nu)+2{\\cal V}_{k2}^{(s)}(\\nu) \\big)\n+\\frac{\\alpha_s^2(m\\nu)}{8}\\big(\\frac{C_F^2}{2}-C_AC_F\\big)\n\\bigg)\\,\n   {}_2F_1({\\textstyle\\frac{1}{2},-m,\\frac{3}{2},2})\n\\,,\n\\nonumber\\\\[2mm]\n\\delta z^{\\rm NLL}_{c,(c,d)} & = &\n-\\frac{\\big({\\cal V}_c^{(s)}(\\nu)\\big)^2}\n   {256\\pi^2}\\,\n   {}_2F_1({\\textstyle\\frac{1}{2},-m,\\frac{3}{2},2}) \n\\,.\n\\end{eqnarray}\nIn App.~\\ref{appendix5} some useful formulae to determine these\nresults are given.\nThe final result for the anomalous dimension can then be obtained by\nusing Eq.~(\\ref{Legendredef}) for the definition of the generalized Legendre\npolynomial. The result is remarkably simple and reads\n\\begin{eqnarray}\n\\delta z^{\\rm NLL}_c & = & \n\\frac{1}{4(2L+1)}\\,\\bigg[\n-\\frac{1}{16\\pi^2}\\,{\\cal V}_c^{(s)}(\\nu)\\,\n\\Big( \\frac{{\\cal V}_c^{(s)}(\\nu)}{4} + {\\cal V}_r^{(s)}(\\nu) \\Big)\n\\nonumber\\\\[2mm] & & \\qquad  +\\,\n\\alpha_s(m\\nu)^2 \\Big(3{\\cal V}_{k1}^{(s)}(\\nu)+2{\\cal V}_{k2}^{(s)}(\\nu) \\Big)\n+\\frac{\\alpha_s^2(m\\nu)}{2}\\big(\\frac{C_F^2}{2}-C_AC_F\\big)\n\\,\\bigg]\n\\nonumber\\\\[2mm] & & \n-\\, \\frac{{\\cal V}_c^{(s)}(\\nu){\\cal V}_2^{(s)}(\\nu)}{64\\pi^2}\\,\\delta_{L0} \n\\,.\n\\label{Zscalarpot}\n\\end{eqnarray}\nThe explicit expressions for the potential Wilson coefficients for quarks and\ncolored scalars can be found in Refs.~\\cite{Hoang3loop} and\n\\cite{HoangRuizSvNRQCD} (see also Ref.~\\cite{Pineda1}).\nFor S- and P-waves the expression agrees with previous results, see\ne.g.~Ref.~\\cite{HoangRuizSvNRQCD}. Written in this way, the \nresult~(\\ref{Zscalarpot}) is valid for  \nboth scalar and fermion singlet currents. Differences in the running \nof the singlet currents for scalars and fermions can only arise at this order\nfrom a different running of the Wilson coefficients for the potentials.  \nSince at this order only the coefficient ${\\cal V}_2^{(s)}$ differs for quarks\nand for scalars, and since ${\\cal V}_2^{(s)}$ only contributes for S-waves, \nthe NLL order anomalous dimensions for quarks and scalars agree for $L\\ge 2$.\nWe note that the NLL anomalous dimension is\nindependent of the scheme used for the currents (or the potentials)\nsince at LL order the currents are not renormalized. Had we\ne.g.\\,used the singlet current with $\\sigma^{ijk}$ the spin-dependent\npotentials could in general contribute to the UV-finite terms at this order,\nbut not to the NLL anomalous dimension. \n\n\n\n\\section{NLL Order anomalous dimensions: Triplet Case} \n\\label{sectionNLLtriplet}\n\n\\subsection*{Spin Dependent Potentials}\n\nBefore determining the NLL order anomalous dimension for\nthe spin triplet currents, it is instructive to briefly discuss the form of the\nspin-dependent potentials from the point of view of scheme-dependences in the\nEFT in $n$ dimensions - despite the fact that the results for the NLL order\nanomalous dimensions of the currents defined above are\nscheme-independent. From the expansion of full theory \nquark-antiquark scattering diagrams  without relying on any $\\sigma$-matrix\nrelation for $n=3$ at intermediate steps, one obtains the tree-level potential  \n($\\mathbf k=\\mathbf p^\\prime-\\mathbf p$)\n\\begin{eqnarray}\n V({\\mathbf p},{\\mathbf p^\\prime}) &  = & \ng_s^2 (T^A \\otimes \\bar T^A) \\bigg[\\!\n \\frac{1}{\\mathbf k^2}\n + \\frac{{\\mathbf p^2 + \\mathbf p^{\\prime 2}}}{2 m^2 \\mathbf k^2} - \\frac{1}{2m^2 } \n + \\frac{k^l k^m}{m^2\\mathbf k^2}\n   \\left[S_p^i,S_p^l\\right]\\left[S_{ap}^i,S_{ap}^m\\right] \n - \\frac{3\\,p^ip^{\\prime j}}{2m^2\\mathbf k^2}\\left[S^i,S^j\\right]\n\\!\\! \\bigg]\n\\nonumber\\\\[2mm] & = &\ng_s^2 (T^A \\otimes \\bar T^A)\\bigg[\n \\frac{1}{\\mathbf k^2}\n + \\frac{{\\mathbf p^2 + \\mathbf p^{\\prime 2}}}{2 m^2 \\mathbf k^2} + \\frac{n-3}{4m^2 } \n\\nonumber\\\\[2mm] & & \\qquad \\qquad\n+ \\frac{\\mathbf S^2}{m^2 } \\left(\\frac{\\mathbf S^2}{n}-1 \\right)\n- \\frac{3}{4 m^2 \\mathbf k^2} \\, S^{[ij]} \\cdot T^{[p^i {p'}^j]}\n+ \\frac{2\\mathbf S^2-n}{2 m^2 \\mathbf k^2} \\, S^{ij} \\cdot T^{ij}(\\mathbf k)\n\\, \\bigg] \\,,\n\\label{ndimpot}\n\\end{eqnarray}\nwhere in Eqs.~(\\ref{ndimpot}) and in the following the symbol for \nthe particle-antiparticle tensor product is indicated only in color\nspace. The spin-dependent terms, displayed in the last equality, have been written\nin terms of irreducible tensors with\n\\begin{eqnarray}\nT^{[p^i {p'}^j]} &=& p^ip^{\\prime\\,j}-p^jp^{\\prime\\,i}\\,,\\nonumber\\\\\nS^{[ij]}=[S^i,S^j] &,&\nS^{ij}=\\frac{1}{2}\\{S^i,S^j\\}-\\frac{\\delta^{ij}}{n}\\mathbf S^2 \\,.\n\\end{eqnarray}\nThe tree-level potential in Eq.~(\\ref{ndimpot}) is consistent with the\n$n$-dimensional expression used in Ref.~\\cite{beneke} accounting for the fact that\nan insertion of the last two terms in the second line of Eq.~(\\ref{ndimpot})\ndoes not contribute for the current\n$j^{S=1}_{J=L+1=1}=\\psi_{\\mathbf p}^\\dagger\\, \\sigma^{i}\\,(i\\sigma_2)\\chi_{-\\mathbf p}^*$  \nat NNLO.\\footnote{\nWhile any number of insertions of the spin-orbit potential\n$\\sim S^{[ij]} T^{[p^i {p'}^j]}$ in the correlator of two\n$j^{S=1}_{J=L+1=1}$ currents vanishes identically in $n$ dimensions, several\ninsertions of the tensor potential $\\sim  S^{ij} T^{ij}(\\mathbf k)$\ncan yield non-vanishing contributions beyond NNLO.}\n\n\nApart from the spin-orbit potential $\\sim [S^i,S^j]p^i {p^\\prime}^j$ which \ntrivially reduces to $\\sim \\mathbf S.(\\mathbf p\\times\\mathbf p^\\prime)$ for $n=3$, the result \ndoes not have a very close resemblance to the well known spin-dependent terms\nin three dimensions~\\cite{TitardYndurain},\n\\begin{eqnarray}\n V^\\prime({\\mathbf p},{\\mathbf p^\\prime}) &  = & \ng_s^2 (T^A \\otimes \\bar T^A)\\! \\bigg[ -\\frac{\\mathbf S^2}{3 m^2 }\n-\\frac{3}{2m^2}\\Lambda({\\mathbf p^\\prime},{\\mathbf p})-\\frac{1}{12m^2} \\, T({\\mathbf k}) \\, \\bigg] \n\\,,\n\\label{spinpotdef1}\n\\end{eqnarray}\nwhere \n\\begin{equation}\n{\\mathbf S} \\, = \\, \\frac{ {{\\mathbf \\mbox{\\boldmath $\\sigma$}}_p + {\\mathbf \\mbox{\\boldmath $\\sigma$}}_{ap}} }{2}\\,,\n \\quad \n \\Lambda({\\mathbf p^\\prime},{\\mathbf p}) \\, = \\, -i \\frac{{\\mathbf S}( {\\mathbf p^\\prime}\n \\times {\\mathbf p}) }{  {\\mathbf k}^2 }\\,,\\quad\n T({\\mathbf k}) \\, = \\, {({\\mathbf \\mbox{\\boldmath $\\sigma$}}_{p}{\\mathbf \\mbox{\\boldmath $\\sigma$}}_{ap}}) - \n\\frac{3\\, { ({\\mathbf \\mbox{\\boldmath $\\sigma$}}_p\\mathbf k})\\,({\\mathbf \\mbox{\\boldmath $\\sigma$}}_{ap}\\mathbf k)}{ {\\mathbf k}^2} \\,.\n\\end{equation}\nThis is because the three-dimensional relation \n$\\sigma_{a}^{ij}\\otimes \\sigma_{ap}^{ik}=\n\\sigma_{a}^k\\otimes\\sigma_{ap}^j-\n\\delta^{jk}\\sigma_{a}^l\\otimes\\sigma_{ap}^l$ \ndoes not hold in this form for $n\\neq 3$. As for the  currents we are,\nhowever, free to use any scheme for the spin-dependent potentials as long as\nthey are SO(n)-scalars and reduce to the known spin-dependent potentials for\n$n=3$. \n\nA suitable scheme choice can be obtained from the $\\sigma$ structures\nshown in Eq.~(\\ref{ndimpot}), or from three-dimensional potentials\nin Eq.\\,(\\ref{spinpotdef1}) through the $n$-dimensional replacements\n\\begin{eqnarray}\n -i\\mathbf S(\\mathbf p'\\times\\mathbf p)\n  &\\longrightarrow &\n [S^i,S^j]\\,p^i {p'}^j \n\\, = \\, \n\\frac{1}{2}\\, S^{[ij]} \\cdot T^{[p^i {p'}^j]}\n\\,,\n\\nonumber\\\\[3mm]\n (\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap})-3\\,\\frac{(\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf k)\\,(\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}\\mathbf k)}\n  {\\mathbf k^2} \n &\\longrightarrow &\n (\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap})-n\\,\\frac{(\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf k)\\,(\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}\\mathbf k)}\n {k^2}\n\\, = \\,\n-2n\\, S^{ij} \\frac{T^{ij}(\\mathbf k)}{\\mathbf k^2}\n\\,.\n\\label{tensordef1}\n\\end{eqnarray} \nWe will use the scheme shown in Eq.~(\\ref{tensordef1}) for the\ncomputations of the anomalous dimensions \nin the following sections. This scheme is very convenient because the\ndifferent potentials are \ntotal contractions of spin- and angular momentum tensors containing only one\nsingle irreducible representation. As a comparison, consider in\ncontrast the scheme choice $(\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}) - 3\n(\\mathbf \\mbox{\\boldmath $\\sigma$}_p\\mathbf k)(\\mathbf \\mbox{\\boldmath $\\sigma$}_{ap}\\mathbf k)\/\\mathbf k^2$ for the \ntensor potential in $n$ dimensions, which is in principle a viable\nscheme as well since it is a scalar and also reduces to the correct\nexpression for $n=3$. Written in terms of irreducible tensors one\nfinds \n\\begin{eqnarray}\n{{\\mathbf \\mbox{\\boldmath $\\sigma$}}_p {\\mathbf \\mbox{\\boldmath $\\sigma$}}_{ap}} - 3\\frac{({\\mathbf \\mbox{\\boldmath $\\sigma$}}_p\\mathbf k)\\, \n ({\\mathbf \\mbox{\\boldmath $\\sigma$}}_{ap}\\mathbf k) }{ {\\mathbf k}^2}\n& = &\n-6\\, S^{ij} \\frac{T^{ij}(\\mathbf k)}{\\mathbf k^2} \n\\, + \\,\n\\frac{2(n-3)}{n}\\,\\mathbf S^2 - (n-3)\n\\,.\n\\label{tensordef2}\n\\end{eqnarray}\nWhile the tensor potential based on Eq.~(\\ref{tensordef1}) is a natural\nminimal extension of the tensor potential in three dimensions, the tensor\npotential based on Eq.~(\\ref{tensordef2}) does implicitly lead to a change of\nscheme for the $\\mathbf S^2\/m^2$ potential and the spin-independent $1\/m^2$\npotential, which in general affects matrix elements, matching conditions and\nanomalous dimensions at non-trivial subleading order, {\\it i.e.} beyond NLL\norder. \n\nWe would like to point out that the spin-dependent potentials in\nEqs.~(\\ref{tensordef1}) cannot produce transitions between the spin triplet\ncurrents with fully symmetric indices introduced in Sec.~\\ref{sectiontriplet}\nand other currents that transform as inequivalent irreducible representations \nof SO(n) for $n\\neq 3$, but that have the same quantum numbers for $n=3$ \n(e.g. built from the tensors $\\widetilde{\\Gamma}^{S=1}_J$ introduced in the\nAppendix.~\\ref{appendix3}). Technically this property\narises because a given current with fully symmetric indices and quantum numbers\n$J$ and $L$ (${\\Gamma}^{S=1}_J$) \nalways differs from its counterpart $\\widetilde{\\Gamma}^{S=1}_J$\nin the number of sigma matrices. For example, the insertion of a spin-orbit\npotential in a current with a spin structure $\\sigma^{i_1\\ldots i_m}$  yields\nthe spin tensor on the LHS of Eq.~(\\ref{sigma3}), where the indices $j$ and\n$\\ell$ are contracted with the initial and final momenta. The RHS of the\nsame equation shows that the resulting spin structure consists of terms with\nthe same number of sigma matrices as the initial current. The same holds\ntrivially for the case of the $\\mathbf S^2$ potential, and also for the tensor\npotential, where the relevant relation is Eq.~(\\ref{sigma4}). The possibility\nthat the spin-dependent $\\mathbf S^2$ and the spin-orbit potentials could induce\ntransitions between currents \ntransforming according to different irreducible representations with the same\n$J$ but different $L$ is also ruled out, as the $\\mathbf S^2$ and the spin-orbit\npotentials cannot change the angular  momentum $L$. On the other hand, the\ntensor potential can generate transitions among the $\\Gamma^{S=1}_{L\\pm 1}$\nfully symmetric currents by two units of $L$. \n\n\n\n\\subsection*{Anomalous Dimensions}\n\nThe spin-orbit and tensor potentials defined in Eq.~(\\ref{tensordef1})\ncontribute to the NLL anomalous dimension of the spin triplet currents through\ninsertions in the three-loop current correlator diagrams in\nFig.\\,\\ref{figspincorrelator}. A single insertion of the spin-orbit potential\n$-i{\\cal V}_{\\Lambda}^{(s)} S^{[ij]} \\cdot T^{[p^i {p'}^j]}\/(2m^2\\mathbf k^2)$ \nyields for any of the spin triplet currents defined in\nEqs.~(\\ref{tripletcurrent1},\\ref{tripletcurrent2},\\ref{tripletcurrent3})\n\n\\begin{figure}[t] \n\\begin{center}\n \\leavevmode\n \\epsfxsize=5cm\n \\leavevmode\n \\epsffile{figs\/3loopspin.eps}\n \\vskip  0.0cm\n \\caption{\nThree-loop current correlator diagrams with insertions of the spin-dependent\npotentials for the computation of the NLL anomalous dimension of the\ncurrents. Combinatorial factors are suppressed.\n \\label{figspincorrelator} }\n\\end{center}\n\\end{figure}\n\\begin{eqnarray}\n\\lefteqn{\n-i\\,m\\,\nN_c \\, {\\cal V}_c^{(s)}\\,{\\cal V}_{\\Lambda}^{(s)}\n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\,\n}\\nonumber\\\\ && \\times\n \\frac{\\mbox{Tr}\\Big[\n\\left( \n\\Gamma_{J}^{S=1}(\\mathbf q_3,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\n\\right)^\\dagger\\,[S^i,S^j]\\,\n\\left(\n \\Gamma_{J}^{S=1}(\\mathbf q_1,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\n\\right)\n\\Big]q_2^i\\,q_3^j}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,,    \n\\label{Vspinorbitcorrelator}\n\\end{eqnarray}\nwhere the proper contraction of spinor indices is in analogy to\nEqs.\\,(\\ref{S2op},\\ref{SLop}). Note that the correlator of two spin triplet \ncurrents with the same $J$ but different $L$ vanishes, as the spin-orbit\npotential commutes with the squared angular momentum operator $\\mathbf L^2$.  To\ncompute the contraction in the numerator of Eq.~(\\ref{Vspinorbitcorrelator})\nit is useful to perform the shift (\\ref{shift}) in the $\\mathbf q_1$\nintegration. This reduces all non-trivial contractions in the numerator to the\nform of Eq.~(\\ref{TyTx}) after use of the relations (\\ref{Trelations}). The\ncomputation then reduces to  \n\\begin{eqnarray}\n&&-i m\\,N_c\\,{\\cal V}_c^{(s)}\\,\n  {\\cal V}_{\\Lambda}^{(s)}\\frac{N_L\\,c_{J,L}}{2L+n-2}\\,\\mbox{Tr}\\,[\\mathbf 1]\\nonumber\\\\ \n&&\\times\n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\, \n\\frac{|\\mathbf q_1|^L|\\mathbf q_2||\\mathbf q_3|^{L+1} P_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_2)\n\\Big(P_{L+1}(n,\\bar\\mathbf q_2.\\bar\\mathbf q_3)-P_{L-1}(n,\\bar\\mathbf q_2.\\bar\\mathbf q_3)\\Big)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\,,\\qquad\n\\label{Vspinorbitcorrelator2}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nc_{J,L}=\\left\\{ \\begin{array}{rl} -L(L+1)(2L+n)\\frac{(L+n-2)}{(2L+n-2)}\\,,  & \n                 \\;\\;\\; J=L+1 \\\\\n         L(n-2)(L+n-2)\\,,  & \\;\\;\\; J=L \\\\\n\t L+n-2\\,, & \\;\\;\\; J=L-1\n\t \\end{array} \\right.\n\\end{eqnarray}\n\nThe calculation of the three-loop current correlator with a tensor potential\ninsertion $-i{\\cal V}_t^{(s)}(-2n)S^{ij}T^{ij}(\\mathbf k)\/(m^2\\mathbf k^2)$ can be\ncarried out similarly. For the case where the triplet currents at the sides\nhave the same $L$ the correlator reads \n\\begin{eqnarray}\n\\lefteqn{\n-im\\,N_c\\, {\\cal V}_c^{(s)}\\,{\\cal V}_{t}^{(s)}(-2n)\n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\, \n}\n\\nonumber\\\\ && \\times\n\\frac{\\mbox{Tr}\\Big[\n\\left(\n \\Gamma_{J}^{S=1}(\\mathbf q_3,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\n\\right)^\\dagger\\,S^{ij}\\,T^{ij}(\\mathbf q_3-\\mathbf q_2)\\,\n\\left(\n \\Gamma_{J}^{S=1}(\\mathbf q_1,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\\right)\n\\Big]}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,(\\mathbf q_2-\\mathbf q_3)^2\\,\n(\\mathbf q_3^2+\\delta)}\\nonumber\\\\[3mm]\n& =& -i m\\,N_c\\,{\\cal V}_c^{(s)}\\,{\\cal V}_{t}^{(s)}N_L\\,\\tilde{c}_{J,L}\\,\n  \\mbox{Tr}\\,[\\mathbf 1]\n \\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\, \n \\frac{|\\mathbf q_1|^L|\\mathbf q_3|^{L} P_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_2)\\,\n  \\chi_L(n;\\mathbf q_2,\\mathbf q_3)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,(\\mathbf q_3^2+\\delta)}\n\\,,\\nonumber\\\\&&\n\\label{Vtensorcorrelator}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\chi_L(n;\\mathbf p,\\mathbf q)&=& (n-2)(L+n-2)\\,P_{L}(n,\\bar\\mathbf p.\\bar\\mathbf q) \\nonumber\\\\[3mm]\n&&-\\,n\\,\\frac{(n-2)(L+n-2)}{2L+n-2}\n\\frac{|\\mathbf p||\\mathbf q|}{(\\mathbf p-\\mathbf q)^2}\n\\Big(P_{L-1}(n,\\bar\\mathbf p.\\bar\\mathbf q)-P_{L+1}(n,\\bar\\mathbf p.\\bar\\mathbf q)\\Big) \\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\n\\tilde{c}_{J,L}=\\left\\{ \\begin{array}{cl} \\frac{2L(L+1)}{2L+n-2}\\,,  & \\;\\;\\; J=L+1 \\\\\n         -2L\\,,  & \\;\\;\\; J=L \\\\\n\t \\frac{2}{2L+n-4}\\,, & \\;\\;\\; J=L-1\n\t \\end{array} \\right.\n\\end{eqnarray}\nThe divergent part of Eqs.~(\\ref{Vspinorbitcorrelator2})\nand~(\\ref{Vtensorcorrelator}) can be extracted from the integrals listed in\nAppendix D. The insertion of a $\\mathbf S^2$ potential is identical to that of the\n${\\cal V}_2\/m^2$ potential, Eq.~(\\ref{V2correlatorexample}), and the result\ncan be read off from  Eq.~(\\ref{Zscalarpot}). The result for the\nrenormalization constant of the currents from the diagrams in\nFig.~\\ref{figspincorrelator} including combinatorial factors thus reads  \n($S\\equiv 1$)\n\\begin{eqnarray}\n\\delta z^{\\rm NLL}_{c,spin} \n& = & \n-\\frac{{\\cal V}_c^{(s)}(\\nu)\\,{\\cal V}_{\\Lambda}^{(s)}(\\nu)}{256\\pi^2}\n\\,\\frac{J(J+1)-L(L+1)-2}{(2L+1)L(L+1)}\\,(1-\\delta_{L0})\\nonumber\\\\\n&&\n-\\frac{3{\\cal V}_{c}^{(s)}(\\nu)\\,{\\cal V}_t^{(s)}(\\nu)}{64\\pi^2}\\,\n \\frac{f_{J,L}}{2L+1}\\,(1-\\delta_{L0})\n-\\, \\frac{{\\cal V}_c^{(s)}(\\nu){\\cal\n    V}_s^{(s)}(\\nu)}{64\\pi^2}\\,\\delta_{L0}\\,S(S+1)\\,,\n\\label{Zspin0}\n\\end{eqnarray}\n\\begin{eqnarray}\nf_{J,L}=\\left\\{ \\begin{array}{cl} -\\frac{1}{J(2J+1)}\\,,  & \\;\\;\\; J=L+1 \\\\\n         \\frac{1}{J(J+1)}\\,, & \\;\\;\\; J=L \\\\\n\t -\\frac{1}{(J+1)(2J+1)}\\,, & \\;\\;\\; J=L-1\n\t \\end{array} \\right.\n\\label{Zspin}\t \n\\end{eqnarray}\nExplicit expressions for the Wilson coefficients of the spin-dependent\npotentials can be found in Ref.~\\cite{amis}. \nFor ${}^{2S+1}L_J={}^3S_1$ the result agrees with the known results \nfrom Refs.~\\cite{LMR,Hoang3loop,amis3,Pineda2}, and for \n${}^{2S+1}L_J={}^3P_{1,0}$ the renormalization constant at the\nmatching scale $\\nu=1$ is consistent with the IR-divergences in the hard\nregion obtained from the threshold expansion in\nRef.~\\cite{2loop_hard}. \n\nThe tensor potential can also change the orbital angular momentum $L$ by two\nunits while conserving the total angular momentum $J$.\\,\\footnote{Note, however,\n  that the spin of the fermion-antifermion system (S=0,1) cannot be changed by\n  the $S=2$ tensor $S^{ij}$. Changes of the angular momentum $L$ by one unit\n  are not allowed due to parity.}\nThus transitions between spin triplet currents with orbital angular momenta\n$L=J\\mp 1$ and $L'=J\\pm 1 =L\\pm 2$  \ndriven by diagrams such as Fig.~\\ref{figmixcorrelator} with a tensor potential\ninsertion  do not vanish and could induce a mixing between the currents\nthrough renormalization. The expression for the three-loop correlator  \nof Fig.~\\ref{figmixcorrelator}, where the tensor potential is inserted next to\nthe annihilation current on the RHS, reads now\n\\begin{figure}[t] \n\\begin{center}\n \\leavevmode\n \\epsfxsize=6cm\n \\leavevmode\n \\epsffile{figs\/3loopmix.eps}\n \\vskip  0.0cm\n \\caption{\nThree-loop current correlator diagram between the currents with\norbital angular momentum $L$ and $L'=L\\pm 2$ and a tensor potential insertion.\n \\label{figmixcorrelator} }\n\\end{center}\n\\end{figure}\n\\begin{eqnarray}\n\\lefteqn{\n-i m\\,N_c\\, {\\cal V}_c^{(s)}\\,{\\cal V}_{t}^{(s)}(-2n)\n\\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\, \n}\n\\nonumber\\\\ && \\times\n\\frac{\\mbox{Tr}\\Big[\n\\left(\n \\Gamma_{J,L'}^{S=1}(\\mathbf q_3,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\n\\right)^\\dagger\\,S^{ij}\\,T^{ij}(\\mathbf q_3-\\mathbf q_2)\\,\n\\left(\n \\Gamma_{J,L}^{S=1}(\\mathbf q_1,\\mathbf \\mbox{\\boldmath $\\sigma$})^{i_1\\dots i_{J}}(i\\sigma_2)\n\\right)\n\\Big]}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\\nonumber\\\\[3mm]\n& = &\n-i m\\,N_c\\,{\\cal V}_c^{(s)}\\,{\\cal\n  V}_{t}^{(s)}(2n\\,J\\,N_{J+1})\\,\\mbox{Tr}\\,[\\mathbf 1]\n\\nonumber\\\\ & & \\times\n \\int D^n \\mathbf q_1\\, D^n \\mathbf q_2\\, D^n \\mathbf q_3 \\, \n \\frac{|\\mathbf q_1|^L|\\mathbf q_3|^{L'} P_L(n,\\bar\\mathbf q_1.\\bar\\mathbf q_2)\\,\n\\tilde{\\chi}_{J,L}(n;\\mathbf q_2,\\mathbf q_3)}\n{(\\mathbf q_1^2+\\delta)\\,(\\mathbf q_1-\\mathbf q_2)^2\\,(\\mathbf q_2^2+\\delta)\\,\n(\\mathbf q_2-\\mathbf q_3)^2\\,(\\mathbf q_3^2+\\delta)}\n\\label{Vtensorcorrelatormix}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\n\\tilde\\chi_{J,L}(n;\\mathbf p,\\mathbf q) &=& \n \\left\\{ \\begin{array}{cl}\n  \\left( \\mathbf p^2 \\,P_{J+1}(n,\\bar\\mathbf p.\\bar\\mathbf q)-2 |\\mathbf p||\\mathbf q| P_{J}(n,\\bar\\mathbf p.\\bar\\mathbf q)\n        + \\mathbf q^2 \\,P_{J-1}(n,\\bar\\mathbf p.\\bar\\mathbf q) \\right)\\,,  & \\;\\;\\; L=J-1 \\nonumber\\\\[3mm]\n  \\left( \\mathbf p^2 \\,P_{J-1}(n,\\bar\\mathbf p.\\bar\\mathbf q)-2 |\\mathbf p||\\mathbf q| P_{J}(n,\\bar\\mathbf p.\\bar\\mathbf q)\n        + \\mathbf q^2 \\,P_{J+1}(n,\\bar\\mathbf p.\\bar\\mathbf q) \\right)\\,, & \\;\\;\\; L=J+1 \n\t \\end{array} \\right. \\,.\\nonumber\\\\\n\\end{eqnarray}\nThe divergent contributions of Eq.~(\\ref{Vtensorcorrelatormix}) can be\nobtained from the results for the relevant integrals given in Appendix D. \nWe find that the UV divergences from the sum of both insertions of the tensor\npotential cancel and that there is no mixing between the $j_{J=L-1}^{S=1}$ and\nthe $j_{J=L+1}^{S=1}$ currents at NLL order. \n\n\n\\section{Solutions and Discussions} \n\\label{sectiondiscussion}\n\n\\subsection*{Scalar Production Currents}\n\nFor scalar fields, only the spin-independent contributions shown in\nEq.~(\\ref{Zscalarpot}) have to be taken into account and we can write the RGE\nfor the Wilson coefficient of the current with quantum number $L$ in terms  \nof the RGE's of the S- and P-wave current Wilson coefficients which\nwere determined in Ref.~\\cite{HoangRuizSvNRQCD}, \n\\begin{eqnarray}\n\\frac{\\partial}{\\partial\\ln \\nu}\\ln[c_L(\\nu)] & = & \n  4\\delta z^{\\rm NLL}_c \\nonumber\\\\ \n&=& \\delta_{L0}\\,\\frac{\\partial}{\\partial\\ln \\nu}\\ln[c_S(\\nu)]+\n\\frac{3}{2L+1}\\left(1-\\delta_{L0}\\right)\\frac{\\partial}{\\partial\\ln\n  \\nu}\\ln[c_P(\\nu)]\\,. \n\\label{scalarRGE1}\n\\end{eqnarray}\nThe solution is thus found\nimmediately and reads \n\\begin{eqnarray}\n\\ln \\frac{c_L(\\nu)}{c_L(1)}&=& B_2\\,\\pi \\alpha_s(m)\\, (1-z) + B_3\\,\\pi \\alpha_s(m)\\, \\ln (z) \n \\nonumber\\\\[3pt] &&   + \\,B_4\\,\\pi \\alpha_s(m)\\,\\Big[1-z^{1-13C_A\/(6\\beta_0)} \\Big] \n  + B_0\\,\\pi \\alpha_s(m)\\Big[z-1-w^{-1}\\ln (w)\\Big]\n\\,,\n\\label{runningscalar}\n\\end{eqnarray}\nwhere $z=\\alpha_s(m\\nu)\/\\alpha_s(m)$,\n$w=\\alpha_s(m\\nu^2)\/\\alpha_s(m\\nu)$, and\n\\begin{eqnarray}\nB_i=\\delta_{L0}\\,b_i+\\frac{3}{2L+1}\\left(1-\\delta_{L0}\\right)\\,d_i\\,,\n\\end{eqnarray}\nand with~\\cite{HoangRuizSvNRQCD}\n\\begin{eqnarray}\nb_2 & = & \\frac{C_F \\,[\\, C_A C_F(9C_A-100C_F)-\\beta_0(26C_A^2+19C_F\n    C_A -32C_F^2)\\,]\\,}{26\\beta_0^2 C_A} \n\\,,\\nonumber\\\\[3mm]\nb_3 & = & \n-\\frac{C_F^2\\,[\\,C_A(9C_A - 100 C_F) - 6 \\beta_0(3C_A - 4C_F)\\,]}{2 \\beta_0^2(6\\beta_0 - 13 C_A)}\n\\,,\\nonumber\\\\[3mm] \nb_4 & = & \\frac{24C_F^2(11C_A-3\\beta_0)(5C_A+8C_F)}{13C_A(6\\beta_0-13C_A)^2} \n\\,,\\nonumber\\\\[3mm] \nb_0 & = & - \\frac{8C_F(C_A+C_F)(C_A+2C_F)}{3\\beta_0^2}\n\\,,\\nonumber\\\\[3mm]\nd_2 & = & -\\frac{ C_F (C_A+2C_F)}{3\\beta_0}\n\\,,\\nonumber\\\\[3mm]\nd_0 & = & -\\frac{8 C_A C_F(C_A+4C_F)}{9\\beta_0^2}\\,.\n\\end{eqnarray}\n\nIn Fig.~{\\ref{singletplot}} we have displayed the NLL running of the \nscalar currents exemplarily for $L=0,1$ and $4$ normalized to their\nmatching values. For the input parameters we have chosen  \ntwo different values for the heavy scalar mass, $m=220$~GeV (solid lines) \n$m=500$~GeV (dashed lines), and $\\alpha_s(m_Z)=0.118$, taking \nleading-logarithmic running for $\\alpha_s$ with $n_f=5$ active massless \nquark flavors and no active massless squarks ($n_s=0$). As is already \nvisible from the form of Eq.~(\\ref{scalarRGE1}), we find that the \n$\\nu$-variation of the currents becomes weaker for larger angular \nmomenta. While the S-wave coefficient increases by about 6\\% for \n$\\nu\\sim\\alpha_s$ compared to the matching value at $\\nu=1$, the\nP-wave coefficient only increases by less than 3\\%. For $L\\le 4$\nthe maximal relative variation is already below 1\\%.\nIt is also conspicuous that for any $L$ the maximum slightly\ndecreases with the heavy scalar mass and also moves towards smaller\nvalues of $\\nu$. As already discussed in Ref.~\\cite{HoangRuizSvNRQCD}, the\nlatter feature can be understood qualitatively from the fact that the average\nvelocity $\\langle  v\\rangle\\sim\\alpha_s(m\\alpha_s)$ decreases with the heavy\nscalar mass $m$.  \n     \n\\begin{figure}\n\\epsfxsize=10.cm \n\\epsffile[80 450 560 720]{figs\/cscalar.eps}\n{\\caption{NLL running of the normalized Wilson coefficient \n    for colored scalars for\n    $L=0,1$ and $4$ for $m=220$~GeV (solid lines) and\n    for $m=500$~GeV (dashed lines).}\n\\label{singletplot}}\n\\end{figure}\n\n\n\n\\subsection*{Fermion Production Currents}\n\nThe RGE of Eq.~(\\ref{rgeCurrent}) for the Wilson coefficients of the currents which\nproduce a pair of quarks with quantum numbers $J,\\,L$ and $S$ is obtained at\nNLL by adding the anomalous dimensions from the diagrams with spin independent\nand spin dependent potentials (Eqs.~(\\ref{Zscalarpot}) and (\\ref{Zspin0})\nrespectively):   \n\\begin{eqnarray}\n\\frac{\\partial}{\\partial\\ln\\nu}\\ln[c_{J,L}^S(\\nu)]& = & \n4\\delta z^{\\rm NLL}_c+ 4\\delta z^{\\rm NLL}_{c,spin}\\nonumber\\\\ \n& = &\n\\delta_{L0}\\,\\frac{\\partial}{\\partial\\ln\\nu}\\ln[c_{J,0}^S(\\nu)]\\nonumber\\\\\n&&+\\,\\frac{(1-\\delta_{L0})}{2L+1}\\bigg\\{\n-\\frac{1}{16\\pi^2}\\,{\\cal V}_c^{(s)}(\\nu)\\,\n\\Big( \\frac{{\\cal V}_c^{(s)}(\\nu)}{4} + {\\cal V}_r^{(s)}(\\nu) \\Big)\n\\nonumber\\\\[2mm] & & \\qquad +\\,\n\\alpha_s(m\\nu)^2 \\Big(3{\\cal V}_{k1}^{(s)}(\\nu)+2{\\cal V}_{k2}^{(s)}(\\nu) \\Big)\n+\\,\n\\frac{\\alpha_s^2(m\\nu)}{2}\\Big(\\frac{C_F^2}{2}-C_AC_F\\Big)\n\\nonumber\\\\[2mm] & & \\qquad  \n -\\,\\frac{{\\cal V}_c^{(s)}(\\nu)\\,{\\cal V}_{\\Lambda}^{(s)}(\\nu)}{64\\pi^2}\n\\,\\frac{J(J+1)-L(L+1)-2}{L(L+1)}\\,\\delta_{S1}\n\\nonumber\\\\[2mm] & & \\qquad \n-\\frac{3{\\cal V}_{c}^{(s)}(\\nu)\\,{\\cal V}_t^{(s)}(\\nu)}{16\\pi^2}\\,\n f_{J,L}\\,\\delta_{S1}\n\\,\\bigg\\}\n\\,.\n\\label{rgeCJL}\n\\end{eqnarray}\nWe have singled out the known RGE for the S-wave\ncurrents~\\cite{LMR,HoangStewartultra,Pineda2}, \n$c_{J,0}^S$, that is relevant for $\\gamma^* \\to q\\bar{q}$ production in the\nspin triplet case, and for  $\\gamma\\gamma \\to q\\bar{q}$ in the spin singlet\ncase. Solving Eq.~(\\ref{rgeCJL}) yields \n($z=\\alpha_s(m\\nu)\/\\alpha_s(m), w=\\alpha_s(m\\nu^2)\/\\alpha_s(m\\nu)$)\n\\begin{eqnarray}\n\\ln \\frac{c_{J,L}^S(\\nu)}{c_{J,L}^S(1)}&=& A_2\\,\\pi \\alpha_s(m)\\, (1-z) + \nA_3\\,\\pi \\alpha_s(m)\\, \\ln (z) \n \\nonumber\\\\[2pt] &&   + \\,A_4\\,\\pi \\alpha_s(m)\\,\\Big[1-z^{1-13C_A\/(6\\beta_0)} \\Big] \n  + \\,A_5\\,\\pi \\alpha_s(m)\\,\\Big[1-z^{1-2C_A\/\\beta_0} \\Big] \n \\nonumber\\\\[3pt] &&   + \\,A_8\\,\\pi \\alpha_s(m)\\,\\Big[1-z^{1-C_A\/\\beta_0} \\Big] \n  + A_0\\,\\pi \\alpha_s(m)\\Big[z-1-w^{-1}\\ln (w)\\Big]\n\\,,\n\\label{runningfermion}\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nA_2 &=& a_2\\,\\delta_{L0} - \\frac{(1-\\delta_{L0})}{2L+1}\\,\n\\bigg\\{\\frac{C_F(C_A+2C_F)}{\\beta_0}\n+\\frac{C_F^2}{4\\beta_0}\\frac{J(J+1)-L(L+1)-2}{L(L+1)}\\,\\delta_{S1}\\,\\bigg\\}\n\\,,\\nonumber\\\\[3mm]\nA_3 &=& a_3\\,\\delta_{L0}\\,,\\nonumber\\\\[3.5mm]\nA_4 &=& a_4\\,\\delta_{L0}\\,,\\nonumber\\\\[3mm]\nA_5 &=& a_5\\,\\delta_{L0} +\n\\frac{(1-\\delta_{L0})}{2L+1}\\frac{f_{J,L}\\,C_F^2}{2(\\beta_0-2C_A)}\\,\\delta_{S1}\n\\,,\\nonumber\\\\ \nA_8 &=&\n\\frac{(1-\\delta_{L0})}{2L+1}\\,\\frac{C_F^2}{\\beta_0-C_A}\\,\n\\frac{J(J+1)-L(L+1)-2}{L(L+1)}\\,\\delta_{S1}\n\\,,\\nonumber\\\\  \nA_0 &=& a_0\\,\\delta_{L0} - \\frac{(1-\\delta_{L0})}{2L+1}\\,\n\\frac{8C_A C_F(C_A+4C_F)}{3\\beta_0^2} \\,.\n\\end{eqnarray}\nThe coefficients $a_i$ for the running of the S-wave Wilson coefficients\n$c_{J,0}^{S=0,1}$ read~\\cite{HoangStewartultra,Pineda2} \n\\begin{eqnarray} \n  a_2 &=& b_2\\,,\\quad \n  a_4 =  b_4\\,,\\quad \n  a_0  =  b_0\n  \\,, \\nonumber\\\\[3mm]\n  a_3 &=&\n    \\frac{C_F^2 }{ \\beta_0^2\\, (6\\beta_0-13C_A) (\\beta_0-2C_A)}\n    \\, \\Big\\{ C_A^2 ( 9 C_A- 100 C_F )\n    + \\beta_0\\,C_A \\Big[ 74 C_F + C_A ( 13 {\\bf S^2} - 42 ) \\Big]\\nonumber \\\\[3mm]\n   &&\\qquad - 6 \\beta_0^2  \\Big[ 2 C_F + C_A ( {\\bf S^2} - 3 ) \\Big]\\,\\Big\\} \n   \\,,\\nonumber\\\\[3mm]\n a_5 &=& \\frac{C_F^2 \\big[ C_A( 15\\! -\\! 14\\,{\\bf S^2} ) \n    \\!+\\! \\beta_0 (4{\\bf S^2}\\!-\\!3) \\big] }{ 6 (\\beta_0\\!-\\!2 C_A)^2 }\\,,\n\\label{acoeffs2}  \n\\end{eqnarray}\nwhere ${\\bf S}^2\\equiv 2$ in the triplet case and \n${\\bf S}^2\\equiv 0$ for the singlet case.\n\nIn Fig.~{\\ref{tripletplot}}a and b we have displayed the NLL running of the \nquark currents for spin singlets and triplets, respectively, for a number of\ncases. For the heavy quark mass we have chosen $m=175$~GeV, and for the strong\ncoupling we employed the convention used in the discussion of the scalar\ncurrents. For the singlet cases we have shown the evolution for\n$L=0,1,2,4$. The dependence of the evolution of the currents on the value of\n$L$ is very similar to the scalar currents.  \nFor the triplet cases we have displayed the evolution for the cases $L=0,1,3$ \nand all possible values of $J$. Here the dependence of evolution on the choice\nof the angular momentum $L$ (for a fixed $J$ value) is similar to the spin\nsinglet cases. But we also find that the evolution for a fixed value of $L$ is\nstronger for the smaller $J$ values.\n     \n\\begin{figure}\n\\epsfxsize=8.1cm \n\\epsffile[80 450 560 720]{figs\/cquarksinglet.eps}\n\\epsfxsize=8.1cm \n\\epsffile[80 454 560 720]{figs\/cquarktriplet.eps}\n{\\caption{Left panel: NLL running of the normalized Wilson coefficient \n    for quarks in a spin singlet configuration for\n    $L=0,1,2$ and $4$ for $m=175$~GeV. Right panel: NLL running for quarks in\n    spin triplet configuration for $L=0,1,3$ and all possible $J$ values for\n    $m=175$~GeV.}\n\\label{tripletplot}}\n\\end{figure}\n\n\n\\section{Comments on Scheme Dependent Results} \n\\label{sectioncomments}\n\nIn this section we comment on scheme dependences of a number of\nresults for nonrelativistic currents that can be found in the\nliterature. The current that has been studied most extensively is the\nvector current $\\psi^\\dagger_{\\mathbf p}\\mathbf \\mbox{\\boldmath $\\sigma$}(i\\sigma_2)\\chi_{-\\mathbf p}^*$, which\ngoverns heavy quark pair production in $e^+e^-$\nannihilation~\\cite{FadinKhoze,Jezabek,Hoangsynopsis} or the large\nHiggs energy endpoint \nregion in $e^+e^-\\to t\\bar t H$~\\cite{Farrell1,Farrell2}. At NNLL\norder the matching condition and the anomalous dimension depend on\nthe scheme used for the spin-dependent interactions and\npotentials. Using dimensional regularization such computations were\ncarried out in\nRefs.~\\cite{Czarnecki1,Beneke4,hmst1,Hoang3loop,HoangStewartultra}.   \n\nIn Refs.~\\cite{hmst1,Hoang3loop,HoangStewartultra}, the NNLL order\nmatching condition of the vector current \n$\\psi^\\dagger_{\\mathbf p}\\mathbf \\mbox{\\boldmath $\\sigma$}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ was determined using\nthree-dimensional $\\sigma$ structures for the spin-dependent\npotentials, as shown in Eq.~(\\ref{spinpotdef1}), and three-dimensional \n$\\sigma$ algebra. Although the use of the three-dimensional $\\sigma$\nalgebra is a priori inconsistent, the results for the NNLL order\nmatching conditions derived in\nRefs.~\\cite{hmst1,Hoang3loop,HoangStewartultra} nevertheless represent \na viable scheme. This scheme is related to the spin-dependent potentials \nin Eqs.~(\\ref{ndimpot}) or (\\ref{tensordef1}), with strict $n$-dimensional\n$\\sigma$ algebra, through multiplicative $n$-dependent factors. For the\nvector current $\\psi^\\dagger_{\\mathbf p}\\mathbf \\mbox{\\boldmath $\\sigma$}(i\\sigma_2)\\chi_{-\\mathbf p}^*$ this\nscheme has, for the tree level potentials, the form \n\\begin{eqnarray}\n V^{\\prime\\prime}({\\mathbf p},{\\mathbf p^\\prime}) & \\simeq & \ng_s^2 (T^A \\otimes \\bar T^A) \\, \\bigg\\{\n \\frac{1}{\\mathbf k^2}\n + \\frac{{\\mathbf p^2 + \\mathbf p^{\\prime 2}}}{2 m^2 \\mathbf k^2} \n + \\left(1-\\frac{5}{12}\\,\\epsilon\\right)\n \\,\\left[\n    \\frac{n-3}{4m^2 } \n   +\\frac{\\mathbf S^2}{m^2 } \\left(\\frac{\\mathbf S^2}{n}-1 \\right)\n \\right]\n\\bigg\\}\n\\nonumber\\\\[2mm] &\\simeq & \ng_s^2 (T^A \\otimes \\bar T^A) \\, \\bigg\\{\n \\frac{1}{\\mathbf k^2}\n + \\frac{{\\mathbf p^2 + \\mathbf p^{\\prime 2}}}{2 m^2 \\mathbf k^2} \n + \\left(1+\\epsilon\\right)\n \\,\\left[\n   -\\frac{\\mathbf S^2}{3\\,m^2}\n \\right]\n \\bigg\\}\n\\,,\n\\label{3dimpot}\n\\end{eqnarray}\nand can be derived from the entries in Tab.~\\ref{tab2}.\nWe have dropped the spin-orbit and tensor potentials that were\ndisplayed in Eqs.~(\\ref{ndimpot}) or (\\ref{tensordef1}) because they\ndo not contribute for the vector current\n$\\psi^\\dagger_{\\mathbf p}\\mathbf \\mbox{\\boldmath $\\sigma$}(i\\sigma_2)\\chi_{-\\mathbf p}^*$. The first relation in\n(\\ref{3dimpot}) allows to compute the difference between the NNLL\nmatching condition obtained in Ref.~\\cite{Hoang3loop} and the scheme where the\nNNLL matching condition is identified with the contributions from the hard\nregion in the threshold expansion~\\cite{Czarnecki1,Beneke4}, which implies the\nuse of the scheme for the potentials shown in Eq.~(\\ref{ndimpot}) (see also\nRef.~\\cite{Pineda:2006ri}). These two schemes only differ in the\ntreatment of the spin-dependent potentials. \nUsing the contribution to the NLL anomalous\ndimension from the spin-dependent potential ${\\cal V}_s^{(s)}$ at the\nhard matching scale from Eq.~(\\ref{Zspin0}), \n$\\delta z_{c,{\\rm spin}}^{\\rm NLL}=\n-{\\cal V}_c^{(s)}(1){\\cal\n  V}_s^{(s)}(1)\/32\\pi^2=C_F^2\\alpha_s^2(m)\/6$~\\cite{amis,amis2},\none finds that the relation between the NNLL matching condition based\non the three-dimensional $\\sigma$ algebra from Ref.~\\cite{Hoang3loop},\n$c_{\\rm NNLL,v}^{\\rm 3\\,dim}(1)$, and the one based on the threshold\nexpansion scheme, $c_{\\rm NNLL,v}^{\\rm thr\\,exp}(1)$, reads\n\\begin{eqnarray}\nc_{\\rm NNLL,v}^{\\rm 3\\,dim}(1) & = &\n c_{\\rm NNLL,v}^{\\rm thr\\,exp}(1) - \\frac{5}{12}\\delta z_{c,{\\rm\n spin}}^{\\rm NLL}\n\\, = \\,\n c_{\\rm NNLL,v}^{\\rm thr\\,exp}(1) - \\frac{5}{72}\\,C_F^2\\alpha_s^2(m)\n\\,,\n\\label{crelations}\n\\end{eqnarray}\nwhich agrees with the results given in\nRefs.~\\cite{Czarnecki1,Beneke4,Hoang3loop}. \nIt is also straightforward to compute such a scheme relation\nfor the spin-dependent non-mixing contributions of the\nNNLL anomalous dimension of the vector current that was also computed \nin Ref.~\\cite{Hoang3loop}.\n\nScheme relations for the NNLL matching conditions analogous to\nEq.~(\\ref{crelations}), connecting the scheme from Ref.~\\cite{Hoang3loop} \nwith the one based on the threshold expansion can also be derived for the \n${}^3P_1$ axial-vector \n($J^i_{av}=\\psi^\\dagger_{\\mathbf p}\\frac{p^\\ell}{m}\\sigma^{\\ell i}(i\\sigma_2)\\chi_{-\\mathbf p}^*$),\n${}^3P_0$ scalar\n($J_{s}=\\psi^\\dagger_{\\mathbf p}\\frac{p^\\ell}{m}\\sigma^{\\ell}(i\\sigma_2)\\chi_{-\\mathbf p}^*$),\nand ${}^1S_0$  pseudo-scalar \n($J_{ps}=\\psi^\\dagger_{\\mathbf p}(i\\sigma_2)\\chi_{-\\mathbf p}^*$)\ncurrents:\n\\begin{eqnarray}\nc_{\\rm NNLL,av}^{\\rm 3\\,dim}(1) & = &\n c_{\\rm NNLL,av}^{\\rm thr\\,exp}(1) - \\frac{1}{16}\\,C_F^2\\alpha_s^2(m)\n\\,,\n\\\\[2mm]\nc_{\\rm NNLL,s}^{\\rm 3\\,dim}(1) & = &\n c_{\\rm NNLL,s}^{\\rm thr\\,exp}(1) - \\frac{1}{24}\\,C_F^2\\alpha_s^2(m)\n\\,,\n\\\\[2mm]\nc_{\\rm NNLL,ps}^{\\rm 3\\,dim}(1) & = &\n c_{\\rm NNLL,ps}^{\\rm thr\\,exp}(1)\n\\,.\n\\end{eqnarray}\nThe respective NNLL matching conditions in the scheme based on the threshold\nexpansion were computed in Ref.~\\cite{2loop_hard}.\n\nFinally, we would like to make a few comments on the conventions for\n$\\gamma_5$. In previous NRQCD literature based on dimensional\nregularization, only the totally anticommuting\nversion with the form $\\gamma_5=(^{0\\,\\mathbf I}_{\\mathbf I\\,0})$ has been\nconsidered~\\cite{hmst1,2loop_hard}, which leads to the nonrelativistic\ncurrents $\\psi^\\dagger_{\\mathbf p}(i\\sigma_2)\\chi_{\\mathbf p}^*$ and\n$\\psi^\\dagger_{\\mathbf p}\\frac{p^\\ell}{m}\\sigma^{\\ell i}(i\\sigma_2)\\chi_{\\mathbf p}^*$ \nfor the pseudo-scalar $\\bar\\psi\\gamma_5\\psi$ and axial-vector\n$\\bar\\psi\\gamma^i\\gamma_5\\psi$ ($i=1,\\dots,n$) full theory currents,\nrespectively.  In particular, in Ref.~\\cite{2loop_hard}\nthe NNLL (two loop) matching conditions for these pseudo-scalar and\naxial-vector currents were identified with the hard contributions\nobtained in the threshold expansion using the form\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3=\n-i\\gamma_0\\gamma_1\\gamma_2\\gamma_3$ for triangle graphs and the \ntotally anticommuting $\\gamma_5$ for the other diagrams. Such an\napproach is viable as long as the\nhard contributions from triangle graphs only lead to IR-finite terms. \nHowever, from the point of view of the full theory it\ncan be important or even mandatory to employ only the fully consistent\ndefinition $\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$. The latter \nform for $\\gamma_5$ \nalso leads to a different from of the EFT currents from the\nnonrelativistic expansion. Writing $\\gamma_5=\n\\frac{i}{3!}\\epsilon^{ijk}\\gamma^0\\gamma^{i}\\gamma^{j}\\gamma^{k}=\n\\frac{i}{3!}\\epsilon^{ijk}\\gamma^0\\gamma^{ijk}$, and\n$\\gamma^i\\gamma_5=\\frac{i}{2}\\epsilon^{ijk}\\gamma^0\\gamma^{jk}$,\nwhere $\\epsilon^{ijk}$ is the usual three-dimensional\n$\\epsilon$-tensor if all indices assume values $1,2$ or $3$, and\nidentically zero otherwise, one obtains for the leading order\nexpansion \n\\begin{eqnarray}\n\\bar\\psi\\gamma_5\\psi \n& \\to &\n -\\frac{i}{3!}\\,\\epsilon^{ijk}\\,\n  \\psi^\\dagger_{\\mathbf p}\\sigma^{ijk}(i\\sigma_2)\\chi_{-\\mathbf p}^*\n\\,,\n\\label{altcur1}\n\\\\[2mm]\n\\bar\\psi\\gamma^i\\gamma_5\\psi \n& \\to &\n -\\frac{i}{2}\\,\\epsilon^{ijk}\\,\n  \\psi^\\dagger_{\\mathbf p}\\left[\\frac{1}{m}(p^j\\sigma^k-p^k\\sigma^j)\\right]\n (i\\sigma_2)\\chi_{-\\mathbf p}^*\n\\,.\n\\label{altcur2}\n\\end{eqnarray} \nThe current in Eq.~(\\ref{altcur1}) was discussed in\nSec.~\\ref{sectiontriplet} and the one in Eq.~(\\ref{altcur2}) is a\nsimple example for the alternative ${}^3L_L$ currents involving the\ntensor $\\widetilde\\Gamma^{S=1}_L$ shown in Eq.~(\\ref{newLcurrent}).\nThe EFT currents on the RHS's of Eqs.~(\\ref{altcur1}) and\n(\\ref{altcur2}) represent a viable scheme and are equivalent to the\ncurrents above for $n=3$, but inequivalent otherwise. So at NNLL order their\nrespective matching conditions and anomalous dimensions differ. Using\n$\\gamma_5=i\\gamma^0\\gamma^1\\gamma^2\\gamma^3$ in the full theory, the 2-loop\nhard contributions from the threshold expansion give the NNLL matching\nconditions of the currents in Eqs.~(\\ref{altcur1}) and (\\ref{altcur2}).\n\n\n\n\\section{Conclusion}\n\\label{sectionconclusion}\nThe construction of non-relativistic interpolating currents for\ndescribing the production and annihilation of a color singlet heavy\nparticle-antiparticle pair with arbitrary quantum numbers\n${}^{2S+1}L_J$ in $n=3-2\\epsilon$\ndimensions has been addressed in this work. Arbitrary angular momentum\nconfigurations are accounted for by the generalization of the\nspherical harmonics in $n$ dimensions, for which we have provided\na representation in terms of cartesian coordinates. The use of the\n$n$-dimensional spherical harmonics is required to maintain SO(n)\nrotational invariance in calculations within dimensional\nregularization.\n\nSimilarly, a consistent description of the spin of a\nfermion-antifermion pair requires the treatment of Pauli\n$\\sigma$-matrices in $n$ dimensions. The antisymmetrized product of\n$\\sigma$-matrices are the proper irreducible tensors with respect to\nSO(n) to build interpolating currents describing an arbitrary\nangular-spin state of a fermion-antifermion pair. In $n$ dimensions\nthe form of irreducible currents in SO(n) which reduce to a specific\nangular-spin state in $n=3$ is not unique. This is related to the\nexistence of evanescent operators that can be built from the\nnonrelativistic field operators and the $\\sigma$-matrices in $n$\ndimensions. Such evanescent structures occur even in simple standard\nprocesses, and in general need to be taken into account consistently\nat subleading order for the renormalization process.\nThe specific choice of a basis for the physical currents defines a\nspecific renormalization scheme. Similar scheme choices are possible\nfor the  spin-dependent interactions and potentials. In this work we\nhave discussed different versions for spin-singlet and spin-triplet\ncurrents, and also for the spin-dependent potential, which are\ninequivalent in general $n$ dimensions, but equivalent for $n=3$.\n\nWe have discussed the latter issues in the framework of NRQCD and have\nshown that at NNLL order transitions between currents containing\ndifferent irreducible $\\sigma$ structures cannot occur. Since for a\nspecific angular-spin state the corresponding leading order currents\ndo not have an anomalous dimension at leading-logarithmic order, the\nscheme-dependence introduced by specific choices for\nthe currents and spin-dependent potentials in NRQCD only affects NNLL\nmatching conditions and anomalous dimensions. The relation between NNLL\nmatching conditions of a number of currents for schemes used in the\nliterature have been determined.\n\nFinally, we have determined the NLL order anomalous dimensions for\nthe leading interpolating currents for quarks-antiquark and colored\nscalar-antiscalar pairs for arbitrary spin and angular momentum\n$(J,L,S)$ quantum numbers. We have shown by explicit computation that \nthe spin-dependent potential do not cause any mixing between currents  \nwith different $L$ at NLL order.\nAn important property of the solution of the respective anomalous\ndimensions is that the scale variation of the currents becomes in\ngeneral weaker for higher angular momentum $L$.\n\n\n\n\\begin{acknowledgments} \nWe thank A.~Buras, D.~Maison, M.~Misiak and U.~Nierste for discussions.\nWe particularly thank P.~Breitenlohner for numerous\nhelpful discussions.\n\n\\end{acknowledgments}\n\n\\mbox{}\n\\vskip 1cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe frequency and properties of multiple star systems are important diagnostics \nfor placing constraints on star formation processes and calibrating stellar \nevolutionary models. This has prompted numerous attempts to characterize the \nproperties of nearby binary systems in the field. Multiplicity surveys of \nsolar-type stars (e.g. Abt \\& Levy 1976; Duquennoy \\& Mayor 1991) found \nrelatively high binary frequencies ($\\ga$60\\%) and a wide range of binary \nseparations ($\\la$$10^4$ AU) and mass ratios (1 to $\\la$0.1). This has led to \nthe common assumption that binary systems are the primary channel for star \nformation. However, multiplicity surveys of lower-mass M dwarfs (Fischer \\& \nMarcy 1992; Reid \\& Gizis 1997) observed marginally lower binary frequencies \n(35-43\\%) and surveys near and below the substellar boundary (Close et al. 2003; \nBouy et al. 2003; Burgasser et al. 2003;  Siegler et al. 2005) found \nsubstantially lower binary frequencies (10-20\\%) and separations (typically \n$\\la$20 AU) and a strong tendency toward mass ratios near unity.\n\nThese results demonstrate that field binary properties depend on mass. \nUnfortunately, binary properties for field stars are reported only for broad \nmass ranges, so they do not place strong constraints on the functional form of \nthis dependence. Various groups interpret the transition in binary properties as \neither a sharp break near the stellar\/substellar boundary (Kroupa et al. 2003;  \nClose et al. 2003) or a smooth mass dependence (Luhman et al. 2004d). Also, \nfield multiplicity surveys cannot constrain the mass dependence of substellar \nbinary properties due to the degeneracy between brown dwarf masses and ages. \nSubstellar companions in the field also tend to be old and intrinsically faint, \nso a limited range of binary mass ratios are accessible to observations. \nFinally, the field represents a composite population drawn from all \nstar-formation regions, so field surveys cannot probe the dependence of binary \nproperties on initial conditions (the stellar density, total mass, or mean Jeans \nmass of the formation region). One solution to these problems is to extend \nmultiplicity surveys to the nearest young uniform stellar populations: OB \nassociations, T associations, and open clusters.\n\nMultiplicity surveys have been conducted for many of the bright members of \nnearby open clusters and associations over the past decade using HST (Martin et \nal. 2000; Luhman et al. 2005), adaptive optics (Patience et al. 2002; Bouy et \nal. 2006b), and speckle interferometry (Kohler et al. 2000; White et al.  \n2006). These surveys have confirmed many trends observed in the field, such as \nthe high binary frequency and separations of solar-type stars (e.g. Ghez et al. \n1993; Kohler et al. 2000) and the low frequency and separation of the \nlowest-mass systems (Martin et al.  2003; Luhman et al. 2005; Kraus et al. 2005, \n2006; Bouy et al. 2006a). However, they have also found some potentially \ninteresting discrepancies. Surveys of different regions have revealed a mass \ndependence in binary frequency that is either smooth (Taurus-Auriga; White et \nal. 2006) or potentially discontinuous near the substellar boundary (Upper \nScorpius; Kohler et al. 2000; Kraus et al. 2005). Several systems with unusually \nwide separations or low mass ratios have also been found (e.g. Luhman et al. \n2004b; Bouy et al. 2006b).\n\nHigh-resolution imaging techniques are typically resource-intensive, so it is \nexpensive to undertake large programs which can sample a wide range of mass with \nsufficient statistical significance to characterize these effects. However, wide \nbinary systems in the nearest associations have angular separations large enough \nto resolve without these techniques. A program which exploits a uniform, \nhigh-quality seeing-limited survey could substantially enhance our understanding \nof the role of mass and environment in binary properties.\n\nIn this paper, we present the results from a search for new young binary systems\nin the Two-Micron All-Sky Survey (2MASS), an all-sky imaging survey conducted in\nthe near-infrared. In Section 2, we describe the selection of our survey sample, \nand we describe our data processing techniques in Section 3. We summarize the \nresults of our search in Section 4. Finally, in Section 5 we compare these \nresults to the standard paradigm of stellar multiplicity and discuss the \nimplications for the processes of multiple star formation.\n\n\\section{Sample Selection}\n\nIn Table 1, we describe the young associations from which we have drawn our\nsample: Taurus-Auriga, Chamaeleon I, and the two proposed subgroups of Upper\nScorpius.  The sample regions have been selected to include all large stellar\npopulations ($\\ga$100 known members) which are not heavily embedded, are\nlocated at distances of $\\la$200 pc, and have ages $\\la$30 Myr. These criteria\nneglect small associations and moving groups which can not contribute\nsignificant statistics (TW Hya, MBM 12, Chamaeleon II, $\\eta$ and $\\epsilon$\nCham, and the Lupus clouds), distant populations for which seeing-limited\nobservations on a small telescope cannot probe sufficiently small separations\n(IC 348 and the subgroups of Orion), embedded populations like the $\\rho$ Oph\ncomplex, and old populations in which the wide binary population may have been\nshaped by dynamical evolution (Praesepe, Pleiades, $\\alpha$ Persei).\n\nIn Table 2, we list the association members which we have adopted as our primary \nsample in the multiplicity search. The regional membership of our sample has been \nconfirmed via low-resolution spectroscopy to verify signatures of youth, so \ncontamination of the primary sample should be negligible. As we discuss in Section \n4.2, the surveys from which we draw our sample are likely to be incomplete due to \nselection biases; many of the new candidate companions found here would have been \nidentified in previous surveys if they were complete and unbiased. This could \npotentially cause us to overestimate the wide binary frequency. Wide binaries \nwould only be excluded from our sample if both components were absent from \nprevious membership surveys, so they are less likely to have been omitted from our \nsample than single members. However, this effect would have been more prevalent \namong faint low-mass systems (where incompleteness is higher). We are testing for \na decline in the binary frequency with mass, and any detection of this trend would \nbe robust against this bias.\n\nSaturation occurs for 2MASS sources brighter than $K\\sim$8, but the images can \nstill be used for sources as bright as $K\\sim$6; we have neglected only the \nhigh-mass association members which are brighter than this limit, corresponding \nto spectral types earlier than G0. These bright stars typically have been \nstudied with adaptive optics (e.g. Kouwenhoven et al. 2005 for the Sco-Cen \ncomplex), so analysis of 2MASS data would not contribute significant new \nresults. We also omit all sources which do not have confirmed spectral types \nsince we can not estimate their mass.  This criterion should eliminate most of \nthe sources which are embedded in massive envelopes and surrounded by resolved \nnebulosity.  Finally, four of our primary sample members are fainter than \nour detection limit for binary companions ($K=14.3$), but we retain them \nin our sample in case they are binary companions to higher-mass \nassociation members which have not yet been identified.\n\nIn the following subsections, we briefly describe each association and summarize \nthe construction of our search sample.\n\n\\begin{deluxetable}{lcclc}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Nearby Young Associations}\n\\tablehead{\\colhead{Name} & \\colhead{Distance} & \n\\colhead{Age} & \\colhead{Type} & \\colhead{Members} \\\\\n\\colhead{} & \\colhead{(pc)} & \\colhead{(Myr)}\n}\n\\startdata\nChamaeleon I&170&1-2&T Assoc.&147\\\\\nTaurus-Auriga&145&1-2&T Assoc.&235\\\\\nUpper Scorpius A&145&5&OB Assoc.&356\\\\\nUpper Scorpius B&145&5&OB Assoc.\\tablenotemark{a}&45\\\\\n\\enddata\n\\tablenotetext{a}{As we discuss in Appendix C, the nature of Upper Sco B is \nstill uncertain.}\n\\end{deluxetable}\n\n\\begin{deluxetable*}{llccrrrccrcl}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Confirmed Members of Nearby Young Associations}\n\\tablehead{\\colhead{Name} & \\colhead{Region} & \\colhead{RA} & \n\\colhead{DEC} & \\colhead{$K$} & \\colhead{$J-K$} & \\colhead{$H-K$} &\n\\colhead{SpT} & \\colhead{Mass} & \\colhead{$\\chi$$_3$\\tablenotemark{a}} &\n\\colhead{$\\mu_{\\alpha}$,$\\mu_{\\delta}$} & \\colhead{References}\n\\\\\n\\colhead{} & \\colhead{} & \\multicolumn{2}{c}{(Eq=2000)} & \\colhead{}\n& \\colhead{} & \\colhead{} & \\colhead{} & \\colhead{($M_{\\sun}$)} & \\colhead{} &\n\\colhead{(mas yr$^{-1}$)}\n}\n\\startdata\nScoPMS005&UScoA&15 54 59.86&-23 47 18.2&7.03&0.54&0.16&G2&1.66&26.57&-28,-38&Walter et al. (1994)\\\\\nScoPMS013&UScoA&15 56 29.42&-23 48 19.8&8.75&0.92&0.23&M1.5&0.54&2.00&16,-42&Walter et al. (1994)\\\\\nScoPMS014&UScoA&15 56 54.97&-23 29 47.8&10.29&0.93&0.30&M3&0.36&1.67&-8,-28&Walter et al. (1994)\\\\\n\\enddata\n\\tablenotetext{a}{The $\\chi$$_3$ statistic is a measure of how well each \nobject is fit by a single point source; see Section 3.2.}\n\\tablecomments{The full table of 783 sample members can be accessed from\nthe website http:\/\/www.astro.caltech.edu\/$\\sim$alk\/ (in PDF, TeX, or\nplain-text formats) and will be posted as online-only content in ApJ.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lccccccccccc} \n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt} \n\\tablecaption{Close Pairs of Confirmed Association Members\\label{tbl3_01}} \n\\tablehead{\\colhead{Name} & \\multicolumn{3}{c}{Primary} & \n\\multicolumn{3}{c}{Secondary} & \\colhead{Projected} & \n\\colhead{Position}\n\\\\\n\\colhead{} & \\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} & \n\\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} &\n\\colhead{Sep(\\arcsec)} & \\colhead{Angle(deg)} \n} \n\\startdata \nDHTau&1.59&0.65&8.18&0.93&0.21&8.39&15.23&126\\\\\nDITau\\\\\nFVTau&2.48&0.88&7.44&1.93&0.62&8.87&12.29&105.7\\\\\nFVTau\/c\\\\\nFZTau&2.55&1.05&7.35&1.93&0.62&8.05&17.17&250.5\\\\\nFYTau\\\\\nGGTau A&1.31&0.45&7.36&1.09&0.42&9.97&10.38&185.1\\\\\nGGTau B\\\\\nGHTau&1.32&0.44&7.79&1.19&0.40&6.96&21.77&15.2\\\\\nV807Tau\\\\\n\\enddata \n\\tablecomments{The full version of this table has been moved to the end of this\nmanuscript (Table 8) to enhance readability.}\n\\end{deluxetable*}\n\n\\subsection{Scorpius-Centaurus}\n\nThe Sco-Cen OB Association consists of three distinct subgroups: Upper Scorpius \n(USco; 5 Myr and 145 pc), Upper Centaurus-Lupus (UCL; 13 Myr and 160 pc), and \nLower Centaurus-Crux (LCC; 10 Myr and 118 pc) (de Geus et al. 1989; de Zeeuw et \nal. 1999). Sco-Cen has been recognized for nearly a century as a moving group of \nearly-type stars (e.g. Kapteyn 1914; Blaauw 1946;  Bertiau 1958; Jones 1971). \nHowever, surveys to identify low-mass stellar members have been undertaken only \nin the past 15 years and have concentrated almost exclusively on USco. Initial \nsurveys (Walter et al. 1994;  Kunkel 1999)  identified candidate members from \nsurveys for X-ray emission, while subsequent surveys (Preibisch et al. 1998; \nArdila et al. 2000; Preibisch et al.  2001; Preibisch et al. 2002; Martin et al. \n2004; Slesnick et al. 2006a) used wide-field optical\/NIR surveys to select \ncandidate members with colors and magnitudes consistent with the assumed age and \ndistance. Membership was confirmed with low- or intermediate-resolution \nspectroscopy to confirm indicators of youth such as lithium absorption, \nH$\\alpha$ emission, or low surface gravity.  Proper-motion member identification \nis typically not possible for faint Sco-Cen members since their proper motions \nare not sufficiently distinct from those of background stars; the only major \neffort has been by Mamajek et al.  (2003), who identified candidate G- and \nK-dwarf members of UCL and LCC based on proper motions, then confirmed their \nmembership with low-resolution spectroscopy.\n\nThe sample sizes for UCL and LCC are marginal ($\\sim$50 members each) and span\na limited range of masses, and the associations' low galactic latitude\n($|b|<20$) results in substantial contamination from reddened background\nstars, so we have chosen to only consider Upper Sco. We select our sample from\nthe surveys of Walter et al. (1994), Preibisch et al. (1998), Kunkel (1999),\nArdila et al. (2000), Preibisch et al. (2001), Preibisch et al. (2002), Martin\net al. (2004), and Slesnick et al. (2006a).\n\nBrandner et al. (1996) noted that some of the objects in these surveys form a\ndistinct subgroup in the southwest, near the border with UCL;  they named the\nmain population Upper Sco A and the subgroup Upper Sco B (hereafter USco-A and\nUSco-B).  A multiplicity survey by Kohler et al. (2000) subsequently found that\nthese two populations might have distinct binary statistics, with a much wider\nmean separation in USco-B. As we show in Appendix B, the members of USco-B also\nappear to have distinct kinematics. These results suggest that USco-B should be\ntreated as a distinct population. Based on the population kinematics and the\nprevious dividing lines adopted by Brandner et al. and Kohler et al., we assign\nall sample members west of 16h and south of -28 deg to USco-B, and all remaining\nmembers to USco-A. It is quite likely that there is some overlap along this\nborder, but the precision of the kinematic data does not allow us to\nunambiguously determine this or to establish the subgroup membership of \nindividual sources.\n\nWe also note that two USco members, ScoPMS008A and ScoPMS008B, are located \n$\\sim$15\\arcsec\\, from an early type USco member, HD 142424 (A8IV\/V; de Zeeuw et \nal. 1999). It is possible that these stars are companions to HD 142424 and not \nindependent primaries; since they fall within our identification range for binary \ncompanions in USco-A ($\\la$20\\arcsec), then we do not treat these sources as \nindependent primaries. Kohler et al. (2000) found (and we verify) that ScoPMS008A \nis itself a binary system with a separation of $\\sim$1.6\\arcsec, which suggests \nthat this could be at least a quadruple system.\n\n\\subsection{Taurus}\n\nThe Taurus-Auriga association (Taurus; 1-2 Myr; 145 pc; Bertout et al. 1999; \nWhite \\& Ghez 2001) has been recognized for more than 60 years as the nearest \nnorthern site of low-mass star formation and is the home of the archetypical \nstar T Tauri. The low-mass stellar population of Taurus-Auriga has been \nclassified gradually over the this time period (e.g. Joy et al. 1945;  Herbig et \nal. 1952; Cohen \\& Kuhi 1979); unlike Sco-Cen, Taurus is largely devoid of stars \nmore massive than 1-2 $M_\\sun$.\n\nA census of known Taurus members was presented in Kenyon \\& Hartmann (1995) and \nhas been supplemented by additional surveys to identify very-low-mass stellar \nand substellar members of Taurus-Auriga by Briceno et al. (1993, 1998, 2002), \nStrom \\& Strom (1994), Martin et al. (2001), Luhman et al. (2003a, 2004d, 2006), \nGuieu et al. (2005), and Slesnick et al. (2006b). Members of Taurus-Auriga have \nalso been confirmed in a followup survey of continuum (heavily veiled) sources \nby White \\& Basri (2003) and a survey for Hyades members by Reid \\& Hawley \n(1999). Finally, it was pointed out by White et al. (2006) that the source FV \nTau\/c2 (Hartigan et al. 1994) was omitted from the compilation of Kenyon \\& \nHartmann. We have constructed our Taurus source list from the Kenyon \\& Hartmann \ncensus, plus all subsequent surveys.\n\n\\subsection{Chamaeleon I}\n\nThe Chamaeleon I complex (ChamI; 1-2 Myr; $\\sim$160-170 pc; Whittet et al. 1997;  \nWichmann et al. 1998; Bertout et al. 1999) is another nearby site of ongoing\nstar formation. Like Taurus-Auriga, it is composed primarily of low-mass stars\nand molecular clouds and possesses few high-mass stars. Much of its stellar\npopulation was identified by optical and near-infrared surveys during the 1970s\nand 1980s (e.g Henize \\& Mendoza 1973; Schwartz 1977; Glass 1979; Baud et al.\n1984).\n\nCarpenter et al. (2002) and Luhman (2004b) have compiled censuses of known \nmembers and candidate members based on these and other surveys, and Luhman \n(2004b) confirmed the membership of many candidate members using optical and NIR \nspectroscopy. An objective prism survey of the entire cloud by Cameron et al. \n(2004) also confirmed 4 additional candidate members and identified 7 new \nmembers. Finally, one candidate substellar member from the survey of Oasa et al. \n(1999) was spectroscopically confirmed as a ChamI member by Luhman et al. \n(2004c). We have constructed our ChamI sample from the 151 confirmed ChamI \nmembers of Luhman (2004b), Luhman et al. (2004c), and Cameron et al. (2004) with \nspectral types later than G0.\n\n\\subsection{Spectroscopically Confirmed Stellar Pairs}\n\nSpectroscopic surveys of these stellar associations have identified many close \n($<$30\\arcsec)  pairs of members. Given the typical low surface density of \nassociation members on the sky, these stars could be gravitationally bound \nbinary companions. We list these candidate binary systems in Table 3. Many \nsystems have projected separations lower than our survey's outer identification \nlimits (Section 4.1); in these cases, we have removed the secondary star in each \npair from our statistical sample. Candidate secondaries at wider separations are \nconsidered to be independent systems for statistical purposes.\n\n\\section{Observations and Data Analysis}\n\n\\subsection{The Two-Micron All-Sky Survey}\n\nThe Two-Micron All-Sky Survey (2MASS; Skrutskie et al. 2006) observed 99.998\\% of\nthe sky in the $J$, $H$, and $K_s$ bands over an interval of 4 years. Each point\non the sky was imaged six times and the coadded total integration time was 7.8s,\nyielding 10$\\sigma$ detection limits of $K=14.3$, $H=15.1$, and $J=15.8$. The\nsaturation levels depend on the seeing and sky background for each image, but \nare typically $J<9$, $H<8.5$, and $K_{s}<8$. The pixel scale of the detector was \n2 arcsec pix$^{-1}$, but acquisition of multiple images allowed for subsampling to \nincrease the effective resolution; the final pixel scale for each processed \nimage is 1 arcsec pix$^{-1}$, which critically samples stellar point-spread functions \n(PSFs) given a typical resolution of 3$\\arcsec$ FWHM. The typical astrometric \naccuracy attained for the brightest unsaturated sources ($K\\sim8$) is $\\sim$100 \nmas, and the photometric zero-points are calibrated to $<$0.02 mag.\n\nThe 2MASS Point Source Catalog (PSC; Cutri et al. 2003) and the processed survey \nimages are available from the 2MASS \nwebsite\\footnote{http:\/\/www.ipac.caltech.edu\/2mass\/}.  We use PSC data to \nidentify all wide ($>$5\\arcsec) visual companions to our sample members. \nHowever, the PSC does not always distinguish multiple point sources in close \nproximity ($\\la$5\\arcsec), instead reporting only the brightest source.  This \nsuggests that wide neighbors to our sample members should be identified in the \nPSC, but most close neighbors are probably absent.\n\nWe address this incompleteness by working directly with the processed survey \nimages to identify close ($\\la$5\\arcsec) companions via PSF-fitting photometry.  \nFrom the 2MASS website, we extracted postage-stamp (60x60$\\arcsec$) and \nwide-field (510x1024$\\arcsec$) images for each of the association members \ndescribed in Section 2. The wide-field images were used to create reference PSFs \nfor each science target, while the postage-stamp images have been used to \nidentify close visual companions. The width of the wide-field images \n(510$\\arcsec$) corresponds to the width of each 2MASS survey tile; any image \nwith larger width would include data taken at different epochs, and therefore \nwith different seeing conditions. The height was chosen to allow for $\\ga$10 PSF \nreference stars brighter than $K\\sim$11 in all fields. The size of the overlap \nregion between adjacent tiles was 60\\arcsec\\, in right ascension and \n8.5\\arcmin\\, in declination, so each science target appeared to be $>$30\\arcsec \naway from the edge in at least one tile.\n\nThe 2MASS survey images were produced by coadding multiple exposures taken in \nsequence, each offset by $\\sim$85\\arcsec\\, in declination, so drawing PSF \nreference stars from several arcminutes away could lead to nonuniform images. \nOnly sources $\\la$40$\\arcsec$\\, north or south of a science target were observed \nin all six exposures that the science target was observed, and sources \n$\\ga$500$\\arcsec$\\, north or south do not share any simultaneous scans. However, \nall of the scans which contribute to a wide-field image were observed within \n$\\sim$30 seconds. We do not expect the seeing-based PSF to change on this short \ntimescale, and we have found that the PSF is usually constant over each entire \nwide-field image ($\\sigma$$_{FWHM}$$\\sim$0.1$\\arcsec$).\n\n\\subsection{Data Reduction and Source Identification}\n\nWe identified candidate companions and measured their fluxes from the \npostage-stamp image of each sample member using the \nIRAF\\footnote{IRAF is distributed by the National Optical \nAstronomy Observatories, which are operated by the Association of Universities \nfor Research in Astronomy, Inc., under cooperative agreement with the National \nScience Foundation.} package DAOPHOT (Stetson 1987), specifically with the \nPSF-fitting photometry routine ALLSTAR. The template PSFs for each \npostage-stamp image were created \nusing the PSTSELECT and PSF tasks. We selected template stars for each source \nfrom the corresponding wide-field image; each PSF was based on the eight \nbrightest, unsaturated stars which appeared to be isolated under visual \ninspection. The appropriate photometric zero-point was extracted from the image \nheaders. We compared PSF-fitting magnitudes for single stars to the \ncorresponding PSC values in order to test our results; there is no systematic \noffset, and the standard deviation of the random scatter in $m_{PSF}-m_{PSC}$ is \n$\\sim$0.03 magnitudes.\n\nAs we have discussed in previous publications (Kraus et al. 2005, 2006), one \nlimitation of ALLSTAR-based PSF photometry is that binaries with very close \n($\\la$$\\theta$$_{FWHM}$) separations are often not identified, even when their \ncombined PSF deviates significantly from that of a true point source. This \nlimitation can be overcome for known or suspected binaries by manually adding a \nsecond point source in approximately the correct location and letting ALLSTAR \nrecenter it to optimize the fit. However, this method requires objective \ncriteria for identifying suspected binaries;  subjective selection methods like \nvisual inspection would not allow us to rigorously choose and characterize a \nstatistically complete sample. We have found that ALLSTAR's $\\chi$$^2$ \nstatistic, which reports the goodness-of-fit between a source and the template \nPSF, is an excellent diagnostic for this purpose. Since there are images in \nthree bandpasses, we use a single diagnostic value, denoted $\\chi$$_3$, which is \nthe sum of the three $\\chi$$^2$ values obtained for each association member when \nfit with a single point source. We list the value of $\\chi$$_3$ for each \nassociation member in Table 2.\n\nIn Figure 1, we plot the values of $\\chi$$_3$ as a function of K-band\nmagnitude for a subset of sample members with no known companions between\n0.5$\\arcsec$ and 15$\\arcsec$ (according to the surveys of Leinert et al.\n1993; Ghez et al.  1993;  Simon et al. 1995; Duchene 1999; Kohler et al.\n2000; Kraus et al.  2005, 2006; White et al. 2006). The goodness of fit\ndegrades rapidly for saturated stars ($K\\la8$), so our technique does not\ndiscriminate betweeen single stars and candidate binaries in this regime.\nHowever, since there are few stars brighter than the saturation limit, we\ndecided not to reject them until we were certain we could not identify any\nbinary systems via other methods. The distribution of $\\chi$$_3$ values for\nunambiguously unsaturated stars ($K>8.5$) is not normally distributed, but\n95\\% of these stars produce fits with $\\chi$$_3$$<$2.5, so we have selected all \nsources with $\\chi$$_3$$\\ge$2.5 as candidate binary systems.\n\nThe mean value of $\\chi$$_3$ for single stars should be $\\sim$3 since it \nrepresents the sum of 3 variables which follow a $\\chi$$^2$ distribution. However, \nwe find that the mean value reported by ALLSTAR for unsaturated single sources is \n$\\sim$1.75. This disagreement is caused by an overestimate of the photometric \nerrors in each observation by ALLSTAR. The coadding and subsampling process used \nin the 2MASS image processing pipeline results in correlated noise between \nadjacent pixels of the final survey images, so the true uncertainties are lower \nthan those estimated solely by Poisson statistics (Skrutskie et al. 2006).\n\nWe identified the candidate binaries in our sample based on this empirically \nmotivated $\\chi$$_3$ selection criterion, and then we attempted to fit each with \na pair of point sources separated initially by the PSF FWHM (3$\\arcsec$) and \nwith position angle corresponding to the angle of maximum elongation of the \nsystem PSF. The ALLSTAR routine optimized the components' separation, position \nangle, and magnitudes to produce the optimal fit; as we further summarize in \nSection 3.3, known binaries were typically fit with consistent positions and \nflux ratios in all three bandpasses while contaminants (such as sources with an \nerroneous template PSF in one filter)  did not produce consistent fits in \nmultiple images. We adopt the criterion that any candidate binary with component \npositions within 1$\\arcsec$ (3$\\sigma$ for astrometry of very close, faint \ncompanions; Section 3.4) in all three filters is a bona-fide visual binary. We \nfound that saturated stars produced fits for erroneous companions at separations \nof 1.0-1.5\\arcsec, so we have rejected all candidate companions to saturated \ntargets ($K_{tot}<8$) with separations of $<$2\\arcsec. Known binaries with wider \nseparations produced consistent fits even in the saturated regime for systems \nfainter than $K\\sim6$, so we adopted this as a maximum brightness limit for our \nsample.\n\nFinally, we compared the location of each candidate companion with the online \ncatalog of 2MASS image artifacts. We found that a candidate companion to \nMHO-Tau-4 was coincident with a persistence artifact flag. Furthermore, a \nprevious high-resolution imaging survey with HST (Kraus et al. 2006) found no \noptical counterpart to a limit of $z'$$\\sim$24, so we removed this candidate \ncompanion from further consideration and treat MHO-Tau-4 as a single star.\n\n\\begin{figure*}\n\\plotone{f1.eps}\n\\caption{A plot of the goodness-of-fit as a function of K-band magnitude for 203 \nobjects with no wide companions (0.5-15$\\arcsec$). The sharp increase in \n$\\chi$$_3$ at $K\\sim$8 is due to the onset of image saturation; the stars in \nthis brightness range are typically late K or early M, so saturation begins \nsimultaneously in all three bands. The solid line at $\\chi$$_3$$=2.5$ denotes \nthe 95\\% confidence interval for nominally single stars; we have selected all \nsample members above this limit as candidate close binaries. We found that our \nfitting algorithm for identifying companions is effective for mildly saturated \nstars, so we include association members up to $K=6$.\n} \n\\end{figure*}\n\n\\subsection{Sensitivity Limits}\n\nWe determined companion detection limits as a function of distance from the \nprimary stars via a Monte Carlo simulation similar to that of Metchev et al. \n(2003).  We used the IRAF task DAOPHOT\/ADDSTAR to add artificial stars at a \nrange of radial separations and magnitudes to the images of FO Tau, MHO-Tau-5, \nKPNO-Tau-8, and KPNO-Tau-9. These four sources have been shown to be single to \nthe limits of high-resolution imaging (Ghez et al. 1993; Kraus et al. 2006) \nand span the full range of brightness in this sample. We then attempted to \nidentify the artificial companions via PSF-fitting photometry. Our photometric \nroutines attempt simultaneous source identification in all three filters in \norder to separate erroneous detections from genuine companions, so we created \nthe same synthetic source in all three filters using colors from the 2 Myr \nBaraffe isochrones (Baraffe et al. 1998).\n\nIn Figure 2, we show our survey's 50\\% detection limits as a function of \nseparation for identifying candidate companions using the same PSF-fitting \nalgorithm as our actual search program. The minimum separation at which we can \ndetect equal-flux companions is $\\sim$1\\arcsec\\, for bright, unsaturated sources \nand $\\sim$1.6\\arcsec\\, for sources just above our adopted $K$ band magnitude \nlimit ($K=14.3$). The 10\\% and 90\\% detection limits are typically $\\sim$0.5 \nmagnitudes below and above the 50\\% limit. The sensitivity of PSF-fitting \nphotometry falls at separations $\\ga$5\\arcsec\\, since objects become cleanly \nresolved and most companion flux falls outside the fitting radius for the \nprimary. However, the PSC is complete to at least $K=14.3$ at larger separations, \nso wider companions will be recovered by our search of the catalog.\n\nWe also show the separation and flux ratio for known binary systems which \nhave been detected in K-band surveys (Kohler et al.  2000; White et al. \n2006) and whether these systems were unambiguously recovered (via either \nPSF-fitting photometry or the PSC), identified as candidate systems based on \nthe $\\chi$$_3$ criterion, or not recovered. The limits between detected and \nnondetected systems are roughly consistent with our empirically determined \nmagnitude limits, but there are few known systems which fall near these \nlimits. There are only two known wide systems among the faintest members of \nour sample ($K>11$), so we can not significantly test the detection limits \nof our search method in this brightness range. However, we identified four \nadditional candidate companions to sources in this brightness range, plus \nnumerous likely background stars, so our survey appears to be sensitive to \ncompanions in this regime.\n\n\\begin{figure} \n\\plotone{f2.eps} \n\\caption{Detection frequencies as a function of separation for \nartificially-introduced companions to four known single objects spanning the \nsurvey sample's brightness range: FO Tau ($K=8.12$), MHO-Tau-5 ($K=10.06$), \nKPNO-Tau-8 ($K=11.99$) and KPNO-Tau-9 ($K=14.19$). The solid lines denote the \n50\\% detection limit for our PSF-fitting photometry. The symbols represent \nknown binary companions from high-resolution K-band multiplicity surveys in \nUpper Scorpius (Kohler et al. 2000) and Taurus (White et al. 2006 and \nreferences therein). Filled circles denote companions which we recovered, open \ncircles denote companions which passed our $\\chi$$^2$ criterion but did not \nproduce significant fits, and crosses denote companions which were not \nrecovered. The dotted line shows the minimum separation at which the PSC will \nidentify all companions bright enough to be considered in our search \n($K<14.3$).\n} \n\\end{figure}\n\n\\begin{figure} \n\\plotone{f3.eps} \n\\caption{\nThe uncertainty in the measured binary companion brightness as a function of \nseparation for simulated binary images spanning the range of primary and \nsecondary brightnesses. The flux ratios shown are $\\Delta$$K=$0, 1, 2, 4, and 6 \n(solid, dotted, short-dashed, long-dashed, dash-dotted lines, respectively). The \nphotometric uncertainties increase sharply at separations of $\\la$3\\arcsec, \nsuggesting that observed photometric colors will not be accurate in \nthis separation range.\n} \n\\end{figure}\n\n\\begin{figure} \n\\plotone{f4.eps} \n\\caption{\nAs in Figure 3, showing uncertainties in binary secondary positions as a \nfunction of separation. \n} \n\\end{figure}\n\n\\subsection{Uncertainties in Binary Properties}\n\nMany of our candidate binaries have separations of $\\la$$\\theta$$_{FWHM}$, \nso our measurements could be subject to significant uncertainties. We \ntested these uncertainties by using a Monte Carlo routine to produce \nsynthetic images for binaries spanning a range of primary brightnesses, \nflux ratios, and separations. Specifically, we used ADDSTAR to construct \nsimulated $JHK$ images, and then we measured the binary fluxes and \nseparations for each set of simulated images using ALLSTAR. For each \ncombination of parameters, we produced 100 sets of synthetic images with \nrandomly distributed position angles. The $J-K$ and $H-K$ colors for the \nsecondaries were drawn from the 2 Myr isochrone of Baraffe et al. (1998) \nin order to determine realistic values for $\\Delta$$K$, $\\Delta$$H$, and \n$\\Delta$$J$.\n\nIn Figure 3, we show the standard deviation in the measured brightness for our \nsimulated binary companions as a function of separation. These simulations \npredict that photometric uncertainties increase significantly at separations of \n$\\la$3\\arcsec, so measured colors may not be reliable at small separations. As \nwe describe in Section 3.5, these colors are necessary at large separations \n($>$5\\arcsec) to distinguish candidate companions from background stars. \nHowever, contamination from background sources should be low at small \nseparations ($\\la$3\\arcsec) due to their low surface density, so we can neglect \nthese selection criteria with only a minor increase in the number of erroneous \nbinary identifications.\n\nIn Figure 4, we show a similar plot of the RMS scatter in the measured position \nof the secondary. The typical standard deviations are $\\la$0.3\\arcsec\\, for all \nbut the faintest companions, so the uncertainties in our measured separations \nshould have similar precision. Given these positional uncertainties, the \ncorresponding uncertainties in position angles range from 1 to 10 degrees, \ndepending on the binary separation. The standard deviations in secondary \nposition for our simulated images are consistent with the scatter between the \nthree filters for each observed binary, so we adopt the results from these \nsimulations as our estimated uncertainties.\n\nWe also conducted Monte Carlo tests to determine the probability of mistakenly\nidentifying a true single star as a binary. We constructed a series of simulated\nimages (100 each for four objects spanning our sample's range of brightness),\nand then tried to fit each object with two point sources. We found that this\nnever produced consistent fits in 3 filters, though faint peaks due to noise\nwere occasionally identified in one of the 3 images. This suggest that the\nprobability of an erroneous binary identification due to statistical errors is\nlow ($<1\\%$). This agrees with our results for known single stars; as we note in\nSection 3.2, 5\\% of known single stars fall above our $\\chi$$_3$ criterion for\nidentifying candidate binaries. However, none of these yielded fits for multiple \npoint sources in all 3 filters.\n\n\\subsection{Field Star Contamination}\n\nThe identification of binary companions based solely on proximity is\ncomplicated by contamination from foreground dwarfs, background giants, and\nreddened early-type background dwarfs. We have not conducted followup\nspectroscopic or astrometric observations to confirm association membership,\nso we must limit the survey to a total area in which the contamination from\nbackground stars is small compared to the number of candidate binary\ncompanions. We estimate the surface density of contaminants for each\nassociation based on the total number of objects within an annulus of\n30-90\\arcsec\\, from all of the association members in our sample. Field\nsurveys (e.g. Duquennoy \\& Mayor 1991; Reid \\& Gizis 1997) have identified few\nbinaries with projected separations of $\\ga$500 AU ($\\ga$30\\arcsec at the\ndistance of our sample members), so this method will also address the\nprobability of chance alignment with other association members.\n\nOur estimate of the contamination could be influenced by variations in\nbackground source counts due to the large angular extent of these associations\nor by variations in galactic latitude or extinction. The result would be a\nsystematic overestimation of the association probability for candidate\ncompanions at points of high contamination and a corresponding underestimation\nat points of low contamination. However, any local deviation from the mean\ncontamination rate should not affect the binary statistics for the association\nas a whole since the ensemble background at 30-90\\arcsec\\, will match the\nensemble background at $<$30\\arcsec. Our subsequent cuts against color, mass\nratio, and separation will also help to homogenize the sample by preferentially\nremoving background stars.\n\nMost previous multiplicity surveys were based on observations in a single \noptical or near-infrared bandpass (e.g. Kohler et al. 2000); in the absence \nof color information, these surveys can only estimate physical association \nprobabilities for candidate companions based on the surface density of \nbackground stars of similar brightness. Since 2MASS includes images in 3 \nfilters, we can reject most background stars by requiring colors consistent \nwith regional membership (Section 4.1).  Specifically, we have plotted \n($K,J-K$) and ($K,H-K$) color-magnitude diagrams for each region and we \nrequire prospective binary companions to fall above a smoothed field main \nsequence (Bessell \\& Brett 1988; Leggett et al. 2001)  for the regional \ndistance in both CMDs. We have chosen to use $K$ as a proxy for luminosity \ninstead of $J$ in order to minimize the effect of extinction for background \nstars. This choice will cause disk-bearing association members to sit \npreferentially higher in our color-magnitude diagrams, but this moves them \nfurther from our selection cutoff, so our results should be robust.\n\nAs a test of these color criteria, we have plotted color-magnitude diagrams for \nthe members of our primary sample. We find that $\\sim$97\\% of the primaries have \ncolors consistent with our definition of association membership, so any \nincompleteness in the selection of binary companions should be negligible. Most \nunselected primaries fall just below our color cuts; the only sample members \nwhich fall well below the association sequences are GSC 06191-00552 and \nUSco-160803.6-181237.  Both of these objects are claimed to be \nspectroscopically-confirmed members of USco-A, but the spectra are not available \nin the literature. We have not detected any binary companions to these objects, \nso their erroneous inclusion in our sample would not significantly change our \nresults. However, it might be prudent to reconsider their membership status with \nadditional spectroscopic observations in the future.\n\n\\begin{deluxetable*}{lllllllllllll}\n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt}\n\\tablecaption{Association Star Counts\\label{tbl2}}\n\\tablehead{\\colhead{} & \n\\multicolumn{3}{c}{Chamaeleon I ($N=147$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{Taurus-Auriga ($N=235$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{USco-A ($N=356$\\tablenotemark{a})} & \n\\multicolumn{3}{c}{USco-B ($N=45$\\tablenotemark{a})}\n\\\\\n\\colhead{Sep} \n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n& \\colhead{Source} & \\colhead{Color} &\\colhead{Bkgd}\n\\\\\n\\colhead{(\\arcsec)} \n& \\colhead{Count\\tablenotemark{b}} & \\colhead{Valid\\tablenotemark{b}} \n&\\colhead{Stars\\tablenotemark{b}}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n& \\colhead{Count} & \\colhead{Valid} &\\colhead{Stars}\n}\n\\startdata\n0-3\\tablenotemark{c}&7&-&0.9&9&-&0.9&15&-&2.0&8&-&0.3\\\\\n3-5&5&5&1.0&6&5&0.7&8&4&0.3&1&0&0.1\\\\\n5-10&8&6&4.8&10&5&3.1&12&5&1.4&6&4&0.4\\\\\n10-15&19&12&8.0&22&11&5.2&32&8&2.4&3&0&0.6\\\\\n15-20&20&13&11.2&23&13&7.2&36&6&3.4&4&0&0.8\\\\\n20-25&34&18&14.4&21&12&9.3&44&6&4.3&5&1&1.1\\\\\n25-30&39&28&17.6&33&16&11.4&60&5&5.3&9&4&1.3\\\\\n30-90&766&461&-&733&298&-&1566&138&-&215&34&-\\\\\n\\enddata\n\\tablenotetext{a}{The total sample size for each region, as \nsummarized in Table 1.}\n\\tablenotetext{b}{The number of unassociated contaminants was estimated \nfrom the surface density of sources which meet our color selection \ncriteria in the 30-90\\arcsec\\, separation range; most of these sources \nshould be foreground stars, background stars, or unbound association \nmembers.}\n\\tablenotetext{c}{We cannot use color criteria at separations of \n$<3$$\\arcsec$\\, due to the poor photometric precision (Section 3.4), so \nthe surface density of unassociated contaminants is higher.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lccccccccccl} \n\\tabletypesize{\\tiny}\n\\tablewidth{0pt} \n\\tablecaption{Candidate Wide Binary Systems\\label{tbl3_02}} \n\\tablehead{\\colhead{Name} & \\multicolumn{3}{c}{Primary} & \n\\multicolumn{3}{c}{Secondary} & \\colhead{Projected} & \n\\colhead{Position} & \\colhead{$\\mu_{\\alpha}$,$\\mu_{\\delta}$\\tablenotemark{a}}\n& \\colhead{Ident} & \\colhead{References}\n\\\\\n\\colhead{} & \\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} & \n\\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} &\n\\colhead{Sep(\\arcsec)} & \\colhead{Angle(deg)} & \\colhead{(mas yr$^{-1}$)} & \n\\colhead{Method}\n} \n\\startdata \n2M11103-7722&2.00&0.68&10.03&2.21&0.77&13.85&9.30&108.8&-&PSC&-\\\\\nC7-1&1.78&0.62&10.55&1.67&0.43&13.32&5.73&214.9&0,0&PSC&-\\\\\nCHSM1715&2.05&0.85&10.90&1.42&0.43&13.94&9.07&30.3&-58,42&PSC&-\\\\\nCHXR26&2.02&0.46&9.92&2.68&1.07&9.98&1.41&215.2&-&PSF&Luhman (2004b)\\\\\nCHXR28&1.17&0.32&8.23&1.53&0.39&8.83&1.78&121.6&-&PSF&Brandner et al. (1996)\\\\\n\\enddata \n\\tablecomments{The full table of sample members can be found in Table 9 at the \nend of this manuscript.}\n\\tablenotetext{a}{An entry of 0,0 denotes a source which was detected by \nthe USNO-B survey, but did not show a significant proper motion. An entry of \"-\" \ndenotes a source which was not detected by the USNO-B survey.}\n\\tablenotetext{b}{ScoPMS052 B is also known as GSC06209-01312; Martin et al. (1998) \nidentified it as a WTTS.}\n\\end{deluxetable*}\n\n\\begin{deluxetable*}{lcccccccccl} \n\\tabletypesize{\\tiny}\n\\tablewidth{0pt} \n\\tablecaption{Ultrawide Visual Companions\\label{tbl3_03}} \n\\tablehead{\\colhead{Name} & \\multicolumn{3}{c}{Primary} & \n\\multicolumn{3}{c}{Secondary} & \\colhead{Projected} & \n\\colhead{Position} & \\colhead{$\\mu_{\\alpha}$,$\\mu_{\\delta}$} &\n\\colhead{References}\n\\\\\n\\colhead{} & \\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} & \n\\colhead{$J-K$} & \\colhead{$H-K$} & \\colhead{$K$} &\n\\colhead{Sep(\\arcsec)} & \\colhead{Angle(deg)} & \\colhead{(mas yr$^{-1}$)}\n} \n\\startdata\nC1-6&3.92&1.68&8.67&1.93&0.80&14.10&27.58&156.0&&OTS12(candidate; Oasa et al. 1999)\\\\\nC1-6&3.92&1.68&8.67&2.26&0.70&13.75&24.51&123.8&-&OTS14(candidate; Oasa et al. 1999)\\\\\nCam2-19&2.40&0.74&10.25&2.36&0.72&13.45&23.13&107.6&-&-\\\\\nCam2-42&2.44&0.73&9.16&2.09&0.50&13.51&27.64&261.7&-&-\\\\\nCam2-42&2.44&0.73&9.16&2.11&0.65&14.14&28.18&180.4&-&-\\\\\n\\enddata \n\\tablecomments{The full table of sample members can be found in Table 10 at the \nend of this manuscript.}\n\\tablenotetext{a}{The source [CCE98] 2-26 is an ultrawide neighbor of both ChaHa7 and CHXR76; its \nphysical association, if any, is uncertain.}\n\\tablenotetext{b}{Haro 6-5 B is a known member of Taurus (Mundt et al. 1984), but was not included as \npart of our statistical sample because its spectral type is uncertain.}\n\\tablenotetext{c}{USco-160428.4-190441 B is also known as GSC06208-00611; Preibisch et al. (1998) \nidentified it as a field star.}\n\\tablenotetext{d}{SCH16075850-20394890 B is also known as T64-2; The (1964) identified it as a strong \nH$\\alpha$ emitter.}\n\n\\end{deluxetable*}\n\n\\begin{figure*} \n\\plotone{f5.eps} \n\\caption{$K,J-K$ and $K,H-K$ color-magnitude diagrams for the four regions in our\nsurvey.  The top panels show the confirmed association members in our\nsurvey, the middle panels show all objects within 5-15\\arcsec\\, of known\nassociation members, and the bottom panels show all objects within the\nbackground annuli (30-90\\arcsec). The solid line shows the main sequence at\nthe association distance and the dashed line shows the isochrone for the\nadopted association age (Table 2). In the top panels, association members are \nshown with filled circles. In all other panels, sources which lie above a \nsmoothed main sequence in both CMDs are shown with open circles and other \nsources are shown with small dots.\n} \n\\end{figure*}\n\n\\section{Results}\n\n\\subsection{Candidate Binary Companions}\n\nWe identified a total of 451 well-resolved visual companions brighter than \n$K=14.3$ within 30\\arcsec\\, of our sample members in the 2MASS PSC (Section 3.1), \nas well as 48 close ($\\la$5\\arcsec) candidate companions based on our PSF-fitting \nphotometry of 2MASS image data (Section 3.2). We have chosen 30\\arcsec\\, \n($\\sim$5000 AU) as an absolute upper limit for for identifying candidate \ncompanions since it corresponds to the maximum separation seen for field binaries \nat the distances of these association members. We also found 3280 visual \ncompanions within 30-90\\arcsec\\, of our sample members. Since the ratio of \nsources at 0-30\\arcsec\\, and 30-90\\arcsec\\, is roughly equal to the ratio of \nareas (1\/8), we expect that most of the sources within 30\\arcsec\\, of our sample \nstars are foreground or background stars having colors inconsistent with \nassociation membership.\n\nIn Figure 5, we present ($K$,$J-K$) and ($K$,$H-K$) color-magnitude diagrams \nfor the four regions showing all confirmed association members in our \nsample and all companions in two separation ranges (5-15\\arcsec\\, and \n30-90\\arcsec) corresponding to likely companions and likely background stars. \nWe summarize the number of objects which pass or fail the color selection \ncriteria (Section 3.5) as a function of separation in Table 4. We also \nestimate the number of contaminants which are expected to pass both selection \ncriteria in each separation range, assuming that the source density at \n30-90\\arcsec\\, represents the contaminant source density.\n\nWe showed in Section 3.4 that the uncertainties in our PSF-fitting photometry \nbecome significant at small separations, so we cannot use color criteria to \nidentify candidate companions inside $\\sim$3\\arcsec. However, given the low \nsurface density of background sources and the faintness of most nonmembers, we \nexpect only a small level of contamination in this separation range. Each of the \n39 candidate companions at separations $<$3\\arcsec\\, has a sufficiently high \nprobability of physical association ($\\ga$80\\%) to merit inclusion in our sample \nwithout using color cuts.\n\nWe have defined the maximum separation at which we identify candidate binary \ncompanions by requiring that the number of sources which pass our color \nselection requirement in each separation bin be $\\ga$2 times the number of \nexpected background companions. The corresponding probability that any \nindividual source inside that separation limit is a background star will be \n$\\la$50\\%. Based on the expected contamination rates and visual companion \ncounts in Table 4, these separations are 10\\arcsec\\, for ChamI, 15\\arcsec\\, for \nTaurus, 20\\arcsec\\, for USco-A, and 30\\arcsec\\, for USco-B. The separation limit \nis lower for regions with higher extinction since a higher fraction of \nbackground stars are reddened into our selection range. We adopt these \nseparation limits as our criteria for identifying candidate binary companions. \nWe note that sources at higher separations still have a nonnegligible \nprobability of association, but the probability that any individual source is a \nbinary companion will be low.\n\nUsing the color and separation cuts described above, we have identified (of 451 \nsources identified in the PSC and 48 sources identified with PSF-fitting \nphotometry) a total of 18 candidate binary companions in ChamI, 32 in Taurus, 40 \nin USco-A, and 17 in USco-B. Of these candidates, 4, 7, 23, and 5, respectively, \nhave not been previously reported in the literature.  We summarize the binary \nproperties of these candidate systems in Table 5. Some of the very wide and very \nfaint companions are likely to be unassociated foreground or background stars, so \nwe will consider a restricted range of separations and mass ratios in our \nsubsequent statistical analysis. In Table 6, we list the other visual companions \nwith separations $<$30\\arcsec\\, (but wider than the association's companion \nidentification limit) which have colors consistent with association membership and \nseparations greater than the limits given above. Many of these sources are \nexpected to be background stars, but additional information (such as optical \nphotometry or kinematic data) could be used in the future to remove additional \ncontaminants and more securely identify any ultrawide binary companions.\n\n\\subsection{Previous Observations}\n\nMany of our candidate companions have been identified previously in the \nliterature, but as we note in Tables 5 and 6, several of our candidates have also \nbeen rejected as association members based on the absence of spectroscopic \nsignatures of youth. Some of the candidates we list have probably been considered \nand rejected in previous work, but most surveys do not publish their catalogue of \nconfirmed field stars, so we cannot assess this number. \n\nWe also find that five members of our sample (USco-160700.1-203309, \nSCH16151115-24201556, and USco80 in USco-A; 2MASSJ04080782+2807280 and \n2MASSJ04414489+2301513 in Taurus) have candidate companions which are \nsignificantly brighter, and thus are likely to be the system primary \n(making the known association member a binary secondary). This result is \nnot surprising for the three Upper Sco members. Upper Sco is thought to \ncontain several thousand low-mass members, and photometric surveys have \nidentified many more candidates than could be confirmed via spectroscopy, \nso there are many more association members awaiting discovery. The two \nTaurus members are located on the edges of the association and were \ndiscovered by the only survey which considered these areas (Luhman 2006). \nOur newly-identified candidate companions are both brighter than the upper \nbrightness limit for this survey ($H=10.75$), so there were no previous \nopportunities for them to have been discovered.\n\nFinally, we find that 5 candidate companions identified in previous surveys have \n2MASS colors inconsistent with association membership: UX Tau B, V819 Tau B, \nHBC355 (HBC354 B), RXJ1524.2-3030B B, and RXJ1559.8-2556 B. Since $\\sim$3\\% of \nthe spectroscopically confirmed association members in our primary star sample \ndid not meet both color cuts, we expect (adopting the same percentage for the \nsecondaries) that only $\\sim$1-2 bona fide binary companions would not be \nselected. However, close pairs of stars have larger photometric errors, which \nincreases the probability that some companions might fall outside our selection \ncuts. Of these five companions, three fall just below the color cuts (UX Tau B, \nHBC 355, and RXJ1524.2-3030B B) in our CMDs and the other two fall significantly \nbelow the color cuts, so we suggest that the first three are erroneous \nrejections, and therefore we keep these objects, while we consider the other two \nto be valid rejections.\\footnote{V819 Tau B has also been classified as a \nbackground star by Woitas et al. (2001) due to its position on a (J,J-K) CMD and \nby Koenig et al. (2001) due to an absence of x-ray emission. UX Tau B and HBC355 \nare spectroscopically confirmed cluster members, and no membership assessments \nare available for the other two sources.}\n\n\\subsection{Inferred Stellar Properties}\n\n\\begin{deluxetable*}{lccccccccl} \n\\tabletypesize{\\scriptsize}\n\\tablewidth{0pt} \n\\tablecaption{Inferred Binary Properties\\label{tbl3_04}} \n\\tablehead{\\colhead{Name} & \\multicolumn{2}{c}{Primary}\n& \\multicolumn{2}{c}{Secondary} & \\colhead{Projected} \n& \\colhead{Mass}\n\\\\\n\\colhead{}  & \\colhead{SpT} & \\colhead{Mass ($M_{\\sun}$)}\n& \\colhead{SpT\\tablenotemark{b} \\tablenotemark{b}} \n& \\colhead{Mass\\tablenotemark{a} \\tablenotemark{b} ($M_{\\sun}$)} & \n\\colhead{Separation(AU)\\tablenotemark{b}} & \n\\colhead{Ratio\\tablenotemark{b} ($q$)}\n} \n\\startdata \n2M11103&M4&0.27&(M8.5)&(0.02)&1535&0.08\\\\\n2M11103(\/ISO250)&M4&0.27&M4.75(M5.5)&0.20(0.15)&1569&0.56\\\\\nC7-1&M5&0.18&(M8)&(0.03)&945&0.18\\\\\nCHSM1715&M4.25&0.25&(M7)&(0.05)&1497&0.18\\\\\nCHXR26&M3.5&0.33&(M5)&(0.19)&233&0.57\\\\\n\\enddata \n\\tablenotetext{a}{Values in parentheses are estimated from the system flux \nratio $\\Delta$$J$ and the spectroscopically determined properties of the \nprimary.}\n\\tablenotetext{b}{Estimated statistical uncertainties are $\\sim$10\\% for \nmass ratios, $\\sim$20\\% for secondary masses, $\\sim$2-3 subclasses for \nspectral types, and $\\sim$10\\% for projected separations.}\n\\tablecomments{The full table of sample members can be found in Table 11 at the \nend of this manuscript.}\n\\end{deluxetable*}\n\nIn Table 2, we list the inferred spectral types and masses for all of the \nassociation and cluster members in our sample. Spectral types are taken from the \nprimary reference and were typically determined via low- or \nintermediate-resolution spectroscopy. We assume that the spectroscopically \ndetermined spectral type and mass for previously-unresolved binary systems \ncorresponds to the primary mass and spectral type. Equal-mass binary components \nshould have similar spectral types and the flux from inequal-mass systems should \nbe dominated by the primary; in either case, spectroscopic observations of the \nunresolved system should have been affected only marginally by the flux from the \nsecondary.\n\nWe estimated the masses of sample members by combining mass-temperature and \ntemperature-SpT relations from the literature. No single set of relations spans \nthe entire spectral type range of our sample, so we have chosen the M dwarf \ntemperature scale of Luhman et al. (2003b), the early-type ($\\le$M0) temperature \nscale of Schmidt-Kaler (1982), the high-mass stellar models of D'Antona \\& \nMazzitelli (1997; DM97), and the low-mass stellar models of Baraffe et al. (1998; \nNextGen). We apply the DM97 mass-temperature models for masses of $\\ga$1 $M_\\sun$ \nand the NextGen models for masses of $\\la$0.5 $M_\\sun$; in the 0.5-1.0 $M_\\sun$ \nregime, we have adopted an average sequence. For each association, we adopt the \nmodels corresponding to the mean age listed in Table 1; this will introduce some \nuncertainty given the unknown age spread for each association. Large systematic \nerrors may be present in these and all pre-main sequence models (e.g. Baraffe et \nal. 2002; Hillenbrand \\& White 2004; Close et al. 2005; Reiners et al. 2005), so \nthey are best used for relative comparison only.\n\nMuch of the uncertainty in theoretical mass-temperature relations can be assessed \nin terms of a zero-point shift in the mass; preliminary observational \ncalibrations by the above authors suggest that theoretical models overestimate \nmasses by 10-20\\% over most of our sample mass range. This suggests that \ntheoretical predictions of relative properties (e.g. mass ratios, \n$q=m_{s}\/m_{p}$) might be more accurate than absolute properties (e.g. individual \ncomponent masses) since the systematic mass overestimates will cancel. Relative \nquantities are also largely independent of age and extinction, which are expected \nto be similar for binary components. We have combined our adopted \nmass-luminosity-SpT relations with the near-infrared colors of Bessell \\& Brett \n(1988) and the K-band bolometric corrections of Leggett et al. (1998, 2000, 2002) \nand Masana et al. (2006) to predict values for $q$ as a function of primary \nbrightness $m$ and flux ratio $\\Delta$$m$ in all three 2MASS filters. Some of our \nsample members could possess K-band excesses due to hot inner disks, so we have \nadopted the $q$ values predicted by the J-band fluxes; this will not eliminate \nthe effect, but should minimize it. We have also combined our derived $q$ values \nwith the estimated primary masses to predict secondary masses, and we use our \nmass-SpT relations to predict the corresponding secondary spectral types.\n\nWe list the derived values for each binary system in Table 7. Some wide \nbinaries have independent SpT determinations for both components, so we \nreport derived quantities with parentheses and measured quantities \nwithout. The typical uncertainties in $q$ are $\\sim$10\\% and represent the \nuncertainties in the photometry and the assigned spectral types, though \nsome systematic effects (e.g.  unresolved multiplicity or different levels \nof extinction) could produce far larger values.  This can be seen in the \ndiscrepancies for some systems (e.g. GG Tau AB, MHO-2\/1) which are known \nto be hierarchical multiple systems.  We can not quantify the unknown \nuncertainties in the theoretical models, but they should be considered \nwhen interpreting these results. The typical uncertainty in physical \nseparation is $\\sim$10\\% and reflects the uncertainty in angular \nseparation and the unknown depth of each system in its association; we \nassume each association has a total depth equal to its extent on the sky \n($\\sim$40 pc for Taurus and Upper Sco, $\\sim$20 pc for ChamI). The \nuncertainty in the mean association distance ($\\sim$5 pc) introduces a \nsystematic uncertainty of $\\pm$3\\%, but this is generally negligible.\n\n\\subsection{Binary Statistics}\n\nMultiplicity surveys typically consider the frequency of binary systems for \nrestricted ranges of parameter space (observed separations and mass ratios) \ncorresponding to the survey completeness limits. For our analysis, we select a \nrange of projected separations (330-1650 AU, set by the inner and outer detection \nlimits of ChamI since those limits are most restrictive)  and flux ratios \n($\\Delta$$K<2$, corresponding to $q\\ga$0.25) that should be complete for all but \nthe lowest-mass brown dwarfs in our sample. The inner separation limit and mass \nratio limit are set by the resolution limit for low-mass sample members \n($K\\sim$12.3) in ChamI, while the outer separation limit is set by the background \ncontamination in ChamI, where our mass ratio cut allows us to choose a 90\\% pure \nsample for separations $<$10\\arcsec.\n\nIn Figure 6, we present plots of the wide binary frequency as a function of \nprimary mass for each region in our sample. The binary fractions plotted \ncorrespond to our designated completeness regime: mass ratios $q>0.25$ and \nprojected separations of 330-1650 AU. In the bottom panel, we show the field \nbinary frequency in the same range of mass ratios and projected separations for \nsolar-type stars (Duquennoy \\& Mayor 1991), early-mid M dwarfs (M0-M6; Reid \\& \nGizis 1997), and brown dwarfs (Bouy et al. 2003; Burgasser et al. 2003). We also \nshow the corresponding frequencies for early-type stars in USco-A and USco-B \n(Kouwenhoven et al. 2005). The bin sizes were chosen to evenly sample the mass \nrange of our survey (0.025-2.50 $M_{\\sun}$ for which the primary targets were \nbrighter than our brightness cutoff ($K=14.3$). For each region in our survey, \nwe also show the expected frequency for foreground and background sources which \npass our color selection criteria and have $\\Delta$$K<2$, assuming a background \nsource count function N(K) matching that shown in Figure 2; in all cases, the \nexpected contamination rate is negligible. USco-A, ChamI, and Taurus all show a \ndecline in the binary frequency with mass, consistent with the results shown for \nfield multiplicity surveys. USco-B does not show a decline, but the \nuncertainties are not small enough to strongly constrain the slope of any \nmass dependence.\n\nThis binary search may not be complete for objects in the lowest-mass bin where \nsome binary companions could have been fainter than the survey detection limits \n($K>14.3$), so the true upper limits may be marginally higher. However, it has \nbeen observationally determined that most very low mass binaries in the field \nhave mass ratios near unity ($q>0.7$) and much smaller separations ($\\la$20 AU), \nso we are unlikely to have missed any wider or lower-mass ratio companions \n(Close et al. 2003; Burgasser et al. 2003; Bouy et al.  2003).\n\nAnother interesting distribution to consider would be the mass ratio\ndistribution for wide binaries as a function of mass and environment.\nUnfortunately, extending our binary results along another axis of parameter\nspace exceeds the statistical limits of our sample, leaving most bins with\nonly 0-1 detections. The best solution for this is to combine all regions into\na single population. In Figure 7, we plot the mass ratio distribution in our\nsurvey separation range (330-1650 AU) for the three highest-mass bins. We also\nshow the best-fit distribution for solar-type stars in the field (Duquennoy \\&\nMayor 1991).\n\nThis result should be treated with caution since it represents an admixture of \nformation environments which likely does not match the composition of the field. \nAs we show in Figure 6, the binary frequency appears to be fundamentally \ndifferent in the dark cloud complexes (Taurus and Chamaeleon) than in USco-A. \nThis distinction suggests that binary formation processes can vary significantly \nbetween different environments, and therefore that analysis of other binary \nproperties should take the environment into account when possible.\n\n\\begin{figure*} \n\\epsscale{0.90} \n\\plotone{f6.eps} \n\\caption{\nThe wide (330-1650 AU) binary frequency as a function of mass for each region and \nas determined from field multiplicity surveys. The higher-mass histogram bins are \nequally sized in $log M$, but the three lowest-mass bins have been combined to \nillustrate the absence of any companions. The error bars are calculated assuming \nbinomial statistics. The highest-mass datapoints for USco-A and USco-B denote the \nresults of Kouwenhoven et al. (2005). The dashed lines show the expected \nfrequency for each bin solely from foreground and background sources and \nunbound association members; they are not distinguishable from zero in most \nbins. Most upper limits for the lowest-mass bins are also very close to zero.\n} \n\\end{figure*}\n\n\\begin{figure} \n\\epsscale{1.00} \n\\plotone{f7.eps} \n\\caption{\nThe mass ratio distribution for wide binaries in the three highest-mass \nbins of our survey, calculated as a frequency among all sample members. \nThe mass ratio distribution function found by Duquennoy \\& Mayor (1991) \nfor field solar-type stars is denoted with a dashed line. These results \nrepresent the sum over all associations in our sample; the binary \nfrequency varies between environments (Figure 6) and our sample represents \na different admixture of formation environments than the field sample, so \nthe sample and field frequencies should be compared with caution.\n} \n\\end{figure}\n\n\\section{Discussion}\n\n\\subsection{The Role of Mass and Environment in Multiplicity}\n\nField multiplicity surveys have established several apparent trends for the mass \ndependence of binary properties. Solar-mass binaries occur at high frequency \n($\\ga$60\\%) and possess high mean and maximum separations (30 AU and 10$^4$ AU) \nand a mass ratio distribution biased toward low-mass companions ($q<0.5$) (e.g. \nDuquennoy \\& Mayor 1991). By contrast, binaries near and below the substellar \nboundary occur at low frequency ($\\sim$10-20\\%) and possess small mean and \nmaximum separations (4 AU and 20 AU), and a mass ratio distribution biased toward \nunity ($q$$\\ga$0.7) (Close et al. 2003; Bouy et al. 2003; Burgasser et al. 2003). \nObservations of intermediate-mass M dwarfs (e.g. Fischer \\& Marcy 1992; Reid \\& \nGizis 1997) suggest that their binary properties are transitional, with an \nintermediate binary frequency and possibly an intermediate separation range, plus \na mass ratio distribution that is nearly flat for $q>0.1$.\n\nThese results have been supported by recent surveys of young open clusters and\nassociations (e.g. Kohler et al. 2000; Patience et al. 2002; Luhman et al. 2005;\nKraus et al. 2005, 2006; White et al. 2006). High-mass stars in these regions\ntypically have higher binary frequencies and wider binary separations than\nlower-mass stars. There is emerging evidence that high-density regions might have\nlower binary frequencies or preferentially smaller binary separations (e.g.\nKohler et al. 2006), but it has not yet been conclusively determined whether this\nis a primordial feature or the result of early dynamical evolution.\n\n\\subsubsection{The Frequency of Wide Binary Formation}\n\nOur results appear to be broadly consistent with the established paradigm of \nmass-dependent multiplicity. Wide (330-1650 AU) binaries are very common among \nstars of $\\ga$1 $M_{\\sun}$ and the frequency appears to decline smoothly with \nmass (Figure 6). We found few wide binaries with primaries less massive than \n$\\sim$0.25 $M_{\\sun}$. Wide binary systems also appear to be common in the \nlow-density T associations (Taurus and Cham I), but comparatively rare in the \nUSco-A OB association. This suggests that the trend against wide binaries in \ndense bound clusters might extend to unbound associations, and therefore may be \nthe result of another initial condition besides stellar density.\n\nThe high frequency of wide binary systems in USco-B also suggests that binary\nformation is not a pure function of stellar density. This population is\nkinematically associated with the Sco-Cen complex and its proper motions most\nclosely match the Upper Centaurus-Lupus OB association, but the wide binary\nfrequency for solar-type stars in USco-B is more consistent with the T\nAssociations in our sample. As we discuss further in Appendix C, this\ncould also be explained if the stars in USco-B represent an evolved low-density\nassociation analogous to the $\\rho$ Oph or Lupus complexes rather than a \nsubgroup of an OB association.\n\n\\subsubsection{The Separation Distribution of Binary Systems}\n\nThe wide binary systems discovered by our survey only represent the outer tail\nof the separation distribution function. The measurement of its functional\nform will require large high-resolution imaging surveys sensitive to the core\nof the separation distribution ($\\sim$10-100 AU for solar-type stars,\ndeclining to $\\sim$1-10 AU for brown dwarfs). The uncertainties in results\nfrom the literature do not allow for strong constraints in this separation\nrange, but our results are consistent with some of the proposed environmental\ntrends. Wide binary systems appear to be significantly less common in USco-A\nthan in USco-B, a fact which was noted by Kohler et al. (2000). Their\nhigh-resolution speckle interferometry survey found many binaries in USco-A\nwith projected separations of 20-300 AU, but most of the binaries they\ndiscovered in USco-B had significantly higher separations. This led them to\nconclude that the binary separation distribution is biased toward tighter \nsystems in USco-A than in USco-B. Numerous multiplicity surveys in Taurus and \nCham I (e.g. Ghez et al. 1993, 1997) have also found a wider separation \ndistribution than in USco-A, which is also consistent with our results.\n\nField multiplicity surveys have shown a probable mass dependence in the\nmaximum binary separation. A census of previous surveys (Reid et al. 2001)\nfound that the maximum field binary separation can be described empirically\nwith an exponential function of the total system mass,\n$a_{max}$$\\propto$$10^{3.3M}$; an extension of this study to the substellar\nregime by Burgasser et al. (2003) found a corresponding quadratic function,\n$a_{max}$$\\propto$$M^2$. Burgasser et al.  demonstrated, using the formalism\nof Weinberg et al. (1987), that this is not due to interactions with field\nstars or giant molecular clouds, but instead must be a feature of the\nformation process or a result of early dynamical evolution in the formation\nenvironment.\n\nThese empirical relations predict maximum separations of 330 AU and 1650 AU for \ntotal system masses of $\\sim$0.4 and 0.6 $M_{\\sun}$, respectively. This \nprediction is consistent with the general minimum primary mass of $\\sim$0.25 \n$M_{\\sun}$ that we have identified among the wide binaries in our sample. The \nimplication is that this limit is indeed set by the T Tauri stage, either as a \nresult of the formation process or during dynamical evolution while these systems \nare still embedded in their natal gas cloud.\n\nHowever, we have identified one candidate system, USco-160611.9-193532, with an \napparent low-mass primary (0.13 $M_{\\sun}$; SpT M5) and a very wide projected \nseparation (10.8\\arcsec; 1550 AU). The USNO-B proper motion for the secondary \n($\\mu_{\\alpha}$,$\\mu_{\\delta}$=-8,-18 mas yr$^{-1}$) suggests that it is a \ngenuine USco member and not a background star. As we will report in a future \npublication, subsequent observations with Laser Guide Star Adaptive Optics on the \nKeck-II telescope also find that the primary is itself a close ($\\sim$0.1\\arcsec)  \nequal-flux pair. If the wide visual companion is gravitationally bound, then this \ntriple system ($M_{tot}\\sim$0.4 $M_{\\sun}$)  does not follow the empirical \nmass-maximum separation relations. There are several other candidate wide binary \nsystems which could potentially violate these relations, but the probability of \nbackground star contamination is high enough in these cases that association \nmembership should be confirmed via spectroscopy before any conclusions are drawn.\n\nFinally, a census of several star-forming regions by Simon (1997) found that\npre-main sequence stars tend to cluster on two length scales, with two-point\ncorrelation functions described by separate power laws. He concluded that\nsmall-scale clustering is a result of binary formation, while clustering on\nlarger scales is a result of the condensation of multiple cores from single\nmolecular clouds. This could potentially explain the excess of wide companions\nin Taurus, where the stars are younger and have not dispersed as far from their\nformation point.  However, Simon found that the transition occurred at\nseparations of $\\sim$$10^4$ AU in Taurus, and our survey truncates at $\\sim$1500\nAU. This suggests that unless his initial estimate was significantly higher than\nthe true transition point, all of our candidate companions fall within the\nbinary regime. \n\n\\subsubsection{The Mass Ratio Distribution of Binary Systems}\n\nField multiplicity surveys have found that the mass ratio distribution\nvaries significantly with primary star mass. Most solar-mass primaries\npossess binary companions with low mass ratios (Duquennoy \\& Mayor 1991),\nearly M dwarf primaries possess companions with a uniform mass ratio\ndistribution (Fischer \\& Marcy 1992), and late-M dwarf and brown dwarf\nprimaries possess companions with mass ratios near unity (e.g. Close et al.\n2003; Siegler et al. 2005).\n\nOur results can not support any strong statistical claims, but they are largely\nconsistent with this pattern. The only exception is that our results for the\nhighest-mass stars (1.16-2.50 $M_{\\sun}$) suggest the presence of a possible\nexcess of wide similar-mass binaries. The excess is not consistent with\nbackground star contamination since the primaries are all very bright and most\nbackground stars should be significantly fainter; many of these binary\ncompanions have been confirmed independently as association members. It is also\nunlikely that we missed a significant number of companions with $0.25<q<0.50$;\nthe detection limits (in $\\Delta$$K$ or $q$) are best for bright stars and\nsignificantly exceed the limits of our statistically complete sample. This mass \nratio distribution is also discrepant with that found by Kouwenhoven et\nal.  (2005) for B-A stars in Sco-Cen; they found a distribution which is very \nsimilar to that of Duquennoy \\& Mayor, with a strong deficit of equal-mass binaries \ncompared to unequal-mass binaries. \n\nThis discrepancy could represent another environmental dependence in multiple\nstar formation. Most of our high-mass binaries are found in the dark-cloud\ncomplexes, while the Kouwenhoven sample was drawn from the three OB associations\nof Sco-Cen. The field sample of Duquennoy \\& Mayor is probably also dominated by\nstars from dissolved OB associations or open clusters since those are thought to\nbe the dominant star formation channel in our galaxy. Since the binary frequency\nand binary separations appear to be affected by environmental conditions, it is\nplausible that the mass ratio distribution could also be affected.\n\n\\subsection{Summary of Implications for Wide Binary Formation}\n\nRecent efforts to model binary formation have typically assumed that a cluster of \n5-10 protostellar embryos form from a single fragmenting cloud core (e.g. Kroupa \n1995; Sterzik \\& Durisen 1998; Kroupa \\& Bouvier 2003; Kroupa et al. 2003; \nDelgado-Donate et al. 2003; Hubber \\& Whitworth 2005); these embryos would then \nundergo accretion and dynamical evolution to form single stars and stable \nmultiple systems. This scenario would provide a convenient explanation for \nvariations in binary properties between unbound associations or bound clusters \nsince the stellar encounter rate would vary with stellar density.\n\nHowever, the frequency of multiple stellar systems has been interpreted by \nGoodwin \\& Kroupa (2005) to mean that the collapse and fragmentation of a cloud \ncore produces only 2 or 3 stars, not 5-10. Larger systems would eject more single \nstars and tight binaries than are observed. Most models also predict that \ndynamical evolution would alter other stellar properties (spatial and velocity \ndispersion, disk lifetime, and accretion frequency), particularly at low masses.  \nThe preponderance of observational evidence shows that these properties are not \nconsistent with strong dynamical evolution and do not vary significantly with \nmass (White \\& Basri 2003; Luhman 2004d; White \\& Hillenbrand 2004; Mohanty et \nal.  2005). However, some updated models of dynamical evolution suggest that not \nall of these features would show strong signatures (e.g. Bate \\& Bonnell \n2005).\n\nThe apparently minor role of dynamical processes suggests that binary properties\nmight be established during the fragmentation of a cloud core, rather than in\nsubsequent dynamical evolution. Sterzik et al. (2003) suggested that the initial\ncloud temperature could play a role in determining the frequency of wide binary\nsystems. They noted that the radius of a cloud core at the end of isothermal\ncollapse is inversely related to the initial cloud temperature. This suggests\nthat regions of low temperature will have larger spatial scales during\nfragmentation, and therefore a wider distribution of binary separations. This\ncould be due either to a higher primordial cloud temperature or due to heating\nfrom other young stars.\n\nWe have found that the wide binary frequency in USco-A is comparable to that of\nopen clusters like Praesepe or the Pleiades (Patience et al. 2002) or young\nclusters like IC-348 and the ONC (Luhman et al. 2005; Kohler et al. 2006). The\nstellar density is much lower in OB associations like USco-A, so the similarity\nbetween their binary populations suggests that another initial condition might\nplay a key role in determining the binary separation distribution (and thus the\nfrequency of wide binary systems). Kohler et al. also found that binary\nproperties do not differ significantly between the core and the periphery of the\nONC, which span a significant range of stellar densities. These trends imply that\nwide binaries only form in the absence of high-mass stars, or equivalently that\nthe presence of high-mass stars suppress wide binary formation. The argument by\nSterzik et al. could provide a natural explanation for this trend; high-mass\nstars irradiate surrounding cloud cores, and the subsequent increase in\ntemperature would decrease the final length scale over which fragmentation would\noccur.\n\n\\section{Conclusions}\n\nWe present the results of a search for wide binary systems among 783 members of\nthree nearby young associations: Taurus-Auriga, Chamaeleon I, and the two\nsubgroups of Upper Scorpius.  This program analyzed near-infrared $JHK$ imagery\nfrom the Two-Micron All-Sky Survey to search for wide (1-30\\arcsec;  \n$\\sim$150-4500 AU) companions to known association members, using color-magnitude\ncuts to reject likely background stars. We identified a total of 131 candidate\nbinary companions in these associations, of which 39 have not been previously\nidentified in the literature. \n\nWe find that the wide binary frequency (330-1650 AU; $q>0.25$) is a function of\nboth mass and environment, with significantly higher frequencies among high-mass\nstars than lower-mass stars and in the T associations than in the OB\nassociation. We discuss the implications for wide binary formation and conclude\nthat the environmental dependence is not a direct result of stellar density or\ntotal association mass, but instead might depend on another parameter like the\ngas temperature of the formation environment.\n\nWe also analyze the mass ratio distribution as a function\nof mass and find that it largely agrees with the\ndistribution seen for field stars. There appears to be a moderate excess\nof similar-mass ($q>0.5$) wide binaries among the highest-mass (1.16-2.50\n$M_{\\sun}$)  stars in our sample, but the number statistics do not support\nany other strong conclusions. The binary populations in these associations\ngenerally follow the empirical mass-maximum separation relation observed\nfor field binaries, but we have found one candidate low-mass system\n(USco-160611.9-193532; $M_{tot}$$\\sim$0.4 $M_{\\sun}$) which has a projected\nseparation (10.8\\arcsec; 1550 AU) much larger than the limit for its mass.\n\nFinally, we find that the binary frequency in the USco-B subgroup is\nsignificantly higher than the USco-A subgroup and is consistent with the measured\nvalues in Taurus and Cham I. This discrepancy, the absence of high-mass stars in\nUSco-B, and its marginally distinct kinematics suggest that it might not be\ndirectly associated with either USco-A or Upper Centaurus-Lupus, but instead\nrepresent an older analogue of the $\\rho$ Oph or Lupus associations.\n\n\n\\acknowledgements\n\nThe authors thank R. White and C. Slesnick for helpful feedback on the manuscript \nand on various ideas presented within.\n\nThis work makes use of data products from the Two Micron All-Sky Survey, which is a\njoint project of the University of Massachusetts and the Infrared Processing and\nAnalysis Center\/California Institute of Technology, funded by the National\nAeronautics and Space Administration and the National Science Foundation. This\nresearch has also made extensive use of the SIMBAD database, operated at CDS,\nStrasbourg, France, and of the USNOFS Image and Catalogue Archive operated by \nthe United States naval Observatory, Flagstaff Station \n(http:\/\/www.nofs.navy.mil\/data\/fchpix\/).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{}\n\n\n\\section{Introduction}\nThe radiative properties of an atom are a function of the available \nelectromagnetic \nvacuum modes, hence they are altered by the presence of an optical resonator. \nCorrespondingly, spontaneous emission of an atom can be enhanced and suppressed by placing it \ninside a cavity depending on the cavity resonance - atom transition detuning \nas has been shown in \\cite{Kleppner1}. For the last 30 years cavity quantum \nelectrodynamics has been attracting attention as an ideal framework to explore\nthe foundations of quantum mechanics, but also with wide \nranging application in metrology (e.g \\cite{Kasevich2006}), quantum computing \n(e.g \\cite{Zoller1}), quantum cryptography and the development of novel cavity \nassisted laser cooling mechanisms \\cite{Domokos2003, Vuletic2001}.\\\\\nIn the far-off-resonance case, the interaction of an atom, or more generally a polarizable particle,\nwith the optical resonator can be understood classically as a position \ndependent refractive index changing the optical pathlength of the cavity \nand therefore its scattering and emission properties. The interaction strength \nbetween atom and cavity depends on the dipole moment $\\mu$ of the scatterer \ncompared to the overall mode volume $V_{mode}$ of the resonator, and the \ncoupling constant is \n$g=\\mu\\sqrt{\\omega_A\/\\left(2 \\hbar \\epsilon_0 V_{mode}\\right)}$, where $\\omega_A$ here is the frequency of the atomic transition.\\\\ \nQuantum mechanically the single-atom single-mode interaction is best described by the \nJaynes-Cummings Hamiltonian \\cite{Jaynes63}. In the new dressed basis the \nexcited state is split into two components due to the vacuum fluctuation of the \nelectro-magnetic field. The so called vacuum Rabi splitting is proportional to\nthe atom-cavity coupling constant and is observable when it is bigger than \nthe cavity linewidth $\\nu_C$ and the atomic transition linewidth $\\Gamma$. The cavity-atom interaction can also be characterized by the \nso-called cooperativity parameter, i.e. the ratio of the photon scattering\/emission rate into the cavity mode $\\Gamma_C$ with respect to the free space scattering\/emission rate $\\Gamma_{FS}$. For a single atom - single mode interaction, the \ncooperativity parameter is given by:\n\t\\[C=\\frac{\\Gamma_C}{\\Gamma_{FS}}=\\frac{g^2}{2 \\kappa \\gamma}~. \\]\nHere $\\kappa=\\nu_C\/2$ is half the cavity loss rate and $\\gamma=\\Gamma\/2$ is half the excited state decay rate.\nIf $g>>\\left(\\kappa+\\gamma\\right)$ and therefore $C>>1$ the coherent cavity-atom excitation exchange is much quicker than the dissipative \ndecay into free space or the decay of the cavity mode. Therefore an emitted photon stays \ninside the cavity, eventually gets reabsorbed by the atom, reemitted into the cavity and so on. \nFinally, the photon will leave the system either into free space or through \none of the mirrors. This regime is called the strong coupling regime.\nDepending on the specific application, strong coupling may be a requirement. For example in the context of cavity cooling it was pointed out in Ref. \\cite{Prospects1} that strong coupling is not essential for cavity-assisted laser cooling of two-level atoms, but it is important for cavity-cooling of multi-level atoms and molecules. There are now two ways to reach the strong coupling regime. One way is to reduce the mode-volume of the cavity and improve coating of the mirrors to produce a very large coupling constant $g$ \nand small cavity loss rate $\\kappa$ as done for example in Refs.\n\\cite{Rempe1, Kimble1, Brennecke2007}. \nA different approach, which is also the one followed in this work, involves increasing the number of particles inside the cavity mode and studying\nthe collective strong coupling regime. For N atoms in the cavity mode the \ncoupling scales as $g_{coll}=\\sqrt{N} g$ \\cite{TavisCummings1968}, \nthe vacuum Rabi splitting scales as \\cite{Transmission1}: \n\\[\n\\Omega=2\\sqrt{{{g_{coll}}^2-\\left(\\frac{\\gamma-\\kappa}{2}\\right)^2}}\n\\]\n\n\nand the collective \ncooperativity becomes\n\\[\nC_{Coll}=\\frac{N g^2}{2\\kappa \\gamma}\n\\]\nThe collective coupling of multiple atoms can increase the interaction \ndramatically. It has been observed in several experiments \n\\cite{Raizen1989,Kasevich2006,Klinner2006,Brennecke2007}. In most experiments \n(except e.g \\cite{Vuletic2003,BadCavity1,BadCavity2}) the coupling constant \n$g$ for the single atom is at least comparable to $\\kappa$. \nIn contrast, our aim is to investigate the coherent interaction between \nmultiple cold atoms and a cavity mode in a lossy optical resonator, where the\nsingle atom-cavity interaction is governed by the incoherent atomic decay rate\ninto free space. The large number of atoms only ensures that our system \nbecomes strongly coupled.\\\\\n\nIn this work  we describe our experimental apparatus and present experimental results for \nthe normal mode splitting of the cavity-atom system, a signature of  the strong coupling regime. \nCollective cooperativity parameters up to $C_{coll}>180$ are measured in our experiments.\\\\ \nA cooperativity parameter much larger than unity will allow us to study \ncollective phenomena in atom-cavity systems, from cavity-assisted cooling \nof large samples of atoms \\cite{Vuletic2003}, a model system for the \ncooling of molecules, to superradiance into the cavity mode and collective \nself-organisation of atoms.\n\n\\section{Experimental setup}\n\nIn the experiment we load a Magneto-Optical Trap (MOT) of Cesium atoms from the background gas directly into the mode of a nearly confocal optical resonator. The cavity and the MOT are depicted in figure \\ref{fig:CavityPump}.\\\\\n\\begin{figure}[h!]\n\t\\centering\n\t\\includegraphics[width=0.25\\textwidth]{Figure1_new2.eps}\n\t\\caption{\\label{fig:CavityPump} A schematic of the cavity arrangement and the MOT. The third pair of MOT beams is orthogonal to the image plane.}\n\\end{figure}\nThe FWHM linewidth of the cavities $TEM_{00}$ mode $\\nu_C$ was measured to be \n$2\\pi\\times\\left(1.60\\pm0.05\\right)$ MHz. For a Free Spectral Range (FSR)\nof $2\\pi\\times\\left(1249.5\\pm0.4\\right)$ MHz this leads to a finesse of $F=\\left(780\\pm15\\right)$.\\\\ With about 12 cm cavity length the single mode-single atom coupling constant $g$ is very small, smaller than the cavity half loss rate $\\kappa$ and much smaller \nthan the atomic excited state half decay rate $\\gamma$. The cavity-atom interaction \nat the single atom level is completely governed by the decay into free space and \nthe influence of the resonator can be treated as a perturbation. This regime is the \nso called \"bad cavity limit\". The relevant parameters for our system  are:\n$\\left(g,\\kappa, \\gamma\\right)=2\\pi\\times\\left(0.12,0.8,2.6\\right)$ MHz.\\\\\nTo increase the coupling as well as the overall mode volume the cavity in the \nexperiment was chosen to be confocal (the length of the resonator equals the \ncurvature radius of the mirrors). In this ideal case the higher order eigenmodes of \nthe cavity are degenerate in frequency with the TEM$_{00}$-order mode, so that the \ncoupling to each mode should contribute to the overall interaction.\\\\\n\nWe notice that the overlap of many transverse modes with slightly different frequencies leads to an empty-cavity linewidth larger than the $TEM_{00}$ value of $2\\pi\\times\\left(1.60\\pm0.05\\right)$ MHz. The degeneracy is lifted due to abberations on the mirrors and a small deviation from the confocality condition. To measure the TEM$_{00}$-order linewidth the contributions of the higher order transverse modes were spatially filtered out \nby a small aperture. For the observation of the vacuum Rabi splitting the filter was then removed. The transmission now contains higher order transversal mode components and appears broader.\\\\\n\\\\\nTwo important parameters need to be controlled separately to study the behaviour of the \ncloud in the mode with frequency $\\omega_C$ illuminated by a pump beam with frequency \n$\\omega_P$. First the cavity pump laser - atomic transition detuning $\\Delta_A=\\omega_P - \\omega_A$  needs to be adjustable and stable on long time scales to enable averaging of several experimental runs. This is done by a home built extended cavity diode laser (ECDL) offset locked to the repumper laser of the MOT. The offset lock \\cite{Weitz2004} \nenables us to gain atomic transition stability at an adjustable detuning  \n$\\Delta_A= 2\\pi\\times\\left(-3 \\rightarrow +2\\right)$ GHz from the Cs $D_2$ $F=4 \\rightarrow F'=5$ \n transition. The other important experimental parameter is the cavity pump laser - cavity resonance detuning $\\Delta_C=\\omega_P - \\omega_C$. Since there are cavity resonances every free spectral range the maximum detuning is $\\Delta_C=\\pm \\left(FSR\/2= 2\\pi \\times624.25\\right) MHz$. The cavity resonance stabilization scheme is described in the next section.\n\n\\subsection{Cavity stabilization}\n\nDuring each experiment it was crucial that the cavity kept a stable, \nconstant resonance frequency $\\omega_C$, both with respect to the atomic transition \nfrequency $\\omega_A$, and with respect to the probe laser $\\omega_P$. We established \nthat by actively stabilizing the cavity to an atomic rubidium resonance. In order to be able to change the cavity resonance frequency, the locking point needed to be frequency offset by an adjustable radio frequency between 500 and 1500 \nMHz provided by a radio-frequency generator. The stabilization schematic can be seen \nin figure \\ref{fig:CavityStabi}.\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.9\\textwidth]{Figure2.eps}\n\t\\caption{\\label{fig:CavityStabi} The schematic of the experimental setup to control the cavity frequency. The beat signal of the two ECDL is mixed down with a function generator from R\\&S (SMA02A). The error signal to stabilize the cavity piezo is generated with an error signal generating circuit (EGC) following the design of Ref. \\cite{Weitz2004}.}\n\\end{figure}\n\nBecause of widespread availability we use a 780nm Extended Cavity Diode Lasers (ECDL) \\cite{ECDL1995} stabilized via FM-Spectroscopy to the Rubidium $F=2\\rightarrow F'=2,3$ crossover line as an atomic reference signal. Another ECDL lasing at a similar wavelength is injected into the cavity on a high-order TEM mode (a so called v-mode). This is possible in nearly confocal cavities and has the benefit of very small mode overlap with the MOT. No influence of this laser onto the cloud could be detected. It is stabilized to the mode using a Pound-Drever-Hall lock (PDH) \\cite{PDH2001} and constantly kept on resonance to it.\\\\ It is basically measuring the length of the cavity very precisely. When the cavity length changes, the cavity resonance $\\omega_C$ moves and the laser follows. To gain atomic transition stability for the cavity length this laser is now offset locked via a side-of-filter \\cite{Weitz2004} technique to the 780 nm reference laser. The error signal is fed back to drive the piezo of the cavity, which can change the length of the cavity by about $17\\mu m$, compensating mostly for slow temperature drifts and low frequency acoustic noise. The offset lock enables us to change the frequency difference of the two ECDLs so that we can position a cavity resonance at arbitrary detunings to both the atomic Cesium transition as well as the cavity probe laser. The lock is stable for the whole day with small drifts well below one cavity linewidth. The overall stabilization bandwidth is limited by the mechanical resonance of the science cavity spacer to about 800Hz.\\\\\n\n\n\\section{Strong Coupling in the bad cavity limit}\n\tOne experimental cycle involved loading the MOT for an arbitrary length of time (between 100 and 3500ms), turning off the MOT cooling beams and the magnetic field followed by pumping the cavity with a weak probe beam. After a short off time, in which the cavity probe laser or the cavity could be moved in frequency the cycle started again. \n\tThe atom number interacting with the mode was controlled by choosing different MOT loading times.\n\tThree different experimental procedures were implemented to observe the normal mode splitting of the cavity-atom system. Two methods involved observing the transmission of the cavity as a function of the probe laser frequency. The third method relied on the collection of the scattered light into free space with a CCD camera during the probe. The observation of the Rabi splitting in the MOT fluorescence confirmed the results, but the data presented in this publication originates solely from transmission measurements.\\\\\n \n \\begin{figure}\n\\begin{center}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure3a_3.eps}}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure3b_3}}\n\\caption{\\label{fig2_a} a) Cavity transmission measurements with a weak incident $\\sigma_+$-probe beam for different atom numbers in the mode. The cavity is positioned at the Cs $D_2$ $F=4 \\rightarrow F'=5$ transition. The red curves are fits with the cavity transmission function from \\cite{Transmission1} with free parameters $\\kappa$ and $g_{coll}$. The scatter of the data can be explained by shot-to-shot atom number fluctuations as well as probe and cavity resonance frequency drifts. A conservative estimate for the atom number fluctuations is 6\\% and $2\\pi\\times0.3$ MHz for the maximum drift of cavity resonance frequency and probe beam frequency during the whole experiment. b) The collective coupling constant displays the typical $\\sqrt{N}$-behaviour indicating the strong collective coupling regime with an effective coupling parameter $g_{eff}=\\textbf{$g$}\/\\sqrt{2}$. The linear fit to the data was used to calibrate the atom number in the mode.\n\\label{sample-figure1}\n\\end{center}\n\\end{figure}\nThe first set of transmission measurements used a constant probe laser frequency for\neach point. The MOT was loaded for a certain time to a specific density, then \nswitched off and a weak cavity probe beam was turned on with a \nmechanical shutter about $250\\mu s$ later. The transmitted peak power for most experiments was measured to be below $\\left(2.0\\pm0.2\\right)$ nW. According to Ref. \\cite{transmission2}, this is much smaller than the critical photon number to cause bistable behaviour and therefore suitable to observe stable Rabi splitting. The shutter was left open for two milliseconds and we averaged the transmission signal on the photodiode over 1 ms. After \nthat, the cavity probe laser was moved in frequency by $2\\pi\\times2$ MHz and the cycle started \nagain. The scan of the probe beam begins at a detuning $\\Delta_A=-2\\pi\\times 50$ MHz and ends at  \n$\\Delta_A=2\\pi\\times 50$ MHz so that for each curve 51 points were taken. The \ncavity resonance was tuned so to have the same frequency as the atomic transition, so that $\\Delta_A-\\Delta_C=\\omega_C-\\omega_A=0$. For a rough alignment the empty cavity transmission was compared to a Doppler-free Cesium spectroscopy while scanning. \nThen the cavity was locked and a first transmission measurement taken. If there \nwas an amplitude difference in the two peaks indicating a detuning between atomic \nand cavity resonance frequency, the cavity was moved with the offset lock until \nthe transmission peaks were similar in strength.\\\\ \nFigure \\ref{fig2_a} shows data taken for the cavity positioned at the \nCs $D_2$ $F=4\\rightarrow F'=5$ transition. With respect to the empty cavity resonance the doublet \nsplits with increasing number of atoms according to the $\\sqrt{N}$ behaviour. \nThe aquired data was fitted with the theoretical curve for the cavity transmission in the linear regime (see e.g Ref. \\cite{Transmission1}) with the collective coupling $g_{coll}$ and the cavity half decay rate $\\kappa$ as free parameters. The collective coupling is then displayed as a function of the square root of the atom number in figure \\ref{fig2_a} b. The fit result for the cavity half loss rate is $\\kappa_{fit}=2\\pi\\times\\left(5.8\\pm0.2\\right)$ MHz for all the data and is therefore about 7 times larger than the measured $\\kappa$ of the TEM$_{00}$ mode. We attribute this fact to the multimode structure of the nearly confocal resonator: higher order TEM modes have a bigger mode volume and therefore a smaller coupling constant g. This causes the higher order normal mode splitting to be smaller which appears as broadening of $\\kappa$.\nTo calibrate the atom number in the mode we plotted the collective coupling constant over the square root of the observed MOT fluorescence first.  The MOT fluorescences was measured with a CCD camera for the same experimental parameters but without probing the cavity mode for each loading time. The shot to shot fluctuations are estimated to be below 6\\%.  \nThe result was fitted with a line through the origin and the horizontal axis of figure \\ref{fig2_a} b rescaled in a way that the gradient corresponded to the effective coupling constant $g_{eff}= g\/\\sqrt{2}=2\\pi \\times 85 kHz$. Since the probe is too weak to trap the atoms they see an averaged coupling during their ballistic movement in the 1 ms probe time, being zero at the nodes and $g$ at the antinodes of the cavity field. This results in the smaller coupling constant \n$g_{eff}$ according to Ref. \\cite{Leslie2004}. The largest observed collective coupling of \n$g_{coll}=2\\pi\\times \\left(27.8\\pm0.4\\right)$ MHz corresponded to $\\left(1.33\\pm0.08\\right)\\times 10^5$ atoms effectively coupled to the cavity mode and a maximum collective cooperativity of $C_{coll}=186\\pm5$. The rescaling factor relating the atom number in the MOT to the atom number coupled to the mode was 1.47. It is bigger than 1 because not all atoms in the MOT are in the center of the mode. \n\n\n\nThe collective behaviour of the vacuum Rabi splitting of the cavity-atom system is clearly demonstrated. The coupling to the resonator dominates the environmental dissipation process indicating the collective strong coupling regime for atom numbers in the mode bigger than about 600 corresponding to $C_{coll}=1$.\\\\\n\\begin{figure}\n\\begin{center}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure4a.eps}}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure4b_1.eps}}\n\\caption{\\label{fig2} a) The transmission of $\\sigma_+$-polarized light through the cavity as a function of the probe laser frequency for an atom number $N=\\left(15\\pm22\\right)\\times10^3$. The different traces (twenty averages each) are taken for different cavity resonance-atomic transition detuning. The error in the detuning is estimated to be below $2\\pi\\times0.3$ MHz. The difference between each trace is about $2\\pi\\times4.6$ MHz (except the middle three, where the frequency difference was $2\\pi\\times3.2$ MHz) starting from the top at $-2\\pi\\times21.5$ MHz. The red line is a double lorenzian fit to those curves where two peaks were observable. b) The center frequencies of the two peaks displayed over the cavity resonance - atomic transition detuning. For an empty cavity the mode position should follow the dashed line. Coupling atoms to the cavity splits the resonance and an avoided crossing can be observed.\n\\label{sample-figure}\n\\end{center}\n\\end{figure}\nThe second set of measurements was taken in a slightly different way. Instead of pumping the mode for a certain time with the same frequency after the release from the MOT, we scanned the probe laser over the atomic transition. The scan speed was about 40 MHz\/ms. Eventhough the ballistic expansion of the cloud during the first 2 ms is negligible, the two peaks are probed one after the other. For high intensities of the probe light, this causes the second peak to become asymmetric in height and position even if the empty cavity resonance equals the atomic transition. For small intensities this was not observable.\nThe traces of the transmission were recorded with an oscilloscope and averaged over 20 shots. They can be seen in figure \\ref{fig2} a. After aquiring the waveform the cavity offset lock was used to move the cavity resonance with respect to the atomic transition. In this way the avoided crossing induced by the collective cavity-atom coupling could be observed. Figure \\ref{fig2} b shows how, for large cavity-transition detunings, the coupling is small, therefore the eigenfrequency of the cavity mode with atoms does not deviate from the case without atoms (indicated by the dashed line). Approaching the atomic resonance causes the coupling to increase and the mode to split.\\\\ \n  \n\\begin{figure}\n\\begin{center}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure5a_2.eps}}\n\\subfigure[]{\n\\resizebox*{6.5cm}{!}{\\includegraphics{Figure5b_2.eps}}\n\\caption{\\label{fig3} a) The transmission of the cavity of a weak $\\pi$-polarized probe beam for each hyperfine transition in comparison with the empty cavity. The atom number in the cavity is $N=\\left(110\\pm7\\right)\\times 10^3$. The stronger the dipole moment of the transition the bigger the splitting. The red line is the theoretical curve for the transmission according to Ref. \\cite{Transmission1}. b) The observed collective coupling constant as a function of the relative transition strength as given in Ref. \\cite{Steck}.}\n\\label{sample-figure}\n\\end{center}\n\\end{figure}\n\nThe last set of measurements involved observing the splitting for the other hyperfine states of cesium as seen in figure \\ref{fig3} a. The cavity was positioned according to the procedure for the 4-5 transition. The MOT loading time and all other parameters are the same for each transmission measurement but the observed splitting is less. \nOur data shows that the splitting is proportional to the relative hyperfine transition \nstrength, consistent with the fact  that the  coupling constant $g$ is proportional \nto the relative transition strength.\\\\\n\\section{Conclusion}\nIn conclusion we have demonstrated the strong coupling of an ensemble of up to $\\left(1.33\\pm0.08\\right) \\times 10^5$ atoms with a lossy nearly confocal cavity leading to a collective \ncooperativity parameter of up to $C_{coll}=186\\pm5$. In contrast to other observations of the avoided crossing and the behaviour of the vacuum Rabi splitting (e.g \\cite{Raizen1989, Kasevich2006}) the single atom - single mode coupling $g$ is very small compared to $\\kappa$ and the relevant excited state half decay rate $\\gamma$. The large number of inner cavity scatterers only enables us to reach the strong coupling regime which is a prerequisite to study certain cavity cooling mechanisms. This system is also suitable for the study of other collective effects such as superradiance or collective selforganization in a lossy cavity.\\\\\n\n\\section{Acknowledgement} This work is supported by the EPSRC grant EP\/H049231\/1. We would like to thank Jon Goldwin for stimulating discussions.\\\\\n\n\\bigskip \n\n\n\\bibliographystyle{tMOP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n \n \n\n \n\nImproper priors and finitely additive probabilities (FAP) are the  two main alternatives to the standard Bayesian Paradigm using proper priors, i.e. countably additive probabilities   in the Kolmogorov  axiomatic. Both alternatives are involved in  paradoxical phenomena such as non-conglomerability,  marginalization paradox, etc.\nAs an heuristic argument, some authors such as  \\citet{stone1982} or  \\citet[][p.218]{kada1986}, consider  proper prior sequences, for example sequence of uniform prior, and derive under two different topologies two kind of  limits: FAPs limits and improper limits. The choice between FAP distribution and improper distribution has been largely  debated in  the Bayesian literature  \\citep[see][p.15]{hart1983}. \n\nThe aim of FAP limits is to preserve the total mass   equal to 1, while sacrificing the countable additivity. This point of view has been mainly defended by \\citet{defi1972}. On the other hand, improper distributions aim to preserve the countable additivity, while  sacrificing  a total mass equal to 1. Improper distribution  appears naturally in the framework of conditional probability, see \\citet{reny1955} \tand more recently \\citet{taralind2010,taralind2016} and \\citet{lindtara2018}. Conditional probability spaces are also related to projective spaces of measures \\citep{reny1970} which have a natural quotient space topology  and a natural convergence mode, named $q$-vague convergence by \\citet{bidr2016}. \n\nIn Bayesian inference some paradoxes such as  non-conglomerability  \\citep{ston1976, stone1982} or the marginalization paradox  \\citep{dastzi1973} occur  with improper or diffuse FAP priors \\citep[][]{dubi1975}, but not with proper priors.  This lead some  authors to  include the  likelihood in the definition of a   convergence mode for the priors, by for instance considering the  convergence of the posterior distribution w.r.t. to an entropy criterion \\citep{akai1980} or an integrated version of this criterion \\citep{bebesu2009} when the posterior is proper. Bayesian inference with  improper posterior is justified by  \\citet{tatuli2018} from a theoretical point of view.  \\citet{ bobidr2018} consider the convergence of proper   to an improper posterior for Bayesian   estimation of abundance   by removal sampling.  \\citet*{tulari2018} propose to adapt MCMC for the  estimation of  improper posterior. \n\nIn this paper, we only  consider convergence of distributions regardless to any statistical model. The implication of our result in Bayesian inference, especially the construction of specific sequences of distribution used in the proof of Theorem \\ref{thm.sameqvnotFAP}   is left for future works. In Section \\ref{section.defcv}, we define  several concept of convergence in the setting of improper and FAP distributions. In Section \\ref{section.uniform} we revisit the notion of uniform distribution on integer. In Section \\ref{section.examples}, we illustrate the fundamental difference between convergence to an improper prior and to an FAP. \n \n \n\\section{Convergence  of probability  sequences}\n\\label{section.defcv}\nWe denote by $\\mathcal C_b$ the set of continuous real-valued  bounded functions, by $\\mathcal C_K$ the set of continuous real-valued functions with compact support.\nFor a $\\sigma$-finite measure $\\pi$, we denote $\\pi(f)=\\int f(\\theta) \\;d\\pi(\\theta)$.\n Consider   a sequence of proper priors $\\{\\pi_n\\}_{n\\in \\mathbb N}$.  The usual   converge mode of  $\\{\\pi_n\\}_{n\\in \\mathbb N}$  to a proper prior $\\pi$ is the narrow   convergence,  also called weak convergence or convergence in law, defined by:\n\\begin{equation}\n\\label{eq.cvnarrow}\n\\pi_n \\xrightarrow[n\\rightarrow +\\infty]{narrowly} \\pi\\iff \\pi_n(f) \\xrightarrow[n\\rightarrow +\\infty]{}\\pi(f)\\quad \\forall f\\in\\mathcal C_b\n\\end{equation}\n\n When it exists, the narrow limit of $\\{\\pi_n\\}_n$ is necessarily unique. In this section, we consider two alternative convergence modes when there is no narrow limit, and especially    when the total  mass tends to concentrate around  the boundary on the domain, more precisely when $\\lim_n \\pi_n(f)=0$ for all $f$ in $\\mathcal C_K$. The idea is to consider a proper prior as a special case of FAP or as a special case of  a Radon measure, and for each case to define in a formalized way, a convergence mode. In both cases the limit is not unique in general but, as a requirement,  must coincide with the narrow convergence when Eq. (\\ref{eq.cvnarrow})  holds.\n\n~\n\n\nIn the following, we consider that $\\Theta$ is a metric, locally compact, second countable Hausdorff space  $\\Theta$.   This is the case, for example, for usual topological finite-dimensional  vector spaces or denumerable sets with the discrete topology.  In the latter case, any  function is continuous and a compact set is a finite set.  \n\n\\subsection{Convergence to an improper distribution}\nTo extend the notion of the narrow limit, we consider here  proper distributions  within the set of projective space of positive Radon measures as follows: we denote by $\\mathcal R$ the set of non-null Radon measures, that is     regular countably additive measures with finite mass  on each compact set.  Note that, in the discrete case  any $\\sigma$-finite measure is a Radon measure. \n\n~\n\n\n\n \nWe define an improper distribution as an unbounded  Radon measure which appear   in parametric Bayesian statistics  \\citep[see, e.g.][]{jeff1983}.\nThe  \\textit{projective space} $\\overline{\\mathcal R}$\n  associated to $\\mathcal R$ is the quotient space for the equivalence relation $\\sim$  defined by $\\pi_1\\sim\\pi_2$ iff $\\pi_2=\\alpha\\, \\pi_1$ for some positive scalar factor $\\alpha$. To a Radon measure $\\pi$, it can be associated a unique equivalence class $\\overline\\pi=\\{\\pi'=\\alpha\\,\\pi\\,;\\,\\alpha>0\\}$. Therefore, a projective space is a space where  objects are defined up to a positive scalar factor. It is natural in Bayesian statistics  to consider such projective space since   two equivalent priors give the same posterior.    The projective space $\\mathcal R$ is also naturally linked with  conditional probability spaces     \\citep\n  {reny1955}. \n All the results presented below on the convergence mode w.r.t. to the projective space $\\overline{\\mathcal R}$ can be found in \\citet{bidr2016}. The usual topology on $\\mathcal R$ is the vague topology defined by \n\\begin{equation}\n\\label{eq.cvvague}\n\\pi_n \\xrightarrow[n\\rightarrow +\\infty]{vaguely} \\pi\\iff \\pi_n(f) \\xrightarrow[n\\rightarrow +\\infty]{}\\pi(f)\\quad \\forall f\\in\\mathcal C_K\n\\end{equation}\nwhere $\\mathcal C_K$ is the set of all real-valued functions on $\\Theta$ with compact support.\n\n\nFrom  the related  quotient topology, we can derive a convergence mode,  called q-vague convergence: a sequence  $\\{\\pi_n\\}_n$ in $\\mathcal R$ converge $q-vaguely$ to a (non-null) improper distribution $\\pi$ in $\\mathcal R$  if $\\overline \\pi_n$ converges to $\\overline\\pi$ w.r.t.  the quotient topology where $\\overline \\pi_n=\\{\\alpha\\pi_n;\\alpha>0\\}$ is the equivalence class associated to $\\pi_n$ and  similarly for $\\overline\\pi$. The  limit $\\overline \\pi$  is unique whereas $\\pi$ is unique only  up to   a positive scalar factor. It is not always tractable  to   check a convergence in the  quotient space. However, there is  an equivalent definition in the initial space $\\mathcal R$:  $\\{\\pi_n\\}_n$   converges $q-vaguely$ to  $\\pi$ if there exists some scalar factors $\\alpha_n$ such that $\\{\\alpha_n\\,\\pi_n\\}_n$ converges vaguely to $\\pi$:\n \\begin{equation}\n \\label{eq.cvqvague2}\n \\pi_n \\xrightarrow[n\\rightarrow +\\infty]{q-vaguely} \\pi\\iff a_n\\pi_n \\xrightarrow[n\\rightarrow +\\infty]{vaguely}\\pi\\quad   \\textrm{for some } a_1,a_2,...>0\n \\end{equation}\n \n \n  The $q$-vague convergence can be considered as an extension of the narrow convergence in the sense that if $\\{\\pi_n\\}_n$ and  $\\pi$  are proper distributions and  $\\{\\pi_n\\}_n$ converge narrowly to $\\pi$ then   $\\{\\pi_n\\}_n$ converge q-vaguely to $\\pi$. Note that the converse part holds if and only if $\\{\\pi_n\\}_n$ is tight \\citep[see][Proposition 2.8]{bidr2016}.\n \n   ~\n   \n   \n   \n   When a sequence $\\{\\pi_n\\}_n$ of proper distributions converges q-vaguely to an improper distribution, then  $\\lim_n\\pi_n(K)=0$ for any compact $K$ \\citep[Prop. 2.11]{bidr2016}.  The following lemma gives an apparently stronger, but in fact equivalent, result. It will be useful to establish the main result and to construct examples in Section \\ref{section.exampleFAPvsqvague}.\n   \n   \\begin{lemma}\\label{lemma.suitecpctslow}\n   \tLet $\\{\\pi_n\\}_n$ be a sequence of proper distributions such that $\\lim_n\\pi_n(K)=0$ for any compact $K$. Then there exists a non-decreasing  sequence of compact sets $K_n$ such that $\\cup_n K_n=\\Theta$ and  $\\lim_n\\pi_n(K_n)=0$. Moreover, $K_n$ may be chosen such that, for any compact $K$, there exists an integer $N$ such that $K\\subset K_N$.\n   \\end{lemma}\n   \\begin{proof}\n   \tLet $\\widetilde K_m$, $m\\geq 1$, be a increasing sequence of compact sets with $\\cup_m \\widetilde  K_m=\\Theta$. For each $m $, $\\lim_n\\pi_n(\\widetilde K_m)=0$, so  there exists an integer $N_m$ such that $N_m>N_{m-1}$ and  $\\pi_n(\\widetilde K_m)\\leq 1\/m$ for $n>N_m$. Consider now such a sequence of integers $N_m$, $m\\geq 1$. For any $n$ there exists a unique integer  $m$ such that  $N_m\\leq n <N_{m+1}$. We define $K_n$ by $K_n=\\widetilde K_m$. So,  $\\pi_n(K_n)=\\pi_n(\\widetilde K_{m})\\leq 1\/m$. Since $m$ increases with $n$, $\\lim_n\\pi_n(K_n)=0$. Futhermore, the sequence $\\widetilde K_m$ can be chosen such that, for any compact $K$, $K$ is a subset of all but finitely many $\\widetilde K_m$, \\citep[see  e.g.][Lemma 29.8]{bauer2001}. By construction, the same property holds for  the sequence $K_n$.\n    \\end{proof}\n   \nNote that,  $\\lim_n\\pi_n(K)=0$ for any compact set $K$ does not imply that $\\{\\pi_n\\}$ converge q-vaguely. See Section \\ref{section.examples} for some examples of such sequences.\n\\subsection{Convergence   to a FAP}\n\\label{section.FAPlimit}\n\nIn this section, we consider a  proper distribution as special cases of FAPs. When a sequence $\\{\\pi_n\\}$ of proper  distribution does not converge narrowly to  a proper prior, we need to define a weaker  convergence mode  so that the sequence can converge to some FAPs.\n This convergence mode will be named FAP convergence, and the corresponding  limits,  the  FAP limits.\n\n~\n\n\nDenote by $\\mathcal F_b$  the set of bounded real-valued measurable  functions on $\\Theta$.  A FAP $\\pi $ can be defined as a linear functional on $\\mathcal F_b$  which is positive, i.e.  $\\pi(f)\\geq 0$ if $f\\geq 0$ and which satisfies  $\\pi(1)=1$.  Therefore, the set of FAPs belongs to the topological dual of $\\mathcal F_b$ \nequipped with the sup-norm. For any measurable set $E\\in  \\Theta$, we define  $\\pi(E)=\\pi(\\mathds 1_E)$, where $\\mathds 1_E(x)=1$ if $x\\in E$ and 0 otherwise. We also denote $\\int f(\\theta) \\;d\\pi(\\theta)=\\pi(f)$. Since      proper distributions are required to  be   special cases of FAPs, we do impose here measurability conditions  in  the definition of  $\\mathcal F_b$, contrary to  many authors \\citep[see, e.g.][]{heathsudd1978}. In the case where  $\\Theta$ is a  denumerable set equipped with the usual discrete topology, any functions or sets are measurable and  therefore this distinction is not relevant. \n\n  ~\n  \n\n\nConsider a sequence $\\{\\pi_n\\}_n$ of FAPs.  The minimal requirement for   $\\pi$ to be  a FAP limit is  that \n $\\lim_n \\pi_n(f) =\\pi(f)$ for any $f\\in \\mathcal S_c$,\n  where $\\mathcal S_c$ is the set of $f\\in\\mathcal F_b$ such that  $\\lim_n \\pi(f_n)$ exists.\n  The existence of  a limit $\\pi$ that satisfies this requirement is guaranteed by the Hahn-Banach theorem \\citep[see][]{rudi1991, huis2016} as follows: define the linear function $\\Phi$ on $\\mathcal S_c$ by $\\Phi(f)=\\lim_n \\pi_n(f)$ and the sublinear functional $p(f)=\\limsup_n \\pi_n(f)$. Then, there exists a linear functional $\\pi$ on $\\mathcal F_b$ that coincides with $\\Phi$ on $\\mathcal S_c$ and that satisfies  $ \\pi(f)\\leq p(f)$ on $\\mathcal F_b$.  \n   The functional $\\pi$ will be called a \\textit{ FAP limit}. \n  The condition $\\pi(f)\\leq p(f)$ implies that    $\\pi$ is a FAP. Conversely, a FAP necessarily  satisfies $\\pi(f)\\leq p(f)$. Replacing    $f$ by  $-f$ gives  $\\pi(f)\\geq \\liminf_n\\pi_n(f)$.  Therefore, a FAP limit can be characterized by the  following lemma:\n\n\n\\begin{lemma}\n\\label{lemma.cvFAP}\nA FAP $\\pi$ is a FAP-limit of the sequence $\\{\\pi_n\\}_n$ if and only if   $\\textrm{for any } f\\in\\mathcal F_b $ \n\\begin{equation}\\label{eq.cvFAP}\n\\liminf_n\\pi_n(f) \\leq \\pi(f)\\leq  \\limsup_n\\pi_n(f) \n\\end{equation}\nor equivalently   if and only if $ \\textrm{for any } \\textrm{ measurable set } E$\n\\begin{equation}\n\\label{eq.cvFAPdiscret}\n \\liminf_n\\pi_n(E) \\leq \\pi(E)\\leq  \\limsup_n\\pi_n(E)  .\n\\end{equation}\n\n\\end{lemma}\n  When $\\mathcal S_c=\\mathcal F_b$, the FAP limit is unique and corresponds to the convergence associated to the   weak$^*$-topology. In the general case, \nthe FAP limit is not unique  and its existence relies on the axiom of choice.  \n  Unlike the q-vague convergence, the FAP convergence cannot be considered as an extension of the narrow convergence. For example,  consider the proper distributions   $\\pi_n=\\delta_{\\sqrt 2\/n}$, where $\\delta$ is the Dirac measure. The sequence $\\{\\pi_n\\}_n$ converges narrowly to $\\pi=\\delta_0$ but $\\pi$ is not a FAP limit of $\\{\\pi_n\\}_n$. To show this, consider  $f(\\theta)=\\mathds 1_{\\mathbb Q} (\\theta)\\in\\mathcal F_b$, with $\\mathbb Q$ the set rational numbers, we have $\\lim_n\\pi_n(f)=0\\neq \\pi(f)=1$. However, in the special case where   $\\Theta$ is a denumerable set, any real-valued function on $\\Theta$ is continuous and therefore  if a sequence of proper distribution $\\{\\pi\\}_n$ converges narrowly to  a proper distribution $\\pi$, then $\\pi$ is a FAP limit. \n\n \n\n ~\n \nAnother way to define  a FAP limit in a formalized way  has been proposed by \\citet{stone1982} for denumerable sets but can be extended to more  general sets considered in this paper. The existence of a limit relies  on the  Banach-Alaoglu-Bourbaki theorem   \\citep[see, e.g.][]{rudi1991}, since   a FAP belongs to the unit ball in  the dual of $\\mathcal F_b$ which  is compact for the weak$^*$-topology. Hence, for any  sequence $\\{ \\pi_n\\}_n$ of  FAPs,  there exists at least one accumulation point $\\pi$ which is defined as a FAP limit.  We recall that $\\pi$ is an accumulation point of $\\{\\pi_n\\}_n$ for the weak$^*$-topology if and only if for any   integer  $p$, any  $f_1,...f_p$ in $\\mathcal F_b$ and any $\\varepsilon>0$, there exists  an  infinite number of  $n$ such \nthat $\\left|\\pi_n(f_i)-\\pi(f_i)\\right|\\leq \\varepsilon$, $i=1,...,p$. Note that, since $\\mathcal F_b$ is not in general first-countable, there does not necessarily exist a subsequence $\\{\\pi_{n_k}\\}_k$ that converges to $\\pi$. We can only say that, for any $f\\in\\mathcal F_b$, there exists a subsequence $\\{\\pi_{n_k}(f)\\}_k$ such that $\\pi_{n_k}(f)$ converges to $\\pi(f)$.\n\nTherefore, the  set of FAP limits of  $\\{\\pi_n\\}_n$ obtained by Stone's approach is included in the set of FAP limits obtained  by using the Hahn-Banach theorem as above  and (\\ref{eq.cvFAP}) or (\\ref{eq.cvFAPdiscret}) still hold but are not sufficient conditions. The converse inclusion is false in general. It is easy to see that  the closed convex hull of the set of FAP limits defined by Stone   is included in the set of FAP limits defined in this paper.  We conjecture that, conversely, the set of FAP limits defined by (\\ref{eq.cvFAP}) is the convex hull for the limits defined by Stone. \nConsider for example $\\pi_{2n}=\\delta_0$ and $\\pi_{2n+1}=\\delta_1$. There are only two  FAP limits $\\delta_0$ and $\\delta_1$  with  Stone's construction, whereas any $\\pi=\\alpha \\delta_0+(1-\\alpha) \\delta_1$, $0\\leq\\alpha\\leq 1$ is a FAP limit with our construction. In Section \\ref{section.limdeltan}, we illustrate the difference between the two convergence modes with another example. \n\n~\n\nEven if the two definitions  of FAP limits are not equivalent,  the main results, especially   Theorem \\ref{thm.sameqvnotFAP}, Corollary \\ref{corol.main},  Proposition \\ref{prop.PoissonlimitFAP}, Lemma \\ref{lemma.FAPlimitcombconvexe} and \\ref{lemma.FAPKNslow}   hold for  both of them. In the following, we consider only the first definition of FAP limits.\n  \n\n \n\n~\n\n\n\\subsection{FAP limits vs q-vague convergence}\n\\label{section.FAPvsqvague}\nThe fact that a sequence of proper distributions have both  improper and  FAP limits may suggest  a connection between the two notions as proposed heuristically by many authors. The following results show  that this is not the case. Roughly speaking, it is shown that any  FAP which is a FAP limit of some proper distribution sequence  can be connected to  any improper prior   by this mean.   \n\n \n\\begin{theorem}\n\t\\label{thm.sameqvnotFAP}\nLet  $\\{\\pi_n\\}_n$ be a  sequence of proper distributions such that  $\\lim_{n}\\pi_n(K)=0 $ for any compact set $K$. Then, for any improper distribution $\\pi$,  it can be constructed a sequence $\\{\\widetilde\\pi_n\\}_n $ which  converges q-vaguely to $\\pi$ and which has the same set of  FAP limits  as      $\\{\\pi_n\\}_n$.\n\\end{theorem}\n\n\\begin{proof}\nFor any FAP or any  proper or improper distribution $\\mu$ we define the distribution  $(\\mathds{1}_A\\, \\mu)$ by $(\\mathds{1}_A\\,\\mu)(f)=\\mu(\\mathds 1_A \\,f)$.\tFrom  Lemma \\ref{lemma.suitecpctslow}, it can be constructed  an  exhaustive increasing  sequence  $K_n$  of compact sets such that $\\lim_n \\pi_n(K_n)=0$.  \n\tPut    $\\gamma_n=\\pi_n(K_n)$  and define the sequence of proper distributions $\\widetilde \\pi_n=\\gamma_n \\frac{1}{\\pi(K_n)} \\mathds 1_{K_n}\\pi$ $+$ $  (1-\\gamma_n)  \\frac{1}{\\pi_n  (K_n^c)}\\mathds{1}_{K_n^c}\\,\\pi_n $, with  $K^c$  the complement of $K$. By Lemma    \\ref{lemma.FAPKNslow} and  \\ref{lemma.FAPlimitcombconvexe} in  Appendix A, $\\widetilde\\pi_n$ has the same FAP limits   as $\\{\\pi_n\\}$. By Lemma \\ref{lemma.cvqvtronque},  $\\widetilde\\pi_n$ converges q-vaguely to  $\\pi$.\n\t\n\\end{proof} \n\n\n\n\\begin{corollary}\n\t\\label{corol.main}\n\tLet $\\{\\pi_n\\}_n$ be a sequence of proper distributions that converges q-vaguely to \nan improper distribution $\\pi^{(1)}$. Then, for any other  improper distribution $ \\pi^{(2)}$, it can be constructed a sequence $\\{ \\widetilde\\pi_n\\}_n$ that converges q-vaguely to $\\pi^{(2)}$ and that has the same FAP limits as $\\{\\pi_n\\}_n$.\n\\end{corollary}\n   The only link that   can be established between improper q-vague limits  and FAP  limits of the same  proper distribution sequence is that the   FAP limits are necessarily  \\textit{diffuse}, i.e.  they assign  a probability 0 to any compact set.\n     \n   \n\n \\section{Uniform distribution on integers} \n \\label{section.uniform}\n In this section, we compare different  notions of uniform distributions on the set of integers $\\mathbb N$.  by using several considerations such as limit of proper uniform distributions.\n \n \n \n \n We illustrate the fact that FAP uniform distributions  are not  well  defined objects \\citep[][pp.122,224]{defi1972}. Contrary to  uniform improper distributions, FAP limits of  uniform distributions    on an  exhaustive sequence of compact sets are highly dependent on the choice of that sequence.\n \n \n \n \n \n \n \n \\subsection{Uniform improper distribution}\nThere are several equivalent  ways  to define a uniform improper  prior on integers. These definitions lead  to a unique, up to a scalar factor,  distribution.  The uniform distribution can be defined directly as a  flat distribution, i.e. $\\pi(k)\\propto 1$ for $k$ integer. It is also the unique (up to a scalar factor) measure that is shift invariant, i.e. such that  $\\pi(k+A)=\\pi(A)$ for any integer $k$ and any set of integers $A$. The uniform distribution is also the q-vague  limit  of the sequence of  uniform proper  distributions on $K_n=\\{0,1,...,n\\}$. More  generally and equivalently, the uniform distribution is the      q-vague limit of any sequence of proper uniform  priors on an exhaustive increasing sequence $\\{K_n\\}_n$ of finite subsets of integers. \n\n\\subsection{Finitely additive uniform distribution}\nThe notion of  uniform   finitely additive probabilities is more complex. Contrary to the improper   case,  there is no explicit definition    since   $\\pi({k})=0$ for any integer $k$. We present here several non equivalent approaches to define a uniform FAP. The first two ones can be found in  \\cite{kadaohagan1995} and   \\citet{schikada2007}.\n \n\\subsubsection{Shift invariant  (SI) uniform distribution}\n\nAs for the improper case, a uniform FAP distribution $\\pi$  can be defined as been any shift invariant FAP, i.e.  FAPs satisfying  $\\pi (A) = \\pi (A+k)$ for any subset of integers  $A$ and any integer $k$. Such a distribution will  be called SI-uniform. In that case, one necessarily has : $\\pi(k_1+k_2\\times \\mathbb N)=k_2^{-1}$, for any $(k_1,k_2) \\in \\mathbb{N}\\times\\mathbb{N}^*$.  \\cite{kadaohagan1995} also  investigate the properties of FAPs  satisfying only  $\\pi(k_1+k_2\\times \\mathbb N)=k_2^{-1}$, where $k_1+k_2\\times \\mathbb N$ are called \\emph{residue classes}. \n\n\n\\subsubsection{Limiting relative frequency (LRF) uniform distributions.}\n\n\\cite{kadaohagan1995} consider a stronger condition to define uniformity. For a  subset  $A$,  define its limiting relative frequency LRF$(A)$  by  $$\\textrm{LRF}(A) = \\lim_{N\\to\\infty} \\frac{\\#\\{k\\leq N, \\; \\text{s.t.}\\; k\\in A\\}}{N+1}, $$\nwhen this limit exists. A  FAP $\\pi$ on $\\mathbb N$ is said to be {\\em LRF-uniform} if  $\\pi (A) = p$  when $\\textrm{LRF}(A)=p$. \n\n Let  $\\pi_n$ be the  uniform proper  distribution on  $K_n=\\{0,1,...,n\\}$, then $\\textrm{LRF}(A) =$   $ \\lim_{n\\to\\infty} \\pi_n(A)$. Therefore a FAP $\\pi$ is LRF uniform if and only if it is a FAP limit of  $\\{\\pi_n\\}_n$. It is worth noting that, unlike the improper case, the   FAP limits are highly dependent on the  choice of the increasing exhaustive  sequence of finite sets $K_n$. Changing the sequence $\\{K_n\\}_n$ changes the notion of uniformity. For example, if $\\widetilde\\pi_n$ is the uniform distribution on  $K_n=\\{2k\\,;\\; 0\\leq k\\leq n^2\\} \\cup \\{2k+1\\,;\\;0\\leq k\\leq n\\}$, then $\\lim_n \\widetilde \\pi_n(2\\mathbb N)= 1$, whereas $\\lim_n\\pi_n(2 \\mathbb N)=1\/2$. \n\n \n\n \n\n\n\\subsubsection{Bernoulli Scheme (BS) uniform distribution}\n\n    \nWe propose here another notion of uniformity that is not dependent of the choice a particular increasing sequence of $K_n$ as for the LRF uniformity. Consider a Bernoulli Scheme, that is a sequence    $\\{X_k\\}_{k\\in\\mathbb N}$ of i.i.d.  Bernoulli distributed  random variables with mean $p\\in[0;1]$. Define the random set $A(X)=\\{k \\in \\mathbb N, \\text{ s.t. } X_k =1 \\}$. A   FAP $\\pi$ is said to be  {\\em BS-uniform } if, for any $p\\in[0;1]$, $\\pi(A(X)) = p$, almost surely. Note that the strong law of large numbers, LRF$(A(X))=p$, almost surely.\n \n\\begin{proposition} \nLet $\\{K_n\\}$ be an    increasing sequence of finite subsets of $\\mathbb{N}$, with $\\cup_{n\\in\\mathbb N} K_n $ being infinite. Then any FAP-limit of the sequence $\\pi_n$ of uniform distributions on $K_n$ is BS-uniform.\n\\end{proposition} \nWhen $\\cup_{n\\in\\mathbb N} K_n =\\mathbb N$, this proposition shows that any FAP limit of uniform distribution is BS-uniform. In particular, a LRF uniform FAP is also BS uniform.    However, if for example $K_n$ is  the set of even numbers less or equal to $n$, then any FAP-limit of the sequence of  uniform distributions on $K_n$  will be BS-uniform, although it is intuitively, certainly not uniform on $\\mathbb N$ but on $2\\mathbb N$. Therefore, BS uniformity looks much more like a necessary condition for a FAP to be uniform, than like a complete definition.  \n\n\n\\section{Comparison of convergence modes on examples}\n\\label{section.examples}\n  \nWe consider here some examples that illustrate the difference between convergence of proper distributions to an improper distribution or a to FAP.\n\n\n\\subsection{FAP limits on ${\\mathbb N}$.}\n\\label{section.limdeltan}\nFor a sequence $\\{\\pi_n\\}$ of proper distribution on $\\mathbb N$, it is known that there does not necessarily exist a q-vague limit,  but if it exists, it is unique in the projective space of Radon measures, i.e. unique up to a scalar factor. At the opposite, we have seen that a FAP limit always exists but is not necessarily unique. \n\nWe illustrate the non-uniqueness of FAP limits with  an extreme case. Consider the  sequence of proper distributions $\\pi_n = \\delta_n$, where $\\delta _n$ is  the Dirac measure on $n$. This sequence has no q-vague limit since $\\pi_n(k)=0$ for $n>k$.  Moreover, for any subset $A$ of ${\\mathbb N}$ so that $A$ and $A^c$ are both infinite:\n$$\\displaystyle\n0=\\liminf_{n\\to\\infty} \\pi_n(A) \\leq \\pi (A) \\leq \\limsup_{n\\to\\infty} \\pi_n(A) =1,\n$$ \nwhereas, for any finite set $A$, $\\lim_n\\pi_n(A)=0=\\pi(A)$ and for any cofinite set $A$,   $\\lim_n\\pi_n(A)=1=\\pi(A)$. Therefore, from  (\\ref{eq.cvFAPdiscret}), the set of FAP limits of $\\pi_n$ is the set of  all diffuse FAPs on $\\mathbb N$. This shows that all the diffuse FAPs are connected through the same sequence $\\pi_n$.\n\n \n ~\n \nLet's examine  the  set of  FAP limits   of $\\pi_n=\\delta_n$ obtained with   Stone's definition of FAP convergence (see Section \\ref{section.FAPlimit}).  For any subset $A$, there exits a subsequence $\\{\\pi_{n_k}\\}$ such that  $\\pi_{n_k}(A)$  convergences to $\\pi(A)$.  So, $\\pi(A)\\in\\{0,1\\}.$ Therefore   the  FAP limits of $\\pi_n$  in Stone's sense  are all  \\textit{remote} FAPs, that is diffuse FAPs $\\pi$ such that $\\pi(A)\\in\\{0,1\\}$, as defined by \\citet[][p.92]{dubi1975}.  This also proves the existence of remote FAPs.   Note that a remote distribution is neither BS uniform nor SI  and therefore cannot be LRF uniform. \n  \n\\subsection{Convergence  of sequence of  Poisson distributions}\n\\label{section.poisson} \n\n\nWe consider  the sequence $\\{\\pi_n\\}_n$ of  Poisson  distributions with mean  $n$. For any finite set $K$, we have $\\lim_n\\pi_n(K)=0$. However this   sequence of proper priors   does not converge   $q$-vaguely  to any improper distribution  \\citep[][\u00a75.2]{bidr2016}. As a remark, let $\\widetilde\\pi_n$ be shifted  measures on $\\mathbb{Z}$, defined by $\\pi_n( B) = \\pi_n(B+n)$, where \n$\\pi_n$ can be seen as a measure on the set of all integers $\\mathbb{Z}$ with $\\widetilde\\pi_n(k)=0$ for $k<0$. Then, using the approximation of the Poisson distribution by a the normal distribution, it can be shown that the sequence $\\widetilde\\pi_n$  converges  $q-$vaguely to the improper uniform measure on $\\mathbb{Z}$.  \n  \nWe consider now the  FAP limits of the sequence    $\\{\\pi_n\\}_n$. The next result shows that the limit have some properties of uniformity described in Section \\ref{section.uniform} but not all of them. The proof is  given in Appendix B.\n\n \n\n\\begin{proposition}\n\t\\label{prop.PoissonlimitFAP}\n\t Any FAP limit $\\pi$ of the sequence $\\{\\pi_n\\}_n$ of Poisson distribution with mean $n$  is   shift invariant and BS-uniform but not necessarily  LRF uniform. \n\\end{proposition}\n\nTherefore, the FAP limits of the  Poisson distribution sequence are examples of SI- and BS-uniform distributions that are not LRF uniform.\n\\citet{kaji2014} give\n another  example of SI but not LFR uniform FAPs using paths of random walks. Even if they consider FAP on a subset of  bounded function, it can be extended to $\\mathcal F_b$ using Hahn-Banach theorem similarly to Section \\ref*{section.FAPlimit}.\n\n\n\n \\subsection{FAP vs q-vague convergence of uniform proper  distributions}\n \\label{section.exampleFAPvsqvague}\n \n\nTo illustrate the fact that any FAP limit can be related with any improper distribution,  consider again the sequence  $\\{\\pi_n\\}_n$ of Poisson distributions with mean $n$  and  any given improper distribution $\\pi^0$ on the integers.  Since  $\\lim_n\\pi_n(K)=0$ for any finite set, the proof of  Lemma \\ref{lemma.suitecpctslow} shows how to construct an exhaustive sequence of finite set $K_n$ such that $\\lim_n \\pi_n(K_n)=0$.  For example,  choose  $K_n = \\{k\\in \\mathbb N, \\;  k \\leq n\/2 \\}$ and define the sequence of proper distributions $\\widetilde \\pi_n$ by :\n\n\n\\begin{equation}\n\\label{eq.seqpoisexample}\n\\widetilde\\pi_n(A)=\\pi_n(K_n) \\frac{\\pi^0(A\\cap K_n)}{\\pi^0(K_n)} + (1-\\pi_n (K_n)) \\frac{\\pi_n(A\\cap K_n^c)}{\\pi_n(K_n^c)} \\vspace{1mm}\n\\end{equation}\nfor any set $A$.  From Theorem \\ref{thm.sameqvnotFAP}, $\\widetilde{\\pi}_n$ converge $q$-vaguely to $\\pi^0$ and has the same FAP limit as the Poisson distribution sequence.\n\n\nAs another example, consider the sequence $\\{\\pi_n\\}_n$ of uniform distribution on $\\{0,1,...,n\\}$ and choose  $K_n = \\{k\\in \\mathbb N, \\;  k \\leq \\sqrt n \\}$. We have $\\lim_n \\pi_n(K_n)=0$. Therefore, for any improper distribution  $\\pi^0$ on the set of integers, the sequence constructed as in  (\\ref{eq.seqpoisexample}) have the same FAP limits as that of sequence of  uniform distributions $\\{\\pi_n\\}_n$ and  converge q-vaguely to $\\pi^0$. This shows again the difficulty to connect   improper and FAP uniform distributions by limits of proper distribution.\n\n\n\\subsection{Convergence of   beta  distributions}\nIn this section, we consider the limit of the sequence of  beta distribution $\\pi_{a_n,b_n}$ $=$ $\\text{Beta}(a_n,b_n)$ defined on $\\Theta=]0,1[$ when $a_n$ and $b_n $ go to 0.  We  see that the FAP limits depend on the way $a_n$ and $b_n$ goes to $0$, which is not the case for the q-vague improper limit. \nThis   illustrates  the difference between FAP limits and q-vague limits of proper distribution sequences.  \n\n \n\n\nThe density of a beta distribution Beta$(a,b)$ is given by  \n$$\n\\pi_{a,b} (x) =\\frac{1}{\\beta(a, b)} \\, x^{a-1} (1-x)^{b-1}  \\text{ for  } \\; x \\in ]0;1[\n$$\n where $\\beta(a,b)$ is the beta function.\n \nFrom \\citet{bidr2016}, the unique (up to a scalar factor)  q-vague limit of  $\\text{Beta} (a_n,b_n)$ when  $a_n$ and $b_n $ go to 0 is  the Haldane improper distribution: $$\\pi_H(x)=\\frac{1}{x(1-x)}  \\text{ for  } \\; x \\in ]0;1[.$$ \n \n\nThe q-vague  limit gives no information on   the relative concentration of the mass around $0$ and $1$: for $0<u<v<1$,   $\\pi_H(]0,u])=\\pi_H(]v,1[])=+\\infty$. To explore this concentration, we   temporally replace  the space $\\Theta$ by $\\widetilde \\Theta=[0,1]$. This has no consequence on the Beta distribution but will change radically the q-vague limit. Put $c_n=a_n\/b_n$ and assume that $\\{c_n\\}_n$ converges to some $c\\in[0,1]$. The sequence  $\\{\\pi_{a_n,b_n}\\}_n$ converges narrowly, and hence q-vaguely, to the proper distribution $\\widetilde\\pi=\\frac{1}{1+c}\\delta_0+\\frac{c}{1+c}\\delta_1$. Contrary to the Haldane prior, $\\widetilde{\\pi}$ shows how the mass concentrate on the boundary of the domain, but gives no information on the behavior of the sequence inside the domain. Note that the Haldane distribution is not a Radon measure on $\\widetilde{\\Theta}$ since  $\\pi_H([0,1])=+\\infty$ where $[0,1]$ is a compact set. Therefore $\\pi_H$ cannot be a candidate for the q-vague limit on $\\widetilde{\\Theta}$.\n\nWe now consider the  FAP limits on $\\Theta=]0,1[$ of $\\pi_{a_n,b_n}$ and we show that they give an information similar  to that given by $\\widetilde{\\pi}$ on the way the mass concentrate on the boundary of the domain.  Again, we assume that $c_n=a_n\/b_n$ converge to some   $c\\in [0,1]$. Easy calculations show that for any  $0<\\varepsilon<1$ $\\lim_n \\pi_{a_n,b_n}(]0,\\varepsilon [) = \\frac{1}{1+c}$ and $\\lim_n \\pi_{a_n,b_n}(]1-\\varepsilon,1 [) = \\frac{c}{1+c}$.\nTherefore, for any FAP-limit $\\pi$  and for any $\\varepsilon \\in ]0,1[$, we have  $\\pi (]0,\\varepsilon [) = \\frac{1}{1+c}$ and $\\pi (]1-\\varepsilon ,1[) =\\frac{c}{1+c}$, whereas $\\pi([u,v])=0$ for $0<u<v<1$.\n\n \n\n\\section{Conclusion and perspectives}\nIn this paper, we have shown that  improper distributions and FAPs limits give quite different  information on the behaviour of the sequence of proper distribution and are therefore complementary. \n\nIn a Bayesian context,  under some mild regularity condition on the  likelihood $f(x|\\theta)$, these results can be applied to the joint distribution $f_n(x,\\theta)=f(x|\\theta)\\pi_n(\\theta)$ and to the marginal distribution $p_n(x)=\\int f(x|\\theta)\\pi_n(d\\theta)$ when it is well defined, both converging to an improper prior if the sequence of prior converge to an improper distribution, similarly for FAP limits.  As limits of proper priors, FAP limits and improper prior limits represent two different features of the sequence :  the first one describes  the behavior of the sequence inside the domain, whereas the second one describes how the mass concentrate on the boundary of the domain. We hope to use these differences in future works to understand some paradoxical phenomenon that occurs in Bayesian statistics.\n \n\n\n\\section*{Appendix A}\nWe  establish some lemmas useful to prove Theorem \\ref{thm.sameqvnotFAP}. The first one is straightforward.\n\n\\begin{lemma}\n\t\\label{lemma.FAPlimitcombconvexe}\n\tLet $\\{\\pi_n^{(1)}\\}_n$ and $ \\{\\pi_n^{(2)}\\}_n$ be two sequences of proper distributions and $0\\leq \\gamma_n\\leq 1$ be a sequence of scalars that converges to $0$. Then, the sequence defined by  $\\widetilde\\pi_n=\\gamma_n\\pi^{(1)}+(1-\\gamma_n)\\pi_n^{(2)}$ has the same FAP limits as $\\{\\pi_n^{(2)}\\}_n$.\n\\end{lemma}\n\n\\begin{proof}\n\tFor any $f\\in \\mathcal F_b$, $ \\limsup_n\\widetilde\\pi_n(f)=\\limsup_n\\pi_n^{(2)}(f)$ and   $ \\liminf_n\\widetilde\\pi_n(f)=$ $\\liminf_n\\pi_n^{(2)}(f)$. The result follows from Lemma \\ref{lemma.cvFAP}.\t\n\\end{proof}\n\n\\begin{lemma}\n\t\\label{lemma.FAPKNslow}\nLet $\\{\\pi_n\\}$ be a sequence of proper priors and  $K_n$ be a non-decreasing sequence of compact sets such that  $\\lim_n \\pi_n(K_n)=0$, then the sequence defined by  $\\widetilde\\pi_n=\\frac{1}{\\pi_n(K_n^c)}\\mathds{1}_{K_n^c}\\pi_n$ has the same FAP limits than $\\{\\pi_n\\}_n$.\n\\end{lemma}\n  \n\\begin{proof}\n\nFirst, note that $\\{\\pi_n\\}_n$ is not defined when $\\pi_n(K_n)=1$,  but this cannot occur more than a finite number of times.\nFor any $f\\in \\mathcal F_b$, $\\pi_n(f)=$ $\\mathds{1}_{K_n}\\pi_n(f)+\\mathds{1}_{K_n^c}\\pi_n(f)=$ $\\pi_n(\\mathds{1}_{K_n}f)+\\pi_n(K_n^c)\\widetilde\\pi_n(f)$. \nSince $f$ is bounded, $\\lim_n\\pi_n(\\mathds{1}_{K_n}f)$ $=$ $0$.  Moreover, $\\lim_n(K_n^c)=1$. \nTherefore, $\\pi_n(f)$ and $\\widetilde\\pi_n(f)$ have the same limit-sup and limit-inf and thus they have the same FAP limits by Lemma \\ref{lemma.cvFAP}. \n\n \\end{proof}\n\nAt the opposite of Lemma \\ref{lemma.FAPKNslow}, the following lemma shows that if we consider the restriction of a sequence $\\{\\pi_n\\}_n$ of a proper or improper distribution on a exhaustive increasing sequence $\\{K_n\\}_n$ of compact sets, we preserve the q-vague limits.\n\\begin{lemma}\\label{lemma.cvqvtronque}\n\tLet $K_n$ be a non-decreasing sequence of compact sets such that $\\cup_n K_n=\\Theta$ and such that, for any compact $K$, there exists $N$ such that $K\\subset K_N$. A sequence   $\\{\\pi_n\\}$  of Radon measures converges q-vaguely  to the Radon measure $\\pi$ if and only if $\\widetilde \\pi_n=\\frac{1}{\\pi_n(K_n)} \\mathds{1}_{K_n}\\pi_n$ converges q-vaguely to $\\pi$.\n\\end{lemma}\n\\begin{proof}\n\tAssume that  $\\pi_n$ converges q-vaguely to $\\pi$, then there exists some positive scalars $\\{a_n\\}_n$ such that for any $f$ in $\\mathcal C_K$, $\\lim_n a_n\\pi_n(f)=\\pi(f)$. Put $\\widetilde a_n= a_n\\,\\pi_n(K_n)$ and denote by $K_f$ a compact set that includes the support of $f$. Then, there exists an integer $N$ such that $K_f\\subset K_n$ for $n>N$.\n\tTherefore, for $n>N$,  $\\widetilde a_n\\widetilde \\pi_n(f)=a_n\\pi_n(f)$. The result and its reciprocal follow. \n\\end{proof}\n \n\n\n\\section*{Appendix B}\n \n  We prove here Proposition \\ref{prop.PoissonlimitFAP} of section\n  \\ref{section.poisson}.\n   \n  In order to show that  $\\pi$ is SI,  we consider   $\\pi_n$ as\n  distribution  on $\\mathbb Z$, extending them by $0$ on the non-positive\n  integers. Define the   $\\pi^{(k)}_n$  by   $\\pi^{(k)}_n (A) = \\pi_n\n  (A+k)$, for any subset of $\\mathbb Z$.  It is known that that\n  $\\|\\pi^{(k)}_n - \\pi_n \\|_{TV} \\leq \\frac{k}{\\sqrt{2\\pi n}}$, where\n  $\\|\\cdot\\|_{TV}$ is the total variation norm. Therefore, for any subset\n  of $\\mathbb N$, $\\lim_{n\\to\\infty} |\\pi_n(A+k) - \\pi_n(A) | =0$. Letting\n  $n$ go to infinity, we deduce that, for any FAP limit $ \\pi$ of $\\pi_n$,\n  and any integer $k$ : $ \\pi (A+k) =  \\pi (A)$.\n  \n  \n  The fact that $\\pi$ is uniform in BS sense comes from an easy\n  adaptation of the Hoeffding inequality in that context. Let $(X_k)_{k\n  \t\\in \\mathbb N}$ be a Bernoulli scheme, of parameter $p$, and denote by\n  $\\mathbb P$ being the associated probability.\n  Hoeffding inequality gives, that, for any $n$~:\n  \n  $$\n  \\mathbb P\n  \\left\\{\n  \\bigg|\n  \\sum_{k=0}^{\\infty} e^{-n}\\frac{k^n}{n!} (X_k(\\omega) -p)\n  \\bigg|\n  \\geq\n  t\n  \\right\\}\n  \\; \\leq \\;\n  2 e^{-2 c \\sqrt{2\\pi n}\\, t^2},\n  $$\n  for some positive constant $c$. The expected conclusion is then\n  obtained thanks to the  Borel-Cantelli lemma.\n  \n  The fact that some of the FAP limits $\\pi$ are not LRF uniform is a\n  direct  consequence of the following lemma.\n  \n  \n  \\begin{lemma}\n  \tFor any $0\\leq p,p'\\leq 1$, there exists a set $A$ and some FAP\n  \tlimits ${\\pi}$ of  $\\{{\\pi}\\}_n$  such that $LRF(A)=p$ and ${\\pi}(A)=p'$.\n  \\end{lemma}\n  \\begin{proof}\n  \tFirst note  that , for any set $A'$, $LRF(A')=p$ if, and only\n  \tif, $\\sharp \\{ k \\leq n, \\, k\\in A'\\} =$ $ pn + o(n)$. Therefore, for any\n  \tset $A$ with $LRF(A)=p$ and for any set $B$ such that $\\sharp \\{k\\leq n\n  \t, \\, k\\in B\\} = o(N)$, one has both $LRF(A\\cup B) = p$ and\n  \t$LRF(A\\setminus B ) = p$. Take now for set $B$ the following~:\n  \t$$\n  \tB = \\bigcup_{k\\in\\mathbb N} \\big\\{ u\\in\\mathbb N \\; :\\; 4^k -2^k k\n  \t\\leq u \\leq  4^k+2^k k \\big\\}.\n  \t$$\n  \tFor that $B$, one has~:\n  \t$$\n  \t\\limsup_{n\\to\\infty} \\frac{\\sharp\\{k\\leq n, \\, k \\in B\\}}{n+1} =\n  \t\\lim_{k \\to\\infty} \\frac{\\sum_{i=0}^{k} 2^{i+1} i }{4^k + 2^k k}\n  \t\\; \\leq \\; \\lim_{k \\to\\infty}  \\frac{(k+1)2^{k+2}}{4^k } = 0,\n  \t$$\n  \tand thus $LRF(B) =0$. However, $\\pi_{4^k} (B)$ converges to $1$.\n  \tIndeed, if $U_k$ is some random variable with law $\\pi_{4^k}$, one has~:\n  \t$$\n  \t\\pi_{4^k}\n  \t\\big(\n  \t\\big\\{ u\\in\\mathbb N \\; :\\; 4^k -2^k k \\leq u \\leq  4^k+2^k k \\big\\}\n  \t\\big)\n  \t=\n  \t\\mathbb P\n  \t\\bigg( \\frac{U_k -4^k}{\\sqrt{4^k}} \\in\n  \t\\big[ - k \\, ; \\, k\n  \t\\big]\n  \t\\bigg).\n  \t$$\n  \tThe rigth-hand side term above converges to $1$ thanks to the\n  \tcentral limit theorem. Hence $LRF(A\\cup B) = LRF(A\\setminus B) = p$\n  \twhile $\\pi_{4^k}(A\\cup B)$ converges to $1$, and $\\pi_{4^k}(A\\setminus\n  \tB) $ converges to $0$. Now, for any $p'\\in[0;1]$, choose two numbers\n  \t$a<b$, so that\n  \t$\n  \n  \tp'=\\int_a^b \\frac{e^{-u^2\/2}}{\\sqrt{2\\pi}} du $. Take the set $B'$\n  \tto be~:\n  \t$$\n  \tB' = \\bigcup_{k\\in\\mathbb N}\n  \t\\big\\{\n  \tu\\in\\mathbb N \\; :\\;\n  \t4^k +2^k \\text{max}(-k,a) \\leq u \\leq  4^k+2^k \\text{min}(k,b)\n  \t\\big\\},\n  \t$$\n  \tthen $LRF(B)=0$ again and $\\pi_{4^k} (B')$ converges to $p'$\n  \t, still thanks to the central limit theorem. Let $A = (A'\\setminus\n  \tB)\\cup B'$. Then $LRF (A) =p $ and\n  \t$\\lim_{k \\to \\infty} \\pi_{4^k} (A) =p'$. Now, any FAP-limit $\\pi$\n  \tof subsequence $\\pi_{4^k}$ is also a FAP-limit of $\\pi_k$ : $\\pi$ is\n  \ttherefore shift uniform and BS uniform,\n  \tbut one has $ \\pi (A)= p'$.\n  \t\\end{proof}\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe study of information theoretic secrecy was initiated by Shannon\nin \\cite{Shannon:1949}. Following Shannon's work, significant\ncontributions were made by Wyner \\cite{WynerWiretap} who established\nthe rate-equivocation region of a degraded broadcast\nchannel. Wyner's result was generalized to the case of a general\nbroadcast channel by Csiszar and Korner \\cite{CsiszarKorner}.\nRecently, there has been a resurgence of activity in studying\nmulti-terminal and vector extensions of \\cite{WynerWiretap},\n\\cite{CsiszarKorner}.\n\nIn this paper, we investigate a secure transmission problem from a source coding perspective.\nIn particular, we first consider a simple setup consisting of four\nterminals. Terminal $1$ (say Alice) observes an\ni.i.d. source $X^{n}$ which it intends to transmit losslessly to terminal $2$\n(say Bob). A malicious but passive user (say Eve) gets to observe\nthe coded output of Alice. In other words, the communication link\nbetween Alice and Bob is public, i.e., insecure. It is clear\nthat since the malicious user gets the same information as the\nlegitimate user, there cannot be any positive secret rate of transmission.\nOn the other hand, if there is a helper, say Helen,  who observes an i.i.d. source $Y^{n}$\nwhich is correlated with the source $X^{n}$ and transmits information\nover a secure rate-limited link to Bob, then one can aim for creating\nuncertainty at the eavesdropper (see Figure $1$\\footnote{In Figures\n  $1$, $2$ and $3$, secure links are shown by {\\bf{bold}} lines.}).\nFor the model shown in Figure $1$, we\ncompletely characterize the rate-equivocation\nregion. From our result, we observe that the classical\nachievablity scheme of Ahlswede and Korner \\cite{AhlswedeKorner:1975}\nand Wyner \\cite{Wyner:1975} for source coding with rate-limited side information is robust in the\npresence of a passive eavesdropper.\n\nSecondly, we consider the model where Alice also has access to the\ncoded output of Helen and completely characterize the\nrate-equivocation region. We will call this model the two-sided\nhelper model (see Figure $2$). From our result, we observe that\nthe availability of additional coded side information at Alice\nallows her to increase uncertainty of the source at Eve even\nthough the rate needed by Alice to transmit the source losslessly\nto Bob remains the same. This observation is in contrast with the\ncase of insecure source coding with side information where\nproviding coded side information to Alice is of no value\n\\cite{AhlswedeKorner:1975}.\n\nWe next generalize the setup of Figure $2$ to the case when there are both secure and insecure\nrate-limited links from Helen and there is additional side\ninformation $W^{n}$ at Bob and additional side\ninformation $Z^{n}$ at Eve. In particular, there is a secure link\nof capacity $R_{sec}$, whose output is available at Alice and Bob and an\ninsecure link, of capacity $R_{ins}$, whose output is available at all\nthree terminals, i.e., at Alice, Bob and Eve (see Figure $3$). The\npresence of both secure and insecure links from Helen can be\ninterpreted as a source-coding analogue of a degraded broadcast\nchannel from Helen where Alice and Bob receive both secure and\ninsecure streams $J_{sec}$ and $J_{ins}$, whereas,\nEve only receives the insecure stream $J_{ins}$. We completely\ncharacterize the rate-equivocation region for this model when\n$Y^{n}\\rightarrow X^{n}\\rightarrow (W^{n},Z^{n})$ forms a Markov chain.\n\n\\begin{figure}[t]\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=0.63]{Helper1.eps}\n\\caption{One-sided helper.}\n\\label{fig:figure1}\n\\end{minipage}\n\\hspace{0.12cm}\n\\begin{minipage}[b]{0.5\\linewidth}\n\\centering\n\\includegraphics[scale=0.63]{Helper2.eps}\n\\caption{Two-sided helper.}\n\\label{fig:figure2}\n\\end{minipage}\n\\end{figure}\n\nWe explicitly compute the rate-equivocation region for the cases of one-sided helper\nand two-sided helper for a pair of binary symmetric sources. We show that\nhaving access to Helen's coded output at Alice yields a strictly larger\nequivocation than the case of one-sided helper.\n\n\\section{Related Work}\nThe secure source coding setup shown in Figure $1$ was considered\nin \\cite{Deniz:ITW08} where it was also assumed\nthat Eve has access to additional correlated side information\n$Z^{n}$. Inner and outer bounds for the rate-equivocation region were\nprovided for this setup, which do not match in general. The\nrate-equivocation region was completely characterized in\n\\cite{Deniz:ITW08} for the case when Bob has\ncomplete uncoded side information $Y^{n}$ and Eve has additional side information\n$Z^{n}$. This result also\nfollows from \\cite{VinodKannan:ITW07} where a similar three terminal\nsetup was studied and the maximum uncertainty at Eve was\ncharacterized under the assumption of no rate constraint in the\nlossless transmission of the source to Bob. A similar model was also\nstudied in \\cite{KundurG:2007} where Bob intends to reconstruct both $X^{n}$ and $Y^{n}$\nlosslessly. It was shown that Slepian-Wolf type coding suffices for\ncharacterizing the rate-equivocation region when the\neavesdropper does not have additional correlated side\ninformation. This setup was generalized in \\cite{Deniz:ISIT08} to the case when the\neavesdropper has additional side information $Z^{n}$, and inner and\nouter bounds were provided, which do not match in general.\n\n\nIn \\cite{Grokop:ISIT05}, a multi-receiver\nsecure broadcasting problem was studied, where Alice intends to\ntransmit a source $X^{n}$ to $K$ legitimate users. The $k$th user\nhas access to a correlated source $Y_{k}^{n}$, where\n$Y_{k}^{n}=X^{n}\\oplus B_{k}^{n}$, for $k=1,\\ldots K$, and the\neavesdropper has access to $Z^{n}$, where $Z^{n}=X^{n}\\oplus\nE^{n}$, and the noise sequences $(B_{1}^{n},\\ldots,\nB_{K}^{n},E^{n})$ are mutually independent and also independent of\nthe source $X^{n}$. Furthermore, it was assumed that Alice also\nhas access to $(Y_{1}^{n},\\ldots Y_{K}^{n})$. For sources with\nsuch modulo-additive structure, it was shown that to maximize the\nuncertainty at the eavesdropper, Alice cannot do any better than\ndescribing the error sequences $(B_{1}^{n},\\ldots,B_{K}^{n})$ to\nthe legitimate users. This model is related to the two-sided helper\nmodel shown in Figure $2$; see Section $5$ for details.\n\nFor the model shown in Figure $3$, when we set $R_{sec}=R_{ins}=0$,\ni.e., in the absence of Helen, we recover the\nresult obtained in \\cite{VinodKannan:ITW07}. Therefore, our result can\nalso be viewed as a generalization of the result obtained in \\cite{VinodKannan:ITW07}.\n\n\\begin{figure}[t]\n  \\centerline{\\epsfig{figure=Helper22L.eps,width=8.5cm}}\n  \\caption{Secure and insecure links from two-sided\n    helper.}\\label{fig8}\n\\vspace{-0.15in}\n\\end{figure}\n\nBy setting $R_{sec}=0$, i.e., in the absence of the secure\nrate-limited link, the resulting model is related to the model\nconsidered in \\cite{CsiszarNarayan} where the aim is to generate a\nsecret key between two terminals via an insecure rate-limited\ntwo-sided helper. In the model studied in this work, the aim is to\nsecurely transmit the source $X^{n}$ to Bob. Note that, when\n$R_{sec}=0$ and $W=\\phi$, both the secret key generation capacity\n\\cite{CsiszarNarayan} and secure transmission rate are zero since\nEve has access to both  $J_{ins}$ and $J_{x}$ along with $Z^{n}$.\nOn the other hand, in the presence of a secure link, i.e., when\n$R_{sec}>0$, even when $W=\\phi$, we can still create uncertainty\nat the eavesdropper. This is possible since Helen can choose not\nto transmit any information on the insecure link and transmit only\na coded description of $Y^{n}$ by using the secure link at the\nrate $R_{sec}$, which plays the role of a correlated key. Furthermore,\nbeing correlated with the source $X^{n}$, the coded description of\n$Y^{n}$ also permits Alice to lower the rate of transmission when\ncompared to the case of using an uncorrelated secret key, where Alice\ntransmits at a rate $H(X)$.\n\n\\section{Summary of Main Results}\nIn Section $4$, we present the rate-equivocation region for the\ncase of one-sided helper. We show that Slepian-Wolf type coding alone\nat Alice is optimal for this case. We present the\nrate-equivocation region for the case of two-sided helper in\nSection $5$.  For the case of two-sided helper, Alice uses a\nconditional rate-distortion code to create an auxiliary $U$ from\nthe source $X$ and the coded output $V$ received from Helen. This\ncode construction is reminiscent of lossy-source coding with\ntwo-sided helper \\cite{KaspiBerger:1982},\\cite{Vasudevan:ISIT07}, \\cite{Helper:ITW09}.\nFor the case of lossy source coding, a conditional rate-distortion\ncode is used where $U$ is selected to satisfy the distortion\ncriterion. On the other hand, the purpose of $U$ in secure\nlossless source coding is to confuse the eavesdropper. From this\nresult, we demonstrate the insufficiency of Slepian-Wolf type coding at\nAlice by explicitly utilizing the side information at Alice. This observation\nis further highlighted in Section $6$ where we compare the\nrate-equivocation regions of two-sided helper and one-sided helper cases for a pair\nof binary symmetric sources. For this example, we show that for\nall $R_{y}>0$, the information leakage to the eavesdropper for the\ntwo-sided helper is strictly less than the case of one-sided\nhelper. We finally generalize the result of two-sided helper to\nthe case when there are both secure and insecure rate-limited\nlinks from the two-sided helper and additional side informations\n$W$ and $Z$, at Bob and Eve, respectively. The presence of secure\nand insecure rate-limited links from Helen can be viewed as a\nsource-coding analogue of a degraded broadcast channel from Helen\nto (Alice, Bob) and Eve. We characterize the rate-equivocation region for this\nmodel when the sources satisfy the condition $Y\\rightarrow X\n\\rightarrow (W,Z)$.\n\n\n\\section{One-Sided Helper}\n\\subsection{System model}\nWe consider the following source coding problem. Alice observes an $n$-length source\nsequence $X^{n}$, which is intended to be transmitted losslessly\nto Bob. The coded output of Alice can be observed by the malicious\nuser Eve. Moreover, Helen observes a correlated source $Y^{n}$ and\nthere exists a noiseless rate-limited channel from Helen to Bob.\nWe assume that the link from Helen to Bob is a secure link and the\ncoded output of Helen is not observed by\nEve (see Figure $1$). The sources $(X^{n},Y^{n})$ are generated\ni.i.d. according to $p(x,y)$ where $p(x,y)$ is defined over the finite\nproduct alphabet $\\mathcal{X}\\times \\mathcal{Y}$. The aim of Alice is\nto create maximum uncertainty at Eve regarding the source $X^{n}$\nwhile losslessly transmitting the source to Bob.\n\nAn $(n, 2^{nR_{x}},2^{nR_{y}})$ code for this model consists of an\nencoding function at Alice, $f_{x}:X^{n}\\rightarrow \\{1,\\ldots,2^{nR_{x}}\\}$,\nan encoding function at Helen, $f_{y}:Y^{n}\\rightarrow \\{1,\\ldots,2^{nR_{y}}\\}$, and a decoding function at Bob, $g: \\{1,\\ldots,2^{nR_{x}}\\} \\times \\{1,\\ldots,2^{nR_{y}}\\} \\rightarrow X^{n}$.\nThe uncertainty about the source $X^{n}$ at Eve is measured by\n$H(X^{n}|f_{x}(X^{n}))\/n$. The probability of error in the reconstruction of\n$X^{n}$ at Bob is defined as\n$P_{e}^{n}=\\mbox{Pr}(g(f_{x}(X^{n}),f_{y}(Y^{n}))\\neq X^{n})$.\nA triple $(R_{x},R_{y},\\Delta)$ is achievable if for any $\\epsilon>0$,\nthere exists a $(n, 2^{nR_{x}}, 2^{nR_{y}})$ code such that $P_{e}^{n}\\leq\n\\epsilon$ and \\break $H(X^{n}|f_{x}(X^{n}))\/n \\geq \\Delta$. We denote\nthe set of all achievable $(R_{x},R_{y},\\Delta)$ rate triples as $\\mathcal{R}_{1-sided}$.\n\n\\subsection{Result}\nThe main result is given in the following theorem.\n\\begin{Theo}\\label{Theorem1} The set of achievable rate triples $\\mathcal{R}_{1-sided}$ for\n  secure source coding with one-sided helper is given as\n\\begin{align}\n\\mathcal{R}_{1-sided}=\\Big\\{(R_{x},R_{y},\\Delta): R_{x}&\\geq H(X|V)\\\\\nR_{y}&\\geq I(Y;V) \\\\\n\\Delta&\\leq I(X;V)\\Big\\}\n\\end{align}\nwhere the joint distribution of the involved random variables is as\nfollows,\n\\begin{align}\np(x,y,v)&=p(x,y)p(v|y)\n\\end{align}\nand it suffices to consider such distributions for which\n$|\\mathcal{V}|\\leq |\\mathcal{Y}|+2$.\n\\end{Theo}\nThe proof of Theorem $1$ is given in the Appendix.\n\nWe note that inner and outer bounds for source coding model considered in this\nsection can be obtained from the results presented in \\cite{Deniz:ITW08} although\nthese bounds do not match in general. These bounds match when Bob has\ncomplete uncoded side information $Y^{n}$, i.e., when $R_{y}\\geq\nH(Y)$.\n\nThe achievability scheme which yields the rate region described in\nTheorem $1$ is summarized as follows:\n\\begin{enumerate}\n\\item Helen uses a rate-distortion\ncode to describe the correlated source $Y$ to Bob.\n\\item Alice performs\nSlepian-Wolf binning of the source $X$ with respect to the coded\nside information at Bob.\n\\end{enumerate}\nTherefore, our result shows that the\nachievable scheme of Ahlswede, Korner \\cite{AhlswedeKorner:1975} and\nWyner \\cite{Wyner:1975} is optimal in the presence of an\neavesdropper. Moreover, on dropping the security\nconstraint, Theorem $1$ yields the result of \\cite{AhlswedeKorner:1975},\\cite{Wyner:1975}.\n\n\\section{Two-Sided Helper}\n\\subsection{System model}\nWe next consider the following generalization of the model considered\nin Section $4$. In this model, Alice also has access to the coded\noutput of Helen besides the source sequence $X^{n}$ (see Figure $2$).\nAn $(n, 2^{nR_{x}},2^{nR_{y}})$ code for this model consists of an\nencoding function at Alice, $f_{x}:X^{n} \\times \\{1,\\ldots,2^{nR_{y}}\\} \\rightarrow \\{1,\\ldots,2^{nR_{x}}\\}$,\nan encoding function at Helen, $f_{y}:Y^{n}\\rightarrow \\{1,\\ldots,2^{nR_{y}}\\}$, and a decoding function at Bob,\n$g: \\{1,\\ldots,2^{nR_{x}}\\} \\times \\{1,\\ldots,2^{nR_{y}}\\} \\rightarrow X^{n}$.\nThe uncertainty about the source $X^{n}$ at Eve is measured by\n$H(X^{n}|f_{x}(X^{n}))\/n$. The probability of error in the reconstruction of\n$X^{n}$ at Bob is defined as\n$P_{e}^{n}=\\mbox{Pr}(g(f_{x}(X^{n},f_{y}(Y^{n})),f_{y}(Y^{n}))\\neq X^{n})$.\nA triple $(R_{x},R_{y},\\Delta)$ is achievable if for any $\\epsilon>0$,\nthere exists a $(n, 2^{nR_{x}}, 2^{nR_{y}})$ code such that $P_{e}^{n}\\leq\n\\epsilon$ and $H(X^{n}|f_{x}(X^{n}))\/n \\geq \\Delta$. We denote\nthe set of all achievable $(R_{x},R_{y},\\Delta)$ rate triples as $\\mathcal{R}_{2-sided}$.\n\\subsection{Result}\nThe main result is given in the following theorem.\n\\begin{Theo}\\label{Theorem2} The set of achievable rate triples $\\mathcal{R}_{2-sided}$ for\nsecure source coding with two-sided helper is given as\n\\begin{align}\n\\mathcal{R}_{2-sided}=\\Big\\{(R_{x},R_{y},\\Delta): R_{x}&\\geq H(X|V)\\\\\n\\vspace{-0.1in}R_{y}&\\geq I(Y;V)\\\\\n\\Delta&\\leq \\min(I(X;V|U),R_{y})\\Big\\}\\label{Theo2}\n\\end{align}\nwhere the joint distribution of the involved random variables is as\nfollows,\n\\begin{align}\np(x,y,v,u)&=p(x,y)p(v|y)p(u|x,v)\n\\end{align}\nand it suffices to consider distributions such that $|\\mathcal{V}|\\leq\n|\\mathcal{Y}|+2$ and $|\\mathcal{U}|\\leq |\\mathcal{X}||\\mathcal{Y}|+2|\\mathcal{X}|$.\n\\end{Theo}\nThe proof of Theorem $2$ is given in the Appendix. We remark here that\nthe proof of converse for Theorem $2$ is closely related to the proof of the converse of the rate-distortion function with a two-sided\nhelper  \\cite{KaspiBerger:1982},\\cite{Vasudevan:ISIT07},\\cite{Helper:ITW09}.\n\n\nThe achievability scheme which yields the rate region described in\nTheorem $2$ is summarized as follows:\n\\begin{enumerate}\n\\item Helen uses a rate-distortion\ncode to describe the correlated source $Y$ to both Bob and\nAlice through a coded output $V$.\n\\item Using the coded output $V$ and\nthe source $X$, Alice jointly quantizes $(X,V)$ to an auxiliary random\nvariable $U$. She subsequently performs Wyner-Ziv coding, i.e., bins\nthe $U$ sequences at the rate $I(X;U|V)$ such that Bob can decode $U$ by using\nthe side information $V$ from Helen.\n\\item  Alice also bins the source $X$\nat a rate $H(X|U,V)$ so that having access to $(U,V)$, Bob can\ncorrectly decode the source $X$. The total rate used by\nAlice is $I(X;U|V)+H(X|U,V)=H(X|V)$.\n\\end{enumerate}\nTherefore, the main difference between the achievability schemes for Theorems $1$\nand $2$ is at the encoding at Alice and decoding at Bob. Also note\nthat selecting a constant $U$ in Theorem $2$ corresponds to\nSlepian-Wolf type coding at\nAlice which resulted in an equivocation of $I(X;V)$\nin Theorem $1$. We will show in the next section through an example that the uncertainty about\nthe source at Eve for the case of two-sided helper can be strictly\nlarger than the case of one-sided helper and selecting $U$ as a\nconstant is clearly suboptimal.\n\nBesides reflecting the fact that the uncertainty at Eve\ncan be strictly larger than the case of a one-sided helper, Theorem $2$\nhas another interesting interpretation. If Alice and Helen can use\nsufficiently large rates to securely transmit the source $X^{n}$ to Bob, then the helper\ncan simply transmit a secret key of entropy $H(X)$ to both Alice\nand Bob. Alice can then use this secret key to\nlosslessly transmit the source to Bob in perfect secrecy by using a\none-time pad \\cite{Shannon:1949}. In other\nwords, when $R_{x}$ and $R_{y}$ are larger than $H(X)$, one can\nimmediately obtain this result from Theorem $2$ by selecting $V$\nto be independent of $(X,Y)$ and uniformly distributed on\n$\\{1,\\ldots,|\\mathcal{X}|\\}$. Finally, selecting $U=X\\oplus V$, we\nobserve that $\\min{(R_{y},I(X;V|U))}=H(X)$, yielding perfect\nsecrecy. We note that, here, $V$ plays the role of a secret key.\n\nNow consider the model where the side information $Y^{n}$ is\nof the form $Y^{n}=X^{n}\\oplus B^{n}$, where\n$|\\mathcal{B}|=|\\mathcal{X}|$, and $B^{n}$ is independent of $X^{n}$.\nMoreover, assume that the side information $Y^{n}$ is available to both Alice and\nBob in an uncoded manner. For this model, it follows from \\cite{Grokop:ISIT05}\nthat, to maximize the uncertainty at the eavesdropper, Alice cannot\ndo any better than describing the error sequence $B^{n}$ to Bob. \nNote that our two-sided helper model differs from this model in two\naspects: first, in our case, the common side information available to\nAlice and Bob is coded and rate-limited, secondly, the sources in our\nmodel do not have to be in modulo-additive form.\n\n\nOur encoding scheme at Alice for the case of two-sided helper \ncomprises of two steps: (a) using the\ncoded side information $V$ and the source $X$, Alice creates $U$ and\ntransmits the bin index of $U$ at a rate $I(X;U|V)$ so that Bob can estimate $U$ using\n$V$, and, (b) Alice bins the source $X$ at a rate $H(X|U,V)$ and transmits\nthe bin index of the source $X$. Note that if for a pair of sources $(X,Y)$, \nthe optimal $V$ is of the form $V=X\\oplus B$, where $|\\mathcal{B}|=|\\mathcal{X}|$ and $B$\nis independent of $X$, then it suffices to choose $U=B$, so that \n$I(X;V|U)=H(X)$ and $H(X|U,V)=0$. In other words, for such sources, step (b) in\nour achievability scheme is not necessary, which is similar to the\nachievability for the case of modulo-additive sources in \\cite{Grokop:ISIT05}.\nOn the other hand, for a general pair of sources $(X,Y)$, the optimal\n$V$ may not be of the form $V=X\\oplus B$ and the optimal $U$ may not\nalways satisfy $H(X|U,V)=0$. Therefore, we need the step (b) for\nour achievability scheme. This differentiates our achievable scheme\nfrom that of \\cite{Grokop:ISIT05}, which holds for modulo-additive\nsources with uncoded side information.\n\nAlso note that if $Y^{n}$ is not of the form $X^{n}\\oplus B^{n}$, and if $R_{y}\n\\geq H(X)$, then Helen can transmit a secret key which will enable\nperfectly secure transmission of $X^{n}$ by using a one-time pad \\cite{Shannon:1949}. \nThis phenomenon does not always occur when the side information $Y^{n}$ \nis available to both Alice and Bob in an uncoded fashion  \\cite{Grokop:ISIT05}.\n\n\\section{An Example: Binary Symmetric Sources}\nBefore proceeding to further generalizations of Theorems $1$ and $2$, we\nexplicitly evaluate Theorems $1$ and $2$ for a pair of binary sources.\n\nLet $X$ and $Y$ be binary sources with $X\\sim \\mbox{Ber}(1\/2), Y\\sim \\mbox{Ber}(1\/2)$ and\n$X=Y\\oplus E$, where $E\\sim \\mbox{Ber}(\\delta)$. For this pair of\nsources, the region described in Theorem $1$ can be completely characterized as,\n\\begin{align}\n\\mathcal{R}_{1-sided}(R_{y})=\\big\\{(R_{x},\\Delta):\\hspace{0.05in}R_{x}&\\geq h(\\delta*h^{-1}(1-R_{y}))\\nonumber\\\\\n\\Delta&\\leq 1-h(\\delta*h^{-1}(1-R_{y}))\\big\\}\\label{regn1}\n\\end{align}\nand the region in Theorem $2$ can be completely characterized as,\n\\begin{align}\n\\mathcal{R}_{2-sided}(R_{y})=\\big\\{(R_{x},\\Delta):\\hspace{0.05in}R_{x}&\\geq h(\\delta*h^{-1}(1-R_{y}))\\nonumber\\\\\n\\Delta&\\leq \\min(R_{y},1)\\big\\}\n\\end{align}\nwhere $h(.)$ is the binary entropy function, and $a*b=a(1-b)+b(1-a)$.\n\nWe start with the derivation of (\\ref{regn1}). Without loss of\ngenerality, we assume that $R_{y}\\leq H(Y)$. Achievability follows by\nselecting $V=Y\\oplus N$, where $N\\sim \\mbox{Ber}(\\alpha)$, where\n\\begin{align}\n\\alpha&=h^{-1}(1-R_{y})\n\\end{align}\nSubstituting, we obtain\n\\begin{align}\nH(X|V)&= h(\\delta*h^{-1}(1-R_{y}))\\\\\nI(X;V)&= 1-h(\\delta*h^{-1}(1-R_{y}))\n\\end{align}\nwhich completes the achievability. Converse follows by simple application of Mrs. Gerber's lemma \\cite{Gerber:1973} as\nfollows. Let us be given $R_{y}\\in (0,1)$. We have\n\\begin{align}\nR_{y}&\\geq I(Y;V)\\\\\n&= H(Y)-H(Y|V)\\\\\n&= 1 - H(Y|V)\n\\end{align}\nwhich implies $H(Y|V)\\geq 1-R_{y}$. Mrs. Gerber's lemma states that for\n$X=Y\\oplus E$, with $E\\sim \\mbox{Ber}(\\delta)$, if $H(Y|V)\\geq \\beta$, then $H(X|V)\\geq\nh(\\delta*h^{-1}(\\beta))$. We therefore have,\n\\begin{align}\nR_{x}&\\geq H(X|V)\\\\\n&\\geq h(\\delta*h^{-1}(1-R_{y}))\n\\end{align}\nand\n\\begin{align}\n\\Delta&\\leq I(X;V)\\\\\n&= H(X)-H(X|V)\\\\\n&= 1- H(X|V)\\\\\n&\\leq 1- h(\\delta*h^{-1}(1-R_{y}))\n\\end{align}\nThis completes the converse.\n\nFor the case of two-sided helper, we compute the equivocation $\\Delta$ as\nfollows. We choose $V$ as $V=Y \\oplus N$ where $N\\sim \\mbox{Ber}(\\alpha)$ as\nin the case of one-sided helper. We choose $U$ as,\n\\begin{align}\nU&=X\\oplus V\n\\end{align}\nWe next compute the term $I(X;V|U)$ as follows,\n\\begin{align}\nI(X;V|U)&=I(X;V|X \\oplus V)\\\\\n&=H(X,X\\oplus V)-H(X\\oplus V)\\\\\n&=H(X)+H(V|X)-H(X\\oplus V)\\\\\n&=H(X)+H(Y\\oplus N|X)-H(X\\oplus Y \\oplus N)\\\\\n&=H(X)+H(X\\oplus E \\oplus N|X)-H(X\\oplus X\\oplus E \\oplus N)\\\\\n&=H(X)\\\\\n&=1\n\\end{align}\nand therefore\n\\begin{align}\n\\min(R_{y},I(X;V|U))&=\\min(R_{y},1)\n\\end{align}\nFor the converse part, we also have that\n\\begin{align}\n\\Delta&\\leq \\min(R_{y},I(X;V|U))\\\\\n&\\leq \\min(R_{y},H(X))\\\\\n&= \\min(R_{y},1)\n\\end{align}\n\nThe rate from Alice, $R_{x}$ and\nthe equivocation $\\Delta$ for the cases of one-sided and two-sided helper are shown in\nFigure $4$ for the case when $\\delta=0.05$. For the one-sided helper, we can observe a trade-off\nin the amount of information Alice needs to send versus the\nuncertainty at Eve. For small values of $R_{y}$, Alice needs\nto send more information thereby leaking out more\ninformation to Eve. The amount of information\nleaked is exactly the information sent by Alice.\nOn the other hand, for the case of two-sided helper, the\nuncertainty at the eavesdropper is always strictly larger than the\nuncertainty in the one-sided case. Also note that for this pair of\nsources, perfect secrecy is possible for the case of two-sided\nhelper when $R_{y}\\geq H(Y)$ which is not possible for the case of\none-sided helper.\n\n\\begin{figure}[t]\n  \\centerline{\\epsfig{figure=comparison.eps,width=11cm}}\\vspace{-0.1in}\n   \\caption{The rate-equivocation region for a pair of binary\n     symmetric sources.}\\label{fig3}\n\\end{figure}\n\n\\section{Secure and Insecure Links from Two-Sided Helper }\n\\subsection{System model}\nIn this section, we consider a generalization of the model considered\nin Section $5$. We consider the case when there are two links from\nHelen (see Figure $3$). One link of rate $R_{sec}$ is secure, i.e., the output of this link is\navailable to only Alice and Bob. The second link of rate $R_{ins}$ is public\nand the output of this link is available to Alice, Bob and\nEve. The sources $(X^{n},Y^{n},W^{n},Z^{n})$ are generated\ni.i.d. according to $p(x,y,w,z)=p(x,y)p(w,z|x)$ where $p(x,y,w,z)$ is defined over the finite\nproduct alphabet $\\mathcal{X}\\times \\mathcal{Y}  \\times \\mathcal{W}\n\\times \\mathcal{Z}$.\n\nA $(n, 2^{nR_{x}}, 2^{nR_{sec}}, 2^{nR_{ins}})$ code for this model consists of an\nencoding function at Helen, $f_{y}:Y^{n}\\rightarrow\nJ_{sec}\\times J_{ins}$, where $|J_{sec}|\\leq 2^{nR_{sec}}$,\n$|J_{ins}|\\leq 2^{nR_{ins}}$, an\nencoding function at Alice, $f_{x}:X^{n}\\times\nJ_{sec}\\times J_{ins}\\rightarrow \\{1,\\ldots,2^{nR_{x}}\\}$,\nand a decoding function at Bob, $g: \\{1,\\ldots,2^{nR_{x}}\\} \\times\nJ_{sec}\\times J_{ins} \\times W^{n} \\rightarrow X^{n}$.\nThe uncertainty about the source $X^{n}$ at Eve is measured by\n$H(X^{n}|f_{x}(X^{n},J_{sec}, J_{ins}),J_{ins},Z^{n})\/n$. The\nprobability of error in the reconstruction of\n$X^{n}$ at Bob is defined as\n$P_{e}^{n}=\\mbox{Pr}(g(f_{x}(X^{n},J_{sec}, J_{ins}),J_{sec}, J_{ins},W^{n})\\neq X^{n})$.\nA quadruple $(R_{x}, R_{sec}, R_{ins}, \\Delta)$ is achievable if for any $\\epsilon>0$,\nthere exists a $(n, 2^{nR_{x}}, 2^{nR_{sec}}, 2^{nR_{ins}})$ code such that $P_{e}^{n}\\leq\n\\epsilon$ and $H(X^{n}|f_{x}(X^{n},J_{sec},J_{ins}),J_{ins}, Z^{n})\/n\n\\geq \\Delta$. We denote the set of all achievable\n$(R_{x},R_{sec},R_{ins},\\Delta)$ rate quadruples as $\\mathcal{R}_{sec-ins}^{W,Z}$.\n\\subsection{ Result}\nThe main result is given in the following theorem.\n\\begin{Theo}\\label{Theorem3} The set of achievable rate quadruples\n  $\\mathcal{R}_{sec-ins}^{W, Z}$ for\nsecure source coding with secure and insecure links from a two-sided helper, additional side\ninformation $W$ at Bob and $Z$ at Eve is given as\n\\begin{align}\n\\mathcal{R}_{sec-ins}^{W,Z}=\\Big\\{(R_{x},R_{sec}, R_{ins}, \\Delta): R_{x}&\\geq H(X|V_{1},V_{2},W)\\\\\nR_{sec}&\\geq I(Y;V_{1}|W)\\\\\nR_{ins}&\\geq I(Y;V_{2}|W,V_{1})\\\\\n\\Delta&\\leq \\min(R_{sec},I(X;V_{1}|U,V_{2},W))\\nonumber\\\\\n&\\hspace{0.18in}+I(X;W|U,V_{2})-I(X;Z|U,V_{2})\\Big\\}\n\\end{align}\nwhere the joint distribution of the involved random variables is as\nfollows,\n\\begin{align}\np(x,y,w,z,v_{1},v_{2},u)&=p(x,y)p(w,z|x)p(v_{1},v_{2}|y)p(u|x,v_{1},v_{2})\n\\end{align}\nand it suffices to consider such distributions for which\n$|\\mathcal{V}_{1}|\\leq |\\mathcal{Y}|+3$, $|\\mathcal{V}_{2}|\\leq\n|\\mathcal{Y}|+4$  and $|\\mathcal{U}|\\leq |\\mathcal{X}||\\mathcal{Y}|^{2}+7|\\mathcal{X}||\\mathcal{Y}|+12|\\mathcal{X}|+2$.\n\\end{Theo}\nThe proof of Theorem $3$ is given in the Appendix.\n\nThe achievability scheme which yields the rate region described in\nTheorem $3$ is summarized as follows:\n\\begin{enumerate}\n\\item Helen first uses the secure\nlink to describe the source $Y$ to both Alice and Bob at a rate\n$I(Y;V_{1}|W)$, where $W$ plays the role of side\ninformation. Subsequently, the insecure link is used to provide\nanother description of the correlated source $Y$ at a rate\n$I(Y;V_{2}|W,V_{1})$, where $(W,V_{1})$ plays the role of side\ninformation. Therefore, the key idea is to first use the secure link\nto build common randomness at Alice and Bob and subsequently use this\ncommon randomness to send information  over the insecure link at a lower rate.\n\\vspace{-0.1in}\n\\item Having access to the coded outputs\n$(V_{1},V_{2})$ from Helen and\nthe source $X$, Alice jointly quantizes $(X,V_{1},V_{2})$ to an auxiliary random\nvariable $U$. She subsequently performs Wyner-Ziv coding, i.e., bins\nthe $U$ sequences at the rate $I(X;U|V_{1},V_{2},W)$ such that Bob can decode $U$ by using\n$W$ and the coded outputs $(V_{1},V_{2})$ from the helper.\n\\item She also bins the source $X$\nat a rate $H(X|U,V_{1},V_{2},W)$ so that having access to $(U,V_{1},V_{2},W)$, Bob can\ncorrectly decode the source $X$. The total rate used by\nAlice is $I(X;U|V_{1},V_{2},W)+H(X|U,V_{1},V_{2},W)=H(X|V_{1},V_{2},W)$.\n\\end{enumerate}\n\nWe note here that using Theorem $3$, we can recover Theorem $2$ by\nsetting $R_{ins}=0$, and selecting $W=Z=V_{2}=\\phi$.\n\nOn setting $R_{sec}=0$, the  resulting model is similar to the one\nconsidered in \\cite{CsiszarNarayan} although the aim in\n\\cite{CsiszarNarayan} is to generate a secret key to be shared by\nAlice and Bob, while we are interested in the secure\ntransmission of the source $X$.\n\nIf $R_{sec}=R_{ins}=0$, then we recover the result of\n\\cite{VinodKannan:ITW07} as a special case by setting\n$V_{1}=V_{2}=\\phi$.\nTherefore, Theorem $3$ can also be viewed as a\ngeneralization of the result of \\cite{VinodKannan:ITW07} where no rate\nconstraint is imposed on the transmission of Alice and the goal is\nto maximize the uncertainty at Eve while losslessly transmitting the\nsource to Bob.\n\n\n\\section{Conclusions}\nIn this paper, we considered several secure source coding problems. We\nfirst provided the characterization of the rate-equivocation region\nfor a secure source coding problem with coded side information at\nthe legitimate user. We next generalized this result for two different\n models with increasing complexity. We characterized the\n rate-equivocation region for the case of two-sided helper. \nThe value of two-sided coded side information is emphasized by\ncomparing the respective equivocations for a pair of binary sources.\nIt is shown for this example that Slepian-Wolf type coding alone is\ninsufficient and using our achievable scheme, one attains strictly\nlarger uncertainty at the eavesdropper than the case of one-sided helper.\nWe next considered the case when there are both secure and insecure\nrate-limited links from the helper and characterized the\nrate-equivocation region.\n\n\n\n\\section{Appendix}\n\\subsection{Proof of Theorem \\ref{Theorem1}}\n\n\\subsubsection{Achievability}\nFix the distribution $p(x,y,v)=p(x,y)p(v|y)$.\n\\begin{enumerate}\n\\item Codebook generation at Helen:\nFrom the conditional probability distribution $p(v|y)$ compute $p(v)=\\sum_{y}p(y)p(v|y)$.\nGenerate $2^{nI(V;Y)}$ codewords $v(l)$ independently\naccording to $\\prod_{i=1}^{n}p(v_{i})$, where $l=1,\\ldots,2^{nI(V;Y)}$.\n\\item Codebook generation at Alice:\nRandomly bin the $x^{n}$ sequences into\n$2^{nH(X|V)}$ bins and index these bins as $m=1,\\ldots,M$, where $M=2^{nH(X|V)}$.\n\\item Encoding at Helen:\nOn observing the sequence $y^{n}$, Helen tries to find a\nsequence $v(l)$ such that $(v(l),y^{n})$ are jointly\ntypical. From rate-distortion theory, we know that there exists one\nsuch sequence as long as $R_{y}\\geq I(V;Y)$. Helen sends the index $l$ of\nthe sequence $v(l)$.\n\\item Encoding at Alice:\nOn observing the sequence $x^{n}$, Alice finds the bin index $m_{X}$\nin which the sequence $x^{n}$ falls and transmits the bin index $m_{X}$.\n\\item Decoding at Bob:\nOn receiving $l$ and the bin index $m_{X}$, Bob tries to find\na unique $x^{n}$ sequence in bin $m_{X}$ such that $(v(l),x^{n})$\nare jointly typical. This is possible since the number of $x^{n}$\nsequences in each bin is roughly $2^{nH(X)}\/2^{nH(X|V)}$ which is\n$2^{nI(X;V)}$ and the existence of a unique $x^{n}$ such that $(v(l),x^{n})$\nare jointly typical is guaranteed by the Markov lemma \\cite{Cover:book}.\n\\item Equivocation:\n\\begin{align}\nH(X^{n}|m_{X})&=H(X^{n},m_{X})-H(m_{X})\\\\\n&=H(X^{n})+H(m_{X}|X^{n})-H(m_{X})\\\\\n&= H(X^{n})-H(m_{X})\\\\\n&\\geq H(X^{n})-\\mbox{log}(M)\\\\\n&=H(X^{n})-nH(X|V)\\\\\n&= n I(X;V)\n\\end{align}\nTherefore,\n\\begin{align}\n\\Delta&\\leq I(X;V)\n\\end{align}\nis achievable. This completes the achievability part.\n\\end{enumerate}\n\n\\subsubsection{Converse}\nLet the output of the helper be $J_{y}$, and the\noutput of Alice be $J_{x}$, i.e.,\n\\begin{align}\nJ_{y}&=f_{y}(Y^{n})\\\\\nJ_{x}&=f_{x}(X^{n})\n\\end{align}\nFirst note that, for noiseless reconstruction of the sequence $X^{n}$ at the legitimate decoder, we have by Fano's inequality\n\\begin{align}\nH(X^{n}|J_{x},J_{y})&\\leq n\\epsilon_{n}\\label{fano}\n\\end{align}\n\nWe start by obtaining a lower bound on $R_{x}$, the rate of Alice, as follows\n\\begin{align}\nnR_{x}&\\geq H(J_{x})\\\\\n&\\geq H(J_{x}|J_{y})\\\\\n&= H(X^{n},J_{x}|J_{y})-H(X^{n}|J_{x},J_{y})\\\\\n&\\geq H(X^{n},J_{x}|J_{y})-n\\epsilon_{n}\\label{e1}\\\\\n&\\geq H(X^{n}|J_{y})-n\\epsilon_{n}\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|X^{i-1},J_{y})-n\\epsilon_{n}\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|V_{i})-n\\epsilon_{n}\\label{e2}\\\\\n&= nH(X_{Q}|V_{Q},Q)-n\\epsilon_{n}\\\\\n&=nH(X|V)-n\\epsilon_{n}\\label{etemp2}\n\\end{align}\nwhere (\\ref{e1}) follows by (\\ref{fano}). In (\\ref{e2}), we have defined\n\\begin{align}\nV_{i}&=(J_{y},X^{i-1})\n\\end{align}\nIn (\\ref{etemp2}), we have defined,\n\\begin{align}\nX=X_{Q}, \\quad V=(Q,V_{Q})\n\\end{align}\nwhere $Q$ is uniformly distributed on $\\{1,\\ldots,n\\}$ and is independent of all\nother random variables.\n\nNext, we obtain a lower bound on $R_{y}$, the rate of the helper,\n\\begin{align}\nnR_{y}&\\geq H(J_{y})\\\\\n&\\geq I(J_{y};Y^{n})\\\\\n&= \\sum_{i=1}^{n}I(J_{y},Y^{i-1};Y_{i})\\\\\n&= \\sum_{i=1}^{n}I(J_{y},Y^{i-1},X^{i-1};Y_{i})\\label{e3}\\\\\n&\\geq \\sum_{i=1}^{n}I(J_{y},X^{i-1};Y_{i})\\\\\n&= \\sum_{i=1}^{n}I(V_{i};Y_{i})\n\\end{align}\n\\begin{align}\n&= nI(V_{Q};Y_{Q}|Q)\\\\\n&=nI(V;Y)\\label{etemp3}\n\\end{align}\nwhere (\\ref{e3}) follows from the Markov chain\n\\begin{align}\nX^{i-1}\\rightarrow (J_{y},Y^{i-1}) \\rightarrow Y_{i}\n\\end{align}\nand in (\\ref{etemp3}), we have defined $Y=Y_{Q}$.\n\nWe now have the main step, i.e., an upper bound on the equivocation\nrate of the eavesdropper,\n\\begin{align}\nH(X^{n}|J_{x})&=\nH(X^{n},J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})\\\\\n&=\nH(J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})+H(X^{n}|J_{x},J_{y})\\\\\n&=\nH(J_{y}|J_{x})-H(J_{y}|X^{n})+H(X^{n}|J_{x},J_{y})\\label{e4}\\\\\n&\\leq\nH(J_{y})-H(J_{y}|X^{n})+H(X^{n}|J_{x},J_{y})\\\\\n&\\leq I(J_{y};X^{n}) + n\\epsilon_{n}\\label{e5}\\\\\n&= \\sum_{i=1}^{n}I(J_{y};X_{i}|X^{i-1}) + n\\epsilon_{n}\\\\\n&= \\sum_{i=1}^{n}I(J_{y},X^{i-1};X_{i}) +\nn\\epsilon_{n}\\\\\n&=\\sum_{i=1}^{n}I(X_{i};V_{i})+\nn\\epsilon_{n}\\\\\n&= nI(X_{Q};V_{Q}|Q)+n\\epsilon_{n}\\\\\n&= nI(X;V)+n\\epsilon_{n}\n\\end{align}\nwhere (\\ref{e4}) follows from the Markov chain\n\\begin{align}\nJ_{x}\\rightarrow X^{n} \\rightarrow J_{y}\n\\end{align}\nand (\\ref{e5}) follows from (\\ref{fano}). This implies\n\\begin{align}\n\\Delta&\\leq I(X;V)\n\\end{align}\n\nAlso note that the following is a Markov chain,\n\\begin{align}\nV\\rightarrow Y \\rightarrow X\n\\end{align}\nTherefore, the joint distribution of the involved random variables is\n\\begin{align}\np(x,y,v)&=p(x,y)p(v|y)\n\\end{align}\nFrom support lemma \\cite{Csiszar:book}, it can be shown that it\nsuffices to consider such joint distributions for which\n$|\\mathcal{V}|\\leq |\\mathcal{Y}|+2$.\n\n\n\\subsection{Proof of Theorem \\ref{Theorem2}}\n\\subsubsection{Achievability}\nFix the distribution $p(x,y,v)=p(x,y)p(v|y)p(u|x,v)$.\n\\begin{enumerate}\n\\item Codebook generation at Helen:\nFrom the conditional probability distribution $p(v|y)$ compute $p(v)=\\sum_{y}p(y)p(v|y)$.\nGenerate $2^{nI(V;Y)}$ codewords $v(l)$ independently\naccording to $\\prod_{i=1}^{n}p(v_{i})$, where $l=1,\\ldots,2^{nI(V;Y)}$.\n\\item Codebook generation at Alice:\nFrom the distribution $p(u|x,v)$, compute $p(u)$.\nGenerate $2^{nI(X,V;U)}$ sequences $u(s)$ independently\naccording to $\\prod_{i=1}^{n}p(u_{i})$, where\n$s=1,\\ldots,2^{nI(X,V;U)}$. Next, bin these sequences uniformly into\n$2^{nI(X;U|V)}$ bins.\n\nAlso, randomly bin the $x^{n}$ sequences into\n$2^{nH(X|U,V)}$ bins and index these bins as $m=1,\\ldots,2^{nH(X|U,V)}$.\n\\item Encoding at Helen:\nOn observing the sequence $y^{n}$, Helen tries to find a\nsequence $v(l)$ such that $(v(l),y^{n})$ are jointly\ntypical. From rate-distortion theory, we know that there exists one\nsuch sequence. Helen sends the index $l$ of\nthe sequence $v(l)$.\n\\item Encoding at Alice:\nThe key difference from the one-sided helper case is in the encoding at\nAlice. On observing the sequence $x^{n}$, Alice first finds the bin index $m_{X}$\nin which the sequence $x^{n}$ falls. Alice also has the sequence $v(l)$\nreceived from Helen. Alice next finds a sequence $u$ such\nthat $(u,v(l),x^{n})$ are jointly typical. Let the\nbin index of this resulting $u$ sequence be $s_{U}$.\n\nAlice transmits the pair $(s_{U},m_{X})$ which is received by Bob and\nEve. The total rate used by Alice is $ I(X;U|V)+H(X|U,V)=H(X|V)$.\n\n\\item Decoding at Bob: On receiving the pair $(s_{U},m_{X})$ from\nAlice and the index $l$ from Helen, Bob first searches the\nbin $s_{U}$ for a sequence $\\hat{u}$ such that $(\\hat{u},v(l))$\nare jointly typical. This is possible since the number of $u$\nsequences in each auxiliary bin is approximately\n$2^{nI(X,V;U)}\/2^{nI(X;U|V)}$ which is $2^{nI(U;V)}$ and therefore\nwith high probability, Bob will be able to obtain the correct $u$\nsequence.\n\nUsing the estimate $\\hat{u}$ and $v(l)$, Bob\nsearches for a unique $x^{n}$ sequence in the\nbin $m_{X}$  such that $(\\hat{u},v(l),x^{n})$ are jointly\ntypical. This is possible since the number of $x^{n}$\nsequences in each bin is approximately $2^{nH(X)}\/2^{nH(X|U,V)}$ which is\n$2^{nI(U,V;X)}$.\n\\item Equivocation:\n\\begin{align}\nH(X^{n}|s_{U},m_{X})&=H(X^{n},m_{X},s_{U})-H(m_{X},s_{U})\\\\\n&=H(X^{n})+H(m_{X},s_{U}|X^{n})-H(m_{X},s_{U})\\\\\n&=H(X^{n})+H(s_{U}|X^{n})-H(m_{X},s_{U})\\label{equi1}\\\\\n&\\geq H(X^{n})+H(s_{U}|X^{n})-H(s_{U})-H(m_{X})\\label{equi2}\\\\\n&= H(X^{n})-H(m_{X})-I(s_{U};X^{n})\\\\\n&\\geq H(X^{n})-nH(X|U,V)-I(s_{U};X^{n})\\label{equi3}\\\\\n&= nI(X;U,V)-I(s_{U};X^{n})\\\\\n&\\geq nI(X;U,V)-I(U^{n};X^{n})\\label{equi4}\\\\\n&\\geq nI(X;U,V)-nI(U;X)\\label{equi5}\\\\\n&= nI(X;V|U) \\\\\n&\\geq n\\min(I(X;V|U),R_{y}) \\label{equi6}\n\\end{align}\nwhere (\\ref{equi1}) follows from the fact that $m_{X}$ is the bin\nindex of the sequence $X^{n}$, (\\ref{equi2}) follows from the fact\nthat conditioning reduces entropy, (\\ref{equi3}) follows from the fact\nthat $H(m_{X})\\leq \\mbox{log}(2^{nH(X|U,V)})$, (\\ref{equi4}) follows\nfrom the fact that $s_{U}$ is the bin index of the sequence $U^{n}$,\ni.e., $s_{U}\\rightarrow U^{n}\\rightarrow X^{n}$ forms a Markov chain\nand subsequently using the data processing inequality. Finally,\n(\\ref{equi6}) follows from the fact that $\\min(I(X;V|U),R_{y})\\leq I(X;V|U)$.\nWe therefore have\n\\begin{align}\n\\Delta&\\leq \\min(I(X;V|U),R_{y})\n\\end{align}\nis achievable. This completes the achievability part for the case of two-sided helper.\n\\end{enumerate}\n\n\\subsubsection{Converse}\nThe only difference in the converse part for the case of two-sided helper\nis when deriving an upper bound on the equivocation\nrate of the eavesdropper:\n\\begin{align}\nH(X^{n}|J_{x})&=\nH(X^{n},J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})\\\\\n&=\nH(J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})+H(X^{n}|J_{x},J_{y})\\\\\n&\\leq\nI(X^{n};J_{y}|J_{x})+n\\epsilon_{n}\\\\\n&=\n\\sum_{i=1}^{n}I(X_{i};J_{y}|J_{x},X^{i-1})+n\\epsilon_{n}\\\\\n&=\n\\sum_{i=1}^{n}I(X_{i};J_{y},X^{i-1}|J_{x},X^{i-1})+n\\epsilon_{n}\\\\\n&=n I(X;V|U) + n \\epsilon_{n}\n\\end{align}\nwhere we have defined\n\\begin{align}\nV_{i}&=(J_{y},X^{i-1})\\label{defV}\\\\\nU_{i}&=(J_{x},X^{i-1})\\label{defU}\n\\end{align}\nand\n\\begin{align}\nX=X_{Q},\\quad Y=Y_{Q}, \\quad V=(Q,V_{Q}), \\quad U=(Q,U_{Q})\n\\end{align}\nWe also have,\n\\begin{align}\nH(X^{n}|J_{x})&=\nH(X^{n},J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})\\\\\n&=\nH(J_{y}|J_{x})-H(J_{y}|X^{n},J_{x})+H(X^{n}|J_{x},J_{y})\\\\\n&\\leq\nH(J_{y}|J_{x})+n\\epsilon_{n}\\\\\n&\\leq\nH(J_{y})+n\\epsilon_{n}\\\\\n&\\leq n R_{y} + n \\epsilon_{n}\n\\end{align}\nThis implies\n\\begin{align}\n\\Delta&\\leq \\min(I(X;V|U),R_{y})\n\\end{align}\n\nThe joint distribution of the involved random variables is as\nfollows,\n\\begin{align}\np^{out}(x,y,v,u)&=p(x,y)p(v|y)p^{out}(u|x,v,y)\\label{outerr}\n\\end{align}\nNote that in the achievability proof of Theorem $2$, joint distributions of the\nfollowing form are permitted,\n\\begin{align}\np^{ach}(x,y,v,u)&=p(x,y)p(v|y)p^{ach}(u|x,v)\\label{innerr}\n\\end{align}\ni.e, we have the Markov chain, $Y\\rightarrow (X,V) \\rightarrow U$.\nWith the definition of $V$ and $U$ as in (\\ref{defV})-(\\ref{defU}), these random variables do not satisfy this\nMarkov chain. This implies that what we have shown so far is the following,\n\\begin{align}\n\\mathcal{R}_{2-sided}\\subseteq \\mathcal{R}_{out}\n\\end{align}\nwhere\n\\begin{align}\n\\mathcal{R}_{out}=\n\\Big\\{(R_{x},R_{y},\\Delta): R_{x}&\\geq H(X|V)\\\\\n\\vspace{-0.1in}R_{y}&\\geq I(Y;V)\\\\\n\\Delta&\\leq \\min(I(X;V|U),R_{y})\\Big\\}\n\\end{align}\nwhere the joint distribution of the involved random variables is as\ngiven in (\\ref{outerr}).\n\nHowever, we observe that the term $I(X;V|U)$ depends only on the\nmarginal $p^{out}(u|x,v)$. Similarly, the terms $H(X|V)$ and $I(X;V)$\ndepend only on the marginal $p(x,v)$. We use these observations\nto show that the region $\\mathcal{R}_{out}$\nis the same when it is evaluated using the joint distributions of the\nform given in (\\ref{innerr}). This is clear by noting that once we are given a distribution of the\nform given in (\\ref{outerr}), we can construct a new distribution of the\nform given in (\\ref{innerr}) with the same rate expressions.\nConsider any distribution $p^{out}(x,y,v,u)$ of the form given in (\\ref{outerr}).\nUsing $p^{out}(x,y,v,u)$, compute the marginal $p^{out}(u|x,v)$ as,\n\\begin{align}\np^{out}(u|x,v)&=\\frac{\\sum_{y}p^{out}(x,y,u,v)}{p(x,v)}\n\\end{align}\nWe now construct a distribution $p^{ach}(x,y,v,u)\\in \\mathcal{P}_{ach}$ as,\n\\begin{align}\np^{ach}(x,y,v,u)&=p(x,y)p(v|y)p^{out}(u|x,v)\n\\end{align}\nsuch that the terms $I(X;V|U)$, $H(X|V)$ and $I(X;V)$ are the same whether they\nare evaluated according to $p^{out}(x,y,v,u)$ or according to $p^{ach}(x,y,v,u)$.\nTherefore, we conclude that it suffices to consider input\ndistributions satisfying the Markov chain $Y\\rightarrow\n(X,V)\\rightarrow U$ when evaluating $\\mathcal{R}_{out}$ and hence $\\mathcal{R}_{out}=\\mathcal{R}_{ach}$.\nThis completes the converse part. We remark here that this observation\nwas useful in obtaining the converse for the rate-distortion function with common coded\nside information \\cite{KaspiBerger:1982},\\cite{Vasudevan:ISIT07},\\cite{Helper:ITW09}.\n\n\n\\subsection{Proof of Theorem \\ref{Theorem3}}\n\\subsubsection{Achievability}\nFix the distribution $p(x,y,w,z,v_{1},v_{2},u)=p(x,y)p(w,z|x)p(v_{1},v_{2}|y)p(u|x,v_{1},v_{2})$.\n\\begin{enumerate}\n\\item Codebook generation at Helen: From the conditional\nprobability distribution $p(v_{1},v_{2}|y)$ compute\n$p(v_{1})=\\sum_{(y,v_{2})}p(y)p(v_{1},v_{2}|y)$ and\n$p(v_{2})=\\sum_{(y,v_{1})}p(y)p(v_{1},v_{2}|y)$. Generate\n$2^{nI(V_{1};Y)}$ sequences $v_{1}(l)$ independently according to\n$\\prod_{i=1}^{n}p(v_{1i})$, where $l=1,\\ldots,2^{nI(V_{1};Y)}$.\nBin these sequences uniformly and independently in\n$2^{n(I(V_{1};Y)-I(V_{1};W))}$ bins. Denote the bin index of the\nsequence $v_{1}(l)$ as $b_{V_{1}}(v_{1}(l))$.\n\nNext generate $2^{nI(V_{2};Y,V_{1})}$ sequences $v_{2}(j)$\nindependently according to $\\prod_{i=1}^{n}p(v_{2i})$, where\n$j=1,\\ldots,2^{nI(V_{2};Y,V_{1})}$. Bin these sequences uniformly\nand independently in \\break\n$2^{n(I(V_{2};Y,V_{1})-I(V_{2};W,V_{1}))}$ bins. Denote the bin\nindex of the sequence $v_{2}(j)$ as $b_{V_{2}}(v_{2}(j))$.\n\n\\item Codebook generation at Alice:\nFrom the distribution $p(u|x,v_{1},v_{2})$, compute $p(u)$.\nGenerate $2^{nI(X,V_{1},V_{2};U)}$ sequences $u(s)$ independently\naccording to $\\prod_{i=1}^{n}p(u_{i})$, where\n$s=1,\\ldots,2^{nI(X,V_{1},V_{2};U)}$. Next, bin these sequences uniformly into\n$2^{n(I(X,V_{1},V_{2};U)-I(W,V_{1},V_{2};U))}$ bins.\n\nAlso, randomly bin the $x^{n}$ sequences into\n$2^{nH(X|U,V_{1},V_{2},W)}$ bins and index these bins as\n$m=1,\\ldots,2^{nH(X|U,V_{1},V_{2},W)}$.\n\n\\item Encoding at Helen:\nOn observing the sequence $y^{n}$, Helen tries to find a\nsequence $v_{1}(l)$ such that $(v_{1}(l),y^{n})$ are jointly\ntypical. From rate-distortion theory, we know that there exists one\nsuch sequence. Helen sends the bin index $b_{V_{1}}(v_{1}(l))$ of\nthe sequence $v_{1}(l)$ on the secure link which is received by Alice and Bob.\n\nHelen also finds a sequence $v_{2}(j)$ such that $(v_{1}(l),v_{2}(j),y^{n})$ are jointly\ntypical. From rate-distortion theory, we know that there exists one\nsuch sequence. Helen sends the bin index $b_{V_{2}}(v_{2}(j))$ of\nthe sequence $v_{2}(j)$ on the insecure link which is received by Alice, Bob and Eve.\n\n\\item Encoding at Alice:\nOn observing the sequence $x^{n}$, Alice first finds the bin index $m_{X}$\nin which the sequence $x^{n}$ falls. Alice also receives the bin indices\n$b_{V_{1}}(v_{1}(l))$ and $b_{V_{2}}(v_{2}(j))$ from Helen. She first looks for a unique $\\hat{v}_{1}(l)$\nin the bin $b_{V_{1}}(v_{1}(l))$ such that $(x^{n}, \\hat{v}_{1}(l))$\nare jointly typical. Alice can estimate the correct sequence\n$v_{1}(l)$ as long as the number of sequences in each bin is\nless than $2^{nI(X;V_{1})}$. Note from the codebook generation step at\nHelen, that the number of\n$v_{1}$ sequences in each bin is approximately\n$2^{nI(V_{1};Y)}\/2^{n(I(V_{1};Y)-I(V_{1};W))}=2^{nI(V_{1};W)}$. Since $V_{1}\\rightarrow X\n\\rightarrow W$ forms a Markov chain, we have $I(V_{1};W)\\leq I(V_{1};X)$ and therefore the number of\n$v_{1}$ sequences in each bin is less than  $2^{nI(X;V_{1})}$ and\nconsequently Alice can estimate the correct sequence $v_{1}(l)$.\n\nUsing $x^{n}$ and $\\hat{v}_{1}(l)$, Alice looks for a unique $\\hat{v}_{2}(j)$\nin the bin $b_{V_{2}}(v_{2}(j))$ such that $(x^{n}, \\hat{v}_{1}(l), \\hat{v}_{2}(j))$\nare jointly typical.  Alice can estimate the correct sequence\n$v_{2}(j)$ as long as the number of sequences in each bin is\nless than $2^{nI(X,V_{1};V_{2})}$.  The number of $v_{2}(j)$\nsequences in each bin is $2^{nI(W,V_{1};V_{2})}$. From the Markov chain\n$W\\rightarrow X\\rightarrow (V_{1},V_{2})$, we have $I(W,V_{1};V_{2})\\leq I(X,V_{1};V_{2})$\nand therefore Alice can correctly estimate the sequence $v_{2}(j)$.\n\nShe next finds a sequence $u$ such that $(u,\\hat{v}_{1}(l),\\hat{v}_{2}(j),x^{n})$ are jointly typical. Let the\nbin index of this resulting $u$ sequence be $s_{U}$.\n\nAlice transmits the pair $(s_{U},m_{X})$ which is received by Bob and\nEve. The total rate used by Alice is $ I(X;U|V_{1},V_{2},W)+H(X|U,V_{1},V_{2},W)=H(X|V_{1},V_{2},W)$.\n\n\\item Decoding at Bob:\nOn receiving the pair $(s_{U},m_{X})$ from Alice and the bin indices\n$b_{V_{1}}(v_{1}(l))$ and $b_{V_{2}}(v_{2}(j))$ from\nHelen, Bob looks for a unique $\\hat{v}_{1}(l)$\nin the bin $b_{V}(v_{1}(l))$ such that $(w^{n}, \\hat{v}_{1}(l))$\nare jointly typical. He can estimate the correct sequence $v_{1}(l)$ with\nhigh probability since the number of $v_{1}$ sequences in each\nbin is approximately $2^{nI(V_{1};W)}$. Using the estimate\n$\\hat{v}_{1}(l)$ and $w^{n}$, he looks for a unique $\\hat{v}_{2}(j)$\nin the bin $b_{V_{2}}(v_{2}(j))$ such that $(w^{n}, \\hat{v}_{1}(l)), \\hat{v}_{2}(j))$\nare jointly typical. With high probability, the correct sequence\n$v_{2}(j)$ can be decoded by Bob since the number of\n$v_{2}$ sequences in each bin is approximately $2^{nI(V_{2};W|V_{1})}$.\n\nHe next searches the bin $s_{U}$ for a sequence $\\hat{u}$ such\nthat $(\\hat{u},\\hat{v}_{1}(l),\\hat{v}_{2}(j),w^{n})$ are jointly\ntypical. Since the number of $u$ sequences in each auxiliary bin\nis approximately\n$2^{nI(X,V_{1},V_{2};U)}\/2^{n(I(X,V_{1},V_{2};U)-I(W,V_{1},V_{2};U))}$\nwhich is $2^{nI(W,V_{1},V_{2};U)}$, with high probability, Bob\nwill be able to obtain the correct $u$ sequence.\n\nUsing the estimates $\\hat{u}$, $\\hat{v}_{1}(l)$, and\n$\\hat{v}_{2}(j)$, Bob\nsearches for a unique $x^{n}$ sequence in the\nbin $m_{X}$  such that $(\\hat{u}, \\hat{v}_{1}(l),\n\\hat{v}_{2}(j), w^{n},x^{n})$ are jointly\ntypical. This is possible since the number of $x^{n}$\nsequences in each bin is approximately\n$2^{nH(X)}\/2^{nH(X|U,V_{1},V_{2},W)}$, i.e.,\n$2^{nI(U,V_{1},V_{2},W;X)}$. Therefore, Bob can correctly decode the\n$x^{n}$ sequence with high probability.\n\n\\item{Equivocation}: \n\\begin{align}\nH(X^{n}|s_{U},m_{X},b_{V_{2}},Z^{n})&=H(X^{n},m_{X},s_{U},b_{V_{2}}|Z^{n})-H(m_{X},s_{U},b_{V_{2}}|Z^{n})\\\\\n&=H(X^{n}|Z^{n})+H(m_{X},s_{U},b_{V_{2}}|X^{n},Z^{n})-H(m_{X},s_{U},b_{V_{2}}|Z^{n})\\\\\n&=H(X^{n}|Z^{n})+H(s_{U},b_{V_{2}}|X^{n},Z^{n})-H(m_{X},s_{U},b_{V_{2}}|Z^{n})\\label{e12L}\\\\\n&\\geq H(X^{n}|Z^{n})+H(s_{U},b_{V_{2}}|X^{n},Z^{n})-H(m_{X}|Z^{n})\\nonumber\\\\&\\hspace{0.17in}-H(s_{U},b_{V_{2}}|Z^{n})\\label{e12Ltem1}\\\\\n&\\geq\nH(X^{n}|Z^{n})+H(s_{U},b_{V_{2}}|X^{n},Z^{n})-H(m_{X})\\nonumber\\\\&\\hspace{0.17in}-H(s_{U},b_{V_{2}}|Z^{n})\\label{e12Ltem2}\\\\\n&\\geq\nH(X^{n}|Z^{n})+H(s_{U},b_{V_{2}}|X^{n},Z^{n})-nH(X|U,V_{1},V_{2},W)\\nonumber\\\\&\\hspace{0.17in}-H(s_{U},b_{V_{2}}|Z^{n})\\label{e22L}\\\\\n&=\nH(X^{n}|Z^{n})-I(s_{U},b_{V_{2}};X^{n}|Z^{n})-nH(X|U,V_{1},V_{2},W)\\\\\n&\\geq\nH(X^{n}|Z^{n})-I(U^{n},V_{2}^{n};X^{n}|Z^{n})-nH(X|U,V_{1},V_{2},W)\\label{e32L}\\\\\n&\\geq\nH(X^{n}|Z^{n})-nI(U,V_{2};X|Z)-nH(X|U,V_{1},V_{2},W)\\\\\n&=\nnH(X|Z)-nI(U,V_{2};X|Z)-nH(X|U,V_{1},V_{2},W)\n\\end{align}\n\\begin{align}\n&=\nn(H(X|U,V_{2},Z)-H(X|U,V_{1},V_{2},W))\\\\\n&=\nn(I(X;V_{1}|U,V_{2},W)+I(X;W|U,V_{2})-I(X;Z|U,V_{2}))\\\\\n&\\geq\nn(\\min(R_{sec},I(X;V_{1}|U,V_{2},W))+I(X;W|U,V_{2})\\nonumber\\\\&\\hspace{0.34in}-I(X;Z|U,V_{2}))\n\\end{align}\nwhere (\\ref{e12L}) follows from the fact that $m_{X}$ is the bin index\nof the sequence $X^{n}$, (\\ref{e12Ltem1}) and (\\ref{e12Ltem2}) follow\nfrom the fact that conditioning reduces entropy, (\\ref{e22L}) follows from the fact that\n$H(m_{X})\\leq \\mbox{log}(2^{nH(X|U,V_{1},V_{2},W)})$, (\\ref{e32L}) follows from\nthe fact that $s_{U}$ is the bin index of the sequence $U^{n}$ and $b_{V_{2}}$\n is the bin index of the sequence $V_{2}^{n}$.\nThis completes the achievability part.\n\\end{enumerate}\n\n\\subsubsection{Converse}\nLet the coded outputs of the helper be denoted as $(J_{sec},J_{ins})$, where\n$J_{sec}$ denotes the coded output of the secure link, $J_{ins}$ denotes the\ncoded output of the insecure link, and the\noutput of Alice be denoted as $J_{x}$, i.e.,\n\\begin{align}\n(J_{sec},J_{ins})=f_{y}(Y^{n})\\qquad \\mbox{and} \\qquad J_{x}=f_{x}(X^{n},J_{sec},J_{ins})\n\\end{align}\nFirst note that, for noiseless reconstruction of the sequence $X^{n}$ at Bob, we have by Fano's inequality\n\\begin{align}\nH(X^{n}|J_{x},J_{sec},J_{ins},W^{n})&\\leq n\\epsilon_{n}\\label{fano2L}\n\\end{align}\n\nWe start by obtaining a lower bound on $R_{x}$, the rate of\nAlice, as follows,\n\\begin{align}\nnR_{x}&\\geq H(J_{x})\\\\\n&\\geq H(J_{x}|J_{sec},J_{ins},W^{n})\\\\\n&= H(X^{n},J_{x}|J_{sec},J_{ins},W^{n})-H(X^{n}|J_{x},J_{sec},J_{ins},W^{n})\\\\\n&\\geq H(X^{n},J_{x}|J_{sec},J_{ins},W^{n})-n\\epsilon_{n}\\label{eX1}\\\\\n&\\geq H(X^{n}|J_{sec},J_{ins},W^{n})-n\\epsilon_{n}\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|X^{i-1},J_{sec},J_{ins},W^{n})-n\\epsilon_{n}\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|J_{sec},J_{ins},X^{i-1},W_{i+1}^{n},W_{i})-n\\epsilon_{n}\\label{eXX1}\\\\\n&\\geq \\sum_{i=1}^{n}H(X_{i}|J_{sec},J_{ins},Y^{i-1},X^{i-1},W_{i+1}^{n},W_{i})-n\\epsilon_{n}\n\\end{align}\n\\begin{align}\n&=\\sum_{i=1}^{n}H(X_{i}|V_{1i},V_{2i},W_{i})-n\\epsilon_{n}\\label{eX2}\\\\\n&=nH(X|V_{1},V_{2},W)-n\\epsilon_{n}\n\\end{align}\nwhere (\\ref{eX1}) follows by (\\ref{fano2L}) and (\\ref{eXX1}) follows\nfrom the following Markov chain,\n\\begin{align}\nW^{i-1}\\rightarrow (J_{sec},J_{ins},X^{i-1},W_{i+1}^{n},W_{i})\\rightarrow X_{i}\n\\end{align}\nIn (\\ref{eX2}), we have defined\n\\begin{align}\nV_{1i}&=(J_{sec},Y^{i-1},X^{i-1},W_{i+1}^{n})\\\\\nV_{2i}&=J_{ins}\n\\end{align}\n\nWe next obtain a lower bound on $R_{sec}$, the rate of the secure link,\n\\begin{align}\nnR_{sec}&\\geq H(J_{sec})\\\\\n&\\geq H(J_{sec}|W^{n})\\\\\n&\\geq I(J_{sec};Y^{n}|W^{n})\\\\\n&=\n\\sum_{i=1}^{n}I(J_{sec},Y^{i-1},W^{i-1},W_{i+1}^{n};Y_{i}|W_{i})\\\\\n&= \\sum_{i=1}^{n}I(J_{sec},Y^{i-1},W_{i+1}^{n};Y_{i}|W_{i})\\label{2L1a}\\\\\n&= \\sum_{i=1}^{n}I(J_{sec},Y^{i-1},X^{i-1},W_{i+1}^{n};Y_{i}|W_{i})\\label{2L1b}\\\\\n&= \\sum_{i=1}^{n}I(V_{1i};Y_{i}|W_{i})\\\\\n&=nI(V_{1};Y|W)\n\\end{align}\nwhere (\\ref{2L1a}) and (\\ref{2L1b}) follow from the Markov chain\n\\begin{align}\n(X^{i-1},W^{i-1})\\rightarrow (J_{sec},Y^{i-1},W_{i+1}^{n}) \\rightarrow Y_{i}\n\\end{align}\n\nNext, we provide a lower bound on $R_{ins}$, the rate of the insecure\nlink,\n\\begin{align}\nnR_{ins}&\\geq H(J_{ins})\\\\\n&\\geq H(J_{ins}|J_{sec},W^{n})\\\\\n&\\geq H(J_{ins}|J_{sec},W^{n})-H(J_{ins}|J_{sec},Y^{n})\\\\\n&= I(J_{ins};Y^{n}|J_{sec})-I(J_{ins};W^{n}|J_{sec})\\\\\n&= \\sum_{i=1}^{n}I(J_{ins};Y_{i}|J_{sec},Y^{i-1},W_{i+1}^{n})-I(J_{ins};W_{i}|J_{sec},Y^{i-1},W_{i+1}^{n})\\label{2L2}\\\\\n&= \\sum_{i=1}^{n}I(J_{ins};Y_{i}|J_{sec},Y^{i-1},X^{i-1}W_{i+1}^{n})-I(J_{ins};W_{i}|J_{sec},Y^{i-1},X^{i-1},W_{i+1}^{n})\\label{2L3}\\\\\n&= \\sum_{i=1}^{n}I(V_{2i};Y_{i}|V_{1i})-I(V_{2i};W_{i}|V_{1i})\\\\\n&= \\sum_{i=1}^{n}I(V_{2i};Y_{i}|W_{i},V_{1i})\\\\\n&=nI(V_{2};Y|W,V_{1})\n\\end{align}\nwhere (\\ref{2L2}) follows from the Csiszar's sum lemma\n\\cite{Csiszar:book}, and (\\ref{2L3}) follows from the following Markov\nchain,\n\\begin{align}\nX^{i-1}\\rightarrow (J_{sec},Y^{i-1},W_{i+1}^{n}) \\rightarrow (J_{ins},Y_{i},W_{i})\n\\end{align}\n\nWe now have the main step, i.e., an upper bound on the equivocation\nrate of the eavesdropper,\n\\begin{align}\nH(X^{n}|J_{x},J_{ins},Z^{n})&=\nH(X^{n},Z^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins})\\\\\n&=H(X^{n},W^{n},J_{sec},Z^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins})\\nonumber\\\\&\\hspace{0.17in}-H(W^{n},J_{sec}|X^{n},Z^{n},J_{x},J_{ins})\\\\\n&=H(W^{n},J_{sec}|J_{x},J_{ins})+H(X^{n}|W^{n},J_{x},J_{sec},J_{ins})+H(Z^{n}|X^{n},W^{n})\\nonumber\\\\&\\hspace{0.17in}-H(Z^{n}|J_{x},J_{ins})-H(W^{n}|X^{n},Z^{n})-H(J_{sec}|X^{n},J_{x},J_{ins},W^{n})\\label{2L4}\\\\\n&=(H(W^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins}))-(H(W^{n}|X^{n},Z^{n})\\nonumber\\\\&\\hspace{0.17in}\n-H(Z^{n}|X^{n},W^{n}))+I(J_{sec};X^{n}|J_{x},J_{ins},W^{n})\\nonumber\\\\&\\hspace{0.17in}+H(X^{n}|W^{n},J_{x},J_{sec},J_{ins})\\\\\n&\\leq(H(W^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins}))-(H(W^{n}|X^{n},Z^{n})\\nonumber\\\\&\\hspace{0.17in}-H(Z^{n}|X^{n},W^{n}))\n+I(J_{sec};X^{n}|J_{x},J_{ins},W^{n})+n\\epsilon_{n}\\label{2L5}\n\\end{align}\nwhere (\\ref{2L4}) follows from the Markov chain $Y^{n}\\rightarrow\nX^{n}\\rightarrow (W^{n},Z^{n})$ and (\\ref{2L5}) follows by (\\ref{fano2L}).\nNow, using Csiszar's sum lemma \\cite{Csiszar:book}, we have\n\\begin{align}\nH(W^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins})&=\\sum_{i=1}^{n}H(W_{i}|Z^{i-1},W_{i+1}^{n},J_{x},J_{ins})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(Z_{i}|Z^{i-1},W_{i+1}^{n},J_{x},J_{ins})\\label{2L6}\n\\end{align}\nand we also have,\n\\begin{align}\nH(W^{n}|X^{n},Z^{n})-H(Z^{n}|X^{n},W^{n})&=H(W^{n}|X^{n})-H(Z^{n}|X^{n})\\\\\n&=\\sum_{i=1}^{n}H(W_{i}|X_{i})-\\sum_{i=1}^{n}H(Z_{i}|X_{i})\\label{2L7a}\\\\\n&=\\sum_{i=1}^{n}H(W_{i}|X_{i},Z^{i-1},W_{i+1}^{n},J_{x},J_{ins})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(Z_{i}|X_{i},Z^{i-1},W_{i+1}^{n},J_{x},J_{ins})\\label{2L7b}\n\\end{align}\nwhere (\\ref{2L7a}) follows from the memorylessness of the sources and\n(\\ref{2L7b}) follows from the Markov chain\n$Y^{n}\\rightarrow X^{n}\\rightarrow (W^{n},Z^{n})$.\nNow, defining,\n\\begin{align}\nU_{i}&=(J_{x},Z^{i-1},W_{i+1}^{n})\n\\end{align}\nand \n\\begin{align}\nX=X_{Q},\\quad Y=Y_{Q},\\quad Z=Z_{Q}, \\quad W=W_{Q}\\\\\nV_{1}=(Q,V_{1Q}),\\quad V_{2}=(Q,V_{2Q}), \\quad U=(Q,U_{Q})\n\\end{align}\nwe have from (\\ref{2L6}) and (\\ref{2L7b}),\n\\begin{align}\nH(W^{n}|J_{x},J_{ins})-H(Z^{n}|J_{x},J_{ins})&=n(H(W|U,V_{2})-H(Z|U,V_{2}))\\label{2L8}\\\\\nH(W^{n}|X^{n},Z^{n})-H(Z^{n}|X^{n},W^{n})&=n(H(W|X,U,V_{2})-H(Z|X,U,V_{2}))\\label{2L9}\n\\end{align}\nNow consider,\n\\begin{align}\nI(J_{sec};X^{n}|J_{x},J_{ins},W^{n})&=\\sum_{i=1}^{n}I(J_{sec};X_{i}|J_{x},J_{ins},W^{n},X^{i-1})\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},W^{n},X^{i-1})-\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},J_{sec},W^{n},X^{i-1})\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},W_{i},X^{i-1},W_{i+1}^{n})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},J_{sec},W_{i},X^{i-1},W_{i+1}^{n})\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},W_{i},X^{i-1},Z^{i-1},W_{i+1}^{n})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},J_{sec},W_{i},X^{i-1},Z^{i-1},W_{i+1}^{n})\\\\\n&\\leq\n\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},W_{i},Z^{i-1},W_{i+1}^{n})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},J_{sec},W_{i},X^{i-1},Z^{i-1},W_{i+1}^{n})\\\\\n&\\leq \\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},W_{i},Z^{i-1},W_{i+1}^{n})\\nonumber\\\\&\\hspace{0.17in}-\\sum_{i=1}^{n}H(X_{i}|J_{x},J_{ins},J_{sec},W_{i},X^{i-1},Y^{i-1},Z^{i-1},W_{i+1}^{n})\\\\\n&=\\sum_{i=1}^{n}H(X_{i}|U_{i},W_{i},V_{2i})-\\sum_{i=1}^{n}H(X_{i}|U_{i},W_{i},V_{2i},V_{1i})\\\\\n&=\\sum_{i=1}^{n}I(X_{i};V_{1i}|W_{i},U_{i},V_{2i})\\\\\n&=nI(X;V_{1}|W,U,V_{2})\n\\end{align}\nWe also have\n\\begin{align}\nI(J_{sec};X^{n}|J_{x},J_{ins},W^{n})&\\leq H(J_{sec})\\\\\n&\\leq nR_{sec}\n\\end{align}\nTherefore, we have\n\\begin{align}\nI(J_{sec};X^{n}|J_{x},J_{ins},W^{n})&\\leq n\\min(R_{sec},I(X;V_{1}|W,U,V_{2}))\\label{2L10}\n\\end{align}\nFinally, on substituting (\\ref{2L8}), (\\ref{2L9}) and (\\ref{2L10}) in (\\ref{2L5}), we arrive\nat\n\\begin{align}\\hspace{-0.1in}\nH(X^{n}|J_{x},J_{ins},Z^{n}) &\\leq n(\\min(R_{sec},I(X;V_{1}|W,U,V_{2}))+I(X;W|U,V_{2})-I(X;Z|U,V_{2})+\\epsilon_{n})\n\\end{align}\nThis implies\n\\begin{align}\n\\Delta&\\leq  \\min(R_{sec},I(X;V_{1}|W,U,V_{2}))+I(X;W|U,V_{2})-I(X;Z|U,V_{2})\n\\end{align}\n\nAlso note that the following is a Markov chain,\n\\begin{align}\n(V_{1},V_{2})\\rightarrow Y \\rightarrow X \\rightarrow (W,Z)\n\\end{align}\nTherefore, the joint distribution of the involved random variables is\n\\begin{align}\np^{out}(x,y,w,z,v_{1},v_{2},u)&=p(x,y)p(w,z|x)p(v_{1},v_{2}|y)p(u|x,v_{1},v_{2},y)\n\\end{align}\nOn the other hand, the joint distribution of the involved random\nvariables in the achievability proof of Theorem $3$ is in the following form,\n\\begin{align}\np^{ach}(x,y,w,z,v_{1},v_{2},u)&=p(x,y)p(w,z|x)p(v_{1},v_{2}|y)p(u|x,v_{1},v_{2})\\label{innner2s}\n\\end{align}\ni.e., they satisfy the Markov chain $Y\\rightarrow\n(X,V_{1},V_{2})\\rightarrow U$. Now using the observation that\n$I(X;W|U,V_{1},V_{2})$ depends on the marginal\n$p(x,w,u,v_{1},v_{2})$ and  $I(X;Z|U,V_{1},V_{2})$ depends on the\nmarginal $p(x,z,u,v_{1},v_{2})$ and using similar arguments used\nin the converse proof of Theorem $2$, it can be shown that it\nsuffices to consider distributions of the form given in\n(\\ref{innner2s})  when evaluating our outer bound. This completes\nthe proof of the converse part.\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Basic notation and information structure}\\label{sec-nota}\n\\section*{Problem formulation and solution}\nConsider a Markov chain $\\{\\eta(k),\\,\\,k=0,1,\\ldots \\}$ \nwith distribution and transition probabilities given respectively by \n\\begin{equation}\\nonumber\n\\begin{aligned}\ns_{ij}(k)&=\\text{Pr}(\\eta(k+1)=j|\\eta(k)=i), \n\\\\ \n\\upsilon_i(k)&=\\text{Pr}(\\eta(k)=i),\\;\\;i,j\\in\\mathbb{N}, k\\geq 0,\n\\end{aligned}    \n\\end{equation}\nwhere $\\mathbb{N}=\\{1,\\ldots,N\\}$ is the state space. \nFor Markov chains and basic probability rules employed in this note, the reader is referred to \n\\cite{Cinlar75,Papoulis}.\nConsider the process $\\{\\theta(k), k=0,1,\\ldots,\\ell\\}$, defined as follows, based on the  Markov chain in a finite time window,\n$$\\theta(k)=\\eta(\\ell-k), \\;\\; \\,k=0,1,\\ldots,\\ell.$$ \nAssume that a cluster observation is available \nin the form $\\{\\theta(0)\\in \\mathbb{C}_0\\}$, $\\mathbb{C}_0\\subseteq \\mathbb{N}$, as well as a cluster observation with anticipation \n$\\{\\theta(\\ell)\\in \\mathbb{C}_\\ell$\\}, \n$\\mathbb{C}_\\ell\\subseteq\\mathbb{N}$.\n$E_{\\text{o}}\\in\\mathcal{F}$ stands for the associated $\\sigma$-algebra. \n\nThe adopted process $\\theta$ and information structure may appear in real-world problems, including problems involving a first-in last-out queue. For example, suppose the Markov chain characterise the presence of certain irregularities in $\\ell$ items arriving at a store, and that the first item in goes through an inspection of these irregularities, so that we know the state of this item (when it enters the queue); following the queue rule, and counting the items when leaving the queue, we have information ``with anticipation'' on the state of the $\\ell$-th item. Note, via this example, that observation with anticipation does not necessarily require prediction of future events.\n\n\nIn this note we seek for formulas \nfor the conditional transition probabilities and distributions \ndefined in (1), based on the Markov chain parameters $\\upsilon_i$ and \\textcolor{black}{$s_{ij}(k)$} and the cluster observations. We shall need some additional notation.\nFor each $i,j\\in\\mathbb{N}$ and  $k=0,1,\\ldots,\\ell-1$, we define\n\\begin{equation}\\label{eq-prob-theta}\n\\begin{aligned}\np_{ij}&=\\text{Pr}(\\theta(k)=j\\,|\\,\\theta(k+1)=i\\,,\\,E_{\\text{o}}),\n\\\\\n\\pi_i(k)&=\\text{Pr}(\\theta(k)=i)|E_{\\text{o}}), \n\\;\\;i,j\\in\\mathbb{N}, 0\\leq k\\leq \\ell.\n\\end{aligned}\n\\end{equation}\nWe write $\\mathbf{P}(k)$ to represent a matrix of dimension $N$ by $N$, whose components are $p_{ij}(k)$, therefore satisfying the Chapman-Kolmogorov equation \n\\begin{equation}\\label{eq-chapman}\n\\pi(k+1)=\\mathbf{P}(k)\\pi(k).\n\\end{equation}\n\n\\noindent{W}e denote, for $i\\in\\mathbb{N}$, and $k=1,\\ldots,\\ell-1$, \n$${e}= \\sum_{r\\in\\mathbb{C}_\\ell}\\sum_{j\\in\\mathbb{C}_{0}}[\\mathbf{S}^{\\ell}]_{rj}\\upsilon_r(\\textcolor{black}{0}),\\quad {g}_i(k)=\\sum_{r\\in\\mathbb{C}_{0}}[\\mathbf{S}^{k+1}]_{ir}.$$\n\n\n\\noindent\\textbf{Lemma 1.} \n\\textit{The following is valid for $1\\leq k\\leq \\ell-2$ and $i,j\\in\\mathbb{N}$; if \\textcolor{black}{$i$ is such that $\\upsilon_i(\\ell-k-1)>0$}, then\n\\begin{equation}\\label{eq-prob_lema3}\n    p_{ij}(k)=\\begin{cases}\n    (g_i(k))^{-1}\\sum_{r\\in\\mathbb{C}_0}[\\mathbf{S}^{k}]_{jr}s_{ij}(k),&\\, g_i(k)\\neq 0,\\\\\n    \\quad\\quad\\quad \\text{arbitrary},&\\, {otherwise};\n    \\end{cases}\n\\end{equation}\n\\textcolor{black}{if  $\\upsilon_i(\\ell-k-1)=0$, then  $p_{ij}(k)$ is arbitrary.} \n\\eqref{eq-prob_lema3} is also valid for $k=\\ell-1$, $i\\in\\mathbb{C}_{\\ell}$ and $j\\in\\mathbb{N}$, as well as for $k=0$, $i\\in\\mathbb{N}$ and $j\\in\\mathbb{C}_{0}$. %\nThe remaining cases regarding $p_{ij}(k)$ are: \n$p_{ij}(\\ell-1)$ \\textcolor{black}{is arbitrary} for $i\\notin\\mathbb{C}_{\\ell}$ and $j\\in\\mathbb{N}$; \n\\textcolor{black}{if $i\\in\\mathbb{C}_\\ell$ and there is \nno Markov state $\\eta(\\ell)\\in\\mathbb{C}_{0}$ that can be reached from $i$ \n(in the sense that $\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(0)=i)=0$) then $p_{ij}(\\ell-1)$ is  arbitrary;} \n$p_{ij}(0)$ is arbitrary for $i\\in\\mathbb{N}$ and $j\\notin\\mathbb{C}_{0}$. Regarding $\\pi_{i}(\\ell)$, for $i\\in\\mathbb{C}_{\\ell}$, \n\\begin{equation}\\label{eq-pi-terminal}\n    \\pi_{i}(\\ell)=\\begin{cases}\n    e^{-1}\\displaystyle\\sum_{j\\in\\mathbb{C}_{0}} [\\mathbf{S}^{\\ell}]_{ij}\\upsilon_i(0),&\\, e\\neq 0,\\\\\n    \\quad\\quad\\quad \\text{arbitrary},&\\, {otherwise}.\n    \\end{cases}\n\\end{equation}\nFor $i\\notin\\mathbb{C}_{\\ell}$, $\\pi_{i}(\\ell)=0$. \nFinally, $\\pi(k)$, $1\\leq k\\leq \\ell-1$, is given by \\eqref{eq-chapman} and the formulas above.\n}\n\n\\textbf{Proof:} for $i\\in\\mathbb{N}$ by definition we have \n\\begin{equation}\\nonumber\n\\begin{aligned}\n \\pi_i(\\ell)&=\\text{Pr}(\\theta(\\ell)=i\\,|\\,\\eta(0)\\in\\mathbb{C}_\\ell,\\,\\eta(\\ell)\\in\\mathbb{C}_0)\\\\\n &=\\text{Pr}(\\eta(0)=i\\,|\\,\\eta(0)\\in\\mathbb{C}_\\ell,\\,\\eta(\\ell)\\in\\mathbb{C}_0).  \n  \\end{aligned}\n\\end{equation}\nOf course,  \n$\\pi_i(\\ell)=0$  whenever $i\\notin\\mathbb{C}_\\ell$, as it is given that \n$\\theta(\\ell)=\\eta(0)$ is in $\\mathbb{C}_\\ell$. If $i\\in\\mathbb{C}_\\ell$, and assuming that \n$\\text{Pr}\\big(\\eta(0)\\in\\mathbb{C}_\\ell\\,,\\,\\eta(\\ell)\\in\\mathbb{C}_0\\big)>0$, we may write \n\\begin{equation*}\n\\begin{aligned}\n\\pi_i(\\ell)&=\\text{Pr}\\big(\\eta(0)=i\\,|\\,\\eta(0)\\in\\mathbb{C}_\\ell\\,,\\,\\eta(\\ell)\\in\\mathbb{C}_0\\big)\\\\\n    &=\\frac{\\text{Pr}\\big(\\eta(\\ell)\\in\\mathbb{C}_0\\,\\,,\\,\\,\\eta(0)=i\\big)}{\\text{Pr}\\big(\\eta(\\ell)\\in\\mathbb{C}_0\\,\\,,\\,\\,\\eta(0)\\in\\mathbb{C}_\\ell\\big)}\\\\\n    &=\\frac{\\displaystyle\\sum_{j\\in\\mathbb{C}_{0}}\\text{Pr}\\big(\\eta(\\ell)=j\\,,\\,\\eta(0)=i\\big)}{\\displaystyle\\sum_{r\\in\\mathbb{C}_{\\ell}}\\sum_{j\\in\\mathbb{C}_0}\\text{Pr}\\big(\\eta(\\ell)=j\\,,\\,\\eta(0)=r\\big)}.\n\\end{aligned}\n\\end{equation*}\nNote that\n\\begin{equation}\\nonumber\n\\begin{aligned}\n\\text{Pr}(\\eta(\\ell)=j,\\,\\eta(0)=r)&=\\text{Pr}(\\eta(\\ell)=j|\\eta(0)=r)\\\\\n&\\cdot\\text{Pr}(\\eta(0)=r)=[\\mathbf{S}^{\\ell}]_{rj}\\upsilon_r(0), \n\\end{aligned}\n\\end{equation}\nand substituting in the above\n$$\\begin{aligned}\n\\pi_i(\\ell)&=\\displaystyle\\sum_{j\\in\\mathbb{C}_{0}}[\\mathbf{S}^{\\ell}]_{i,j}\\upsilon_i(0)\\left(\\displaystyle\\sum_{r\\in\\mathbb{C}_\\ell}\\sum_{j\\in\\mathbb{C}_{0}}[\\mathbf{S}^{\\ell}]_{r,j}\\upsilon_r(0)\\right)^{-1}\n\\\\& \n=e^{-1}\\displaystyle\\sum_{j\\in\\mathbb{C}_{0}} [\\mathbf{S}^{\\ell}]_{ij}\\upsilon_i(0)\n\\end{aligned}$$\nIf $i\\in\\mathbb{C}_\\ell$ and \n$\\text{Pr}\\big(\\eta(0)\\in\\mathbb{C}_\\ell\\,,\\,\\eta(\\ell)\\in\\mathbb{C}_0\\big)=0.$ \n(in which case, the inverse in the above equation does \nnot exist), then $\\pi_i(\\ell)$ \nis conditioned on an event of probability zero \ntherefore $\\pi_i(\\ell)$ is arbitrary,\nthus completing the demonstration of \\eqref{eq-pi-terminal}.\nNow we turn our attention to $\\mathbf{P}$.\n\\begin{equation}\\nonumber\n\\begin{aligned}\np_{ij}(\\ell-1)&=\\text{Pr}(\\theta(\\ell-1)=j|\\theta(\\ell)=i,\\eta(0)\\in\\mathbb{C}_\\ell,\\eta(\\ell)\\in\\mathbb{C}_0)\\\\\n&=\\text{Pr}(\\eta(1)=j|\\eta(0)=i,\\eta(0)\\in\\mathbb{C}_\\ell,\\eta(\\ell)\\in\\mathbb{C}_0).\n\\end{aligned}\n\\end{equation}\nIt is clear that, if $i\\notin\\mathbb{C}_\\ell$, then $p_{ij}(\\ell-1)$ \nis conditioned  on an empty set, therefore an event of probability zero, making \n$p_{ij}(\\ell-1)$ arbitrary, for any $j\\in\\mathbb{N}$. \nIf $i\\in\\mathbb{C}_\\ell$ and the event $\\{\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(0)=i\\}$ \nis not of probability zero, then using the total probability law we write\n\\begin{equation}\\nonumber\n\\begin{aligned}\np_{ij}(\\ell-1)\n&=\\frac{\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(1)=j,\\eta(0)=i)\\cdot s_{ij}(k)} {\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(0)=i)}\\\\\n&=\\frac{\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(1)=j)\\cdot  s_{ij}(k)}\n {\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(0)=i)}\\\\\n&=\\displaystyle\\sum_{r\\in\\mathbb{C}_{\\ell}}[\\mathbf{S}^{\\ell-1}]_{jr} s_{ij}(k) \\left(\\displaystyle\\sum_{r\\in\\mathbb{C}_{\\ell}}[\\mathbf{S}^{\\ell}]_{ir}\\right)^{-1},  j\\in\\mathbb{N},\n\\end{aligned}\n\\end{equation}\nwhere the second inequality is due to the Markov property. \nIf $i\\in\\mathbb{C}_\\ell$ and the event $\\{\\eta(\\ell)\\in\\mathbb{C}_{0}\\,|\\,\\eta(0)=i\\}$ \nis an empty set, then $p_{ij}(\\ell-1)$ is conditional on an event of zero probability, hence it is arbitrary. \nRegarding $p_{ij}(k)$ with $1\\leq k\\leq \\ell-2$ and $i,j\\in\\mathbb{N}$, \ndenoting \n$A = \\{\\eta(\\ell-k-1)=i\\}$ for a better visual diagramming of \nthe next equation, \nassuming $\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0},A)>0$ and using the Markov property, we write \n\\begin{equation}\\nonumber\n\\begin{aligned}\n& p_{ij}(k)=\\text{Pr}(\\eta(\\ell-k)=j|A,\\eta(0)\\in\\mathbb{C}_\\ell,\\eta(\\ell)\\in\\mathbb{C}_0)\\\\\n& \\;\\;=\\text{Pr}(\\eta(\\ell-k)=j|A,\\eta(\\ell)\\in\\mathbb{C}_{0})\\\\\n& \\;\\;=\\frac{\\text{Pr}(\\eta(\\ell-k)=j,\\eta(\\ell)\\in\\mathbb{C}_{0},A)}{\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0},A)}\\\\\n& \\;\\;=\\frac{\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}|\\eta(\\ell-k)=j,A)\\text{Pr}(\\eta(\\ell-k)=j|A)P(A)} {\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}|A)P(A)}\\\\\n& \\;\\;=\\frac{\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}|\\eta(\\ell-k)=j)\\text{Pr}(\\eta(\\ell-k)=j|A)} {\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0}|A)},\n\\end{aligned}\n\\end{equation}\nyielding\n$$\\begin{aligned}\np_{ij}(k)&=\\sum_{r\\in\\mathbb{C}_{0}}[\\mathbf{S}^{k}]_{jr} s_{ij}(k) \\left(\\displaystyle\\sum_{r\\in\\mathbb{C}_{0}}[\\mathbf{S}^{k+1}]_{ir}\\right)^{-1}\n\\\\&\n=(g_i(k))^{-1}\\sum_{r\\in\\mathbb{C}_0}[\\mathbf{S}^{k}]_{jr}s_{ij}(k).\n\\end{aligned}\n$$\n\\textcolor{black}{Note that the requirement $\\text{Pr}(\\eta(\\ell)\\in\\mathbb{C}_{0},A)>0$ is equivalent to say: (i) considering the Markov chain $\\{\\eta_k, k\\geq 0\\}$,  \nthe set $\\mathbb{C}_{0}$ is reachable from state $i$ in $k+1$ steps \n(in which case $g_i(k)=0$), and (ii) $\\text{Pr}(\\eta(\\ell-k-1)=i)=\\upsilon_i(\\ell-k-1)>0$. If any of (i) or (ii) is false, or both are false, then then $p_{ij}(k)$  is conditional on an event of probability zero, \nhence it is \\textcolor{black}{arbitrary}.}\nIt only remains to find the formula for $p_{ij}(0)$.\n\\begin{equation}\\nonumber\n    \\begin{aligned}\np_{ij}(0)&=\\text{Pr}(\\theta(0)=j|\\theta(1)=i,\\eta(0)\\in\\mathbb{C}_\\ell,\\eta(\\ell)\\in\\mathbb{C}_0)\\\\\n&=\\text{Pr}(\\eta(\\ell)=j|\\eta(\\ell-1)=i,\\eta(0)\\in\\mathbb{C}_\\ell,\\eta(\\ell)\\in\\mathbb{C}_0).\n    \\end{aligned}\n\\end{equation}\nIf  $j\\notin\\mathbb{C}_{0}$ \nthen $p_{ij}(0)=0$ because it is given \nthat $\\eta(0)\\in\\mathbb{C}_{0}$; otherwise, we have\n\\begin{equation}\\nonumber\n\\begin{aligned}\np_{ij}(0)&=\\text{Pr}\\big(\\eta(\\ell)=j\\,|\\,\\eta(\\ell-1)=i\\,,\\,\\eta(\\ell)\\in\\mathbb{C}_0\\big)\\\\\n\\end{aligned}\n\\end{equation}\nso that, when the conditional event is not of probability zero, \n\\begin{equation}\\nonumber\n\\begin{aligned}\np_{ij}(0)&=\\frac{\\text{Pr}\\big(\\eta(\\ell)=j,\\eta(\\ell)\\in\\mathbb{C}_0\\,|\\,\\eta(\\ell-1)=i\\big)}{\\text{Pr}\\big(\\eta(\\ell)\\in\\mathbb{C}_0\\,|\\,\\eta(\\ell-1)=i\\big)}\\\\\n&=\\frac{\\text{Pr}\\big(\\eta(\\ell)=j\\,|\\,\\eta(\\ell-1)=i\\big)}{\\text{Pr}\\big(\\eta(\\ell)\\in\\mathbb{C}_0\\,|\\,\\eta(\\ell-1)=i\\big)}\\\\\n&=s_{ij}(k) \\left(\\displaystyle\\sum_{r\\in\\mathbb{C}_{0}}s_{ir}\\right)^{-1},\\forall\\,i\\in\\mathbb{N}\\,\\,\\text{and}\\,\\,j\\in\\mathbb{C}_{0}, \n\\end{aligned}\n\\end{equation}\nand, when the conditional event $\\{\\eta(\\ell-1)=i,\\,\\eta(\\ell)\\in\\mathbb{C}_0\\}$ is of probability zero \n(in which case the inverse in the above equation does not exist), then $p_{ij}(0)$ is \\textcolor{black}{arbitrary}.\n  \nFinally, $\\pi(k)$, $1\\leq k\\leq \\ell-1$, is given by \\eqref{eq-chapman} and the formulas above.\n\\hfill Q.E.D.\n\n\\medskip\n{\\it Remark 1.} All arbitrary values in Lemma 1 can be set to zero. This is a suitable choice \nin some cases, as in \\cite{Daniel-duality,PachasSubm}, where a Riccati-like equation is computed for every $i,k$ such that $\\pi_i(k)>0$, so that, choosing $\\pi_i(k)=0$ avoids unnecessary computations. \n\n\\medskip\n{\\it Remark 2.} In view of \\eqref{eq-prob_lema3},\nthe transition probabilities of the process $\\theta$ depend on $k$ even if the Markov chain is time-homogeneous. \nFor $\\mathbf{P}$ to be irrespective of $k$, it would be necessary that the Markov chain is time-homogeneous \\emph{and} there is no observation of $\\theta(0)$,  that is, $\\mathbb{C}_0=\\mathbb{N}$; in this case, \n\\eqref{eq-prob_lema3} reduces to $p_{ij}=s_{ij}$.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\n\\section{Introduction}\\label{sec:introduction} Visual odometry\n(VO)~\\cite{nister2004visual}, commonly referred to as ego-motion\nestimation, is a fundamental capability that enables robots to reliably\nnavigate its immediate environment. With the wide-spread adoption of\ncameras in various robotics applications, there has been an evolution\nin visual odometry algorithms with a wide set of variants including\nmonocular VO~\\cite{nister2004visual,konolige2010large}, stereo\nVO~\\cite{howard2008real,kitt2010visual} and even non-overlapping\n\\textit{n}-camera VO~\\cite{hee2013motion,kneip2013using}. Furthermore, each of\nthese algorithms has been custom tailored for specific camera optics\n(pinhole, fisheye, catadioptric) and the range of motions observed by\nthese cameras mounted on various\nplatforms~\\cite{scaramuzza20111}.\n\n\\begin{figure}[t]\n  \\includegraphics[width=\\columnwidth]{graphics\/mdn-vae.pdf}\n  \\caption{\\textbf{Visual Ego-motion Learning Architecture: } We\n    propose a visual ego-motion learning architecture that maps optical\n    flow vectors (derived from feature tracking in an image sequence) to\n    an ego-motion density estimate via a Mixture Density Network (MDN). By\n    modeling the architecture as a Conditional Variational Autoencoder\n    (C-VAE), our model is able to provide introspective reasoning and\n    prediction for scene-flow conditioned on the ego-motion estimate and\n    input feature location.}\n  \\label{fig:egomotion-architecture}\n  \\vspace{-5mm}\n\\end{figure}\n\nWith increasing levels of model specification for each domain, we\nexpect these algorithms to perform differently from others while\nmaintaining lesser generality across various optics and camera\nconfigurations. Moreover, the strong dependence of these algorithms on\ntheir model specification limits the ability to actively monitor and\noptimize their intrinsic and extrinsic model parameters in an online\nfashion. In addition to these concerns, autonomous systems today use\nseveral sensors with varied intrinsic and extrinsic properties that\nmake system characterization tedious. Furthermore, these\nalgorithms and their parameters are fine-tuned on specific datasets\nwhile enforcing little guarantees on their generalization performance\non new data.\n\nTo this end, we propose a fully trainable architecture for visual\nodometry estimation in generic cameras with varied camera optics\n(\\textit{pinhole}, \\textit{fisheye} and \\textit{catadioptric}\nlenses). In this work, we take a geometric approach by posing the\nregression task of ego-motion as a density estimation problem. By\ntracking salient features in the image induced by the ego-motion (via\nKanade-Lucas-Tomasi\/KLT feature tracking), we learn the mapping from\nthese tracked flow features to a probability mass over the range of\nlikely ego-motion. We make the following contributions:\n\\begin{itemize}\n  \\item \\textbf{A fully trainable ego-motion estimator}: We\nintroduce a fully-differentiable density estimation model for visual\nego-motion estimation that robustly captures the inherent ambiguity\nand uncertainty in relative camera pose estimation (See\nFigure~\\ref{fig:egomotion-architecture}).\n\\item \\textbf{Ego-motion for generic camera optics}: Without imposing\nany constraints on the type of camera optics, we propose an approach\nthat is able to recover ego-motions for a variety of camera models\nincluding \\textit{pinhole}, \\textit{fisheye} and \\textit{catadioptric}\nlenses.\n\\item\\textbf{Bootstrapped ego-motion training and refinement}: We propose a\nbootstrapping mechanism for autonomous systems whereby a robot\nself-supervises the ego-motion regression task. By fusing information\nfrom other sensor sources including GPS and INS (Inertial Navigation\nSystems), these indirectly inferred trajectory estimates serve as\nground truth target poses\/outputs for the aforementioned regression\ntask. Any newly introduced camera sensor can now leverage this\ninformation to learn to provide visual ego-motion estimates without\nrelying on an externally provided ground truth source.\n\\item\\textbf{Introspective reasoning via scene-flow predictions}: We\ndevelop a generative model for optical flow prediction that can be\nutilized to perform outlier-rejection and scene flow reasoning. \n\\end{itemize}\nThrough experiments, we provide a thorough analysis of ego-motion\nrecovery from a variety of camera models including pinhole, fisheye\nand catadioptric cameras. We expect our general-purpose approach to be\nrobust, and easily tunable for accuracy during\noperation. We illustrate the robustness and generality of our approach\nand provide our findings in Section~\\ref{sec:experiments}.\n\n\n\n\n\n\\section{Related Work}\\label{sec:related-work} Recovering relative\ncamera poses from a set of images is a well studied problem under the\ncontext of Structure-from-Motion\n(SfM)~\\cite{triggs1999bundle,hartley2003multiple}. SfM is usually\ntreated as a non-linear optimization problem, where the camera poses\n(extrinsics), camera model parameters (intrinsics), and the 3D scene\nstructure are jointly optimized via non-linear\nleast-squares~\\cite{triggs1999bundle}.\n\n\n\\textbf{Unconstrained VO}: Visual odometry, unlike incremental\nStructure-from-Motion, only focuses on determining the 3D camera pose\nfrom sequential images or video imagery observed by a monocular\ncamera. Most of the early work in VO was done primarily to determine\nvehicle\negomotion~\\cite{moravec1980obstacle,matthies1989dynamic,olson2000robust}\nin 6-DOF, especially in the Mars planetary rover. Over the years\nseveral variants of the VO algorithm were proposed, leading up to the\nwork of Nister et al.~\\cite{nister2004visual}, where the authors\nproposed the first real-time and scalable VO algorithm. In their work,\nthey developed a 5-point minimal solver coupled with a RANSAC-based\noutlier rejection scheme~\\cite{fischler1981random} that is still\nextensively used today. Other researchers~\\cite{corke2004omnidirectional}\nhave extended this work to various camera types including catadioptric\nand fisheye lenses.\n\n\\textbf{Constrained VO}: While the classical VO objective does not\nimpose any constraints regarding the underlying motion manifold or\ncamera model, it however contains several failure modes that make it\nespecially difficult to ensure robust operation under arbitrary scene\nand lighting conditions. As a result, imposing egomotion constraints has\nbeen shown to considerably improve accuracy, robustness, and run-time\nperformance. One particularly popular strategy for VO estimation in\nvehicles is to enforce planar homographies during matching features\non the ground\nplane~\\cite{liang2002visual,ke2003transforming},\nthereby being able to robustly recover both relative orientation and\nabsolute scale. For example, Scaramuzza et\nal.~\\cite{scaramuzza20111,scaramuzza2009real} introduced a novel\n1-point solver by imposing the vehicle's non-holonomic motion\nconstraints, thereby speeding up the VO estimation up to 400Hz.\n\n\n\\textbf{Data-driven VO}: While several model-based methods have been\ndeveloped specifically for the VO problem, a few have attempted to\nsolve it with a data-driven approach. Typical approaches have\nleveraged dimensionality reduction techniques by learning a\nreduced-dimensional subspace of the optical flow vectors induced by\nthe\negomotion~\\cite{roberts2009learning}. In~\\cite{ciarfuglia2014evaluation},\nCiarfuglia et al. employ Support Vector Regression (SVR) to recover\nvehicle egomotion (3-DOF). The authors further build upon their\nprevious result by swapping out the SVR module with an end-to-end\ntrainable convolutional neural network~\\cite{costante2016exploring}\nwhile showing improvements in the overall performance on\nthe KITTI odometry benchmark~\\cite{Geiger2012CVPR}. Recently, Clarke\net al.~\\cite{wen2016vinet} introduced a visual-inertial odometry\nsolution that takes advantage of a neural-network architecture to learn\na mapping from raw inertial measurements and sequential imagery to\n6-DOF pose estimates.  By posing visual-inertial odometry (VIO) as a\nsequence-to-sequence learning problem, they developed a neural network\narchitecture that combined convolutional neural networks with Long\nShort-Term Units (LSTMs) to fuse the independent sensor measurements\ninto a reliable 6-DOF pose estimate for ego-motion. Our work closely\nrelates to these data-driven approaches that have recently been\ndeveloped. We provide a qualitative comparison of how our approach is\npositioned within the visual ego-motion estimation landscape in\nTable~\\ref{table:vo-landscape}. \n\n\n\\begin{table}[h]\n  \\centering\n  \\scriptsize\n  \\rowcolors{2}{gray!25}{white}\n  {\\renewcommand{\\arraystretch}{1}\n    {\\setlength{\\tabcolsep}{0.2mm}\n      \\begin{tabular}{lM{1cm}M{1cm}M{1cm}M{1.25cm}}\n        \\toprule\n        \\centering\n       \n        \\textbf{Method Type} & \\textbf{Varied Optics} & \\textbf{Model Free} & \\textbf{Robust} \n        & \\textbf{Self Supervised} \\\\ \\midrule\n        \\textit{Traditional VO~\\cite{scaramuzza2011visual}}\n                        & $\\text{\\ding{55}}$ & $\\text{\\ding{55}}$ & $\\text{\\ding{51}}$ & $\\text{\\ding{55}}$ \\\\ \n        \\textit{End-to-end VO~\\cite{costante2016exploring,wen2016vinet}}\n                        & $\\text{\\ding{55}}$ & $\\text{\\ding{51}}$ & $\\text{\\ding{51}}$ & $\\text{\\ding{55}}$ \\\\ \n                       \n        \\textit{This work}\n                        & $\\text{\\ding{51}}$ & $\\text{\\ding{51}}$ & $\\text{\\ding{51}}$ & $\\text{\\ding{51}}$ \\\\\n        \\bottomrule\n      \\end{tabular}}}\n\\caption{\\textbf{Visual odometry landscape}: A qualitative comparison of how our approach is\n  positioned amongst existing solutions to ego-motion\n  estimation.}\n\\label{table:vo-landscape}\\vspace{-5mm}\n\\end{table}\n\n\n\n\n\n\\section{Ego-motion regression}\\label{sec:procedure}\n\nAs with most ego-motion estimation solutions, it is imperative to\ndetermine the minimal parameterization of the underlying motion\nmanifold. In certain restricted scene structures or motion manifolds,\nseveral variants of ego-motion estimation are\nproposed~\\cite{scaramuzza20111,liang2002visual,ke2003transforming,scaramuzza2009real}.\nHowever, we consider the case of modeling cameras with varied optics\nand hence are interested in determining the full range of ego-motion,\noften restricted, that induces the pixel-level optical flow. This\nallows the freedom to model various unconstrained and partially\nconstrained motions that typically affect the overall robustness of\nexisting ego-motion algorithms.  While model-based approaches have\nshown tremendous progress in accuracy, robustness, and run-time\nperformance, a few recent data-driven approaches have been shown to\nproduce equally compelling\nresults~\\cite{costante2016exploring,wen2016vinet,konda2015learning}. An\nadaptive and trainable solution for relative pose estimation or\nego-motion can be especially advantageous for several reasons: (i) a\ngeneral-purpose end-to-end trainable model architecture that applies to a\nvariety of camera optics including pinhole, fisheye, and catadioptric\nlenses; (ii) simultaneous and continuous optimization over both ego-motion estimation\nand camera parameters (intrinsics and extrinsics that are implicitly\nmodeled); and (iii) joint reasoning over resource-aware\ncomputation and accuracy within the same architecture is amenable.  We\nenvision that such an approach is especially beneficial in the context\nof bootstrapped (or weakly-supervised) learning in robots, where the\nsupervision in ego-motion estimation for a particular camera can be\nobtained from the fusion of measurements from other robot\nsensors (GPS, wheel encoders etc.).\n\n\n\nOur approach is motivated by previous minimally parameterized\nmodels~\\cite{scaramuzza20111,scaramuzza2009real} that are able to\nrecover ego-motion from a \\textit{single tracked feature}. We find\nthis representation especially appealing due to the simplicity and\nflexibility in~\\textit{pixel-level} computation. Despite the reduced\ncomplexity of the input space for the mapping problem, recovering the\nfull 6-DOF ego-motion is ill-posed due to the inherently\nunder-constrained system. However, it has been previously shown that\nunder non-holonomic vehicle motion, camera ego-motion may be fully\nrecoverable up to a sufficient degree of accuracy using a single\npoint~\\cite{scaramuzza20111,scaramuzza2009real}.\n\nWe now focus on the specifics of\nthe ego-motion regression objective. Due to the under-constrained\nnature of the prescribed regression problem, the pose estimation is\nmodeled as a density estimation problem over the range of possible\nego-motions\\footnote{\\scriptsize Although the parametrization is\n  maintained as $SE(3)$, it is important to realize\nthat the nature of most autonomous car datasets involve a\nlower-dimensional ($SE(2)$) motion manifold}, conditioned on the\ninput flow features. It is important to note that the output of the\nproposed model is a density estimate\n$p(\\hat{\\mathbf{z}}_{t-1,t}\\vert\\mathbf{x}_{t-1,t})$ for every feature tracked\nbetween subsequent frames.\n\n\n\n\\subsection{Density estimation for\n  ego-motion}\\label{subsec:density-egomotion}\nIn typical associative mapping problems, the joint probability density\n$p(\\mathbf{x},\\mathbf{z})$ is decomposed into the product of two terms: (i) $p(\\mathbf{z}\n\\vert \\mathbf{x})$: the conditional density of the target pose $\\mathbf{z} \\in\nSE(3)$ conditioned on the input feature correspondence $\\mathbf{x} = (\\bm{x},\n\\Delta\\bm{x})$ obtained from sparse optical flow (KLT)~\\cite{birchfield2007klt} (ii)\n$p(\\mathbf{x})$: the unconditional density of the input data $\\mathbf{x}$. While we\nare particularly interested in the first term $p(\\mathbf{z}\\vert\\mathbf{x})$ that\npredicts the range of possible values for $\\mathbf{z}$ given new values of\n$\\mathbf{x}$, we can observe that the density $p(\\mathbf{x}) = \\sum_z p(\\mathbf{x},\\mathbf{z})\nd\\mathbf{z}$ provides a measure of how well the prediction is captured by the\ntrained model.\n\nThe critical component in estimating the ego-motion belief is the\nability to accurately predict the conditional probability distribution\n$p(\\mathbf{z}\\vert\\mathbf{x})$ of the pose estimates that is induced by the given\ninput feature $\\bm{x}$ and the flow $\\Delta\\bm{x}$. Due to its powerful and\nrich modeling capabilities, we use a \\textit{Mixture Density Network}\n(MDN)~\\cite{bishop1994mixture} to parametrize the conditional density\nestimate. MDNs are a class of end-to-end trainable\n(fully-differentiable) density estimation techniques that leverage\nconventional neural networks to regress the parameters of a generative\nmodel such as a finite Gaussian Mixture Model (GMM). The powerful representational\ncapacity of neural networks coupled with rich probabilistic modeling\nthat GMMs admit, allows us to model multi-valued or multi-modal\nbeliefs that typically arise in inverse problems such as visual\nego-motion.\n\nFor each of the $F$ input flow features $\\mathbf{x}_i$ extracted via KLT, the\nconditional probability density of the target pose data $\\mathbf{z}_i$\n(Eqn~\\ref{eq:cdf}) is\nrepresented as a convex combination of $K$ Gaussian components,\n\\vspace{-1mm}\n\\begin{align}\\label{eq:cdf}\n  p(\\mathbf{z}_i \\mid \\mathbf{x}_i) = \\sum_{k=1}^{K} \\pi_k(\\mathbf{x}_i) \\mathcal{N}(\\mathbf{z} \\mid \\mu_k(\\mathbf{x}_i), \\sigma_{k}^2(\\mathbf{x}_i))\n\\end{align}where $\\pi_k(\\mathbf{x})$ is the mixing coefficient for the $k$-th component\nas specified in a typical GMM. The Gaussian\nkernels are parameterized by their mean vector $\\mu_k(\\mathbf{x})$ and\ndiagonal covariance $\\sigma_{k}(\\mathbf{x})$. It is important to note that\nthe parameters $\\pi_k(\\mathbf{x}), \\mu_k(\\mathbf{x})$, and $\\sigma_k(\\mathbf{x})$ are\ngeneral and continuous functions of $\\mathbf{x}$. This allows us to model\nthese parameters as the output ($a^{\\pi}$, $a^{\\mu}$,\n$a^{\\sigma}$) of a conventional neural network which takes $\\mathbf{x}$ as\nits input. Following~\\cite{bishop1994mixture}, the outputs of the\nneural network are constrained as follows: (i) The mixing coefficients\nmust sum to 1, i.e. $\\sum_{K}\\pi_k(\\mathbf{x}) = 1$ where $0 \\leq \\pi_k(\\mathbf{x})\n\\leq 1$. This is accomplished via the \\textit{softmax} activation as\nseen in Eqn~\\ref{eq:mdn-pi}. (ii) Variances $\\sigma_k(\\mathbf{x})$ are\nstrictly positive via the \\textit{exponential} activation\n(Eqn~\\ref{eq:mdn-sigma}).\n\\begin{align}\n  &\\pi_k(\\mathbf{x}) = \\frac{\\exp(a_k^\\pi)} { \\sum_{l=1}^{K} \\exp(a_l^\\pi) }\\label{eq:mdn-pi} \\\\\n  &\\sigma_k(\\mathbf{x}) = \\exp(a_k^\\sigma), \\hspace{4mm}\n    \\mu_{k}(\\mathbf{x}) = a_{k}^\\mu \\label{eq:mdn-sigma} \\vspace{4mm}\\\\\n  \\mathcal{L_{MDN}} = -\n  &\\sum_{n=1}^{N} \\ln \\Bigg\\{ \\sum_{k=1}^{K} \\pi_k(\\mathbf{x}_n) \\mathcal{N}(\\mathbf{z} \\mid\n    \\mu_k(\\mathbf{x}_n), \\sigma_{k}^2(\\mathbf{x}_n)) \\Bigg\\} \\label{eq:nll}\n\\end{align}\nThe proposed model is learned end-to-end by maximizing the data\nlog-likelihood, or alternatively minimizing the negative log-likelihood\n(denoted as $\\mathcal{L_{MDN}}$ in Eqn~\\ref{eq:nll}), given the $F$ input feature tracks ($\\mathbf{x}_1\\dots\\mathbf{x}_F$) and expected\nego-motion estimate $\\mathbf{z}$. The resulting ego-motion density estimates\n$p(\\mathbf{z}_i\\vert\\mathbf{x}_i)$\nobtained from\neach individual flow vectors $\\mathbf{x}_i$ are then fused by taking the\nproduct of their densities. However, to\nmaintain tractability of density products, only the mean and\ncovariance corresponding to the largest mixture coefficient (i.e. most\nlikely mixture mode) for each feature is considered for subsequent trajectory\noptimization (See Eqn~\\ref{eq:density-products}).\n\\begin{align}\\label{eq:density-products}\n  p(\\mathbf{z} \\vert \\mathbf{x}) \\simeq \\prod_{i=1}^{F} \\max_{k} \\Big\\{\\pi_k(\\mathbf{x}_i)\n  \\mathcal{N}(\\mathbf{z}_i \\mid \\mu_k(\\mathbf{x}_i), \\sigma_{k}^2(\\mathbf{x}_i)) \\Big\\}\n\\end{align}\n\n\n\\begin{figure}[!t]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{graphics\/losses-graphic.pdf}\n  \\caption{\\textbf{Windowed trajectory optimization}: An illustration\n    of the losses introduced for training frame-to-frame ego-motion\n    (\\textit{local}) and windowed ego-motion (\\textit{global}) by\n    compounding the poses determined from each of the individual\n    frame-to-frame measurements.}\n  \\label{fig:losses-illustration}\n  \\vspace{-6mm}\n\\end{figure}\n\n\\subsection{Trajectory\n  optimization}\\label{sec:trajectory-optimization} While minimizing the MDN loss\n($\\mathcal{L}_{MDN}$) as described above provides a reasonable\nregressor for ego-motion estimation, it is evident that optimizing\nframe-to-frame measurements do not ensure long-term consistencies in\nthe ego-motion trajectories obtained by integrating these regressed\nestimates. As one expects, the integrated trajectories are sensitive\nto even negligible biases in the ego-motion regressor.\n\n\\textbf{Two-stage optimization}: To circumvent the aforementioned issue, we introduce a\nsecond optimization stage that jointly minimizes the \n\\textit{local} objective ($\\mathcal{L}_{MDN}$) with a \\textit{global} objective that\nminimizes the error incurred between the overall trajectory and the\ntrajectory obtained by integrating the regressed pose estimates\nobtained via the \\textit{local} optimization. This allows the\n\\textit{global} optimization stage to have a warm-start with an almost\ncorrect initial guess for the network parameters.\n\nAs seen in Eqn~\\ref{eq:losses}, $\\mathcal{L}_{TRAJ}$ pertains to the\noverall trajectory error incurred by integrating the individual\nregressed estimates over a batched window (we typically consider 200\nto 1000 frames). This allows us to fine-tune the regressor to predict\nvalid estimates that integrate towards accurate long-term ego-motion\ntrajectories. As\nexpected, the model is able to roughly learn the curved trajectory\npath, however, it is not able to make accurate predictions when\nintegrated for longer time-windows (due to the lack of the\n\\textit{global} objective loss term in Stage\n1). Figure~\\ref{fig:losses-illustration} provides a high-level\noverview of the input-output relationships of the training procedure,\nincluding the various network losses incorporated in the ego-motion\nencoder\/regressor. For illustrative purposes only, we refer the reader to\nFigure~\\ref{fig:two-stage-illustration} where we validate this\ntwo-stage approach over a simulated dataset~\\cite{Zhang2016ICRA}. \n\nIn Eqn~\\ref{eq:losses}, $\\hat{\\mathbf{z}}_{{\\scriptsize t-1,t}}$ is the\nframe-to-frame ego-motion estimate and the regression target\/output of the MDN\nfunction $F$, where $F:\\mathbf{x} \\mapsto \\Big(\\mu({\\mathbf{x}}_{{\\scriptsize\nt-1,t}}), \\sigma(\\mathbf{x}_{{\\scriptsize t-1,t}}), \\pi(\\mathbf{x}_{{\\scriptsize\nt-1,t}})\\Big)$. $\\hat{\\mathbf{z}}_{1,t}$ is the overall trajectory\npredicted by integrating the individually regressed frame-to-frame\nego-motion estimates and is defined by $\\hat{\\mathbf{z}}_{1,t} = \\hat{\\mathbf{z}}_{1,2} \\oplus \\hat{\\mathbf{z}}_{2,3} \\oplus\n\\dots \\oplus \\hat{\\mathbf{z}}_{t-1,t}$.\n\\begin{equation}\n  \\begin{aligned}\n    \\mathcal{L_{\\text{ENC}}} =\n    \\underbrace{\\sum_{t} \\mathcal{L}^t_{{\\scriptsize MDN}}\\Big( F(\\mathbf{x}),\n    \\mathbf{z}_{{\\scriptsize t-1,t}}\\Big)}_{\\text{MDN Loss}} + \n    \\underbrace{\\sum_{t} \\mathcal{L}^t_{{\\scriptsize TRAJ}}(\\mathbf{z}_{1,t} \\ominus\n      \\hat{\\mathbf{z}}_{1,t})}_{\\text{Overall Trajectory Loss}}\n  \\end{aligned}\n  \\label{eq:losses}           \n\\end{equation}\n\\vspace{-5mm} \n\n\n\\begin{figure}[!t]\n  \\centering\n  \\centering \n  {\\renewcommand{\\arraystretch}{0.4}\n    {\\setlength{\\tabcolsep}{0.2mm}\n      \\begin{tabular}{cccc}\n       \n       \n       \n       \n        \\includegraphics[width=0.25\\columnwidth]{graphics\/learning\/rpg\/map_02.pdf} \n        &\\includegraphics[width=0.25\\columnwidth]{graphics\/learning\/rpg\/map_04.pdf}\n        &\\includegraphics[width=0.25\\columnwidth]{graphics\/learning\/rpg\/map_08.pdf}\n        &\\includegraphics[width=0.25\\columnwidth]{graphics\/learning\/rpg\/map_18.pdf}\\\\\n        {\\scriptsize \\textbf{Stage 1}} & {\\scriptsize \\textbf{Stage 2}}\n        &{\\scriptsize \\textbf{Stage 2}} & {\\scriptsize \\textbf{Stage 2}}\\\\\n        {\\scriptsize (Final)} & {\\scriptsize (Epoch 4)}\n        &{\\scriptsize (Epoch 8)} & {\\scriptsize (Epoch 18)}\n      \\end{tabular}}}\n  \\caption{\\textbf{Two-stage Optimization}: An illustration of the\n    two-stage optimization procedure. The \\textit{first} column shows\n    the final solution after the first stage. Despite the\n    minimization, the integrated trajectory is clearly biased and\n    poorly matches the expected result. The \\textit{second},\n    \\textit{third} and \\textit{fourth} column shows the gradual improvement of the\n    second stage (global\n    minimization) and matches the expected ground truth trajectory\n    better (i.e. estimates the regressor biases better).}\n  \\label{fig:two-stage-illustration}\n  \\vspace{-5mm}\n\\end{figure}\n\n\n\n\n\n\n\n\n\\subsection{Bootstrapped learning for ego-motion\nestimation}\\label{sec:proc-bootstrapped} Typical robot navigation\nsystems consider the fusion of visual odometry estimates with other\nmodalities including estimates derived from wheel encoders, IMUs, GPS\netc. Considering odometry estimates (for e.g. from wheel encoders)\nas-is, the uncertainties in open-loop chains grow in\nan unbounded manner. Furthermore, relative pose estimation may also be\ninherently biased due to calibration errors that eventually contribute\nto the overall error incurred. GPS, despite being noise-ridden,\nprovides an absolute sensor reference measurement that is especially\ncomplementary to the open-loop odometry chain maintained with odometry\nestimates. The probabilistic fusion of these two relatively\nuncorrelated measurement modalities allows us to recover a\nsufficiently accurate trajectory estimate that can be directly used\nas ground truth data $\\mathbf{z}$ (in Figure~\\ref{fig:egomotion-regression-illustration}) for our supervised regression problem.\n\n\n\n\n\n\\begin{figure}[!b]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{graphics\/egomotion-regression-graphic.pdf}\n  \\caption{\\textbf{Bootstrapped Ego-motion Regression}:\n    Illustration of the bootstrap mechanism whereby a robot\n    self-supervises the proposed ego-motion regression task in a new\n    camera sensor by fusing information from other sensor sources\n    such as GPS and INS.}\n  \\label{fig:egomotion-regression-illustration}\n  \\vspace{-4mm}\n\\end{figure}\n\n\\begin{figure}[!b]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{graphics\/egomotion-deployment-graphic.pdf}\n  \\caption{\\textbf{Learned Ego-motion Deployment}:\n    During model deployment, the learned visual-egomotion model\n    provides valuable relative pose constraints to augment the standard\n    navigation-based sensor fusion (GPS\/INS and wheel encoder odometry\n    fusion).}\n  \\label{fig:egomotion-deployment-illustration}\n  \\vspace{-2mm}\n\\end{figure}\n\nThe indirect recovery of training data from the fusion of other sensor\nmodalities in robots falls within the \\textit{self-supervised or\nbootstrapped} learning paradigm. We envision this capability to be\nespecially beneficial in the context of life-long learning in future\nautonomous systems. Using the fused and optimized pose estimates $\\mathbf{z}$\n(recovered from GPS and odometry estimates), we are able to recover\nthe required input-output relationships for training visual ego-motion\nfor a completely new sensor (as illustrated in\nFigure~\\ref{fig:egomotion-regression-illustration}). Figure~\\ref{fig:egomotion-deployment-illustration}\nillustrates the realization of the learned model in a typical\nautonomous system where it is treated as an additional sensor\nsource. Through experiments~\\ref{sec:bootstrap-exp}, we illustrate\nthis concept with the recovery of ego-motion in a robot car equipped\nwith a GPS\/INS unit and a single camera.\n\n\n\n\\subsection{Introspective Reasoning for Scene-Flow Prediction} Scene\nflow is a fundamental capability that provides directly measurable\nquantities for ego-motion analysis. The flow observed by sensors\nmounted on vehicles is a function of the inherent scene depth, the\nrelative ego-motion undergone by the vehicle, and the intrinsic and\nextrinsic properties of the camera used to capture it. As with any\nmeasured quantity, one needs to deal with sensor-level noise\npropagated through the model in order to provide robust\nestimates. While the input flow features are an indication of\nego-motion, some of the features may be corrupted due to lack of or\nambiguous visual texture or due to flow induced by the dynamics of\nobjects other than the ego-motion itself. Evidently, we observe that\nthe dominant flow is generally induced by ego-motion itself, and it is\nthis flow that we intend to fully recover via a conditional\nvariational auto-encoder (C-VAE). By inverting the regression problem,\nwe develop a generative model able to predict the most-likely flow\n$\\hat{\\Delta x}$ induced given an ego-motion estimate $\\mathbf{z}$, and\nfeature location $x$. We propose a scene-flow specific autoencoder\nthat encodes the implicit egomotion observed by the sensor, while\njointly reasoning over the latent depth of each of the individual\ntracked features. \n\n\n\\begin{equation}\n  \\begin{aligned}\n    \\mathcal{L_{\\text{CVAE}}} =& \\mathbb{E}{\\big [} \\log p_{\\theta}(\\Delta x\n    | \\mathbf{z},x) {\\big]}\\\\\n                              &- D_{KL}\\big[q_{\\phi}(\\mathbf{z} |\n                              x,\\Delta x) ||\n                              p_{\\theta}(\\mathbf{z}|x)\\big] \n  \\end{aligned}\n  \\label{eq:cvae}                              \n\\end{equation}\n\nThrough the proposed denoising autoencoder model, we are also able to\nattain an introspection mechanism for the presence of outliers. We\nincorporate this additional module via an auxiliary loss as specified\nin Eqn~\\ref{eq:cvae}. An illustration of these flow\npredictions are shown in Figure~\\ref{fig:evaluation-flow-prediction}.\n\n\n\n\n\n\\section{Discussion}\\label{sec:discussion} The initial results in\nbootstrapped learning for visual ego-motion has motivated new\ndirections towards life-long learning in autonomous robots. While our\nvisual ego-motion model architecture is shown to be sufficiently\npowerful to recover ego-motions for non-linear camera optics such as\nfisheye and catadioptric lenses, we continue to investigate further\nimprovements to match existing state-of-the-art models for these lens\ntypes. Our current model does not capture distortion effects yet,\nhowever, this is very much a future direction we would like to\ntake. Another consideration is the resource-constrained setting, where\nthe optimization objective incorporates an additional regularization\nterm on the number of parameters used, and the computation load\nconsumed. We hope for this resource-aware capability to transfer to\nreal-world limited-resource robots and to have a significant impact on\nthe adaptability of robots for long-term autonomy.\n\n\\section{Conclusion}\\label{sec:conclusion} While many visual\nego-motion algorithm variants have been proposed in the past decade,\nwe envision that a fully end-to-end trainable algorithm for generic\ncamera ego-motion estimation shall have far-reaching implications in\nseveral domains, especially autonomous systems. Furthermore, we expect\nour method to seamlessly operate under resource-constrained situations\nin the near future by leveraging existing solutions in model reduction\nand dynamic model architecture tuning. With the availability of\nmultiple sensors on these autonomous systems, we also foresee our\napproach to bootstrapped task (visual ego-motion) learning to\npotentially enable robots to learn from experience, and use the new\nmodels learned from these experiences to encode redundancy and\nfault-tolerance all within the same framework.\n\n\n\\section{Experiments}\\label{sec:experiments} In this section, we\nprovide detailed experiments on the performance, robustness and\nflexibility of our proposed approach on various datasets. Our approach\ndifferentiates itself from existing solutions on various fronts as\nshown in Table~\\ref{table:vo-landscape}. We evaluate the performance\nof our proposed approach on various publicly-available datasets\nincluding the KITTI dataset~\\cite{Geiger2012CVPR}, the Multi-FOV synthetic\ndataset~\\cite{Zhang2016ICRA} (pinhole, fisheye, and catadioptric\nlenses), an omnidirectional-camera\ndataset~\\cite{schonbein2014omnidirectional}, and on the Oxford Robotcar\n1000km Dataset~\\cite{maddern20161}.\n\nNavigation solutions in autonomous systems today typically fuse\nvarious modalities including GPS, odometry from wheel encoders and INS\nto provide robust trajectory estimates over extended periods of\noperation. We provide a similar solution by leveraging the learned\nego-motion capability described in this work, and fuse it with\nintermittent GPS updates\\footnote{\\scriptsize For evaluation purposes\nonly, the absolute ground truth locations were added as weak priors on\ndatasets without GPS measurements}\n(Secion~\\ref{sec:performance}). While maintaining similar performance\ncapabilities (Table~\\ref{tab:trajectory-errors}), we re-emphasize the benefits\nof our approach over existing solutions:\n\\begin{itemize}\n\\item \\textbf{Versatile}: With\na fully trainable model, our approach is able to\nsimultaneously reason over both ego-motion and implicitly modeled camera parameters\n(\\textit{intrinsics} and \\textit{extrinsics}). Furthermore, online calibration and\nparameter tuning is implicitly encoded within the same learning\nframework.\n\\item \\textbf{Model-free}: Without imposing any constraints\non the type of camera optics, our approach is able to recover\nego-motions for a variety of camera models including \\textit{pinhole},\n\\textit{fisheye} and \\textit{catadioptric}\nlenses. (Section~\\ref{sec:model-free})\n\\item \\textbf{Bootstrapped training and refinement}: We illustrate a bootstrapped\nlearning example whereby a robot self-supervises the proposed\nego-motion regression task by fusing information from other sensor sources\nincluding GPS and INS (Section~\\ref{sec:bootstrap-exp})\n\\item \\textbf{Introspective reasoning for scene-flow prediction}: Via the C-VAE\ngenerative model,\nwe are able to reason\/introspect over the predicted flow vectors in\nthe image given an ego-motion estimate. This provides an obvious\nadvantage in \\textit{robust} outlier detection and identifying dynamic\nobjects whose flow vectors need to be disambiguated from the\nego-motion scene flow (Figure~\\ref{fig:evaluation-flow-prediction})\n\\end{itemize}\n\n\\input{tex\/fig-qualitative-results}\n\n\n\\subsection{Evaluating ego-motion performance with sensor fusion}\n\\label{sec:performance} In this section, we evaluate our approach\nagainst a few state-of-the-art algorithms for monocular visual\nodometry~\\cite{kitt2010visual}. On the KITTI\ndataset~\\cite{Geiger2012CVPR}, the pre-trained estimator is used to\nrobustly and accurately predict ego-motion from KLT features tracked\nover the dataset image sequence. The frame-to-frame ego-motion\nestimates are integrated for each session to recover the full\ntrajectory estimate and simultaneously fused with intermittent GPS\nupdates (incorporated every 150 frames). In\nFigure~\\ref{fig:egomotion-fusion}, we show the qualitative performance\nin the overall trajectory obtained with our method.  The entire\npose-optimized trajectory is compared against the ground truth\ntrajectory. The\ntranslational errors are computed for each of the ground truth and\nprediction pose pairs, and their median value is reported in\nTable~\\ref{tab:trajectory-errors} for a variety of datasets with\nvaried camera optics.\n\n\n\\subsection{Varied camera optics}\\label{sec:model-free} Most of the\nexisting implementations of VO estimation are restricted to a class of\ncamera optics, and generally avoid implementing a general-purpose VO estimator for\nvaried camera optics. Our approach on the other hand, has shown the ability to provide\naccurate VO with intermittent GPS trajectory estimation while\nsimultaneously being applicable to a varied range of camera\nmodels. In\nFigure~\\ref{fig:egomotion-evaluation-varied-optics}, we compare with \nintermittent GPS trajectory estimates for all three camera models, and\nverify their performance accuracy compared to ground truth. In our\nexperiments, we found that while our proposed solution was\nsufficiently powerful to model different camera optics, it was\nsignificantly better at modeling pinhole lenses as compared to fisheye\nand catadioptric cameras (See Table~\\ref{tab:trajectory-errors}). In\nfuture work, we would like to investigate further extensions that\nimprove the accuracy for both fisheye and catadioptric lenses.\n\n\\input{tex\/tab-trajectory-prediction-performance}\n\n\\begin{figure}[h]\n  \\centering \n  {\\renewcommand{\\arraystretch}{0.1}\n    {\\setlength{\\tabcolsep}{0.1mm}\n      \\begin{tabular}{ccc}\n        \\includegraphics[width=0.33\\columnwidth]{graphics\/trajectory-prediction\/rpg\/rpg_urban_pinhole_map_500_update.pdf}&\n        \\includegraphics[width=0.33\\columnwidth]{graphics\/slam\/rpg-fisheye-vo-slam\/rpg_urban_fisheye_map_500_update.pdf}&\n        \\includegraphics[width=0.33\\columnwidth]{graphics\/slam\/rpg-cata-vo-slam\/rpg_urban_cata_map_500_update.pdf}\\\\\n        \\scriptsize \\textbf{Pinhole} & \\scriptsize \\textbf{Fisheye} & \\scriptsize \\textbf{Catadioptric}\n      \\end{tabular}}}\n  \\caption{\\textbf{Varied camera optics:} An illustration of the\n    performance of our general-purpose approach for varied camera optics\n    (pinhole, fisheye, and catadioptric lenses) on the\n    Multi-FOV synthetic dataset~\\cite{Zhang2016ICRA}. Without any prior\n    knowledge on the camera optics, or the mounting configuration\n    (extrinsics), we are able to robustly and accurately recover the full\n    trajectory of the vehicle (with intermittent GPS updates every 500\n    frames).}\n  \\label{fig:egomotion-evaluation-varied-optics}\n  \\vspace{-6mm}\n\\end{figure}\n\n\\subsection{Self-supervised Visual Ego-motion Learning in Robots}\n\\label{sec:bootstrap-exp} We\nenvision the capability of robots to self-supervise tasks such as\nvisual ego-motion estimation to be especially beneficial in the\ncontext of life-long learning and autonomy. We experiment and validate\nthis concept through a concrete example using the 1000km Oxford Robot\nCar dataset~\\cite{maddern20161}. We train the task of visual\nego-motion on a new camera sensor by leveraging the fused GPS and INS\ninformation collected on the robot car as ground truth trajectories\n(6-DOF), and extracting feature trajectories (via KLT) from image\nsequences obtained from the new\ncamera sensor. The timestamps from the cameras are synchronized\nwith respect to the timestamps of the fused GPS and INS information,\nin order to obtain a one-to-one mapping for training purposes. We\ntrain on the \\texttt{stereo\\_centre} \\textit{(pinhole)} camera dataset\nand present our results in Table~\\ref{tab:trajectory-errors}. As seen\nin Figure~\\ref{fig:egomotion-fusion}, we are able to achieve\nconsiderably accurate long-term state estimates by fusing our proposed\nvisual ego-motion estimates with even sparser GPS updates (every 2-3\nseconds, instead of 50Hz GPS\/INS readings). This allows the robot to\nreduce its reliance on GPS\/INS alone to perform robust, long-term\ntrajectory estimation.\n\n\n\\begin{figure}[!t]\n  \\centering \n  {\\renewcommand{\\arraystretch}{0.4}\n    {\\setlength{\\tabcolsep}{0.4mm}\n      \\begin{tabular}{cccc}\n        \\rotatebox[x=-6mm]{90}{\\scriptsize \\textbf{Image}}& \n          \\includegraphics[width=0.29\\columnwidth,frame={\\fboxrule}]{graphics\/flow-prediction\/pinhole-img0250_0.png}&\n        \\includegraphics[width=0.29\\columnwidth,frame={\\fboxrule}]{graphics\/flow-prediction\/fisheye-img0250_0.png}&\n        \\includegraphics[width=0.29\\columnwidth,frame={\\fboxrule}]{graphics\/flow-prediction\/cata-img0250_0.png}\\\\\n        \\rotatebox[x=-5.5mm]{90}{\\scriptsize \\textbf{Forward}}&\n          \\includegraphics[width=0.3\\columnwidth]{graphics\/flow-prediction\/pinhole-forward-flow.pdf}&\n        \\includegraphics[width=0.3\\columnwidth]{graphics\/flow-prediction\/fisheye-forward-flow.pdf}&\n        \\includegraphics[width=0.3\\columnwidth]{graphics\/flow-prediction\/cata-forward-flow.pdf}\\\\\n       \n       \n       \n       \n       \n       \n       \n       \n        & \\scriptsize \\textbf{(a) Pinhole} & \\scriptsize \\textbf{(b) Fisheye} & \\scriptsize \\textbf{(c) Catadioptric}\\\\\n      \\end{tabular}}}\n  \\caption{\\textbf{Introspective reasoning for scene-flow prediction}:\n    Illustrated above are the dominant flow vectors corresponding to\n    scene-flow given the corresponding ego-motion. While this module is\n    not currently used in the ego-motion estimation, we expect it to be\n    critical in outlier rejection. \\textbf{Row 1}: Sample image from\n    camera, \\textbf{Row 2}: Flow induced by forward motion\n   \n   \n  }\n  \\label{fig:evaluation-flow-prediction}\n  \\vspace{-6mm}\n\\end{figure}\n\n\n\n\n\\subsection{Implementation Details}\\label{sec:implementation}\nIn this section we describe the details of our proposed model,\ntraining methodology and parameters used. The input $\\mathbf{x} = (\\bm{x},\n\\Delta\\bm{x})$ to the density-based ego-motion estimator are feature\ntracks extracted via (Kanade-Lucas-Tomasi) KLT feature tracking over\nthe raw camera image sequences. The input feature positions and flow\nvectors are normalized to be the in range of $[-1,1]$ using the\ndimensions of the input image. We evaluate sparse LK (Lucas-Kanade)\noptical flow over 7 pyramidal scales with a scale factor of\n$\\sqrt{2}$. As the features are extracted, the corresponding robot\npose (either available via GPS or GPS\/INS\/wheel odometry sensor\nfusion) is synchronized and recorded in $SE(3)$ for training\npurposes. The input KLT features, and the corresponding relative pose\nestimates used for training are parameterized as $\\mathbf{\\mathbf{z}} =\n(\\mathbf{t},\\mathbf{r}) \\in \\mathbb{R}^6$, with a Euclidean\ntranslation vector $\\mathbf{t} \\in \\mathbb{R}^3$ and an Euler rotation\nvector $\\mathbf{r} \\in \\mathbb{R}^3$.\n\n\\textbf{Network and training:} The proposed architecture\nconsists of a set of fully-connected stacked layers (with 1024, 128\nand 32 units) followed by a Mixture Density Network with 32 hidden\nunits and 5 mixture components ($K$). Each of the initial\nfully-connected layers implement \\textit{tanh} activation after it,\nfollowed by a dropout layer with a dropout rate of 0.1. The final\noutput layer of the MDN ($a^{\\pi}$, $a^{\\mu}$,\n$a^{\\sigma}$) consists of $(O + 2) * K$ outputs where $O$ is the\ndesired number of states estimated. \n\nThe network is trained (in Stage 1) with loss weights of 10, 0.1, 1\ncorresponding to the losses $\\mathcal{L}_{MDN}, \\mathcal{L}_{TRAJ},\n\\mathcal{L}_{CVAE}$ described in previous sections. The training data\nis provided in batches of 100 frame-to-frame subsequent image pairs,\neach consisting of approximately 50 randomly sampled feature matches\nvia KLT. The learning rate is set to $1\\mathrm{e}{-3}$ with Adam as\nthe optimizer. On the synthetic Multi-FOV dataset and the KITTI\ndataset, training for most models took roughly an hour and a half\n(3000 epochs) independent of the KLT feature extraction step.\n\n\\textbf{Two-stage optimization}: We found the one-shot joint\noptimization of the \\textit{local} ego-motion estimation and\n\\textit{global} trajectory optimization to have sufficiently low\nconvergence rates during training. One possible explanation is the\nhigh sensitivity of the loss weight parameters that is used for tuning\nthe local and global losses into a single objective. As previously addressed in\nSection~\\ref{sec:trajectory-optimization}, we\nseparate the training into two stages thereby alleviating the\naforementioned issues, and maintaining fast convergence rates in Stage\n1. Furthermore, we note that during the second stage, it only requires\na few tens of iterations for sufficiently accurate ego-motion\ntrajectories.  In order to optimize over a larger time-window in stage\n2, we set the batch size to 1000 frame-to-frame image matches, again\nrandomly sampled from the training set as before. Due to the large integration\nwindow and memory limitations, we train this stage purely on the CPU\nfor only 100 epochs each taking roughly 30s per epoch. Additionally,\nin stage 2, the loss weights for $\\mathcal{L}_{TRAJ}$ are increased to\n100 in order to have faster convergence to the \\textit{global}\ntrajectory. The remaining loss weights are left unchanged.\n\n\\textbf{Trajectory fusion}: We use\nGTSAM\\footnote{\\scriptsize\\url{http:\/\/collab.cc.gatech.edu\/borg\/gtsam}}\nto construct the underlying factor graph for pose-graph\noptimization. Odometry constraints obtained from the frame-to-frame\nego-motion are incorporated as a 6-DOF constraint parameterized in $SE(3)$ with $1*10^{-3}$~rad rotational noise\nand $5*10^{-2}$~m translation noise. As with typical autonomous\nnavigation solutions, we expect measurement updates in the form of GPS\n(absolute reference updates) in order to correct for the long-term\ndrift incurred in open-loop odometry chains. We incorporate absolute\nprior updates only every 150 frames, with a weak translation prior of $0.01$~m. The constraints\nare incrementally added and solved using iSAM2~\\cite{kaess2012isam2} as\nthe measurements are streamed in, with updates performed every 10\nframes.\n\nWhile the proposed MDN is parametrized in Euler angles, the\n\\textit{trajectory integration module} parameterizes the rotation\nvectors in quaternions for robust and unambiguous long-term trajectory\nestimation. All the rigid body transformations are implemented\ndirectly in Tensorflow for pure-GPU training support.\n\n\\textbf{Run-time performance}: We are particularly interested in the\nrun-time \/ test-time performance of our approach on CPU architectures\nfor mostly resource-constrained settings. Independent of the KLT feature\ntracking run-time, we are able to recover ego-motion estimates at\nroughly 3ms on a consumer-grade Intel(R) Core(TM)\ni7-3920XM CPU @ 2.90GHz.\n\n\n\\textbf{Source code and Pre-trained weights}: We implemented the\nMDN-based ego-motion estimator with Keras and Tensorflow, and trained\nour models using a combination of CPUs and GPUs (NVIDIA Titan X). All\nthe code was trained on an server-grade Intel(R) Xeon(R) CPU E5-2630\nv3 @ 2.40GHz and tested on the same consumer-grade machine as\nmentioned above to\nemulate potential real-world use-cases.  The source code and\npre-trained models used will be made available\nshortly\\footnote{\\scriptsize\nSee~\\url{http:\/\/people.csail.mit.edu\/spillai\/learning-egomotion}\nand~\\url{https:\/\/github.com\/spillai\/learning-egomotion}}.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe study on defaultable corporate bond and credit risk is now one of the most promising areas of cutting edge in financial mathematics. There are two main approaches to modeling credit risk and pricing defaultable corporate bonds; one is the {\\it structural approach} and the other one is the {\\it reduced form approach}. In the structural method, we think that the default event occurs when the firm value is not enough to repay debt, that is, the firm value reaches a certain lower threshold ({\\it default barrier}) from the above. Such a default can be expected and thus we call it {\\it expected default}. In the reduced-form approach, the default is treated as an unpredictable event governed by a default intensity process. In this case, the default event can occur without any correlation with the firm value and such a default is called {\\it unexpected default}. In the reduced-form approach, if the default probability in time interval  $[t,~t+\\Delta t]$ is , then $\\lambda \\Delta t$ is $\\lambda$ called {\\it default intensity} \\cite{ohc, OW, OWR, wil}.\n\nThe two approaches have got their own advantages and shortcomings (\\cite{BaP, OW}) and therefore the use of unified models of structural approach and reduced-form approach is a trend. (See \\cite{BaP, BiB, CaE1, CaE2, ohc, OW, OWR, rea}.) Cathcart et al \\cite{CaE1} studied a pricing of corporate bonds in the case when the default intensity is a linear function of the interest rate and gave semi-analytical pricing formulae. Cathcart et al \\cite{CaE2} studied a valuation model in the case when the default intensity (hazard rate) is a linear function of the state variable and the interest rate. Realdon \\cite{rea} studied a pricing of corporate bonds in the case with constant default intensity and gave pricing formulae of the bond using PDE method. Some authors studied the pricing model of defaultable bonds in which the default intensity is given as a stochastic process \\cite{BaP, BiB, OW}.  In \\cite{OW}, the authors provided analytical pricing formula of corporate defaultable bond with both expected and unexpected default in the case when stochastic default intensity follows Wilmott model where drift and volatility are linear of state variables \\cite{wil}. Bi et all \\cite{BiB} got the similar result with \\cite{OW} in the case when stochastic default intensity follows CIR-like model. Ballestra et al \\cite{BaP} proposed a model to price defaultable bonds where default intensity follows Vasicek-like model or CIR-like model coupled with the process of the firm's asset value and provided a closed-form approximate solution to their model. In \\cite{BaP, BiB, CaE1, CaE2, OW, rea} expected default barrier is given in the whole lifetime of the bond. \n\nOn the other hand, in \\cite{ohc, OWR} the author studied the pricing problem for defaultable corporate bond under the assumption that we only know the firm value and the default barrier at 2 fixed discrete announcing dates, we don't know about any information of the firm value in another time and the default intensity between the adjoined two announcing dates is a constant determined by its announced firm value at the former announcing date. The computational error in \\cite{OWR} is corrected in \\cite{ohc}. The approach of \\cite{ohc, OWR} is a kind of study of defaultable bond under {\\it insufficient information} about the firm and it is interesting to note that Agliardi et al \\cite{AA} studied bond pricing problem under {\\it imprecise information} with the technique of fuzzy mathematics. The approach of \\cite{ohc, OWR} can be seen as a {\\it unified model} of structural model and reduced form model. Agliardi \\cite{agl} studied a {\\it structural model} for defaultable bond with several (discrete) coupon dates where the default can occur only when the firm value is not large enough to pay its debt and coupon in those {\\it discrete coupon dates}.   \n\nSpeaking on default recovery, most of authors including \\cite{AA, BaP, BiB, CaE1, ohc, OW, OWR} have studied the case of {\\it exogenous} default recovery which is independent on firm value whereas \\cite{agl} have studied the case of {\\it endogenous} recovery which is related to firm value, and \\cite{rea} studied both cases of exogenous and endogenous recovery. \n\nHere we study the problem of pricing defaultable bond with discrete default intensity and barrier under constant risk free short rate using higher order binary options and their integrals. In our credit risk model, the default event occurs in an expected manner when the firm value reaches a certain lower threshold - the default barrier at predetermined discrete announcing dates or in an unexpected manner at the first jump time of a Poisson process with given default intensity given by a step function of time variable, respectively. We consider both {\\it endogenous} and {\\it exogenous} default recovery. Our pricing problem is derived to a solving problem of {\\it inhomogeneous} or {\\it homogeneous Black-Scholes PDEs} with {\\it different coefficients} and terminal value of binary type in every subinterval between the two adjacent announcing dates. In order to deal with the difference of coefficients in subintervals we use a {\\it relation} between prices of higher order binaries with different coefficients. In our model, due to the inhomogenous term related to endogenous recovery, our pricing formulae are represented by not only the prices of higher binary options but also the integrals of them. So we consider a {\\it special binary} option called {\\it integral of i-th binary} or {\\it nothing} and then we obtain the pricing formulae of our defaultable corporate bond by using the pricing formulae of higher binary options and integrals of them.\n\nOur approach to model credit risk is similar with the one of \\cite{OWR, ohc}. One of the different points of our model from \\cite{ohc} is that we here consider arbitrary number of announcing dates but \\cite{ohc} consider only 2 announcing dates. Another different point from \\cite{ohc} is that we use constant risk free rate, the purpose of which is to show the applicability of higher order binaries to the pricing of defaultable bonds in the simplest way. Unlikely in \\cite{ohc} we here consider discrete default intensity independent on firm value and it can be seen incompatible with reality but we think our analytical pricing formulae can help the further study on the more realistic situation with discrete default intensity dependent on firm value.\n\nThe remainder of the article is organized as follows. In section 2 we give some preliminary knowledge on prices of higher order binary options and their integral on the last expiry date. In section 3 we set our problem for corporate defaultable bonds, provide the pricing formulae in both cases of endogenous and exogenous default recovery and analyze the credit spread. In section 4 we derive the pricing formulae using and higher order binary options and their integral. \n\n\\section{Preliminaries and Notes on Binary Options and their Integrals}\nFirst, we introduce the concept of higher order bond and asset binaries with risk free rate $r$, dividend rate $q$ and volatility $\\sigma$ and their pricing formulae \\cite{buc, OK1, OK2}.\n\\begin{equation}\\label{eq2-1}\n\\frac{\\partial V}{\\partial t}+\\frac{\\sigma^{2}}{2}x^2\\frac{\\partial^2 V}{\\partial x}+(r-q)x\\frac{\\partial V}{\\partial x}-rV=0,\\quad 0\\leq t<T,~0<x<\\infty,      \n\\end{equation}\n\\begin{equation}\\label{eq2-2}\nV(x,~T)=x\\cdot 1(sx>s\\xi),\n\\end{equation}\n\\begin{equation}\\label{eq2-3}\nV(x,~T)=1(sx>s\\xi).\n\\end{equation}\nThe solution to the problem \\eqref{eq2-1} and \\eqref{eq2-2} is called the {\\it asset-or-nothing} binaries (or {\\it asset} binaries) and denoted by $A^s_\\xi (x,t;T)$. The solution to the problem \\eqref{eq2-1} and \\eqref{eq2-3} is called the {\\it cash-or-nothing} binaries (or {\\it bond} binaries) and denoted by $B^s_\\xi (x,t;T)$. Asset binary and bond binary are called the {\\it first order binary} options. If necessary, we will denote by $A^s_\\xi (x,t;T; r,q,\\sigma)$ or $B^s_\\xi (x,t;T; r,q,\\sigma)$ where the coefficients $r$, $q$ and $\\sigma$ of Black-Scholes equation \\eqref{eq2-1} are explicitly included in the notation.\n\nLet assume that $0<T_0<T_1<\\cdots<T_{n-1}$ and the $(n-1)$-th order (asset or bond) binary options $A^{s_1\\cdots s_{n-1}}_{\\xi_1\\cdots \\xi_{n-1}}(x,t;T_1,\\cdots,T_{n-1})$ and $B^{s_1\\cdots s_{n-1}}_{\\xi_1\\cdots \\xi_{n-1}}(x,t;T_1,\\cdots$ $,T_{n-1})$ are already defined. Let \n\\begin{equation}\\label{eq2-4}\nV(x,~T_0)=A^{s_1\\cdots s_{n-1}}_{\\xi_1\\cdots \\xi_{n-1}}(x,T_0;T_1,\\cdots,T_{n-1})\\cdot 1(s_0x>s_0\\xi_0),\n\\end{equation}\n\\begin{equation}\\label{eq2-5}\nV(x,~T_0)=B^{s_1\\cdots s_{n-1}}_{\\xi_1\\cdots \\xi_{n-1}}(x,T_0;T_1,\\cdots,T_{n-1})\\cdot 1(s_0x>s_0\\xi_0).\n\\end{equation}\nThe solution to the problem \\eqref{eq2-1} and \\eqref{eq2-4} is called the {\\it n-th order asset binaries} and denoted by $A^{s_0s_1\\cdots s_{n-1}}_{\\xi_0\\xi_1\\cdots \\xi_{n-1}}(x,t;T_0,T_1,\\cdots,T_{n-1})$. The solution to the problem \\eqref{eq2-1} and \\eqref{eq2-5} is called the {\\it n-th order bond binaries} and denoted by $B^{s_0s_1\\cdots s_{n-1}}_{\\xi_0\\xi_1\\cdots \\xi_{n-1}}(x,t;T_0,T_1,\\cdots,T_{n-1})$.\\\\\n\n{\\bf Lemma 1}. (The pricing formulae of higher order binary options) \\cite{buc, OK1, OK2} {\\it The prices of higher order bond and asset binaries with risk free rate $r$, dividend rate $q$ and volatility $\\sigma$ are as follows}.\n\\begin{align}\\label{eq2-6}\n\\nonumber &A^s_\\xi (x,t;T; r,q,\\sigma)=xe^{-q(T-t)}N(sd^+),\\\\\n&B^s_\\xi (x,t;T; r,q,\\sigma)=e^{-r(T-t)}N_1(sd^-), s=+~\\mathtt{or}~-.\n\\end{align}\n{\\it Here}\n\\begin{align*}\n&N_1(x)=(\\sqrt{2\\pi})^{-1}\\int_{-\\infty}^{x}exp(-y^2\/2)dy,\\\\\n&d^{\\pm}=(\\sigma\\sqrt{T-t})^{-1}[\\ln(x\/K)+(r-q\\pm\\sigma^2\/2)(T-t)].\n\\end{align*}\n\\begin{align}\\label{eq2-7}\n\\nonumber &A^{s_1~s_2}_{K_1K_2}(x,t;T_1,T_2; r,q,\\sigma)=xe^{-q(T_2-t)}N_2(s_1d^+_1,s_2d^+_2;s_1s_2\\rho),\\\\\n&B^{s_1~s_2}_{K_1K_2}(x,t;T_1,T_2; r,q,\\sigma)=e^{-r(T_2-t)}N_2(s_1d^-_1,s_2d^-_2;s_1s_2\\rho), s_1,s_2=+~\\mathtt{or}~-.\n\\end{align}\n{\\it Here}\n\\begin{align*}\n&N_2(a,b;\\rho)=\\int_{-\\infty}^{a}\\int_{-\\infty}^{b}(2\\pi\\sqrt{1-\\rho^2})^{-1}e^{-\\frac{y^2-2\\rho yz+z^2}{2(1-\\rho^2)}}dydz,\\\\\n&d^{\\pm}_i=(\\sigma\\sqrt{T_i-t})^{-1}[\\ln(x\/K_i)+(r-q\\pm\\sigma^2\/2)(T_i-t)],i=1,2,\\\\\n&\\rho=\\sqrt{(T_1-t)\/(T_2-t)}.\n\\end{align*}\n{\\it If $m>2$ and $s_i=+$ or $-$, $i=1,\\cdots,m$, then we have}\n\\begin{align}\\label{eq2-8}\n\\nonumber &A^{s_1\\cdots s_m}_{K_1\\cdots K_m}(x,t;T_1,\\cdots,T_m; r,q,\\sigma)=xe^{-q(T_m-t)}N_m(s_1d^+_1,\\cdots,s_md^+_m;A_{s_1\\cdots s_m}),\\\\\n&B^{s_1\\cdots s_m}_{K_1\\cdots K_m}(x,t;T_1,\\cdots,T_m; r,q,\\sigma)=e^{-r(T_m-t)}N_m(s_1d^-_1,\\cdots,s_md^-_m;A_{s_1\\cdots s_m}).\n\\end{align}\n{\\it Here}\n\\begin{align}\\label{eq2-9}\n\\nonumber &N_m(a_1,\\cdots,a_m;A)=\\int_{-\\infty}^{a_1}\\cdots\\int_{-\\infty}^{a_m}(\\sqrt{2\\pi})^{-m}\\sqrt{\\det A}\\exp\\left(-\\frac{1}{2}y^\\mathsf{T} Ay\\right)dy,\\\\\n\\nonumber &d^{\\pm}_i=(\\sigma\\sqrt{T_i-t})^{-1}[\\ln(x\/K_i)+(r-q\\pm\\sigma^2\/2)(T_i-t)],i=1,\\cdots,m,\\\\\n&A_{s_1\\cdots s_m}=(s_is_ja_{ij})_{i,j=1}^{m},~y^\\mathsf{T}=(y_1,\\cdots,y_m),\n\\end{align}\nand the matrix $(a_{i,j})_{i,j=1}^{m}$ is given as follows:\n\\begin{align}\\label{eq2-10}\n\\nonumber &a_{11}=(T_2-t)\/(T_2-T_1),~~~a_{mm}=(T_m-t)\/(T_m-T_{m-1}),\\\\\n\\nonumber &a_{ii}=(T_i-t)\/(T_i-T_{i-1})+(T_i-t)\/(T_{i+1}-T_{i}),~2\\leq i\\leq m-1,\\\\\n\\nonumber &a_{i,i+1}=a_{i+1,i}=-\\sqrt{(T_i-t)(T_{i+1}-t)}\/(T_{i+1}-T_{i}),~1\\leq i\\leq m-1,\\\\\n&a_{ij}=0~\\mathtt{~for~another~}~ i,j =1,\\cdots,m.\n\\end{align}\n{\\it Note} that $N_2(a,b;\\rho)$ is the {\\it cumulative distribution function of bivariate normal distribution} with a mean vector $[0, 0]$ and a {\\it covariance matrix} $[1, \\rho; \\rho, 1]$  (symbols in {\\bf Mat lab}), and $N_m(a_1,\\cdots,a_m;A)$ is the {\\it cumulative distribution function of m-variate normal distribution} with {\\bf zero} {\\it mean vector} and a {\\it covariance matrix} $A^{-1}=(r_{ij})_{i,j=1}^{m}$  where $r_{ij}=\\sqrt{(T_i-t)\/(T_j-t)},~r_{ji}=r_{ij}, i\\leq j$. Such special functions can easily be calculated by standard functions supplied in software for mathematical calculation (for example, {\\bf Mat lab}). Note that $(A_{s_1\\cdots s_m})^{-1}=(s_is_jr_{ij})_{i,j=1}^m$.\n\nSecond, we consider a relation between prices of higher order binaries with different risk free rates and dividend rates. From the formulae \\eqref{eq2-6}, \\eqref{eq2-7} and \\eqref{eq2-8}, we can easily know that the {\\it following relations} between prices of {\\it higher order binaries} with {\\it different} risk free rates and dividend rates hold:\n\\begin{align}\\label{eq2-11}\n\\nonumber F^{s_1\\cdots s_m}_{K_1\\cdots K_m}&(x,t;T_1,\\cdots,T_m; r_1,r_1+b,\\sigma)=\\\\\n&=e^{-(r_1-r_2)(T_m-t)}F^{s_1\\cdots s_m}_{K_1\\cdots K_m}(x,t;T_1,\\cdots,T_m; r_2,r_2+b,\\sigma).\n\\end{align}\nHere $F=A$ or $F=B$.\n\nNext, we will discuss {\\it integrals} of the prices of higher order binary options on {\\it the last expiry date variable}. Let consider \\eqref{eq2-1} with the following two terminal conditions:\n\\begin{equation}\\label{eq2-12}\nV(x,~T)=f(x,~\\tau),\n\\end{equation}\n\\begin{equation}\\label{eq2-13}\nV(x,~T)=F(x):=\\int_C^Df(x,~\\tau)d\\tau.\n\\end{equation}\n   \n{\\bf Lemma 2}. {\\it Assume that there exist non negative constants $M$ and $\\alpha$ such that     $|f(x,\\tau)|\\leq M\\cdot x^{\\alpha\\ln x},~x>0$ and $f(x,\\tau)$ is a continuous function of $\\tau\\in [C,~D]$. Then the solution $V_F(x,t)$ to the problem \\eqref{eq2-1} and \\eqref{eq2-13} is given by the integral of the solution $V_f(x,t;\\tau)$ to the problem \\eqref{eq2-1} and \\eqref{eq2-12}}:\n\\begin{equation}\\label{eq2-14}\nV_F(x,~t)=\\int_C^DV_f(x,~t;~\\tau)d\\tau.\n\\end{equation}\n\n{\\bf Proof}: If we use the proposition 1 at page 249 in \\cite{OK1} and the continuity of $f$ on $\\tau$, we can easily get \\eqref{eq2-14}.(QED)   \\\\\n\nNow let consider a {\\it special} binary option called {\\it integral of i-th binary or nothing}.\\\\\n\n{\\bf Corollary.} Let $g(\\tau)$ be a continuous function of $\\tau\\in[T_{i-1},T]$ and\n\\begin{equation}\\label{eq2-15}\nV(x,~T_0)=1(s_0x>s_0K_0)\\int_{T_{i-1}}^Tg(\\tau)F^{s_1\\cdots s_{i-1}~s_i}_{K_1\\cdots K_{i-1}K_i}(x,T_0;T_1,\\cdots,T_{i-1},\\tau)d\\tau.\n\\end{equation}\n{\\it Then the solution of \\eqref{eq2-1} and \\eqref{eq2-15} is given as follows}:\n\\begin{equation}\\label{eq2-16}\nV(x,~t)=\\int_{T_{i-1}}^Tg(\\tau)F^{s_0~s_1\\cdots s_{i-1}~s_i}_{K_0K_1\\cdots K_{i-1}K_i}(x,t;T_0,T_1,\\cdots,T_{i-1},\\tau)d\\tau,~t<T_0.\n\\end{equation}\n{\\it Here $F=A$ or $F=B$}.\n\n   {\\bf Proof}: We will prove only for bond binary in the case when $i =1$. The proofs for other cases are the same. By the proposition 1 at page 249 in \\cite{OK1}, the solution to \\eqref{eq2-1} with \n\\begin{equation*}\nV(x,~T_0)=1(s_0x>s_0K_0)\\int_{T_{0}}^Tg(\\tau)B^{s_1}_{K_1}(x,T_0;\\tau)d\\tau\n\\end{equation*}\nis given as follows:\n\\begin{align*}\n&U(x,~t)=\\\\\n=&\\frac{e^{-r(T_0-t)}}{\\sigma \\sqrt{2 \\pi (T_0-t)}}\\int_{0}^{\\infty} \\frac{1}{z}e^{-\\frac{[\\ln \\frac{x}{z}+(r-q- \\frac{\\sigma^{2}}{2})(T_0-t)]^2}{2 \\sigma^2 (T_0-t)}}1(s_0z>s_0K_0)\\int_{T_{0}}^Tg(\\tau)B^{s_1}_{K_1}(z,T_0;\\tau)d\\tau dz\\\\\n=&\\int_{T_{0}}^Tg(\\tau)\\frac{e^{-r(T_0-t)}}{\\sigma \\sqrt{2 \\pi (T_0-t)}}\\int_{0}^{\\infty} \\frac{1}{z}e^{-\\frac{[\\ln \\frac{x}{z}+(r-q- \\frac{\\sigma^{2}}{2})(T_0-t)]^2}{2 \\sigma^2 (T_0-t)}}B^{s_1}_{K_1}(z,T_0;\\tau)1(s_0z>s_0K_0)dzd\\tau\\\\\n=&\\int_{T_{0}}^Tg(\\tau)B^{s_0~s_1}_{K_0K_1}(x,t;T_0,\\tau)d\\tau. ~~~\\mathtt{(QED)}\n\\end{align*}\n\n\\section{The Problem of Defaultable Bonds and The Pricing Formulae}\n\\subsection{The Problem with Endogenous Recovery}\nLet Assume the followings: \n\n1) Short rate $r$ is a constant.\n\n2) $0=t_0<t_1<\\cdots<t_{N_1}<t_N=T$ are announcing dates and $T$ is the maturity of our corporate bond with face value 1 (unit of currency). For every $i=0,\\cdots,N-1$  the unexpected default probability in the interval $[t,t+dt]\\cap(t_i,t_{i+1})$ is $\\lambda_idt$. Here the {\\it default intensity} $\\lambda_i$ is a constant.\n\n3) The firm value $V(t)$ follows a geometric Brownian motion \n\\begin{equation}\\label{eq3-1}\ndV(t)=(r-b)dtV(t)dt+s_V\\cdot V(t)dW(t)\n\\end{equation}\nunder the risk neutral martingale measure and the firm continuously pays out dividend in rate $b$ (constant) for a unit of firm value. Like in \\cite{OW}, the firm value $V_t$ is assumed to consist of $m$ shares of stock $S$ and $n$ sheets of corporate bonds $C_t$:\n\\begin{equation}\\label{eq3-2}\nV_t=mS_t+nC_t.\n\\end{equation}\n\n4) The expected default barrier is only given at time $t_i$ and the expected default event occurs when\n\\begin{equation}\\label{eq3-3}\nV(t_i)\\leq K_ie^{-r(T-t_i)},~i=1,\\cdots,N.\n\\end{equation}\nHere $K_i$ is a constant reflecting the quantity of debt and $e^{-r(T-t_i)}$ is default free zero coupon bond price.\n\n5) The default recovery $R_d$ is given as the form of endogenous face value\n\\begin{equation}\\label{eq3-4}\nR_d=\\min\\{e^{-r(T-t)}, R\\cdot V\/n\\}.\n\\end{equation}\nHere {\\it recovery rate} $0\\leq R\\leq 1$ is a constant.\n\n   6) In the subinterval $(t_i,t_{i+1})$, the price of our corporate bond is given by a sufficiently smooth function $C_i(V,~t)$ ($i=0,\\cdots,N-1$).\n\n{\\bf Problem}: {\\it Find the representation of the price function $C_i(V,~t)$ ($i=0,\\cdots,N-1$) under the above assumptions}. \\\\\n\n{\\bf The Pricing Model.}\nAccording to \\cite{wil}, under the above assumptions the price $C$ of defaultable bond with a constant default intensity $\\lambda$ and default recovery $R_d$ satisfies the following PDE:\n\\begin{equation*}\n\\frac{\\partial C}{\\partial t}+\\frac{s_V^{2}}{2}V^2\\frac{\\partial^2 C}{\\partial V^2}+(r-b)V\\frac{\\partial C}{\\partial V}-(r+\\lambda)C+\\lambda R_d=0.          \n\\end{equation*}\nTherefore if we let $C_N(V,  t)\\equiv 1$ , then the {\\it price model} of our bond is given as follows:\n\\begin{align}\\label{eq3-5}\n\\nonumber &\\frac{\\partial C_i}{\\partial t}+\\frac{s_V^{2}}{2}V^2\\frac{\\partial^2 C_i}{\\partial V^2}+(r-b)V\\frac{\\partial C_i}{\\partial V}-(r+\\lambda_i)C_i+\\lambda_i\\min\\{e^{-r(T-t)}, \\frac{RV}{n}\\}=0,~t_i<t<t_{i+1}, \\\\\n&C_i(t_{i+1})=C_{i+1}(t_{i+1})1(V>K_{i+1}e^{-r(T-t_{i+1})})+\\min\\{e^{-r(T-t_{i+1})}, \\frac{RV}{n}\\}1(V\\leq K_{i+1}e^{-r(T-t_{i+1})}).\n\\end{align}\nHere $i=0,1,\\cdots,N-1$.\\\\\n\n{\\bf The Pricing Formulae.}\nUnder the assumptions 1)-- 6), we have the following pricing formulae:\\\\\n\n{\\bf Theorem 1}. (endogenous recovery) i) {\\it Assume that $K_i\\leq n\/R, i=1,\\cdots,N$. Under the assumptions 1)--6), the price of our bond, that is, the solution of \\eqref{eq3-5} is represented as follows}: \n\\begin{equation}\\label{eq3-6}\nC_i(V, t)=e^{-r(T-t)}u_i(V\/e^{-r(T-t)}, t), t_i\\leq t<t_{i+1},~i=0,\\cdots,N-1.\n\\end{equation}\n{\\it Here}\n\\begin{align}\\label{eq3-7}\n\\nonumber &u_i(x,t)=e^{-\\lambda_i(t_{i+1}-t)}\\left\\{e^{-\\sum_{k=i+1}^{N-1}\\lambda_k(t_{k+1}-t_k)}B^{+\\quad\\cdots~+}_{K_{i+1}\\cdots K_N}(x,t;t_{i+1},\\cdots,t_N)+\\right. \\\\\n\\nonumber &\\quad+\\frac{R}{n}\\sum_{m=i}^{N-1}e^{-\\sum_{k=i+1}^{m}\\lambda_k(t_{k+1}-t_k)}A^{+\\quad\\cdots~+\\quad-}_{K_{i+1}\\cdots K_mK_{m+1}}(x,t;t_{i+1},\\cdots,t_m,t_{m+1}) \\\\\n\\nonumber&\\quad+\\sum_{m=i+1}^{N-1}\\lambda_me^{-\\sum_{k=i+1}^{m-1}\\lambda_k(t_{k+1}-t_k)}\\int_{t_m}^{t_{m+1}}e^{-\\lambda_m(\\tau-t_m)}\\left[B^{+\\quad\\cdots~+~+}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,\\tau) \\right.\\\\\n\\nonumber&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+\\left.\\frac{R}{n}\\left.A^{+\\quad\\cdots~+~~-}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,\\tau)\\right]d\\tau\\right\\}\\\\\n&+\\lambda_i\\int_{t}^{t_{i+1}}e^{-\\lambda_i(\\tau-t)}\\left[B^{+}_{\\frac{n}{R}}(x,t;\\tau; 0,b,s_V)+\\frac{R}{n}A^{-}_{\\frac{n}{R}}(x,t;\\tau; 0,b,s_V)\\right]d\\tau.\n\\end{align}\nii) {\\it Assume that $K_i>n\/R, i=1,\\cdots,N$. Under the assumptions 1)--6), the price of our bond, that is, the solution of \\eqref{eq3-5} is represented by \\eqref{eq3-6} with the following} $u_i(x,t)$:  \n\\begin{align}\\label{eq3-8}\n\\nonumber &u_i(x,t)=\\\\\n\\nonumber &=e^{-\\lambda_i(t_{i+1}-t)}\\left\\{ \\sum_{m=i}^{N-1}e^{-\\sum_{k=i+1}^{m}\\lambda_k(t_{k+1}-t_k)}\\left[B^{+\\quad\\cdots~+~+}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,t_{m+1})\\right. \\right. \\\\\n\\nonumber &\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+\\left.\\frac{R}{n}A^{+\\quad\\cdots~+\\quad-}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,t_{m+1})\\right]\\\\\n\\nonumber &\\quad-\\sum_{m=i}^{N-2}e^{-\\sum_{k=i+1}^{m}\\lambda_k(t_{k+1}-t_k)}B^{+\\quad\\cdots~+\\quad+}_{K_{i+1}\\cdots K_mK_{m+1}}(x,t;t_{i+1},\\cdots,t_m,t_{m+1})\\\\\n\\nonumber&\\quad+\\sum_{m=i+1}^{N-1}\\lambda_me^{-\\sum_{k=i+1}^{m-1}\\lambda_k(t_{k+1}-t_k)}\\int_{t_m}^{t_{m+1}}e^{-\\lambda_m(\\tau-t_m)}\\left[B^{+\\quad\\cdots~+~+}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,\\tau) \\right.\\\\\n\\nonumber&\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad\\quad+\\left.\\frac{R}{n}\\left. A^{+\\quad\\cdots~+~~-}_{K_{i+1}\\cdots K_m\\frac{n}{R}}(x,t;t_{i+1},\\cdots,t_m,\\tau)\\right]d\\tau \\right\\}\\\\\n&+\\lambda_i\\int_{t}^{t_{i+1}}e^{-\\lambda_i(\\tau-t)}\\left[B^{+}_{\\frac{n}{R}}(x,t;\\tau; 0,b,s_V)+\\frac{R}{n}A^{-}_{\\frac{n}{R}}(x,t;\\tau; 0,b,s_V)\\right]d\\tau.\n\\end{align}\n{\\it Here $B^{s_1\\cdots s_m}_{K_1\\cdots K_m}(x,t;t_1,\\cdots,t_m)$ and $A^{s_1\\cdots s_m}_{K_1\\cdots K_m}(x,t;t_1,\\cdots,t_m)$ are respectively the prices of m-th order bond and asset binaries with $0$-risk free rate, $b$-dividend rate and $s_V$-volatility}. (See lemma 1.)\n\n    The proof is not difficult but somewhat complicated. We will prove it in the section 4. \n\n{\\bf Remark 1}. In this theorem, the {\\it financial meaning} of $u_i(x,t)$ is that it is the {\\it relative price} of our bond in a subinterval {\\it with respect to the risk free zero coupon bond}. We can derive the pricing formulae of our bond under other assumptions on the relations between  $K_i(i=1,\\cdots,N)$ and $n\/R$ using the same method.\n\n\\subsection{The Problem with Exogenous Recovery}\n\nInstead of the assumption 5) let assume the following:\n\n7) The default recovery $R_d$ is given as the form of exogenous face value \n\\begin{equation}\\label{eq3-9}\nR_d =Re^{-r(T-t)}\\quad (0\\leq R\\leq 1 \\mathtt{~is~a~constant}.)\n\\end{equation}\nThen under the assumptions 1), 2), 3), 4), 6) and 7) the pricing model of our bond is given as follows:\n\\begin{align}\\label{eq3-10}\n\\nonumber &\\frac{\\partial C_i}{\\partial t}+\\frac{s_V^{2}}{2}V^2\\frac{\\partial^2 C_i}{\\partial V^2}+(r-b)V\\frac{\\partial C_i}{\\partial V}-(r+\\lambda_i)C_i+\\lambda_i Re^{-r(T-t)}=0,~t_i<t<t_{i+1}, \\\\\n&C_i(t_{i+1})=C_{i+1}(t_{i+1})1(V>K_{i+1}e^{-r(T-t_{i+1})})+Re^{-r(T-t_{i+1})}1(V\\leq K_{i+1}e^{-r(T-t_{i+1})}).\n\\end{align}\nHere $i=0,1,\\cdots,N-1$ and $C_N(V,  t)\\equiv 1$.\\\\\n\n{\\bf Theorem 2}. (exogenous recovery) {\\it Under the assumptions 1), 2), 3), 4), 6) and 7) the price of our bond, that is, the solution of \\eqref{eq3-10} is represented as follows}: \n\\begin{align}\\label{eq3-11}\n\\nonumber C_i(V, t)=W_i(V\/e^{-r(T-t)},&~t)e^{-r(T-t)}+[1-W_i(V\/e^{-r(T-t)},~t)]Re^{-r(T-t)}, \\\\\n& t_i\\leq t<t_{i+1},~i=0,\\cdots,N-1.\n\\end{align}\n{\\it Here}\n\\begin{align}\\label{eq3-12}\nW_i(x,t)=e^{-\\lambda_i(t_{i+1}-t)-\\sum_{k=i+1}^{N-1}\\lambda_k(t_{k+1}-t_k)}B^{+\\quad\\cdots~+}_{K_{i+1}\\cdots K_N}(x,t;t_{i+1},\\cdots,t_N;0,b,s_V).\n\\end{align}\nThe proof is done by the same way with that of theorem 1. See the section 4.\n\n{\\bf Remark 2}. The {\\it financial meaning} of the pricing formulae \\eqref{eq3-11} is similar with that of \\cite{OW}: the price of our defaultable bond at time $t$ can be seen as a probabilistic mean value of the current value $e^{-r(T-t)}$ of the bond in the case when there is no default after time $t$ and the value $Re^{-r(T-t)}$ of the bond in the case when default occurs after time $t$. So $W_i(V\/e^{-r(T-t)},~t)$ is the survival probability after the time $t\\in[t_i,t_{i+1})$, that is, the probability with which no default event occurs in the interval $(t,~T]$ and $1-W_i(V\/e^{-r(T-t)},~t)$ is the ruin probability after the time $t\\in[t_i,t_{i+1})$, that is, the probability with which default event occurs in the interval $(t,~T]$ when $t_i\\leq t<t_{i+1}$. The formulae \\eqref{eq3-11} can be written as follows:  \n\\begin{align}\\label{eq3-13}\n\\nonumber C_i(V, t)=Re^{-r(T-t)}+&(1-R)W_i(V\/e^{-r(T-t)},~t)e^{-r(T-t)}, \\\\\n& t_i\\leq t<t_{i+1},~i=0,\\cdots,N-1.\n\\end{align}\nThe {\\it financial meaning} of \\eqref{eq3-13} is as follows: the {\\bf first term} of \\eqref{eq3-13} is the current price of the part to be given to bond holder {\\it regardless of default occurs or not}, and the {\\bf second term} is the {\\it allowance} dependent on the survival probability after time $t$. If after some moment $t$, the default is certain ($W = 0$), then the price of the bond at $t$ is exactly the current price of default recovery $Re^{-r(T-t)}$. If the default recovery rate is zero, that is, $R = 0$, then the ratio of the defaultable bond price and default free zero coupon bond price is the very the survival probability after time $t$. If the default recovery rate is full, that is, $R = 1$, then default event does not effect to the bond price and defaultable bond price is the same with default free zero coupon bond price. \n\n\\subsection{Illustration of the Effect of Parameters on the Bond Price}\nIn this subsection we illustrate the effect of several parameters including recovery rate $R$, volatility $s_V$ of firm value, $x = V\/e^{-r(T-t)}$ (relative price of firm value), default boundary $K$ and default intensity $\\lambda$ on the price of the defaultable bonds.  Let $N=2,~t_1=3, t_2=6$ (annum).\n\nBasic data for calculation are as follows: Short rate $r = 0.1$; Firm value: dividend rate $b = 0.05$, volatility $s_V =1.0$, $x = V\/e^{-r(T-t)}= 200$; $\\lambda_0=0.002,~\\lambda_1=0.005$ are respectively default intensities in the intervals $(0,t_1)$, $(t_1,t_2)$; $K_1=K_2=100$ is default barrier at time $t_1,~t_2$; recovery rate $R = 0.5$.\n\nWe will analyze $(t, C)$-plot changing one of $R, s_V, x, K$ and $\\lambda$ under keeping the remainder of data on as the above. See the following figures \\ref{fig01}--\\ref{fig09}.\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.63\\textwidth]{figure01}\n   \\caption{Plot $(t : C)$ when $R= 0.2, 0.5, 0.95$}\n  \\label{fig01}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.87\\textwidth]{figure02}\n   \\caption{Plot $(t : C)$ when $s_V = 0.5, 1.0, 1.5$}\n  \\label{fig02}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.87\\textwidth]{figure03}\n   \\caption{Plot $(t : C)$ when $x = V\/e^{-r(T-t)}= 200, 350, 500$}\n  \\label{fig03}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure04}\n   \\caption{Plot $(t : C)$ when $(K_1,K_2)=(50,50), (100,100), (150, 150)$}\n  \\label{fig04}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure05}\n   \\caption{Plot $(t : C)$ when $(K_1,K_2)=(50,150), (100,100), (150, 50)$}\n  \\label{fig05}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure06}\n   \\caption{Plot $(t : C)$ when $K_1=100, K_2 =50,100, 150$}\n  \\label{fig06}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure07}\n   \\caption{Plot $(t : C)$ when $(\\lambda_0,\\lambda_1)=(0.001,0.002), (0.01,0.02), (0.1, 0.2)$}\n  \\label{fig07}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure08}\n   \\caption{Plot $(t : C)$ when $(\\lambda_0,\\lambda_1)=(0.001,0.2), (0.01,0.02), (0.1, 0.002)$}\n  \\label{fig08}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure09}\n   \\caption{Plot $(t : C)$ when $\\lambda_0=0.01, \\lambda_1=0.002,0.02,0.2$}\n  \\label{fig09}\n\\end{figure}\n\nNote that figure \\ref{fig01} shows that increase of recovery rate results in increase of bond price. Figure \\ref{fig02} shows that increase of volatility of firm value results in decrease of bond price. The reason is that when $s_V$ increases, the firm value fluctuates more seriously and there are more risks of default, which results in decrease of bond price. Figure \\ref{fig03} shows that increase of firm value results in increase of bond price. Figures \\ref{fig04}--\\ref{fig09} show the effect of default barrier and default intensity on bond price. In particular, in the figure \\ref{fig05} (or \\ref{fig08}) we can see the mixed effect of increase of $K_1$ (or $\\lambda_0$) and decrease of $K_2$ (or $\\lambda_1$) in the subinterval $[0, 3]$.\n\n\n\\subsection{Credit Spread Analysis}\n\nIn this subsection, we illustrate the effect of several parameters including recovery rate $R$, volatility $s_V$ of firm value, $x=V\/e^{-r(T-t)}$, default boundary $K$ and default intensity $\\lambda$ on credit spreads. The {\\it credit spread} is defined using the difference between the yields of the defaultable bond $C$ and the default-free bond $e^{-r(T-t)}$ and is given by the following expression:\n\\begin{equation*}\nCS =-\\frac{\\ln[C\/e^{-r(T-t)}]}{T-t}.\n\\end{equation*}\n \nFor simplicity, we only consider the case of exogenous default recovery (theorem 2). Then, the credit spread is differently given in every subinterval as follows: \n\\begin{equation} \\label{eq3-14}\nCS_i=-\\frac{\\ln[R+(1-R)W_i(V\/e^{-r(T-t)},t)]}{T-t}, t_i\\leq t<t_{i+1},i=0,\\cdots,N-1.\n\\end{equation}\nLet $N=2, t_1=3, t_2=T=6$ (annum) as in the above.\n\nBasic data for calculation of $CS$ are as follows: Short rate $r = 0.1$; Firm value: dividend rate $b = 0.05$, volatility $s_V =1.0$, $x = V\/e^{-r(T-t)}=200$; $\\lambda_0=0.002,~\\lambda_1=0.005$ are respectively default intensities in the intervals $(0,t_1)$, $(t_1,t_2)$; $K_1=K_2=100$ is default barrier at time $t_1,~t_2$; recovery rate $R = 0.5$.\n\nWe will analyze $(t , CS)$-plot changing one of $R, s_V, x, K$ and $\\lambda$ under keeping the remainder of data on as the above. In what follows, the figure \\ref{fig10} shows that increase of recovery rate results in decrease of credit spread. Figure \\ref{fig11} shows that increase of volatility of firm value results in increase of credit spread. The reason is that when $s_V$ increases, the firm value fluctuates more seriously and there are more risks of default, which results in increase of credit spread. Figure \\ref{fig12} shows that increase of firm value results in decrease of credit spread. Figures \\ref{fig13}--\\ref{fig18} show the effect of default barrier and default intensity on credit spread. In particular, in the figure \\ref{fig14} (or \\ref{fig16}) we can see the mixed effect of increase of $K_1$ (or $\\lambda_0$) and decrease of $K_2$ (or $\\lambda_1$) in the subinterval $[0, 3]$.\n\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure10}\n   \\caption{Plot $(t : CS)$ when $R= 0.2, 0.5, 0.95$}\n  \\label{fig10}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure11}\n   \\caption{Plot $(t : CS)$ when $s_V = 0.5, 1.0, 1.5$}\n  \\label{fig11}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure12}\n   \\caption{Plot $(t : CS)$ when $x = V\/e^{-r(T-t)}= 200, 350, 500$}\n  \\label{fig12}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure13}\n   \\caption{Plot $(t : CS)$ when $(K_1,K_2)=(50,50), (100,100), (150, 150)$}\n  \\label{fig13}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure14}\n   \\caption{Plot $(t : CS)$ when $(K_1,K_2)=(50,150), (100,100), (150, 50)$}\n  \\label{fig14}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure15}\n   \\caption{Plot $(t : CS)$ when $K_1=100, K_2 =50,100, 150$}\n  \\label{fig15}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure16}\n   \\caption{\\small Plot$(t:CS)$ when $(\\lambda_0,\\lambda_1)=(0.001,0.002),(0.01,0.02),(0.1,0.2)$}\n  \\label{fig16}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.9\\textwidth]{figure17}\n   \\caption{\\small Plot$(t:CS)$ when $(\\lambda_0,\\lambda_1)=(0.001,0.2),(0.01,0.02),(0.1, 0.002)$}\n  \\label{fig17}\n\\end{figure}\n\\begin{figure}[h]\n   \\centering\n   \\includegraphics[width=0.89\\textwidth]{figure18}\n   \\caption{\\small Plot $(t : CS)$ when $\\lambda_0=0.01, \\lambda_1=0.002,0.02,0.2$}\n  \\label{fig18}\n\\end{figure}\n\n\n\\section{The Proofs of The Pricing Formulae}\n\n{\\bf The Proof of Theorem 1}. i) In \\eqref{eq3-5}, we use change of numeraire\n\\begin{equation} \\label{eq4-1}\nx=\\frac{V}{e^{-r(T-t)}},~u_i(x,t)=-\\frac{C_i(V,t)}{e^{-r(T-t)}}, t_i\\leq t<t_{i+1},i=0,\\cdots,N-1.\n\\end{equation}\nSubstituting \\eqref{eq4-1} into \\eqref{eq3-5} we get\n\\begin{align}\\label{eq4-2}\n\\nonumber &\\frac{\\partial u_i}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u_i}{\\partial x^2}-bx\\frac{\\partial u_i}{\\partial x}-\\lambda_i u_i+\\lambda_i\\min\\{1, \\frac{R}{n}x\\}=0,~t_i<t<t_{i+1}, x>0,\\\\\n&u_i(x,t_{i+1})=u_{i+1}(x,t_{i+1})1(x>K_{i+1})+\\min\\{1, \\frac{R}{n}x\\}1(x\\leq K_{i+1}),i=0,\\cdots,N-1.\n\\end{align}\nHere $u_N(x,t)\\equiv 1$. From the assumption\n\\begin{equation} \\label{eq4-3}\nK_i\\leq n\/R, i=1,\\cdots,N\n\\end{equation}\nIf $V(t_i)\\leq K_ie^{-r(T-t_i)}$, that is, if the default event occurs at time $t_i$, then $\\min\\{e^{-r(T-t_i)}, RV(t_i)\/n\\}=RV(t_i)\/n$ and we have\n\\begin{equation} \\label{eq4-4}\n\\min\\{1, \\frac{R}{n}x\\}1(x\\leq K_{i+1})=\\frac{R}{n}x\\cdot 1(x\\leq K_{i+1}).\n\\end{equation}\nThen the problem \\eqref{eq4-2} is changed into the following one.\n\\begin{align}\\label{eq4-5}\n\\nonumber &\\frac{\\partial u_i}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u_i}{\\partial x^2}-bx\\frac{\\partial u_i}{\\partial x}-\\lambda_i u_i+\\lambda_i\\min\\{1, \\frac{R}{n}x\\}=0,~t_i<t<t_{i+1}, x>0,\\\\\n&u_i(x,t_{i+1})=u_{i+1}(t_{i+1})1(x>K_{i+1})+\\frac{R}{n}x\\cdot 1(x\\leq K_{i+1}),i=0,\\cdots,N-1.\n\\end{align}\nWhen $i=N-1$, \\eqref{eq4-5} is as follows: \n\\begin{align}\\label{eq4-6}\n\\nonumber &\\frac{\\partial u_{N-1}}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u_{N-1}}{\\partial x^2}-bx\\frac{\\partial u_{N-1}}{\\partial x}-\\lambda_{N-1} u_{N-1}+\\lambda_{N-1}\\min\\{1, \\frac{R}{n}x\\}=0,\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~t_{N-1}<t<T, x>0,\\\\\n&u_{N-1}(x,T)=1(x>K_N)+\\frac{R}{n}x\\cdot 1(x\\leq K_N),~~\\qquad x>0.\n\\end{align}\nThis is a terminal value problem for an inhomogenous Black-Scholes equation with coefficients $r=\\lambda_{N-1},~q=\\lambda_{N-1}+b,~\\sigma=s_V$. Let $L_{N-1}$ be the Black-Scholes partial differential operator with coefficients $r=\\lambda_{N-1},~q=\\lambda_{N-1}+b, \\sigma=s_V$, that is, \n\\[L_{N-1}u=\\frac{\\partial u}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u}{\\partial x^2}-bx\\frac{\\partial u}{\\partial x}-\\lambda_{N-1}u.\\]\nThen the solution of \\eqref{eq4-6} is provided by sum of the solutions $U_1$ and $U_2$ to the two following problems:\n\\begin{align}\\label{eq4-7}\n\\nonumber &L_{N-1}U_1=0,\\qquad\\qquad\\qquad\\qquad\\quad~t_{N-1}<t<T, x>0,\\\\\n&U_1(x,T)=1(x>K_N)+\\frac{R}{n}x\\cdot 1(x\\leq K_N),\\qquad x>0.\n\\end{align}\n\\begin{align}\\label{eq4-8}\n\\nonumber &L_{N-1}U_2+\\lambda_{N-1}\\min\\{1, \\frac{R}{n}x\\}=0,\\quad t_{N-1}<t<T, x>0,\\\\\n&U_2(x,T)=0,\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad x>0.\n\\end{align}\nThe terminal payoff of \\eqref{eq4-7} is linear combination of the terminal payoffs of bond and asset binaries (refer to section 2) and thus the solution to \\eqref{eq4-7} is given as follows:\n\\[U_1=B_{K_N}^+(x,t;T;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)+\\frac{R}{n}A_{K_N}^-(x,t;T;\\lambda_{N-1},\\lambda_{N-1}+b,s_V),~t_{N-1}\\leq t<T.\\]\nThe problem \\eqref{eq4-8} is a 0-terminal value problem of an inhomogeneous equation and thus we can use the {\\it Duhamel's principle} to solve it. Fix $\\tau\\in(t_{N-1},T]$ and let $W(x,t;\\tau)$ be the solution to the following terminal value problem:\n\\begin{align*}\n&L_{N-1}W=0,\\qquad\\qquad\\qquad t_{N-1}<t<\\tau, x>0,\\\\\n&W(x,\\tau;\\tau)=\\lambda_{N-1}\\min\\{1, \\frac{R}{n}x\\},\\qquad x>0.\n\\end{align*}\nSince $\\lambda_{N-1}\\min\\{1, \\frac{R}{n}x\\}=\\lambda_{N-1}\\left[1(x>n\/R)+\\frac{R}{n}x\\cdot 1(x<n\/R)\\right]$, the solution is given as follows:\n\\[W(x,t;\\tau)=\\lambda_{N-1}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right.+\\]\n\\[\\qquad\\qquad\\qquad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right],~t_{N-1}\\leq t<\\tau,~x>0.\\]\nThen the solution $U_2$ to \\eqref{eq4-8} is given as follows:\n\\begin{align*}\nU_2&=\\int_{t}^{T}W(x,t;\\tau)d\\tau=\\\\\n&=\\lambda_{N-1}\\int_{t}^{T}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right.+\\\\\n&\\qquad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right]d\\tau,~t_{N-1}\\leq t<T,~x>0.\n\\end{align*}\nThus the solution to \\eqref{eq4-6} is provided by $u_{N-1}(x,t)=U_1+U_2$, that is, \n\\begin{align}\\label{eq4-9}\n\\nonumber u&_{N-1}(x,t)=\\\\\n\\nonumber &=B_{K_N}^+(x,t;T;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)+\\frac{R}{n}A_{K_N}^-(x,t;T;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)+\\\\\n\\nonumber &+\\lambda_{N-1}\\int_{t}^{T}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right.+\\\\\n&\\quad\\qquad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-1},\\lambda_{N-1}+b,s_V)\\right]d\\tau,~t_{N-1}\\leq t<T,~x>0.\n\\end{align}\nFor our further purpose, using the relations \\eqref{eq2-11} we rewrite \\eqref{eq4-9} by the price of bond and asset binaries with the coefficients $r=0, q=b, \\sigma=s_V$: \n\\begin{align}\\label{eq4-10}\n\\nonumber u&_{N-1}(x,t)=e^{-\\lambda_{N-1}(T-t)}\\left[B_{K_N}^+(x,t;T;0,b,s_V)+\\frac{R}{n}A_{K_N}^-(x,t;T;0,b,s_V)\\right]+\\\\\n\\nonumber &+\\lambda_{N-1}\\int_{t}^{T}e^{-\\lambda_{N-1}(\\tau-t)}\\left[B_{n\/R}^+(x,t;\\tau;0,b,s_V)+\\frac{R}{n}A_{n\/R}^-(x,t;\\tau;0,b,s_V)\\right]d\\tau,\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~t_{N-1}\\leq t<T,~x>0.\n\\end{align}\nNow solve \\eqref{eq4-5} when $i=N-2$. In this case \\eqref{eq4-5} is as follows:\n\\begin{align}\\label{eq4-11}\n\\nonumber &\\frac{\\partial u_{N-2}}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u_{N-2}}{\\partial x^2}-bx\\frac{\\partial u_{N-2}}{\\partial x}-\\lambda_{N-2} u_{N-1}+\\lambda_{N-2}\\min\\{1, \\frac{R}{n}x\\}=0,\\\\\n\\nonumber &\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad~t_{N-2}<t<t_{N-1}, x>0,\\\\\n&u_{N-2}(x,t_{N-1})=u_{N-1}(x,t_{N-1})1(x>K_{N-1})+\\frac{R}{n}x\\cdot 1(x\\leq K_{N-1}).\n\\end{align}\nThis is a terminal value problem of the inhomogeneous Black-Scholes equation with coefficients $r=\\lambda_{N-2},~q=\\lambda_{N-2}+b,~\\sigma=s_V$.\n\n{\\bf Remark 3}. If we consider \\eqref{eq4-9}, then the expiry payoff of \\eqref{eq4-11} is the linear combination of first order binaries or zero and integrals of first order binaries or zero and therefore you could think that it is natural to solve \\eqref{eq4-11} using the pricing formulae of second order binaries and their integrals. But we must {\\it note} that the coefficients of \\eqref{eq4-11} {\\it are different} from those of \\eqref{eq4-6} and \\eqref{eq4-9} and thus we can't directly apply the pricing formulae of second order binaries here. Fortunately, the {\\it differences} between risk free rates and dividend rates in adjacent subintervals are {\\it all a constant} $-b$ and {\\it volatility is not changed} in whole time interval and thus we can carefully use the pricing formulae of second order binaries with \\eqref{eq2-11} together to give a representation of the solution to \\eqref{eq4-11}.  \n\nIf we rewrite the terminal payoff of \\eqref{eq4-11} into prices of binaries with the coefficients $r=\\lambda_{N-2},~q=\\lambda_{N-2}+b,~\\sigma=s_V$ using \\eqref{eq2-11}, then from \\eqref{eq4-9} we get:\n\\begin{align*}\nu_{N-1}&(x,t_{N-1})=e^{-(\\lambda_{N-1}-\\lambda_{N-2})(T-t_{N-1})}\\left[B_{K_N}^+(x,t;T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\quad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+\\frac{R}{n}\\left.A_{K_N}^-(x,t;T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]+\\\\\n&+\\lambda_{N-1}\\int_{t}^{T}e^{-(\\lambda_{N-1}-\\lambda_{N-2})(\\tau-t_{N-1})}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau.\n\\end{align*}\nLet $L_{N-2}$ be the Black-Scholes partial differential operator with coefficients $r=\\lambda_{N-2},~q=\\lambda_{N-2}+b,~\\sigma=s_V$. Then the solution to \\eqref{eq4-11} is the sum $U_1+U_2+U_3$ of the solutions to the following three problems:\n\\begin{align}\\label{eq4-12}\n\\nonumber L_{N-2}U_1=0&,\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad~t_{N-2}<t<t_{N-1}, x>0,\\\\\n\\nonumber U_1(x,t_{N-1})&=e^{-(\\lambda_{N-1}-\\lambda_{N-2})(T-t_{N-1})}\\left[B_{K_N}^+(x,t_{N-1};T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\quad+\\frac{R}{n}\\left.A_{K_N}^-(x,t_{N-1};T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]1(x>K_{N-1})+\\\\\n&+\\frac{R}{n}x\\cdot 1(x\\leq K_{N-1}),\n\\end{align}\n\\begin{align}\\label{eq4-13}\n\\nonumber &L_{N-2}U_2=0,\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad~t_{N-2}<t<t_{N-1}, x>0,\\\\\n\\nonumber &U_2(x,t_{N-1})=\\\\\n\\nonumber &=\\lambda_{N-1}\\int_{t_{N-1}}^{T}e^{-(\\lambda_{N-1}-\\lambda_{N-2})(\\tau-t_{N-1})}\\left[B_{\\frac{n}{R}}^+(x,t_{N-1};\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\qquad\\quad+\\frac{R}{n}\\left.A_{\\frac{n}{R}}^-(x,t_{N-1};\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau\\cdot 1(x>K_{N-1}).\n\\end{align}\n\\begin{align}\\label{eq4-14}\n\\nonumber &L_{N-2}U_3+\\lambda_{N-2}\\min\\{1, \\frac{R}{n}x\\}=0,\\qquad\\qquad\\qquad t_{N-2}<t<t_{N-1}, x>0,\\\\\n&U_3(x,t_{N-1})=0,\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad x>0.\n\\end{align}\nUsing the prices of first and second order binaries \\eqref{eq2-6} and \\eqref{eq2-7}, the solution to \\eqref{eq4-12} is given as follows:\n\\begin{align}\\label{eq4-15}\n\\nonumber U_1&(x,t)=\\\\\n\\nonumber &=e^{-(\\lambda_{N-1}-\\lambda_{N-2})(T-t_{N-1})}\\left[B_{K_{N-1}K_N}^{+\\quad+}(x,t;t_{N-1},T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\quad+\\frac{R}{n}\\left.A_{K_{N-1}K_N}^{+\\quad -}(x,t;t_{N-1},T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]+\\\\\n&+\\frac{R}{n}A_{K_{N-1}}^{-}(x,t;t_{N-1};\\lambda_{N-2},\\lambda_{N-2}+b,s_V), ~t_{N-2}\\leq t<t_{N-1}.\n\\end{align}\nFrom the corollary of Lemma 2, the solution to \\eqref{eq4-13} is given as follows:\n\\begin{align}\\label{eq4-16}\n\\nonumber &U_2(x,t)=\\\\\n\\nonumber &=\\lambda_{N-1}\\int_{t_{N-1}}^Te^{-(\\lambda_{N-1}-\\lambda_{N-2})(\\tau-t_{N-1})}\\left[B_{K_{N-1}\\frac{n}{R}}^{~+\\quad +}(x,t,t_{N-1},\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\qquad+\\frac{R}{n}\\left.A_{K_{N-1}\\frac{n}{R}}^{~+\\quad -}(x,t;t_{N-1},\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau,~t_{N-2}\\leq t<t_{N-1}.\n\\end{align}\n\\eqref{eq4-14} is a 0-terminal value problem of an inhomogeneous equation just like \\eqref{eq4-8}, so its solution is given by\n\\begin{align}\\label{eq4-17}\n\\nonumber &U_3(x,t)=\\lambda_{N-2}\\int_{t}^{t_{N-1}}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\qquad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau,~t_{N-2}\\leq t<t_{N-1}.\n\\end{align}\nThus we obtain the representation of $u_{N-2}(x,t)=U_1+U_2+U_3$, that is,     \n\\begin{align}\\label{eq4-18}\n\\nonumber &u_{N-2}(x,t)=\\\\\n\\nonumber &=e^{-(\\lambda_{N-1}-\\lambda_{N-2})(T-t_{N-1})}\\left[B_{K_{N-1}K_N}^{+\\quad+}(x,t;t_{N-1},T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\quad+\\frac{R}{n}\\left.A_{K_{N-1}K_N}^{+\\quad -}(x,t;t_{N-1},T;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]+\\\\\n\\nonumber &+\\frac{R}{n}A_{K_{N-1}}^{-}(x,t;t_{N-1};\\lambda_{N-2},\\lambda_{N-2}+b,s_V) \\\\\n\\nonumber &+\\lambda_{N-1}\\int_{t_{N-1}}^Te^{-(\\lambda_{N-1}-\\lambda_{N-2})(\\tau-t_{N-1})}\\left[B_{K_{N-1}\\frac{n}{R}}^{~+\\quad +}(x,t,t_{N-1},\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\quad+\\frac{R}{n}\\left.A_{K_{N-1}\\frac{n}{R}}^{~+\\quad -}(x,t;t_{N-1},\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau \\\\\n\\nonumber &+\\lambda_{N-2}\\int_{t}^{t_{N-1}}\\left[B_{n\/R}^+(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right.+\\\\\n&\\qquad\\quad+\\frac{R}{n}\\left.A_{n\/R}^-(x,t;\\tau;\\lambda_{N-2},\\lambda_{N-2}+b,s_V)\\right]d\\tau,~t_{N-2}\\leq t<t_{N-1}.\n\\end{align}\nFor our further purpose, using the relations \\eqref{eq2-11} we rewrite \\eqref{eq4-18} by the price of bond and asset binaries with the coefficients $r=0,~q=b,~\\sigma=s_V$ to get\n\\begin{align}\\label{eq4-19}\n\\nonumber &u_{N-2}(x,t)=\\\\\n\\nonumber &=e^{-\\lambda_{N-2}(T-t)-(\\lambda_{N-1}-\\lambda_{N-2})(T-t_{N-1})}\\left[B_{K_{N-1}K_N}^{+\\quad+}(x,t;t_{N-1},T;0,b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+\\frac{R}{n}\\left.A_{K_{N-1}K_N}^{+\\quad -}(x,t;t_{N-1},T;0,b,s_V)\\right]+\\\\\n\\nonumber &+e^{-\\lambda_{N-2}(t_{N-1}-t)}\\frac{R}{n}A_{K_{N-1}}^{-}(x,t;t_{N-1};0,b,s_V)+ \\\\\n\\nonumber &+\\lambda_{N-1}\\int_{t_{N-1}}^Te^{-\\lambda_{N-2}(\\tau-t)-(\\lambda_{N-1}-\\lambda_{N-2})(\\tau-t_{N-1})}\\left[B_{K_{N-1}\\frac{n}{R}}^{~+\\quad +}(x,t,t_{N-1},\\tau;0,b,s_V)\\right.+\\\\\n\\nonumber &\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad+\\frac{R}{n}\\left.A_{K_{N-1}\\frac{n}{R}}^{~+\\quad -}(x,t;t_{N-1},\\tau;0,b,s_V)\\right]d\\tau+\\\\\n\\nonumber &+\\lambda_{N-2}\\int_{t}^{t_{N-1}}e^{-\\lambda_{N-2}(\\tau-t)}\\left[B_{\\frac{n}{R}}^+(x,t;\\tau;0,b,s_V)\\right.+\\frac{R}{n}\\left.A_{\\frac{n}{R}}^-(x,t;\\tau;0,b,s_V)\\right]d\\tau,\\\\\n&\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad t_{N-2}\\leq t<t_{N-1}.\n\\end{align}\n\nBy induction we can obtain the representations of all $u_i(x,t)(i=0,\\cdots,N-1)$. If in every representation of $u_i(x,t)$ we replace the higher order binaries with the coefficients $r=\\lambda_i,~q=\\lambda_i+b,~\\sigma=s_V$ into the higher order binaries with the coefficients $r=0,~q=b,~\\sigma=s_V$ using the relation \\eqref{eq2-11} and arrange the exponents properly, we soon obtain \\eqref{eq3-7}. If we return to the original variable $V$ and the unknown function $C$ using \\eqref{eq4-1}, then we soon obtain \\eqref{eq3-6}. \n\n   The proof of ii) is the same. (QED)\\\\\n\n{\\bf The proof of Theorem 2}. In \\eqref{eq3-10}, if we use change of numeraire \\eqref{eq4-1}, then we have\n\\begin{align}\\label{eq4-20}\n\\nonumber &\\frac{\\partial u_i}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 u_i}{\\partial x^2}-bx\\frac{\\partial u_i}{\\partial x}-\\lambda_i u_i+\\lambda_i R=0,~t_i<t<t_{i+1}, x>0,\\\\\n&u_i(x,t_{i+1})=u_{i+1}(x,t_{i+1})1(x>K_{i+1})+R\\cdot 1(x\\leq K_{i+1}),i=0,\\cdots,N-1.\n\\end{align}\nHere $u_N(x,t)\\equiv 1$. We use the change of unknown function \n\\begin{equation}\\label{eq4-21}\nu_i=(1-R)W_i+R, i=0,\\cdots,N-1.\n\\end{equation}\nThen the problem \\eqref{eq4-20} is changed into the following one.\n\\begin{align}\\label{eq4-22}\n\\nonumber &\\frac{\\partial W_i}{\\partial t}+\\frac{s_V^{2}}{2}x^2\\frac{\\partial^2 W_i}{\\partial x^2}-bx\\frac{\\partial W_i}{\\partial x}-\\lambda_i W_i=0,~t_i<t<t_{i+1}, x>0,\\\\\n&W_i(x,t_{i+1})=W_{i+1}(x,t_{i+1})1(x>K_{i+1}),~x>0,~i=0,\\cdots,N-1.\n\\end{align}\nHere  $W_N(x,t)\\equiv 1$. These equations are simpler than ones in theorem 1 (note that \\eqref{eq4-22} are {\\it homogenous} Black-Scholes equations) and we can easily solve them with the same method in the above to get \\eqref{eq3-12} and \\eqref{eq3-11}.\n\n\\section{Conclusions}\n\nIn this paper we studied the pricing of defaultable bond with discrete default intensity and barrier under constant risk free short rate using higher order binary options (\\cite{buc, OK1, OK2}) and their integrals. We considered both {\\it endogenous} and {\\it exogenous} default recovery. Our pricing problem is derived to a solving problem of {\\it inhomogeneous} or {\\it homogeneous Black-Scholes PDEs} with {\\it different coefficients} and terminal value of binary type in every subinterval between the two adjacent announcing dates. See \\eqref{eq3-5} and \\eqref{eq3-10}. In order to deal with the difference of coefficients in subintervals we used a {\\it relation} \\eqref{eq2-11} between prices of higher order binaries with different coefficients. In our model, due to the inhomogenous term related to endogenous recovery, our bond prices are represented by {\\it not only} the prices of higher binary options {\\it but also} the {\\it integrals} of them. See the formulae \\eqref{eq3-7} and \\eqref{eq3-8}(3.8). So first we provided the pricing formulae (corollary of lemma 2) of a {\\it special binary option} called {\\it integral of i-th binary or nothing} and then we obtain the pricing formulae of our defaultable corporate bond by using the pricing formulae of higher binary options and integrals of them and provided illasration of the effect of parameters on the price of corporate bond and the credit spread.\n\n{\\bf Acknowledgment} Authors thank anonymous arXiv moderators for strict note which helps to make this version better and more complete.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Effects of even-tag exclusion}\n\\label{sec:eventag}\n\n\\begin{table}\n\t\\centering\n\t\\sisetup{\n\t\ttable-figures-uncertainty = 1,\n\t\ttable-number-alignment = center,\n\t\tseparate-uncertainty\n\t}\n\t\\begin{tabular}{ r c }\n\t\t\\toprule\n\t\t\\multicolumn{1}{c}{$\\Pp\\Pp\\to\\PWp\\PH$} & $\\sigma_{\\NNLO}\\, [\\si{\\fb}]$ \\\\\n\t\t\\midrule\n\t\tEven-tag exclusion                     & \\num{20.6828(55)}             \\\\\n\t\tOriginal flavour-$k_t$                 & \\num{20.7093(63)}             \\\\\n\t\t\\midrule\n\t\t\\textbf{Ratio}                         & \\textbf{99.87\\%}              \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{Fiducial cross sections for $\\PWp\\PH$ at NNLO for both the original flavour-$k_t$ algorithm and our modified version where all even-tagged jets are excluded from the list of $\\Pqb$-jets.\n\t\tThe values are shown only for the central scales and their error represents the statistical uncertainty of the Monte Carlo integrations.}\n\t\\label{tab:even_vs_vanilla}\n\\end{table}\n\n\\begin{figure}\n\t\\centering\n\t\\includegraphics[width=0.7\\textwidth]{compare.pdf}\n\t\\caption{Bin-by-bin ratio between distributions that were calculated with the even-tag-excluded and the original variants of the flavour-$k_t$ algorithm for three observables of $\\PWp\\PH$: the $p_{\\bot,\\Pqb}$, $p_{\\bot,\\Pqb\\Pqb}$, and $\\Delta R_{\\Pqb\\Pqb}$ distributions at NNLO for central scale values.}\n\t\\label{fig:even_vs_vanilla}\n\\end{figure}\n\nAs discussed in section~\\ref{sec:jet}, the clustering outcome of the algorithm can be altered according to the criterion used to define the flavour of (pseudo)-jets. Our results have been presented with the criterion that the flavour of (pseudo)-jets is assigned as the net flavour of its constituents modulo two, which we believe is more motivated from an experimental point of view as discussed in section~\\ref{sec:jet_vh}.\n\nTo investigate the impact of this ``Even-tag exclusion'' on the fixed-order predictions, we have re-computed the fiducial cross-section and distributions reported in section~\\ref{sec:fiducial} and~\\ref{sec:results_dist} without the additional ``modulo two'' criterion --- we refer to these results as ``Original flavour-$k_t$''.\nThis impact of the choice of this criterion is visualized in the case of the $\\PWp\\PH$ process in figure~\\ref{fig:even_vs_vanilla} for the $p_{\\bot,\\Pqb}$, $p_{\\bot,\\Pqb\\Pqb}$, and $\\Delta R_{\\Pqb\\Pqb}$ distributions.\nIn that figure, the ratio of the two NNLO central values are divided bin-by-bin, demonstrating that this choice has no overall effect on the shape of these distributions. The small variation between bins can be attributed to statistical fluctuations.\nThis behaviour is also confirmed at the level of the fiducial cross section as reported in table~\\ref{tab:even_vs_vanilla}, where the results are consistent within statistical uncertainties.\nThis supports our claim that no significant portion of the events are discarded by switching to the even-tag-excluded version of flavour-$k_t$ in our fixed-order predictions.\n\n\\section{Comparison with previous formulations}\n\\label{sec:fixbr}\n\nAs mentioned in section~\\ref{sec:genframe}, NNLO-accurate observables for associated Higgs production have also been presented in~\\cite{Campbell:2016jau,Ferrera:2017zex,Caola:2017xuq}.\nHowever, the cross section in these calculations is assembled in a different manner compared with our expression in eq.~\\eqref{eq:VH2}.\nSpecifically, the Higgs decay at the different orders had been scaled up to a fixed value of the accurately known branching ratio of the $\\PH\\to\\Pqb\\Paqb$ process.\nUp to NNLO, the cross sections in this formulation is assembled as follows:\n\\begin{align}\n\t\\label{eq:VH_br1}\n\t\\rd \\sigma_{\\LO}^{\\text{scaled}}\n\t & = \\rd \\sigma_{V\\PH}^{(0)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(0)},                                                                                     \\\\[10pt]\n\t\\label{eq:VH_br2}\n\t\\rd \\sigma_{\\NLO}^{\\text{scaled}}\n\t & = \\rd \\sigma_{V\\PH}^{(0)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} + \\rd \\sigma^{(1)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(1)} \\nonumber                                       \\\\\n\t & + \\rd \\sigma_{V\\PH}^{(1)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(0)},                                                                                     \\\\[10pt]\n\t\\label{eq:VH_br3}\n\t\\rd \\sigma_{\\NNLO}^{\\text{scaled}}\n\t & = \\rd \\sigma_{V\\PH}^{(0)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} + \\rd \\sigma^{(1)}_{\\PH\\to\\Pqb\\Paqb}  + \\rd \\sigma^{(2)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(2)} \\nonumber \\\\\n\t & + \\rd \\sigma_{V\\PH}^{(1)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} + \\rd \\sigma^{(1)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(1)} \\nonumber                                       \\\\\n\t & + \\rd \\sigma_{V\\PH}^{(2)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\times K^{(0)} .\n\\end{align}\nHere, the scaling factors $K^{(i)}$ contain the branching ratio and are given by\n\\begin{equation}\n\tK^{(i)} = \\frac{\\text{Br}(\\PH\\to\\Pqb\\Paqb) \\, \\Gamma_{\\PH}}{\\sum_{j=0}^{i}  \\Gamma_{\\PH\\to\\Pqb\\Paqb}^{(j)}} .\n\\end{equation}\nThe branching ratio $\\text{Br}(\\PH\\to\\Pqb\\Paqb)$ is kept fixed and is not a subject to an $\\alphas$ expansion.\n\nIn the following, we elaborate on possible drawbacks that this prescription entails, in particular concerning theory uncertainties as estimated through scale variations.\n\nFirstly, the scaling factors effectively divide out the Yukawa coupling $\\overline{y}_{\\Pqb}(\\muR^{\\text{dec.}}) \\propto \\overline{m}_{\\Pqb}(\\muR^{\\text{dec.}})$ from the prediction.\nAs a result, any running of the mass as induced by the \\MSbar scheme exactly cancel in the final result.\nThis can lead to underestimating the uncertainties, which is especially apparent at LO where the scale dependence of the Yukawa coupling otherwise dominates the uncertainties.\n\nSecondly, analysing the structure of the scaled cross sections at NLO~\\eqref{eq:VH_br2} and NNLO~\\eqref{eq:VH_br3}, it is apparent that they are assembled as a sum of terms where different scaling factors $K^{(i)}$ accompany the different perturbative coefficients of the production cross section $\\rd\\sigma_{V\\PH}^{(j)}$.\nThis mismatch can interfere with the compensation mechanism between terms of different orders, possibly distorting the theory error estimate obtained through variations of the production scale $\\muR^{\\text{prod.}}$.\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{r @{\\qquad} c c c}\n\t\t\\toprule\n\t\t                                                     & $\\mathrm{W^+H} $                            & $\\mathrm{W^-H} $                            & $ \\mathrm{ZH} $                            \\\\\n\t\t\\midrule\n\t\t$\\sigma_{\\text{LO}}^{\\text{scaled}} \\, [\\si{\\fb}]$   & $\\num{22.52}\\,^{+\\num{0.63}}_{-\\num{0.80}}$ & $\\num{14.91}\\,^{+\\num{0.42}}_{-\\num{0.54}}$ & $\\num{6.02}\\,^{+\\num{0.17}}_{-\\num{0.21}}$ \\\\\n\t\t$\\sigma_{\\text{NLO}}^{\\text{scaled}} \\, [\\si{\\fb}]$  & $\\num{22.87}\\,^{+\\num{0.76}}_{-\\num{0.87}}$ & $\\num{15.11}\\,^{+\\num{0.51}}_{-\\num{0.58}}$ & $\\num{6.06}\\,^{+\\num{0.20}}_{-\\num{0.23}}$ \\\\\n\t\t$\\sigma_{\\text{NNLO}}^{\\text{scaled}} \\, [\\si{\\fb}]$ & $\\num{20.93}\\,^{+\\num{0.61}}_{-\\num{0.73}}$ & $\\num{13.80}\\,^{+\\num{0.41}}_{-\\num{0.49}}$ & $\\num{6.10}\\,^{+\\num{0.31}}_{-\\num{0.31}}$ \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{The scaled fiducial cross sections for all $V\\PH$ processes according to the setup of appendix~\\ref{sec:fixbr} at each perturbative order up to $\\cO(\\alphas^2)$.}\n\t\\label{tab:br}\n\\end{table}\n\nTo quantify the differences between the two approaches, in table~\\ref{tab:br} we report the fiducial cross sections obtained according to~\\eqref{eq:VH_br1}--\\eqref{eq:VH_br3} using $\\mathrm{Br}(\\PH\\to\\Pqb\\Paqb) = \\num{58.09}\\%$~\\cite{deFlorian:2016spz}.\nComparing these predictions with those given in table~\\ref{tab:fid} using the unscaled cross section formul\\ae~\\eqref{eq:VH1}, we observe that the central value of the LO prediction is substantially improved in the scaled predictions thanks to absorbing higher-order effects to the $\\PH\\to\\Pqb\\Paqb$ decay through the branching ratio.\nAt NLO, however, the scaled prediction appears to slightly overestimate the cross section, while the associated theory uncertainties are comparable in size between the two formulations.\nAt NNLO, both prescriptions agree well in their respective central values, however, sizeable differences can be seen in their associated uncertainties.\nThe scaled predictions at NNLO show almost no reduction in scale uncertainties --- even increasing for $\\PZ\\PH$ --- compared to the respective NLO number, whereas our formulation~\\eqref{eq:VH1} exhibits a substantial reduction in scale uncertainties when going from NLO to NNLO.\nThis difference can be attributed to the aforementioned compensation of scale dependences, which is spoiled by the different rescaling factors used in eq.~\\eqref{eq:VH_br3}.\n\nThe effects of dividing out the Yukawa coupling in the decay and the scaling factor mismatch during the assembly of production cross sections are apparent as the theoretical uncertainties of the NNLO cross section barely change compared to their NLO values.\nIn our opinion, the consistent treatment of theoretical uncertainties outweighs the precision gain that one might (or might not) get by scaling to a fixed branching ratio, especially in the case of NNLO-accurate observables.\nThis further motivates our initial and simpler formulation we presented in eq.~\\eqref{eq:VH1} of section~\\ref{sec:genframe} where no scaling factors are applied.\n\n\\section{Introduction}\n\\label{sec:intro}\n\nOne of the highest priorities of the LHC physics programme is the detailed exploration of the mechanism of electroweak symmetry breaking that predicts the existence of the Higgs boson and its interactions with the fermions and gauge bosons of the Standard Model (SM).\nIn July 2012, the ATLAS and CMS Collaborations at the LHC reported the discovery of a resonance with a mass close to \\SI{125}{\\GeV}~\\cite{Aad:2c012tfa,Chatrchyan:2012xdj}.\nAt the current level of accuracy, the discovered particle proves to be consistent with the Higgs boson predicted by the SM but the limited precision of some of the measurements still leaves room for possible alternative interpretations beyond the SM.\nMeasurements of various properties of the Higgs boson have been carried out since then.\nOne of the main goals of the completed Run~II  at $\\sqrt{s} = \\SI{13}{\\TeV}$ and the future Run~III at $\\sqrt{s} = \\SI{14}{\\TeV}$ of the LHC is to test the coupling strength of the discovered Higgs-like particle to known SM particles through the study of a variety of processes at these increased luminosity and collisions energies.\n\nThe production of a Higgs boson ($\\PH$) in association with either a $\\PWpm$ or a $\\PZ$ boson and possible hadronic jets --- also known as \\emph{Higgs Strahlung} --- is among the most promising class of channels that can lead to the accurate determination of the Higgs-boson couplings.\nThese were also the channels that were mainly probed during the search for a light Higgs boson at the Tevatron; and the observation of excess events at the Tevatron turned out to be consistent with the observed Higgs boson at the LHC~\\cite{Aaltonen:2012qt}.\n\nAt LHC energies the $V\\PH$ processes are the third ($V=\\PWpm$) and fourth ($V=\\PZ$) largest production channels after the dominant gluon--gluon and vector-boson-fusion ones.\nThese classes of Higgs production modes provide the opportunity to probe the gauge-boson--Higgs vertex ($VV\\PH$) separately for $V=\\PWpm$ and $V=\\PZ$.\nMoreover, a second and particularly relevant feature of the $\\Pp\\Pp \\to V\\PH$ process is the possibility to study the decay of a Higgs boson into a pair of bottom--antibottom quarks.\nThis decay is extremely important to measure since it provides a direct measurement of the Higgs coupling to fermions, thereby testing the mechanism of fermion mass generation in the SM.\nFurthermore, since this decay mode dominates the total width of the Higgs boson, the uncertainty on this branching ratio enters into other studies as well, for instance in measurements of the decay of the Higgs boson to invisible final states, which are relevant for dark matter\nsearches~\\cite{Aad:2014iia}.\nSuch a decay is hard to measure in inclusive Higgs production through the leading production modes like the gluon--gluon or vector-boson-fusion channels due to the presence of enormous QCD backgrounds.\nIn the Higgs Strahlung process the presence of a vector boson decaying leptonically provides a clean experimental signature and means experimental analyses related to $V\\PH$ production have a manageable background.\n\nDirect searches for the SM Higgs boson through $V\\PH$ production and $\\PH\\to\\Pqb\\Paqb$ decay has been carried out at the LHC at centre-of-mass energies of \\SI{7}{\\TeV}, \\SI{8}{\\TeV}, and \\SI{13}{\\TeV}.\nWhile the use of Run~I data at $\\sqrt{s} =$ \\SIlist{7;8}{\\TeV} by the ATLAS and CMS Collaborations was not able to firmly establish the discovery of the SM-like Higgs boson through this channel~\\cite{Aad:2012gxa,Chatrchyan:2013zna}, the use of Run~II data at $\\sqrt{s} = \\SI{13}{\\TeV}$ enabled to do so.\nIn 2017, The LHC experiments~\\cite{Aaboud:2018zhk,Sirunyan:2018kst} announced the observation of a SM Higgs-like particle decaying to a pair of bottom--antibottom quarks precisely through this Higgs Strahlung production channel with a significance of \\numlist{5.6;5.3} standard deviations for CMS and ATLAS respectively.\n\nIn view of prospective measurements of Higgs Strahlung final states including data from Run~II and~III at the LHC, it is of crucial importance to have precise theoretical predictions for cross sections and differential distributions in the kinematic regions probed by the experiments.\nThis includes in particular QCD effects in both the production and in the decay of the Higgs boson into a bottom-quark pair.\n\nThe present status of theoretical predictions for observables related to $V\\PH$ production with a vector boson decaying leptonically and a Higgs boson decaying into a bottom--antibottom quark pair can be summarised as follows:\n\nThe total inclusive cross section for associated $V\\PH$ production is known at NNLO QCD precision.\nIt is available through the numerical program VH@NNLO~\\cite{Brein:2012ne} whose ingredients have been reported in refs.~\\cite{Harlander:2002wh,Brein:2011vx}.\nThe electroweak corrections to the total cross section are known at NLO~\\cite{Ciccolini:2003jy,Denner:2011id}.\nDifferential distributions have also been computed at NNLO QCD, including the computation of $\\PH\\to\\Pqb\\Paqb$ decay at different orders.\nIn refs.~\\cite{Ferrera:2014lca,Ferrera:2011bk,Campbell:2016jau}, the Higgs decay has been included at NLO while it is included up to NNLO in refs.~\\cite{Ferrera:2017zex,Caola:2017xuq}.\nIn addition, the fully differential decay rate for $\\PH\\to\\Pqb\\Paqb$ known so far at NNLO QCD~\\cite{Anastasiou:2011qx,DelDuca:2015zqa} has recently been computed at N$^3$LO accuracy in ref.~\\cite{Mondini:2019gid}, although jet-flavour is not identified in this calculation.\nThe combination of fixed-order QCD computations with parton showers have also been the subject of phenomenological studies~\\cite{Hamilton:2012rf,Luisoni:2013kna,Astill:2018ivh}.\n\nFully differential NNLO predictions for $V\\PH$ observables obtained via the combination of Higgs production and decay to bottom--antibottom processes have been presented in ref.~\\cite{Ferrera:2017zex} (for $V=\\PZ, \\PWp$) and in ref.~\\cite{Caola:2017xuq} (for $V=\\PWm$).\nThese computations have essential features in common: at parton level, both consider massless $\\Pqb$-quarks except in the Higgs Yukawa coupling and use the same flavour-$k_t$ algorithm~\\cite{Banfi:2006hf} to define $\\Pqb$-jets.\nFurthermore, the Higgs decay is treated in the narrow-width approximation and the Higgs Yukawa coupling $y_\\Pqb$ is computed at fixed scale $\\mu=m_{\\PH}$.\nScale variations are only considered in the production sub-process using the central scale choice $\\mu=M_{V\\PH}$.\n\nThe aforementioned computations differ instead in the theoretical framework employed to regulate infrared divergences at NNLO level: in ref.~\\cite{Ferrera:2017zex} the $q_T$-subtraction formalism~\\cite{Catani:2007vq} is used for the $V\\PH$ production cross section combined with the CoLoRFulNNLO subtraction method~\\cite{DelDuca:2016ily} for the $\\PH\\to\\Pqb\\Paqb$ decay.\nIn ref.~\\cite{Caola:2017xuq} the nested soft-collinear subtraction scheme~\\cite{Caola:2017dug,Caola:2019pfz} is used (an extension of the residue subtraction scheme~\\cite{Czakon:2014oma}) in both production and decay sub-processes.\n\nIt is the aim of this paper to present a computation of $V\\PH$ observables for all three processes $(V=\\PZ,\\PWpm)$ including NNLO corrections to both production and decay sub-processes.\nOur goal  is to yield a fully differential description of the final states, i.e.\\ including the decays of the vector boson into leptons and the Higgs boson into bottom quarks with off-shell propagators of the vector- and Higgs-boson.\nThe NNLO corrections to both production and decay sub-processes are calculated using the antenna subtraction formalism~\\cite{GehrmannDeRidder:2005cm,GehrmannDeRidder:2005aw,GehrmannDeRidder:2005hi,Daleo:2006xa,Daleo:2009yj,Boughezal:2010mc,Gehrmann:2011wi,GehrmannDeRidder:2012ja,Currie:2013vh} implemented within the \\NNLOJET framework~\\cite{Currie:2018oxh}.\n\nThe structure of this paper is as follows: in section~\\ref{sec:flavour}, we provide an overview explaining how flavour-dependent observables are computed at fixed-order accuracy within the parton level event generator \\NNLOJET.\nA detailed description of the jet-algorithm used to achieve this goal, as well as its application to the $V\\PH$ process are also specified.\nIn section~\\ref{sec:details}, we present the details of the $V\\PH$ calculation giving explicitly the different ingredients appearing in production and decay sub-processes up to NNLO level.\nSection~\\ref{sec:results} contains our results for the fiducial cross sections and differential observables related to $V\\PH$ production in $\\Pp\\Pp$ collisions at $\\sqrt{s}=\\SI{13}{\\TeV}$.\nThose include, for the first time, scale uncertainty estimations related to the separate variation of production and decay scales at each order in $\\alphas$.\nTwo appendices are enclosed: in appendix~\\ref{sec:eventag} the impact of different criteria for defining the net flavour of jets is studied for a number of relevant NNLO distributions in $V\\PH$ production.\nAppendix~\\ref{sec:fixbr} is dedicated to a comparison of results obtained using two different expressions of the cross section, including either a fixed branching ratio $\\text{Br}(\\mathrm{H\\to \\Pqb \\bar{b}})$ as used previously in refs.~\\cite{Caola:2017xuq,Ferrera:2017zex,Campbell:2016jau}, or not, as in this paper.\n\n\\section{Flavour tagging of jets}\n\\label{sec:flavour}\n\nThe goal of this work is to provide fixed-order predictions for the hadron-level process $\\Pp\\Pp\\to\\Pl\\Pal\\,+2\\,\\Pqb\\text{-jets}+X$, i.e.\\ yielding a final state which contains flavour-tagged bottom-quark jets (\\Pqb-jets) and (charged) leptons.\nThe presence of two identified \\Pqb-jets with a combined invariant mass consistent with $m_\\PH$ allows us to associate this final state with the underlying process $\\Pp\\Pp \\to V\\PH \\,+\\, X \\to \\Pl\\Pal\\;\\Pqb\\Paqb \\,+\\, X$.\nThe identification of jet flavour is an essential component of any experimental analysis of this process, which is required to reduce otherwise overwhelming background processes.\nIt is therefore also an integral part of the requirements needed to obtain the corresponding theoretical predictions.\n\nThe computation of such observables at fixed order requires the application of a flavour-sensitive jet algorithm that --- besides reconstructing flavour-insensitive properties such as four-momenta --- identifies the flavour of the reconstructed jets based on some well-defined (infrared-safe) criteria~\\cite{Banfi:2006hf}.\nThe application of such an algorithm requires a tracking of the flavour of individual partons, which appear in the partonic cross section at each perturbative order.\n\nIn the following, we provide a description of how this is achieved within the parton-level event generator \\NNLOJET.\nThe discussion is however not specific to the use of the antenna subtraction formalism to regulate infrared divergences occurring in partonic sub-processes beyond LO.\nIn addition, as the application of a flavour-sensitive jet algorithm is not standard (although required from the point of view of massless fixed-order computations) for either theory or experimental communities, we also give a brief overview of the algorithm used for these computations.\nThis section is concluded by providing specific details of the jet clustering implementation relevant for the results presented here regarding the computation of NNLO observables for $V\\PH$ production.\n\n\n\\subsection{Flavour dressing}\n\nThe first step towards computing flavour-dependent jet observables is to ensure that the jet algorithm has access to both momentum and flavour information when evaluating the contributions from matrix elements and subtraction terms.\nTo address this issue within \\NNLOJET, an additional ``flavour-dressing\" layer that tracks the flavours of all amplitudes as well as reduced matrix elements appearing in subtraction terms has been implemented.\n\nTo illustrate how this proceeds, we consider the construction of a generic NLO-type cross section for an $n$-body final state initiated by the two partons $i$ and $j$. Following the notation of ref.~\\cite{Currie:2013vh}, we may write the contribution to the partonic cross section as\n\\begin{align} \\label{eq:NLO}\n\t\\rd\\hat{\\sigma}_{ij,\\NLO} & =\n\t\\int_{n+1} \\left[ \\rd\\hat{\\sigma}^{R}_{ij,\\NLO} - \\rd\\hat{\\sigma}^{S}_{ij,\\NLO}\\right]\n\t+ \\int_{n} \\left[ \\rd\\hat{\\sigma}^{V}_{ij,\\NLO} - \\rd\\hat{\\sigma}^{T}_{ij,\\NLO}\\right] ,\n\\end{align}\nwhere the superscripts $R$, $S$, $V$, $T$ indicate the real, real-subtraction, virtual, and virtual-subtraction terms, respectively.\n\nAs an example of the flavour-dressing procedure for the amplitudes, we consider the real-emission cross section (omitting the sum over potential colour orderings) which takes the general form\n\\begin{align}\n\t\\rd \\hat{\\sigma}^R_{ij,\\NLO} & =\n\t\\mathcal{N}^R_{\\NLO} \\;\\rd \\Phi_{n+1} \\left(\\left\\{p_3,\\ldots,p_{n+3}\\right\\}; p_{1}, p_{2}\\right) \\; \\frac{1}{S_{n+1}} \\nonumber \\\\\n\t                             & \\times \\left[\n\tM^0_{n+3}   \\left(\\left\\{p_{n+3}\\right\\}, \\left\\{f_{n+3}\\right\\} \\right) \\;\n\tJ_n^{(n+1)}\\! \\left(\\left\\{p_{n+1}\\right\\}, \\left\\{f_{n+1}\\right\\} \\right) \\right] .\n\\end{align}\nWe denote the final-state symmetry factor by $S_{n+1}$, the normalisation factor (which includes constants, couplings, colour factors) by $\\mathcal{N}^R_{\\NLO}$, the $2\\to n+1$ particle phase space by $\\rd \\Phi_{n+1}$, and the momentum of the partons $i,j$ by $p_{1,2}$.\nThe partial squared amplitude $M^0_{n+3}$ is evaluated with the momentum set $\\left\\{p_{n+3}\\right\\}$ and a corresponding flavour set $ \\left\\{f_{n+3}\\right\\}$.\nThe flavour-sensitive jet algorithm $J_n^{(n+1)}$ builds $n$ jets from $n+1$ final-state partons which carry momentum and flavour labelled by the sets $\\left\\{p_{n+1}\\right\\}$ and $\\left\\{f_{n+1}\\right\\}$ respectively.\n\nThe real subtraction cross section can be written in a similar fashion:\n\\begin{align}\n\t\\rd \\hat{\\sigma}^S_{ij,\\NLO} & =\n\t\\mathcal{N}^R_{\\NLO} \\,\\sum_{k} \\rd \\Phi_{n+1} \\left(\\left\\{p_3,\\ldots,p_{n+3}\\right\\}; p_{1}, p_{2}\\right) \\; \\frac{1}{S_{n+1}} \\nonumber \\\\\n\t                             & \\times \\left[\n\tX_3^0(\\cdot,k,\\cdot) \\;\n\tM^0_{n+2} \\left(\\{\\tilde{p}_{n+2}\\}, \\{\\tilde{f}_{n+2}\\} \\right) \\;\n\tJ_n^{(n)}\\! \\left(\\{\\tilde{p}_n\\}, \\{\\tilde{f}_n\\}         \\right)  \\right] ,\n\\end{align}\nwhere the index $k$ runs over all possible unresolved partons in $\\rd \\hat{\\sigma}^R_{ij,\\NLO}$ and $X_3^0(\\cdot,k,\\cdot)$ denotes the three-parton antenna function that factorises from the associated reduced squared matrix-element $M^0_{n+2}$.\nIn this case, the jet algorithm acts upon mapped final-state momentum and flavour sets $\\{\\tilde{p}_n\\}$ and $\\{\\tilde{f}_n\\}$ associated with the reduced squared matrix element $M^0_{n+2}$.\nAs the total subtraction cross section must take into account all possible unresolved limits of parton $k$, this cross section may be composed of multiple flavour structures.\nThe subtraction method is only effective if the evaluation of flavour-dependent observables in both the real and real-subtraction cross sections match in all possible unresolved limits. This is only ensured if an infrared-safe flavour-sensitive jet algorithm is applied.\n\nTo construct the NLO cross section according to eq.~\\eqref{eq:NLO}, a similar procedure must also be applied to both virtual and virtual-subtraction (in the antenna formalism, these include integrated subtraction and mass-factorisation contributions) cross sections.\nThis construction is obtained in a similar fashion, by tracking both the momentum and flavour sets associated to all partial squared amplitudes and reduced squared matrix elements appearing in these contributions and then applying the flavour-sensitive jet algorithm to the subset of final-state particles within these sets.\nTo allow the computation of flavour-dependent jet observables at NNLO, the same ideas extend to one order higher and this flavour-dressing procedure is applied to all NNLO-type parton level contributions and their corresponding subtraction terms.\n\n\n\\subsection{Flavoured-jet algorithm}\n\\label{sec:jet}\n\nThroughout this work jets are reconstructed with the flavour-$k_t$ algorithm, which provides an infrared-safe definition of jet flavour.\nThe main difference with respect to a native jet algorithm is that the clustering of particles relies on both momentum and flavour information of the input pseudo-jets.\nFor completeness, we summarise the main steps of the algorithm for hadron--hadron collisions originally presented in ref.~\\cite{Banfi:2006hf} (also summarised in ref.~\\cite{Banfi:2007gu}).\n\nThe algorithm proceeds by assigning a net flavour to all pseudo-jets or jets based on their quark flavour content, attributing $+1$ ($-1$) if a quark (antiquark) of the flavour under consideration is present.\nIn an experimental context, the presence of a quark flavour could be inferred from a fully\/partially reconstructed hadron.\nA criterion is then applied to these objects to determine if they carry flavour, possible examples being: the net flavour (sum of quarks and antiquarks); or the net flavour modulo two.\nObjects are considered to carry flavour if they carry non-zero values of this criterion.\nThe algorithm then proceeds by constructing distance measures for pairs of all final-state pseudo-jets $i$ and $j$ ($d_{ij}$) as well as beam distances ($d_{iB}$ and $d_{i\\bar{B}}$).\nThese (flavour-dependent) distances are defined as\n\\begin{equation}\n\t\\label{eq:flavourkt_meas}\n\td_{ij} = \\frac{\\Delta y_{ij}^2 + \\Delta \\phi_{ij}^2}{R^2}\n\t\\begin{cases}\n\t\t\\max(k_{ti}, k_{tj})^{\\alpha}\\,\\min(k_{ti}, k_{tj})^{2-\\alpha} & \\text{softer of } i, j \\text{ is flavoured,}   \\\\\n\t\t\\min(k_{ti}, k_{tj})^{\\alpha}                                  & \\text{softer of } i, j \\text{ is unflavoured,} \\\\\n\t\\end{cases}\n\\end{equation}\nand\n\\begin{equation}\n\t\\label{eq:flavourkt_beam}\n\td_{i\\parenbar{B}} =\n\t\\begin{cases}\n\t\t\\max(k_{ti}, k_{t\\parenbar{B}}(y_i))^{\\alpha}\\,\\min(k_{ti}, k_{t\\parenbar{B}}(y_i))^{2-\\alpha} & \\text{softer of } i, j \\text{ is flavoured,}   \\\\\n\t\t\\min(k_{ti}, k_{t\\parenbar{B}}(y_i))^{\\alpha}                                                  & \\text{softer of } i, j \\text{ is unflavoured.} \\\\\n\t\\end{cases}\n\\end{equation}\nIn these definitions, $k_{ti}$ and $k_{tj}$ are the transverse momentum of the pseudo-jets $i$ and $j$, and the rapidity difference and azimuthal angular separation between these pseudo-jets is given by $\\Delta y_{ij}$ and $\\Delta \\phi_{ij}$, respectively.\nThe parameters $R$ and $\\alpha$ define a class of measures for the algorithm.\nThe (rapidity-dependent) transverse momentum of the beam $B$ at positive rapidity $k_{tB}$, and beam $\\bar{B}$ at negative rapidity $k_{t\\bar{B}}$, are defined as:\n\\begin{align}\n\tk_{tB}(y)       & = \\sum_i k_{ti} \\left( \\Theta(y_i - y) +  \\Theta(y - y_i) \\; \\re^{y_i-y} \\right), \\label{eq:ktb}    \\\\\n\tk_{t\\bar{B}}(y) & = \\sum_i k_{ti} \\left( \\Theta(y - y_i) +  \\Theta(y_i-y)   \\; \\re^{y-y_i} \\right), \\label{eq:ktbbar}\n\\end{align}\nwith $ \\Theta(0) = 1\/2$ and the index $i$ going over all pseudo-jets.\n\nWhile this flavour-aware jet algorithm is substantially more complex than the flavour-blind anti-$k_t$ algorithm~\\cite{Cacciari:2008gp}, its use is unavoidable in fixed-order computations based on massless quarks.\nAt NLO, the flavour criterion of a pseudo-jet ensures that a collinear splitting of the form $\\Pg\\to\\Pq\\Paq$ is indistinguishable from a gluon (or flavourless) jet.\nThe subtraction formalism presented in eq.~\\eqref{eq:NLO} would already be spoiled without this criterion.\nAt NNLO, the flavour-dependent distance measure in eq.~\\eqref{eq:flavourkt_meas} ensures that a pair of flavoured quarks originating from a wide-angle gluon splitting is clustered into a pseudo-jet \\emph{before} being combined with any other (harder) pseudo-jets.\nThis avoids the situation where one of these soft quarks may be clustered with a hard pseudo-jet that carries zero flavour, which would lead to a definition of jet flavour sensitive to soft physics.\nThese are issues which are otherwise insurmountable for fixed-order computations involving massless quarks.\n\nThe flavour-$k_t$ algorithm described above is available in the \\NNLOJET framework and has been validated against an independent implementation using FastJet~\\cite{Cacciari:2005hq,Cacciari:2011ma}.\n\n\n\\subsection{Jet clustering for the \\texorpdfstring{$V\\PH$}{VH} process}\n\\label{sec:jet_vh}\n\nThe discussion of flavour dressing and the jet algorithm presented in this section are quite general and are applicable to all processes implemented within \\NNLOJET.\nHere we discuss a few specific points related to the application of the jet algorithm to the $V\\PH$ process, which will be relevant to the results presented in later sections of this paper.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\VHjets{0}{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:vh_jets_a}\n\t\\end{subfigure}\n\t\\hspace{1em}\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\VHjets{1}{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:vh_jets_b}\n\t\\end{subfigure}\n\t\\caption{Possible event configurations characterised by the presence of two hard \\Pqb-jets at LO in~(\\subref{fig:vh_jets_a}) and at NNLO in~(\\subref{fig:vh_jets_b}) where multiple \\Pqb-jets and light jets can be emitted from the production or the decay sides.}\n\t\\label{fig:vh_jets}\n\\end{figure}\n\nThe first point is that we wish to apply the flavour-$k_t$ algorithm to the partonic process $\\Pq\\Paq\\to V\\PH\\to \\Pl\\Pal\\;\\Pqb\\Paqb$, including NNLO QCD corrections which will be discussed in section~\\ref{sec:details}.\nWhen higher-order corrections are considered, additional light or \\Pqb-quark partons can be emitted from both production and the decay sides, as illustrated in figure~\\ref{fig:vh_jets_b}.\nThe jet clustering is performed by considering \\Pqb-quarks to be flavoured (all other partons carrying zero flavour) and fully accounting for emissions from both production and decay during the jet clustering process.\nWhile our calculation focusses on the decay sub-process $\\PH\\to\\Pqb\\Paqb$, it has been implemented in such a way that predictions for the hadronic process $\\Pp\\Pp\\to\\Pl\\Pal\\,+2\\,\\Pqc\\text{-jets}+X$ can also be produced.\nThis may be interesting in view of possible future measurements by the LHCb Collaboration~\\cite{LHCb-CONF-2016-006}.\n\nThe second point is that the definition of the transverse momentum of the beam is altered to account for the presence of a leptonically decaying gauge-boson.\nThis is done by modifying eq.~\\eqref{eq:ktb} according to\n\\begin{align}\n\t\\tilde{k}_{tB}(y) & = k_{tB}(y) + E_{t,V} \\left( \\Theta(y_V - y) +  \\Theta(y - y_V)  e^{y_V-y} \\right),\n\\end{align}\nwhere $E_{t,V}$ and $y_V$ are the transverse energy and rapidity of the reconstructed gauge-boson.\nA similar modification to the beam transverse momentum at negative rapidity~\\eqref{eq:ktbbar} is assumed.\nThis modification is introduced to provide a better estimate of the hardness of the beam, which can affect the clustering outcome.\nOne could alternatively modify the beam hardness by including the charged leptons, which may be necessary in experimental situations where the gauge-boson cannot be fully reconstructed.\n\nThe final point is related to our choice of flavour criterion during the clustering process.\nWe have chosen to define the flavour of pseudo-jets to be the net-flavour of its constituents modulo two, which means that all pseudo-jets which contain an even flavour content are considered to have zero net-flavour.\nThe motivation for this choice is that, in our opinion, it is the most feasible realisation of the flavour-$k_t$ algorithm experimentally.\nFocussing on the case of \\Pqb-jets, the main consideration is that most experimental approaches to flavour tagging are sensitive only to the absolute flavour~\\cite{Aaij:2015yqa,Aad:2015ydr,Sirunyan:2017ezt} (and do not additionally charge tag the jets).\nAll implementations of the algorithm must consider the combination of a $\\Pqb\\Paqb$-quark pair (or equivalently a $\\PB\\PBb$-hadron pair) as carrying zero flavour, as required to guarantee its infrared safety as discussed above.\nTherefore, in the absence of charge tagging, any (pseudo)-jet which contains the presence of an even number of \\Pqb (\\PB) and\/or \\Paqb (\\PBb) quarks (hadrons) should also be considered to carry zero flavour, as experimentally these signatures are indistinguishable.\n\nThe charge-tagging of flavoured jets is also possible~\\cite{Aaij:2014ywa}, for example in the presence of semi-leptonic \\PB-hadron decays.\nHowever, the drawback is a large reduction in event statistics (roughly an order of magnitude for each \\Pqb-jet, as the branching fraction ${\\mathrm{Br}(\\PB\\to \\Pl+X)} \\approx 10\\%$) with little informational gain.\nAccordingly, to present our results for NNLO observables related to $V\\PH$ production in section~\\ref{sec:results}, we shall use the version of the flavour-$k_t$ algorithm where all even-tagged (pseudo)-jets carry zero flavour. We further provide an examination of the impact of the even-tag exclusion in the shape and normalisation of flavour sensitive observables in appendix~\\ref{sec:eventag}.\n\n\\section{Details of the calculation}\n\\label{sec:details}\n\nIn this section we present the main ingredients that enter the calculation of the Higgs Strahlung process at NNLO.\nWe establish how those building blocks are assembled to express the cross section in a factorised form in terms of production and decay sub-processes.\n\n\n\\subsection{General framework}\n\\label{sec:genframe}\n\nWe consider the process $\\Pp\\Pp\\to V \\PH \\,+\\,X \\to \\Pl\\Pal\\;\\Pqb\\Paqb \\,+\\,X$ where the vector boson ($V$) decays leptonically and the Higgs boson (\\PH) decays into a pair of bottom quarks $\\Pqb\\Paqb$.\nWe compute NNLO QCD observables related to these reactions by including corrections up to order $\\alphas^2$ in both production and decay sub-processes.\nThis enables us to express the fully differential cross section at the $k$th order in a factorised form given as\n\\begin{align}\n\t\\label{eq:VH1}\n\t\\rd\\sigma^{\\N{k}\\LO} & =\n\t\\sum_{\\substack{i,j=0    \\\\ i + j \\le k}}^{k}\n\t\\rd \\sigma^{(i)}_{V\\PH} \\times \\rd \\sigma^{(j)}_{\\PH\\to\\Pqb\\Paqb} \\,.\n\\end{align}\nThe term $\\rd\\sigma^{(i)}_{V\\PH}$, which corresponds to the production part, comprises the vector propagator and the leptonic decay $V\\to\\Pl\\Pal$, including spin correlations between the initial-state partons and the final-state leptons.\nThe term denoted by $\\rd\\sigma^{(j)}_{\\PH\\to\\Pqb\\Paqb}$ corresponds to the decay part and includes the Higgs propagator and its subsequent decay to a bottom--antibottom quark pair.\nWe treat all light quarks as massless including the bottom quark with the exception of the Yukawa coupling mediating the $\\PH\\to\\Pqb\\Paqb$ decay.\nIn the decay the bottom quark Yukawa coupling to the Higgs boson is renormalised in the \\MSbar scheme at the scale $\\mu^{\\text{dec.}}$, taken to be proportional to the Higgs-boson mass $m_\\PH$.%\n\\footnote{\n\tIt is known from the computation of the inclusive cross section that this choice of regularisation scheme leads to a reduction of the size of the QCD corrections.\n}\nNote that the factorised form of the associated Higgs production cross section~\\eqref{eq:VH1} does not include interferences between production and decay sub-processes.\nThis is a valid approximation owing to the smallness of the Higgs decay width, which further formed the basis of the narrow-width approximation (NWA) as used in previous calculations.\n\nUp to $\\cO(\\alphas^2)$, the cross section may then be written as\n\\begin{align}\n\t\\rd \\sigma^{\\NNLO}\n\t & = \\rd \\sigma_{V\\PH}^{(0)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} + \\rd \\sigma^{(1)}_{\\PH\\to\\Pqb\\Paqb}  + \\rd \\sigma^{(2)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\nonumber \\\\\n\t & + \\rd \\sigma_{V\\PH}^{(1)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} + \\rd \\sigma^{(1)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\nonumber                                       \\\\\n\t & + \\rd \\sigma_{V\\PH}^{(2)} \\times \\left( \\rd \\sigma^{(0)}_{\\PH\\to\\Pqb\\Paqb} \\right) \\, .\n\t\\label{eq:VH2}\n\\end{align}\nWe note that this formulation of the NNLO cross section does not contain the Higgs-boson branching ratio into \\Pqb quarks given as $\\text{Br}(\\PH\\to\\Pqb\\Paqb) = \\Gamma_{\\PH\\to\\Pqb\\Paqb}\/\\Gamma_{\\PH}$.\nThis is in contrast to previous calculations for the $V\\PH$ process at NNLO, either considering the decay sub-process at NLO~\\cite{Ferrera:2014lca,Ferrera:2011bk,Campbell:2016jau} or NNLO~\\cite{Ferrera:2017zex,Caola:2017xuq}, which all employed a scaled variant of the cross section incorporating the Higgs-boson branching ratio at a fixed value and thus not subject to an $\\alphas$ expansion.\nIt is worth mentioning that this scaled variant of the cross section was essential in describing the data using fixed-order predictions at LO and NLO~\\cite{Ferrera:2011bk,Ferrera:2014lca}.\nWith this formulation, the LO predictions have the correct normalisation; NLO corrections are kept small and have a small residual theoretical uncertainty.\nIf computed up to order $\\alphas^2$, we here argue that the need of such scaling factors in the formulation of the cross section becomes questionable.\n\nIn appendix~\\ref{sec:fixbr}, we will further elaborate on this matter and compare the results obtained with both approaches for the fiducial cross sections up to order $\\alphas^2$.\nWe find that a consistent treatment of theoretical uncertainties outweighs the precision gain that one might obtain by scaling the cross section to a fixed branching ratio, if the cross section is computed including NNLO corrections in both production and decay parts.\nThis further motivates the simpler formulation of the cross section given above in eq.~\\eqref{eq:VH2} where no scaling factors are applied.\nThis will be our default setup throughout this work and for the results presented in section~\\ref{sec:results}.\n\nAs a validation of our calculation, we performed a comparison to the results of ref.~\\cite{Caola:2017xuq} by adopting their calculational setup and found perfect agreement with the reported values for the total cross sections in Table~I at each perturbative order in $\\alphas$.\n\n\n\\subsection{Production and decay parts up to \\texorpdfstring{$\\cO(\\alphas^2)$}{O(alphas**2)} }\n\nBased on our master formula~\\eqref{eq:VH2} for the $V\\PH$ process at NNLO, we specify the individual ingredients of the production and decay parts in the following and describe how they are assembled to the final prediction for the Higgs Strahlung process.\n\n\n\\subsubsection{Production parts}\n\nUp to order $\\alphas$, only one type of contribution enters the associated Higgs production cross section, which is given by Drell--Yan-like diagrams with a subsequent Higgs emission from the gauge-boson leg.\nStarting from $\\cO(\\alphas^2)$, additional quark-loop induced contributions arise.\nThese can be treated independently from the aforementioned Drell--Yan-type ones as the relevant Feynman amplitudes are individually gauge invariant.\nIn the following, we describe these two production modes one after the other.\n\n\\paragraph{Drell--Yan-type:}\nThese contributions arise from the Drell--Yan-like production of a virtual \\PWpm or \\PZ boson, which then splits into a real vector boson and a Higgs particle.\nIn our calculation we include them up to $\\cO(\\alphas^2)$ using off-shell amplitudes that effectively treat both the directly produced vector boson and the vector boson that decays leptonically as virtual particles.\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\DrellYan{0}{0}{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:drell-yan_a}\n\t\\end{subfigure}\n\t%\n\t\\hspace{1em}\n\t%\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\DrellYan{1}{1}{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:drell-yan_b}\n\t\\end{subfigure}\n\t\\caption{Examples of Feynman diagrams entering the Drell--Yan type contributions at~(\\subref{fig:drell-yan_a}) LO and at~(\\subref{fig:drell-yan_b}) NNLO.\n    Production and decay parts have an additional final state gluon in~(\\subref{fig:drell-yan_b}) compared to case~(\\subref{fig:drell-yan_a}).\n\t\tBoth vector bosons and the Higgs boson are treated in an off-shell manner, as explained in the main text.}\n\t\\label{fig:drell-yan}\n\\end{figure}\nRepresentative Feynman diagrams for this production mode are illustrated in figure~\\ref{fig:drell-yan} at LO~(\\subref{fig:drell-yan_a}) and NNLO~(\\subref{fig:drell-yan_b}).\n\nThese contributions only involve the square of Drell--Yan-like amplitudes and the infrared singularities are dealt with using the NNLO antenna subtraction formalism~\\cite{GehrmannDeRidder:2005cm}.\nThe subtraction terms can be readily constructed based on the NNLO calculation for the Drell--Yan processes, which are available within the \\NNLOJET framework.\n\n\n\\paragraph{Top-quark-loop induced:}\nStarting from $\\cO(\\alphas^2)$, new types of diagrams induced by quark loops must be taken into account for the $V\\PH$ production process.\nDepending on whether the gauge boson and\/or the Higgs boson couple to the quark loop, these contributions can be classified into three categories:\n\\begin{enumerate}\n\t\\myitem{(a)}\\label{item:ri} A class of amplitudes with no vector boson coupling to the quark loop.\n\t        As such, this class contributes to all Higgs Strahlung processes involving either \\PZ or \\PWpm bosons.\n\t        %\n\t\\myitem{(b)}\\label{item:self-zh-i} A second class of amplitudes that is only present for $\\PZ\\PH$ production where the gauge boson as well as the Higgs boson directly couple to the quark loop.\n\t        %\n\t\\myitem{(c)}\\label{item:self-zh-ii} Finally, the class of amplitudes where only the \\PZ boson attaches to the quark loop while the Higgs boson is emitted from the external massive gauge-boson leg.\n\\end{enumerate}\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\TopRI{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:ri}\n\t\\end{subfigure}\n\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\TopSelfZHI{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:top-self-zh-i}\n\t\\end{subfigure}\n\t%\n\t\\hspace{1em}\n\t%\n\t\\begin{subfigure}[t]{0.45\\textwidth}\n\t\t\\centering\n\t\t\\TopSelfZHII{1.3cm}{0}\n\t\t\\caption{}\n\t\t\\label{fig:top-self-zh-ii}\n\t\\end{subfigure}\n\t%\n\t\\caption{Representative Feynman diagrams of the heavy-quark-loop-induced amplitudes at $\\cO(\\alphas^2)$ included in our calculation.\n\t\tFigure~(\\subref{fig:ri}) depicts an $R_I$-type amplitude, which is present for both $\\PZ\\PH$ and $\\PW\\PH$ production channels.\n\t\tFigures~(\\subref{fig:top-self-zh-i},\\subref{fig:top-self-zh-ii}) illustrate representative gluon--gluon induced heavy quark loop amplitudes, which are only present for $\\PZ\\PH$ production.}\n\t\\label{fig:top-loop-inc}\n\\end{figure}\nRepresentative Feynman diagrams for each part are shown in figures~\\ref{fig:top-loop-inc}(\\subref{fig:ri}--\\subref{fig:top-self-zh-ii}), respectively, where we have omitted the Higgs decay for clarity.\nThe contribution to the cross section either arises through the square of these diagrams (e.g.\\ for the gluon--gluon-induced channels) or though the interference with Drell--Yan-type amplitudes.\nNote that all quark-loop-induced contributions are both infrared and ultraviolet finite and thus no subtraction procedure is needed in their evaluation.\n\nFor cases~\\ref{item:ri} and \\ref{item:self-zh-i} where the Higgs boson directly couples to the quark loop, we only consider the top quark running inside the loop.\nThey constitute the dominant contribution in this class and are proportional to the second power of the top Yukawa coupling $y_\\Pqt$.\nThe omission of the light-quark --- including the bottom-quark --- amplitudes is justified by their much smaller Yukawa couplings.\n\nIn case~\\ref{item:self-zh-ii} on the other hand, the Higgs boson does not couple directly to the quark loop and we have to consider all quark flavours inside the loop.\nFor the quarks of the first two generations: $\\Pq=\\Pqu,\\,\\Pqd,\\,\\Pqs,\\,\\Pqc$, the corresponding amplitudes cancel due to the fact that an equal count of up- and down-type quark flavours are evaluated.\nThis cancellation is spoiled in the case of the third generation due to the non-vanishing mass of the top quark.\nAs a result, both the top- and bottom-loop components are included.\n\nThe complete $\\cO(\\alphas^2)$ top-loop-induced contributions were computed for on-shell vector bosons in ref.~\\cite{Brein:2011vx}, relying in some cases on the infinite-top-mass approximation.\nIn our NNLO calculation we include those that are known in the exact theory and numerically sizeable but omit those which are only known in the infinite-top-mass limit.\nSpecifically, for the NNLO contributions associated with diagrams~\\ref{item:ri}, we include diagrams with top-quark-loop insertions onto an external gluon line.\nThe related amplitudes are referred to as $R_I$ in ref.~\\cite{Brein:2011vx} and have been included in the previous computations~\\cite{Ferrera:2017zex,Caola:2017xuq}.%\n\\footnote{\n\tWe did not include the two-loop amplitudes from this class as they are currently not known in the full theory but only in the infinite-top-mass limit.\n\tDiagrammatically, these would be given by\n\t%\n\t\\newline\\bigskip\n\t\\centerline{\\TopVI{1cm}{0}}\n\t%\n\tand are referred to as $V_I$ in ref.~\\cite{Brein:2011vx}.\n\tThe numerical impact to the total NNLO cross section is estimated to be below the percent level.\n\tThis contribution was omitted in ref.~\\cite{Ferrera:2017zex} but kept in ref.~\\cite{Caola:2017xuq}.\n}\nRegarding the amplitudes of class~\\ref{item:self-zh-i} and~\\ref{item:self-zh-ii}, which are exclusively present in $\\PZ\\PH$ production, we only include the gluon--gluon-induced channels shown in figures~\\ref{fig:top-loop-inc}(\\subref{fig:top-self-zh-i},\\subref{fig:top-self-zh-ii}).\nPhenomenologically, they represent the dominant component among the top-loop-induced contributions due to the large gluon luminosity at the LHC and were also considered in the previous calculations at NNLO.%\n\\footnote{\n\tThe contributions that we omitted from this class are are given by diagrams of the following type:\n\t%\n\t\\newline\\bigskip\n\t\\centerline{\\TopRII{1cm}{0} \\hspace{2em} \\TopVII{1cm}{0.25cm}}\n\t%\n\tThey are denoted as $R_{II}$ and $V_{II}$ respectively in ref.~\\cite{Harlander:2002wh}.\n\tThe one-loop amplitude $R_{II}$ is known in the full theory but it merely constitutes a sub-permille effect.\n\tThe two-loop amplitude $V_{II}$ is currently only known in the large-top-mass limit but its impact is estimated to be at the sub-percent level.\n\tThese contributions were also omitted in ref.~\\cite{Ferrera:2017zex}.\n}\n\nThe heavy-quark-loop-induced contributions included in our calculation have been either independently rederived or implemented using known results, in particular those given in ref.~\\cite{Campbell:2016jau}.\nA validation of the implementation was performed against OpenLoops amplitudes~\\cite{Cascioli:2011va} and full agreement was found in all cases.\n\n\n\\subsubsection{Decay parts}\n\nFor the decay sub-process $\\PH\\to\\Pqb\\Paqb$, we required the corrections up to $\\cO(\\alphas^2)$ as indicated in our master formula~\\eqref{eq:VH2}.\nThe corresponding amplitudes at one- and two-loop level were obtained from the analytic expressions of refs.~\\cite{Anastasiou:2011qx,DelDuca:2015zqa} and were decomposed into different colour levels according to antenna formalism conventions.\nA validation of all one-loop amplitudes was performed against the OpenLoops library~\\cite{Cascioli:2011va}, yielding full agreement.\nIn addition, subtraction terms capturing the infrared singularities are required.\nThose have been constructed for the Higgs decay up to order $\\cO(\\alphas^2)$ for the present computation.\nChecks for the correct divergent behaviour in all single- and double-unresolved limits have been performed in order to ensure the proper cancellation of singularities in the real-emission corrections as well as the cancellation of poles against the virtual amplitudes.\n\nThe decay sub-process up to NNLO only enters in eq.~\\eqref{eq:VH2} when combined with the Drell--Yan-type production parts.\nFor the top-quark-loop induced contributions, which are already of $\\cO(\\alphas^2)$, the decay only needs to be considered at tree level.\n\n\\section{Numerical results}\n\\label{sec:results}\n\nIn this section we present phenomenological results obtained for the different $V\\PH$ processes using our implementation in the parton-level event generator~\\NNLOJET.\nWe first summarise the general setup in section~\\ref{sec:setup} and move on to discuss the integrated fiducial cross sections obtained within this setup in section~\\ref{sec:fiducial}.\nWe devote section~\\ref{sec:scale} to validating the scale dependence of our numerical results and present differential distributions for flavour-sensitive observables in section~\\ref{sec:results_dist}.\n\n\\subsection{General setup}\n\\label{sec:setup}\n\nWe provide predictions for proton--proton collisions at $\\sqrt{s} = \\SI{13}{\\TeV}$ using the parton distribution function \\verb|NNPDF31_nnlo_as_0118| provided via the LHAPDF library~\\cite{Buckley:2014ana}.\nEach event was required to contain at least two \\Pqb-jets with transverse momentum $p_{\\bot,\\Pqb} > \\SI{25}{\\GeV}$ and rapidity $|y_{\\Pqb}| < \\num{2.5}$.\nCharged leptons were required to have a transverse momentum above $p_{\\bot,\\Pl} > \\SI{15}{\\GeV}$ and for their rapidity to satisfy $|y_{\\Pl}| < \\num{2.5}$.\nFor the $\\PWpm\\PH$ processes, we additionally demanded a minimum missing transverse energy of $E_{\\bot,\\text{miss}} > \\SI{15}{\\GeV}$ to identify the neutrino in the final state.\nWe used the flavour-$k_t$ algorithm with an even-tag exclusion to cluster \\Pqb-jets as described in sections~\\ref{sec:jet} and \\ref{sec:jet_vh} with the parameters $R=0.5$ and $\\alpha=2$.\n\nWe employed the $G_\\mu$-scheme for the electroweak input parameters and the full set of independent parameters entering the computation are given by\n\\begin{align}\n\tm_{\\PZ}              & = \\SI{91.1876}{\\GeV} ,           &\n\tm_{\\PW}              & = \\SI{80.385}{\\GeV} ,            &\n\tm_{\\PH}              & = \\SI{125.09}{\\GeV} ,\n\t\\nonumber                                                 \\\\\n\t\\Gamma_{\\PZ}         & = \\SI{2.4952}{\\GeV} ,            &\n\t\\Gamma_{\\PW}         & = \\SI{2.085}{\\GeV} ,             &\n\t\\Gamma_{\\PH}         & = \\SI{4.1}{\\MeV} ,\n\t\\\\\n\t\\overline{m}_{\\Pqb}\n\t                     & = \\SI{4.18}{\\GeV} ,              &\n\tm_{\\Pqt}^\\text{pole} & = \\SI{173.21}{\\GeV} ,            &\n\tG_\\text{F}           & = \\SI{1.1663787e-5}{\\GeV^{-2}} .\n\t\\nonumber\n\\end{align}\nThe running of the strong coupling ($\\alphas$) was evaluated using the LHAPDF library with the associated PDF set, while the \\MSbar mass of the bottom quark ($\\overline{m}_{\\Pqb}$) was directly computed within \\NNLOJET.\nFinally, in the case of $\\PWpm\\PH$ production, we assumed a diagonal CKM matrix for the vector-boson--quark couplings.\n\nFor the unphysical scales appearing in the calculation, we chose to set and vary them independently for the production and decay parts.\nThe central factorisation and renormalisation scales of the production sub-processes were chosen to the invariant mass of the $V\\PH$ system~$M_{V\\PH}$, whereas the central renormalisation scale of the decay was set to the Higgs-boson mass~$m_{\\PH}$.\nWe evaluate the differential cross section for a total of 21 different scale settings that are obtained from all possible combinations of\n\\begin{align}\n\t\\muF              & = M_{V\\PH} \\, \\left[ 1, \\tfrac{1}{2}, 2 \\right], &\n\t\\muR^\\text{prod.} & = M_{V\\PH} \\, \\left[ 1, \\tfrac{1}{2}, 2 \\right], &\n\t\\muR^\\text{dec.}  & = m_{\\PH}  \\, \\left[ 1, \\tfrac{1}{2}, 2 \\right],\n\t\\label{eq:scales}\n\\end{align}\nwith the additional constraint $\\tfrac{1}{2}\\leq \\muF\/\\muR^\\text{prod.} \\leq 2$ following the conventional 7-point scale variation for the production sub-process.\n\n\n\\subsection{Fiducial cross section}\n\\label{sec:fiducial}\n\n\\begin{table}\n\t\\centering\n\t\\begin{tabular}{r @{\\qquad} c c c}\n\t\t\\toprule\n\t\t                                    & $\\PWp\\PH $                                  & $\\PWm\\PH $                                  & $ \\PZ\\PH $                                 \\\\\n\t\t\\midrule\n\t\t$\\sigma_{\\text{LO}} \\, [\\si{\\fb}]$  & $\\num{18.06}\\,^{+\\num{2.87}}_{-\\num{2.41}}$ & $\\num{11.96}\\,^{+\\num{1.90}}_{-\\num{1.60}}$ & $\\num{4.83}\\,^{+\\num{0.77}}_{-\\num{0.65}}$ \\\\\n\t\t$\\sigma_{\\text{NLO}} \\, [\\si{\\fb}]$ & $\\num{21.52}\\,^{+\\num{0.88}}_{-\\num{1.08}}$ & $\\num{14.21}\\,^{+\\num{0.58}}_{-\\num{0.71}}$ & $\\num{5.71}\\,^{+\\num{0.22}}_{-\\num{0.28}}$ \\\\\n\t\t$\\sigma_{\\text{NNLO}}\\, [\\si{\\fb}]$ & $\\num{20.68}\\,^{+\\num{0.16}}_{-\\num{0.46}}$ & $\\num{13.64}\\,^{+\\num{0.11}}_{-\\num{0.31}}$ & $\\num{5.92}\\,^{+\\num{0.13}}_{-\\num{0.16}}$ \\\\\n\t\t\\bottomrule\n\t\\end{tabular}\n\t\\caption{The fiducial cross sections for all $V\\PH$ processes according to the setup of section~\\ref{sec:setup}.\n\t\tThe error on the values represents the theoretical uncertainty which was obtained by taking the minimum and maximum values of the 21-point scale variation.}\n\t\\label{tab:fid}\n\\end{table}\n\nThe cross-section predictions including fiducial cuts for the different $V\\PH$ processes are summarised in table~\\ref{tab:fid} at the various orders in \\alphas.\n\nRegarding the $\\PWpm\\PH$ fiducial values, we observe an $\\cO(20\\%)$ increase in the cross section from the NLO corrections and a slight $\\cO(5\\%)$ decrease when going from NLO to NNLO.\nThe minimum and the maximum values of the 21-point scale variations yield the theoretical uncertainties of our predictions, which are $\\cO(15\\%)$ at LO, $\\cO(5\\%)$ at NLO, and reduce to only $\\cO(2\\%)$ at NNLO with a three-fold asymmetry between the lower and upper bounds of the latter values.\nThe decrease in the size of the theoretical uncertainty is apparent at each of these orders, demonstrating the perturbative convergence of these results in a satisfying manner.\nThis will be further accentuated for flavour-sensitive jet observables in section~\\ref{sec:results_dist}.\n\nFor the $\\PZ\\PH$ fiducial values we see a different behaviour beyond NLO: the gluon--gluon-induced $\\PZ\\PH$-only top loop contributions of figures~\\ref{fig:top-self-zh-i} and~\\ref{fig:top-self-zh-ii} dominate the NNLO coefficient such that there is an $\\cO(4\\%)$ increase going from NLO to NNLO, contrasting the decrease seen for the $\\PWpm\\PH$ case.\nThe $\\PZ\\PH$-exclusive channels open up at NNLO, and therefore the theoretical uncertainty does not exhibit such a strong reduction in size but remains around~$\\cO(3\\%)$.\n\nNote that the reduction of scale uncertainties observed here is spoiled in all cases when a rescaling prescription is employed that incorporates a fixed branching ratio for the $\\PH\\to\\Pqb\\Paqb$ decay, as is commonly done in previous calculations for the $V\\PH$ processes.\nA comparison of our results in table~\\ref{tab:fid} to such a rescaled cross-section prediction is presented in appendix~\\ref{sec:fixbr}.\n\n\n\\subsection{Scale variations}\n\\label{sec:scale}\n\nThe dependence on the renormalisation scales $\\muR^{\\text{prod.}\/\\text{dec.}}$ can serve as a non-trivial check of the final results obtained from the numerical computation.\nTo this end, we ensure that the different scale settings of eq.~\\eqref{eq:scales} are correctly reproduced by the analytic renormalisation-group running starting from the central scale choice.%\n\\footnote{\n\tFor processes involving just a production part, the analytic expressions have been explicitly given in ref.~\\cite{Currie:2018xkj}.\n}\nThis is of particular importance for the calculation at hand, as the independent variation of scales for the different sub-processes was for the first time implemented in the \\NNLOJET framework for the present work.\n\n\\begin{figure}\n\t\\centering\n\t\\renewcommand\\thesubfigure{a}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=1]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_a}\n\t\\end{subfigure}\n\t\\hfill\n\t\\renewcommand\\thesubfigure{d}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=4]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_d}\n\t\\end{subfigure}\n\t\\newline\n\t\\renewcommand\\thesubfigure{b}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=2]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_b}\n\t\\end{subfigure}\n\t\\hfill\n\t\\renewcommand\\thesubfigure{e}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=5]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_e}\n\t\\end{subfigure}\n\t\\newline\n\t\\renewcommand\\thesubfigure{c}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=3]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_c}\n\t\\end{subfigure}\n\t\\hfill\n\t\\renewcommand\\thesubfigure{f}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth,page=6]{{var_plot_paper}.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:scales_f}\n\t\\end{subfigure}\n\t%\n\t\\caption{Numerical versus analytical scale variation of the $\\PWp\\PH$ process at NLO~(left) and NNLO~(right) for the bin $\\SI{220}{\\GeV} \\le M_{\\PW\\PH} \\le \\SI{230}{\\GeV}$ and three different slices in the $(\\muR^\\text{prod.},\\muR^\\text{dec.})$ plane.\n\t}\n\t\\label{fig:scales}\n\\end{figure}\n\nThe comparison between the analytic evolution and the 21 points obtained from the numerical computation using \\NNLOJET are shown in figure~\\ref{fig:scales} for the case of the $\\PWp\\PH$ process at NLO~(\\subref{fig:scales_a}--\\subref{fig:scales_c}) and NNLO~(\\subref{fig:scales_d}--\\subref{fig:scales_f}).\nWe performed a scan in the two-dimensional $(\\muR^\\text{prod.},\\muR^\\text{dec.})$ space by choosing three different slices that cover the combinations in eq.~\\eqref{eq:scales} where the three choices in the factorisation scale $\\muF=M_{\\PW\\PH}  \\;\\left[ 1, \\tfrac{1}{2}, 2 \\right ]$ are shown as separate curves:\n\\begin{itemize}\n\t\\item[(a,d)] We keep the decay renormalisation scale fixed to $\\muR^{\\text{dec.}} = m_{\\PH}$ and vary the scale in the production sub-process according to\n\t      \\begin{align}\n\t\t      \\muR^{\\text{prod.}} & = K_\\mathrm{R}^{\\text{prod.}} \\times M_{\\PW\\PH} & \\text{with } K_\\mathrm{R}^{\\text{prod.}} \\in \\left[ \\tfrac{1}{2}, 2 \\right ].\n\t      \\end{align}\n\t\\item[(b,e)] We keep the production renormalisation scale fixed to $\\muR^{\\text{prod.}} = M_{\\PW\\PH}$ and vary the scale in the decay sub-process according to\n\t      \\begin{align}\n\t\t      \\muR^{\\text{dec.}} & = K_\\mathrm{R}^{\\text{dec.}} \\times m_{\\PH} & \\text{with } K_\\mathrm{R}^{\\text{dec.}} \\in \\left[ \\tfrac{1}{2}, 2 \\right ].\n\t      \\end{align}\n\t\\item[(c,f)] We choose a diagonal slice in the $(\\muR^\\text{prod.},\\muR^\\text{dec.})$ plane setting $K_\\mathrm{R}^{\\text{prod.}} = K_\\mathrm{R}^{\\text{dec.}} \\equiv K_\\mathrm{R}$ with the individual scales given as\n\t      \\begin{align}\n\t\t      \\muR^{\\text{prod.}} & = K_\\mathrm{R} \\times M_{\\PW\\PH} , &\n\t\t      \\muR^{\\text{dec.}}  & = K_\\mathrm{R} \\times m_{\\PH}      &\n\t\t      \\text{with } K_\\mathrm{R} \\in \\left[ \\tfrac{1}{2}, 2 \\right ].\n\t      \\end{align}\n\\end{itemize}\nNote that the invariant mass $M_{\\PW\\PH}$ constitutes a dynamical quantity that varies on an event-by-event basis.\nThe curves in figure~\\ref{fig:scales} are obtained by picking a specific bin $M_{\\PW\\PH} \\in [220,230]~\\GeV$ to assign a value to the production scale, where the width of the bands in the smooth curves correspond to the uncertainty that arises from the finite bin width.\n\nWe observe an excellent agreement between the numerical results from \\NNLOJET and the curves predicted from the renormalisation group equations.\nThe dramatic reduction in scale uncertainties can be further appreciated by contrasting the vertical ranges in the figures at NLO~(left) and NNLO~(right).\nWe carried out the same tests also for the $\\PWm\\PH$ and the $\\PZ\\PH$ processes as well as for other individual $M_{V\\PH}$ bins in the distributions and found that the scale variation of the numerical results match the analytical formul\\ae\\ in all cases.\n\n\n\\subsection{Distributions}\n\\label{sec:results_dist}\n\nIn figures~\\ref{fig:wph_dist}--\\ref{fig:zh_dist} we present differential distributions of flavour-sensitive observables for the three different associated Higgs boson production processes $\\PWp\\PH$, $\\PWm\\PH$, and $\\PZ\\PH$:%\n\\footnote{\n\tWe focus on this set of observables in order to allow for a qualitative comparison with refs.~\\cite{Ferrera:2017zex,Caola:2017xuq}.\n}\n\\begin{enumerate}[label=(\\alph*)]\n\t\\item the transverse momentum $p_{\\bot,\\Pqb}$ of the leading \\Pqb-jet,\n\t\\item the transverse momentum $p_{\\bot,\\Pqb\\Pqb}$ of the pair of two \\Pqb-jets,\n\t\\item the angular separation $\\Delta R_{\\Pqb\\Pqb} = \\sqrt{\\Delta \\eta_{\\Pqb\\Pqb}^2 + \\Delta \\phi_{\\Pqb\\Pqb}^2}$ of two \\Pqb-jets,\n\t\\item and the invariant mass $m_{\\Pqb\\Pqb}$ of two \\Pqb-jets,\n\\end{enumerate}\nwhere in~(b--d) the two \\Pqb-jets are selected whose invariant mass is closest to $m_\\PH$ in order to identify the candidate pair that is most likely to originate from the Higgs decay.\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{flav_ptj1-wph.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wph_ptb1}\n\t\\end{subfigure}\n\t%\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{ptbb-wph.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wph_ptbb}\n\t\\end{subfigure}\n\n\t\\vspace{3em}\n\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{dRbb-wph.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wph_dRbb}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{mbb-wph.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wph_mbb}\n\t\\end{subfigure}\n\t\\caption{Flavour-sensitive jet distributions for the $\\PWp\\PH$ process showing (\\subref{fig:wph_ptb1})~the transverse momentum of the leading \\Pqb-jet, (\\subref{fig:wph_ptbb})~the transverse momentum of the \\Pqb-jet pair, (\\subref{fig:wph_dRbb})~the angular separation of the \\Pqb-jet pair, and~(\\subref{fig:wph_mbb})~the invariant mass of the \\Pqb-jet pair closest to the Higgs boson mass.\n\t\tThe upper panel contains the absolute values while the lower panel shows the bin-by-bin ratios with respect to the previous order evaluated at the central scale.}\n\t\\label{fig:wph_dist}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{flav_ptj1-wmh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wmh_ptb1}\n\t\\end{subfigure}\n\t%\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{ptbb-wmh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wmh_ptbb}\n\t\\end{subfigure}\n\n\t\\vspace{3em}\n\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{dRbb-wmh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wmh_dRbb}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{mbb-wmh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:wmh_mbb}\n\t\\end{subfigure}\n\t\\caption{Flavour-sensitive jet distributions for the $\\PWm\\PH$ process showing (\\subref{fig:wmh_ptb1})~the transverse momentum of the leading \\Pqb-jet, (\\subref{fig:wmh_ptbb})~the transverse momentum of the \\Pqb-jet pair, (\\subref{fig:wmh_dRbb})~the angular separation of the \\Pqb-jet pair, and~(\\subref{fig:wmh_mbb})~the invariant mass of the \\Pqb-jet pair closest to the Higgs boson mass.\n\t\tThe upper panel contains the absolute values while the lower panel shows the bin-by-bin ratios with respect to the previous order evaluated at the central scale.}\n\t\\label{fig:wmh_dist}\n\\end{figure}\n\n\\begin{figure}\n\t\\centering\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{flav_ptj1-zh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:zh_ptb1}\n\t\\end{subfigure}\n\t%\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{ptbb-zh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:zh_ptbb}\n\t\\end{subfigure}\n\n\t\\vspace{3em}\n\n\t%\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{dRbb-zh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:zh_dRbb}\n\t\\end{subfigure}\n\t\\begin{subfigure}[h]{.48\\textwidth}\n\t\t\\includegraphics[width=\\textwidth]{mbb-zh.pdf}\n\t\t\\subcaption{}\n\t\t\\label{fig:zh_mbb}\n\t\\end{subfigure}\n\t\\caption{Flavour-sensitive jet distributions for the $\\PZ\\PH$ process showing (\\subref{fig:zh_ptb1})~the transverse momentum of the leading \\Pqb-jet, (\\subref{fig:zh_ptbb})~the transverse momentum of the \\Pqb-jet pair, (\\subref{fig:zh_dRbb})~the angular separation of the \\Pqb-jet pair, and~(\\subref{fig:zh_mbb})~the invariant mass of the \\Pqb-jet pair closest to the Higgs boson mass.\n\t\tThe upper panel contains the absolute values while the lower panel shows the bin-by-bin ratios with respect to the previous order evaluated at the central scale.}\n\t\\label{fig:zh_dist}\n\\end{figure}\n\nUp to NLO, all three production modes of $\\PWp\\PH$, $\\PWm\\PH$, and $\\PZ\\PH$ show similar qualitative behaviour for all four investigated distributions.\nHowever, there are significant phenomenological differences between the $\\PWpm\\PH$ and $\\PZ\\PH$ distributions at NNLO.\n\nNNLO corrections to the $\\PWpm\\PH$ cases lead to substantial stabilisation of the predictions for the first three distributions shown in figures~\\ref{fig:wph_dist}--\\ref{fig:wmh_dist}, parts~(\\subref{fig:wph_ptb1}--\\subref{fig:wph_dRbb}): size and shape are only slightly modified at NNLO compared to the NLO predictions; the scale-variation bands, however, are reduced considerably.\nIn contrast, the first three of the $\\PZ\\PH$ distributions show an excess of events in the central regions throughout figure~\\ref{fig:zh_dist}, parts~(\\subref{fig:zh_ptb1}--\\subref{fig:zh_dRbb}).\nThis behaviour is attributed to top-quark-loop threshold effects in the dominant gluon--gluon-induced $\\PZ\\PH$-exclusive amplitudes of figures~\\ref{fig:top-self-zh-i} and~\\ref{fig:top-self-zh-ii}.\nAs mentioned earlier, these channels first contributed at NNLO, which also explains the widening of the theoretical uncertainty bands around the threshold regions of these distributions.\n\nConcerning the invariant mass distribution of all three production modes shown in figures~\\ref{fig:wph_mbb}--\\ref{fig:zh_mbb}, the features previously noted in refs.~\\cite{Ferrera:2017zex,Caola:2017xuq} can be confirmed by our predictions as well:\ndue to the very narrow width of the Higgs boson, the $m_{\\Pqb\\Pqb}$ distribution has a natural kinematic threshold at $m_{\\PH}=\\SI{125.09}{\\GeV}$ and the phase space away from this value is barely populated at LO.\nConsequently, NNLO corrections are effectively NLO-accurate for most of the bins, which explains the large corrections and relatively larger uncertainty bands for these distributions.\nThe left shoulder below $m_{\\PH}$ is mainly the result of radiation from the decay, whereas the shoulder above $m_{\\PH}$ is due to radiative corrections to the production.\nFixed-order predictions at the threshold region of $m_{\\Pqb\\Pqb} \\approx m_{\\PH}$, however, should not be trusted as they are prone to Sudakov-type instabilities.\nA proper treatment of this region would require the inclusion of resummation effects.\nIn our case, the binning is sufficiently coarse so that such instabilities only manifest in larger uncertainty bands for the $m_{\\Pqb\\Pqb} = m_{\\PH}$ bin and not as an explicit divergence.\n\n\\section{Summary and conclusions}\n\\label{sec:summary}\n\nWe reported on the calculation of NNLO corrections to the Higgs Strahlung processes $\\PWp\\PH$, $\\PWm\\PH$, and $\\PZ\\PH$ including the off-shell leptonic decay of the gauge boson as well as the Higgs decaying into a bottom--antibottom pair.\nThe calculation consistently takes into account NNLO corrections to the production and decay sub-processes and fully retains the differential information on the final state.\n\nThe study of $V\\PH$ ($\\PH\\to\\Pqb\\Paqb$) processes critically relies on the tagging of bottom jets in order to isolate the candidate pairs associated to the Higgs boson.\nWe described our independent implementation of the infrared-safe flavour-$k_t$ algorithm in the \\NNLOJET parton-level event generator and the necessary modifications this entails in the framework of the antenna subtraction formalism.\n\nA detailed account was given on the residual theory uncertainties by allowing the scales in the production and decay sub-processes to vary independently.\nThis conservative approach resulted into taking the envelope of 21 scale variations for the full process but allowed for a more comprehensive view into the impact of higher orders on the reduction of scale uncertainties.\nThe NNLO corrections to the fiducial cross section were found to exhibit a good perturbative convergence with residual uncertainties at the percent level.\nWe contrasted our na\u00efve perturbative expansion of the cross section with a more commonly employed rescaling procedure using the branching ratio $\\mathrm{BR}(\\PH\\to\\Pqb\\Paqb)$, where we observed the latter to overestimate the residual scale uncertainties.\nThis was attributed to a miscancellation in the scale dependence among the terms that receive different rescaling factors and lead us to advocate the simpler prescription to be more reliable beyond NLO.\n\nFlavour-sensitive observables were studied by investigating differential distributions where a similar stabilisation of the perturbative series was found as in the cross sections.\nLarger effects from higher-order corrections were seen in the invariant mass distributions of two \\Pqb-jets, which can be attributed to this observables being only NLO-accurate away from $m_{\\Pqb\\Pqb}\\sim m_\\PH$.\nA comparison between the $\\PWpm\\PH$ and $\\PZ\\PH$ processes showed a qualitatively similar behaviour but also emphasised the phenomenologically sizeable impact that arise from the gluon--gluon-induced top-quark loop amplitudes.\n\nThe study of flavour-sensitive jet observables with fixed-order predictions, such as those associated to \\Pqb-jets in the present work, must be performed in an infrared-safe way. For calculations based on massless QCD this can only be achieved with a flavour-aware jet algorithm (such as flavour-$k_t$), while for a massive calculation this is achievable with a flavour-blind algorithm (such as anti-$k_t$).\nIn many cases the corresponding massive calculation may not be available, or the massless calculation may actually be preferred due to the presence of large logarithmic corrections which may be easily resummed via PDF evolution.\nFuture comparisons to measurements are only viable if a similar prescription is also employed in the experiment, and the application of the even-tag exclusion here was mainly motivated to facilitate the experimental implementation.\nThe use of flavour-sensitive jet algorithms is not only important to the $V\\PH$ process class but we expect it to be of relevance for \\emph{any flavour-sensitive jet observable}, such as the associated production of the flavoured jet with a gauge boson.\nSuch studies will be left for future work.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\nThe paper by \\cite{NelderWedderburn1972} introduced\nthe class of generalised linear models (GLMs) and showed that a large\nvariety of non-normal data may be analysed by a simple general\ntechnique \\citep[see, for example,][]{McCu:Neld:1989,\nDobsonBarnett2008}. The GLMs were originally developed for the\nexponential family of distributions, but the main ideas\nwere extended to a wider class of models called dispersion models (DMs)\nin such a way that most of their good properties were preserved.\nThis class of models was introduced by \\cite{Jorgensen1987a} and\nstudied in details in \\cite{Jorgensen1997a}. Some recent references\nabout DMs are \\cite{Kokonendji-et-al-2004},\n\\cite{Jorgensen-et-al-2010}, \\cite{Simas-et-al-2010} and\n\\cite{Rocha-et-al-2010}.\n\nThe class of DMs with position parameter $\\theta$ (which vary in an interval\nof the real line) and precision parameter $\\phi>0$ has\nprobability density function of the form\n\\begin{equation}\\label{dens1}\n\\pi(y;\\theta,\\phi) = \\exp\\{\\phi t(y,\\theta) + c(y,\\phi)\\}, \n\\end{equation}\nwhere $t(\\cdot,\\cdot)$ and $c(\\cdot,\\cdot)$ are known functions.\nIf $Y$ is continuous, $\\pi$ is assumed to be a density with\nrespect to the Lebesgue measure, while if $Y$ is discrete, $\\pi$\nis assumed to be a density with respect to the counting measure.\nThe parameter $\\theta$ may be generally interpreted as a kind of location parameter,\nnot necessarily the mean of the distribution. Several\nmodels of the form (\\ref{dens1}) are discussed by \\cite{Jorgensen1987a,\nJorgensen1987b, Jorgensen1997a}, who also examined\ntheir statistical properties. It is evident that\nsome special cases arise from (\\ref{dens1}). Exponential dispersion models (EDMs)\nrepresent a special case of DMs with $t(y,\\theta)=y\\theta-b(\\theta)$,\nwhere $\\Es(Y)=\\dd b(\\theta)\/\\dd\\theta$; see \\cite{Jorgensen1992}.\nAn important subclass of DMs of\nspecial interest, called proper dispersion models (PDMs),\narise when $c(y,\\phi)$ is additive,\ni.e.~$c(y,\\phi)=a_1(y)+a_2(\\phi)$, where $a_1(\\cdot)$\nand $a_2(\\cdot)$ are known functions \\citep[see, for instance,][]{Jorgensen1997b}.\nThe class of PDMs covers important distributions which are not covered by the EDMs,\nsuch as the log-gamma distribution, the McCullagh distribution\n\\citep{McCullagh1989}, the reciprocal inverse Gaussian distribution\nand the simplex distribution, which is suitable for modeling continuous proportions\n\\citep{BNJorgensen1991}. The von\nMises distribution, which also belongs to the class of PDMs and does\nnot belong to the EDMs, is particularly useful for the analysis\nof circular data; see \\cite{MardiaJupp2000}. The PDMs have two important\ngeneral properties. First, the distribution of the statistic $T = t(Y,\\theta)$\ndoes not depend on $\\theta$ when $\\phi$ is known, that is, $T$\nis a pivotal quantity for $\\theta$. Second, (\\ref{dens1}) is an\nexponential family with canonical statistic $T$ when $\\theta$\nis known.\n\nLarge-sample tests, such as the likelihood ratio, Wald and Rao score tests,\nare usually employed for testing hypotheses in parametric models.\nA new criterion for testing hypotheses, referred to as the  {\\it gradient test},\nwas proposed in \\cite{Terrell2002}.\nIts statistic is very simple to compute when compared with\nthe other three classic statistics. Here, it is worthwhile to quote \n\\cite{Rao2005}: ``The suggestion by Terrell is attractive as it is simple to compute.\nIt would be of interest to investigate the performance of the [gradient] statistic.''\nAlso, Terrell's statistic shares the same first\norder asymptotic properties with the\nlikelihood ratio, Wald and score statistics.\nThat is, to the first order of approximation, the likelihood ratio,\nWald, score and gradient statistics have the same asymptotic\ndistributional properties either under the null hypothesis or under a sequence of\nPitman alternatives, i.e.~a sequence of local\nalternatives that shrink to the null hypothesis at a convergence rate $n^{-1\/2}$.\nAdditionally, it is known that, up to an error of order $n^{-1}$, the likelihood ratio,\nWald, score and gradient tests have the\nsame size properties but their local powers differ in the $n^{-1\/2}$ term.\nTherefore, a meaningful comparison among the criteria can be performed by comparing\nthe nonnull asymptotic expansions to order $n^{-1\/2}$ of the distribution functions\nof these statistics under a sequence of Pitman alternatives.\n\nIn this paper, our main objective is to derive\nnonnull asymptotic expansions to order $n^{-1\/2}$\nof the distribution functions of the likelihood ratio, Wald, score and\ngradient statistics under a sequence of local alternatives\nand to compare the local power of the corresponding tests in the class of DMs.\nIn order to compare the finite-sample performance of these tests in this class of models we\nalso perform a Monte Carlo simulation study.\nAs far as we know, there is no mention in the\nstatistical literature on the use of the gradient test in DMs.\n\nThe nonnull asymptotic expansions up to order $n^{-1\/2}$\nfor the distribution functions of the likelihood ratio and Wald statistics were derived by\n\\cite{Hayakawa1975}, while an analogous result for the\nscore statistic was obtained by \\cite{HarrisPeers1980}.\nThe asymptotic expansion up\nto order $n^{-1\/2}$ for the distribution functions of the gradient statistic was derived\nby \\cite{LemFer2010}. The expansions are\nvery general, although being difficult or even impossible to\nparticularize their formulas for specific regression models.\nAs we shall see below, we have been capable to apply their results for DMs.\nIn particular, we derive closed-form expressions for the\ncoefficients that define the nonnull asymptotic expansions\nof these statistics in this class of models and show\nthat there is no uniform superiority\nof one test with respect to the others for testing a subset of\nregression parameters.\n\nThe rest of the paper is organized as follows. Section \\ref{tests} briefly describes the\nlikelihood ratio, Wald, score and gradient tests. We present the class\nof DMs in Section \\ref{DMs}. In Section \\ref{main_result} we derive\nthe nonnull asymptotic expansions of the likelihood ratio,\nWald, score and gradient statistics for testing a subset\nof regression parameters in DMs. The local power of the\nlikelihood ratio, Wald, score and gradient tests are compared in\nSection \\ref{power_comparison}. In Section \\ref{test_phi}\nwe consider hypothesis testing on the precision parameter.\nMonte Carlo simulation results are addressed in Section \\ref{simulations}.\nWe consider an empirical application in Section \\ref{applications}\nfor illustrative purposes. Section \\ref{conclusions} closes the\npaper with some concluding remarks.\n\n\\section{Background}\\label{tests}\n\nLet $\\ell(\\bm{\\theta})$, $\\bm{U}_{\\bm{\\theta}}$ and $\\bm{K}_{\\bm{\\theta}}$\ndenote the total log-likelihood function, the score\nfunction and the information matrix for the parameter vector\n$\\bm{\\theta}=(\\theta_{1},\\ldots,\\theta_{k})^\\top$ of dimension $k$, respectively.\nLet $\\bm{K}_{\\bm{\\theta}}^{-1}$ denote the inverse of $\\bm{K}_{\\bm{\\theta}}$.\nConsider the partition $\\bm{\\theta} = (\\bm{\\theta}_{1}^{\\top}, \\bm{\\theta}_{2}^{\\top})^{\\top}$,\nwhere the dimensions of $\\bm{\\theta}_{1}$ and $\\bm{\\theta}_{2}$\nare $q$ and $k-q$, respectively.\nSuppose the interest lies in testing the composite null hypothesis\n$\\mathcal{H}_{0}:\\bm{\\theta}_{2} = \\bm{\\theta}_{20}$\nagainst $\\mathcal{H}_{1}:\\bm{\\theta}_{2}\\neq\\bm{\\theta}_{20}$,\nwhere $\\bm{\\theta}_{20}$ is a specified vector.\nHence, $\\bm{\\theta}_{1}$ acts as a vector of nuisance parameters.\nThe likelihood ratio ($S_1$), Wald ($S_2$), score ($S_3$) and gradient ($S_4$) statistics for\ntesting $\\mathcal{H}_{0}$ versus $\\mathcal{H}_{1}$ are given, respectively, by\n\\[\nS_{1} = 2\\bigl\\{\\ell(\\widehat{\\bm{\\theta}}) - \\ell(\\widetilde{\\bm{\\theta}})\\bigr\\},\n\\qquad\nS_{2} = (\\widehat{\\bm{\\theta}} - \\widetilde{\\bm{\\theta}})^{\\top}\\widehat{\\bm{K}}_{\\bm{\\theta}}\n(\\widehat{\\bm{\\theta}} - \\widetilde{\\bm{\\theta}}),\n\\]\n\\[\nS_{3} = \\widetilde{\\bm{U}}_{\\bm{\\theta}}^\\top\\widetilde{\\bm{K}}_{\\bm{\\theta}}^{-1}\\widetilde{\\bm{U}}_{\\bm{\\theta}},\n\\qquad\nS_{4} = \\widetilde{\\bm{U}}_{\\bm{\\theta}}^\\top(\\widehat{\\bm{\\theta}} - \\widetilde{\\bm{\\theta}}),\n\\]\nwhere $\\widehat{\\bm{\\theta}}=(\\widehat{\\bm{\\theta}}_1^\\top,\\widehat{\\bm{\\theta}}_2^\\top)^\\top$ and\n$\\widetilde{\\bm{\\theta}}=(\\widetilde{\\bm{\\theta}}_1^\\top,\\bm{\\theta}_{20}^{\\top})^\\top$\ndenote the maximum likelihood estimators of\n$\\bm{\\theta} =(\\bm{\\theta}_{1}^{\\top}, \\bm{\\theta}_{2}^{\\top})^{\\top}$ under\n$\\mathcal{H}_{1}$ and $\\mathcal{H}_{0}$, respectively,\n$\\widehat{\\bm{K}}_{\\bm{\\theta}}=\\bm{K}_{\\bm{\\theta}}(\\widehat{\\bm{\\theta}})$,\n$\\widetilde{\\bm{K}}_{\\bm{\\theta}}=\\bm{K}_{\\bm{\\theta}}(\\widetilde{\\bm{\\theta}})$ and\n$\\widetilde{\\bm{U}}_{\\bm{\\theta}} = \\bm{U}_{\\bm{\\theta}}(\\widetilde{\\bm{\\theta}})$.\nThe limiting distribution of $S_1$, $S_2$, $S_3$ and $S_4$ is $\\chi_{k-q}^2$ under\n$\\mathcal{H}_{0}$ and $\\chi_{k-q,\\lambda}^{2}$, i.e.~a noncentral chi-square\ndistribution with $k-q$ degrees of freedom and an appropriate noncentrality parameter $\\lambda$,\nunder $\\mathcal{H}_{1}$. The null hypothesis is rejected for a given nominal level, $\\gamma$ say,\nif the test statistic exceeds the upper $100(1-\\gamma)\\%$ quantile of the $\\chi_{k-q}^{2}$\ndistribution.\n\nFrom the partition of $\\bm{\\theta}$, we have the corresponding partitions\n\\[\n\\bm{U}_{\\bm{\\theta}} = (\\bm{U}_{\\bm{\\theta}_1}^\\top, \\bm{U}_{\\bm{\\theta}_2}^\\top)^\\top,\n\\quad\n\\bm{K}_{\\bm{\\theta}} =\n\\begin{bmatrix}\n\\bm{K}_{11} & \\bm{K}_{12} \\\\\n\\bm{K}_{21} & \\bm{K}_{22}\n\\end{bmatrix},\n\\quad\n\\bm{K}_{\\bm{\\theta}}^{-1} =\n\\begin{bmatrix}\n\\bm{K}^{11} & \\bm{K}^{12} \\\\\n\\bm{K}^{21} & \\bm{K}^{22}\n\\end{bmatrix}.\n\\]\nThus, the statistics $S_2$, $S_3$ and $S_4$ can be rewritten as\n\\[\nS_{2} = (\\widehat{\\bm{\\theta}}_{2} - \\bm{\\theta}_{20})^{\\top}\\widehat{\\bm{K}}^{22^{-1}}\n(\\widehat{\\bm{\\theta}}_{2} - \\bm{\\theta}_{20}),\n\\quad\nS_{3} = \\widetilde{\\bm{U}}_{\\bm{\\theta}_2}^\\top\\widetilde{\\bm{K}}^{22}\\widetilde{\\bm{U}}_{\\bm{\\theta}_2},\n\\quad\nS_{4} = \\widetilde{\\bm{U}}_{\\bm{\\theta}_2}^\\top(\\widehat{\\bm{\\theta}}_{2} - \\bm{\\theta}_{20}),\n\\]\nwhere $\\widehat{\\bm{K}}^{22}=\\bm{K}^{22}(\\widehat{\\bm{\\theta}})$,\n$\\widetilde{\\bm{K}}^{22}=\\bm{K}^{22}(\\widetilde{\\bm{\\theta}})$ and\n$\\widetilde{\\bm{U}}_{\\bm{\\theta}_2} = \\bm{U}_{\\bm{\\theta}_2}(\\widetilde{\\bm{\\theta}})$.\n\nNoticed that $S_{4}$ has a very simple form and does not involve\nthe information matrix, neither expected nor observed, unlike $S_{2}$ and $S_{3}$.\n\\cite{Terrell2002} points out that the\ngradient statistic ``is not transparently non-negative, even\nthough it must be so asymptotically.'' His Theorem 2 implies that\nif the log-likelihood function is concave and is differentiable at\n$\\widetilde{\\bm{\\theta}}$, then $S_{4}\\ge 0$.\n\nRecently, \\cite{LemonteFerrari2011}\nobtained the nonnull asymptotic expansions of the\nlikelihood ratio, Wald, score and gradient statistics in\nBirnbaum--Saunders regression models \\citep{RiekNedelman91}.\nAn interesting finding is that, up to an error of order $n^{-1}$,\nthe four tests have the same local power in this class of models.\nTheir simulation study evidenced that the score and the gradient tests\nperform better than the likelihood ratio and Wald tests in small and\nmoderate-sized samples and hence they concluded that the\ngradient test is an appealing alternative to the three\nclassic asymptotic tests in Birnbaum--Saunders regressions.\n\n\n\\section{Dispersion models}\\label{DMs}\n\nWe assume that the random variables $y_1,\\ldots, y_n$ are independent\nand each $y_l$ has a probability density function of the form\n\\begin{equation}\\label{dens2}\n\\pi(y_{l};\\theta_{l},\\phi) = \\exp\\{\\phi t(y_{l},\\theta_{l}) + c(y_{l},\\phi)\\},\n\\quad l = 1,\\ldots,n.\n\\end{equation}\nThe mean of $Y_l$ will be denoted by $\\mu_{l}$, and is not\nnecessary equal to $\\theta_{l}$, the parameter of interest.\nIn order to introduce a regression structure in the class of models in (\\ref{dens2}),\nwe assume that\n\\begin{equation}\\label{sistpart}\nd(\\theta_{l}) = \\eta_{l} = f(\\bm{x}_l;\\bm{\\beta}),\\quad l = 1,\\ldots,n,\n\\end{equation}\nwhere $d(\\cdot)$ is a known one-to-one differentiable link function,\n$\\bm{x}_{l}=(x_{l1},\\ldots,x_{lm})^\\top$ is an $m$-vector of nonstocastic\nvariables associated with the $l$-th response, $\\bm{\\beta}=(\\beta_1,\\ldots,\\beta_p)^\\top$\nis a set of unknown parameters to be estimated ($m\\leq p<n$), and $f(\\cdot;\\cdot)$ is a possible\nnonlinear twice continuous differenciable function with respect to $\\bm{\\beta}$.\nThe regression structure links the covariates $\\bm{x}_l$ to the parameter\nof interest $\\theta_{l}$. The\n$n\\times p$ matrix of derivatives of $\\bm{\\eta}=(\\eta_1,\\ldots,\\eta_n)^\\top$ with\nrespect to $\\bm{\\beta}$, specified by $\\bm{X}^*=\\partial\\bm{\\eta}\/\\partial\\bm{\\beta}^\\top$,\nis assumed to be of full rank, i.e.~rank$(\\bm{X}^*)=p$ for all $\\bm{\\beta}$.\nFurther, it is assumed that the precision parameter is\nunknown and it is the same for all observations. It is also assumed that the\nusual regularity conditions for maximum likelihood estimation and large sample\ninference hold; see \\citet[Ch.~9]{CoxHinkley1974}.\n\nThe class of regression models defined by\n(\\ref{dens2}) and (\\ref{sistpart}) extends the class of generalised\nlinear models discussed by \\cite{McCu:Neld:1989} in two directions.\nFirst and as noted before, it includes important distributions\nwhich are not exponential family models. Second, it allows for a nonlinear\nstructure in $\\bm{\\eta}$. The class of models in\n(\\ref{dens2})-(\\ref{sistpart}) is also a natural extension\nof the exponential family nonlinear models (EFNLMs)\nintroduced by  \\cite{CordPaula1989}, which in turn\nextends the well-known GLMs by allowing the regression\nstructure to be nonlinear. The EFNLMs are defined by\nequations (\\ref{dens2}) and (\\ref{sistpart}), with $t(y_{l},\\theta_{l})=y_{l}\\theta_{l}-b(\\theta_{l})$\nand $c(y_{l},\\phi)=a_1(y_{l})+a_2(\\phi)$ in (\\ref{dens2}).\n\nLet $\\ell=\\ell(\\bm{\\beta},\\phi)=\\sum_{l=1}^{n}\\{\\phi t(y_{l},\\theta_{l}) + c(y_{l},\\phi)\\}$\nbe the total log-likelihood function\nfor $\\bm{\\beta}$ and $\\phi$, where $\\theta_{l}$ is related to $\\bm{\\beta}$ by (\\ref{sistpart}). We define\n$D_{il}=D_{il}(\\theta_{l},\\phi)=\\Es\\{\\partial^{i}t(Y_l,\\phi)\/\\partial\\theta_{l}^i\\}$,\nfor $i=1,2,3$ and $l=1,\\ldots,n$. From regularity conditions we have that $D_{1l}=0$,\nfor $l=1,\\ldots,n$. Table \\ref{tab1}\nlists $D_{2l}$ and $D_{3l}$ for some dispersion models.\nThe total score function and the total Fisher information matrix for $\\bm{\\beta}$\nare given, respectively, by\n$\\bm{U}_{\\bm{\\beta}}=\\phi\\bm{X}^{*\\top}\\dot{\\bm{t}}$ and\n$\\bm{K}_{\\bm{\\beta}} = \\phi\\bm{X}^{*\\top}\\bm{W}\\bm{X}^*$,\nwhere $\\dot{\\bm{t}}=\\dot{\\bm{t}}(\\bm{y},\\bm{\\theta})=(\\dot{t}_1,\\ldots,\\dot{t}_n)^\\top$ \nis an $n\\times 1$ vector with $\\dot{t}_l=\\partial t(y_l,\\theta_{l})\/\\partial\\theta_{l}$,\n$\\bm{y}=(y_1,\\ldots,y_n)^\\top$,\n$\\bm{\\theta}=(\\theta_{1},\\ldots,\\theta_{n})^\\top$ and\n$\\bm{W}=\\diag\\{w_{1},\\ldots,w_{n}\\}$ with $w_{l} = -D_{2l}(\\dd\\theta_{l}\/\\dd\\eta_{l})^2$.\nA simple calculation shows that\n$\\Es(\\partial^2\\ell\/\\partial\\bm{\\beta}\\partial\\phi)=\\bm{0}$ and then the parameters\n$\\bm{\\beta}$ and $\\phi$ are globally orthogonal \\citep{CoxReid1987}.\nLet $\\alpha_i=\\sum_{l=1}^{n}\\Es\\{\\partial^ic(Y_l,\\phi)\/\\partial\\phi^i\\}\n=\\sum_{l=1}^{n}\\Es\\{c^{(i)}(Y_l,\\phi)\\}$, for $i=1,2,3$. The derivatives\nof the $\\alpha_i$'s with respect to $\\phi$ are written with primes,\ni.e.~$\\alpha_{i}' = \\dd\\alpha_i\/\\dd\\phi$ and so on. We have that\nthe joint information matrix for $(\\bm{\\beta}^\\top,\\phi)^\\top$ is given by\n$\\diag\\{\\bm{K}_{\\bm{\\beta}},-\\alpha_2\\}$.\n\\begin{table}[!htp]\n{\\footnotesize\n\\begin{center}\n\\caption{Expressions of $D_{2l}$ and $D_{3l}$ ($l=1,\\ldots,n$) for some dispersion models.$^{\\dagger}$}\\label{tab1}\n\\begin{tabular}{lcc}\\hline\nModel            &  $D_{2l}$                                   & $D_{3l}$     \\\\\\hline\nNormal           &  $-1$                          & 0    \\\\\nInverse Gaussian &  $-(-2\\theta_{l})^{-3\/2}$                          & $-3(-2\\theta_{l})^{-5\/2}$   \\\\\nReciprocal inverse Gaussian &   $-1\/\\theta_{l}$                          & 0   \\\\\nGamma            &  $-1\/\\theta_{l}^2$ & $2\/\\theta_{l}^3$ \\\\\nReciprocal gamma &  $-1\/\\theta_{l}^2$ & $2\/\\theta_{l}^3$ \\\\\nLog-gamma        &  $-1$ & 1 \\\\\nvon Mises        &  $-I_1(\\phi)\/I_0(\\phi)$ & 0\\\\\ngeneralised hyperbolic secant & $2\/(\\theta_l^2 + 1)^3$ & $(2\\theta_l^3 + 10\\theta_l)\/(\\theta_l^2 + 1)^3$\\\\\\hline\n\\multicolumn{3}{l}{{\\footnotesize $^{\\dagger}$$I_j(\\phi)$ is the modified Bessel function of the first kind and order $j$.}}\n\\end{tabular}\n\\end{center}    }\n\\end{table}\n\nThe maximum likelihood estimate\n(MLE) $\\widehat{\\bm{\\beta}}$ of $\\bm{\\beta}$ can be obtained iteratively using\nstandard reweighted least squares method \\citep{Jorgensen1983,Jorgensen1984}:\n\\[\n\\bm{X}^{*(m)\\top}\\bm{W}^{(m)}\\bm{X}^{*(m)}\\bm{\\beta}^{(m+1)} = \\bm{X}^{*(m)\\top}\\bm{W}^{(m)}\\bm{y}^{*(m)},\n\\quad m = 0,1,\\ldots,\n\\]\nwhere $\\bm{y}^{*(m)} = \\bm{X}^{*(m)}\\bm{\\beta}^{(m)} + \\bm{N}^{(m)}\\dot{\\bm{t}}^{(m)}$\nis an adjusted dependent variable and $\\bm{N}$ is a diagonal matrix\ngiven by $\\bm{N}=-\\diag\\{D_{21}^{-1}(\\dd\\theta_{1}\/\\dd\\eta_{1})^{-1}, \\ldots,\nD_{2n}^{-1}(\\dd\\theta_{n}\/\\dd\\eta_{n})^{-1}\\}$. The estimate $\\widehat{\\bm{\\beta}}$ depends\ndirectly on the distribution only through the function $D_{2l}$ and\ndoes not depend on the parameter $\\phi$. The maximum likelihood\nestimate $\\widehat{\\phi}$ of $\\phi$ is the solution of\n\\begin{equation}\\label{est_phi}\n\\sum_{l=1}^{n}\\{t(y_l,\\widehat{\\theta}_l) + c^{(1)}(y_l,\\widehat{\\phi})\\} = 0.\n\\end{equation}\nThe maximum likelihood estimators $\\widehat{\\bm{\\beta}}$\nand $\\widehat{\\phi}$ are asymptotically independent due to\ntheir asymptotic normality and the block diagonal structure of the joint\ninformation matrix. If the model is a PDM the $\\alpha_{i}$'s\ncan be expressed as functions of $\\phi$ only, namely $\\alpha_{i}=na_2^{(i)}(\\phi)$\nfor $i=1,2,3$, where $a_2^{(i)}(\\phi)$ is the $i$-th derivative of $a_2(\\phi)$ with\nrespect to $\\phi$. In this case, the $(p+1,p+1)$-th element of the joint\ninformation matrix is simply $-na_2^{(2)}(\\phi)$ and equation\n(\\ref{est_phi}) reduces to $a_2^{(1)}(\\widehat{\\phi})=-\\sum_{l=1}^{n}t(y_l,\\widehat{\\theta}_l)\/n$.\n\nIn what follows, we shall consider tests based on the\nlikelihood ratio ($S_1$), Wald ($S_2$), Rao score ($S_3$) and gradient ($S_4$) statistics\nin the class of DMs for testing a composite null hypothesis\n$\\mathcal{H}_{0}: \\bm{\\beta}_{2} = \\bm{\\beta}_{20}$.\nThis hypothesis will be tested against the alternative hypothesis\n$\\mathcal{H}_{1}:\\bm{\\beta}_{2}\\neq\\bm{\\beta}_{20}$,\nwhere $\\bm{\\beta}$ is partitioned as $\\bm{\\beta} = (\\bm{\\beta}_{1}^{\\top},\n\\bm{\\beta}_{2}^{\\top})^{\\top}$, with\n$\\bm{\\beta}_{1} = (\\beta_{1}, \\dots,\\beta_{q})^{\\top}$ and\n$\\bm{\\beta}_{2} = (\\beta_{q+1},\\dots,\\beta_{p})^{\\top}$. Here,\n$\\bm{\\beta}_{20}$ is a fixed column vector of dimension $p-q$. The\npartition of the parameter vector $\\bm{\\beta}$ induces the corresponding partitions\n$\\bm{U}_{\\bm{\\beta}} = (\\bm{U}_{\\bm{\\beta}_1}^\\top, \\bm{U}_{\\bm{\\beta}_2}^\\top)^\\top$,\nwith $\\bm{U}_{\\bm{\\beta}_1}=\\phi\\bm{X}_1^{*\\top}\\dot{\\bm{t}}$ and\n$\\bm{U}_{\\bm{\\beta}_2}=\\phi\\bm{X}_2^{*\\top}\\dot{\\bm{t}}$,\n\\[\n\\bm{K}_{\\bm{\\beta}} =\n\\begin{bmatrix}\n\\bm{K}_{\\bm{\\beta}11} & \\bm{K}_{\\bm{\\beta}12} \\\\\n\\bm{K}_{\\bm{\\beta}21} & \\bm{K}_{\\bm{\\beta}22}\n\\end{bmatrix} = \\phi\n\\begin{bmatrix}\n\\bm{X}_1^{*\\top}\\bm{W}\\bm{X}_1^{*} & \\bm{X}_1^{*\\top}\\bm{W}\\bm{X}_2^{*} \\\\\n\\bm{X}_2^{*\\top}\\bm{W}\\bm{X}_1^{*} & \\bm{X}_2^{*\\top}\\bm{W}\\bm{X}_2^{*}\n\\end{bmatrix},\n\\]\nwith the matrix $\\bm{X}^*$ partitioned\nas $\\bm{X}^* = \\bigl[\\bm{X}_{1}^*\\ \\ \\bm{X}_{2}^*\\bigr]$,\n$\\bm{X}_{1}^*$ being $n\\times q$ and $\\bm{X}_{2}^*$ being $n\\times (p-q)$.\nLet $(\\widehat{\\bm{\\beta}}_{1}, \\widehat{\\bm{\\beta}}_{2}, \\widehat{\\phi})$ and\n$(\\widehat{\\bm{\\beta}}_{1}, \\bm{\\beta}_{20}, \\widetilde{\\phi})$ be the\nunrestricted and restricted MLEs of $(\\bm{\\beta}_{1}, \\bm{\\beta}_{2}, \\phi)$, respectively.\nThe likelihood ratio, Wald, score and gradient statistics for testing $\\mathcal{H}_{0}$\ncan be expressed, respectively, as\n\\[\nS_{1} = 2\\bigl\\{\\ell(\\widehat{\\bm{\\beta}}_{1}, \\widehat{\\bm{\\beta}}_{2},\\widehat{\\phi})\n- \\ell(\\widetilde{\\bm{\\beta}}_{1}, \\bm{\\beta}_{20},\\widetilde{\\phi})\\bigr\\},\n\\qquad\nS_{2} = \\widehat{\\phi}(\\widehat{\\bm{\\beta}}_{2}- \\bm{\\beta}_{20})^{\\top}\n(\\widehat{\\bm{R}}^{\\top}\\widehat{\\bm{W}}\\widehat{\\bm{R}})\n(\\widehat{\\bm{\\beta}}_{2} - \\bm{\\beta}_{20}),\n\\]\n\\[\nS_{3} = \\widetilde{\\bm{s}}^{\\top}\\widetilde{\\bm{W}}^{1\/2}\\widetilde{\\bm{X}}_{2}^*\n(\\widetilde{\\bm{R}}^{\\top}\\widetilde{\\bm{W}}\\widetilde{\\bm{R}})^{-1}\n\\widetilde{\\bm{X}}_{2}^{*\\top}\\widetilde{\\bm{W}}^{1\/2}\\widetilde{\\bm{s}},\\qquad\nS_{4} = \\widetilde{\\phi}^{1\/2}\\widetilde{\\bm{s}}^{\\top}\\widetilde{\\bm{W}}^{1\/2}\\widetilde{\\bm{X}}_{2}^*\n(\\widehat{\\bm{\\beta}}_{2} - \\bm{\\beta}_{20}),\n\\]\nwhere $\\bm{s} = (s_1,\\ldots,s_n)^\\top$ with $s_l=\\phi^{1\/2}\\dot{t}_{l}(-D_{2l})^{-1\/2}$ and\n$\\bm{R} = \\bm{X}_{2}^* - \\bm{X}_{1}^*(\\bm{X}_{1}^{*\\top}\\bm{W}\\bm{X}_{1}^*)^{-1}\n\\bm{X}_{1}^{*\\top}\\bm{W}\\bm{X}_{2}^*$.\nHere, tildes and hats indicate evaluation at the restricted and unrestricted MLEs,\nrespectively. The limiting distribution of all these statistics\nunder $\\mathcal{H}_{0}$ is $\\chi_{p-q}^2$. Note that,\nunlike the Wald and score statistics, the gradient statistic\ndoes not involve any matrix inversion.\n\n\n\\section{Nonnull asymptotic distributions in DMs}\\label{main_result}\n\nWe present in this section expressions for the nonnull asymptotic\nexpansions up to order $n^{-1\/2}$ for the nonnull distribution of\nthe likelihood ratio, Wald, score and gradient statistics for\ntesting a subset of regression parameters in DMs.\nIt should be mentioned that the general nonnull asymptotic expansions derived in\n\\cite{Hayakawa1975}, \\cite{HarrisPeers1980} and \\cite{LemFer2010}\nwere developed for continuous distributions. It implies that the results derived in this\nsection are only valid for continuous DMs. Here, we shall assume the following local alternative\nhypothesis $\\mathcal{H}_{1n}:\\bm{\\beta}_{2}=\\bm{\\beta}_{20} + \\bm{\\epsilon}$,\nwhere $\\bm{\\epsilon} = (\\epsilon_{q+1}, \\ldots,\\epsilon_{p})^{\\top}$ with\n$\\epsilon_{r} = O(n^{-1\/2})$ for $r=q+1,\\ldots,p$.\n\nWe introduce the following quantities:\n\\[\n\\bm{\\epsilon}^{*} = \\begin{bmatrix}\n\\bm{K}_{\\bm{\\beta}11}^{-1}\\bm{K}_{\\bm{\\beta}12}\\\\\n-\\bm{I}_{p-q}\n\\end{bmatrix}\\bm{\\epsilon},\n\\quad\n\\bm{A} = \\begin{bmatrix}\n                       \\bm{K}_{\\bm{\\beta}11}^{-1} & \\bm{0}\\\\\n                       \\bm{0} & \\bm{0}\n                      \\end{bmatrix},\n\\quad\n\\bm{M} = \\bm{K}_{\\bm{\\beta}}^{-1} - \\bm{A},\n\\]\nwhere $\\bm{I}_{p-q}$ is a $(p-q)\\times(p-q)$ identity matrix.\nAdditionally, let $\\bm{Z} = \\bm{X}^*(\\bm{X}^{*\\top}\\bm{W}\\bm{X}^*)^{-1}\\bm{X}^{*\\top} = \\{z_{lm}\\}$,\n$\\bm{Z}_{1} = \\bm{X}_{1}^*(\\bm{X}_{1}^{*\\top}\\bm{W}\\bm{X}_{1}^*)^{-1}\\bm{X}_{1}^{*\\top} = \\{z_{1lm}\\}$,\n\\[\n\\bm{X}_{l}^* = \\biggl\\{\\frac{\\partial^2\\eta_{l}}{\\partial\\beta_{r}\\partial\\beta_{s}}\\biggr\\} =\n\\begin{bmatrix}\n\\bm{X}_{11l}^* & \\bm{X}_{12l}^* \\\\\n\\bm{X}_{21l}^* & \\bm{X}_{22l}^*\n\\end{bmatrix},\\quad r,s=1,\\ldots,p,\\quad l =1,\\ldots,n,\n\\]\n$\\bm{Z}_{d} = \\diag\\{z_{11},\\ldots, z_{nn}\\}$,\n$\\bm{Z}_{1d} = \\diag\\{z_{111},\\ldots, z_{1nn}\\}$,\n$\\bm{F} = \\diag\\{f_{1},\\ldots,f_{n}\\}$, $\\bm{G} = \\diag\\{g_{1},\\ldots,g_{n}\\}$,\n$\\bm{E} = \\diag\\{e_{1},\\ldots,e_{n}\\}$,\n$\\bm{t} = (t_{1},\\ldots,t_{n})^{\\top} = \\bm{X}^*\\bm{\\epsilon}^{*}$,\n$\\bm{b} = (b_{1},\\ldots,b_{n})^{\\top} = \\bm{X}_{2}^*\\bm{\\epsilon}$,\n$\\bm{T} = \\diag\\{t_{1},\\ldots,t_{n}\\}$,\n$\\bm{T}^{(2)} = \\bm{T}\\odot\\bm{T}$, $\\bm{T}^{(3)} = \\bm{T}^{(2)}\\odot\\bm{T}$\nand $\\bm{B} = \\diag\\{b_{1},\\ldots,b_{n}\\}$, where ``$\\odot$'' denotes the Hadamard (direct)\nproduct of matrices, and\n\\[\nf_l = -\\frac{\\dd \\theta_l}{\\dd \\eta_l}\\frac{\\dd ^2\\theta_l}{\\dd \\eta_l^2}D_{2l}\n            - \\biggl(\\frac{\\dd\\theta_l}{\\dd \\eta_l}\\biggr)^{3}D_{3l},\\quad\ng_l = -\\frac{\\dd\\theta_l}{\\dd \\eta_l}\\frac{\\dd ^2\\theta_l}{\\dd\\eta_l^2}D_{2l},\\quad\ne_l = - \\biggl(\\frac{\\dd\\theta_l}{\\dd \\eta_l}\\biggr)^{3}D_{2l}',\n\\quad l=1,\\ldots,n,\n\\]\nwhere $D_{2l}'$ denotes the first derivative of\n$D_{2l}$ with respect to $\\theta_{l}$, for $l=1,\\ldots,n$.\n\nThe nonnull distributions of $S_1$, $S_2$, $S_3$ and $S_4$\nunder Pitman alternatives for testing $\\mathcal{H}_{0}:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$\nin DMs can be expressed as\n\\begin{equation*}\\label{expansion}\n\\Pr(S_{i}\\leq x) = G_{p-q,\\lambda}(x) + \\sum_{k=0}^{3}b_{ik}G_{p-q+2k,\\lambda}(x) + O(n^{-1}),\n\\quad i=1,2,3,4,\n\\end{equation*}\nwhere $G_{m,\\lambda}(x)$ is the cumulative distribution function of\na non-central chi-square variate with $m$ degrees of freedom\nand non-centrality parameter $\\lambda$. Here,\n$\\lambda = \\phi\\tr\\{\\bm{K}_{22.1}\\bm{\\epsilon}\\bm{\\epsilon}^\\top\\}\/2$,\nwhere $\\bm{K}_{22.1} = \\bm{K}_{\\bm{\\beta}22} - \\bm{K}_{\\bm{\\beta}21}\\bm{K}_{\\bm{\\beta}11}^{-1}\n\\bm{K}_{\\bm{\\beta}12}$ and $\\tr(\\cdot)$ denotes the trace operator.\nThe coefficients $b_{ik}$'s ($i=1,2,3,4$ and $k=0,1,2,3$) can be written\nin matrix notation, after extensive algebra, as\n\\begin{align*}\nb_{11} &= \\frac{\\phi}{2}\\tr\\{(\\bm{E} + 2\\bm{G})\\bm{B}\\bm{T}^{(2)}\n+ (2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\\\\\n&\\quad+\\frac{1}{2}\\tr\\{(2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{Z}_{1d}\\bm{T}+\\bm{W}\\bm{J}\\bm{T}\\},\n\\end{align*}\n\\[\nb_{12} = -\\frac{\\phi}{6}\\tr\\{(3\\bm{E}-2\\bm{F} + 2\\bm{G})\\bm{T}^{(3)}\\},\n\\quad\nb_{13} = 0,\n\\]\n\\begin{align*}\nb_{21} &= \\frac{\\phi}{2}\\tr\\{(\\bm{E} + 2\\bm{G})\\bm{B}\\bm{T}^{(2)}\n+ (2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\} \\\\\n&\\quad + \\frac{1}{2}\\tr\\{(2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{Z}_{d}\\bm{T}\n+ 2(\\bm{F}-\\bm{E})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}+\\bm{W}(\\bm{U}\\bm{T} + 2\\bm{H})\\},\n\\end{align*}\n\\begin{align*}\nb_{22} = \\frac{\\phi}{2}\\tr\\{(\\bm{F}-\\bm{E})\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\n-\\frac{1}{2}\\tr\\{(\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}+\\bm{W}\\bm{T}(\\bm{U} - \\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\end{align*}\n\\[\nb_{23} = -\\frac{\\phi}{6}\\tr\\{(\\bm{F} + 2\\bm{G})\\bm{T}^{(3)}+3\\bm{W}\\bm{T}\\bm{C}\\},\n\\]\n\\begin{align*}\nb_{31} &= \\frac{\\phi}{2}\\tr\\{(\\bm{E} + 2\\bm{G})\\bm{B}\\bm{T}^{(2)}\n+ (2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\\\\\n&\\quad + \\frac{1}{2}\\tr\\{(2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{Z}_{1d}\\bm{T}\n+ (3\\bm{E}-2\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}+\\bm{W}\\bm{T}\\bm{J}\\},\n\\end{align*}\n\\[\nb_{32} = -\\frac{1}{2}\\tr\\{(3\\bm{E}-2\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\quad\nb_{33} = -\\frac{\\phi}{6}\\tr\\{(3\\bm{E}-2\\bm{F} + 2\\bm{G})\\bm{T}^{(3)}\\},\n\\]\n\\begin{align*}\nb_{41} &= \\frac{\\phi}{2}\\tr\\{(\\bm{E}+2\\bm{G})\\bm{B}\\bm{T}^{(2)}\n+ (2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\\\\\n&\\quad + \\frac{1}{4}\\tr\\{(6\\bm{G}-\\bm{F} + 4\\bm{E})\\bm{Z}_{1d}\\bm{T}\n- (\\bm{F} + 2\\bm{G})\\bm{Z}_{d}\\bm{T}+\\bm{W}\\bm{T}(3\\bm{J}-\\bm{U}) - 2\\bm{W}\\bm{H}\\},\n\\end{align*}\n\\begin{align*}\nb_{42} &= -\\frac{\\phi}{4}\\tr\\{(2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\\\\\n&\\quad +\\frac{1}{4}\\tr\\{(\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}+\\bm{W}\\bm{T}(\\bm{U}-\\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\end{align*}\n\\[\nb_{43} = \\frac{\\phi}{12}\\tr\\{(\\bm{F} + 2\\bm{G})\\bm{T}^{(3)}+3\\bm{W}\\bm{T}\\bm{C}\\},\n\\]\nwhere\n$\\bm{U}=\\diag\\{u_1,\\ldots,u_n\\}$ with $u_l = \\tr\\{\\bm{X}_{l}^*(\\bm{X}^{*\\top}\\bm{W}\\bm{X}^{*})^{-1}\\}$,\n$\\bm{J}=\\diag\\{j_1,\\ldots,j_n\\}$ with $j_l = \\tr\\{\\bm{X}_{11l}^*(\\bm{X}_{1}^{*\\top}\\bm{W}\\bm{X}_{1}^{*})^{-1}\\}$,\n$\\bm{C}=\\diag\\{c_1,\\ldots,c_n\\}$ with $c_{l}=\\tr\\{\\bm{X}_l^*\\bm{\\epsilon}^*\\bm{\\epsilon}^{*\\top}\\}$,\n$\\bm{P}=\\diag\\{p_1,\\ldots,p_n\\}$ with $p_l = \\tr\\{\\bm{X}_{l}^*\\bm{\\epsilon}^*\\bm{\\delta}^\\top\\}$,\n$\\bm{H}=\\diag\\{h_1,\\ldots,h_n\\}$ with $h_l = \\phi\\tr\\{\\bm{M}\\bm{X}_{l}^*\\bm{\\epsilon}^*\\bm{x}_l^{*\\top}\\}$,\n$\\bm{\\delta}^\\top = (\\bm{0}^\\top,\\bm{\\epsilon}^\\top)$ and\n$\\bm{x}_l^{*\\top}$ is the $l$th line of $\\bm{X}^*$.\nThe coefficients $b_{i0}$ are obtained from $b_{i0} = -(b_{i1} + b_{i2} + b_{i3})$,\nfor $i=1,2,3,4$. The $b_{ik}$'s are of order $n^{-1\/2}$ and all quantities except $\\bm{\\epsilon}$\nare evaluated under the null hypothesis $\\mathcal{H}_0$. The detailed derivation of these expressions\nis long and extremely tedious but may be obtained from the authors upon request.\n\nIt is interesting to note that the $b_{ik}$'s are functions of the local\nderivative matrix and of the (possibly unknown) precision parameter.\nThese coefficients depend on the second derivative of the (possibly nonlinear)\nfunction $f(\\bm{x}_{l};\\bm{\\beta})$ and\ninvolve the link function and its first and second derivatives.\nUnfortunately, they are very difficult to interpret. The\nmatrices $\\bm{C}$, $\\bm{H}$, $\\bm{J}$, $\\bm{P}$ and $\\bm{U}$\nmay be considered the nonlinear contribution of the dispersion model since\nthey vanish if the regression model is linear.\nObviously, these coefficients depend heavily on the particular dispersion\nmodel under consideration. In particular, these coefficients do not change for the\nclass of PDMs, since the only difference between PDMs and DMs\nis the form of the function $c(\\cdot,\\cdot)$, which can be\ndecomposed as $c(y,\\phi)=a_1(y)+a_2(\\phi)$ for PDMs. \nBy replacing $\\bm{E}$ by $\\bm{F}-\\bm{G}$ in these coefficients, we obtain the\nnonnull asymptotic distributions of the four statistics in\nthe class of EFNLMs \\citep[see][]{Lemonte2011}. \n\nSome simplifications in the coefficients $b_{ik}$ ($i=1,2,3,4$ and $k=0,1,2,3$)\ncan be achieved by examining special cases. For example,\nconsider the null hypothesis $\\mathcal{H}_0:\\bm{\\beta}=\\bm{\\beta}_0$ (i.e.~$q=0$) and\nan identity link function ($d(\\theta_{l})=\\theta_{l}$),\nwhich implies that $f_l=-D_{3l}$, $g_l=0$ and $e_l = -D_{2l}'$\n($l=1,\\ldots,n$). Therefore, the $b_{ik}$'s can be written as\n\\[\nb_{11} = \\frac{\\phi}{2}\\tr\\{\\bm{E}\\bm{B}\\bm{T}^{(2)} + (2\\bm{E}-\\bm{F})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+\\frac{1}{2}\\tr\\{\\bm{W}\\bm{J}\\bm{T}\\},\n\\]\n\\[\nb_{12} = b_{33} = -\\frac{\\phi}{6}\\tr\\{(3\\bm{E}-2\\bm{F})\\bm{T}^{(3)}\\},\n\\quad\nb_{13} = 0,\n\\quad\nb_{32} = -\\frac{1}{2}\\tr\\{(3\\bm{E}-2\\bm{F})\\bm{Z}_d\\bm{T}\\},\n\\]\n\\begin{align*}\nb_{21} &= \\frac{\\phi}{2}\\tr\\{\\bm{E}\\bm{B}\\bm{T}^{(2)} + (2\\bm{E}-\\bm{F})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\} \\\\\n&\\quad + \\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_{d}\\bm{T}+\\bm{W}(\\bm{U}\\bm{T} + 2\\bm{H})\\},\n\\end{align*}\n\\[\nb_{22} = \\frac{\\phi}{2}\\tr\\{(\\bm{F}-\\bm{E})\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\n-\\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}(\\bm{U} - \\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nb_{23} = -2b_{43} =-\\frac{\\phi}{6}\\tr\\{\\bm{F}\\bm{T}^{(3)}+3\\bm{W}\\bm{T}\\bm{C}\\},\n\\]\n\\begin{align*}\nb_{31} &= \\frac{\\phi}{2}\\tr\\{\\bm{E}\\bm{B}\\bm{T}^{(2)} + (2\\bm{E}-\\bm{F})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\\\\\n&\\quad + \\frac{1}{2}\\tr\\{(3\\bm{E}-2\\bm{F})\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}\\bm{J}\\},\n\\end{align*}\n\\begin{align*}\nb_{41} &= \\frac{\\phi}{2}\\tr\\{\\bm{E}\\bm{B}\\bm{T}^{(2)} + (2\\bm{E}-\\bm{F})\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\\\\\n&\\quad + \\frac{1}{4}\\tr\\{-\\bm{F}\\bm{Z}_{d}\\bm{T}+\\bm{W}\\bm{T}(3\\bm{J}-\\bm{U}) - 2\\bm{W}\\bm{H}\\},\n\\end{align*}\n\\[\nb_{42} = -\\frac{\\phi}{4}\\tr\\{(2\\bm{E}-\\bm{F})\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\n+\\frac{1}{4}\\tr\\{\\bm{F}\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}(\\bm{U}-\\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\]\nand  $b_{i0} = -(b_{i1} + b_{i2} + b_{i3})$, for $i=1,2,3,4$. For the\nlog-gamma model, the above coefficients reduce to\n\\[\nb_{11} = \\frac{\\phi}{2}\\tr\\{-\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+\\frac{1}{2}\\tr\\{\\bm{W}\\bm{J}\\bm{T}\\},\n\\quad\nb_{12} = b_{33} = \\frac{\\phi}{3}\\tr\\{\\bm{F}\\bm{T}^{(3)}\\},\n\\quad\nb_{13} = 0,\n\\]\n\\[\nb_{21} = \\frac{\\phi}{2}\\tr\\{-\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+ \\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_{d}\\bm{T}+\\bm{W}(\\bm{U}\\bm{T} + 2\\bm{H})\\},\n\\]\n\\[\nb_{22} =\\frac{\\phi}{2}\\tr\\{\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\n-\\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}(\\bm{U} - \\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nb_{23} = -2b_{43} = -\\frac{\\phi}{6}\\tr\\{\\bm{F}\\bm{T}^{(3)}+3\\bm{W}\\bm{T}\\bm{C}\\},\n\\quad\nb_{32} = \\tr\\{\\bm{F}\\bm{Z}_d\\bm{T}\\},\n\\]\n\\[\nb_{31} = \\frac{\\phi}{2}\\tr\\{-\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+ \\frac{1}{2}\\tr\\{-2\\bm{F}\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}\\bm{J}\\},\n\\]\n\\[\nb_{41} = \\frac{\\phi}{2}\\tr\\{-\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+ \\frac{1}{4}\\tr\\{-\\bm{F}\\bm{Z}_{d}\\bm{T}+\\bm{W}\\bm{T}(3\\bm{J}-\\bm{U}) - 2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nb_{42} = -\\frac{\\phi}{4}\\tr\\{-\\bm{F}\\bm{T}^{(3)}+\\bm{W}\\bm{T}\\bm{C}\\}\n+\\frac{1}{4}\\tr\\{\\bm{F}\\bm{Z}_d\\bm{T}+\\bm{W}\\bm{T}(\\bm{U}-\\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\]\nAlso, for the von Mises model we have\n\\[\nb_{11} = b_{31} = \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+\\frac{1}{2}\\tr\\{\\bm{W}\\bm{J}\\bm{T}\\},\n\\quad\nb_{12} = b_{13} = b_{32} = b_{33} =0,\n\\]\n\\[\nb_{21} = \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+ \\frac{1}{2}\\tr\\{\\bm{W}(\\bm{U}\\bm{T} + 2\\bm{H})\\},\n\\quad\nb_{23} = -2b_{43} = -\\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}\\bm{C}\\},\n\\]\n\\[\nb_{22}  = -2b_{42} = \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}\\bm{C}\\}\n-\\frac{1}{2}\\tr\\{\\bm{W}\\bm{T}(\\bm{U} - \\bm{J}) + 2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nb_{41} = \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}(\\bm{C}+2\\bm{P})\\}\n+ \\frac{1}{4}\\tr\\{\\bm{W}\\bm{T}(3\\bm{J}-\\bm{U}) - 2\\bm{W}\\bm{H}\\},\n\\]\nNote that for the von Mises linear regression model,\nthe $b_{ij}$'s above vanish and hence we can write\n\\[\n\\Pr(S_{i}\\leq x) = G_{p,\\lambda}(x) + O(n^{-1}),\\qquad i=1,2,3,4.\n\\]\nThis is a very interesting\nresult, which implies that the likelihood ratio, score, Wald\nand gradient tests for testing the null hypothesis\n$\\mathcal{H}_{0}:\\bm{\\beta}=\\bm{\\beta}_{0}$\nhave exactly the same local power up to an error of order $n^{-1}$\nwhen we consider an identity link function.\nIt should be noticed that  this result also happens for testing the\ncomposite null hypothesis $\\mathcal{H}_{0}:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$,\ni.e~$\\Pr(S_{i}\\leq x) = G_{p-q,\\lambda}(x) + O(n^{-1})$, for $i=1,2,3,4$.\n\nNow, we present the coefficients that define the nonnull asymptotic distributions\nof the likelihood ratio, Wald, score and gradient statistics\nfor testing the composite null hypothesis\n$\\mathcal{H}_{0}:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$\nin GLMs. We have $t(y_{l},\\theta_{l})=y_{l}\\theta_{l}-b(\\theta_{l})$\nand $\\mu_{l} = \\Es(Y_{l})=\\dd b(\\theta_{l})\/\\dd\\theta_{l}$.\nThe class of GLMs is characterized by its variance function\n$V_{l} = \\dd\\mu_{l}\/\\dd\\theta_{l}$, which plays a key role in the study of\nits mathematical properties and estimation.\nThe variance of $Y_{l}$ can be written as var$(Y_{l}) = \\phi^{-1}V_{l}$.\nFor the GLMs, we have $D_{2l}=-V_l^{-1}$ and $D_{3l}=2V_{l}^{-1}(\\dd V_l\/\\dd\\mu_l)$\nand hence we can rewrite\n\\[\nf_l = \\frac{1}{V_l}\\frac{\\dd \\mu_l}{\\dd \\eta_l}\\frac{\\dd ^2\\mu_l}{\\dd \\eta_l^2},\\quad\ng_l = \\frac{1}{V_l}\\frac{\\dd \\mu_l}{\\dd \\eta_l}\\frac{\\dd ^2\\mu_l}{\\dd \\eta_l^2}\n-\\frac{1}{V_l^2}\\frac{\\dd V_l}{\\dd \\mu_l}\\biggl(\\frac{\\dd \\mu_l}{\\dd \\eta_l}\\biggr)^3,\n\\quad l=1,\\ldots,n,\n\\]\nand redefine the matrices $\\bm{F}$ and $\\bm{G}$ given before.\nAdditionally, the link function is $d(\\mu_{l})=\\eta_{l}=\\bm{x}_l^\\top\\bm{\\beta}$\nwith $m=p$. Also, $\\bm{\\eta}=\\bm{X}\\bm{\\beta}$ with $\\bm{X}=(\\bm{x}_1,\\ldots,\\bm{x}_n)^\\top$,\ni.e.~here $\\bm{X}^*=\\bm{X}$. Hence, in this class of models we have\n\\[\nb_{11} = \\frac{\\phi}{2}\\tr\\{(\\bm{F} + \\bm{G})\\bm{B}\\bm{T}^{(2)} + \\bm{F}\\bm{T}^{(3)}\\}\n    + \\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_{1d}\\bm{T}\\},\n\\quad\nb_{12} = b_{33} = -\\frac{\\phi}{6}\\tr\\{(\\bm{F} - \\bm{G})\\bm{T}^{(3)}\\},\n\\]\n\\[\nb_{21} = \\frac{\\phi}{2}\\tr\\{(\\bm{F} + \\bm{G})\\bm{B}\\bm{T}^{(2)} + \\bm{F}\\bm{T}^{(3)}\\}\n    + \\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_{d}\\bm{T} + 2\\bm{G}(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\n\\[\nb_{22} = \\frac{\\phi}{2}\\tr\\{\\bm{G}\\bm{T}^{(3)}\\}-\\frac{1}{2}\\tr\\{(\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\quad\nb_{13} = 0,\n\\]\n\\[\nb_{23}  = -2b_{43} = -\\frac{\\phi}{6}\\tr\\{(\\bm{F} + 2\\bm{G})\\bm{T}^{(3)}\\},\n\\quad\nb_{32} = -\\frac{1}{2}\\tr\\{(\\bm{F} - \\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\n\\[\nb_{31} = \\frac{\\phi}{2}\\tr\\{(\\bm{F} + \\bm{G})\\bm{B}\\bm{T}^{(2)} + \\bm{F}\\bm{T}^{(3)}\\}\n    + \\frac{1}{2}\\tr\\{\\bm{F}\\bm{Z}_{1d}\\bm{T} + (\\bm{F}-\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\n\\[\nb_{41} = \\frac{\\phi}{2}\\tr\\{(\\bm{F}+\\bm{G})\\bm{B}\\bm{T}^{(2)} + \\bm{F}\\bm{T}^{(3)}\\}\n    + \\frac{1}{4}\\tr\\{(3\\bm{F} + 2\\bm{G})\\bm{Z}_{1d}\\bm{T} - (\\bm{F} + 2\\bm{G})\\bm{Z}_{d}\\bm{T}\\},\n\\]\n\\[\nb_{42} = -\\frac{\\phi}{4}\\tr\\{\\bm{F}\\bm{T}^{(3)}\\}\n        +\\frac{1}{4}\\tr\\{(\\bm{F} + 2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\nBy considering the identity link function, these coefficients reduce to\n\\[\nb_{11} = \\frac{\\phi}{2}\\tr\\{\\bm{G}\\bm{B}\\bm{T}^{(2)}\\},\n\\quad\nb_{12} = b_{33} = \\frac{\\phi}{6}\\tr\\{\\bm{G}\\bm{T}^{(3)}\\},\n\\quad\nb_{32} = b_{42} = \\frac{1}{2}\\tr\\{\\bm{G}(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\n\\[\nb_{13} = 0,\n\\quad\nb_{23} = -2b_{43} = -2b_{12}, \n\\quad\nb_{21} = b_{11} + 2b_{32},\n\\quad\nb_{22} = 3b_{12}- 2b_{32},\n\\quad\nb_{31} = b_{41} = b_{11}-b_{32}.\n\\]\nAs expected, the above coefficients vanish for the normal model since\nthe nonnull distributions of all the four criteria agree with the\n$\\chi_{p-q,\\lambda}^{2}$ distribution.\n\n\n\\section{Power comparisons}\\label{power_comparison}\n\nIt is known that, to the first order of approximation, the likelihood ratio,\nWald, score and gradient statistics have the same asymptotic\ndistributional properties either under the null hypothesis or under a sequence of\nlocal alternatives.\nOn the other hand, up to an error of order $n^{-1}$ the corresponding criteria have the\nsame size properties but their local powers differ in the $n^{-1\/2}$ term. A meaningful\ncomparison among the criteria can then be performed by comparing\nthe nonnull asymptotic expansions to order $n^{-1\/2}$,\ni.e.~ignoring terms or order less than $n^{-1\/2}$.\n\nIn what follows, we shall compare the local powers of the rival tests\nbased on the general nonnull asymptotic expansions derived in Section \\ref{main_result}\nfor testing the null hypothesis $\\mathcal{H}_0:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$\nin the class of DMs. Let $\\Pi_{i}$ be the power function, up to order $n^{-1\/2}$, of the test that uses\nthe statistic $S_{i}$, for $i=1,2,3,4$. We have\n\\begin{equation}\\label{diff_power}\n\\Pi_{i} - \\Pi_{j} = \\sum_{k=0}^{3}(b_{jk} - b_{ik})G_{p-q+2k,\\lambda}(x),\n\\end{equation}\nfor $i\\neq j$. It is well known that\n\\begin{equation}\\label{diff_G}\nG_{m,\\lambda}(x) - G_{m+2,\\lambda}(x) = 2g_{m+2,\\lambda}(x),\n\\end{equation}\nwhere $g_{\\nu,\\lambda}(x)$ is the probability density\nfunction of a non-central chi-square random variable\nwith $\\nu$ degrees of freedom and non-centrality parameter $\\lambda$.\nFrom~(\\ref{diff_power}) and (\\ref{diff_G}) we have after some algebra\n\\begin{align}\\label{eq:power}\n\\begin{split}\n\\Pi_{1} - \\Pi_{4} &= k_{1}g_{p-q+4,\\lambda}(x) + k_{2}g_{p-q+6,\\lambda}(x),\\quad\n\\Pi_{2} - \\Pi_{4}  = k_{3}g_{p-q+4,\\lambda}(x) + k_{4}g_{p-q+6,\\lambda}(x),\\\\\n\\Pi_{3} - \\Pi_{4} &= k_{5}g_{p-q+4,\\lambda}(x) + k_{6}g_{p-q+6,\\lambda}(x),\\quad\n\\Pi_{1} - \\Pi_{2}  = k_{7}g_{p-q+4,\\lambda}(x) + k_{8}g_{p-q+6,\\lambda}(x),\\\\\n\\Pi_{1} - \\Pi_{3} &= k_{9}g_{p-q+4,\\lambda}(x) + k_{10}g_{p-q+6,\\lambda}(x),\\quad\n\\Pi_{2} - \\Pi_{3}  = k_{11}g_{p-q+4,\\lambda}(x) + k_{12}g_{p-q+6,\\lambda}(x),\n\\end{split}\n\\end{align}\nwhere\n\\[\nk_1 = -\\frac{1}{2}\\tr\\{(\\bm{F}+2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\}\n+ \\frac{1}{2}\\tr\\{\\bm{W}\\bm{T}(\\bm{J}-\\bm{U})-2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nk_2 = -\\frac{\\phi}{6}\\tr\\{(\\bm{F}+2\\bm{G})\\bm{T}^{(3)}\\} - \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}\\bm{C}\\},\n\\quad\nk_3 = 3k_1,\n\\quad\nk_4 = 3k_2,\n\\]\n\\[\nk_5 = k_1 - \\tr\\{(3\\bm{E}-2\\bm{F}+2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\]\n\\[\nk_6 = - \\frac{\\phi}{2}\\tr\\{(2\\bm{E}-\\bm{F}+2\\bm{G})\\bm{T}^{(3)}\\}- \\frac{\\phi}{2}\\tr\\{\\bm{W}\\bm{T}\\bm{C}\\},\n\\]\n\\[\nk_7 = -2k_1,\n\\quad\nk_8 = -2k_2,\n\\quad\nk_9 = k_1 - k_5,\n\\quad\nk_{10} = \\frac{\\phi}{3}\\tr\\{(3\\bm{E}-2\\bm{F}+2\\bm{G})\\bm{T}^{(3)}\\},\n\\]\n\\[\nk_{11} = -3\\tr\\{(\\bm{F}-\\bm{E})(\\bm{Z}_d- \\bm{Z}_{1d})\\bm{T}\\}\n-\\tr\\{\\bm{W}\\bm{T}(\\bm{U}-\\bm{J})+2\\bm{W}\\bm{H}\\},\n\\]\n\\[\nk_{12} = -\\phi\\tr\\{(\\bm{F}-\\bm{E})\\bm{T}^{(3)}\\}\n-\\phi\\tr\\{\\bm{W}\\bm{T}\\bm{C}\\}.\n\\]\n\nFor proper dispersion models, the above expressions are the same.\nReplacing $\\bm{E}$ by $\\bm{F}-\\bm{G}$\nwe obtain these quantities for exponential family nonlinear models.\nFrom equations~(\\ref{eq:power}) we have $\\Pi_{1}>\\Pi_{3}$ if $k_{9}\\geq 0$ and $k_{10}\\geq 0$\nwith $k_{9}+k_{10}>0$, and if $k_{9}\\leq 0$ and $k_{10}\\leq 0$ with $k_{9}+k_{10}<0$, we have\n$\\Pi_{1}<\\Pi_{3}$. Also, $\\Pi_{1}=\\Pi_{3}$ if $k_{9}=k_{10}=0$, i.e.~$\\bm{F}=\\bm{G}$\nand $\\bm{E}=\\bm{0}$, which occurs only for von Mises and normal models with any link function.\nAdditionally, equations (\\ref{eq:power})\nshow that with the exception of the likelihood ratio\nand score tests, is not possible to have any other\nequality among the power functions in the class of DMs for\ntesting the null hypothesis $\\mathcal{H}_0:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$.\nThe reason is that $\\bm{C}$, $\\bm{H}$, $\\bm{J}$ and $\\bm{U}$,\nwhich may be considered as the nonlinear contribution of the dispersion model,\nvanish only for linear regression models.\nIt implies that only strict inequality holds for any other power comparison\namong the power functions of the tests that are based on the\nstatistics $S_1$, $S_2$, $S_3$ and $S_4$. For example,\nfrom (\\ref{eq:power}) we have $\\Pi_{1}>\\Pi_{4}$ ($\\Pi_{1}<\\Pi_{4}$)\nif $k_{1}\\geq 0$ and $k_{2}\\geq 0$ with $k_{1}+k_{2}>0$\n(if $k_{1}\\leq 0$ and $k_{2}\\leq 0$ with $k_{1}+k_{2}<0$), and so on.\n\nWe now move to the class of GLMs, in which \n$\\bm{C}=\\bm{H}=\\bm{J}=\\bm{P}=\\bm{U}=\\bm{0}$. By using the coefficients derived\nfor this class of models in Section \\ref{main_result}, the quantities that define\nequation (\\ref{eq:power}) reduce to\n\\[\nk_1 = -\\frac{1}{2}\\tr\\{(\\bm{F}+2\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\quad\nk_2 = -\\frac{\\phi}{6}\\tr\\{(\\bm{F}+2\\bm{G})\\bm{T}^{(3)}\\},\n\\quad\nk_3 = 3k_1,\n\\]\n\\[\nk_5 = k_1 - \\tr\\{(\\bm{F}-\\bm{G})(\\bm{Z}_d - \\bm{Z}_{1d})\\bm{T}\\},\n\\quad\nk_6 = - \\frac{\\phi}{2}\\tr\\{\\bm{F}\\bm{T}^{(3)}\\},\n\\quad\nk_4 = 3k_2,\n\\]\n\\[\nk_7 = -2k_1,\n\\quad\nk_8 = -2k_2,\n\\quad\nk_9 = k_1 - k_5,\n\\quad\nk_{10} = \\frac{\\phi}{3}\\tr\\{(\\bm{F}-\\bm{G})\\bm{T}^{(3)}\\},\n\\]\n\\[\nk_{11} = -3\\tr\\{\\bm{G}(\\bm{Z}_d- \\bm{Z}_{1d})\\bm{T}\\},\n\\quad\nk_{12} = -\\phi\\tr\\{\\bm{G}\\bm{T}^{(3)}\\}.\n\\]\nFor GLMs with canonical link ($\\bm{G}=\\bm{0}$), we have\n$k_{11}=k_{12}=0$ and hence $\\Pi_{2}=\\Pi_{3}$.\nIt is possible to show that $\\Pi_{1}=\\Pi_{2}=\\Pi_{4}$ if $\\bm{F}=-2\\bm{G}$, that is\n\\[\n\\frac{\\dd^2\\mu_{l}}{\\dd\\eta_l^2} = \\frac{2}{3V_l}\\biggl(\\frac{\\dd\\mu_{l}}{\\dd\\eta_l}\\biggr)^2,\n\\quad l=1,\\ldots,n.\n\\]\nThe GLMs for which this equality holds have the link function\ndefined by $\\eta_{l}=\\int V_l^{-3\/2}\\dd\\mu_{l}$ ($l=1,\\ldots,n$). For the gamma\nmodel this function is $\\eta_{l}=\\mu_{l}^{-1\/3}$ ($l=1,\\ldots,n$).\nAdditionally, we have that $\\Pi_{3}=\\Pi_{4}$ for any GLM with identity link function,\ni.e.~$\\bm{F}=\\bm{0}$. Also, $\\Pi_{1}=\\Pi_{3}$ if $k_{9}=k_{10}=0$,\ni.e.~$\\bm{F}=\\bm{G}$, which occurs only for normal models\nwith any link. Finally, the equality $\\Pi_{1}=\\Pi_{2}=\\Pi_{3}=\\Pi_{4}$ holds\nonly for normal models with identity link function.\n\nWe can conclude that there is no uniform superiority\nof one test with respect to the others for\ntesting the null hypothesis $\\mathcal{H}_{0}:\\bm{\\beta}_{2}=\\bm{\\beta}_{20}$\nin the class of DMs. Hence, if the sample size is large, all tests\ncould be recommended, since their type I error probabilities\ndo not significantly deviate from the true nominal level\nand their local powers are approximately equal.\nThe natural question is how these tests perform when the sample size is\nsmall or of moderate size, and which one is the most reliable.\nIn Section \\ref{simulations}, we shall use Monte Carlo simulations\nto shed some light on this issue.\n\n\n\\section{Tests for the precision parameter}\\label{test_phi}\n\nIn this section we derive asymptotic expansions for the nonnull\ndistribution of the four statistics for testing the precision\nparameter $\\phi$ in DMs. We are interested in testing the\nnull hypothesis $\\mathcal{H}_{0}:\\phi=\\phi_{0}$ against\na two-sided alternative hypothesis\n$\\mathcal{H}_{1}:\\phi\\neq\\phi_{0}$, where $\\phi_{0}$ is a positive specified value\nfor $\\phi$. Here, $\\bm{\\beta}$ acts as a nuisance parameter. The likelihood\nratio, Wald, score and gradient\nstatistics are expressed as follows:\n\\[\nS_1 = \\sum_{l=1}^{n}\\{(\\widehat{\\phi}-\\phi_{0})t(y_l,\\widehat{\\theta}_{l})\n+ c(y_l,\\widehat{\\phi}) - c(y_l,\\phi_{0})\\},\n\\quad\nS_{2} = (\\widehat{\\phi} - \\phi_{0})^2\\{-\\alpha_{2}(\\widehat{\\phi})\\},\n\\]\n\\[\nS_{3} = \\{-\\alpha_{2}(\\phi_{0})\\}^{-1}\\Biggl[\\sum_{l=1}^n\\{t(y_l,\\widehat{\\theta}_{l})+c^{(1)}(y_l,\\phi_{0})\\}\\Biggr]^2,\n\\quad\nS_{4} = (\\widehat{\\phi} - \\phi_{0})\\sum_{l=1}^n\\{t(y_l,\\widehat{\\theta}_{l})+c^{(1)}(y_l,\\phi_{0})\\}.\n\\]\nFor PDMs, these statistics can be expressed as\n\\[\nS_{1} = 2n\\{a_{2}(\\widehat{\\phi}) - a_{2}(\\phi_{0})\n- (\\widehat{\\phi} - \\phi_{0})a_2^{(1)}(\\widehat{\\phi})\\},\n\\quad\nS_{2} = -n(\\widehat{\\phi} - \\phi_{0})^2a_{2}^{(2)}(\\widehat{\\phi}),\n\\]\n\\[\nS_{3} = -\\frac{n\\{a_{2}^{(1)}(\\widehat{\\phi}) - a_{2}^{(1)}(\\phi_{0})\\}^2}{a_{2}^{(2)}(\\phi_{0})},\n\\quad\nS_{4} = n\\{a_{2}^{(1)}(\\phi_{0}) - a_{2}^{(1)}(\\widehat{\\phi})\\}(\\widehat{\\phi} - \\phi_{0}).\n\\]\nFor example, for the von Mises model $a_2(\\phi) = -\\log\\{I_0(\\phi)\\}$. Also,\n$a_2^{(1)}(\\phi) = -r(\\phi)$ and $a_2^{(2)}(\\phi) = r(\\phi)^2 + r(\\phi)\/\\phi-1$, where\n$r(\\phi) = I_1(\\phi)\/I_0(\\phi)$. Thus, we can write\n\\[\nS_{1} = 2n[\\log\\{I_0(\\phi_0)\/I_0(\\widehat{\\phi})\\}\n+ (\\widehat{\\phi} - \\phi_{0})r(\\widehat{\\phi})],\n\\quad\nS_{2} = -n(\\widehat{\\phi} - \\phi_{0})^2\\{r(\\widehat{\\phi})^2 + r(\\widehat{\\phi})\/\\widehat{\\phi}-1\\},\n\\]\n\\[\nS_{3} = -\\frac{n\\{r(\\phi_{0})-r(\\widehat{\\phi})\\}^2}{r(\\phi_{0})^2 + r(\\phi_{0})\/\\widehat{\\phi_{0}}-1},\n\\quad\nS_{4} = n\\{r(\\widehat{\\phi})-r(\\phi_{0})\\}(\\widehat{\\phi} - \\phi_{0}).\n\\]\nAlso, for normal and inverse Gaussian models we have $a_{2}(\\phi) = \\log(\\phi)\/2$. Hence\n\\[\nS_{1} = 2n\\biggl\\{\\log\\biggl(\\frac{\\widehat{\\phi}}{\\phi_{0}}\\biggr)\n- \\biggl(\\frac{\\widehat{\\phi} - \\phi_{0}}{\\widehat{\\phi}}\\biggr)\\biggr\\},\n\\quad\nS_{2} = S_{3} = \\frac{n}{2}\\biggl\\{\\frac{\\widehat{\\phi} - \\phi_{0}}{\\widehat{\\phi}}\\biggr\\}^2,\n\\quad\nS_{4} = \\frac{n}{2}\\biggl\\{\\frac{\\widehat{\\phi} - \\phi_{0}}{\\phi_{0}}\n- \\frac{\\widehat{\\phi} - \\phi_{0}}{\\widehat{\\phi}}\\biggr\\}.\n\\]\nWe have $a_2(\\phi)=\\phi\\log(\\phi) - \\log\\{\\Gamma(\\phi)\\}$ for the gamma model\nand therefore these statistics reduce to\n\\[\nS_1 = 2n\\biggl\\{\\phi_{0}\\log\\biggl(\\frac{\\widehat{\\phi}}{\\phi_{0}}\\biggr)\n-\\log\\biggl(\\frac{\\Gamma(\\widehat{\\phi})}{\\Gamma(\\phi_{0})}\\biggr)\n-(\\widehat{\\phi} - \\phi_{0})(1-\\psi(\\widehat{\\phi}))\\biggr\\},\n\\]\n\\[\nS_2 = n\\{\\widehat{\\phi}\\psi'(\\widehat{\\phi}) - 1\\}\n\\frac{(\\widehat{\\phi} - \\phi_{0})^2}{\\widehat{\\phi}},\n\\quad\nS_3 = \\frac{n\\phi_{0}\\{\\log(\\widehat{\\phi}\/\\phi_{0}) - (\\psi(\\widehat{\\phi}) - \\psi(\\phi_{0}))\\}}\n{\\phi_{0}\\psi'(\\phi_{0}) - 1}\n\\]\nand\n\\[\nS_{4} = n(\\widehat{\\phi}-\\phi_{0})\\biggl\\{\\log\\biggl(\\frac{\\widehat{\\phi}}{\\phi_{0}}\\biggr)\n+\\psi(\\widehat{\\phi}) - \\psi(\\phi_{0})\\biggr\\},\n\\]\nwhere $\\Gamma(\\cdot)$, $\\psi(\\cdot)$ and $\\psi'(\\cdot)$ are the gamma, digamma and\ntrigamma functions, respectively.\n\nThe nonnull asymptotic distributions of $S_1$, $S_2$, $S_3$ and\n$S_4$ for testing $\\mathcal{H}_{0}:\\phi=\\phi_{0}$ in DMs under the\nlocal alternative $\\mathcal{H}_{1n}:\\phi=\\phi_{0}+\\epsilon$,\nwhere $\\epsilon=\\phi-\\phi_{0}$ is assumed to be $O(n^{-1\/2})$, is\n\\[\n\\Pr(S_{i}\\leq x) = G_{1,\\lambda}(x) + \\sum_{k=0}^{3}b_{ik}G_{1+2k,\\lambda}(x) + O(n^{-1}),\n\\qquad i=1,2,3,4.\n\\]\nThe noncentrality parameter is given by $\\lambda = -\\alpha_{2}\\epsilon^2$\nand the the coefficients $b_{ik}$'s can be written as\n\\[\nb_{11} = \\frac{(\\alpha_2'-\\alpha_3)\\epsilon^3}{2}+\\frac{p\\epsilon}{2\\phi},\n\\quad\nb_{12} =  \\frac{(2\\alpha_3-3\\alpha_2')\\epsilon^3}{6},\n\\quad\nb_{13} = 0,\n\\]\n\\[\nb_{21} = \\frac{(\\alpha_2'-\\alpha_3)\\epsilon^3}{2}-\\frac{\\alpha_{3}\\epsilon}{2\\alpha_2}+\\frac{p\\epsilon}{2\\phi},\n\\quad\nb_{22} = -\\frac{(\\alpha_2'-\\alpha_3)\\epsilon^3}{2}+\\frac{\\alpha_{3}\\epsilon}{2\\alpha_2},\n\\quad\nb_{23} = -\\frac{\\alpha_3\\epsilon^3}{6},\n\\]\n\\[\nb_{31} = \\frac{(\\alpha_2'-\\alpha_3)\\epsilon^3}{2} + \\frac{(2\\alpha_3-3\\alpha_2')\\epsilon}{2\\alpha_2}\n+\\frac{p\\epsilon}{2\\phi},\n\\quad\nb_{32} = -\\frac{(2\\alpha_3-3\\alpha_2')\\epsilon}{2\\alpha_2},\n\\quad\nb_{33} = \\frac{(2\\alpha_3-3\\alpha_2')\\epsilon^3}{6},\n\\]\n\\[\nb_{41} = \\frac{(\\alpha_2'-\\alpha_3)\\epsilon^3}{2} + \\frac{\\alpha_3\\epsilon}{4\\alpha_2}\n+\\frac{p\\epsilon}{2\\phi},\n\\quad\nb_{42} = -\\frac{(2\\alpha_2'-\\alpha_3)\\epsilon^3}{4} - \\frac{\\alpha_3\\epsilon}{4\\alpha_2},\n\\quad\nb_{43} = \\frac{\\alpha_3\\epsilon^3}{12},\n\\]\nwith $b_{i0} = -(b_{i1}+b_{i2}+b_{i3})$, for $i=1,2,3,4$.\nIt should be noticed that the above expressions depend on the parameter\n$\\phi$ and depend on the local derivative matrix $\\bm{X}^*$ only through its rank $p$.\nSince $\\alpha_2'=\\alpha_3=na_{2}^{(3)}(\\phi)$ for PDMs, these coefficients reduce to\n\\[\nb_{11} =  \\frac{p\\epsilon}{2\\phi},\n\\quad\nb_{12} = b_{23} = b_{33} = -\\frac{na_{2}^{(3)}(\\phi)\\epsilon^3}{6},\n\\quad\nb_{13} = 0,\n\\quad\nb_{21} = b_{31} = \\frac{p\\epsilon}{2\\phi} - \\frac{a_{2}^{(3)}(\\phi)\\epsilon}{2a_{2}^{(2)}(\\phi)},\n\\]\n\\[\nb_{22} = b_{32} = b_{11}-b_{21}, \n\\quad\nb_{41} = b_{11} + \\frac{1}{2}(b_{11}-b_{21}),\n\\quad\nb_{42} = -\\frac{1}{2}(b_{11}-b_{21}-3b_{12}),\n\\quad\nb_{43} = -\\frac{b_{12}}{2},\n\\]\nwith $b_{i0} = -(b_{i1}+b_{i2}+b_{i3})$, for $i=1,2,3,4$.\nThese coefficients do not change for the class of GLMs.\n\nIn what follows, we present an analytical comparison\namong the local powers of the four tests for testing\nthe null hypothesis $\\mathcal{H}_{0}:\\phi=\\phi_{0}$.\nWe have\n\\[\n\\Pi_{i} - \\Pi_{j} = \\sum_{k=0}^{3}(b_{jk} - b_{ik})G_{1+2k,\\lambda}(x).\n\\]\nAfter some algebra, we can write\n\\[\n\\Pi_{1}-\\Pi_{2} = -\\frac{\\alpha_3\\epsilon}{\\alpha_2} g_{5,\\lambda}(x)\n+ \\frac{\\alpha_3\\epsilon^3}{3} g_{7,\\lambda}(x),\n\\]\n\\[\n\\Pi_{1}-\\Pi_{3} = \\frac{(2\\alpha_3-3\\alpha_2')\\epsilon}{\\alpha_2} g_{5,\\lambda}(x)\n-\\frac{(2\\alpha_3-3\\alpha_2')\\epsilon^3}{3} g_{7,\\lambda}(x),\n\\]\n\\[\n\\Pi_{1}-\\Pi_{4} = \\frac{\\alpha_3\\epsilon}{2\\alpha_2} g_{5,\\lambda}(x)\n- \\frac{\\alpha_3\\epsilon^3}{6} g_{7,\\lambda}(x),\n\\]\n\\[\n\\Pi_{2}-\\Pi_{3} = \\frac{3(\\alpha_3-\\alpha_2')\\epsilon}{\\alpha_2} g_{5,\\lambda}(x)\n-(\\alpha_3-\\alpha_2')\\epsilon^3 g_{7,\\lambda}(x),\n\\]\n\\[\n\\Pi_{2}-\\Pi_{4} = \\frac{3\\alpha_3\\epsilon}{2\\alpha_2} g_{5,\\lambda}(x)\n- \\frac{\\alpha_3\\epsilon^3}{2} g_{7,\\lambda}(x),\n\\]\n\\[\n\\Pi_{3}-\\Pi_{4} = -\\frac{3(\\alpha_3-2\\alpha_2')\\epsilon}{\\alpha_2} g_{5,\\lambda}(x)\n+\\frac{(\\alpha_3-2\\alpha_2')}{2}\\epsilon^3 g_{7,\\lambda}(x).\n\\]\nFrom the above expressions, we can obtain the following general conclusions.\nBy assuming $\\phi>\\phi_{0}$ (opposite inequalities hold if $\\phi<\\phi_{0}$),\nwe have that $\\Pi_{3} < \\Pi_{2} < \\Pi_{1} < \\Pi_{4}$ if $\\alpha_3>0$ with\n$\\alpha_2'>\\alpha_3$. Also, $\\Pi_{2} = \\Pi_{3} < \\Pi_{1} < \\Pi_{4}$ if\n$\\alpha_2'=\\alpha_3>0$. For example, for normal and inverse Gaussian models\nwe have $a_2(\\phi) = \\log(\\phi)\/2$,\nwhich implies that $a_2^{(1)}(\\phi) = 1\/(2\\phi)$,\n$a_2^{(2)}(\\phi) = -1\/(2\\phi^2)$ and $a_2^{(3)}(\\phi) = 1\/\\phi^3$.\nSince $\\alpha_2'=\\alpha_3=n\/\\phi^3>0$,\nwe arrive at the following inequalities:\n$\\Pi_{2} = \\Pi_{3} < \\Pi_{1} < \\Pi_{4}$  if $\\phi>\\phi_{0}$,  and\n$\\Pi_{2} = \\Pi_{3} > \\Pi_{1} > \\Pi_{4}$ if $\\phi<\\phi_{0}$.\n\n\n\\section{Monte Carlo simulation}\\label{simulations}\n\nIn this section we conduct Monte Carlo simulations in order\nto compare the performance of the likelihood ratio, Wald,\nscore and gradient tests in small- and moderate-sized samples.\n\nWe consider the von Mises regression model, which is quite useful\nfor modeling circular data; see \\cite{Fisher1993} and \\cite{MardiaJupp2000}.\nHere,\n\\[\n\\pi(y;\\theta,\\phi)=\\frac{\\exp\\{\\phi\\cos(y-\\theta)\\}}{2\\pi I_0(\\phi)},\n\\quad y\\in(-\\pi,\\pi),\n\\]\nwhere $\\theta\\in(-\\pi,\\pi)$ and $\\phi>0$. This density function\nis symmetric around $y=\\theta$, which is the mode and the circular mean of\nthe distribution. Also, $\\phi$ is a precision parameter in the sense that\nthe larger the value of $\\phi$ the more concentrated the density\nfunction around $\\theta$. It is evident the density function above\nis a special case of (\\ref{dens1}) with $t(y,\\theta)=\\cos(y-\\theta)$ and\n$c(y,\\phi)=-\\log(I_0(\\phi))$.\n\nWe assume that\n\\[\n\\tan(\\theta_{l}\/2) = \\eta_{l} = \\beta_{1}x_{i1} + \\beta_{2}x_{i2} + \\cdots + \\beta_{p}x_{ip},\n\\]\nwhere $x_{i1} = 1$ and $\\theta_{l}=2\\arctan(\\eta_{l})$, $l = 1, \\ldots, n$.\nThe covariate values were selected as random draws from the $\\mathcal{U}(0,1)$\ndistribution and for fixed $n$ those values were kept constant\nthroughout the experiment. The number of Monte Carlo replications was 10,000, the nominal levels\nof the tests were $\\gamma$ = 10\\%, 5\\% and 1\\%, and all simulations were carried\nout using the {\\tt Ox} matrix programming language \\citep{DcK2007}.\n{\\tt Ox} is freely distributed for academic\npurposes and available at http:\/\/www.doornik.com.\n\nFirst, the null hypothesis is $\\mathcal{H}_{0}:\\beta_{p-1} =\\beta_{p} = 0$,\nwhich is tested against a two-sided alternative. The sample size is $n=50$,\n$\\phi = 1.5, 2.5, 4$ and $p = 3, 4, \\ldots, 8$. The values of the\nresponse were generated using $\\beta_{1} = \\cdots=\\beta_{p-2} = 1$.\nThe null rejection rates of the four tests are presented in Table~\\ref{tab1}.\nIt is clear that the likelihood ratio ($S_1$) and Wald ($S_2$) tests are\nmarkedly liberal, more so as the number of regressors increases.\nThe score ($S_3$) and gradient ($S_4$) tests  are also liberal in most of\nthe cases, but much less size distorted than the likelihood ratio\nand Wald tests in all cases. For instance, when $\\phi=2.5$,\n$p=4$ and $\\gamma = 5\\%$, the rejection rates\nare 7.05\\% ($S_1$), 8.28\\% ($S_2$),  5.15\\% ($S_3$) and 6.30\\% ($S_4$).\nWe note that the score test is much less liberal than the\nlikelihood ratio and Wald tests and slightly less liberal than the\ngradient test. Additionally, the Wald test is much more liberal than\nthe other tests. Note that as $\\phi$ increases the tests become less size\ndistorted, as expected, since the\nvon Mises distribution approaches a normal distribution as $\\phi$\nincreases.\n\\begin{table}[!htp]\n{\\footnotesize\n\\begin{center}\n\\caption{Null rejection rates (\\%); $\\phi$ = 1.5, 2.5 and 4, with $n = 50$.}\\label{tab2}\n\\begin{tabular}{c c c c c| c c c c| c c c c  }\\hline\n  & \\multicolumn{12}{c}{$\\phi = 1.5$} \\\\\\cline{2-13}\n  & \\multicolumn{4}{c|}{$\\gamma = 10\\%$} & \\multicolumn{4}{c|}{$\\gamma = 5\\%$}\n  & \\multicolumn{4}{c}{$\\gamma = 1\\%$}\\\\\\cline{2-13}\n  $p$   & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\\\\\\hline\n    3   & 13.31 & 15.42 & 10.12 & 10.42 &  6.90 &  9.93 & 4.65 & 5.04 & 1.75 & 4.13 & 0.79 & 1.20 \\\\\n    4   & 14.48 & 16.31 & 10.26 & 12.49 &  7.75 & 10.86 & 4.83 & 6.83 & 1.93 & 4.62 & 0.59 & 2.08 \\\\\n    5   & 16.65 & 19.34 & 10.92 & 12.46 &  9.55 & 12.36 & 5.05 & 6.62 & 2.67 & 4.87 & 0.84 & 1.83 \\\\\n    6   & 19.04 & 21.93 & 11.94 & 14.81 & 11.78 & 15.00 & 5.90 & 8.26 & 3.62 & 6.50 & 1.03 & 2.40 \\\\\n    7   & 22.09 & 26.39 & 12.44 & 15.94 & 13.71 & 18.12 & 6.12 & 8.87 & 4.27 & 7.67 & 1.27 & 2.21 \\\\\n    8   & 24.16 & 26.58 & 13.03 & 17.66 & 15.87 & 17.42 & 6.63 & 9.82 & 5.23 & 6.82 & 1.39 & 2.76 \\\\ \\hline\n\n  & \\multicolumn{12}{c}{$\\phi = 2.5$} \\\\\\cline{2-13}\n  & \\multicolumn{4}{c|}{$\\gamma = 10\\%$} & \\multicolumn{4}{c|}{$\\gamma = 5\\%$}\n  & \\multicolumn{4}{c}{$\\gamma = 1\\%$}\\\\\\cline{2-13}\n  $p$   & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\\\\\\hline\n    3   & 12.02 & 12.96 & 10.56 & 10.50 &  6.21 &  7.35 & 5.17 & 5.29 & 1.39 & 2.31 & 0.78 & 1.04 \\\\\n    4   & 12.97 & 13.66 & 11.05 & 11.77 &  7.05 &  8.28 & 5.15 & 6.30 & 1.73 & 3.05 & 0.90 & 1.52 \\\\\n    5   & 14.28 & 16.38 & 10.97 & 11.68 &  7.96 & 10.31 & 4.94 & 6.25 & 2.11 & 4.28 & 0.85 & 1.65 \\\\\n    6   & 14.83 & 15.33 & 11.90 & 13.02 &  8.36 &  9.82 & 5.71 & 7.27 & 2.09 & 3.85 & 1.01 & 1.80 \\\\\n    7   & 15.93 & 18.00 & 12.60 & 13.87 &  9.20 & 11.30 & 6.66 & 7.60 & 2.72 & 3.71 & 1.53 & 1.87 \\\\\n    8   & 18.12 & 19.53 & 13.45 & 16.12 & 11.16 & 12.29 & 7.02 & 9.38 & 3.31 & 4.79 & 1.55 & 2.68 \\\\ \\hline\n\n  & \\multicolumn{12}{c}{$\\phi = 4$} \\\\\\cline{2-13}\n  & \\multicolumn{4}{c|}{$\\gamma = 10\\%$} & \\multicolumn{4}{c|}{$\\gamma = 5\\%$}\n  & \\multicolumn{4}{c}{$\\gamma = 1\\%$}\\\\\\cline{2-13}\n  $p$   & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\\\\\\hline\n    3   & 11.99 & 12.59 & 10.72 & 10.81 & 6.32 &  7.19 & 5.02 & 5.25 & 1.37 & 2.20 & 0.82 & 1.12 \\\\\n    4   & 13.15 & 14.48 & 11.49 & 11.74 & 7.19 &  8.66 & 5.50 & 5.83 & 1.67 & 2.89 & 0.84 & 1.13 \\\\\n    5   & 13.59 & 13.67 & 11.87 & 12.26 & 7.21 &  7.64 & 5.72 & 6.25 & 1.68 & 2.50 & 0.96 & 1.35 \\\\\n    6   & 14.08 & 15.60 & 11.85 & 12.65 & 7.57 &  9.04 & 5.88 & 6.30 & 1.73 & 2.88 & 1.00 & 1.21 \\\\\n    7   & 15.16 & 16.42 & 12.79 & 13.52 & 8.34 &  9.55 & 6.42 & 7.03 & 2.28 & 3.16 & 1.43 & 1.71 \\\\\n    8   & 16.14 & 17.36 & 13.53 & 14.57 & 9.28 & 10.31 & 7.13 & 7.84 & 2.42 & 2.96 & 1.28 & 1.61 \\\\ \\hline\n\\end{tabular}\n\\end{center}    }\n\\end{table}\n\nTable~\\ref{tab3} reports results for $\\phi = 3$, $p=4$ and sample sizes ranging\nfrom 20 to 150. As expected, the null rejection rates of all the tests approach the\ncorresponding nominal levels as the sample size grows. Again,\nthe score and gradient tests present the best performances.\nIn Table \\ref{tab4} we present the first two moments\nof $S_1$, $S_2$, $S_3$ and $S_4$ and the corresponding moments\nof the limiting $\\chi^2$ distribution. Note that the\ngradient and score statistics present a good\nagreement between the true moments (obtained by simulation)\nand the moments of the limiting distribution.\n\\begin{table}[!htp]\n{\\footnotesize\n\\begin{center}\n\\caption{Null rejection rates (\\%); $\\phi = 3$,\n        $p = 4$ and different sample sizes.}\\label{tab3}\n\\begin{tabular}{c c c c c| c c c c| c c c c  }\\hline\n  & \\multicolumn{4}{c|}{$\\gamma = 10\\%$} & \\multicolumn{4}{c|}{$\\gamma = 5\\%$}\n  & \\multicolumn{4}{c}{$\\gamma = 1\\%$}\\\\\\cline{2-13}\n  $n$   & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\n        & $S_1$  & $S_2$  & $S_3$ & $S_4$\\\\\\hline\n   20   & 17.33 & 19.18 & 13.71 & 13.89 & 10.50 & 11.95 & 6.92 & 7.04 & 3.33 & 4.38 & 1.16 & 1.14 \\\\\n   30   & 15.04 & 16.33 & 11.65 & 12.76 &  8.29 & 10.19 & 5.10 & 6.66 & 2.05 & 4.14 & 0.75 & 1.50 \\\\\n   40   & 13.49 & 15.23 & 11.44 & 11.44 &  7.56 &  9.43 & 5.72 & 5.96 & 1.81 & 3.07 & 0.92 & 1.18 \\\\\n   50   & 12.51 & 13.78 & 10.77 & 11.05 &  6.65 &  7.79 & 5.40 & 5.59 & 1.66 & 2.31 & 1.02 & 1.25 \\\\\n   70   & 12.01 & 12.46 & 11.00 & 11.17 &  6.20 &  6.90 & 5.41 & 5.58 & 1.48 & 2.18 & 1.12 & 1.28 \\\\\n  100   & 11.30 & 12.13 & 10.74 & 10.69 &  5.86 &  6.65 & 4.92 & 5.44 & 1.22 & 2.04 & 0.94 & 1.07 \\\\\n  150   & 10.51 & 11.01 & 10.02 & 10.10 &  5.05 &  6.03 & 4.59 & 4.63 & 1.08 & 1.66 & 0.94 & 0.95 \\\\ \\hline\n\\end{tabular}\n\\end{center}  }\n\\end{table}\n\\begin{table}[!htp]\n{\\footnotesize\n\\begin{center}\n\\caption{Moments; $\\phi = 2$, $n = 35$, $p=4$.}\\label{tab4}\n\\begin{tabular}{l c c c c c}\\hline\n  & $S_1$ & $S_2$ & $S_3$ & $S_4$ & $\\chi_{2}^{2}$ \\\\\\hline\nMean     & 2.50  & 2.68 & 2.16 & 2.23  &  2.0 \\\\\nVariance & 6.23  & 8.73 & 4.14 & 4.63  &  4.0 \\\\ \\hline\n\\end{tabular}\n\\end{center}   }\n\\end{table}\n\nWe also performed Monte Carlo simulations considering hypothesis testing on $\\phi$.\nTo save space, the results are not shown. The score and gradient tests exhibited superior\nbehaviour than the likelihood ratio and Wald tests. For example, when $n=35$, $p=3$,\n$\\gamma = 10\\%$ and $\\mathcal{H}_{0}:\\phi = 2$, we\nobtained the following null rejection rates: 13.23\\%\n($S_1$), 14.75\\% ($S_2$), 10.61\\% ($S_3$) and 9.97\\% ($S_4$).\nAgain, the best performing tests are the score and gradient tests.\n\nOverall, in small to moderate-sized samples the best performing tests are the\nscore and the gradient tests. They are less size distorted than the other two.\nHence, these tests may be recommended for testing hypotheses on the regression parameters\nin the von Mises regression model.\nThe gradient test has a slight advantage over the score test because\nthe gradient statistic is simpler to calculate than the\nscore statistic for testing a subset of regression parameters.\nIn particular, no matrix needs to be inverted; see Section \\ref{DMs}.\n\n\n\\section{Application}\\label{applications}\n\nIn this section we shall illustrate an application of \nthe likelihood ratio, Wald, score and gradient tests in a real data set.\nWe consider the data described in \\cite{FL1992} regarding the distance traveled by\n31 small blue periwinkles ({\\it Nodilittorina unifasciata}) after they have moved down-shore\nfrom the height at which they normally live. Following \\cite{FL1992} we assume\na von Mises distribution for the animals' path, but with the assumption of\nconstant dispersion and link function\n\\[\n\\tan(\\theta_{l}\/2) = \\beta_{1} + \\beta_{2}x_l,\\quad l=1,\\ldots,31,\n\\]\nwhere $\\theta_{l}=2\\arctan(\\beta_{1} + \\beta_{2}x_l)$ denotes the mean direction for a given distance\nmoved $x_{l}$ (cm). These data have been previously analysed by\n\\cite{Paula1996} and \\cite{SouzaPaula2002} with emphasis on local influence and\nresidual analysis, respectively. The angular responses were transformed\nto the range $(-\\pi,\\pi)$. The maximum likelihood estimates of the parameters\n(asymptotic standard errors in parentheses) are: $\\widehat{\\beta}_{1} = -0.323\\,(0.151)$,\n$\\widehat{\\beta}_{2} = -0.013\\,(0.004)$ and $\\widehat{\\phi}=3.265\\,(0.726)$.\nThe values of the likelihood ratio ($S_1$), Wald ($S_2$), score ($S_3$) and\ngradient ($S_4$) statistics for testing the null hypothesis\n$\\mathcal{H}_{0}:\\beta_{2}=0$ are 9.526\\,($p$-value: 0.002),\n11.031\\,($p$-value: 0.001), 7.126\\,($p$-value: 0.008) and 8.280\\,($p$-value: 0.004),\nrespectively. At any usual significance level, all tests lead to the same\nconclusion, i.e.~the null hypothesis should be rejected.\n\nNow, we consider different values for $\\beta_{20}$ and we wish to test\n$\\mathcal{H}_0:\\beta_{2}=\\beta_{20}$ against $\\mathcal{H}_1:\\beta_{2}\\neq\\beta_{20}$.\nTable \\ref{tab_new} lists the observed values of the different test statistics and the corresponding\n$p$-values for $\\beta_{20} = -0.026, -0.024, -0.022, -0.020$ and $-0.018$.\nThe asterisks indicate that the null hypothesis is\nrejected at respectively the 1\\% (***), the 5\\% (**) or at the 10\\% (*) significance level.\nNotice that the same decision is reached by all the tests when\n$\\beta_{20} = -0.018$ but not when\n$\\beta_{20} = -0.026, -0.024, -0.022$ and $-0.020$. In all cases considered here, the\nscore and gradient tests lead to the same conclusion. Additionally, the\nlikelihood ratio and Wald tests display the smallest $p$-values\nin all cases, in accordance with their\nliberal behaviours observed in our simulation study.\n\\begin{table}[!htp]\n{\\footnotesize\n\\begin{center}\n\\caption{Test statistics for $\\mathcal{H}_0:\\beta_{2}=\\beta_{20}$ against $\\mathcal{H}_1:\\beta_{2}\\neq\\beta_{20}$\n($p$-values between parentheses).}\\label{tab_new}\n\\begin{tabular}{cccccc}\\hline\n          &      \\multicolumn{5}{c}{$\\beta_{20}$}\\\\  \\cline{2-6}\nstatistic &$-0.026$        & $-0.024$       &  $-0.022$      & $-0.020$       & $-0.018$      \\\\ \\hline\n$S_1$     &$ 7.314\\,(0.007)^{***}$ & $5.606\\,(0.018)^{**}$ & $4.011\\,(0.045)^{**}$ & $2.591\\,(0.107)$ & $1.411\\,(0.235)$\\\\\n$S_2$     &$11.409\\,(0.001)^{***}$ & $8.193\\,(0.004)^{***}$ & $5.509\\,(0.019)^{**}$ & $3.355\\,(0.067)^{*}$ & $1.733\\,(0.188)$\\\\\n$S_3$     &$ 5.872\\,(0.015)^{**}$ & $4.636\\,(0.031)^{**}$ & $3.407\\,(0.065)^{*}$ & $2.251\\,(0.134)$ & $1.249\\,(0.264)$\\\\\n$S_4$     &$ 5.728\\,(0.017)^{**}$ & $4.611\\,(0.032)^{**}$ & $3.458\\,(0.063)^{*}$ & $2.332\\,(0.127)$ & $1.321\\,(0.250)$\\\\ \\hline\n\\end{tabular}\n\\end{center}\n}\n\\end{table}\n\nNotice that the sample size is $n=31$, but if $n$ were smaller,\nthe tests could lead to different conclusions. To\nillustrate this, a randomly chosen subset of the data set with $n=10$ was drawn.\nThe null hypothesis to be tested is $\\mathcal{H}_{0}:\\beta_{2}=0$.\nThe observed value of the test statistics\nare $S_1 = 2.939$ ($p$-value: 0.086), $S_2 = 2.980$\n($p$-value: 0.084), $S_3 = 2.491$ ($p$-value: 0.114) and\n$S_4 = 2.682$ ($p$-value = 0.101).  Hence, at the 10\\% significance level, the score and gradient\ntests do not reject the null hypothesis unlike the likelihood ratio\nand Wald tests, which are much more oversized than the score\nand gradient tests as evidenced by our simulation results.\n\n\n\\section{Concluding remarks}\\label{conclusions}\n\nThe dispersion models (DMs) extend the well-known generalised\nlinear models \\citep{NelderWedderburn1972} and also the exponential family\nnonlinear models \\citep{CordPaula1989}.\nAdditionally, the class of DMs covers a comprehensive range of\nnon-normal distributions. In this paper, we dealt with\nthe issue of performing hypothesis testing in DMs. We considered the three\nclassic tests, likelihood ratio, Wald and score tests, and a recently\nproposed test, the gradient test. We have derived formulae for the asymptotic\nexpansions up to order $n^{-1\/2}$ of the distribution functions\nof the likelihood ratio, Wald, score and gradient\nstatistics, under a sequence of Pitman alternatives,\nfor testing a subset of regression  parameters and for\ntesting the dispersion parameter.\nThe formulae derived are simple to be used analytically\nto obtain closed-form expressions for these expansions in\nspecial models. Also, the power of all four\ncriteria, which are equivalent to first order,\nwere compared under specific conditions based on second order\napproximations. Additionally, we present Monte Carlo simulations in order to\ncompare the finite-sample performance of these tests. From the simulation results\nwe can conclude that the score and gradient tests should be preferred.\nFinally, we present an empirical application for illustrative purposes.\n\n\n\\section*{Acknowledgments}\n\nWe gratefully acknowledge the financial support of FAPESP and CNPq (Brazil).\n\n\n{\\small\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWe work over the field $\\mathbb{C}$ of complex numbers.\n\nIt all began with a famous result, proved by Camille Jordan in the year 1878, asserting that for any field $k$ of characteristic $0$ and any positive integer $n$, the (linear) automorphism group $\\operatorname{GL}_n(k)$ of an $n$-dimensional vector space has the {\\it Jordan property}:\nthere is a Jordan constant $J = J(n)$ such that every finite subgroup $H \\le \\operatorname{GL}_n(k)$\nhas an abelian subgroup $H_1$ of index $[H : H_1] \\le J(n)$.\n\nPeople then wondered whether the same Jordan property is shared by other automorphism groups, for instance, (not necessarily linear) general automorphism groups or even birational automorphism groups of varieties.\n\nMore than a century having passed, it is Popov who first asked whether the group $\\operatorname{Aut}(X)$ (resp. $\\operatorname{Bir}(X)$) of all automorphisms (resp. all birational automorphisms) of an algebraic variety $X$ is Jordan\n(cf.~\\cite[Question 2.30-2.31]{Pop11}).\nPopov himself proved that for a projective surface $X$, the group $\\operatorname{Bir}(X)$ is Jordan unless\n$X$ is birational to the product $\\mathbb{P}^1 \\times E$ with $E$ an elliptic curve, and late on Zarhin confirmed that\n$\\operatorname{Bir}(\\mathbb{P}^1 \\times E)$ is not Jordan while $\\operatorname{Aut}(X)$ is still Jordan when $X$ is projective and birational to $\\mathbb{P}^1 \\times E$ (cf.~\\cite[\\S 2.2]{Pop11}, \\cite[Theorem 5.3]{Ser09}, \\cite[Theorem 1.2]{Zar14}, \\cite[Theorem 1.3]{Zar15}).\n\nFor quasi-projective varieties, Bandman and Zarhin proved that $\\operatorname{Aut}(X)$ is Jordan when $\\dim(X)=2$ or $X$ is birational to the product $\\mathbb{P}^1 \\times A$ with $A$ having no rational curve\n(cf.~\\cite[Theorem 1.7]{BZ15}, \\cite[Theorem 4]{BZ19}).\n\nFor algebraic varieties of higher dimensions, with the help of the minimal model program, Prokhorov and Shramov \\cite[Theorem 1.8]{PS14} confirmed the Jordan property of the group $\\operatorname{Bir}(X)$ for an algebraic variety $X$, assuming either $X$ is non-uniruled,\nor $X$ has vanishing irregularity as well as the (then) outstanding\nBorisov-Alexeev-Borisov conjecture about the bounded-ness of terminal Fano varieties\nwhich has been affirmatively confirmed soon after by Birkar; see \\cite[Theorem 1.1]{Bir}.\nIn particular, the Cremona groups have the Jordan property, confirming a conjecture of Serre.\n\nThe first and third authors proved that $\\operatorname{Aut}(X)$ is Jordan when $X$ is a projective variety (cf.~\\cite{MZ18}).\nThis result is extended to compact normal K\\\"ahler spaces by J.~Kim \\cite{Kim18}, while Popov offered a much simpler proof by reducing the Jordan property to (real) Lie groups; see \\cite[Theorem 5]{Pop18} and also Theorem \\ref{thm-kim-popov}.\n\nIn non-algebraic cases, compact complex surfaces still behave well. Indeed, Prokhorov and Shramov \\cite[Theorems 1.6 and 1.7]{PS} showed that $\\operatorname{Aut}(X)$ (and even the group $\\operatorname{Bim}(X)$ of all bimeromorphic automorphisms of $X$) are Jordan for any non-projective compact complex surface $X$.\n\nHowever, one cannot generalize Popov's question further to the settings of non-compact complex manifolds, or diffeomorphism groups of compact Riemmannian manifolds.\nWe refer to \\cite{CPS}, \\cite{Zar19} and \\cite{Pop18} for the counter examples; see also \\cite{BZ17}, \\cite{BZ20} and \\cite{Mun19} for positive cases.\n\nIn higher dimensions, it remains unknown whether the biholomorphic or biregular automorphism group $\\operatorname{Aut}(X)$ is Jordan for any {\\it compact} complex manifold $X$ or {\\it non-projective} algebraic variety $X$, respectively.\n\nWe refer to Mundet i Riera \\cite[\\S 1]{Mun19} for an excellent survey of more related results.\n\nAs in \\cite{BZ19} and \\cite{PS19},\na group $G$ is called {\\it strongly Jordan} if it is Jordan and if there is a constatnt $N = N(G)$ such that any finite abelian subgroup is generated by at most $N$ elements.\nBy \\cite[Theorem 2.5]{MS63}, the group $\\operatorname{Aut}(X)$ being Jordan automatically implies that it is strongly Jordan for any compact complex manifold $X$ (or any compact complex space by taking an equivariant projective resolution, due to Hironaka).\n\nTheorem \\ref{main-thm} below is our main result. The assumption of $X$ being smooth can be weakened to being irreducible  by taking an equivariant projective resolution.\nWe refer to \\cite[Definition 1.1 and Lemma 1.1]{Fuj78} for equivalent definitions of Fujiki's class $\\mathcal{C}$.\n\n\\begin{theorem}\\label{main-thm}\nLet $X$ be a connected complex manifold and let $Z$ be a compact complex subspace in Fujiki's class $\\mathcal{C}$.\nThen the automorphism group $\\operatorname{Aut}(X, Z)$ of all biholomorphic automorphisms of $X$ preserving $Z$ is strongly Jordan.\n\\end{theorem}\n\nThe following result is immediately obtained by taking an equivariant projective resolution  to reduce to the smooth case and then applying Theorem 1.1 with $Z=X$.\nIn particular, it answers the question for the Moishezons by Prokhorov and Shramov, who proved the case of Moishezon threefolds by using a quite different method (cf.~\\cite{PS19}).\n\n\\begin{corollary} \\label{Cor1}\nLet $X$ be a reduced compact complex space.\nThen $\\operatorname{Aut}(X)$ is strongly Jordan in the following cases (where (1) is a special case of (2)):\n\n\\begin{enumerate}\n\\item $X$ is Moishezon, i.e., $X$ is bimeromorphic to a projective variety.\n\\item $X$ is in Fujiki's class $\\mathcal{C}$, i.e., $X$ is the meromorphic image of a compact K\\\"ahler manifold.\n\\end{enumerate}\n\\end{corollary}\n\nOur approach to Theorem \\ref{main-thm} is based on a very simple idea: make use of the non-K\\\"ahler locus of a big $(1,1)$ class $[\\alpha]$ on $Z$ to find some invariant K\\\"ahler submanifold $Z_1 \\subseteq Z \\subseteq X$. Then we focus on the (linear) automorphism group of the normal bundle $\\mathcal{N}_{Z_1\/X}$ as inspired by Mundet i Riera \\cite{Mun19}; see Lemma \\ref{lem-copy}. This way, we reduce the question on (strongly) Jordan property to the case for compact K\\\"ahler manifolds by an equivariant compactification of $\\mathcal{N}_{Z_1\/X}$.\nWe refer to \\cite[\\S 2.4]{Tos18} for the definition and properties of the non-K\\\"ahler locus of a big class.\n\nWe end the introduction with the following two questions.\n\n\\begin{question}\\label{Q2}\nLet $X$ be a compact complex manifold.\nSuppose $X$ is Moishezon or is in Fujiki's class $\\mathcal{C}$.\nIs $\\operatorname{Aut}_{\\tau}(X):=\\{g\\in \\operatorname{Aut}(X)\\,|\\, g^*|_{H^2(X,\\mathbb{Q})}=\\operatorname{id}\\}$ a finite-index extension of\nthe neutral connected component $\\operatorname{Aut}_0(X)$ of $\\operatorname{Aut}(X)$?\n\\end{question}\n\n\\begin{question}\\label{Q1}\nLet $X$ be a compact complex manifold.\nIf $X$ is Moishezon (or in Fujiki's class $\\mathcal{C}$), can one find a bimeromorphic model $\\widetilde{X}$ of $X$\nsuch that $\\operatorname{Aut}(X)$ lifts to $\\widetilde{X}$, and $\\widetilde{X}$ is projective (or K\\\"ahler)?\n\\end{question}\n\nQuestion \\ref{Q2} has a positive answer when $X$ is a compact K\\\"ahler manifold (cf.~\\cite{Fuj78}, \\cite{Lie78}).\n\nA positive answer to Question \\ref{Q1} implies a positive answer to Question \\ref{Q2}, by making use\nof Fujiki \\cite[Theorem 4.8]{Fuj78} or Lieberman \\cite[Proposition 2.2]{Lie78} and the norm criterion \\cite[Proposition 2.9]{MZ18} for the pseudo-effective cone and nef cone.\n\nA positive answer to Question \\ref{Q2} will render an alternative proof to Corollary \\ref{Cor1} by applying Minkowski Theorem \\ref{thm-Mink} to $\\operatorname{GL}(H^2(X, \\mathbb{Q}))$ in order to reduce to the case for $\\operatorname{Aut}_0(X)$ (a Lie group)\nwhich is a known case (cf.~\\cite{Pop18}).\n\n\\par \\vskip 1pc \\noindent\n{\\bf Acknowledgement.}\n\nThe first author would like to thank Doctor Xueyuan Wan for answering several questions on currents.\nThe third author would like to thank Professor Mihai P\\u{a}un for the helpful discussions on Question \\ref{Q1}.\nThe first author is supported by a Research Fellowship of KIAS (MG075501).\nThe second author is supported by the national projects\nPRIN 2015EYPTSB-PE1 ``Geometria delle variet\\`a algebriche''\nand 2017SSNZAW 005-PE1 ``Moduli Theory and Birational Classification'',\nby the research group GNSAGA of INDAM and by FRA 2018 of the University of Trieste.\nThe third author is supported by an ARF of NUS.\n\n\\section{Preliminaries}\n\nWe use the following notation throughout this paper.\n\\begin{notation}\\label{notation2.1}\nLet $X$ be a connected complex space.\n\n\\begin{enumerate}\n\\item $\\operatorname{Aut}(X)$ is the group of all biholomorphic automorphisms of $X$.\n\\item $\\operatorname{Aut}(X,Z):=\\{g\\in \\operatorname{Aut}(X)\\,|\\, g(Z)=Z\\}$ for a subset $Z$ of $X$.\n\\item $\\operatorname{Aut}(\\mathcal{E}\\to X):=\\{g\\in \\operatorname{Aut}(\\mathcal{E})\\,|\\, g \\text{ maps every bundle fibre linearly to some bundle fibre}\\}$ for a vector bundle $\\mathcal{E}$ over $X$.\n\n\\par \\noindent\nIn the following, $X$ is further assumed to be smooth and compact.\n\\item $\\operatorname{Aut}_0(X)$ is the neutral connected component of $\\operatorname{Aut}(X)$.\n\\item $\\operatorname{Aut}_{\\tau}(X):=\\{g\\in \\operatorname{Aut}(X)\\,|\\, g^*|_{H^2(X,\\mathbb{Q})}=\\operatorname{id}\\}$.\nClearly, $\\operatorname{Aut}_{\\tau}(X) \\supseteq \\operatorname{Aut}_0(X)$.\n\\end{enumerate}\n\\end{notation}\n\n\\begin{notation}\\label{notation2.2}\nLet $X$ be a connected complex manifold and $Z$ a connected complex submanifold.\nLet $\\mathcal{T}_X$ be the tangent bundle of $X$ and $\\mathcal{N}_{Z\/X}$ the normal bundle.\nLet $g\\in \\operatorname{Aut}(X, Z)$.\nDenote by $\\mathcal{T}_g$ the induced tangent automorphism of $\\mathcal{T}_X$.\nThen we have the following commutative diagram:\n\\[\n\\xymatrix{\n0\\ar[r]&\\mathcal{T}_Z\\ar[r]\\ar[d]^{\\mathcal{T}_{g|_Z}}&\\mathcal{T}_X|_Z\\ar[r]\\ar[d]^{{\\mathcal{T}_g}|_Z}&\\mathcal{T}_X|_Z\/\\mathcal{T}_Z\\ar[r]\\ar[d]^{\\mathcal{N}_g}&0\\\\\n0\\ar[r]&\\mathcal{T}_Z\\ar[r]&\\mathcal{T}_X|_Z\\ar[r]&\\mathcal{T}_X|_Z\/\\mathcal{T}_Z\\ar[r]&0\n}\n\\]\nwhere $\\mathcal{N}_g$ is the induced automorphism of the normal bundle $\\mathcal{N}_{Z\/X} = (\\mathcal{T}_X|_Z)\/\\mathcal{T}_Z$.\nIn particular, we have a group homomorphism\n$$\\mathcal{N}: \\operatorname{Aut}(X, Z) \\to \\operatorname{Aut}(\\mathcal{N}_{Z\/X}\\to Z)$$\nvia $g\\mapsto \\mathcal{N}_g$.\n\\end{notation}\n\nThe following result is very important in the proof of Theorem \\ref{main-thm}.\n\n\\begin{lemma}\\label{lem-copy}\nLet $X$ be a connected complex manifold and $Z$ a connected complex submanifold.\nThen the kernel of the natural homomorphism $\\mathcal{N}: \\operatorname{Aut}(X,Z)\\to \\operatorname{Aut}(\\mathcal{N}_{Z\/X}\\to Z)$\ncontains no non-trivial subgroup of finite order. In particular,\n$\\operatorname{Aut}(\\mathcal{N}_{Z\/X}\\to Z)$ contains an isomorphic copy of every finite subgroup of $\\operatorname{Aut}(X,Z)$.\n\\end{lemma}\n\n\\begin{proof}\nSuppose $g\\in \\operatorname{Ker} \\mathcal{N}$ has finite order.\nThen $g|_Z=\\operatorname{id}$ (and hence $\\mathcal{T}_{g|_Z}=\\operatorname{id}$) and $\\mathcal{N}_g=\\operatorname{id}$.\nIn particular, $\\mathcal{T}_{g}|_{z}$ is a unipotent action for each point $z\\in Z$ by the commutative diagram in Notation \\ref{notation2.2}.\nNote that $\\mathcal{T}_{g}$ has finite order.\nThen $\\mathcal{T}_{g}|_{z}=\\operatorname{id}$ for $z\\in Z$.\nThis, together with $g$ having finite order and $X$ being connected, imply $g=\\operatorname{id}$ (cf.~\\cite[Lemma 2.1(2)]{Mun19}).\nThe lemma is proved.\n\\end{proof}\n\n\nA group $G$ has {\\it bounded finite subgroups} if there is a constant $N = N(G)$ such that any finite subgroup $H\\le G$ has order $|H| \\le N$.\n\nFor the (strongly) Jordan property, we may always replace the group by its normal subgroup with the quotient group having bounded finite subgroups.\n\n\\begin{lemma}\\label{lem-exact}\nConsider the exact sequence of groups\n$$1 \\to G_1 \\to G \\to G_2 \\to 1.$$\nSuppose $G_1$ is Jordan (resp. strongly Jordan) and $G_2$ has bounded finite subgroups.\nThen $G$ is Jordan (resp. strongly Jordan).\n\\end{lemma}\n\nTo obtain the quotient group having bounded finite subgroups, we have the following wonderful theorem of Minkowski which allows us to make use of rational representations of geometric automorphisms; see \\cite[Theorem 5, and \\S4.3]{Ser07}.\n\n\\begin{theorem}[Minkowski]\\label{thm-Mink}\n$\\operatorname{GL}_n(K)$ has bounded finite subgroups, whenever $K$ is a number field. The bound depends only on $n$ and\nthe field extension degree $[K:\\mathbb{Q}]$.\n\\end{theorem}\n\n\nThere are two ways to obtain the strongly Jordan property for compact K\\\"ahler manifolds.\nWe adopt a shorter one here.\n\n\\begin{theorem}\\textup{(\\cite[Theorem 1.1]{Kim18}, \\cite[Theorem 2]{Pop18})}\\label{thm-kim-popov}\nLet $X$ be a compact K\\\"ahler manifold.\nThen $\\operatorname{Aut}(X)$ is strongly Jordan.\n\\end{theorem}\n\n\\begin{proof}\nThe pullback action of $\\operatorname{Aut}(X)$ on $H^2(X, \\mathbb{Q})$ gives a faithful rational representation\n$$\\operatorname{Aut}(X)\/\\operatorname{Aut}_\\tau(X)\\hookrightarrow \\operatorname{GL}(H^2(X, \\mathbb{Q}))$$\nwith the latter group having bounded finite subgroups by Theorem \\ref{thm-Mink}.\nNote that $\\operatorname{Aut}_\\tau(X)\/\\operatorname{Aut}_0(X)$ is a finite group by \\cite[Proposition 2.2]{Lie78} or \\cite[Theorem 4.8]{Fuj78}.\nTherefore, $\\operatorname{Aut}(X)\/\\operatorname{Aut}_0(X)$ has bounded finite subgroups.\nBy Lemma \\ref{lem-exact},\nit suffices to show $\\operatorname{Aut}_0(X)$ is strongly Jordan.\nBy the proof of \\cite[Theorem 2]{Pop18},\nthere exists some $n$ such that $\\operatorname{GL}_n(\\mathbb{R})$ contains an isomorphic copy of every finite subgroup of\n$\\operatorname{Aut}_0(X)$.\nNote that $\\operatorname{GL}_n(\\mathbb{R})$ is strongly Jordan (cf.~e.g.~\\cite[Lemmas 2.3 and 2.4]{MZ18}).\nThe theorem follows.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{main-thm}, and another open question}\n\nIn this section, we prove Theorem \\ref{main-thm}, ask Question \\ref{Q3} and give Remark \\ref{rem_Q3}\nillustrating the usefulness of the latter. We begin with the following.\n\n\\begin{theorem}\\label{thm-jordan-bundle}\nLet $X$ be a compact K\\\"ahler manifold and $\\mathcal{E}$ a vector bundle of finite rank $r$.\nThen $\\operatorname{Aut}(\\mathcal{E}\\to X)$ is strongly Jordan.\n\\end{theorem}\n\n\\begin{proof}\nLet $\\mathcal{O} = X \\times \\mathbb{C}$\nbe the trivial line bundle over $X$.\nLet $T:=\\mathbb{C}^*$ act on $\\mathcal{E} \\oplus \\mathcal{O}$\nby the natural scalar multiplication.\nThen there is a natural $T$-equivariant monomorphism\n$$\\phi:\\operatorname{Aut}(\\mathcal{E}\\to X)\\to \\operatorname{Aut}(\\mathcal{E}\\oplus \\mathcal{O} \\to X)$$\nvia $g\\mapsto g\\oplus (g|_X \\, \\times \\, \\operatorname{id}_{\\mathbb{C}})$.\nHence we have the following $\\operatorname{Aut}(\\mathcal{E}\\to X)$-equivariant commutative diagram\n\\[\n\\xymatrix{\n\\mathcal{E}\\ar@{^{(}->}[r]\\ar[d] & \\mathcal{E}\\oplus \\mathcal{O}\\ar@{.>}[r]\\ar[d] & \\mathbb{P}(\\mathcal{E}\\oplus \\mathcal{O})\\ar[d]\\\\\nX\\ar@{=}[r]&X\\ar@{=}[r]&X\n};\n\\]\nhere  $\\mathbb{P}(\\mathcal{E}\\oplus \\mathcal{O}):=(\\mathcal{E}\\oplus \\mathcal{O})\/T$ is an analytic $\\mathbb{P}^{r}$-bundle over $X$; hence it is again a compact K\\\"ahler manifold.\n\nConsider the homomorphism\n$$\\psi:\\operatorname{Aut}(\\mathcal{E}\\oplus \\mathcal{O} \\to X)\\to \\operatorname{Aut}(\\mathbb{P}(\\mathcal{E}\\oplus \\mathcal{O}))$$\ninduced by the $T$-quotient.\nNote that $\\psi\\circ\\phi$ is still a monomorphism.\nBy Theorem \\ref{thm-kim-popov}, $\\operatorname{Aut}(\\mathbb{P}(\\mathcal{E}\\oplus \\mathcal{O}))$ and hence $\\operatorname{Aut}(\\mathcal{E}\\to X)$ are strongly Jordan.\n\\end{proof}\n\nWe wonder whether we can weaken the assumption of $X$\nbeing compact K\\\"ahler in Theorem \\ref{thm-jordan-bundle} to just $\\operatorname{Aut}(X)$ being (strongly) Jordan.\n\n\\begin{question}\\label{Q3}\nLet $X$ be a connected complex manifold and $\\mathcal{E}$ a vector bundle of finite rank.\nWill $\\operatorname{Aut}(\\mathcal{E}\\to X)$ or $\\operatorname{Aut}(\\mathbb{P}(\\mathcal{E}\\oplus \\mathcal{O}))$ be (strongly) Jordan if so is $\\operatorname{Aut}(X)$?\n\\end{question}\n\nWe can make use of Lemma \\ref{lem-copy} and Theorem \\ref{thm-jordan-bundle} to prove the following:\n\n\\begin{theorem}\\label{thm-jordan-Z}\nLet $X$ be a connected complex manifold and $Z\\subseteq X$ a connected compact K\\\"ahler submanifold.\nThen $\\operatorname{Aut}(X,Z)$ is strongly Jordan.\n\\end{theorem}\n\n\\begin{proof}\nBy Lemma \\ref{lem-copy}, $\\operatorname{Aut}(\\mathcal{N}_{Z\/X}\\to Z)$ contains an isomorphic copy of every finite subgroup of $\\operatorname{Aut}(X,Z)$.\nBy Theorem \\ref{thm-jordan-bundle}, $\\operatorname{Aut}(\\mathcal{N}_{Z\/X}\\to Z)$ is strongly Jordan.\nThe theorem follows.\n\\end{proof}\n\n\\begin{remark}\\label{rem_Q3}\nTogether with \\cite[Theorem 1.6]{PS}, a positive answer to Question \\ref{Q3} for compact complex surfaces will deduce the (stongly) Jordan property of $\\operatorname{Aut}(V)$ for every compact complex threefold $V$ with $\\operatorname{Sing}(V)\\neq \\emptyset$.\nIndeed, just take an equivariant projective log resolution and apply the same proof of Theorem \\ref{thm-jordan-Z} for $X$ being any (smooth) exceptional prime divisor which is a compact complex surface.\n\\end{remark}\n\n\nNote that the $\\partial\\bar{\\partial}$-lemma holds for compact complex manifolds in Fujiki's class $\\mathcal{C}$.\nSo it is free for us to use the equivalent Bott-Chern ($\\partial\\bar{\\partial}$),  Dolbeault ($\\bar{\\partial}$) and De Rham ($d$) cohomologies.\nMoreover, Hodge decomposition holds true.\nWe refer to \\cite[Lemma 5.15 and Proposition 5.17]{DGMS75} and \\cite[Proposition 1.6 and Corollary 1.7]{Fuj78} for the details.\n\nNow we are ready for:\n\n\\begin{proof}[Proof of Theorem \\ref{main-thm}]\nWe take the reduced structure of $Z$.\nWe first let $G:=\\operatorname{Aut}(X,Z)$.\nTo show $G$ is strongly Jordan,\nwe shrink $Z$ and $G$ by running the Main Program (several times) such that $Z$ is a $G$-invariant (non-empty) connected compact K\\\"ahler submanifold of $X$ (which could be a single point or remain unchanged).\nNote that the Fujiki's class $\\mathcal{C}$ is closed under closed subspaces and compact meromorphic images.\n\n\\vskip 1pc \\noindent\n\\textbf{Main Program.}\n\nIf $Z$ is smooth connected compact K\\\"ahler, then we stop.\n\nIf $Z$ is not connected, we run Step A and restart the Main Program; else we continue.\n\nIf $Z$ is singular, we run Step B and restart the Main Program; else we continue.\n\nElse: assume that $Z$ is smooth connected compact in $\\mathcal{C}$ but not K\\\"ahler.\nLet $$G_{\\tau}:=\\{g\\in G\\,|\\, (g|_Z)^*|_{H^2(Z,\\mathbb{Q})}=\\operatorname{id}\\}.$$\nBy\nthe Hodge decomposition\n(which still exists for those in\n$\\mathcal{C}$), $G_{\\tau}$ acts trivially, via pullback, on $H^{1,1}_{\\partial\\bar{\\partial}}(X,\\mathbb{R})$.\nSince $Z\\in \\mathcal{C}$, there is a big real $(1,1)$-class $[\\alpha]\\in H^{1,1}_{\\partial\\bar{\\partial}}(Z,\\mathbb{R})$ by \\cite[Theorem 0.7]{DP04}.\nNote that its non-K\\\"ahler locus $E_{nK}(\\alpha)$ is a $G_{\\tau}$-invariant non-empty closed analytic subset of $Z$ with $\\dim(E_{nK}(\\alpha))<\\dim(Z)$ (cf.~\\cite[\\S2.4]{Tos18}).\nWe replace $Z$ by $E_{nK}(\\alpha)$ (with reduced structure) and $G$ by its finite-index subgroup $G_{\\tau}$\n(cf.~Theorem \\ref{thm-Mink}).\nThen we restart the Main Program.\n\n\\vskip 1pc \\noindent\n\\textbf{Step A.}\nWe replace $G$ by its\nsubgroup of finite index such that $G$ fixes all (finitely many) connected components of $Z$.\nWe replace $Z$ by one of its connected component (with reduced structure).\n\n\\vskip 1pc \\noindent\n\\textbf{Step B.}\nIf $Z$ is singular, then its singular locus $\\operatorname{Sing}(Z)$ is a $G$-invariant non-empty closed analytic subset of $Z$ with $\\dim(\\operatorname{Sing}(Z))<\\dim(Z)$.\nWe then replace only $Z$ by $\\operatorname{Sing}(Z)$ (with reduced structure).\n\n\\vskip 1pc \\noindent\n\\textbf{End of the proof:}\n\nNote that our (finitely many) replacements of $G$ always fit the assumption in Lemma \\ref{lem-exact}.\nSo it suffices to show that the finally chosen $G$ is still strongly Jordan.\nBy Theorem \\ref{thm-jordan-Z}, $\\operatorname{Aut}(X,Z)$ is strongly Jordan, because our finally chosen $Z$ is smooth compact K\\\"ahler.\nSince our current $G$ is contained in $\\operatorname{Aut}(X, Z)$, the theorem follows.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{introduction}\n\\setcounter{equation}{0}\n\nLocalized bases and frames allow to decompose functions and distributions\nin terms of building blocks of simple nature and have numerous advantages\nover other means of representation.\nIn particular, they enable one to encode smoothness and other norms\nin terms of the coefficients of the decompositions.\nMeyer's wavelets \\cite{Meyer} and the $\\varphi$-transform of\nFrazier and Jawerth \\cite{FJ1, FJ2, FJW} provide such building\nblocks for decomposition of Triebel-Lizorkin and Besov spaces\nin the classical case on $\\mathbb{R}^d$.\n\nThe aim of this article is to develop similar tools\nfor decomposition of weighted Triebel-Lizorkin and Besov spaces\non the unit ball $B^d$ in ${\\mathbb R}^d$ ($d>1$) with weights\n$$\n{w_\\mu}(x):=(1-|x|^2)^{\\mu-1\/2}, \\quad \\mu \\ge 0,\n$$\nwere $|x|$ is the Euclidean norm of $x\\in B^d$.\nThese include $L_p(B^d,{w_\\mu})$, the Hardy spaces $H_p(B^d,{w_\\mu})$, and\nweighted Sobolev spaces.\nFor our purposes we develop localized frames which can be viewed as an analogue\nof the $\\varphi$-transform of Frazier and Jawerth on $B^d$.\n\nFor the construction of our frame elements we shall use\northogonal polynomials in the weighted space $L_2({w_\\mu}):= L_2(B^d, {w_\\mu})$.\nDenote by $\\Pi_n$ the space of all algebraic polynomials of degree $n$\nin $d$ variables and by $V_n$ the subspace of all polynomials of degree $n$\nwhich are orthogonal to lower degree polynomials in $L_2({w_\\mu} )$.\nThese are eigenspaces of the differential operator\n\\begin{equation}\\label{def-diff-oper}\nD_\\mu:= -\\Delta +\\langle x, \\nabla \\rangle^2\n+(2\\mu+d-1)\\langle x, \\nabla \\rangle.\n\\end{equation}\nMore precisely (see e.g. \\cite{DXu}),\n\\begin{equation}\\label{eigen-func}\nD_\\mu P=n(n+d+2\\mu -1)P\n\\quad\\mbox{for } P\\inV_n.\n\\end{equation}\nWe have the orthogonal polynomial decomposition\n\\begin{equation}\\label{L2:decomp}\nL_2({w_\\mu}) = \\bigoplus_{n=0}^\\infty V_n, \\qquad\nV_n \\subset \\Pi_n.\n\\end{equation}\nNote that $\\dim V_n = \\binom{n+d-1}{n}\\sim n^{d-1}$.\nAs is shown in \\cite{X99} the orthogonal projector\n$\\operatorname{Proj}_n: L_2({w_\\mu}) \\mapstoV_n$\ncan be written as\n\\begin{equation}\\label{def-Pn}\n(\\operatorname{Proj}_n f)(x) = \\int_{B^d} f(y){\\mathsf {P}}_n(x,y) {w_\\mu}(y) dy,\n\\end{equation}\nwhere, for $\\mu > 0$, the kernel ${\\mathsf {P}}_n(x,y)$ has the representation\n\\begin{align} \\label{compactPn}\n{\\mathsf {P}}_n(x,y)& = b_d^\\mu b_1^{\\mu-\\frac{1}{2}}\\frac{n+\\lambda}{\\lambda} \\\\\n&\\times  \\int_{-1}^1 C_n^\\lambda \\left(\\langle x,y\\rangle +\nu\\sqrt{1-|x|^2} \\sqrt{1-|y|^2}\\right) (1-u^2)^{\\mu-1}du.  \\notag\n\\end{align}\nHere $\\langle x, y \\rangle$ is the Euclidean inner product in $\\mathbb{R}^d$,\n$C_n^\\lambda$ is the $n$-th degree Gegenbauer polynomial,\n\\begin{equation}\\label{def-lambda}\n   \\lambda = \\mu + \\frac{d-1}{2},\n\\end{equation}\nand the constants\n$b_d^\\mu$, $ b_1^{\\mu-\\frac{1}{2}}$ are defined by\n$(b_d^\\gamma)^{-1} := \\int_{B^d} (1-|x|^2)^{\\gamma-1\/2} dx$.\nFor a~representation of ${\\mathsf {P}}_n(x, y)$ in the limiting case $\\mu=0$,\nsee (4.2) in \\cite{PX2}.\n\nEvidently,\n\\begin{equation}\\label{def:Kn}\nK_n(x,y):= \\sum_{j=0}^n {\\mathsf {P}}_j(x,y)\n\\end{equation}\nis the kernel of the orthogonal\nprojector of $L_2({w_\\mu})$ onto the space\n$\\bigoplus_{\\nu=0}^n V_\\nu$.\n\nA key role in this study will play the fact (established in \\cite{PX2})\nthat if the coefficients on the right in (\\ref{def:Kn})\nare ``smoothed out\" by sampling a compactly supported $C^\\infty$ function,\nthen the resulting kernel has nearly exponential localization\naround the main diagonal $y=x$ in $B^d\\times B^d$.\nMore precisely, let\n\\begin{equation} \\label{def-Ln}\nL_n(x,y):= \\sum_{j=0}^\\infty \\widehat a\\Big(\\frac{j}{n}\\Big) {\\mathsf {P}}_j(x,y),\n\\end{equation}\nwhere the ``smoothing\" function $\\widehat a$ is admissible in the sense of the following definition:\n\n\n\\begin{definition} \\label{defn:admissible}\nA function $\\widehat a \\in C^\\infty[0, \\infty)$\nis called admissible of type\n\n$(a)$ if\n$\\operatorname{supp} \\widehat a \\subset [0,2]$ and $\\widehat a(t)=1$ on $[0, 1]$,\nand of type\n\n$(b)$ if $\\operatorname{supp} \\widehat a \\subset [1\/2,2]$.\n\\end{definition}\n\\noindent\nWe introduce the distance\n\\begin{equation} \\label{eq:distant}\nd(x,y):= \\arccos \\left\n\\{ \\langle x,y\\rangle + \\sqrt{1-|x|^2}\\sqrt{1-|y|^2} \\right \\}\n\\quad \\mbox{on $B^d$}\n\\end{equation}\nand set\n\\begin{equation} \\label{def-WW}\n{W_\\mu}(n;x)\n:= \\left(\\sqrt{1-|x|^2} + n^{-1}\\right)^{2\\mu}, \\quad x \\in B^d.\n\\end{equation}\nOne of our main results in \\cite[Theorem 4.2]{PX2} asserts that for any $k >0$\nthere exists a constant $c_k>0$ depending only on $k$, $d$, $\\mu$, and $\\widehat a$\nsuch that\n\\begin{equation} \\label{eq:main-estI}\n|L_n(x,y)| \\le c_k \\frac{n^d }{\\sqrt{{W_\\mu}(n;x)} \\sqrt{{W_\\mu}(n;y)}\n(1 + n\\,d(x,y))^k},\n\\quad x, y\\in B^d.\n\\end{equation}\n\nThe kernels $L_n$ are our main ingredient in constructing\n{\\it analysis} and {\\it synthesis} needlet systems\n$\\{{\\varphi}_\\xi\\}_{\\xi\\in{\\mathcal X}}$ and $\\{\\psi_\\xi\\}_{\\xi\\in{\\mathcal X}}$ here,\nindexed by a multilevel set ${\\mathcal X}=\\cup_{j=0}^\\infty {\\mathcal X}_j$ (\\S\\ref{def-needlets}).\nThis is a pair of dual frames whose elements have nearly exponential localization on $B^d$\nand provide representation of every distribution $f$ on $B^d$:\n\\begin{equation}\\label{eq:decomposition}\nf=\\sum_{\\xi \\in {\\mathcal X}}\\langle f, {\\varphi}_\\xi \\rangle \\psi_\\xi.\n\\end{equation}\nThe superb localization of the frame elements prompted us to term them {\\em needlets}.\n\nOur main interest lies with distributions in the weighted Triebel-Lizorkin ($F$-spaces) and\n Besov spaces ($B$-spaces) on $B^d$.\nThese spaces are naturally defined via spectral decompositions\n(see \\cite{Pee, T1} for the general idea).\nTo be specific, let\n$$\n\\Phi_0(x, y) := 1\n\\quad\\mbox{and}\\quad\n\\Phi_j(x, y) := \\sum_{\\nu=0}^\\infty \\widehat a\n\\Big(\\frac{\\nu}{2^{j-1}}\\Big){\\mathsf {P}}_\\nu(x,y), \\quad j\\ge 1,\n$$\nwhere ${\\mathsf {P}}(\\cdot, \\cdot)$ is from \\eqref{compactPn} and\n$\\widehat a$ is admissible of type (b) (see Definition~\\ref{defn:admissible})\nsuch that $|\\widehat a|>0$ on $[3\/5, 5\/3]$.\n\nThe $F$-space ${F_{pq}^{s{\\rho}}}$ with\n$s, {\\rho} \\in \\mathbb{R}$, $0<p<\\infty$, $0<q\\le\\infty$,\nis defined (\\S\\ref{Tri-Liz})\nas the space of all distributions $f$ on $B^d$ such that\n\\begin{equation}\\label{Tri-Liz-norm0}\n\\norm{f}_{{F_{pq}^{s{\\rho}}}}:=\\nnorm{\\biggl(\\sum_{j=0}^{\\infty}\n(2^{sj}{W_\\mu}(2^j; \\cdot)^{-{\\rho}\/d}|\\Phi_j\\ast f(\\cdot)|)^q\\biggr)^{1\/q}}_{{L_p({w_\\mu})}} <\\infty,\n\\end{equation}\nwhere $\\Phi_j*f(x):= \\langle f, \\overline{\\Phi(x, \\cdot)}\\rangle$\n(see Definition~\\ref{Def-conv}).\n\nThe corresponding scales of weighted Besov spaces ${B_{pq}^{s{\\rho}}}$\nwith $s, {\\rho} \\in \\mathbb{R}$, $0<p, q\\le\\infty$, are\ndefined (\\S\\ref{Besov}) via the (quasi-)norms\n\\begin{equation}\\label{Besov-norm0}\n\\|f\\|_{{B_{pq}^{s{\\rho}}}} :=\n\\Big(\\sum_{j=0}^\\infty \\Big(2^{s j}\n\\|{W_\\mu}(2^j; \\cdot)^{-{\\rho}\/d}\\Phi_j*f(\\cdot)\\|_{{L_p({w_\\mu})}}\\Big)^q\\Big)^{1\/q}.\n\\end{equation}\n\nUnlike in the classical case on $\\mathbb{R}^d$, we have introduced an additional\nparameter~${\\rho}$, which allows considering different scales\nof Triebel-Lizorkin and Besov spaces.\nTo~us most natural are the spaces\n\\begin{equation}\\label{Tri-Liz-Besov}\n{F_{pq}^{s}}:=F_{pq}^{s s}\\quad \\mbox{and}\\quad {B_{pq}^{s}}:=B_{pq}^{s s},\n\\end{equation}\nwhich scale (are embedded) correctly with respect to the smoothness parameter~$s$.\nA ``classical\" choice would be to consider the spaces $F_{pq}^{s 0}$ and $B_{pq}^{s 0}$,\nwhere the weight ${W_\\mu}(2^j;\\cdot)$ is excluded from\n\\eqref{Tri-Liz-norm0}-\\eqref{Besov-norm0}.\nThe introduction of the parameter ${\\rho}$ enables us to treat these\nspaces simultaneously.\n\n\nOne of the main results of this paper is the characterization of  the $F$-spaces\nin terms of the size of the needlet coefficients in the decomposition (\\ref{eq:decomposition}),\nnamely,\n$$\n\\norm{f}_{{F_{pq}^{s{\\rho}}}}\n\\sim \\nnorm{\\biggl(\\sum_{j=0}^\\infty 2^{sjq}\n\\sum_{\\xi\\in {\\mathcal X}_j}|\\ip{f, \\varphi_\\xi}|{W_\\mu}(2^j;\\xi)^{-{\\rho}\/d}|\\psi_\\xi(\\cdot)|^q\\biggr)^{1\/q}}_{{L_p({w_\\mu})}}.\n$$\nSimilarly for the Besov spaces ${B_{pq}^{s{\\rho}}}$ we have the characterization (\\S\\ref{Besov})\n$$\n\\norm{f}_{{B_{pq}^{s{\\rho}}}}\n\\sim \\Big(\\sum_{j=0}^\\infty2^{sjq}\n\\Bigl[\\sum_{\\xi\\in {\\mathcal X}_j}\n\\Big({W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}\\norm{\\ip{f,\\varphi_\\xi}\\psi_\\xi}_{{L_p({w_\\mu})}}\\Big)^p\\Bigr]^{q\/p}\\Bigr)^{1\/q}.\n$$\n\nFurther, the weighted Besov spaces are applied to nonlinear $n$-term approximation from\nneedlets on $B^d$ (\\S\\ref{Nonlin-app}).\n\n\nThis is a follow-up paper of \\cite{PX2}, where the localization \\eqref{eq:main-estI}\nis established and the construction and basic properties of a single system of needlets are given.\nOur development here is a part of a broader undertaking for needlet characterization of\nTriebel-Lizorkin and Besov spaces on nonclassical domains,\nincluding the multidimensional unit sphere \\cite{NPW1, NPW2}, ball,\nand cube (interval \\cite{KPX, PX1}) with weights.\nThe~results in this paper generalize the results in the univariate case from \\cite{KPX}\n(with $\\alpha=\\beta$), where needlet characterizations of $F$- and $B$-spaces on\nthe interval are obtained.\n\nThe organization of the paper is the following:\nIn \\S\\ref{preliminaries} the needed results from \\cite{PX2} and some background material\nare given, including localized polynomial kernels, the maximal operator,\ndistributions on $B^d$, and cubature formula on $B^d$.\nThe definition and some basic properties of needlets are given in \\S\\ref{def-needlets}.\nIn \\S\\ref{Tri-Liz} the weighted Triebel-Lizorlin space on $B^d$ are introduced\nand characterized via needlets,\nwhile the weighted Besov spaces are explored in \\S\\ref{Besov}.\nIn \\S\\ref{Nonlin-app} Besov spaces are applied to nonlinear $n$-term approximation\nfrom needlets.\nSection~\\ref{proofs} contains the proofs of various lemmas from previous sections.\n\n\n\nThroughout the paper we use the following notation:\n$$\n\\|f\\|_{p}:=\\Big(\\int_{B^d}|f(x)|^p{w_\\mu}(x)dx\\Big)^{1\/p},\n\\quad 0<p<\\infty, \\quad\n\\|f\\|_{\\infty}:={\\rm ess \\; sup}_{x\\in B^d}|f(x)|.\n$$\nFor a measurable set $E\\subset B^d$,\n$|E|$ denotes the Lebesgue measure of $E$,\n$m(E):= \\int_E{w_\\mu}(x)dx$,\n${\\mathbbm 1}_E$ is the characteristic function of $E$, and\n${\\tilde{\\mathbbm 1}}_E:=|m(E)|^{-1\/2}{\\mathbbm 1}_E$ is the $L_2({w_\\mu})$ normalized\ncharacteristic function of $E$.\nPositive constants are denoted by $c$, $c_1, c_*, \\dots$ and they\nmay vary at every occurrence; $A\\sim B$ means $c_1A\\le B\\le c_2 A$.\n\n\n\n\\section{Preliminaries}\\label{preliminaries}\n\\setcounter{equation}{0}\n\n\\subsection{Localized polynomial kernels on \\boldmath $B^d$}\\label{Local-kernels}\n\nThe polynomial kernels $L_n(x,y)$ introduced in (\\ref{def-Ln})\nwill be our main vehicle in developing needlet systems.\nHere we give come additional properties of these kernels.\n\nWe have\n\\begin{equation} \\label{eq:LpUB}\n\\|L_n(x, \\cdot)\\|_{{p}}\n\\le c\\Bigl(\\frac{n^{d}}{{W_\\mu}(n; x)}\\Bigr)^{1-1\/p},\n\\quad x\\in B^d,\n\\quad 0<p\\le\\infty.\n\\end{equation}\nThis estimate is an immediate consequence of \\eqref{eq:main-estI}\nand the following lemma (see \\cite[Lemma 4.6]{PX2}),\nwhich will be instrumental in several proofs below.\n\n\n\n\\begin{lemma}\\label{lem:instrumental}\nIf $\\sigma > d\/p + 2\\mu|1\/p-1\/2|$, $\\mu\\ge 0$, $0 < p < \\infty$, then\n\\begin{equation} \\label{est-inst}\n\\int_{B^d} \\frac{{w_\\mu}(y) dy}\n{{W_\\mu}(n;y)^{p\/2} (1+n d(x,y))^{\\sigma p}}\n\\le c\\,n^{-d}{W_\\mu}(n;x)^{1-p\/2}.\n\\end{equation}\n\\end{lemma}\n\nWe now establish a matching lower bound estimate.\n\n\n\\begin{theorem}\\label{thm:est-Lp-norm}\nLet $\\widehat a$ be admissible and $|\\widehat a(t)| \\ge c_* > 0$ for $t \\in [3\/5,5\/3]$.\nThen for $0<p\\le \\infty$ and $n\\ge 2$\n\\begin{equation} \\label{eq:est-Lp-norm}\n\\|L_n(x, \\cdot)\\|_{{p}}\n\\ge c\\Bigl(\\frac{n^{d}}{{W_\\mu}(n; x)}\\Bigr)^{1-1\/p},\n\\quad x \\in B^d.\n\\end{equation}\nHere the constant $c>0$ depends only on $d$, $\\mu$, $p$, and $c_*$.\n\\end{theorem}\n\nThe proof of this theorem is given in \\S\\ref{proofsA}.\n\n\nThe kernels $L_n(x,y)$\nare in a sense Lip1 functions in both variables with respect\nto the distance $d(\\cdot,\\cdot)$ from \\eqref{eq:distant}:\nLet $\\xi, y\\in B^d$ and $c^*>0$, $n\\ge 1$.\nThen for all $x, z\\in B_\\xi(c^*n^{-1})$ and an arbitrary $k$, we have\n\\begin{equation} \\label{Lip}\n|L_n(x,y)-L_n(\\xi,y)|\n\\le c_k \\frac{n^{d+1}d(x, \\xi)} {\\sqrt{{W_\\mu}(n; y)}\\sqrt{{W_\\mu}(n; z)}(1+nd(y,z))^k},\n\\end{equation}\nwhere $c_k$ depends only on $k$, $\\mu$, $d$, $\\widehat a$, and $c^*$ (see\n\\cite[Proposition 4.7]{PX2}).\n\n\n\\medskip\nWe shall also need the following inequality from \\cite[Lemma 4.1]{PX2}:\n\\begin{equation} \\label{norm-dist2}\n\\Big|\\sqrt{1-|x|^2}-\\sqrt{1-|y|^2}\\Big| \\le \\sqrt{2}\\, d(x, y),\n\\quad x,y\\in B^d,\n\\end{equation}\nwhich yields\n\\begin{equation}\\label{eq:useful}\n{W_\\mu}(n;x)\\le 2^\\mu{W_\\mu}(n;y)(1+nd(x,y))^{2\\mu},\n\\quad x,y\\in B^d.\n\\end{equation}\n\n\n\\subsection{Reproducing polynomial kernels and applications}\\label{kernels}\n\nTo simplify our notation we introduce the following\n``convolution\": For functions $\\Phi: B^d\\times B^d \\to {\\mathbb C}$ and\n$f: B^d \\to {\\mathbb C}$, we write\n\\begin{equation}\\label{convolution}\n\\Phi*f(x) := \\int_{B^d} \\Phi(x, y)f(y){w_\\mu}(y)\\,dy.\n\\end{equation}\nWe denote by $E_n(f)_p$ the best approximation\nof $f \\in {L_p({w_\\mu})}$ from $\\Pi_n$, i.e.\n\\begin{equation}\\label{def-En}\nE_n(f)_p := \\inf_{g \\in \\Pi_n}\\|f-g\\|_{p}.\n\\end{equation}\n\n\n\\begin{lemma}\\label{lem:Ker-n}\nLet $L_n$ be the kernel from $(\\ref{def-Ln})$,\nwith $\\widehat a$ admissible of type $(a)$. Then\n\n$(i)$ ${L}_n*g =g$ for $g \\in \\Pi_n$, i.e.\n${L}_n$ is a reproducing kernel for\n$\\Pi_n$, and\n\n$(ii)$ for any $f \\in {L_p({w_\\mu})}$, $1\\le p \\le \\infty$, we have\n${L}_n*f \\in \\Pi_{2n}$,\n\\begin{equation}\\label{Ker-n3}\n\\|{L}_n*f\\|_{p}\\le c \\|f\\|_{p}, \\quad\\mbox{and}\\quad\n\\|f-{L}_n*f\\|_{p} \\le cE_n(f)_p.\n\\end{equation}\n\\end{lemma}\n\nThis lemma follows readily by the definition of $L_n$\n(see also Definition~\\ref{defn:admissible}) and (\\ref{eq:LpUB})\n(see \\cite[Proposition 4.8]{PX2}).\n\nLemma~\\ref{lem:Ker-n} (i) and (\\ref{eq:LpUB}) are instrumental\nin relating weighted norms of polynomials.\n\n\n\\begin{proposition}\\label{Nikolski}\nFor $0 < q \\le p \\le \\infty$ and $g \\in \\Pi_n$, $n\\ge 1$,\n\\begin{equation}\\label{norm-relation}\n\\|g\\|_p \\le cn^{(d+2\\mu)(1\/q-1\/p)}\\|g\\|_q,\n\\end{equation}\nand for any $\\gamma\\in{\\mathbb R}$\n\\begin{equation}\\label{norm-relation2}\n\\|{W_\\mu}(n;\\cdot)^\\gamma g(\\cdot)\\|_p \\le\ncn^{d(1\/q-1\/p)}\\|{W_\\mu}(n;\\cdot)^{\\gamma+1\/p-1\/q} g(\\cdot)\\|_q.\n\\end{equation}\n\\end{proposition}\n\nThe proof of this proposition is quite similar to the proof of Proposition 2.6\nfrom~\\cite{KPX}; for completeness it is given in \\S\\ref{proofsA}.\n\n\\subsection{Maximal operator}\\label{Max-iequal}\n\nWe denote by $B_\\xi(r)$ the ball centered at $\\xi\\in B^d$ of\nradius $r>0$ with respect to the distance $d(\\cdot,\\cdot)$ on\n$B^d$, i.e.\n\\begin{equation}\\label{def-ball}\nB_\\xi(r)= \\{x\\in B^d: d(x, \\xi)< r\\}.\n\\end{equation}\nIt is straightforward to show that (see \\cite[Lemma 5.3]{PX2})\n\\begin{equation}\\label{ball-measure}\n|B_\\xi(r)|\\sim r^d\\sqrt{1-|\\xi|^2}\n\\end{equation}\nand\n\\begin{equation}\\label{ball-size}\nm(B_\\xi(r)):=\\int_{B_\\xi(r)}{w_\\mu}(x)\\, dx\n\\sim r^d(r+\\sqrt{1-|\\xi|^2})^{2\\mu}\n\\sim r^d(r+ d(\\xi,\\partial B^d))^{2\\mu},\n\\end{equation}\nwhere $\\partial B^d$ is the boundary of $B^d$,\ni.e. the unit sphere in $\\mathbb{R}^d$.\n\n\nThe maximal operator ${\\mathcal M}_t$ ($t>0$) is defined by\n\\begin{equation}\\label{def: max-op}\n{\\mathcal M}_tf(x):=\\sup_{B\\ni x}\\left(\\frac1{m(B)}\\int_B|f(y)|^t{w_\\mu}(y)\\, dy\\right)^{1\/t},\n\\quad x\\in B^d,\n\\end{equation}\nwhere the sup is over all balls  $B\\subset B^d$ (with respect to $d(\\cdot, \\cdot)$)\ncontaining~$x$.\n\nIt follows by \\eqref{ball-size} that the measure\n$m(E):=\\int_E{w_\\mu}(x)\\, dx$\nis a doubling measure on $B^d$,\ni.e. for $\\xi\\in B^d$ and $0<r\\le \\pi$\n\\begin{equation}\\label{eq:doub-vol}\nm(B_\\xi(2r))\\le c m(B_\\xi(r)).\n\\end{equation}\nConsequently, the general theory of maximal operators applies\nand the Fefferman-Stein vector-valued maximal inequality is valid (see \\cite{Stein}):\nIf $0<p<\\infty, 0<q\\le\\infty$, and\n$0<t<\\min\\{p,q\\}$ then for any sequence of functions\n$\\{f_\\nu\\}_\\nu$ on $B^d$\n\\begin{equation}\\label{max-ineq}\n\\nnorm{\\Bigl(\\sum_{\\nu=1}^\\infty|{\\mathcal M}_tf_\\nu(\\cdot)|^q\\Bigr)^{1\/q}}_{p}\n\\le \\nnorm{\\Bigl(\\sum_{\\nu=1}^\\infty|\nf_\\nu(\\cdot)|^q\\Bigr)^{1\/q}}_{p}.\n\\end{equation}\n\nWe need to estimate ${\\mathcal M}_t {\\mathbbm 1}_{B}$ for an arbitrary ball $B\\subset B^d$.\n\n\\begin{lemma}\\label{lem:B-maximal}\nLet $\\xi \\in B^d$ and $0<r\\le \\pi$.\nThen for $x\\in B^d$\n\\begin{equation}\\label{B-max1}\n({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi(r)})(x)\n\\sim \\Big(1+\\frac{d(\\xi , x)}{r}\\Big)^{-d\/t}\n     \\Big(1+\\frac{d(\\xi , x)}{r+d(\\xi , \\partial B^d)}\\Big)^{-2\\mu\/t}\n\\end{equation}\nand hence\n\\begin{equation}\\label{B-max3}\nc'\\Big(1+\\frac{d(\\xi , x)}{r}\\Big)^{-(2\\mu+d)\/t}\n\\le ({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)\n\\le c\\Big(1+\\frac{d(\\xi , x)}{r}\\Big)^{-d\/t}.\n\\end{equation}\nHere the constants depend only on $d$, $\\mu$, and $t$.\n\\end{lemma}\n\n\\begin{proof}It is easy to see that\n$$\n({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)\n=\\sup_{B\\ni x}\\left(\\frac{m(B\\cap B_{\\xi}(r))}{m(B)}\\right)^{1\/t},\n\\quad x\\in B^d,\n$$\nwhere the sup is taken over all the balls $B\\subset B^d$\n(with respect to $d(\\cdot, \\cdot)$) containing~$x$.\nThis immediately leads to $({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)\\sim 1$ if $d(x, \\xi)\\le 2r$\nand hence (\\ref{B-max1}) holds in this case.\n\nSuppose $d(\\xi,x)>2r$. Then evidently\n$$\n({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)^{1\/t} \\ge \\left(\\frac{m(B_{\\xi}(r))}{m(B_{\\xi}(d(x,\\xi)))}\\right)^{1\/t}.\n$$\nFor the other direction, suppose $B_z(r^*)\\subset B^d$ is the smallest ball such that\n$x\\in \\overline{B_z(r^*)}$ and $\\overline{B_z(r^*)}\\cap \\overline{B_\\xi(r)}\\ne \\emptyset$.\nA simple application of the triangle inequality shows that\n$B_\\xi(d(\\xi,x))\\subset B_z(5r^*)$.\nThus using (\\ref{eq:doub-vol})\n$$\n({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)\\le \\left(\\frac{m(  B_{\\xi}(r))}{m(B_z(r^*))}\\right)^{1\/t}\n\\le c \\left(\\frac{m(B_{\\xi}(r))}{m(B_{\\xi}(d(x,\\xi))}\\right)^{1\/t}.\n$$\nTherefore, using \\eqref{ball-size}\n$$\n({\\mathcal M}_t {\\mathbbm 1}_{B_\\xi (r)})(x)\n\\sim \\left(\\frac{m(B_{\\xi}(r))}{m(B_{\\xi}(d(x,\\xi))}\\right)^{1\/t}\n\\sim \\left(\\frac{r^d(r+d(\\xi, \\partial B^d))^{2\\mu}}\n                {d(x,\\xi)^d(d(x,\\xi)+d(\\xi, \\partial B^d))^{2\\mu} }\\right)^{1\/t},\n$$\nwhich implies (\\ref{B-max1}) since $d(\\xi,x)>2r$. Estimate\n(\\ref{B-max3}) is immediate from  (\\ref{B-max1}).\n\\end{proof}\n\n\n\n\n\\subsection{Distributions on \\boldmath $B^d$}\\label{distributions}\n\n\nTo define distributions on $B^d$ we shall use as\ntest functions the set $\\mathcal{D}:=C^{\\infty}(B^d)$ of all infinitely continuously differentiable\ncomplex valued functions on $B^d$ such that\n\\begin{equation}\\label{norms}\n\\|\\phi\\|_{W_\\infty^k}\n:=\\sum_{|\\alpha|\\le k}\\|\\partial^\\alpha \\phi\\|_\\infty<\\infty\n\\quad\\mbox{for } k=0, 1, \\dots.\n\\end{equation}\nWe assume that the topology in $\\mathcal{D}$ is defined by these norms.\n\nEvidently all polynomials belong to $\\mathcal{D}$.   More importantly, the space $\\mathcal{D}$\nof test functions $\\phi$ can be completely characterized by their orthogonal\npolynomial expansions. Denote\n\\begin{equation}\\label{D-norms}\n\\mathcal{N}_k(\\phi):=\\sup_{n\\ge 0}\\, (n+1)^k \\|\\operatorname{Proj}_n \\phi\\|_2.\n\\end{equation}\n\n\n\\begin{lemma}\\label{lem:char-D}\n$(a)$\n$\\phi\\in\\mathcal{D}$ if and only if\n$\\|\\operatorname{Proj}_n \\phi\\|_2 =\\mathcal{O}(n^{-k})$ for all $k$.\n\n$(b)$\nFor each $\\phi\\in \\mathcal{D}$,\n$\n\\phi=\\sum_{n=0}^\\infty \\operatorname{Proj}_n \\phi,\n$\nwhere the convergence is in the topology of $\\mathcal{D}$.\n\n$(c)$\nThe topology in $\\mathcal{D}$ can be equivalently defined by the norms\n$\\mathcal{N}_k(\\cdot)$, $k=0, 1, \\dots$.\n\\end{lemma}\n\n\n\\begin{proof}Let $\\phi\\in\\mathcal{D}$. Assume that $Q_{n-1}\\in\\Pi_{n-1}$ ($n\\ge\n1$) is the polynomial of best $L_2({w_\\mu})$-approximation to $\\phi$,\ni.e. $\\|\\phi-Q_{n-1}\\|_2=E_{n-1}(\\phi)_2$. Since ${\\mathsf {P}}_n(x, \\cdot)$\nis orthogonal to $\\Pi_{n-1}$,\n$$\n|\\operatorname{Proj}_n \\phi (x)| = |\\langle \\phi, {\\mathsf {P}}_n(x,\\cdot) \\rangle|\n=|\\langle \\phi-Q_{n-1}, {\\mathsf {P}}_n(x,\\cdot)\\rangle|\n\\le E_{n-1}(\\phi)_2 {\\mathsf {P}}_n(x,x)^{1\/2}.\n$$\nBy the Jackson type estimate from \\cite{X05}, for any $k\\ge 1$,\n$$\nE_n(\\phi)_2\n\\le c_k n^{-2k} \\|D_\\mu^k\\phi\\|_{2}\n\\le c n^{-2k} \\|D_\\mu^k\\phi\\|_\\infty\n\\le c n^{-2k}\\sum_{|\\alpha|\\le 2k}\\|\\partial^\\alpha \\phi\\|_\\infty\n= c n^{-2k}\\|\\phi\\|_{W_\\infty^{2k}}.\n$$\nHere $D_\\mu$ is the differential operator from \\eqref{def-diff-oper}.\nIt is easy to see that\n$$\n\\|{\\mathsf {P}}_n(x,x)^{1\/2}\\|_2^2 = \\binom{n+d-1}{n} \\sim n^{d-1}.\n$$\nAll of the above leads to\n$$\n\\|\\operatorname{Proj}_n \\phi\\|_2  \\le c_k n^{-2 k+ (d-1)\/2} \\|\\phi\\|_{W_\\infty^{2k}},\n\\quad n\\ge 1,\n\\quad \\mbox{for any } k\\ge 1.\n$$\nTherefore, for any $m\\ge 0$\n$$\n\\mathcal{N}_m(\\phi) \\le c\\|\\phi\\|_{W_\\infty^{2k}}\n\\quad\\mbox{if \\; $k\\ge m\/2+(d-1)\/4$.}\n$$\n\nIn the other direction, by Markov's inequality (see \\cite{Kellogg}) and \\eqref{norm-relation},\nit follows that\n$$\n\\|\\partial^\\alpha \\operatorname{Proj}_n \\phi \\|_\\infty\n\\le n^{2|\\alpha|}\\|\\operatorname{Proj}_n \\phi\\|_\\infty\n\\le cn^{2|\\alpha|+d\/2+\\mu}\\|\\operatorname{Proj}_n \\phi\\|_2.\n$$\nConsequently, if $\\|\\operatorname{Proj}_n \\phi\\|_2 =\\mathcal{O}(n^{-k})$ for all $k$,\nthen\n$\n\\partial^\\alpha \\phi=\\sum_{n=0}^\\infty  \\partial^\\alpha \\operatorname{Proj}_n \\phi\n$\nfor all multi-indices $\\alpha$ with the series converging uniformly and\n$$\n\\|\\phi\\|_{W_\\infty^k}\n\\le c\\sum_{|\\alpha|\\le k}\\sum_{n=0}^\\infty n^{2 |\\alpha|+d\/2+\\mu} \\|\\operatorname{Proj}_n \\phi \\|_2\n\\le c \\mathcal{N}_{m}(\\phi),\n\\quad m\\ge 2k+d\/2+\\mu+2.\n$$\nThis completes the proof of the lemma.\n\\end{proof}\n\n\\medskip\n\nThe space $\\mathcal{D}':=\\mathcal{D}'(B^d)$ of distributions on $B^d$ is defined\nas the set of all continuous linear functionals on $\\mathcal{D}$.\nThe pairing of $f\\in \\mathcal{D}'$ and $\\phi\\in\\mathcal{D}$ will be denoted by\n$\\langle f, \\phi \\rangle:= f(\\overline{\\phi})$, which will be shown\nto be consistent with the inner product\n$\n\\langle f, g \\rangle :=\\int_{B^d} f(x)\\overline{g(x)} {w_\\mu}(x)dx\n$\nin $L_2({w_\\mu})$.\n\nWe now extend the definition of the nonstandard ``convolution\" from\n\\eqref{convolution} to distributions.\n\n\n\\begin{definition}\\label{Def-conv}\nLet $f \\in \\mathcal{D}'$ and assume that $\\Phi: B^d\\times B^d \\mapsto {\\mathbb C}$ is such that\n$\\Phi(x, \\cdot)\\in \\mathcal{D}$ for all $x\\in B^d$.\nWe define\n$$\n( \\Phi \\ast f)(x) := \\langle f, \\overline{\\Phi(x,\\cdot)}\\rangle,\n$$\nwhere on the right $f$ acts on $\\overline{\\Phi(x, y)}$ as a function of $y$.\n\\end{definition}\n\nFor later use we next record some simple properties of this ``convolution\".\n\n\n\\begin{lemma}\\label{lem:prop-conv}\n$(i)$ If $f\\in \\mathcal{D}'$ and $\\Phi(\\cdot,\\cdot)\\in C^\\infty(B^d\\times B^d)$, then\n$\\Phi*f\\in \\mathcal{D}$,\nand in particular ${\\mathsf {P}}_n*f \\in V_n$.\nWe define $\\operatorname{Proj}_n f:={\\mathsf {P}}_n*f$.\n\n$(ii)$\nIf $f\\in \\mathcal{D}'$ and $\\Phi(\\cdot,\\cdot)\\in C^\\infty(B^d\\times B^d)$, then\n$$\n\\langle \\Phi*f, \\phi \\rangle = \\langle f, \\overline{\\Phi}*\\phi\\rangle,\n\\quad \\phi\\in\\mathcal{D}.\n$$\n\n$(iii)$\nLet $\\Phi(\\cdot,\\cdot), \\Psi(\\cdot, \\cdot)\\in C^\\infty(B^d\\times B^d)$,\nand $\\Phi(x, y)=\\Phi(y, x)$ and $\\Psi(x, y)=\\Psi(y, x)$ for $x, y\\in B^d$.\nThen for any $f\\in\\mathcal{D}'$ and $x\\in B^d$\n$$\n\\Psi*\\overline{\\Phi}*f(x)=\\langle\\Psi(x, \\cdot), \\Phi(\\cdot, \\cdot)\\rangle*f.\n$$\n\\end{lemma}\n\nThe proof of this lemma is standard and will be omitted.\n\nWe next give the representation of distributions from $\\mathcal{D}'$\nin terms of orthogonal polynomials on $B^d$.\n\n\n\\begin{lemma}\\label{lem:dec-D1}\n$(a)$\nA linear functional $f\\in\\mathcal{D}'$\nif and only if there exists $k\\ge 0$ such that\n\\begin{equation}\\label{D1}\n|\\langle f, \\phi\\rangle|\\le c_k\\mathcal{N}_k(\\phi)\n\\quad \\mbox{for all } \\quad \\phi \\in \\mathcal{D},\n\\end{equation}\nHence, for $f \\in \\mathcal{D}'$ there exits $k\\ge 0$ such that\n\\begin{equation}\\label{D2}\n\\|\\operatorname{Proj}_n f\\|_2= \\|{\\mathsf {P}}_n*f\\|_2\n\\le c_k(n+1)^k, \\quad n=0, 1, \\dots.\n\\end{equation}\n\n$(b)$\nEvery $f\\in\\mathcal{D}'$ has the representation\n$\nf=\\sum_{n=0}^\\infty \\operatorname{Proj}_n f\n$\nin distributional sense, i.e.\n\\begin{equation}\\label{D3}\n\\langle f, \\phi\\rangle\n=\\sum_{n=0}^\\infty  \\langle \\operatorname{Proj}_n f, \\phi  \\rangle\n=\\sum_{n=0}^\\infty  \\langle \\operatorname{Proj}_n f, \\operatorname{Proj}_n\\phi  \\rangle\n\\quad\\mbox{for all}\n\\quad \\phi\\in\\mathcal{D},\n\\end{equation}\nwhere the series converges absolutely.\n\\end{lemma}\n\n\\noindent\n\\begin{proof}(a) Part (a) follows immediately by the fact that the\ntopology in $\\mathcal{D}$ can be defined by the norms $\\mathcal{N}_k(\\cdot)$\ndefined in \\eqref{D-norms}.\n\n(b) Using Lemma~\\ref{lem:char-D} (b) we get for $\\phi\\in\\mathcal{D}$,\n$$\n\\langle f, \\phi \\rangle\n=\\lim_{N\\to\\infty}\\Big\\langle f, \\sum_{n=0}^N \\operatorname{Proj}_n \\phi \\Big\\rangle\n=\\lim_{N\\to\\infty}\\sum_{n=0}^N  \\langle f, \\operatorname{Proj}_n \\phi\\rangle\n=\\sum_{n=0}^\\infty \\langle \\operatorname{Proj}_n f, \\operatorname{Proj}_n \\phi \\rangle,\n$$\nwhere the last equality is justified by using (\\ref{D2}) and the\nrapid decay of $\\|\\operatorname{Proj}_n \\phi\\|_2$.\n\\end{proof}\n\n\n\n\n\\subsection{Cubature formula and subdivision of \\boldmath $B^d$}\n\\label{cubature}\nFor the construction of our building blocks (needlets)\nwe shall  utilize the positive cubature formula given in \\cite{PX2}. This formula is based\non almost equally distributed knots on $B^d$ with respect to the distance $d(\\cdot,\\cdot)$.\n\n\n\n\\begin{definition}\\label{def.eq-dist-p}\nWe say that a set ${\\mathcal X}_{\\varepsilon} \\subset B^d$, along with an associated\npartition ${\\mathcal R}_{\\varepsilon}$ of $B^d$ consisting of measurable subsets of $B^d$,\nis a {\\em set of almost uniformly ${\\varepsilon}$-distributed points} on $B^d$\nif\n\\begin{enumerate}\n\\item[(i)]\n$B^d = \\bigcup_{R \\in {\\mathcal R}_{\\varepsilon}} R$ and the sets in ${\\mathcal R}_{\\varepsilon}$\ndo not overlap\n$(R_1^\\circ \\cap R_2^\\circ = \\emptyset$ if $R_1 \\ne R_2)$.\n\\item[(ii)]\nFor each $R \\in {\\mathcal R}_{\\varepsilon}$ there is a unique $\\xi \\in {\\mathcal X}_{\\varepsilon}$\nsuch that\n$\nB_\\xi(c^*{\\varepsilon}) \\subset R \\subset B_\\xi({\\varepsilon}).\n$\n\\end{enumerate}\nHence\n$\\# {\\mathcal X}_{\\varepsilon} = \\# {\\mathcal R}_{\\varepsilon} \\le c^{**}{\\varepsilon}^{-d}$.\nHere the constant $c^*>0$, depending only on $d$, is fixed but sufficiently small, so that the existence\nof sets of almost uniformly ${\\varepsilon}$-distributed points on $B^d$\nis guaranteed $($see the next lemma$)$.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:points} \\cite{PX2}\nFor a sufficiently small constant $c^*>0$, depending only on $d$,\nand an arbitrary $0<{\\varepsilon} \\le \\pi$ there exists\na set ${\\mathcal X}_{\\varepsilon} \\subset B^d$  of almost uniformly ${\\varepsilon}$-distributed points on $B^d$,\nwhere the associated partition ${\\mathcal R}_{\\varepsilon}$ of $B^d$\nconsists of projections of spherical simplices.\n\\end{lemma}\n\nAn important element in the construction of needlets will be\nthe cubature formula given in \\cite[Corollary 5.10]{PX2}:\n\n\n\\begin{proposition}\\label{prop:cubature}\nThere exists a constant $c^\\diamond >0$ {\\rm(}depending only on\n$d${\\rm)} and a sequence $\\{{\\mathcal X}_j\\}_{j=0}^\\infty$ of almost\nuniformly ${\\varepsilon}_j$-distributed points on $B^d$ with\n${\\varepsilon}_j:=c^\\diamond 2^{-j}$, and there exist positive coefficients\n$\\{\\lambda_\\xi\\}_{\\xi \\in {\\mathcal X}_j}$ such that the cubature formula\n\\begin{equation}\\label{eq:cubature}\n\\int_{B^d} f(x){w_\\mu}(x)\\,dx \\sim \\sum_{\\xi \\in {\\mathcal X}_j} \\lambda_\\xi\nf(\\xi)\n\\end{equation}\nis exact for all polynomials of degree $\\le 2^{j+2}$. In addition,\n\\begin{equation}\\label{eq:coeff}\n\\lambda_\\xi \\sim 2^{-jd}{W_\\mu}(2^j;\\xi)\n\\sim m(B_\\xi(2^{-j}))\n\\end{equation}\nwith constants of equivalence depending only on $\\mu$ and $d$.\n\\end{proposition}\n\n\nIt follows from above that\n\\begin{equation}\\label{eq:mR}\nm(R_\\xi)\\sim 2^{-jd}{W_\\mu}(2^j;\\xi)\\sim \\lambda_\\xi,\n\\quad \\xi \\in {\\mathcal X}_j,\n\\end{equation}\nwhile\n\\begin{equation}\\label{mes-R-xi}\n|R_\\xi|\\sim 2^{-jd}(\\sqrt{1-|\\xi|^2}+2^{-j}),\n\\quad \\xi \\in {\\mathcal X}_j.\n\\end{equation}\n\n\n\\section{Localized building blocks (Needlets) on \\boldmath $B^d$}\\label{def-needlets}\n\\setcounter{equation}{0}\n\nWe utilize the ideas from \\cite{NPW2, KPX} in constructing a pair of sequences\nof ``analysis\" and ``synthesis\" needlets on $B^d$.\nLet $\\widehat a,\\widehat b$ satisfy the conditions\n\\begin{align}\n&\\quad \\widehat a,\\widehat b\\in C^\\infty[0, \\infty),\n\\quad \\operatorname{supp} \\widehat a,\\widehat b \\subset [1\/2, 2], \\label{ha-hb1}\\\\\n&\\quad  |\\widehat a(t)|, |\\widehat b(t)| >c>0, \\quad \\text{if  } t \\in [3\/5, 5\/3],\\label{ha-hb2}\\\\\n&\\quad \\overline{\\widehat a(t)}\\, \\widehat b(t) + \\overline{\\widehat a(2t)}\\,\\widehat b(2t)  =1,\\label{ha-hb3}\n\\quad\\text{if } t \\in [1\/2, 1].\n\\end{align}\nHence,\n\\begin{equation}\\label{part-unity}\n\\sum_{\\nu=0}^\\infty \\overline{\\widehat a(2^{-\\nu}t )}\\, \\widehat b(2^{-\\nu}t)=1,\n\\quad t\\in [1,\\infty).\n\\end{equation}\nIt is easy to see that\nif $\\widehat a$ satisfies (\\ref{ha-hb1})-(\\ref{ha-hb2}), then\nthere exists $\\widehat b$ satisfying (\\ref{ha-hb1})-(\\ref{ha-hb2})\nsuch that (\\ref{ha-hb3}) is valid (see e.g. \\cite{FJ2}).\n\n\\smallskip\n\nLet $\\widehat a$, $\\widehat b$ satisfy (\\ref{ha-hb1})-(\\ref{ha-hb3}).\nWe define\n$\\Phi_0(x, y)= \\Psi_0(x, y):=1$,\n\\begin{equation}\\label{def.Phi-j}\n\\Phi_j(x, y) := \\sum_{\\nu=0}^\\infty\n\\widehat a\\Big(\\frac{\\nu}{2^{j-1}}\\Big) {\\mathsf {P}}_\\nu(x,y), \\quad j\\ge 1,\n\\end{equation}\n\\begin{equation}\\label{def-Psi-j}\n\\Psi_j(x, y) := \\sum_{\\nu=0}^\\infty\n\\widehat b\\Big(\\frac{\\nu}{2^{j-1}}\\Big) {\\mathsf {P}}_\\nu(x,y), \\quad j\\ge 1.\n\\end{equation}\n\nAssume that ${\\mathcal X}_j$ is the set of knots and $\\lambda_\\xi$'s are the coefficients of\nthe  cubature formula \\eqref{eq:cubature}.\nWe define the $j$th level {\\em needlets} by\n\\begin{equation}\\label{def-needlets1}\n{\\varphi}_\\xi(x) := \\lambda_\\xi^{1\/2}\\Phi_j(x, \\xi)\n\\quad\\mbox{and}\\quad\n\\psi_\\xi(x) := \\lambda_\\xi^{1\/2}\\Psi_j(x, \\xi),\n\\qquad \\xi \\in {\\mathcal X}_j.\n\\end{equation}\nNotice that for $\\xi\\in{\\mathcal X}_1$, we have\n${\\varphi}_\\xi(x)=\\widehat a(1){\\mathsf {P}}_1(x, \\xi)$ and $\\psi_\\xi(x)=\\widehat b(1){\\mathsf {P}}_1(x, \\xi)$,\nbut ${\\mathsf {P}}_1(\\cdot, \\xi)\\equiv 0$ if and only if $\\xi=0$.\nSo, to prevent $\\psi_\\xi\\equiv 0$ and $\\psi_\\xi\\equiv 0$ for $\\xi\\in{\\mathcal X}_1$,\nwe (may) assume that $0\\notin {\\mathcal X}_1$.\n\nWe set ${\\mathcal X} := \\cup_{j = 0}^\\infty {\\mathcal X}_j$,\nwhere equal points from different levels ${\\mathcal X}_j$ are considered\nas distinct elements of ${\\mathcal X}$, so that ${\\mathcal X}$ can be used as an index set.\nWe define the {\\em analysis} and {\\em synthesis} needlet systems\n$\\Phi$ and $\\Psi$ by\n\\begin{equation}\\label{def-needlets2}\n\\Phi:=\\{{\\varphi}_\\xi\\}_{\\xi\\in{\\mathcal X}}, \\quad \\Psi:=\\{\\psi_\\xi\\}_{\\xi\\in{\\mathcal X}}.\n\\end{equation}\n\n\nEstimate (\\ref{eq:main-estI}) yields the rapid decay of needlets,\nnamely, for $x\\in B^d$,\n\\begin{equation}\\label{local-Needlets2}\n|\\Phi_j(\\xi, x)|, |\\Psi_j(\\xi, x)| \\le \\frac{c_k 2^{jd}}\n{\\sqrt{{W_\\mu}(2^j; \\xi)}\\sqrt{{W_\\mu}(2^j; x)}(1+2^{j}d(\\xi, x))^k} \\quad\n\\forall k,\n\\end{equation}\nand hence\n\\begin{equation}\\label{local-Needlets21}\n|{\\varphi}_\\xi(x)|, |\\psi_\\xi(x)|\n\\le \\frac{c_k 2^{jd\/2}}{\\sqrt{{W_\\mu}(2^j;x)}(1+2^{jd}d(\\xi, x))^k}\n\\quad \\forall k.\n\\end{equation}\nNote that on account of \\eqref{eq:useful} $x$ in the term $\\sqrt{{W_\\mu}(2^j; x)}$\nin \\eqref{local-Needlets21} can be replaced by~$\\xi$.\n\nThe needlets are {\\rm Lip 1} functions in the following sense:\nLet $\\xi\\in{\\mathcal X}_j$, $j\\ge 0$, $c^*>0$, and $\\omega\\in B^d$. Then for each $x\\in B_\\omega(c^*2^{-j})$\n\\begin{equation}\\label{Lip-Needlets}\n|{\\varphi}_\\xi(x)-{\\varphi}_\\xi(\\omega)|, |\\psi_\\xi(x)-\\psi_\\xi(\\omega)|\n\\le \\frac{c_k 2^{j(d\/2+1)}d(\\omega, x)}{\\sqrt{{W_\\mu}(2^j;\\xi)}(1+2^{jd}d(\\xi, \\omega))^k}\n\\quad \\forall k.\n\\end{equation}\nThis estimate follows readily from \\eqref{Lip}.\n\nWe shall need estimates of the norms of the needlets.\nBy \\eqref{eq:LpUB}, \\eqref{eq:est-Lp-norm}, and since $0\\notin {\\mathcal X}_1$,\nwe have for $0<p\\le\\infty$,\n\\begin{equation}\\label{norm-Needlets}\n\\|{\\varphi}_\\xi\\|_{p} \\sim \\|\\psi_\\xi\\|_{p} \\sim \\|{\\tilde{\\mathbbm 1}}_{R_\\xi}\\|_{p}\n\\sim \\Big(\\frac{2^{jd}}{{W_\\mu}(2^j; \\xi)}\\Big)^{1\/2-1\/p},\n\\quad \\xi\\in{\\mathcal X}_j.\n\\end{equation}\nFurthermore, there exist constants $c^*, c >0$ such that\n\\begin{equation}\\label{norm-Needlets2}\n\\|{\\varphi}_\\xi\\|_{L_\\infty(B_\\xi(c^*2^{-j}))},\\;\n\\|\\psi_\\xi\\|_{L_\\infty(B_\\xi(c^*2^{-j}))}\n\\ge c \\Big(\\frac{2^{jd}}{{W_\\mu}(2^j; \\xi)}\\Big)^{1\/2},\n\\quad \\xi\\in{\\mathcal X}_j.\n\\end{equation}\nThe proof of (\\ref{norm-Needlets2}) is given in \\S\\ref{proofsA}.\nNotice that if $\\widehat a$, $\\widehat b$ are real valued, then Lemma~\\ref{L2-lower-bound} bellow\nyields\n$$\n|{\\varphi}_\\xi(\\xi)|, |\\psi_\\xi(\\xi)|\\ge c \\Big(\\frac{2^{jd}}{{W_\\mu}(2^j; \\xi)}\\Big)^{1\/2},\n\\quad \\xi\\in{\\mathcal X}_j.\n$$\n\n\n\n\n\nOur first step in implementing needlets is to establish needlet decompositions\nof $\\mathcal{D}'$ and ${L_p({w_\\mu})}$.\n\n\n\\begin{proposition}\\label{prop:needlet-rep}\n$(a)$ For any $f \\in \\mathcal{D}'$,\n\\begin{equation}\\label{Needle-rep}\nf = \\sum_{j=0}^\\infty\n\\Psi_j*\\overline{\\Phi}_j*f\n\\quad\\mbox{in} ~ \\mathcal{D}'\n\\end{equation}\nand\n\\begin{equation}\\label{needlet-rep1}\nf = \\sum_{\\xi \\in {\\mathcal X}}\n\\langle f, {\\varphi}_\\xi\\rangle \\psi_\\xi\n\\quad\\mbox{in} ~ \\mathcal{D}'.\n\\end{equation}\n\n$(b)$ For $f \\in {L_p({w_\\mu})}$, $1\\le p \\le \\infty$,\n$(\\ref{Needle-rep})-(\\ref{needlet-rep1})$ hold in ${L_p({w_\\mu})}$.\nMoreover, if $1 < p < \\infty$, then the convergence in\n$(\\ref{Needle-rep})-(\\ref{needlet-rep1})$ is unconditional.\n\\end{proposition}\n\n\n\\begin{proof}\nBy Definition~\\ref{Def-conv} and\n\\eqref{def.Phi-j} we have, for $f\\in\\mathcal{D}'$,\n\\begin{equation}\\label{Phi-f}\n\\overline{\\Phi}*f = \\sum_{\\nu=0}^{2^j}\\overline{\\widehat a\\Big(\\frac{\\nu}{2^{j-1}}\\Big)}{\\mathsf {P}}_\\nu*f\n\\end{equation}\nand using Lemma~\\ref{lem:prop-conv} and that ${\\mathsf {P}}_\\nu*{\\mathsf {P}}_\\nu(\\cdot, y)={\\mathsf {P}}_\\nu(\\cdot, y)$\n\\begin{equation}\\label{Psi-Phi-f}\n\\Psi*\\overline{\\Phi}*f\n= \\sum_{\\nu=0}^{2^j}\\overline{\\widehat a\\Big(\\frac{\\nu}{2^{j-1}}\\Big)}\\widehat b\\Big(\\frac{\\nu}{2^{j-1}}\\Big){\\mathsf {P}}_\\nu*f.\n\\end{equation}\nThen \\eqref{Needle-rep} follows from the above, \\eqref{part-unity}, and Lemma~\\ref{lem:dec-D1}.\n\nNote that $\\Psi_j(x, y)\\overline{\\Phi(y, z)}$ belongs to $\\Pi_{2^{j+1}-1}$\nas a function of $y$ and, therefore, employing the cubature formula from\nProposition~\\ref{prop:cubature} we get\n\\begin{align*}\n\\Psi_j*\\overline{\\Phi_j(\\cdot, z)}\n&=\\int_{B^d}\\Psi_j(x, y)\\overline{\\Phi(y, z)}{w_\\mu}(y)dy\\\\\n&=\\sum_{\\xi\\in{\\mathcal X}_j}\\lambda_\\xi \\Psi_j(x, \\xi)\\overline{\\Phi(\\xi, z)}\n=\\sum_{\\xi\\in{\\mathcal X}_j}\\psi_\\xi(x)\\overline{{\\varphi}_\\xi(z)},\n\\end{align*}\nwhich leads to\n$$\n\\Psi_j*\\overline{\\Phi}_j*f=\\sum_{\\xi\\in{\\mathcal X}_j}\\langle f, {\\varphi}_\\xi\\rangle \\psi_\\xi.\n$$\nCombining this with \\eqref{Needle-rep} yields \\eqref{needlet-rep1}.\n\nThe convergence of \\eqref{Needle-rep} and \\eqref{needlet-rep1} in\n${L_p({w_\\mu})}$ for $f\\in {L_p({w_\\mu})}$ follows in a similar fashion (see also\n\\cite[Proposion 3.1]{KPX}). The unconditional convergence in\n${L_p({w_\\mu})}$, $1<p<\\infty$, follows by\nTheorem~\\ref{thm:Fnorm-equivalence} and\nProposition~\\ref{prop:ident} below.\n\\end{proof\n\n\n\n\\section{Weighted Triebel-Lizorkin  spaces on \\boldmath $B^d$}\\label{Tri-Liz}\n\\setcounter{equation}{0}\n\nFollowing the general idea of using spectral decompositions (see e.g. \\cite{Pee, T1}),\nwe next employ orthogonal polynomials to introduce\nweighted Triebel-Lizorkin spaces on $B^d$.\nTo this end we define a sequence of kernels $\\{\\Phi_j\\}$ by\n\\begin{equation}\\label{def-Phi-j}\n\\Phi_0(x, y) := 1\n\\quad\\mbox{and}\\quad\n\\Phi_j(x, y) := \\sum_{\\nu=0}^\\infty \\widehat a\n\\Big(\\frac{\\nu}{2^{j-1}}\\Big){\\mathsf {P}}_\\nu(x,y), ~~~j\\ge 1,\n\\end{equation}\nwhere $\\{{\\mathsf {P}}_\\nu(x,y)\\}$ are from \\eqref{def-Pn}-\\eqref{compactPn} and\n$\\widehat a$ obeys the conditions\n\\begin{align}\n&\\quad \\widehat a\\in C^\\infty[0, \\infty),\n\\quad \\operatorname{supp} \\, \\widehat a \\subset [1\/2, 2], \\label{ha1}\\\\\n&\\quad  |\\widehat a(t)|>c>0, \\quad \\text{if } t \\in [3\/5, 5\/3].\\label{ha2}\n\\end{align}\n\n\n\n\\begin{definition}\nLet  $s, {\\rho} \\in \\mathbb{R}$, $0<p<\\infty$, and $0<q\\le\\infty$. Then the\nweighted Triebel-Lizorkin space ${F_{pq}^{s{\\rho}}}:={F_{pq}^{s{\\rho}}}({w_\\mu})$ is defined as\nthe set of all $f\\in \\mathcal{D}'$ such that\n\\begin{equation}\\label{Tri-Liz-norm}\n\\|f\\|_{{F_{pq}^{s{\\rho}}}}:=\\Big\\|\\Big(\\sum_{j=0}^{\\infty}\n\\Big[2^{sj}{W_\\mu}(2^j;\\cdot)^{-{\\rho}\/d}|\\Phi_j*f(\\cdot)|\\Big]^q\\Big)^{1\/q}\\Big\\|_{{p}} <\\infty\n\\end{equation}\nwith the usual modification when $q=\\infty$.\n\\end{definition}\n\n\nObserve that the above definition is independent of the choice\nof $\\widehat a$ as long as it satisfies\n(\\ref{ha1})-(\\ref{ha2}) (see Theorem~\\ref{thm:Fnorm-equivalence} below).\n\n\n\\begin{proposition}\\label{Fsrpq-embedding}\nFor all $s,{\\rho} \\in \\mathbb{R}$, $0<p<\\infty$, and $0<q\\le\\infty$,\n${F_{pq}^{s{\\rho}}}$ is a quasi-Banach space which is continuously embedded in $\\mathcal{D}'$.\n\\end{proposition}\n\n\\begin{proof}The completeness of the space ${F_{pq}^{s{\\rho}}}$\nfollows easily (see e.g. \\cite{T1}, p. 49) by the continuous\nembedding of ${F_{pq}^{s{\\rho}}}$ in $\\mathcal{D}'$, which we establish next.\n\nLet $\\{\\Phi_j\\}$ be the kernels from the definition of ${F_{pq}^{s{\\rho}}}$ with $\\widehat a$ obeying\n(\\ref{ha1})-(\\ref{ha2}) that are the same as (\\ref{ha-hb1})-(\\ref{ha-hb2}).\nAs already indicated there exists a function $\\widehat b$ satisfying (\\ref{ha-hb1})-(\\ref{ha-hb3}).\nWe use this function to define $\\{\\Psi_j\\}$ as in (\\ref{def-Psi-j}).\nThen by Proposition~\\ref{prop:needlet-rep}\n$f=\\sum_{j=0}^\\infty \\Psi_j*\\Phi_j*f$ in $\\mathcal{D}'$ and hence\n$$\n\\langle f, \\phi\\rangle=\\sum_{j=0}^\\infty \\langle \\Psi_j*\\overline{\\Phi}_j*f, \\phi\\rangle,\n\\quad \\phi\\in\\mathcal{D}.\n$$\nWe now employ \\eqref{Phi-f}-\\eqref{Psi-Phi-f} and the Cauchy-Schwarz inequality\nto obtain, for $j\\ge 2$,\n\\begin{align*}\n&|\\langle \\Psi_j\\ast\\overline{\\Phi}_j\\ast f, \\phi\\rangle|^2\n=\\Big|\\sum_{\\nu=2^{j-2}+1}^{2^j}\n\\overline{\\widehat a\\Big(\\frac{\\nu}{2^{j-1}}\\Big)}\\widehat b\\Big(\\frac{\\nu}{2^{j-1}}\\Big)\n\\langle \\operatorname{Proj}_\\nu f, \\operatorname{Proj}_\\nu \\phi\\rangle\\Big|^2\\\\\n&\\qquad \\le\\sum_{\\nu=2^{j-2}+1}^{2^j}\n\\Big|\\widehat a\\Big(\\frac{\\nu}{2^{j-1}}\\Big)\\Big|^2\\|\\operatorname{Proj}_\\nu f\\|_2^2\n\\sum_{\\nu=2^{j-2}+1}^{2^j}\n\\Big|\\widehat b\\Big(\\frac{\\nu}{2^{j-1}}\\Big)\\Big|^2\\|\\operatorname{Proj}_\\nu \\phi\\|_2^2\\\\\n&\\qquad\\le 2^j\\|\\Phi_j*f\\|_2^2\n\\max_{2^{j-2}< \\nu\\le 2^j}\\|\\operatorname{Proj}_\\nu \\phi\\|_2^2.\n\\end{align*}\nUsing inequality \\eqref{norm-relation} we get\n\\begin{align*}\n\\|\\Phi_j*f\\|_2 \\le c2^{j(d+2\\mu)\/p}\\|\\Phi_j*f\\|_p\n\\le c2^{j((d+2\\mu)\/p+2\\mu|{\\rho}|\/d-s)}\\|2^{sj}{W_\\mu}(2^j;\\cdot)^{-{\\rho}\/d}\\Phi_j*f(\\cdot)\\|_p.\n\\end{align*}\nFrom the above estimates we infer\n\\begin{align*}\n|\\langle \\Psi_j\\ast\\overline{\\Phi}_j\\ast f, \\phi\\rangle|\n\\le c 2^{-j}\\|f\\|_{{F_{pq}^{s{\\rho}}}} 2^{jk} \\max_{2^{j-2}< \\nu\\le 2^j}\\|\\operatorname{Proj}_\\nu f\\|_2\n\\le c 2^{-j}\\|f\\|_{{F_{pq}^{s{\\rho}}}}\\mathcal{N}_k(\\phi)\n\\end{align*}\nfor $k\\ge (d+2\\mu)\/p+2\\mu|{\\rho}|\/d +3\/2-s$.\nA similar estimate trivially holds for $j=0, 1$.\nSumming up we get\n$$\n|\\langle f, \\phi\\rangle| \\le c\\|f\\|_{{F_{pq}^{s{\\rho}}}}\\mathcal{N}_k(\\phi),\n$$\nwhich completes the proof.\n\\end{proof\n\n\\medskip\n\nAs a companion to ${F_{pq}^{s{\\rho}}}$ we now introduce the sequence spaces ${f_{pq}^{s{\\rho}}}$.\nHere we assume that $\\{{\\mathcal X}_j\\}_{j=0}^\\infty$ is a sequence of almost\nuniformly ${\\varepsilon}_j$-distributed points on $B^d$\n$({\\varepsilon}_j:=c^\\diamond 2^{-j})$\nwith associated neighborhoods $\\{R_\\xi\\}_{\\xi\\in{\\mathcal X}_j}$,\ngiven by Proposition~\\ref{prop:cubature}.\nJust as in the definition of needlets in \\S\\ref{def-needlets},\nwe set ${\\mathcal X}:=\\cup_{j\\ge 0} {\\mathcal X}_j$.\n\n\n\\begin{definition}\nSuppose $s, {\\rho} \\in \\mathbb{R}$, $0<p<\\infty$, and $0<q\\le\\infty$. Then~${f_{pq}^{s{\\rho}}}$\nis defined as the space of all complex-valued sequences\n$h:=\\{h_{\\xi}\\}_{\\xi\\in {\\mathcal X}}$ such that\n\\begin{equation}\\label{def-f-space}\n\\norm{h}_{{f_{pq}^{s{\\rho}}}} :=\\nnorm{\\biggl(\\sum_{j=0}^\\infty\n2^{sjq}\\sum_{\\xi \\in {\\mathcal X}_j}\n(|h_{\\xi}|{W_\\mu}(2^j;\\xi)^{-{\\rho}\/d}{\\tilde{\\mathbbm 1}}_{R_\\xi}(\\cdot))^q\\biggr)^{1\/q}}_{{p}} <\\infty\n\\end{equation}\nwith the usual modification for $q=\\infty$. Recall that\n${\\tilde{\\mathbbm 1}}_{R_\\xi}:=m(R_\\xi)^{-1\/2}{\\mathbbm 1}_{R_\\xi}$.\n\\end{definition}\n\n\nIn analogy to the classical case on $\\mathbb{R}^d$ we introduce ``analysis\" and ``synthesis\" operators by\n\\begin{equation}\\label{anal_synth_oprts}\nS_\\varphi: f\\rightarrow \\{\\ip{f, \\varphi_\\xi}\\}_{\\xi \\in {\\mathcal X}}\n\\quad\\text{and}\\quad\nT_\\psi: \\{h_\\xi\\}_{\\xi \\in {\\mathcal X}}\\rightarrow \\sum_{\\xi\\in \\mathcal{X}}h_\\xi\\psi_\\xi.\n\\end{equation}\n\n\nWe now give our main result on weighted Triebel-Lizorkin spaces.\n\n\n\\begin{theorem}\\label{thm:Fnorm-equivalence}\nLet $s, {\\rho}\\in \\mathbb{R}$, $0< p< \\infty$ and $0<q\\le \\infty$. Then the operators\n$S_\\varphi:{F_{pq}^{s{\\rho}}}\\rightarrow{f_{pq}^{s{\\rho}}}$ and $T_\\psi:{f_{pq}^{s{\\rho}}}\\rightarrow {F_{pq}^{s{\\rho}}}$\nare bounded and $T_\\psi\\circ S_\\varphi=Id$ on ${F_{pq}^{s{\\rho}}}$.\nConsequently, $f\\in {F_{pq}^{s{\\rho}}}$ if and only if\n$\\{\\ip{f, \\varphi_\\xi}\\}_{\\xi \\in {\\mathcal X}}\\in {f_{pq}^{s{\\rho}}}$.\nFurthermore,\n\\begin{align}\\label{Fnorm-equivalence-1}\n\\norm{f}_{{F_{pq}^{s{\\rho}}}} &\n\\sim  \\norm{\\{\\ip{f,\\varphi_\\xi}\\}}_{{f_{pq}^{s{\\rho}}}}\n\\sim \\nnorm{\\biggl(\\sum_{j=0}^\\infty 2^{sjq}\n\\sum_{\\xi\\in {\\mathcal X}_j}(|\\ip{f, \\varphi_\\xi}|{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}|\\psi_\\xi(\\cdot)|)^q\\biggr)^{1\/q}}_{{p}}.\n\\end{align}\nIn addition, the definition of ${F_{pq}^{s{\\rho}}}$ is independent of the particular selection of\n$\\widehat a$ satisfying $(\\ref{ha1})$--$(\\ref{ha2})$.\n\\end{theorem}\n\n\nThe proof of this theorem relies on several lemmas whose proofs are given\nin Section \\ref{proofsB}.\nIn the following we assume that $\\{\\Phi_j\\}$ are from the definition\nof weighted Triebel-Lizorkin spaces,\nwhile $\\{{\\varphi}_\\xi\\}_{\\xi\\in{\\mathcal X}}$ and $\\{\\psi_\\xi\\}_{\\xi\\in{\\mathcal X}}$\nare needlet systems defined as in (\\ref{def-needlets1})\nwith no connection between the functions $\\widehat a$'s from (\\ref{def-Phi-j}) and (\\ref{def.Phi-j}).\n\n\n\\begin{lemma}\\label{lem:Phi*psi}\nFor any $k > 0$ there exists a constant $c_k>0$ such that\n\\begin{equation}\\label{Phi*psi}\n|\\Phi_j\\ast \\psi_\\xi(x)| \\le c_k\\frac{2^{jd\/2}}{\\sqrt{ {W_\\mu}(2^j;x)\n}(1+2^jd(x, \\xi))^k}, \\quad \\xi\\in{\\mathcal X}_\\nu, \\quad  j-1\\le \\nu\\le j+1,\n\\end{equation}\nand $\\Phi_j\\ast \\psi_\\xi\\equiv 0$ for $\\xi\\in{\\mathcal X}_\\nu$ if $\\nu\\ge j+2$ or\n$\\nu\\le j-2$. Here ${\\mathcal X}_\\nu:=\\emptyset$ if $\\nu < 0$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:Max-needl}\nFor any $t>0$ and $\\xi\\in {\\mathcal X}_j,$ $j\\ge 0$,\n\\begin{equation}\\label{Max-needl1}\n|{\\varphi}_\\xi(x)|, |\\psi_\\xi(x)| \\le c ({\\mathcal M}_t{\\tilde{\\mathbbm 1}}_{R_\\xi})(x), \\quad\nx\\in B^d, \\quad\\mbox{and}\n\\end{equation}\n\\begin{equation}\\label{Max-needl2}\n{\\tilde{\\mathbbm 1}}_{R_\\xi}(x) \\le c({\\mathcal M}_t {\\varphi}_\\xi)(x), c({\\mathcal M}_t \\psi_\\xi)(x),\n\\quad x\\in B^d.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\begin{definition}\\label{def-h-star}\nFor any set of complex numbers\n$\\{h_{\\xi}\\}_{\\xi\\in {\\mathcal X}_j}$ $(j\\ge 0)$ we define\n\\begin{equation}\\label{def.h-star}\nh^{\\ast}_\\xi:=\\sum_{\\eta\\in\n{\\mathcal X}_j}\\frac{|h_\\eta|}{(1+2^jd(\\eta,\\xi))^\\sigma},\n\\quad \\xi\\in {\\mathcal X}_j,\n\\end{equation}\nwhere $\\sigma>1$ is a sufficiently large constant that will be selected later on.\n\\end{definition}\n\n\n\\begin{lemma}\\label{lem:weak_inequality}\nLet $P\\in \\Pi_{2^j}$, $j\\ge 0$, and denote\n$a_\\xi:=\\max_{x\\in R_\\xi}|P(x)|$ for $\\xi\\in{\\mathcal X}_j$.\nThere exists $r\\ge 1$, depending only on $\\sigma$, $\\mu$, and $d$\nsuch that if\n$$\nb_\\xi:=\\max\\{\\min_{x\\in R_\\eta }|P(x)|:\\eta\\in {\\mathcal X}_{j+r},\nR_\\xi\\cap R_\\eta\\ne\\emptyset \\}, \\quad \\xi\\in{\\mathcal X}_j,\n$$\nthen\n\\begin{equation}\\label{weak_inequality}\na_\\xi^\\ast\\sim b_\\xi^\\ast\n\\end{equation}\nwith constants of equivalence independent of $P$, $j$, and $\\xi$.\n\\end{lemma}\n\n\n\\begin{lemma}\\label{lem:disc-max}\nAssume $t>0$, $\\gamma \\in \\mathbb{R}$, and let $\\{b_\\xi\\}_{\\xi\\in {\\mathcal X}_j}$\n$(j\\ge 0)$ be a set of complex numbers. Also, let $\\sigma$ in the\ndefinition $(\\ref{def.h-star})$ of $b_\\xi^*$ obey\n$\\sigma>d+(d+2\\mu)\/t+2\\mu|\\gamma|$.\nThen for any $\\xi \\in {\\mathcal X}_j$\n\\begin{equation}\\label{disc-max}\nb_\\xi^{\\ast}{W_\\mu}(2^j; \\xi)^{\\gamma}{\\mathbbm 1}_{R_\\xi}(x)\n\\le c {\\mathcal M}_t\\Big(\\sum_{\\eta\\in {\\mathcal X}_j}|b_\\eta|{W_\\mu}(2^j; \\eta)^{\\gamma}{\\mathbbm 1}_{R_\\eta}(\\cdot)\\Big)(x),\n\\quad x\\in R_\\xi.\n\\end{equation}\n\\end{lemma}\n\n\n\n\n\\noindent {\\em Proof of Theorem $\\ref{thm:Fnorm-equivalence}$.}\nChoose $0<t<\\min\\{p,q\\}$ and let $\\sigma$ in\nDefinition~\\ref{def-h-star} obey $\\sigma\n>d+(d+2\\mu)\/t+2\\mu|{\\rho}|\/d$. Now, choose $k\\ge\\sigma+2\\mu|{\\rho}|\/d$.\nObserve first that the right-hand side equivalence in\n(\\ref{Fnorm-equivalence-1}) follows immediately from Lemma\n\\ref{lem:Max-needl} and the maximal inequality (\\ref{max-ineq}).\n\nLet $\\{\\Phi_j\\}$  be a sequences of kernels as in\nthe definition of weighted Triebel-Lizorkin spaces,\ni.e. $\\Phi_j$ is defined by (\\ref{def-Phi-j})\nwith $\\widehat a$ satisfying (\\ref{ha1})-(\\ref{ha2}),\nthe same as (\\ref{ha-hb1})-(\\ref{ha-hb2}).\nAs already mentioned, there exists a function $\\widehat b$ satisfying\n(\\ref{ha-hb1})-(\\ref{ha-hb2}) such that (\\ref{ha-hb3}) holds.\nLet $\\Psi_j$ be defined by (\\ref{def-Psi-j}) with this $\\widehat b$.\nIn addition, let $\\{{\\varphi}_\\xi\\}_{\\xi\\in{\\mathcal X}}$ and $\\{\\psi_\\xi\\}_{\\xi\\in{\\mathcal X}}$\nbe the associated needlet systems defined as in (\\ref{def-needlets1})\nusing these $\\widehat a$ and $\\widehat b$.\n\nExactly in the same way, let $\\{\\widetilde\\Phi_j\\}$ and $\\{\\widetilde\\Psi_j\\}$ be two sequences\nof kernels defined as above using completely different functions $\\widehat a$ and $\\widehat b$.\nAlso, assume that $\\{\\widetilde{\\varphi}_\\xi\\}$, $\\{\\widetilde\\psi_\\xi\\}$\nare the associated needlet systems, defined as in\n(\\ref{def.Phi-j})-(\\ref{def-needlets1}).\nAs a result, we have two completely different systems of kernels and associated needlet systems.\n\nLet us first prove the boundedness of the operator\n$T_{\\widetilde\\psi}:{f_{pq}^{s{\\rho}}}\\rightarrow{F_{pq}^{s{\\rho}}}$,\ndefined similarly as in~(\\ref{anal_synth_oprts}) with $\\{\\psi_\\xi\\}$\nreplaced by $\\{\\widetilde{\\psi}_\\xi\\}$.\nHere we assume that space ${F_{pq}^{s{\\rho}}}$ is defined by $\\{\\Phi_j\\}$.\nLet $h:=\\{h_\\xi\\}_{\\xi\\in {\\mathcal X}}$ be an arbitrary finitely supported sequence and\n$f:=\\sum_{\\xi}h_\\xi\\widetilde{\\psi}_\\xi$.\nUsing Lemma~\\ref{lem:Phi*psi} we have, for $x\\in B^d$,\n\\begin{align*}\n|\\Phi_j\\ast f(x)| &=\\Big|\\sum_{\\xi\\in {\\mathcal X}}  h_\\xi\n\\Phi_j\\ast\\widetilde\\psi_{\\xi}(x)\\Big| \\le \\sum_{j-1\\le \\nu\\le\nj+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu} |h_\\xi |\n|\\Phi_j\\ast\\widetilde\\psi_{\\xi}(x)|\\\\\n&\\le c2^{jd\/2} \\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu} \\frac{\n|h_\\xi| }{\\sqrt{ {W_\\mu}(2^\\nu;x)}(1+2^\\nu d(\\xi, x))^k}.\n\\end{align*}\nFor $\\eta\\in{\\mathcal X}_j$, denote\n$\\Gamma_\\eta:=\\{w\\in{\\mathcal X}_{j-1}\\cup{\\mathcal X}_j\\cup{\\mathcal X}_{j+1}: R_w\\cap R_\\eta\\ne \\emptyset\\}$.\nHere ${\\mathcal X}_{-1}:=\\emptyset$.\nNote first that $\\# \\Gamma_\\eta \\le c$.\nSecondly, for $x\\in R_\\eta$ and $w\\in\\Gamma_\\eta$, we have $d(x, w) \\le c2^{-j}$ and\nusing inequality \\eqref{eq:useful}\n$$\n{W_\\mu}(2^j; x)^{-{\\rho}\/d} \\le c{W_\\mu}(2^j; w)^{-{\\rho}\/d}\n\\le c{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}(1+2^jd(\\xi, \\omega))^{2\\mu|{\\rho}|\/d}.\n$$\nWe use the above estimates to obtain, for $x\\in R_\\eta$,\n\\begin{align*}\n&{W_\\mu}(2^j; x)^{-{\\rho}\/d}|\\Phi_j\\ast f(x)|\\\\\n&\\qquad\\qquad \\le c2^{jd\/2} \\sum_{j-1\\le \\nu\\le j+1}\n\\sum_{\\omega\\in\\Gamma_\\eta\\cap{\\mathcal X}_\\nu}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n\\frac{|h_\\xi|{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\omega}(x)}\n{\\sqrt{{W_\\mu}(2^\\nu;\\omega)}(1+2^\\nu d(\\xi, \\omega))^{k-2\\mu|{\\rho}|\/d}}\\\\\n&\\qquad\\qquad \\le c2^{jd\/2}\n\\sum_{\\omega\\in\\Gamma_\\eta}\n\\frac{H_\\omega^*{\\mathbbm 1}_{R_\\omega}(x)}{\\sqrt{{W_\\mu}(2^j;\\omega)}}\n\\le c\\sum_{\\omega\\in\\Gamma_\\eta}H_\\omega^*{\\tilde{\\mathbbm 1}}_{R_\\omega}(x),\n\\end{align*}\nwhere\n$H_\\omega:= h_\\omega{W_\\mu}(2^j; \\omega)^{-{\\rho}\/d}$.\nHere we used that $k-2\\mu|{\\rho}|\/d \\ge \\sigma$\nand \\eqref{eq:mR}.\nWe insert the above in (\\ref{Tri-Liz-norm}) and use Lemma~\\ref{lem:disc-max} (with $\\gamma=0$)\nand the maximal inequality (\\ref{max-ineq}) to obtain\n\\begin{equation}\n\\begin{aligned}\\label{F-less-f}\n\\norm{f}_{{F_{pq}^{s{\\rho}}}}\n&\\le c\\nnorm{\\Bigr(\\sum_{j=0}^\\infty \\Big[2^{sj}\\sum_{\\eta\\in {\\mathcal X}_j}\\sum_{\\omega\\in \\Gamma_\\eta}\nH_\\omega^*{\\tilde{\\mathbbm 1}}_{R_\\omega}(\\cdot)\\Big]^q\\Big)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigr(\\sum_{j=0}^\\infty \\Big[2^{sj}\\sum_{\\xi\\in {\\mathcal X}_j}\nH_\\xi^*{\\tilde{\\mathbbm 1}}_{R_\\xi}(\\cdot)\\Big]^q\\Big)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigr(\\sum_{j=0}^\\infty \\Big[{\\mathcal M}_t\\Big(\\sum_{\\xi\\in {\\mathcal X}_j}\n2^{sj}|H_\\xi|{\\tilde{\\mathbbm 1}}_{R_\\xi}\\Big)(\\cdot)\\Big]^q\\Big)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigr(\\sum_{j=0}^\\infty \\Big[\\sum_{\\xi\\in {\\mathcal X}_j}\n2^{sj}|H_\\xi|{\\tilde{\\mathbbm 1}}_{R_\\xi}(\\cdot)\\Big]^q\\Big)^{1\/q}}_{{p}}\n\\le c\\norm{\\{h_\\xi\\} }_{{f_{pq}^{s{\\rho}}}},\n\\end{aligned}\n\\end{equation}\nwhere in  the second inequality  above we used that\n$\\#\\Gamma_\\eta\\le c$.\nThis establishes the desired result for finitely supported\nsequences. Using  the continuous  embedding of ${F_{pq}^{s{\\rho}}}$ in $\\mathcal{D}'$\n(Proposition~\\ref{Fsrpq-embedding}) and the density of finitely\nsupported sequences in ${f_{pq}^{s{\\rho}}}$ it follows from  (\\ref{F-less-f})\nthat for every $h\\in {f_{pq}^{s{\\rho}}}$, $T_{\\widetilde\\psi} h:=\\sum_{\\xi\\in\n{\\mathcal X}}h_\\xi\\widetilde\\psi_\\xi$ is a well defined distribution in $\\mathcal{D}'$.\nThen a standard density argument shows that\n$T_{\\widetilde\\psi}:{f_{pq}^{s{\\rho}}}\\rightarrow{F_{pq}^{s{\\rho}}}$ is bounded.\n\n\n\nAssume now that the space ${F_{pq}^{s{\\rho}}}$ is defined in terms of $\\{\\overline{\\Phi}_j\\}$\nin place of $\\{\\Phi_j\\}$.\nUsing this definition we shall prove the boundedness of the operator\n$S_\\varphi:{F_{pq}^{s{\\rho}}}\\rightarrow {f_{pq}^{s{\\rho}}}$.\n\nLet $f\\in {F_{pq}^{s{\\rho}}}$. Then $\\overline{\\Phi}_j\\ast f\\in \\Pi_{2^j}$.\nFor $\\xi \\in {\\mathcal X}_j$, we define\n$$\na_\\xi:=\\max_{x\\in R_\\xi}|\\overline{\\Phi}_j\\ast f(x)|, \\quad\nb_\\xi:=\\max\\{\\min_{x\\in R_\\eta }|\\overline{\\Phi}_j\\ast f(x)|:\\eta\\in\n{\\mathcal X}_{j+r}, R_\\xi\\cap R_\\eta\\ne\\emptyset \\},\n$$\nwhere $r\\ge 1$ is from Lemma \\ref{lem:weak_inequality}.\nThen by the same lemma $a_\\xi^*\\sim b_\\xi^*$.\nHence, using \\eqref{eq:mR},\n$$\n|\\langle f, {\\varphi}_\\xi\n\\rangle|=\\lambda_\\xi^{1\/2}|\\overline{\\Phi}_j*f(\\xi)| \\le c\nm(R_\\xi)^{1\/2}a_\\xi \\le c m(R_\\xi)^{1\/2}a_\\xi^* \\le c\nm(R_\\xi)^{1\/2}b_\\xi^*.\n$$\nFrom this, recalling that\n${\\tilde{\\mathbbm 1}}_{R_\\xi}:=m(R_\\xi)^{-1\/2}{\\mathbbm 1}_{R_\\xi}$, we get\n\\begin{equation}\\label{asdfasf}\n\\begin{aligned}\n\\norm{\\{\\ip{f, \\varphi_\\xi}\\}}_{{f_{pq}^{s{\\rho}}}}\n&=\\nnorm{\\Bigl(\\sum_{j=0}^\\infty 2^{jsq}\n\\sum_{\\xi\\in {\\mathcal X}_j}\n[|\\ip{f, \\varphi_\\xi}|{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}{\\tilde{\\mathbbm 1}}_{R_\\xi}(\\cdot)]^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty 2^{jsq}\n\\sum_{\\xi\\in {\\mathcal X}_j}[b_\\xi^*{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}\n{\\mathbbm 1}_{R_\\xi}(\\cdot)]^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty 2^{jsq}\n\\Big[{\\mathcal M}_t\\Big(\\sum_{\\xi\\in {\\mathcal X}_j}\nb_\\xi{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big)(\\cdot)\\Big]^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty2^{jsq}\n\\Big[\\sum_{\\xi\\in {\\mathcal X}_j}\nb_\\xi {W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big]^q\\Bigr)^{1\/q}}_{{p}}.\n\\end{aligned}\n\\end{equation}\nHere for the second inequality above we used Lemma~\\ref{lem:disc-max}\nand for the third one the maximal inequality (\\ref{max-ineq}).\n\nDenote $m_\\eta:=\\min_{x\\in R_\\eta}|\\overline{\\Phi}_j\\ast f(x)|$ for\n$\\eta\\in {\\mathcal X}_{j+r}$\nand\n$$\n{\\mathcal X}_{j+r}(\\xi):=\\{w\\in {\\mathcal X}_{j+r}:R_w\\cap R_\\xi \\ne \\emptyset\\}\n\\quad\\mbox{for $\\xi\\in{\\mathcal X}_j$}.\n$$\nEvidently $\\#{\\mathcal X}_{j+r}(\\xi)\\le c(r,d)$.\nFurther, for $w, \\eta\\in {\\mathcal X}_{j+r}(\\xi)$ we have\n$d(w,\\eta)\\le c2^{-j}$ and hence\n$$\nm_w\\le c\\frac{m_w}{(1+2^{j+r}d(w,\\eta))^\\sigma}\\le cm^\\ast_\\eta,\n\\quad c=c(r, \\sigma, d).\n$$\nTherefore, for any $\\eta\\in {\\mathcal X}_{j+r}(\\xi)$,\n$\nb_\\xi=\\max_{w\\in{\\mathcal X}_{j+r}(\\xi)}m_w \\le cm^\\ast_\\eta.\n$\nand hence\n\\begin{equation}\\label{b-xi-m-eta}\nb_\\xi{\\mathbbm 1}_{R_\\xi}\\le \\sum_{\\eta\\in {\\mathcal X}_{j+r}(\\xi)}  m^\\ast_\\eta {\\mathbbm 1}_{R_\\eta}.\n\\end{equation}\nClearly,\n${W_\\mu}(2^j; \\xi) \\sim {W_\\mu}(2^{j+r}; \\eta)$ for $\\eta\\in{\\mathcal X}_{j+r}(\\xi)$.\nThis along with \\eqref{b-xi-m-eta} leads to\n\\begin{equation}\\label{W-b-xi-m-eta}\nb_\\xi{W_\\mu}(2^j;\\xi)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\xi}\n\\le c\\sum_{\\eta\\in {\\mathcal X}_{j+r}(\\xi)}  m^*_\\eta{W_\\mu}(2^{j+r}; \\eta)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\eta}.\n\\end{equation}\nUsing  this estimate in (\\ref{asdfasf}) we get\n\\begin{equation*}\n\\begin{aligned}\n\\norm{\\{\\ip{f, \\varphi_\\xi}\\}}_{{f_{pq}^{s{\\rho}}}}\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty 2^{jsq}\n\\Bigl(\\sum_{\\eta\\in {\\mathcal X}_{j+r}}  m^*_\\eta{W_\\mu}(2^{j+r}; \\eta)^{-{\\rho}\/d}\n{\\mathbbm 1}_{R_\\eta}(\\cdot) \\Bigr)^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty 2^{jsq}\n\\Big[{\\mathcal M}_t\\Big(\\sum_{\\eta\\in {\\mathcal X}_{j+r}}m_\\eta{W_\\mu}(2^{j+r}; \\eta)^{-{\\rho}\/d}\n{\\mathbbm 1}_{R_\\eta}\\Big)(\\cdot)\\Big]^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty \\Bigl(2^{js}\n\\sum_{\\eta\\in {\\mathcal X}_{j+r}} m_\\eta{W_\\mu}(2^{j+r}; \\eta)^{-{\\rho}\/d}{\\mathbbm 1}_{R_\\eta}(\\cdot) \\Bigr)^q\\Bigr)^{1\/q}}_{{p}}\\\\\n&\\le c\\nnorm{\\Bigl(\\sum_{j=0}^\\infty (2^{js}\n {W_\\mu}(2^{j}; \\cdot)^{-{\\rho}\/d} |\\overline{\\Phi}_j*f(\\cdot)|)^q\\Bigr)^{1\/q}}_{{p}}\n =c\\|f\\|_{{F_{pq}^{s{\\rho}}}}.\n\\end{aligned}\n\\end{equation*}\nHere for first inequality we used that $\\# {\\mathcal X}_{j+r}(\\xi) \\le c$,\nfor the second inequality we used Lemma~\\ref{lem:disc-max},\nand for third one the maximal inequality \\eqref{max-ineq}.\nWe also use that ${W_\\mu}(2^{j+r}; \\eta)\\sim {W_\\mu}(2^{j}; x)$ if $x\\in \\mathbb{R}_\\eta$, $\\eta\\in{\\mathcal X}_{j+r}$.\nThus the boundedness of $S_\\varphi:{F_{pq}^{s{\\rho}}}\\rightarrow {f_{pq}^{s{\\rho}}}$ is established.\n\n\nThe identity $T_\\psi\\circ S_\\varphi=Id$ follows by Proposition~\\ref{prop:needlet-rep}.\n\nIt remains to show that ${F_{pq}^{s{\\rho}}}$ is independent of the particular selection\nof $\\widehat a$ in the definition of  $\\{\\Phi_j\\}$.\nDenote by $\\|\\cdot\\|_{{F_{pq}^{s{\\rho}}}(\\Phi)}$ the F-norm defined by $\\{\\Phi_j\\}$.\nThen by the above proof it follows that\n$$\n\\|f\\|_{{F_{pq}^{s{\\rho}}}(\\Phi)}\n\\le c\\|\\{\\langle f, \\widetilde{\\varphi}_\\xi\\rangle\\}\\|_{{f_{pq}^{s{\\rho}}}}\n\\quad\\mbox{and}\\quad\n\\|\\{\\langle f, {\\varphi}_\\xi\\rangle\\}\\|_{{f_{pq}^{s{\\rho}}}}\n\\le c\\|f\\|_{{F_{pq}^{s{\\rho}}}(\\overline{\\Phi})}\n$$\nand hence\n$$\n\\|f\\|_{{F_{pq}^{s{\\rho}}}(\\Phi)}\n\\le c\\|\\{\\langle f, \\widetilde{\\varphi}_\\xi\\rangle\\}\\|_{{f_{pq}^{s{\\rho}}}}\n\\le c\\|f\\|_{{F_{pq}^{s{\\rho}}}(\\overline{\\widetilde\\Phi})}.\n$$\nNow the desired independence follows by interchanging the roles of\n$\\{ \\Phi_j\\}$,$\\{\\widetilde\\Phi_j\\}$, and  their complex conjugates.\n$\\qed$\n\n\n\n\\medskip\n\nIn a sense the spaces $\\F sspq$ are more natural than the spaces\n${F_{pq}^{s{\\rho}}}$ with ${\\rho}\\ne s$ since they scale (are embedded)\n``correctly\" with respect to the smoothness index $s$.\n\n\\begin{proposition}\\label{F-embedding}\nLet $0<p<p_1<\\infty$, $0<q, q_1\\le\\infty$, and\n$-\\infty<s_1<s<\\infty$. Then we have the continuous embedding\n\\begin{equation}\\label{F-embed}\n\\F sspq \\subset \\F {s_1}{s_1}{p_1}{q_1} \\quad\\mbox{if}\\quad\ns\/d-1\/p=s_1\/d-1\/p_1.\n\\end{equation}\n\\end{proposition}\n\nThe proof of this embedding result can be carried out similarly as\nin the classical case on ${\\mathbb R}^n$ using inequality\n(\\ref{norm-relation2}) and Theorem~\\ref{thm:Fnorm-equivalence}\n(see e.g. \\cite{T1}, page 129). It~will be omitted.\n\n\n\\medskip\n\n\nFinally, we would like to link the weighted Triebel-Lizorkin spaces ${F_{pq}^{s{\\rho}}}$\nto ${L_p({w_\\mu})}$ and weighted potential space\n(generalized weighted Sobolev space) on $B^d$.\n\nWe define the weighted potential space\n$H_p^s := H_p^s({w_\\mu})$,\n$s >0$, $1\\le p \\le \\infty$,  on $B^d$\nas the set of all $f\\in \\mathcal{D}'$ such that\n\\begin{equation}\\label{def-H}\n\\|f\\|_{H_p^s}:=\n\\Big\\|\\sum_{n=0}^\\infty (n+1)^s \\operatorname{Proj}_n f\\Big\\|_{p}\n< \\infty,\n\\end{equation}\nwhere\n$\\operatorname{Proj}_n f:={\\mathsf {P}}_n*f$.\n\nWe have the following identification of certain weighted Triebel-Lizorkin spaces.\n\n\n\\begin{proposition}\\label{prop:ident}\nWe have\n$$\n\\F s0p2 \\sim H_p^s,\n\\quad s >0, ~1 < p < \\infty,\n$$\nand\n$$\n\\F 00p2\\sim {L_p({w_\\mu})},\n\\quad 1 < p < \\infty,\n$$\nwith equivalent norms.\nConsequently, for any $f\\in{L_p({w_\\mu})}$, $1<p<\\infty$,\n$$\n\\|f\\|_p\n\\sim \\nnorm{\\biggl(\\sum_{j=0}^\\infty\n\\sum_{\\xi\\in {\\mathcal X}_j}(|\\ip{f, \\varphi_\\xi}||\\psi_\\xi(\\cdot)|)^2\\biggr)^{1\/2}}_{{p}}.\n$$\n\\end{proposition}\n\nThe proof of this proposition uses the multipliers from \\cite[Theorem 5.2]{DX}\nand can be carried out exactly as in the case of spherical\nharmonic expansions in \\cite[Proposition~4.3]{NPW2}.\nWe omit it.\n\n\n\n\\section{Weighted Besov spaces on \\boldmath $B^d$}\\label{Besov}\n\\setcounter{equation}{0}\n\n\nFor the definition of weighted Besov spaces on $B^d$ we use the sequence\nof kernels $\\{\\Phi_j\\}$ defined in (\\ref{def-Phi-j}) with $\\widehat a$\nobeying (\\ref{ha1})-(\\ref{ha2})\n(see \\cite{Pee, T1} for the general idea of using spectral decompositions).\n\n\n\\begin{definition}\nLet $s, {\\rho}\\in \\mathbb{R}$ and $0<p,q \\le \\infty$. The weighted Besov space\n${B_{pq}^{s{\\rho}}} := {B_{pq}^{s{\\rho}}}({w_\\mu})$ is defined\nas the set of all $f \\in \\mathcal{D}'$ such that\n\\begin{equation}\\label{def-Besov-sp}\n\\|f\\|_{{B_{pq}^{s{\\rho}}}} :=\n\\Big(\\sum_{j=0}^\\infty \\Big(2^{s j}\n\\|{W_\\mu}(2^j; \\cdot)^{-{\\rho}\/d}\\Phi_j*f(\\cdot)\\|_{{p}}\\Big)^q\\Big)^{1\/q}\n< \\infty,\n\\end{equation}\nwhere the $\\ell_q$-norm is replaced by the sup-norm if $q=\\infty$.\n\\end{definition}\n\nObserve that as in the case of weighted Triebel-Lizorkin spaces the above definition\nis independent of the particular choice of $\\widehat a$ obeying  (\\ref{ha1})-(\\ref{ha2})\n(see Theorem~\\ref{thm:Bnorm-eq}).\nAlso, as for ${F_{pq}^{s{\\rho}}}$ the Besov space ${B_{pq}^{s{\\rho}}}$ is a quasi-Banach\nspace which is continuously embedded in $\\mathcal{D}'$. We skip the details.\n\n\\smallskip\n\nWe next introduce the sequence spaces ${b_{pq}^{s{\\rho}}}$\nassociated to the weighted Besov spaces ${B_{pq}^{s{\\rho}}}$.\nTo this end, we assume that $\\{{\\mathcal X}_j\\}_{j=0}^\\infty$ is a sequence of almost\nuniformly ${\\varepsilon}_j$-distributed points on $B^d$\n$({\\varepsilon}_j:=c^\\diamond 2^{-j})$\nwith associated neighborhoods $\\{R_\\xi\\}_{\\xi\\in{\\mathcal X}_j}$,\ngiven by Proposition~\\ref{prop:cubature}.\nAs before we set ${\\mathcal X}:=\\cup_{j\\ge 0} {\\mathcal X}_j$.\n\n\n\\begin{definition}\nLet $s, {\\rho}\\in \\mathbb{R}$ and $0<p,q \\le \\infty$. Then ${b_{pq}^{s{\\rho}}}$\nis defined to be the space of all complex-valued sequences\n$h:=\\{h_{\\xi}\\}_{\\xi\\in {\\mathcal X}}$ such that\n\\begin{equation}\\label{def-berpq}\n\\norm{h}_{{b_{pq}^{s{\\rho}}}} :=\\Bigl(\\sum_{j=0}^\\infty\n2^{j(s-d\/p+d\/2)q}\n\\Bigl[\\sum_{\\xi\\in {\\mathcal X}_j}\\Big({W_\\mu}(2^j;\\xi)^{-{\\rho}\/d+1\/p-1\/2}|h_\\xi|\\Big)^p\n\\Bigr]^{q\/p}\\Bigr)^{1\/q}\n\\end{equation}\nis finite,\nwith the usual modification for $p=\\infty$ or $q=\\infty$.\n\\end{definition}\n\nOur main result in this section is the following characterization of\nweighted Besov spaces, which employs the operators $S_{\\varphi}$ and\n$T_\\psi$ defined in (\\ref{anal_synth_oprts}).\n\n\n\\begin{theorem}\\label{thm:Bnorm-eq}\nLet $s, {\\rho}\\in \\mathbb{R}$ and  $0< p,q\\le \\infty$.\nThe operators\n$S_\\varphi:{B_{pq}^{s{\\rho}}}\\rightarrow{b_{pq}^{s{\\rho}}}$ and\n$T_\\psi:{b_{pq}^{s{\\rho}}}\\rightarrow {B_{pq}^{s{\\rho}}}$\nare bounded and $T_\\psi\\circ S_\\varphi=Id$ on ${B_{pq}^{s{\\rho}}}$.\nConsequently, for $f\\in \\mathcal{D}'$ we have that $f\\in {B_{pq}^{s{\\rho}}}$ if and\nonly if $\\{\\ip{f, \\varphi_\\xi}\\}_{\\xi \\in {\\mathcal X}}\\in {b_{pq}^{s{\\rho}}}$.\nMoreover,\n\\begin{align}\\label{Bnorm-equivalence-1}\n\\norm{f}_{{B_{pq}^{s{\\rho}}}} & \\sim  \\norm{\\{\\ip{f,\\varphi_\\xi}\\}}_{{b_{pq}^{s{\\rho}}}}\n\\sim \\Big(\\sum_{j=0}^\\infty2^{sjq} \\Bigl[\\sum_{\\xi\\in {\\mathcal X}_j}\n\\Big({W_\\mu}(2^j; \\xi)^{-{\\rho}\/d}\\norm{\\ip{f,\\varphi_\\xi}\\psi_\\xi}_{{p}}\\Big)^p\\Bigr]^{q\/p}\\Bigr)^{1\/q}.\n\\end{align}\nIn addition, the definition of ${B_{pq}^{s{\\rho}}}$ is independent of the particular\nselection of $\\widehat a$ satisfying $(\\ref{ha1})$--$(\\ref{ha2})$.\n\\end{theorem}\n\n\nFor the proof of this theorem we shall utilize some of the lemmas from \\S\\ref{Tri-Liz}\nas well as the following additional lemma\nwhose proof is given in Section~\\ref{proofsB}.\n\n\n\n\n\n\\begin{lemma}\\label{l:half_shannon}\nLet $0<p\\le \\infty$ and $\\gamma\\in \\mathbb{R}$.\nThen for any $P\\in \\Pi_{2^j}, j\\ge 0$,\n\\begin{equation}\n \\Big(\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma\\max_{x\\in R_\\xi}|P(x)|^p m(R_\\xi)\\Big)^{1\/p}\n \\le c\\norm{{W_\\mu}(2^j; \\cdot)^\\gamma P(\\cdot)}_{{p}}.\n\\end{equation}\n\\end{lemma}\n\n\n\n\\noindent {\\em Proof of Theorem~$\\ref{thm:Bnorm-eq}$.} We first\nnote that the right-hand side of (\\ref{Bnorm-equivalence-1})\nfollows immediately from (\\ref{norm-Needlets}).\n\nJust as in the proof of Theorem~\\ref{thm:Fnorm-equivalence},\nwe assume that $\\{\\Phi_j\\}$ are kernels defined by (\\ref{def-Phi-j}),\nwith $\\widehat a$ satisfying (\\ref{ha1})-(\\ref{ha2}).\nNext, suppose $\\{\\Psi_j\\}$ are defined by (\\ref{def-Psi-j})\nwith $\\widehat b$ obeying (\\ref{ha-hb1})-(\\ref{ha-hb3}).\nAlso, let $\\{{\\varphi}_\\xi\\}_{\\xi\\in{\\mathcal X}}$ and $\\{\\psi_\\xi\\}_{\\xi\\in{\\mathcal X}}$\nbe the associated needlet systems defined as in (\\ref{def-needlets1}).\nFurther, assume that $\\{\\widetilde\\Phi_j\\}$, $\\{\\widetilde\\Psi_j\\}$,\n$\\{\\widetilde{\\varphi}_\\xi\\}$, $\\{\\widetilde\\psi_\\xi\\}$ is a second (completely different) set of kernels and\nneedlets.\n\nOur first step is to prove the boundedness of the operator\n$T_{\\widetilde\\psi}:{b_{pq}^{s{\\rho}}}\\rightarrow{B_{pq}^{s{\\rho}}}$ defined as in \\eqref{anal_synth_oprts}\nwith $\\{\\psi_\\xi\\}$ replaced by $\\{\\widetilde\\psi_\\xi\\}$;\nwe assume that ${B_{pq}^{s{\\rho}}}$ is defined by $\\{\\Phi_j\\}$.\n\nPick $0<t<\\min\\{p,1\\}$ and $k \\ge 2\\mu|{\\rho}|\/d+\\mu+(2\\mu+d)\/t$.\nLet $h=\\{h_\\xi\\}_{\\xi\\in{\\mathcal X}_j}$ be a finitely supported sequence and\n$f:=\\sum_{\\xi\\in {\\mathcal X}}h_\\xi\\widetilde{\\psi}_\\xi$.\nSimilarly as in the proof of Theorem~\\ref{thm:Fnorm-equivalence},\nwe use Lemmas~\\ref{lem:B-maximal} and \\ref{lem:Phi*psi}, and\n(\\ref{eq:useful}) to obtain\n\\begin{align*}\n{W_\\mu}(2^j; x)^{-{\\rho}\/d}|\\Phi_j\\ast f(x)|\n&\\le c\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi|{W_\\mu}(2^j; x)^{-{\\rho}\/d}|\\Phi_j\\ast\\widetilde\\psi_{\\xi}(x)|\\\\\n&\\le c\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi| \\frac{2^{jd\/2} {W_\\mu}(2^j; x)^{-{\\rho}\/d}}{\\sqrt{ {W_\\mu}(2^j;x)}(1+2^jd(\\xi, x))^k } \\\\\n&\\le c2^{jd\/2}\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi| \\frac{{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d-1\/2}}{(1+2^jd(\\xi, x))^{k-2\\mu|{\\rho}|\/d-\\mu} } \\\\\n&\\le c2^{jd\/2}\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu} |h_\\xi|\n{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d-1\/2}{\\mathcal M}_t({\\mathbbm 1}_{R_\\xi})(x),\n\\end{align*}\nwhere ${\\mathcal X}_{-1}:=\\emptyset$ and\nin the fourth inequality we used that\n$k \\ge 2\\mu|{\\rho}|\/d+\\mu+(2\\mu+d)\/t$.\nNow employing the maximal inequality (\\ref{max-ineq}) we get\n\\begin{equation*}\n\\begin{aligned}\n&\\norm{{W_\\mu}(2^j; \\cdot)^{-{\\rho}\/d}\\Phi_j*f(\\cdot)}_{{p}}\\\\\n&\\qquad\\qquad\\qquad \\le c2^{jd\/2}\\Big\\|\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi|{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d-1\/2}{\\mathcal M}_t({\\mathbbm 1}_{R_\\xi})(\\cdot)\\Big\\|_{{p}}\\\\\n&\\qquad\\qquad\\qquad \\le c2^{jd\/2}\\Big\\|\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi|{W_\\mu}(2^j; \\xi)^{-{\\rho}\/d-1\/2}{\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big\\|_{{p}}\\\\\n&\\qquad\\qquad\\qquad \\le c2^{jd(1\/2-1\/p)}\\Big(\\sum_{j-1\\le \\nu\\le j+1}\\sum_{\\xi\\in {\\mathcal X}_\\nu}\n|h_\\xi|^p{W_\\mu}(2^j; \\xi)^{-({\\rho}\/d-1\/p+1\/2)p}\\Big)^{1\/p}.\n\\end{aligned}\n\\end{equation*}\nUsing this in Definition~\\ref{def-Besov-sp} we obtain\n$\\norm{f}_{{B_{pq}^{s{\\rho}}}}\\le c\\norm{\\{h_\\xi\\}}_{{b_{pq}^{s{\\rho}}}}$.\n\nFurther, we extend this result to an arbitrary sequence $h=\\{h_\\xi\\}\\in {b_{pq}^{s{\\rho}}}$\nsimilarly as in the Triebel-Lizorkin case\nby using the embedding of ${B_{pq}^{s{\\rho}}}$ in $\\mathcal{D}'$\nand the density of finitely supported sequences in ${b_{pq}^{s{\\rho}}}$.\n\n\\smallskip\n\n\nWe next prove the boundedness of the operator\n$S_\\varphi:{B_{pq}^{s{\\rho}}}\\rightarrow {b_{pq}^{s{\\rho}}}$,\nassuming that the space ${B_{pq}^{s{\\rho}}}$ is defined in terms\nof $\\{\\overline{\\Phi}_j\\}$ in place of $\\{\\Phi_j\\}$.\nObserve first that\n$$\n|\\ip{f,\\varphi_\\xi}|\n\\sim m(R_\\xi)^{1\/2}|\\overline{\\Phi}_j*f(\\xi)|\n\\sim 2^{-jd\/2}{W_\\mu}(2^j; \\xi)^{1\/2}|\\overline{\\Phi}_j*f(\\xi)|,\n\\quad \\xi \\in {\\mathcal X}_j.\n$$\nSince $\\overline{\\Phi}_j\\ast f\\in \\Pi_{2^j}$, Lemma~\\ref{l:half_shannon} yields\n\\begin{align*}\n&\\sum_{\\xi\\in {\\mathcal X}_j}\\Big({W_\\mu}(2^j;\\xi)^{-{\\rho}\/d+1\/p-1\/2}|\\ip{f,\\varphi_\\xi}|\\Big)^p\\\\\n&\\qquad\\qquad \\le c2^{-jd(p\/2-1)}\n\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j;\\xi)^{-{\\rho} p\/d}|\\overline{\\Phi}_j*f(\\xi)|^p m(R_\\xi)\\\\\n&\\qquad\\qquad \\le c2^{-jd(p\/2-1)}\\norm{{W_\\mu}(2^j;\\xi)^{-{\\rho}\/d}\\overline{\\Phi}_j\\ast f}_{{p}}^p.\n\\end{align*}\nThis at once yields\n$\n\\|\\{\\langle f, {\\varphi}\\rangle\\}\\|_{{b_{pq}^{s{\\rho}}}} \\le c\\norm{f}_{{B_{pq}^{s{\\rho}}}}.\n$\n\nThe identity $T_\\psi\\circ S_\\varphi=Id$ follows by Proposition~\\ref{prop:needlet-rep}.\n\nThe independence of ${B_{pq}^{s{\\rho}}}$ of the particular selection of $\\widehat a$\nin the definition of  $\\{\\Phi_j\\}$ follows from above exactly as\nin the Triebel-Lizorkin case (see the proof of\nTheorem~\\ref{thm:Fnorm-equivalence}). $\\qed$\n\n\nThe parameter ${\\rho}$ in the definition of the Besov spaces ${B_{pq}^{s{\\rho}}}$ allow to\nconsider different scales of spaces.\nA ``classical\" choice of ${\\rho}$ would be ${\\rho}=0$.\nHowever, we maintain that most natural are the spaces $\\B sspq$ (${\\rho}=s$).\nThe main advantages of the spaces\n$\\B sspq$  over $\\B s\\rho pq$ with $\\rho\\ne s$ are\nthat, first, they scale (are embedded) ``correctly\" with respect\nto the smoothness index $s$, and secondly, the right smoothness\nspaces in nonlinear n-term weighted approximation from needles are\ndefined in terms of spaces $\\B sspq$ (see \\S\\ref{Nonlin-app}\nbelow).\n\n\\begin{proposition}\\label{B-embedding}\nLet $0<p\\le p_1<\\infty$, $0<q\\le q_1\\le \\infty$, and\n$-\\infty<s_1\\le s<\\infty$. Then we have the continuous embedding\n\\begin{equation}\\label{B-embed}\n\\B sspq \\subset \\B {s_1}{s_1}{p_1}{q_1} \\quad\\mbox{if}\\quad\ns\/d-1\/p=s_1\/d-1\/p_1.\n\\end{equation}\n\\end{proposition}\nThis embedding result follows immediately by applying inequality\n(\\ref{norm-relation2}).\n\n\\medskip\n\n\nWe finally want to link the weighted Besov spaces\nto best polynomial approximation in ${L_p({w_\\mu})}$.\nAs in (\\ref{def-En}), let $E_n(f)_p$ denote the best approximation of $f \\in {L_p({w_\\mu})}$\nfrom $\\Pi_n$.\n\n\n\\begin{proposition}\\label{prop:character-Besov}\nLet $s > 0$, $1 \\le p \\le \\infty$, and $0 < q \\le \\infty$.\nThen $f \\in \\B s0pq$ if and only if\n\\begin{equation}\\label{character-Besov1}\n\\|f\\|_{B^{s 0}_{pq}}^A\n:= \\|f\\|_{p} +\n\\Big(\\sum_{j=0}^\\infty (2^{s j}E_{2^j}(f)_p)^q\\Big)^{1\/q}\n< \\infty.\n\\end{equation}\nMoreover,\n\\begin{equation}\\label{character-Besov2}\n\\|f\\|_{\\B s0pq}^A \\sim \\|f\\|_{\\B s0pq}.\n\\end{equation}\n\\end{proposition}\n\nThe proof of this proposition is similar to the proof of\nProposition 5.3 in \\cite{NPW2} and Proposition 6.2 in \\cite{KPX}.\nWe omit it.\n\n\n\n\\section{Application of weighted Besov spaces to nonlinear approximation}\\label{Nonlin-app}\n\\setcounter{equation}{0}\n\n\nLet us consider nonlinear n-term approximation for a needlet system\n$\\{\\psi_\\eta\\}_{\\eta\\in {\\mathcal X}}$ defined as in (\\ref{def.Phi-j})-(\\ref{def-needlets2})\nwith $\\widehat b = \\widehat a$, $\\widehat a\\ge 0$. Thus ${\\varphi}_\\eta=\\psi_\\eta$ are real-valued.\nThen by Proposition~\\ref{prop:needlet-rep}, for any $f\\in{L_p({w_\\mu})}$, $1\\le p\\le \\infty$,\n$$\nf=\\sum_{\\xi\\in{\\mathcal X}} \\langle f, \\psi_\\xi\\rangle \\psi_\\xi\n\\quad \\mbox{in } {L_p({w_\\mu})}.\n$$\n\nSuppose $\\Sigma_n$ is the nonlinear set of all functions $g$ of the form\n$$\ng = \\sum_{\\xi \\in \\Lambda} a_\\xi \\psi_\\xi,\n$$\nwhere $\\Lambda \\subset {\\mathcal X}$, $\\#\\Lambda \\le n$,\nand $\\Lambda$ may vary with $g$.\nLet $\\sigma_n(f)_p$ denote the error of best ${L_p({w_\\mu})}$-approximation to\n$f \\in {L_p({w_\\mu})}$ from $\\Sigma_n$, i.e.\n$$\n \\sigma_n(f)_p := \\inf_{g \\in \\Sigma_n} \\|f - g\\|_p.\n$$\nWe consider approximation in ${L_p({w_\\mu})}$, $0<p<\\infty$.\nSuppose $s>0$ and let $1\/\\tau := s\/d+1\/p$.\nDenote briefly\n$$\n{B_\\tau^{s}}:=\\B ss\\tau\\tau.\n$$\nFrom Theorem~\\ref{thm:Bnorm-eq} and \\eqref{norm-Needlets} one derives the following\nrepresentation of the norm in ${B_\\tau^{s}}$:\n\\begin{equation}\\label{Btau-norm}\n\\|f\\|_{{B_\\tau^{s}}}\\sim\n\\Big(\\sum_{\\xi\\in{\\mathcal X}} \\|\\langle f, \\psi_\\xi \\rangle \\psi_\\xi\\|_p^\\tau\\Big)^{1\/\\tau}.\n\\end{equation}\n\nThe following embedding result shows the importance of the spaces ${B_\\tau^{s}}$ fot\nnonlinear approximation from needlets.\n\n\n\\begin{proposition}\\label{prop:embed-Lp}\nIf $f \\in {B_\\tau^{s}}$, then $f$ can be identified as a function\n$f\\in {L_p({w_\\mu})}$ and\n\\begin{equation}\\label{embedding}\n\\|f\\|_p \\le\n\\Big\\|\\sum_{\\xi\\in{\\mathcal X}}|\\langle f, \\psi_\\xi\\rangle\\psi_\\xi(\\cdot)|\\Big\\|_p\n\\le c\\|f\\|_{{B_\\tau^{s}}}.\n\\end{equation}\n\\end{proposition}\nFor the proof one proceeds exactly as in the proof of the embedding result\nfrom \\cite[Theorem 3.3]{KP} (see also \\cite[Proposition 8.1]{KPX}).\nThe proof will be omitted.\n\n\nWe now give the main result of this section.\n\n\n\\begin{theorem}\\label{thm:jackson} {\\rm [Jackson estimate]}\nIf $f \\in {B_\\tau^{s}}$, then\n\\begin{equation}\\label{jackson}\n\\sigma_n(f)_p \\le c n^{-s}\\|f\\|_{{B_\\tau^{s}}}.\n\\end{equation}\n\\end{theorem}\n\nThe proofs of this theorem can be carried out exactly as the proof of\nTheorem~3.4 in \\cite{KP} or \\cite[Theorem~8.2]{KPX} and will be omitted.\n\nHere the main open problem is to prove the companion to (\\ref{jackson})\nBernstein estimate:\n\\begin{equation}\\label{bernstein}\n\\|g\\|_{{B_\\tau^{s}}} \\le c n^s \\|g\\|_p\n\\quad \\hbox{for}\\quad g \\in \\Sigma_n,\n\\quad 1<p<\\infty.\n\\end{equation}\nThis estimate would allow to characterize the rates\nof nonlinear n-term approximation in ${L_p({w_\\mu})}$ ($1<p<\\infty$) from needlet systems.\n\n\n\n\n\n\n\\section{Proofs}\\label{proofs}\n\\setcounter{equation}{0}\n\n\n\\subsection{Proofs for Sections~\\ref{preliminaries}-\\ref{def-needlets}}\\label{proofsA} ${}$\n\n\\medskip\n\n\\noindent {\\em Proof of Theorem~$\\ref{thm:est-Lp-norm}$.} We shall\nfirst establish (\\ref{eq:est-Lp-norm}) for $p=2$.\nFrom the definition of the kernels ${\\mathsf {P}}_n(x,y)$\n(see (\\ref{def-Pn})-(\\ref{compactPn}))\nit follows that\n$$\n\\int_{B^d}{\\mathsf {P}}_n(x,y){\\mathsf {P}}_m(x,y){w_\\mu}(y)\\, dy=\\delta_{n,m}{\\mathsf {P}}_n(x,x)\n$$\nand hence\n\\begin{equation}\\label{Rep-Ln}\n \\int_{B^d} |L_n(x,y)|^2 {w_\\mu}(y)dy  = \\sum_{k=0}^{2n}\n   \\Big|\\widehat a\\Big(\\frac{k}{n}\\Big)\\Big|^2 {\\mathsf {P}}_k(x, x).\n\\end{equation}\nTherefore, for $p=2$ estimate (\\ref{eq:est-Lp-norm}) will follow by\nthe following lemma.\n\n\n\\begin{lemma}\\label{L2-lower-bound}\nFor any ${\\varepsilon}>0$\n\\begin{equation} \\label{lowerbd}\n\\sum_{j=n}^{n+[{\\varepsilon} dn]} {\\mathsf {P}}_j(x,x)\n\\ge \\frac{cn^{d}}{{W_\\mu}(n;x)},\n\\qquad x \\in B^d, \\quad n\\ge 1\/{\\varepsilon},\n\\end{equation}\nwhere $c>0$ depends only on ${\\varepsilon}$, $\\mu$, and $d$.\n\\end{lemma}\n\n\\begin{proof}Assume $\\mu > 0$. We shall utilize\nrepresentation (\\ref{compactPn}) of ${\\mathsf {P}}_n(x, y)$. The case\n$\\mu=0$ is easier and will be omitted (in this case one uses\nrepresentation (4.2) of ${\\mathsf {P}}_n(x, y)$ from \\cite{PX2}).\n\n\nFrom \\eqref{compactPn} it is obvious that ${\\mathsf {P}}_n(x,x)$ depends only on $|x|$.\nFor the rest of the proof, we denote\n${\\mathsf {P}}_{n,d}(r) := {\\mathsf {P}}_n(x,x)$, where $r:=|x|$, and\n$\n\\Lambda_{n,d}(r):= \\sum_{j=n}^{n+[{\\varepsilon} dn]} {\\mathsf {P}}_{j, d}(r).\n$\nSumming up the well known recurrence relation \\cite[(4.7.29)]{Sz}\n$$\nC_n^\\lambda(x)-C_{n-2}^\\lambda(x)=\\frac{n+\\lambda-1}{\\lambda-1}C_n^{\\lambda-1}(x),\n\\quad \\hbox{where}\\quad\nC_{-1}^\\lambda(x)=C_{-2}^\\lambda(x):=0,\n$$\nwe get\n$$\nC_n^\\lambda(x)\n=\\sum_{0\\le 2j\\le n}\\frac{n-2j+\\lambda-1}{\\lambda-1}C_{n-2j}^{\\lambda-1}(x).\n$$\nCombining this with \\eqref{compactPn} we arrive at\n$$\n{\\mathsf {P}}_{n, d}(r)= \\frac{b_d^\\mu}{b_{d-2}^\\mu}\\frac{n+\\lambda}{\\lambda}\n\\sum_{0\\le 2j\\le n}{\\mathsf {P}}_{n-2j, d-2}(r).\n$$\nHence\n\\begin{align*}\n\\Lambda_{n,d}(r)\n&=  \\sum_{k=n}^{n+[{\\varepsilon} dn]} {\\mathsf {P}}_{k,d}(r)\n= \\frac{b_d^\\mu}{b_{d-2}^\\mu}\n\\sum_{k=n}^{n+[{\\varepsilon} dn]}\n\\frac{k+\\lambda}{\\lambda} \\sum_{0 \\le 2j \\le k} {\\mathsf {P}}_{k-2j,d-2}(r) \\\\\n& \\ge  c \\, n^2 \\sum_{k=n}^{n+[{\\varepsilon} (d-2)n]}\n{\\mathsf {P}}_{k,d-2}(r) = c \\, n^2 \\Lambda_{n, d-2}(r).\n\\end{align*}\nHere $c>0$ depends only on ${\\varepsilon}$, $\\mu$, and $d$; we used that $n\\ge 1\/{\\varepsilon}$.\n\nEvidently, the above estimate leads to  \\eqref{lowerbd} using induction on $d$,\nprovided we prove \\eqref{lowerbd} for $d =1$ and $d=2$.\nHowever, the case $d =1$ is already established in \\cite[Proposition 2.4]{KPX},\nnamely,\n\\begin{equation}\\label{est-Lambda1}\n\\Lambda_{n, 1}(r)\\ge \\frac{cn}{{W_\\mu}(n;r)}.\n\\end{equation}\n\n\nIt remains to prove \\eqref{lowerbd} in the case $d =2$.\nThe proof relies on the well known identity \\cite[p. 59]{Askey}\n\\begin{equation}\\label{rel-Gegen}\nC_n^\\lambda(x) = \\sum_{0 \\le 2k \\le n}\n\\frac{\\Gamma(\\mu)(n-2k+\\mu)\\Gamma(k+\\lambda-\\mu)\\Gamma(n-k+\\lambda)}\n     {\\Gamma(\\lambda)\\Gamma(\\lambda-\\mu)k!\\Gamma(n-k+\\mu+1)}\\,\n     C_{n-2k}^\\mu(x)\n\\end{equation}\nand the product formula of Gegenbauer polynomials\n\\cite[Vol I, Sec. 3.15.1, (20)]{Edelyi}:\n\\begin{equation}\\label{prod-Gegen}\n \\frac{C_n^\\mu(s) C_n^\\mu(t)} {C_n^\\mu (1)}\n = b_1^{\\mu-1\/2}\\int_{-1}^1C_n^\\mu\\left(s t + u\\sqrt{1-s^2}\\sqrt{1-t^2}\\right)\n           (1-u^2)^{\\mu-1} du.\n\\end{equation}\nUsing\n\\eqref{rel-Gegen}  (with $\\lambda=\\mu+1\/2$)\nalong with \\eqref{compactPn} and then \\eqref{prod-Gegen}, we obtain\n\\begin{align}\\label{est-Pn2}\n{\\mathsf {P}}_{n,2}(r)\n&= b_2^\\mu \\frac{n+\\mu +1\/2}{\\mu+1\/2} \\sum_{0 \\le 2k \\le n}\nc_{k,n} \\frac{n-2k+\\mu}{\\mu} \\frac{[C_{n-2k}^\\mu(r)]^2}{C_{n-2k}^\\mu(1)} \\notag\\\\\n&= \\frac{b_2^\\mu}{b_1^\\mu} \\frac{n+\\mu +1\/2}{\\mu+1\/2} \\sum_{0 \\le 2k \\le n}\nc_{k,n} {\\mathsf {P}}_{n-2k,1}(r),\n\\end{align}\nwhere\n$$\nc_{k,n} = \\frac{\\Gamma(\\mu+1)\\Gamma(k+1\/2)\\Gamma(n-k + \\mu +1\/2)}\n               {\\Gamma(\\mu+1\/2)\\Gamma(1\/2)\\Gamma(n-k + \\mu +1) k!}.\n$$\nHere we used that the $L_2({w_\\mu})$-normalized Gegenbauer polynomial\n$\\widetilde{C}_n^\\mu$ can be written in the form\n$\\widetilde{C}_n^\\mu(x)=h_n^{-1\/2}C_n^\\mu(x)$ with\n$h_n:= (b_1^{\\mu})^{-1}\\frac{\\mu}{n+\\mu}C_n^\\mu(1)$,\nwhich is a matter of simple verification,\nand hence\n$$\n{\\mathsf {P}}_{n, 1}(r)\n=[\\widetilde{C}_n^\\mu(r)]^2\n=b_1^{\\mu}\\frac{n+\\mu}{\\mu}\\frac{[C_n^\\mu(r)]^2}{C_n^\\mu(1)}.\n$$\nIt is straightforward to verify that if $0 \\le k \\le n\/2$,\nthen $c_{k,n} \\sim (kn)^{-1\/2}$\nand hence $c_{k,n} \\ge c n^{-1}$.\nTherefore, from \\eqref{est-Pn2}\n\\begin{align*}\n\\Lambda_{n,2}(r)\n&= \\sum_{k=n}^{n+[2{\\varepsilon} n]} {\\mathsf {P}}_{k,2}(r)\n= \\frac{b_2^\\mu}{b_1^\\mu}\\sum_{k=n}^{n+[2{\\varepsilon} n]}\\frac{k+\\mu +1\/2}{\\mu+1\/2}\n\\sum_{0 \\le 2j \\le k}c_{j,k} {\\mathsf {P}}_{k-2j,1}(r) \\\\\n& \\ge c  \\sum_{k=n}^{n+[2{\\varepsilon} n]}\n\\sum_{0 \\le 2j \\le k}  {\\mathsf {P}}_{k-2j,1}(r)\n\\ge  c\\, n  \\Lambda_{n,1}(r).\n\\end{align*}\nThis combined with \\eqref{est-Lambda1} yields \\eqref{lowerbd} for\n$d=2$.\n\\end{proof\n\n\\medskip\n\nWe now continue with the proof of Theorem~\\ref{thm:est-Lp-norm}.\nApplying \\eqref{lowerbd} with ${\\varepsilon}=2\/3d$ yields\n$\n\\|L_n(x, \\cdot)\\|_2 \\ge cn^d{W_\\mu}(n;x)^{-1}\n$\nfor $n\\ge 2d$.\nIf~$2\\le n < 2d$, then as in the proof of Lemma~\\ref{L2-lower-bound} it follows that\n$$\n\\|L_n(x, \\cdot)\\|_2^{1\/2} \\ge c({\\mathsf {P}}_n(x, x) + {\\mathsf {P}}_{n+1}(x,x))\n\\ge c(C_n^\\mu(|x|) + C_{n+1}^\\mu(|x|)) >c>0\n$$\nfor all $x\\in B^d$,\nwhere we used the fact that the polynomials $C_n^\\mu$ and $C_{n+1}^\\mu$ have no common zeros.\nTaking into account that ${W_\\mu}(n;x) \\sim 1$ when $n\\le 2d$, the above leads again to\n$\n\\|L_n(x, \\cdot)\\|_2 \\ge cn^d{W_\\mu}(n;x)^{-1}.\n$\nThis completes the proof of estimate \\eqref{eq:est-Lp-norm}  for $p=2$.\n\n\n\nNow one easily derives\n\\eqref{eq:est-Lp-norm} for\n$p\\ne 2$ from the same estimate for $p=2$ and the upper bound estimate\n\\eqref{eq:LpUB}.\nIndeed, for $2<p<\\infty$ applying H\\\"{o}lder's inequality we get\n\\begin{align*}\n\\frac{cn^d}{{W_\\mu}(n,x)}\n&\\le \\int_{B^d}|L_n(x, y)|^2{w_\\mu}(y)dy\n\\le \\|L_n(x, \\cdot)\\|_p\\|L_n(x, \\cdot)\\|_{p'}\\\\\n&\\le c_1\\|L_n(x, \\cdot)\\|_p\\Big(\\frac{n^d}{{W_\\mu}(n,x)}\\Big)^{1-1\/p'}\n\\quad (1\/p+1\/p'=1),\n\\end{align*}\nwhich implies \\eqref{eq:est-Lp-norm}.\nOne proceeds similarly whenever $p=\\infty$.\n\nIf $0<p<2$, using \\eqref{eq:est-Lp-norm} for $p=2$ and\n\\eqref{eq:LpUB} for $p=\\infty$, we get\n\\begin{align*}\n\\frac{cn^d}{{W_\\mu}(n,x)}\n&\\le \\int_{B^d}|L_n(x, y)|^2{w_\\mu}(y)dy\n\\le \\int_{B^d}|L_n(x, y)|^p{w_\\mu}(y)dy\\|L_n(x, \\cdot)\\|_\\infty^{2-p}\\\\\n&\\le c_1\\int_{B^d}|L_n(x, y)|^p{w_\\mu}(y)dy\\Big(\\frac{n^d}{{W_\\mu}(n,x)}\\Big)^{2-p},\n\\end{align*}\nThis again leads to \\eqref{eq:est-Lp-norm}. The proof of\nTheorem~\\ref{thm:est-Lp-norm} is complete. $\\qed$\n\n\\medskip\n\n\n\\noindent {\\em Proof of Proposition~$\\ref{Nikolski}$.}\nLet $g\\in\\Pi_n$.\nAssume $1<q<\\infty$ and let $L_n$ be the kernel from $(\\ref{def-Ln})$,\nwith $\\widehat a$ admissible of type $(a)$.\nBy Lemma~\\ref{lem:Ker-n}, $g=L_n*g$.\nWe~use this, H\\\"older's inequality,\n\\eqref{eq:LpUB}, and that ${W_\\mu}(n;x)\\ge n^{-2\\mu}$\nto obtain\n$$\n   |g(x)| \\le \\|g\\|_q\\left(\\frac{n^d}{{W_\\mu}(n;x)}\\right)^{1\/q}\n         \\le c n^{(d+2 \\mu) \/q}\\|g\\|_q,\n\\quad x\\in B^d,\n$$\nand hence\n\\begin{equation}\\label{Nik1}\n\\|g\\|_\\infty \\le c n^{(d+2\\mu)\/q} \\|g\\|_q,\n\\quad 1<q \\le \\infty.\n\\end{equation}\nLet $0<q\\le 1$. The above inequality with $q=2$ yields\n$$\n\\|g\\|_\\infty^2\n\\le c n^{d+2\\mu} \\int_{-1}^1 |g(y)|^{2-q} |g(y)|^q {w_\\mu}(y)dy\n   \\le c n^{d+ 2\\mu}\\|g\\|_\\infty^{2-q}\\|g\\|_q^q.\n$$\nTherefore, (\\ref{Nik1}) holds for $0<q\\le 1$ as well.\n\nLet $0<q<p<\\infty$. Using (\\ref{Nik1}) we have\n\\begin{align*}\n\\|g\\|_p  & = \\left(\\int_{B^d} |g(x)|^{p-q} |g(x)|^q {w_\\mu}(x) dx \\right)^{1\/p}\\\\\n   & \\le c n^{(d+2\\mu)(\\frac{1}{q} - \\frac{1}{p})} \\|g\\|_q^{\\frac{p-q}{p} }\n     \\|g\\|_q^{\\frac{q}{p}}\n    =  c n^{(d+2\\mu )(\\frac{1}{q} - \\frac{1}{p})} \\|g\\|_q.\n\\end{align*}\nThus we have proved \\eqref{norm-relation}.\n$\\qed$\n\n\nWe next prove (\\ref{norm-relation2}). Assume first that $1<q<\\infty$.\nUsing again that $g=L_n*g$, H\\\"older's inequality ($1\/q + 1\/q'=1$),\nand \\eqref{eq:main-estI} we obtain for $x\\in B^d$,\n\\begin{align*}\n|g(x)|\n& \\le \\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q\n\\left( \\int_{B^d} \\Big|L_n(x,y) {W_\\mu}(n;y)^{-\\gamma-\\frac{1}{p}+\n\\frac{1}{q}}\\Big|^{q'} {w_\\mu}(y) dy \\right)^{1\/q'} \\\\\n& \\le c \\frac{n^d} {{W_\\mu}(n;x)^{1\/2}} \\left (\\int_{B^d}\n\\frac{{w_\\mu}(y)dy}{ {W_\\mu}(n;y)^{\\frac{q'}{2} + \\beta} (1+ n d(x,y))^\\sigma } \\right)^{1\/q'}\n\\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q,\n\\end{align*}\nwhere $\\beta = q'(\\gamma + \\frac{1}{p}-\\frac{1}{q})$.\nThe last integral can be estimated by using \\eqref{eq:LpUB}, yielding\n$$\n |g(x)|  \\le  c \\frac{n^{d\/q}}{{W_\\mu}(n;x)^{\\gamma +1\/p}}\n     \\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q.\n$$\nHence\n\\begin{equation} \\label{aaa2}\n\\|{W_\\mu}(n;\\cdot)^{\\gamma+1\/p}g(\\cdot)\\|_\\infty \\le c n^{d\/q}\n\\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q,\n\\quad 1<q\\le \\infty.\n\\end{equation}\nLet $0 < q \\le 1$. Then by (\\ref{aaa2}) with $q=2$ we have\n\\begin{align*}\n \\|{W_\\mu}(n;\\cdot)^{\\gamma+1\/p}g(\\cdot)\\|_\\infty\n&\\le c n^{d\/2}\n \\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{2}} g(\\cdot)\\|_2 \\\\\n & \\le c n^{d\/2}  \\| {W_\\mu}(n;\\cdot)^{\\gamma+1\/p}g(\\cdot)\\|_\\infty^{1-q\/2}\n \\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q^{q\/2}.\n\\end{align*}\nTherefore, \\eqref{aaa2} holds for $0 < q \\le 1$ as well.\n\nLet $p<\\infty$.\nUsing (\\ref{aaa2}), we have\n\\begin{align*}\n\\|{W_\\mu} & (n;\\cdot)^\\gamma g(\\cdot)\\|_p\n= \\left( \\int_{B^d} \\left[{W_\\mu}(n;x)^\\gamma g(x)\\right]^{p-q}\n\\left[{W_\\mu}(n;x)^\\gamma g(x)\\right]^{q}{w_\\mu}(x) dx \\right)^{1\/p} \\\\\n\\le & c n^{d(\\frac{1}{q} - \\frac{1}{p})}\n\\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q^{p-q}\n\\left( \\int_{B^d} \\frac{\\left[{W_\\mu}(n;x)^\\gamma g(x)\\right]^{q}}\n{{W_\\mu}(n;x)^{\\frac{p-q}{p}}} {w_\\mu}(x) dx \\right)^{1\/p} \\\\\n= & c n^{d(\\frac{1}{q}-\\frac{1}{p})}\n\\|{W_\\mu}(n;\\cdot)^{\\gamma +\\frac{1}{p} - \\frac{1}{q}} g(\\cdot)\\|_q.\n\\end{align*}\nHence \\eqref{norm-relation2} holds for $p < \\infty$. If $p =\n\\infty$ \\eqref{norm-relation2} follows from \\eqref{aaa2}.\n $\\qed$\n\n\\bigskip\n\n\n\n\\noindent {\\em Proof of $(\\ref{norm-Needlets2})$.} From\n\\eqref{local-Needlets21} with $k$ sufficiently large ($k >d+2\\mu$\nwill do), and (\\ref{norm-Needlets}), we infer for $0<r\\le\\pi$\n\\begin{equation*}\n\\begin{aligned}\n0&<c_1\\le \\norm{{\\varphi}_\\xi}_2\\\\\n&\\le \\norm{{\\varphi}_\\xi}_{L_\\infty(B_\\xi(r))}m(B_\\xi(r))\n+c 2^{jd}\\int_{B^d\\setminus B_\\xi(r)}\\frac{{w_\\mu}(y)}{{W_\\mu}(2^j;y)(1+2^jd(\\xi,y))^{2k}}\\, dy\\\\\n&\\le \\norm{{\\varphi}_\\xi}_{L_\\infty(B_\\xi(r))}m(B_\\xi(r))\n+c\\frac{2^{jd}}{(1+2^j r)^k}\\int_{B^d}\\frac{{w_\\mu}(y)}{{W_\\mu}(2^j;y)(1+2^jd(\\xi,y))^k}\\, dy\\\\\n&\\le \\norm{{\\varphi}_\\xi}_{L_\\infty(B_\\xi(r))} m(B_\\xi(r))+ \\frac{c_2}{1+2^j r},\n\\end{aligned}\n\\end{equation*}\nwhere $c_2$ depends only on $k$, $d$, and $\\mu$.\nFor the last inequality we used Lemma~\\ref{lem:instrumental} with $p=2$.\nLet $r:=c^* 2^{-j}$, where $c^*>0$ is selected so that\n$ c_2\/(1+2^j r) = c_2\/(1+c^*) < c_1\/2$. Then from above\n$$\n\\norm{{\\varphi}_\\xi}_{L_\\infty (B_\\xi(c^*2^{-j}))}\n\\ge \\frac{c}{m(B_\\xi(c^*2^{-j}))}\\ge c \\Big(\\frac{2^{jd}}{{W_\\mu}(2^j;\\xi)}\\Big)^{1\/2}.\n$$\nA similar estimate holds for $\\psi_\\xi$ as well.  $\\qed$\n\n\n\n\\subsection{Proofs for Sections 4-5}\\label{proofsB}${}$\n\n\\medskip\n\n\\noindent {\\em Proof of Lemma $\\ref{lem:Phi*psi}$.}\nUsing the orthogonality of the subspaces $\\mathcal{V}_n^d$, we have\n$\\Phi_j\\ast \\psi_\\xi(x)=0$ if $\\xi\\in{\\mathcal X}_\\nu$ for $\\nu\\ge j+2$ or\n$\\nu\\le j-2$.\n\nLet $\\xi \\in {\\mathcal X}_\\nu$, $j-1\\le \\nu\\le j+1$.\nFrom the localization of the kernels $\\Phi_j$, given in (\\ref{local-Needlets2}),\nand the needlet localization from (\\ref{local-Needlets21}) it follows that for any $k>0$\nthere is a constant $c_k>0$ such that\n\\begin{equation*}\n\\begin{aligned\n&|\\Phi_j\\ast \\psi_\\xi(x)|\n\\le  c_k \\frac{2^{j3d\/2}}{\\sqrt{{W_\\mu}(2^j;x)}}\\int_{B^d}\\frac{{w_\\mu}(y)}\n{\\sqrt{ {W_\\mu}(2^j;y)}(1+2^jd(x,y))^k(1+2^jd(y,\\xi))^k}\\, dy.\n\\end{aligned}\n\\end{equation*}\nDenote\n$$\n\\Omega_\\xi:=\\{y\\in B^d: d(y,\\xi)\\ge d(x,\\xi)\/2\\} \\text{ and }\n\\Omega_x:=\\{y\\in B^d: d(x, y)\\ge d(x,\\xi)\/2\\}.\n$$\nEvidently, $B^d=\\Omega_{\\xi}\\cup \\Omega_x$ and hence\n\\begin{align*}\n|\\Phi_j\\ast \\psi_\\xi(x)|\n&\\le c_k\\frac{2^{j3d\/2}}{\\sqrt{ {W_\\mu}(2^j;x)}(1+2^jd(x, \\xi))^k} \\int_{\\Omega_\\xi}\n \\frac{{w_\\mu}(y)}{{W_\\mu}(2^j; y)(1+2^jd(x,y))^k}\\, dy\\\\\n %\n&+ c_k\\frac{2^{j3d\/2}}{\\sqrt{ {W_\\mu}(2^j;x) }(1+2^jd(x, \\xi))^k}\n\\int_{\\Omega_x} \\frac{ {w_\\mu}(y)}{{W_\\mu}(2^j; y)(1+2^jd(y,\\xi))^k}\\, dy\\\\\n&=:J_1+J_2.\n\\end{align*}\nWe may assume that $k>d$. Then employing Lemma~\\ref{lem:instrumental} with $p=2$,\nwe get\n\\begin{align*}\n\\int_{\\Omega_\\xi}\n\\frac{ {w_\\mu}(y)}{{W_\\mu}(2^j; y)(1+2^jd(x,y))^k}\\, dy\\le \\int_{B^d}\n\\frac{ {w_\\mu}(y)}{{W_\\mu}(2^j; y)(1+2^jd(x,y))^k}\\, dy\\le c2^{-jd},\n\\end{align*}\nwhich yields\n$$\nJ_1\\le c\\frac{2^{jd\/2}}{\\sqrt{{W_\\mu}(2^j;x)}(1+2^jd(x, \\xi))^k}.\n$$\nOne similarly estimates $J_2$. This completes the proof of the\nlemma.  $\\qed$\n\n\\medskip\n\n\n\\noindent {\\em Proof of Lemma~$\\ref{lem:Max-needl}$.}\n Estimate \\eqref{Max-needl1} follows readily from the localization of\n the needlets (see (\\ref{local-Needlets21})) and the lower bound estimate from (\\ref{B-max3})\n taking into account that\n $R_\\xi \\subset B_\\xi(c^\\diamond 2^{-j})$ for $\\xi\\in{\\mathcal X}_j$.\n\nWe now prove (\\ref{Max-needl2}).\nBy the lower bound estimate (\\ref{norm-Needlets2}) it follows that there exists\n$\\omega\\in B_\\xi(c^*2^{-j})$ such that\n\\begin{equation}\\label{eq:needlUB}\n|{\\varphi}_\\xi(\\omega)|\\ge c  \\frac{2^{jd\/2}}{\\sqrt{{W_\\mu}(2^j; \\xi)}}.\n\\end{equation}\nAlso, by (\\ref{Lip-Needlets}) it follows that for every $x\\in B_\\omega(2^{-j})$\n\\begin{equation}\\label{eq:needl-smooth}\n |{\\varphi}_\\xi(\\omega)-{\\varphi}_\\xi(x)|\n \\le c\\frac{2^{j(d\/2+1)}d(\\omega,x)}{\\sqrt{{W_\\mu}(2^j; \\xi)}}.\n\\end{equation}\nBy (\\ref{eq:needlUB})-(\\ref{eq:needl-smooth}) it follows that\nfor a sufficiently small constant $\\hat{c}>0$\n$$\n|{\\varphi}_\\xi(x)|\n\\ge |{\\varphi}_\\xi(\\omega)|-|{\\varphi}_\\xi(\\omega)-{\\varphi}_\\xi(x)|\n\\ge c\\frac{2^{jd\/2}}{\\sqrt{{W_\\mu}(2^j; \\xi)}}\n\\ge c{\\tilde{\\mathbbm 1}}_{B_\\omega(\\hat{c}2^{-j})}(x),\n\\quad x\\in B_\\omega(\\hat{c}2^{-j}),\n$$\nwhich yields\n\\begin{equation*}\n({\\mathcal M}_t{\\varphi}_\\xi)(x)\n\\ge c ({\\mathcal M}_t{\\tilde{\\mathbbm 1}}_{B_\\omega(\\hat{c}2^{-j})})(x)\n\\ge c{\\tilde{\\mathbbm 1}}_{B_\\xi(2^{-j})}(x)\n\\ge c{\\tilde{\\mathbbm 1}}_{R_\\xi}(x),\n\\quad x\\in B^d,\n\\end{equation*}\nwhere in the second inequality we used (\\ref{B-max3}).\n\nOne similarly shows that ${\\mathcal M}_t\\psi_\\xi \\ge c{\\tilde{\\mathbbm 1}}_{R_\\xi}$.\n $\\qed$\n\n\\medskip\n\n\n\n\n\\noindent {\\em Proof of Lemma~$\\ref{lem:weak_inequality}$.} For\nthe proof of this lemma we need a couple of additional lemmas.\n\n\n\\begin{lemma}\\label{lem:sums}\nLet $k>d$ and $j\\ge 0$. Then\n\\begin{equation} \\label{eq:sum1}\n\\sum_{\\xi\\in {\\mathcal X}_j} \\frac{1}{(1+2^jd(x,\\xi))^k}\\le c,\n\\quad x\\in B^d,\n\\end{equation}\nand for any $\\xi,\\eta\\in B^d$\n\\begin{equation}\\label{eq:sum2}\n\\sum_{w\\in {\\mathcal X}_j}\\frac{1}{(1+2^jd(\\xi,w))^k(1+2^jd(\\eta,w))^{k}}\n\\le c\\frac{1}{(1+2^jd(\\xi,\\eta))^{k}}.\n\\end{equation}\n\\end{lemma}\n\n\\begin{proof}Fix $\\xi\\in{\\mathcal X}_j$. Evidently, $1+2^jd(x,\n\\xi)\\sim 1+2^jd(x, y)$ for $y \\in R_\\xi$, and by\n\\eqref{norm-dist2}\n$$\n|\\sqrt{1-|\\xi|^2}-\\sqrt{1-|y|^2}|\\le \\sqrt{2}\\,d(\\xi, y)\\le c2^{-j},\n\\quad y\\in R_\\xi,\n$$\nwhich implies\n$$\n|R_\\xi|\\sim 2^{-jd}(\\sqrt{1-|\\xi|^2}+2^{-j})\n\\sim 2^{-jd}(\\sqrt{1-|y|^2}+2^{-j}),\n\\quad y\\in R_\\xi.\n$$\nWe use the above to obtain\n\\begin{align*}\n\\sum_{\\xi\\in{\\mathcal X}_j}\\frac{1}{(1+2^jd(x, \\xi))^k}\n&\\le c\\sum_{\\xi\\in{\\mathcal X}_j}\\frac{1}{|R_\\xi|} \\int_{R_\\xi}\\frac{1}{(1+2^jd(x,y))^k}\\,dy\\\\\n&\\le c2^{jd}\\int_{B^d}\\frac{1}{(\\sqrt{1-|y|^2}+2^{-j})(1+2^jd(x,y))^k}\\,dy\n\\le c.\n\\end{align*}\nHere for the last inequality we used Lemma~\\ref{lem:instrumental}\nwith $p=2$ and $\\mu=1\/2$.\n\n\nFor the proof of \\eqref{eq:sum2}, assume that $\\xi\\ne \\eta$ and denote\n$$\n{\\mathcal X}_j(\\xi):=\\{w\\in {\\mathcal X}_j: d(\\xi,w)\\ge d(\\xi,\\eta)\/2\\}, \\qua\n{\\mathcal X}_j(\\eta):=\\{w\\in {\\mathcal X}_j: d(\\eta,w)\\ge d(\\xi,\\eta)\/2\\}.\n$$\nThen\n\\begin{align*}\n&\\sum_{w\\in {\\mathcal X}_j}\\frac{1}{(1+2^jd(\\xi,w))^k(1+2^jd(\\eta,w))^{k}}\n\\le c\\frac{1}{(1+2^jd(\\xi,\\eta))^k}\n\\sum_{w\\in {\\mathcal X}_j(\\xi)}\\frac{1}{(1+2^jd(\\eta,w))^{k}}\\\\\n&\\qquad\\qquad\\quad + c\\frac{1}{(1+2^jd(\\xi,\\eta))^k}\n\\sum_{w\\in {\\mathcal X}_j(\\eta)}\\frac{1}{(1+2^jd(\\xi,w))^{k}}\\\\\n&\\qquad\\qquad\\quad \\le c\\frac{1}{(1+2^jd(\\xi,\\eta))^k}\n\\Big(\\sum_{w\\in {\\mathcal X}_j}\\frac{1}{(1+2^jd(\\eta,w))^{k}}\n+\\sum_{w\\in {\\mathcal X}_j}\\frac{1}{(1+2^jd(\\xi,w))^{k}}\\Big)\\\\\n&\\qquad\\qquad\\quad \\le c\\frac{1}{(1+2^jd(\\xi,\\eta))^k},\n\\end{align*}\nwhere for the last inequality we used \\eqref{eq:sum1}.\n \\end{proof\n\n\n\n\\begin{lemma}\\label{lem:quasi-Lip}\nAssume $P\\in \\Pi_{2^j}$ $(j \\ge 0)$, $\\xi\\in {\\mathcal X}_j$, and\nlet $x_1,x_2\\in B^d$ and\n$d(x_\\nu,\\eta)\\le \\tilde{c}2^{-j}$, $\\nu=1,2$.\nFor any $k>0$\n$$\n|P(x_1)-P(x_2)|\\le c 2^j d(x_1, x_2)\n\\sum_{\\xi\\in{\\mathcal X}_j}\\frac{|P(\\xi)|}{(1+2^j d(\\eta, \\xi))^k},\n$$\nwhere $c>0$ depends only on $d$, $k$, $\\mu$, and $\\tilde{c}$.\n\\end{lemma}\n\n\\begin{proof}Fix $P\\in\\Pi_{2^j}$ and assume that\n$L_{2^j}$ is the reproducing kernel from Lemma~\\ref{lem:Ker-n}\nwith $n=2^j$. Then, $L_{2^j}*P=P$. Since $L_{2^j}(x,\n\\cdot)P(\\cdot) \\in \\Pi_{2^{j+2}}$, and the cubature formula\n(\\ref{eq:cubature}) is exact for all polynomials from\n$\\Pi_{2^{j+2}}$ we have\n$$\nP(x)=\\int_{B^d} L_{2^j}(x,y)P(y){w_\\mu}(y)dy\n=\\sum_{\\xi\\in{\\mathcal X}_j}\\lambda_\\xi L_{2^j}(x,\\xi)P(\\xi),\n\\quad x\\in B^d.\n$$\nWe use \\eqref{Lip} to obtain\nfor $x_1, x_2\\in B^d$ with $d(x_\\nu, \\eta)\\le \\tilde{c}2^{-j}$, $\\nu=1, 2$,\n\\begin{align*}\n|P(x_1)-P(x_2)|\n&=\\Big|\\int_{B^d}[{L}_{2^j}(x_1,y)-{L}_{2^j}(x_2,y)]P(y){w_\\mu}(y) \\, dy\\Big|\\\\\n&\\le\\sum_{\\xi\\in {\\mathcal X}_j}\n|\\lambda_{\\xi}||{L}_{2^j}(x_1,\\xi)- {L}_{2^j}(x_2,\\xi)||P(\\xi)|\\\\\n&\\le c2^{j} d(x_1,x_2) \\sum_{\\xi\\in {\\mathcal X}_j}\n\\Big(\\frac{{W_\\mu}(2^j;\\xi)}{{W_\\mu}(2^j;\\eta)}\\Big)^{1\/2}\n\\frac{|P(\\eta)|}{(1+2^jd(\\xi,\\eta))^{k}}\\\\\n&\\le c2^j d(x_1,x_2) \\sum_{\\eta\\in {\\mathcal X}_j}\n\\frac{|P(\\eta)|}{(1+2^jd(\\xi,\\eta))^{k-2\\mu}}.\n\\end{align*}\nHere we used that $\\lambda_\\xi\\sim 2^{-jd}{W_\\mu}(2^j;\\xi)$ and\nfor the last inequality we used (\\ref{eq:useful}).\nTaking into account that $k>0$ can be arbitrarily large the result follows.\n \\end{proof\n\n\n\n\n\\medskip\n\n\\noindent {\\em Completion of the proof of Lemma\n$\\ref{lem:weak_inequality}$.} Since $b_\\xi\\le a_\\xi,$ it trivially\nfollows that $b_\\xi^*\\le a_\\xi^*$.\n\nFor the other direction let\n$$\nd_\\xi:=\\max\\{|P(x_1)-P(x_2)|: x_1\\in R_\\xi, d(x_1,x_2)\\le  2^{-j-r}\\}.\n$$\nObviously $a_\\xi\\le b_\\xi +d_\\xi$.\nNow Lemma \\ref{lem:quasi-Lip} yields\n$$\nd_\\xi\\le c2^{-r} \\sum_{\\eta\\in{\\mathcal X}_j}\\frac{|P(\\eta)|}{(1+2^j d(\\xi, \\eta))^k},\n\\quad \\xi\\in {\\mathcal X}_j.\n$$\nFrom the definition of $d_\\xi^*$ in (\\ref{def.h-star}) we infer\n\\begin{align*}\nd_\\xi^\\ast&\\le c 2^{-r} \\sum_{w\\in {\\mathcal X}_j}\\sum_{\\eta\\in{\\mathcal X}_j}\n\\frac{|P(\\eta)|}{(1+2^j d(w,\\eta))^k(1+2^jd(\\xi,w))^k}\\\\\n&\\le c 2^{-r}\\sum_{\\eta\\in {\\mathcal X}_j}\\frac{|P(\\eta)|}{(1+2^j d(\\eta, \\xi))^k}\n\\le c 2^{-r}a_\\xi^*,\n\\end{align*}\nwhere for the second inequality we interchanged the order of summation and\nused Lemma \\ref{lem:sums}.\nHence,\n$a_\\xi^*\\le b_\\xi^*+d_\\xi^* \\le b_\\xi^*+c2^{-r}a_\\xi^*$ with $c>0$\nindependent of $r$.\nBy selecting $r$ sufficiently large we get\n$a_\\xi^*\\le cb_\\xi^\\ast$.\n $\\qed$\n\n\\medskip\n\n\n\n\\noindent {\\em Proof of Lemma $\\ref{lem:disc-max}$.} We first\nprove Lemma \\ref{lem:disc-max} in the case ${\\rho}=0$. We fix\n$\\xi\\in{\\mathcal X}_j$ and define $ S_0:=\\{\\eta\\in {\\mathcal X}_j: d(\\eta,\\xi)\\le\nc^\\diamond 2^{-j}\\} $ and\n$$\nS_m :=\\{\\eta\\in {\\mathcal X}_j: c^\\diamond 2^{-j+m-1}\n< d(\\eta,\\xi)\\le c^\\diamond 2^{-j+m}\\}, \\quad m\\ge 1,\n$$\nwhere $c^\\diamond$ is the constant from Proposition \\ref{prop:cubature}.\nBy Definition~\\ref{def.eq-dist-p} it follows that $\\# S_m\\le c2^{md}$.\nLet us also set\n$$\nB_m := B_\\xi(c^\\diamond(2^m+1)2^{-j}), \\quad m\\ge 0.\n$$\nEvidently,\n$R_\\eta\\subset B_m$ for $\\eta\\in S_\\nu$, $0\\le \\nu\\le m$.\nMoreover, if $\\eta\\in S_m$, then\n$$\nd(\\xi,\\partial B^d)\\le d(\\xi,\\eta)+d(\\eta,\\partial B^d)\n\\le c^\\diamond 2^{-j+m}+d(\\eta,\\partial B^d).\n$$\nHence, using \\eqref{ball-size}, we get\n\\begin{equation}\\label{quotient}\n\\begin{aligned}\n\\frac{m(B_m)}{m(R_\\eta)}\n&\\le 2^{md}\\left(\\frac{d(\\xi,\\partial B^d)+ 2^{-j+m}}{d(\\eta,\\partial B^d)+ 2^{-j}}\\right)^{2\\mu}\\\\\n&\\le c 2^{md}\\left(\\frac{d(\\eta,\\partial B^d)+ 2^{-j+m}}{d(\\eta,\\partial B^d)+ 2^{-j}}\\right)^{2\\mu}\n\\le c 2^{m(d+2\\mu)}.\n\\end{aligned}\n\\end{equation}\n\nSet $\\gamma:=\\max \\{0, 1-\\frac1{t}\\}<1$.\nUsing H\\\"older's inequality if $t> 1$ and\nthe $t$-triangle inequality if $0<t\\le1$, we get\n\\begin{align*}\nb_\\xi^{\\ast}=\n\\sum_{\\eta\\in{\\mathcal X}_j}\\frac{|b_\\eta|}{(1+2^jd(\\eta,\\xi))^\\sigma}\n\\le c\\sum_{m\\ge 0}2^{-m\\sigma}\\sum_{\\eta\\in S_m} |b_\\eta|\n\\le c\\sum_{m\\ge 0}2^{-m(\\sigma-d\\gamma)}(\\sum_{\\eta\\in S_m} |b_\\eta|^t)^{1\/t}.\n\\end{align*}\nWe now use (\\ref{quotient}) to obtain, for $x\\in R_\\xi$,\n\\begin{align*}\nb_\\xi^{\\ast}\n&=c \\sum_{m=0}^\\infty 2^{-m(\\sigma-d)}\\Bigl(\\int_{B^d}\n\\Big[\\sum_{\\eta\\in S_m}\n|b_\\eta|m(R_\\eta)^{-1\/t}{\\mathbbm 1}_{R_\\eta}(y)\\Big]^t{w_\\mu}(y)\\, dx\\Bigr)^{1\/t}\\\\\n&\\le c \\sum_{m=0}^\\infty 2^{-m(\\sigma-d)}\\Bigl(\\frac1{m(B_m)}\\int_{B_m}\n\\Big[\\sum_{\\eta\\in S_m}\n\\Big(\\frac{m(B_m)}{m(R_\\eta)}\\Big)^{1\/t}|b_\\eta|{\\mathbbm 1}_{R_\\eta}(y)\\Big]^t{w_\\mu}(y)\\,dy\\Bigr)^{1\/t}\\\\\n&\\le c\\sum_{m\\ge 0}2^{-m(\\sigma-d-(d+2\\mu)\/t)}\n\\Bigl(\\frac1{m(B_m)}\\int_{B_m}\n\\Big[\\sum_{\\eta\\in S_m}|b_\\eta|{\\mathbbm 1}_{R_\\eta}(y)\\Big]^t{w_\\mu}(y)\\, dy\\Bigr)^{1\/t}\\\\\n&\\le c{\\mathcal M}_t\\Big(\\sum_{w\\in {\\mathcal X}_j}|b_w|{\\mathbbm 1}_{R_\\omega}\\Big)(x),\n\\end{align*}\nwhere for the last inequality we used that $\\sigma>d+(d+2\\mu)\/t$.\n\nConsider now the general case.\nUsing (\\ref{eq:useful}) we have for $\\xi\\in{\\mathcal X}_j$\n\\begin{align*}\n{W_\\mu}(2^j;\\xi)^{\\gamma}b_\\xi^*\n&\\le \\sum_{\\eta\\in{\\mathcal X}_j}\\frac{{W_\\mu}(2^j;\\xi)^{\\gamma}|b_\\eta|}{(1+2^jd(\\xi, \\eta))^\\sigma}\n\\le c\\sum_{\\eta\\in{\\mathcal X}_j}\\frac{{W_\\mu}(2^j;\\eta)^{\\gamma}|b_\\eta|}\n                              {(1+2^jd(\\xi, \\eta))^{\\sigma-2\\mu |\\gamma|}}\\\\\n&\\le c\\Big({W_\\mu}(2^j;\\xi)^{\\gamma}|b_\\xi|\\Big)^*,\n\\end{align*}\nwhere we used that $\\sigma>d+(d+2\\mu)\/t+2\\mu |\\gamma|$. Now\n(\\ref{disc-max}) in the general case follows by the same\ninequality in the case $\\rho=0$ established above.  $\\qed$\n\n\n\\medskip\n\n\n\\noindent {\\em Proof of Lemma $\\ref{l:half_shannon}$.} For any\n$\\xi\\in{\\mathcal X}_j$, we denote $a_\\xi:=\\max_{x\\in R_\\xi}|P(x)|$,\n$$\nm_\\xi:=\\min_{x\\in R_\\xi}|P(x)|,\n\\quad \\mbox{ and }\\quad\nb_\\xi:=\\max\\{\\min_{x\\in R_w }|P(x)|:w\\in {\\mathcal X}_{j+r},R_w\\cap R_\\xi\\ne\\emptyset \\},\n$$\nwhere $r\\ge 1$ is the constant from Lemma \\ref{lem:weak_inequality}.\n\nChoose $0<t<p$. By Lemma ~\\ref{lem:weak_inequality} we have $a_\\xi^* \\le cb_\\xi^*$.\nWe use this, Lemmas~\\ref{lem:disc-max}, and the maximal inequality \\eqref{max-ineq}\nto obtain\n\\begin{equation}\n\\begin{aligned}\\label{Wa<Wb}\n\\Big(\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma a_\\xi^p m(R_\\xi)\\Big)^{1\/p}\n&=\\Big\\|\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma a_\\xi {\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big\\|_{{p}}\\\\\n\\le c\\Big\\|\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma b_\\xi^\\ast {\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big\\|_{{p}}\n&\\le c\\Big\\|{\\mathcal M}_t\\Big(\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma b_\\xi  {\\mathbbm 1}_{R_\\xi}\\Big)(\\cdot)\\Big\\|_{{p}}\\\\\n&\\le c\\Big\\|\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma b_\\xi {\\mathbbm 1}_{R_\\xi}(\\cdot)\\Big\\|_{{p}}.\n\\end{aligned}\n\\end{equation}\nNow, exactly as in the proof of Theorem~\\ref{thm:Fnorm-equivalence} (see \\eqref{W-b-xi-m-eta})\nwe have\n\\begin{equation}\\label{eq:important}\nb_\\xi{W_\\mu}(2^j; \\xi)^\\gamma{\\mathbbm 1}_{R_\\xi}\n\\le \\sum_{\\eta\\in {\\mathcal X}_{j+r}(\\xi)}  m^\\ast_\\eta {W_\\mu}(2^{j+r}; \\eta)^\\gamma {\\mathbbm 1}_{R_\\eta}.\n\\end{equation}\nwhere\n${\\mathcal X}_{j+r}(\\xi):=\\{w\\in {\\mathcal X}_{j+r}:R_w\\cap R_\\xi\\ne \\emptyset\\}$.\nCombining this with (\\ref{Wa<Wb}) and using that $\\# {\\mathcal X}_{j+r}(\\xi)\\le c$,\nLemmas~\\ref{lem:disc-max}, and the maximal inequality \\eqref{max-ineq},\nwe get\n\\begin{equation*}\n\\begin{aligned}\n\\Big(\\sum_{\\xi\\in {\\mathcal X}_j}{W_\\mu}(2^j; \\xi)^\\gamma a_\\xi^p m(R_\\xi)\\Big)^{1\/p}\n&\\le c\\Big\\|\\sum_{\\eta\\in {\\mathcal X}_{j+r}}m^\\ast_\\eta {W_\\mu}(2^{j+r}; \\eta)^\\gamma{\\mathbbm 1}_{R_\\eta}(\\cdot)\\Big\\|_{p}\\\\\n\\le c\\Big\\|{\\mathcal M}_t\\Big(\\sum_{\\eta\\in {\\mathcal X}_{j+r}}m_\\eta{W_\\mu}(2^{j+r}; \\eta)^\\gamma{\\mathbbm 1}_{R_\\eta}\\Big)(\\cdot)\\Big\\|_{p}\n&\\le c\\Big\\|\\sum_{\\eta\\in {\\mathcal X}_{j+r}}m_\\eta {W_\\mu}(2^{j+r}; \\eta)^\\gamma {\\mathbbm 1}_{R_\\eta}(\\cdot)\\Big\\|_{p}\\\\\n&\\le c\\norm{P}_{p}.\n\\end{aligned}\n\\end{equation*}\nHere for the forth inequality we used that ${W_\\mu}(2^{j+r}; \\eta)\\sim\n{W_\\mu}(2^{j}; x)$ if $x\\in \\mathbb{R}_\\eta$, $\\eta\\in{\\mathcal X}_{j+r}$. $\\qed$\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:introduction}}\n\\section{Introduction}\n\\label{sec:introduction}\n\\begin{figure*}[!t]\n\\center\n\\includegraphics[width=0.95\\textwidth]{Figures\/CamVidTeaser.pdf}\n\\caption{\\footnotesize{SegNet predictions on urban and highway scene test samples from the wild. The class colour codes can be obtained from Brostow \\emph{et al.} \\cite{GabeDataset}. To try our system yourself please see our online web demo at \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/}. }}\n\\label{Teaser}\n\\end{figure*} \n\nSemantic segmentation has a wide array of applications ranging from scene understanding, inferring support-relationships among objects to autonomous driving. Early methods that relied on low-level vision cues have fast been superseded by popular machine learning algorithms. In particular, deep learning has seen huge success lately in handwritten digit recognition, speech, categorising whole images and detecting objects in images \\cite{DBLP:journals\/corr\/SzegedyLJSRAEVR14, Simonyan}. Now there is an active interest for semantic pixel-wise labelling \\cite{FarabetPAMI} \\cite{raey},  \\cite{socher2011parsing},\\cite{FCN},\\cite{NohDeconvNets},\\cite{CRFRNN},\\cite{ParseNetRabinovich},\\cite{SegNetarXiv},\\cite{eigen2014predicting}, \\cite{DeepLab}. \nHowever, some of these recent approaches have tried to directly adopt deep architectures designed for category prediction to pixel-wise labelling \\cite{FarabetPAMI}. The results, although very encouraging, appear coarse \\cite{DeepLab}. This is primarily because max pooling and sub-sampling reduce feature map resolution. Our motivation to design SegNet arises from this need to map low resolution features to input resolution for pixel-wise classification. This mapping must produce features which are useful for accurate boundary localization. \n\nOur architecture, SegNet, is designed to be a \\textit{core segmentation engine} for pixel-wise semantic segmentation. It is primarily motivated by road scene understanding applications which require the ability to model appearance (road, building), shape (cars, pedestrians) and understand the spatial-relationship (context) between different classes such as road and side-walk. In typical road scenes, the majority of the pixels belong to large classes such as road, building and hence the network must produce smooth segmentations. The engine must also have the ability to delineate moving and other objects based on their shape despite their small size. Hence it is important to retain boundary information in the extracted image representation. From a computational persepective, it is necessary for the network to be efficient in terms of both memory and computation time during inference. It must also be able to train end-to-end in order to jointly optimise all the weights in the network using an efficient weight update technique such as stochastic gradient descent (SGD) \\cite{Bottou}. Networks that are trained end-to-end or equivalently those that do not use multi-stage training \\cite{FCN} or other supporting aids such as region proposals \\cite{NohDeconvNets} help establish benchmarks that are more easily repeatable. The design of SegNet arose from a need to match these criteria.\n\nThe encoder network in SegNet is topologically identical to the convolutional layers in VGG16 \\cite{simonyan2014very}. We remove the fully connected layers of VGG16 which makes the SegNet encoder network significantly smaller and easier to train than many other recent architectures \\cite{FCN,NohDeconvNets,ParseNetRabinovich,DecoupledNet}. The key component of SegNet is the decoder network which consists of a hierarchy of decoders one corresponding to each encoder. Of these, the appropriate decoders use the max-pooling indices received from the corresponding encoder to perform non-linear upsampling of their input feature maps. This idea was inspired from an architecture designed for unsupervised feature learning \\cite{Ranzato}. Reusing max-pooling indices in the decoding process has several practical advantages; (i) it improves boundary delineation , (ii) it reduces the number of  parameters enabling end-to-end training, and (iii) this form of upsampling can be incorporated into any encoder-decoder architecture such as \\cite{FCN,CRFRNN} with only a little modification. \n\nOne of the main contributions of this paper is our analysis of the SegNet decoding technique and the widely used Fully Convolutional Network (FCN) \\cite{FCN}. This is in order to convey the practical trade-offs involved in designing segmentation architectures. Most recent deep architectures for segmentation have identical encoder networks, i.e VGG16, but differ in the form of the decoder network, training and inference. Another common feature is they have trainable parameters in the order of hundreds of millions and thus encounter difficulties in performing end-to-end training \\cite{NohDeconvNets}. The difficulty of training these networks has led to multi-stage training \\cite{FCN}, appending networks to a pre-trained core segmentation engine such as FCN \\cite{CRFRNN}, use of supporting aids such as region proposals for inference \\cite{NohDeconvNets}, disjoint training of classification and segmentation networks \\cite{DecoupledNet} and use of additional training data for pre-training \\cite{ParseNetRabinovich} \\cite{mottaghi2014role} or for full training \\cite{CRFRNN}. In addition, performance boosting post-processing techniques \\cite{DeepLab} have also been popular. Although all these factors improve performance on challenging benchmarks \\cite{Pascal}, it is unfortunately difficult from their quantitative results to disentangle the key design factors necessary to achieve good performance. We therefore analysed the decoding process used in some of these approaches \\cite{FCN,NohDeconvNets} and reveal their pros and cons.\n\nWe evaluate the performance of SegNet on PascalVOC12 salient object(s) segmentation \\cite{Pascal}, \\cite{hariharan2011semantic} and scene understanding challenges such as CamVid road scene segmentation \\cite{GabeDataset} and SUN RGB-D indoor scene segmentation \\cite{song2015sun}. Pascal VOC12 \\cite{Pascal} is  the benchmark for segmentation due to its size and challenges, but the majority of this task has one or two foreground classes surrounded by a highly varied background. This implicitly favours techniques used for detection as shown by the recent work on a decoupled classification-segmentation network \\cite{DecoupledNet} where the classification network can be trained with a large set of weakly labelled data and the independent segmentation network performance is improved. The method of \\cite{DeepLab} also use the feature maps of the classification network with an independent CRF post-processing technique to perform segmentation. The performance can also be boosted by the use additional inference aids such as region proposals \\cite{NohDeconvNets}, \\cite{zitnick2014edge}. Therefore, it is different from scene understanding where the idea is to exploit co-occurences of objects and other spatial-context to perform robust segmentation. To demonstrate the efficacy of SegNet, we present a real-time online demo of road scene segmentation into 11 classes of interest for autonomous driving (see Fig. \\ref{Teaser}). Some example test results produced on randomly sampled road scene images from Google are shown in Fig. \\ref{Teaser}. \n\nThe remainder of the paper is organized as follows. In Sec. \\ref{LitReview} we review related recent literature. We describe the SegNet architecture and its analysis in Sec. \\ref{Architecture}. In Sec. \\ref{Benchmarking} we evaluate the performance of SegNet on several benchmark datasets. This is followed by a general discussion regarding our approach with pointers to future work in Sec. \\ref{Discussion}. We conclude in Sec. \\ref{Conclusion}. \n\n\\section{Literature Review}\n\\label{LitReview}\n\nSemantic pixel-wise segmentation is an active topic of research, fuelled by challenging datasets \\cite{GabeDataset,silberman2012indoor,GeigerKITTI,Pascal,song2015sun}. Before the arrival of deep networks, the best performing methods mostly relied on hand engineered features classifiying pixels independently. Typically, a patch is fed into a classifier \\textit{e.g.} Random Forest \\cite{Jamie2,Brostow} or Boosting \\cite{Sturgess,LadickyECCV} to predict the class probabilities of the center pixel. Features based on appearance \\cite{Jamie2} or SfM and appearance \\cite{Brostow,Sturgess, LadickyECCV} have been explored for the CamVid road scene understanding test \\cite{GabeDataset}. These per-pixel noisy predictions (often called \\textit{unary} terms) from the classifiers are then smoothed by using a pair-wise or higher order CRF \\cite{Sturgess,LadickyECCV} to improve the accuracy. More recent approaches have aimed to produce high quality unaries by trying to predict the labels for all the pixels in a patch as opposed to only the center pixel. This improves the results of Random Forest based unaries \\cite{kontschieder2011structured} but thin structured classes are classfied poorly. Dense depth maps computed from the CamVid video have also been used as input for classification using Random Forests \\cite{zhang2010semantic}. Another approach argues for the use of a combination of popular hand designed features and spatio temporal super-pixelization to obtain higher accuracy \\cite{tighe2013superparsing}. The best performing technique on the CamVid test \\cite{LadickyECCV} addresses the imbalance among label frequencies by combining object detection outputs with classifier predictions in a CRF framework. The result of all these techniques indicate the need for improved features for  classification.  \n\nIndoor RGBD pixel-wise semantic segmentation has also gained popularity since the release of the NYU dataset \\cite{silberman2012indoor}. This dataset showed the usefulness of the depth channel to improve segmentation. Their approach used features such as RGB-SIFT, depth-SIFT and pixel location as input to a neural network classifier to predict pixel unaries. The noisy unaries are then smoothed using a CRF. Improvements were made using a richer feature set including LBP and region segmentation to obtain higher accuracy \\cite{ren2012rgb} followed by a CRF. In more recent work \\cite{silberman2012indoor}, both class segmentation and support relationships are inferred together using a combination of RGB and depth based cues. Another approach focusses on real-time joint reconstruction and semantic segmentation, where Random Forests are used as the classifier \\cite{Hermans14ICRA}. Gupta \\emph{et~al.}  \\cite{gupta2013perceptual} use boundary detection and hierarchical grouping before performing category segmentation. The common attribute in all these approaches is the use of hand engineered features for classification of either RGB or RGBD images. \n\nThe success of deep convolutional neural networks for object classification has more recently led to researchers to exploit their feature learning capabilities for structured prediction problems such as segmentation. There have also been attempts to apply networks designed for object categorization to segmentation, particularly by replicating the deepest layer features in blocks to match image dimensions \\cite{FarabetPAMI,FarabetPurityCover,Grangier,Gatta}. However, the resulting classification is blocky \\cite{Grangier}. Another approach using recurrent neural networks \\cite{pinheiro2014recurrent} merges several low resolution predictions to create input image resolution predictions. These techniques are already an improvement over hand engineered features \\cite{FarabetPAMI} but their ability to delineate boundaries is poor. \n\nNewer deep architectures \\cite{FCN,NohDeconvNets,eigen2014predicting,DecoupledNet,CRFRNN} particularly designed for segmentation have advanced the state-of-the-art by learning to decode or map low resolution image representations to pixel-wise predictions. The encoder network which produces these low resolution representations in all of these architectures is the VGG16 classification network \\cite{simonyan2014very} which has $13$ convolutional layers and $3$ fully connected layers. This encoder network weights are typically pre-trained on the large ImageNet object classification dataset \\cite{ImageNet}. The decoder network varies between these architectures and is the part which is responsible for producing multi-dimensional features for each pixel for classification.\n\nEach decoder in the Fully Convolutional Network (FCN) architecture \\cite{FCN} learns to upsample its input feature map(s) and combines them with the corresponding encoder feature map to produce the input to the next decoder. It is a core segmentation engine which has a large number of trainable parameters in the encoder network (134M) but a very small decoder network (0.5M). The overall large size of this network makes it hard to train end-to-end on a relevant task. Therefore, the authors use a stage-wise training process. Here each decoder in the decoder network is progressively added to an existing trained network. The network is grown until no further increase in performance is observed. This growth is stopped after three decoders thus ignoring high resolution feature maps can certainly lead to loss of edge information \\cite{NohDeconvNets}. Apart from training related issues, the need to reuse the encoder feature maps in the decoder makes it memory intensive in test time. We study this network in more detail as it the core of other recent architectures \\cite{ParseNetRabinovich, CRFRNN}.\n\nThe predictive performance of FCN has been improved further by appending the FCN with a recurrent neural network (RNN) \\cite{CRFRNN} and fine-tuning them on large datasets \\cite{Pascal},\\cite{COCO}. The RNN layers mimic the sharp boundary delineation capabilities of CRFs while exploiting the feature representation power of FCN's. They show a significant improvement over FCN-8 but also show that this difference is reduced when more training data is used to train FCN-8. The main advantage of the CRF-RNN is when it is jointly trained with a core segmentation engine such as the FCN-8. The fact that joint training helps is also shown in other recent results \\cite{UrtasunSegmentation}, \\cite{lin2015efficient}. Interestingly, the deconvolutional network \\cite{NohDeconvNets} performs significantly better than FCN although at the cost of a more complex training and inference. This however raises the question as to whether the perceived advantage of the CRF-RNN would be reduced as the core feed-forward segmentation engine is made better. In any case, the CRF-RNN network can be appended to any core segmentation engine including SegNet.\n\nMulti-scale deep architectures are also being pursued \\cite{eigen2014predicting,lin2015efficient}. They come in two flavours, (i) those which use input images at a few scales and corresponding deep feature extraction networks, and (ii) those which combine feature maps from different layers of a single deep architecture \\cite{Hypercolumns} \\cite{ParseNetRabinovich}. The common idea is to use features extracted at multiple scales to provide both local and global context \\cite{mostajabi2014feedforward} and the using feature maps of the early encoding layers retain more high frequency detail leading to sharper class boundaries. Some of these architectures are difficult to train due to their parameter size \\cite{eigen2014predicting}. Thus a multi-stage training process is employed along with data augmentation. The inference is also expensive with multiple convolutional pathways for feature extraction. Others \\cite{lin2015efficient} append a CRF to their multi-scale network and jointly train them. However, these are not feed-forward at test time and require optimization to determine the MAP labels.\n\nSeveral of the recently proposed deep architectures for segmentation are not feed-forward in inference time \\cite{NohDeconvNets}, \\cite{DeepLab}, \\cite{DecoupledNet}. They require either MAP inference over a CRF \\cite{lin2015efficient}, \\cite{UrtasunSegmentation} or aids such as region proposals \\cite{NohDeconvNets} for inference. We believe the perceived performance increase obtained by using a CRF is due to the lack of good decoding techniques in their core feed-forward segmentation engine. SegNet on the other hand uses decoders to obtain features for accurate pixel-wise classification.\n\nThe recently proposed Deconvolutional Network \\cite{NohDeconvNets} and its semi-supervised variant the Decoupled network \\cite{DecoupledNet} use the max locations of the encoder feature maps (pooling indices) to perform non-linear upsampling in the decoder network. The authors of these architectures, independently of SegNet (first submitted to CVPR 2015 \\cite{SegNetarXiv}), proposed this idea of decoding in the decoder network. However, their encoder network consists of the fully connected layers from the VGG-16 network which consists of about $90\\%$ of the parameters of their entire network. This makes training of their network very difficult and thus require additional aids such as the use of region proposals to enable training. Moreover, during inference these proposals are used and this increases inference time  significantly. From a benchmarking point of view, this also makes it difficult to evaluate the performance of their core segmentation engine (encoder-decoder network) without other aids. In this work we discard the fully connected layers of the VGG16 encoder network which enables us to train the network using the relevant training set using SGD optimization. Another recent method \\cite{DeepLab} shows the benefit of reducing the number of parameters significantly without sacrificing performance, reducing memory consumption and improving inference time. \n\\begin{figure*}[!ht]\n\\center\n\\includegraphics[width=0.95\\textwidth]{Figures\/segnet_softmax.pdf}\n\\caption{\\footnotesize{An illustration of the SegNet architecture. There are no fully connected layers and hence it is only convolutional. A decoder upsamples its input using the transferred pool indices from its encoder to produce a sparse feature map(s). It then performs convolution with a trainable filter bank to densify the feature map. The final decoder output feature maps are fed to a soft-max classifier for pixel-wise classification.}}\n\\label{SegNetArchitecture}\n\\end{figure*}\n\nOur work was inspired by the unsupervised feature learning architecture proposed by Ranzato \\emph{et al.} \\cite{Ranzato}. The key learning module is an encoder-decoder network. An encoder consists of convolution with a filter bank, element-wise tanh non-linearity, max-pooling and sub-sampling to obtain the feature maps. For each sample, the indices of the max locations computed during pooling are stored and passed to the decoder. The decoder upsamples the feature maps by using the stored pooled indices. It convolves this upsampled map using a trainable decoder filter bank to reconstruct the input image. This architecture was used for unsupervised pre-training for classification. A somewhat similar decoding technique is used for visualizing trained convolutional networks\\cite{zeiler2010deconvolutional} for classification. The architecture of Ranzato \\emph{et al.} mainly focussed on layer-wise feature learning using small input patches. This was extended by Kavukcuoglu et. al. \\cite{KorayUnsup} to accept full image sizes as input to learn hierarchical encoders. Both these approaches however did not attempt to use \\textit{deep encoder-decoder} networks for unsupervised feature training as they discarded the decoders after each encoder training. Here, SegNet differs from these architectures as the deep encoder-decoder network is trained jointly for a supervised learning task and hence the decoders are an integral part of the network in test time.\n\nOther applications where pixel wise predictions are made using deep networks are image super-resolution \\cite{dong2014learning} and depth map prediction from a single image \\cite{eigen2014depth}. The authors in \\cite{eigen2014depth} discuss the need for learning to upsample from low resolution feature maps which is the central topic of this paper.  \n\n\\section{Architecture}\n\\label{Architecture}\nSegNet has an encoder network and a corresponding decoder network, followed by a final pixelwise classification layer. This architecture is illustrated in Fig. \\ref{Architecture}. The encoder network consists of $13$ convolutional layers which correspond to the first $13$ convolutional layers in the VGG16 network \\cite{simonyan2014very} designed for object classification. We can therefore initialize the training process from weights trained for classification on large datasets \\cite{ImageNet}. We can also discard the fully connected layers in favour of retaining higher resolution feature maps at the deepest encoder output. This also reduces the number of parameters in the SegNet encoder network  significantly (from 134M to 14.7M) as compared to other recent architectures \\cite{FCN}, \\cite{NohDeconvNets} (see. Table \\ref{PascalQuant}). Each encoder layer has a corresponding decoder layer and hence the decoder network has $13$ layers. The final decoder output is fed to a multi-class soft-max classifier to produce class probabilities for each pixel independently. \n\nEach \\textit{encoder} in the encoder network performs convolution with a filter bank to produce a set of feature maps. These are then batch normalized \\cite{BN}). Then an element-wise rectified-linear non-linearity (ReLU) $max(0,x)$ is applied. Following that, max-pooling with a $2\\times2$ window and stride $2$ (non-overlapping window) is performed and the resulting output is sub-sampled by a factor of $2$. Max-pooling is used to achieve translation invariance over small spatial shifts in the input image. Sub-sampling results in a large input image context (spatial window) for each pixel in the feature map. While  several layers of max-pooling and sub-sampling can achieve more translation invariance for robust classification correspondingly there is a loss of spatial resolution of the feature maps. The increasingly lossy (boundary detail) image representation is not beneficial for segmentation where boundary delineation is vital. Therefore, it is necessary to \\textit{capture and store} boundary information in the encoder feature maps before sub-sampling is performed. If memory during inference is not constrained, then all the encoder feature maps (after sub-sampling) can be stored. This is usually not the case in practical applications and hence we propose a more efficient way to store this information. It involves storing only the max-pooling \\textit{indices}, i.e, the locations of the maximum feature value in each pooling window is memorized for each encoder feature map. In principle, this can be done using 2 bits for each $2\\times2$ pooling window and is thus much more efficient to store as compared to memorizing feature map(s) in float precision. As we show later in this work, this lower memory storage results in a slight loss of accuracy but is still suitable for practical applications.\n\nThe appropriate \\textit{decoder} in the decoder network upsamples its input feature map(s) using the memorized max-pooling indices from the corresponding encoder feature map(s). This step produces sparse feature map(s). This SegNet decoding technique is illustrated in Fig. \\ref{Upsampling}. These feature maps are then convolved with a trainable decoder filter bank to produce dense feature maps. A batch normalization step is then applied to each of these maps. Note that the decoder corresponding to the first encoder (closest to the input image) produces a multi-channel feature map, although its encoder input has 3 channels (RGB). This is unlike the other decoders in the network which produce feature maps with the same number of size and channels as their encoder inputs. The high dimensional feature representation at the output of the final decoder is fed to a trainable soft-max classifier. This soft-max classifies each pixel independently. The output of the soft-max classifier is a K channel image of probabilities where K is the number of classes. The predicted segmentation corresponds to the class with maximum probability at each pixel.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures\/PoolingExp\/decoderVariantsNew.pdf}\n\\caption{\\footnotesize{An illustration of SegNet and FCN \\cite{FCN} decoders. $a,b,c,d$ correspond to values in a feature map. SegNet uses the max pooling indices to upsample (without learning) the feature map(s) and convolves with a trainable decoder filter bank. FCN upsamples by learning to deconvolve the input feature map and adds the corresponding encoder feature map to produce the decoder output. This feature map is the output of the max-pooling layer (includes sub-sampling) in the corresponding encoder. Note that there are no trainable decoder filters in FCN.}}\n\\label{Upsampling}\n\\end{figure*}\n\n\n\\begin{table*}[t]\n\\footnotesize\n\\centering\n\\tabcolsep=1pt\n\\begin{tabular}{c|c|c|c|c|ccc|ccc|c|ccc|ccc}\n\\multicolumn{5}{c}{} &  \\multicolumn{6}{c}{Median frequency balancing} & \\multicolumn{1}{c}{} & \\multicolumn{6}{c}{Natural frequency balancing} \\\\\n\\hline\n& & Encoder & Infer & & \\multicolumn{3}{c|}{Test } &  \\multicolumn{3}{c|}{Train } &  & \\multicolumn{3}{c|}{Test } & \\multicolumn{3}{c|}{Train }\\\\\nVariant                   &Params (M) &  storage (MB) &  time (ms) & Iter(K)  & G       & C       & I\/U        &  G & C & I\/U  & Iter(K) & G & C & I\/U  & G & C & \\multicolumn{1}{c|}{I\/U}               \\\\ \\hline \\hline\n\\multicolumn{18}{c}{Fixed upsampling} \\\\ \\hline\nBilinear-Interpolation           & 0.625 & 0 & 31 &  6 & 78.7      &    62.3     &     45.6    & 89.1 & 90.2 &  82.7     & 6 & 82.7 & 52.5 & 35.9 &93.5&74.1 &         59.9 \\\\ \\hline                                                                                                   \n\\multicolumn{18}{c}{Upsampling using max-pooling indices}    \\\\                                                                                              \n \\hline\nSegNet-Basic                    & 1.425 & 1x & 80 &   4 & 82.8       &   62.4      &   46.3   & 94.7 & 96. 2 & 92.7   & 5 & 84.1 & 55.0 & 38.0 & 96.1 & 83.9 &73.3                   \\\\ \\hline\nSegNet-Basic-EncoderAddition    &1.425 & 64x  & 80 &  5 & 83.6  &     \\textbf{64.0 }   &      \\textbf{48.0 }         & 94.3 &  95.8 &  92.0 & 6 & \\textbf{84.}2 & 56.5 &          39.8 & 95.3&80.9 & 68.9\\\\ \\hline\nSegNet-Basic-SingleChannelDecoder& 0.625 &  1x & 68 &   6 &  81.4     &    61.1     &     44.6         & 93.2 & 94.8 &   90.3    & 2 & 83.7 & 54.0 & 37.0 & 92.6 & 68.4 & 52.8        \\\\ \\hline\n\\multicolumn{18}{c}{Learning to upsample (bilinear initialisation)}                                                                                                         \n \\\\ \\hline\nFCN-Basic                     & 0.65 & 11x  & 31.3 &     4 &   82.0    &     62.9    &    46.4        & 92.8 & 93.6 &  88.1    &2 & 83.9 & 55.6 & 38.6 & 92.0 & 66.8 & 50.7       \\\\ \\hline\nFCN-Basic-NoAddition           &0.65 & n\/a &31 &   6 &   80.7    &    59.0     &     42.7         & 92.5 & 93.0 &    87.2 & 4 & 82.3 & 54.1 & 37.0 & 93.1& 72.8 & 57.6         \\\\ \\hline\nFCN-Basic-NoDimReduction                    &1.625 & 64x & 52.6 & 5  &  \\textbf{84.31}     &  63.8       &   47.6          & \\textbf{95.1} & \\textbf{96.5}  &  \\textbf{93.2}   & 6 &        83.6 & \\textbf{57.7} & \\textbf{40.4} &\\textbf{97.2} & \\textbf{91.7} & \\textbf{84.8}\n\\\\ \\hline\nFCN-Basic-NoAddition-NoDimReduction                    &1.625 & 0 & 52.2 &    4   & 80.6      &   62.0        & 45.2 &   92.5 & 94.6 & 89.9 &   6 &  83.8 & 54.8 & 38.2 & 95.0 & 80.2 & 67.8      \\\\ \\hline \n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Comparison of decoder variants. We quantify the performance using global (G), class average (C) and mean of intersection over union (I\/U) metrics. The testing and training accuracies are shown as percentages for both natural frequency and median frequency balanced training loss function. SegNet-Basic performs at the same level as FCN-Basic but requires only storing max-pooling indices and is therefore more memory efficient during inference. Note that the theoretical memory requirement reported is based only on the size of the first layer encoder feature map. Networks with larger decoders and those using the encoder feature maps in full perform best, although they are least efficient.}}\n\\label{PoolingQuantCAFFE}\n\\end{table*}\n\n\\subsection{Decoder Variants}\n\\label{Variants}\nMany segmentation architectures \\cite{FCN, NohDeconvNets, DeepLab} share the same encoder network and they only vary in the form of their decoder network. \nOf these we choose to compare the SegNet decoding technique with the widely used Fully Convolutional Network (FCN) decoding technique \\cite{FCN, CRFRNN}. \n\nIn order to analyse SegNet and compare its performance with FCN (decoder variants) we use a smaller version of SegNet, termed \\textbf{SegNet-Basic} \\footnote{SegNet-Basic was earlier termed SegNet in a archival version of this paper \\cite{SegNetarXiv}}, which has 4 encoders and 4 decoders. All the encoders in SegNet-Basic perform max-pooling and sub-sampling and the corresponding decoders upsample its input using the received max-pooling indices. Batch normalization is used after each convolutional layer in both the encoder and decoder network. No biases are used after convolutions and no ReLU non-linearity is present in the decoder network. Further, a constant kernel size of $7\\times7$ over all the layers is chosen to provide a wide context for smooth labelling \\textit{i.e.} a pixel in the deepest layer feature map (layer $4$) can be traced back to a context window in the input image of $106\\times106$ pixels. This small size of SegNet-Basic allows us to explore many different variants (decoders) and train them in reasonable time. Similarly we create \\textbf{FCN-Basic}, a comparable version of FCN for our analysis which shares the same encoder network as SegNet-Basic but with the FCN decoding technique in the corresponding decoders.  \n\nOn the left in Fig. \\ref{Upsampling} is the decoding technique used by SegNet (also SegNet-Basic), where there is no learning involved in the upsampling step. However, the upsampled maps are convolved with trainable multi-channel decoder filters to densify the sparse inputs. Each decoder filter has the same number of channels as the number of upsampled feature maps. A smaller variant is one where the decoder filters are single channel, i.e they only convolve their corresponding upsampled feature map. This variant (\\textbf{SegNet-Basic-SingleChannelDecoder}) reduces the number of trainable parameters and inference time significantly.\n\nOn the right in Fig. \\ref{Upsampling} is the FCN (also FCN-Basic) decoding technique. The important design element of the FCN model is dimensionality reduction step of the encoder feature maps. This \\textit{compresses} the encoder feature maps which are then used in the corresponding decoders. Dimensionality reduction of the encoder feature maps, say of 64 channels, is performed by convolving them with $1\\times 1\\times 64\\times K$ trainable filters, where $K$ is the number of classes. The compressed $K$ channel final encoder layer feature maps are the input to the decoder network. In a decoder of this network, upsampling is performed by convolution using a trainable \\textit{multi-channel upsampling kernel}. We set the kernel size to $8\\times8$. This manner of upsampling is also termed as \\textit{deconvolution}. Note that in SegNet the multi-channel convolution using trainable decoder filters is performed after upsampling to densifying feature maps. The upsampled feature map in FCN is then added to the corresponding resolution encoder feature map to produce the output decoder feature map. The upsampling kernels are initialized using bilinear interpolation weights \\cite{FCN}. \n\nThe FCN decoder model requires storing encoder feature maps during inference. This can be memory intensive, for e.g. storing 64 feature maps of the first layer of FCN-Basic at $180\\times240$ resolution in 32 bit floating point precision takes 11MB. This can be made smaller using dimensionality reduction to the 11 feature maps which requires $\\approx$ 1.9MB storage. SegNet on the other hand requires almost negligible storage cost for the pooling indices ($.17$MB if stored using 2 bits per $2\\times2$ pooling window). We can also create a variant of the FCN-Basic model which discards the encoder feature map addition step and only learns the upsampling kernels (\\textbf{FCN-Basic-NoAddition}). \n\nIn addition to the above variants, we study upsampling using fixed bilinear interpolation weights which therefore requires no learning for upsampling (\\textbf{Bilinear-Interpolation}). At the other extreme, we can add 64 encoder feature maps at each layer to the corresponding output feature maps from the SegNet decoder to create a more memory intensive variant of SegNet (\\textbf{SegNet-Basic-EncoderAddition}). Another and more memory intensive FCN-Basic variant (\\textbf{FCN-Basic-NoDimReduction}) is where there is no dimensionality reduction performed for the encoder feature maps. Finally, please note that to encourage reproduction of our results we release the Caffe implementation of all the variants \\footnote{See \\url{ http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/} for code and web demo.}. \n\nWe also tried other generic variants where feature maps are simply upsampled by \\textit{replication} \\cite{FarabetPAMI}, or by using a fixed (and sparse) array of indices for upsampling. These performed quite poorly in comparison to the above variants. A variant without max-pooling and sub-sampling in the encoder network (decoders are redundant) consumes more memory, takes longer to converge and performs poorly. \n\n\\subsection{Training}\n\\label{Training}\nWe use the CamVid road scenes dataset to benchmark the performance of the decoder variants. This dataset is small, consisting of 367 training and 233 testing RGB images (day and dusk scenes) at $360\\times480$ resolution. The challenge is to segment $11$ classes such as road, building, cars, pedestrians, signs, poles, side-walk etc. We perform local contrast normalization \\cite{Jarrett} to the RGB input.\n\nThe encoder and decoder weights were all initialized using the technique described in He \\emph{et~al.} \\cite{he2015delving}. To train all the variants we use stochastic gradient descent (SGD) with a fixed learning rate of 0.1 and momentum of 0.9 \\cite{Bottou} using our Caffe implementation of SegNet-Basic \\cite{jia2014caffe}. We train the variants for a maximum of $200$ epochs where each epoch is one pass through the full dataset. Before each epoch, the training set is shuffled and each mini-batch (12 images) is then picked in order thus ensuring that each image is used only once in an epoch. We use no validation set to stop training as the training set is small and also because this enables us to compare these variants against other methods (see Table \\ref{CamVidQuant}). \n\n\nWe use the cross-entropy loss \\cite{FCN} as the objective function for training the network. The loss is summed up over all the pixels in a mini-batch. When there is large variation in the number of pixels in each class in the training set (e.g road, sky and building pixels dominate the CamVid dataset) then there is a need to weight the loss differently based on the true class. This is termed \\textit{class balancing}. We use \\textit{median frequency balancing } \\cite{eigen2014predicting} where the weight assigned to a class in the loss function is the ratio of the median of class frequencies computed on the entire training set divided by the class frequency. This implies that larger classes in the training set have a weight smaller than $1$ and the weights of the smallest classes are the highest. We also experimented with training the different variants without class balancing or equivalently using \\textit{natural frequency balancing}.\n\n\\subsection{Analysis}\n\\label{Analysis}\nTo compare the quantitative performance of the different decoder variants, we use three commonly used performance measures: global accuracy (G) which measures the percentage of pixels correctly classified in the dataset, class average accuracy (C) is the mean of the predictive accuracy over all classes and mean intersection over union (I\/U) over all classes as used in the Pascal VOC12 challenge \\cite{Pascal}. The mean I\/U metric is the hardest metric since it penalizes false positive predictions unlike class average accuracy. However, I\/U metric is not optimized for directly through the class balanced cross-entropy loss. \n\nWe test each variant at five points, $2K,3K,4K,5K,6K$ iterations of optimization, on the CamVid test set. With a training mini-batch size of 12 this corresponds to testing approximately at $65,98,131,163,196$ epochs (passes) through the training set. We select the iteration wherein the global accuracy is highest amongst the five evaluations and report all the three measures of performance at this point. Although we use class balancing while training the variants, it is still important to achieve high global accuracy to result in an overall smooth segmentation. Another reason is that the contribution of segmentation towards autonomous driving is mainly for delineating classes such as roads, buildings, side-walk, sky. These classes dominate the majority of the pixels in an image and a high global accuracy corresponds to good segmentation of these important classes. We also observed that reporting the numerical performance when class average is highest can often correspond to low global accuracy indicating a perceptually noisy segmentation output.\n\nIn Table \\ref{PoolingQuantCAFFE} we report the numerical results of our analysis. We also show the size of the trainable parameters, highest resolution feature map or pooling indices storage memory, i.e, of the first layer feature maps after max-pooling and sub-sampling, average time for one forward pass based on $50$ measurements using a $360\\times480$ input and also the iterations required to get the best global accuracy. We show the results for both testing and training for all the variants at the selected iteration. The results are also tabulated without class balancing (natural frequency) for training and testing accuracies. Below we analyse the results with class balancing.\n\nFrom the Table \\ref{PoolingQuantCAFFE}, we see that bilinear interpolation based upsampling without any learning performs the worst based on all the three measures of accuracy. All the other methods which either use learning for upsampling (FCN-Basic and variants) or learning decoder filters after upsampling (SegNet-Basic and its variants) perform significantly better. This emphasizes the need to learn decoders for segmentation. This is also supported by experimental evidence gathered by other authors when comparing FCN with SegNet-type decoding techniques\\cite{NohDeconvNets}.\n\nWhen we compare SegNet-Basic and FCN-Basic we see that both perform equally well on this test over all the three measures of accuracy. The difference is that SegNet uses less \\textbf{memory} during inference since it only stores max-pooling indices. On the other hand FCN-Basic stores \\textbf{encoder feature maps} in full which consumes much more memory (11 times more). SegNet-Basic has a decoder with 64 feature maps in each decoder layer. In comparison FCN-Basic, which uses dimensionality reduction, has fewer (11) feature maps in each decoder layer. This reduces the number of convolutions in the decoder network and hence FCN-Basic is faster during inference (forward pass). From another perspective, the decoder network in SegNet-Basic makes it overall a larger network than FCN-Basic. This endows it with more flexibility and hence achieves higher training accuracy than FCN-Basic for the same number of iterations. Overall we see that SegNet-Basic has an advantage over FCN-Basic when memory during inference is constrained but where inference time can be compromised to an extent. \n\nSegNet-Basic is most similar to FCN-Basic-NoAddition in terms of their decoders, although the decoder of SegNet is larger. Both learn to produce dense feature maps, either directly by learning to perform deconvolution as in FCN-Basic-NoAddition or by first upsampling and then convolving with trained decoder filters. \nThe performance of SegNet-Basic is superior, in part due to its larger decoder size. Now, the accuracy of FCN-Basic-NoAddition is also lower as compared to FCN-Basic. This shows that it is important to capture the information present in the encoder feature maps for better performance. This can also explain the part of the reason why SegNet-Basic outperforms FCN-Basic-NoAddition. \n\nThe size of the FCN-Basic-NoAddition-NoDimReduction model is slightly larger than SegNet-Basic and this makes it a fair comparison. The performance of this FCN variant is poorer than SegNet-Basic in test but also its training accuracy is lower for the same number of training epochs. This shows that using a larger decoder is not enough but it is also important to capture encoder feature map information to learn better. Here it is also interesting to see that SegNet-Basic has a competitive training accuracy when compared to larger models such FCN-Basic-NoDimReduction. \n\nAnother interesting comparison between FCN-Basic-NoAddition and SegNet-Basic-SingleChannelDecoder shows that using max-pooling indices for upsampling and an overall larger decoder leads to better performance. This also lends evidence to SegNet being a good architecture for segmentation, particularly when there is a need to find a compromise between storage cos, accuracy versus inference time. In the best case, when both memory and inference time is not constrained, larger models such as FCN-Basic-NoDimReduction and SegNet-EncoderAddition are both more accurate than the other variants. Particularly, discarding dimensionality reduction in the FCN-Basic model leads to the best performance amongst the FCN-Basic variants. This once again emphasizes the trade-off involved between memory and accuracy in segmentation architectures.\n\nThe last two columns of Table \\ref{PoolingQuantCAFFE} show the result when no class balancing is used (natural frequency). Here, we can observe that without weighting the results are poorer for all the variants, particularly for class average accuracy and mean I\/U metric. The global accuracy is the highest without weighting since the majority of the scene is dominated by sky, road and building pixels. Apart from this all the inference from the comparative analysis of variants holds true for natural frequency balancing too. SegNet-Basic performs as well as FCN-Basic and is better than the larger FCN-Basic-NoAddition-NoDimReduction. The bigger but less efficient models FCN-Basic-NoDimReduction and SegNet-EncoderAddition perform better than the other variants.\n\nWe can now summarize the above analysis with the following general points.\n\\begin{enumerate}\n\\item The best performance is achieved when encoder feature maps are stored in full.\n\\item When memory during inference is constrained, then compressed forms of encoder feature maps (dimensionality reduction, max-pooling indices) can be stored and used with an appropriate decoder (e.g. SegNet type) to improve performance.\n\\item Larger decoders increase performance for a given encoder network.\n\\end{enumerate}\n\n\n\n\\section{Benchmarking}\n\\label{Benchmarking}\nWe quantify the performance of SegNet on three different benchmarks using our Caffe implementation \\footnote{\\label{webdemo} Our web demo and Caffe implementation is available for evaluation at \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/}}. Through this process we demonstrate the efficacy of SegNet for various scene segmentation tasks which have practical applications. In the first experiment, we test the performance of SegNet on the CamVid road scene dataset (see Sec. \\ref{Training} for more information about this data). We use this result to compare SegNet with several methods including Random Forests \\cite{Jamie2}, Boosting \\cite{Jamie2,Sturgess} in combination with CRF based methods \\cite{LadickyECCV}.  We also trained SegNet on a larger dataset of road scenes collected from various publicly available datasets \\cite{GabeDataset,LabelMe,SpanishKITTI} and show that this leads to a large improvement in accuracy.\n\nSUN RGB-D \\cite{song2015sun} is a very challenging and large dataset of indoor scenes with $5285$ training and $5050$ testing images. The images are captured by different sensors and hence come in various resolutions. The task is to segment $37$ indoor scene classes including wall, floor, ceiling, table, chair, sofa etc. This task is made hard by the fact that object classes come in various shapes, sizes and in different poses. There are frequent partial occlusions since there are typically many different classes present in each of the test images. These factors make this one of the hardest segmentation challenges. We only use the RGB modality for our training and testing. Using the depth modality would necessitate architectural modifications\/redesign \\cite{FCN}. Also the quality of depth images from current cameras require careful post-processing to fill-in missing measurements. They may also require using fusion of many frames to robustly extract features for segmentation. Therefore we believe using depth for segmentation merits a separate body of work which is not in the scope of this paper. We also note that an earlier benchmark dataset NYUv2 \\cite{silberman2012indoor} is included as part of this dataset.\n\n\nPascal VOC12 \\cite{Pascal} is a RGB dataset for segmentation with $12031$ combined training and validation images of indoor and outdoor scenes. The task is to segment $21$ classes such as bus, horse, cat, dog, boat from a varied and large background class. The foreground classes often occupy a small part of an image. The evaluation is performed remotely on $1456$ images.\n\nIn all three benchmark experiments, we scale the images to $224\\times224$ resolution for training and testing. We used SGD with momentum to train SegNet. The learning rate was fixed to 0.001 and momentum to 0.9. The mini-batch size was 4. The optimization was performed for 100 epochs and then tested.\n\n\\subsection{CamVid Road Scenes}\n\\label{CamVid}\nA number of outdoor scene datasets are available for semantic parsing \\cite{gould2009decomposing,russell2008labelme,GabeDataset,GeigerKITTI}. Of these we choose to benchmark SegNet using the CamVid dataset \\cite{GabeDataset} as it contains video sequences. This enables us to compare our proposed architecture with those which use motion and structure \\cite{LadickyECCV,Sturgess,Brostow} and video segments \\cite{tighe2013superparsing}. We also use an ensemble of training images from some of these datasets totalling 3433 images to train SegNet for an additional benchmark. For a web demo (see the link \\ref{webdemo}) of road scene segmentation, we include the CamVid test set to this larger dataset.   \n\nThe qualitative comparisons of SegNet-Basic and SegNet predictions with several well known algorithms (unaries, unaries+CRF) are shown in Fig. \\ref{CamVidQualy} along with the input modalities used to train the methods. The qualitative results show the ability of the proposed architectures to segment small (cars, pedestrians, bicyclist) classes while producing a smooth segmentation of the overall scene. The unary only methods like Random Forests, Boosting which are trained to predict the label of the center-pixel of a small patch produce low quality segmentations. Smoothing unaries with CRF's improve the segmentation quality considerably with higher order CRF's performing best. Although the CRF based results appear smooth, upon close scrutiny, we see that shape segmentation of smaller but important classes such as bicyclists, pedestrians are poor. In addition, natural shapes of classes like trees are not preserved and the details like the wheels of cars are lost.  More dense CRF models \\cite{koltun2011efficient} can be better but with additional cost of inference. SegNet-Basic and SegNet (without large dataset training) clearly indicate their ability to retain the natural shapes of classes such as bicyclists, cars, trees, poles etc better than CRF based approaches. The overall segmentation quality is also smooth except for the side-walk class. This is because the side-walk class is highly varied in terms of size and texture and this cannot be captured with a small training set. Another explanation could be that the size of the receptive fields of the deepest layer feature units are smaller than their theoretical estimates \\cite{zhou2014object,ParseNetRabinovich} and hence unable to group all the side-walk  pixels into one class. Illumination variations also affect performance on cars in the dusk examples. However, several of these issues can be ameliorated by using larger amounts of training data. In particular, we see that smaller classes such as pedestrians, cars, bicyclists, column-pole are segmented better than other methods in terms of shape retention. The side-walk class is also segmented smoothly.\n\nThe quantitative results in Table \\ref{CamVidQuant} show SegNet-Basic, SegNet obtain competitive results even without CRF based processing. This shows the ability of the deep architecture to extract meaningful features from the input image and map it to accurate and smooth class segment labels. SegNet is better in performance than SegNet-Basic although trained with the same (small) training set. This indicates the importance of using pre-trained encoder weights and choice of the optimization technique . Interestingly, the use of the bigger and deeper SegNet architecture improves the accuracy of the larger classes as compared to SegNet-Basic and not the smaller classes as one might expect. We also find that SegNet-Basic \\cite{SegNetarXiv} trained in a layer-wise manner using L-BFGS \\cite{Nocedal} also performs competitively and is better than SegNet-Basic trained with SGD (see Sec. \\ref{Training}). This is an interesting training approach but needs further research in order for it scale to larger datasets.\n\nThe most interesting result is the approximately $15\\%$ performance improvement in class average accuracy that is obtained when a large training dataset is utilized to train SegNet. The mean of intersection over union metric is also very high. Correspondingly, the qualitative results of SegNet (see Fig. \\ref{CamVidQualy}) are clearly superior to the rest of the methods. It is able to segment both small and large classes well. In addition, there is an overall smooth quality of segmentation much like what is typically obtained with CRF post-processing. Although the fact that results improve with larger training sets is not surprising, the percentage improvement obtained using pre-trained encoder network and this training set indicates that this architecture can potentially be deployed for practical applications. Our random testing on urban and highway images from Google (see Fig. \\ref{Teaser}) demonstrates that SegNet can \\textit{absorb} a large training set and generalize well to unseen images. It also indicates the contribution of the prior (CRF) can be lessened when sufficient amount of training data is made available.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.8\\textwidth]{Figures\/CamVidQualitative.pdf}\n\\caption{\\footnotesize{Results on CamVid day and dusk test samples. The evolution of results from various patch based predictions \\cite{Jamie2,Brostow}, then combined with CRF smoothing models  \\cite{LadickyECCV, Sturgess}. Unlike CRF based methods, SegNet-Basic and SegNet predictions retain the shape of small categories such as poles (column 2,4),  bicyclist (column 3), far side side-walk (column 2) better. Large sized side-walk is not smoothly segmented possibly because the empirical size of the receptive fields of deep feature units is not large enough \\cite{zhou2014object,ParseNetRabinovich}. When SegNet is trained on a large dataset it produces the most accurate predictions with an overall smooth appearance.\n}}\n\\label{CamVidQualy}\n\\end{figure*}\n\n\\begin{table*}[tch]\n \\resizebox{\\textwidth}{!}{\n\\small{\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|ccc}\n\n\\multicolumn{1}{c}{Method}                   & \\multicolumn{1}{c}{\\rotatebox{90}{Building}} & \\multicolumn{1}{c}{\\rotatebox{90}{Tree}} & \\multicolumn{1}{c}{\\rotatebox{90}{Sky}}  & \\multicolumn{1}{c}{\\rotatebox{90}{Car}}  & \\multicolumn{1}{c}{\\rotatebox{90}{Sign-Symbol}} & \\multicolumn{1}{c}{\\rotatebox{90}{Road}} & \\multicolumn{1}{c}{\\rotatebox{90}{Pedestrian}} & \\multicolumn{1}{c}{\\rotatebox{90}{Fence}} & \\multicolumn{1}{c}{\\rotatebox{90}{Column-Pole}} & \\multicolumn{1}{c}{\\rotatebox{90}{Side-walk}} & \\multicolumn{1}{c}{\\rotatebox{90}{Bicyclist}} & \\multicolumn{1}{c}{\\rotatebox{90}{Class avg.}} & \\multicolumn{1}{c}{\\rotatebox{90}{Global avg.}} & \\multicolumn{1}{c}{\\rotatebox{90}{Mean I\/U}}\\\\ \\hline \\hline\n\n\nSfM+Appearance  \\cite{Brostow}           & 46.2     & 61.9 & 89.7 & 68.6 & 42.9        & 89.5 & 53.6       & 46.6  & 0.7         & 60.5     & 22.5      & 53.0       & 69.1  & n\/a      \\\\ \\hline\n\nBoosting    \\cite{Sturgess}              & 61.9     & 67.3 & 91.1 & 71.1 & 58.5        & 92.9 & 49.5       & 37.6  & 25.8        & 77.8     & 24.7      & 59.8       & 76.4  & n\/a      \\\\ \\hline\n\nDense Depth Maps   \\cite{zhang2010semantic}          & 85.3     & 57.3 & 95.4 & 69.2 & 46.5        & \\textbf{98.5} & 23.8       & 44.3  & 22.0        & 38.1     & 28.7      & 55.4       & 82.1    & n\/a    \\\\ \\hline\n\nStructured Random Forests \\cite{kontschieder2011structured}& \\multicolumn{11}{c}{n\/a}                                                                          & 51.4       & 72.5    & n\/a    \\\\ \\hline\n\nNeural Decision Forests \\cite{BuloNeural}  & \\multicolumn{11}{c}{n\/a}                                                                          & 56.1       & 82.1   & n\/a     \\\\ \\hline\n\nLocal Label Descriptors  \\cite{yang2012local}  & 80.7     & 61.5 & 88.8 & 16.4 & n\/a         & 98.0 & 1.09       & 0.05  & 4.13        & 12.4     & 0.07      & 36.3       & 73.6    & n\/a    \\\\ \\hline\n\nSuper Parsing   \\cite{tighe2013superparsing}           & 87.0     & 67.1 & 96.9 & 62.7 & 30.1        & 95.9 & 14.7       & 17.9  & 1.7         & 70.0     & 19.4      & 51.2       & 83.3    & n\/a    \\\\ \\hline\n\n\nSegNet-Basic          &  80.6    & 72.0  & 93.0 & 78.5 & 21.0   & 94.0 &   62.5     & 31.4  &     36.6    &  74.0   & 42.5      &   62.3    & 82.8   &   46.3   \\\\ \\hline\nSegNet-Basic (layer-wise training \\cite{SegNetarXiv})         & 75.0     & 84.6 & 91.2 & 82.7 & 36.9        & 93.3 & 55.0       & 37.5  & 44.8        & 74.1     & 16.0      & 62.9       & 84.3   & n\/a     \\\\ \\hline\nSegNet    &  \\textbf{88.0 }   & 87.3 & 92.3  & 80.0 & 29.5   & 97.6 &  57.2      & 49.4  &  27.8       &  84.8   &  30.7     &   65.9    &  \\textbf{88.6}  & 50.2     \\\\ \\hline\nSegNet (3.5K dataset training)   & 73.8  & \\textbf{90.7}  & 90.1  & \\textbf{83.0}  &  \\textbf{83.9}  & 95.21 &   \\textbf{ 86.8}   & \\textbf{68.0}  &   \\textbf{74.6}\t   &  \\textbf{95.3}\t   &  \\textbf{53.0}    & \\textbf{81.3}  & 86.8 &       \\textbf{69.1}\\\\ \\hline\n\n\\multicolumn{15}{c}{CRF based approaches}                                                                                                       \\\\ \\hline\n\nBoosting + pairwise CRF  \\cite{Sturgess} & 70.7     & 70.8 & 94.7 & 74.4 & 55.9        & 94.1 & 45.7       & 37.2  & 13.0        & 79.3     & 23.1      & 59.9       & 79.8  & n\/a      \\\\ \\hline\n\nBoosting+Higher order \\cite{Sturgess}    & 84.5     & 72.6 & \\textbf{97.5} & 72.7 & 34.1        & 95.3 & 34.2       & 45.7  & 8.1         & 77.6     & 28.5      & 59.2       & 83.8    & n\/a    \\\\ \\hline\n\nBoosting+Detectors+CRF \\cite{LadickyECCV}   & 81.5     & 76.6 & 96.2 & 78.7 & 40.2        & 93.9 & 43.0       & 47.6  & 14.3        & 81.5     & 33.9      & 62.5       & 83.8    & n\/a    \\\\ \\hline\n\\end{tabular}\n}}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Quantitative results on CamVid \\cite{GabeDataset} consisting of 11 road scene categories. SegNet outperforms all the other methods, including those using depth, video and\/or CRF's. In comparison with the CRF based methods SegNet predictions are more accurate in 8 out of the 11 classes. It also shows a good $\\approx 15\\%$ improvement in class average accuracy when trained on a large dataset of 3.5K images and this sets a new benchmark for the majority of the individual classes. Particularly noteworthy are the significant improvements in accuracy for the smaller\/thinner classes. \n}}\n\\label{CamVidQuant}\n\\end{table*}\n\n\\subsection{SUN RGB-D Indoor Scenes}\n\\label{SUNRGBD}\nRoad scene images have limited variation, both in terms of the classes of interest and their spatial arrangements, especially when captured from a moving vehicle. In comparison, images of indoor scenes are more complex since the view points can vary a lot and there is less regularity in both the number of classes present in a scene and their spatial arrangement. Another difficulty is caused by the widely varying sizes of the object classes in the scene. Some test samples from the recent SUN RGB-D dataset \\cite{song2015sun} are shown in Fig. \\ref{SUNRGBDQualy}. We observe some scenes with few large classes and some others with dense clutter (bottom row and right). The appearance (texture and shape) can also widely vary in indoor scenes. Therefore, we believe this is the hardest challenge for segmentation architectures and methods in computer vision. Other challenges such as Pascal VOC12 \\cite{Pascal} salient object segmentation have occupied researchers more, but indoor scene segmentation is more challenging and has more practical applications such as in robotics. To encourage more research in this direction we set a new benchmark on the large SUN RGB-D dataset.\n\nThe qualitative results of SegNet on some images of indoor scenes of different types such as bedroom, kitchen, bathroom, classroom etc. are shown in Fig. \\ref{SUNRGBDQualy}. We see that SegNet obtains sharp boundaries between classes when the scene consists of reasonable sized classes but even when view point changes are present (see bed segmentation from different view points). This is particularly interesting since the input modality is only RGB. It indicates the ability of SegNet to extract features from RGB images which are useful for view-point invariant segmentation provided there is sufficient training data (here 5285 images). RGB images are also useful to segment thinner structures such as the legs of chairs and tables, lamps which is difficult to achieve using depth images from currently available sensors. It is also useful to segment decorative objects such as paintings on the wall. \n\nIn Table \\ref{SUNRGBDtest} we report the quantitative results on the $37$ class segmentation task. We first note here that the other methods that have been benchmarked are not based on deep architectures and they only report class average accuracy. The existing top performing method \\cite{ren2012rgb} relies on hand engineered features using colour, gradients and surface normals for describing super-pixels and then smooths super-pixel labels with a CRF. For SegNet we achieve a high global accuracy which correlates with an overall smooth segmentation. This is also an indicator that largest classes such as wall, floor, bed, table, sofa are segmented well in spite of view-point changes and appearance variations. However, the class average accuracy and mean I\/U metric are poor, but at the same level as the hand engineered method which also includes the depth channel as input. This shows that smaller and thinner classes which have lesser training data are not segmented well. The individual class accuracies are reported in Table \\ref{SUNRGBDClassavg}. From these we see that there is a clear correlation between the size and natural frequency of occurrence of classes and their individual accuracies. It is also informative to note RGB input is useful to segment widely varying (shape, texture) categories such as wall, floor, ceiling, table, chair, sofa with reasonable accuracy.\n\n\\begin{table*}[t]\n\\centering\n\\tabcolsep=3pt\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\multicolumn{1}{c|}{Wall} & \\multicolumn{1}{c|}{Floor} & \\multicolumn{1}{c|}{Cabinet} & \\multicolumn{1}{c|}{Bed} & \\multicolumn{1}{c|}{Chair} & \\multicolumn{1}{c|}{Sofa} & \\multicolumn{1}{c|}{Table} & \\multicolumn{1}{c|}{Door} & \\multicolumn{1}{c|}{Window} & \\multicolumn{1}{c|}{Bookshelf} & \\multicolumn{1}{c|}{Picture} & \\multicolumn{1}{c|}{Counter} & \\multicolumn{1}{c}{Blinds} \\\\ \\hline\n86.6                       & 92.0                       & 52.4                         & 68.4                     & 76.0                       & 54.3                      & 59.3                       & 37.4                      & 53.8                        & 29.2                           & 49.7                         & 32.5                         & 31.2                        \\\\ \\hline\nDesk                       & Shelves                    & Curtain                      & Dresser                  & Pillow                     & Mirror                    & Floor mat                  & Clothes                   & Ceiling                     & Books                          & Fridge                       & TV                           & Paper                       \\\\ \\hline\n17.8                       & 5.3                        & 53.2                         & 28.8                     & 36.5                       & 29.6                      & 0.0                        & 14.4                      & 67.7                        & 32.4                           & 10.2                         & 18.3                         & 19.2                        \\\\ \\hline\nTowel                      & Shower curtain             & Box                          & Whiteboard               & Person                     & Night stand               & Toilet                     & Sink                      & Lamp                        & Bathtub                        & Bag                          & \\multicolumn{2}{c}{\\multirow{2}{*}{}}                     \\\\ \\cline{1-11}\n11.5                       & 0.0                        & 8.9                          & 38.7                     & 4.9                        & 22.6                      & 55.6                       & 52.7                      & 27.9                        & 29.9                           & 8.1                          & \\multicolumn{2}{c}{}                                      \\\\ \\cline{1-11}\n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{Class average accuracy of SegNet predictions for the 37 indoor scene classes in the SUN RGB-D benchmark dataset.}\n\\label{SUNRGBDClassavg}\n\\end{table*}\n\n\\begin{table}[t]\n\\centering\n\\begin{tabular}{c|ccc}\n{Method} & {Global avg.} & {Class avg.} & {Mean I\/U} \\\\ \\hline \\hline\n\\multicolumn{4}{c}{RGB}                                                        \\\\ \\hline\nLiu \\emph{et~al.}  \\cite{SIFT_flow}      & n\/a                  & 9.3                & n\/a               \\\\ \\hline\nSegNet            & \\textbf{70.3}                & 35.6               & \\textbf{22.1 }            \\\\ \\hline\n\\multicolumn{4}{c}{RGB-D}                                                       \\\\ \\hline\nLiu \\emph{et~al.}   \\cite{SIFT_flow}    & n\/a                  & 10.0               & n\/a               \\\\ \\hline\nRen et. al \\cite{ren2012rgb}     & n\/a                  & \\textbf{36.3}               & n\/a               \\\\ \\hline\n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{Quantitative comparison on the SUN RGB-D dataset which consists of 5050 test images of indoor scenes with 37 classes. SegNet RGB based predictions have a high global accuracy and also matches the RGB-D based predictions \\cite{ren2012rgb} in terms of class average accuracy.}\n\\label{SUNRGBDtest}\n\\end{table}\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures\/SUNRGBDQualy2.pdf}\n\\caption{\\footnotesize{Qualitative assessment of SegNet predictions on RGB indoor test scenes from the recently released SUN RGB-D dataset \\cite{song2015sun}. In this hard challenge, SegNet predictions delineate inter class boundaries well for object classes in a variety of scenes and their view-points. The segmentation quality is good when object classes are reasonably sized (rows (c,f)) but suffers when the scene is more cluttered (last two samples in row (i)). Note that often parts of an image of a scene do not have ground truth labels and these are shown in black colour. These parts are not masked in the corresponding SegNet predictions that are shown.}}\n\\label{SUNRGBDQualy}\n\\end{figure*}\n\n\\subsection{Pascal VOC12 Segmentation Challenge}\n\\label{Pascal}\nThe Pascal VOC12 segmentation challenge \\cite{Pascal} consists of segmenting a few salient object classes from a widely varying background class. It is unlike the segmentation for scene understanding benchmarks described earlier which require learning both classes and their spatial context. A number of techniques have been proposed based on this challenge which are increasingly more accurate and complex \\footnote{\\label{pascallb} See the leader board at \\url{http:\/\/host.robots.ox.ac.uk:8080\/leaderboard}}. Our efforts in this benchmarking experiment have not been diverted towards attaining the top rank by either using multi-stage training \\cite{FCN}, other datasets for pre-training such as MS-COCO \\cite{COCO, CRFRNN}, training and inference aids such as object proposals \\cite{zitnick2014edge, NohDeconvNets} or post-processing using CRF based methods \\cite{DeepLab, NohDeconvNets}.  Although these supporting techniques clearly have value towards increasing the performance it unfortunately does not reveal the true performance of the deep architecture which is the \\textit{core segmentation engine}. It however does indicate that some of the large deep networks are difficult to train end-to-end on this task even with pre-trained encoder weights. Therefore, to encourage more controlled benchmarking, we trained SegNet end-to-end without other aids and report this performance. \n\nIn Table \\ref{PascalQuant} we show the class average accuracy for some recent methods based on deep architectures. To the best of our ability, we have tried to gather the performance measures of the competing methods for their runs using minimum supporting techniques. We also specify when a method reports the performance of its core engine on the smaller validation set of $346$ images \\cite{DeepLab}. We find the performance on the full test set to be approximately  $1\\%$ less as compared to the smaller validation set. \n\nFrom the results in Table \\ref{PascalQuant} we can see that the best performing networks are either very large (and slow) \\cite{NohDeconvNets} and\/or they use a CRF \\cite{CRFRNN}. The CRF encourages large segments with a single label and this suits the Pascal challenge wherein there are one or two salient objects in the center of the image. This prior also has a larger impact when training data is limited. This is shown in the experiments using CRF-RNN  \\cite{CRFRNN} wherein the core FCN-8 model predictions are less accurate without extra training data. \n\nIt is interesting that the DeepLab \\cite{DeepLab} architecture which is simply upsampling the FCN encoder features using bilinear interpolation performs reasonably well (on the validation set). The fact that a coarse segmentation is enough to produce this performance shows that this challenge is unlike scene understanding wherein many classes of varying shape and size need to be segmented.\n\nMethods using object proposals during training and\/or inference \\cite{NohDeconvNets,Hypercolumns} are very slow in inference time and it is hard to measure their true performance. These aids are necessitated by the very large size of their deep network \\cite{NohDeconvNets} and also because the Pascal data can also be processed by a detect and segment approach. In comparison, SegNet is smaller by virtue of discarding the fully connected layers in the VGG16 \\cite{simonyan2014very}. The authors of DeepLab \\cite{DeepLab} have also reported little loss in performance by reducing the size of the fully connected layers. The smaller size of SegNet makes end-to-end training possible for benchmarking. Although it may be argued that larger networks perform better, it is at the cost of a complex training mechanism, increased memory and inference time. This makes them unsuitable for real-time applications such as road scene understanding. \n\nThe individual class accuracies for SegNet predictions on Pascal VOC12 are shown in Table \\ref{PascalClassavg}. From this we once again see larger and more frequently occurring classes such as aeroplane, bus, cat etc. have higher accuracy and smaller\/thinner classes such as potted plant, bicycle are poorly segmented. We believe more training data \\cite{COCO} can help improve the performance of SegNet.\n\n\n\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{Figures\/PascalQualy2.pdf}\n\\caption{\\footnotesize{Qualitative assessment of SegNet predictions on test samples from Pascal VOC12 \\cite{Pascal} dataset. SegNet performs competitively (row (b) on several object classes of varying shape and appearance. However, it lacks smoothness particularly on large objects (see row(d)). This can be perhaps be attributed to the smaller empirical size of the receptive field of the feature units in the deepest encoder layer size \\cite{zhou2014object}.}}\n\\label{PascalQualy}\n\\end{figure*}\n\n\n\\begin{table*}[]\n\\centering\n\\tabcolsep=2pt\n\\begin{tabular}{c|c|c|c|c|c|c}\n\\hline\nMethod       & Encoder size(M) & Decoder size (M) & Total size (M) & Class avg. acc. & Inference $500$ (ms) & Inference $224$ (ms) \\\\ \\hline \\hline\nDeepLab \\cite{DeepLab} (validation set)  & n\/a & n\/a & $<$ 134.5 & 58   & n\/a & n\/a       \\\\ \\hline\nFCN-8 \\cite{FCN}  (multi-stage training)  & 134 & 0.5 & 134.5 & 62.2  & 210 & n\/a       \\\\ \\hline\nHypercolumns \\cite{Hypercolumns}  (object proposals) & n\/a & n\/a & $>$ 134.5 & 62.6  & n\/a & n\/a            \\\\ \\hline\nDeconvNet \\cite{NohDeconvNets} (object proposals) &  138.35 & 138.35 & 276.7 & 69.6  & n\/a & 92.26 ($\\times$ 50)        \\\\ \\hline  \nCRF-RNN \\cite{CRFRNN} (multi-stage training)                  & n\/a & n\/a & $>$ 134.5 & \\textbf{69.6} & n\/a & n\/a     \\\\ \\hline\nSegNet       & 14.725       & 14.725 & 29.45 & 59.1   & \\textbf{193.4}  & \\textbf{48.4}  \\\\ \\hline\n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Quantitative comparison on Pascal VOC12 dataset. The accuracies for the competing architectures are gathered for their inference run using the least number of supporting training and inference techniques. However, since they are not trained end-to-end like SegNet and use aids such as object proposals, we have added corresponding qualifying comments. The first three columns show the number of trainable parameters in the encoder, decoder and full  network. Many of the models are approximately the same size as FCN. In comparison, SegNet is considerably smaller but achieves a competitive accuracy without resorting to supporting training or inference aids. SegNet is also the fastest in terms of inference time.}}\n\\label{PascalQuant}\n\\end{table*}\n\n\n\\begin{table*}[t]\n\\centering\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\n\\hline\n\\multicolumn{1}{c|}{Aeroplane} & \\multicolumn{1}{c|}{Bicycle} & \\multicolumn{1}{c|}{Bird} & \\multicolumn{1}{c|}{Boat} & \\multicolumn{1}{c|}{Bottle} & \\multicolumn{1}{c|}{Bus} & \\multicolumn{1}{c|}{Car} & \\multicolumn{1}{c|}{Cat} & \\multicolumn{1}{c|}{Chair} & \\multicolumn{1}{c|}{Cow} & \\multicolumn{1}{c}{Dining table} \\\\ \\hline\n74.5                           & 30.6                        & 61.4                     & 50.8                     & 49.8                       & 76.2                    & 64.3                    & 69.7                    & 23.8                      & 60.8                    & 54.7                             \\\\ \\hline\nDog                             & Horse                        & Motor bike                & Person                    & Potted plant                & Sheep                    & Sofa                     & Train                    & TV                         & \\multicolumn{1}{c|}{Background}             & \\multirow{2}{*}{}                 \\\\ \\cline{1-10}\n62.0                           & 66.4                        & 70.2                     & 74.1                     & 37.5                       & 63.7                    & 40.6                    & 67.8                    & 53.0                      &    \\multicolumn{1}{c|}{88.6}                      &                                   \\\\ \\cline{1-10}\n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Individual class accuracies of SegNet predictions on the Pascal VOC12 segmentation benchmark consisting of 21 object classes.}}\n\\label{PascalClassavg}\n\\end{table*}\n\n\\section{Discussion and future work}\n\\label{Discussion}\nThe success of deep learning is due largely because of the availability of massive datasets and to a lesser extent the use of very deep (and often big) models. The use of millions of training parameters in these networks has resulted in performance gains and this effect can be observed in our analysis of various models (see Sec. \\ref{Analysis}). However, less attention has been paid to smaller and more efficient models for real-time applications (of segmentation) such as road scene understanding. This was the primary motivation behind the proposal of SegNet, which is smaller and more efficient than other competing architectures but efficient, for tasks such as road scene understanding. \n \nAlthough in some tasks it may be sufficient to simply replicate the encoder features and post-process with a CRF, it is less elegant and does not exploit the potential of deep learning for feed-forward segmentation. Using SegNet as a candidate architecture, we have shown the need for designing trainable decoders which can learn to map the output of the encoder network to input resolution feature maps for classification. Our controlled analysis reveals the trade-offs involved in designing architectures for segmentation, mostly involving inference memory, time and accuracy. In particular, transferring more (feature) information to the decoders can lead to simpler decoding process but with additional storage costs. When compressed forms of encoder feature maps are stored and transferred to the decoders then, to achieve similar performance, the decoders need to be more complex (more training parameters).\n\nOur experience with benchmarking experiments have shown the efficacy of our proposed architecture for segmentation on both indoor and outdoor scenes. We outperform all the state-of-the-art methods for road scene understanding. Indoor scene understanding is very challenging and SegNet has set a new benchmark on a large dataset for RGB based scene understanding. An important issue we faced while conducting these experiments is the lack of controlled baselines for many of the recent architectures. Many of these architectures have used a host of supporting techniques to arrive at high accuracies on datasets but this makes it difficult to gather conclusive evidence about their true performance. This is further worsened by the fact that different variants of SGD is used to train these architectures and the learning rate is tuned manually by observing the progress of the loss. To avoid this and in the interest of future research, we trained SegNet end-to-end using SGD with a fixed learning rate and momentum throughout the training process for a predefined number of epochs. We also provide our Caffe implementation of SegNet-Basic (and its variants), SegNet and a web demo for evaluation of the real-time road scene segmentation. For the future, we would like to exploit our understanding of segmentation architectures gathered from our analysis to design more efficient architectures for real-time applications. We are also interested in experimenting with Dropout \\cite{gal2015dropout} during training and testing. This can help estimate the uncertainty of the predictions for scene understanding.\n\n\n\n\n\n\n\n\n\\section{Conclusion}\n\\label{Conclusion}\nWe presented SegNet, a deep convolutional network architecture for semantic segmentation. The main motivation behind SegNet was the need to design an efficient architecture for road scene understanding which is efficient both in terms of memory and computational time. We analysed SegNet and compared it with other important variants to reveal the trade-offs involved in designing architectures for segmentation. Those which store the encoder network feature maps in full perform best but consume more memory during inference time. SegNet on the other hand is more efficient since it only stores the max-pooling indices of the feature maps and uses them in its decoder network to achieve good performance. However, the inference time is increased since the decoder network is required to be more larger. On large and well known datasets SegNet performs competitively on challenges such as indoor scene understanding and Pascal VOC12. It sets a new benchmark for road scene understanding when trained on a large dataset. With its efficient architecture and competitive performance SegNet is well suited for scene understanding applications. In future, we would also like to estimate the uncertainty of the labelling using techniques such as Dropout.\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\\bibliographystyle{ieeetr}\n\n\\section{Introduction}\\label{sec:introduction}}\n\\section{Introduction}\n\\label{sec:introduction}\n\\begin{figure*}[!t]\n\\center\n\\includegraphics[width=0.95\\textwidth]{Figures\/CamVidTeaserwithIndoorScenes.pdf}\n\\caption{\\footnotesize{SegNet predictions on road scenes and indoor scenes. To try our system yourself, please see our online web demo at \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/}. }}\n\\label{Teaser}\n\\end{figure*} \n\nSemantic segmentation has a wide array of applications ranging from scene understanding, inferring support-relationships among objects to autonomous driving. Early methods that relied on low-level vision cues have fast been superseded by popular machine learning algorithms. In particular, deep learning has seen huge success lately in handwritten digit recognition, speech, categorising whole images and detecting objects in images \\cite{szegedy2015going, Simonyan}. Now there is an active interest for semantic pixel-wise labelling \\cite{FarabetPAMI} \\cite{raey},  \\cite{socher2011parsing},\\cite{FCN},\\cite{noh2015learning},\\cite{zheng2015conditional},\\cite{ParseNetRabinovich},\\cite{SegNetarXiv},\\cite{eigen2015predicting}, \\cite{liang2015semantic}, \\cite{papandreou2015weakly},\\cite{yu2015multi}, \\cite{ronneberger2015u}. \nHowever, some of these recent approaches have tried to directly adopt deep architectures designed for category prediction to pixel-wise labelling \\cite{FarabetPAMI}. The results, although very encouraging, appear coarse \\cite{liang2015semantic}. This is primarily because max pooling and sub-sampling reduce feature map resolution. Our motivation to design SegNet arises from this need to map low resolution features to input resolution for pixel-wise classification. This mapping must produce features which are useful for accurate boundary localization. \n\nOur architecture, SegNet, is designed to be an efficient architecture  for pixel-wise semantic segmentation. It is primarily motivated by road scene understanding applications which require the ability to model appearance (road, building), shape (cars, pedestrians) and understand the spatial-relationship (context) between different classes such as road and side-walk. In typical road scenes, the majority of the pixels belong to large classes such as road, building and hence the network must produce smooth segmentations. The engine must also have the ability to delineate objects based on their shape despite their small size. Hence it is important to retain boundary information in the extracted image representation. From a computational perspective, it is necessary for the network to be efficient in terms of both memory and computation time during inference. The ability to train end-to-end in order to jointly optimise all the weights in the network using an efficient weight update technique such as stochastic gradient descent (SGD) \\cite{Bottou} is an additional benefit since it is more easily repeatable. The design of SegNet arose from a need to match these criteria.\n\nThe encoder network in SegNet is topologically identical to the convolutional layers in VGG16 \\cite{simonyan2014very}. We remove the fully connected layers of VGG16 which makes the SegNet encoder network significantly smaller and easier to train than many other recent architectures \\cite{FCN,noh2015learning,ParseNetRabinovich,hong2015decoupled}. The key component of SegNet is the decoder network which consists of a hierarchy of decoders one corresponding to each encoder. Of these, the appropriate decoders use the max-pooling indices received from the corresponding encoder to perform non-linear upsampling of their input feature maps. This idea was inspired from an architecture designed for unsupervised feature learning \\cite{Ranzato}. Reusing max-pooling indices in the decoding process has several practical advantages; (i) it improves boundary delineation , (ii) it reduces the number of  parameters enabling end-to-end training, and (iii) this form of upsampling can be incorporated into any encoder-decoder architecture such as \\cite{FCN,zheng2015conditional} with only a little modification. \n\nOne of the main contributions of this paper is our analysis of the SegNet decoding technique and the widely used Fully Convolutional Network (FCN) \\cite{FCN}. This is in order to convey the practical trade-offs involved in designing segmentation architectures. Most recent deep architectures for segmentation have identical encoder networks, i.e VGG16, but differ in the form of the decoder network, training and inference. Another common feature is they have trainable parameters in the order of hundreds of millions and thus encounter difficulties in performing end-to-end training \\cite{noh2015learning}. The difficulty of training these networks has led to multi-stage training \\cite{FCN}, appending networks to a pre-trained architecture such as FCN \\cite{zheng2015conditional}, use of supporting aids such as region proposals for inference \\cite{noh2015learning}, disjoint training of classification and segmentation networks \\cite{hong2015decoupled} and use of additional training data for pre-training \\cite{ParseNetRabinovich} \\cite{mottaghi2014role} or for full training \\cite{zheng2015conditional}. In addition, performance boosting post-processing techniques \\cite{liang2015semantic} have also been popular. Although all these factors improve performance on challenging benchmarks \\cite{Pascal}, it is unfortunately difficult from their quantitative results to disentangle the key design factors necessary to achieve good performance. We therefore analysed the decoding process used in some of these approaches \\cite{FCN,noh2015learning} and reveal their pros and cons.\n\nWe evaluate the performance of SegNet on two scene segmentation tasks, CamVid road scene segmentation \\cite{GabeDataset} and SUN RGB-D indoor scene segmentation \\cite{song2015sun}. Pascal VOC12 \\cite{Pascal} has been the benchmark challenge for segmentation over the years. However, the majority of this task has one or two foreground classes surrounded by a highly varied background. This implicitly favours techniques used for detection as shown by the recent work on a decoupled classification-segmentation network \\cite{hong2015decoupled} where the classification network can be trained with a large set of weakly labelled data and the independent segmentation network performance is improved. The method of \\cite{liang2015semantic} also use the feature maps of the classification network with an independent CRF post-processing technique to perform segmentation. The performance can also be boosted by the use additional inference aids such as region proposals \\cite{noh2015learning}, \\cite{zitnick2014edge}. Therefore, it is different from scene understanding where the idea is to exploit co-occurrences of objects and other spatial-context to perform robust segmentation. To demonstrate the efficacy of SegNet, we present a real-time online demo of road scene segmentation into 11 classes of interest for autonomous driving (see link in Fig. \\ref{Teaser}). Some example test results produced on randomly sampled road scene images from Google and indoor test scenes from the SUN RGB-D dataset \\cite{song2015sun} are shown in Fig. \\ref{Teaser}. \n\nThe remainder of the paper is organized as follows. In Sec. \\ref{LitReview} we review related recent literature. We describe the SegNet architecture and its analysis in Sec. \\ref{Architecture}. In Sec. \\ref{Benchmarking} we evaluate the performance of SegNet on outdoor and indoor scene datasets. This is followed by a general discussion regarding our approach with pointers to future work in Sec. \\ref{Discussion}. We conclude in Sec. \\ref{Conclusion}. \n\n\\section{Literature Review}\n\\label{LitReview}\n\nSemantic pixel-wise segmentation is an active topic of research, fuelled by challenging datasets \\cite{GabeDataset,silberman2012indoor,GeigerKITTI,Pascal,song2015sun}. Before the arrival of deep networks, the best performing methods mostly relied on hand engineered features classifying pixels independently. Typically, a patch is fed into a classifier \\textit{e.g.} Random Forest \\cite{Jamie2,Brostow} or Boosting \\cite{Sturgess,LadickyECCV} to predict the class probabilities of the center pixel. Features based on appearance \\cite{Jamie2} or SfM and appearance \\cite{Brostow,Sturgess, LadickyECCV} have been explored for the CamVid road scene understanding test \\cite{GabeDataset}. These per-pixel noisy predictions (often called \\textit{unary} terms) from the classifiers are then smoothed by using a pair-wise or higher order CRF \\cite{Sturgess,LadickyECCV} to improve the accuracy. More recent approaches have aimed to produce high quality unaries by trying to predict the labels for all the pixels in a patch as opposed to only the center pixel. This improves the results of Random Forest based unaries \\cite{kontschieder2011structured} but thin structured classes are classified poorly. Dense depth maps computed from the CamVid video have also been used as input for classification using Random Forests \\cite{zhang2010semantic}. Another approach argues for the use of a combination of popular hand designed features and spatio-temporal super-pixelization to obtain higher accuracy \\cite{tighe2013superparsing}. The best performing technique on the CamVid test \\cite{LadickyECCV} addresses the imbalance among label frequencies by combining object detection outputs with classifier predictions in a CRF framework. The result of all these techniques indicate the need for improved features for  classification.  \n\nIndoor RGBD pixel-wise semantic segmentation has also gained popularity since the release of the NYU dataset \\cite{silberman2012indoor}. This dataset showed the usefulness of the depth channel to improve segmentation. Their approach used features such as RGB-SIFT, depth-SIFT and pixel location as input to a neural network classifier to predict pixel unaries. The noisy unaries are then smoothed using a CRF. Improvements were made using a richer feature set including LBP and region segmentation to obtain higher accuracy \\cite{ren2012rgb} followed by a CRF. In more recent work \\cite{silberman2012indoor}, both class segmentation and support relationships are inferred together using a combination of RGB and depth based cues. Another approach focuses on real-time joint reconstruction and semantic segmentation, where Random Forests are used as the classifier \\cite{Hermans14ICRA}. Gupta \\emph{et~al.}  \\cite{gupta2013perceptual} use boundary detection and hierarchical grouping before performing category segmentation. The common attribute in all these approaches is the use of hand engineered features for classification of either RGB or RGBD images. \n\nThe success of deep convolutional neural networks for object classification has more recently led researchers to exploit their feature learning capabilities for structured prediction problems such as segmentation. There have also been attempts to apply networks designed for object categorization to segmentation, particularly by replicating the deepest layer features in blocks to match image dimensions \\cite{FarabetPAMI,FarabetPurityCover,Grangier,Gatta}. However, the resulting classification is blocky \\cite{Grangier}. Another approach using recurrent neural networks \\cite{pinheiro2014recurrent} merges several low resolution predictions to create input image resolution predictions. These techniques are already an improvement over hand engineered features \\cite{FarabetPAMI} but their ability to delineate boundaries is poor. \n\nNewer deep architectures \\cite{FCN,noh2015learning,eigen2015predicting,hong2015decoupled,zheng2015conditional} particularly designed for segmentation have advanced the state-of-the-art by learning to decode or map low resolution image representations to pixel-wise predictions. The encoder network which produces these low resolution representations in all of these architectures is the VGG16 classification network \\cite{simonyan2014very} which has $13$ convolutional layers and $3$ fully connected layers. This encoder network weights are typically pre-trained on the large ImageNet object classification dataset \\cite{ImageNet}. The decoder network varies between these architectures and is the part which is responsible for producing multi-dimensional features for each pixel for classification.\n\nEach decoder in the Fully Convolutional Network (FCN) architecture \\cite{FCN} learns to upsample its input feature map(s) and combines them with the corresponding encoder feature map to produce the input to the next decoder. It is an architecture which has a large number of trainable parameters in the encoder network (134M) but a very small decoder network (0.5M). The overall large size of this network makes it hard to train end-to-end on a relevant task. Therefore, the authors use a stage-wise training process. Here each decoder in the decoder network is progressively added to an existing trained network. The network is grown until no further increase in performance is observed. This growth is stopped after three decoders thus ignoring high resolution feature maps can certainly lead to loss of edge information \\cite{noh2015learning}. Apart from training related issues, the need to reuse the encoder feature maps in the decoder makes it memory intensive in test time. We study this network in more detail as it the core of other recent architectures \\cite{ParseNetRabinovich, zheng2015conditional}.\n\n\\begin{figure*}[!ht]\n\\center\n\\includegraphics[width=0.95\\textwidth]{Figures\/segnet_softmax.pdf}\n\\caption{\\footnotesize{An illustration of the SegNet architecture. There are no fully connected layers and hence it is only convolutional. A decoder upsamples its input using the transferred pool indices from its encoder to produce a sparse feature map(s). It then performs convolution with a trainable filter bank to densify the feature map. The final decoder output feature maps are fed to a soft-max classifier for pixel-wise classification.}}\n\\label{SegNetArchitecture}\n\\end{figure*}\n\nThe predictive performance of FCN has been improved further by appending the FCN with a recurrent neural network (RNN) \\cite{zheng2015conditional} and fine-tuning them on large datasets \\cite{Pascal},\\cite{COCO}. The RNN layers mimic the sharp boundary delineation capabilities of CRFs while exploiting the feature representation power of FCN's. They show a significant improvement over FCN-8 but also show that this difference is reduced when more training data is used to train FCN-8. The main advantage of the CRF-RNN is revealed when it is jointly trained with an architecture such as the FCN-8. The fact that joint training helps is also shown in other recent results \\cite{UrtasunSegmentation}, \\cite{lin2015efficient}. Interestingly, the deconvolutional network \\cite{noh2015learning} performs significantly better than FCN although at the cost of a more complex training and inference. This however raises the question as to whether the perceived advantage of the CRF-RNN would be reduced as the core feed-forward segmentation engine is made better. In any case, the CRF-RNN network can be appended to any deep segmentation architecture including SegNet.\n\nMulti-scale deep architectures are also being pursued \\cite{eigen2015predicting,lin2015efficient}. They come in two flavours, (i) those which use input images at a few scales and corresponding deep feature extraction networks, and (ii) those which combine feature maps from different layers of a single deep architecture \\cite{hariharan2015hypercolumns} \\cite{ParseNetRabinovich}. The common idea is to use features extracted at multiple scales to provide both local and global context \\cite{mostajabi2015feedforward} and the using feature maps of the early encoding layers retain more high frequency detail leading to sharper class boundaries. Some of these architectures are difficult to train due to their parameter size \\cite{eigen2015predicting}. Thus a multi-stage training process is employed along with data augmentation. The inference is also expensive with multiple convolutional pathways for feature extraction. Others \\cite{lin2015efficient} append a CRF to their multi-scale network and jointly train them. However, these are not feed-forward at test time and require optimization to determine the MAP labels.\n\nSeveral of the recently proposed deep architectures for segmentation are not feed-forward in inference time \\cite{noh2015learning}, \\cite{liang2015semantic}, \\cite{hong2015decoupled}. They require either MAP inference over a CRF \\cite{lin2015efficient}, \\cite{UrtasunSegmentation} or aids such as region proposals \\cite{noh2015learning} for inference. We believe the perceived performance increase obtained by using a CRF is due to the lack of good decoding techniques in their core feed-forward segmentation engine. SegNet on the other hand uses decoders to obtain features for accurate pixel-wise classification.\n\nThe recently proposed Deconvolutional Network \\cite{noh2015learning} and its semi-supervised variant the Decoupled network \\cite{hong2015decoupled} use the max locations of the encoder feature maps (pooling indices) to perform non-linear upsampling in the decoder network. The authors of these architectures, independently of SegNet (first submitted to CVPR 2015 \\cite{SegNetarXiv}), proposed this idea of decoding in the decoder network. However, their encoder network consists of the fully connected layers from the VGG-16 network which consists of about $90\\%$ of the parameters of their entire network. This makes training of their network very difficult and thus require additional aids such as the use of region proposals to enable training. Moreover, during inference these proposals are used and this increases inference time  significantly. From a benchmarking point of view, this also makes it difficult to evaluate the performance of their architecture (encoder-decoder network) without other aids. In this work we discard the fully connected layers of the VGG16 encoder network which enables us to train the network using the relevant training set using SGD optimization. Another recent method \\cite{liang2015semantic} shows the benefit of reducing the number of parameters significantly without sacrificing performance, reducing memory consumption and improving inference time. \n\nOur work was inspired by the unsupervised feature learning architecture proposed by Ranzato \\emph{et al.} \\cite{Ranzato}. The key learning module is an encoder-decoder network. An encoder consists of convolution with a filter bank, element-wise tanh non-linearity, max-pooling and sub-sampling to obtain the feature maps. For each sample, the indices of the max locations computed during pooling are stored and passed to the decoder. The decoder upsamples the feature maps by using the stored pooled indices. It convolves this upsampled map using a trainable decoder filter bank to reconstruct the input image. This architecture was used for unsupervised pre-training for classification. A somewhat similar decoding technique is used for visualizing trained convolutional networks\\cite{zeiler2010deconvolutional} for classification. The architecture of Ranzato \\emph{et al.} mainly focused on layer-wise feature learning using small input patches. This was extended by Kavukcuoglu et. al. \\cite{KorayUnsup} to accept full image sizes as input to learn hierarchical encoders. Both these approaches however did not attempt to use \\textit{deep encoder-decoder} networks for unsupervised feature training as they discarded the decoders after each encoder training. Here, SegNet differs from these architectures as the deep encoder-decoder network is trained jointly for a supervised learning task and hence the decoders are an integral part of the network in test time.\n\nOther applications where pixel wise predictions are made using deep networks are image super-resolution \\cite{dong2014learning} and depth map prediction from a single image \\cite{eigen2014NIPS}. The authors in \\cite{eigen2014NIPS} discuss the need for learning to upsample from low resolution feature maps which is the central topic of this paper.  \n\n\\section{Architecture}\n\\label{Architecture}\nSegNet has an encoder network and a corresponding decoder network, followed by a final pixelwise classification layer. This architecture is illustrated in Fig. \\ref{Architecture}. The encoder network consists of $13$ convolutional layers which correspond to the first $13$ convolutional layers in the VGG16 network \\cite{simonyan2014very} designed for object classification. We can therefore initialize the training process from weights trained for classification on large datasets \\cite{ImageNet}. We can also discard the fully connected layers in favour of retaining higher resolution feature maps at the deepest encoder output. This also reduces the number of parameters in the SegNet encoder network  significantly (from 134M to 14.7M) as compared to other recent architectures \\cite{FCN}, \\cite{noh2015learning} (see. Table \\ref{TimeBenchmark}). Each encoder layer has a corresponding decoder layer and hence the decoder network has $13$ layers. The final decoder output is fed to a multi-class soft-max classifier to produce class probabilities for each pixel independently. \n\nEach \\textit{encoder} in the encoder network performs convolution with a filter bank to produce a set of feature maps. These are then batch normalized \\cite{BN},\\cite{badrinarayananunderstanding}). Then an element-wise rectified-linear non-linearity (ReLU) $max(0,x)$ is applied. Following that, max-pooling with a $2\\times2$ window and stride $2$ (non-overlapping window) is performed and the resulting output is sub-sampled by a factor of $2$. Max-pooling is used to achieve translation invariance over small spatial shifts in the input image. Sub-sampling results in a large input image context (spatial window) for each pixel in the feature map. While  several layers of max-pooling and sub-sampling can achieve more translation invariance for robust classification correspondingly there is a loss of spatial resolution of the feature maps. The increasingly lossy (boundary detail) image representation is not beneficial for segmentation where boundary delineation is vital. Therefore, it is necessary to \\textit{capture and store} boundary information in the encoder feature maps before sub-sampling is performed. If memory during inference is not constrained, then all the encoder feature maps (after sub-sampling) can be stored. This is usually not the case in practical applications and hence we propose a more efficient way to store this information. It involves storing only the max-pooling \\textit{indices}, i.e, the locations of the maximum feature value in each pooling window is memorized for each encoder feature map. In principle, this can be done using 2 bits for each $2\\times2$ pooling window and is thus much more efficient to store as compared to memorizing feature map(s) in float precision. As we show later in this work, this lower memory storage results in a slight loss of accuracy but is still suitable for practical applications.\n\nThe appropriate \\textit{decoder} in the decoder network upsamples its input feature map(s) using the memorized max-pooling indices from the corresponding encoder feature map(s). This step produces sparse feature map(s). This SegNet decoding technique is illustrated in Fig. \\ref{Upsampling}. These feature maps are then convolved with a trainable decoder filter bank to produce dense feature maps. A batch normalization step is then applied to each of these maps. Note that the decoder corresponding to the first encoder (closest to the input image) produces a multi-channel feature map, although its encoder input has 3 channels (RGB). This is unlike the other decoders in the network which produce feature maps with the same number of size and channels as their encoder inputs. The high dimensional feature representation at the output of the final decoder is fed to a trainable soft-max classifier. This soft-max classifies each pixel independently. The output of the soft-max classifier is a K channel image of probabilities where K is the number of classes. The predicted segmentation corresponds to the class with maximum probability at each pixel.\n\nWe add here that two other architectures, DeconvNet \\cite{NohDeconvNets} and U-Net \\cite{ronneberger2015u} share a similar architecture to SegNet but with some differences. DeconvNet has a much larger parameterization, needs more computational resources and is harder to train end-to-end (Table \\ref{TimeBenchmark}), primarily due to the use of fully connected layers (albeit in a convolutional manner) We report several comparisons with DeconvNet later in the paper Sec. \\ref{Benchmarking}. \n\nAs compared to SegNet, U-Net \\cite{ronneberger2015u} (proposed for the medical imaging community) does not reuse pooling indices but instead transfers the entire feature map (at the cost of more memory) to the corresponding decoders and concatenates them to upsampled (via deconvolution) decoder feature maps. There is no conv5 and max-pool 5 block in U-Net as in the VGG net architecture. SegNet, on the other hand, uses all of the pre-trained convolutional layer weights from VGG net as pre-trained weights.\n\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{Figures\/PoolingExp\/decoderVariantsNew.pdf}\n\\caption{\\footnotesize{An illustration of SegNet and FCN \\cite{FCN} decoders. $a,b,c,d$ correspond to values in a feature map. SegNet uses the max pooling indices to upsample (without learning) the feature map(s) and convolves with a trainable decoder filter bank. FCN upsamples by learning to deconvolve the input feature map and adds the corresponding encoder feature map to produce the decoder output. This feature map is the output of the max-pooling layer (includes sub-sampling) in the corresponding encoder. Note that there are no trainable decoder filters in FCN.}}\n\\label{Upsampling}\n\\end{figure*}\n\n\n\n\\begin{table*}[t]\n\\footnotesize\n\\centering\n\\tabcolsep=1pt\n\\begin{tabular}{c|c|c|c|cccc|ccc|cccc|ccc}\n\\multicolumn{4}{c}{} &  \\multicolumn{7}{c}{Median frequency balancing} & \\multicolumn{7}{|c}{Natural frequency balancing} \\\\\n\\hline\n& & Storage & Infer & \\multicolumn{4}{c|}{Test } &  \\multicolumn{3}{c|}{Train } &   \\multicolumn{4}{c|}{Test } & \\multicolumn{3}{c}{Train }\\\\\nVariant                   &Params (M) &  multiplier   &  time (ms)  & G       & C       & mIoU   &BF     &  G & C & mIoU   & G & C & mIoU  & BF & G & C & \\multicolumn{1}{c}{mIoU}               \\\\ \\hline \\hline\n\\multicolumn{16}{c}{Fixed upsampling} \\\\ \\hline\nBilinear-Interpolation           & 0.625 & 0 & 24.2 & 77.9 & 61.1 & 43.3 &20.83 & 89.1 & 90.2 &  82.7     & 82.7 & 52.5 & 43.8 &23.08 &93.5&74.1 &         59.9 \\\\ \\hline                                                                                                   \n\\multicolumn{16}{c}{Upsampling using max-pooling indices}    \\\\                                                                                              \n \\hline\nSegNet-Basic                    & 1.425 & 1 & 52.6 & 82.7       &   62.0      &   47.7  &35.78  & 94.7 & 96. 2 & 92.7 &  84.0 & 54.6 & 46.3 & 36.67 & 96.1 & 83.9 &73.3                   \\\\ \\hline\nSegNet-Basic-EncoderAddition    &1.425 & 64  & 53.0 & 83.4  &     \\textbf{63.6 }   &      48.5    &35.92    & 94.3 &  95.8 &  92.0 & \\textbf{84.2} & 56.5 & \\textbf{47.7} & 36.27 & 95.3&80.9 & 68.9\\\\ \\hline\nSegNet-Basic-SingleChannelDecoder& 0.625 &  1 & 33.1 &  81.2     &    60.7     &     46.1   & 31.62     & 93.2 & 94.8 &   90.3  & 83.5 & 53.9 & 45.2 & 32.45 &92.6 & 68.4 & 52.8        \\\\ \\hline\n\\multicolumn{16}{c}{Learning to upsample (bilinear initialisation)}                                                                                                         \n \\\\ \\hline\nFCN-Basic                     & 0.65 & 11  & 24.2  &   81.7    &     62.4    &    47.3  &\\textbf{38.11}      & 92.8 & 93.6 &  88.1  & 83.9 & 55.6 & 45.0 & \\textbf{37.33} & 92.0 & 66.8 & 50.7       \\\\ \\hline\nFCN-Basic-NoAddition           &0.65 & n\/a & 23.8 &   80.5    &    58.6     &     44.1    &31.96     & 92.5 & 93.0 &    87.2 & 82.3 & 53.9 & 44.2 & 29.43 &93.1& 72.8 & 57.6         \\\\ \\hline\nFCN-Basic-NoDimReduction                    &1.625 & 64 & 44.8  &  \\textbf{84.1}     &  63.4       &   \\textbf{50.1}    &37.37      & \\textbf{95.1} & \\textbf{96.5}  &  \\textbf{93.2} &        83.5 & \\textbf{57.3} & 47.0 & 37.13 &\\textbf{97.2} & \\textbf{91.7} & \\textbf{84.8}\n\\\\ \\hline\nFCN-Basic-NoAddition-NoDimReduction                    &1.625 & 0 & 43.9   & 80.5      &   61.6        & 45.9 & 30.47 &  92.5 & 94.6 & 89.9 &  83.7 & 54.8 & 45.5 & 33.17 & 95.0 & 80.2 & 67.8      \\\\ \\hline \n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Comparison of decoder variants. We quantify the performance using global (G), class average (C), mean of intersection over union (mIoU) and a semantic contour measure (BF). The testing and training accuracies are shown as percentages for both natural frequency and median frequency balanced training loss function. SegNet-Basic performs at the same level as FCN-Basic but requires only storing max-pooling indices and is therefore more memory efficient during inference. Note that the theoretical memory requirement reported is based only on the size of the first layer encoder feature map. FCN-Basic, SegNet-Basic, SegNet-Basic-EncoderAddition all have high BF scores indicating the need to use information in encoder feature maps for better class contour delineation. Networks with larger decoders and those using the encoder feature maps in full perform best, although they are least efficient in terms of inference time and memory.}}\n\\label{PoolingQuantCAFFE}\n\\end{table*}\n\n\\subsection{Decoder Variants}\n\\label{Variants}\nMany segmentation architectures \\cite{FCN, noh2015learning, liang2015semantic} share the same encoder network and they only vary in the form of their decoder network. \nOf these we choose to compare the SegNet decoding technique with the widely used Fully Convolutional Network (FCN) decoding technique \\cite{FCN, zheng2015conditional}. \n\nIn order to analyse SegNet and compare its performance with FCN (decoder variants) we use a smaller version of SegNet, termed \\textbf{SegNet-Basic} \\footnote{SegNet-Basic was earlier termed SegNet in a archival version of this paper \\cite{SegNetarXiv}}, which has 4 encoders and 4 decoders. All the encoders in SegNet-Basic perform max-pooling and sub-sampling and the corresponding decoders upsample its input using the received max-pooling indices. Batch normalization is used after each convolutional layer in both the encoder and decoder network. No biases are used after convolutions and no ReLU non-linearity is present in the decoder network. Further, a constant kernel size of $7\\times7$ over all the encoder and decoder layers is chosen to provide a wide context for smooth labelling \\textit{i.e.} a pixel in the deepest layer feature map (layer $4$) can be traced back to a context window in the input image of $106\\times106$ pixels. This small size of SegNet-Basic allows us to explore many different variants (decoders) and train them in reasonable time. Similarly we create \\textbf{FCN-Basic}, a comparable version of FCN for our analysis which shares the same encoder network as SegNet-Basic but with the FCN decoding technique (see Fig. \\ref{Upsampling}) used in all its decoders.  \n\nOn the left in Fig. \\ref{Upsampling} is the decoding technique used by SegNet (also SegNet-Basic), where there is no learning involved in the upsampling step. However, the upsampled maps are convolved with trainable multi-channel decoder filters to densify its sparse inputs. Each decoder filter has the same number of channels as the number of upsampled feature maps. A smaller variant is one where the decoder filters are single channel, i.e they only convolve their corresponding upsampled feature map. This variant (\\textbf{SegNet-Basic-SingleChannelDecoder}) reduces the number of trainable parameters and inference time significantly.\n\nOn the right in Fig. \\ref{Upsampling} is the FCN (also FCN-Basic) decoding technique. The important design element of the FCN model is dimensionality reduction step of the encoder feature maps. This \\textit{compresses} the encoder feature maps which are then used in the corresponding decoders. Dimensionality reduction of the encoder feature maps, say of 64 channels, is performed by convolving them with $1\\times 1\\times 64\\times K$ trainable filters, where $K$ is the number of classes. The compressed $K$ channel final encoder layer feature maps are the input to the decoder network. In a decoder of this network, upsampling is performed by \\textit{inverse convolution} using a fixed or trainable \\textit{multi-channel upsampling kernel}. We set the kernel size to $8\\times8$. This manner of upsampling is also termed as \\textit{deconvolution}. Note that, in comparison, SegNet the multi-channel convolution using trainable decoder filters is performed after upsampling to densifying feature maps. The upsampled feature map in FCN has $K$ channels. It is then added element-wise to the corresponding resolution encoder feature map to produce the output decoder feature map. The upsampling kernels are initialized using bilinear interpolation weights \\cite{FCN}. \n\nThe FCN decoder model requires storing encoder feature maps during inference. This can be memory intensive for embedded applications; for e.g. storing 64 feature maps of the first layer of FCN-Basic at $180\\times240$ resolution in 32 bit floating point precision takes 11MB. This can be made smaller using dimensionality reduction to the 11 feature maps which requires $\\approx$ 1.9MB storage. SegNet on the other hand requires almost negligible storage cost for the pooling indices ($.17$MB if stored using 2 bits per $2\\times2$ pooling window). We can also create a variant of the FCN-Basic model which discards the encoder feature map addition step and only learns the upsampling kernels (\\textbf{FCN-Basic-NoAddition}). \n\nIn addition to the above variants, we study upsampling using fixed bilinear interpolation weights which therefore requires no learning for upsampling (\\textbf{Bilinear-Interpolation}). At the other extreme, we can add 64 encoder feature maps at each layer to the corresponding output feature maps from the SegNet decoder to create a more memory intensive variant of SegNet (\\textbf{SegNet-Basic-EncoderAddition}).\nHere both the pooling indices for upsampling are used, followed by a convolution step to densify its sparse input. This is then added element-wise to the corresponding encoder feature maps to produce a decoders output.\n\nAnother and more memory intensive FCN-Basic variant (\\textbf{FCN-Basic-NoDimReduction}) is where there is no dimensionality reduction performed for the encoder feature maps. This implies that unlike FCN-Basic the final encoder feature map is not compressed to $K$ channels before passing it to the decoder network. Therefore, the number of channels at the end of each decoder is the same as the corresponding encoder (i.e $64$). \n\nWe also tried other generic variants where feature maps are simply upsampled by \\textit{replication} \\cite{FarabetPAMI}, or by using a fixed (and sparse) array of indices for upsampling. These performed quite poorly in comparison to the above variants. A variant without max-pooling and sub-sampling in the encoder network (decoders are redundant) consumes more memory, takes longer to converge and performs poorly. \nFinally, please note that to encourage reproduction of our results we release the Caffe implementation of all the variants \\footnote{See \\url{ http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/} for our SegNet code and web demo.}. \n\n\\subsection{Training}\n\\label{Training}\nWe use the CamVid road scenes dataset to benchmark the performance of the decoder variants. This dataset is small, consisting of 367 training and 233 testing RGB images (day and dusk scenes) at $360\\times480$ resolution. The challenge is to segment $11$ classes such as road, building, cars, pedestrians, signs, poles, side-walk etc. We perform local contrast normalization \\cite{Jarrett} to the RGB input.\n\nThe encoder and decoder weights were all initialized using the technique described in He \\emph{et~al.} \\cite{he2015delvingICCV}. To train all the variants we use stochastic gradient descent (SGD) with a fixed learning rate of 0.1 and momentum of 0.9 \\cite{Bottou} using our Caffe implementation of SegNet-Basic \\cite{jia2014caffeACM}. We train the variants until the training loss converges. Before each epoch, the training set is shuffled and each mini-batch (12 images) is then picked in order thus ensuring that each image is used only once in an epoch. We select the model which performs highest on a validation dataset.\n\nWe use the cross-entropy loss \\cite{FCN} as the objective function for training the network. The loss is summed up over all the pixels in a mini-batch. When there is large variation in the number of pixels in each class in the training set (e.g road, sky and building pixels dominate the CamVid dataset) then there is a need to weight the loss differently based on the true class. This is termed \\textit{class balancing}. We use \\textit{median frequency balancing } \\cite{eigen2015predicting} where the weight assigned to a class in the loss function is the ratio of the median of class frequencies computed on the entire training set divided by the class frequency. This implies that larger classes in the training set have a weight smaller than $1$ and the weights of the smallest classes are the highest. We also experimented with training the different variants without class balancing or equivalently using \\textit{natural frequency balancing}.\n\n\\subsection{Analysis}\n\\label{Analysis}\nTo compare the quantitative performance of the different decoder variants, we use three commonly used performance measures: global accuracy (G) which measures the percentage of pixels correctly classified in the dataset, class average accuracy (C) is the mean of the predictive accuracy over all classes and mean intersection over union (mIoU) over all classes as used in the Pascal VOC12 challenge \\cite{Pascal}. The mIoU metric is a more stringent metric than class average accuracy since it penalizes false positive predictions. However, mIoU metric is not optimized for directly through the class balanced cross-entropy loss. \n\nThe mIoU metric otherwise known as the Jacard Index is most commonly used in benchmarking. However,  Csurka et al. \\cite{csurka2013good} note that this metric does not always correspond to human qualitative judgements (ranks) of good quality segmentation. They show with examples that mIoU favours region smoothness and does not evaluate boundary accuracy, a point also alluded to recently by the authors of FCN \\cite{FCNnew}. Hence they propose to complement the mIoU metric with a boundary measure based on the Berkeley contour matching score commonly used to evaluate unsupervised image segmentation quality \\cite{martin2004learning}. Csurka et al. \\cite{csurka2013good} simply extend this to semantic segmentation and show that the measure of semantic contour accuracy used in conjunction with the mIoU metric agrees more with human ranking of segmentation outputs. \n\nThe key idea in computing a semantic contour score is to evaluate the F1-measure \\cite{martin2004learning} which involves computing the precision and recall values between the predicted and ground truth class boundary given a pixel tolerance distance. We used a value of $0.75\\%$ of the image diagonal as the tolerance distance. The F1-measure for each class that is present in the ground truth test image is averaged to produce an image F1-measure. Then we compute the whole test set average, denoted the boundary F1-measure (BF) by average the image F1 measures.\n\nWe test each architectural variant after each $1000$ iterations of optimization on the CamVid validation set until the training loss converges. With a training mini-batch size of 12 this corresponds to testing approximately every 33 epochs (passes) through the training set. We select the iteration wherein the global accuracy is highest amongst the evaluations on the validation set. We report all the three measures of performance at this point on the held-out CamVid test set. Although we use class balancing while training the variants, it is still important to achieve high global accuracy to result in an overall smooth segmentation. Another reason is that the contribution of segmentation towards autonomous driving is mainly for delineating classes such as roads, buildings, side-walk, sky. These classes dominate the majority of the pixels in an image and a high global accuracy corresponds to good segmentation of these important classes. We also observed that reporting the numerical performance when class average is highest can often correspond to low global accuracy indicating a perceptually noisy segmentation output.\n\nIn Table \\ref{PoolingQuantCAFFE} we report the numerical results of our analysis. We also show the size of the trainable parameters and the highest resolution feature map or pooling indices storage memory, i.e, of the first layer feature maps after max-pooling and sub-sampling. We show the average time for one forward pass with our Caffe implementation, averaged over $50$ measurements using a $360\\times480$ input on an NVIDIA Titan GPU with cuDNN v3 acceleration. We note that the upsampling layers in the SegNet variants are not optimised using cuDNN acceleration. We show the results for both testing and training for all the variants at the selected iteration. The results are also tabulated without class balancing (natural frequency) for training and testing accuracies. Below we analyse the results with class balancing.\n\nFrom the Table \\ref{PoolingQuantCAFFE}, we see that bilinear interpolation based upsampling without any learning performs the worst based on all the measures of accuracy. All the other methods which either use learning for upsampling (FCN-Basic and variants) or learning decoder filters after upsampling (SegNet-Basic and its variants) perform significantly better. This emphasizes the need to learn decoders for segmentation. This is also supported by experimental evidence gathered by other authors when comparing FCN with SegNet-type decoding techniques\\cite{noh2015learning}.\n\nWhen we compare SegNet-Basic and FCN-Basic we see that both perform equally well on this test over all the measures of accuracy. The difference is that SegNet uses less \\textbf{memory} during inference since it only stores max-pooling indices. On the other hand FCN-Basic stores \\textbf{encoder feature maps} in full which consumes much more memory (11 times more). SegNet-Basic has a decoder with 64 feature maps in each decoder layer. In comparison FCN-Basic, which uses dimensionality reduction, has fewer (11) feature maps in each decoder layer. This reduces the number of convolutions in the decoder network and hence FCN-Basic is faster during inference (forward pass). From another perspective, the decoder network in SegNet-Basic makes it overall a larger network than FCN-Basic. This endows it with more flexibility and hence achieves higher training accuracy than FCN-Basic for the same number of iterations. Overall we see that SegNet-Basic has an advantage over FCN-Basic when inference time memory is constrained but where inference time can be compromised to some extent. \n\nSegNet-Basic is most similar to FCN-Basic-NoAddition in terms of their decoders, although the decoder of SegNet is larger. Both learn to produce dense feature maps, either directly by learning to perform deconvolution as in FCN-Basic-NoAddition or by first upsampling and then convolving with trained decoder filters. \nThe performance of SegNet-Basic is superior, in part due to its larger decoder size. The accuracy of FCN-Basic-NoAddition is also lower as compared to FCN-Basic. This shows that it is vital to capture the information present in the encoder feature maps for better performance. In particular, note the large drop in the BF measure between these two variants. This can also explain the part of the reason why SegNet-Basic outperforms FCN-Basic-NoAddition. \n\nThe size of the FCN-Basic-NoAddition-NoDimReduction model is slightly larger than SegNet-Basic since the final encoder feature maps are not compressed to match the number of classes $K$. This makes it a fair comparison in terms of the size of the model. The performance of this FCN variant is poorer than SegNet-Basic in test but also its training accuracy is lower for the same number of training epochs. This shows that using a larger decoder is not enough but it is also important to capture encoder feature map information to learn better, particular the fine grained contour information (notice the drop in the BF measure). Here it is also interesting to see that SegNet-Basic has a competitive training accuracy when compared to larger models such FCN-Basic-NoDimReduction. \n\nAnother interesting comparison between FCN-Basic-NoAddition and SegNet-Basic-SingleChannelDecoder shows that using max-pooling indices for upsampling and an overall larger decoder leads to better performance. This also lends evidence to SegNet being a good architecture for segmentation, particularly when there is a need to find a compromise between storage cost, accuracy versus inference time. In the best case, when both memory and inference time is not constrained, larger models such as FCN-Basic-NoDimReduction and SegNet-EncoderAddition are both more accurate than the other variants. Particularly, discarding dimensionality reduction in the FCN-Basic model leads to the best performance amongst the FCN-Basic variants with a high BF score. This once again emphasizes the trade-off involved between memory and accuracy in segmentation architectures.\n\nThe last two columns of Table \\ref{PoolingQuantCAFFE} show the result when no class balancing is used (natural frequency). Here, we can observe that without weighting the results are poorer for all the variants, particularly for class average accuracy and mIoU metric. The global accuracy is the highest without weighting since the majority of the scene is dominated by sky, road and building pixels. Apart from this all the inference from the comparative analysis of variants holds true for natural frequency balancing too, including the trends for the BF measure. SegNet-Basic performs as well as FCN-Basic and is better than the larger FCN-Basic-NoAddition-NoDimReduction. The bigger but less efficient models FCN-Basic-NoDimReduction and SegNet-EncoderAddition perform better than the other variants.\n\nWe can now summarize the above analysis with the following general points.\n\\begin{enumerate}\n\\item The best performance is achieved when encoder feature maps are stored in full. This is reflected in the semantic contour delineation metric (BF) most clearly.\n\\item When memory during inference is constrained, then compressed forms of encoder feature maps (dimensionality reduction, max-pooling indices) can be stored and used with an appropriate decoder (e.g. SegNet type) to improve performance.\n\\item Larger decoders increase performance for a given encoder network.\n\\end{enumerate}\n\n\n\n\\section{Benchmarking}\n\\label{Benchmarking}\nWe quantify the performance of SegNet on two scene segmentation  benchmarks using our Caffe implementation \\footnote{\\label{webdemo} Our web demo and Caffe implementation is available for evaluation at \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/}}. The first task is road scene segmentation which is of current practical interest for various autonomous driving related problems. The second task is indoor scene segmentation which is of immediate interest to several augmented reality (AR) applications. The input RGB images for both tasks were $360\\times480$. \n\nWe benchmarked SegNet against several other well adopted deep architectures for segmentation such as FCN \\cite{FCN}, DeepLab-LargFOV \\cite{liang2015semantic} and DeconvNet \\cite{noh2015learning}. Our objective was to understand the performance of these architectures when trained end-to-end on the same datasets. To enable end-to-end training we added batch normalization \\cite{BN} layers after each convolutional layer. For DeepLab-LargeFOV, we changed the max pooling 3 stride to 1 to achieve a final predictive resolution of $45\\times60$. We restricted the feature size in the fully connnected layers of DeconvNet to $1024$ so as to enable training with the same batch size as other models. Here note that the authors of DeepLab-LargeFOV \\cite{liang2015semantic} have also reported little loss in performance by reducing the size of the fully connected layers.\n\n In order to perform a controlled benchmark we used the same SGD solver \\cite{Bottou} with a fixed learning rate of $10^{-3}$ and momentum of $0.9$. The optimization was performed for more than 100 epochs through the dataset until no further performance increase was observed. Dropout of $0.5$ was added to the end of deeper convolutional layers in all models to prevent overfitting (see \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/tutorial.html} for example caffe prototxt). For the road scenes which have $11$ classes we used a mini-batch size of $5$ and for indoor scenes with $37$ classes we used a mini-batch size of $4$. \n\n\n\n\n\n\\begin{figure*}\n\t\\centering\n\t\\includegraphics[width=0.8\\textwidth]{CamVidQualitativeDeep.pdf}\n\t\\caption{\\footnotesize{Results on CamVid day and dusk test samples.\n\t\t\tSegNet shows superior performance, particularly with its ability to delineate boundaries, as compared to some of the larger models when all are trained in a controlled setting. DeepLab-LargeFOV is the most efficient model and with CRF post-processing can produce competitive results although smaller classes are lost. FCN with learnt deconvolution is clearly better. DeconvNet is the largest model with the longest training time, but its predictions loose small classes. Note that these results correspond to the model corresponding to the highest mIoU accuracy in Table \\ref{CamvidDeepBenchmark}.\n\t\t\n\t\t\n\t}}\n\t\\label{CamVidQualy}\n\\end{figure*}\n\n\n\n\\subsection{Road Scene Segmentation}\n\\label{CamVid}\nA number of road scene datasets are available for semantic parsing \\cite{gould2009decomposing,russell2008labelme,GabeDataset,GeigerKITTI}. Of these we choose to benchmark SegNet using the CamVid dataset \\cite{GabeDataset} as it contains video sequences. This enables us to compare our proposed architecture with those which use \\textit{motion and structure} \\cite{LadickyECCV,Sturgess,Brostow} and video segments \\cite{tighe2013superparsing}. We also combine \\cite{gould2009decomposing,russell2008labelme,GabeDataset,GeigerKITTI} to form an ensemble of 3433 images to train SegNet for an additional benchmark. For a web demo (see footnote \\ref{webdemo}) of road scene segmentation, we include the CamVid test set to this larger dataset. \nHere, we would like to note that another recent and independent segmentation benchmark on road scenes has been performed for SegNet and the other competing architectures used in this paper \\cite{cordts2016cityscapes}. However, the benchmark was not controlled, meaning that each architecture was trained with a separate recipe with varying input resolutions and sometimes with a validation set included. Therefore, we believe our more controlled benchmark can be used to complement their efforts.\n\n\n\\begin{table*}[th]\n\t\\resizebox{\\textwidth}{!}{\n\t\t\\small{\n\t\t\t\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|cccc}\n\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\t\\multicolumn{1}{c}{Method}                   & \\multicolumn{1}{c}{\\rotatebox{90}{Building}} & \\multicolumn{1}{c}{\\rotatebox{90}{Tree}} & \\multicolumn{1}{c}{\\rotatebox{90}{Sky}}  & \\multicolumn{1}{c}{\\rotatebox{90}{Car}}  & \\multicolumn{1}{c}{\\rotatebox{90}{Sign-Symbol}} & \\multicolumn{1}{c}{\\rotatebox{90}{Road}} & \\multicolumn{1}{c}{\\rotatebox{90}{Pedestrian}} & \\multicolumn{1}{c}{\\rotatebox{90}{Fence}} & \\multicolumn{1}{c}{\\rotatebox{90}{Column-Pole}} & \\multicolumn{1}{c}{\\rotatebox{90}{Side-walk}} & \\multicolumn{1}{c}{\\rotatebox{90}{Bicyclist}} & \\multicolumn{1}{c}{\\rotatebox{90}{Class avg.}} & \\multicolumn{1}{c}{\\rotatebox{90}{Global avg.}} & \\multicolumn{1}{c}{\\rotatebox{90}{mIoU}}&\\multicolumn{1}{c}{\\rotatebox{90}{BF}}\\\\ \\hline \\hline\n\t\t\t\t\n\t\t\t\n\t\t\t\t\n\t\t\t\tSfM+Appearance  \\cite{Brostow}           & 46.2     & 61.9 & 89.7 & 68.6 & 42.9        & 89.5 & 53.6       & 46.6  & 0.7         & 60.5     & 22.5      & 53.0       & 69.1  & \\multicolumn{2}{c}{n\/a$^{*}$}      \\\\ \\hline\n\t\t\t\t\n\t\t\t\tBoosting    \\cite{Sturgess}              & 61.9     & 67.3 & 91.1 & 71.1 & 58.5        & 92.9 & 49.5       & 37.6  & 25.8        & 77.8     & 24.7      & 59.8       & 76.4  & \\multicolumn{2}{c}{n\/a$^{*}$}      \\\\ \\hline\n\t\t\t\t\n\t\t\t\tDense Depth Maps   \\cite{zhang2010semantic}          & 85.3     & 57.3 & 95.4 & 69.2 & 46.5        & \\textbf{98.5} & 23.8       & 44.3  & 22.0        & 38.1     & 28.7      & 55.4       & 82.1    & \\multicolumn{2}{c}{n\/a$^{*}$}    \\\\ \\hline\n\t\t\t\t\n\t\t\t\tStructured Random Forests \\cite{kontschieder2011structured}& \\multicolumn{11}{c|}{n\/a}                                                                          & 51.4       & 72.5    &  \\multicolumn{2}{c}{n\/a$^{*}$}    \\\\ \\hline\n\t\t\t\t\n\t\t\t\tNeural Decision Forests \\cite{BuloNeural}  & \\multicolumn{11}{c|}{n\/a}                                                                          & 56.1       & 82.1   & \\multicolumn{2}{c}{n\/a$^{*}$}    \\\\ \\hline\n\t\t\t\t\n\t\t\t\tLocal Label Descriptors  \\cite{yang2012local}  & 80.7     & 61.5 & 88.8 & 16.4 & n\/a         & 98.0 & 1.09       & 0.05  & 4.13        & 12.4     & 0.07      & 36.3       & 73.6    & \\multicolumn{2}{c}{n\/a$^{*}$}    \\\\ \\hline\n\t\t\t\t\n\t\t\t\tSuper Parsing   \\cite{tighe2013superparsing}           & 87.0     & 67.1 & 96.9 & 62.7 & 30.1        & 95.9 & 14.7       & 17.9  & 1.7         & 70.0     & 19.4      & 51.2       & 83.3    & \\multicolumn{2}{c}{n\/a$^{*}$}     \\\\ \\hline\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\n\t\t\t\tSegNet (3.5K dataset training - 140K)          & \\textbf{89.6}     & \\textbf{83.4}  & 96.1 & \\textbf{87.7} & 52.7 & 96.4 & \\textbf{62.2} & \\textbf{53.45} & \\textbf{32.1}  & \\textbf{93.3} & \\textbf{36.5}  & \\textbf{71.20}      & \\textbf{90.40}   &60.10    &  46.84 \\\\ \\hline\n\t\t\t\t\\multicolumn{15}{c}{CRF based approaches}                                                                                                       \\\\ \\hline\n\t\t\t\t\n\t\t\t\tBoosting + pairwise CRF  \\cite{Sturgess} & 70.7     & 70.8 & 94.7 & 74.4 & 55.9        & 94.1 & 45.7       & 37.2  & 13.0        & 79.3     & 23.1      & 59.9       & 79.8  & \\multicolumn{2}{c}{n\/a$^{*}$}       \\\\ \\hline\n\t\t\t\t\n\t\t\t\tBoosting+Higher order \\cite{Sturgess}    & 84.5     & 72.6 & \\textbf{97.5} & 72.7 & 34.1        & 95.3 & 34.2       & 45.7  & 8.1         & 77.6     & 28.5      & 59.2       & 83.8    & \\multicolumn{2}{c}{n\/a$^{*}$}     \\\\ \\hline\n\t\t\t\t\n\t\t\t\tBoosting+Detectors+CRF \\cite{LadickyECCV}   & 81.5     & 76.6 & 96.2 & 78.7 & 40.2        & 93.9 & 43.0       & 47.6  & 14.3        & 81.5     & 33.9      & 62.5       & 83.8    & \\multicolumn{2}{c}{n\/a$^{*}$}     \\\\ \\hline\n\t\t\t\\end{tabular}\n\t}}\n\t\\vspace*{0.1cm}\n\t\\caption{\\footnotesize{Quantitative comparisons of SegNet with traditional methods on the CamVid 11 road class segmentation problem \\cite{GabeDataset}. SegNet outperforms all the other methods, including those using depth, video and\/or CRF's on the majority of classes. In comparison with the CRF based methods SegNet predictions are more accurate in 8 out of the 11 classes. It also shows a good $\\approx 10\\%$ improvement in class average accuracy when trained on a large dataset of 3.5K images. Particularly noteworthy are the significant improvements in accuracy for the smaller\/thinner classes. * Note that we could not access predictions for older methods for computing the mIoU, BF metrics.\n\t}}\n\t\\label{CamVidQuant}\n\\end{table*}\n\nThe qualitative comparisons of SegNet predictions with other deep architectures can be seen in Fig. \\ref{CamVidQualy}. The qualitative results show the ability of the proposed architecture to segment smaller classes in road scenes while producing a smooth segmentation of the overall scene. Indeed, under the controlled benchmark setting, \nSegNet shows superior performance as compared to some of the larger models. DeepLab-LargeFOV is the most efficient model and with CRF post-processing can produce competitive results although smaller classes are lost. FCN with learnt deconvolution is clearly better than with fixed bilinear upsampling. DeconvNet is the largest model and the most inefficient to train. Its predictions do not retain small classes.\n\n\nWe also use this benchmark to first compare SegNet with several non deep-learning methods including Random Forests \\cite{Jamie2}, Boosting \\cite{Jamie2,Sturgess} in combination with CRF based methods \\cite{LadickyECCV}. This was done to give the user a perspective of the improvements in accuracy that has been achieved using deep networks compared to classical feature engineering based techniques. \n\nThe results in Table \\ref{CamVidQuant} show SegNet-Basic, SegNet obtain competitive results  when compared with methods which use  CRFs. This shows the ability of the deep architecture to extract meaningful features from the input image and map it to accurate and smooth class segment labels. The most interesting result here is the large  performance improvement in class average and mIOU metrics that is obtained when a large training dataset, obtained by combining \\cite{gould2009decomposing,russell2008labelme,GabeDataset,GeigerKITTI}, is used to train SegNet. Correspondingly, the qualitative results of SegNet (see Fig. \\ref{CamVidQualy}) are clearly superior to the rest of the methods. It is able to segment both small and large classes well. We remark here that we used median frequency class balancing \\cite{eigen2014NIPS} in training SegNet-Basic and SegNet. In addition, there is an overall smooth quality of segmentation much like what is typically obtained with CRF post-processing. Although the fact that results improve with larger training sets is not surprising, the percentage improvement obtained using pre-trained encoder network and this training set indicates that this architecture can potentially be deployed for practical applications. Our random testing on urban and highway images from the internet (see Fig. \\ref{Teaser}) demonstrates that SegNet can \\textit{absorb} a large training set and generalize well to unseen images. It also indicates the contribution of the prior (CRF) can be lessened when sufficient amount of training data is made available.\n\nIn Table \\ref{CamvidDeepBenchmark} we compare SegNet's performance with now widely adopted fully convolutional architectures for segmentation. As compared to the experiment in Table \\ref{CamVidQuant}, we did not use any class blancing for training any of the deep architectures including SegNet. This is because we found it difficult to train larger models such as DeconvNet with median frequency balancing. We benchmark performance at 40K, 80K and $>$80K iterations which given the mini-batch size and training set size approximately corresponds to $50,100$ and $>$100 epochs. For the last test point we also report the maximum number of iterations (here atleast 150 epochs) beyond which we observed no accuracy improvements or when over-fitting set in. We report the metrics at three stages in the training phase to reveal how the metrics varied with training time, particularly for larger networks. This is important to understand if additional training time is justified when set against accuracy increases. Note also that for each evaluation we performed a complete run through the dataset to obtain batch norm statistics and then evaluated the test model with this statistic (see \\url{http:\/\/mi.eng.cam.ac.uk\/projects\/segnet\/tutorial.html} for code.). These evaluations are expensive to perform on large training sets and hence we only report metrics at three time points in the training phase. \n\nFrom Table \\ref{CamvidDeepBenchmark} we immediately see that SegNet, DeconvNet achieve the highest scores in all the metrics as compared to other models. DeconvNet has a higher boundary delineation accuracy but  SegNet is much more efficient as compared to DeconvNet. This can be seen from the compute statistics in Table \\ref{TimeBenchmark}. FCN, DeconvNet which have  fully connected layers (turned into convolutional layers) train much more slowly and have comparable or higher forward-backward pass time with reference to SegNet. Here we note also that over-fitting was not an issue in training these larger models, since at comparable iterations to SegNet their metrics showed an increasing trend. \n\nFor the FCN model learning the deconvolutional layers as opposed to fixing them with bi-linear interpolation weights improves performance particularly the BF score. It also achieves higher metrics in a far lesser time. This fact agrees with our earlier analysis in Sec. \\ref{Analysis}. \n\nSurprisingly, DeepLab-LargeFOV which is trained to predict labels at a resolution of $45\\times60$ produces competitive performance given that it is the smallest model in terms of parameterization and also has the fastest training time as per Table \\ref{TimeBenchmark}. However, the boundary accuracy is poorer and this is shared by the other architectures. DeconvNet's BF score is higher than the other networks when trained for a very long time. Given our analysis in Sec. \\ref{Analysis} and the fact that it shares a SegNet type architecture. \n\nThe impact of dense CRF \\cite{koltun2011efficient} post-processing can be seen in the last time point for DeepLab-LargeFOV-denseCRF. Both global and mIoU improve but class average diminshes. However a large improvement is obtained for the BF score. Note here that the dense CRF hyperparameters were obtained by an expensive grid-search process on a subset of the training set since no validation set was available.\n\n\\begin{table*}[t]\n\\centering\n\\tabcolsep=2pt\n\\begin{tabular}{|c|c|c|c|c||c|c|c|c||c|c|c|c|c|}\n\\hline\nNetwork\/Iterations       & \\multicolumn{4}{c||}{40K} & \\multicolumn{4}{c||}{80K} & \\multicolumn{4}{c|}{$>$80K} & Max iter \\\\ \\hline \\hline\n  & G & C & mIoU & BF & G & C & mIoU & BF & G & C & mIoU & BF & \\\\ \\hline \n\nSegNet   & 88.81 & 59.93 & 50.02 & 35.78 & 89.68 & 69.82 &   57.18 & 42.08 &90.40 & 71.20  &60.10  & 46.84 & 140K\\\\ \\hline\nDeepLab-LargeFOV\\cite{liang2015semantic} & 85.95 & 60.41 & 50.18 & 26.25 & 87.76 & 62.57 & 53.34 & 32.04 & 88.20 & 62.53 & 53.88 & 32.77 & 140K\\\\ \\hline\nDeepLab-LargeFOV-denseCRF\\cite{liang2015semantic} & \\multicolumn{8}{c|}{not computed} & 89.71 & 60.67 & 54.74 & 40.79 & 140K\\\\ \\hline\nFCN & 81.97 & 54.38 & 46.59 & 22.86  &82.71 & 56.22 & 47.95 & 24.76& 83.27 & 59.56 & 49.83 & 27.99 & 200K \\\\  \\hline\nFCN (learnt deconv) \\cite{FCN} &83.21 &56.05 & 48.68 &27.40 & 83.71 & 59.64 & 50.80 & 31.01 & 83.14 & 64.21 & 51.96 & 33.18 & 160K \\\\ \\hline\nDeconvNet \\cite{noh2015learning} & 85.26 & 46.40 & 39.69 & 27.36 & 85.19 & 54.08 &43.74 & 29.33 & 89.58 & 70.24 & 59.77 & 52.23 & 260K \\\\ \\hline \n\\end{tabular}\n\\vspace*{0.1cm}\n\\caption{\\footnotesize{Quantitative comparison of deep networks for semantic segmentation on the CamVid test set when trained on a corpus of 3433 road scenes \\textit{without class balancing}. When end-to-end training is performed with the same and fixed learning rate, smaller networks like SegNet learn to perform better in a shorter time. The BF score which measures the accuracy of inter-class boundary delineation is significantly higher for SegNet, DeconvNet as compared to other competing models. DeconvNet matches the metrics for SegNet but at a much larger computational cost.  Also see Table \\ref{CamVidQuant} for individual class accuracies for SegNet.}}\n\\label{CamvidDeepBenchmark}\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\centering\n\t\\tabcolsep=2pt\n\t\\begin{tabular}{|c|c|c|c|c||c|c|c|c||c|c|c|c|c|}\n\t\t\\hline\n\t\tNetwork\/Iterations       & \\multicolumn{4}{c||}{80K} & \\multicolumn{4}{c||}{140K} & \\multicolumn{4}{c|}{$>$140K} & Max iter  \\\\ \\hline \\hline\n\t\t& G & C & mIoU & BF & G & C & mIoU & BF & G & C & mIoU & BF  &\\\\ \\hline \n\t\t\n\t\tSegNet   &70.73  & 30.82 & 22.52 & 9.16  &71.66  &37.60  &27.46  & 11.33 & 72.63 & 44.76 & 31.84 & 12.66 & 240K\\\\ \\hline\n\t\tDeepLab-LargeFOV \\cite{liang2015semantic}&70.70 &41.75 &30.67 &7.28 &71.16 & 42.71 & 31.29 & 7.57 & 71.90 & 42.21 & 32.08 & 8.26 & 240K\\\\ \\hline\n\t\tDeepLab-LargeFOV-denseCRF \\cite{liang2015semantic}& \\multicolumn{8}{c|}{not computed} & 66.96 & 33.06 & 24.13& 9.41 & 240K \\\\ \\hline\n\t\tFCN (learnt deconv) \\cite{FCN}& 67.31 & 34.32 & 24.05 & 7.88 & 68.04 & 37.2 & 26.33 & 9.0 & 68.18 & 38.41 & 27.39 & 9.68 & 200K \\\\  \\hline\n\t\tDeconvNet \\cite{noh2015learning}& 59.62 & 12.93 & 8.35 & 6.50 & 63.28 & 22.53 & 15.14 & 7.86 & 66.13 & 32.28 & 22.57 & 10.47 & 380K \\\\ \\hline \n\t\\end{tabular}\n\t\\vspace*{0.1cm}\n\t\\caption{\\footnotesize{Quantitative comparison of deep architectures on the SUNRGB-D dataset when trained on a corpus of 5250 indoor scenes. Note that only the RGB modality was used in these experiments. In this complex task with $37$ classes all the architectures perform poorly, particularly because of the smaller sized classes and skew in the class distribution. DeepLab-Large FOV, the smallest and most efficient model has a slightly higher mIoU but SegNet has a better G,C,BF score. Also note that when SegNet was trained with \\textit{median frequency class balancing} it obtained 71.75, 44.85, 32.08, 14.06 (180K) as the metrics.}}\n\t\\label{SUNRGBBenchmark}\n\\end{table*}\n\n\\subsection{SUN RGB-D Indoor Scenes}\n\\label{SUNRGBD}\nSUN RGB-D \\cite{song2015sun} is a very challenging and large dataset of indoor scenes with $5285$ training and $5050$ testing images. The images are captured by different sensors and hence come in various resolutions. The task is to segment $37$ indoor scene classes including wall, floor, ceiling, table, chair, sofa etc. This task is made hard by the fact that object classes come in various shapes, sizes and in different poses. There are frequent partial occlusions since there are typically many different classes present in each of the test images. These factors make this one of the hardest segmentation challenges. We only use the RGB modality for our training and testing. Using the depth modality would necessitate architectural modifications\/redesign \\cite{FCN}. Also the quality of depth images from current cameras require careful post-processing to fill-in missing measurements. They may also require using fusion of many frames to robustly extract features for segmentation. Therefore we believe using depth for segmentation merits a separate body of work which is not in the scope of this paper. We also note that an earlier benchmark dataset NYUv2 \\cite{silberman2012indoor} is included as part of this dataset.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.9\\textwidth]{SUNRGBQualitativeDeep.pdf}\n\\caption{\\footnotesize{Qualitative assessment of SegNet predictions on RGB indoor test scenes from the recently released SUN RGB-D dataset \\cite{song2015sun}. In this hard challenge, SegNet predictions delineate inter class boundaries well for object classes in a variety of scenes and their view-points. Overall rhe segmentation quality is better when object classes are reasonably sized but is very noisy when the scene is more cluttered. Note that often parts of an image of a scene do not have ground truth labels and these are shown in black colour. These parts are not masked in the corresponding deep model  predictions that are shown. Note that these results correspond to the model corresponding to the highest mIoU accuracy in Table \\ref{SUNRGBBenchmark}.}}\n\\label{SUNRGBDQualy}\n\\end{figure*}\n\nRoad scene images have limited variation, both in terms of the classes of interest and their spatial arrangements. When captured from a moving vehicle where the camera position is nearly always parallel to the road surface limiting variability in view points. This makes it easier for deep networks to learn to segment them robustly. In comparison, images of indoor scenes are more complex since the view points can vary a lot and there is less regularity in both the number of classes present in a scene and their spatial arrangement. Another difficulty is caused by the widely varying sizes of the object classes in the scene. Some test samples from the recent SUN RGB-D dataset \\cite{song2015sun} are shown in Fig. \\ref{SUNRGBDQualy}. We observe some scenes with few large classes and some others with dense clutter (bottom row and right). The appearance (texture and shape) can also widely vary in indoor scenes. Therefore, we believe this is the hardest challenge for segmentation architectures and methods in computer vision. Other challenges, such as Pascal VOC12 \\cite{Pascal} salient object segmentation have occupied researchers  more \\cite{liu2015semantic}, but we believe indoor scene segmentation is more challenging and has more current practical applications such as in AR and robotics. To encourage more research in this direction we compared well known deep architectures on the large SUN RGB-D dataset.\n\n\n\nThe qualitative results of SegNet on samples of indoor scenes of different types such as bedroom, living room, laboratory, meeting room, bathroom are shown in Fig. \\ref{SUNRGBDQualy}. We see that SegNet obtains reasonable predictions when the size of the classes are large under different view points. This is particularly interesting since the input modality is only RGB.\nRGB images are also useful to segment thinner structures such as the legs of chairs and tables, lamps which is difficult to achieve using depth images from currently available sensors. This can be seen from the results of SegNet, DeconvNet in Fig. \\ref{SUNRGBDQualy}. It is also useful to segment decorative objects such as paintings on the wall for AR tasks. However as compared to outdoor scenes the segmentation quality is clearly more noisy. The quality drops significantly when clutter is increased (see the result sample in the middle column).\n\nThe quantitative results in Table \\ref{SUNRGBBenchmark} show that all the deep architectures share low mIoU and boundary metrics. The global and class averages (correlates well with mIou) are also small. SegNet outperforms all other methods in terms of G,C, BF metrics and has a slightly lower mIoU than DeepLab-LargeFOV. As a stand alone experiment we trained SegNet with median frequency class balancing \\cite{eigen2014predicting} and the metrics were higher (see Table \\ref{SUNRGBBenchmark}) and this agrees with our analysis in Sec. \\ref{Analysis}. Interestingly, using the grid search based optimal hyperparameters for the dense-CRF worsened all except the BF score metric for DeepLab-LargeFOV-denseCRF. More optimal settings could perhaps be found but the grid search process was too expensive given the large inference time for dense-CRFs. \n\nOne reason for the overall poor performance is the large number of classes in this segmentation task, many of which occupy a small part of the image and appear infrequently. The accuracies reported in Table \\ref{SUNRGBDClassavg} clearly show that larger classes have reasonable accuracy and smaller classes have lower accuracies. This can be improved with larger sized datasets and class distribution aware training techniques. Another reason for poor performance could lie in the inability of these deep architectures (all are based on the VGG architecture \\cite{Simonyan}) to large variability in indoor scenes . This conjecture on our part is based on the fact that the smallest model DeepLab-LargeFOV produces the best accuracy in terms of mIoU and in comparison, larger parameterizations in DeconvNet, FCN did not improve perfomance even with much longer training (DeconvNet). This suggests there could lie a common reason for poor performance across all architectures. More controlled datasets \\cite{handa2015scenenet} are needed to verify this hypothesis.\n\n\n\n\n\n\n\\section{Discussion and future work}\n\\label{Discussion}\nDeep learning models have often achieved increasing success due to the availability of massive datasets and expanding model depth and parameterisation. However, in practice factors like memory and computational time during training and testing are important factors to consider when choosing a model from a large bank of models. \nTraining time becomes an important consideration particularly when the performance gain is not commensurate with increased training time as shown in our experiments. Test time memory and computational load are important to deploy models on specialised embedded devices, for example, in AR applications. \nFrom an overall efficiency viewpoint, we feel less attention has been paid to smaller and more memory, time efficient models for real-time applications such as road scene understanding and AR. This was the primary motivation behind the proposal of SegNet, which is significantly smaller and faster than other competing architectures, but which we have shown to be efficient for tasks such as road scene understanding. \n \nSegmentation challenges such as Pascal \\cite{Pascal} and MS-COCO \\cite{COCO} are object segmentation challenges wherein a few classes are present in any test image.  Scene segmentation is more challenging due to the high variability of indoor scenes and a need to segment a larger number of classes simultaneously. The task of outdoor and indoor scene segmentation are also more practically oriented with current applications such as autonomous driving, robotics and AR. \n\n The metrics we chose to benchmark various deep segmentation architectures like the boundary F1-measure (BF) was done to complement the existing metrics which are more biased towards region accuracies. It is clear from our experiments and other independent benchmarks \\cite{cordts2016cityscapes} that outdoor scene images captured from a moving car are easier to segment and deep architectures perform robustly. We hope our experiments will encourage researchers to engage their attention towards the more challenging indoor scene segmentation task.\n \nAn important choice we had to make when benchmarking different deep architectures of varying parameterization was the manner in which to train them. Many of these architectures have used a host of supporting techniques and multi-stage training recipes to arrive at high accuracies on datasets but this makes it difficult to gather evidence about their true performance under time and memory constraints. Instead we chose to perform a controlled benchmarking where we used batch normalization to enable end-to-end training with the same solver (SGD). However, we note that this approach cannot entirely disentangle the effects of model versus solver  (optimization) in achieving a particular result. This is mainly due to the fact that training these networks involves gradient back-propagation which is imperfect and the optimization is a non-convex problem in extremely large dimensions. Acknowledging these shortcomings, our hope is that this controlled analysis complements other benchmarks \\cite{cordts2016cityscapes} and reveals the practical trade-offs involved in different well known architectures.\n\nFor the future, we would like to exploit our understanding of segmentation architectures gathered from our analysis to design more efficient architectures for real-time applications. We are also interested in estimating the model uncertainty for predictions from deep segmentation architectures \\cite{gal2015dropout},  \\cite{kendall2015bayesian}.\n\n\\begin{table*}[t]\n\t\\centering\n\t\\tabcolsep=3pt\n\t\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c}\n\t\n\t\t\\multicolumn{1}{c|}{Wall} & \\multicolumn{1}{c|}{Floor} & \\multicolumn{1}{c|}{Cabinet} & \\multicolumn{1}{c|}{Bed} & \\multicolumn{1}{c|}{Chair} & \\multicolumn{1}{c|}{Sofa} & \\multicolumn{1}{c|}{Table} & \\multicolumn{1}{c|}{Door} & \\multicolumn{1}{c|}{Window} & \\multicolumn{1}{c|}{Bookshelf} & \\multicolumn{1}{c|}{Picture} & \\multicolumn{1}{c|}{Counter} & \\multicolumn{1}{c}{Blinds} \\\\ \\hline\n\t\t83.42                    & 93.43                       & 63.37                        & 73.18                     & 75.92                      & 59.57                      &       64.18                 & 52.50                      & 57.51                        & 42.05                           & 56.17                         & 37.66                        & 40.29                      \\\\ \\hline\n\t\tDesk                       & Shelves                    & Curtain                      & Dresser                  & Pillow                     & Mirror                    & Floor mat                  & Clothes                   & Ceiling                     & Books                          & Fridge                       & TV                           & Paper                       \\\\ \\hline\n\t\t11.92                      & 11.45                        & 66.56                         & 52.73                     & 43.80                       & 26.30                     & 0.00                       & 34.31                      & 74.11                        & 53.77                           & 29.85                         & 33.76                         & 22.73                       \\\\ \\hline\n\t\tTowel                      & Shower curtain             & Box                          & Whiteboard               & Person                     & Night stand               & Toilet                     & Sink                      & Lamp                        & Bathtub                        & Bag                          & \\multicolumn{2}{c}{\\multirow{2}{*}{}}                     \\\\ \\cline{1-11}\n\t\t19.83                       & 0.03                        & 23.14                          & 60.25                     & 27.27                       & 29.88                     & 76.00                       & 58.10                     & 35.27                        & 48.86                           & 16.76                          & \\multicolumn{2}{c}{}                                      \\\\ \\cline{1-11}\n\t\\end{tabular}\n\t\\vspace*{0.1cm}\n\t\\caption{Class average accuracies of SegNet predictions for the 37 indoor scene classes in the SUN RGB-D benchmark dataset. The performance correlates well with size of the classes in indoor scenes. Note that class average accuracy has a strong correlation with mIoU metric.}\n\t\\label{SUNRGBDClassavg}\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\centering\n\t\\tabcolsep=2pt\n\t\\begin{tabular}{|c|c|c|c|c|c|}\n\t\t\\hline\n\t\tNetwork       & Forward pass(ms) & Backward pass(ms) & GPU training memory (MB) & GPU inference memory (MB) & Model size (MB)  \\\\ \\hline \\hline\n\t\tSegNet  & 422.50 & 488.71 & 6803 & \\textbf{1052} & 117\\\\ \\hline\n\t\tDeepLab-LargeFOV \\cite{liang2015semantic} & \\textbf{110.06} & \\textbf{160.73}& \\textbf{5618} & 1993 & 83 \\\\ \\hline\n\t\tFCN (learnt deconv)\\cite{FCN} & 317.09 & 484.11 & 9735 & 1806 & 539 \\\\ \\hline\n\t\tDeconvNet \\cite{noh2015learning} & 474.65 & 602.15 & 9731 & 1872 & 877 \\\\ \\hline\n\t\\end{tabular}\n\t\\vspace*{0.1cm}\n\t\\caption{\\footnotesize{A comparison of computational time and hardware resources required for various deep architectures. The caffe time command was used to compute time requirement averaged over 10 iterations with mini batch size 1 and an image of $360\\times480$ resolution We used nvidia-smi unix command to compute memory consumption. For training memory computation we used a mini-batch of size 4 and for inference memory the batch size was 1. Model size was the size of the caffe models on disk. SegNet is most memory efficient during inference model.}}\n\t\\label{TimeBenchmark}\n\\end{table*}\n\n\\section{Conclusion}\n\\label{Conclusion}\nWe presented SegNet, a deep convolutional network architecture for semantic segmentation. The main motivation behind SegNet was the need to design an efficient architecture for road and indoor scene understanding which is efficient both in terms of memory and computational time. We analysed SegNet and compared it with other important variants to reveal the practical trade-offs involved in designing architectures for segmentation, particularly training time, memory versus accuracy. Those architectures which store the encoder network feature maps in full perform best but consume more memory during inference time. SegNet on the other hand is more efficient since it only stores the max-pooling indices of the feature maps and uses them in its decoder network to achieve good performance. On large and well known datasets SegNet performs competitively, achieving high scores for road scene understanding. End-to-end learning of deep segmentation architectures is a harder challenge and we hope to see more attention paid to this important problem.\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\\bibliographystyle{ieeetr}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nShor's quantum algorithm for integer factorization \\cite{Shor} is a striking case of superpolynomial speed-up promised by a quantum computer over the best-known classical algorithms.\nSince Shor's original paper, many explicit circuit constructions over qubits for performing the algorithm have been developed and analyzed.\nThis includes automated synthesis of the underlying quantum circuits for the binary case (see the following and references therein:\n\\cite{BeckmanEtAl,Beauregard,CleveWatrous,Markov2012,Markov2013,MeterItoh,TakaKuni,VedralEtAl,Zalka}).\n\nIt has been previously noted that arithmetic encoding systems beyond binary may yield more natural embeddings for some computations and potentially lead to more efficient solutions.\n(A brief history note on this line of thought can be found in section 4.1 of \\cite{Knuth}.)\nExperimental implementation of computation with ternary logic, for example with Josephson junctions, dates back to 1989 \\cite{morisue1989jctl,morisue1998memory}. More recently, multi-valued logic has been\nproposed for linear ion traps \\cite{Muthu2000mvl}, cold atoms \\cite{smith2013cs}, and entangled photons \\cite{malik2016multiPhoton}.\nIn topological quantum computing it has been shown that metaplectic non-Abelian anyons \\cite{CuiWang} naturally align with ternary, and not binary, logic.\nThese anyons offer a natively topologically protected universal set of quantum gates (see, for example, \\cite{NayakFreedman}), in turn requiring little to no quantum error correction.\n\nIt is also interesting to note that qutrit-based computers are in certain sense space-optimal among all the qudit-based computers with varying local quantum dimension. Thus in\n\\cite{GreentreeEtAl} an argument is made that, as the dimension of the constituent qudits increases, the cost of maintaining a qudit in fully entangled state also increases and the optimum cost\nper Hilbert dimension is attained at local dimension of $\\lceil e \\rceil = 3$.\n\nTransferring the wealth of multi-qubit circuits to multi-qutrit framework is not straightforward. Some of the binary primitives, for example the binary Hadamard gate and the two-qubit $\\mbox{CNOT}$ gate, do\nnot remain Clifford operations in the ternary case.  Therefore, they cannot be emulated by ternary Clifford circuits. We resolve this complication by developing efficient non-Clifford circuits for a\n\\emph{generic} ternary quantum computer first.  We then extend the solution to the Metaplectic Topological Quantum Computer (MTQC) platform \\cite{CuiWang}, which further reduces the cost of implementation.\n\nA generic ternary framework that supports the full ternary Clifford group, measurement, and classical control  \\cite{CampbellEtAl}, also supports a distillation protocol that prepares magic states for the\n$P_9$ gate:\n\\begin{equation} \\label{eq:def:P:9}\nP_9 = \\omega_9^{-1}\\,|0\\rangle \\langle 0|+|1\\rangle \\langle 1|+\\omega_9\\,|2\\rangle \\langle 2|, \\, \\omega_9 = e^{2 \\pi \\, i\/9}.\n\\end{equation}\nThe Clifford+$P_9$ basis is universal for quantum computation and serves a similar functional role in ternary logic as the Clifford+$T$ basis in binary logic (see \\cite{CampbellEtAl,howard2012qudit} for the\nmore general qudit context).\n\nWe show in more detail further that the primitive $R$ gate available in MTQC is more powerful in practice than the $P_9$ gate.\n\nArguably, a natural choice to implement Shor's algorithm on a ternary quantum computer is to translate the entire arithmetic into ternary form. We do so by using ternary arithmetic tools developed in\n\\cite{BCRS} (with some practical improvements). We also explore alternative approach: emulation of binary version of Shor's period finding algorithm on ternary processor. Emulation has notable practical advantages in some contexts. For example, as shown in section \\ref{subsec:ripple:carry}, using a binary ripple-carry additive shift consumes fewer clean $P_9$\nmagic states than the corresponding  ternary ripple-carry additive shift (cf. table \\ref{table:binary:vs:ternary}).\n\nWe also show that on a metaplectic ternary computer the magic state coprocessor is asymptotically smaller than a magic state distillation coprocessor, such as the one developed in\n\\cite{CampbellEtAl} for the generic ternary quantum computer. Another benefit of the MTQC is the ability to approximate desired non-Clifford reflections directly to the required fidelity, thus eliminating the need for magic states. The tradeoff is an increase in the depth of the emulated Shor's period finding circuit by a logarithmic factor, which is tolerable for the majority of instances.\n\n\nThe cost benefits of using exotic non-Abelian anyons for integer factorization has been previously noted, for example in \\cite{BarabanEtAl}, where hypothetical\nFibonacci anyons were used. It is worthwhile noting that neither binary nor ternary logic is native to Fibonacci anyons, so the $\\mbox{NOT}$, $\\mbox{CNOT}$ or Toffoli gates are\nmuch harder to emulate there than on a hypothetical metaplectic anyon computer.\n\nThe paper is organized as follows.\nIn Section \\ref{sec:backgroud} we state the definitions and equations pertaining the two ternary architectures used, and gaive a quick overview of the Shor's period finding function.\nIn Section \\ref{sec:Emulation} we perform a detailed analysis of reversible classical circuits for modular exponentiation. We compare two designs of the modular\nexponentiation arithmetic. One is emulation of binary encoding of integers combined with ternary arithmetic gates. The other uses ternary encoding of integers with ternary gates.\n\nIn Section \\ref{sec:reflections} we develop circuits for the  key arithmetic gates based on designs from \\cite{BCRS} with further optimizations.\n\nIn Section \\ref{sec:specific:resource:counts} we compare the resource cost of performing modular exponentiation. An interesting feature of\nternary arithmetic circuits is the fact that the denser and more compact ternary encoding of integers does not necessarily lead to more resource-efficient period finding solutions compared to binary encoding.\nThe latter appears to be better suited in practice for low-width arithmetic circuit designs (hence, e.g., for smaller quantum computers).\n\n\nWe also compare the magic state preparation requirements. We highlight the huge advantage of the metaplectic topological computer. Magic state\npreparation requires width that is linear in $\\log(n)$ on an MTQC, whereas it requires width in $O(\\log^3(n))$ on a generic ternary quantum computer. \\footnote{It requires width in\n$O(\\log^{\\gamma}(n))$ in the binary Clifford+T architecture, where $\\gamma$ can vary between $\\log_2(3)$ and $\\log_3(15)$ depending on practically applicable distillation protocol.}\n\nAll the circuit designs and resource counts are done under assumption of fully-connected multi-qutrit network. Factorization circuitry optimized for sparsely connected networks, such as nearest-neighbor for example, is undeniably interesting (cf. \\cite{PhamSvore}) but we had to set this topic aside in the scope of this paper.\n\n\n\\section{Background and Notation} \\label{sec:backgroud}\n\n\n\nA common assumption for a multi-qudit quantum computer architecture is the availability of quantum gates generating the full multi-qudit Clifford group (see \\cite{howard2012qudit},\\cite{CampbellEtAl}). In this\nsection we describe a \\emph{generic} ternary computer, where the full ternary Clifford group is postulated; we also describe the more specific Metaplectic Topological Quantum Computer (MTQC) where the required\nClifford gates are explicitly implemented by braiding non-Abelian anyons \\cite{CuiHongWang, CuiWang}. For purposes of this paper, each braid corresponds to a unitary operation on qutrits. Braids are considered relatively inexpensive and tolerant to local noise. Universal quantum computation on MTQC is achieved by adding a single-qutrit phase flip gate\n($\\mbox{Flip}$ in \\cite{CuiWang}, $R_{|2\\rangle}$ in \\cite{BCKW} and our Subsection \\ref{subsec:metaplectic:basis}). In contrast with the binary phase flip $Z$, which is a Pauli gate, the ternary phase flip is\nnot only non-Clifford, but it does not belong to \\emph{any} level of Clifford hierarchy (see, for example, \\cite{BCRS}). Intuitively one should expect this gate to be very powerful. Level $\\mathcal{C}_3$ of the ternary Clifford hierarchy is emulated quite efficiently on MTQC architecture, while the converse is quite expensive: implementing phase flip in terms of $\\mathcal{C}_3$ requires\nseveral ancillas and a number of repeat-until-success circuits.\n\n\\subsection{Ternary Clifford group}\nLet $\\{|0\\rangle,|1\\rangle,|2\\rangle\\}$ be the standard computational basis for a qutrit.\nLet $\\omega_3 = e^{2 \\pi \\, i\/3}$ be the third primitive root of unity.\nThe ternary \\emph{Pauli} group is generated by the \\emph{increment} gate\n\\begin{equation} \\label{eq:INC}\n\\mbox{INC}= |1\\rangle \\langle 0|+|2\\rangle \\langle 1|+|0\\rangle \\langle 2|,\n\\end{equation}\n\\noindent and the ternary $Z$ gate\n\\begin{equation} \\label{eq:Z}\nZ= |0\\rangle \\langle 0|+\\omega_3 |1\\rangle \\langle 1|+\\omega_3^2 |2\\rangle \\langle 2|.\n\\end{equation}\n\nThe ternary \\emph{Clifford} group stabilizes the Pauli group is obtained by adding the ternary Hadamard gate $H$,\n\\begin{equation} \\label{eq:H}\nH = \\frac{1}{\\sqrt{3}} \\sum \\omega_3^{j\\, k} |j\\rangle \\langle k|,\n\\end{equation}\n\\noindent the $Q$ gate\n\\begin{equation} \\label{eq:Q}\nQ= |0\\rangle \\langle 0|+|1\\rangle \\langle 1|+\\omega_3 |2\\rangle \\langle 2|,\n\\end{equation}\n\\noindent and the two-qutrit $\\mbox{SUM}$ gate,\n\\begin{equation} \\label{eq:SUM}\n\\mbox{SUM} |j,k\\rangle = |j,j+k \\; {\\rm mod} \\; 3\\rangle, j,k \\in \\{0,1,2\\}\n\\end{equation}\n\\noindent to the set of generators of the Pauli group.\n\nCompared to the binary Clifford group, $H$ is the ternary counterpart of the binary Hadamard gate, $Q$ is the counterpart of the phase gate $S$, and $\\mbox{SUM}$ is an analog of the $\\mbox{CNOT}$ (although,\nintuitively it is a ``weaker'' entangler than $\\mbox{CNOT}$, as described below).\n\nFor any $n$, ternary Clifford gates and their various tensor products generate a finite subgroup of $U(3^n)$; therefore they are not sufficient for universal quantum computation.\nWe consider and compare two methods of building up quantum universality: by implementing the $P_9$ gate as per Eq. (\\ref{eq:def:P:9}) and by expanding into the metaplectic basis (Subsection\n\\ref{subsec:metaplectic:basis}).\nGiven enough ancillae, these two bases are effectively and efficiently equivalent in principle (see Appendix \\ref{app:sec:R:2}), and the costs in ancillae create practical tradeoffs depending on the given\napplication.\n\n\\subsection{Binary and ternary control}\n\nGiven an $n$-qutrit unitary operator $U$ there are different ways of expanding it into an $(n+1)$-qutrit unitary using the additional qutrit as ``control''.\nLet $|c\\rangle$ be a state of the control qutrit and $|t\\rangle$ be a state of the $n$-qutrit register. We define\n\\[\nC_{\\ell}(U)|c\\rangle |t\\rangle = |c\\rangle \\otimes (U^{\\delta_{c,\\ell}}) |t\\rangle,\\, \\ell \\in \\{0,1,2\\},\n\\]\n\\noindent wherein $\\delta$ denotes the Kronecker delta symbol.\nWe refer to this operator as a \\emph{binary-controlled} unitary $U$ and denote it in circuit diagrams as\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=1.000000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (24.000000, 22.500000);\n\\draw[color=black] (0.000000,15.000000) -- (24.000000,15.000000);\n\\draw[color=black] (0.000000,0.000000) -- (24.000000,0.000000);\n\\draw (12.000000,15.000000) -- (12.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 0.000000) circle(6.000000pt);\n\\clip (12.000000, 0.000000) circle(6.000000pt);\n\\draw (12.000000, 0.000000) node {U};\n\\end{scope}\n\\filldraw (12.000000, 15.000000) circle(1.500000pt);\n\\draw (16.000000, 19.000000) node {$\\ell$};\n\\end{tikzpicture}\n\\;\\raisebox{2.3mm}{.}\n\\end{figure}\n\nWe omit the label $\\ell$ when $\\ell=1$. We also define the \\emph{ternary-controlled} extension of $U$ by\n\\[\n\\Lambda(U)|c\\rangle |t\\rangle = |c\\rangle \\otimes (U^c\\, |t\\rangle)\n\\]\n\\noindent and denote it in circuit diagrams as\n\n\\begin{figure}[h!]\n\\centering\n\\begin{tikzpicture}[scale=1.000000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (24.000000, 22.500000);\n\\draw[color=black] (0.000000,15.000000) -- (24.000000,15.000000);\n\\draw[color=black] (0.000000,0.000000) -- (24.000000,0.000000);\n\\draw (12.000000,15.000000) -- (12.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 0.000000) circle(6.000000pt);\n\\clip (12.000000, 0.000000) circle(6.000000pt);\n\\draw (12.000000, 0.000000) node {U};\n\\end{scope}\n\\draw[fill=white] (12.000000, 15.000000) circle(1.500000pt);\n\\end{tikzpicture}\n\\;\\raisebox{2.3mm}{.}\n\\end{figure}\n\nIt is paramount to keep in mind that\n\\[\n\\mbox{SUM} = \\Lambda(\\mbox{INC})\n\\]\n\\noindent (see equations (\\ref{eq:INC}) and (\\ref{eq:SUM})).\nAnother useful observation is that for any unitary $U$ we have that $\\Lambda(U) = C_1(U) \\, (C_2(U))^2$.\n\nMore detail can be found in \\cite{BCRS}.\n\n\n\\subsection{The $P_9$ gate and its corresponding magic state} \\label{subsec:P9:gate}\n\nIt is easy to see that the $P_9$ gate in Eq. (\\ref{eq:def:P:9}) is not a Clifford gate, e.g., it does not stabilize the ternary Pauli group. However, it can be realized by a certain deterministic measurement-assisted\ncircuit given a copy of the \\emph{magic} state\n\\begin{equation} \\label{eq:def:mu}\n\\mu = \\omega_9^{-1} |0\\rangle + |1\\rangle+\\omega_9 |2\\rangle, \\, \\omega_9 = e^{2 \\pi \\, i\/9}.\n\\end{equation}\n\nAn appropriate deterministic magic state injection circuit, as proposed in Ref.~\\cite{CampbellEtAl}, is shown in Figure \\ref{fig:mu:state:injection}.\n\\begin{figure}[h!]\n\\begin{tikzpicture}[scale=2.000000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (72.000000, 22.500000);\n\\draw[color=black] (0.000000,15.000000) -- (36.000000,15.000000);\n\\draw[color=black] (36.000000,14.500000) -- (60.000000,14.500000);\n\\draw[color=black] (36.000000,15.500000) -- (60.000000,15.500000);\n\\draw[color=black] (0.000000,15.000000) node[left] {$|\\mu\\rangle$};\n\\draw[color=black] (0.000000,0.000000) -- (72.000000,0.000000);\n\\draw[color=black] (0.000000,0.000000) node[left] {$|\\mbox{input}\\rangle$};\n\\draw (12.000000,15.000000) -- (12.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 15.000000) circle(6.000000pt);\n\\clip (12.000000, 15.000000) circle(6.000000pt);\n\\draw (12.000000, 15.000000) node {$\\mbox{INC}^{\\dagger}$};\n\\end{scope}\n\\draw[fill=white] (12.000000, 0.000000) circle(1.500000pt);\n\\draw[fill=white] (30.000000, 9.000000) rectangle (42.000000, 21.000000);\n\\draw[very thin] (36.000000, 15.600000) arc (90:150:6.000000pt);\n\\draw[very thin] (36.000000, 15.600000) arc (90:30:6.000000pt);\n\\draw[->,>=stealth] (36.000000, 9.600000) -- +(80:10.392305pt);\n\\draw (59.500000,15.000000) -- (59.500000,0.000000);\n\\draw (60.500000,15.000000) -- (60.500000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (60.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (60.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (60.000000, -0.000000) node {$C_{\\mu,m}$};\n\\end{scope}\n\\filldraw (60.000000, 15.000000) circle(1.500000pt);\n\\draw[color=black] (72.000000,0.000000) node[right] {$P_9 |\\mbox{input}\\rangle$};\n\\end{tikzpicture}\n\\caption{\\label{fig:mu:state:injection} Exact representation of the $P_9$ gate by state injection. $C_{\\mu,m}$ stands for a certain precompiled ternary Clifford gate, classically predicated by the measurement\nresult $m$. }\n\\end{figure}\nFor completeness,  $C_{\\mu,m}=(P_9 \\, \\mbox{INC} \\, P_9^{\\dagger})^{-m}\\, \\mbox{INC}^m$. Note that $P_9 \\, \\mbox{INC} \\, P_9^{\\dagger}$ is a Clifford gate, since $P_9$ is at level 3 of the ternary Clifford\nhierarchy (cf.~\\cite{BCRS}).\n\nSuch magic state naturally exists in any multi-qudit framework with qudits of prime dimension \\cite{CampbellEtAl}. When the framework supports the full multi-qudit Clifford group, projective measurements and\nclassical control, then it also supports stabilizer protocols for magic state distillation based on generalized Reed-Muller codes. In particular, a  multi-qutrit framework supports a distillation protocol that\nrequires\n$O(\\log^3(1\/\\delta))$ raw magic states of low fixed fidelity in order to distill a copy of the magic state $\\mu$ at fidelity $1-\\delta$.\nThe distillation protocol is iterative and converges to that fidelity in $O(\\log(\\log(1\/\\delta)))$ iterations.\nThe protocol performance is analogous to the magic state distillation protocol for the $T$ gate in the Clifford+T framework \\cite{BravyiKitaev}.\n\n One architectural design is to split the actual computation into ``online'' and ``offline'' components where the main part of quantum processor runs the target quantum circuit whereas the (potentially rather\n large) ``offline'' coprocessor distills copies of a magic state that are subsequently injected into the main circuit by a deterministic widget of constant depth.\nDiscussing the details of the distillation protocol for the magic state $\\mu$ is beyond the scope of this paper and we refer the reader to Ref. \\cite{CampbellEtAl}.\n\n\\subsection{Metaplectic quantum basis} \\label{subsec:metaplectic:basis}\n\nThe ternary \\emph{metaplectic} quantum basis is obtained by adding the \\emph{single qutrit axial reflection} gate\n\n\\begin{equation} \\label{eq:def:R:2}\nR_{|2\\rangle} = |0\\rangle \\langle 0| + |1\\rangle \\langle 1| - |2\\rangle \\langle 2|\n\\end{equation}\n\n\\noindent to the ternary Clifford group.\nIt is easy to see that $R_{|2\\rangle}$ is a non-Clifford gate and that Clifford+$R_{|2\\rangle}$ framework is universal for quantum computation.\n\nIn Ref.~\\cite{CuiWang} this framework has been realized with certain weakly integral non-Abelian anyons called \\emph{metaplectic anyons} which explains our use of the ``metaplectic'' epithet in the name of\nthis universal basis.\nIn Ref.~\\cite{CuiWang}, $R_{|2\\rangle}$ is produced by injection of the magic state\n\\begin{equation} \\label{eq:def:psi}\n|\\psi\\rangle = |0\\rangle - |1\\rangle +|2\\rangle.\n\\end{equation}\nThe injection circuit is coherent probabilistic, succeeds in three iterations on average and consumes three copies of the magic state $|\\psi\\rangle$ on average.\n\nFor completeness we present the logic of the injection circuit on Figure \\ref{fig:R2:RUS}. Each directed arrow in the circuit is labeled with the result of standard measurement of the first qutrit in the state\n$\\mbox{SUM}_{2,1} \\, (|\\psi\\rangle \\otimes |input\\rangle)$. On $m=0$ the sign of the third component of the input is flipped; on $m=1,2$ the sign of the first or second component respectively is flipped.\n\n\\begin{figure*}\n\n\\tikzstyle{state}=[rectangle,thick,rounded corners=10,minimum size=1cm,draw=black!80,fill=black!10]\n\n\\begin{tikzpicture}[>=latex,text height=1.5ex,text depth=0.25ex]\n  \n   \\node (i_1) [state] {$\\pm (a \\ket{0} + b \\ket{1} + c \\ket{2})$};\n   \\node (i_2) [state,below left=10mm and 20mm of i_1] {$\\pm (-a \\ket{0} + b \\ket{1} + c \\ket{2})$};\n     \\node (i_3) [state,below right=10mm and 20mm of i_1] {$\\pm (a \\ket{0} - b \\ket{1} + c \\ket{2})$};\n     \\node (i_4) [state,accepting,below right=10mm and 20mm of i_2] {$\\pm (a \\ket{0} + b \\ket{1} - c \\ket{2})$};\n\n  \n  \\path[->]\n\t(i_1) edge[thick] node [above=2mm] {$m=1$} (i_2)\n\t(i_2) edge[thick] node [above] {} (i_1)\n\t(i_1) edge[thick] node [above=2mm] {$m=2$} (i_3)\n\t(i_3) edge[thick] node [above] {} (i_1)\n\t(i_1) edge[thick] node [left]  {\\raisebox{1cm}{$m=0$}} (i_4)\n\t(i_2) edge[thick] node [below=2mm] {$m=2$} (i_4)\n\t(i_3) edge[thick] node [below=2mm] {$m=1$} (i_4)\n\t(i_2) edge[thick] node [below] {} (i_3)\n\t(i_3) edge[thick] node [above] {\\hspace*{3cm} $m=0$} (i_2);\n\\end{tikzpicture}\n\\caption{\\label{fig:R2:RUS} Markov chain for repeat-until-success implementation of the injection of the $R_{|2\\rangle}$ gate \\cite{CuiWang}. Starting point is a general input $a\\ket{0}+b\\ket{1}+c\\ket{2}$,\nwhere $a,b,c \\in \\mathbb C$. Arrows indicate transitions between single-qutrit states. Each arrow represent a single trial including measurement and consumption of the resource state $\\ket{\\psi}$, where each of the\ntransitions is labeled with the measurement result. The absorbing state corresponds to successful implementation of the $R_{|2\\rangle}$ gate and is denoted by double borders.}\n\\end{figure*}\n\nIn the original anyonic framework the $|\\psi\\rangle$ state is produced by a relatively inexpensive protocol that uses topological measurement and consequent intra-qutrit projection (see \\cite{CuiWang}, Lemma\n5). This protocol requires only three qutrits and produces an exact copy of $|\\psi\\rangle$ in $9\/4$ trials on average. This is much better than any state distillation method, especially because it produces a\ncopy of $|\\psi\\rangle$ with fidelity $1$.\n\nIn \\cite{BCKW} we have developed effective compilation methods to compile efficient circuits in the metaplectic basis Clifford+$R_{|2\\rangle}$.\nIn particular, given an arbitrary two-level Householder reflection $r$ and a desired target precision $\\varepsilon$, then $r$ is effectively approximated by a metaplectic circuit of $R$-count at most\n$8\\,\\log_3(1\/\\varepsilon)+O(\\log(\\log(1\/\\varepsilon)))$, where $R$-count is the number of occurrences of non-Clifford axial reflections in the circuit.\nThis allows us to approximate the $\\mbox{CNOT}$ and Toffoli gates very tightly and at low cost over the metaplectic basis (see Section \\ref{subsec:reflection:metaplectic}).\nMoreover if we wanted constant-depth high-fidelity widgets for $\\mbox{CNOT}$ and Toffoli we can do so by emulating, rather than distilling the magic state $|\\mu\\rangle$ of (\\ref{eq:def:mu}) by a metaplectic\ncircuit and thus obtain a high fidelity emulation of the $P_9$ gate at constant online depth (see Section \\ref{subsec:classical:with:P9}).\n\nAs we show in Appendix \\ref{app:sec:R:2}, the converse also works. With available ancillas and enough reversible classical gates we can prepare the requisite magic state $|\\psi\\rangle$ exactly\n\non a generic ternary computer. The particular method in the appendix is probabilistic circuit for the magic state $|\\psi\\rangle$ of (\\ref{eq:def:psi}) using the classical non-Clifford gate\n$C_2(\\mbox{INC})$. Our current method for the latter gate is to implement it as a ancilla-free circuit with three $P_9$ gates.\n\n\\subsection{Top-level view of Shor's integer factorization algorithm} \\label{subsec:shor:top:level}\n\nThe polynomial-time algorithm for integer factorization originally developed in Ref. \\cite{Shor} is a hybrid algorithm that combines a quantum circuit with classical preprocessing and post-processing.\nIn general, the task of factoring an integer can be efficiently reduced classically to a set of hard cases. A hard case of the factorization problem comprises factoring a large integer $N$ that is odd, square-free\nand composite.\n\n Let $a$ be a randomly picked integer that is relatively prime with $N$. By Euler's theorem, $a^{\\phi(N)} = 1 \\; {\\rm mod} \\; N$, where $\\phi$ is the Euler's totient function,  and thus the modular\n exponentiation function $e_a: x \\mapsto a^x \\; {\\rm mod} \\; N$ is periodic with period $\\phi(N) < N$.\nLet now $0 < r < N$ be a period of the $e_a(x)$ function ($e_a(x+r)=e_a(x), \\forall x$) and suppose, additionally that $r$ is even and $a^{r\/2} \\neq -1 \\; {\\rm mod} \\; N$. Then the $\\mbox{gcd}(a^{r\/2}-1,N)$\nmust be a non-trivial divisor of $N$. The greatest common divisor is computed efficiently by classical means and it can be shown that the probability of satisfying the conditions $r = 0 \\; {\\rm mod} \\; 2$ and\n$a^{r\/2} \\neq -1 \\; {\\rm mod} \\; N$ is rather high when $a$ is picked at random.\nTherefore in Shor's algorithm a quantum circuit is only used for finding the small period $r$ of $e_a(x)$ once an appropriate $a$ has been randomly picked.\n\nOne quantum circuit to solve for $r$ consists of three stages:\n\\begin{enumerate}\n\\item Prepare quantum state proportional to the following superposition:\n\\begin{equation} \\label{eq:Shor:superposition}\n\\sum_{k = 0}^{N^2} |k\\rangle |a^k \\; {\\rm mod} \\; N\\rangle.\n\\end{equation}\n\\item Perform in-place quantum Fourier transform of the first register.\n\\item Measure the first register.\n\\end{enumerate}\n\nThe process is repeated until a classical integer state $j$ obtained as the result of measurement in step 3 enables recovery of a small period $r$ by efficient classical post-processing.\n\nShor has shown  \\cite{Shor} that the probability of successful recovery of $r$ in one of the iterations is in\n$\\Omega(1\/\\log(\\log(N)))$. Therefore we will succeed ``almost certainly'' in finding a desired small period $r$ in $O(\\log(\\log(N)))$ trials.\n\nGiven the known efficiency of the quantum Fourier transform, most of the quantum complexity of this solution falls in 1, where the state (\\ref{eq:Shor:superposition}) is prepared. Specific quantum circuits for\npreparing this superposition have been proposed (cf. \\cite{BeckmanEtAl,Beauregard,CleveWatrous,Markov2012,Markov2013,MeterItoh,TakaKuni,VedralEtAl,Zalka,Zalka2006}).\n\nIn the context of this paper, distinguish between two types of period-finding circuits. One type, as in Ref.~\\cite{Beauregard}, is width-optimizing and uses approximate\narithmetic. These circuits interleave multiple quantum Fourier transform and inverse Fourier transform blocks into modular arithmetic circuits, which in practice leads to significant\ndepth overhead.  We forego the analysis of circuits of this type for the lack of space leaving such analysis for future research.\n\nThe second type are framed as exact reversible arithmetic circuits. Their efficient ternary emulation amounts to efficient emulation of $\\mbox{CNOT}$ and Toffoli gates, possibly\nafter some peephole optimization.\nWe discuss two typical circuits of this kind in detail in Section \\ref{sec:Emulation}, and briefly touch upon a number of alternatives in Appendix \\ref{app:sec:alternative:circuits}.\n\nIt is important to note that, with a couple of exceptions\nthe multi-qubit designs for Shor state preparation assumed ideal $\\mbox{CNOT}$ and Toffoli gates.\nHowever, in Clifford+T framework, for example, the Toffoli gate is often only as ideal as the $T$ gate.\nThe question of the required fidelity of $\\mbox{CNOT}$ and Toffoli gates for the quantum period finding loop to work is an important one.\n\nIf the superposition (\\ref{eq:Shor:superposition}) is prepared imperfectly, with fidelity $1-\\varepsilon$ for some $\\varepsilon$ in $o(1\/\\sqrt{\\log(\\log(N)})$,\nthen the probability of obtaining one of the ``useful'' measurements will be asymptotically the same as with the ideal superposition, i.e., in $\\Omega(1\/\\log(\\log(N)))$.  (For completeness, we spell out\nthe argument in Appendix \\ref{app:sec:fidelity}.) Therefore, if $d$ is the depth of the corresponding quantum circuit preparing the state, then the bound on the required precision of the individual gates in\nthe circuit may be in\n$o(1\/(d\\,\\sqrt{\\log(\\log(N)}))$ in the context of Shor's algorithm.\n\nIn the rest of the paper we explore ternary emulations of binary period-finding circuits and compare them to truly ternary period-finding circuits with ternary encoding of integers. We demonstrate\nthat the fidelity and non-Clifford cost of such ternary circuits are reduced to those of the $C(\\mbox{INC})$ gates. We also demonstrate that efficient emulation of binary period finding requires mostly binary\nToffoli gates with some use of $C(\\mbox{INC})$.\n\n\\section{Multi-Qutrit and Multi-Qubit Arithmetic on Generic Ternary Quantum Computer} \\label{sec:Emulation}\n\nWe explore two options for cost-efficient integer arithmetic over the ternary Clifford+$P_9$ basis: (a) by emulating arithmetic on binary-encoded data; and, (b) by performing arithmetic on\nternary-encoded data, based on tools developed in \\cite{BCRS}.\n\nCircuits for reversible ternary adders have been explored earlier. (See, for example, \\cite{MillerEtAl,SatohEtAl,KhanPerkowski}). Since this field has been\nin early stages so far, there is a lot of divergence in terminology: however in \\cite{MillerEtAl,SatohEtAl,KhanPerkowski} the key non-Clifford tool\nfor the circuitry is an equivalent of the $C_f(\\mbox{INC})$ gate, in our notation. As pointed out in \\cite{BCRS}, our use of this tool is more efficient,\n mainly due to\nthe design of ``generalized'' carry gates and other reflection-based operations.\n\nOur ternary circuits for emulated binary encoding of integers are new, as far as we know.\n\nThe emulated binary and genuine ternary versions of integer arithmetic have different practical bottlenecks, although they are asymptotically equivalent in terms of cost. With the ripple-carry\nadders,  the emulated binary encoding wins, in practice, in both width and depth over the ternary encoding, whereas with carry-lookahead adders the ternary encoding achieves smaller width but\nyields no notable non-Clifford depth advantage in the context of modular exponentiation.\n\nTo give the study a mathematical form, let us agree to take into account only non-Clifford gates used with either encoding and let us agree\nto count a stack of non-Clifford gates performed in parallel in one time slice as a single \\emph{unit of non-Clifford depth}.\nWe call the number of units of non-Clifford depth in a circuit the \\emph{non-Clifford depth} of the circuit.\n\nThroughout the rest of the paper we use the following\n\n\\begin{defin}\nFor integer $n>0$ let $|j\\rangle$, $|k\\rangle$ be two different standard basis vectors in the $n$-qudit Hilbert space. We call the classical gate\n\\begin{equation} \\label{eq:def:tau}\n\\tau_{|j\\rangle,|k\\rangle} = I^{\\otimes n} -  |j\\rangle \\langle j| -  |k\\rangle \\langle k| +  |j\\rangle \\langle k| + |k\\rangle \\langle j|\n\\end{equation}\na \\emph{two-level axial reflection} in $n$ qudits.\n\\end{defin}\n\nAs a motivation for this term, note that $\\tau_{|j\\rangle,|k\\rangle}$ can be rewritten as the two-level Householder reflection\n\\[\n I^{\\otimes n} -  2\\, |u\\rangle \\langle u|,  |u\\rangle = (|j\\rangle - |k\\rangle)\/\\sqrt{2}.\n\\]\nClearly, in binary encoding, the $\\mbox{CNOT}$, the Toffoli and any variably controlled Toffoli gate is a two-level axial reflection in the corresponding number of dimensions.\n\n\n\\subsection{Ternary circuit for binary ripple-carry additive shift} \\label{subsec:ripple:carry}\n\nWe discuss emulating an additive shift circuit improving on a quantum ripple-carry adder from \\cite{Cuccaro}.\nare cast in {\\bf bold} font below.\n\n\\begin{comment}\nThat adder has simple structure and requires only two ancillas. In the context of Shor's algorithm it would be very practical to use the adder when factorizing medium-size numbers. However, since for $n$-bit\nnumbers the depth of the circuit is $O(n)$ , it is asymptotically inferior to lookahead additive shift described in the next subsection and may become unusable in practice when factoring very large numbers.\nEither way, the simpler ripple carry adder gives an excellent opportunity to introduce the emulation concepts.\n\\end{comment}\n\nLet $a$  be a classically known $n$-bit integer and $b$ be a quantumly-stored $n$-qubit basis state. We are looking for a quantum implementation of the function $|b\\rangle \\mapsto |a+b\\rangle$.\nMore specifically, we are looking for a pre-compiled quantum circuit $C_a$ parameterized by $a$ which is known at compilation time.\nConsider the well-known quantum ripple-carry adder from \\cite{Cuccaro} (in particular, the circuit illustrated on Figure 4 for $n=6$ there that is copied, for completeness into our Fig.\n\\ref{fig:original:cuccaro}).\n\n\\begin{figure}[ht]\n\\input{\"Adder_CDKM.tikz\"}\n\\caption{\\label{fig:original:cuccaro} Ripple-carry adder from \\cite{Cuccaro}.}\n\\end{figure}\n\nThe adder uses $2n+2$ qubits. It performs a ladder of $n$ $\\mbox{MAJ}$ gates to compute all the carry bits, including the top one. The top carry bit is copied unto the last qubit and\nthe ladder of $n$ $\\mbox{UMA}$ gates is performed. Each $\\mbox{UMA}$ gate uncomputes the corresponding  $\\mbox{MAJ}$ function and performs the three-way $\\mathbb{Z}_2$ addition\n$a_i \\oplus b_i \\oplus c_i$.\n\nIt is somewhat hard to fold in the classically known $a$ in the multi-qubit framework using this design. Note however, that a solution along these lines is offered in \\cite{TakaKuni}.\nHowever it is easy to fold in $a$ in ternary emulation using the third basis state of the qutrit. We show that it takes exactly $n+2$ qutrits to emulate the binary shift\n$|b\\rangle \\mapsto |a+b\\rangle$.\n\n\n\\begin{comment}\nIt is helpful for our purposes that the $\\; {\\rm mod} \\; 2$ additive shifts $b_i \\oplus$ and $a_i \\oplus$ can be taken out of the $\\mbox{UMA}$ blocks, pushed into the end of the circuit and all the additive\nshifts can be done in parallel in depth at most two there.\n\\end{comment}\n\nConsider $n+2$ qutrits where the top and the bottom ones are prepared in $|0\\rangle$ state and the remaining $n$ encode the {\\emph binary bits} of the  $|b\\rangle$.\n\nWe will be looking for reversible two-qutrit gates $Y_0, Y_1$ such that\n\\begin{equation} \\label{eq:for:Y:gate}\nY_{a_j} \\, |c_j,b_j\\rangle = |c'_j,c_{j+1}\\rangle\n\\end{equation}\n\\noindent where $c_{j+1}$ is the correct carry bit for $c_j+a_j+b_j$ and $c'_j$ is an appropriate trit.\n\n\\begin{comment}\nConsider the two two-qutrit gates parameterized by $\\mathbb{Z}_2$ : $Y_0 = \\tau_{|01\\rangle,|20\\rangle}$  and $Y_1 = \\tau_{|10\\rangle,|21\\rangle}$.\nAssuming $a$, $b$ and all the carries are encoded as binary numbers, it is easily verified by direct computation that\n$Y_{a_j} \\, |c_j,b_j\\rangle = |c'_j,c_{j+1}\\rangle$, where $c_{j+1}$ is the correct carry bit for the $a_j+b_j+c_j$ addition and $c'_j$ is a trit that can assume the value of $2$ in two out of the eight\ncases.\n\\end{comment}\n\nSince all the bits of $a$ are known we can precompile a ladder of $Y$ gates that correctly computes the top carry bit $c_n$ and puts the modified carry trit $c'_j$ on each $b_j$ wire. Having copied $c_n$ onto\nthe last qutrit, we sequentially undo the $Y$ gates in lockstep with computing partial $\\mathbb{Z}_2$-sums $b_j\\oplus c_j$ on all the $b_j$ wires using gates of $\\mbox{CNOT}$ type.\n\nWe note that $Y_0,Y_1$ are ternary gates, used however in a narrow context of a truth table with just four columns. One would intuitively expect that their restriction to the context can be emulated\nat a relatively small expense.\n\nIndeed:\n\n\\begin{prop} \\label{prop:Ys:improved}\nLabel the $c_i$ wire by $0$ and $b_i$ wire by $1$\nIn the context of binary data the gates\n\\[\n\\begin{split}\nY_0 = C_2(\\mbox{INC})_{0,1}^{\\dagger} \\mbox{SUM}_{1,0} (\\tau_{|0\\rangle,|1\\rangle} \\otimes I)\n\\end{split}\n\\]\n\\noindent and\n\\[\n\\begin{split}\nY_1  = C_2(\\mbox{INC})_{0,1} (I \\otimes \\tau_{|0\\rangle,|1\\rangle} ) \\mbox{SUM}_{1,0} (I \\otimes \\tau_{|0\\rangle,|1\\rangle} )\n\\end{split}\n\\]\n\\noindent satisfy the condition (\\ref{eq:for:Y:gate}).\n\nHere the $C_2(\\mbox{INC})$ is the binary-controlled increment\n$C_2(\\mbox{INC}): |j,k\\rangle \\mapsto |j,k+\\delta_{j,2}\\rangle$\n\n\\end{prop}\n\\begin{proof}\nBy direct computation. Note, that we do not care, what either of these two circuits does outside of the binary data subspace as long as the action is reversible.\n\\end{proof}\n\n\nThe $C_2(\\mbox{INC})$ gate is also denoted as $C_2(X)$ in Ref. \\cite{BCRS}, where its cost and utility is discussed in detail (see also further discussion in section \\ref{sec:reflections}). The non-Clifford cost of\neither $Y_j$ gate is equal to the non-Clifford cost of $C_2(\\mbox{INC})$ which is known to be $3$ $P_9$ gates. Allowing one ancillary qutrit, the $C_2(\\mbox{INC})$ is represented by a circuit of $P_9$-depth of\n$1$ and $P_9$-width of $3$.\n\n\nBesides the generalized carry computation, the additive shift circuit also needs to perform the bitwise $\\; \\mod 2\\;$ addition by emulated gates of $\\mbox{CNOT}$ type.\n\nRecall that $\\mbox{CNOT}$  gate cannot be exactly represented by a ternary Clifford circuit (cf. \\cite{BCRS}, Appendix A). As shown further in Proposition \\ref{prop:CNOT:ancilla-free}, the non-Clifford cost of\nternary-emulated $\\mbox{CNOT}$ on binary data only is an equivalent of two $C_2(\\mbox{INC})$.\nThus the additive shift takes roughly ${12\\, n}$ $P_9$ gates to complete (not counting the Clifford scaffolding). With one ancilla this can be done at $P_9$-depth of $4\\, n$ and $P_9$-width of $3$.\n\nHowever, Shor's period funding functions relies on controlled and doubly-controlled versions of the additive shift. It suffices to control only the bitwise addition gates. Thus adding one level of control\nproduces $n$ additional Toffoli gates and adding the second level of control turns these gates into controlled Toffolis.\nThis is the bottleneck of the emulated solution: as per Corollaries \\ref{corol:Toffoli:the:12} and \\ref{corol:ctrl:Toffoli:18} in section below, an emulated Toffoli takes $12$ $P_9$ and the\nbinary-controlled Toffoli takes $18$ $P_9$ gates respectively.\n\nThus overall the controlled shift takes ${18 \\, n}$ $P_9$  and the doubly-controlled shift takes ${24 \\, n}$ $P_9$ gates. Allowing, again, an ancillary qutrit the $P_9$-depths of the corresponding\ncircuits can be made $6\\,n$ and $8\\,n$ respectively.\n\nFor what it is worth, the $P_9$-counts in this solution are similar (and in fact marginally lower) than the $T$-counts required for running the original binary adder  \\cite{Cuccaro} on the more common binary\nClifford+T platform. Indeed each of the $\\mbox{MAJ}$ and $\\mbox{UMA}$ gates shown on figure \\ref{fig:original:cuccaro} is Clifford-equivalent to a Toffoli gate that takes $7$ $T$ gates to implement. Adding one\nlevel of control to the adder increases the non-Clifford complexity by an additional $n$ Toffoli gates to the total $T$-count of $21\\,n$. Adding the second level of control, conservatively, brings in $2\\,n$\nadditional modified Toffoli gates to yield the total $T$-count of $29\\,n$.\n\nWe also note that the width of the ternary emulation circuit is equal to $n+2$ qutrits, whereas the original purely binary design appears to require $2\\, n + 2$ qubits.\n\nThe construction of Corollaries \\ref{corol:Toffoli:the:12} and \\ref{corol:ctrl:Toffoli:18} requires 1 and 2 ancillas respectively. These ancillae can be shared along the depth of the circuit inflating the overall\nwidth by 2 qutrits.\n\n\\subsection{Ternary circuit for ternary ripple-carry additive shift} \\label{subsec:ternary:ternary:ternary}\nConsider ripple-carry implementation of the quantum function $|b\\rangle \\mapsto |a+b\\rangle$, where $|b\\rangle$ is quantumly encoded integer and $a$ is an integer that is classically known.\nSuppose $a$ and $b$ are encoded as either bit strings with at most $n$ bits or as trit strings with at most $m=\\lceil \\log_3(2) n \\rceil$ trits (with $\\log_3(2) \\approx 0.63093$).\nSince $a$ is classically known, we strive to improve on the ternary ripple-carry adder of \\cite{BCRS} by folding in the trits of $a$. However, we are no longer able to encode all of the quantum information for\n$b$ and the carry on the same qutrit. The additive shift thus requires  roughly $2\\,m-w_1(a)$  qutrits to run (where $w_1(a)$ is the number of trits equal to $1$ in the ternary expansion of $a$).\n\nThe ternary additive shift in this design has somewhat higher non-Clifford time cost compared to the emulated binary shift of section \\ref{subsec:ripple:carry}.\n\nFor the classical additive shift we do not physically encode the trits of $a$ and instead pre-compile different\ngeneralized carry circuits for different values of these trits. Tables \\ref{table:ai:1} and \\ref{table:ai:0} show the truth tables for the consecutive carry $c_{i+1}$ given, respectively, $a_i = 1$ and $a_i=0$\n(the case of $a_i=2$ is symmetric to the case $a_0$ and yields the came conclusions).\n\n\\begin{table*}\n    \\begin{tabular}{l@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c} \\hline\\hline\\\\[-1.5ex]\n    $c_i$                                  & $0$                         & $0$ & $0$                         & $1$ & $1$                         & $1$\\\\[1ex]\n    $b_i$        & $0$                         & $1$ & $2$                         & $0$ & $1$                         & $2$  \\\\\n    $c_{i+1}$        & $0$                         & $0$ & $1$                         & $0$ & $1$                         & $1$\\\\[1ex]\n\\hline\\hline\n    \\end{tabular}\n    \\caption{Truth table for $c_{i+1}$ given $a_i = 1$} \\label{table:ai:1}\n  \\end{table*}\n\n\\begin{table*}\n    \\begin{tabular}{l@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c@{\\qquad\\qquad}c} \\hline\\hline\\\\[-1.5ex]\n    $c_i$                                  & $0$                         & $0$ & $0$                         & $1$ & $1$                         & $1$\\\\[1ex]\n    $b_i$        & $0$                         & $1$ & $2$                         & $0$ & $1$                         & $2$  \\\\\n    $c_{i+1}$        & $0$                         & $0$ & $0$                         & $0$ & $0$                         & $1$\\\\[1ex]\n\\hline\\hline\n    \\end{tabular}\n    \\caption{Truth table for $c_{i+1}$ given $a_i = 0$} \\label{table:ai:0}\n  \\end{table*}\n\nThe case of $a_i = 1$ does not require any ancillary qutrits since the $c_{i+1}$ is a balanced binary function that can be produced reversibly on the pair of qutrits encoding $c_i$ and $b_i$ by ternary\n$\\mbox{SWAP}$ gate followed by $|01\\rangle \\leftrightarrow |20\\rangle$.\n\nHowever, in the case of $a_i = 0$ the $c_{i+1}=0$ in five cases (respectively, for $a_i = 2$ the $c_{i+1}=1$ in five cases) and such five basis vectors cannot be represented in two-qutrit state space. These cases\nthus require an ancillary qutrit to encode $c_{i+1}$.\n\nIn the case of $a_i = 0$ we simply take the ancilla in the $|0\\rangle$ state and apply doubly-controlled $\\mbox{INC}$ gate with the ternary control on $c_i$ and binary control on $b_i$. In the case of $a_i = 2$ it\nsuffices to additionally use the Clifford $\\tau_{|0\\rangle,|1\\rangle}$ gate on the $c_{i+1}$.\n\nAssuming $a$ is generic with $w_1(a) \\approx m\/3$ we get an average width of the additive shift circuit of roughly $5\/3 \\, m$ which eliminates the space savings afforded by denser ternary\nencoding ($5\/3 \\log_3(2) \\approx 1.05$).\n\nLet us now make case for the second observation.\nproven practically optimal).\nWe start by assessing the clean magic state counts for simple uncontrolled additive shift. We note that for any classical value of the $a_i$ trit the non-Clifford cost of the carry gate is the same and equals\n$15$ clean magic states. Indeed, depending on $a_i$ and in terminology of \\cite{BCRS} we either need one gate of $S_{01,10}$ type or one gate of $C_0(\\mbox{SUM})$ type. In subsection 5.1 of the \\cite{BCRS}\nboth types are reduced to $5$ binary-controlled increments and consequently to $15$ $P_9$ gates. The concluding trit-wise addition is done by Clifford $\\mbox{SUM}$s at negligible cost. Thus the overall cost of\nthe circuit is roughly  ${30\\,m \\approx 19 \\, n}$ $P_9$ gates. Allowing an ancillary qutrit, the $P_9$-depth of the circuit can be made equal to ${10\\,m > 6 \\, n}$.\n\nAdding one ternary control to the circuit turns all the finalizing $\\mbox{SUM}$s into``Horner'' gates $\\Lambda(\\mbox{SUM})$ that overall takes $4\\, m$ additional $P_9$ gates to the total non-Clifford cost of\n${34\\,m > 21 \\, n}$ $P_9$ gates.\n\nA subtle point discussed in section \\ref{subsec:mod:exponetiation} below is that the second control that is routinely added to the additive shift gate $S_a$ is in fact strict control that turns it into a\n$C_f(S_a)$ gate $f\\in \\{1,2\\}$ where $S_f$ is activated only by the control basis state $|f\\rangle$. This turns each of the the $m$ ``Horner'' gates into a four-qutrit  $C_f(\\Lambda(\\mbox{SUM}))$ gate. We do\nnot have an available ancilla-free design for a synthesis of this gate.\n\nOur best design described in Proposition \\ref{prop:C:Lambda:SUM} sets the non-Clifford cost at $23$ $P_9$ gates given one clean ancilla.\n\nThus adding the required second (strict) control inflates the overall cost of the ternary circuit to ${53 \\, m > 33 \\,n}$ $P_9$ gates.\n\nAgain, with available ancilla the circuit can be restacked to $P_9$-width of $3$ reducing the $P_9$-depth by the factor of $3$ (to roughly ${19 \\, m}$ in case of doubly-controlled additive shift).\nis less than the non-Clifford cost of the emulated binary $n$-bits doubly-controlled additive shift.\n\nThe comparative cost of the binary and ternary options is summarized in table \\ref{table:binary:vs:ternary}.\n\n\\begin{table*}\n    \\begin{tabular}{l@{\\qquad\\qquad}l@{\\qquad\\qquad}l} \\hline\\hline\\\\[-1.5ex]\n\\multicolumn{1}{c}{\\hspace*{-2cm} Circuits}\n           &  \\multicolumn{1}{l}{$\\# P_9$: emulated binary} &  \\multicolumn{1}{l}{$\\# P_9$: ternary}\\\\[0.5ex]\n    \\hline\\\\[-1.5ex]\n   Simple additive shift                                 & $12\\,n$                       & $19 \\,n$ \\\\[1ex]\n    Controlled additive shift       & $18\\,n$    & $> 21\\,n $  \\\\[1ex]\n    Doubly-controlled additive shifts        & $24\\,n$   & $> 33 \\, n$ \\\\[1ex]\n\\hline\\hline\n    \\end{tabular}\n    \\caption{Cost of ripple-carry additive shift: ternary vs. emulated binary. $n$ is the bit size of the arguments.} \\label{table:binary:vs:ternary}\n  \\end{table*}\n\n\nWe demonstrate in the Section \\ref{subsec:modular:shifts} that the best-known ternary-controlled modular shift circuit requires $4$ instead of $3$ additive shift blocks on roughly half of the\nmodular addition cases, so, in the context of the required modular addition, the emulated binary encoding appears to be a practical win-win when a low width ripple-carry adder is used.\n\n\n\n\\subsection{Circuits for carry lookahead additive shift} \\label{subsec:carry:lookahead}\nThe resource layout is different for known carry lookahead solutions. For the sake of space we forego detailed analysis and only sketch the big picture.\n\nWe assume, that the integers $a$ and $b$ are encoded as either bit strings with at most $n$ bits or as trit strings with at most $m=\\lceil \\log_3(2) n \\rceil$ trits.\nWe use carry lookahead additive shifts based on the in-place multi-qubit carry lookahead adder \\cite{DraperSvore} and the in-place multi-qutrit carry lookahead adder \\cite{BCRS}.\n\nThe non-Clifford depths of the corresponding circuits are $4\\, \\log_2(n)$ and $4\\, \\log_2(m)$  respectively up to small additive constants.\n\nBecause $\\log_2(m) = \\log_2(n) + \\log_2(\\log_3(2))$, there is no substantial difference in non-Clifford depths. The non-Clifford layers of the binary adder are populated with Toffoli gates and for the ternary\nadder they are populated with carry status merge\/unmerge widgets (the $\\mathcal{M}$ and $\\mathcal{M}^{\\dagger}$ widgets of \\cite{BCRS}). The cost of ancilla-free emulation of the former or,\nrespectively, execution of the latter is identical with $15$ $P_9$ gates.\n\nWhen levels of control are added to the shift circuit, putting ternary control on ternary widgets is more expensive than building multi-controlled Toffoli gates, as discussion in Section\n\\ref{subsec:ripple:carry} implies. But in the context of carry lookahead circuits the multi-controlled gates are located in just two layers out of $O(\\log(n))$ thus the impact of this cost distinction is both\nasymptotically and practically negligible.\n\nNote that the widths of the binary and ternary circuits are roughly proportional to $n$ and $m=\\lceil \\log_3(2) n \\rceil$, respectively. This means that the purely ternary solution has roughly $m\/n \\approx \\log_3(2)$ smaller\nwidth.\n\n\ndepth overhead percentage is moderate, we should prefer purely ternary encoding when implementing Shor's period finding on small quantum computer.\n\n\\begin{comment}\n\nIn this subsection we review and optimize a circuit for the carry lookahead additive shift based on the in-place binary carry lookahead adder (in place BCLAA) of the \\cite{DraperSvore}.\nBCLAA for two $n$-bit numbers is a classical reversible circuit with the depth $O(\\log(n))$ and $O(n)$ clean ancillary qubits. It computes all the carry bits in $O(\\log(n))$ steps and performs all the bitwise\nsummations in parallel. In order to do that, BCLAA computes a binary tree of ``carry status indicators'' (CSIs) that are stored on some of the ancillary qubits.\nThe notion and formal definition of CSI are given in Section 3 of \\cite{DraperSvore} and specifically on Figures 2 and 3 there. An example of in-place BCLAA for $n=10$ is presented on Figure 5 which we copy\nhere for completeness (our Fig. \\ref{fig:draper:svore:qcla}).\n\\begin{figure*\n\\includegraphics[width=5.5in]{InPlaceDraperSvoreQCLA.pdf}\n\\caption{\\label{fig:draper:svore:qcla} In-place QCLA adder for 10 bits from \\cite{DraperSvore}. The carry status indicators are computed and uncomputed by the\n$P,G$ (resp. $P^{-1},G^{-1}$) rounds shown in blue and red. The concluding $C$ rounds are shown in green. }\n\\end{figure*}\n\nAs easily seen, $O(\\log(n))$ steps in that circuit are spent computing the CSIs, while only $O(1)$ steps compute (and, occasionally, uncompute) the bitwise sums.\nFor us it is important that the $O(\\log(n))$ CSI time steps are populated with the (non-Clifford) Toffoli gates and only $O(1)$ time steps are populated with gates that are Clifford in binary framework but may\nbe non-Clifford in ternary framework. From this it is already clear that the non-Clifford inefficiency factor due to ternary processing is upper-bounded by $(1+O(1\/\\log(n)))$.\nLet us make sure that this bound can be practically specified and, in fact, improved on when folding in the classically known shift value $a$ resulting in the desired circuit for $|b\\rangle \\mapsto\n|a+b\\rangle$.\n\nThe purpose of constant folding is to remove all the wires (qubits) that contain the encoded values of $a_0, \\ldots, a_{n-1}$ from the circuit by pre-compiling all the gates where these wires serve as control\nwires. Thus a Toffoli gate on $\\{A_i,B_i, \\mbox{ancilla}\\}$ resolves into identity when $a_i=0$ or into $\\mbox{CNOT}$ on $\\{B_i,\\mbox{ancilla}\\}$ when $a_i=1$; and $\\mbox{CNOT}$ on $\\{A_i,B_i\\}$ resolves into\nidentity when $a_i=0$ or into $\\mbox{NOT}$ on $B_i$ otherwise. Since none of the Toffoli gates in the $P^{\\pm1}, G^{\\pm1},C$ rounds are controlled by any of the $A_i$s, then these Toffoli gates are unaffected\nby the constant folding;\nthe Toffoli gates that can fold into $\\mbox{CNOT}$ occur only in the opening step (that computes $C_{i,i+1}$) and its mirror counterpart; on the other hand the constant folding immediately converts\n$\\mbox{CNOT}$ gates in three distinct time steps into identities or $\\mbox{NOT}$s. Thus the number of time steps where $\\mbox{CNOT}$ gates can occur has decreased by $1$ and stands at $3$ so far.\n\nFor further practical benefit let us transform the circuit we have obtained so far somewhat by removing uncontrolled $\\mbox{NOT}$ gates from the second half (the ancilla clean-up part) of the folded circuit.\nTo this end we will allow some of the Toffoli gates proper to morph into variably controlled Toffoli gates.\nIf $\\gamma_1,\\gamma_2 \\in \\mathbb{Z}_2$, let\n$\\mbox{Tof}^{[\\gamma_1,\\gamma_2]}: |c_1,c_2,t\\rangle \\mapsto |c_1,c_2,t\\oplus(\\delta_{c_1,\\gamma_1}\\delta_{c_2,\\gamma_2})\\rangle$.\nAs we already noted such variably controlled Toffoli gate is Clifford-equivalent to the regular Toffoli gate in both binary and ternary frameworks.\n\nObviously $(\\mbox{NOT}\\otimes I \\otimes I) \\mbox{Tof}^{[\\gamma_1,\\gamma_2]} = \\mbox{Tof}^{[\\sim \\gamma_1,\\gamma_2]} (\\mbox{NOT}\\otimes I \\otimes I)$ and\n$(I \\otimes \\mbox{NOT} \\otimes I) \\mbox{Tof}^{[\\gamma_1,\\gamma_2]} = \\mbox{Tof}^{[\\gamma_1,\\sim \\gamma_2]} (I \\otimes \\mbox{NOT} \\otimes I)$\nThus all the uncontrolled $\\mbox{NOT}$ gates in the second half of the folded circuit can be commuted through the control sites of the Toffoli gates possibly morphing the Toffoli gates into variably controlled\nToffoli gates.\nIn order to illustrate this analysis, in Figure \\ref{fig:qcla:folded} we show a folded precompiled version of the circuit from Fig. \\ref{fig:draper:svore:qcla} for specific constant $a_0 \\ldots a_9 =\n1010010011$.\n\n\\begin{figure*\n\\input{\"folded.tikz\"}\n\\caption{\\label{fig:qcla:folded} In-place QCLA additive shift, precompiled for the specific 10-bit shift value  $a_0 \\ldots a_9 = 1010010011$.}\n\\end{figure*}\n\nThe $a_j \\oplus$ shift necessary for producing the bits of the sum of $a+b$ can be precompiled as a sequence of $w(a)$ $\\mbox{NOT}$ gates, where $w(a)$ is the Hemming weight of $a$, that can be done in\nparallel as the last step of the circuit. The remaining $\\mbox{CNOT}$s can be corralled into at most three time slices.\nAs per \\cite{DraperSvore}, Section 4.2 the non-Clifford depth of the circuit exceeds $4\\, \\log_2(n)$, therefore  $\\mbox{CNOT}$ layers which amount to at most $3\/(4\\, \\log_2(n))$ fraction of the circuit depth.\nBut it is important to tally the $\\mbox{CNOT}$ layers in the binary controlled version of the circuit.\n\nFor optimal size we need to control only those parts of the circuit that compute the bits of the $a+b$. Those are limited to one layer of $\\mbox{CNOT}$ gates and the final layer of $\\mbox{NOT}$ gates. We\nobserve that adding a layer of control in this fashion, the $\\mbox{CNOT}$ gates turn into Toffoli gates and the $\\mbox{NOT}$ gates turn into $\\mbox{CNOT}$ gates. Thus the number of layers populated solely by\n$\\mbox{CNOT}$s does not change.\nIn summary the ternary emulation of either uncontrolled or controlled version of the additive shift circuit inflates the non-Clifford depth by a factor not exceeding $(1+3\/(4\\,\\log_2(n)))$\n\n\\begin{comment}\nIn order to add one level of binary control to the folded optimized carry lookahead additive shift circuit it suffices to add binary control to $\\mbox{NOT}$ and $\\mbox{CNOT}$ gates in just two time slices,\nwhere $b_i$ is being converted into the $s_i$ bit. As the result, all the $\\mbox{NOT}$ gates are turned into $\\mbox{CNOT}$s, but all the $\\mbox{CNOT}$s are turned into Toffoli gates. Thus the number of time\nsteps where $\\mbox{CNOT}$s can occur has not changed and still stands at $3$.\n\nIt follows from \\cite{DraperSvore}, Section 4.2 that the non-Clifford depth of the circuit exceeds $4\\, \\log_2(n)$. Thus in the ternary emulation having three time slices populated with $\\mbox{CNOT}$s inflates\nthe non-Clifford depth by a factor not exceeding $(1+3\/(4\\,\\log_2(n)))$\n\n\n\\subsection{Circuits for comparison to a classical threshold}\n\nLet $a$ be a classically known integer and $b$ a quantumly encoded integer. Assume, for simplicity that both are $n$-bit integers.\nWe want to implement a Boolean function that returns $1$ when $b < a$ and returns $0$ otherwise.\nWe observe that for $b'=2^n-1-b$ the result of such comparison function is equal to the top carry bit for the sum $b'+a$.\nTherefore the implementation is achieved by modifying the classical shift circuit. Before doing anything further, the comparison circuit applies $\\mbox{NOT}$ gates in parallel to all the $B_i$ wires\n(correspondingly, at the very end of the circuit those $\\mbox{NOT}$s are undone, also in parallel.)\n\nThe circuit proceeds to compute all the carries, including, most importantly, the top one. The top carry is copied onto the result ancilla. Then we uncompute all the carries, but skip the actual bitwise\nsummation on the $B_i$ wires.\nThis way we end up with the $A$ and $B$ wires and all the clean carries restored to the original state. And the result ancilla in the desired result state.\nThe details differ somewhat between the ripple carry additive shift and the carry lookahead shift templates.\nIn the former it simply suffices to literally skip the bitwise summation gates. This actually eliminates uncontrolled $\\mbox{CNOT}$ gates in the singly-controlled version of the circuit. Thus singly controlled\ncomparison circuit ends up being populated with Toffoli gates only.\n\nIn the carry lookahead template, we must be cognizant that the $a_i \\oplus b_i$ part of the bitwise addition is dual purpose and cannot be dropped; it\nmust be undone before the end of the circuit (which, again, is executed in parallel in one time slice). But the time slice performing all the $\\oplus c_i$ bitwise additions can be just dropped. This reduces\nthe non-Clifford depth of the comparison circuit by $1$ compared to the carry lookahead additive shift.\nIt follows that in ternary emulation the non-Clifford depth of the circuit would get inflated by a factor not exceeding\n$(1+2\/(4\\,\\log_2(n))) < (1+3\/(4\\,\\log_2(n)))$.\n\\end{comment}\n\n\\subsection{Circuits for modular additive shifts} \\label{subsec:modular:shifts}\n\nWe review layout for modular additive shift and controlled modular additive shift in both emulated binary and genuine ternary setups.\n\nLet $N >> 0$ and $a < N$ be classically known integers.\nThe commonly used scheme to compute the quantum modular additive shift $|b\\rangle \\mapsto |(a+b) \\; {\\rm mod} \\; N\\rangle$ is to compute $|a+b\\rangle$, figure out whether $a+b <N$ and, if not, then subtract\n$N$.\nIn order to do it coherently without measurement we need to\n\\begin{enumerate}\n\\item Speculatively compute the $|(a-N)+b\\rangle$ shift; structure it so that the top carry bit $c_{n+1}$ is $1$ iff $(a-N)+b <0$.\n\\item Copy $c_{n+1}$ to a clean ancilla $x$.\n\\item Apply the shift by $+N$ controlled by the ancilla $x$.\n\\item Clean up the ancilla $x$.\n\\end{enumerate}\n\nSurprisingly, the last step is less than trivial.\nWe need to compare the encoded integer $|y\\rangle$ after step 3) to $a$. Then $y \\geq a$ if and only if $c_{n+1}=1$. Therefore we must flip the ancilla if and only if $y \\geq a$. We do this by taking a circuit\nfor comparison to classical threshold and wiring the $\\mbox{NOT}\\, x$ into it in place of the top carry qubit. It is easy to see that performing the comparison circuit has the exactly the desired effect on the\nancilla $x$.\nA top level layout of the modular additive shift is shown in Figure \\ref{fig:modular:shift}. We note that the three-stage layout shown in the Figure is not entirely new. It is very similar to designs proposed\nin \\cite{MeterItoh} and \\cite{TakaKuni}.\nClearly the non-Clifford depth of this scheme is roughly triple the non-Clifford depth of the additive shift circuit in either binary or ternary framework.\n\n\\begin{figure*\n\\begin{tikzpicture}[scale=3.50000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (90.000000, 37.500000);\n\\draw[color=black] (0.000000,30.000000) -- (90.000000,30.000000);\n\\draw[color=black] (0.000000,30.000000) node[left] {$|b\\rangle$};\n\\draw[color=black] (0.000000,15.000000) -- (71.000000,15.000000);\n\\draw[color=black] (85.000000,15.000000) -- (90.000000,15.000000);\n\\draw[color=black] (0.000000,15.000000) node[left] {$|0\\rangle$};\n\\draw[color=black] (0.000000,0.000000) -- (90.000000,0.000000);\n\\draw[color=black] (0.000000,0.000000) node[left] {$|0\\rangle$};\n\\draw (12.000000,30.000000) -- (12.000000,15.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and\n19.091883pt) -- cycle;\n\\clip (12.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and 19.091883pt) --\ncycle;\n\\draw (12.000000, 22.500000) node {$+(a-N)$};\n\\end{scope}\n\\draw (33.000000,15.000000) -- (33.000000,0.000000);\n\\filldraw (33.000000, 15.000000) circle(1.000000pt);\n\\begin{scope}\n\\draw[fill=white] (33.000000, 0.000000) circle(3.000000pt);\n\\clip (33.000000, 0.000000) circle(3.000000pt);\n\\draw (30.000000, 0.000000) -- (36.000000, 0.000000);\n\\draw (33.000000, -3.000000) -- (33.000000, 3.000000);\n\\end{scope}\n\\draw (54.000000,30.000000) -- (54.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (54.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and\n19.091883pt) -- cycle;\n\\clip (54.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and 19.091883pt) --\ncycle;\n\\draw (54.000000, 22.500000) node {$+N$};\n\\end{scope}\n\\filldraw (54.000000, 0.000000) circle(1.000000pt);\n\\draw (78.000000,30.000000) -- (78.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (78.000000, 15.000000) +(-45.000000:8.485281pt and 29.698485pt) -- +(45.000000:8.485281pt and 29.698485pt) -- +(135.000000:8.485281pt and 29.698485pt) -- +(225.000000:8.485281pt and\n29.698485pt) -- cycle;\n\\clip (78.000000, 15.000000) +(-45.000000:8.485281pt and 29.698485pt) -- +(45.000000:8.485281pt and 29.698485pt) -- +(135.000000:8.485281pt and 29.698485pt) -- +(225.000000:8.485281pt and 29.698485pt) --\ncycle;\n\\draw (78.000000, 15.000000) node {$\\geq a ?$};\n\\end{scope}\n\\end{tikzpicture}\n\\caption{\\label{fig:modular:shift} Top-level layout of modular additive shift for binary encoding. }\n\\end{figure*}\n\nIn the context of ternary encoding of integers and allowing for ternary control the logic turns out to be more  involved.\nDepending on whether $2 \\, a < N$ or not, which is known at compilation time, we need to compile two different circuits.\nWhen $2 \\, a < N$ we need to speculatively precompute $b+ c\\, a -N$ where $c$ is the quantum value of the control trit. This is different from adding ternary control to the additive shift $+(a-N)$. A\nstraightforward way to do this is by taking the controlled shift $+c\\, (a-N)$ followed by strictly controlled shift $C_2(+N)$. Aside from this additional shift box,  the circuit in Figure\n\\ref{fig:modular:shift} still works as intended, which is easy to establish: the speculative $b+ c\\, a -N$ is corrected back to $b+ c\\, a$ if and only if the eventual result is $\\geq c\\,a$ which is the\ncondition for the ancilla cleanup.\n\nWhen $ 2\\, a > N$ we can precompile ternary control on the entire $+(a-N)$ box, which then precomputes the\n$y=b+ c (a -N)$ for us. However, here we still get some overhead compared to the binary encoding context. Indeed, we need to correct the speculative state $y$ to $y = b + c (a -N) +N$ when $y<0$ and it is\neasily seen that the result is $\\geq c (a -N) +N$ if and only if $y$ was negative and the correction happened. Thus the ancilla cleanup threshold is $t = c (a -N) +N$ on this branch. Since $c$ is the quantum\ncontrol trit, the comparison to $t$ is somewhat more expensive to engineer than comparison to $c\\,a$.\n\nTo summarize, a purely ternary modular shift circuit allowing for ternary control would be similar to one shown in Figure \\ref{fig:modular:shift:ternary}, where the extra dashed $C_2(+N)$ box is inserted at\ncompilation time when $2 \\, a < N$. The latter case constitutes the critical path where we have to use an equivalent of $4$ additive shifts instead of $3$.\n\n\n\\begin{figure*\n\\begin{tikzpicture}[scale=3.50000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (90.000000, 37.500000);\n\\draw[color=black] (0.000000,30.000000) -- (100.000000,30.000000);\n\\draw[color=black] (0.000000,30.000000) node[left] {$|b\\rangle$};\n\\draw[color=black] (0.000000,15.000000) -- (81.000000,15.000000);\n\\draw[color=black] (95.000000,15.000000) -- (100.000000,15.000000);\n\\draw[color=black] (0.000000,15.000000) node[left] {$|0\\rangle$};\n\\draw[color=black] (0.000000,0.000000) -- (100.000000,0.000000);\n\\draw[color=black] (0.000000,0.000000) node[left] {$|0\\rangle$};\n\\draw (12.000000,30.000000) -- (12.000000,15.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and\n19.091883pt) -- cycle;\n\\clip (12.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and 19.091883pt) --\ncycle;\n\\draw (12.000000, 22.500000) node {$c(a-N)$};\n\\end{scope}\n\\draw[dashed] (32.000000,30.000000) -- (32.000000,15.000000);\n\\begin{scope}\n\\draw[dashed][fill=white] (32.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and\n19.091883pt) -- cycle;\n\\clip (32.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and 19.091883pt) --\ncycle;\n\\draw (32.000000, 22.500000) node {$C_2(+N)$};\n\\end{scope}\n\n\\draw (43.000000,15.000000) -- (43.000000,0.000000);\n\\filldraw (43.000000, 15.000000) circle(1.000000pt);\n\\begin{scope}\n\\draw[fill=white] (43.000000, 0.000000) circle(3.000000pt);\n\\clip (43.000000, 0.000000) circle(3.000000pt);\n\\draw (40.000000, 0.000000) -- (46.000000, 0.000000);\n\\draw (43.000000, -3.000000) -- (43.000000, 3.000000);\n\\end{scope}\n\\draw (64.000000,30.000000) -- (64.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (64.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and\n19.091883pt) -- cycle;\n\\clip (64.000000, 22.500000) +(-45.000000:8.485281pt and 19.091883pt) -- +(45.000000:8.485281pt and 19.091883pt) -- +(135.000000:8.485281pt and 19.091883pt) -- +(225.000000:8.485281pt and 19.091883pt) --\ncycle;\n\\draw (64.000000, 22.500000) node {$+N$};\n\\end{scope}\n\\filldraw (64.000000, 0.000000) circle(1.000000pt);\n\\draw (88.000000,30.000000) -- (88.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (88.000000, 15.000000) +(-45.000000:8.485281pt and 29.698485pt) -- +(45.000000:8.485281pt and 29.698485pt) -- +(135.000000:8.485281pt and 29.698485pt) -- +(225.000000:8.485281pt and\n29.698485pt) -- cycle;\n\\clip (88.000000, 15.000000) +(-45.000000:8.485281pt and 29.698485pt) -- +(45.000000:8.485281pt and 29.698485pt) -- +(135.000000:8.485281pt and 29.698485pt) -- +(225.000000:8.485281pt and 29.698485pt) --\ncycle;\n\\draw (88.000000, 15.000000) node {$\\geq t?$};\n\\end{scope}\n\\end{tikzpicture}\n\\caption{\\label{fig:modular:shift:ternary} Top-level layout of ternary modular additive shift. In case $2\\,a < N$ the circuit is compiled with the additional $C_2(+N)$ shift controlled on $c=2$ and using the\nthreshold $t=c\\, a$. In case $2\\, a > N$ the additional shift is not needed, but the threshold $t=c(a-N) + N$.}\n\\end{figure*}\n\n\n\\begin{comment}\nIn ternary framework we propose that each of the three components of the modular shift circuit described above emulates the corresponding binary component as described in the preceding sections. Given this\ndesign, the non-Clifford depth overhead factor due to ternary emulation (compared to binary solution) is asymptotically insignificant when $n \\rightarrow \\infty$ and in practice does not exceed\n$(1+3\/(4\\,\\log_2(n)))$.\nThe same applies to the controlled versions of the modular shift circuit.\n\\end{comment}\n\n\\subsection{Circuits for modular exponentiation} \\label{subsec:mod:exponetiation}\n\nFor modular exponentiation $|k\\rangle |1\\rangle \\mapsto |k\\rangle |a^k \\; {\\rm mod} \\; N\\rangle$ we follow the known implementation proposed in the first half of Ref. \\cite{Zalka}. Our designs are also\nmotivated in part by Ref. \\cite{CleveWatrous}.\n\nWe denote by $d$ the dimension of the single qudit. $d$ is assumed to be either $2$ or $3$ where it matters.\nSuppose that $a,N$ are classically known integers $a < N$, and $n$ is an integer approximately equal to $\\log_d(N)$.\n\nSuppose $|k\\rangle$ is quantumly encoded, $k=\\sum_{j=0}^{2 \\, n-1} k_j \\, d^j$ is base-$d$ expansion of $k$, where $k_j$ are the corresponding qudit states.\nFirst, we observe that\n\\begin{equation} \\label{eq:mod:exp:product}\na^k \\; {\\rm mod} \\; N = \\prod_{j=0}^{2\\,n-1} (a^{d^j} \\; {\\rm mod} \\; N)^{k_j}  \\; {\\rm mod} \\; N.\n\\end{equation}\nNote that $(a^{d^j} \\; {\\rm mod} \\; N)$ are $2\\,n$ classical values that are known and easily pre-computable at compilation time.\nThus $|a^k \\; {\\rm mod} \\; N \\rangle$ is computed as a sequence of modular multiplicative shifts, each quantumly controlled by the $|k_j\\rangle$.\n\nSuppose we have computed the partial product\n\\[\np_{k,m} = \\prod_{j=0}^{m} (a^{d^j} \\; {\\rm mod} \\; N)^{k_j}  \\; {\\rm mod} \\; N,\n\\]\nand let\n\\[\np_{k,m} = \\sum_{\\ell=0}^{n-1} p_{k,m,\\ell} d^{\\ell}\n\\]\nbe the base-$d$ expansion of $p_{k,m}$.\nThen\n\\[\np_{k,m+1} = \\sum_{\\ell=0}^{n-1} p_{k,m,\\ell} (d^{\\ell} a^{d^{m+1}} \\; {\\rm mod} \\; N)^{k_{m+1}} \\; {\\rm mod} \\; N.\n\\]\n\nObserve, again, that\n\n\\begin{equation} \\label{eq:d:ary:shifts}\n\\{(d^{\\ell} a^{d^{m+1}} \\; {\\rm mod} \\; N)^f \\; {\\rm mod} \\; N | f \\in [1..d-1]\\}\n\\end{equation}\n\n\\noindent is the set of fewer than $d$ pre-computable classical values known a priori. Therefore, promoting $p_{k,m}$ to $p_{k,m+1}$ is performed as a sequence of modular additive shifts, controlled by\n$p_{k,m,\\ell}$ and $k_{m+1}$.\n\nHerein lies a subtle difference between the case of $d=2$ and the case of $d>2$ (e.g. $d=3$).\nIn the case of $d=2$ we do the modular shift by $2^{\\ell} a^{2^{m+1}} \\; {\\rm mod} \\; N$ if and only if $p_{k,m,\\ell} = k_{m+1} = 1$. Thus the corresponding gate is simply the doubly-controlled modular\nadditive shift.\n\nIn case of $d>2$ the $d-1$ basis values of $k_{m+1}$ lead to modular additive shift by one of the $d-1$ potentially different values listed in the equation (\\ref{eq:d:ary:shifts}). Thus we need a $(d-1)$-way\nquantum switch capable of selection between the listed values. Let $S_f,f \\in [1..d-1]$ be the modular additive shift by the $f$-th value in (\\ref{eq:d:ary:shifts}). Then the desired switch can be realized\ncoherently as the product $C_1(S_1) \\cdots C_{d-1}(S_{d-1})$ where $C_f(S_f)$ is the $S_f$ activated only by the basis state $k_{m+1} = |f\\rangle$.\n\nThis implies the following difference in the circuit makeup between the case of $d=2$ and the case of $d=3$.\n\nFor $d=2$ modular exponentiation takes roughly $2\\, n^2$ doubly-controlled modular additive shifts; for $d=3$ it takes roughly $4\\, m^2$ doubly-controlled modular additive shifts (where $m$ is the trit size of\nthe arguments), each with one ternary and one strict control on one of the two ternary values.\n\nWhen comparing the option of performing the circuit in emulated binary encoding against the option of running it in true ternary encoding we find a practical dead heat between the two options in terms of\ncircuit depth. Indeed in counting the number of doubly-controlled additive shift boxes we find that $4\\, m^2 = 2\\,(\\log_3(2))^2 \\, (2\\,n^2) \\approx 0.796 \\times (2\\,n^2)$. But we should be aware of possible\nfactor $4\/3$ overhead in the number of additive shifts per a ternary modular shift as suggested, for example, by Figure \\ref{fig:modular:shift:ternary}. (Of course $4\/3 \\times 0.796 \\approx 1.06$.)\n\nTo summarize, solutions based on emulation of binary ripple-carry adders are still win-win over the comparable true ternary ripple-carry designs in the context of the modular exponentiation; when carry\nlookahead adders are used, the two options have nearly identical non-Clifford depth numbers, but there is notable width reduction advantage (factor of $\\log_3(2)$) of using true ternary solution over the\nemulated binary one.\n\n\n\\subsection{Circuits for quantum Fourier transform} \\label{subsec:Fourier:transform}\n\nIn the solutions for period finding discussed so far, the quantum cost is dominated by the cost of modular exponentiation represented by an appropriate reversible classical circuit. In this context\njust a fraction of the cost falls onto the quantum Fourier transform.\nNevertheless, for the sake of completeness we discuss some designs for emulating binary quantum Fourier transform on ternary computers and implementing ternary Fourier transform directly in ternary logic.\n\nOdd radix Fourier transforms appeared in earlier quantum algorithm literature. In particular \\cite{Zalka2006} outlines the benefits of ``trinary'' (ternary) Fourier for low-width Shor\nfactorization circuits and also briefly sketches how ternary Fourier transform can be emulated in multi-qubit framework. On a more general level, Ref. \\cite{HallgrenHales} describes quantum Fourier transform over\n$\\mathbb{Z}_p$. In Subsection \\ref{subsubsec:true:ternary:Fourier} we develop specific circuitry for a version of such a transform over $\\mathbb{Z}_p$ where $p$ is some integer power of $3$.\n\n\\subsubsection{The case of emulated binary}\nA familiar binary circuit for approximate Fourier transform in dimension $2^n$ with precision $\\delta$ consists of roughly $\\Theta(n\\,\\log(n\/\\delta))$ controlled phases and $n$ binary Hadamard gates (see\n\\cite{IkeAndMike2000}, Section 5).\nIn known fault-tolerant binary frameworks, the phases  $e^{\\pi \\, i\/2^k}, k \\in \\mathbb{Z}$ occurring in the Fourier transform have to be treated just like generic phases.\nOf all the possible ways to emulate a controlled phase gate we will focus on just one with minimal parametric cost.\nThis is the one with one clean ancilla, two Toffoli gates and one uncontrolled phase gate. (It is not clear when exactly this design has been invented, but c.f. \\cite{Diagonal}, Section 2 for a more recent\ndiscussion.)\n\nGiven the control qubit $|c\\rangle$ and target qubit $|t\\rangle$ the controlled phase gate $C(P(\\phi)), |\\phi|=1$ is emulated by applying $\\mbox{Toffoli} (I\\otimes I\\otimes P(\\phi)) \\, \\mbox{Toffoli}$ to the\nstate $|c\\rangle |t\\rangle |0\\rangle$.\nTernary emulation of Toffoli gate is discussed in detail in Section \\ref{sec:reflections}. Somewhat surprisingly, ternary emulation of uncontrolled phase gates in practice incurs larger overhead than emulation\nof classical gates.\nAlso the binary Hadamard gate is a Clifford gate in the binary framework, but cannot be emulated by a ternary Clifford circuit. This introduces additional overhead factor of $(1+\\Theta(1\/\\log(1\/\\delta)))$.\n\n\\subsubsection{The case of true ternary} \\label{subsubsec:true:ternary:Fourier}\n\nWe develop our own circuitry for QFT over $\\mathbb{Z}_{3^n}$ based on the textbook Cooley Tukey procedure.\n\nQuantum Fourier transform in the $n$ qutrit state space is given by the unitary matrix\n\n\\begin{equation}\n\\mbox{QFT}_{3^n} = [\\zeta_{3^n}^{j\\,k}]\n\\end{equation}\n\\noindent where $\\zeta_{3^n}^{j\\,k}$ is the $3^n$-th root of unity.\nIn particular the $\\mbox{QFT}_{3}$ coincides with the ternary (Clifford) Hadamard gate.\n\nThe following recursion for $n>1$ is verified by straightforward direct computation:\n\n\\begin{equation}\n\\mbox{QFT}_{3^n} = \\Pi_n \\mbox{QFT}_{3^{n-1}} (\\Lambda(D_n)) \\mbox{QFT}_{3}\n\\end{equation}\n\\noindent where $\\Pi_n$ is a certain $n$-qutrit permutation,\n\n\\begin{equation}\nD_n = \\mbox{diag}(1, \\zeta_{3^n}, \\ldots, \\zeta_{3^n}^{3^{n-1} - 1})\n\\end{equation}\n\\noindent and where $\\Lambda$ is the ternary control.\n\nBy further direct computation we observe that\n\\begin{equation}\nD_n = \\prod_{k=0}^{n-2} \\mbox{diag}(1,\\zeta_{3^n}^{3^k},\\zeta_{3^n}^{2 \\times 3^k}).\n\\end{equation}\n\nThe permutation gate $\\Pi_n$ is not computationally important, since it amounts to $O(n)$ qutrit swaps which are all ternary Clifford.\n\nAside of this tweak we have decomposed $\\mbox{QFT}_{3^n}$ recursively into $\\Theta(n^2)$ gates of the form $\\Lambda(\\mbox{diag}(1,\\zeta_{3^m}^{3^k},\\zeta_{3^m}^{2 \\times 3^k}))$ which are ternary analogs of\nfamiliar controlled phase gates.\n\nSimilar to the binary case, it is known in general (cf. \\cite{HallgrenHales}) that once we are allowed to approximate the QFT to some fidelity $1-\\delta$, we can compute the approximate QFT with $\\Theta(n \\,\n\\log(n\/\\delta)+\\log(1\/\\delta)^2)$ gates. This is because controlled phase gates with phases in some $O(\\delta\/n)$ can be dropped from the circuit without compromising the fidelity.\n\n\n\\subsubsection{Implementation of binary and ternary controlled phase gates in the Clifford+$R_{|2\\rangle}$ basis}\nIn ternary framework a $P(\\phi)=|0\\rangle \\langle 0| + \\phi \\, |1\\rangle \\langle 1|, |\\phi|=1$ can be emulated exactly by the balanced two-level gate\n$P'(\\phi)=|0\\rangle \\langle 0| + \\phi \\, |1\\rangle \\langle 1| + \\phi^{-1} \\, |2\\rangle \\langle 2|$ which is a composition of the Clifford reflection $H^2$ and the non-Clifford reflection\n$P''(\\phi)=|0\\rangle \\langle 0| + \\phi \\, |1\\rangle \\langle 2| + \\phi^{-1} \\, |2\\rangle \\langle 1|$.\nAlso, the binary Hadamard gate $h=(|0\\rangle \\langle 0| + |0\\rangle \\langle 1|+|1\\rangle \\langle 0|-|1\\rangle \\langle 1|)\/\\sqrt{2}$ is a two-level Householder reflection.\nAs per \\cite{5isNew8},\\cite{BCKW}, both $P''(\\varphi)$ and $h$ can be effectively approximated to precision $\\delta$ by Clifford+$R_{|2\\rangle}$ circuits with $R$-counts $\\leq C \\, \\log_3(1\/\\delta) +\nO(\\log(\\log(1\/\\delta)))$ and the constant $C$ in between $5$ and $8$.\nFor reference, in the Clifford+T framework the $T$-count of $\\delta$-approximation of a generic phase gate is in\n$3 \\, \\log_2(1\/\\delta) + O(\\log(\\log(1\/\\delta)))$.\n\n\\begin{comment}\nAn easy variation of Lemma \\ref{lem:dual:binary:control} below shows that a controlled phase gate can be emulated at the cost of two $C_2(\\mbox{INC})$ gates (a total of two $P_9$ gates) and one clean ancilla.\nFor reference, in Clifford+T framework ancilla-assisted emulation of controlled phase gate requires two additional modified Toffoli gates (a total of $8$ $T$ gates).\n\\end{comment}\n\nThus, emulation of the binary circuit for a binary Fourier transform incurs no surprising costs.\n\nIn pure ternary encoding we need to implement the ternary analog of controlled phase gate: gates of the form $\\Lambda(\\mbox{diag}(1,\\phi,\\phi^2)), \\, |\\phi|=1$. This is not difficult after some\nalgebraic manipulation:\n\n\\begin{prop}\nGiven a phase factor $\\phi,|\\phi|=1$  and an arbitrarily small $\\delta > 0$ the gate $\\Lambda(\\mbox{diag}(1,\\phi,\\phi^2))$ can be effectively approximated to precision $\\delta$ by a metaplectic circuit with at\nmost $40\\, (\\log_3(1\/\\delta) + O(\\log(\\log(1\/\\delta))))$ $R_{|2\\rangle}$ gates.\n\nAlternatively such a $\\delta$-approximation can be effectively achieved by a metaplectic circuit with at most $24\\, (\\log_3(1\/\\delta) + O(\\log(\\log(1\/\\delta))))$ $R_{|2\\rangle}$ gates and a fixed-cost widget\nwith at most $30$ $P_9$ gates.\n\\end{prop}\n\\begin{proof}\nWe note that\n\n\\begin{equation}\n\\begin{split}\n\\Lambda(\\mbox{diag}(1,\\phi,\\phi^2)) = \\phi\\, \\mbox{diag}(1,1,1,\\phi^*,1,\\phi,1,1,1)\\\\\n \\mbox{diag}(1,1,1,1,1,1,(\\phi^*)^2,1,\\phi^2) (\\mbox{diag}(\\phi^*,1,\\phi) \\otimes I).\n\\end{split}\n\\end{equation}\n\nEach of the three factors in this decomposition is a product of two two-level reflections. It is also notable that one particular reflection, the $\\tau_{|0\\rangle,|2\\rangle}$ coming from\n$\\mbox{diag}(\\phi^*,1,\\phi) = \\tau_{|0\\rangle,|2\\rangle} (\\phi\\, |0\\rangle \\langle 2| + |1\\rangle \\langle 1| + \\phi^*\\, |2\\rangle \\langle 0|)$ is in fact ternary Clifford. Therefore we are having a total of\nfive non-Clifford reflections in this decomposition, two of which are non-parametric classical reflections.\n\nAs per \\cite{BCKW} any two-level reflection can be effectively $(\\delta\/5)$-approximated by  metaplectic circuit with at most $8\\, (\\log_3(1\/\\delta) + O(\\log(\\log(1\/\\delta))))$ $R_{|2\\rangle}$ gates, and this\ncan be applied to all five non-Clifford reflections.\nAlternatively, each of the two classical ones can be represented exactly as per \\cite{BCRS} using five $C_2(\\mbox{INC})$ or, respectively, $15$ $P_9$ gates.\n\\end{proof}\n\nThus implementation of either version of QFT circuit is never a cost surprise in the metaplectic Clifford+$R_{|2\\rangle}$ basis.\n\nAlthough numerologically the $R$-depth of the required approximation circuits is a good factor higher than the $T$-depth of corresponding circuits required in the Clifford+T framework, we need to keep in mind\nthat the $R_{|2\\rangle}$ is significantly easier to execute on a natively metaplectic computer since, unlike the $T$ gate it does not require magic state distillation.\n\n\\subsubsection{Implementation of binary and ternary controlled phase gates in the Clifford+$P_9$ basis}\n\nAt the time of this writing emulation of QFT circuits on a generic ternary computer is not entirely straightforward.\n\nFirst of all, we currently do not know an efficient direct circuit synthesis method for Householder reflections in the Clifford+$P_9$ basis.\nIf follows from \\cite{Bourgain} that any ternary unitary gate can be also approximated to precision $\\delta$ by an ancilla-free Clifford+$P_9$ circuit of depth in $O(\\log(1\/\\delta))$; but  we do not have a\ngood effective procedure for finding ancilla-free circuits of this sort, neither do we have a clear idea of the practical constant hidden in the $O(\\log(1\/\\delta))$.\n\nAs a bridge solution, we show in Appendix \\ref{app:sec:R:2} that the requisite magic state $|\\psi\\rangle$ (see eq. (\\ref{eq:def:psi})) for the gate $R_{|2\\rangle}$ can be emulated exactly and coherently\nby a set of effective repeat-until-success circuits with four ancillary qutrits and expected average $P_9$-count of $27\/4$.\nThus we can approximate a required uncontrolled phase gate with an efficient Clifford+$R_{|2\\rangle}$ circuit and then transcribe the latter into a corresponding ancilla-assisted probabilistic circuit over\nthe Clifford+$P_9$ basis. In order to have a good synchronization with the Clifford+$R_{|2\\rangle}$ circuit execution it would suffice to have the magic state preparation coprocessor of width somewhat greater than\n$27$. Since the controlled phase gates and hence the approximating Clifford+$R_{|2\\rangle}$ circuits are performed sequentially in the context of the QFT, this coprocessor is shared across the QFT circuit and\nthus the width overhead is bound by a constant.\n\nOn the balance, we conclude that ternary execution of the QFT is likely to be more expensive in terms of required non-Clifford units, than, for example, comparable Clifford+T implementation. However the\nnon-Clifford depth overhead factor over Clifford+T is upper bounded by an $(\\alpha+\\Theta(1\/\\log(1\/\\delta)))$ where $\\alpha$ is a small constant.\nSuch overhead becomes practically valid, however, when hosting period-finding solutions that make heavy use of Fourier transform, such as for example the Beauregard circuit \\cite{Beauregard} (see Appendix\n\\ref{app:sec:alternative:circuits} for a further brief discussion).\n\n\\subsection{Comparative cost of ternary emulation vs. true ternary arithmetic}\nWith the current state of the art ternary arithmetic circuits, modular exponentiation (and hence Shor's period finding) is practically less expensive with emulated binary encoding in low width (e.g. small\nquantum computer); however, when $O(m^2 \\log(m))$ depth is desired, pure ternary arithmetic allows for width reduction by a factor of $\\log_3(2)$ compared to emulated binary circuits, while requiring\nessentially the same non-Clifford depth.\n\n\\section{Implementing Reflections on Generic Ternary and Metaplectic Topological Quantum Computers} \\label{sec:reflections}\n\nState of the art implementation of the three-qubit binary Toffoli gate assumes the availability of the Clifford+T basis \\cite{IkeAndMike2000}. It has been known for quite some time cf. \\cite{AmyEtAl} that a\nToffoli gate can be implemented ancilla-free using a network of $\\mbox{CNOT}$s and $7$ $T^{\\pm 1}$ gates. It has been shown in \\cite{Tcount}  that this is the minimal $T$-count for ancilla-free implementation\nof the Toffoli gate.\n\nIn Section \\ref{subsec:classical:with:P9} we develop emulations of classical two-level reflections (which generalize Toffoli and Toffoli-like gates) on generic ternary computer endowed with the\nClifford+$P_9$ basis as described in Section \\ref{subsec:P9:gate}. We also introduce purely ternary tools necessary for implementing controlled versions of key gates for ternary arithmetic proposed in \\cite{BCRS}.\nancillas. This implies of course an emulation of the three-qubit Toffoli gate with $6$ $P_9$ gates and one clean ancilla.\n\nIn Section \\ref{subsec:reflection:metaplectic}\n we reevaluate the emulation cost assuming a \\emph{metaplectic topological quantum computer} (MTQC) with Clifford+$R_{|2\\rangle}$ basis as described in Section \\ref{subsec:metaplectic:basis}.\nIn that setup we get two different options both for implementing non-Clifford classical two-way transpositions (including the Toffoli gate) and for circuitizing key gate for proper ternary arithmetic.\n\nOne is direct approximation using Clifford+$R_{|2\\rangle}$ circuits. The other is based on the $P_9$ gate but it uses \\emph{magic state preparation} in the Clifford+$R_{|2\\rangle}$ basis instead of magic state\ndistillation. This is explained in detail in Subsection \\ref{subsec:reflection:metaplectic}.\nThe first option might be ideal for smaller quantum computers. It allows circuits of fixed widths but creates implementation circuits for Toffoli gates with the $R$-count of approximately\n$8\\,\\log_3(1\/\\delta)$ when $1-\\delta$ is the desired fidelity of the Toffoli gate.\nThe second option supports separation of the cost of the $P_9$ gate into the ``online'' and ``offline'' components (similar to the Clifford+T framework) with the ``online'' component depth in $O(1)$ and the\n``offline'' cost offloaded to a state preparation part of the computer, which has the width of roughly $9 \\, \\log_3(1\/\\delta)$ qutrits but does not need to remain always coherent.\n\n\\subsection{Implementing classical reflections in the Clifford+$P_9$ basis} \\label{subsec:classical:with:P9}\n\nThe synthesis described here is a generic ternary counterpart of the exact,  constant $T$-count representation of the three-qubit Toffoli gate in the Clifford+T framework.\n\nOne distinction of the ternary framework from the binary one is that not all two-qutrit classical gates are Clifford gates.\nIn particular the $\\tau_{|10\\rangle,|11\\rangle}$ reflection which is a strict emulation of the binary $\\mbox{CNOT}$ is not a Clifford gate; neither is the $\\tau_{|10\\rangle,|01\\rangle}$ which which is a\nstrict emulation of the binary $\\mbox{SWAP}$. However, while binary $\\mbox{SWAP}$ can be emulated simply as a restriction of the (Clifford) ternary swap on binary subspace, the $\\mbox{CNOT}$ cannot be so\nemulated.\n\nA particularly important two-qutrit building block is the following non-Clifford gate\n\\[\nC_1(\\mbox{INC}) |j\\rangle |k\\rangle = |j\\rangle |(k + \\delta_{j,1}) \\; {\\rm mod} \\; 3\\rangle.\n\\]\n\nA peculiar phenomenon in multi-qudit computation (in dimension greater than two) is that a two-qudit classical non-Clifford gate (such as $C_1(\\mbox{INC})$) along with the $\\mbox{INC}$ gate is universal for\nthe ancilla-assisted reversible classical computation, cf. \\cite{Brennen}, whereas a three-qubit gate, such as Toffoli is needed for the purpose in multi-qubit case.\n\nThe following is a slight variation of a circuit from \\cite{BCRS}:\n\\begin{equation*}\\nonumber \\label{eq:five:C1INC}\n\\begin{split}\n\\tau_{|02\\rangle, |2,0\\rangle} =\n\\mbox{TSWAP}\\, C_1(\\mbox{INC})_{2,1} \\,  C_1(\\mbox{INC})_{1,2} \\\\  \\,  C_1(\\mbox{INC})_{2,1}\n\\,  C_1(\\mbox{INC})_{1,2} \\,  C_1(\\mbox{INC})_{2,1},\n\\end{split}\n\\end{equation*}\nwhere $\\mbox{TSWAP}$ is the ternary (Clifford) swap gate. This suggests using $5$ copies of $C_1(\\mbox{INC})$ gate for implementing a two-level two-qutrit reflection. However, this is\ninefficient when we only need to process binary data.\n\n\n\\begin{prop} \\label{prop:CNOT:ancilla-free}\nThe following classical circuit is an exact emulation of the binary $\\mbox{CNOT}$ gate on the binary data:\n\\begin{equation} \\label{eq:two:C1INC}\n\\begin{split}\n \\mbox{SUM}_{2,1} (\\tau_{|1\\rangle,|2\\rangle} \\otimes \\tau_{|1\\rangle,|2\\rangle})\n \\,\\mbox{TSWAP} \\,   C_1(\\mbox{INC})_{2,1} \\\\ \\,  C_1(\\mbox{INC}^{\\dagger})_{1,2}\n \\, (\\tau_{|1\\rangle,|2\\rangle} \\otimes \\tau_{|1\\rangle,|2\\rangle}) \\mbox{SUM}^{\\dagger}_{2,1}\n\\end{split}\n\\end{equation}\n\\end{prop}\n\\begin{proof}\nBy direct computation.\n\\end{proof}\n\n\nThe two non-Clifford gates in this circuit are the $C_1(\\mbox{INC})$ and $C_1(\\mbox{INC}^{\\dagger})$. (To avoid confusion, note that the gate as per Eq. (\\ref{eq:two:C1INC}) is no longer an axial reflection on ternary data.)\n\nThe $C_1(\\mbox{INC})$ is Clifford-equivalent to the $C_1(Z) = diag(1,1,1,1,\\omega_3,\\omega_3^2,1,1,1)$ gate ($\\omega_3 = e^{2 \\pi\\, i\/3}$), and the latter gate is represented exactly by the network\nshown in Figure \\ref{fig:CZ:from:P9} (up to a couple of local $\\tau_{|0\\rangle |1\\rangle}$ gates and a local $Q$ gate).\n\n\\begin{figure*\n\\begin{tikzpicture}[scale=2.000000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (195.000000, 22.500000);\n\\draw[color=black] (0.000000,15.000000) -- (195.000000,15.000000);\n\\draw[color=black] (0.000000,0.000000) -- (195.000000,0.000000);\n\\draw (12.000000,15.000000) -- (12.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 0.000000) circle(6.000000pt);\n\\clip (12.000000, 0.000000) circle(6.000000pt);\n\\draw (12.000000, 0.000000) node {Z};\n\\end{scope}\n\\filldraw (12.000000, 15.000000) circle(1.500000pt);\n\\draw[fill=white,color=white] (30.000000, -6.000000) rectangle (45.000000, 21.000000);\n\\draw (37.500000, 7.500000) node {$\\sim$};\n\\begin{scope}\n\\draw[fill=white] (63.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (63.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (63.000000, -0.000000) node {$P_9$};\n\\end{scope}\n\\draw (87.000000,15.000000) -- (87.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (87.000000, 0.000000) circle(6.000000pt);\n\\clip (87.000000, 0.000000) circle(6.000000pt);\n\\draw (87.000000, 0.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (87.000000, 15.000000) circle(1.500000pt);\n\\begin{scope}\n\\draw[fill=white] (111.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (111.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (111.000000, -0.000000) node {$P_9$};\n\\end{scope}\n\\draw (135.000000,15.000000) -- (135.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (135.000000, 0.000000) circle(6.000000pt);\n\\clip (135.000000, 0.000000) circle(6.000000pt);\n\\draw (135.000000, 0.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (135.000000, 15.000000) circle(1.500000pt);\n\\begin{scope}\n\\draw[fill=white] (159.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (159.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (159.000000, -0.000000) node {$P_9$};\n\\end{scope}\n\\draw (183.000000,15.000000) -- (183.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (183.000000, 0.000000) circle(6.000000pt);\n\\clip (183.000000, 0.000000) circle(6.000000pt);\n\\draw (183.000000, 0.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (183.000000, 15.000000) circle(1.500000pt);\n\\end{tikzpicture}\n\\caption{\\label{fig:CZ:from:P9} Exact representation of $C_1(Z)$ in terms of $P_9$ gates. }\n\\end{figure*}\n\nPlugging in corresponding representations of $C_1(\\mbox{INC})$ and $C_1(\\mbox{INC}^{\\dagger})$ into the circuit (\\ref{eq:two:C1INC}) we obtain an exact emulation of\n$\\mbox{CNOT}$ that uses $6$ instances of the $P_9^{\\pm 1}$ gate.\n\\begin{remark} \\label{remark:C1Z:depth:one}\nBy using an available clean ancilla, we can exactly represent the $C_1(Z)$ in $P_9$-depth one. The corresponding circuit is equivalent to one shown in Figure \\ref{fig:C1Z:depth:one}. Thus the $\\mbox{CNOT}$\ngate can be emulated on binary data using a clean ancilla in $P_9$-depth two.\n\\end{remark}\n\n\\begin{figure*\n\\begin{tikzpicture}[scale=2.000000,x=1pt,y=1pt]\n\\filldraw[color=white] (0.000000, -7.500000) rectangle (168.000000, 37.500000);\n\\draw[color=black] (0.000000,30.000000) -- (168.000000,30.000000);\n\\draw[color=black] (0.000000,30.000000) node[left] {$$$$};\n\\draw[color=black] (0.000000,15.000000) -- (168.000000,15.000000);\n\\draw[color=black] (0.000000,15.000000) node[left] {$$$$};\n\\draw[color=black] (0.000000,0.000000) -- (168.000000,0.000000);\n\\draw[color=black] (0.000000,0.000000) node[left] {$|0\\rangle$};\n\\draw (12.000000,30.000000) -- (12.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (12.000000, 0.000000) circle(6.000000pt);\n\\clip (12.000000, 0.000000) circle(6.000000pt);\n\\draw (12.000000, 0.000000) node {$\\mbox{INC}^{\\dagger}$};\n\\end{scope}\n\\draw[fill=white] (12.000000, 30.000000) circle(1.500000pt);\n\\draw (36.000000,30.000000) -- (36.000000,15.000000);\n\\begin{scope}\n\\draw[fill=white] (36.000000, 30.000000) circle(6.000000pt);\n\\clip (36.000000, 30.000000) circle(6.000000pt);\n\\draw (36.000000, 30.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (36.000000, 15.000000) circle(1.500000pt);\n\\draw (60.000000,15.000000) -- (60.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (60.000000, 0.000000) circle(6.000000pt);\n\\clip (60.000000, 0.000000) circle(6.000000pt);\n\\draw (60.000000, 0.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (60.000000, 15.000000) circle(1.500000pt);\n\\begin{scope}\n\\draw[fill=white] (84.000000, 30.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (84.000000, 30.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (84.000000, 30.000000) node {$P_9$};\n\\end{scope}\n\\begin{scope}\n\\draw[fill=white] (84.000000, 15.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (84.000000, 15.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (84.000000, 15.000000) node {$P_9$};\n\\end{scope}\n\\begin{scope}\n\\draw[fill=white] (84.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt)\n-- cycle;\n\\clip (84.000000, -0.000000) +(-45.000000:8.485281pt and 8.485281pt) -- +(45.000000:8.485281pt and 8.485281pt) -- +(135.000000:8.485281pt and 8.485281pt) -- +(225.000000:8.485281pt and 8.485281pt) -- cycle;\n\\draw (84.000000, -0.000000) node {$P_9$};\n\\end{scope}\n\\draw (108.000000,15.000000) -- (108.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (108.000000, 0.000000) circle(6.000000pt);\n\\clip (108.000000, 0.000000) circle(6.000000pt);\n\\draw (108.000000, 0.000000) node {$\\mbox{INC}^{\\dagger}$};\n\\end{scope}\n\\draw[fill=white] (108.000000, 15.000000) circle(1.500000pt);\n\\draw (132.000000,30.000000) -- (132.000000,15.000000);\n\\begin{scope}\n\\draw[fill=white] (132.000000, 30.000000) circle(6.000000pt);\n\\clip (132.000000, 30.000000) circle(6.000000pt);\n\\draw (132.000000, 30.000000) node {$\\mbox{INC}^{\\dagger}$};\n\\end{scope}\n\\draw[fill=white] (132.000000, 15.000000) circle(1.500000pt);\n\\draw (156.000000,30.000000) -- (156.000000,0.000000);\n\\begin{scope}\n\\draw[fill=white] (156.000000, 0.000000) circle(6.000000pt);\n\\clip (156.000000, 0.000000) circle(6.000000pt);\n\\draw (156.000000, 0.000000) node {$\\mbox{INC}$};\n\\end{scope}\n\\draw[fill=white] (156.000000, 30.000000) circle(1.500000pt);\n\\draw[color=black] (168.000000,0.000000) node[right] {$|0\\rangle$};\n\\end{tikzpicture}\n\\caption{\\label{fig:C1Z:depth:one} Exact representation of $C_0(Z)$ in $P_9$-depth one. }\n\\end{figure*}\n\nThus when depth is the optimization goal, a clean ancilla can be traded for triple compression in non-Clifford depth of ternary emulation of the $\\mbox{CNOT}$. (This rewrite is similar in nature to the one\nemployed in \\cite{Jones} for the binary Margolus-Toffoli gate.)\n\n\n\\begin{prop} \\label{prop:Toffoli:the:15}\nA three-qubit Toffoli gate can be emulated, ancilla-free, by the following three-qutrit circuit:\n\n\\begin{equation} \\label{eq:Toffoli:the:15}\n(\\mbox{SUM}^{\\dagger} \\otimes I) (I \\otimes \\tau_{|20\\rangle,|21\\rangle}) (\\mbox{SUM} \\otimes I)\n\\end{equation}\nThis circuit requires $15$ $P_9$ gates to implement.\n\\end{prop}\n\\begin{proof}\nThe purpose of the emulation is perform the $|110\\rangle \\leftrightarrow |111\\rangle$ reflection in the binary data subspace.\n\nHaving applied the rightmost $\\mbox{SUM} \\otimes I$ we find that $(\\mbox{SUM} \\otimes I) |110\\rangle = |120\\rangle$, $(\\mbox{SUM} \\otimes I) |111\\rangle = |121\\rangle$ and we note that the latter two are the\nonly two transformed binary basis states to have the second trit equal to 2. Therefore the $I \\otimes \\tau_{|20\\rangle,|21\\rangle}$ operator affects only these two transformed states. By uncomputing the\n$\\mbox{SUM} \\otimes I$ we conclude the emulation.\n\\end{proof}\n\nImportantly and typically we can reduce the emulation cost by using a clean ancilla. To this end we first prove the following\n\\begin{lem} \\label{lem:dual:binary:control}\nLet $U$ be $n$-qubit unitary and let the binary-controlled $(n+1)$-qubit unitary $C(U)$ be emulated in the binary subspace of an $m$-qutrit register $m>n$. Then one level of binary control can be added to emulate\n$C(C(U))$ in an $(m+2)$-qutrit register using $6$ additional $P_9$ gates; one of the new qutrits is a clean ancilla in state $|0\\rangle$ and the other new qutrit emulates the binary control.\n\nWith one more ancilla the additional $P_9$ gates can be stacked to  $P_9$-depth $2$.\n\\end{lem}\n\\begin{proof}\nWe prove the lemma by explicitly extending the emulation circuit.\nLet $c_1$ be a label of the qutrit emulating the control wire of $C(U)$. Let $c_2$ be a label of the new qutrit to emulate the new control wire. Let $a$ be the label of the new clean ancilla.\n\nApply the sequence of gates $C_2(INC)_{c2,a} \\mbox{SUM}_{c1,c2}$ (right to left) then use the ancilla $a$ as the control in the known emulation of $C(U)$, then unentangle:\n$\\mbox{SUM}_{c1,c2}^{\\dagger} C_2(INC)_{c2,a}^{\\dagger}$.\n\nThe circuit applies correct emulation to the binary subspace of the $(m+2)$-qutrit register.\nThe correctness is straightforward: within the binary subspace $\\mbox{SUM}_{c1,c2}$ generates $|2\\rangle$ on the $c_2$ wire. The $C_2(INC)_{c2,a}$ promotes the ancilla to $|1\\rangle$ if and only if\n$|c_1,c_2\\rangle = |11\\rangle$. Therefore $U$ is triggered only by the latter basis element, which is the definition of the dual binary control.\n\nThe cost estimate follows from the fact that $C_2(INC)_{c2,a}$ and its inverse take $3$ $P_9$ gates each.\n\\end{proof}\n\n\\begin{corol} \\label{corol:Toffoli:the:12}\nThree-qubit Toffoli gate can be emulated in four qutrits (allowing one clean ancilla) with $12$ $P_9$ gates at $P_9$-depth of $4$.\n\\end{corol}\n\nIndeed $\\mbox{Toffoli} = CC(\\mbox{NOT})$ and $C(\\mbox{NOT})$ takes $6$ $P_9$ gates with no ancillas to emulate as per Proposition \\ref{prop:CNOT:ancilla-free}.\n\n\\begin{corol} \\label{corol:ctrl:Toffoli:18}\nFour-qubit binary-controlled Toffoli gate $CCC(\\mbox{NOT})$ can be emulated\n\n1) in six qutrits (allowing two clean ancillas) with $18$ $P_9$ gates at $P_9$-depth of $6$.\n\n2) in five qutrits (allowing one clean ancilla) with $21$ $P_9$ gates.\n\\end{corol}\n\n\\begin{proof}\n\nFor 1),  we emulate using Lemma \\ref{lem:dual:binary:control} and Corollary \\ref{corol:Toffoli:the:12}.\n\nFor 2), we emulate using Lemma \\ref{lem:dual:binary:control} and Proposition \\ref{prop:Toffoli:the:15}\n\\end{proof}\n\n\n\nWe will further use the three-qutrit ``Horner'' gate $\\Lambda(\\mbox{SUM})$:\n\\[\n\\Lambda(\\mbox{SUM}) |i,j,k\\rangle = |i,j,k+i\\,j \\; \\mod 3 \\;\\rangle, \\, i,j,k \\in \\{0,1,2\\}\n\\]\n\\noindent as a tool for adding levels of control to emulated binary and true ternary gates.\n\nRecall from \\cite{BCRS}, Figure 18 and discussion, that the best-known non-Clifford cost of $\\Lambda(\\mbox{SUM})$ is ${4}$ $P_9$ gates at $P_9$-depth ${2}$.\n\n\n\n\n\nWe now proceed to implement the completely ternary four-qutrit gate $\\Lambda \\Lambda (\\mbox{SUM})$ using the same constuction as above\n\n\\begin{prop} \\label{prop:Lambda:Lambda:SUM}\nLabel primary qutrits with $1,2,3,4$ and label a clean ancillary qutrit in state $|0\\rangle$ with $5$.\nThen the following circuit implements the $\\Lambda \\Lambda (\\mbox{SUM})$ gate on the primary qutrits:\n\\begin{equation} \\label{eq:Lambda:Lambda:SUM}\n\\Lambda(\\mbox{SUM})_{1,2,5}^{\\dagger} \\Lambda(\\mbox{SUM})_{3,5,4} \\Lambda(\\mbox{SUM})_{1,2,5}\n\\end{equation}\nThis circuit requires $12$ $P_9$ gates to implement.\n\\end{prop}\n\nHowever, as follows from discussion in Sections \\ref{subsec:carry:lookahead} and \\ref{subsec:ripple:carry}, controlled ternary modular exponentiation also relies on another form of the doubly-controlled\n$\\mbox{SUM}$ gate: the strictly controlled Horner gate $C_f(\\Lambda(\\mbox{SUM})), f\\in \\{0,1,2\\}$ where the Horner gate  $\\Lambda(\\mbox{SUM})$ is activated only by the basis state $|f\\rangle$ of the\ntop qutrit.\n\nA certain implementation of the $C_f(\\mbox{SUM})$ has been developed in \\cite{BCRS} costing $15$ $P_9$ gates.\n\nThe following Proposition explains how to insert another level of ternary control using a cascade of Horner gates again\n\n\\begin{prop} \\label{prop:C:Lambda:SUM}\nLabel primary qutrits with $1,2,3,4$ and label a clean ancillary qutrit in state $|0\\rangle$ with $5$.\nThen the following circuit implements the $C_f(\\Lambda(\\mbox{SUM})))$ gate on the primary qutrits:\n\\begin{equation} \\label{eq:C:Lambda:SUM}\n\\Lambda(\\mbox{SUM})_{2,3,5}^{\\dagger} C_f(\\mbox{SUM})_{1,5,4} \\Lambda(\\mbox{SUM})_{2,3,5}\n\\end{equation}\nThis circuit takes $23$ $P_9$ gates to implement.\n\nWith one additional ancilla the circuit can be restacked to have $P_9$-depth of $9$.\n\\end{prop}\n\nLet us give a direct proof for transparency\n\\begin{proof}\nBy definition, given a four-qutrit state $|j,k,\\ell,m\\rangle$, we must have $C_f(\\Lambda(\\mbox{SUM})))|j,k,\\ell,m\\rangle = |j,k,\\ell,m+\\delta_{j,f} \\, k \\, \\ell\\rangle$.\n\nAfter applying the rightmost Horner gate to the clean ancilla we have the ancilla in the $|k \\, \\ell\\rangle$ state. The correctness of (\\ref{eq:C:Lambda:SUM}) now follows from the definition of\n$C_f(\\mbox{SUM})_{1,5,4}$.\n\nThe best known circuitry for the components yield the cost of $15+2\\times 4 = 23$ $P_9$ gates.\n\\end{proof}\n\n\\begin{comment}\nTODO:THESE MIGHT BE STILL IMPORTANT\nWe start with some observations, that codify some useful commutants involving the $P_9$ gate:\n\n\\begin{observ} \\label{observ:P9:commute}\n1) $\\mbox{INC}\\,P_9 = P_9 \\, (Q_0 \\, \\mbox{INC} $ up to global phase\n\n2) $\\tau_{|1\\rangle,|2\\rangle} \\, P_9 = P_9^2 \\,(Q_2^{\\dagger} \\, \\tau_{|1\\rangle,|2\\rangle}$ up to global phase\n\n3)  $\\mbox{INC}^{\\dagger} \\,P_9^2  = P_9^2 \\, (Q_2^{-2} \\, \\mbox{INC}^{\\dagger}$ up to global phase\n\\end{observ}\nThese relations are established by direct computation.\n\\end{comment}\n\n\n\n\n\n\\subsection{Implementing classical reflections in metaplectic Clifford+$R_{|2\\rangle}$ basis} \\label{subsec:reflection:metaplectic}\n\nIt has been shown in \\cite{BCKW} that, given a small enough $\\delta>0$ any $n$-qutrit two-level Householder reflection can be approximated effectively and efficiently to precision $\\delta$ by a\nClifford+$R_{|2\\rangle}$ circuit containing at most $8\\,\\log_3(1\/\\delta) + O(\\log(\\log(1\/\\delta)))+ O((2+\\sqrt{5})^n)$ instances of the $R_{|2\\rangle}$ gate.\nIn particular, when $n=1$ the asymptotic term $O((2+\\sqrt{5})^n)$ resolves to exactly $1$ and when $n=2$ it resolves to exactly $4$. In both cases it is safe to merge this term with the\n$O(\\log(\\log(1\/\\delta)))$ term.\n\nThe single-qutrit $P_9$ gate is the composition of the ternary Clifford gate $\\tau_{|0\\rangle,|2\\rangle}$ and the Householder reflection\n$\\omega_9 \\, |0\\rangle \\langle 2| + |1\\rangle \\langle 1| + \\omega_9^{-1} \\, |2\\rangle \\langle 0|$.\nThe two-qutrit gate  $\\mbox{CNOT} = \\tau_{|10\\rangle, |11\\rangle}$ is by itself a two-level Householder reflection $R_{(|10\\rangle- |11\\rangle)\/\\sqrt{2}}$.\nSimilarly, $\\mbox{Toffoli} = \\tau_{|110\\rangle, |111\\rangle} = R_{(|110\\rangle- |111\\rangle)\/\\sqrt{2}}$.\nTherefore, our results apply and we have efficient strict emulations of $P_9$,  $\\mbox{CNOT}$ and Toffoli gates at depths that are logarithmic in $1\/\\delta$ and in practice are roughly\n$8\\,\\log_3(1\/\\delta)$ in depth.\n\nWe note that the direct metaplectic approximation of classical reflections is significantly more efficient than the circuits expressed in $C_f(\\mbox{INC})$ gates (as each of the latter have to be approximated).\n\nLet us briefly review such direct approximation in the context of ternary arithmetic in ternary encoding. As per \\cite{BCRS}, the generalized carry gate of the ternary ripple-carry additive shift contains two\nclassical non-Clifford reflections (\\cite{BCRS}, Fig. 5) that can be represented at fidelity $1-\\delta$ by a metaplectic circuit of $R$-count at most ${16}\\,\\log_3(1\/\\delta)$.\n\nThe same source implies that the carry status merge widget $\\mathcal{M}$ which is key in the carry lookahead additive shift is Clifford-equivalent to a $C_f(\\mbox{SUM})$ which is easily decomposed in four\nclassical two-level reflections and thus can be represented at fidelity $1-\\delta$ by a metaplectic circuit of $R$-count at most ${32}\\,\\log_3(1\/\\delta)$.\n\nA sufficient per-gate precision $\\delta$ may be found in $O(1\/(d\\,\\log(n)))$ where $d$ is the depth of the modular exponentiation circuit expressed in non-Clifford reflections. Therefore, injecting metaplectic\ncircuits in place of reflections creates an overhead factor in $\\Theta(\\log(d) \\log(\\log(n)))$. While being asymptotically moderate, such overhead could be a deterrent when factoring very large numbers.\nThis motivates us to explore constant-depth approximations of classical reflections as in the next section.\n\n\\subsection{Constant-depth implementation of $\\mbox{CNOT}$ and $C_f(\\mbox{INC})$ on ternary quantum computers.}\n\nWe demonstrate that integer arithmetic on a ternary quantum computer can be efficient both asymptotically and in practice.\nWe build on Section \\ref{subsec:classical:with:P9} that describes exact emulation of $\\mbox{CNOT}$ with $6$ instances of the $P_9$ gate.\nA core result in \\cite{CampbellEtAl} implies that the $P_9$ gate can be executed exactly by a deterministic state injection circuit using one ancilla, one measurement and classical feedback,\n\\emph{provided} availability of the ``magic'' ancillary state\n\\[\n\\mu = \\omega_9^{-1} \\, |0\\rangle  + |1\\rangle + \\omega_9 \\, |2\\rangle.\n\\]\nThe state injection circuit is given in Figure \\ref{fig:mu:state:injection}.\n\nAssuming, hypothetically, that the magic state $\\mu$ can be prepared in a separate ancillary component of the computer (then teleported), we get, a separation of the quantum complexity into ``online'' and ``offline'' components -\nsimilar to one employed in the binary Clifford+T network.\n\nWe call these components the \\emph{execution} and \\emph{preparation} components. We use the term execution depth somewhat synonymously to ``logical circuit depth''.\nThe execution part of the $P_9$ state injection, hence $\\mbox{CNOT}$,  Toffoli emulations as well as implementation of $C_f(\\mbox{INC})$ are constant depth. The magic preparation can run separately in parallel\nwhen the preparation code is granted enough width.\n\n\nIn the context of the binary Clifford+T network, assuming the required fidelity of the $T$ gate is $1-\\delta, \\delta>0$, there is a choice of magic state distillation solutions. For comparison we have selected\na particular one protocol described in \\cite{BravyiKitaev}. At the top level, it can be described as a quantum code of depth in $O(\\log(\\log(1\/\\delta)))$ and width of approximately\n$O(\\log^{\\log_3(15)}(1\/\\delta))$.\nThe newer protocol in \\cite{BravyiHaah} achieves asymptotically smaller width in\n$O(\\log^{\\gamma(k)}(1\/\\delta))$ where $k$ is an error correction hyperparameter, and $\\gamma(k) \\rightarrow \\log_2(3)$ when $k \\rightarrow \\infty$. However the \\cite{BravyiHaah} is a tradeoff rather than a win-win over\n\\cite{BravyiKitaev} in terms of practical width value.\n\nIn comparison, the magic state distillation for a generic ternary quantum computer, described in \\cite{CampbellEtAl} maps onto quantum processor of depth in $O(\\log(\\log(1\/\\delta)))$ and width of\n$O(\\log^{3}(1\/\\delta))$.\nTherefore the preparation of a magic state by distillation requires asymptotically larger width than the one for Clifford+T basis.\n\nWe observe that the prepartion width is asymptotically better at $O(\\log(1\/\\delta))$ and significantly better in practice when the target ternary computer is MTQC. Since the MTQC implements the universal Clifford+$R_{|2\\rangle}$\nbasis that does not require magic state distillation, the instances of the magic state $\\mu$ can be prepared on a much smaller scale.\n\\begin{observ} {(see \\cite{5isNew8}, Section 4)}\nAn instance of magic state $\\mu$ can be prepared at fidelity $1-\\delta$ by a  Clifford+$R_{|2\\rangle}$ circuit of non-Clifford depth in $r(\\delta) = 6\\, \\log_3(1\/\\delta) + O(\\log(\\log(1\/\\delta)))$.\n\\end{observ}\n\nTo synchronize with the $P_9$ gates in the logical circuit we need to pipeline $r(\\delta)$ instances of the magic state preparation circuit, so we always have a magic state at\nfidelity $1-\\delta$ ready to be injected into the $P_9$ protocol.\n\nOne important consequence of the synchronization requirement is that higher parallelization of non-Clifford operations reflects proportionally in an increase in width of the preparation coprocessor.\n\nIn particular, when we employ low-width circuits for Shor's period finding, such as based on ripple-carry additive shifts, then it suffices to produce a constant number of clean magic states per time step. For\nexample, if the ternary $C_f(\\mbox{INC})$ is taken as the base classical gate and its realization shown in figure \\ref{fig:C1Z:depth:one} then we need three clean magic states per a time step.\n\nSuppose now we employ an $n$-bit quantum carry lookahead adder in the same context. In order to preserve the logarithmic time cost advantage we should be able to perform up to $n$ base reflection gates\n (such as Toffolis)  in parallel\nor at least $O(n\/\\log(n))$ such gates in parallel on average. Thus the preparation component must deliver at least $O(n\/\\log(n))$  clean magic states per time step and widens the preparation component by\nthat factor.\n\n\\section{Platform specific resource counts} \\label{sec:specific:resource:counts}\n\nIn a more conventional circuit layout for Shor's period finding, the $\\approx N^2$ modular exponentiations $|a^k \\; \\mod N \\;\\rangle , k \\in [1..N^2]$, are done in superposition over $k$ and the width of\nsuch superposition trivially depends on the integer representation radix. Thus the purely ternary encoding has the width advantage with a factor of $\\log_3(2)$.\n\nHowever, on a small quantum computer platform a more practical approach is to use a single control (cf. \\cite{ParkerPlenio} or \\cite{Beauregard}, Section 2.4), which allows to iterate through\nthe modular exponentiations using only one additional qubit (resp. qutrit).\n\nWith this method in mind our principal focus is on the \\emph{overall cost of modular exponentiation}.\n\n\\begin{comment}\nThis is however not what is important in the inter-platform comparison; hence for simplicity we will be assuming that register has the fixed size of $2\\,n$ bits. Under certain assumptions on measurements and\nQFT compilation,\n there is however an available ``one control qubit trick'' (cf. \\cite{ParkerPlenio} or \\cite{Beauregard}, Section 2.4), and that register can be reduced to one qubit (and, respectively, emulated with one\n qutrit). In either case, this component of the width is fixed and can be taken out of the comparison tables, which only need to contain the width comparisons for the modular exponentiation circuits.\n\\end{comment}\n\nWe assume that for bitsize $n$, the $\\varepsilon = 1\/\\log(n)$ is a sufficient end-to-end precision of the\nperiod-finding circuit.\nThen the atomic precision $\\delta$ per individual gate, or rather\nper individual clean magic state within the circuit depends on circuit size. The circuits under comparison differ asymptotically in depth but not in size, which is in $O(n^3)$ (disregarding the slower $O(\\log(n))$ terms).\nWe observe that $\\log(1\/\\delta)$\nis roughly $3\\,\\log(n)$ for the required $\\delta$. It follows that the\ndistillation width\nfor one clean magic state scales like $(3\\,\\log_2(n))^3$ in the ternary context. In case on magic\nstate preparation in the metaplectic basis one needs at most $6 \\times 3\\, \\log_3(n)= 18 \\, \\log_3(n)$ $R$-gate per a clean $P_9$ magic state.\n There has been a wide array of magic state distillation protocols for the Clifford+T benchmark. For practical reasons and for simplicity we have selected the Bravyi-Kitaev protocol (\\cite{BravyiKitaev}) where\n the raw magic state consumption scales like $O(\\log(1\/\\mbox{precision})^{\\log_3(15)})$; $\\log_3(15) \\approx 2.465$. This scaling is shown in the ``preparetion width'' cells in the resource tables below.\n An attractive alternative would be the Bravyi-Haah protocol ( \\cite{BravyiHaah}).  The protocol is defined by the hyperparameter $k$ of the underlying $[n,k,d]$ error correction code and requires preparation\n width in $O((\\log_2(n))^{\\gamma(k)})$ where  $\\gamma(k) \\approx \\log_2((3\\,k+8)\/k)$. In particular for $k=8$ the protocol distills $8$ magic states simultaneously and $\\gamma(k)\\approx 2$. Unfortunately this\n protocol is more sensitive to the fidelity of the raw magic states and this is one of the reasons we decided not to cost it out at this time. One needs to be mindful that the scaling exponent $\\gamma(k)$ can\n in principle be made smaller than $2$ under certain circumstances.\n\nTables \\ref{table:ripple:carry} and \\ref{table:carry:lookahead} contain comparative resource estimates for the modular exponentiation circuits based, respectively, on the ripple-carry additive shift and the carry lookahead\nadditive shift.\nFor simplicity, only \\emph{asymptotically dominating} terms are represented. An actual resource bound may differ by\nterms of lower order w.r.t. $\\log(n)$. \n \n In addition to resource counts for ternary processing we have provided the same for Clifford+T solutions as a backdrop. In the Clifford+T basis,  resource estimate in Table \\ref{table:ripple:carry} for low-width modular exponentiation on a binary quantum computer is based on \\cite{HaenerEtAl} in which an implementation was given that uses $2n+2$ logical qubits. The Toffoli depth of the circuit in \\cite{HaenerEtAl} can be analyzed to be bounded by $160 n^3$. Note that the Toffoli depth is equal to $T$-depth, provided that $4$ additional ancillas are available, leading to an overall circuit width of $2n+6$. The two resource estimates in Table \\ref{table:carry:lookahead} for reduced-depth modular exponentiation are based on \\cite{Kutin2006} and \\cite{DraperSvore}: in \\cite{Kutin2006} an implementation for an arbitrary coupling architecture was given that uses $3n+6 \\log_2(n) + O(1)$ qubits and has a total depth of $12 n^2 + 60n \\log^2_2(n) + O(n \\log(n))$. This implementation is based on a gate set that has arbitrary rotations. To break this further into Clifford$+T$ operations, we require an increase in terms of depth of $4 \\log_2(1\/\\varepsilon) = 12 \\log_2(n)$ as each rotation has to be approximated with accuracy $\\varepsilon \\approx 1\/n^3$. Up to leading order, this leads to the estimate of the circuit depth of $144 n^2 \\log_2(n)$ given in the table. In \\cite{DraperSvore} a Toffoli based circuit to implement an adder in depth $4 \\log_2{n}$ was given that needs $4n-\\omega_1(n)$ qubits, where $\\omega_1$ denotes the Hamming weight of the integer $n$. As there are $O(n)$ Toffoli in parallel in this circuit, we use the implementation of a Toffoli gate in $T$-depth $3$ from \\cite{AmyEtAl} to implement a single addition in $T$-depth $12 \\log_2(n)$. The modular addition can be implemented then using $3$ integer additions. To implemented Shor's algorithm, we need $2n^2$ modular additions, leading to an overall $T$-depth estimate of $72n^2 \\log_2(n)$.\n\n\nThe rightmost column of either table lists counts proportional to either the number of raw magic states or, in the case of MTQC to the number of metaplectic magic states required per a time step\nof the circuit.\n\n\\begin{comment}\nUNTODO: THIS IS DEPRECATED by Krysta\nThe inter-platform comparisons availed by the tables are quite informative as far as the size asymptotics is concerned. The comparison of ``practical'' estimates between the platform should perhaps be takes\nwith grain of salt as the low level cost factors are certainly dependant on specific physical devices. One might assume that the fidelity and cost of raw resource states for the $T$ gate on one side and $P_9$\ngate on the other side depend on the physics of particular qubits\/qutrits being used and that the fidelity and cost of the metaplectic magic states depend on a specific engineering realization of metaplectic\nanyons. Discussing the physics and engineering of the actual devices is beyond the scope and mission of this paper.\n\\end{comment}\n\n\\begin{table*}\n    \\begin{tabular}{l@{\\qquad}l@{\\qquad}l@{\\qquad}l} \\hline\\hline\\\\[-1.5ex]\n\\multicolumn{1}{c}{\\hspace*{-2cm} Platforms}\n           &  \\multicolumn{1}{l}{Circuit width} &  \\multicolumn{1}{l}{Circuit depth ($P_9$\/$R_{|2\\rangle}\/T$)} & \\multicolumn{1}{l}{Preparation width}\\\\[0.5ex]\n    \\hline\\\\[-1.5ex]\n    Emulated binary, metaplectic, via $P_9$& $n+4$ & $48 \\, n^3$ & $54 \\times \\log_3(n)$  \\\\[1ex]\n    Section \\ref{subsec:ripple:carry}, emulated binary, via $P_9$& $n+4$ & $48 \\, n^3$ & $3  (3 \\, \\log_2(n))^3$  \\\\[1ex]\n    Ternary, metaplectic, via $P_9$& $2\\,m - \\omega_1(m)$  & $\\approx 76.35 \\, n^3$ & $54\\times \\log_3(n)$ \\\\[1ex]\n    Section \\ref{subsec:ripple:carry}, ternary, via $P_9$& $2\\,m - \\omega_1(m)$ & $\\approx 76.35 \\, n^3$ & $3  (3 \\, \\log_2(n))^3$  \\\\[1ex]\n    Emulated binary, MTQC inline& $n+4$  & $432\\, n^3\\, \\log_3(n)$ & $3$ \\\\[1ex]\n    Ternary, MTQC inline& $2\\,m - \\omega_1(m)$  & $\\approx 506.3\\, n^3 \\log_3(n)$ & $3$ \\\\[1ex]\n     \\hline\\\\[-1.5ex]\n    Haener et al. \\cite{HaenerEtAl}, Takahashi \\cite{TakaKuni}                                  & $2\\, n +6$ (qubits)    &         $160\\,n^3$           &\n    \\footnote{Here $\\log_2(3) < \\gamma \\leq \\log_3(15)$ depending on practically applicable distillation protocol. $n \\times$ reflects the worst case bound on the logical width of the circuit. } $\\sim n \\times (6 \\log_2(n))^{\\gamma}$ \\\\[1ex]\n\\hline\\hline\n    \\end{tabular}\n    \\caption{Size comparison for low-widths modular exponentiation circuits. $n$ is the bitsize, $m = \\lceil \\log_3(2) \\, n \\rceil$, $\\omega_1(.)$ is the number of $1$s in corresponding ternary or binary\n    expansion. } \\label{table:ripple:carry}\n  \\end{table*}\n\n\n\\begin{table*}\n    \\begin{tabular}{l@{\\qquad}l@{\\qquad}l@{\\qquad}l} \\hline\\hline\\\\[-1.5ex]\n\\multicolumn{1}{c}{\\hspace*{-2cm} Circuits}\n           &  \\multicolumn{1}{l}{Circuit width} & \\multicolumn{1}{l}{Circuit depth ($P_9$\/$R_{|2\\rangle}\/T$)} & \\multicolumn{1}{l}{p width}\\\\[0.5ex]\n    \\hline\\\\[-1.5ex]\n    Emulated binary, metaplectic, via $P_9$& $4\\,n -\\omega_1(n)$ & $120\\, n^2 \\, \\log_2(n)$ & $54\\times n\\,\\log_3(n)$ \\\\[1ex]\n    Section \\ref{subsec:carry:lookahead}, emulated binary, via $P_9$& $4\\,n -\\omega_1(n)$& $120\\,n^2 \\,\\log_2(n)$  & $12\\,n\\,  (3 \\, \\log_2(n))^3$  \\\\[1ex]\n    Ternary, metaplectic, via $P_9$& $4\\,m -\\omega_1(m)$ & $\\approx 127.4 \\, n^2\\, \\log_2(n)$ & $54\\times m\\,  \\log_3(m)$ \\\\[1ex]\n    Section \\ref{subsec:carry:lookahead}, ternary, via $P_9$& $4\\,m -\\omega_1(m)$& $\\approx 127.4 \\, n^2\\, \\,\\log_2(n)$  & $12\\,n\\,  (3 \\, \\log_2(n))^3$  \\\\[1ex]\n\n    Emulated binary, MTQC inline& $3\\,n -\\omega_1(n)$ & $384\\,n^2 \\,\\log_3(2) (\\log_2(n))^2$ & $3\\, n$ \\\\[1ex]\n    Ternary, MTQC inline& $3\\,m -\\omega_1(m)$  & $\\approx 1630.5\\,n^2 \\, \\log_3(2) (\\log_2(n))^2$  & $3\\,m$ \\\\[1ex]\n     \\hline\\\\[-1.5ex]\n    Binary, via Clifford+T, \\cite{DraperSvore} & $4\\,n -\\omega_1(n)$ (qubits)    &      $72\\, n^2\\, \\log_2(n)$             &\n    \\footnote{Here $\\log_2(3) < \\gamma \\leq \\log_3(15)$ depending on practically applicable distillation protocol.} $3\\,n\\,(6 \\log_2(n))^{\\gamma}$ \\\\[1ex]\n   \n    Binary, via Clifford+T, \\cite{Kutin2006} & $3\\,n +6\\,\\log_2(n)$ (qubits)    &      $144\\, n^2 \\log_2(n)$ & $3\\,n\\,(6 \\log_2(n))^{\\gamma}$ \\\\[1ex]\n\\hline\\hline\n    \\end{tabular}\n    \\caption{Sizes for reduced-depth modular exponentiation circuits. $n$ is the bitsize, $m = \\lceil \\log_3(2) \\, n \\rceil$, $\\omega_1(.)$ is the number of $1$s in corresponding ternary or binary expansion.}\n    \\label{table:carry:lookahead}\n  \\end{table*}\n\n\n\nThe logarithmic\nexecution\ndepth for integer addition\nis achieved by using\ncarry lookahead additive shift circuit.\nHowever this comes at significant width cost, as the circuit performs in parallel up to $n$ (in the worst case) or roughly $n\/\\log_2(n)$ (on average) reflection gates.\nThis requires a corresponding number of magic states or metaplectic registers simultaneously, and therefore the preparation width numbers in the last column of Table \\ref{table:carry:lookahead}\nare multiplied by the corresponding bit size, or, respectively, trit size. This represents the critical path bound on the magic state preparation width of the solution.\n\nIn both tables the\npreparation width bound for ternary processing is dominated by the width of the $C_f(\\mbox{INC})$. The Tables do not exhaust the vast array of possible depth\/width\ntradeoffs. We have chosen to represent $C_f(\\mbox{INC})$ with non-Clifford depth one as shown in Fig. \\ref{fig:C1Z:depth:one}. This circuit has the\n$P_9$-width of ${3}$ and requires a clean ancillary qutrit. For the ripple-carry solution the ancillary qutrit is reused and has minimal impact. For the carry lookahead\nsolution up to $n$ (respectively, up to $m=\\lceil \\log_3(2) \\, n$) ancillas must be available in parallel which inflates the online width by more than 30 percent.\n\nThe fifth and sixth rows show tradeoff based on direct approximation of Toffoli gates and (controlled) $C_f(\\mbox{INC})$ gates, respectively, by topological metaplectic\ncircuits to precision $\\Theta(1\/n^3)$. The topological metaplectic $R_{|2\\rangle}$ gates are executed sequentially for each individual arithmetic gate. This nearly eliminates the need for the magic state\npreparation, as only $3$ topological ancillas are needed at a time in the injection circuit for the $R_{|2\\rangle}$ (Figure \\ref{fig:R2:RUS}). This tradeoff introduces the online depth of a\nsubcircuit for a Toffoli gate of roughly $24 \\, \\log_3(n)$. A corresponding subcircuit for a $C_f(\\mbox{INC})$ then has online depth of $48 \\, \\log_3(m)$ (two two-level reflections). For $C_f(\\Lambda\n\\Lambda(\\mbox{INC}))$, it is $192 \\, \\log_3(m)$ (eight two-level reflections).\n\nWe estimate the number of required controlled integer additive shifts in a modular exponentiation circuit as $6\\,n^2$ ($2\\,n^2$ controlled modular additions) when binary emulation is used and as $16\\,m^2$\n($4\\,m^2$ controlled modular additions) when ternary encoding is used. These bounds define the\nexecution depth columns in both tables.\n\nThe most significant distinction\nin the Tables \\ref{table:ripple:carry}, \\ref{table:carry:lookahead} is the asymptotical advantage in the\nmagic state\npreparation width with the MTQC.\n\nThere is also a tradeoff between emulated binary encoding and true ternary encoding on a ternary quantum processor. It is seen from Table\n\\ref{table:ripple:carry} that with ripple-carry adders (e.g., when targeting a small quantum computer) we get a moderate practical advantage in non-Clifford circuit depth when\nemulating binary encoding and a small advantage in width compared to the use of true ternary encoding. This is true even accounting for the fact that the trit size $m$ is smaller than the bit size $n$ by the\nfactor of $\\log_3(2)$.\n\nOn the other hand when carry lookahead adders are used, the difference in the overall non-Clifford circuit depth between the two encoding scenarios is insignificant, unless inline metaplectic circuits with MTQC are compiled.\nBut the use of true ternary encoding yields the width advantage by a factor of roughly $\\log_3(2)$.\nIn the fifth and sixth lines of Table \\ref{table:carry:lookahead}, the use of emulated binary encoding is practically better than the use of ternary encoding.\nIntuitively, this is because the metaplectic circuits are reflection-oriented and best suited for direct approximation of the (controlled) Toffoli gates that are two-level reflections, whereas\nternary arithmetic gates such as $C_f(\\mbox{INC})$ or Horner have to be first decomposed into several two-level reflections.\n\n\nThe resource bounds shown in the tables provide a great deal of flexibility in selecting a resource balance\nappropriate for a specific ternary quantum computer. On a generic ternary quantum computer where\nuniversality is achieved by distillation of magic states for the $P_9$ gate the choice of encoding and arithmetic\ncircuits is likely to be dictated by the size of the actual computer. When native metaplectic\ntopological resources are available, magic states for the $P_9$ gate are prepared asymptotically more efficiently.\nMetaplectic also offers the third choice: that of bypass the $P_9$ gate\naltogether and using inline metaplectic circuits instead at the cost of a factor in $O(log(\\mbox{bitsize}))$\nin circuit depth expansion. In this scenario using emulated binary encoding of integers is always more efficient in\npractice than the use of true ternary encoding.\n\n\n\n\\section{Conclusion}\n\nWe have investigated implementations of Shor's period finding function  (\\cite{Shor}) in quantum ternary logic.\nWe performed comparative resource cost analysis targeting two prospective quantum ternary platforms.\nThe ``generic'' platform uses magic state distillation as described in \\cite{CampbellEtAl}\nfor universality. The other, one referred to as MTQC (metaplectic topological quantum computer), is a non-Abelian anyonic platform, where universality is achieved by a relatively inexpensive protocol based on\nanyonic braiding and interferomic measurement \\cite{CuiHongWang},\\cite{CuiWang}.\n\nOn each of these platforms we considered two different logical solutions for the modular exponential circuit of Shor's period finding function: one where the integers are encoded using the binary subspace\nof the ternary state space and ternary optimizations of known binary arithmetic circuits are employed;\nthe other ternary encoding of integers and arithmetic circuits\nstemming from \\cite{BCRS} are used.\n\nOn the MTQC platform we additionally consider semi-classical metaplectic expansions of arithmetic circuits; the non-Clifford depth of such a circuit is larger than the non-Clifford depth of the\ncorresponding classical arithmetic circuit by a factor of $O(\\log(\\mbox{bitsize}))$.\nNotably, circuits of this type bypass the need for magic states and the $P_9$ gate entirely.\n\nWe have derived both asymptotic and practical bounds on the quantum resources consumed by the Shor's period finding function for practically interesting combinations of platform, integer encoding and modular\nexponentiation. For evaluation purposes we have derived such bounds for widths and non-Clifford depths of the logical\ncircuits as well as for sizes of the state preparation resources that either\ndistill or prepare necessary magic states with the required fidelity.\n\n\nWe find significant asymptotic and practical advantages of the MTQC platform compared to other platforms. In particular this platform\nallows factorization of an $n$-bit number using the smallest possible number of $n+7$ logical qutrits at the cost of inflating the depth of the logical circuit by a logarithmic factor. In scenarios where\nincreasing the depth is undesirable, the MTQC platform still exhibits significant advantage in the size of the\n magic state preparation component that is linear in the bitsize of the\ntarget fidelity (compared to cubic or near-cubic for a generic magic state distillation).\n\n An interesting feature of our ternary arithmetic circuits is the fact that the denser and more compact ternary encoding\n of integers does not necessarily lead to more resource-efficient period finding solutions compared to binary encoding.\n As a rule of a thumb: if low-width circuits are desired, then binary encoding of integers combined with ternary arithmetic\n gates appears more efficient both in\n terms of width and depth than a pure ternary solution. However, even a moderate ancilla-assisted depth compression,\n such as provided by carry lookahead additive shifts, tips the balance in favor of ternary\n encoding and ternary arithmetic gates.\n\n In summary, having a variety of encoding and logic options, provides flexibility when choosing period finding solutions for ternary quantum computers of varying sizes.\n\n\n\n\\acknowledgements\nThe Authors are grateful to Jeongwan Haah for useful references. We would also like to thank Tom Draper and Sandy Kutin for providing $\\langle q | pic \\rangle$ \\cite{DraperKutin} which we used for typesetting\nmost of the figures in this paper. We are thankful to an anonymous Reviewer for insightful comments that inspired us to rewrite the paper into its current, more comprehensive format.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe calibration of the electric drift field in large volume liquid argon time projection chambers (LAr TPC) is required to account and correct for field disuniformities that can be mainly caused by two different mechanisms. First, the sensitive volume of a detector located underground at a shallow depth or at ground surface is subject to permanent ionization by cosmic rays, leading to the accumulation of positive argon ions in the detector drift region. These ions can produce a substantial electrostatic field that distorts the initially uniform drift field, hence affecting the reconstruction of particle tracks. The deviation of the reconstructed ionization track coordinates from the true ones can be as large as $\\approx$10~cm after a 2.5~m drift distance\\footnote{LAr TPC for MicroBooNE experiment, www-microboone.fnal.gov}. A second cause of field disuniformities originates from an initial non-uniform potential distribution over the field-shaping rings in the drift region. \n\nDepending on the aspect ratio (drift length to field cage width) of the detector, one of the two distortion causes can dominate over the other. Building an accurate three-dimensional map of the resulting electric field in the detector allows to efficiently compensate for these distortions and to reconstruct the true track geometry.\n\nThe map of the drift field can be derived by means of straight ionization tracks covering the entire sensitive detector volume, $e.g.$\\ induced by high energy cosmic ray muons. However, to achieve the required granularity of the field map, a sufficiently large statistics is needed. Furthermore, the larger is the detector, the smaller is the high energy fraction of the muon spectrum that can be used for this kind of analysis. The limitation originates from the significant Coulomb multiple scattering of muons in the relatively dense liquid argon medium. A method to produce a large number of straight ionization tracks in a controlled way which are neither subject to Coulomb multiple scattering nor to $\\delta$-ray emission would significantly simplify the task of obtaining such a drift field map in large volume LAr TPCs.\n\nThe ARGONTUBE detector is a liquid argon time projection chamber that allowed to achieve for the first time a 5~m long drift distance for ionization electrons in liquid argon~\\cite{ARGONTUBE0,ARGONTUBE1,ARGONTUBE2}. Given the high aspect ratio of about 25 for the detector sensitive volume (narrow field cage), the mirror charge induced on the surface of the field-shaping rings leads to a relatively fast removal of the positive ions. This  brings the distortion due to ion space charge to a negligibly low level.\n\nTherefore, the dominating field distortion in ARGONTUBE originates from the non-uniform potential distribution on the field-shaping rings created by the undercharged high voltage generator. In order to demonstrate the feasibility of calibrating the drift field by means of straight ionization tracks, the detector is equipped with a high-power pulsed UV laser beam. The technique is based on the process of multi-photon ionization of argon atoms in liquid by a narrow ($\\approx$ 2 mm diameter) beam of UV radiation with a wavelength of 266~nm~\\cite{Badhrees:2010zz}.\n\n\n\\section{Drift field in ARGONTUBE}\nFor a detailed description of the ARGONTUBE detector and of its subsystems and, in particular, laser we refer to previous publications~\\cite{ARGONTUBE0,ARGONTUBE1,ARGONTUBE2}. Electrostatic simulations of the TPC field cage indicated a high uniformity of the electric drift field within the sensitive volume for the chosen field cage geometry~\\cite{ARGONTUBE2}. However, during detector runs it became evident that the drift field was not as uniform as expected from simulations. The ionization tracks produced by cosmic ray muons had apparent strong curvature at the readout plane.\nThe curvature was not only much stronger than expected from Coulomb multiple scattering, but also systematic. A similar effect was observed for straight laser-induced ionization tracks which are free of Coulomb multiple scattering (Figure~\\ref{fig:lastrack}). Such a behavior can be attributed to substantial longitudinal and transverse drift field distortions. \n\n\\begin{figure}[ht]\n\\centering\t\n\\includegraphics[width=1\\linewidth]{laserComb_NEW.pdf}\n\\caption{A typical laser-induced ionization track in ARGONTUBE entering the detector from the top (left in the figure) and crossing the entire sensitive volume longitudinally. The red rectangle marks the electron cloud released from the detector cathode by photoelectric effect at about 5 mm drift distance.}\n\\label{fig:lastrack}\n\\end{figure}\n\nThe high voltage needed to set up the electric drift field in ARGONTUBE is directly generated inside the cryostat by means of a Greinacher voltage multiplier \\cite{Greinacher}. It is a circuit consisting of capacitors and diodes, which is driven by an alternate current voltage source (Figure~\\ref{fig:scheme}). 119 multiplier stages are installed in ARGONTUBE to reach the required high voltage. The longitudinal component (w.r.t TPC drift direction) of the observed field distortions can be explained by the Greinacher multiplier being in an only partially charged state. Reaching fully charged state would take an infinite time. In the case of ARGONTUBE, the charging is stopped when the monitored input AC current drops down to noise level. The transverse parasitic component of the drift field, on the other hand, is mostly due to the fact that the Greinacher circuit is installed on the inner surface of the ARGONTUBE field cage (Figure~\\ref{fig:grein}). \n\n\\begin{figure}[ht]\n\\centering\t\n\\includegraphics[width=0.85\\linewidth]{schemGrein_NEW.pdf}\n\\caption{Electric scheme showing the first two stages of the Greinacher circuit installed in ARGONTUBE. It consists of an alternate current voltage source (VAC), capacitors (C), diodes (D) and current-limiting resistors (R).}\n\\label{fig:scheme}\n\\end{figure}\n\n\\begin{figure}[ht]\n\\centering\t\n\\includegraphics[width=0.6\\linewidth]{GreinCage.png}\n\\caption{The ARGONTUBE field cage seen from the top. The Greinacher circuit is mounted on the inner side of the field cage.}\n\\label{fig:grein}\n\\end{figure}\n\n\\subsection{Model of the Greinacher voltage generator}\nIn the first approximation, the behavior of the potentials at the Greinacher circuit stages can be studied analytically. Let $N$ be the total number of stages of the multiplier and $U(n,\\,t)$ the output voltage at stage $n\\in[1,N]$ after a charging time $t$, and be $U(n,\\,t\\!=\\!0) = 0\\,\\,\\,\\forall n$. Considering a single-stage circuit, $i.e.$\\ $N=1$, the output voltage as a function of $t$ is given by \n\\begin{equation}\n  U(n\\!=\\!1,\\,t) = U_\\infty \\left[1 - e^{-t\/\\tau_1}\\right], \\label{eq:SingleStage}\n\\end{equation} \nwhere $U_\\infty$ denotes asymptotic value of the output voltage at the stage after an infinite charging time, and is referred to as the nominal voltage. $\\tau_1$ is defined by  $R$ and $C$, the resistance and the capacitance of the single-stage circuit, .  \nFor a Greinacher multiplier with an arbitrary number $N$ of identical stages and a characteristic time constant $\\tau$, the output voltage at stage $n$ is determined by replacing $\\tau_1$ in Equation~\\ref{eq:SingleStage} by $\\tau_n$, to take into account all the resistive and capacitive components from stage $1$ to stage $n$. Since the individual stages are assumed to be identical, $\\tau_n$ can be expressed as a fraction of $\\tau$. Precise determination of  $\\tau$ is not required for the presented analysis. The parameter, that defines how close is the circuit to its asymptotic state after charging time $t$ is defined by $t\/\\tau$ ratio.\n\\begin{equation}\n \\tau_n =  \\frac{n}{N} \\; \\tau.\n\\end{equation}\nThe potential distribution is then defined by\n\\begin{equation}\n  U(n, \\,t) = U_\\infty^n \\; \\left[1 - \\exp\\left(-\\frac{N}{n \\tau}\\;t\\right)\\right], \\hspace{1cm} \\text{where} \\hspace{1cm} U_\\infty^n = \\frac{n}{N} \\; U_\\infty \\label{eq:nStage}\n\\end{equation}\ndenotes the nominal voltage at stage $n$.\n\nIn ARGONTUBE one stage is added to the multiplier cascade for every additional field cage ring electrode. To describe the potential in a continuous way, we linearly interpolate the electric potential between each two consecutive rings along the drift spatial coordinate $z$ and replace  in Equation~\\ref{eq:nStage} the integer numbers $n$ and $N$ by $z$ and $z_c$, respectively. \nThe quantity $z_c$ corresponds to the location of the TPC cathode and $z=0$ defines the position of the charge readout plane. The function of the electric potential given in Equation~\\ref{eq:nStage} is reformulated to\n\\begin{equation}\n  U(z, \\,t) = U_\\infty \\; \\frac{z}{z_c} \\; \\left[1 - \\exp\\left(-\\frac{z_c}{z \\tau}\\;t\\right)\\right], \\label{eq:GreinModel}\n\\end{equation}\nwhere $U_\\infty$ can be interpreted as the cathode nominal voltage. Since the readout plane is connected to ground, $U(z\\!=\\!0, \\,t) = 0\\,\\,\\,\\forall t$. \nThe longitudinal electric field $E_L$ is given by\n\\begin{equation}\n  E_L(z, \\,t) = \\frac{\\partial U(z,\\,t)}{\\partial z} = \\frac{U_\\infty}{z_c} \\left[1 - \\exp\\left(-\\frac{z_c}{z \\tau}\\;t\\right) - \\frac{z_c t}{z \\tau}\\,\\, \\exp\\left(-\\frac{z_c}{z \\tau}\\;t\\right)\\right]. \\label{eq:GreinModelE}\n\\end{equation}\n\nIf $t\\!\\rightarrow\\!\\infty$, the exponential terms in Equations~\\ref{eq:GreinModel} and~\\ref{eq:GreinModelE} vanish and the potential follows a linear behavior $U(z, \\,t\\!\\rightarrow\\!\\infty) = U_\\infty\\cdot z\/z_c$ with respect to $z$, corresponding to a constant and uniform drift field $E_L(z, \\,t\\!\\rightarrow\\!\\infty) = U_\\infty\/z_c$ along $z$. This is the desired situation for operating a LAr TPC. It is clarified by the two graphs shown in Figure~\\ref{fig:GreinacherModel}. The dark blue curves correspond to a fully charged voltage multiplier and illustrate the case with $t\\!\\rightarrow\\!\\infty$ both for $U$ (top) and for $E_L$ (bottom). The parameter $t\/\\tau$ that appears in the exponential terms of Equations~\\ref{eq:GreinModel} and~\\ref{eq:GreinModelE} is the quantity describing the state to which the circuit is charged. Apart from the situation of a fully charged high voltage multiplier, given in dark blue, five intermediate charging states, reaching from $t\/\\tau = 0.1$ (red) to $t\/\\tau = 2.0$ (light blue), are shown in Figure~\\ref{fig:GreinacherModel}. They clarify how far the longitudinal electric field is from being uniform in case the circuit is not fully charged.\n\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=0.99\\textwidth]{IllustrCombined_NEW_2.pdf}\n  \\caption[Illustrations of the 119 stages Greinacher model.]{Illustrations of the voltage (top) and the longitudinal electric field (bottom, here denoted by $E$) obtained with the Greinacher multiplier for different charging states $t\/\\tau$ as a function of the drift coordinate $z$. The graphs result from Equations~\\ref{eq:GreinModel} and~\\ref{eq:GreinModelE}. The dark blue curves correspond to a fully charged Greinacher circuit.}\n   \\label{fig:GreinacherModel}\n\\end{figure}\n\n\n\n\\subsection{Drift field calibration}\nFrom straight ionization tracks produced by UV laser beams one can extract information about the strength and direction of the electric field in the drift volume of the TPC. These tracks can be used to decouple the longitudinal from the transverse electric field components and to estimate the strength of the parasitic transverse field in ARGONTUBE. \n\nGiven the longitudinal electric field, the drift time $t_d(z)$ needed for a test charge released at the drift coordinate $z$ to reach the readout plane ($z=0$), is obtained by integrating along the drift direction\n\\begin{equation}\n  t_d(z) = \\int_0^{z} \\frac{1}{v_L(E_L(z'))} \\,\\,dz', \\label{eq:intTd}\n\\end{equation}\nwhere $v_L\\left(E_L\\left(z'\\right)\\right)$ denotes the longitudinal drift velocity in the interval $[z', z'+dz']$.\n\n\nEquation~\\ref{eq:intTd} combined with experimentally determined dependence of the drift velocity $v_L$ on the electric field strength~\\cite{wal99} defines the drift time of the electrons produced at any point in the drift cage for a given longitudinal field distribution. The total drift time can be measured for electrons released from the cathode by photoelectric effect (see the charge cloud at the track end in Figure~\\ref{fig:lastrack}). For derivation of the longitudinal field this is the only experimental constraint for the model. The $v_L(E)$ dependence from~\\cite{wal99} is a monotonically increasing function of the electric field. The dependence of the electric field on time  $E_L(z, \\,t)$ at any given $z$ is also a monotonically increasing function. Therefore, the integral \\ref{eq:intTd} is, as well, a monotonic function of $t\/\\tau$. Hence, the constraint $t_{d}(z_c) = t_c$, where $t_c$ is the measured total drift time across the cathode-anode distance $z_c$, gives unambiguous solution for the only free parameter $t\/\\tau$.\nThis solution can be found numerically by successive binary division approximation for a given charging state of the Greinacher circuit. Since the measurement error of $t_c$ is negligibly small, the main contribution to the error on $t\/\\tau$  is given by uncertainty of $v_L(E)$  parametrization from~\\cite{wal99}. The resulting error on the drift field $E_L(z)$ can be derived from \\ref{eq:GreinModelE} by Gaussian error propagation.\n\nWith a typical cathode nominal voltage $U_\\infty = 130\\,$kV, a total drift length of $z_c = 4.76\\,$m, and the measured total drift time $t_{d}(z_c) = 6.616\\,$ms (compare Figure~\\ref{fig:lastrack}), this value is found to be $t\/\\tau = 1.09 \\pm 0.01$. Considering the distributions in Figure~\\ref{fig:GreinacherModel}, it is clear that the longitudinal electric field is indeed highly non-uniform in ARGONTUBE.\n\nThe derived value for $t\/\\tau$ could, in principle, be compared to the expected one, derived from the parameters of the Greinacher circuit capacitors and resistors. Practically it is not done for two reasons. The first is the difficulty of taking into account all parasitic capacitances in the circuit. The second is related to the typical ARGONTUBE operation sequence. In order to reach required argon purity \na time period of the order of a day is required. Estimation of argon purity is based on observation of tracks in the detector, so having non-zero drift field is crucial. During this time Greinacher circuit is charged in consecutive steps to avoid electrical discharges. There is no tool in ARGONTUBE to fully discharge the partly charged Greinacher circuit. Therefore, the charging time $t$ for estimation of $t\/\\tau$ can not be accurately measured. \n\n\nOnce the longitudinal coordinate of the laser track shown in Figure~\\ref{fig:lastrack} is corrected using the inferred $E_L(z)$ distribution (red line in Figure~\\ref{fig:Elcorrected}), the residual track curvature can be entirely attributed to the transverse field distortion $E_T(z)$. To determine the latter, one performs a straight line fit (shown in blue in Figure~\\ref{fig:Elcorrected}) for $z \\in [0, 300]\\,$mm to the laser track corrected for $E_L(z)$ assuming that $E_T(z) = 0$ in the specified range of $z$. By analyzing the residual track deviation from the extrapolated laser track, one can determine the transverse drift velocity $v_{T}$ at each point along the laser-induced track and, therefore, derive the distribution of $E_T(z)$ in this specific region of the detector sensitive volume.\n\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{ElCorrLaserTrack_combined_cut_NEW_2.pdf}\n  \\caption[test]{The red line represents a polynomial fit to the laser track (green markers) shown in Figure~\\ref{fig:lastrack} corrected for longitudinal disuniformities of the drift field using the Greinacher model and measurements described in the text. The blue line shows the extrapolated laser track determined by fitting the green markers with a straight line for $z \\in [0, 300]\\,$mm assuming $E_T(z) = 0$ in the specified range of $z$. The plot features two different scales along the abscissa, drift time $t_d$ (top axis) and drift coordinate $z$ (bottom axis) which are related by the function $t_d(z)$.}\n  \\label{fig:Elcorrected}\n\\end{figure}\n\n\nThe final result of the analysis is shown in Figure \\ref{fig:ElEt}. The top plot represents the distribution of the longitudinal component (blue) and a projection of transverse component (red) of the drift field along the laser-induced track region. \nDeriving second projection in a similar way will result in full reconstruction of the $\\vec{E}$ vector along the track. This possibility can not be realized on ARGONTUBE data because of the detector induction-collection charge readout scheme ~\\cite{ARGONTUBE0,ARGONTUBE1,ARGONTUBE2}.\nWhile the amplified signal from the charge collection plane used for this analysis is proportional to the input current in a wide bandwidth (1\\%\/ms of droop in response to a rectangular current pulse), the induction signal is proportional to its derivative on time. Therefore, in the situation with long track at a small angle w.r.t. drift direction, the induction plane signal does not allow to reconstruct second projection of laser-induced track in detail. This issue does not exist for detectors with a drift field transverse to a dominant track direction.\nFull reconstruction of drift field vector would allow to cross-check the assumption of negligible contribution of the space charge by testing the condition $\\nabla\\cdot\\vec{E}=0$.\n\nWhile the longitudinal field component is valid for the whole sensitive volume of the detector, the transverse component is only correct along the laser path of the specific track and is found to vary within the detector volume, depending on the location w.r.t the Greinacher circuit components.\nSteering the laser beam across the sensitive volume allows to reconstruct $E_T$ for several beam locations and hence to build up a full 3D map of the drift field components distribution.\n\nThe drift field data obtained for several runs of ARGONTUBE were used as a basis for the calibration of the detector response to the ionization charge (using the field-dependent charge recombination factor) and for charge diffusion measurements. These analyses will be presented in forthcoming publications.\n\n\\begin{figure}[ht]\n  \\centering\n  \\includegraphics[width=1\\textwidth]{ElEt_Combined_NEW_2.pdf}\n  \\caption{Top: the two electric field components in the ARGONTUBE detector drift volume. Bottom: Ratio of the absolute transverse to the longitudinal field strengths along the drift direction.}\n  \\label{fig:ElEt}\n\\end{figure}\n\n\n\\section{Conclusions}\nThe analysis of the apparent curvature of the ionization tracks in the ARGONTUBE LAr TPC produced with a narrow UV laser beam with $\\lambda$=266~nm in combination with an analytic model of the Greinacher voltage multiplier circuit allows to decouple the longitudinal and the transverse components of the drift field. The distribution of the two components along the track volume can be effectively derived. This distribution sets the basis for further analyses of the detector response to ionization and of excess electron diffusion along the $\\approx$5~m long drift in liquid argon.\n\n\n\\section{Acknowledgement}\nWe express our gratitude to engineering and technical staff in Bern for the design, manufacturing and assembly of the detector. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe  {\\it Cellular Automaton Interpretation} of Quantum Mechanics (QM) has recently \nbeen proposed by G.~'t\\,Hooft.\\,\\cite{tHooft2014} \nInterest in redesigning the foundations of quantum theory in accordance with essentially classical concepts -- above all, determinism and existence of ontological states \nof reality --  is founded on the observation that quantum \nmechanical features arise in a large variety of deterministic and, in some sense, ``classical'' \nmodels. {\\it E.g.}, the Born rule and collapse of quantum mechanical states  in measurement processes find a surprising and intuitive explanation here, \nwhere quantum states are superpositions of ontological \n(micro) states, while classical ones are ontological (macro) states, refering to vastly \ndifferent scales in nature.\\,\\cite{tHooft2014} \n\nWhile practically all of these models have been exceptional in that they cannot easily be \ngeneralized to cover real phenomena, incorporating interactions and relativity, Cellular Automata (CA) may provide the necessary versatility, as we shall presently continue to \ndiscuss.\\,\\cite{PRA2014,EmQM13} For an incomplete list of various earlier attempts in \nthis field, see, for example,  \nRefs.\\,\\cite{H1,H2,H3,Kleinert,Elze,Groessing,Khrennikov,Margolus,Jizba,Mairi,Isidro,DAriano} \nand  references therein. \n\nThe linearity of quantum mechanics (QM) is a fundamental feature  of unitary dynamics \nembodied in the Schr\\\"odinger equation. This linearity does not depend on the \nparticular object  under study, provided it is sufficiently isolated from \nanything else. It is naturally reflected in the superposition principle and   \nentails interference effects and the possibility of non-factorizable states \nof composite objects, {\\it i.e.} entanglement in multipartite systems.    \n\nThe linearity of QM has been questioned repeatedly and nonlinear modifications \nhave been proposed -- not only as suitable approximations for complicated \nmany-body dynamics, but especially in order to test experimentally the  \nrobustness of QM against such {\\it nonlinear deformations}. \nThis has been thoroughly discussed by T.F. Jordan who presented a `proof \nfrom within' quantum theory that the theory has to be linear, given the \nessential {\\it separability}   \nassumption ``... that the system we are considering can be described as\npart of a larger system without interaction with the rest of the larger \nsystem.''\\,\\cite{Jordan}\n\nRecently, we have considered a seemingly unrelated {\\it discrete} \ndynamical theory, {\\it i.e.}, which deviates drastically from quantum theory, \nat first sight. \nHowever, we have shown with the help of Sampling Theory \nthat the deterministic mechanics of the class of discrete {\\it Hamiltonian CA} can be \nmapped one-to-one to continuum models pertaining to nonrelativistic QM, however, \nmodified by the presence of a fundamental time scale.\\,\\cite{PRA2014,EmQM13} \n\nFor this construction of an intrinsically linear relation between CA \nand QM with a nonzero discreteness scale, \nthe consistency of the action principle underlying the discrete dynamics on one side \nand the required locality of the continuum description on the other are compatible only with the linearity of both theories.\\,\\cite{Discrete14}  \n  \nThe purpose of the present note, in particular, is to study composite objects formed \nfrom CA subsystems.  -- Clearly, QM is special in that it is characterized not only by interference \neffects, like any classical wave theory would be, but also by the tensor product structures applying for composite systems, which entail the possibility of entanglement. -- \nIt is not obvious that CA can form composites which conform with QM, in the limit of negligible discreteness scale. \nThis is due to the fact that the state space of Hamiltonian CA is not a complex projective \nspace and that the norm of the analogue of state vectors is not conserved by the dynamics; \ninstead there is a conserved two-time correlation function, as we shall see, which becomes \nthe familiar norm only in the continuum limit.   \n\nIn Section\\,2., we will briefly review earlier results concerning individual CA which will be useful in the following. One way of composing CA, which is compatible \nwith QM, will be shown in Section\\,3. Such outside perspective based on CA should eventually lead to additional insight in regard to interference and entanglement.  Concluding remarks are presented \nin Section\\,4.  \n\n\\section{From action to conservation laws for Hamiltonian CA}\nWe describe the dynamics of classical Hamiltonian CA with countably many degrees \nof freedom in terms of {\\it complex integer-valued} state variables \n$\\psi_n^\\alpha$, where \n$\\alpha\\in {\\mathbf N_0}$ denote different degrees of freedom and $n\\in {\\mathbf Z}$  \ndifferent states labelled by this discrete {\\it clock variable}.  \nVarious equivalent forms of the action for such CA can conveniently be chosen, as indicated  \nearlier.\\,\\cite{PRA2014} We will employ a particularly compact form \nhere, which will be useful in the following, when we  construct composite CA \nin analogy with multipartite QM systems.   \n   \nLet $\\hat H:=\\{  H^{\\alpha\\beta}\\}$ denote a self-adjoint complex integer-valued matrix \nthat will play the role of the Hamilton operator shortly. Furthermore, we introduce \nthe suggestive notation $\\dot O_n:=O_{n+1}-O_{n-1}$, for any quantity $O_n$ depending \non the clock variable $n$. Then, with an \nimplicit summation convention for Greek indices, \n$r^\\alpha s^\\alpha\\equiv\\sum_\\alpha r^\\alpha s^\\alpha$, we will often simplify the \nnotation further by suppressing them altogether, for example, writing \n$\\psi_n^{*\\alpha}H^{\\alpha\\beta}\\psi_n^\\beta\\equiv\\psi_n^*\\hat H\\psi_n$. \n\nIncorporating these conventions, an integer-valued CA action  \n${\\cal S}$ is defined by:  \n\\begin{equation}\\label{action} \n{\\cal S}[\\psi ,\\psi^*]\\; :=\\;\n\\sum_n\\big [\\frac{1}{2i}(\\psi_n^*\\dot\\psi_n-\\dot\\psi_n^*\\psi_n)+\\psi_n^*\\hat H\\psi_n\\big ] \n\\;\\equiv\\;\\psi^*\\hat{\\cal S}\\psi \n\\;\\;, \\end{equation} \nwith $\\psi_n^\\alpha$ and $\\psi_n^{*\\alpha}$ as independent variables; the \noperator $\\hat{\\cal S}$ will be a useful abbreviation, {\\it cf.} Section\\,3. \nFor the purpose of setting up a variational principle, we introduce    \n{\\it integer-valued} variations $\\delta f$ which are applied to a polynomial $g$ \nas follows: \n\\begin{equation}\\label{variation} \n\\delta_{f}g(f):=[g(f+\\delta f)-g(f-\\delta f)]\/2\\delta f \n\\;\\; , \\end{equation} \nand $\\delta_fg\\equiv 0$, if $\\delta f=0$. -- \nWe remark that variations  of terms that are \n{\\it constant, linear, or quadratic} in integer-valued variables  yield analogous results as  \nstandard infinitesimal variations of corresponding expressions in the continuum.  --  \nMaking use of these ingredients, we postulate the variational principle:      \n\\vskip 0.15cm \\noindent \n({\\it CA Action Principle}) \\hskip 0.15cm   \nThe discrete evolution of a CA is determined by stationarity of its \naction under arbitrary integer-valued variations of all \ndynamical variables, $\\delta {\\cal S}=0$.\\hfill $\\bullet$ \\vskip 0.15cm  \n\nIt is worth emphasizing several characteristics of this {\\it CA Action Principle}:  \n\\\\ \\noindent  \n{\\bf i)} While infinitesimal variations do not conform with integer valuedness, \nthere is {\\it a priori} no restriction of integer variations. Hence {\\it arbitrary}  \ninteger-valued variations must be admitted. \\\\ \\noindent \n{\\bf ii)} One could imagine contributions to the action (\\ref{action})  which are of \nhigher than second order in $\\psi_n$ or $\\psi_n^*$. \nHowever, in view of arbitrary  variations \n$\\delta \\psi_n^\\alpha$ and $\\delta \\psi_n^{*\\alpha}$, \nsuch additional contributions to the action \nmust be absent for consistency.  \nOtherwise the number of equations of motion \ngenerated by variation of the action, according to Eq.\\,(\\ref{variation}), \nwould exceed the number of variables. Yet a  \nlimited number of such remainder terms, which are nonzero only for some fixed values \nof $n$, could serve to encode  \nthe {\\it initial conditions} for the CA evolution.    \n\nWe have shown earlier that these features of the {\\it CA Action Principle} \nare essential in constructing a map between Hamiltonian CA and equivalent quantum mechanical continuum models.\\,\\cite{PRA2014} -- In addition, generalizations of the \nvariations defined in Eq.\\,(\\ref{variation}) have been considered, which allow  \nhigher than \nsecond order polynomial terms in the action. However, while leading to consistent \ndiscrete equations of motion, these equations are beset with undesirable \nnonlocal features in the corresponding continuum model \ndescription.\\,\\cite{Discrete14}   \n\n\\subsection{The equations of motion} \nIt is straightforward now to obtain the equations of motion determined by the \n {\\it CA Action Principle} for the action ${\\cal S}$ given by Eq.\\,(\\ref{action}) with \nthe definition of variations provided in Eq.\\,(\\ref{variation}). Namely, variations  \n$\\delta\\psi_n^*$ and $\\delta\\psi_n$, respectively, yield \ndiscrete analogues of the Schr\\\"odinger equation and its adjoint: \n\\begin{eqnarray}\\label{delpsistar} \n\\dot\\psi_n&=&\\frac{1}{i}\\hat H\\psi_n \n\\;\\;, \\\\ [1ex] \\label{delpsi} \n\\dot\\psi_n^*&=&-\\frac{1}{i}(\\hat H\\psi_n)^* \n\\;\\;, \\end{eqnarray} \nrecalling that $\\hat H=\\hat H^\\dagger$ and $\\dot\\psi_n =\\psi_{n+1}-\\psi_{n-1}$, {\\it etc.} \nNote that the action ${\\cal S}$ vanishes when \nevaluated for solutions of these equations. \n\nWe remark that by setting $\\psi_n^\\alpha =:x_n^\\alpha +ip_n^\\alpha$, with real \ninteger-valued variables $x_n^\\alpha$ and $p_n^\\alpha$, and suitably separating real and imaginary parts of Eqs.\\,(\\ref{delpsistar})--(\\ref{delpsi}), the equations \nassume a form that resembles Hamilton's equations for a network of coupled \ndiscrete classical oscillators:\\,\\cite{Heslot85,Skinner13}   \n\\begin{equation}\\label{xdotCA} \n\\dot x_n^\\alpha\\;=\\;h_S^{\\alpha\\beta}p_n^\\beta +h_A^{\\alpha\\beta}x_n^\\beta  \n\\;\\;,\\;  \n\\;\\;\\dot p_n^\\alpha \\;=\\;-h_S^{\\alpha\\beta}x_n^\\beta +h_A^{\\alpha\\beta}p_n^\\beta  \n\\;\\;, \\end{equation}\nwhere we split the self-adjoint matrix $\\hat H$ \ninto real integer-valued symmetric and antisymmetric parts, respectively, \n$H^{\\alpha\\beta}=:h_S^{\\alpha\\beta}+ih_A^{\\alpha\\beta}$. -- \nThe appearance of these equations has suggested the name \nHamiltonian CA.\\,\\cite{PRA2014} \n\n\\subsection{The conservation laws} \nThe time-reversal invariant equations of motion that we have obtained \ngive rise to conservation laws which are in {\\it one-to-one correspondence} with those of \nthe related Schr\\\"odinger equation in the continuum. It is straightforward to verify \nthe validity of the following theorem. \n\n\\vskip 0.15cm \\noindent \n({\\it Theorem\\,A}) \\hskip 0.15cm For any matrix $\\hat G$ that commutes with \n$\\hat H$, $[\\hat G,\\hat H]=0$, there \nis a {\\it discrete conservation law}: \n\\begin{equation}\\label{Gconserv} \n \\psi_n^{\\ast\\alpha}G^{\\alpha\\beta}\\dot\\psi_n^\\beta +\n\\dot\\psi_n^{\\ast\\alpha}G^{\\alpha\\beta}\\psi_n^\\beta =0 \n\\;\\;. \\end{equation}  \nFor self-adjoint $\\hat G$,  with complex integer elements, \nthis relation concerns real integer quantities.\\hfill $\\bullet$  \n\nBy rearranging Eq.\\,(\\ref{Gconserv}), we can read off the corresponding \nconserved quantity $q_{\\hat G}$ (using matrix notation, as before): \n\\begin{equation}\\label{qG} \nq_{\\hat G}:=\\psi_n^*\\hat G\\psi_{n-1}+\\psi_{n-1}^*\\hat G\\psi_n\n=\\psi_{n+1}^*\\hat G\\psi_n+\\psi_n^*\\hat G\\psi_{n+1}   \n\\;\\;, \\end{equation}  \n{\\it i.e.} a real integer-valued two-point correlation function which is invariant \nunder a shift $n\\rightarrow n+m$, $m\\in\\mathbf{Z}$. -- In particular, for \n$\\hat G:=\\hat 1$, the corresponding conservation law amounts to a constraint on \nthe state variables: \n\\begin{equation}\\label{normal}  \nq_{\\hat 1}=2\\mbox{Re}\\;\\psi_n^*\\psi_{n-1}\n=2\\mbox{Re}\\;\\psi_{n+1}^*\\psi_n=\\mbox{const}   \n\\;\\;, \\end{equation}  \nwhich we anticipate to play a similar role for discrete CA as the familiar normalization \nof state vectors in continuum QM.  \n  \nFor later convenience, we also define the following symmetrized quantity: \n\\begin{equation}\\label{Q} \n\\psi_n^*\\hat{\\cal Q}\\psi_n :=\\frac{1}{2}\\mbox{Re}\\;\\psi_n^*(\\psi_{n+1}+\\psi_{n-1})  \n\\equiv\\frac{1}{2}\\mbox{Re}\\;\\psi_n^{*\\alpha}(\\psi_{n+1}^\\alpha +\\psi_{n-1}^\\alpha )  \n\\;\\;, \\end{equation} \nwhich, by Eq.\\,(\\ref{normal}), is conserved as well.   \n\n\n\\subsection{The continuum representation}\nPreviously we have constructed a one-to-one invertible \nmap between the dynamics of discrete Hamiltonian CA and continuum QM in the \npresence of a fundamental time scale.\\,\\cite{PRA2014,EmQM13,Discrete14} \nSuch a finite discreteness scale $l$ implies that continuous time wave functions must be \n{\\it bandlimited}, {\\it i.e.}, their Fourier transforms have only finite support \nin frequency space, $\\omega\\in [-\\pi \/l,\\pi \/l]$. \nUnder these circumstances Sampling Theory  \ncan be applied, in order to reconstruct continuous time signals, \nwave functions $\\psi^\\alpha (t)$, from their representative discrete samples, the CA state variables $\\psi_n^\\alpha$, and {\\it vice versa}.\\,\\cite{Shannon,Jerri,Kempf}\n\nInstead of going through the argument,\\,\\cite{PRA2014,Discrete14} we  \ngive the simple mapping rules that result from the reconstruction formula \nprovided by {\\it Shannon's Theorem}:\\,\\cite{Shannon,Jerri}   \n\\begin{eqnarray}\\label{psit} \n\\psi_n^\\alpha &\\longmapsto &\\;\\psi^\\alpha (t)\n\\;\\;, \\\\ [1ex] \\label{npm1}\n\\psi_{n\\pm 1}^\\alpha &\\longmapsto &\n\\;\\exp\\big [\\mp l\\frac{\\mbox{d}}{\\mbox{d}t}\\big ]\\psi^\\alpha (t)\n=\\psi^\\alpha (t\\mp l) \n\\;\\;, \\\\ [1ex] \\label{samp} \n\\psi^\\alpha (nl)&\\longmapsto &\\;\\psi_n^\\alpha \n\\;\\;, \\end{eqnarray} \nkeeping in mind that the continuum wave function is bandlimited. \n\nWith the help of these results, one can map the CA equations of \nmotion, in particular Eqs.\\,(\\ref{delpsistar})--(\\ref{delpsi}) to the appropriate \ncontinuum version. Corresponding to Eqs.\\,(\\ref{Gconserv})--(\\ref{Q}), there exist \nanalogous conservation laws and conserved quantities, which can be \nfound by applying the mapping rules separately to all wave function factors that appear. \nFor example, we obtain from Eq.\\,(\\ref{Q}) the conserved quantity: \n\\begin{eqnarray}\\label{Qcont1}  \n\\mbox{const}=\\psi_n^*\\hat{\\cal Q}\\psi_n&\\longmapsto&\\; \n\\psi^*(t)\\hat{\\cal Q}\\psi (t)\n=\\mbox{Re}\\;\\psi^*(t)\\cosh\\big [l\\frac{\\mbox{d}}{\\mbox{d}t}\\big ]\\psi (t)\n\\\\ [1ex] \\label{Qcont2} \n&\\;&=\n\\psi^{*\\alpha}(t)\\psi^\\alpha (t)\n+\\frac{l^2}{2}\\mbox{Re}\\;\\psi^{*\\alpha}(t)\\frac{\\mbox{d}^2}{\\mbox{d}t^2}\\psi^\\alpha (t) \n+\\mbox{O}(l^4)   \n\\;\\;, \\end{eqnarray} \nwhich shows the $l$-dependent corrections to the continuum limit, which here \namounts to the usual conserved normalization \n$ \\psi^{*\\alpha}\\psi^\\alpha =\\mbox{const}$\\,. \nSimilarly, the Schr\\\"odinger equation and its finite-$l$ corrections are \nobtained.\\,\\cite{PRA2014}  \n\nThis completes our considerations of single Hamiltonian CA, which form the \nbasis for the study of multipartite systems.    \n\n\\section{Composing multipartite CA}  \nHere we address the important question how discrete CA would combine to form \ncomposite multipartite systems. In particular, two requirements appear naturally, when \ndiscussing possible constructions. \n\nRecalling the similarities with QM that we have found, so far, one may wonder whether not only the {\\it linearity} of the evolution law  but \nalso the {\\it tensor product structure} of composite wave functions finds its analogue  here. These are fundamental ingredients of the usual continuum theory, which are reflected in a spectacular manner in interference and entanglement, \nrespectively. Which should be recovered, at least, in the continuum limit \n($l\\rightarrow 0$) of the CA picture. -- Furthermore, when the discreteness scale $l$ is truly finite, the dynamics of composites of CA which do not interact among each other should lead to {\\it no spurious correlations} among them. Such a principle of ``no correlations without interactions'' is  respected more or less explicitly by all known physical theories.\\,\\cite{Jordan}   \n \nWe begin by pointing out obstacles which seem to prevent \nsatisfying the above requirements, when trying to form composites of Hamiltonian CA.  \n\nThe want-to-be \ndiscrete time derivative introduced before, $\\dot O_n:=O_{n+1}-O_{n-1}$, for any quantity $O_n$ depending on the clock variable $n$, which appears all over in the \nCA equations of motion and conservation laws, does not obey the product rule or  \n{\\it Leibniz's rule}: \n\\begin{equation}\\label{Leibniz} \n\\dot {[A_nB_n]}=\n\\dot A_n\\textstyle{\\frac{B_{n+1}+B_{n-1}}{2}}+\n\\textstyle{\\frac{A_{n+1}+A_{n-1}}{2}}\\dot B_n \n\\neq \\dot A_nB_n +A_n\\dot B_n \n\\;\\;. \\end{equation} \nSimilar observations can be expected for other definitions one might come up with. \nLet us ignore this for a moment and, \nby way analogy with the single-CA Eq.\\,(\\ref{delpsistar}),  \nlook at the following multi-CA equation of motion: \n\\begin{equation}\\label{PSIeq} \n\\dot\\Psi_n=\\frac{1}{i}\\hat H_0\\Psi_n \n\\;\\;, \\end{equation} \nwhere $\\hat H_0$  may describe a block-diagonal Hamiltonian in the absence of \ninteractions among the CA. Then, through Eq.\\,(\\ref{Leibniz}),  the \nexpected {\\it factorization} \nof Eq.\\,(\\ref{PSIeq}) is hindered on the left-hand side, since unphysical correlations  \nwill be produced among the components of a factorized wave function, such as   \n\\begin{equation}\\label{PSI} \n\\Psi_n^{\\alpha\\beta\\gamma\\cdots}=\n\\psi_n^\\alpha\\phi_n^\\beta\\kappa_n^\\gamma\\cdots \n\\;\\;, \\end{equation}  \nand, correspondingly, for a superposition of such factorized terms. \nThus, for a bipartite system we have:  \n$\\dot \\Psi_n^{\\alpha\\beta}=\n\\dot\\psi_n^\\alpha (\\phi_{n+1}^\\beta+\\phi_{n-1}^\\beta )\/2+ \n\\psi\\leftrightarrow\\phi \n\\neq\\dot\\psi_n^\\alpha\\phi_n^\\beta +\\psi_n^\\alpha\\dot\\phi_n^\\beta$. \n\nFurthermore, applying the mapping rules of Section\\,2.3, before taking the \nlimit $l\\rightarrow 0$, we find that the bilinear terms here do not \nconverge to the appropriate QM expression.  \nOf course, it should be   \n$\\partial_t(\\psi^\\alpha\\phi^\\beta )=(\\partial_t\\psi^\\alpha )\\phi^\\beta +\\psi^\\alpha \\partial_t\\phi^\\beta$, in order to allow   \nthe decoupling of two subsystems that do not interact.  \n\nHowever, this latter problem is a general one of nonlinear terms in the equations \nof motion of discrete CA, \nwhich we discussed before:\\,\\cite{Discrete14}  {\\it The linear map provided by \nShannon's Theorem does not commute with the multiplication implied by the \nnonlinearities.}  In particular, the map of a bilinear term is not equal to the bilinear \nterm of its mapped entries, symbolically: $$A_nB_n\\equiv C_n\\mapsto C(t)\\;\\neq\\; A(t)B(t)\\;\\;,$$ \nwhere $A_n\\mapsto A(t)$ and $B_n\\mapsto B(t)$, as follows from the \nexplicit reconstruction formula (or any variant thereof that is linear).\\,\\cite{PRA2014,Shannon,Jerri} \nIn fact, this problem arises also on the right-hand side of Eq.\\,(\\ref{PSIeq}), \nwhen trying to map a factorized wave function to its continuous time description.     \n\n\\subsection{The many-time formulation} \nIt appears that \nthe  difficulties arise from the implicit assumption that the components of a multipartite CA are {\\it synchronized} to the extent that they share a common clock \nvariable $n$. We consider a radical way out of the impasse encountered \nby resorting to the {\\it many-time} formalism, which means giving up synchronization \namong parts of the composite CA by introducing a set of clock variables, \n$\\{ n(1),\\;\\dots ,\\;n(m)\\}$, one for each one out of $m$ components.  \n\nThis may come as a surprise in the present nonrelativistic context, since the \nmany-time formalism has been introduced by Dirac, Tomonaga, and Schwinger in \ntheir respective formulations of relativistically covariant many-particle \nQM or quantum field theory, where a global synchronization cannot be \nmaintained.\\,\\cite{Dirac,Tomonaga,Schwinger}  \n  \nReplacing the single-CA action of Eq.\\,(\\ref{action}), we define here\nthe integer-valued multipartite-CA action by: \n\\begin{equation}\\label{maction} \n{\\cal S}[\\Psi ,\\Psi^*] :=\\Psi^*\\big (\n\\sum_{k=1}^m\\hat{\\cal S}_{(k)}\\;+\\;\\hat{\\cal I}\\big )\\Psi\n\\;\\;, \\end{equation} \nwith $\\Psi :=\\Psi^{\\alpha_1\\dots\\alpha_m}_{n_1\\dots n_m}$ and, correspondingly,  $\\Psi^*$ as independent {\\it complex integer-valued} variables; \nthe self-adjoint operator $\\hat{\\cal I}$ incorporates interactions between different CA; \nwhereas\n$\\hat{\\cal S}_{(k)}$ is as introduced in Eq.\\,(\\ref{action}), with the subscript \n$_{(k)}$ indicating that it acts {\\it exclusively} on the pair of indices pertaining to \nthe $k$-th single-CA subsystem:   \n\\begin{equation}\\label{Sopk} \n\\Psi^*\\hat{\\cal S}_{(k)}\\Psi:=\n\\sum_{\\{ n_k\\} }\\big[ (\\mbox{Im}\\;\n\\Psi_{\\dots n_{k}\\dots}^{*\\dots\\alpha_{k}\\dots}\n\\;\\dot\\Psi_{\\dots n_k\\dots}^{\\dots\\alpha_k\\dots}\n\\;+\\;\\Psi_{\\dots n_{k}\\dots}^{*\\dots\\alpha_{k},\\dots}\\;\nH_{(k)}^{\\alpha_k\\beta_k}\\Psi_{\\dots n_k\\dots}^{\\dots\\beta_k\\dots}\n\\big ]\n\\;\\;, \\end{equation} \nwith summation over {\\it all} clock variables (summation over \ntwice appearing Greek \nindices remains understood); the $\\dot{\\phantom .}$-operation, however, acts only with \nrespect to the explicitly indicated $n_k$, $\\dot f(n_k):=f(n_k+1)-f(n_k-1)$, while the \nsingle-CA Hamiltonian, $\\hat H_{(k)}$, requires a matrix multiplication, as before.  \n\nObviously, we can apply the {\\it CA Action Principle} of Section\\,2. to the present situation as well, with the generalized action of Eq.\\,(\\ref{maction}), in particular. This results in the following discrete equations of motion: \n\\begin{equation}\\label{mEoM} \n\\sum_{k=1}^m\\dot\\Psi_{\\dots n_k\\dots}^{\\dots\\alpha_k\\dots}\n\\;=\\;\\frac{1}{i}\\big (\\sum_{k=1}^mH_{(k)}^{\\alpha_k\\beta_k}\n\\Psi_{\\dots n_k\\dots}^{\\dots\\beta_k\\dots} \n\\;+\\;{\\cal I}^{\\dots\\alpha_k\\dots\\;\\beta_1\\dots\\beta_m}\n\\Psi_{\\dots n_k\\dots}^{\\beta_1\\dots\\beta_m} \n\\big )\n\\;\\;, \\end{equation}  \ntogether with the adjoint equations; here the interaction $\\hat{\\cal I}$, \nlike $\\hat H_{(k)}$, is assumed to be independent of the clock variables and    \nthe $\\dot{\\phantom .}$-operation acts only with respect to \n$n_k$ in the $k$-th term on the left-hand side.   \n\nLet us verify that the many-time formulation presented here avoids  \nthe problems of a single-time multi-CA equation, such as Eq.\\,(\\ref{PSIeq}), which we \npointed out. \n\nFirst of all, in the absence of interactions  \nbetween CA subsystems, $\\hat{\\cal I}\\equiv 0$, it is {\\it sufficient} for a solution of  \nEqs.\\,(\\ref{mEoM}) that the multi-CA wave function factorizes:  \n\\begin{equation}\\label{PSIfact} \n\\Psi =\\prod_{k=1}^m\\psi_{n_k}^{\\alpha_k} \n\\;\\;, \\end{equation} \nwhich differs from Eq.\\,(\\ref{PSI}) by the presence of an individual clock variable \nfor each  component CA,  $\\{ n_k,\\;k=1,\\dots ,m\\}$, or is a superposition of \nsuch factorized wave functions, and that each factor solves the \nappropriate single-CA equation of motion (as before, cf. Section\\,2.1.): \n\\begin{equation}\\label{sEoM} \n\\dot\\psi_{n_k}^{\\alpha_k}=\\frac{1}{i}H_{(k)}^{\\alpha_k\\beta_k}\n\\psi_{n_k}^{\\beta_k}\\;\\;,\\;\\;\\;k=1,\\dots ,m\n\\;\\;. \\end{equation}  \nThus, {\\it no unphysical correlations} are introduced among independent CA subsystems \nwhich do not interact with each other.  \n\nSecondly, the \n{\\it continuous multi-time equations} corresponding to Eqs.\\,(\\ref{mEoM}) are obtained \nby applying the mapping rules given in Section\\,2.3. to the discrete equations, as \ndetermined by Sampling Theory. Presently, there arises no problem of incompatibility between multiplication according to nonlinear terms {\\it vs.} linear mapping  according to \n{\\it Shannon's Theorem}, since a separate mapping has to be applied for each one of the \nindividual clock variables. This effectively replaces $n_k\\rightarrow t_k,\\;k=1,\\dots,m$, \nwhere $t_k$ is a continuous real time variable.  In this way, the following {\\it modified \nmulti-time Schr\\\"odinger equation} is obtained:  \n\\begin{equation}\\label{mSchroed} \n\\sum_{k=1}^m\\sinh\\big [l\\frac{\\mbox{d}}{\\mbox{d}t_k}\\big ] \\Psi_{\\dots t_k\\dots}^{\\dots\\alpha_k\\dots}\n\\;=\\;\\frac{1}{i}\\big (\\sum_{k=1}^mH_{(k)}^{\\alpha_k\\beta_k}\n\\Psi_{\\dots t_k\\dots}^{\\dots\\beta_k\\dots} \n\\;+\\;{\\cal I}^{\\dots\\alpha_k\\dots\\;\\beta_1\\dots\\beta_m}\n\\Psi_{\\dots t_k\\dots}^{\\beta_1\\dots\\beta_m} \n\\big )\n\\;\\;, \\end{equation}  \nwhere an overall factor of two from the left-hand side has been absorbed into the matrices  on the right.  Note that the wave function $\\Psi$ is bandlimited, by construction,  \nwith respect to each variable $t_k$. \n\nPerforming the continuum limit, $l\\rightarrow 0$, we arrive at the  \nmulti-time Schr\\\"odinger equation (one power of $l^{-1}$ providing the \nphysical dimension of $\\hat H_{(k)}$ and $\\hat {\\cal I}$) considered by Dirac and \nTomonaga.\\,\\cite{Dirac,Tomonaga} However, when $l$ is fixed and finite, modifications \nin the form of powers of $l\\mbox{d}\/\\mbox{d}t_k$ arise on its left-hand side.     \n\nFurthermore, in the present nonrelativistic context, it may be appropriate to identify  \n$t_k\\equiv t,\\;k=1,...,m$, in which case the operator on the left-hand side of \nEq.\\,(\\ref{mSchroed}), for $l\\rightarrow 0$, can be simply replaced by \n$\\mbox{d}\/\\mbox{d}t$, which results in the usual (single-time) \n{\\it many-body Schr\\\"odinger equation}.  \n\n\\subsubsection{The conservation laws of multipartite CA} \nSymbolically, the equivalent many-time equations (\\ref{mEoM}) and (\\ref{mSchroed}) \nare obviously both of the form:  \n\\begin{equation}\\label{msymb} \n\\hat {\\cal D}\\Psi =\\frac{1}{i}(\\hat {\\cal H}+\\hat {\\cal I})\\Psi \n\\;\\;, \\end{equation} \nto be complemented by corresponding adjoint equations. Then, for any operator \n$\\hat {\\cal G}$, such that $[ \\hat {\\cal G},\\hat {\\cal H}+\\hat {\\cal I}]=0$, we find \nimmediately the generalization of {\\it Theorem\\,A} of Section\\,2.2., \nnamely the {\\it discrete conservation law for multipartite CA}: \n\\begin{equation}\\label{mTheoremA} \n\\Psi^*\\hat {\\cal G}\\hat {\\cal D}\\Psi +(\\hat {\\cal D}\\Psi^*)\\hat {\\cal G}\\Psi =0 \n\\;\\;, \\end{equation}  \nvalid for the discrete {\\it and} continuous time descriptions with the obvious \nexplicit form of ${\\cal D}\\Psi^{(*)}$ inserted, respectively, according to the left-hand \nsides of Eqs.\\,(\\ref{mEoM}) and (\\ref{mSchroed}). \n\nThis, in turn, leads to conserved quantities, to be compared with \nEqs.\\,(\\ref{qG})--(\\ref{normal}) before. Here we are particularly interested in the case \n$\\hat {\\cal G}:=\\hat 1$, which yields as conserved quantity: \n\\begin{eqnarray}\\label{mQ} \n\\Psi^*\\hat{\\cal Q}\\Psi &:=&  \n\\sum_{k=1}^m\\Psi^{*\\alpha_1\\dots\\alpha_m}_{n_1\\dots n_m}\\hat{\\cal Q}_{(k)}\n\\Psi^{\\alpha_1\\dots\\alpha_m}_{n_1\\dots n_m}  \n\\\\ [1ex] \\label{mQc} \n&=&\\mbox{Re}\\sum_{k=1}^m\\Psi^{*\\alpha_1\\dots\\alpha_m}_{t_1\\dots t_m} \n\\cosh\\big [l\\frac{\\mbox{d}}{\\mbox{d}t_k}\\big ]\\Psi^{\\alpha_1\\dots\\alpha_m}_{t_1\\dots t_m}\n\\\\ [1ex]\\label{mQcl}   \n&\\stackrel{l\\rightarrow 0}{\\longrightarrow}&\\;\nm\\cdot\\Psi^{*\\alpha_1\\dots\\alpha_m}_{t_1\\dots t_m}\n\\Psi^{\\alpha_1\\dots\\alpha_m}_{t_1\\dots t_m}= m\\cdot\n|\\Psi_{t_1\\dots t_m}|^2  \n\\;\\;, \\end{eqnarray} \nwhere subscript $\\phantom ._{(k)}$ serves to indicate on which one of the discrete \nclock variables, namely $n_k$, the operator $\\hat{\\cal Q}$ acts, which has been introduced in Eq.\\,(\\ref{Q}); the second and third equalities, respectively, present the corresponding \ncontinuous multi-time quantity and its continuum limit, cf. \nEqs.\\,(\\ref{Qcont1})--(\\ref{Qcont2}).  This is the \n{\\it wave function normalization} in the multi-time formulation\\;\\cite{Tomonaga};  \nwhen it is appropriate to identify  \n$t_k\\equiv t,\\;k=1,...,m$, the usual many-body wave function normalization follows.  \nIncluded here is, of course, also the case of a factorized wave function as in \nEq.\\,(\\ref{PSIfact}).  \n\n\\subsubsection{The Superposition Principle in composite Hamiltonian CA}   \nThe equivalent discrete or continuous many-time equations (\\ref{mEoM}) and \n(\\ref{mSchroed}) are both linear in the CA wave function $\\Psi$. Therefore, superpositions \nof solutions of these equations also present solutions. Thus, the {\\it Superposition \nPrinciple} holds  not only for single but for multipartite Hamiltonian CA as well.    \n\nAs in the case of single CA, this entails the fact that already these discrete systems \n -- with all variables, parameters, {\\it etc.} being (complex) integer-valued -- can \nproduce {\\it interference} effects as in quantum mechanics. Even more interesting, their \ncomposites can also show {\\it entanglement}, which is deemed an essential feature of QM.    \nThis follows from the form of the equations of motion, which allow for superpositions \nof factorized states, {\\it cf.} Eq.\\,(\\ref{PSIfact}). \n\nFor example, in the bipartite case \n($k=1,2$), assuming that the individual CA are characterized by two degrees of freedom \n($\\alpha_k=0,1$), a time dependent analogue of one of the the well known \n{\\it Bell states}, the totally antisymmetric one, is given by: \n\\begin{equation}\\label{Bell} \n\\Psi\\propto\\psi_{n_1}^{\\alpha_1=0}\\psi_{n_2}^{\\alpha_2=1}\n-\\psi_{n_1}^{\\alpha_1=1}\\psi_{n_2}^{\\alpha_2=0} \n\\;\\;, \\end{equation} \nwhich may be a solution of appropriate discrete equations of motion. \n  \nHowever, a word of warning is in order here. We have freely used expressions familiar \nin QM, such as ``wave functions'' and ``states'', in particular. These are usually taken to \ninvoke the notion of vectors in a {\\it Hilbert space}, which becomes a complex projective space upon normalization of the vectors. \n\nAs we have seen already in Section\\,2.2., \nsee Eqs.\\,(\\ref{normal})--(\\ref{Q}), or Section\\,3.1.1., see Eqs.\\,(\\ref{mQ})--(\\ref{mQcl}), as \nlong as the CA are truly discrete ($l\\neq 0$), the normalization (squared) of vectors is not among the conserved quantities, hence not applicable, but is replaced by a conserved \n(many-)time correlation function instead. \n\nFurthermore, despite \nclose resemblance, the envisaged space of states strictly speaking is not a \nHilbert space, since it fails in two respects: the vector-space and completeness \nproperties are missing.  --  First of all, \nthe relevant {\\it Gaussian integers} (complex integer-valued numbers) are not \ncomplete. Hence the completeness property of the space of states is lacking, which is \nbuilt here with these integers as underlying scalars. Secondly, the integer numbers {\\it only} \nfeaturing in all aspects of the CA do not form a \n{\\it field} but a {\\it commutative ring} (for the multiplication of vectors by such scalars \nthere is no multiplicative inverse, such as exists, {\\it e.g.}, for rational,  \nreal, or complex numbers). Therefore, we cannot form a vector space over a field, as \nusual in QM, \nbut have to replace it  by a more general structure. This is known as a \nmodule over a  ring, in the present case a {\\it module over the commutative ring of \nGaussian integers}. It allows the construction of a linear space endowed with an \ninteger-valued scalar product, {\\it i.e.} a {\\it unitary space}. Taking its incompleteness \ninto account, then, the space of states in the presented CA theory can be classified as a \n{\\it pre-Hilbert module over the commutative ring of Gaussian integers}.\\,\\footnote{We \nthank a referee \nfor his constructive criticism of our earlier presentation of this point, helping to clarify \nthe issues involved.}\n\nWe conclude that superpositions of states, interference effects, and entanglement, as \nin quantum mechanics, all find their correspondents already on the ``primitive'' level \nof the presently considered natural Hamiltonian CA, discrete single or multipartite systems which are characterized by (complex) integer-valued variables and couplings.    \n\n\\section{Conclusion} \nWe have presented a brief review of earlier work which has demonstrated surprising quantum features arising in integer-valued, hence ``natural'', {\\it Hamiltonian cellular \nautomata}.\\,\\cite{PRA2014,EmQM13,Discrete14,WignerSymp13}  The study of this particular class of CA is motivated by 't\\,Hooft's {\\it Cellular Automaton \nInterpretation of QM} \nelaborated in Ref.\\,\\cite{tHooft2014} and various recent attempts to construct models \nwhich serve to illustrate indeed that QM (or, at least, essential features thereof) can \nbe understood to emerge from pre-quantum deterministic dynamics beneath.    \n\nThe single CA we considered previously allowed practically for the first time to \nreconstruct quantum mechanical models  with nontrivial Hamiltonians in terms of \nsuch deterministic systems with a finite discreteness scale. \n\nPresently, we have  extended \nthis study by describing {\\it multipartite systems}, analogous to many-body QM. \nNot only is this useful for the construction of more complex models {\\it per se} (with \na richer structure of energy spectra, in particular), but it is also necessary, in order to \nresearch the equivalent of the {\\it Superposition Principle} of QM, if any, on the CA level. \nThus, we find that it can be introduced already there to the fullest extent, \ncompatible with a tensor product structure of multipartite states, \nwhich entails not only the possibility of their {\\it interference} but also of their \n{\\it entanglement}. \n\nSurprisingly, we have been forced -- in our approach employing Sampling Theory for the map between CA and an equivalent continuum picture -- to introduce a \nmany-time formulation, which only appeared in relativistic quantum mechanics before, in \nthe way introduced by Dirac, Tomonaga, and Schwinger.\\,\\cite{Dirac,Tomonaga,Schwinger} \nThis may point towards a crucial further step in these developments, which is still missing, \nnamely a CA model of {\\it interacting quantum fields}. It is hard to envisage \nsuch a picture of dynamical fields spread out in spacetime without the possibility of  \nmultipartite CA with quantumlike features, which we have presently constructed. \nYet further conceptual advances seem necessary, in order to  \narrive at a relativistic quantum field theory departing from pre-quantum \ncellular automata. \n\n\\section*{Acknowledgments}\nIt is a pleasure to thank A.\\,Khrennikov for inviting me to the conference ``Quantum Theory: from foundations to technologies'' (Vaxjoe, June 2015), where part of this work was presented, N.\\,Margolus for correspondence, and G.\\,'t~Hooft for discussions.   \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nSystems of partial differential equations with cross-diffusion have gained a lot of interest in recent years \\cite{Kuefner1996,Chen2004,Chen2006,Lepoutre2012,Juengel2015bddness}. Such systems\nappear in many applications, for instance the modelling of population dynamics of multiple species~\\cite{Burger2015}, \ncell sorting or chemotaxis-like applications~\\cite{Painter2002,Painter2009} or predator-swarm model~\\cite{Chen2014}, and have been studied in different contexts~\\cite{Aman1990,Le2006,Griepentrog2010,Griepentrog2004,chapman1,chapman2}. \n\n\\medskip\n\nIn this work we focus our attention to a particular multi-species cross-diffusion system which reads as follows. Let $T>0$, $n,d\\in \\mathbb{N}^*$ and $\\Omega \\subset \\mathbb{R}^d$ be a bounded regular domain. For $t\\in (0,T)$ and $x\\in \\Omega$, \nwe consider $(u_0(t,x),\\ldots,u_n(t,x))$ to be a solution to the system of $n+1$ equations\n\\begin{equation}\\label{eq:sys}\n    \\partial_t u_i - \\nabla \\cdot \\left[\\sum_{j=0, j\\neq i}^n K_{ij} (u_j\\nabla u_i-u_i\\nabla u_j)\\right] = 0  \\text{ in }\\quad (0,T) \\times \\Omega,\\quad i=0,\\ldots, n,\n\\end{equation}\nsupplemented with no-flux boundary conditions and some initial data, where $K_{ij}\\geq 0$ for all $0\\leq i \\neq j \\leq n$.\n\n\nSystem~\\eqref{eq:sys} can be seen as the (formal) limit of a microscopic stochastic lattice based model (see the Appendix for more details) and models the evolution of the local volumic fractions of a system composed of $n+1$ different species. The function $u_i(t,x)$ represents the value at some \ntime $t\\in [0,T]$ and point $x\\in \\Omega$ of the density or volumic fraction of the $i^{th}$ entity. \nIn terms of modelling this means that the particles whose densities are given by the functions $u_i$ have a finite, positive size so that there is a maximal number of particles per given volume. \nThis is often referred to as size exclusion (or exclusion process). From a modelling point of view, one is therefore interested in considering solutions to (\\ref{eq:sys}) which satisfy\n\\begin{equation}\\label{eq:volume}\n\\forall 0\\leq i \\leq n, \\quad u_i(t,x)\\geq 0 \\quad \\mbox{ and } \\quad \\sum_{i=0}^n u_i(t,x) =1, \\quad \\mbox{ a.e. in }(0,T)\\times \\Omega.  \n\\end{equation}\n\n\\medskip\n\nFrom an analysis point of view, it is not easy to prove the existence of solutions to cross-diffusion systems satisfying (\\ref{eq:volume}), and uniqueness results are even harder to obtain. \nRecently, cross-diffusion systems which exhibit a (formal) gradient flow structure (see~\\cite{Juengel2015},~\\cite{Burger2010}, or~\\cite{Schlake2011}) have drawn particular interest from mathematicians. \nIndeed, such a structure allows to show the existence of weak solutions in many situations, using the dissipation of the corresponding entropy to get a priori bounds which are enough to pass to the limit in a suitable approximation. \nThis often also relies on the introduction of so-called entropy variables which can be used as a substitute for maximum principles which are not available for such systems. \nSee e.g. \\cite{Juengel2012}, \\cite{Juengel2015} and \\cite{Ehrlacher2017} for examples of this strategy. Also note that due to the degenerate structure of \\eqref{eq:sys}, solutions sometimes have less regularity than in the usual parabolic case (e.g. for $n=2,K_{10}=K_{20}=1$ and $K_{12}=K_{21}=0,$ the solutions $u_i$ are only $L^2$ in space, not $H^1$, see \\cite{Burger2010} for details). \n\n\\medskip\n\nThe existence of weak solutions to (\\ref{eq:sys}) is proved in~\\cite{Ehrlacher2017}, using a general result of~\\cite{Juengel2015bddness}, under the assumption that the \ncross-diffusion coefficients $(K_{ij})_{0\\leq i \\neq j \\leq n}$ are assumed to be positive and to satisfy $K_{ij} = K_{ji}$ for all $0\\leq i \\neq j \\leq n$. \nMost importantly in the context of our work, the existence of strong solutions and uniqueness of weak solutions was\nso far only available in a very special situation, i.e. when all the self-diffusion coefficients $K_{ij}$ are equal to a constant $K$. In this case, system~\\eqref{eq:sys} boils down to a system of $n+1$ independent heat equations, \nthe analysis of which is easy. In the general case, to the best of our knowledge, no such results are available so far.\n\n\\medskip\n\nThe object of the present article is based on the following observation: The system \\eqref{eq:sys} can be considered as a perturbation of a system of heat equations if the coefficients $K_{ij}$ are not too different \nfrom a fixed constant $K$. Indeed, we have \n\\begin{align}\\label{eq:heatperturbation}\n \\partial_t u_i - K \\Delta u_i = \\mbox{\\rm div} \\left[ \\sum_{j=0 }^n (K_{ij}-K) (u_j\\nabla u_i-u_i\\nabla u_j)\\right]. \n\\end{align}\nUnder the assumption that the quantities $\\vert K_{ij} - K \\vert$ are sufficiently small, we prove the existence \\REVV{and uniqueness} of strong solutions to \\eqref{eq:sys} \\REVV{satisfying (\\ref{eq:volume}) in dimension $d\\leq 3$}. \n\\REVV{In addition, we prove} a weak-strong stability estimate \nin dimension 1 which implies the uniqueness \nof weak solutions. A key issue in the proof is to construct approximations that preserve nonnegativity and the volume constraint.\n\n\nThis paper is organized as follows: in Section~\\ref{sec:main} we state our main results. \nSection~\\ref{sec:proofex} is devoted to the proof of the existence \\REVV{ and uniqueness} of strong solutions to the cross-diffusion system we consider \\REVV{in dimension $d\\leq 3$}. \nLastly, Section~\\ref{sec:stability} details the proof of the weak-strong stability result we obtain in dimension 1. Let us mention that a weak-strong stability result is proved in~\\cite{Chen2018} \nfor a system similar to (\\ref{eq:sys}), but with different assumptions on the coefficients $(K_{ij})_{0\\leq i\\neq j \\leq n}$.\n\n\n\\section{Main results}\\label{sec:main}\n\n\\subsection{Notation and preliminaries}\n\n Let $T>0$, $n,d\\in \\mathbb{N}^*$ and $\\Omega \\subset \\mathbb{R}^d$ be a bounded regular domain. For $t\\in (0,T)$ and $x\\in \\Omega$, \nwe consider\n$u(t,x):=(u_0(t,x),\\ldots,u_n(t,x))$ solution to the system of $n+1$ equations:\n\\begin{equation}\\label{eq:crossnplusone}\n  \\left\\{\n    \\begin{array}{rll}\n    \\partial_t u_i - \\nabla \\cdot \\left[\\sum_{j=0, j\\neq i}^n K_{ij} (u_j\\nabla u_i-u_i\\nabla u_j)\\right]& = 0  & \\text{ in }\\quad (0,T) \\times \\Omega,\\\\\n    \\left[\\sum_{j=0, j\\neq i}^n K_{ij}(u_j\\nabla u_i-u_i\\nabla u_j)\\right] \\cdot \\textbf{n} & = 0 & \\text{ on }\\quad (0,T) \\times \\partial\\Omega, \\\\\n   \n  \\end{array} \\right.\n  \\quad i=0,\\ldots, n,\n\\end{equation}\nwhere $\\textbf{n}$ denotes the unit outward pointing normal to the domain $\\Omega$, and $(K_{ij})_{0\\leq i \\neq j \\leq n}$ are non-negative coefficients. System~\\eqref{eq:crossnplusone} is supplemented with the initial condition $u^0:=(u_0^0,\\ldots,u_n^0)\\in \\left(L^1(\\Omega)\\right)^{n+1}$ \n\n\\medskip\n\nWe make the following assumption on the values of the cross-diffusion coefficients $(K_{ij})_{0\\leq i \\neq j \\leq n}$: \n\\begin{assumption}\\label{ass:A1}\nFor all $ 0 \\leq i \\neq j \\leq n$, $K_{ij}>0$ and $K_{ij} = K_{ji}$.\n\\end{assumption}\n\n\\medskip\n\nAs mentioned in the introduction, such a system models the evolution of the local volumic fractions of a system composed of $n+1$ different species and we expect the nonnegativity and volume constraint \\eqref{eq:volume} to hold.\n\n\\medskip\n\nLet us denote by\n$$\n\\mathcal P:= \\left\\{ u:=(u_0,\\ldots,u_n) \\in (0,+\\infty)^{n+1}, \\sum_{i=0}^n u_i = 1\\right\\} \\; \\mbox{ and } \\; \\mathcal D:= \\left\\{ U:=(u_1,\\ldots,u_n) \\in (0,+\\infty)^{n}, \\sum_{i=1}^n u_i < 1\\right\\}.\n$$\nWe point out that for all $u:=(u_0,\\ldots, u_n)\\in \\mathbb{R}^{n+1}$, $u$ belongs to $\\mathcal P$ if and only if $U:=(u_1,\\ldots,u_n)$ belongs to $\\mathcal D$. Similarly, $u$ belongs to $\\overline{\\mathcal P}$ if and only if \n$U$ belongs to $\\overline{\\mathcal{D}}$. Let us mention that condition (\\ref{eq:volume}) can be equivalently rewritten as $u(t,x)\\in \\overline{\\mathcal P}$ for almost all $(t,x)\\in (0,T)\\times \\Omega$.  \nIn what follows, we assume that the initial condition $u^0$ satisfies the following constraint:\n\\begin{equation}\\label{ass:init}\n u^0(x) \\in \\overline{\\mathcal P} \\mbox{ for almost all }x\\in\\Omega.  \n\\end{equation}\nUnder Assumptions~\\ref{ass:A1} and~\\ref{ass:init}, it is easy to see (at least formally) that the dynamics of the system preserves the volume constraint, i.e. \n\\begin{equation}\\label{eq:volume2}\n \\sum_{i=0}^n u_i(t,x)=1  \\quad  \\text{ a.e. in }(0,T) \\times \\Omega.\n\\end{equation}\nHowever, proving the existence of (weak or strong) solutions to system~(\\ref{eq:crossnplusone}) so that $u_i(t,x)\\geq 0$ for all $0\\leq i \\leq n$ and almost all $(t,x)\\in (0,T)\\times \\Omega$ is an intricate task \nfrom an analysis point of view. \\\\\nThe existence of weak solutions to system~(\\ref{eq:crossnplusone}) satisfying (\\ref{eq:volume}) is proved in~\\cite{Ehrlacher2017} under Assumptions~\\ref{ass:A1} and~\\ref{ass:init} and is actually \na consequence of Theorem~2 of~\\cite{Juengel2015bddness}. Let us recall this result and the main arguments of its proof below.\nUsing (\\ref{eq:volume2}) to express $u_0$ as $1 - \\sum_{i=1}^n u_i$, system (\\ref{eq:crossnplusone}) can be equivalently rewritten as a system of $n$ equations of the form\n\\begin{equation}\\label{eq:cross}\n\\left\\{\n  \\begin{array}{rll}\n    \\partial_t U - \\nabla \\cdot (A(U) \\nabla U) &= 0 & \\text{ on }\\quad (0,T) \\times \\Omega,\\\\\n    (A(U)\\nabla U) \\cdot \\textbf{n}&= 0 & \\text{ on }\\quad (0,T) \\times \\partial\\Omega,\\\\\n   \n  \\end{array}\n  \\right.  \n\\end{equation}\nwhere $U := (u_1, \\ldots, u_n)$. The diffusion matrix $A$ is defined by\n\\begin{equation}\\label{eq:defA1}\nA: \\left\\{\n\\begin{array}{ccc}\n\\mathbb{R}^n & \\to & \\mathbb{R}^{n\\times n}\\\\\nU & \\mapsto & A(U) := \\left( A_{ij}(U)\\right)_{1\\leq i,j \\leq n},\\\\\n\\end{array}\n\\right.\n\\end{equation}\nwith, for all $U:=(u_1,\\ldots,u_n)\\in \\mathbb{R}^n$, \n\\begin{equation}\\label{eq:defA}\n\\begin{array}{lll}\nA_{ii}(U)&= \\sum_{j=1, j\\neq i}^n (K_{ij}-K_{i0})u_j+K_{i0,} & \\quad i=1,\\ldots n,\\\\\nA_{ij}(U)&= -(K_{ij}-K_{i0})u_i, & \\quad i,j=1,\\ldots, n,\\; i \\neq j.\\\\\n\\end{array}\n\\end{equation}\n\n\n\nTheorem~2 of~\\cite{Juengel2015bddness} gives sufficient conditions on the diffusion matrix $A$ for a general cross-diffusion system to have a weak solution so that $U(t,x)\\in \\overline{\\mathcal{D}}$ for almost all $(t,x)\\in (0,T)\\times \\Omega$. \nMore precisely, Theorem~2 of~\\cite{Juengel2015bddness} can be stated as follows.\n\n\n\\begin{theorem}[Theorem~2 of~\\cite{Juengel2015bddness}]\\label{th:Jungel}\nLet $A \\in \\mathcal{C}^0(\\overline{D}; \\mathbb{R}^{n\\times n})$ be a continuous matrix-valued field defined on $\\overline{\\mathcal{D}}$ satisfying the following assumptions:\n\\begin{itemize}\n \\item[(H1)] there exists a bounded from below convex function $h\\in \\mathcal{C}^2(\\mathcal{D}, \\mathbb{R})$ such that its derivative $Dh: \\mathcal{D} \\to \\mathbb{R}^n$ is invertible on $\\mathbb{R}^n$;\n \\item[(H2)] there exists $\\alpha >0$, and for all $1\\leq i \\leq n$ there exists $1\\geq m_i >0$ such that, for all $z:=(z_1,\\ldots,z_n)\\in \\mathbb{R}^n$ and all $U:=(u_1,\\ldots,u_n)\\in\\mathcal{D}$, \n $$\n z^T D^2h(U)A(U) z \\geq \\alpha \\sum_{i=1}^n u_i^{2m_i-2}z_i^2.\n $$\n\\end{itemize}\nLet $U^0\\in L^1(\\Omega; \\mathcal{D})$ such that $w^0:= Dh(U^0) \\in L^\\infty(\\Omega; \\mathbb{R}^n)$. Then, there exists a weak solution $U$ with initial condition $U^0$ to\n\\begin{equation}\\label{eq:crossdiffgen}\n\\left\\{\n  \\begin{array}{rll}\n    \\partial_t U - \\nabla \\cdot (A(U) \\nabla U) &= 0 & \\text{ on }\\quad (0,T) \\times \\Omega,\\\\\n    (A(U)\\nabla U) \\cdot \\textbf{n} &= 0 & \\text{ on }\\quad (0,T) \\times \\partial\\Omega,\\\\\n  \\end{array}\n  \\right.\n\\end{equation}\nsuch that for almost all $(t,x)\\in (0,T)\\times \\Omega$, $U(t,x)\\in \\overline{D}$ with \n$$\nU\\in L^2_{\\rm loc}((0,T); H^1(\\Omega; \\mathbb{R}^n)) \\quad \\mbox{ and } \\partial_t U \\in L^2_{\\rm loc}((0,T); (H^1(\\Omega;\\mathbb{R}^n))').\n$$\n\\end{theorem}\n\nThe proof of Theorem~\\ref{th:Jungel} relies on the fact that a system of the form~\\eqref{eq:crossdiffgen} with a matrix-valued field $A$ satisfying conditions (H1)-(H2)\nexhibits a (formal) gradient flow structure, which we detail below for the particular case of~\\eqref{eq:cross} with $A$ defined by~\\eqref{eq:defA}. \n\nTo this end, let us introduce the entropy density $h$ given by\n\\begin{equation}\\label{eq:defh}\n   h: \\left\\{\n   \\begin{array}{ccc}\n    \\bar{\\mathcal{D}} & \\to & \\mathbb{R} \\\\\n    U:= (u_i)_{1\\leq i \\leq n} & \\mapsto & \\sum_{i=1}^n u_i \\log u_i + (1-\\sum_{i=1}^n u_i)\\log (1-\\sum_{i=1}^n u_i).\\\\\n   \\end{array} \\right.\n\\end{equation}\nand the corresponding entropy functional\n\\begin{equation}\\label{eq:defE}\n   \\mathcal{E}:\n   \\begin{array}{ccc}\n L^{\\infty}(\\Omega,\\bar{\\mathcal{D}}) & \\to & \\mathbb{R} \\\\   \n     U & \\mapsto&  \\int_{\\Omega} h(U(x))\\, dx .\\\\\n   \\end{array}\n\\end{equation}\nIt is proved in Lemma~2.3 of~\\cite{Ehrlacher2017} that the function $h$ defined by (\\ref{eq:defh}) satisfies conditions (H1) and (H2) of Theorem~\\ref{th:Jungel} for the matrix-valued function $A$ defined by (\\ref{eq:defA}) \nwith $m_i = \\frac{1}{2}$ for every $1\\leq i  \\leq n$ and $\\alpha=\\min_{1\\leq i\\neq j\\leq n}K_{ij}$. Furthermore, we can rewrite the system \\eqref{eq:cross} as\n\\begin{align*}\n\\partial_t U - \\nabla \\cdot (A(U)(D^2h(U))^{-1} \\nabla D\\mathcal{E}(U)) &= 0 \\text{ on }\\quad (0,T) \\times \\Omega, \\\\\n(A(U)(D^2h(U))^{-1}\\nabla D\\mathcal{E}(U)) \\cdot \\normalfont\\textbf{n} &= 0 \\text{ on }\\quad (0,T) \\times \\partial\\Omega, \\\\\nU(0,x)&=U^0(x) \\quad \\text{ a.e. in } \\Omega.\n\\end{align*}\nIn this formulation, it becomes clear that the entropy functional $\\mathcal{E}$ is a Lyapunov function for system (\\ref{eq:cross}).\n\n\\medskip\n\nThe existence of weak solutions to~\\eqref{eq:cross} satisfying $U(t,x)\\in \\overline{\\mathcal D}$ almost everywhere is then a direct consequence of Theorem~2 of \\cite{Juengel2015bddness} and Lemma~2.3 of~\\cite{Ehrlacher2017}. More precisely, we have the following proposition\n\\begin{proposition}[Existence of weak solutions]\\label{prop:weak}\n Let $u^0\\in L^1(\\Omega; \\mathcal{P})$ and $U^0:=(u_1^0, \\ldots, u_n^0)$. Let us assume in addition that $w^0:= Dh(U^0) \\in L^\\infty(\\Omega; \\mathbb{R}^n)$ with $h$ defined by (\\ref{eq:defh}). \n Then, there exists a weak solution $u$ with initial condition $u^0$ to \\eqref{eq:crossnplusone} such that\n \\begin{itemize}\n  \\item[(i)] $\\displaystyle u\\in L^2_{\\rm loc}((0,T); H^1(\\Omega; \\mathbb{R}^{n+1}))$ and $\\displaystyle \\partial_t u \\in L^2_{\\rm loc}((0,T); (H^1(\\Omega;\\mathbb{R}^{n+1}))')$;\n  \\item[(ii)] for almost all $(t,x)\\in (0,T)\\times \\Omega$, $u(t,x)\\in \\overline{\\mathcal{P}}$. \n \\end{itemize}\n\\end{proposition}\n\n\n\n\\subsection{Main results}\n\n\nThe aim of this work is to prove the existence \\REVV{ and uniqueness} of strong solutions to system (\\ref{eq:cross}) satisfying (\\ref{eq:volume}) under additional assumptions on the cross-diffusion coefficients $(K_{ij})_{0\\leq i \\neq j \\leq n}$. \nSuch a result holds \\REVV{ in dimension $d\\leq 3$}. For the particular case when $d=1$, we can also prove a weak-strong stability result which \nimplies that there exists a unique weak solution to system~(\\ref{eq:crossnplusone}) satisfying (\\ref{eq:volume}) and that this solution is strong.\n\n\nBefore stating our main results, let us make a preliminary remark on the no-flux boundary conditions imposed on $U$ in (\\ref{eq:cross}) which will be useful in the sequel. It is shown in \\cite[Lemma~5]{Juengel2012} that the matrix \n$A(U)$ is invertible for all $U\\in \\overline{\\mathcal{D}}$. Besides, for all $U\\in L^1(\\Omega; \\overline{\\mathcal{D}})$, $(A(U) \\nabla U)\\cdot \\textbf{n} = A(U) \\left( \\nabla U \\cdot \\textbf{n}\\right)$ on $\\partial \\Omega$. \nThis implies that a solution \n$U$ to~\\eqref{eq:cross} is equivalently a solution to the system\n\\begin{equation}\\label{eq:cross2}\n\\left\\{\n  \\begin{array}{rll}\n    \\partial_t U - \\nabla \\cdot (A(U) \\nabla U) &= 0 & \\text{ on }\\quad (0,T) \\times \\Omega,\\\\\n    \\nabla U \\cdot \\textbf{n}&= 0 & \\text{ on }\\quad (0,T) \\times \\partial\\Omega,\\\\\n   \n  \\end{array}\n  \\right.\n\\end{equation}\nand, denoting by $u_0: = 1 - \\sum_{i=1}^n u_i$, $u:=(u_0, \\ldots, u_n)$ is then equivalently a solution to\n\\begin{equation}\\label{eq:crossnplusone2}\n  \\left\\{\n    \\begin{array}{rll}\n    \\partial_t u_i - \\nabla \\cdot \\left[\\sum_{j=0, j\\neq i}^n K_{ij} (u_j\\nabla u_i-u_i\\nabla u_j)\\right]& = 0  & \\text{ in }\\quad (0,T) \\times \\Omega,\\\\\n    \\nabla u_i \\cdot \\textbf{n} & = 0 & \\text{ on }\\quad (0,T) \\times \\partial\\Omega, \\\\\n   \n  \\end{array} \\right.\n  \\quad i=0,\\ldots, n.\n\\end{equation}\nProving the existence of strong solutions to system~(\\ref{eq:cross}) is then equivalent to proving the existence of strong solutions to system~\\eqref{eq:crossnplusone2} and it will be more convenient for our analysis to consider the latter formulation in the sequel. \n\nTo obtain this strong existence result, we make an additional assumption on the cross-diffusion coefficients $(K_{ij})_{0\\leq i \\neq j \\leq n}$ which we detail hereafter.  For all \n$0\\leq i \\leq n$, let \n\\REVTWO{\n\\begin{equation}\\label{eq:defKi}\nK^+:= \\max_{0\\leq j \\neq i \\leq n} K_{ij}, \\quad K^-:= \\min_{0\\leq j \\neq i \\leq n} K_{ij}, \\quad K:= \\frac{K^+ + K^-}{2} \\quad \\mbox{ and } \\kappa:= \\frac{K^+ - K^-}{2}. \n\\end{equation}\n}\n\nThe additional assumption which we make from now on and in all the sequel reads as follows: \n\n\\begin{assumption}\\label{ass:A2}\n$\\displaystyle K \\; > \\; 2n\\kappa$. \n\\end{assumption}\n\n\\medskip\n\nIn other words, Assumption~\\ref{ass:A2} means that all the coefficients $K_{ij}$ should be sufficiently close to one another. The motivation for considering such a situation stems from the following observation: if there exists a constant $K>0$ such that \nfor all $0\\leq i \\neq j \\leq n$, $K_{ij} = K$, then $\\kappa = 0$ and system~(\\ref{eq:crossnplusone2}) boils down to a system of $n+1$ independent heat equations for which the existence and uniqueness of strong solutions satisfying (\\ref{eq:volume}) is obvious. \\REVTWO{Let us point out direct consequences of Assumption \\ref{ass:A2} that will be used frequently. We have \n\t\\begin{align}\n\t |K_{ij} -K| \\le \\kappa\\quad \\text{ as well as }\\quad K- 2n\\kappa - \\varepsilon > 0\n\t\\end{align}\n\tfor $\\varepsilon > 0$ sufficiently small.\n}\n\n\\medskip\n\nWe are now in position to state our two main results. \n\n\n\n\\REVTWO{This lemma ensures that provided a strong solution exists, it is unique. Existence is dealt with in the following result.}\n\\begin{theorem}(Existence and uniqueness of strong solutions)\\label{thm:existencestrong}\nLet us assume \\REVV{that $d\\leq 3$ and} that Assumptions~\\ref{ass:A1} and~\\ref{ass:A2} hold. Then, for every initial datum $u^0\\in [H^1(\\Omega)]^{n+1}$, with $u^0(x) \\in\\overline{\\mathcal{P}}$ for almost all $x\\in\\Omega$, \nthere exists a \\REV{unique} strong solution $u$ to \\eqref{eq:crossnplusone2} (or equivalently to \\eqref{eq:crossnplusone}) such that\n\\begin{itemize}\n \\item [(i)] $\\displaystyle u \\in [L^2((0,T), H^2(\\Omega)) \\cap H^1((0,T), L^2(\\Omega))]^{n+1}$;\n \\item[(ii)] $u(t,x)\\in \\overline{\\mathcal{P}}$ for almost all $(t,x)\\in (0,T)\\times \\Omega$.\n\\end{itemize}\n\\end{theorem}\n\n\\begin{theorem}(Weak-strong stability estimate in $d=1$)\\label{thm:stability}\nLet us assume that Assumptions~\\ref{ass:A1} and~\\ref{ass:A2} hold. Let $\\tilde u$ be a weak solution to~\\eqref{eq:crossnplusone2} (or equivalently to \\eqref{eq:crossnplusone}) in the sense of \nProposition~\\ref{prop:weak}, and let $u$ be a strong solution in the sense of \nTheorem~\\ref{thm:existencestrong}.\nThen, there exists a constant $C>0$ such that the following stability estimate holds for all $0< t\\leq T$:\n\\begin{equation}\\label{eq:estimate}\n\\|u(t,\\cdot)-\\tilde{u}(t,\\cdot)\\|_{L^2(\\Omega)}^2 \\leq e^{C\\| \\nabla u\\|_{L^2(0,t; L^\\infty(\\Omega))}^2}\\|u(0,\\cdot)-\\tilde{u}(0,\\cdot)\\|_{L^2(\\Omega)}^2.\n\\end{equation}\n\\end{theorem}\n\nA direct corollary of Theorem~\\ref{thm:stability} is the weak-strong uniqueness of solutions to \\eqref{eq:crossnplusone2} in dimension~1 for regular initial data, which can be stated as follows.\n\\begin{corollary}\\label{cor:wsuniq}\nLet us assume that $d=1$ and let $u^0 \\in H^1(\\Omega)^{n+1}$ such that $u^0(x)\\in\\overline{\\mathcal P}$ for almost all $x\\in \\Omega$.\nLet $\\tilde u$ be a weak solution to \\eqref{eq:crossnplusone2} (or equivalently to \\eqref{eq:crossnplusone}) in the sense of Proposition \\ref{prop:weak} and $u\\in L^2((0,T), H^2(\\Omega)) \\cap H^1((0,T), L^2(\\Omega))$ \nbe a strong solution in the sense of Theorem \\ref{thm:existencestrong}.\\\\\nThen, if the corresponding initial data $u(0, \\cdot)$ and $\\tilde{u}(0,\\cdot)$ agree a.e. on $\\Omega$, we also have\n\\begin{align*}\nu = \\tilde u \\text{ a.e. in } \\Omega \\times (0,T).\n\\end{align*}\n\\end{corollary}\n\n\\REVV{For the proof of Theorem~\\ref{thm:stability} and Corollary~\\ref{cor:wsuniq}}, we are restricted to one spatial dimension due to the fact that we need the embedding $H^2(\\Omega) \\hookrightarrow W^{1,\\infty}(\\Omega)$, \nso that a strong solution $u \\in \\left[L^2((0,T), H^2(\\Omega)) \\cap H^1((0,T), L^2(\\Omega))\\right]^{n+1}$ in the sense of Theorem~\\ref{thm:existencestrong}, satisfies $\\nabla u \\in \\left[L^2(0,T; L^\\infty(\\Omega))^d\\right]^{n+1}$.\n\n\n\\section{Proof of Theorem~\\ref{thm:existencestrong}}\\label{sec:proofex}\n\\selectlanguage{english}\nThe aim of this section is to prove Theorem~\\ref{thm:existencestrong} which states the existence \\REVV{and uniqueness} of strong solutions to system~(\\ref{eq:crossnplusone2}). \n\n\nThe proof is based on a fixed-point argument: we will show existence and uniqueness of strong solutions to a linearized system and subsequently apply Brouwer's fixed point theorem. \n\\REV{Uniqueness will be shown seperately}. For convenience let us define\n\\begin{align}\\label{eq:def_W}\nW:= L^2((0,T);H^2(\\Omega)) \\cap H^1((0,T);L^2(\\Omega)),\n\\end{align}\n\\REVV{\nand \n$$\nW_n:= \\left\\{ u \\in W, \\; \\nabla u \\cdot \\textbf{n} = 0 \\mbox{ a.e. on } (0,T)\\times \\partial \\Omega\\right\\}. \n$$\nIt can be easily checked that $W_n$ is a Banach space when endowed with the norm defined by\n$$\n\\forall u\\in W_n, \\quad \\|u\\|^2_{W_n}:= \\|\\partial_t u \\|^2_{L^2(0,T;L^2(\\Omega))} + \\|\\Delta u \\|^2_{L^2(0,T; L^2(\\Omega))} + \\|u(t=0,\\cdot)\\|^2_{H^1(\\Omega)}.\n$$\n}\n\n\\normalfont\n\n\\subsection{Auxiliary \\REV{lemmata}}\n\nIn this section, we start by stating some auxiliary results which are needed in the sequel. \n\n\n\\begin{lemma}\\label{lem:timederiv} \nLet $u \\in W$. Then, it holds that \n\\begin{itemize}\n \\item[(i)] $u \\in \\mathcal{C}([0,T];H^1(\\Omega))$;\n \\item[(ii)] there exists a constant $C>0$ which only depends on $T$ and $\\Omega$ such that\n\\begin{equation}\\label{eq:est1}\n \\mathop{\\max}_{0\\leq t \\leq T} \\|u(t,\\cdot)\\|_{H^1(\\Omega)} \\leq C\\left( \\|u\\|_{L^2(0,T; H^2(\\Omega))} + \\|\\partial_t u\\|_{L^2(0,T; L^2(\\Omega)}\\right);\n\\end{equation}\n\\item[(iii)] if in addition \\REVV{$u\\in W_n$} (i.e. if $\\nabla u \\cdot \\textbf{n} = 0$ a.e. in $\\partial \\Omega \\times (0,T)$), the mapping $(0,T)\\ni t \\mapsto \\|\\nabla u(t, \\cdot)\\|_{L^2(\\Omega)}^2$ is absolutely continuous, with \n$$\n\\frac{d}{dt}\\|\\nabla u(t, \\cdot)\\|_{L^2(\\Omega)}^2 = -2 \\langle \\partial_t u(t,\\cdot), \\Delta u(t,\\cdot) \\rangle_{L^2(\\Omega)}, \\quad \\mbox{ for a.e. }t \\in (0,T).\n$$\n\n\\normalfont\n\\end{itemize}\n\\end{lemma}\n\n\\begin{proof}\nItems (i) and (ii) are direct applications of [Theorem~4, Section~5.9.2] of~\\cite{Evans}. Let us now turn to the proof of (iii). \nWe extend $u$ by zero to a function $\\overline{u}$ defined for all $t \\in \\mathbb{R}$ and define, for all $\\delta >0$ and $t\\in \\mathbb{R}$, $u^\\delta(t,\\cdot) := \\int_\\mathbb{R} \\eta_\\delta(t-s)\\overline{u}(s,\\cdot)\\,ds$, \nwhere $\\eta_\\delta$ is a standard mollifier. It then holds that $u^\\delta \\in \\mathcal{C}^\\infty(0,T; H^2(\\Omega))$ and that for all $t\\in (0,T)$, $\\nabla u^\\delta(t, \\cdot) \\cdot \\textbf{n} = 0$ a.e. in $\\partial \\Omega$. \n\n\nLet $0< t <T$. Then, we have\n\\begin{align*}\n\\frac{d}{dt} \\left( \\|\\nabla u^\\delta(t,\\cdot)\\|_{L^2(\\Omega)}^2\\right) = 2\\langle\\partial_t \\nabla u^\\delta(t,\\cdot), \\nabla u^\\delta(t, \\cdot)\\rangle_{L^2(\\Omega)} = 2\\int_\\Omega \\partial_t\\nabla u^\\delta(t,\\cdot) \\cdot \\nabla u^\\delta(t,\\cdot),\n\\end{align*}\nwhere $\\partial_t \\nabla u^\\delta$ is the weak time derivative of $\\nabla u^\\delta$. For all $0\\leq t \\leq T$, the following convergence holds strongly in $L^2(\\Omega)$\n$$\n\\mathop{\\lim}_{h\\to 0 }\\frac{\\nabla u^\\delta (t+h,\\cdot) - \\nabla u^\\delta(t,\\cdot)}{h} = \\partial_t \\nabla u^\\delta(t,\\cdot), \n$$\nand\n\\begin{equation}\\label{eq:boundunif}\n\\left\\|\\frac{\\nabla u^\\delta (t+h, \\cdot) - \\nabla u^\\delta(t, \\cdot)}{h}\\right\\|_{L^2(\\Omega)} = \\left\\|\\frac{1}{h}\\int_{t}^{t+h} \\partial_s \\nabla u^\\delta(s, \\cdot)\\,ds\\right\\|_{L^2(\\Omega)} \\leq \\mathop{\\sup}_{0 \\leq s \\leq T} \\|\\partial_s \\nabla u^{\\delta}(s, \\cdot)\\|_{L^2(\\Omega)}.\n\\end{equation}\nInequality (\\ref{eq:boundunif}) implies that the difference quotient $\\displaystyle \\left( \\frac{\\nabla u^\\delta (\\cdot,t+h) - \\nabla u^\\delta(\\cdot,t)}{h}\\right)_{h>0}$ is uniformly bounded in $L^2(\\Omega)$ as $h$ goes to $0$, so that \n$\\displaystyle \\left( \\frac{\\nabla u^\\delta (\\cdot,t+h) - \\nabla u^\\delta(\\cdot,t)}{h}\\cdot \\nabla u^\\delta \\right)_{h>0}$ is uniformly bounded in $L^1(\\Omega)$. As a consequence, applying Lebesgue's dominated convergence theorem, we obtain\n\\begin{align*}\n 2\\int_\\Omega \\partial_t\\nabla u^\\delta(t,\\cdot) \\cdot \\nabla u^\\delta(t,\\cdot) & = \\mathop{\\lim}_{h\\to 0} \\int_\\Omega  \\frac{\\nabla u^\\delta (\\cdot,t+h) - \\nabla u^\\delta(\\cdot,t)}{h}\\cdot \\nabla u^\\delta(\\cdot, t).\n\\end{align*}\nBesides, since $\\nabla u^\\delta(s, \\cdot) \\cdot \\textbf{n} = 0$ on $\\partial \\Omega$ for all $s \\in (0,T)$, it holds that for all $h>0$, \n$$\n\\int_\\Omega  \\frac{\\nabla u^\\delta (\\cdot,t+h) - \\nabla u^\\delta(\\cdot,t)}{h}\\cdot \\nabla u^\\delta = - \\int_\\Omega \\frac{u^\\delta (\\cdot,t+h) - u^\\delta(\\cdot,t)}{h} \\Delta u^\\delta(\\cdot, t).\n$$\nApplying again Lebesgue's convergence theorem, we obtain\n$$\n\\mathop{\\lim}_{h\\to 0} \\int_\\Omega \\frac{u^\\delta (\\cdot,t+h) - u^\\delta(\\cdot,t)}{h} \\Delta u^\\delta(\\cdot, t) = \\int_\\Omega \\partial_t u^\\delta (\\cdot,t) \\Delta u^\\delta(\\cdot, t), \n$$\nin $L^1(\\Omega)$. Thus, for all $\\delta >0$, we have\n\\begin{equation}\\label{eq:delta}\n\\frac{d}{dt} \\left( \\|\\nabla u^\\delta(t,\\cdot)\\|_{L^2(\\Omega)}^2\\right) = - 2\\langle \\partial_t u^\\delta(t,\\cdot), \\Delta u^\\delta(t,\\cdot)\\rangle_{L^2(\\Omega)}.  \n\\end{equation}\nAs $\\delta$ goes to $0$, the convergences $u^\\delta \\to u,\\, \\nabla u^\\delta \\to \\nabla u$ and $\\Delta u^\\delta \\to \\Delta u$ hold strongly in $L^2(0,T; L^2(\\Omega))$ (since $u \\in L^2((0,T);H^2(\\Omega))$.\nWe finally obtain the result by passing to the limit $\\delta \\to 0$ in (\\ref{eq:delta}).\n\\end{proof}\n\n\\REVV{\nLet us now introduce the space\n$$\nW_0 := \\{ u \\in W\\; | \\; \\nabla u \\cdot \\textbf{n} = 0 \\text{ a.e. on } \\partial\\Omega \\times (0,T),\\, u=0 \\text{ a.e. on } \\Omega \\times \\{t=0\\} \\},\n$$\nwhich is a closed subspace of $W_n$. For all $u \\in W_0$ and all $0\\leq i \\leq n$, let us define\n$$\n\\| u \\|_{W_0}^2 :=\\int_0^T \\| \\partial_t u(\\cdot,t) - K \\Delta u(\\cdot,t)\\|_{L^2(\\Omega)}^2\\;dt\n$$\nand \n$$\n\\| u \\|_{\\widetilde W_0}^2 :=\\int_0^T \\|\\partial_t u(\\cdot,t)^2\\|_{L^2(\\Omega)}^2 + K^2 \\|\\Delta u(\\cdot,t)\\|_{L^2(\\Omega)}^2\\;dt.\n$$\n}\n\nWe then have the following result (see also \\cite{Breden2017} for a similar argument).\n\\begin{lemma}[Equivalence of norms]\\label{lem:equi}\nThe two applications $\\|\\cdot\\|_{W_0}$ and $\\|\\cdot\\|_{\\widetilde W_0}$ define norms on $W_0$ which are equivalent, and equivalent to the norm $\\|\\cdot\\|_{W_n}$.\n\\end{lemma}\n\\begin{proof}\nLet $u\\in W_0$. On the one hand, \n\\begin{align} \\nonumber\n\\|u \\|_{W_0}^2 & = \\int_0^T\\int_\\Omega (\\partial_t u - K \\Delta u)^2 = \\int_0^T\\int_\\Omega (\\partial_t u)^2 -2K\\partial_t u \\Delta u + K^2( \\Delta u)^2\\\\ \\label{eq:normestimate}\n&\\ge \\int_0^T\\int_\\Omega (\\partial_t u)^2 + K^2( \\Delta u)^2 = \\|u\\|_{\\widetilde W_0}^2\\\\\\nonumber\n\\end{align}\nsince\n\\begin{align*}\n\\int_0^T\\int_\\Omega 2K\\partial_t u \\Delta u = - \\int_0^T\\int_\\Omega 2K\\partial_t \\nabla u \\cdot \\nabla u = - K \\int_\\Omega |\\nabla u|^2(T) \\le 0.\n\\end{align*}\nOn the other hand, we always have \n\\begin{equation}\n\\|u \\|_{W_0}^2 = \\int_0^T\\int_\\Omega (\\partial_t u - K \\Delta u)^2 \\le 2\\int_0^T\\int_\\Omega (\\partial_t u)^2 + K^2( \\Delta u)^2= 2 \\|u\\|_{\\widetilde W_0}.\n\\end{equation}\nHence the equivalence of the norms. The fact that the norm $\\|\\cdot\\|_{\\widetilde W_0}$ is equivalent to the norm $\\|\\cdot\\|_{W_n}$ on $W_0$ is obvious and yields the desired result.\n\\end{proof}\nWe will make use of these norms in the proof of Lemma~\\ref{lem:linear}. Lastly, we introduce the following Lemma which is used in the proof of Lemma~\\ref{lem:strong_uniqueness}.\n\\begin{lemma}\\label{lem:reg}\nLet us assume that $d\\leq 3$. For all $\\gamma >0$, there exists a constant $C_\\gamma >0$ such that \n\\begin{equation}\\label{eq:ineqimport}\n\\forall v \\in H^2(\\Omega), \\quad \\|v\\|^2_{L^\\infty(\\Omega)} \\leq \\gamma \\|\\Delta v \\|^2_{L^2(\\Omega)} + C_\\gamma\\left( \\|v\\|_{L^2(\\Omega)}^2 + \\|\\nabla v \\|_{L^2(\\Omega)}^2\\right).\n\\end{equation}\n\\end{lemma}\n\\begin{proof}[Proof of Lemma~\\ref{lem:reg}]\nTo prove (\\ref{eq:ineqimport}), we use the continuity of the embeddings $H^1(\\Omega) \\hookrightarrow  L^{p}(\\Omega)$ for all $p\\leq 6$ and $W^{1,3 + \\delta}(\\Omega) \\hookrightarrow L^\\infty(\\Omega)$ for all $\\delta >0$. \nThus, for all $v\\in H^2(\\Omega)$, it holds that $v \\in  W^{1,3 + 1\/4}(\\Omega)$. Throughout the proof, $C$ will denote a positive constant, which is independent of $v$, and may change along computatations.\nIt holds that\n\\begin{align*}\n\\|v\\|_{L^{\\infty}(\\Omega)}^2 & \\leq C \\|v\\|_{W^{1,3 + 1\/4}(\\Omega)}^2\\\\\n&= C \\left( \\|v\\|_{L^{3 + 1\/4}(\\Omega)} + \\|\\nabla v\\|_{L^{3 + 1\/4}(\\Omega)}\\right)^2\\\\\n& \\leq C \\left( \\|v\\|_{H^1(\\Omega)} +  \\|\\nabla v\\|_{L^{3+1\/4}(\\Omega)}\\right)^2.\\\\\n\\end{align*} \nTo estimate the term $\\|\\nabla v\\|_{L^{3 +1\/4}(\\Omega)}$, we now use the Gagliardo-Nirenberg-Sobolev inequality \\cite[Thm 13.54]{Leoni2009} to obtain that\n$$\n\\|\\nabla v\\|_{L^{3 +1\/4}(\\Omega)} \\leq C \\left( \\|\\Delta v\\|_{L^2(\\Omega)}^{1\/2} \\|v\\|^{1\/2}_{L^{13\/3}(\\Omega)}  + \\|v\\|_{L^{13\/3}(\\Omega)}\\right) \n\\leq C \\left( \\|\\Delta v\\|_{L^2(\\Omega)}^{1\/2} \\|v\\|^{1\/2}_{H^1(\\Omega)} + \\|v\\|_{H^1(\\Omega)}\\right).\n$$\nThus, we obtain that for all $\\gamma >0$, \n\\begin{align*}\\label{eq:w13estimate}\n\\|v\\|_{L^\\infty(\\Omega)}^2 &\\le C\\left( \\|v\\|_{H^1(\\Omega)}+ \\|v\\|_{H^1}^{1\/2}\\|\\Delta v \\|_{L^2(\\Omega)}^{1\/2}) \\right)^2\\\\\n&\\le C\\left( \\|v\\|_{H^1(\\Omega)}+ \\frac{C}{2\\gamma}\\|v\\|_{H^1} + \\frac{\\gamma}{2C}\\|\\Delta v \\|_{L^2(\\Omega)} \\right)^2. \\\\\n&\\le C_\\gamma (\\|\\nabla v\\|_{L^2(\\Omega)}^2+\\|v\\|_{L^2(\\Omega)}^2)+ \\gamma\\|\\Delta v \\|_{L^2(\\Omega)}^2,\\\\\n\\end{align*}\nwhere $C_\\gamma>0$ is a positive constant which depends on $\\gamma$ but is independent of $v$. Hence the desired result.\n\\end{proof}\n\n \n\n\\REV{\\subsection{Uniqueness of strong solutions}}\n\n\nTo \\REVTWO{improve} readability, we first show the uniqueness of strong solutions. \n\\begin{lemma}[Uniqueness of strong solutions]\\label{lem:strong_uniqueness}\n\tLet \\REVV{$d\\leq 3$ and let }us assume that Assumptions~\\ref{ass:A1} and~\\ref{ass:A2} hold. Let $u^0\\in [H^1(\\Omega)]^{n+1}$, with $u^0(x) \\in\\overline{\\mathcal{P}}$ for almost all $x\\in\\Omega$. \n\tIf there exists at least one strong solution $u$ to \\eqref{eq:crossnplusone2} (or equivalently to \\eqref{eq:crossnplusone}) such that\n\t\\begin{itemize}\n\t\t\\item [(i)] $\\displaystyle u \\in [L^2((0,T), H^2(\\Omega)) \\cap H^1((0,T), L^2(\\Omega))]^{n+1}$,\n\t\t\\item[(ii)] $u(t,x)\\in \\overline{\\mathcal{P}}$ for almost all $(t,x)\\in (0,T)\\times \\Omega$,\n\t\\end{itemize}\n\tthen it is unique.\n\\end{lemma}\n\n \n\\REVV{\n\\begin{proof}\nLet $\\left(u_{i,1}\\right)_{0\\leq i \\leq n}$ and $\\left(u_{i,2}\\right)_{0\\leq i \\leq n}$ be two strong solutions to \\eqref{eq:crossnplusone2} satisfying (i) and (ii) with initial datum $u^0$ \nand let us denote by $\\bar u_i = u_{i,1} - u_{i,2}$. \\REVTWO{The equation for the $i^{th}$ read as \n\\begin{equation}\\label{eq:icross}\n\\partial_t u_{i,k}= \\sum_{j=0, j\\neq i}^n K_{ij}(u_{j,k}\\Delta u_{i,k}- u_{i,k} \\Delta u_{j,k}), \\quad k=1,2.\n\\end{equation}\nUsing the fact that $\\sum_{j=0}^n u_{j,} = 1$, \\REV{which implies that\n\t\\begin{align*}\n\t\\Delta u_{i,k} = \\sum_{j=0}^n u_{j,k} \\Delta u_{i,k} = \\sum_{j=0}^n (u_{j,k} \\Delta u_{i,k} - u_{i,k} \\Delta u_{j,k}) = \\sum_{j=0, j \\neq i}^n (u_{j,k} \\Delta u_{i,k} - u_{i,k} \\Delta u_{j,k}),\n\t\\end{align*}\n}\nand subtracting the equations for $k=1,2$ we obtain that for all $0\\leq i \\leq n$},\n\\begin{equation}\\label{eq:suniqueness_diff}\n\\partial_t \\bar u_i - K \\Delta \\bar u_i = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)(u_{j,2}\\Delta \\bar u_i + \\bar u_j \\Delta u_{i,1} - (u_{i,2} \\Delta \\bar u_j + \\bar u_i \\Delta u_{j,1})).\n\\end{equation}\nAs a first step we multiply (\\ref{eq:suniqueness_diff}) by $-\\Delta \\bar u_i$ and integrate over $\\Omega$. This yields, almost everywhere in $(0,T)$ and for all $\\eta>0$,\n\\begin{align*}\n &\\frac{d}{dt}\\int_\\Omega |\\nabla \\bar u_i|^2 + K \\int_\\Omega(\\Delta \\bar u_i)^2   = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)\\int_\\Omega (u_{j,2}(\\Delta \\bar u_i)^2 + \\bar u_j \\Delta u_{i,1}\\Delta \\bar u_i - u_{i,2} \\Delta \\bar u_j\\Delta \\bar u_i \n - \\bar u_i \\Delta u_{j,1}\\Delta \\bar u_i)\\\\\n &\\le n\\kappa \\int_\\Omega (\\Delta \\bar u_i)^2 + \\frac{\\kappa}{2\\eta}\\sum_{j=0,j\\neq i}^n \\|\\bar u_j\\|_{L^\\infty(\\Omega)}^2 + \\frac{n\\kappa\\eta}{2}\\|\\Delta u_{i,1}\\|_{L^2(\\Omega)}^2\\|\\Delta \\bar u_i\\|_{L^2(\\Omega))}^2\\\\\n  &+\\kappa  \\left(\\frac{n}{2}\\|\\Delta \\bar u_i\\|_{L^2(\\Omega)}^2 + \\sum_{j=0,j\\neq i }^n\\frac{1}{2}\\|\\Delta \\bar u_j\\|_{L^2(\\Omega)}^2 + \\frac{n}{2\\eta}\\|\\bar u_i\\|_{L^\\infty(\\Omega)}^2 + \\frac{\\eta}{2} \\|\\Delta \\bar u_i\\|_{L^2(\\Omega)}^2 \\sum_{j=0,j\\neq i }^n\\|\\Delta u_{j,1}\\|_{L^2(\\Omega)}^2\\right).\n \\end{align*}%\n Let us now choose $\\varepsilon >0$ small enough and define $\\eta_\\varepsilon(t) = \\frac{\\varepsilon}{1+ n\\kappa \\sum_{i=0}^n \\|\\Delta u_{i,1}(t,\\cdot)\\|_{L^2(\\Omega)}^2}$. \n Choosing $\\eta = \\eta_\\varepsilon(t)$ in the above inequality and summing over $i=0,\\ldots, n$, we obtain that for almost all $t\\in (0,T)$:\n \\begin{equation}\\label{eq:suniqueness_1}\n  \\sum_{i=0}^n\\left(\\frac{d}{dt}\\int_\\Omega |\\nabla \\bar u_i|^2\\;dx + (K- 2n\\kappa - \\varepsilon) \\int_\\Omega(\\Delta \\bar u_i)^2\\;dx\\right)  \\le \\frac{n\\kappa}{\\eta_\\varepsilon} \\sum_{i=0}^n\\|\\bar u_i\\|_{L^\\infty(\\Omega)}^2.\n \\end{equation\n Let us point out that $\\frac{1}{\\eta_\\varepsilon} \\in L^1(0,T)$ by definition. To estimate the terms on the right hand side, we make use of Lemma~\\ref{lem:reg} and obtain that for all $\\gamma >0$, there exists $C_\\gamma >0$ such that for all $0\\leq i \\leq n$, \n \\begin{equation}\\label{eq:auxineq}\n \\|\\bar u_i\\|_{L^{\\infty}(\\Omega)}^2 \\leq \\gamma \\|\\Delta \\bar u_i \\|_{L^2(\\Omega)}^2 + C_\\gamma \\left( \\|\\nabla \\bar u_i \\|_{L^2(\\Omega)}^2 + \\|\\bar u_i\\|_{L^2(\\Omega)}^2\\right)\n \\end{equation}\n Thus, choosing $\\gamma = \\varepsilon$ in (\\ref{eq:auxineq}), we obtain \n \\begin{equation}\\label{eq:suniqueness_2}\n  \\sum_{i=0}^n\\left(\\frac{d}{dt}\\int_\\Omega |\\nabla \\bar u_i|^2 + (K- 2n\\kappa - 2\\varepsilon) \\int_\\Omega(\\Delta \\bar u_i)^2\\right)  \\le g_\\varepsilon \\sum_{i=0}^n\\left(\\|\\bar u_i\\|_{L^2(\\Omega)}^2 + \\|\\nabla \\bar u_i\\|_{L^2(\\Omega)}^2 \\right),\n \\end{equation}%\nfor some function $g_\\varepsilon\\in L^1(0,T)$. In order to control the $\\|\\bar u_i\\|_{L^2(\\Omega)}^2$-term on the right hand side, we consider the weak form of \\eqref{eq:suniqueness_diff} and chose $\\bar u_i$ as a test function. We obtain\n \\begin{align*}\n   & \\frac{d}{dt} \\int_\\Omega \\left|{\\bar u_i}\\right|^2\\;dx + K \\int_\\Omega |\\nabla \\bar u_i|^2\\;dx = \\\\\n   & \\sum_{j=0, j\\neq i}^n(K_{ij}-K)\\int_\\Omega 2 \\bar u_i\\nabla u_{j,1} \\cdot \\nabla\\bar  u_i + \\nabla \\bar u_j \\cdot (\\bar u_i \\nabla u_{i,2} + u_{i,2} \\nabla \\bar u_i) \\\\\n   & - \\sum_{j=0, j\\neq i}^n(K_{ij}-K)\\int_\\Omega \\nabla \\bar u_i\\cdot(\\bar u_i \\nabla u_{j,2} + u_{j,2} \\nabla \\bar u_i) + \\nabla u_{i,1} \\cdot (\\bar u_i \\nabla \\bar u_j + \\bar u_j \\nabla \\bar u_i )\\\\\n    & = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)\\int_\\Omega \\bar u_i\\nabla u_{j,1} \\cdot \\nabla\\bar  u_i  - u_{j,2} |\\nabla \\bar u_i|^2 + u_{i,2} \\nabla \\bar u_i \\cdot \\nabla \\bar u_j - \\bar u_j \\nabla u_{i,1} \\cdot \\nabla \\bar u_i\\\\\n \\end{align*}\n Arguing as above and using again Lemma~\\ref{lem:reg}, we eventually obtain that for all $\\varepsilon>0$, there exists a function $\\widetilde{g}_\\varepsilon\\in L^1(0,T)$ such that\n \\begin{align}\\label{eq:suniqueness_3}\n  &\\sum_{i=0}^n\\left(\\frac{d}{dt}\\int_\\Omega \\left|\\bar u_i\\right|^2 + (K- 2n\\kappa - \\varepsilon) \\int_\\Omega|\\nabla  \\bar u_i|^2 - \\varepsilon \\int_\\Omega |\\Delta \\bar u_i|^2\\right) \\le \n  \\widetilde{g}_\\varepsilon \\sum_{i=0}^n \\left(\\|\\bar u_i\\|_{L^2(\\Omega)}^2 + \\|\\nabla \\bar u_i\\|_{L^2(\\Omega)}^2\\right).\n \\end{align}\n Adding \\eqref{eq:suniqueness_3} and \\eqref{eq:suniqueness_2} and choosing $\\varepsilon$ small enough thus imply that\n \\begin{align*}\n  &\\sum_{i=0}^n\\left(\\frac{d}{dt}\\int_\\Omega \\bar u_i^2\\;dx + \\frac{d}{dt}\\int_\\Omega |\\nabla\\bar u_i|^2\\;dx\\right) \\le h \\sum_{i=0}^n \\left(\\|\\bar u_i\\|_{L^2(\\Omega)}^2 + \\|\\nabla \\bar u_i\\|_{L^2(\\Omega)}^2\\right), \n \\end{align*}\n for some function $h \\in L^1(0,T)$. As the initial data of $(u_{i,1})_{0\\leq i \\leq n}$ and $(u_{i,2})_{0\\leq i \\leq n}$ coincide in $H^1(\\Omega)$, Gronwall's lemma yields the assertion.\n \\end{proof}\n }\n\n\n\n\n\\subsection{Existence for a linear problem}\n\nTo prove Theorem~\\ref{thm:existencestrong}, we begin by proving the existence of a strong solution to a truncated linearized approximate problem, which we present hereafter. \n\n\\medskip\n\n\\REVV{\nIn this section and the following one, we fix the initial condition $u^0 = (u_0^0, \\cdots, u_n^0)\\in H^1(\\Omega)^{n+1}$, and define for all $0\\leq i \\leq n$\n$$\nZ_i:= \\left\\{ u\\in W, \\; \\nabla u \\cdot \\textbf{n} = 0, \\; u(0) = u_i^0 \\right\\},\n$$\nand $Z:= Z_0\\times \\cdots \\times Z_n$. \n}\n\n\\medskip\n\n\nLet us assume for now that there exists a smooth solution $u:=(u_0, \\ldots, u_n)$ to~\\eqref{eq:cross} satisfying $\\sum_{j=0}^n u_j = 1$.\nUsing again the fact that $\\sum_{j=0}^n u_j = 1$, the equation for each component (\\ref{eq:icross}) can be rewritten as\n\\begin{equation} \\label{eq:strongicross} \n \\partial_t u_i - K \\Delta u_i = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)(u_j\\Delta u_i- u_i \\Delta u_j),\n\\end{equation}\nwhere the \\REV{the positive constant $K$ is }defined in (\\ref{eq:defKi}).\n\n\\medskip\n\n\nLet $\\tilde u :=(\\tilde u_0, \\tilde u_1, \\ldots, \\tilde u_n)\\in \\REVV{Z}$. We consider the following linear, regularized problem:\n\n\\REV{\\begin{equation}\\label{eq:cross_truncated}\n\\left\\{\n\\begin{array}{rll}\n\\partial_t u_i - K\\Delta u_i & = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)({\\tilde u_j^\\diamond}\\Delta u_i- {\\tilde u_i^\\diamond} \\Delta u_j),& \\\\\n\\nabla u_i \\cdot \\textbf{n} & = 0 ,& \\\\\n\\end{array} \\right.\n\\quad i=0,\\ldots, n,\n\\end{equation}\nwith $x^\\diamond:= \\max(0, \\min(1,x))$. Note that this implies $0 \\le x^\\diamond \\le 1$ for every $x \\in \\mathbb{R}$.\n}\nAlso note that even though we are dealing with a linear problem, we employ a fixed point strategy as in \\cite{Breden2017}.\n\\begin{lemma}[Existence of a strong solution to the linearized problem]\\label{lem:linear}\n\\REVV{For all $\\tilde u:=(\\tilde u_0,\\ldots, \\tilde u_n) \\in Z$, there exists a unique solution \n$u:=(u_0,\\ldots, u_n) \\in Z$ to \\eqref{eq:cross_truncated}.} In addition, the three following a priori estimates hold\n\\begin{align}\\label{eq:ineq1}\n\\sum_{i=0}^n\\mathop{\\sup}_{0\\leq t \\leq T} \\|\\nabla u_i(\\cdot, t)\\|_{L^2(\\Omega)}^2 + \\sum_{i=0}^n\\int_0^T\\int_\\Omega (\\Delta u_i)^2 &\n\\leq C_0,\\\\ \\label{eq:ineq2}\n\\mathop{\\sup}_{0\\leq s \\leq T}\\sum_{i=0}^n \\left\\| u_i(t,\\cdot)\\right\\|^2_{L^2(\\Omega)} & \\leq C_1,\\\\ \\label{eq:ineq3}\n\\sum_{i=0}^n \\left\\| \\partial_t u_i\\right\\|^2_{L^2(0,T; L^2(\\Omega))} & \\leq C_2,\\\\ \\nonumber\n\\end{align}\nwith \n\\begin{align} \\label{eq:C0}\n C_0 & := 2 \\max\\left(1, \\frac{1}{K - 2n\\kappa }\\right) \\sum_{i=0}^n\\|\\nabla u_i^0\\|_{L^2(\\Omega)}^2,\\\\ \\label{eq:C1}\n C_1 & := e^{2n \\kappa  T}\\left( \\sum_{i=0}^n \\left\\| u_i^0\\right\\|^2_{L^2(\\Omega)} + 2n \\kappa  C_0 \\right),\\\\ \\label{eq:C2}\n C_2& := \\left( K + 2n\\kappa \\right) C_0.\n\\end{align}\n\\end{lemma}\n\n\n\\begin{proof}\n\\REV{\n\\itshape Step 1 (Existence): \\normalfont \nLet $\\tilde u:=(\\tilde u_0,\\ldots, \\tilde u_n) \\in Z$. For all $(\\overline u_0, \\ldots, \\overline u_n) \\in \\REVV{Z}$, consider the problem \n\\begin{equation}\\label{eq:cross_truncated2}\n\\left\\{\n\\begin{array}{rll}\n\\partial_t u_i - K\\Delta u_i & = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)({\\tilde u_j^\\diamond}\\Delta \\overline u_i- {\\tilde u_i^\\diamond} \\Delta \\overline u_j),& \\\\\n\\nabla u_i \\cdot \\textbf{n} & = 0 ,& \\\\\n\\end{array} \\right.\n\\quad i=0,\\ldots, n.\n\\end{equation}\nNow as $\\Delta \\overline u_i \\in L^2((0,T);L^2(\\Omega))$ and $\\tilde u_i^\\diamond \\in L^\\infty((0,T);L^\\infty(\\Omega))$ for $i=0,\\ldots, n$, standard theory for linear parabolic equations yields \nthe existence of a \\REVV{unique} solution $(u_i)_{0\\leq i \\leq n} \\in \\REVV{Z}$. Let us denote by\n$$\nF: \\left\\{\n\\begin{array}{lll}\n Z & \\to & Z\\\\\n (\\bar u_{i})_{0\\leq i \\leq n} & \\mapsto & (u_i)_{0\\leq i \\leq n}\\\\\n\\end{array}\n\\right.\n$$\nthe application such that $(u_i)_{0\\leq i \\leq n} \\in Z$ is the unique solution to (\\ref{eq:cross_truncated2}). \n}\n\n\\REV{\nLet now $(\\overline u_i^1)_{0\\leq i \\leq n},(\\overline u_i^2)_{0\\leq i \\leq n}\\in Z$ and let $(u_i^1)_{0\\leq i \\leq n}:= F\\left( (\\bar u_i^1)_{0\\leq i \\leq n}\\right)$ and $(u_i^2)_{0\\leq i \\leq n}:= F\\left( (\\bar u_i^2)_{0\\leq i \\leq n}\\right)$.\nThen, we have\n\\begin{equation*}\n\\left\\{\n\\begin{array}{rll}\n\\partial_t (u_i^1-u_i^2) - K\\Delta (u_i^1-u_i^2) & = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)({\\tilde u_j^\\diamond}\\Delta  (\\overline u_i^1-\\overline u_i^2)- {\\tilde u_i^\\diamond} \\Delta (\\overline  u_j^1-\\overline u_j^2)),& \\\\\n\\nabla (u_i^1-u_i^2) \\cdot \\textbf{n} & = 0 ,& \\\\\n(u_i^1-u_i^2)(0,\\cdot) & = 0\n\\end{array} \\right.\n\\quad i=0,\\ldots, n.\n\\end{equation*}\nTaking the $L^2((0,T);L^2(\\Omega))$-norm on both sides, noting that both $\\bar u_i^1 - \\bar u_i^2 \\in W_0$ and $u_i^1 - u_i^2 \\in W_0$, and (\\ref{eq:normestimate}), yields\n\\begin{align*}\n\\|u_i^1 - u_i^2 \\|_{\\widetilde W_0} \\le \\|u_i^1 - u_i^2 \\|_{W_0} &\\le \\kappa n \\|\\Delta(\\overline u_i^1 - \\overline u_i^2)\\|_{L^2(\\Omega \\times (0,T))} + \\kappa \\sum_{j=0, j\\neq i}^n \\|\\Delta (\\overline  u_j^1-\\overline u_j^2)\\|_{L^2(\\Omega \\times (0,T))}\\\\\n&= \\frac{\\kappa n}{K} K\\|\\Delta(\\overline u_i^1 - \\overline u_i^2)\\|_{L^2(\\Omega \\times (0,T))} + \\sum_{j=0, j\\neq i}^n \\frac{\\kappa }{K}K\\|\\Delta (\\overline  u_j^1-\\overline u_j^2)\\|_{L^2(\\Omega \\times (0,T))},\n\\end{align*}\n\\REVTWO{where we used that by Assumption \\ref{ass:A2} we have $|K_{ij} - K| \\le \\kappa$ and by definition $\\tilde u_i^\\diamond \\le 1$.} Summing over $i=0,\\ldots,n$, we obtain\n\\begin{align}\n\\begin{split}\n\\sum_{i=0}^n \\|u_i^1 - u_i^2 \\|_{\\widetilde W_0} &\\le  \\sum_{i=0}^n\\left(\\frac{\\kappa n}{K} K\\|\\Delta(\\overline u_i^1 - \\overline u_i^2)\\|_{L^2(\\Omega \\times (0,T))} + \\sum_{j=0, j\\neq i}^n \\frac{\\kappa }{K}K\\|\\Delta (\\overline  u_j^1-\\overline u_j^2)\\|_{L^2(\\Omega \\times (0,T))}\\right)\\\\ \n&\\le \\underbrace{\\frac{2\\kappa n}{K}}_{< 1} \\sum_{i=0}^n \\|\\bar u_i^1 - \\bar u_i^2\\|_{\\widetilde W_0}.\n\\end{split}\\label{eq:contraction}\n\\end{align}\n\n\\REVV{\nLet us now introduce the distance $d:Z\\times Z \\to \\mathbb{R}_+$ defined by\n$$\n\\forall u:=(u_i)_{0\\leq i \\leq n},\\, v:=(v_i)_{0\\leq i \\leq n} \\in Z, \\quad d(u,v):= \\sum_{i=0}^n \\|u_i - v_i\\|_{\\widetilde W_0}.\n$$\nThen, $(Z,d)$ is a complete metric space, and (\\ref{eq:contraction}) implies that the map $F$ is a contraction with respect to $d$. \nBanach's fixed point theorem then ensures the existence and uniqueness of a strong solution $ u = (u_0,\\ldots, u_n) \\in Z$ to the equation \\eqref{eq:cross_truncated}.}\n}\n\n\\medskip\n\n\\itshape Step 2 (A priori estimates): \\normalfont \nDenoting again by $u:=(u_0, \\ldots, u_n)$, we can rewrite the system (\\ref{eq:cross_truncated}) as follows:\n\\begin{align}\\label{eq:matrix}\n\\partial_t u = (P-B(\\tilde u) )\\Delta u\n\\end{align}\nwhere $P= KI$ with $I$ the identity matrix in $\\mathbb{R}^{(n+1)\\times (n+1)}$ and\n\\begin{align*}\nB(\\tilde u):=\\begin{pmatrix}\n\\displaystyle \\sum_{j=0, j\\neq 0}^n(K_{0j}-K)\\tilde{u}_j^\\diamond & \\dots & -(K_{0n}-K)\\tilde{u}_0^\\diamond\\\\\n\\vdots & \\ddots & \\vdots \\\\\n-(K_{n0}-K)\\tilde{u}_n^\\diamond & \\dots &   \\sum_{j=0, j\\neq n}^n(K_{nj}-K)\\tilde{u}_j^\\diamond\\\\\n\\end{pmatrix}.\n\\end{align*}\nFor any $\\xi \\in \\mathbb{R}^{n+1}$, we have, using Assumption \\ref{ass:A2},\n\\begin{align*}\n \\xi^T (P-B(\\tilde u)) \\xi &= \\sum_{i=0}^n K \\xi_i^2 + \\sum_{i=0}^n\\sum_{j=0,i\\neq j}^n (K_{ij}-K)\\left( \\tilde u_j^\\diamond \\xi_i^2 - \\tilde u_i^\\diamond \\xi_i\\xi_j\\right)\\\\\n &\\ge \\left(\\displaystyle K - n\\kappa \\right) \\|\\xi\\|_2^2 - \\sum_{i=0}^n\\sum_{j=0,i\\neq j}^n (K_{ij}-K) \\tilde u_i^\\diamond \\xi_i\\xi_j.\\\\\n\\end{align*}\nBesides, using again Assumption \\ref{ass:A2}, it holds that\n\\begin{align*}\n &\\left| \\sum_{i=0}^n \\sum_{j=0,i\\neq j}^n (K_{ij} - K)\\tilde u_i^\\diamond \\xi_i\\xi_j\\right| \\le \\sum_{i=0}^n \\sum_{j=0,i\\neq j}^n\\kappa |\\tilde u_i^\\diamond| |\\xi_i\\xi_j|\\\\\n &\\le\\frac{1}{2}\\sum_{i=0}^n\\sum_{j=0,i\\neq j}^n\\kappa \\left(\\xi_i^2+\\xi_j^2\\right)\n \\le \\frac{n}{2}\\kappa  \\|\\xi\\|_2^2 + \\frac{n}{2}\\kappa  \\|\\xi\\|_2^2.\n\\end{align*}\nWe thus obtain\n\\begin{align}\\label{eq:coercivity}\n \\xi^T (P-B(\\tilde u)) \\xi \\ge \\left( K \\; - 2n\\kappa  \\right) \\|\\xi\\|_2^2.\n\\end{align}\nNow multiplying \\eqref{eq:matrix} by the vector $(-\\Delta u_0, \\ldots, -\\Delta u_n)^T$ and integrating over $\\Omega$, we obtain that for almost all $t\\in (0,T)$,\n\\begin{align*}\n-\\sum_{i=0}^n(\\partial_t u_i, \\Delta u_i)_{L^2(\\Omega)} +  \\left( K - 2n\\kappa  \\right) \\sum_{i=0}^n\\int_\\Omega (\\Delta u_i)^2\\;dx \\le 0,\n\\end{align*}\nwhich implies, using Lemma \\ref{lem:timederiv} and integrating in time between $0$ and $t$, \n$$\n\\sum_{i=0}^n\\|\\nabla u_i(\\cdot, t)\\|_{L^2(\\Omega)}^2 + \\left( K - 2n\\kappa  \\right)\\sum_{i=0}^n\\int_0^t\\int_\\Omega (\\Delta u_i)^2\\;dxdt\n\\le \\sum_{i=0}^n\\|\\nabla u_i^0\\|_{L^2(\\Omega)}^2.\n$$\nWe thus get\n\\begin{align}\\label{eq:h2linear}\n\\sum_{i=0}^n\\sup_{0\\leq t \\leq T} \\|\\nabla u_i(\\cdot, t)\\|_{L^2(\\Omega)}^2 + &\\left( K - 2n\\kappa  \\right)\\sum_{i=0}^n\\int_0^T\\int_\\Omega (\\Delta u_i)^2\\;dxdt\n\\le 2 \\sum_{i=0}^n\\|\\nabla u_i^0\\|_{L^2(\\Omega)}^2, \n\\end{align}\nwhich immediately yields (\\ref{eq:ineq1}).\nOn the other hand, multiplying \\eqref{eq:matrix} by the vector $(u_0, \\ldots, u_n)^T$ and integrating over $\\Omega$, we obtain that for almost all $t\\in (0,T)$,\n\\begin{align*}\n\\sum_{i=0}^n\\frac{d}{dt}\\left( \\left\\| u_i(t,\\cdot)\\right\\|^2_{L^2(\\Omega)}\\right) +\\sum_{i=0}^n\\int_\\Omega K |\\nabla u_i|^2\\;dx & = \\sum_{i=0}^n \\sum_{j=0, j\\neq i}^n (K_{ij}-K)\\int_\\Omega ({\\tilde u_j^\\diamond}u_i \\Delta u_i- {\\tilde u_i^\\diamond}u_i \\Delta u_j)\\;dx \\\\\n& \\leq  2n \\kappa  \\sum_{i=0}^n \\left( \\|\\Delta u_i(\\cdot, t)\\|_{L^2(\\Omega)}^2 +  \\|u_i(t,\\cdot)\\|_{L^2(\\Omega)}^2 \\right).\\\\\n\\end{align*}\nApplying Gronwall's lemma, we thus obtain that \n\\begin{equation}\n\\mathop{\\sup}_{0\\leq s \\leq T}\\sum_{i=0}^n \\left\\| u_i(t,\\cdot)\\right\\|^2_{L^2(\\Omega)} \\leq e^{2n \\kappa  T}\\left( \\sum_{i=0}^n \\left\\| u_i^0\\right\\|^2_{L^2(\\Omega)} + 2n \\kappa  \\|\\Delta u_i(\\cdot, t)\\|_{L^2(0,T; L^2(\\Omega))}^2 \\right),\n\\end{equation}\nwhich yields (\\ref{eq:ineq2}).\nLastly, using (\\ref{eq:cross_truncated}), we obtain that\n$$\n\\sum_{i=0}^n \\left\\| \\partial_t u_i \\right\\|_{L^2(0,T, L^2(\\Omega))}^2 \\leq \\left(  K + 2n \\kappa  \\right) \\sum_{i=0}^n \\left\\| \\Delta u_i \\right\\|^2_{L^2(0,T; L^2(\\Omega))}, \n$$\nwhich immediately yields estimate (\\ref{eq:ineq3}).\n\\end{proof}\n\n\n\n\n\\subsection{Proof of Theorem \\ref{thm:existencestrong}}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm:existencestrong}]\n\n\nLet us now assume that $u^0:=(u_0^0, u_1^0, \\cdots, u_n^0) \\in H^1(\\Omega)^{n+1}$ satsifies $u^0(x)\\in \\overline{P}$ for almost all $x\\in \\Omega$ and use the same notation as in the preceding section. \n\n\\medskip\n\n\\REV{Let us denote by $\\mathcal{M}$ the set of functions $(u_0, \\ldots, u_n)\\in \\REVV{Z}$ satisfying (\\ref{eq:ineq1}), (\\ref{eq:ineq2}) and (\\ref{eq:ineq3})\nwith constants $C_0$, $C_1$ and $C_2$ defined by (\\ref{eq:C0}), (\\ref{eq:C1}) and (\\ref{eq:C2}) respectively. For all $(\\tilde u_0, \\ldots, \\tilde u_n)\\in \\mathcal{M}$, let us denote by \n$S\\left( (\\tilde u_0, \\ldots, \\tilde u_n) \\right):= (u_0, \\ldots, u_n)\\in \\REVV{Z}$, where $(u_0, \\ldots, u_n)$ is the unique strong solution of (\\ref{eq:cross_truncated}). \n}\n\n\\medskip\n\nIn view of Lemma \\ref{thm:existencestrong}, the operator $S: \\mathcal{M} \\to \\REVV{Z}$ is well-defined and self-mapping, i.e. $S(\\mathcal{M}) \\subset \\mathcal{M}$. \nMoreover, due to the Aubin-Lions lemma \\cite[Theorem~5.1, p. 58]{lions1969quelques}, the set $\\mathcal{M}$ is a convex compact subset of $L^2((0,T);L^2(\\Omega))$. \n\\REV{In order to apply Brouwer's fixed point theorem, it remains to show that $S$ is continuous. We consider a sequence $(\\tilde u^\\delta_0, \\ldots, \\tilde u_n^\\delta)_{\\delta >0} \\subset \\mathcal{M}$ which strongly \nconverges in $L^2((0,T);L^2(\\Omega))$ to some $(\\tilde u_0, \\ldots, \\tilde u_n)\\in \\mathcal{M}$. Thus if for all $\\delta >0$, we denote by $(u_0^\\delta, \\ldots, u_n^\\delta)\\in \\REVV{Z}$ the unique solution to\n\\begin{align}\\label{eq:cross_truncated_regularized}\n\\begin{split}\n\\partial_t u^\\delta_i(x,t) - K\\Delta u^\\delta_i&= \\sum_{j=0, j\\neq i}^n(K_{ij}-K)(({\\tilde u^\\delta_j)^\\diamond}\\Delta u^\\delta_i- ({\\tilde u^\\delta_i})^\\diamond \\Delta u^\\delta_j),\\\\\n\\nabla u^\\delta_i \\cdot \\textbf{n} &= 0,\\\\ \n\\end{split}\n\\quad \ni=0,\\ldots,n.\n\\end{align}\nwith initial condition $(u_0^0, \\ldots, u_n^0)$. Using the a priori estimates \\eqref{eq:ineq1}, \\eqref{eq:ineq2} and \\eqref{eq:ineq2} we  obtain that the sequence\n$(u_0^\\delta, \\ldots, u_n^\\delta)_{\\delta >0}$ is thus bounded in $W^{n+1}$. Up to the extraction of a subsequence, there exists $(u_0, \\ldots,u_n)\\in  W^{n+1}$ such that $(u_i^\\delta)_{\\delta>0}$ weakly \nconverges in $W$ to $u_i$ for all $0\\leq i \\leq n$. In addition we have that $(\\tilde u^\\delta_i)^\\diamond \\to \\tilde u_i^\\diamond$ as the mapping $u \\mapsto u^\\diamond$ is Lipschitz continuous with Lipschitz constant $1$. \nThis allows us to pass to the limit $\\delta \\to 0$ in (\\ref{eq:cross_truncated_regularized}) in the distributional sense, and yields the continuity of $S$.\n}\n\\medskip\n\nThus, we can apply Brouwer's fixed point theorem and conclude to the existence of a strong solution $(u_0,\\ldots, u_n) \\in W^{n+1}$ to the regularized system\n\\REV{\n\\begin{equation}\\label{eq:cross_truncated3}\n\\left\\{\n\\begin{array}{rll}\n\\partial_t u_i - K\\Delta u_i & = \\sum_{j=0, j\\neq i}^n(K_{ij}-K)(u_j^\\diamond\\Delta u_i- u_i^\\diamond \\Delta u_j),& \\\\\n\\nabla u_i \\cdot \\textbf{n} & = 0 ,& \\\\\nu_i(0,\\cdot) & = u_i^0,  \\\\\n\\end{array} \\right.\n\\quad i=0,\\ldots, n,\n\\end{equation}\nwhich satisfies the a priori estimates\n\\begin{align*}\n\\sum_{i=0}^n\\mathop{\\sup}_{0\\leq t \\leq T} \\|\\nabla u_i(\\cdot, t)\\|_{L^2(\\Omega)}^2 + \\sum_{i=0}^n\\int_0^T\\int_\\Omega (\\Delta u_i)^2 &\n\\leq C_0,\\\\ \n\\mathop{\\sup}_{0\\leq s \\leq T}\\sum_{i=0}^n \\left\\| u_i(t,\\cdot)\\right\\|^2_{L^2(\\Omega)} & \\leq C_1,\\\\ \n\\sum_{i=0}^n \\left\\| \\partial_t u_i\\right\\|^2_{L^2(0,T; L^2(\\Omega)}) & \\leq C_2,\\\\ \n\\end{align*}\nwhere $C_0$, $C_1$ and $C_2$ are defined respectively in (\\ref{eq:C0}), (\\ref{eq:C1}) and (\\ref{eq:C2}). \n}\n\n\\medskip\n\nTo end the proof, it remains to show that $u_i\\geq 0$ almost everywhere in $(0,T)\\times \\Omega$ for all $0\\leq i \\leq n$. \n\\REV{First note that $(u_0,\\cdots,u_n)$ satisfies the system of equations\n\\begin{align}\\label{eq:strong_linear}\n \\left\\{\n \\begin{array}{l}\n  \\partial_t u_i - A_i(x,t) \\Delta u_i = B_i(x,t) u_i^\\diamond,\\\\\n  \\nabla u_i \\cdot \\textbf{n} = 0 \\; \\mbox{ on }(0,T)\\times \\partial \\Omega,\\\\\n  u_i(t=0,\\cdot) = u_i^0,\\\\ \n \\end{array}\n \\right.\n\\; i =0,\\ldots, n,\n\\end{align}\nwith \n\\begin{align}\nA_i(x,t) := \\left(K- \\sum_{j=0, j\\neq i}^n(K_{ij}-K)u^\\diamond_j\\right)\\text{ and } B_i(x,t) :=\\left(-\\sum_{j=0, j\\neq i}^n(K_{ij}-K) \\Delta u_j\\right).\n\\end{align}\nIn particular, if we consider $(A_i)_{0\\leq i \\leq n}$ and $(B_i)_{0\\leq i \\leq n}$ as given coefficients, \\REVV{it holds that there exists a unique solution $(u_0, \\cdots, u_n)\\in Z$ to \\eqref{eq:strong_linear}. \nIndeed, this can be shown using the same arguments as in the proof} of Lemma~\\ref{lem:strong_uniqueness}, i.e. testing both with the difference of two solutions \nas well as the Laplace of that difference and using again Lemma~\\ref{lem:reg}. \n}\n\n\\REV{\nIn order to show the desired non-negativity, we regularise the coefficients $A_i(x,t)$ (with respect to to the $x$ variable) by convolving it with a smooth kernel. \\REVV{More precisely, \nlet $\\eta \\in \\mathcal{C}^\\infty(\\mathbb{R}^d)$ be a standard non-negative mollifier so that $\\int_{\\mathbb{R}^d}\\eta  = 1$ and for all $\\varepsilon>0$, let us denote by $\\eta_\\varepsilon(x):=\\frac{1}{\\varepsilon^d}\\eta(x\/\\varepsilon)$. \nNote that $K + n \\kappa \\geq A_i(x,t) \\geq K - n\\kappa$ a.e. in $(0,T)\\times \\Omega$. We extend $A_i(t,x)$ to a function defined over $(0,T)\\times \\mathbb{R}^d$ by defining\n$$\n\\overline{A}_i(x,t):=\\left\\{\n\\begin{array}{ll}\n A_i(t,x) & \\mbox{ if } x\\in \\Omega,\\\\\n K & \\mbox{ otherwise}.\\\\\n\\end{array}\n\\right.\n$$\nWe then define for all $\\varepsilon>0$,\n$$\nA_i^\\varepsilon(t,x):= \\int_{\\Omega} \\overline{A}_i(t,y) \\eta_\\varepsilon(x-y)\\,dy, \\quad \\forall (t,x)\\in (0,T)\\times \\Omega.\n$$\nThen, it holds that for all $\\varepsilon >0$, $K + n \\kappa \\geq A^\\varepsilon_i(x,t) \\geq K - n\\kappa$ a.e. in $(0,T)\\times \\Omega$ and for almost all $t\\in (0,T)$, \n$\\displaystyle A_i^\\varepsilon(t,\\cdot) \\mathop{\\longrightarrow}_{\\varepsilon \\to 0} A_i(t,\\cdot)$ strongly in $L^p(\\Omega)$ for all $0<p<\\infty$. }\n}\n\nWe also denote by $(u_i^\\varepsilon)_{0\\leq i \\leq n}$ \nthe (unique) solution in $Z$ to\n\\begin{align}\\label{eq:linear_regularised}\n\\partial_t u_i^\\varepsilon - A_i^\\varepsilon \\Delta u_i^\\varepsilon = B_i(x,t) u_i^{\\varepsilon, \\diamond},\\; i =0,\\ldots, n.\n\\end{align}\nThe existence and uniqueness of strong solutions in $Z$ to (\\ref{eq:linear_regularised}) can be obtained again using similar arguments as above.\nWe claim that for all $0\\leq i \\leq n$, $u_i^\\varepsilon \\ge 0$. Indeed, multiplying (\\ref{eq:linear_regularised}) with the negative part $(u_i^\\varepsilon)_-$ and integrating over the spatial domain $\\Omega$ gives\n\\begin{align*}\n \\frac{d}{dt} \\int_\\Omega \\left|(u_i^\\varepsilon)_-\\right|^2  + C\\int_\\Omega |\\nabla (u_i^\\varepsilon)_-|^2 \\le  \\underbrace{\\int_\\Omega B_i u_i^{\\diamond,\\varepsilon} (u_i^\\varepsilon)_- }_{=0} + \\int_\\Omega |\\nabla A_i^\\varepsilon| |\\nabla (u_i^\\varepsilon)_-| (u_i^\\varepsilon)_-,\n\\end{align*}\ni.e.\n\\begin{align*}\n \\frac{d}{dt} \\int_\\Omega |(u_i^\\varepsilon)_-|^2  + (C-\\gamma)\\int_\\Omega |\\nabla (u_i^\\varepsilon)_-|^2 \\le  \\frac{1}{4\\gamma} \\|\\nabla A_i^\\varepsilon \\|_{L^\\infty(\\Omega)}\\int_\\Omega |(u_i^\\varepsilon)_-|^2,\n\\end{align*}\nfor any $\\gamma >0$, so that Gronwall's lemma implies $u_i^\\varepsilon \\ge 0$ a.e. as $(u_i^0)_- = 0$ a.e. in $(0,T)\\times \\Omega$. Besides, there exists  $(w_i)_{0\\leq i \\leq n}\\in W_n^{n+1}$ such that, up to the extraction of a subsequence,\n\\begin{align*}\n \\partial_t u_i^\\varepsilon \\rightharpoonup \\partial_t w_i \\text{ in } L^2(0,T;L^2(\\Omega))\\;\\text{ and }\\; \\Delta u_i^\\varepsilon  \\rightharpoonup \\Delta w_i \\text{ in } L^2(0,T;L^2(\\Omega)),\n \\end{align*}\n as well as \n \\begin{align*}\n A_i^\\varepsilon &\\to A_i \\text{ in } L^p((0,T)\\times \\Omega),\\quad \\text{ for every } p < \\infty, \\text{ since } A_i\\in L^\\infty((0,T)\\times \\Omega),\\\\\n u_i^\\varepsilon &\\to w_i \\text{ in } L^2(0,T; L^2(\\Omega)), \\quad \\text{by compactness of the  embedding}.\n\\end{align*}\nThe last convergence implies $u_i^{\\varepsilon,\\diamond} \\to w_i^\\diamond$ in $L^2(0,T; L^2(\\Omega))$. Thus testing \\eqref{eq:linear_regularised} with $C^\\infty_0$-functions \n(which are dense in $L^2(\\Omega)$) we can pass to the limit $\\varepsilon \\to 0$ in (\\ref{eq:linear_regularised}) and conclude that \n\\REVV{\n\\begin{itemize}\n \\item $(w_i)_- = 0$ a.e. in $\\Omega$ for a.e. $t \\in (0,T)$ for all $0\\leq i \\leq n$;\n \\item $w_i(t=0,\\cdot) = u_i^0$;\n \\item $(w_0,\\cdots,w_n)$ is a solution in $Z$ to (\\ref{eq:strong_linear}).\n\\end{itemize}\n}\nUsing the uniqueness of strong solutions in $Z$ to (\\ref{eq:strong_linear}), we thus obtain that $w_i = u_i$ for all $0\\leq i \\leq n$, which implies that $u_i\\geq 0$. \\REVTWO{Finally, to show the upper bound on the $u_i$ we note that the sum $\\bar u = \\sum_{i=0}^n u_i$ satisfies the heat equation\n\t\\begin{align}\\label{eq:heat}\n\t\\left\\{\\begin{array}{l}\n\t\\partial_t \\bar u - K\\Delta \\bar u = 0,\\\\\n\t\\nabla \\bar u \\cdot \\mathbf{n} = 0,\\\\\n\t\\bar u(x,0) = \\sum_{i=0}^n u_i^0.\n\t\\end{array}\\right.\n\t\\end{align}\nAs the initial data was such that $\\sum_{i=0}^n u_i^0 = 1$, the unique solution to \\eqref{eq:heat} is $\\bar u(x,t)=1$ for a.e. $x \\in \\Omega$, $t \\in (0,T)$ and thus $u(t,x)\\in \\overline{\\mathcal{P}}$ almost everywhere.}\n\\end{proof}\n\n\\section{Weak strong stability}\\label{sec:stability}\n\n\nThis section is devoted to the proof of Theorem~\\ref{thm:stability}, which provides a weak-strong stability result provided that there exists a strong solution $u$ to the system of interest which satisfies the additional regularity property $\\nabla u \\in L^2(0,T; L^\\infty(\\Omega))$. \n\\begin{proof}[Proof of Theorem~\\ref{thm:stability}]\nWe start by rewriting the $i^{th}$ component of \\eqref{eq:crossnplusone2} as \n\\begin{align*}\n \\int_\\Omega \\partial_t u_i\\varphi \\;dx + K\\int_\\Omega \\nabla u_i \\cdot \\nabla \\varphi\\;dx = \\int_\\Omega [  \\sum_{j=1,i\\neq j}^n (K_{ij}-K)(u_j\\nabla u_i - u_i\\nabla u_i)] \\cdot \\nabla \\varphi\\;dx,\n\\end{align*}\nfor all $\\varphi \\in H^1(\\Omega)$. Denoting by \n\\begin{align*}\nD(v):=\\begin{pmatrix}\n \\sum_{j=1}^n(K_{0j}-K)v_j& \\dots & -(K_{0n}-K)v_0\\\\\n\\vdots & \\ddots & \\vdots \\\\\n-(K_{n0}-K) {v_n} & \\dots &   \\sum_{j=0 }^{n-1}(K_{nj}-K)v_j\\\\\n\\end{pmatrix}.\n\\end{align*}\nfor all $v:=(v_i)_{0\\leq i \\leq n}\\in \\overline{\\mathcal{P}}$, we obtain that\n\\begin{equation}\\label{eq:weakC}\n \\int \\partial_tu \\Phi\\;dx + \\int_\\Omega K\\nabla u\\cdot \\nabla \\Phi \\;dx = \\int_{\\Omega} D(u)\\nabla u \\cdot \\nabla \\Phi\\;dx,\\text{ for all } \\Phi \\in [H^1(\\Omega)]^{n+1},\n\\end{equation}\nSince we know that $\\sum_{i=0}^n u_i = 1$ and that $u_i \\ge 0$ for $i=0,\\ldots, n$, we immediately obtain that \n\\begin{align*}\n \\|D(u)\\|_{L^\\infty(\\Omega)} \\le 2 n\\kappa ,\n\\end{align*}\nin the sense of the spectral matrix norm. In addition, $D: \\overline{\\mathcal{P}} \\to \\mathbb{R}^{(n+1)\\times (n+1)}$ is Lipschitz continuous, with Lipschitz constant $2n \\kappa$. \nNow we consider the difference of the respective weak formulations \\eqref{eq:weakC} for $u$ and $\\tilde u$ and obtain\n\\begin{align*}\n\\int_\\Omega \\partial_t (u-\\tilde u)\\Phi \\;dx - K \\int_\\Omega  \\left(\\nabla u - \\nabla \\tilde u\\right)\\cdot \\nabla \\Phi \\;dx = \\int_\\Omega \\left[D(u)\\nabla u - D(\\tilde u)\\nabla\\tilde u\\right]\\cdot \\nabla \\Phi\\;dx.\n\\end{align*}\nTaking  $\\Phi = (u-\\tilde{u})(t, \\cdot)$ (which belongs to $H^1(\\Omega)$ for almost all $t\\in (0,T)$) yields\n\\begin{align*}\n&\\frac{d}{dt}\\frac{1}{2}\\|u-\\tilde{u}\\|_{L^2(\\Omega)}^2+K \\|\\nabla(u-\\tilde{u})\\|_{L^2(\\Omega)}^2\\\\\n&= - \\int_\\Omega (D(u)-D(\\tilde{u}))\\nabla u\\cdot \\nabla(u-\\tilde{u}) \\;dx-\\int_\\Omega D(\\tilde{u})\\nabla(u-\\tilde{u})\\cdot \\nabla(u-\\tilde{u})\\;dx\n\\end{align*}\nUsing the fact that $\\|D(\\tilde{u})\\|_{L^{\\infty}(\\Omega)} \\leq 2n \\kappa $ on the second term of the right hand side, we obtain\n\\begin{align*}\n&\\frac{d}{dt}\\frac{1}{2}\\|u-\\tilde{u}\\|_{L^2(\\Omega)}^2+(K-2n\\kappa) \\|\\nabla(u-\\tilde{u})\\|_{L^2(\\Omega)}^2 \\leq - \\int_\\Omega (D(u)-D(\\tilde{u}))\\nabla u \\cdot \\nabla(u-\\tilde{u})\\;dx.\n\\end{align*}\nSince $d=1$, it holds that $L^\\infty(\\Omega) \\subset H^1(\\Omega)$ with continuous injection, which implies that $\\|\\nabla u \\|_{L^\\infty(\\Omega)} \\in L^2(0,T)$ (since $u\\in L^2((0,T), H^2(\\Omega))$. \nThus, applying the weighted Young's inequality with $0 < \\epsilon <(K-2n\\kappa)$ and using the Lipschitz continuity of $D$ yield \n\\begin{align*}\n& \\frac{d}{dt}\\frac{1}{2}\\|u-\\tilde{u}\\|_{L^2(\\Omega)}^2+(K-2n\\kappa-\\epsilon) \\|\\nabla(u-\\tilde{u})\\|_{L^2(\\Omega)}^2\\leq \\frac{1}{4\\epsilon} \\|(D(u)-D(\\tilde{u}))\\nabla u\\|_{L^2(\\Omega)}^2 \\\\\n& \\leq \\frac{1}{4\\epsilon}\\|\\nabla u\\|_{L^\\infty(\\Omega)}^2 \\|D(u)-D(\\tilde{u})\\|_{L^2(\\Omega)}^2 \\\\\n& \\leq \\frac{2n\\kappa}{4\\epsilon}\\|\\nabla u\\|_{L^\\infty(\\Omega)}^2 \\|u-\\tilde{u}\\|_{L^2(\\Omega)}^2.\\\\ \n\\end{align*}\n\nApplying the differential form of the Gronwall lemma then implies that there exists $C'>0$ such that for all $t\\in (0,T)$, \n\\begin{align*}\n\\|u(t,\\cdot)-\\tilde{u}(t,\\cdot)\\|_{L^2(\\Omega)}^2 \\leq e^{C'\\|\\nabla u\\|^2_{L^2((0,t), L^\\infty(\\Omega))}}\\|u(0,\\cdot)-\\tilde{u}(0,\\cdot)\\|_{L^2(\\Omega)}^2,\n\\end{align*}\nwith $C' = \\frac{2n\\kappa}{4\\epsilon}$. Hence the result.\n\\end{proof}\n\\begin{remark} Let us remark that in dimension one, a strong solution $u$ in the sense of Theorem~\\ref{thm:existencestrong} necessarily satisfies $\\nabla u \\in L^2(0,T; L^\\infty(\\Omega))$ since the injection $H^2(\\Omega) \\hookrightarrow W^{1,\\infty}(\\Omega)$ is continuous. To extend this results in higher dimension, one would need to prove the existence of solutions with this additional regularity property, for instance with more regular initial data.\n\\end{remark}\n\n\\section*{Appendix: Microscopic interpretation}\nFollowing \\cite{Burger2010}, we briefly describe a lattice based modelling approach and a formal way to obtain a \\eqref{eq:cross} in the limit. We start with a one-dimensional lattice on which particles of $i=1,\\ldots n$ \nspecies can jump to neighbouring sites. Let $\\mathcal{T}_h$ denote an equidistant\ngrid of mesh size $h$, where a cell is either empty or can be occupied by at most one particle. We denote the probability to find a particle of species $i$ at location $x$ and time $t$ by\n\\begin{align*}\nc_i(x,t)&=P(\\text{particle of species $i$ at position $x$ at time $t$}),\n\\end{align*}\nand assume that the motion of these particles is due to two different effects: Diffusion and exchange (switching) of particles of different species. To this end, we introduce the rates\n\\begin{align}\\label{eq:rate1}\n\\Pi^{+}_{c_i}&=P(\\text{jump of $c_i$ from position $x$ to $x+h$ in ($t,t+\\Delta t)$})\\\\\n&=K_{i0}(1-\\rho) + \\sum_{j=1,\\,i\\neq j}^n K_{ij}c_j,\\\\\\nonumber\n\\Pi^{-}_{c_i}&=P(\\text{jump of $c_i$ from position $x$ to $x-h$ in ($t,t+\\Delta t)$})\\\\\n&=K_{i0}(1-\\rho) + \\sum_{j=1,\\,i\\neq j}^n K_{ij}c_j.\n\\end{align}\nHere $K_{i0}$ is a diffusion coefficient which controls the tendency of a particle to jump to a neighboring site. Since we restrict to at most one particle per site, \nthis has to be modified by a factor of $(1-\\rho)$, i.e. the particle can only jump if the target site is empty. On the other hand, in order to exchange places with a particle from a different species, \nthe target site has to be occupied and thus, for the second term we have to multiply the rate $K_{ij}$ with $c_j$. \n\nNow we consider the following cases: If $K_{i0} \\gg K_{ij}$, then the probability of switching is small compared to that of diffusion and the effect of size exclusion will be essential. \nIf, on the other hand $K_{i0} \\ll K_{ij}$, switching will dominate and size exclusion will not play a role anymore. Note that in this case, $\\rho$, which is the sum of all densities, remains constant.\n\nOur subsequent analysis deals with the case when $K_{i0} \\approx K_{ij}$, which is the most interesting. In fact, let us rewrite \\eqref{eq:rate1} as follows:\n\\begin{align*}\n\\Pi^{+}_{c_i}&=K_{i0}(1-\\sum_{j=1,i\\neq j}^n c_j - c_i) + \\sum_{j=1,\\,i \\neq j}^n K_{ij}c_j,\\\\\\nonumber\n&= K_{i0}(1- c_i) + \\sum_{j=1,\\,i\\neq j}^n (K_{ij}-K_{i0})c_j.\n\\end{align*}\nNow if $K_{i0} \\approx K_{ij}$, the switching will effectively aneal the size exclusion effect. In other words, it does not make a difference whether a target site is occupied by a particle of species $j$ or if it is empty \nsince in both cases, the particle at the source site can reach this target: Either by jumping to the empty cell or by switching positions. The resulting PDE can be written as \n\\begin{align*}\n\\partial_t c_i &= \\nabla \\cdot (K_{i0}((1-c_i)\\nabla c_i + c_i\\nabla c_i + \\sum_{j=1,i\\neq j}^n (K_{i0}-K_{ij})(c_j \\nabla c_i - c_i \\nabla _j))\\\\\n&=\\nabla \\cdot (K_{i0}c_i + \\sum_{j=1,i\\neq j}^n (K_{i0}-K_{ij})(c_j \\nabla c_i - c_i \\nabla c_j)),\\quad i=1,\\ldots, n.\n\\end{align*}\nwhich reveals that we are dealing with a perturbation of the heat equations, as already entailed in \\eqref{eq:heatperturbation} in the introduction.\n\n\n\\section*{Acknowledgements}\nThe work of MB has been supported by ERC via Grant EU FP 7 - ERC Consolidator Grant 615216 LifeInverse. MB and JFP acknowledge support by the German Science Foundation DFG via EXC 1003 Cells in Motion Cluster of Excellence, M\\\"unster. VE acknowledges support by the ANR via the ANR JCJC COMODO project. VE and JFP are grateful to the DAAD\/ANR for their support via the project 57447206. Furthermore, the authors would like to thank Robert Haller-Dintelmann (TU Darmstadt) for useful discussions. We would also like to thank the anonymous referee for his very useful comments and suggestions.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\\label{sec:intro}\n\\setcounter{equation}{0}\n\nRegge calculus is a coordinate free geometric formalism of gravitation \non triangulated piecewise linear manifolds \\cite{Regge:1961aa, MTW}. \nIt is envisaged from classical to quantum as an approach of \nEinstein gravity to problems where analytic methods cannot be reachable. \nThough Regge theory or its evolved ones have brought considerable \nprogress in our understanding of quantum gravity, in particular in \ntwo and three dimensions, efforts to develop the formalism are \nvigorously continued to overcome the conceptual and technical \ndifficulties \\cite{BOW:2018aa}. \n\nAs in continuum gravity, Regge calculus allows exact solutions \nfor systems, where the numbers of variables are largely reduced \nby some symmetry. They are expected not only to play a role of \na test tube to examine the validity of Regge calculus but to \nexpose origins of intriguing geometrical properties of gravitation \nsuch as dynamical behaviors of space-time and black hole singularities. \nAlong this line of thought Regge calculus has been applied \nto spherically symmetric static geometries such as the Schwarzschild \nspace-time \\cite{Wong:1971} and the \nFriedmann--Lema\\^itre--Robertson--Walker (FLRW) \nuniverse \\cite{CW:1973aa,Brewin:1987aa,LW:2015aa, LW:2015ab, LW:2015ac}. \nMost researches assume realistic four dimensions and\napplication of Regge calculus to higher dimensions have not been \ntargeted so far. \n\nIn this paper we investigate vacuum solution of a discretized \nclosed FLRW universe with a positive cosmological constant in \nan arbitrary dimensions via\nRegge calculus. In the previous \npapers \\cite{TF:2016aa, TF:2020aa} we have analyzed \nthe FLRW universe in three and four dimensions within the \nframework of Collins--Williams (CW) formalism\\cite{CW:1973aa}. \nIt is base on $3+1$ decomposition of space-time similar to \nArnowitt--Deser--Misner (ADM) formalism in General \nRelativity \\cite{AD:1959aa, ADM:1959aa}. Three-dimensional \nspherical Cauchy surfaces are replaced by regular polytopes \nand truncated world-tubes are \ntaken as the fundamental building blocks of the discretized \nFLRW universe. \nRegge calculus describes qualitative properties \nof the continuum solution during the period small enough \ncompared with the characteristic time scale $\\sim1\/\\sqrt{\\Lambda}$, \nthe inverse square root of the cosmological constant. \nThe deviation from the continuum theory becomes apparent as time \npasses. In three dimensions the universe expands to infinity \nin a finite time, whereas it repeats expansions and contractions\nperiodically in four dimensions. \n\nIn order for Regge calculus to approximate continuum theory \nquantitatively edge lengths must be sufficiently small \ncompared both with the curvature radius and\n$1\/\\sqrt{\\Lambda}$. \nThis cannot be satisfied for regular \npolytopes since the edge lengths and their circumradii \nare of same order,\nand the minimum edge lengths are of order \n$1\/\\sqrt{\\Lambda}$. To improve the approximation we must \nintroduce nonregular polytopes with shorter edge lengths. \nA natural construction of such polytopes is geodesic dome. \nRegge calculus for them, however, becomes impractical as \nthe number of cells increases. This can be bypassed by \nworking with the pseudo-regular polytopes introduced \nin \\cite{TF:2016aa, TF:2020aa}. \nThey can be simply defined by extending the Schl\\\"afli \nsymbol of the original regular polytope to fractional or \nnoninteger one corresponding to the geodesic dome. \nWe will extend the results obtained in three and four \ndimensions to arbitrary dimensions. \n\nThis paper is organized as follows; in the next section we set up the \nregular polytopal universe by the CW formalism in arbitrary dimensions\nand formulate the Regge action in the continuum time limit. In \nSect. \\ref{sec:req} we give gauge fixed Regge equations in Lorentzian \nsignature. We describe the evolution of the polytopal universe in \ndetail. Comparison with the continuum solutions is made. \nIn Sect. \\ref{sec:prpt} we consider the pseudo-regular polytope \nhaving a $D$-cube as the parent regular polytope and define the \nfractional Schl\\\"afli symbol. Taking the infinite frequency limit, \nwe argue that the pseudo-regular polytope model can reproduce the \ncontinuum FLRW universe. Sect. \\ref{sec:sum} is devoted to summary \nand discussions. In Appendix \\ref{sec:cada}, we describe \ncircumradii and dihedral angles of regular polytopes in \narbitrary dimensions.  Appendices \\ref{sec:dadpf} and \\ref{sec:pnu} \nare to explain some technicalities. \n\n\n\n\n\n\n\\section{Regge action for a regular $D$-polytopal universe}\n\n\\label{sec:ra}\n\\setcounter{equation}{0}\n\nIn the beginning we would like to briefly summarize the FLRW universe in General Relativity.\nThe continuum gravitational action with a cosmological constant in $D$ dimensions is given by \n\\begin{align}\n  \\label{eq:cEHa}\n  S=\\frac{1}{16\\pi}\\int d^Dx\\sqrt{-g}(R-2\\Lambda).\n\\end{align}\nThe FLRW metric\n\\begin{align}\n  \\label{eq:FLRWm}\n  ds^2=-dt^2+a(t)^2\\left[\\frac{dr^2}{1-kr^2}+r^2 \\sigma_{AB} dx^A dx^B \\right]\n\\end{align}\nis an exact solution of Einstein's field equations, where $\\sigma_{AB}$ is \nthe metric tensor on $(D-2)$-dimensional unit sphere. It describes \nan expanding or contracting universe of homogeneous and isotropic \nspace. All the time dependence of the metric is included in \n$a\\left(t\\right)$, known as scale factor in cosmology.\nEinstein equations for the metric (\\ref{eq:FLRWm}) derive the \nFriedmann equations as differential equations of scale factor\n\\begin{align}\n\\label{eq:Feq}\n&\\ddot{a}= \\Lambda_D a , \\quad \\dot{a}^2 = \\Lambda_D a^2 - k ,\n\\end{align}\nwhere we have introduced $\\Lambda_D$ by\n\\begin{align}\n  \\label{eq:LamD}\n \\Lambda_D = \\frac{2\\Lambda}{ \\left( D-1 \\right) \\left( D-2 \\right) }.\n\\end{align}\nThe curvature parameter $k = 1, 0, -1$ corresponds to space being \nspherical, Euclidean, or hyperbolic, respectively. The relations \nbetween the solutions and curvature parameter are summarized in \nTable \\ref{tab:flrw} with the proviso that the behaviors of the \nuniverses are restricted to expanding at the beginning for the \ninitial condition $a\\left(0\\right)=\\min a\\left(t\\right)$. \nNote that we have assumed $a (0) = \\frac{1}{ \\sqrt{\\Lambda_D} }$ \nfor the case of $k=0$ and $ \\Lambda > 0 $.\n\n\n\n\\begin{table}[t]\n\\begin{align*}\n   \\begin{array}{cccc} \\hline\n   & k=1 & k=0 & k=-1 \\\\ \\hline\n   \\Lambda>0 & a=\\frac{1}{\\sqrt{\\Lambda_D}}\n   \\cosh\\left(\\sqrt{\\Lambda_D}t\\right) \n   & a=\\frac{1}{\\sqrt{\\Lambda_D}}\\exp\\left(\\sqrt{\\Lambda_D}t\\right) \n   & a=\\frac{1}{\\sqrt{\\Lambda_D}}\\sinh\\left(\\sqrt{\\Lambda_D}t\\right) \\\\\n   \\Lambda=0 & \\mbox{no solution} & a = \\mbox{const.} & a=t \\\\\n   \\Lambda<0 & \\mbox{no solution} & \\mbox{no solution} \n   & a=\\frac{1}{\\sqrt{-\\Lambda_D}}\\sin\\left(\\sqrt{-\\Lambda_D}t\\right) \\\\ \\hline\n   \\end{array}\n\\end{align*}\n\\caption{Solutions of the Friedmann equations.}\n\\label{tab:flrw}\n\\end{table}\n\n\n\n\\begin{table}[t]\n\\centering\n  \\begin{tabular}{clll}\\hline \n    & Name & $\\left\\{p_1,p_2,p_3,\\cdots,p_D,p_0\\right\\}$ & \n    $[D,\\kappa_D,\\lambda_D,\\mu_D,\\zeta_D]$\\\\ \\hline\n   \n    0-polytope & Point & $\\left\\{2\\right\\}$ & $[0,3,3,3,3]$\\\\\n   \n    1-polytope & Line segment & $\\left\\{2,2\\right\\}$ & $[1,3,3,3,3]$ \\\\\n   \n    2-polytope & $n$-sided polygon & $\\left\\{2,n,2\\right\\}$ & $[2,n,3,3,3]$ \\\\\n    \\\\\n   \n    \\multirow{5}{*}{3-polytope} & Tetrahedron & $ \\left\\{2,3,3,2\\right\\}$\n    & $[3,3,3,3,3]$ \\\\ \n    & Cube & $ \\left\\{2,4,3,2\\right\\}$ & $[3,4,3,3,3]$ \\\\ \n    & Octahedron & $ \\left\\{2,3,4,2\\right\\}$ & $[3,3,4,3,3]$ \\\\ \n    & Dodecahedron & $ \\left\\{2,5,3,2\\right\\}$ & $[3,5,3,3,3]$ \\\\ \n    & Icosahedron & $ \\left\\{2,3,5,2\\right\\}$ & $[3,3,5,3,3]$ \\\\\n    \\\\\n   \n    \\multirow{6}{*}{4-polytope} & 5-cell & $ \\left\\{2,3,3,3,2\\right\\}$ \n    & $[4,3,3,3,3]$\\\\\n    & 8-cell & $ \\left\\{2,4,3,3,2\\right\\}$ & $[4,4,3,3,3]$ \\\\\n    & 16-cell & $ \\left\\{2,3,3,4,2\\right\\}$ & $[4,3,3,4,3]$ \\\\\n    & 24-cell & $ \\left\\{2,3,4,3,2\\right\\}$ & $[4,3,4,3,3]$ \\\\\n    & 120-cell & $ \\left\\{2,5,3,3,2\\right\\}$ & $[4,5,3,3,3]$ \\\\\n    & 600-cell & $ \\left\\{2,3,3,5,2\\right\\}$ & $[4,3,3,5,3]$ \\\\ \n    \\\\\n   \n    \\multirow{3}{*}{\\begin{tabular}{l} $n$-polytope \\\\ \n        $\\left(n\\geq 5\\right)$ \\end{tabular}} & $n$-simplex $\\alpha_n$ & \n    $\\left\\{2,3^{n-1},2\\right\\}$ & $[n,3,3,3,3]$ \\\\\n    & $n$-orthoplex $\\beta_n$ & $\\left\\{2,3^{n-2},4,2\\right\\}$ & $[n,3,3,3,4]$ \\\\ \n    & $n$-cube $\\gamma_n$ & $\\left\\{2,4,3^{n-2},2\\right\\}$ & $[n,4,3,3,3]$ \\\\\n    \\hline\n  \\end{tabular}\n  \\caption{Extended Schl\\\"afli symbols for regular polytopes. \n    The symbol $\\{2,3^4,2\\}$ is an abbreviation of \n  $\\{2,3,3,3,3,2\\}$.\n  By H. M. S. Coxeter the $n$-simplex, $n$-orthoplex, and $n$-cube are labeled as $\\alpha_n$, $\\beta_n$, and $\\gamma_n$, respectively \\cite{Coxeter}.\n  The parameter set $ \\left[ D, \\kappa_D, \\lambda_D, \\mu_D, \\zeta_D \\right] $ is another way to specify a regular $D$-polytope introduced in Sect. \\ref{sec:req}.\n  }\n  \\label{tab:ssfrpt}\n\\end{table}\n\n\n\nAs preparation for the investigation of polytopal universes, \nwe work in Euclidean space-time for the time being and explain an \nepitome of Regge calculus; in Regge calculus, the discrete gravitational \naction is given by the Regge action\\cite{Miller:1997aa}\n\\begin{align}\n  \\label{eq:ract}\n  S_{\\rm Regge}=\\frac{1}{8\\pi}\\left(\\sum_{i\\in\\rm \\{hinges\\}}\n    \\varepsilon_iA_i-\\Lambda\\sum_{i\\in\\rm \\{blocks\\}} V_i\\right),\n\\end{align}\nwhere $A_i$ is the volume of a hinge, $\\varepsilon_i$ the deficit \nangle around the hinge of volume $A_i$, and $V_i$ the volume of a building \nblock of the piecewise linear manifold.\nThe fundamental variables in Regge calculus are the edge lengths $l_i$. \nVarying the Regge action with respect to $l_i$, we obtain the Regge \nequations\n\\begin{align}\n  \\label{eq:regeq}\n  \\sum_{i\\in \\rm \\{hinges\\}}\\varepsilon_i\\frac{\\partial A_i}{\\partial l_j}\n  -\\Lambda\\sum_{i\\in \\rm \\{ blocks \\}}\\frac{\\partial V_i}{\\partial l_j}=0.\n\\end{align}\nNote that there is no need to carry out the variation of the deficit \nangle owing to the Schl\\\"afli \nidentity\\cite{Schlafli:1858aa,HHKL:2015aa}\n\\begin{align}\n\\sum_{i\\in\\rm\\{hinges\\}}A_i\\frac{\\partial\\varepsilon_i}{\\partial l_j}=0.\n\\end{align}\n\nWe now turn to polytopal universe. According to CW formalism we \nreplace $(D-1)$-dimensional hyperspherical Cauchy surface in FLRW \nuniverse by a fixed type of regular $D$-polytope. In general a \nregular $D$-polytope for $D \\geq 2$ is characterized by a set of \n$D-1$ integer parameters $ \\left\\{p_2,p_3,\\cdots,p_D\\right\\}$, known \nas Schl\\\"afli symbol\\cite{Coxeter,Hitotsumatsu}. In this paper we \nintroduce $p_0=p_1=2$ to include the cases of $D=0,~1$ and write \nthe Schl\\\"afli symbol as $\\left\\{p_1,p_2 , p_3\\cdots,p_D,p_0\\right\\}$, which \nwill be referred to as extended Schl\\\"afli symbol. \nEach regular $D$-polytope has a corresponding dual polytope represented by the extended Schl\\\"afli symbol in reverse order $\\left\\{ p_0 , p_n , p_{n-1} , \\cdots , p_1 \\right\\}$.\nNote that there are only three types of regular polytopes in dimensions larger than \nfour: the $n$-simplex, $n$-orthoplex, and $n$-cube being, \nrespectively, higher dimensional analogs of the tetrahedron, octahedron, and \ncube in three dimensions. In \nTable \\ref{tab:ssfrpt}\\cite{TF:2020aa} we summarize all  possible \nregular polytopes in arbitrary dimensions.\n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=1]{fig_frustum.eps}\n  \\caption{The $i$-th frustum as the fundamental building block \n    of the 5-polytopal universe for $\\left\\{2,3^4,2\\right\\}$.\n    A lower cell like ABCDE for $ \\left\\{2,3,3,3,2\\right\\} $ with \n    edge length $l_i$ at the time $t_i$ evolves into an upper one \n    A$^\\uparrow$B$^\\uparrow$C$^\\uparrow$D$^\\uparrow$E$^\\uparrow$ \n    with $l_{i+1}$ at $t_{i+1}$. The 3-frustum \n    ABC-A$^\\uparrow$B$^\\uparrow$C$^\\uparrow$ having 2-simplices \n    $\\left\\{2,3,2\\right\\}$ as base faces is a temporal hinge, and \n    the 3-simplex ABCD for $\\left\\{2,3,3,2\\right\\}$ a spatial hinge.}\n  \\label{fig:fbb}\n\\end{figure}\n\n\n\nIn the present polytopal universe the fundamental building blocks \nof space-time are world-tubes of $D$-dimensional frustums with the \nregular $(D-1)$-polytopes $\\left\\{p_1,\\cdots,p_{D-1},p_0\\right\\} $ as the \nupper and lower cells. We will refer to them as $D$-frustums. \nIn Figure \\ref{fig:fbb} we give, as an illustration, a depiction of \na 5-frustum with 4-simplices as base cells. \nWe assume that the \nupper and lower cells of a block lie in two consecutive time-slices \nseparately and every strut between them has equal length. \nWe denote the volume of the $i$-th $D$-frustum by $V_i$. It \ncontains two types of the fundamental variables: the edge lengths \n$l_i$ and $l_{i+1}$ of the lower and upper $(D-1)$-polytopes, and the \nlengths of the struts $m_i$. In a $D$-dimensional piecewise linear \nmanifold, hinges are ($D-2$)-dimensional objects, where curvature \nis concentrated. \nThere are two types of hinges. One is temporally \nextended $(D-2)$-frustums with regular $(D-3)$-polytopes \n$\\left\\{p_1,\\cdots,p_{D-3},p_0\\right\\}$ as the base cells, like the frustum \n$\\mathrm{ABC\\hbox{-}A}^\\uparrow\\mathrm{B}^\\uparrow\\mathrm{C}^\\uparrow$ \nin Figure \\ref{fig:fbb}. We call them ``temporal hinges'' and denote by \n$A_i^{({\\rm t})}$ the volume of a temporal hinge between the $i$-th \nand $(i+1)$-th Cauchy surfaces. The other is spatially traversed\nregular $(D-2)$-polytopes $\\left\\{p_1,\\cdots,p_{D-2},p_0\\right\\}$ as \nfacets of a Cauchy cell, or equivalently ridges of Cauchy surface, \nsuch as $\\mathrm{ABCD}$. \nNote that in geometry a $(D-1)$-, $(D-2)$-, and $(D-3)$-dimensional face of \n$D$-polytope are also called a facet, ridge, and peak, respectively.  \nWe call the codimension two polytopes ``spatial hinges'' and denote by \n$A_i^{({\\rm s})}$ the volume of the hinge lying in the $i$-th time-slice. \n\nWe are able to write the Regge action for the polytopal universe\nby counting the numbers of temporal hinges lying \nbetween two consecutive time-slices, spatial hinges \nin a time-slice, and $D$-frustums. They are just the numbers of peaks, ridges, and facets of the $D$-polytope, respectively. \nLet $N^{(D)}_n$ be the number of $n$-dimensional faces of \na regular $D$-polytope, then the Regge action (\\ref{eq:ract}) \ncan be written as\n\\begin{align}\n  \\label{eq:regact}\n  S_\\mathrm{Regge}=\\frac{1}{8\\pi}\\sum_i\\left(N^{(D)}_{D-3}A^{({\\rm t})}_i\n    \\varepsilon_i^{\\rm (t)}+N^{(D)}_{D-2} A^{({\\rm s})}_i\\varepsilon_i^{\\rm (s)}\n    -N^{(D)}_{D-1}\\Lambda V_i\\right),\n\\end{align}\nwhere $\\varepsilon^{\\rm ({\\rm t})}_i$ and $\\varepsilon_i^{\\rm ({\\rm s})}$ are \nthe deficit angles around a temporal hinge of volume $A_i^{(\\rm t)}$ and \na spatial hinge of volume $A_i^{(\\rm s)}$, respectively. The summation is \ntaken over the time-slices. The volume of the frustum, those of hinges, \nand deficit angles can be expressed in terms of the fundamental \nvariables $l$'s and $m$'s. \n\nFor the purpose it is convenient to introduce the circumradius $\\hat R_n$ and \nvolume $\\hat{\\cal V}^{(n)}$ of a regular $n$-polytope \n$\\Pi_n= \\{p_1,p_2,\\cdots,p_n,p_0\\} $ with unit edge length. In \nAppendix \\ref{sec:cada} we give a general formula for $\\hat R_n$. \nSee (\\ref{eq:hatRn}) and (\\ref{eq:cfsinphi}). \nThe normalized \nvolume $\\hat{\\cal V}^{(n)}$ can be obtained from the recurrence \nrelation\n\\begin{align}\n  \\label{eq:rr_rpv}\n  \\hat{\\cal{V}}^{(n)}=\\frac{N^{(n)}_{n-1}\\sqrt{\\hat{R}_n^2-\\hat{R}_{n-1}^2}}{n} \n  \\hat{\\cal{V}}^{(n-1)}, \\quad \\hat{\\cal{V}}^{(0)}=1,\n\\end{align}\nwhere $\\hat R_0=0$ is assumed.\nIt is now straightforward to write the volumes $V_i$ and \n$A_i^{(\\mathrm{s,t})}$ as  \n\\begin{align}\n  \\label{eq:VAA}\n   V_i &=\\frac{1}{D} \\hat{\\cal{V}}^{(D-1)} \n   \\sqrt{m_i^2-\\hat{R}_{D-1}^2 \\delta l_i^2}\n   \\frac{l^D_{i+1}-l^D_i}{l_{i+1}-l_i}, \\\\\n   A_i^{({\\rm s})}&=\\hat{\\cal{V}}^{(D-2)}l_i^{D-2}, \\\\\n   A_i^{({\\rm t})}&=\\frac{1}{D-2}\\hat{\\cal{V}}^{(D-3)} \n   \\sqrt{m_i^2-\\hat{R}_{D-3}^2\\delta l_i^2}\n   \\frac{l^{D-2}_{i+1}-l^{D-2}_i}{l_{i+1}-l_i},\n\\end{align}\nwhere we have introduced the difference of edge \nlength $ \\delta l_i = l_{i+1} - l_i $.\n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=.9]{fig_temporal.eps}\n  \\caption{(a) Two lateral cells $c^{(\\rm l)}_{{\\rm D}i}$ and \n    $c^{(\\rm l)}_{{\\rm E}i}$ are meeting at a temporal hinge \n    $h^{(\\rm t)}_i$, (b)  $\\theta^{(4)}_i$ is the dihedral \n    angle between these cells, and (c) $\\varepsilon^{(\\rm t)}_i$ \n    the deficit angle around the hinge $h^{(\\rm t)}_i$ made by \n    $p_5$ frustums $\\left(V_i\\right)_1,\\cdots,\\left(V_i\\right)_{p_5}$ \n    having $ h^{(\\rm t)}_i $ as a lateral cell in common.}\n  \\label{fig:tda}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=.9]{fig_spatial.eps}\n  \\caption{ (a) Two spatial hinges $h^{(\\rm s)}_i$ and $h^{(\\rm s)}_{i+1}$ in the $i$-th frustum,\n  (b) dihedral angles $\\phi^{(4)\\uparrow}_i$ and $\\phi^{(4)\\downarrow}_{i+1}$,\n  and (c) deficit angle $ \\varepsilon^{(\\rm s)}_i $.\n    }\n  \\label{fig:sda}\n\\end{figure}\n\n\n\nTo find the deficit angle around a hinge we need a dihedral \nangle between two adjacent cells jointed at the hinge. As an example \nconsider the hinges of a 5-frustum with regular 4-polytopal bases as \nlaid out in Figure \\ref{fig:fbb}. At the temporal hinge \n$h^{(\\rm t)}_i=\\mathrm{ABC\\hbox{-}A^\\uparrow B^\\uparrow C^\\uparrow}$ in \nFigure \\ref{fig:tda}, the dihedral angle $\\theta^{(4)}_i$ is made by two \nlateral cells \n$c^{(\\rm l)}_{{\\rm D}i}=\\mathrm{ABCD\\hbox{-}A^\\uparrow B^\\uparrow C^\\uparrow D^\\uparrow}$ \nand $c^{(\\rm l)}_{{\\rm E}i} =\\mathrm{ABCE\\hbox{-}A^\\uparrow B^\\uparrow C^\\uparrow E^\\uparrow}$.\nOn the other hand, $\\phi^{(4)\\uparrow}_i$ is the dihedral angle at the hinge \n$h^{(\\rm s)}_i =\\mathrm{ABCD}$ between the lateral cell $c^{(\\rm l)}_{{\\rm D}i}$ \nand the lower base cell $c^{(\\rm b)}_{{\\rm E}i}=\\mathrm{ABCDE}$ as illustrated \nin Figure \\ref{fig:sda}, and similarly $\\phi^{(4)\\downarrow}_{i+1}$ the \none between $c^{(\\rm l)}_{{\\rm D}i}$ and \n$c^{(\\rm b)}_{{\\rm E}i+1}=\\mathrm{A^\\uparrow B^\\uparrow C^\\uparrow D^\\uparrow E^\\uparrow}$ \nat $h^{(\\rm s)}_{i+1} =\\mathrm{A^\\uparrow B^\\uparrow C^\\uparrow D^\\uparrow }$.\nFor a $D$-frustum with $(D-1)$-polytopal bases, the dihedral angles \n$\\theta^{(D-1)}_i$ and $\\phi^{(D-1)\\downarrow}_{i+1}$ can be written as\n\\begin{align}\n   \\label{eq:theta_ptu}\n   \\theta_i^{(D-1)} &=2\\arccos\\left[\\sqrt{\\frac{m_i^2-\\hat R^2_{D-1}\\delta l_i^2}{%\n         m_i^2-\\hat R^2_{D-2}\\delta l_i^2}}\\cos\\frac{\\vartheta_{D-1}}{2}\\right], \\\\\n   \\label{eq:phi_down_ptu}\n   \\phi_{i+1}^{(D-1)\\downarrow} &=\\arccos\\left[\\sqrt{\\frac{\\hat R^2_{D-1}-\\hat R_{D-2}^2}{%\n         m_i^2-\\hat R^2_{D-2}\\delta l_i^2}}\\:\\delta l_i\\right],\n\\end{align}\nwhere $\\vartheta_n$ is the dihedral angle of a regular $n$-polytope $\\Pi_n$. \nSince the upper cell of the $D$-frustum is parallel to the lower, \n$\\phi^{(D-1)\\uparrow}_i$ and $\\phi^{(D-1)\\downarrow}_{i+1}$ satisfy\n\\begin{align}\n\\phi^{(D-1)\\uparrow}_i + \\phi^{(D-1)\\downarrow}_{i+1} = \\pi.\n\\end{align}\nIn Appendix \\ref{sec:cada} we give a short account of dihedral angles of \nregular polytopes. Derivations of (\\ref{eq:theta_ptu}) and (\\ref{eq:phi_down_ptu}) \nare given in Appendix \\ref{sec:dadpf}. \n\nTaking it into the consideration that $p_D$ \nfrustums have a temporal hinge in common as in Figure \\ref{fig:tda}(c), \nthe deficit angle $\\varepsilon_i^{({\\rm t})}$ is given by\n\\begin{align}\n  \\label{eq:dati}\n  \\varepsilon_i^{({\\rm t})}=2\\pi- p_D \\theta_i^{(D-1)}.\n\\end{align}\nOn the other hand the spatial hinge $h^{(\\rm s)}_i$ is always \nshared by four frustums as illustrated in Figure \\ref{fig:sda}(c): \ntwo adjacent blocks of volume $V_i$ in the future side and two $V_{i-1}$ in the \npast side. Thus the deficit angle $\\varepsilon_i^{({\\rm s})}$ is \nexpressed as\n\\begin{align}\n  \\label{eq:dasi}\n  \\varepsilon_i^{({\\rm s})}=2 \\pi-2\\left(\\phi_i^{(D-1)\\uparrow}\n    +\\phi_i^{(D-1)\\downarrow}\\right)\n  =2\\delta\\phi_i^{(D-1)\\downarrow},\n\\end{align}\nwhere $\\delta\\phi_i^{(D-1)\\downarrow}\n=\\phi_{i+1}^{(D-1)\\downarrow}-\\phi_i^{(D-1)\\downarrow}$. \n\nA facet of regular $D$-polytope is a $(D-1)$-polytope having \n$N^{(D-1)}_{D-2}$ ridges of the $D$-polytope and a ridge is shared \nby two facets, so that $N^{(D)}_{D-1}$, $N^{(D-1)}_{D-2}$, and \n$N^{(D)}_{D-2}$ satisfy \n$N^{(D-1)}_{D-2}N^{(D)}_{D-1}=2N^{(D)}_{D-2}$. \nLikewise, a ridge has $N^{(D-2)}_{D-3}$ peaks of the $D$-polytope and a peak joints two \nridges in a facet, so a facet has $\\dfrac{N^{(D-1)}_{D-2}N^{(D-2)}_{D-3}}{2}$ \npeaks. Taking it into account of the fact that a peak connects $p_D$ \nfacets, we find a relation\n$\\dfrac{N^{(D-1)}_{D-2} N^{(D-2)}_{D-3} }{2}N^{(D)}_{D-1}\n=p_DN^{(D)}_{D-3}$. These constraints together with (\\ref{eq:rr_rpv})\nlead to \n\\begin{align}\n  \\label{eq:RatioAs}\n  \\frac{N^{(D)}_{D-2}\\hat{\\cal V}^{(D-2)}}{N^{(D)}_{D-3}\\hat{\\cal V}^{(D-3)}}\n  =&\\frac{p_D}{D-2}\n  \\sqrt{\\hat R_{D-2}^2-\\hat R_{D-3}^2}, \\\\\n  \\label{eq:RatioV}\n  \\frac{N^{(D)}_{D-1}\\hat{\\cal V}^{(D-1)}}{N^{(D)}_{D-3}\\hat{\\cal V}^{(D-3)}} \n  =&\\frac{2p_D}{(D-1)(D-2)}\n  \\sqrt{(\\hat R_{D-1}^2-\\hat R_{D-2}^2)(\\hat R_{D-2}^2-\\hat R_{D-3}^2)} \\nonumber \\\\\n  =&\\frac{2p_D}{(D-1)(D-2)}\n  (\\hat R_{D-2}^2-\\hat R_{D-3}^2)\\tan\\frac{\\vartheta_{D-1}}{2},\n\\end{align}\nwhich can be used to factor out the three couplings \nappearing \nin the action (\\ref{eq:regact}). As for the second equality in \n(\\ref{eq:RatioV}), use has been made of (\\ref{eq:Rnrecr}). We thus obtain \n\\begin{align}\n  \\label{eq:regactd}\n  S_\\mathrm{Regge}=&\\frac{N^{(D)}_{D-3}\\hat{\\cal{V}}^{(D-3)}}{8\\pi}\\sum_i\n  \\Biggl(\\frac{1}{D-2} \n  \\sqrt{m_i^2-\\hat{R}_{D-3}^2\\delta l_i^2}\n  \\frac{l^{D-2}_{i+1}-l^{D-2}_i}{l_{i+1}-l_i}\n  \\varepsilon_i^{\\rm (t)} \\nonumber\\\\\n  &+\\frac{2p_D}{D-2}\n  \\sqrt{\\hat R_{D-2}^2-\\hat R_{D-3}^2}l_i^{D-2}\n  \\delta\\phi_i^{(D-1)\\downarrow} \\nonumber\\\\\n  &-\\frac{p_D\\Lambda_D}{D}(\\hat R_{D-2}^2-\\hat R_{D-3}^2)\n  \\sqrt{m_i^2-\\hat{R}_{D-1}^2 \\delta l_i^2}\n  \\frac{l^D_{i+1}-l^D_i}{l_{i+1}-l_i}\\tan\\frac{\\vartheta_{D-1}}{2}\\Biggr).\n\\end{align}\nIn later sections we are interested in the continuum time limit. \nWe replace $l_i$ and $m_i$ by $l(\\tau)$ and $n(\\tau)\\delta \\tau$, where \n$\\tau$ is an arbitrary parameter and $n(\\tau)$ can be regarded as lapse \nfunction in ADM formalism. The continuum limit $\\delta\\tau\\rightarrow d \\tau$ \nof the action can easily be obtained from (\\ref{eq:regactd}) as  \n\\begin{align}\n  \\label{eq:ctregact}\n  S_\\mathrm{Regge}=&\\frac{N^{(D)}_{D-3}\\hat{\\cal{V}}^{(D-3)}}{8\\pi}\n  \\int d\\tau\n  \\Biggl( \n  \\sqrt{n^2-\\hat{R}_{D-3}^2\\dot l^2}\\:l^{D-3}\n  \\varepsilon^{\\rm (t)}\n  -2p_D\n  \\sqrt{\\hat R_{D-2}^2-\\hat R_{D-3}^2}\\:l^{D-3}\\dot l\n  \\phi^{(D-1)\\downarrow} \\nonumber\\\\\n  &-p_D\\Lambda_D(\\hat R_{D-2}^2-\\hat R_{D-3}^2)\n  \\sqrt{n^2-\\hat{R}_{D-1}^2\\dot l^2}\\:l^{D-1}\\tan\\frac{\\vartheta_{D-1}}{2}\\Biggr),\n\\end{align}\nwhere $\\dot l=\\dfrac{dl}{d\\tau}$ and total $\\tau$ derivative \nterms are suppressed. We have also introduced continuum limits \nof (\\ref{eq:theta_ptu}), (\\ref{eq:phi_down_ptu}), and (\\ref{eq:dati})\nby \n\\begin{align}\n  \\label{eq:ctheta_ptu}\n  \\varepsilon^{(\\mathrm{t})}&=2\\pi-p_D\\theta^{(D-1)} \\quad\n  \\hbox{with} \\quad\n  \\theta^{(D-1)}=2\\arccos\\left[\\sqrt{\\frac{n^2-\\hat R^2_{D-1}\\dot l^2}{%\n         n^2-\\hat R^2_{D-2}\\dot l^2}}\\cos\\frac{\\vartheta_{D-1}}{2}\\right], \\\\\n   \\label{eq:cphi_down_ptu}\n   \\phi^{(D-1)\\downarrow} &=\\arccos\n   \\left[\\sqrt{\\frac{\\hat R^2_{D-1}-\\hat R_{D-2}^2}{%\n         n^2-\\hat R^2_{D-2}\\dot l_i^2}}\\:\\dot l\\right].\n\\end{align}\nThe Regge action (\\ref{eq:ctregact}) is invariant under an\narbitrary reparameterization\n\\begin{align}\n  \\label{eq:rep}\n  \\tau \\to \\tau'=f(\\tau), \\quad\n  n (\\tau) \\to n'(\\tau')=\\frac{n(\\tau)}{\\dot f(\\tau)}, \\quad\n  l (\\tau) \\to l'(\\tau')=l(\\tau).\n\\end{align}\nThis can be used to fix the lapse function. \n\n\n\n\n\n\n\\section{Regge equations}\n\\label{sec:req}\n\\setcounter{equation}{0}\n\nThe Regge equations can be obtained by taking variations of \nthe Regge action with respect to $n$ and $l$. The equations \nof motion possess the local symmetry (\\ref{eq:rep}). \nWe must fix it by imposing some condition on the dynamical \nvariables. Furthermore, the action is based on the piecewise \nlinear manifold with Euclidean signature. We must carry out \ninverse Wick rotation to recover Lorentzian signature. As for \nfixing the local invariance we impose the following gauge \ncondition on the lapse function \n\\begin{align}\n  \\label{eq:gconn}\n  n(\\tau)=1.\n\\end{align}\nWe then carry out inverse Wick rotation by $\\tau=it$, where $t$ \ncan be regarded as the time of a clock fixed \nat a vertex of the  polytopal universe. The time axis is taken \nto be parallel to a strut. It is not orthogonal to Cauchy cells. \nIf we consider nonregular polytopes with shorter edge lengths \nand more cells such as geodesic domes \\cite{TF:2016aa}, we \nwould have a better approximation of a smooth hypersphere. \nThe orthogonality of the time axis with the spatial ones as \nin the FLRW universe can be restored in the limit of smooth \nhypersphere. We thus obtain the Regge equations \n\\begin{align}\n  &2\\pi-p_D\\theta^{(D-1)}\n  =p_D\\Lambda_D(\\hat{R}_{D-2}^2-\\hat{R}_{D-3}^2)\n  \\sqrt{\\frac{1+\\hat{R}_{D-3}^2\\dot{l}^2}{1+\\hat{R}_{D-1}^2\\dot{l}^2}}\n    \\:l^2\\tan\\frac{\\vartheta_{D-1}}{2}, \n  \\label{eq:chc_ptu} \\\\\n \n  &\\frac{\\ddot l}{1+\\hat R_{D-2}^2\\dot l^2}\n    =\\Lambda_D l\\left[1+\\hat R_{D-3}^2\\dot l^2\n    -\\frac{(\\hat R_{D-1}^2-\\hat R_{D-3}^2)l\\ddot l}{%\n    2(1+\\hat R_{D-1}^2\\dot l^2)}\\right],\n\\label{eq:cev_ptu}\n\\end{align}\nwhere the dots on $l$ stand for $t$ derivatives and \n$\\theta^{(D-1)}$ in lorentzian signature is given by\n\\begin{align}\n  \\label{eq:lstheta}\n  \\theta^{(D-1)}=2\\arccos\\left[\\sqrt{\\frac{1+\\hat R^2_{D-1}\\dot l^2}{%\n         1+\\hat R^2_{D-2}\\dot l^2}}\\cos\\frac{\\vartheta_{D-1}}{2}\\right].\n\\end{align}\nEq. (\\ref{eq:chc_ptu}) is known as the Hamiltonian constraint in \nADM formalism of canonical General Relativity. The equation of \nmotion for $l$ is referred to as the evolution equation. \nWe have simplified the evolution equation by using the Hamiltonian \nconstraint. It is straightforward to show that the evolution \nequation can be obtained as the consistency of the Hamiltonian \nconstraint with the time-development. We also mention that \n(\\ref{eq:chc_ptu}) and (\\ref{eq:cev_ptu}) reproduce the results \nof Refs. \\cite{TF:2016aa,TF:2020aa} in three and four dimensions.\n\nIt is convenient to express the solution to the Regge equations \nin terms of the dihedral angle $\\theta=\\theta^{(D-1)}$. Solving \n(\\ref{eq:chc_ptu}) and (\\ref{eq:lstheta}) with respect to \n$l^2$ and $\\dot l^2$, we obtain \n\\begin{align}\n  \\label{eq:lsqr}\n  l=&\\sqrt{\\frac{(2\\pi-p_D\\theta)\\cot\\frac{\\theta}{2}}{%\n    p_D\\Lambda_D(\\hat R_{D-2}^2-\\hat R_{D-3}^2)}}, \\\\\n  \\label{eq:dsqr}\n  \\dot l=&\\pm\\frac{1}{\\hat R_{D-2}}\\sqrt{\\frac{\\cos\\theta-\\cos\\theta_0}{%\n      \\cos\\theta_\\mathrm{c}-\\cos\\theta}},\n\\end{align}\nwhere $\\theta_0=\\vartheta_{D-1}$ stands for the dihedral angle of a Cauchy cell $ \\left\\{ p_1 , p_2 , \\cdots , p_{D-1} , p_0 \\right\\} $ and determines the minimum size of \nthe universe.\n$\\theta_\\mathrm{c}$ is defined by\n\\begin{align}\n  \\label{eq:jc}\n  \\theta_{\\mathrm{c}}=2\\arcsin\\left[\\frac{\\hat R_{D-3}}{\\hat R_{D-2}}\n    \\sin\\frac{\\vartheta_{D-1}}{2}\\right]. \n\\end{align}\nThe velocity $\\dot l$ diverges for $\\theta=\\theta_\\mathrm{c}$, where \nthe edge length becomes maximum. In three dimensions $\\theta_\\mathrm{c}=0$ \nsince $\\hat R_0=0$. \nIt matches $\\vartheta_1$ the dihedral angle of a 1-polytope $ \\left\\{ p_1 , p_0 \\right\\} $.\nSee (\\ref{eq:vth1}).\nIn dimensions larger than three $\\theta_\\mathrm{c}$  \nequals a dihedral angle of a regular polytope corresponding to extended \nSchl\\\"afli symbol $\\{p_1,p_3,\\cdots,p_{D-1},p_0\\}$, \nwhich is a vertex figure of a Cauchy cell. \nFor the vertex figure, see Appendix \\ref{sec:cada}.\n\n\n\nEliminating $l$ from (\\ref{eq:lsqr}) and (\\ref{eq:dsqr}),\nwe can derive the differential equation for $\\theta$\n\\begin{align}\n  \\dot\\theta&=\\mp\\frac{2\\sqrt{p_D\\Lambda_D(2\\pi-p_D\\theta)\\sin\\theta}}{%\n    2\\pi-p_D(\\theta-\\sin \\theta)}\n  \\frac{\\sin\\frac{\\theta}{2}}{\\sin\\frac{\\theta_0}{2}}\n  \\sqrt{\\frac{(\\cos\\theta_{\\mathrm{c}}-\\cos\\theta_0) \n      (\\cos\\theta-\\cos\\theta_0)}\n    {\\cos\\theta_{\\mathrm{c}}-\\cos\\theta}}.\n  \\label{eq:tde_ptu}\n\\end{align}\nThe upper sign corresponds to expanding universe and the lower \nto shrinking one. This leads to an integral \nrepresentation\n\\begin{align}\n  \\label{eq:time}\n  t \\left( \\theta \\right) =\\pm\\frac{1}{2\\sqrt{p_D\\Lambda_D}}\n  \\int_\\theta^{\\theta_0} du\\frac{2\\pi-p_D(u-\\sin u)}{%\n    \\sqrt{(2\\pi-p_Du)\\sin u}}\\frac{\\sin\\frac{\\theta_0}{2}}{%\n    \\sin\\frac{u}{2}}\n  \\sqrt{\\frac{\\cos\\theta_{\\mathrm{c}}-\\cos u}{%\n      (\\cos\\theta_{\\mathrm{c}}-\\cos\\theta_0)(\\cos u-\\cos\\theta_0)}},\n\\end{align}\nwhere $\\theta_{\\mathrm c} \\leq \\theta\\leq\\theta_0$. We have assumed the initial \ncondition\n\\begin{align}\n  \\label{eq:initc}\n  \\theta(0)=\\theta_0.\n\\end{align}\nAs a function of $t$, the dihedral angle $\\theta$ \nis even \nand monotonically decreasing from $\\theta_0$ to \n$\\theta_\\mathrm{c}$ for $0\\leq t\\leq\\tau_\\mathrm{p}\/2$, where $\\tau_\\mathrm{p}$ is given by \n$\\tau_\\mathrm{p}=2t(\\theta_\\mathrm{c})$. We can extend $\\theta(t)$ as a \ncontinuous periodic function for arbitrary $t$ by\n\\begin{align}\n  \\label{eq:tau}\n  \\theta(t+\\tau_\\mathrm{p})=\\theta(t). \n\\end{align}\nThe edge length (\\ref{eq:lsqr}) is also a periodic function of $t$. \nIt is continuous for $D\\geq4$, while $l$ diverges for \n$\\theta \\left( \\tau_\\mathrm{p}\/2 \\right) =\\theta_{\\mathrm{c}}=0$ \nin three dimensions. \nNote that $\\dot l\/l$ not only diverges \nbut also has a discontinuity at $t=\\pm\\tau_\\mathrm{p}\/2,\n~\\pm3\\tau_\\mathrm{p}\/2,~\\cdots$. At present it is only an \nassumption that the polytopal universe in four or more dimensions \njumps from expansion to contraction when it reaches the maximum size. \n\n\n\nIn dimensions larger than four there are only three types of regular \npolytopes. As can easily be seen from Table \\ref{tab:ssfrpt} any regular \npolytope can be characterized by $p_2$, $p_3$, $p_D$, and $D$. It is \npossible to write the circumradii $\\hat R_{D-k}$ ($k=1,2,3$) and \ndihedral angles $\\vartheta_{D-1}$ appearing \nin (\\ref{eq:chc_ptu})--(\\ref{eq:lstheta}) in more tractable forms by noting (\\ref{eq:hatRD}) \nand (\\ref{eq:vTD}). To this end we define a set of parameters \n$\\kappa_n$, $\\lambda_n$, $\\mu_n$, and $\\zeta_n$ by\n\\begin{align}\n  \\kappa_n &= 3 \\sum_{j=0}^1\\delta_{j,n}+p_2\\sum_{j=2}^\\infty\\delta_{j,n}, \\\\\n  \\lambda_n&=3\\sum_{j=0}^2\\delta_{j,n}+p_3\\sum_{j=3}^\\infty\\delta_{j,n},\\\\\n  \\mu_n&=3\\sum_{j=0}^3\\delta_{j,n}+p_4\\sum_{j=4}^\\infty\\delta_{j,n},\\\\\n  \\zeta_n&=3\\sum_{j=0}^4\\delta_{j,n}+p_n\\sum_{j=5}^\\infty\\delta_{j,n},\n\\end{align}\nwhere $\\delta_{j,k}$ is the Kronecker delta. \nObviously, $\\kappa_n=p_2$, $\\lambda_n=p_3$, $\\mu_n=p_4$, and $\\zeta_n=p_n$ for $n\\geq5$. We assign a regular \n$D$-polytope to a set of five parameters $\\left[D,\\kappa_D,\n\\lambda_D,\\mu_D,\\zeta_D \\right]$. In Table \\ref{tab:ssfrpt} we summarize the correspondence \nbetween regular polytopes and the symbol $\\left[D,\\kappa_D,\n\\lambda_D,\\mu_D,\\zeta_D\\right]$. \nThis allows us to express the normalized circumradius $\\hat{R}_D$ and the dihedral angle $\\vartheta_D$ in the closed forms as\n\\begin{align}\n  \\label{eq:hRD}\n\\hat{R}_D &= \\frac{1}{2} \\sqrt{ \\frac{ \\left[ 1 - \\left( D-4 \\right) \\cos \\frac{ 2 \\pi }{ \\zeta_D } \\right] \\sin^2 \\frac{\\pi}{\\lambda_D} - 2 \\left[ 1 - \\left( D - 5 \\right) \\cos \\frac{ 2 \\pi }{ \\zeta_D } \\right] \\cos^2 \\frac{ \\pi }{ \\mu_D } }{%\n\\left[ 1 - \\left( D-4 \\right) \\cos \\frac{2 \\pi}{\\zeta_D} \\right] \\left( \\sin^2 \\frac{\\pi}{\\lambda_D} - \\cos^2 \\frac{\\pi}{\\kappa_D}  \\right) \n- \n2 \\left[ 1 - \\left( D-5 \\right) \\cos \\frac{2 \\pi}{\\zeta_D} \\right] \\sin^2 \\frac{\\pi}{\\kappa_D} \\cos^2 \\frac{ \\pi}{\\mu_D} \n} }, \\\\\n\\label{eq:vthD}\n\\vartheta_D &= 2 \\arcsin \\left( \\sqrt{ 2 \\frac{ \\sin^2 \\frac{ \\pi }{ \\kappa_D } \\left[ 1 - \\left( D - 5 \\right) \\cos \\frac{ 2 \\pi }{ \\mu_D } \\right] - \\left( D - 4 \\right) \\cos^2 \\frac{\\pi}{ \\lambda_D } }{%\n\\sin^2 \\frac{ \\pi }{ \\kappa_D } \\left[ 1 - \\left( D - 4 \\right) \\cos \\frac{ 2 \\pi }{ \\mu_D } \\right] - \\left( D - 3 \\right) \\cos^2 \\frac{\\pi}{ \\lambda_D }\n} } \\cos \\frac{\\pi}{ \\zeta_D } \\right).\n\\end{align}\nThe circumradius (\\ref{eq:hRD}) is applicable in $D \\geq 0$,\nwhereas the dihedral angle (\\ref{eq:vthD}) is valid in the dimensions larger than zero.\nNote that $\\vartheta_0$ is undetermined.\nIn particular the fact that $\\mu_{D-k}=\\zeta_{D-k}=3$ with $ 1 \\leq k \\leq D $ for any regular polytope \nenables us to write the following equalities \n\\begin{align}\n  \\label{eq:RDk}\n  \\hat{R}_{D-k}\n   &=\\displaystyle \\frac{1}{2} \\sqrt\n     \\frac{\\left(D-1- k\\right)-2\\left(D-2-k\\right)\n       \\cos^2\\frac{\\pi}{\\lambda_{D-1}}}\n     {%\n      \\left(D-1-k\\right)\\sin^2\\frac{\\pi}{p_2}-2\\left(D-2-k\\right)\n      \\cos^2\\frac{\\pi}{\\lambda_{D-1}}}} \\quad \\left(k=1,2,3\\right), \\\\\n  \\label{eq:cosvthD}\n  \\cos\\vartheta_{D-1}\n  &=\\frac{\\sin^2 \\frac{\\pi}{p_2}-2\\cos^2\\frac{\\pi}{\\lambda_{D-1}}}{%\n    \\left(D-3\\right)\\sin^2\\frac{\\pi}{ p_2}-2\\left(D-4\\right)\n    \\cos^2\\frac{\\pi}{\\lambda_{D-1}}}.\n\\end{align}\nThe Regge equations (\\ref{eq:chc_ptu}) and \n(\\ref{eq:cev_ptu}) give descriptions of the time-development of \nthe universe with a regular polytopal Cauchy surface for the parameter \nset $\\left[D,\\kappa_D,\\lambda_D,\\mu_D,\\zeta_D\\right]$. \n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=1]{fig_da_simp.eps}\n  \\caption{Plots of the dihedral angles of the simplicial polytope \n    models for $3\\leq D\\leq7$.}\n  \\label{fig:da_simp}\n\\end{figure}\n\n\n\nTime-development of the dihedral angle $\\theta$ can be obtained \nby integrating (\\ref{eq:tde_ptu}) numerically for the initial \ncondition (\\ref{eq:initc}). We give plots of the dihedral angles \nof simplicial polytope models for $D=3,4,\\cdots,7$ and \n$0\\leq t\\leq \\tau_\\mathrm{p}\/2$ in Figure \\ref{fig:da_simp}.\n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=1]{fig_sf_simp.eps}\n  \\caption{Plots of the scale factors of the simplicial polytope \n    models for $3\\leq D\\leq7$.\n  The broken curve corresponds to the $D$-dimensional FLRW universe.\n    }\n  \\label{fig:sf_simp}\n\\end{figure}\n\n\n\nTo compare the polytopal universe with the continuum, we must \nintroduce a Regge calculus analog of the scale factor. \nThere are, however, ambiguities in defining a radius of a\nregular polytope. Here we simply introduce it as the radius of \nthe circumsphere of the regular polytope\n\\begin{align} \n  a_{\\mathrm{R}}(t)\n      &=\\hat{R}_Dl(t).\n      \\label{eq:sf_ptu}\n\\end{align}\n\nInserting the solutions of (\\ref{eq:tde_ptu}) into (\\ref{eq:sf_ptu}), \nwe obtain the time-developments of the scale factors of polytopal \nuniverses. Figure \\ref{fig:sf_simp} shows the behaviors of the simplicial \nuniverses. The broken curve corresponds to the $D$-dimensional FLRW \nsolution. The 3-simplicial model expands faster than the continuum \none and diverges at $t=\\tau_\\mathrm{p} \/ 2$. For $D \\geq 4$, after \narriving at the maximum scale $a(\\tau_\\mathrm{p}\/2)$ the universe begins to \ncontract to the initial minimum size $a(0)=a(\\tau_\\mathrm{p})$. Then the \nuniverse repeats expanding and contracting with a period $\\tau_\\mathrm{p}$. \nOne easily sees that the $D$-simplices are too crude to approximate the \ncontinuum solution. The larger the space-time dimensions, the bigger \ndifference we have. The situation is somewhat improved by considering \n$D$-orthoplices or $D$-cubes in this order. For fixed space-time \ndimensions the deviation from the continuum FLRW universe become \nsmaller as the number of vertices increases. \n\nIn closing this section we comment on the case of $D$-polytopal universe \nwithout cosmological constant. In this case the Hamiltonian constraint \n(\\ref{eq:chc_ptu}) yields $\\theta^{(D-1)}=\\dfrac{2\\pi}{p_D}$. We obtain \nfrom  (\\ref{eq:lstheta}) \n\\begin{align}\n  \\label{eq:hcwc_ptu}\n  \\dot{l}^2=-\\frac{\\cos^2\\frac{\\vartheta_{D-1}}{2}-\\cos^2\\frac{\\pi}{p_D}}{%\n    \\hat R_{D-1}^2\\cos^2\\frac{\\vartheta_{D-1}}{2}-\\hat R_{D-2}^2\n    \\cos^2\\frac{\\pi}{p_D}}\n  =-\\frac{1}{\\hat R_D^2}.\n\\end{align}\nThere is no convex regular polytope satisfying this. The Hamiltonian \nconstraint, however, admits infinite honeycomb lattices in flat Euclidean \nspace. For any space-filling honeycomb the circumradius $\\hat R_D$ \ndiverges and the dihedral angle is given by \n$\\vartheta_D=\\pi$, which immediately yields \n\\begin{align}\n  \\label{eq:condsfhl}\n  \\cos\\frac{\\vartheta_{D-1}}{2}=\\cos\\frac{\\pi}{p_D}. \n\\end{align}\nSee (\\ref{eq:pnvth}). In Table \\ref{tab:ssfm} we summarize \nspace-filling honeycomb lattices in arbitrary dimensions. \nIt is straightforward to verify (\\ref{eq:condsfhl}).\nWe thus obtain static solutions \n$l=\\mathrm{const}$. They correspond to the Minkowski \nspace-time. In addition, in the case of $\\dot{l}^2 > 0$, \nSchl\\\"afli symbol \nsatisfying this inequality stands for a regular lattice of open \nCauchy surface of constant negative curvature. These results are \nconsistent with solutions of the Friedmann equations (\\ref{eq:Feq}).\nSee Table \\ref{tab:flrw}. \n\n\n\n\\begin{table}[t]\n\\centering\n  \\begin{tabular}{llll}\\hline \n    Dimensions $D$ & Name & Extended Schl\\\"afli symbol & $[D,\\kappa_D,\\lambda_D,\\mu_D,\\zeta_D]$  \\\\ \\hline\n    %\n    2 & Apeirogon & $\\left\\{2,\\infty , 2 \\right\\}$ & $[2,\\infty,3,3,3]$ \\\\\n    \\\\\n   \n    & Triangular tiling & $\\left\\{2,3,6,2\\right\\}$ & $[3,3,6,3,3]$ \\\\ \n    3 & Square tiling & $\\left\\{2,4,4,2\\right\\}$ & $[3,4,4,3,3]$ \\\\ \n    & Hexagonal tiling & $\\left\\{2,6,3,2\\right\\}$ & $[3,6,3,3,3]$ \\\\ \n    \\\\\n   \n    4 & Cubic honeycomb & $\\left\\{2,4,3,4,2\\right\\}$ & $[4,4,3,4,3]$ \\\\ \n    \\\\\n   \n    & 8-cell honeycomb & $\\left\\{2,4,3,3,4,2\\right\\}$ & $[5,4,3,3,4]$ \\\\\n    5 & 16-cell honeycomb & $\\left\\{2,3,3,4,3,2\\right\\}$ & $[5,3,3,4,3]$ \\\\\n    & 24-cell honeycomb & $\\left\\{2,3,4,3,3,2\\right\\}$ & $[5,3,4,3,3]$ \\\\ \n    \\\\\n    %\n    $n+1\\geq6$ & $n$-cubic honeycomb $ \\delta_{n+1} $ & $\\left\\{2,4,3^{n-2},4,2\\right\\}$\n    & $[n+1,4,3,3,4]$ \\\\ \\hline\n  \\end{tabular}\n\\caption{Space-filling lattices in Euclidean $\\left(D-1\\right)$-space.\n    The lattices for $ D\\geq3 $ are  \n    corresponding to Minkowski space-time. \n    The $n$-cubic honeycomb is named by Coxeter as $ \\delta_{n+1} $ \\cite{Coxeter},\n    which has the extended Schl\\\"afli symbol $ \\left\\{ 2,4,3^{n-2},4,2 \\right\\} $.\n    The only misfit is $ \\delta_2 = \\left\\{ 2,\\infty,2 \\right\\} $.\n    }\n  \\label{tab:ssfm}\n\\end{table}\n\n\n\n\n\n\n\\section{Fractional Schl\\\"afli symbol and pseudo-regular \\\\ $D$-polytopal universes}\n\n\\label{sec:prpt}\n\\setcounter{equation}{0}\n\nSo far we have investigated evolution of regular polytopes \nas a discretized FLRW universe. To go beyond the approximation \nby regular polytopes, we must introduce polytopes with more cells.  \nOne way to implement this is to employ geodesic domes \\cite{TF:2016aa}. \nHypercube is the only type of regular polytope having subdivisions \nof facets in arbitrary dimensions by the same type of polytopes with\nthe parent facets. In this section we consider hypercube-based \ngeodesic domes as Cauchy surfaces of the universe. \n\n\n\n\\begin{figure}[t]\n  \\centering\n\\includegraphics[scale=1.0]{fig_cube_decomp.eps}\n  \\caption{Subdivision of a 3-cube as a cell of a 4-cube for \n    (a) $\\nu=2$, (b) $\\nu=3$, and (c) $\\nu=4$.\n    In four dimensions the peaks are the edges.\n    Solid lines are the three-way connectors and broken lines \n    the four-way connectors.\n    }\n  \\label{fig:cube_decomp}\n\\end{figure}\n\n\n\nA hypercube in $D$ dimensions has $(D-1)$-cubes as its facets. \nTo define a geodesic dome for the hypercube we first divide \neach facet into $\\nu^{D-1}$ pieces of $(D-1)$-cubes of edge \nlength $l\/\\nu$ as depicted in Figure \\ref{fig:cube_decomp}, \nwhere $\\nu$ is the level of the division, called frequency. \nWe then radially project the tessellated hypercube on the \ncircumsphere of the original hypercube. This results in a \ntessellation of the circumsphere. The geodesic dome $\\Gamma_\\nu$ \ncan be obtained by replacing each circular arc of the tessellated \ncircumsphere with a line segment jointing its end points. \nIn general each facet of $\\Gamma_\\nu$ thus constructed \nis not a flat $(D-1)$-space. We can always decompose these \nfacets into flat $(D-1)$-polytopes by adding extra edges. \nThe deviations from the flat $(D-1)$-spaces, however, become\nnegligible as $\\nu$ increases. We can effectively regard the \nfacets of $\\Gamma_\\nu$ as flat $(D-1)$-cubes and see any \npolytopal data of the geodesic dome such as the numbers of \nfacets, ridges, etc. from the tessellated $D$-cube. \n\nWe can apply Regge calculus \nto $\\Gamma_\\nu$ as the polyhedral model in Ref. \\cite{TF:2016aa}. \nIn the infinite frequency limit $ \\nu \\to \\infty $, geodesic \ndome reproduces a smooth sphere. So the model universe approaches \nthe FLRW universe in the limit $\\nu\\rightarrow\\infty$.\nIn practice the larger the frequency, the more cumbersome the Regge \ncalculus for geodesic domes becomes. We avoid this complexity by \nintroducing pseudo-regular polytopes as in Refs. \n\\cite{TF:2016aa,TF:2020aa}.\n\nLet us denote the pseudo-regular polytope corresponding to $\\Gamma_\\nu$ \nby $\\tilde\\Gamma_\\nu$. We assign it a fractional Schl\\\"afli symbol \n\\begin{align}\n  \\label{eq:SSprP}\n  \\{2,4,3^{D-3},p(\\nu),2\\},\n\\end{align}\nwhere \n$p(\\nu)$ is the averaged number of facets sharing a peak of $\\Gamma_\\nu$\nand the other $D$ integers are the Schl\\\"afli symbol of the facets of $\\Gamma_\\nu$.\nThere are two types of peaks of $\\Gamma_\\nu$ as illustrated in \nFigure \\ref{fig:cube_decomp} for a cell of 4-cube. One is \nshared by three facets. These come from the peaks of the \noriginal $D$-cube. The other connects four facets.\nThey are generated in subdividing the facets of the original \n$D$-cube. We refer to the former type as ``three-way connector''\nand the later ``four-way connector''. Counting the numbers of \neach type of connectors and averaging the number \nof facets around a peak in $\\Gamma_\\nu$, we find\n\\begin{align}\n  \\label{eq:pnu}\n  p(\\nu)=\\frac{12\\nu^2}{3\\nu^2+1}.\n\\end{align}\nSee Appendix \\ref{sec:pnu} for details. The result is independent \nof $D$. Furthermore, the fractional Schl\\\"afli symbol approaches the \none of $(D-1)$-cubic honeycomb in the limit $\\nu\\rightarrow\\infty$. \n\n\n\nThe basic approach of pseudo-regular polytope is to regard \n$\\tilde \\Gamma_\\nu$ as a regular polytope of edge length $l$ \nwith the fractional Schl\\\"afli symbol (\\ref{eq:SSprP}) and \nto assume that the model universe is described by the Regge \nequations (\\ref{eq:chc_ptu}) and (\\ref{eq:cev_ptu}).  \nThe symbol (\\ref{eq:SSprP}) corresponds to the assignment \n\\begin{align}\n  p_2=4, \\quad \\lambda_{D-1}=3, \\quad p_D=p(\\nu).\n\\end{align}\nIn particular the normalized circumradii (\\ref{eq:RDk}) and \nthe dihedral angle (\\ref{eq:cosvthD}) coincide with those of \nthe regular $D$-cube. They are independent of the frequency $\\nu$. \nThe differential equation for the dihedral angle $\\theta(t)$ can be \nwritten explicitly as\n\\begin{align}\n  \\label{eq:tde_prpt}\n  \\dot{\\theta}(t)&=\\mp\\frac{2}{%\n    2\\pi-p(\\nu)\\left( \\theta(t)-\\sin\\theta(t)\\right)}\n  \\sqrt{\\frac{p(\\nu)\\Lambda_D\\left( 2\\pi-p(\\nu)\\theta(t)\\right) \\sin2\\theta(t)}\n    {1-\\left(D-2\\right)\\cos\\theta(t)}}\\sin\\frac{\\theta(t)}{2}.\n\\end{align}\nNote that the initial dihedral angle is $\\theta(0)=\\theta_0\n=\\vartheta_{D-1}=\\pi\/2$. Both $\\theta_0$ and $\\theta_{\\mathrm{c}}\n=\\arccos\\dfrac{1}{D-2}$ do not depend on $\\nu$. \n\nThe scale factor $a_{\\mathrm{R}}$ for the pseudo-regular \n$D$-polytopal universe can be defined similarly as the regular \npolytopal models as \n\\begin{align}\n  a_{\\rm R}\\left(t\\right)\n  \\label{eq:sf_prpt_theta}\n  &=\\hat R_D(\\nu)l(t),\n\\end{align}\nwhere the edge length $l(t)$ for $\\tilde\\Gamma_\\nu$ can be \nfound from (\\ref{eq:lsqr}) as \n\\begin{align}\n  \\label{eq:elprp}\n  l(t)=\\frac{2}{\\sqrt{\\Lambda_D}}\n  \\sqrt{ \\left(\\frac{2\\pi}{p(\\nu)}-\\theta(t)\\right)\\cot\\frac{\\theta(t)}{2}}.\n\\end{align}\nThe normalized circumradius $\\hat R_D(\\nu)$ also depends on $p_D=p(\\nu)$ \nand can be obtained from (\\ref{eq:hRD}) as  \n\\begin{align}\n\\label{eq:Rhat_dcube}\n  \\hat R_D(\\nu)=\\frac{1}{2}\n  \\sqrt{D-2-\\sec\\frac{2\\pi}{p(\\nu)}}.\n\\end{align}\nFor $\\nu=1$ this coincides with the circumradius of a regular $D$-cube \nof unit edge length. It grows with the frequency $\\nu$ and diverges linearly\nfor $\\nu\\rightarrow\\infty$. \nIn fact Eq. (\\ref{eq:Rhat_dcube}) can be approximated for large frequency by\n\\begin{align}\n\\hat{R}_D (\\nu) \\approx \\sqrt{ \\frac{3}{2 \\pi} } \\nu.\n\\end{align}\nOn the other hand the edge length \n(\\ref{eq:elprp}) decreases roughly inversely with $\\nu$ and approaches \nzero as $\\nu\\rightarrow\\infty$. This can be seen explicitly for the \ninitial edge length\n\\begin{align}\n  \\label{eq:initelprp}\n  l(0)=\\sqrt{\\frac{2\\pi}{3\\Lambda_D}}\\frac{1}{\\nu}.\n\\end{align}\nThe scale factor (\\ref{eq:sf_prpt_theta}), however, remains finite for \n$\\nu\\rightarrow\\infty$. \nNoting that $\\hat R_{D-k}$ ($k=1,2,3$) \nare independent of $\\nu$ as given by (\\ref{eq:RDk}),\nit is \nstraightforward to verify that the Regge equations (\\ref{eq:chc_ptu}) \nand (\\ref{eq:cev_ptu}) for $\\tilde\\Gamma_\\nu$ reduce to the Friedmann \nequations (\\ref{eq:Feq}) in the limit $\\nu\\rightarrow\\infty$. \n\nTo see the dependences on $\\nu$ we give plots of the dihedral angles\nin Figure \\ref{fig:da_prpt} and those of the scale factors in \nFigure \\ref{fig:sf_prpt} for $D=5$, $\\nu\\leq 5$, and \n$0\\leq t\\leq\\tau_\\mathrm{p}(\\nu)\/2$, where $\\tau_\\mathrm{p}(\\nu)$ is \nthe period of the \noscillation of $\\tilde\\Gamma_\\nu$. One might think that $D$-cube-based \npseudo-regular polytopes are too crude to approximate $D$-spheres. \nAs can be seen from Figure \\ref{fig:sf_prpt}, the scale factor \napproaches rapidly the continuum one as $\\nu$ increases. \nAs mentioned above, the geodesic dome $\\Gamma_\\nu$ becomes impractical to \ncarry out Regge calculus for large $\\nu$. The advantage of the approach \nof pseudo-regular polytopes is its applicability to arbitrarily large \nfrequency without effort. The scale factor for $\\nu = 100$ is shown in \nFigure \\ref{fig:sf_prpt_nu100}. Coincidence with the continuum theory \nis excellent for $\\sqrt{\\Lambda_5}t\\sim4$. The edge length becomes \ncomparable with $1\/\\sqrt{\\Lambda_5}$ at around $\\sqrt{\\Lambda_5}t\\sim4$,   \nonset of the deviation from the continuum solution.  \n\n\n\n\\begin{figure}[t]\n  \\centering\n  \\includegraphics[scale=1]{fig_da_prpt.eps}\n  \\caption{Plots of the dihedral angles of the pseudo-regular \n    5-polytopal universes for $\\nu\\leq5$.}\n  \\label{fig:da_prpt}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n  \\centering\n\\includegraphics[scale=1]{fig_sf_prpt.eps}\n  \\caption{Plots of the scale factors of the pseudo-regular 5-polytopal universes for $\\nu \\leq 5$.\n  The broken curve corresponds to the five-dimensional FLRW universe.\n    }\n  \\label{fig:sf_prpt}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n  \\centering\n\\includegraphics[scale=1]{fig_sf_prpt_nu100.eps}\n  \\caption{Plot of the scale factor of the pseudo-regular 5-polytopal universe for $\\nu=100$.\n  The broken curve stands for the exact solution of the continuum theory.\n    }\n  \\label{fig:sf_prpt_nu100}\n\\end{figure}\n\n\n\n\n\n\n\\section{Summary and discussions}\n\n\\label{sec:sum}\n\\setcounter{equation}{0}\n\nFollowing the CW formalism, we have carried out Regge calculus \nfor closed FLRW universe with a positive cosmological constant \nin arbitrary dimensions.  The geometrical characterization of \nregular polytopes by the Schl\\\"afli symbol has turned out to be \nvery efficient in describing systematically the discrete FLRW \nuniverse in spite of there being only three types of regular \npolytopes in dimensions more than four. We have given the Regge \naction in closed form in the continuum time limit. It possesses \na reparameterization invariance of time variable to ensure \ncoordinate independence of the formalism. The Regge equations \nare the Hamiltonian constraint and the evolution equation as \nthe continuum theory, describing the time development of the \ndiscrete FLRW universe. They coincide with the previous results \nin three and four dimensions \\cite{TF:2016aa,TF:2020aa}. In \nparticular under the gauge choice (\\ref{eq:gconn}) the \ncircumsphere of the regular polytope repeats periodically \nexpansion and shrinking in any dimensions larger than four\nas the four dimensional case. The Regge equations have more or \nless the same structures in dimensions greater than three. \nIt is only in three dimensions where the edge length diverges \nin finite time. \n\nAs we have shown in Sect. \\ref{sec:req} the approximation by \nregular polytopes is not so accurate even for $\\sqrt{\\Lambda_D}t\\ll 1$. \nThe situation gets worse as the dimensions increase. \nThis is contrasted with the cases of dodecahedron in three \ndimensions and 120-cell in four dimensions, which describe \nthe continuum FLRW universe rather well until $t$ becomes \ncomparable with $1\/\\sqrt{\\Lambda_D}$. The difference basically \ncomes from that of the number of vertices in a polytope. A \n120-cell has six hundred vertices, whereas a 4-cube does only sixteen. \nIn five or more dimensions there are no such special \npolytopes. \nOne must refine the tessellation of the Cauchy \nsurface by \nnonregular polytopes with smaller cells\nto have better \napproximations. Though this can be done by extending the \ngeodesic domes in three dimensions, we have analyzed \npseudo-regular polytopes with the expectation that the Regge \nequations for the pseudo-regular polytopes approximate well \nthe Regge calculus of the corresponding geodesic domes. \nWe stress that the pseudo-regular polytope is a substitute \nof the corresponding geodesic domes characterized by \nthe frequency $\\nu$, not the continuum hypersphere. \nThe Regge equations (\\ref{eq:chc_ptu}) and (\\ref{eq:cev_ptu}) \ntherefore should be considered as an effective description \nof the Regge equations for the geodesic dome, not of the \ncontinuum Freedman equations. The approach of pseudo-regular \npolytopes can be applied to an arbitrary $\\nu$. \nIn particular \nwe can infer the validity of Regge calculus for geodesic \ndomes. Because of this, the pseudo-regular polytope universe \nbegins to deviate from the continuum solution when the edge \nlength becomes larger than $1\/\\sqrt{\\Lambda_D}$. \n\nIn this paper we have considered vacuum universes without \nmatters. Incorporating gravitating matter sources is worth\ninvestigation. In General Relativity, Friedmann equations \nhave a solution for a negative cosmological constant. It \ndescribes hyperbolic Cauchy surfaces expanding or contracting \nwith time. Applying the method of pseudo-regular polytope to \nsuch non-compact universe be interesting. We will address \nthese issues elsewhere. \n\n\n\n\\vskip .5cm\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAfter a successful experimental detection of Bose-Einstein condensates (BEC) of\ndilute trapped bosonic alkali-metal atoms $^7$Li, $^{23}$Na, and $^{87}$Rb\n\\cite{review,books} at ultra-low temperatures, there have been intense\ntheoretical activities in studying properties of the condensate using the\ntime-dependent mean-field Gross-Pitaevskii (GP) equation under different trap\nsymmetries. Among many possibilities, the following traps have been used in\nvarious studies: three-dimensional (3D) spherically-symmetric, axially-symmetric\nand anisotropic harmonic traps, two-dimensional (2D) circularly-symmetric and\nanisotropic harmonic traps, and one-dimensional (1D) harmonic trap. The\ninter-atomic interaction leads to a nonlinear term in the GP equation, which\ncomplicates its accurate numerical solution, specially for a large nonlinearity.\nThe nonlinearity is large for a fixed harmonic trap when either the number of\natoms in the condensate or the atomic scattering length is large and this is\nindeed so under  many experimental conditions. Special care is needed \nfor the\nsolution of the time-dependent GP equation with large nonlinearity and there has\nbeen an extensive literature on this topic\n\\cite{Tiwari_Shukla,Bao_Tang,Schneider_Feder,chio1,chio2,chang,num1,num2,num3,num4,num5,num6,num7,num8,num9,num10,num11,num12,num13,num14,num15,num16,num17,num18,num19,num20,num21,num22,num23,num24,num25,num26,num27,num28,num29,num30,num31,num32,num33,num34,num35,aq,xyz1,burnett,holland,baer}.\n\n\nThe time-dependent GP equation is a partial differential equation in space and\ntime variables involving first-order time and second-order space derivatives\ntogether with a harmonic and a nonlinear potential term, and has the structure\nof a nonlinear Schr\\\"odinger equation with a harmonic trap. { One\ncommonly used} procedure for the solution of the time-dependent GP equation makes\nuse of a discretization of this equation in space and time  and subsequent\nintegration and time propagation of the discretized equation. From a knowledge\nof the solution of this equation at a specific time, this procedure finds the\nsolution after a small time step by solving the discretized equation.\n{A commonly used} discretization scheme for the GP equation is the\nsemi-implicit Crank-Nicolson discretization scheme \\cite{koonin,ames,dtray}\nwhich has certain advantages and will be used in this work.\n\n\nIn the simplest one-space-variable form of the GP  equation, the solution\nalgorithm is executed in two steps. In the first step, using a known initial\nsolution, an intermediate solution  after a small interval of time $\\Delta$ is\nfound   neglecting the harmonic and  nonlinear potential terms. The effect of\nthe potential terms is then included by a first-order time integration to obtain\nthe final solution after time $\\Delta$. In case of two or three spatial\nvariables, the space derivatives are dealt with in two or three steps and the\neffect of the potential terms are included next. As the time evolution is\nexecuted  in different steps it is called a split-step real-time propagation\nmethod. This method is equally applicable to stationary ground and \nexcited states as well as non-stationary states, although in this paper \nwe do not consider stationary excited states. \nThe virtues of the semi-implicit Crank-Nicolson scheme\n\\cite{koonin,ames,dtray} are that it is  unconditionally stable and preserves the\nnormalization  of  the solution under real-time propagation. A simpler \nand\nefficient variant of the {scheme} called the split-step\nimaginary-time propagation method obtained by replacing the time variable by an\nimaginary time is also considered. (The GP equation involves complex\n{variables}. However, after replacing the time variable by an\nimaginary time the resultant partial differential equation is real, and hence\nthe imaginary-time propagation method involves real {variables}\nonly. {This trick\nleads to  an \nimaginary-time operator which results in exponential decay of all states \nrelative to the   ground state and   can then be applied to any \ninitial \ntrial wave function to compute an approximation to the actual ground \nstate rather accurately. We shall use imaginary-time propagation to \ncompute the ground state in this paper.)} The split-step \nimaginary-time propagation method involving real variables\n yields very precise result at low computational cost (CPU time) and is very\nappropriate for the solution of  stationary problems involving the \nground state. The split-step \nreal-time propagation method uses complex {quantities} and yields\nless precise results for stationary problems; however, they are appropriate for\nthe study of non-equilibrium dynamics in addition to stationary problems \ninvolving excited states also.\n\n\nMost of the previous studies\n\\cite{Tiwari_Shukla,Bao_Tang,Schneider_Feder,chang,num1,num4,num10,num12,num13,num16,num17,num25,num26,num27,num31,xyz1,burnett}\non the numerical solution of the GP equation are confined to a consideration of\nstationary states only. Some used specifically the imaginary-time propagation\nmethod \\cite{chio1,num23,aq,xyz1}. There are few studies\n\\cite{num22,num29,num30,num33} for the numerical solution of the time-dependent\nGP equation using  the Crank-Nicolson method \\cite{koonin,ames,dtray}. Other\nmethods for numerical solution of the time-dependent GP equation have also\nappeared in the literature\n\\cite{chio2,num6,num8,num15,num18,num19,num20,num21,num24,num34,num35,holland,baer}.\nThese time-dependent methods can be used for studying non-equilibrium dynamics\nof the condensate involving non-stationary states.\n\n\nThe purpose of the present paper is to develop {a simple and\nefficient} algorithm for the numerical solution of the GP equation using  time\npropagation together with the semi-implicit Crank-Nicolson discretization\nscheme \\cite{koonin,ames,dtray} specially useful to newcomers in this field\ninterested in obtaining a numerical solution of the time-dependent GP equation.\nEasy-to-use Fortran 77  programs for different  trap symmetries with adequate\nexplanation are also included. In case of two and three space variables, Fortran\n90\/95 programs are more compact in nature and these programs are also included.\nWe include programs using both real- and imaginary-time propagation. For\nstationary ground states the imaginary-time method has a much quicker \nconvergence rate\ncompared to the real-time method and should be used for the calculation of\nchemical potential, energy  and root-mean-square (rms) sizes. We calculate the chemical\npotential and rms sizes of the condensate for stationary problems and compare\nthese results with those previously obtained by other workers for different trap\nsymmetries. These results can be easily calculated in a decent PC using the\nFortran programs provided. In addition to the results for the stationary states,\nthe real-time propagation routines can also be used to study the \nnon-stationary\ntransitions,  as in  collapse dynamics \\cite{ska5} and non-equilibrium\noscillation \\cite{num22}.\n\n\n{This paper is organized as follows.} In Sec. \\ref{gpe} we present\nthe GP equations with different traps that  we consider in this paper, e.g., the\n3D spherically-symmetric, 2D circularly-symmetric  and 1D harmonic traps\ninvolving one space variable, the anisotropic 2D  and axially-symmetric 3D\nharmonic traps in two space variables and the fully anisotropic 3D harmonic trap\nin three space variables. In Sec. \\ref{CN1D} we elaborate the numerical\nalgorithm for solving the GP equation in one space variable (the 1D, \ncircularly-symmetric 2D, and spherically-symmetric 3D cases) \nand for calculating\nthe chemical potential, energy and rms sizes  employing both the real- and\nimaginary-time propagation methods. \nIn Sec. \\ref{CN23D} we present the same  for\nsolving the GP equation in two and three space variables (the anisotropic  2D\nand axially-symmetric and anisotropic 3D cases). In Sec. \\ref{FOR} we present a\ndescription of the Fortran programs, an explanation about how to use them, and\nsome sample outputs. In Sec. \\ref{NUM} we present the numerical results for\nchemical potential, rms size, value of the wave function at the center, for the\nground-state\nproblem using the imaginary-time propagation routines and compare our\nfinding with previous results for different trap symmetries in 1D, 2D, and 3D. \nWe also present a study of non-stationary oscillation in some of these cases\nusing the real-time propagation routines when the nonlinear coefficient in the\nGP equation with a stationary solution was suddenly reduced to half its value.\nFinally, in Sec. \\ref{SUM} we present a brief summary of our study.\n\n\n\\section{Nonlinear Gross-Pitaevskii Equation}\n\\label{gpe}\n\nAt zero temperature, the time-dependent Bose-Einstein condensate wave function\n$\\Psi \\equiv \\Psi({\\bf r};\\tau)$ at position ${\\bf r}$ and time $\\tau $ may be\ndescribed by the following  mean-field nonlinear GP equation \\cite{review}\n\\begin{eqnarray}\n\\mbox{i}\\hbar\\frac{\\partial \\Psi({\\bf r};\\tau) }{\\partial \\tau}  =\n\\left[-\\frac{\\hbar^2\\nabla\n^2}{2m} + V({\\bf r}) + gN\\vert\\Psi({\\bf r};\\tau) \\vert^2\n\\right]\\Psi({\\bf r};\\tau),\n\\label{eqn:gp0}\n\\end{eqnarray}\nwith $\\mbox{i}=\\sqrt{-1}$. Here $m$ is the mass of an atom and   $N$ the number\nof atoms in the condensate, $g=4\\pi\\hbar^2 a\/m $ the strength of inter-atomic\ninteraction, with $a$ the atomic scattering length. The normalization condition\nof the wave function is $\\int d{\\bf r} \\vert \\Psi({\\bf r};\\tau)\\vert^2 = 1$.\n\n\\subsection{Spherically-symmetric GP equation in 3D}\n\nIn this case  the trap potential is given by $V({\\bf r}) =\\frac{1}{2}m \\omega ^2\n\\tilde r^2$, where $\\omega$ is the angular frequency and $\\tilde r$ the radial\ndistance. After a partial-wave projection the radial part $\\psi$ of \nthe wave function $\\Psi$ can be written as $\\Psi({\\bf \nr};\\tau) =\\psi(\\tilde\nr,\\tau)$. After a transformation of variables to  dimensionless quantities\ndefined by $r =\\sqrt 2 \\tilde r\/l$, $t=\\tau \\omega$, $l\\equiv \\sqrt\n{(\\hbar\/m\\omega)} $ and $\\phi(r;t) \\equiv \\varphi(r;t)\/r =\\psi(\\tilde r,\\tau)[\nl^3\/(2\\sqrt 2)]^{1\/2}$, the GP equation (\\ref{eqn:gp0}) in this case becomes\n\\begin{eqnarray}\n\\left[-\\frac{\\partial^2}\n{\\partial r^2}+\\frac{r^2}{4}+\\aleph           \n\\left| \\frac{\\varphi(r;t)}{r}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0,\n\\label{sph}\n\\end{eqnarray}\nwhere $\\aleph\n=8\\sqrt 2\\pi  N a\/l$. {The purpose of \nchanging the wave function from $\\psi$ to $\\varphi=r\\psi$ is a matter of \ntaste and  it has certain advantages.\n First, this transformation removes the first derivative \n$\\partial\/\\partial r$ from the differential equation (\\ref{sph}) \nand thus results in a simpler equation  \\cite{num26}. \nSecondly, at the origin $r=0$, $\\psi$ is a constant, or \n$\\partial \\psi\/\\partial r=0$. But the new variable satisfies \n$\\varphi(0,t)=0$. Hence, while solving the\ndifferential equation (\\ref{sph}),\nwe can implement the \nsimple boundary condition \nthat as $r \\to 0 $ or $\\infty$, $\\varphi$ vanishes. \nThe boundary condition for the \ndifferential \nequation in $\\psi$ will be  a mixed one, e.g., the function $\\psi$ \nshould \nvanish at infinity  and its first space derivative should vanish at \nthe origin. }     \nThe normalization condition for the\nwave function is\n\\begin{align}\n\\label{n1}\n4\\pi \\int_0^\\infty dr \\vert\\varphi(r;t)\\vert^2=1.\n\\end{align}\n\nHowever, Eq. (\\ref{sph}) is not the unique form of dimensionless GP equation in\nthis case. Other forms of dimensionless equations have been obtained and used by\ndifferent workers. For example, using  the transformations\n$r =\\tilde r\/l$,\n$t=\\tau \\omega$, $l\\equiv \\sqrt {(\\hbar\/m\\omega)} $ and $\\phi(r;t)\n\\equiv \\varphi(r;t)\/r =\\psi(\\tilde r,\\tau)l^{3\/2}$, the GP equation\n(\\ref{eqn:gp0}) becomes\n\\begin{align}\n\\left[-\\frac{1}{2}\\frac{\\partial^2}\n{\\partial r^2}+\\frac{1}{2}{r^2}+\\aleph       \n\\left| \\frac{\\varphi(r;t)}{r}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0,\n\\label{sph2}\n\\end{align}\nwhere $\\aleph\n=4\\pi  N a\/l$ with normalization (\\ref{n1}).\nFinally, using\nthe transformations\n$r =\\tilde r\/l$,\n$t=\\tau \\omega\/2$, $l\\equiv \\sqrt {(\\hbar\/m\\omega)} $ and $\\phi(r;t)\n\\equiv \\varphi(r;t)\/r =\\psi(\\tilde r,\\tau)l^{3\/2}$, the GP equation\n(\\ref{eqn:gp0}) becomes\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial r^2}+{r^2}+\\aleph \n\\left| \\frac{\\varphi(r;t)}{r}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0,\n\\label{sph3}\n\\end{align}\nwhere $  \\aleph\n=8\\pi  N a\/l$ with normalization (\\ref{n1}). These three sets of\ndimensionless GP equations have been {widely} used in the\nliterature and will be considered here. Equations (\\ref{sph}), (\\ref{sph2}), and\n(\\ref{sph3}) allow stationary solutions $\\varphi(r;t)\\equiv\n\\varphi(r)\\exp(-\\mbox{i}\\mu t)$ where  $\\mu$ is the chemical potential. The\nboundary conditions for the solution of these equations are $\\varphi(0,t)=0$ and\n$\\lim_{r\\to \\infty}\\varphi(r,t)=0 $ \\cite{koonin}.\n\n\n\n\\subsection{Anisotropic GP equation in 3D}\n\nThe three-dimensional trap potential is given by $V({\\bf r})\n=\\frac{1}{2}m \\omega^2(\\nu^2\\bar x^2+\\kappa^2\\bar\ny^2+\\lambda^2\\bar\nz^2)$,\nwhere $\\omega_x \\equiv \\nu \\omega$, $\\omega_y\\equiv \\omega\\kappa$, and\n$\\omega_z\\equiv \\omega\\lambda$ are the angular frequencies in the\n$x$,\n$y$ and $z$ directions, respectively, and ${\\bf r}\\equiv (\\bar x,\\bar\ny,\\bar z)$ is the radial vector. In terms of dimensionless variables\n$x=\\sqrt 2 \\bar x\/l, y=\\sqrt 2 \\bar y\/l,z=\\sqrt 2 \\bar z\/l, t=\\tau\n\\omega, l=\\sqrt{\\hbar\/(m\\omega))}$, and $\\varphi(x,y,z;t)=\\sqrt{\nl^3\/(2\\sqrt 2)}\\Psi({\\bf r};\\tau)$, the GP equation (\\ref{eqn:gp0})\nbecomes\n\\begin{align}\n\\left[\n-\\frac{\\partial^2}{\\partial  x^2}\n-\\frac{\\partial^2}{\\partial  y^2}\n-\\frac{\\partial^2}{\\partial  z^2}\n+\\frac{1}{4} \\biggr(\\nu^2x^2+\\kappa^2 y^2+\\lambda^2 z^2  \\biggr)\n+   \\aleph\n\\left\\vert\\varphi(x,y,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial  t} \\right]\n\\varphi(x,y,z;t)= 0,\\label{ani}\n\\end{align}\nwith $\\aleph\n=8\\sqrt 2 \\pi aN\/l$ and normalization\n\\begin{align}\\label{n3}\n\\int_{-\\infty}^{\\infty}dx\n\\int_{-\\infty}^{\\infty}dy\n\\int_{-\\infty}^{\\infty}dz\n|\\varphi(x,y,z;t)|^2 =1.\n\\end{align}\n\nSimilarly, using $x=\\bar x\/l, y=\\bar y\/l,z= \\bar\nz\/l, t=\\tau\n\\omega, l=\\sqrt{\\hbar\/(m\\omega)}$, and $\\varphi(x,y,z;t)=\\sqrt{\nl^3}\\Psi({\\bf r};\\tau)$, the GP equation (\\ref{eqn:gp0}) becomes\n\\begin{eqnarray}\n\\left[\n-\\frac{1}{2}\\frac{\\partial^2}{\\partial  x^2}\n-\\frac{1}{2}\\frac{\\partial^2}{\\partial  y^2}\n-\\frac{1}{2}\\frac{\\partial^2}{\\partial  z^2}\n+\\frac{1}{2} \\biggr(\\nu^2x^2+\\kappa^2 y^2+\\lambda^2 z^2  \\biggr)\n+\\aleph \n\\left\\vert\\varphi(x,y,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial\nt} \\right]\n\\varphi(x,y,z;t)= 0,\\label{ani2}\n\\end{eqnarray}\nwith $\\aleph\n=4 \\pi aN\/l$ and normalization (\\ref{n3}). Now \nwith\nscaling $t\\to 2t$, Eq. (\\ref{ani2}) can be rewritten as\n\\begin{eqnarray}\n\\left[\n-\\frac{\\partial^2}{\\partial  x^2}\n-\\frac{\\partial^2}{\\partial  y^2}\n-\\frac{\\partial^2}{\\partial  z^2}\n+ \\biggr(\\nu^2x^2+\\kappa^2 y^2+\\lambda^2 z^2  \\biggr)\n+\\aleph \n\\left\\vert\\varphi(x,y,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial\nt} \\right]\n\\varphi(x,y,z;t)= 0,\\label{ani3}\n\\end{eqnarray}\nwith $\\aleph \n=8\\pi aN\/l$.\nThe boundary conditions for solution are $\\lim_{x\\to \\pm \\infty}\n\\varphi(x,y,z;t)=0, \\lim_{y\\to \\pm \\infty}\n\\varphi(x,y,z;t)=0,\\lim_{z\\to \\pm \\infty}\n\\varphi(x,y,z;t)=0$  \\cite{koonin}.\n\n\\subsection{Axially-symmetric GP equation in 3D}\n\nIn the special case of axial symmetry ($\\nu = \\kappa$) Eqs.\n(\\ref{ani}), (\\ref{ani2}) and (\\ref{ani3}) can be simplified considering\n${\\bf r}\\equiv (\\rho,z)$ where $\\rho=\\sqrt{x^2+y^2}$ is the radial\ncoordinate and $z$\nis the axial coordinate. Then Eq. (\\ref{ani}) becomes\n\\begin{align}\n\\left[\n-\\frac{\\partial^2}{\\partial  \\rho^2}\n-\\frac{1}{\\rho}\\frac{\\partial}{\\partial  \\rho}\n-\\frac{\\partial^2}{\\partial  z^2}\n+\\frac{1}{4} \\biggr(\\kappa^2 \\rho^2+\\lambda^2 z^2  \\biggr)\n+\\aleph  \n\\left\\vert\\varphi(\\rho,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial  t} \\right]\n\\varphi(\\rho,z;t)= 0,\\label{axi}\n \\end{align}\nwith $\\aleph \n=8\\sqrt 2 \\pi aN\/l$ and normalization\n$2\\pi \\int_{0}^{\\infty}\\rho d\\rho\n\\int_{-\\infty}^{\\infty}dz\n|\\varphi(\\rho,z;t)|^2 =1.$\nSimilarly, Eqs. (\\ref{ani2}) and (\\ref{ani3}) can be written as\n\\begin{eqnarray}\n\\left[\n-\\frac{1}{2}\\frac{\\partial^2}{\\partial  \\rho^2}\n-\\frac{1}{2\\rho}\\frac{\\partial}{\\partial  \\rho}\n-\\frac{1}{2}\\frac{\\partial^2}{\\partial  z^2}\n+\\frac{1}{2} \\biggr(\\kappa^2\\rho^2+\\lambda^2 z^2  \\biggr)\n+\\aleph\n\\left\\vert\\varphi(\\rho,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial\nt} \\right]\n\\varphi(\\rho,z;t)= 0,\\label{axi2}\n\\end{eqnarray}\nwith $\\aleph \n=4 \\pi aN\/l$ and\n\\begin{eqnarray}\n\\left[\n-\\frac{\\partial^2}{\\partial  \\rho^2}\n-\\frac{1}{\\rho}\\frac{\\partial}{\\partial  \\rho}\n-\\frac{\\partial^2}{\\partial  z^2}\n+ \\biggr(\\kappa^2 \\rho^2+\\lambda^2 z^2  \\biggr)\n+\\aleph \n\\left\\vert\\varphi(\\rho,z;t)\\right\\vert^2 - \\mbox{i}\n\\frac{\\partial}{\\partial\nt} \\right]\n\\varphi(\\rho,z;t)= 0,\\label{axi3}\n\\end{eqnarray}\nwith $\\aleph  \n=8\\pi aN\/l$. In this case $\\varphi(\\rho=0,z;t)$ \nis not\nzero but a constant.\nConvenient boundary conditions for solution\nin this case are $\\lim_{z\\to \\pm \\infty}\\varphi(\\rho,z;t)=0, \\lim_{\\rho\\to\n\\infty}\\varphi(\\rho,z;t)=0,$ and \n$\\partial \\varphi(\\rho,z;t)\/\\partial \\rho|_{\\rho=0}=0$\n\\cite{num4}.\n\n\n\n\\subsection{One-dimensional GP equation}\n\nIn case of an elongated cigar-shaped trap, which is essentially an\naxially-symmetric trap with strong transverse  confinement,  Eq.\n(\\ref{ani}) reduces to a quasi one-dimensional form. This is achieved by\nassuming that the system remains confined to the ground state in the\ntransverse direction.  In this case the wave function of Eq.\n(\\ref{ani}) can be written as $\\varphi(x,y,z;t) =\\tilde \\varphi(x;t)\n\\phi_0(y)\n\\phi_0(z)\\exp[-i(\\lambda+\\kappa)t\/2]$ with $\\phi_0(y)=\n[\\kappa\/(2\\pi)]^{1\/4} \\exp(-\\kappa y^2\/4)$ and $\\phi_0(z)=\n[\\lambda\/(2\\pi)]^{1\/4} \\exp(-\\lambda z^2\/4)$ the respective ground state\nwave functions in $y$ and $z$ directions. Using this ansatz in Eq.\n(\\ref{ani}), multiplying by $\\phi_0(y)\\phi_0(z)$, integrating over\n$y$ and $z$,  dropping the tilde over $\\varphi$, and setting $\\nu=1$  we\nobtain\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}+\\frac{x^2}{4}+\\aleph\n\\left| {\\varphi(x;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x;t)=0  ,\n\\label{1d}\n\\end{align}\nwith $\\aleph\n= 2a N \\sqrt {2\n\\lambda \\kappa} \/l$ and\n normalization\n\\begin{align}\\label{n5}\n\\int_{-\\infty}^\\infty dx |\\varphi(x;t)|^2 =1.\n\\end{align}\nInstead if we employ\n $\\varphi(x,y,z;t) =\\tilde  \\varphi(x;t) \\phi_0(y)\n\\phi_0(z)\\exp[-i(\\lambda+\\kappa)t\/2]$ with $\\phi_0(y)=\n(\\kappa\/\\pi)^{1\/4} \\exp(-\\kappa y^2\/2)$ and $\\phi_0(z)=\n(\\lambda\/\\pi)^{1\/4} \\exp(-\\lambda z^2\/2)$\n in Eq.\n(\\ref{ani2}), in a similar fashion\n we obtain\n\\begin{align}\n\\left[-\\frac{1}{2}\\frac{\\partial^2}\n{\\partial x^2}+\\frac{x^2}{2}+\\aleph \n\\left| {\\varphi(x;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x;t)=0  ,\n\\label{1d2}\n\\end{align}\nwith $\\aleph \n= 2a N \\sqrt {\n\\lambda \\kappa} \/l$ and normalization (\\ref{n5}). Now with scaling\n$t\\to 2t$ Eq. (\\ref{1d2}) can be rewritten as\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}+{x^2}+\\aleph \n\\left| {\\varphi(x;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x;t)=0  ,\n\\label{1d3}\n\\end{align}\nwith $\\aleph \n= 4a N \\sqrt {\n\\lambda \\kappa} \/l$ and normalization (\\ref{n5}).\nFor numerical solution we take $\\lim_{x\\to \\pm \\infty}\\varphi(x,t)=0$.\n\n\n\n\\label{1D}\n\n\n\\subsection{Anisotropic GP equation in 2D}\n\n\\label{2.5}\n\nIn case of a disk-shaped trap, which is essentially an\nanisotropic\ntrap in two dimensions\nwith strong axial binding Eq.\n(\\ref{ani}) reduces to a two-dimensional form. This is achieved by\nassuming that the system remains confined to the ground state in the\naxial direction.  In this case the wave function of Eq.\n(\\ref{ani}) can be written as $\\varphi(x,y,z;t) = \\tilde \\varphi(x,y;t)\n\\phi_0(z)\\exp[-i\\lambda t\/2]$ with  $\\phi_0(z)=\n[\\lambda\/(2\\pi)]^{1\/4} \\exp(-\\lambda z^2\/4)$ the  ground state\nwave function in  $z$ direction. Using this ansatz in Eq.\n(\\ref{ani}), multiplying by $\\phi_0(z)$, integrating over\n$z$, dropping the tilde over $\\varphi$ and setting $\\nu =1$  we obtain\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}-\\frac{\\partial^2}\n{\\partial y^2}\n+\\frac{x^2+\\kappa y^2}{4}+\\aleph\n\\left| {\\varphi(x,y;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x,y;t)=0  ,\n\\label{2d}\n\\end{align}\nnow with $\\aleph\n= 4a N \\sqrt {2\n\\pi \\lambda} \/l$ and\n normalization\n\\begin{align}\\label{n6}\n\\int_{-\\infty}^\\infty dx\n\\int_{-\\infty}^\\infty dy\n|\\varphi(x,y;t)|^2 =1.\n\\end{align}\n\nInstead if we use in Eq. (\\ref{ani2})\n$\\phi(x,y,z;t) =\\tilde  \\varphi(x,y;t)\n\\phi_0(z)\\exp[-i\\lambda t\/2]$ with  $\\phi_0(z)=\n[\\lambda\/\\pi]^{1\/4} \\exp(-\\lambda z^2\/2)$,  then\nin a similar fashion  we obtain\n\\begin{align}\n\\left[-\\frac{1}{2}\\frac{\\partial^2}\n{\\partial x^2}-\\frac{1}{2}\\frac{\\partial^2}\n{\\partial y^2}\n+\\frac{x^2+\\kappa y^2}{2}+\\aleph \n\\left| {\\varphi(x,y;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x,y;t)=0  ,\n\\label{2d2}\n\\end{align}\nnow with $\\aleph \n= 2a N \\sqrt {2\n\\pi \\lambda} \/l$ and normalization (\\ref{n6}). Finally, with scaling\n$t\\to 2t$, Eq. (\\ref{2d2}) can be written as\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}-\\frac{\\partial^2}\n{\\partial y^2}\n+{x^2+\\kappa y^2}+\\aleph\n\\left| {\\varphi(x,y;t)}\n\\right| ^2\n-\\mbox{i}\\frac{\\partial }{\\partial t}\n\\right] \\varphi (x,y;t)=0  ,\n\\label{2d3}\n\\end{align}\nwith $\\aleph\n= 4a N \\sqrt {2\n\\pi \\lambda} \/l$ and normalization (\\ref{n6}).\nFor numerical solution we take $\\lim_{x\\to \\pm \\infty}\\varphi(x,y,t)=0$\nand $\\lim_{y\\to \\pm \\infty}\\varphi(x,y,t)=0$.\n\n\n\n\\subsection{Circularly-symmetric GP equation in 2D}\n\n\n\nIn the special case of circular symmetry the equations of Sec. \\ref{2.5}\ncan be written in one-dimensional form. In this case $\\kappa =1$, and\nwe introduce the radial variable ${\\bf r}\\equiv (x,y)$, and rewrite the\nwave function as $\\varphi(r)$. Then the GP equation (\\ref{2d}) become\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial r^2}-\\frac{1}{r}\\frac{\\partial}\n{\\partial r} +\\frac{r^2}{4}+\\aleph \n\\left| {\\varphi(r;t)}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0.\n\\label{cir}\n\\end{align}\n The normalization of  the\nwave function is\n$2\\pi \\int_0^\\infty dr r\\vert\\varphi(r;t)\\vert^2=1.$\n\nIn the circularly-symmetric case Eq. (\\ref{2d2}) becomes\n\\begin{align}\n\\left[-\\frac{1}{2}\\frac{\\partial^2}\n{\\partial r^2}-\\frac{1}{2r}\\frac{\\partial}\n{\\partial r}+\n\\frac{1}{2}{r^2}+\\aleph \n\\left| {\\varphi(r;t)}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0.\n\\label{cir2}\n\\end{align}\nFinally, Eq. (\\ref{2d3}) can be written as\n\\begin{align}\n\\left[-\\frac{\\partial^2}\n{\\partial r^2}-\\frac{1}{r}\\frac{\\partial}\n{\\partial r}\n+{r^2}+\\aleph  \n\\left| {\\varphi(r;t)}\n\\right| ^2 -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0.\n\\label{cir3}\n\\end{align}\nThe convenient boundary condition in this case is $\\lim_{r\\to \\infty }\n\\varphi\n(r;t)=0$ and $d\\varphi\n(r;t)\/dr|_{r=0}=0$ \\cite{num4}.\n\n\nIn this section we have exhibited GP equations for different\ntrap symmetries. In the next section we illustrate the Crank-Nicolson\nmethod for the GP equation in one space variable, which is then\nextended to\nother types of equations in Sec. \\ref{CN23D}.\n\n\n\n\\section{Split-Step Crank-Nicolson Method for the GP\nEquation in one Space Variable}\n\\label{CN1D}\n\n\n\\subsection{The GP Equation in the 1D and radially-symmetric 3D\ncases}\n\nTo introduce the Crank-Nicolson Method \\cite{koonin,ames,dtray}\nfor GP equation we consider\nfirst the one-dimensional case of Sec. \\ref{1D}. The\n nonlinear GP equation (\\ref{1d})\nin this\ncase can be expressed in the following form:\n\\begin{align}\n\\mbox{i}\\frac{\\partial }{\\partial t} \\varphi (x;t) & =\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}+\\frac{x^2}{4}+ \\aleph\n\\left| {\\varphi(x;t)}\n\\right| ^2\\right] \\varphi (x;t)  , \\notag \\\\\n & \\equiv H  \\varphi (x;t)\n\\label{e1d}\n\\end{align}\nwhere the Hamiltonian $H$ contains the different linear and nonlinear terms\nincluding the spatial derivative. (The spherically-symmetric GP equation in 3D\nhas a similar structure and can be treated similarly.) We solve this equation by\ntime iteration \\cite{koonin,ames,dtray}. A given  trial input solution is\npropagated in time over small time steps until a stable final  solution is\nreached. The GP equation is discretized in space and time using the finite\ndifference scheme. This procedure results in a set of algebraic equations which\ncan be solved by time iteration using an input solution consistent with the\nknown boundary condition. In the present split-step method \\cite{ames} this\niteration is conveniently done in few steps by breaking up the full Hamiltonian\ninto different derivative and non-derivative parts.\n\n\n\n\\subsubsection{Real-time propagation} \\label{RT}\n\nThe time iteration is performed by splitting $H$ into two parts:\n$H=H_1+H_2$, with\n\\begin{align}\\label{h1}\nH_1 & = \\left[ \\frac{x^2}{4}+\\aleph \n\\left|\n{\\varphi(x;t)}\n\\right| ^2 \\right], \\;\\; \\\\ \\label{h2}\nH_2 & = -\\frac {\\partial ^2}{\\partial x^2}.\n\\end{align}\n{\nEssentially we split Eq.~(\\ref{e1d}) into\n\\begin{align}\n\\mbox{i}\\frac{\\partial }{\\partial t} \\varphi (x;t) & =\n\\left[ \\frac{x^2}{4}+\\aleph \n\\left\\vert {\\varphi(x;t)}\n\\right\\vert ^2\\right] \\varphi (x;t)  \\equiv H_1  \\varphi (x;t)\n\\label{e1d_1} \\\\\n\\mbox{i}\\frac{\\partial }{\\partial t} \\varphi (x;t) & =\n-\\frac{\\partial^2}\n{\\partial x^2}\\varphi (x;t)   \\equiv H_2  \\varphi (x;t)\n\\label{e1d_2} \n\\end{align}\nWe first solve Eq.~(\\ref{e1d_1}) with a initial value $\\varphi (x;t_0)$ \nat $t = t_0$ to obtain an intermediate solution at $t = t_0 + \\Delta$, where\n$\\Delta$ is the time step. \nThen this intermediate solution is used as initial value to solve \nEq.~(\\ref{e1d_2}) yielding the final solution at $t = t_0 + \\Delta$ as \n$\\varphi (x;t_0 + \\Delta)$. This procedure is repeated $n$ times to get \nthe final\nsolution at a given time $t_{\\text{final}}=t_0+n\\Delta $}. \n\nThe time variable is discretized as $t_n=n\\Delta$ where $\\Delta$ is\nthe time step. The solution is advanced first over the time step\n$\\Delta$ at time $t_n$ by solving the GP equation (\\ref{e1d}) with\n$H=H_1$  to produce an intermediate solution $\\varphi^{n+1\/2}$ from\n$\\varphi^n$, where $\\varphi^n$ is the discretized wave  function at\ntime $t_n$. As there is no derivative in $H_1$, this propagation is\nperformed essentially exactly for small $\\Delta$ through the\noperation\n\\begin{align}\\label{al1}\n\\varphi^{n+1\/2}\n& = {\\bigcirc}_{\\mathrm{nd}}(H_1) \\varphi^n \\equiv  e^{-\n\\mbox{i}\\Delta H_1}\n\\varphi^n,\n\\end{align}\nwhere ${\\bigcirc}_{\\mathrm{nd}} (H_1)$ denotes time-evolution operation\nwith $H_1$ and the suffix `nd' denotes non-derivative. Next we perform\nthe time propagation corresponding to the operator $H_2$ numerically\nby the semi-implicit Crank-Nicolson scheme (described below)\n\\cite{koonin}:\n\\begin{align}\\label{gp2}\n\\frac{ \\varphi^{n+1}- \\varphi^{n+1\/2}}{-\\mbox{i}\\Delta } =\n\\frac{1}{2}H_2(\n\\varphi^{n+1} + \\varphi^{n+1\/2}).\n\\end{align}\nThe formal solution to (\\ref{gp2}) is\n\\begin{align}\\label{gp3}\n \\varphi^{n+1}= {\\bigcirc}_{\\mathrm{CN}}(H_2) \\varphi^{n+1\/2}\n\\equiv\n\\frac{1-\\mbox{i}\\Delta H_2\/2  }{ 1+\\mbox{i}\\Delta\nH_2\/2 }\n\\varphi^{n+1\/2},\n\\end{align}\nwhich combined with Eq. (\\ref{al1}) yields\n\\begin{align}\\label{gp4}\n \\varphi^{n+1}={\\bigcirc}_{\\mathrm{CN}}(H_2)   \n{\\bigcirc}_{\\mathrm{nd}}(H_1)\n\\varphi^n,\n\\end{align}\nwhere ${\\bigcirc}_{\\mathrm{CN}} $ denotes time-evolution operation with \n$H_2$\nand   the suffix `CN' refers to the  Crank-Nicolson algorithm. Operation \n${\\bigcirc}_{\\mathrm{CN}} $ is used to propagate the intermediate \nsolution  $ \n\\varphi^{n+1\/2} $ by time step $\\Delta$  to generate the solution $ \n\\varphi^{n+1}$ at the next time step $t_{n+1}=(n+1)\\Delta$.\n\nThe advantage of the above split-step method with small time step $\\Delta$ is\ndue to the following three factors \\cite{ames,dtray}. First, all iterations\nconserve normalization of the wave function. Second, the error involved in\nsplitting the Hamiltonian is proportional to $\\Delta^2$ and can be neglected and\nthe method preserves the {symplectic} structure of the Hamiltonian formulation. \nFinally, as a major part of the Hamiltonian including the nonlinear term is\ntreated fairly accurately without mixing with the delicate Crank-Nicolson\npropagation, the method can deal with an arbitrarily large nonlinear term and\nlead to stable and accurate converged result.\n\nNow we describe explicitly the semi-implicit\nCrank-Nicolson algorithm.\nThe GP\nequation is mapped onto  $N_x$ one-dimensional spatial grid points in\n$x$.\nEquation (\\ref{e1d}) is discretized with $H=H_2$ of (\\ref{h2}) by the\nfollowing Crank-Nicolson scheme \\cite{koonin,ames,dtray}:\n\\begin{align}\\label{kn1}\n\\frac{\\mbox{i}(\\varphi_{i}^{n+1}-\n\\varphi_{i}^{n+1\/2})}{\\Delta}=-\\frac{1}{2h^2}\\biggr[(\\varphi^{n+1}_{i+1}-\n2\\varphi^{n+1}_i\n+\\varphi^{n+1}_{i-1})\n+(\\varphi^{n+1\/2}_{i+1}-2\\varphi_{i}^{n+1\/2}\n+\\varphi^{n+1\/2}_{i-1})\\biggr],\n\\end{align}\nwhere for the spherically-symmetric Eq. (\\ref{sph})\n$\\varphi_i^n=\\varphi(x_i;t_n)$ refers to  $x\\equiv x_i=ih,$\n$i=0,1,2,...,N_x$  and $h$ is the space \nstep.\nIn the case of\nthe \n1D Eq. (\\ref{1d}), we  choose  $x\\equiv x_i=-N_xh\/2+  ih,$\n$i=0,1,2,...,N_x$.\n {(The choice $-$ even $N_x$ $-$ has the\nadvantage\nof taking an equal number of space points on both sides of $x=0$ in the\n1D  case setting the point $N_x\/2$ at $x=0$.)} \nEquation (\\ref{kn1}) is the\nexplicit form of the formal Eq. (\\ref{gp2}). This scheme is\nconstructed by approximating $\\partial \/\\partial t$ by a two-point\nformula connecting the present ($n+1\/2$) to future ($n+1$). The\nspatial partial derivative\n$\\partial^2 \/\\partial x^2$ is\napproximated  by a three-point formula averaged over the present and\nthe future time grid points.\nThis\nprocedure results in a series of tridiagonal sets of equations\n(\\ref{kn1}) in $\\varphi^{n+1}_{i+1}$,  $\\varphi^{n+1}_{i}$, and\n$\\varphi^{n+1}_{i-1}$ at time $t_{n+1}$, which are solved using the\nproper boundary conditions.\n\n\nThe Crank-Nicolson scheme (\\ref{kn1}) possesses certain properties\nworth mentioning \\cite{ames,dtray}. The error in this scheme is both\nsecond order in space and time steps so that for small $\\Delta$ and\n$h$ the error is negligible. This scheme is also unconditionally\nstable \\cite{dtray}. The boundary condition at infinity is preserved for\nsmall\nvalues of $\\Delta\/h^2$\\cite{dtray}.\n\nThe tridiagonal equations emerging from Eq. (\\ref{kn1}) are written\nexplicitly as \\cite{koonin}\n\\begin{align}\\label{eq1}\nA_i^-\\varphi^{n+1}_{i-1}+A_i^0\\varphi^{n+1}_{i}+\nA_i^+\\varphi^{n+1}_{i+1}= b_i,\n\\end{align}\nwhere\n\\begin{align}\nb_i=\\frac{\\mbox{i}\\Delta}{2h^2}(\\varphi^{n+1\/2}_{i+1}-2\\varphi_{i}^{n+1\/2}\n+\\varphi^{n+1\/2}_{i-1})+\\varphi_i^{n+1\/2},\n\\end{align}\nand $A_i^-=A_i^+=  -\\mbox{i}\\Delta\/(2h^2), A_i^0 = 1+\\mbox{i}\n\\Delta\/h^2$. All quantities in $b_i$ refer to time step $t_{n+1\/2}$\nand are considered known. The only unknowns in Eq. (\\ref{eq1}) are the\nwave forms $\\varphi^{n+1}_{i\\pm 1}$ and $\\varphi^{n+1}_{i}$ at time\nstep $t_{n+1}$. To solve Eq. (\\ref{eq1}), we assume the one-term\nforward recursion relation\n\\begin{align}\\label{eq2}\n\\varphi^{n+1}_{i+1}=\\alpha_i\\varphi^{n+1}_{i}+\\beta_i,\n\\end{align}\nwhere $\\alpha_i$ and $\\beta_i$ are coefficients to be determined.\nSubstituting Eq. (\\ref{eq2}) in Eq. (\\ref{eq1}) we obtain\n\\begin{align}\\label{eq3}\nA_i^-\\varphi^{n+1}_{i-1}+A_i^0\\varphi^{n+1}_{i}+\nA_i^+(\\alpha_i \\varphi^{n+1}_{i}+\\beta_i)= b_i,\n\\end{align}\nwhich leads to the solution\n\\begin{align}\\label{eq4}\n\\varphi_i^{n+1}=\\gamma_i(A_i^- \\varphi_{i-1}^{n+1}+A_i^+\\beta_i-b_i),\n\\end{align}\nwith\n\\begin{align}\\label{eq41}\n\\gamma_i=-1\/(A_i^0+A_i^+\\alpha_i).\n\\end{align}\nFrom Eqs. (\\ref{eq2}) and (\\ref{eq4}) we obtain the following backward\nrecursion relations for the coefficients $\\alpha_i$ and $\\beta_i$\n\\begin{align}\\label{eq5}\n\\alpha_{i-1}=\\gamma_iA_i^-, \\;\\;\\; \\beta_{i-1}=\n\\gamma_i(A_i^+\\beta_i-b_i).\n\\end{align}\nWe shall use the recursion relations (\\ref{eq4}), (\\ref{eq41}) and (\\ref{eq5})\nin a backward sweep of the lattice to determine $\\alpha_i$ and $\\beta_i$ for $i$\nrunning from $N_x-2$ down to 0. The initial values chosen are $\\alpha_{N_x-1}=0,\n\\beta_{N_x-1} = \\varphi^{n+1}_{N_x}.$ This ensures the correct value of\n$\\varphi$ at the last lattice point. After determining the coefficients\n$\\alpha_i, \\beta_i$ and $\\gamma_i$, we can use the recursion relation\n(\\ref{eq2}) from $i=0$ to $N_x-1$ to determine the solution for the entire space\nrange using the starting value $\\varphi^{n+1}_0$  (=0) known from the boundary\nconditions. The value at the last lattice point is also taken to be known (= 0).\nThus we have determined the solution by using two sets of recursion relations\nacross the lattice each involving about $N_x$ operations.\n\nIn the numerical implementation of CN real-time propagation the\ninitial state at $t=0$ is usually chosen to be the analytically known\nsolution of the harmonic potential with zero nonlinearity: \n$\n\\aleph \n=0$. In the course of time iteration the nonlinearity is slowly\nintroduced until the desired final nonlinearity is attained. This\nprocedure will lead us to the final solution of the problem.\n\n\n\\subsubsection{Imaginary-time propagation}\n\\label{IT}\n\nAlthough, real-time propagation as described above has many\nadvantages, in this approach one has to deal with complex variables\nfor a complex wave function for non-stationary states. For stationary\nground\nstate the wave function is essentially real- and the imaginary-time\npropagation method dealing with real variables  seems to be\nconvenient. In this approach time $t$ is replaced by an imaginary\nquantity $t=- \\mbox{i}\\bar t$ and Eq. (\\ref{e1d}) now becomes\n\\begin{align}\n-\\frac{\\partial }{\\partial \\bar  t} \\varphi (x;\\bar t) & =\n\\left[-\\frac{\\partial^2}\n{\\partial x^2}+\\frac{x^2}{4}+\\aleph \n\\left|\n{\\varphi(x;\\bar t)}\n\\right| ^2\\right] \\varphi (x;\\bar t), \\notag \\\\\n& \\equiv H  \\varphi (x;\\bar t)\n\\label{imt}\n\\end{align}\nIn this equation $\\bar t$ is just  a mathematical parameter. \n\n{From Eq. (\\ref{imt}) we see that an eigenstate \n$\\varphi_i$ of eigenvalue $E_i$ \nof $H$ satisfying $H\\varphi_i=E_i\\varphi_i$ behaves under imaginary-time \npropagation as $\\partial \\varphi_i(\\bar t) \/\\partial \\bar t= -E_i \n\\varphi_i(\\bar t)$, so \nthat  $\\varphi_i(\\bar t)= \\exp(-E_i\\bar t)\\varphi_i( 0)$. Hence,  if we \nstart \nwith as arbitrary initial  $\\varphi (x;\\bar t)$ which can be taken as a \nlinear combination of all eigenfunctions of $H$, then upon \nimaginary-time propagation all the eigenfunctions will decay \nexponentially with \ntime. However, all the excited states with larger $E_i$ will decay \nexponentially faster compared to the ground state with the smallest \neigenvalue. Consequently, after some time only the ground state \nsurvives. As all the states are decaying with time during \nimaginary-time propagation, we need to multiply the wave function by a \nnumber \ngreater than unity to preserve its normalization so that the solution \ndoes not go to zero.}\n\nThe imaginary-time iteration is performed by splitting $H$ into two parts as\nbefore: $H=H_1+H_2$, with $H_1$ and $H_2$ given by Eqs. (\\ref{h1}) and\n(\\ref{h2}). It is realized that the entire analysis of Sec. \\ref{RT} remains\nvalid provided we replace $\\mbox{i}$ by 1 in Eq. (\\ref{al1}) and by $-1$ in the\nremaining equations. However, there appears one trouble.  The CN real-time\npropagation preserves the normalization of the wave function, whereas the CN\nimaginary-time propagation does not preserve the normalization. This problem can\nbe circumvented by restoring the normalization of the wave function after each\noperation of Crank-Nicolson propagation. Once this is done the imaginary-time\npropagation method for  stationary ground state problems yields very \naccurate result at\nlow computational cost.\n\nCompared to the real-time propagation method, the imaginary-time\npropagation method is very robust. The initial solution in the \nimaginary-time method \ncould be any reasonable solution and not the analytically known solution\nof a related problem as in the real-time method. { Also, the\nfull  nonlinearity can be added in a small number of time steps or even\nin a single step and not in a large number of  steps as in the real-time \nmethod.} In the programs using the imaginary-time method  we include the \n nonlinearity in a single step.  These two \nadded features  together with\nthe use of real algorithm make the imaginary-time propagation method\nvery accurate with quick convergence for stationary ground states as we \nshall\nsee below.\n\n\n\\subsection{The GP Equation in the circularly-symmetric 2D\ncase}\n\\label{gp2dc}\n\n\nThe Crank-Nicolson discretization for real- and imaginary-time propagation for\nthe circularly-symmetric GP equation (\\ref{cir}) is performed in a similar\nfashion as for Eq. (\\ref{e1d}), apart from the difference that here we also have\na first derivative in space variable in addition to the second derivative.\nAnother difference is that the wave function is not zero at $r=0$. In the case\nof Eq. (\\ref{e1d}) we took the boundary condition as $\\varphi(x;t)=0$ at the\nboundaries. For the circularly-symmetric case, the convenient boundary\nconditions are $\\lim_{r\\to \\infty}\\varphi(r;t)=0$ and $d\\varphi\n(r;t)\/dr|_{r=0}=0$.\n\nWe describe below the Crank-Nicolson discretization and the solution\nalgorithm in this case for the following equation\n\\begin{equation}\n\\left[-\\frac{\\partial^2}\n{\\partial r^2}-\\frac{1}{r}\\frac{\\partial}\n{\\partial r}  -\n\\mbox{i}\\frac{\\partial }{\\partial t}\\right] \\varphi\n(r;t)=0,\n\\label{CNCIR}\n\\end{equation}\nrequired for the solution of Eq. (\\ref{cir}). The remaining procedure is\nsimilar to that described in detail above in the one-dimensional case.\n\nEquation (\\ref{CNCIR}) is discretized by the\nfollowing Crank-Nicolson scheme as in Eq.\n(\\ref{kn1}) \\cite{koonin,ames,dtray}:\n\\begin{align}\\label{knx}\n\\frac{\\mbox{i}(\\varphi_{i}^{n+1}-\n\\varphi_{i}^{n+1\/2})}{\\Delta}=-\\frac{1}{2h^2}\\biggr[(\\varphi^{n+1}_{i+1}-\n2\\varphi^{n+1}_i\n+\\varphi^{n+1}_{i-1})\n+(\\varphi^{n+1\/2}_{i+1}-2\\varphi_{i}^{n+1\/2}\n+\\varphi^{n+1\/2}_{i-1})\\biggr]\\nonumber \\\\ -\\frac{1}{4r_ih}\n\\left[(\\varphi^{n+1}_{i+1}-\\varphi^{n+1}_{i-1})+(\\varphi^{n+1\/2}_{i+1}-\n\\varphi^{n+1\/2}_{i-1})\n\\right],\n\\end{align}\nwhere again\n$\\varphi_i^n=\\varphi(r_i;t_n)$,   $r\\equiv r_i=ih,$\n$i=0,1,2,...,N_r$  and $h$ is the space step.\nThis scheme is\nconstructed by approximating $\\partial \/\\partial x$ by a two-point\nformula averaged over  present and  future time grid points.\nThe discretization of the first-order time and second-order space\nderivatives is done as in Eq. (\\ref{kn1}).\nThis\nprocedure results in the tridiagonal sets of equations\n(\\ref{knx}) in $\\varphi^{n+1}_{i+1}$,  $\\varphi^{n+1}_{i}$, and\n$\\varphi^{n+1}_{i-1}$ at time $t_{n+1}$, which are solved using the\nproper boundary conditions.\n\n\n\nThe tridiagonal equations emerging from Eq. (\\ref{knx}) are written\nexplicitly as\n\\begin{align}\\label{ep1}\nA_i^-\\varphi^{n+1}_{i-1}+A_i^0\\varphi^{n+1}_{i}+\nA_i^+\\varphi^{n+1}_{i+1}= b_i,\n\\end{align}\nwhere\n\\begin{align}\nb_i=\\frac{\\mbox{i}\\Delta}{2h^2}(\\varphi^{n+1\/2}_{i+1}-2\\varphi_{i}^{n+1\/2}\n+\\varphi^{n+1\/2}_{i-1})+\\varphi_i^{n+1\/2}+\\frac{\\mbox{i}\\Delta}{4r_ih}\n(\\varphi^{n+1\/2}_{i+1}-\\varphi^{n+1\/2}_{i-1}),\n\\end{align}\nand $A_i^-= \\mbox{i}\\Delta[1\/(4hr_i) -1\/(2h^2)],\nA_i^+= -\\mbox{i}\\Delta[1\/(4hr_i) +1\/(2h^2)],\nA_i^0 =\n1+\\mbox{i}\n\\Delta\/h^2$. All quantities in $b_i$ refer to time step $t_{n+1\/2}$\nand are considered known. The only unknowns in Eq. (\\ref{ep1}) are the\nwave forms $\\varphi^{n+1}_{i\\pm 1}$ and $\\varphi^{n+1}_{i}$ at time\nstep $t_{n+1}$. To solve Eq. (\\ref{ep1}), we assume the one-term\nbackward recursion relation\n\\begin{align}\\label{ep2}\n\\varphi^{n+1}_{i-1}=\\alpha_i\\varphi^{n+1}_{i}+\\beta_i,\n\\end{align}\nwhere $\\alpha_i$ and $\\beta_i$ are coefficients to be determined.\nSubstituting Eq. (\\ref{ep2}) in Eq. (\\ref{ep1}) we obtain\n\\begin{align}\\label{ep3}\nA_i^+\\varphi^{n+1}_{i+1}+A_i^0\\varphi^{n+1}_{i}+\nA_i^-(\\alpha_i \\varphi^{n+1}_{i}+\\beta_i)= b_i,\n\\end{align}\nwhich leads to the solution\n\\begin{align}\\label{ep4}\n\\varphi_i^{n+1}=\\gamma_i(A_i^+ \\varphi_{i+1}^{n+1}+A_i^-\\beta_i-b_i),\n\\end{align}\nwith\n\\begin{align}\\label{ep41}\n\\gamma_i=-1\/(A_i^0+A_i^-\\alpha_i).\n\\end{align}\nFrom Eqs. (\\ref{ep2}) and (\\ref{ep4}) we obtain the following\nforward\nrecursion relations for the coefficients $\\alpha_i$ and $\\beta_i$\n\\begin{align}\\label{ep5}\n\\alpha_{i+1}=\\gamma_iA_i^+, \\;\\;\\; \\beta_{i+1}=\n\\gamma_i(A_i^-\\beta_i-b_i).\n\\end{align}\nWe shall use the recursion relations (\\ref{ep4}), (\\ref{ep41}) and (\\ref{ep5})\nin a forward sweep of the lattice to determine $\\alpha_i$ and $\\beta_i$ for $i$\nrunning from $1$  to $N_r-1$. The initial values chosen are $\\alpha_{1}=1,\n\\beta_{1} =0.$ This ensures the correct value of the space derivative of\n$\\varphi(r;t)=0$ at $r=0$. After determining the coefficients $\\alpha_i,\n\\beta_i$ and $\\gamma_i$, we can use the recursion relation (\\ref{ep2}) from\n$i=N_r$ to $1$ to determine the solution for the entire space range using the\nstarting value $\\varphi^{n+1}(N_r)$  (=0) known from the boundary condition.\nThus we have determined the solution by using two sets of recursion relations\nacross the lattice each involving about $N_r$ operations.\n\n\n\n\\subsection{Chemical potential}\n\n\\label{CH}\n\n\nFor stationary states the wave functions for the  1D case\nhave the trivial time\ndependence $\\varphi(x;t) \\equiv \\hat  \\varphi(x) \\exp(-\\mbox{i}\\mu\nt)$,\nwhere\n$\\mu $ is the chemical potential. Substituting this condition in Eq.\n(\\ref{1d}) we obtain\n\\begin{align} \\label{mueq}\n\\left[-\\frac{d^2}\n{d x^2}+\\frac{x^2}{4}+\\aleph \n{\\hat \\varphi^2(x)}\n\\right] \\hat \\varphi (x)\n=\n  \\mu \\hat \\varphi (x).\n\\end{align}\nAssuming that the wave form is normalized to unity\n$\\int_{-\\infty}^\\infty \\hat \\varphi^2(x) dx =1$, the chemical\npotential can\nbe calculated from the following expression obtained by multiplying\nEq. (\\ref{mueq}) by $\\hat \\varphi (x)$ and integrating over all space\n\\begin{align}\\label{muz}\n\\mu = \\int_{-\\infty}^\\infty \\left[\\biggr(\\frac{d\\hat\n\\varphi(x)}{dx}\n\\biggr)^2 +\\hat \\varphi^2(x)\n\\left(\n\\frac{x^2}{4}+\\aleph\n{\\hat \\varphi^2(x)}\n\\right) \\right] dx,\n\\end{align}\nwhere the second derivative has been simplified by an integration by\nparts.\n\nAll the programs also calculate the many-body energy, which is of\ninterest. The analytical expression for energy is the\nsame as that of the chemical potential but with the nonlinear term\nmultiplied by 1\/2, e.g., \\cite{review}\n\\begin{align}\\label{energy}\nE = \\int_{-\\infty}^\\infty \\left[\\biggr(\\frac{d\\hat\n\\varphi(x)}{dx}\n\\biggr)^2 +\\hat \\varphi^2(x)\n\\left(\n\\frac{x^2}{4}+\\frac{\\aleph} \n{2} {\\hat \\varphi^2(x)}\n\\right) \\right] dx.\n\\end{align}\nThe programs will write the value of energy as output. However, we shall\nnot study or tabulate the results for energy and we shall not write the\nexplicit algebraic  expression of energy in the case of other trap\nsymmetries.\n\n\n\nThe GP equation (\\ref{sph}) with spherically-symmetric potential  is\nalso an one-variable equation quite similar in structure to the\none-dimensional equation (\\ref{1d}) considered above. Hence the entire\nanalysis of Secs. \\ref{RT}, \\ref{IT}, and \\ref{CH} will be applicable\nin this case\nwith $\\varphi(x;t)$ replaced by $\\varphi(r;t)\/r$ in the\nnonlinear term. Now if we consider stationary states of the form\n$\\varphi(r,t)\\equiv \\hat \\varphi(r)\\exp (-\\mbox{i}\\mu t\/\\hbar)$,\n the\nexpression for the chemical potential\nbecomes\n\\begin{align}\\label{mueq2}\n\\mu =4\\pi  \\int_{0}^\\infty \\left[\\biggr(\\frac{d\\hat \\varphi(r)}{dr}\n\\biggr)^2 +\\hat \\varphi^2(r)\n\\left(\n\\frac{r^2}{4}+\\aleph\n\\frac{\\hat \\varphi^2(r)}{r^2}\n\\right) \\right] dr.\n\\end{align}\nWe shall use Eqs. (\\ref{muz}) and (\\ref{mueq2}) for the calculation of\nchemical potential from Eqs. (\\ref{1d}) and (\\ref{sph}). The energy will\nbe calculated from Eq. (\\ref{energy}).\nThe expressions\nfor chemical potential in other cases can be written down in a\nstraight-forward fashion.\n\n\n\\section{Split-Step Crank-Nicolson method in two and three space\nvariables}\n\\label{CN23D}\n\n\\subsection{Anisotropic GP equation in 2D}\n\n\\label{anp}\n\nIn this case the GP equation (\\ref{2d}) can be written as\n\\begin{eqnarray}\n\\mbox{i}\n\\frac{\\partial}{\\partial t}\\varphi(x,y;t) &  =&\n\\biggr[\n-\\frac{\\partial^2}{\\partial  x^2}\n-\\frac{\\partial^2}{\\partial  y^2}\n+ \\frac{1}{4} \\biggr( x^2 + \\kappa^2 y^2   \\biggr)\n+\\aleph \n|\\varphi(x,y;t)|^2  \\biggr]\n\\varphi(x,y;t) \\nonumber \\\\ & \\equiv & H \\varphi(x,y;t)\n,\\label{a2}\n\\end{eqnarray}\nwith $\\aleph \n= 4\\sqrt {2 \\pi\\lambda} {Na}\/{l}$.\nThe Hamiltonian $H$ can be conveniently broken\ninto three pieces $H=H_1+H_2+H_3$, where\n\\begin{align}\n& H_1= \\frac{1}{4} \\biggr(x^2+\\kappa^2 y^2  \\biggr)\n+\\aleph \n|\\varphi(x,y;t)|^2,  \\\\\n& H_2=-\\frac{\\partial^2}{\\partial x^2}, \\;\\;\nH_3=-\\frac{\\partial^2}{\\partial y^2}. \\;\\;\n\\end{align}\nNow we adopt a policy quite similar to that elaborated in Sec.\n\\ref{RT} where the Hamiltonian was broken into two parts and where the\ntime propagation over time step $\\Delta$ using the two parts were\ncarried out alternatively. The same procedure will be adopted in the\npresent case where we perform the time propagation using the pieces\n$H_1$, $H_2$, and $H_3$ of the Hamiltonian successively in\nthree\nindependent time sub-steps $\\Delta$ to complete a single time\nevolution over time step $\\Delta$ of the entire GP Hamiltonian $H$.\nThe time propagation over $H_1$ is performed as in Eq. (\\ref{al1}) and\nthose over $H_2$ and $H_3$  as in Eqs. (\\ref{gp3}) and\n(\\ref{kn1}).\nThe chemical potential in this case for the stationary state\n$\\varphi(x,y;t)\\equiv \\hat \\varphi(x,y)\\exp(-\\mbox{i}\\mu t)$\ncan be\nwritten as\n\\begin{align}\n\\mu =\n\\int_{-\\infty}^\\infty dx \\int_{-\\infty}^\\infty dy\n \\left[ \\biggr(\\frac{{\\partial}\\hat \\varphi}{\n{\\partial}x} \n\\biggr)^2 +\n\\biggr(\\frac{{\\partial}\\hat \n\\varphi}{{\\partial}y}\n\\biggr)^2\n + \\hat \\varphi^2 \\left(\n\\frac{x^2+\\kappa^2 y^2}{4}+\\aleph \n{\\hat \\varphi^2} \\right)\n\\right] ,\n\\end{align}\n{\nwhere we have performed integrations by parts \nto obtain this form for the chemical potential in terms of first \nderivatives only.}\n\n\n\\subsection{Axially-symmetric GP equation in 3D}\n\nIn this case the GP equation (\\ref{axi}) can be written as\n\\begin{eqnarray}\n\\mbox{i}\n\\frac{\\partial}{\\partial t}\\varphi(x,y;t) &  =&\n\\biggr[\n-\\frac{\\partial^2}{\\partial  \\rho^2}\n-\\frac{1}{\\rho}\\frac{\\partial}{\\partial  \\rho}\n-\\frac{\\partial^2}{\\partial  z^2}\n+ \\frac{1}{4} \\biggr( \\kappa^2\\rho^2 + \\lambda^2 z^2   \\biggr)\n+\\aleph \n|\\varphi(\\rho,z;t)|^2  \\biggr]\n\\varphi(\\rho,z;t) \\nonumber \\\\ & \\equiv & H \\varphi(\\rho,z;t).\n\\label{ax2}\n\\end{eqnarray}\nThe Hamiltonian $H$ can be conveniently broken\ninto three pieces $H=H_1+H_2+H_3$, where\n\\begin{align}\n& H_1= \\frac{1}{4} \\biggr(  \\rho^2 + \\lambda^2 z^2   \\biggr)\n+\\aleph \n|\\varphi(\\rho,z;t)|^2,  \\\\\n& H_2=-\\frac{\\partial^2}{\\partial\n\\rho^2}-\\frac{1}{\\rho}\\frac{\\partial}{\\partial  \\rho}, \\;\\;\nH_3=-\\frac{\\partial^2}{\\partial z^2}. \\;\\;\n\\end{align}\nNow we adopt a policy quite similar to that elaborated in Sec.\n\\ref{anp} where the Hamiltonian was broken into three parts and where\nthe\ntime propagation over time step $\\Delta$ using the three parts were\ncarried out alternatively.\nThe time propagation over $H_1$ is performed as in Eq. (\\ref{al1}) and\nthose over $H_2$ and $H_3$  as in Eqs. (\\ref{knx}) and\n(\\ref{kn1}).\nThe chemical potential in this case for the stationary state\n$\\varphi(\\rho,z;t)\\equiv \\hat \\varphi(\\rho,z)\\exp(-\\mbox{i}\\mu t)$\ncan be\nwritten as\n\\begin{align}\n\\mu = 2\\pi\n\\int_{0}^\\infty \\rho d\\rho \\int_{-\\infty}^\\infty dz\n \\left[ \\biggr(\\frac{{\\partial}\n\\hat \\varphi}{{\\partial}\\rho} \n\\biggr)^2 +\n\\biggr(\\frac{{\\partial}\\hat \n\\varphi}{{\\partial}z}\n\\biggr)^2\n + \\hat \\varphi^2 \\left(\n\\frac{\\kappa^2 \\rho^2+\\lambda^2 z^2}{4}+  \\aleph\n{\\hat \n\\varphi^2}\n\\right)\n\\right] ,\n\\end{align}{\nwhere we have again used integrations by parts to simplify the\nfinal expression.}\n\n\n\n\\subsection{Anisotropic GP equation in 3D}\n\nIn this case the GP equation (\\ref{ani}) can be written as\n\\begin{eqnarray}\n\\mbox{i}\n\\frac{\\partial}{\\partial t}\\varphi(x,y,z;t) &  =&\n\\biggr[\n-\\frac{\\partial^2}{\\partial  x^2}\n-\\frac{\\partial^2}{\\partial  y^2}\n-\\frac{\\partial^2}{\\partial  z^2}\n+ \\frac{1}{4} \\biggr(\\nu^2 x^2 + \\kappa^2 y^2 + \\lambda^2 z^2  \\biggr)\n+\\aleph  \n|\\varphi(x,y,z;t)|^2  \\biggr]\n\\varphi(x,y,z;t) \\nonumber \\\\ & \\equiv & H \\varphi(x,y,z;t)\n,\\label{a3}\n\\end{eqnarray}\nwith $\\aleph\n= 8\\sqrt 2 \\pi {aN}\/{l}$.\nThe Hamiltonian $H$ can be conveniently broken\ninto four pieces $H=H_1+H_2+H_3+H_4$, where\n\\begin{align}\n& H_1= \\frac{1}{4} \\biggr(\\nu^2 x^2 + \\kappa^2 y^2 + \\lambda^2 z^2\n\\biggr)\n+\\aleph \n|\\varphi(x,y,z;t)|^2,  \\\\\n& H_2=-\\frac{\\partial^2}{\\partial x^2}, \\;\\;\nH_3=-\\frac{\\partial^2}{\\partial y^2}, \\;\\;\nH_4=-\\frac{\\partial^2}{\\partial z^2}.\n\\end{align}\nNow we adopt a policy quite similar to that elaborated in Sec.\n\\ref{RT} where the Hamiltonian was broken into two parts and where the\ntime propagation over time step $\\Delta$ using the two parts were\ncarried out alternatively. The same procedure will be adopted in the\npresent case where we perform the time propagation using the pieces\n$H_1$, $H_2$, $H_3$ and $H_4$ of the Hamiltonian successively in four\nindependent time sub-steps $\\Delta$ to complete a single time\nevolution over time step $\\Delta$ of the entire GP Hamiltonian $H$.\nThe time propagation over $H_1$ is performed as in Eq. (\\ref{al1}) and\nthose over $H_2$, $H_3$ and $H_4$ as in Eqs. (\\ref{gp3}) and (\\ref{kn1}).\nThe chemical potential in this case for the stationary state\n$ \\varphi(x,y,z;t)\\equiv \\hat \\varphi(x,y,z)\\exp(-\\mbox{i}\\mu t)$\ncan\nbe\nwritten as\n\\begin{align}\n\\mu =\n\\int_{-\\infty}^\\infty dx \\int_{-\\infty}^\\infty dy \\int_{-\\infty}^\\infty\ndz \\left[ \\biggr(\\frac{{\\partial}\n\\hat \\varphi}{{\\partial}x} \n\\biggr)^2 +\n\\biggr(\\frac{{\\partial}\\hat \n\\varphi}{{\\partial}y}\n\\biggr)^2\n+ \\biggr(\\frac{{\\partial}\\hat \\varphi}\n{{\\partial}z} \\biggr)^2 + \n\\hat \\varphi^2 \\left(\n\\frac{\\nu^2x^2+\\kappa^2 y^2+\\lambda^2 z^2}{4}+\\aleph\n{\\hat \n\\varphi^2}\n\\right)\n\\right] ,\n\\end{align}{\nwhere we have again used integrations by parts to simplify the\nfinal expression.}\n\n\n\n\\section{Description of Numerical Programs}\n\\label{FOR}\n\n\n\\subsection{GP equation in one space variable}\n\nIn this subsection we describe six  Fortran  codes involving GP equation in one\nspace variable. These are programs for solving the 1D GP equation (program {\\bf\nimagtime1d.F}), the circularly-symmetric 2D GP equation (program {\\bf\nimagtimecir.F}) and the radially-symmetric 3D  GP equation  (program {\\bf\nimagtimesph.F}) using imaginary-time propagation. The similar routines using\nreal-time propagation are {\\bf realtime1d.F}, {\\bf realtimecir.F} and {\\bf\nrealtimesph.F}, respectively. These real- and imaginary-time routines in \none\nspace variable have similar structures. However, the wave function is real in\nimaginary-time propagation, whereas it is complex in real-time propagation. To\naccommodate this fact many variables in the real-time propagation routines are\ncomplex.\n\n\nThe principal variables employed in the MAIN program are:  N = number of space\nmesh points, NSTP = number of time iterations during which the nonlinearity is\nintroduced in real-time propagation, (in imaginary-time propagation the\nnonlinearity is introduced in one step,) NPAS = number of subsequent time\niterations with fixed nonlinearity, NRUN =  number of final time iterations with\n(a) fixed nonlinearity in imaginary-time propagation and (b) modified\nnonlinearity to study dynamics in real-time propagation, X(I) = space mesh,\nX2(I) = X(I)*X(I), V(I) = potential, CP(I) = wave function at space point X(I),\nG, G0 = coefficient of nonlinear term, MU = chemical potential, EN = energy,\nZNORM = normalization of wave function, RMS = rms size or radius, DX = space\nstep, DT = time step, OPTION and XOP decide which equation to solve. Also\nimportant is the subroutine INITIALIZE, where the space mesh X(I), potential  \nV(I), and the initial wave function CP(I), are calculated.  (An advanced user\nmay need to change the variables V(I) and CP(I) so as to adopt the program to\nsolve  a different equation with different nonlinearity.) The functions and\nvariables not listed above are auxiliary variables, that the user should not\nneed to modify.\n\n\nNow we describe the function of the subroutines, which  the user should not need\nto change. The subroutine NORM calculates by Simpson's rule the normalization \nof the wave function and sets the normalization to unity, the preassigned value.\nThe real-time propagation preserves the normalization of the wave function and\nhence the subroutine NORM is not used during time propagation. The subroutines\nRAD and CHEM calculate the rms size (length or radius), and chemical \npotential (and energy) of\nthe system. The subroutines COEF and LU together implement the time propagation\nwith the spatial and temporal time derivative terms.  The subroutine NU performs\nthe time propagation with the nonlinear term and the potential.  In the\nimaginary-time program the action of the subroutine LU does not preserve\nnormalization and hence each time the subroutine LU is called, the subroutine\nNORM has to be called to set the normalization of the wave function back to\nunity. (This is not necessary in the real-time programs which preserve the\nnorm.) The subroutine NONLIN calculates the nonlinear term. The subroutine DIFF\ncalculates the space derivative of the wave function by Richardson's\nextrapolation formula needed for the computation of the chemical potential and\nenergy. The function SIMP does the numerical space integrations with the\nSimpson's rule.\n\n\n\nThe programs implement the splitting method described in Sec.~\\ref{RT} and\n\\ref{gp2dc} and calculate the wave function, chemical potential, size,\nnormalization, etc. The number of points in the one-dimensional space grid\nrepresented by the integer variable N has to be chosen consistent with space\nstep DX such that the total space covered N$\\times$DX is significantly larger\nthan the size of the condensate so that at the boundaries the wave \nfunction \nattains the asymptotic limits (e.g., the absolute value of the wave \nfunction  or its space derivative becomes \nless than 10$^{-10}$ or so \nfor imaginary-time\npropagation and less than 10$^{-7}$ for real-time propagation. Note \nthat in the 1D and spherically-symmetric 3D problems we are using the \nasymptotic condition that the wave function is zero at both \nboundaries. In case of the circularly-symmetric 2D problem we use a mixed \nboundary condition, e.g.,  at origin the \nderivative of the wave function is zero and at infinity the wave \nfunction is zero.) In the\nimaginary-time routine the total space covered should be about 1.5  times the\nextension of the condensate. In the real-time routines, to obtain good \nprecision the total space covered\nshould be at least 2  times the extension of the condensate. \nThis is because the\nimaginary-time routine is more precise and the solution attains its \nlimiting asymptotic value at the\nboundary very rapidly as the total space covered is increased. The real-time\nroutine is less accurate  and  one has to go to a larger distance before the\nsolution or its derivative drops to zero. A couple of runs with a \nsufficiently large N and large\nDX are recommended to have an idea of the size\/extension  of the system. \nA\nsmaller value of DX leads to a more accurate result provided an appropriate DT\nis chosen.\n\n\nThe Crank-Nicolson method, described, for example, by Eq. (\\ref{kn1}), is\nunconditionally stable for all $\\Delta\/h^2$. Nevertheless, for a numerical\napplication to a specific problem one has to fix the time step DT  ($=\\Delta$) and\nspace step DX ($=h$) for good convergence. Space and time steps are given in the\nMAIN program,   as ``DATA DX\/0.0025D0\/, DT\/0.00002D0\/\" with correlated DX and DT\nvalues obtained by trial. (If the user wants to use other values of DX and DT, a\nset of correlated values obtained by trial is given in Table \\ref{table1}.)  The\ntotal number of space points N in calculation has to be fixed in the line,\ne.g., ``PARAMETER (N = 6000, N2 = N\/2, NX = N-1)\",  in the MAIN program, so\nthat the final wave function is within the space range and its value is\nnegligibly small at the boundaries.  The total numbers of time iterations are\nfixed in the line, e.g., ``PARAMETER (NSTP = 500000, NPAS = 10000, NRUN =\n20000)\", in the MAIN program as described below in Sec. \\ref{howto}.\n\n\nThe integer parameter OPTION should be set 1, 2 or 3 in the MAIN routine for\nsolution of equations of type  (\\ref{sph3}),  (\\ref{sph2}), or (\\ref{sph}), [or\nfor solution of equations of type (\\ref{1d3}), (\\ref{1d2}), or (\\ref{1d}),]\nrespectively. The difference between these three types of equations is in the\nvalues of the coefficients in the first two terms. The nonlinear term calculated\nin the subroutine   NONLIN is of the form $|\\varphi(r,t)|^2$ with coefficient G.\nIf a different type of nonlinear term (with different functional dependence on\nthe wave function) is to be introduced, it should be done in the subroutine\nNONLIN. Otherwise, this subroutine should not be changed. A different type of\nnonlinear term is appropriate for a Fermi super-fluid \\cite{ska1,ska2} or a\nTonks-Girardeau gas \\cite{tg}.\n\n\\subsection{GP equation in two and three space variables}\n\n\nIn addition to the programs in one space variable we have six programs in two\nand three space variables. The programs {\\bf imagtime2d.F} and {\\bf\nrealtime2d.F} apply to 2D Cartesian space using imaginary- and real-time\npropagations, respectively. Similarly, {\\bf imagtime3d.F} and {\\bf realtime3d.F}\napply to 3D Cartesian space using imaginary- and real-time propagation,\nrespectively. Finally, programs {\\bf imagtimeaxial.F} and {\\bf realtimeaxial.F}\napply to an axially symmetric trap in 3D. The Fortran 90\/95 versions of these\nprograms are somewhat more condensed and provide some advantage. Hence we also\ninclude these programs as {\\bf imagtime2d.f90},  {\\bf realtime2d.f90}, \n{\\bf\nimagtime3d.f90},  {\\bf realtime3d.f90}, {\\bf imagtimeaxial.f90}, and \n{\\bf\nrealtimeaxial.f90}. The output of the Fortran 90\/95 programs are \nidentical with\ntheir corresponding Fortran 77 versions. All these programs are written using a\nvery similar logic used in the programs in one space variable. So most of the\nconsiderations described earlier also applies to these cases. We describe the\nprincipal differences below.\n\nIn case of total number of space points N, one now has the variables NX, NY, and\nNZ for total number of space points in X, Y and Z directions in case of three\nspace variables. In case of two space variables the Z component is absent. These\nvariables\nshould be chosen equal to each other for a nearly  symmetric case. However, if\nthe problem is anisotropic in space, these variables can and should be chosen\ndifferently. Similarly, the space variable X(I) is now replaced by space\nvariables X(I), Y(J), and Z(K) in three directions. Now there are space steps\nDX, DY, and DZ in place of DX in one space variable. The variables DX, DY, and\nDZ can now be chosen differently in three directions in case of an anisotropic\nproblem. However, they should  be chosen together with NX, NY, and NZ so that\nthe wave function lies entirely inside the chosen space and becomes negligibly\nsmall at the boundaries. If the spatial extension of the wave function is much\nsmaller in one space direction than another, one should take a smaller space\nstep in the direction in which the spatial extension of the wave function is\nsmall. The potential V and wave function CP      are now functions of 2 or 3\nspace variables and are represented by matrices in place of a column in the case\nof one space variable. The variables AL (or KAP) and BL (or LAM) now define the\nanisotropy of the trap and are used in the subroutine INITIALIZE to define the\ntrap. The subroutine LU is now replaced by LUX, LUY, and LUZ to implement the\nsolution in different space directions. In the imaginary-time propagation each\ntime the subroutines LUX, LUY, and LUZ  operate, one has to set the\nnormalization of the wave function to unity by calling the subroutine NORM.\n\n\n\\subsection{Instruction to use the programs}\n\\label{howto}\n\nThe programs, as supplied, solve the GP equations for a specific value of\nnonlinearity and write the wave function, chemical potential, energy, rms size\nor radius, wave function at the center, and nonlinearity, for  specific values\nof space and time steps. The real- and imaginary-time programs for a \nspecific\nequation, for example, spherically-symmetric 3D equation, employ similar set of\nparameters like space and time steps. The real-time programs use a larger value\nof N, so that the discretization covers a larger region in space. In all cases\nthe supplied programs solve an equation of type (\\ref{sph2}) with a factor of\n1\/2 in front of the space derivative, selected by setting the integer parameter\nOPTION = 2 in the MAIN routine.  Other types of equations can be obtained by\nsetting OPTION = 1 for equations of type  (\\ref{sph3}) or = 3 for equations of\ntype (\\ref{sph}).\n\nFor solving a stationary ground state problem, the imaginary-time \nprograms  are far more\naccurate and should be used. The real-time programs should be used for studying\nnon-equilibrium problems often using  an initial wave function calculated by the\nimaginary-time program. The non-equilibrium problems include the study of\nsoliton dynamics \\cite{ska3}, expansion \\cite{ska4}, collapse dynamics\n\\cite{ska5}, and other types of problems.\n\nEach program is preset at a fixed nonlinearity G0 (= G), correlated DX-DT values\nand NSTP, NPAS, and NRUN.  Smaller the steps DX and DT, more accurate will be\nthe result.  The correlated values of DX and DT on a data line should be found\nby trial to obtain good convergence. \nEach supplied program produces result up-to a desired precision consistent\nwith the parameters employed $-$ G0, DX, DT, N, NSTP, NPAS, and NRUN. If the\nnonlinearity G0 is increased, one might need to increase N to achieve similar\nprecision.\nIn many cases one may need an approximate solution (with lower\naccuracy) involving less CPU time or one may need to solve the GP\nequation for a different value of nonlinearity. \n\n(a) If G0 is reduced, just\nchange the card defining G0. However, if G0 is increased, changing the\nvalue of G0 may not be enough.  For an increased G0, the wave function\nextends to a larger region in space. One may need to increase the\n``Number of space mesh points\". For a new G0, just plot the output file\nfort.3 for all programs except realtime3d.F, realtime3d.f90,\nimagtime3d.F and imagtime3d.f90, where one should plot output files\nfort.11, fort.12, and fort.13 and see that the wave function is fully\nand adequately accommodated in the space domain. If not, one needs to\nincrease the number in input cards ``Number of space mesh points\" until\nthe wave function is fully and adequately accommodated in the space\ndomain. \n\n(b)The CPU time involved can be reduced by sacrificing the precision.\nThis can be done by increasing the space step(s) and time step and\nreducing the \"Number of space mesh points\". If the space step DX is\nincreased by a factor of f = 2, the number of space mesh points should\nbe reduced by the same factor. The time step DT should be increased by a\nlarger factor, more like f**2. The optimum increase in time step should\nbe determined by some experimentation. (A set of\ncorrelated DX-DT\nvalues is given in Tables \\ref{table1} and \\ref{table4} for the \nsolution of corresponding equations.)\nThe CPU time is the largest in\nthe case of programs realtime3d.F, realtime3d.f90, imagtime3d.F and\nimagtime3d.f90 and we give an example of changes below in these cases\nto reduce the CPU time and precision. For example, for realtime3d.F, \njust change the \nlines  15, 16, 17, and 39.  The new\nset of lines should be\n\n      PARAMETER (NX=40, NXX = NX-1, NX2 = NX\/2)\n\n      PARAMETER (NY=32, NYY = NY-1, NY2 = NY\/2)\n\n      PARAMETER (NZ=24, NZZ = NZ-1, NZ2 = NZ\/2)\n\n      DATA DX \/0.5D0\/, DY \/0.5D0\/,  DZ \/0.5D0\/, DT\/0.05D0\/\n\nFor imagtime3d.F, just change the lines  15, 16, 17, and 37.  The new\nset of lines should be\n\n      PARAMETER (NX=24, NXX = NX-1, NX2 = NX\/2)\n\n      PARAMETER (NY=20, NYY = NY-1, NY2 = NY\/2)\n\n      PARAMETER (NZ=16, NZZ = NZ-1, NZ2 = NZ\/2)\n\n      DATA DX \/0.5D0\/, DY \/0.5D0\/,  DZ \/0.5D0\/, DT\/0.04D0\/\n\nPlease verify, by running the corresponding programs,  that the new \nresults so obtained are quite similar to the \nexisting results. \n\n\nThe integer parameter NSTP refers to the number of time iterations during which\nthe nonlinear term is slowly introduced in the real-time propagation. This\nnumber should be large (typically more than 100,000 for small nonlinearity, for\nlarger nonlinearity could be 1000,000 ) for good convergence; this means that\nthe nonlinearity should be introduced in small amounts over a large number of\ntime iterations. In real-time propagation, NPAS refers to certain number of time\niterations with the constant nonlinear term already introduced in NSTP  and\nshould be small (typically 1000). NRUN refers to time iterations with a \nmodified\nnonlinearity so as to generate a non-equilibrium dynamics. In the imaginary-time\npropagation the parameters NPAS and NRUN refer to certain number of time\niterations with the constant nonlinear term already introduced in one step and\nshould be large (typically NPAS = 200,000 or more) for good convergence.\n\n\n\n\n\\subsection{Output files}\n\n\nThe programs write via statements WRITE(1,*), WRITE(2,*) WRITE(3,*)  in Files 1,\n2, and 3, respectively, the initial stationary wave function  and that after\nNSTP, and NPAS time iterations for real-time propagation and after NPAS and NRUN\ntime iterations for imaginary-time propagation. File 3  gives the final\nstationary wave function of the calculation. However, in the case of the\nanisotropic 3D programs realtime3d.F and imagtime3d.F, sections of the wave\nfunction as plotted in Fig. \\ref{fig4} (b) are written in Files 1, 2, and 3\nbefore  NSTP (realtime3d.F) and after NPAS  (imagtime3d.F) iterations, \nand in Files 11,\n12, and 13 after NPAS (realtime3d.F) and NRUN  (imagtime3d.F) iterations.\n\nIn the real-time program a non-stationary oscillation is initiated by suddenly\nmodifying the nonlinearity from G to G\/2 after NPAS time iterations. During NRUN\ntime iterations the non-stationary dynamics is studied. The real-time programs\nwrite on File 8  the running time and  rms size during non-stationary\noscillation.\n\n\n\nIn addition, these programs write in File 7 the values of nonlinearity G, space\nsteps DX, DY, DZ,  time step DT, number of space mesh points N,  number of time\niterations NSTP, NPAS, and NRUN together with the values of normalization of the\nwave function, chemical potential, energy, rms size, value of the wave function\nat center, nonlinearity coefficient G.\n\n\n\nBelow we provide some sample output listed on File 7 from the different programs\nusing OPTION = 2. File 7 (fort.7) represents the comprehensive result in each\ncase.\n\n\n(1)  Program {\\bf imagtime1d.F}\n\n\\begin{verbatim}\n\n  OPTION =   2\n\n# Space Stp N =     8000\n# Time Stp :  NPAS =    200000, NRUN =     20000\n  Nonlinearity G =      62.74200000\n  Space Step DX =     0.002500, Time Step DT =     0.000020\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <x>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000    0.500000    0.500000    0.70711    0.75113\nAfter NPAS iter.:  0.9996   10.369462    6.256976    2.04957    0.40606\nAfter NRUN iter.:  0.9996   10.369462    6.256976    2.04957    0.40606\n                   ----------------------------------------------------\n\n\\end{verbatim}\n(2)  Program {\\bf imagtimesph.F}\n\n\\begin{verbatim}\n\n  OPTION =   2\n\n# Space Stp N =     3000\n# Time Stp : , NPAS =    200000, NRUN =     20000\n  Nonlinearity G =     125.48400000\n  Space Step DX =     0.002500, Time Step DT =     0.000020\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000    1.500000    1.500000    1.22474    0.42378\nAfter NPAS iter.:  0.9998    4.014113    3.070781    1.88214    0.17382\nAfter NRUN iter.:  0.9998    4.014113    3.070781    1.88214    0.17382\n                   ----------------------------------------------------\n\\end{verbatim}\n\n(3)  Program {\\bf imagtimecir.F}\n\\begin{verbatim}\n\n  OPTION =   2\n\n# Space Stp N =     2000\n# Time Stp : , NPAS =    200000, NRUN =    200000\n  Nonlinearity G =      -2.50970000\n  Space Step DX =     0.002500, Time Step DT =     0.000020\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000    1.000000    1.000000    1.00000    0.56419\nAfter NPAS iter.:  1.0000    0.499772    0.770107    0.87758    0.67535\nAfter NRUN iter.:  1.0000    0.499772    0.770107    0.87758    0.67535\n                   ----------------------------------------------------\n\n\\end{verbatim}\n\n\n(4)  Program {\\bf imagtime2d.F and imagtime2d.f90}\n\\begin{verbatim}\n\n\n  OPTION =   2\n  Anisotropy AL =     2.000000\n\n# Space Stp NX =      800, NY =      800\n# Time Stp : NPAS =     30000, NRUN =      5000\n  Nonlinearity G =      12.54840000\n  Space Step DX =   0.020000, DY =   0.020000\n  Time Step  DT =   0.000100\n\n                   -----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0,0)\n                   -----------------------------------------------------\nInitial :          1.0000    1.500000    1.500000    0.86603    0.67094\nAfter NPAS iter.:  0.9999    3.254878    2.490493    1.17972    0.46325\nAfter NRUN iter.:  0.9999    3.254878    2.490493    1.17972    0.46325\n                   -----------------------------------------------------\n\n\\end{verbatim}\n\n\n(5)  Program {\\bf imagtimeaxial.F and imagtimeaxial.f90}\n\\begin{verbatim}\n  OPTION =   2\n  Anisotropy KAP =     1.000000, LAM =     4.000000\n\n# Space Stp NX =      500, NY =      500\n# Time Stp : NPAS =     100000, NRUN =     20000\n  Nonlinearity G =      18.81000000\n  Space Step DX =   0.020000, DY =   0.020000\n  Time Step  DT =   0.000040\n\n            ------------------------------------------------------------------\n             Norm    Chem      Energy    <rho>      <z>       <r>     psi(0)\n            ------------------------------------------------------------------\nInitial   : 1.000   3.00000   3.00000   1.00000   0.35355   1.06066   0.59883\nNPAS iter : 1.000   4.36113   3.78228   1.32490   0.38049   1.37846   0.38129\nNRUN iter : 1.000   4.36113   3.78228   1.32490   0.38049   1.37846   0.38129\n            ------------------------------------------------------------------\n\n\\end{verbatim}\n\n\n(6) Program {\\bf imagtime3d.F and imagtime3d.f90}\n\\begin{verbatim}\n OPTION =   2\n  Anisotropy AL =     1.414214, BL = 2.000000\n\n# Space Stp NX =      240, NY =      200, NZ =      160\n# Time Stp : NPAS =      5000, NRUN =       500\n  Nonlinearity G =      44.90700000\n  Space Step DX =   0.050000, DY =   0.050000, DZ =   0.050000\n  Time Step  DT =   0.000400\n\n                  ------------------------------------------------------\n                    Norm      Chem        Ener        <r>     Psi(0,0,0)\n                  ------------------------------------------------------\nInitial         :  1.0000      2.2071      2.2071     1.0505     0.5496\nAfter NPAS iter.:  0.9997      4.3446      3.4862     1.4583     0.2888\nAfter NRUN iter.:  0.9997      4.3446      3.4862     1.4583     0.2888\n                  ------------------------------------------------------\n\n\n\\end{verbatim}\n\n\n(7)  Program {\\bf realtime1d.F}\n\\begin{verbatim}\n  OPTION =   2\n\n# Space Stp N =     5000\n# Time Stp : NSTP  1000000 , NPAS =     1000, NRUN =     40000\n  Nonlinearity G =      62.74200000\n  Space Step DX =     0.010000, Time Step DT =     0.000100\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <x>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000       0.500       0.500      0.707      0.751\nAfter NSTP iter.:  1.0000      10.368       6.257      2.050      0.406\nAfter NPAS iter.:  1.0000      10.375       6.257      2.047      0.406\n                   ----------------------------------------------------\n\n\n\n\\end{verbatim}\n\n(8)  Program {\\bf realtimesph.F}\n\\begin{verbatim}\n  OPTION =   2\n\n# Space Stp N =     2000\n# Time Stp : NSTP =   1000000, NPAS =      1000, NRUN =     40000\n  Nonlinearity G =     125.48400000\n  Space Step DX =     0.010000, Time Step DT =     0.000100\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000       1.500       1.500      1.225      0.424\nAfter NSTP iter.:  1.0000       4.015       3.071      1.881      0.174\nAfter NPAS iter.:  1.0000       4.011       3.071      1.884      0.174\n                   ----------------------------------------------------\n\n\\end{verbatim}\n\n(9)  Program {\\bf realtimecir.F}\n\\begin{verbatim}\n  OPTION =   2\n\n# Space Stp N =     2000\n# Time Stp : NSTP =   1000000, NPAS =      1000, NRUN =     40000\n  Nonlinearity G =      12.54840000\n  Space Step DX =     0.010000, Time Step DT =     0.000100\n\n                   ----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0)\n                   ----------------------------------------------------\nInitial :          1.0000       1.000       1.000      1.000      0.564\nAfter NSTP iter.:  1.0000       2.255       1.708      1.308      0.391\nAfter NPAS iter.:  1.0000       2.255       1.708      1.308      0.391\n                   ----------------------------------------------------\n\\end{verbatim}\n\n\n\n(10)  Program {\\bf realtime2d.F and realtime2d.f90}\n\\begin{verbatim}\n\n  OPTION =   2\n  Anisotropy AL =     1.000000\n\n# Space Stp NX =      200, NY =      200\n# Time Stp : NSTP =    100000, NPAS =      1000, NRUN =      5000\n  Nonlinearity G =      12.54840000\n  Space Step DX =   0.100000, DY =   0.100000\n  Time Step  DT =   0.001000\n\n                   -----------------------------------------------------\n                    Norm       Chem        Ener       <r>       Psi(0,0)\n                   -----------------------------------------------------\nInitial :           1.000       1.000       1.000      1.000      0.564\nAfter NSTP iter.:   1.000       2.256       1.708      1.307      0.392\nAfter NPAS iter.:   1.000       2.257       1.708      1.305      0.392\n                   -----------------------------------------------------\n\n\n\n\n\n\\end{verbatim}\n\n\n\n(11)  Program {\\bf realtimeaxial.F and realtimeaxial.f90}\n\\begin{verbatim}\n\n  OPTION =   2\n  Anisotropy KAP =     1.000000, LAM =     4.000000\n\n# Space Stp NX =      130, NY =      130\n# Time Stp : NSTP =    100000, NPAS =      1000, NRUN =     20000\n  Nonlinearity G =      18.81000000\n  Space Step DX =   0.100000, DY =   0.100000\n  Time Step  DT =   0.001000\n\n            -------------------------------------------------------\n              Norm    Chem     Energy    <rho>     <z>       psi(0)\n            -------------------------------------------------------\nInitial :   1.000     3.000     3.000     1.000     0.354     0.587\nNSTP iter : 0.999     4.362     3.782     1.323     0.381     0.376\nNPAS iter : 1.000     4.362     3.782     1.327     0.379     0.376\n            -------------------------------------------------------\n\n\n\\end{verbatim}\n\n\n(12)  Program {\\bf realtime3d.F and realtime3d.f90}\n\\begin{verbatim}\n  OPTION =   2\n  Anisotropy AL =     1.414214, BL =     2.000000\n\n# Space Stp NX =      200, NY =      160, NZ =      120\n# Time Stp : NSTP =     60000, NPAS =      1000, NRUN =      4000\n  Nonlinearity G =      22.45400000\n  Space Step DX =   0.100000, DY =   0.100000, DZ =   0.100000\n  Time Step  DT =   0.002000\n\n                  ------------------------------------------------------\n                    Norm      Chem        Ener        <r>     Psi(0,0,0)\n                  ------------------------------------------------------\nInitial :           1.000      2.207      2.207     1.051     0.550\nAfter NSTP iter.:   1.000      3.572      2.992     1.321     0.347\nAfter NPAS iter.:   1.000      3.572      2.992     1.320     0.347\n                  ------------------------------------------------------\n\n\n\\end{verbatim}\n\n\n\\label{5.3}\n\n\n\n\\section{Numerical Results}\n\\label{NUM}\n\\subsection{Stationary Problem}\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Convergence of result in the   1D (X) and\nradially-symmetric  3D ($r$) cases for  nonlinearities $\\aleph \n=627.42$ and\n627.4, calculated\nusing the imaginary-time propagation programs imagtime1d.F  [Eq.\n(\\ref{1d2}), OPTION 2]\nand imagtimesph.F [Eq. (\\ref{sph2}), OPTION 2]\nrespectively, for various space step DX and time step DT.\n}\n\\label{table1}\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\n$\\aleph $ \n& DX  &  DT &\n$\\varphi(0)\/\\phi(0)$ &  $x_{\\mathrm{rms}}\/r_{\\mathrm{rms}}$ &\n{$\\mu$ } \\\\\n\\hline\n  627.42(X) & 0.08  & 0.005   & 0.276647 &   4.384825 & 48.024062  \\\\\n  627.42(X) & 0.04  & 0.001  &0.276648 &4.384744 & 48.024389  \\\\\n  627.42(X) & 0.02  &0.0005   &0.276649&     4.384734  & 48.024429  \\\\\n  627.42(X) & 0.01 &  0.0001  &0.276649&    4.384726 &  48.024462  \\\\\n  627.42(X) & 0.005  & 0.00005  &0.276649   &     4.384725 &\n48.024466  \\\\\n  627.42(X) & 0.0025  & 0.00002  &0.276649   &     4.384724 &\n48.024468  \\\\\n  627.42(X) & 0.001  &   0.00001  & 0.276649   &     4.384724 &\n48.024468\n\\\\\n\\hline\n  627.4($r$) &0.08   &   0.005   &   0.106655 &      2.506348   &\n7.247479\n\\\\\n  627.4($r$) &0.04   &    0.001  & 0.106679    &   2.505886    &\n7.248206\n\\\\\n  627.4($r$) & 0.02  & 0.0005  & 0.106684&       2.505833  &  7.248292\n\\\\\n  627.4($r$) & 0.01& 0.0001  &0.106686&        2.505785     & 7.248365\n\\\\\n  627.4($r$) & 0.005  &  0.00005  &0.106686 &       2.505780  & 7.248374\n\\\\\n  627.4($r$) & 0.0025  &0.00002  &0.106686 &       2.505776    &\n7.248380  \\\\\n  627.4($r$) & 0.001  & 0.00001  & 0.106686   &    2.505776    &\n7.248380  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\begin{figure}[tbp] \\begin{center}\n{\\includegraphics[width=.8\\linewidth]{Fig1.eps}}\n\\end{center}\n\\caption{(Color online) Relative percentage error\nas a function of\nspace step\nfor one-dimensional (1D)  and spherically-symmetric  3D (radial)\nmodels with nonlinearity $\\aleph\n=627.42$ and 627.4, respectively,\ncalculated using the imaginary-time propagation programs imagtime1d.F\n[Eq. (\\ref{1d2}), OPTION 2]\nand imagtimesph.F [Eq. (\\ref{sph2}), OPTION 2]\nwith the\ndata of\nTable \\ref{table1}.\n}\n\\label{fg1}\n\\end{figure}\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{The chemical potential $\\mu$, rms radius {$r_{\\mathrm{rms}}$}, and the wave function\n$\\phi(0)$ at the center  for various\nnonlinearities in the radially symmetric 3D case calculated using the\nprogram imagtimesph.F [Eq. (\\ref{sph2}), OPTION 2]. Table completed with space step\nDR $\n\\le 0.0025$\nand DT $=0.00002$. }\n\\label{table2}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\n\\hline\n{$ \\aleph $\n} &   {$\\phi(0)$} &\n$\\phi(0)$ \\cite{Bao_Tang}&\n{$r_{\\mathrm{rms}}$}      &\n$r_{\\mathrm{rms}}$ \\cite{Bao_Tang}&\n{$\\mu$}  &{$\\mu$ \\cite{Tiwari_Shukla,Bao_Tang}} \\\\\n\\hline\n-3.1371 & 0.48792(1) &  0.4881 & 1.51213(1)& 1.1521& 1.265184(2) &\n$1.2652$ \\\\\n       0 &  0.42378 & 0.4238& 1.22474& 1.2248 & 1.500000 &1.5000\n\\\\\n    3.1371 &0.38425(1)& 0.3843 &1.27857(1)   &1.2785 & 1.677451(1)  &\n1.6774\\\\\n   12.5484 & 0.31800(1)  & 0.3180 &  1.39211(1)  & 1.3921 &\n2.065018(1) &\n2.0650 \\\\\n  31.371 & 0.25810(1)&  0.2581  & 1.53561(1)& 1.5356  &  2.586116(1) &\n2.5861 \\\\\n125.484 &  0.17382(1) &  0.1738 &  1.88215(1) &1.8821 &  4.014113(2)\n&4.0141 \\\\\n  627.4 & 0.10669(1) &0.1066 & 2.50578(1) &2.5057 & 7.248380(3)  &\n7.2484\n\\\\\n    3137.1 &0.06559(1) &0.0655 & 3.41450(1) &3.4145 &  13.553403(4)  &\n13.553\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nIn this subsection we present results for the stationary ground \nstate problem\ncalculated with the imaginary-time programs.\nAll numerical results presented in this paper are for the\nnonlinear equations with a factor of 1\/2 in front of the gradient term\nobtained by choosing OPTION = 2 in the MAIN.\nFirst we consider the numerical result for the simplest cases $-$ the\n1D and the radially-symmetric 3D problems by\nimaginary-time propagation with nonlinearities $\\aleph\n=$ 627.42 \nand\n627.4, respectively. The calculations were performed with different\nspace and time steps DX and DT, respectively. (As DX is reduced, DT\nshould be reduced also to have good convergence. The correlated DX-DT\nvalues were obtained by trial to achieve good convergence.)\nIn each case a\nsufficiently large number of space points N is to be taken, so that the\nspace domain of integration covers the extension of the wave function adequately.\nWe exhibit in Table \\ref{table1}, the results for the wave function at\ncenter, rms\nsize, and chemical potential in these two cases for a fixed\nnonlinearity for different DX and DT.\nWe find that convergence is achieved up to six significant digits after the\ndecimal point\nwith space step DX = 0.0025 and time step DT = 0.00002.\nIn Fig. \\ref{fg1} we plot the relative percentage error in chemical\npotential $\\mu$ for various space steps. The percentage error rapidly\nreduces as space step is reduced.\n\nIn Table \\ref{table2} we exhibit the chemical potential, rms radius, and\nthe wave function at the center for various nonlinearities  in the\nspherically symmetric 3D case using space step\nDX = 0.0025 and time step DT = 0.00002 in the imaginary-time\npropagation program. From the\ntable we find that the results are in agreement with those calculated\nin Refs. \\cite{Tiwari_Shukla,Bao_Tang}.\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{The chemical potential $\\mu$,\nrms size {$x_{\\mathrm{rms}}$}, and the wave function\n$\\varphi(0)$ at the center  for various\nnonlinearities\n in the  1D case calculated using the program imagtime1d.F\n[Eq. (\\ref{1d2}), OPTION 2]. Table\ncompleted with DX $\\le 0.0025$ and\nDT $=0.00002$.\n}\n\\label{table3}\n\\begin{tabular}{|r|r|r|r|r|r|r|}\n\\hline\n{$\\aleph$}\n&   {$\\varphi(0)$} &\n$\\varphi(0)$ \\cite{Bao_Tang}&\n{$x_{\\mathrm{rms}}$}      &\n$x_{\\mathrm{rms}}$ \\cite{Bao_Tang}&\n{$\\mu$}  &{$\\mu$ \\cite{Bao_Tang}} \\\\\n\\hline\n  -2.5097 & 0.91317(1) &  0.9132 & 0.51334(1)& 0.5133& $-0.80623(3)$ &\n$-0.8061$ \\\\\n   0 &  0.75112 & 0.7511& 0.70711& 0.7071 & 0.500000 &0.5000    \\\\\n    3.1371 &0.64596(1)& 0.6459 &0.89602(1)&0.8960 &$1.526593(3)$  &\n1.5265\\\\\n   12.5484 & 0.52975(1)& 0.5297 & 1.24549(1)& 1.2454 &   3.596560(2) &\n3.5965 \\\\\n  31.371 & 0.45567(1)&0.4556  & 1.64170(1)&1.6416 &  $6.552682(2)$ &\n6.5526 \\\\\n62.742&  0.40606(1)& 0.4060 &  2.04957(1) & 2.0495 & $10.369462(2)$ &\n10.369\n\\\\\n156.855 &  0.34856(1) &  0.3485 &  2.76794(1) &2.7679 &$19.070457(2) $\n&19.0704 \\\\\n   313.71 & 0.31053(1) & 0.3105 &3.48237(1) &3.4823 & 30.259178(3) &\n30.259\n\\\\\n  627.42 & 0.27665(1) &0.2766 & 4.38472(1) &4.3847 & 48.024468(3)  &\n48.024\n\\\\\n     1254.8 & 0.24647(1) &0.2464 & 5.52282(1) &5.5228 & 76.226427(3)&\n76.226\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\n\nIn Table \\ref{table3} we exhibit the chemical potential, rms size, and\nthe wave function at the center for various nonlinearities  in the\n 1D  case using space step\nDX = 0.0025 and time step DT = 0.00002. From the\ntable we find that the results are in agreement with those calculated\nin Ref. \\cite{Bao_Tang}.\n\n\n\n\n\\begin{figure}[tbp] \\begin{center}\n{\\includegraphics[width=.49\\linewidth]{fig2a.ps}}\n{\\includegraphics[width=.49\\linewidth]{fig2b.ps}}\n\\end{center}\n\\caption{(Color online) Plot of wave function profile for the (a)\nradially-symmetric 3D case [Eq. (\\ref{sph2}), using imagtimesph.F,\nOPTION 2]  and (b)  1D\ncase [Eq. (\\ref{1d2}),  using imagtime1d.F, OPTION 2].  The\ncurves are\nlabeled by their respective nonlinearities as tabulated in Tables\n\\ref{table2} and \\ref{table3}, respectively.}\n\\label{fig2}\n\\end{figure}\n\n\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{Convergence of results for chemical potential $\\mu$, rms size $r_{\\mathrm{rms}}$ and\nthe wave function $\\phi(0)$ at the center\nin the Cartesian 2D\n case for a nonlinearity $\\aleph\n=12.5484$ and anisotropy\n$\\kappa =1$ for different space steps $h\\equiv$ DX = DY obtained from\nthe program imagtime2d.F\n[using Eq. (\\ref{2d2}), OPTION 2].}\n\\label{table4}\n\\begin{tabular}{|r|r|r|r|r|r|}\n\\hline\n$\\aleph\n$ & DX=DY  &  DT &\n$\\varphi(0)$ &\n$r_{\\mathrm{rms}}$ &\n{$\\mu$ } \\\\\n\\hline\n  12.5484 & 0.10 & 0.01  &0.39189(2) &     1.30678(3) &   2.2559  \\\\\n  12.5484 & 0.08 & 0.005  &0.39190(2) &     1.30685(3) &   2.2559  \\\\\n  12.5484 & 0.06 & 0.003  &0.39190(2) &     1.30690(3) &   2.25582(3)\n\\\\\n  12.5484 & 0.04  & 0.001   & 0.39190(2)   &    1.30693(2) &\n2.25579(2)\n\\\\\n  12.5484 & 0.02  & 0.0005  &0.39190(2)  &    1.30687(2) &\n2.25583(2)\n\\\\\n  12.5484 & 0.015  & 0.0003  &0.39190(2)  &    1.30687(2) &\n2.25583(2)\n\\\\\n  12.5484 & 0.01  & 0.0001  &0.39190(2)  &    1.30687(2) &\n2.25583(2)\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\nIn Figs. \\ref{fig2} (a) and (b) we plot the wave function profiles for\nthe radially-symmetric 3D and  1D cases calculated using the\nprograms imagtimesph.F and imagtime1d.F, respectively,  for\ndifferent nonlinearities presented in Tables   \\ref{table2}  and\n\\ref{table3}. As the nonlinearity is increased the system becomes more\nrepulsive and the wave function extends to a larger domain in space.\n\n\n\nIn Table \\ref{table4}\nwe present  the results for the wave function at\ncenter, rms\nradius, and chemical potential for the Cartesian 2D case with\nnonlinearity $\\aleph \n$ = 12.5484 using the imaginary-time \npropagation\nprogram imagtime2d.F.\nWe find that desired convergence is achieved with space step DX =\n0.02. (Note that the converged result in this case is less accurate\nthan those in Tables \\ref{table1}, \\ref{table2} and \\ref{table3} as we\nhave used a larger space step in order to keep the CPU time small. A\nfiner mesh will increase the accuracy requiring a larger CPU time.)\n\n\n\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{The chemical potential $\\mu$, rms size $r_{\\mathrm{rms}}$,\nand the wave function\n$\\varphi(0)$ at the center  for various\nnonlinearities in the anisotropic  2D case obtained using the programs\nimagtime2d.F [Eq. (\\ref{2d2}), OPTION 2] and imagtimecir.F [Eq.\n(\\ref{cir2}), OPTION 2]. The\ncase $\\kappa=1$\nrepresents\ncircular symmetry and $\\kappa\\ne 1$ corresponds to anisotropy.\nTable completed with\nDX = DY $\n\\le 0.02$\nand DT $=0.0001$ for imagtime2d.F and space step DR $\\le 0.0025$ and DT $=0.00002$ for\nimagtimecir.F. }\n\\label{table5}\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n$\\kappa$ &\n{$\\aleph$ \n} &\n\\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{$\\varphi(0)$} &\n      &\n\\multicolumn{1}{c}{}      &\n\\multicolumn{1}{c}{$r_{\\mathrm{rms}}$} &\n&\n\\multicolumn{1}{c}{}  &\n\\multicolumn{1}{c}{$\\mu $}   &\n \\\\\n\\hline\n&   & anisotropic &circular &\\cite{Bao_Tang} &anisotropic &circular\n&\\cite{Bao_Tang}\n&anisotropic &circular & \\cite{Bao_Tang}\\\\\n\\hline\n1&$-2.5097$ & 0.6754&0.67532(3) &  0.6754 & 0.87759(2)& 0.87758(1) &\n0.8775  &\n0.49978(3) &0.49978(1)\n&\n0.4997\\\\\n1&       0 &  0.5642& 0.56419(1)  & 0.5642& 1.00000 & 1.00000\n&1.0000 & 1.00000 & 1.000000 &\n1.0000\n\\\\\n1&    3.1371 &0.4913& 0.49128(1) & 0.4913 &1.10513(2) & 1.10515(1)\n&1.1051 & 1.42005(1)\n&1.420054(3)  &\n1.4200\\\\\n1&   12.5484 & 0.3919& 0.39190(2) & 0.3919 &  1.30687(2)& 1.30686(1)\n& 1.3068 &\n2.25583(1) &\n   2.255840(3)&\n2.2558 \\\\\n$\\sqrt 2$&   12.5484 & 0.4267&  &  &  1.22054(2) & &  &\n2.69607(1) &  &  \\\\\n$ 2$&   12.5484 & 0.4633(1)&  &  &  1.17972(2)  &  & &\n3.25488(1) &  &  \\\\\n1&  62.742 & 0.2676&  0.26760(3) &0.2676  & 1.78817(2)&1.78816(1)\n&1.7881  &  4.60982(1)\n&  4.609831(3)  &\n4.6098 \\\\\n$1\/\\sqrt 2$&  62.742 & 0.2453&  &  & 1.99987(2)& &  &  3.88210(2) &\n&\n\\\\\n1\/ 2&  62.742 & 0.2249&  &  & 2.34157(2)&  & &  3.27923(2) & &    \\\\\n1&313.71 &  0.1787 & 0.17872(3) &0.1787 &  2.60441(2) & 2.60441(1)\n&2.6044 &\n10.06825(3)&10.068262(5)\n&10.068 \\\\\n1&  627.42 & 0.1502 &0.15024(3) &0.1502 & 3.08453(2)  &3.08453(2)\n&3.0845 &\n14.18922(3)\n& 14.189228(5)&\n14.1892\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\\begin{figure}[tbp] \\begin{center}\n{\\includegraphics[width=.49\\linewidth]{fig3a.ps}}\n{\\includegraphics[width=.49\\linewidth]{fig3b.ps}}\n\\end{center}\n\\caption{(Color online) Plot of wave function profile for the (a)\ncircularly-symmetric [Eq. (\\ref{cir2}), OPTION 2]\nand\n(b)\nanisotropic 2D cases [Eq. (\\ref{2d2}), OPTION 2]\nwith nonlinearity $\\aleph\n=12.5484$ and anisotropy\n$\\kappa=2$ using programs imagtimecir.F and imagtime2d.F, respectively.\nCurves in (a) are labeled by the respective nonlinearities\nas in Table  \\ref{table5}.\n}\n\\label{fig3}\n\\end{figure}\n\n\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{The chemical potential $\\mu$, rms sizes,\nand the wave function\n$\\varphi(0)$ at the center  for various\nnonlinearities in the axially-symmetric  3D case\nfor $^{87}$Rb atoms in an axial trap\nwith $\\lambda=4,\\kappa=1$\nobtained using the\nprogram\nimagtimeaxial.F [Eq. (\\ref{axi2}), OPTION 2]. The axial frequency is\ntaken as\n$\\omega_z=80\\pi $ Hz,\n$m(^{87}$Rb$)=1.44\\times 10^{-25}$ kg,\n$l=\\sqrt{\\hbar\/m\\omega_x}=0.3407\\times 10^{-5}$ m, $a=5.1$ nm,\nand the ratio between scattering and oscillator length\n$4\\pi a\/l=0.01881$.\nTable completed with DZ\n$ \\le 0.02$, D$\\rho \\le 0.02$ and\nand DT $=0.00004$.}\n\\label{table6}\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n$N$        &\n${\\aleph}$  &\n$\\varphi(0)$ &\n$\\varphi(0)$\\cite{Bao_Tang} &\n$ \\rho _{\\mathrm{rms}}$  &\n$\\rho_{\\mathrm{rms}}$\\cite{Bao_Tang} &\n$ z_{\\mathrm{rms}}$ &\n$  z_{\\mathrm{rms}}$\\cite{Bao_Tang} &\n$\\mu $  &$\\mu $ \\cite{Bao_Tang}\\\\\n\\hline\n0& 0  &     0.5993(1) &  0.602  & 1.0000 & 1.000 &  0.3536 & 0.3539 &\n3.0000  & 3.0000 \\\\\n1000& 18.81  &       0.3813(2)  &0.3824 &1.3249  &1.325 &0.3805 & 0.3807\n& 4.3611  & 4.362\\\\\n5000& 94.05  &    0.2474     &0.2477 & 1.7742 &1.7742  &0.4212  &0.4214\n& 6.6797  & 6.680 \\\\\n10000& 188.1  &  0.2021       &0.2023 &2.0411 &2.041  & 0.4496  &0.4497\n& 8.3671  & 8.367 \\\\\n50000& 940.5  &   0.1247      &0.1248  &2.8424 & 2.842 &0.5531 & 0.5532\n& 14.9487  &14.95  \\\\\n100000& 1881  &   0.1011      &0.1012  &3.2758 & 3.276 &0.6173 & 0.6174\n&  19.4751 & 19.47 \\\\\n400000& 7524  &  0.0666       & 0.0666 &4.3408 &4.341  &0.7881 & 0.7881\n& 33.4677  & 33.47 \\\\\n800000& 15048  &      0.0540   & 0.0540 &4.9922 &4.992  & 0.8976 &\n0.8976\n& 44.0234  & 43.80 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[tbp] \\begin{center}\n{\\includegraphics[width=.49\\linewidth]{fig4a.ps}}\n{\\includegraphics[width=.49\\linewidth]{fig4b.ps}}\n\\end{center}\n\\caption{(Color online) Plot of wave function profile for the (a)\naxially-symmetric 3D case with nonlinearity $\\aleph\n=1881$\nanisotropy\n$\\lambda=4, \\kappa=1$  using imagtimeaxial.F [Eq. (\\ref{axi2}), OPTION\n2]\nand (b) anisotropic 3D case with nonlinearity $\\aleph\n=22.454,\n359.26,$ and 11496.3 and anisotropy $\\nu=1, \\lambda =\\sqrt 2$ and $\\kappa =2$\n using imagtime3d.F [Eq. (\\ref{ani2}), OPTION 2].\nIn the 3D case only the sections $\\varphi(x,0,0), \\varphi(0,y,0)$ and\n$\\varphi(0,0,z)$ of the wave functions are plotted.}\n\\label{fig4}\n\\end{figure}\n\n\n\n\n\n\n\\begin{table}[!ht]\n\\begin{center}\n\\caption{The chemical potential $\\mu,$ rms sizes, and wave function \n$\\varphi(0)$ at the center  for various number $N$ of\ncondensate of Na\natoms. The constants used are $m$(Na) $= 38.175\\times 10^{-27}$ kg,\n$a$(Na)\n = 2.75 nm. In all cases the input to numerical calculation was the\nnonlinearity\ncoefficient $\\aleph\n=4\\pi Na\/l$ shown below.\nFor the\nspherically-symmetric\ncase, we solved the radially symmetric program imagtimesph.F [Eq.\n(\\ref{sph2}), OPTION 2],\n(in\naddition to the 3D\nanisotropic program imagtime3d.F\nsetting equal frequencies in all three directions\nwith DX = DY =DZ = 0.05 and DT = 0.0004)\nusing \\cite{Schneider_Feder,hau}\n$\\omega_0^S=87$ rad\/s,\n$\nl=\\sqrt{\\hbar\/m\\omega_0^S}= 5.635$   $\\mu$m, DR $\\le 0.0025$ and\nDT = 0.00002.\nFor the fully\nanisotropic case we used the program imagtime3d.F [Eq.\n(\\ref{ani2}), OPTION 2] with \\cite{Schneider_Feder,kozuma}\n$\\omega_x\\equiv \\omega_0 ^A=354 \\pi$ rad\/s,\n$\\omega_y=\\sqrt 2\n\\omega_x,\n\\omega_z=2\\omega_x$, $l=\\sqrt{\\hbar\/m\\omega_0^A}=1.576$ $\\mu$m,\nDX = DY =DZ = 0.05 and DT = 0.0004.}\n\\label{table7}\n\\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|}\n\\hline\n   & \\multicolumn{1}{c}{} &\n\\multicolumn{1}{c}{Spherical} &  \\multicolumn{1}{c}{}&{}\n& \\multicolumn{1}{c}{}\n& \\multicolumn{1}{c}{}\n&\\multicolumn{1}{c}{anisotropic}\n& \\multicolumn{1}{c}{}\n& {}\n\\\\\n\\hline\n$N $\n& {$\\aleph $} \n& $\\mu$(sph)&$\\mu$(ani)  &$\\mu $ \n\\cite{Schneider_Feder}\n& {$\\aleph$}&$r_{\\mathrm{rms}}$   &$\\varphi(0)$&  $\\mu$(ani) & $\\mu $\n\\cite{Schneider_Feder} \\\\\n\\hline\n0 &  0   &1.500000    & 1.5000 &1.500\n& 0&1.0505 &0.5496 &\n2.2071 &\n2.207\n\\\\\n    1024 &  6.2798   &1.824546(1)& 1.8245   &  1.825 &\n22.454&1.3211\n&\n0.3471& 3.5718&\n3.572\\\\\n    2048 &    12.5597 & 2.065406(1) & 2.0654 &2.065    &\n44.907&\n 1.4584\n& 0.2888&\n4.3446\n&4.345 \\\\\n    4096 &    25.1194 & 2.434526(1)  & 2.4345 &2.435  &\n89.81\n&1.6328\n&0.2363 &5.4253\n&5.425 \\\\\n    8192 &   50.239  &  2.970180(1)  & 2.9702 &2.970  &\n179.63&1.8460\n&0.1919\n&6.9042\n&6.904 \\\\\n   16384 &    100.477 & 3.719211(1)  & 3.7192 &3.719  &\n359.26\n&2.0999\n&0.1555 &8.9003\n&8.900\\\\\n   32768 &   200.955 & 4.743445(2)  & 4.7434 &4.743  &\n718.52&\n2.3979\n&0.1260 &11.5718\n&11.572\\\\\n   65536 &   401.91 &6.123751(2)  & 6.1238 &6.124  &\n1437.03&2.7447\n&0.1022\n&  15.1284\n&15.128\n\\\\\n  131072 &   803.82 & 7.970154(2)  & 7.9702 &7.970  &\n2874.06 &  3.1460\n&0.0829 & 19.8475& 19.847\n\\\\\n  262144 &  1607.64 & 10.426912(3)     & 10.4269  & 10.427  &\n5748.13 &\n3.6092&\n0.0673&\n26.0961&26.096\n\\\\\n  524288 &  3215.28 & 13.685486(3)  &13.6855 &13.685  &\n11496.3\n&4.1426\n&0.0546 & 34.3590 &\n34.358\n\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nIn Table \\ref{table5} we present results for the wave function at the\ncenter, rms size, and chemical potential for different nonlinearities\nin the anisotropic  and circularly-symmetric 2D case calculated using\nthe imaginary-time routine\nimagtime2d.F and imagtimecir.F,  respectively.  In the anisotropic\ncase we used\n space step 0.02 and time step 0.0001 and in the circularly-symmetric\ncase we used  space step 0.0025 and time step 0.00002. We also compare\nthese\nresults with those of Ref. \\cite{Bao_Tang} and establish more accurate\nresults in the present numerical calculation.\n\n\nIn  Figs. \\ref{fig3} (a) and (b) we plot the wave function profiles for\nthe circularly-symmetric and  anisotropic 2D  cases using the programs\nimagtimecir.F\n and\nimagtime2D.F, respectively.\nIn the anisotropic 2D case the anisotropy $\\kappa =2$ and nonlinearity\n$\\aleph\n=12.4584$. Because of anisotropy the wave function in \nFig. \\ref{fig3}\n(b)\nis compressed in\nthe $y$ direction (note the different scales in $x$ and $y$ directions\nof the plot.)\n\n\n\n\nIn Table \\ref{table6} we present results for the wave function at the\ncenter, rms sizes, and chemical potential for different nonlinearities\nin the axially-symmetric 3D  case with $\\kappa=1$ and $\\lambda =4 $  calculated using\nthe imaginary-time routine\nimagtimeaxial.F and  space step $D\\rho=DZ=0.02$ and time step $DT=$ 0.0004. We\ncompare with the results of Ref. \\cite{Bao_Tang} and establish more\naccurate\nresults.\n\n\nIn Figs. \\ref{fig4} (a) and (b) we plot the wave function profiles for\nthe axially-symmetric and fully anisotropic 3D cases using the programs\nimagtimeaxial.F  and  imagtime3d.F, respectively.\nIn the axially-symmetric 3D case the anisotropy $\\kappa=1,$ $\\lambda =4$ and\nnonlinearity $\\aleph\n=1881$ were employed.\nIn the  fully anisotropic 3D case the anisotropies  $\\nu=1$,  $\\kappa\n=\\sqrt 2$, and\n$\\lambda =2$ and nonlinearities $ \\aleph\n=22.454, 359.26$  and \n11496.3 are used. The effect of the anisotropy\nis\nexplicit in the 3D case in generating different profiles of the\nwave function along $x$, $y$, and $z$ directions for a fixed\nnonlinearity.\n\n\n\nNext we consider  the fully anisotropic case\nin three dimensions and compare our results with those of Ref.\n\\cite{Schneider_Feder} calculated using the program imagtime3d.F. This\ncase mimics a realistic case of\nexperimental\ninterest with Na atoms. The completely anisotropic trap considered here\nis time-orbiting potential (TOP) trap with angular frequencies in the\nnatural ratio  ($\\omega_x,\\omega_y,\\omega_z)=\\omega_0^A(1,\\sqrt\n2,2)$, with $\\omega_0^A=354\\pi $ rad\/s. BEC in such a system has been\nobserved by Kozuma {\\it et al.} \\cite{kozuma}. We also consider\nthe\nspherical potential with $\\omega_0^S=87$ rad\/s \\cite{hau}. The s-wave\nscattering\nlength of Na is taken as $a=52a_0\\approx 2.75$ nm, with $a_0= 0.5292$\n\\AA \\-  the Bohr\nradius\n\\cite{Schneider_Feder}.\n\nIn Table \\ref{table7} we exhibit the results for our calculations with\nthe fully anisotropic potential  together with those for the\nspherically-symmetric potential in three dimensions. The results for the\nspherically-symmetric potential are also calculated using the\none-dimensional\nradially symmetric imaginary-time program imagtimesph.F\nin addition to the fully\nanisotropic program imagtime3d.F.  In the anisotropic case the\npresent results are consistent with those of Ref.\n\\cite{Schneider_Feder}.\nIn the spherical case the two sets of the present results (calculated\nwith\nthe  spherically-symmetric and anisotropic programs) as well as those of\nRef. \\cite{Schneider_Feder} are consistent with each other.\n\n\n\\begin{figure}[tbp] \\begin{center}\n{\\includegraphics[width=.45\\linewidth]{fig5a.ps}}\n{\\includegraphics[width=.45\\linewidth]{fig5b.ps}}\n{\\includegraphics[width=.45\\linewidth]{fig5c.ps}}\n{\\includegraphics[width=.45\\linewidth]{fig5d.ps}}\n\\end{center}\n\\caption{(Color online) Plot of rms size vs. time for non-stationary\noscillation of the system obtained by running the real-time programs for\n(a) 1D  case (using program realtime1d.F with\n$\\aleph\n= 62.742$), (b)  spherically-symmetric case (using\nprogram realtimesph.F with\n$\\aleph \n= 125.484$), (c) 2D circularly-symmetric case (using\nprogram realtime2d.F\nwith $\\aleph\n= 12.5484$ and $\\kappa=1$),  and\n(d) 3D axially-symmetric case (using program realtimeaxial.F  with\n $\\aleph\n= 18.81$ and $\\kappa=1, \\lambda=4$).\n The oscillation is started during time evolution\nby suddenly reducing the\nnonlinearity $\\aleph$ \nto half after the formation of the \nstationary\ncondensate.}\n\\label{fig5}\n\\end{figure}\n\n\n\n\n\nThe input to our calculation is the nonlinearity coefficient $ \\aleph$, \nwhich is related to the scattering length $a$, number of atoms $N$ and\nharmonic oscillator length $l$ via $\\aleph\n=4\\pi a N\/l$ in Eq.\n(\\ref{ani2}). We provide the nonlinearity values of our calculations.\nAlthough the present results are in agreement with those of Ref.\n\\cite{Schneider_Feder}, a\nvery precise comparison of the two calculations\n is not to the point as Schneider and Feder  did not provide the\nnonlinearity coefficient $\\aleph\n$ used in their calculation.\n\n\n\n\n\n\n\\subsection{Non-stationary Oscillation}\n\n\nThe real-time propagation programs calculate the stationary states under\ndifferent trap symmetries. However, they are less efficient than the\nimaginary-time propagation programs in this task requiring more CPU\ntime and producing less accurate results. However, unlike the\nimaginary-time propagation programs, the real-time programs  can produce\ntime\nevolution of non-stationary states also and next we present results of\nsuch time evolution using the real-time propagation programs under\ndifferent trap symmetries.\n\nIn this subsection we present results for non-stationary oscillation\nobtained with the use of the real-time programs. After the calculation\nof the  stationary profile, the nonlinearity is suddenly reduced to half.\nThe wave function is no longer an eigenstate of the new nonlinear\nequation. This sets the system into non-stationary oscillation which\ncontinues for ever.  In Fig. \\ref{fig5} we plot the rms size of the wave\nfunction vs. time $t$ showing this oscillation using the output from\nFile 8  for (a)  1D case (using program realtime1d.F), (b)\nradially-symmetric 3D case (using program realtimesph.F), (c)\nCartesian 2D case with anisotropy $\\kappa=1$ (using program\nrealtime2d.F), and (d) 3D axially-symmetric case (using program \nrealtimeaxial.F)  with  respective\nnonlinearities $\\aleph=$\n62.742, 125.484, 12.5484, and 18.81.\nThe rms size at $t=0$\nis the rms size of the stationary wave function obtained after NPAS time\niterations.\n\nBecause of transverse instability, the real-time program \nrealtime3d.F in 3D does not lead to stable sinusoidal oscillation as in \nother\ncases for a large change in nonlinearity (nonlinearity reduced to half \nof \nits initial value) as shown in Fig. \\ref{fig5}.\nOnly for small perturbation a sinusoidal oscillation is observed.\nHowever, we do not present a systematic study of such oscillation.\n\n\n\n\n\n\\section{Summary and Conclusion}\n\\label{SUM}\n\nIn this paper we describe a split-step method for the numerical solution\nof the time-dependent nonlinear GP equation under the action of a\ngeneral anisotropic 3D trap using real- and imaginary-time propagation.\nSimilar methods for 1D and anisotropic 2D traps are also described. The\ntime propagation is carried with an initial input. The full Hamiltonian\nis split into several spatial derivative and a non-derivative parts. The\nspatial derivative parts are treated by the Crank-Nicolson method.\nDifferent spatial derivative and non-derivative parts are dealt in\nindependent steps. This, so called  split-step, method leads to highly\nstable and accurate results.\n\nWe considered two types of time iterations $-$ real-time propagation and\nimaginary-time propagation. In the real-time propagation\ntime evolution  is performed with the original complex equation. The\nnumerical algorithm in this case requires the use of complex variable\nbut produces solution of non-stationary problems. In the imaginary-time\npropagation, the time variable is replaced by i ($=\\sqrt{-1})$ times a\nnew time variable, consequently the GP equation becomes real. The\nnumerical solution of this equation  can no longer yield the solution of\nnon-stationary problems; but yields very accurate solution of stationary \nground state \nproblems only, requiring  much smaller CPU time.\n\nWe provide the numerical algorithm in detail in 1D, 2D, and 3D for real-\nand imaginary-time propagations.  We consider six different harmonic \noscillator trap\nsymmetries, e.g., a  1D trap, a circularly-symmetric 2D trap, a\nradially-symmetric 3D trap, an anisotropic trap in 2D, an\naxially-symmetric 3D trap, and an anisotropic trap in 3D. Each of these\ncases are treated with real- and imaginary-time propagation algorithms\nresulting in twelve different Fortran 77 programs supplied. We use the\nimaginary-time propagation programs to provide results for different\nstationary properties of the condensate (chemical potential, rms size,\netc) in 1D, 2D, 3D, for different nonlinearities $\\aleph $\nand compare\nwith previously obtained results \\cite{Bao_Tang,Schneider_Feder}. In\naddition we study a non-stationary oscillation initiated by suddenly\naltering the nonlinearity to half its initial value on these preformed\ncondensates using the real-time propagation programs. In addition six\nFortran 90\/95 programs are supplied in the case of two and three space\nvariables.\n\nAlthough the present programs are valid for   the standard GP equation\nwith cubic nonlinearity in a harmonic potential, they can be easily\nadopted for other types of\nbosonic \\cite{tg} or fermionic equations \\cite{ska1,ska2} with different\nnonlinearities and under\ndifferent types of potentials. To change the potential one should change\nthe variable V in the subroutine INITIALIZE and the change in the\nnonlinearity can be performed in the subroutine NONLIN.\n\n\n\n\n\\ack\n\nWe thank Dr. A. Gammal for helpful comments regarding the\nsolution of the GP equation in the circularly-symmetric and\naxially-symmetric cases.\nWe thank Prof. W. Bao for the hospitality at the National University of\nSingapore when this project was started. The research was partially\nsupported by the CNPq and FAPESP of Brazil, and the Institute for\nMathematical Sciences of the National University of Singapore.\nPM thanks  the Third World Academy of Sciences (TWAS-UNESCO\nAssociateship at the Center of Excellence in the South), and Department\nof Science and Technology, Government of India for partial\nsupport.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec:Intro}\nIn his seminal 1948 paper~\\cite{shawea49}, Claude Shannon gave a formula for the increase in differential entropy per degree of freedom that a continuous-time, band-limited random process $\\rvau(t)$ experiences after passing through a linear time-invariant (LTI) continuous-time filter.\nIn this formula, if the input process is band-limited to a frequency range $[0,B]$, has differential entropy rate (per degree of freedom) $\\bar{h}(\\rvau)$, and the LTI filter has frequency response $G\\jw$, then the resulting differential entropy rate of the output process $\\rvay(t)$  is given by%\n\\cite[Theorem~14]{shawea49}\n\\begin{align}\\label{eq:hgain_Shannon}\n \\bar{h}(\\rvay) = \\bar{h}(\\rvau)  + \\frac{2}{B}\\Intfromto{0}{B} \\log \\abs{G\\jw}d\\w.\n\\end{align}\nThe last term on the right-hand side (RHS) of~\\eqref{eq:hgain_Shannon} \ncan be understood as the \\textit{entropy gain} (entropy amplification or entropy boost)\nintroduced by the filter~$G\\jw$.\nShannon proved this result by arguing that an LTI filter can be seen as a linear operator that selectively scales its input signal \nalong infinitely many frequencies, each of them representing an orthogonal component of the source.\nThe result is then obtained by writing down the determinant of the Jacobian of this operator as the product of the frequency response of the filter over $n$ frequency bands, applying logarithm and then taking the limit as the number of frequency components tends to infinity.\n\n\n\nAn analogous result can be obtained for discrete-time input $\\procu$ and output $\\procy$ processes,  and an LTI discrete-time filter $G(z)$ by relating them to their continuous-time counterparts, which yields\n\\begin{align}\\label{eq:hgain_discrete}\n \\bar{h}(\\procy) = \\bar{h}(\\procu) + \\frac{1}{2\\pi}\\intfromto{-\\pi}{\\pi}\\log\\abs{G\\ejw}d\\w,\n\\end{align}\nwhere \n$$\n\\bar{h}(\\procu)\\eq \\lim_{n\\to\\infty} \\tfrac{1}{n} h(\\rvau(1),\\rvau(2),\\ldots,\\rvau(n))\n$$\nis the differential entropy rate of the process $\\procu$.\nOf course the same formula can also be obtained by applying the frequency-domain proof technique that Shannon followed in his derivation of~\\eqref{eq:hgain_Shannon}.\n\n\nThe rightmost term in~\\eqref{eq:hgain_discrete}, which corresponds to the entropy gain of $G(z)$, can be related to the structure of this filter. \nIt is well known that if $G$ is causal with a rational transfer function $G(z)$ such that \n$\\lim_{z\\to\\infty}|G(z)|=1$ (i.e., such that the first sample of its impulse response has unit magnitude), then  \n\\begin{align}\\label{eq:Jensen}\n\\frac{1}{2\\pi}\\intfromto{-\\pi}{\\pi}\\log\\abs{G\\ejw}d\\w \n = \\Sumover{c_{i}\\notin\\mathbb{D}}\\log\\abs{\\rho_{i}},  \n\\end{align}\nwhere $\\set{\\rho_{i}}$ are the zeros of $G(z)$ and \n${\\mathbb{D}}\\eq \\set{z\\in\\mathbb{C}: \\abs{z} <1}$ is the open unit disk on the complex plane.\nThis provides a straightforward way to evaluate the entropy gain of a given LTI filter with rational transfer function $G(z)$.\nIn addition,~\\eqref{eq:Jensen} shows that, if $\\lim_{z\\to\\infty}|G(z)|=1$, then such gain is greater than one if and only if $G(z)$ has zeros outside $\\mathbb{D}$.\nA filter with the latter property is said to be \\textit{non-minimum phase} (NMP); conversely, a filter with all its zeros inside $\\mathbb{D}$ is said to be \\textit{minimum phase} (MP)~\\cite{serbra97}.\n\nNMP filters appear naturally in various applications. \nFor instance, any unstable LTI system stabilized via linear feedback control will yield transfer functions which are NMP~\\cite{serbra97,googra00}.\nAdditionally, NMP-zeros also appear when a discrete-time with ZOH (\\emph{zero order hold}) equivalent system is obtained from a plant whose number of poles exceeds its number of zeros by at least 2, as the sampling rate increases~\\cite[Lemma 5.2]{yuzgoo14}. \nOn the other hand, all linear-phase filters, \nwhich are specially suited for audio and image-processing applications, are NMP~\\cite{hayes-96,vaidya93}. \nThe same is true for any all-pass filter, which is an important building block in signal processing applications~\\cite{smith-07,hayes-96}.\n\n\n\nAn alternative approach for obtaining the entropy gain of LTI filters is to work in the time domain; obtain $\\rvay_{1}^{n}\\eq \\set{\\rvay_{1},\\rvay_{1},\\ldots,\\rvay_{n}}$\nas a function of $\\rvau_{1}^{n}$, for every $n\\in\\Nl$,\nand evaluate the limit \n$\n \\lim_{n\\to\\infty}\\frac{1}{n}\\left(h(\\rvay_{1}^{n}) - h(\\rvau_{1}^{n}) \\right)\n$.\nMore precisely, for a filter $G$ with impulse response $g_{0}^{\\infty}$, we can write\n\\begin{align}\\label{eq:y_of_u_matrix}\n \\rvey^{1}_{n} = \n\\underbrace{\\begin{pmatrix}\n g_{0} & 0    &\\cdots& 0\\\\\n g_{1} & g_{0}&\\cdots &0\\\\\n \\vdots & &\\ddots &\\vdots\\\\\n g_{n-1}& g_{n-2}&\\cdots &g_{0}\n\\end{pmatrix}}_{\\bG_{n}}\n\\rveu^{1}_{n},\n\\end{align}\nwhere $\\rvey^{1}_{n} \\eq [\\rvay_{1}\\ \\rvay_{1}\\,\\cdots \\ \\rvay_{n}]^{T}$ and the random  vector $\\rveu^{1}_{n}$ is defined likewise. \nFrom this, it is clear that \n\\begin{align}\\label{eq:hy=hu+logdet}\n h(\\rvey^{1}_{n}) = h(\\rveu^{1}_{n}) + \\log|\\det(\\bG_{n})|,\n\\end{align}\nwhere $\\det(\\bG_n)$ (or simply $\\det \\bG_n$) stands for the determinant of $\\bG_n$. Thus, \n\\begin{align}\\label{eq:f0=1_andtherest}\n|g_{0}|=1\\Longrightarrow |\\det(\\bG_{n})|=1, \\;\\forall n\\in\\Nl \\Longleftrightarrow\nh(\\rvey^{1}_{n}) = h(\\rveu^{1}_{n}),\\;\\forall n\\in\\Nl\n\\Longrightarrow\n \\lim_{n\\to\\infty}\\frac{1}{n}\\left[h(\\rvey_{1}^{n}) - h(\\rveu_{1}^{n})\\right] =0,\n\\end{align}\nregardless of whether $G(z)$ (i.e., the polynomial $g_{0} + g_{1}z^{-1}+\\cdots $) has zeros with magnitude greater than one, \\textbf{which clearly\ncontradicts~\\eqref{eq:hgain_discrete} and~\\eqref{eq:Jensen}}. \nPerhaps surprisingly, the above contradiction not only has been overlooked in previous works (such as~\\cite{aarmcd67,zanigl03}), but the time-domain formulation  in the form of~\\eqref{eq:y_of_u_matrix} has been utilized as a means to prove or disprove~\\eqref{eq:hgain_discrete} (see, for example, the reasoning in~\\cite[p.~568]{papou91}).\n\n\n\n\n\n\n\n\nA reason for why the contradiction between~\\eqref{eq:hgain_discrete},~\\eqref{eq:Jensen} and~\\eqref{eq:f0=1_andtherest} arises \ncan be obtained from the analysis developed in~\\cite{mardah08} for an LTI system $P$ within a noisy feedback loop, as the one depicted in Fig.~\\ref{fig:fbksystem}.\nIn this scheme, $C$ represents a causal feedback channel which combines the output of $P$ with an exogenous (noise) random process $\\rvac_{1}^{\\infty}$ to generate its output.\nThe process $\\rvac_{1}^{\\infty}$ is assumed independent of the initial state of $P$, represented by the random vector $\\rvex_{0}$, which has finite differential entropy.\n\\begin{figure}[t\n\\centering\n\\input{fbksystem.pstex_t}\n\\caption{Left: LTI system $P$ within a noisy feedback loop. Right: equivalent system when the feedback channel is noiseless and has unit gain.}\n\\label{fig:fbksystem}\n\\end{figure}\nFor this system, it is shown in~\\cite[Theorem 4.2]{mardah08} that \n\\begin{subequations}\\label{eq:martins_both}\n \\begin{align}\\label{eq:martins}\n \\bar{h}(\\rvay_{1}^{\\infty}) \\geq \\bar{h}(\\rvau_{1}^{\\infty}) + \\lim_{n\\to\\infty}\\frac{1}{n}I(\\rvex_{0}; \\rvay_{1}^{n}),\n\\end{align}\nwith equality if $\\rvaw$ is a deterministic function of $\\rvav$.\nFurthermore, it is shown in~\\cite[Lemma 3.2]{mardah05} that if $|h(\\rvex_{0})|<\\infty$ and the steady state variance of system $P$ remains asymptotically bounded as $k\\to\\infty$, then  \n\\begin{align}\\label{eq:martins_I_bound}\n\\lim_{n\\to\\infty}\\frac{1}{n}I(\\rvex_{0}; \\rvay_{1}^{n})\n\\geq \\sumover{p_{i}\\notin\\mathbb{D}}\\log\\abs{p_{i}},\n\\end{align}\n\\end{subequations}\nwhere $\\set{p_{i}}$ are the poles of $P$.\nThus, for the (simplest) case in which $\\rvaw=\\rvav$, the output $\\rvay_{1}^{\\infty}$ is the result of filtering $\\rvau_{1}^{\\infty}$ by a filter $G=\\frac{1}{1+P}$ (as shown in Fig.~\\ref{fig:fbksystem}-right), and the resulting entropy rate of $\\procy$ will exceed that of $\\procu$ only if there is a random initial state with bounded differential entropy (see~\\eqref{eq:martins}). \nMoreover, under the latter conditions,~\\cite[Lemma 4.3]{mardah08} implies that if $G(z)$ is stable and $|h(\\rvex_{0})|<\\infty$, then this entropy gain will be lower bounded by the \\textit{right-hand side} (RHS) of~\\eqref{eq:Jensen}, which is greater than zero if and only if $G$ is NMP. \nHowever, the result obtained in~\\eqref{eq:martins_I_bound} does not provide conditions under which the equality in the latter equation holds.\n\nAdditional results and intuition related to this problem can be obtained from in~\\cite{kim-yh10}.\nThere it is shown that if $\\procy$ is a two-sided Gaussian stationary random process generated by a state-space recursion of the form \n\\begin{subequations}\\label{subeq:State_Space_YHKIM}\n\\begin{align}\n \\rves_{k+1}& = (\\bA - \\bg\\bh^{H}) \\rves_{k} - \\bg \\rvau_{k},\\\\\n \\rvay_{k}  & = \\bh^{H} \\rves_{k} + \\rvau_{n},\n\\end{align}\n\\end{subequations}\nfor some $\\bA\\in\\mathbb{C}^{M\\times M}$, $\\bg\\in\\mathbb{C}^{M\\times 1}$, $\\bh\\in\\mathbb{C}^{M\\times 1}$,\nwith unit-variance Gaussian i.i.d. innovations $\\rvau_{-\\infty}^{\\infty}$, then its entropy rate will be exactly\n$\\frac{1}{2}\\log(2\\pi\\expo{})$ (i.e., the differential entropy rate of $\\rvau_{-\\infty}^{\\infty}$) plus the RHS of~\\eqref{eq:Jensen} (with $\\set{\\rho_{i}}$ now being the eigenvalues of $\\bA$ outside the unit circle).\nHowever, as noted in~\\cite{kim-yh10}, if the same system with zero (or deterministic) initial state is excited by a one-sided infinite Gaussian i.i.d. process $\\rvau_{1}^{\\infty}$ with unit sample variance, then the (asymptotic) entropy rate of the output process $\\rvay_{1}^{\\infty}$ is just~$\\frac{1}{2}\\log(2\\pi\\expo{})$ (i.e., there is no entropy gain).\nMoreover, it is also shown that if $\\rvav_{1}^{\\ell}$ is a Gaussian random sequence with positive-definite covariance matrix and $\\ell\\geq M$, then the entropy rate of $\\rvay_{1}^{\\infty}+\\rvav_{1}^{\\ell}$ also exceeds that of $\\rvau_{1}^{\\infty}$ by the RHS of~\\eqref{eq:Jensen}.\nThis suggests that for an LTI system which admits a state-space representation of the form~\\eqref{subeq:State_Space_YHKIM}, the entropy gain \nfor a single-sided Gaussian i.i.d. input is zero, and that the entropy gain from the input to the output-plus-disturbance is~\\eqref{eq:Jensen}, for any Gaussian disturbance of length $M$ with positive definite covariance matrix (no matter how small this covariance matrix may be).\n\n\nThe previous analysis suggests that it is the absence of a random initial state or a random additive output disturbance that makes the time-domain formulation~\\eqref{eq:y_of_u_matrix} yield a zero entropy gain.\nBut, how would the addition of such finite-energy exogenous random variables to~\\eqref{eq:y_of_u_matrix} actually produce an increase in the differential entropy rate which asymptotically equals the RHS of~\\eqref{eq:Jensen}? \nIn a broader sense, it is not clear from the results mentioned above what the necessary and sufficient conditions \nare under which  \nan entropy gain equal to the RHS of~\\eqref{eq:Jensen} arises (the analysis in~\\cite{kim-yh10} provides only a set of sufficient conditions and relies on second-order statistics and Gaussian innovations to derive the results previously described). \nAnother important observation to be made is the following: it is well known that the entropy gain introduced by a linear mapping is independent of the input statistics~\\cite{shawea49}. \nHowever, \nthere is no reason to assume such independence \nwhen this entropy gain arises as the result of adding a random signal to the input of the mapping, i.e., when the mapping by itself does not produce the entropy gain. \nHence, it remains to characterize the largest set of input statistics which yield an entropy gain, and the magnitude of this gain.\n\nThe first part of this paper provides answers to these questions. \nIn particular, in Section~\\ref{sec:geometric_interpretation} explain how and when the entropy gain arises (in the situations described above), starting with input and output sequences of finite length,\nin a time-domain analysis similar to~\\eqref{eq:y_of_u_matrix},\nand then taking the limit as the length tends to infinity.\nIn Section~\\ref{sec:entropy_gain_output_disturb} it is shown that, in the output-plus-disturbance scenario, \nthe entropy gain is \\emph{at most} the RHS of~\\eqref{eq:Jensen}. \nWe show that, for a broad class of input processes (not necessarily Gaussian or stationary), this maximum entropy gain\nis reached only when the disturbance has bounded differential entropy and its length is at least equal to the number of non-minimum phase zeros of the filter.\nWe provide upper and lower bounds on the entropy gain if the latter condition is not met.\nA similar result is shown to hold when there is a random initial state in the system (with finite differential entropy).\nIn addition, in Section~\\ref{sec:entropy_gain_output_disturb} we study the entropy gain between the \\emph{entire output sequence} that a filter yields as response to a shorter input sequence (in Section~\\ref{sec:effective_entropy}).\nIn this case, however, it is necessary to consider a new definition for differential entropy, named \\emph{effective differential entropy}.\nHere we show that \n an effective entropy gain equal to the RHS of~\\eqref{eq:Jensen} is obtained provided the input has finite differential entropy rate, even when there is no random initial state or output disturbance.\n\n\nIn the second part of this paper (Section\\ref{sec:implications}) we apply the conclusions obtained in the first part to three problems, namely, \nnetworked control,\nthe rate-distortion function for non-stationary Gaussian sources, \nand the Gaussian channel capacity with feedback.\nIn particular, we show that equality holds in~\\eqref{eq:martins_I_bound} \nfor the feedback system in Fig.~\\ref{fig:fbksystem}-left \nunder very general conditions (even when the channel $C$ is noisy).\nFor the problem of finding the quadratic rate-distortion function for non-stationary auto-regressive Gaussian sources, previously solved in~\\cite{gray--70,hasari80,grahas08}, we provide a simpler proof based upon the results we derive in the first part.\nThis proof extends the result stated in~\\cite{hasari80,grahas08} to a broader class of non-stationary sources. \nFor the feedback Gaussian capacity problem, we show that capacity results based on \nusing a short random sequence as channel input and relying on a feedback filter which boosts the entropy rate of the end-to-end channel noise (such as the one proposed in~\\cite{kim-yh10}), crucially depend  upon the complete absence of any additional disturbance anywhere in the system.\nSpecifically, we show that the information rate of such capacity-achieving schemes drops to zero in the presence of any such additional disturbance.\nAs a consequence, the relevance of characterizing the robust (i.e., in the presence of disturbances) feedback capacity of Gaussian channels, which appears to be a fairly unexplored problem, becomes evident.\n\nFinally, the main conclusions of this work are summarized in Section~\\ref{sec:conclusions}.\n\nExcept where present, all proofs are presented in the appendix.\n\n\\subsection{Notation}\nFor any LTI system $G$, the transfer function $G(z)$ corresponds to the $z$-transform of the impulse response $g_{0},\\, g_{1}, \\ldots$, i.e., $G(z) = \\sumfromto{i=0}{\\infty} g_{i}z^{-i}$.\nFor a transfer function $G(z)$, we denote by $\\bG_{n}\\in\\Rl^{n\\times n}$ the lower triangular Toeplitz matrix having $[g_{0}\\ \\cdots \\ g_{n-1}]^{T}$ as its first column.\nWe write $x_{1}^{n}$ as a shorthand for the sequence $\\set{x_{1},\\ldots, x_{n}}$ and, when convenient, we write $x_{1}^{n}$ in vector form as  $\\bx^{1}_{n}\\eq [x_{1}\\ x_{2}\\ \\cdots \\ x_{n}]^{T}$, where $()^{T}$ denotes transposition.\nRandom scalars  (vectors) are denoted using non-italic characters, such as $\\rvax$ (non-italic and boldface characters, such as $\\rvex$).\nFor matrices we use upper-case boldface symbols, such as $\\bA$.\nWe write $\\lambda_{i}(\\bA)$\nto the note the $i$-th smallest-magnitude eigenvalue of $\\bA$. \nIf $\\bA_{n}\\in\\mathbb{C}^{n\\times n}$, then $\\bA_{i,j}$ denotes the entry in the intersection between the $i$-th row and the $j$-th column.\nWe write $[\\bA_{n}]^{i_{1}}_{i_{2}}$, with $i_{1}\\leq i_{2}\\leq n$, to refer to the matrix formed by selecting the rows $i_{1}$ to $i_{2}$ of $\\bA$.\nThe expression ${^{m_{1}}}\\![\\bA]_{m_{2}}$ corresponds to the square sub-matrix along the main diagonal of $\\bA$, with its top-left and bottom-right corners on $\\bA_{m_1,m_1}$ and $\\bA_{m_{2},m_{2}}$, respectively.\nA diagonal matrix whose entries are the elements in $\\Dsp$ is denoted as $\\diag\\Dsp$\n\n\\section{Problem Definition and Assumptions}\nConsider the discrete-time system depicted in Fig.~\\ref{fig:general}.\nIn this setup, the input $\\rvau_{1}^{\\infty}$ is a random process\nand the block $G$ is a causal, linear and time-invariant system with random initial state vector $\\rvex_{0}$ and random output disturbance $\\rvaz_{1}^{\\infty}$.\n\\begin{figure}[t\n \\centering\n\\input{system.pstex_t}\n\\caption{Linear, causal, stable and time-invariant system $G$ with input and output processes, initial state and output disturbance.}\n\\label{fig:general}\n\\end{figure}\nIn vector notation,\n\\begin{align}\\label{eq:system_equation}\n \\rvey^{1}_{n}\\eq \\bG_{n}\\rveu^{1}_{n} + \\bar{\\rvey}^1_n + \\rvez^1_n,\\fspace n\\in\\Nl,\n\n\\end{align}\nwhere \n$\\bar{\\rvey}^1_n$\nis the natural response of $G$ to the initial state $\\rvex_{0}$.\nWe make the following further assumptions about $G$ and the signals around it:\n\\begin{assu}\\label{assu:G_Factorized}\n$G(z)$ is a causal, stable and rational transfer function of finite order, whose impulse response $g_{0},g_{1},\\ldots$ satisfies\n$ g_{0}=1$.\n\\finenunciado\n %\n\\end{assu}\nIt is worth noting that there is no loss of generality in considering $g_{0}=1$, since otherwise one can write $G(z)$ as $G'(z)=g_{0}\\cdot G(z)\/g_{0}$,\n  and thus the entropy gain introduced by $G'(z)$ would be $\\log g_{0}$ plus the entropy gain due to  $G(z)\/g_{0}$, which has an impulse response where the first sample equals $1$.\n  \n\n  \n\n\\begin{assu}\nThe random initial state $\\rvex_{0}\n$ is independent of $\\rvau_{1}^{\\infty}$.\n\\end{assu}\n\\begin{assu}\\label{assu:z}\nThe disturbance $\\rvaz_{1}^{\\infty}$ is independent of $\\rvau_{1}^{\\infty}$ and  belongs to a $\\kappa$-dimensional linear subspace, for some finite $\\kappa\\in\\Nl$.\nThis subspace is spanned by the $\\kappa$ orthonormal columns of a matrix $\\boldsymbol{\\Phi}\\in\\Rl^{|\\Nl|\\times \\kappa}$ (where $|\\Nl|$ stands for the countably infinite size of $\\Nl$), such that \n$|h(\\boldsymbol{\\Phi}^{T}\\rvez^{1}_{\\infty})|<\\infty$.  \nEquivalently, $\\rvez^{1}_{\\infty} = \\boldsymbol{\\Phi} \\rves^{1}_{\\kappa}$, where the random vector \n$\\rves^{1}_{\\kappa}\\eq \\boldsymbol{\\Phi}^{T}\\rvez^{1}_{\\infty}$ has finite differential entropy and is independent of $\\rveu^{1}_{\\infty}$.\n\\end{assu}\n\n\n\n\n\n\n\n\n\nAs anticipated in the Introduction, we are interested in characterizing the entropy gain $\\Gsp$ of $G$ in the presence (or absence) of the random inputs\n$\\rvau_{1}^{\\infty},\\rvex_{0},\\rvaz_{1}^{\\infty}$, denoted by \n\\begin{align}\\label{eq:Gsp_def}\n\\Gsp(G,\\rvex_{0},\\rvau_{1}^{\\infty},\\rvaz_{1}^{\\infty})\\eq \\lim_{n\\to\\infty}\n\\frac{1}{n}\n\\left(h(\\rvay_{1}^{n})- h(\\rvau_{1}^{n})\\right).\n\\end{align}\nIn the next section we provide geometrical insight into the behaviour of $\\Gsp(G,\\rvex_{0},\\rvau_{1}^{\\infty},\\rvaz_{1}^{\\infty})$ for the situation where there is a random output disturbance and no random initial state. \nA formal and precise treatment of this scenario is then presented in Section~\\ref{sec:entropy_gain_output_disturb}. \nThe other scenarios are considered in the subsequent sections.\n\n\n\n\n\n\\section{Geometric Interpretation}\\label{sec:geometric_interpretation}\nIn this section we provide an intuitive geometric interpretation of how and when the entropy gain defined in~\\eqref{eq:Gsp_def} arises. \nThis understanding will justify the introduction of the notion of an entropy-balanced random process (in Definition~\\ref{def:entropy_balanced} below), which will be shown to play a key role in this and in related problems. \n\n\\subsection{An Illustrative Example}\nSuppose for the moment that $G$ in Fig.~\\ref{fig:general} is an FIR filter with impulse response $g_{0}=1,\\ g_{1}=2,\\ g_{i}=0,\\, \\forall i\\geq 2$.\nNotice that this choice yields $G(z)= (z-2)\/z$, and thus $G(z)$ has one non-minimum phase zero, at $z=2$.\nThe associated matrix $\\bG_{n}$ for $n=3$ is\n$$\n\\bG_{3}=\\begin{pmatrix}\n 1&0&0\\\\\n 2&1&0\\\\\n0&2&1\n\\end{pmatrix},\n$$ \nwhose determinant is clearly one (indeed, all its eigenvalues are $1$).\nHence, as discussed in the introduction, \n$h(\\bG_{3}\\rveu^{1}_{3})=h(\\rveu^{1}_{3})$, and thus $\\bG_{3}$ (and $\\bG_{n}$ in general) does not introduce an entropy gain by itself.\nHowever, an interesting phenomenon becomes evident by looking at the singular-value decomposition (SVD) of $\\bG_{3}$, given by \n$\\bG_{3}=\\bQ_{3}^{T}\\bD_{3}\\bR_{3}$, \nwhere $\\bQ_{3}$ and $\\bR_{3}$ are unitary matrices and $\\bD_{3}\\eq \\diag\\set{d_{1},d_{2},d_{3}}$. \nIn this case, $\\bD_3 = \\diag\\set{ 0.19394,\\, 1.90321, \\, 2.70928}$, and thus one of the singular values of $\\bG_{3}$ is much smaller than the others (although the product of all singular values yields $1$, as expected).\nAs will be shown in Section~\\ref{sec:entropy_gain_output_disturb}, for a stable $G(z)$ such uneven distribution of singular values arises only when $G(z)$ has non-minimum phase zeros.\nThe effect of this can be visualized by looking at the image of the cube $[0,1]^{3}$ through $\\bG_{3}$ shown in Fig.~\\ref{fig:cube}.\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width = 8 cm, trim= 0 70 0 70, clip]{cube_shear.eps}\n\\caption{Image of the cube $[0,1]^{3}$ through the square matrix with columns \n$[1\\; 2 \\; 0]^{T}$,\n$[0\\; 1 \\; 2]^{T}$ and\n$[0\\; 0 \\; 1]^{T}$.\n}\n\\label{fig:cube}\n\\end{figure}\nIf the input $\\rveu^{1}_{3}$ were uniformly distributed over this cube (of unit volume), then $\\bG_{3}\\rveu^{1}_{3}$ would distribute uniformly over the unit-volume parallelepiped depicted in Fig.~\\ref{fig:cube}, and hence\n$h(\\bG_{3}\\rveu^{1}_{3})=h(\\rveu^{1}_{3})$. \n\nNow, if we add to $\\bG_{3}\\rveu^{1}_{3}$ a disturbance $\\rvez^{1}_{3}=\\boldsymbol{\\Phi} \\rvas$, with scalar $\\rvas$ uniformly distributed over $[-0.5,\\ 0.5]$ independent of $\\rveu^{1}_{3}$, and with $\\boldsymbol{\\Phi}\\in\\Rl^{3\\times 1}$, the effect would be to ``thicken'' the support over which the resulting random vector \n$\\rvey^{1}_{3}=\\bG_{3}\\rveu^{1}_{3}+\\rvez^{1}_{3}$ is distributed, along the direction pointed by $\\boldsymbol{\\Phi}$.\nIf $\\boldsymbol{\\Phi}$ is aligned with the direction along which the support of $\\bG_{3}\\rveu^{1}_{3}$ is thinnest\n(given by $\\bq_{3,1}$, the first row of $\\bQ_{3}$), then the resulting support would have its volume significantly increased, which can be associated with a large increase in the differential entropy of $\\rvey^{1}_{3}$ with respect to $\\rveu^{1}_{3}$.\nIndeed, a relatively small variance of $\\rvas$ and an approximately aligned $\\boldsymbol{\\Phi}$ would still produce a  significant entropy gain.\n\nThe above example suggests that the entropy gain from $\\rveu^{1}_{n}$ to $\\rvey^{1}_{n}$ appears as a combination of two factors.\nThe first of these is the uneven way in which the random vector $\\bG_{n}\\rveu^{1}_{n}$ is distributed over $\\Rl^{n}$.\nThe second factor is the alignment of the disturbance vector $\\rvez^{1}_{n}$ with respect to the \nspan of the subset $\\set{\\bq_{n,i}}_{i\\in\\Omega_{n}}$ of columns of $\\bQ_{n}$, associated with smallest singular values of $\\bG_{n}$, indexed by the elements in the set $\\W_n$.\nAs we shall discuss in the next section, if $G$ has $m$ non-minimum phase zeros, then, as $n$ increases, there will be $m$ singular values of $\\bG_{n}$ going to zero exponentially.\nSince the product of the singular values of $\\bG_{n}$ equals $1$ for all $n$, it follows that $\\prod_{i\\notin \\W_{n}}d_{n,i}$ must grow exponentially with $n$, where $d_{n,i}$ is the $i$-th diagonal entry of $\\bD_n$.\nThis implies that $\\bG_{n}\\rveu^{1}_{n}$ expands with $n$ along the span of $\\set{\\bq_{n,i}}_{i\\notin\\W_{n}}$, compensating its shrinkage along the span of $\\set{\\bq_{n,i}}_{i\\in\\W_{n}}$, thus keeping $h(\\bG_{n}\\rveu^{1}_{n})=h(\\rveu^{1}_{n})$ for all $n$.\nThus, as $n$ grows, any small disturbance distributed over the span of $\\set{\\bq_{n,i}}_{i\\in\\W_{n}}$, added to $\\bG_{n}\\rveu^{1}_{n}$, will keep the support of the resulting distribution from shrinking along this subspace.\nConsequently, the expansion of \n$\\bG_{n}\\rveu^{1}_{n}$ with $n$ along the span of $\\set{\\bq_{n,i}}_{i\\notin\\W_{n}}$ is no longer compensated, yielding an entropy increase proportional to $\\log(\\prod_{i\\notin\\W_{n}} d_{n,i})$.\n\nThe above analysis allows one to anticipate a situation in which no entropy gain would take place even when some singular values of $\\bG_{n}$ tend to zero as $n\\to\\infty$. \nSince the increase in entropy is made possible by the fact that, as $n$ grows, the support of the distribution of $\\bG_{n}\\rveu^{1}_{n}$ shrinks along the span of $\\set{\\bq_{n,i}}_{i\\in\\W_{n}}$, no such entropy gain should arise if the support of the distribution of the input $\\rveu^{1}_{n}$ expands accordingly along the directions pointed by the rows $\\set{\\br_{n,i}}_{i\\in\\W_{n}}$ of $\\bR_{n}$. \n\nAn example of such situation can be easily constructed as follows: Let $G(z)$ in Fig.~\\ref{fig:general} have non-minimum phase zeros and \nsuppose that $\\rvau_{1}^{\\infty}$ is generated as $G^{-1}\\tilde{\\rvau}_{1}^{\\infty}$, where $\\tilde{\\rvau}_{1}^{\\infty}$ is an i.i.d. random process with bounded entropy rate.\nSince the determinant of $\\bG_{n}^{-1}$ equals $1$ for all $n$, we have that $h(\\rveu^{1}_{n})=h(\\tilde{\\rveu}^{1}_{n})$, for all $n$.\nOn the other hand,\n$\\rvey^{1}_{n}\n=\n\\bG_{n}\\bG_{n}^{-1}\\tilde{\\rveu}^{1}_{n} + \\rvez^{1}_{n}\n=\n\\tilde{\\rveu}^{1}_{n} + \\rvez^{1}_{n}\n$.\nSince $\\rvez^{1}_{n}=[\\boldsymbol{\\Phi}]^{1}_{n}\\rves^{1}_{\\kappa}$ for some finite $\\kappa$ (recall Assumption~\\ref{assu:z}), it is easy to show that \n$\n\\lim_{n\\to\\infty}\\frac{1}{n} h(\\rvey^{1}_{n})\n= \n\\lim_{n\\to\\infty}\\frac{1}{n} h(\\tilde{\\rveu}^{1}_{n})\n= \n\\lim_{n\\to\\infty}\\frac{1}{n} h({\\rveu}^{1}_{n})\n$,\nand thus no entropy gain appears.\n\nThe preceding discussion reveals that the entropy gain produced by $G$ in the situation shown in Fig.~\\ref{fig:general} \\textbf{depends on the distribution of the input and on the support and distribution of the disturbance}.\nThis stands in stark contrast with the well known fact that the increase in differential entropy produced by an invertible linear operator depends only on its Jacobian, and not on the statistics of the input~\\cite{shawea49}. \nWe have also seen that the distribution of a random process along the different directions within the Euclidean space which contains it plays a key role as well.\nThis motivates the need to specify a class of random processes which distribute more or less evenly over all directions.\nThe following section introduces a rigorous definition of this class and characterizes a large family of processes belonging to it.\n\n\\subsection{Entropy-Balanced Processes}\\label{subsec:entropy_balanced}\nWe begin by formally introducing the notion of an ``entropy-balanced'' process\n $\\rvau_{1}^{\\infty}$, being one in which, for every finite $\\nu\\in\\Nl$, the differential entropy rate of the orthogonal projection of $\\rvau_{1}^{n}$ into any subspace of dimension $n-\\nu$ equals the entropy rate of $\\rvau_{1}^{n}$ as $n\\to\\infty$.\nThis idea is precisely in the following definition.\n\\begin{defn}\\label{def:entropy_balanced}\n A random process $\\set{\\rvav(k)}_{k=1}^{\\infty}$ is said to be entropy balanced if, for every $\\nu\\in\\Nl$, \n\\begin{subequations}\\label{eq:the_painful_assumption}\n\\begin{align}\n\\lim_{n\\to\\infty}\\frac{1}{n}& \n\\left( \nh(\\boldsymbol\\Phi_{n}\\rvev^{1}_{n}  )- h(\\rvev^{1}_{n})\n\\right) =0, \n\\end{align} \n\\end{subequations} \nfor every sequence of matrices $\\set{\\boldsymbol{\\Phi}_{n}}_{n=\\nu+1}^{\\infty}$, $\\boldsymbol{\\Phi}_n\\in\\Rl^{(n-\\nu)\\times n}$, with orthonormal rows. \n\\finenunciado\n\\end{defn}\nEquivalently, a random process $\\set{\\rvav(k)}$ is entropy balanced if every unitary transformation on $\\rvav_{1}^{n}$ yields a random sequence $\\rvay_{1}^{n}$ such that\n$\n \\lim_{n\\to \\infty}\\frac{1}{n}|h(\\rvay_{n-\\nu+1}^{n}|\\rvay_{1}^{n-\\nu})| =0\n$.\nThis property of the resulting random sequence $\\rvay_{1}^{n}$ means that one cannot predict its last $\\nu$ samples with arbitrary accuracy by using its previous $n-\\nu$ samples, even if $n$ goes to infinity.\n\nWe now characterize a large family of entropy-balanced random processes and establish some of their properties.\nAlthough intuition may suggest that most random processes (such as i.i.d. or stationary processes) should be entropy balanced, that statement seems rather difficult to prove.\nIn the following, we show that the entropy-balanced condition is met by i.i.d. processes with per-sample \\textit{probability density function} (PDF) being  uniform, piece-wise constant or Gaussian.\nIt is also shown that adding to an entropy-balanced process an independent random processes independent of the former yields another entropy-balanced process, and that filtering an entropy-balanced process by a stable and minimum phase filter yields an entropy-balanced process as well.\n\n\\begin{prop}\\label{prop:gaussian_is_entropy_balanced}\nLet $\\rvau_{1}^{\\infty}$ be a Gaussian i.i.d. random process with positive and bounded per-sample variance.\n Then $\\rvau_{1}^{\\infty}$ is entropy balanced.\\finenunciado\n\\end{prop}\n\\begin{lem}\\label{lem:piecewiseconstant}\n Let $\\rvau_{1}^{\\infty}$ be an i.i.d. process with finite differential entropy rate, in which each $\\rvau_i$ is distributed according to a piece-wise constant PDF in which each interval where this PDF is constant has measure greater than $\\epsilon$, for some bounded-away-from-zero constant $\\epsilon$. \n Then $\\rvau_{1}^{\\infty}$ is entropy balanced.\\finenunciado\n\\end{lem}\n\n\n\\begin{lem}\\label{lem:sum_yields_entropy_balanced}\n Let $\\rvau_{1}^{\\infty}$ and $\\rvav_{1}^{\\infty}$ be mutually independent random processes.\n If $\\rvau_{1}^{\\infty}$ is entropy balanced, then $\\rvaw_{1}^{\\infty}\\eq \\rvau_{1}^{\\infty} + \\rvav_{1}^{\\infty}$ is also entropy balanced.\\finenunciado\n\\end{lem}\nThe working behind this lemma can be interpreted intuitively by noting that adding to a random process another independent random process can only increase the ``spread'' of the distribution of the former, which tends to balance the entropy of the resulting process along all dimensions in Euclidean space.\nIn addition, it follows from Lemma~\\ref{lem:sum_yields_entropy_balanced} that all i.i.d. processes having a per-sample PDF which can be constructed by convolving uniform, piece-wise constant or Gaussian PDFs as many times as required are entropy balanced.\nIt also implies that one can have non-stationary processes which are entropy balanced, since Lemma~\\ref{lem:sum_yields_entropy_balanced} imposes no requirements for the process $\\rvav_{1}^{\\infty}$.\n\nOur last lemma related to the properties of entropy-balanced processes shows that filtering by a stable and minimum phase LTI filter preserves the entropy balanced condition of its input.\n\\begin{lem}\\label{lem:filtering_preserves_entropy_balance}\n Let $\\rvau_{1}^{\\infty}$ be an entropy-balanced process and $G$ an LTI stable and minimum-phase filter.\n Then the output $\\rvaw_{1}^{\\infty}\\eq G\\rvau_{1}^{\\infty}$ is also an entropy-balanced process.\\finenunciado\n\\end{lem}\nThis result implies that any stable moving-average auto-regressive process constructed from entropy-balanced innovations is also entropy balanced, provided the coefficients of the averaging and regression correspond to a stable MP filter.\n\nWe finish this section by pointing out two examples of processes which are non-entropy-balanced, namely, \nthe output of a NMP-filter to an entropy-balanced input and the output of an unstable filter to an entropy-balanced input. \nThe first of these cases plays a central role in the next section.\n\n\n\n\n\n\\section{Entropy Gain due to External Disturbances}\\label{sec:entropy_gain_output_disturb}\nIn this section we formalize the ideas which were qualitatively outlined in the previous section.\nSpecifically, \nfor the system shown in Fig.~\\ref{fig:general}\nwe will characterize the entropy gain $\\Gsp(G,\\rvex_{0},\\rvau_{1}^{\\infty},\\rvaz_{1}^{\\infty})$ defined in~\\eqref{eq:Gsp_def} for the case in which the initial state $\\rvex_{0}$ is zero (or deterministic) and there exists an  output random disturbance of (possibly infinite length) $\\rvaz_{1}^{\\infty}$ which satisfies Assumption~\\ref{assu:z}.\nThe following lemmas will be instrumental for that purpose.\n\n\n\\begin{lem}\\label{lem:singular_values_bounded}\n Let $A(z)$ be a causal, finite-order, stable and minimum-phase rational transfer function with impulse response $a_{0},a_{1},\\ldots$ such that $a_{0}=1$.\n Then \n $\\lim_{n\\to\\infty}\\lambda_{1}(\\bA_{n}\\bA^{T}_{n})>0$\n and \n $\\lim_{n\\to\\infty}\\lambda_{n}(\\bA_{n}\\bA^{T}_{n})<\\infty$.\n \\finenunciado\n\\end{lem}\n\n\n\n\n\n\\begin{lem}\\label{lem:gap_with_two_terms}\nConsider the system in Fig.~\\ref{fig:general}, and suppose $\\rvaz_{1}^{\\infty}$ satisfies Assumption~\\ref{assu:z}, and that the input process $\\rvau_{1}^{\\infty}$ is entropy balanced.\n Let $\\bG_{n}=\\bQ_{n}^{T}\\bD_{n}\\bR_{n}$ be the SVD of $\\bG_{n}$, where $\\bD_{n}=\\diag\\set{d_{n,1},\\ldots,d_{n,n}}$ are the singular values of $\\bG_{n}$, with $d_{n,1}\\leq d_{n,2}\\leq\\cdots \\leq  d_{n,n}$, such that $|\\det\\bG_{n}| = 1$ $\\forall n$. \nLet $m$ be the number of these singular values which tend to zero exponentially as $n\\to\\infty$.\nThen\n \\begin{align}\\label{eq:gap_with_two_terms}\n  \\lim_{n\\to\\infty}\\frac{1}{n}\\left(h(\\rvay_{1}^{n}) -h(\\rvau_{1}^{n})\\right) \n  =\n  \\lim_{n\\to\\infty}\\frac{1}{n}\\left( -\\Sumfromto{i=1}{m}   \\log d_{n,i} + h\\left([\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\rvez^{1}_{n} \\right) \\right).\n \\end{align}\n \\finenunciado\n\\end{lem}\n(The proof of this Lemma can be found in the Appendix, page~\\pageref{proof:lem_gap_with_two_terms}).\n\nThe previous lemma precisely formulates the geometric idea outlined in Section~\\ref{sec:geometric_interpretation}.\nTo see this, notice that no entropy gain is obtained if the output disturbance vector $\\rvez^{1}_{n}$ is orthogonal to the space spanned by the first $m$ columns of $\\bQ_{n}$.\nIf this were the case, then the disturbance would not be able fill the subspace along which $\\bG_{n}\\rveu^{1}_{n}$ is shrinking exponentially.\nIndeed, if $[\\bQ_{n}]^{1}_{n}\\rvez^{1}_{n}=0$ for all $n$, then\n$\nh([\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\rvez^{1}_{n} )\n=\nh({^{1}}\\![\\bD_{n}]_{m}[\\bR_{n}]^{1}_{m}\\rveu^{1}_{n})\n=\n\\sum_{i=1}^{m} \\log d_{n,i}+h([\\bR_{n}]^{1}_{m}\\rveu^{1}_{n})\n$,\nand the latter sum cancels out the one on the RHS of~\\eqref{eq:gap_with_two_terms}, while \n$\\lim_{n\\to\\infty}\\frac{1}{n}h([\\bR_{n}]^{1}_{n}\\rveu^{1}_{n})=0$ since $\\rvau_{1}^{\\infty}$ is entropy balanced.\nOn the contrary (and loosely speaking), if the projection of the\nsupport of $\\rvez^{1}_{n}$ onto the subspace spanned by the first $m$ rows of $\\bQ_{n}$ is of dimension $m$, then\n$h([\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\rvez^{1}_{n} )$ remains bounded for all $n$, and the entropy limit of the sum \n$\\lim_{n\\to\\infty}\\frac{1}{n}( -\\sumfromto{i=1}{m}   \\log d_{n,i})$ on the RHS of~\\eqref{eq:gap_with_two_terms} yields the largest possible entropy gain.\nNotice that \n$\n-\\sumfromto{i=1}{m}   \\log d_{n,i} \n=\n\\sumfromto{i=m+1}{n}   \\log d_{n,i}\n$ (because $\\det(\\bG_{n})=1$), \nand thus this entropy gain stems from the uncompensated expansion of $\\bG_{n}\\rveu^{1}_{n}$ along the space spanned by the rows of $[\\bQ_{n}]^{m+1}_{n}$.\n\nLemma~\\ref{lem:gap_with_two_terms} also yields the following corollary, which states that \nonly a filter $G(z)$ with zeros outside the unit circle (i.e., an NMP transfer function) can introduce entropy gain.\n\\begin{coro}[Minimum Phase Filters do not Introduce Entropy Gain]\\label{coro:MP_filters_no_EG}\nConsider the system shown in Fig.~\\ref{fig:general} and\nlet $\\rvau_{1}^{\\infty}$ be an entropy-balanced random process with bounded entropy rate.\nBesides Assumption~\\ref{assu:G_Factorized}, suppose that $G(z)$ is minimum phase.\n Then \n \\begin{align}\n\\lim_{n\\to\\infty}\\frac{1}{n}\\left(h(\\rvay_{1}^{n}) -h(\\rvau_{1}^{n})\\right) =0.  \n \\end{align}\n \\finenunciado\n\\end{coro}\n\\begin{proof}\nSince $G(z)$ is minimum phase and stable, it follows from Lemma~\\ref{lem:singular_values_bounded} that the number of singular values of $\\bG_{n}$ which go to zero exponentially, as $n\\to\\infty$, is zero.\nIndeed, all the singular values vary polynomially with $n$.\nThus $m=0$ and Lemma~\\ref{lem:gap_with_two_terms} yields directly that the entropy gain is zero (since the RHS of~\\eqref{eq:gap_with_two_terms} is zero).\n\\end{proof}\n\n\n\n\n\\subsection{Input Disturbances Do Not Produce Entropy Gain}\nIn this section we show that random disturbances satisfying Assumption~\\ref{assu:z},\nwhen added to the \\textit{input} $\\rvau_{1}^{\\infty}$ (i.e., before $G$), do not introduce entropy gain.\nThis result can be obtained from Lemma~\\ref{lem:gap_with_two_terms}, as stated in the following theorem:\n\\begin{thm}[Input Disturbances do not Introduce Entropy Gain]\nLet $G$ satisfy Assumption~\\ref{assu:G_Factorized}. \nSuppose that $\\rvau_{1}^{\\infty}$ is entropy balanced and consider the output \n \\begin{align}\n  \\rvay_{1}^{\\infty} = G\\  ( \\rvau_{1}^{\\infty} +  \\rvab_{1}^{\\infty}).\n \\end{align}\n where \n$\n \\rveb^{1}_{\\infty} = \\boldsymbol{\\Psi} \\rvea^{1}_{\\nu},\n$\nwith $\\rvea^{1}_{\\nu}$ being a random vector satisfying $h(\\rvea^{1}_{\\nu})<\\infty$, and where $\\boldsymbol{\\Psi}\\in\\Rl^{|\\Nl|\\times \\nu}$ has orthonormal columns.\nThen,\n\\begin{align}\n \\lim_{n\\to\\infty}\\frac{1}{n}\\left(h(\\rvay_{1}^{n}) -h(\\rvau_{1}^{n})\\right) =0\n\\end{align}\n\\end{thm}\n\n\\begin{proof}\n In this case, the effect of the input disturbance in the output is the forced response of $G$ to it.\n This response can be regarded as an output disturbance $\\rvaz_{1}^{\\infty} = G \\rvab_{1}^{\\infty}$. \n Thus, the argument of the differential entropy on the RHS of~\\eqref{eq:gap_with_two_terms} is \n %\n \\begin{align}\n  [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\rvez^{1}_{n}\n  &\n  =\n  [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m} \\bQ_{n}^{T}\\bD_{n}\\bR_{n}\\rveb^{1}_{n}\n\\\\&\n=\n  [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bD_{n}]^{1}_{m}\\bR_{n}\\rveb^{1}_{n}\n\\\\&\n=\n  \\block[\\bD_{n}]{1}{m} \\rows[\\bR_{n}]{1}{m}\\left( \\rveu^{1}_{n} + \\rveb^{1}_{n}\\right).\n  \\end{align}\n  %\nTherefore,\n\\begin{align}\n  h([\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\rvez^{1}_{n})\n  &\n  =\n  h(\\block[\\bD_{n}]{1}{m} \\rows[\\bR_{n}]{1}{m}\\left( \\rveu^{1}_{n} + \\rveb^{1}_{n}\\right))\n  \\\\&\n  =\n  \\sumfromto{i=1}{m}\\log d_{n,i} + h(\\rows[\\bR_{n}]{1}{m}\\left( \\rveu^{1}_{n} + [\\boldsymbol{\\Psi}]^{1}_{n}\\rvea^{1}_{\\nu}\\right) ).\n\\end{align}\nThe proof is completed by substituting this result into the RHS of~\\eqref{eq:gap_with_two_terms} and noticing that $$\\lim_{n\\to\\infty}\\frac{1}{n}h\\left([\\bR_{n}]^{1}_{m}(\\rveu^{1}_{n} +[\\boldsymbol{\\Psi}]^{1}_{n}\\rvea^{1}_{\\nu})\\right)=0.$$\n\\end{proof}\n\n\\begin{rem}\n An alternative proof for this result can be given based upon the properties of an entropy-balanced sequence, as follows.\n Since $\\det(\\bG_{n})=1,\\ \\forall n$, we have that \n $\n h(\\bG_{n}(\\rveu^{1}_{n}+\\rveb^{1}_{n}))\n= h(\\rveu^{1}_{n}+\\rveb^{1}_{n})$.\nLet $\\boldsymbol{\\Theta}_{n}\\in\\Rl^{\\nu\\times n}$ and  $\\overline{\\boldsymbol{\\Theta}}_{n}\\in\\Rl^{(n-\\nu)\\times n}$ be a matrices with orthonormal rows which satisfy  \n$\\overline{\\boldsymbol{\\Theta}}_{n}[\\boldsymbol{\\Psi}]^{1}_{n}=\\bzero$ and such that\n $[\\boldsymbol{\\Theta}_{n}^{T} \\,|\\, \\overline{\\boldsymbol{\\Theta}}_{n}^{T}]^{T}$ is a unitary matrix.\n Then\n %\n\\begin{align}\n h([\\boldsymbol{\\Theta}_{n}^{T} \\,|\\, \\overline{\\boldsymbol{\\Theta}}_{n}^{T}]^{T}\\left(\\rveu^{1}_{n}+\\rveb^{1}_{n}\\right))\n =\n h(\\boldsymbol{\\Theta}_{n} \\rveu^{1}_{n}+ \\boldsymbol{\\Theta}_{n}[\\boldsymbol{\\Psi}]^{1}_{n}\\rvea^{1}_{\\nu} \\,|\\, \\overline{\\boldsymbol{\\Theta}}_{n}\\rveu^{1}_{n})\n +\n h(\\overline{\\boldsymbol{\\Theta}}_{n}\\rveu^{1}_{n}),\n\\end{align}\nwhere we have applied the chain rule of differential entropy.\nBut\n\\begin{align}\n h(\\boldsymbol{\\Theta}_{n} \\rveu^{1}_{n}+ \\boldsymbol{\\Theta}_{n}[\\boldsymbol{\\Psi}]^{1}_{n}\\rvea^{1}_{\\nu} | \\overline{\\boldsymbol{\\Theta}}_{n}\\rveu^{1}_{n})\n \\leq  \n h(\\boldsymbol{\\Theta}_{n} \\rveu^{1}_{n}+ \\boldsymbol{\\Theta}_{n}[\\boldsymbol{\\Psi}]^{1}_{n}\\rvea^{1}_{\\nu} )\n\\end{align}\nwhich is upper bounded for all $n$ because $h(\\rvea^{1}_{n})<\\infty$ and $h(\\boldsymbol{\\Theta}_{n} \\rveu^{1}_{n})<\\infty$, the latter due to $\\rvau_{1}^{\\infty}$ being entropy balanced.\nOn the other hand, since $\\rveb^{1}_{n}$ is independent of $\\rveu^{1}_{n}$, it follows that $h(\\rveu^{1}_{n}+\\rveb^{1}_{n})\\geq h(\\rveu^{1}_{n})$, for all $n$.\nThus \n$\n\\lim_{n\\to\\infty}\\frac{1}{n}(h(\\rvey^{1}_{n}) -h(\\rveu^{1}_{n})) \n=\n\\lim_{n\\to\\infty}\\frac{1}{n}(h(\\overline{\\boldsymbol{\\Theta}}_{n}\\rveu^{1}_{n}) -h(\\rveu^{1}_{n}))\n=0$,\nwhere the last equality stems from the fact that $\\rvau_{1}^{\\infty}$ is entropy balanced.\n\\finenunciado\n\\end{rem}\n\n\n\\subsection{The Entropy Gain Introduced by Output Disturbances when $G(z)$ has NMP Zeros}\nWe show here that the entropy gain of a transfer function with zeros outside the unit circle is at most the sum of the logarithm of the magnitude of these zeros.\nTo be more precise, the following assumption is required. \n\n\\begin{assu}\\label{assu:zeros_of_G}\nThe filter $G$ satisfies Assumption~\\ref{assu:G_Factorized} and its transfer function $G(z)$\nhas $p$ poles and $p$ zeros,  $m$ of which are NMP-zeros.\nLet $M$ be the number of distinct NMP zeros, given by\n$\\{\\rho_i\\}_{i=1}^M$, i.e., such that $|\\rho_1|>|\\rho_2|>\\dots>|\\rho_{M}|>1$, with \n$\\ell_i$ being the multiplicity of the $i$-th distinct zero.\nWe denote by $\\iota(i)$, where $\\iota:\\set{1,\\ldots,m}\\to\\set{1,\\ldots,M}$, the distinct zero of $G(z)$ associated with the $i$-th non-distinct zero of $G(z)$, i.e.,\n\\begin{align}\\label{eq:iota_def}\n \\iota(k) &\\eq \n \\min\\set{\\iota  :\\sumfromto{i=1}{\\iota}\\ell_i\\geq k}.\n\\end{align}\n\\finenunciado \n\\end{assu}\n\n\nAs can be anticipated from the previous results in this section, we will need to characterize the asymptotic behaviour of the singular values of $\\bG_{n}$.\nThis is accomplished in the following lemma, which relates these singular values to the zeros of $G(z)$.\nThis result is a generalization of the unnumbered lemma in the proof of~{\\cite[Theorem~1]{hasari80}} (restated in the appendix as Lemma~\\ref{lem:hashimoto}), which holds for FIR transfer functions, to the case of \\emph{infinite-impulse response} (IIR) transfer functions (i.e., transfer functions having poles).\n\\begin{lem}\\label{lem:hashimoto_IIR}\nFor a transfer function $G$ satisfying Assumption~\\ref{assu:zeros_of_G}, it holds that\n\\begin{align}\n \\lambda_{l}(\\bG_n\\bG_n^T) \n = \n \\begin{cases}  \n \\alpha_{n,l}^{2}(\\rho_{ \\iota(l)})^{-2n}\t&, \\text{if } l\\leq m,\\\\\n \\alpha_{n,l}^{2}\t\t\t\t&, \\text{otherwise },\n \\end{cases}\n\\end{align}\nwhere the elements in the sequence $\\set{\\alpha_{n,l} }$ are positive and increase or decrease at most polynomially with $n$. \\finenunciado\n\\end{lem}\n(The proof of this lemma can be found in the appendix, page~\\pageref{proof:lem_hashimoto_IIR}).\n\n\n\n\nWe can now state the first main result of this section.\n\\begin{thm}\\label{thm:eg_n_instate_w_disturb_ineq}\nIn the system of Fig.~\\ref{fig:general}, suppose that $\\rvau_{1}^{\\infty}$ is entropy balanced and that\n$G(z)$ and $\\rvaz_{1}^{\\infty}$ satisfy assumptions~\\ref{assu:zeros_of_G}  and~\\ref{assu:z}, respectively.\nThen\n\\begin{align}~\\label{eq:eg_n_instate_w_disturb_ineq}\n0\\leq \\lim_{n\\to\\infty}\\frac{1}{n}\\left(h(\\rvay_{1}^{n}) -h(\\rvau_{1}^{n})\\right)\n \\leq \n \\Sumover{i=1}^{\\bar{\\kappa}}\\log |\\rho_{\\iota(i)}|,\n\\end{align}\nwhere\n$\\bar{\\kappa}\\eq \\min\\set{\\kappa,m}$ and\n$\\kappa$ is as defined in Assumption~\\ref{assu:z}.\nBoth  bounds are tight.\nThe upper bound is achieved if \n$\\lim_{n\\to\\infty}\\det(\n\\rows[\\bQ_{n}]{1}{\\bar{\\kappa}} \\rows[\\boldsymbol{\\Phi}]{1}{n}(\\rows[\\bQ_{n}]{1}{\\bar{\\kappa}} \\rows[\\boldsymbol{\\Phi}]{1}{n})^{T})>0$,\nwhere the unitary matrices $\\bQ_{n}^{T}\\in\\Rl^{n\\times n}$ hold the left singular vectors of $\\bG_{n}$.\n\\finenunciado\n\\end{thm}\n\\begin{proof}\nSee Appendix, page~\\pageref{proof:thm_eg_n_instate_w_disturb_ineq}. \n\\end{proof}\n\n\n\nThe second main theorem of this section is the following:\n\\begin{thm}\\label{thm:eg_n_instate_w_disturb}\nIn the system of Fig.~\\ref{fig:general}, suppose that $\\rvau_{1}^{\\infty}$ is entropy balanced and that\n$G(z)$ satisfies Assumption~\\ref{assu:zeros_of_G}.\nLet $\\rvaz_{1}^{\\infty}$ be a random output disturbance, such that $\\rvaz(i)=0,\\, \\forall i > m$, and that $|h(\\rvaz_{1}^{m})|<\\infty$.\nThen\n\\begin{align}\n \\lim_{n\\to\\infty}\n \\frac{1}{n}\n \\left(h(\\rvay_{1}^{n}) - h(\\rvau_{1}^{n}) \\right) = \\Sumfromto{i=1}{m}\\log |\\rho_{\\iota(i)}|.\n\\end{align}\n\\finenunciado\n\\end{thm}\n\n\\begin{proof} \nSee Appendix, page~\\pageref{proof:thm:eg_n_instate_w_disturb}.\n\\end{proof}\n\n\n\n\n\n\n\\section{Entropy Gain due to a Random Initial Sate}\\label{sec:entropy_gain_initial_stat}\n\nHere we analyze the case in which there exists a random initial state $\\rvex_{0}$ independent of the input $\\rvau_{1}^{\\infty}$, \nand zero (or deterministic) output disturbance.\n\nThe effect of a random initial state appears in the output as the natural response of $G$ to it, namely the sequence \n$\\bar{\\rvay}_{1}^{n}$.\nThus, $\\rvay_{1}^{n}$ can be written in vector form as\n\\begin{align}\n \\rvey^{1}_{n} = \\bG_{n}\\rveu^{1}_{n} + \\bar{\\rvey}^{1}_{n}.\n\\end{align}\nThis reveals that the effect of a random initial state can be treated as a random output disturbance, which allows us to apply the results from the previous sections.\n\n\nRecall from Assumption~\\ref{assu:zeros_of_G} that $G(z)$ is a stable and biproper rational transfer function with $m$ NMP zeros.\nAs such, it can be factored as \n \\begin{align}\\label{eq:G_Factorized_as_tilde_G_F}\n  G(z) = P(z)N(z),\n  \\end{align}\nwhere $P(z)$ is a biproper filter containing only all the poles of $G(z)$, and $N(z)$ is a FIR biproper filter, containing all the zeros of $G(z)$.\n\nWe have already established (recall Theorem~\\ref{coro:MP_filters_no_EG}) that the entropy gain introduced by the minimum phase system $P(z)$ is zero.\nIt then follows that the entropy gain can be introduced only by the NMP-zeros of $N(z)$ and an appropriate output disturbance $\\bar{\\rvay}_1^\\infty$.\nNotice that, in this case, the input process $\\rvaw_1^\\infty$ to $N$ (i.e., the output sequence of $P$ due to a random input $\\rvau_1^\\infty$) is independent of $\\bar{\\rvay}_1^\\infty$ (since we have placed the natural response $\\bar{\\rvay}_{1}^{\\infty}$ after the filters $P$ and $N$, hose initial state is now zero).\nThis condition allows us to directly use Lemma~\\ref{lem:gap_with_two_terms} in order to analyze the entropy gain that $\\rvau_1^\\infty$ experiences after being filtered by $G$, which coincides with $\\bar{h}(\\rvay_1^\\infty)-\\bar{h}(\\rvaw_1^\\infty)$.\nThis is achieved by the next theorem.\n\n\n\n\\begin{thm}\\label{thm:eg-due-to-random-xo}\n Consider a stable $p$-th order biproper filter $G(z)$ having $m$ NMP-zeros, and with a random initial state $\\rvex_0$, such that $|h(\\rvex_0)|<\\infty$. \n Then, the entropy gain due to the existence of a random initial state is \n %\n \\begin{align}\n  \\lim_{n\\to\\infty} \\frac{1}{n}( h(\\rvay_1^n) - h(\\rvau_1^n) ) = \\sumfromto{i=1}{m} \\log\\abs{\\rho_{\\iota(i)}}.\n \\end{align}\n\\end{thm}\n\n\\begin{proof}\\label{proof:thm_eg-due-to-random-xo}\nBeing a biproper and stable rational transfer function, $G(z)$ can be factorized as\n\\begin{align}\n G(z) = P(z)N(z),\n\\end{align}\nwhere $P(z)$ is a stable biproper transfer function containing only all the poles of $G(z)$ and with all its zeros at the origin,\nwhile $N(z)$ is stable and biproper FIR filter, having all the zeros of $G(z)$.\nLet $\\tilde{\\bC}_n\\rvex_0$ and $\\bC_{n}\\rvex_0$ be the natural responses of the systems $P$ and $N$ to their common random initial state $\\rvex_{0}$, respectively, where $\\tilde{\\bC}_n, \\bC_n \\in \\Rl^{n\\times p}$.\nThen we can write \n\\begin{align}\n \\rvey^{1}_{n} \n = \n \\bG_{n}\\rveu^{1}_{n} + \\bar{\\rvey}^{1}_{n}\n =\n  \\bN_{n}\\underbrace{\\bP_{n}\\rveu^{1}_{n}}_{\\eq \\rvew^{1}_{n}} + \\bar{\\rvey}^{1}_{n}\n  =\n  \\bN_{n}\\rvew^{1}_{n} + \\bar{\\rvey}^{1}_{n}.\n\\end{align}\nSince $P(z)$ is stable and MP, it follows from Corollary~\\ref{coro:MP_filters_no_EG} that \n$h(\\rvew^{1}_{n})=h(\\rveu^{1}_{n})$ for all $n$, and therefore \n\\begin{align}\n h(\\rvey^{1}_{n}) - h(\\rveu^{1}_{n}) \n = \n h(\\rvey^{1}_{n}) - h(\\rvew^{1}_{n}).\n\\end{align}\nTherefore, we only need to consider the entropy gain introduced by the (possibly) non-minimum filter $N$ due to a random output disturbance \n$\\rvez^{1}_{n} \n=\n\\bar{\\rvey}^{1}_{n}\n= \n\\bN_{n}\\tilde{\\bC}_{n}\\rvex_{0}\n+\n\\bC_{n}\\rvex_{0}\n$, \nwhich is independent of the input $\\rvew^{1}_{n}$.\nThus, the conditions of Lemma~\\ref{lem:gap_with_two_terms} are met considering $\\bG_{n}=\\bN_{n}$,\nwhere now\n$\\bN_{n} = \\bQ_{n}^{T}\\bD_{n}\\bR_{n}$ is the SVD for $\\bN_{n}$, and $d_{n,1}\\leq d_{n,2}\\leq \\cdots \\leq d_{n,n}$.\nConsequently, it suffices to consider the differential entropy on the RHS of~\\eqref{eq:gap_with_two_terms}, whose argument is\n\\begin{align}\n [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\bar{\\rvey}^{1}_{n}  \n &\n =\n [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m} \n \\left( \\bN_{n}\\tilde{\\bC}_{n}\\rvex_{0} + \\bC_{n}\\rvex_{0}\\right)\n \\\\&\n  =\n[\\bD_{n}]^{1}_{m}\\bR_{n} \\left(\\rveu^{1}_{n} + \\tilde{\\bC}_{n}\\rvex_{0}\\right)\n+[\\bQ_{n}]^{1}_{m} \\bC_{n}\\rvex_{0}\n \\\\&\n  =\n[\\bD_{n}]^{1}_{m}\\bR_{n} \\rvev^{1}_{n} \n+[\\bQ_{n}]^{1}_{m} \\bC_{n}\\rvex_{0},\\label{eq:the_term_of_v_and_x0}\n\\end{align}\nwhere $\\rvev^{1}_{n}\\eq \\rveu^{1}_{n} + \\tilde{\\bC}_{n}\\rvex_{0}$ has bounded entropy rate and is entropy balanced (since $\\tilde{\\bC}_{n}\\rvex_{0}$ is the natural response of a stable LTI system and because of Lemma~\\ref{lem:sum_yields_entropy_balanced}).\nWe remark that, in~\\eqref{eq:the_term_of_v_and_x0}, $\\rvev^{1}_{n}$ is not independent of $\\rvex_{0}$, which precludes one from using the proof of Theorem~\\ref{thm:eg_n_instate_w_disturb_ineq} directly.\n\n\nOn the other hand, since $N(z)$ is FIR of order (at most) $p$, we have that\n$\\bC_{n}= [ \\bE_{p}^T \\,|\\, \\bzero^T\\,]^T$, \nwhere $\\bE_p\\in\\Rl^{p\\times p}$ is a non-singular upper-triangular matrix independent of $n$.\nHence, $\\bC_{n}\\rvex_{0}$ can be written as $[\\boldsymbol{\\Phi}]^{1}_{n}\\rves^{1}_{p}$, where \n$[\\Phi]^{1}_{n} = [\\bI_p^T \\,|\\,\\bzero^T]^T$ \nand $\\rves^1_p \\eq \\bE_p\\rvex_0$.\nAccording to~\\eqref{eq:the_term_of_v_and_x0}, the entropy gain in~\\eqref{eq:eg_n_instate_w_disturb_ineq} arises as long as $h([\\bQ_n]^1_m\\bC_n\\rvex_0)$ is lower bounded by a finite constant (or if it decreases sub-linearly as $n$ grows).\nThen, we need $[\\bQ_n]^1_m[\\boldsymbol{\\Phi}]^1_n$ to be a full row-ranked matrix in the limit as $n\\to\\infty$.\nHowever,\n\\begin{align}\n \\det \\left([\\bQ_{n}]^{1}_{m}[\\boldsymbol{\\Phi}_{n}]^{1}_{n} ([\\bQ_{n}]^{1}_{m}[\\boldsymbol{\\Phi}_{n}]^{1}_{n})^{T}\\right)\n &=\n \\det \\left([\\bQ_{n}^{(p)}]^{1}_{m}([\\bQ_{n}^{(p)}]^{1}_{m})^{T}\\right),\n\\end{align}\nwhere $[\\bQ_n^{(p)}]^1_m$ denotes the first $p$ columns of the first $m$ rows in $\\bQ_n$.\nWe will now show that these determinants do not go to zero as $n\\to\\infty$.\nDefine the matrix $\\overline{\\bQ}_n\\in\\Rl^{m\\times (p-m)}$ such that  $[\\bQ_n^{(p)}]^1_m = [{}^1[\\bQ_n]_m \\,|\\,\\, \\overline{\\bQ}_n]$.\nThen, it holds that $\\forall\\bx\\in\\Rl^n$,\n\\begin{align}\\label{eq:lowerbound_Qp}\n \\norm{([\\bQ_n^{(p)}]^1_m)^T \\bx}^2 &= \\norm{({}^1[\\bQ_n]_m)^T\\bx}^2 + \\norm{(\\overline{\\bQ}_n)^T \\bx}^2 \\\\\n &\\geq \\norm{({}^1[\\bQ_n]_m)^T\\bx}^2 \\\\\n &\\geq \\left(\\lambda_{\\text{min}}({}^1[\\bQ_n]_m({}^1[\\bQ_n]_m)^T)\\right)^{2}.\n\\end{align}\nHence, the minimum singular value of $[\\bQ_n^{(p)}]^1_m$ is lower bounded by the smallest singular value of ${}^1[\\bQ_n]_m$, for all $n\\geq m$.\nBut it was shown in the proof of Theorem~\\ref{thm:eg_n_instate_w_disturb} (see page~\\pageref{proof:thm:eg_n_instate_w_disturb}) that\n$\\lim_{n\\to\\infty}\\lambda_{\\text{min}}({}^1[\\bQ_n]_m({}^1[\\bQ_n]_m)^T)>0$.\nUsing this result in~\\eqref{eq:lowerbound_Qp} and taking the limit, we arrive to \n\\begin{align}\n\\lim_{n\\to\\infty}\\det \\left([\\bQ_{n}^{(p)}]^{1}_{m}([\\bQ_{n}^{(p)}]^{1}_{m})^{T}\\right)\n >\n 0.\n\\end{align}\nThus\n\\begin{align\n h\\left( [\\bD_{n}]^{1}_{m}\\bR_{n}\\rveu^{1}_{n} +[\\bQ_{n}]^{1}_{m}\\bar{\\rvey}^{1}_{n} \\right) \n &\n =\n h\\left([\\bD_{n}]^{1}_{m}\\bR_{n}\\rvev^{1}_{n} + [\\bQ_{n}]^{1}_{m} [\\boldsymbol{\\Phi}]^{1}_{n}\\rves^{1}_{p}\\right)\n \\end{align}\n %\nis upper and lower bounded by a constant independent of $n$  because $\\rvav_{1}^{\\infty}$ is entropy balanced, $[\\bD_{n}]^{1}_{m}$ has decaying entries, and $h(\\rvas_1^{p})<\\infty$, which means that the entropy rate in the RHS of~\\eqref{eq:gap_with_two_terms} decays to zero.\nThe proof is finished by invoking Lemma~\\ref{lem:hashimoto_IIR}.\n\\end{proof}\n\nTheorem~\\ref{thm:eg-due-to-random-xo} allows us to formalize the effect that the presence or absence of a random initial state has on the entropy gain using arguments similar to those utilized in Section~\\ref{sec:entropy_gain_output_disturb}.\nIndeed, if the random initial state $\\rvex_0\\in\\Rl^p$ has finite differential entropy, then the entropy gain achieves~\\eqref{eq:Jensen}, since the alignment between $\\rvex_0$ and the first $m$ rows of $\\bQ_n$ is guaranteed.\nThis motivates us to characterize the behavior of the entropy gain (due only to a random initial state), when the initial state $\\rvex_0$ can be written as $[\\boldsymbol{\\Phi}]^1_p\\rves^1_\\tau$, with $\\tau\\leq p$, which means that $\\rvex_0$ has an undefined (or $-\\infty$) differential entropy.\n\n\\begin{coro}\\label{coro:eg_due_to_xo_ineq}\n Consider an FIR, $p$-order filter $F(z)$ having $m$ NMP-zeros, such that its random initial state can be written as $\\rvex_0 = \\boldsymbol{\\Phi} \\rves^1_\\tau$, \n where $|h(\\rvas_1^\\tau)|<\\infty$ and $\\boldsymbol{\\Phi}\\in\\Rl^{p\\times \\tau}$ contains orthonormal rows .\n Then,\n  \\begin{align}\n  \\lim_{n\\to\\infty} ( h(\\rvay_1^n) - h(\\rvau_1^n) ) \\leq \\Sumfromto{i=1}{\\bar{\\tau}} \\log\\abs{\\rho_{\\iota(i)}}, \\label{eq:eg_due_to_xo_ineq}\n \\end{align}\n where $\\set{\\bar{\\tau}} \\eq \\min\\set{m,\\tau}$.\n The upper bound in~\\eqref{eq:eg_due_to_xo_ineq} is achieved when $[\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi}([\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi})^T$ is a non-singular matrix, with $\\bC_n$ defined by $\\bar{\\rvey}^1_n = \\bC_n\\rvex_0$ (as in Theorem~\\ref{thm:eg-due-to-random-xo}).\n\\end{coro}\n\n\\begin{proof}\nThe effect of the random initial state to the output sequence $\\rvay_1^\\infty$ can be written as $\\rvey^1_n = \\bC_n\\rvex_0$, where $\\bC_n = [\\bE_p^T \\,|\\, \\bzero^T]^T \\in \\Rl^{n\\times p}$.\nTherefore, if $\\bQ^T_n\\bD_n\\bR_n$ is an SVD for $\\bF_n$, it holds that \n\\begin{align}\nh([\\bD_n]^1_n\\bR_n\\rveu^1_n + [\\bQ_n]^1_m \\bC_n\\boldsymbol{\\Phi}\\rves^1_\\tau) \\label{eq:car}\n\\end{align}\nremains bounded, for $n\\to\\infty$, if and only if $\\lim_{n\\to\\infty}\\det([\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi}([\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi})^T)>0$.\n\nDefine the rank of $[\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi}$ as $\\tau_n\\in\\set{1,\\ldots,\\bar{\\tau}}$.\nIf $\\det([\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi}([\\bQ_n]^1_m\\bC_n\\boldsymbol{\\Phi})^T)=0$, then the lower bound is reached by inserting~\\eqref{eq:car} in~\\eqref{eq:gap_with_two_terms}.\nOtherwise, there exists $L$ large enough such that $\\tau_n \\geq 1$, $\\forall n\\geq L$.\n\nWe then proceed as the proof of Theorem~\\ref{thm:eg_n_instate_w_disturb_ineq}, by considering a unitary $(m\\times m)$-matrix $\\bH_n$, and a $(\\tau_n\\times m)$-matrix $\\bA_n$ such that\n\\begin{align}\n \\bH_{n}[\\bQ_{n}]^{1}_{m} \\bC_n\\boldsymbol{\\Phi}\n =\n \\begin{pmatrix}\n  \\bA_{n}[\\bQ_{n}]^{1}_{m} \\bC_n\\boldsymbol{\\Phi}\n  \\\\\n  \\bzero\n \\end{pmatrix}, \\fspace n\\geq L.\n\\end{align}\n\nThis procedure allows us to conclude that $h([\\bD_n]^1_n\\bR_n\\rveu^1_n + [\\bQ_n]^1_m \\bC_n\\boldsymbol{\\Phi}\\rves^1_\\tau) \\leq \\sumfromto{i=\\tau_n + 1}{m}\\log d_{n,i}$, and that the lower limit in the latter sum equals $\\bar{\\tau}+1$ when $[\\bQ_n]^1_m \\bC_n\\boldsymbol{\\Phi}\\rves^1_\\tau$ is a full row-rank matrix. \nReplacing the latter into~\\eqref{eq:gap_with_two_terms} finishes the proof.\n\\end{proof}\n\n\\begin{rem}\n If the random initial state $\\rvex_0 = \\boldsymbol{\\Phi}\\rves^1_\\tau$ is generated with $\\tau\\geq p-m$, then the entropy gain introduced by an FIR minimum phase filter $F$ is at least $\\log \\rho_1$.\n Otherwise, the entropy gain could be identically zero, as long as the columns of $\\bE_n\\boldsymbol{\\Phi}(\\bE_n\\boldsymbol{\\Phi})^T$ fill only the orthogonal space to the span of the row vectors in $[\\bQ_n^{(p)}]^1_m$, where $\\bE_n$, $\\boldsymbol{\\Phi}$ and $[\\bQ_n^{(p)}]^1_m$ are defined as in the proof of Theorem~\\ref{thm:eg-due-to-random-xo}.\n\\end{rem}\n\nBoth results, Theorem~\\ref{thm:eg-due-to-random-xo} and Corollary~\\ref{coro:eg_due_to_xo_ineq}, reveal that the entropy gain arises as long as the effect of the random initial state aligns with the first rows of $\\bQ_n$, just as in the results of the previous section.\n\n\n\\section{Effective Entropy Gain due to the Intrinsic Properties of the Filter}\\label{sec:effective_entropy}\nIf there are no disturbances and the initial state is zero, then the first $n$ output samples to an input $\\rvau_{1}^{n}$ is given by~\\eqref{eq:y_of_u_matrix}.\nTherefore, the entropy gain in this case, as defined in~\\eqref{eq:Gsp_def}, is zero, regardless of whether or not $G$ is NMP.\n\nDespite the above, there is an interesting question which, to the best of the authors' knowledge, has not been addressed before:\nSince in any LTI filter the entire output is longer than the input, what would happen if one compared the differential entropies of the complete output sequence to that of the (shorter) input sequence?\nAs we show next, a proper definition of this question requires recasting the problem in terms of a new definition of differential entropy.\nAfter providing a geometrical interpretation of this problem, we prove that the (new) entropy gain in this case is exactly~\\eqref{eq:Jensen}.\n\n\\subsection{Geometrical Interpretation} \nConsider the random vectors \n$\\rveu\\eq [\\rvau_1\\ \\rvau_2]^{T}$ \nand \n$\\rvey\\eq [\\rvay_1\\ \\rvay_2 \\ \\rvay_3]^{T}$  \nrelated via\n\\begin{align}\\label{eq:little_example}\n \\begin{bmatrix}\n  \\rvay_1\\\\\n  \\rvay_2\\\\\n  \\rvay_3\n \\end{bmatrix}\n=\n\\underbrace{\\begin{pmatrix}\n 1 & 0\\\\\n 2 & 1\\\\\n 0 & 2\n\\end{pmatrix}}\n_{\\eq\\breve{\\bG}_{2}}\n \\begin{bmatrix}\n  \\rvau_1\\\\\n  \\rvau_2\n \\end{bmatrix}.\n\\end{align}\nSuppose $\\rveu$ is uniformly distributed over $[0 ,1]\\times[0,1]$.\nApplying the conventional definition of differential entropy of a random sequence, we would have that\n\\begin{align}\n h(\\rvay_1,\\rvay_2,\\rvay_3)\n &\n =\n h(\\rvay_1,\\rvay_2) + h(\\rvay_3|\\rvay_1,\\rvay_2)\n =\n -\\infty,\n\\end{align}\nbecause $\\rvay_3$ is a deterministic function of $\\rvay_1$ and $\\rvay_2$:\n$$\n\\rvay_3 = [0\\;\\; 2][\\rvau_1 \\; \\rvau_2]^{T}=\n[0\\;\\; 2]\n\\begin{pmatrix}\n 1 & 0\\\\\n 2 & 1\\\\\n\\end{pmatrix}^{-1} \n \\begin{bmatrix}\n  \\rvay_1\\\\\n  \\rvay_2\n \\end{bmatrix}.\n$$\nIn other words, the problem lies in that although the output is a  three dimensional vector, it only has two degrees of freedom, i.e., it is restricted to a 2-dimensional subspace of $\\Rl^{3}$.\nThis is illustrated in Fig.~\\ref{fig:square_shear}, where the set $[0,1]\\times[0,1]$ is shown (coinciding with the \\texttt{u}-\\texttt{v} plane), together with its image through $\\breve{\\bG}_{2}$ (as defined in~\\eqref{eq:little_example}).\n\\begin{figure}[htbp]\n \\centering\n \\includegraphics[width = 8 cm]{square_shear.eps}\n \\caption{Support of $\\rveu$ (laying in the \\texttt{u}-\\texttt{v} plane) compared to that of $\\rvey=\\breve{\\bG}\\rveu$ (the rhombus in $\\Rl^3$).}\\label{fig:square_shear}\n\\end{figure}\n\nAs can be seen in this figure, the image of the square $[0,1]^{2}$ through $\\breve{\\bG}_{2}$ is a $2$-dimensional rhombus over which \n$\\set{\\rvay_1,\\rvay_2,\\rvay_3}$ distributes uniformly.\nSince the intuitive notion of differential entropy of an ensemble of random variables (such as how difficult it is to compress it in a lossy fashion) relates to the size of the region spanned by the associated random vector, one could argue that the differential entropy of $\\set{\\rvay_1,\\rvay_2,\\rvay_3}$, far from being $-\\infty$, should be somewhat larger than that of $\\set{\\rvau_1,\\rvau_2}$ (since the rhombus $\\breve{\\bG}_{2}[0,1]^{2}$ has a larger area than $[0,1]^{2}$). \nSo, what does it mean that (and why should) $h(\\rvay_1,\\rvay_2,\\rvay_3)=-\\infty$?\nSimply put, the differential entropy relates to the volume spanned by the support of the probability density function.\nFor $\\rvey$ in our example, the latter (three-dimensional) volume is clearly zero.\n\nFrom the above discussion, the comparison between the differential entropies of $\\rvey\\in\\Rl^{3}$ and $\\rveu\\in\\Rl^{2}$ of our previous example should take into account that $\\rvey$ actually lives in a two-dimensional subspace of $\\Rl^{3}$.\nIndeed, since the multiplication by a unitary matrix does not alter differential entropies, we could consider the differential entropy of \n\\begin{align}\n\\begin{bmatrix}\n\\tilde{\\rvey}\n\\\\\n0\n\\end{bmatrix}\n \\eq \n \\begin{pmatrix}\n \\breve{\\bQ} \n \\\\\n \\bar{\\bq}^{T}\n \\end{pmatrix}\n  \\rvey,  \n\\end{align}\nwhere $\\breve{\\bQ}^T$ is the $3\\times2$ matrix with orthonormal rows in the singular-value decomposition of $\\breve{\\bG}_{2}$\n\\begin{align}\n \\breve{\\bG}_{2} = \n  \\breve{\\bQ}^T \\breve{\\bD}\\, \\breve{\\bR}.\n\\end{align}\nand $\\bar{\\bq}$ is a unit-norm vector orthogonal to the rows of $\\breve{\\bQ}$ (and thus orthogonal to $\\rvey$ as well).\nWe are now able to compute the differential entropy in $\\Rl^2$ for $\\tilde{\\rvey}$, corresponding to the rotated version of $\\rvey$ such that its support is now aligned with $\\Rl^2$.\n\nThe preceding discussion motivates the use of a modified version of the notion of differential entropy for a random vector $\\rvey\\in\\Rl^{n}$ which considers the number of dimensions actually spanned by $\\rvey$ instead of its length.\n\n\\begin{defn}[The Effective Differential Entropy]\\label{def:BMD_entropy}\nLet $\\rvey\\in\\Rl^\\ell$ be a random vector. \nIf $\\rvey$ can be written as a linear transformation $\\rvey = \\bS \\rveu$, for some $\\rveu\\in\\Rl^n$ ($ n\\leq \\ell$), $\\bS \\in \\Rl^{\\ell\\times n}$, then the effective differential entropy of $\\rvey$ is defined as\n\\begin{align}\n \\breve{h}(\\rvey) \\eq h(\\bA\\rvey),\n\\end{align}\nwhere $\\bS = \\bA^{T}\\bT\\bC$ is an SVD for $\\bS$, with $\\bT\\in\\Rl^{n\\times n}$.\n\\finenunciado\n\\end{defn}\nIt is worth mentioning that Shannon's differential entropy\nof a vector $\\rvey\\in\\Rl^{\\ell}$, whose support's $\\ell$-volume is greater than zero, arises from considering it as the difference between its (absolute) entropy and that of a random variable uniformly distributed over an $\\ell$-dimensional, unit-volume region of $\\Rl^{\\ell}$.\nMore precisely, if in this case the \\textit{probability density function} (PDF) of $\\rvey = [\\rvay_{1}\\;\\rvay_{2}\\; \\cdots \\; \\rvay_{\\ell}]^{T}$ is Riemann integrable, then~\\cite[Thm.~9.3.1]{covtho91},\n\\begin{align}\\label{eq:h_as_limit_covtho06}\n h(\\rvey) = \\lim_{\\Delta\\to 0} \\left[H(\\rvey^{\\Delta}) + \\ell\\log\\Delta\\right], \n\\end{align}\nwhere $\\rvey^{\\Delta}$ is the discrete-valued random vector resulting when $\\rvey$ is quantized using an $\\ell$-dimensional uniform quantizer with $\\ell$-cubic quantization cells with volume $\\Delta^{\\ell}$.\nHowever, if we consider a variable $\\rvey$ whose support belongs to an $n$-dimensional subspace of \n$\\Rl^\\ell$, $n < \\ell$ (i.e., $\\rvey=\\bS\\rveu=\\bA^{T}\\bT\\bC\\rveu$, as in Definition~\\ref{def:BMD_entropy}), \nthen the entropy of its quantized version in $\\Rl^\\ell$, say \n$H_\\ell(\\rvey^{\\Delta})$, is distinct from \n$H_n((\\bA\\rvey)^{\\Delta})$,\nthe entropy of \n$\\bA\\rvey$ in \n$\\Rl^n$.\nMoreover, it turns out that, in general, \n\\begin{align}\\label{eq:unequal}\n \\lim_{\\Delta\\to 0}\\left(H_\\ell(\\rvey^{\\Delta}) -H_n( (\\bA\\rvey)^{\\Delta}) \\right) \\neq 0,\n\\end{align}\ndespite the fact that $\\bA$ has orthonormal rows.\nThus, the definition given by~\\eqref{eq:h_as_limit_covtho06} does not yield consistent results for the case wherein a random vector has a support's dimension (i.e., its number of degrees of freedom) smaller that its length\\footnote{The mentioned inconsistency refers to~\\eqref{eq:unequal}, which reveals that the asymptotic behavior $H_\\ell(\\rvey^{\\Delta})$ changes if $\\rvey$ is rotated.} \n(If this were not the case, then we could redefine~\\eqref{eq:h_as_limit_covtho06} replacing $\\ell$ by $n$, in a spirit similar to the one behind Renyi's $d$-dimensional entropy~\\cite{renyi-59}.)\nTo see this, consider the case in which $\\rveu\\in\\Rl$ distributes uniformly over $[0,1]$ and \n$\\rvey = [1 \\quad 1]^{T}\\rveu\/\\hsqrt{2}$.\nClearly, $\\rvey$ distributes uniformly over the unit-length segment connecting the origin with the point $(1,1)\/\\hsqrt{2}$.\nThen \n\\begin{align}\n H_{2}(\\rvey^{\\Delta}) \n = \n -\n \\left\\lfloor \\tfrac{1}{\\Delta \\hsqrt{2}}  \\right\\rfloor \n  \\Delta\\hsqrt{2}\n  \\log \\left( \\Delta\\hsqrt{2}   \\right)\n  - \n  \\left(1- \\left\\lfloor \\tfrac{1}{\\Delta \\hsqrt{2}}  \\right \\rfloor \\hsqrt{2}\\Delta \\right)\n  \\log   \\left(1- \\left\\lfloor \\tfrac{1}{\\Delta \\hsqrt{2}}  \\right \\rfloor \\hsqrt{2}\\Delta \\right).\n\\end{align}\nOn the other hand, since in this case $\\bA\\rvey=\\rveu$, we have that\n\\begin{align}\n H_{1}((\\bA\\rvey)^{\\Delta})\n =\n H_{1}(\\rveu^{\\Delta})\n =\n -\n \\left\\lfloor \\tfrac{1}{\\Delta }  \\right\\rfloor \n \\Delta \\log\\Delta \n -\n (1-\\left\\lfloor \\tfrac{1}{\\Delta }  \\right\\rfloor \\Delta)\n \\log (1-\\left\\lfloor \\tfrac{1}{\\Delta }  \\right\\rfloor \\Delta).\n\\end{align}\nThus \n\\begin{align}\n \\lim_{\\Delta\\to 0}\n &\n \\left(\n H_{1}((\\bA\\rvey)^{\\Delta})\n -\n H_{2}(\\rvey^{\\Delta}) \n  \\right)\n  =\n  \\lim_{\\Delta \\to 0}\n  \\left( \n  \\left\\lfloor \\tfrac{1}{\\Delta \\hsqrt{2}}  \\right\\rfloor \n  \\Delta\\hsqrt{2}\n  \\log \\left( \\Delta\\hsqrt{2}   \\right)\n  -\n \\left\\lfloor \\tfrac{1}{\\Delta }  \\right\\rfloor \n \\Delta \\log\\Delta \n  \\right)\n  =\\log\\hsqrt{2}.\n\\end{align}\n\n\nThe latter example further illustrates why the notion of effective entropy is appropriate in the setup considered in this section, where\nthe effective dimension of the random sequences does not coincide with their length \n(it is easy to verify that the effective entropy of $\\rvey$ does not change if one rotates $\\rvey$ in $\\Rl^{\\ell}$).\nIndeed, we will need to consider only sequences which can be constructed by multiplying some random vector $\\rveu\\in\\Rl^{n}$, with bounded differential entropy, by a tall matrix $\\breve{\\bG}_n\\in\\Rl^{n\\times (n+\\eta)}$, with $\\eta>0$ (as in~\\eqref{eq:little_example}), which are precisely the conditions required by Definition~\\ref{def:BMD_entropy}. \n\n\\subsection{Effective Entropy Gain}\nWe can now state the main result of this section:\n\\begin{thm}\\label{thm:BMD_entropy_gain}\n Let the entropy-balanced random sequence $\\rvau_{1}^{\\infty}$ be the input of an LTI filter $G$, and let $\\rvay_{1}^{\\infty}$ be its output.\n Assume that $G(z)$ is the $z$-transform of the $(\\eta+1)$-length sequence $\\set{g_k}_{k=0}^{\\eta}$.\n Then \n \\begin{align}\n  \\lim_{n\\to \\infty} \\frac{1}{n}\\left(\\breve{h}(\\rvay_{1}^{n+\\eta}) -\\breve{h}(\\rvau_{1}^{n}) \\right)\n  =\\intpipi{\\log\\abs{G\\ejw} }.\n \\end{align}\n \\finenunciado\n\\end{thm}\n\nTheorem~\\ref{thm:BMD_entropy_gain} states that, when considering the full-length output of a filter, the effective entropy gain is introduced by the filter itself, without  requiring the presence of external random disturbances or initial states.\nThis may seem a surprising result, in view of the findings made in the previous sections, where the entropy gain appeared only when such random exogenous signals were present. \nIn other words, when observing the full-length output and the input, the (maximum) entropy gain of a filter can be recasted in terms of the ``volume'' expansion yielded by the filter as a linear operator, provided we measure effective differential entropies instead of Shannon's differential entropy.\n\n\n\\begin{proof}[Proof of Theorem~\\ref{thm:BMD_entropy_gain}]\nThe total length of the output $\\ell$, will grow with the length $n$ of the input, if $G$ is FIR, and will be infinite, if $G$ is IIR.\nThus, we define the \\textit{output-length function}\n\\begin{align}\n \\ell(n) \\eq \\text{length of $\\rvey$ when input is $\\rveu^{1}_{n}$}\n =\n \\begin{cases}\n  n+\\eta & , \\text{ if $G$ is FIR with i.r. length $\\eta+1$,}\\\\\n  \\infty& , \\text{ if $G$ is IIR.}\n \\end{cases}\n\\end{align}\nIt is also convenient to define the sequence of matrices $\\set{\\breve{\\bG}_{n}}_{n=1}^{\\infty}$, where\n$\\breve{\\bG}_{n}\\in\\Rl^{\\ell(n)\\times n}$ is Toeplitz with \n$\\left[\\breve{\\bG}_{n}\\right]_{i,j}=0,\\forall i<j$,\n$\\left[\\breve{\\bG}_{n}\\right]_{i,j}=g_{i-j},\\forall i\\geq j$.\nThis allows one to write\nthe \\emph{entire} output $\\rvay_1^{\\ell}$ of a causal LTI filter $G$ with impulse response $\\set{g_{k}}_{k=0}^{\\eta}$ to an input $\\rvau_1^\\infty$ as \n\\begin{align}\\label{eq:yFu_tall}\n\\rvey^{1}_{\\ell(n)} \n= \n\\breve{\\bG}_{n}\n\\rveu^{1}_{n}.\n\\end{align}\nLet the SVD of $\\breve{\\bG}_{n}$ be\n$\\breve{\\bG}_{n} = \\breve{\\bQ}^T_n\\breve{\\bD}_n\\breve{\\bR}_n$, \nwhere $\\breve{\\bQ}_n\\in\\Rl^{n\\times \\ell(n)}$ has orthonormal rows,\n$\\breve{\\bD}_n\\in\\Rl^{n\\times n}$ is diagonal with positive elements, \nand \n$\\breve{\\bR}_n\\in\\Rl^{n\\times n}$ is unitary.\n\nThe effective differential entropy of $\\rvay_1^{n(\\ell)}$ exceeds the one of $\\rvau_1^{n}$ by \n\\begin{align}\\label{eq:hybreve_of_logdetD}\n\\breve{h}(\\rvey^1_{n(\\ell)}) - \\breve{h}(\\rveu^1_{n}) &= h(\\breve{\\bQ}_n \\breve{\\bG}_n\\rveu^1_{n}) - h(\\rveu^1_{n}) \\\\\n&= h(\\breve{\\bD}_n\\breve{\\bR}_n\\rveu^1_n) - h(\\rveu^1_n) \\\\\n&= \\log \\det \\breve{\\bD}_n,\n\\end{align}\nwhere the first equality follows from the fact that $\\rveu^1_n$ can be written as $\\bI_n\\rveu^1_n$, which means that \n$\\breve{h}(\\rveu^1_n) = h(\\rveu^1_n)$.\nBut\n\\begin{align}\n \\breve{\\bG}_{n}^{T}\\breve{\\bG}_{n} \n = (\\breve{\\bQ}^T_n\\breve{\\bD}_n\\breve{\\bR}_n)^{T}(\\breve{\\bQ}^T_n\\breve{\\bD}_n\\breve{\\bR}_n)\n = \\breve{\\bR}_n^T\\breve{\\bD}_n\\breve{\\bQ}_n\\breve{\\bQ}_n^T\\breve{\\bD}_n\\breve{\\bR}_n\n = \\breve{\\bR}_n^T\\breve{\\bD}_n^2\\breve{\\bR}_n. \\label{eq:casicasi}\n\\end{align}\nSince $\\breve{\\bR}_n$ is unitary, it follows that \n$\\det\\breve{\\bD}_n^{2}= \\det\\breve{\\bG}_{n}^{T}\\breve{\\bG}_{n}$\n, which means that  \n$\\det\\breve{\\bD}_n = \\frac{1}{2}\\det\\breve{\\bG}_{n}^{T}\\breve{\\bG}_{n}$.\nThe product \n$\\bH_{n} \\eq  \\breve{\\bG}_{n}^{T} \\breve{\\bG}_{n}$\nis a symmetric Toeplitz matrix, with its first column,  $[h_{0}\\, h_{1}\\, \\cdots \\,h_{n-1}]^{T}$,  given by\n\\begin{align}\n h_{i} & =  \\Sumfromto{k=0}{n}g_{k}g_{k-i}.\n\\end{align}\nThus, the sequence $\\set{h_i}_{i=0}^{n-1}$ corresponds to the samples $0$ to $n-1$ of those resulting from the complete convolution $g*g^{-}$, even when the filter $G$ is IIR, where $g^{-}$ denotes the time-reversed (perhaps infinitely large) response $g$.\nConsequently, using the Grenander and Szeg{\\\"o}'s theorem~\\cite{gresze58}, it holds that\n\\begin{align}\\label{eq:GSz}\n\\lim_{n\\to\\infty} \\log\\left( \\det(\\breve{\\bG}^T_{n}\\breve{\\bG}_n)^{1\/n}\\right) = \\Intpipi{\\log \\abs{G\\ejw} },\n\\end{align}\nwhere $G\\ejw$ is the discrete-time Fourier transform of $\\set{g_{k}}_{k=0}^{\\ell}$.\n\nIn order to finish the proof, we divide~\\eqref{eq:hybreve_of_logdetD} by $n$, take the limit as $n\\to\\infty$, and replace~\\eqref{eq:GSz} in the latter.\n\\end{proof}\n\n\n\n\n\n\\section{Some Implications}\\label{sec:implications}\n\\subsection{Rate Distortion Function for Non-Stationary Processes}\nIn this section we obtain a simpler proof of a result by \nGray, \nHashimoto and Arimoto~\\cite{gray--70,hasari80,grahas08},\nwhich compares the rate distortion function (RDF) of \na non-stationary auto-regressive Gaussian process $\\rvax_{1}^{\\infty}$ (of a certain class) to that of a corresponding stationary version, under MSE distortion.\nOur proof is based upon the ideas developed in the previous sections, and extends the class of non-stationary sources for which the results in~\\cite{gray--70,hasari80,grahas08} are valid.\n\nTo be more precise,\nlet\n$\\set{a_i}_{i=1}^{\\infty}$ \nand \n$\\set{\\tilde{a}_i}_{i=1}^{\\infty}$ \nbe the impulse responses of two linear time-invariant filters $A$ and $\\tilde{A}$ with rational transfer functions\n\\begin{align}\n A(z) &= \\frac{z^{M}}{\\prod_{i=1}^{M}(z-p_{i})}\\\\\n \\tilde{A}(z) &= \\frac{z^{M}}{\\prod_{i=1}^{M}|p_{i}^{*}|(z-1\/p_{i}^{*})},\n\\end{align}\nwhere $\\abs{p_{i}}>1$, $\\forall i=1,\\ldots, M$.\nFrom these definitions it is clear that $A(z)$ is unstable, $\\tilde{A}(z)$ is stable, and\n\\begin{align}\n |A\\ejw| = |\\tilde{A}\\ejw|, \\fspace \\forallwinpipi.\n\\end{align}\nNotice also that \n$\\lim_{\\abs{z}\\to\\infty}A(z)=1$\nand\n$\\lim_{\\abs{z}\\to\\infty}\\tilde{A}(z)=1\/\\prod_{i=1}^{M}|p_{i}|$, and thus \n\\begin{align}\n a_{0}=1, && \\tilde{a}_{0}= \\prod_{i=1}^{M}|p_{i}|^{-1}.\n\\end{align}\n\nConsider the non-stationary random sequences (source) \n$\\rvax_{1}^{\\infty}$ and\nthe asymptotically stationary source\n$\\tilde{\\rvax}_{1}^{\\infty}$\ngenerated by passing  a stationary Gaussian process $\\rvaw_{1}^{\\infty}$ through $A(z)$ and $\\tilde{A}(z)$, respectively, which can be written as\n\\begin{align}\n \\rvex^{1}_{n} &= \\bA_{n}\\rvew^{n}_{1}, \\fspace n=1,\\ldots,\\label{eq:x_def}\\\\\n \\tilde{\\rvex}^{1}_{n} &= \\tilde{\\bA}_{n}\\rvew^{n}_{1}, \\fspace n=1,\\ldots.\\label{eq:tildex_def}\n\\end{align}\n(A block-diagram associated with the construction of $\\rvax$ is presented in Fig.~\\ref{fig:rdfns}.)\n\\begin{figure}[t]\n\\centering\n\\input{rdfns.pstex_t}\n\\caption{Block diagram representation of how the non-stationary source $\\rvax_{1}^{\\infty}$ is built and then reconstructed as $\\rvay=\\rvax+\\rvau$.}\n\\label{fig:rdfns}\n\\end{figure}\nDefine the rate-distortion functions for these two sources  as \n\\begin{align}\nR_{\\rvax}(D) &\\eq  \\lim_{n\\to\\infty}  R_{\\rvax,n}(D),\n&\nR_{\\rvax,n}(D)&\\eq \\min \\frac{1}{n}I(\\rvax_{1}^{n};\\rvax_{1}^{n}+\\rvau_{1}^{n}),\n\\\\\nR_{\\tilde{\\rvax}}(D) &\\eq  \\lim_{n\\to\\infty}  R_{\\tilde{\\rvax},n}(D),\n&\nR_{\\tilde{\\rvax},n}(D)&\\eq \\min \\frac{1}{n}I(\\tilde{\\rvax}_{1}^{n};\\tilde{\\rvax}_{1}^{n}+\\tilde{\\rvau}_{1}^{n}),\n\\end{align}\nwhere, for each $n$, the minimums are taken over all the conditional probability density functions \n$f_{\\rvau_{1}^{n}|\\rvax_{1}^{n}}$ \nand\n$f_{\\tilde{\\rvau}_{1}^{n}|\\tilde{\\rvax}_{1}^{n}}$ \nyielding \n $\\Expe{\\norm{\\rveu^{1}_{n}}^{2}}\/n\\leq D$ and\n $\\Expe{\\norm{\\tilde{\\rveu}^{1}_{n}}^{2}}\/n\\leq D$,\nrespectively.\n\nThe above rate-distortion functions have been characterized in~\\cite{gray--70,hasari80,grahas08} for the case in which $\\rvaw_{1}^{\\infty}$ is an i.i.d. Gaussian process.\nIn particular, it is explicitly stated in~\\cite{hasari80,grahas08} that, for that case, \n\\begin{align}\n R_{\\rvax}(D)- \n R_{\\tilde{\\rvax}}(D)\n =\n \\intpipi{\\log|A^{-1}\\ejw|}\n =\n \\sumfromto{i=1}{M}\\log|p_{i}|.\\label{eq:gap}\n\\end{align}\nWe will next provide an alternative and simpler proof of this result, and extend its validity for general (not-necessarily stationary) Gaussian $\\rvaw_{1}^{\\infty}$, using the entropy gain properties of non-minimum phase filters established in Section~\\ref{sec:entropy_gain_output_disturb}.\nIndeed, the approach in~\\cite{gray--70,hasari80,grahas08} is based upon asymptotically-equivalent Toeplitz matrices in terms of the signals' covariance matrices.\nThis restricts $\\rvaw_{1}^{\\infty}$ to be Gaussian and i.i.d. and $A(z)$ to be an all-pole unstable transfer function, and then, the only non-stationary allowed is that arising from unstable poles.\nFor instance, a cyclo-stationarity innovation followed by an unstable filter $A(z)$ would yield a source which cannot be treated using Gray and Hashimoto's approach.\nBy contrast, the reasoning behind our proof lets $\\rvaw_1^\\infty$ be any Gaussian process, and then let the source be $A\\rvaw$, with $A(z)$ having unstable poles (and possibly zeros and stable poles as well).\n\nThe statement is as follows:\n\\begin{thm}\\label{thm:RDF_non_stat}\n Let $\\rvaw_{1}^{\\infty}$ be any Gaussian stationary process with bounded differential entropy rate, and let \n $\\rvax_{1}^{\\infty}$ and $\\tilde{\\rvax}_{1}^{\\infty}$ be as defined in~\\eqref{eq:x_def} and~\\eqref{eq:tildex_def}, respectively.\n Then~\\eqref{eq:gap} holds.\n \\finenunciado\n\\end{thm}\n\nThanks to the ideas developed in the previous sections, it is possible to give an intuitive outline of the proof of this theorem (given in the appendix, page~\\pageref{proof:RDF_non_stat}) by using a sequence of block diagrams.\nMore precisely, consider the diagrams shown in Fig.~\\ref{fig:rdfnsproof}.\n\\begin{figure}[t]\n\\centering\n\\input{rdfns2.pstex_t}\n\\caption{Block-diagram representation of the changes of variables in the proof of Theorem~\\ref{thm:RDF_non_stat}.}\n\\label{fig:rdfnsproof}\n\\end{figure}\nIn the top diagram in this figure, suppose that $\\rvay=C\\rvax+\\rvau$ realizes the RDF for the non-stationary \nsource $\\rvax$.\nThe sequence $\\rvau$ is independent of $\\rvex$, and the linear filter $C(z)$ is such that the error $(\\rvay-\\rvax)\\Perp \\rvay$ \n(a necessary condition for minimum MSE optimality).\nThe filter $B(z)$ is the Blaschke product of $A(z)$ (see~\\eqref{eq:B_def} in the appendix) (a stable, NMP filter with unit frequency response magnitude such that \n$\\tilde{\\rvax} = B \\rvax$).\n\nIf one now moves the filter $B(z)$ towards the source, then the middle diagram in Fig.~\\ref{fig:rdfnsproof} is obtained.\nBy doing this, the stationary source $\\tilde{\\rvax}$ appears with an additive error signal \n$\\tilde{\\rvau}$ that has the same asymptotic variance as $\\rvau$, reconstructed as \n$\\tilde{\\rvay}=C\\tilde{\\rvax} + \\tilde{\\rvau}$.\nFrom the invertibility of $B(z)$, it also follows that the mutual information rate between \n$\\tilde{\\rvax}$ and $\\tilde{\\rvay}$\nequals that between \n$\\rvax$ and $\\rvay$.\nThus, the channel \n$\\tilde{\\rvay}=C\\tilde{\\rvax}+\\tilde{\\rvau}$ \nhas the same rate and distortion as the channel\n$\\rvay =C\\rvax+\\rvau$. \n\n\nHowever, if one now adds a short disturbance $\\rvad$ to the error signal $\\tilde{\\rvau}$ \n(as depicted in the bottom diagram of Fig.~\\ref{fig:rdfnsproof}),\nthen the resulting additive error term $\\bar{\\rvau}=\\tilde{\\rvau}+\\rvad$ will be independent of $\\tilde{\\rvax}$ and\nwill have the same asymptotic variance as $\\tilde{\\rvau}$.\nHowever, the differential entropy rate of $\\bar{\\rvau}$ will exceed that of $\\tilde{\\rvau}$ by the RHS of~\\eqref{eq:gap}.\nThis will make the mutual information rate between \n$\\tilde{\\rvax}$ and $\\bar{\\rvay}$ to be less than that between \n$\\tilde{\\rvax}$ and $\\tilde{\\rvay}$ by the same amount.\nHence, \n$R_{\\tilde{\\rvex}}(D)$ be at most \n$R_{\\rvex}(D) - \\sumfromto{i=1}{M}\\log\\abs{p_{i}}$.\nA similar reasoning can be followed to prove that \n$R_{\\rvex}(D)-R_{\\tilde{\\rvex}}(D)\\leq  \\sumfromto{i=1}{M}\\log\\abs{p_{i}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{Networked Control}\nHere we revisit the setup shown in Fig.~\\ref{fig:fbksystem} and discussed in Section~\\ref{sec:Intro}.\nRecall  from~\\eqref{eq:martins_I_bound} that, for this general class of networked control systems, it was shown in~\\cite[Lemma 3.2]{mardah05} that\n\\begin{align}\n \\lim_{n\\to\\infty}\\frac{1}{n}I(\\rvex_{0}; \\rvay_{1}^{n})\n \\geq \\Sumover{\\abs{p_{i}}>1}\\log\\abs{p_{i}},\n\\end{align}\nwhere $\\set{p_{i}}_{i=1}^{M}$ are the poles of $P(z)$ (the plant in Fig.~\\ref{fig:fbksystem}).\n\nBy using the results obtained in Section~\\ref{sec:entropy_gain_initial_stat} we show next that equality holds in~\\eqref{eq:martins_I_bound} provided the feedback channel \nsatisfies the following assumption:\n\n\\begin{assu}\\label{assu:fbck_channel}\nThe feedback channel in Fig.~\\ref{fig:fbksystem} can be written as\n\\begin{align}\n \\rvaw= AB \\rvav + BF(\\rvac), \\label{eq:channel}\n\\end{align}\nwhere\n\\begin{enumerate}\n \\item $A$ and $B$ are stable rational transfer functions such that $AB$ is biproper,  $ABP$  has the same unstable poles as $P$, and the feedback $AB$ stabilizes the plant $P$.\n \\item $F$ is any (possibly non-linear) operator such that $\\tilde{\\rvac}\\eq F(\\rvac)$ satisfies \n $\\frac{1}{n}h(\\tilde{\\rvac}_{1}^{n})<K$, for all $n\\in\\Nl$, and\n \\item $\\rvac_{1}^{\\infty}\\Perp (\\rvex_{0},\\rvau_{1}^{\\infty})$.\n \\finenunciado\n \\end{enumerate}\n \\end{assu}\n\n An illustration of the class of feedback channels satisfying this assumption is depicted on top of Fig.~\\ref{fig:fbkplant}.\n Trivial examples of channels satisfying Assumption 5 are a Gaussian additive channel preceded and followed by linear operators~\\cite{elia-04}.\n Indeed, when $F$ is an LTI system with a strictly causal transfer function, the feedback channel that satisfies Assumption~\\ref{assu:fbck_channel} is widely known as a \\emph{noise shaper with input pre and post filter}, used in, e.g.~\\cite{silder11,silder11b,yuksel12,fremid11}.\n %\n \\begin{figure}[t]\n  \\centering\n  \\input{fbkplant.pstex_t}\n  \\caption{Top: The class of feedback channels described by Assumption~\\ref{assu:fbck_channel}. Bottom: an equivalent form.}\n  \\label{fig:fbkplant}\n \\end{figure}\n\n \n \\begin{thm}\\label{thm:equality_in_martins}\n  In the networked control system of Fig.~\\ref{fig:fbksystem}, suppose that the feedback channel satisfies Assumption~\\ref{assu:fbck_channel} and that the input $\\rvau_{1}^{\\infty}$ is entropy balanced.\n  If the random initial state of the plant $P(z)$, with poles $\\set{p_{i}}_{i}^{M}$, satisfies $|h(\\rvex_{0})|<\\infty$, then\n  %\n  \\begin{align}\n   \\lim_{n\\to\\infty}\\frac{1}{n}I(\\rvex_{0}; \\rvay_{1}^{n})\n   = \\sumover{\\abs{p_{i}}>1}\\log\\abs{p_{i}}.\n  \\end{align}\n\\finenunciado\n \\end{thm}\n %\n %\n\\begin{proof\n Let \n $P(z)=N(z)\/\\Lambda (z)$ and \n $T(z)\\eq A(z)B(z)=\\Gamma(z)\/\\Theta(z)$.\n Then, from Lemma~\\ref{lem:initial_states} (in the appendix), \nthe output $\\rvey^1_n$ can be written as\n \\begin{align}\n  \\rvay = \\underbrace{\\Lambda}_{\\text{init. state $\\rvex_{0}$}} \\cdot \n\t  \\underbrace{\\frac{\\Theta }{\\Theta \\Lambda + \\Gamma N}}_{\\eq \\tilde{G}\\text{, init. state $\\set{\\rvex_{0},\\rves_{0}}$}} \\tilde{\\rvau},\n\t  \\label{eq:y_as_Gu}\n \\end{align}\nwhere $\\rves_{0}$ is the initial state of $T(z)$ and \n\\begin{align}\n \\tilde{\\rvau} \\eq u + B \\tilde{\\rvac}.\n\\end{align}\n(see Fig.~\\ref{fig:fbkplant} Bottom).\nThen \n\\begin{align}\n I(\\rvex_{0};\\rvey^{1}_{n}) \n &\n =\n h(\\rvey^{1}_{n}) -   h(\\rvey^{1}_{n}|\\rvex_{0})\n \\\\&\n =\n h(\\rvey^{1}_{n}) -   h (\\boldsymbol{\\Lambda}_{n} [\\tilde{\\bG}_{n}\\tilde{\\rveu}^{1}_{n} + \\tilde{\\bC}_{n}\\rves_{0} ]  )\n \\\\&\n =\n h(\\boldsymbol{\\Lambda}_{n} [\\tilde{\\bG}_{n}\\tilde{\\rveu}^{1}_{n} + \\tilde{\\bC}_{n}\\rves_{0} +\\bar{\\bC}_{n}\\rvex_{0}] + \\bC_{n}\\rvex_{0}) -   h(\\boldsymbol{\\Lambda}_{n} [\\tilde{\\bG}_{n}\\tilde{\\rveu}^{1}_{n} + \\tilde{\\bC}_{n}\\rves_{0} ] )\n \\\\&\n =\n h(\\boldsymbol{\\Lambda}_{n} [\\tilde{\\bG}_{n}\\tilde{\\rveu}^{1}_{n} + \\tilde{\\bC}_{n}\\rves_{0} +\\bar{\\bC}_{n}\\rvex_{0}] + \\bC_{n}\\rvex_{0}) -   \n h(\\tilde{\\bG}_{n}\\tilde{\\rveu}^{1}_{n} + \\tilde{\\bC}_{n}\\rves_{0}  ),\n \\label{eq:I_as_h}\n\\end{align}\n %\n where $\\tilde{\\bC}_{0}$ maps the initial state $\\rves_{0}$ to $\\rvey^{1}_{n}$,\n $\\bar{\\bC}_{n}$ maps the initial state $\\rvex_{0}$ to the output of $\\tilde{G}(z)$,\n and $\\bC_{n}$ maps the initial state $\\rvex_{0}$ (of $\\Lambda(z)$) to $\\rvey^{1}_{n}$.\nSince $\\rvau_{1}^{\\infty}$ is entropy balanced and $\\tilde{\\rvac}_{1}^{\\infty}$ has finite entropy rate, it follows from Lemma~\\ref{lem:sum_yields_entropy_balanced} that $\\tilde{\\rvau}_{1}^{\\infty}$ is entropy balanced as well.\nThus, we can proceed as in the proof of Theorem~\\ref{thm:eg-due-to-random-xo} to conclude that\n %\n \\begin{align}\n\\lim_{n\\to\\infty}\\frac{1}{n}I(\\rvex_{0}; \\rvay_{1}^{n})\n=\n\\sumover{\\abs{p_{i}}>1}\\log\\abs{p_{i}}.\n\\label{eq:I_as_sum_log}\n \\end{align}\n %\n This completes the proof.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\\subsection{The Feedback Channel Capacity of (non-white) Gaussian Channels}\nConsider a non-white additive Gaussian channel of the form\n\\begin{align}\n \\rvay_{k}= \\rvax_{k} + \\rvaz_{k},\n\\end{align}\nwhere the input $\\rvax$ is subject to the power constraint\n\\begin{align}\n \\lim_{n\\to\\infty}\\frac{1}{n}\\expe{\\norm{\\rvex^{1}_{n}}^{2}}\\leq P,\n\\end{align}\nand\n$\\rvaz_{1}^{\\infty}$ is a stationary Gaussian process. \n\nThe feedback information capacity of this channel is realized by a Gaussian input $\\rvax$, and is given by \n\\begin{align}\n C_{\\text{FB}} = \\lim_{n\\to\\infty} \n \\max_{\\bK_{\\rvex^{1}_{n}}: \\frac{1}{n}\\tr{\\bK_{\\rvex^{1}_{n}}}\\leq P}\n I(\\rvex^{1}_{n};\\rvey^{1}_{n}),\n\\end{align}\nwhere $\\bK_{\\rvex^{1}_{n}}$ is the covariance matrix of $\\rvex^{1}_{n}$ and, for every $k\\in\\Nl$, the input $\\rvex_{k}$ is allowed to depend upon the channel outputs $\\rvay_{1}^{k-1}$ (since there exists a causal, noise-less feedback channel with one-step delay).\n\nIn~\\cite{kim-yh10}, it was shown that \nif $\\rvaz$ is an auto-regressive moving-average process of $M$-th order, then\n$C_{\\text{FB}}$ can be achieved by the scheme shown in Fig.~\\ref{fig:Kim_FBK_Cap_system}.\nIn this system, $B$ is a strictly causal and stable finite-order filter and $\\rvav_{1}^{\\infty}$ is Gaussian with $\\rvav_{k}=0$ for all $k>M$ and such that\n$\\rvev^{1}_{n}$ is Gaussian with a positive-definite covariance matrix $\\bK_{\\rvev^{1}_{M}}$.\n\\begin{figure}[t]\n\\centering\n \\input{v2Kim_FBK_Cap_system.pstex_t}\n \\caption{Block diagram representation a non-white Gaussian channel $\\rvay=\\rvax+\\rvaz$ and the coding scheme considered in~\\cite{kim-yh10}.}\n \\label{fig:Kim_FBK_Cap_system}\n\\end{figure}\n\nHere we use the ideas developed in Section~\\ref{sec:entropy_gain_output_disturb} to show that \\textbf{the information rate achieved by the capacity-achieving scheme proposed in~\\cite{kim-yh10} drops to zero if there exists any additive disturbance of length at least $M$ and finite differential entropy affecting the output, no matter how small}.\n\nTo see this, notice that, in this case, and for all $n>M$, \n\\begin{align}\n I(\\rvax_{1}^{n};\\rvay_{1}^{n}) \n &= I(\\rvav_{1}^{M};\\rvay_{1}^{n})\n =\n h(\\rvey^{1}_{n}) \n - \n h(\\rvey^{1}_{n}|\\rvev^{1}_{n}) \n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} + \\rvev^{1}_{n}|\\rvev^{1}_{M}) \n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} |\\rvev^{1}_{M})\n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n}) \n =\n h(\\rvey^{1}_{n}) \n - \n h(\\rvez^{1}_{n})\n \\\\&\n =\n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} + \\rvev^{1}_{n})\n -\n h(\\rvez^{1}_{n}),\n\\end{align}\nsince $\\det(\\bI_{n}+\\bB_{n})=1$.\nFrom Theorem~\\ref{thm:eg_n_instate_w_disturb}, this gap between differential entropies is precisely the entropy gain introduced by $\\bI_{n}+\\bB_{n}$ to an input $\\rvez^{1}_{n}$ when the output is affected by the disturbance $\\rvev^{1}_{M}$.\nThus, from Theorem~\\ref{thm:eg_n_instate_w_disturb}, the capacity of this scheme will correspond to\n$\\intpipi{\\log\\abs{1+B\\ejw}}=\\sumover{\\abs{\\rho_{i}}>1}\\log\\abs{\\rho_{i}}$, where $\\set{\\rho_{i}}_{i=1}^{M}$ are the zeros of  $1+B(z)$, which is precisely the result stated in~\\cite[Theorem 4.1]{kim-yh10}.\n\nHowever, if the output is now affected by an additive disturbance $\\rvad_{1}^{\\infty}$ not passing through $B(z)$ such that $\\rvad_{k}=0$, $\\forall k>M$ and $|h(\\rved^{1}_{M})|<\\infty$, with $\\rvad_{1}^{\\infty}\\Perp (\\rvav_{1}^{M},\\rvaz_{1}^{\\infty})$, then we will have \n\\begin{align}\n \\rvey^{1}_{n} = \\rvev^{1}_{n} + (\\bI_{n} +\\bB_{n})\\rvez^{1}_{n} + \\rved^{1}_{n}.\n\\end{align}\nIn this case, \n\\begin{align}\n I(\\rvax_{1}^{n};\\rvay_{1}^{n}) \n &= I(\\rvav_{1}^{M};\\rvay_{1}^{n})\n =\n h(\\rvey^{1}_{n}) \n - \n h(\\rvey^{1}_{n}|\\rvev^{1}_{n}) \n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} + \\rvev^{1}_{n} + \\rved^{1}_{n}|\\rvev^{1}_{M}) \n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} + \\rved^{1}_{n}|\\rvev^{1}_{M})\n \\\\&\n =\n h(\\rvey^{1}_{n}) \n - \n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n}+ \\rved^{1}_{n}) .\n\\end{align}\nBut $\\lim_{n\\to\\infty}\\frac{1}{n}( h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n} + \\rvev^{1}_{n}+ \\rved^{1}_{n})\n -\n h(( \\bI_{n}+\\bB_{n})\\rvez^{1}_{n}+ \\rved^{1}_{n}))=0,$\n which follows directly from applying Theorem~\\ref{thm:eg_n_instate_w_disturb} to each of the differential entropies.\nNotice that this result holds irrespective of how small the power of the disturbance may be.\n\nThus, the capacity-achieving scheme proposed in~\\cite{kim-yh10} (and further studied in~\\cite{ardfra12}), although of groundbreaking theoretical importance, would yield zero rate in any practical situation, since every real signal is unavoidably affected by some amount of noise. \n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\nThis paper has provided a geometrical insight and rigorous results for characterizing the increase in differential entropy rate (referred to as entropy gain) introduced by passing an input random sequence through a discrete-time linear time-invariant (LTI) filter $G(z)$ such that the first sample of its impulse response has unit magnitude.\nOur time-domain analysis allowed us to explain and establish under what conditions the entropy gain coincides with what was predicted by Shannon, who followed a frequency-domain approach to a related problem in his seminal 1948 paper.\nIn particular, we demonstrated that the entropy gain arises only if $G(z)$ has zeros outside the unit circle (i.e., it is non-minimum phase, (NMP)).\nThis is not sufficient, nonetheless, since letting the input and output be $\\rvau$ and $\\rvay=G\\rvau$, the difference $h(\\rvay_{1}^{n})-h(\\rvau_{1}^{n})$ is zero for all $n$, yielding no entropy gain.\nHowever, if \nthe distribution of the input process $\\rvau$ satisfies a certain regularity condition (defined as being ``entropy balanced'') and the output \nhas the form $\\rvay=G\\rvau + \\rvaz$, with $\\rvaz$ being an output disturbance with bounded differential entropy, we have shown that the entropy gain can range from zero to the sum of the logarithm of the magnitudes of the NMP zeros of $G(z)$, depending on how $\\rvaz$ is distributed.\nA similar result is obtained if, instead of an output disturbance, we let $G(z)$ have a random initial state.\nWe also considered the difference between the differential entropy rate of the \\textit{entire} (and longer) output of $G(z)$ and that of its input, i.e., $h(\\rvay_{1}^{n+\\eta}) -h(\\rvau_{1}^{n})$, where $\\eta+1$ is the length of the impulse response of $G(z)$.\nFor this purpose, we introduced the notion of ``effective differential entropy'', which can be applied to a random sequence whose support has dimensionality smaller than its dimension.\nInterestingly, the effective differential entropy gain in this case, which is intrinsic to $G(z)$, is also the sum of the logarithm of the magnitudes of the NMP zeros of $G(z)$, without the need to add disturbances or a random initial state.\nWe have illustrated some of the implications of these ideas in three problems.\nSpecifically, we used the fundamental results here obtained to provide a simpler and more general proof to characterize the rate-distortion function for Gaussian non-stationary sources and MSE distortion.\nThen, we applied our results to provide sufficient conditions for equality in an information inequality of significant importance in networked control problems.\nFinally, \nwe showed that the information rate of the capacity-achieving scheme proposed in~\\cite{kim-yh10} for the autoregressive Gaussian channel with feedback drops to zero in the presence of any additive disturbance in the channel input or output of sufficient (finite) length, no matter how small it may be.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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\\vphantom{#1}}\n\\newcommand{\\widedots}[1]{\\overset{\\mskip1mu\\dotfill\\mskip1mu}{#1}\n                \\vphantom{#1}}\n\\newcommand{\\llr}{\\Longleftrightarrow}\n\n\\newcommand{\\tem}{{\\bf S}}\n\\newcommand{\\tnem}{{\\bf NS}}\n\n\\numberwithin{equation}{section}\n\n\\hyphenation{Tim-mes-feld}\n\n\\begin{document}\n\\title[A non-split sharply $2$-transitive group]{A sharply $2$-transitive group without a non-trivial abelian normal subgroup}\n\\author[Eliyahu Rips, Yoav Segev, Katrin Tent]{Eliyahu Rips$^1$\\qquad Yoav Segev\\qquad Katrin Tent}\n\n\\address{Eliyahu Rips\\\\\n       Einstein Institute of Mathematics\\\\\n        Hebrew University \\\\\n        Jerusalem 91904\\\\\n        Israel}\n\\email{eliyahu.rips@mail.huji.ac.il}\n\\thanks{$^1$This research was partially supported by the Israel Science Foundation}\n\n\\address{Yoav Segev \\\\\n         Department of Mathematics \\\\\n         Ben-Gurion University \\\\\n         Beer-Sheva 84105 \\\\\n         Israel}\n\\email{yoavs@math.bgu.ac.il}\n\n\\address{Katrin Tent \\\\\n         Mathematisches Institut \\\\\n         Universit\\\"at M\\\"unster \\\\\n\t Einsteinstrasse 62\\\\\n         48149 M\\\"unster \\\\\n         Germany}\n\\email{tent@wwu.de}\n\n\n\\keywords{sharply $2$-transitive, free product, HNN extension, malnormal}\n\\subjclass[2010]{Primary: 20B22}\n\n\\begin{abstract} \nWe show that any group $G$ is contained in some sharply 2-transitive \ngroup $\\calg$ without a non-trivial\nabelian normal subgroup.\nThis answers a long-standing open question.  The involutions\nin the groups $\\calg$  that we construct have no fixed points.\n\\end{abstract}\n\n\\date{\\today}\n\\maketitle\n \n\\section{Introduction}\n\n \nThe \\emph{finite} sharply $2$-transitive groups were classified by Zassenhaus\nin 1936 \\cite{Z} and it is known that any finite\nsharply $2$-transitive group contains a non-trivial abelian normal subgroup.\n \nIn the infinite situation no classification is known (see \\cite[Problem 11.52, p.~52]{MK}).\nIt was a long standing open problem\nwhether every infinite sharply $2$-transitive group contains a non-trivial abelian normal subgroup.\nIn \\cite{Ti} \nTits proved that this holds for locally compact connected sharply $2$-transitive groups.\nSeveral other papers showed that under certain special conditions the assertion holds\n(\\cite{BN, GMS, GlGu, M, T2, Tu, W}). \nThe reader may wish to consult Appendix \\ref{app A} for more detail,\nand for a description of our main results using permutation group theoretic language.\n\nAn equivalent formulation to the above problem is \nwhether every near-domain is a near-field (see \\cite{Hall, K, SSS} and  \nAppendix \\ref{app A} below).     \n\nWe here show that this is not the case.  We construct a sharply $2$-transitive infinite group\nwithout a non-trivial abelian normal subgroup.\nIn fact, the construction is similar in flavor to the free completion of partial generalized polgyons \\cite{T1}.\n\nWe are grateful to Joshua Wiscons for pointing out an instructive counterexample\nto a first version of this paper, and for greatly simplifying parts of the proof \nin a later version.  We are also grateful to Avinoam Mann for greatly simplifying \nthe proof of Proposition \\ref{prop A1 fp} and for drawing our attention to a point in the\nproof that needed correction. We thank all the referees \nof this paper for carefully reading the manuscript\nand making very useful remarks that helped to improve the exposition.\n\nRecall that a proper subgroup $A$ of a group\n$G$ is {\\it malnormal} in $G$ if\\linebreak\n$A\\cap g^{-1}Ag=1,$ for all $g\\in G\\sminus A$.\n \n\n\\begin{thm}\\label{thm main}\nLet $G$ be a group with a malnormal subgroup $A$ and an involution $t\\in G\\sminus A$ such that\n$A$ contains no involutions.\nThen for any two elements $u, v\\in G$ with $Au\\ne Av$ there exist\n\\begin{itemize}\n\\item[(a)]\nan extension $G\\le G_1$;\n\n\\item[(b)]\na malnormal subgroup $A_1$ of $G_1$ such that $A_1$ does not contain involutions and satisfies $A_1\\cap G=A$;\n\n\\item[(c)] an element $f\\in G_1$ such that $A_1f=A_1u$ and $A_1tf=A_1v.$\n\\end{itemize}\n\\end{thm}\n\n\\begin{remark}\nIt is easy to see (see \\S2) that in Theorem \\ref{thm main}\nwe may assume that $u=1, v\\notin AtA$  and that either: \n(1)\\ $v^{-1}\\notin AvA$ or: (2)\\ $v$ is an involution.\nIf case (1) holds we take $G_1=G*\\lan f\\ran$\nto be the free product of $G$ with an infinite cyclic group generated by $f,$\nand $A_1=\\lan A, f, tfv^{-1}\\ran$.  If case (2) holds we take\n$G_1=\\lan G, f\\mid f^{-1}tf=s\\ran$ and HNN extension and $A_1=\\lan A, f\\ran$.\n\\end{remark}\n\nAs a corollary to Theorem \\ref{thm main} we get the following.\n\n\\begin{thm}\\label{thm s2t in char 2}\nLet $G$ be a group with a malnormal subgroup $A$ such that\n$A$ contains no involutions.\nAssume further that $G$ is \\texttt{not} sharply $2$-transitive on the set of right cosets $A\\backslash G$.\nThen $G$ is contained in a group $\\calg$ having a malnormal subgroup $\\cala$ such that\n\\begin{enumerate}\n\\item\n$\\cala\\cap G=A;$\n\n\\item\n$\\calg$ is sharply $2$-transitive on the set of right cosets $X:=\\cala\\backslash\\calg;$\n\n\\item\n$\\cala$ contains no involutions (i.e.~$\\calg$ is of permutational characteristic $2$);\n\n\\item\n$\\calg$ does not contain a non-trivial abelian normal subgroup;\n\n\\item\nif $G$ is infinite then $G$ and $\\calg$ have the same cardinality (similarly \nfor $X$ and $A\\backslash G$).\n\\end{enumerate}\n\\end{thm}\n\n\nAs an immediate consequence of Theorem \\ref{thm s2t in char 2} we have\n\n\\begin{thm}\\label{thm eg}\nAny group $G$ is contained in a group $\\calg$ acting sharply $2$-transitively\non a set $X$ such that each involution in $\\calg$ has no fixed point in $X,$ and such that\n$\\calg$ does not contain a non-trivial abelian normal subgroup.\n\\end{thm}\n\\begin{proof}\nFor $|G|=1,2$ this is obvious.  Otherwise take $A=1$ in Theorem \\ref{thm s2t in char 2}.  \n\\end{proof}\n\\noindent\nIn fact there are many other ways to obtain a group $\\calg$ having a malnormal subgroup \n$\\cala$ and satisfying (2)--(4) of Theorem \\ref{thm s2t in char 2},\ne.g., take $G=\\lan t\\ran* A,$ where $t$ is an involution,\nand $A$ a non-trivial group without involutions, and apply Theorem \\ref{thm s2t in char 2}.\n(Here the free product guarantees that $A$ is malnormal in $G$.)\n\n\nTheorem \\ref{thm eg} shows that there exists a sharply $2$-transitive\ngroup $\\calg$ of {\\it characteristic $2$} (see Definition \\ref{def char} in Appendix \\ref{app A})\nsuch that $\\calg$ does not contain a non-trivial abelian normal subgroup.  Further\nas noted in Appendix \\ref{app A}, if $G$ is sharply $2$-transitive of characteristic $3,$\nthen $G$ contains a non-trivial abelian normal subgroup.  The cases where $\\charc(G)$ is distinct from\n$2$ and $3$ remain open.\n\nFinally we mention that the hypothesis that $A$ does not contain involutions\nin Theorem \\ref{thm main} is used only in the case where we take $G_1$ to be\nan HNN extension of $G$, and then, it is used only\nin the proof of the malnormality of $A_1$ in $G_1$.\n\n \n\\section{Some preliminaries regarding Theorem \\ref{thm main}}\\label{sect explanation}\n\nThe following observations and remarks are here in order to explain \nto the reader the way we intend to prove Theorem \\ref{thm main}, and\nto explain the main division between the two cases  we deal with in \\S\\ref{sect nonhnn} and \\S\\ref{sect hnn}.\n\nIn fact Lemma \\ref{lem hyp}(3) and Lemma \\ref{lem fuv} below, together with Remark \\ref{rem pf thm 1.1}, show that we may\nassume throughout this paper that hypothesis \\ref{hyp main} holds; and that hypothesis naturally\nleads to the division of the two cases dealt with in \\S\\ref{sect nonhnn} and \\S\\ref{sect hnn}.  \n\n\n\\begin{lemma}\\label{lem hyp}\nLet $A$ be a malnormal subgroup of a group $G$ and let $g\\in G\\sminus A$. Then\n\\begin{enumerate}\n\\item\n$C_G(a)\\le A,$ for all $a\\in A,\\ a\\ne 1;$ \n\n\\item\n$\\lan g\\ran\\cap A=1;$\n\n\\item\n$AgA$ contains an involution iff $g^{-1}\\in AgA.$\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof}\n(1): Let $a\\in A$ with $a\\ne 1,$ and let $h\\in C_G(a)$. Then $a\\in A\\cap A^h$. \nSo $h\\in A,$ since $A$ is malnormal in $G$.\n\\medskip\n\n\\noindent\n(2):  Since $g\\in C_G(g^k)$ for all integers $k,$  part (2) follows from (1).\n\\medskip\n\n\\noindent\n(3):\\quad\nIf $g^{-1}\\notin AgA,$ then clearly $AgA$ does not contain an involution.\nConversely, assume that $g^{-1}\\in AgA$.  Then $g^{-1}=agb,$ for some $a, b\\in A,$  \nso $(ag)^2=ab^{-1}\\in A$.  Then, by (2), either $(ag)^2=1$ or $ag\\in A$.  But $g\\notin A,$\nso $ag\\notin A,$ and we have $(ag)^2=1$.  Hence $AgA$ contains the involution $ag$.\n\\end{proof}\n \nWe now make the following observation (and introduce the following notation):\n \n\\begin{lemma}\\label{lem fuv}\nLet $G$ be a group with a malnormal subgroup $A$ and an involution $t\\in G\\sminus A$.\nLet $G_1$ be an extension of $G,$ such that\n$G_1$ contains a malnormal subgroup $A_1$ with $A_1\\cap G=A$. Let $r, s\\in G$ be such that $Ar\\ne As$. Then\n\\begin{itemize}\n\\item[(1)]\nthere is at most one element $f'\\in G_1$ with $A_1f'=A_1r$ and $A_1tf'=A_1s$, which\nwe denote by $f'=f_{r,s}$ (if it exists).\n\\end{itemize}\nThe convention in $(2)$--$(4)$ below is that the left side exists if and only if the right side does\nand then they are equal:\n\\begin{itemize}\n\\item[(2)]\n$f_{r,s}g=f_{rg, sg}$ for any $g\\in G.$\n\n\\item[(3)]\n$tf_{r,s}=f_{s,r}.$\n\n\\item[(4)]\n$f_{a_1r, a_2s}=f_{r,s}$ for all $a_1, a_2\\in A$.\n\\end{itemize}  \n\\end{lemma}\n\\begin{proof}\n(1):\\quad\nLet $f_1, f_2\\in G_1$ such that $A_1f_1=A_1f_2=A_1r$ and $A_1tf_1=A_1tf_2=A_1s$.  Then $f_1f_2^{-1}\\in A_1$ \nand $tf_1f_2^{-1}t\\in A_1$.  Since $t\\in G_1\\sminus A_1,$ and  since\n$A_1$ is malnormal in $G_1,$  we obtain that $f_1f_2^{-1}=1,$ so $f_1=f_2$.\n\\medskip\n\n\\noindent\n(2):\\quad \n$A_1f_{r,s}g=A_1rg=A_1f_{rg, sg}$ and $A_1tf_{r,s}g=A_1sg=A_1f_{rg,sg}$.\nSo, by (1), $f_{rg, sg}=f_{r,s}g$.\n\\medskip\n\n\\noindent\n(3):\\quad\n$A_1tf_{r,s}=A_1s,$ and $A_1ttf_{r,s}=A_1f_{r,s}=A_1r$. So, by (1), $tf_{r,s}=f_{s,r}$.\n\\medskip\n\n\\noindent\n(4):\\quad\n$A_1f_{a_1r, a_2s}=A_1a_1r=A_1r,$ and $A_1tf_{a_1r,a_2s}= A_1a_2s=A_1s$. So, by (1), $f_{a_1r, a_2s}=f_{r,s}$. \n\\end{proof}\n\n\n\\begin{remark}\\label{rem pf thm 1.1}\nLet the notation be as in Theorem \\ref{thm main}.  Notice that if there is an element \n$f\\in G$ such that $Af=Au$ and $Atf=Av,$ we can just take  $G_1=G$\nand $A_1=A$ and there is nothing to prove in Theorem \\ref{thm main}.  \n\nHence we may assume throughout this paper that this is not the case. \nIn view of (2) and (4)\nof Lemma \\ref{lem fuv}, $f_{u,v}=f_{1,vu^{-1}}u,$ and $f_{1,a'va}=f_{a^{-1},a'v}a=f_{1,v}a,$ for $a, a'\\in A$.\nHence we may assume that $u=1$ (and hence $v\\notin A$) and replace $v$ by any element of\nthe double coset $AvA$.  By Lemma \\ref{lem hyp}(3), we may assume that either $v^{-1}\\notin AvA,$\nor $v$ is an involution.  Further, since $f_{1,t}=1$ and since $t$ is an involution, we may assume\nthat $v\\notin AtA$ and $v^{-1}\\notin AtA$.\n\\end{remark}\n\nHence it suffices to prove Theorem \\ref{thm main} under the following\nhypothesis which we assume for the rest of the paper.\n\n\\begin{hypothesis}\\label{hyp main}\nIn the setting of Theorem \\ref{thm main}, assume $u=1, v, v^{-1}\\notin AtA$ and either $v^{-1}\\notin AvA$ or $v$ is an involution.\n\\end{hypothesis} \n\n\n\n\n\\section{The case  $v^{-1}\\notin AvA$}\\label{sect nonhnn}\n\nThe purpose of this section is to prove Theorem \\ref{thm main} of\nthe introduction  in the case where $v^{-1}\\notin AvA$. We refer\nthe reader to Hypothesis \\ref{hyp main} and to its explanation in \\S\\ref{sect explanation}.\nThus, throughout this section we assume that $v^{-1}\\notin AvA$.  \nAlso, throughout\nthis section we use the notation and hypotheses of Theorem \\ref{thm main}.\n\nLet $\\lan f_1\\ran$ be an infinite cyclic group.\nWe let\n\\[\nG_1=G*\\lan f_1\\ran,\\quad f_2=tf_1v^{-1},\\quad A_1=\\lan A, f_1, f_2\\ran.\n\\]\n\nIn this section we will prove the following theorem.\n\n\\begin{thm}\\label{thm main nonhnn}\nWe have\n\\begin{enumerate}\n\\item\n$A_1=A*\\lan f_1\\ran*\\lan f_2\\ran,$ with $f_1, f_2$ of infinite order;\n\n\\item\n$A_1$ is malnormal in $G_1$.\n\\end{enumerate}\n\\end{thm}\n\nSuppose Theorem \\ref{thm main nonhnn} is proved.  We now prove Theorem \\ref{thm main}\nin the case where $v^{-1}\\notin AvA$.\n\\medskip\n\n\\begin{proof}[Proof of Theorem \\ref{thm main} in the case where $v^{-1}\\notin AvA$]\\hfill\n\nLet $f:=f_1$.  Then $A_1f=A_1f_1=A_1,$ and \n\\[\nA_1tf=A_1tf_1=A_1tf_1v^{-1}v=A_1f_2v=A_1v.\n\\]\nBy Theorem  \\ref{thm main nonhnn}(2), $A_1$ is malnormal in $G_1$.  \nBy Theorem \\ref{thm main nonhnn}(1),  \n$A_1\\cap G=A,$ and $f_2$ is of infinite order.  \nSince $A_1=A*\\lan f_1\\ran*\\lan f_2\\ran,$ and $A$ does not contain involutions, $A_1$\ndoes not contain involutions.  \n\\end{proof}\n \n\n\\begin{prop}\\label{prop A1 fp}\n$f_2$ is of infinite order in $G_1,$ and $A_1=A* \\lan f_1\\ran*\\lan f_2\\ran.$\n\\end{prop}\n\\begin{proof}\nWe first show that $f_2$ is of infinite order.  Indeed let $h:=f_2^n,$ for some\n$n\\in\\zz,$ and write $h$ in terms of $f_1$ and elements of $G$.  If $n>0,$\nthen $h$ starts with $t$ and ends with $v^{-1},$ while if $n<0,$ then\n$h$ starts with $v$ and ends with $t$.  In particular $f_2$ has infinite order.\n\nNext let $F:=\\lan f_1, f_2\\ran$.  Then any element of $F$ is a product of\nalternating powers of $f_1$ and $f_2$.  As we saw in the previous paragraph of the proof,\nany non-zero power of $f_2$ starts with $t$ or $v$ and ends with $t$ or $v^{-1}$.\nSince $G_1=G*\\lan f_1\\ran$ there will be no cancellation between powers of $f_1$ and  powers of $f_2$.\nIt follows that $F$ is a free group.\n\nNow consider an element in $A_1=\\lan A, F\\ran$.  It is an alternating product\nof elements of $A$ and elements of $F$.  \nWhen we express it as an element of $G_1=G*\\lan f_1\\ran,$ $f_2$ is written as $tf_1v^{-1}$\nand $f_2^{-1}$ is written as $vf_1^{-1}t$.  Accordingly, an element $1\\ne a\\in A$ in this alternating\nproduct is multiplied with $1, v^{-1}$ or $t$ on the left, and with $1, t$ or $v$ on the right.  \nThe possibilities\nare: \n\\begin{itemize}\n\\item\n$v^{-1}a,\\ ta,\\ at,\\ av:$ all are distinct from $1$ since $t$ and $v$ are not in $A$.\n\n\\item\n$tat,\\ v^{-1}av:$ all are distinct from $1$ since they are conjugate to $a$.\n\n\\item\n$tav,\\ v^{-1}at:$ all are distinct from $1$ since $v\\notin AtA$.\\qedhere\n\\end{itemize}\n\\end{proof}\n\n\\begin{prop}\\label{prop malnor nonhnn}\n$A_1$ is a malnormal subgroup of $G_1$.\n\\end{prop}\n\\begin{proof}\nWe will show that the existence of elements $a, b\\in A_1,$ and $g\\in G_1\\sminus A_1,$\nsuch that $a\\ne 1$ and $g^{-1}ag=b$ leads to a contradiction.\n\nLet \n\\[\na=a_1f_{\\gd_1}^{\\gre_1}a_2f_{\\gd_2}^{\\gre_2}\\cdots a_nf_{\\gd_n}^{\\gre_n}a_{n+1},\\ a\\ne 1,\\qquad\\text{and}\n\\]\n\\[\nb=b_1f_{\\gc_1}^{\\mu_1}b_2f_{\\gc_2}^{\\mu_2}\\cdots b_{\\ell}f_{\\gc_{\\ell}}^{\\mu_{\\ell}}b_{\\ell+1},\n\\]\nwhere $a_i, b_j\\in A,\\ \\gre_i, \\mu_j=\\pm 1,\\ \\gd_i, \\gc_j\\in\\{1,2\\},$\n and if $\\gd_i=\\gd_{i-1}$ and $\\gre_i=-\\gre_{i-1}$ then\n$a_i\\ne 1$ (i.e.~there are no $f_i$-cancellations in $a$),\nand similarly there are no $f_i$-cancellations in $b$.\nWrite\n\\[\ng=g_1f_1^{\\gl_1}g_2f_1^{\\gl_2}\\cdots g_mf_1^{\\gl_m}g_{m+1}\\in G_1\\sminus A_1,\n\\]\nwhere $g_i\\in G,\\ \\gl_i=\\pm 1,$ and there are no $f_1$-cancellations in $g$.\n\nAssume that $m$ is the least possible.\nWe have the picture as in Figure \\ref{fig9} below.\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild9.pspdftex}}\n    \\caption{}\n    \\label{fig9}\n\\end{figure} \n\n\\noindent\n{\\bf Case 1.}\\ $m=n=0$.\n\nIn this case $b=g^{-1}a_1g \\in A_1\\cap G$. By Proposition \\ref{prop A1 fp},  $A_1\\cap G=A,$ so $b\\in A,$\nand we get a contradiction to the malnormality of $A$ in $G$.\n \n\\medskip\n\n\\noindent  \n\\smallskip\n\nThe next case to consider is: \n\\smallskip\n\n\\noindent   \n{\\bf Case 2.}\\ $m=0,$ and $n>0$.\n\nSince $G_1=G*\\lan f_1\\ran,$ we must have $n=\\ell$.\nConsider Figure \\ref{fig9}.  \nBy an analysis of the normal form in the free\nproduct $G*\\lan f_1\\ran$ we see that\nthe only way we can get the equality $g_1^{-1}ag_1=b$ is \nif both $\\gre_1=\\mu_1$ and $\\gre_n=\\mu_n$.  \nWe distinguish a number of cases as follows.\n\\begin{itemize}\n\\item[(i)]\n$\\gd_1=\\gc_1$ or $\\gd_n=\\gc_n$.\n\n\\item[(ii)]\n$\\gd_1\\ne\\gc_1$ and $\\gd_n\\ne \\gc_n$.\n\\begin{itemize}\n\\item[(a)] \n$n=1$.\n\\item[(b)]\n$n>1$.\n\\end{itemize}\n\\end{itemize}\n{\\bf Case (i).}\\ \nBy symmetry we may consider only the case where $\\gd_1=\\gc_1$.  \nIn this case, regardless of whether $\\gre_1=1$ or $-1$ and whether $\\gd_1=1$ or $2,$\nwe get that $g_1=a_1b_1^{-1}\\in A,$ a contradiction.  \n\\medskip\n\n\\noindent\n{\\bf Case (iia).}\\\nBy symmetry we may assume that $\\gd_1=1$ and $\\gc_1=2$.\n\nSuppose first that $\\gre_1=\\mu_1=1$.\nThen from the left side of Figure \\ref{fig9} we get\n$a_1^{-1}g_1b_1t=1,$ and from the right side we get $a_2g_1b_2^{-1}v=1$.\nThis implies that $t\\in Ag_1A$ and $v^{-1}\\in Ag_1A$.  But then $v^{-1}\\in AtA,$\na contradiction.\n\nSuppose next that $\\gre_1=\\mu_1=-1$.  Then, from the left side of Figure \\ref{fig9} we get\n$a_1^{-1}g_1b_1v=1,$ and from the right side we get $a_2g_1b_2^{-1}t=1$.\nAgain this implies that $v^{-1}\\in AtA,$\na contradiction.\n\\medskip\n\n\\noindent\n{\\bf Case (iib).}\\ \nBy symmetry, we may assume without loss of generality that \n\\[\n\\gd_1=1\\text{ and }\\gc_1=2.\n\\]\nSuppose first that \n\\[\n\\gre_1=\\mu_1=1.\n\\]\nWe may further assume that \n\\[\na_1^{-1}g_1b_1t=1\\quad\\text{and}\\quad\\gre_2=\\mu_2.\n\\]\nWe now separate the discussion according to the following cases: \n\\begin{itemize}\n\\item\n$\\gd_2=\\gc_2$.  In this case, regardless of\nthe sign of $\\gre_2=\\mu_2$ and whether $\\gd_2=\\gc_2=1$ or $2,$ we get that $a_2^{-1}v^{-1}b_2=1,$\nwhich is false since $v\\notin A$.\n\n\\item\n$\\gre_2=\\mu_2=1,\\ \\gd_2=1,\\ \\gc_2=2$.  We get $a_2^{-1}v^{-1}b_2t=1,$ contradicting $v\\notin AtA$.\n\n\\item\n$\\gre_2=\\mu_2=-1,\\ \\gd_2=1,\\ \\gc_2=2$.  We get $a_2^{-1}v^{-1}b_2v=1$ with $b_2\\ne 1$.\nBut this contradicts the malnormality of $A$ in $G$.\n\n\n\\item\n$\\gre_2=\\mu_2=1,\\gd_2=2,\\ \\gc_2=1$.  We get $ta_2^{-1}v^{-1}b_2=1,$ contrary to $v^{-1}\\notin AtA$.\n\n\\item\n$\\gre_2=\\mu_2=-1,\\ \\gd_2=2,\\ \\gc_2=1$. We get $v^{-1}a_2^{-1}v^{-1}b_2=1$.  This implies that $v^{-1}\\in AvA,$\ncontrary to our hypotheses.\n\\end{itemize}\n\nSuppose next that\n\\[\n\\gre_1=\\mu_1=-1.\n\\]\nWe may further assume that \n\\[\na_1^{-1}g_1b_1v=1\\quad\\text{and}\\quad \\gre_2=\\mu_2.\n\\]\nAgain we separate the discussion according to the following cases: \n\\begin{itemize}\n\\item\n$\\gd_2=\\gc_2$.  In this case, regardless of\nthe sign of $\\gre_2=\\mu_2$ and whether $\\gd_2=\\gc_2=1$ or $2,$ we get that $a_2^{-1}tb_2=1,$\nwhich is false since $t\\notin A$.\n\n\\item\n$\\gre_2=\\mu_2=1,\\ \\gd_2=1,\\ \\gc_2=2$.  We get $a_2^{-1}tb_2t=1,$ and $b_2\\ne 1$.  This contradicts \nthe malnormality of $A$ in $G$.\n\n\\item\n$\\gre_2=\\mu_2=-1,\\ \\gd_2=1,\\ \\gc_2=2$.  We get $a_2^{-1}tb_2v=1,$ impossible, as above.\n\n\n\\item\n$\\gre_2=\\mu_2=1,\\gd_2=2,\\ \\gc_2=1$.  We get $ta_2^{-1}tb_2=1$.\nThis case forces $a_2=b_2=1$ (because $A$ is malnormal in $G$) .  If $n=2$ we get \n$v^{-1}a_3g_1b_3^{-1}=1$.  But this together with $a_1^{-1}g_1b_1v=1$\nimplies that $v^{-1}\\in AvA,$ contrary to our hypotheses.  Thus $n\\ge 3$.  But now,\nwe must have $\\gre_3=\\mu_3,$ and arguing exactly as in the previous cases, \nfor all choices of $\\gre_3=\\mu_3, \\gd_3$ and $\\gc_3,$ we get a contradiction as in one of the cases above.\n\n\\item\n$\\gre_2=\\mu_2=-1,\\ \\gd_2=2,\\ \\gc_2=1$. We get $v^{-1}a_2^{-1}tb_2=1,$ impossible, as above.\n\\end{itemize}\n\\smallskip\n\n\nNext we consider:\n\\smallskip\n\n\\noindent\n{\\bf Case 3.}\\ $n=0=\\ell$ and $m > 0$.\n\n\\noindent\nNotice that in this case there will be no cancellations in Figure \\ref{fig9},\nsince otherwise we must either have $g_1^{-1}a_1g_1=1,$ or $g_{m+1}^{-1}b_1^{-1}g_{m+1}=1,$\nwhich is false.\n\nHence we may assume that either $n>0$ or $\\ell>0$ or both.\nBy symmetry we may consider the following case:\n\\smallskip\n\n\\noindent\n{\\bf Case 4.}\\ $m > 0$ and $n >0$.\n\nNotice that $f_i$-cancellations have to occur in the product $g^{-1}agb^{-1},$ since\nit is equal to $1$.  Now $f_i$-cancellations can occur\nonly if  one of the following cases occurs:\n\\begin{itemize}\n\\item[(i)]\nThe product $f_1^{-\\gl_1}g_1^{-1}a_1f_{\\gd_1}^{\\gre_1}$ equals $1,\\ v^{-1},$ or $t$.\n\\item[(ii)]\nThe product $f_{\\gd_n}^{\\gre_n}a_{n+1}g_1f_1^{\\gl_1}$ equals $1,\\ t$ or $v$.\n\\item[(iii)]\nThe product $f_1^{\\gl_m}g_{m+1}b_1f_{\\gc_1}^{\\mu_1}$ equals $1,\\ v^{-1}$ or $t$.\n\\item[(iv)]\nThe product $f_{\\gc_{\\ell}}^{\\mu_{\\ell}}b_{\\ell+1}g_{m+1}^{-1}f_1^{-\\gl_m}$\nequals $1,\\ t$ or $v$.\n\\end{itemize}\n\\medskip\n\nBy symmetry, we may consider only case (i).\nIf $f_1^{-\\gl_1}g_1^{-1}a_1f_{\\gd_1}^{\\gre_1}=1,$\nthen $g_1f_1^{\\gl_1}=a_1f_{\\gd_1}^{\\gre_1}$.  Let \n$h:=g_2f_1^{\\gl_2}\\cdots f_1^{\\gl_m}g_{m+1},$ and\n\\[\na'=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}g_1f_1^{\\gl_1}=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}a_1f_{\\gd_1}^{\\gre_1}\\in A_1.\n\\]\nNotice that $a'$ is conjugate to $a,$ so $a'\\ne 1$.\nAlso $h=f_1^{-\\gl_1}g_1^{-1}g,$ and $h\\notin A_1,$ since $f_1^{-\\gl_1}g_1^{-1}\\in A_1,$ while $g\\notin A_1$.\nWe get (see Figure \\ref{fig9}) $g^{-1}ag=h^{-1}a'h\\in A_1,$ contradicting the minimality of $m$.\n\\smallskip\n\nIf $f_1^{-\\gl_1}g_1^{-1}a_1f_{\\gd_1}^{\\gre_1}=v^{-1},$ then $g_1f_1^{\\gl_1}=a_1f_{\\gd_1}^{\\gre_1}v$.\nLet $h:=vg_2f_1^{\\gl_2}\\cdots f_1^{\\gl_m}g_{m+1},$ and \n\\[\na'=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}g_1f_1^{\\gl_1}v^{-1}=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}a_1f_{\\gd_1}^{\\gre_1}\\in A_1.\n\\]\nAs above, $1\\ne a'\\in A_1,$ and if $h\\in A_1,$ then \n$g=g_1f_1^{\\gl_1}v^{-1}h=a_1f_{\\gd_1}^{\\gre_1}h\\in A_1,$\nwhich is false.\nWe again get $g^{-1}ag=h^{-1}a'h\\in A_1,$ which contradicts the minimality of $m$.\n\nFinally if $f_1^{-\\gl_1}g_1^{-1}a_1f_{\\gd_1}^{\\gre_1}=t,$ then $g_1f_1^{\\gl_1}=a_1f_{\\gd_1}^{\\gre_1}t$.\nLet $h:=tg_2f_1^{\\gl_2}\\cdots f_1^{\\gl_m}g_{m+1},$ and \n\\[\na'=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}g_1f_1^{\\gl_1}t=a_2f_{\\gd_2}^{\\gre_2}\\cdots f_{\\gd_n}^{\\gre_n}a_{n+1}a_1f_{\\gd_1}^{\\gre_1}\\in A_1.\n\\]\nAs above we get $1\\ne a'\\in A_1,$ and $h\\notin A_1,$\nand  again we get the same contradiction.\n\nNote that if $\\ell=0,$ then no cancellation of the type (iii) or (iv) above can occur.\n\\end{proof}\n\\begin{proof}[Proof of Theorem \\ref{thm main nonhnn}]\\hfill\n\\medskip\n\n\\noindent\nBy Proposition \\ref{prop A1 fp}, part (1) holds, and by Proposition \\ref{prop malnor nonhnn} part (2) holds.\n\\end{proof}\n\\section{The case $v$ is an involution and $v\\notin AtA$}\\label{sect hnn}\nThe purpose of this section is to prove Theorem \\ref{thm main} of\nthe introduction  in the case where $v$ is an involution. We refer\nthe reader to Hypothesis \\ref{hyp main} and to its explanation in \\S\\ref{sect explanation}.\nThus, throughout this section we assume that $v$ is an involution and that $v\\notin AtA$.  Further, throughout\nthis section we use the notation and hypotheses of Theorem \\ref{thm main}.\n\nLet $\\lan f\\ran$ be an infinite cyclic group.\nWe define an HNN extension\n\\[\nG_1=\\lan G, f\\mid f^{-1}tf=v\\ran,\\qquad A_1=\\lan A, f\\ran.\n\\]\n\nIn this section we will prove the following theorem.\n \n\\begin{thm}\\label{thm main hnn}\nWe have\n\\begin{enumerate}\n\\item\n$A_1=A*\\lan f\\ran$;\n\n\\item\n$A_1$ is malnormal in $G_1.$\n\\end{enumerate}\n\\end{thm}\nSuppose Theorem \\ref{thm main hnn} is proved.  We now use it to prove Theorem \\ref{thm main}\nin the case where $v$ is an involution. \n\\smallskip\n\n\\noindent\n\\begin{proof}[Proof of Theorem \\ref{thm main} in the case where $v$ is an involution]\\hfill\n\\medskip\n\n\\noindent\nWe have $A_1f=A_1$ and $A_1tf=A_1fv=A_1v$.  By Theorem \\ref{thm main hnn}(2), $A_1$\nis malnormal in $G_1$.  By Theorem \\ref{thm main hnn}(1), $A_1\\cap G=A$. \nAlso $A_1$ does not\ncontain involutions since $A_1=A*\\lan f\\ran,$ and $A$ does not contain involutions.\n\\end{proof}\n\n\\begin{remark}\\label{rem hnn}\n\\begin{comment}\nAny element of $G_1$ has the form \n\\[\ng=g_0f^{\\gd_1}g_1\\cdots g_{m-1}f^{\\gd_m}g_m,\n\\]\nwhere $g_i\\in G,\\ i=0,\\dots m,\\ \\gd_i=\\pm 1,\\ i=1,\\dots m$.\nAccording to Britton's lemma we say that there are {\\it no $f$-cancellations in $g$} if the equality\n$\\gd_i=-\\gd_{i-1}$ implies that if $\\gd_i=1,$ then $g_i\\ne 1, t,$ while if $\\gd_i=-1,$ then\n$g_i\\ne 1,v$.\n \nFurther let $g$ be as above, let $h\\in G_1,$ and write:\n\\[\nh=h_0f^{\\eta_1}h_1\\cdots h_{k-1}f^{\\eta_k}h_k,\n\\]\nwhere $h_j\\in G,\\ j=0,\\dots m,\\ \\eta_j=\\pm 1,\\ j=1,\\dots k,$  \nand there are no $f$-cancellations in $g$ and $h$.\n\nThen $g=h$    \nif and only if $m=k,\\ \\gd_i=\\eta_i,\\ i=1,\\dots, m,$\nand there are elements $w_0,z_1,w_1,z_2,w_2,\\dots,z_m,w_m,z_{m+1}$ such that\nfor every oriented loop in Figure \\ref{fig1} the product of edges is $1,$ that is:\n \n\n\\begin{itemize}\n\\item[(a)]\n$h_i=w_ig_iz_{i+1},\\ i=0,\\dots, m;$\n\n\\item[(b)]  $w_0=1,\\ z_{m+1}=1;$\n\n\\item[(c)]  If $\\gd_i=1,$ then either $z_i=1,\\ w_i=1,$ or $z_i=t,\\ w_i=v;$\n\n\\item[(d)]  If $\\gd_i=-1,$ then either $z_i=1,\\ w_i=1,$ or $z_i=v,\\ w_i=t.$\n\\end{itemize}\n\\bigskip\n\n\\noindent\n\\end{comment}\nAny element of $G_1$ has the form \n\\[\ng=g_1f^{\\gd_1}g_2\\cdots g_mf^{\\gd_m}g_{m+1},\n\\]\nwhere $g_i\\in G,\\ i=1,\\dots m+1,\\ \\gd_i=\\pm 1,\\ i=1,\\dots m$.\nAccording to Britton's lemma we say that there are {\\it no $f$-cancellations in $g$} if the equality\n$\\gd_i=-\\gd_{i-1}$ implies that if $\\gd_i=1,$ then $g_i\\ne 1, t,$ while if $\\gd_i=-1,$ then\n$g_i\\ne 1,v$.\n \nFurther let $g$ be as above, let $h\\in G_1,$ and write:\n\\[\nh=h_1f^{\\eta_1}h_2\\cdots h_kf^{\\eta_k}h_{k+1},\n\\]\nwhere $h_j\\in G,\\ j=1,\\dots k+1,\\ \\eta_j=\\pm 1,\\ j=1,\\dots k,$  \nand there are no $f$-cancellations in $g$ and $h$.\n\nThen $g=h$    \nif and only if $m=k,\\ \\gd_i=\\eta_i,\\ i=1,\\dots, m,$\nand there are elements $w_0,z_1,w_1,z_2,w_2,\\dots,z_m,w_m,z_{m+1}$ such that\nfor every oriented loop in Figure \\ref{fig1} the product of edges is $1,$ that is:\n \n\n\\begin{itemize}\n\\item[(a)]\n$h_i=w_{i-1}g_iz_i,\\ i=1,\\dots, m+1;$\n\n\\item[(b)]  $w_0=1,\\ z_{m+1}=1;$\n\n\\item[(c)]  if $\\gd_i=1,$ then either $z_i=1,\\ w_i=1,$ or $z_i=t,\\ w_i=v;$\n\n\\item[(d)]  if $\\gd_i=-1,$ then either $z_i=1,\\ w_i=1,$ or $z_i=v,\\ w_i=t.$\n\\end{itemize}\n\\clearpage\n\n\n\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild8.pspdftex}}\n    \\caption{}\n    \\label{fig1}\n\\end{figure}  \n\\end{remark}\n\\begin{lemma}\\label{A1 is a free product}\n$A_1=A*\\lan f\\ran$.\n\\end{lemma}\n\\begin{comment}\n\\begin{proof}\nSuppose that\n\\[\ng_1f^{\\gd_1}g_2\\cdots g_mf^{\\gd_m}g_{m+1}=h_1f^{\\gd_1}h_2\\cdots h_mf^{\\gd_m}h_{m+1},\n\\]\n\\[\ng_0f^{\\gd_1}g_1\\cdots g_{m-1}f^{\\gd_m}g_m=h_0f^{\\gd_1}h_1\\cdots h_{m-1}f^{\\gd_m}h_m,\n\\]\nand $h_i, g_i\\in A,\\ i=0,\\dots, m$. By Remark \\ref{rem hnn}, $h_0=g_0z_1,$\nhence, by (a)--(d) of Remark \\ref{rem hnn}, since $t, v\\notin A,$ we have $z_1=1,$ \nso $h_0=g_0,$ and then, by Remark \\ref{rem hnn} (c) and (d), $w_1=1$.\n\nAssume $w_i=1$.  Then $h_i=w_ig_iz_{i+1}=g_iz_{i+1}$.  Since $t, v\\notin A,$ this implies $z_{i+1}=1,$ and\nthen $w_{i+1}=1$.  So $g_i=h_i$ for $i=0,\\dots, m$.  Hence $A_1=A*\\lan f\\ran$.\n\\end{proof}\n\\bigskip\n\n\\noindent\n\\end{comment}\n\\begin{proof}\nSuppose that\n\\[\ng_1f^{\\gd_1}g_2\\cdots g_mf^{\\gd_m}g_{m+1}=h_1f^{\\gd_1}h_2\\cdots h_mf^{\\gd_m}h_{m+1},\n\\]\nand $h_i, g_i\\in A,\\ i=1,\\dots, m+1$. By Remark \\ref{rem hnn}, $h_1=g_1z_1,$\nhence, by (a)--(d) of Remark \\ref{rem hnn}, since $t, v\\notin A,$ we have $z_1=1,$ \nso $h_1=g_1,$ and then, by Remark \\ref{rem hnn} (c) and (d), $w_1=1$.\n\nAssume $w_i=1$.  Then $h_{i+1}=w_ig_{i+1}z_{i+1}=g_{i+1}z_{i+1}$.  \nSince $t, v\\notin A,$ this implies $z_{i+1}=1,$ and\nthen $w_{i+1}=1$.  So $g_i=h_i$ for $i=1,\\dots, m+1$.  Hence $A_1=A*\\lan f\\ran$.\n\\end{proof}\n \n\n\\begin{prop}\\label{prop malnormal hnn}\n$A_1$ is malnormal in $G_1$. \n\\end{prop}\n\\begin{proof}\nWe will show that the existence of elements $a, b\\in A_1,\\ g\\in G_1\\sminus A_1$\nsuch that $a\\ne 1$ and $g^{-1}ag=b$ leads to a contradiction. Let\n\\[\na=a_1f^{\\ga_1}a_2\\cdots a_mf^{\\ga_m}a_{m+1},\\qquad b=b_1f^{\\gb_1}b_2\\cdots b_nf^{\\gb_n}b_{n+1},\n\\]\nwhere $a_i, b_i\\in A,\\ \\ga_i, \\gb_i=\\pm 1,$ and if $\\ga_i=-\\ga_{i-1},$ then $a_i\\ne 1,$\nand if $\\gb_i=-\\gb_{i-1},$ then $b_i\\ne 1$.  Recall that by Lemma \\ref{A1 is a free product}, $A_1=A*\\lan f\\ran,$\nand therefore in the above expressions for $a$ and $b$ there are no $f$-cancellations.  We also have\n\\[\ng=g_1f^{\\gd_1}g_2\\cdots g_kf^{\\gd_k}g_{k+1},\n\\]\nwhere $g_i\\in G,\\ \\gd_i=\\pm 1,$ and $\\gd_i=-\\gd_{i-1}$ implies that if $\\gd_i=1,$ then $g_i\\ne 1, t,$\nand if $\\gd_i=-1,$ then $g_i\\ne 1, v$.\n\nWe assume that $k$ is the least possible.\n\\medskip\n\n\\noindent\n{\\bf Case 1.}\\ $k=0$.\n\nThen $g=g_1,$ so we have \n\\[\ng_1^{-1}a_1f^{\\ga_1}a_2\\cdots a_mf^{\\ga_m}a_{m+1}g_1=b_1f^{\\gb_1}b_2\\cdots b_nf^{\\gb_n}b_{n+1}.\n\\]  \nWe conclude that $n=m,\\ \\ga_i=\\gb_i,$ for $i=1,2,\\dots, m$.  If $m=n=0,$ then $a=a_1\\ne 1,\\ b=b_1,$ so \n$g_1^{-1}a_1g_1=b_1$ which is impossible because $A$ is malnormal in $G$.\n\nLet $m=n>0$.  We obtain Figure \\ref{fig2} below, where\n\\begin{gather}\\label{eq prop 4.4}\n\\text{if }\\ga_i=1,\\text{ then either }p_i=q_i=1,\\text{ or }p_i=t,\\ q_i=v,\\\\\\notag\n\\text{and if }\\ga_i=-1,\\text{ then either }p_i=q_i=1,\\text{ or }p_i=v,\\ q_i=t.\n\\end{gather}\n\n\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild1.pspdftex}}\n    \\caption{}\n    \\label{fig2}\n\\end{figure}  \n\n\\noindent\nWe have $p_1=a_1^{-1}g_1b_1\\notin A$ since $g_1\\notin A$.  Now \nassume $p_i\\notin A$.  Then $q_i\\notin A$ by \\eqref{eq prop 4.4} \nand by Britton's Lemma $p_{i+1}=a_{i+1}^{-1}q_ib_{i+1}$ is not in $A$ either.\nIn Particular $p_i, q_i\\ne 1$ for all $i\\le m$.\n\nIf $m=n\\ge 2,$ consider Figure \\ref{fig3}:\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild2.pspdftex}}\n    \\caption{}\n    \\label{fig3}\n\\end{figure} \n\n\\noindent\nWe now use equation \\eqref{eq prop 4.4}.\nIf $\\ga_1=1,\\ \\ga_2=1,$ then $q_1=v,\\ p_2=t,$ so $v=a_2tb_2^{-1}\\in AtA,$ a contradiction.\n\nIf $\\ga_1=1,\\ \\ga_2=-1,$ then $a_2\\ne 1,\\ q_1=v,\\ p_2=v$.  Then $va_2v=b_2,$\ncontradicting the malnormality of $A$ in $G$.\n\nIf $\\ga_1=-1,\\ \\ga_2=1,$ then $a_2\\ne 1,\\ q_1=t, p_2=t,$ and $ta_2t=b_2,$ again contradicting the malnormality of $A$ in $G$.\n\nIf $\\ga_1=-1,\\ \\ga_2=-1,$ then $q_1=t,\\ p_2=v,$ and $v=a_2^{-1}tb_2\\in AtA,$\na contradiction.\n\nSo we are left with the possibility $m=n=1$.  In Figure \\ref{fig2} above, after cutting and pasting\nwe obtain the following figure \\ref{fig4}:\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild3.pspdftex}}\n    \\caption{}\n    \\label{fig4}\n\\end{figure} \n \n\n\\noindent\nIf $\\ga_1=1,$ then $p_1=t,\\ q_1=v,$ and if $\\ga_1=-1,$ then $p_1=v,\\ q_1=t$.  In\nboth cases $v\\in AtA,$ contrary to  the choice of $v$.\n\\medskip\n\n\\noindent\n{\\bf Case 2.} $k>0$.\n\nConsider Figure \\ref{fig5} below.\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild4.pspdftex}}\n    \\caption{}\n    \\label{fig5}\n\\end{figure} \n \nNotice that $f$-cancellations have to occur in the product $g^{-1}agb^{-1},$ since\nit is equal to $1$. Therefore,   at least one of the following cases must happen:\n\\begin{enumerate}\n\\item\n$m=0,\\ a=a_1,$ and $f^{-\\gd_1}$ cancels with $f^{\\gd_1}$ in the product $f^{-\\gd_1}g_1^{-1}a_1g_1f^{\\gd_1};$\n\n\\item\n$n=0,\\ b=b_1$ and $f^{\\gd_k}$ cancels with $f^{-\\gd_k}$ in the product $f^{\\gd_k}g_{k+1}b_1g_{k+1}^{-1}f^{-\\gd_k};$\n\n\\item\n$m>0,$ and $f^{-\\gd_1}$ cancels with $f^{\\ga_1}$ in the product $f^{-\\gd_1}g_1^{-1}a_1f^{\\ga_1};$\n\n\\item\n$m>0,$ and $f^{\\ga_m}$ cancels with $f^{\\gd_1}$ in the product $f^{\\ga_m}a_{m+1}g_1f^{\\gd_1};$\n\n\\item\n$n>0,$ and $f^{\\gd_k}$ cancels with $f^{\\gb_1}$ in the product $f^{\\gd_k}g_{k+1}b_1f^{\\gb_1};$\n\n\\item\n$n>0,$ and $f^{\\gb_n}$ cancels with $f^{-\\gd_k}$ in the product $f^{\\gb_n}b_{n+1}g_{k+1}^{-1}f^{-\\gd_k}.$ \n\\end{enumerate}  \nIn case (1), $a=a_1\\ne 1,$ so $g_1^{-1}a_1g_1=t\\text{ or }v$. Hence $a_1$\nis conjugate to an involution, which is impossible, as $A$ does not contain involutions.\n\nSimilarly, in case (2) $b=b_1\\ne 1,$ so $g_{k+1}b_1g_{k+1}^{-1}=t\\text{ or }v,$ again a contradiction.\n\nIn case (3) we have Figure \\ref{fig6} below,\n\\begin{figure}[h]\n    \\centering\n\\scalebox{0.70}{\\input{bild5a.pspdftex}}\n    \\caption{}\n    \\label{fig6}\n\\end{figure} \n\n\\noindent\nwhere $p,q\\in\\{1, t, v\\}$ by Britton's Lemma.  We define \n\\[\na'=a_2\\cdots a_mf^{\\ga_m}a_{m+1}a_1f^{\\ga_1}\\quad\\text{ and }\\quad h=qg_2\\cdots g_kf^{\\gd_k}g_{k+1}.\n\\]\nWe have $h^{-1}a'h=b,\\ a'$ is conjugate to $a$. So $a\\ne 1$ implies $a'\\ne 1$.  Also the $f$-length\nof $h$ is $k-1$.  \nNotice that\n$h=f^{-\\ga_1}a_1^{-1}g,$ and $h\\notin A_1$ since $f^{-\\ga_1}a_1^{-1}\\in A_1,$ and $g\\notin A_1$.\nWe obtained a contradiction to the minimality of $k$.\n\nThe remaining cases are handled in entirely the same way. \n\\end{proof}\n\\smallskip\n\n\\begin{proof}[Proof of Theorem \\ref{thm main hnn}]\\hfill\n\\medskip\n\n\\noindent\nBy Lemma \\ref{A1 is a free product}, part (1) holds, and by Proposition \\ref{prop malnormal hnn},\npart (2) holds.\n\\end{proof}\n\n \n \n\\section{The proof of Theorem \\ref{thm s2t in char 2}}\n\nIn this section we show how Theorem \\ref{thm s2t in char 2} of the\nintroduction follows from Theorem \\ref{thm main}.\n\nLet $G$ be a group with a malnormal subgroup $A$ such that $A$ contains no involutions.   \nAssume that $G$ is {\\it not}\\, $2$-transitive on the set of right cosets $A\\backslash G$.\nIf there exists an involution $t\\in G\\sminus A,$ set $G_0:=G,\\ A_0:=A$.\nOtherwise, let $G_0:=G*\\lan t\\ran,$ where $t$ is an involution, and let $A_0=A$.\nThen, by  \\cite[Corollary 4.1.5]{MaKS}, $G$ is malnormal in $G_0,$ and\nthen since $A$ is malnormal in $G,$ it is malnormal in $G_0$.\n\nWe now construct a sequence of groups $G_i$ and of subgroups\n$A_i\\le G_i,\\ i=0,1,2\\dots,$ having the following properties for all $i\\ge 0$:\n\\begin{enumerate}\n\\item\n$G_i\\le G_{i+1},$ and $A_i\\le A_{i+1};$\n\n\\item\n$A_i$ is malnormal in $G_i$ and $t\\in G_i\\sminus A_i;$\n\n\\item\n$A_i$ does not contain involutions;\n\n\\item\n$A_{i+1}\\cap G_i=A_i$;\n\n\\item\nfor each $v\\in G_i\\sminus A_i$ there\nexists an element $f_v\\in A_{i+1}$ such that  $A_{i+1}tf_v=A_{i+1}v.$\n\\end{enumerate}\n\nIn order to construct $G_{i+1}, A_{i+1}$ from $G_i, A_i$\nwe enumerate the set\\linebreak $G_i\\sminus A_i=\\{v_\\alpha:\\alpha< \\rho\\}$\nfor some ordinal $\\rho$.  For each ordinal $\\ga<\\gr$ we construct the pair \n$G_i^{\\ga},\\ A_i^{\\ga}$ and the element $f_{v_{\\ga}}\\in A_i^{\\ga}$ having\nthe following properties:\n\\begin{itemize}\n\\item[(i)]  $G_i^{\\gb}\\le G_i^{\\ga},$ for all ordinals $\\gb<\\ga;$\n\n\\item[(ii)] $A_i^{\\ga}$ is malnormal in $G_i^{\\ga}$ and $t\\in G_i^{\\ga}\\sminus A_i^{\\ga};$\n\n\\item[(iii)]  $A_i^{\\ga}$ contains no involutions;\n\n\\item[(iv)]  $A_i^{\\ga}\\cap G_i^{\\gb}=A_i^{\\gb}$ for all $\\gb<\\ga;$\n\n\\item[(v)]   $f_{v_{\\ga}}\\in A_i^{\\ga}$ and $A_i^{\\ga}tf_{v_{\\ga}}=A_i^{\\ga}v_{\\ga}$.\n\\end{itemize}\nWe let $G_i^0=G_i$ and $A_i^0=A_i$.  If $\\ga=\\gb+1,$\nwe construct $(G_i^{\\alpha},\\ A_i^{\\alpha}, f_{v_{\\ga}})$ from  $(G_i^{\\gb},\\ A_i^{\\gb})$\nas follows:\nIf there is some $f\\in A_i^{\\gb}$ with\n$A_i^\\gb tf = A_i^\\gb v_\\alpha$ we let $G_i^{\\alpha}=G_i^{\\gb},\\ $\n$A_i^{\\alpha}= A_i^{\\gb}$ and $f_{v_{\\ga}}=f$.\nOtherwise apply Theorem \\ref{thm main} to  $G_i^\\gb,\\ A_i^\\gb$ with $u=1$ and $v=v_\\alpha$\nto obtain the groups  $G_i^{\\alpha},\\ A_i^{\\alpha}$ and the element $f_{v_{\\ga}}\\in A_i^{\\ga}$.\nOf course, by construction, $A_i^{\\ga}$ contains no involutions\nand $A_i^{\\ga}\\cap G_i^{\\gb}=A_i^{\\gb}$. So (i)--(v) hold.\n\nFor a limit ordinal $\\alpha$ we put $G_i^{(\\ga,1)}=\\bigcup_{\\beta<\\alpha}\nG_i^\\beta,\\ A_i^{(\\ga,1)}=\\bigcup_{\\beta<\\alpha} A_i^\\beta$.\nWe now show that  when $\\ga$ is a limit ordinal $A_i^{(\\ga,1)}$ is malnormal in $G_i^{(\\ga,1)}$.\nNotice that for each ordinal $\\gb<\\ga$ and each $g\\in G_i^{\\gb}\\sminus A_i^{\\gb},$ \nwe have $g\\in G_i^{(\\ga,1)}\\sminus A_i^{(\\ga,1)}$.\nIndeed else take the minimal $\\gc<\\ga$ such that $g\\in A_i^{\\gc}$.  \nThen, by definition, $\\gc$ is not a limit ordinal,\nand $g\\in G_i^{\\gc-1}\\sminus A_i^{\\gc-1}$.\nSo $g\\in A_i^{\\gc}\\cap G_i^{\\gc-1}=A_i^{\\gc-1},$ \na contradiction.  This means that $A_i^{(\\ga,1)}\\cap G_i^{\\gb}=A_i^{\\gb},$\nfor all ordinals $\\gb<\\ga$.\n \n\nSuppose  now that $g^{-1}ag=b$ with $g\\in G_i^{(\\ga,1)}\\sminus A_i^{(\\ga,1)}$ and\n$a, b\\in A_i^{(\\ga,1)}$. Then, by the previous paragraph, there exists $\\gb<\\ga$ \nso that $a,b\\in A_i^{\\gb}$\nand $g\\in G_i^{\\gb}\\sminus A_i^{\\gb}$ and then we get a contradiction\nto the malnormality of $A_i^{\\gb}$ in $G_i^{\\gb}$.  Clearly\n$A_i^{(\\ga,1)}$ contains no involutions.  Next if there exists $f\\in A_i^{(\\ga,1)}$\nsuch that $A_i^{(\\ga,1)}tf=A_i^{(\\ga,1)}u_{\\ga}$ then we let \n$G_i^{\\ga}=G_i^{(\\ga,1)},\\ A_i^{\\ga}=A_i^{(\\ga,1)}$ and $f_{v_{\\ga}}=f$.\nElse we construct $G_i^{\\ga},\\ A_i^{\\ga}$ and $f_{v_{\\ga}}$ from \n$G_i^{(\\ga,1)},\\ A_i^{(\\ga,1)}$ using Theorem \\ref{thm main} with\n$u=1$ and $v=v_{\\ga}$ (just as in the construction above in the case\nof a non-limit ordinal).\nAgain we see that (i)--(v) hold.\n\nFinally put \n\\[\nG_{i+1}=\\bigcup_{\\alpha<\\rho} G_i^\\alpha,\\quad A_{i+1}=\\bigcup_{\\alpha<\\rho} A_i^\\alpha,\n\\]\n\\begin{center}\n\n\\end{center}\nand set \n\\[\n\\calg=\\bigcup_{i<\\omega}G_i,\\quad \\cala=\\bigcup_{i<\\omega}A_i\\quad\\text{and}\\quad X= \\cala\\backslash \\calg.\n\\]\nAs in the construction of $G_i^{(\\ga,1)},\\ A_i^{(\\ga,1)}$\nin the case where $\\ga$ is a limit ordinal, we see that $\\cala$ is malnormal in $\\calg$\nand that $\\cala\\cap G_i=A_i,$ for each $i<\\omega$.\nTo see that the action of $G$ on $X$ is $2$-transitive\njust note that any $v\\in \\calg\\sminus \\cala$ is contained in some $G_i$\nso that there is some $f_v\\in A_{i+1}\\subseteq\\cala$ with $A_{i+1} tf_v = A_{i+1}v$.\nSince $A_{i+1}\\le \\cala$ we see that $\\cala tf_v=\\cala v$ as required.\nSince $\\cala$ is malnormal in $\\calg$ the action of $\\calg$ on $X$\nis sharply $2$-transitive.  By construction, $\\cala$ contains no involutions.\n\nFinally, as is well known, if $\\calg$ contains a non-trivial abelian\nnormal subgroup, then necessarily all involutions in $\\calg$\ncommute with each other (see, e.g., \\cite[Remark 4.4]{GMS}).\nBut, by our construction, this is not the case in $\\calg$.\nIndeed, if $G_1=G_0*\\lan f_1\\ran$ is a free product, then $t$ does not\ncommute with $f_1^{-1}tf_1$.  \nSuppose that $G_1=\\lan G, f\\mid f^{-1}tf=v\\ran$ is\nan HNN extension. Let $s\\in G$ be an involution distinct from $t$\n(notice that $t$ is not in the center of $G$ since $A$ is malnormal in $G,$\nso such $s$ exists).\nThen $sf^{-1}sf$ and $f^{-1}sfs$ are in canonical form, so they are distinct,\nand the involutions $s$ and $f^{-1}sf$ do not commute\\footnotemark.\n\\footnotetext{Note that we could start with a group $G_0$  \nwhich already contains an involution that does not commute with $t$.  Then\nit would immediately follow that $\\calg$ does not split. We thank Uri Bader\nfor pointing this out.}\nThis completes the proof of Theorem \\ref{thm s2t in char 2}.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIn the standard cosmological paradigm, the formation of large-scale\nstructure is driven by the gravitational amplification of small\ninitial density fluctuations (e.g. Peebles 1980).  In addition to\ngravity, hydrodynamical processes can influence the formation and\nevolution of galaxies, groups and clusters of galaxies.  But since\nhydrodynamical effects play a minor role on scales larger than the\nsize of galaxy clusters, gravitational instability theory alone can\ndirectly relate the present day large-scale structure to the initial\ndensity field and provide the framework within which the observations\ncan be analyzed and interpreted.  Gravitational instability is a\nnonlinear process, making numerical methods an essential tool for\nunderstanding the observed large-scale structure.~\\footnote{\nPresent address~: Racah Institute of Physics, The Hebrew University, \nJerusalem Israel}\n\nThere are two complementary numerical approaches to studying\ncosmological structure.  The first relies on $N$-body techniques\ndesigned to solve an initial value problem in which the evolution of a\nself-gravitating system of massive particles is determined by forward\nnumerical integration of the Newtonian differential equations.\nBecause the exact initial conditions are unknown, comparisons between\nthese simulations and observations are mainly concerned with general\nstatistical properties.\n\nThe second approach works in the opposite direction, deriving from the\nobserved present-day distribution and peculiar motions of galaxies,\nand independently of the nature of the dark matter, certain features\nof the dynamics at earlier times.  The numerical action method (NAM)\nbelongs to this second category of approaches.  It arises from the\nobservation that the present-day distribution of galaxies, combined\nwith the reasonable assumption that their peculiar velocities vanish\nat early times, presents a boundary value problem that naturally lends\nitself to an application of Hamilton's principle in which stationary\nvariations of the action are found subject to the boundary conditions.\nThe result is a prediction of the full orbit histories of individual\ngalaxies, either with real space boundary conditions (Peebles 1989,\n1990, 1994, 1995) or, after a coordinate transformation, in redshift\nspace (Peebles {\\it et al.\\ } 2001, Phelps 2002).\n\nThe potential of NAM as a probe of galaxy dynamics and of cosmological\nparameters has been explored in a number of studies following the\nintroduction of the method in Peebles 1989.  Possible applications\ninclude the full nonlinear analysis of orbit histories of nearby\ngalaxies (Peebles 1990, 1994, 1995; Sharpe {\\it et al.\\ } 2001), recovering the\ninitial power spectrum of density fluctuations (Peebles 1996),\npredicting the values of cosmological parameters (Shaya {\\it et al.\\ } 1995),\nand estimating the proper motions of nearby galaxies (Peebles {\\it et al.\\ }\n2000).  Concerning the latter application, ground and space-based\nobservations will soon make possible the measurement of the full\nthree-dimensional velocities of many nearby galaxies and promise both\na rigourous test of NAM predictions and, given the additional\ndynamical constraints on galaxy motions, the possiblity of using NAM\nas a probe of individual masses of nearby galaxies.\n\nSince a central result of NAM, the past orbit histories of galaxies,\ncannot be confirmed by direct observations, $N$-body simulations\nprovide an important test of NAM and its key assumption that\ngalaxies can be approximated as discrete, non-merging objects\nthroughout their history.  It is desirable then to test NAM in a\nscenario which approximates the complexity of the observational\nsituation but where all of the relevant physical quantities are known.\nPrevious tests of NAM using $N$-body simulations have either been\nconfined to a few dark matter haloes at the scale of the Local Group\n(Branchini \\& Carlberg 1994, Dunn \\& Laflamme 1995), traced the paths\nof individual dark matter particles rather than extended haloes\n(Nusser \\& Branchini 2000), or used simulations which demonstrate in\nprincipal the ability of NAM to recover particle orbits to a high\ndegree of accuracy but which do not reproduce the full complexity of\nextended mass distributions (Phelps 2002).\n\nIn this paper we extend the tests of NAM to simulations at a scale\napproaching that of the local supercluster with a catalogue containing\nseveral hundred extended objects modelled as particles.  We begin with\nan overview of the relevant properties of the $N$-body simulation and\nthe halo catalogue we derived from it, and follow with details of the\nversion of NAM used here, which includes a novel approach to the\nassignment of halo masses.  We will then test the sensitivity of NAM\nboth as a probe of the total mass as well as of the linear bias, and\nexamine in some detail a representative solution, focusing on the\ncomparison between the NAM predictions and the actual halo orbits.\n\n\n\n\\section{The simulation}\n\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{omega.eps}}\n\\caption{The average matter density $\\Omega(<r)$ in a sphere of radius\n$r$ from the center of mass of the simulation, as a function of\nredshift. The arrows mark the nominal initial boundary ($r=26\\ {\\rm {\\it h}^{-1}Mpc}$)\nof the high-resolution region. The average density within $26\\ {\\rm {\\it h}^{-1}Mpc}$ is\nabout 10 per cent larger than the cosmic mean $\\om{m}=0.3$ at $z=0$.}\n\\label{fig:omega}\n\\end{figure}\n\nWe will test NAM in a $N$-body simulation of a $\\Lambda$CDM cosmology\nwith matter density $\\om{m}=0.3$, vacuum energy density\n$\\om{\\Lambda}=0.7$ (in units of the critical density $\\rho_{\\rm c}$),\nand a Hubble constant $h=0.7$ (in units of 100$\\ {\\rm km\\,s^{-1}\\,Mpc^{-1}}$). The\nsimulation has an initial fluctuation power spectrum of spectral index\n$n=1$ and a present-day normalization amplitude $\\sigma_8=0.9$. It was\nrun with the GADGET $N$-body code (Springel, Yoshida \\& White\n2001). It is part of a suite of simulations which zoom in on a\nspherical region of initial comoving radius 26$\\ {\\rm {\\it h}^{-1}Mpc}$ selected from a\nlarger parent simulation (see Stoehr 2003 for details).  This\n``zooming'' technique has the advantage of preserving the large-scale\npower of the parent simulation while allowing the inner region to be\nresolved at smaller scales.  In particular, the mass of the 168436\n``high-resolution'' particles which sample the inner region is\n$\\mhr=6.82\\times 10^{10}\\ {\\rm M_\\odot\/{\\it h}}$, in contrast to the 602272\n``low-resolution'' particles (each having a mass $\\gg \\mhr$) which\nsample the outer region.\n\nFig.~\\ref{fig:omega} shows the average matter density $\\Omega(<r)$\naveraged over a sphere of radius $r$ as a function of redshift.  At\nthe initial time step ($z=20$), the mean density in a sphere of radius\n$r=26\\ {\\rm {\\it h}^{-1}Mpc}$, $\\Omega(<26)$, is very close to $\\om{m}$, but increases\nby about 10 per cent from initial to final time.  The net mass inflow\nacross the region boundary has the potential to disrupt the\nreconstruction at the edge of the region and provides a further\nopportunity to test the sensitivity of NAM to the total mass.\n\n\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{hmass.eps}}\n\\caption{The friends-of-friends differential mass function of dark\nmatter haloes (in logarithmic bins $\\Delta\\log_{10}m=0.2$) computed\nfrom the snapshot at $z=0$ (histogram) compared  to the theoretical\nprediction of Sheth \\& Tormen (dashed curve).  The arrow indicates\n$M_\\star\\sim 10^{13}\\ {\\rm M_\\odot\/{\\it h}}$, or the typical mass of haloes collapsing\nat the present time. }\n\\label{fig:haloes}\n\\end{figure}\n\n\\subsection{The halo catalogue}\n\nThe principal objective of this paper is to test the ability of NAM to\nreconstruct the orbits of simulated dark matter (DM) haloes which resemble\nthe galaxy + halo distributions of the ``real world''. To this end we\nneed to specify the positions, velocities and masses of a {\\it fixed}\nnumber of haloes as a function of time, since our implementation of NAM\nconserves particle number.  However, haloes do not form\nat any single redshift, and mergers potentially play an important role in the\ndynamics. \n\nIn order to create a catalogue of a fixed number of haloes from the\ninner region of the simulation, we first identify all the DM haloes\nabove a given mass threshold $M_{\\rm min}$ in the final simulation\ntime step $z=0$, using a friends-of-friends (FOF) group-finding\nalgorithm based on the method given in Davis {\\it et al.\\ } (1985). Only groups\ncontaining at least 20 DM particles are classified as haloes,\nresulting in a mass threshold $M_{\\rm min}=1.36\\times 10^{12}\\ {\\rm M_\\odot\/{\\it h}}$. We\nused the standard linking length of 20 per cent of the mean\ninterparticle distance (Jenkins {\\it et al.\\ } 2001).  Using haloes defined\nwith a linking length of 30 per cent we found a significant\nimprovement in the accuracy of NAM reconstructions, which is not\nunexpected since this probes a more linear regime.  In this paper\nhowever we confine our analysis to the standard linking length since\nit most closely reproduces the expected mass function and thus gives\nthe best approximation to real galaxy haloes.\n\nThe center of mass (CM) positions and velocities of all those DM\nparticles which are associated with haloes at $z=0$ define the\nresulting catalogue of 576 objects which is the input to the numerical\naction code. The orbit histories of these objects are then defined as\nthe CM positions and velocities of the DM particles which belong to\nthe corresponding halo at $z=0$.  For our purposes the redshift\noutputs at $z$=20, 4, 3, 2, 1 and 0 were sufficient for comparison of\nhalo orbits with the action reconstruction.\n\nTo model the tidal field from the outer region of the simulation, we\ncreated an auxiliary particle catalogue from a random sample of 1000\nlow-resolution DM particles, multiplying the masses by the appropriate\nfactor of $277.9$ to bring the mass density in the outer region to\n$\\om{m} = .3$.  To reduce the computational burden, the positions of\nthese tidal field particles were fixed in the NAM reconstructions and\ntheir masses were evolved according to linear perturbation theory, as\ndescribed in Shaya {\\it et al.\\ } (1995).  Parallel sets of NAM solutions were\ncreated with and without this tidal field.\n\nThe final step is to identify one of the haloes near the center of the\nsimulation as the reference halo, and to build an input catalogue to\nNAM using the angular positions and redshifts relative to this halo.\nThe analysis below was repeated for two haloes within 8 Mpc\/h of the\nCM of the high-resolution region, as a check on the sensitivity to the\nposition and velocity of the reference halo, and since the results\nwere consistent we will confine our analysis to a single reference halo.\n\nIn Fig.~\\ref{fig:haloes} we show as a histogram the FOF differential\nmass function of DM haloes in the high-resolution region at $z=0$. The\narrow marks $M_\\star\\sim 10^{13}\\ {\\rm M_\\odot\/{\\it h}}$, the typical mass of haloes that have\njust formed by redshift $z=0$. As can be seen, the halo masses span\nmore than two orders of magnitude, with the largest  haloes having a\nmass $M\\gsim 10^{14}\\ {\\rm M_\\odot\/{\\it h}}$.  The dashed curve is the theoretical\nprediction of Sheth \\& Tormen (1999) for a $\\Lambda$CDM cosmology\nmatching that of the simulation. The good agreement shows that the\nhigh-resolution region of the simulation is a fair  reproduction of\nthe expected mass function.\n\nIn the top panel of Fig.~\\ref{fig:bias} we compare the\nauto-correlation  of DM particles (solid) and haloes (dashed), $\\xm$\nand $\\xh$ respectively.  In the lower panel we plot the bias\n$b^2\\equiv\\xh\/\\xm$. The dotted horizontal line marks $b=1$. Results are\nshown at redshift $z=0$ (square) and 1 (triangle).  Note that, on the\nscale $r\\gsim 2\\ {\\rm {\\it h}^{-1}Mpc}$, DM haloes are as biased as the matter, i.e.\n$b\\simeq 1$. This follows from the fact that at $z=0$ most of the\nhaloes have a mass $M\\lsim M_\\star\\approx 10^{13}\\ {\\rm M_\\odot\/{\\it h}}$ (see\nFig.~\\ref{fig:haloes}), where $M_\\star(z)$ is the typical mass of\nhaloes which collapse at redshift $z$.  The bias factor inferred from\nthe {\\it unsmoothed}  $\\xh$ and $\\xm$ characterizes  the clustering of\nDM  haloes.  We will propose below that it is possible using NAM to\ndirectly measure this bias factor when the total mass is known.\n\n\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{biasr.eps}}\n\\caption{ The auto-correlation of haloes (dashed\ncurve) and dark matter (solid curve) obtained from the unsmoothed\nhalo and matter distribution at $z=0$ (square) and $z=2$ (triangle). \nThe bias $b^2=\\xh\/\\xm$ is plotted in the lower panel. The horizontal\ndashed line shows $b=1$.}\n\\label{fig:bias}\n\\end{figure}\n\n\n\\section{The action reconstruction}\n\nThe implementation of NAM used in this paper is fully described in\nPeebles {\\it et al.\\ } (2001) and Phelps (2002). We refer the reader to\nappendix A for details.  NAM solves for the orbits of mass tracers\ngiven the boundary conditions that the object positions in redshift\nspace are fixed at the present time and that at the initial time $a^2\nm_i d {\\mathbf{x_i}}\/dt \\rightarrow 0$.  While some implementations of\nNAM expand the orbits in terms of suitable trial functions (Peebles\n1989, 1990, 1994; Nusser \\& Branchini 2000), the present\nimplementation relies on matrix inversion of a discrete form of the\nequations of motion (Peebles 1995, Phelps 2002), which allows the\ncreation of more complex orbits.  In this formulation the initial\nvelocities $a d {\\mathbf{x_i}}\/dt$ are not mathematically constrained to\nvanish.  Our approach to this problem will be discussed in the\nfollowing section.\n\nThe generation of solutions begins with a set of randomized halo\norbits with 10 time steps and relaxes to a target in the gradient of\nthe action many orders of magnitude below that of the initial guess by\niterated per-particle matrix inversion.  Different choices of initial\nrandomized orbits may yield different solutions, since uniqueness is\nnot guaranteed, and so for each choice of cosmological parameters a\nsuite of 8 solutions was generated and the solution with the lowest\n$\\chi^2$ (see below) was used.  Orbits with negative radial distances\nare allowed mathematically but in such cases the orbit is\nre-randomized and the matrix inversion procedure repeated until all of\nthe particles have positive distances.  The resulting orbit paths\nsatisfy the standard equations of motion, as confirmed by a simple\nleapfrog integration forward in time from the first time step of the\nsolution.  Solutions with intersecting orbits, which are also\nmathematically allowed, are effectively suppressed by modelling each\nparticle as a constant density sphere with radius proportional to the\ncube root of the mass, normalized so that a halo of mass $2.4\\times\n10^{11}\\rm M_\\odot$ has a radius of 180 kpc.\n\nTo more closely parallel the observational situation, where redshifts\nare known to greater accuracy than the distances, the standard action\nwas recast with a partial transformation of coordinates, as described\nin Phelps 2002, so that the redshift at $z=0$ is fixed as the boundary\ncondition instead of the radial distance.  The distances then emerge as\npredictions, from which a standard $\\chi^2$ measure is defined to\nmeasure the goodness of fit of the solutions:\n\\begin{equation}\n\\chi^2 = \\sum_i \\frac{ (\\mu_i^{mod} - \\mu_i^{cat})^2}{\\sigma_i^2},\n\\end{equation}\nwhere $\\mu_i^{mod}$ and $\\mu_i^{cat}$ are the predicted and catalogue\ndistance moduli for each particle and $\\sigma_i$ is the dispersion in\nthe observed distance moduli.  Since there are no observational errors\nin the simulation, we added to the input distance moduli a gaussian\nrandom error with a dispersion of $\\sigma_i=.2$ (10 per cent distance\nerrors) when computing contours in $\\chi^2$.\nAs the mean $\\chi^2$ may be dominated by a few particles with\nlarge distances errors that have settled on the wrong side of a\ntriple-value zone, we exclude the ten highest per-particle $\\chi^2$s in\nthe computation of the mean.  Excluding the tail of the\ndistribution was found to affect the magnitude but not the\nlocation of the minimum $\\chi^2$.\n\nSince our goal is the comparison of predicted and actual halo orbits,\nit was useful to define two further quantitative measures of the\nquality of the orbits, the first, ${\\overline{ \\Delta \\theta}}$, the\n``directional error'', being defined as the per-particle average angle\nbetween the predicted and actual halo velocity vectors at $z=0$, and\nthe second, ${\\overline{ \\Delta d}}$, the ``initial displacement error'',\nbeing the average per-particle distance between the actual and\npredicted halo positions at the first time step.\n\n\n\\subsection{Halo mass assignment:  background mass and linear growth factor}\n\nWhile each halo has a well-defined mass according to the FOF\nalgorithm, the total mass in haloes is only 37 per cent of the total\nmass in the simulation catalogue.  If each halo were simply assigned\nits bare FOF mass, then the mean density in the catalogue region as input\nto NAM would be too low, and in the NAM reconstruction the region\nwould behave dynamically like a local void. Since the boundary\ncondition in the action is the redshift, and since particles at a\ngiven distance would have higher redshifts in the reconstruction owing\nto the repulsive effect of the void, NAM would predict\nhalo distances that are too small.\n\nIn observational catalogues, where the galaxy luminosities are given\nand where $\\om{m}$ is assumed to be unknown, the standard approach is\nto multiply each galaxy by some constant factor, its mass to light\nratio M\/L, which may be a function of galaxy type.  The minimum in\n$\\chi^2$ as a function of M\/L will then be governed by a combination\nof two factors: the degree to which the total mass in the catalogue\nregion matches the background density set by $\\om{m}$, and the\ngravitational dynamics of galaxies, groups and clusters.  In practice\nthe dominant contribution to $\\chi^2$ is the former factor, which is\nlargely independent of the dynamics of the haloes themselves.\n\nIn an $N$-body simulation we define the ``halo mass multiplier''\n$\\mathcal{M}$, the analogue of M\/L in the observational catalogues, as\nthe global factor by which we multiply the halo masses, which in our\nsimulation gives a mass density of $\\om{m} = .3$ when ${\\mathcal{M}} =\n2.7$.  Since $\\om{m}$ is a known quantity, and since it is also known\nwhether the region under study is close to the global mean density, it\nis possible in this case to employ NAM as a probe not only of the\ntotal mass but also of the relative distribution of mass in haloes\nrelative to the background, i.e. the bias.  We do this by simply\nadding a smooth constant density component to the value of $\\om{m}$\nwhich is input to NAM so that, for any choice of $\\mathcal{M}$, the\ntotal mass density in the catalogue region is matched to the input\n$\\om{m}$.  When the total dark matter density in the catalogue region\nis close to the global mass density, as it is in our simulation, this\nis equivalent to multiplying $\\om{m}$ as it appears in eq. A15 by the\nfactor ${\\mathcal{M}}\/2.7$.  When ${\\mathcal{M}} = 2.7$ the smooth\ncomponent vanishes and all of the mass is located in the haloes as\nbefore.  With this reassignment of the catalogue mass, a search for a\nminimum in $\\chi^2$ in the space of $\\mathcal{M}$ becomes a test of\nthe linear bias relation between the underlying mass density and the\nhalo distribution.  To reiterate this important point: By assuming\nprior knowledge of $\\om{m}$, we can use NAM to probe the relative\ndistribution of dark matter in the haloes vs. the background.  We will\nmotivate this further in the following subsection.\n\n\nA further modification to the halo masses is demanded by the absence\nof an analytical zero velocity constraint at the initial time step.\nSince in addition the haloes are modeled as non-evolving point masses\nthroughout their histories, NAM solutions involving close interactions\nbetween haloes at high redshift are commonly seen.  While we have not\nfound a way, within the framework of the matrix inversion\nimplementation of the action, to write down an analytical initial-time\nzero-velocity constraint, we have found that initial velocities are\neffectively suppressed by the procedure, already employed in modeling\nthe tidal field, of rescaling the halo masses in time according to\nlinear perturbation theory.  We thus multiply the halo masses at each\ntime step by the standard linear growth factor (see Peebles 1980,\neq. 11.15), which is zero at $t=0$ and unity at $z=0$.  The value of\n$\\om{m}$ in eq. A15 is further rescaled by the same factor to maintain\nparity at each time step with the halo mass density.  This ad hoc\nprocedure forces the initial velocities to vanish, eliminates hard\ninteractions at early times, and significantly reduces the\nnonuniqueness in the solutions, but at the cost of producing halo\norbits with typical total displacements some 20 per cent shorter than\nthe actual paths in the simulation.  We conclude that there is room\nfor improvement in our method of defining halo masses, and anticipate\nthat some variation on the above theme, or the discovery of an\nanalytical zero-velocity constraint, will open the way to further\nimprovements in the reconstructions presented below.\n\n\n\\subsection{Biasing and the mass to light ratio}\n\nConsider a discrete distribution of objects (haloes) sampling the\nunderlying mass density field in a region of volume $V_0$ in the\nuniverse.  Let $n({\\bf x})$ be the number density of haloes and $\\rho_{\\rm\nm}({\\bf x})$ be the mass density field, at comoving position ${\\bf x}$.\nDenote the mean values of $n({\\bf x})$ and $\\rho_{\\rm m}({\\bf x})$ inside\n$V_0$ by $\\overline n$ and ${\\overline \\rho}_{\\rm m}$, respectively.  Further,\nchoose $V_0$ to be large enough such that ${\\overline \\rho}_{\\rm m}$ is\nvery close to the universal value $\\om{m}\\rho_{\\rm c} $.  Assume that\nthe halo and DM density fields are related by means of a\nlinear bias relation between the the density contrasts $\n\\delta_{\\rm h}({\\bf x})=n({\\bf x})\/{\\overline n}-1$ and $\\delta_{\\rm m}({\\bf x})= \\rho_{\\rm m}({\\bf x})\/ {\\overline\n\\rho}_{\\rm m} -1$.  We write this biasing relation as\n\\begin{equation}\n\\delta_{\\rm h}=b\\delta_{\\rm m} \\; .\n\\label{eq:bias}\n\\end{equation}\n\nWe will express the bias factor $b$ in terms of the assumed $\\mathcal{M}$\nof haloes as it is invoked in the formalism of NAM presented\nhere.  This is useful in connecting this formalism to standard\napplications of the linear theory of gravitational instability as well\nas to other implementations of NAM (e.g. Nusser \\& Branchini 2000).  A\nnatural way to proceed is to use the Poisson equation and the bias\nrelation (\\ref{eq:bias}) to write the peculiar acceleration, ${\\bf g}$, of an\nobject in terms of $\\delta_{\\rm h}$ and to compare that with the corresponding\nexpression derived from NAM.  The Poisson equation is\n\\begin{equation}\n{\\bf \\nabla}\\cdot {\\bf g}=-4\\pi G{\\overline \\rho}_{\\rm m}\\delta_{\\rm m} \\; ,\n\\label{poisson}\n\\end{equation}\nwhich upon integration and substituting  $\\delta_{\\rm h}=b\\delta_{\\rm m}$ gives\n\\begin{equation}\n{\\bf g}({\\bf x})=-G{\\overline \\rho}_{\\rm m} b^{-1}\\int{\\rm d}^3x' \\delta_{\\rm h}({\\bf x}'){\\bf K}({\\bf x},{\\bf x}')\n\\label{vg1}\n\\end{equation}\nwhere ${\\bf K}({\\bf x},{\\bf x}')=({\\bf x}-{{\\bf x}}')\/|{\\bf x}-{{\\bf x}}'|^3$.\nIn discrete form,\n\\begin{equation}\n{\\bf g}({\\bf x})=-G\\frac{{\\overline \\rho}_{\\rm m}} {{\\overline n}}b^{-1}\n\\sum_i {\\bf K}({\\bf x},{\\bf x}_i)+\\frac{4\\pi}{3}G{\\overline \\rho}_{\\rm m} {b}^{-1}{\\bf x}\n\\label{vg2}\n\\end{equation}\nNAM gives, when $a=1$,\n\\begin{equation}\n{\\bf g}_{\\rm a}=-G\n\\sum_i m_i {\\bf K}({\\bf x},{\\bf x}_i)+\\frac{4\\pi}{3}G{\\overline \\rho}_{\\rm mc} {\\bf x}\n\\label{vg3}\n\\end{equation}\nwhere $m_i$ is the mass assigned to object $i$ and $ {\\overline \\rho}_{\\rm\nmc}=\\sum_i m_i\/V_0$ is the mean mass density estimated from the\nobjets.  Define $m_{0i}$ such that for $m_i=m_{0i}$ we have ${\\overline\n\\rho}_{\\rm mc}={\\overline \\rho}_{\\rm m}$. In this case, a comparison\nbetween (\\ref{vg2}) and (\\ref{vg3}) implies $m_{0i}={\\overline \\rho}_{\\rm\nm}\/\\overline n$ and $b=1$.  Consider now ${\\overline \\rho}_{\\rm mc}\\ne {\\overline\n\\rho}_{\\rm m}$. The difference between the mean mass densities must be\nattributed to a uniformly distributed mass component which can be\nassociated with low (luminosity) objects that are not included in the\ncatalogue. Low mass objects are typically less biased than higher mass\nones (Mo \\& White 1996).  Comparison between the uniform terms in\n(\\ref{vg2}) and (\\ref{vg3}) implies\n\\begin{equation}\nb=\\frac{{\\overline \\rho}_{\\rm m}}{{\\overline \\rho}_{\\rm mc}} \\; , \\quad {\\rm and}\n\\quad\nm_i=b^{-1} m_{0i} \\; .\n\\label{b1}\n\\end{equation}\n\nIn most of the applications of NAM to real data, the masses $m_i$ are\nestimated from the luminosities, $L_i$, by $m_i={\\mathcal{M}} L_i$\nwhere $\\mathcal{M}$ is a global mass to light ratio. Assume that\n${\\overline \\rho}_{\\rm mc}={\\overline \\rho}_{\\rm m}$ is obtained for\n${\\mathcal{M}}={\\mathcal{M}}_0$. \nThen, for a biased distribution of objects, the relations\n(\\ref{b1}) imply that the same peculiar acceleration is obtained with\nthe global mass to light ratio\n\\begin{equation}\n{\\mathcal{M}}=b^{-1}{\\mathcal{M}}_0\\; .\n\\label{b3} \n\\end{equation}\nIn our simulation, since $b \\simeq 1$ on the scale $r\\gsim 2\\ {\\rm {\\it h}^{-1}Mpc}$ and\nsince ${\\mathcal{M}}_0 = 2.7$, we will expect to find the\nbest solutions when ${\\mathcal{M}} \\simeq 2.7$, i.e. when all the\nmass is located in the haloes.\n\n\n\n\\section{Results}\n\nWe searched for a minimum in $\\chi^2$ in the two-dimensional parameter\nspace of $\\om{m}$ and $\\mathcal{M}$, holding $H_0$ constant at 70, by\nfinding NAM solutions on a grid of 56 points and interpolating a\ncontour map in $\\chi^2$.  The results are shown in\nFig.~\\ref{fig:cont3}.  At 1$\\sigma$ the best solutions are found at\n$.27 < \\om{m} < .45$ and $1.7 < {\\mathcal{M}} < 3.9$, consistent with\nthe simulation parameter $\\om{m} = .3$ and the predicted\n${\\mathcal{M}} = 2.7$ (since $b \\simeq 1$).  A second set of solutions\nwas computed without the external tidal field, and its absence had\nlittle effect on the quality of the solutions or the location of the\nminimum in $\\chi^2$.  In an idealized scenario where the\nexact halo masses are known, and where the region being reconstructed\nis a fair sample of the global mean, NAM can thus correctly recover\nboth $\\om{m}$ and $b$ but with an uncertainty of about 50 per cent in\nboth values.  In an actual observational situation with a comparable\nnumber of particles, where significant uncertainties in the galaxy +\nhalo masses must additionally be taken into account, it is expected\nthat the uncertainties in the predicted values of $\\om{m}$ and $b$\nwill be even larger.  It should be remarked that the tightness of\nthese constraints is partially a function of the size of the\ncatalogue, since $sigma$ in a standard $\\chi^2$ measure is an inverse\nsquare-root function of the number of particles entering into the\ncomputation of $\\chi^2$.\n\n\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{contours.eps}}\n\\caption{Mean $\\chi^2$ contours showing the sensitivity of NAM to\n$\\om{m}$ and, through $\\mathcal{M}$, to the linear bias $b$.  Solid\ncontours lines are from solutions including the external tidal field\nand dotted contour lines are from a parallel set of solutions computed\nwithout the tidal field.  Contour levels in both cases are .06 or\n$1\\sigma$.  The lowest computed mean $\\chi^2$ of 1.3 (including the\ntidal field) was found at $\\om{m}=.3$, ${\\mathcal{M}} = 2.7$\n(corresponding to $b=1$).  At each grid point a set of 16 solutions\nwas computed, half with and half without the tidal field. For values\nof ${\\mathcal{M}}$ different from 2.7 a smooth component is added to\nensure that the total mass density in the catalog matches the\nbackground density.  The filled square at center shows the global\nsimulation parameters, assuming a linear bias of unity, and the nearby\nfilled circle the parameters in the catalog region, which is slightly\noverdense due to mass inflow across the catalog boundary.}\n\\label{fig:cont3}\n\\end{figure}\n\n\nA series of closer looks at the best obtainable NAM solution\n(Figs.~\\ref{fig:radialerror} through \\ref{fig:all_errors}), all\nplotted from the same solution with input parameters $\\om{m} = .3$,\n${\\mathcal{M}} = 2.7$, gives a good feel for the strengths and\nlimitations of the method.  Note that for this solution all of\nthe mass is located within the haloes.  Fig.~\\ref{fig:radialerror}\nshows in x-y projection the final positions of the haloes in the\nhigh-density region of the simulation (open circles, with radii\nproportional to the mass) and the error in the predicted positions\n(line segments).  Note that the distance errors are purely radial due\nto the boundary condition in the action.  The reconstruction tends to\nbe less accurate in the vicinity of massive haloes, which is due to\nthe presence of triple-value regions where NAM is easily confused.\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{radialerror.eps}}\n\\caption{catalogue vs. predicted halo positions at $z=0$ in x-y projection,\nfor the best case solution at $\\om{m}=.3$, ${\\mathcal{M}} = 3.2$.\nThe dotted outer circle (in green) marks the nominal $26\\ {\\rm {\\it h}^{-1}Mpc}$ boundary of the\nhigh-density region of the simulation.\nActual halo positions at the final time step are marked by circles with radii \nproportional to $m_i^{1\/3}$.  The free end of the line segments mark the\npositions predicted by NAM (red when the predicted distances are larger\nand blue when they are smaller); the length of the line segment thus reveals \nthe radial distance error.}\n\\label{fig:radialerror}\n\\end{figure}\nThis can be seen more clearly in Fig.~\\ref{fig:viewfromhome}, which\nshows a view of the ``sky'' from the reference halo, highlighting\nthose haloes with the largest $\\chi^2$.  Some of these haloes are\nfound near the edge of the catalog region, where interactions with\nparticles in the low-resolution region, which is only approximately\nmodelled by NAM, may disrupt the reconstruction.  The majority\nof the remaining haloes with high $\\chi^2$ are often found within\na few degrees of the line of sight to a massive halo; these have\ntypically settled on the wrong side of a triple-value region.  Haloes\nwith high $\\chi^2$ also tend to be found close to each other,\nsuggesting that if the dynamics of one influential halo are not\ncorrectly modelled it may disrupt the accurate reconstruction of\nseveral other nearby haloes.  Finally, a few small haloes with large\ndistances errors can be seen in relative isolation, indicating that\nthe interaction of haloes with unassociated dark matter particles, and the\ncomplex formation and merger history of the haloes themselves, may\nprevent accurate NAM reconstruction even in the absence of more\nobvious disruptive factors.\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{viewfromhome.eps}}\n\\caption{A view of the ``sky'' from the perspective of the reference\nhalo near the center of the simulation, highlighting the location of\nhaloes with poorly recovered distances in the best NAM solution.\nPoint size is proportional to the halo mass.  Filled points indicate\nhaloes with $\\chi^2 > 1$ (radial distance error greater than 10 per\ncent).  For haloes located towards the edge of the catalogue ($> 24\nMpc\/h$) these are indicated by filled squares, and towards the center\nof the catalogue ($< 24 Mpc\/h$), by filled circles.}\n\\label{fig:viewfromhome}\n\\end{figure}\n\nThe top panel of Fig.~\\ref{fig:scatter} shows the error in the\npredicted distances as a function of the distance from the reference\nhalo.  Here the excellent overall prediction of halo distances, with\ntypical errors of less than 3 per cent, can be seen most clearly.\nAny mismatch in the halo mass density relative to $\\om{m}$\nwould be revealed here by a tilt in the distance errors as a function\nof distance from the reference halo (with a positive slope indicating\nan overdensity and a negative slope an underdensity).  As the distance\nerrors trace a line with vanishing slope, this confirms that the total\nmass for a choice of ${\\mathcal{M}}=2.7$ is well matched to $\\om{m}$.\nThe bottom panel of Fig.~\\ref{fig:scatter} compares the distance\nerrors with those obtained assuming zero peculiar velocities and\nHubble-flow distances ($d = cz_i H_0$).  In the latter case the average\ndistance errors are 5 per cent.  The difference in the\nsharpness of the peaks in the two histograms gives an indication of\nthe ability of NAM to correctly model the interparticle dynamics.\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{ddscatter.eps}}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{histogram.eps}}\n\\caption{\n{\\it Top panel}~: Scatterplot of the error in radial distance predictions \nfor the best NAM solution, showing good overall recovery of the distances.\nPoint size is proportional to halo mass.\n{\\it Bottom panel}~: Histogram in $\\chi \\equiv (\\mu_i^{mod} - \\mu_i^{cat})\/\\sigma_i$\nof the same solution.  For comparison the dotted line shows the errors \nobtained when Hubble-flow distances are used in place of $\\mu_i^{cat}$.}\n\\label{fig:scatter}\n\\end{figure}\n\n\nFig.~\\ref{fig:orbitcomp2} compares the NAM-reconstructed halo orbits\nwith the actual halo orbits, the latter being defined previously as\nthe CM motion of the dark matter particles comprising the halo at\n$z=0$.  The reconstruction is more accurate for the heaviest haloes\n(top panel) than for the rest (bottom panel): for the former the\ndirectional error $\\overline {\\Delta \\theta} = 41^o$, while for the latter\n$\\overline {\\Delta \\theta} = 48^o$.  By comparison, the chance\nthat a random orbit will have a directional error of $48^o$ or less,\nthat is, that a point placed at random within a sphere will fall within the\nvolume of the cone swept out by an opening angle of $2 * 48^o$, is about 17 per cent.\nAccording to the directional error the quality\nof the orbit reconstructions, like $\\chi^2$, is not a particularly\nsensitive function of ${\\mathcal{M}}$: far from the $\\chi^2$ minimum\nat the same $\\om{m}$ and ${\\mathcal{M}} = 1$, $\\overline {\\Delta \\theta_i}\n\\simeq 50^o$ for the entire catalogue.  Similarly, the average initial\ndisplacement error $\\overline {\\Delta d_i}$, was 2.5 Mpc\/h for the best\nsolution at ${\\mathcal{M}} = 2.7$ , while far from the minimum at\n${\\mathcal{M}} = 1$ it was 2.9 Mpc\/h.  A further indication of the\napproximate character of the predicted orbits is that the magnitude of\nthe initial displacement error is comparable to the total distance\ntravelled by the typical halo orbit: The centre of mass of the average\nhalo in the simulation travelled 3.2 Mpc\/h, while the reconstructed\nhaloes travelled an average of 2.6 Mpc\/h, the shorter path lengths in\nthe reconstruction being a feature of our halo mass assignment scheme\nas discussed in section 3.1.  While this error may seem large, the\nchance that a random orbit with a total displacement of 2.6 Mpc\/h\nwill end up within 2.5 Mpc\/h of the actual initial position is only\n16 per cent (this is the volume overlap of two spheres of equal radii \nwhose centers are separated by a distance equal to their radii).\nA trial NAM\nreconstruction without the linear growth factor, assigning the full\nhalo mass at early times, gave similar values for the late-time\nmeasures $\\chi^2$ and $\\overline {\\Delta \\theta_i}$, while the early-time\nmeasure $\\overline{\\Delta d_i} \\sim 4.9 Mpc\/h$, or about twice the error.\nThe average total distance travelled by the haloes in these solutions\nwas 6.3 Mpc\/h, nearly twice as long as the actual halo paths and\nillustrative of the instability of the solutions when the haloes are\nallowed to keep their full masses in the initial time steps.\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{orbitcomphi-lpt.eps}}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{orbitcomplo-lpt.eps}}\n\\caption{x-y projection of actual (solid lines) and reconstructed\norbits (dotted lines) for the heaviest haloes (top panel) and for all\nother haloes (bottom panel).  Dotted lines indicate the actual\nhalo paths to $z=20$, while solid lines mark the paths\npredicted by NAM.  Actual halo positions are indicated by circles of\nradii proportional to the halo mass.  Straight radial dotted line\nsegments connect these positions to those predicted by NAM, as\nin Fig.~\\ref{fig:radialerror}.  Heavier haloes tend to have more\naccurately reconstructed orbits: For the heaviest haloes $\\overline {\\Delta\n\\theta_i} = 41^o$ and for the remaining haloes $\\overline {\\Delta \\theta_i}\n= 48^o$. }\n\\label{fig:orbitcomp2}\n\\end{figure}\n\nFinally, Fig.~\\ref{fig:all_errors} compares three measures of errors\nin NAM reconstructions: $\\overline{\\Delta d_i}$ and $\\overline {\\Delta\n\\theta_i}$ are plotted on the x and y axes, respectively, while the\npoint size is proportional to $\\chi^2$.  Haloes with $\\chi^2 > 1$ are\nshown as filled circles.  $\\chi^2$ is fairly well correlated to\n$\\overline{\\Delta d_i}$, indicating that reconstructed orbits which begin\nat positions well removed from their actual starting positions in the\nsimulation are likely to end with inaccurately modeled distances.\nSignificantly, $\\chi^2$ is poorly correlated to $\\overline {\\Delta\n\\theta_i}$.  This may have been expected since, in the absence of\nnearby haloes constraining the dynamics, the motion of a given halo in\nthe plane of the sky relative to the reference halo should be fully\ndegenerate.  The extent to which this degeneracy is broken and the\ntransverse orbital motions at the present epoch are correctly\nrecovered is a measure of the sensitivity of NAM to the dynamics\nbetween haloes.  The weak correlation between $\\chi^2$ and $\\overline\n{\\Delta \\theta_i}$ is the clearest evidence that the predicted halo\ndistances alone are not a sufficient discriminator of the quality of\nreconstructed orbits.  \n\n\n\n\\begin{figure}\n\\resizebox{0.48\\textwidth}{!}{\\includegraphics{errors.eps}}\n\\caption{Scatterplot showing errors in the direction of the\nreconstructed final-time velocities, as well as errors in the early\nand late time predicted positions, for each of the 533 particles in\nthe best NAM solution.  The x-axis shows $\\overline{\\Delta d_i}$, the\nerror in the initial placement of the haloes at the first timestep.\nThe y-axis shows $\\overline {\\Delta \\theta_i}$, the error in the\ndirection of the velocity vector at the final timestep.  Halo orbits\nwhich most accurately predict the direction of the halo orbit at the\npresent epoch are thus found at the top of the graph.  Point size is\nproportional $\\chi^2$.}\n\\label{fig:all_errors}\n\\end{figure}\n\n\n\\section{Discussion and conclusions}\n\nWe have shown, using a catalogue of over 500 dark matter haloes\nderived from a large $N$-body simulation at the scale of the local\nsupercluster, that it is possible with the numerical action method to\nreconstruct the full dynamical histories of dark matter haloes given\nthe masses and the redshift space coordinates at the present epoch.\nThe reconstruction is most successful in recovering the halo distances\nat the present epoch, with typical errors of less than 3 per cent.\nIndividual orbits paths, including the initial positions as well as\nthe direction of motion of the haloes at the present epoch, are\npredicted with less accuracy.  By varying the relative contributions\nto the total mass from the haloes and the background, we have also\nfound a way to use NAM to directly measure the linear bias when the\ntotal mass density is known, although with an uncertainty of about 50\nper cent.\n\nGiven the dynamical complexity of millions of interacting particles,\nand the sweeping nature of NAM's central simplifying assumption that\ngalaxy haloes can be approximated as discrete, non-merging point\nmasses throughout their evolution, it is remarkable how successfully\nthe dynamics of a many-body system can be reconstructed on the basis\nof an incomplete catalogue of facts.  The successes of NAM as it has\nbeen implemented here are of course partially offset by their\nweaknesses.  Among these is the relatively poor quality of the\nreconstruction in the vicinity of massive haloes, clearly seen in\nFig.~5, which shows a breakdown in the non-linear regime where NAM,\nwhich is itself a fully non-linear method, might have potentially\noffered the most insight.  In these regions $\\chi^2$ is a good\nindicator of poorly reconstructed orbits, but preliminary attempts to\nuse this information to nudge haloes into the correct orbits while not\nimposing any additional formal constraints have so far been\nunsuccessful.  A second concern is the inability of NAM in many cases\nto isolate, on the basis of $\\chi^2$ alone, predicted halo orbits which are\nmoving in the wrong direction at the present epoch.  Fig.~9 shows, for\nexample, 34 haloes moving more than $90^o$ in the wrong direction but\nwith good distances at the present epoch and thus low $\\chi^2$.\n\nThe above innacuracies may in part arise from\nthe details of our implementation, such as our ad hoc procedure of\nscaling of halo masses according to linear theory, and there is\ncertainly room here for improvement.  The scale of the catalog is also\na factor to consider, and in particlar the density of mass tracers\nwithin it.  This analysis should be repeated at the scale of the Local\nGroup, where a larger number of mass tracers acting within a smaller\nvolume may better constrain the dynamics and permit more accurate\norbit reconstructions.  It is also possible that, in dynamical systems\nof this complexity, the angular positions, redshifts and masses are by\nthemselves insufficient to lift the degeneracies in the halo orbits,\nand that the full three-dimensional velocities at the present epoch\nwill be needed to accomplish this.  This again is work to be\nundertaken at Local Group scales, where next-generation observations\nfrom SIM and GAIA hold out the promise of multiple galaxy proper\nmotion measurements with which to test the NAM predictions.  One\nrelated concern is that part of the proper motion data may be needed\nto recover accurate orbits, leaving fewer remaining free parameters to\nassist with the more weighty problem of constraining individual galaxy\nhalo masses, although it is possible that only one of the two\ncomponents of the tangential velocity will be sufficient to break the\norbtial degeneracy.  Finally, inaccuracies in the NAM predictions are\ndoubtless due at least in part to intrinsic limitations of the method\nand its assumptions, although we do not wish to suggest at this stage,\ngiven the work that remains to be done, that that an upper\nlimit on NAM accuracy in orbit reconstruction has yet been reached.\n\nWe anticipate that work on NAM in the near term will lie principally\nin two directions. The first is an extension of the above analysis, with\nfurther improvements in the implementation, to a high-resolution\nsimulation at the scale of the Local Group, where present-day\nthree-dimensional velocities can provide significant additional\ndynamical constraints.  The second is a direct comparison of\nNAM with other reconstruction methods, both in real\nspace (e.g, Nusser \\& Dekel 1992, Gramman 1993, Croft \\& Gazta\\~{n}aga\n1998, Frisch {\\it et al.\\ } 2002, Mohayaee {\\it et al.\\ } 2005) and redshift space\n(e.g., Narayanan \\& Weinberg 1998, Monaco \\& Efstathiou 1999, Mohayaee\n\\& Tully 2005), that help to bridge the present-day observations of\nlarge-scale structure with the initial conditions prevailing in the\nearly universe.\n\nWe acknowledge the support of the Asher Space Research Institute.  We\nwould like to thank Felix Stoehr for providing us with the snapshots\nfrom his simulation.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\nDue to its high error threshold and since it requires only nearest-neighbor gates to be performed, the surface code \\cite{Dennis2002,Raussendorf2007} is the most promising platform for scalable, fault-tolerant, and universal quantum computation \\cite{Fowler2012a}.\nIn order to test its resilience and benchmark the performance of classical algorithms for quantum error correction (QEC), the surface code is often studied with simplistic stochastic error models, where an error is an unphysical event that happens instantaneously at a specified point in space-time.\nFurthermore, it is usually assumed that these errors are not spatially correlated (see, e.g., Refs.~\\cite{Duclos2009,Wang2011,Fowler2012b,Wootton2012,Hutter2013,Fowler2013,Duclos2014}).\nIt is thus of importance to study to what degree these assumptions are satisfied for realistic models of a physical environment, and in case they are not, what the resilience of the surface code against the resulting effective error model is.\n\nIn this work, we will consider a surface code coupled to a thermal bath of freely propagating modes. A pair of recent articles \\cite{Novais2013,Jouzdani2013} studied the fidelity of the surface code in this setup (at zero temperature). \nThey showed that, under the assumption of a trivial error syndrome (all stabilizer operators of the code still yield a $+1$ eigenvalue), there is a sharp transition between maximal and minimal surface code fidelity as the coupling strength to the bath is increased. \nThis transition provides an upper bound to the resilience of the surface code, since a logical error with a trivial error syndrome certainly cannot be corrected.\n\nBy contrast, our goal here is to find the actual time when QEC in the surface code breaks down as a function of coupling and bath parameters.\nThis is the time at which an error correction algorithm is no longer able to pair the surface code defects in a way that leads to a trivial operation performed on the code subspace.\n\nIn order to find these times, we follow a three-step strategy. First, calculate the error rate on each individual qubit as a function of time and physical parameters.\nThere are three different physical mechanisms contributing to this error rate -- the direct interaction of each qubit with the bath, subluminal interactions mediated by the bath as well as superluminal ones.\nSecond, study numerically how spatial correlations between such errors affect the threshold error rate of the surface code.\nThird, solve for the times for which the single-qubit error rate reaches the modified threshold error rates.\n\nWhen deriving actual threshold estimates, Refs.~\\cite{Novais2013,Jouzdani2013} resort to the case of nearest-neighbor correlations only. \nHowever, we show that both subluminal and superluminal mediated long-range interactions can actually be the dominant error mechanism at the time for which the error rate reaches critical values.\n\n\n\n\\section{Problem and Overview}\\label{sec:problem}\n\nWe consider a surface code each qubit of which is coupled to a bosonic bath at thermal equilibrium. \nIn accordance with Refs.~\\cite{Novais2013,Jouzdani2013}, we only consider bit-flip errors here ($\\sigma^x$) and make the simplifying assumption that the bath is in thermal equilibrium at the beginning of each QEC cycle, i.e., \nthat bath correlations between different QEC cycles are negligible. Physically, this can be thought of as the bath thermalizing with an even larger bath during one QEC period. \nHowever, we generalize the discussion in Refs.~\\cite{Novais2013,Jouzdani2013} to the case of finite temperature.\n\nSums and  products with a tilde on top run over all surface code qubit indices $i$, while sums without a tilde are over bath modes $\\textbf{k}$.\nLet $H=H_0+V$ with \n\\begin{align}\\label{eq:H0}\nH_0 = H_{\\m{bos}} = \\sum_{\\bf{k}}\\omega_{\\bf{k}} a_{\\bf{k}}^{\\dagger} a_{\\bf{k}},\n\\end{align}\nand \n\\begin{align}\\label{eq:V}\nV=\\tilde{\\sum}_i\\sigma^x_i\\otimes\\frac{\\lambda}{\\sqrt{N}}\\sum_{\\bf{k}}|\\textbf{k}|^r\\left(e^{i{\\bf k}{\\bf R}_i}a_{\\bf{k}}+e^{-i{\\bf k}{\\bf R}_i}a_{\\bf{k}}^{\\dagger}\\right),\n\\end{align}\nwhere $a_{\\bf{k}}^{\\dagger}$ ($a_{\\bf{k}}$) are the standard creation (annihilation) operators obeying bosonic commutation relations.\nHere, $\\textbf{R}_i$ is the spatial location of qubit $i$ and $N=\\sum\\kk1$ is the number of bosonic modes of the bath. \nPhysically interesting are the cases $r=0,\\pm\\frac{1}{2}$ \\cite{Jouzdani2013}.\nWe consider a linear dispersion of the bath modes, $\\omega_{\\bf{k}}=v|{\\bf k}|$, as is accurate for acoustic phonons, spin-waves in an antiferromagnet, or electromagnetic waves. Here, $v$ is the corresponding velocity of the modes.\n\nLet the initial qubit density matrix be given by $\\rho_q$ and the thermal state of the bath by $\\rho_B\\propto\\exp(-\\beta H_{\\m{bos}})$,\nwhere $T=1\/\\beta$ is the bath temperature.\nThe surface code requires a set of commuting many-qubit Pauli operators, called \\emph{stabilizer operators}, to yield a $+1$ eigenvalue. \nAll of these operators are measured at the end of each QEC cycle.\nStabilizer measurements can be performed either by applying entangling gates between code and auxiliary qubits \\cite{Dennis2002,Fowler2012a} or by direct measurement of the corresponding many-qubit parity operators \\cite{DiVincenzo2012,Nigg2013}.\nEigenvalues $-1$ signal that an error has occurred and are interpreted as the presence of an \\emph{anyon}.\nQuantum information is stored in the subspace for which all stabilizers yield a $+1$ eigenvalue.\nCorrespondingly, the state $\\rho_q$ is restricted to this subspace, i.e., $\\rho_q$ is an anyon-free state.\nQEC is successful if the anyons are paired in a way which is homologically equivalent to the way they have been created.\nFinding such a pairing is the task of a classical error correction algorithm \\cite{Duclos2009,Wang2011,Fowler2012b,Wootton2012,Hutter2013,Fowler2013,Duclos2014}, one of which we will encounter in Sec.~\\ref{sec:errcorr}.\nFor more details about the surface code, see Ref.~\\cite{Fowler2012a}.\n\nThe decoherent evolution of the qubits is given by\n\\begin{align}\n \\rho_q \\mapsto \\Phi_d(\\rho_q) = \\mathop{\\mathrm{tr}}\\nolimits_B\\left\\lbrace e^{-iHt}(\\rho_q\\otimes\\rho_B)e^{+iHt}\\right\\rbrace\\ .\n\\end{align}\nAt the end of each QEC cycle, after some time $t$, we perform a measurement of all surface code stabilizer operators, which is described by the quantum channel \n\\begin{align}\n \\Phi_m(\\sigma) = \\sum_aP_a\\sigma P_a\\ .\n\\end{align}\nHere, $P_a$ projects onto the space with anyon configuration $a$ and the sum runs over all possible anyon configurations $a$.\n\nFinally, we study the state $\\rho_i(t)=\\mathop{\\mathrm{tr}}\\nolimits_{\\bar{i}}\\circ\\,\\Phi_m\\circ\\Phi_d(\\rho_q)$ of one particular qubit. Here, $\\mathop{\\mathrm{tr}}\\nolimits_{\\bar{i}}$ denotes a partial trace over all qubits except qubit $i$.\nSince $\\rho_q$ is an anyon-free state and the stabilizer measurement projects the density matrix of the qubits to the spaces with well-defined anyon numbers, \n$\\rho_i(t)$ has no contributions of terms $\\sigma^x_i\\rho_i$ or $\\rho_i\\sigma^x_i$ (here, $\\rho_i=\\mathop{\\mathrm{tr}}\\nolimits_{\\bar{i}}\\rho_q$).\nWe can thus write $\\rho_i(t)=(1-p_x(t))\\rho_i+p_x(t)\\sigma^x_i\\rho_i\\sigma^x_i$.\n\n\n\\setlength{\\unitlength}{0.75mm}\n\\begin{picture}(60,40)\n\\thicklines\n\n\\put(15,20){\\large $\\rho_q$}\n\n\\put(27,22){decoherence}\n\\put(26,20){\\vector(1,0){30}}\n\\put(37,15){$\\Phi_d$}\n\n\\put(65,27){syndrome}\n\\put(65,22){measurement}\n\\put(65,20){\\vector(1,0){30}}\n\\put(75,15){$\\Phi_m$}\n\n\\put(107,27){restrict to }\n\\put(107,22){$i$-th qubit}\n\\put(104,20){\\vector(1,0){30}}\n\\put(114,15){$\\mathop{\\mathrm{tr}}\\nolimits_{\\bar{i}}$}\n\n\\put(140,20){\\large $\\rho_i(t)=\\mathop{\\mathrm{tr}}\\nolimits_{\\bar{i}}\\circ\\,\\Phi_m\\circ\\Phi_d(\\rho_q)$}\n\\put(152.5,11){\\large $=(1-p_x(t))\\rho_i+p_x(t)\\sigma^x_i\\rho_i\\sigma^x_i$}\n\n\\end{picture}\n\n\nOur first goal is to calculate $p_x(t)$ as a function of the time $t$, the parameters in $H$, and the bath temperature $T=1\/\\beta$, which is what we carry out in Sec.~\\ref{sec:sc}. \nUsing the results from Sec.~\\ref{sec:sc}, we calculate in Sec.~\\ref{sec:entangling} the exact evolution of the density matrix of two qubits coupled to the bath and discuss the use of this bath coupling as an entangling gate.\n\nSecondly, we discuss what implications such an error rate has for surface code error correction. \nError correction will inevitably break down once the error rate $p_x$ on each qubit surpasses a certain critical value $p_c$.\nThis critical value depends on the spatial correlations between errors in the code, on the classical algorithm that is employed in order to find a pairing of the anyons, and on the probability $p_m$ with which a syndrome measurement fails.\nIn the symmetric case of $p_x=p_m$ and for uncorrelated errors, efficient error correction algorithms are able to perform successful error correction up to a critical value of $1.9\\% - 2.9\\%$ \\cite{Harrington2004,Duclos2014}.\nIn a more involved, circuit-based modelling of syndrome extraction, critical error rates are around $1\\%$ \\cite{Raussendorf2007,Wang2011,Fowler2012b}.\n\nThe higher $p_m$, the lower the probability of error $p_x$ for which successful correction is possible.\nFollowing Refs.~\\cite{Novais2013,Jouzdani2013}, we consider in the following the perfect measurement case $p_m=0$ for definiteness and simplicity.\nGeneralization to the more realistic case of $p_m>0$ is straightforward; it merely corresponds to replacing $p_c$ (or $\\tilde{p}_c$, see below) by a lower value.\n\nIf the errors on different qubits are independent from each other and stabilizer measurements are flawless ($p_m=0$), error correction inevitably breaks down if $p_x(t)>p_c=10.9\\%$ \\cite{Dennis2002}. \nFor $p_x(t)<p_c$ the probability of an error is exponentially small in $L$, the linear size of the code, if quantum error correction is performed optimally.\nThe problem of performing error correction in the surface code with perfect syndrome measurements can be mapped to the classical Ising model with erroneous qubits corresponding to antiferromagnetic bonds.\nThe critical value $p_c$ corresponds to an order-disorder transition in this model \\cite{Dennis2002}.\n\nFor uncorrelated errors, the maximal duration $\\tau$ of one QEC  cycle can thus be obtained by simply inverting $p_x(\\tau)=p_c$, which we exemplify for an Ohmic bath in Sec.~\\ref{sec:uncorr}. \nAlternative bath types are discussed in Appendix~\\ref{app:different}.\nWhen the errors on different qubits are not independent, the breakdown of error correction will in general occur at a single-qubit error probability $\\tilde{p}_c$ different from $p_c$.\nIf the correlations between the errors on different qubits are ignored, $\\tilde{p}_c$ may be lower than $p_c$.\nOn the other hand, taking knowledge about such correlations properly into account can even increase $\\tilde{p}_c$ beyond $p_c$.\nWe present an efficient algorithm that is capable of doing this for a specific kind of correlations in Appendix~\\ref{app:algo}.\nHowever, we do not know the value of $\\tilde{p}_c$ for the kind of correlations between errors that arise from coupling to the bosonic bath.\nStill, solving $p_x(\\tau)=\\tilde{p}_c$ for $\\tau$ will provide us with the correct scaling of $\\tau$ as a function of physical parameters like the bath temperature.\n\nFurthermore, in Sec.~\\ref{sec:errcorr} we numerically find values for $\\tilde{p}_c$ for different kinds of spatial correlations between errors and provide heuristic evidence \nthat the value of $\\tilde{p}_c$ for the errors arising due to the bath coupling does not differ drastically from $p_c$.\nWe also show that for correlated two-qubit errors the surface code can, due to being a degenerate code, be used to perform error correction in regimes where the entropy in the noise exceeds the information obtained from stabilizer measurements\n -- which is in contrast to the uncorrelated case.\n\nThe resulting maximal QEC cycle times $\\tau$ for the more general case of correlated errors are derived in Sec.~\\ref{sec:qectime}.\nThe obtained expressions for $\\tau$ for a variety of different parameter regimes are summarized in Sec.~\\ref{sec:summary}.\nWe conclude in Sec.~\\ref{sec:conclusions}.\n\n\n\n\n\\section{The single-qubit error rate $p_x(t)$}\\label{sec:sc}\n\nIn this section, we calculate exactly the joint unitary dynamics of the qubits in the surface code and the modes in the bosonic bath.\nFrom this, we derive the probability of an error on each qubit $p_x(t)$ as a function of time, taking into account all correlations with errors affecting other qubits.\n\nWe have \n\\begin{align}\\label{eq:Phid}\n \\Phi_d(\\rho_q) &= \\mathop{\\mathrm{tr}}\\nolimits_B\\left\\lbrace e^{-iHt}(\\rho_q\\otimes\\rho_B)e^{+iHt}\\right\\rbrace \\nonumber\\\\\n&= \\mathop{\\mathrm{tr}}\\nolimits_B\\left\\lbrace e^{iH_0t}e^{-iHt}(\\rho_q\\otimes\\rho_B)e^{+iHt}e^{-iH_0t}\\right\\rbrace \\nonumber\\\\\n&= \\mathop{\\mathrm{tr}}\\nolimits_B\\left\\lbrace U(t)(\\rho_q\\otimes\\rho_B)U(t)^{\\dagger}\\right\\rbrace\\ ,\n\\end{align}\nwhere $U(t)=e^{iH_0t}e^{-iHt}=\\mathcal{T}e^{-i\\int_0^t\\m{d}t'\\,V(t')}$ denotes the evolution operator in the interaction picture.\nIt follows directly from the Magnus expansion (cf.\\,Ref.~\\cite[Appendix A]{Jouzdani2013}) \nand the fact that $\\left[V(t_1),[V(t_2),V(t_3)]\\right]=0$ that\n\\begin{align}\\label{eq:Ut}\n U(t) &= \\exp\\left\\lbrace -i\\int_0^t\\m{d}t_1\\,V(t_1) -\\frac{1}{2}\\int_0^t\\m{d}t_1\\int_0^{t_1}\\m{d}t_2\\,[V(t_1),V(t_2)] \\right\\rbrace \\nonumber\\\\\n&=: \\exp\\left\\lbrace \\tilde{\\sum}_i\\sigma^x_i\\otimes X_i(t) \\right\\rbrace \\exp\\left\\lbrace -\\frac{i}{2}\\tilde{\\sum}_{ij}J_{ij}(t)\\sigma^x_i\\otimes\\sigma^x_j\\right\\rbrace\\ .\n\\end{align} \nWe have defined\n\\begin{align}\n X_i(t) = \\frac{\\lambda}{\\sqrt{N}}\\sum_{\\bf{k}}\\frac{|\\textbf{k}|^r}{\\omega_{\\bf{k}}}\\left(e^{i{\\bf k}{\\bf R}_i}(e^{-i\\omega_{\\bf{k}} t}-1)a_{\\bf{k}}-e^{-i{\\bf k}{\\bf R}_i}(e^{i\\omega_{\\bf{k}} t}-1)a_{\\bf{k}}^{\\dagger}\\right)\n\\end{align}\nand \n\\begin{align}\\label{eq:Jij}\n J_{ij}(t) &= -i\\frac{\\lambda^2}{N}\\sum_{\\bf{k}}|{\\bf k}|^{2r}\\int_0^t\\m{d}t_1\\int_0^{t_1}\\m{d}t_2\\left\\lbrace e^{i{\\bf k}({\\bf R}_i-{\\bf R}_j)}e^{-i\\omega_{\\bf{k}}(t_1-t_2)} - \\m{c.c.}\\right\\rbrace \\nonumber\\\\\n&= 2\\lambda^2\\int\\m{d}{\\bf k}\\,\\frac{|{\\bf k}|^{2r}}{\\omega_{\\bf{k}}^2}\\cos\\left({\\bf k}({\\bf R}_i-{\\bf R}_j)\\right)\\left(\\sin(\\omega_{\\bf{k}} t)-\\omega_{\\bf{k}} t\\right)\\ . \n\\end{align}\nIn Appendix~\\ref{sec:induced}, we provide the functions $J_{ij}(t)$ for different bath types (i.e., different combinations of spatial dimension, $D=2,3$, and bath coupling, $r=0,\\pm\\frac{1}{2}$).\n\nIt is straightforward to show that $[X_i(t),X_j(t)]=0$ and thus we can also write \n\\begin{align}\\label{eq:U}\n U(t) &= \\tilde{\\prod}_i\\exp\\left\\lbrace\\sigma^x_i\\otimes X_i(t) \\right\\rbrace\\tilde{\\prod}_{\\lbrace i,j\\rbrace}\\exp\\left\\lbrace -iJ_{ij}(t)\\sigma^x_i\\otimes\\sigma^x_j\\right\\rbrace \\nonumber\\\\\n&= \\tilde{\\prod}_i\\left(\\cosh(X_i(t))+\\sigma^x_i\\otimes\\sinh(X_i(t))\\right)\\tilde{\\prod}_{\\lbrace i,j\\rbrace}\\left(\\cos(J_{ij}(t))-i\\sin(J_{ij}(t))\\sigma^x_i\\otimes\\sigma^x_i\\right)\\ .\n\\end{align}\nThe product $\\tilde{\\prod}_{\\lbrace i,j\\rbrace}$ is over all pairs $\\lbrace i,j\\rbrace$, i.e., without double-counting.\n\nWe will refer to the first factor in Eqs.~(\\ref{eq:Ut}) and (\\ref{eq:U}) as the \\emph{decoherent} part of the evolution, and to the second part as the \\emph{coherent} part.\nNote that only the decoherent part of the evolution will lead to a dependence of the evolution of the code on the state of the bath (in particular its temperature). \nThe coherent part is, in principle, reversible and does not lead to a transfer of quantum information from the code qubits into the bath.\n\n\nInserting \\eqref{eq:U} into \\eqref{eq:Phid} and expanding the products can only be done if the number of qubits coupled to the bath is small.\nIn Sec.~\\ref{sec:entangling} we consider the case of two qubits coupled to the same bath and calculate the exact evolution of the two-qubit density matrix. \nHowever, if the number of qubits coupled to the bath is large, we need to follow a different route.\nNote that we are only interested in whether a net-error (i.e., an odd number of $\\sigma^x$-errors) occurs on qubit $i$ after application of $\\Phi_d$ and $\\Phi_m$.\nThis probability can be found with an inductive argument over $N_q$, the number of qubits in the code.\n\nSince $\\rho_q$ is a state with no anyons, the syndrome measurement $\\Phi_m$ eliminates all terms in $\\Phi_d(\\rho_q)$ that apply a different tensor product of Pauli errors `to the left' and `to the right' of $\\rho_q$.\nFormally, let $\\ell$ label the $2^{N_q}$ possible configurations of $\\sigma^x$ errors on the code and let $\\xi_{\\ell}$ denote the $\\ell$-th error configuration. \nThen,\n\\begin{align}\n \\Phi_m\\left(\\xi_{\\ell_1}\\rho_q\\xi_{\\ell_2}^{\\dagger}\\right) = \\delta_{\\ell_1\\ell_2}\\xi_{\\ell_1}\\rho_q\\xi_{\\ell_1}^{\\dagger}\\ .\n\\end{align}\nLet us call terms which have the same tensor products of Pauli operators on the left and on the right and hence survive application of $\\Phi_m$ `valid' terms.\n\nConsider first the case $N_q=1$.\nThen we simply have\n\\begin{align}\n \\Phi_d(\\rho_q) = \\left\\langle\\cosh^2\\left(X_i(t)\\right)\\right\\rangle \\rho_q - \\left\\langle\\sinh^2\\left(X_i(t)\\right)\\right\\rangle \\sigma^x\\rho_q\\sigma^x\\ .\n\\end{align}\n(Note that $X_i(t)$ is anti-Hermitian, so $\\left(\\sinh\\left(X_i(t)\\right)\\right)^{\\dagger}=-\\sinh\\left(X_i(t)\\right)$.)\nWe have introduced the notation $\\langle O\\rangle = \\mathop{\\mathrm{tr}}\\nolimits_B\\lbrace O\\rho_{B}\\rbrace$.\nLet us thus define the single-qubit decoherence rate by $p_d(t) = -\\left\\langle\\sinh^2\\left(X_i(t)\\right)\\right\\rangle$.\n\n\nLet $p_x(t)$ denote the error probability on qubit $1$. In the case of the surface code, this is then up to boundary effects the error probability on all other qubits as well.\nThe error probability $p_x(t)$ is the total probabilistic weight of all valid terms that apply an odd number of errors to qubit $1$.\nLet $p_x(t)_{N_q}$ denote the probability of an error on qubit $1$ if there is a total number of $N_q$ qubits in the code.\nClearly, we have $p_x(t)_1=p_d(t)$. When increasing $N_q\\mapsto N_q+1$, the parity of errors on qubit $1$ is only changed if a pair of errors is applied to qubit $1$ and qubit $N_q+1$.\nThe weight of this happening is $\\sin^2(J_{1,N_q+1}(t))$, while the weight of it not happening is  $\\cos^2(J_{1,N_q+1}(t))$. This leads to the recursive formula\n\\begin{align}\n p_x(t)_{N_q+1} = \\cos^2(J_{1,N_q+1}(t))p_x(t)_{N_q} + \\sin^2(J_{1,N_q+1}(t))(1-p_x(t)_{N_q})\\ .\n\\end{align}\nLet sums and products with a prime run over all qubits except qubit $1$, i.e., from $2$ to $N_q$.\nThe solution is then evidently given by\n\\begin{align}\\label{eq:pxt}\n p_x(t)_{N_q} &= \\prod_i\\!^{'}\\cos^2(J_{1i}(t)) \\times \\nonumber\\\\\n&\\qquad\n\\left\\lbrace\n  p_d(t)\\sum_{\\substack{m_i\\in\\lbrace0,1\\rbrace\\\\ \\sum'_im_i\\equiv\\,0\\,(\\m{mod}\\, 2)}} \\prod_i\\!^{'}(\\tan^2(J_{1i}(t)))^{m_i} \n + (1-p_d(t))\\sum_{\\substack{m_i\\in\\lbrace0,1\\rbrace\\\\ \\sum'_im_i\\equiv\\,1\\,(\\m{mod}\\, 2)}} \\prod_i\\!^{'}(\\tan^2(J_{1i}(t)))^{m_i}\n\\right\\rbrace\\ .\n\\end{align}\n\n\n\n\n\\section{Evolution of a two-qubit density matrix coupled to the bath}\\label{sec:entangling}\n\nConsider two qubits $i$ and $j$ at locations ${\\bf R}_i$ and ${\\bf R}_j$, respectively, that are coupled to a bosonic bath.\nWe assume them to be uncorrelated with the bath at $t=0$, $\\rho(0)=\\rho_{ij}\\otimes\\rho_B$.\nThe evolution of the two-qubit density matrix can be found using Eqs.~(\\ref{eq:Phid}) and (\\ref{eq:U}).\nWe keep the technicalities in Appendix~\\ref{app:coupling} and present here the final result for the state of the two-qubit density matrix after some time $t$.\nWe have\n\\begin{align}\\label{eq:2qubitfinal}\n \\rho_{ij}(t) &= \\left(\\frac{1}{4}(1+e^{-4\\Lambda(t)}\\cosh(4C_{ij}(t))) +\\frac{1}{2}e^{-2\\Lambda(t)}\\cos(2J_{ij}(t))  \\right) \\times \\rho_{ij} \\nonumber\\\\&\\quad\n+\\left(\\frac{1}{4}(1-e^{-4\\Lambda(t)}\\cosh(4C_{ij}(t))\\right) \\times (\\sigma^x_i\\rho_{ij}\\sigma^x_i + \\sigma^x_j\\rho_{ij}\\sigma^x_j) \\nonumber\\\\&\\quad\n+\\left(\\frac{1}{4}(1+e^{-4\\Lambda(t)}\\cosh(4C_{ij}(t))) -\\frac{1}{2}e^{-2\\Lambda(t)}\\cos(2J_{ij}(t))  \\right) \\times \\sigma^x_i\\sigma^x_j\\rho_{ij}\\sigma^x_i\\sigma^x_j \\nonumber\\\\&\\quad\n+\\left(-\\frac{i}{2}e^{-2\\Lambda(t)}\\sin(2J_{ij}(t))\\right) \\times \\left(\\sigma^x_i\\sigma^x_j\\rho_{ij}-\\rho_{ij}\\sigma^x_i\\sigma^x_j\\right) \\nonumber\\\\&\\quad\n+\\left(\\frac{1}{4}e^{-4\\Lambda(t)}\\sinh(4C_{ij}(t))\\right) \\times \\left(\\sigma^x_i\\sigma^x_j\\rho_{ij}+\\rho_{ij}\\sigma^x_i\\sigma^x_j-\\sigma^x_i\\rho_{ij}\\sigma^x_j-\\sigma^x_j\\rho_{ij}\\sigma^x_i\\right)\\ .\n\\end{align}\nHere, $J_{ij}(t)$ is as defined in \\eqref{eq:Jij} and\n\\begin{align}\n C_{ij}(t) = \\langle X_i(t)X_j(t)\\rangle = -\\frac{\\lambda^2}{N}\\sum_{\\bf{k}}|{\\bf k}|^{2r}\\cos\\left({\\bf k}({\\bf R}_i-{\\bf R}_j)\\right)\\coth(\\beta\\omega_{\\bf{k}}\/2)\\frac{\\sin^2(\\omega_{\\bf{k}} t\/2)}{(\\omega_{\\bf{k}}\/2)^2}\\ .\n\\end{align}\nFurthermore, we introduced the non-negative function\n\\begin{align}\n \\Lambda(t) = - C_{ii}(t) \\geq 0\\ .\n\\end{align}\nIt characterizes the decoherence of each individual qubit due to its coupling to the bath and will be discussed in more detail in the next section.\n\nUnlike the functions $J_{ij}(t)$, the functions $C_{ij}(t)$ depend on temperature. \nFor $i\\neq j$, they are in general hard to evaluate at finite temperature.\nAt zero temperature, they have been calculated for 2D baths in Ref.~\\cite{Jouzdani2013}.\nIn the rest of this work, we will follow Ref.~\\cite{Novais2013} and focus on a bath with $r=0$, $D=2$, corresponding to an Ohmic bath.\nFor this case, the correlator $C_{ij}(t)$ evaluates \\emph{at zero temperature} to \n\\begin{align}\n C_{ij}(t) = -\\frac{\\lambda^2}{\\pi v^2} \\theta(vt-R)\\text{arccosh}(vt\/R)\\ .\n\\end{align}\nAs it turns out, however, the single-qubit error rate $p_x(t)$ depends only on the functions $\\Lambda(t)$ and $J_{ij}(t)$, but not on $C_{ij}(t)$ for $i\\neq j$.\nFor example, one easily verifies that the partial trace $\\rho_i(t)$ of \\eqref{eq:2qubitfinal} is independent of $C_{ij}(t)$ and, using \\eqref{eq:pd} below, that the probability for a $\\sigma^x$-error agrees with \\eqref{eq:pxt} for $N_q=2$.\nThis allows us in the following sections to evaluate $p_x(t)$ without knowing the functions $C_{ij}(t)$ for $i\\neq j$.\n\n\n\\subsection{Bath coupling as an entangling gate}\nRecently, the idea of perfoming entangling gates between two qubits by coupling them to an ordered ferromagnet (which can be seen as a ``magnon bath'') and exploiting the mediated interaction has been studied in Ref.~\\cite{Trifunovic2013}.\nThe availability of entangling gates between nearest-neighbor qubits is crucial for the circuit-based implementation of the surface code \\cite{Dennis2002,Wang2011,Fowler2012a}.\nUsing the above result, it is straightforward to evaluate the fidelity of such a gate.\nFor concreteness, let us study the fidelity of maximally entangled two-qubit states (ebits) obtained using such a gate.\n\nConsider the initial state $\\rho_{ij}=\\proj{0}_i\\otimes\\proj{0}_j$ and the maximally entangled states $\\ket{\\psi^\\pm}=\\frac{1}{\\sqrt{2}}(\\ket{0}_i\\ket{0}_j\\pm i\\ket{1}_i\\ket{1}_j)$.\nThen,\n\\begin{align}\n \\bra{\\psi^\\pm}\\rho_{ij}(t)\\ket{\\psi^\\pm} = \\frac{1}{4}(1+e^{-4\\Lambda(t)}\\cosh(4C_{ij}(t))) \\mp\\frac{1}{2}e^{-2\\Lambda(t)}\\sin(2J_{ij}(t))\\ .\n\\end{align}\nAt times for which $J_{ij}(t)$ is an odd multiple of $\\pi\/4$, we obtain ebits with fidelity $\\frac{1}{4}(1+e^{-4\\Lambda(t)}\\cosh(4C_{ij}(t)))+\\frac{1}{2}e^{-2\\Lambda(t)}$.\nFor nearby qubits, $C_{ij}(t)\\simeq-\\Lambda(t)$, such that the fidelity simplifies to $\\frac{3}{8}+\\frac{1}{8}e^{-8\\Lambda(t)}+\\frac{1}{2}e^{-2\\Lambda(t)}$.\nHigh-fidelity ebits can thus only be obtained for times $t$ such that $\\Lambda(t)\\ll1$. The gate is only useful if $J_{ij}(t)$ reaches $\\pi\/4$ in such times.\n\nNote that the magnon bath considered in Ref.~\\cite{Trifunovic2013} has a dispersion which is parabolic rather than linear, as assumed in this work.\nFor a 2D Ohmic bath ($r=0$, $D=2$), the function $J_{ij}(t)$ can be calculated as described in Ref.~\\cite[Appendix C]{Jouzdani2013} and evaluates to\n\\begin{align}\\label{eq:Jijspec}\n J_{ij}(t) =\n \\frac{\\lambda^2}{2\\pi^2v^2}\\left(\\theta(R-vt)\\arcsin(vt\/R) + \\theta(vt-R)\\frac{\\pi}{2}\\right)\\ , \n\\end{align}\nwhere we have defined $R:=|{\\bf R}_i-{\\bf R}_j|$. \nNote that $J_{ij}(t)$ reaches a stationary value of $\\frac{\\lambda^2}{4\\pi v^2}$ for times $t$ such that $vt>|{\\bf R}_i-{\\bf R}_j|$.\nChoosing $\\lambda=\\pi v$ thus produces ebits with fidelity $\\simeq1-2\\Lambda(t)$ for times such that $vt>|{\\bf R}_i-{\\bf R}_j|$.\nHigh-fidelity ebits are obtained in the time-interval for which $vt>|{\\bf R}_i-{\\bf R}_j|$ and $\\Lambda(t)\\ll1$, if this interval exists.\n\nBaths in 3D behave very differently in this respect: for all values of $r=0,\\pm\\frac{1}{2}$, $J_{ij}(t)$ grows linearly with $t$ for $t>R\/v$ in 3D (see Appendix~\\ref{sec:induced}).\nSimilarly, $J_{ij}(t)$ grows linearly with $t$ for large enough $t$, see Sec.~\\ref{sec:parabolic}.\nIn these cases, ebits can be obtained by maintaining the bath-coupling for a certain amount of time.\n\n\n\n\n\n\n\\section{Maximal QEC cycle time for uncorrelated errors}\\label{sec:uncorr}\n\nLet us now first consider the simple case where the noise on the different qubits is uncorrelated, which is relevant if the qubits are sufficiently far apart from each other such that each qubit effectively couples to its ``private bath''.\nNote that for the noise to be uncorrelated, it is not enough to require that $J_{ij}(t)$ vanish for all $i$ and $j$.\nThe decoherent part of the evolution, too, leads to correlations between the errors on different qubits, which can be quantified by correlators $\\langle X_i(t)X_j(t)\\ldots X_m(t)\\rangle$.\nUncorrelated noise requires that both $J_{ij}(t)\\approx0$ and $C_{ij}(t)=\\langle X_i(t)X_j(t)\\rangle\\approx0$ for all $i\\neq j$.\nIn this case, we simply have $p_x(t)=p_d(t)$ for each qubit.\n\nSince $X_i(t)$ is linear in the creation\/annihilation operators of the bath, we can apply Wick's theorem to calculate thermal expectation values of products of the operators $X_i(t)$.\nI.e., \n\\begin{align}\n \\left\\langle X_i(t)^{2k}\\right\\rangle = \\frac{(2k)!}{2^kk!}\\left\\langle X_i(t)^2\\right\\rangle^k\\ ,\n\\end{align}\nwhere $(2k-1)\\times(2k-3)\\times\\ldots\\times3\\times1=\\frac{(2k)!}{2^kk!}$ is the number of possible contractions.\nWe thus find\n\\begin{align}\\label{eq:pd}\n p_d(t) &:= -\\left\\langle\\sinh^2\\left(X_i(t)\\right)\\right\\rangle \\nonumber\\\\\n&= -\\sum_{n,m=0}^\\infty\\frac{1}{(2n+1)!}\\frac{1}{(2m+1)!}\\left\\langle X_i(t)^{2n+2m+2}\\right\\rangle \\nonumber\\\\\n&= -\\sum_{n,m=0}^\\infty\\frac{1}{(2n+1)!}\\frac{1}{(2m+1)!}\\frac{(2n+2m+2)!}{2^{n+m+1}(n+m+1)!}\\left\\langle X_i(t)^2\\right\\rangle^{n+m+1} \\nonumber\\\\\n&= -\\sum_{k=0}^\\infty\\left\\langle X_i(t)^2\\right\\rangle^{k+1}\\frac{(2k+2)!}{2^{k+1}(k+1)!}\\times\\underbrace{\\sum_{n=0}^k\\frac{1}{(2n+1)!}\\frac{1}{(2k-2n+1)!}}_{2^{2k+1}\/(2k+2)!} \\nonumber\\\\\n&= \\frac{1}{2}\\left(1-\\exp\\left\\lbrace2\\left\\langle X_i(t)^2\\right\\rangle\\right\\rbrace\\right) \\nonumber\\\\\n&= \\frac{1}{2}\\left(1-\\exp\\left\\lbrace-2\\Lambda(t)\\right\\rbrace\\right)\\ ,\n\\end{align}\nwhere we have defined $k=n+m$ and $\\Lambda(t)=-\\left\\langle X_i(t)^2\\right\\rangle\\geq0\\,$.\n\nDifferent baths are characterized by their spectral density function\n\\begin{align}\n J(\\omega) = \\frac{\\lambda^2}{N}\\sum_{\\bf{k}}|{\\bf k}|^{2r}\\delta(\\omega-\\omega_{\\bf{k}}) = \\alpha\\omega^s\\omega_0^{1-s}e^{-\\omega\/\\omega_c}\\ .\n\\end{align}\nHere, $\\alpha$ is a dimensionless bath strength, $\\omega_0$ is a characteristic frequency of the bath, and $\\omega_c$ is a high-frequency cut-off.\nA bath with $s<1$ is called sub-Ohmic, one with $s=1$ is called Ohmic, and one with $s>1$ is called super-Ohmic.\n\nThe function $\\Lambda(t)$ depends only on the spectral density function of the bath and its temperature, namely we have\n\\begin{align}\\label{eq:integral}\n \\Lambda(t) = \\int_0^\\infty\\m{d}\\omega\\, J(\\omega)\\coth\\left(\\beta\\omega\/2\\right)\\frac{\\sin^2\\left(\\omega t\/2\\right)}{(\\omega\/2)^2}\\ .\n\\end{align}\nWe see that for $s\\geq1$ a finite $\\omega_c$ is necessary to ensure the convergence of \\eqref{eq:integral}.\nWith a linear dispersion, $\\omega_{\\bf{k}}=v|{\\bf k}|$, and a $D$-dimensional bath, we have $s=D+2r-1$.\n\nFor uncorrelated errors, surface code error correction breaks down if $p_x(t)>p_c=10.9\\%$ \\cite{Dennis2002}. Inverting \\eqref{eq:pd}, we thus find the maximal time $\\tau$ of one error correction cycle from\n\\begin{align}\\label{eq:tauEq}\n \\Lambda(\\tau) = \\frac{1}{2}\\log\\frac{1}{1-2p_c} \\simeq 0.123\\ .  \n\\end{align}\nThis solves the problem up to evaluation of the integral in \\eqref{eq:integral} and inversion of \\eqref{eq:tauEq}.\n\n\n\nFollowing Ref.~\\cite{Novais2013}, we restrict in the main text to the case $D=2$ and $r=0$, corresponding to an Ohmic bath.\nThe dimensionless bath strength parameter evaluates in this case to $\\alpha=\\frac{\\lambda^2}{2\\pi v^2}$. \nThe functions $\\Lambda(t)$ for the remaining combinations of $D=2,3$ and $r=0,\\pm\\frac{1}{2}$ are presented in Appendix~\\ref{sec:decoherence}.\n\nFor the integral in \\eqref{eq:integral}, we find with $s=1$ and $\\beta\\omega_c\\gg1$, using $\\coth(x)=1+2\\sum_{n=1}^\\infty e^{-2nx}$,\n\\begin{align}\\label{eq:Lambda}\n \\Lambda(t) &= \\int_0^\\infty\\m{d}\\omega\\, \\alpha\\omega e^{-\\omega\/\\omega_c}\\coth\\left(\\beta\\omega\/2\\right)\\frac{\\sin^2\\left(\\omega t\/2\\right)}{(\\omega\/2)^2} \\nonumber\\\\\n&= \\int_0^\\infty\\m{d}\\omega\\, \\alpha\\omega e^{-\\omega\/\\omega_c}\\frac{\\sin^2\\left(\\omega t\/2\\right)}{(\\omega\/2)^2} + 2\\sum_{n=1}^\\infty\\int_0^\\infty\\m{d}\\omega\\, \\alpha\\omega e^{-\\omega\/\\omega_c}e^{-2n\\beta\\omega\/2}\\frac{\\sin^2\\left(\\omega t\/2\\right)}{(\\omega\/2)^2} \\nonumber\\\\\n&= \\alpha\\log\\left[1+\\omega_c^2t^2\\right] + 2\\alpha\\sum_{n=1}^\\infty\\log\\left[1+\\frac{\\omega_c^2t^2}{(1+n\\beta\\omega_c)^2}\\right] \\nonumber\\\\\n&\\simeq \\alpha\\log\\left[1+\\omega_c^2t^2\\right] + 2\\alpha\\sum_{n=1}^\\infty\\log\\left[1+\\frac{t^2}{n^2\\beta^2}\\right] \\nonumber\\\\\n&= \\alpha\\log\\left[1+\\omega_c^2t^2\\right] + 2\\alpha\\log\\left[\\frac{\\beta}{\\pi t}\\sinh(\\frac{\\pi t}{\\beta})\\right]\\ .\n\\end{align}\nInserting this into \\eqref{eq:pd} yields\n\\begin{align}\n p_d(t)=\\frac{1}{2}-\\frac{1}{2}\\left[(1+\\omega_c^2t^2)\\frac{\\sinh^2(\\pi t\/\\beta)}{(\\pi t\/\\beta)^2}\\right]^{-2\\alpha}\\ ,\n\\end{align}\nwhich for non-vanishing times ($t\\gg\\frac{1}{\\omega_c}$) is well-approximated by\n\\begin{align}\\label{eq:pdFinal}\n p_d(t)=\\frac{1}{2}-\\frac{1}{2}\\left[\\frac{\\beta\\omega_c}{\\pi}\\sinh(\\frac{\\pi t}{\\beta})\\right]^{-4\\alpha}\\ .\n\\end{align}\nInverting $p_d(\\tau)=p_c$ leads to our final solution\n\\begin{align}\\label{eq:tauUncorr}\n \\tau = \\frac{\\beta}{\\pi}\\text{arcsinh}\\left[\\frac{\\pi}{\\beta\\omega_c}(1-2p_c)^{-1\/4\\alpha}\\right]\\ .\n\\end{align}\n\n\n\n\\section{Surface code error correction for spatially correlated errors}\\label{sec:errcorr}\n\nThe form of the evolution operator derived in \\eqref{eq:U} reveals that the state $\\Phi_m\\circ\\Phi_d(\\rho_q)$ contains correlations between the errors on arbitrary numbers of qubits.\nThe coherent part of the evolution affects each pair $\\lbrace i,j\\rbrace$ of qubits by a two-qubit error with probability $\\sin^2(J_{ij}(t))$, while any set $\\lbrace1,2,\\ldots,m\\rbrace$ of $m$ qubits suffers an $m$-qubit error with probability\n$(-1)^{m}\\left\\langle\\sinh^2(X_1(t))\\ldots\\sinh^2(X_m(t))\\right\\rangle$\ndue to the decoherent evolution. If the decoherent evolution were uncorrelated, this probability would be given by $(-1)^{m}\\left\\langle\\sinh^2(X_1(t))\\right\\rangle\\ldots\\left\\langle\\sinh^2(X_m(t))\\right\\rangle$.\nThe difference between the two terms implies the presence of correlations: if a qubit suffers an error, nearby qubits have a higher chance of also being affected by an error than one would expect from the single-qubit error rate \\eqref{eq:pxt} alone.\n\nThe threshold error rate of $p_c=10.9\\%$ derived in Ref.~\\cite{Dennis2002} applies in the case of uncorrelated errors.\nThe correlations mentioned above will change this value to an unknown threshold $\\tilde{p}_c$.\nA recent work studied the effect of clusters of errors on surface code correction when the probability of a certain cluster size is exponentially or polynomially suppressed \\cite{Fowler2014}.\nThresholds were not studied in terms of the single-qubit error rate $p_x$ but in terms of an over-all probability $p$ for single-qubit errors and clusters of errors.\nIf the probability of a large cluster decays sufficiently slowly, any $p>0$ will lead to $p_x\\rightarrow\\frac{1}{2}$ for large enough $L$.\nThis makes a direct application of the results of Ref.~\\cite{Fowler2014} to our problem impossible.\n\nIn the following, we thus want to investigate how different kinds of spatial correlations between errors affect the threshold error rate for the single-qubit error rate $p_x$.\nThe modified threshold error rate $\\tilde{p}_c$ strongly depends on the type of correlations that are present between the errors.\n\nFig.~\\ref{fig:errorCorrelations} summarizes our results. A worst case is given by ballistically propagating anyons, leaving a linear trail of errors behind. In this case, $\\tilde{p}_c$ can be smaller than $p_c$ by an order of magnitude or more.\nTo understand this, note that the task of error correction is to pair the anyons in a way that is homologically equivalent to the way they have been created.\nError correction breaks down if choosing the right homology class becomes ambiguous. This is achieved with the smallest number of errors if the anyons in each pair propagate into opposite directions.\n\n\\begin{figure}\n  \\setlength{\\unitlength}{0.8\\textwidth}\n  \\begin{picture}(0.65,0.5)\n\t\\thicklines\n\t\\put(-0.15,0.32){\\includegraphics[width=0.15\\textwidth]{linearConfiguration.png}}\n\t\\put(0.15,0.32){\\includegraphics[width=0.15\\textwidth]{diffusiveConfiguration.png}}\n\t\\put(0.45,0.32){\\includegraphics[width=0.15\\textwidth]{indepConfiguration.png}}\n\t\\put(0.45,0.0){\\includegraphics[width=0.15\\textwidth]{clusterConfiguration.png}}\n\t\\put(-0.20,0.24){\\vector(1,0){0.9}}\n\t\\put(-0.09,0.28){ballistic}\n\t\\put(0.21,0.28){diffusive}\n\t\\put(0.49,0.28){uncorrelated}\n\t\\put(0.50,0.17){clustered}\n\t\\put(-0.05,0.22){\\line(0,1){0.04}}\n\t\\put(0.25,0.22){\\line(0,1){0.04}}\n\t\\put(0.55,0.22){\\line(0,1){0.04}}\n\t\\put(0.465,0.21){\\large $\\underbrace{\\qquad\\qquad\\qquad}$}\n\t\\put(0.73,0.245){threshold}\n\t\\put(0.73,0.220){error rate}\n  \\end{picture}\n  \\caption{Different kinds of spatial correlations between errors in the surface code and how they affect its threshold error rate.}\n  \\label{fig:errorCorrelations}\n\\end{figure}\n\nIf anyons perform a diffusive random walk in the toric code, the modified threshold error rate $\\tilde{p}_c$ can also be significantly smaller than $p_c$.\nThis scenario is physically relevant if there is a non-trivial surface (or toric) code Hamiltonian that energetically penalizes the creation, but not the propagation of anyons.\nThe error model of diffusive errors and its effect on error correction have been studied in this context in Refs.~\\cite{Chesi2010,Hutter2012}.\n\nFor both ballistic propagation and a diffusive random walk of anyons, there is a tendency for errors to form string-like patterns.\nBy contrast, the correlations discussed at the beginning of this section favor a clustering of errors (i.e., it is more likely than in the uncorrelated case that errors are spatially close to each other) but there is no mechanism that favors string-like error configurations.\n\nWe do not expect clustering of errors to strongly harm the threshold error rate $p_c$.\nMost clusters of nearby errors do not form string-like patterns and thus do not help to bring pairs of anyons apart from each other and make a homologically correct pairing ambiguous.\nFor a fixed single-qubit error rate $p_x$, the presence of regions with a high density of errors implies the presence of regions with a low density of errors. The latter help to avoid ambiguities.\n\nIn the following subsections we study the modified threshold error rate $\\tilde{p}_c$ for different kinds of spatial correlations between surface code errors by use of Monte Carlo simulations.\nIn agreement with our expectations, we find that clustering of errors leads to at most a mild decrease of the threshold error rate -- and can even be beneficial in the strongly correlated regime.\n\nWe conclude that even in the presence of spatial correlations between errors \\emph{without a mechanism that prefers string-like arrangements} the modified threshold error rate $\\tilde{p}_c$ does not differ drastically from $p_c$.\nHeuristically, we expect correlations between errors arising from coupling the code to the bath not to be of the string-like type.\nWe will thus in the following section invert the equation $p_x(\\tau)=\\tilde{p}_c$ without knowing the exact value of $\\tilde{p}_c$, and simply assume that it is of the same order of magnitude as $p_c$.\n\n\n\n\\subsection{Ballistic propagation of anyons}\\label{sec:ballistic}\n\nIn the following subsections, we study the impact of correlated errors on the correctability of the surface code by use of Monte Carlo simulations.\nThat is, we produce a large number of error configurations using a certain error model, and see whether we are able to find a pairing of the resulting anyon configuration that is homologically equivalent to the actual one.\nFinding such a paring is the task of a classical decoding algorithm. Only if unrealistic computing power is available can we hope to actually perform correction up to the theoretical threshold of $p_c=10.9\\%$ (in the uncorrelated case).\nTherefore, an efficient approximate error correction algorithm is needed in practice.\nWe will employ minimum-weight perfect matching (MWPM) \\cite{Edmonds1965}, which, for a graph with weighted edges and an even number of vertices provides the matching of minimal weight.\nHere, the vertices correspond to the anyons found as a result of the stabilizer measurements, \nand the weight of an edge connecting two anyons is simply given by the minimal number of qubits that have to suffer an error in order to create that pair from the anyonic vacuum (i.e., their Manhattan distance).\nWe employ the library \\texttt{Blossom V} \\cite{Kolmogorov2009} to perform MWPM.\nUsing MWPM for performing error correction in the surface code reduces the threshold error rate to $10.2\\%$ \\cite{Chesi2010,Fowler2012b}.\n\nFor our first ``worst case'' error model, we envision anyons that after creation start to ballistically propagate into a certain direction.\nMore precisely, we specify the error model by two parameters $f$ and $l$.\nFirst, we draw a number $n$ at random from a Poisson distribution with mean $2fL^2$.\nThen, we perform $n$ times the following. Choose one of the $L^2$ anyon locations and an angle $\\phi\\in\\left[0 , 2\\pi\\right)$ at random. \n(Recall that we consider one type of error only, so for a surface code of linear size $L$ with periodic boundary conditions, there are $L^2$ anyon locations of the relevant type.)\nDraw random numbers $l_h$ and $l_v$ from Poisson distributions with mean $l|\\cos(\\phi)|$ and $l|\\sin(\\phi)|$, respectively (the expectation value for $l_h+l_v$ is thus $\\frac{4}{\\pi}l$).\nStarting from the initial anyon location, apply $l_h$ errors horizontally and $l_v$ errors vertically, with the directions  given by the sign of the trigonometric functions.\nAfter doing this $n$ times, perform error correction by means of MWPM.\n\nFor each value of $l$, there is a threshold value $f_c$ such that for $f<f_c$ the logical error rate decreases exponentially with $L$ and for $f>f_c$ the logical error rate approaches $\\frac{1}{2}$.\nFor each triple of $l$, $f$, and $L$, we generate a number $N$ of error configurations which is such that error correction fails $10^4$ times. The logical error rate can then be estimated as $10^4\/N$.\nThe threshold values $f_c$ are then determined for each value of $l$ by comparing the logical error rates for code sizes up to $L=60$.\nFinally, once we know the threshold value $f_c$ we can determine the threshold $\\tilde{p}_c$ for the single-qubit error rate $p_x$ by determining the fraction of qubits that suffer an error for the given pair of $l$ and $f_c$.\nAn even number of errors on the same qubit count as no error, and on odd number as one. \nIf the errors are sufficiently sparse such that the probability of several errors happening on the same qubit is negligible, we have $p_x=\\frac{4}{\\pi}l\\times 2fL^2\/(2L^2)=\\frac{4}{\\pi}lf$, while otherwise it will be smaller.\n\nThe single-qubit threshold error rates $\\tilde{p}_c$ as a function of $l$ are illustrated by the purple squares in Fig.~\\ref{fig:stringlike}.\nWhile for $l=\\frac{1}{2}$ the threshold is still comparable with the value of $10.2\\%$ for the uncorrelated case, it decreases strongly as $l$ is increased.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.6\\textwidth]{stringlike.png}\n\t\\caption{Single-qubit error rate $\\tilde{p}_c$ for which error correction breaks down for two error models that lead to string-like error patterns: ballistic and diffusive propagation of anyons.}\n\t\\label{fig:stringlike}\n\\end{figure}\n\n\\subsection{Diffusive propagation of anyons}\n\nIn the case where anyons perform a random walk, the simulation works in much the same way as described in the previous subsection.\nFor each initial anyon location, we draw a random number from a Poisson distribution with mean $l$, and then perform a random walk whose length is given by this number.\nThe resulting thresholds are displayed by the blue circles in Fig.~\\ref{fig:stringlike}. Threshold error values are, for a given value of $l$, significantly higher than in the ballistic case though significantly lower than in the uncorrelated case.\n\n\\subsection{Clustered errors}\n\nHere, we study a family of error models that describe clustering of errors in the surface code.\nFor $l\\leq m^2$, we define the error model $m$-$l$-cluster as follows: from each square of $m\\times m$ qubits in the surface code, pick $l$ qubits at random and apply an error to all of them with probability $f$.\nThe resulting single-qubit error rate is $p_x\\lesssim fl$. (Note that the same qubit can suffer several errors and an even number corresponds to no error at all, leading to $p_x<fl$.)\nThe modified critical error rates $\\tilde{p}_c$ are again determined as described in Sec.~\\ref{sec:ballistic}.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.6\\textwidth]{clusterBreakdown.png}\n\t\\caption{Single-qubit error rate $\\tilde{p}_c$ for which error correction breaks down in the $m$-$l$-cluster error models.}\n\t\\label{fig:cluster}\n\\end{figure}\n\nFig.~\\ref{fig:cluster} shows our results.\nIf $l\\ll m$, errors are essentially uncorrelated and the threshold values for $\\tilde{p}_c$ are close to $10.2\\%$, the threshold for MWPM-based error correction in the uncorrelated case.\nFor $l\\lesssim m$, $\\tilde{p}_c$ falls slightly below $10\\%$, though the decrease is not dramatic.\nThis decrease is due to the possibility of forming string-like patterns of length $l$, which leads to a smaller number of errors being necessary for correction to become ambiguous.\nFinally, for $l>m$ the threshold increases significantly beyond $p_c$. Additional errors now make it easier to recognize the cluster and increase the probability that several errors together form a (partial) stabilizer operator and therefore do no harm to the code.\nFor instance, in the $2$-$4$-cluster case the threshold error rate is as high as $\\tilde{p}_c=29.0\\%$, since half of all errors combine to a stabilizer operator.\nIn reality, we do of course not expect the environment to apply exclusively $2\\times2$ squares of errors, but to find ourselves in the regime where the clustering of errors leads to a slight reduction of the single-qubit threshold error rate.\n\n\n\\subsection{Correlated two-qubit errors}\n\nLet us now study the case where there are correlations between errors on pairs of qubits only.\nNote that the coherent part of the evolution is able to produce such correlations only.\nThe regime considered here is thus relevant if correlations between error events on more than two qubits due to the decoherent evolution are weak.\n\nThe study of correlated two-qubit errors is simplified by the fact that there is a clear worst-case, namely a two-qubit error on a pair of nearest-neighbor qubits.\nWe assume that each qubit in the code suffers an error with probability $p_1$ and that, furthermore, each pair of nearest neighbors in the code suffers a pair of errors with probability $p_2$.\nWe expect and have verified in numerical simulations (see below) that correlated errors on pairs of qubits which are not nearest neighbors have, for a fixed single-qubit error rate $p_x$, less of an effect on error correction than correlated errors on nearest neighbor qubits.\nStudying this particular case thus allows us to find the maximal impact of correlated two-qubit error events.\n\nWith the above parameters, and since each qubit in the code has four nearest neighbors, the single-qubit error rate $p_x$ can be calculated in analogy to \\eqref{eq:pxt} as\n\\begin{align}\\label{eq:px}\n p_x &= p_1\\sum_{k\\text{ even}}\\binom{4}{k}p_2^k(1-p_2)^{4-k} + (1-p_1)\\sum_{k\\text{ odd}}\\binom{4}{k}p_2^k(1-p_2)^{4-k}\\nonumber\\\\\n&= \\frac{1}{2}-\\frac{1}{2}(1-2p_1)(1-2p_2)^4\\ .\n\\end{align}\n\nWe can make two estimates for where error correction will break down in the above model.\nFirst, we can simply assume that the correlations do neither help nor derogate the correctability of the code. In this case, the breakdown occurs for $p_x = p_c$ (or, with MWPM correction, for $p_x=10.2\\%$), independently of $p_2$.\nA second estimate is of entropic nature. It is obtained by studying whether it is at all possible that the stabilizer measurements provide us with enough information to infer what errors have happened.\nAssume that there are $n$ qubits in the code. There are $2n$ pairs of nearest neighbors and $n\/2$ plaquette stabilizers that can give us information about bit-flip errors.\nFor large $n$, the total information contained in the noise can be compressed to $nh(p_1)+2nh(p_2)$ bits, where $h(p)=-p\\log_2(p)-(1-p)\\log_2(1-p)$ is the binary entropy function. On the other hand, the plaquette stabilizers give us at most $n\/2$ bits of information.\nError correction will thus break down if\n\\begin{align}\\label{eq:entropicEstimate}\n 2h(p_1)+4h(p_2)=1\\ .\n\\end{align}\nIf we needed to know exactly which qubits have suffered a bit-flip, \\eqref{eq:entropicEstimate} would put a rigorous upper bound on the correctability of the surface code.\nHowever, we only need to know the error pattern \\emph{modulo application of stabilizer operators}. For this reason, \\eqref{eq:entropicEstimate} should rather be seen as an estimate of an upper bound.\nSuch an entropic estimate predicts the unavoidable breakdown of surface code error correction to high accuracy for both uncorrelated bit-flip errors \\cite{Dennis2002} (i.e., $2h(p_c)\\simeq1$) and depolarizing noise \\cite{Bombin2012}.\nRef.~\\cite{Roethlisberger2012} shows that variations of the surface code tailored for stability against biased noise ($p_x\\neq p_z$) give thresholds that fall only a few percents short of the ones suggested by such entropic arguments\n -- even with error correction performed by an efficient approximate algorithm.\n\nWe will use two different algorithms for performing error correction for the above error model.\nBoth of them are based on MWPM, but they differ in the weights they assign to the edges.\nThe first one is the algorithm used in the previous subsections. It ignores correlations and assigns the Manhattan distance between two anyons to the edge connecting them.\nThe second algorithm, described in more detail in Appendix~\\ref{app:algo}, uses a more sophisticated assignment of edge weights that allows it to take spatial correlations between the errors into account.\n\n\\begin{figure}[h]\n\t\\centering\n\t\t\\includegraphics[width=0.8\\textwidth]{ignoreNot.png}\n\t\\caption{Each qubit is independently subjected to an error with probability $p_1$. Furthermore, each pair of nearest-neighbor qubits is subjected to a a pair of errors with probability $p_2$.\n  The blue lines correspond to a constant value of $p_x$, calculated according to \\eqref{eq:px}, while the red line shows the entropic bound \\eqref{eq:entropicEstimate}.\n  Diamonds represent threshold error rates $(p_1,p_2)$ when error correction is performed with MWPM and correlations are ignored.\n  Squares represent threshold error rates for an algorithm that takes correlations into account.\n  Threshold error rates have been determined to accuracy $10^{-3}$, by comparing logical error rates for code sizes between $10$ and $50$ (periodic boundary conditions).\n  For each combination of error rates and code sizes, the logical error rates were obtained from as many error configurations as were necessary to obtain $10^4$ logical errors.}\n\t\\label{fig:breakdown}\n\\end{figure}\n\nFig.~\\ref{fig:breakdown} compares the above estimates with the resulting combinations $(p_1,p_2)$ for which error correction breaks down in actual numerical simulations, when the two algorithms described above are used for performing error correction.\nIf the Manhattan distance between two anyons is used as the edge weight and correlations between the errors are ignored, error correction breaks down for $p_x=10.2\\%$ for $p_2\\ra0$, slightly below the value of $p_c$ for perfect error correction.\nIn the maximally correlated regime, $p_1\\ra0$, error correction already breaks down for $p_x=9.6\\%$ -- a pretty insignificant decrease.\nWe have obtained similar data to the one displayed in Fig.~\\ref{fig:breakdown} for correlated errors that happen on pairs of qubits which are further away from each other than nearest neighbors.\nIn this case, the deviations from the line $p_x=10.2\\%$ are smaller.\nAlready for pairs of qubits that are three lattice constants away from each other, the obtained threshold error rates are indistinguishable (to accuracy $10^{-3}$) from this line.\n\nFor the second, improved algorithm, error correction breaks down for $p_x=10.6\\%$ in the uncorrelated case ($p_2\\ra0$), close to the theoretical value of $p_c$, and for $p_x=18.6\\%$ in the maximally correlated case ($p_1\\ra0$). \nThe threshold error rates $(p_1,p_2)$ approximately follow that of the two above estimates wich predicts the higher threshold value and significantly beat both estimates in some regimes.\nBeyond the red line in Fig.~\\ref{fig:breakdown}, it is information-theoretically impossible that we learn from the stabilizer measurements what errors have happened.\nThat it is possible to error correct beyond that line shows that due to its degenerate nature (i.e., different error configurations can lead to the same syndrome) the surface code is able to take care of some of the entropy in the noise itself.\n\nIn conclusion, ignoring during error correction that pairs of qubits can be affected by correlated errors hardly affects the single-qubit threshold error rate of the surface code.\nIf an algorithm takes these correlations into account, the single-qubit threshold error rate can be significantly boosted in the strongly correlated regime.\nDue to its degenerate nature, the surface code is able to correct in regimes where it is information-theoretically impossible that we learn what errors the code has suffered.\n\n\n\\section{Maximal QEC cycle time for correlated errors}\\label{sec:qectime}\n\nAssuming that the form of spatial correlations between errors that will be present in $\\Phi_m\\circ\\Phi_d(\\rho_q)$ does not lead to a threshold error rate $\\tilde{p}_c$ that differs drastically from $p_c$,\nthe single-qubit error rate $p_x(t)$ in \\eqref{eq:pxt} contains already all the information we need in order to predict the maximal QEC period $\\tau$.\nA great advantage of \\eqref{eq:pxt} is that it depends only on $p_d(t)$ and the coherent interaction strengths $J_{ij}(t)$, but not on the temperature-dependent correlators $C_{ij}(t)$ for $i\\neq j$.\n\nOur goal is thus to solve the equation $p_x(\\tau)=\\tilde{p}_c$ for $\\tau$, where $p_x(t)$ is given by \\eqref{eq:pxt}.\nSince $\\tilde{p}_c$ is an order of magnitude smaller than $1$, we can approximate $p_x(t)$ by its leading-oder contributions,\n\\begin{align}\\label{eq:pxApprox}\n p_x(t)\\simeq p_d(t) + \\sum_i\\!^{'}\\sin^2(J_{1i}(t))\\ .\n\\end{align}\nWe follow again Ref.~\\cite{Novais2013} and study an Ohmic bath ($r=0$, $D=2$). The function $J_{ij}(t)$ for this bath type has been provided in \\eqref{eq:Jijspec}.\nNote that $J_{ij}(t)$ decays inversely with distance outside of the light-cone.\nTherefore, the second summand in \\eqref{eq:pxApprox} diverges logarithmically with the code size $L$ at any non-zero time (up to constant prefactors of order $1$, we have $\\sum_i\\!^{'}\\frac{1}{|{\\bf R}_1-{\\bf R}_i|^2}\\sim\\int_1^{L\/2}\\frac{1}{r^2}r\\m{d}r\\sim\\log(L)$).\nCorrespondingly, the maximal QEC period vanishes in the thermodynamic limit (though it does so very slowly, see below).\nFor all other combinations of $D=2,3$ and $r=0,\\pm\\frac{1}{2}$, $J_{ij}(t)$ decays stronger than $|{\\bf R}_i-{\\bf R}_j|^{-1}$ outside of the light-cone (see Appendix~\\ref{sec:induced}).\nThe maximal QEC period remains thus finite in the thermodynamic limit for all other bath types.\n\nSetting the lattice constant of the surface code to unity and assuming a linear code size $L$, we can estimate\n\\begin{align}\n \\sum_i\\!^{'}\\sin^2(J_{1i}(t)) \\simeq 2\\pi\\int_0^{L\/2}\\m{d}R\\,R\\sin^2\\left[\\frac{\\lambda^2}{2\\pi^2v^2}\\left(\\theta(R-vt)\\arcsin(vt\/R) + \\theta(vt-R)\\frac{\\pi}{2}\\right)\\right]\\ .\n\\end{align}\nSince we are interested in times where this sum is (still) sufficiently smaller than $1$, in particular each summand has to be much smaller than $1$.\nDefining $m(t)=\\min\\lbrace L\/2,vt\\rbrace$, we find\n\\begin{align}\\label{eq:mediatedError}\n  \\sum_i\\!^{'}\\sin^2(J_{1i}(t)) &\\simeq 2\\pi\\int_{m(t)}^{L\/2}\\m{d}R\\,\\frac{1}{R}\\left(\\frac{\\lambda^2t}{2\\pi^2v}\\right)^2 + 2\\pi\\int_0^{m(t)}\\m{d}R\\,R\\left(\\frac{\\lambda^2}{4\\pi v^2}\\right)^2 \\nonumber\\\\\n&=\\frac{\\lambda^4t^2}{2\\pi^3v^2}\\log(\\frac{L\/2}{m(t)}) + \\frac{\\lambda^4}{16\\pi v^4}m^2(t)\\ .\n\\end{align}\nCombining Eqs.~(\\ref{eq:pdFinal}), (\\ref{eq:pxApprox}), and (\\ref{eq:mediatedError}) we conclude that for times $t$ which are small enough such that $p_x(t)\\ll1$ we have\n\\begin{align}\n p_x(t) \\simeq \\underbrace{\\frac{1}{2} - \\frac{1}{2}\\left[\\frac{\\beta\\omega_c}{\\pi}\\sinh(\\frac{\\pi t}{\\beta})\\right]^{-2\\lambda^2\/\\pi v^2}}_{A(t)} + \\underbrace{\\frac{\\lambda^4t^2}{2\\pi^3v^2}\\log(\\frac{L\/2}{m(t)})}_{B(t)} + \\underbrace{\\frac{\\lambda^4}{16\\pi v^4}m^2(t)}_{C(t)}\\ .\n\\end{align}\n\nWe can recognize three different mechanisms contributing to the single-qubit error rate $p_x(t)$. \nSummand $A(t)$ describes errors due to each qubit coupling individually to the bath. Correspondingly, this term is independent of $L$.\nIt is the only term that depends on temperature and the only term that contributes if the qubits do not interact via the bath.\nSummand $B(t)$ describes errors due to superluminal interactions between the qubits mediated by the bath.\nIt diverges logarithmically with $L$ for short enough times but vanishes once all qubits are within their mutual light-cones.\nFinally, summand $C(t)$ describes errors due to subluminal interactions between the qubits. Once all qubits are within their mutual light-cones, this term reaches a time-independent constant which is proportional to the number of qubits in the code.\n\nWe have already studied the times $\\tau_d$ which are necessary for summand $A(t)$ to reach critical levels ($A(\\tau_d)\\simeq p_c$) in Sec.~\\ref{sec:uncorr}.\nThe only question that remains is whether $B(t)$ or $C(t)$ reach critical levels before $A(t)$ and if so, on what time-scales.\nAs shown in Fig.~\\ref{fig:times}, each of the three summands can be the dominant force leading to the breakdown of error correction.\nA higher temperature increases the weight of summand $A(t)$, while a larger code size increases the weight of summands $B(t)$ and $C(t)$.\n\n\n\n\\begin{figure}[htb]\n\\setlength{\\unitlength}{\\textwidth}\n\\begin{picture}(0.8,0.62)\n  \\put(-0.11,0.32){\\includegraphics[width=0.51\\textwidth]{breakdown2.png}}\n  \\put(0.4,0.32){\\includegraphics[width=0.50\\textwidth]{breakdown3.png}}\n  \\put(-0.1,-0.01){\\includegraphics[width=0.50\\textwidth]{breakdown4.png}}\n\n  \\put(0.24,0.42){\\Large $L=10^2$}\n  \\put(0.74,0.42){\\Large $L=10^3$}\n  \\put(0.24,0.095){\\Large $L=10^4$}\n\n  \\put(-0.04,0.605){$p_c$}\n  \\put(0.00,0.595){$p_x(t)$}\n  \\put(0.00,0.56){$A(t)$}\n  \\put(-0.025,0.43){$B(t)$}\n  \\put(0.00,0.495){$C(t)$}\n\n  \\put(0.61,0.545){$p_c$}\n  \\put(0.515,0.60){$p_x(t)$}\n  \\put(0.61,0.50){$A(t)$}\n  \\put(0.60,0.47){$B(t)$}\n  \\put(0.565,0.57){$C(t)$}\n\n  \\put(0.125,0.185){$p_c$}\n  \\put(0.04,0.245){$p_x(t)$}\n  \\put(0.13,0.150){$A(t)$}\n  \\put(0.07,0.222){$B(t)$}\n  \\put(0.09,0.20){$C(t)$}\n\n\n\\end{picture}\n  \\caption{The three summands $A(t)$, $B(t)$, and $C(t)$ and their sum $p_x(t)$ compared with $p_c$ for code sizes $L=10^2$, $L=10^3$, and $L=10^4$. We have used paramters $v=1$, $\\lambda=0.1$, $T=0.01$, and $\\omega_c=30$.\nNote that the assumptions $\\beta\\omega_c\\gg1$ and $\\tau_d\\gg1\/\\omega_c$ made during the derivation of $A(t)$ are well-satisfied.}\n\\label{fig:times}\n\\end{figure}\n\nIn order to find the maximal QEC period $\\tau$, we make the simplifying assumption that the breakdown is due to the dominant mechanism alone, i.e., we approximate $p_x(t)\\simeq\\max\\lbrace A(t),B(t),C(t)\\rbrace$.\nNote that for times much smaller than $L\/v$, we have $B(t)>C(t)$, while for times of order $L\/v$ or larger, we have $B(t)<C(t)$.\nFor times larger than $L\/2v$, $C(t)$ reaches its maximal value $\\frac{\\lambda^4L^2}{64\\pi v^4}$. \nTherefore, for $L>8\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}$ the term $C(t)$ will reach critical values ($\\tilde{p}_c$) in a time\n\\begin{align}\\label{eq:tauSub}\n \\tau_{\\text{sub}}=\\frac{4\\sqrt{\\pi \\tilde{p}_c}v}{\\lambda^2}\\ ,\n\\end{align}\nwhile otherwise it will never do so.\nElementary calculus shows that the maximal value, which $B(t)$ can achieve while still being larger than $C(t)$, is $\\frac{e^{-\\pi^2\/4}}{64\\pi}\\frac{\\lambda^4}{v^4}L^2$, and that $B(t)$ is monotonically increasing until it reaches this value.\nTherefore, $B(t)$ reaches $\\tilde{p}_c$ before $C(t)$ if and only if $L>8e^{\\pi^2\/8}\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}$.\nThe (relevant) solution to $B(\\tau_{\\text{super}})=\\tilde{p}_c$ is given by\n\\begin{align}\\label{eq:tauSuper}\n \\tau_{\\text{super}} = 2\\pi\\sqrt{\\pi \\tilde{p}_c}\\frac{v}{\\lambda^2} \\left|W_{-1}(-16\\pi^3\\tilde{p}_cv^4\/\\lambda^4L^2)\\right|^{-1\/2}\\ ,\n\\end{align}\nwhere $W_{-1}$ is the lower branch of the Lambert $W$ function \\cite{Lambert}.\nFor $z\\ra0^{-}$, we have $W_{-1}(z)\\simeq\\log|z|$, showing that the available QEC time vanishes in the thermodynamic limit $L\\rightarrow\\infty$ like $\\tau\\sim1\/\\sqrt{\\log(L)}$, that is, very slowly.\n\n\n\n\n\n\n\\subsection{Summary of results}\\label{sec:summary}\n\nLet us summarize our results for a 2D Ohmic bath. There are three different mechanisms that contribute to the error rate on each qubit and hence put limits on the maximal QEC period $\\tau$: \nthe individual coupling of each qubit to the bath, superluminal interactions between the qubits mediated by the bath as well as subluminal ones.\n\nThe direct interaction of each qubit with the bath puts an upper bound $\\tau_d$ on the maximal time for which error correction can suceed.\nThis time is given by \\eqref{eq:tauUncorr}, for which we find a high- and a low-temperature value \n\\begin{align}\n \\tau_d &= \\frac{1}{\\pi T}\\text{arcsinh}\\left[\\frac{\\pi T}{\\omega_c}(1-2\\tilde{p}_c)^{-\\pi v^2\/2\\lambda^2}\\right] \\nonumber\\\\\n&\\simeq\n\\begin{cases}\n  \\frac{1}{\\omega_c}\\exp(cv^2\/\\lambda^2) &\\text{if}\\quad T<\\frac{\\omega_c}{\\pi}\\exp(-cv^2\/\\lambda^2) \\\\\n  \\quad\\\\\n  \\frac{1}{\\pi T}\\left(cv^2\/\\lambda^2 - \\log\\left[\\frac{\\omega_c}{2\\pi T}\\right]\\right) &\\text{if}\\quad T>\\frac{\\omega_c}{\\pi}\\exp(-cv^2\/\\lambda^2)\\ .\n \\end{cases}\n\\end{align}\nHere, $c=\\frac{\\pi}{2}\\log\\frac{1}{1-2\\tilde{p}_c}$. Assuming $\\tilde{p}_c\\simeq p_c$, we find $c\\simeq0.4$.\n\nThe interaction between the qubits mediated by the bath is a further source of errors, both due to subluminal and superluminal interactions.\nErrors due to mediated interactions can only reach critical values if the linear code size $L$ is large enough; if $L<8\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}$, neither the error strength due to sub- nor due to super-luminal interactions will ever reach $\\tilde{p}_c$.\nFor $8\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}<L<8e^{\\pi^2\/8}\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}$, errors due to sub-luminal interaction reach a critical strength in a time $\\tau_{\\text{sub}}\\sim v\/\\lambda^2$.\nIf errors due to superluminal interactions also reach criticality, they will do so on times larger than $\\tau_{\\text{sub}}$ for these values of $L$.\nFinally, if $L>8e^{\\pi^2\/8}\\sqrt{\\pi \\tilde{p}_c}\\frac{v^2}{\\lambda^2}$, superluminally meadiated errors reach criticality before subluminal ones, and they do so in a time $\\tau_{\\text{super}}\\sim v\/\\lambda^2\\sqrt{\\log L}$.\nThis time vanishes very slowly in the thermodynamic limit. These results are summarized in the following table (assuming $\\tilde{p}_c\\simeq p_c$).\n\n\\begin{center}\n    \\begin{tabular}{ | l | l | l | }\n    \\hline\n    Code size & Breakdown in a time & Dominant mechanism  \\\\ \\hline\\hline\n    $L<4.7\\frac{v^2}{\\lambda^2}$ & $\\tau_d$ & direct bath coupling \\\\ \\hline\n    $4.7\\frac{v^2}{\\lambda^2}<L<16.1\\frac{v^2}{\\lambda^2}$ & $\\min\\lbrace\\tau_d,\\tau_{\\text{sub}}\\rbrace$ & direct bath coupling or subluminal interactions \\\\ \\hline\n    $L>16.1\\frac{v^2}{\\lambda^2}$ & $\\min\\lbrace\\tau_d,\\tau_{\\text{super}}\\rbrace$ & direct bath coupling or superluminal interactions \\\\ \\hline\n    \\end{tabular}\n\\end{center}\n\n\n\n\n\n\n\n\\section{Conclusions}\\label{sec:conclusions}\n\nQuantum information is fragile and can only be maintained if the accumulation of entropy in the information-bearing degrees of freedom of a storage device can be suppressed -- either by preventing entropy from entering or by removing it at a sufficient pace.     \nAny possible measure to achive this can only succeed for certain classes of system-environment couplings.\nCorrespondingly, a proposal that promises stability of quantum information is only as valuable as the error source against which it protects is realistic.\n\nIn this work, we have investigated how long the surface code is able to protect a quantum state against noise emerging from a physically relevant type of environment -- a bath of freely propagating bosonic modes.\nWe have seen that there are two very distince kinds of error mechanisms: the individual decoherence of each qubit, and induced interactions between the code qubits.\nBoth mechanisms lead to spatial and temporal correlations between the errors happening in the code.\nHowever, we have shown that a tendency of errors to cluster without a tendency to form string-like configurations does not strongly derogate the correctability of the surface code -- even when these correlations are ignored during error correction.\n\nWe have managed to express the time before the error rates in the code reach critical values in terms of code size ($L$), accidental coupling strength ($\\lambda$), mode velocity ($v$), and bath temperature ($T$) across a wide range of different parameter regimes.\nTwo further parameters that determine the physical character of the qubits' decoherence mechanism are the spatial dimension of the medium in which the modes propagate ($D$) and the nature of the coupling to the bath ($r$).\nWe have focused our discussion on the specific combination ($D=2$, $r=0$) investigated in Ref.~\\cite{Novais2013}, which corresponds to an Ohmic bath. \nThis combination is of particular interest since it is the only one for which the maximal QEC time vanishes (very slowly) in the thermodynamic limit.\nFor all other combinations of $D=2,3$ and $r=0,\\pm\\frac{1}{2}$, this time remains finite.\n\nFollowing Refs.~\\cite{Novais2013,Jouzdani2013}, we have made several simplifying assumptions to make the actual problem analytically tractable. \nThese are: a trivial Hamiltonian for the qubits; undamped and non-interacting bath modes; no residual bath correlations between different QEC periods; one type of errors only (bit-flips); immediate and flawless syndrome measurement and error correction \n(including no time cost for efficient classical computations).\nRelaxing these assumptions opens a wide field of additional challenges. \nFor instance, fully fault-tolerant syndrome extraction and error correction are discussed in Ref.~\\cite{Wang2011,Duclos2014}.\nA finite probability of syndrome measurement failure will lead to a lower value of $\\tilde{p}_c$ and hence necessitate shorter QEC periods.\nMoreover, we have in this work been concerned exclusively with spatial and temporal correlations between errors in the surface code.\nIf there are non-commuting error types on the same qubit (bit- and phase-flips), a further type of correlation in the noise emerges, namely correlations between different error types on the same qubit.\nSuch correlations are present in the often-used error model of depolarizing noise. How they can be taken into account during error correction is studied in Refs.~\\cite{Duclos2009,Wootton2012,Hutter2013,Fowler2013}.\nFinally, adding an energy splitting $-\\frac{\\Delta}{2}\\tilde{\\sum}_i\\sigma^z_i$ for the code qubits would transform the problem into a many-spin generalization of the well-studied spin-boson problem.\nFor a single spin-qubit coupled to an Ohmic bath, the spin-boson problem has been solved within the Born approximation in Ref.~\\cite{DiVincenzo2005}. \nHowever, the generalization of this problem to the many-qubit case may well be analytically intractable \\cite{Terhal2005}.\n\n\n\n\\section{Acknowledgements}\n\nWe would like to thank J.~R. Wootton, P. Jouzdani, B.~M. Terhal, and A.~G. Fowler for helpful discussions. This work was supported by the Swiss NF, NCCR QSIT, and IARPA.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction and positioning}\nDuring the last decade, several projects have proposed to use\nintelligent virtual agents in digital games for user empowerment\n\\cite{Marsella2003, Paiva2004, Pareto2009, tardis2013,\n  Batrinca2013}. The work presented in this paper considers the use of\nvirtual agents in job interview simulation games for young unemployed\npeoples, a.k.a NEETs\\footnote{NEET is a government acronym for young\n  people not in employment, education or training. According to\n  Eurostat, in march 2012, 5.5 million of European youngster (16 to 25\n  years old) were unemployed meaning that $22.6\\%$ of the youngster\n  global population in European union is unemployed. This unemployment\n  percentage is 10 points superior to the whole population showing\n  that the employment of NEETs is a real problem in Europe.}. Current\nresearch reveals that NEETs often lack self-confidence and the\nessential social skills needed to seek and secure employment\n\\cite{Bynner2002}. Training with a virtual agent can help them acquire\nself-confidence and improve their social skills. Indeed, it has\nalready been proven that training at job interviews with a virtual\nagent could improve the performance \\cite{hoque2013}.\n\nThe role of the virtual agent in such training games is to be able to\nreact in a coherent manner: based on the non-verbal inputs (smiles,\nemotion expressions, body movements), the agent must select relevant\nverbal and non-verbal responses. In this context, several work\nillustrated the role of emotion regulation in the context of job\ninterviews. For instance, in \\cite{Sieverding2009}, a study shows that\npeople who tried to suppress or hide negative emotions during a job\ninterview are considered more competent by evaluators. Similarly,\nTiedens \\cite{Tiedens2001} shows that anger and sadness play an\nimportant role in job interviews. For this reason, credible\nsimulation of emotions appears as a key issue when it comes to using\nvirtual agents in job interview simulations.\n\nMost existing models for virtual agents rely on a reactive approach,\nin which the system does not manipulate or reason on the mental states\nof the interlocutor \\cite{hoque2013,Paiva2004,Pareto2009}. However, in human psychology, \\textit{Theory of\n  Mind} (ToM) refers to the ability of human beings and primates to\ninterpret, predict and even influence others' behavior\n\\cite{baron1997mindblindness}. Such an ability is a key feature in the\ndevelopment of intelligent virtual agents in the context of tutoring\nand training systems. In this paper, we propose a new model of ToM for\nvirtual agent in the context of job interview simulation.\n\nThe next section briefly discusses existing research that serves as a\nbasis to our work. Sections \\ref{sect:archi} and\n\\ref{sect:logic} present the general architecture and the logical\nframework for our ToM. Section \\ref{sect:implem} describes our implementation of\nthis model in the context of job interviews simulation. An outline of\nthe preliminary evaluation we conducted is given in Section\n\\ref{sect:eval}. Finally, results and perspectives are discussed in\nSection \\ref{sect:discuss}.\n\\section{Related work}\n\nIn order to be able to reason on the affective dimension of the\ninteraction, conveyed by the non-verbal behaviour of both\ninterlocutors, several models rely on the cognitive structure of\nemotions and appraisal theories such as CPM \\cite{scherer2010emotion}\nor OCC \\cite{Ortony1990}. These theories provide domain-independant\ndescriptions of triggering conditions of emotions, that are required\nfor the development the affective aspect of the ToM reasoner. For\ninstance, \\cite{Adam2009,Dastani2012} are BDI-based implementation of\nthe OCC theory. FAtiMA's double appraisal model \\cite{aylett2008if},\nalthough not implemented using a BDI framework, also encodes the OCC\nmodel. However, in these models, the inference mechanism itself\nencodes the chosen Appraisal Theory. On the contrary, in our model, we\npropose a theory-independant ToM reasoner. While our experiments were\nconducted using an OCC-based model, the corresponding rules (described\nin equation \\ref{eq:emo}) could be easily replaced by another theory.\n\nTo support such adaptability, we propose to rely on the BDI model.\nSeveral computational models of emotions have already been proposed\n(e.g. \\cite{herzig2002logic,Dastani2012}) that show that\nBDI is a good basis to represent and to reason about the\ninterlocutor's mental state. Our aim is thus to define a logical model\nof emotions and ToM in BDI.\n\nFrom the philosophical point of view, a debate about\nhow ToM is processed by human adults opposes two theories. The\n\\textit{theory-theory}(TT) argues for a folk-psychology reasoning,\ni.e. a set of rules one acquires regarding human mind\nfunctioning. \\cite{botterill1999philosophy}. The\n\\textit{simulation-theory} (ST) \\cite{goldman2006simulating} defends a\nmirroring or projection process allowing for taking someone else's\nperspective. Various research demonstrated that neither pure TT nor\npure ST were realistic \\cite{vogeley2001mind} and both theorists and\nsimulationists turn toward more hybrid models \\cite{botterill1999philosophy}\\cite{goldman2006simulating}.\n\n\n\n\n\nExisting computational ToM models either imply a choice between the TT\nand ST theories (\\textit{e.g.} \\cite{aylett2008if} that relies on a ST\napproach, or \\cite{bosse2011recursive} and \\cite{pynadath2013you} that\nposition in the TT) or implement them separately as in\n\\cite{harbers2011explaining}. In our work, we propose a hybrid\napproach that relies on theory-theory to model the agent's mental\nstates and commonsense rules, but also on simulation-theory to others'\nperspective by projecting attributed mental states on its own\ninference engine. Both models are integrated in the same reasoner.\n\n\n\n\n\n\n\n\n\n\n\n\\section{Architecture Overview}\n\\label{sect:archi}\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[width=\\linewidth]{ArchiToM2.PNG}\n\\caption{General architecture allowing for modeling hybrid Theory of Mind. }\n\\label{fig:archi}\n\\end{figure}\nOur ToM reasoning architecture consists of two main components, as presented on Figure \\ref{fig:archi}.\n\nThe \\textbf{agent's mental states} contains beliefs, attitudes, goals and intentions. Beliefs represent knowledge about general facts, rules of the world (\\textit{i.e.} commonsense knowledge from the theory-theory) and mental states of self or other's (\\textit{i.e.} attributed mental states). Attitudes represent appreciations of the current state of affairs and, by extension, desires (what the agent wants to be true in the future) and ideals (what the agents would like to be always true). Goals and intentions form the deliberative aspect, limited to immediate actions.\n\nThe \\textbf{agent's inference engine} contains three parts. The folk-psychology deliberative reasoner is responsible for intention generation (according to the agent's beliefs and attitudes) and updating mental state. The Commonsense reasoner's role it to enrich the agent's beliefs base using commonsense rules and facts. Finally, the emotional inference engine computes emotion based on the appraisal theory.\n\nOur hybrid ToM modeling relies on: 1) a TT approach based on folk-psychology and commonsense to reason about others, and 2) a ST approach consisting in projecting their attributed mental states on the agent's own inference engine. The following section details this logical model.\n\n\\section{Logical framework}\n\\label{sect:logic}\n\nIn the following, $\\stackrel{\\mathrm{def}}{=}$ and\n$\\stackrel{\\mathrm{def}}{\\Longrightarrow}$ respectively mean\n\\textit{equals by definition} and \\textit{implies by definition}. The\nformer is used to define new operators as functions of others and the\nlatter to express inference rules.\n\n\n\\subsection{Syntax}\nAssume finite sets of atomic propositions $ATM$, physical actions\n$ACT$, illocutionary (speech) acts $ILL$, agents $AGT $, emotions\n$EMO$ (which is a subset of the twenty two OCC emotions in our model),\nand the intervals of real numbers $DEG = [-1,1]$ and $DEG^+=\n[0,1]$. $ATM$ describes facts or assertions (\\textit{e.g.}\n\\textit{salary\\_is\\_bad}, \\textit{picnic\\_is\\_fun}) or external events\nsuch as \\textit{rain\\_starts\\_falling}. $ACT$ describes actions that\nthe agents or humans ($AGT$) may perform,\ne.g. \\textit{introduce\\_itself} or \\textit{have\\_a\\_picnic}.\n\nOur model defines events as acts in which at least one of the actors\nof the interaction take part. Elements in $EVT$ are tuples in\n$AGT\\times AGT\\times (ACT\\cup ILL(ATM))$ where the first element is\nthe actor that performs the action, the second is a passive agent and\nthe act can be either an actions ($ACT$) or a speech act ($ILL$). This\nrepresentation is similar to the one in \\cite{ochs2009simulation}\nexcept we associate a subjective \\textit{degree of plausibility} as is\nusually done in BDI models and we do not distinguish actions from\ncommuncation. Illocutionnary speech acts have the form\n$\\varsigma(\\varphi)$ and mean \\textit{\\textquotedblleft actor utters\n  $\\varphi$ to recipient through the illocutionary act $\\varsigma$\"}.\n\nThe language we define is the set of formulas described by the\nfollowing BNF (Backus-Naur-Form):\n\\begin{equation}\n\\begin{split}\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\t\\\\\nEvt \t&: \\epsilon\t::= \\langle a, (a|\\varnothing), \\alpha \\rangle \t\\,|\\,\t\\langle a, a, Spk(\\varsigma ,\\varphi) \\rangle\t\\\\\nPrp\t&: \\pi\t::=\t p \\,|\\, \\epsilon \\,|\\, Like_{a,b}^k \\,|\\, Dom_{a,b}^k \t\t\t\t\t\t\t\\\\\nFml\t\t&: \\varphi ::=  \\pi \\,|\\, Bel_a^l(\\varphi) \\,|\\, Att_a^k(\\varphi) \\,|\\, Int_a(\\varphi) \\,|\\, Emo_{a,(b|\\varnothing)}^i(\\varepsilon,\\varphi) \\,| \\\\\n&\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\,\\, N(\\varphi) \\,|\\, U(\\varphi,\\varphi) \\,|\\, \\neg\\varphi \\,|\\, \\varphi \\wedge \\varphi\t\t\t\n\\end{split}\n\\label{eq:bnf}\n\\end{equation}\nwhere $a,b \\in AGT$, $\\alpha \\in ACT$, $p \\in ATM$, $\\epsilon \\in\nEVT$, $\\varepsilon\\in EMO$, $\\varsigma\\in ILL$, $l,i \\in DEG^+$, $k\n\\in DEG$. $Like$, $Dom$, $Bel$, $Att$ and $Int$ are modal operators\nand $N$, and $U$ are temporal operators \\textit{Next} and \\text{Until}\nfrom LTL and CTL* \\cite{pnueli1977temporal}. The other temporal operators $F$ and $G$ and\nboolean conditions $\\top$, $\\bot$, $\\vee$ and $\\Rightarrow$ are\ndefined in the standard way. Moreover, in the events' representation,\nwe use ``$-$'' as the \\textit{any} operator.\n\nFor the representation of social relation \\cite{leary1957interpersonal}, $Like_{a,b}^k$\ndetermines the level of liking agent $a$ has for agent $b$, while\n$Dom_{a,b}^k$ represents the degree of dominance.\n\n$Bel_a^l(\\varphi)$ is a graded belief, in a similar manner to \\cite{Dastani2012}, and has to be read \\textit{\\textquotedblleft a believes that $\\varphi$ with certainty l\"}. For instance, $Bel_a^{1}(\\varphi)$ means \\textit{\\textquotedblleft a is sure that $\\varphi$\"} and $Bel_a^0(\\varphi)$ can be read \\textit{\\textquotedblleft For a, $\\varphi$ is not plausible at all\"}.\n\nSimilarly, $Att_a^k(\\varphi)$ is a graded attitude that has to be read\n\\textit{\\textquotedblleft a appreciates\/values the fact that $\\varphi$\n  with a degree l\"}. In our context, this operator will be used to\ncover various notions, such as \\textit{Desires}, \\textit{Ideals} and\n\\textit{Goals} that are represented with distinct modal operators in\nother work such as \\cite{Adam2009} and \\cite{Guiraud2011}. We define\ndesires as a positive attitude toward future facts and ideals as what\nthe agents would like to be always true:\n\\begin{equation}\n\\begin{split}\nDes_a^k(\\varphi) &\\stackrel{\\mathrm{def}}{=} Att_a^k(F(\\varphi)) \\\\\nIdeal^{k>0}_a(\\varphi) &\\stackrel{\\mathrm{def}}{=} Att^{k>0}_a(G(\\varphi)) = Des^{-k<0}_a(\\neg \\varphi)\n\\label{eq:desideal}\n\\end{split}\n\\end{equation}\nThe definition of goals through attitudes will be presented in the\nnext subsection. \n\nNote that in our model, the subject of an attitude can as well be\n\\textit{preserving\\_forest}, \\textit{being\\_nice\\_to\\_others},\n\\textit{hiring\\_new\\_employee} or $Bel_b^l(\\langle a, c,\ngive\\_sandwich\\rangle)$, eventually encapsulated in temporal\noperators.\n\nAs in classical BDI, $Int_a(\\varphi)$ represents an agent's plan \\cite{rao1991modeling} and has to be read \\textit{\"a intends to make $\\varphi$ true\"} (with $\\varphi$ being an event in the general case).\n\n$Emo_{a,(b|\\varnothing)}^i(\\varepsilon,\\varphi)$ represent emotions. Following classical literature \\cite{frijda1986emotions}, our emotions are related to facts and can be directed toward an agent. $Emo_{a,(b|\\varnothing)}^i(\\varepsilon,\\varphi)$ has to be read \\textit{\\textquotedblleft a feels $\\varepsilon$, eventually for\/towards b, with intensity i, regarding the fact that $\\varphi$\"} with $\\varepsilon\\in EMO$. In the following sections, it will be simplified into $\\varepsilon_{a,(b|\\varnothing)}^i(\\varphi)$.\n\nFor the sake of readability, we introduce new operators to represent agents' involvement in an event. $Resp_a$ expresses a \\textit{direct responsibility}. Unlike \\cite{Adam2009,Guiraud2011}, we do not consider an agent responsible for a situation it could have avoided. $Wit_a$ means that the agent witnessed the occurrence of the event:\n\\begin{equation}\n\\begin{split}\nResp_a(\\epsilon) &\\stackrel{\\mathrm{def}}{=} (\\epsilon = \\langle a,-,-\\rangle) \\\\\nWit_a(\\epsilon) &\\stackrel{\\mathrm{def}}{=} (\\epsilon = \\langle a,-,-\\rangle) \\vee (\\epsilon = \\langle -,a,-\\rangle)\n\\label{eq:respwit}\n\\end{split}\n\\end{equation}\n\n\n\n\\subsection{Semantics}\nBased on possible world semantics, we define a frame\n$\\mathcal{F} = \\langle\\mathcal{W,B,D,I,E}\\rangle$ as a tuple where:\n\\begin{itemize}\n\\item $W$ is a nonempty set of possible worlds,\n\\item $\\mathcal{B}:AGT\\rightarrow (W\\rightarrow 2^W)$ is the function that associates each agent $a\\in AGT$ and possible world $w\\in W$ to the set of belief-accessible worlds $\\mathcal{B}_a(w)$,\n\\item $\\mathcal{D}:AGT\\rightarrow (W\\times DEG^+\\rightarrow 2^W)$ is the function that associates each agent $a\\in AGT$ and possible world $w\\in W$ with a level of desirability $l\\in DEG^+$ to the set of desire-accessible worlds $\\mathcal{D}_a(w,l)$,\n\\item $\\mathcal{I}:AGT\\rightarrow (W\\rightarrow 2^W)$ is the function that associates each agent $a\\in AGT$ and possible world $w\\in W$ to the set of intention-accessible worlds $\\mathcal{I}_a(w)$, and\n\\item $\\mathcal{E}:EVT\\rightarrow W$ is the function that associates each event $\\epsilon \\in EVT$ to the resulting possible world.\n\\end{itemize}\nThen, a model $\\mathcal{M=\\langle F,V\\rangle}$ is a couple where $\\mathcal{F}$ is a frame and $\\mathcal{V}:W\\rightarrow ATM$ a valuation function.  \n\\par Given a model $\\mathcal{M}$ we note $\\mathcal{M},w \\models \\varphi$ a formula $\\varphi$ that is true in a world $w$. Truth conditions of formulas are defined by induction in the classical way:\n\\begin{itemize}\n\\item $\\mathcal{M},w \\models p$ iff $p\\in \\mathcal{V}(w)$;\n\\item $\\mathcal{M},w \\models \\neg\\varphi$ iff not $\\mathcal{M},w \\models \\varphi$;\n\\item $\\mathcal{M},w \\models \\varphi \\wedge \\psi$ iff $\\mathcal{M},w \\models \\varphi$ and $\\mathcal{M},w \\models \\psi$ ;\n\\item $\\mathcal{M},w \\models Bel_a^l(\\varphi)$ iff $\\frac{card(\\mathcal{GB}_a(w))}{card(\\mathcal{B}_a(w))}=l\\;$ \\\\where $\\mathcal{GB}_a(w) = \\{v\\in\\mathcal{B}_a(w)$ ; $\\mathcal{M},v \\models \\varphi \\}$ ;\n\\item $\\mathcal{M},w \\models Des_a^l(\\varphi)$ iff $\\mathcal{M},v \\models \\varphi$ $\\forall v\\in\\mathcal{D}_a(w,l)$;\n\\item $\\mathcal{M},w \\models Int_a(\\varphi)$ iff $\\mathcal{M},v \\models \\varphi$ $\\forall v\\in\\mathcal{I}_a(w)$;\n\\item $\\mathcal{M},w \\models \\epsilon$ iff $\\mathcal{M},v \\models \\top$ $\\forall v\\in\\mathcal{E}(\\epsilon)$;\n\\end{itemize}\nThe truth condition of $Bel_a^l(\\varphi)$ states that the level of plausibility of $\\varphi$ is the proportion of belief-accessible worlds where $\\varphi$ is true. The next subsections describe the rules for the inference engines presented in section \\ref{sect:archi}. When required, the computation of believability, desirability and intensity degrees will be represented by a $f$ function that is part of the implementation and will not be detailed in this section (see section \\ref{sect:implem} instead).\n\n\n\\subsection{Folk-psychology reasoner}\n\\subsubsection{Graded beliefs}\nFollowing \\cite{Dastani2012} and \\cite{Adam2009}, all accessibility relations $\\mathcal{B}$ are transitive and euclidean, which ensures that the agent is aware of its own beliefs\\footnote{If $w\\mathcal{R}v$ and $v\\mathcal{R}u$, then successively by transitivity, euclidianity and transitivity again: $w\\mathcal{R}v$ and $v\\mathcal{R}v$.}:\n\\begin{equation}\nBel_a^l(\\varphi) \\stackrel{\\mathrm{def}}{\\Longrightarrow} Bel_a^1(Bel_a^l(\\varphi))\n\\label{eq:beltreuclid}\n\\end{equation}\nWe generalize (\\ref{eq:beltreuclid}) so that agents are aware of their own mental states, social relations and involvement.\n\nHowever, unlike other models \\cite{Adam2009} \\cite{Dastani2012}, $\\mathcal{B}$ is not serial\\footnote{A relation $\\mathcal{R}$ is serial iff $\\forall w,\\;\\exists v$ so that $w\\mathcal{R}v$. }. Only $\\mathcal{GB}$ is. This represents the fact that the agent generally has uncertainty about states of affairs. Intuitively:\n\\begin{equation}\nBel_a^l(\\varphi) \\stackrel{\\mathrm{def}}{\\Longrightarrow} Bel_a^{1-l}(\\neg\\varphi)\n\\label{eq:belnotphi}\n\\end{equation}\n\nFor convenience, we define two thresholds $mod\\_th$ and $str\\_th$ wich $0.5 < mod\\_th < str\\_th$. They correspond to situations where the agent \\textit{moderately} ($Bel_a^{l>mod\\_th}(\\varphi)$) and \\textit{strongly} ($Bel_a^{l>str\\_th}(\\varphi)$) believes something.\n\nFinally, if an agent believes a state of affairs to possibly cause another, it will deduce a belief about it:\n\\begin{equation}\nBel_a^l(\\psi)\\wedge Bel_a^{l'}(\\psi \\Rightarrow \\varphi) \\stackrel{\\mathrm{def}}{\\Longrightarrow} Bel_a^{f(l,l')}(\\varphi)\n\\label{eq:newbel}\n\\end{equation}\n\n\\subsubsection{Graded attitudes}\nAttitudes can be positive or negative and we assume that agents hold consistent desires:\n\\begin{equation}\n\\mathcal{M},w \\models (Att_a^{k}(\\varphi) \\wedge Att_a^{k'}(\\neg\\varphi)) \\,\\mathrm{iff}\\, k = -k'\n\\end{equation}\n\nHowever, \\textit{indirect} inconsistency is still possible: an agent might want something that can possibly lead to or be caused by (the occurrence of) the negation of another desire of his. The consistency is then preserved at the level of desire adoption:\n\\begin{equation}\n\\begin{split}\nDes^k_a(\\varphi) \\wedge Bel^{l>str\\_th}_a(\\psi \\Rightarrow F(\\varphi)) \\wedge \\neg IncDes^k_a(\\psi) \\\\ \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Des^k_a(\\psi))\n\\label{eq:newdes}\n\\end{split}\n\\end{equation}\nwith $IncDes^k_a(\\varphi)$ representing inconsistent desires:\n\\begin{equation}\n\\begin{split}\nIncDes^k_a(\\varphi) \\stackrel{\\mathrm{def}}{=} \n&(Bel^{l>str\\_th}_a(\\varphi \\Rightarrow \\neg\\psi) \\wedge Des^{k'>0}_a(\\psi)) \\\\ \\vee &(Bel^{l>str\\_th}_a(\\varphi \\Rightarrow \\psi) \\wedge Des^{k'<0}_a(\\psi)) \n\\label{eq:inconsdes}\n\\end{split}\n\\end{equation}\n\nThis means that desiring $\\varphi$ is inconsistent when the agent\nstrongly beliefs it might lead to an undesirable $\\psi$. We allow for\nadopting indirectly inconsistent desires only when the agent only\nbelieves \\textit{moderately} that there can be a certain\nincompatibility with existing ones.\n\nWe also define a weaker case of inconsistency where $\\varphi$ leads to\nan undesirable state of affairs of a higher level:\n\\begin{equation}\n\\begin{split}\nWIncDes^k_a(\\varphi) \\stackrel{\\mathrm{def}}{=} Bel^{l>str\\_th}_a(\\varphi \\Rightarrow \\neg\\psi)\n\\\\ \\wedge Des^{k';|k'| > |k|}_a(\\psi)\n\\end{split}\n\\label{eq:winconsdes}\n\\end{equation}\n\n\\subsubsection{Goals}\nFollowing the BDI model \\cite{rao1991modeling}, goals are defined as\ndesires that are consistent -- at least weakly, in our case -- and\nbelieved to be achievable. To this purpose, we introduce a new\nthreshold $des\\_th$:\n\\begin{equation}\nGoal^{k>0}_a(\\varphi) \\stackrel{\\mathrm{def}}{=} Des^{k>des\\_th}_a(\\varphi) \\wedge Bel^{l}_a(F(\\varphi)) \\wedge \\neg WIncDes^k_a(\\varphi)\n\\label{eq:goal}\n\\end{equation}\n\nGoals are then turned into intentions either because the agent can achieve it:\n\\begin{equation}\nGoal^{k>0}_a(\\epsilon) \\wedge Resp_a(\\epsilon) \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Int_a(\\epsilon))\n\\label{eq:goalintevt}\n\\end{equation}\nor, similarly to \\cite{bosse2011recursive}, because the agent strongly believes there is -- at least -- one mean to achieve it:\n\\begin{equation}\n\\begin{split}\nGoal^{k>0}_a(\\varphi) &\\wedge Bel^{l>str\\_th}_a(\\psi \\Rightarrow F(\\varphi)) \\wedge \\neg WIncDes^k_a(\\psi) \n\\\\ &\\wedge Bel^{l'}_a(F(\\psi)) \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Int_a(\\psi))\n\\label{eq:goalintphi}\n\\end{split}\n\\end{equation}\nWe leave it to the implementation phase (section~\\ref{sect:implem}) to decide how intentions are ordered when several possible known $\\psi$ can be used to achieve a goal.\n\n\\subsubsection{Intentions and acts}\nSince intentions are generated from desires all accessibility relations $\\mathcal{I}$ are serial: $\\mathcal{M},w \\not\\models Int_a(\\neg\\varphi)$ if $\\mathcal{M},w \\models Int_a(\\varphi)$\n\nIf an agent intends a state of affairs and knows a means to achieve it, it will also intend the latter:\n\\begin{equation}\nInt_a(\\varphi) \\wedge Bel^{l>str\\_th}_a(\\psi \\Rightarrow F(\\varphi)) \\stackrel{\\mathrm{def}}{\\Longrightarrow} Int_a(\\psi)\n\\label{eq:newint}\n\\end{equation}\nAdditionally, if an agent intends an act which it is responsible for, it will perform it in the next step:\n\\begin{equation}\nInt_a(\\epsilon) \\wedge Resp_a(\\epsilon) \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(\\epsilon)\n\\label{eq:intevt}\n\\end{equation}\n\nFurthermore, when an event occurs, we propagate responsibility to all the states of affairs it is believed to have caused:\n\\begin{equation}\n\\begin{split}\nBel_a^d(\\psi)\\wedge Bel_a^l(Resp_b(\\psi)) \\wedge Bel_a^{l'}(\\varphi) \\wedge Bel_a^{l''}(\\psi \\Rightarrow F(\\varphi)) \n\\\\\\stackrel{\\mathrm{def}}{\\Longrightarrow} Bel_a^{f(l,l',l'')}(Resp_b(\\varphi))\n\\label{eq:rspphi}\n\\end{split}\n\\end{equation}\n\\par Finally, as far as accessibility relations $\\mathcal{E}$ are\nconcerned, we consider any witness believes with degree 1 that the\nevent happened and that the other witness also believes it. Note that\nwhen an event occurs, the belief that it happened remains true\nafterwards.\n\n\\subsubsection{Updating attitudes}\nBeliefs are updated as new events occur (except for ideals that are\nconstant and hold globally). In order for the agent to react to\nsituation change, attitudes about new states of affairs have to be\ntriggered. In our model, following\n\\cite{ochs2009simulation,castelfranchi1998modelling,aylett2008if}, the\nattitude is influenced not only by new beliefs, but also by the\nattitude of others and the social relation. Formally:\n\\begin{equation}\n\\begin{split}\nBel^{l>str\\_th}_a(\\varphi) \\wedge Att^k_a(F(\\varphi)) \\wedge Bel^{l'}_a(Att^{k'}_b(F(\\varphi))) \n\\\\ \\wedge Like^{h}_{a,b} \\wedge Dom^{h'}_{a,b} \\stackrel{\\mathrm{def}}{\\Longrightarrow} Att^{f(k,k',h,h')}_a(\\varphi) \\\\\nBel_a^{l>str\\_th}(Des_b^{k}(\\varphi)) \\wedge Like_{a,b}^{k'>0} \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Des_a^{f(k,k')}(\\varphi)))\n\\end{split}\n\\label{eq:attcurr}\n\\end{equation}\n\n\\subsubsection{Speech acts and social interaction}\nBeliefs can also be updated through communication. Although our work\nmostly focus on non-verbal communication, we consider a limited set of\nillocutionnary acts \\cite{searle1969speech} $ILL = \\{Assert, Request,\nCommit, Express\\}$.\n\nBased on similar work in speech acts formalization \\cite{herzig2002logic,Guiraud2011}, we define trigering rules for our speech acts. For instance:\n\\begin{equation}\n\\neg Bel_a^1(Int_b(\\varphi)) \\wedge Int_a(Int_b(\\varphi)) \\stackrel{\\mathrm{def}}{\\Longrightarrow} Request_{a,b}(\\varphi)\n\\label{eq:spact}\n\\end{equation}\n\n\nIn turn, these events will lead to new mental states for the recipient agent, similarly to classical FIPA semantics and existing work on social interaction modeling \\cite{herzig2002logic,castelfranchi1998modelling}. For the sake of conciseness, we only present these two examples here:\n\\begin{equation}\n\\begin{split}\nAssert_{b,a}(\\varphi) \\wedge Like_{a,b}^{k}\\wedge Dom_{a,b}^{k'} &\\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Bel_a^{f(k,k')}(\\varphi))\\\\\nRequest_{b,a}(\\varphi) \\wedge Dom_{a,b}^{k<0} &\\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Int_a(\\varphi)))\n\\end{split}\n\\label{eq:perloc}\n\\end{equation}\n\n\n\n\n\n\n\\subsection{Emotional inference engine}\nThe emotional inference engine consists of a set of appraisal rules\nfor emotion categories $EMO$. In this implementation, we have used an OCC-based model, highly inspired by\n\\cite{Adam2009,Guiraud2011,Dastani2012}. Here are some examples of triggering conditions for each group of emotions.\n\n\\begin{equation}\n\\begin{split}\nBel_a^l(\\gamma) \\wedge Att_a^{k>0}(\\gamma) &\\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Joy_a^{i=f(l,k)}(\\gamma)) \\\\\nBel_a^l(F(\\gamma)) \\wedge Des_a^{k<0}(\\gamma) & \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Fear_a^{i=f(l,k)}(\\gamma))\\\\\nBel_a^d(\\gamma)  \\wedge Bel_a^l(Att_b^{k<0}&(\\gamma)) \\wedge Like_{a,b}^{k'<0} \\\\ & \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Gloating_{a,b}^{i=f(l,k,k',d)}(\\gamma)) \\\\\nBel_a^l(\\gamma) \\wedge Ideal_a^k(\\gamma) \\wedge &Bel_a^{l'}(Rsp_b(\\gamma)) \\\\ & \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Admiration_{a,b}^{i=f(l,l',k)}(\\gamma))  \\\\\nBel_a^l(\\gamma) \\wedge Ideal_a^k(\\gamma) \\wedge &Bel_a^{l'}(Rsp_b(\\gamma)) \\wedge Goal_a^{k'}(\\gamma)\\\\ & \\stackrel{\\mathrm{def}}{\\Longrightarrow} N(Gratitude_{a,b}^{i=f(l,l',k, k')}(\\gamma)) \n\\label{eq:emo}\n\\end{split}\n\\end{equation}\n\nPlease note that $\\gamma$ is a proposition that do not involve any temporal\noperator. Besides, the intensity of an emotion is a\ncombination of the degree of certainty of beliefs and the degree of\ndesirability in attitudes. Depending on the appraisal model, the\ndegree of certainty can represent the \\textit{sense of reality}, the\n\\textit{unexpectedness}, the \\textit{likelihood} and the\n\\textit{realization}. The degree of desirability can correspond to\n\\textit{desirability-for-self} but also to \\textit{praiseworthiness}\n\\cite{Adam2009}.\n\n\n\\par In OCC \\cite{Ortony1990}, \\textit{Gratification}, \\textit{Remorse},\n\\textit{Gratitude} and \\textit{Anger} are defined as\nWell-being\/Attribution emotions, triggered when one focuses both on\nthe praiseworthiness of an action and on its desirability. However, in\nour model, these two notions overlap since ideals are deduced from\nattitude. Nevertheless, similarly to \\cite{Guiraud2011} we think that\none might distinguish \\textit{Gratitude} and \\textit{Anger} from\n\\textit{Admiration} and \\textit{Reproach} if the triggering state of\naffairs corresponds to a goal, that is to say it is not only\npraiseworthy but is also desirable and consistent enough to generate\nan intention of achievement.\n\n\n\n\n\\subsection{Commonsense reasoner}\nThe commonsence reasoner allows the agent to acquire new beliefs based\non a set of commonsense rules. It is mostly domain-dependent. Section\n\\ref{sec:job} describes how we used it to implement a job interview\nsimulation scenario. Here is a simple example of how this reasoner can\ncombine with the folk-psychology inference engines\npresented above.\n\\subsubsection{Example} \nConsider two friends John (J) and Mary (M) having a conversation about\ntheir holidays. Mary is going to her hometown (ht). The fact that she\nis going to visit her father is a detail she could either mention or\nnot:\n\\begin{equation}\n\\nonumber\n\\begin{tabular*}{\\linewidth}{l @{\\extracolsep{\\fill}} r}\n$Des_{M}^{0.77}(talking\\_about\\_holidays)$ & (input)\\\\\n$Bel_{M}^{0.8}(\\langle M,J,visiting\\_ht\\_and\\_dad \\rangle$ \\\\$\\hfill\\Longrightarrow F(talk\\_about\\_holidays))$ & (input)\\\\\n$Bel_{M}^{0.8}(\\langle M,J,visiting\\_ht \\rangle $\\\\$\\hfill\\Longrightarrow F(talk\\_about\\_holidays))$ & (input)\n\\end{tabular*}\n\\end{equation}\nNevertheless Mary remembers John recently lost his father and thus supposes it is a sensitive topic:\n\\\\\\begin{equation}\n\\nonumber\n\\begin{tabular*}{\\linewidth}{l @{\\extracolsep{\\fill}} r}\n$Bel_{M}^{1}(J\\_lost\\_his\\_dad)$ & (input)\\\\\n$Bel_{M}^{0.76}(J\\_lost\\_his\\_dad \\Longrightarrow Ideal_{J}^{0.8}(\\neg \\langle -,J,dad \\rangle) )$ & (input)\\\\\n$\\hspace{0.5cm} (\\Longrightarrow) Bel_{M}^{l}(Ideal_{J}^{0.8}(\\neg\\langle -,J,dad \\rangle))$ & (\\ref{eq:newbel})\n\\end{tabular*}\n\\end{equation}\nOf course, Mary knows that saying she is going to visit her father implies actually talking about her father:\n\\begin{equation}\n\\nonumber\n\\begin{tabular*}{\\linewidth}{l @{\\extracolsep{\\fill}} r}\n$Bel_{M}^{0.8}(\\langle M,-,visiting\\_ht\\_and\\_dad \\rangle \\hfill\\Longrightarrow \\langle M,-,dad \\rangle )$ & (input)\n\\end{tabular*}\n\\end{equation}\nAnd, knowing that John wants to avoid this topic, she does too. Hence, she is will not mention the fact that she is visiting her father when talking about her holidays:\n\\begin{equation}\n\\nonumber\n\\begin{tabular*}{\\linewidth}{l @{\\extracolsep{\\fill}} r}\n$\\hspace{0.5cm} (\\Longrightarrow) Ideal_{M}^{k}(\\neg\\langle -,J,dad \\rangle)$ & (\\ref{eq:newdes})\\\\\n$\\hspace{0.5cm} (\\Longrightarrow)\nWIncDes_{M}^{0.77}(\\langle M,J,visiting\\_ht\\_and\\_dad \\rangle)$ & (\\ref{eq:winconsdes}) \\\\\n$\\hspace{0.5cm} (\\Longrightarrow)\nGoal_{M}^{0.77}(\\langle M,J,visiting\\_ht \\rangle)$ & (\\ref{eq:goal}) \n\\end{tabular*}\n\\end{equation}\n\n\n\\section{Implementation}\n\\label{sect:implem}\n\nThe theoretical model we presented in previous sections is aimed to be domain-independent. Yet, the purpose of our current work in the TARDIS project \\cite{tardis2013} is to develop a training game in order to facilitate NEETs' access to employment. Therefore, we propose to implement it in the context of job interview simulation. Indeed, this sort of application appears as a promising way to increase applicant's self-confidence \\cite{hoque2013}. Additionally, job interviews are a good example of semi-structured dyadic interactions where recruiters have several opportunities to reason about candidates' mental and affective states. Nevertheless, to our knowledge, existing models do not include a Theory of Mind. \n\nOur implementation was done in SWI-Prolog for the inference engine and\nthe logical framework. This reasoner was embedded in a C++ program\nthat handles the reasoning loop and the communication between the\nmodules.\n\n\n\\subsubsection*{Reasoning loop}\nFollowing the classical BDI interpreter, at every cycle, the agent\ninterpret external events to generate a list of potential actions,\ndeliberate to select one of them, update its intentions and then\nexecute them:\n\\begin{algorithm}\n\\caption{ToM Reasoning loop}\n\\label{algo:loop}\n\\begin{algorithmic}\n\\small{\n\\Loop\n\t\\State Execute\\_intentions()\n\t\\State Simulate\\_others\\_emotions()\n\t\\State Update\\_beliefs\\_and\\_attitudes()\n\t\\State \\hspace{\\algorithmicindent} Update\\_beliefs\\_with\\_new\\_SoA()\n\t\\State \\hspace{\\algorithmicindent} Handle\\_operators\\_equivalence()\n\t\\State \\hspace{\\algorithmicindent} Adopt\\_new\\_desires()\n\t\\State \\hspace{\\algorithmicindent} Order\\_goals()\n\t\\State Adopt\\_new\\_intentions()\n\t\\State \\hspace{\\algorithmicindent} Adopt\\_new\\_intentions\\_from\\_goals()\n\t\\State \\hspace{\\algorithmicindent} Adopt\\_new\\_intentions\\_from\\_intentions()\n\\EndLoop\n}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection*{Thresholds and level functions}\nThe implementation of the model requires to instanciate all thresholds\n($th$) and combination function ($f$) for degrees of believability and\ndesirability of new mental state or the intensity of emotions.\n\nIn our implementation, $mod\\_th = 0.5$, $str\\_th = 0.75$ and $des\\_th\n= 0.7$.\n\nThe combination function cannot be given in detail in this\npaper but we consider two families:\n\n-- For attitude dynamics and credibility (\\textit{e.g.} equation\n  \\ref{eq:attcurr}), we use simple average functions on the relevant\n  interval:\n\\begin{equation}\n\\nonumber\nf(k,k') = ((k + k')\/4)+0.5\n\\end{equation}\n\n-- For emotions (see equation \\ref{eq:emo}), we combine the linear\n  influence from attitude (for instance, joy has been chosen to be\n  linearly correlated to the attitude toward the fact) with a\n  logarithmic influence of the degree of certainty. This way, we get\n  to trigger more salient emotions even with relatively weak\n  beliefs. Nevertheless, let us remind here that we only consider\n  beliefs which levels are greater than a certain threshold\n  ($mod\\_thld=0.5$ in our implementation).\n\n\\begin{equation}\n\\nonumber\nf(l,k) = \\frac{k}{2} \\times \\frac{Log (2l-1) - min}{min}+0.5\n\\end{equation}\nwhere $min$ represents the smallest value coded by the machine\n(\\textit{i.e.} the value of $Log(x)$ when $x\\rightarrow 0$). The\n$2l-1$ facto is used to adjust the value in [0,1] before we compute\nthe intensity, which is then readjusted in [0.5,1] to get significant\nvalues.\n\n\n\\subsubsection*{Job interview simulation}\n\\label{sec:job}\nThe course of the job interview is handled in the commonsense module:\nwe define a series of topics that must be adressed by the agent\nthrough speech acts (\\textit{e.g.} questions about the salary, the\nexperience...). Moreover, each topic is associated with some\nexpectations about the impact of the question. Based on the current\n\\textit{goals} (in terms of affective state for the interlocutor), the\nagent will select a question (a speech act) or another.\n\nMoreover, the agent computes beliefs about the interlocutor's\nself-confidence, motivation and qualification, based on its reaction\nto the questions and simple TT-rules. For instance, hesitating in the\njob description topic can indicate they are not qualified enough while\nbeing focused when introducing themselves denotes a good\nself-confidence level. The perception of ``hesitation'' and\n``focused'' is done by another module of the TARDIS platform which is not part of this paper (see \\cite{tardis2013}).\n\n\n\n\\section{Preliminary evaluation}\n\\label{sect:eval}\nIn this section, we describe a preliminary evaluation aiming to assess\nthe functioning of our model and its possible contribution in the\ncontext of job interview simulation.\n\n\nSubjects play the role of an unemployed youngster lacking work\nexperience and applying for the job of sales department secretary. The\nvirtual recruiter utterances are predefined for each possible speech act and situation in the model. No\nconstraint were given about a supposed personality, level of education\nof professional background of the role-played interviewee.\n\n\n\\subsection{Method}\nWe recruited 30 volunteers -- 11 females and 19 males --, 19 of them working or having an internship at our university. All the subjects were aged over 24, had gone to university and were familiar with computers. 18 of the participants are native speakers and the remaining have at least an intermediate level. Since we are not interested in verbal communication, this is sufficient so that the participants understand what the recruiter says.\n\nThe recruiter's utterances were given in a very simple Graphical User Interface (GUI). The valence of the agent's affective state and its runtime evaluation of the candidate's self-confidence, motivation and qualification (values in [-1,1]) were represented by slide-bars. A text field allows the subject to type his\/her answer to the virtual recruiter's questions. Besides, a series of 8 sliders (values in [0,1]) gives them the possibility to express their affective states to the recruiter as combinations of the following affects: relieved (REL), embarrassed (EMB), hesitating (HES), stressed (STR), ill at ease (IAE), focused (FOC), aggressive (AGG) and bored (BOR).\\footnote{In the full TARDIS project's setting, these sliders are replaced by automatic recognition of user affects using the SSI system \\cite{wagner2013social}.}\n\n\nSubjects faced one agent out of 3 possible recruiter profiles: one that tries to make the candidates feel at ease (PROFILE\\_A), one asking regular questions, with no specific goal on the user's mental state (PROFILE\\_B) and one that, asks embarassing questions (PROFILE\\_C). This is simply done by varying their goals regarding the emotional reaction they want to elicit in our model. All three agents use the same ToM reasoner described in previous sections.\n\n\\par \\textbf{Hypothesis:} The profile variation will have an impact on the participants' emotional states as expressed through the slidebars.\n\\par \\textbf{Measures:} In this paper, we focus on measures extracted from the interaction history. They refer to the average intensity of relief, embarrassment, hesitation, stress, uneasiness, concentration, aggressiveness and boredom expressed by the participants as well as the total emotional expressiveness (TOT). More specifically, we measure the mean amount of information the candidates gave about their affective states.\n\n\\subsection{Results}\nShapiro-Wilks test shows that none of our measures follows a normal distribution. Besides, Kruskal-Wallis test reveals a main effect of PROFILE on TOT ($Chi^2(2,629)=11.435; p<0.01$) and particularly EMB ($Chi^2(2,629)=6.231; p<0.05$) and FOC ($Chi^2(2,629)=9.218; p<0.01$). This means that the profile of the recruiter (comprehensive, neutral or challenging) has an effect on the affects assessed (and possibly expressed) by the user, and that this effect is particularly important for embarassement and concentration.\n\nA Mann-Whitney test then shows that participants that interact with PROFILE\\_A (comprehensive recruiter) express more affects in general ($U = 20; p<0.05$), more embarrassment ($U = 20; p<0.05$) and more concentration ($U = 21; p<0.05$) than those who interact with PROFILE\\_B. Likewise, PROFILE\\_C (challenging recruiter) elicits more affects ($U = 6; p<0.01$) and in particular stress ($U = 18; p<0.05$), uneasiness ($U = 24; p<0.05$) and concentration ($U = 10; p<0.01$) than PROFILE\\_B. We also note that in this case, no effect appears regarding embarrassment ($U = 26; p=0.069$). Finally, no significant effect is revealed between PROFILE\\_A and PROFILE\\_C. See Figure \\ref{fig:resH2}.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=\\linewidth]{ResH2.PNG}\n\\caption{Effect of PROFILE factor. This figure shows the average affects intensity expressed by the participants. The error bars represent the standard error. Significant effect appears on the embarrassment, stress, uneasiness, concentration and the total emotional intensity.}\n\\label{fig:resH2}\n\\end{figure}\n\n\n\\subsection{Discussion}\n\\label{sect:discuss}\n\n\n\n\n\n\nTheory of Mind is a complex process that relies on various other cognitive and perceptual processes. It is not only hard to model but also to assess. Thus, a simple protocol such as the one we used in this study is not sufficient to fully evaluate the impact of our ToM model on the quality of the training. First, the GUI we used is not user-friendly and does not allow for user's immersion in the scenario. We assume that using the full TARDIS project's setting \\cite{tardis2013} would enhance the interaction credibility and help highlight the virtual agent's reasoning and reactivity. Besides, the evaluation process should be based on richer measures (e.g. thorough post-hoc questionnaire) in order to evaluate the effect of our model. Yet, such a specific evaluation protocol for affective and interaction-oriented ToM still has to be defined.\n\nIn the litterature, there are validated methods to evaluate whether subjects -- generally children -- have ToM abilities and use it \\cite{blijd2008measuring}. Nevertheless, there is no such test that integrates a strong interactional aspect to our knowledge. From the computational point of view, \\cite{harbers2011explaining} points out the issue of evaluating a ToM model. In this work, the course of events and the agent's actions and explanations are specified in advance for different scenarios. Thus, the ToM models are evaluated based on whether they match these specifications. Similarly, \\cite{pynadath2013you} builds expectations about user's actions -- based on formal models in the specific context of wartime negotiations -- in order to model a simplified theory of mind and then compare them with the actual user's behavior. These two approaches are not applicable in our human\/agent interaction situation, because they rely on a model of the task which is difficult to describe when it comes to afective non-verbal behaviour.\n\nNevertheless, the study we present in this paper shows promising results regarding the contribution of our ToM model in the context of a training game. Although all recruiters' profiles benefit from the ToM reasoner, only PROFILE\\_A and PROFILE\\_C use it to select questions according to a reasoning about the mental and emotional states they could induce. The more recruters ask such questions, the more mindreading they perform. The study shows that this kind of ToM-based behavior indeed has an impact on the users' reactions. It also demonstrates the benefit of implementing several profiles in the enrichment of the coaching scenarios.\n\n\n\n\\section{Conclusion \\& perspectives}\n\\label{sect:conc}\nIn this paper, we proposed an affective and interaction-oriented Theory of Mind model to support the development of intelligent agents that are able to represent and reason about their human interlocutors' mental states. It relies on a hybrid mindreading approach mixing theory-theory and simulation-theory paradigms. This model is domain-independent, which means it can potentially be used in different context of application, including social coaching.\n\nIt has been implemented and evaluated in the context of job interview simulation in which the virtual recruiter both evaluates the human candidates based on their affective reactions, and reacts emotionally according to its desires and ideals. This study demonstrates the influence of the implementation of various recruiter profiles on the enrichment of the system's efficiency. In addition, the explanatory capability of our reasoning model is a key feature for the users to benefit from a rich post-interview feedback. While evaluating such a complex cognitive process as ToM remains a difficult task, we are currently working on the integration of this model in the TARDIS platform in order to perform mental states evaluation using signal processing. We plan to evaluate the impact of such a ToM model on the credibility of the virtual recruiter.\n\\bibliographystyle{abbrv}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nA restricted Boltzmann machine (RBM) is a generative neural-network model used to represent the distribution of random observations. In RBM, the independence structure between its variables is modeled through the use of latent variables (see Figure \\ref{fig:rbmtopology}). RBMs were introduced in \\cite{smolensky_information_1986}; although it was not until Hinton proposed contrastive divergence (CD) as a training technique in \\cite{hinton_training_2002}, that their true potential was unveiled.\n\n\\begin{figure}[h]\n\t\\centering\n\t\\def\\layersep{1.5cm}\n\t\\def\\numvis{4}\n\t\\def\\numhid{3}\n\t\\begin{tikzpicture}[\n\tnode distance=\\layersep,\n\tline\/.style={-}\n\t]\n\t\\tikzstyle{neuron}=[draw,circle,fill=black!25,minimum size=21pt,inner sep=0pt];\n\t\\tikzstyle{visible neuron}=[neuron, fill=white];\n\t\\tikzstyle{hidden neuron}=[neuron, fill=white];\n\t\\tikzstyle{annot}=[text width=4em];\n\n\n\t\\foreach \\name \/ \\y in {1,...,\\numvis}\n\t\\node[visible neuron] (V\\name) at (\\y,0) {$v_\\y$};\n\n\n\t\\foreach \\name \/ \\y in {1,...,\\numhid}\n\n\n\t\\pgfmathparse{\\y + (\\numvis - \\numhid) * 0.5}\n\t\\node[hidden neuron] (H\\name) at (\\pgfmathresult, \\layersep) {$h_\\y$};\n\n\n\t\\foreach \\source in {1,...,\\numvis}\n\t\\foreach \\dest in {1,...,\\numhid}\n\t\\draw[line] (V\\source) -- (H\\dest);\n\n\n\t\\ifthenelse{\\numvis > \\numhid}\n\t{\n\t\t\\node[annot,left of=V1, node distance=1cm] (hl) {Visible};\n\t\t\\node[annot,above of=hl] {Hidden};\n\t}\n\t{\n\t\t\\node[annot,left of=H1, node distance=1cm] (hl) {Hidden};\n\t\t\\node[annot,below of=hl] {Visible};\n\t}\n\t\\end{tikzpicture}\n\t\\caption[RBM Topology]{Example of an RBM  with \\numvis{} visible units and \\numhid{} hidden units.}\n\t\\label{fig:rbmtopology}\n\\end{figure}\n\n\nRestricted Boltzmann machines have proven powerful enough to be effective in diverse settings. Applications of RBM include: collaborative filtering \\cite{salakhutdinov_restricted_2007}, acoustic modeling \\cite{dahl_phone_2010}, human-motion modeling \\cite{taylor_modeling_2006,taylor_factored_2009}, and music generation \\cite{boulanger-lewandowski_modeling_2012}.\n\n\nRestricted Boltzmann machine variations have gained popularity over the past few years. Two variations relevant to this work are: the RNN-RBM \\cite{dahl_phone_2010} and the conditional RBM \\cite{taylor_factored_2009}. The RNN-RBM estimates the density of multivariate time series data by pre-training an RBM and then training a recurrent neural network (RNN) to make predictions; this allows the parameters of the RBM to be kept constant serving as a prior for the data distribution while the biases are allowed to be modified by the RNN to convey temporal information. Conditional RBMs, on the other hand, estimate the density of multivariate time series data by connecting past and present units to hidden variables. However, this makes the conditional RBM unsuitable for traditional CD training \\cite{mnih_conditional_2012}.\n\nIn this paper, we focus on the use of a modified RBM model that does not keep its parameters constant in each time step (unlike the RNN-RBM); and that adds hidden units for past interactions and lacks connections between past and future visible units (unlike the conditional RBM). Our model is advantageous because of two factors: (1) the topology of the model can be changed easily because it is controlled by a single set of parameters; and (2) the structure of our model allow the modeling of auto-correlation within a time series and correlation between multiple time series. These two factors allow many models to be readily tested and compared.\n\nWe show the performance of our model by applying it to the problem of forecasting stock market directions, i.e. predicting if the value of a stock will rise or fall after a pre-defined period of time. Previous work on the problem include \\cite{huang_forecasting_2005} where a support vector machine model was trained on the  NIKKEI 225 index, the Japanese analog of the Dow Jones Industrial Average index.\n\n\nThe rest of the paper is organized as follows. Section II presents a review of RBMs describing their energy function and training method trough CD. Section III introduces our proposed model, called the $p$-RBM, which can be viewed as an ensemble of RBMs with the property of recalling past interactions. In section IV we apply our model to 100 stocks of the NASDAQ-100 index and show its prediction results. Conclusions and future research directions are provided in Section V.\n\n\n\\section{The Restricted Boltzmann Machine Model}\n\n\nRestricted Boltzmann machines are formed by $n$ visible units, which we represent  as $\\textbf{v} \\in \\{ 0,1\\}^{n}$; and  $m$ hidden units, which we represent as $\\textbf{h} \\in \\{ 0,1\\}^{m}$. The joint probability of these units is modeled as\n\\begin{equation}\n\\begin{aligned}\np(\\textbf{v}, \\textbf{h} )=\\frac{1}{Z} e^{-  E(\\textbf{v}, \\textbf{h})};\n\\end{aligned}\n\\end{equation}\nwhere the energy function $E(\\textbf{v}, \\textbf{h})$ is given by\n\\begin{equation}\n\\begin{aligned}\nE(\\textbf{v}, \\textbf{h})= -\\textbf{a}^{\\intercal}\\textbf{v} -\\textbf{b}^{\\intercal} \\textbf{h} -\\textbf{v}^{\\intercal} \\textbf{W} \\textbf{h};\n\\end{aligned}\n\\end{equation}\nmatrix $\\textbf{W} \\in \\mathbb{R}^{n \\times m}$ represents the interaction between visible and hidden units; $\\textbf{a} \\in \\mathbb{R}^{n}$ and $\\textbf{b} \\in \\mathbb{R}^{m}$ are the biases for the visible an hidden units, respectively; and $Z$ is the partition function defined by\n\\begin{equation}\n\\begin{aligned}\nZ(\\textbf{a}, \\textbf{b}, \\textbf{W})=\\sum_{\\textbf{v}, \\textbf{h}}e^{-  E(\\textbf{v}, \\textbf{h})} .\n\\end{aligned}\n\\end{equation}\n\nThe bipartite structure of an RBM is convenient because it implies that the visible units are conditionally independent given the hidden units and vice versa. This ensures that the model is able to capture the statistical dependencies between the visible units, while remaining tractable.\n\nTraining an RBM is done via CD, which yields a learning rule equivalent to subtracting two expected values: one with respect to the data and the other with respect to the model. For instance, the update rule for $\\textbf{W}$ is\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:update_rule}\n \\Delta \\textbf{W}=\\langle \\textbf{v}\\textbf{h}^\\intercal  \\rangle_{\\mbox{Data}}-\\langle \\textbf{v}\\textbf{h}^\\intercal \\rangle_{\\mbox{Model}}.\n\\end{aligned}\n\\end{equation}\nThe first term in equation \\eqref{eq:update_rule} is the expected value with respect to the data and the second is the expected value with respect to the model. For a detailed introduction to RBMs the reader is referred  to \\cite{fischer_training_2014}.\n\n\n\\section{The $p$-RBM Model}\n\n\nWe generalized the RBM model by constructing an ensemble of RBMs, each one representing the state of the system at $p$ connected moments in time. One way to visualize this is to think of it as $p$ distinct RBMs connected together so as to model the correlation between moments in time. Each RBM contains a representation of the object of interest at different time steps, for example pixels of a video frame, or the value of a dynamic economic index. We then added connections between all the visible and hidden units across time to model their autocorrelation (see Figure \\ref{fig: p_rbm}).\n\nOur model resembles a Markov chain of order $p$, because the RBM at time $t$ is conditionally independent of the past, given the previous $p$ RBMs. We showed that even with these newly added time connections the model remains tractable and can be trained in a similar fashion to that of a single RBM.\n\nFor convenience, we bundled the visible and hidden units in block vectors denoted $\\tilde{\\textbf{v}}$ and $\\tilde{\\textbf{h}}$, respectively; we also included a vector of ones of appropriate size, to account for biases interactions; giving\n\\begin{equation}\n\\begin{aligned}\n\\tilde{\\textbf{v}}=\\left[\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n\\textbf{v}_t &\\textbf{v}_{t-1} & \\cdots & \\textbf{v}_{t-p} & \\mathbf{1}\n\\end{array}\\right]^\\intercal, \\mbox{ and}\n\\end{aligned}\n\\end{equation}\n\\begin{equation}\n\\begin{aligned}\n\\tilde{\\textbf{h}}=\\left[\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n\\textbf{h}_t &\\textbf{h}_{t-1} & \\cdots & \\textbf{h}_{t-p} & \\mathbf{1}\n\\end{array}\\right]^\\intercal.\n\\end{aligned}\n\\end{equation}\nThe parameters can be bundled in a block matrix as follows\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:bmatrix}\n\\resizebox{0.5\\textwidth}{!}{$\n\\tilde{\\textbf{W}}=\\left[\n\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n\\textbf{W}^{v_t,h_t} & \\textbf{W}^{v_t,h_{t-1}}&\\cdots& \\textbf{W}^{v_t,h_{t-p}}& \\textbf{W}^{v_t} \\\\ \\hdashline\n\\textbf{W}^{v_{t-1},h_t} & \\textbf{W}^{v_{t-1},h_{t-1}} &\\cdots & \\textbf{W}^{v_{t-1},h_{t-p}}& \\textbf{W}^{v_{t-1}} \\\\ \\hdashline\n\\vdotswithin{} & \\vdotswithin{} &\\vdotswithin{} & \\vdotswithin{} & \\vdotswithin{} \\\\ \\hdashline\n\\textbf{W}^{v_{t-p},h_t} & \\textbf{W}^{v_{t-p},h_{t-1}}&\\cdots& \\textbf{W}^{v_{t-p},h_{t-p}}& \\textbf{W}^{v_{t-p}} \\\\ \\hdashline\n\\textbf{W}^{h_t} & \\textbf{W}^{h_{t-1}} &\\cdots & \\textbf{W}^{v_{t-p}}& \\textbf{0}\n\\end{array}\\right]$}.\n\\end{aligned}\n\\end{equation}\n\nWe added a new hyperparameter $\\textbf{A}\\in[0,1]^{p+2,p+2}$ to the model, that acts as a forgetting rate, allowing the model to prioritize connections that are closed in time. While this new hyperparameter could be learned through, for instance, cross-validation or bayesian-optimization methods, in this work it is imposed.\n\nWe proposed the following structure for matrix $\\textbf{A}$\n\\begin{equation}\n\\begin{aligned}\n \\textbf{A}= \\left\\{\n \\begin{array}{ll}\n \\alpha^{|i-j|} & i,j \\leq p  \\\\\n 1 & \\mbox{elsewhere} \\\\\n \\end{array}\n \\right.\n \\end{aligned},\n\\end{equation}\nfor fixed $\\alpha \\in [0,1]$; or expressed in matrix form as\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:Alpha}\n\\textbf{A}=\\left[\n\\begin{array}{ccccc}\n\\alpha^{0} & \\alpha^{1}&\\cdots&  \\alpha^{p}& 1 \\\\\n \\alpha^{1} &  \\alpha^{0} &\\cdots &  \\alpha^{p-1}& 1 \\\\\n\\vdotswithin{} & \\vdotswithin{} &\\vdotswithin{} & \\vdotswithin{} & \\vdotswithin{} \\\\\n \\alpha^{p} &  \\alpha^{p-1}&\\cdots&  \\alpha^{0}& 1 \\\\\n1 & 1 &\\cdots & 1& 1\n\\end{array}\\right].\n\\end{aligned}\n\\end{equation}\nFor $\\alpha=1$ the model becomes fully connected, with all connections having the same significance. For $\\alpha=0$  the model is completely disconnected. Thus, matrix $\\textbf{A}$ has some control on the topology of the model. \\\\\n\\begin{figure*}[ht!]\n\\centering\n\\includegraphics[width=0.65\\linewidth]{p_rbm.pdf}\n\\caption{Example of a $p$-RBM with $p=3$ steps in the past.}\n\\label{fig: p_rbm}\n\\end{figure*}\n\nThe energy function for our $p$-RBM model is given by\n\\begin{equation}\n\\begin{split}\n\\begin{aligned}\n\\label{eq:model}\nE(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}}) ={}&-\\tilde{\\textbf{v}}^\\intercal (\\textbf{A} \\circ \\tilde{\\textbf{W}}) \\tilde{\\textbf{h}}\\\\\n={} &  -\\sum_{i=0}^{p} \\sum_{j=0}^{p} \\alpha^{|i-j|} \\textbf{v}_{t-i}^\\intercal \\textbf{W}^{v_{t-i},h_{t-j}}\\textbf{h}_{t-j}\\\\\n&  -\\sum_{i=0}^{p}\\textbf{v}_{t-i}^\\intercal  \\textbf{W}^{v_{t-i}} - \\sum_{j=0}^{p} \\textbf{h}_{t-j}^\\intercal \\textbf{W}^{h_{t-j}} ,\n\\end{aligned}\n\\end{split}\n\\end{equation}\nwhere $\\circ$ denotes the Hadamard product, indicating element-wise multiplication between matrices.\n\nThe energy function given in equation \\eqref{eq:model} considers the effect of the previous interaction. It provides the model with the capacity to include the past, allowing for greater flexibility than that of a typical RBM. As a consequence, the model can, for instance, model high-dimensional time series and non-linear dynamical systems, or reconstruct video from incomplete or corrupted versions.\n\nThe joint probability distribution under the model is given by the following Boltzmann-like distribution\n\\begin{equation}\n\\begin{aligned}\np(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})=\\frac{1}{Z(\\tilde{\\textbf{W}})} e^{-  E(\\tilde{\\textbf{v}},\\tilde{\\textbf{h}})},\n\\end{aligned}\n\\end{equation}\nwhere $E(\\tilde{\\textbf{v}},\\tilde{\\textbf{h}})$ is given in equation \\eqref{eq:model}, and  $Z(\\tilde{\\textbf{W}})$  is the partition function defined as\n\\begin{equation}\n\\begin{aligned}\nZ(\\tilde{\\textbf{W}})=\\sum_{\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}}} e^{-  E(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})}.\n\\end{aligned}\n\\end{equation}\n\nThe model structure induces the following relation of conditional independence\n\\begin{equation}\n\\begin{aligned}\n p(\\tilde{\\textbf{h}}|\\tilde{\\textbf{v}})= & p(\\textbf{h}_t|\\tilde{\\textbf{v}})  p(\\textbf{h}_{t-1}|\\tilde{\\textbf{v}}) \\cdots  p(\\textbf{h}_{t-p}|\\tilde{\\textbf{v}}) .\n\\end{aligned}\n\\end{equation}\n\n\\subsubsection{Derivative of the log-likelihood of the model}\nGiven  a single training example $\\tilde{\\textbf{v}}=\\{\\textbf{v}_t, \\textbf{v}_{t-1}, \\cdots, \\textbf{v}_{t-p} \\}$, the log-likelihood of our model is given by\n\\begin{equation}\n\\begin{aligned}\n\\mbox{ln} \\mathcal{L}(\\tilde{\\textbf{W}} | \\tilde{\\textbf{v}}) & =  \\mbox{ln}  P(\\tilde{\\textbf{v}} | \\tilde{\\textbf{W}}) \\\\& = \\mbox{ln} \\frac{1}{Z(\\tilde{\\textbf{W}})} \\sum_{\\tilde{\\textbf{h}}} e^{- E(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})} \\\\\n\t& =  \\mbox{ln} \\sum_{\\tilde{\\textbf{h}}} e^{- E(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})}- \\mbox{ln} \\sum_{\\tilde{\\textbf{v}},\\tilde{\\textbf{h}}} e^{- E(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})}.\n\\end{aligned}\n\\end{equation}\nIn what follows, we replace $E(\\tilde{\\textbf{v}}, \\tilde{\\textbf{h}})$ with $E$.\n\nLet $w$ be a parameter in $\\tilde{\\textbf{W}}$, then the derivative of the log-likelihood w.r.t. $w$ becomes\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:loglik}\n\\frac{\\partial \\mbox{ln} \\mathcal{L}(\\tilde{\\textbf{W}} | \\tilde{\\textbf{v}})}{\\partial w} &= -  \\sum_{\\tilde{\\textbf{h}}}  p(\\tilde{\\textbf{h}}|\\tilde{\\textbf{v}}) \\frac{\\partial E}{\\partial w} + \\sum_{ \\tilde{\\textbf{h}}, \\tilde{\\textbf{v}}}  p(\\tilde{\\textbf{h}}, \\tilde{\\textbf{v}}) \\frac{\\partial E}{\\partial w}.\n\\end{aligned}\n\\end{equation}\nEquation \\eqref{eq:loglik} shows that, as with the RBM, the derivative of the log-likelihood of the $p$-RBM can be written as the sum of two expectations. The first term is the expectation of the derivative of the energy under the conditional distribution of the hidden variables given an example $\\{\\textbf{v}_t, \\textbf{v}_{t-1}, \\cdots, \\textbf{v}_{t-p} \\}$ from the training set $\\mathcal{T}$. The second term is the expectation of the derivative of the energy under the $p$-RBM distribution \\cite{fischer_training_2014}.\n\nNote that equation \\eqref{eq:loglik} can be written as\n\\begin{equation}\n\\label{eq:update}\n\\begin{aligned}\n\\Delta w = \\frac{\\partial\\mbox{ln}\\mathcal{L}(w |\\mathcal{T} )}{\\partial w} = \\bigg\\langle \\frac{\\partial E}{\\partial w}\\bigg\\rangle_{\\mbox{Data}}-\\bigg\\langle \\frac{\\partial E}{\\partial w} \\bigg\\rangle_{\\mbox{Model}}.\n\\end{aligned}\n\\end{equation}\nEquation \\eqref{eq:update} is the update step corresponding to the $p$-RBM model and is analogous to equation \\eqref{eq:update_rule}.\n\\subsubsection{Contrastive divergence for the model}\nWhen considering our model, the $k$-step contrastive divergence ($\\mbox{CD}_k$) becomes\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:cdk}\n\\mbox{CD}_k(w, \\tilde{\\textbf{v}}^{(0)})= - \\sum_{\\tilde{\\textbf{h}}^{(0)}} p(\\tilde{\\textbf{h}}^{(0)} |\\tilde{\\textbf{v}}^{(0)} )\\frac{\\partial E}{\\partial w} + \\sum_{\\tilde{\\textbf{h}}^{(k)}} p(\\tilde{\\textbf{h}}^{(k)} |\\tilde{\\textbf{v}}^{(k)}) \\frac{\\partial E}{\\partial w},\n\\end{aligned}\n\\end{equation}\nwhere a Gibbs chain has been run $k$ times on the second term in order to approximate the expectation of the derivative of the energy under the model, given in equation \\eqref{eq:loglik}.\n\\subsubsection{Training the model}\nIn order to define a training rule for the model, we propose a way to sample for the CD. The sampling of a block vector of hidden variables can be done from the following block vector of probabilities\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:prob}\n\\mathbb{P}(\\tilde{\\textbf{h}}|\\tilde{\\textbf{v}})=\\left[\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n p(\\textbf{h}_t |\\tilde{\\textbf{v}}) & \\cdots &p(\\textbf{h}_{t-p} | \\tilde{\\textbf{v}}) & \\textbf{1}\n\\end{array}\\right]^\\intercal.\n\\end{aligned}\n\\end{equation}\nTo sample a vector of visible variables, we construct a block matrix similar to equation \\eqref{eq:bmatrix} as\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:bmatrix2}\n\\mathbb{P}(\\tilde{\\textbf{v}}|\\tilde{\\textbf{h}})=\\left[\n\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n p(\\textbf{v}_t |\\textbf{h}_t)  &\\cdots&  p(\\textbf{v}_t |\\textbf{h}_{t-p})&  p(\\textbf{v}_t |\\tilde{\\textbf{h}}) \\\\ \\hdashline\n p(\\textbf{v}_{t-1} |\\textbf{h}_t) &\\cdots&  p(\\textbf{v}_{t-1} |\\textbf{h}_{t-p})&  p(\\textbf{v}_{t-1} |\\tilde{\\textbf{h}}) \\\\ \\hdashline\n\\vdotswithin{}  &\\vdotswithin{} & \\vdotswithin{} & \\vdotswithin{} \\\\ \\hdashline\n p(\\textbf{v}_{t-p} |\\textbf{h}_t)&\\cdots&  p(\\textbf{v}_{t-p} |\\textbf{h}_{t-p})&  p(\\textbf{v}_{t-p} |\\tilde{\\textbf{h}}) \\\\ \\hdashline\n\\textbf{1} &\\cdots & \\textbf{1} & \\textbf{0}\n\\end{array}\\right].\n\\end{aligned}\n\\end{equation}\n\n\\begin{figure*}[ht!]\n        \\centering\n        \\begin{subfigure}[h]{.47\\textwidth}\n                \\includegraphics[width=1\\textwidth]{loss_k_1.pdf}\n                \\caption{Negative log-likelihood.}\n                \\label{fig: energy_loss_k_1_a}\n        \\end{subfigure}\n        ~~~\n         \n        \\begin{subfigure}[h]{.47\\textwidth}\n                \\includegraphics[width=1\\textwidth]{error_k_1.pdf}\n                \\caption{Misclassification rate.}\n                \\label{fig: energy_loss_k_1_b}\n        \\end{subfigure}\n    \\caption{Evaluation of a $p$-RBM model on 100 stocks of the NASDAQ-100 index, with $p=30$, mixing time $k=1$ and $\\alpha=0.5$.}\\label{fig: energy_loss_k_1}\n\\end{figure*}\nWe may obtain the expected value of the hidden vector by repeatedly sampling from equation \\eqref{eq:prob}. For our purposes it is better to place them in a block matrix with the same dimension as $\\tilde{\\textbf{W}}$. We call this matrix $\\mathbb{E}[\\tilde{\\textbf{h}}| \\tilde{\\textbf{v}}]$ and it is formed by vertically stacking $p+2$ block vectors of the form\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:expec}\n\\left[\\begin{array}{c;{2pt\/2pt}c;{2pt\/2pt}c;{2pt\/2pt}c}\n\\mathbb{E}[\\textbf{h}_t| \\tilde{\\textbf{v}}] & \\cdots &\\mathbb{E}[\\textbf{h}_{t-p}| \\tilde{\\textbf{v}}] & \\textbf{1}\n\\end{array}\\right].\n\\end{aligned}\n\\end{equation}\nThe derivatives of $E$ can be written as\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:derivative}\n\\frac{\\partial E}{\\partial \\tilde{\\textbf{W}}}= \\textbf{A} \\circ \\tilde{\\textbf{v}} \\circ \\tilde{\\textbf{h}}.\n\\end{aligned}\n\\end{equation}\nCombining a $\\mbox{CD}_k$, as in equation \\eqref{eq:cdk}, with equations \\eqref{eq:expec} and \\eqref{eq:derivative}, equation \\eqref{eq:loglik} becomes\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:delta}\n\\Delta \\tilde{\\textbf{W}}= \\textbf{A} \\circ (\\tilde{\\textbf{v}}^{(0)} \\circ \\mathbb{E}[\\tilde{\\textbf{h}}^{(0)}| \\tilde{\\textbf{v}}^{(0)}]-\\tilde{\\textbf{v}}^{(k)}\\circ \\mathbb{E}[\\tilde{\\textbf{h}}^{(k)}| \\tilde{\\textbf{v}}^{(k)}]).\n\\end{aligned}\n\\end{equation}\nThe learning rule is then given by\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:learnrule}\n\\tilde{\\textbf{W}}^{(\\tau+1)}=\\tilde{\\textbf{W}}^{(\\tau)}+ \\eta \\Delta \\tilde{\\textbf{W}},\n\\end{aligned}\n\\end{equation}\nwhere $\\eta>0$ is the learning rate.\n\nNote that each item in the block matrix can be learned independently, once  the CD step is finalized. This results in a model that is highly suitable for parallelization during training, specially on a GPU where its multicore structure allows the concurrent handling of multiple computational threads.\n\nAlgorithm \\ref{alg:model} describes the detailed procedure for training the $p$-RBM model.\n\n\\begin{algorithm}[H]\n\\caption{$k$-step contrastive divergence for the $p$-RBM model}\n\\label{alg:model}\n\\begin{algorithmic}[1]\n\\Require{Training set $ \\mathcal{T} = \\{\\textbf{v}_t, \\textbf{v}_{t-1}, \\cdots, \\textbf{v}_{t-p}\\}^T_{t=1}$, and a value for $\\alpha$}\n\\Ensure{Updates of all parameters in the model, contained in the block matrix from equation \\eqref{eq:bmatrix}}\n\\ForAll{$\\{ \\textbf{v}_t, \\textbf{v}_{t-1}, \\cdots, \\textbf{v}_{t-p}\\} \\in \\mathcal{T} $}\n\t\\State $\\textbf{v}_t^{(0)},\\textbf{v}_{t-1}^{(0)}, \\cdots, \\textbf{v}_{t-p}^{(0)} \\gets  \\tilde{\\textbf{v}}$\n\t\\For{$k=1,\\dots,K-1$}\n\t\t\\State  sample $\\tilde{\\textbf{h}}^{(k)} \\sim \\mathbb{P}(\\tilde{\\textbf{h}}|\\tilde{\\textbf{v}}^{(k-1)})$ as in equation \\eqref{eq:prob}\n\t\t\\State  sample $\\tilde{\\textbf{v}}^{(k)} \\sim \\mathbb{P}(\\tilde{\\textbf{v}}| \\tilde{\\textbf{h}}^{(k)}) $ as in equation  \\eqref{eq:bmatrix2}\n\t\\EndFor\n\t\\For{$t=1, \\dots, n$}\n\t\t\\State Calculate  $\\mathbb{E}[\\tilde{\\textbf{h}}| \\tilde{\\textbf{v}}]$ according to equation  \\eqref{eq:expec}\n\t\t\\State Update $\\tilde{\\textbf{W}}$ according to equation  \\eqref{eq:delta}\n\t\\EndFor\n\\EndFor\n\\State \\textbf{Return} $\\tilde{\\textbf{W}}$\n\\end{algorithmic}\n\\end{algorithm}\n\n\\section{Numerical Results and Analysis}\n\nStocks in the stock market are correlated (similar sectors influence each other) and autocorrelated (dependent on their previous states) \\cite{king_transmission_1990}. This causes the stock market to present itself as chaotic, noisy, unstructured, and non-stationary \\cite{abu-mostafa_introduction_1996}. Accurate forecasting of stocks is of interest to organizations trying to make sense of the market data. Traditional approaches to stock-market forecasting include the ARIMA and ARCH models \\cite{bontempi_machine_2013,zhang_applying_2009}; while less-traditional approaches include, for instance, hidden Markov models that make use of the partitioning properties to generate fuzzy logic rules  that have been reported to yield better results than those of the ARIMA  model \\cite{hassan_combination_2009}.\n\n\n\nMore recently, machine learning techniques  have been used to predict the stock market. For example, in \\cite{gradojevic_non-linear_2007} an artificial neural network (ANN) was combined with fuzzy logic to generate trading strategies, shown to perform better than common buy-and-hold strategies; in \\cite{armano_hybrid_2005} various ANNs formed a population of experts that evolves according to genetic programming rules to make forecasts and create investment strategies on the S\\&P 500 index.\n\nLikewise, the use of recurrent neural networks (RNNs) is an attractive approach to time-series modeling because they provide a sense of order when handling data, allowing the network to abstract the temporal context of the  series. In \\cite{saad_comparative_1998} various neural networks techniques were compared and cases where RNNs outperform feed-forward neural networks were shown.\n\nWe trained our model in 100 stocks selected from the NASDAQ-100 index. We chose this set of stocks because most of them belong to the technological sector, a highly interdependent sector. This allows our model to demonstrate its ability to extract hidden relations between stocks.\n\nOur setup consisted on extracting the values of the stocks from a web resource\\footnote{\\url{https:\/\/www.google.com\/finance}}, using intervals of 5 minutes during a period of 20 days from June 5, 2017 to June, 25, 2017. The direction of the stock was calculated by subtracting its closing value from its opening value. The obtained set was split in two: 80\\% for the training set  and the remaining 20\\% for the validation set. We chose the network to remember $p=30$ periods of 5-minute steps. Our model considered a hidden layer with $m=1000$ nodes, $\\alpha=0.5$, and $\\eta=0.001$. Additionally, we used the mixing rate of $k=1$, since it has been argued in the literature that this choice is enough to appropriately approximate the direction of the gradient \\cite{hinton_practical_2012}. Our model was implemented using TensorFlow library version 1.1 \\cite{tensorflow2015-whitepaper}.\n\nWe studied our model's performance at predicting unseen observations by means of its misclassification error rate, defined as\n\\begin{equation}\n\\begin{aligned}\n\\label{eq:error}\n\\mbox{Loss}=\\frac{1}{2n}\\frac{1}{|\\mathcal{V}|} \\sum_{\\tilde{\\textbf{v}} \\in \\mathcal{V}} \\sum_{i=1}^{n} |1-\\tilde{\\textbf{v}}_i f(\\tilde{\\textbf{v}}_i)|,\n\\end{aligned}\n\\end{equation}\nwhere $\\mathcal{V}$ is the validation set, $|\\mathcal{V}|$ denotes its cardinality, $\\tilde{\\textbf{v}}_i$ is the real direction of the stock $i$, and $f(\\tilde{\\textbf{v}}_i)$ is the direction predicted by our model. The loss function used penalizes misclassification cases in the validation set and is solely used for evaluation purposes.\n\n\\begin{table*}[ht!]\n\\centering\n\\caption{Comparison of errors obtained for different models (less is better).}\n\\label{table:comparison}\n\\begin{tabular}{lccccccc}\n                      & \\multicolumn{5}{c}{\\textbf{Iteration}}                                                                                                                             & \\multicolumn{1}{l}{}              & \\multicolumn{1}{l}{}              \\\\ \\cline{2-8}\n\\textbf{Model}        & \\multicolumn{1}{l}{\\textbf{1}} & \\multicolumn{1}{l}{\\textbf{2}} & \\multicolumn{1}{l}{\\textbf{3}} & \\multicolumn{1}{l}{\\textbf{4}} & \\multicolumn{1}{l}{\\textbf{5}} & \\multicolumn{1}{l}{\\textbf{Mean}} & \\multicolumn{1}{l}{\\textbf{Std.}} \\\\ \\hline\n\\textbf{3 layer LSTM} & 0.4386       & 0.4029                            &  0.4158                   & 0.4495                  &                               0.4432 & 0.4300                                  &  0.0177                                 \\\\\n\\textbf{VAR(1)}       & 0.4571                        & 0.4404                      &0.4588                  &0.4582                     &                               0.4587 & 0.4546                                  & 0.0072                                  \\\\\n\\textbf{1 layer LSTM} &   0.4542              &   0.4542                 &  0.4572                  &   0.4563                 &                               0.4560 &  0.4556                               & 0.0012                                \\\\\n\\textbf{p-RBM}        & 0.4450                    & 0.4945                               &0.4638                             &                               0.4979 &  0.4830                              & 0.4768                                  &  0.0222                                 \\\\\n\\textbf{RW}        & 0.4798                         & 0.4790                          & 0.4759                         & 0.5289                         & 0.4780                      & 0.4883                            & 0.0203                            \\\\\n\\hline\n\\end{tabular}\n\\end{table*}\n\nDuring training, the negative log-likelihood of the model decreases in the training set and in the validation set (see Figure \\ref{fig: energy_loss_k_1_a}); this indicates that the model is learning the distribution of the data. The misclassification rate of our model decreased in the training set and exhibited a cyclic behavior in the validation set. Our model seems to have not overfitted the data; as shown by the fact that the gap between the negative log-likelihood on the training set and on the validation set does not seem to increase over iterations.\n\nWe also evaluated the performance of our model by constructing  a confusion matrix on the validation set, shown in TABLE \\ref{table:error_table_k_1}.\n\\begin{table}[h]\n\t\\caption{Confusion matrix predicted directions for 100 stocks in the validation set.}\n\t\\label{table:error_table_k_1}\n\t\\centering\n\t\\resizebox{0.30\\textwidth}{!}{\n\t\t\\begin{tabular}{l c c }\n\t\t\t&\\multicolumn{2}{c}{\\textbf{Predicted direction}}\\\\\n\t\t\t\\cline{2-3}\n\t\t\t\\textbf{Real direction} & \\textbf{Up} & \\textbf{Down}\\\\\n\t\t\t\\cline{1-3}\n\t\t    \\textbf{Up} & $5,756$ & $6,246$ \\\\\n\t\t\t\\textbf{Down}& $5,057$ & $8,341$ \\\\\n\\hline\n\t\t\\end{tabular}}.\n\t\\end{table} \\\\\nThe overall prediction error on the validation set is $0.445$. Considering sequential buy-or-sell decisions, one every 5 minutes, we can construct a simple trading strategy on this set of 100 stocks. Applying the strategy on the validation set---consisting of $25,400$ total trading decisions---resulted in $14,097$ winning and $11,303$ losing trades: a win-to-loss ratio of $1.2472$, and a return of $11 \\%$ given that we buyed and sold an equal amount of each stock. More complex trading strategies, based on the obtained predictions, may provide improved results.\n\n\\subsection{Comparison to other models}\nWe compared the forecasting performance of the $p$-RBM model with four other models: a random walk (RW), a vector auto-regression (VAR), and a 1- and 3-layer long short term memory (LSTM).\n\nThe RW model produces one-step ahead predictions and is defined as follows\n\\[\ny_{t+1}=y_{t}+\\alpha+e_{t},\n\\]\nwhere $e_t$ is a normal i.i.d. error term and $\\alpha$  is the  drift term.\n\nThe VAR model is a generalization of the autoregressive model that aims to capture linear interdependencies among several time series. More specifically, a VAR($p$) model is one with $p$ lag if the evolution of the series is a linear function of only their $p$ past values; the forecast is obtained as follows\n\\[\n    \\mathbf{y}_t = \\mathbf{c} + \\mathbf{A}_1 \\mathbf{y}_{t-1} + \\mathbf{A}_2 \\mathbf{y}_{t-2} + \\cdots + \\mathbf{A}_p \\mathbf{y}_{t-p} + \\mathbf{e}_t,\n\\]\nwhere $\\mathbf{A}_i$ is a time-invariant  matrix and $\\mathbf{e}_t$ is a  vector representing the error term. Here we use a VAR(1) model.\n\nFinally, an LSTM model is a type of recurrent neural network with a series of gates that control information flow within the internal states of the network. This gives the model the ability to remember long and short-term dependencies.\n\nEach of the forecasting models considered here was fitted using the training set. The evaluation procedure was made on the validation set and the relative performance was measured by the misclassification error rate. The results obtained are summarized in TABLE \\ref{table:comparison} in descending order from best to worst.\n\nThe RW model produced an overall error of $0.4883$, having the worst performance due to, probably, the fact that this model assumes that all the information necessary to predict the future is summarized in the current value only. It also assumes that the increments---up or down---are symmetric, that is, with an expected value of zero. This means that the RW model cannot make predictions different than the current trend.\n\nLSTM comes out first with an overall error of $0.4300$. One reason LSTM performs better than the other methods is its ability to discover hidden structures within the data. Specifically, LSTM may have outperformed the other methods because the data is structurally non-linear. Neither the VAR, that had an overall error of $0.4546$, nor the $p$-RBM, that had an overall error of $0.4768$, can fully capture this non-linearity in the data.\n\nThe $p$-RBM comes out in fifth place, just outperforming the RW with drift. We note that the $p$-RBM had a higher variance when compared to the other methods; this might be due to the sampling method used. In fact, we noted that when sampling with a hard-threshold rule, the $p$-RBM achieved a lower variance, but its accuracy worsened.\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Conclusion and Future Work}\nIn this paper we proposed an extension to the RBM model, called the $p$-RBM. We tested our model in the problem of forecasting stock market directions with positive results. Experimentation with real data showed that our model is suitable for problems involving high-dimensional correlated time series. When compared with other models, however, the $p$-RBM showed larger prediction errors than that of 1- and 3-layer LSTM models.\n\nThe following are possible research directions for our work: (1) combining our model with reinforcement learning to develop agents that discover optimal trading strategies; (2) discovering the most suitable topology for a problem by exploring the space of hyperparameter matrix $\\textbf{A}$; and (3) extending our model to consider continuous visible units.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\n\\section{Introduction}\nCommonly used modulation schemes like square quadrature amplitude modulation or phase-shift keying are characterized by a bijective symbol constellation.\nThus, $N$ bits are mapped onto $M=2^N$ symbols.\nIn contrast, there also exist non-bijective modulation schemes like superposition modulation (SM) \\cite{bib:sm_maf}, which typically produce a symbol alphabet with a cardinality less than $2^N$.\nThis reduction in symbol cardinality can be exploited to achieve power efficiency, bandwidth efficiency, and\/or to reduce the detection complexity.\nSM is of particular importance, because active signal shaping can be avoided, which makes it naturally near-capacity achieving.\nFurthermore the complexity of the optimal detector can be reduced from $\\mathcal{O}(2^N)$ down to $\\mathcal{O}(N)$ by exploiting the tree-based non-bijective modulation structure.\nIn the case of non-bijective modulation, redundancy due to channel coding is mandatory.\nAccording to the state-of-the-art, extrinsic information is exchanged between the demodulator (detector) and the channel decoder.\nHence a soft-output detector is mandatory in iterative receivers.\nWhen performing soft-input soft-output (SISO) detection, the a posteriori probability (APP) algorithm provides the optimal solution.\nA common simplification of APP detection provides the max-log APP detector \\cite{bib:maxlogapp}.\nHowever, these detection algorithms historically have been derived for bijective symbol constellations.\nWhen considering non-bijective modulation schemes, there are important facts to notice when applying APP and max-log APP detection.\nEspecially for the max-log APP detector, these facts are essential with respect to the performance.\nEven if they are rather simple, to the best knowledge of the authors, they have not been published yet.\nFurthermore there is no conventional detection method for max-log APP detection of non-bijective modulation schemes in the literature.\nIn this concise letter the main differences between bijective and non-bijective modulation schemes for implementing the APP detector or max-log APP detector are highlighted, \nand a conventional detection method for the max-log APP detector for non-bijective modulation schemes is proposed.\n\n\n\\section{A Posteriori Probability Detection\\label{sec:app}}\nFor SISO detection, the APP detector provides the optimal solution.\nConsidering a memoryless channel, the extrinsic log-likelihood ratio (LLR) $L_n$ of bit $b_n$ can be obtained via the well-known formula\n\\begin{align}\nL_n &\\doteq \\log\\frac{p(y|b_n=0)}{p(y|b_n=1)}\\\\\n&=\\log\\frac{\\sum\\limits_{\\*b_{\\sim n}}P(\\*b_{\\sim n})p(y|\\*b_{\\sim n},b_n=0)}{\\sum\\limits_{\\*b_{\\sim n}}P(\\*b_{\\sim n})p(y|\\*b_{\\sim n},b_n=1)}\\label{eq:ellr_bit}\\\\\n&=\\log\\frac{\\sum\\limits_{x\\in\\mathcal{X}^{(0)}_n}P(z(x))p(y|x)}{\\sum\\limits_{x\\in\\mathcal{X}^{(1)}_n}P(z(x))p(y|x)} \\label{eq:ellr_sym} \\textnormal{,}\n\\end{align}\nwhere $y$ represents the channel observation,\n$\\*b_{\\sim n}$ denotes the bit set excluding $b_n$,\n$x$ is  a symbol defined over the alphabet $\\mathcal{X}$,\n$\\mathcal{X}_n^{(b)}$ stands for the symbol subset of $\\mathcal{X}$ with $b_n = b \\in \\{0,1\\}$,\nand $z(x)$ represents the set of all those $\\*b_{\\sim n}$ that will lead in combination with $b_n=b$ to the symbol $x\\in\\mathcal{X}_n^{(b)}$.\nThe terms $P(\\*b_{\\sim n})$ in (\\ref{eq:ellr_bit}) and $P(z(x))$ in (\\ref{eq:ellr_sym}) represent a priori information,\nand hence are time-varying in the case of iterative processing.\nBoth (\\ref{eq:ellr_bit}) and (\\ref{eq:ellr_sym}) are applicable independent of whether the symbol constellation is bijective or non-bijective.\nNevertheless, there is a difference in the number of summands between them for non-bijective symbol constellations.\nWhile each sum in (\\ref{eq:ellr_bit}) consists of $2^{N-1}$ summands, each sum in (\\ref{eq:ellr_sym}) consists of just $M \\leq 2^{N-1}$ summands.\n\n\n\\section{Max-log APP Detection\\label{sec:maxlogapp}}\nThe max-log APP detector \\cite{bib:maxlogapp} is a common simplification of the APP algorithm, which avoids the high complexity of exponential and logarithmic operations in APP detection.\nEspecially in high-SNR scenarios the detector achieves an APP-like performance.\nNevertheless, the complexity still remains to be $\\mathcal{O}(2^N)$.\nThe max-log APP concept is based on the APP detector in the logarithmic domain:\n\\begin{align}\nL_n &= \\log\\frac{\\sum\\limits_{\\*b_{\\sim n}}\\ensuremath{\\textnormal{e}}^{\\log P(\\*b_{\\sim n}) + \\log p(y|\\*b_{\\sim n},b_n=0)}}{\\sum\\limits_{\\*b_{\\sim n}}\\ensuremath{\\textnormal{e}}^{\\log P(\\*b_{\\sim n}) + \\log p(y|\\*b_{\\sim n},b_n=1)}}\\label{eq:ellr_log_bit}\\\\\n&= \\log\\frac{\\sum\\limits_{x\\in\\mathcal{X}_n^{(0)}}\\ensuremath{\\textnormal{e}}^{\\log P(z(x)) + \\log p(y|x)}}{\\sum\\limits_{x\\in\\mathcal{X}_n^{(1)}}\\ensuremath{\\textnormal{e}}^{\\log P(z(x)) + \\log p(y|x)}} \\label{eq:ellr_log_sym}\\textnormal{.}\n\\end{align}\nMotivated by \n\\begin{align}\n\\max \\phantom{}^*\\left(a,b\\right) &\\doteq \\log(e^a+e^b) \\\\\n&\\doteq \\max\\left(a,b\\right) + \\log\\left(1+\\ensuremath{\\textnormal{e}}^{-|a-b|}\\right) \\textnormal{,} \\label{eq:correction}\n\\end{align}\nthe $\\max\\phantom{}^*$-operation can be approximated as ${\\max\\phantom{}^*(a,b)\\approx\\max(a,b)}$ \\cite{bib:maxlogapp,bib:logapp}.\nThus, (\\ref{eq:ellr_log_bit}) and (\\ref{eq:ellr_log_sym}) can be approximated by:\n\\begin{align}\n\\begin{split}\nL_n &\\approx \\max\\limits_{\\*b_{\\sim n}}\\left\\{\\log P(\\*b_{\\sim n})+\\log p(y|\\*b_{\\sim n},b_n=0)\\right\\} \\\\\n&- \\max\\limits_{\\*b_{\\sim n}}\\left\\{\\log P(\\*b_{\\sim n}) + \\log p(y|\\*b_{\\sim n},b_n=1)\\right\\}\\label{eq:ellr_bit_ml}\\end{split} \\\\\n\\begin{split}\nL_n &\\approx \\max\\limits_{x\\in\\mathcal{X}_n^{(0)}}\\left\\{ \\log P(z(x)) + \\log p(y|x)\\right\\} \\\\\n&- \\max\\limits_{x\\in\\mathcal{X}_n^{(1)}}\\left\\{\\log P(z(x)) + \\log p(y|x)\\right\\} \\label{eq:ellr_sym_ml} \\textnormal{.} \\end{split}                                                                                                                                                                                                   \n\\end{align}\nWhile (\\ref{eq:ellr_bit_ml}) and (\\ref{eq:ellr_sym_ml}) leads to the same results for bijective symbol constellations, \nthere is an important difference considering non-bijective symbol constellations.\nDue to the fact that there is at least one symbol consisting of more than one bit set $\\*b_{\\sim n}$,\nthe results of (\\ref{eq:ellr_bit_ml}) and (\\ref{eq:ellr_sym_ml}) might no longer be equal.\nIn fact, (\\ref{eq:ellr_sym_ml}) outperforms (\\ref{eq:ellr_bit_ml}) and provides a reasonable approximation of the APP algorithm, \nas it will be shown in Section~\\ref{sec:numresults}.\nThus, (\\ref{eq:ellr_sym_ml}) should be applied as the conventional detection method for non-bijective modulation schemes.\nIt is also possible to transform (\\ref{eq:ellr_bit_ml}) into (\\ref{eq:ellr_sym_ml}).\nTherefore, all summands corresponding to a bit set $\\*b_{\\sim n}$, which leads in combination with $b_n$ to the identical symbol $x$, have to be summed up in (\\ref{eq:ellr_bit_ml}),\nbefore applying the $\\max$-operation:\n\\begin{equation}\n\\begin{split}\nL_n &\\approx \\max\\limits_{x\\in\\mathcal{X}_n^{(0)}}\\left\\{\\log\\left( \\sum\\limits_{\\*b_{\\sim n} \\to x} P(\\*b_{\\sim n})\\right) + \\log p(y|x)\\right\\} \\\\\n&- \\max\\limits_{x\\in\\mathcal{X}_n^{(1)}}\\left\\{\\log \\left( \\sum\\limits_{\\*b_{\\sim n} \\to x} P(\\*b_{\\sim n}) \\right) + \\log p(y|x)\\right\\} \\\\\n&\\approx \\max\\limits_{x\\in\\mathcal{X}_n^{(0)}}\\left\\{\\max\\limits_{\\*b_{\\sim n} \\to x}\\phantom{}^* \\left(\\log P(\\*b_{\\sim n})\\right) + \\log p(y|x)\\right\\} \\\\\n&- \\max\\limits_{x\\in\\mathcal{X}_n^{(1)}}\\left\\{\\max\\limits_{\\*b_{\\sim n} \\to x}\\phantom{}^* \\left(P(\\*b_{\\sim n}) \\right) + \\log p(y|x)\\right\\} \\textnormal{.}\\label{eq:ellr_bitsym_ml}\n\\end{split}\n\\end{equation}\nIn comparison, (\\ref{eq:ellr_bitsym_ml}) is more complex than (\\ref{eq:ellr_bit_ml}), \nbecause the $\\max\\phantom{}^*$-operator has to be applied.\nHowever, non-bijective symbol constellations often imply a potential for complexity reduction, when exploiting the non-bijectivity in calculation of (\\ref{eq:ellr_sym_ml}).\nIn detail, the construction of $z(x)$ can be visualized as a tree diagram as it is done in \\cite{bib:sm_maf}.\nAssuming that the complexity of determining $\\log P(z(x))$ is proportional to the number of branches in the tree diagram,\nthe reduction in complexity compared to bijective computation becomes clear.\nIn the worst case $z(x)$ is bijective and the complexity of both (\\ref{eq:ellr_bit_ml}) and (\\ref{eq:ellr_sym_ml}) are the same.\nThe amount of branches in this case is given by $2^N - 2$.\nIn contrast, if $z(x)$ is non-bijective, the complexity of determining $\\log P(z(x))$ can be reduced by reusing several states in the tree diagram for different $\\log P(z(x))$.\nWhen applying direct superposition modulation with equal power allocation, the amount of branches needed for $\\log P(z(x))$ can be reduced from $2^N - 2$ (for bijective calculation) to $N^2 - N$ (for non-bijective calculation).\nThus (\\ref{eq:ellr_sym_ml}) implies a significant potential of complexity reduction compared to (\\ref{eq:ellr_bit_ml}).\n\nGiven (\\ref{eq:ellr_bit_ml}) and (\\ref{eq:ellr_sym_ml}), true APP detection is obtained by (re-)substituting all $\\max$-operations by $\\max\\phantom{}^*$-operations.\nHence, the extra computational complexity of APP detection only depends on how to implement the correction term $\\log(1+\\ensuremath{\\textnormal{e}}^{-|a-b|})$, cf. (\\ref{eq:correction}).\nTowards this goal, numerous solutions exist, ranging from a simple table look-up \\cite{bib:maxlogapp} to the direct implementation of the complex correction term.\n\n\\section{Numerical Results\\label{sec:numresults}}\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\\begin{axis}[width=8.5cm, grid=both,\ngrid style={dotted},\nxmin=-2.0, xmax=2.0, ymin=-4, ymax=4,\nlegend style={legend pos=north west},\nlegend cell align=left,\nxlabel={$\\mbox{Re}\\{y\\}$}, ylabel={$L_n$}]\n\\addplot [black] table[x=y, y=APP] {.\/pic\/DSM_EPA_LA0.dat};\n\\addplot [blue, dashed] table[x=y, y=BML] {.\/pic\/DSM_EPA_LA0.dat};\n\\addplot [red] table[x=y, y=ML] {.\/pic\/DSM_EPA_LA0.dat};\n\\legend{\\small{APP}, \\small{Max-Log APP (\\ref{eq:ellr_bit_ml})}, \\small{Max-Log APP (\\ref{eq:ellr_sym_ml})}}\n\\end{axis}\n\\end{tikzpicture}\n\\end{center}\n\\caption{Relationship between extrinsic log-likelihood ratio $L_n$ and the real part of the channel observation $y$ ($N=16$ DSM-EPA, $\\textnormal{SNR}=12\\,\\textnormal{dB}$).}\n\\label{fig:yllr}\n\\end{figure}\n\\begin{figure}\n\\begin{center}\n\\begin{tikzpicture}\n\\usetikzlibrary{plotmarks}\n\\begin{semilogyaxis}[width=8.5cm, grid=both,\ngrid style={dotted},\nxmin=11.8, xmax=13.3, ymin=0.0001,\nlegend style={legend pos=south west},\nlegend cell align=left,\nxlabel={$\\textnormal{SNR}$ ($\\textnormal{dB}$)}, ylabel={$\\textnormal{BER}$}]\n\\addplot [black, mark=star, mark repeat=1] table[x=SNR, y=BER] {.\/pic\/app.dat};\n\\addplot [blue, dashed] table[x=SNR, y=BERsoft] {.\/pic\/main_dsm8_real_maxlogapp.old};\n\\addplot [red, mark=square, mark repeat=1] table[x=SNR, y=BER] {.\/pic\/maxlogapp.dat};\n\\legend{\\small{APP}, \\small{Max-Log APP (\\ref{eq:ellr_bit_ml})}, \\small{Max-Log APP (\\ref{eq:ellr_sym_ml})}}\n\\end{semilogyaxis}\n\\end{tikzpicture}\n\\end{center}\n\\caption{Bit error rate performance for DSM-EPA ($N=16$).}\n\\label{fig:ber}\n\\end{figure}\nSo far, the derivation is general and not restricted to modulation schemes.\nIn fact, it can be exploited in many applications like multiple-input multiple-output detection or multi-user detection.\nHowever, the numerical results of this letter are focused on modulation schemes, especially on direct superposition modulation with equal power allocation (DSM-EPA) \\cite{bib:sm_maf}.\nFor the following simulations, the additive white Gaussian noise channel is assumed.\nThus, a channel observation can be written as $y=x+w$,\nwhere $w \\sim \\mathcal{N}(0,\\sigma^2)$ is a zero-mean white Gaussian noise process with the variance $\\sigma^2$.\nConsequently, the conditional probability is given as\n\\begin{equation}\np(y|x)=\\frac{1}{\\pi \\sigma^2}\\ensuremath{\\textnormal{e}}^{-\\frac{|y-x|^2}{\\sigma^2}} \\textnormal{.} \n\\end{equation}\n\nFig.~\\ref{fig:yllr} shows the relationship between the extrinsic log-likelihood ratio and the real part of the channel observations for DSM-EPA with $N=16$ layers for the case that no a priori information is available.\nDSM-EPA is strongly non-bijective and leads to $N^2\/4+N+1 \\ll 2^N$ symbols.\nThe APP detector from Section~\\ref{sec:app} is visualized by the black curve and provides the optimum performance.\nThe max-log APP detector is realized for the two different implementations discussed in Section~\\ref{sec:maxlogapp}: \nEquation (\\ref{eq:ellr_bit_ml}) corresponds to the blue curve and (\\ref{eq:ellr_sym_ml}) corresponds to the red curve.\nAs it can be seen from Fig.~\\ref{fig:yllr}, (\\ref{eq:ellr_sym_ml}) performs a better approximation of the APP detector than (\\ref{eq:ellr_bit_ml}) in the case of a non-bijective symbol constellation.\nThis statement is still true, if a priori information is available.\n\n\\begin{table}[!t]\n\\renewcommand{\\arraystretch}{1.3}\n\\caption{Parameter set of the irregular convolutional code used in the simulation results.\nThe subcodes are obtained from a recursive systematic convolutional code by puncturing and repetitions, respectively.\nThe code polynomials and the puncturing\/repetition table is given in \\cite{bib:ircc}.}\n\\label{tab:ircc}\n\\centering\n\\begin{tabular}{|c||c|c|}\n\\hline\n$j$ & ${R_j}$ & $\\alpha_j$\\\\\n\\hline\\hline\n1 & $0.10$ & $0.254042$ \\\\\n2 & $0.15$ & $0.292594$ \\\\\n3 & $0.20$ & $0.003651$ \\\\\n4 & $0.25$ & $0.133594$ \\\\\n5 & $0.30$ & $0.054518$ \\\\\n6 & $0.35$ & $0.032276$ \\\\\n7 & $0.40$ & $0.092666$ \\\\\n8 & $0.45$ & $0.000000$ \\\\\n9 & $0.50$ & $0.000000$ \\\\\n10 & $0.55$ & $0.105838$ \\\\\n11 & $0.60$ & $0.030820$ \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\nConcerning bit error rate (BER) simulations, a bit-interleaved coded modulation \\cite{bib:bicm} system with iterative processing has been implemented.\nIn order to achieve a near-capacity performance (at least for APP detection) without active signal shaping, an irregular convolutional code%\n\\footnote{An irregular convolutional code $C$ of rate $R$ consists of $J$ punctured convolutional codes $C_j$ of rates $R_j$ implemented in parallel,\nwhere $R=\\sum\\limits_{j=1}^{J}\\alpha_j R_j$ and $\\sum\\limits_{j=1}^{J}\\alpha_j=1$.\nThe EXIT chart characteristic of the irregular convolutional code can be shaped by optimizing the parameter set $\\alpha_j$, $1\\leq j \\leq J$.}\naccording to \\cite{bib:ircc} with a code rate of about $R\\approx1\/4$ has been matched by means of an EXIT chart design to DSM-EPA employing $N=16$ layers, cf. Table \\ref{tab:ircc}.\nThe bandwidth efficiency is $R\\cdot N=4$~bits\/symbol, which can theoretically be achieved by $2^{16}$-ary DSM-EPA at an SNR of $12.0\\,\\textnormal{dB}$. \nThe information word length has been chosen to $100\\,000$ bits and $500$ iterations between the detector and the channel decoder have been performed.\nBER simulation results are visualized in Fig.~\\ref{fig:ber}.\nThe top blue curve shown in broken lines is obtained, when all $\\max\\phantom{}^*$-operations in an APP detector matched to the symbol alphabet of size $2^N$ are replaced by the $\\max$-operation,\nwhich corresponds to (\\ref{eq:ellr_bit_ml}). \nAlthough this is usually exactly what is done quite often in the case of bijective modulation schemes (and referred to as max-log APP detection \\cite{bib:maxlogapp}),\nit completely fails in the case of non-bijective modulation schemes like DSM: \nThe iterative receiver does not converge in the interesting SNR range.\nThe max-log APP detector according to (\\ref{eq:ellr_sym_ml}), which performs the $\\max$-operation over the symbol alphabet of cardinality $N^2\/4+N+1$, provides a much better performance.\nCompared to the APP detector, it degrades only by about $0.56\\,\\textnormal{dB}$ in the area of the turbo cliff, and the iterative receiver converges.\nThus, (\\ref{eq:ellr_sym_ml}) clearly outperforms (\\ref{eq:ellr_bit_ml}) in the case of non-bijective symbol constellations.\nThe error floor shown in Fig.~\\ref{fig:ber} can be reduced or avoided by means of doping, which is beyond the scope of this letter, however.\n\n\\section{Conclusion}\nIt is shown that there are important facts to notice when applying APP and max-log APP detection for non-bijective symbol constellations.\nIn APP detection, there is no need to distinguish between bijectivity or non-bijectivity, \nbut in max-log APP detection, it is important to distinguish between them for achieving the best performance for non-bijective symbol constellations.\nThe main differences with respect to non-bijective modulation schemes are highlighted and supported by numerical results.\nStarting off from an APP detector and replacing all $\\max\\phantom{}^*$-operations by $\\max$-operations, as done quite frequently, completely fails for non-bijective modulation schemes,\nthus a conventional detection method for max-log APP detection of non-bijective modulation schemes is proposed.\n\n\n\\bibliographystyle{sty\/IEEEtranTCOM}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Supplementary Information}\n\n\\section{\\label{SM}Samples and experimental techniques}\nFor this work we used commercially obtained SrTiO$_{3}$ and Sr$_{1-x}$Ca$_{x}$TiO$_{3}$ ($x=0.0022$, 0.0045 and 0.009) single crystals. The nominal calcium concentration of two samples was checked using the Secondary Ion Mass Spectrometry (SIMS) analysis technique as detailed previously\\cite{DeLima:2015}. The oxygen content has been changed by heating the samples in vacuum (pressure $10^{-6}-10^{-7}$ mbar) to temperatures of $775-1100$ $^\\circ$C. In order to attain carrier densities above $\\sim 4\\times 10^{18}$ cm$^{-3}$, a piece of titanium has been placed next to the sample during heating. Ohmic contacts have been realized prior to oxygen removal by evaporation of gold contact pads.\\\\\nThe electrical measurements have been performed in a Quantum Design Physical Property Measurement System (PPMS) between 1.8 and 300 K as well as in a 17 T dilution refrigerator with a base temperature of 26 mK. Detailed electrical transport information on all samples presented in the main text is listed in Tab. S\\ref{tab:Samples} of this supplement.\\\\\nThe ac magnetic susceptibility was measured in a homemade set-up, comprising a primary coil and a compensating pick-up coil with two sub-coils with their turns in opposite direction. A Lock-in amplifier was utilized to supply the exciting ac current and pick up the induced voltage signal. The applied ac field was as low as 10 mG with a frequency of 16 kHz.\\\\\nThe dielectric permittivity measurements were performed employing a frequency-response analyzer ({\\sc Novocontrol} Alpha-Analyzer). Using silver paint, the plate-like samples were prepared as capacitors with typical electrode dimensions of $3 \\times 3$~mm$^2$ and a typical thickness of 0.5-0.85~mm. For the evaluation of the as-measured data $C_{meas}$, passivated surface layers were assumed, as described in \\cite{Aso:1976}. Such layers can be considered as additional capacitors $C_{surf}$ in series to the remaining bulk specimen $C_{bulk}$, which limits the total capacitance data. Therefore the data were corrected assuming a temperature independent surface contribution $C_{bulk}{-1}=C_{meas}{-1}-C_{surf}{-1}$, which results in $\\varepsilon(T)$ curves comparable to literature data on surface-etched samples \\cite{Aso:1976}. Measurements of $P(E)$-hysteresis loops were performed using the same setup with an additional high-voltage module ({\\sc Novocontrol} HVB1000). The actual field dependent polarization was calculated from the non-linear dielectric permittivities up to the tenth order as described in \\cite{Niermann:2014}. The thermo-remanent polarization data was gained from the integrated pyro-current as collected with an electrometer (Keithley 6517) after cooling in a poling field of approximately 120~V\/mm.\\\\\nA home-built capacitance dilatometer has been used to detect the uniaxial length changes $\\Delta L(T)$ while continuously heating the crystal from about 5 to 150~K with a rate of about $0.1$~K\/min. Here, $\\Delta L(T)$ was measured along the [100] directions of Sr$_{1-x}$Ca$_{x}$TiO$_{3-\\delta}$ single crystals with total lengths $L_0\\simeq 2$~mm and the uniaxial thermal expansion coefficient $\\alpha=1\/L_0 \\partial \\Delta L\/\\partial T$ has been derived numerically.\\\\\nThe Raman measurements were performed using the 532 nm line of Diode Pumped Solid State (DPSS) laser. An incident power of 5mW was focused on a spot of dimension 50 $\\times$ 80 $\\mu$m approximately. Power dependance measurements at low temperature indicated negligible laser heating for this incident power. The inelastically scattered photons were analyzed using a triple grating spectrometer working in subtractive configuration and equipped with a nitrogen cooled Charge Coupled Device (CCD) camera. The spectral resolution was about 1.5 cm$^{-1}$. All spectra were recorded with linearly polarized and parallel incoming and outgoing photons. The crystals were cooled using a close-cycle optical cryostat with a base temperature of 3 K.\\\\\n\n\\begin{table}\n\\centering\n\\caption{\\label{tab:Samples} Details of the Sr$_{1-x}$Ca$_{x}$TiO$_{3-\\delta}$ samples. Hall carrier density ($n_H$), room temperature ($\\rho_{\\textnormal{300K}}$) and 2 K ($\\rho_{\\textnormal{2K}}$) resistivity, the ratio RRR$=\\rho_{\\textnormal{300K}}\/\\rho_{\\textnormal{2K}}$, the Hall mobility at 2 K ($\\mu_{H-\\textnormal{2K}}$) and the Curie temperature seen by resistivity are specified.}\n\\footnotesize\n\\begin{tabular}{ccccccc}\n\\hline\n$x$ \t\t& $n_H$  \t\t\t& $\\rho_{\\textnormal{300K}}$ & $\\rho_{\\textnormal{2K}}$ \t& RRR & $\\mu_{H-\\textnormal{2K}}$ & $T_{Curie,{\\rho}_{xx}}$ \\\\\n\t& $10^{18}$cm$^{-3}$  \t& m$\\Omega$cm \t\t\t\t& m$\\Omega$cm  \t\t\t\t\t\t\t& \t\t& cm$^2$\/Vs & K\\\\\n\\hline\n\\hline\n\t0.0022 & 0.72 & 1840 & 2.30 & 800 & 3770 & $9.6 \\pm 0.4$ \\\\\n\t0.0022 & 1.1 & 1380 & 2.31 & 600 & 2456 & $ 7.8\\pm 0.5$ \\\\\n\t0.0022 & 3.3 & 553 & 1.08 & 512 & 1751  & $ 6.8\\pm 0.5$ \\\\\n\t0.0022 & 4.4 & 344 & 0.847 & 406 & 1675  & $ 3.7\\pm 0.4 $\\\\\n\t0.0022 & 6.8 & 87 & 0.625 & 139 & 1469 &  0\\\\\n\t\\hline\n\t0.0045 & 0.66 & 1830 & 3.43 & 535 & 2760 & $ 19.1\\pm 1 $ \\\\\n\t\\hline\n\t0.009 & 0.83 & 1470 & 6.37 & 231 & 1183 & $24.5\\pm 1$\\\\\n\t0.009 & 1.5 & 857 &\t4.17 & 206 & 1005 & $24.4 \\pm 1$\\\\\n\t0.009 & 3.8 & 394 & 2.27 & 174 & 724 & $19.8 \\pm 0.4$\\\\\n\t0.009 & 4.4 & 346 & 2.17 & 159 & 654 & $ 18.1 \\pm 0.4$\\\\\n\t0.009 & 7.2 & 217 & 1.48 & 147 & 586 & $17.7 \\pm 0.8$\\\\\n\t0.009 & 13 & 132 &\t0.941 & 140 & 510 & $9.1 \\pm 0.4$\\\\\n\t0.009 & 20 & 91.3 & 0.656 & 139 & 476 & 0\\\\\n\t0.009 & 26 & 70.4 & 0.542 & 130 & 443 & 0\\\\\n\t\\hline\n\\end{tabular}\n\\normalsize\n\\end{table}\n\n\\section{\\label{Tdep}Temperature dependence of resistivity}\nFigures S\\ref{FigSI1} and S\\ref{FigSI2} plot the resistivity $\\rho_{xx}$ as well as its derivative $d\\rho_{xx}\/dT$ measured on Sr$_{1-x}$Ca$_{x}$TiO$_{3-\\delta}$ samples with Ca contents of $x=0.0022$ and 0.009, respectively, as a function of temperature $T$. The arrows mark the temperatures associated with the resistivity anomaly and the Curie temperature $T_{Curie,\\rho_{xx}}$ seen in resistivity plotted in Fig. 2c of the main text (see also Tab. S\\ref{tab:Samples}). The temperatures $T_{Curie,\\rho_{xx}}$ have been taken as the temperature at which $d\\rho_{xx}\/dT$ shows a kink.\\\\\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{FigureSI3a.pdf}\n\\caption{Temperature dependence of resistivity $\\rho_{xx}$ and its first derivative $d\\rho_{xx}\/dT$ in metallic Sr$_{0.9978}$Ca$_{0.0022}$TiO$_{3-\\delta}$. The arrows mark the temperatures associated with the resistivity anomaly and the Curie temperature $T_{Curie,\\rho_{xx}}$ seen in resistivity.}\n\\label{FigSI1}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.6\\textwidth]{FigureSI3b.pdf}\n\\caption{Temperature dependence of resistivity $\\rho_{xx}$ and its first derivative $d\\rho_{xx}\/dT$ in metallic Sr$_{0.991}$Ca$_{0.009}$TiO$_{3-\\delta}$.}\n\\label{FigSI2}\n\\end{figure}\n\n\\clearpage\n\n\\section{\\label{US}Temperature dependence of thermo-remanent polarization}\nFig. S\\ref{FigSI5tp} displays the thermo-remanent polarization in the system Sr$_{0.991}$Ca$_{0.009}$TiO$_{3}$. A  theoretical description of the quantum ferroelectric regime according to the transverse Ising model yields values for the saturation polarization reaching up to 20\\,mC\/m$^2$ for a comparable Ca concentration \\cite{Guo:2012}. However, as we are dealing only with the remanant polarization, the maximum value at low temperatures lies around 20\\,mC\/m$^2$, which is smaller due to domain formation but is still of the same order of magnitude. Upon heating above the ferroelectric ordering temperature $T_{Curie}$ near 25\\,K $P(T)$ shows a steep fall, but does not vanish completely. Above $T_{Curie}$, there is a finite frozen polarization, as seen by the opening of the $P(E)$ hysteresis loops  shown in the inset of Fig. S\\ref{FigSI5tp}. Obviously, polar entities like ferroelectric clusters or polar structural domain walls persist to temperatures well above the ferroelectric transition. The latter is in accord to results gained from ultrasound experiments even in pure STO \\cite{Salje:2013}. A corresponding characterization of the oxygen-reduced samples obviously cannot be made as the macroscopic polarization features are shielded by the metallic background.\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[width=0.4\\textwidth]{Fig_STO09CaPyro.pdf}\n\\caption{Thermo-remanent polarization measured in an insulating sample of Sr$_{0.991}$Ca$_{0.009}$TiO$_{3}$ after cooling in a poling field of 120\\,V\/mm. The inset shows $P(E)$ loops measured at various temperatures between 5 and 250\\,K.}\n\\label{FigSI5tp}\n\\end{figure}\n\n\\section{\\label{US}Detection of the phase transition with sound-velocity measurements}\nThe sound velocity was measured in transmission geometry using longitudinal 10~MHz PZT-transducers as emitter and detector. A network analyzer (Rohde \\& Schwarz ZVB4) with time domain option was used to determine the transit time trough plate-like samples with a length of typically 5 mm\\cite{Balashova:1996}. The FFT-representation of the response carries an absolute time resolution of the reciprocal resonance frequency, i.e., approximately 100 nS. However, relative changes of the transit time can be determined with much higher resolution.\\\\\nThe sound velocity as a function of temperature in an insulating ($\\delta = 0$) and metallic ($\\delta \\neq 0$) Sr$_{0.991}$Ca$_{0.009}$TiO$_{3-\\delta}$ sample is shown in Fig. S\\ref{FigSI7}. In both cases the phase transition gives rise to an anomaly near the Curie temperature.\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FigureSI7.pdf}\n\\caption{Sound velocity as a function of temperature in insulating ($\\delta = 0$) and metallic ($\\delta \\neq 0$, $n=1.4 \\times 10^{18}$ cm$^{-3}$) Sr$_{0.991}$Ca$_{0.009}$TiO$_{3-\\delta}$ samples. The arrow depicts the Curie temperature.}\n\\label{FigSI7}\n\\end{figure}\n\n\\section{\\label{QO}Quantum oscillations}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FigureSI5.pdf}\n\\caption{Quantum oscillations detected at low temperatures on metallic Sr$_{0.991}$Ca$_{0.009}$TiO$_{3-\\delta}$. Each panel plots the oscillating part $\\Delta\\rho_{xx}$ of the magnetoresistance, obtained after subtracting a smooth background, as a function of inverse magnetic field $1\/B$.}\n\\label{FigSI3}\n\\end{figure}\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.4\\textwidth]{FigureSI6.pdf}\n\\caption{Oscillation period F detected on Sr$_{0.991}$Ca$_{0.009}$TiO$_{3-\\delta}$ as a function of carrier density together with data obtained on SrTiO$_3$ \\cite{Lin:2014}. The doping levels $n_{c1}$ and $n_{c2}$ correspond to the threshold for the filling of a second and third band.}\n\\label{FigSI4}\n\\end{figure}\nQuantum oscillations in SrTiO$_{3-\\delta}$ have been systematically studied for a large range of carrier densities ($10^{17} - 10^{20}$ cm$^{-3}$) using electric and thermoelectric measurements \\cite{Lin:2013,Lin:2014}. Fig. S\\ref{FigSI3} plots the oscillating part $\\Delta\\rho_{xx}$ of the magnetoresistance, obtained after subtracting a smooth background, as a function of inverse magnetic field $1\/B$ for metallic Sr$_{1-x}$Ca$_{x}$TiO$_{3-\\delta}$ ($x=0.0091$) samples. The detected oscillation periods $F$ are plotted as a function of carrier density in Fig. S\\ref{FigSI4} and compared with data obtained on SrTiO$_{3-\\delta}$ \\cite{Lin:2014}. Lin et al. identified two critical doping levels $n_{c1}\\approx 1.5\\times 10^{18}$ and $n_{c2}\\approx 3 \\times 10^{19}$ cm$^{-3}$ that correspond to the threshold for the filling of a second and third band and which are associated with the appearance of a second or third oscillation period, respectively. As seen in Fig. S\\ref{FigSI4}, the detected periods obtained on Sr$_{1-x}$Ca$_{x}$TiO$_{3-\\delta}$ agree well with those obtained on Ca-free STO. However, due to the $3-5$ times lower mobility in the Ca-doped samples compared to pure STO, only the lowest oscillation period could be resolved with certainty for $n>n_{c1}$.\\\\\nEven for low carrier concentrations below $n_{c1}$ the Fermi surface of n-doped SrTiO$_{3-\\delta}$ is not a perfect sphere but an ellipsoid squeezed along the c-axis due to the tetragonal distortion of the lattice. As done in \\cite{Lin:2014} we estimated the carrier concentration $n_{SdH}$ from the oscillations using the magnitude of the tetragonal distortion reported by Allen et al. \\cite{Allen:2013}. The values of $n_{SdH}$ given in the main text were calculated as $n_{SdH}=1.26\\times n_{sphere}$ with $n_{sphere}=k_{F}^{3}\/3\\pi^2$ and $k_{F}=\\sqrt{2eF\/\\hbar}$ with the oscillation period $F$. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\\label{sec:intro}\n\nUltra-luminous X-ray sources (or ULXs) are extra-galactic sources first discovered by \\cite{1989ARA&A..27...87F} that have been observed to have X-ray luminosities of $L_\\mathrm{X}\\gtrsim10^{39}$~erg~s$^{-1}$.  X-ray variability  suggests that ULXs are binaries with a non-degenerate star, referred to as a donor, transferring mass onto a compact object, the accretor \\citep{1976MNRAS.175..395B, 2009MNRAS.397.1061H}. These objects often dominate the total X-ray emission of their host galaxy. They are generally too bright to be low-mass X-ray binaries (LMXBs; for which $L_\\mathrm{X}\\lesssim10^{37}$~erg~s$^{-1}$), and too dim when compared to active galactic nuclei (AGNs; for which $L_\\mathrm{X}\\gtrsim10^{41}$~erg~s$^{-1}$) with their position being off-centre in their host galaxy \\cite[see][for a review on the observational properties of ULXs]{2017ARA&A..55..303K}.\n\nEddington luminosity is the limit above which any accretion onto the compact object is stopped by outgoing radiation. In general, ULX luminosities far exceed the Eddington limit of stellar-mass black holes (which is of the order of $10^{39}$~erg~s$^{-1}$) provided the emission is assumed to be isotropic. Initial suggestions as to the nature of the accretor include intermediate-mass black holes (IMBHs; with masses $\\gtrsim100$~M$_{\\odot}$) accreting at sub-Eddington rates \\citep{1999ApJ...519...89C,2001ApJ...562L..19E, 2006ASPC..352..121M, 2007Natur.445..183M}, or stellar-mass black holes (SMBHs) accreting at super-Eddington rates. For the latter, in order not to exceed the Eddington limit locally while still producing the observed luminosity, magnetic fields and geometric X-ray beaming effects have been proposed as an explanation \\citep[e.g.][]{2001ApJ...552L.109K,2014arXiv1411.5434C}. \n\nSince the first gravitational wave event, GW150914, detected by LIGO \\citep{2016PhRvD..93l2003A}, there has been vast renewed interest in studying the formation of double compact objects. Many of the proposed formation channels for these double compact objects predict that the binary will go through one or more phases of super-Eddington accretion onto a compact object.  \\cite{2017A&A...604A..55M} explored how some observed ULXs might indeed be progenitors of coalescing double compact object binaries, specifically through the chemically homogeneous channel. In this type of binary evolution, the star would avoid large expansion of the envelope, and therefore the binary would avoid a common-envelope (CE) phase \\citep{2016MNRAS.458.2634M}. \\cite{2017MNRAS.472.3683F} compared observation trends in the number of ULXs per unit of star-formation rate and as a function of metallicity of the host galaxy to the merger rate of binary BHs. These latter authors found that the majority of ULXs could be progenitors of binary BH mergers. Therefore, studying the formation of ULXs could shed light on merging binary compact objects, and vice versa.\n\nIn recent decades, debate over the nature of ULXs has focused on whether they are SMBHs or IMBHs, and neutron stars (NSs) have not been considered. However, theoretical studies have extended ULX binary models to include NSs \\citep[e.g.][]{2001ApJ...552L.109K, 2009MNRAS.393L..41K}. That is until \\cite{2014Natur.514..202B} reported the first ever observations of X-ray pulsations in the M82 galaxy with a period of about 1.37~seconds and a 2.52-day sinusoidal modulation in the ULX~X-2. Detection of X-ray pulsations in ULXs suggests the presence of NSs as accretors instead of BHs, at least in a certain fraction of the ULX population where such pulses have been observed. This is because X-ray pulsations are characteristic of accreting NSs with radiation being emitted along their magnetic poles as they rotate about their axes. Therefore, the question of accreting NSs in binaries with relatively massive donors was raised. This discovery was followed by the detection of X-ray pulsations in several other ULXs \\citep{2017MNRAS.466L..48I, 2017Sci...355..817I, 2018MNRAS.476L..45C,2018NatAs...2..312B, 2019MNRAS.tmpL.104S,2019ApJ...879...61Z,2019arXiv190604791R}. \n\nThe two most prominent questions regarding the nature of NS ULXs refer to the emission mechanism and the formation channel.\nFor the first question, a number of mechanisms are invoked to explain super-Eddington luminosities. With the apparent extremely high mass-transfer rate, a lot of the transferred mass could be blown away by strong radiation outflows \\citep{1973A&A....24..337S}. \\cite{2006MNRAS.370..399B} applied this idea to SS433 where observations of massive outflows suggest that the source is a ULX seen from the side. The absorption features associated with these outflows have been observed in some but not all ULXs; see for example Holmberg IX X-1 and NGC 1313 X-1  \\citep{2012MNRAS.426..473W}. However, this does not invalidate the theory, as beamed X-ray emission would not be visible unless the observer had a direct line of sight to the accreting compact object, down the collimating funnel, in which direction the outflows are limited. \\cite{1973A&A....24..337S}, \\cite{2002ApJ...568L..97B}, and \\cite{2007MNRAS.377.1187P} explored the idea of the presence of strong optically thick outflows that blow away some part of the disc from where radiation can escape. \\cite{1973A&A....24..337S} and \\cite{2007MNRAS.377.1187P} also suggested the formation of  geometrically thick accretion discs. This structure would cause the emission to be beamed, and therefore the observed isotropic-equivalent luminosity would be much higher than the intrinsic one. \\cite{2001ApJ...552L.109K} postulated that based on the assumption of mild beaming, intermediate- and high-mass X-ray binaries (IMXBs and HMXBs) undergoing mass transfer on a thermal timescale would be the best candidates for ULXs. The effect of beaming on the emission has, in general, been explored \\citep{2009MNRAS.393L..41K,2017MNRAS.468L..59K,2019ApJ...875...53W}. In addition to that, NSs with strong magnetic fields (around $10^{14}$~G) reduce the electron-scattering cross-section and could anchor the infalling matter to accretion columns above the magnetic poles and thereby produce sufficiently high luminosities \\citep{1976MNRAS.175..395B, 2018NatAs...2..312B}. \\cite{2017ApJ...845L...9T} carried out general relativistic radiation magnetohydrodynamics simulations of super-Eddington accretion onto a non-rotating, magnetised NS and found a spin-up rate of $\\sim -{10}^{-11}~{\\rm{s}}~{{\\rm{s}}}^{-1}$, which is consistent with observations. In contrast, \\cite{2019MNRAS.485.3588K} suggested that the observed ULX properties are explained by NSs with normal magnetic fields and not by the presence of magnetars. For the specific case of M82~X-2, \\cite{2014arXiv1410.8745L} suggested the presence of an optically thick accretion disc that acts as a curtain and shields some of the outgoing radiation, thus allowing for super-Eddington luminosities driven by Roche-lobe overflow (RLO). Finally, \\cite{2011ApJ...739...42W} studied accretion discs in weakly magnetised NSs, finding that at super-Eddington rates, the magnetic field has little effect on the accretion disc. \n\nDespite a lot of research done in the field, the question surrounding how ULXs attain super-Eddington luminosities is still open and an active field of research. Regarding a possible formation channel for NS ULXs,  a lot of work has been carried out to address evolution and mass transfer in X-ray binaries. \\cite{1999A&A...350..928T} performed non-conservative mass-transfer calculations of low-mass X-ray binaries with a $1.3$~M$_{\\odot}$ NS and found that the binaries can undergo very high mass-transfer rates (super-Eddington by a factor of $\\sim 10^4$) for donors with deep convective envelopes (mass range $1.6$--$2.0$~M$_{\\odot}$). An important X-ray source in the context of high mass-transfer rates is Cygnus X-2, which is an X-ray binary containing a NS. Cygnus X-2 has a mass ratio of $\\approx 0.34~(=M_{\\mathrm{donor}}\/M_{\\mathrm{acc}})$, so for an estimated accretor mass of about $1.78$~M$_{\\odot}$ the donor has a mass of $0.6$~M$_{\\odot}$ \\citep{1998ApJ...493L..39C, 1999MNRAS.305..132O}. \\cite{1999MNRAS.309..253K} argued that the donor star lost a lot of mass ($\\sim 3.0$~M$_{\\odot}$) in an intense mass-transfer phase on a thermal timescale. \\cite{1999MNRAS.309..253K,2000ApJ...529..946P,2000MNRAS.317..438K} showed that Cygnus X-2 observations can be explained by case B mass transfer from a donor of mass $3.5$~M$_{\\odot}$; that is, the progenitor for the source was an IMXB. \n\nContrary to the general understanding of the stability of mass-transfer in X-ray binaries, there is similar evidence in the literature that IMXBs can undergo stable mass transfer with a NS and avoid CE. \\cite{2000ApJ...530L..93T} carried out numerical calculations of IMXBs with $2.0$--$6.0$~M$_\\odot$ donor and $1.3$~M$_\\odot$ accretor masses using an updated version of the Eggleton code \\citep{1971MNRAS.151..351E,1972MNRAS.156..361E,1998MNRAS.298..525P}. The authors studied the initial parameter space for producing binary millisecond pulsars with a heavy carbon--oxygen (CO) white dwarf companion, and demonstrated for the first time the full stability of IMXBs. \\cite{2009MNRAS.393L..41K} stated that NSs would have lower accretion rates for the same mass-transfer rates because NSs would have stronger beaming as their Eddington limit would be lower than that of BHs. Calculations similar to those of \\cite{2000ApJ...530L..93T} were carried out by \\cite{2012ApJ...756...85S} using the Eggleton code. These latter authors studied the initial parameter space for binary pulsars while considering orbital angular momentum losses from gravitational wave radiation, magnetic braking, and mass lost from the system. \\cite{2017ApJ...846..170T} compared the orbital evolution between IMXBs and HMXBs in studying the connection between ULXs and double NS systems and concluded that the orbital period evolution of IMXBs makes them more likely to be NS ULXs than HMXBs. \n\n\\cite{2015ApJ...802L...5F}   studied the origin of the NS ULX M82~X-2 specifically, by combining parametric population synthesis calculations (using BSE; \\citealt{2002MNRAS.329..897H}) with detailed binary evolution calculations (using MESA; \\citealt{2011ApJS..192....3P,2013ApJS..208....4P}). Assuming highly non-conservative mass transfer and that a significant fraction of the mass lost from the binary carries the specific orbital angular momentum of the donor star, these latter authors found that the most probable parameters to form a NS ULX are donors with initial masses in the range of $3.0$--$8.0$~M$_{\\odot}$ , and initial orbital periods of  $1.0$--$3.0$~days. \\cite{2015ApJ...802..131S} suggested that NS ULXs, in general, might represent a higher contribution to the general ULX population than BH ULXs, and a significant portion of those would be IMXBs. Similarly, \\cite{2015ApJ...810...20W,2017ApJ...846...17W}, using parametric population synthesis calculations, showed that ULXs are more likely to be NS ULXs than BH ULXs, especially in solar metallicity environments. These latter authors found a typical NS ULX to have a $\\sim 1.0$\n~M$_{\\odot}$ red giant. However, they found that extreme NS ULXs ($L_\\mathrm{X}\\gtrsim10^{42}$~erg~s$^{-1}$) typically have evolved, low-mass, stripped helium-star donors ($\\sim 2.0$~M$_{\\odot}$). \\cite{2019ApJ...875...53W} found that most BH ULXs emit X-rays isotropically, while NS ULXs are generally beamed. Therefore, even though NS ULXs might be intrinsically more numerous than BH ULXs, these latter authors predict that BH accretors dominate the observed ULX population.\n\nAnother discussed property of pulsating ULXs is the very high NS spin-up rate (i.e. the rate of change of spin period) in some observed pulsars. NGC~5907~ULX1 shows a change from a spin period of $1.43$~s to $1.13$~s in about 10 years \\citep{2017Sci...355..817I} with an inferred spin-up rate of $-8\\pm0.1\\times10^{-10}$s~s$^{-1}$. The spin period of NGC~300~ULX1 went from $31.71$~s to $31.54$~s in $4$~days \\citep{2018MNRAS.476L..45C} with a spin-up rate $\\sim-5.56\\times10^{-7}$s~s$^{-1}$, which is the highest rate observed so far for a NS ULX. However, the rate of change of spin period given by  \n\\begin{equation}\n    \\dot{P} = \\dot{\\bigg(\\frac{1}{{\\nu}}\\bigg)} = - \\frac{\\dot{\\nu}}{\\nu^2},\n\\end{equation}\nwhere $\\nu$ is the NS spin frequency, does not directly reflect the mass accretion rate. Rather, the rate of change of $\\nu$ is roughly proportional to the mass accretion rate \\citep[for instance see Eq.~(2) in][]{2017MNRAS.468L..59K}. Nevertheless, we  continue to mention $\\dot{P}$ values when talking about observations (Table \\ref{tab:obs_data}) as they are the ones most often reported in the literature.\n\nHigh rates of frequency increase could suggest efficient spin up from very high accretion rates onto the NS. This is because high mass accretion provides the NS with enough torque to spin it up to such high rates \\citep{2001ASPC..229..423R, 2006csxs.book..623T}. However, there is a caveat in calculating accretion rates from spin-up frequency as they might not represent the secular average accretion rate over an evolutionary timescale of the donor star. In some cases, the extremely high spin-up rate would grossly overestimate the amount of matter accreted by the NS. For example, M82~X-2 showed a high spin-up rate of $-2\\times10^{-10}$s~s$^{-1}$ when X-ray pulses were first discovered \\citep{2014Natur.514..202B}. However, later, \\cite{2019arXiv190506423B} observed an average spin down of $-5\\times 10^{-11}$~Hz~s$^{-1}$ over a period of two years. These latter authors suggested that the source might be close to spin equilibrium and is alternating between phases of spin up and spin down. \\cite{2017MNRAS.468L..59K} estimated the spin-up timescale for three pulsating ULXs and concluded that we observe them close to equilibrium. \\cite{2019A&A...626A..18C} performed semi-analytical calculations for accretion onto a magnetised NS using the NS being close to spin equilibrium as a boundary condition. The reason for this is that even though $\\dot{\\nu}$ gives an estimate of the instantaneous accretion rate, it should not be compared directly to long-term average estimates that binary evolution models are giving, as many of these systems might be close to their spin equilibrium.\n\nIn our work, we do not investigate X-ray pulses and the super-Eddington emission mechanism. Instead, we focus on ULX formation and long-term evolution. Therefore, in the entirety of this study, we refer to LMXBs\/IMXBs with NS accretors that drive super-Eddington mass-transfer rates as NS ULXs. Pulsating ULXs are a subset of NS ULXs as pulsations are not a necessary outcome of super-Eddington mass transfer but rather they are a product of the presence of a relatively strongly magnetised NS. In this paper, we investigate how these NS ULXs could be formed and try to explain the physical properties involved using numerical computations. In doing so, we study how the stability of binaries is affected by spin-orbit coupling, and by a higher accretor mass ($2.0$~M$_\\odot$ instead of a typical NS mass of $\\sim 1.3$~M$_\\odot$). We also assume that there is no precession of the accretion disc or absorption of X-ray flux by optically thick material around the source that might cause the luminosity to vary. In the following section (Section~\\ref{sec:observations}), we summarise the properties of the currently observed pulsating ULX sample. In Section~\\ref{sec:numericaltools}, we discuss the numerical methods and physics employed for the simulations, while in Section~\\ref{sec:results} we present the results from our simulations, highlighting the allowed initial parameter space for NS ULX formation and the properties of the formed population. Section~\\ref{sec:discussion} discusses the observed NS ULXs in the context of our results and how the angular momentum exchange between spin and orbit affects the result through tides. Finally, we end with concluding remarks in Section~\\ref{sec:conclusion}.\n\n\\section{Currently observed pulsating ULX sample}\\label{sec:observations}\n\nIn this section, we discuss some of the observed and predicted parameters for the NS ULXs present so far in the literature. In most cases, the observables are not well constrained, and therefore large uncertainties are involved and assumptions about the physical properties have been made. In the entirety of this paper we refer to the mass of the donor as $M_{\\mathrm{donor}}$, that of the accretor as $M_{\\mathrm{acc}}$, the orbital separation as $a$, and the orbital period as $P_{\\mathrm{orb}}$. For our work, the most important pulsating ULX is M82~X-2 as it has the most well-constrained observational parameters, even though we comment on other pulsating ULXs observed as well. We summarise all the relevant properties of the currently known sample of pulsating ULXs in Table~\\ref{tab:obs_data}.\n\n\\begin{table*}[htp]\n\\begin{center}\n\\begin{threeparttable}\n\\begin{tabular}{lllllll}\n\\hline\nNS ULX & $L_\\mathrm{X}$ (erg~s$^{-1}$) & $M_{\\mathrm{donor}}$ (M$_{\\odot}$) & $P_\\mathrm{orb}$ (days) & $P_\\mathrm{spin}$ (s) & $\\dot{P}_\\mathrm{spin}$ (s s$^{-1}$) \\\\ \\hline\nM82~X-2\\tnote{1} & $1.8\\times 10^{40}$ & $\\gtrsim 5.2$ & $2.52$ & $1.37$ & $-2.0\\times 10^{-10}$ \\\\\nNGC~7793~P13\\tnote{2} & $5.0\\times 10^{39}$ & $18.0$--$23.0$\\tnote{3} & $64.0$\\tnote{4} & $0.417$ & $-3.5\\times 10^{-11}$ \\\\\nNGC~5907~ULX1\\tnote{5} & $\\sim 10^{41}$ & $2.0$--$6.0$ & $5.3^{+2.0}_{-0.9}$ &  $1.137$ & $-8.1\\times 10^{-10}$ \\\\\nNGC~300~ULX1\\tnote{6} & $4.7\\times 10^{39}$ & $\\gtrsim 8.0$--$10.0$\\tnote{7} & $\\gtrsim 1.0$~yr\\tnote{8} &$31.6$ & $-5.56\\times 10^{-7}$ \\\\\nM51~ULX-8\\tnote{9} & $2.0\\times 10^{39}$ & - & - & - & - \\\\\nNGC~1313~X-2\\tnote{10} & $1.5\\times 10^{40}$ & $\\lesssim 12.0$\\tnote{11} & - & $1.5$ & $-1.2\\times 10^{-10}$ \\\\\nSwift~J0243.6+6124\\tnote{12} & $\\sim 10^{39}$ & - & $28.3$\\tnote{13} & $9.86$\\tnote{14} & $-2.2\\times 10^{-8}$ \\\\\nM51~ULX-7\\tnote{15} & $10^{39}$--$10^{40}$ & $\\gtrsim 8.0$ & 2.0 & $2.8$ & $-10^{-9}$ \\\\\n\\hline\n\\end{tabular}\n\\begin{tablenotes}\n\\item[1]\\cite{2014Natur.514..202B},\\item[2]\\cite{2017MNRAS.466L..48I, 2016ApJ...831L..14F},\\item[3]\\cite{2011AN....332..367M},\\item[4]\\cite{2014Natur.514..198M},\\item[5]\\cite{2017Sci...355..817I},\\item[6]\\cite{2018MNRAS.476L..45C},\\item[7]\\cite{2019arXiv190902171H},\\item[8]\\cite{2018A&A...620L..12V,2019ApJ...879..130R},\\item[9]\\cite{2018NatAs...2..312B},\\item[10]\\cite{2019MNRAS.tmpL.104S},\\item[11]\\cite{2008A&A...486..151G},\\item[12]\\cite{2019ApJ...879...61Z}, \\item[13]\\cite{2018A&A...613A..19D,2017ATel10907....1G},\\item[14]\\cite{2017ATel10809....1K,2017ATel10812....1J},\\item[15]\\cite{2019arXiv190604791R}.\n\\end{tablenotes}\n\\end{threeparttable}\n\\end{center}\n\\caption{Observed and inferred parameters of NS ULXs from the literature. $L_\\mathrm{X}$ is the X-ray luminosity, $M_{\\mathrm{donor}}$ the donor mass, $P_{\\mathrm{orb}}$ the orbital period, $P_{\\mathrm{spin}}$ the NS spin period, and $\\dot{P}_\\mathrm{spin}$ is the time derivative of the NS spin period.}\n\\label{tab:obs_data}\n\\end{table*}\n\n\\subsection{M82~X-2}\\label{m82x2}\nThis source is one of the most studied NS ULXs. It is observed in the core of M82 galaxy and was discovered to show X-ray pulsations by \\cite{2014Natur.514..202B}. It has a peak X-ray luminosity of $1.8\\times10^{40}$erg~s$^{-1}$. Using the observed orbital period of $2.52$~days and inclination of $<60\\degree$, the binary mass function for M82~X-2 was calculated as $2.1$~M$_{\\odot}$. Assuming that the NS mass is $1.4$~M$_{\\odot}$, the donor mass was estimated to be $\\gtrsim5.2$~M$_\\odot$.\n\n\\subsection{NGC~7793~P13}\\label{7793}\nThis ULX source appears to be a HMXB containing an accreting NS in galaxy NGC 7793, with a luminosity of about $5.0\\times 10^{39}$erg~s$^{-1}$. Earlier, \\cite{2011AN....332..367M}  identified the donor in the then-not-discovered ULX system as a late B-type super-giant star with a mass of $18.0$~M$_{\\odot}<M_{\\mathrm{donor}}<23.0$~M$_{\\odot}$.  \\cite{2014Natur.514..198M} estimated the orbital period to be 64 days using optical modulation in observations. The NS was found to be present after pulsations were detected from the source \\citep{2017MNRAS.466L..48I, 2016ApJ...831L..14F}. \n\n\\subsection{NGC~5907~ULX1}\nDiscovered to have X-ray pulsations by \\cite{2017Sci...355..817I}, this ULX is observed in the galaxy NGC 5907. Its X-ray luminosity, at around $10^{41}$erg~s$^{-1}$, makes it the most luminous ULX discovered to date. This source does not have as well-constrained parameters as M82 X\u20132. However, some estimates could be made. The authors derived constraints on the orbital period and arrive at a value of $5.3^{+2.0}_{-0.9}$~days using a projected semi-major axis of $a\\,\\sin i =2.5^{+4.3}_{-0.8}$~lt-s (light-seconds). They also compared  HMXBs, LMXBs, and IMXBs as possible explanations and found them all to be consistent with the upper limits derived from HST images \\citep{2013MNRAS.434.1702S}.\n\n\\subsection{NGC~300~ULX1}\nThis X-ray source, located in the galaxy NGC 300,  was recently discovered to show pulsations \\citep{2018MNRAS.476L..45C}. The authors assumed an orbital period in the range $1$--$3$~days and used a projected semi-major axis of $\\lesssim4.6\\times 10^{-5}$~lt-s and arrived at a binary mass function of $0.8\\times 10^{-3}$~M$_{\\odot}$. The X-ray luminosity for ULX1 was found to be $4.7\\times 10^{39}$erg~s$^{-1}$.\n\\cite{2019arXiv190902171H} identified the donor as a red super-giant star ($\\gtrsim 8.0$~M$_{\\odot}$). They reported that the observed effective temperature and luminosity of the donor is consistent with evolutionary tracks of singles stars of initial masses of $8.0$--$10.0$~M$_{\\odot}$. \\cite{2019MNRAS.488.5225V} observed that the flux of the source decreased by a factor of $\\sim 50$ over a few months, while its spin-up remained constant at $4.0\\times10\n^{-10}$~Hz~s$^{-1}$. This could result from absorption from optically thick material nearby (perhaps from outflows) or from a precessing accretion disc.\n\n\\subsection{M51~ULX-8}\n\\cite{2018NatAs...2..312B} identified M51~ULX-8 as a NS ULX on the basis of the detection of cyclotron resonance scattering features instead of direct observations of X-ray pulses. The cyclotron features observed can be translated into magnetic field strengths, which in this case correspond to that of a highly magnetized pulsar ($\\sim 10^{15}$~G). \n\n\\subsection{NGC~1313~X-2}\\label{1313}\n\\cite{2019MNRAS.tmpL.104S} discovered X-ray pulsations in the ULX source NGC~1313 X-2 with a pulse period of $\\sim 1.5$~s. \\cite{2008A&A...486..151G} estimated the mass of the donor star to be $\\lesssim 12.0$~M$_\\odot$ assuming it to be part of a metal-poor star cluster with an age of $2.0$~Myr. \\cite{2008AIPC.1010..303P} estimated the ULX lifetime using observations of the large bubble nebulae around the source, which is $\\sim 1.0$~Myr. Based on studies showing that very few NS ULXs would last as long as $1.0$~Myr \\citep[e.g.][]{2017ApJ...846...17W}, \\cite{2019MNRAS.tmpL.104S} suggested that NGC~1313~X-2 is in its last stages of mass transfer.\n\n\\subsection{Swift~J0243.6+6124}\nDiscovered by \\cite{2017ATel10809....1K} as a transient source, Swift~J0243.6+6124 was known to have an accreting NS with a spin period of $9.86$~s. \\cite{2018A&A...613A..19D} estimated an orbital period of $28.3$~days and semi-major axis of $a$ sin$ i =140^{+3}_{-3}$~lt-s. \\cite{2019ApJ...879...61Z} studied {\\it{Insight}}-HXMT data for this source and estimated its magnetic field as $\\sim 10^{13}$~G and an X-ray luminosity of $>10^{39}$erg~s$^{-1}$, confirming the source to be the first Galactic ULX with an NS accretor. Furthermore, these latter authors calculated a high spin-up rate of $-2.2\\times 10^{-8}$~s~s$^{-1}$. \\cite{2019ApJ...873...19T} reported on the spectral behaviour of this source and suggested that if it were located in an external galaxy it would have a similar appearance to the other pulsating ULXs observed.\n\n\\subsection{M51~ULX-7}\n\\cite{2019arXiv190604791R} discovered $2.8$~s X-ray pulses in observations of ULX-7 in galaxy M51, therefore finding another pulsating ULX following ULX-8 in the same galaxy. The secular spin-up rate was measured as $-10^{-9}$~s~s$^{-1}$ and a variable X-ray luminosity in the range of $10^{39}$--$10^{40}$~erg~s$^{-1}$. The authors used the projected semi-major axis of $28.0$~lt-s and an assumed accretor mass of $1.4$~M$_{\\odot}$ to infer a donor of mass $>8.0$--$13.0$~M$_{\\odot}$. \\cite{2020MNRAS.491.4949V} studied the X-ray light curves of the source and estimated a magnetic field of the NS (rotating near spin equilibrium) of $2.0$--$7.0\\times 10\n^{13}$~G. They also assumed that the NS is freely precessing and estimated the magnetic field to be $3.0$--$4.0\\times 10\n^{13}$~G, agreeing with their previous estimate.\n\n\n\\section{Numerical tools and calculations}\\label{sec:numericaltools}\n\nIn this section we discuss the numerical code used for the binary evolution calculations along with the adopted model parameters and the code modifications that we introduced. \n\n\\subsection{Numerical stellar evolution code and progenitor binary}\n\\label{code}\n\nTo simulate the evolution of the binaries we use MESA (version 10108; MESASDK version 20180127) which is a stellar structure and binary evolution code developed by \\cite{2011ApJS..192....3P,2013ApJS..208....4P,2015ApJS..220...15P,2018ApJS..234...34P,2019ApJS..243...10P}.\n\nAll LMXBs\/IMXBs are calculated as systems with an initially zero-age main-sequence (ZAMS) donor and a point-mass NS accretor. It is assumed that the NS was formed during a previous evolutionary stage which we do not study, and the binary survived a possible NS natal kick from the supernova explosion that formed the NS. We compute a grid of models spanning $0.92$--$8.0$~M$_{\\odot}$ in initial donor mass, and $0.5$--$100$~days in initial orbital period. For reference, initial parameters refer to the orbital parameters at the onset of RLO.\nThe calculation of the mass-transfer rate during RLO is done implicitly using the scheme proposed by \\cite{1990A&A...236..385K}.\n\nWe use $1.3$~M$_{\\odot}$ and $2.0$~M$_{\\odot}$ for the mass of the accretor;  $1.3$~M$_{\\odot}$ is close to the post-supernova Chandrasekhar mass limit for the formation of a NS and $2.0$~M$_{\\odot}$ is on the high-mass end of the NS mass distribution \\citep{2010arXiv1012.3208L,2013Sci...340..448A, 2018ApJ...852L..25R}. The NS companion to the pulsar J0453+1559 has a mass of $1.174^{0.004}_{0.004}$~M$_{\\odot}$ \\citep{2015ApJ...812..143M}\\footnote{See \\cite{2019ApJ...886L..20T} for an alternative possibility} and therefore NS masses below $1.3$~M$_{\\odot}$ have indeed been observed. However, we take $1.3$~M$_{\\odot}$ as the standard NS mass.\n\nFor the radiative efficiency of the accretion onto the NS with initial mass $M^i_{\\rm acc}$ (i.e. the release of gravitational energy of the infalling material in the form of radiation; in units of rest-mass energy) we use,\n\n\\begin{equation}\\label{eta}\n    \\eta = \\frac{G M^i_{\\rm acc}}{c^2 R_{\\rm acc}},\n\\end{equation}\nwhere, $c$ is the speed of light and $R_{\\rm acc}$ is the NS radius which we take as 11.0~km.\nFor the Eddington limit of an accretor with initial mass $M^i_{\\rm acc}$,\n\\begin{equation}\\label{Ledd}\n    L_{\\rm Edd} = \\frac{4\\pi G M^i_{\\rm acc} c }{\\kappa},\n\\end{equation}\nwhere, $\\kappa$ is the opacity. Using Eq.~(\\ref{eta}) and simplifying Eq.~(\\ref{Ledd}) (for accretion of pure ionised hydrogen), we get\n\\begin{equation}\\label{Eq:Edd}\n    \\dot{M}_\\mathrm{Edd}=1.5\\times 10^{-8}~\\bigg(\\frac{M^i_{\\mathrm{acc}}}{1.3~ \\mathrm{M}_{\\odot}}\\bigg) ~\\mathrm{M}_{\\odot}~\\mathrm{yr}^{-1}.\n\\end{equation}\n\nWe fix the Eddington limit and the radiative efficiency to initial values as they would not change significantly from the amount of mass that the NSs accrete in our grids. If the mass-transfer rate goes beyond this value, the radiation pressure will prevent any excess material from being accreted. Mass from the donor is transferred conservatively to the accretor and a fraction of this transferred mass is lost from the vicinity of the accretor as an isotropic fast wind or jet with the specific angular momentum of the accretor \\citep[see][for a detailed explanation]{2006csxs.book..623T}. The efficiency of accretion ($\\epsilon$) by the NS is $\\epsilon = 1 - (\\alpha + \\beta + \\delta)$, where $\\alpha$ is the fractional mass lost directly from the donor, $\\beta$ is the fractional mass lost from the vicinity of the accretor, and $\\delta$ is the fractional mass lost from a circumbinary toroid. We take the values, $\\alpha =\\delta= 0$, and $\\beta = {\\it max}~\\{0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}\\}$. \n\nWe consider orbital angular momentum (J$_{\\mathrm{orb}}$) losses via gravitational wave radiation, spin-orbit coupling due to tidal effects (Section~\\ref{sync_time}), and mass lost from the system. We also include effects due to magnetic braking following the prescription by \\cite{1983ApJ...275..713R} for donor masses that develop an outer convective envelope at any point. We take the eccentricity to be negligible, as tidal forces would circularise the orbit of a semi-detached binary with a giant star on a relatively short timescale of $10^4$~years \\citep{1995A&A...296..709V}. This is orders of magnitude shorter than the main sequence lifetime of intermediate-mass stars which are of the order of $10^8$--$10^9$~years (main sequence lifetime of low-mass stars would be even longer). Furthermore, tidal forces aim to synchronise the stars with the orbit. We assume the orbit is synchronised by the time RLO begins since the tidal forces would cause the stars to synchronise with the orbit on a relatively short timescale \\citep{2000ScChA..43..331H}.\n\nWe assume solar metallicity, that is Z$_{\\odot}$=0.02, and that any layer in the donor interior is stable against convection if the Ledoux criteria for convection is fulfilled \\citep{1947ApJ...105..305L}. At the edges of convective zones, we account for overshooting because the convective material slightly enters non-convective zones due to inertia. To describe overshooting we follow the exponential overshooting efficiencies used in the MIST models for low- and intermediate-mass stars, which are $f_{\\mathrm{ov, core}} = 0.0160$ in the core calibrated from properties of the open cluster M67 and $f_{\\mathrm{ov, en}} = 0.0174$ in the envelope calibrated from solar properties \\citep{2016ApJS..222....8D, 2016ApJ...823..102C}. For stellar winds, we use the cool red giant branch wind scheme described by \\cite{1975psae.book..229R} with a scaling factor of 0.1. In cases where\nwe get a stripped helium (He) star, we use the prescription for Wolf-Rayet stars by \\cite{2000A&A...360..227N}, included in MESA under the Dutch hot wind scheme. \n\nThe mass-transfer calculations are carried out until one of the following conditions are met: \\textit{(i)} the donor forms a white dwarf (WD), \\textit{(ii)} the age of the donor star  exceeds the Hubble time, \\textit{(iii)} the radius of the donor star extends so far beyond its Roche lobe that L$_{2}$ overflow is initiated and the system becomes dynamically unstable (see Section~\\ref{rl2}), \\textit{or (iv)} the number of computational steps exceeds a limit of 300,000 (this value was chosen based on previous grid runs). We include condition \\textit{(iv)} for those systems where MESA runs into converging problems and cannot find a solution. This happens for only two types of binaries in our numerical calculations. In the first case, the mass transfer cannot properly remove the last bit of the envelope from the donor. In the second case, MESA is not able to solve for the donor radius which extends quite far beyond the Roche lobe but not enough to trigger the condition of L$_2$ overflow. Both these cases occur at the end of the mass-transfer phase, and therefore we accept that the binary was stable until the end of its evolution.\n\n\\subsection{Super-Eddington accretion onto the NS}\\label{acc}\n\nFor non-conservative super-Eddington mass transfer the amount of mass that is accreted is less than that transferred per unit time ($0.3\\times $ mass transferred, as per our assumptions) until the Eddington limit prevents a further increase in accretion rate. Cygnus X-2 is an example of a system observed to have survived super-Eddington mass transfer while presumably having accreted comparatively less. This source suggested that high mass transfer onto a NS (or BH) could avoid a CE phase with the NS ejecting most of the mass that was transferred at super-Eddington rates \\citep{1999MNRAS.309..253K, 1999ApJ...519L.169K, 2000ApJ...529..946P, 2001ApJ...552L.109K}.\n\n\\cite{1973A&A....24..337S} studied the observational characteristics of accretion discs in sub-Eddington and super-Eddington regimes of mass transfer in the case of black-hole binaries. In this picture, as the mass transfer approaches the  Eddington limit, the structure of the disc changes from a slim disc to a disc with an inner geometrically thick component and an outer thin component. As matter that is transferred to the accretion disc at super-Eddington rates moves radially inwards (transporting angular momentum outwards), strong outflows begin at a certain radius which remove a fraction of the matter, thereby also taking away excess angular momentum \\cite[see][for an application of the super-Eddington disc model]{2013AstBu..68..139V}. The spherisation radius is defined as the radius at which the accretion luminosity first reaches the Eddington limit and strong outflows begin, and one can approximate it as \n\\begin{equation}\n     R_\\mathrm{sph}= \\dot{M}_{\\mathrm{donor}}\\frac{G M_\\mathrm{acc}}{L_\\mathrm{Edd}}.\n\\end{equation}\nOutside $R_\\mathrm{sph}$ the disc emits X-rays with luminosity $L_\\mathrm{Edd}$ which depends on the accretion rate following the equation,\n\\begin{equation}\n    L_\\mathrm{Edd}= \\eta \\dot{M}_{\\mathrm{Edd}}c^2.\n\\end{equation}\n\nInside $R_\\mathrm{sph}$, the outflowing matter has a velocity which depends on the difference between inward gravity and outward radiation pressure. The velocity of the outflow increases inward which in turn decreases the mass-accretion rate at each radius. This keeps the disc locally Eddington limited. The mass-accretion rate within $R_{\\mathrm{sph}}$ at each point in the disc can be described by the following equation:\n\\begin{equation}\n    \\dot{M}^{\\mathrm{local}}_{\\mathrm{acc}}(R) = \\dot{M}_{\\mathrm{donor}} \\frac{R}{R_\\mathrm{sph}}.\n\\end{equation}\nSince the radiation pressure is balanced by the gravitational pressure at each point, the accretion disc inside the spherisation radius has a thickness of the order of the distance from the accretor and emits X-rays with luminosity $L_\\mathrm{Edd}\\times\\ln{\\dot{m}}$ \\citep[where \n$\\dot{m} \\equiv \\dot{M}_\\mathrm{donor}\/\\dot{M}_\\mathrm{Edd}$; ][]{1973A&A....24..337S}. For $\\dot{m}>1$, the total radiated luminosity from the accretor can exceed the Eddington limit by\n\\begin{equation}\\label{edd_eq}\n    L_{\\mathrm{acc}} = L_{\\mathrm{Edd}}(1 + \\ln{\\dot{m}}).\n\\end{equation}\n\nFor highly super-Eddington mass-transfer rates another effect could come into play because of the geometrically thick accretion disc.\nA narrow funnel forms along the rotation axis of the accretor from where radiation can escape as a collimated jet. The observed isotropic-equivalent accretion luminosity as described by \\cite{2001ApJ...552L.109K} and \\cite{2009MNRAS.393L..41K} is then as follows:\n\n\\begin{equation}\\label{new_edd}\n    L^{\\mathrm{iso}}_{\\mathrm{acc}} = \\frac{L_{\\mathrm{Edd}}}{b}(1 + \\ln{\\dot{m}}),\n\\end{equation}\nwhere $b$ is the beaming factor describing the amount of collimation to the outgoing radiation. The approximated value of $b$ is\n\\begin{equation}\\label{eq:beaming}\n        b= \n\\begin{dcases}\n    \\frac{73}{\\dot{m}^2},& \\text{if } \\dot{m}> 8.5,\\\\\n    1,              & \\text{otherwise}.\n\\end{dcases}\n\\end{equation}\nBecause the beaming factor is an approximation, we apply an upper limit ($\\sim 10^{42}$~erg~s$^{-1}$) in calculating the isotropic-equivalent accretion luminosities so that we do not get unphysically high values.\n\n\\subsection{Spin-orbit coupling and synchronisation timescales}\n\\label{sync_time}\n\nAs mentioned before, tidal forces synchronise the stars with the orbit on a timescale which is relatively short. Therefore, whenever there is a change in either the spin angular momentum or the orbital angular momentum, tides will work to synchronise the system again. This action of tides on the orbit may affect the stability and evolution of the binary \\citep{2001nsbh.conf..337T}.\n\nWhen mass is lost from the donor star during mass transfer it also removes spin angular momentum from the star which is supplied to the orbit. If the spin angular momentum removed from the donor and returned to the orbit is non-negligible, then it has a widening effect on the orbit. This is followed by mass loss from the accretor's vicinity, with the mass lost carrying away the  specific angular momentum of the accretor. This competing effect tends to shrink the orbit. Mass that is leaving the system from the accretor's vicinity carries with it relatively high specific orbital angular momentum (when the accretor is the less massive binary component), having a shrinking effect on the orbit. The donor then becomes sub-synchronous with the orbit and angular momentum has to be transferred from the orbit to the star in order to spin it up, causing the orbit to shrink even further. The interplay between the orbital shrinking and widening effects can reveal how spin-orbit coupling affects the stability of mass transfer. In the presence of strong winds, there is additional loss of angular momentum from the donor. However, for stars with masses of $0.92$--$8.0$~M$_{\\odot}$, stellar winds are too weak to cause significant orbital change. \n\nThe tidal synchronisation timescale is defined as follows \\citep{1977A&A....57..383Z,1981A&A....99..126H}:\n\\begin{equation}\n    \\frac{1}{T_\\mathrm{sync}} = 3 \\frac{K}{T} \\bigg(\\frac{q}{r_\\mathrm{g}}\\bigg)^2 \\bigg(\\frac{R_\\mathrm{donor}}{a}\\bigg)^6,\n\\end{equation}\nwhere $q$ is the mass ratio (we define $q\\equiv M_{\\mathrm{acc}}\/{M_{\\mathrm{donor}}}$) and $r_\\mathrm{g}$ is the gyration radius of the star. ${K}\/{T}$ is the spin-orbit coupling parameter and is described in two ways for a star with an outer radiative envelope as $({K}\/{T})_{\\mathrm{rad}}$, and a star with an outer convective envelope as $({K}\/{T})_{\\mathrm{conv}}$. For the former case we use,\n\\begin{align}\n    \\bigg(\\frac{K}{T}\\bigg)_{\\mathrm{rad}} &= \\bigg(\\frac{GM_{\\mathrm{donor}}}{R_\\mathrm{donor}^3}\\bigg)^{1\/2}(1+q)^{5\/6}E_{2}\\bigg(\\frac{R_\\mathrm{donor}}{a}\\bigg)^{5\/2},\\\\\n    E_2 &= 10^{-0.42}\\bigg(\\frac{R_\\mathrm{conv}}{R_\\mathrm{donor}}\\bigg)^{7.5},\n\\end{align}\nwhere $R_\\mathrm{conv}$ is the radius of the convective core. The value for $E_2$ is computed from fitting formulae for H-rich stars derived by \\citet{2018A&A...616A..28Q}. For $({K}\/{T})_{\\mathrm{conv}}$, the definition is taken from \\cite{2002MNRAS.329..897H} to be\n\\begin{equation}\n    \\bigg(\\frac{K}{T}\\bigg)_{\\mathrm{conv}} = \\frac{2}{21}\\frac{f_\\mathrm{conv}}{\\tau_\\mathrm{conv}}\\frac{M_\\mathrm{env}}{M_{\\mathrm{donor}}} \\mathrm{yr}^{-1},\n\\end{equation}\nwhere $f_\\mathrm{conv} = {\\it min}~\\{1.0, (P_{\\rm tidal}\/(2\\tau_{\\rm conv}))^{2}\\}$ is a numerical factor and $P_{\\rm tidal}$ is the tidal pumping time-scale defined by $\\lvert 1\/P_{\\rm orb} - 1\/P_{\\rm spin}\\rvert$. Here, $\\tau_\\mathrm{conv}\\approx(MR^2\/L)^{1\/3}$ is the eddy turnover timescale in units of years \\citep{1996ApJ...470.1187R} and $M_\\mathrm{env}$ is the convective envelope mass. More details about our applied orbital angular momentum evolution, with spin-orbit coupling including tides, are discussed in Section~\\ref{ls-coup} and Appendix~\\ref{append1}.\n\n\n\\subsection{Angular momentum loss from the second Lagrangian point}{\\label{rl2}}\n\n\\begin{figure}[!ht]\n\\centering\n\\includegraphics[width=\\linewidth]{countour.pdf}\n\\caption{Equipotential lines of the Roche potential for a binary consisting of stars with mass ratio $q=0.26$ and binary separation $a$. The equipotential lines passing through Lagrangian points L$_1$, L$_2$ , and L$_3$ are shown.}\n\\label{contour}\n\\end{figure} \n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=\\linewidth]{countour_f.pdf}\n\\caption{Equipotential lines of the Roche potential for a binary consisting of stars with mass ratio $q=1.85$ and binary separation $a$ to illustrate the swapping of the positions of L$_2$ and L$_3$ when $M_{\\mathrm{donor}}\\leq M_{\\mathrm{acc}}$ (compared to Fig.~\\ref{contour}).}\n\\label{contour_f}\n\\end{figure} \n\n\\begin{figure*}[ht]\n\\centering\n\\includegraphics[scale=0.6]{L2_R2.pdf}\n\\caption{Our results form the volume integration of L$_2$ equipotential surfaces with respect to different mass ratios $q$. All distance values are presented in units of $R_{\\mathrm{L}_{1}}$. Here, $R_{\\mathrm{L}_2}$ is the radius of a sphere with volume equal to that of the L$_2$ equipotential surface (solid orange line). $D_{\\mathrm{L}_2}$ is the distance of the L$_2$ point from $M_{\\mathrm{donor}}$ (solid blue line). The dashed black line shows the orbital separation in units of $R_{\\mathrm{L}_{1}}$. Systems where the donor's radius exceeds any of these limiting radii are considered as undergoing dynamical instability.}\n\\label{L2_R2}\n\\end{figure*} \n\nLagrangian points are equilibrium points in space where the gravitational and centrifugal forces in the system cancel each other out. L$_1$, L$_2$, and L$_3$ are unstable equilibrium points from where a test particle, upon small displacement, would move further away. Figure~\\ref{contour} shows these unstable Lagrangian points for a system with a mass ratio of $q=0.26$.\n\nIn most cases of X-ray binaries, analysis has been done for mass transfer via L$_{1}$. L$_1$ lies in between the two stars (hence also known as the inner Lagrangian point) and the equipotential surface passing L$_1$ is known as the Roche lobe. When a star fills its Roche lobe, any material that crosses L$_1$ from one star will fall towards the other star. This transfer of matter either decreases the radius (for radiative envelopes) of the donor or increases it (for convective envelopes). In some cases of extreme binary mass ratio, the RLO might not be enough to provide efficient mass-transfer rate and the donor might extend far beyond its Roche lobe to reach the equipotential surface passing through L$_2$. However, contrary to when mass passes through L$_1$, the mass that crosses L$_2$ takes away a large amount of angular momentum from the binary. Once the outer layers of the donor reach L$_2$ (or the donor obtains a volume equivalent to that of the equipotential lobe passing through L$_2$; see below) it is expected that the binary orbit will shrink rapidly. We consider this the onset of dynamical instability. For illustration, the L$_2$ potential surface is shown in Fig.~\\ref{contour}. It is the peanut-shaped surface enclosing both binary mass components and passing through the point L$_2$. \n\n\\cite{1983ApJ...268..368E} calculated the stellar radius needed in order to initiate mass transfer from the inner Lagrangian point of a binary by calculating the radius of a sphere that will have the same volume as the Roche lobe. This radius for the donor star is referred to as the Roche-lobe radius ($R_{\\mathrm{donor},\\mathrm{L}_1}$). We take a similar approach to quantify overflow from L$_2$. In cases where mass transfer via the L$_1$ point is not sufficient to keep the donor star confined within its Roche lobe we assume that the expanding star needs to fill the entire volume enclosed by the L$_2$ equipotential surface before the onset of dynamical instability. $R_{\\mathrm{L}_2}$ is the volume equivalent radius for the equipotential surface passing through the L$_{2}$ point.\n\nAnother possibility of mass loss from L$_2$ point occurs when the radius of the donor star reaches the point L$_2$ before the donor volume overfills the L$_2$ potential surface. This case applies only when $q\\ge 1$ (i.e. $M_{\\mathrm{donor}}\\le M_{\\mathrm{acc}}$) as the point L$_2$ is much closer to the donor. We refer to the distance between the centre of the donor and L$_2$ as $D_{\\mathrm{L}_2}$. This case is illustrated in Fig.~\\ref{contour_f} which shows the equipotential surfaces  with mass ratio, $q=1.85$. In our simulations we assume that binaries experience stable mass transfer when the donor star does not cross any of the two limits discussed above at any point during each evolution (i.e. for stable RLO: $R_{\\rm donor}< {\\it min}~\\{R_{\\mathrm{L}_2},~D_{\\mathrm{L}_2}\\}$ for the entire binary evolution).\n\nTo calculate the L$_2$ volume via numerical integration, we begin by finding the Lagrange points (L$_1$, L$_2$ and L$_3$) for a particular mass ratio. Along the axis joining the centres of the $M_{\\mathrm{donor}}$ and $M_{\\mathrm{acc}}$, the entire volume is assumed to be a summation of thin discs. The volume of each disc slice is calculated going from one boundary end to the other using the boundaries of the L$_2$ equipotential surface, and the subsequent disc volumes are added together to cover the entire volume. In these calculations we consider two mass ratio regimes, $q < 1$ and $q\\ge 1$.\n\nWe use both $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ normalised to R$_{\\mathrm{donor},\\mathrm{L}_1}$ in order to remove the dependence on the binary orbital separation. Once these radii are calculated we find a fit of ${R_{\\mathrm{L}_2}}\/{R_{\\mathrm{donor},\\mathrm{L}_1}}$ and ${D_{\\mathrm{L}_2}}\/{R_{\\mathrm{donor},\\mathrm{L}_1}}$ on the mass ratio ($q=M_{\\rm acc}\/M_{\\rm donor}$) of the binary. For $q < 1$, we find that the $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ follow monotonically increasing trends toward $q = 1$ (shown in Fig.~\\ref{L2_R2} in the left panel), which is fitted by the following functions,\n\\begin{equation}\\label{eq1}\n    \\frac{R_{\\mathrm{L}_2} (q < 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 0.784~q^{1.05} e^{-0.188~q} + 1.004,\n\\end{equation}\n\\begin{align}\\label{eq2}\n    \\frac{D_{\\mathrm{L}_2}(q < 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 3.334~q^{0.514} e^{-0.052~q} + 1.308.\n\\end{align}\nHere, $R_{\\mathrm{donor},\\mathrm{L}_1}$ is also calculated from the volume of the L$_{1}$ equipotential surface using the method described above and the result is consistent with the calculations from \\cite{1983ApJ...268..368E}. \n\nFor the second mass ratio regime, we calculate $R_{\\mathrm{L}_2}(q\\ge 1)$  using\n\n\\begin{equation}\n    R_{\\mathrm{L}_2}(q) = R_{\\mathrm{L}_2}\\bigg(\\frac{1}{q}\\bigg) \\qquad \\forall\\; q>0.\n\\end{equation}\nWe find $D_{\\mathrm{L}_2}(q\\ge 1)$ using the distance between L$_2$ and $M_{\\mathrm{acc}}$ from the case $q < 1$. As seen in Fig.~\\ref{L2_R2} on the right, there is a sudden jump in the values of $D_{\\mathrm{L}_2}$: when crossing $q=1,$ it goes from around $4.45$ for $q=0.997$ to around $1.82$ for $q=1.003$. We fitted functions to the calculated $R_{\\mathrm{L}_2}$ and $D_{\\mathrm{L}_2}$ values as follows,\n\\begin{equation}\\label{eq3}\n    \\frac{R_{\\mathrm{L}_2}(q \\ge 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = 0.290~q^{0.829}~e^{-0.016~q} + 1.362,\n\\end{equation}\n\\begin{equation}\\label{eq4}\n    \\frac{D_{\\mathrm{L}_2}(q \\ge 1)}{R_{\\mathrm{donor},\\mathrm{L}_1}} = -0.040~q^{0.866}~e^{-0.040~q} + 1.883.\n\\end{equation}\n\nThe relative errors between our calculations and the corresponding fits are less than 1\\%. \n\nVolume equivalent L$_{2}$ radii calculations have also been done by \\cite{2016A&A...588A..50M} for $q<1$ but with an entirely different approach as these latter authors considered the case of overcontact binaries. They calculated the L$_{2}$ volume assuming that both the stars expand and fill their respective L$_2$ sub-volume. In order to test our numerical volume-integrating scheme against that of \\cite{2016A&A...588A..50M}, we split the L$_2$ volume at the L$_1$ point and compared the resulting calculations to their $R_{\\mathrm{L}_{2}}$~radii, finding good agreement. However, we should stress again that the approach by \\cite{2016A&A...588A..50M} is only applicable to overcontact binaries. Our work was followed by \\cite{2020arXiv200600774G} who, in a different context, derived fits to the volume equivalent L$_{2}$ radius using a similar approach.\n\n\n\\section{Results}\\label{sec:results}\n\nWe explore the evolution of LMXBs\/IMXBs with different initial conditions,  taking into account the physics described in the earlier sections. In both our grids (for NS masses of 1.3~M$_{\\odot}$ and 2.0~M$_{\\odot}$ respectively) the binaries that interacted via mass transfer can  undergo either stable or unstable mass transfer, excluding the systems with P$^i_\\mathrm{orb} \\lesssim 0.50$~days where the donor star already overflows its Roche lobe at ZAMS which we do not further evolve. The superscript $i$ stands for initial values which corresponds to the orbital parameters at the onset of RLO. We flag systems as `stable' when either the donor has detached from its Roche lobe at the end of mass transfer or the hydrogen in the outer layer has been almost completely removed (remaining hydrogen in the outer layer $<0.005$~M$_\\odot$). Donors in stable binaries formed a WD at the end of the mass-transfer sequence resulting in a neutron star--white dwarf (NS--WD) binary. The `unstable' binaries underwent the onset of L$_{2}$ overflow.\n\n\\begin{figure}[t]\n    \\vfill\n        \\includegraphics[width=\\linewidth]{rlo_cases1.pdf}\n        \\caption{Allowed initial parameter space for LMXBs\/IMXBs to undergo stable mass transfer with $1.3$~M$_\\odot$ accretors (orange squares) and $2.0$~M$_\\odot$ accretors (blue squares). Grey squares correspond to systems that encountered dynamical instability. The dark red squares towards the left edge of the figure correspond to systems that never initiated RLO. The lower dashed white line separates systems undergoing case A mass transfer from those undergoing case B mass transfer (same boundary in both grids). The middle dotted white line separates early case B (where the donor has yet to form a deep convective envelope) from late case B RLO (where donor has formed a deep convective envelope at onset of RLO). The upper dashed white curve encloses systems that undergo case C mass transfer (same boundary in both grids). Green stars correspond to those systems that undergo a second mass-transfer phase from a stripped Helium-giant star, i.e. case BB RLO. The three red crosses in case BB correspond to systems that terminated due to numerical issues.}\n        \\label{rlo_cases1}\n\\end{figure}\n\nIn Fig.~\\ref{rlo_cases1} we present this allowed initial parameter space for LMXBs\/IMXBs to undergo stable mass transfer for grids containing $1.3$~M$_\\odot$ (orange squares) and $2.0$~M$_\\odot$ NS accretors (blue squares) along with the unstable sequences (grey squares). The dark red squares towards the left are the systems that never initiated RLO. The general shape of the stable region resembles the work done by \\cite{2000ApJ...530L..93T} where they explored the allowed parameter space to form binary millisecond pulsars while avoiding a CE phase. The different dashed white lines separate the grids based on the type of mass-transfer phase that the system underwent. Case A is when the donor is on the main sequence at the onset of mass loss, that is, it is burning hydrogen in the core. Case B is when the donor has exhausted H in its core and H-shell burning phase (post-main sequence). The threshold between the two cases A and B (lower dashed white line in figure) depends more on the initial orbital period than the donor mass; the limit for case A being in the range $2.0$--$2.8$~days for both grids. The middle dashed white line separates two subsets of case B RLO: early case B and late case B. This threshold depends almost linearly on both the initial orbital period and the initial donor mass. At RLO onset, if the donor has a radiative envelope it is termed early case B and if the donor has developed a convective envelope it is termed late case B RLO. The upper dashed white curve encloses systems that undergo case C RLO, which means the donor has exhausted He in its core at the onset of RLO. \n\nThe stability region increases for higher accretor mass to include higher donor masses and orbital periods. A similar effect of increase in the parameter space with increasing the accretor mass was obtained by \\cite{2012ApJ...756...85S} for the formation of recycled pulsars from IMXBs and LMXBs. The overall shape of the parameter space for both the grids depends a lot on the structure of the donor envelope and the response of the donor to mass loss. A radiative envelope would shrink on mass loss and contribute to the stability of the binary while a convective envelope would expand rapidly on mass loss and make the system increasingly unstable. \\cite{1999ApJ...519L.169K} showed that mass transfer on a thermal timescale would avoid the CE phase in a binary as long as the envelope is mostly radiative. Case A and early case B have radiative envelopes, with binaries of initial mass ratio greater than $\\sim 0.28,$ undergoing stable mass-transfer depending on the initial orbital period. In contrast, late case B and case C have convective envelopes at RLO with the stability region being defined by a fixed critical mass ratio which we find to be at around $\\sim 0.5$. \\cite{2014A&A...571A..45I} studied the binary evolution in close LMXBs and also found an increase in the depth of the convective envelope with increasing P$^i_\\mathrm{orb}$ (Fig.~4 in their paper).\n\nIn Fig.~\\ref{rlo_cases1}, green stars correspond to systems that go through case~BB RLO, which is when a binary, after having lost its hydrogen envelope in a case~B mass-transfer phase, detaches and evolves as a stripped helium star and initiates a second RLO phase during the helium-shell burning stage. This subset of case B occurs for only a small part of the parameter space (and only for the grid with a $2.0$~M$_{\\odot}$ accretor). This is because low-mass helium stars ($<0.8\\;M_\\odot$) do not expand by any significant amount  \\citep[e.g.][]{heu16,2018MNRAS.481.1908K} and thus only donor stars $\\ga 5.8\\;M_\\odot$ leave behind stripped helium stars massive enough to eventually lead to case~BB RLO. The three red crosses in case BB correspond to systems that terminated due to numerical issues.\n\nDuring the RLO phase, all LMXBs\/IMXBs are expected to be bright X-ray binaries, and in many cases the mass-transfer rate from the donor star to the vicinity of the NS can exceed the Eddington limit ($L_{\\rm Edd}$) significantly. As described in Section~\\ref{acc}, we only allow accretion onto the NS up to the Eddington limit, but we do consider the transition from a thin accretion disc to an inner thick disc model when the mass-transfer rate supplied from the donor star exceeds the Eddington limit \\citep{1973A&A....24..337S}, as well as the geometric beaming model proposed by \\cite{2001ApJ...552L.109K}. \n\n\\begin{figure}[t]\n    \\vfill\n    \\centering\n        \\includegraphics[width=\\linewidth]{mdot.pdf}\n        \\caption{Estimated observed X-ray luminosity for a binary with M$^i_\\mathrm{donor} =2.77$~M$_{\\odot}$, M$^i_\\mathrm{acc} =1.30$~M$_{\\odot}$, and P$^i_\\mathrm{orb} =5.38$~days under different assumptions. The solid green curve is the accretion luminosity corresponding to the mass-accretion rate onto the NS, assuming a fixed radiative efficiency ($\\eta=0.1$). The solid orange curve is the accretion luminosity corresponding to a super-Eddington accretion disc following the \\cite{1973A&A....24..337S} disc model given by Eq.~(\\ref{edd_eq}). The solid blue curve is the isotropic-equivalent accretion luminosity corresponding to beamed super-Eddington emission following the \\cite{2001ApJ...552L.109K} geometric beaming model (limited at $10^{42}$~erg~s$^{-1}$) given by Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming}). Four reference luminosity values, corresponding to $L_{\\rm Edd}$ (solid black line), $10\\,L_{\\rm Edd}$ (dashed black line),  $100\\,L_{\\rm Edd}$ (dotted black line), and $1000\\,L_{\\rm Edd}$ (dot-dashed black line), are also plotted. The initial properties of this binary are highlighted with a magenta star in Fig.~\\ref{peak_acc}.}\n        \\label{mdot}\n\\end{figure}\n \nFigure~\\ref{mdot} shows an example of a stable system with M$^i_\\mathrm{donor} =2.77$~M$_{\\odot}$, M$^i_\\mathrm{acc} =1.30$~M$_{\\odot}$, P$^i_\\mathrm{orb} =5.38$~days, where we demonstrate the estimated observed X-ray luminosity under different assumptions. The solid green curve is the accretion luminosity corresponding to the amount of mass accreted by the NS (i.e. capped at exactly the Eddington limit), assuming a radiative efficiency following Eq.~(\\ref{eta}). The solid orange curve is the accretion luminosity corresponding to a super-Eddington accretion disc following Eq.~(\\ref{edd_eq}), where although the Eddington limit is locally satisfied at every point in the disc, the integrated luminosity of the disc can exceed the Eddington limit by a small factor. Finally, the solid blue curve is the isotropic-equivalent accretion luminosity corresponding to beamed super-Eddington emission following Eq.~(\\ref{new_edd}) where the estimated observed luminosity can exceed the Eddington limit by up to a few orders of magnitude. \nFor comparison, we mark with horizontal lines the luminosity of ULXs corresponding to $L_{\\rm Edd}$ (solid black line), $10\\,L_{\\rm Edd}$ (dashed black line),  $100\\,L_{\\rm Edd}$ (dotted black line), and $1000\\,L_{\\rm Edd}$ (dot-dashed black line). For simplicity, we use Eq.~(\\ref{Eq:Edd}) which is fixed for an initial accretor mass, because the amount of mass accreted does not change the accretion luminosity significantly. We note that since the prescription used for beamed emission (Section~\\ref{acc}) is an approximation, we fix an upper limit for the calculated isotropic-equivalent accretion luminosities at $10^{42}$~erg~s$^{-1}$ in order to avoid unphysically high values. In any case, the time that our binaries spend at such high luminosities combined with the inferred very small beaming factors make binaries on that phase effectively non-detectable. In the remainder of the paper, we use the beamed model (i.e. equivalent to the blue curve based on Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming})) as our estimate of the observed X-ray luminosity of the binaries.\n\n\\begin{figure*}[t]\n\\centering\n    \\vfill\n        \\includegraphics[width=1.0\\textwidth]{accreted_integrated.pdf}\n        \\caption{(Left) Time-averaged isotropic-equivalent accretion luminosities ($\\langle L_{\\rm acc}^{\\rm iso}\\rangle$) in LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS accretor that reached an instantaneous luminosity above $10\\,L_{\\rm Edd}$. These luminosities were calculated based on Eqs.~(\\ref{new_edd}) and (\\ref{eq:beaming}) for each binary and at each time-step, and averaged over the entire RLO lifetime. Grey colour denotes the sequences that never achieved an instantaneous luminosity above $10\\,L_{\\rm Edd}$. The systems that reach the highest accretion luminosities correspond to unstable systems, on the higher donor mass ends in both panels, before the onset of dynamical instability. The stable systems (enclosed by the solid black boundary) have a relatively low accretion luminosity  on average. The magenta star corresponds to the binary shown in Fig.~\\ref{mdot}. (Right) Same as left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs. The three red crosses correspond to systems that terminated due to numerical issues.}\n        \\label{peak_acc}\n\\end{figure*}\n\nFigure~\\ref{peak_acc} shows the time-averaged isotropic-equivalent accretion luminosities with respect to the initial parameters in both the grids. The magenta star is the binary shown in Fig.~\\ref{mdot}. For accreting NSs of either $1.3\\;$M$_\\odot$ or $2.0\\;$M$_\\odot$, we find time-averaged isotropic-equivalent X-ray luminosities of $\\langle L_{\\rm acc}^{\\rm iso}\\rangle \\simeq 10^{36}-10^{41}\\;{\\rm erg\\,s}^{-1}$ (in some cases even up to $10^{42}\\;{\\rm erg\\,s}^{-1}$).\nThese luminosities were calculated following Eq.~(\\ref{new_edd}) and averaged over the entire RLO lifetime for systems that reached instantaneous luminosity above $10\\;L_{\\rm Edd}$. The stable systems are enclosed within the solid black boundary (for this and all subsequent figures). Comparing Fig.~\\ref{rlo_cases1} with Fig.~\\ref{peak_acc}, even the systems that undergo dynamical instability (L$_2$ overflow) are included. This is because, when using detailed binary evolution calculation, we are able to resolve the onset of the dynamical instability which is not instantaneous and is most often preceded by a short but intense phase of mass transfer. In fact, the systems that reach the highest luminosities correspond to the unstable systems on the higher donor-mass end in both panels of Fig.~\\ref{peak_acc}. Therefore, binaries that would suffer dynamical instabilities and coalesce (likely producing Thorne-\\.Zytkow objects hypothesised by \\cite{1977ApJ...212..832T}) could also be observed as NS ULXs earlier in their evolution. However, the lifetimes of these binaries as ULXs is very short, and so this would act against their detectability.\n\n\\begin{figure*}[t]\n    \\vfill\n    \\centering\n        \\includegraphics[width=\\linewidth]{ulx_time1.pdf}\n        \\caption{(Top row) ULX lifetime for LMXB\/IMXB systems with a NS accretor mass of $1.3$~M$_{\\odot}$. Left, middle, and right panels show the time that systems spent with $L_{\\rm acc}^{\\rm iso}$ above $10$, $100$, and $1000\\,L_{\\rm Edd}$, respectively. White arrows (middle panels) enclose the potential properties of M82~X-2 at the onset of RLO (see Sections~\\ref{m82x2} and \\ref{m82x2b}). (Bottom row) Same as the top row but for LMXB\/IMXB systems with a NS accretor mass of $1.3$~M$_{\\odot}$.}\n        \\label{result_1}\n\\end{figure*}\n\nWe define three X-ray luminosity ranges  >$10\\,L_{\\rm Edd}$, >$100\\,L_{\\rm Edd}$, and >$1000\\,L_{\\rm Edd}$, and calculate how long each system spends in each luminosity range. We refer to this time duration as ULX lifetime. The defined luminosity ranges are based on the observed pulsating ULX luminosities, which are in the range of $10$--$1000\\, L_{\\rm Edd}$ (Section \\ref{sec:observations}). We compare the isotropic-equivalent accretion luminosities with these luminosity ranges. Figure~\\ref{result_1} presents these results for LMXBs\/IMXBs with $1.3$~M$_{\\odot}$ and $2.0$~M$_{\\odot}$ accretors. In systems with $1.3$~M$_{\\odot}$ NSs, the longest ULX lifetime is $1.6\\times 10^6$~years, corresponding to $M^i_\\mathrm{donor} = 2.3$~M$_\\odot$ and $P^i_{\\mathrm{orb}}=2.16$~days, for observed luminosities of >$10\\,L_{\\rm Edd}$. In systems with $2.0$~M$_{\\odot}$ NSs, the longest ULX lifetime is $1.1\\times 10^6$~years, corresponding to $M^i_\\mathrm{donor}=2.77$~M$_\\odot$ and $P^i_{\\mathrm{orb}}= 2.16$~days, again for observed luminosities of >$10\\,L_{\\rm Edd}$. The upper limits to ULX lifetimes are comparable to the ULX age estimate of $\\sim~1$~Myr for NGC~1313~X-2 by \\cite{2008AIPC.1010..303P} (Section~\\ref{1313}). Looking at similar initial donor masses and initial orbital periods in both sets of LMXBs\/IMXBs (going from top to bottom row in Fig.~\\ref{result_1}), higher accretor mass corresponds to a much higher ULX lifetime as the stability area increases. As an example, for $M^i_\\mathrm{donor}=5.0$~M$_\\odot$ and $P^i_{\\mathrm{orb}}=1.0$~days, the ULX time increases from $3.0\\times 10^4$~years to $3.0\\times 10^5$~years (going from lower to higher NS accretor mass).\n\nIn Fig.~\\ref{result_1} (top row), from the leftmost panel to the rightmost (from >$10\\,L_{\\rm Edd}$ to >$1000\\,L_{\\rm Edd}$) the systems which initially have long ULX lifetimes ($M^i_\\mathrm{donor}\\sim 2.3$~M$_\\odot$ and $P^i_{\\mathrm{orb}}\\sim 2.0$~days) either no longer appear on the plot or have a smaller ULX lifetime. Their isotropic-equivalent luminosities barely reach the higher cutoff values. Similarly, in the bottom row of Fig.~\\ref{result_1}, most LMXBs\/IMXBs with long ULX lifetimes ($M^i_\\mathrm{donor}\\sim 2.77$~M$_\\odot$ and $P^i_{\\mathrm{orb}}\\sim 2.0$~days) for >$10\\,L_{\\rm Edd}$, do not reach luminosities >$1000\\,L_{\\rm Edd}$. This implies that binaries that are in the ULX phase for the longest time do not always achieve the highest luminosities. This effect is not seen in the unstable systems, which maintain their short ULX lifetime across the different ULX criteria. There is a part of the stable parameter space where the ULX lifetime decreases significantly, the ULX lifetime going from about $10^5$~years for systems outside this region to $10^1$--$10^2$~years (the boundary corresponding to $M^{i}_{\\rm donor}<3.0$~M$_{odot}$ and $P ^{i}_{\\rm orb}<2.0$). Their accretion luminosity is sub-Eddington for almost the entire mass-transfer phase. After a mass-transfer episode, these systems are left with a He core and thin H envelope which expands briefly during H-shell burning, which is enough to increase their luminosity to exceed the Eddington limit albeit for a very short period of time.\n\nMost NS ULXs observed so far have been in the higher luminosity range ($100$--$1000\\;L_{\\rm Edd}$) potentially due to selection effects as more luminous ULXs have a higher chance of being observed. On the other hand, the short lifespans of NS ULXs compared to the age of the universe ($\\sim 1.4\\times 10^{10}$~yr) implies that it is unlikely to observe these systems in large numbers, which is consistent with the existing small NS ULX sample. The extra physical argument that the emission is highly beamed, especially for very luminous ULXs, further decreases the chances of detecting a large number of NS ULXs.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=\\linewidth]{likelihood.pdf}\n    \\caption{(Top row) Relative likelihood ($\\mathscr{L}$) to observe a system as a ULX, as described by Eq.~(\\ref{beam_eq}), for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS. The panels are arranged as in Fig.~\\ref{result_1}. Going from the leftmost to the rightmost panel  ($>10$ to >$1000\\,L_{\\rm Edd}$), $\\mathscr{L}$ is higher for stable binaries with lower luminosities as their emission is not as strongly beamed. There is in addition some effect by the transition from case~A to case~B RLO. All values below $10^{-6}$ are shown in black as they correspond to insignificant likelihood. (Bottom row) Same as the top row but for LMXBs\/IMXBs with a $2.0$~M$_\\odot$ NS.}\n    \\label{likelihood}\n\\end{figure*}\n\nAs alluded to above, the probability of observing any of these LMXB\/IMXB sources as ULXs depends on the beaming factor, the ULX lifetime, and the probability of the particular system being formed. We evaluate the first two factors in the following equation describing the likelihood of a ULX observation, which is the beaming factor integrated over the three defined ULX lifetimes:\n\\begin{equation}\\label{beam_eq}\n    \\mathscr{L} = \\int_{\\mathrm{ULX}} b(t) dt.\n\\end{equation}\nCalculating the likelihood using this equation for both the grids and for all three ULX lifetime criteria, we obtain a measure of the relative chance of observing each system in Fig.~\\ref{likelihood} for the $1.3$~M$_\\odot$ NS (top row) and the $2.0$~M$_\\odot$ NS (bottom row) grids. All values below $10^{-6}$ are shown in black as they correspond to insignificant likelihood. The highest likelihood is for the stable systems with lower luminosities ($10\\;L_{\\rm Edd}$), where the mass-transfer rate never increases to extreme values and thus collimation is very small to almost negligible. Figure~\\ref{likelihood} shows that the chance of a system being observed as a ULX decreases at higher luminosities. This is due to the fact that for higher luminosities the mass-accretion rates (and the mass-transfer rates) are extremely high which causes the emission from the system to be highly beamed. Also, the probability depends on the structure of the donor at the onset of RLO. Case A systems have lower mass-transfer rates and more isotropic emission and thus slightly higher probabilities of being observed than case~B. As mentioned before, there is a part in the stable parameter space where the ULX lifetime drops by many orders of magnitude (Fig.\n~\\ref{result_1}). Looking at the same systems in Fig.~\\ref{likelihood}, the likelihood to observe them is negligible. These systems are super-Eddington for a very brief moment in time which is to the detriment of their detectability. Looking at Figs.~\\ref{result_1} and \\ref{likelihood}, it is evident that even though LMXBs are included in the initial parameter space, IMXBs (with donor masses $\\gtrsim 2.0$~M$_{\\odot}$) are better candidates for NS ULXs.\n\nOne caveat is that we assume that all systems have an equal probability of formation whereas in reality many systems might be formed at a higher rate than others. To account for these effects we would need to do population synthesis studies which would include exploring the formation probability of close NS + main sequence binaries and the distribution of their binary parameters. However, this is outside the scope of this study. We intend to study the formation rate of these systems in a future work.\n\nFurthermore, one should always be careful when directly comparing our results to the observed population as our models give predictions about the whole population of ULXs with NS accretors, and not only the pulsating ones. The conditions for a NS to produce coherent pulses while also reaching super-Eddington luminosities are still unclear. If for example a strong magnetic field is required then perhaps the pulses are only observable at the beginning of the mass-transfer phase before any significant amount of material is accreted onto the NS, which could bury the magnetic field.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{wd.pdf}\n\\caption{(Left) Resulting distribution of final WD masses for LMXBs\/IMXBs with $1.3$~M$_\\odot$ NSs. The He~WDs are the least massive, followed by hybrid WDs, and the CO~WDs are the most massive. The outermost black boundary encloses the stable systems and the inner boundaries differentiate between the different WD types. (Right) Same as left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs.}\n\\label{wd}\n\\end{figure*}\n\nIn cases where the mass-transfer sequence is stable throughout the binary evolution, we expect a NS--WD binary to be formed. Figure~\\ref{wd} shows the type of white dwarf formed at the end and the final white dwarf masses for the $1.3$~M$_\\odot$ NS (left panel) and the $2.0$~M$_\\odot$ NS (right panel) grids. The superscript $f$ stands for final values. Overall the white dwarfs resulting from the $1.3$~M$_{\\odot}$ NS grid are in the mass range of $0.23$--$0.71$~M$_{\\odot}$. For systems with $2.0$~M$_\\odot$ NS accretors,  the white dwarfs have an overall mass range of $0.25$--$0.95$~M$_{\\odot}$. We used final total mass fractions of carbon to distinguish between different WD types. We define WDs with $>95\\%$ carbon mass fraction as CO white dwarfs, $0.01$--$95\\%$ as hybrid white dwarfs, and $<0.01\\%$ as He white dwarfs. The initially higher donor masses and longer orbital periods result in a degenerate CO core with negligible helium on the surface. For some donor masses and orbital periods, the donor forms a degenerate CO core with a relatively large helium-rich envelope leading to hybrid WDs. In systems with low donor masses and short orbital periods, the donors end up as helium WD systems. The reason for the final fate of the latter class of systems is that  due to deep envelope stripping early in the evolution of the donor, combined in some cases with the low initial mass of the donor, the final stripped helium cores are not massive enough to ignite helium.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{ns.pdf}\n\\caption{(Left) Mass accreted by the NS ($M_{\\rm acc}^{f}-M_{\\rm acc}^I$) for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS and $\\beta = \\mathrm{max}[0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}]$. Binaries producing the lowest WD masses show the highest accretion on the NS. (Right) Same as left but for LMXBs\/IMXBs with a $2.0$~M$_\\odot$ NS. The systems below the white boundary are those where the accretor has accreted enough mass to exceed the maximum mass limit for a NS, here assumed to be $2.17\\;M_\\odot$.}\n\\label{ns}\n\\end{figure*}\n\nFigure~\\ref{ns} shows the mass accreted by the NSs, for systems with $1.3$~M$_\\odot$ (left) and $2.0$~M$_{\\odot}$ NS (right) accretors. We should reiterate here our assumption of non-conservative mass transfer with $\\beta = \\mathrm{max}[0.7, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}]$ and that the accretion by the NS is Eddington limited. The $1.3$~M$_\\odot$ NSs end up accreting $0.004$--$0.415$~M$_{\\odot}$ of mass by the end of the mass-transfer phase, the systems with values around $M_{\\mathrm{donor}}^i = 2.0$~M$_\\odot$ and $P_{\\rm{orb}}^i = 0.87$~day accreting the most. The amount of mass accreted is not enough to collapse the NS. The $2.0$~M$_{\\odot}$ NSs accrete mass in the range of $0.002$--$0.585$~M$_{\\odot}$, with the systems with values around $M_{\\mathrm{donor}}^i = 3.0$~M$_\\odot$ and $P_{\\rm{orb}}^i = 3.1$~days accreting the highest amount. We compare our results to the maximum NS mass known. The most massive precisely measured NS is PSR~J0348+0432 with a mass of $2.01\\pm0.04\\;M_\\odot$ \\citep{2013Sci...340..448A}. However, recently, a candidate with a higher NS mass of $2.17^{+0.11}_{-0.10}\\;M_\\odot$ was announced \\citep[PSR~J0740+6620;][]{2019arXiv190406759C}. Although the error bar of the latter source is relatively large, we take $2.17\\;M_\\odot$ as our assumed upper limit. This value is also supported by constraints on GW170817 based on combined gravitational wave and electromagnetic observations \\citep{2017ApJ...850L..19M}.\n\nLooking at the final NS masses in Fig.~\\ref{ns} (right panel), we see a certain population enclosed by a white boundary. In these systems the accretor has accreted enough material to overcome the neutron degeneracy pressure that is supporting the star against gravitational collapse (assuming an upper NS mass limit of $2.17\\;M_\\odot$). Therefore, according to the maximum NS mass limit assumed, these NSs will collapse to form BHs. \n\nNet accretion on the NS can be broadly described by,\n\\begin{equation}\n    \\Delta M_{\\mathrm{acc}} = \\langle\\dot{M}_{\\mathrm{acc}}\\rangle \\times \\Delta t_{\\mathrm{X}},\n\\end{equation}\nwhere $\\Delta t_{\\mathrm{X}}$ is the lifetime as an X-ray binary (or the overall mass-transfer phase) and $\\langle\\dot{M}_{\\mathrm{acc}}\\rangle$ is the average accretion rate onto the NS. A high amount of accreted mass is achieved with a combination of both a large accretion rate and a long time duration over which it occurs. \nFor LMXBs\/IMXBs that reach up to Eddington mass-transfer rates for only a small part of the mass-transfer phase, to zeroth order, $\\Delta M_{\\mathrm{acc}}\\simeq 0.3\\times \\Delta M_{\\rm donor}$. In contrast, for the brightest ULXs, where most of the mass-transfer happens at a highly super-Eddington rate, $\\langle\\dot{M}_{\\mathrm{acc}}\\rangle \\simeq \\dot{M}_{\\rm Edd}$ and $\\Delta M_{\\mathrm{acc}} << \\Delta M_{\\rm donor}$. Even if the accretion onto the NS was allowed to reach a few times the Eddington limit, the net amount of accreted material by the NS in the brightest ULXs would still be small.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=\\linewidth]{pd.pdf}\n\\caption{(Left) Final orbital periods $(P^{f}_{\\mathrm{orb}})$ for LMXBs\/IMXBs with a $1.3$~M$_\\odot$ NS. The longest final orbital periods result from initial periods $\\gtrsim 10.0$~days and donors with masses less than about 2.75~M$_{\\odot}$. The shortest final orbital periods are for systems that formed the heaviest WDs, originating from donors of $\\gtrsim 3.5$~M$_{\\odot}$. (Right) Same \nas left but for LMXBs\/IMXBs with $2.0$~M$_\\odot$ NSs. }\n\\label{pd}\n\\end{figure*}\n\nThe final orbits in most systems that underwent stable mass transfer are wide because during the mass-transfer phase, after the orbit shrinks initially, the donor evolves to become less massive than the accretor causing the system to widen significantly. These widened orbits can be seen in Fig.~\\ref{pd} which shows that the final orbital periods can be as large as $5$--$7$ times the initial orbital periods. Some of these binaries may be observable by Gaia. \\cite{2019ApJ...886...68A} postulate that Gaia can detect and measure the properties of hidden wide binaries with WD and NS components using astrometric observations. The comparison of NS--WD observations to final orbital parameters from our calculations might help in understanding which binaries have undergone a super-Eddington mass-transfer phase in the past and could have been observed as ULXs, as well as constrain binary evolution physics such as the accretion efficiency.\n\n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection{Comparison to observations of M82~X-2}\\label{m82x2b}\nSince M82~X-2 has fairly well constrained parameters (see Section~\\ref{m82x2} and Table~\\ref{tab:obs_data}), we can compare it to our results and see if our results help to explain the observations. We look at the middle panels in both rows in Figs.~\\ref{result_1} and \\ref{likelihood}. Since the currently observed donor is more massive than the accretor, the orbit will shrink as mass is lost following the orbital angular momentum balance equation as described in Eq.~(\\ref{analytical}) for a non-conservative mass-transfer phase. Therefore, we use the observed parameters as lower limits of the initial binary configuration.\n\nWe only consider the luminosity range of $100\\,L_{\\rm Edd}$ because it is comparable to the observed luminosity of about $1.8\\times 10^{40}$~erg~s$^{-1}$. \\cite{2014Natur.514..202B} assumed an accretor of $1.4$~M$_{\\odot}$ and estimated the donor to be $\\gtrsim 5.2$~M$_{\\odot}$. For a $1.3$~M$_{\\odot}$ accretor mass, the donor mass is $\\gtrsim 5.1$~M$_{\\odot}$, using the same binary mass function. The initial parameter estimates are enclosed by the white arrows in Figs.~\\ref{result_1} and \\ref{likelihood} (middle panels in both rows). The potential initial systems have ULX lifetimes as long as $0.7\\times 10^4$ years and a relative peak likelihood of $0.7\\times 10^{-4}$. However, if we assume a higher accretor mass of $2.0$~M$_{\\odot}$, the donor mass is $\\gtrsim 5.83$~M$_{\\odot}$. The longest ULX lifetime in this case for the potential initial systems is about $1.1\\times 10^5$ years with a peak likelihood of $0.7\\times 10^{-2}$. Taking these numbers at face value, it is clear that an initially heavy NS is preferred in order to explain the currently observed properties of M82~X-2. This cannot be excluded as the $1.4$~M$_{\\odot}$ NS used by \\cite{2014Natur.514..202B} was an assumption. We should note, however, that for both NS masses, the peak of the relative likelihood does not lie very close to the observed limits we have for the current properties of M82~X-2. This is not necessarily problematic, as in order to calculate the actual probability density distribution of what the properties of NS ULXs ought to be, based on our model, we need to multiply the relative likelihood shown in Fig.~\\ref{likelihood} with the `prior' probability of forming a NS binary with these initial conditions. This convolution might significantly shift the peak of the resulting probability distribution. We leave this calculation for future work. \\cite{2015ApJ...802L...5F}, who followed such an approach, estimated the most probable initial donor mass of any NS ULX to be $3.0$--$8.0$~M$_{\\odot}$, and the initial orbital period to be $1.0$--$3.0$~days. This parameter space lies within our results.\n\n\\subsubsection{Comparison with high-mass X-ray binary ULXs}\n\nWith our work we aim to explore the possibility of an LMXB or IMXB origin for pulsating ULXs. However, there are at least three known NS ULXs which are HMXBs, namely NGC~7793~P13, M51~ULX-7, and more recently NGC 300 ULX1.\n\nNGC~7793~P13 has a luminosity of about $100\\;L_{\\rm Edd}$. With the presence of an observed high-mass donor ($>18\\;M_\\odot$, Section~\\ref{7793} and Table~\\ref{tab:obs_data}), this system cannot be explained by our LMXB\/IMXB models. In our NS ULXs, extreme mass ratios produce unstable binaries which reach the ULX luminosity observed by the source for a timescale of $\\sim 10^2$--$10^4 $~years before initiating L$_2$ overflow. However, this system is a HMXB with a donor of mass $18.0$~M$_{\\odot}<M_{\\mathrm{donor}}<23.0$~M$_{\\odot}$ and is outside the range of our simulations. \n\nM51~ULX-7 is another HMXB in a ULX phase and containing an accreting NS. \\cite{2019arXiv190604791R} suggest that the binary consists of an OB giant star ($M_{\\mathrm{donor}}\\lesssim 8.0$--$13.0$~M$_{\\odot}$) and a NS with a magnetic field of $10^{12}$--$10^{13}$~G. The most recent NS ULX observed to have a super-giant donor is NGC 300 ULX1, with a donor star mass of $M_{\\mathrm{donor}}\\gtrsim 8.0$~M$_{\\odot}$ \\citep{2019arXiv190902171H}. These two systems are also not satisfactorily explained by our calculations, since we find such systems have a ULX lifetime of $\\lesssim10^4$~years (for donors of $8.0$~M$_{\\odot}$).\n\n\\cite{2019arXiv190304995Q} explored the stability of super-giant X-ray binaries with a very high donor mass  compared to accretor mass (donors up to 20 times more massive than the accretors), and found that the ULX phase could be long lasting ($\\sim 0.4\\times 10^{6}$~years) for such an extreme mass ratio if the super-giant star has a H\/He gradient in the layers beneath its surface. These latter authors demonstrated that such systems can evolve on a nuclear timescale with a BH or NS accretor, even for binaries where the donor mass is up to 20 times the accretor mass. In their binary evolution models, the donor stars rapidly decrease their thermal equilibrium radius and can therefore cope with the inevitably strong orbital contraction imposed by such a high mass ratio.\nThese binaries could be post-CE systems, where the super-giant donor star has lost most of its hydrogen-rich envelope and the remaining one is enriched in helium. Recent 1D hydrodynamical simulations of a CE phase between a super-giant donor and a NS predict the formation of such binary configurations \\citep{2019ApJ...883L..45F}.\n\n\\subsection{Effect of spin-orbit coupling on orbital evolution}\\label{ls-coup}\n\nAs we already described in Section~\\ref{sync_time}, our binary evolution calculations take into account the effect of spin-orbit coupling by modelling the tidal interactions, internal rotation, and mass transfer and\/or loss. Here we want to quantify the effect of spin--orbit coupling on the orbital evolution and hence on the stability of mass transfer. We perform tests to calculate the orbital evolution in some limiting cases  and compare our numerical calculations with others available in the literature, as well as analytical solutions.\n\nFor simplicity, we assume a fully non-conservative mass transfer ($\\beta = 1.0$) in the following calculations, i.e. no mass is accreted by the NS. Also, we assume that the binary remains always in synchronous rotation with its components so that the spin period of the donor is the same as the orbital period. This assumption corresponds to an infinitely efficient spin--orbit coupling. Under these assumptions, we obtain the analytical solution for the orbital evolution as follows (see Appendix~\\ref{append1} for the detailed derivation):\n\\begin{align}\\label{analytical}\n    \\nonumber \\frac{\\dot{a}}{a} = \\bigg[\\frac{M_{\\mathrm{donor}} M_{\\mathrm{acc}} a^{2}}{M_{\\mathrm{donor}} M_{\\mathrm{acc}} a^{2} - 3 I_{\\mathrm{donor}} M}\\bigg]\\times \\bigg[\\dot{M}_{\\mathrm{donor}} \\bigg(\\frac{2M_{\\mathrm{donor}}}{M_{\\mathrm{acc}}M} - \\\\ \\frac{2}{M_{\\mathrm{donor}}} +\n    \\frac{1}{M}\\bigg) - \\bigg(\\frac{2\\dot{I}_\\mathrm{donor}M + I_\\mathrm{donor}\\dot{M}_{\\mathrm{donor}}}{M_{\\mathrm{donor}} M_{\\mathrm{acc}} a^{2}}\\bigg)\\bigg],\n\\end{align}\nwhere $I_{\\rm donor}$ is the moment of inertia of the donor.\nWhen the system loses mass from the vicinity of the accretor, according to Eq.~(\\ref{analytical}) the orbit shrinks. This decrease occurs as long as $M_{\\mathrm{donor}}\\gtrsim 1.28\\;M_{\\mathrm{acc}}$ \\citep{2001nsbh.conf..337T}. The orbit expands if mass is being lost from the approximate vicinity of the more massive star.\n\nWe run models using MESA, including angular momentum losses from spin--orbit coupling and mass lost from the system, while ignoring gravitational wave radiation and magnetic braking. The results of this test are shown in Fig.~\\ref{test1} (blue and orange curves), and they are compared to the analytical solution of Eq.~(\\ref{analytical}) (solid and dashed black curves), where $I_\\mathrm{donor}$ and $\\dot{I}_\\mathrm{donor}$ are taken from numerical stellar structure calculations. For two extreme cases of spin--orbit coupling, we first consider the case where the donor star is an extended body ($I_\\mathrm{donor}\\neq 0$) and there is efficient coupling of donor spin and orbit and thus an efficient transfer of angular momentum between spin and orbit (orange curve in the figure). This keeps the spin of the donor synchronised with the orbit throughout the binary evolution and the binary is `tidally locked'. The second case is where both binary components are approximated as point masses and there is no exchange of angular momentum between the donor (blue curve in the figure) and the orbit ($I_\\mathrm{donor}=0$). The two cases described above are extremes of spin--orbit coupling; any realistic solution would be in-between them. The MESA calculations are consistent with an analytic solution and the coupling does have a slight effect on the orbital evolution.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{ls.pdf}\n\\caption{Binary separation evolutionary sequences for a system with initially $M_{\\mathrm{donor}}=4$~M$_{\\odot}$, $M_{\\mathrm{acc}}=1.3$~M$_{\\odot}$ and $8$~days orbital period ($\\beta = 1.0$). Different cases showing spin--orbit and no spin--orbit coupling are shown in comparison with the analytical equation calculations (described by Eq.~(\\ref{analytical})).}\n\\label{test1}\n\\end{figure} \n\nWe also compare two MESA runs with the results of \\cite{2001nsbh.conf..337T} (their Fig.~2). In this comparison we assume that $\\beta = \\mathrm{max}[0, 1-{\\dot{M}_{\\mathrm{Edd}}}\/{\\dot{M}_{\\mathrm{donor}}}]$, and therefore that mass is transferred conservatively until the Eddington limit is reached. Furthermore, gravitational wave radiation and magnetic braking are included. This comparison is shown in Fig.~\\ref{test2}. We again consider two cases, one with a point mass approximation (blue and dashed black curves) and the other one with efficient spin-orbit coupling (orange and solid black curves). While the evolutionary sequences for the point mass approximation (blue curve) match those from \\citet{2001nsbh.conf..337T} (dashed black curve), the sequences corresponding to an efficient spin--orbit coupling show a discrepancy. The effect of angular momentum transport between the donor star and the orbit seems to be much larger  in \\citet{2001nsbh.conf..337T} (solid black curve) compared to our calculated MESA tracks and analytic solutions (orange curve). In an attempt to understand this discrepancy, we also considered the case of highly inefficient synchronisation during RLO. In this scenario, during the high mass-loss phase, the donor loses mass (and angular momentum) which causes its spin to decrease. Since the synchronisation is not efficient, the orbit does not regulate the spin of the star as quickly as angular momentum is being lost from the star. \\cite{1976ApJ...204L..29P} explore this case as a means to have a wider orbit and we also present this run in Fig.~\\ref{test2} as the solid red line. This model has highly efficient spin--orbit coupling until the point the donor almost overflows its Roche lobe and begins mass transfer after which the spin--orbit coupling is highly inefficient. However, this reasoning does not seem to explain the described discrepancy either. Unfortunately, it is impossible to further investigate this apparent discrepancy, as the original calculations and code setup from \\citet{2001nsbh.conf..337T} were not available.\n\nIn conclusion, due to the fact that the widening effect of spin--orbit coupling is weak in our calculations, we infer that spin--orbit coupling can affect the mass-transfer stability but is a second-order effect.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=\\linewidth]{tauris2000.pdf}\n\\caption{Binary separation evolutionary sequences for a system with initially $M_{\\mathrm{donor}}=4$~M$_{\\odot}$, $M_{\\mathrm{acc}}=1.3$~M$_{\\odot}$ and $8$~days orbital period ($\\beta =  {\\it max}~\\{0, 1-\\dot{M}_{\\mathrm{Edd}}\/\\dot{M}_{\\mathrm{donor}}\\}$). Different cases showing efficient spin--orbit and point mass approximation are shown in comparison with similar tracks from \\cite{2001nsbh.conf..337T} (Fig.~2 in their paper).}\n\\label{test2}\n\\end{figure} \n\n\\section{Conclusions}\\label{sec:conclusion}\n\nThere is much intrigue surrounding the nature of ULXs because they challenge our understanding of accretion onto compact objects at high rates and of the evolution of binaries containing compact objects. After the discovery of the first pulsating ULX (M82~X-2), NS~X-ray binary systems were known to constitute a (possibly major) sub-population of ULXs. \n\nNS IMXBs, with donor masses of $2.0-10.0$~M$_{\\odot}$, have been proposed in the past as possible ULXs \\citep{2000ApJ...530L..93T, 2015ApJ...802L...5F}. Their extreme mass ratio leads to intense mass transfer, which can become  highly super-Eddington. Due to changes in the structure of the accretion disc as the mass-transfer rate exceeds the Eddington limit,  face-on observers can infer super-Eddington isotropic-equivalent accretion luminosities, even if the accretion onto the NS is actually limited to the Eddington rate. This is because the increased radiation pressure causes the inner parts of the disc to puff up, creating a funnel, which in turn can cause beaming of the outgoing radiation. These effects can increase the observed isotropic-equivalent luminosities of some accreting NSs to a few orders of magnitude above the Eddington limit. However, the higher the accretion luminosity the lower the chances of actually observing the source since the emission is highly beamed and depends on the angle of viewing. \n\nIn this work, we attempt to explain super-Eddington luminosities observed in NS~ULXs by calculating extensive grids of binary evolution models of LMXBs\/IMXBs with donor stars in the mass range of $0.92$--$8.0\\;$M$_\\odot$ using the MESA code. We specifically investigate the stability and rate of mass transfer, taking into account rotation and tidal effects, as well as the observable properties and the final outcomes of these binaries. The results we obtain can be summarised as follows:\n\n\\begin{itemize}\n   \\item It has been demonstrated previously that the stability of mass transfer in a binary system primarily depends on the structure of the envelope of the donor star at the onset of RLO. From our numerical modelling, including stellar rotation and tides, we find that the most extreme mass ratios ($q=M_{\\rm acc}\/M_{\\rm donor}$) that are able to evolve through long-term stable mass transfer are $0.5$ for donors with convective envelopes and 0.28 for donors with radiative envelopes.\n   \\item For more extreme mass ratios, mass transfer via the first Lagrangian point (L$_1$) cannot keep the donor star contained within its Roche lobe, as the Roche-lobe radius is shrinking faster than the radius of the donor star. In these cases, the donor star expands well outside its Roche lobe, and its volume can even exceed the volume enclosed by the equipotential surface that passes from the second Lagrangian point (L$_2$). At this stage, mass loss from the L$_2$ point will occur, which causes significant loss of orbital angular momentum, thereby accelerating the shrinkage of the orbit and leading to the onset of a CE event. We have calculated L$_2$ lobe volumes and derived fitting formulae (accurate to errors less than 1\\% for all values of $q$) to calculate the volume-equivalent radii of donor stars at the onset of mass loss via L$_2$. In our calculations, we self-consistently identify the onset of the L$_2$ overflow and consider this to be the beginning of an irreversible dynamical instability. \n   \\item We demonstrate that LMXBs\/IMXBs can produce NS-ULXs with typical time-averaged isotropic-equivalent X-ray luminosities between $10^{39}$ and $10^{41}\\;{\\rm erg\\,s}^{-1}$ on a timescale of up to $\\sim\\!1.0\\;{\\rm Myr}$ for the lower luminosities. We present plots of their likelihood of detection, taking into account beaming effects. Based on these results, IMXBs are better candidates for NS ULXs than LMXBs.\n   \\item We find that the beamed super-Eddington accretion disc model provides the very high isotropic-equivalent X-ray luminosities that have been observed for NS ULXs. The model also explains the low detectability of these sources as highly beamed emission and short ULX lifetimes are common properties of the sources.\n   \\item LMXBs\/IMXBs cannot explain the origin of all NS~ULXs given that some of their donor stars are massive (i.e. therefore HMXBs). However, we confirm that IMXBs, specifically, are strong candidates for a large fraction of NS~ULXs. For the most well-studied NS~ULX, M82~X-2, our LMXB\/IMXB mass-transfer sequences with $2.0$~M$_{\\odot}$ NSs better explain the observations compared to systems with $1.3$~M$_{\\odot}$ NSs.\n   \\item We obtain three populations of WDs at the end of the stable mass-transfer phase: He~WDs, CO~WDs, and hybrid WDs which have a CO core and a significant amount of He in the envelope. Comparing observations of NS--WD binaries could help to infer whether or not a binary underwent a ULX phase in the past.\n   \\item The tidal coupling between the spin and the orbit has a second-order effect on the stability of the binary. Our models contradict some previous results in the literature.\n\\end{itemize}\n\n\\begin{acknowledgements}\nThe authors thank the anonymous referee for their constructive comments that helped improve the manuscript. The authors are also grateful to Pablo Marchant, Vasilopoulos Georgios, and Andrew King for their useful discussions. This work was supported by the Swiss National Science Foundation Professorship grant (project number PP00P2 176868). This project has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie RISE action, grant agreement no. 691164 (ASTROSTAT). The authors acknowledge the International Space Science Institute (ISSI) for supporting and funding the international teams program \"Ultraluminous X-ray Sources: From the Local Group to the Very First Galaxies\".\nD.M. thanks the LSSTC Data Science Fellowship Program, which is funded by LSSTC, NSF Cybertraining Grant \\#1829740, the Brinson Foundation, and the Moore Foundation; her participation in the program has benefited this work.\nT.M.T. acknowledges an AIAS-COFUND Senior Fellowship funded by the European Union's Horizon 2020 Research and Innovation Programme (grant no. 754513) and the Aarhus University Research Foundation. E.Z. acknowledges support from the Federal Commission for Scholarships for Foreign Students for the Swiss Government Excellence Scholarship (ESKAS No. 2019.0091) for the academic year 2019-2020.\n\n\n\\end{acknowledgements}\n\n\\bibliographystyle{aa}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLow energy supersymmetry (SUSY) is a promising candidate for\nphysics beyond the standard model at TeV scale\n\\cite{Nilles:1983ge}. In addition to solving the naturalness\nproblem associated with the weak to Planck scale hierarchy $M_{\\rm\nweak}\/M_{Pl}\\sim 10^{-16}$,\nlow energy SUSY  provide a natural candidate for the cold dark\nmatter (DM) in the universe \\cite{lspdm}, the neutralino being the\nlightest supersymmetric particle (LSP), under the assumption of\n$R$-parity conservation. The assumption that the LSP neutralino\nconstitutes DM constrains SUSY model in various ways. The first\nconstraint comes from the requirement that the LSP of the model\nshould be a neutralino. Although not severe, the current limits on\nthe direct or indirect detection of DM also provide an additional\nconstraint on the model. Finally, if the mechanism of cosmological\nneutralino production is specified, the model parameters should be\nin the range giving the correct DM abundance measured by WMAP\n\\cite{wmap}: \\begin{eqnarray} 0.085< \\Omega_{DM}h^2< 0.119. \\end{eqnarray}\n\nRecent development in  string moduli stabilization \\cite{gkp,kklt}\nhas led to a new pattern of soft SUSY breaking terms which has not\nbeen explored before \\cite{choi1}. In KKLT-type moduli\nstabilization scenario \\cite{kklt}, the volume modulus $T$ is\nstabilized at SUSY AdS vacuum by non-perturbative effect,\nand this SUSY  AdS vacuum of $T$ is lifted to a phenomenologically\nviable dS or Minkowski vacuum by SUSY breaking brane.\n  Such set-up\nleads to a mass pattern \\cite{choi1}: \\begin{eqnarray} \\frac{F^T}{T} \\,\\sim\\,\n\\frac{m_{3\/2}^2}{m_T}\\,\\sim\\,\n\\frac{m_{3\/2}}{\\ln(M_{Pl}\/m_{3\/2})},\\end{eqnarray} where $m_T$ and $F^T$\ndenote the modulus mass and $F$-component, respectively. If the\nSUSY breaking brane is sequestered from the visible sector, the\nsoft masses of visible fields are determined by  the modulus\nmediation \\cite{modulus} of ${\\cal O}(F^T\/T)$ and the anomaly\nmediation \\cite{anomaly} of ${\\cal O}(m_{3\/2}\/8\\pi^2)$ which are\ncomparable to each other if the gravitino mass $m_{3\/2}\\sim 10$\nTeV as required to give the weak scale size of soft masses. An\ninteresting consequence of this mixed mediation is that soft\nmasses are unified at a mirage messenger scale \\cite{Choi:2005uz}\n\\begin{eqnarray} M_{\\rm\nmir}=M_{GUT}\\left(\\frac{m_{3\/2}}{M_{Pl}}\\right)^{\\alpha\/2},\\end{eqnarray}\n where\n$\\alpha$ is a parameter of order unity  which represents the\nanomaly to modulus mediation ratio\\footnote{Here we are following\nthe notations of Ref.~\\cite{Choi:2005uz}. We warn the readers that\nsome subsequent works \\cite{falkowski05,baer06} are using a\ndifferently defined $\\alpha$ which corresponds to $\\alpha_{\\rm\nRef.\\cite{falkowski05}}=16\\pi^2\/[\\alpha_{\\rm\nours}\\ln(M_{Pl}\/m_{3\/2})]\\simeq 4.9\/\\alpha_{\\rm ours}$. We stress\nthat our definition of $\\alpha$ is more convenient for  matching\nthe mirage mediation to underlying moduli stabilization model. For\ninstance, the original KKLT moduli stabilization gives\n$\\alpha_{\\rm ours}=1$, and many of its generalizations give a\nrational value of $\\alpha_{\\rm ours}$.}. This feature named as\nmirage mediation \\cite{ratz} is due to the particular property of\nanomaly mediation which is closely related to the renormalization\ngroup (RG) evolution of soft parameters. KKLT-type moduli\nstabilization typically gives a positive $\\alpha={\\cal O}(1)$, and\nthus a mirage messenger scale hierarchically lower than $M_{GUT}$.\nThe corresponding low energy superparticle masses have a quite\ndifferent pattern from those in other SUSY breaking scenarios such\nas mSUGRA, gauge mediation and anomaly mediation\n\\cite{Choi:2005uz,endo05,falkowski05,baer06}.\n\nIn this paper, we wish to study  some  phenomenological aspects of\nneutralino DM in mirage mediation, particularly examine the\nparameter range giving LSP neutralino, the relic DM density under\nthe assumption of standard thermal production, and the prospect of\ndirect or indirect detection of DM neutralino. As we will see,\nwhen the anomaly to modulus mediation ratio $\\alpha$ increases\nfrom zero to a positive value of order unity, or equivalently the\nmirage messenger scale $M_{\\rm\nmir}=M_{GUT}(m_{3\/2}\/M_{Pl})^{\\alpha\/2}$ decreases from $M_{GUT}$\nto a hierarchically lower scale, the nature of LSP neutralino is\nchanged from Bino-like to Higgsino-like via Bino-Higgsino mixing\nregion. The enhanced Higgsino component can be understood by the\ngluino, Wino and Bino mass ratios in mirage mediation: \\begin{eqnarray}\nM_3:M_2:M_1\\simeq\n(1-0.3\\alpha)g_3^2:(1+0.1\\alpha)g_2^2:(1+0.66\\alpha)g_1^2,\\end{eqnarray}\nwhere $\\alpha\\neq 0$ represents  the effects of anomaly mediation.\nThus, compared to the mSUGRA-type pure modulus mediation\n($\\alpha=0$), the gluino to Bino mass ratio for $\\alpha\\sim 1$ is\nsignificantly reduced. This results in smaller $|m_{H_u}^2|\/M^2_1$\n at the weak scale,\n and thus a smaller Higgsino to Bino mass\nratio when the electroweak symmetry breaking condition is imposed.\n(Here $m^2_{H_u}$ is the soft SUSY breaking mass-square of the\nup-type Higgs doublet $H_u$.) As a consequence, for $\\alpha\\sim\n1$, the LSP neutralino has a sizable Higgsino component over a\nlarge fraction of the parameter space, and this Higgsino component\neventually becomes dominant for a larger value of $\\alpha$.  For\nthe parameter region leading to Bino-like LSP, the thermal\nproduction mechanism can give a right amount of relic DM density\nthrough the stop\/stau-neutralino coannihilation process or the\npseudo-scalar Higgs resonance effect, depending upon the value of\n$\\tan\\beta=\\langle H_u\\rangle\/\\langle H_d\\rangle$. In overall,\ncompared to mSUGRA scenario, a significantly larger fraction of\nthe parameter space of mirage mediation can give the WMAP DM\ndensity under the assumption of the conventional thermal\nproduction of neutralino LSP, while satisfying all known\nphenomenological constraints.  As for the DM detection, neutralino\nDM in galactic halo might be detected by near future direct\ndetection experiments in the case of Bino-Higgsino mixed LSP which\ncan be  obtained over a large fraction of the parameter space. The\ngamma ray flux from galactic center might be detectable also if\nthe DM density profile takes a cuspy shape.\n\n Some phenomenological and\ncosmological aspects of mirage mediation have been investigated by\nseveral groups\n\\cite{Choi:2005uz,endo05,falkowski05,baer06,baer,yama}. In\nparticular, the properties of neutralino DM have been studied in\nRef.~\\cite{baer06}.\nThe sub-GUT CMSSM model of Ref.~\\cite{ellis06} also shares a\ncommon feature with  mirage mediation as it assumes the\nunification scale of soft parameters below the GUT scale. In this\npaper, we provide a more extensive analysis of  neutralino DM for\ncertain range of model parameters which are expected to be\nobtained in KKLT-type moduli stabilization scenarios together with\na more detailed study of  the direct and indirect DM detections.\n Our results agree qualitatively\nwell with Ref.~\\cite{baer06} when the considered parameter space\noverlaps.\n\n\n\nRecent study of moduli cosmology \\cite{yama} in KKLT-type moduli\nstabilization\n indicates that if the early universe underwent a period\ndominated by the coherent oscillation of moduli, there can be too\nmany gravitinos and neutralino LSPs produced by moduli decays,\nwhich would spoil the Big-Bang nucleosynthesis or overclose the\nuniverse. This requires a mechanism to dilute the primordial\nmoduli oscillation, e.g. a thermal inflation as proposed in\n\\cite{stewart}. In the presence of such thermal inflation, the\nneutralino DM can be produced either by the conventional  thermal\nproduction mechanism or by the decays of flaton triggering thermal\ninflation, depending upon the details of reheat procedure.\nThroughout this paper, we will focus on the relic density of\nneutralino DM produced  by the conventional thermal production\nmechanism while leaving an additional non-thermal production\n\\cite{yama,nonthermal} as an open possibility.\n\nIn fact, if the anomaly to modulus mediation ratio $\\alpha$ takes\na value giving a mirage messenger scale $M_{\\rm\nmir}=M_{GUT}(m_{3\/2}\/M_{Pl})^{\\alpha\/2}$  much lower than the\nintermediate scale, the squark\/slepton mass-squares have negative\nvalues at certain {\\it high} renormalization point $\\mu\\gtrsim\n\\mu_c$, although they become positive at lower renormalization\npoint around TeV \\cite{Choi:2005uz,falkowski05}. As long as the\nlow energy squark\/slepton mass-squares are positive\\footnote{It\nhas been argued that a negative stop mass-square at high\nrenormalization point is helpful for ameliorating the fine-tuning\nfor the electroweak symmetry breaking in the MSSM \\cite{hdkim}.},\nthe model has a correct color\/charge preserving (but electroweak\nsymmetry breaking) vacuum with squark\/slepton fields $\\phi=0$.\nStill, tachyonic squark\/slepton mass-squares at high\nrenormalization point $\\mu\\gtrsim \\mu_c$ might give a deeper\ncolor\/charge breaking (CCB) minimum or a unbounded from below\n(UFB) direction in the effective potential at $\\phi\n> \\mu_c$ \\cite{munoz}, which would make the color\/charge conserving vacuum metastable.\n In the presence of such CCB minimum or UFB direction, there are two\npoints to be clarified to make sure that the model is\nphenomenologically viable. One first needs a cosmological scenario\nwhich allows our universe to be settled down at the correct\nvacuum, not at the CCB minimum or UFB direction. The second is\nthat the color\/charge preserving vacuum should be stable enough\nagainst the tunnelling into CCB minimum or UFB direction. The\nfirst point might depend on the detailed history of the early\nuniverse. However in view of that squark\/sleptons get large\npositive mass-squares in the high temperature limit, it is a\nrather plausible assumption that squark\/sleptons are settled down\nat the color\/charge preserving minimum after the inflation\n\\cite{kuzenko}. As for the vacuum stability, it has been noticed\nthat the potential barrier between $\\phi=0$ and $\\phi\\gtrsim\n\\mu_c$ gives a tunnelling rate much less than the Hubble expansion\nrate as long as $\\mu_c\\gtrsim 10$ TeV \\cite{riotto}. In mirage\nmediation with positive $\\alpha={\\cal O}(1)$, once one requires a\nsuccessful electroweak symmetry breaking, $\\mu_c$ is  higher than\n$10^3$ TeV, thus satisfies safely the stability condition.  In\nthis paper, we do not take the existence of CCB minimum or UFB\ndirection at large squark\/slepton value $|\\phi|\\gg 10$ TeV as a\nserious problem of the model as long as a good color\/charge\npreserving and electroweak symmetry breaking vacuum exists, and\nfocus on the phenomenology of the model under the assumption that\nwe are living in the color\/charge preserving local minimum which\nis stable enough to have a lifetime longer than the age of the\nuniverse.\n\n\n\n\n\n\nThe organization of this paper is as follows. In section II, we\nreview the basic features of mirage mediation. In section III, we\nexamine the prospect of neutralino DM in  intermediate scale\nmirage mediation scenario  for several different choices of the\nmatter\/Higgs modular weights. In section IV, we extend the\nanalysis to general values of the mirage messenger scale. Section\nV is the conclusion, and Appendix A contains a summary of our\nconvention and notation.\n\n\n\\section{Mirage mediation}\n\n\nMirage mediation is a natural consequence of the KKLT-type moduli\nstabilization scenario satisfying the following two  assumptions:\n(i) the modulus $T$ (or dilaton) which determines the standard\nmodel gauge couplings is stabilized by non-perturbative effects\nand (ii) SUSY is broken by a brane-localized source which is\nsequestered from the visible sector. A well known example of such\nset-up is the KKLT moduli stabilization \\cite{kklt} in type IIB\nstring theory\\footnote{As was noticed in \\cite{choi3,hebecker},\ndue to the effect of throat vector multiplet, the sequestering\nmight not be precise enough in the case of KKLT compactification\nof type IIB string theory. The size of non-sequestered soft scalar\nmass induced by the exchange of throat vector multiplet is quite\nsensitive to the unknown details of compactification, however\nthere exists a reasonable parameter limit in which the\nnon-sequestered effects can be safely ignored \\cite{choi3}.} A\nsimilar but simpler example would be 5D brane model with a flat\ninterval in one side and an warped interval in other side, in\nwhich  SUSY breaking brane is introduced at the IR fixed point of\nthe warped interval.\n\nUnder these two assumptions, the 4D effective action of the\nvisible sector fields and the gauge coupling modulus $T$ is given\nby\n \\begin{eqnarray}\n \\label{superspace}\n \\int d^4\\theta \\left[-3CC^*e^{-K\/3}\n-C^2C^{*2}{\\cal P}_{\\rm\nlift}\\theta^2\\bar{\\theta}^2\\right]+\\left(\\int d^2\\theta \\left[\n\\frac{1}{4}f_aW^{a\\alpha}W^a_\\alpha+C^3W\\right]+{\\rm h.c.}\n\\right), \\end{eqnarray} where $C=C_0+F^C\\theta^2$ is the chiral compensator\nsuperfield, $f_a$ are the holomorphic gauge kinetic function of\nthe visible sector gauge fields, $K$ and $W$ are the effective\nK\\\"ahler potential and superpotential of the visible matter\nsuperfields $\\Phi_i$ and the gauge coupling modulus $T$, which\nwould be obtained by integrating out heavy moduli. As long as the\nSUSY breaking brane is sequestered from the visible gauge and\nmatter superfields, its low energy consequence can be described by\na simple spurion operator of the form ${\\cal P}_{\\rm\nlift}\\theta^2\\bar{\\theta}^2$, independently of the detailed\n dynamics on the SUSY-breaking brane. Assuming an axionic shift\nsymmetry: \\begin{eqnarray} \\label{shift} U(1)_T:\\quad {\\rm Im}(T)\\rightarrow\n{\\rm Im}(T)+\\mbox{real constant}\\end{eqnarray} which is broken by\nnon-perturbative term in the superpotential, the model is given by\n\\begin{eqnarray}\n\\label{u1tmodel}K&=&K_0(T+T^*)+Z_i(T+T^*)\\Phi^*_i\\Phi_i, \\nonumber \\\\\nW &=&  w-Ae^{-aT}+\\frac{1}{6}\\lambda_{ijk} \\Phi_i\\Phi_j\\Phi_k,\n\\nonumber\n\\\\\nf_a&=& kT+\\Delta f_a, \\nonumber \\\\\n{\\cal P}_{\\rm lift}&=& {\\cal P}_{\\rm lift}(T+T^*), \\end{eqnarray} where $a$\nand $k$ are (discrete) real constants, while $\\Delta f_a, w, A$\nand $\\lambda_{ijk}$ are  complex effective constants obtained\nafter heavy moduli are integrated out. The axionic symmetry\n(\\ref{shift}) ensures that $K_0$,\n $Z_i$ and ${\\cal P}_{\\rm lift}$ depend only on $T+T^*$, and $a$ and $k$  are real constant.\nWith these features, the resulting gaugino masses and trilinear\n$A$ parameters preserve CP as was pointed out in \\cite{susycp}.\n\nThere might be various ways to generate the modulus superpotential\n$W_0=w-Ae^{-aT}$ stabilizing  $T$. Generically the\nnon-perturbative term $e^{-aT}$ can be induced by either  a\ngaugino condensation of $T$-dependent hidden gauge interaction or\na stringy instanton whose Euclidean action is controlled by $T$.\nAs for the constant term $w$, it might be induced by a fine-tuned\nconfiguration of fluxes as in the original KKLT scenario\n\\cite{kklt}, or alternatively by $T$-independent non-perturbative\neffect whose strength is controlled by heavy moduli \\cite{luty}.\nAs we will see, the non-perturbative stabilization of $T$ by $W_0$\ngenerates a little hierarchy between the modulus mass and the\ngravitino mass: \\begin{eqnarray} \\frac{m_T}{m_{3\/2}}\\,\\sim\\, aT\\,\\sim\\,\n\\ln(M_{Pl}\/m_{3\/2}), \\end{eqnarray} which results in a little suppression of\nthe modulus $F$-component: \\begin{eqnarray} \\frac{F^T}{T} \\,\\sim\n\\,\\frac{m_{3\/2}^2}{m_T} \\,\\sim\\,\n\\frac{m_{3\/2}}{\\ln(M_{Pl}\/m_{3\/2})}.\\end{eqnarray} Then,  $F^T\/T$ naturally\nhas a size comparable to the anomaly mediated soft mass of ${\\cal\nO}(m_{3\/2}\/4\\pi^2)$ for the gravitino mass around TeV. If the SUSY\nbreaking source is sequestered from the visible sector, the soft\nterms of visible fields are determined by the modulus mediation of\n${\\cal O}(F^T\/T)$ and the anomaly mediation of ${\\cal\nO}(m_{3\/2}\/4\\pi^2)$ which are comparable to each other. This leads\nto a mirage unification \\cite{Choi:2005uz} of soft masses at a\nscale hierarchically different from the gauge coupling unification\nscale $M_{GUT}$.\n\n\n\nIn the Einstein frame, the modulus potential from\n(\\ref{superspace}) is given by\n \\begin{eqnarray}\n \\label{moduluspotential}\n V_{\\rm TOT}\n=e^{K_0}\\left[(\\partial_T\\partial_{\\bar{T}}K_0)^{-1}|D_TW_{0}|^2-3|W_{0}|^2\\right]+V_{\\rm\nlift}, \\end{eqnarray} where\n\\begin{eqnarray}\n W_0=w-Ae^{-aT}, \\quad V_{\\rm lift}= e^{2K_0\/3}{\\cal P}_{\\rm\nlift}.\\end{eqnarray} The superspace lagrangian density (\\ref{superspace})\nalso determines the auxiliary components of $C$ and $T$ as\n\\begin{eqnarray}\n\\label{approx-F} \\frac{F^C}{C_0}&=\n&\\frac{1}{3}\\partial_TK_0F^T+m_{3\/2}^*,\n\\nonumber \\\\\nF^T&=\n&-e^{K_0\/2}\\left(\\partial_T\\partial_{T^*}K_0\\right)^{-1}\\left(D_T\nW_{0}\\right)^*, \\end{eqnarray} where $m_{3\/2}= e^{K_0\/2}W_0$. Note that one\ncan always make both $w$ and $A$ real by appropriate $U(1)_R$ and\n$U(1)_T$ transformations. In such field basis, the $U(1)_T$\ninvariance of $K_0$ assures that both $m_{3\/2}$ and $F^T$ are\nreal. In the following, we will use this field basis in which the\nCP invariance of soft parameters is easier to be recognized.\n\n\nTo stabilize $T$ at a reasonably large value while having\n$m_{3\/2}$ hierarchically smaller than $M_{Pl}$, one needs to\nassume that $w$  is hierarchically smaller than $A$ in the unit\nwith $M_{Pl}=1$. Since $w\\sim m_{3\/2}$ and one needs $m_{3\/2}\\sim\n10$ TeV to get the weak scale superparticle masses, $\\ln(A\/w)$\ntypically has a value of ${\\cal O}(4\\pi^2)$. It is then\nstraightforward to compute the vacuum values of $T$ and $F^T$ by\nminimizing the modulus potential (\\ref{moduluspotential}) under\nthe fine tuning condition $\\langle V_{\\rm TOT}\\rangle =0$. At\nleading order in $\\epsilon=1\/\\ln(A\/w)$,\none finds \\cite{choi1} \\begin{eqnarray} \\label{vev} a T &\\simeq &\\ln(A\/w),\\nonumber \\\\\n \\frac{F^T}{T+T^*}\n &\\simeq& \\frac{m_{3\/2}}{\\ln(A\/w)}\\left(1+\\frac{3\\partial_T\\ln({\\cal P}_{\\rm\nlift})}{2\\partial_TK_0}\n \\right)\\,,\\end{eqnarray}\nwhich shows that $F^T\/T$ is indeed of the order of\n$m_{3\/2}\/4\\pi^2$ for $\\ln(A\/w)\\sim 4\\pi^2$.\n\nLet us consider the soft terms of canonically normalized visible\nfields derived from the 4D effective action (\\ref{superspace}):\n\\begin{eqnarray}\n{\\cal L}_{\\rm\nsoft}&=&-\\frac{1}{2}M_a\\lambda^a\\lambda^a-\\frac{1}{2}m_i^2|\\phi_i|^2\n-\\frac{1}{6}A_{ijk}y_{ijk}\\phi_i\\phi_j\\phi_k+{\\rm h.c.},\n\\end{eqnarray}\nwhere $\\lambda^a$ are gauginos, $\\phi_i$ are the scalar component\nof $\\Phi_i$ and $y_{ijk}$ are the canonically normalized Yukawa\ncouplings: \\begin{eqnarray}\ny_{ijk}=\\frac{\\lambda_{ijk}}{\\sqrt{e^{-K_0}Z_iZ_jZ_k}}. \\end{eqnarray} For\n$F^T\/T\\sim m_{3\/2}\/4\\pi^2$,\n the soft parameters  at energy scale just below  $M_{GUT}$ are\n determined by the modulus-mediated and\n anomaly-mediated contributions which are comparable to each\n other.\n One then finds \\cite{choi1}\n\\begin{eqnarray}\n\\label{soft1} M_a&=& M_0 +\\frac{m_{3\/2}}{16\\pi^2}\\,b_ag_a^2,\n\\nonumber \\\\\nA_{ijk}&=&\\tilde{A}_{ijk}-\n\\frac{m_{3\/2}}{16\\pi^2}\\,(\\gamma_i+\\gamma_j+\\gamma_k),\n\\nonumber \\\\\nm_i^2&=& \\tilde{m}_i^2-\\frac{m_{3\/2}}{16\\pi^2}M_0\\,\\theta_i\n-\\left(\\frac{m_{3\/2}}{16\\pi^2}\\right)^2\\dot{\\gamma}_i\n\\label{eq:bc}\n\\end{eqnarray}\nwhere $M_0$, $\\tilde{A}_{ijk}$ and $\\tilde{m}_i$ are the pure\nmodulus-mediated gaugino mass, trilinear $A$-parameters and\nsfermion masses which are given by\n\\begin{eqnarray}\n\\label{tmediation} M_0&=&F^T\\partial_T\\ln{\\rm Re}(f_a),\n\\nonumber \\\\\n\\tilde{m}_i^2 &=& -F^TF^{T*}\\partial_T\\partial_{\\bar{T}}\n\\ln(e^{-K_0\/3}Z_i),\n\\nonumber \\\\\n\\tilde{A}_{ijk}&=& -F^T\\partial_T\\ln\n             \\left(\\frac{\\lambda_{ijk}}{e^{-K_0}Z_iZ_jZ_k}\\right)\n             \\,=\\,\n             F^T\\partial_T\\ln(e^{-K_0}Z_iZ_jZ_k)\n\\nonumber \\\\\n              &=&\\tilde{A}_i+\\tilde{A}_j+\\tilde{A}_k\n               \\quad\\,\n               \\mbox{for}\\quad\\,\\tilde{A}_i=F^T\\partial_T\\ln(e^{-K_0\/3}Z_i).\n\\end{eqnarray} Here we have used that the holomorphic Yukawa couplings\n$\\lambda_{ijk}$ are $T$-independent constants as a consequence of\nthe axionic  shift symmetry $U(1)_T$, and the one-loop beta\nfunction coefficient  $b_a$, the anomalous dimension $\\gamma_i$\nand its derivative $\\dot{\\gamma}_i$, and $\\theta_i$ are defined as\n\\begin{eqnarray}\nb_a&=&-3{\\rm tr}\\left(T_a^2({\\rm Adj})\\right)\n        +\\sum_i {\\rm tr}\\left(T^2_a(\\phi_i)\\right),\n\\nonumber \\\\\n\\gamma_i&=&2\\sum_a g^2_a C^a_2(\\phi_i)-\\frac{1}{2}\\sum_{jk}|y_{ijk}|^2,\n\\nonumber \\\\\n\\dot{\\gamma}_i&=&8\\pi^2\\frac{d\\gamma_i}{d\\ln\\mu},\\nonumber \\\\\n\\theta_i&=& 4\\sum_a g^2_a C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2\n\\frac{\\tilde{A}_{ijk}}{M_0},\n \\end{eqnarray} where the\nquadratic Casimir $C^a_2(\\phi_i)=(N^2-1)\/2N$ for a fundamental\nrepresentation $\\phi_i$ of the gauge group $SU(N)$,\n$C_2^a(\\phi_i)=q_i^2$ for the $U(1)$ charge $q_i$ of $\\phi_i$, and\n$\\omega_{ij}=\\sum_{kl}y_{ikl}y^*_{jkl}$ is assumed to be diagonal.\nIn Appendix A, we provide a summary of our convention and\nnotation.\n\nFor our later discussion, it is convenient to define\n\\begin{eqnarray}\n\\alpha\\,\\equiv\\,\\frac{m_{3\/2}}{M_0\\ln(M_{Pl}\/m_{3\/2})},\\quad\n a_i\\,\\equiv\\,\\frac{\\tilde{A}_{i}}{M_0}, \\quad\n c_i\\,\\equiv\\, \\frac{\\tilde{m}_i^2}{M_0^2},\n\\label{eq:def}\n \\end{eqnarray}\n where $\\alpha$ represents the anomaly to modulus mediation\n ratio, while $a_{i}$ and $c_i$ parameterize the pattern of the pure modulus mediated soft masses.\nThen the boundary values of soft parameters at $M_{GUT}$ are given\nby\n\\begin{eqnarray}\nM_a&=& M_0 \\Big[\\,1+\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2} b_a\ng_a^2\\alpha\\,\\Big],\\nonumber \\\\\nA_{ijk}&=&M_0\\Big[\\,(a_i+a_j+a_k)\n-\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}(\\gamma_i+\\gamma_j+\\gamma_k)\\alpha\\,\\Big],\n\\nonumber \\\\\nm_i^2&=&M_0^2\\Big[\\,c_i-\\,\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}\n\\theta_i\\alpha-\\left(\\frac{\\ln(M_{Pl}\/m_{3\/2})}{16\\pi^2}\\right)^2\\dot{\\gamma}_i\\alpha^2\\,\\Big],\n\\label{eq:bc1}\n\\end{eqnarray}\nwhere \\begin{eqnarray}\n\\theta_i=4\\sum_a g^2_a C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2(a_i+a_j+a_k).\\end{eqnarray}\nIn this prescription, generic mirage mediation is parameterized by\n\\begin{eqnarray} M_0,\\,\\, \\alpha,\\,\\, a_i,\\,\\, c_i,\\,\\, \\tan\\beta,\\end{eqnarray} where\nwe have replaced the Higgs mass parameters  $\\mu$ and $B$ by\n$\\tan\\beta=\\langle H_u\\rangle\/\\langle H_d\\rangle$ and $M_Z$ as\nusual. As we will see, this parameterization of mirage mediation is\nparticularly convenient when one compute the mirage mediation\nparameters from underlying SUGRA model. For instance, $\\alpha$,\n$a_i$ and $c_i$ are given by simple rational numbers in minimal\nKKLT-type moduli stabilization.\n\n\n\n\nTaking into account the 1-loop RG evolution, the soft masses of\n(\\ref{soft1}) at $M_{GUT}$ leads to low energy soft masses which\ncan be described in terms of the mirage messenger scale: \\begin{eqnarray}\nM_{\\rm mir}=\\frac{M_{GUT}}{(M_{Pl}\/m_{3\/2})^{\\alpha\/2}}. \\end{eqnarray} For\ninstance, the low energy gaugino masses are given by\n\\cite{Choi:2005uz}\\begin{eqnarray} \\label{lowgaugino} M_a(\\mu)=M_0\\left[\\,\n1-\\frac{1}{8\\pi^2}b_ag_a^2(\\mu)\\ln\\left(\\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right] =\\frac{g_a^2(\\mu)}{g_a^2(M_{\\rm\nmir})}M_0, \\end{eqnarray} showing that the gaugino masses are unified at\n$M_{\\rm mir}$, while the gauge couplings are unified at $M_{GUT}$.\nThe low energy values of $A_{ijk}$ and $m_i^2$ generically depend\non the associated Yukawa couplings $y_{ijk}$. However if $y_{ijk}$\nare small enough or \\begin{eqnarray}\\label{con1} a_i+a_j+a_k=c_i+c_j+c_k=1\n\\quad \\mbox{for} \\quad y_{ijk}\\sim 1,\\end{eqnarray} their low energy values\nare given by \\cite{Choi:2005uz}\\begin{eqnarray} \\label{lowsfermion}\nA_{ijk}(\\mu)&=& M_0\\left[\\,a_i+a_j+a_k+\n\\frac{1}{8\\pi^2}(\\gamma_i(\\mu)\n+\\gamma_j(\\mu)+\\gamma_k(\\mu))\\ln\\left(\\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right], \\nonumber \\\\\nm_i^2(\\mu)&=&M_0^2\\left[\\,c_i-\\frac{1}{8\\pi^2}Y_i\\left(\n\\sum_jc_jY_j\\right)g^2_Y(\\mu)\\ln\\left(\\frac{M_{GUT}}{\\mu}\\right)\\right.\n\\nonumber \\\\\n&+&\\left.\\frac{1}{4\\pi^2}\\left\\{\n\\gamma_i(\\mu)-\\frac{1}{2}\\frac{d\\gamma_i(\\mu)}{d\\ln\\mu}\\ln\\left(\n\\frac{M_{\\rm mir}}{\\mu}\\right)\\right\\}\\ln\\left( \\frac{M_{\\rm\nmir}}{\\mu}\\right)\\,\\right], \\end{eqnarray} where $Y_i$ is the $U(1)_Y$\ncharge of $\\phi_i$. Quite often, the modulus-mediated squark and\nslepton masses have a common value, i.e.\n$c_{\\tilde{q}}=c_{\\tilde{\\ell}}$. Then, according to the above\nexpression of low energy sfermion mass, the 1st and 2nd generation\nsquark and slepton masses are unified again at the mirage\nmessenger scale $M_{\\rm mir}$.\n\n\n\nIn  KKLT compactification of type IIB string theory \\cite{kklt},\n$T$ corresponds to the Calabi-Yau  volume modulus, and the\nuplifting brane is located at the end of warped throat. In this\ncase, the uplifting operator is sequestered from $T$ \\cite{choi1}:\n\\begin{eqnarray} \\label{sequestered}\\partial_T {\\cal P}_{\\rm lift}=0. \\end{eqnarray}\nAlso, the minimal  KKLT compactification of IIB theory gives \\begin{eqnarray}\n\\label{minimal} K_0&=&-3\\ln(T+T^*), \\quad\nZ_i\\,=\\,\\frac{1}{(T+T^*)^{n_i}}, \\nonumber \\\\\nf_a&=& kT, \\quad W_0=w-Ae^{-aT} \\quad (A\\,=\\, {\\cal O}(1)),\\end{eqnarray}\nwhere the modular weight $n_i$ is a rational number depending on\nthe origin of matter superfield $\\Phi_i$. Then from\n(\\ref{tmediation}) and (\\ref{vev}), one immediately finds \\begin{eqnarray}\n\\label{benchmark} \\alpha\\,=\\, 1,\\quad a_i\\,=\\, c_i\\,=\\, 1-n_i,\n\\end{eqnarray} giving an intermediate mirage messenger scale: \\begin{eqnarray} M_{\\rm\nmir}\\,\\sim\\, 3\\times 10^9 \\,\\, \\mbox{GeV}. \\end{eqnarray}\n If $\\Phi_i$ lives on the entire world-volume of $D7$ brane\nfrom which the visible gauge bosons originate, the corresponding\nmodular weight $n_i=0$. However, if $\\Phi_i$ is confined in the\nintersections of $D7$ branes, $n_i$ has a positive value, e.g.\n$n_i=1\/2$ or 1. In Fig.~\\ref{fig:rge}, we depict the RG evolution\nof gauge couplings and soft masses in intermediate scale mirage\nmediation scenario with $n_i=0$, i.e.  $\\alpha=a_i=c_i=1$, which\nshows that indeed the gaugino masses and the 1st and 2nd\ngenerations of squarks and slepton masses are unified at $M_{\\rm\nmir}\\sim 3\\times 10^9$ GeV as indicated by (\\ref{lowgaugino}) and\n(\\ref{lowsfermion}). Note that still the gauge couplings are\nunified at the conventional GUT scale $M_{GUT}\\sim 2\\times\n10^{16}$ GeV. In view of its minimality, intermediate scale mirage\nmediation can be considered as a benchmark scenario, thus we\nperform a detailed analysis of neutralino DM in intermediate scale\nmirage mediation in the next section.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{gauge_tb10M0800t173dmkklt.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt3_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{kklt2_tb10M0800t173.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{RG evolution of (a) gauge couplings\n$\\alpha_i$, (b) gaugino masses $M_a$, (c) sfermion and Higgs masses\n$m_i$, (d) trilinear $A$ parameters in intermediate scale mirage\nmediation with $a_i=c_i=1$. Here we choose $M_0=800$ GeV and\n$\\tan\\beta=10$.} \\label{fig:rge}\n\\end{figure}\n\nIt is in fact easy to generalize the compactification to get\ndifferent values of the mirage mediation parameters $\\alpha$,\n$a_i$ and $c_i$. For instance, the compactification  can be\ngeneralized to have a dilaton-modulus mixing in gauge kinetic\nfunctions \\cite{ck,lust} and\/or the non-perturbative\nsuperpotential \\cite{abe}. For the case of type IIB\ncompactification, a nonzero gauge flux on $D7$ branes can generate\nsuch dilaton-modulus mixing in $D7$ gauge kinetic functions\n\\cite{lust}, which would result in \\begin{eqnarray} f_a\\,=\\,kT+lS_0, \\quad\nW_0\\,=\\,w-Ae^{-(aT+bS_0)}\\quad (A={\\cal O}(1)),\\end{eqnarray} where $S_0$\ndenotes the vacuum value of the string dilaton $S$ which is\nassumed to get superheavy mass from RR and NS-NS 3-form fluxes,\nand $k,l,a$ and $b$ are real parameters. In such IIB\ncompactifications, the coefficients of $T$ in $D7$ gauge kinetic\nfunctions and non-perturbative superpotential, i.e. $k$ and $a$,\nare positive, however the coefficients of $S$, i.e. $l$ and $b$,\nmight have both signs under the conditions:\\begin{eqnarray}\n\\label{condition}k{\\rm Re}(T)+l{\\rm Re}(S_0)&\\simeq&\n\\frac{1}{g_{GUT}^2}\\,\\simeq\\, 2,\n\\nonumber \\\\\na{\\rm Re}(T)+b{\\rm Re}(S_0)&\\simeq& \\ln(M_{Pl}\/m_{3\/2})\\,\\simeq\\,\n4\\pi^2.\\end{eqnarray} Assuming that $e^{-K_0\/3}Z_i$ has the same form as the\nminimal model (\\ref{minimal}), it is then straightforward to find\nthat the mirage mediation\nparameters are given by \\begin{eqnarray}\\label{general} \\alpha&=&\\frac{1+R_1}{(1+R_2)(1+R_3)},\\nonumber \\\\\na_i&=& (1-n_i)(1+R_1),\\nonumber \\\\\nc_i&=& (1-n_i)(1+R_1)^2 \\end{eqnarray} where \\begin{eqnarray} R_1=\\frac{l{\\rm\nRe}(S_0)}{k{\\rm Re}(T)}, \\quad R_2=\\frac{b{\\rm Re}(S_0)}{a{\\rm\nRe}(T)},\\quad R_3=\\frac{3\\partial_T\\ln({\\cal P}_{\\rm\nlift})}{2\\partial_T K_0}.\\end{eqnarray}\n Again, for ${\\cal P}_{\\rm lift}$\ninduced by an uplifting brane at the end of warped throat,\n$R_3=0$. However, $1+R_{1}$ and $1+R_2$ can have a variety of\n(positive) values under the condition (\\ref{condition}). As a\nresult, the anomaly to modulus mediation ratio $\\alpha$ can easily\nhave any (positive) value within the range of order unity.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mirage3_tb10M0800t173.eps}\n\\includegraphics[height=7cm,width=7cm]{mirage_tb10M0800t173.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{RG evolution of (a) gaugino masses and (b)\nsfermion and Higgs masses in TeV scale mirage mediation. Here we\nfixed $M_0=800$ GeV, $a_{\\rm H}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm\nM}=1\/2$.} \\label{fig:rge2}\n\\end{figure}\n\n\nA particularly interesting model is the TeV scale mirage mediation\nwith \\begin{eqnarray} \\alpha\\,=\\,2,\\quad a_{H_u}\\,=\\, c_{H_u}\\,=\\, 0, \\quad\na_{Q_3}+a_{U_3}=c_{Q_3}+c_{U_3}=1, \\end{eqnarray} where\n$Q_3$ and $U_3$ denote the left-handed and right handed top-quark\nsuperfields, respectively. This particular model has been claimed\nto minimize the fine tuning for the electroweak symmetry breaking\nin the MSSM \\cite{tevmirage}. In Fig.~\\ref{fig:rge2}, we depict\nthe RG evolution of soft masses in TeV scale mirage mediation\nmodel with $a_{\\rm H}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$,\nwhere the subscripts H and  M stands for the MSSM Higgs doublets\n$H_{u,d}$ and the quark\/lepton matter superfields, respectively.\n\nNote that the squark\/slepton mass-squares renormalized  at high\nenergy scale, e.g. at a scale near $M_{GUT}$,  are negative in\nthis model, while the values at low energy scale below $10^6$ GeV\nare positive.\n In fact,\nsuch tachyonic {\\it high energy} squark\/slepton mass-squares is a\ngeneric feature of mirage mediation for $\\alpha>\\alpha_c$ where\nthe precise value of $\\alpha_c$ depends on $a_i$ and $c_i$, but\nnot significantly bigger than 1 in most cases. As long as the low\nenergy squark\/slepton mass-squares are positive, the model has a\ncorrect color\/charge preserving (but electroweak symmetry\nbreaking) vacuum. For instance, the TeV scale mirage mediation\nmodel of Fig.~\\ref{fig:rge2} has a such vacuum which is a local\nminimum of the scalar potential over the squark\/slepton values\n$|\\phi|\\lesssim 10^6$ GeV. On the other hand, tachyonic  squark\nmass-squares at the RG point $\\mu > 10^6$ GeV indicates that there\nmight be a deeper CCB minimum color\/charge breaking (CCB) or an\nunbounded from below (UFB) direction at $|\\phi|> 10^6$ GeV. One\nthen needs a cosmological scenario which allows our universe to be\nsettled down at the correct vacuum with $\\phi=0$. In view of that\nthe squarks and sleptons get large positive mass-squares in the\nhigh temperature limit, it is rather plausible assumption that\nsquark\/sleptons are settled down at the color\/charge preserving\nminimum after the inflation \\cite{kuzenko}. One still needs to\nconfirm that the color\/charge preserving vacuum is stable enough\nagainst the decay into CCB vacuum. It has been noticed that the\ncorresponding tunnelling rate is  small enough, i.e. less than the\nHubble expansion rate, as long as the RG points of vanishing\nsquark\/slepton mass-squares are all higher than $10^4$ GeV\n\\cite{riotto,kuzenko}, which is satisfied safely by the TeV scale\nmirage mediation of Fig.~\\ref{fig:rge2}.\n\n\n\n\n\n\\section{Neutralino DM in intermediate scale mirage mediation}\n\n\nIn this section, we examine the prospect of neutralino DM in\nintermediate scale mirage mediation scenario. As was noticed in\nthe previous section, the minimal KKLT-type model (\\ref{minimal})\nwith a sequestered uplifting brane  gives $\\alpha=1$, thus an\nintermediate mirage messenger scale \\begin{eqnarray} M_{\\rm mir}\\sim\nM_{GUT}(m_{3\/2}\/M_{Pl})^{1\/2}\\sim 3\\times 10^9\\,\\, {\\rm GeV}.\\end{eqnarray}\nIn this minimal set-up, the discrete parameters $a_i$ and $c_i$\ndescribing the modulus mediated $A$-parameters and sfermion masses\nare determined to be $a_i=c_i=1-n_i$, where $n_i$ denote the\nmodular weights of matter and Higgs superfields. Throughout this\npaper, we will assume $a_i=c_i$ and consider the following four\ndifferent cases: \\begin{eqnarray} \\label{4choices} (a_{\\rm H}=c_{\\rm H},\\,\na_{\\rm M}=c_{\\rm M})=(1,\\,1), \\quad (\\frac{1}{2},\\,\\frac{1}{2}),\n\\quad (0,\\,1), \\quad (0,\\,\\frac{1}{2}). \\end{eqnarray} We also choose the\nHiggsino mass parameter $\\mu > 0$ in light of the experimental\nvalue of the muon anomalous magnetic moment which favors positive\n$\\mu$ \\cite{g-2}, and treat $\\tan\\beta=\\langle H_u\\rangle\/\\langle\nH_d\\rangle$ as a free parameter without specifying the origin of\nthe corresponding $\\mu$ and $B$ parameters. We then obtain the\nparameter range of the model for which the LSP is the lightest\nneutralino as well as the relic neutralino DM abundance under the\nassumption of thermal production, and finally  the direct and\nindirect detection rates of the neutralino LSP using the DarkSUSY\nroutine \\cite{darksusy}.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{t-m0.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq05-nh05.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq0-nh1.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{t-m0-nq05-nh1.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{ Parameter region of neutralino LSP and the\nthermal relic density depicted on the plane of $(\\tan\\beta, M_0)$\nin intermediate scale mirage mediation models with $(a_i,c_i)$\nspecified in Eq.~(3.2).\n}\n\\label{fig:density}\n\\end{figure}\n\n\\subsection{Parameter region of neutralino LSP and thermal relic density}\n\nIn Fig.~\\ref{fig:density}, we show the neutralino LSP region and\nthe thermal neutralino relic density in intermediate scale mirage\nmediation scenario on the ($\\rm{tan}\\beta$,$M_0$)-plane for the\nvalues of\n $a_i$ and $c_i$ specified in Eq.~(\\ref{4choices}).\nWe computed  the sparticle mass spectrum at the electroweak  scale\nby solving the RG equations with the boundary condition\n(\\ref{soft1}) at $M_{GUT}$. Our results show that in all cases\nthere is a large parameter region for which the LSP is given by\nthe lightest neutralino.\n\n\nFig.~\\ref{fig:density}.a is the result for the case in which\n$a_i=c_i=1$ for both matter and Higgs multiplets. In this case,\nlarge $\\rm{tan}\\beta >34$ (grey color) for which the tau Yukawa\ncouplings becomes sizable gives stau LSP (see also Fig.\n\\ref{fig:higgs}.a), while small $M_0\\lesssim0.5~ \\rm TeV$ (green color)\ngives stop LSP. In the remaining region, the LSP is the lightest\nneutralino which turns out to be Bino-like. Small\n$\\tan\\beta\\lesssim 3$ is excluded by the Higgs mass limit $m_h >\n114$ GeV.\nUnder the assumption that the DM neutralinos are produced purely\nby the conventional thermal production mechanism, the magenta stripe corresponds to\nthe parameter region giving a relic neutralino density consistent with   the recent WMAP observation\n\\cite{wmap}:\n\\begin{eqnarray}\n0.085 < \\Omega_{\\rm DM} h^2 < 0.119 ~~(2\\sigma~ \\rm level).\n\\label{eq:wmap}\n\\end{eqnarray}\nIn the region below the magenta stripe, $\\Omega_\\chi h^2 < 0.085$,\nwhile $\\Omega_\\chi h^2 > 0.119$ for the upper region. Thus the\n(cyan) region below the magenta stripe (but above the stop LSP\nregion) can be phenomenologically viable if additional DM\nneutralinos were produced by non-thermal mechanism such as the\ndecays of flaton in thermal inflation \\cite{stewart}.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mass.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-mass.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-mass.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-mass.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{ Particle masses as a function of $\\tan\n\\beta$ in intermediate scale mirage mediation models with\n$(a_i,c_i)$ specified in Eq.~(3.2).\nHere, we fixed $M_0=800$ GeV.}\n\\label{fig:higgs}\n\\end{figure}\n\n\n\n\n\nAlthough the neutralino LSP  is Bino-like in this particular\nintermediate scale mirage mediation, the WMAP mass density is\nobtained for a rather heavy neutralino mass $m_{\\chi^0}\\gtrsim\n450$ GeV. This can be understood by Fig.~\\ref{fig:higgs}.a which\nshows the masses of the lightest neutralino, lighter stop and\nstau, and also the pseudo-scalar Higgs boson as a function of\n$\\tan\\beta$ for the model with $\\alpha=1$, $a_i=c_i=1$ and\n$M_0=800$ GeV. Around $\\tan\\beta\\sim 20$, the pseudoscalar Higgs\nmass  becomes same as $2m_{\\chi^0}$, leading to a resonant\nenhancement of neutralino annihilation through the s-channel\npseudo-scalar Higgs exchange. For other values of $\\tan\\beta$, the\nneutralino mass is somewhat close to the stop mass  (or to the\nstau mass at $\\tan\\beta\\sim 34$), making the stop-neutralino  or\nstau-neutralino coannihilation process becomes efficient. In\nFig.~\\ref{fig:coan}, we depicted $\\Omega_\\chi h^2$  as a function\nof $\\rm tan\\beta$ for $M_0 = 800$ GeV, which shows clearly the\neffect of Higgs resonance at $\\tan\\beta\\sim 20$ and also the\neffect of stop\/stau coannihilation effects for other values of\n$\\tan\\beta$. On the plot, the dotted line corresponds to the relic\ndensity computed without including coannihilation effects. It\nindicates that the stop\/stau-neutralino coannihilation plays a\ncrucial role for the relic neutralino density to have the WMAP\nvalue (\\ref{eq:wmap}) for $M_0=700\\sim 800$ GeV and $\\tan\\beta$\noutside the Higgs resonance region.  Note that in intermediate\nscale mirage mediation with $a_i=c_i=1$, the pseudoscalar Higgs\nresonance condition $m_A \\simeq 2 m_\\chi$ is satisfied for smaller\nvalue of $\\rm tan\\beta$ compared to the mSUGRA case. This can be\nunderstood by noting that the low energy gaugino masses in mirage\nmediation are more compressed compared to mSUGRA, e.g. $M_3 \/M_1\n\\sim 2.3$ in the intermediate scale mirage mediation with $\\alpha\n= 1$, while $M_3 \/M_1 \\sim 6$ in mSUGRA. For a given value of\n$M_1$, smaller $M_3$ gives smaller $\\mu$ and $m_A^2 \\sim m_{H_d}^2\n+ \\mu^2$ at the electroweak scale, thus the pseudoscalar Higgs\nresonance appears at smaller value of $\\rm tan\\beta$ compared to\nmSUGRA case.\n\n\n\\vskip 0.8cm\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{oh2.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{$\\Omega_\\chi h^2$ as a function of\n$\\tan\\beta$ in intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. Here the dotted line corresponds to\nthe result computed without including the stop\/stau coannihilation\neffects.} \\label{fig:coan}\n\\end{figure}\n\n\n\n\n\n\nSo far, we have been focusing on the specific intermediate scale\nmirage mediation model with $a_i=c_i=1$ which might be obtained\nwhen all modular weights $n_i=0$. However as anticipated in the\nprevious section, $a_i=c_i=1$ is not necessarily a more favored\nchoice than the other values of $(a_i,c_i)$ in\nEq.~(\\ref{4choices}). Different values of $a_i$ and $c_i$, e.g.\nsmaller but still non-negative values, are also equally plausible.\nObviously, for a fixed value of $M_0$, the gaugino masses are not\naffected by changing $a_i$ and $c_i$.\nHowever the low energy stop, stau and Higgs masses are somewhat\nsensitive to the values of $a_i$ and $c_i$. They depend on $a_i$\nand $c_i$ either through their boundary values at $M_{GUT}$, or\nthrough their RG evolutions, or through the mass-mixing induced by\nthe low energy $A$-parameters.\n\nThe effects of changing $a_i$ and $c_i$ on the RG evolution can be\nread off from the following one-loop RG equations for the Higgs\nand third generation sfermion mass-squares:\n\\begin{eqnarray}\n\\label{rgequation} 16\\pi^2{d\\over dt} m_{H_u}^2 &=& 3 X_t -6 g_2^2\n|M_2|^2\n- {6\\over 5} g_1^2 |M_1|^2 ,\\nonumber \\\\\n16\\pi^2{d\\over dt} m_{H_d}^2 &=& 3 X_b + X_\\tau-6 g_2^2 |M_2|^2\n- {6\\over 5} g_1^2 |M_1|^2 ,\\nonumber \\\\\n16\\pi^2{d\\over dt} m_{Q_3}^2 &=& X_t+X_b-{32\\over 3} g_3^2 |M_3|^2\n-6 g_2^2 |M_2|^2-{2\\over 15} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{U_3}^2 &=& 2 X_t-{32\\over 3} g_3^2 |M_3|^2\n-{32\\over 15} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{L_3}^2 &=& X_\\tau\n-6 g_2^2 |M_2|^2-{3\\over 5} g_1^2 |M_1|^2, \\nonumber \\\\\n16\\pi^2{d\\over dt} m_{E_3}^2 &=& 2 X_\\tau -{24\\over 5} g_1^2\n|M_1|^2,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nX_t &=& 2 y_t^2 (m_{H_u}^2+m_{Q_3}^2+m_{U_3}^2+ A_{H_uQ_3U_3}^2), \\nonumber \\\\\nX_b &=& 2 y_b^2 (m_{H_d}^2+m_{Q_3}^2+m_{D_3}^2+ A_{H_dQ_3D_3}^2), \\nonumber \\\\\nX_\\tau &=& 2 y_\\tau^2 (m_{H_d}^2+m_{L_3}^2+m_{E_3}^2+\nA_{H_dL_3E_3}^2).\n\\end{eqnarray}\nThese RG equations show that smaller $X_I$ ($I=t,b,\\tau$) increase\nthe low energy soft mass-squares.\n Since $a_i$ and $c_i$ determine\n the modulus-mediated trilinear $A$\nparameters and soft mass-squares at $M_{GUT}$ as\n$\\tilde{A}_{ijk}=(a_i+a_j+a_k)M_0$ and $\\tilde{m}_i^2=c_iM_0^2$,\nsmaller $a_{\\rm H}=c_{\\rm H}$ give smaller $X_I$  without\naffecting the boundary values of squark and slepton masses at\n$M_{GUT}$, eventually making the stop and stau masses at TeV scale\nlarger. On the other hand, the consequence of smaller $a_{\\rm\nM}=c_{\\rm M}$ is more complicate as it depends on the relative\nimportance of the Yukawa-induced RG evolution. It turns out that\nchanging $a_{\\rm M}=c_{\\rm M}$ to smaller value makes the stop\nmass larger, while the stau mass smaller.\n\nIn Figs.~\\ref{fig:density}.b and \\ref{fig:higgs}.b, we depict the\nresults for the case in which  $a_i=c_i=1\/2$ for both the matter\nand Higgs multiplets. As can be understood from the above\ndiscussion, this intermediate scale mirage mediation\ndoes not contain any parameter region of stop LSP, while having a\nlarger parameter region of stau LSP (see Fig.~\\ref{fig:higgs}.b).\nAnother important feature is that the weak scale value of\n$|\\,m_{H_u}^2|$  becomes smaller compared to the case of\n$a_i=c_i=1$, which is mainly due to smaller $X_t$.\n This results in smaller $\\mu$ and $m_A$. Smaller $\\mu$ makes the\nneutralino LSP have a sizable Higgsino component, while smaller\n$m_A$ makes the pseudo-scalar resonance region disappear. Again\nthe magenta region in Fig. \\ref{fig:density}.b corresponds to the\nparameter region giving the WMAP DM density (\\ref{eq:wmap}) under\nthe assumption of pure thermal production. In this case, the\nneutralino pair annihilation into gauge boson pair becomes\nefficient due to the enhanced Higgsino component of neutralino\nLSP. Finally, the brown region is excluded by giving the Br($b \\to\ns \\gamma$) smaller than the allowed range.\n\nFigs.~\\ref{fig:density}.c and \\ref{fig:higgs}.c are the result for\nthe case with $a_{\\rm M}=c_{\\rm M}=1$ and $a_{\\rm H}=c_{\\rm H}=0$,\nwhile Figs.~\\ref{fig:density}.d and \\ref{fig:higgs}.d are for the\ncase with $a_{\\rm M}=c_{\\rm M}=1\/2$ and $a_{\\rm H}=c_{\\rm H}=0$.\nThe case of $a_{\\rm M}=c_{\\rm M}=1\/2$ and $a_{\\rm H}=c_{\\rm H}=0$\nis quite similar to the case of $a_i=c_i=1\/2$: LSP is the lightest\nneutralino with a sizable Higgsino component for\n$\\tan\\beta\\lesssim 20$ (see Figs.~\\ref{fig:higgs}.b and\n\\ref{fig:higgs}.d). On the other hand, the case of $a_{\\rm\nM}=c_{\\rm M}=1$ and $a_{\\rm H}=c_{\\rm H}=0$ is somewhat\ndistinctive since there is no parameter region of stop or stau LSP\nand the WMAP DM density is obtained for a light neutralino mass\n$m_{\\chi^0}\\sim 250$ GeV, while in other cases the WMAP DM density\nis obtained for heavier $m_{\\chi^0}\\gtrsim 350$ GeV. Again, the\nbrown region is excluded by giving the Br($b \\to s \\gamma$)\nsmaller than the allowed range.\n\n\n\n\n\\subsection{Dark matter detections}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mn-sigsi.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-si.eps}\n\\vskip 1.2cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-si.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-si.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Scatter plot of the spin-independent\nneutralino-proton scattering cross section vs. $m_\\chi$ in\nintermediate scale mirage mediation with $(a_i,c_i)$ specified in\nEq.~(3.2).\n}\n\\label{fig:sigsip}\n\\end{figure}\n\nIf neutralino LSP is the main component of the matter budget in\nthe Milky Way, it might be detected through the elastic scattering\nwith terrestrial nuclear target \\cite{goodman,lspdm}. In the MSSM,\n$t$-channel Higgs boson and $s$-channel squark exchange processes\ncontribute to the spin-independent (scalar) scattering between\nneutralino and nuclei. In many cases, dominant contribution to the\nscalar cross section comes from the Higgs exchange process which\nbecomes  bigger for larger $\\tan\\beta$, smaller Higgs masses, and\nmixed Bino-Higgsino LSP.\n\n\n\n\nIn the specific intermediate scale mirage mediation model with\n$a_i=c_i=1$, the neutralino LSP is Bino-like and the mass of heavy\nCP-even Higgs boson is rather large when we require the neutralino\nto be LSP. It is thus expected that the elastic scattering cross\nsection between neutralino DM and nuclei is rather small. In\nFig.~\\ref{fig:sigsip}.a, we depict spin-independent (scalar) cross\nsection $\\sigma_{SI}$ of neutralino-proton scattering as a\nfunction of the LSP neutralino mass in this specific intermediate\nscale mirage mediation. Here we have imposed the experimental\nbounds on the Higgs\/sparticle masses and $b \\rightarrow s \\gamma$\nbranching ratio, and required that the lightest neutralino is the\nLSP. Red points in the figure correspond to the parameter values\ngiving the WMAP DM density (\\ref{eq:wmap}) under the assumption of\npure thermal production,  while the cyan points represent the\nparameter values for which the thermal production mechanism gives\na smaller relic density. As expected, the cross section in the\ncase of $a_i=c_i=1$ is quite small: $\\sigma_{SI}\\lesssim 5 \\times\n10^{-9}$ pb, which is much smaller than the current experimental\nupper bound. It is even smaller than the sensitivity of future\nexperiment such as SuperCDMS \\cite{supercdms} which would reach\nnear $10^{-9}$ pb level. On the other hand, intermediate scale\nmirage mediations with different values of $(a_i,c_i)$ have a\nquite better prospect for direct detection. As can be seen from\nFig.~\\ref{fig:sigsip}, most of the (red) WMAP points are above the\nsensitivity of SuperCDMS for the other three cases  of different\n$(a_i,c_i)$. This is mainly due to the enhanced Higgsino component\nof the neutralino LSP and the reduced Higgs mass.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{mn-gamma.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-alpha1-gamma.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-alpha1-gamma.eps}\n\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-alpha1-gamma.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Scatter plot of the continuum gamma ray\nflux vs. $m_\\chi$ in intermediate scale mirage mediation with\n$(a_i,c_i)$ specified in Eq.~(3.2).\n}\n\\label{fig:gamma}\n\\end{figure}\n\n\nLet us now examine gamma ray signals  from DM annihilation in the\ngalactic center,  providing another feasible but indirect\ndetection method for dark matter. The integrated gamma ray flux\ndepends on the quantity $\\bar{J} (\\Delta\\Omega)$, which is a\nmeasure of the cuspiness of the galactic halo density profile over\na spherical region of solid angle $\\Delta\\Omega$. In this paper,\nwe use a conservative galactic halo model (isothermal halo density\nprofile) which gives $\\bar{J} \\sim 30$ with the detector angular\nresolution $\\Delta\\Omega = 10^{-3}$ sr and set $E_{thr} = 1$ GeV\nfor gamma ray energy threshold. Fig.~\\ref{fig:gamma} shows\ncontinuum gamma ray flux from the galactic center in intermediate\nscale mirage mediation scenarios under consideration, where red\npoints give the WMAP value (\\ref{eq:wmap}) of the relic DM\ndensity. Here the four different choices of $a_i=c_i$ do not lead\nto a dramatic difference in the gamma ray flux. The maximal value\nof flux given by the most favored WMAP (red) points is about ${\\rm\nfew}\\times 10^{-11} \\rm{cm^{-2} s^{-1}}$ which is somewhat below\nthe expected reach ($\\sim 10^{-10}\\rm{cm^{-2} s^{-1}}$) of GLAST,\nalthough the (cyan) points giving smaller relic density can give a\nlarger flux around $10^{-10}\\rm{cm^{-2} s^{-1}}$. However, it\nshould be noticed that our calculation for the gamma ray flux is\nbased on a conservative halo density profile. If one uses an\nextreme halo model like the spiked profile \\cite{moore}, the\nresulting gamma ray flux increases by a factor of $\\sim 10^4$. In\nthis case, the gamma ray signals can be detected for a significant\nportion of the parameter space. A caveat is that the continuum\ngamma ray signals suffer from unknown astrophysical background.\nRecent observations of a bright gamma ray source in the direction\nof galaxy center by the Air Cherenkov Telescopes such as H.E.S.S.\n\\cite{hess} might be explained by an astrophysical process rather\nthan the dark matter annihilation \\cite{hooper}.\n\nWe finally notice an interesting enhancement of the gamma ray flux\ndue to the Higgs resonance effect. Fig.~\\ref{fig:gammas} shows the\ngamma ray flux from the galactic center as a function of $\\rm\ntan\\beta$ in the specific intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. One can see a clear enhancement of\nthe flux around $\\rm tan\\beta \\sim 22$ for which $m_A \\sim 2\nm_\\chi$. In this case, neutralino annihilation to heavy quarks is\ndominated and the subsequent quark hadronization produces many\ngamma rays.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{gamma.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Continuum gamma ray flux as a function of\n$\\tan\\beta$ in the intermediate scale mirage mediation with\n$a_i=c_i=1$ and $M_0=800$ GeV. Note the resonant peak due to the\npsuedo-scalar Higgs resonance.} \\label{fig:gammas}\n\\end{figure}\n\n\n\n\n\\section{Neutralino DM for generic mirage messenger scale}\n\n\nIn the previous section, we have examined the prospect of\nneutralino DM in intermediate scale mirage mediation models\n($\\alpha=1$). As was discussed in section 2, in string\ncompactifications with non-trivial dilaton-modulus mixing, the\nanomaly to modulus mediation ratio $\\alpha$ can have a more\nvariety of values. In fact, the nature of neutralino LSP is\nsomewhat sensitive to the value of $\\alpha$, typically it changes\nfrom Bino-like to Higgsino-like via Bino-Higgsino mixing region\nwhen $\\alpha$ is increased from zero to a value of order unity.\nThis feature is essentially due to the following behavior of the\ngaugino masses  as a function of $\\alpha$: \\begin{eqnarray} M_3:M_2:M_1\\simeq\n (1-0.3\\alpha)g_3^2:(1+0.1\\alpha)g_2^2:(1+0.66\\alpha)g_1^2,\n \\end{eqnarray}\nIf $\\alpha$ increases from zero, the gluino mass decreases as\n$M_3\\propto (1-0.3\\alpha)$. Smaller $M_3$ then weakens  the\nradiative electroweak symmetry breaking mechanism as it gives a\nsmaller stop mass-square, thus leads to smaller $|m_{H_u}|^2$ and\n$|\\mu|$ at the weak scale. On the other hand,\n the Bino mass increases as $M_1\\propto\n(1+0.66\\alpha)$, thus the lightest neutralino changes from\nBino-like to Higgsino-like when $\\alpha$ is varying from zero to a\npositive value of order unity. If $\\alpha$ is further increased,\neventually the model does not allow electroweak symmetry breaking.\nIn this section, we extend the analysis of the previous section to\nthe range of $\\alpha$ from zero to the value at which the\nelectroweak symmetry starts to be restored.\n\n\\subsection{Parameter region of neutralino LSP and thermal relic\nDensity}\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{tb10-mass.eps}\\hskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb35-mass.eps}\n\\vskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb10.eps}\\hskip 0.5cm\n\\includegraphics[height=7cm,width=7cm]{tb35.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Sparticle masses vs. $\\alpha$ for\n$\\tan\\beta=10$ and $\\rm tan\\beta=35$ in case with $a_i=c_i=1$. The\nlower figures show the parameter space of neutralino LSP and its\nthermal relic density on the plane of ($\\alpha$, $M_0$).}\n\\label{fig:density-tb35}\n\\end{figure}\n\n\nAgain, let us first consider the case with $a_i=c_i=1$. We will\ntreat $M_0$ and $\\alpha$ as free parameters, while focusing on\n$\\tan\\beta=10$ and 35. Figs.~\\ref{fig:density-tb35}.a and\n\\ref{fig:density-tb35}.b  show how some of the superparticle\nmasses vary as a function of $\\alpha$ for a fixed $M_0 = 800$ GeV.\nFor $\\alpha\\lesssim 1$, the LSP is the lightest neutralino which\nis mostly Bino, and thus its mass varies as $m_{\\chi^0}\\simeq\nM_1\\propto (1+0.66\\alpha)$. In the range of\n$1\\lesssim\\alpha\\lesssim 1.8$, stau or stop becomes the LSP. For\n$1.8\\lesssim \\alpha\\lesssim 2$, the lightest neutralino which is\nnow mostly Higgsino becomes  the LSP. If $\\alpha$ increases\nfurther, the model does not allow  electroweak symmetry breaking.\n\nIn Figs.~\\ref{fig:density-tb35}.c and \\ref{fig:density-tb35}.d,\nthe two distinct magenta regions seperated by stop\/stau LSP\nregions give the WMAP DM density, $0.085<\\Omega_{DM} h^2<0.119$,\nunder the assumption that all neutralino DMs are produced by the\nconventional thermal production mechanism. Below (above) these\nmagenta regions, $\\Omega_\\chi h^2 < 0.085$  ($> 0.119$). In the\nBino-like LSP region, stop-neutralino coannihialtion plays a\ncrucial role to get the WMAP DM density for $\\rm tan\\beta=10$,\nwhile stau-neutralino coannihilation or pseudoscalar Higgs\nresonance processes are important for $\\rm tan\\beta=35$. For\nHiggsino-like LSP, the charged Higgsino $\\chi_1^\\pm$ and two\nneutral Higgsinos $\\chi_1^0, \\chi_2^0$ are nearly degenerate. Then\nthe dominant annihilation processes are the neutralino pair\nannihilation into gauge bosons, and the neutralino-chargino\nco-annihilation into fermion pair \\cite{coanil}. These\nannihilations of Higgsino-like LSP are very efficient, so that the\nrelic mass density is too small unless $m_\\chi^0$ is quite heavy.\nIndeed, from Figs.~\\ref{fig:density-tb35}.c and\n\\ref{fig:density-tb35}.d, we can see that the WMAP DM density is\nobtained only for $M_0\\gtrsim 2.2$ TeV in the Higgsino LSP region\naround $\\alpha\\sim 1.8$. However it should be stressed that the\ncyan regions of Figs.~\\ref{fig:density-tb35}.c and\n\\ref{fig:density-tb35}.d can be allowed  if some part of DM were\nproduced by non-thermal mechanism. Such parameter region contains\n$\\alpha\\sim 1.8$ and $M_0\\sim 1$ TeV for which the neutral\nHiggsino with $m_{\\chi^0}\\sim 200$ GeV is the LSP and the stop is\nrather light as $m_{\\tilde{t}_1}\\sim 250$ GeV.  \nThe brown regions are excluded by the $b \\to s \\gamma$ constraint. On the brown\nregion in Fig. 9.c, the chargino loop\ncontribution to $b\\to s\\gamma$ dominates, which results in Br$(b\n\\to s \\gamma)$ smaller than the experimentally allowed range. \n\n\\vskip 0.8cm\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb35-mass.eps}\n\\vskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb10.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw-tb35.eps}\n\\end{center}\n\\vskip -0.4cm\n\\caption{The results for the case in which $a_i=c_i=1\/2$ for both\nthe Higgs and matter multiplets.} \\label{fig:density-nzero}\n\\end{figure}\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb35-mass.eps}\\vskip\n1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb10.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw2-tb35.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{The results for the case in which $a_{\\rm\nH}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1$.}\n\\label{fig:density-nzero2}\n\\end{figure}\n\nLet us now consider the case with  $a_i=c_i=1\/2$. Obviously,\nsmaller $(a_i,c_i)$ give smaller stop\/stau mass-squares at\n$M_{GUT}$. However, as was anticipated in the previous section,\n$X_I$ ($I=t,b,\\tau$) which govern the RG evolution of stop\/stau\nmass-squares (see Eq.~\\ref{rgequation}) become smaller also, which\nwould increase the stop\/stau masses at the weak scale. Together\nwith the reduction of $A_{H_uQ_3U_3}$, this effect on the RG\nevolution eventually makes the physical lighter stop mass\n$m_{\\tilde t_1}$ larger compared to the case with $a_i=c_i=1$. On\nthe other hand, stau masses are more affected by the change of the\nboundary values, thus their weak scale values become lighter\ncompared to the case of $a_i=c_i=1$.\n Smaller $X_t$ leads to\nalso a smaller $|m_{H_u}^2|$ at the weak scale, resulting the\nreduction of the Higgsino mass $\\mu$ and the pseudoscalar Higgs\nboson mass $m_A$. Figs.~\\ref{fig:density-nzero}.a and\n\\ref{fig:density-nzero}.b show all of these features.  Again, as\n$\\alpha$ increases, the neutralino LSP changes from Bino-like to\nHiggsino-like. Comparing to Fig.~\\ref{fig:density-tb35}, the\nlighter stop becomes heavier, while the lighter stau and the\npseudoscalar Higgs become lighter. As a consequence, the stop LSP\nregion disappears, but the stau LSP region at large $\\tan\\beta$\nbecomes larger. The magenta regions of\nFigs.~\\ref{fig:density-nzero}.c and \\ref{fig:density-nzero}.d\ncorrespond to the parameter region giving the WMAP DM density\nunder the assumption of pure thermal production. They clearly show\nthe Higgsino-like LSP at $\\alpha>1$ and also the pseudoscalar\nHiggs resonance effect for the Bino-like LSP at smaller $\\alpha$.\n\n\\vskip 0.8cm\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=7cm,width=7cm]{nzmw3-tb10-mass.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{nzmw3-tb35-mass.eps} \\vskip\n1cm\n\\includegraphics[height=7cm,width=7cm]{tb10-nq05-nh1.eps}\\hskip 1cm\n\\includegraphics[height=7cm,width=7cm]{tb35-nq05-nh1.eps}\n\\end{center}\n\\vskip -0.3cm \\caption{The results for the case in which $a_{\\rm\nH}=c_{\\rm H}=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$.}\n\\label{fig:density-nq05-nh1}\n\\end{figure}\n\n\n\nFig.~\\ref{fig:density-nzero2} shows the results for the case in\nwhich $a_{\\rm H}=c_{\\rm H}=0$ for the Higgs multiplets, while\n$a_{\\rm M}=c_{\\rm M}=1$ for the quark\/lepton matter multiplets. A\ncharacteristic feature of this case is that the lightest\nneutralino is the LSP over the entire region of parameter space\nallowing the electroweak symmetry breaking. Compared to the case\nin which  $a_i=c_i=1$ for both the Higgs and matter multiplets,\n$X_I$ ($I=t,b,\\tau$) for the RG evolution (\\ref{rgequation})  have\nsmaller values, while the boundary values of stop\/stau\nmass-squares remain the same. This results in heavier stop and\nstau at the weak scale. Except for the absence of stop\/stau LSP\nregion, other features are somewhat similar to other cases.\nThe brown regions are excluded by the $b \\to s \\gamma$ constraint. \nOn the brown region with small $M_0$ in Fig. 11.c, the chargino loop\ncontribution to $b\\to s\\gamma$ dominates, which results in Br$(b\n\\to s \\gamma)$ smaller than the experimentally allowed range. \nOn the other hand, the charged Higgs boson loop becomes significant\nin the large $M_0$ region, making  the predicted Br$(b \\to s\n\\gamma)$ exceed the experimental bound. The region between those\ntwo brown regions is allowed due to the cancellation between the\nchargino and charged Higgs boson loop contributions.\n\nFinally, Fig.~\\ref{fig:density-nq05-nh1} is for the case with\n$a_H=c_H=0$ and $a_{\\rm M}=c_{\\rm M}=1\/2$. The results are quite\nsimilar to the case in which $a_i=c_i=1\/2$ for both the Higgs and\nmatter multiplets. The $\\alpha=2$ region of this case corresponds\nto the TeV scale mirage mediation model proposed in\n\\cite{tevmirage} as a model to minimize the fine tuning for the\nelectroweak symmetry breaking in the MSSM.\n\n\n\n\n\n\n\\subsection{Dark matter detections} \\vspace{0.5cm}\n\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{tb10-si.eps}\\hskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{tb35-si.eps}\\vskip 0.5cm\n\\end{center}\n\\vskip -0.5cm \\caption{Spin-independent neutralino and proton\nscattering cross section in case with $a_i=c_i=1$.}\n\\label{fig:sigsip-tb35}\n\\end{figure}\n\n\nTo see the prospect of direct DM detection, spin-independent cross\nsection  of the neutralino-proton scattering is presented in\nFig.~\\ref{fig:sigsip-tb35} for the case with $a_i=c_i=1$. Here, we\nimposed the experimental bounds on sparticle and Higgs masses, and\n$b\\rightarrow s\\gamma$ branching ratio. In the figures, the red\npoints give the WMAP DM density: $0.085 < \\Omega_\\chi h^2 <\n0.119$, the cyan  corresponds to the region giving  $\\Omega_\\chi\nh^2< 0.085$, and the rest gives $\\Omega_\\chi h^2 > 0.119$, under\nthe assumption of pure thermal production of neutralino DM.\nOne can notice that there are two distinct branches  of the WMAP\npoints which correspond to the Bino branch and the Higgsino branch, respectively.\nIn our scan, Higgsino-like LSP gives a larger $\\sigma_{SI}$ for a given\n$m_\\chi$. The dominant contribution to $\\sigma_{SI}$ usually comes\nfrom the Higgs exchange process which becomes\n significant if the LSP neutralino is a mixed Bino-Higgsino state.\nOn the other hand, for $a_i=c_i=1$, the LSP neutralino is either\nBino-like or Higgsino-like since the mixed Bino-Higgsino region\ngives a stop or stau LSP.  Therefore, it is expected that the\ncross section is rather small for the case with $a_i=c_i=1$.\nIndeed, Fig.~\\ref{fig:sigsip-tb35} shows that the predicted values\nare all less than the current and near future experimental\nsensitivity.\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{nzmw-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw-tb35-si.eps}\\vskip\n0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw2-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw2-tb35-si.eps}\\vskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-si.eps}\\hskip 1cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-si.eps}\\vskip 0.5cm\n\\end{center}\n\\vskip -0.5cm \\caption{Spin-independent neutralino and proton\nscattering cross section for other values of $(a_i,c_i)$ giving a\nmixed Bino-Higgsino LSP over a significant fraction of the\nparameter space.} \\label{fig:sigsip-tb36}\n\\end{figure}\n\n\nHowever, the prospect of direct DM detection is dramatically\nchanged if one considers other choices of $a_i$ and $c_i$.\nFig.~\\ref{fig:sigsip-tb36} shows the predictions for\nspin-independent cross section of the neutralino-proton scattering\nfor the three other choices of $(a_i,c_i)$  giving a mixed\nBino-Higgsino LSP over a significant fraction of the parameter\nspace and also a reduced value of the pseudoscalar Higgs mass.\nThese values of $a_i$ and $c_i$ give  heavier stop, thereby the\n$b\\rightarrow s\\gamma$ constraint becomes less significant\ncompared to the case with $a_i=c_i=1$. Again, the red points\nrepresent the parameter values giving the WMAP DM density $ 0.085\n< \\Omega_\\chi h^2 < 0.119$, the cyan points give $\\Omega_\\chi\nh^2 < 0.085$, and the rest stands for $\\Omega_\\chi h^2 > 0.119$,\nunder the assumption of thermal production of neutralino LSP.\nAs expected, the scattering cross sections are largely enhanced\ncompared to the case with $a_i=c_i=1$. Now, much of the WMAP\npoints give $\\sigma_{SI}$ exceeding the sensitivity limit of the\nplanned SuperCDMS experiment. If one includes the cyan points,\nthe cross section can be much bigger, reaching even at the current\nCDMS sensitivity limit.\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{tb10-gamma.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{tb35-gamma.eps}\\vskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{tb10-mono.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{tb35-mono.eps}\n\\end{center}\n\\vskip -0.5cm \\caption{Continuum (a and b) and monochromatic (c\nand d) gamma ray flux from the Galactic Center vs. $m_\\chi$ for\nthe case with $a_i=c_i=1$ and $\\rm{tan}\\beta=10$ or $35$.}\n\\label{fig:fluxgam-tb35}\n\\end{figure}\n\n\nGamma rays induced by  neutralino annihilation in  Galactic Center\nmight provide an indirect detection  of neutralino DM.\nFig.~\\ref{fig:fluxgam-tb35} shows the predicted continuum (a and\nb) and monochromatic (c and d) gamma ray fluxes from the Galactic\nCenter  as a function of the LSP neutralino mass for the case with\n$a_i=c_i=1$. Here we chose the same halo density profile as the\nprevious section, giving $\\bar J(\\Delta\\Omega=10^{-3} {\\rm sr})\n\\sim 30$. The red points in the figures give the WMAP DM density,\nwhile the cyan and the rest give $\\Omega_\\chi h^2<0.085$ and\n$\\Omega_\\chi h^2>0.119$, respectively, under the assumption of\npure thermal production. Again the WMAP points have two distinct\nbranches, the Bino-branch and the Higgsino-branch. \nIncluding the cyan points\ngiving smaller thermal relic DM density, the case with $a_i=c_i$\ncan give a continuum\ngamma ray flux up to $2\\times 10^{-11}  {\\rm cm^{-2} s^{-1}}$ and\n$10^{-10} {\\rm cm^{-2} s^{-1}}$ for $\\tan\\beta = 10$ and 35,\nrespectively.\n This maximum flux of the continuum gamma rays barely touch\nthe expected reach of GLAST. However, the real gamma ray flux can\nbe much bigger than these predictions if the actual halo density\nprofile is denser  than the assumed profile. For Higgsino LSP,\nunsuppressed annihilation into W or Z boson pair is the major\nsource of continuum gamma rays. As can be noticed from\nFig.~\\ref{fig:fluxgam-tb35}, for some parameter values, the gamma\nray flux from Bino LSP is largely enhanced by the pseudoscalar\nHiggs resonance effect.\n\n\n\n\\begin{figure}[ht!]\n\\begin{center}\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-gamma.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-gamma.eps}\\vskip 1.0cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb10-mono.eps}\\hskip 0.5cm\n\\includegraphics[height=6cm,width=6cm]{nzmw3-tb35-mono.eps}\n\\end{center}\n\\vskip -0.4cm \\caption{Continuum (a and b) and monochromatic (c\nand d) gamma ray flux from the Galactic Center vs. $m_\\chi$ for\nthe case in which $a_{\\rm H}=c_{\\rm H}=0$, $a_{\\rm M}=c_{\\rm\nM}=1\/2$, and $\\rm{tan}\\beta=10$ or 35.} \\label{fig:fluxgam-nzmw3}\n\\end{figure}\n\n\n\n\n\nIn fact, it is quite nontrivial to discriminate the continuum\ngamma rays produced by neutralino annihilation from the diffuse\ngalactic gamma ray backgrounds. On the other hand, the\nmonochromatic gamma ray line from $\\chi\\chi \\rightarrow\n\\gamma\\gamma$ or $\\gamma Z$ can be considered as a 'smoking gun'\nsignal of WIMP dark matter. Figs.~\\ref{fig:fluxgam-tb35}.c and\n\\ref{fig:fluxgam-tb35}.d  show the gamma ray line flux produced by\nneutralino pair annihilation  in Galactic Center for the case with\n$a_i=c_i$. One can notice that there is a clear distinction\nbetween the Bino and Higgsino LSP regions. The gamma ray line flux\nranges from $10^{-19}$ to $10^{-16}\\, {\\rm cm^{-2} s^{-1}}$ for\nthe Bino LSP branch of WMAP points,  while it ranges from\n$10^{-16}$ to $10^{-15}\\, {\\rm cm^{-2} s^{-1}}$ for the Higgsino\nLSP branch. For Higgsino-like LSP,  the gamma ray line flux comes\ndominantly from the $W^\\pm \\chi_1^\\mp$ loop diagrams resulting in\na large cross section for $\\chi\\chi \\rightarrow \\gamma\\gamma$ or\n$\\gamma Z$ \\cite{higgsinoDM}. While GLAST will probe the photon\nenergies only up to 300 GeV with a low energy threshold,\nAtmospheric Cherenkov Telescopes (ACT) such as H.E.S.S. will be\nable to cover higher photon energy ranges and probe the gamma ray\nflux down to $10^{-14} {\\rm cm^{-2} s^{-1}}$. The predicted\nmonochromatic fluxes in Figs.~\\ref{fig:fluxgam-tb35}.c and\n\\ref{fig:fluxgam-tb35}.d  are still below this sensitivity limit.\nHowever, as we have stressed,  these results are based on a rather\nconservative halo density profile. In view of that the predicted\nflux can increase even by a factor of $10^4$ if one uses an\nextreme halo model like the spiked profile, the monochromatic\ngamma ray signal for the Higgsino dark matter might be measurable\nin case of a cuspy halo density profile.\n\n\n\n\nAs we have anticipated, other values of $(a_i,c_i)$ specified in\n(\\ref{4choices}) allow a mixed Bino-Higgsino LSP over a\nsignificant fraction of parameter space. It is thus expected that\nthose other cases can give a larger gamma ray flux compared to the\ncase with $a_i=c_i=1$. In Fig.~\\ref{fig:fluxgam-nzmw3}, we\ndepicted the results for the case with $a_{\\rm H}=c_{\\rm H}=0$ and\n$a_{\\rm M}=c_{\\rm M}=1\/2$. Indeed, this case gives a  larger flux,\nalthough not dramatically different. The red WMAP points can give\na continuum gamma ray flux up to $3 \\times 10^{-11} {\\rm cm^{-2}\ns^{-1}}$, while the cyan points giving  smaller thermal relic DM\ndensity can reach up to $2\\times 10^{-10} {\\rm cm^{-2} s^{-1}}$.\nThe maximal flux of monochromatic gamma ray is about $10^{-15}\n{\\rm cm^{-2} s^{-1}}$ for the red WMAP points and about  $7 \\times\n10^{-15} {\\rm cm^{-2} s^{-1}}$ for the cyan points. Again, these\nresults are obtained for the conservative halo density profile\ngiving $\\bar J(\\Delta\\Omega=10^{-3} {\\rm sr}) \\sim 30$. The real\ngamma ray flux can be significantly bigger than these predictions\nif the actual halo density profile is denser  than the assumed\nprofile.\n\n\n\n\n\n\\section{Conclusions}\n\nIn this paper, we have examined the prospect of  neutralino dark\nmatter in mirage mediation scenario of SUSY breaking in which soft\nmasses receive comparable contributions from modulus mediation and\nanomaly mediation. Depending upon the model parameters, especially\nthe anomaly to modulus mediation ratio, the nature of the lightest\nneutralino changes from Bino-like to Higgsino-like via\nBino-Higgsino mixing region. For Bino-like LSP, the conventional\nthermal production mechanism can give a right amount of relic DM\ndensity, i.e. the WMAP observation $0.085 < \\Omega_{DM}h^2 <\n0.119$, through the stop\/stau-neutralino coannihilation process or\nthe pseudo-scalar Higgs resonance effect. In overall, compared to\nthe mSUGRA scenario, a significantly larger fraction of the\nparameter space  can give the WMAP DM density under the assumption\nof thermal production, while satisfying all known phenomenological\nconstraints. This is partly because the lightest neutralino is a\nmixed Bino-Higgsino over a sizable fraction of the parameter\nspace.\n\n\nWe also studied the detection possibilities of neutralino dark\nmatter in mirage mediation.  For the parameter region giving the\nWMAP density of Bino-like or Higgsino-like LSP, direct detection\nvia elastic scattering between neutralino DM and nuclear target\nturns out to be mostly under the sensitivity of near future\nexperiments. However the other parameter region giving the WMAP\ndensity of mixed Bino-Higgsino LSP predicts typically a cross\nsection above the expected sensitivity limit of SuperCDMS. The\ncontinuum and monochromatic gamma ray fluxes from neutralino\nannihilation in Galactic Center have been analyzed also.\nGenerically, Higgsino-like LSP gives a larger gamma ray flux than\nBino-like LSP, however the continuum gamma ray flux from Bino LSP\ncan be significantly enhanced for some particular parameter values\ndue to the pseudo-scalar Higgs resonance effect. Although the\ngamma ray fluxes predicted within a conservative halo model are\nbelow the sensitivity of ongoing and planned experiments, it might\nbe detectable if the actual halo density  is denser  than the\nconservative profile used in our analysis.\n\n\n\\bigskip\n\n\n\n\\acknowledgments We thank Kwang-Sik Jeong for helpful discussions\nand also for clarifying various conventions for soft terms. This\nwork is supported by the KRF Grant KRF-2005-201-C00006 funded by\nthe Korean Government (K.C. and Y.S.), the KOSEF Grant\nR01-2005-000-10404-0 (K.C. and Y.S.), the Center for High Energy\nPhysics of Kyungpook National University (K.C.), the BK21 program\nof Ministry of Education (K.Y.L.), and the Astrophysical Research\nCenter for the Structure and Evolution of the Cosmos funded by the\nKOSEF (Y.G.K.). K.O. has been supported by the grant-in-aid for\nscientific research on priority areas (No. 441): \"Progress in\nelementary particle physics of the 21 century through discoveries\nof Higgs boson and supersymmetry\" (No. 16081209) from the Ministry\nof Education, Culture, Sports, Science and Technology of Japan.\nK.O. and Y.S. thank Yukawa Institute in Kyoto University for the\nuse of Altix3700 BX2 by which much of the numerical calculation\nhas been made. Y.S. also thanks the Particle Theory and Cosmology\nGroup at Tohoku University for the use of the computer facility.\n\n\n\n\n\n\n\\section*{Appendix A.}\n\n\\vskip 0.5cm In this appendix, we summarize  the notations and\nconventions used in this paper. The quantum effective action in\n$N=1$ superspace is given by \\begin{eqnarray} && \\int d^4\\theta\n\\left[-3CC^*e^{-K\/3} +\\frac{1}{16}\\left(\nG_aW^{a\\alpha}\\frac{D^2}{\\partial^2}W^a_\\alpha+{\\rm\nh.c.}\\right)\\right] +\\left(\\,\\int d^2\\theta\\, C^3W+{\\rm h.c.}\\, \\right) \\nonumber \\\\\n&=& \\int d^4\\theta\n\\left[\\,-3CC^*e^{-K_0\/3}+CC^*e^{-K_0\/3}Z_i\\Phi^*_ie^{2V_aT_a}\\Phi_i\n +\\frac{1}{16}\\left(\\,\n G_aW^{a\\alpha}\\frac{D^2}{\\partial^2}W^a_\\alpha+{\\rm\n h.c.}\\,\\right)\\,\\right]\\nonumber \\\\\n &&+\\,\\left(\\,\\int d^2\\theta\\,\n C^3\\left[\\,W_0+\\frac{1}{6}\\lambda_{ijk}\\Phi_i\\Phi_j\\Phi_k\\,\\right]+{\\rm h.c.}\n\\,\\right)+..., \\end{eqnarray} where the gauge kinetic terms are written as a\n$D$-term operator to accommodate the radiative corrections to\ngauge couplings, and the ellipsis stands for the irrelevant higher\ndimensional operators. The K\\\"ahler potential $K$ is expanded as\n\\begin{eqnarray} K=K_0(T_A,T_A^*)+Z_i(T_A,T_A^*)\\Phi^*_ie^{2V_aT_a}\\Phi_i+...,\n\\end{eqnarray} where $V_a$ and $\\Phi_i$ denote the visible gauge and matter\nsuperfields given by \\begin{eqnarray}\n\\Phi^i&=&\\phi^i+\\sqrt{2}\\,\\theta\\psi^i+\\theta^2F^i,\\nonumber \\\\\nV^a &=& -\\theta\\sigma^\\mu\\bar\\theta A^a_\\mu\n-i\\bar\\theta^2\\theta\\lambda^a + i\\theta^2\\bar\\theta\\bar\\lambda^a +\n\\frac{1}{2}\\theta^2\\bar\\theta^2 D^a, \\end{eqnarray}  and $T_A=(C,T)$ are the\nSUSY breaking messengers including the conformal compensator\nsuperfield $C=C_0+\\theta^2F^C$ and the modulus superfield\n$T=T_0+\\sqrt{2}\\theta\\tilde{T}+\\theta^2F^T$. The radiative\ncorrections due to renormalizable gauge and Yukawa interactions\ncan be encoded in the matter K\\\"ahler metric $Z_i$ and the gauge\ncoupling superfield $G_a$ which is given by\n  \\begin{eqnarray} G_a\\,=\\,{\\rm Re}(f_a)+\\Delta G_a, \\end{eqnarray} where\n$f_a$ is the holomorphic gauge kinetic function and $\\Delta G_a$\nincludes  the $T_A$-dependent radiative correction to gauge\ncoupling. The superpotential is expanded as \\begin{eqnarray}\nW=W_0(T)+\\frac{1}{6}\\lambda_{ijk}(T)\\Phi_i\\Phi_j\\Phi_k+..., \\end{eqnarray}\nwhere $W_0(T)$ is the modulus superpotential stabilizing $T$. Here\nwe do not specify the mechanism to generate the MSSM Higgs\nparameters $\\mu$ and $B$, and treat them as free parameters\nconstrained only by the electroweak symmetry breaking condition.\nFor a discussion of $\\mu$ and $B$ in mirage mediation, see\nRef.~\\cite{Choi:2005uz}.\n\n\nFor the canonically normalized component fields, the above\nsuperspace action gives the following form of the running gauge\nand Yukawa couplings, the supersymmetric gaugino-matter fermion\ncoupling ${\\cal L}_{\\lambda\\psi}$, and the soft SUSY breaking\nterms: \\begin{eqnarray} \\frac{1}{g_a^2}&=& {\\rm Re}(G_a),\\quad y_{ijk}\\, =\\,\n\\frac{\\lambda_{ijk}}{\\sqrt{e^{-K_0}Z_iZ_jZ_k}},\\nonumber \\\\\n {\\cal\nL}_{\\lambda\\psi}&=&i\\sqrt{2}\\left(\\phi_i^* T^a\\psi_i\\lambda^a\n-\\bar\\lambda^aT^a\\phi_i\\bar\\psi_i \\right), \\nonumber\n\\\\\n{\\cal L}_{\\rm soft}&=&-m^2_i\\phi^i\\phi^{i*}\n-\\left(\\,\\frac{1}{2}M_a\\lambda^a\\lambda^a\n+\\frac{1}{6}A_{ijk}y_{ijk}\\phi^i\\phi^j\\phi^k +{\\rm h.c.}\\right),\n\\end{eqnarray} where \\begin{eqnarray} M_a &=& F^A\\partial_A\\ln ({\\rm Re}(G_a)),\n\\nonumber \\\\\nA_{ijk} &=& -F^A\\partial_A \\ln\\left(\n\\frac{\\lambda_{ijk}}{e^{-K_0}Z_iZ_jZ_k}\\right),\n\\nonumber \\\\\nm^2_i &=&  -F^AF^{B*}\\partial_A\\partial_{\\bar B} \\ln\\left(\ne^{-K_0\/3}Z_i\\right) \\end{eqnarray} for \\begin{eqnarray} F^T&=&\n-e^{K_0\/2}(\\partial_T\\partial_{T^*})^{-1}(D_TW_0)^*,\\nonumber \\\\\nF^C&=&m_{3\/2}^*+\\frac{1}{3}\\partial_TK_0F^T \\quad(m_{3\/2}=\ne^{K_0\/2}W_0). \\end{eqnarray} In the approximation ignoring the off-diagonal\ncomponents of $w_{ij}=\\sum_{pq}y_{ipq}y^*_{jpq}$, the 1-loop RG\nevolution of soft parameters is determined by \\begin{eqnarray}\n{16\\pi^2}\\frac{dM_a}{d\\ln\\mu}&=& 2 \\left[-3\\,{\\rm\ntr}\\Big(T_a^{2}({\\rm Adj})\\Big) +\\sum_i {\\rm\ntr}\\Big(T_a^{2}(\\phi^i)\\Big) \\right] g^2_aM_a,\n\\nonumber \\\\\n{16\\pi^2}\\frac{dA_{ijk}}{d\\ln\\mu} &=& \\left[\n\\sum_{p,q}|y_{ipq}|^2A_{ipq} - 4 \\sum_a g^2_aC_2^a(\\phi^i) M_a\n\\right] + \\Big[i \\leftrightarrow j\\Big] + \\Big[i \\leftrightarrow\nk\\Big],\n\\nonumber \\\\\n{16\\pi^2}\\frac{d m^2_i}{d\\ln\\mu} &=&\n\\sum_{j,k}|y_{ijk}|^2\\left(m^2_i+m^2_j+m^2_k+|A_{ijk}|^2\\right)\n\\nonumber \\\\ &-& 8 \\sum_a g^2_aC_2^a(\\phi^i)|M_a|^2 +2g_1^2q_i\n\\sum_j q_j m^2_j,\n\\end{eqnarray} where the quadratic Casimir $C^a_2(\\phi_i)=(N^2-1)\/2N$ for a\nfundamental representation $\\phi_i$ of the gauge group $SU(N)$,\n$C_2^a(\\phi_i)=q_i^2$ for the $U(1)$ charge $q_i$ of $\\phi_i$.\n\n\n\nIn mirage mediation, soft terms at $M_{GUT}$ are determined by the\nmodulus mediation of ${\\cal O}(F^T\/T)$ and the anomaly mediation\nof ${\\cal O}(F^C\/8\\pi^2 C_0)$ which are comparable to each other.\nIn the presence of the axionic shift symmetry \\begin{eqnarray} U(1)_T: \\quad\n{\\rm Im}(T)+ \\mbox{real constant}\\end{eqnarray} which is broken by the\nnon-perturbative term in the modulus superpotential \\begin{eqnarray}\nW_0=w-Ae^{-aT},\\end{eqnarray} one can always make that $m_{3\/2}$ and $F^T$\nare simultaneously real. Also since $F^T\/T\\sim m_{3\/2}\/4\\pi^2$, we\nhave \\begin{eqnarray} \\frac{F^C}{C_0}= m_{3\/2}\\left(\\,1+{\\cal\nO}\\left(\\frac{1}{8\\pi^2}\\right)\\,\\right). \\end{eqnarray} Then, upon ignoring\nthe parts of ${\\cal O}(F^T\/8\\pi^2 T)$, the resulting soft\nparameters at $M_{GUT}$ are given by\n\\begin{eqnarray}\n M_a&=& M_0 +\\frac{m_{3\/2}}{16\\pi^2}\\,b_ag_a^2,\n\\nonumber \\\\\nA_{ijk}&=&\\tilde{A}_{ijk}-\n\\frac{m_{3\/2}}{16\\pi^2}\\,(\\gamma_i+\\gamma_j+\\gamma_k),\n\\nonumber\\\\\nm_i^2&=& \\tilde{m}_i^2-\\frac{m_{3\/2}}{16\\pi^2}M_0\\,\\theta_i\n-\\left(\\frac{m_{3\/2}}{16\\pi^2}\\right)^2\\dot{\\gamma}_i,\n\\end{eqnarray} where \\begin{eqnarray} M_0&=&F^T\\partial_T\\ln{\\rm Re}(f_a),\n\\nonumber\n\\\\\n\\tilde{A}_{ijk}&\\equiv& (a_i+a_j+a_k)M_0\n\\,=\\,F^T\\partial_T\\ln(e^{-K_0}Z_iZ_jZ_k),\\nonumber \\\\\n\\tilde{m}_i^2&\\equiv& c_iM_0^2\\,=\\,\n-|F^T|^2\\partial_T\\partial_{\\bar{T}} \\ln(e^{-K_0\/3}Z_i),\\end{eqnarray} and\n  \\begin{eqnarray} b_a&=&-3{\\rm tr}\\left(T_a^2({\\rm\nAdj})\\right)+\\sum_i {\\rm tr}\\left(T^2_a(\\phi_i)\\right),\n\\nonumber \\\\\n\\gamma_i&=&2\\sum_a\ng_a^2C^a_2(\\phi_i)-\\frac{1}{2}\\sum_{jk}|y_{ijk}|^2, \\nonumber\n\\\\\n\\theta_i&=& 4\\sum_a g_a^2 C^a_2(\\phi_i)-\\sum_{jk}|y_{ijk}|^2\n(a_i+a_j+a_k), \\nonumber\n\\\\\n\\dot{\\gamma}_i&=&8\\pi^2\\frac{d\\gamma_i}{d\\ln\\mu},\\end{eqnarray} where\n$\\omega_{ij}=\\sum_{kl}y_{ikl}y^*_{jkl}$ is assumed to be diagonal.\nHere we have used that $\\lambda_{ijk}$ are $T$-independent\nconstant as ensured by the axionic shift symmetry $U(1)_T$.\n\n\nLet us now summarize our conventions for the MSSM. The\nsuperpotential of canonically normalized matter superfields is\ngiven by \\begin{eqnarray} W &=& y_DH_d\\cdot QD^c+y_LH_d\\cdot LE^c-y_UH_u\\cdot\nQU^c - \\mu H_d\\cdot H_u, \\end{eqnarray} where the $SU(2)_L$ product is\n$H\\cdot Q=\\epsilon_{ab}H^aQ^b$ with\n$\\epsilon_{12}=-\\epsilon_{21}=1$, and color indices are\nsuppressed.  Then the chargino and neutralino mass matrices are\ngiven by \\begin{eqnarray} -\\frac{1}{2}\\,\\tilde\\psi^{-T}{\\cal M}_C\\tilde\\psi^+\n-\\frac{1}{2}\\,\\tilde\\psi^{0T}{\\cal M}_N\\tilde\\psi^0 + {\\rm h.c.},\n\\end{eqnarray} where \\begin{eqnarray} {\\cal M}_C &=& \\left(\n\\begin{array}{cc}\n- M_2\\,\\,\n& g_2 \\langle H^0_u \\rangle \\\\\ng_2 \\langle H^0_d \\rangle & \\mu\n\\end{array}\n\\right),\\nonumber \\\\\n{\\cal M}_N &=& \\left(\n\\begin{array}{cccc}\n-M_1 & 0 & -\\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_d \\rangle\n& \\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_u \\rangle \\\\\n0 & -M_2 & \\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_d \\rangle\n& -\\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_u \\rangle \\\\\n  -\\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_d \\rangle\n& \\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_d \\rangle\n& 0 & -\\mu  \\\\\n  \\frac{1}{\\sqrt 2}\\,g_Y \\langle H^0_u \\rangle\n& -\\frac{1}{\\sqrt 2}\\,g_2 \\langle H^0_u \\rangle & -\\mu & 0\n\\end{array}\n\\right), \\end{eqnarray} in the field basis \\begin{eqnarray} \\tilde\\psi^{+T} &=&\n-i\\left(\\tilde W^+,\\, i\\tilde H^+_u \\right), \\quad \\tilde\\psi^{-T}\n\\,=\\, -i\\left(\\tilde W^-,\\, i\\tilde H^-_d \\right),\n\\nonumber \\\\\n\\tilde\\psi^{0T} &=& -i\\left( \\tilde B,\\,\\tilde W^3,\\, i\\tilde\nH^0_d,\\,i\\tilde H^0_u \\right), \\end{eqnarray} for $\\tilde W^{\\pm}=(\\tilde\nW^1\\mp i \\tilde W^2)\/\\sqrt 2.$\n\n\nThe one-loop beta function coefficients $b_a$ and anomalous\ndimension $\\gamma_i$ in the MSSM are given by \\begin{eqnarray} b_3&=&-3, \\qquad\nb_2=1,\\qquad b_1=\\frac{33}{5},\n\\nonumber \\\\\n\\gamma_{H_u} &=& \\frac{3}{2}g_2^2+\\frac{1}{2}g_Y^2 -3y_t^2,\n\\nonumber \\\\\n\\gamma_{H_d} &=& \\frac{3}{2}g_2^2+\\frac{1}{2}g_Y^2 - 3 y_b^2 -  y_\\tau^2\n\\nonumber \\\\\n\\gamma_{Q_a} &=& \\frac{8}{3} g_3^2 + \\frac{3}{2} g_2^2\n                +\\frac{1}{18} g_Y^2 - (y_t^2 + y_b^2) \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{U_a} &=& \\frac{8}{3} g_3^2  + \\frac{8}{9} g_Y^2\n                - 2 y_t^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{D_a} &=& \\frac{8}{3} g_3^2 + \\frac{2}{9} g_Y^2\n                - 2 y_b^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{L_a} &=& \\frac{3}{2} g_2^2 + \\frac{1}{2} g_Y^2\n                - y_\\tau^2 \\delta_{3a},\n\\nonumber \\\\\n\\gamma_{E_a} &=& 2 g_Y^2 - 2 y_\\tau^2 \\delta_{3a}, \\end{eqnarray} where\n$g_2$ and $g_Y=\\sqrt{3\/5}g_1$ denote the $SU(2)_L$ and $U(1)_Y$\ngauge couplings.\n The\n$\\theta_i$ and $\\dot{\\gamma}_i$ which determine the soft scalar\nmasses at $M_{GUT}$ are given by \\begin{eqnarray} \\theta_{H_u} &=&\n3g_2^2+g_Y^2 -6y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}), \\nonumber \\\\\n\\theta_{H_d} &=& 3g_2^2+g_Y^2 - 6y_b^2(a_{H_d}+a_{Q_3}+a_{D_3}) -\n2y_\\tau^2(a_{H_d}+a_{L_3}+a_{E_3})\n\\nonumber \\\\\n\\theta_{Q_a} &=& \\frac{16}{3} g_3^2 + 3 g_2^2\n                +\\frac{1}{9} g_Y^2 - 2\\Big(y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}) + y_b^2(a_{H_d}+a_{Q_3}+a_{D_3})\\Big) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{U_a} &=& \\frac{16}{3} g_3^2  + \\frac{16}{9} g_Y^2\n                - 4y_t^2(a_{H_u}+a_{Q_3}+a_{U_3}) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{D_a} &=& \\frac{16}{3} g_3^2 + \\frac{4}{9} g_Y^2\n                - 4y_b^2(a_{H_d}+a_{Q_3}+a_{D_3}) \\delta_{3a},\n\\nonumber \\\\\n\\theta_{L_a} &=& 3 g_2^2 + g_Y^2\n                - 2y_\\tau^2 (a_{H_d}+a_{L_3}+a_{E_3})\\delta_{3a},\n\\nonumber \\\\\n\\theta_{E_a} &=& 4 g_Y^2 - 4 y_\\tau^2(a_{H_d}+a_{L_3}+a_{E_3})\n\\delta_{3a}, \\end{eqnarray} and \\begin{eqnarray} \\dot\\gamma_{H_u} &=& \\frac{3}{2} g_2^4\n+ \\frac{11}{2} g_Y^4\n                     - 3 y_t^2 b_{y_t},\n\\nonumber \\\\\n\\dot\\gamma_{H_d} &=& \\frac{3}{2} g_2^4 + \\frac{11}{2} g_Y^4\n                    - 3 y_b^2 b_{y_b} - y_\\tau^2 b_{y_\\tau},\n\\nonumber \\\\\n\\dot \\gamma_{Q_a} &=&  -8 g_3^4 + \\frac{3}{2} g_2^4 + \\frac{11}{18} g_Y^4\n                  -(y_t^2 b_{y_t} + y_b^2  b_{y_b}) \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{U_a} &=& - 8 g_3^4  +  \\frac{88}{9} g_Y^4\n                   - 2 y_t^2 b_{y_t} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{D_a} &=& - 8 g_3^4 + \\frac{22}{9} g_Y^4\n                     - 2 y_b^2 b_{y_b} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{L_a} &=& \\frac{3}{2}g_2^4 + \\frac{11}{2} g_Y^4\n                     - y_\\tau^2 b_{y_\\tau} \\delta_{3a},\n\\nonumber \\\\\n\\dot\\gamma_{E_a} &=& 22 g_Y^4 - 2 y_\\tau^2 b_{y_\\tau} \\delta_{3a},\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nb_{y_t} &=& - \\frac{16}{3} g_3^2 - 3 g_2^2 - \\frac{13}{9} g_Y^2\n             + 6 y_t^2 + y_b^2,\n\\nonumber \\\\\nb_{y_b} &=& - \\frac{16}{3} g_3^2 - 3 g_2^2 - \\frac{7}{9} g_Y^2\n             + y_t^2 + 6 y_b^2 + y_\\tau^2,\n\\nonumber \\\\\nb_{y_\\tau} &=& - 3 g_2^2 - 3 g_Y^2 + 3 y_b^2  + 4 y_\\tau^2. \\end{eqnarray}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nTwo classfiles namely \\file{cas-sc.cls} and \\file{cas-dc.cls} were\nwritten for typesetting articles submitted in journals of Elsevier's\nComplex Article Service (CAS) workflow.\n\n\\subsection{Usage}\n\\begin{enumerate}\n\\item \\file{cas-sc.cls} for single column journals. \n\n\\begin{vquote}\n \\documentclass[<options>]{cas-sc}\n\\end{vquote}\n\\item \\file{cas-dc.cls} for single column journals. \n\n\\begin{vquote}\n \\documentclass[<options>]{cas-dc}\n\\end{vquote}\n\\end{enumerate}\nand have an option longmktitle to handle long front matter. \n\n\\section{Front matter}\n\n\\begin{vquote}\n\\title [mode = title]{This is a specimen $a_b$ title}                      \n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n   project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer text \n   matter to fill through the whole text width and overflow into\n   another line in the footnotes area of the first page.}\n\n\\author[1,3]{CV Radhakrishnan}[type=editor,\n                        auid=000,bioid=1,\n                        prefix=Sir,\n                        role=Researcher,\n                        orcid=0000-0001-7511-2910]\n\\cormark[1]\n\\fnmark[1]\n\\ead{cvr_1@tug.org.in}\n\\ead[url]{www.cvr.cc, cvr@sayahna.org}\n\\end{vquote}\n\n\\begin{vquote}\n\n\\credit{Conceptualization of this study, Methodology, \n        Software}\n\n\\address[1]{Elsevier B.V., Radarweg 29, 1043 NX Amsterdam, \n            The Netherlands}\n\n\\author[2,4]{Han Theh Thanh}[style=chinese]\n\n\\author[2,3]{CV Rajagopal}[%\n   role=Co-ordinator,\n   suffix=Jr,\n   ]\n\\fnmark[2]\n\\ead{cvr3@sayahna.org}\n\\ead[URL]{www.sayahna.org}\n\n\\credit{Data curation, Writing - Original draft preparation}\n\n\\address[2]{Sayahna Foundation, Jagathy, Trivandrum 695014, \n            India}\n\n\\author[1,3]{Rishi T.}\n\\cormark[2]\n\\fnmark[1,3]\n\\ead{rishi@stmdocs.in}\n\\ead[URL]{www.stmdocs.in}\n\n\\address[3]{STM Document Engineering Pvt Ltd., Mepukada,\n    Malayinkil, Trivandrum 695571, India}\n\n\\cortext[cor1]{Corresponding author}\n\\cortext[cor2]{Principal corresponding author}\n\\fntext[fn1]{This is the first author footnote. but is common \n  to third author as well.}\n\\fntext[fn2]{Another author footnote, this is a very long \n  footnote and it should be a really long footnote. But this \n  footnote is not yet sufficiently long enough to make two lines \n  of footnote text.}\n\\end{vquote}\n\n\\begin{vquote}\n\\nonumnote{This note has no numbers. In this work we \n  demonstrate $a_b$ the formation Y\\_1 of a new type of \n  polariton on the interface between a cuprous oxide slab \n  and a polystyrene micro-sphere placed on the slab.\n  }\n\n\\begin{abstract}[S U M M A R Y]\nThis template helps you to create a properly formatted \n \\LaTeX\\ manuscript.\n\n\\noindent\\texttt{\\textbackslash begin{abstract}} \\dots \n\\texttt{\\textbackslash end{abstract}} and\n\\verb+\\begin{keyword}+ \\verb+...+ \\verb+\\end{keyword}+ \nwhich contain the abstract and keywords respectively. \nEach keyword shall be separated by a \\verb+\\sep+ command.\n\\end{abstract}\n\n\\begin{keywords}\nquadrupole exciton \\sep polariton \\sep \\WGM \\sep \\BEC\n\\end{keywords}\n\n\\maketitle\n\\end{vquote}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{sc-sample.pdf}\n\\caption{Single column output (classfile: cas-sc.cls).}\n\\end{figure}\n\n\\begin{figure}\n\\includegraphics[width=\\textwidth]{dc-sample.pdf}\n\\caption{Double column output (classfile: cas-dc.cls).}\n\\end{figure}\n\n\\subsection{Title}\n\n\\verb+\\title+ command have the below options:\n\\begin{enumerate}\n\\item \\verb+title:+ Document title\n\\item \\verb+alt:+ Alternate title\n\\item \\verb+sub:+ Sub title\n\\item \\verb+trans:+ Translated title\n\\item \\verb+transsub:+ Translated sub title\n\\end{enumerate}\n\n\\begin{vquote}\n \\title[mode=title]{This is a title}\n \\title[mode=alt]{This is a alternate title}\n \\title[mode=sub]{This is a sub title}\n \\title[mode=trans]{This is a translated title}\n \\title[mode=transsub]{This is a translated sub title}\n\\end{vquote}\n\n\n\\subsection{Author}\n\\verb+\\author+ command have the below options: \n\n\\begin{enumerate}\n\\item \\verb+auid:+ Author id\n\\item \\verb+bioid:+ Biography id\n\\item \\verb+alt:+ Alternate author\n\\item \\verb+style:+ Style of author name chinese\n\\item \\verb+prefix:+ Prefix Sir\n\\item \\verb+suffix:+ Suffix\n\\item \\verb+degree:+ Degree\n\\item \\verb+role:+ Role\n\\item \\verb+orcid:+ ORCID\n\\item \\verb+collab:+ Collaboration\n\\item \\verb+anon:+ Anonymous author\n\\item \\verb+deceased:+ Deceased author\n\\item \\verb+twitter:+ Twitter account\n\\item \\verb+facebook:+ Facebook account\n\\item \\verb+linkedin:+ LinkedIn account\n\\item \\verb+plus:+ Google plus account\n\\item \\verb+gplus:+ Google plus account\n\\end{enumerate}\n\n\\begin{vquote}\n\\author[1,3]{Author Name}[type=editor,\n    auid=000,bioid=1,\n    prefix=Sir,\n    role=Researcher,\n    orcid=0000-0001-7511-2910,\n    facebook=<facebook id>,\n    twitter=<twitter id>,\n    linkedin=<linkedin id>,\n    gplus=<gplus id>]\n\\end{vquote}\n\n\\subsection{Various Marks in the Front Matter}\n\nThe front matter becomes complicated due to various kinds\nof notes and marks to the title and author names. Marks in\nthe title will be denoted by a star ($\\star$) mark;\nfootnotes are denoted by super scripted Arabic numerals,\ncorresponding author by of an Conformal asterisk (*) mark.\n\n\\subsubsection{Title marks}\n\nTitle mark can be entered by the command, \\verb+\\tnotemark[<num>]+\nand the corresponding text can be entered with the command\n\\verb+\\tnotetext[<num>]+ \\verb+{<text>}+. An example will be:\n\n\\begin{vquote}\n\\title[mode=title]{Leveraging social media news to predict\n                      stock index movement using RNN-boost}\n\n\\tnotemark[1,2]\n\n\\tnotetext[1]{This document is the results of the research\n   project funded by the National Science Foundation.}\n\n\\tnotetext[2]{The second title footnote which is a longer \n   text matter to fill through the whole text width and \n   overflow into another line in the footnotes area of \n   the first page.}\n\\end{vquote}\n\n\\verb+\\tnotetext+ and \\verb+\\tnotemark+ can be anywhere in\nthe front matter, but shall be before \\verb+\\maketitle+ command.\n\n\\subsubsection{Author marks}\n\nAuthor names can have many kinds of marks and notes:\n\n\\begin{vquote}\n    footnote mark : \\fnmark[<num>]\n    footnote text : \\fntext[<num>]{<text>}\n    affiliation mark : \\author[<num>]\n    email : \\ead{<emailid>}\n    url : \\ead[url]{<url>}\n    corresponding author mark : \\cormark[<num>]\n    corresponding author text : \\cortext[<num>]{<text>}\n\\end{vquote}\n\n\\subsubsection{Other marks}\n\nAt times, authors want footnotes which leave no marks in\nthe author names. The note text shall be listed as part of\nthe front matter notes. Class files provides\n\\verb+\\nonumnote+ for this purpose. The usage\n\n\\begin{vquote}\n\\nonumnote{<text>}\n\\end{vquote}\n\n\\noindent and should be entered anywhere before the \\verb+\\maketitle+\ncommand for this to take effect. \n\n\\subsection{Abstract and Keywords}\n\nAbstract shall be entered in an environment that starts\nwith \\verb+\\begin{abstract}+ and ends with\n\\verb+\\end{abstract}+. Longer abstracts spanning more than\none page is also possible in Class file even in double\ncolumn mode. We need to invoke longmktitle option in the\nclass loading line for this to happen smoothly.\n\nThe key words are enclosed in a \\verb+{keyword}+\nenvironment.\n\n\\begin{vquote}\n\\begin{abstract}\n This is a abstract. \\lipsum[3]\n\\end{abstract}\n\n\\begin{keywords}\n First keyword \\sep Second keyword \\sep Third \n    keyword \\sep Fourth keyword\n\\end{keywords}\n\\end{vquote}\n\n\\section{Main Matter}\n\\subsection{Tables}\n\\subsubsection{Normal tables}\n\n\\begin{vquote}\n\\begin{table}\n  \\caption{This is a test caption.}\n  \\begin{tabular*}{\\tblwidth}{@{} LLLL@{} }\n   \\toprule\n    Col 1 & Col 2\\\\\n   \\midrule\n    12345 & 12345\\\\\n    12345 & 12345\\\\\n    12345 & 12345\\\\\n   \\bottomrule\n  \\end{tabular*}\n\\end{table}\n\\end{vquote}\n\n\\subsubsection{Span tables}\n\n\\begin{vquote}\n\\begin{table*}[width=.9\\textwidth,cols=4,pos=h]\n  \\caption{This is a test caption.}\n  \\begin{tabular*}{\\tblwidth}{@{} LLLLLL@{} }\n   \\toprule\n    Col 1 & Col 2 & Col 3 & Col4 & Col5 & Col6 & Col7\\\\\n   \\midrule\n    12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n    12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n    12345 & 12345 & 123 & 12345 & 123 & 12345 & 123 \\\\\n   \\bottomrule\n  \\end{tabular*}\n\\end{table*}\n\\end{vquote}\n\n\\subsection{Figures}\n\\subsubsection{Normal figures}\n\\begin{vquote}\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[scale=.75]{Fig1.pdf}\n\t\\caption{The evanescent light - $1S$ quadrupole coupling\n\t($g_{1,l}$) scaled to the bulk exciton-photon coupling\n\t($g_{1,2}$). The size parameter $kr_{0}$ is denoted as $x$ and\n\tthe \\PMS is placed directly on the cuprous oxide sample ($\\delta\n\tr=0$, See also Fig. \\protect\\ref{FIG:2}).}\n\t\\label{FIG:1}\n\\end{figure}\n\\end{vquote}\n\n\\subsubsection{Span figures}\n\n\\begin{vquote}\n\\begin{figure*}\n\t\\centering\n\t  \\includegraphics[width=\\textwidth,height=2in]{Fig2.pdf}\n\t\\caption{Schematic of formation of the evanescent polariton on\n\tlinear chain of \\PMS. The actual dispersion is determined by \n  the ratio of two coupling parameters such as exciton-\\WGM \n  coupling and \\WGM-\\WGM coupling between the microspheres.}\n  \\label{FIG:2}\n\\end{figure*}\\end{vquote}\n\n\\subsection{Theorem and theorem like environments}\n\nCAS class file provides a few hooks to format theorems and\ntheorem like environments with ease. All commands the\noptions that are used with \\verb+\\newtheorem+ command will work\nexactly in the same manner. Class file provides three\ncommands to format theorem or theorem like environments:\n\n\\begin{enumerate}\n\\item \\verb+\\newtheorem+ command formats a theorem in\n\\LaTeX's default style with italicized font for theorem\nstatement, bold weight for theorem heading and theorem\nnumber typeset at the right of theorem heading. It also\noptionally accepts an argument which will be printed as an\nextra heading in parentheses. Here is an example coding and\noutput:\n\n\\begin{vquote}\n\\newtheorem{theorem}{Theorem}\n\\begin{theorem}\\label{thm}\n The \\WGM evanescent field penetration depth into the \n cuprous oxide adjacent crystal is much larger than the \n \\QE radius: \n \\begin{equation*}\n  \\lambda_{1S}\/2 \\pi \\left({\\epsilon_{Cu2O}-1}\n    \\right)^{1\/2} = 414 \\mbox{ \\AA} \\gg a_B = 4.6 \n    \\mbox{ \\AA}  \n \\end{equation*}\n\\end{theorem}\n\\end{vquote}\n\n\\item \\verb+\\newdefinition+ command does exactly the same\nthing as with except that the body font is up-shape instead\nof italic. See the example below:\n\n\\begin{vquote}\n\\newdefinition{definition}{Definition}\n\\begin{definition}\n The bulk and evanescent polaritons in cuprous oxide\n are formed through the quadrupole part of the light-matter\n interaction:\n \\begin{equation*}\n  H_{int} = \\frac{i e }{m \\omega_{1S}} {\\bf E}_{i,s} \n    \\cdot {\\bf p}\n \\end{equation*}\n\\end{definition}\n\\end{vquote}\n\n\\item \\verb+\\newproof+ command helps to define proof and\ncustom proof environments without counters as provided in\nthe example code. Given below is an example of proof of\ntheorem kind.\n\n\\begin{vquote}\n\\newproof{pot}{Proof of Theorem \\ref{thm}}\n\\begin{pot}\n The photon part of the polariton trapped inside the \\PMS\n moves as it would move in a micro-cavity of the effective\n modal volume $V \\ll 4 \\pi r_{0}^{3} \/3$. Consequently, it\n can escape through the evanescent field. This evanescent\n field essentially has a quantum origin and is due to\n tunneling through the potential caused by dielectric\n mismatch on the \\PMS surface. Therefore, we define the\n \\emph{evanescent} polariton (\\EP) as an evanescent light -\n \\QE coherent superposition.\n\\end{pot}\n\\end{vquote}\n\n\\end{enumerate}\n\n\\subsection{Enumerated and Itemized Lists}\n\nCAS class files provides an extended list processing macros\nwhich makes the usage a bit more user friendly than the\ndefault LaTeX list macros. With an optional argument to the\n\\verb+\\begin{enumerate}+ command, you can change the list\ncounter type and its attributes. You can see the coding and\ntypeset copy. \n\n\\begin{vquote}\n\\begin{enumerate}[1.]\n  \\item The enumerate environment starts with an optional\n        argument `1.' so that the item counter will be suffixed\n        by a period as in the optional argument.\n  \\item If you provide a closing parenthesis to the number in the\n        optional argument, the output will have closing \n        parenthesis for all the item counters.\n  \\item You can use `(a)' for alphabetical counter and `(i)' for\n        roman counter.\n  \\begin{enumerate}[a)]\n    \\item Another level of list with alphabetical counter.\n    \\item One more item before we start another.\n    \\begin{enumerate}[(i)]\n      \\item This item has roman numeral counter.\n\\end{vquote}\n\n\\begin{vquote}\n      \\item Another one before we close the third level.\n    \\end{enumerate}\n    \\item Third item in second level.\n  \\end{enumerate}\n  \\item All list items conclude with this step.\n\\end{enumerate}\n\n\\section{Biography}\n\n\\verb+\\bio+ command have the below options:\n\\begin{enumerate}\n \\item \\verb+width:+ Width of the author photo (default is 1in).\n \\item \\verb+pos:+ Position of author photo.\n\\end{enumerate}\n\n\\begin{vquote}\n\\bio[width=10mm,pos=l]{tuglogo.jpg}\n \\textbf{Another Biography:}\n  Recent experimental \\cite{HARA:2005} and theoretical\n  \\cite{DEYCH:2006} studies have shown that the \\WGM can travel\n  along the chain as \"heavy photons\". Therefore the \\WGM \n  acquires the spatial dispersion, and the evanescent \n  quadrupole polariton has the form (See Fig.\\ref{FIG:3}):\n\\endbio\n\\end{vquote}\n\n\\section[CRediT...]{CRediT authorship contribution statement}\n\nGive the authorship contribution after each author as \n\n\\begin{vquote}\n \\credit{Conceptualization of this study, Methodology, \n         Software}\n\\end{vquote}\n\nTo print the details use \\verb+\\printcredits+ \n\n\\begin{vquote}\n \\author[1,3]{V. {{\\=A}}nand Rawat}[auid=000,\n                   bioid=1,\n                   prefix=Sir,\n                   role=Researcher,\n                   orcid=0000-0001-7511-2910]\n\\end{vquote}\n\n\\begin{vquote}\n  \\cormark[1]\n  \\fnmark[1]\n  \\ead{cvr_1@tug.org.in}\n  \\ead[url]{www.cvr.cc, www.tug.org.in}\n\n  \\credit{Conceptualization of this study, Methodology, \n          Software}\n\n  \\address[1]{Indian \\TeX{} Users Group, Trivandrum 695014, \n       India}\n\n  \\author[2,4]{Han Theh Thanh}[style=chinese]\n\n  \\author[2,3]{T. Rishi Nair}[role=Co-ordinator,\n                   suffix=Jr]\n  \\fnmark[2]\n  \\ead{rishi@sayahna.org}\n  \\ead[URL]{www.sayahna.org}\n\n  \\credit{Data curation, Writing - Original draft preparation}\n\n  . . .\n  . . .\n  . . .\n  \\printcredits\n\\end{vquote}\n\n\\section{Bibliography}\n\nFor CAS categories, two reference models are recommended.\nThey are \\file{model1-num-names.bst} and \\file{model2-names.bst}.\nFormer will format the reference list and their citations according to\nnumbered scheme whereas the latter will format according name-date or\nauthor-year style. Authors are requested to choose any one of these\naccording to the journal style. You may download these from \n\nThe above bsts are available in the following location for you to\ndownload:\n\n\\url{https:\/\/support.stmdocs.in\/wiki\/index.php?title=Model-wise_bibliographic_style_files} \n\\hfill $\\Box$\n\n\\end{document}\n\n\n\\section{Introduction}\n\nThe inherent property of wetting refers to the preferential affinity of a fluid that is immersed in another immiscible fluid to coat a solid material \\cite{de1985wetting,bonn2009wetting,de2013capillarity}. Due to the ubiquitous existence of colloidal and interfacial phenomena in nature and applications, understanding the role of wetting is a key principal of interest. This intriguing interest in wetting behavior is motivated by numerous advanced technologies in nanotechnology, biological engineering, material science and geosciences \\cite{powell2011electric,xu2014proteins,blossey2003self,bartels2017oil}. For instance, the design of bio-inspired fluidics, directional fluid transportation, composite functional materials, nano\/micro-fluidics and energy storage systems require a precise and effective way to describe the wetting state therein. \n\nSince 1805, Young's equation, based on thermodynamic laws, is firmly established to infer the wettability of a flat and chemically homogeneous solid surface at equilibrium \\cite{young1805iii}, $\\cos \\theta_Y  = \\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}}$, where the subscripts denote the immiscible fluids (liquid $l$ and vapor $v$) and solid ($s$) for the associated surface free energies. $\\theta_Y$ is the contact angle formed at the microscopic contact point on the surface, as depicted in Fig. \\ref{fig1}. Wetting hysteresis due to the contribution of surface heterogeneity and contact line dynamics has been studied in detail over the last century \\cite{wenzel1936resistance,cassie1944wettability,decker1997contact}. However, despite theoretical and experimental investigations through the last 70 years \\cite{de1985wetting,bonn2009wetting,yu2015wetting,turmine2000thermodynamic,blunt2019thermodynamically,mahani2015kinetics}, several fundamental challenges in characterizing wetting state of complex multiphase systems remain unsolved and are currently pending. \n\nFor disordered and complex porous geometries ranging from catalysts in fuel cells to lungs in the respiratory system to porous glass filters to subsurface rocks, there are multiple length scales, various surface free energies, and surface heterogeneity involved. The convoluted interplay of physicochemical properties with multi-scale complexities therein presents significant spatial variability of contact angles along the three-phase contact line \\cite{sun2020probing,holtzman2015wettability}, which leads to the resulting wetting hysteresis behavior and pinning effects. Accordingly, these features trigger a formidable challenge to characterizing wetting behavior, which are far from being solved. It is therefore controversial whether Young's law is still applicable for these disordered and complex solids \\cite{garfi2019fluid}. \n\nIn the past, contact angle (and curvature) measurements were mainly based on two-dimensional (2D) projections, and in configurations that are optically transparent such as a typical sessile drop setup \\cite{kwok1997contact}. However, three-dimensional (3D) imaging provides the possibility to determine contact angles within opaque porous media \\cite{scanziani2017automatic,alratrout2017automatic,ibekwe2020automated,dalton2018methods}. Possible 3D imaging technologies are 3D X-ray microcomputed tomography (micro-CT) \\cite{wildenschild2013x} and confocal microscopy \\cite{sundberg2007contact}. Given the rapid developments in 3D imaging technologies, there is an opportunity for interface science to make use of this methodology where the wetting of various porous domains would be of interest. The first community that started adopting this new technology is the porous media community, out of pure necessity to measure the wettability of immiscible fluids in geological materials, which are naturally opaque \\cite{andrew2014pore,alratrout2018wettability,tudek2017situ}. These efforts rely on micro-CT images of porous rocks saturated with immiscible phases \\cite{wildenschild2013x}. The image voxels of a 3D system are segmented into respective phases \\cite{schluter2014image} followed by the identification of the three-phase contact line to measure spatially distributed apparent contact angles, $\\theta_{app}$, for each microscopic three-phase contact point along the contact line. As shown in the right side of Fig. \\ref{fig1}, the method measures the angle between the local tangential plane of liquid\/vapor interface and the solid surface in the vicinity of three-phase contact points. Thus, the approach represents \\textit{in situ} $\\theta_{app}$ directly along the contact line, which refers to microscopic wetting. However, the simplicity of $\\theta_{app}$ computed by the local method is susceptible to errors due to image quality and pixelation-related segmentation errors \\cite{armstrong2012linking,klise2016automated}. In particular, the identification of the three-phase contact line and measurement of an angle over a few voxels is inherently error-prone, as will be investigated herein. Ultimately, these systematic errors complicate the process of quantifying the wetting state of a porous multiphase system. \n\nAnother issue arises by using for the purpose of wetting characterization the concept of contact angle, which is subject to hysteresis. Due to hysteresis that is emphasized by complex geometries, surface heterogeneity, and interface dynamics, it is very questionable whether contact angle is a representative measure for characterizing wetting in such systems \\cite{wenzel1936resistance,cassie1944wettability,johnson1964contact,morrow1975effects,priest2007asymmetric}. In addition, the line tension and disjoining\/cojoining pressure occur along the contact line. Contact line tension, which is the excess free energy per unit length, is a dominant parameter in microscopic wetting \\cite{indekeu1994line}. It contributes to intermolecular force balance, resulting in contact line pinning and the associated hysteresis loop of contact angles \\cite{de1985wetting,bonn2009wetting,de2013capillarity}. Therefore, the variations on contact line curvature and the associated $\\theta_{app}$ vary from a microscopic point of view due to the effects of surface heterogeneity and flow dynamics, which yields unexpected wetting behavior. Despite these parameters influencing microscopic wetting and hysteresis behavior, the question arises as to whether $\\theta_{app}$ can capture enough information to represent the wetting state of the system and provide sufficient guidance for the design of functional surfaces?\n\n\\begin{figure}\n\\hspace*{-0.2cm}\\includegraphics[width=0.51\\textwidth]{FIG1.png}\n\\caption{The schematic illustration of topological principle based on integral geometry \\cite{sun2020probing} and Young's equation based on thermodynamics at a microscopic contact point when a sessile droplet is deposited on a flat solid substrate exposed to vapor phase. The schematic illustration of apparent contact angle $\\theta_{app}$ extraction by local measurement along the contact line.}\n\\label{fig1}\n\\end{figure}\n\nPrevious studies demonstrate that $\\theta_{app}$ alone provides an incomplete description of wetting where the contact line is asymmetric \\cite{rabbani2018pore}; especially for characterizing the macroscopic wetting behavior of multiphase systems, which has remained unexplored from a theoretical perspective. The wetting behavior synergistic affects on the phase topology and contact area with the solid surface, which must also be considered. It can be representative of the macroscopic wetting of the system and captures the complete microscopic information related to the thermodynamics. To this end, in our previous works \\cite{sun2020linking,sun2020probing}, we developed a theory based on topological principles that can effectively describe wetting behavior, which links across various length scales pertinent to wetting phenomena. Herein, we aim to quantitatively and qualitatively unravel the effective and robust nature of the proposed theory by providing in-depth analysis and comparison to other recent methods for characterizing the wettability of multiphase systems.  \n\n\\section{Theoretical Concepts}\n\\subsection{Deficit Curvature from Gaussian Curvature}\nTopological principle is applied to characterize the wetting state of a given multiphase system by the link of the Gauss-Bonnet theorem. For a fluid droplet $D$ that has a closed interface in the three-phase system, where liquid ($l$), vapor ($v$) and solid ($s$) are present, the total curvature of the fluid surface and its global topology are related by the Gauss-Bonnet theorem \\cite{chern1944simple,sun2020linking}. As a consequence, the Euler characteristic $\\chi$ and its total curvature for the surface $I$ of the droplet $D$ obey the following expression,\n\n\\begin{equation} \\label{eq:1}\n 2\\pi \\chi(D)=\\int_{I} \\kappa_G dA + \\int_{\\partial I} \\kappa_g dC,\n\n\n\n\\end{equation}\nwhere $dA$ and $dC$ are the droplet interfacial area element along the surface $I$ and the line element along the contact line $\\partial I$, respectively. $\\kappa_G$ and $\\kappa_g$ are Gaussian curvature along the droplet interface and geodesic curvature along the contact line $\\partial I$, respectively. \n\\begin{figure*}\n\\centering\\includegraphics[width=0.85\\textwidth]{FIG2}\n\\caption{Schematic diagram of 3D droplets on contact with a complex solid (transparent) with labeled definitions where grey color denotes the liquid\/vapor interface ($I_{lv}$), and blue color denotes the liquid\/solid interface ($I_{ls}$). The contact line loops are formed by three-phase contact points (yellow circles). From left to right, there is an increasing complexity of droplet morphology.}\n\\label{fig2}\n\\end{figure*}\n\nFor a 3D droplet, we can arrive at a generalized form of the expression by subdividing the fluid surface into liquid\/vapor ($lv$) and liquid\/solid ($ls$) interfaces,\n\n\\begin{equation} \\label{eq:2}\n\\begin{split}\n 4\\pi\\chi(D) &=2 \\pi \\chi(I_{lv}) + 2 \\pi \\chi (I_{ls}) \\\\\n &=\\int_{I_{lv}} \\kappa_{G}dA+\\int_{I_{ls}} \\kappa_{G}dA +\\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC. \n\\end{split}\n\\end{equation}\n\n\\noindent It indicates that the Euler characteristic of the droplet always remains constant and the geodesic curvature source term, $\\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC$, will change accordingly based on the contribution of total surface curvature due to the wetting behavior of the system. The deficit curvature, $\\Theta$, is defined as the summation of geodesic curvatures along the contact line relative to the tangential plane for each interface and corresponds to a total angle of change. By considering a droplet deposited on a flat, smooth and homogeneous surface as shown in the top of Fig.\\ref{fig1}, $\\Theta$ along the contact line can be expressed by the labeled notations as,\n\n\\begin{equation} \\label{eq:3}\n\\begin{split}\n\\Theta &= \\int_{\\partial I}(\\kappa_{g_{lv}}+\\kappa_{g_{ls}})dC \\\\\n&=\\int_{\\partial I} \\kappa_{lvs}\\mathbf{n}_{lvs}\\cdot \\big[\\mathbf{n}_s \\sin \\theta_Y + \\mathbf{n}_{ls} (1-\\cos \\theta_Y) \\big]dC,\n\\end{split}\n\\end{equation}\n\n\\noindent where $\\mathbf{n}_{lvs}$ is the normal vector to the contact line, which points in the direction of the curvature for the contact line. $\\mathbf{n}_{lv}$ is the outward normal vector along the contact line relative to the droplet interface. $\\mathbf{n}_{ls}$ is the outward normal vector along the contact line relative to the liquid\/solid interface. From Eq. (\\ref{eq:3}), it is evident that $\\Theta$ obtained by applying the topological principle is an explicit average of intrinsic contact angle, i.e. Young's angle ($\\theta_Y$) for this situation. Therefore, $\\Theta$ can be interpreted in terms of fluid morphology in a way that is not affected by the contact angle hysteresis due to surface heterogeneity, geometry complexity, and interface dynamics \\cite{sun2020probing}.\n\nFor solid surfaces that contain disordered and complex geometries or even with confined domains as shown in Fig. \\ref{fig2}, we can revisit Eq. (\\ref{eq:2}) to obtain $\\Theta$. Macroscopic contact angle $\\theta_{macro}$ can be obtained by using a normalizing factor $\\lambda$ to scale the desired contact angle interval in $\\theta_{macro}$ $\\in [0,\\pi]$,\n\n\\begin{equation} \\label{eq:4}\n\\begin{split}\n\\theta_{macro} &= \\lambda \\Theta\\\\\n&= \\lambda \\Big[4\\pi\\chi(D) - \\Big(\\int_{I_{lv}} \\kappa_{G}dA+\\int_{I_{ls}} \\kappa_{G}dA \\Big)\\Big].\n\\end{split}\n\\end{equation}\nThe nature of this formulation resolves a few pressing issues. Firstly, the angle along the contact line is not directly computed as an average from the sequence of local geometric contact angle point measurements. $\\theta_{macro}$ is inferred from interfacial curvature and area measurements along with a topological measurement. While these measures are still susceptible to pixelization effects, it is expected that these effects would be less than that resulting from local measurements along the three-phase contact line (as tested herein). Secondly, $\\theta_{macro}$ accounts for the complete geodesic curvature of the contact line, which is the curvature of the three-phase contact line relative to both the solid surface and liquid\/vapor interface. Therefore, the formulation captures wetting effects as evident by line tension and $\\theta_{app}$, which are both known to influence wetting behavior. \n\n\\subsection{Derivation of Young's Equation}\nIn this section, we show the direct link between the topological and thermodynamic concepts for determining the wetting state of the system by applying surface energy minimization and variational principles \\cite{seo2015re}. The Gauss-Bonnet theorem allows us to express the total curvature of a droplet $D$ in terms of deficit curvature, corresponding average curvature and surface area, \n\n\\begin{align}\n4 \\pi \\chi(D) = \\Theta + \\kappa_{lv} A_{lv} + \\kappa_{ls} A_{ls} \\;.\n\\label{eq:5}\n\\end{align}\n\nHere, we consider a variational principle of the internal energy as applied in thermodynamic approaches. The variation of the internal energy is given by Euler's homogeneous function theorem,\n\n\\begin{align}\n\\delta U = T \\delta S - p_v \\delta V_v - p_l\\delta V_l + \\sigma_{lv} \\delta A_{lv}\n\\nonumber\\\\ + \\sigma_{ls}\\delta A_{ls} + \\sigma_{vs} \\delta A_{vs} \\;.\n\\label{eq:6}\n\\end{align}\nBased on this, we consider a closed system with $\\delta U=0$. Droplet rearrangements can occur provided that the volume is not changed, meaning that $\\delta V_l = \\delta V_v = 0$. Therefore, entropy production in the system is strictly linked to the minimization of the surface energy,\n\n\\begin{align}\n\\delta S = - \\frac 1 T \\Big[ \\sigma_{lv} \\delta A_{lv}\n+ \\sigma_{ls}\\delta A_{ls} + \\sigma_{vs} \\delta A_{vs} \\Big] \\ge 0 \\;.\n\\label{eq:7}\n\\end{align}\nThe total surface area of the solid substrate is constant, then   \n\n\\begin{align}\n\\delta A_s = \\delta A_{vs} + \\delta A_{ls} = 0 \\;,\n\\label{eq:8}\n\\end{align}\nwhich can be used to eliminate one of the surface areas from Eq. (\\ref{eq:7}). The topological constraint from Eq. (\\ref{eq:5}) determines the condition that should be imposed to ensure that geometric variation occurs without changing the topology,\n\n\\begin{equation} \\label{eq:9}\n\\begin{split}\n\\delta \\big[ 4 \\pi \\chi(D)\\big] & =  \\delta \\Theta + \n\\kappa_{lv} \\delta A_{lv} + A_{lv} \\delta \\kappa_{lv} \\\\ &\\quad\n\\kappa_{ls} \\delta A_{ls} + A_{ls} \\delta \\kappa_{ls} \\\\ & = 0\n \\;.\n\\end{split}\n\\end{equation}\n\nWe shall impose Eq. (\\ref{eq:9}) as a constraint on Eq. (\\ref{eq:7}) using the method of Lagrange multipliers, also using Eq. (\\ref{eq:8}) to eliminate $A_{vs}$. Now, we can express the entropy production as\n\n\\begin{equation}\n\\begin{split}\n\\delta S &= - \\frac 1 T \\Big[\n\\underbrace{\\Big( \\sigma_{ls} - \\sigma_{vs}  + \\frac{\\sigma_{lv}\\kappa_{ls}}{\\kappa_{lv}} \\Big)\n}_{\\mbox{surface area variation}}\\delta A_{ls} \\\\\n &\\quad + \\frac{\\sigma_{lv}}{\\kappa_{lv}} \n\\underbrace{\\big( \\delta \\Theta + \n A_{lv} \\delta \\kappa_{lv}  + A_{ls} \\delta \\kappa_{ls} \\big)}_{\\mbox{total curvature variation}} \\Big]\n \\;. \n\\end{split}\n \\label{eq:10}\n\\end{equation}\nThis expression separates terms associated with the variation of the surface area from terms that redistribute curvature along the cluster boundary. Put another way, the second term corresponds to a redistribution of the total curvature that occurs at constant surface area due to the deformation of the droplet. One of the ways to redistribute the total curvature is by altering the deficit curvature, which changes the contact angle. Since $T > 0$, the inequality can be re-expressed in the form,\n\n\\begin{equation}\n\\Big( \\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}} \\kappa_{lv}  + \\kappa_{ls} \\Big) \\delta A_{ls} \n+ \\delta \\Theta +  A_{lv} \\delta \\kappa_{lv} + A_{ls} \\delta \\kappa_{ls}  \\le 0\\;,\n \\ \n\\label{eq:11}\n\\end{equation}\nwhich is equivalent to stating that the surface energy of the system must decrease.\n\n\\noindent Since the solid surface is flat, which means $\\kappa_{ls} = 0$ and $\\delta \\kappa_{ls} = 0$. The variations can be computed directly based on expressions for a spherical cap with droplet radius $R$ and droplet height $h$ to the solid surface, where\n\n\\begin{equation}\n\\begin{split}\n\\Theta &= 2 \\pi (1-\\cos \\theta)  \\\\\n\\kappa_{lv} &= \\frac{1}{R^2} \\\\\nA_{lv} &= 2 \\pi R h \\\\\nA_{ls} &= \\pi h (2R-h)\n\\end{split}\n\\label{eq:12} \n\\end{equation}\n\nand the associated variations\n\n\\begin{equation} \\label{eq:13}\n\\begin{split}\n\\delta \\Theta &= \\delta \\big [ 2 \\pi (1-\\cos \\theta) \\big ] \n             = 2 \\pi (h R^{-2}\\delta R - R^{-1} \\delta h )  \\\\\n        \\delta \\kappa_{lv} &= \\delta \\big [ R^{-2} \\big] = -2 R^{-3} \\delta R \\\\\n\\delta A_{ls} &= \\delta \\big [\\pi h (2R-h) \\big] =\n                    \\pi \\big( 2 h \\delta R + 2 R \\delta h - 2 h \\delta h \\big) \n\\end{split}\n\\end{equation}\nThe volume of a spherical cap is $V = \\frac 1 3 \\pi h^2 \\big( 3 R-h \\big)$, and setting $\\delta V=0$ imposes the relationship $\\delta R = \\Big( 1 - \\frac{2R}{h} \\Big) \\delta h$. Inserting this into expressions above, we obtain\n\n\\begin{equation}\n\\begin{split}\n\\delta \\Theta &= 2 \\pi \\Big(\\frac{h}{R^2} - \\frac{3}{R} \\Big) \\delta h \n\\\\\n\\delta \\kappa_{lv} &= \\Big(\\frac{4}{R^2h} - \\frac{2}{R^3} \\Big) \\delta h \\\\\n\\delta A_{ls} &=  -2 \\pi R \\delta h\n\\end{split}\n\\label{eq:14}\n\\end{equation}\nInserting Eqs. (\\ref{eq:12}) and (\\ref{eq:14}) into Eq. (\\ref{eq:11}) and rearranging terms gives\n\n\\begin{eqnarray}\n \\frac{2\\pi \\delta h }{R} \\Bigg\\{\n1 - \\frac{h}{R}  -\\frac{\\sigma_{cs} - \\sigma_{as}}{\\sigma_{ca}} \n\\Bigg\\}\\le 0\\;.\n\\end{eqnarray}\nNoting that $cos \\theta = 1 - h\/R$, we observe that for a general variation $\\delta h$\nand $R>0$, the entropy maximum will be obtained based on the condition\n\n\\begin{align}\n    \\cos \\theta -\\frac{\\sigma_{ls} - \\sigma_{vs}}{\\sigma_{lv}}  = 0\\;,\n\\end{align}\nwhich is Young's equation for the contact angle. The classical result is thereby obtained when the shape of the contact line is radially symmetric. This outcome provides evidence to demonstrate that the proposed topological concept is a general explicit geometric statement and also has a direct link to classical thermodynamics. \n\n\\section{Computation of Macroscopic Contact Angle}\n\\begin{algorithm}\n    \\caption{Implementation in computation of macroscopic contact angles for droplets in multiphase system.}\n    \\label{codel}\n    \\begin{algorithmic}\n    \\For{each droplet in the system}\n        \\State 2D surface manifold generation by marching cubes algorithm\n        \\State Gaussian smooth for the manifold:\n        \\State $v_i' = v_i +\\alpha \\sum w_{ij}(v_j - v_i)$\n        \\For{each triangle on the manifold}\n            \\State Compute triangle area:\n            \\State $A_i = \\frac{1}{2}|v_{12} \\times v_{13}|$\n            \\State Compute principal curvatures $\\kappa_1$ and $\\kappa_2$\n            \\State Compute Gaussian curvature:\n            \\State $\\kappa_{Gi} = \\kappa_1 \\kappa_2$\n            \\State Compute mean curvature:\n            \\State $\\kappa_{Mi} = \\frac{\\kappa_1 + \\kappa_2}{2}$\n            \\If {$\\kappa_M < 0$}\n                \\State $\\kappa_{Gi} A_i = - \\kappa_{Gi} A_i$\n            \\EndIf\n        \\EndFor\n        \\State Compute $\\theta_{macro}$ by normalizing deficit curvature with the number of contact line loop $N$:\n        \\State $\\theta_{macro} = \\lambda \\Theta = \\frac{4 \\pi - \\sum \\kappa_{Gi} A_i}{4N}$\n    \\EndFor\n    \\end{algorithmic}\n\\end{algorithm}\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.5\\textwidth]{FIG3}\n\\caption{The distribution of (a) - (c) Gaussian curvatures and (d) - (f) mean curvatures on the droplet interface for various degrees of $n_{Layer}$, which determines the number of triangles considered to be neighbours of a given point. Consequently, it determines the quadratic form equation of the surface patch. As the value of $n_{Layer}$ increases, the number of the surrounding triangles to be considered as neighbours increases, which leads to more accurate results.}\n\\label{fig3}\n\\end{figure}\n\nTo explain the implementation of the proposed topological principle to characterize the wetting state of the system, we first chose a droplet in a complex and confined domain from the segmented image for illustration. By modeling the whole droplet surface manifold using a generalized marching cubes algorithm \\cite{hege1997generalized}, we performed a triangular approximation of the interface for each region (e.g., liquid\/vapor and liquid\/solid) to preserve the droplet topology. To obtain accurate results and remove extreme outliers of curvatures, a simplification of the triangles by applying an edge collapse algorithm for the droplet surface was required to omit small non-smooth regions. For instance, the small concave spots in the convex regions will be degenerated during the triangle contraction process. The area of each triangle $A_i$ can be computed by the cross product of the two adjacent vectors ($v_{12}$, $v_{13}$) formed by the vertices ($v_1$, $v_2$ and $v_3$), which is $A_i = \\frac{1}{2}|v_{12} \\times v_{13}|$. Further, we smoothed the surface manifold by shifting its vertices to minimize the voxelization effects and segmentation errors. Each new vertex position ($v_i'$) was shifted towards the average position of its neighbours ($v_j$) by the weights ($w_{ij}$). Therefore, the vector average for the vertex can be obtained,\n\n\\begin{equation}\n    \\Delta v_i=\\sum w_{ij}(v_j - v_i),\n\\end{equation}\nand the updated vertex position will be\n\n\\begin{equation}\n    v_i' = v_i +\\alpha \\Delta v_i,\n\\end{equation}\nwhere $\\alpha$ represents a scale factor ranging from $0$ to $1$. We then computed the magnitude and direction of the principal curvatures ($\\kappa_1$ and $\\kappa_2$) by fitting a quadratic form equation on the desired surface patch and obtaining its corresponding eigenvalues and eigenvectors \\cite{armstrong2012linking}. The Gaussian curvature ($\\kappa_{Gi}$) and mean curvature ($\\kappa_{Mi}$) for each triangle can thereby be obtained in terms of the principal curvatures,\n\n\\begin{equation}\n\\begin{split}\n    \\kappa_{Gi} &= \\kappa_1 \\kappa_2,\\\\\n    \\kappa_{Mi} &= \\frac{\\kappa_1 + \\kappa_2}{2}.\n\\end{split}\n\\end{equation}\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.53\\textwidth]{FIG4}\n\\caption{Measured contact angles for different simulated droplets with increasing window size. It shows that an optimal window size of $60$ pixels is sufficient to accurately measure the apparent contact angle by the 2D local method.}\n\\label{fig4}\n\\end{figure}\nThe values of mean curvature provide the information of droplet interface, where the positive values of mean curvature indicate the convex interface and negative values indicate the concave interface as shown in Fig. \\ref{fig3}d-f. In addition, there is a negative sign assigned to $\\kappa_{Gi}A_i$ for concave interfaces and a value of $0$ assigned for flat interfaces. Finally, we can obtain the macroscopic contact angle, $\\theta_{macro}$, for the droplet by normalizing the deficit curvature $\\Theta$ as illustrated in Eq. (\\ref{eq:4}),\n\n\\begin{equation}\n    \\theta_{macro} = \\lambda \\Theta = \\frac{4 \\pi - \\sum \\kappa_{Gi} A_i}{4N}\n\\end{equation}\nwhere $N$ is the count of contact line loops. For more information, please see \\cite{sun2020probing}. For multiphase systems where a large number of droplets are present, such as porous media and subsurface rocks, the proposed implementation procedure as described in Algorithm \\ref{codel} enables the computation of macroscopic contact angles on a droplet-by-droplet basis.\n\n\\section{Results and Discussion}\n\\subsection{Wetting in Flat and Ideally-smooth Surfaces}\n\nTo unravel the effective and robust nature of the developed theory, we first compare the macroscopic contact angle $\\theta_{macro}$ and the apparent microscopic contact angles $\\theta_{app}$ on a flat and smooth surface. We performed a quasi-steady-state simulation of 3D sessile oil droplets immersed in immiscible ambient water on various flat surfaces that have different intrinsic contact angles defined by Young's equation. It is a more ideal solution for performing simulation than experiments where the droplet after deposition achieves a different contact angle, i.e. wetting hysteresis, when compared to the intrinsic contact angle. This is due to the difficulties in attaining experimental equilibrium condition and collecting ideally-flat surface \\cite{arjmandi2017kinetics}. The equilibrium topologies of the oil droplets on the surface were simulated by minimizing the overall system energy and were subjected to constraints such as surface tension and gravitational energy. In the system, the overall energy is the sum of its interfacial potential energy and is governed by the Young-Laplace equation. For the interfacial potential energy of the droplet ($E$), it can be expressed as the sum of the respective interfacial energies,\n\n\\begin{equation}\n    E=\\int\\int_{A_{ls}}\\sigma_{ls}dA+\\int\\int_{A_{lv}}\\sigma_{lv}dA+\\int\\int_{A_{vs}}\\sigma_{vs}dA,\n\\label{eq:21}\n\\end{equation}\nwhere $\\sigma$ and $A$ represent interfacial tension and interfacial area, respectively. By applying Young's equation, Eq. (\\ref{eq:21}) becomes\n\n\\begin{equation}\n    \\frac{E}{\\sigma_{lv}}=A_{lv}-\\int\\int_{A_{ls}}\\cos \\theta_Y dA.\n\\end{equation}\n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.8\\textwidth]{FIG5}\n\\caption{The comparison diagram of $\\theta_{macro}$ and the contact angle distributions of the 3D local measurement by \\cite{alratrout2017automatic} and modified 2D local measurement by \\cite{scanziani2017automatic} for the simulated droplets with different intrinsic contact angles.}\n\\label{fig5}\n\\end{figure*}\n\nAs shown in Fig. \\ref{fig5}, the equilibrium topologies of oil droplets were simulated on a surface that gradually change from oleophobic to oleophilic. To obtain the microscopic apparent contact angles along the contact line, we implemented the 2D local method similar to \\cite{scanziani2017automatic} and the 3D local method developed by \\cite{alratrout2017automatic}. For the 2D local method, the contact points were smoothed using a moving average to determine the position and direction of the contact line. A 2D image was then extracted normal to the direction of the contact line for each contact point. In Fig. \\ref{fig4}, we computed the contact angle measurement with different window (slice) sizes for each simulated droplet. The results demonstrate that a window (slice) size of $60$ pixels is sufficient to provide an accurate contact angle. In contrast to \\cite{scanziani2017automatic}, both circular and linear regressions were applied to best fit the solid surface. The best fit of either the circular or linear regression was determined by the lower of the two root-mean-square deviations. If the circle approximation was selected as the best fit, a line tangent to the circle at the contact point was calculated to represent the slope of the surface at that point. In addition, a constant curvature of the liquid\/vapor interface was assumed under the assumption that the system is at equilibrium. Consequently, the apparent contact angle was measured as the angle between the solid surface and liquid\/vapor interface tangent lines \\cite{meisenheimer2020optimizing}. For the 3D local method, Gaussian smoothing was applied to the droplet surface. Then, the two vectors that have a direction perpendicular to liquid\/vapor and liquid\/solid interface were identified for each contact point along the contact line. The apparent contact angle was thereby computed from the dot product of these vectors.  \n\n\\begin{figure*}\n\\centering\\includegraphics[width=0.9\\textwidth]{FIG6}\n\\caption{Top: The comparison of droplet topology as down-sampling the image resolution with measured local contact angles for each three-phase contact points. Bottom: The corresponding macroscopic contact angle ($\\theta_{macro}$) and 3D local measurements ($\\theta_{app}$) associated with their change by comparing with the original resolution. As deficit curvature is less sensitive to resolution effects, it is particularly attractive as a way to capture wetting on rough surfaces where sub-resolution heterogeneity can have a demonstrated influence on the measured macroscopic contact angle.}\n\\label{fig6}\n\\end{figure*}\n\nIn Fig. \\ref{fig5}, we compare the macroscopic contact angle $\\theta_{macro}$ distributions from the developed topological principle to the apparent contact angle distributions by the two local methods for each intrinsic contact angle. It is shown that the 2D local measurement has a higher standard deviation compared with the 3D local measurement. However, the mean values of the contact angle distribution for the 2D local measurement are closer to the intrinsic contact angle $\\theta_{in}$. Furthermore, the contact angle distribution calculated by 3D local method tends to have higher contact angles for more oleophobic surfaces and lower contact angles for more oleophilic surfaces, which provides a systematic bias (error) towards intermediate-wet conditions. Conversely, $\\theta_{macro}$ is slightly less than the intrinsic contact angle for oleophobic surfaces and slightly greater than the intrinsic contact angle for oleophilic surfaces. The trend observed for $\\theta_{macro}$ has a theoretical explanation since the total deficit curvature accounts for both local contact angles and the deformation of the contact line along the solid surface (line tension). The theoretical development for this particular aspect is explained in detail by Sun et al. \\cite{sun2020probing}. Overall, contact angles measured for flat and homogeneous surfaces are comparable for all three tested methods, which supports the theoretical developments in Section 2.2 since the proposed topological approach under these ideal conditions can be simplified to Young's equation. \n\n\\subsection{Wetting in Complex Geometries}\n\\begin{figure}\n\\centering\\includegraphics[width=0.5\\textwidth]{FIG7}\n\\caption{The contact angle distributions of different image resolution for sintered glass by applying (b) 3D local method for each contact point and (c) the topological approach on a droplet-by-droplet basis. The plot of (d) mean and (e) standard deviation of the distribution for each image resolution.}\n\\label{fig7}\n\\end{figure}\n\nTo further investigate the wetting state on heterogeneous surfaces where surface chemistry and roughness are present, we performed two quasi-static capillary forces dominated flow experiments for sintered glass at the Swiss Light Source and Bentheimer sandstone at the Australian National University \\cite{dataset}. The details of the experimental setup and image processing can be found in \\cite{schluter2016pore} and \\cite{sun2020probing}. The sintered glass has a relatively lower degree of surface geometry with smooth surfaces and a lower degree of surface chemistry. While for Bentheimer sandstone, the surface chemical heterogeneity, surface geometry and roughness are relatively higher. In Table \\ref{tab}, a summary of experimental setup and image processing is given for both sintered glass and Bentheimer sandstone. The 3D fluid configurations and spatial arrangements for both samples are displayed in Figs. \\ref{fig7}a and \\ref{fig8}a.\n\n\\begin{table*}\n   \\centering\n   \\caption{Summary of experimental setup and image processing.}\n   \\begin{tabular}{L{2.7cm}C{5cm}C{5cm}}\n      \\toprule \n        & Sintered Glass & Bentheimer Sandstone\\\\\n      \\midrule\n      Porosity & $31.8 \\%$ & $24.1 \\%$\\\\\n      Permeability & $21.5 \\pm 2$ $\\rm D$ & $4.3$ $\\rm D$\\\\\n      Sample diameter & $4$ $\\rm mm$ & $4.9$ $\\rm mm$\\\\\n      Sample length & $10$ $\\rm mm$ & $10$ $\\rm mm$\\\\\n      Non-wetting phase & n-decane & ambient air\\\\\n      Wetting phase & brine & brine\\\\\n      Injection rate & $0.1$ $\\rm \\mu L\/min$ & $0.3$ $\\rm \\mu L\/min$\\\\\n      Brine saturation & $78 \\%$ & $93 \\%$\\\\\n      Imaging source & fast synchrotron-based tomography & bench-top helical micro-tomography\\\\\n      Image resolution & $4.2$ $\\rm \\mu m$ & $4.95$ $\\rm \\mu m$\\\\\n      Time step & $30$ $\\rm s$ & $1.35$ $\\rm hr$\\\\\n    \\bottomrule\n    \\end{tabular}\n   \\label{tab}\n\\end{table*}\n\n\\begin{figure}\n\\centering\\includegraphics[width=0.5\\textwidth]{FIG8}\n\\caption{The contact angle distributions of different image resolution for Bentheimer sandstone by applying (b) 3D local method for each contact point and (c) the topological approach on a droplet-by-droplet basis. The plot of (d) mean and (e) standard deviation of the distribution for each image resolution.}\n\\label{fig8}\n\\end{figure}\n\nTo investigate the resolution effects, we first performed a resolution study by picking a well-resolved droplet from the dataset of Bentheimer sandstone. The original dimension of image for the droplet is $47 \\times 51 \\times 48$ voxels. We then down-sample the image of the droplet by halving the resolution twice as shown in Fig. \\ref{fig6}a-c. It is intuitive in Fig. \\ref{fig6} how the down-sampling impacts the droplet interfacial curvature and contact line loops. The results demonstrate that the macroscopic contact angle based on the deficit curvature is less sensitive to resolution effects than local contact angle measurements, since the change in the mean value of $\\theta_{app}$ is larger than the change in $\\theta_{macro}$. Local contact angle measurements cannot be performed without resolving the contact line region. It is clear that the resolution necessary to measure the angles along the contact line is lost before the topological structure of the contact line loops is destroyed. In Fig. \\ref{fig6}a-c, we investigate that the local contact angles vary significantly as the resolution decreases. For the topological approach based on the Gauss-Bonnet theorem, there remains a basis to obtain deficit curvature when the topological structure of the contact line loops is captured, even if only a single voxel represents the contact line loop as shown on the bottom right of Fig. \\ref{fig6}c.\n\nFurther, we performed the contact angle distribution measurements for both samples on a collection of droplets by applying the topological approach for each droplet and 3D local method for each contact point. By gradually decreasing the image resolution as shown in Figs. \\ref{fig7} and \\ref{fig8}, the changes in the mean and standard deviation of the contact angle distributions for the topological approach are notably less than that of the local measurements. For sintered glass, the mean of the contact angle distribution decreases by an additional $7 \\%$ while there is an additional $31 \\%$ reduction in the standard deviation for the topological approach. Similar results for the topological method are investigated in the Bentheimer sandstone with a decrease in mean contact angle and standard deviation of $12 \\%$ and $27 \\%$, respectively. Overall, at sufficiently high resolution, the topological approach and local measurements are comparable. While at lower resolutions, the topological approach deviates less from the high resolution results than the local contact angle measurements. \n\nLastly, we performed contact angle measurements for both samples by applying the topological approach, 2D and 3D local methods with the highest image resolution obtained as shown in Fig. \\ref{fig9}. It reveals that the 2D local measurement provides a lower mean and a higher standard deviation of the contact angle distribution. However, the mean value of the 3D local measurement is the highest, which can be explained in the simulation of droplets results for the flat and homogeneous surfaces. For more oleophobic (water-wet) surfaces, contact angle measurements computed by the 3D local method is larger than the intrinsic contact angles. The $\\theta_{macro}$ distribution computed by the topological approach falls between the two local methods. Overall, the topological and 3D local methods are comparable at sufficiently high resolution, suggesting that microscopic wetting information is captured. \n\n\\begin{figure}\n\\hspace*{-0.5cm}\\includegraphics[width=0.5\\textwidth]{FIG9}\n\\caption{The contact angle distributions for (a) sintered glass and (b) Bentheimer sandstone by applying topological approach, 2D and 3D local methods with the highest resolution obtained together with the associated mean and standard deviation.}\n\\label{fig9}\n\\end{figure}\n\n\\section{Conclusions}\n\nRecent methods for characterizing wetting behavior rely on restrictive thermodynamic laws such as Young's equation for flat homogeneous surfaces, and Wenzel and Cassie-Baxter models for heterogeneous surfaces \\cite{young1805iii,wenzel1936resistance,cassie1944wettability}. In this work, we developed a new theory based on topological principles to determine the wetting state of multiphase systems. Specifically, the concept of thermodynamic equilibrium is not necessary since the formulation is a geometrical statement between the total surface curvature and global topology of a fluid object. In this regard, the proposed methodology does not require visualization of the three-phase contact line. It thus would be advantageous for applications where a large fluid body is resolved while the contact line pinning occurs at a smaller length scale. This could be particularly informative for problems where the droplet is connected to a thin film, such as a droplet slipping along a declining plane and\/or the corner flow mechanism observed in porous media flows. We also highlight that the proposed theory is consistent with Young's equation for flat and homogeneous surfaces by performing a variational analysis.\n\nBy comparing with traditional local contact angle measurements \\cite{alratrout2017automatic,scanziani2017automatic,klise2016automated}, we assess the sensitivity and robustness of the proposed topological approach. It is found that the proposed theory has less sensitivity to resolution effects than the tested local methods. While at sufficiently high image resolution, the results are comparable. As observed in Fig. \\ref{fig6}, the resolution necessary to measure the contact line is lost before the topological structure of the contact line loops are destroyed. Contact angles cannot be measured without resolving the contact line region. However, the contact line region must always form a loop even if it is captured by only one voxel. Therefore, there remains a basis to measure $\\theta_{macro}$ even at low image resolution since topology and interfacial curvatures of fluid surface are still captured. Another way to think about why  $\\theta_{macro}$ works well at low resolution is because fluid blobs are physically larger than the contact lines, and we can use information embedded in the fluid interface structure to infer the wetting state.\n\nOverall, the results demonstrate that the proposed theory provides an accurate macroscopic wetting description based on microscopic wetting information that is less susceptible to resolution-based errors due to its robust nature of measuring interfacial area and curvature rather than angles along the contact line. The theoretical links of the proposed theory explored in other recent publications \\cite{sun2020linking,sun2020probing} are further highlighted by the variational analysis presented herein. The particular example was provided for a simple sessile droplet system. However, more complex systems with morphologically and mineralogically heterogeneous geometries could be considered leading to a more general condition for the constrained entropy inequality. This could be particularly important for the development of multiphase flow models that explicitly account for wettability and will be the focus of further work. \n\n\\section*{Acknowledgments}\nA part of this work was performed at the Swiss Light Source, Paul Scherrer Institute, Villigen, Switzerland. C. S. acknowledges an Australian Government Research Training Program Scholarship. A. H. acknowledges ARC DE180100082 and the ANU\/UNSW Digicore Research Consortium. J. M. acknowledges an award of computer time provided by the Department of Energy Early Science Program. This work was supported by the U.S. National Science Foundation, Hydrologic Sciences Program under award No.1344877. This research also used resources of the Oak Ridge Leadership Computing Facility, which is a DOE Office of Science User Facility supported under Contract DE-AC05-00OR22725.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe Galactic interstellar medium (ISM) has a multiphase structure \nwith neutral hydrogen being distributed between the \ncold neutral (CNM), warm neutral (WNM) and warm \nionized (WIM) media. A large fraction of the gas is also found in \ndiffuse, translucent and dense molecular clouds. Newly formed stars\nare associated with these dense molecular clouds and \nstrongly influence the physical state of  the rest of the gas in different\nforms through radiative and mechanical inputs. \nThe physical conditions in the multiphase ISM depend on the UV background \nradiation field, metallicities, dust content and the\ndensity of cosmic rays \n\\citep[see  Figs.~5, 6 and 7 in][]{Wolfire95}.\nIn addition, \nthe filling factor of the different phases depends sensitively\non the supernova rate \\citep{deavillez2004}.\nTherefore, detecting and studying the multiphase ISM\nin external galaxies has great importance for our understanding of \ngalaxy evolution.\n\nDamped Lyman-$\\alpha$ systems (DLAs) are the highest H~{\\sc i} column\ndensity  absorbers seen in QSO spectra, with \n$N$(H~{\\sc i})$\\ge 2\\times 10^{20}$~cm$^{-2}$.\nThese absorbers trace the bulk of the neutral hydrogen at $2\\le z\\le 3$ \n\\citep [][]{Prochaska05,Noterdaeme09dla} and have long been identified as \nrevealing the interstellar medium of the high-redshift precursors\nof present day galaxies \\citep[for a review see,][]{wolfe05}.\n\nThe typical dust-to-gas ratio of DLAs, is less than one tenth of that observed \nin the local ISM, and only a small fraction ($<\u223c10$\\%) of DLAs show detectable\namounts of molecular hydrogen \\citep{Petitjean00,Ledoux03,Noterdaeme08} with\nthe detection rate being correlated to the dust content of the gas \n\\citep{Petitjean06}. \nThe estimated temperature and molecular fraction in these \nsystems are consistent with them \noriginating from the CNM \\citep{Srianand05}.\nIt has been shown recently that strong C~{\\sc i} absorbers detected in \nlow-resolution Sloan Digital Sky Survey (SDSS) \nspectra are good candidates for H$_2$ bearing systems.\nIndeed these absorbers\nhave yielded the first detections of CO molecules in high-$z$ DLAs\n\\citep{Srianand08,Noterdaeme09co,Noterdaeme10co,Noterdaeme11}.\nThe properties of these absorbers are similar to those of translucent \nmolecular clouds. \nThe fact that no DLA is found to be associated with a dense molecular \ncloud, a fundamental ingredient \nof star-formation, is most certainly related to \nthe large extinction that these clouds are expected to produce and\/or\nthe small size of such regions \\citep{Zwaan06} making \ndetections difficult.\n\nThus, most DLAs detected in optical spectroscopic surveys seem to \nprobe the diffuse H~{\\sc i} \ngas \\citep{Petitjean00}. However, \nabout 50\\% of the DLAs show detectable C~{\\sc ii}$^*$ absorption\n\\citep{Wolfe08}, and \\citet{Wolfe03b} argued that a \nconsiderable fraction of the\nC~{\\sc ii}$^*$ absorption in DLAs originates from CNM \ngas \\citep[see however][]{Srianand05}. \n{ Detection of 21-cm absorption is the best way to estimate the CNM\nfraction of DLAs as it is sensitive to both $N$(H~{\\sc i}) and\nthermal state of the gas \\citep{Kulkarni88}.}\n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 40.0mm, clip, scale=0.33,angle=90.0]{J0733+2721c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J0801+4725c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J0816+4823c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J0839+2002c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J0852+2431c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1017+6116c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J1223+5037c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1237+4708c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 50.0mm, clip, scale=0.33,angle=90.0]{J1242+3720c.ps}\n}\n\\hbox{\n\\includegraphics[trim= 25.0mm 50.0mm 70.0 40.0mm, clip, scale=0.33,angle=90.0]{J1406+3433c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J1413+4505c.ps}\n\\includegraphics[trim= 25.0mm 50.0mm 90.0 50.0mm, clip, scale=0.33,angle=90.0]{J1435+5434c.ps}\n}\n}}\n\\caption[]{\nSDSS spectra showing the \\lya lines for 12 DLAs in our sample. \nThe best fitted Voigt profiles (solid curves) together with the \nassociated 1$\\sigma$ errors (shaded regions) are over-plotted. The dotted\ncurve gives the best fitted continuum (in some cases the continuum fit\nincludes the emission lines also). We have used VLT UVES spectra\nto get $N$(H~{\\sc i}) in the case of \\zabs = 3.1745 system towards J1337+3152\n\\citep[see][]{Srianand10}\nand \\zabs = 2.595 and  2.622 systems towards J0407$-$4410 \\citep[CTS 247, see][]{Ledoux03}.\n}\n\\label{dlafits}\n\\end{figure*}\nThis is why it is important to search for 21-cm absorption in DLAs  \nover a wide redshift range. While a good fraction of DLAs\/sub-DLAs \npreselected through\nMg~{\\sc ii} absorption seems to show 21-cm absorption at $z\\sim 1.3$\n\\citep[see for example][]{Gupta09, Kanekar09mg2},\nsearches for 21-cm absorption in DLAs at $z_{\\rm abs}\\ge2$ have\nmostly resulted in null detections \\citep[see][]{Kanekar03,Curran10}\nwith only four detections reported till now \\citep[see][]{Wolfe85,Kanekar06,Kanekar07,York07}. \n{\nThe low detection rate of 21-cm absorption in high-$z$ DLAs can be\nrelated to either the gas being warm (i.e high spin temperature, $T_{\\rm S}$, \nas suggested by \\citet{Kanekar03}) \nand\/or the low value of covering factor ($f_c$)\nthrough high-z geometric effects\n\\citep{Curran06}. \n\nThe best way to address the covering factor issue is to perform \nmilliarcsecond scale spectroscopy in the redshifted 21-cm line \nusing very long baseline interferometry (VLBI) to measure the extent \nof absorbing gas \\citep{Lane00}. Unfortunately due to limited \nfrequency coverage and sensitivity of the receivers available with \nVLBI such studies cannot be extended to high redshift DLAs.  \nAlternatively, the core fraction measured in the  milliarcsecond scale\nimages can be used to get an estimate of the covering factor\n\\citep[see][] {Briggs89, Kanekar09vlba}. Here one assumes that the \nabsorbing gas completely covers at least the emission from the milliarcsec\nscale core.\nTherefore, to address this issue, \none needs, not only to increase the number of systems searched for \n21-cm absorption but also to perform milliarcsecond scale imaging of\nthe background radio sources.\n\nWe report here the results of a search for 21-cm  absorption in\n10 DLAs at $z>2$ we have carried out using GBT and GMRT, \ncomplemented by L-band VLBA images of the background QSOs. \n\\begin{table}\n\\caption{Log of GBT and GMRT observations to search for 21-cm absorption}\n\\begin{center}\n\\begin{tabular}{lccccc}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Source}& Tele-    & Date       & Time    &  BW      &    Ch.         \\\\\n\\multicolumn{1}{c}{name}  & scope    &            &         &          &                \\\\\n            &          & yy-mm-dd           & (hr)    & (MHz)    &                \\\\\n\\multicolumn{1}{c}{(1)} & (2)      &  (3)       &  (4)    &   (5)    &    (6)         \\\\\n\\hline\n\nJ0407$-$4410 &  GBT  &  2006-10-20  &   4.7   &  0.625   &    512         \\\\  \n(CTS247)  &       &  2007-01-05  &         &          &                \\\\  \n             &       &  2007-01-06  &         &          &                \\\\  \n             &       &  2007-01-08  &         &          &                \\\\  \n J0733+2721  &  GBT  &  2007-12-05  &  10.7   &  1.25    &    512         \\\\  \n             &       &  2007-12-06  &         &          &                \\\\  \n J0801+4725  &  GMRT &  2006-12-22  &  10.8   &  1       &    128         \\\\  \n             &       &  2006-11-23  &         &          &                \\\\\n J0852+2431  &  GBT  &  2009-08-07  &   5.6   &  1.25    &    1024        \\\\  \n             &       &  2009-08-08  &         &          &                \\\\  \n             &       &  2009-09-06  &         &          &                \\\\  \n J1017+6116  &  GBT  &  2008-10-15  &   4.5   &  1.25    &    1024        \\\\  \n             &       &  2008-10-16  &         &          &                \\\\  \n             &       &  2008-10-18  &         &          &                \\\\\n J1242+3720  &  GMRT &  2007-06-08  &   6.1   &  0.5     &    128         \\\\  \n             &       &  2007-06-10  &         &          &                \\\\  \n J1337+3152  &  GMRT &  2009-01-13  &   6.2   &  1       &    128         \\\\  \n             &       &  2009-03-17  &   7.8   &  0.25    &    128         \\\\  \n J1406+3433  &  GBT  &  2009-03-05  &   8.0   &  1.25    &    1024        \\\\  \n             &       &  2009-03-06  &         &          &                \\\\  \n             &       &  2009-03-07  &         &          &                \\\\  \n             &       &  2009-03-08  &         &          &                \\\\  \n             &       &  2009-04-07  &         &          &                \\\\  \n J1435+5435  &  GMRT &  2007-06-10  &   6.7   &  1       &    128         \\\\  \n             &       &                 &         &          &                \\\\  \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: Telescope used for 21-cm absorption search. \nColumn 3: Date of observations. \nColumn 4: Total time on source i.e. after excluding telescope set-up time and calibration overheads. \nColumns 5 and 6: Spectral setup for the observations i.e. bandwidth (BW) and number of \nspectral channels respectively.  \n\\end{flushleft}\n\\label{obslog}\n\\end{table}\nThis survey has resulted in the detection of 21-cm absorption in \nthe \\zabs = 3.1745\nDLA towards J1337+3152. A detailed analysis of this system and\ntwo sub-DLAs close to this system are presented in \\citet{Srianand10}.\nSection~\\ref{samp} presents the details of our sample. \nIn Section 3 we present the details of  GBT and GMRT spectroscopic \nobservations, VLBA continuum observations, and data reduction. \nThe detection rate of 21-cm absorption in DLAs is discussed in \nSection~\\ref{detect}. In Section~\\ref{gencor} we study the\ncorrelations between the parameters derived from 21-cm observations, \n$N$(H~{\\sc i}), metallicity\nand redshift. In Section~\\ref{mole} we study the relation between\n21-cm and \\h2 absorption. The results  are summarized in \nSection~\\ref{results}. In this work we assume a flat Universe with\n$H_0$~=~71~\\kms~Mpc$^{-1}$, $\\Omega_{\\rm m}$~=~0.27 and $\\Omega_\\Lambda$~=~0.73.\n\n\\section{The sample of DLAs}\n\\label{samp}\n\n\\begin{table*}\n\\caption{GMRT low-frequency flux density measurements for the DLAs observed with the GBT \n}\n\\begin{center}\n\\begin{tabular}{lcccc}\n\\hline\n\\hline\n\\multicolumn{1}{c}{Source name}       & $S_{\\rm 610MHz}$ &   Date              & $S_{\\rm 325MHz}$ &    Date          \\\\\n         &                &                       &                &                  \\\\\n             &  (mJy)         &   yy-mm-dd       &   (mJy)        &   yy-mm-dd               \\\\\n\\multicolumn{1}{c}{(1)}  &   (2)          &    (3)                &    (4)         &     (5)          \\\\\n\\hline                                                                                  \nJ0407$-$4410 &  124           &  2006-05-23        &    52          & 2007-11-24     \\\\\nJ0733$+$2721 &  248           &  2007-11-04        &    549         &      ''          \\\\\nJ0852$+$2431 &  198           &  2008-12-26        &    237         & 2009-03-17     \\\\\nJ1017$+$6116 &  274           &     ''                &    266         &      ,,          \\\\\nJ1406$+$3433 &  165           &     ''                &    185$^\\dag$  &     (WENSS)      \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name.  \nColumns 2 and 4: GMRT flux density measurements at 610 and 325\\,MHz respectively. \nColumns 3 and 5: Dates of the 610 and 325\\,MHz observations respectively. \\\\\n$^\\dag$ From the WENSS catalog.\\\\ \n\\end{flushleft}\n\\label{flux}\n\\end{table*}\n\\begin{table*}\n\\caption{Results from the GBT and GMRT observations}\n\\begin{center}\n\\begin{tabular}{lccccccccc}\n\\hline\n\\hline\nSource name &   \\zem   & \\zabs  &  log\\,$N$(H~{\\sc i})   &S$_{1.4\\,GHz}$& $\\delta $  &Spectral rms & S$_{\\nu_{abs}}$ & $\\int\\tau$dv  & {${ T}_{\\rm s}\\over f_{\\rm c}$}   \\\\\n            &          &        &        (cm$^{-2}$)    &   (mJy)      & (km\\,s$^{-1}$) &(mJy\\,b$^{-1}$\\,ch$^{-1}$)  & (mJy)&  (km\\,s$^{-1}$)&(K)  \\\\\n~~~~~~~~(1) & (2)      &  (3)   &  (4)            &   (5)        &     (6)          &    (7)        &    ~~(8)     &  (9)  & (10) \\\\\n\\hline\n\\\\\n J040718$-$441013    & 3.020 &2.595  & 21.05$\\pm$0.10 &   -     & 3.7   & 5.9    &   67  & $<$1.61  &  $>$382    \\\\  \n J040718$-$441013    &  ''   &2.622  & 20.45$\\pm$0.10 &   -     & ''    & 7.1    &   67  & $<$1.93  &  $>$81     \\\\  \n J073320.49+272103.5 & 2.938 &2.7263 & 20.25$\\pm$0.20 &  240    & 3.8   & 3.4    &  451  & $<$0.14  &  $>$692    \\\\  \n J080137.68+472528.2 & 3.276 &3.2235 & 20.80$\\pm$0.15 &   78    & 7.0   & 1.5    &  164  & $<$0.22  &  $>$1563   \\\\  \n J085257.12+243103.2 & 3.617 &2.7902 & 20.70$\\pm$0.20 &  160    & 3.9   & 3.9    &  228  & $<$0.32  &  $>$850    \\\\  \n J101725.89+611627.5 & 2.805 &2.7681 & 20.60$\\pm$0.15 &  477    & 3.9   & 4.2    &  268  & $<$0.29  &  $>$758    \\\\  \n J124209.81+372005.7 & 3.839 &3.4135 & 20.50$\\pm$0.30 &  662    & 3.6   & 3.6    &  615  & $<$0.11  &  $>$1567   \\\\  \n J133724.69+315254.5 & 3.174 &3.1745 & 21.36$\\pm$0.10 &   83    & 6.9   & 1.3    &   69  &2.08$\\pm$0.17 &  600$^{+220}_{-160}$\\\\  \n J140653.84+343337.4 & 2.566 &2.4989 & 20.20$\\pm$0.20 &  167    & 3.6   & 3.0    &  178  & $<$0.31  & $>$356    \\\\  \n J143533.78+543559.4 & 3.811 &3.3032 & 20.30$\\pm$0.20 &   96    & 7.1   & 1.5    &  145  & $<$0.26  & $>$418    \\\\  \n                     &       &       &                &         &       &        &       &          &           \\\\  \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: QSO emission redshift. \nColumn 3: Absorption redshift of DLAs as determined from the metal absorption lines. \nColumn 4: H~{\\sc i} column density.  \nColumn 5: Flux density at 1.4\\,GHz from the FIRST catalog.\nColumns 6 and 7: Spectral resolution and rms for the survey spectrum.  \nColumn 8: Continuum flux density of source at the redshifted 21-cm absorption frequency. \nColumn 9: Integrated 21-cm optical depth or 3-sigma upper limit to $\\int \\tau dv$ for the equivalent spectral resolution of \n10\\,km\\,s$^{-1}$. \nColumn 10: Ratio of spin temperature and covering factor of absorbing gas. \n\\\\\n\\end{flushleft}\n\\label{dlasamp}\n\\end{table*}\nTo construct our sample, we cross-correlated the overall sample of \nDLA-bearing QSO sightlines from SDSS-DR7 \n\\citep[][including systems that are not part\nof the published statistical sample used to measure $\\Omega_{\\rm HI}$]{Noterdaeme09dla} with the VLA FIRST catalog to\nidentify DLAs in front of radio sources brighter than 50~mJy at 1.4 GHz.  \nWe excluded radio sources with the DLA 21-cm absorption frequencies redshifted \ninto GBT and GMRT frequency ranges known to be affected by \nstrong radio frequency interference (RFI). There are 13 DLAs from the SDSS-DLA catalog that satisfy these conditions.  \nIn addition, there are 4 DLAs along the sight line towards J0407$-$4410\n(also known as CTS\\,247) at \\zabs~=~1.913, 2.550, 2.595 \\& 2.622. Two of these,\nat \\zabs~=~2.595 and 2.622, have redshifted 21-cm absorption frequency in the \nrelatively RFI free frequency range of GBT. Including these two DLAs towards \nCTS\\,247, we have a sample of 15 DLAs for which a search\nfor 21-cm absorption was carried-out using either GMRT or GBT. We observed 14 DLAs, (the exception is the \\zabs = 3.079 system towards \nJ1413+4505), but obtained useful spectra for only 10 DLAs.\nIn addition, we have obtained milliarcsecond scale \nimages at 1.4\\,GHz for all the QSOs except CTS\\,247 to understand the role of \nradio structure in detectability of 21-cm absorption in DLAs. \nThe details of the GBT, GMRT and VLBA observations are given below.  \n\nThe Lyman-$\\alpha$ profiles for 12 DLAs selected from the SDSS-DLA catalog \nare shown in the Fig.~\\ref{dlafits}.  The H~{\\sc i} column density for \neach of these DLAs has been estimated using Voigt profile fits to the \nLyman-$\\alpha$ absorption line.  The QSO continuum was approximated by a \nlower order spline using absorption free regions on both sides of \nthe H~{\\sc i} trough (dotted curves in each panel). In addition, \nspecial care was taken to fit the emission line profiles whenever \nthe \\lya absorption is close to QSO emission lines. \nFor the remaining three DLAs in our sample, \\zabs=3.1745 DLA towards \nJ1337+3152 and the two DLAs towards CTS\\,247, we use the column \ndensities measured by \n\\citet{Srianand10} and \\citet{Ledoux03} respectively from high \nresolution VLT UVES spectra.\n\n\n\n\\section{Details of Observations and data reduction}\n\\subsection{The GBT and GMRT observations}\n\nWe observed our sample of 14 DLAs using the GBT prime focus receivers \nPF1-340\\,MHz and PF1-450\\,MHz, and the GMRT P-band receiver.  Although \nwe selected DLAs such that the redshifted 21-cm absorption frequencies were \nnot affected by strong RFI, no useful data could be obtained for \n4 absorption systems either due to \nRFI or other technical reasons.  The observing log for the remaining \n10 DLAs and the spectral set-up used for these observations are \nprovided in Table~\\ref{obslog}.  \nGBT observations were performed in the standard position-switching mode with \ntypically 5\\,min spent on-source and 5\\,min spent off-source. \nThe data were acquired in the orthogonal polarization \nchannels XX and YY. We used the \nGBT spectral processor as the backend for these observations.  \nThe two DLAs towards \nCTS\\,247 were observed simultaneously using two bands of 0.625\\,MHz split \ninto 512 channels.\nFor the GMRT observations, typically a  \nbandwidth of 0.5 or 1\\,MHz \nsplit into 128 frequency channels was used. The data were acquired in the two \northogonal polarization channels RR and LL.  \nFor the flux density\/bandpass calibration of GMRT \ndata, standard flux density calibrators were observed for 10-15\\,min every \ntwo hours.  A phase calibrator was also observed for 10\\,min every \n$\\sim$45\\,min to get reliable phase solutions.    \n\nWe used NRAO's GBTIDL package to develop a pipeline to automatically \nanalyse the GBT spectral-line data sets. After excluding time ranges for \nwhich no useful data were obtained, the data were processed through this \npipeline. The pipeline calibrates each data record individually and flags \nthe spectral channels with deviations larger than 5$\\sigma$ as \naffected by RFI. After subtracting a second order baseline these data \nare averaged to produce baseline (i.e. continuum) subtracted spectra for \nXX and YY. The baseline fit and statistics \nfor the flagging are determined using the spectral region that excludes the\ncentral 25\\% and last \n10\\% channels at both ends of the spectrum. If necessary, a first-order cubic spline was \nfitted to the averaged XX and YY spectra obtained from the pipeline, which were then combined \nto produce the Stokes-I spectrum. The spectrum was then shifted to the heliocentric frame.  \nThe multi-epoch spectra for a source were then resampled onto the same frequency scale and \ncombined to produce the final spectrum.\n\n\nThe GMRT data were reduced using the NRAO AIPS package \nfollowing the standard procedures  described in \\citet{Gupta06}.  \nSpecial care was taken to exclude the baselines and time stamps affected \nby RFI.  \nThe spectra at the quasar positions were extracted from the RR and LL \nspectral cubes and compared for consistency. If necessary, a first-order \ncubic-spline was fitted to remove the residual continuum from the spectra.  \nThe two polarization channels were then combined to get the stokes I spectrum \nwhich was then shifted to the heliocentric frame.\n\n\n\\begin{figure*}\n\\centerline{\\hbox{\n\\psfig{figure=\"dlaspec.ps\",height=16.0cm,width=17.0cm,angle=0}\n}}\n\\caption[]{GBT and GMRT spectra of DLAs in our sample. \nShaded regions mark features that are due to RFI. The arrows in the case of non-detections indicate the expected\npositions of the 21-cm absorption. In the case of \\zabs = 3.1745 system towards\nJ1337+3152 we show the high resolution 21-cm absorption spectrum only. Two arrows in the case of\nCTS247a indicate the expected position of 21-cm absorption from the two H$_2$ components.\n}\n\\label{21cm}\n\\end{figure*}\nThe FWHM of the GBT beam at 400\\,MHz is 30$^\\prime$ and the rms confusion is \n500\\,mJy. This is comparable  \nto the flux densities of the background radio sources observed with the GBT. \n{Therefore, to  \ncorrect for the effect of other confusing sources in the GBT beam and determine \nthe QSO flux densities at the redshifted 21-cm frequency, we observed these with the GMRT at 610 and 325\\,MHz. For these observations we \nhave used 32\\,MHz bandwidth.  Details of these GMRT observations \nand the measured flux densities are provided in Table~\\ref{flux}.}  For J1406+3433, the 325\\,MHz \nflux density is taken from the Westerbork Northern Sky Survey (WENSS). We interpolate these flux density measurements to determine \nthe flux densities at redshifted 21-cm frequencies for the quasars observed with the GBT.  \nSince, flux densities for these 5 QSOs are not measured at the same epoch as the \nGBT spectroscopic observations, in principle, radio flux density \nvariability can affect \nour estimates of 21-cm optical depths for the corresponding DLAs.  However, \nthis effect is much smaller than the error caused by confusion from other sources in the beam and should not \naffect the statistical results derived later in the paper.  \n\nThe GBT and GMRT observations of our DLA sample have resulted in useful \n21-cm absorption spectra for 10 DLAs.  These spectra are presented \nin Fig.~\\ref{21cm}. GBT spectra, typically acquired at a spectral \nresolution, of $\\sim$1\\,\\kms \nhave been smoothed to $\\sim$4\\,\\kms for presentation. \nThe 21-cm absorption is detected only for one DLA (i.e. \\zabs~=~3.1745 DLA towards J1337+3152)\nand a detailed analysis of this system is presented in \\citet{Srianand10}.\nNone of the other ``absorption-like features'', marked as shaded regions, \nare reproduced in  spectra from different polarizations and epochs, \nbut are due to RFI.  For CTS247b (i.e for \\zabs = 2.622 DLA towards\nCTS247)  these \nfeatures are present only in one polarization at certain times.  For J0852+2431  \nand J1017+6116, using a combination of high spectral resolution ($\\sim$1\\,\\kms) and\/or \nmulti-epoch observations we rule out the possibility of these features \nbeing real 21-cm absorption.  \nDetails of the optical depth measurements and other observational results for all the \n10 DLAs are summarized in Table~\\ref{dlasamp}. \n\n\n\n\\begin{table}\n\\caption{Details of phase-referencing calibrators used for the VLBA observations }\n\\begin{center}\n\\begin{tabular}{lcc}\n\\hline\n\\hline\nSource      & Calibrator    & Separation        \\\\\n            &               &(degrees)\\\\\n~~~~~~~~(1) & (2)           &  (3)              \\\\\n\\hline\n J0733+2721 & J0732+2548    &   1.5             \\\\  \n J0801+4725 & J0754+4823    &   1.5             \\\\  \n J0816+4823 & J0808+4950    &   1.9             \\\\\n J0839+2002 & J0842+1835    &   1.6             \\\\\n J0852+2431 & J0856+2111    &   3.5             \\\\  \n J1017+6116 & J1031+6020    &   2.0             \\\\  \n J1223+5037 & J1227+4932    &   1.3             \\\\\n J1237+4708 & J1234+4753    &   0.9             \\\\\n J1242+3720 & J1242+3751    &   0.5             \\\\  \n J1337+3152 & J1329+3154    &   1.6             \\\\  \n J1406+3433 & J1416+3444    &   1.9             \\\\  \n J1413+4505 & J1417+4607    &   1.2             \\\\\n J1435+5435 & J1429+5406    &   1.0             \\\\  \n\\hline\n\\end{tabular}\n\\end{center}\n\\begin{flushleft}\nColumn 1: Source name. \nColumn 2: Phase-referencing calibrator. \nColumn 3: Separation between the radio source and phase-referencing calibrator. \n\\end{flushleft}\n\\label{vlbalog}\n\\end{table}\n\n\\subsection{Continuum observations with VLBA}\nThe sample of quasars presented here was observed as part of a \nlarger VLBA survey to obtain milliarcsecond scale images for QSOs \nwith foreground DLAs and Mg~{\\sc ii} systems,  \nand understand the relationship between radio structure \nand detectability of 21-cm \nabsorption.   We have observed using VLBA 21-cm receiver for 11~hrs and  18~hrs\non 21\/02\/2010 and 10\/06\/2010 respectively.\nWe used eight 8\\,MHz baseband channels, i.e. the total bandwidth of 64\\,MHz. \nEach baseband channel was split into 32 spectral points.  Both the right \nand left-hand circular polarization channels were recorded.  Two bit \nsampling and a post-correlation time resolution of 2 seconds were used. \n\nThe observations were done using nodding-style phase-referencing with a cycle \ntime of $\\sim$5\\,min, i.e. 3\\,min on the source and $\\sim$1.5\\,min on the \nphase-referencing calibrator. The phase-referencing calibrators were selected \nfrom the VLBA calibrator survey (VCS) at 2.3 and 8.6\\,GHz \n(Table~\\ref{vlbalog}).  \nIn order to improve the uv-coverage, the total observing time was split \ninto snapshots over a number of different hour angles. Each source, except \nCTS247 which was excluded due to observational constraints, was typically \nobserved for a total of $\\sim$30\\,min. During both observing runs, strong \nfringe finders\/bandpass calibrators such as J0555+3948, J0927+3902, \nJ1800+3848 and J2253+1608 were also observed every $\\sim$3\\,hr for 4-5\\,min. \n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\psfig{figure=\"J0733MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0801MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0816MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J0839MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J0852MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1017MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J1223MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1237MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1242MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n\\hbox{\n\\psfig{figure=\"J1337MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1406MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n\\psfig{figure=\"J1413MAP_NOLABELS.PS\",height=5.0cm,width=5.5cm,angle=-90}\n}\n}\n}\n\\caption[]{Contour plots of VLBA images at 1.4\\,GHz. The rms in the images are listed \nin Table~\\ref{vlbares}. \n{At the bottom of each image the restoring beam is shown as an ellipse, \nand the first contour level (CL) in mJy\\,beam$^{-1}$ and FWHM are noted}.  \nThe contour levels are plotted as CL$\\times$($-$1, 1, 2, 4, 8,...)\\,mJy\\,beam$^{-1}$. {Depending upon the detailed structure of the radio sources,\nthe emission could be more extended at the redshifted\n21-cm frequencies.}\n}\n\\vskip -22.3cm\n\\begin{picture}(400,400)(0,0)\n\\put(-019,373){\\small      J0733$+$2721}\n\\put( 084,272){\\tiny       CL = 1.5}\n\\put( 043,265){\\tiny       B:0.011$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$13$^\\circ$ }\n\\put( 144,373){\\small      J0801$+$4725}\n\\put( 245,272){\\tiny       CL = 0.8}\n\\put( 205,265){\\tiny       B:0.012$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$21$^\\circ$ }\n\\put( 304,373){\\small      J0816$+$4823}\n\\put( 405,272){\\tiny       CL = 0.6}\n\\put( 364,265){\\tiny       B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$ 1$^\\circ$ }\n\\put(-017,230){\\small      J0839$+$2002}\n\\put( 084,129){\\tiny       CL = 0.7}\n\\put( 043,122){\\tiny       B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$-$ 1$^\\circ$ }\n\\put( 144,230){\\small      J0852$+$2431}\n\\put( 244,129){\\tiny       CL = 0.6}\n\\put( 204,122){\\tiny       B:0.010$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$ 2$^\\circ$ }\n\\put( 304,230){\\normalsize J1017$+$6116}\n\\put( 404,129){\\tiny       CL = 2.5}\n\\put( 363,122){\\tiny       B:0.014$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$46$^\\circ$ }\n\\put(-017,087){\\normalsize J1223+5037}\n\\put( 084,-13){\\tiny       CL = 2.5}\n\\put( 043,-20){\\tiny       B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$12$^\\circ$ }\n\\put( 144,087){\\normalsize J1237$+$4708}\n\\put( 244,-13){\\tiny       CL = 0.7}\n\\put( 204,-20){\\tiny       B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$+$ 7$^\\circ$ }\n\\put( 304,087){\\normalsize J1242$+$3720}\n\\put( 404,-13){\\tiny       CL = 4.0}\n\\put( 363,-20){\\tiny       B:0.008$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$+$ 4$^\\circ$ }\n\n\\put(-017,-56){\\normalsize J1337$+$3152}\n\\put( 084,-156){\\tiny      CL = 0.5}\n\\put( 043,-163){\\tiny      B:0.009$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$11$^\\circ$ }\n\\put( 144,-56){\\normalsize J1406$+$3433}\n\\put( 244,-156){\\tiny      CL = 1.5}\n\\put( 204,-163){\\tiny      B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$+$ 8$^\\circ$ }\n\\put( 304,-56){\\normalsize J1413$+$4505}\n\\put( 404,-156){\\tiny      CL = 0.7}\n\\put( 363,-163){\\tiny      B:0.010$^{\\prime\\prime}$$\\times$0.004$^{\\prime\\prime}$,$-$10$^\\circ$ }\n\\end{picture}\n\\vskip +9.5cm\n\\label{vlbamap}\n\\end{figure*}\n\\begin{figure}\n\\addtocounter{figure}{-1}\n\\centerline{\n\\vbox{\n\\hbox{\n\\psfig{figure=\"J1435MAP_NOLABELS.PS\",height=6.0cm,width=6.5cm,angle=-90}\n}\n}\n}\n\\caption[]{ {\\sl Continued}. \n}\n\\vskip -7.1cm\n\\begin{picture}(400,400)(0,0)\n\\put( 050,373){\\normalsize J1435$+$5435}\n\\put( 172,252){\\tiny       CL = 0.8}\n\\put( 131,245){\\tiny       B:0.013$^{\\prime\\prime}$$\\times$0.005$^{\\prime\\prime}$,$-$19$^\\circ$ }\n\\end{picture}\n\\vskip -7.1cm\n\\label{vlbamap}\n\\end{figure}\n\n\\begin{sidewaystable*}[]\n\\vskip -7 in\n\\vskip 0.3in\n\\centering\n\\begin{tabular}{lcccccccccccccc}\n\\multicolumn{15}{l}{{\\bf Table 5}: ~Results from the VLBA data. }\n\\\\\n\\hline\n\\hline\nSource      &$z_{\\rm abs}$&  Right       &   Declination       & rms &  Comp. &  S  &  r  &$\\theta$&  a & b\/a &$\\phi$& $S_{\\rm T}$ & $f_{\\rm VLBA}$ &LLS \\\\\nname        &       &  ascension     &                     &     &        &     &     &        &    &     &      &       &       &    \\\\\n            &       &   (J2000)      &   (J2000)&(mJy\\,beam$^{-1}$)&        &(mJy)&(mas)& ($^\\circ$)&(mas)& &($^\\circ$)& (mJy) &       &(pc)\\\\\n (1)        &  (2)  &      (3)       &       (4)           & (5) &  (6)   & (7) & (8) & (9)    &(10)&(11) &(12)  & (13)  &  (14) &(15)\\\\\n\\hline\nJ0733+2721  & 2.7263& 07 33 20.4830 & +27 21 03.430  & 0.3 &   1    & 97  &  0  &   -    &4.52 &0.05 &-88   & 240   & 0.62  &348 \\\\%\n            &       &               &                &     &   2    & 3   & 9.1 &-129    &13.24&0.00 &-18   &       &       &   \\\\%\n            &       &               &                &     &   3    & 19  &32.7 &-140    &5.98 &0.26 & 32   &       &       &   \\\\%\n            &       &               &                &     &   4    & 29  &10.8 &  51    &8.37 &0.44 &-17   &       &       &   \\\\%\nJ0801+4725  & 3.2235& 08 01 37.6930 & +47 25 28.082  & 0.2 &   1    & 28  &  0  &   -    &6.77 &0.00 & 72   & 78    & 0.67  &342 \\\\%\n            &       &               &                &     &   2    & 8   &9.34 &-112    &7.09 &0.00 & 72   &       &       &   \\\\%\n            &       &               &                &     &   3    & 16  &44.7 &-112    &8.63 &0.22 & 64   &       &       &   \\\\%\nJ0816+4823  &       & 08 16 19.0044 & +48 23 28.490  & 0.2 &   1    & 42  &  0  &   -    &4.59 &0.24 & 87   & 69    & 0.75  &68   \\\\%\n            &       &               &                &     &   2    & 10  & 9.0 &  132   &8.75 &0.43 &-63   &       &       &    \\\\%\nJ0839+2002  &       & 08 39 10.8970 & +20 02 07.391  & 0.2 &   1    & 113 &  0  &   -    &4.91 &0.24 & 84   & 130   & 0.87  &$\\le$39\\\\%\nJ0852+2431  & 2.7902& 08 52 57.1211 & +24 31 03.271  & 0.2 &   1    & 78  &  0  &   -    &4.92 &0.31 & 71   & 160   & 0.55  & 175\\\\%\n            &       &               &                &     &   2    & 10  & 21.9&  42    &15.2 &0.16 & 53   &       &       &   \\\\%\nJ1017+6116  &2.7681 & 10 17 25.8865 & +61 16 27.414  & 0.5 &   1    &388  &  0  &   -    &1.50 &0.68 & 55   & 477   & 0.86  &38\\\\%\n            &       &               &                &     &   2    & 24  & 4.7 &  145   &2.45 &0.00 & 70   &       &       &   \\\\%\nJ1223+5037  &       & 12 23 43.1740 & +50 37 53.344  & 0.5 &   1    & 96  &  0  &   -    &2.50 &0.00 & 78   & 229   & 0.60  &554   \\\\%\n            &       &               &                &     &   2    & 16  &13.6 &  80    &3.01 &0.83 &-19   &       &       &   \\\\%\n            &       &               &                &     &   3    & 25  &71.4 &  78    &8.61 &0.00 & 89   &       &       &   \\\\%\nJ1237+4708  &       & 12 37 17.4413 & +47 08 06.964  & 0.2 &   1    & 64  &  0  &   -    &3.17 &0.20 &-74   & 80    & 0.80  & $\\le27$\\\\%\nJ1242+3720  &3.4135 & 12 42 09.8121 & +37 20 05.692  & 0.6 &   1    & 848 &  0  &   -    &1.93 &0.76 & 22   & 662   & 1.00  & $\\le14$  \\\\%\nJ1337+3152  &3.1747 & 13 37 24.6931 & +31 52 54.642  & 0.2 &   1    & 83  &  0  &   -    &3.85 &0.38 & 74   & 83    & 1.00  &$\\le30$   \\\\%\nJ1406+3433  &2.4989 & 14 06 53.8532 & +34 33 37.339  & 0.4 &   1    &127  &  0  &   -    &3.24 &0.22 &-23   & 167   & 0.87  & 153  \\\\%\n            &       &               &                &     &   2    & 18  & 18.7& -30    &23.79&0.23 &-25   &       &       &   \\\\%\nJ1413+4505  &       & 14 13 18.8652 & +45 05 22.990  & 0.2 &   1    & 105 &  0  &   -    &2.19 &0.42 &-67   & 140   & 0.88  & 216  \\\\%\n            &       &               &                &     &   2    & 12  & 3.4 & -77    &5.37 &0.00 &-86   &       &       &   \\\\%\n            &       &               &                &     &   3    & 6   &27.9 & -72    &6.28 &0.03 &-67   &       &       &   \\\\%\nJ1435+5435  &3.3032 & 14 35 33.7812 & +54 35 59.312  & 0.2 &   1    & 31  &  0  &   -    &2.66 &0.17 &-29   & 96    & 0.55  &155   \\\\%\n            &       &                  &                   &     &   2    & 17  &20.4 &  155   &4.00 &0.47 &-36   &       &       &   \\\\%\n            &       &                  &                   &     &   3    & 5   & 7.6 &  153   &5.52 &0.00 &-7    &       &       &   \\\\%\n            &       &                  &                   &     &        &     &     &        &     &     &      &       &       &   \\\\%\n\\hline\n\\end{tabular}\n\\begin{flushleft}\nColumn 1:~Source name. Column 2:~absorption redshift. \nColumns 3 and 4:~right ascension and declination of component-1 (see column 6)\nfrom the multiple Gaussian fit to the source, respectively. \nColumn 5:~rms in the map in mJy\\,beam$^{-1}$. \nColumn 6:~component id. \nColumn 7:~flux density of the component in mJy.\nColumns 8 and 9:~radius and position angle of the component with respect to component-1, respectively.\n{Columns 10, 11 and 12: major axis, axial ratio and position angle \nof the deconvolved Gaussian component, respectively.}\nColumn 13:~flux density in mJy from FIRST\/NVSS. \nColumn 14:~$c_f$ is the ratio of 1.4 GHz flux density in \nVLBA image to that in the FIRST image, \nColumn 15:~largest projected linear size in pc. \\\\\n\\end{flushleft}\n\\label{vlbares}\n\\end{sidewaystable*}\n\n\nData were calibrated and imaged using AIPS and DIFMAP in a standard way. \nGlobal fringe fitting was performed on the phase-referencing calibrators.  \nThe delays, rates and phases estimated from these were transferred to the \nsources which were then self-calibrated until the final images were obtained. \nRadio sources were characterised by fitting Gaussian models to the self-calibrated \nvisibilities. \nVLBA maps of the 13 QSOs are shown in Fig.~\\ref{vlbamap} \nand the results of model fitting are listed in columns\\,\\#\\,6-12 of Table~\\ref{vlbares}.  \n\nNon-detection of 21-cm absorption in a DLA could be due to the small covering \nfactor of the absorbing gas. The typical spatial resolution achieved in our \nVLBA observations is $\\sim$8\\,mas. \nIf the extent of absorbing gas is of the order of the scales probed by our \nVLBA observations (i.e. $>$20pc) then we expect the detectability of \n21-cm absorption to depend on the fraction and spatial extent of radio \nflux density detected in these images.  In column\\,\\#\\,14 of Table~\\ref{vlbares} \nwe give the ratio of total flux densities detected \nin the VLBA and FIRST images at 20cm, i.e. $f_{\\rm VLBA}$. \\\nThe last column of this table \ngives the largest linear size (LLS), i.e the separation between the farthest radio components, of the \nradio source at the redshift of the DLA.        \nOut of the 13 QSOs presented in Fig.~\\ref{vlbamap} that have DLAs along \ntheir line of sight, we have 21-cm absorption \nspectra for only 8 DLAs.  For the DLA towards \nJ0816+4823, we use the 21-cm absorption measurement from   \n\\citet{Curran10}.  Thus we have a sample of 9 DLAs with both 21-cm \nabsorption measurements and VLBA 21-cm maps for the background QSOs. \nThe $f_{\\rm VLBA}$ for this sample ranges from 0.6 to 1, and LLS from \n$<$15\\,pc to 340\\,pc.  \n\n\n\n\nIn the absence of VLBI spectroscopy at the redshifted 21-cm line\nfrequency, the ratio of VLBA core flux density to the total flux density measured in the arcsecond scale\nimages (called core fraction $c_f$) has been used as an\nindicator of the covering factor $f_c$ \\citep[see][]\n{Briggs89, Kanekar09vlba}. \nHere we use the term `core' to refer to the flat spectrum\nunresolved radio component coincident with\nthe optical QSO in the VLBA image.\n\nFor J1242+3720 and J1337+3152 the radio source is modelled \nas a single unresolved component and within the uncertainties all \nthe flux density in the arcsecond scale \nFIRST images is recovered in our VLBA images (Fig.~\\ref{vlbamap}). \nBoth of these sources,  have \nflat spectra\n{suggesting the radio emission even at lower frequencies\noriginates predominantly from the compact core.\nTherefore we take $f_c$=1 for this case.}\n\nThe radio source J1017+6116 has an inverted radio spectrum ($\\alpha$=$-$0.4) \n and 86\\% of the \nflux density in the FIRST image is recovered in the VLBA image, 94\\% of which is contained in the main unresolved\ncomponent (Fig.~\\ref{vlbamap}).  \nFor another 3 flat-spectrum sources with 21-cm absorption measurements, i.e. \nJ0816+4823, J0852+2431, and J1406+3433, more than 80\\% of the VLBA flux \ndensity is present in a \nsingle unresolved component.    \nFor these 4 sources, based on the flat spectral index, the dominant\ncomponent in the VLBA image can be identified with radio core\/optical\nQSO.  \nTherefore for the 6 sources mentioned above, we have estimated \nthe core fraction, \n$c_f$, and used it as the covering factor, $f_c$, of the gas}. \nThe remaining three sources, J0733+2721, J0801+4725 and J1435+5435, exhibit multiple components in their \nVLBA 20-cm images.  The identification of the component coincident with the \nQSO, \nand the estimation of their $c_f$ from VLBA images for these three\nsources are highly uncertain.          \n\n\n\n\\section{Detectability of 21-cm absorption}\n\\label{detect}\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.4,angle=0.0]{fig6.ps}\n}\n\\caption{ The allowed range of fraction of DLAs having harmonic mean\nspin temperature $T_{\\rm S}$ greater than a limiting value $T_{\\rm S}^l$\nas a function of $T_{\\rm S}^l$. We use only those DLAs for which\n$c_f$ measurements are available.  The lower envelope of the shaded \nregion is obtained considering all the lower limits \non $T_{\\rm S}$  as measurements. The upper envelope is obtained \nassuming all the lower limits as measurements with \n$T_{\\rm S}\\ge T_{\\rm S}^l$.\n}\n\\label{fwnm}\n\\end{figure}\n\nIn this Section we investigate the detectability of 21-cm absorption in DLAs \nand the implication of non-detections for the physical state of the H~{\\sc i} \ngas.\nIt is clear from the last column of Table~\\ref{dlasamp}, \nthat for most of the DLAs, our data has good sensitivity to detect \n$T_{\\rm S}\/f_{\\rm c}$$\\sim$100~K gas.\n\nThe H~{\\sc i} 21-cm absorption is detected only in the \\zabs~=~3.1745 system\ntowards SDSS~J1337+3152. {This is one of the weakest radio sources in\nour sample (with a 3$\\sigma$ $\\int\\tau dv$ limit of 0.4 \\kms).  However, \nthanks to high $N$(H~{\\sc i}) our spectrum is sensitive enough to detect\nany gas with $T_{\\rm S}\\le 3100$ K.\n}\n This source is unresolved in our\nVLBA observations (see Fig~\\ref{vlbamap}). The L-band flux density\nmeasured in our VLBA image is consistent with that\nmeasured by the FIRST survey. Therefore,  the core fraction is,\n$c_{\\rm f} \\sim 1$, and the size of the VLBA beam is less than 30~pc at \nthe redshift of the absorber.\nThe spin-temperature, measured from the ratio of 21-cm optical depth and\nthe $N$(H~{\\sc i}) column density derived from the Lyman-$\\alpha$ trough,\nis 600$^{+220}_{-160}$~K which is consistent with the upper limit on $T_{\\rm S}$ \nobtained from the width of the single component Gaussian fit to the \n21-cm absorption \\citep{Srianand10}. \n\n\\begin{table*}\n\\addtocounter{table}{1}\n\\caption{Summary of 21-cm searches in $z\\ge2$ DLAs. Column 1: QSO. \nColumn 2: absorption redshitft.\nColumn 3: log $N$(H~{\\sc  i}). Column 4: Integrated optical depth. \nColumn 5: Reference for  $\\int\\tau dv$\ngiven in column 4. Column 6: the core fraction $c_f$. Column 9: the fraction\nof CNM (see the text for its definition). Column 10: the H$_2$ fraction\nand Column 11: References for $N$(H~{\\sc i}) and\/or f(H$_2$) measurements.\n}\n\\begin{tabular}{lcccccccccc}\n\\hline\n\\multicolumn{1}{c}{QSO} & \\zabs & log$N$(H~{\\sc i}) &$\\int \\tau dv$ & Refs&$c_f$ & $T_{\\rm S}\/f_{\\rm  c}$ & $T_{\\rm S}$     & $f({\\rm CNM})$& log f(\\h2) & Refs\\\\\n\\multicolumn{3}{c}{} & (\\kms) &\\multicolumn{2}{c}{} & (K) & (K) \\\\ \n\\multicolumn{1}{c}{(1)}        &  (2)  &      (3)       &       (4)           & (5) &  (6)   & (7) & (8) & (9)    &(10)&(11)  \\\\\n\\hline\n\\hline\n\\multicolumn{10}{c}{\\bf 21-cm detections}\\\\\nJ020346.6+113445& 3.38714& 21.26$\\pm$ 0.08 & 0.71$\\pm$0.02 & 1         &0.76     &1397 &1062&$\\sim$0.19 &$-6.2,-4.6$&a\\\\  \nJ031443.6+431405& 2.28977& 20.30$\\pm$ 0.11 & 0.82$\\pm$0.09 & 2         &....     &133  &....&$\\sim$1.00 &....       &d\\\\\nJ044017.2$-$433309& 2.34747& 20.78$\\pm$ 0.10 & 0.22$\\pm$0.03 & 3         &0.59     &1493 & 881&$\\sim$0.23 &....       &e\\\\  \nJ050112.8$-$015914& 2.03955& 21.70$\\pm$ 0.10 & 7.02$\\pm$0.16 & 4         &....     &390  & ...&....       &$\\le -6.40$ &b\\\\  \nJ133724.6+315254& 3.17447& 21.36$\\pm$ 0.10 & 2.08$\\pm$0.17 & 5         &1.00     &600& 600&$\\sim$0.33&$-7.00$      &c\\\\  \n\\multicolumn{10}{c}{\\bf 21-cm non-detections having metallicity measurements}  \\\\\nJ033755.7$-$120412& 3.1799 & 20.65$\\pm$ 0.10&$\\le$0.06       & 6         & 0.62&$\\ge$4057 &$\\ge$2515 &$\\le 0.08$ &$\\le -5.10$  &b\\\\  \nJ033901.0$-$013318& 3.0619 & 21.10$\\pm$ 0.10&$\\le$0.06       & 6         & 0.68&$\\ge$11435&$\\ge$7775 &$\\le 0.03$ &$\\le -6.90$  &b\\\\  \nJ040733.9$-$330346& 2.569  & 20.60$\\pm$ 0.10&$\\le$0.12       & 7         & 0.44&$\\ge$1807 &$\\ge$795  &$\\le 0.25$ & ....& e\\\\  \nJ040718.0$-$441013& 2.59475& 21.05$\\pm$ 0.10&$\\le$1.61       & 8         & ....&$\\ge$380  &....      &     ....  &$-2.61^{+0.17}_{-0.20}$&b\\\\  \nJ040718.0$-$441013& 2.62140& 20.45$\\pm$ 0.10&$\\le$1.93       & 8         & ....&$\\ge$80   &....      &   ....    &$\\le -6.20$&b\\\\  \nJ043404.3$-$435550& 2.30197& 20.95$\\pm$ 0.10&$\\le$0.33      & 7         & ....&$\\ge$1471 &....      &   ....    &$\\le -5.15$&b\\\\  \nJ053007.9$-$250330& 2.81115& 21.35$\\pm$ 0.07&$\\le$0.58      & 9$^\\dag$  & 0.94&$\\ge$2103 &$\\ge$1977&$\\le 0.10$  &$-2.83^{+0.18}_{-0.19}$&b\\\\  \nJ091551.7+000713& 2.7434 & 20.74$\\pm$ 0.10&$\\le$0.37      & 7         & ....&$\\ge$809  &....      &  ....     &.... &e\\\\  \nJ135646.8$-$110129& 2.96680& 20.80$\\pm$ 0.10&$\\le$0.33      & 6         & ....&$\\ge$1042  &....      &  ....     &$\\le -6.75$&b\\\\  \nJ135706.1$-$174402& 2.77990& 20.30$\\pm$ 0.15&$\\le$0.14      & 7         & ....&$\\ge$777  &....      &  ....     &$\\le -5.99$ &a\\\\  \nJ142107.7$-$064356& 3.44828& 20.50$\\pm$ 0.10&$\\le$0.14      & 7         & 0.69&$\\ge$1231 &$\\ge$849  &$\\le 0.24$ &$\\le -5.69$&b\\\\  \nJ234451.2+343348& 2.90910& 21.11$\\pm$ 0.10&$\\le$0.21      & 6         & 0.71&$\\ge$3343 &$\\ge$2373 &$\\le 0.08$ &$\\le -6.19$ &a\\\\  \n\\multicolumn{10}{c}{\\bf 21-cm non-detections having no metallicity measurements}  \\\\                                          \nJ053954.3$-$283956& 2.9742 & 20.30$\\pm$ 0.11&$\\le$0.06      & 6         & 0.47 & $\\ge$1812&$\\ge851$ &$\\le 0.23$ & ....&e\\\\  \nJ081618.9+482328& 3.4358 & 20.80$\\pm$ 0.20&$\\le$1.43      & 9         & 0.60 & $\\ge$240 &$\\ge144$ &$\\le 1.00$ & ....&a\\\\  \nJ073320.4+272103& 2.7263 & 20.25$\\pm$ 0.20&$\\le$0.14      & 8         & .... & $\\ge$692 &....     &.... & ....&a\\\\   \nJ080137.6+472528& 3.2235 & 20.80$\\pm$ 0.15&$\\le$0.22      & 8         & .... & $\\ge$1563&....     &.... & ....&a\\\\  \nJ085257.1+243103& 2.7902 & 20.70$\\pm$ 0.20&$\\le$0.32      & 8         & 0.49 & $\\ge$854 &$\\ge418$ &$\\le 0.48$ & ....&a\\\\\nJ101725.8+611627& 2.7681 & 20.60$\\pm$ 0.15&$\\le$0.29      & 8         & 0.81 & $\\ge$748 &$\\ge606$ &$\\le 0.33$ & ....&a\\\\  \nJ124209.8+372005& 3.4135 & 20.50$\\pm$ 0.30&$\\le$0.11      & 8         & 1.00  & $\\ge$1566&$\\ge1566$&$\\le 0.13$ & ....&a\\\\\nJ140501.1+041536& 2.708  & 21.07$\\pm$ 0.24&$\\le$0.19      & 9         & .... & $\\ge$3369& ....    &.... & ....&f\\\\  \nJ140501.1+041536& 2.485  & 20.20$\\pm$ 0.20&$\\le$0.08      & 9         & .... & $\\ge$1080 & ....    &.... & ....&f\\\\  \nJ140653.8+343337& 2.4989 & 20.20$\\pm$ 0.20&$\\le$0.31      & 8         & 0.76 & $\\ge$279 &$\\ge212$ &$\\le 0.94$ & ....&a\\\\  \nJ143533.7+543559& 3.3032 & 20.30$\\pm$ 0.20&$\\le$0.26      & 8         & .... & $\\ge$418 & ....    &.... & ....&a\\\\  \n\\hline\n\\end{tabular}\n\\begin{flushleft}\n{\nReferences in column \\#5: 1) \\citet{Kanekar07}, 2) \\citet{York07}, 3) \\citet{Kanekar06}, 4) \\citet{Briggs89}, 5) \\citet{Srianand10}, \n6) \\citet{Kanekar03}, 7) \\citet{Kanekar09ts}, 8) This paper, and 9) \\citet{Curran10}. $^\\dag$ Archival data from GBT08A\\_003 (PI: Curran) was processed \nthrough our pipeline. See text for details. \n\\\\}\n{References for $N$(H~{\\sc i}) and\/or f(H$_2$) measurements (column \\# 11): a) This paper, b) \\citet{Noterdaeme08}, c) \\citet{Srianand10}, d) \\citet{Ellison08}, e) \\citet{Akerman05}\nf) \\citet{Curran10}.}\n\\end{flushleft}\n\\label{tablesum}\n\\end{table*}                                      \n\nIn Table~\\ref{tablesum} we provide various details of our measurements \ntogether with the previous measurements at $z\\ge 2$ from the literature. \nWe present the results dividing the sample into three groups. These are systems\n with 21-cm detections (five systems), systems with 21-cm absorption upper limits with (twelve systems) and without (eleven systems) high-resolution optical \nspectra from which to derive accurate metallicities. The first\ntwo groups are used to investigate the connection between UV measurements\nand 21-cm optical depth. \nIn all cases the 3$\\sigma$ upper limits on the integrated\n21-cm optical depth are computed assuming a line width of 10~\\kms. \n\nThe 21-cm detection rate from our sample, \nwithout putting any sensitivity limit, \nis 10\\%. This is 13\\% when we restrict to $\\int \\tau dv$ limit\nof 0.4 \\kms ( the limit achieved in the case of J1337+3152\nwhere we have 21-cm detection).  Taken at face value, \nthe extended sample listed in Table~\\ref{tablesum}\ngives a 21\\% detection rate for  $\\int \\tau dv$ limit of 0.4 \\kms.\nFor a $\\int \\tau dv$ limit of 0.2 \\kms we get the detection rate of\n28\\%.\nHowever, these may not be representative values as the list of \nsystems compiled from the literature may be biased towards detections \nas some authors may not have reported their non-detections systematically\n\nSince we know $N$(H~{\\sc i}) from the damped Lyman-$\\alpha$ line, the \ndetection limit on the integrated optical depth implies a lower limit on the \nratio $T_{\\rm S}\/f_{\\rm c}$. \nThe  $T_{\\rm S}\/f_{\\rm c}$ measurements are reported in column 7 \nof Table~\\ref{tablesum}. \nIn column 6 of this table, we give the core fraction $c_{ f}$. \nAs mentioned above, $c_f$ is basically the ratio of flux density in the unresolved core seen in\nVLBA images to the total flux density measured in the arcsecond scale FIRST images. \nFor the objects from the literature we use the $c_f$ values given in\n\\citet{Kanekar09vlba}. These measurements were made at 327~MHz, close\nto the redshifted 21-cm frequencies. \nFollowing \\citet{Kanekar09vlba} we use core fraction ($c_f$)\nas the estimate of the covering factor ($f_c$).\nThe $T_{\\rm S}$ measurements given in column 8 of Table~\\ref{tablesum}\nare obtained by assuming $f_c = c_f$.\n\nIn Fig.~\\ref{fwnm} we plot the  percentage of DLAs having $T_{\\rm S}$ \ngreater than a limiting value $T_{\\rm S}^l$ as a function of \n$T_{\\rm S}^l$ for systems with $T_{\\rm S}$ measurements given in column\n8 of Table~\\ref{tablesum}. The lower envelope of the shaded region is \nobtained considering all the lower limits on $T_{\\rm S}$  as \nmeasurements. The upper envelope is obtained assuming\nall the lower limits as measurements with $T_{\\rm S}\\ge T_{\\rm S}^l$.\nIt is clear from the figure that\nmore than 50\\% of the DLAs have  $T_{\\rm S}\\ge 700 K$. Remember that the $T_{\\rm S}$ measured in an individual DLA\nis the harmonic mean temperature of different phases that contribute to the observed\n$N$(H~{\\sc i}).\nAssuming that the gas is simply a two phase medium with similar \ncovering factors the\nfraction of H~{\\sc i} in the CNM (called $f{\\rm (CNM)}$) can be written as,\n\\begin{equation}\nf{\\rm (CNM)} = {1 \\over T_{\\rm S}^W} \\bigg{[} {T_{\\rm S}^{\\rm C}T_{\\rm S}^W \\over T_{\\rm S}} - T_{\\rm S}^{\\rm C}\\bigg{]} \n\\end{equation}\nwhere,  ${T_{\\rm S}^{\\rm C}}$ and ${T_{\\rm S}^{\\rm W}}$ are the spin-temperature of the CNM and WNM\nrespectively.\n\\citet{Srianand05} have noticed that the H~{\\sc i} phase traced by the H$_2$ \nabsorption has temperature typically in the range 100-200~K. Thus we\nconsider the CNM temperature to be 200 K (instead of 70 K as\nseen in CNM of the\nGalaxy) so that the $f$(CNM) we get will be\na conservative upper limit.  Assuming ${T_{\\rm S}^{\\rm C}}\\sim200$~K and  ${T_{\\rm S}^{\\rm W}}\\sim10^4$~K,\n$T_S = 700$ K can be obtained for a combination of $f{\\rm (CNM)}$ = 0.27 and   \n$f{\\rm (WNM)}$ = 0.73. Therefore  $f{\\rm (CNM)}$ is less than  0.27 in \nat least 50\\% of the DLAs. Note that choosing  ${T_{\\rm S}^{\\rm W}}\\sim8000$~K\n(as suggested for the Galactic ISM) instead of the 10$^4$ K used here, \ndoes not change the results appreciably. \n\nWe estimate $f{\\rm (CNM)}$ for the four 21-cm detections\n(excluding J0501-0159 (B0458-020)  for which we do not have the covering factor value). \nApart from J0314+4314 (3C082) which seems to be a special case \\citep{York07}, \nthe CNM seems to represent roughly 20 to \n30\\% of the total $N$(H~{\\sc i}) measured in these DLAs. \nFor individual non-detections, we can calculate conservative upper limits of the \nfraction of $N$(H~{\\sc i}) in the CNM phase assuming $T_{\\rm S}^{\\rm C}$~=~200~K.\nThe values of $f$(CNM) are given in column \\#9 of Table~\\ref{tablesum} for systems with \n$f_{\\rm c}$ measurements. The upper limits vary between 0.10 and 1.0 with a\nmedian value of 0.23. \nThus the analysis presented here, under the assumption that $f_c = c_f$, \nsuggests that most of the neutral hydrogen in \nhigh-$z$ DLAs is warm. This is very much consistent with the conclusion of \\citet{Petitjean00} \nbased on the lack of \\h2 detections in most high-$z$ DLAs.\n\\begin{figure}\n\\centerline{\n\\includegraphics[scale=0.4,angle=0.0]{fig_metvsts.ps}\n}\n\\caption{Metallicity vs $T_S\/f_c$. The vertical dashed line marks the\nmedian metallicity measured in our sample. }\n\\label{tsvmet}\n\\end{figure}\n\n\\section{Results of Correlation analysis}\n\\label{gencor}\n\\begin{figure*}\n\\centerline{\n\\vbox{\n\\hbox{\n\\includegraphics[scale=0.4,angle=0.0]{fig1.ps}\n\\includegraphics[scale=0.4,angle=0.0]{fig2.ps}}\n\\hbox{\n\\includegraphics[scale=0.4,angle=0.0]{fig3.ps}\n\\includegraphics[scale=0.4,angle=0.0]{fig4.ps}\n}\n}}\n\\caption{Properties of 21-cm absorptions vs redshift and $N$(H~{\\sc i}).\nThe vertical dashed lines give the median value of the quantity plotted\nin the x-axis.}\n\\label{figcor}\n\\end{figure*}\n\nIn this Section we explore correlations between the 21-cm\noptical depth and other observable parameters. As we have only\na few 21-cm detections and mostly \nupper limits we use survival analysis and in particular the generalised rank \ncorrelation test \\citep{Isobe86}. For this purpose we use\nthe Astronomical SURVival analysis (ASURV) package.\n\n\\subsection{Metallicity vs T$_s$\/$f_c$}\nFirstly we study the importance of the metallicity of the gas.\nOnly 3 DLAs in our sample have measurements of metallicity from\nhigh resolution optical spectroscopy. In the extended sample\nat $z\\ge2$ (see Table~\\ref{tablesum}) there are 17 systems with metallicity measurements\nand 21-cm spectra. In Fig.~\\ref{tsvmet}, we plot $T_{\\rm S}\/f_{\\rm c}$ versus\nmetallicity.  The vertical long-dashed line marks the median metallicity of the points \nplotted in the figure. The only detection found in the low metallicity half is for \n\\zabs~=~3.1745 towards J1337+3152 reported from\nour survey.  The other four detections are from the high metallicity half. \nThe non-parametric generalized Kendall rank correlation test \nsuggests only a weak correlation between Z and $T_{\\rm S}\/f_{\\rm c}$ (at the 1.42$\\sigma$ level) with the probability that \nit can arise due to chance being 0.15. The significance is even lower (i.e 0.9$\\sigma$ with\na chance probability of 0.37) when we use $T_{\\rm S}$ (instead\nof $T_{\\rm S}\/f_{\\rm c}$) for cases where we have estimated \n$f_{\\rm c}$ measurements. \nWe wish to point out that a correlation between $T_s\/f_c$ and metallicity\nis reported in the literature \\citep[][]{Curran07,Kanekar09ts,Curran10}.\nThe lack of correlation in our sample (with systems in a restricted\nredshift range) could either reflect redshift evolution of the relationship\nor small range in metallicity covered by the sample.\nMetallicity measurements for the remaining 11 systems in Table~\\ref{tablesum} would \nallow us to address this issue in a statistically significant manner.\n\nWe also looked at the possible correlation between the 21-cm\noptical depth and the velocity width ($\\Delta v$) of the low \nionization lines. This information is available for 14 sources.\nAgain we find no statistically significant correlation \nbetween the two.  This is inconsistent with the 2.2 to 2.8 $\\sigma$ level\nweak correlation between $W$(Mg~{\\sc ii}) and $\\Delta v$\nreported by \\citet{Curran07}.\nHowever, this is consistent with the finding of\n\\citet{Gupta09}, that\n$W$(Mg~{\\sc ii}) \nand 21-cm optical depth are not correlated for a sample of 33 strong Mg~{\\sc ii} systems at \n1.10$\\le z \\le 1.45$ \\citep[see also][]{Kanekar09mg2}. \n\n\\subsection{Redshift dependence}\n\nIn the left hand side panels of Fig.~\\ref{figcor} we plot \n$T_s\/f_c$ and integrated 21-cm optical depth  vs redshift.\nNo clear correlation is evident in this figure. The non-parametric\nKendall test finds no significant correlation between $\\int \\tau dv$ \n(or T$_S$) and $z$. Note our sample probes only  a restricted \nredshift range in terms of time interval probed.\nHowever, the lack of correlation found here is consistent\nwith the near constancy of T$_s$\/$f_c$ as a function of redshift\nfound by \\citet{Curran10}. Understanding the redshift dependence of\n$T_S$ is very important in particular to address whether there is \nany evolution in $T_s$ \\citep[][]{Kanekar03} or\ngeometric effects \\citep[][]{Curran06}. \nTo make an unbiased comparison we need to have 21-cm measurements \nat low $z$ for a well defined sample of DLAs detected \nbased on \\lya\\ absorption.\n \n\n\\subsection{Dependence on $N$(H~{\\sc i})}\n\nRecently \\citet{Curran10} have found a 3$\\sigma$ level\ncorrelation between $N$(H~{\\sc i}) and $T_s\/f_c$. To check whether\nthis correlation holds at $z>2$, we plot, in the top  panels \nof Fig.~\\ref{figcor}, the integrated\n21-cm optical depth as a function of redshift and $N$(H~{\\sc i}).\nWe note\nthat there is a tendency for more 21-cm detections in DLAs\nwith higher $N$(H~{\\sc i}). \nHowever, the non-parametric\nKendall test finds no significant correlation  between\n$\\int \\tau dv$ and  $N$(H~{\\sc i}).\nIn the bottom right panel we plot $T_{\\rm S}\/f_{\\rm c}$ against log~$N$(H~{\\sc i}). \nThe Kendall test does not show any significant relation between the two \nquantities (1.28$\\sigma$ with a probability of 0.2 for this to be due to chance). \nThus we do not find any evidence for the 21-cm optical depth to depend on\n$N$(H~{\\sc i}) in our sample.\n\n\\section{21-cm absorption and H$_2$}\n\\label{mole}\nAs 21-cm absorption and H$_2$ molecules can give complementary information\non the physical state of the gas. In this Section, we study the relationship\nbetween these two indicators. \nThere are 13 DLAs in our extended sample for which\nthe expected optical wavelength range of redshifted \\h2 \nabsorptions has been observed at high spectral resolution. \nNine of these sources are part of the UVES sample of \\citet{Noterdaeme08}. \n\\citet{Srianand10} have reported the detection of \\h2 in J1337+3152 \nand here we report the search \nfor \\h2 in the remaining three DLAs (\\zabs = 3.3871 towards \nJ0203+1134, \\zabs = 2.7799 towards J1357$-$1744\nand \\zabs = 2.9091 towards J2344+3433). In the 10th column of \nTable~\\ref{tablesum}, we summarize \nthe molecular fraction \n$f$(\\h2)~=~2$N$(\\h2)\/(2$N$(\\h2)+$N$(H~{\\sc i}))] derived for these\n13 systems.\n\nIn 8 systems, neither 21-cm absorption nor H$_2$ molecules are detected\nwith typical upper limits of the order of 10$^{-6}$ for $f$(\\h2).\nApart from the system at \\zabs~=~2.6214 towards J0407$-$4410, the lower\nlimits on $T_{\\rm S}\/f_{\\rm c}$ for the remaining 7 systems are higher \nthan 700~K.  \nThere are 4 cases where $f_{\\rm c}$ measurements are available. In three\ncases (\\zabs~=~3.1799 towards J0337$-$1204, \\zabs~=~3.0619\ntowards J0339$-$0133 and \\zabs~=~2.9019 towards J2344+3433), the \nlower limit on $T_{\\rm S}$ is more than 2000~K. These are in line\nwith the suggestion by \\citet{Petitjean00} that the absence of\n\\h2 in most of the DLAs is due to the low density and high\ntemperature of the gas.\n\nIn two cases (\\zabs~=~2.5947 towards J0407$-$4410 (CTS 247) and \\zabs~=~2.8112 \ntowards J0530$-$2503 (PKS 0528-250)), strong H$_2$ absorption is detected with \nrotational excitations consistent with the \\h2-bearing gas being a CNM.\nHowever, 21-cm absorption is not detected in either case. We\ndiscuss these two systems in detail below.  \n\nAmong the five 21-cm absorbers, high resolution UVES spectra covering \nthe expected wavelength range of \\h2 absorption are available for four \nsystems. The exception is the \\zabs~=~2.28977 \nsystem towards J0314+4314 (B0311+430). For the \\zabs~=~2.3474 system towards \nJ0440$-$4333 (B0438$-$436) the continuum flux in the expected wavelength range is removed \nby high ionization lines from an associated system, as well as by a high-$z$ Lyman \nlimit system present along the line of sight.\nBelow we discuss the five systems where simultaneous\nanalysis of \\h2 and 21-cm absorption is possible.\n\n\\begin{figure*}\n\\centerline{\n\\includegraphics[scale=0.6,angle=270]{b0201_h2fit.ps}\n}\n\\caption{Voigt profile fits to H$_2$ Lyman and Werner band\nabsorption lines in the \\zabs~=~3.3868 DLA system towards J0203+1134 (PKS~0201+113).\nThe zero of the velocity scale is defined at $z = 3.38716$. The two\nvertical lines at $v = 0$ and $-$68~\\kms~ show the locations of\ntwo 21-cm absorption components reported by \\citep{Kanekar07}. \nThe vertical dotted line indicates the location of \\h2 absorption.\n In each panel we also show the error spectrum with dotted curves.\n}\n\\label{h2b0201}\n\\end{figure*}\n\n\n\\subsection{\\zabs = 3.3868 DLA towards  J0203+1134 (PKS 0201+113)}\n\nSearches for 21-cm absorption in this system  have yielded\nconflicting results \\citep{DeBruyn96, Briggs97}.\nBased on GMRT spectra taken at three different epochs, \n\\citet{Kanekar07} reported the detection of 21-cm absorption \nin two components at \\zabs~=~3.387144(17) and 3.386141(45). \nUsing $N$(\\ion{H}{i})$\\sim$ (1.8$\\pm$0.3)$\\times 10^{21}$~cm$^{-2}$ they \nobtained $T_{\\rm S} = [955\\pm160](f_{\\rm c}\/0.69)$~K. \nUsing high resolution optical spectrum,\n\\citet{Ellison01Q0201} have\nfound a gas phase metallicity of 1\/20 of solar with\nvery little dust depletion.\nThe gas cooling rate,\nlog~$l_{\\rm c} = -26.67\\pm0.10$~erg s$^{-1}$ Hz$^{-1}$,\nderived using C~{\\sc ii*} absorption is consistent with\nthis DLA being part of high-cool population defined by\n \\citet{Wolfe03a}. \nFrom \\citet{Kanekar07} we \ncan see that the strongest 21-cm absorption does not correspond to the strongest velocity\ncomponent in  either C~{\\sc ii$^*$} or Fe~{\\sc ii}. \n\nHere we report the detection of  \nH$_2$ absorption from J=0 and J=1 levels originating \nfrom both Lyman and Werner bands \n(see Fig.~\\ref{h2b0201}). A single component Voigt profile fit reproduces the data well. As the \nLyman-$\\alpha$ forest is dense and the spectral signal-to-noise ratio is not very high due to the\nfaintness of the QSO, we considered a range of $b$ values (i.e between 1 and 5~\\kms) to get the best\nfit values of log[$N$(H$_2$, J=0)] in the range 16.10$-$14.48 and  log[$N$(H$_2$, J=1)] = 16.03$-$14.57. \nWe estimated the kinetic temperature using the ortho-to-para ratio (i.e $T_{01}$) \nand found it to be in the range 48$-$108~K for the range of $b$ parameters\nconsidered above.  We note that for $b$ parameters greater than 2~\\kms, the H$_2$ lines are \nmainly in the linear portion of the curve of growth and the column density estimate is insensitive to the \nassumed value. \nThe average molecular fraction, $log~f({\\rm H_2})$, in the range, \n$-4.6\\le log~f({\\rm H_2})\\le -6.2$. \n\nDespite the gas being cold, there is no 21-cm absorption\ndetected at the position of the H$_2$ component (at z = 3.38679) \nwhich is well separated from the 21-cm absorption components \nIf we use $f_c=0.76$, as found by \\citet{Kanekar09vlba} using 326 MHz\nobservations, we find\nlog~$N$(H~{\\sc i}) $\\le19.12$. This is less than 1\\% of the total H~{\\sc i} column density measured in this system.\n\nUnlike most of the strong \\h2 systems, this system does not  show detectable C~{\\sc i} absorption. This means we do not have,\nunfortunately, an independent estimate of the density \nfrom fine-structure excitation.\n\n\\subsection{\\zabs = 2.5948 towards J0407-4410 (CTS 247)}\n\n As the radio source is faint, our GBT spectrum only gives \na weak limit on the spin temperature, $T_{\\rm S} \\ge 380$~ K when we use a \nline width of 10 \\kms.\n\\citet{Srianand05} have reported log~$N$(C~{\\sc ii$^*$})~=~13.66$\\pm$0.13. This, together with log~$N$(H~{\\sc i})~=~21.05$\\pm$0.10, \ngives a gas cooling rate of  log~$l_{\\rm c} = 26.92\\pm0.16$. This is very close to  the value $l_{\\rm c}^{\\rm crit}$\nthat seems to demarcate between the high and low cool systems defined by \\citet{Wolfe08}. \n\n\\citet{Ledoux03} reported the detection of H$_2$ from this system.\nThe H$_2$ absorption is well fitted with two components at \\zabs~=~2.59471 and 2.49486\nwith log~$N$(H$_2$)~=~18.14 and 15.51 respectively \\citep{Srianand05}. These components have $T_{01}$~=~121$\\pm$10 \nand 91$\\pm$6~K respectively. The average molecular fraction, log~$f$(H$_2$), is found to be $-2.42^{+0.07}_{-0.12}$ with \nan average metallicity of $-1.02\\pm0.12$ and moderate dust depletion \\citep{Ledoux03}. \n\nThe absence of 21-cm  absorption from this system is intriguing\nas H$_2$ components have T$\\sim$100 K.\nWith the same $b$ parameters as used to fit the \\h2 lines and \nthe rms from the GBT spectrum, we get a 3$\\sigma$ upper limit\nof $\\int \\tau dv = 0.88$ \\kms.  This translates to a constraint, $f_{\\rm c}\\times$$N$(H~{\\sc i}) $\\le$ 2$\\times 10^{20}$~cm$^{-2}$\nin the H$_2$ components where we have assumed $T_{\\rm S}$~=~$T_{01}$.\nUnfortunately we do not have a VLBA image of this source\nand it is difficult to constrain the covering factor of the gas. \nIf we assume $f_{\\rm c}\\sim 1$ then the upper limit on $N$(H~{\\sc i}) implies \nthat the H$_2$ component is a sub-DLA with\nlog~$f$(\\h2) $>-1.85$.\n\n From the column densities of the C~{\\sc i} fine-structure lines,\n\\citet{Srianand05} have constrained the particle density in the gas to be in the range \n$4.5<n_{\\rm H} ({\\rm cm}^{-3})<57.3$. For $n_{\\rm H} = 4.5~{\\rm cm}^{-3}$ and $f_{\\rm c} = 1$, we estimate \nthe thickness of the \\h2 cloud (i.e $N$(H~{\\sc i})\/$n_{\\rm H}$ ) \nalong the line of sight to be $\\le 15$ pc.\n\n\\subsection{\\zabs = 2.0395 towards J0501$-$0159 (B0458-020)}\n\n21-cm absorption in two velocity components was reported by \\citet{Wolfe85}. The background radio source shows structure \nover a wide range of scales and the absorbing cloud seems to cover most of these components \\citep{Briggs89}. \nThe estimated extent of the H~{\\sc i} absorber is $\\sim$8~kpc and the spin-temperature of the system is 390~K.\n\nC~{\\sc ii$^*$} absorption is detected  and the measured cooling rate is $l_c = -26.41\\pm0.10$ \\citep{Wolfe08}. This is consistent with the high cool population. \n\\h2 molecules are not detected with an upper limit of the molecular fraction of log~$f$(H$_2$)~$\\le -6.52$. This is one\nof the rare DLAs to show Lyman-$\\alpha$ emission in the middle of the DLA absorption \\citep{Moller04,Heinmuller06}.\n\nBased on the star formation rate derived from the Lyman-$\\alpha$ flux, \\citet{Heinmuller06} argued that the ambient radiation field\nis 10 times higher than the Galactic UV background. This excess radiation could be the main reason for the absence of \\h2 in the\ngas. \nThis picture is also confirmed by the lack of associated C~{\\sc i} absorption.\n\n\\subsection{\\zabs = 2.8111 towards  J0530-2503 (PKS 0528$-$250)}\n\n\\citet{Carilli96} searched for 21-cm absorption in this system using the WSRT and did not find any significant absorption. \nRecently \\citet{Curran10} have reported  GBT observations of this source with a better \nsignal-to-noise. They obtain a lower limit of the spin temperature of 700~K for $f_{\\rm c} = 1$, \nagain assuming a width of 10 \\kms. This source was found to be compact in the low frequency VLBA images with $f_c = 0.94$ \\citep{Kanekar09vlba}.\n\nNo measurement of cooling rate is possible for this system as the \nthe C~{\\sc ii$^*$} absorption is blended with other saturated lines.\n\nThis system has log~$N$(H~{\\sc i})~=~21.35$\\pm$0.07, $Z$~=~$-0.91\\pm0.07$ and \\zabs~higher than \\zem.\nThis is the first high redshift DLA where \\h2 absorption has been detected \\citep{Levshakov85,Srianand98, Srianand05}.\nThe high resolution UVES spectrum reveals two strong \\h2 components at \\zabs~=~2.81100 and \n2.81112 with log~$N$(\\h2) ~= ~17.93$^{+0.14}_{-0.20}$ and 17.90$^{+0.11}_{-0.14}$, and \n$ T_{01}$~=~167$\\pm$7 and 138$\\pm$12 K respectively.\n\n\n\nWe re-reduced the GBT data of \\citet{Curran10} in the same way as our GBT data and found the rms in the frequency range \nexpected for the 21-cm absorption associated with the \\h2 components to be 6.7~mJy.  This, together with the temperature \n$T_{01}$ and the \\h2 $b$ parameter give a constraint of log~$N$(H~{\\sc i})~$\\le$~20 in the \\h2 components. \n\nC~{\\sc i} absorption is detected only \nin one of the \\h2 components (i.e at \\zabs~=~2.81112)\\citep{Srianand05}.  From the C~{\\sc i}  level populations \nand for the above temperature, we derive that the particle density is in the range $25<n_{\\rm H} ({\\rm cm}^{-3})<270$. \nIf we use the $N$(H~{\\sc i}) limit of 10$^{20}$~cm$^{-2}$ derived from the 21-cm absorption, and use the constraint we get \nfor the hydrogen density in the case of the \\zabs = 2.81112 component, we get a limit on the thickness of the gas along\nthe line of sight to be 1.33~pc. \n\n\\subsection{\\zabs= 3.1745 towards J1337+3152}\n\nDetection of 21-cm and \\h2 in this system was reported in \\citet{Srianand10}. The weak \\h2 absorption is  \nwell aligned with the strongest metal line component but is\nslightly shifted by 2.5 \\kms with respect to the 21-cm absorption component.\nThis again suggests that the two transitions do not originate exactly from the same gas and that \nthe absorptions arise from an inhomogeneous absorbing region.\n\nAs the \\h2 column density is very low and the ratio of $N$(\\h2,J=0) to $N$(\\h2,J=1) is close to the maximum value\nallowed by the Boltzmann distribution,  we cannot constrain the kinetic temperature of the gas.\nC~{\\sc ii$^*$} absorption is detected with a total column density log~$N$(C~{\\sc ii}$^*$)~=~13.61$\\pm$0.08.\nUsing log~$N$(\\hi)~=~21.36$\\pm$0.10 we derive a gas cooling rate of log $l_{\\rm c} = -27.28\\pm0.13$. Thus the system \nbelongs to the low cool part of the DLA population.\nThe quasar is unresolved in our 1420 MHz VLBA image with a total flux density consistent with the measurement from FIRST\nobservations. This suggests that most of the emission is in the unresolved component. Using Gaussian fits we\nestimate the limit on the largest angular size to be $\\le$ 3.8 mas  (or $\\le$~30 pc at \\zabs).\nGiven the compact nature of the radio source it is  intriguing to see the difference between the \n\\h2 and 21-cm absorptions.  \nThis clearly shows that the absorbing gas contains a mixture of different phases at parsec scales.\n\n\n\\section{Results and discussion}\n\\label{results}\n\nWe have carried out a systematic search for 21-cm absorption in 10 \nDLAs at \\zabs$>2$ using GMRT and GBT. \nWe detect 21-cm absorption in only one of them. From our sample we \nfind the 21-cm detection rate is 13\\% \nfor a $\\int \\tau dv$ limit of 0.4 km\/s (the detection limit reached in \nthe case of J1337+3152).\nWe also obtained 1420~MHz VLBI images for the  sources in our sample.\n\n\nThe 21-cm detection at $z\\ge 2$ seems to favour systems with high \nmetallicity  and\/or  high $N$(H~{\\sc i}) \\citep[see also][]{Kanekar09ts, Curran10}.\nThis basically means that the probability of detecting cold components that can \nproduce detectable 21-cm absorption is higher in systems with high values \nof $N$(H~{\\sc i}) and Z.  However, we do not\nfind any correlation between the integrated optical depth \n(or T$_{\\rm S}$\/$f_{\\rm c}$) and $N$(H~{\\sc i}) or metallicity. \n\n\nIt is important to address the covering factor issue before drawing\nany conclusions on $T_S$. Ideally one should do high spatial resolution\nVLBA spectroscopy for this purpose \\citep[see for example][]{Lane00}.\nHowever, this is not possible at present specially for $z\\ge2$ absorbers.\nTherefore, we proceed by  assuming that the core fraction  found in the VLBA \nimages\nas the covering factor of the absorbing gas \\citep[as in the case of][]{Kanekar09vlba}.\nWe find that more than 50\\% of DLAs have weighted mean spin  temperature \n($T_{\\rm S}$) in excess of 700 K. For the  assumed temperature\nof the CNM gas $T_{\\rm S}^C = 200$ K (as seen in \\h2 components in high-z DLAs) \nwe find that more than 73\\% of H~{\\sc i} in such systems is\noriginating from WNM. The median value CNM fraction (i.e $f$(CNM)) \nobtained for the detections and the\nmedian value of upper limits in the case of non-detections are in the range 0.2 to 0.25. \n\nWe study the connection between 21-cm and \\h2 absorption in a sub-sample of 13 DLAs where both these species can be searched for.\nWe report the detection and detailed analysis of \\h2 molecules in the \\zabs=3.3871 DLA system towards J0203+1134 where 21-cm \nabsorption is also detected. For a $b$ parameter in the range 1-5~\\kms\\  we find 14.57$\\le$log~$N$(\\h2)$\\le$16.03. The inferred kinetic\ntemperature is in the range 48-108~K based on $T_{01}$ of H$_2$. However no 21-cm absorption is detected at the very position of \nthis \\h2\\  component. This suggests that the H~{\\sc i} column density associated with this component is $\\le$ 10$^{19}$~cm$^{-2}$. \nHowever, the lack of proper coincidence between 21-cm and any of the strong \nUV absorption components may also mean that \nthe radio and optical sight lines probe different volumes of the gas.\n\nIn the case of 8 DLAs, neither 21-cm nor H$_2$ are detected. Typical upper \nlimits on the molecular fraction  ($f_{\\rm H_2}$) in these systems are \n$\\le 10^{-6}$. \nThe lack of \\h2 in DLAs can be explained if the H~{\\sc i} gas originates\nfrom low density regions photoionized by the metagalactic UV\n\\citep[see for example,][]{Petitjean92,Petitjean00,Hirashita05}.\nThis also indicates that the volume filling  factor of \\h2 in DLAs \nis small  \\citep{Zwaan06}.  Typical limits obtained for  $T_{\\rm S}$\nin these systems are consistent with only a small fraction of\nthe H~{\\sc i} gas originating from the CNM phase as suggested by the\nlack of \\h2 absorption.\n\nIn two cases strong \\h2 absorption is detected and kinetic temperatures are in the range 100-200~K, but 21-cm absorption is not \ndetected.  Even in two cases where both the species are  detected they do not originate from the same velocity component. \nThe lack of 21-cm absorption directly associated with \\h2\\  indicates that only a small fraction (typically $\\le$ 10\\%)\nof the neutral hydrogen seen in the DLA \nis associated with the \\h2 components \\citep[see also][]{Noterdaeme10co}. This implies that the molecular fractions $f$(\\h2)\nreported from the \\h2 surveys should be considered as conservative lower limits for the \\h2 components.\n\nFor two of the \\h2-bearing DLAs with density measurements based on \nC~{\\sc i} fine-structure excitation we derive an upper limit on the line of \nsight thickness of $\\le 15$~pc. { This is consistent with the size estimate for the H$_2$-bearing gas in \\zabs = 2.2377 DLA\ntowards Q1232+082 based on partial coverage \\citep{Balashev11}.}\n\nIn principle, the presence of \\h2 and 21-cm absorptions in a single component provides  \na unique combination to simultaneously constrain the variation\nof the fine-structure constant ($\\alpha$), the electron-to-proton mass ratio ($\\mu$) and the proton G-factor. \nAs shown here, DLAs with 21-cm \nand \\h2 detections are rare. Even in these cases the presence of multiphase structure at parsec scale is evident, \nintroducing velocity shifts\nbetween the different absorption components that will affect the constraints on the variation of constants.\n\n\\section{acknowledgements}\nWe thank GBT, GMRT, VLBA and VLT staff for their support\nduring the observations and the anonymous referee for some\nuseful comments. We acknowledge the use of SDSS\nspectra from the archive (http:\/\/www.sdss.org\/).   \nThe National Radio Astronomy Observatory is a facility of the\nNational Science Foundation operated under cooperative agreement by\nAssociated Universities, Inc. \nVLBA data\nwere correlated using NRAO's implementation of the DiFX\nsoftware correlator that was developed as part of the Australian\nMajor National Research Facilities Programme and operated under\nlicence.\nRS and PPJ gratefully acknowledge support from the Indo-French\nCentre for the Promotion of Advanced Research (Centre Franco-Indien pour\nla promotion de la recherche avanc\\'ee) under Project N.4304-2.\n\n\\def\\aj{AJ\n\\def\\actaa{Acta Astron.\n\\def\\araa{ARA\\&A\n\\def\\apj{ApJ\n\\def\\apjl{ApJ\n\\def\\apjs{ApJS\n\\def\\ao{Appl.~Opt.\n\\def\\apss{Ap\\&SS\n\\def\\aap{A\\&A\n\\def\\aapr{A\\&A~Rev.\n\\def\\aaps{A\\&AS\n\\def\\azh{AZh\n\\def\\baas{BAAS\n\\def\\bac{Bull. astr. Inst. Czechosl.\n\\def\\caa{Chinese Astron. Astrophys.\n\\def\\cjaa{Chinese J. Astron. Astrophys.\n\\def\\icarus{Icarus\n\\def\\jcap{J. Cosmology Astropart. Phys.\n\\def\\jrasc{JRASC\n\\def\\mnras{MNRAS\n\\def\\memras{MmRAS\n\\def\\na{New A\n\\def\\nar{New A Rev.\n\\def\\pasa{PASA\n\\def\\pra{Phys.~Rev.~A\n\\def\\prb{Phys.~Rev.~B\n\\def\\prc{Phys.~Rev.~C\n\\def\\prd{Phys.~Rev.~D\n\\def\\pre{Phys.~Rev.~E\n\\def\\prl{Phys.~Rev.~Lett.\n\\def\\pasp{PASP\n\\def\\pasj{PASJ\n\\def\\qjras{QJRAS\n\\def\\rmxaa{Rev. Mexicana Astron. Astrofis.\n\\def\\skytel{S\\&T\n\\def\\solphys{Sol.~Phys.\n\\def\\sovast{Soviet~Ast.\n\\def\\ssr{Space~Sci.~Rev.\n\\def\\zap{ZAp\n\\def\\nat{Nature\n\\def\\iaucirc{IAU~Circ.\n\\def\\aplett{Astrophys.~Lett.\n\\def\\apspr{Astrophys.~Space~Phys.~Res.\n\\def\\bain{Bull.~Astron.~Inst.~Netherlands\n\\def\\fcp{Fund.~Cosmic~Phys.\n\\def\\gca{Geochim.~Cosmochim.~Acta\n\\def\\grl{Geophys.~Res.~Lett.\n\\def\\jcp{J.~Chem.~Phys.\n\\def\\jgr{J.~Geophys.~Res.\n\\def\\jqsrt{J.~Quant.~Spec.~Radiat.~Transf.\n\\def\\memsai{Mem.~Soc.~Astron.~Italiana\n\\def\\nphysa{Nucl.~Phys.~A\n\\def\\physrep{Phys.~Rep.\n\\def\\physscr{Phys.~Scr\n\\def\\planss{Planet.~Space~Sci.\n\\def\\procspie{Proc.~SPIE\n\\let\\astap=\\aap\n\\let\\apjlett=\\apjl\n\\let\\apjsupp=\\apjs\n\\let\\applopt=\\ao\n\\bibliographystyle{mn2e}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\begin{figure*}[htbp]\n\t\n\t\\includegraphics[width=\\textwidth]{fig\/motivation}\n\t\\caption{Illustration of a toy sequential recommendation example, where the blue node is the target user, the green nodes are the neighbors of the target user in the social network, and the pink nodes denote the items. The black dashed lines denote the anticipated recommended results between items and target user $u_1$. (a)-(b) Existing methods, which rarely consider the temporally dependent dynamics of the heterogeneous graph, lead to inaccurate prediction results. (c) On the contrary, the proposed \\textbf{TEA} method simultaneously aggregates the historical user behavior sequence and the dynamic heterogeneous graph, thus resulting in more accurate predictions than existing methods.}\n\t\\label{fig:motivation}\n\\end{figure*}\n\nThe sequential recommendation system is achieving more and more attention because of its practicality and effectiveness \\cite{eskandanian2019modeling, he2017translation, tang2013exploiting, tang2016recommendation}. \\textcolor{black}{In a sequential recommendation system,\nthe users access different items at different time stamps and frequently interact with each other.} The difficulties of sequential recommendation mainly come from two aspects:\n\\textcolor{black}{the temporal dependency of historical behaviors and the nonstationarity of users} \\textcolor{black}{The temporal dependency of historical behaviors means that the decision of a user is influenced by the historical behaviors.\nAnd the nonstationarity of users means that the decision of a user is influenced by the social relationship with the neighbors and the user-item interactions of the neighbors.}\nTherefore, one important challenge is how to effectively leverage the historical behaviors and the social relationship between users.\n\nFocusing on the above challenge, numerous sequential recommendation algorithms have been proposed in recent years, which leverage the user behavior sequence and employ the Markov Chains to model the transition of items. Eskandanian et al. \\cite{eskandanian2019modeling} mine the user preference and identify change-points\nin the sequence of user interactions by using Hidden Markov Model (as shown in Figure \\ref{fig:motivation} (a)). \nHe et al. \\cite{he2017translation} model the personalized sequential behavior by using the personalized translation vectors and the previous item embedding to predict the next item. \\textcolor{black}{These transition-based methods assume that the users are independent of each other, which ignores the influence between the users. Considering that the behavior of a user is easily affected by the neighbors, ignoring the dependence between users will suffer from limited performance in sequential recommendation.} \n\nAnother kind of recommendation algorithm \\cite{fan2019graph, ma2011recommender, tang2016recommendation,jamali2010matrix} focuses on analyzing the social relationships between users and user-item interactions in a static user-item graph.\nThe typical methods include the traditional Collaborative Filtering (CF) methods \\cite{hu2008collaborative, koren2008factorization, rendle2009bpr}, the deep learning enhanced approaches \\cite{liu2020deep, wang2019multi, deng2019deepcf, covington2016deep}, the recently developed graph neural networks based methods \\cite{battaglia2018relational}, as well as the social-network-based methods \\cite{fan2019graph,song2019session,yu2020enhance}. These methods reveal that both the interactions among users in social networks and the user-item bipartite graphs are beneficial to the performance of recommendation system. \nHowever, almost all the aforementioned methods assume the heterogeneous graphs of the users and the items are static, which ignores the dynamical influence \\textcolor{black}{of the temporal interaction between items and social networks}, and further results in the suboptimal performance of recommendation systems. Take Figure \\ref{fig:motivation} (b) for a toy example. The existing methods without considering the dynamic user-item heterogeneous graph might recommend the $v_1$ in preference to $v_2$ since more friends of user $U_1$ choose $v_1$. \n\n\n\nThus, it is essential to devise a unified framework to take advantage of both the historical behaviors of a user and dynamic interactions between the neighbors and items.  Figure \\ref{fig:motivation} (c) illustrates our main idea that models the temporally user-item heterogeneous graphs and generates a more accurate prediction.\nIn the figure, the decision of whether a user $u_i$ will choose a given item $v_{t+1}$ is controlled by two important factors: (1) the historical interactions between him or her and items; (2) the temporal dynamic heterogeneous graph, including the interactions between the neighbors and items. Hence, the goal of the proposed method is to estimate the conditional probability of $v_{t+1}$ given a user $u_i$, the historical accessed item sequence $v_{1:t}$, as well as the heterogeneous graph sequence $H_{1:t+1}$, which can be formulated as $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$.\n\nBased on the above idea, we propose the \\textbf{T}emporally \\textbf{E}volving \\textbf{A}ggregation (\\textbf{TEA} in short) framework for sequential recommendation by aggregating the user behavior sequence as well as the dynamic user-item heterogeneous graph. Inspired by the sequence labeling in natural language processing \\cite{panchendrarajan2018bidirectional, hao2021semi} to model the joint probability distribution), we adopt CRF to model the item decision sequence and estimate $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$. In order to alleviate the issue of the large item space, we use the pseudo likelihood method to approximate the aforementioned conditional probability. \n\\textcolor{black}{By doing this, the training procedure can be performed by estimating the unary score and transition score in CRF, which are implemented by our designed modules.}\n{Technically, we design a \\textit{Time-Restricted User Behavior Sequence Aggregation Module} to estimate the transition score of \\text{CRF}, and a \\textit{Temporal Dynamic Heterogeneous Graphs Aggregation Module }to estimate the Unary Scores of \\text{CRF}.}\nWe further provide two different practical implementations based on the proposed framework. Extensive experimental studies demonstrate that our \\textbf{TEA} framework outperforms the state-of-the-art recommendation methods on two published datasets and one real-world WeChat official accounts dataset.\n\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:related}, we review related researches into recommendation systems, including social recommendation and sequential recommendation. In Section \\ref{sec:model}, we define the problem of sequential recommendation under the dynamical heterogeneous graph and further derive the objective function based on the conditional random field. In Section \\ref{implementation}, we provide the implementation details of the proposed \\textbf{TEA} model. We further analyze the connection to existing methods in Section \\ref{connection}. And then, we present our experimental results based on two standard benchmarks and one real-world dataset in Section \\ref{sec:exp}. Finally, we give our conclusion of the proposed method. \n\n\n\\section{Related Works} \\label{sec:related}\nIn this section, we mainly discuss the existing techniques on social recommendation and sequential recommendation.\n\nIn order to effectively mine the deep demands of users, researchers set their sights on social relations, hence social recommendation has received more and more attention. One of the most important methods is Matrix Factorization (MF) \\cite{mnih2007probabilistic,baltrunas2011matrix,he2016fast}. Based on the traditional matrix factorization methods, Hao et al. \\cite{ma2008sorec} proposed a co-factorization method, which shares a common latent user-feature matrix factorized by both ratings and social relations. With the development of deep learning methods, He et al. \\cite{he2017neural} propose NeuMF by replacing the inner product with a neural architecture that can learn an arbitrary function from data. \nFan et al. \\cite{fan2018deep} propose a deep neural network-based model to learn non-linear features of each user from social relations and to integrate them into probabilistic matrix factorization for the social recommendation.\nDeng et al. \\cite{deng2016deep} propose a two-phase recommendation process to utilize deep learning to calculate the impact of community effect from the interests of users' trusted friends for recommendations.\n\nRecently, graph neural networks (GNNs) \\cite{battaglia2018relational,kipf2016semi} are widely used to aggregate node information and topological structure from social networks, hence GNNs are employed to address the social recommendation problem. In order to well aggregate the heterogeneous information, Fan et al. \\cite{fan2019graph} propose the GraphRec for the social recommendation. Fu et al. \\cite{fu2020magnn} leverage the metapaths \\cite{shi2018heterogeneous} to obtain the heterogeneous graph embedding. Considering that the influences in the social network may be context-dependent, Song et al. \\cite{song2019session} address the session-based social recommendation by using a dynamic-graph-attention neural network architecture. \nHowever, the aforementioned methods rarely consider the fact that different friends in social networks choose different items. In this work, considering the fact that social influence and user behaviors are time-dependent, the proposed \\textbf{TEA} method focuses on aggregating the temporally evolving social influence and the user behavior sequence.\n\nSince users usually access the items in chronological order, the users are likely to choose the items that are closely relevant to those they just accessed. Many works on sequential recommendation follow this assumption. Aiming to model the item-item transition probabilities, some traditional works borrow the idea of the Markov chain. Rendle et al. \\cite{rendle2010factorizing} bridge the Matrix Factorization (MF) and Markov Chains (MC). He et al. \\cite{he2017translation} propose TransRec to model such third-order relationships \\textcolor{black}{(e.g. the relationships among a user, the previously accessed item and the next item)} for large-scale sequential prediction. Motivated by the advantages of sequence learning in natural language processing, many neural network-based methods are proposed to learn the sequential dynamics. Tang et al. \\cite{tang2018personalized} leverage convolutional neural networks to encode the sequences into the embeddings. Other works \\cite{hidasi2018recurrent, quadrana2017personalizing} leverage recurrent neural networks and their variants to model the sequences of items. Kang et al. \\cite{kang2018self} further leverage attention-mechanism and propose the SASRec to balance the goal of MC-based methods and RNNs based methods. Moreover, Sun et al. \\cite{sun2019bert4rec} argue that such left-to-right unidirectional models are sub-optimal. So they propose BERT4Rec, which employs deep bidirectional self-attention to model user behavior sequences.\n\\textcolor{black}{In this paper, the proposed \\textbf{TEA} leverage the Conditional Random Field (CRF) to model the translation of items, which calculates the transition score and the unary score by respectively aggregating the user behavior sequence information as well as the dynamic user-item heterogeneous graph.}\n\n\n\n\\section{Model}\\label{sec:model}\n\\newcommand{\\tabincell}[2]{\\begin{tabular}{@{}#1@{}}#2\\end{tabular}}  \n\\begin{table}\n\\caption{Notation and Descriptions.}\n\t\\centering\n\t\\begin{tabular}{c|c}\n\t\t\\hline\n\t\t\\small{Notations}  & Descriptions \\\\\n\t\t\\hline\n\t\t$U,V$            & User and item set. \\\\\n\t\t\\hline\n\t\t$m,n$            & The size of user set and item set. \\\\\n\t\t\\hline\n\t\t$\\mathcal{G}^b, \\mathcal{G}^b_t$  & \\tabincell{c}{The bipartite graph only includes the user-item \\\\ interaction and that at the $t$-th timestamp.} \\\\\n\t\t\\hline\n\t\t$\\mathcal{E}^b, \\mathcal{E}^b_t$  & \\tabincell{c}{The edges set of bipartite graph \\\\ and that at the $t$-th  time-step.} \\\\\n\t\t\\hline\n\t\t$\\mathcal{G}^s$  & The social networks. \\\\\n\t\t\\hline\n\t\t$\\mathcal{E}^s$  & The edges among users in social network\\\\\n\t\t\\hline\n\t\t$\\mathbf{H}_t$  & \\tabincell{c}{The heterogeneous graph that includes the \\\\ social network $\\mathcal{G}^s$ and the bipartite graph $\\mathcal{G}^b_t$ \\\\ at $t$-th time-step.} \\\\ \n\t\t\\hline\n\t\t$\\mathbf{p}_i$   & The embedding of user $u_i$. \\\\\n\t\t\\hline\n\t\t$\\mathbf{q}_j$   & The embedding of item $v_j$. \\\\\n\t\t\\hline\n\t\t$\\mathbf{k}_j$   & The embedding of the $j$-th position in item sequences.\\\\\n\t\t\\hline\n\t\t$\\mathbf{W}, \\mathbf{b}$ & Weights and biases in neural networks. \\\\\n\t\t\\hline\n\t\t$d$              & The dimension number of representation. \\\\\n\t\t\\hline\n\t\t$\\Theta_f$       & The parameters of unary scores function. \\\\\n\t\t\\hline\n\t\t$\\Theta_g$       & The parameters of transition scores function. \\\\\n\t\t\\hline\n\t\t$\\oplus$         & The concatenation operator of any two vectors. \\\\\n\t\t\\hline\n\t\t$\\bm{x}$         & The observed item sequence.\\\\\n\t\t\\hline\n\t\t$\\bm{y}$         & The label item sequence of $\\bm{x}$ \\\\\n\t\t\\hline\n\t\t$\\mathcal{N}(u_i)$    & The 1st-order neighbourhood of user $u_i$ \\\\\n\t\t\\hline\n\t\t$\\mathcal{I}_t({u_i})$    & The accessed items of user $u_i$ at $t$-th time-step. \\\\\n\t\t\\hline\n\t\t$\\tau$    & \\tabincell{c}{Time window for selecting the walks in the duration \\\\ of [t-$\\tau$, t+$\\tau$].} \\\\\n\t\t\\hline\n\t\t$d$    & The dimension of user and item embedding. \\\\\n\t\t\\hline\n\t\\end{tabular}\n\t\\label{tab:notation}\n\\end{table}\nIn this section, we begin with the problem definition of sequential recommendation. Then we derive the unified objective function under conditional probability $P(v_{t+1}|u_i, \\mathcal{H}_{1:t+1}; v_{1:t})$.\n\n\\subsection{Problem Definition}\nLet $U=\\{u_1, u_2, \\cdots, u_n\\}$ and $V=\\{v_1, v_2, \\cdots, v_m\\}$ denote the sets of users and items respectively, in which $n$ is the number of users and $m$ is the number of items. For user-item interactions, we let $\\mathcal{G}^b=\\{U \\cup V, \\mathcal{E}^b\\}$ be the user-item bipartite graph with edges $(u_i, v_j) \\in \\mathcal{E}^b$. As for user-user relations, we let $\\mathcal{G}^s=\\{U, \\mathcal{E}^s\\}$ be the social graph with edges $(u_i, u_j) \\in \\mathcal{E}^s$. If we combine the bipartite graph and the social graph, we obtain the following heterogeneous graph $\\mathbf{H}=\\{U \\cup V, \\mathcal{E}^b \\cup \\mathcal{E}^s\\}$. Let $v_{1:t}$ be the user behaviors sequence for $u_i$. Since we consider the temporal evolving social influence, we let $\\mathcal{H}_{t+1}=\\{\\mathbf{H}_1, \\mathbf{H}_2, \\cdots, \\mathbf{H}_{t+1}\\}$ be the heterogeneous graph sequence, where $\\mathbf{H}_t=\\{U \\cup V, \\mathcal{E}^b_t \\cup \\mathcal{E}^s\\}$ and $\\mathcal{E}^b_t$ is the user-item interactions in $t$-th time-step. For user $u_i$, given the behavior sequence $v_{1:t}$ and the heterogeneous graph sequence $\\mathcal{H}_{t+1}$ as well as the item $v_{t+1}$, our goal is to estimate the conditional probability of $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$. The mathematical notation and the corresponding descriptions are summarized in Table \\ref{tab:notation}.\n\n\n\n\\subsection{Methodology}\nWe begin with the traditional Conditional Random Field (CRF), which is a probabilistic graphical model widely used in sequence labeling \\cite{panchendrarajan2018bidirectional}. CRF has shown to be very effective since it can jointly model the label decision by capturing the dependencies across adjacent labels. Considering the general definition of CRF, let $\\bm{x}=\\{x_1, \\cdots,x_t,\\cdots, x_T\\}$ and $\\bm{y}=\\{y_1, \\cdots, y_t, \\cdots, y_T\\}$ denote the observed sequence and its corresponding labels respectively. Formally, the conditional distribution $p(\\bm{y}|\\bm{x})$ of Linear Chain CRF\\cite{ma2016end} is given by:\n\\begin{equation}\n\\label{equ:crf}\n\\begin{split}\np(\\bm{y}|\\bm{x}) &= \\frac{1}{Z(\\bm{x})}\\exp(\\sum_{t=1}^{T}f(x_t,y_t;\\Theta_f) + \\sum_{t=1}^{T-1}g(y_t, y_{t-1};\\Theta_g)), \\\\\nZ(\\bm{x}) &= \\sum_{\\bm{y'}}\\exp(\\sum_{t=1}^{T}f(x_t,y'_t;\\Theta_f) + \\sum_{t=1}^{T-1}g(y'_t, y'_{t-1};\\Theta_g)),\n\\end{split}\n\\end{equation}\nin which $\\Theta_f$ and $\\Theta_g$ are the trainable parameters.\n\nThere are three important components in the above CRF model: the partition function $Z(\\bm{x})$, the unary scores function $f(x_t,y_t)$ and the transition scores function $g(y_t, y_{t-1})$. The partition function $Z(\\bm{x})$ is a normalization factor in order to obtain a probability. The unary scores function $f(x_t,y_t)$ is used to estimate the probability of $y_t$ given the observed $x_t$. And the transition scores function $g(y_t, y_{t-1})$ is used to estimate the probability of $y_t$ given $t_{t-1}$.\n\nThe three components framework provides us a unified solution to aggregate both the historical behaviors of users and the dynamic social influence from the social networks. Following the formulation of CRF, the purpose of our model is to estimate the conditional distribution as follow:\n\\begin{equation}\n\\label{equ:social_crf}\n\\begin{split}\nP(v_{1:t+1}|u_i, \\mathcal{H}_{t+1}) &= \\\\\n\\frac{1}{Z(\\mathcal{H}_{t+1}, u_i)}\\exp(&\\sum_{t=1}^{T}f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)\\\\&+\\sum_{t=1}^{T-1}g(v_{t+1}, v_{t};\\Theta_g)), \\\\\nZ(\\mathcal{H},u_i)=\\sum_{S'^{u_i}_{t+1}}\\exp(&\\sum_{t=1}^{T}f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)\\\\&+\\sum_{t=1}^{T-1}g(v_{t+1}, v_{t};\\Theta_g)),\n\\end{split}\t\n\\end{equation}\nin which $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ denotes the aggregation of temporal evolving social influence, $g(v_{t+1}, v_{t};\\Theta_g)$ denotes the aggregation of user behaviors. In specific, $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ describes the relationship between the the dynamic heterogeneous graph $\\mathbf{H}_{t+1}$ and the available item $v_{t+1}$ and $g(v_{t+1}, v_{t};\\Theta_g)$ models the dependency between the available item $v_{t+1}$ and the user behavior sequence.\n\nHowever, it is almost impossible to calculate $Z(\\mathcal{H}, u_i)$ since the sequence length is too large. In order to address this issue, we employ the pseudo likelihood method as an effective approximation method \\cite{besag1975statistical,ma2018cgnf}, and further derive the following estimation of the conditional probability:\n\\begin{equation}\n\\label{equ:social_crf_app}\n\\begin{split}\nP(v_{1:t+1}|u_i,\\mathcal{H}_{t+1}) \\approx PL(v_{1:t+1}|u_i,\\mathcal{H}_{t+1}) =\\\\ \\prod \\limits_{t} P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}).\n\\end{split}\n\\end{equation}\n\nCombining Equation (\\ref{equ:social_crf}) and Equation (\\ref{equ:social_crf_app}), we further derive the following estimation of the conditional probability $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$:\n\\begin{equation}\n\\begin{split}\nP(v_{t+1}|v_{1:t}&,u_i,\\mathcal{H}_{t+1})=\\\\\n&\\frac{\\exp(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)+g(v_{t+1}, v_{1:t};\\Theta_g))}\n{\\sum_{v_j\\in V}\\exp(f(\\mathbf{H}_{t+1}, u_i, v_j;\\Theta_f)+g(v_j, v_{1:t};\\Theta_g))}.\n\\end{split}\n\\end{equation}\n\nFinally, we can obtain the objective function of our proposed model as follows:\n\\begin{equation}\n\\mathcal{L}_{crf} = \\frac{1}{n}\\sum_{i=1}^{n} \\sum_{t=1}^{T}\\log P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}).\n\\end{equation}\n\nThe aforementioned objective function is usually impractical because the size of the item set is very large and the computation cost is unaffordable. \nInspired by \\cite{mikolov2013distributed}, we employ the negative sampling strategy to obtain the tractable unified objective function of sequential recommendation as follows:\n\\begin{equation}\n\\label{equ:final_loss}\n\\begin{split}\n\\mathcal{L}_{crf} =&\\frac{1}{n}\\sum_{i=1}^{n}\\sum_{t=1}^{T-1} \\log\\sigma(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f) \\\\&+ g(v_{t+1}, v_{1:t};\\Theta_g)) +\\\\\n& \\sum_{k=1}^{n_s}[\\log\\sigma(-f(\\mathbf{H}_{t+1}, u_i, v_{k};\\Theta_f) \\\\&- g(v_{k}, v_{1:t};\\Theta_g))],\n\\end{split}\n\\end{equation}\nwhere $\\sigma$ is the sigmoid activation function and $v_k$ is the negative item uniformly sampled from the whole item set $V$.\n\nThe objective function enjoys an \\textcolor{black}{appealing} physical meaning.\nIt provides the insights of how to design the model for sequential recommendation:  $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$ models the information of temporal evolving heterogeneous graph in the forms of the unary energy function; meanwhile  $g(v_{t+1}, v_{1:t};\\Theta_g)$ not only models the alternative item $v_t$ but also the user behavior sequence in the form of the pairwise energy function. \n\n\\section{Implementation of Temporally Evolving Aggregation Framework}\\label{implementation}\n\n\\begin{figure*}[htbp]\n\t\\includegraphics[width=\\textwidth]{fig\/model}\n\t\\caption{The framework of the temporally evolving aggregation model for the sequential recommendation. (a) The overview of the proposed model, the temporally dependent heterogeneous graphs aggregated representation $h_{t+1}^u$, the user behavior aggregated representation $h_{t+1}^v$ and the item embedding $\\mathbf{q}_{t+1}$ are fed into the CRF layer and $P(v_{t+1}|u_i, \\mathcal{H}_{t+1}; v_{1:t})$ is estimated. (b) The time-restricted user behavior sequence aggregation block is based on the user behavior sequence aggregation and the time-restricted aggregation. Note that the GRU used in this module is different from that in (a). (c) The dynamic temporally heterogeneous graph aggregation block, which is based on the bipartite graph aggregation and social network aggregation, takes $\\mathcal{H}_t$ as input, the arrows denote the message passing direction.}\n\t\\label{fig:model}\n\\end{figure*}\nIn this section, we provide the implementation details of the proposed temporally evolving aggregation model. As illustrated in Figure \\ref{fig:model}(a), our implementation takes both the aggregation of user behavior sequences and the aggregation of temporally dependent heterogeneous graphs into consideration and employs the GRU cells \\cite{cho2014learning} and CRF layers to predict the final results. The details of the two aggregation modules are presented in Figure \\ref{fig:model} (b) and Figure \\ref{fig:model} (c) respectively. We will give detailed descriptions of these two aggregation modules in the following subsections. \n\n\\subsection{Time-Restricted User Behavior Sequence Aggregation for the Transition Scores}\nIn this subsection, we will introduce the technical details of $g(v_{t+1}, v_{1:t};\\Theta_g)$. Given user $u_i$ and the corresponding behavior sequence $v_{1:t}$, we aim to calculate the user-specific item transition score. \n\n\\subsubsection{User Behavior Sequence Aggregation}\nConsidering that the future behavior of a user is not only influenced by the latest accessed items but also the items that the user has accessed before, the user behavior sequence aggregation block should consider both the transition between items and the long-term dependency of items. Inspired by the great success of the self-attention mechanism \\cite{vaswani2017attention} in various tasks like machine translation, we propose an extension of the self-attention mechanism to model the personalized item transition and long-term dependency by simultaneously leveraging the item information and the position information. Formally, given the $j$-th candidate item, we calculate the weights of each historical item as follows:\n\\begin{equation}\n\\begin{split}\n    a_{\\tau j} = \\text{softmax}(\\frac{\\mathbf{W}_{Q}(\\mathbf{q}_j+\\mathbf{k}_j) \\left(\\mathbf{W}_{K} (\\mathbf{q}_\\tau + \\mathbf{k}_\\tau) \\right)^{\\mathsf{T}}}{\\sqrt{d}}), &\\tau < j ,\n\\end{split}\n\\end{equation}\nwhere $\\mathbf{q}_j$ is the embedding of item $v_j$, $\\mathbf{k}_j$ is the position embedding at $j$-th position of the input sequence, $\\mathbf{W}_Q, \\mathbf{W}_K$ are trainable projection parameters and $\\sqrt{d}$ is the scaling factor, and $d$ is the dimension of the embedding. As a result, we can calculate the historical item aggregated representation as follows:\n\\begin{equation}\n\\mathbf{z}_j = \\sum_{\\tau=1}^{\\tau=j-1} a_{\\tau j}\\mathbf{W}_V\\left(\\mathbf{q}_\\tau + \\mathbf{k}_\\tau\\right),\n\\end{equation}\nin which $\\mathbf{W}_V$ are trainable projection parameters. \n\n\\subsubsection{Time-Restricted Aggregation}\nSince the temporal interactions between users and items are very sparse, for the users that contain limited social relationships and items interactions, it is hard to obtain a ideal user embedding for the sparse social substructure, and it is also difficult to obtain a debiased item embedding. Therefore, it is a challenging task to well aggregate the information from the users to the items and vice verse. Fortunately, we find that the users that select the same items usually share the same interests and intent. Inspired by this intuition, we further proposed the time-restricted aggregation module.\n\nFirst, we selected the walk with three nodes (e.g., USER-ITEM-USER) with the restriction of time window $\\tau$. In detail, given the interaction $(u_i, v_t)$, we find the other users that select the same item in the time window of $[t-\\tau, t+\\tau]$, where $\\tau$ is the window size. In our experimental implementation, we choose $\\tau=60$ days. Therefore, we can collect the $\\tau-$restricted walks for example $u_i-v_t-u'$. Sequentially, we employ another GRU to aggregate the information from the dense substructures to the sparse substructures, which can be formalized as follow:\n\\begin{equation}\n    \\mathbf{h}_{u_i}, \\mathbf{h}_{v_t}, \\mathbf{h}_{u'} = \\text{GRU}(\\mathbf{p}_i,\\mathbf{q}_t,\\mathbf{p}';\\mathcal{W}_R),\n\\end{equation}\nin which we take the walk $u_i-v_t-u'$ as input and $\\mathbf{h}_{u_i}, \\mathbf{h}_{v_t}, \\mathbf{h}_{u'}$ are the output of GRU of each timestamp; $\\mathcal{W}_R$ are the trainable parameters.\n\n\\subsubsection{Calculate the Transition Scores}\nIn order to well perform the personalized user behavior sequence aggregation, we further add the user embedding $\\mathbf{p}_i$ into the transformed item representation. Formally, we can calculate the transition score $s_t$ as follows:\n\\begin{equation}\n\\begin{split}\ns_t &= \\left(\\mathbf{W}_g^{(3)}\\left[\\mathbf{h}_t^{v_j}\\oplus \\bm{h}_{u_i} \\oplus  \\bm{h}_{v_t}\\oplus \\mathbf{p}_i \\right]\\right)^\\mathsf{T} \\bm{q}_j, \\\\\n\\mathbf{h}_t^{v_j} &= \\mathbf{p}_i + \\mathbf{W}_g^{(2)}\\left(\\text{ReLU}(\\mathbf{W}_g^{(1)}\\mathbf{z}_j  + \\mathbf{b}_g^{(1)}) \\right) + \\mathbf{b}_g^{(2)},\n\\end{split}\n\\end{equation}\nin which $\\mathbf{W}_g^{(1)}, \\mathbf{W}_g^{(2)}, \\mathbf{b}_g^{(1)}, \\mathbf{b}_g^{(2)}$ are the trainable parameters. For convenience, we let $\\Theta_g=\\{\\mathbf{W}_Q, \\mathbf{W}_K, \\mathbf{W}_V, \\mathbf{W}_g^{(1)}, \\mathbf{W}_g^{(2)}, \\mathbf{W}_g^{(3)},\\mathbf{b}_g^{(1)}, \\mathbf{b}_g^{(2)}, \\mathbf{p}, \\mathbf{q}, \\mathbf{k},\\bm{\\omega}_R\\}$ be the trainable parameters of $g(v_{t+1}, v_{1:t};\\Theta_g)$.\n\n\\subsection{Dynamic Temporal Heterogeneous Graphs Aggregation for the Unary Scores}\nIn this part, we will introduce the details of the dynamic temporally heterogeneous graphs aggregation $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$, which is used to calculate the unary scores. The dynamic temporally heterogeneous graphs aggregation contains the bipartite graph aggregation and the social network aggregation. \n\n\\subsubsection{Bipartite Graph Aggregation} In this part, we aim to obtain the aggregated of the bipartite graph at $t$-th time-step. Given user $u_i$ and the heterogeneous graph sequence $\\mathcal{H}_{t+1}$, we first obtain the user-specific representation $\\hat{\\mathbf{h}_{t}}$ of $\\mathcal{H}_t$. Specifically, we employ two different aggregated strategies and raise two variants of the proposed method: the GraphSAGE \\cite{hamilton2017inductive} based method (named TEA-S) and the graph attention networks \\cite{velivckovic2017graph} based method (named TEA-A). More experimental details will be introduced in the next section.\n\nAs for the TEA-S variation, we can obtain the user-specific representation $\\hat{\\mathbf{h}_{t}}$ as follows: \n\\begin{equation}\n\\hat{\\mathbf{h}_{t}} = \\text{ReLU} \\left({\\mathbf{W}_{A}} \\operatorname{MEAN}\\left(\\mathbf{q}_{k}, \\forall k \\in \\mathcal{I}_t(\\mathcal{N}(u_i))\\right)\\right) ,\n\\end{equation}\nwhere $\\mathbf{W}_{A}$ are the trainable parameters and $\\mathcal{I}_t(\\mathcal{N}(u_i))$ denotes the items interacted by $u_i$'s neighbors at between $t$-th and $t+1$-th time-step; and $\\operatorname{MEAN}$ denotes the average pooling operation.\n\nAs for the TEA-A variation, we aggregate the item information to the user with the help of the graph attention mechanism, which can be formulated as: \n\\begin{equation}\n\\begin{split}\n&\\qquad\\qquad\\hat{\\mathbf{h}_{t}} = \\text{ReLU} \\left(\\sum_{j \\in \\mathcal{I}_t(\\mathcal{N}(u_i)) } \\alpha_{i j}\\mathbf{q}_{j}\\right),\\\\\n\\end{split}\n\\end{equation}\n{where $\\alpha_{ij}$ is the weight of user $u_i$ and item $v_j$ and is defined as}\n\\begin{equation}\n\\small\n\\begin{split}\n\\alpha_{i j} = &\\frac{\\exp \\left(\\operatorname{LeakyReLU}\\left({\\mathbf{w}_{A}}^{\\mathsf{T}}\\left[ {\\mathbf{W}_{A}\\mathbf{q}_{t}} \\oplus  \\mathbf{W}_{A}\\mathbf{q}_{j}\\right]\\right)\\right)}{\\sum_{k \\in \\mathcal{I}_t(\\mathcal{N}(u_i)) } \\exp \\left(\\operatorname{LeakyReLU}\\left({\\mathbf{w}_{A}}^{\\mathsf{T}}\\left[ \\mathbf{W}_{A}\\mathbf{q}_{t} \\oplus  \\mathbf{W}_{A}\\mathbf{q}_{k}\\right]\\right)\\right)}\n,\n\\end{split}\n\\end{equation}\nin which $\\mathbf{q}_t$ is the embedding of the item interacted by $u_i$ at $t$-th time-step and $\\oplus$ is the concatenation operation. $\\mathbf{w}_{A}$ and $\\mathbf{W}_{A}$ are trainable parameters. And $LeakyReLU$ is the leaky version of a rectified linear unit.\n\nIn order to model temporally dependent heterogeneous graphs propagation, we feed $\\hat{\\mathbf{h}_t}$ into the Gated Recurrent Unit \\cite{cho2014learning}. The GRU cell operation at the $t$-th time-step can be formulated as: \n\\begin{equation}\n\\mathbf{h}_t = \\text{GRU}(\\hat{\\mathbf{h}_t}, \\mathbf{h}_{t-1}; \\mathcal{\\bm{W}}_G),\n\\end{equation}\nin which $\\mathcal{\\bm{W}}_G$ denotes all trainable parameters of the GRU cell. \\\\ \n\\subsubsection{Social Network Aggregation}\nTo propagate the information of neighbors' interests, we further aggregate the information from the social network. For simplicity, we only formulate the GraphSAGE aggregation as follows: \n\\begin{equation}\n\\mathbf{h}_{s} = \\text{ReLU}\\left(\\mathbf{W}_{S} \\operatorname{MEAN}\\left( \\mathbf{p}_{k}, \\forall k \\in \\mathcal{N}(u_i) \\right)\\right),\n\\end{equation}\nwhere $\\mathbf{W}_{S}$ is the trainable parameters. \n\n\n\\subsubsection{Calculate the Uunary Scores}\nBased on the aforementioned aggregation, we fuse the time-dependent representation $\\mathbf{h}_t$ and time-independent representation $\\mathbf{h}_s$ into one vector and calculate the social influence score $s_f$, i.e., the output of unary scores function $f(\\cdot)$. It is formulated as: \n\\begin{equation}\n\\begin{split}\ns_f &= {\\mathbf{h}_t^{u_i}}^\\mathsf{T} \\mathbf{q}_j,\\\\\n\\mathbf{h}_t^{u_i} = \\mathbf{W}_f^{(2)}\\text{ReLU}&(\\mathbf{W}_f^{(1)} [\\mathbf{h}_{t} \\oplus \\mathbf{h}_{s} ]  + \\mathbf{b}_f^{(1)}) + \\mathbf{b}_f^{(2)},\n\\end{split}\n\\end{equation}in which $\\mathbf{W}_f^{(1)}, \\mathbf{W}_f^{(2)}, \\mathbf{b}_f^{(1)}, \\mathbf{b}_f^{(2)}$ are trainable parameters. In summary, we let $\\Theta_f=\\{ \\mathbf{W}_{A}, \\mathbf{W}_{S}, \\bm{\\omega}_{G},\\mathbf{W}_f^{(1)}, \\mathbf{W}_f^{(2)}, \\mathbf{b}_f^{(1)}, \\mathbf{b}_f^{(2)}, \\mathbf{p}, \\mathbf{q}\\}$ be the trainable parameters of $f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\Theta_f)$.\n\n\\subsection{Model Summarization}\n\n\nThe total loss of our proposed model is summarized as follow:\n\\begin{equation}\n\\mathcal{L} = \\mathcal{L}_{crf} + \\gamma \\mathcal{L}_{reg},\n\\end{equation}\nwhere $\\mathcal{L}_{reg}$ is the L2 normalization on all parameters and $\\gamma$ is a trade-off hyper-parameter. \n\nBased on this objective function, our model is trained by the following procedure:\n\\begin{equation}\n(\\hat{\\Theta_g},\\hat{\\Theta_f}) = \\underset{\\Theta_g, \\Theta_f}{\\arg \\min }\\mathcal{L}.\n\\end{equation}\nAll parameters are jointly optimized using the Adam\\cite{kingma2014adam} algorithm. \n\nIn the testing, we estimate the probability of $P(v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1})$ as follows:\n\\begin{equation}\n\\begin{split}\nP(&v_{t+1}|v_{1:t},u_i,\\mathcal{H}_{t+1}) =\\\\&\\sigma(f(\\mathbf{H}_{t+1}, u_i, v_{t+1};\\hat{\\Theta_f})+g(v_{t+1}, v_{1:t};\\hat{\\Theta_g})).\n\\end{split}\n\\end{equation}\n\n\n\\section{Connections to Existing Models}\\label{connection}\nWe will discuss the connections to the existing transition-based sequential recommendation methods. Most of the existing works of transition-based sequential recommendation methods \\cite{he2017translation,rendle2010factorizing} are based on Markov Chains. These methods mainly consider two important factors: (1) the interactions between users and items to capture the inherent intent of users, (2) the sequential dynamics between items to capture the relationships between items. Thus, we find that our method is more general and some of the existing works can be taken as special cases of ours. The detailed discussions for each work are as follows. \n\nRegarding the work FPMC \\cite{rendle2010factorizing}, it simplifies the huge state space problem by introducing the basket of items and consequently ignores the sequence information of historical items in each basket. In the contrast, our method utilizes the historical item sequence by using the self-attention mechanism with position embedding and is more general than FPMC.\n\nRegarding the work TransRec\n\\cite{he2017translation}, it models the personalized sequential behavior by using the personalized translation vectors and the previous item embedding to predict the next items but ignores the long-term dependencies since it only considers the relationships between any two items. Moreover, TransRec addresses the problem of the huge state space of items by introducing the subspace, while our method utilizes the negative sampling strategy. Thus, our method is more feasible and efficient to capture the dynamic social influence of the target users.\n\n\n\n\n\\section{Experiment}\\label{sec:exp}\n\nIn this section, we experimentally evaluate the performance of our method on three datasets against the state-of-the-art compared methods. The preprocessed scripts and the source code can be found at \\footnote{{https:\/\/github.com\/DMIRLAB-Group\/TEA}}.\n\\begin{table}[t]\n\t\\caption{Statistics of the datasets.}\n\t\\label{tab:stat}\n\t\\centering\n\t\\scalebox{1.0}{\n\t\t\\begin{tabular}{p{2.0cm}p{1.5cm}p{1.5cm}p{1.5cm}}\n\t\t\t\\toprule\n\t\t\tDataset & \\textbf{Epinions} & \\textbf{Yelp} & \\textbf{Wechat}\\\\\n\t\t\t\\midrule\n\t\t\t\\# users & 22,167 & 270,770 & 568,100 \\\\\n\t\t\t\\# items & 296,278 & 184,134 & 242,702 \\\\\n\t\t\t\\# interactions & 798,620 & 3,602,495 & 9,422,722 \\\\\n\t\t\t\\# social links & 355,813 & 5,974,526 & 5,667,864 \\\\\n\t\t\tdensity & 0.0121\\% & 0.0072\\% & 0.0068\\% \\\\\n\t\t\tsocial density & 0.0724\\%\t& 0.0081\\% & 0.0018\\% \\\\\n\t\t\t\\bottomrule\n\t\t\\end{tabular}\n\t}\n\\end{table}\n\\subsection{Datasets}\nWe evaluate our proposed TEA framework on two public datasets (Epinions and Yelp) and a large-scale private dataset (WeChat Official Accounts Dataset). \nThe statistics of datasets are summarized in Table \\ref{tab:stat}. The brief information of the datasets is as follows: \n\\begin{itemize}\n\t\\item {Epinions}\\footnote{{http:\/\/www.trustlet.org\/extended\\_epinions.html}}: A benchmark dataset for the recommendation. In Epinions, a user can rate and give comments on items. Besides, a user can also select other users as their trusters, and we use the trust graphs \\textcolor{black}{(which are composed of the trust relationships)} as the network information. \n\t\\item {Yelp}\\footnote{{https:\/\/www.kaggle.com\/yelp-dataset\/yelp-dataset}}: An online review platform where users review local businesses (e.g., restaurants and shops). The user-item interactions and the social networks are extracted in the same way as Epinions.  \n\t\\item {WeChat Official Accounts Dataset}: WeChat is a Chinese multi-purpose messaging, social media, and mobile payment application developed by Tencent. And WeChat official accounts dataset is one of the functions. On the WeChat Official Account platform, users can read and share articles. This dataset is constructed by user-article clicking records and user-user social networks on this platform. \n\\end{itemize}\n\nWe preprocess the datasets following the approach in \\cite{he2017translation}. \\textcolor{black}{Specifically, for all these datasets, we follow the previous works \\cite{kang2018self,sun2019bert4rec} and treat a rating or review as implicit feedback.} We further use the timestamps to determine the sequence order of actions. We discard users and items with fewer than 5 associated actions. In cases where star ratings are available, we take the item with a rating higher than 3 as users' positive feedback. \n\nFor data splitting, we employ the widely used leave-one-out evaluation \\cite{rendle2009bpr, he2017neural}. We hold out the latest interaction of each user as the test set and select the second latest interaction as the validation set. The remaining data are used for training. \n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on Epinions dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:epin}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|lcccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models}            & HR@5           & NDCG@5         & HR@10          & NDCG@10        & HR@20          & {NDCG@20}       \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}}                        & 38.72$\\pm$0.10 & 29.66$\\pm$0.12 & 47.53$\\pm$0.10 & 32.50$\\pm$0.07 & 57.21$\\pm$0.22 & 34.95$\\pm$0.13  \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}}                        & 41.35$\\pm$0.59 & 31.13$\\pm$0.69 & 51.15$\\pm$0.43 & 34.31$\\pm$0.64 & 60.93$\\pm$0.34 & 36.78$\\pm$0.59  \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}}                     & 41.78$\\pm$0.16 & 32.57$\\pm$0.29 & 50.01$\\pm$0.18 & 35.23$\\pm$0.29 & 58.23$\\pm$0.14 & 37.31$\\pm$0.25  \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}}                       & 40.81$\\pm$0.33 & 31.14$\\pm$0.30 & 49.61$\\pm$0.16 & 33.99$\\pm$0.24 & 58.42$\\pm$0.19 & 36.22$\\pm$0.25  \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}}                     & 39.50$\\pm$0.35 & 30.16$\\pm$0.27 & 48.94$\\pm$0.42 & 33.21$\\pm$0.21 & 58.87$\\pm$0.29 & 35.72$\\pm$0.20  \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}}                     & 42.59$\\pm$0.07 & 32.20$\\pm$0.09 & 51.92$\\pm$0.08 & 35.22$\\pm$0.07 & 60.54$\\pm$0.09 & 37.41$\\pm$0.08  \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}}                        & 40.36$\\pm$0.25 & 30.52$\\pm$0.16 & 49.67$\\pm$0.14 & 33.53$\\pm$0.15 & 59.26$\\pm$0.19 & 35.95$\\pm$0.15 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}}                     & 35.15$\\pm$0.27 & 27.06$\\pm$0.33 & 45.01$\\pm$0.06 & 30.23$\\pm$0.24 & 55.85$\\pm$0.27 & 32.98$\\pm$0.30 \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}}                     & 44.79$\\pm$0.12 & 36.09$\\pm$0.21 & 52.51$\\pm$0.11 & 38.58$\\pm$0.17 & 60.98$\\pm$0.11 & 40.72$\\pm$0.07 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}}                          & 43.32$\\pm$0.20 & 33.97$\\pm$0.20 & 51.88$\\pm$0.20 & 36.74$\\pm$0.20 & 60.31$\\pm$0.20 & 38.87$\\pm$0.18  \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASAS \\cite{manotumruksa2020sequential}}                         & 44.97$\\pm$0.34 & 35.59$\\pm$0.29 & 53.44$\\pm$0.29 & 38.33$\\pm$0.27 & 61.41$\\pm$0.29 & 40.35$\\pm$0.28  \\\\ \n\t\t\\hline\n\t\tOurs \n\t\t& \\multicolumn{1}{l|}{TEA-A}                      & 47.84$\\pm$0.04 & 38.40$\\pm$0.41 & 55.99$\\pm$0.04 & 41.04$\\pm$0.41 & 63.51$\\pm$0.29 & 42.95$\\pm$0.38  \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S}                & \\textbf{48.13}$\\pm$0.25 & \\textbf{38.65}$\\pm$0.18 & \\textbf{56.10}$\\pm$0.17 & \\textbf{41.24}$\\pm$0.15 & \\textbf{63.58}$\\pm$0.08 & \\textbf{43.13}$\\pm$0.11  \\\\\n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\n\\subsection{Implementation Details}\nWe use PyTorch to implement our model and deploy it on RTX 2080 GPU. Hyper-parameter settings for all three datasets are as follows: embedding dimension $d=64$, batch size $B=1024$, dropout rate $p_\\text{drop}=0.5$, L2 regularization weight $\\gamma$=5e-4, negative sampling size $n_s=50$, sequence truncation length $L_s=50$, neighbor truncation length $L_n=20$, and learning rate $\\eta=0.01$. We train all the methods with five different random seeds and report the means and standard deviations.  \n\n\\subsection{Evaluation Metrics}\nWe evaluate all the models with two widely used Top-N metrics: Hit Rate@$K$ (HR@$K$) and Normalized Discounted Cumulative Gain@$K$ (NDCG@$K$).\nHR measures the percentage that recommended items contain at least one correct item interacted by the user, while NDCG considers the positions of correct recommended items. \nIn the context of sequential recommendation, since we only test on the latest item in a user behavior sequence, HR is identical to recall and proportional to precision \\cite{kang2018self}. \n\nSince it is time-consuming to rank all items for each user during the evaluation, we followed the strategy in \\cite{kang2018self}. Specifically, for each user, we randomly sample 100 negative items and rank these items with the ground-truth item. HR and NDCG are estimated based on the ranking results. We report the experiment results for $K=5\/10\/20$. \n\n\n\\subsection{Compared Methods}\nWe compare our proposed models (TEA-S and TEA-A) based on TEA framework with three kinds of baselines: the matrix factorization based models, the graph neural networks based models, and the sequence recommendation methods. \\\\\n\\textbf{Matrix Factorization based Methods}:\n\\begin{itemize}\n\t\\item BPRMF\\cite{rendle2009bpr}: A general learning framework for personalized ranking recommendation uses implicit feedback. \n\t\\item NeuMF\\cite{he2017neural}: It replaces the inner product with a multilayer perception (MLP) to learn the user-item interaction function.\n\t\\item SocialMF \\cite{jamali2010matrix}: It considers the social information and propagation of social information into the matrix factorization model.\n\t\\item SoRec\\cite{ma2008sorec}: It performs co-factorization on the user-item rating matrix and user-user social relations matrix.\n\\end{itemize}\n\\textbf{Graph Neural Network based Methods}:\n\\begin{itemize}\n\t\\item GraphRec\\cite{fan2019graph}: It uses the graph neural network to combine user behavior information and social network information into the recommendation task. For fairness, we discard the opinion\/rate embedding in our implementation. \n\n\t\\item LightGCN \\cite{he2020lightgcn}: A state-of-the-art graph-based collaborative filtering method. It explicitly integrates a bipartite graph structure into the embedding learning process to model the high-order connectivity in the user-item interaction graph\n\t\\item DGRec \\cite{song2019session}: A session-based recommendation method that combines the user action-temporal information and the social information via recurrent neural networks and dynamic graph attention networks.\n\\end{itemize}\n\\textbf{Sequential Recommendation Methods:}\n\\begin{itemize}\n\t\\item TransRec\\cite{he2017translation}: A sequential recommendation method that models each user as a translation vector to capture the transition from the current item to the next.\n\t\\item SASRec \\cite{kang2018self}: It leverages the Transformer\\cite{vaswani2017attention} to capture users' sequential behaviors.\n\t\\item ASASRec \\cite{manotumruksa2020sequential}: An improved version of SASRec with an adversarial training strategy. \n\t\\item DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}: It effectively utilizes the sequential data by segmenting the overall behavior sequence and maintaining the long-term interests of users. \n\\end{itemize}\n\\textbf{Model Variants:}\n\\begin{itemize}\n\t\\item TEA-S: We use the GraphSAGE based aggregation method in the bipartite graph aggregation. \n\t\\item TEA-A: We use the Graph Attention mechanism based aggregation method in the bipartite graph aggregation.\n\t\\item TEA-RS: We remove the time-restricted aggregation and use the GraphSAGE based aggregation method in the bipartite graph aggregation. \n\t\\item TEA-RA: We remove the time-restricted aggregation and use the Graph Attention mechanism based aggregation method in the bipartite graph aggregation. \n\\end{itemize}\n\n\\subsection{Results}\n\n\n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on Yelp dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:yelp}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|lcccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models}            & HR@5           & NDCG@5         & HR@10          & NDCG@10        & HR@20          & {NDCG@20}       \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}}                        & 66.33$\\pm$0.27 & 52.46$\\pm$0.16 & 76.51$\\pm$0.26 & 55.77$\\pm$0.16 & 84.59$\\pm$0.22 & 57.82$\\pm$0.15  \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}}                        & 70.38$\\pm$0.26 & 56.14$\\pm$0.28 & 79.35$\\pm$0.12 & 59.06$\\pm$0.24 & 86.14$\\pm$0.12 & 60.79$\\pm$0.22  \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}}                     & 64.82$\\pm$0.24 & 49.69$\\pm$0.24 & 76.27$\\pm$0.28 & 53.42$\\pm$0.21 & 84.99$\\pm$0.28 & 55.63$\\pm$0.19  \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}}                        & 70.41$\\pm$0.10 & 54.55$\\pm$0.10 & 81.45$\\pm$0.04 & 58.15$\\pm$0.07 & 89.03$\\pm$0.04 & 60.08$\\pm$0.06  \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}}                     & 68.37$\\pm$0.23 & 51.44$\\pm$0.27 & 81.55$\\pm$0.17 & 55.74$\\pm$0.18 & 90.61$\\pm$0.17 & 58.05$\\pm$0.16  \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}}                     & 73.04$\\pm$0.21 & 57.10$\\pm$0.21 & 84.39$\\pm$0.07 & 60.80$\\pm$0.19 & 92.08$\\pm$0.07 & 62.76$\\pm$0.17  \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}}                        & 76.22$\\pm$0.24 & 60.18$\\pm$0.28 & 86.57$\\pm$0.18 & 63.55$\\pm$0.26 & 92.93$\\pm$0.08 & 65.18$\\pm$0.16 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}}                     & 72.93$\\pm$0.33 & 57.45$\\pm$0.16 & 83.64$\\pm$0.34 & 60.94$\\pm$0.29 & 91.03$\\pm$0.25 & 62.82$\\pm$0.26 \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}}                     & 75.81$\\pm$0.15 & 60.63$\\pm$0.16 & 80.19$\\pm$0.20 & 64.00$\\pm$0.15 & 93.13$\\pm$0.12 & 65.78$\\pm$0.15 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}}                          & 69.28$\\pm$0.39 & 53.18$\\pm$0.43 & 81.66$\\pm$0.08 & 57.21$\\pm$0.37 & 90.36$\\pm$0.08 & 59.43$\\pm$0.34  \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASASRec \\cite{manotumruksa2020sequential}}                         & 72.97$\\pm$0.13 & 56.76$\\pm$0.10 & 84.53$\\pm$0.04 & 60.53$\\pm$0.09 & 92.18$\\pm$0.04 & 62.48$\\pm$0.07  \\\\ \n\\hline\n\t\tOurs\n\t\t& \\multicolumn{1}{l|}{TEA-A}                      & 80.38$\\pm$0.25 & 65.42$\\pm$0.36 & \\textbf{88.99}$\\pm$0.14 & 68.23$\\pm$0.33 & \\textbf{94.11}$\\pm$0.10 & 69.54$\\pm$0.21 \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S}                & \\textbf{80.43}$\\pm$0.18 & \\textbf{65.59}$\\pm$0.26 & 88.97$\\pm$0.08 & \\textbf{68.37}$\\pm$0.23 & 94.09$\\pm$0.07 & \\textbf{69.68}$\\pm$0.21 \\\\ \n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\\begin{table*}[htb]\n\t\\caption{The performance evaluation of the compared methods on WeChat dataset. The value presented are averaged over 5 replicated with different random seeds.}\n\t\\label{tab:Wechat}\n\t\\centering\n\t\\scalebox{1.1}{\n\t\\begin{tabular}{|l|ccccccc|}\n\t\t\\hline\n\t\tModel Class & \\multicolumn{1}{l|}{Models}            & HR@5            & NDCG@5          & HR@10           & NDCG@10          & HR@20          & {NDCG@20}        \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{BPRMF \\cite{rendle2009bpr}}                        & 62.33$\\pm$0.12 & 56.38$\\pm$0.12 & 68.55$\\pm$0.18 & 58.38$\\pm$0.14 & 75.60$\\pm$0.10 & 60.16$\\pm$0.14 \\\\\n\t\t& \\multicolumn{1}{l|}{NeuMF \\cite{he2017neural}}                        & 68.58$\\pm$0.16 & 61.66$\\pm$0.16 & 75.26$\\pm$0.18 & 63.82$\\pm$0.16 & 82.53$\\pm$0.14 & 65.65$\\pm$0.15 \\\\\n\t\t& \\multicolumn{1}{l|}{SocialMF \\cite{jamali2010matrix}}                     & 68.25$\\pm$0.13 & 61.30$\\pm$0.42 & 74.62$\\pm$0.07 & 63.36$\\pm$0.39 & 81.44$\\pm$0.04 & 65.09$\\pm$0.37 \\\\\n\t\t\\multirow{-4}{*}{\\begin{tabular}[c]{@{}l@{}}Matrix \\\\ Factorization \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{SoRec \\cite{ma2008sorec}}                       & 73.66$\\pm$0.04 & 66.43$\\pm$0.11 & 79.60$\\pm$0.02 & 68.36$\\pm$0.10 & 85.56$\\pm$0.03 & 69.86$\\pm$0.09 \\\\ \\hline\n\t\t\n\t\t& \\multicolumn{1}{l|}{GraphRec \\cite{fan2019graph}}                     & 66.04$\\pm$0.33 & 52.17$\\pm$0.29 & 76.80$\\pm$0.25 & 55.66$\\pm$0.26 & 85.72$\\pm$0.18 & 57.93$\\pm$0.24 \\\\\n\t\t& \\multicolumn{1}{l|}{LightGCN \\cite{he2020lightgcn}}                     & - & - & - & - & - & - \\\\\n\t\t\\multirow{-2}{*}{\\begin{tabular}[c]{@{}l@{}}Graph Neural \\\\ Network based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{DGRec \\cite{song2019session}}                        & 74.99$\\pm$0.22 & 63.94$\\pm$0.24 & 82.29$\\pm$0.14 & 66.31$\\pm$0.24 & 88.52$\\pm$0.09 & 67.89$\\pm$0.20 \\\\ \\hline\n\t\t& \\multicolumn{1}{l|}{DMAN \\cite{Tan_Zhang_Liu_Huang_Yang_Zhou_Hu_2021}}                     & - & - & - & - & - & - \\\\\n\t\t& \\multicolumn{1}{l|}{TransRec \\cite{he2017translation}}                     & 73.54$\\pm$0.17 & 64.87$\\pm$0.16 & 80.06$\\pm$0.14 & 66.99$\\pm$0.15 & 86.03$\\pm$0.14 & 68.50$\\pm$0.14 \\\\\n\t\t& \\multicolumn{1}{l|}{SASRec \\cite{kang2018self}}                          & 76.37$\\pm$0.35 & 65.97$\\pm$0.35 & 83.76$\\pm$0.21 & 68.37$\\pm$0.29 & 89.94$\\pm$0.11 & 69.94$\\pm$0.23 \\\\\n\t\t\\multirow{-3}{*}{\\begin{tabular}[c]{@{}l@{}}Sequence \\\\ based\\end{tabular}}\n\t\t& \\multicolumn{1}{l|}{ASASRec \\cite{manotumruksa2020sequential}}                         & 78.28$\\pm$0.23 & 68.13$\\pm$0.25 & 85.19$\\pm$0.21 & 70.38$\\pm$0.19 & 90.81$\\pm$0.14 & 71.80$\\pm$0.13 \\\\ \n        \\hline\n\t\tOurs\n\t\t& \\multicolumn{1}{l|}{TEA-A}                      & 82.06$\\pm$0.25 & 73.47$\\pm$0.26 & 87.61$\\pm$0.16 & 75.28$\\pm$0.21 & 92.18$\\pm$0.17 & 76.44$\\pm$0.13 \\\\\n\t\t& \\multicolumn{1}{l|}{TEA-S}                & \\textbf{83.23}$\\pm$0.19 & \\textbf{76.12}$\\pm$0.21 & \\textbf{88.12}$\\pm$0.13 & \\textbf{77.70}$\\pm$0.18 & \\textbf{92.42}$\\pm$0.17 & \\textbf{78.79}$\\pm$0.16 \\\\\n\t\t\\hline \n\t\\end{tabular}}\n\\end{table*}\n\n\n\n\n{Tables \\ref{tab:epin}, \\ref{tab:yelp} and \\ref{tab:Wechat} present the recommendation performance of all the methods on the three datasets, respectively. We do not report the performance of LightGCN, DMAN on WeChat Official Accounts dataset because of the limitation of memory.}\n\nFirst, by modeling social influence, the performances of social-aware methods (SocialMF, SoRec, GraphRec, and DGRec) are improved compared with that of BPRMF in most cases, which is consistent with previous works. This {observation} indicates that social information reflects users' interests effectively. \nSecond, the sequence based methods (DGRec, TransRec, SAS, and ASAS) also perform comparably well. These improvements reflect the importance of temporal information on recommendation tasks.  \nThird, DGRec and our proposed methods (including TEA-S and TEA-A) that combine social information and temporal information achieve much better performance, especially on large datasets. \nAt last, our proposed TEA-S and TEA-A consistently outperform all the {compared methods} on both public and real-world datasets with an average improvement of 3.15\\% on HR@10 and 8.38\\% on NDCG@10 against the best {competitor}. The significant improvements validate the effectiveness of aggregating the user behavior sequence and the the influence between the users. We also observe that performance of TEA-A is slightly {lower} than that of TEA-S, indicating that the graph attention mechanism is difficult to handle the high sparsity of temporally evolving heterogeneous graphs.\n\n\\begin{figure}[t]\n\t\\label{fig:hypm_emb}\n\t\\centering\n\t\\scalebox{0.48}\n\t{\\includegraphics[width=\\textwidth]{fig\/hypm_emb}}\n\t\\caption{The sensitivity of the embedding dimension $d$.}\n\\end{figure}\n\\subsection{The Sensitivity of Hyper-parameters}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[htbp]{0.47\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{fig\/ns_line_hr10}\n\\caption{The sensitivity of the negative sampling size $n_s$.}\n\\end{minipage}\n\\begin{minipage}[htbp]{0.47\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{fig\/ns_line_nd10}\n\\caption{The sensitivity of the negative sampling size $n_s$.}\n\\end{minipage}\n\\end{figure}\n\n\\begin{figure}[htbp]\n\\centering\n\\begin{minipage}[t]{0.24\\textwidth}\n\\centering\n\\label{fig:ablation_yelp}\n\\includegraphics[width=\\textwidth]{fig\/yelp_albation_hr20}\n\\caption{The Experiment results of ablation studies on Yelp dataset.}\n\\end{minipage}\n\\begin{minipage}[t]{0.24\\textwidth}\n\\centering\n\\label{fig:ablation_epinion}\n\\includegraphics[width=\\textwidth]{fig\/epinion_albation_hr20}\n\\caption{The Experiment results of ablation studies on Epinion dataset.}\n\\end{minipage}\n\\end{figure}\n\n\\subsubsection{Embedding Dimension.}\n\n\n\n\nIn {Figure} 3 we analyze the sensitivity of the embedding dimension $d$ by showing HR@10 and NDCG@10 of our proposed {TEA-S} with $d$ varying from 8 to 64. We can observe that our model significantly benefits from a larger dimension when the dataset is large. {A small embedding dimension ($d=16$) is enough for {TEA-S} to achieve the best performance on Epinions.} \n\n\n\n\n\n\\subsubsection{Sensitivity of the Number of Negative Samples.}\n{Figures} 4 and 5 shows the sensitivity of the number of negative samples $n_s$ in Equation (\\ref{equ:final_loss}) by showing HR@10 and NDCG@10 of our proposed {TEA-S} with $n_s$ varying from 1 to 100. The variant with $n_s=5$ performs comparably well, though using $n_s \\geq 10$ still boosts performance especially on the large-scale dataset, which means that using more negative samples is helpful to estimate the item transition probability. \nThe variant with $n_s=100$ achieves similar performance to the default setting $n_s=50$, which indicats that our model is stable with $n_s$. \n\n\\subsection{Ablation Study}\nIn order to evaluate the effectiveness of the time-restricted aggregation, we remove the aforementioned aggregation module and obtain the variants \\textbf{TEA-RS} and \\textbf{TEA-RA}. {experimental results are} shown in {Figures} 6 and 7. {From these results}, we can find that the models with time-restricted aggregation achieve a better performance, especially the results on the Yelp dataset. We also find that the promotion in Epinion dataset is not so remarkable, this is {since} the social networks in Epinion are much denser than that of Yelp. To some extent, the experiment results reflect that the proposed time-restricted aggregation can mitigate the drawbacks of sparse social networks and user-item interactions.\n\n\n\n\\section{Conclusion}\\label{sec:conclu}\nThis paper presents a temporally evolving aggregations framework for the sequential recommendation. Beginning from the original conditional random field, we derive the unified objective function for the sequential recommendation, \\textcolor{black}{which leverages the social influence between users and the dynamic user-item heterogeneous graph.} The proposed framework provides the insights and principles of designing the sequential recommendation model. We further provide two different implementations of the proposed framework. Experimental results on three real-world datasets show that the \\textbf{TEA} framework outperforms state-of-the-art methods.\n\n\\section{Acknowledgments}\n\nWe would like to thank Lingling Yi and Li Li from WeChat for their help and supports on this work.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\ifCLASSOPTIONcaptionsoff\n  \\newpage\n\\fi\n\n\n\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Baryonic two-point functions}\\label{sec:bar2pt}\n\nAs we are aiming for the determination of baryon masses, we consider baryonic two-point functions consisting of zero-momentum projected baryonic creation and annihilations operators $\\overline{\\mathcal{B}}=\\overline{B}\\overline{\\Psi}\\overline{\\Psi}\\overline{\\Psi}$ and $\\mathcal{B}=B\\Psi\\Psi\\Psi$. \nIn the following, we assume that $\\overline{\\mathcal{B}}$ and $\\mathcal{B}$ do not depend on $e$, i.e. $\\overline{\\mathcal{B}}=\\overline{\\mathcal{B}}{}^{(0)}$ and $\\mathcal{B}=\\mathcal{B}^{(0)}$. \nIn particular, we may apply QCD-covariant but not QCD+QED-covariant operator smearing. \nThe latter would lead to contributions from $\\overline{\\mathcal{B}}{}^{(\\frac{1}{2})}$, $\\overline{\\mathcal{B}}{}^{(1)}$, $\\mathcal{B}^{(\\frac{1}{2})}$ and $\\mathcal{B}^{(1)}$ and hence to additional diagrams.\n\nAs an example, the expansion of baryonic correlation functions is shown in \\cref{fig:diagrams}, in which sequential propagators are used in the first-order terms.\n\\begin{figure}[htb]\n\\begin{align*}\n\\langle\\mathcal{B}\\overline{\\mathcal{B}}\\rangle_S &=\n\\begin{aligned}[t]\n\\Big\\langle&\\vcenter{\\hbox{\\includegraphics[width=3cm]{figures\/bar2pt_0}}}+\\sum_f\\Delta m_f\\vcenter{\\hbox{\\includegraphics[width=3cm]{figures\/bar2pt_1_detf}}}  \\\\\n&+ e^2\\qty(\\vcenter{\\hbox{\\includegraphics[width=3cm]{figures\/bar2pt_1_exch}}} + \\vcenter{\\hbox{\\includegraphics[width=3cm]{figures\/bar2pt_1_tad}}} + \\vcenter{\\hbox{\\includegraphics[width=3cm]{figures\/bar2pt_1_bow}}}) \\\\\n    &+\\cdots\\Big\\rangle_{S_\\text{eff}^{(0)}}\n\\end{aligned}\n\\end{align*}\n\\caption{Diagrammatic expansion of QCD+QED correlation functions in terms of isospin-breaking parameters around correlators in $\\text{QCD}_{\\text{iso}}$. The index $f$ counts the different quark flavors.}\n\\label{fig:diagrams}\n\\end{figure}\nThe full expansion also contains diagrams with quark-disconnected parts and corrections acting in the sea quark sector.\nHowever, in this project we neglect isospin-breaking effects in the sea quarks (for the time being) and we set $\\Delta\\beta$ to 0.\nAs a consequence, only quark-connected contributions remain and $R=1$ in \\cref{eq:reweighting}.\nNonetheless, we calculate the diagrams with a photon line connected to one of the valence quarks in case we decide to include the disconnected contributions at a later stage since they are needed for diagrams in which a sea quark electromagnetically interacts with a valence quark.\nNote that in \\cref{sec:optimizations} only contributions shown in \\cref{fig:diagrams} are discussed. \nHowever, the additional diagrams can trivially be incorporated into the optimizations described, which moreover remove any significant overhead arising from the additional computation of these diagrams.\n\nIn order to extract the mass of the lowest  state incorporated in a baryonic two-point function, we consider the asymptotic time dependence $C(t_2,t_1)= c e^{-m(t_2-t_1)}$. \nExpanding the latter in terms of the isospin-breaking parameters $X=X^{(0)}+\\sum_i\\Delta\\varepsilon_iX_i+\\mathcal O(\\Delta\\varepsilon^2)$ with $X\\in\\{c,m,C\\}$, one finds the zeroth- and first-order contributions\n\\begin{align*}\nC^{(0)}(t_2,t_1)=&c^{(0)}e^{-m^{(0)}(t_2-t_1)}, \\\\\nC_i^{(1)}(t_2,t_1)=&\\qty(c^{(1)}_i-c^{(0)}m^{(1)}_i(t_2-t_1))e^{-m^{(0)}(t_2-t_1)}.\n\\end{align*}\n$m^{(0)}$ can then be reconstructed from the usual definition of the effective mass\n\\[\n    (am_\\text{eff})^{(0)}(t_{2},t_{1}):=\\log\\frac{C^{(0)}(t_2,t_1)}{C^{(0)}(t_2+a,t_1)},\n\\]\nwith the analogous definition for the first-order terms\n\\[\n    (am_\\text{eff})_{i}^{(1)}(t_{2},t_{1}):=\\frac{C_i^{(1)}(t_2,t_1)}{C^{(0)}(t_2,t_1)}-\\frac{C_i^{(1)}(t_2+a,t_1)}{C^{(0)}(t_2+a,t_1)}.\n\\]\n\n\\section{Inclusion of Perturbative Isospin-breaking Effects by Reweighting}\\label{sec:ib}\n\nThe procedure for the calculation of isospin-breaking corrections to the hadron masses follows the perturbative approach introduced by the RM123 collaboration \\cite{deDivitiis:2011eh,deDivitiis:2013xla}.\nHere, the QCD+QED action $S$ described by the set of parameters $\\varepsilon=(\\beta,e^2,m_u,m_d,m_s)$ (the inverse QCD coupling, the squared QED coupling, and the masses of the up, down, and strange quarks) is expanded around the isospin-symmetric action $S^{(0)}$ with parameters $\\varepsilon^{(0)}=(\\beta^{(0)},0,m_{ud}^{(0)},m_{ud}^{(0)},m_s^{(0)})$ in terms of these parameters.\n$S^{(0)}[U,\\psi,\\bar\\psi]=S_g^{(0)}[U]+S_q^{(0)}[U,\\psi,\\bar\\psi]$ here denotes an isospin-symmetric action consisting of the L\u00fcscher-Weisz action $S_g^{(0)}$ for the gauge fields and an $O(a)$ improved action $S_q^{(0)}$ for Wilson fermions with $N_f=2+1$ flavors \\cite{Bruno:2014jqa}. In this work we use gauge ensembles simulated by the CLS\ncollaboration.\nThe QCD+QED action can be divided into three parts, a QCD gauge action, a QED gauge action, and a quark action:\n\\[\n    S[U,A,\\psi,\\bar\\psi]=S_g[U]+S_\\gamma[A]+S_q[U,A,\\psi,\\bar\\psi]\n\\]\nFor the QED gauge action in finite volume, we use the non-compact QED$_{\\text{L}}$ prescription \\cite{Hayakawa:2008an} in Coulomb gauge, which introduces an infrared regularisation by eliminating the zero-momentum modes of the photon field by setting\n\\[\n    \\sum_{\\vb x}A^{\\vb x,t}\\equiv0\n\\]\non every timeslice $t$.\n\nIn this setup, expectation values for operators in the full theory, i.e.\n\\begin{align}\n\\begin{aligned}\n    \\ev{\\mathcal O}_S=&\\frac1{Z}\\int\\mathcal DU\\mathcal DA\\mathcal D\\psi\\mathcal D\\bar\\psi\\,\\mathcal O[U,A,\\psi,\\bar\\psi]e^{-S[U,A,\\psi,\\bar\\psi]} \\\\\n    =&\\frac1Z\\int\\mathcal DU\n    \\underbrace{\\qty(\\frac1{Z_{q\\gamma}[U]}\\int\\mathcal DA\\mathcal D\\psi\\mathcal D\\bar\\psi\\,\\mathcal O[U,A,\\psi,\\bar\\psi]e^{-S_\\gamma[A]-S_q[U,A,\\psi,\\bar\\psi]})}_{=:\\ev{\\mathcal O}_{S_{q\\gamma}}[U]}\n    \\underbrace{Z_{q\\gamma}[U]e^{-S_g[U]}}_{=:\\exp(-S_\\text{eff}[U])}\n\\end{aligned}\\label{eq:ev}\n\\end{align}\ncan be expressed in terms of those in isosymmetric QCD by reweighting~\\cite{Risch:2017xxe,Risch:2018ozp}:\n\\begin{align}\n\\begin{aligned}\n    \\ev{\\mathcal O}_{S}=&\\frac{\\frac1{Z_\\text{eff}^{(0)}}\\int\\mathcal DU\\ev{\\mathcal O}_{S_{q\\gamma}}[U]\\frac{\\exp(-S_\\text{eff}[U])}{\\exp(-S_\\text{eff}^{(0)}[U])}e^{-S_\\text{eff}^{(0)}[U]}}{\\frac1{Z_\\text{eff}^{(0)}}\\int\\mathcal DU\\,\\frac{\\exp(-S_\\text{eff}[U])}{\\exp(-S_\\text{eff}^{(0)}[U])}e^{-S_\\text{eff}^{(0)}[U]}}=\\frac{\\ev{R\\ev{\\mathcal O}_{S_{q\\gamma}}}_{S_\\text{eff}^{(0)}}}{\\ev R_{S_\\text{eff}^{(0)}}} \n\\end{aligned}\\label{eq:reweighting} ,\n\\end{align}\nwhere $R$ denotes the reweighting factor\n\\begin{align}\n    R=\\frac{\\exp(-S_\\text{eff})}{\\exp(-S_\\text{eff}^{(0)})}=\\frac{\\exp(-S_g)\\,Z_{q\\gamma}}{\\exp(-S_g^{(0)})\\,Z_q^{(0)}},\n\\end{align}\nwhich replaces the Boltzmann weight associated with the effective action of QCD$_{\\mathrm{iso}}$ by its counterpart in QCD+QED.\nThe effective actions are defined as\n\\begin{align*}\n    S_\\text{eff}[U]=S_g[U]-Z_{q\\gamma}[U]=&S_g[U]-\\log(\\int\\mathcal DA\\mathcal D\\psi\\mathcal D\\bar\\psi\\,e^{-S_\\gamma[A]-S_q[U,A,\\psi,\\bar\\psi]}) \\\\\n    S_\\text{eff}^{(0)}=S_g^{(0)}[U]-Z_q^{(0)}[U]=&S_g^{(0)}[U]-\\log(\\int\\mathcal D\\psi\\mathcal D\\bar\\psi\\,e^{-S_q^{(0)}[U,\\psi,\\bar\\psi]}).\n\\end{align*}\n\nTo evaluate the expectation value $\\ev{\\mathcal O}_{\\mathrm{q\\gamma}}$ and the reweighting factor $R$ in \\cref{eq:reweighting} we use perturbation theory and expand the latter expressions in terms of the parameters\n\\begin{align*}\n\\Delta\\varepsilon=&\\varepsilon-\\varepsilon^{(0)} = (\\Delta\\beta,e^2,\\Delta m_u,\\Delta m_d,\\Delta m_s) \\\\\n=&(\\beta-\\beta^{(0)},e^2,m_u-m_{ud}^{(0)},m_d-m_{ud}^{(0)},m_s-m_s^{(0)})\n\\end{align*}\naround $\\varepsilon^{(0)}$. Operators that depend on the QED gauge links $\\exp(\\mathrm{i}aeQA)$, where $Q$ denotes the matrix of quark charges, also have to be expanded in $e$, i.e. $\\mathcal{O}=\\mathcal{O}^{(0)}+e\\mathcal{O}^{(\\frac{1}{2})}+\\frac{1}{2}e^{2}\\mathcal{O}^{(1)}+O(e^{3})$.\n\n\\section{Introduction}\\label{sec:introduction}\n\nWe are in an era of precision lattice QCD physics, where contributions from QED and strong-isospin-breaking can no longer be ignored. \nAn example where these contributions are of significant importance is in the lattice determination of the anomalous magnetic moment of the muon $(g-2)_\\mu$. \nCommonly, one of the largest uncertainties in any lattice determination of $(g-2)_\\mu$ comes from the scale setting and it is of great importance that a calculation has the physical scale set accurately (below $1\\%$) incorporating QED and isospin-breaking effects\n\nAs ensembles for lattice QCD are often generated for the isosymmetric theory $\\text{QCD}_{\\text{iso}}$, perturbative methods were developed to incorporate isospin-breaking effects into calculations on these isosymmetric ensembles \\cite{deDivitiis:2011eh,deDivitiis:2013xla}.\nThe goal of this project is to reduce the contribution of isospin-breaking effects to the uncertainty of the lattice scale $a$ for $N_f=2+1$ CLS ensembles. \n\nIn ref. \\cite{Bruno:2016plf}, the lattice scale for the of CLS ensembles was determined using a combination of pion and kaon decay constants.\nWhile the final result has a total error at the level of 1\\%, the incorporation of isospin-breaking corrections turns out to be quite difficult \\cite{Carrasco:2015xwa}.\nIn this project, we investigate the prospects of precision scale setting using the masses of the lowest-lying baryon octet and decuplet, for which isospin-breaking effects are simpler to incorporate.\n\n\n\\section{Baryon Operators}\\label{sec:operators}\n\nThe operator bases used for the various baryons considered in this work were first introduced in Ref.~\\cite{Basak:2005ir}. \nThese operators are obtained from a group-theoretical construction of the coefficients $\\lambda$ for a baryonic interpolator\n\\[\n    \\mathcal O_B=\\sum_{a,b,c,f_i,\\mu_j}\\varepsilon_{abc}\\lambda^{\\mu_1,\\mu_2,\\mu_3}_{f_1,f_2,f_3}q_{\\mu_1}^{f_1,a}q_{\\mu_2}^{f_2,b}q_{\\mu_3}^{f_3,c},\n\\]\nwhere $f_i\\in{u,d,s}$ label the flavor of the quarks, $a,b,c$ are color indices, and $\\mu_i$ are Dirac-spinor indices. \nIn this project we use Wuppertal-smeared fields, i.e. $q=W\\Psi$, where $\\Psi$ is a point source and $W$ is an APE-smeared smearing operator.\n\nThis construction distinguishes different baryons by their symmetries w.r.t the quark flavors $f_i$ and the flavors themselves.\nFor each flavor symmetry, a set of operators based on the third spin component and parity are constructed from a tensor product of Weyl spinors.\nThe only differences between the operators we use and those in \\cite{Basak:2005ir} are their normalization and that we do not make use of the totally symmetric $\\Sigma$ and $\\Xi$ operators.\nFurthermore, we do not use the H-irreps of the nucleon and (in the isosymmetric case) $\\Lambda$, as well as the $G_1$-irreps of the $\\Delta$ and $\\Omega$ states since we are only interested in the ground state energies.\n\nThese operators are constructed in the Dirac-Pauli basis, in which the parity operator only acts on the first two indices of a Dirac spinor and the spin-z operator only acts on the last two spinor indices, which allows for a convenient construction of the operators based on parity and spin-z eigenvalues in terms of Weyl spinors $\\chi,\\xi$ such that the Dirac spinor is given by $\\psi=\\chi\\otimes\\xi$:\n\\begin{align*}\n    \\mathcal P\\psi(\\vb x,t)=&\\gamma_0\\psi(-\\vb x,t)=((\\sigma_3\\chi)\\otimes\\xi)(-\\vb x,t) , \\\\ \n    S_z\\psi(\\vb x,t)=&-i\\gamma_1\\gamma_2\\psi(\\vb x,t)=\\chi\\otimes(\\sigma_3\\xi)(\\vb x,t) .\n\\end{align*}\nUsing this fact, baryonic operators are constructed from three Weyl spinors on which the parity operator acts and three Weyl spinors defining the spin of the baryon.\nThe tensor product of different symmetrizations of these combinations of three Weyl spinors then define the baryon operators. \nSince the baryon operators are already antisymmetric w.r.t their color indices, the combined symmetry of spinor and flavor indices has to be chosen such that the operators are antisymmetric under the exchange of two quarks.\nAs the different baryons are classified according to their flavor symmetries, the Dirac indices thus have to make the operators symmetric or mixed-symmetric under simultaneous exchange of flavor and spin indices.\n\nIn total, this procedure results in 116 different operators: 58 for each parity eigenvalue.\nBecause more than one of these operators are expected to have overlap with the same ground state, this allows us to perform a GEVP in order to better control the excited states \\cite{Blossier:2008tx,Blossier:2009kd}.\nThe sizes of the correlator matrices for the different baryons are listed in \\cref{tab:corr_sizes}, with each baryon having one correlator matrix for each spin-z eigenvalue.\n\n\\begin{table}[htb]\n\\centering\n\\caption{Sizes of the correlator matrices for each particle.}\n\\label{tab:corr_sizes}\n\\begin{tabular}{r|c|c|c|c|c|c|c}\n    & $N$ & $\\Lambda$ & $\\Sigma\/\\Xi$ & $\\Delta\/\\Omega$ & $\\Sigma^*\/\\Xi^*$ & $\\Sigma$-$\\Lambda$-mixing & $\\Sigma^*$-$\\Lambda$-mixing \\\\\n    \\hline\n    Correlator size & $3\\times3$ & $4\\times4$ & $3\\times3$ & $2\\times2$ & $2\\times2$ & $7\\times7$ & $2\\times2$ \n\\end{tabular}\n\\end{table}\n\n\\section{Computational Optimizations}\\label{sec:optimizations}\n\nWe compute correlation functions using the operators described in \\cref{sec:operators} with a program written in \\texttt{C++} making use of the libraries \\texttt{OpenQCD} for quark propagator inversions and \\texttt{QDP++} for the remaining computations. For better statistics for the same amount of computing time, we apply all-mode averaging\\ \\cite{Shintani:2014vja}.\n\nAs the chosen set of operators described in \\cref{sec:operators} yields a large number (528) of non-vanishing correlator matrix entries, the contractions can easily become a bottleneck in the computation of the different correlators, especially in the case of isospin-breaking corrections.\nTherefore, a number of optimizations have been applied to the production code in order to keep the computational costs small compared to the inversions.\n\nFirst, the contractions are simplified algebraically by identifying the up and down quark propagators with the light quark $l$ used in the simulations, which is possible as we use isospin-symmetric ensembles.\nA similar simplification is done for up and down propagators including a photon vertex:\nIn the contractions, these terms come with a factor $e_fe$ for each three-point-vertex or $(e_fe)^2$ in the case of a four-point-vertex, where $e$ is the electromagnetic coupling and \n\\[\n    e_f=\\begin{cases} \\frac23 & f=u \\\\ -\\frac13 & f\\in\\{d,s\\}\\end{cases}. \n\\]\nThus, in a diagram with photon interactions, the product of the fractional charges of the quarks at each photon vertex yields simply a prefactor for the contractions with $S_d$ and $S_u$ replaced by $S_l$.\nNote, that the charge multiplicity $e_f$ is already absorbed into the definition of the vertices in \\cref{fig:diagrams}.\n\nThis leaves 101580 individual, color-contracted terms of the form:\n\\begin{align}\n    T_{\\mu_1\\mu_2\\mu_3}^{f_1f_2f_3}=\\sum\\limits_{\\smqty{a,b,c \\\\ a',b',c'}}\\varepsilon_{abc}\\varepsilon_{a'b'c'}\\lambda_{f_1f_2f_3}^{\\mu_1\\mu_2\\mu_3}\\lambda_{f_1f_2f_3}^{\\mu_4\\mu_5\\mu_6}S^{f_1,aa'}_{\\mu_1\\mu_4}S^{f_2,bb'}_{\\mu_2\\mu_5}S^{f_3,cc'}_{\\mu_3\\mu_6}, \\label{eq:terms}\n\\end{align}\nwhich divide into 8304 isosymmetric contributions, 38316 contributions from mass detuning, and 54960 from QED corrections.\n\nHowever, many of these terms appear in multiple contractions (possibly differing in the coefficients $\\lambda$), enabling a reduction of the computational cost by reusing these terms after their first computation.\nThis reduces the number of terms to be computed to 10104 unique terms of the form\n\\[\n    \\tilde T_{\\mu_1\\mu_2\\mu_3}^{f_1f_2f_3}=\\sum\\limits_{\\smqty{a,b,c \\\\ a',b',c'}}\\varepsilon_{abc}\\varepsilon_{a'b'c'}S^{f_1,aa'}_{\\mu_1\\mu_4}S^{f_2,bb'}_{\\mu_2\\mu_5}S^{f_3,cc'}_{\\mu_3\\mu_6}\n\\]\nsuch that all correlators can be expressed as linear combinations of these unique terms.\nThis, however, leads to the next challenge: since each term resembles a complex field on the lattice, this is still an unreasonable amount of data to hold in memory at once, especially on large ensembles.\nThis problem can be circumvented by making use of the fact that each correlator has a fixed combination of flavor indices $(f_1,f_2,f_3)$, where we count the sequential propagators from isospin-breaking contributions as separate flavors so that there are a total of 25 different flavor combinations.\nOne can therefore split the set of correlators into much smaller subsets that can be computed in one go without the need of keeping the terms in memory for any other subset.\nThese small subsets then contain at most 640 unique terms, which is much more manageable.\n\nFurther optimizations were necessary due to the nature of the \\texttt{QDP++} library used for the computation of the contractions.\nThis library is optimized for matrix-based computations on the lattice, but has rather slow routines for retrieving data based on color or spin indices.\nTo avoid these routines, propagators are saved as \\texttt{std::vector<std::vector<LatticeColorMatrix>>}, so that \\texttt{QDP++}'s \\texttt{peekSpin}-routine only needs to be called 16 times per propagator in order to store the propagator in this format.\n\nOnce all propagators for a given flavor combination are calculated, a lookup table for all color-contracted terms is constructed for that flavor combination.\nThis lookup table is finally used to calculate the contractions.\nIt was found to be beneficial, performance wise, to use precompiled functions that return the complete contraction as a single expression in terms of the elements of the lookup table.\nAn example for such a function calculating the contraction for a simple isospin-symmetric nucleon correlator would take references to a \\texttt{std::map<std::tuple<int, int, int, int, int, int>, LatticeComplex> map} and a \\texttt{LatticeComplex result} and perform the following computation:\n\\begin{verbatim}\n    result =  3.0 * map.at(std::make_tuple(0, 0, 0, 0, 1, 1))\n            - 3.0 * map.at(std::make_tuple(0, 0, 0, 1, 1, 0));\n\\end{verbatim}\nThe factors \\texttt{3.0} and \\texttt{-3.0} correspond to the product of the $\\lambda$-coefficients in \\cref{eq:terms}, \\texttt{map} is the above mentioned lookup table which uses keys in the form of a tuple containing the six spinor indices identifying the unique term needed for the contraction.\nThe flavor indices are not mentioned here, since the correlators are already split up according to the flavor index combination, for each of which a set of such functions is defined.\n\n\\section{Conclusion and Outlook} \\label{sec:outlook}\n\nWe have presented our calculational framework for the inclusion of isospin-breaking effects for baryon correlators using isosymmetric CLS ensembles, for which we employ a perturbative method \\cite{deDivitiis:2011eh,deDivitiis:2013xla}.\nFor the baryon operators, we use a construction based on parity and symmetries in flavor and spinor indices \\cite{Basak:2005ir} which gives rise to several correlator matrices for the different baryons.\nWe have implemented several optimizations to deal with the vast amount\nof correlators to be computed.\n\nWe have tested our code on an ensemble (A654) of size $48\\times24^3$ with antiperiodic temporal boundary conditions. \nPreliminary results for the single nucleon correlator suggest that we expect a statistical uncertainty below 1\\%.\n\nOver the course of the next months, we intend to perform a spectroscopic analysis of the different correlators we have for A654 including the use of the GEVP method.\nMoreover, we are going to generate correlator data for larger $128\\times64^3$ ensembles, namely D450 and D452, to perform similar analyses on these ensembles.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe aim of this paper is to the study of the nonlinear Volterra integral\nequation\n\\begin{equation}\n  \\label{eq:vol1}\n  u\\left( x\\right) =\\int_0^xk\\left( x,s\\right) g\\left( u\\left( s\\right) \\right) \\,ds,\n\\end{equation}\nthat will be denoted by $\\left( k,g\\right) $. We will assume that\nthe following conditions are held.\n\\begin{enumerate}\n\\item [$\\mathbf{K}_1$.] The kernel $k:\\R ^2\\to\\R ^+$ is a\n  locally bounded function, such that $k\\left( x,s\\right) =0$ whenever $s>x$.\n\\item [$\\mathbf{K}_2$.] For every $x\\in\\R $, the map $s\\to\n  k\\left( x,s\\right) $ is locally integrable, and $K\\left( x\\right) =\\int_0^xk\\left( x,s\\right) \\,ds$ is a\n  strictly increasing function.\n\\item [$\\mathbf{G}_1$.] The nonlinearity $g$ is a\n  strictly increasing continuous function, vanishing on $(-\\infty,0]$, and such\n  that $g'>0$ almost everywhere.\n\\end{enumerate}\nFrom now on, these conditions will be referred to as (GC).\n\nSolutions of an equation $\\left( k,g\\right) $ are fixed points of the operator\n$T_{kg}$, defined as\n\\begin{equation}\n  \\label{eq:op1}\n  T_{kg}f\\left( x\\right) :=\\int_0^xk\\left( x,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds.\n\\end{equation}\nThe monotone behaviour of $T_{kg}$ is an immediate consequence of $\\mathbf{G}_1$ and the strictly increasing behaviour of the integral operator; i.e.,  if $f_1\\leq f_2$, then\n$T_{kg}f_1\\leq T_{kg}f_2$. Moreover, since $g\\left( 0\\right) =0$, the\nzero function is a solution of (\\ref{eq:vol1}), known as \\textit{the\ntrivial solution}.\n\nThe following two lemmas allow us to consider only bounded solutions on a certain interval $[0,\\delta]$, for some positive $\\delta$. This kind of solutions will be referred to as \\textit{bounded near zero} functions.\n\n\\begin{lemma}\n\\label{lema:1}\nLet $k$ be a kernel satisfying the following inequality,\n  \\begin{equation}\n\\label{prop:szwarc}\nk\\left( x,s\\right) \\leq k\\left( y,s\\right) ,\\qquad \\forall x\\leq y,\n\\end{equation}\nfor each $s\\in \\R$. Then, the operator $T_{kg}$ transforms positive functions into\nincreasing functions.\n\\end{lemma}\n\\begin{proof}\n  Let $f$ be a positive function, and let $x\\leq y$. From\n  $\\mathbf{K}_1$, we have\n\\begin{equation*}\n  \\begin{split}\n    T_{kg}f\\left( x\\right) =&\\int_0^xk\\left( x,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds=\\int_0^yk\\left( x,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds\\\\\n\\leq &\\int_0^yk\\left( y,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds=T_{kg}f\\left( y\\right) .\\\\\n  \\end{split}\n\\end{equation*}\n\\end{proof}\n\n\\begin{lemma}\n\\label{lastlemma}\n  Let $f$ be a positive function. Then, for every $x$ in its domain of\n  definition, $T_{kg}f$ is bounded on $[0,x]$.\n\\end{lemma}\n\\begin{proof}\nLet us define the auxiliary kernel,\n\\begin{equation*}\n\\overline{k}\\left( x,s\\right) =\\max\\left\\{ k\\left( t,s\\right) :\\quad 0\\leq t\\leq x\\right\\} .\n\\end{equation*}\nThe kernel $\\overline{k}$ verifies the condition (\\ref{prop:szwarc}) and $k\\leq\\overline{k}$. Then, if $f$ is a positive function, $T_{kg}f\\leq T_{\\overline{k}g}f$. From Lemma \\ref{lema:1}, it follows that $T_{\\overline{k}g}f$ is an increasing function. Thus, for every $x$ where $T_{kg}f$ is defined, $T_{kg}f$ is bounded by $T_{\\overline{k}g}f\\left( x\\right) $ on $[0,x]$.\n\\end{proof}\n\nTaking into account Lemma \\ref{lastlemma}, positive solutions for\nequation (\\ref{eq:vol1}) are \\textit{bounded near zero}. Unless otherwise stated, any function considered in this paper\nwill be \\textit{bounded near zero}.\n\nA particular case of equation $\\left( k,g\\right) $ is the well known \\textit{convolution equation},\n\\begin{equation}\n  \\label{eq:vol2}\n  u\\left( x\\right) =\\int_0^x\\phi\\left( x-s\\right) g\\left( u\\left( s\\right) \\right) \\,ds.\n\\end{equation}\nHere, the kernel is $k\\left( x,s\\right) =\\phi\\left( x-s\\right) $, being $\\phi$ a locally bounded\nfunction of one real variable. This kind of kernels are known as \\textit{convolution kernels}.\n\nThe existence of a nontrivial solution for convolution equations is equivalent to the existence of a nontrivial\n\\textit{subsolution}; i.e., a function $v$ such that $v\\leq T_{\\phi g}v$ \\cite{Ari00,AriCas98,BusOkra90,Grip81,Myd91,Zeid90}.\nMoreover, if a positive solution of (\\ref{eq:vol2}) exists, then it is unique, strictly increasing, continuous and a global attractor\nof any positive and measurable function $f$ (see, for instance, \\cite{Ari00,AriBen01,AriCas99,Szw92}. Recall that a solution is a global attractor of a positive measurable function $f$ if the sequence $(T^n_{\\phi g}f)_{n\\in \\N }$ converges to that solution, where $T_{\\phi g}^n$ denotes the composition of $T_{\\phi g}$ with itself $n$ times.\n\nSzwarc, in \\cite{Szw92}, presented several results about existence, uniqueness, and attracting behaviour of solutions for nonconvolution Volterra integral equations. In that paper, the author uses different techniques and ideas which appear in many results concerning the existence, uniqueness, and attracting behaviour of solutions for convolution equations. Our aim in this paper is the same. That is, to study how the results known for the\nconvolution equation (\\ref{eq:vol2}), can be used in order to\nobtain properties for the solutions of the nonconvolution equation\n(\\ref{eq:vol1}). The hypotheses considered in this paper are weaker than those considered by Szwarc in \\cite{Szw92}.\n\n\n\\section{Existence of solutions}\n\\label{sec2}\n\nAs mentioned above, for nonlinear Volterra integral equations of convolution type there is a strong relation between the existence of subsolutions and the existence of nontrivial solutions. First we will show that, also for\nnonconvolution equations, the existence of solutions and the existence of subsolutions are equivalent.\n\nThroughout this section, we will assume that equation $\\left( k,g\\right) $ verifies conditions (GC).\n\n\\begin{theorem}\\label{teo:subsol}\n  There is a solution for the equation (\\ref{eq:vol1}) if and only if equation (\\ref{eq:vol1}) admits a subsolution.\n\\end{theorem}\n\\begin{proof}\nThe sufficient condition is immediate, because every solution of equation (\\ref{eq:vol1}) is a subsolution.\n\nTo prove the necessary condition, let us consider a positive subsolution of (\\ref{eq:vol1}), $v$. First, we want to note that, by Lemma \\ref{lastlemma}, subsolutions of (\\ref{eq:vol1}) are necessarily \\textit{bounded near zero}. So, there\nexist positive $\\delta_1$ and $M$, such that\n\\begin{equation}\n\\label{eq:des1}\nv\\leq M,\\qquad \\text{ on } [0,\\delta_1].\n\\end{equation}\nNow, we need to prove that $M$ is a \\textit{supersolution near zero}, which is equivalent to prove the existence of a positive $\\delta _2$, such that\n\\begin{equation}\n\\label{eq:des2}\nT_{kg}M\\leq M,\\qquad \\text{on }[0,\\delta_2].\n\\end{equation}\nTaking into account conditions $\\mathbf{K}_1$ and $\\mathbf{K}_2$, we have that $K\\left( 0\\right) =0$ and $\\lim_{x\\to 0+}K\\left( x\\right) =0$. Therefore, since $T_{kg}M\\left( x\\right) =g\\left( M\\right) K\\left( x\\right) $, the existence of $\\delta_2$ is guaranteed.\n\nLet us define $\\delta=\\min\\left\\{ \\delta_1,\\delta_2\\right\\} $. From\n  (\\ref{eq:des1}) and (\\ref{eq:des2}), we have\n  \\begin{equation*}\n    v\\leq T_{kg}v\\leq T_{kg}M\\leq M,\\qquad \\text{on }[0,\\delta].\n  \\end{equation*}\n  Note that $(T^n_{kg}v)_{n\\in \\N }$ is a nondecreasing sequence\n  bounded from above by $M$. Thus, we can define the pointwise limit\n  \\begin{equation*}\n    u\\left( x\\right) :=\\lim_{n\\to \\infty }T^n_{kg}v\\left( x\\right) ,\\qquad \\forall x\\in [0,\\delta].\n  \\end{equation*}\n  For each $x\\in [0,\\delta]$, we consider the sequence $\\left( \\phi_n\\right) _{n\\in \\N }$, where\n  \\begin{equation*}\n    \\phi_n\\left( s\\right) =k\\left( x,s\\right) g\\left( T^n_{kg}v\\left( s\\right) \\right) .\n  \\end{equation*}\n  By the monotone convergence theorem, the function\n\\begin{equation*}\nu\\left( x\\right) = \\lim_{n\\to \\infty }\\int_{\\R }\\phi _n ^x \\left( s\\right) \\,ds\n\\end{equation*}\nexists on $[0,\\delta]$ and is a solution of equation (\\ref{eq:vol1}).\n\\end{proof}\n\nNote that the necessary condition of last lemma remains true when you assume just the existence of a subsolution \\textit{near zero}, i.e., the existence of a function $v$ and a positive $\\delta _0$ such that\n\\begin{equation*}\nv\\leq T_{kg}v,\\qquad \\text{ on }[0,\\delta _0].\n\\end{equation*}\nIn this case, it only would be necessary to change, in the proof of the necessary condition, the definition of $\\delta $; the new definition would be $\\delta =\\min \\left\\{ \\delta _0,\\delta _1,\\delta _2\\right\\} $.\n\nAs we mentioned in the introduction, Volterra integral equations of convolution kind are a particular case of equation (\\ref{eq:vol1}). There are many results about the existence and uniqueness of solutions for convolution Volterra integral equations \\cite{Ari00,BusOkra90,Myd91,Zeid90,AriBen03a,Myd99}. Some of the foremost techniques to study Volterra integral equations are comparison techniques \\cite{Zeid90,Ask91,Kar00}. The rest of this section is devoted to the use of such techniques in order to establish a relation between existence results for \\textit{convolution equations} and for equation (\\ref{eq:vol1}). To do it, we will need to show that any locally bounded kernel can be bounded from above and below by \\textit{convolution kernels}, on every bounded region of $\\R ^2$.\n\nSince our interest is to relate equation (\\ref{eq:vol1}) with \\textit{Volterra integral equations of convolution kind}, at a first stage, it would be natural to consider kernels $k:\\R^2\\rightarrow \\R^+$ verifying $k\\left( x,s\\right) =k\\left( x+\\lambda,s+\\lambda\\right) $ for all $\\lambda \\in \\R $ and $\\left( x,s\\right) \\in \\R^2$. Such kernels will be referred to as \\textit{invariant kernels}. Note that convolution kernels are invariant because there is a function $\\phi:\\R \\rightarrow \\R^+$ such that $k\\left( x,s\\right) =\\phi\\left( x-s\\right) $. Next, we are going to see that any \\textit{invariant kernel} is a \\textit{convolution kernel}. Let $k$ be an invariant kernel, then\n$k\\left( x,s\\right) =k\\left( x-s,0\\right) $, for all $\\left( x,s\\right) \\in \\R^2$; so defining $\\phi\\left( x\\right) =k\\left( x,0\\right) $, we have\n$k\\left( x,s\\right) =\\phi\\left( x-s\\right) $. Thus, both families, invariant and convolution kernels are the same.\n\nNow, let us consider a kernel $k$ satisfying $\\mathbf{K}_1$, and let us\nstudy equation (\\ref{eq:vol1}) in an interval $[0,x_0]$, for a given $x_0>0$. First, we define a couple of auxiliary functions,\n\\begin{equation}\\label{eq:func_min}\n  \\phi _{x_0}\\left( x\\right) =\\min \\left\\{ k\\left( \\left( 1-\\lambda \\right) x+\\lambda x_0,\\lambda \\left( x_0-x\\right) \\right) :\n\\lambda \\in [0,1]\\right\\}\n\\end{equation}\n\n\\noindent and\n\n\\begin{equation}\\label{eq:func_max}\n    \\psi_{x_0}\\left( x\\right) =\\max \\left\\{ k\\left( \\left( 1-\\lambda \\right) x+\\lambda x_0,\\lambda\\left( x_0-x\\right) \\right) :\n\\lambda \\in[0,1]\\right\\}.\n\\end{equation}\n\nLet $\\mathcal{T}_{x_0}$ be the right triangle determined by $\\left( 0,0\\right) $,\n$\\left( x_0,0\\right) $ and $\\left( x_0,x_0\\right) $. For every $x\\in [0,x_0]$, $\\phi\\left( x\\right) $ and\n$\\psi\\left( x\\right) $ are the minimum and the maximum, respectively, of $k$ on the\nsegment $l_x$, determined by the intersection of $\\mathcal{T}_{x_0}$ and the graph of $y\\left( s\\right) =s-x$. So, we have\n\\begin{equation}\n\\label{eq9}\n\\phi_{x_0}\\left( x_1-s_1\\right) \\leq k\\left( x_1,s_1\\right) \\leq\\psi_{x_0}\\left( x_1-s_1\\right) ,\n\\end{equation}\nfor any $\\left( x_1,s_1\\right) \\in \\mathcal{T}_{x_0}$, because $\\left( x_1,s_1\\right) $ is on $l_{x_1-s_1}$.\n\nFrom (\\ref{eq9}) and Theorem \\ref{teo:subsol}, it\nfollows that the existence of a solution for a equation\n$\\left( \\phi_{x_0},g\\right) $ implies the existence of solutions for equation\n$\\left( k,g\\right) $ and\n$\\left( \\psi_{x_0},g\\right) $. In general, the converse is not true. But if we assume the existence of a positive constant $c$ such that $\\psi_{x_0}\\leq c\\phi_{x_0}$, the following inequalities hold,\n\\begin{equation*}\n  \\phi_{x_0}\\left( x-s\\right) \\leq k\\left( x,s\\right) \\leq c\\phi_{x_0}\\left( x-s\\right) ;\n\\end{equation*}\nand, therefore, by Theorem \\ref{teo:subsol}, the existence of solutions for $\\left( k,g\\right) $ is\nequivalent to the existence of solutions for $(\\phi_{x_0},g)$. There are different cases in which such constant can be found. For instance, when\n\\begin{equation}\n\\label{cond10}\n  \\lim_{x\\to 0^+}\\frac{\\psi_{x_0}\\left( x\\right) }{\\phi_{x_0}\\left( x\\right) }=l\\in[0,+\\infty).\n\\end{equation}\n\nWhat we have proved in the last part of this section is the following result.\n\\begin{theorem}\nLet $\\left( k,g\\right) $ be a nonconvolution equation satisfying (GC), and let $\\phi_{x_0}$ and $\\psi_{x_0}$ be defined as in (\\ref{eq:func_min}) and (\\ref{eq:func_max}). Then, the existence of a solution for equation $\\left( \\phi_{x_0},g\\right) $ implies the existence of a solution for equation $\\left( k,g\\right) $. Moreover, if condition(\\ref{cond10}) holds, the equation $\\left( k,g\\right) $ has a solution if and only if either equation $\\left( \\phi_{x_0},g\\right) $ or $\\left( \\psi_{x_0},g\\right) $ have a solution.\n\\end{theorem}\n\nLet us see a couple of examples about how to use the techniques described in this section to prove the existence of solutions for $\\left( k,g\\right) $.\n\n\\noindent\\subparagraph{Example 1.}\nLet the equation $\\left( k,g\\right) $ be\n\\begin{equation}\\label{eq:ex1}\nu\\left( x\\right) =\\int_0^x\\left( a^{x+s}+1\\right) \\sqrt{2u\\left( s\\right) }\\,ds,\\qquad a>0,\\quad x\\geq 0.\n\\end{equation}\n\nLet us consider an arbitrary positive constant $x_0>0$, and restrict the problem to the interval $[0,x_0]$.\nConsider the triangle\n\\begin{equation*}\n\\mathcal{T}_{x_0}=\\left\\{ \\left( x,s\\right) \\in\\R^2:0\\leq x\\leq x_0,\\,0\\leq s\\leq x\\right\\} .\n\\end{equation*}\n\nSince the kernel is increasing with respect both variables, the functions $\\phi$ and $\\psi$, defined in (\\ref{eq:func_min}) and (\\ref{eq:func_max}), are $\\phi\\left( x\\right) =a^x+1$ and $\\psi\\left( x\\right) =a^{2x_0-x}+1$.\n\nWe also have\n\\begin{equation*}\n\\lim_{x\\to 0^+}\\frac{\\psi\\left( x\\right) }{\\phi\\left( x\\right) }=\\frac{a^{2x_0}+1}{2}\\in [0,+\\infty );\n\\end{equation*}\nthus, condition (\\ref{cond10}) holds, and therefore, the existence of solutions for equation (\\ref{eq:ex1}) is equivalent to the existence of a solution for the equation\n\\begin{equation}\\label{eq:convex1}\nu\\left( x\\right) =\\int _0^x \\left( a^{x-s}+1\\right) \\sqrt{2u\\left( s\\right) }\\,ds,\\qquad x\\in[0,x_0].\n\\end{equation}\nIt can be easily checked that (\\ref{eq:convex1}) verifies some conditions for the existence of solutions for convolution equations given in \\cite{Ari00}. Hence, the nonconvolution equation $\\left( k,g\\right) $ has a solution.\n\n\\noindent\\subparagraph{Example 2.} Let the equation $\\left( k,g\\right) $ be\n\\begin{equation*}\nu\\left( x\\right) =\\int _0^x x\\left( x-s\\right) u\\left( s\\right) ^{\\beta }\\, ds,\\qquad x\\in [0,L],\\quad \\beta \\in \\left( 0,1\\right) .\n\\end{equation*}\n\nIn this case, the kernel is $k\\left( x,y\\right) = x\\left( x-y\\right) $. As in the last example, it is possible to find the expressions of the functions $\\phi $ and $\\psi $ when we restrict the problem to the region $\\mathcal{T}_{x_0}$. Here, we have $\\phi \\left( x\\right) =x^2$ and $\\psi \\left( x\\right) =x_0x$. For such functions, we find that\n\\begin{equation*}\n\\lim _{x\\to 0^+}\\frac{\\psi\\left( x\\right) }{\\phi \\left( x\\right) }=\\lim_{x\\to 0^+}\\frac{x_0}{x}=+\\infty .\n\\end{equation*}\nHence, condition (\\ref{eq:convex1}) does not hold. Nevertheless, it is immediate to check that both equations, $\\left( x^2,x^\\beta \\right) $ and $\\left( x_0x,x^\\beta \\right) $ have a solution. Indeed, it is possible to obtain the solutions in closed form. The functions\n\\begin{equation*}\nu_{\\phi }\\left( x\\right) =B\\left( 3,\\frac{3\\beta +1}{1-\\beta }\\right) ^{1\/\\left( 1-\\beta \\right) } x^{3\/\\left( 1-\\beta \\right) }\n\\end{equation*}\nand\n\\begin{equation*}\nu_{\\psi }\\left( x\\right) =\\left( x_0B\\left( 2,\\frac{2\\beta +1}{1-\\beta }\\right) \\right)^{1\/\\left( 1-\\beta \\right) } x^{2\/\\left( 1-\\beta \\right) }\n\\end{equation*}\nare the solutions for equations $\\left( x^2,x^\\beta \\right) $ and $\\left( x_0x,x^\\beta \\right) $ respectively. Therefore, every solution for equation $\\left( k,g\\right) $ lies between $u_{\\phi }$ and $u_{\\psi }$.\n\n\\section{Uniqueness}\nFor convolution equations with locally bounded kernels, under very weak assumptions, nontrivial solutions are unique, see \\cite{AriBen01}. Our aim in this section is to prove the uniqueness of nontrivial solutions for nonconvolution equations. To do it, we will consider the following additional hypotheses on the kernel.\n\n\\begin{enumerate}\n\\item[$\\mathbf{K}_3$.] The function $K\\left( x\\right) =\\int _0^x k\\left( x,s\\right) \\, ds$ is continuous.\n\\item[$\\mathbf{K}_4$.] For every $\\left( x,s\\right) \\in \\R ^2$ and $\\lambda \\geq 0$, $k\\left( x,s\\right) \\leq k\\left( x+\\lambda ,s+\\lambda \\right) $.\n\\end{enumerate}\n\n\\begin{lemma}\\label{lemma:2}\nLet us suppose that, in addition to (GC), equation $\\left( k,g\\right) $ also verifies $\\mathbf{K}_3$. Then, the operator $T_{kg}$ transforms\n  bounded functions into continuous functions.\n\\end{lemma}\n\\begin{proof}\nLet $f$ be a positive function bounded from above by $M$. Let\n $x_1\\leq x_2$, then, since $k\\left( x,s\\right) =0$ whenever $s>x$, we have\n\\begin{eqnarray*}\nT_{kg}f\\left( x_2\\right) -T_{kg}f\\left( x_1\\right) & = & \\int_0^{x_2}k\\left( x_2,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds-\\int_0^{x_1}\nk\\left( x_1,s\\right) g\\left( f\\left( s\\right) \\right) \\,ds \\\\\n& = & \\int_0^{x_2}\\left( k\\left( x_2,s\\right) -k\\left( x_1,s\\right) \\right) g\\left( f\\left( s\\right) \\right) \\,ds \\\\\n& \\leq & g\\left( M\\right) \\int_0^{x_2}k\\left( x_2,s\\right) -k\\left( x_1,s\\right) \\,ds \\\\\n& = & g\\left( M\\right) \\left( K\\left( x_2\\right) -K\\left( x_1\\right) \\right) . \\\\\n\\end{eqnarray*}\nThe continuity of $T_{kg}f$ is immediate from the continuity of $K$.\n\\end{proof}\n\nThe next corollary is followed from Lemma \\ref{lastlemma} and the last result.\n\n\\begin{corollary}\\label{cor:1}\nEvery solution of equation $\\left( k,g\\right) $ is a continuous function.\n\\end{corollary}\n\nThe proof of the next lemma has been adapted from a paper due to Mydlardzyc \\cite{Myd99}, where a similar result was proved for Abel integral equations. Here, we have used the ideas presented in \\cite{Myd99}, and extended them to nonconvolution equations.\n\n\\begin{lemma}\\label{lemma:3}\nLet us suppose that, in addition to (GC), equation $\\left( k,g\\right) $ also verifies $\\mathbf{K}_3$ and $\\mathbf{K}_4$. Then, every continuous subsolution of equation $\\left( k,g\\right) $ is bounded from above by any solution of equation $\\left( k,g\\right) $.\n\\end{lemma}\n\\begin{proof}\nLet $v$ and $u$ be a subsolution and a solution of equation $\\left( k,g\\right) $,\n respectively. First, we will show that, for every $c>0$, the function\n\\begin{equation*}\nv_c\\left( x\\right) =\\begin{cases}\n0,&\\text{if }x\\in [0,c]\\\\\nv\\left( x-c\\right) ,&\\text{if } x>c, \\\\\n       \\end{cases}\n\\end{equation*}\nis also a subsolution of equation $\\left( k,g\\right) $.\nFor $x\\in [0,c]$, this is trivial since $v_c\\left( x\\right) =T_{kg}v_c\\left( x\\right) =0$. For $x>c$,\n\\begin{equation}\n\\label{eq14}\nv_c\\left( x\\right) =v\\left( x-c\\right) \\leq T_{kg}v\\left( x-c\\right) =\\int_0^{x-c}k\\left( x-c,s\\right) g\\left( v\\left( s\\right) \\right) \\,ds.\n\\end{equation}\nSince $k$ verifies $\\mathbf{K}_4$, making the change of variable $t=s+c$ in the last integral, (\\ref{eq14}) takes the form\n\\begin{eqnarray*}\n  v_c\\left( x\\right) & \\leq & \\int _c^x k\\left( x-c,t-c\\right) g\\left( v\\left( t-c\\right) \\right) \\,dt \\\\\n& \\leq & \\int _c^x k\\left( x,t\\right) g\\left( v_c\\left( t\\right) \\right) \\, dt=\\int _0^x k\\left( x,t\\right) g\\left( v_c\\left( t\\right) \\right) \\, dt = T_{kg}v_c\\left( x\\right) .\\\\\n\\end{eqnarray*}\nNow, let us compare $v_c$ and $u$. For $0<x<c$, it is obvious that\n$0=v_c\\left( x\\right) <u\\left( x\\right) $. Since $v_c$ and $u$ are continuous, there exists an interval\n$[0,x_0)$, with $x_0>c$, where $v_c\\leq u$. Then,\n\\begin{eqnarray*}\nu\\left( x_0\\right) -v_c\\left( x_0\\right) & \\geq & \\int _0^{x_0} k\\left( x_0,s\\right) \\left[ g\\left( u\\left( s\\right) \\right) -g\\left( v_c\\left( s\\right) \\right) \\right] \\, ds \\\\\n& > & \\int _0^c k\\left( x_0,s\\right) \\left[ g\\left( u\\left( s\\right) \\right) -g\\left( v_c\\left( s\\right) \\right) \\right] \\, ds \\\\\n& = & \\int _0^c k\\left( x_0,s\\right) g\\left( u\\left( s\\right) \\right) \\, ds>0.\n\\end{eqnarray*}\nAnalogously, it can be assured that $v_c<u$ in the whole domain of the solution.\n\nFinally, since $v_c<u$ for every positive $c$, taking limits as $c\\to 0^+$, we obtain that $v\\leq u$.\n\\end{proof}\n\nA consequence of this Lemma is the uniqueness of positive solutions for the\n equation $\\left( k,g\\right) $.\n \n\\begin{theorem}\nUnder the hypotheses of Lemma \\ref{lemma:3}, equation $\\left( k,g\\right) $ has at most\n one positive solution.\n\\end{theorem}\n\\begin{proof}\nSince every solution can be considered as a special\ncase of continuous subsolution, this proof is trivial.\n\nLet $u_1$ and $u_2$ be two solutions of $\\left( k,g\\right) $.\nConsidering $u_1$ as a continuous subsolution, by Lemma \\ref{lemma:3} $u_1\\leq u_2$;\nand considering $u_2$ as a continuous subsolution, $u_2\\leq u_1$, so\n$u_1\\equiv u_2$.\n\\end{proof}\n\n\\section{Attracting behavior}\n\nIn this section, we are going to study\nthe attracting behaviour of the solutions for\nthe equations $\\left( k,g\\right) $ verifying conditions (GC).\n\nRecall that in Section \\ref{sec2}, for a kernel $k$ satisfying $\\mathbf{K}_1$, there were defined the functions\n\\begin{equation*}\n  \\phi _{x_0}\\left( x\\right) =\\min \\left\\{ k\\left( \\left( 1-\\lambda \\right) x+\\lambda x_0,\\lambda \\left( x_0-x\\right) \\right) :\n\\lambda \\in [0,1]\\right\\}\n\\end{equation*}\n\n\\noindent and\n\n\\begin{equation*}\n    \\psi_{x_0}\\left( x\\right) =\\max \\left\\{ k\\left( \\left( 1-\\lambda \\right) x+\\lambda x_0,\\lambda\\left( x_0-x\\right) \\right) :\n\\lambda \\in[0,1]\\right\\}.\n\\end{equation*}\nIn that section, nonconvolution equations $\\left( k,g\\right) $ were studied in some arbitrary interval $[0,x_0]$ using the auxiliary convolution equations $\\left( \\phi _{x_0},g\\right) $ and $\\left( \\psi _{x_0},g\\right) $.\n\nIn order to simplify the notation, unless otherwise stated, $\\phi _{x_0}$ and $\\psi _{x_0}$ will be referred to as $\\phi $ and $\\psi $, respectively.\n\nThe proofs of the results presented in this section are mainly based on the attracting character of the solutions for the equations $\\left( \\phi ,g\\right) $ and $\\left( \\psi ,g\\right) $, and some standard comparison techniques. Throughout this section, we will assume the existence of solutions for equations $\\left( \\phi ,g\\right) $ and $\\left( \\psi ,g\\right) $, that will be denoted by $u_{\\phi }$ and $u_{\\psi }$ respectively.\n\nNote that $u_{\\phi }$ and $u_{\\psi }$ are unique and global attractors of all positive and measurable functions (see \\cite{AriBen01}). Moreover, as we saw in Section \\ref{sec2}, from (\\ref{eq9}) and Theorem \\ref{teo:subsol}, the existence of a solution for an equation $\\left( \\phi ,g\\right) $ implies the existence of solutions for $\\left( k,g\\right) $. These solutions are comparable functions, as we will see in the next result.\n\n\\begin{lemma}\\label{prop:u<upsi}\nLet $u$ be a solution of the nonconvolution equation $\\left( k,g\\right) $. Then  $u_{\\phi }\\leq u\\leq u_{\\psi }$.\n\\end{lemma}\n\\begin{proof}\nNote that\n\\begin{equation}\n\\label{eq15}\nT_{\\phi g}\\leq T_{kg}\\leq T_{\\psi g},\n\\end{equation}\nbecause $\\phi \\leq k\\leq \\psi $. Thus, $T_{\\phi g}u\\leq u=T_{kg}u\\leq T_{\\psi g}u$, and then, for every natural $n$,\n\\begin{equation*}\nT^n_{\\phi g}u\\leq u\\leq T^n_{\\psi g}u.\n\\end{equation*}\n\nSince both, $u_{\\phi }$ and $u_{\\psi }$, are global attractors, the sequences\n  $(T^n_{\\phi g}u)_{n\\in \\N }$ and $(T^n_{\\psi g}u)_{n\\in \\N }$ converge to $u_{\\phi }$ and $u_{\\psi }$,\n  respectively. Thus, taking limits as $n$ tends to $\\infty $, we have $u_{\\phi }\\leq u\\leq u_{\\psi }$.\n\\end{proof}\n\n\\begin{proposition}\n\\label{prop:upsi}\n  The sequence $(T^n_{kg}u_{\\psi })_{n\\in \\N }$ converges to the maximum solution of the equation $\\left( k,g\\right) $.\n\\end{proposition}\n\\begin{proof}\n  By Lemma \\ref{prop:u<upsi}, $u\\leq u_\\psi$. Thus, from the monotony of the operators $T_{k g}$\n  and $T_{\\psi g}$, it follows that\n  \\begin{equation*}\n    u=T_{kg}u\\leq T_{kg}u_{\\psi }\\leq T_{\\psi g}u_{\\psi }=u_{\\psi }.\n  \\end{equation*}\n  Hence, for every $x\\geq 0$, the decreasing sequence\n  $(T^n_{kg}u_{\\psi }\\left( x\\right) )_{n\\in \\N }$ is bounded from below by $u\\left( x\\right) $, so it converges pointwisely to a function\n  \\begin{equation*}\n    u_{\\max }\\left( x\\right) :=\\lim_{n\\to \\infty }T^n_{kg}u_{\\psi }\\left( x\\right) =\\inf \\left\\{ T^n_{kg}u_{\\psi }\\left( x\\right) :n\\in\\N\\right\\} .\n  \\end{equation*}\n  By the monotone convergence theorem, we can assure that $u_{\\max }$ is a solution of the equation $\\left( k,g\\right) $;\n  moreover from the way of constructing $u_{\\max }$, it is immediate that it is the maximum solution.\n\\end{proof}\n\nWith a similar proof we obtain an analogous result for the minimum solution.\n\\begin{proposition}\n\\label{prop:uphi}\n  The sequence $(T^n_{kg}u_{\\phi })_{n\\in \\N }$ converges to the minimum solution of the equation $\\left( k,g\\right) $.\n\\end{proposition}\n\nNow we are in position to give a result on the attracting character of the maximum and minimum solutions of equation $\\left( k,g\\right) $.\n\n\\begin{theorem}\nThe maximum (resp. minimum) solution of the equation $\\left( k,g\\right) $ attracts globally any measurable function bounded fom below (resp. above) by the maximum (resp. minimum) solution.\n\\end{theorem}\n\\begin{proof}\nWe shall prove the theorem for the maximum solution. For the minimum solution, there can be used analogous arguments.\n\nLet $u_{\\max }$ denote the maximum solution of the equation $\\left( k,g\\right) $, and let $f$ be a measurable function such that $u_{\\max }\\leq f$. We have to show that $(T^n_{kg}f)_{n\\in \\N }$ converges to $u_{\\max }$.\n\nFrom (\\ref{eq15}) and the increasing character of the operators $T_{kg}$ and $T_{\\psi g}$, we obtain\n\\begin{equation*}\nu_{\\max }=T^n_{kg}u_{\\max }\\leq T^n_{kg}f\\leq T^n_{\\psi g}f,\\qquad \\forall n\\in \\N .\n\\end{equation*}\nThus, for every $x\\geq 0$, the sequence $(T^n_{kg}f\\left( x\\right) )_{n\\in \\N }$ is bounded from below by $u_{\\max }\\left( x\\right) $, and from above by the  sequence $(T^n_{\\psi g}f\\left( x\\right) )_{n\\in \\N }$, which, as said above, converges to $u_\\psi \\left( x\\right) $. Then, the set of accumulation points of the sequence $(T^n_{kg}f\\left( x\\right) )_{n\\in \\N }$, denoted by $\\Omega _f\\left( x\\right) $, verifies $u_{\\max }\\left( x\\right) \\leq \\Omega_f\\left( x\\right) \\leq u_\\psi \\left( x\\right) $.\n\nTo finish the proof it suffices to show that $\\Omega_f\\left( x\\right) =\\left\\{ u_{\\max }\\left( x\\right) \\right\\} $. This is obvious, because $\\Omega_f\\left( x\\right) $ is invariant under $T_{kg}$ and, by Proposition \\ref{prop:upsi}, the sequence $(T^n_{kg}u_\\psi )_{n\\in \\N }$ converges to $u_{\\max }$. Hence, $u_{\\max }\\left( x\\right) \\leq \\Omega_f\\left( x\\right) \\leq u_{\\max}\\left( x\\right) $.\n\\end{proof}\n\n\\begin{remark}\nNote that if we could assure the uniqueness of solutions for the nonconvolution equation $\\left( k,g\\right) $, then the maximum and the minimum solutions are the same. In that case, a simple comparison reasoning guarantees that the unique solution is a global attractor of any  positive and measurable function.\n\\end{remark}\n\n\\section{Final Remarks}\nFor convolution equations, there are a lot of results about existence and uniqueness of continuous solutions with no other assumptions on the kernel than the local integrability. Just mention, for example, the theory of Abel integral equations.\n\nWhen nonconvolution equations are considered, a wide range of situations appears. The aim of the following examples is to illustrate such variety. In the first example, it is shown that if $\\mathbf{K}_3$ does not hold, then the only continuous solution for equation $\\left( k,g\\right) $ is the trivial one. In the second example, we will see that if the kernel does not verify $\\mathbf{K}_1$, then we cannot guarantee the uniqueness of solutions.\n\n\\subsection{An Equation with Discontinuous Solutions}\n\nLet us consider equation $\\left( k,g\\right) $ with $g$ verifying $\\mathbf{G}_1$ and $k$ a strictly increasing function of the variable $x$, that is, for any fixed $s$, the function $x\\mapsto k\\left( x,s\\right) $ is strictly increasing. We also assume that $k\\left( x,s\\right) $ has a simple discontinuity at $x_0$ in the following sense. Let us define\n\\begin{equation*}\nk_0^-\\left( s\\right) :=\\lim _{x\\to x_0 \\,\\!^-}k\\left( x,s\\right) \\qquad \\text{and}\\qquad k_0^+\\left( s\\right) :=\\lim _{x\\to x_0 \\,\\!^+}k\\left( x,s\\right) ;\n\\end{equation*}\nthen, for every $s$, $k_0^-\\left( s\\right) <k_0^+\\left( s\\right) $. Let $u$ be a solution for equation $\\left( k,g\\right) $. If $x<x_0$, then\n\\begin{equation*}\nu\\left( x\\right) =\\int _0^x k\\left( x,s\\right) g\\left( u\\left( s\\right) \\right) \\, ds<\\int _0^{x_0}k_0^-\\left( s\\right) g\\left( u\\left( s\\right) \\right) \\, ds.\n\\end{equation*}\nIn a similar way, for $x>x_0$,\n\\begin{equation*}\nu\\left( x\\right) =\\int _0^x k\\left( x,s\\right) g\\left( u\\left( s\\right) \\right) \\, ds>\\int _0^{x_0}k_0^+\\left( s\\right) g\\left( u\\left( s\\right) \\right) \\, ds.\n\\end{equation*}\nHence, taking lateral limits, we obtain\n\\begin{equation*}\n\\lim _{x\\to x_0 \\,\\!^-}u\\left( x\\right) \\leq \\int _0^{x_0}k_0^-\\left( s\\right) g\\left( u\\left( s\\right) \\right) \\, ds < \\int _0^{x_0}k_0^+\\left( s\\right) g\\left( u\\left( s\\right) \\right) \\, ds \\leq \\lim _{x\\to x_0 \\,\\!^+} u\\left( x\\right) .\n\\end{equation*}\nThus, $u$ also has a simple discontinuity at $x_0$. Note that with an analogous proof, we can show that the function $K\\left( x\\right) =\\int _0^x k\\left( x,s\\right) \\, ds$ has a simple discontinuity at $x_0$. So, condition $\\mathbf{K}_3$ does not hold.\n\n\\subsection{An Equation with Multiple Solutions}\n\nLet us consider the equation\n\\begin{equation}\n\\label{eq16}\nu\\left( x\\right) = \\int _0^x \\left( x-s\\right) ^{\\alpha }s^{-\\alpha -1}g\\left( u\\left( s\\right) \\right) \\, ds,\\qquad \\alpha \\in (-1,0).\n\\end{equation}\nIt is an equation of type (\\ref{eq:vol1}), where $k\\left( x,s\\right) = \\left( x-s\\right) ^{\\alpha }s^{-\\alpha -1}$. We are considering a nonlocally bounded kernel; hence, condition $\\mathbf{K}_1$ does not hold. As $-\\alpha -1<0$,\n\\begin{equation*}\nk\\left( x+\\lambda ,s+\\lambda \\right) =\\left( x-s\\right) ^{\\alpha }\\left( s+\\lambda \\right) ^{-\\alpha -1}<\\left( x-s\\right) ^{\\alpha }s^{-\\alpha -1}=k\\left( x,s\\right) ,\n\\end{equation*}\nfor every $\\lambda >0$. Thus, condition $\\mathbf{K}_4$ is not verified either.\n\nNow, let us suppose that equation (\\ref{eq16}) has a positive constant solution, $u\\left( x\\right) =M$, for some $M$. Then,\n\\begin{eqnarray*}\nM & = & \\int _0^x \\left( x-s\\right) ^{\\alpha }s^{-\\alpha -1}g\\left( M\\right) \\, ds=g\\left( M\\right) \\int _0^x \\left( x-s\\right) ^{\\alpha }s^{-\\alpha -1}\\, ds \\\\\n& = & g\\left( M\\right) B\\left( \\alpha +1,-\\alpha \\right) .\n\\end{eqnarray*} \nTherefore, $u$ is a solution for equation (\\ref{eq16}) if and only if $M$ is a root of the scalar equation\n\\begin{equation}\n\\label{eq17}\nM-g\\left( M\\right) B\\left( \\alpha +1,-\\alpha \\right) =0.\n\\end{equation}\nEquation (\\ref{eq17}) depends on $g$. Then, the number of its roots also depends on $g$. It is not difficult to find nonlinearities in order to obtain any fixed number of roots for (\\ref{eq17}). For instance, let\n\\begin{equation*}\ng\\left( x\\right) =\\frac{1}{B\\left( \\alpha +1,-\\alpha \\right) }\\left( x^3 -3x^2 +3x\\right) ,\n\\end{equation*}\n(note that $g$ verifies $\\mathbf{G}_1$). In this case, equation (\\ref{eq17}) becomes\n\\begin{equation*}\nx^3 -3x^2 +2x=0.\n\\end{equation*}\nThe roots of this equation are: $0$, $1$ and $2$; and the positive constant functions $u_1\\left( x\\right) =1$ and $u_2\\left( x\\right) =2$ are two solutions for equation (\\ref{eq16}).\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn everyday life, we promptly adapt our movements to the different properties, e.g., weight, size, shape, or temperature, of the objects we interact with. By observing others manipulating objects, we easily infer their properties. Thanks to the product of motor resonance, observing an action triggers the same set of neurons, providing a common ground for understanding others \\cite{Rizzolatti}.\nAction understanding enables humans to adapt to their partners during the interaction, and it correlates with the ability to interpret and send implicit signals for cooperation. A robot should learn how to interpret such implicit signals to achieve seamless collaboration with humans \\cite{legibility}.\nMany studies have been conducted to estimate the physical properties of handled objects, particularly for tasks where humans and robots are expected to collaborate and interact physically, e.g., handovers. \nIt has been discussed how the kinematics of the movements correlate with object weight \\cite{velWeight,weightFlanagan}, and that it is possible to estimate the object weight by observing another person \\cite{sciutti:weightChildren} or a humanoid robot \\cite{sciutti:weight} lifting it.\nIn this study, we focus on another property which significantly influences human movements, namely the \\textit{carefulness}. We define it as the caution and attention that humans exercise when handling an object. This qualitative property is influenced both by the object's physical characteristics, e.g., the object fragility, and by other factors such as emotional attachment or economic value. \nLet us imagine a robot which is asked to receive a glass of water from a human: it should recognize the human carefulness to manipulate the glass without spilling water.\nThe carefulness has been explored in studies of human-human handovers to teach robots how to correctly transfer objects \\cite{billard:careful,otherCareful}, monitoring human movements with motion capture sensors. In a previous study, we demonstrated that it is possible to train a classifier to distinguish between \\textit{careful} and \\textit{non careful} human motions using only data from a low-resolution camera \\cite{HFR}. However, our carefulness recognition method was tested offline on precisely segmented data, with a single experimental scenario. To overcome these limitations, we propose: (i) an online implementation of our method for carefulness recognition, (ii) a study to demonstrate its online performance, and (iii) a study to evaluate the generalization of the method to new scenarios. Although we are aware that carefulness only partially accounts for all possible properties of an object, we believe that this work is an important step towards a global approach for robots to interpret human movements relying solely on vision.\n\n\\section{Methods} \\label{sec:methods}\nThe objective of this paper is to prove that a robot, and in particular iCub can use our previously published approach to distinguish online and in different scenarios whether a human is performing a Careful (C) transport motion or a Not Careful (NC) motion. To this extent, we developed, using the YARP middleware \\cite{yarp}, the software architecture presented in Figure \\ref{fig:diag}. The iCub's camera captures images with a resolution of $320 \\times 240$ pixels. Then, the following module computes the optical flow (OF) using a dense approach \\cite{farn:OF}, and applies a threshold on the OF magnitude to consider only the parts of the image where the change is significant. This choice introduces the strong assumption that, in the robot's field of view, relevant motions are the ones that generate the largest OF. Furthermore, choosing the OF to characterize the human motion grants the system robustness to small changes in the point of view. The components of the motion velocity (horizontal \\textit{u} and vertical \\textit{v}) are extracted from the OF, as described by Vignolo \\textit{et al}. \\cite{vignolo:OF}, and used to compute the norm of the tangential velocity, as in Eq. \\ref{eq:normVel}. The architecture extracts this feature with a frequency of 15 Hz.\n\n\\begin{equation} \\label{eq:normVel}\n    V(t)=\\sqrt{u(t)^{2}+v(t)^{2}+\\Delta _{t}^{2}}\n\\end{equation}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=\\textwidth]{images\/new_schema.png}\n    \\caption{The system's architecture structure gathering images from the robot camera to discriminate between careful (C) and non careful (NC) motions.}\n    \\label{fig:diag}\n\\end{figure}\n\nThe segmentation module implements an heuristic threshold mechanism to consider only significant data: it detects the start of a motion when the velocity $V(t)$ overcomes a threshold $\\tau$ and the end when the velocity becomes lower than $\\tau$. Once the end of the movement is detected, the segmentation module has two alternatives. If the temporal length is below $1$ second, the motion is discarded. Otherwise, the temporal sequence of size $1 \\times K$ is fed to the classifier. As anticipated, the classifier model is inspired by our previous work where a Long-Short Term Memory (LSTM) neural network showed promising results for the classification of temporal sequences of tangential velocity between careful and non careful motions \\cite{HFR}.\nIn this study, we adopted a neural network with one hidden layer followed by an output layer. The hidden layer is a 32-neuron bidirectional LSTM, while the output layer has two neurons and a sigmoidal activation function. The training has been performed using the ADAM optimization algorithm, binary cross-entropy loss function, exponential decay of the learning rate, and a batch size of 30. An early stopping condition on the validation loss, i.e., patience set to 5, has been introduced to prevent over-fitting. A zero-padding and masking technique has been adopted for the training to handle sequences with different temporal lengths.\n\nThe dataset, used to train and preliminarily test the model, had been collected asking 14 volunteers to displace a glass filled to the brim with water, for careful motions, or half full with coins, for non careful ones, in front of iCub (for more detail about the data collection process, refer to Lastrico \\textit{et al}., 2020 \\cite{HFR}). This dataset contains 878 segmented sequences, 438 for each class. Preserving the class balance, we used $72\\%$ of the data for the training, $8\\%$ for the validation, and $20\\%$ for the test. The trained model on the segmented data of the test set got an accuracy of $95.14\\%$,  in line with the results of our previous work ($90.5 \\%$) \\cite{HFR}. Furthermore, thanks to statistical analysis on this dataset, we determine the threshold value $\\tau$ for the segmentation module as $5.25\\,pixels\/s$.\n\n\\section{System evaluation}\nGiven the system presented in Section \\ref{sec:methods} for the discrimination of careful and non careful motions, we performed new experiments to test its performance. In particular, the objectives to assess are:\n\n\\begin{itemize}\n    \\item[{O1}] The possibility for the system to work online, providing the C\/NC label when a human completes a transportation motion.\n    \\item[{O2}] The ability of the system to generalize over unknown human subjects.\n    \\item[{O3}] The possibility for the system to generalize over new kinds of transportation motions.\n\\end{itemize}\n\nEleven healthy subjects, members of our organizations, voluntarily agreed to participate in the data collection (7 females, 4 males, age: $28.0\\pm2.4$); none of them is author of this research. Only one participant was left-handed. All participants used their dominant hand in the experiment. We divided the volunteers into two groups $G1$ (4 females, 2 males, age $27.8\\pm3.6$, one left-handed) and $G2$ (3 females, 3 males, age $28.2\\pm1.3$). We purposely chose different participants from those included in our training set to grant a wider variability in the new data collection and assess O2.\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{.32\\textwidth}\n        \\includegraphics[width=1\\textwidth]{images\/shelves_setup.png}\n        \\caption{Shelves}\n        \\label{fig:shelves}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{.32\\textwidth}\n        \\includegraphics[width=1\\textwidth]{images\/table1_setup.png}\n        \\caption{Simple Table}\n        \\label{fig:table1}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{.32\\textwidth}\n        \\includegraphics[width=1\\textwidth]{images\/table2_setup.png}\n        \\caption{Advanced Table}\n        \\label{fig:table2}\n    \\end{subfigure}\n    \\caption{Setups of the different scenarios explored for the system evaluation. The Shelves scenario replicates the training condition (\\ref{fig:shelves}). Simple Table (\\ref{fig:table1}) and Advanced Table (\\ref{fig:table2}) scenarios are introduced to evaluate the generalization performance.}\n    \\label{fig:setup}\n\\end{figure}\n\nThe experiment consisted of a series of structured transportation movements of four glasses performed by the participants seated at a table in front of iCub. In all the experiments iCub is passive and its role is limited to observe the scene. We use glasses identical to those in the training set divided into two classes, namely C and NC. The glasses in the first class are filled to the brim with water to impose careful transportation, but one also contains coins to increase its weight. Instead, glasses in the second class contain only coins to match the water weight. The four objects are grouped by their weight into two classes, namely light (167 gr) and heavy (667 gr). The weight difference has been introduced to increase the dataset variance. \n\nThroughout the experiment, a synthetic voice instructs the participant on which object to grasp and where to place it. Placing poses in the scenario are identified with letters (see Figure \\ref{fig:setup}). To receive instruction on the next transportation, the participant presses a key on a keyboard with their non-dominant hand. In between each transport motion, the volunteer rests the hand on the table. To investigate the system's ability to generalize over new transportation trajectories (O3), we have designed three experimental scenarios, namely: \\textit{Shelves}, \\textit{Simple Table} and \\textit{Advanced Table}.\n\\paragraph{\\textbf{Shelves.}} The first scenario replicates the one used to collect the training set. This scenario allows for testing the online performance of the classifier (O1) and the generalization of the system over new subjects (O2). The objects are transported back and forth from a fixed position on the table, delimited by a scale, to two shelves located on the right and left hand side of the table (see Figure \\ref{fig:shelves}). Eight positions where the objects can be grasped or placed are defined on the two shelves. Both $G1$ and $G2$ completed the experiment in this scenario, and each participant performed 32 transport movements (16 careful and 16 non careful).\n\\paragraph{\\textbf{Simple Table.}} This scenario is aimed at assessing the system's capability to generalize on a new set of movements (O3) and has been performed only by the $G1$ group. The glasses are moved from the scale in front of the participant to four positions on the table, delimited by a container, or vice-versa (seen Figure \\ref{fig:table1}). Each volunteer performed 32 transport movements (16 for each class).\n\\paragraph{\\textbf{Advanced Table.}} This scenario tests the system's capability to generalize over more ample and complex transport movements (O3). In this scenario, the glasses are moved between poses defined on the table, i.e., the scale is removed. In this way, the transportation motion is no more toward and away from the volunteer. Three containers are placed on the table, with two possible positions each, and columns are mounted on their frontal corners (see Figure \\ref{fig:table2}). The columns obstacle the transportation, making the experiment more challenging. Only volunteers from $G2$ experimented with this scenario, and each volunteer performed 16 transport movements (8 for each class).\n \n\\section{Results}\nThroughout all the experiments described in the previous Section, the recognition architecture described in Section \\ref{sec:methods} was running, recognizing careful and non careful motions. We analyze these results for each scenario, focusing on the system accuracy and the recognition time (i.e., the time between the motion end and the system recognition). Furthermore, we performed a statistical analysis of the velocities extracted from the OF to highlight possible differences between the three scenarios.\n\n\\subsection{Shelves} \\label{res:shelves}\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/shelvesAll_CM_excel1.png}\n    \\caption{\\textit{Shelves}. The confusion matrix of the classification results over $G2$ participants. The bottom-right cell shows the overall accuracy.}    \n    \\label{fig:shelvesCM}\n\\end{figure}\n\n\\begin{figure}\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{images\/dur_shelvesAll.png}\n        \\caption{MD}\n        \\label{fig:durShelves}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[trim={0 0.1cm 0 0},clip,width=0.7\\textwidth]{images\/FB_shelvesAll.png}\n        \\caption{AD \/ MD}\n        \\label{fig:FBShelves}\n    \\end{subfigure}\n    \\caption{\\textit{Shelves}. Box plots of the movement duration (\\ref{fig:durShelves}) and asymmetry of the velocity profiles (\\ref{fig:FBShelves}) for careful (C) and non careful (NC) transport motions. The red lines represent the medians, the blue rectangles limit the 25\\textsuperscript{th} and 75\\textsuperscript{th} percentiles, and $*$ indicates a significant difference according to the Wilcoxon test.}\n    \\label{fig:distanceShelves}\n\\end{figure}\n  \nSince all the 11 participants experience this scenario we reported in Figure \\ref{fig:shelvesCM} the overall confusion matrix, with a F1-Score of $72.9\\%$ ($G1$ $73.7\\%$ - $G2$ $72.3\\%$). In this scenario, the classifier has been invoked correctly for all the 352 transport movements (32 movements by 11 volunteers) with a recognition time of 150 $ms$ ($G1$: $140.4\\pm21.1$, $G2$: $133.7\\pm15.2$, $All$: $136.6\\pm18.8$ $ms$ - median and median absolute deviation). However, because of the system design, the classifier was called every time a velocity above the threshold persisted for more than one second, not only for transport movements. In fact, 300 more movements were detected and classified as NC $89.3\\%$ of the times. \nThese movements are those that the volunteer performs to reach the glass and to reach back to the resting position. Since these movements are not transportations, it is reasonable that the majority of them are classified as NC. Finally, we characterized the velocity profiles using two metrics, i.e., the transport movement duration (MD, proposed as significant to investigate the carefulness by \\cite{otherCareful}), and the asymmetry of the velocity peak (AD\/MD, see Eq. \\ref{eq:fB}). This last metric is expressed as the acceleration duration (AD) over the movement duration (MD), and it is widely used to characterize arm movements \\cite{asymmetry,asymmetry2}.\n\n\\begin{equation} \\label{eq:fB}\n    AD\/MD=\\frac{index_{Vmax}}{MD}\n\\end{equation}\n\nSince the populations were not normally distributed, in order to test if these two metrics showed any significant differences between C and NC motions, we used a Wilcoxon Signed Rank test. We report for the MD $p-values$: $G1, G2, All$:$\\,<.01$, while for the AD\/MD $p-values$: $G1$:$\\,<.01, G2$:$\\,> .05, All$:$\\,< .05$. Only the difference in the asymmetry between C and NC velocities of $G2$ was not found statistically significant\nsince the prolonged deceleration phase typical of careful movements was not detectable. In Figure \\ref{fig:distanceShelves} are shown duration and asymmetry ranges for the 11 participants.\n\n\\subsection{Simple Table} \\label{res:table1}\nThis scenario entailed movements that differed from those included in the training set, and only $G1$ experienced it. The online classifier did not achieve a good performance. We report an F1-Score of $66.09\\%$ with $96.25\\%$ recall and $50.33\\%$ precision values. The system tended to classify as not careful most movements, correctly identifying only $2.5\\%$ of the careful trials. However, the classifier was rightfully called at the end of every one of the 160 transport movements, with a median recognition time of $137.8\\pm21.4\\,ms$. Regarding the motions detected beyond the transport ones, the classifier was called 77 times, giving an NC label in $96.1\\%$ of the cases.  \n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{images\/dur_Table1.png}\n        \\caption{MD}\n        \\label{fig:durTable1}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{images\/FB_Table1.png}\n        \\caption{AD\/MD}\n        \\label{fig:FBTable1}\n    \\end{subfigure}\n    \\caption{\\textit{Simple Table}. Box plots of the movement duration (\\ref{fig:durTable1}) and asymmetry of the velocity profiles (\\ref{fig:FBTable1}) for careful (C) and non careful (NC) transport motions. The graphical conventions are the same as in Figure \\ref{fig:distanceShelves}.}\n    \\label{fig:distanceTable1}\n\\end{figure}\n\nInterestingly, analyzing the MD and AD\/MD metrics (see Figure \\ref{fig:distanceTable1}), which we use as distance measures between the careful and not careful movements, the Wilcoxon Rank Signed test reported \\textit{p-values} $> .2$ for both. Thus, according to the chosen metrics, no significant difference in the velocity profiles was detected between the C and NC groups in this scenario. These results suggest that for short transportations (about $40\\,cm$) with no obstacles, the kinematics properties do not change significantly between careful and non careful motions.\n\n\\begin{figure}[t]\n    \\centering\n    \\includegraphics[width=0.7\\textwidth]{images\/table2_CM_excel1.png}\n    \\caption{\\textit{Advanced Table}. Confusion matrix for the classification of the transport movements performed by $G2$. The dark grey cell shows the overall accuracy.}    \n    \\label{fig:table2CM}\n\\end{figure}\n\n\\begin{figure}[t]\n    \\centering\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{images\/dur_Table2.png}\n        \\caption{MD}\n        \\label{fig:durTable2}\n    \\end{subfigure}\n    \\begin{subfigure}[b]{0.45\\textwidth}\n        \\centering\n        \\includegraphics[width=0.7\\textwidth]{images\/FB_Table2.png}\n        \\caption{AD\/MD}\n        \\label{fig:FBTable2}\n    \\end{subfigure}\n    \\caption{\\textit{Advanced Table}. Box plots of the movement duration (\\ref{fig:durTable2}) and asymmetry of the velocity profiles (\\ref{fig:FBTable2}) for careful (C) and non careful (NC) transport motions. The graphical conventions are the same as in Figure \\ref{fig:distanceShelves}.}\n    \\label{fig:distanceTable2}\n\\end{figure}\n\n\\subsection{Advanced Table}\nThis scenario was designed to further test the generalization capability of the model. Glasses handling has been made more difficult by introducing obstacles and forcing longer paths between the grasping and release positions. In Figure \\ref{fig:table2CM} the confusion matrix for this scenario is shown, in which our system reaches an F1-Score of $82.4\\%$. The classifier output was available for every one of the 96 glass manipulations with a recognition time of $145.3\\pm16.3\\,ms$ (median and median absolute deviation). Regarding the 143 other movements that the classifier evaluated, the given label was NC for $97.9\\%$ of them.\nFinally, concerning the parametric measures (shown in Figure \\ref{fig:distanceTable2}), both differences between C and NC were statistically significant ($MD$: $p<.01,\\,AD\/MD$: $p<.05$).\n\n\\section{Discussion}\nWith this work, we claim that a robot can recognize online motion carefulness with low-resolution cameras. To this extent, the usage of optical flow as motion descriptor is quite suitable since it gives a global evaluation of the whole movement and should be robust to small and quick occlusions as the ones posed by the shelves (see Figure \\ref{fig:diag}). However, when the motions are slow, as it happens with the glasses full of water, the image obstructions might be prolonged and have a greater impact. The proposed architecture generated a classifier output for every glass transportation, i.e., no transport movements went undetected. The model output was readily available at the end of the transportation, with a median recognition time of $135.9\\pm17.9\\,ms$. The system detected other movements beyond the transport ones. These motions were related to the reaching and departing requested to grasp the glass or return to the resting position. Since, in these instances, no object was being carried, it is reasonable that the classifier returned a not careful label in the 92.7\\% of the occurrences. This result implies that when the system returns the \\textit{careful} label, this label has high confidence. \n\nIn the Shelves scenario, which replicates the training conditions, the performance of the overall online classifier are lower than those obtained with offline testing. However, given the novel testing conditions, i.e., different light and perspective, these results can suggest that our system is capable of working online (O1) while generalizing over new subjects (O2).\nAt the same time, in the Simple Table scenario, our architecture did not obtain a good classification performance. We ascribe this to the setup design. Indeed, comparing it to the Shelves and Advanced Table scenarios (see Figure \\ref{fig:setup} for reference), the Simple Table scenario requires shorter movement without any obstacles. This result can lead us to hypothesize that the carefulness effect can be stressed by the boundary conditions of the external environment. Therefore, in a more complex scenario, it is easier to detect the presence of carefulness.\nThis hypothesis is supported by the analysis of the distance metrics of the velocity profiles, presented in Figure \\ref{res:table1}. Indeed, in the Simple Table scenario, no significant difference was found in movements duration (MD) or in the asymmetry of the velocity peaks (AD\/MD). \nThese results leave us with two possible answers: (i) in the Simple Table scenario, volunteers did not act with particular care when transporting the glasses full of water, or (ii) the tangential velocity is not sufficient to discriminate between careful and non careful motions, and additional data are required, e.g., the actor's gaze pattern.\n\nFinally, our system obtained the best results when monitoring a completely novel scenario (see Figure \\ref{fig:table2CM}). As we hypothesized previously, this result is linked to the additional care that the volunteer needs to transport the glass of water in a more complex scenario. To further corroborate this hypothesis, we observe the striking difference for the MD and AD\/MD metrics (see Figure \\ref{fig:distanceTable2}) between the two classes. Nevertheless, these results support the capability of our system to work online (O1) and to generalize over new subjects (O2). Furthermore, we infer that the system can generalize over new scenarios if the transportation carefulness is evident (O3).\n\n\\section{Conclusions}\nWith the proposed approach, a robot can identify whether the object is handled with care or not, simply observing the human movements. A robot may be able to exploit this capability to select its subsequent manipulations to match the observed carefulness, with no need for \\textit{a priori} knowledge of the object or visual detection of its physical properties. It is worth noting that we tested our system with non-interactive actions (i.e., participants perform the task alone, with the robot acting as an observer). An interactive context might facilitate carefulness recognition, inducing participants to convey, more explicitly, this information as it happens in human signaling \\cite{legibility,signaling}. For this reason, future works should include interactive scenarios together with a more in-depth validation.\n\\bibliographystyle{splncs04}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\\n\nLet $\\mathcal{A}$ be a unital algebra and  $\\mathcal{M}$ be a unital\n$\\mathcal{A}$-bimodule. We denote $C(\\mathcal{A},\\mathcal{M})=\\{M\\in\n\\mathcal{M}: AM=MA~ \\textrm{for~ every}~A\\in \\mathcal{A}\\}$ and\n$L(\\mathcal{A},\\mathcal{M})$ the set of all linear mappings from\n$\\mathcal{A}$ to $\\mathcal{M}$. When $\\mathcal{M}=\\mathcal{A}$, we\nrelabel $L(\\mathcal{A},\\mathcal{M})$ as $L(\\mathcal{A})$. Let\n$\\delta\\in L(\\mathcal{A},\\mathcal{M})$. $\\delta$ is called a\n\\textit{derivation} if $\\delta(AB)=\\delta(A)B+A\\delta(B)$ for all\n$A,B\\in \\mathcal{A}$; it is a \\textit{Jordan derivation} if\n$\\delta(AB+BA)=\\delta(A)B+A\\delta(B)+\\delta(B)A+B\\delta(A)$ for all\n$A,B\\in \\mathcal{A}$; it is a \\textit{generalized derivation}\nif there exists an $M_{\\delta}\\in C(\\mathcal{A},\\mathcal{M})$ such\nthat\n$\\delta(AB)=\\delta(A)B+A\\delta(B)-M_{\\delta}AB$\nfor all $A,B\\in \\mathcal{A}$. For any fixed $M\\in \\mathcal{M}$, each\nmapping of the form $\\delta_{M}(A)=MA-AM$ for every $A\\in\n\\mathcal{A}$ is called an \\textit{inner derivation}. Clearly each\ninner derivation is a derivation and each derivation in a Jordan\nderivation. But the converse is not true in general. The questions\nof characterizing derivations and Jordan derivations have received\nconsiderable attention from several authors, who revealed the\nrelations among derivations, Jordan derivations as well as inner\nderivations (see for example\n\\cite{CHDM,semiprime,prime,Johnson,LU2,Moore,JDOTA}, and the\nreferences therein).\n\n\nIn general there are two directions in the study of the local\nactions of derivations of operator algebras. One is the well known\nlocal derivation problem (see for example\n\\cite{local3,local2,local1,Zhu6,Zhu5}). The other is to study\nconditions under which derivations of operator algebras can be\ncompletely determined by the action on some subsets of operators\n(see for example \\cite{CHDM,Chebotar, LI1,Hou2,  ZHOU, ZHUJUN2}). A\nmapping $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ is called a\n\\textit{Jordan derivable mapping at $C\\in \\mathcal{A}$} if\n$\\delta(AB+BA)=\\delta(A)B+A\\delta(B)+\\delta(B)A+B\\delta(A)$ for all\n$A,B\\in \\mathcal{A}$ with $AB=C$. It is obvious that a linear\nmapping is a Jordan derivation if and only if it is Jordan derivable\nat all points. It is natural and interesting to ask the question\nwhether or not a linear mapping is a Jordan derivation if it is\nJordan derivable only at one given point. If such a point exists, we\ncall this point a Jordan all-derivable point. To be more precise, an\nelement $C\\in \\mathcal{A}$ is called a \\textit{Jordan all-derivable\npoint} of $L(\\mathcal{A},\\mathcal{M})$ if every Jordan derivable\nmapping at $C$ is a Jordan derivation. It is quite surprising that\nthere do exist Jordan all-derivable points for some algebras. An and\nHou \\cite{Hou} show that under some mild conditions on unital prime\nring or triangular ring $\\mathcal{A}$, $I$ is a Jordan all-derivable\npoint of $L(\\mathcal{A})$. Jiao and Hou \\cite{Hou3} study Jordan\nderivable mappings at zero point on nest algebras. Zhao and Zhu\n\\cite{ZHUJUN} prove that $0$ and $I$ are Jordan all-derivable points\nof the triangular algebra. In \\cite{submit}, the authors study some\nderivable mappings in the generalized matrix algebra $\\mathcal{A}$,\nand show that $0$, $P$ and $I$ are Jordan all-derivable points,\nwhere $P$ is the standard non-trivial idempotent. In \\cite{ZHUJUN},\nZhao and Zhu prove that every element in the algebra of all $n\\times\nn$ upper triangular matrices over the complex field $\\mathbb{C}$ is a Jordan all-derivable point. In\nSection 2, we give some general characterizations of Jordan\nderivable mappings, which will be used to determine Jordan\nall-derivable points for some general bimodules.\n\nLet $\\mathcal{A}$ be a unital algebra and $\\mathbb{N}$ be the set of non-negative integers. A sequence of mappings\n$\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is called a\n\\textit{higher derivation} if $d_{n}(AB)=\\sum_{i+j=n}d_i(A)d_j(B)$\nfor all $A,B\\in \\mathcal{A}$; it is called a \\textit{Jordan higher\nderivation} if\n$d_{n}(AB+BA)=\\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all\n$A,B\\in \\mathcal{A}$. With the development of derivations, the study\nof higher and Jordan higher derivations has attracted much attention\nas an active subject of research in operator algebras, and the local\naction problem ranks among in the list. A sequence of mappings\n$\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is called\n\\textit{Jordan higher derivable at $C\\in \\mathcal{A}$} if\n$d_{n}(AB+BA)=\\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all\n$A,B\\in \\mathcal{A}$ with $AB=C$. An element $C\\in \\mathcal{A}$ is\ncalled a \\textit{Jordan higher all-derivable point} if every\nsequence of Jordan higher derivable mappings at $C$ is a Jordan\nhigher derivation. In Section 3, we generalize the results in\nSection 2 to the case of Jordan higher derivable mappings.\nMeanwhile, we find the connection between Jordan all-derivable\npoints (all-derivable points, S-Jordan all derivable points,\nrespectively) and Jordan higher all-derivable points (higher\nall-derivable points, S-Jordan higher all-derivable points,\nrespectively). We also discuss the automatic continuity property of\n(Jordan) higher derivations.\n\nLet $X$ be a complex Banach space and $B(X)$ be the set of all\nbounded linear operators on $X$. For any non-empty subset\n$L\\subseteq X$, $L^\\perp$ denotes its annihilator, that is,\n$L^\\perp=\\{f\\in X^*: f(x)=0~\\mathrm{for}~\\mathrm{all}~x\\in L\\}$. By\na \\textit{subspace lattice} on $X$, we mean a collection\n$\\mathcal{L}$ of closed subspaces of $X$ with (0) and $X$ in\n$\\mathcal{L}$ such that for every family $\\{M_r\\}$ of elements of\n$\\mathcal{L}$, both $\\cap M_r$ and $\\vee M_r$ belong to\n$\\mathcal{L}$. For a subspace lattice $\\mathcal{L}$ of $X$, let\nalg$\\mathcal{L}$ denote the algebra of all operators in $B(X)$ that\nleave members of $\\mathcal{L}$ invariant. A totally ordered subspace\nlattice is called a \\textit{nest}. If $\\mathcal{L}$ is a nest, then\nalg$\\mathcal{L}$ is called a \\textit{nest algebra}, see \\cite{NEST}\nfor more on nest algebras. When $X$ is a separable Hilbert space\nover the complex field $\\mathbb{C}$, we change it to $H$. In a\nHilbert space, we disregard the distinction between a closed\nsubspace and the orthogonal projection onto it. An immediate but\nnoteworthy application of our main result shows that for a nest\n$\\mathcal{N}$ on a Banach $X$ with the associated nest algebra\n$alg\\mathcal{N}$, if there exists a non-trivial element in\n$\\mathcal{N}$ which is complemented in $X$, then every $C\\in\nalg\\mathcal{N}$ is a Jordan all-derivable point of\n$L(alg\\mathcal{N}, B(X))$ and a Jordan higher all-derivable point of\n$L(alg\\mathcal{N})$.\n\n\n\n\\section{Jordan derivable mappings}\\\n\nWe start with Peirce decomposition of algebras and its bimodules.\n\nLet $\\mathcal{A}$ be a unital algebra and $\\mathcal{M}$ be a unital\n$\\mathcal{A}$-bimodule.  For any idempotent $E_1\\in \\mathcal{A}$,\nlet $E_2=I-E_1$. For $i,j\\in\\{1,2\\},$ define\n$\\mathcal{A}_{ij}=E_i\\mathcal{A}E_j$, which gives the Peirce\ndecomposition of $\\mathcal{A}: A=A_{11}+A_{12}+A_{21}+A_{22}.$\nSimilarly, we define $\\mathcal{M}_{ij}=E_i\\mathcal{M}E_j$. We say\n$\\mathcal{A}_{ij}$ is \\textit{left faithful} with respect to\n$\\mathcal{M}$ if for any $M\\in \\mathcal{M}$, the condition\n$M\\mathcal{A}_{ij}=\\{0\\}$  implies $ME_i=0$ and $\\mathcal{A}_{ij}$\nis \\textit{right faithful} with respect to $\\mathcal{M}$ if the\ncondition $\\mathcal{A}_{ij}M=\\{0\\}$  implies $E_jM=0.$ We say\n$\\mathcal{A}_{ij}$ is \\textit{faithful} with respect to\n$\\mathcal{M}$ if it is both left faithful and right faithful. In\nthis paper, we will always use the notations $P=E_1$ and\n$Q=E_2=I-E_1$ for convenience.\n\nIn this section, we will assume $\\mathcal{A}$ is a unital\nalgebra over a  field $\\mathbb{F}$ of characteristic not equal to $2$ and $|\\mathbb{F}|\\geq4$,\n$\\mathcal{M}$ is a unital $\\mathcal{A}$-bimodule and $\\mathcal{A}$\nhas a non-trivial idempotent $P=E_1\\in \\mathcal{A}$ such that the\ncorresponding Peirce decomposition has the following property: Every\nelement of $A_{11}$ is a linear combination of invertible elements\nof $A_{11}$ and every element of $A_{22}$ is a linear combination of\ninvertible elements of $A_{22}$. Algebras satisfying these\nassumptions include all finite-dimensional unital algebras over an\nalgebraically closed field and all unital Banach algebras.\n\nFor any $A,B\\in \\mathcal{A}$, define $A\\circ B=AB+BA$; similarly, for any\n$A\\in \\mathcal A$ and $M\\in \\mathcal M$, define $A\\circ M=AM+MA.$\nFor $A,B, D, E\\in \\mathcal{A}$, we say any $\\delta \\in L(\\mathcal A, \\mathcal M)$\n\\textit{differentiates} $A\\circ B$ if $\\delta(A\\circ B)=\\delta(A)\\circ B+A\\circ \\delta(B)$\n; we say $\\delta $ \\textit{differentiates} $A\\circ B+ C\\circ D$ if\n$\\delta (A\\circ B+ C\\circ D)=\\delta (A)\\circ B+A\\circ \\delta (B)+\\delta (D)\\circ E+D\\circ \\delta (E)$.\nWe see that a mapping $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ is a Jordan derivation if and only if\n$\\delta $ differentiates $A\\circ B$ for all\n$A,B\\in \\mathcal{A}$, and $\\delta$ is Jordan derivable at $C\\in \\mathcal{A}$\nif and only if $\\delta $ differentiates $A\\circ B$ for all\n$A,B\\in \\mathcal{A}$ with $AB=C$.\n\nThe following proposition is elementary, we omit the proof.\n\n\\begin{proposition}\\label{0000}\nLet $\\mathcal{V}$ be a vector space over a field $\\mathbb{F}$ with\n$|\\mathbb{F}|>n$.  For any fixed $v_i\\in \\mathcal{V}, i=0, 1, \\cdots , n$, define\n$p(t)=\\sum_{i=0}^n v_i t^i $ for $t\\in \\mathbb{F}$. If $p(t)=0$ has\nat least $n+1$ distinct solutions in $\\mathbb{F}$, then $v_i=0, \\ i=0, 1, \\cdots , n$.\n\\end{proposition}\n\nA simple application of Proposition $2.1$ yields the following proposition, which will be used repeatedly in this paper.\n\n\\begin{proposition}\\label{0001}\nSuppose $A, B, D, E, K, L \\in \\mathcal{A}$ and $\\delta\\in L(\\mathcal{A},\\mathcal{M})$.\n\n(a) If $\\delta $ differentiates $(tA+B)\\circ (tD+E)$ for at least three $t\\in \\mathbb{F}$, then\n$\\delta $ differentiates $A\\circ D$,  $B\\circ E$, and $ A\\circ E+B\\circ D$; in particular, if $A=0$ then $\\delta $\ndifferentiates $B\\circ D$.\n\n(b) If $\\delta $ differentiates  $A\\circ (tD+E)+B\\circ (tK+L)$ for at least two $t\\in \\mathbb{F}$, then\n $\\delta $ differentiates  $A\\circ D+B\\circ K$ and  $A\\circ E+B\\circ L$.\n\\end{proposition}\n\n\nNow we characterize Jordan-derivable mappings in terms of Peirce decomposition as follows.\n\n\\begin{theorem}\\label{0002}\nFor any $C\\in\\mathcal{A}$ such that $C_{21}=0$, if $\\Delta\\in\nL(\\mathcal{A},\\mathcal{M})$ is  Jordan-derivable at $C$, then there\nexists a $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ such that\n$\\Delta-\\delta$ is an inner derivation and the following hold:\n\n(a)~~$\\delta(P)A_{12}=A_{12}\\delta(Q)$ for any $A_{12}\\in \\mathcal{A}_{12}$.\\\n\n(b)~~$A_{12}\\delta(A_{12})=\\delta(A_{12})A_{12}=0$ for any $A_{12}\\in \\mathcal{A}_{12}$.\\\n\n(c)~~$\\delta(\\mathcal{A}_{11})\\subset \\mathcal{M}_{11}$, $\\delta(\\mathcal{A}_{22})\\subset \\mathcal{M}_{22}$.\n\nIf $\\mathcal{A}_{12}$ is left faithful, then\n\n(d)~~$\\delta(P)\\in C(\\mathcal{A}_{11},\\mathcal{M})$.\n\n(e)~~$\\delta|_{\\mathcal{A}_{11}}$ is a generalized derivation from $\\mathcal{A}_{11}$ to $\\mathcal{M}_{11}$.\n\nIf $\\mathcal{A}_{12}$ is right faithful, then\n\n(f)~~$\\delta(Q)\\in C(\\mathcal{A}_{22},\\mathcal{M})$.\n\n(g)~~$\\delta|_{\\mathcal{A}_{22}}$ is a generalized derivation from $\\mathcal{A}_{22}$ to $\\mathcal{M}_{22}$.\n\\end{theorem}\n\n\\begin{proof}\nLet $M=P\\Delta(Q)Q-Q\\Delta(Q)P$ and define\n$\\delta(A)=\\Delta(A)-(MA-AM)$ for every $A\\in \\mathcal{A}$.  Then\n$\\delta$ is Jordan-derivable at any $G\\in \\mathcal{A}$ if and only\nif $\\Delta$ is Jordan-derivable at $G$; moreover $\\delta(Q)\\in\n\\mathcal{M}_{11}+\\mathcal{M}_{22}$ by direct computation. Write $C=\nC_{11}+C_{12}+C_{22}$. Fix any $A_{11}\\in\\mathcal{A}_{11}$ that is\ninvertible in $\\mathcal{A}_{11}$ with $A_{11}^{-1}\\in\n\\mathcal{A}_{11}$ and  $Z_{22}, W_{22}\\in \\mathcal{A}_{22}$ such\nthat $Z_{22}W_{22}=C_{22}$. Note that we can take any $W_{22}$ that\nis invertible in $\\mathcal{A}_{22}$ with $W_{22}^{-1}\\in\n\\mathcal{A}_{22}$ and $Z_{22}=C_{22}W_{22}^{-1}$ to satisfy\n$Z_{22}W_{22}=C_{22}$. For any $0\\neq t\\in \\mathbb{F}$,  $s\\in \\mathbb{F}$,  and\n$A_{12}\\in \\mathcal{A}_{12}$, a routine computation yields\n$$[A_{11}+t(sA_{11}A_{12}+Z_{22})][(A_{11}^{-1}C-sA_{12}W_{22})+t^{-1}W_{22}]=C.$$\nSince $\\delta$ is Jordan derivable at C, $\\delta $ differentiates\n$$[A_{11}+t(sA_{11}A_{12}+Z_{22})]\\circ [(A_{11}^{-1}C-sA_{12}W_{22})+t^{-1}W_{22}].$$\nThus $\\delta $ differentiates\n$[A_{11}+t(sA_{11}A_{12}+Z_{22})]\\circ [t(A_{11}^{-1}C-sA_{12}W_{22})+W_{22}].$\nBy Proposition \\ref{0001}$(a)$, we get $(i)$  $\\delta $ differentiates $A_{11}\\circ W_{22}$, \\ \\\n$(ii)$   $\\delta $ differentiates $(sA_{11}A_{12}+Z_{22})\\circ (A_{11}^{-1}C-sA_{12}W_{22})$, and\n$(iii)$   $\\delta $ differentiates  $A_{11}\\circ (A_{11}^{-1}C-sA_{12}W_{22})+(sA_{11}A_{12}+Z_{22})\\circ W_{22}.$\n\nBy $(i)$, we get\n\\begin{eqnarray}\\delta (A_{11})\\circ W_{22}+A_{11}\\circ \\delta (W_{22})=\\delta (A_{11}\\circ W_{22})=0 \\label{2000}\n\\end{eqnarray}\n\nBy $(ii)$ and Proposition \\ref{0001}$(a)$, we have $\\delta $  differentiates $(A_{11}A_{12})\\circ (A_{12}W_{22})$, i.e.\n\\begin{eqnarray}\n\\delta (A_{11}A_{12})\\circ (A_{12}W_{22}) +(A_{11}A_{12})\\circ \\delta (A_{12}W_{22})\n=\\delta [(A_{11}A_{12})\\circ (A_{12}W_{22})]=0\\label {2002}\n\\end{eqnarray}\n\nBy $(iii)$ and Proposition \\ref{0001}$(b)$, we have\n\\begin{eqnarray*}\n&0=\\delta [A_{11}\\circ (-A_{12}W_{22})+(A_{11}A_{12})\\circ W_{22}]\n=\\delta (A_{11})\\circ (-A_{12}W_{22}) +A_{11}\\circ \\delta (-A_{12}W_{22}) \\\\\n&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\\delta (A_{11}A_{12})\\circ W_{22}+(A_{11}A_{12})\\circ \\delta (W_{22})\n\\end{eqnarray*}\n\nThus\n\\begin{eqnarray}\n\\delta (A_{11}A_{12})\\circ W_{22}+(A_{11}A_{12})\\circ \\delta (W_{22})\n-\\delta (A_{11})\\circ (A_{12}W_{22}) - A_{11}\\circ \\delta (A_{12}W_{22})=0 \\label {2003}\n\\end{eqnarray}\nSince $\\delta (Q)\\in \\mathcal M_{11}+\\mathcal M_{22}$, $A_{11}\\circ \\delta (Q)\\in \\mathcal M_{11}$.\nSetting $W_{22}=Q$ in Eq. $(2.1)$ gives\n\\begin{eqnarray*} \\delta (A_{11})\\circ Q+A_{11}\\circ \\delta (Q)=0\n\\end{eqnarray*}\nThus\n$A_{11}\\circ \\delta (Q)=\\delta (A_{11})\\circ Q=0$. It follows $\\delta (A_{11})Q=Q\\delta (A_{11})=0.$\nHence $\\delta (A_{11})\\in \\mathcal M_{11},$ and $\\delta (A_{11})\\circ W_{22}=0$. By Eq. $(2.1)$ again, we get\n$A_{11}\\circ \\delta (W_{22})=0$; in particular, $P\\circ \\delta (W_{22})=0$.\nIt follows that $\\delta (W_{22})\\in \\mathcal M_{22}$,\n which proves $(c)$.\n\nTaking $A_{11}=P$ in Eq. $(2.3)$ yields\n\\begin{eqnarray}\n\\delta (A_{12})\\circ W_{22}+A_{12}\\circ \\delta (W_{22})\n-\\delta (P)\\circ (A_{12}W_{22}) - P\\circ \\delta (A_{12}W_{22})=0 \\label {2004}\n\\end{eqnarray}\nMultiplying $P$ from both sides of Eq. $(2.4)$ gives\n$P \\delta (A_{12}W_{22})P=0$. In particular,\n\\begin{eqnarray}\nP \\delta (A_{12})P=0 \\label {2005}\n\\end{eqnarray}\nMultiplying $P$ from the left of Eq. $(2.4)$ and applying Eq. $(2.5)$, we get\n\\begin{eqnarray}\nP\\delta (A_{12}) W_{22}+A_{12}\\delta (W_{22})\n-\\delta (P)A_{12}W_{22} - P\\delta (A_{12}W_{22})=0\n\\end{eqnarray}\nSetting $W_{22}=Q$ in Eq. $(2.6)$ and combining with Eq. $(2.5)$ leads to\n\\begin{eqnarray}\nA_{12}\\delta (Q)=\\delta (P)A_{12} \\label {2006}\n\\end{eqnarray}\nThis proves $(a)$.\n\nTaking $A_{11}=P$ and $W_{22}=Q$ in Eq. $(2.2)$, we get $A_{12}\\circ \\delta (A_{12})=0$, i.e.\n\\begin{eqnarray}\nA_{12}\\delta (A_{12})+\\delta (A_{12})A_{12}=0 \\label {2007}\n\\end{eqnarray}\nMultiplying $P$ from the left of Eq. $(2.8)$ and applying Eq. $(2.5)$, yields $A_{12}\\delta (A_{12})=0$; which gives\n$\\delta (A_{12})A_{12}=0$, when applied to Eq. $(2.8)$. This proves $(b)$.\n\nTaking $W_{22}=Q$ in Eq. $(2.3)$ yields\n\\begin{eqnarray}\n\\delta (A_{11}A_{12})\\circ Q+(A_{11}A_{12})\\circ \\delta (Q)\n-\\delta (A_{11})\\circ A_{12} - A_{11}\\circ \\delta (A_{12})=0 \\label {2008}\n\\end{eqnarray}\nMultiplying $Q$ from both sides of Eq. $(2.9)$ gives\n$Q\\delta (A_{11}A_{12})Q=0$. In particular,\n\\begin{eqnarray}\nQ \\delta (A_{12})Q=0 \\label {2009}\n\\end{eqnarray}\nMultiplying $Q$ from the right of Eq. $(2.9)$ and applying Eq. $(2.10)$ gives\n\\begin{eqnarray*}\n\\delta (A_{11}A_{12})Q+A_{11}A_{12}\\delta (Q)\n-\\delta (A_{11})A_{12} - A_{11}\\delta (A_{12})Q=0\n\\end{eqnarray*}\nCombining this with Eq. $(2.5)$ yields\n\\begin{eqnarray}\n\\delta (A_{11}A_{12})Q=\n\\delta (A_{11})A_{12} + A_{11}\\delta (A_{12}) - A_{11}A_{12}\\delta (Q) \\label {2010}\n\\end{eqnarray}\nReplacing $A_{11}$ with $A_{11}U_{11}$ in Eq. $(2.11)$ gives\n\\begin{eqnarray}\n\\delta (A_{11}U_{11}A_{12})Q=\n\\delta (A_{11}U_{11})A_{12} + A_{11}U_{11}\\delta (A_{12}) - A_{11}U_{11}A_{12}\\delta (Q) \\label {2011}\n\\end{eqnarray}\nOn the other hand, applying Eq. $(2.11)$ twice gives\n\\begin{align}\n \\delta (A_{11}U_{11}A_{12})Q &=\nA_{11}\\delta (U_{11}A_{12}) + \\delta (A_{11})U_{11}A_{12}  - A_{11}U_{11}A_{12}\\delta (Q) \\nonumber \\\\\n& = A_{11}\\delta (U_{11}A_{12})Q + \\delta (A_{11})U_{11}A_{12} - A_{11}U_{11}A_{12}\\delta (Q) \\nonumber \\\\\n& = A_{11}[\\delta (U_{11})A_{12} + U_{11}\\delta (A_{12}) - U_{11}A_{12}\\delta (Q)]  \\nonumber \\\\\n&~~~~ + \\delta (A_{11})U_{11}A_{12} - A_{11}U_{11}A_{12}\\delta (Q) \\label {2012}\n\\end{align}\nBy Eqs. $(2.12)$, $(2.13)$, and $(2.7)$, we have\n\\begin{eqnarray*}\n\\delta (A_{11}U_{11})A_{12}=\n[\\delta (A_{11})U_{11} + A_{11}\\delta (U_{11}) - \\delta (P)A_{11}U_{11}]A_{12}\n\\end{eqnarray*}\nIf $\\mathcal A_{12}$ is left faithful,\n\\begin{eqnarray}\n\\delta (A_{11}U_{11})=\n\\delta (A_{11})U_{11} + A_{11}\\delta (U_{11}) - \\delta (P)A_{11}U_{11} \\label {2013}\n\\end{eqnarray}\nTaking $U_{11}=P$ in Eq. $(2.14)$ gives $A_{11}\\delta (P) = \\delta (P)A_{11}$, that is,\n$\\delta (P)\\in C(\\mathcal A_{11}, \\mathcal M)$. This proves $(d)$ and now $(e)$ follows directly\nfrom Eq. $(2.14)$.\n\nSince $\\delta(P)A_{12}=A_{12}\\delta(Q)$ for any $A_{12}\\in \\mathcal{A}_{12}$, we\nhave $A_{12}\\delta(Q)A_{22}=\\delta(P)A_{12}A_{22}=A_{12}A_{22}\\delta(Q)$, then faithfulness\nof $\\mathcal{A}_{12}$ leads to $\\delta(Q)A_{22}=A_{22}\\delta(Q)$, that is,\n $\\delta(Q)\\in C(\\mathcal{A}_{22},\\mathcal{M})$. This proves $(f)$.\n\nBy Eqs. $(2.6)$ and $(2.10)$,\n\\begin{eqnarray}\nP\\delta (A_{12}W_{22})=\\delta (A_{12}) W_{22}+A_{12}\\delta (W_{22})\n-\\delta (P)A_{12}W_{22}\n\\end{eqnarray}\nReplacing $W_{22}$ with $V_{22}W_{22}$ in Eq. $(2.15)$ gives\n\\begin{eqnarray}\nP\\delta (A_{12}V_{22}W_{22})=\\delta (A_{12}) V_{22}W_{22}+A_{12}\\delta (V_{22}W_{22})\n-\\delta (P)A_{12}V_{22}W_{22}\n\\end{eqnarray}\nOn the other hand, applying Eq. $(2.15)$ twice gives\n\\begin{align}\nP\\delta (A_{12}V_{22}W_{22})&=\\delta (A_{12}V_{22}) W_{22}+A_{12}V_{22}\\delta (W_{22})\n-\\delta (P)A_{12}V_{22}W_{22} \\nonumber \\\\\n&=P\\delta (A_{12}V_{22}) W_{22}+A_{12}V_{22}\\delta (W_{22})\n-\\delta (P)A_{12}V_{22}W_{22} \\nonumber \\\\\n&=[\\delta (A_{12}) V_{22}+A_{12}\\delta (V_{22})\n-\\delta (P)A_{12}V_{22}]W_{22} \\nonumber \\\\\n&~~~~+A_{12}V_{22}\\delta (W_{22})\n-\\delta (P)A_{12}V_{22}W_{22}\n\\end{align}\nBy Eqs. $(2.16)$, $(2.17)$, and $(2.7)$,\n\\begin{eqnarray*}\nA_{12}\\delta (V_{22}W_{22})=A_{12}[\\delta (V_{22})W_{22}+V_{22}\\delta (W_{22})\n-\\delta (Q)V_{22}W_{22}]\n\\end{eqnarray*}\nSince $\\mathcal A_{12}$ is left faithful,\n\\begin{eqnarray}\n\\delta (V_{22}W_{22})=\\delta (V_{22})W_{22}+V_{22}\\delta (W_{22})\n-\\delta (Q)V_{22}W_{22}\n\\end{eqnarray}\nThis proves $(g)$.\n\\end{proof}\n\nSuppose $\\mathcal{B}$ is an algebra containing $\\mathcal{A}$ and shares the same identity with $\\mathcal{A}$, then\n$\\mathcal{B}$ is an $\\mathcal{A}$-bimodule with respect to the\nmultiplication and addition of $\\mathcal{B}$. Let $\\mathcal{T}_{\\mathcal{A}}= \\{A\\in \\mathcal A: A_{21}=0\\}$.\n The following proposition is contained in  \\cite[Theorem 3.3]{PAN}, we include a proof here for completeness.\n\n\\begin{proposition}\\label{2.4}\nSuppose $\\mathcal{A}_{12}$ is faithful to $\\mathcal{B}$, $C(\\mathcal{T}_{\\mathcal{A}},\\mathcal{B})=\\mathbb{F}I$, and $B\\in \\mathcal B$.\nIf $T_{12}BT_{12}=0$ for every $T_{12}\\in \\mathcal{A}_{12}$, then $QBP=0$.\n\\end{proposition}\n\\begin{proof}\nSuppose $T_{12}BT_{12}=0$ for every $T_{12}\\in \\mathcal{A}_{12}$. For any non-zero $A_{12},T_{12}\\in\\mathcal{A}_{12}$, we have $T_{12}BT_{12}=0$, $A_{12}BA_{12}=0$ and $(A_{12}+T_{12})B(A_{12}+T_{12})=0$. It follows that\n\\begin{eqnarray}\nA_{12}BT_{12}+T_{12}BA_{12}=0.\\label{2023}\n\\end{eqnarray}\nFor any $A_{11}\\in \\mathcal{A}_{11}$, replacing $A_{12}$ in Eq. $(2.19)$ with $A_{11}A_{12}$ gives\n\\begin{eqnarray}\nA_{11}A_{12}BT_{12}+T_{12}BA_{11}A_{12}=0.\\label{2024}\n\\end{eqnarray}\nMultiplying $A_{11}$ from the left of Eq. $(2.19)$ gives\n\\begin{eqnarray}\nA_{11}A_{12}BT_{12}+A_{11}T_{12}BA_{12}=0.\\label{2025}\n\\end{eqnarray}\nBy Eq. (\\ref{2024}) and Eq. (\\ref{2025}), we have\n\\begin{eqnarray*}\nT_{12}BA_{11}A_{12}=A_{11}T_{12}BA_{12}.\n\\end{eqnarray*}\nSince $A_{12}$ is arbitrary and $\\mathcal{A}_{12}$ is faithful, we have\n\\begin{eqnarray}\nT_{12}BA_{11}=A_{11}T_{12}BP.\\label{2026}\n\\end{eqnarray}\nSimilarly, we have\n\\begin{eqnarray}\nA_{22}QBT_{12}=QBT_{12}A_{22}.\\label{2027}\n\\end{eqnarray}\nLet $\\widetilde{B}=T_{12}BP-QBT_{12}$. It follows from Eqs. (\\ref{2023}),\n(\\ref{2026}) and  (\\ref{2027}) that $\\widetilde{B}$ commutes with $A_{12}$,\n$A_{11}$ and $A_{22}$, that is, $\\widetilde{B}\\in C(\\mathcal{T}_{\\mathcal{A}},\\mathcal{B})$.\nHence there exists a $k\\in \\mathbb{F}$ such that $\\widetilde{B}=kI$. It follows\n$T_{12}BP=kP$. Now $T_{12}BT_{12}=0$ leads to $kT_{12}=0$. Hence\n$k=0$ and $T_{12}BP=0$. Since $T_{12}$ is arbitrary and\n$\\mathcal{A}_{12}$ is faithful, we have $QBP=0$.\n\\end{proof}\n\n\\begin{theorem}\\label{2.5}\nSuppose $\\mathcal{A}_{12}$ is faithful to $\\mathcal{B}$ and $C(\\mathcal{T}_{\\mathcal{A}},\\mathcal{B})=\\mathbb{F}I$.\nIf $\\delta\\in L(\\mathcal{A},\\mathcal{B})$ is Jordan derivable at $C\\in \\mathcal{T}_{\\mathcal{A}}$ then\n$\\delta |_{\\mathcal{T}_{\\mathcal{A}}}$ is a derivation from $\\mathcal{T}_{\\mathcal{A}}$ to $\\mathcal{B}$.\n\\end{theorem}\n\\begin{proof}\nSubstracting an\ninner derivation from $\\delta$ if necessary, we can assume $\\delta$\nsatisfies the properties of Theorem $2.3$.\nThus, for any $A_{12}$ and $T_{12}$ in $\\mathcal{A}_{12}$, we have $\\delta(A_{12})A_{12}=0$,\n$\\delta(T_{12})T_{12}=0$ and\n$\\delta(A_{12}+T_{12})(A_{12}+T_{12})=0$. It follows that\n$\\delta(A_{12})T_{12}+\\delta(T_{12})A_{12}=0.$ Multiplying $T_{12}$\nfrom the left we obtain $T_{12}\\delta(A_{12})T_{12}=0$. Since\n$T_{12}$ is arbitrary, $Q\\delta(A_{12})P=0$, by Proposition $2.4$. This, together with Eqs.\n$(2.5)$ and  $(2.10)$, yields $\\delta(A_{12})\\in\n\\mathcal{B}_{12}$.\n\nFor any $A_{11}\\in \\mathcal A_{11}$ and $A_{22}\\in \\mathcal A_{22}$, by Theorem $2.3$\n$\\delta (A_{11})\\in \\mathcal B_{11}$ and $\\delta (A_{22})\\in \\mathcal B_{22}$.\n\nSince $\\delta(P)\\in\\mathcal{B}_{11}$ and\n$\\delta(Q)\\in\\mathcal{B}_{22}$, by Theorem $2.3$\n$\\delta(I)=\\delta(P)+\\delta(Q)$ commutes with $\\mathcal{A}_{11}$,\n$\\mathcal{A}_{12}$ and $\\mathcal{A}_{22}$, whence $\\delta(I)\\in\nC(\\mathcal{T}_{\\mathcal{A}},\\mathcal{B})$. Thus $\\delta(I)=\\lambda I$. By the fact\nthat $\\delta$ is Jordan derivable at $C$, we have\n$\\delta(IC+CI)=\\delta(I)C+I\\delta(C)+C\\delta(I)+\\delta(C)I$, which\nimplies $\\lambda C=0$. If $C\\neq0$, $\\lambda=0$. Hence\n$\\delta(P)=\\delta(Q)=0$. If $C=0$, then the fact that\n$A_{12}A_{11}=0$ holds for every $A_{11}\\in \\mathcal{A}_{11}$ and\n$A_{12}\\in \\mathcal{A}_{12}$ implies that\n$\\delta(A_{11}A_{12})=\\delta(A_{12})A_{11}+A_{12}\\delta(A_{11})+A_{11}\\delta(A_{12})+\\delta(A_{11})A_{12}=A_{11}\\delta(A_{12})+\\delta(A_{11})A_{12}$,\nwhich together with faithfulness of $\\mathcal{A}_{12}$ leads to\n$\\delta(A_{11}U_{11})=\\delta(A_{11})U_{11}+A_{11}\\delta(U_{11})$ for\nevery $A_{11}, U_{11}\\in \\mathcal{A}_{11}$. Comparing with\nEq. $(2.14)$, we have that $\\delta(P)=\\delta(Q)=0$.\n\nTo see $\\delta |_{\\mathcal{T}_{\\mathcal{A}}}$ is a derivation, it suffices to show that for any $A_{ij}, A_{kl}\\in \\mathcal{T}_{\\mathcal{A}}$\n$$\\delta(A_{ij}A_{kl})=\\delta(A_{ij})A_{kl}+A_{ij}\\delta(A_{kl}).$$\nWe will label each case as Case $(ij,kl)$. Since $\\delta(A_{11})\\in\\mathcal{B}_{11}$, $\\delta(A_{12})\\in\\mathcal{B}_{12}$,\nand $\\delta(A_{22})\\in\\mathcal{B}_{22}$, we only need to check cases for $j=k$. There are only 4 cases.\n\nCase $(11,11)$ follows from Eq. $(2.14)$.\n\nCase $(11,12)$ follows from Eq. $(2.11)$.\n\nCase $(12,22)$ follows from Eq. $(2.15)$.\n\nCase $(22,22)$ follows from Eq. $(2.18)$.\n\\end{proof}\n\n\\begin{corollary}\\label{0002}\nSuppose $\\mathcal{A}_{12}$ is faithful to $\\mathcal{B}$, $\\mathcal{A}_{21}=\\{0\\}$, and\n$C(\\mathcal{A},\\mathcal{B})=\\mathbb{F}I$. If $\\delta\\in L(\\mathcal{A},\\mathcal{B})$ is Jordan derivable\nat $C\\in \\mathcal A$ then $\\delta $ is a derivation. In particular, every $C\\in \\mathcal A$ is a Jordan all-derivable point\nof $L(\\mathcal A, \\mathcal B)$ and every Jordan derivation is a derivation.\n\\end{corollary}\n\nAs a consequence of Corollary \\ref{0002}, similar to \\cite[Theorem 4.4]{PAN} we have\n\n\\begin{theorem}\\label{0003}\nLet $\\mathcal{L}$ be a subspace lattice on a Banach space $X$ and\n$\\mathcal{A}=alg\\mathcal{L}$. Suppose there exists a non-trivial\nidempotent $P\\in \\mathcal{A}$ such that $ran(P)\\in \\mathcal{L}$ and\n$PB(X)(I-P)\\subseteq \\mathcal{A}$. If $\\delta\\in L(\\mathcal{A},B(X))$ is Jordan derivable\nat $C\\in \\mathcal A$ then $\\delta $ is a derivation. In particular, every $C\\in \\mathcal A$ is a Jordan all-derivable point\nof $L(\\mathcal A, B(X))$.\n\\end{theorem}\n\n\\begin{proof}\nWe will apply Corollary \\ref{0002} with $\\mathcal{B}=B(X)$. Let\n$Q=I-P$. The condition $ran(P)\\in \\mathcal{L}$ implies\n$\\mathcal{A}_{21}=Q\\mathcal{A}P=\\{0\\}$. The condition\n$PB(X)Q\\subseteq \\mathcal{A}$ implies $\\mathcal{A}_{12}=PB(X)Q$ is\nfaithful. To see $C(\\mathcal{A},B(X))=\\mathbb{C}I$, take any $B\\in\nC(\\mathcal{A},B(X))$, from $BP=PB$ we get $PBQ=0$. From $BQ=QB$, we\nhave $QBP=0$. Thus $B=B_{11}+B_{22}$. For any $x\\in ran(P)$ and\n$f\\in X^*$, $x\\otimes f Q\\in \\mathcal{A}_{12}$. It follows from\n$Bx\\otimes f Q=x\\otimes f Q B$ that $B_{11}x\\otimes f Q=x\\otimes f Q\nB_{22}$, which leads to $B_{11}x\\in \\mathbb{C}x$. Since $x\\in\nran(P)$ is arbitrary, it follows $B_{11}=kP$ for some $k\\in\n\\mathbb{C}$. Hence $x\\otimes f(kQ-B_{22})=0$, and we have\n$B_{22}=kQ$ and $B=kI$. Now the conclusion follows from Corollary\n\\ref{0002}.\n\\end{proof}\n\n\nAs an immediate but noteworthy application of Theorem \\ref{0003}, we\nhave the following corollary which generalizes the main result in\n\\cite{ZHUJUN}.\n\n\\begin{corollary}\nLet $\\mathcal{N}$ be a nest on a Banach space $X$ and\n$\\mathcal{A}=alg\\mathcal{N}$. Suppose there exists a non-trivial\nidempotent $P\\in \\mathcal{A} $ such that $ran(P) \\in \\mathcal{N}$.\nIf $\\delta\\in L(\\mathcal{A},B(X))$ is Jordan derivable\nat $C\\in \\mathcal A$ then $\\delta $ is a derivation.\nIn particular, every $C\\in \\mathcal A$ is a Jordan all-derivable point\nof $L(\\mathcal A, B(X))$.\n\\end{corollary}\n\\begin{proof}\nLet $Q=I-P$.  Then $PB(X)Q\\subseteq \\mathcal{A}$. Now applying Theorem \\ref{0003} completes the proof.\n\\end{proof}\nFor an algebra $\\mathcal{A}$ and a left $\\mathcal{A}$-module $\\mathcal{M}$,\n we call a subset $\\mathcal{B}$ of $\\mathcal{A}$ \\emph{separates} $\\mathcal{M}$\n if for every $M\\in \\mathcal{M}$, $\\mathcal{B}M={0}$ implies $M=0$.\nLet $[\\mathcal A_{11}, \\mathcal A_{11}]=\\{A_{11}B_{11}-B_{11}A_{11}: \\ A_{11}, B_{11}\\in \\mathcal A_{11}\\}$.\n\n\\begin{theorem} \\label{0008}\nSuppose $\\mathcal{A}_{12}$ is faithful to $\\mathcal{B}$, $C(\\mathcal{T}_{\\mathcal{A}},\\mathcal{B})=\\mathbb{F}I$, and\n$[\\mathcal A_{11}, \\mathcal A_{11}]$ separates $\\mathcal B_{12}$.  If $\\delta \\in L(\\mathcal A, \\mathcal B)$ is Jordan derivable at some\n$C\\in \\mathcal A_{11}+\\mathcal A_{12}$ then $\\delta $ is a derivation. In particular, every $C\\in \\mathcal A_{11}+\\mathcal A_{12}$\nis a Jordan all-derivable point of $L(\\mathcal A, \\mathcal B).$\n\\end{theorem}\n\\begin{proof} Let $C\\in \\mathcal A_{11}+\\mathcal A_{12}$ and\n$\\delta\\in L(\\mathcal{A},\\mathcal{B})$ be Jordan derivable at $C$.\nSubstracting an inner derivation from $\\delta$ if necessary, we can assume $\\delta$\nsatisfies the properties of Theorem $2.3$.\nLet $Q=I-P$ then $QC=0$.\nBy Theorem $2.3$ and Proposition \\ref{2.4}, $\\delta (\\mathcal A_{11})\\subseteq \\mathcal B_{11}$,\n$\\delta (\\mathcal A_{22})\\subseteq \\mathcal B_{22}$, and $\\delta (\\mathcal A_{12})\\subseteq \\mathcal B_{12}$;\nmoreover, $\\delta (I)=\\delta (P)=\\delta (Q)=0.$\nFor any $t\\in \\mathbb F$, $A_{11}\\in \\mathcal A_{11}$ that is invertible in $\\mathcal A_{11}$ with\n$A_{11}^{-1}\\in \\mathcal A_{11}$, and $A_{21}\\in \\mathcal A_{21}$, clearly $A_{11}(A_{11}^{-1}C+tA_{21})=C$.\nSince $\\delta $ is Jordan derivable at $C$, $\\delta $ differentiates $A_{11}\\circ (A_{11}^{-1}C+tA_{21})$.\nBy Proposition $2.2$,\n\\begin{eqnarray}\n\\delta (A_{11}\\circ A_{21}) = \\delta (A_{11})\\circ A_{21}+A_{11}\\circ \\delta (A_{21})\n\\end{eqnarray}\nMultiplying $Q$ from the right of Eq. $(2.24)$ gives\n\\begin{eqnarray}\nA_{11}\\delta (A_{21})Q = \\delta (A_{21}A_{11})Q\n\\end{eqnarray}\nFor any $U_{11}\\in \\mathcal A_{11}$, by Eq. $(2.25)$ we get\n\\begin{eqnarray*}\nA_{11}U_{11}\\delta (A_{21})Q = \\delta (A_{21}A_{11}U_{11})Q\n\\end{eqnarray*}\nOn the other hand, applying Eq. $(2.25)$ twice gives\n\\begin{eqnarray*}\nU_{11}A_{11}\\delta (A_{21})Q = U_{11}\\delta (A_{21}A_{11})Q = \\delta (A_{21}A_{11}U_{11})Q\n\\end{eqnarray*}\nIt follows $[A_{11}U_{11}-U_{11}A_{11}]\\delta (A_{21})Q=0$.\nSince $[\\mathcal A_{11}, \\mathcal A_{11}]$ separates $\\mathcal B_{12}$, $P\\delta (A_{21})Q=0$.\nMultiplying $Q$ from the left of Eq. $(2.25)$ gives $Q\\delta (A_{21}A_{11})Q = 0$. In particular,\n$Q\\delta (A_{21})Q = 0$.\nMultiplying $P$ from both sides of Eq. $(2.24)$ and setting $A_{11}=P$ leads to\n$P\\delta (A_{21})P = 0$. Thus $\\delta (A_{21})\\in \\mathcal B_{21}$.\n\nTo see $\\delta $ is a derivation, it suffices to show that for any $A_{ij}\\in \\mathcal A_{ij}, A_{kl}\\in \\mathcal A_{kl}$\n$$\\delta(A_{ij}A_{kl})=\\delta(A_{ij})A_{kl}+A_{ij}\\delta(A_{kl})$$\nWe will again label each case as Case $(ij,kl)$. Since $\\delta(A_{ij})\\in\\mathcal{B}_{ij}$, for all $i, j=1, 2,$\nwe only need to check cases for $j=k$. There are 8 cases.\n\nCase $(11,11)$ follows from Eq. $(2.14)$.\n\nCase $(11,12)$ follows from Eq. $(2.11)$.\n\nCase $(12,22)$ follows from Eq. $(2.15)$.\n\nCase $(22,22)$ follows from Eq. $(2.18)$.\n\nCase $(21,11)$ follows from Eq. $(2.24)$. \\\\\nIt remains to show Cases $(12, 21)$, $(21, 12)$, and $(22, 21)$.\n\nFor any $s, t \\in \\mathbb F$, a routine computation shows $(P+sA_{12})[t(A_{21}-sA_{12}A_{21})-sA_{12}+C+Q]=C$.\nSince $\\delta $ is Jordan derivable at $C$, $\\delta $ differentiates\n $(P+sA_{12})\\circ [t(A_{21}-sA_{12}A_{21})-sA_{12}+C+Q].$ By  Proposition $2.2$,\n$\\delta $ differentiates  $(P+sA_{12})\\circ (A_{21}-sA_{12}A_{21})$. Applying   Proposition $2.2$ again , we see\n $\\delta $ differentiates $P\\circ (-A_{12}A_{21})+A_{12}\\circ A_{21}$. Case $(11,11)$ implies $\\delta $  differentiates $P\\circ (-A_{12}A_{21})$,\nIt follows that $\\delta $ differentiates $A_{12}\\circ A_{21},$ i.e.\n\\begin{eqnarray}\n\\delta (A_{12}\\circ A_{21}) = \\delta (A_{12})\\circ A_{21}+A_{12}\\circ \\delta (A_{21})\n\\end{eqnarray}\nMultiplying $P$ from both sides of Eq. $(2.26)$ gives Case $(12, 21)$ and multiplying $Q$ from both sides of Eq. $(2.26)$ gives Case $(21, 12)$.\n\nApplying Case $(21, 12)$, we obtain\n\\begin{eqnarray}\n\\delta (A_{22}A_{21}A_{12})=\\delta (A_{22}A_{21})A_{12}+A_{22}A_{21}\\delta (A_{12})\n\\end{eqnarray}\n\nUsing Cases $(22, 22)$ and $(21, 12)$, we have\n\\begin{eqnarray} & \\delta (A_{22}A_{21}A_{12}) =\\delta (A_{22})A_{21}A_{12}+A_{22}\\delta (A_{21}A_{12})\\nonumber \\\\\n&~~~~~~~~~~= \\delta (A_{22})A_{21}A_{12}+A_{22}\\delta (A_{21})A_{12}+A_{22}A_{21}\\delta (A_{12})\n\\end{eqnarray}\n\nBy $(2.27)$ and $(2.28)$,\n$\\delta (A_{22}A_{21})A_{12} =  \\delta (A_{22})A_{21}A_{12}+A_{22}\\delta (A_{21})A_{12}.$ Since $\\mathcal A_{12}$ is faithful, we get\n$\\delta (A_{22}A_{21})= \\delta (A_{22})A_{21}+A_{22}\\delta (A_{21}),$ completing the proof of Case $(21, 11)$.\n\\end{proof}\n\n\n\\begin{corollary} Suppose $H$ is a Hilbert space and $C\\in B(H)$ such that $\\ker (C)\\neq 0$ or $\\ker (C^{\\ast})\\neq 0.$ If\n$\\delta \\in L(B(H), B(H))$ is Jordan derivable at $C$  then $\\delta $ is a derivation. In particular,\n$C$ is a Jordan all-derivable point of $L(B(H), B(H)).$\n\\end{corollary}\n\\begin{proof}\nIf  $\\ker (C^{\\ast})\\neq 0$, then there exists a proper orthogonal projection $P\\in B(H)$ such that $ran(C)\\subseteq PH$.\nLet $Q=I-P$ then $QC=0$. Take $\\mathcal A=\\mathcal B =B(H)$, the one can check that all hypotheses of\nTheorem $2.9$ are satisfied and the conclusions follow.\n\nIf $\\ker (C)\\neq 0$, we can define $\\delta ^{\\ast}\\in L(B(H), B(H))$\n by $\\delta ^{\\ast}(A)=(\\delta(A^{\\ast}))^{\\ast}$ for every $A\\in B(H)$.\n  Since $\\delta$ is Jordan derivable at $C$, we have $\\delta ^{\\ast}$ is\n  Jordan derivable at $C^{\\ast}$. Now by the argument in the first paragraph\n  we have $\\delta ^{\\ast}$ is a derivation, and in turn $\\delta$ is a derivation.\n   This completes the proof.\n\\end{proof}\n\n\n\\section{Jordan higher derivable mappings}\\\n\nIn this section, we assume that $\\mathcal{A}$ is an algebra over a field\n$\\mathbb{F}$ of characteristic zero. Before stating our main results\nin this section, we first need a proposition that characterizes\nJordan higher derivations in terms of Jordan derivations. Since the\nproof is similar to the proof of \\cite[Theorem 2.5]{Higher}, we omit\nit here.\n\n\\begin{proposition}\\label{0005}\nLet $\\mathcal{A}$ be an algebra, $\\{d_i\\}_{i\\in\\mathbb{N}}$ be a\nsequence of mappings on $\\mathcal{A}$ with $d_0=I_\\mathcal{A}$ and\n$\\{\\delta_i\\}_{i\\in\\mathbb{N}}$ be the a sequences of (Jordan)\nderivations on $\\mathcal{A}$ with $\\delta_0=0$. If the following\nrecursive relation holds: \\begin{eqnarray*}\nnd_n=\\sum_{k=0}^{n-1}\\delta_{k+1}d_{n-1-k}\n\\end{eqnarray*}\nfor $n\\geq1$, then $\\{d_i\\}_{i\\in\\mathbb{N}}$ is a (Jordan) higher derivation.\n\\end{proposition}\n\nLet $\\mathcal R(\\mathcal A)$ be a relation on $\\mathcal A$, i.e. $\\mathcal R(\\mathcal A)$ is a\nnonempty subset of $\\mathcal A \\times \\mathcal A$. We say $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ is\n\\textit{derivable on} $\\mathcal R(\\mathcal A)$ if $\\delta(AB)=\\delta(A)B+A\\delta(B)$ for all\n$(A,B)\\in \\mathcal R(\\mathcal{A})$. We say $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ is\n\\textit{Jordan derivable on} $\\mathcal R(\\mathcal A)$ if $\\delta(AB+BA)=\\delta(A)B+A\\delta(B)+\\delta(B)A+B\\delta(A)$ for all\n$(A,B)\\in \\mathcal R(\\mathcal{A})$. A sequence of mappings\n$\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is called\n\\textit{higher derivable on} $\\mathcal R(\\mathcal A)$  if\n$d_{n}(AB)=\\sum_{i+j=n}d_i(A)d_j(B)$ for all\n$(A,B)\\in \\mathcal R(\\mathcal{A})$. A sequence of mappings\n$\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is called\n\\textit{Jordan higher derivable on} $\\mathcal R(\\mathcal A)$  if\n$d_{n}(AB+BA)=\\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all\n$(A,B)\\in \\mathcal R(\\mathcal{A})$. We say $\\mathcal R(\\mathcal A)$ is \\textit{(Jordan) derivational}\nfor $L(\\mathcal{A},\\mathcal{M})$ if every (Jordan) derivable\nmapping on $\\mathcal R(\\mathcal A)$ is a (Jordan) derivation.\nWe say $\\mathcal R(\\mathcal A)$ is \\textit{(Jordan) higher derivational}\nfor $L(\\mathcal{A})$ if every (Jordan) higher derivable\nmapping on $\\mathcal R(\\mathcal A)$ is a (Jordan) higher derivation.\n\n\\begin{remark}\n\\emph{The above definitions allow us to unify some\nof the notions in the literature. For example,\nin literature, there are two definitions of Jordan derivable\nmappings, one is what we use in this paper (see for example \\cite{chen, ZHUJUN} and\nreferences therein), and the other\n(see for example \\cite{Hou, Hou3, ZHUJUN1}) is what we call S-Jordan\nderivable mappings (S stands for standard). A mapping $\\delta\\in\nL(\\mathcal{A},\\mathcal{M})$ is called a \\textit{S-Jordan derivable\nmapping at $C\\in \\mathcal{A}$} if\n$\\delta(AB+BA)=\\delta(A)B+A\\delta(B)+\\delta(B)A+B\\delta(A)$ for all\n$A,B\\in \\mathcal{A}$ with $AB+BA=C$. An element $C\\in \\mathcal{A}$\nis called a \\textit{S-Jordan all-derivable point} if every S-Jordan\nderivable mapping at $C$ is a Jordan derivation. A sequence of\nmappings $\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is\ncalled \\textit{S-Jordan higher derivable at $C\\in \\mathcal{A}$} if\n$d_{n}(AB+BA)=\\sum_{i+j=n}(d_i(A)d_j(B)+d_i(B)d_j(A))$ for all\n$A,B\\in \\mathcal{A}$ with $AB+BA=C$. An element $C\\in \\mathcal{A}$\nis called a \\textit{S-Jordan higher all-derivable point} if every\nsequence of S-Jordan higher derivable mappings at $C$ is a Jordan\nhigher derivation. The above two notions of Jordan derivable mappings at $C$ are special case of\nJordan derivable mappings on $\\mathcal R(\\mathcal A)$,\nwhere $\\mathcal R(\\mathcal A)=\\{(A, B)\\in \\mathcal A \\times \\mathcal A: AB=C\\}$ and\n$\\mathcal R(\\mathcal A)=\\{(A, B)\\in \\mathcal A \\times \\mathcal A: AB+BA=C\\}$, respectively.}\n\\end{remark}\n\n\\begin{theorem} \\label{0006}\nIf $\\mathcal{A}$ is an algebra such that $\\mathcal R(\\mathcal A)$ is (Jordan) derivational for $L(\\mathcal{A})$,\nthen $\\mathcal R(\\mathcal A)$ is  (Jordan) higher derivational.\n\\end{theorem}\n\n\\begin{proof}\nFirst, suppose $\\mathcal R(\\mathcal A)$ is Jordan derivational and\n$\\{d_i\\}_{i\\in\\mathbb{N}}$ is a sequence of mappings in $L(\\mathcal{A})$\nJordan higher derivable on $\\mathcal R(\\mathcal A)$. Let $\\delta_1=d_1$ and\n$\\delta_n=nd_n-\\sum_{k=0}^{n-2}\\delta_{k+1}d_{n-1-k}$ for every\n$n(\\geq2)\\in \\mathbb{N}.$ We will show $\\{\\delta_i\\}_{i\\in\\mathbb{N}}$ is a sequence\nof Jordan derivations, and in turn $\\{d_i\\}_{i\\in\\mathbb{N}}$ is a Jordan higher\nderivation by Proposition \\ref{0005}. We prove by induction.\n\nWhen $n=1$, since $\\mathcal R(\\mathcal A)$ is Jordan derivational, we have that $\\delta_1$ is a Jordan derivation.\n\nNow suppose $\\delta_k$ is defined as above and is a Jordan\nderivation for $k\\leq n$. For $(A,B)\\in \\mathcal R(\\mathcal{A})$, we\nhave\n\\begin{eqnarray*}\n&&\\delta_{n+1}(A\\circ B)=(n+1)d_{n+1}(A\\circ B)-\\sum_{k=0}^{n-1}\\delta_{k+1}d_{n-k}(A\\circ B)\\nonumber\\\\\n&&~~~~~~~~~~~~~~~=(n+1)\\sum_{k=0}^{n+1}\\{d_k(A)\\circ d_{n+1-k}(B)\\}\n-\\sum_{k=0}^{n-1}\\delta_{k+1}\\sum_{l=0}^{n-k}\\{d_l(A)\\circ d_{n-k-l}(B)\\}.\\nonumber\\\\\n\\end{eqnarray*}\nBy induction we have\n\\begin{eqnarray*}\n&&\\delta_{n+1}(A\\circ B)\n=\\sum_{k=0}^{n+1}kd_k(A)\\circ d_{n+1-k}(B)+\\sum_{k=0}^{n+1}d_k(A)\\circ (n+1-k)d_{n+1-k}(B)\\nonumber\\\\\n&&~~~~~~~~~~~~~-\\sum_{k=0}^{n-1}\\sum_{l=0}^{n-k}\\{\\delta_{k+1}(d_l(A))\\circ d_{n-k-l}(B)+d_l(A)\\circ \\delta_{k+1}(d_{n-k-l}(B))\\}.\\nonumber\\\\\n\\end{eqnarray*}\nSet\n\\begin{eqnarray*}\n&&K_1=\\sum_{k=0}^{n+1}kd_k(A)\\circ d_{n+1-k}(B)-\\sum_{k=0}^{n-1}\\sum_{l=0}^{n-k}\\delta_{k+1}(d_l(A))\\circ d_{n-k-l}(B),\\\\\n&&K_2=\\sum_{k=0}^{n+1}d_k(A)\\circ (n+1-k)d_{n+1-k}(B)-\\sum_{k=0}^{n-1}\\sum_{l=0}^{n-k}d_l(A)\\circ \\delta_{k+1}(d_{n-k-l}(B)).\\\\\n\\end{eqnarray*}\nThen $\\delta_{n+1}(A\\circ B)=K_1 +K_2$.\nLet us compute $K_1$ and $K_2$. If we put $r=k+l$ in the summation $\\sum_{k=0}^{n-1}\\sum_{l=0}^{n-k}$, then we may write it as $\\sum_{r=0}^{n}\\sum_{0\\leq k\\leq r, k\\neq n}$. Hence\n\\begin{eqnarray*}\nK_1=\\sum_{k=0}^{n+1}kd_k(A)\\circ d_{n+1-k}(B)-\\sum_{r=0}^{n}\\sum_{0\\leq k\\leq r, k\\neq n}\\delta_{k+1}(d_{r-k}(A))\\circ d_{n-r}(B).\n\\end{eqnarray*}\nPutting $r+1$ instead of $k$ in the first summation, we have\n\\begin{eqnarray*}\n&&K_1+\\sum_{k=0}^{ n-1}\\delta_{k+1}(d_{n-k}(A))\\circ B\\\\\n&&~~~=\\sum_{r=0}^{n}(r+1)d_{r+1}(A)\\circ d_{n-r}(B)-\\sum_{r=0}^{n-1}\\sum_{k=0}^r\\delta_{k+1}(d_{r-k}(A))\\circ d_{n-r}(B)\\\\\n&&~~~=\\sum_{r=0}^{n-1}\\{(r+1)d_{r+1}(A)-\\sum_{k=0}^r\\delta_{k+1}(d_{r-k}(A))\\}\\circ d_{n-r}(B)+(n+1)d_{n+1}(A)\\circ B.\n\\end{eqnarray*}\nBy our assumption $(r+1)d_{r+1}(A)=\\sum_{k=0}^r\\delta_{k+1}(d_{r-k}(A))$ for $r=0, \\ldots, n-1,$ we obtain\n\\begin{eqnarray*}\n&&K_1=(n+1)d_{n+1}(A)\\circ B-\\sum_{k=0}^{ n-1}\\delta_{k+1}(d_{n-k}(A))\\circ B=\\delta_{n+1}(A)\\circ B.\\\\\n\\end{eqnarray*}\nSimilary, we may deduce that\n\\begin{eqnarray*}\n&&K_2=(n+1)A\\circ d_{n+1}(B)-\\sum_{k=0}^{ n-1}A\\circ \\delta_{k+1}(d_{n-k}(B))=A\\circ \\delta_{n+1}(B).\\\\\n\\end{eqnarray*}\nTherefore, $\\delta_{n+1}$ is Jordan derivable on $\\mathcal R(\\mathcal A)$.\nSince $\\mathcal R(\\mathcal A)$ is Jordan derivational, we have that\n$\\delta_{n+1}$ is a Jordan derivation.\n\nSimilarly, we can prove the case when $\\mathcal R(\\mathcal A)$ is assumed to be derivational by changing ``$\\ \\circ $\" to\nthe normal multiplication of $\\mathcal A$.\n\\end{proof}\n\nRecall that a mapping $\\delta\\in L(\\mathcal{A},\\mathcal{M})$ is called \\textit{derivable at $C\\in \\mathcal{A}$}\nif $\\delta(AB)=\\delta(A)B+A\\delta(B)$ for all $A,B\\in \\mathcal{A}$ with $AB=C$. An element $C\\in \\mathcal{A}$ is called an \\textit{all-derivable point} if every derivable mapping at $C$ is a derivation. A sequence of mappings $\\{d_i\\}_{i\\in\\mathbb{N}}\\in L(\\mathcal{A})$ with $d_0=I_\\mathcal{A}$ is called \\textit{higher derivable at $C\\in \\mathcal{A}$} if $d_{n}(AB)=\\sum_{i+j=n}d_i(A)d_j(B)$ for all $A,B\\in \\mathcal{A}$ with $AB=C$. An element $C\\in \\mathcal{A}$ is called a \\textit{higher all-derivable point} if every sequence of higher derivable mappings at $C$ is a higher derivation.\n\n\\begin{remark}\n\\emph{Several authors (see for example \\cite{Hou5,Hou6, Hou4,\nLI7,LI5,LU1, Hou2,Zhu2,Zhu3,Zhu4,ZHUJUN2})  investigate derivable\nmappings at 0, invertible element, left (right) separating point,\nnon-trivial idempotent, and the unit $I$ on certain algebras. By\nTheorem \\ref{0006}, we can generalize these results to the higher\nderivation case. Many authors also study (S-)Jordan derivable\nmappings (see for example \\cite{ Hou, chen,Hou3,ZHUJUN1,ZHUJUN}) at\nthese points. Theorems \\ref{0006}\nalso allow us to generalize these results to the (S-)Jordan higher derivation\ncase. }\n\\end{remark}\n\n\nCombining Theorem \\ref{0006} with Corollary \\ref{0002}, we have\n\n\\begin{corollary}\\label{0007}\nSuppose $\\mathcal{A}$, $\\mathcal{B}$ are as in Corollary \\ref{0002} with $\\mathcal{B}=\\mathcal{A}$.\nThen every $C\\in \\mathcal{A}$\nis a Jordan higher all-derivable point.\n\\end{corollary}\n\nCombining Theorem \\ref{0006} with Theorem \\ref{0003}, we have\n\n\\begin{corollary}\nLet $\\mathcal{L}$ be a subspace lattice on a Banach space $X$ and\n$\\mathcal{A}=alg\\mathcal{L}$. If there exists a non-trivial\nidempotent $P\\in \\mathcal{A}$ such that $ran(P)\\in \\mathcal{L}$ and\n$PB(X)(I-P)\\subseteq \\mathcal{A}$, then every $C \\in \\mathcal{A}$ is\na Jordan higher all-derivable point.\n\\end{corollary}\n\n\nCombining Theorem \\ref{0006} with \\cite[Theorem 3.3]{PAN}, we have\n\n\\begin{corollary}\nSuppose $\\mathcal{A}$, $\\mathcal{B}$ are as in Corollary \\ref{0002} with $\\mathcal{B}=\\mathcal{A}$. Then every $0\\neq C\\in \\mathcal{A}$ is a higher all-derivable point .\n\\end{corollary}\nWe say that $W$ in an algebra $\\mathcal{A}$ is a left (or right) separating point of $\\mathcal{A}$ if $WA=0$ (or $AW=0$) for $A \\in \\mathcal{A}$ implies $A=0.$ In \\cite[Remark 1]{LI2}, the authors point out that if every Jordan derivation on a unital Banach algebra $\\mathcal{A}$ is a derivation, then every linear mapping on $\\mathcal{A}$ which is derivable at an arbitrary left or right separating point of $\\mathcal{A}$ is a derivation. Together with Theorem \\ref{0006}, we may generalize this result to the higher derivation case.\n\n\\begin{theorem}\nLet $\\mathcal{A}$ be a unital Banach algebra such that every\nJordan derivation on $\\mathcal{A}$ is a derivation. Suppose that $W$\nin $\\mathcal{A}$ is a left or right separating point. If\n$D=(d_i)_{i\\in\\mathbb{N}}$ is a family of linear mappings higher\nderivable at $W$, then $D=(d_i)_{i\\in\\mathbb{N}}$ is a higher\nderivation.\n\\end{theorem}\n\n\nFor any non-zero vectors $x\\in X$ and $f\\in X^*$, the rank one\noperator $x\\otimes f$ is defined by $x\\otimes f(y)=f(y)x$ for $y\\in\nX$.\n\n\\begin{lemma}\\label{3002}\nIf $\\mathcal{A}$ is a norm-closed subalgebra of $B(X)$\nsuch that  $\\vee \\{x: x\\otimes f\\in \\mathcal A\\}=X$ and\n$\\wedge \\{\\emph{ker}(f): x\\otimes f\\in \\mathcal A\\}=(0)$, then every derivation $\\delta$ from\n$\\mathcal{A}$ into $B(X)$ is bounded.\n\\end{lemma}\n\\begin{proof}\nBy the closed graph theorem, it is sufficient to show if $A_n\\rightarrow A$ and $\\delta(A_n)\\rightarrow B$,\nas $n\\rightarrow \\infty$, then $\\delta(A)= B$.\n\nFor any $x\\otimes f, \\ y\\otimes g \\in \\mathcal{A}$,\nsince\n\\begin{align*}\n\\delta(x\\otimes f A_n y\\otimes g)&=f(A_ny)\\delta(x\\otimes g)\\\\\n&=x\\otimes f\\delta(A_n y\\otimes g)+\\delta(x\\otimes f)(A_n y\\otimes g)\\\\\n&=x\\otimes f(\\delta(A_n)y\\otimes g+A_n \\delta(y\\otimes g))+\\delta(x\\otimes f)(A_n y\\otimes g),\\\\\n\\end{align*}\nwe have\n\\begin{eqnarray}\n(x\\otimes f)\\delta(A_n)(y\\otimes g)=f(A_ny)\\delta(x\\otimes g)-(x\\otimes f) A_n \\delta(y\\otimes g)-\\delta(x\\otimes f)(A_n y\\otimes g).\\label{3001}\n\\end{eqnarray}\nTaking limit in (\\ref{3001}) yields\n\\begin{eqnarray*}\n(x\\otimes f) B (y\\otimes g)=(x\\otimes f)\\delta(A)(y\\otimes g).\n\\end{eqnarray*}\nHence $f(By)=f(\\delta(A))$. Thus $\\delta(A)= B$.\n\\end{proof}\n\nBy \\cite{higher2}, if $\\{d_i\\}_{i\\in\\mathbb{N}}$ is a Jordan higher derivation on an\nalgebra $\\mathcal{A}$, then there is a\nsequence $\\{\\delta_i\\}_{i\\in\\mathbb{N}}$ of Jordan derivations on $\\mathcal{A}$ such that\n\\begin{eqnarray*}\nd_n=\\sum_{i=1}^n\\left(\\sum_{\\sum_{j=1}^i r_j=n}\\left(\\prod_{j=1} ^ i \\frac{1}{r_j+\\cdots r_i} \\right)\\delta_{r_1}\\ldots\\delta_{r_i} \\right),\n\\end{eqnarray*}\nwhere the inner summation is taken over all positive integers $r_j$\nwith $\\sum_{j=1}^i r_j=n$. This together with Lemma \\ref{3002}\nleads to the following Theorem.\n\n\\begin{theorem}\\label{3003}\nIf $\\mathcal{A}$ is a norm-closed subalgebra of $B(X)$\nsuch that  $\\vee \\{x: x\\otimes f\\in \\mathcal A\\}=X$ and\n$\\wedge \\{\\emph{ker}(f): x\\otimes f\\in \\mathcal A\\}=(0)$, then every Jordan\nhigher derivation on $alg\\mathcal{L}$ is bounded.\n\\end{theorem}\n\\begin{proof}\nSince every Jordan derivation on $\\mathcal{A}$ is a derivation by \\cite[Theorem 4.1]{LI7}.\n\\end{proof}\n\nFor a subspace lattice $\\mathcal{L}$ of a Banach space $X$ and for $E\\in \\mathcal{L}$, define\n$$E_-=\\vee\\{F\\in \\mathcal{L}: F\\nsupseteq E\\}.$$\nPut $$\\mathcal{J}(\\mathcal{L})=\\{K\\in \\mathcal{L}: K\\neq(0)~\\mathrm{and}~K_-\\neq X\\}.$$\n\n\n\n\\begin{remark}\n\\emph{ It is well known (see \\cite{ORO}) that $x\\otimes f\\in\n\\textrm{alg}\\mathcal{L}$ if and only if there exists some $K\\in\n\\mathcal{J}(\\mathcal{L})$ such that $x\\in K$ and $f\\in K_-^\\perp$.\nIt follows that if a subspace lattice $ \\mathcal{L}$ satisfies\n$\\vee\\{K: K\\in \\mathcal{J}(\\mathcal{L})\\}=H$ and\n$\\wedge\\{K_-: K\\in \\mathcal{J}(\\mathcal{L})\\}=(0)$, then  $alg\\mathcal{L}$\nsatisfies the hypothesis of  Theorem \\ref{3003}. Such subspace lattices include\ncompletely distributive subspace lattices, $\\mathcal{J}$-subspace lattices, and\nsubspace lattices with $H_-\\neq H$ and $(0)_+\\neq (0)$.\nRecall that (see \\cite{SRL}), a subspace lattice $\\mathcal{L}$ is called\n\\textit{completely~distributive} if $L=\\vee\\{E\\in \\mathcal{L}:\nE_-\\ngeq L\\}$ and $L=\\wedge\\{E_-: E\\in\n\\mathcal{L}~\\mathrm{and}~E\\nleq L\\}$  for all $L\\in \\mathcal{L}$. It\nfollows that completely distributive subspace lattices satisfy the\nconditions $\\vee\\{K: K\\in \\mathcal{J}(\\mathcal{L})\\}=H$ and\n$\\wedge\\{K_-: K\\in \\mathcal{J}(\\mathcal{L})\\}=(0)$. A subspace lattice $\\mathcal{L}$ is called a\n\\textit{$\\mathcal{J}$-subspace lattice} on $H$ if $\\vee\\{K: K\\in\n\\mathcal{J}(\\mathcal{L})\\}=H$, $\\wedge\\{K_-: K\\in\n\\mathcal{J}(\\mathcal{L})\\}=(0)$, $K\\vee K_-=H$ and $K\\wedge\nK_-=(0)~\\mathrm{for~any}~K\\in \\mathcal{J}(\\mathcal{L})$.}\n\\end{remark}\n\n\n\n\nOn October 22, 2011, in the Third Operator Theory and Operator\nAlgebras Conference of China, the first author reported main results\nof the paper.\n\n\\section*{Acknowledgement}\nThis work is supported by NSF of China and the Ruth and Ted Braun\nFellowship from the Saginaw Community Foundation.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction} \n\nThe direct detection of gravitational waves (GW) from black hole and neutron star mergers \n\\cite{Abbott:2016blz,Abbott:2016nmj,TheLIGOScientific:2017qsa}, together with the observation of an  electromagnetic counterpart \\cite{Monitor:2017mdv}, have opened a new window for observational cosmology. Interestingly, these individual two-body sources are just one of the many signals that could be detected by present and future experiments.\nA stochastic background of GWs could be produced, for instance, during  inflation and (re)heating \\cite{Starobinsky:1979ty,Rubakov:1982df,GarciaBellido:2007af} or be associated with exotic post-inflationary physics such as phase transitions or topological defects \\cite{Grojean:2006bp,Weir:2017wfa,Vilenkin:2000jqa,Copeland:2009ga,Dufaux:2010cf,Figueroa:2012kw}.\nThe discovery of any of these backgrounds would provide an extremely valuable piece of information on energy scales much beyond the reach of any particle physics accelerator.\n\nTopological defects are created whenever there is a spontaneous symmetry breaking, usually associated with some critical temperature. A temporary network of these objects is also  expected in the presence of non-minimal couplings to gravity, provided the existence of a kinetic dominated  era with stiff  equation-of-state parameter $w>1\/3$  \\cite{Bettoni:2018utf}.\nIn this paper, we argue that a stochastic background of GWs could be produced by a temporary network of cosmic string appearing in quintessential inflationary models with non-oscillatory runaway potentials. The main ingredient of our proposal is a \\textit{subdominant} non-minimally coupled $U(1)$ scalar field on top of the inflaton and matter sectors.\nThe non-minimal coupling to gravity renders heavy the $U(1)$ field during inflation, preventing the generation of dangerous isocurvature perturbations \\cite{Bettoni:2018utf}. \n\nAfter inflation, where the violation of the slow-roll conditions together with the absence of a minimum for the inflaton field leads to the onset of a kinetic-dominated era, the Ricci scalar turns negative and with it the effective mass of the $U(1)$ field. The symmetry becomes then spontaneously broken and the field develops a new vacuum at large field values, leading to the formation of a cosmic string network by the standard Kibble mechanism \\cite{Copeland:2009ga}.\nThe oscillations of the strings within the Hubble volume generate a GWs spectrum with an energy density that becomes cumulatively enhanced with respect to the rapidly decreasing energy density of the kinetic-dominated background. The production of GWs continues till the onset of the hot big bang era, where the Ricci scalar vanishes and the negative mass term enforcing the symmetry breaking effectively disappears. When that happens, the initial symmetry is restored and the origin becomes again the absolute minimum of the potential. The complex $U(1)$ field starts spiraling around it, destroying the cosmic string network and halting the GW production. Since the spectrum of GWs is frozen during the long-lasting radiation-dominated era, the final energy density in GWs remains essentially unchanged up to the present cosmological epoch.\n\nThis paper is organized as follows. In Section \\ref{sec:Q-inflation} we recapitulate the most important aspects of quintessential inflation. The dynamics of a subdominant non-minimally coupled $U(1)$ field in this cosmological background is described in Section \\ref{sec:spectator}. Section \n\\ref{sec:GW_production} discuss the formation of cosmic strings and the salient features of the associated GW spectrum. Section \\ref{sec:GW_today} is devoted to the computation of the present day GW spectrum and to the discussion of the constraints that present and future experiments could cast on the model parameters and early universe physics. Finally, our conclusions are drawn in Section \\ref{sec:conclusions}.\n\n\\section{Quintessential inflation}\\label{sec:Q-inflation}\n\nInflation and dark energy are usually understood as two independent epochs  in the history of the Universe. Note, however, that there is no fundamental reason for this to be the case. The common features of these cosmological periods could be related, for instance, to some underlying principle, such as the explicit \\cite{GarciaBellido:2011de,Karananas:2016kyt,Casas:2017wjh} or emergent realization of scale invariance in the vicinity of non-trivial fixed points \\cite{Wetterich:1987fm,Wetterich:1994bg, Wetterich:2014gaa,Rubio:2017gty}. A framework that fits well with the last possibility is quintessential inflation \\cite{Peebles:1998qn,Spokoiny:1993kt,Brax:2005uf,Hossain:2014xha,Agarwal:2017wxo,Geng:2017mic,Dimopoulos:2017zvq,Rubio:2017gty}. In its simplest terms, this paradigm makes use of a single degree of freedom --- dubbed \\textit{cosmon} \\cite{Peccei:1987mm} --- to provide a unified description of inflation and dark energy. Although certain parametrizations involving a non-canonical cosmon field are certainly preferred to highlight the aforementioned connection to scale symmetry \\cite{Wetterich:1987fm,Wetterich:1994bg, Wetterich:2014gaa,Rubio:2017gty}, we will follow here the standard approach and describe quintessential inflation in terms of a canonically normalized cosmon field $\\phi$ with Lagrangian density \n\\begin{equation}\\label{eq:cosmon}\n\\frac{\\cal L}{\\sqrt{-g}}=\\frac{M_{\\rm P}^2}{2}R -\\frac{1}{2}\\partial_\\mu \\phi \\partial^\\mu \\phi -U(\\phi)\\,,\n\\end{equation}\nwhere $M_{\\rm P}=(8\\pi G)^{-1\/2}$ is the reduced Planck mass and $R$ is the Ricci scalar.\nA particular model within the paradigm is specified by a choice of the potential $U(\\phi)$, which is required to be of the \\textit{runaway form} in order to support inflation and dark energy. Several quintessential potentials have been proposed in the literature \\cite{Wetterich:1987fm,Wetterich:1994bg,Wetterich:2013jsa,Wetterich:2014gaa,Hossain:2014xha,Rubio:2017gty,Wang:2018kly}, each of them leading to slightly different predictions for the inflationary and dark energy observables. Despite numerical dissimilarities, the background evolution of the Universe is rather insensitive to the precise form of $U(\\phi)$ and involves always the same sequence of cosmological epochs. \nAt early times, the scalar field is generically displaced at large field values, allowing for inflation with the standard chaotic initial conditions. The inflationary epoch will end, as usual, when the kinetic energy density of the cosmon starts to dominate over the potential counterpart, signaling the break-down of the slow-roll approximation.  In the absence of a potential minimum, the Universe will inevitably enter a \\textit{kination} or \\textit{deflation} era with effective equation-of-state parameter $w=1$ \\cite{Spokoiny:1993kt}. The duration of this unusual cosmological epoch depends on the efficiency of the heating process.\n\nA plethora of heating mechanisms within quintessential inflation have been proposed in the literature \\cite{Ford:1986sy,Damour:1995pd,Peebles:1998qn,Felder:1999pv,Feng:2002nb,BuenoSanchez:2007jxm,Rubio:2017gty,Dimopoulos:2018wfg,Nakama:2018gll}. All these mechanisms share a common feature: the decay of the inflaton field does not need to be complete. Even if the initial particle creation is small, the rapid decrease of the cosmon energy density during kinetic domination ($\\rho_\\phi\\sim a^{-6}$) as compared to the redshift of the created plasma $(\\rho_{\\rm R}\\sim a^{-4})$ will eventually lead to radiation domination. A useful way of parametrizing the duration of this transition period is to make use of a \\textit{heating efficiency} parameter \\cite{Rubio:2017gty}\n\\begin{equation}\\label{thetadef}\n\\Theta\\equiv \\frac{\\rho_{\\rm R}^{\\rm kin}}{\\rho^{\\rm kin}_\\phi}=\\left(\\frac{a_{\\rm kin}}{a_{\\rm rad}}\\right)^2\\,,\n\\end{equation}\nwhere $\\rho^{\\rm kin}_\\phi$ and $\\rho_{\\rm R}^{\\rm kin}$ stand for the energy density of the cosmon and the heating products at the beginning of kinetic domination and $a_{\\rm kin}$ and $a_{\\rm rad}$ denote respectively the values of the scale factor at the onset of kinetic and radiation domination. For a fiducial Hubble rate $H_{\\rm kin }\\sim 10^{11}\\,{\\rm GeV}$,\nthe typical heating efficiencies per degree of freedom vary between $\\Theta\\sim 10^{-19}$ in gravitational heating scenarios \\cite{Ford:1986sy,Damour:1995pd,Peebles:1998qn} and $\\Theta\\sim {\\cal O}(1)$ in heating scenarios involving matter fields \\cite{Felder:1999pv,Rubio:2017gty}.\nThe minimal value of $\\Theta$ is restricted by big bang nucleosynthesis, where the standard hot big bang history should be definitely recovered. In particular, quintessential inflation generates a \\textit{primordial} gravitational wave background that becomes blue tilted during kinetic domination \\cite{Sahni:1990tx,Rubio:2017gty}, \n\\begin{equation}\n\\Omega_{\\rm GW}(k)\\sim k \\,,\n\\end{equation}\nand may excessively contribute to the effective number of relativistic degrees of freedom at big bang nucleosynthesis, modifying with it the light elements' abundance. The (integrated) nucleosynthesis constraint on the GW density fraction \\cite{Maggiore:1999vm,Caprini:2018mtu},\n\\begin{equation}\\label{GWbound1}\nh^2\\int_{k_{\\rm BBN}}^{k_{\\rm end}}\\Omega_{\\rm GW}(k)\\, d\\ln k \\lesssim 1.12\\times 10^{-6}\\,, \n\\end{equation}\nwith $h=0.678$ and $k_{\\rm end}$ and $k_{\\rm BBN}$ the momenta associated to the horizon scale at the end  of inflation and at big bang nucleosynthesis, \ntranslates into a lower bound on the heating efficiency \\cite{Rubio:2017gty}\n\\begin{eqnarray}\\label{GWbound2}\n&&\\Theta\\gtrsim          10^{-16}\\left(\\frac{H_{\\rm kin}}{10^{11}\\,{\\rm GeV}}\\right)^2\\,,\n\\end{eqnarray}\nwhich should be compared with the efficiency of a given heating scenario.\n\nBeyond the onset of radiation domination, the background follows the usual hot big bang evolution ~\\cite{Rubio:2017gty}. \nAt the early stages of radiation domination, the cosmon field  \\textit{freezes} to a constant value, allowing for the resurgence of its potential \nenergy density. When this component re-approaches the decreasing energy density of the radiation fluid, the evolution of the system settles down to a scaling solution in which the scalar energy density \\textit{tracks} the background evolution \\cite{Wetterich:2007kr,Amendola:2018ltt}. Eventually the cosmon will exit the tracking regime, leading to a dark energy dominated era ~\\cite{Wetterich:2007kr,Amendola:2007yx}.  \n\n\\section{Non-minimally coupled spectator field dynamics}\\label{sec:spectator}\n\nLet us study the dynamics of a \\textit{subdominant} non-minimally coupled $U(1)$ scalar field in the quintessential inflation scenario discussed in the previous section. To this end, we  consider a Lagrangian density\n\\begin{equation}\\label{LBL}\n\\frac{\\mathcal{L}_\\chi}{\\sqrt{-g}} = -\\partial_\\mu\\chi^\\dagger \\partial^\\mu\\chi - \\left(m_\\chi^2+\\xi R\\right)\\chi^\\dagger \\chi - \\lambda \\frac{(\\chi^\\dagger \\chi)^{\\frac{n}{2}}}{\\Lambda^{n-4}}\\,,\n\\end{equation}\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{sketch_paper.pdf}\n\\caption{Schematic behaviour of the $U(1)$ potential and of the $\\chi$ field evolution\nduring the different cosmological epochs in quintessential inflationary scenarios. The horizontal lines correspond to the zero value of the corresponding quantity.}\n\\label{fig:sketch}\n\\end{center}\n\\end{figure}\nwith $m_\\chi^2$ a bare mass parameter, $\\xi$ is a positive dimensionless parameter, $\\lambda$ a positive dimensionless coupling, $n\\geq 4$ and  $\\Lambda$ a cutoff scale (\\textit{a priori} unknown). The non-minimal coupling to gravity is motivated by quantum field computations in curved spacetime \\cite{Birrell:1982ix} and its associated phenomenology has been widely studied in the literature, with applications in baryogenesis \\cite{Bettoni:2018utf}, (re)heating \\cite{Dimopoulos:2018wfg} and dark matter production \\cite{Fairbairn:2018bsw,Alonso-Alvarez:2018tus}. \n\nRather than specifying a particular quintessential inflation potential in Eq.~\\eqref{eq:cosmon}, we will follow here a model-independent approach and describe the background evolution of the Universe in  terms  of  the Ricci scalar behaviour at the different cosmological epochs. For a flat Friedmann-Lemaitre-Robertson-Walker \\textit{background} metric $g_{\\mu\\nu}={\\rm diag}(-1,a^2(t)\\,\\delta_{ij})$, the Ricci scalar in Eq.~\\eqref{LBL} can be written as \n\\begin{equation}\nR=3(1-3w) H^2\\,,\n \\end{equation}\nwith $H=\\dot a\/a$ the Hubble rate and $w$ the equation-of-state of  the dominant energy component. \n\nThe evolution of the $U(1)$ potential is summarized in Fig. \\ref{fig:sketch}. During the inflationary stage, the parameter $w$ approaches the de Sitter value $w\\simeq -1$ and the Ricci scalar is constant and positive,\n\\begin{equation}\\label{Rinf}\nR=12 H^2\\,.\n\\end{equation}\nThe mass term for the $\\chi$ field in Eq.~\\eqref{LBL} is then also positive definite and the ground state of the system is located at the origin of the effective potential, $ \\vert\\chi_{\\rm min}\\vert=0$. This implies, not only the $U(1)$ symmetry is preserved, but also that the $U(1)$ field is heavy during inflation and no sizable isocurvature perturbations are generated \\cite{Bettoni:2018utf}.\n\nAs soon as the slow-roll conditions are violated, the system enters a deflation period with stiff equation-of-state parameter $w=1$. During this era, the Ricci scalar flips sign \n\\begin{equation}\\label{Rkin}\nR=-6 H^2\\,.\n\\end{equation}\nIf the associated Hubble-induced mass in Eq.~\\eqref{LBL} exceeds the bare mass of the $\\chi$ field at that time, the $U(1)$ symmetry becomes spontaneously broken.  \nWhen that happens, the field starts rolling down from the origin to the new minimum of the potential, which is time-dependent and located at\n\\begin{equation}\\label{eq:chimint}\n\\vert \\chi_{\\rm min}(a)\\vert = \\vert \\chi_{\\rm min}(a_{\\rm kin})\\vert \\left(\\frac{a}{a_{\\rm kin}}\\right)^{-\\frac{6}{n-2}}\\,, \\hspace{10mm} \\vert \\chi_{\\rm min}(a_{\\rm kin})\\vert \\simeq       \\left[\\frac{12\\,\\xi}{n\\lambda }\\left(\\frac{H_{\\rm kin}}{M_{P}}\\right)^2\\left(\\frac{\\Lambda}{M_{P}}\\right)^{n-4}\\right]^{\\frac{1}{n-2}}M_{P}\\,.\n\\end{equation}\n During the first stages of kinetic domination, the higher dimensional operators in  Eq.~\\eqref{LBL} can be safely neglected. In this case, the evolution of the field in its way to the time-dependent minimum \\eqref{eq:chimint} can be well-described by the approximate solution \\cite{Dimopoulos:2018wfg,Bettoni:2018utf}\n\\begin{equation}\\label{eq:chi_roll}\n\\vert \\chi (a) \\vert \\simeq \\vert \\chi (a_{\\rm kin})\\vert  \\left(\\frac{a}{a_{\\rm kin}}\\right)^{\\sqrt{6\\xi}} \\,,\\hspace{15mm} \\vert \\chi (a_{\\rm kin})\\vert  \\equiv \\frac{1+\\sqrt{6\\xi}}{2\\sqrt{2} \\pi} H_{\\rm kin}\\,,\n\\end{equation} \nwhere we have neglected the subleading mass term $m_\\chi$ and assumed standard vacuum fluctuations as initial conditions.\nAn estimate of the number of e-folds $\\Delta N_{\\rm tr}$  needed to reach \\eqref{eq:chimint}  can be obtained by equating $\\vert\\chi(a_{\\rm tr})\\vert=\\vert\\chi_{\\rm min}(a_{\\rm tr})\\vert$ with $a_{\\rm tr}$ the value of the scale factor at the transition time,\n\\begin{equation}\\label{eq:DN}\n \\Delta N_{\\rm tr}\\equiv \\ln \\left(\\frac{a_{\\rm tr}}{a_{\\rm kin}}\\right)=\\frac{n-2}{(n-2)\\sqrt{6\\xi}+6}\\ln\\left[\\frac{ \\vert \\chi_{\\rm min}(a_{\\rm kin})\\vert }{\\vert \\chi (a_{\\rm kin})\\vert }\\right]\\,,\n\\end{equation}\n\nOnce in the minimum, the field tracks its evolution until the $U(1)$ symmetry is restored.  Then, the origin becomes again the ground state of the system and the spectator field $\\chi$ starts oscillating around it, provided that the bare mass $m_\\chi$ exceeds the instantaneous Hubble rate at the time. If this is not the case, the field will stay frozen at its restoration value until the Hubble parameter has decreased enough to allow for the oscillations. \n\n\\section{Cosmic string network and GW production \\label{sec:GW_production}}\n\nAt the onset of kinetic domination the phases of the $U(1)$ field are randomly distributed, with a correlation length of the order of the Hubble parameter at that time \\cite{Zeldovich:1974uw}. As soon as the horizon expands, multiple patches with different phases enter in causal contact, leading to the formation of a cosmic string network \\cite{Kibble:1976sj}. The spontaneous breaking of the global $U(1)$ symmetry comes associated with a massless Goldstone boson, which mediates a long-range interaction among the strings and keeps their energy density finite by effectively introducing a radial cut-off.~\\footnote{Note that in a global $U(1)$ scenario like the one under consideration, the energy density of an infinite and isolated string should be expected to diverge since there is no gauge field to compensate the variation of the phase at large distances \\cite{Hindmarsh:1994re}.}\nThe width of the cosmic strings is inversely proportional to the expectation value of the $\\chi$ field, which, as shown in Eq~\\eqref{eq:chi_roll},  rapidly increases during the first stages of kinetic domination. Since the Hubble rate is the only relevant scale in our model, we have ``fat'' strings with a width $\\sim H^{-1}$ soon after the spontaneous symmetry breaking.\n\nFat strings like the ones under consideration have been extensively studied in the context of axion dark matter \\cite{Moore:2016itg,Gorghetto:2018myk,Vaquero:2018tib}, finding an average of one string per Hubble volume. In our case, this is even more expected since the  width of the string is already proportional to the Hubble rate at the time of formation.  Note that this unusual property restricts the formation of cosmic string loops, since the length of these objects cannot be --- for obvious reasons --- shorter than its width. This allows us to neglect the (usually dominant) GW production by loops \\cite{Damour:2001bk,Siemens:2006yp,Kawasaki:2011dp} and estimate the total energy density of the network by computing the energy density stored in a single string of Hubble length  $L\\propto H^{-1}$ within a Hubble volume $H^3$, namely\\footnote{\nFor simplicity, we disregard the time dependence that logarithmic corrections to the energy density may induce \\cite{Moore:2016itg}. The estimates presented in this paper are expected to be robust upon the inclusion of this subleading effect.}\n\\begin{equation}\n \\rho_{\\rm cs} \\approx \\mu(t) \\times L \\times H^{3} \\approx \\mu(t) H^2\\,,\n\\end{equation}\nwith\n\\begin{equation}\n \\mu(t) \\sim \n \\vert{\\chi}(t)\\vert^2\\,,\n\\end{equation}\nthe energy of the string per unit length.\\footnote{We omit here a numerical prefactor depending logarithmically on the ratio between the width and the radial cut-off of the string \\cite{Hindmarsh:1994re,Moore:2016itg}.}\nNote that the relative energy density of the strings as compared to the background component depends on time,\n\\begin{equation}\\label{noscaling}\n\\Omega_{\\rm cs}\\equiv \\frac{\\rho_{\\rm cs}}{3H^2M_{\\rm P}^2}\\sim\\frac{\\mu(t)}{M_{\\rm P}^2}\n\\sim \\frac{\\vert \\chi(t)\\vert^2}{M^2_{\\rm P}}\\,,\n\\end{equation}\nmeaning that an exact scaling regime \\cite{Bennett:1989ak,Allen:1990tv,BlancoPillado:2011dq,Ringeval:2005kr} is never achieved.  This is not necessarily a problem as long as the energy density  of cosmic strings stays subdominant with respect to the background component, i.e. $\\Omega_{\\rm cs}\\ll 1$. This consistency condition translates into a limit on the maximum value that can be explored by the $U(1)$ field, restricting it be sub-Planckian,\n\\begin{equation}\\label{cons_cond}\n\\vert \\chi(t_{\\rm tr})\\vert\\ll M_P\\,.\n\\end{equation}\nand setting a maximum on the number of e-folds \\eqref{eq:DN} required to arrive to the time-dependent minimum \\eqref{eq:chimint},\n\\begin{equation}\\label{eq:DN_crit}\n\\Delta N_{\\rm tr} \\ll \\frac{1}{\\sqrt{6\\xi}}\\left[18+\\ln\\left(\\frac{1}{1+\\sqrt{6\\xi}}\\right)+\\ln\\left(\\frac{10^{11}\\,{\\rm GeV}}{H_{\\rm kin}}\\right)\\right]\\,.   \n\\end{equation}\n\nIn the presence of a scaling regime, the evolution of the cosmic strings' energy momentum tensor in the vicinity of the horizon  leads to the production of  a scale-invariant gravitational wave spectrum \\cite{Krauss:1991qu} (for a recent review see Ref.~\\cite{Caprini:2018mtu}). Note, however, that the vacuum expectation value of the scalar field in our scenario evolves with time. We expect therefore a residual scale dependence in the spectrum, measured exactly by the change of this quantity. The amount of GWs produced by a vibrating string of length $L\\propto H^{-1}$ can be estimated by approximating the associated quadrupole $Q$ by that of a cylinder of mass $M\\approx \\mu\\, L$ and width $R\\propto H^{-1}$ \\cite{Krauss:1991qu,Maggiore:1999vm}, namely \n\\begin{equation}\n Q(t) \\propto MR^2 \\sim \\mu(t) H^{-3}\\,,\n\\end{equation}\nwith the precise proportionality constant depending on the level of non-sphericity of the configuration.\nAssuming that the relaxation time is of the order of $H$, the luminosity in GWs becomes\n\\begin{equation}\\label{eq:luminosity}\n \\mathcal{L}(t) \\sim M_{\\rm P}^{-2}\\, \\dddot{Q}(t)^2 \\sim M_{\\rm P}^{-2} (H^3(t) Q(t))^2\\sim \\mu^2(t)M_{\\rm P}^{-2}\\,,\n\\end{equation}\nwhere we have replaced time derivatives in favor of the Hubble parameter $H$. At a given instant $t$, the emission of GWs is peaked around the characteristic scale of the system at that time, decaying as a power-law for  shorter and longer distances. The relative power emitted by the string network in a Hubble time (i.e. per $dN\\equiv Hdt$) is given by \\cite{Kamada:2015iga}\n\\begin{equation}\\label{k_dep}\n \\frac{\\Delta P_{\\rm GW}(t,k)}{\\Delta N}\\equiv\\frac{1}{3H^2M_{\\rm P}^2}\\frac{d\\rho_{\\rm GW}(t,k)}{d\\log k}\n \\simeq \n\\begin{cases}\nP_{\\rm peak}(t)\n \\left(\\frac{k}{k_{\\rm peak}(t)} \\right)^{\\alpha} \\hspace{7mm}{\\rm for}\\hspace{5mm}  k \\lesssim k_{\\rm peak}\\,, \\\\ \nP_{\\rm peak}(t)\n \\left(\\frac{k}{k_{\\rm peak}(t)} \\right)^{-\\bar\\alpha} \\hspace{5mm}{\\rm for}\\hspace{5mm}  k > k_{\\rm peak}\\,, \\\\ \n\\end{cases}\n\\end{equation}\nwith $k_{\\rm peak}(t)\\sim aH$ the peak emission wavenumber\\footnote{We omit here a (larger than one)  proportionality constant ensuring causality \\cite{Kamada:2015iga}.} and \n\\begin{eqnarray}\\label{eq:P_peak}\nP_{\\rm peak}(t)   \n &\\sim& \n \\frac{L^{-2}\\times\\mathcal{L}}{H^2 M_{\\rm P}^2}  \\sim  \\frac{\\mathcal{L}}{M_{\\rm P}^2}\n \\sim \n \\left( \\frac{\\vert \\chi(t)\\vert}{M_{\\rm P}} \\right)^4 \\,.\n\\end{eqnarray} \nThe exponents $\\alpha$ and $\\bar \\alpha$ in Eq.~\\eqref{k_dep} parametrize our ignorance about the exact momentum dependence. While a scaling $\\alpha=2$  can be inferred from simple causality arguments~\\footnote{The modes with $ k \\lesssim k_{\\rm peak}$ at a given time correspond to super-horizon scales.} \\cite{Dufaux:2007pt,Kamada:2015iga}, the value of $\\bar \\alpha$ must be derived from simulations. Note, however, that this exponent is intimately related to the dynamics of small scales and it should be therefore rather independent of the specific expansion history. This observation allows us to benefit from existing numerical results to set a fiducial value $\\bar\\alpha\\sim 2$ ~\\cite{Kamada:2015iga}.\n \n\\subsection{GW spectrum in a generic background}\n\nThe precise form of the accumulated GW spectrum at large momenta is obtained by integrating Eq.~\\eqref{k_dep} from the time $t_{\\rm i}$ at which the strings start to form to an arbitrary time $t$ that will be later associated with the moment at which strings decay and the production of GWs is halted. Since the emission peak in our scenario is set by the comoving horizon, $k_{\\rm peak} \\sim a H=\\tau^{-1}$, it is convenient to work in conformal time $\\tau$. In terms of this time coordinate, the integrated GW spectrum takes the form\n\\begin{eqnarray} \\label{eq:OmegaGW-gen}\n && \\Omega_{\\rm GW}\\left( \\tau,k \\right) \n =\\int^{\\tau}_{\\tau_{\\rm i}} d \\log \\tau' \n \\frac{\\Delta P_{\\rm GW} (\\tau',k)}{ \\Delta \\log \\tau' }\\left(\\frac{a(\\tau')}{a(\\tau)} \\right)^b \\,,\n \\end{eqnarray} \n where we have used the fact that $dN\\propto d\\log\\tau$ and accounted for the evolution of the GW energy density. For the sake of completeness and in order to facilitate the comparison with other works in the literature, \nwe have considered a generic power-law evolution $a\\sim t^p\\sim \\tau^{p\/(1-p)}$ with $p$ a constant.~\\footnote{The choice of  constant $p$ is indeed a very good approximation \\textit{within a given cosmological epoch}. Although the formalism presented here could be easily extended to account for smooth transitions among different cosmological eras by simply promoting $p$ to $p(\\tau)$, this will not be necessary since we will focus here on the GW production during kinetic domination.}\nThe scaling power $b$ in Eq.~\\eqref{eq:OmegaGW-gen} is related to $p$ via $b\\equiv 4-2\/p$ and takes values $b=-2$ and $b=0$ during kination and radiation domination.\n\nWe can distinguish two possible scenarios depending on the relation between the bare mass $m_\\chi$ and the Hubble rate $H_{\\rm rad}$ at the time of heating: \n\\begin{enumerate}\n\\item If $m_\\chi \\geq H_{\\rm rad}$ the symmetry is restored before radiation domination and the $U(1)$ scalar field is free to oscillate around the origin, leading to the evanescence of the cosmic string network and halting the GW production. \n\\item If $m_\\chi<H_{\\rm rad}$ the $U(1)$ field remains frozen at its vacuum expectation value after symmetry restoration. Although additional strings are no longer generated, the existing ones will continue producing GWs after reentering the expanding horizon. This GW production will stop when the instantaneous Hubble rate becomes comparable to the mass of the field, $H\\lesssim m_\\chi$. From there on, the field $\\chi$ will start oscillating around the origin and the strings will consequently decay. \n\\end{enumerate} \nSince the use of numerical simulations is most probably required in the latest scenario, we will focus here on the $m_\\chi \\geq H_{\\rm rad}$ case.\nThe total power in GWs produced by the cosmic string network can be then computed by integrating Eq.~\\eqref{eq:OmegaGW-gen} from the formation time $\\tau_i$ to the time $\\tau = \\tau_{\\rm osc}$ at which the symmetry is restored. Taking into account Eq.~\\eqref{k_dep} we can rewrite this expression as \\cite{Kamada:2015iga}\n\\begin{eqnarray}\\label{eq:integral_split}\\nonumber\n\\Omega_{\\rm GW}\\left( \\tau_{\\rm osc},k \\right)  &\\simeq&\n \\int^{\\tau_k}_{\\tau_{\\rm i}} \\frac{d\\tau' }{\\tau'}P_{\\rm peak}(\\tau') \n\\left( \\frac{k}{k_{\\rm peak}(\\tau')} \\right)^{\\alpha }\\left(\\frac{a(\\tau')}{a(\\tau)} \\right)^b \\\\  &+& \n \\int^{\\tau_{\\rm osc}}_{\\tau_k} \\frac{d\\tau' }{\\tau'}P_{\\rm peak}(\\tau')  \n\\left( \\frac{k}{k_{\\rm peak}(\\tau')} \\right)^{-\\bar \\alpha } \\left(\\frac{a(\\tau')}{a(\\tau)} \\right)^b \\,,\n\\end{eqnarray}\nwhere we have defined $\\tau_k$ as the time at which the peak emission mode coincides with the mode $k$, i.e. $k_{\\rm peak}(\\tau_k)=k$. This splitting takes into account the amount of time that each mode spends in the blue or red part of the spectrum given that the peak is evolving with time. Also notice that, since we require that $\\tau_k>\\tau$, the integral is only valid for $k>k_{\\rm peak}(\\tau)$ \\cite{Kamada:2015iga}. \n\nThe integrated GW spectrum in Eq.~\\eqref{eq:integral_split} has three main contributions. On the one hand, we have two associated with the blue-shifting of the power spectrum \nitself, which give either a $k^{-\\bar \\alpha}$ or a $k^\\alpha$ scaling depending on how much time did the wavenumber $k$ spend in each slope.\nOn the other hand, we have the contribution coming from the time at which the mode $k$ coincides with the generic time-dependent peak emission mode $k_{\\rm peak}(\\tau_k)$. In that case the power spectrum is evaluated at $k\\,\\tau_k\\sim1$. Assuming a generic power-law dependence  $P_{\\rm peak}\\sim \\tau^{-\\gamma}$ with constant parameter  $\\gamma$,~\\footnote{As we will see in the next section, the precise value of $\\gamma$ depends on the $U(1)$ field dynamics and on the model parameters. Nevertheless, for an easier comparison with the literature, we report here that for the field evolution in  Eq.~\\eqref{eq:chimint}, we have $\\gamma= -\\frac{8}{(n-2)(1-p)}$. Using this expression, we can easily recover the results of Ref.~\\cite{Kamada:2014qja} by simply setting $p=2\/3$ and $n\\geq 6$, cf.~also Section \\ref{sec:GW_today}.  } this leads to a scaling $\\Omega_{\\rm GW}\\sim k^\\beta$ with \n\\begin{equation}\\label{eq:beta_gamma}\n\\beta\\equiv \\gamma -\\frac{4\\, p-2}{1-p}\\,.\n\\end{equation}\nThe above contributions compete in the integrated form of Eq.~\\eqref{eq:integral_split} yielding~\\footnote{Notice that for $\\beta-\\alpha=0$, the spectrum has a different solution, namely\n\\begin{equation}\n\\Omega_{\\rm GW}(\\tau_{\\rm osc},k)\\simeq P_{\\rm peak}(\\tau_{\\rm osc})\\left(\\frac{k}{k_{\\rm peak}(\\tau_{\\rm osc})}\\right)^\\beta\\left[\\log\\left(\\frac{k_{\\rm peak}(\\tau_{\\rm i})}{k}\\right)+\\frac{1}{\\beta+\\bar \\alpha}\\left(1-\\left(\\frac{k_{\\rm peak}(\\tau_{\\rm osc})}{k}\\right)^{\\bar\\alpha+\\beta}\\right)\\right]\\,.\\nonumber\n\\end{equation}\nHence, its power law behavior for intermediate frequencies is $\\Omega_{\\rm GW}(\\tau_{\\rm osc},k)\\propto k^\\beta \\log(k_{\\rm peak}(\\tau_{\\rm i})\/k)$ rather than just $k^\\beta$. A similar logarithmic dependence can be found in the $\\bar \\alpha+\\beta=0$ case.}\n \\begin{eqnarray} \\label{eq:OmegaGW_early}\\nonumber\n\\Omega_{\\rm GW}\\left( \\tau_{\\rm osc},k \\right)  &\\simeq &\nP_{\\rm peak}(\\tau_{\\rm osc}) \n\\left( \\frac{k}{k_{\\rm peak} (\\tau_{\\rm osc})} \\right)^{\\beta}\\times\\\\\n&\\times&\\left[ \\frac{1}{\\beta- \\alpha}\\left( \n\\left( \\frac{k_{\\rm peak} (\\tau_{\\rm i})}{k}\\right)^{\\beta-\\alpha}  -1\\right)\n\\frac{1}{\\bar \\alpha + \\beta} \n\\left(1 -\\left( \\frac{k_{\\rm peak} (\\tau_{\\rm osc})}{k} \\right)^{\\bar\\alpha  + \\beta} \\right)\\right]\\,.\n\\end{eqnarray} \nThis analytical expression explores a range of frequencies comprised between $k_{\\rm peak}(\\tau_{\\rm osc})\\le k \\le k_{\\rm peak}(\\tau_{\\rm i})$. For intermediate frequencies well within the integration limits, $k_{\\rm peak}(\\tau_{\\rm osc})\\ll k \\ll k_{\\rm peak}(\\tau_{\\rm i})$, we can identify three regimes depending on the sign of $\\beta-\\alpha$ and $\\beta+\\bar\\alpha$, namely \n\\begin{equation}\\label{eq:power_index}\n  \\Omega_{\\rm GW}\n \\propto \n\\begin{cases}\nk^\\alpha \\hspace{7mm}{\\rm for}\\hspace{5mm}  \\beta-\\alpha>0\\,, \\\\ \nk^\\beta \\hspace{7mm}{\\rm for}\\hspace{5mm}  \\beta-\\alpha<0 \\quad \\text{and}\\quad  \\bar\\alpha+\\beta>0\\,, \\\\ \nk^{-\\bar\\alpha} \\hspace{5mm}{\\rm for}\\hspace{5mm}  \\beta-\\alpha<0 \\quad \\text{and}\\quad  \\bar\\alpha+\\beta<0\\,. \\\\\n\\end{cases}\n\\end{equation}\nSince $\\alpha>0$, the first case corresponds to a blue-tilted spectrum while the second one can be either blue or red depending on the sign of $\\beta$, which is ultimately determined by the evolution of the $\\chi $ field and the background, cf. Eq.~\\eqref{eq:beta_gamma}.  Finally, the last case corresponds to a red tilted spectrum since $\\bar\\alpha>0$ \\cite{Kamada:2015iga}.\n\n\\subsection{GW spectrum in quintessential inflation}\\label{sec:Int_spec_Kin}\n\nThe formalism presented in the previous section is quite general and holds for any background and temporal dependence of the emission peak, as long as this can be parametrized by a power law  $P_{\\rm peak}\\sim \\tau^{-\\gamma}$ with $\\gamma$ approximately constant for some temporal interval $\\Delta\\log\\tau$. In this section, we particularize our results to the kinetic-dominated era appearing in our quintessential inflation scenario, determining the precise form of $\\gamma$ and the power spectrum as a function of the model parameters. \n\nThe background evolution during kinetic domination is specified by taking $p=1\/3$ and hence $b=-2$. During this epoch, the dynamics of the scalar field $\\chi$ can be split into a \\textit{rolling} and a \\textit{minimum phase}, each of them leading to different characteristic shapes in the GWs spectrum. The first phase is associated with the initial stages of kination, where the $U(1)$ scalar field is still rolling down the origin towards the Hubble-induced minimum of the potential. The second phase describes the tracking of the minimum once the field reaches it, if it does. We refer the interested reader to the Appendix \\ref{sec:ais}, where a detailed calculation of the GW spectrum is performed.\\footnote{It should noted that we are implicitly assuming that the cosmic string network forms and starts to emit gravitational waves right after the symmetry breaking. To the best of our knowledge, such transition has not been studied in numerical simulations but we expect  our order of magnitude estimates to stay valid.} Here we will simply discuss the shape of the spectrum and its dependence on the parameters.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{lccccccc}    \\hline\\hline\nPhase & \\hspace{2mm}  $\\alpha$ & $\\hspace{2mm}\\bar \\alpha$ & \\hspace{2mm} $\\beta$ &  $\\hspace{5mm}\\gamma$  & \\hspace{5mm}$\\beta-\\alpha$ &$\\bar\\alpha+\\beta$& \\hspace{2mm}  tilt range \\Tstrut\\Bstrut \\\\\n \\hline\n\\textit{Rolling} & \\hspace{2mm}  2 & $\\hspace{2mm}\\sim 2$\n & \\hspace{2mm} $1-2\\sqrt{6 \\xi}$ & $\\hspace{2mm}-2\\sqrt{6 \\xi}$  &\\hspace{2mm} $-1-2\\sqrt{6 \\xi}$& \\hspace{2mm} $3-2\\sqrt{6 \\xi}$ & \\hspace{2mm}  $1$ to $-2$ \\Tstrut\\Bstrut\\\\ \n \n\\textit{Minimum} &\\hspace{2mm}   2 & $\\hspace{2mm} \\sim 2$& \\hspace{2mm} $\\dfrac{10+n}{n-2}$ &  \\hspace{2mm} $\\dfrac{12}{n-2}$ & \\hspace{2mm} $\\dfrac{14-n}{n-2}$ & \\hspace{2mm} $3\\dfrac{n+2}{n-2}$ & \\hspace{2mm} $1$ to $2$ \\Tstrut\\Bstrut \\\\   \n\\hline\\hline \n\\end{tabular}\n\\end{center}\n \\caption{Values of the spectrum powers during the \\textit{rolling} and the \\textit{minimum phase}. The value of $\\gamma$ has been extracted from Eqs.~\\eqref{eq:chimint}, \\eqref{eq:chi_roll} into \\eqref{eq:P_peak}. The tilt range has been computed using Eq.~\\eqref{eq:power_index} taking the \\textit{formal} limits $\\xi\\rightarrow 0$, $\\xi\\rightarrow \\infty$ and $n\\rightarrow 4$, $n\\rightarrow \\infty$, respectively.}\\label{tab:table2}\n\\end{table}\n  In particular, we report in Table \\ref{tab:table2} the values of the power law indexes during the various stages. Note that the parameter $\\gamma$ is negative during the \\textit{rolling phase} $(\\tau<\\tau_{\\rm tr})$ and positive definite during the \\textit{minimum phase} $(\\tau>\\tau_{\\rm tr})$. As shown in the last column, the spectrum of GWs  during the \\textit{rolling phase} can be both blue and red depending on the values of the parameter $\\xi$, namely\n\\begin{align}\\label{eq:Om_scaling_roll}\n\\Omega_{\\rm GW}(\\tau,k>k_{\\rm tr})\\propto\n\\begin{cases}\nk^{1+\\gamma} \\hspace{5mm}{\\rm if}\n\\hspace{5mm} \\xi<3\/8\\,, \\\\ \nk^{-\\bar\\alpha} \\hspace{7mm}{\\rm if}\n\\hspace{5mm} \\xi>3\/8 \\,.\n\\end{cases}\n\\end{align}\nOn the other hand, the spectrum  during the \\textit{minimum phase} can only be blue, with the precise tilt depending on the order of the higher-order operators in Eq.~\\eqref{LBL},\n \\begin{align}\\label{eq:Om_scaling_min}\n\\Omega_{\\rm GW}(\\tau,k<k_{\\rm tr})\\propto\n\\begin{cases}\nk^{1+\\gamma} \\hspace{7mm}{\\rm if }\\hspace{5mm} n>14\\,, \\\\ \nk^{\\alpha}   \\hspace{10mm}{\\rm if} \\hspace{5mm}  n<14 \\,.\n\\end{cases}\n\\end{align}\nThe integrated GW spectrum when both phases are present can be generically written as an amplitude times a $k$-dependent form function, namely\n\\begin{equation}\\label{eq:spectrum_formal}\n \\Omega_{\\rm GW}(\\tau_{\\rm osc},k)\\sim P_{\\rm peak}(\\tau_{\\rm kin})\\Theta^{-1}\\left(\\frac{a_{\\rm osc}}{a_{\\rm rad}}\\right)^2 F\\left(\\kappa,s_{\\rm osc},s_{\\rm tr}\\right)\\,,\n\\end{equation}\nwhere the blue-shifting factor $a_{\\rm osc}\/a_{\\rm rad}$ appears because we factored out $\\Theta^{-1}$ and we have defined the dimensionless variables\n\\begin{equation}\\label{eq:s_and_x}\n s\\equiv \\frac{\\tau}{\\tau_{\\rm kin}}\\hspace{12mm} \\text{and}\\hspace{12mm}\\kappa\\equiv \\frac{k}{a_{\\rm kin}H_{\\rm kin}}\\,.\n\\end{equation}\nThe functional $F$ encodes the momentum behavior of the spectrum as a function of the model parameters. A detailed computation of this quantity assuming  a sudden transition between the \\textit{rolling} and the \\textit{minimum phases} can be found in the Appendix \\ref{sec:ais}.\\footnote{Note that this slightly overestimates the size of spectrum since the transition from the \\textit{rolling} to the \\textit{minimum phase} is expected to be smooth rather than instantaneous.} \n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.9]{F_vs_x.pdf}\n\\caption{Form of the function $F$ in Eq.~\\eqref{eq:spectrum_formal} for various values of $\\xi$. The shaded area represents the asymptotic values reported in Table~\\ref{tab:table2}. Here we have assumed $a_{\\rm osc}=a_{\\rm rad}$ and $n=4$.}\n\\label{fig:F_vs_x}\n\\end{center}\n\\end{figure}\nAs illustrated in Figure~\\ref{fig:F_vs_x}, the shape of the spectrum is determined by the values of the $\\alpha$, $\\bar \\alpha$ and $\\beta$ parameters, according to the discussion in Eq.~\\eqref{eq:power_index}. In particular, the two stages of evolution give rise to two spectral indexes, hence to an inflection point in the spectrum. \nWe can distinguish three characteristic $k$ values, namely $k_j\\equiv a(\\tau_j) H(\\tau_j)$ with $j={\\rm kin},{\\rm tr},{\\rm rad}$ standing respectively for the symmetry breaking time, the onset of the \\textit{minimum phase} and that of radiation domination.\\footnote{\nWe could additionally estimate what would happen if the universe enter the radiation domination regime before the $U(1)$ field decays. Assuming that the field stays frozen at the value of restoration for a while -- i.e. $P_{\\rm peak}\\approx {\\rm constant}$ -- and taking into account that the background energy density scales in that case as the one in GWs, the only contribution will be that from $k\\,\\tau_k\\sim 1$, leading then to a scale-invariant spectrum. If that were the case, there would be a plateau for small $k<k_{\\rm rad}$ in the power spectrum with a cut-off when the field decays. We will not further pursue this possibility in this paper, but it is worth noting the possibility.} \nFor each of these modes the amplitude of the power spectrum can be determined as follows: using Eq.~\\eqref{eq:F_min} one can get the amplitude for $k=k_{\\rm kin}$ and then use the scaling in Eqs.~\\eqref{eq:Om_scaling_roll} and \\eqref{eq:Om_scaling_min} to compute the amplitude for the other two modes. Assuming for simplicity  that both phases are present and that  they have a duration of several e-folds, i.e.  $k_{\\rm rad}\\ll k_{\\rm tr}\\ll k_{\\rm kin}$, i.e.  \nwe get\n\\begin{align}\\label{eq:Omega_amplitude_B}\n\\hspace{-1mm}\\Omega_{\\rm GW}(\\tau_{\\rm osc},k_{\\rm kin}) \\propto \n\\left(\\frac{\\vert \\chi_{\\rm min}(a_{\\rm kin})\\vert}{M_{\\rm P}}\\right)^4 \\left(\\frac{a_{\\rm osc}}{a_{\\rm rad}}\\right)^2 \\Theta^{-1}\n\\begin{cases}\n1 \\hspace{30mm}{\\rm if}\n\\hspace{3mm} \\bar\\alpha +\\beta>0\\,, \\\\ \n \\left(\\frac{a_{\\rm rad}}{a_{\\rm tr}}\\right)^{2(\\bar\\alpha+\\beta)}\\Theta^{\\bar \\alpha+\\beta} \\hspace{3mm}{\\rm if}\n\\hspace{3mm} \\bar\\alpha+\\beta<0 \\,,\n\\end{cases}\n\\end{align}\nand \n\\begin{align}\n\\label{eq:Omega_amplitude_eq}\n\\Omega_{\\rm GW}(\\tau_{\\rm osc},k_{\\rm tr})&=\\Omega_{\\rm GW}(k_{\\rm kin})\\left(\\frac{a_{\\rm kin}H_{\\rm kin}}{a_{\\rm tr} H_{\\rm tr}}\\right)^{c}\\,,\n\\\\\n\\label{eq:Omega_amplitude_R}\n\\Omega_{\\rm GW}(\\tau_{\\rm osc},k_{\\rm rad})&= \\,\\Omega_{\\rm GW}(k_{\\rm tr})\\left(\\frac{a_{\\rm rad}H_{\\rm rad}}{a_{\\rm tr} H_{\\rm tr}}\\right)^{d}\\,,\n\\end{align}\nwhere in the $\\bar \\alpha+\\beta<0$ case there is an extra $a_{\\rm tr}\/a_{\\rm rad}$ term because we factored out $\\Theta$ and we have defined exponents $c=\\rm{max}(-\\bar \\alpha, 1+\\gamma)$ and $d=\\rm{max}(\\alpha, 1+\\gamma)$.  Note that the heating efficiency governs the range of modes over which the spectrum extends,\n\\begin{equation}\\label{eq:delta_k}\n\\frac{k_{\\rm  peak}(\\tau_{\\rm kin})}{k_{\\rm peak}(\\tau_{\\rm osc})} = \\Theta^{-1}\\left(\\frac{a_{\\rm osc}}{a_{\\rm rad}}\\right)^2\\,,\n\\end{equation}\n becoming smaller for increasing $\\Theta$.\n\n\\section{GW spectrum today}\\label{sec:GW_today}\n\nOnce the symmetry is restored, the strings decay and the production of GWs terminates. Since the energy density of the residual GW background scales as radiation, the GWs spectrum at the present cosmological epoch can be well approximated by~\\footnote{ In deriving this expression we have neglected the variation of the effective number of degrees of freedom. This effect can be easily incorporated by replacing\n\\begin{equation}\n \\Omega_{\\rm GW} ( \\tau_{\\rm rad})\\,\\,\\,\\rightarrow\\,\\,\\, \\left( \\frac{g_s (\\tau_0)}{g_s(\\tau_{\\rm rad})} \\right)^{4\/3} \n \\left( \\frac{g_*(\\tau_{\\rm rad})}{g_* (\\tau_0)} \\right) \n \\Omega_{\\rm GW} ( \\tau_{\\rm rad}) \\,, \\nonumber\n\\end{equation} \nwith $g_s$ and $g_*$ the entropic and relativistic degrees of freedom at a given time\/temperature. Note, however, that, for the order of magnitude estimates presented in this paper, the additional factors play almost no role.~Indeed, for $g_s\\sim g_*$ the correction to Eq.~\\eqref{Omeganow} is  ${\\cal O}(1)$.}  \n\\begin{eqnarray}\\label{Omeganow}\n\\Omega_{\\rm GW} (\\tau_0,k)h^2 &=&  \\Omega_{\\rm R} h^2 \n\\left( \\frac{a_{\\rm rad}}{a_{\\rm osc}}  \\right)^{2\n } \\,\n \\Omega_{\\rm GW} ( \\tau_{\\rm osc},k)\\,,\n\\end{eqnarray}\nwith $\\tau_0$ the current conformal time, $\\Omega_{\\rm R}  h^{2}\\simeq 4.15 \\times 10^{-5}$ the radiation energy density nowadays and $h=0.678$. The momentum dependence of this expression can be made explicit by combining it with  Eq.~\\eqref{eq:spectrum_formal},\n\\begin{equation}\\label{Omeganowk}\n \\Omega_{\\rm GW} (\\tau_0,k)h^2 = \\Omega_{\\rm R} h^2  \n  P_{\\rm peak}(\\tau_{\\rm kin})\\Theta^{-1}F(\\kappa,s_{\\rm osc},s_{\\rm tr})\\,.\n\\end{equation}\nIn order to connect the mode interval in Eq.~\\eqref{Omeganowk} with the present-day frequency range, we make use of the standard relation among frequencies and wavenumbers, namely\n\\begin{equation}\n f_0=\\frac{1}{2 \\pi}\\frac{k}{a_0}\\,,   \n\\end{equation}\nwith the subscript $0$ referring to present day quantities. Taking into account the cosmological evolution during radiation domination, we can rewrite this expression as \n\\begin{equation}\\label{eq:x_to_f}\n\\kappa=\\frac{2 \\pi f_0}{\\sqrt{H_0 H_{\\rm kin}}}\\Theta^{1\/4}\\,,\n\\end{equation}\nwith $\\kappa$ the dimensionless variable in Eq.~\\eqref{eq:s_and_x} and  $H_0\\simeq 100\\, h \\,{\\rm km}\/{\\rm s\\, Mpc}^{-1}$ the present Hubble rate.\nCombining this result with \nEq.~\\eqref{eq:delta_k} we find that the range of frequencies generated by our scenario lays between\n\\begin{equation} \nf_0^{\\rm kin}\\sim 3\\times 10^{11} \\,{\\rm Hz} \\left(\\frac{H_{\\rm kin}}{10^{11} {\\rm GeV}}\\right)^{1\/2}\\, \\left(\\frac{\\Theta}{10^{-14}}\\right)^{-1\/4}\\,,\n\\end{equation}\nand\n\\begin{equation}\nf_0^{\\rm osc}\\sim 3\\times 10^{-3} \\,{\\rm Hz} \\left(\\frac{H_{\\rm kin}}{10^{11} {\\rm GeV}}\\right)^{1\/2}\\, \\left(\\frac{\\Theta}{10^{-14}}\\right)^{3\/4}\\left(\\frac{a_{\\rm rad}}{a_{\\rm osc}}\\right)^2 \\,,\n\\end{equation}\nwith the superscripts kin and rad referring respectively to the evaluation of the spectrum at $k=k_{\\rm peak}(\\tau_{\\rm kin})$ and $k=k_{\\rm peak}(\\tau_{\\rm osc})$. Within this frequency interval, we can distinguish the transition between the \\textit{rolling phase} and the \\textit{minimum phase} (cf.~Section \\ref{sec:Int_spec_Kin}), namely\n\\begin{equation}\n f_0^{\\rm tr}\\sim 0.2 \\,{\\rm Hz} \\left(\\frac{H_{\\rm kin}}{10^{11} {\\rm GeV}}\\right)^{1\/2}\\, \\left(\\frac{\\Theta}{10^{-14}}\\right)^{-1\/4}  \\exp\\left(\\frac{\\Delta N_{\\rm tr}}{14}\\right)\\,,\n\\end{equation}\nwith the superscript tr denoting the evaluation of the spectrum at $k=k_{\\rm peak}(\\tau_{\\rm tr})$ and $\\Delta N_{\\rm tr}$ given by Eq.~\\eqref{eq:DN}.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[scale=0.7]{OGW0PI_vs_f.pdf}\n\\caption{Comparison among the GW spectrum produced by the global cosmic string network during kinetic domination and the  power-law integrated sensitivity curves of various GW experiments \\cite{Thrane:2013oya,Breitbach:2018ddu}.\nHere we have assumed that the strings are immediately destroyed at the onset of radiation domination ($a_{\\rm osc}=a_{\\rm rad}$) and chosen fiducial values $H_{\\rm kin}=10^{11}$ GeV and $\\xi = 0.2$. For this choice of parameters the backreaction of the cosmic string network is always small and the big bang nucleosynthesis constraint \\eqref{GWbound1} is satisfied.\n}\n\\label{fig:Om_vs_f2}\n\\end{center}\n\\end{figure}\n\nGround-based interferometers such as LIGO \\cite{adv_ligo}, VIRGO \\cite{virgo} or the Einstein Telescope \\cite{et} are already sensitive to frequencies around $f\\sim 100$ Hz, while space interferometers such as eLISA \\cite{lisa} or DECIGO \\cite{decigo} will be able to cover a $0.1$ mHz-$10$ Hz frequency window. The GW spectrum predicted by our mechanism is confronted with the  power-law integrated sensitivity curves\\footnote{ These curves take into account the enhancement in detector sensitivity following the integration over frequency on top of the integration over time.} \nof these experiments\nin Fig.~\\ref{fig:Om_vs_f2}. As shown in this plot, a potentially observable window within the scope of  future surveys can be easily found for not too large values of $H_{\\rm kin}$ and relatively small heating efficiencies. \nIt is important to notice that the produced spectrum is not restricted by CMB \\textit{anisotropies}. In particular, the production of GWs by Hubble-induced cosmic strings happens i) well-inside the horizon, ii) has a limited duration and iii) is restricted to highly energetic frequencies significantly exceeding those associated to the Hubble  scale at decoupling, namely  $3.4\\times 10^{-19} \\,{\\rm Hz}<f_{\\rm CMB} < 2.1\\times 10^{-17}\\, {\\rm Hz}$ \\cite{Caprini:2018mtu}. The only restriction comes from its (integrated) contribution to the \\textit{background} energy density in Eq.~\\eqref{GWbound1}. Note also that although our results are based on pure dimensional analysis and could therefore differ from the actual values by a few orders of magnitude (see for instance Ref.~\\cite{Figueroa:2012kw}) such uncertainty would not necessarily affect the observability of the model. In particular, a potential change in the amplitude of the spectrum could be easily compensated by a change on the number of e-folds spent in the \\textit{rolling phase} and on the duration of the kinetic dominated era.\n \n\nIt is interesting to compare our findings with previous results in the literature.  On general grounds, our mechanism makes use of two basic ingredients: a non-minimal coupling to gravity\\footnote{The use of non-minimal couplings in the context of topological defects goes back to Ref.~\\cite{Yokoyama:1988zza}, where they were introduced to avoid the generation of cosmic strings during inflation. Note, however, that the post-inflationary cosmic string formation described in this reference happens due to an explicit symmetry-breaking potential and not to the Ricci scalar itself.} and a period of kinetic domination. The fact that we are dealing with a Hubble-induced symmetry breaking pattern has far reaching consequences. First, the width of the strings in our scenario is proportional to the Hubble rate at the time of formation. Second, the temporal dependence in Eq.~\\eqref{noscaling} forbids the appearance of an exact scaling regime. Third, the eventual restoration of the symmetry at the onset of radiation domination rescues the model from the limitations associated to (long-lived) non-scaling topological defects. All these properties are genuinely new, marking significant differences with the usual global string scenarios \\cite{Hagmann:1990mj,Yamaguchi:1998gx,Yamaguchi:1999yp,Yamaguchi:1999dy}. This applies also to other studies in the literature involving also a kinetic dominated era \\cite{Cui:2017ufi,Cui:2018rwi,Artymowski:2017pua}. Interestingly, a restoration mechanism similar to the one presented in this paper was used in  Ref.~\\cite{Kamada:2015iga} in a matter dominated era.\nAlthough the shape of the spectrum in the latest scenario shares some similarities with that in Fig.~\\ref{fig:Om_vs_f2}, the associated physics is completely different, in particular: \n\\begin{enumerate}[i)]\n\\item The energy density parameter $\\Omega_{\\rm GW}$  grows during kinetic domination rather than decreasing.\n\\item The knee in the spectrum in our scenario is not due to the transition from matter to radiation domination, but rather to the evolution of the $U(1)$ field from a \\textit{rolling phase} in which the energy density of the strings increases to a \\textit{minimum phase} in which the energy density of the strings decreases. \n\\item   The spectral tilt at large frequencies depends in our case on  the non-minimal coupling to gravity, rather than on the dimension of higher-dimensional operators. \n\\item Contrary to what happens in matter domination, the low frequency band of the spectrum is not always proportional to $k^{2}$, but rather depends on the detailed structure of the scalar potential. \n\\end{enumerate}\n\nAlthough the existence of a \\textit{rolling phase} is crucial for the GWs produced in our scenario to be within the observable band, its absence is also interesting by itself.   In particular, the spectrum generated by an the \\textit{minimum phase} alone is \\textit{always} blue during kinetic domination, meaning that the leading contribution at the dominant high-frequency end $k_{\\rm kin}$ can be easily constrained by the integrated bound in Eq.~\\eqref{GWbound1}, namely\n\\begin{align}\n\\Omega_{\\rm GW}(k_{\\rm kin})\\sim 10^{-5}\\,\\Theta^{-1}\\left(\\frac{\\vert \\chi_{\\rm min}(a_{\\rm kin})\\vert}{M_{\\rm P}}\\right)^4 < 10^{-6}\\,.\n\\end{align}\nThis translates into a strong restriction on the presence of non-minimal couplings and on the duration of a kinetic dominated stage. Either the field stays sub-Planckian or the heating efficiency is high. In both cases, the GWs spectrum would never hit the observational band due to its  strong blue tilt. \n\n\\section{Conclusions}\\label{sec:conclusions}\n\nThe free propagation of gravitational waves upon production makes them a perfect cosmological probe for retrieving information on energy scales and epochs that are hardly accessible by any other means. Violent processes such as inflation, (re)heating or phase transitions could naturally produce stochastic GW backgrounds within the reach of  present and future GW interferometers.\n\nIn this paper we have considered the GW spectrum generated by a network of short-lived cosmic strings appearing in runaway quintessential inflation scenarios displaying a non-minimally coupled $U(1)$ spectator field. In this class of models, the Universe enters a kinetic-dominated regime soon after the end of inflation. During this unusual cosmological epoch the Ricci scalar turns negative,  breaking spontaneously  the $U(1)$ symmetry  and triggering the formation of global cosmic strings. These topological defects emit GWs until the onset of radiation domination, where the Ricci scalar vanishes and the symmetry is restored. At this point, the strings decay, halting the GW production and leaving behind a GW background within the scope of near future observational campaigns. Interestingly, the spectrum displays some characteristic features that might allow to discriminate the proposed scenario from other mechanisms in the literature. On the one hand, its tilts are directly related to the non-minimal coupling to gravity and to the structure of the higher dimensional operators ensuring vacuum stability during kinetic domination. On the other hand, the amplitude at peak position is associated to the transition between \na \\textit{rolling phase} associated with the initial stages of kinetic domination and a \\textit{minimum phase} in which the field tracks the minimum of the effective potential. In the event of a detection, these features provide a way of testing the structure of the scalar sector and the existence of non-minimal couplings to gravity, seeding some light on the underlying particle physics theory describing  the early Universe. We have also argued that in the absence of a long \\textit{rolling phase}, the spectrum is only blue-tilted, leading to a strong restriction on non-minimal couplings to gravity and the duration of a kinetic dominated epoch.\n \nThe treatment presented in this paper is rather general and not linked to any particular realization of the quintessential inflationary paradigm or to the details of the entropy production process. Among other possibilities, our scenario could take place, for instance, in $\\alpha$-attractor models \\cite{Dimopoulos:2017zvq,Dimopoulos:2017tud,Akrami:2017cir,Garcia-Garcia:2018hlc} or quantum gravity frameworks. \\cite{Wetterich:2014gaa,Hossain:2014xha,Rubio:2017gty}.\nThe formalism could be additionally extended to accommodate a richer phenomenology, such as the generation of different \\textit{short-lived} topological defects or the creation of primordial black holes \\cite{Amendola:2017xhl}. An interesting possibility along this line of thinking would be to identify the $U(1)$ charge with the baryonic number, while allowing for a small and explicit breaking of the $U(1)$ symmetry at large field values \\cite{Bettoni:2018utf}. This framework could generate a non-vanishing global baryon asymmetry together with a \\textit{correlated} GW background, opening the possibility of testing baryogenesis with GWs.  Determining the precise dynamics of this appealing scenario would most probably require the use of numerical simulations. We postpone the detailed analysis to a future work.\n\n\\section*{Acknowledgments}\nDB and JR acknowledge support from DFG through the project TRR33 ``The Dark Universe'' during the first stages of this work. GD is partially supported by DFG Collaborative Research center SFB 1225 (ISOQUANT). JR thanks Asier Lopez-Eiguren and David Weir for useful discussions. GD would like to thank A.~Kamada, M.~Sasaki and T.~Sekiguchi for useful discussions.  We finally thank the anonymous referee for useful and constructive comments.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section*{INTRODUCTION} \\setlength{\\arraycolsep}{0.1em}\n Quantum coherence (QC) originating from quantum superposition plays a central role in quantum mechanics,\n     and is a key resource in quantum information processing scenarios such as quantum reference frames \\cite{01,02,03}, transport in\n     biological systems \\cite{04,05,06} and quantum thermodynamics \\cite{07,08,09}. How to measure QC  is an important problem in both\n     quantum theory and quantum information science, and has attracted much attention recently \\cite{10}. A  framework to quantify  QC  of\n     quantum\n    states in the resource theories has been recently proposed by Baumgratz et. al \\cite {10}. By following this framework, a number of\nQC  measures based on various physical contexts have been put forward.  Norm of coherence and the relative entropy of coherence were first suggested as two QC measures based on distance measures \\cite{10}. The QC  measures based on entanglement \\cite{11}, operation \\cite{12a,13a} and convex-roof construction \\cite{14a,15a} were subsequently proposed. With QC  measures, various properties of QC, such as the relations between QC and other quantum resources \\cite{11,16f,16a}, the QC in infinite-dimensional systems \\cite{16b,16c}, the complementarity relations of QC \\cite{16d}, and the measure of macroscopic coherence\\cite{16e}, have been discussed. The quantification of QC  promoted in a resource-theoretic framework thus encouraged many further considerations about QC \\cite{17a,17b,17c}.\n\n   We would like to point out that, until now,  most of these studies on quantifying QC are not very perfect and are not applicable in\n   certain situations. For example, the quantification of QC depends on a particular reference basis, namely, these QC measures are\n   basis-dependent. As we known, the physical laws and properties of physical systems do not change with the different choice of reference\n   basis. Therefore, QC should also be a physical property of the quantum state itself in quantum  system.  Previous basis-dependent QC\n   measures cannot reflect physical reality related to the quantum state itself in quantum system. In this paper, we try to solve this\n   problem, i.e., to establish a basis-independent QC  measure. By refining an accurate and unique zero-coherence state as the complete\n   incoherent state, we propose a new QC  measure, called as an \\textit{intrinsic basis-independent quantum coherence measure} (IBIQCM),\n   which not only satisfies all the conditions in the resource-theoretic framework of quantifying QC, but also is basis-independent and can\n   reflects the physical essence of the quantum state itself. Moreover, the new QC measure also simplifies the QC calculation.\n\n\\section*{Results and Discussion} \\subsection*{Basis-dependent QC measures} \\setlength{\\arraycolsep}{0.1em}\n  First we review some previous basis-dependent approaches for QC measures  \\cite{10}. Fixing a reference basis in a $d$-dimensional Hilbert\n  space $\\mathcal{H}$, say $\\{ |i\\rangle \\}_{i=0}^{d-1}$, any quantum state can be expressed by $\\rho=\\sum_{i,j}\\rho_{i,j}|i\\rangle\\langle\n  j|$. Label the set of incoherent states by $\\mathcal{I} \\subset \\mathcal{H}$, where the incoherent states are diagonal states in the\n  reference basis, i.e.,\n\\begin{equation} \\delta =\\sum_{i=1}^d \\delta_i |i\\rangle\\langle i| \\label{incostate} \\end{equation} The incoherence operators are defined to fulfil $K_n \\mathcal{I} K_n^{\\dagger} \\subset \\mathcal{I}$ for all $n$, where quantum operations are specified by a set of Kraus operators $\\{ K_n\\}$ satisfying $\\sum_n K_n  K_n^{\\dagger} = \\mathbf{1}$. Following the approach to entanglement measurement, any proper measure of coherence for the state $\\rho$ requires a set of conditions:\n\n\\begin{description} \\item[(C1)] $C(\\delta) = 0$ iff $\\rho$ belongs to the set of incoherent states $\\mathcal{I}$ (i.e., $\\delta \\subset \\mathcal{I}$). \\item[(C2)] Monotonicity under incoherent completely positive and trace preserving maps $\\Phi_{ICPTP} = \\sum_{n} K_n \\rho K_n^{\\dagger}$,\n    i.e.,  $C(\\rho) \\ge C(\\Phi_{ICPTP} (\\rho)$, and under selective measurements on average, i.e., $C(\\rho) \\ge \\sum_n p_n C(\\rho_n)$,\n    where the retained state corresponding to outcome $n$ after a measurement $\\rho_n =  K_n \\rho K_n^{\\dagger} \/p_n$ with probability $p_n\n    = tr[K_n \\rho K_n^{\\dagger} ]$.\n\\item[(C3)] Nonincreasing under missing of quantum states (convexity), i.e., $\\sum_n p_n C(\\rho_n) \\ge C(\\sum_n p_n\\rho_n)$ for any set of\n    states $\\rho_n$ and any $p_n \\ge 0$ with $\\sum_n p_n = 1$.\n\\end{description}\n\n\nSeveral quantities have been proposed as candidates for measuring QC \\cite{10,11,12a,13a,14a,15a}. For example, the candidate coherence measurement can be defined according to the distance measure $\\mathcal{D}$ between quantum states\n \\begin{equation} C_{\\mathcal{D}}(\\rho) = \\min_{\\delta \\in \\mathcal{I}} \\mathcal{D}(\\rho, \\delta) \\label{C_D} \\end{equation}\n Therefore, two proper measures of QC \\cite{10} are the $l_1$ norm of coherence\n    \\begin{equation}\\label{06}\n    C_{n}(\\rho)=\\sum_{i,j(i\\neq j)}  |\\rho_{i,j}|.\n    \\end{equation} \nand the relative entropy of coherence which describe coherence of quantum states:\n    \\begin{equation}\\label{07}\n    C_{re}(\\rho)=S(\\rho_{diag})-S(\\rho),\n   \\end{equation}\n    where $S(\\rho) = -tr(\\rho \\log_2 \\rho)$ is Von Neumann entropy, and $\\rho_{diag}=\\sum_{i}\\rho_{i,i}|i\\rangle\\langle i|$ denotes the\ndiagonal part of $\\rho$. It should be pointed out that a striking feature of the relative entropy of coherence is that it  connectes the QC with the purity of quantum states via von Neunnman entropy. From the inequation $C_{re}(\\rho)\\leq S(\\rho_{diag})\\leq \\log(d)$ and the definition of relative entropy of coherence, it can be see that, the smaller $S(\\rho)$, the higher purity of quantum states, the stronger $C_{re}(\\rho)$. \n\nAlthough these measures satisfy the above conditions (C1),(C2) and (C3),  all they are defined in a particular reference basis. Therefore, for the same quantum state $\\rho$,  by choosing different reference basis, the results obtained by these QC  measures are completely different.  For example, for the diagonal state $\\varrho_z$ of a qubit in Pauli operator $\\sigma_z$ basis\n    \\begin{eqnarray}\\label{03}\n     \\varrho_z=\\left(\n       \\begin{array}{ccc}\n         \\cos^{2}\\alpha & 0  \\\\\n         0 & \\sin^{2}\\alpha \\\\\n       \\end{array}\n     \\right),\n    \\end{eqnarray}\nboth the relative entropy of coherence $ C_{re}(\\varrho_z)$ and the norm of\n    coherence $ C_n(\\varrho_z)$ are zero because all off-diagonal elements of the density matrix $\\varrho_z$ are zero.\nBut if $\\varrho_z$ is written down in $\\sigma_x$ basis, the density matrix will be\n    \\begin{eqnarray}\\label{04}\n     & & \\varrho_x= H\\varrho_{z}H=\\frac{1}{2} \\left(\n       \\begin{array}{ccc}\n         1 & \\cos(2\\alpha)  \\\\\n         \\cos(2\\alpha) & 1\\\\\n       \\end{array}\n     \\right),  \\\\\n    & & H=\\frac{1}{\\sqrt{2}}\\left(\n       \\begin{array}{ccc}\n          1 & 1  \\\\\n         1 & -1\\\\\n       \\end{array}\n     \\right), \\nonumber\n     \\end{eqnarray}\n  where $H$ is Hadamard gate, which is unitary operator. When $\\alpha$ is not $\\frac{\\pi}{4}$ or $\\frac{3\\pi}{4}$, the off-diagonal elements\n  of $\\varrho_x$ are nonzero. Then, both $ C_{re}(\\varrho_x)$ and $C_{n}(\\varrho_x)$ are also nonzero. It is obvious that, the\n    two QC  measures depend on the chosen reference basis. Meanwhile, physically, basis-dependent QC measures can also lead to a conclusion\n    that does not accord with physical reality: according to the conclusion obtained by basis-dependent QC measures, before the quantum state\n    of quantum system is measured, one can change the measuring basis to affect the QC  of quantum state itself. Therefore,\n    \\emph{Basis-dependent QC measures cannot reflect the intrinsic QC of the quantum state itself} and are not good QC measures. A good QC\n    measure, thus, requires basis-independent to overcome the shortcoming.\n\n      In order to quantify QC  precisely, it is important to exactly define the zero-coherence state, namely, we must find a complete\n      incoherent state as the zero-coherence state. In basis-dependent QC measures, the incoherent states are defined as\n      Eq.(\\ref{incostate}), i.e., the coherence of all the states $\\delta$ is zero. But we find that \\emph{the definition of incoherent\n      states in basis-dependent QC measures is not accurate.} In fact, the states defined by Eq.(\\ref{incostate}) are not complete incoherent\n      states, but partial incoherent states, except for all $\\delta_i$ take the same constant.  This stems from the fact that, the QC  of the\n      quantum state cannot be determined uniquely by the off-diagonal elements of the density matrix, which means that, there may be a\n      certain QC  for a quantum state described by the density matrix with the zero off-diagonal elements. If we choose the state defined by\n      Eq.(\\ref{incostate}) as zero-coherence state, this will lead to a result that zero-coherence state depend on the choice of the\n      reference basis, that is, the change of the reference basis will change their zero-coherence property. Therefore, in order to find an\n      intrinsic basis-independent QC measure, one must re-define the zero-coherence state which is independent of the reference basis as the\n      complete incoherent state.\n\\subsection*{Defining the zero-coherence state}\n  \\setlength{\\arraycolsep}{0.1em}\n   Since the feature of QC is impossible within classical physics, we must consider a maximum classical mixture state as the accurate and\n   unique zero-coherence state.  According to the density matrix expression method  of quantum state, a  maximum classical mixture state can\n   be described by the density matrix  with equal diagonal elements (equal probability) and zero off-diagonal elements. Thus we define the\n   maximum classical mixed state $\\delta_0$ as the $complete$ $incoherent$ state\n    \\begin{equation}\\label{14}\n    \\delta_0=\\frac{1}{d}\\sum_{i=1}^{d}|i\\rangle\\langle i|,\n    \\end{equation}\nwhich is basis-independent and does not possess any QC in $d$-dimensional Hilbert space. Obviously, the polarization state of natural light, which is incoherence light in experiment of interference of polarized light, can be described by the maximum classical mixed state.\n\n Given a $d$-dimensional Hilbert space, there is only one $complete$ $incoherent$ state $\\delta_0$, which  is different from the $incoherent$\n state defined  by Eq.(\\ref{incostate}). It is easy to prove the $complete$ $incoherent$ state $\\delta_0$ is invariant under arbitrary\n unitary transformation, which means that, there is not any QC in the $complete$ $incoherent$ state in arbitrary basis. Seeing that the\n $complete$ $incoherent$ state $\\delta_0$ possesses both the properties of the uniqueness and the basis-independence, an accurate and unique\n \\emph{zero-coherence} state can be determined by it. Obviously, for arbitrary completely positive and trance preserving map $\\Phi_{ICPTP}$,\n $\\delta_{0}=\\Phi_{ICPTP}(\\delta_{0})$ is always satisfied.\n\n\n \\subsection*{Basis-independent QC measures: IBIQCM}\n After introducing the accurate and unique  \\emph{zero-coherence} state $\\delta_0$, we can define a new basis-independent QC measure by the\n distance measure Eq. (\\ref{C_D}), i.e., by comparing the distinction between a quantum state $\\rho$ and the zero-coherence state $\\delta_0$\n to quantify QC. Here we still use the relative entropy \\cite{17} for the distance measure. Therefore, this new QC measure also describes the coherence of quantum states by using purity of quantum states and is defined by\n      \\begin{eqnarray}\\label{08}\n   C_{IBIQC}(\\rho)&=&-tr[\\rho\\log\\delta_{0}]+tr[\\rho\\log\\rho], \\nonumber \\\\\n   &=&-tr[\\rho\\log(\\frac{1}{d}\\sum_{i=1}^{d}|i\\rangle\\langle i|)]+tr[\\rho\\log\\rho] \\nonumber \\\\\n   &=& \\log d-S(\\rho)\n    \\end{eqnarray}\n    by substituting $\\delta_0$ given by Eq.(\\ref{14}), where $S(\\rho)$ is still Von Neumann entropy.  Equation (\\ref{08}) is the central\n    result of this paper.\n    We call this QC  measure as $intrinsic$ $basis$-$independent$ $quantum$ $coherence$ $measure$ (IBIQCM) and denote it by\n    $C_{IBIQC}(\\rho)$.\n\nAccording to the central result, we make the following discussion:\n\n(1) \\emph{IBIQCM is a kind of explicit, succinct and basis-independent coherence measure.}\n\nIt is obvious from Eq.(\\ref{08}) that, although IBIQCM relates to  the purity of quantum state via the Von Neumann entropy as same as  the relative entropy of coherence, it is independent on the choice of the reference basis due to the first term $ \\log d$ in Eq.(\\ref{08}) only depend on the dimension of Hilbert space. Obviously, the basic characteristics of IBIQCM is able to be obtained from basic the quantum entropy\\cite{17aa}. Moreover, since IBIQCM are basis-independent, arbitrary unitary operators are strictly incoherence operators in our scheme. However, we should note that the unitary operator can be either incoherence operator or coherence operator for different quantum states in framework of basis-dependent QC measures, for instance, Hadamard gate $H$ is coherence operator for $\\rho_z$, but it is incoherence operator for $\\delta_0$. Thus categorizations of incoherence operators and coherence operators are ambiguous in framework of basis-dependent QC measures.\n\nIn IBIQCM, since there is only one $complete$ $incoherent$ state  in $d$-dimensional quantum system, the optimization procedure  of distance measurement $\\mathcal{D}(\\rho,\\delta)$ can be avoided, which leads to a great simplification for the  calculation of QC measure.\n\nBy the similar procedure to the relative entropy of coherence $C_{re}(\\rho) $ \\cite{10},  one can prove generally that $C_{IBIQC}(\\rho)$ satisfies all the conditions (C1) (C2) and (C3) in the resource-theoretic framework of quantifying coherence, and an additional condition \\begin{description} \\item[(C0)] $C(\\rho)$ is invariant under unitary transformations, namely $C(\\rho)$ is basis-independent. \\end{description}\n\n\n(2) \\emph{IBIQCM is intrinsic QC  measure reflecting the physical essence of QC of the quantum state itself.}\n\n Considering that Von Neumann entropy $S(\\rho)$ is quantum information entropy about quantum systems \\cite{08}, it is apparent from\n Eq.(\\ref{08}) that $C_{IBIQC}(\\rho)$ is a measure of the information of quantum state $\\rho$ which we have obtained.\n  If  quantum system  is in a pure state, we would have zero Von Neumann entropy and the maximum $C_{IBIQC}(\\rho) = \\log d$, namely we can\n  obtain all quantum information from system without any losing in a certain time $t$, and the corresponding state is just a complete\n  coherence state. When quantum system is in the maximum classical mixture state which loses all quantum information about system, we would\n  have the maximum Von Neumann entropy $\\log d$ and zero $C_{IBIQC}(\\rho)$, namely we know nothing about system in a certain $t$, and the\n  corresponding state is just the complete incoherent state.\n\n\n\n\n   In order to further  examine the validity of IBIQCM, we return to the problem of $\\varrho_z$ and $\\varrho_x$ mentioned above. Substituting\n   $\\varrho_z$ and $\\varrho_x$ into Eq.(\\ref{08}), one obtains\n   \\begin{eqnarray}\\label{09}\n    C_{IBIQC}(\\varrho_z)\n   &=&\\log 2+\\cos^2\\alpha\\log\\cos^2\\alpha\\nonumber\\\\\n   &+&\\sin^2\\alpha\\log\\sin^2\\alpha,\\nonumber \\\\\n   C_{IBIQC}(\\varrho_x)&=&\\log 2+\\cos^2\\alpha\\log\\cos^2\\alpha\\nonumber\\\\\n    &+&\\sin^2\\alpha\\log\\sin^2\\alpha.\n  \\end{eqnarray}\n  Obviously, $C_{IBIQC}(\\varrho_z)$ is equal to $C_{IBIQC}(\\varrho_x)$, namely, quantum states $\\varrho_z$ and $\\varrho_x$ possess the same\n  QC. Therefore, IBIQCM reflects the intrinsic property of the quantum state according with physical reality whatever the reference basis is.\n  However, substituting $\\varrho_z$ and $\\varrho_x$ into Eq.(\\ref{07}), one gets\n  \\begin{eqnarray}\\label{10}\n    C_{re}(\\varrho_z)&=0 \\nonumber\\\\\n    C_{re}(\\varrho_x)&=&\\log 2+\\cos^2\\alpha\\log\\cos^2\\alpha\\nonumber\\\\\n    &+& \\sin^2\\alpha\\log\\sin^2\\alpha.\n  \\end{eqnarray}\n$C_{re}(\\varrho_x)$ is not equal to $ C_{re}(\\varrho_z)$, which means that the relative entropy of coherence is basis-dependent.\n\n In addition, from Eqs. (\\ref{09}) and  (\\ref{10}) , it can be seen that in $\\sigma_x$ basis, the results are the same for both the\n IBIQCM and the relative entropy of coherence, which means that the correct result can be obtained for the relative entropy of coherence only\n under some particular reference basis.\n\n In order to further illustrate the differences and connections between IBIQCM and the relative entropy of coherence, we design a polarized-light interference experiment device composed of a crystal wave plate with a phase delay and a linear polarizer.\n\n Here, we discuss two cases: Firstly, let a coherent plane polarized state project to vertically the crystal wave plate in Angle of $\\frac{\\pi}{4}$ between its vibration direction and the principal section of crystal wave plate (corresponding to a base vector selection).  After passing the crystal wave plate, the coherent plane polarized state is decomposed into two mutually vertical vibration components with the fixed phase relationship. Then, the two components project to the direction of the transmission light of the polarizer, and generate two mutually parallel vibration coherent components which can produce interference phenomenon behind the polarizer. If let a coherent plane polarized state pass the crystal wave plate in which the vibration direction is parallel to the direction of the principal section of the crystal wave plate, (corresponding to another base vector selection), there is only one coherent component behind crystal wave plate. After this coherent component passing the polarizer, there is still one coherent component and interference phenomenon cannot be produced. It is obvious that, the generation of the interference phenomenon requires not only the coherent source, but also the reasonable interference experiment design (corresponding to a reasonable base vector selection). If the interference experiment is not designed correctly (corresponding to the base vector selection is incorrect), even if there is coherent source, interference phenomenon can not be still produced; Second, let an incoherent nature light project to vertically the crystal wave plate, behind the crystal wave plate, although two mutually vertical vibration components can be produced, there is not fixed phase relationship between the two vibration components. Through the polarizer projection, the two mutually parallel vibration components naturally cannot produce any interference phenomenon.\n\n The above discussion shows that, the coherence of coherence source is independent on the design of interference experiment device, namely the coherence of quantum state itself is independent on base vector selection, while generation of interference phenomenon is dependent on the both the coherence of coherence source and design of interference experiment, namely, generation of interference phenomenon is dependent on the both the coherence of quantum state itself and experiment base vector selection.\n\n IBIQCM is an essential measure and description for the coherence of the quantum state itself, but it cannot provide judgment on whether the interference phenomenon can occur. While the relative entropy of coherence can provide a judgment for the possibility of experimental interference phenomenon occur if the coherent source is given, but it cannot provide the essential measurement and description for the coherence of quantum state itself.\n\nOn the other hand, A maximally coherent state, which is defined by literature\\cite{10}, can be written as\n   \\begin{eqnarray}\\label{11}\n    |\\Psi\\rangle=\\frac{1}{\\sqrt{d}}\\sum_{i=1}^{d}|i\\rangle.\n  \\end{eqnarray}\n  According to the theory of coherence of electromagnetic\n  field\\cite{18b}, coherence state $|\\alpha\\rangle=exp(-\\frac{1}{2}|\\alpha|^{2})\\sum_{n=0}^{\\infty}\\frac{\\alpha^{n}}{\\sqrt{n!}}|n\\rangle$ is\n  totally coherent, but it is not maximally coherent state obviously in accordance with (\\ref{11}). However in our scheme, coherence state\n  $|\\alpha\\rangle$ is a maximally coherent state, thus we have reason to believe that our scheme is better than previous schemes.\n\n\\section*{Conclusions}\n In summary, we have proposed a new intrinsic basis-indpendent QC measure (IBIQCM) which reflects physical reality of the quantum state\n itself. More in details,  we have discussed some defects in previous basis-dependent QC measures, such as that basis-dependent QC measures\n cannot reflect the intrinsic QC of the quantum state itself, and the definition of the zero-coherence states is not accurate. For the\n resource-theoretic framework of quantifying coherence, in order to find better QC measures, an additional condition that intrinsic QC\n measures must be basis-independent should be supplemented. By redefining\n  the maximum classical mixture state as a complete incoherent state, i.e., it is the accurate and unique zero-coherence state for the finite\n  dimensional Hilbert space,  we have presented an IBIQCM, which can reflect the intrinsic QC property of the quantum state itself, by using\n  the relative entropy. The IBIQCM satisfies all the conditions of the resource-theoretic framework of quantifying coherence. Moreover, the\n  new IBIQCM reveals physical essence of quantum coherence, and simplifies the calculation of QC measure. Based on the importance of\n  quantifying coherence in QC investigations, our research result is an important progress.\n\n\\section*{Outlook} First, in definition of  IBIQCM,  since  the relative entropy  plays a role of  distance measures between any one quantum state and the complete incoherent state,  there are other potential candidates for distance measures, which we have not discussed here, such as fidelity, trace norm \\cite{18a} and so on.\n  one can also define many other QC measures  similar to IBIQCM by using these potential candidates. Secondly, it is  meaningful for\n  studying or restudying  various properties of quantum coherence by using IBIQCM,\nsuch as the relations between quantum coherence and other quantum resources \\cite{11,16f,16a}, the complementarity relations of quantum coherence \\cite{16d}, cohering, decohering power of quantum channels \\cite{19} and so on. Thirdly, it is important for generalizing IBIQCM to situation of infinite dimensional systems \\cite{16b,16c}, because of many quantum states and quantum physical systems are discussed in infinite dimensional systems necessarily. Finally, in our scheme, owing to the reason that zero-coherence state is unique in $d$-dimensional Hilbert space, the optimization procedure of distance measure can be avoided. Therefore, it is valuable  to consider that whether other quantum resource theory, such as entanglement and quantum discord, has this advantage that can greatly simplify the calculation.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nKlein's quartic curve, $w^3z + z^3 + w = 0$, admits a tesselation by 336 hyperbolic triangles, and has $\\text{PSL}_2(7)$ as its full group of conformal  automorphisms  \\cite{K79}.  It is characterized as the curve of smallest genus  realizing the upper bound $84(g-1)$  on the order of a group of conformal  automorphisms of a curve of genus $g>1$, given by A. Hurwitz \\cite{H93} in 1893.      Around the same time, A. Wiman characterized the curves $w^2 = z^{2g+1}-1$ and $w^2=z(z^{2g}-1)$, $g>1$,   \nas the unique curves of genus $g$ admitting cyclic automorphism groups of largest and second-largest  possible order ($4g+2$ and $4g$, respectively) \\cite{W1895}.      Two other curve families, identified much later, share a similar characterization:  the Accola-Maclachlan and Kulkarni curves of genus $g$  realize the sharp lower bound $8g+8$ on the maximum order of an automorphism group of a curve of genus $g>1$ (\\cite{A68}, \\cite{M69}, \\cite{K91}).   Collectively, we call these curves (together with one other exceptional curve discovered by Wiman) the {\\it classical curves}.   Further details are given in Section~\\ref{S:classical}.  \n\nThe triangular tesselation on the Klein quartic is an example of a {\\it map}, or imbedding of a finite connected graph on a compact oriented  surface,  so that  the complement  is a union of simply connected faces.    Maps are a special type of {\\it dessin d'enfant} (\"child's drawing\") as defined by G. Grothendieck \\cite{G84}.   The topological surface underlying a map has a canonical complex structure,  making it a compact Riemann surface, and hence,  a complex algebraic curve (see, e.g., \\cite{S57}, \\cite{JS87}).    It turns out that any curve with a map (or dessin) admits a Bely\\u\\i\\  function (a meromorphic function with at most three critical values) and is therefore, by Bely\\u\\i's theorem (\\cite{B80}, \\cite{K04}, \\cite{W97}),  defined over an algebraic number field.    Conversely, any curve defined over a number field admits a map which is geometric in the sense that its edges are geodesics in a canonical complex structure \\cite{JS96}.   The absolute Galois group acts faithfully on maps (or dessins) via its action on the coefficients of the equations defining the curves and the formulae for the  Bely\\u\\i\\  functions \\cite{JS97}.    \n\nIn this paper, we show that the classical curves can be uniformly characterized as curves admitting a strictly edge-transitive one-vertex map whose automorphisms are a proper subgroup of the full group of conformal automorphisms of the curve (see Section~\\ref{SS:regular} for definitions).  The main result is Theorem~\\ref{T:main}.  The paper includes three expository sections (Sections~\\ref{S:combinatorics}, \\ref{S:conformal} and \\ref{S:extendability}) and two sections containing new results (Sections~\\ref{S:onevertex} and \\ref{S:largeonevertex}).      Section~\\ref{S:combinatorics} gives the definition of a map and its automorphisms purely combinatorially in terms of permutation groups.  Section~\\ref{S:conformal} constructs the canonical complex structure on a surface with a map, making the map geometric and its automorphisms  conformal.    Section~\\ref{S:extendability} discusses conformal group actions on Riemann surfaces, and the question of whether and how  a finite cyclic action can extend to the action of a larger group.  Details on the classical curves, all of which admit large cyclic automorphism groups,  are also given there.   In Section~\\ref{S:onevertex} we enumerate equivalence classes of one-vertex maps according to the size of the group of map automorphisms,  staying within the purely combinatorical framework.         In Section~\\ref{S:largeonevertex} we specialize to regular and strictly edge-transitive one-vertex maps, whose automorphism groups are, respectively, maximal and second-maximal (with respect to the number of edges).   The classical curves (and some others) arise when the  second-maximal  group of map automorphisms is not the full group of conformal automorphisms of the curve (Theorem~\\ref{T:main}).\n\nOne-face or unicellular maps are  the duals of one-vertex maps, and  have been studied from various points of view.   They were enumerated by means of a recurrence relation on the number of maps of a given genus  \\cite{HZ86}, and more recently by a  different combinatorial identity \\cite{Ch11}.   Neither enumeration gives any explicit information about the automorphism groups, as ours does.    In \\cite{S01}, D. Singerman characterized Riemann surfaces admitting strictly edge-transitive  uniform unicellular dessins, a less general class of maps than the one we focus on.  One-face maps of genus $0$ (plane trees) have been extensively studied (see, e.g., \\cite{JS97}, \\cite{LZ04} and references therein), since they are easily visualized, and the action of the  absolute Galois group, restricted to these simple dessins, remains faithful.    The final paragraph of the paper summarizes the work most closely related to our main result.\n\nThe recent book \\cite{LZ04} is an excellent general reference on dessins; see also \\cite{JS96}.     The first complete exposition of the theory of maps on surfaces was given by G.A.  Jones and D. Singerman \\cite{JS78}, and it remains an excellent reference.  The paper~\\cite{MP98} describes an analytic approach to maps using quadratic differentials.\n\n  \n\\section{Maps and permutation groups}\\label{S:combinatorics}\n\nA {\\it map}  is  a finite connected graph embedded on a compact oriented  surface so that the complement is a union of simply connected {\\it faces}.   Loops and multiple edges are allowed.   A directed edge is called a {\\it dart}, symbolized by an arrowhead on the edge pointing toward one of the two incident vertices.   We usually assume that all edges carry two darts, although it is possible and sometimes useful to relax this assumption.      The set of  darts  pointing toward a given vertex receives a counter-clockwise cyclic ordering from the orientation of the ambient surface.  If the map  has $k$ edges, and the darts are labelled arbitrarily with the symbols $0,1, \\dots, 2k-1$, the cyclic ordering of the darts at each vertex gives a collection of disjoint cycles in the symmetric group $S_{2k}$; we define  $x \\in S_{2k}$ to be the product of these cycles.  We define $y \\in S_{2k}$ to be the permutation that interchanges the dart labels on each edge.  With the assumption that all edges carry two darts, $y$ is a free involution.    The {\\it monodromy group} $G_{\\cal M}$ of a map ${\\cal M}$ is  the subgroup $ \\langle x,y \\rangle \\leq S_{2k}$.  By the connectedness of the underlying graph, $G_{\\cal M}$ acts transitively on the darts.  \n \n Figure~\\ref{F:torusmap} shows a map with $k=24$ darts imbedded on the torus  of modulus $e^{\\pi i\/3}$.   (The lower left corner of the parallelogram is  $0 \\in {\\mathbb C}$ and the upper left is $e^{\\pi i \/3}$.) There are $8$ vertices, $12$ edges, and $4$ hexagonal faces.  Evidently \n \\begin{gather}\\notag\n x=(0\\ 1\\ 2)\\ (3\\ 4\\ 5)\\ (6\\ 7\\ 8)\\ (9\\ 10\\ 11)\\ (12\\ 13\\ 14)\\ (15\\ 16\\ 17)\\ (18\\ 19\\ 20)\\ (21\\ 22\\ 23), \\\\ \\notag\n y=(0\\ 10)\\ (1\\ 17)\\ (2\\ 3)\\ (4\\ 6)\\ (5\\ 13)\\ (7\\ 23)\\ (8\\ 9)\\ (11\\ 19)\\ (12\\ 22)\\ (14\\ 15)\\ (16\\ 18)\\ (20\\ 21) \\in S_{24}.\n \\end{gather}\n We will revisit this example throughout the paper.\n \n \\begin{figure}\n \\begin{center}\n \\psset{unit=1.5}\n\\begin{pspicture}(0,0)(8,4)\n\\SpecialCoor\n\\degrees[6]\n\\rput{5.5}(0,0)\n{\n\n\n \n\n\n\n\n  \n\n  \n\n  \n \n   \\rput(1;0){\\psdots[dotstyle=o,dotscale=1](1;2)}\n   \\rput(2;0){\\psdots[dotstyle=o,dotscale=1](3;2)}\n  \\rput(3;0){\\psdots[dotstyle=o,dotscale=1](2;2)(5;2)}\n     \\rput(4;0){\\psdots[dotstyle=o,dotscale=1](4;2)}\n   \\rput(5;0){\\psdots[dotstyle=o,dotscale=1](3;2)(6;2)}\n      \\rput(6;0){\\psdots[dotstyle=o,dotscale=1](5;2)}\n    \\rput(7;0){\\psdots[dotstyle=o,dotscale=1](7;2)}\n    \n   \n    \\psdots[dotstyle=*,dotscale=1](2;1)(6;1)\n    \n    \\rput(3;0){\\psdots[dotstyle=*,dotscale=1](3;2)(4;2)}\n    \\rput(4;0){\\psdots[dotstyle=*,dotscale=1](3;2)(5;2)}\n     \\rput(5;0){\\psdots[dotstyle=*,dotscale=1](4;2)(5;2)}\n    \n\n\\rput(1;1){\\psline[linestyle=dashed, dash= 2pt  5pt, dash= 2pt  5pt](0,0)(!0 3 sqrt 2 mul)}   \n\\rput(2;0){\\rput(3;1){\\psline[linestyle=dashed](0,0)(!0 3 sqrt 2 mul)}}\n\n\\rput{.5}(1;1){\\psline[linestyle=dashed, dash= 2pt  5pt](0,0)(!3 sqrt 2 mul 0)} \n\n\\rput{3.5}(7;1){\\psline[linestyle=dashed, dash= 2pt  5pt](0,0)(!3 sqrt 2 mul 0)} \n\n\\rput(2;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\\rput(5;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\n\\rput{1}(3;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\\rput{5}(3;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\n\\rput{2}(5;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\\rput{4}(5;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}\n\n\\rput(1;0){\\rput(3;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}}\n\\rput(-1;0){\\rput(4;1){\\psline[linewidth=2pt, arrows=<->](0,0)(1;1)}}\n\n\\rput{3}(2;1){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}\n\\rput{5}(2;1){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}\n   \\rput{0}(6;1){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}\n\\rput{2}(6;1){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}\n\n\\rput(1;0){\\rput{0}(4;1){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}}\n\\rput(3;0){\\rput{3}(4;2){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}}\n\n\\rput(4;0){\\rput{5}(3;2){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}}\n\\rput(4;0){\\rput{2}(5;2){\\psline[linewidth=2pt, arrows=<-](0,0)(.5,0)}}\n\n}\n\\uput{7 pt}[3.1](2;.5){\\tiny 0}\n\\uput{7 pt}[5.1](2;.5){\\tiny 1}\n\\uput{7 pt}[1.1](2;.5){\\tiny 2}\n\n\\uput{7 pt}[2.9](3;.5){\\tiny 3}\n\\uput{7 pt}[4.9](3;.5){\\tiny 4}\n\\uput{7 pt}[0.9](3;.5){\\tiny 5}\n\n\n\\uput{7 pt}[3.1](5;.5){\\tiny 18}\n\\uput{7 pt}[5.1](5;.5){\\tiny 19}\n\\uput{7 pt}[1.1](5;.5){\\tiny 20}\n\n\\uput{7 pt}[2.9](6;.5){\\tiny 21}\n\\uput{7 pt}[4.9](6;.5){\\tiny 22}\n\\uput{7 pt}[0.9](6;.5){\\tiny 23}\n\n\n\\rput(4; .5){\n\\uput{7 pt}[3.1](1;4.5){\\tiny 6}\n\\uput{7 pt}[5.1](1;4.5){\\tiny 7}\n\\uput{7 pt}[1.1](1;4.5){\\tiny 8}\n\\uput{7 pt}[2.9](1;1.5){\\tiny 15}\n\\uput{7 pt}[4.9](1;1.5){\\tiny 16}\n\\uput{7 pt}[0.9](1;1.5){\\tiny 17}\n\n\\uput{7 pt}[2.9](1;5.5){\\tiny 9}\n\\uput{7 pt}[4.9](1;5.5){\\tiny 10}\n\\uput{7 pt}[0.9](1;5.5){\\tiny 11}\n\\uput{7 pt}[3.1](1;2.5){\\tiny 12}\n\\uput{7 pt}[5.1](1;2.5){\\tiny 13}\n\\uput{7 pt}[1.1](1;2.5){\\tiny 14}\n\n}\n\n\n\\uput[4.5](1;.5){A}\n\\rput(1;.5){\\uput[3](1.75;1){C}}\n\\rput(1;.5){\\uput[1.5](3.5;1){A}}  \n\n\n\\uput[1.5](7;.5){A}\n\\rput(7;.5){\\uput[0](1.75;4){C}}\n\\rput(7;.5){\\uput[4.5](3.5;4){A}}  \n\n\n\\rput(7;.5){\\uput[1.5](1.75;3){B}}\n\\rput(7;.5){\\rput(1,75;3){\\uput[0](1.75;4){D}}}\n\\rput(7;.5){\\rput(1,75;3){\\uput[4.5](3.5;4){B}}} \t\n\n\\end{pspicture}\n\n\n \\caption{A map on a torus.  Opposite sides of the parallelogram are identified.}\\label{F:torusmap}\n\\end{center}\n\\end{figure}\n \n For any map with monodromy group $\\langle x, y \\rangle$, the cycles  of $(xy)^{-1}=yx^{-1}$ describe closed oriented circuits (paths of darts arranged tip-to-tail) which bound the faces of the map.     To see this, start with a  dart $\\delta_1$ pointing toward a given vertex $v_1$.   $x^{-1}\\delta_1$ is the next dart after $\\delta_1$ in the {\\it clockwise} cyclic ordering of the darts pointing toward $v_1$.  Let $\\delta_2 = yx^{-1}\\delta_1$,   the ``reversal\" of $x^{-1}\\delta_1$, pointing along the same edge,  toward an adjacent vertex, $v_2$.   (It may happen that $v_2 = v_1$;  in this case the edge carrying $\\delta_2$  is a loop at $v_1$).  Repeat the process staring with $\\delta_2$, obtaining $\\delta_3 =y x^{-1}\\delta_2$,  pointing toward an adjacent vertex, $v_3$ (possibly $v_3 = v_2$ or $=v_1$).   After finitely many steps, say, $n\\geq 1$ of them,   $\\delta_n=\\delta_1$, that is, we arrive again at the original dart.    At no step does an  edge incident with a vertex $v_i$ intervene between the edge carrying  $\\delta_i$ and the edge carrying  $\\delta_{i+1}$.  Thus the  circuit $\\delta_1, \\delta_2, \\dots, \\delta_{n-1}$ traverses the boundary of a unique face of ${\\cal M}$, namely, the face which is on the left from the point of a view of a traveller following the circuit.   With this convention, each dart belongs to the boundary of a unique face.    For the map in Figure~\\ref{F:torusmap},\n $$yx^{-1}=(0\\ 3\\ 13\\ 22\\ 20\\ 11)\\ (1\\ 10\\ 8\\ 23\\ 12\\ 15)\\ (2\\ 17\\ 18\\ 21\\ 7\\ 4)\\ (5\\ 6\\ 9\\ 19\\ 16\\ 14) \\in S_{24}.$$\n  The four cycles specify the boundaries of the hexagonal faces centered at $C$, $A$, $B$, $D$, respectively.\n  \n The topological genus $g$ of the surface underlying a map is easily determined from the formula for the Euler characteristic, $2g-2 =| \\{\\text{vertices}\\}| -|\\{\\text{edges}\\}| + |\\{\\text{faces}\\}|$, where the number of vertices  is   the number of cycles in $x$, and the number of faces is  the number of cycles in $(xy)^{-1}$ ($1$-cycles are counted).  The number of edges is the number of $2$-cycles in a free involution on $2k$ symbols, that is, $k$.   \n \n \n     If there is a permutation in $S_{2k}$ which {\\it simultaneously} conjugates the monodromy generators $x_1,y_1 \\in G_{{\\cal M}_1}$ of a map ${\\cal M}_1$ to, respectively,  the monodromy generators $x_2,y_2 \\in G_{{\\cal M}_2}$ of another map ${\\cal M}_2$,   we say that the groups $G_{{\\cal M}_1}$, $G_{{\\cal M}_2}$ are {\\it strongly conjugate}.\n     \\begin{dfn} Two maps  ${\\cal M}_1$, ${\\cal M}_2$ are {\\em equivalent} if their monodromy groups are strongly conjugate. \n       \\end{dfn}\n    \\noindent  A map $\\cal M$ is (non-trivially) equivalent to itself if and only if there is a permutation in $S_{2k}$,  not contained in $G_{\\cal M}$,  which commutes with both monodromy generators and hence centralizes $G_{\\cal M}$.    \n     Since conjugate subgroups have conjugate centralizers, the {\\it automorphism group\\,} of a map,  \n$$\\text{Aut}({\\cal M})=\\text{Cent}_{S_{2k}}(G_{\\cal M}) = \\{\\pi \\in S_{2k} \\mid \\pi g = g \\pi \\text{\\ for all\\ } g \\in G_{\\cal M}\\},$$\n is well-defined (up to isomorphism) on equivalence classes  of maps.   \n     \n            The equivalence relation we adopt is  appropriate  for studying the faithful action of the absolute Galois group on maps:   if maps from distinct equivalence classes  are in the same Galois orbit, their monodromy groups are conjugate but not strongly conjugate (see, e.g., \\cite{JS97}).  We do not pursue this since we focus on maps on surfaces defined over ${\\mathbb Q}$, which are fixed by the Galois action.  \n              \n\\subsection{Regular and edge-transitive maps}\\label{SS:regular}\n   \\begin{dfn} A map ${\\cal M}$ has {\\em type} $(n,r)$, if $n$ is the lcm  (least common multiple) of the vertex valencies and $r$ is the lcm of the face  valencies.  (The valence of a face is the number of darts in its bounding circuit.)  The map is {\\em uniform} if all vertices have valence equal to $n$ and all faces have valence equal to $r$.\n   \\end{dfn} \n\n   The map ${\\cal M}$ is called {\\it regular} if $\\text{Aut}({\\cal M})$ acts transitively on the set $D$ of darts.    This implies that the map is uniform, and in particular,  that every dart has the same local incidence relations.     The map in Figure~\\ref{F:torusmap} is uniform of type $(3,6)$.  It is also regular: $\\text{Aut}({\\cal M}) \\simeq {\\mathbb Z}_6 \\ltimes ({\\mathbb Z}_2 \\oplus {\\mathbb Z}_2)$, where  ${\\mathbb Z}_2 \\oplus {\\mathbb Z}_2$ is the factor group generated by translations carrying $A\\mapsto B$ and $A\\mapsto C$, modulo the normal subgroup generated by their squares.    The study of regular maps goes back at least to Klein and Dyck, and perhaps as far as Kepler (see \\cite{CM72}, Chapter 8); for more recent work see, e.g., \\cite{C09}, \\cite{JSW10}.\n   \n   A map for which $\\text{Aut}({\\cal M})$ is  transitive on edges (but not necessarily on darts) is called {\\it edge-transitive}.  In this case there are at most two sets of local incidence relations since a dart might not be equivalent (via an automorphism) to its reversal.   Thus there are at most two distinct face-valences, and at most two distinct vertex valences.    We call a map which is edge-transitive but not regular  {\\it strictly\\,} edge transitive (\"half-regular\" in \\cite{S01}). \n \n    \n\n\n  \n\n\\section{Uniformization of maps and surfaces}\\label{S:conformal}\n\n  Let $G=G_{\\cal M}=\\langle x, y \\rangle$ be  the monodromy group of a map ${\\cal M}$ of type $(n,r)$.  Let $\\Gamma=\\Gamma(n,r)$ be the  group with presentation \n\\begin{equation}\\label{E:gammapres}\n\\Gamma(n,r)= \\langle \\xi_1,\\xi_2, \\xi_3 \\mid \\xi_1^n = \\xi_2^2 = \\xi_3^{r} = \\xi_1\\xi_2\\xi_3=1 \\rangle.\n\\end{equation}\nThere is an obvious surjective homomorphism $\n\\theta: \\Gamma \\rightarrow G$, defined by  \n\\begin{equation}\\label{E:theta}\n \\theta: \\xi_1 \\mapsto x, \\quad \\xi_2 \\mapsto y,\\quad \\xi_3 \\mapsto (xy)^{-1}.\n \\end{equation}\nFor $\\delta \\in D$,  where $D$ is the set of darts of ${\\cal M}$, let $G_\\delta$ denote the isotropy subgroup $\\{g \\in G \\mid \\delta g = \\delta\\}$.  \n \\begin{dfn}\\label{D:mapsubgp} \nThe subgroup $M = \\theta^{-1}(G_\\delta)\\leq \\Gamma$,  is called the {\\rm canonical map subgroup} for ${\\cal M}$.\n\\end{dfn}\n\\noindent Transitivity of the permutation group $(G, D)$  implies all $G_\\delta$ are conjugate, so $M$ is well-defined (up to conjugacy) independent of $\\delta$.     The {\\it core} of $M$ in $\\Gamma$ is the normal subgroup\n$$M^\\ast = \\bigcap_{\\gamma \\in \\Gamma}\\gamma^{-1}M\\gamma.$$\n$\\Gamma\/M^\\ast$ acts on the set $| \\Gamma\/M |$ of  cosets  $M$ in $\\Gamma$, as follows:   \n $M^*\\gamma_1:  M\\gamma \\mapsto M\\gamma\\gamma_1 $, $ \\gamma,\\gamma_1 \\in \\Gamma.$  \n One may verify that the action is well-defined, faithful and transitive.\n\n\n \\begin{lem}\\label{L:mapsubgroup}\n The permutation groups $(G, D)$ and $(\\Gamma\/M^*,| \\Gamma\/M |)$ are isomorphic.\n\\end{lem}\n\\begin{proof} It suffices to verify that the diagram \\begin{equation}\\notag\n\\begin{matrix} G &\\times& D& \\rightarrow & D \\\\\n\\overset{i \\big\\downarrow}{\\ } & & \\overset{b \\big\\downarrow}{\\ } && \\overset{b \\big\\downarrow}{\\ } \\\\\n\\displaystyle{\\frac{\\Gamma}{M^\\ast}} &\\times & \\displaystyle{\\left| \\frac{\\Gamma}{M} \\right|}&\\rightarrow  &\\displaystyle{\\left| \\frac{\\Gamma}{M} \\right|}\n\\end{matrix}\n\\end{equation}\nis commutative, where arrows denote group actions, and the maps $i$ and $b$ are defined as follows:  $i: G \\rightarrow \\Gamma\/M^\\ast$ maps $g \\mapsto M^\\ast\\theta^{-1}(g)$;   for fixed but arbitrary $\\delta \\in D$,   $b: D \\rightarrow | \\Gamma\/M |$  maps  $b:\\delta g \\mapsto   M\\theta^{-1}(g)$, $g \\in G$.  To show that $i$  is a well-defined isomorphism, one first verifies that $M^\\ast = \\text{ker}(\\theta)$; then $i$ is simply the inverse of the canonical isomorphism $\\Gamma \/ \\text{ker}(\\theta) \\rightarrow G$.  $b$ is a well-defined bijection according to the following argument, whose steps are reversible:    for $g_1, g_2 \\in G$, $\\delta g_1 =\\delta  g_2  \\iff g_2g_1^{-1} \\in G_\\delta$ $\\iff \\theta^{-1}(g_2g_1^{-1}) \\subseteq M$  $\\iff M\\theta^{-1}(g_2)(\\theta^{-1}(g_1))^{-1} = M$ $\\iff M\\theta^{-1}(g_1) = M\\theta^{-1}(g_2)$.  \n\\end{proof}\n \n \\subsection{Groups with signature}  \n \n The map subgroup $M$, and its overgroup $\\Gamma$,  are examples of  {\\it groups with signature}.  These groups act  properly discontinuously by conformal isometries on  one of the three simply connected Riemann surfaces:   the Riemann sphere (${\\mathbb P}^1$), the complex plane (${\\mathbb C}$), or the Poincar\\'e upper half plane (${\\mathbb H}$).  This fact allows us to obtain a canonical complex structure on the topological surface underlying a map.\n \n  \nA group $\\Lambda=\\Lambda(\\sigma)$ with {\\it signature} $\\sigma=(h; r_1, r_2, \\dots, r_s)$ has presentation\n\\begin{equation}\\label{E:sigpres}\n\\Lambda(\\sigma)=\\langle a_1, b_1, \\dots, a_h, b_h, \\xi_1, \\dots \\xi_s \\mid \\xi_1^{r_1} = \\dots = \\xi_s^{r_s} = \\prod_{i=1}^ha_i^{-1}b_i^{-1}a b \\prod_{j=1}^s \\xi_j =1 \\rangle,   \n\\end{equation}  \nand acts by conformal isometries on  ${\\mathbb P}^1$,  ${\\mathbb C}$, or  ${\\mathbb H}$,  depending on whether the number\n$$\\mu(\\sigma) = 2h-2 + \\sum_{i=1}^s\\bigl(1-\\frac{1}{r_i}\\bigr)$$\nis, respectively, negative, $0$, or positive.      If $s>0$, the $r_i$  are called the {\\it periods} of  $\\Lambda$.  All periods are $>1$, and the number of occurrences of a given period is the number of conjugacy classes of elements of that order in $\\Lambda$.  Signatures which differ only by a permutation of the periods determine isomorphic groups and are considered the same.   If $s=0$, $\\Lambda$ is a torsion-free {\\it surface group}, with signature $(h; -)$, isomorphic to the fundamental group of a surface of genus $h$.\n  If $\\Delta$  is any subgroup of finite index $n$ in $\\Lambda(\\sigma)$,  then $\\Delta$ is also a group with signature $\\sigma'$ and   the {\\it Riemann-Hurwitz relation} states that  $\\mu(\\sigma') = n \\mu(\\sigma)$. \n\n\n  The orbit space  $\\,{\\cal U}\/\\Lambda$, where $\\,{\\cal U}$ denotes ${\\mathbb P}^1$,  ${\\mathbb C}$, or  ${\\mathbb H}$ as appropriate,  is a compact Riemann surface of genus $h$ with $s$ distinguished  points, over which the projection $\\,{\\cal U} \\rightarrow {\\cal U}\/\\Lambda$ branches.    The complex structure on ${\\cal U}\/\\Lambda$ is the one which makes the branched covering map $\\,{\\cal U} \\rightarrow {\\cal U}\/\\Lambda$ holomorphic.     The uniformization theorem of Klein, Poincar\\'e and Koebe states that  every compact surface of genus $h$ can be  obtained in this way; moreover one may choose the uniformizing group $\\Lambda$ to be a surface group, so that there are no distinguished points, and the covering is regular.    \n  \n    When $\\mu(\\sigma)>0$, which is the case except for finitely many signatures, $\\Lambda(\\sigma)$ is a co-compact {\\it Fuchsian group} (discrete subgroup of $\\text{PSL}(2,{\\mathbb R})$ containing no parabolic elements) and $2\\pi\\mu(\\sigma)$ is the hyperbolic area of a fundamental domain for the action of $\\Lambda(\\sigma)$ on ${\\mathbb H}$.   The normalizer $N_{\\text{PSL}(2,{\\mathbb R})}(\\Lambda)$ of a Fuchsian group $\\Lambda$  is itself a Fuchsian group, containing $\\Lambda$ with finite index.  \n    \\begin{lem}\\label{L:confaut} The automorphism group of  $\\,{\\mathbb H}\/\\Lambda$  is isomorphic to the finite quotient group $N_{\\text{PSL}(2, {\\mathbb R})}(\\Lambda)\/\\Lambda$.\n    \\end{lem}\n For proofs and further details, see, e.g.,  \\cite{JS78},  \\cite{JS87}, \\cite{SK92}. \n\n\n\\subsection{The canonical Riemann surface of a map}\\label{SS:universalmaps}\n\nLet $M \\leq \\Gamma=\\Gamma(n,r)$ be the canonical map subgroup of a map ${\\cal M}$ of type $(n,r)$.     Since $M$ is a group with signature, the quotient space \n$\\,{\\cal U}\/M$  is a compact Riemann surface, known as the {\\em canonical Riemann surface} for ${\\cal M}$.  The canonical Riemann surface contains  the map ${\\cal M}$ {\\it geometrically},  that is,  the edges of ${\\cal M}$ are geodesics, faces are regular geodesic polygons,  and face-centers are well-defined points.   To see this,  we pass to the {\\it universal map} $\\hat{\\cal M}_{n,r}$ of type $(n,r)$.  $\\hat{\\cal M}_{n,r}$ is a regular map  imbedded on one of  the simply connected Riemann surfaces $\\,{\\cal U}$ $={\\mathbb P}^1$,  ${\\mathbb C}$, or  ${\\mathbb H}$.  It can be constructed as follows.  \nLet $T \\in {\\cal U}$ be a geodesic triangle with vertices $a,b,c$,  at which the  interior angles are $\\pi\/n$, $\\pi\/2$, $\\pi\/r$ respectively (see Figure~\\ref{F:universalmap}).   Reflections in the sides of $T$ generate a discrete group of isometries of $\\,{\\cal U}$ whose sense-preserving subgroup  (of index $2$) is isomorphic to $\\Gamma=\\Gamma(n,r)$.    The generators $\\xi_1$, $\\xi_2$, $\\xi_3$ $\\in \\Gamma$ correspond to the rotations about  $a$, $b$, $c$ through angles of, respectively,  $2\\pi\/n$, $\\pi$, $2\\pi\/r$;  the product of the three rotations  (considered as a product of six side reflections)  is easily seen to be trivial.    Let ${\\bf \\hat e}$ be the geodesic segment   $\\overline{ab} \\cup \\xi_2(\\overline{ab})$, directed toward the vertex $a$.  The map with dart set   $\\hat D=\\{\\gamma({\\bf \\hat e}) \\mid \\gamma \\in \\Gamma\\}$ and vertex set    $\\hat V=\\{\\gamma(a) \\mid \\gamma \\in \\Gamma\\}$ is the universal map $\\hat{\\cal M}_{n,r}$.  \n   Darts meet in sets of $n$ at each vertex and the angle between any two consecutive edges is  $2\\pi\/n$.   The faces are regular geodesic $r$-gons centered at the $\\Gamma$-images of $c$.      \n The permutation group $(\\Gamma, \\hat D)$ is isomorphic to $(\\Gamma,  |\\Gamma|)$, the  right regular representation of $\\Gamma$ on itself.  Thus the map is regular and $\\text{Aut}(\\hat{\\cal M}_{n,r}) = \\text{Aut}(\\Gamma, |\\Gamma |) = \\Gamma$.  \n \n  If the number $\\dfrac 12 - \\dfrac 1n -\\dfrac 1r$    is negative, $0$, or positive, $\\hat{\\cal M}_{n,r}$ lies on ${\\mathbb P}^1$,  ${\\mathbb C}$, or  ${\\mathbb H}$, respectively.  In the negative case, $\\hat{\\cal M}_{n,r}$ is a  central projection of one of the Platonic solids onto the circumscribed sphere; in the positive case, it is one of infinitely many regular tesselations of the hyperbolic plane.  In the $0$ case, it is a tesselation of ${\\mathbb C}$ by squares, hexagons, or equilateral triangles.  (Figure~\\ref{F:torusmap} is a portion of $\\hat{\\cal M}_{3,6}$.)   See, e.g., \\cite{CM72}.\n \n \n \n \\begin{figure}\n \\begin{center}\n \\psset{unit=1.4}\n \\begin{pspicture}(0,.5)(7,4)\n\n\\psarc(-3,0){5}{22.4}{37}\n\\psarc(1,0){2}{72}{90}\n\\psline(1,2)(1,3)\n\\uput[90](1,3){a}\n\\uput[180](1,2){b}\n\\uput[315](1.6, 1.9){c}\n\n\\uput[0](.8, 2.6){ \\tiny $\\dfrac{\\pi}{n}$}\n\\uput[0](1.1,2.15){ \\tiny $\\dfrac{\\pi}{r}$}\n\\pspolygon(1,2)(1.1,2)(1.1,2.1)(1, 2.1)\n\n\\uput[0](1.1, 1.4){$T$}\n\n\\psarc[linestyle=dashed](0,0){5}{22.4}{37}\n\\psarc[linestyle=dashed](4,0){2}{72}{90}\n\\psline[linewidth=1.5pt, arrows=->](4,1)(4,3)\n\n\\uput[90](4,3){a}\n\\uput[315](4.6, 1.9){c}\n\\uput[180](4,1){a$\\xi_2$}\n\n\\psarc[linewidth=1.5pt, arrows=->](4,.5){2.5}{37}{90}\n\\uput[315](6, 2){a$\\xi_2\\xi_1$}\n\n\n\\uput[0](3.8, 2.6){ \\tiny $\\dfrac{\\pi}{n}$}\n\n\\uput[0](4.15, 2.7){ \\tiny $\\dfrac{\\pi}{n}$}\n\n\n\\pspolygon(4,2)(4.1,2)(4.1,2.1)(4, 2.1)\n\n\\uput[0](3.5,2){$\\bf \\hat e$}\n\n\\uput[0](5.2,2.8){${\\bf \\hat e} \\xi_1$}\n\n\\psdots(4,1)(4,3)(6,2)\n\n\\psdots[dotstyle=o,dotscale=1](4.6, 1.9)\n\\end{pspicture}\n\\caption{Construction of $\\hat{\\cal M}_{n,r}$, the universal map of type $(n,r)$}\\label{F:universalmap}\n\\end{center}\n\\end{figure}\n\n  The orbifold map ${\\cal U} \\rightarrow {\\cal U}\/M$ is a local isometry.   Hence    \nthe image of $\\hat D$ in the canonical Riemann surface $\\,{\\cal U}\/M$  is  a set of directed geodesic segments.  These segments are in bijection with the set $|\\Gamma\/M|$ of right cosets of $M$ in $\\Gamma$, and   $\\Gamma$ acts by right multiplication on $|\\Gamma\/M|$.   If $\\gamma \\in \\Gamma$ fixes {\\it every} coset, it belongs to  the core $M^\\ast$ of $M$ in $\\Gamma$.   Hence the universal map $\\hat{\\cal M}_{n,r}$ projects to a map $\\,\\hat {\\cal M}_{n,r}\/M$ on $\\,{\\cal U}\/M$ -- the promised geometric map -- with  monodromy group    $(\\Gamma\/M^*,| \\Gamma\/M |)$.   With Lemma~\\ref{L:mapsubgroup}, we conclude that {\\it every} map of type $(n,r)$ is a quotient of the universal map $\\,\\hat{\\cal M}_{n,r}$,  by a subgroup $M$ of finite index in the group $\\Gamma=\\Gamma(n,r)$, and may be assumed to lie on its canonical Riemann surface.\n  \n  \n\nThe general theory of covering spaces specializes to the category of maps of type $(n,r)$.  Thus, if ${\\cal M}_1$ and ${\\cal M}_2$ are two  finite  maps with canonical map subgroups $M_1, M_2 \\leq \\Gamma=\\Gamma(n,r)$, then ${\\cal M}_1$ is a finite covering of ${\\cal M}_2$ if and only if  $M_1$ is conjugate within $\\Gamma$ to a subgroup of finite index in $M_2$.    \n   In particular, if $M_2=N_\\Gamma(M_1)$, the normalizer of $M_1$ in $\\Gamma$, then $\\text{Aut}({\\cal M}_1) \\simeq M_2\/M_1$.  This leads to the following Lemma.   \n\n\n   \\begin{lem}\\label{L:mapaut} Let ${\\cal M}$ be a map of type $(n,r)$ with canonical map subgroup $M \\leq \\Gamma = \\Gamma(n,r)$.  Then $\\text{Aut}({\\cal M}) \\leq \\text{Aut}(X)$, where $X ={\\cal U}\/M$ is the canonical Riemann surface for ${\\cal M}$,  and $\\text{Aut}(X)$ is the group of conformal automorphisms of $X$.\n    \\end{lem}\n  \\begin{proof}   $\\text{Aut}({\\cal M})$ is isomorphic to  $N_\\Gamma(M)\/M$.  \n Since $\\Gamma$, $M$ and $N_\\Gamma(M)$ are discrete subgroups of $\\text{Isom}^+({\\cal U})$, the sense-preserving isometries of  ${\\,\\cal U}$,  $$ \\frac{N_\\Gamma(M)}{M} \n\\leq \\frac{N_{\\text{Isom}^+({\\cal U})}(M)}{M} =\\text{Aut}({\\,\\cal U}\/{M}).$$ \nThe equality on the right is a consequence of Lemma~\\ref{L:confaut} if ${\\cal U}={\\mathbb H}$.\n  \\end{proof}\n  \n      In the local geometry of the canonical Riemann surface, an automorphism $\\alpha \\in \\text{Aut}({\\cal M})$ which  preserves a  face acts   as a rotation about the fixed face-center through an angle $2\\pi\/r'$, where $r'$ is a divisor of the face-valence (and a divisor of $r$ if the type is $(n,r)$).     Similarly, if $\\alpha$  preserves an edge while transposing its darts, the geometric action is a rotation through an angle $\\pi$ about the fixed edge midpoint.     These are consequences  of corresponding facts about the  universal map $\\hat{\\cal M}_{n,r}$,  together with the fact that the covering projection ${\\cal U} \\rightarrow {\\cal U}\/M$ is a local isometry.  See \\cite{JS78}.\n\n\n\n\n   \n\n \n \\section{One-vertex maps with automorphisms}\\label{S:onevertex}\n \n We now enumerate one-vertex maps, up to equivalence, according to the size of their automorphism groups.\n\n\n Let ${\\cal M}$ be a map with one vertex and $k$  edges.    The monodromy generator $x \\in S_{2k}$  is a  $2k$-cycle specifying the counterclockwise cyclic ordering of the  darts (arbitrarily labelled $0,1,2, \\dots, 2k-1$) around the vertex.  The other monodromy generator, $y\\in S_{2k}\n $, is a free involution whose  $k$ disjoint transpositions specify how the darts pair off into edges.   \nSince all $2k$-cycles are conjugate in $S_{2k}$,   we may assume, up to map equivalence,  that $x$ is the standard $2k$-cycle\n\\begin{equation}\\label{E:standardcycle}\n(0\\ 1\\ 2\\   \\dots\\  2k-1).\n\\end{equation}\nThen ${\\cal M}$  is determined  \nby the free involution $y$, \nwhich can be chosen in \n$$\\prod_{i=0}^{k-1}\\begin{pmatrix} 2k-2i \\\\2 \\end{pmatrix} =  (2k-1)\\cdot (2k-3)\\cdot \\dots \\cdot 3 \\cdot 1 = (2k-1)!!$$ \nways.   Figure~3 shows three of the $15$ one-vertex maps with three edges.  \n\nThe number of equivalence classes of one-vertex maps with $k$ edges is smaller than $(2k-1)!!$,  since some of the maps have non-trivial automorphisms.   \n\n\\begin{theorem}\\label{T:MapAuts} Let ${\\cal M}$ be a one-vertex map with $k$ edges and monodromy group $G_{\\cal M} = \\langle x,y \\rangle$,  where $x$ is a $2k$-cycle and  and $y$ is a free involution.  Then:  \n(i) $\\text{Aut}({\\cal M}) \\leq \\langle x \\rangle \\simeq {\\mathbb Z}_{2k}.$   (ii) If $\\text{Aut}({\\cal M})=\\langle x^p \\rangle$, $p$ a divisor of $2k$,  the monodromy groups\n$\\langle x, y_s \\rangle,$  where $y_s=x^{-s}yx^s$, $s=0, \\dots p-1$, determine distinct but equivalent maps.  Conversely, if ${\\cal M}$ is equivalent to ${\\cal M}^\\prime$ with monodromy group $G_{{\\cal M}^\\prime }= \\langle x, y^\\prime \\rangle$,  then $\\text{Aut}({\\cal M}) \\simeq \\text{Aut}({\\cal M}^\\prime) = \\langle x^p \\rangle$ for some divisor $p$ of $2k$, and $y^\\prime = y_s$ for some $s$, $0 \\leq s \\leq p-1$.\n \\end{theorem}\n \\begin{proof}   The centralizer of $G_{\\cal M}$  is contained in the centralizer of each of its generators;  in particular, it is contained in the centralizer of $\\langle x \\rangle$.  The latter is an abelian transitive permutation group which, by a well-known result in the theory of permutation groups, is its own centralizer (see, e.g., \\cite{S87}, Section 10.3).    It follows that $\\text{Aut}({\\cal M}) \\leq \\langle x \\rangle \\simeq {\\mathbb Z}_{2k}.$  Now suppose $p$ is a divisor of $2k$ and $\\text{Aut}({\\cal M})=\\langle x^p \\rangle$.   The permutations $x^s$, $s=1, \\dots, p-1$ do not commute with $y$ (otherwise they would be automorphisms of ${\\cal M}$), hence there are $p$ distinct free involutions $y_s=x^{-s}y{x^s}$,  $s=0, 1, \\dots, p-1$,  belonging to strongly conjugate monodromy groups $\\langle x, y_s \\rangle$, determining distinct but equivalent maps.  Conversely, if ${\\cal M}$ is equivalent to ${\\cal M}^\\prime$, the monodromy groups $G_{{\\cal M}}= \\langle x, y \\rangle$ and $G_{{\\cal M}^\\prime }= \\langle x, y^\\prime \\rangle$ are strongly conjugate, with conjugate centralizers and hence isomorphic automorphism groups.    Since both automorphism groups are contained in $\\langle x \\rangle$,  $\\text{Aut}({\\cal M}) = \\text{Aut}({\\cal M}^\\prime)=\\langle x^p \\rangle$ for some divisor $p$ of $2k$.  By strong conjugacy of $G_{\\cal M}$ and $G_{{\\cal M}^\\prime}$,  there exists a permutation $\\sigma \\in S_{2k}$, such that $\\sigma^{-1}x\\sigma = x$ and $\\sigma^{-1}y\\sigma = y'$.  In particular $\\sigma \\in \\text{Cent}_{S_{2k}}(\\langle x \\rangle) = \\langle x \\rangle$.  If $y^\\prime \\ne y$, $\\sigma \\in \\langle x \\rangle \\setminus \\langle x^p \\rangle$.  Thus $\\sigma = x^d$, for $d$ a non-multiple of $p$.  If $d>p$,  $\\sigma^{-1} y\\sigma={\\sigma^\\prime}^{-1}y{\\sigma^\\prime}$, where $\\sigma^\\prime = x^{d-p}$, since $x^p$ commutes with $y$.   It follows that all possible $y^\\prime$ are obtained by taking $\\sigma \\in \\{x, x^2, \\dots, x^{p-1}\\}$.\n \\end{proof}\n \n\n \n\\subsection{Free involutions with prescribed commutation property}\\label{SS:prescribedcommuters}\n\n If $x \\in S_{2k}$ is a $2k$-cycle, and $p$ is a divisor of $2k$, then $x^p$ is a product of $p$ cycles of length $d=2k\/p$.  Assuming $x$ is the standard $2k$-cycle \\eqref{E:standardcycle},  \n the cycles are\n \\begin{equation}\\label{E:p-cycles}\n C_j = (j\\  j+p\\  \\dots\\ j+2k-p),\\qquad j=0, 1, \\dots, p-1.\n \\end{equation}\n  Let $c_y: S_{2k} \\rightarrow S_{2k}$  denote conjugation by a free involution $y \\in S_{2k}$.   If  $y$ commutes with $x^p$,  then  $\\langle c_y \\rangle \\simeq{\\mathbb Z}_2$   block-permutes the $p$ $d$-cycles at \\eqref{E:p-cycles},  while preserving their internal cyclic orderings.     The permutation $c_y$ either  preserves a given cycle  or interchanges two distinct cycles in a block-orbit of length $2$.     Suppose $c_y$ preserves the cycle  $C_j$, as well as its internal cyclic order.  Then for some positive integer $t$ and for every symbol $a \\in\\  C_j$, $c_y(a) = a+tp \\pmod{2k}$.  Because $c_y$ has order $2$,  $2tp \\equiv 0 \\pmod{2k}$ or, equivalently, \n  $tp \\equiv 0 \\pmod k$, which has the unique (mod $k$) solution $t=k\/p=d\/2$, implying that {\\it $d$ is even}.      On the other hand, \n  if $c_y: S_{2k}\\rightarrow S_{2k}$  interchanges two cycles $C_i$ and $C_j$, $i<j$, preserving their internal cyclic orders,  there exists an integer $t \\geq 0$ such that  every symbol $i+ap \\in\\  C_i$, is mapped to $c_y(i+ap) = j+(a+t)p$, and, reciprocally,  every symbol  $j + bp \\in \\ C_j$, is mapped to $c_y(j+bp) = (i + (b-t)p$ (all symbols are reduced mod $2k$).  The {\\it shift parameter} $t$ is not uniquely determined, but may take any value $0, 1, \\dots, d-1$.  There is no requirement that $d$ be even.   It follows that a free involution commuting with $x^p$ is uniquely determined by the following data:\n  \\begin{itemize}\n  \\item  the set of cycles $C_j$ fixed by $c_y$ (empty if $d=2k\/p$ is odd);\n  \\item the (possibly empty) set of cycle pairs  $[C_i, C_j]$, $i<j$, transposed by $c_y$;\n  \\item for each transposed pair, a choice of shift parameter $t$, $0 \\leq t \\leq d-1$.\n  \\end{itemize}\nLet $q=\\lfloor p\/2\\rfloor$, the greatest integer less than or equal to $p\/2$.  If $c_y$ transposes $m \\leq q$ cycle pairs, those pairs can be chosen, up to reordering, in\n$$\\dfrac{1}{m!}\\begin{pmatrix}p \\\\ 2 \\end{pmatrix} \\begin{pmatrix}p-2 \\\\ 2 \\end{pmatrix}  \\dots \\begin{pmatrix}p-(2m-2) \\\\ 2 \\end{pmatrix}= \\biggl(\\frac 12 \\biggr)^m \\frac{p!}{m!(p-2m)!}$$\nways.    Each pair requires a choice of shift parameter  from among $0,1,2,\\dots, d-1$, so the number of choices is multiplied by $d^m$.  \nIf $d$ is odd, there are no unpaired cycles and $m=p\/2=q$, an integer since $p$ is even.   If $d$ is even,  unpaired cycles are possible so the number of choices is a sum over all possible $m$.  This yields the following Lemma.  \n\\begin{lem}\\label{L:freecommuters} Let $\\overline{\\nu}_p$ be the number of free involutions in $S_{2k}$ commuting with $x^p$, where $x$ is a $2k$-cycle and  $p$ a divisor of $2k$.   Then\n\\begin{equation}\\notag\n\\overline{\\nu}_p = \n\\begin{cases} \\displaystyle{\\sum_{m=0}^{ q } \\biggl(\\frac d2 \\biggr)^m \\frac{p!}{m!(p-2m)!}} &\\text{if $d$ is even} \\\\\n\\displaystyle{\\biggl(\\frac d2 \\biggr)^q\\  \\frac{p!}{q!} }&\\text{if $d$ is odd},\n\\end{cases}\n\\end{equation}\nwhere $d=2k\/p$, and  $q=\\lfloor \\frac p2 \\rfloor$. \n\\end{lem}\n\\noindent \nThe Lemma yields $\\overline{\\nu}_{2k} = (2k-1)!!$,  as it should, since every free involution commutes with $x^{2k}=\\{1\\}$, and $\\overline{\\nu}_{1} = 1$, since  $x^k$ is the unique (free) involution in $\\text{Cent}_{S_{2k}}(x) = \\langle x \\rangle$.  \n\n\n\\subsection{One-vertex maps with prescribed automorphism}\n\nIf $y$ commutes with $x^p$, but no lower power of $x$, then \n$\\langle x^p \\rangle$ is the {\\it full} automorphism group of a one-vertex map whose monodromy group is strongly conjugate to $\\langle x, y \\rangle$.  \n\n\n\n\\begin{theorem}\\label{T:fullautcount} Let $\\nu_p$ be the number of one-vertex maps with $k$ edges whose full automorphism group is $\\langle x^p \\rangle$, $p$ a divisor of $2k$.   Then  \n$\\nu_p = \\overline{\\nu}_p + \\sum_{i=1}^s (-1)^i\\sigma_i$,\nwhere \n$$\\sigma_1 = \\sum_{1 \\leq j \\leq s}\\overline{\\nu}_{p\/p_j},\\quad \\sigma_2 = \\sum_{1\\leq j <k \\leq s}\\overline{\\nu}_{p\/p_jp_k}, \\quad \\sigma_3 = \\sum_{1 \\leq j<k<l \\leq s}\\overline{ \\nu}_{p\/p_jp_kp_l},  \\quad \\dots, $$\nand $p_i$, $i=1, \\dots, s$, are the prime divisors of $p$.\n\\end{theorem}\n\\begin{proof}\n Let $p'$ be any proper divisor of $p$.  If $p$ is a prime, $p'=1$ and $\\overline{\\nu}_1=1$.  Hence $\\nu_p = \\overline{\\nu}_p-1 = \\overline \\nu_p-\\sigma_1$, proving the theorem in this case.  If $p$ is not prime, $p=p' p_1^{\\epsilon_1}p_2^{\\epsilon_2}\\dots p_s^{\\epsilon_s}$, where  at least one of the $\\epsilon_i > 0$.  Reordering the $p_i$ if necessary we may assume that the first $t\\leq s$ of the $\\epsilon_i > 0$, while the remaining $s-t$ are $0$.  Hence,  without loss of generality, $p=p'p_1^{\\epsilon_1}p_2^{\\epsilon_2}\\dots p_t^{\\epsilon_{t}}$,  with all $\\epsilon_i > 0,$ $i=1,2, \\dots, t$.  \n  Each map with automorphism group $\\langle x^{p'} \\rangle$ contributes $1$ to $\\overline\\nu_p$; it suffices to show that such a map contributes $-1$ to$ \\sum_{i=1}^s (-1)^i\\sigma_i$, so that its total contribution to the formula in the theorem is $0$.  We use a standard inclusion\/exclusion argument.   If $t=1$, the map contributes $-1$ to $-\\sigma_1$ and $0$ to the remaining summands.  If $t=2$, the map contributes $-2$ to $-\\sigma_1$, $1$ to $\\sigma_2$, and $0$ to the remaining summands.  In general, for arbitrary $t\\leq s$, the contribution is $-(\\begin{smallmatrix} t \\\\ 1\\end{smallmatrix})$ to $-\\sigma_1$,  $(\\begin{smallmatrix} t \\\\ 2\\end{smallmatrix})$ to $\\sigma_2$, etc.  It follows from the binomial theorem that the total contribution is\n  $$1 + \\sum_{i=1}^t(-1)^i\\begin{pmatrix}t \\\\ i\\end{pmatrix}=(1-1)^t=0.$$  Thus the formula counts only those maps whose automorphism group is equal to $\\langle x^p \\rangle$.   \n  \\end{proof}\n \\begin{cor}\\label{C:equivcount} The number of equivalence classes of one-vertex maps with $k$ edges whose full automorphism group is  $\\langle x^p \\rangle$, $p$ a divisor of $2k$, is $\\nu_p\/p$.\n\\end{cor}\n\\begin{proof}  Theorem~\\ref{T:MapAuts}(ii) combined with Theorem~\\ref{T:fullautcount}. \n  \\end{proof}\n\n  In particular, the number of equivalence classes of strictly edge-transitive,  one-vertex maps with $k$ edges is\n\\begin{equation}\\label{E:edgetranscount}\n\\nu_2\/2 = \\begin{cases}\nk\/2 &\\text{if $k$ is even;}\\\\\n(k-1)\/2 &\\text{if $k$ is odd.}\n\\end{cases}\n\\end{equation}\nSince $\\nu_1 = 1$, there is a unique regular one-vertex map with $k$ edges.  Figure~\\ref{F:k=3} illustrates the case $k=3$.\n\n\\begin{figure}\n\\begin{center}\n\\begin{pspicture}(0,-2)(13,2.5)\n\\SpecialCoor\n\\degrees[6]\n\\rput(1;0){\\psdots(1;1)}\n\\rput(6;0){\\psdots(1;1)}\n\\rput(10;0){\\psdots(1;1)}\n\\rput(1;0){\\psarc[linewidth=0.04](1;1){1}{0}{1}}\n\\rput(1;0){\\psarc[linewidth=0.04](1;1){1}{2}{3}}\n\\rput(1;0){\\psarc[linewidth=0.04](1;1){1}{4}{5}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;0)}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;1)}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;2)}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;3)}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;4)}}\n\\rput(1;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;5)}}\n\n\n\\rput(6;0){\\psarc[linewidth=0.04](1;1){1.2}{0}{3}}\n\\rput(6;0){\\psarc[linewidth=0.04](1;1){1}{1}{4}}\n\\rput(6;0){\\psarc[linewidth=0.04](1;1){.8}{2}{5}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1.2;0)}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1.2;3)}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;1)}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;4)}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(0.8;2)}}\n\\rput(6;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(0.8;5)}}\n\n\n\n\n\n\\rput(10;0){\\psarc[linewidth=0.04](1;1){1}{1}{2}}\n\\rput(10;0){\\psarc[linewidth=0.04](1;1){1}{3}{4}}\n\\rput(10;0){\\psarc[linewidth=0.04](1;1){1}{5}{6}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;1)}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;2)}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;3)}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;4)}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;5)}}\n\\rput(10;0){\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(1;6)}}\n\n\\rput(1;0){\n\\uput{9 pt}[0.4](1;1){\\tiny 0}\n\\uput{9 pt}[1.4](1;1){\\tiny 1}\n\\uput{9 pt}[2.4](1;1){\\tiny 2}\n\\uput{9 pt}[3.4](1;1){\\tiny 3}\n\\uput{9 pt}[4.4](1;1){\\tiny 4}\n\\uput{9 pt}[5.4](1;1){\\tiny 5}\n}\n\n\\rput(6;0){\n\\uput{9 pt}[0.4](1;1){\\tiny 0}\n\\uput{9 pt}[1.4](1;1){\\tiny 1}\n\\uput{9 pt}[2.4](1;1){\\tiny 2}\n\\uput{9 pt}[3.4](1;1){\\tiny 3}\n\\uput{9 pt}[4.4](1;1){\\tiny 4}\n\\uput{9 pt}[5.4](1;1){\\tiny 5}\n}\n\n\n\\rput(10;0){\n\\uput{9 pt}[0.4](1;1){\\tiny 0}\n\\uput{9 pt}[1.4](1;1){\\tiny 1}\n\\uput{9 pt}[2.4](1;1){\\tiny 2}\n\\uput{9 pt}[3.4](1;1){\\tiny 3}\n\\uput{9 pt}[4.4](1;1){\\tiny 4}\n\\uput{9 pt}[5.4](1;1){\\tiny 5}\n}\n\n\\uput[0](-.5, -.6){$x= (0\\ 1\\ 2\\ 3\\ 4\\ 5)$}\n\\uput[0](-.5,-1){$y=  (0\\ 1)(2\\ 3)(4\\ 5)$}\n\\uput[0](-.5, -1.4){$yx^{-1} = (0\\ 4\\ 2)\\ (1)\\ (3)\\ (5)$}\n\n\\uput[0](4.5, -.6){$x= (0\\ 1\\ 2\\ 3\\ 4\\ 5)$}\n\\uput[0](4.5,-1){$y=  (0\\ 3)(1\\ 4)(2\\ 5)$}\n\\uput[0](4.5, -1.4){$yx^{-1} = (0\\ 2\\ 4)\\ (1\\ 3\\ 5)$}\n\n\\uput[0](9, -.6){$x= (0\\ 1\\ 2\\ 3\\ 4\\ 5)$}\n\\uput[0](9,-1){$y=  (1\\ 2)(3\\ 4)(5\\ 0)$}\n\\uput[0](9, -1.4){$yx^{-1} = (0)\\ (2)\\ (4)\\ (1\\ 3\\ 5)$}\n\n\n\\uput[0](-.8,1.6){(a)}\n\\uput[0](4,1.6){(b)}\n\\uput[0](8.6,1.6){(c)}\n\\end{pspicture}\n\\caption{One-vertex maps with three edges.  (a) and (c) are equivalent and strictly edge-transitive;  (b) is regular.}\\label{F:k=3}\n \\end{center}\n \\end{figure}\n  \n \\section{Conformal automorphisms of Riemann surfaces}\\label{S:extendability}\n \n A finite group G acts by conformal automorphisms on a compact Riemann surface $X$ of genus $g$ if and only if  there is a group $\\Lambda(\\sigma)$, with signature  $\\sigma$,  containing a normal surface subgroup $M_g$, with signature $(g; -)$,  and a {\\it surface-kernel epimorphism\\,} $\\rho: \\Lambda \\rightarrow G$ making the sequence of homomorphisms\n \\begin{equation}\\label{E:shortexact}\n \\{1\\} \\rightarrow M_g \\hookrightarrow \\Lambda(\\sigma) \\overset{\\rho}{\\rightarrow} G \\rightarrow \\{1\\}\n \\end{equation}\n exact (see \\cite{M61}, or \\cite{JS87}, Theorem 5.9.5).   The surface $X$ is conformally equivalent to ${\\,\\cal U}\/M_g$, where ${\\,\\cal U}$ is the universal covering space.   $G$ is isomorphic to $\\Lambda(\\sigma)\/M_g$,  and the Riemann-Hurwitz relation requires $2g-2=|G|\\mu(\\sigma)$.    We say that {\\it $G$ acts in genus $g$ with signature $\\sigma$}.     \n \\begin{dfn}  A {\\em $\\sigma$-generating vector  for a finite group $G$} is  an ordered set of elements, one for each generator of $\\Lambda(\\sigma)$,   which generate $G$ and fulfill  the corresponding relations in $\\Lambda(\\sigma)$ (and possible other relations).  \n \\end{dfn}\n\\noindent A $\\sigma$-generating vector for $G$ determines a surface kernel epimorphism $\\rho$ satisfying \\eqref{E:shortexact}, and conversely.  Hence \n\\begin{lem}$G$ acts in genus $g$ with signature $\\sigma$ if and only if $G$ has a $\\sigma$-generating vector.\n\\end{lem}\n  \n  \n Let  $\\sigma=(0;n, m, r)$, so that\n$\\Lambda(\\sigma)$ is a {\\it triangle group\\,}, and let $G={\\mathbb Z}_n$.  By a general theorem of Maclachlan \\cite{M65} (see also Harvey \\cite{H66}),  an abelian group of order $n$ has an $(0;n,m,r)$-generating vector  if an only if $\\text{lcm}(m,r)=n$.  We establish a normal form for such a generating vector.  Let ${\\mathbb Z}_n = \\langle x \\mid x^n=1 \\rangle$, and let  $\\langle x^a, x^b, x^c \\rangle$ be a $(0;n,m,r)$-generating vector.  Then $a+b+c \\equiv 0 \\pmod n$ since the product of the elements must be the identity.   For the elements to have the appropriate orders,  $n\/(a,n) =n$, $n\/(b,n) = m$, and $n\/(c,n) = r$, where $(\\ ,\\ )$ denotes the gcd (greatest common divisor) of two integers.    If $\\rho$ in \\eqref{E:shortexact} is post-composed with an automorphism of $G$, an equivalent generating vector, specifying a topologically equivalent action, is obtained (see \\cite{B90}, Prop. 2.2).  Hence we may assume $a=1$.   Writing ${\\mathbb Z}_n$  additively, with the generator  $1$,  we define \n\\begin{dfn} The {\\em normal form} for an {\\it $(0;n,m,r)$-generating vector\\,} for ${\\mathbb Z}_n = \\langle \\{0,1,\\dots, n\\}, + \\rangle$ is  \n \\begin{equation}\\label{E:ZnGenVector}\n \\langle\\  1,\\  b,\\  c\\  \\rangle, \\quad 1+b+c \\equiv 0 \\ (\\text{mod\\ }n), \\quad n\/(b,n)=m, \\quad n\/(c,n)=r.\n \\end{equation}\n \\end{dfn}\n\n\\subsection{Finite maximality and extendability}\n \\begin{dfn} A group with signature is {\\em finitely maximal} if it is not a subgroup of finite index in any other group with signature.\n \\end{dfn}\n\\noindent    If $\\Lambda(\\sigma)$  in \\eqref{E:shortexact} is finitely maximal, then $\\Lambda(\\sigma)\/M_g$ is necessarily the {\\it full\\,} automorphism group of $X={\\,\\cal U}\/M_g$. \n    \n      Finite maximality is not generally a property of the abstract group $\\Lambda(\\sigma)$, but depends on the particular imbedding of the group in $\\text{Isom}^+({\\cal U})$.  However,  there are some pairs of signatures $(\\sigma, \\sigma')$ such that every imbedding of a group $\\Lambda(\\sigma)$ with signature $\\sigma$ in $\\text{Isom}^+({\\cal U})$ extends to an imbedding of an overgroup $\\Lambda(\\sigma')$.  Signature pairs of this type were considered by Greenberg in \\cite{G74}, and fully classified by Singerman \\cite{S72}.    For each such pair, there arises a delicate problem in finite group theory:   for each subgroup-overgroup pair $(G,G')$, with $[G':G]=[\\Lambda(\\sigma'):\\Lambda(\\sigma)]$, and such that $G'$ has a $\\sigma'$-generating vector and $G$ has a $\\sigma$-generating vector, determine conditions under which the $G'$-action is an extension of the $G$-action {\\em on the same surface}.  Equivalently, determine conditions under which the surface-kernel epimorphism $\\rho$ in \\eqref{E:shortexact} is the restriction of a surface-kernel epimorphism (with the same kernel) $\\rho':\\Lambda(\\sigma') \\rightarrow G'$.  This problem is treated comprehensively in the papers \n  \\cite{BCC02} and \\cite{BC99}.   The signature pairs $(\\sigma, \\sigma')$ where both signatures are triangular and the first is {\\it cyclic-admissable} -- that is, $\\Lambda(\\sigma)$ admits surface-kernel epimorphisms onto a cyclic $G$ -- are given in Table~\\ref{Ta:cyclicadmissible}, which uses the case nomenclature established in \\cite{BCC02}.   The index $[\\Lambda(\\sigma'):\\Lambda(\\sigma)]=[G':G]$ is  easily computed using the Riemann-Hurwitz relation and the fact that $M_g$ is a subgroup of both groups.  In cases N6 and N8 in the table, $\\Lambda(\\sigma)$ is a normal subgroup of  $\\Lambda(\\sigma')$.   We give complete descriptions of these cases below.  The necessary numerical conditions on  $k$ are implicit but not stated in \\cite{BCC02}.\n  \\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{| r | c | c | c | c |}\n\\hline\nCase & $\\sigma$ & $\\sigma'$ & $[\\Lambda(\\sigma') :\\Lambda(\\sigma)]$& Conditions\\\\\n\\hline\nN6 & $(0;k,k,k)$ & $(0;3,3,k)$ & $3$& $k\\geq 4$ \\\\\nN8 & $(0;k,k,u)$ & $(0;2,k,2u)$ & $2$ &$u | k$, $k\\geq 3$ \\\\\nT1 & $(0;7,7,7)$ &$(0;2,3,7)$ & $24$& -\\\\\nT4 & $(0;8,8,4)$ & $(0;2,3,8)$ & $12$& - \\\\\nT8 & $(0;4k,4k,k)$ & $(0;2,3,4k)$ & $6$ & $k\\geq 2$ \\\\\nT9 & $(0;2k,2k, k)$ & $(0;2,4,2k)$ & $4$ &$k\\geq 3$  \\\\\nT10 & $(0;3k,k,3)$ & $(0;2,3,3k)$ & $4$&$k\\geq 3$  \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Cyclic-admissible signatures ($\\sigma$) and possible extensions ($\\sigma'$)}\\label{Ta:cyclicadmissible}\n\\end{table}\n\n\n\\begin{lem}[Case N6]\\label{L:N6}\n A ${\\mathbb Z}_k$ action with signature $(0;k,k,k)$  has an extension of type {\\rm N6} to a $G$ action with signature $(0;3,3,k)$  only if $3^2 \\nmid k$ and $p \\equiv 1 \\pmod 3$ for every prime divisor $p\\ne 3$ of $k$.   The numerical conditions imply that ${\\mathbb Z}_k$ has an automorphism $\\beta$ of order $3$, acting non-trivially on each $p$-Sylow subgroup, $p \\ne 3$.  Then $G \\simeq {\\mathbb Z}_3 \\ltimes_\\beta {\\mathbb Z}_k$ and the normal form of the generating vector for the ${\\mathbb Z}_k$ action is $\\langle 1, \\beta(1), \\beta^2(1)\\rangle$.\n\\end{lem}\n\\begin{proof}  Since ${\\mathbb Z}_k$ is a normal subgroup, $G$ has the semi-direct product structure ${\\mathbb Z}_3 \\ltimes_{\\beta}{\\mathbb Z}_k$, where $\\beta$ is a (possibly trivial) automorphism of ${\\mathbb Z}_k$ of order a divisor of $3$.  In addition, \n any non-trivial subgroup of $G$, being  a homomorphic image of the group with signature $(0;3,3,k)$, is generated by at most two elements of order $3$.  Let $^pG$ denote a $p$-Sylow subgroup of $G$.   $^3G$ has the structure ${\\mathbb Z}_3 \\ltimes_{\\beta}{\\mathbb Z_{3^s}}$, $s \\geq 0$, but since it is generated by a most two elements of order $3$, $s \\leq 1$.   Hence $3^2 \\nmid k\n $.  For a prime divisor $p\\ne 3$  of $k$, $^pG$ is cyclic, and $\\beta$ induces an automorphism $\\beta_p$ on $^pG$.  The subgroup ${\\mathbb Z}_3 \\ltimes_{\\beta_p}(^pG) \\leq G$  is  generated by two elements of order $3$ only if  $\\beta_p$ is a non-trivial automorphism of order $3$.  This implies that $p \\equiv 1 \\pmod 3$, and yields the stated necessary conditions on $k$.    If $1$ is the additive generator of $^pG$,  then the element $1 + \\beta_p(1) + \\beta_p^2(1) \\in ^pG$ is fixed  by $\\beta_p$ and hence must be the identity (equivalently, $1 + \\beta_p(1) + \\beta_p^2(1) \\equiv 0 \\pmod {|^pG|}$).  Now,   ${\\mathbb Z}_k$ is  the direct sum of its $p$-Sylow subgroups; hence $\\beta = \\oplus_p \\beta_p$,  where the sum is over the prime divisors of $k$.  ($\\beta_p$ is trivial if $p=3$.)  It follows that $\\langle \\beta \\rangle$ is a subgroup of order $3$ in the automorphism group of ${\\mathbb Z}_k$ which acts non-trivially on each $p$-Sylow subgroup, $p \\ne 3$.  Taking  $1$ as the additive generator of ${\\mathbb Z}_k$,  we have $1+\\beta(1) + \\beta^2(1) \\equiv 0 \\pmod k$.   Thus $\\langle 1, \\beta(1), \\beta^2(1) \\rangle$ is the normal form of a $(0;k,k,k)$ generating vector for a ${\\mathbb Z}_k$ action extendable to a $G$ action.\n\\end{proof}\n\n\n\n\\begin{lem}[Case N8]\\label{L:N8}\n A ${\\mathbb Z}_k$ action with signature $(0;k,k,u)$ has an  extension of type {\\rm N8} to a $G$ action with signature $(0;2,k,2u)$ if and only if either\n \\begin{enumerate} \n \\item $k$ is odd, $G\\simeq  {\\mathbb Z}_{2k}$,  $u=k$, and the normal form of the ${\\mathbb Z}_k$ generating vector is $\\langle 1, 1, k-2 \\rangle$; \n \\item $k$ is even, $G={\\mathbb Z}_2 \\oplus {\\mathbb Z}_k$,  $u=k\/2$,  and the normal form of the ${\\mathbb Z}_k$ generating vector is $\\langle 1, 1, k-2 \\rangle$; \n\\item $k \\ne 2,4, p^s, 2p^s$, ($p$ an odd prime),  $G \\simeq {\\mathbb Z}_2\\ltimes_{\\alpha}  {\\mathbb Z}_k$,  (non-abelian, non-dihedral),  where $\\alpha$ is an automorphism of  ${\\mathbb Z}_k$ of order $2$,  $\\alpha(1) \\ne -1$, and the normal form of the ${\\mathbb Z}_k$ generating vector is $\\langle 1, \\alpha(1), k-(1+\\alpha(1))\\rangle$.  \n \\end{enumerate}\n \\end{lem}\n \\begin{proof} The group $G$ has the semi-direct product structure ${\\mathbb Z}_2 \\ltimes_{\\alpha}{\\mathbb Z}_k$, where $\\alpha$ is a (possibly trivial) automorphism of ${\\mathbb Z}_k$ of order a divisor of $2$.   If $G$ is non-abelian, $\\alpha$ is non-trivial and the generating vector is of the form $\\langle 1, \\alpha(1), c \\rangle$.   $\\alpha (1) \\ne -1$, for then $c=0$, and $u = 1$.  Such an $\\alpha$ exists if and only if $k \\ne 2,4, p^s, 2p^s$, ($p$ an odd prime) \\cite{IR82}.  $G$ has the (non-dihedral) semi-direct product structure ${\\mathbb Z}_2\\ltimes_{\\alpha}  {\\mathbb Z}_k$.  If  $G$ is abelian, $\\alpha$ is trivial, and by Maclalchlan's lcm condition \\cite{M65}, either $u=k$ and $G \\simeq {\\mathbb Z}_{2k}$,  or $u=k\/2$ and $G={\\mathbb Z}_2 \\oplus {\\mathbb Z}_k$.   \n \\end{proof}\n\n\\subsection{Cyclic actions on the classical curves}\\label{S:classical}\n\nThe classical curves admit automorphism groups which are ``large'' with respect to the genus $g$;  in particular, the maximal cyclic subgroups of automorphisms have order at least $\\geq 2g+1$.   \nThe paper \\cite{K97} gives a  precise definition of ``large,''  and has   interesting points of contact with our work.   (See, e.g., Section~3 of \\cite{K97}, where cyclic group actions with a fixed point are described in terms of  side-pairings of hyperbolic polygons; being one-face maps, the polygons can be viewed as the duals of one-vertex maps in which  side-pairings correspond to dart-pairings on edges.)  \n\n\n\nThe Wiman  curves of genus $g>1$ are\n\\begin{gather}\\notag\nw^2 = z^{2g+1}-1 \\qquad\\text{(type  I)} \\\\ \\notag\nw^2=z(z^{2g}-1) \\qquad\\text{(type  II)} \\\\ \\notag\nw^3 = z^4 + 1 \\quad(g=3) \\qquad\\text{(type III)}.\n\\end{gather}\nTheir maximal cyclic automorphism groups  are  \n\\begin{gather}\\notag\n(w,z) \\mapsto (-w, ze^{2\\pi i\/2g+1})\\quad \\text{of order $4g+2$}\\quad\\text{(type I)} \\\\ \\notag\n(w,z) \\mapsto (we^{\\pi i\/2g}, ze^{\\pi i\/g}) \\quad \\text{of order $4g$}\\quad \\text{(type II)}\\\\ \\notag\n(w,z) \\mapsto (we^{2\\pi i\/3}, ze^{2\\pi i\/4}) \\quad \\text{of order $4g=12$} \\quad\\text{(type III)}.\n\\end{gather}\n  Signatures for the cyclic actions can be deduced by analyzing the ramifications of the branched covering projection  \n   $P_z: (w,z) \\mapsto z$.    For example, for the Wiman curve of type I,  $P_z: (w,z) \\mapsto z$ has degree $4g+2$ with ramification of order $2g+1$ over $z=0$, order $4g+2$ over $z=\\infty$, and order $2$ over the $(2g+1)$st  roots of unity.   The unique normal form of a generating vector for ${\\mathbb Z}_{4g+2}$ with a $(0;4g+2, 2g+1, 2)$ signature is easily seen to be $\\langle 1, 2g, 2g+1\\rangle$.  By similar arguments one sees that the signatures of the Type II and III actions are $(0;4g,4g,2)$ and $(0;12,4,3)$, respectively.   Corresponding generating vectors, unique in normal form,  are given in Table~\\ref{Ta:largecyclic}.   The type I signature is finitely maximal (it does not appear in Table~\\ref{Ta:cyclicadmissible}), so ${\\mathbb Z}_{4g+2}$ is the full automorphism group of the type I curve.  The  type II and type III signatures are not:  the cyclic actions extend to larger conformal actions according to cases N8 and T10 of Table~\\ref{Ta:cyclicadmissible}, with $k=4g$, $k=4$, respectively.    The full\\footnote{If $g=2$,  $SD_{16}$ extends  to an action of $\\text{GL}_2({\\mathbb Z}_3)$, of order $48$,  with signature $(0; 2,3,8)$ (Case T11 in \\cite{BCC02}).} automorphism groups are  \n  \\begin{gather}\\notag\n  \\text{SD}_{8g} = \\langle a,b \\mid  a^{4g}=b^2 = 1,\\quad b^{-1}ab = a^{2g-1} \\rangle\\quad\\text{acting with signature $(0;2,4g,4)$}\\quad\\text{(type II)}\\\\ \\notag\n  \\text{H}_{48} = \\langle\\text{Kulkarni's group of order $48$}\\rangle \\quad  \\text{acting with signature $(0;2,3,12)$}\\quad\\text{(type III)}.\n  \\end{gather}\nA presentation of $H_{48}$ is given in \\cite{K97}.  \n\n  In the late 1960's R.D.M. Accola \\cite{A68} and C. Maclachlan \\cite{M69} independently showed that the genus $g$ curve with equation  \n    $w^2=z^{2g+2}-1$ \n          has full automorphism group \n           \\begin{equation}\\notag\n   \\text{AM}_{8g+8} = \\langle a, b \\mid a^{2g+2}=b^4=1, \\quad (ab)^2=[a,b^2]=1\\rangle,  \n   \\end{equation}\n   of order $8g+8$, and that, for infinitely many $g \\geq 2$, this curve realizes the maximal order of an automorphism group of a surface of genus $g$.   (A. M. Macbeath had shown in 1961 \\cite{M61a} that the Hurwitz bound $84(g-1)$ is not attained for  infinite sequences of genera.)        \n       Thus, for every $g \\geq 2$, there exists a surface $X_g$ of genus $g$ such that $|\\text{Aut}(X_g)| \\geq 8g+8$, and the bound is sharp for infinitely many $g$.  \n  If  $g \\equiv -1 \\pmod 4$, \n  there is a second surface of genus $g$, identified by Kulkarni in 1991 \\cite{K91}, having full automorphism group of order $8g+8$, acting with the same signature as $\\text{AM}_{8g+8}$.    The full\\footnote{If $g=3$,  $K_{32}$ extends to an action by a group of order $96$ acting with signature $(0;2,3,8)$ (Case T11 in \\cite{BCC02}).}  automorphism group of the Kulkarni surface is\n \\begin{equation}\\notag\n \\text{K}_{8g+8} = \\langle a, b \\mid a^{2g+2}=b^4=1, \\quad (ab)^2=1,\\quad b^2ab^2=a^{g+2}\\rangle.  \n   \\end{equation}\n  Both $\\text{AM}_{8g+8}$ and $\\text{K}_{8g+8}$   act with signature $(0;2,4, 2g+2)$ and contain the cyclic subgroup  $\\langle a \\mid a^{2g+2}=1\\rangle$ of index $4$.\n  \n  The distinct normal forms of the  generating vectors for the cyclic actions, given in Table~\\ref{Ta:largecyclic}, show that the Accola-Maclachlan and Kulkarni curves, when they exist in the same genus, are distinct.    \n The index $4$ extensions to $\\text{AM}_{8g+8}$ and $\\text{K}_{8g+8}$ realize Case T9 in Table~\\ref{Ta:cyclicadmissible}. \n\n  The Klein quartic is uniquely determined by the normal form of the generating vector for its maximal cyclic group of automorphisms (${\\mathbb Z}_7$) given in Table~\\ref{Ta:largecyclic}.  This action has an index $3$ extension (type N6) to the group ${\\mathbb Z}_3 \\ltimes_\\beta {\\mathbb Z}_7$, $\\beta(1)=2$, and a further extension (type T1) to   $\\text{PSL}_2(7)$.  It is worth noting that there is another ${\\mathbb Z}_7$ action in genus $3$, with the same signature $(0;7,7,7)$ as the one determining the Klein quartic.  This action has generating vector $\\langle 1,1,5 \\rangle$ and determines the Wiman curve of type I via an N8 extension, with $u=k=2g+1=7$.   \n  \n   \\begin{table}[h]\n\\begin{center}\n\\begin{tabular}{| l | l | l | l | }\n\\hline\nCurve & Group & Signature & Gen. Vector \\\\\n\\hline\nWiman type I& $ {\\mathbb Z}_{4g+2}$ & $(0; 4g+2, 2g+1, 2)$ & $\\langle 1, 2g, 2g+1 \\rangle$   \\\\\nWiman type II & $ {\\mathbb Z}_{4g}$ &$ (0; 4g,\\  4g,\\  2)$ & $\\langle 1, 2g-1, 2g \\rangle$ \\\\\nAccola-Maclachlan & $ {\\mathbb Z}_{2g+2}$ & $(0;2g+2, 2g+2, g+1)$ & $\\langle 1,\\  1,\\  2g \\rangle$  \\\\\nKulkarni & $ {\\mathbb Z}_{2g+2}$ & $(0;2g+2, 2g+2, g+1)$ & $\\langle 1, g+2, g-1 \\rangle$  \\\\\nWiman type III &  ${\\mathbb Z}_{12}$ & $(0; 12,\\ 4\\ ,3)$ & $\\langle 1, \\ 3,\\  8\\rangle$\\\\\nKlein & ${\\mathbb Z}_{7}$ & $(0;7,\\ 7,\\ 7)$ & $\\langle 1,\\  2,\\ 4 \\rangle$   \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Maximal cyclic actions on the classical curves}\\label{Ta:largecyclic}\n\\end{table}\n\n\n\\section{The classical curves and edge-transitive one-vertex maps}\\label{S:largeonevertex}\n\nWe have seen that a one-vertex map ${\\cal M}$ with $k$ edges has $\\text{Aut}({\\cal M})=\\langle x^p \\rangle \\leq {\\mathbb Z}_{2k}=\\langle x \\rangle$, where $p$ is a divisor of $2k$.   If ${\\cal M}$ is edge-transitive, $\\text{Aut}({\\cal M})$ must have at least $k$ elements, hence $p=1$ or $p=2$.  If $p=1$, ${\\cal M}$ is regular.  If $p=2$, ${\\cal M}$ is {\\it strictly\\,} edge transitive, i.e., edge transitive but not regular.  \n\nThe following theorem (stated  in terms of unifacial dessins) was proved in \\cite{S01}, and completely characterizes regular one-vertex maps.  \n\n\\begin{theorem}\\label{T:regular1vertex}  Let $\\,{\\cal M}$ be a regular, one-vertex map on a surface $X$   of genus   $g>0$.  Then $X$ is the Wiman curve of type I  and ${\\cal M}$ has $2g+1$ edges, or $X$ is the Wiman curve of type II and $\\,{\\cal M}$ has $2g$ edges.  \n\\end{theorem}\n\n\n\n\n\n The regular one-vertex maps in genus $1$ are shown in Figure~\\ref{F:reggenus1}.  The type I map on the left has two faces, centered at $A$  and $B$, and is the geometrization of the map in Figure~\\ref{F:k=3} (b).   It lies on the elliptic curve of modulus $e^{2\\pi i\/3}$.  The type II map on the right has one face, centered at $A$, and lies on the elliptic curve of modulus $i$.    These curves are the natural genus $1$ analogues of the Wiman curves of type I,\\, II\\,,  characterized (up to conformal equivalence) by admitting an automorphism of order greater than $2$  ($6$, $4$, respectively) which {\\em fix a point}.   Note that the map in Figure~\\ref{F:torusmap} also lies on the type I elliptic curve.  \n\n\\begin{figure}\n\\begin{center}\n\\begin{pspicture}(-.6,-.4)(5.6,2.2)\n\\SpecialCoor\n\\degrees[6]\n\n\\rput(1;1){\\pspolygon[linestyle=dashed](1;1)(1;2)(1;3)(1;4)(1;5)(1;6)}\n\\psdots(1;1)\n\\psline[linewidth=1.5 pt, arrows=->](.5,0)(1;1)\n\\rput(!0 3 sqrt 2 div){\\psline[linewidth=1.5 pt, arrows=<-](.5,0)(1;1)}\n\n\\rput{1}(1;0)\n{\n\\psline[linewidth=1.5 pt, arrows=->](.5,0)(1;1)\n\\rput(!0 3 sqrt 2 div){\\psline[linewidth=1.5 pt, arrows=<-](.5,0)(1;1)}\n}\n\n\\rput(1;0){\n\\rput{2}(1;1)\n{\n\\psline[linewidth=1.5 pt, arrows=->](.5,0)(1;1)\n\\rput(!0 3 sqrt 2 div){\\psline[linewidth=1.5 pt, arrows=<-](.5,0)(1;1)}\n}\n}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;1)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;2)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;3)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;4)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;5)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](1;6)}\n\\rput(1;1){\\uput[1.5](1;2){A}}\n\\rput(1;1){\\uput[4.5](1;4){A}}\n\\rput(1;1){\\uput[0](1;0){A}}\n\n\\rput(1;1){\\uput[1.5](1;1){B}}\n\\rput(1;1){\\uput[3](1;3){B}}\n\\rput(1;1){\\uput[4.5](1;5){B}}\n\n\n\\uput{8 pt}[0.8](1;1){\\tiny 0}\n\\uput{8 pt}[1.8](1;1){\\tiny 1}\n\\uput{8 pt}[2.8](1;1){\\tiny 2}\n\\uput{8 pt}[3.8](1;1){\\tiny 3}\n\\uput{8 pt}[4.8](1;1){\\tiny 4}\n\\uput{8 pt}[5.8](1;1){\\tiny 5}\n\n\\rput(4;0)\n{\n\\rput(1;1){\\pspolygon[linestyle=dashed](.8,.8)(-.8,.8)(-.8,-.8)(.8,-.8)}\n\\rput(1;1){\\psdots[dotstyle=o,dotscale=1](.8,.8)(-.8,.8)(-.8,-.8)(.8,-.8)}\n\\rput(1;1){\\uput[1.5](.8,.8){A}}\n\\rput(1;1){\\uput[1.5](-.8,.8){A}}\n\\rput(1;1){\\uput[4.5](-.8,-.8){A}}\n\\rput(1;1){\\uput[4.5](.8,-.8){A}}\n\\psdots(1;1)\n\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=->](-.8,0)(0,0)}\n\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(.8,0)}\n\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(0,-.8)}\n\\rput(1;1){\\psline[linewidth=1.5 pt, arrows=<-](0,0)(0,.8)}\n\\uput{8 pt}[0.3](1;1){\\tiny 0}\n\\uput{8 pt}[1.8](1;1){\\tiny 1}\n\\uput{8 pt}[3.3](1;1){\\tiny 2}\n\\uput{8 pt}[4.8](1;1){\\tiny 3}\n}\n\n\\end{pspicture}\n\\caption{Regular one-vertex maps in genus $1$}\\label{F:reggenus1}\n\\end{center}\n\\end{figure}\n\n \\subsection{Strictly edge-transitive one-vertex  maps}\n  If ${\\cal M}$ is strictly edge-transitive with $k$ edges, the edge midpoints are permuted regularly in a $k$-cycle (with trivial isotropy subgroup) and the face centers fall into two orbits,  each having isotropy subgroup of order equal to the corresponding face-valence.   By Wiman's bound on the order of an automorphism  (or an automorphism with a fixed point if $g=1$), $k \\leq 4g+2$.  $k=4g+1$ is not possible, by the Riemann-Hurwitz relation.  \n  \n     \\begin{theorem}\\label{T:edgetransaction} Let $\\,{\\cal M}$ be strictly edge-transitive one-vertex map on a surface $X$   of genus   $g\\geq 1$.  The action of $\\text{Aut}({\\cal M})\\simeq{\\mathbb Z}_k$ has signature and generating vector\n\\begin{equation}\\label{E:edgetransaction}\n{\\mathbb Z}_k \\quad (0;k, l_1, l_2) \\quad \\langle 1, t, k-(t+1) \\rangle, \\quad  l_1 = {k}\/{(t,k)}, \\ l_2 = {k}\/{(t+1,k)}, \n\\end{equation}\nfor some $t$,  $0 < t <k\/2$, $t\\ne (k-1)\/2$.  The notation $(\\ ,\\ )$ indicates the gcd (greatest common divisor) of two integers.\n\n\\end{theorem}\n\n\\begin{proof}  In the monodromy group $G_{\\cal M}=\\langle x, y \\rangle$, the free involution $y$ commutes with $x^2$ but not with $x$, hence it is obtained from a pairing of the two $k$-cycles \n$$C_0 = (0\\  2\\  \\dots\\ 2k-2) \\qquad\\text{and}\\qquad C_1=(1\\ 3\\ \\dots\\ 2k-1)$$\nwith a shift parameter $t$, $0 \\leq t \\leq k-1$ (cf. Section~\\ref{SS:prescribedcommuters}).  If $k$ is odd, $t = (k-1)\/2$ is excluded, since in that case, $y=x^k$ and the map would be  regular.  The permutation $(xy)^{-1}=yx^{-1}$ consists of cycles of at most two different lengths.  If  a cycle of $yx^{-1}$ contains an odd symbol $2c+1$, then \n$$2c+1 \\overset{x^{-1}}{\\quad\\mapsto\\quad}2c \\overset{y}{\\quad\\mapsto\\quad}2c + 2t + 1, $$\nwhich implies that the cycle has length $l_1$ where $l_1$ is the minimal positive integer for which\n$ 2l_1t \\equiv 0 \\pmod {2k}.$  \nIt follows that \n$ l_1 = {k}\/{(t,k)}.$    Similarly, if a cycle contains an even symbol $2c$, then \n$$2c \\overset{x^{-1}}{\\quad\\mapsto\\quad}2c-1 \\overset{y}{\\quad\\mapsto\\quad}2c -2- 2t,$$\nwhich implies that the cycle has length $l_2$, where $l_2$ is the minimal positive integer such that $-l_2(2+2t) \\equiv 0 \\pmod {2k}$.  This yields\n$ l_2 = {k}\/{(t+1, k)}.$  \nIt follows that $(xy)^{-1}$ is a product of $(t,k)$ cycles of length $l_1$ and $(t+1, k)$ cycles of length $l_2$.           The single vertex of ${\\cal M}$ is fixed by $\\text{Aut}({\\cal M})$.  The edge midpoints are permuted regularly in a $k$-cycle, and the centers of the two face types are permuted in orbits of lengths $(t,k)$ and $(t+1, k)$, with isotropy subgroups of orders $l_1$ and $l_2$, respectively.  It follows that $\\text{Aut}({\\cal M})$ acts with signature $(0;k, l_1, l_2)$.   By Theorem~\\ref{T:MapAuts}, $\\langle x, y \\rangle$ is strongly conjugate to $\\langle x, y_1\\rangle$, where $y_1=x^{-1}yx$.  Moreover $y_1$ is obtained from $y$ by replacing the shift parameter $t$ with the shift parameter $k-(t+1)$.  We note that since $(t+1,k) = (k-(t+1),k)$, the signature of the ${\\mathbb Z}_k$ action is not changed if $t$ is replaced by $k-(t+1)$.   It follows that up to map equivalence, we may assume $t<k\/2$.     If $t=0$,  the signature reduces to $(0;k,k)$ which places the map (and the ${\\mathbb Z}_k$ action) on   the Riemann sphere.    The lcm of $l_1$ and $l_2$ is $k$, so a $(k, l_1, l_2)$-generating vector for ${\\mathbb Z}_k$ exists and may be taken in the normal form $\\langle 1, t, k-(t+1)\\rangle$ (cf. \\eqref{E:ZnGenVector}).  \n \\end{proof}\n\n\n\nIn most cases, if a curve $X$ of genus $g>1$ has  a strictly edge-transitive, one-vertex map ${\\cal M}$, then $\\text{Aut}({\\,\\cal M})=\\text{Aut}(\\,X)$.  The exceptions are characterized in our final theorem, below.   If $\\text{Aut}{\\cal M} = \\text{Aut}(\\,X)$, an equation for $X$ is $w^k=z^{k\/l_1}(z-1)^{k\/ l_2}$, where $\\text{Aut}({\\cal M})\\simeq {\\mathbb Z}_k$  is generated by $(w,z) \\mapsto (w, e^{2\\pi i\/k}z)$.   \n\n\n\\begin{theorem}\\label{T:main} Let ${\\cal M\\,}$ be a strictly edge-transitive one-vertex map with $k$ edges, where $\\text{Aut}(\\,{\\cal M})={\\mathbb Z}_k$ has signature and generating vector \\eqref{E:edgetransaction}.  Let $X$ be the  canonical Riemann surface of genus $g>1$.     If $\\text{Aut}(\\,X) >  \\text{Aut}(\\,{\\cal M})$,   then \n\\begin{enumerate}\n\\item\n $k=12$, $t=3$, and $X$ is the Wiman type III  curve;\n\\item\n$k=2g+1$, $t=1$, and $X$ is the Wiman type I curve;\n\\item\n $k=2g+2$, $t=1$, and $X$ is the Accola-Maclachlan curve;\n\\item\n $k=2g+1$, $t=\\beta(1)$,   and $\\text{Aut}(X) \\simeq {\\mathbb Z}_3\\ltimes_{\\beta}  {\\mathbb Z}_k$ as in Lemma~\\ref{L:N6}; except\n\\begin{itemize}\n\\item\nif $k=7$, $t=\\beta(1)=2$,  $X$ is the Klein quartic.\n\\end{itemize}\n\\item\n $2g+2\\leq k\\leq 4g$, $t=\\alpha(1)$,  and $\\text{Aut}(X)\\simeq {\\mathbb Z}_2 \\ltimes_{\\alpha} {\\mathbb Z}_k$ as in Lemma~\\ref{L:N8}; except\n \\begin{itemize}\n   \\item\n     if $k=2g+2$, $g \\equiv -1 \\pmod 4$,  $\\alpha(1)=g+2$,  $X$ is the Kulkarni curve; or  \n   \\item\n    if $k=12$ or $24$, $g=4$ or $10$, $\\alpha(1)=7$ or $19$ (resp.),   $\\text{Aut}(X)$ contains $ {\\mathbb Z}_2 \\ltimes_{\\alpha} {\\mathbb Z}_k$ with index $3$. \n    \\end{itemize}\n\\end{enumerate}\nIn case 5, with $k=4g$ and $\\alpha(1) = 2g-1$, ${\\mathbb Z}_2 \\ltimes_{\\alpha} {\\mathbb Z}_k \\simeq \\text{SD}_{8g}$ and $X$ is the Wiman curve of type II.\n \\end{theorem}\n\n\n \\begin{proof}  If $\\text{Aut}({\\,X}) > \\text{Aut}({\\,\\cal M})$,  the signature in \\eqref{E:edgetransaction} must coincide with one of the cyclic-admissible signatures $\\sigma$ in Table~\\ref{Ta:cyclicadmissible}.  For an extension of type N6, by  the Riemann-Hurwitz relation, $k=2g+1$; if $g=3$, the N6 extension is subsumed by the T1 extension, which yields the Klein quartic.  For an extension of type N8, by the Riemann-Hurwitz relation, $k= 2g(u\/u-1)$.  For an abelian extension, there are two possibilities:  (i) $u=k\/2$ (equivalently, $k=2g+2$) and $t=1$; or (ii) $u=k$ (equivalently, $k=2g+1$) and $t=1$.  In the first case we obtain the ${\\mathbb Z}_k$ action (see Table~\\ref{Ta:largecyclic})  which determines the Accola-Maclachlan curve.   (The N8 extension to ${\\mathbb Z}_2 \\oplus {\\mathbb Z}_{2g+2}$ is subsumed by the T9 extension.)   In the second case, we obtain the cyclic action which determines the Wiman curve of type I, extending to  ${\\mathbb Z}_{4g+2}$.   For a non-abelian extension of type N8, $u \\leq k\/2$ (otherwise the extension would be cyclic of order $2k$), hence $2g+2 \\leq k \\leq 4g$.   Both the lower and upper bounds, which occur when $l_2=k\/2$, $l_2 = 2$, respectively, satisfy the necessary conditions on $k$ given in Lemma~\\ref{L:N8} for a nonabelian, non-dihedral extension to ${\\mathbb Z}_2 \\ltimes_{\\alpha} {\\mathbb Z}_k$. \n    If $g \\equiv -1 \\pmod 4$, $\\alpha(1)=g+2$ determines an automorphism of ${\\mathbb Z}_{2g+2}$ of order $2$ since $(g+2)^2 \\equiv g^2 \\equiv 1 \\pmod{2g+2}$.  Taking $k=2g+2$, we have  $k-(1 + \\alpha(1)) = g-1$ which has has order $g+1$ (mod $k$) since $(g+1)(g-1)=(2g+2)(g-1)\/2  \\equiv 0 \\pmod k.$  This yields the ${\\mathbb Z}_k$ action which determines the Kulkarni surface.  (The N8 extension is subsumed by the T9 extension, or by the T4 extension if $g=3$).  In the last two exceptional cases ($k=12$, $24$) the N8 extension is subsumed by the T8 extension.  The Wiman type III curve arises from the T10 extension. \n \\end{proof}\n\nSubsets of the curves mentioned in Theorem~\\ref{T:main} have been characterized in ways related to our characterization.   A. Meleko\\u glu and D. Singerman (\\cite{MS08}, Section~6) characterized \nthe Wiman curves of type I and II, and the Accola-Maclachlan curves  (all of which are hyperelliptic)  as the unique curves of genus $g>1$  admitting {\\it  double-star maps}.  These are two-sheeted covers (via the hyperelliptic involution) of a one-vertex map on the sphere having only free edges (carrying a single dart).  We note that  double-star maps are the ``exceptional'' ribbon graphs of \\cite{MP98} (Definition 1.9).  Meleko\\u glu and  Singerman (\\cite{MS08}, Theorems~8.2 and 8.3) also characterized the  Accola-Maclachlan  and Wiman type II curves  as the unique Platonic $M$- and $(M-1)$-surfaces, respectively.  These are curves of genus $g>1$ admitting an anti-conformal involution with the maximal and second-maximal ($g+1$, resp., $g$) number of fixed simple closed curves.    D. Singerman (\\cite{S01}, Theorem~6.2) classified the surfaces admitting {\\it uniform} strictly edge-transitive one-vertex maps; this is the special case $l_1=l_2$ in our Theorem~\\ref{T:edgetransaction}. \n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n \n ","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\nCosmological observations based on electromagnetic waves (EMWs) have unveiled countless secrets of the cosmos. They have provided us with information about the history of the Universe and the dynamics of astrophysical objects within it. This knowledge has allowed us to test General Relativity (GR) and conclude that it can suitably describe our cosmos assuming the existence of two dark cosmic components. In 2016 the observations come from a different domain for the very first time and GR passed one of its latest checks due to the detection of gravitational waves (GWs) \\cite{LIGOScientific:2016aoc}. This detection was just the beginning of the era of gravitational wave measurements increasing the possibilities of multi-messenger astronomy.\n\nDespite the great success of GR, this theoretical framework may be challenged. On the one hand, at large scales it is necessary to assume the existence of dark matter and dark energy to describe the observational data in the framework of GR.\nAs it is well known, most of the matter of the Universe has to be dark and the nature of the dark matter particle remains elusive. Furthermore, it is not clear that the final description of dark energy, which should drive the current accelerated cosmic expansion being almost the 70\\% of the cosmic energetic content, is just a cosmological constant.\nOn the other hand, in the microscopic world of particles and high energies GR is thought to be inadequate. Any attempts to quantize and unify gravity with the Standard Model has been unsuccessful.\nSo, in the recent years alternative theories of gravity (ATGs) \\cite{Clifton:2011jh,EuclidTheoryWorkingGroup:2012gxx,CANTATA:2021ktz} have been explored in great detail as possible candidates to describe gravitational phenomena at all scales.\nA popular set of ATGs are scalar-tensor theories of gravity, which introduce a non-minimally coupled scalar field into the action. In this framework Horndeski theory is the most general family of models leading to second-order differential equations \\cite{Horndeski,Deffayet:2011gz,Kase:2018aps}. Theories known as bimetric gravity, which introduce a second metric tensor in the formulation of the theory, $\\widetilde{g}_{\\mu\\nu}$, can reproduce massive gravity with a single massive spin-2 field \\cite{deRham:2014zqa}.  Other well-studied theories are the $f(R)$ theories of gravity \\cite{Nojiri:2006gh,Sotiriou:2008rp}. These theories introduce higher-order terms to the action of Einstein-Hilbert, with a generic function of Ricci's scalar. (We do not intend to provide an exhaustive list of ATGs in this introduction, the interested reader can check, for example, reference  \\cite{CANTATA:2021ktz}.)\n\n\nThere is an interesting theoretical prediction of GR regarding the interaction of GWs and EMWs that has long been known: graviton-photon conversion in a stationary electromagnetic background \\cite{Gertsenshtein,Lupanov}. This is a phenomenon of conversion between GWs and EMWs that is produced when the waves cross an external non-evolving electromagnetic field. In addition, when the waves propagate through a (real or effective) dielectric medium, there is no complete conversion, but graviton-photon oscillation or mixing \\cite{Zeldovich,Raffelt}, analogous to neutrino oscillations for example. This is, the evolution of the system at first order in perturbations is such that gravitational and electromagnetic degrees of freedom are mixed as a single superposed state throughout its propagation. From a cosmological point of view it is of special interest the phenomenon that can take place due to the propagation of primordial GWs through the cosmic magnetic field \\cite{Dolgov:2012be,Dolgov:2013pwa} (see also references \\cite{Ejlli:2018hke,Ejlli:2020fpt} for more recent studies). The hypothetical detection of EMWs generated in this process would allow us to measure indirectly GWs that are beyond the scope of current detectors. Moreover, the no observation of these waves when they are predicted may indicate the absence of the corresponding primordial GWs or point towards some modification of the gravitational theory. In this respect, the predictions about the efficiency of the conversion process in GR indicate that the sensibility of the instruments could be too low to measure the effect of laboratory experiments \\cite{Boccaletti}, although it could left footprints in the radiation cosmic background (excluding their relevance in the microwave) \\cite{Dolgov:2012be}.\n\n\nWe want to stress that this phenomenon is predicted in GR even in regions where the refractive index equals unity, in which case there is just conversion between EMWs and GWs. This is the case when we just consider classical electrodynamics and GR. Therefore, in order to get a mixture in GR and not just conversion, it is necessary to take into account terms of greater perturbative order to the Maxwell Lagrangian, such as the Euler-Heisenberg Lagrangian \\cite{Raffelt}. This term describes nonlinear corrections and induces an effective refractive index for EMWs \\cite{Dolgov:2012be,Dolgov:2013pwa}.\nOn the other hand, following the spirit of reference \\cite{Cembranos:2018jxo} one can also consider that the GWs propagate with an effective refractive index. This situation does not appear in GR for vacuum solutions, since GWs propagate in vacuum at the speed of light. When they propagate in a FLRW spacetime, however, the expansion of the Universe generate attenuation in the amplitude of the wave \\cite{Caprini:2018mtu}. Considering that the gravitational phenomena are describe by a given ATG, the situation can be much more varied. Propagation of GWs in ATGs can include, for example, a mass term, an attenuation term, or variations in the speed of propagation \\cite{Saltas:2014dha,Bettoni:2016mij}. As in electromagnetism these terms arise when EMWs propagate through a dielectric media, by analogy we can think in a medium formed by the additional degrees of freedom with respect to GR in which GWs propagate. These effective media are called diagravitational media \\cite{Cembranos:2018lcs}. Multi-messenger astronomy \\cite{LIGOScientific:2017vwq} has already allowed to conclude that GWs propagate today at the speed of light, ruling out a large number of ATGs as dark energy mimickers \\cite{Lombriser:2016yzn,Ezquiaga:2017ekz,Creminelli:2017sry,Sakstein:2017xjx}. However, other theories are still compatible with this constraint. \nFor instance, theories of massive gravity induce a dispersive medium \\cite{deRham:2014zqa}. An attenuation term may also appear in the refractive index, which may imply wave amplification or absorption.  This term, which is typically linked to variations in some fundamental quantity of the ATG \\cite{Barrow:1993ad}, combines with the cosmic attenuation in cosmological backgrounds.\nIn addition, Lorentz violating ATGs entail modified dispersion relations\n\\cite{Mirshekari:2011yq}. All these different terms can be included in a refractive index for the diagravitational medium  \\cite{Cembranos:2018lcs}. Therefore, the predictions for the phenomenon of graviton-photon oscillations may be affected in ATGs due to the existence of this effective medium.\nGraviton-photon mixing in ATGs was first considered in reference \\cite{Cembranos:2018jxo}, paying particular attention to the case of a real refractive index when obtaining the conversion probability. In this work we shall investigate a more general framework in detail, taking into account in discussing the effects of a cosmic background.\n\nThis work can be outline as follows: In section \\ref{secII} we present the results that we have obtained for the phenomenon of graviton-photon mixing following a procedure based in studying the mixing probability. In particular, first we focus in GR, in section \\ref{secIIA}, and obtain the mixing matrix of the graviton-photon system in a cosmological scenario, extending the results of reference \\cite{Dolgov:2012be} to propagation in a FLRW spacetime. In section \\ref{secIIB} we discuss how the system is modified when considering theories beyond GR, as in reference \\cite{Cembranos:2018jxo} but for a cosmological background. In section \\ref{secIIC} we obtain the mixing probability of primordial gravitational and electromagnetic waves and estimate the effect in some ATGs. \nOn the other hand, in section \\ref{secIII}, we consider the density matrix formalism taking into account effects of decoherence on the photons. We consider for the very first time decoherence effects affecting gravitational radiation and quantify the modification of the predictions of graviton-photon mixing for some ATGs. Finally, in section \\ref{secIV}, we summarize our results. \n\n\n\n\\section{Graviton-photon mixing in a cosmological background}\\label{secII}\nThe action contains a gravitational part, which is the Einstein--Hilbert action for GR but has a different form for ATGs.\nWe assume that the action of the electromagnetic part is given by the Maxwell action plus a second order term known as the Euler-Heisenberg Lagrangian\n\\begin{equation}\\label{eq2}\n\\begin{split}\n    S_{em}=&-\\dfrac{1}{4}\\int d^4x \\sqrt{-g}F_{\\mu\\nu}F^{\\mu\\nu}\\\\\n    &+\\dfrac{\\alpha^2}{90m_e^4} \\int d^4x \\sqrt{-g}[(F_{\\mu\\nu}F^{\\mu\\nu})^2+\\dfrac{7}{4}(\\widetilde{F}_{\\mu\\nu}F^{\\mu\\nu})^2],\n    \\end{split}\n\\end{equation}\nwhere we have introduced the usual electromagnetic field tensor $F_{\\mu\\nu}=\\partial_{\\mu}A_{\\nu}-\\partial_{\\nu}A_{\\mu}$.\nHere, $\\alpha=e^2\/4\\pi$ is the fine structure constant, $m_e$ is the electron mass and $\\widetilde{F}_{\\mu\\nu}=\\dfrac{1}{2}\\epsilon_{\\mu\\nu\\rho\\sigma}F^{\\rho\\sigma}$ is the dual of $F_{\\mu\\nu}.$ The second order term describe quantum corrections to the classical electromagnetic action and it is only valid at the low frequencies regime, $\\omega<<m_e.$ We will take frequencies for the cosmological gravitational waves up to $O(10^5)$ eV so we can assume this approximation is valid.\n\n\nOn the other hand, we consider that for the metric there is a background (or order zero) contribution given by the cosmological flat FLRW metric and a small perturbation, $h_{\\mu\\nu}$, such that the complete metric is\n\\begin{equation}\\label{eq1}\n    ds^2=a(\\eta)^2[-d\\eta^2+(\\delta_{ij}+\\kappa h_{ij})dx^idx^j],\n\\end{equation}\nwhere we have introduced $\\kappa=\\sqrt{16\\pi G}$ in order to have a dimensionless metric perturbation, that is $\\kappa h_{ij}$. \nWe are working in the conformal time coordinate related with the cosmological time as $dt=a(\\eta)\\,d\\eta$. We have a gauge freedom for the perturbation components so we work in the Trace-Transverse gauge (TT) that is, $h_{0\\mu}=\\partial_{i}h_{ij}=h_i{}^i=0$, so only the spatial components are non trivial. In addition, we assume that the electromagnetic field tensor is composed by a background external contribution plus a variable small perturbation, $F_{\\mu\\nu}=F^{(e)}_{\\mu\\nu}+f_{\\mu\\nu}$. The term $f_{\\mu\\nu}$ gives rise to the photon field if $|f_{\\mu\\nu}|<<|F^{(e)}_{\\mu\\nu}|$.\n\nIn order to get the electromagnetic equations of motions we expand metric (\\ref{eq1}) in the action (\\ref{eq2}) up to second order in $h_{\\mu\\nu}$ and $f_{\\mu\\nu}$. As we are interested only in  terms which give us the evolution of the graviton and photon field, we ignore the terms of order zero of the action, that define the Einstein equation (or modified Einstein equation) of the background. The electromagnetic action including only first and second order terms in perturbations is\n\\begin{equation}\\label{eq3}\n\\begin{split}\n    S_{em}^{(1)}&=-\\dfrac{1}{4}\\int d^4x f_{\\mu\\nu}f_{\\rho\\sigma}\\eta^{\\mu\\rho}\\eta^{\\nu\\sigma}+\\dfrac{\\kappa}{2}\\int d^4xh^{\\mu\\nu}T^{em}_{\\mu\\nu}\\\\\n    &+\\dfrac{\\alpha^2}{90m_e^4} \\int d^4x \\sqrt{-g}[4(F^{(e)}_{\\mu\\nu}f^{\\mu\\nu})^2+7(\\widetilde{F}^{(e)}_{\\mu\\nu}f^{\\mu\\nu})^2],\n    \\end{split}\n\\end{equation}\nwhere we introduce the leading order of the external background electromagnetic energy-momentum tensor, $T^{\\rm em}_{\\mu\\nu}=F_{\\mu\\rho}F_{\\sigma\\nu}\\eta^{\\rho\\sigma}-\\frac{1}{4}\\eta_{\\mu\\nu}F_{\\rho\\sigma}F^{\\rho\\sigma}$. We neglect higher orders terms in the Euler-Heisenberg Lagrangian and a term proportional to the Maxwell Lagrangian that is subdominant with respect to the first one\\footnote{In Minkowski spacetime one asumes that the external field is static (or varies slowly), so we can perform this approximation. In a FLRW this is still true since $F_{\\mu\\nu}$ is independent of $a(t)$ for a freely propagating electromagnetic field, due to conformal invariance \\cite{Durrer:2013pga}.}.\n\n\n\\subsection{Equations of motion in GR}\\label{secIIA}\nWe start by considering that the gravitational phenomena are described by GR. Thus, the action for the metric field is given by the Einstein-Hilbert action\n\\begin{equation}\\label{eq4}\n    S_g=\\dfrac{1}{\\kappa^2}\\int d^4x \\sqrt{-g}R,\n\\end{equation}\nwhere $R$ is the curvature scalar, $g$ is the determinant of the metric $g={\\rm det}(g_{\\mu\\nu})$, and we take the signature $\\eta_{\\mu\\nu}={\\rm diag}(-1,1,1,1)$.\n\n\nVarying the action with respect to the $h_{\\mu\\nu}$ and $A_{\\mu}$ fields (considering now $A_{\\mu}$ to be photon potential associated to $f_{\\mu\\nu}$), one gets the coupled equations of motion for the graviton and the photon. For the $h_{\\mu\\nu}$ field, the variation of  the action leads to \\cite{Caprini:2018mtu}\n\\begin{equation}\n\\begin{split}\n    a^2\\Box \\bar{h}_{\\mu\\nu}+2g^{\\alpha\\beta}\\partial_{\\alpha}a^2\\nabla_{\\beta}\\bar{h}_{\\mu\\nu}&\\\\\n    +\\partial_{\\alpha}\\partial^{\\alpha}a^2\\bar{h}_{\\mu\\nu}-2a^2R^{\\lambda}_{\\mu\\nu}{}^{\\sigma}\\bar{h}_{\\lambda\\sigma}&=-\\kappa T_{\\mu\\nu}^{(1)},\n\\end{split}\n\\end{equation}\nwhich is the Einstein equation with the electromagnetic energy momentum tensor as the source up to first order in the perturbation. We have defined the trace-reversed metric perturbation, $\\bar{h}_{\\mu\\nu}= h_{\\mu\\nu}-\\frac{1}{2}\\eta_{\\mu\\nu}h$, which will be the new field representing the graviton. \nFrom now on we will omit the bar. The Riemann tensor is defined as $R^{\\lambda}_{\\mu\\nu\\sigma}=\\partial_{\\nu}\\Gamma^{\\lambda}_{\\mu\\sigma}-\\partial_{\\sigma}\\Gamma^{\\lambda}_{\\mu\\nu}+\\Gamma^{\\lambda}_{\\alpha\\nu}\\Gamma^{\\alpha}_{\\mu\\sigma}-\\Gamma^{\\lambda}_{\\alpha\\sigma}\\Gamma^{\\alpha}_{\\mu\\nu}$. \nConsidering again that we are working on the TT gauge, only the spatial components are not-zero, so developing the left hand side of the equation it is easy to get\n\\begin{equation}\\label{eq6}\n    h_{ij}^{''}-\\nabla^2h_{ij}+2\\mathcal{H}h_{ij}^{'}=\\kappa T_{ij}^{(1)},\n\\end{equation}\nwhere $\\mathcal{H}=\\dfrac{a'}{a}$ is the conformal Hubble rate and the prime denotes derivatives with respect to conformal time. It is a wave equation with a dumping term that is governed by the Hubble expansion rate. On the right hand side, we have the electromagnetic energy-momentum tensor as the source. While its background components are associated with the Einstein equations to zero order, in this equation we write its first-order perturbative components, $ O(f_{\\mu\\nu}) $. Taking its definition, since the external field in the chosen coordinates has only a magnetic component, this energy-momentum tensor can be written as $T_{ij}^{(1)}=B_{(i}^eB_{j)}$, where the parentheses indicate symmetrization, and the index $e$ indicate the component of the external magnetic field.\n\n\nThe evolution equation for the electromagnetic field is obtained by varying action (\\ref{eq3}) with respect to $ A_{\\mu} $. So, one has\n\\begin{equation}\\label{eq7}\n\\begin{split}\n  &  \\nabla_{\\mu}\\left( f^{\\mu\\nu}-\\dfrac{4\\alpha^2}{45m_e^4}[4F_{\\alpha\\beta}^{(e)}f^{\\alpha\\beta}F^{(e)\\mu\\nu}+7\\widetilde{F}_{\\alpha\\beta}^{(e)}   f^{\\alpha\\beta}\\widetilde{F}^{(e)\\mu\\nu}] \\right) \n    =    \\\\\n&\\kappa\\nabla_{\\mu}\\left( h^{\\mu\\beta}F^{(e)\\nu}_{\\beta}- h^{\\nu\\beta}F^{(e)\\mu}_{\\beta}\\right).\n    \\end{split}\n\\end{equation}\nNow, we take into account that only the field $f_{\\mu\\nu}$ is variable while the external field $F_{\\mu\\nu}^{(e)}$ remains constant \\cite{Durrer:2013pga} and only has spatial components, or temporal components for its dual. In terms of $ A_{\\mu} $ the Maxwell equation is just a wave equation, without the absorption term associated with the Hubble expansion since we are in conformal time. The additional term of order $\\alpha^2$ results in an effective mass term that is included in the wave equation. Following reference \\cite{Dolgov:2012be}, one can define the refractive index from this term as\n\\begin{equation}\nm_{\\gamma}^2=\\omega^2-k^2=\\omega^2(1-n_{\\gamma}^2).\n\\end{equation}\nThe right hand side of equation (\\ref{eq7})  is the source term, which contains the metric perturbation and the external field. Taking into account these consideratons the equation for the photon can be reduced to \\cite{Dolgov:2012be}\n\\begin{equation}\\label{eq9}\n    \\left[\\Box -m_{\\gamma}^2\\right]A^j=\\kappa\\partial_{i}\\left(  h^{ik}F^{(e)j}_k-h^{jk}F^{(e)i}_k  \\right),\n\\end{equation}\nwhere the refractive index has been defined for the polarizations transverse and parallel to the external magnetic field as\n\\begin{eqnarray}\n    n_{\\perp}^2-1&=&4\\dfrac{\\alpha}{45\\pi}\\dfrac{e^2}{m_e^4}B_T^2a^{-4}(\\eta),\\label{eq10}\\\\\n    n_{\\parallel}^2-1&=&7\\dfrac{\\alpha}{45\\pi}\\dfrac{e^2}{m_e^4}B_T^2a^{-4}(\\eta),\\label{eq11}\n\\end{eqnarray}\nwith $ B_T $ the transverse component of the magnetic field  to the direction of the wave. These expressions arise when expanding the components of the $F_{\\mu\\nu}^{(e)}$ field, giving the transverse term, and its dual $\\widetilde{F}_{\\mu\\nu}^{(e)}$, giving the parallel term, in equation (\\ref{eq7}) as is was shown in reference \\cite{Dolgov:2012be}. As each term gives rise to different indices of refraction in the presence of an external magnetic field, birefringence effects appear in the propagation of the photon. As it can be seen in equations (\\ref{eq10}) and (\\ref{eq11}), there is also an extra dependence on the refractive index with the scale factor due to the cosmological background. According to reference \\cite{Dolgov:2012be}, the effective mass of the photon in Minkowsky background, $\\widetilde{m}_{\\gamma}$, is related to the cosmological as $\\widetilde{m}_{\\gamma}=m_{\\gamma}a^4(\\eta)$, what agrees with our result.\n\n\n\nEquations (\\ref{eq6}) and (\\ref{eq9}) describe the evolution of the graviton-photon system. Before getting the final equations some considerations are in order. We redefined the gravitational wave tensor as $H_{ij}=ah_{ij}$. This allows us to remove the absorption term from equation (\\ref{eq6}). We assume that gravitational and electromagnetic waves expand in their Fourier modes such that\n\\begin{eqnarray}\n    H_{ij}(z,\\eta)&=&\\sum_{\\lambda}H_{\\lambda}(\\eta)e^{\\lambda}_{ij}e^{ikz},\\label{eq12}\\\\\n    A_{j}(z,\\eta)&=&i\\sum_{\\lambda}A_{\\lambda}(\\eta)e^{\\lambda}_{j}e^{ikz},\\label{eq13}\n\\end{eqnarray}\nwith $\\lambda={\\times,+}$, for the gravitational wave tensor, and  $\\lambda={\\perp,\\parallel}$, for the electromagnetic wave vector,  denoting polarization modes, where $\\perp,\\parallel$ denote perpendicular and parallel to the external magnetic field.  Assuming that the waves propagate in the direction of the z-axis, the non-zero elements of the tensor modes are defined such that $e^{+}_{xx}=-e^{+}_{yy}=1$ and $e^{\\times}_{xy}=e^{\\times}_{yx}=1$. Introducing the definitions (\\ref{eq12}) and (\\ref{eq13}) in equations (\\ref{eq6}) and (\\ref{eq9}), we obtain\n\\begin{eqnarray}\\label{eq12}\n\\left[ \\partial_{\\eta}^2+\\left( k^2-\\dfrac{a^{''}}{a}\\right) \\right]H_{\\lambda}=-\\kappa akA_{\\lambda}B_{T},\\\\\n        \\left(\\partial_{\\eta}^2+k^2  \\right)A_{\\lambda}-\\left(\\omega^2-k^2  \\right)A_{\\lambda}=-\\dfrac{\\kappa}{a}kH_{\\lambda}B_T.\\label{eq13}\n\\end{eqnarray}\nAssuming that the Euler--Heisenberg term introduces small corrections, the refractive index is close to unity, $|n_\\gamma-1|<<1$, so $\\omega\\approx k$. In this way we can approximate the d'Alembert operator as, $(\\partial_{\\eta}^2+k^2)A_{\\lambda}=(i\\partial_{\\eta}+k)(-i\\partial_{\\eta}+k)A_{\\lambda}\\simeq 2\\omega(\\omega+i\\partial_{\\eta})A_{\\lambda}$, where we have also assumed $-i\\partial_{\\eta}A_{\\lambda}\\simeq \\omega A_{\\lambda}$.  In addition the effective mass term can be approximated as $(\\omega^2-k^2)\\approx 2\\omega^2(1-n_\\gamma)$. On the other hand, for GWs we consider only the sub-Hubble modes, those that satisfy, $k\\eta >> 1$, so we can eliminate the term that goes with the second derivative in the scale factor of equation (\\ref{eq12}). We also assume that the propagation of GWs do not differ too much from the general relativistic prediction, that is $-i\\partial_{\\eta}H_{\\lambda}\\simeq \\omega H_{\\lambda}$.\nWith these approximations we get\n\\begin{eqnarray}\\label{eq14}\n        (\\omega+i\\partial_{\\eta})H_{\\lambda}=-\\dfrac{\\kappa}{2} aA_{\\lambda}B_T,\\\\\n    (\\omega+i\\partial_{\\eta})A_{\\lambda}+\\omega(n-1)A_{\\lambda}=-\\dfrac{\\kappa}{2}\\dfrac{H_{\\lambda}}{a}B_{T}.\\label{eq15}\n\\end{eqnarray}\nIt can be noted that the mixing of the modes in this system formed by equation (\\ref{eq14}) and (\\ref{eq15}) is not symmetric. However, if we consider again the physical mode $h(\\lambda)$, we can obtain symmetric mixing. So, in order to obtain a system that is coupled by a symmetric matrix, we have to consider $A_{\\lambda}$ and $h_{\\lambda}$ as the degrees of freedom. We get\n\\begin{eqnarray}\\label{eq16}\n   (\\omega+i\\partial_{\\eta}+i\\mathcal{H})h_{\\lambda}=-\\dfrac{\\kappa}{2}A_{\\lambda}B_T,\\\\\n    (\\omega+i\\partial_{\\eta})A_{\\lambda}+\\omega(n-1)A_{\\lambda}=-\\dfrac{\\kappa}{2}h_{\\lambda}B_{T}.\\label{eq17}\n\\end{eqnarray}\nGWs and EMWs are coupled by these equations. There is an imaginary term in equation (\\ref{eq16}) proportional to the Hubble parameter, which affects the propagation of gravitational modes of $h$. It acts as an effective gravitational refractive index term that causes absorption or amplification in the amplitude of the wave.\n\n\nNow, regrouping the modes $A_{\\lambda}$ and $h_{\\lambda}$ in the state vector $\\psi^{T}=\\left[ A_{\\lambda},h_{\\lambda}\\right]$, we can write the equations in matrix form and, therefore, identify the mixing matrix\n  \\begin{equation}\\label{eq18}\n      \\left[  (\\omega+i\\partial_{\\eta})+  \\left(  \\begin{matrix}\n  \\omega(n^{\\lambda}_{\\gamma}-1)  &\n   \\dfrac{\\kappa B_T}{2}  \\\\\n   \\dfrac{\\kappa B_T}{2} &\n    \\omega(n_{g }-1)\n   \\end{matrix}\\right)  \\right] \\begin{bmatrix}\n    A_{\\lambda} \\\\\n    h_{\\lambda} \n  \\end{bmatrix}=0,\n  \\end{equation}\nwhere we define the effective refractive index of the gravitational wave as \n  \\begin{equation}\nn_g=1+i\\dfrac{\\mathcal{H}}{\\omega},\n  \\end{equation} \nbeing the origin of this effective refractive index the expansion of the cosmological background.   It should be noted that equation (\\ref{eq18}) reduces to that obtained in reference \\cite{Dolgov:2012be} for propagation in a Minkowski background, that is $a=1$. In addition, it should be noted that in the case that the Euler-Heisenberg corrections are negligible, then $n_{\\gamma}=1,$ and we recover graviton-photon conversion.\n\n\n\\subsection{Equations of motion beyond GR}\\label{secIIB}\nNow we consider propagation of GWs in a cosmological background when the gravitational phenomena are described by an ATG. At this point we do not restrict attention to a particular theory. So, we consider a general parametrization for the linear transverse-traceless perturbations for the tensor modes\\footnote{This parametrization does not take into account theories that produce oscillations of gravitons with additional gravitational degrees of freedom present in the theory, as those treated in reference \\cite{Jimenez:2019lrk}. Lorentz violating theories will be consider later in this section.}. This is \\cite{Saltas:2014dha}\n\\begin{equation}\\label{eq20}\n    h^{''}_{ij}+2\\mathcal{H}(1+\\nu)h^{'}_{\\ij}+(c_T^2k^2+\\mu^2)h_{ij}=0.\n\\end{equation}\nThe parameter $\\mu$ is the effective mass of the graviton, $c_T$ is the speed of GWs, and $\\nu$ is a ``friction'' term responsible for variations in the amplitude of the tensor modes.  The presence of these parameters in equation (\\ref{eq20}) will depend on parameters of the theory. In some literature, the damping factor is called the running of the Planck mass, $\\nu=\\alpha_M=\\mathcal{H}^{-1}d\\log M_*^2\/dt$, with $M_*$ the effective Planck mass. This term appears, for example, in theories where the extra fields are non-minimally coupled to gravity. On the other hand, it has to be emphasized that according to reference \\cite{LIGOScientific:2017vwq} the most accurate measurements of the speed of gravitational wave have an approximate value $c_T\\simeq 1$ for the present Universe, so we will ignore possible changes in the propagation speed. \n\n\n\nAs discussed in reference \\cite{Cembranos:2018jxo} for the Minkowski case, when we compare equation (\\ref{eq20}) with the equation of the graviton-photon system in GR, that is equation (\\ref{eq6}), we can argue that in an ATG the graviton-photon system is also described by equation (\\ref{eq18}) with the gravitational refractive index of the graviton given by\n\\begin{equation}\n    n_g^2=1+i\\dfrac{2\\mathcal{H}(1+\\nu)}{\\omega}-\\dfrac{\\mu^2}{\\omega^2}.\n\\end{equation}\nFurthermore, we can also add a term $Ak^{\\alpha}$ to this expression, taking into account a gravitational refractive index describing terms that modify the dispersion relation connected to Lorentz violations \\cite{Mirshekari:2011yq}. Now, focusing again our attention to GWs which propagation does not differ too much from that predicted by GR, that is $n_g\\approx 1$, we can write\n\\begin{equation}\\label{eq22}\n    n_g=1+i\\dfrac{\\mathcal{H}(1+\\nu)}{\\omega}-\\dfrac{\\mu^2}{2\\omega^2}-\\dfrac{A}{2}\\omega^{\\alpha-2}.\n\\end{equation}\nThe particular ATG will fix the value of the different terms appearing in this expression. \nIn order to get a graviton-photon evolution equation of the form of equation (\\ref{eq18}), we shall assume that the parameters $\\nu$ and $\\mu$ vary slowly or are constant in the interval under study. Moreover, as discussed in reference \\cite{Cembranos:2018jxo}, when an ATG modifies the effective Plank mass (and, therefore, induces an attenuation parameter $\\nu$) it changes the coupling with the photon modes in the wave equation of the graviton; so, we need to redefine the gravitational modes \n\\begin{equation}\\label{eq23}\n\\widetilde{h}_{\\lambda}=\\dfrac{\\kappa}{\\kappa_{eff}}h_{\\lambda},\n\\end{equation}\nwith $\\kappa_{eff}=M_{*}^{-1}$, to obtain a symmetric mixing matrix. Thus, we finally get\n\\begin{equation}\\label{eq24}\n      \\left[  (\\omega+i\\partial_{\\eta})+ \\left(\\begin{matrix}\n  \\omega(n^{\\lambda}_{\\gamma}-1)  &\n   \\dfrac{\\kappa_{eff} B_T}{2}  \\\\\n   \\dfrac{\\kappa_{eff} B_T}{2} &\n    \\omega(n_{g }-1)\n   \\end{matrix}\\right)  \\right] \\begin{bmatrix}\n    A_{\\lambda} \\\\\n   \\widetilde{h}_{\\lambda}\n  \\end{bmatrix}=0,\n  \\end{equation}\nwhere the refraction index $n_g$ can be described by equation (\\ref{eq22}). From now on we will omit the tilde in the gravitational mode. \n\n\\subsection{Conversion probability in GR and beyond}\\label{secIIC}\nEquation (\\ref{eq24}) describes the mixing of the $A$ and $h$ modes in the presence of an external magnetic field. This mixture depends on the propagation of the waves by means of the refractive indexes that appear in the mixing matrix. We have seen how to include different terms in the refractive index of the graviton that arise both from the cosmic expansion and from possible modifications of the general relativistic predictions. \nAs the different polarizations do not mix with each other, the system can be reduced from four to two dimensions. In such a case, the equation to solve is\n\\begin{equation}\\label{eq25}\n\\left[(\\omega+i\\partial_{\\eta})+\\mathcal M\\right]\\psi=0,\n\\end{equation}\nwith\n\\begin{eqnarray}\\label{eq26}\n\\mathcal M=\n\\left({\\begin{array}{cc}\n   \\Delta_ {\\gamma} & \\Delta_M  \\\\\n   \\Delta_M & \\Delta_{\\rm g} \\\\\n  \\end{array} } \\right),\\quad\n\\psi=\\left(\\begin{array}{c}\nA_{\\lambda}\\\\  h_{\\lambda}\n\\end{array}\\right),\n\\end{eqnarray} \nwhere the elements of the matrix are defined as\n\\begin{eqnarray}\\label{eq27}\n\\Delta_M&=&B_T\/M_{*},\\nonumber \\\\\n\\Delta_{\\gamma}&=&\\omega(n_{\\gamma}-1), \\,\\,  \\Delta_g=\\omega(n_g-1).\n\\end{eqnarray}\nThe refractive index given by the effects of the Euler-Heisenberg term for the photon is defined in equation (\\ref{eq10}), whereas the gravitational refractive index is given by equation (\\ref{eq22}), which can include an imaginary part that give rise to wave attenuation. So,  we will decompose the gravitational refractive index  in its real and imaginary parts, that is $\\Delta_g=\\Delta_g^R+i\\Delta_g^I$. Now, in order to solve the system we have to diagonalize the $\\mathcal M$ matrix. As it is a symmetric matrix, this diagonalization is possible as $\\mathcal M'=O^T\\mathcal M O$, $\\Psi'=O^T\\Psi$, being $O$ a complex\northogonal matrix or complex rotation.  So, the diagonal elements for the $\\mathcal M'$ matrix, which are the eigenvalues of \n$\\mathcal M$, and the angle of rotation will be complex in general:\n\\begin{eqnarray}\n&m_{1,2}&=\\frac{1}{2}\\left[\\Delta_{\\rm g}+\\Delta_{\\gamma}\\pm\\sqrt{4\\Delta_{\\rm M}^2+\\left(\\Delta_{\\rm g}-\\Delta_{\\gamma}\\right)^2}\\right],\\qquad\\\\\n&\\tan&(2\\theta)=\\dfrac{2\\Delta_{\\rm M}}{\\Delta_{\\rm g}-\\Delta_{\\gamma}}.\n\\end{eqnarray}\nIn this section we assume $\\mathcal M$ to be approximately constant in the interval under consideration, in order to do an order of magnitude approximation. Solving the equations we get the solutions\n\\begin{equation}\n\\begin{split}\nA(\\eta)=&\\left[\\cos^2\\theta e^{im_1\\eta}+\\sin^2\\theta e^{im_2\\eta}\\right] A(0) \\\\&-\\sin\\theta\\cos\\theta\\left[e^{im_1\\eta}-e^{im_2\\eta}\\right] h(0),\\\\\n\\end{split}\n\\end{equation}\n\\begin{equation}\n\\begin{split}\n h(\\eta)=&\\sin\\theta\\cos\\theta\\left[e^{im_1\\eta}-e^{im_2\\eta}\\right]A(0)\\\\\n&\\left[\\sin^2\\theta e^{im_1\\eta}+\\cos^2\\theta e^{im_2\\eta}\\right] h(0),\n\\end{split}\n\\end{equation}\nabsorbing a phase $ e^{i\\eta\\omega}$ in the definition of the fields. Interpreting the results as wave functions, the probability of finding a photon after a time $\\eta$ if the initial state is formed by gravitons is given by $P_{{\\rm g}\\rightarrow{\\gamma}}(\\eta)=|A(\\eta)|^2 =A(\\eta)A^*(\\eta)$, where we have to keep in mind that the parameter $ \\Delta_g $ can be complex. Defining the parameters\n\\begin{eqnarray}\n\\alpha=\\dfrac{1}{2}\\operatorname{Re}\\left(\\sqrt{4\\Delta_{\\rm M}^2+\\left(\\Delta_{\\rm g}-\\Delta_{\\gamma}\\right)^2}\\right)\\\\\n\\beta=\\dfrac{1}{2}\\operatorname{Im}\\left(\\sqrt{4\\Delta_{\\rm M}^2+\\left(\\Delta_{\\rm g}-\\Delta_{\\gamma}\\right)^2}\\right),\n\\end{eqnarray}\n the expression of probability takes the form\n\\begin{equation}\\label{eq34}\nP_{{\\rm g}\\rightarrow{\\gamma}}(\\eta)=\n\\dfrac{ \\Delta_M^2 e^{-\\Delta_g^I\\eta}   }{\\alpha^2+\\beta^2}\n\\left[\\sinh^2\\left(\\beta\\eta\\right)+\\sin^2\\left(\\alpha\\eta\\right)\\right].\n\\end{equation}\nTaking $\\Delta_g=0$ in this expression, we recover the predictions for GR in a Minkowski background \\cite{Dolgov:2012be}. In that case resonance is reached for $\\Delta_{\\gamma}=0$, and the regime of weak mixing corresponds to $\\Delta_{\\gamma}\\gg\\Delta_{M} $. \nEven in GR we will have modifications of the vacuum probability for cosmic backgrounds, since in this case one has $\\Delta_g=i\\mathcal H$. So, for an expanding universe the probability will be damped. \nOn the other hand, regarding ATGs, equation (\\ref{eq34}) also reduces to the conversion probability obtained in reference \\cite{Cembranos:2018jxo} for a real gravitational refractive index, that is $\\Delta_g^I=0$ and $\\beta=0$. In this case resonance is achieved for a value of the mixing angle $\\theta=\\pi\/4$, which implies $\\Delta_{\\gamma}=\\Delta_g^R$, and $\\Delta_g^I=0$; in a cosmological background this is only possible if $\\nu = -1$, counteracting the effect of the cosmic expansion.\n\nWe emphasize that the exponential factor of the conversion probability depends only on the attenuation part of $n_g$. For sufficiently low negative values of $\\Delta_g^I$ the probability may grow up beyond unity. However, it has to be noted that this probability is normalized to the amplitude of the initial GWs and, therefore, the decoherence due to the interaction with the possible extra degrees of freedom of the ATG (or with the contraction of the universe if it were the case) can increase the probability to values larger than one. This effect can be better described with the density matrix formalism used in section \\ref{secIII}.\n\nThe coupling term of the mixing matrix (\\ref{eq26}) is determined by the present value of the cosmic magnetic field. In order to get an estimation, we assume as a first approximation that $M_*\\simeq (16 \\pi G)^{-1\/2}$ in equation (\\ref{eq27}). Then, we express the coupling term as in reference \\cite{Dolgov:2012be}\n\\begin{equation}\\label{eq35}\n    \\Delta_M =2.4\\cdot10^{-15} \\left[ \\dfrac{B_T}{1G}\\right]\\qquad s^{-1}.\\end{equation}\nIn reference \\cite{Barrow:1997mj}, with measurements from the COBE experiment, a bound on the present value of the large-scale magnetic fields is found to be $B_T < 5\\cdot10^{-9}$ G.\nOn the other hand, for the photon term we have written the refractive index for the different polarizations in equation (\\ref{eq10}). However, to have a correct description of the propagation of the photon one must consider an effective mass that arises from the effects of the electron primordial plasma on the photon, which depends on the plasma frequency as $ m_ {\\rm plasma} = -\\frac{\\omega_{\\rm plasma}^2}{2\\omega}$. This effective mass is added to the QED contribution, such that, with the approximation made, $(n^2-1)\\simeq 2(n-1)$, and for the values of the electromagnetic constants, $\\Delta_{\\gamma}$ in terms of $B_T$ can be expressed as \\cite{Dolgov:2012be}\n\\begin{equation}\\label{eq36}\n\\begin{split}\n    \\Delta_{\\gamma}=&\\Big| 4\\cdot10^{-17}\\left[ \\dfrac{\\omega}{1 eV}\\right]\\left[ \\dfrac{B_T}{1G}\\right]^2a^{-4}-\n    \\\\\n    &\n    -1.05\\cdot10^{-6}\\left[ \\dfrac{1 eV}{\\omega}\\right]\\left[ \\dfrac{n_e}{cm^{-3}}\\right]^2\\Big| \\qquad s^{-1},\n\\end{split}\n\\end{equation}\nwhere $n_e$ is the electron numerical density. \nWe can compute the probability today considering the present values of cosmological parameters. For the present electron density we take $n_e=2.47\\cdot10^{-7} {\\rm cm}^{-3} $ and for the present magnetic field, we  use the upper bound  $B_0=5\\cdot10^{-9}$ G. We consider a primordial gravitational wave that propagates from the start of the domination of dark energy in the Universe until the present time. It means a total propagation time $\\eta_0\\simeq 3.7\\cdot10^{17} $ s, during which we will assume the conformal Hubble rate today $\\mathcal{H}\\simeq 2.2\\cdot10^{-18}  {\\rm s}^{-1}$. So, the probability calculated for GR from (\\ref{eq34}) is\n\\begin{equation}\\label{eq37}\n    P_{{\\rm g}\\rightarrow{\\gamma}}^{\\rm GR}(\\eta_0)=8.6\\cdot10^{-12},\n\\end{equation}\nwhere we take $\\Delta_g=i\\mathcal{H}$  for the graviton.  For the frequency spectrum we have chosen the largest frequency, $\\omega=10^5$ eV, within the valid range of the Euler-Heisenberg approximation, $w <m_e$ with $m_e=5\\cdot10^5$ eV. This is the frequency for which the probability has the maximal value. \nAs it can be seen in figure \\ref{fig1}, the mixing probability oscillates and increases with frequency.\n\nWe have to emphasize that this probability has been calculated keeping the parameters of the mixing matrix constant in the evolution, as a first approximation. This rude approximation allow us to understand the dependence of the probability with modifications  beyond GR, as we will discuss now for some examples.\n \n\n \\subsubsection{Massive Gravity}\nLet us consider a theory of massive gravity. In this case the gravitational refractive index has an additional real term apart from the imaginary part coming from the cosmic evolution. \nSo, the gravitational term appearing in the mixing matrix can be written as\n\\begin{equation}\\label{eq38}\n    \\Delta_g=-2.4\\cdot10^{14}\\left[\\dfrac{\\mu}{1\\, {\\rm eV}}\\right]^2\\left[\\dfrac{1\\,{\\rm eV}}{\\omega}\\right]+i\\left[\\dfrac{\\mathcal{H}}{1\\,{\\rm s}^{-1}}\\right] \\quad {\\rm s}^{-1}.\n\\end{equation}\nNowadays there are several bounds on the mass of the graviton coming from different observations. Supernova data put the limit in $\\mu <10^{-23}$ eV and LIGO GWs measurements lower it to $\\mu<10^{-29}$ eV \\cite{Will:2014kxa}. It has to be noted that these bounds do not take into account the new kinds of polarization that arise with a mass for the graviton. In massive gravity there are (at least) 5 polarizations that, although they cannot be detected by current interferometers, can have effects on gravity tests in the Solar System or in gravitational lensing. As stated in reference \\cite{deRham:2016nuf}, the precession of the perihelion of the lunar orbit establishes $\\mu<10^{-30}$ eV. Other measurements of the Earth-Moon system establish the bound in $\\mu<10^{-32}$ eV and for the moment the lowest limit taken by gravitational lensing is $\\mu<10^{-33}$ eV. \n\nAssuming a massive gravity theory that saturates the lowest bound, that is $\\mu=10^{-33}$ eV, and taking the present values for the parameters already discussed in the GR case, the probability calculated with equation (\\ref{eq38}) is\n\\begin{equation}\\label{eq38}\nP_{{\\rm g}\\rightarrow{\\gamma}}(\\eta_0)=8.6\\cdot10^{-12},\n\\end{equation} \nwhich is the same value as the one obtained for GR, equation (\\ref{eq37}).  This means that the established limits to the mass of the graviton, do not have a direct effect in the graviton-photon mixing probability.\n\n\n\\subsubsection{Theories modifying the attenuation term}\nA simple example of theories introducing an attenuation term for the propagation of GWs are $f (R)$ theories of gravity. This term introduces an imaginary part for the index of refraction (\\ref{eq22}). In this case, there is neither mass term nor Lorentz violation, so the index of refraction of the graviton is pure imaginary. We keep the same parameters of the matrix $\\mathcal M$ in equations (\\ref{eq35}) and (\\ref{eq36}), and we write the graviton term as\n\\begin{equation}\\label{eq40}\n    \\Delta_g=i(1+\\nu)\\mathcal{H}\\qquad s^{-1},\n\\end{equation}\nwhich is independent of $\\omega$. In reference \\cite{Belgacem:2018lbp} the authors consider a nonlocal theory theory that it only affects the propagation of GWs in a $\\nu$  term, as a typical $f(R)$ theory. The value of $\\nu$ calculated from measurements of the Hubble parameter of GW170817 in LIGO is  $\\nu=5.1_{-20}^{+11}$  \\cite{Belgacem:2018lbp} . For illustrative purposes, we can compute the values of the probability implied by a theory that saturates the upper observational bound and another saturating the lower bound. Those are \n\\begin{equation}\n  P_{{\\rm g}\\rightarrow{\\gamma}}(\\eta_0)=2.09\\cdot10^{-13},\n\\end{equation}\nfor the upper bound and,\n\\begin{equation}\n         P_{{\\rm g}\\rightarrow{\\gamma}}(\\eta_0)=1.67,\n\\end{equation}\nfor the lower one. As we can see, the values differ 13 orders of magnitude. We have taken again the frequency $\\omega=10^5$ eV. \n\nIn figure \\ref{fig1} we show the probability in terms of the frequency spectrum for different values of $\\nu$. The oscillation are stronger for values closer to $\\nu=-1$, reaching zero conversion. For higher negative values of $\\nu$ the probability increases but with suppressed oscillation. Note that $\\Delta_g\/\\omega<<1$ for $\\nu_{max}$ and, therefore, we are in the range of the Euler-Heissenberg approximation.\n  \\begin{figure}[!htb]\n\\begin{center}\n\\includegraphics[width=3in]{nu.pdf}\n\\caption{The probability in logarithmic scale is represented as a function of the frequency, in the range of $0-10$ keV. We compute it for different values of $\\nu$ and we also include the case of GR ($\\nu=0$). The values of the parameters are assumed constant as $B_0 =5\\cdot10^{-9}$ B, $n_e=2.47\\cdot10^{-7} {\\rm cm}^{-3} $ and $\\mathcal{H}=2.2\\cdot10^{-18}  {\\rm s}^{-1}$.}\n\\label{fig1} \n\\end{center}\n\\end{figure}\nIt should be noted that the estimations of the parameter $ \\nu $ are not completely rigorous, since they are obtained with a very low redshift and with a single source, and the range of the measurement is too long for the value obtained to be useful, except to study its behavior with $\\nu$. The probability is quite sensitive to $\\nu$, reaching very high quantities for very large negative values. It must be remembered that the probability is not normalized and can rise above unity, due to the fact that the amplitude is very amplified. Furthermore, the cases of an imaginary $n_g$ are more properly studied with the density matrix formalism, as we cannot approximate the system as being closed (that is, independent of the interaction with other degrees of freedom or even with the cosmic evolution). So, the dependence of the probability with $\\nu$  will be better understood with density matrix formalism. More importantly, with that formalism we will not approximate $\\mathcal{H}$ as a constant parameter.\n\n\n\n \n \n\n \n \n \n\n\\section{Density matrix formalism}\\label{secIII}\nIn previous systems we have treated the graviton-photon system as a closed system.\nHowever, we know that in the primitive universe the medium was composed of a dense plasma that can interact with EMWs, reducing their interaction with GWs in our system.       \nIn the recombination epoch, at redshift $z\\simeq1090$, the collisions of the photons with free electrons were decreasing, but still some effects must be considered. These collisions would produce decoherence effects and the system would become an open system. \nWhen the length of oscillation is less than the average free path of the photons, it can be neglected; nevertheless, it has to be taken into account while the expansion of the universe keeps them interacting.\nIn order to introduce the decoherence effects in the system, we will apply the density matrix formalism in this section. In this framework, the decoherence can be introduced by inserting an antihermitic matrix into the Hamiltonian \\cite{Dolgov:2012be}. \n\nOn the other hand, it has to be noted that GWs can interact with additional fields. In particular, we have shown that the cosmic expansion and the additional degrees of freedom of some ATGs introduce a complex term in $n_g$. This term give rise to attenuation effects and can be interpreted as a signal of decoherence analogous to a ``plasma\" for gravitational radiation. So, gravitational decoherence should also be taken into account in the density matrix formalism.\n \n \n\n \\subsection{Formalism}\\label{secIIIA}\n The evolution of the density matrix is derived from the Schr\\\"odinger equation and is given by the Liouville-von Neumann equation\n \\begin{equation}\n     i\\dfrac{d\\rho}{dt}=\\left[  \\hat H,\\rho  \\right]\\,.\n \\end{equation}\n  We take equation (\\ref{eq25}) as our Schr\\\"odinger equation and define the density matrix as $\\rho=\\psi\\psi^{\\dagger}$, with $\\psi^{T}=\\left[ A_{\\lambda},h_{\\lambda}\\right]$. The Hamiltonian of the system is given by the mixing matrix $\\mathcal M$. We introduce the photon decoherence effects into the system defining the hermitian matrix, $ \\Gamma $, such that the new Hamiltonian takes the form\n  \\begin{equation}\n     H =\\mathcal M-i\\dfrac{1}{2}\\Gamma,\n  \\end{equation}\n  where $ \\Gamma $ is given by the collision integrals of the Boltzmann equation. It only interacts with the electromagnetic part; so, at first linear order, the matrix is written as\n    \\begin{equation}\n    \\Gamma=\\left({\\begin{array}{cc}\n   \\Gamma_{\\gamma} &0 \\\\\n   0 & 0 \\\\\n  \\end{array} } \\right),\n  \\end{equation} \nbeing $\\Gamma_{\\gamma}=\\sigma_Tn_e$ the inverse of the mean free path of photons due to Thompson scattering with electrons and $\\sigma_T $ the Thompson cross section. Decoherence is represented as an antihermitic part of the Hamiltonian, so we have to take into account that in the matrix $\\mathcal M$ we already had an antihermitic part contained in the complex parameter $ \\Delta_g $. Then, taking it into account, from equation (\\ref{eq24}), we write the evolution of the density matrix as\n  \\begin{equation}\\label{eq48n}\n           i\\dfrac{d\\rho}{d\\eta}=\\left[  \\mathcal M_{(+)},\\rho  \\right]+\\{  \\mathcal M_{(-)}-\\dfrac{i}{2}\\Gamma,\\rho  \\},\n  \\end{equation}\n  where $ \\mathcal M_{(+)}$ and $\\mathcal M_{(-)}$ are the hermitic and antihermitic part of $\\mathcal M$, respectively, $\\mathcal M_{(-)}$ being given by the complex part of $n_g$.\n As in the case of $\\Gamma$ for EMWs, the antihermitic part of $\\mathcal M$ is equivalent to a damping factor for GWs due to the interaction with external degrees of freedom. \n  \n  \nIn both matrices, $\\mathcal M$ and $\\Gamma$, there are parameters that depend on the scale factor. While in the wave function formalism we have calculated the probability by treating them as approximately constant in a short time interval, in this formalism we will write their dependence explicitly and calculate the evolution in terms of the scale factor. With this aim, we first express equation (\\ref{eq48n}) using the scale factor $a$ instead of the conformal time $\\eta$; so that the evolution of $\\rho$ is now given by\n\\begin{equation}\\label{eq47}\n i\\mathcal{H}a\\dfrac{d\\rho}{da}=\\left[  \\mathcal M_{(+)},\\rho  \\right]+\\{\\mathcal M_{(-)}-\\dfrac{i}{2}\\Gamma,\\rho  \\}.\n\\end{equation}\nWe describe the elements outside the diagonal in terms of their real and imaginary parts $\\rho_{\\gamma g}=\\rho_{g \\gamma}^*=R+iI$,  while the terms of the diagonal $\\rho_{\\gamma}$ and $\\rho_{g}$ are real numbers.   The diagonal elements of $\\rho$ represent the probabilities of finding a photon or a graviton, repectively. Then, the probability of graviton to photon conversion will be given by the value of $\\rho_{\\gamma}$ defining an initial state without photons. From equation (\\ref{eq47}) we obtain the following system of differential equations\n\\begin{eqnarray}\\label{eq48}\n\\dfrac{d\\rho_{\\gamma}}{da}&=&\\dfrac{2\\Delta_M I+2\\Gamma_{\\gamma}\\rho_{\\gamma}}{\\mathcal{H}a},\\\\\n\\dfrac{d\\rho_{g}}{da}&=&-\\dfrac{2\\Delta_M I+2\\Delta_g^I \\rho_g}{\\mathcal{H}a},\\label{eq49}\\\\\n\\dfrac{dR}{da}&=&\\dfrac{(\\Delta^R_{g}-\\Delta_{\\gamma})I-(\\Delta_g^I+\\Gamma_{\\gamma}) R}{\\mathcal{H}a},\\label{eq50}\\\\\n\\dfrac{dI}{da}&=&\\dfrac{(\\rho_g-\\rho_{\\gamma})\\Delta_M+(\\Delta_{\\gamma}-\\Delta^R_{g})R-(\\Delta_g^I+\\Gamma_{\\gamma}) I}{\\mathcal{H}a}. \\label{eq51}\\qquad\n\\end{eqnarray}\n\n\nNow we must consider the dependence in the scale factor of the parameters we have introduced in these equations, which are the damping factor and the plasma mass of the photon. These are\n  \\begin{eqnarray}\\label{eq52}\n    \\Delta_{\\gamma}(a)&=&3.36\\cdot10^{-4}X_e(a)\\left[ \\dfrac{1 {\\rm eV}}{\\omega_i}\\right]\\left[ \\dfrac{n_{ie}}{{\\rm cm}^{-3}}\\right]^2\\left( \\dfrac{a_i}{a}\\right)^2, \\qquad\\\\\n    \\Gamma_{\\gamma}(a)&=&6.36\\cdot10^{-12}X_e(a)\\left( \\dfrac{a_i}{a}\\right)^3 \\quad s^{-1},\\label{eq53}\n\\end{eqnarray}\n  where the subscript $i$ refers to the initial states at the time of recombination. We take as reference the scale factor $a_i = 1$. So the scale factor at present would be $ a_0 = 1090 $. We have introduced the fraction of electron ionization in the definition of electronic density, $n_e(a)=X_e(a)n_{Bi}(\\frac{a_i}{a})^3$, with $n_B$ the baryonic density, which in recombination has a value of $n_{Bi}=320 $  ${\\rm cm}^{-3}$. On the other hand, the ionization fraction is predicted by the knowledge of atomic physics and cosmology. Its evolution is governed by a differential equation which is solved  numerically \\cite{Dolgov:2012be}. \n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=3in]{ionization.pdf}\n\\caption{Decrease in the fraction of free electron density with respect to the baryonic density as a function of the scale factor.  It is calculated numerically as in reference \\cite{Dolgov:2012be}.}\n\\label{fig2} \n\\end{center}\n\\end{figure}\nIn figure \\ref{fig2} the fraction of free electron density with respect to the baryonic density is depicted for values up to the reionization period, which starts with the birth of the first stars at $z \\sim 20$. At that moment the fraction of ionization increases rapidly up to $X_e \\sim 1$, leaving the medium again ionized due to the radiation of the stars, and so it remains until the present time. \n\n  \n\nOn the other hand, since the mass of the graviton is limited to very low values and we have seen in the previous section that it does not have an important effect in graviton-photon mixing, we set $\\Delta_g^R =0$. So, in this section we just focus on the attenuation part of the gravitational refractive index neglecting also potential Lorentz violating effects.\nThat is, $\\Delta_g$ is given by equation (\\ref{eq40}), that is\n\\begin{equation}\\label{eq55}\n    \\Delta_g=i(1+\\nu)\\mathcal{H}\\qquad s^{-1}.\n\\end{equation}\nIt should be noted that, in principle, the termr $\\nu$ can depend on $a$ through a dependence on the additional degrees of freedom of the ATG.\n\nFinally, the Hubble parameter can be given assuming a background evolution compatible with a $\\Lambda$CDM model\\footnote{It should be noted that in ATG the background cosmic evolution could differ from the standard $\\Lambda$CDM model. However, we assume that potential discrepancies are not observable at the background level.}. We express it in terms of the recombination epoch as\n  \\begin{equation}\\label{eq55}\n      \\mathcal{H}(a)=8\\cdot10^{-14}\\left( \\dfrac{\\Omega_M}{a}+7.7\\cdot10^{-10}\\Omega_{\\Lambda}a^2 \\right)^{1\/2}\\quad {\\rm s}^{-1},\n  \\end{equation}\nwhere we have taken $\\Omega_M=0.3$ and $\\Omega_{\\Lambda}=0.7$ as approximate values for the density parameters. \n\n  \n  \n \\subsection{Phenomenology}\\label{secIIIB}\nWe already have all the terms needed to solve the graviton-photon mixing system. We cannot obtain an analytic solution for $\\rho_{\\gamma}$ but we can solve system (\\ref{eq48})-(\\ref{eq51}) numerically taking into account equations (\\ref{eq40}), (\\ref{eq52}), (\\ref{eq53}), and (\\ref{eq55}) with values of phenomenological interest.\n\nLet us first discuss the results for GR, that are obtained taking $ \\nu = 0 $ in equation (\\ref{eq40}). The evolution of the conversion probability in GR as a function of the cosmological scale factor can be seen in figures \\ref{fig3} and \\ref{fig4} (dashed line). Two phases can be clearly differentiated during the evolution of the conversion probability characterized by the shape of the ionization fraction $ X_e $.  The first period corresponds to the decrease in the amount of free electrons in the Universe due to the recombination with the protons, figure \\ref{fig3}. During this phase the probability increases with time, until the period of reionization begins. From that moment, when the medium is again ionized, oscillations begin to be observed in the probability, \\ref{fig4}.  These oscillations have an amplified amplitud due to the imaginary gravitational refractive index characterizing cosmological backgrounds in GR. Its effect dominates over the damping of $\\Gamma_{\\gamma}$, which tends to attenuate the probability. Such amplification is not observed in Minkowski's spacetime \\cite{Dolgov:2012be}. \n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=3in]{a1.pdf}\n\\caption{Graviton-photon conversion probability as a function of the scale factor for a graviton with frequency $\\omega=10^5$ eV and external magnetic field $B=3\\cdot10^3$ G. We define the initial values $\\rho_g(1)=1$ and $\\rho_{\\gamma}(1)=R(1)=I(1)=0$.  The period of evolution is from the initial state $a=1$ at recombination to $a=51,9$, when reionization begins. It has been calculated solving the system (\\ref{eq48})-(\\ref{eq51}) numerically. The dashed line correspond to the GR case with $\\nu=0$, the green line corresponds to  $\\nu=0.06$ and the orange line to $\\nu=-0.06$.}\n\\label{fig3} \n\\end{center}\n\\end{figure}\n\\begin{figure}[h]\n\\begin{center}\n \\includegraphics[width=3in]{a2.pdf}\n\\caption{This figure represents the continuation of figure 3 for the evolution of probability. Three cases are depicted: GR that is $\\nu=0$ (dashed), $\\nu=0.06$ (green), and $\\nu=-0.06$ (orange). The period shown begins at reionization, $a = 51.9$, and ends at the present time, $ a = 1090 $.}\n\\label{fig4} \n\\end{center}\n\\end{figure}\n\nFrom the previous results for GR and the estimation in ATG using the wave function formalism discussed in previous sections, we can deduce that the probability is sensitive to the amplification\/attenuation term of the graviton.  In order to integrate the system for ATGs, we will consider that $\\nu$ is constant or varies slowly in the interval of interest, so that it can be treated as a parameter characteristic of the ATG. The value of $\\nu$ will bounded by observations  \\cite{Belgacem:2018lbp}, as already discussed in previous sections. In figures \\ref{fig3} and \\ref{fig4} we represent the evolution of probability for different values of $\\nu$. Comparing with the GR case, we observe that the same phases in evolution are differentiated.   For values of $\\nu<0$ we see that the probability is amplified, starting with a faster rise in the first phase and then, in the second phase, an increase in the amplitude of oscillations, as opposed to what happens for values of $ \\nu> 0 $.\nAs we have already discussed using the wave function formalism, the probability is highly sensitive to the value of $\\nu$.\n\n\n\n\\section{Conclusion}\\label{secIV}\nWe have studied the phenomenon of graviton-photon oscillation in a cosmological background in GR. \nWe have argued that the prediction for the mixing probability of GWs after recombination (with energies of the order of $10^5$ eV at that time) is of order  $P_{g\\longrightarrow\\gamma}\\sim10^{-11}$. This value is in agreement with that obtained in reference \\cite{Dolgov:2012be}, even if we take into account the cosmic expansion. \n\n\nWe have also considered the possibility that an ATG could describe the gravitational phenomena. The new parameters induced by the ATG can be introduced in the graviton-photon system through an effective refractive index for the graviton.  \nApplying the wave function formalism we have concluded that a mass term for the graviton does not alter the general relativistic predictions for graviton-photon oscillation. On the other hand, we have shown that the mixing probability has a strong dependence on the attenuation term, which only affects the imaginary part of the gravitational refractive index.  We have compared the mixing probability for different values of the attenuation term $ \\nu$ and concluded that it decreases for $\\nu>0$ and increases for $\\nu<0$, being the amplitude of the oscillations amplified for lower negative values of  $ \\nu $. It can be noted that for $ \\nu = -1 $ one has $ \\Delta_g^I=0 $, as if decoherence be effectively removed. This particular case corresponds to resonance.\n\nFurthermore, we have discussed the consequences of having an open graviton-photon system using the density matrix formalism. With the intuition acquired using the wave function formalism, we have focused on the most interesting case of ATGs leading to purely imaginary deviations of the GR gravitational refractive index. Assuming an approximately constant attenuation term, we have integrated the system numerically and confirmed the amplification of the effects for theories with $\\nu<0$.\n\nAlthough there is no doubt of the qualitative results presented in this work, a more rigorous treatment should take into account the effect of the cosmic expansion on the parameters of the model.\nFor example, a particular ATG would predict a concrete evolution for the attenuation term, which can be substituted when integrating the system for a particular model.\nWhereas we have preferred to discuss graviton-photon mixing in a model independent way, setting out bounds corresponding to the allowed range of values for the parameters, that study may serve to establish new constraints on the parameters of different ATGs.\n \n\n\n\\begin{acknowledgments}\nThis work was partially supported by the MICINN (Ministerio de Ciencia e Innovacion, Spain) project PID2019-107394GB-I00\/AEI\/10.13039\/501100011033 (AEI\/FEDER, UE). This article is based upon work from COST Action COSMIC WISPers CA21106, supported by COST (European Cooperation in Science and Technology). JARC acknowledges support by Institut Pascal at Universite Paris-Saclay during the Paris-Saclay Astroparticle Symposium 2022, with the support of the P2IO Laboratory of Excellence (program ``Investissements d'avenir\" ANR-11-IDEX-0003-01 Paris-Saclay and ANR-10-LABX-0038), the P2I axis of the Graduate School of Physics of Universite Paris-Saclay, as well as IJCLab, CEA, APPEC, IAS, OSUPS, and the IN2P3 master projet UCMN.\n\\end{acknowledgments}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLTE has succeeded in meeting its goals and objectives as a global cellular standard since its first release (Release 8). Through its evolution to 5G New Radio (5G NR), it has acquired enhancements across all layers, with new capabilities such as support for vehicular communications \\cite{Uhlemann_CV2X_VTM_2017}, device to device (D2D) and the Internet of Things (Narrowband-IoT) \\cite{Maldonado_2017_NBIoT}. Because of its success in the commercial and civilian sectors, LTE is being targeted to be used in public safety \\cite{Kumbhar_PubSafe_2016}, military networks, and control of the smart grid.\n\nHowever, LTE is not without its shortcomings. It has been shown to be vulnerable to protocol-aware attacks across all layers \\cite{Lichtman_LTEJS_ComMag_2016},  especially the physical layer \\cite{Raghu_MILCOM_2017}, \\cite{Vuk_VTC_2017}. A particular class of control channel vulnerabilities that is well known is \\emph{pilot\/reference signal interference} \\cite{Clancy_PilJam_2011}. Since pilot signals are used for channel estimation and coherent demodulation, localized interference on pilot resource elements in OFDM-based systems result in a significantly lower SINR than that in the case of \\textit{non-pilot interference}.\n\nShahriar et al. \\cite{Shahriar_PilRand_2013} proposed and evaluated pilot-tone randomization as a strategy to mitigate pilot interference. The key motivation behind their scheme is to facilitate accurate channel estimation by changing pilot locations in a pseudo-random manner, so that uncorrupted data symbols are decoded and the resulting bit error rate (BER) is much lower than that in pilot interference. Karlsson et al. \\cite{Karlsson_TDD_Massive_MIMO_Jam_2017} theoretically analyzed the vulnerability of TDD-massive MIMO to pilot interference. Xu et al. \\cite{Xu_Ind_Code_Check_2019} develop a novel coding scheme to authenticate the channel training phase using an independence-checking coding (ICC)-based protocol. In our prior work in \\cite{Raghu_MILCOM_2017}, we have demonstrated the vulnerability of LTE to pilot interference through software-defined radio (SDR)-based experiments. \n\nIn this paper, we demonstrate the vulnerability of cellular link adaptation mechanisms to \\textit{non-pilot interference} (NPI).\nLink adaptation operates by adapting the transmission mode (e.g. SISO, MIMO, diversity, beamforming) and modulation and coding scheme (MCS) as a function of the SINR. Since modern cellular standards such as LTE and NR implement \\textit{pilot-aided SINR estimation}, these estimates will be accurate as long as the interference and noise statistics on the pilots and data resources are the same. Unfortunately, NPI does not obey this condition, since pilots will be interference-free while non-pilots will be interference-corrupted. We analyze the impact of NPI on link adaptation considering the LTE downlink as an example, and observe significant degradation of block error rate (BLER) and throughput through experimental and simulation results. In addition, we define \\textit{`retransmission-induced latency'}, a metric that quantifies the latency only due to retransmissions, and derive an approximate expression relating it with the BLER. We develop useful insights through simulation results, comment on the accuracy of our expressions, and highlight the detrimental impact of NPI on low-latency communications. \n\nThe rest of the paper is organized as follows: Section \\ref{Sec2_Overview} provides a brief overview of link adaptation and CSI feedback in cellular systems. Section \\ref{Sec3_Non_Pil_Interf} introduces the general structure of NPI and its different types. Section \\ref{Sec4_Results_Thpt_BLER} describes our experimental setup, simulation methodology and the key results. Section \\ref{Sec5_Retx_ind_latency} derives an approximate relation between the BLER and the average retransmission-induced latency $(\\bar{\\tau}_{\\mathtt{retx}})$, and provides the analysis and simulation results of the impact of NPI on $\\bar{\\tau}_{\\mathtt{retx}}$. Section \\ref{Sec6_Conc} concludes the paper.\n\n\\section{Overview of Link Adaptation and CSI Feedback in Cellular Systems} \\label{Sec2_Overview}\nLink adaptation is the process of adapting the transmission parameters and hence, the data rate, as a function of the SINR at the receiver. In modern cellular systems, such as 4G LTE and 5G NR, the transmission parameters include the transmission mode (TM) and modulation and coding scheme (MCS). Example of TMs include single-input-single-output (SISO), multiple-input-multiple-output (MIMO), diversity and beamforming. Examples of  MCS include a combination of digital modulation schemes such as BPSK\/QPSK\/QAM, and different forward error correction (FEC) coding schemes and rates \\cite{sesia2011lte}. The channel state information at the receiver (CSIR) is periodically fed back to the transmitter to successfully implement link adaptation schemes. In LTE and NR, the transmitter sends pilot\/reference signals to the receiver to help in estimating the CSIR, which is possible since they are known to both the transmitter and reciever. The receiver quantizes the pilot-aided SINR estimate to limit CSI feedback overhead, and this quantization function depends on the cellular standard. Up to Third Generation Partnership Project (3GPP) Release 14, CSI consists of the following parameters:\n\\begin{enumerate}\n\\item  Channel Quality Indicator (CQI): A 4-bit value that is mapped to a 5-bit MCS value.\n\\item Precoding Matrix Indicator (PMI): An index from a fixed library of precoding matrix elements.\n\\item Rank Indicator (RI): Index of transmission rank that the user can support.\n\\end{enumerate}\nThe 3GPP standard specifies that for the downlink, up to two LTE codewords can be transmitted for each user, where\nthe MCS value for each codeword must be chosen to keep the $\\text{BLER below } 10\\%$ \\cite{sesia2011lte}. In practical deployments, a lookup table-based approach is used for each TM to map the SINR to a CQI value \\cite{Rupp_Sys_LEv_Sim_VTC_2010}.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.8in]{Fig2_time_freq_non_pil_interf.pdf}\n\\caption{Illustration of non-pilot interference (NPI) on the LTE resource grid, and its different types: Frequency-domain NPI and Time-domain NPI.}\n\\label{Fig1}\n\\end{figure}\n\n\\section{Non-Pilot Interference}\\label{Sec3_Non_Pil_Interf}\nFor pilot-aided CSI estimates, the CSIR is accurate as long as the noise and interference statistics on the pilot and the non-pilot resources are the same. While this is true in typical multi-cellular co-channel interference, it will not be the case in the presence on interference that \\textit{only affects non-pilot resources}. We term such interference as \\textit{`non-pilot interference'}.  \n\nIn typical cellular signals, pilots are sparsely allocated resources in time, frequency and spatial layers. Fig. \\ref{Fig1} shows the example of cell-specific reference signals (CRS), which are pilot resource elements in LTE \\cite{sesia2011lte}. In later releases of LTE and 5G NR, there are additional pilots such as demodulation reference signals (DMRS) and CSI-Reference Signals (CSI-RS), which are used for coherent demodulation and CSI estimation for more advanced multi-antenna techniques \\cite{sesia2011lte}. Similar to CRS, these reference signals are also sparsely located in the OFDM resource grid.\n\nHence an interferer can leverage the sparsity of pilot resources to localize interference on non-pilot resources, as shown in Fig. \\ref{Fig1}, to \\textit{contaminate the CSI estimated by the receiver}. In contrast to pilot interference (PI) which contaminates the channel estimates at the receiver \\cite{Lichtman_LTEJS_ComMag_2016}, non-pilot interference (NPI) contaminates the CSIR (SINR estimates). Hence, the CSI reports (CQI, PMI and PI) fed back to the transmitter will be inaccurate in the case of NPI.\n\nBased on the localization of power in time and frequency, non-pilot interference can be broadly classified into:\n\\begin{enumerate}\n\\item Time-domain non-pilot interference (TD-NPI): The interference can take a pulsed form targeting OFDM symbols between two pilots, as shown in Fig. \\ref{Fig1}. Example: On-off jammers, and pulsed-radar signals.\n\\item Frequency-domain non-pilot interference (FD-NPI): Interference can be localized in between pilot subcarriers, as shown in Fig. \\ref{Fig1}. Example: Multi-tone NPI.  \n\\end{enumerate}\nUnintentional interference such as a pulsed radar signal may not exclusively affect non-pilot resources \\textit{all the time}. However, for large pilot spacing (in time) or large radar pulse repetition intervals, the probability of radar affecting non-pilot resources is high.\n\n\\section{Impact of Non-Pilot Interference on LTE}\\label{Sec4_Imp_Non_Pil_Interf} \\label{Sec4_Results_Thpt_BLER}\nWe consider the LTE downlink with an eNodeB serving a single user equipment (UE) in the presence of multi-tone interference. We consider three scenarios: \n\\begin{enumerate}\n\\item Pilot interference (PI): The interference is localized on pilot subcarriers, and non-pilot subcarriers are unaffected. Such interference has been studied in \\cite{Lichtman_LTEJS_ComMag_2016}-\\cite{Xu_Ind_Code_Check_2019}.\n\\item Non-Pilot Interference (NPI): In contrast to pilot interference, the interference is localized exclusively on non-pilot subcarriers. \n\\item Barrage Jamming: This is a common technique of wideband interference, where all subcarriers are affected.\n\\end{enumerate}\nThese interference strategies are shown in Fig. \\ref{Fig3_JammingStrategies}. Prior work on OFDM pilot jamming has shown that the detriment in performance can be mitigated by pseudo-randomly changing the pilot locations to evade the jammer \\cite{Clancy_PilJam_2011, Shahriar_PilRand_2013}. It is interesting to note that the resulting scenario is equivalent to NPI.\nAs our results will show, evading pilot interference will not provide performance improvements predicted by the theoretical analysis in \\cite{Clancy_PilJam_2011}, \\cite{Shahriar_PilRand_2013}.\n\nWe consider a multi-tone non-pilot interferer with equal spacing between adjacent frequency tones, with equal power allocation across all the targeted OFDM subcarriers. Below, we describe the experimental and simulation methodology to evaluate the impact of these interference strategies.\n\n\\begin{figure}[t]\n\t\\centering\n\t\\includegraphics[width=3.0in]{Fig3_JammingStrategies.pdf}\n\t\\caption{Schematic of the multi-tone interference strategies considered: PI, NPI and barrage jamming. For the same power per subcarrier, barrage jamming needs three times the power than PI and NPI.}\n\t\\label{Fig3_JammingStrategies}\n\\end{figure}\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=3.0in]{Fig4_Expt_setup.pdf}\n\\caption{Schematic of the LTE downlink multi-tone interference experiments using the Virginia Tech LTE-CORNET Testbed.}\n\\label{Fig4_Expt_setup}\n\\end{figure}\n\n\\begin{table}[t]\n\\renewcommand{\\arraystretch}{1.0}\n\\caption{Experimental parameters}\n\\label{Tab1_experiment_param}\n\\centering\n\\begin{tabular}{|l|l|}\n\\hline\n\\textbf{Parameter} & \\textbf{Description}\\\\\n\\hline\nLTE Release & 3GPP Release 10\\\\\n\\hline \nFrequency band & Band 7 \\\\\n\\hline\nBandwidth & $10 \\text{ MHz}$ \\\\\n\\hline\nReference signal received & -72 dBm \\\\\npower (RSRP) & \\\\\n\\hline \nTransmission Mode & TM 0 (SISO) from Port 0 \\cite{sesia2011lte} \\\\\n\\hline\nChannel & Cabled setup with SNR $>30$ dB.\\\\\n\\hline \nCSI feedback mode & Periodic and Wideband \\\\\n\\hline\nCSI periodicity & $10 \\text{ ms}$\\\\\n\\hline\nHARQ mode & Asynchronous with upto 4 retransmissions \\\\\n\\hline\n\\end{tabular} \n\\end{table}\n\n\\subsection{Experimental Setup}\\label{Sec4_Expt_Setup}\nThe experimental setup to evaluate LTE's performance in multi-tone interference is shown in Fig. \\ref{Fig4_Expt_setup}. We used Amarisoft\\texttrademark , a proprietary SDR-based LTE eNodeB, and LTE test UEs of Virginia Tech's LTE-CORNET Testbed \\cite{Vuk_Rao_WCNC_2017}. We used a cabled setup enclosed in a Faraday cage to isolate the experiment to\/from external RF signals. The Amarisoft eNB was interfaced with a Universal Software Radio peripheral (USRP) to generate the LTE downlink signal. Commercial LTE UE dongles are connected to a laptop through a USB interface, to emulate a practical scenario. We monitor the LTE downlink and the jammer spectrum using the CMW 500 Rohde and Schwarz\\texttrademark\\ LTE test equipment, and control the LTE and multi-tone interference power using variable attenuators as shown in Fig. \\ref{Fig4_Expt_setup}. For each SINR value, full buffer traffic was generated using an iPerf client at the PC running Amarisoft, and a jPerf server running on the laptop connected to the LTE UE. We generated custom multi-tone interference waveforms using GNURadio\\texttrademark . The experimental parameters are shown in Table \\ref{Tab1_experiment_param}. For each SINR value and interference scenario (PI\/NPI\/barrage), the average throughput and the BLER is measured for $10^5$ frames of LTE downlink traffic to the UE.\n\n\\subsection{Link-level Simulation Study}\\label{Link_lvl_sim_study}\nWe simulated the LTE downlink using a link-level simulator using the MATLAB\\texttrademark\\ LTE Toolbox, consisting of an eNB serving a single UE in the presence of multi-tone interference in a doubly selective (time and frequency) fading channel. We used the extended pedestrian-A (EPA) channel model to simulate the multipath fading characteristics with a Doppler frequency of $20$ Hz and $\\text{SNR} \\geq 25$ dB. The rest of the parameters are the same as shown in Table \\ref{Tab1_experiment_param}.\n \n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[t]{0.5\\textwidth}\n\t\\label{3a}\n\t\\centering\n\t\\includegraphics[width=2.8in]{Fig6_Expt_BLER_all-eps-converted-to.pdf}\n\t\\caption{}\n\\end{subfigure}\n\n \\begin{subfigure}[t]{0.5\\textwidth}\n\t\\label{3b}\n\t\\centering\n\t\\includegraphics[width=2.8in]{Fig8_BLER_All_High_Int_Free_SINR-eps-converted-to.pdf}\n\t\\caption{}\n\\end{subfigure}\n\\caption{BLER as a function of SINR for PI, NPI and barrage interference, measured during (a) hardware experiments, and (b) link-level simulations.}\n\\label{Fig6_Expt_BLER_all}\n\\end{figure}\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.8in]{Fig9_SINR_actual_vs_reported-eps-converted-to.pdf}\n\\caption{Comparison of the actual SINR ($\\mathtt{SINR_{act}}$) versus the SINR inferred from the median user-reported CQI ($\\mathtt{SINR^{(med)}_{CQI}}$), during our hardware experiments for PI, NPI and barrage interference.}\n\\label{Fig9_SINR_actual_vs_reported}\n\\end{figure}\n\n\n\\begin{figure}[t]\n\\centering\n\\begin{subfigure}[t]{0.5\\textwidth}\n\t\\label{3a}\n\t\\centering\n\t\\includegraphics[width=2.8in]{Fig5_Expt_thpt_all-eps-converted-to.pdf}\n\t\\caption{}\n\\end{subfigure}\n\n\\begin{subfigure}[t]{0.5\\textwidth}\n\t\\label{3b}\n\t\\centering\n\t\\includegraphics[width=2.8in]{Fig7_Thpt_All_High_Int_Free_SINR-eps-converted-to.pdf}\n\t\\caption{}\n\\end{subfigure}\n\\caption{Throughput as a function of SINR for PI, NPI and barrage interference, measured during (a) hardware experiments, and (b) link-level simulations.}\n\\label{Fig5_Expt_thpt_all}\n\\end{figure}\n\n\n\\subsection{Results}\nIn the absence of multi-tone interference, we measured an average throughput of $36.5$ Mbps, which is very close to the theoretical maximum throughput of $37.5$ Mbps achievable with a  10 MHz LTE system. In addition, the measured BLER was consistently under $10 \\%$ in the absence of any interference, indicating successful link adaptation. Fig. \\ref{Fig6_Expt_BLER_all}(a) shows the measured BLER performance for different multi-tone interference scenarios, in our hardware experiment from Fig. \\ref{Fig4_Expt_setup}. We observe that the BLER in non-pilot interference is above $50 \\%$ which \\textit{indicates link adaptation failure}, since successful link adaptation has a $\\text{BLER} \\leq 10\\%$. On the other hand, the BLER in pilot and barrage interference fluctuates around $10\\%$ up to an SINR of $0$ dB, and then gradually begins to deteriorate. We observe a similar trend in Fig. \\ref{Fig6_Expt_BLER_all}(b), which shows the BLER performance in our link-level simulations. \n\nDuring our hardware experiments described in \\ref{Sec4_Expt_Setup}, the actual SINR ($\\mathtt{SINR_{act}}$) was measured for each interference scenario using a spectrum analyzer before each downlink transmission. The \\textit{SINR estimated by the UEs} was inferred by monitoring the CQI reports sent by the UE to the eNB for all interference scenarios. Most UE manufacturers use a lookup table-based approach to calculate CQI from the pilot-aided SINR estimate. Assuming a SINR-to-CQI mapping given by\\footnote{We obtained this mapping from MATLAB's LTE Toolbox. The SINR-to-CQI mapping is not unique, but varies from vendor to vendor. Similar SINR-to-CQI mappings have been reported in the literature, for e.g. in \\cite{Rupp_Sys_LEv_Sim_VTC_2010}. } $\\mathtt{SINR_{CQI}} \\text{ (dB)}= 2.11 \\mathtt{CQI} - 9$, the SINR inferred from the median user-reported CQI ($\\mathtt{SINR^{(med)}_{CQI}}$) was calculated. Fig. \\ref{Fig9_SINR_actual_vs_reported} shows the actual SINR versus the quantized SINR reported by the UE to the eNB in its periodic CQI reports. While $|\\mathtt{SINR_{act}} - \\mathtt{SINR^{(med)}_{CQI}}| \\approx 3 \\text{ dB}$ for PI and barrage jamming, for NPI the value of $|\\mathtt{SINR_{act}} - \\mathtt{SINR^{(med)}_{CQI}}|$ ranges from $7-10$ dB. The steep decline in $\\mathtt{SINR^{(med)}_{CQI}}$ at $\\mathtt{SINR_{act}} = -3$ dB for NPI is an artifact of the SINR-to-CQI mapping. When the UE experiences outage after detaching from the eNB, the CQI is taken to be 0 by default, for which the corresponding $\\mathtt{SINR^{(med)}_{CQI}}=-9$ dB.\n\nSuch overly optimistic SINR estimates with NPI leads to the use of a higher MCS which is not supported by the downlink data channel. Wrongly decoded blocks are retransmitted and when the BLER is high, retransmissions are more frequent. This means that the link adaptation procedure optimized for 3GPP channel models fail when the channel is impaired by multitone interference, especially when pilot signals are not impaired but the data channel is corrupted by interference.\n\nFig. \\ref{Fig5_Expt_thpt_all}(a) shows the measured throughput of the LTE downlink for different interference scenarios. We observe that the throughput performance is close for PI and NPI. However, we observe that outage occurs in NPI at a higher SINR when compared to PI. This behavior can also be observed in our link-level simulation throughput results in Fig. \\ref{Fig5_Expt_thpt_all}(b). We also observe that barrage jamming requires about $5$ dB more power than PI\/NPI to cause the same throughput detriment.\n\n\n\\section{Impact of Non-Pilot Interference on Low-latency Communication Systems} \\label{Sec5_Retx_ind_latency}\nThe end-to-end latency in a wireless link encompasses contributions from various sources such as propagation, queuing, scheduling and signal processing. Here, we are particularly interested in the \\textit{retransmission-induced latency} ${\\tau}_{\\mathtt{retx}}$, which we define as the the latency to a user \\textit{only due to retransmissions}. The the initial transmission is excluded from this latency metric, so that $\\tau_{\\mathtt{retx}}=0$ if the initial transmission succeeds. \n\nTo develop useful insights on the impact of NPI on latency, we use the following system model to derive an approximate relation between the $\\mathtt{BLER}$ and `average' $\\tau_{\\mathtt{retx}}$ ($\\bar{\\tau}_{\\mathtt{retx}}$)\n\\begin{enumerate}\n\\item Each user is allocated resources in entities of \\textit{blocks}, in which the bits are interleaved and encoded\\footnote{In LTE, this is called a transport block which is sent over a transmission time interval (TTI) of 1 ms \\cite{sesia2011lte}.}. \n\\item For each user, the channel and interference is assumed to be quasi-static. The outcome of each scheduling interval forms an i.i.d. sequence $\\mathcal{B}_N=\\{X_1,X_2,\\cdots,X_N \\}$ of Bernoulli trials $X_i$ ($i=1,2,\\cdots,N$), each having a \\textit{probability of success} $p$. \n\\item `Success' is defined as the successful decoding of the data block. \n\\item Random process $\\mathcal{B}_N$ is \\textit{ergodic}. Hence $p=(1-\\mathtt{BLER})$ for $N \\rightarrow \\infty$, since $(1-\\mathtt{BLER})$ represents the fraction of blocks successfully decoded.\n\\end{enumerate}\nIn reality the channel is non-stationary and non-ergodic due to time-varying SINR. In addition, LTE allows a maximum of 4 HARQ retransmissions. Accounting for these factors is beyond the scope of this paper, and interested readers are referred to \\cite{Zhuang_HARQ_Delay_SPecEff_2017} for more details. Below, we demonstrate with our results that the simplified system model is adequate to develop good insights into the impact of NPI on $\\bar{\\tau}_{\\mathtt{retx}}$.\n\nThe number of retransmissions $N_{\\mathtt{retx}} \\in \\{0\\} \\cup \\mathbb{N}$, required to successfully decode the transport block is a \\emph{geometric random variable}. Therefore, its mean $\\bar{N}_{\\mathtt{harq}} = \\mathbb{E}[N_{\\mathtt{harq}}]$ is given by $\\bar{N}_{\\mathtt{harq}} = \\frac{1 - p}{p} = \\frac{\\mathtt{BLER}}{1 - \\mathtt{BLER}}$.\nLet ${\\tau}_{\\mathtt{wait}}$ be the waiting time between consecutive retransmissions to the same user, and $\\bar{\\tau}_{\\mathtt{wait}}$ its average. Then $\\mathbb{E}[\\tau_{\\mathtt{retx}}]=\\bar{\\tau}_{\\mathtt{retx}}$ will be\n\\begin{align}\n\\label{tau_HARQ_appendix}\n\\bar{\\tau}_{\\mathtt{retx}} = \\mathbb{E}[N_{\\mathtt{harq}} \\tau_{\\mathtt{wait}}] = \\bar{N}_{\\mathtt{harq}} \\bar{\\tau}_{\\mathtt{wait}} = \\tfrac{\\mathtt{BLER} \\times \\bar{\\tau}_{\\mathtt{wait}}}{1 - \\mathtt{BLER}}.\n\\end{align}\nWe notice that if $\\mathtt{BLER} \\rightarrow 1$, then $\\bar{\\tau}_{\\mathtt{retx}} \\rightarrow \\infty$ always. In contrast, for perfect link adaptation with $\\mathtt{BLER}=0$, we have $\\bar{\\tau}_{\\mathtt{retx}} = 0$. 4G LTE and 5G NR specify that the TM and MCS should be chosen such that $\\mathtt{BLER} \\leq 10\\%$. Substituting this target $\\mathtt{BLER}$ in (\\ref{tau_HARQ_appendix}), we get $\\bar{\\tau}_{\\mathtt{retx}} \\leq 0.89 \\text{ ms}$.\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width=2.8in]{Fig10_Retx_induced_latency_all-eps-converted-to.pdf}\n\\caption{Comparison of theoretical (solid lines) versus numerical (dashed lines) values of $\\bar{\\tau}_{\\mathtt{retx}}$ as a function of SINR for PI, NPI and barrage interference. The numerical values are obtained from our link-level simulations.}\n\\label{Fig10_Retx_induced_latency_all}\n\\end{figure}\n\nFig. \\ref{Fig10_Retx_induced_latency_all} shows the variation of theoretical and simulated values of $\\bar{\\tau}_{\\mathtt{retx}}$ with SINR, for different interference scenarios. We considered $\\bar{\\tau}_{\\mathtt{wait}}= 8\\text{ ms}$ in our link-level simulations, which is the typical round-trip time in LTE \\cite{sesia2011lte}. We observe that there is a good agreement between the theoretical and simulated values of $\\bar{\\tau}_{\\mathtt{retx}}$. In the case of PI, the non-monotonic behavior of $\\bar{\\tau}_{\\mathtt{retx}}$ is a direct consequence of the non-monotonic behavior of $\\mathtt{BLER}$ in Fig. \\ref{Fig6_Expt_BLER_all}(b). Our model overestimates $\\bar{\\tau}_{\\mathtt{retx}}$ for high values of BLER, which is due to the assumption of ergodicity. The intuition behind this trend is the following. Since the instantaneous BLER can vary due to time fading in the channel, a high BLER in the denominator of (\\ref{tau_HARQ_appendix}) makes $\\bar{\\tau}_{\\mathtt{retx}}$ very sensitive to perturbations in the SINR. However for low to moderate values of BLER, our results suggest that ergodicity is a reasonable approximation, since small perturbations in $\\mathtt{BLER}$ won't cause large deviations in $\\bar{\\tau}_{\\mathtt{retx}}$. \n\nWe observe that NPI has an order of magnitude higher $\\bar{\\tau}_{\\mathtt{retx}}$ when compared to the other interference scenarios. It results in additional latencies of 10-1000 ms for the considered SINRs, whereas a balanced 4G LTE system would have $\\bar{\\tau}_{\\mathtt{retx}} < 1 \\text{ ms}$, irrespective of the SNR. Even for high SINR in NPI, $\\bar{\\tau}_{\\mathtt{retx}} \\geq 10$ ms which is detrimental for low-latency applications, particularly vehicle-to-vehicle communications and ultra-reliable low latency communications in 5G NR.\n\n\\section{Conclusion} \\label{Sec6_Conc}\nIn this paper, we highlighted and analyzed the problem of link adaptation failure in cellular systems due to non-pilot interference (NPI). Using LTE as an example, we demonstrated through experiments and link-level simulations that pilot-aided SINR estimates in NPI are inaccurate and severely degrades link adaptation performance, especially the BLER and throughput. We also derived approximate expressions that relate BLER and \\emph{retransmission-induced latency}. Our results indicate that the links affected by NPI becomes unreliable for vehicular communications and low-latency applications of 4G LTE and 5G NR. Further research into robust SINR estimation and link adaptation is necessary to mitigate this problem in current and future cellular networks, and applications such as virtual\/augmented reality and connected and autonomous vehicles.\n\\balance\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{An example}\n\\label{sec:6}\n\\begin{figure}[H]\n\n\\includegraphics[scale=.6]{schematic}\n\\centering\n\\caption{ A sketch of the example constructed in this section. The manifold decomposes into three pieces, left and right caps and a middle region. The loop drawn on the boundary of the left cap only bounds surfaces of high topological complexity that are at least partly contained in the right cap. This ensures the isoperimetric ratio of that loop is very small.}\n\\label{fig:example_sketch}\n\\end{figure}\n\nThe aim of this section is to show that the first positive eigenvalue of the 1-form Laplacian can vanish exponentially fast in relation to volume. This contrasts the behaviour of the first positive eigenvalue of the Laplacian on functions.\n\nOur construction is similar to that in \\cite{BD}. Essentially, we choose a hyperbolic 3-manifold with totally geodesic boundary and glue it to itself using a particular psuedoAnosov with several useful properties. By \\cite{BMNS}, this family has geometry that up to bounded error can be understood in terms of a simple model family. Using this model family, we show that one can find curves with uniformly bounded length whose stable commutator length grows exponentially in the volume. We then use the spectral gap upper bound in Theorem A to conclude the first positive eigenvalue vanishes exponentially fast.\n\nThroughout this section, we need to compare geodesic lengths in different submanifolds of a given manifold $M$. Let $|\\cdot|_{X}$ denote the geodesic length of a homotopy class of curves relative endpoints in a manifold $X$ and $\\text{length}(\\cdot)$ be the length in $M$ of the curve. Similarly, when we compute stable commutator length for the fundamental group of a manifold $X$, which may or may not be a submanifold of $M$, we denote it $\\scl_X$.\n\nWe will need that for certain curves, $\\scl$ is comparable to length. We begin with a simple but essential technical lemma.\n\\begin{figure}[H]\n\\labellist\n\\small\\hair 2pt\n \\pinlabel {$a_0$} [ ] at 830 1100\n \\pinlabel {$a_1$} [ ] at 1300 1100\n \\pinlabel {$b_0$} [ ] at 830 1250\n \\pinlabel {$b_1$} [ ] at 1300 1250\n \\pinlabel {$t_1$} [ ] at 380 1290\n \\pinlabel {$t_0$} [ ] at 380 1120\n\\endlabellist\n\\centering\n\\includegraphics[width = 15cm]{lemdiagram}\n\\caption{ Illustrating Lemma \\ref{lem:6.1} , the rectangular base of the figure is part of the totally geodesic surface $S$ and the box is the corresponding part of the tubular neighborhood $N_\\varepsilon(S)$ foliated by surfaces $S_t$ parallel to $S$. Drawn in the box is the surface $\\Sigma$, which is transverse the foliation except at isolated points. The multicurve $a_0\\cup a_1$ is part of a single level set $S_{t_0} \\cap \\Sigma$, but only $a_0$ is part of the curve $c_{t_0}$ described in the lemma, whereas the multicurve $b_0\\cup b_1$ forms the multicurve $c_{t_1}$ in the lemma.}\n\\label{fig:box}\n\\end{figure}\n\n\n\\begin{lem} \\label{lem:6.1} Let $M$ be a compact hyperbolic 3-manifold with totally geodesic boundary $ \\d M = S$. Let $\\varepsilon$ be smaller than the injectivity radius of $M$ and such that $N_\\varepsilon(S)$ is an embedded tubular neighborhood. Let $\\{S_t\\}$ be the leaves of the foliation of $N_\\varepsilon(S)$ by surfaces equidistant from $S$.  Let $\\Sigma$ be a smooth incompressible proper not necessarily immersed surface in $M$ that is transverse to the foliation $\\{S_t\\}$ except at isolated points. Let $c = \\d \\Sigma$. By transversality, for generic $t$ the multicurve $c_t$ given by the part of $S_t\\cap \\Sigma$ that cobounds a subsurface of $\\Sigma$ with $c = c_0$ is a smooth multicurve.  Let $T$ be the set (of full measure) of all $t\\in[0,\\varepsilon)$ such that $c_t$ is a smooth multicurve. Since $S$ is totally geodesic, each multicurve $c_t$ is homotopic to a possibly degenerate geodesic multicurve $\\gamma_t$ in $S$. Let $\\Sigma_\\varepsilon$ be the part of $\\Sigma$ contained in $N_\\varepsilon(S)$. Then for $C = 1\/\\varepsilon$, we have that $$\\inf\\limits_{t\\in T} |\\gamma_t|_{S} \\leq C \\emph{Area}(\\Sigma_\\varepsilon).$$\n\\end{lem}\n\n\\begin{proof} The coarea formula implies the inequality $\\inf\\limits_{t\\in T} |c_t|_{S_t} \\leq C \\text{Area}(\\Sigma_\\varepsilon)$.  Since $S$ is totally geodesic, for all $t\\in T$, we have $|\\gamma_t|_S\\leq |c_t|_{S_t}.$  \\end{proof}\nNote that in the previous lemma, when $\\inf\\limits_{t\\in T} |\\gamma_t|_S$ is zero, because $\\Sigma$ is incompressible and any loop with length less than $\\inj(M)$ bounds a disk, every component of $\\Sigma$ can either be homotoped to be disjoint from $N_\\varepsilon(S)$ or be contained in $S$.\n\nThe next proposition requires a notion of geometric complexity for homology classes. For any compact Riemannian manifold $M$ one can define the stable norm on the first homology of $M$ (see \\cite{Gromovmetric} Section 4C). The mass of a Lipschitz 1-chain $\\alpha = \\sum_i t_i\\alpha_i$ in $M$ is defined to be $\\text{mass}(\\alpha) = \\sum_i |t_i|\\text{length}(\\alpha_i).$ The mass of a class $a\\in H_1(M)$ is then the infimal value of the mass of a chain $\\alpha$ representing $a$.\nFor a class $a\\in H_1(M)$, the stable norm of $a$ is then given by $$||a||_{s,M} = \\inf\\limits_{m>0}\\frac{\\text{mass}(m a)}{m}.$$\n\nStable commutator length can also be generalized to geodesic multicurves (see Section 2.6 of \\cite{Calegari}), which can naturally be viewed as Lipschitz chains. Suppose $\\gamma_i\\in \\pi_1 M$ and $ \\sum_i n [\\gamma_i] = 0$ in $H_1(M)$. Let $\\gamma$ be the geodesic multicurve, which is not necessarily simple, consisting of the geodesic loops determined by $\\gamma_i$. Say a surface $f:S\\to M$ is admissible of degree $n(S)$ if it has no closed components and $\\d S$ is a union of circles $S^1_i$ with $f|_{S^1_i}$ a degree $n(S)$ cover of $\\gamma_i$.\nThen we define stable commutator length of $\\gamma$ to be $$\\scl(\\gamma) = \\inf\\limits_{S \\text{ admissible}} \\frac{\\chi_-(S)}{2n(S)}.$$ When $\\gamma$ is a single loop, this definition agrees with the usual definition of stable commutator length.\n\n \\begin{prop} \\label{prop:6.2} Let $M$ be a compact oriented hyperbolic 3-manifold with totally geodesic boundary $\\d M = S$. Let $\\gamma$ be a geodesic multicurve in $S$ that is rationally nullhomologous in $M$. Then there is a constant $D> 0$ depending only on $M$ such that $$||[\\gamma]||_{s,S}\\leq D\\scl_M(\\gamma).$$ \\end{prop}\n\n \\begin{proof} If $\\gamma$ is nullhomologous in $S$, then the left hand side is zero and the inequality holds. Assume now that $[\\gamma]\\neq 0\\in H_1(S)$. Fix $\\delta>0$. Let $\\Sigma_m$ be an incompressible admissible surface for $\\gamma$ of degree $m = n(S)$ such that $\\chi_-(\\Sigma_m)\/2m -\\scl(\\gamma) < \\delta$. We can triangulate $\\Sigma_m$ so that there is a single vertex on each boundary component. This triangulation has $4g +3b - 4$ faces, where $g$ is the genus of $\\Sigma_m$ and $b$ the number of boundary components. We can then straighten this triangulation to obtain a piecewise totally geodesic triangulated surface. Replace $\\Sigma_m$ with this surface. Since every face of this triangulation of $\\Sigma_m$ is geodesic, every face has area at most $\\pi$. Since there are $4g+3b - 4$ faces and $\\chi_-(\\Sigma_m) = 2g-2 + b$, we can estimate  $$\\text{Area}(\\Sigma_m) \\leq 3\\pi\\chi_-(\\Sigma_m).$$\n\nWe can perturb $\\Sigma_m$ to obtain a smooth surface $\\Sigma'_m$ that it is transverse the foliation of $N_\\varepsilon(S)$ except at isolated points and in doing so increase the area by less than $\\delta$. Let $\\gamma_t$ be the family of multicurves in Lemma \\ref{lem:6.1}  applied to $\\Sigma'_m$. Since each curve $\\gamma_t$ cobounds a surface in $S$ with $\\d \\Sigma_m$, they are homologous, thus $||[\\gamma_t]||_{s,S} = m||[\\gamma]||_{s,S}$. Since $||[\\gamma_t]||_{s,S}\\leq |\\gamma_t|_S$,\nLemma \\ref{lem:6.1}  implies that $$m||[\\gamma]||_{s,S}\\leq C\\text{Area}(\\Sigma_m') \\leq C \\text{Area}(\\Sigma_m) + C\\delta \\leq 3C\\pi\\chi_-(\\Sigma_m) + C\\delta.$$\nFrom this we get  $$||[\\gamma]||_{s,S}\\leq 6C\\pi\\chi_-(\\Sigma_m)\/2m + C\\delta\/m\\leq 6C\\pi\\scl_M(\\gamma) + 6C\\pi\\delta +C\\delta\/m.$$\nSince the stable commutator length of a nontrivial rational commutator is bounded away from zero by a constant only depending on $M$, by Theorem 3.9 in \\cite{Calegari}, we can replace $C$ with a larger constant $D$ such that $$||[\\gamma]||_{s,S}\\leq D\\scl_M(\\gamma),$$ as desired. \\end{proof}\n\n We now introduce the family of manifolds that we use in our construction. The family $\\{W_n\\}$ of manifolds we study are easily understood using the model manifold theory of \\cite{BMNS}. In particular, there is a $K$-biLipschitz map between $W_n$ and a model manifold $M_n$, where $K$ is independent of $n$. The base of the construction is Thurston's tripus manifold $W$ (see \\cite{thurstonbook}, Section 3.3.12), a hyperbolic manifold with totally geodesic boundary, and a psuedoAnosov homeomorphism $f$ of the boundary surface $\\d W$. The model manifold $ M_n$ is a degree $n$ cyclic cover of the mapping torus $M_{f}$ cut open along a fiber with two oppositely oriented copies of $W$, denoted $W^+$ and $W^-$ glued as described in \\cite{BMNS} Section 2.15 to the two boundary components of the cut open mapping torus. This decomposes $W_n$ into three pieces, a product region $S\\times [0, n]$ and the caps $W^+$ and $W^-$ in a metrically controlled way. It will be convenient to set $M^+ = W^+\\subset M_n$ and $M^- = W^-\\subset M_n$ when talking about the caps of the model manifold $M_n$ for fixed $n$, and to let $W^+$ and $W^-$ denote the images of these spaces under the natural inclusion into $W_n$.\n\n \\vspace{1cm}\n\\begin{figure}[H]\n\\labellist\n\\small\\hair 2pt\n \\pinlabel {$M^+$} [ ] at 950 1100\n \\pinlabel {$M^-$} [ ] at 2000 1100\n \\pinlabel {$S\\times[0,n]$} [ ] at 1500 1600\n\\endlabellist\n\\centering\n\\includegraphics[scale=.15]{modelscheme}\n\\caption{ A schematic picture of the model manifold $M_n$ with caps $M^+$ and $M^-$ two oppositely oriented copies of the tripus manifold.}\n\\label{fig:caps_schematic}\n\\end{figure}\n\nGiven a multicurve $c$ in $M^{\\pm}$, we say $c$ \\textbf{bounds on both sides} if there are incompressible surfaces $S^+$ in $M^+$ and $S^-$ in $M^-$ both with boundary homotopic to $c$.\n\nWe encode the construction and its essential properties in the following proposition.\n\n\\begin{prop}  \\label{prop:6.3}There is a family $\\{W_n\\}$ of closed hyperbolic 3-manifolds with injectivity radius uniformly bounded below and volume growing linearly in $n$ constructed from the tripus and a pseudoAnosov $f$ as described above. Each manifold $W_n$ is $K$-biLipschitz equivalent to the model manifold $M_n$ for some constant $K$ independent of $n$. Any homologically nontrivial loop in $H_1(\\d W^{\\pm})$ that bounds a surface in $M^{\\pm}$ cannot bound on both sides. The pseudoAnosov $f$ is such that for any nonzero class $a\\in H_1(\\d W^+)$, the stable norm of $f_*^n(a)$ grows exponentially.\n\\end{prop}\n\n\\begin{proof}\nLet $W$ be Thurston's tripus manifold, a compact hyperbolic 3-manifold with totally geodesic boundary a genus 2 surface for which the inclusion map $H_1(\\d W;{\\mathbb Z})\\to H_1(W;{\\mathbb Z})$ is onto.\nThe homology of the boundary $\\d W$ decomposes as the direct sum of rank 2 submodules $U$ and $V$, where $V\\subset H_1(\\d W)$ is the image of the boundary map $\\d : H_2(W,\\d W;{\\mathbb Z})\\to H_1(\\d W;{\\mathbb Z})$ (which is also the kernel of the inclusion $H_1(\\d W)\\to H_1(W)$) and $U$ is a compliment of $V$ (note that the inclusion map $H_1(\\d W)\\to H_1(W)$ restricted to $U$ is an isomorphism).\nLet $S$ be a genus 2 surface, which we will use to mark the boundaries of $W^{+}$ and $W^-$. Assume $H_1(S;{\\mathbb Z})$ is generated by $e_1,~e_2,~e_3,~e_4$. Choose a marking $S\\to \\d W^+$ so in $W^+$ one has $U = \\langle e_1,e_2\\rangle $ and $V = \\langle e_3, e_4\\rangle$.\nSimilarly, choose a marking $S\\to \\d W^-$ so that in $W^-$ one has $V = \\langle e_1,e_2\\rangle $ and $U = \\langle e_3, e_4\\rangle$. We then define $$W_n = W^+\\cup_{f^n}W^-$$ where $f:S\\to S$ is a pseudo-Anosov that acts on $H_1(S)$ by the symplectic matrix\\[ F = \\begin{pmatrix}\n 2 &  1 & 0 & 0 \\\\\n 1 & 1 & 0 & 0 \\\\\n 0 & 0 & 1 & -1\\\\\n 0 & 0 & -1 & 2\n\\end{pmatrix} \\] For the existence of such a pseudoAnosov mapping class, see the proof Lemma 7.1 in \\cite{BD}. This matrix preserves the subspace decomposition above, and so ensures that every curve in $\\d W^{\\pm}$ that is not nullhomologous in $\\d W^{\\pm}$ but bounds a surface in $M^{\\pm}$ cannot bound on both sides.\n\nThe mapping class $f$ acts as an Anosov matrix on $U$ and $V$. This ensures the standard Euclidean $\\ell^2$-norm $||F^n(a)||_E$ of an element $a\\in H_1(S)$ grows exponentially in $n$ (indeed, for our choice of $F$, it grows like $(\\frac{3+\\sqrt{5}}{2})^n$). Since norms on finite dimensional real vector spaces are comparable, there is a constant comparing the stable norm induced by the metric inherited from $W$ to the standard Euclidean $\\ell^2$-norm on $H_1(S)$.\n\nLemma 7.3 in \\cite{BD} explains how Theorem 8.1 in \\cite{BMNS} implies that for large $n$ the manifolds $W_n$ admit a $K$-biLipschitz diffeomorphism $\\mu$ from the model manifold $ M_n$ as described above. After increasing $K$, we can drop the large $n$ condition. This then also implies the linear volume growth and injectivity radius bounds.\n\\end{proof}\n\n\\begin{remark}\nUsing the model manifold, one can easily estimate the Cheeger constant of $W_n$, which will decay like $1\/n$.\n\\end{remark}\n\n\n\\begin{mainthm}  \\label{thm:C} The family $W_n$ of closed hyperbolic 3-manifolds from Proposition \\ref{prop:6.3}  has 1-form Laplacian spectral gap that vanishes exponentially fast in relation to volume:\n$$\\sqrt{\\lambda(W_n)}\\leq B\\vol(W_n)e^{-r\\vol(W_n)},$$\nwhere $r$ and $B$ are positive positive constants and $\\lambda(W_n)$ is the first positive eigenvalue of the 1-form Laplacian on $W_n$.\n\\end{mainthm}\n\n\n\\begin{proof}\nWe continue using notation introduced in the previous propositions. Take $\\gamma$ in $\\d M^+\\subset M_n$ to be an embedded geodesic loop representing the class $e_1 \\in U \\subset H_1(\\d W^+)$. Recall from the proof of Proposition \\ref{prop:6.3} that $\\gamma\\subset \\d M^+$ does not bound a surface in $M^+$ but that $f^n(\\gamma)\\subset \\d M^-$ bounds a surface in $M^-$. Let $\\alpha_n = f^n(\\gamma)\\subset \\d M^- \\subset M_n$. Note that $\\alpha_n$ and $\\gamma$ are isotopic in $M_n$.\n\nFix $\\delta>0$. Consider some positive integer $m$ and  incompressible surface $\\Sigma_m$ bounding $\\alpha_n^m$ in $M_n$ with $\\chi_-(\\Sigma_m)\/2m - \\scl_{M_n}(\\gamma) < \\delta$ and which minimizes $\\chi_-$ among surfaces with boundary $\\alpha_n^m$. We can then replace $\\Sigma_m$ with a homotopic surface that pushes the boundary of $\\Sigma_m$ into the interior of $M^-$ and which intersects $\\d M^-$ transversely and essentially in both $\\d M^-$ and $\\Sigma_m$.\nWe can then attach an annulus to $\\Sigma_m$ cobounding $\\alpha_n^m$ and the boundary of the modified surface $\\Sigma_m$. This new $\\Sigma_m$ bounds $\\alpha_n^m$ with a collar neighborhood of the boundary contained entirely in $M^-$ and intersects $\\d M^-$ transversely in a union of loops essential in both $\\Sigma_m$ and $\\d M^-$.\n\nWe focus on the portion of $\\Sigma_m$ that lies in $M^-$. Define $\\Sigma^-_m =\\Sigma_m\\cap M^-$. If $\\Sigma_m$ is contained in $M^-$, then Proposition \\ref{prop:6.2}  applied to $\\alpha_n^m$ in $M^-$ implies that\n$$||[\\alpha_n^m]||_{s,\\d M^-} = m||[\\alpha_n]||_{s,\\d M^-} \\leq m\\scl_{M^-}(\\alpha_n)\\leq D\\chi_-(\\Sigma_m^-) = D\\chi_-(\\Sigma_m) ,$$\nwhere $||\\cdot||_{s, \\d M^-}$ is the stable norm of $H_1(\\d M^-)$. Since $\\chi_-(\\Sigma_m)\/m- \\scl_{M_n}(\\gamma)\\leq \\delta$, we conclude $$||[\\alpha_n]||_{s,\\d M^-}\\leq D\\scl_{M_n}(\\gamma) + D\\delta.$$\n\nOur goal now is to get this same estimate for the other possible ways $\\Sigma_m$ sits in $M_n$.\n\\vspace{1cm}\n\\begin{figure}[H]\n\\labellist\n\\small\\hair 2pt\n \\pinlabel {$M^+$} [ ] at 900 1560\n \\pinlabel {$S\\times[0,n]$} [ ] at 1480 1650\n \\pinlabel {$M^-$} [ ] at 2000 1560\n \\pinlabel {$\\alpha$} [ ] at 1750 1050\n\\endlabellist\n\\centering\n\\includegraphics[scale=.2]{schematic5}\n\\caption{ A schematic picture of the simplest case of a surface bounding $\\alpha$ in $M^-$.}\n\\label{fig:alpha_bounds}\n\\end{figure}\n\nConsider the case that $\\Sigma_m$ does not lie entirely in $M^-$. There are two possibilities. The first involves the surface $\\Sigma_m$ passing into the product region but not intersecting $M^+$. In this case the surface can be homotoped to lie in $M^-$, so that Proposition \\ref{prop:6.2} applies, giving the desired estimate as in the previous case.\n\\vspace{2cm}\n\\begin{figure}[H]\n\\labellist\n\\small\\hair 2pt\n \\pinlabel {$M^+$} [ ] at 900 1560\n \\pinlabel {$S\\times[0,n]$} [ ] at 1480 1650\n \\pinlabel {$M^-$} [ ] at 2000 1560\n \\pinlabel {$\\alpha$} [ ] at 1750 850\n\\endlabellist\n\\centering\n\n\\includegraphics[scale=.2]{schematic3}\n\\caption{ A schematic picture of a surface bounding $\\alpha$ that passes back into $M^-$ but does not pass into $M^+$.}\n\\label{fig:pass_back_schematic}\n\\end{figure}\n\n\\vspace{2cm}\n\\begin{figure}[H]\n\\labellist\n\\small\\hair 2pt\n \\pinlabel {$M^+$} [ ] at 900 1560\n \\pinlabel {$S\\times[0,n]$} [ ] at 1480 1650\n \\pinlabel {$M^-$} [ ] at 2000 1560\n \\pinlabel {$\\alpha$} [ ] at 1750 850\n \\pinlabel {$c_1$} [ ] at 1750 1285\n \\pinlabel {$c_0$} [ ] at 1750 1070\n\n\\endlabellist\n\\centering\n\n\\includegraphics[scale=.2]{schematic6}\n\\caption{ A schematic picture of a surface bounding $\\alpha$ that passes back into $M^+$. Notice the multicurve $c^- = c_0\\cup c_1$ bounds surfaces in $M^+$ and $M^-$, so is homologically trivial. }\n\\label{fig:bounds_both_sides}\n\\end{figure}\n\n\n\nThe second possibility concerns the surface $\\Sigma_m$ crossing through the product region into $M^+$ with an essential intersection with $\\d \\Sigma^+$. In this case, we will see that the surface $\\Sigma_m^-$ has boundary homologous to $\\alpha_n^m$, which will allow us to apply Proposition \\ref{prop:6.2} to obtain the desired estimate. By construction, a sufficiently small collar $C$ of the boundary $\\d \\Sigma_m$ in $\\Sigma_m$ maps into $M^-$, so in particular, a subsurface of $\\Sigma_m^-$ has some boundary component that maps to $\\alpha_n^m$. That boundary component can be closed by attaching a surface $S^-$ that bounds $\\alpha_n^m$ in $M^-$ to $\\Sigma_m$.\nFrom this, we see that the multicurve $c^- = \\d \\Sigma^-_m -\\alpha_n $ bounds surfaces in $M^+$ and $M^-$.\nThus by Proposition \\ref{prop:6.3}, $c^-$ must be homologically trivial in $\\d M^{-}$. Let $x = \\d\\Sigma_m^- = c^- + \\alpha_n^m$. Since $c^-$ is nullhomologous, $||[x]||_{s,\\d M^-} = ||[\\alpha_n^m]||_{s,\\d M^-} = m||[\\alpha_n]||_{s,\\d M^-}.$\nBy Proposition \\ref{prop:6.2} , $||[x]||_{s,\\d M^-}\\leq D\\scl_{M^-}(x).$ Since $x$ is essential in $\\Sigma_m$, we get that $\\chi_-(\\Sigma_m^-) \\leq \\chi_-(\\Sigma_m)$, then using that $\\chi_-(\\Sigma_m)\/2m - \\delta \\leq \\scl_{M_n}(\\alpha_n)$,\nwe obtain $\\scl_{M^-}(x) \\leq \\chi_-(\\Sigma_m^-)\/2 \\leq m\\scl_{M_n}(\\alpha_n) + \\delta m$.\nPutting this all together and dividing by $m$, we get that $$||[\\alpha_n]||_{s,\\d M^-}\\leq D\\scl_{M_n}(\\alpha_n) + D\\delta.$$\n\nWe therefore have in each case that there is a constant $D$ independent of $n$ such that $$||[\\alpha_n]||_{s,\\d M^-}\\leq D\\scl_{M_n}(\\alpha_n) + D\\delta.$$ By Proposition \\ref{prop:6.3} , $||[\\alpha_n]||_{s,\\d M^-} = ||[f^n(\\gamma)]||_{s,\\d M^+}$ grows exponentially in $n$.\nThus for some constants $B>0$ and $r >0$, we have $$Be^{rn}\\leq D\\scl_{M_n}(\\gamma) + D\\delta,$$ where we use that $\\gamma$ and $\\alpha_n$ are homotopic in $M_n$. Using the injectivity radius lower bound and Theorem 3.9 of \\cite{Calegari}, we can increase $D$ and drop the additive constant in this inequality. By Proposition \\ref{prop:6.3} , the volume growth of the $W_n$ is proportional to $n$ so there is a constant $C$ such that $\\vol(W_n)\\leq Cn$.\nAdditionally, using the $K$-biLipschitz comparison of Proposition \\ref{prop:6.3} , the length of $\\gamma$ in $W_n$ is bounded from above by $2K|\\gamma|_{W}$, where $W$ is the tripus. As a result, Theorem A implies that the spectral gap for the 1-form Laplacian of the manifolds $W_n$ vanishes exponentially fast in $n$.\nIn particular, we have \\[\\sqrt{\\lambda(W_n)} \\leq A\\vol(W_n)\\frac{|\\gamma|_{W_n}}{\\scl_{W_n}(\\gamma)} \\leq 2KACB^{-1}D|\\gamma|_Wne^{-rn},\\] so the result holds after redefining $B$ to be $2KACB^{-1}D|\\gamma|_W.$\n\n\\end{proof}\n\n\\section{Introduction}\nThe spectrum of the Hodge Laplacian is a fundamental and well studied geometric invariant of Riemannian manifolds. The Hodge theorem partitions the positive spectrum into exact and coexact eigenvalues. For differential forms of degree one, the exact eigenvalues contain exactly the data of the Laplacian acting on functions and are well understood. The coexact spectrum however is considerably more mysterious. Recently, the first coexact eigenvalue of the Hodge Laplacian of a closed hyperbolic 3-manifold has been related to other aspects of its geometry and topology. For the function Laplacian, the first eigenvalue is known to be comparable to the square of the isoperimetric Cheeger constant. In this paper, we derive a similar estimate for the first coexact eigenvalue, building on work of Lipnowski and Stern in \\cite{LS} motivated by torsion growth in finite covers. We use this new estimate to construct the first examples of hyperbolic 3-manifolds with coexact eigenvalues exponentially small compared to volume.\n\nGiven a hyperbolic 3-manifold $M$, it is natural to try to extract information about $M$ from its finite covers. A deep and interesting conjecture of Bergeron-Venkatesh, L{\\^{e}, and L\\\"uck (see \\cite{BV}, \\cite{thang}, and \\cite{wolf}) asks in part whether the volume of $M$ can be found by studying the torsion in the homology of a family of finite covers of $M$. In studying this question, Bergeron, \\c Seng\\\"un, and Venkatesh in \\cite{BSV} relate the growth rate of the cardinality of the torsion in the first homology of a tower of covers of a closed \\textit{arithmetic} hyperbolic 3-manifold to the spectrum of the Laplacian on 1-forms. In particular, they prove the following theorem, where the technical definitions are given below.\n\n\\begin{thm} (\\cite{BSV}) If a sequence $M_n\\to M_0$ of congruence covers of an arithmetic hyperbolic 3-manifold $M_0$ satisfies the few small eigenvalues, small Betti numbers, and simple cycles conditions, then the log torsion growth rate is proportional to the volume. In particular, one has \\[\\lim\\limits_{n\\to\\infty} \\frac{\\log\\left|H_1(M_n;{\\mathbb Z})_{\\emph{torsion}}\\right|}{\\vol(M_n)} = \\frac{1}{6\\pi}.\\]\n\\end{thm}\n\n\nThe conditions appearing in Theorem 1.1. are:\n\n\\begin{enumerate}\n    \\item (Few small eigenvalues) For all $\\varepsilon>0$ there is a $c>0$ such that $$\\limsup_{n\\to\\infty} \\frac{1}{\\vol(M_n)}\\sum\\limits_{0<\\lambda<c}|\\log\\lambda| \\leq \\varepsilon,$$\nwhere $\\lambda$ runs over the eigenvalues of the 1-form Laplacian on $M_n$.\n\n    \\item  (Small Betti numbers) $b_1(M_n) = o\\left(\\frac{\\vol(M_n)}{\\log \\vol(M_n)}\\right)$.\n\n    \\item (Simple cycles) There exists a constant $C$ depending on $M_0$ such that $H_2(M_n;{\\mathbb Z})$ admits a basis of surfaces $[S_i]$ with bounded Thurston norm $||[S_i]||_{Th}\\ll \\vol(M)^C$.\n\\end{enumerate}\n\nBergeron, \\c{S}eng\\\"un, and Venkatesh's theorem provides a version of L\\\"uck approximation for the limiting $L^2$-torsion of sequences of manifolds that satisfy these conditions. However, no examples of such a sequence are known to exist. Encouraged by the Betti number approximation theorem of \\cite{samurai}, one would hope the above conditions also imply the convergence of $L^2$-torsion for families of manifolds that Benjamini-Schramm converge to $\\H^3$. Examples of Brock and Dunfield show that Benjamini-Schramm convergence itself is insufficient \\cite{BDE}.\n\nPrevious work has focused on the simple cycles condition. In their paper, Bergeron, \\c Seng\\\"un, and Venkatesh conjecture the simple cycles condition is satisfied for all arithmetic congruence covers, and ask in contrast if there are families of closed hyperbolic 3-manifolds with injectivity radius bounded below and volume going to infinity that do not satisfy it. This question was answered by Brock and Dunfield in \\cite{BD}, where they construct a sequence of closed hyperbolic 3-manifolds $W_n$ with injectivity radius bounded below and volume going to infinity such that $H_2(W_n;{\\mathbb Z})\\cong {\\mathbb Z}$ for every $n$ and for which the Thurston norm of the generator grows exponentially in the volume.\n\nIn this paper, we continue the study of the few small eigenvalues condition initiated by Lipnowski and Stern in \\cite{LS}. There, it is shown that for a family of covers of a triangulated closed hyperbolic $n$-manifold, the first positive eigenvalue of the Laplacian acting on 1-forms is comparable to a certain isoperimetric ratio, where the comparison constants depend on the volume of the cover, the geometry of the base, and the specific triangulation. In this paper, we prove that for closed hyperbolic 3-manifolds satisfying a uniform lower bound on injectivity radius, such a comparison can be done with universal constants and polynomial dependence on volume (Theorems A and B, with only Theorem A requiring the restriction to dimension 3). We then leverage the comparison in Theorem A to show that a specific family of closed hyperbolic 3-manifolds have first positive eigenvalue of the 1-form Laplacian vanishing exponentially fast in the volume (Theorem C). The manifolds in Theorem C are not covers of a fixed base, so the estimates of \\cite{LS} do not apply.\n\nAnother motivation for this work comes from recent work of Lin and Lipnowski in \\cite{LL}, where for closed rational homology 3-spheres they leverage a relationship between the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms and irreducible solutions to the Seiberg-Witten equations to determine if certain spaces are $L$-spaces. In particular, if a closed rational homology 3-sphere $M$ is not an $L$-space, then the first eigenvalue $\\lambda$ of the Hodge Laplacian acting on coexact 1-forms satisfies $\\lambda\\leq 2$. Using a version of the Selberg trace formula, they relate this to the complex length spectrum of $M$. Numerical methods can then be used to verify if $\\lambda \\leq 2$. While the constants in our eigenvalue estimates are rather opaque, so that the isoperimetric ratio we study is not, at least presently, capable of certifying that $\\lambda\\leq 2$, the relationship between the stable isoperimetric ratio and whether a space is an $L$-space remains tantalizing.\n\n\\subsection{Results}\nThe isoperimetric ratio we study relates the topological complexity of a surface with boundary to the geometric length of its boundary. One can view the extremal value of this ratio as an analogue of the two-dimensional Cheeger constant. The topological complexity measure is given by stable commutator length. The stable commutator length of a nullhomologous loop $\\gamma$ in a manifold $M$, denoted $\\scl(\\gamma)$, is defined to be \\[\\scl(\\gamma) = \\inf\\limits_{m\\geq 1}\\frac{{\\tt{cl}}(\\gamma ^{m})}{m},\\] where ${\\tt{cl}}(\\gamma)$ is the word length of $\\gamma$ in the commutator subgroup of $\\pi_1M$ with generating set all commutators. Topologically, stable commutator length corresponds to the stable complexity of a surface bounding a nullhomologous loop. Denote the subgroup of rationally nullhomologous loops by $\\Gamma_{{\\mathbb Q}}' = \\ker\\left(\\pi_1M\\to H_1(M;{\\mathbb Q})\\right)$.\nThen, for $\\gamma\\in \\Gamma'_{{\\mathbb Q}}$, one has $$\\scl(\\gamma) = \\inf\\left\\{\\frac{\\chi_-(S)}{2m}~:~S \\text{ with }\\d S = \\gamma^m,~S\\text{ is connected}\\right\\},$$ where for connected surfaces $\\chi_-(S) = \\max\\{0,-\\chi(S)\\}$. One can think of stable commutator length as a relative version of the Thurston norm. See the monograph \\cite{Calegari} for a detailed exposition of $\\scl$.\nStable commutator length is closely related to area, by Gauss-Bonnet, providing some justification for the following nomenclature. Define the stable isoperimetric constant of $M$ to be \\[\\rho(M) = \\inf\\limits_{\\gamma\\in \\Gamma'_{\\mathbb Q}\\setminus\\{1\\}}\\frac{|\\gamma|}{\\scl(\\gamma)}.\\]\n\nOne can also define the stable area of a rationally nullhomologous loop $\\gamma\\in \\Gamma'_{{\\mathbb Q}}$ to be the infimal normalized area of a surface bounding a power of $\\gamma$: $$\\text{sArea}(\\gamma) = \\inf\\limits_{\\d S = \\gamma^n}\\frac{\\text{Area}(S)}{n}.$$ This leads to another notion of stabilized isoperimetric ratio using stable area in place of stable commutator length: $$\\rho_{\\text{Area}}(M) = \\inf\\limits_{\\gamma\\in\\Gamma'_{{\\mathbb Q}}}\\frac{|\\gamma|}{\\text{sArea}(\\gamma)}.$$\nStable area is related to stable commutator length by $$\\text{sArea}(\\gamma) \\leq 4\\pi\\scl(\\gamma).$$ Thus, $$\\rho(M) \\leq 4\\pi \\rho_{\\text{Area}}(M).$$\nVery little is known generally about how geodesic length relates to stable commutator length in hyperbolic manifolds, though estimates relating length, stable area, and stable commutator length for short curves in hyperbolic manifolds have been obtained by Calegari in \\cite{length}.\n\nOur first theorem relates the coexact spectral gap to stable isoperimetric ratios of arbitrary nullhomologous curves.\n\n\\begin{bigthm}\\label{thm:A}\nLet $M$ be a closed hyperbolic 3-manifold with injectivity radius bounded below by $\\varepsilon>0$ and let $\\lambda$ denote the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms. Then there is a constant $A = A(\\varepsilon)$ such that for any nontrivial element $\\gamma \\in \\Gamma'_{\\mathbb Q}$, one has $$\\sqrt{\\lambda} \\leq A \\vol(M) \\frac{|\\gamma|}{ \\scl( \\gamma)},$$ where $|\\gamma|$ denotes the geodesic length of $\\gamma$.\n\\end{bigthm}\n\nTheorem A has the following obvious corollary:\n\n\n\\begin{cor}\n  Let $M$ be a hyperbolic $3$-manifold with injectivity radius bounded below by $\\varepsilon>0$ and let $\\lambda$ be the first eigenvalue of the Hodge Laplacian acting on coexact 1-forms. Then for the constant $A = A(\\varepsilon)$ from Theorem A, $$ \\sqrt{\\lambda} \\leq A \\vol(M)\\rho(M).$$\n\\end{cor}\n\nThe analogue of Theorem A in \\cite{LS} studies a cochain version of the Hodge Laplacian introduced by Dodziuk in \\cite{Dodziuk} for triangulated manifolds. This chochain Laplacian is called the Whitney Laplacian, and is induced by the Hodge Laplacian by embedding cochains into the $L^2$-de Rham complex.\n\n\\begin{thm} (\\cite{LS} Theorem 1.4) Let $M_0$ be a closed hyperbolic $n$-manifold. Let $K_0$ be a sufficiently fine triangulation. Let $M$ be a finite cover of $M_0$. Let $\\lambda_W(M)_{d^*}$ be the first coexact eigenvalue for the Whitney cochain Laplacian associated to the pullback of the triangulation $K_0$ to $M$. Then if some multiple of $\\gamma\\in\\pi_1(M)$ bounds a surface, then $$\\left(\\frac{\\scl(\\gamma)}{|\\gamma|}\\right)^2 \\leq W_{M_0}\\frac{\\vol(M)}{\\lambda_W(M)_{d^*}},$$\nfor a constant $W_{M_0}$ depending on the triangulation $K_0$.\n\\end{thm}\n\n\nUnder a well behaved sequence of subdivisions, Dodziuk and Patodi in \\cite{dP} showed the spectrum of the Whitney Laplacian converges to the spectrum of the Hodge Laplacian. For a fixed but very fine triangulation, the eigenvalue comparison is somewhat delicate. In this setting, Lipnowski and Stern relate the Whitney Laplacian's first coexact eigenvalue to the Hodge Laplacian's first coexact eigenvalue in the following way:\n\n\\begin{thm} (\\cite{LS} Theorem 1.5)\t Let $M_0$ be a closed hyperbolic $n$-manifold. Let $K_0$ be a sufficiently fine triangulation of $M_0$. Let $M$ be a finite cover of $M_0.$ Let $\\lambda_W(M)_{d^*}$ be the first coexact eigenvalue for the Whitney cochain Laplacian associated to the pullback of the triangulation $K_0$ to $M$. Then, $$\\frac{1}{\\lambda_W(M)_{d^*}}\\leq \\max\\left\\{\\frac{4G_{M_0}^2\\vol(M)}{\\lambda_{d^*}(M)}, G_{M_0}^2C_{M_0}^2\\vol(M)\\right\\}.$$\nThe constants $C_{M_0}$ and $G_{M_0}$ depend only on $K_0$.\n\\end{thm}\n\nIn the course of proving Theorem A, we too require a comparison of this sort. By using a smoothed version of the Whitney embedding of cochains into the de Rham complex and triangulations with uniformly controlled geometry (these are called deeply embedded triangulations and are introduced in Section 2), we prove the following Whitney-Hodge eigenvalue comparison.\n\n\\begin{thm} Let $M$ be a closed hyperbolic 3-manifold with $\\inj(M)>\\varepsilon$. Let $\\lambda$ denote the first coexact eigenvalue for the Hodge Laplacian acting on $1$-forms and let $\\lambda_W$ denote the first coexact eigenvalue for the smoothened Whitney Laplacian on 1-cochains associated to a deeply embedded triangulation. There is a constant $G = G(\\varepsilon)$ such that $$\\lambda \\leq G \\vol(M)\\lambda_W.$$\n\\end{thm}\n\nThe smoothened version of the Whitney map needed for the above proposition is obtained by replacing the barycentric coordinates associated to a triangulation with certain smooth partitions of unity indexed by the vertices of a triangulation (an idea of Dodziuk's \\cite{dodzuik2}), which we call barycentric partitions of unity. One can then show that the forms built from these pieces have well behaved Hodge decompositions. This analysis leads to the above eigenvalue comparison for the smoothened Whitney Laplacian constructed from a barycentric partition of unity.\n\nOur second theorem uses the isoperimetric constant $\\rho(M)$ to provide a lower bound on the first coexact eigenvalue of the 1-form Laplacian.\n\n\\begin{bigthm} Let $M$ be a closed hyperbolic $n$-manifold with $\\inj(M) > \\varepsilon$. Let $\\lambda$ be the first positive eigenvalue for the Hodge Laplacian acting on coexact 1-forms and let $H > \\lambda$. Then there is a constant $P(H,\\varepsilon,n)>0$ such that $$ \\frac{P\\rho(M)}{\\vol(M)^{7\/2+1\/n}}\\leq\\sqrt{\\lambda}.$$\n\\end{bigthm}\n\nTheorem B corresponds to Theorem 1.5 below from \\cite{LS}, which uses stable area in place of stable commutator length. Note that Theorem B above remains true if one replaces $\\rho(M)$ with $\\rho_{\\text{Area}}(M)$.\n\n\\begin{thm}(\\cite{LS} Theorem 1.3 )  Let $M_0$ be a closed hyperbolic $n$-manifold and let $M$ be a finite cover of $M_0.$ Then there are constants $A_0$ and $C$, where $A_0$ is depends only on $M_0$ and $C$ is a constant that is uniformly bounded when the injectivity radius of $M$ is bounded below and $\\lambda_1^1(M)$ is bounded above, for which $$\\frac{1}{\\lambda_1^1(M)_{d^*}}\\leq A_0C^2\\vol(M)^{3\/2}\\diam(M)^2 \\left(1+\\rho_{\\emph{Area}}(M)^{-1}\\right).$$\n\\end{thm}\n\nOur approach to proving Theorems A and B is grounded in the following dual characterizations of $\\scl$. By Bavard duality, stable commutator length is related to the defect norm for quasimorphisms and the Gersten filling norm for singular 1-chains. The Gersten filling norm for a nullhomologous loop $\\gamma$ is given by the infimal $\\ell^1$-norm of a singular 2-chain whose boundary is a fundamental cycle for $\\gamma^m$, normalized by $m$. A quasimorphism for a group $\\Gamma$ is a map from $\\Gamma\\to \\mathbb R$ that is nearly a homomorphism in the sense that its coboundary is a bounded map on $\\Gamma^2$. The defect of a quasimorphism is the sup norm of its coboundary. Bavard duality says $$\\scl(\\gamma) = 4\\fill(\\gamma) = \\frac{1}{2}\\sup\\limits_q \\frac{q(\\gamma)}{D(q)},$$ where the supremum is over all quasimorphisms $q$ and $D(q)$ is the defect of $q$.\n\nOne can therefore use the characterization of $\\scl$ as a filling norm when bounding it from above and, similarly, the quasimorphism point of view when bounding it from below. For Theorem A, we relate the filling norm to the spectrum of the Hodge Laplacian via the Whitney Laplacian and Poincar\\'e duality (which forces us to restrict to dimension 3). For Theorem B, we use de Rham quasimorphisms, which are given by integrating coclosed forms over geodesics. Studying the de Rham quasimorphism of a coexact eigenform gives the connection to the spectrum of the Hodge Laplacian.\n\nOur methods primarily differ from \\cite{LS} in that instead of studying covers of a fixed manifold with a specific triangulation, we use that closed hyperbolic manifolds with injectivity radius greater than some $\\varepsilon>0$ can all be triangulated so that the simplices come from a compact collection, and instead of using an $L^2$ discretization of the eigenvalue problem, we use a smooth discretization. The local structure of these triangulations can then be compared in a uniform way, thereby allowing us to relate various combinatorial and geometric norms. By working in the smooth setting instead of the $L^2$ setting, we are able to make use of geometric estimates that require higher regularity. This leads to the more direct Whitney-Hodge eigenvalue comparison of Proposition 1.4.\n\nAs an application of the spectral gap estimate of Theorem A, we modify the construction in \\cite{BD} to construct a family of closed hyperbolic manifolds for which we have control over the stable isoperimetric constant and which have injectivity radius uniformly bounded below.\n\n\\begin{bigthm}\\label{thm:C} There is a family $\\{W_n\\}$ of closed hyperbolic 3-manifolds with injectivity radius bounded below by some $\\varepsilon>0$ and volume growing linearly in $n$ such that the 1-form Laplacian spectral gap vanishes exponentially fast in relation to volume: $$\\sqrt{\\lambda(W_n)}\\leq B \\vol(W_n)e^{-r\\vol(W_n)}$$ where $r$ and $B$ are positive constants and $\\lambda(W_n)$ is the first positive eigenvalue of the 1-form Laplacian on $W_n$.\n\\end{bigthm}\n\nThe manifolds $W_n$ in Theorem $C$ are obtained by taking a hyperbolic 3-manifold with totally geodesic boundary and gluing it to itself using a particular psuedo-Anosov with several useful properties. By \\cite{BMNS}, this family has geometry that up to bounded error can be understood in terms of a simple model family and therefore has the desired injectivity radius lower bound and linear volume growth.\n\nUsing this model family, we show that one can find curves with uniformly bounded length whose stable commutator length grows exponentially in the volume. This task is rather delicate, as it is quite difficult to ensure that the \\emph{stable} commutator length of a given loop is large--- one typically does not know if passing to a power of the loop causes a drastic simplification in commutator length. To overcome this, we require an algebraic condition on the gluing map to control which surfaces bound the loops we study. We then use the following homological isoperimetric estimate to control stable commutator length.\n\n \\begin{thm} Let $M$ be a compact oriented hyperbolic 3-manifold with totally geodesic boundary $\\d M = S$. Let $\\gamma$ be a geodesic multicurve in $S$ that is rationally nullhomologous in $M$. Let $||\\cdot||_{s,S}$ be the stable norm on $H_1(S)$ induced by the Riemannian metric on $S.$ Then there is a constant $D> 0$ depending only on $M$ such that $$||[\\gamma]||_{s,S}\\leq D\\scl_M(\\gamma).$$ \\end{thm}\n\n\n Theorem A then implies the first positive eigenvalue vanishes exponentially fast. We stress that while the manifolds $W_n$ are constructed from the same basic building blocks, they are not otherwise related to one another. In particular, they are not finite covers of some fixed base manifold, so the estimates of \\cite{LS} do not apply; it is therefore vital that the constants appearing in Theorem A depend only on an injectivity radius lower bound.\n\n\\subsection{Brief Outline} In \\hyperref[sec:2]{Section 2}, we show existence and study the local properties of the triangulations we use throughout the paper. We also introduce the smooth Whitney cochain map used to define the approximation of the Laplacian. In \\hyperref[sec:3]{Section 3}, we relate various chain and cochain norms and compare these to various geometric norms.  In \\hyperref[sec:4]{Section 4}, we compare the eigenvalues of the approximation of the Laplacian to the genuine eigenvalues of the Laplacian, then use this comparison and the estimates from \\hyperref[sec:3]{Section 3} to  prove Theorem A. In \\hyperref[sec:5]{Section 5}, we prove Theorem B. Finally, in \\hyperref[sec:6]{Section 6}, we compare the homological length and stable commutator length of certain curves and construct the example of Theorem C. \\hyperref[sec:6]{Section 6} makes use of Theorem A, but is otherwise independent from the rest of the paper.\n\n\\begin{remark}\nThroughout the paper, numerous constants are used. Constants defined inside proofs have no meaning outside the local setting of the proof. The letter $C$ is repeatedly reused in Sections 2 and 3 to denote a constant coming from a Sobolev type estimate and can at any time be taken to be the maximum among all constants denoted by $C$. Such constants depend only on injectivity radius and local choices of things like bump functions, unless specifically noted.\n\\end{remark}\n\n\\section{The lower bound}\n\\label{sec:5}\n\nWe now turn to proving the lower bound on the first coexact eigenvalue of the 1-form Laplacian that constitutes Theorem B. Unlike Theorem A, we prove this eigenvalue comparison without a dimension constraint. The line of proof follows that of Theorem 1.3 in \\cite{LS}.\n\nIn \\cite{LS}, the authors obtain the following estimate controlling the $L^2$-norm of coclosed forms.  Note that this estimate does not depend on the fundamental domain coming from a deeply embedded triangulation.\n\n\\begin{prop} \\label{prop:5.1} (Proposition 5.4 in \\cite{LS})\nLet $\\eta$ be a 1-form on $M$ and $\\mathcal D\\subset \\H^n$ any fundamental domain. Then,\n$$||\\eta||_2^2 \\leq \\emph{Area}(\\d \\mathcal D)||\\eta||_{\\infty}\\left(3\\pi||d\\eta||_{\\infty} + \\max_i\\left|\\int_{\\gamma_i}\\eta\\right|\\right) + \\frac{1}{2}||d\\eta||_{\\infty}||\\eta||_2\\sqrt{\\vol(M)},$$\nwhere the $\\gamma_i$ are the geodesics in the homotopy class of the loops representing the side pairing transformations of the fundamental domain $\\mathcal D$.\n\\end{prop}\n\n\nStudying the terms in the estimate of Proposition \\ref{prop:5.1} for a coexact $\\lambda$-eigenform provides a lower bound on $\\lambda$ given later as Theorem B. The essential idea is that after applying an $L^2$-$L^{\\infty}$ norm comparison, all but one summand on the right-hand side (the integral term), has a $||d\\eta||_2$ term. In particular, if $\\eta$ is a unit norm eigenform, the right-hand side almost has a $\\sqrt{\\lambda}$ term in every summand. Our aim, then, is to replace the integral term with something that looks like $||d\\eta||_{\\infty} (\\rho(M)^{-1}+\\text{stuff})$, where the stuff is polynomial in the volume of $M$ with constants that depend only on the lower bound on injectivity radius.\n\n\\begin{lem} \\label{lem:5.2}Let $a$ be the lift to $\\H^n$ of the cellular approximation of a geodesic loop and let $\\gamma$ be the (oriented) lift of the same geodesic loop. If $\\tilde \\eta$ is the pullback to $\\H^n$ of a 1-form $\\eta\\in \\Omega^1(M)$, then $$\\left|\\int_{a}\\tilde\\eta - \\int_{\\gamma}\\tilde \\eta\\right| \\leq \\pi||d\\eta||_{\\infty}||a||_G.$$ \\end{lem}\n\\begin{proof}\nLet $x$ be the starting point of $\\gamma$, let $a_i$ be the geodesic arcs of $a$ (so that the word corresponding to the cellular path $a$ is the word $a_1\\cdots a_{||a||_G}$), and let $y$ be the end point of $a_{||a||_G}$. Let $Q$ be the piecewise geodesic $(||a||_G+3)$-gon obtained by taking the union of the triangles $\\text{convex hull}(a_i,x)$ and the triangle $\\text{convex hull}(\\gamma,y)$. Since $Q$ is the union of $(||a||_G + 1)$ geodesic triangles of area bounded by $\\pi$, we have the upper bound of $(||a||_G+1)\\pi$ for the area of $Q$.\nThen, since  $$\n\\left|\\int_a\\tilde\\eta - \\int_{\\gamma}\\tilde\\eta\\right| = \\left|\\int_{\\d Q} \\tilde\\eta\\right|  \\leq \\pi||d\\eta||_{\\infty}(||a||_G+1),$$ where the first equality follows from the deck transformation invariance of $\\tilde \\eta$, which makes the integral over $\\d Q\\setminus(a\\cup\\gamma)$ vanish.\nIf one then considers the cellular\ns $ma$ and the geodesic $\\gamma^m$ integrated over the form $\\frac{1}{m}\\tilde\\eta$, one gets\n$$ \\left|\\int_a\\tilde\\eta - \\int_{\\gamma}\\tilde\\eta\\right| = \\frac{1}{m}\\left|\\int_{ma}\\tilde\\eta - \\int_{\\gamma^m}\\tilde \\eta\\right|\\leq \\frac{1}{m}\\pi||d\\eta||_{\\infty}(||ma||_G+1).$$\nTaking the limit as $m\\to\\infty$ then results in  $$\\left|\\int_{a}\\tilde\\eta - \\int_{\\gamma} \\tilde\\eta\\right|\\leq\\pi||d\\eta||_{\\infty}||a||_G,$$ proving the lemma.\n\\end{proof}\n\n\\begin{lem}\\label{lem:5.3}\nThere is a constant $B_0 = B_0(\\varepsilon)$ such that $\\diam(M)\\leq B_0\\vol(M)$.\n\\end{lem}\n\\begin{proof}\n\tFirst note that there is a constant $T = T(\\varepsilon)$ such that the number of simplices in $M$ is bounded by $T\\vol(M)$ and that each simplex from a deeply embedded triangulation has bounded diameter, say bounded by $C_0$. With $B_0 = C_0T$, one has that $B_0\\vol(M)$ bounds the diameter of $M$, as desired.\n\\end{proof}\n\n\\begin{lem}\\label{lem:5.4}\nLet $\\eta\\in\\Omega^1(M)$ be a 1-form and $\\gamma$ a rationally nullhomologous loop in $M$. Then integrating over the geodesic in the free homotopy class satisfies $$\\left|\\int_{\\gamma}\\eta \\right|\\leq 2 \\pi||d\\eta||_{\\infty}\\scl(\\gamma).$$\n\\end{lem}\n\\begin{proof}\n\nThis follows from Bavard duality and the fact that $\\int_{\\gamma}\\eta$, where the integral is over the geodesic in the free homotopy class of $\\gamma$, is a quasimorphism with defect bounded by $\\pi||d\\eta||_{\\infty}$ (see \\cite{Calegari}, page 21).\n\\end{proof}\n\nThe key estimate allowing us to replace the integral term with one involving the stable isoperimetric constant $\\rho(M)$ is the following proposition; compare with Proposition 5.24 in \\cite{LS}.\n\n\\begin{prop} \\label{prop:5.5} Let $\\eta$ be a 1-form on $M$. Then there is a harmonic form $h$ and a constant $L_0 = L_0(\\varepsilon)>0$ such that for every closed geodesic $\\alpha$ in $M$, one has $$\\left|\\int_{\\alpha}(\\eta - h)\\right|\\leq |\\alpha| L_0\\vol(M)^{3\/2} ||d\\eta||_{\\infty}\\left(\\rho(M)^{-1} + 1 \\right).$$\n\\end{prop}\n\\begin{proof}\n\nIf $M$ is a ${\\mathbb Q}$-homology sphere, $\\alpha$ is rationally nullhomologous and $h$ can only be 0. Lemma \\ref{lem:5.4} gives $$\\left|\\int_{\\alpha}\\eta\\right|\\leq 2\\pi\\scl(\\alpha)||d\\eta||_{\\infty}.$$\nBy multiplying the right-hand side by $|\\alpha|\/|\\alpha|$, this becomes $$ \\left|\\int_{\\alpha}\\eta\\right|\\leq 2\\pi |\\alpha|\\frac{\\scl(\\alpha)}{|\\alpha|}||d\\eta||_{\\infty}\\leq 2\\pi|\\alpha|\\rho(M)^{-1}||d\\eta||_{\\infty}.$$\nIf $W$ is the minimal volume hyperbolic $n$-manifold (in dimension 3, this is the Weeks manifold, see\\cite{minvol}, more generally it is know that in dimension $n\\geq4$ that the set of hyperbolic volumes is discrete in $\\mathbb R$, see \\cite{minvol2}), the claim follows with $L_0 = \\frac{2\\pi}{\\vol(W)^{3\/2}}$.\n\nThus, we assume $M$ has nontrivial real homology classes. Fix a basepoint $x_0\\in M$. Take a basis $c_1,\\dots, c_n$ of harmonic 1-chains for  $C_1(K^*)$ using the Euclidean inner product on $C_1(K^*)$. Harmonic chains are norm minimizing for the induced $\\ell^2$-norm. Let $||\\cdot||_E$ denote this norm. Let $h$ be the (unique) harmonic form that satisfies $\\int_{c_i}\\eta - h = 0$ for each $i$. Let $a$ be a cellular path in $K^*$ approximating $\\alpha$, as in Proposition \\ref{prop: length comparison}, so $||a||_G \\leq L|\\alpha|$ and identify the cellular path $a$ with the chain it represents.\n\nThen, using the Hodge decomposition induced by the Euclidean inner product, we get $a = a_h^E + \\d S,$ where $a_h^E$ is harmonic with respect to the Euclidean inner product on the chain complex $C_1(K^*)$ and $S$ is some 2-chain. Since $\\d$ and the Euclidean adjoint $\\d^*_E$ have integral bases, and since $a$ is integral, $\\d S$ is a rational 2-chain. Recall that there is a constant $T=T(\\varepsilon)$ such that the number of 2-simplices in $K^*$ is bounded by $T\\vol(M)$. A short computation then shows,\n\\begin{align*}\n||\\d S||_G &= ||a - a_h^E||_G\\\\\n\t\t&\\leq ||a||_G+||a_h^E||_G\\\\\n\t\t&\\leq ||a||_G+\\sqrt{T\\vol(M)}||a_h^E||_E, \\text{~by the Euclidean $\\ell^1$-$\\ell^2$ norm comparison,}\\\\\n\t\t&\\leq ||a||_G + \\sqrt{T\\vol(M)}||a||_E, \\text{~ as $a_h^E$ is $\\ell^2$-norm minimizing in its class},\\\\\n\t\t&\\leq ||a||_G + \\sqrt{T\\vol(M)}||a||_G, \\text{~by the Euclidean $\\ell^1$-$\\ell^2$ norm comparison},\\\\\n\t\t&= (\\sqrt{T\\vol(M)}+1 )||a||_G.\n\\end{align*}\nBecause there is a minimal volume hyperbolic 3-manifold, we can increase $T$ so that we can write the above as $||\\d S||_G\\leq T\\sqrt{\\vol(M)}||a||_G$. Additionally, since there is a universal upper bound on the length of an edge in $K^*$, there is a constant $E>0$, such that the geodesic length $|\\gamma|$ of a loop $\\gamma$ satisfies $|\\gamma|\\leq E\\len(c)$ for any cellular path $c$ in $K^*$ homotopic to $\\gamma$.\n\nSince $\\d S$ is a rational cycle, take $N>0$ to be an integer so that $N\\d S$ is integral. Then one can glue together oriented copies of the edges on which $\\d S$ is supported along their boundaries to obtain a (non unique) collection of closed cellular loops $b_1,\\dots,b_m$ whose union represents the cycle $N\\d S$.  Fix a vertex $v_i$ in each loop $b_i$. Note that by construction, $\\len(b_i) = ||b_i||_G$ for each $i$. Let $\\tau_i$ be the geodesic arc connecting the basepoint $x_0$ to $v_i$ and $\\tau_i^{-1}$ the oppositely oriented geodesic arc. Define the curve $b$ to be the path $$\\tau_1b_1\\tau_1^{-1}\\tau_2b_2\\tau_2^{-1}\\cdots \\tau_mb_m\\tau_m^{-1}.$$\nLet $\\beta$ be the geodesic loop through $x_0$ homotopic to $b$. Notice $\\sum_i ||b_i||_G = ||b||_G$, where $||b||_G$ is meant in the sense of the norm on singular chains, where the $\\tau^{\\pm 1}$ terms cancel. This gives a possibly trivial element of $\\Gamma_{\\mathbb Q}'$ whose length is bounded as follows:\n\n\\begin{align*}|\\beta| \\leq |b| &= \\sum_i (2|\\tau_i| + |b_i|) \\\\ &\\leq 2\\diam(M)m + E||b||_G \\\\ &\\leq (2\\diam(M) + E)||b||_G \\\\ &\\leq (2B_0\\vol(M) + E)||b||_G,\n\\end{align*}\nwhere we use that $m \\leq ||b||_G$, the diameter bound of Lemma \\ref{lem:5.3}, along with the remarks in the above discussion.\n\nSince $$\\frac{1}{N}||b||_G = ||\\d S||_G \\leq T\\sqrt{\\vol(M)}||a||_G,$$ and  $||a||_G\\leq L|\\alpha|$, we obtain $$\\frac{||b||_G}{N} \\leq TL\\sqrt{\\vol(M)}|\\alpha|.$$ As a result, $$\\frac{|\\beta|}{N}\\leq TL(2B_0\\vol(M)+E)\\sqrt{\\vol(M)}|\\alpha|.$$\nWe compute,\n\\begin{align*}\n\\left|\\int_{\\alpha}\\eta-h\\right| &= \\left|\\int_{\\alpha-a_h} \\eta-h \\right|\\text{~since $a_h$ is in the span of the $c_i$, and $\\int_{c_i}\\eta-h$ = 0,}\\\\\n&\\leq\\left| \\int_{\\d S}\\eta- h\\right| + \\left|\\left(\\int_{\\alpha} \\eta - h\\right) - \\left(\\int_{a}\\eta-h\\right)\\right| \\\\\n&\\leq\\left| \\int_{\\d S}\\eta- h\\right| + \\pi||d\\eta||_{\\infty}||a||_G,\\text{~by Lemma \\ref{lem:5.2},} \\\\\n&= \\frac{1}{N}\\left|\\int_{N\\d S}\\eta-h\\right| + \\pi||d\\eta||_{\\infty}||a||_G\\\\\n&= \\frac{1}{N}\\left|\\int_{b}\\eta-h\\right| + \\pi||d\\eta||_{\\infty}||a||_G,\\text{~since $b$ abelianizes to $N\\d S$,}\\\\\n&\\leq \\frac{1}{N}\\left(\\left|\\int_{\\beta}\\eta-h\\right| + \\left|\\int_b(\\eta-h) - \\int_{\\beta}(\\eta-h)\\right|\\right) + \\pi||d\\eta||_{\\infty}||a||_G\\\\\n&\\leq \\frac{1}{N}\\left|\\int_{\\beta}\\eta-h\\right| + \\frac{1}{N}\\pi||d\\eta||_{\\infty}||b||_G + \\pi||d\\eta||_{\\infty}||a||_G,\\text{~by Lemma \\ref{lem:5.2}}.\n\\end{align*}\n\nIf $\\beta$ is trivial, then the integral term $|\\int_{\\beta}\\eta-h| $ vanishes, and we can replace that term with $$TL\\pi |\\alpha| ||d\\eta||_{\\infty}\\sqrt{\\vol(M)}\\rho(M)^{-1} $$ to obtain (after using our estimate for $||b||_G\/N$ and $||a||_G \\leq L|\\alpha|$)\n\n\\begin{align*}\n\\left|\\int_{\\alpha}\\eta-h\\right| &\\leq TL\\pi|\\alpha|||d\\eta||_{\\infty}\\sqrt{\\vol(M)}\\left(\\rho(M)^{-1}+1\\right) + \\pi L ||d\\eta||_{\\infty}|\\alpha|\\\\\n&\\leq TL\\pi|\\alpha|||d\\eta||_{\\infty}\\sqrt{\\vol(M)}\\left(\\rho(M)^{-1}+1\\right) + \\frac{\\vol(M)^{3\/2}}{\\vol(W)^{3\/2}}\\pi L ||d\\eta||_{\\infty}||\\alpha|\\\\\n&\\leq TL\\pi|\\alpha|||d\\eta||_{\\infty}\\frac{\\vol(M)^{3\/2}}{\\vol(W)}\\left(\\rho(M)^{-1}+1\\right) + \\frac{\\vol(M)^{3\/2}}{\\vol(W)^{3\/2}}\\pi L ||d\\eta||_{\\infty}|\\alpha|,\n\\end{align*}\nwhere in the last line we again use the minimal volume hyperbolic 3-manifold $W$ to replace $\\sqrt{\\vol(M)}$ with $\\vol(M)^{3\/2}.$\nSetting $$L_0 = 2\\max\\{\\frac{2\\pi BDL}{\\vol(W)},\\frac{\\pi L}{\\vol(W)^{3\/2}}\\}$$ and factoring gives the result.\n\nAssume now that $\\beta$ is nontrivial. Combining the above estimates yields\n\n\\begin{align*}\n\\left|\\int_{\\alpha}\\eta-h\\right| &\\leq \\frac{|\\beta|}{N} \\frac{1}{|\\beta|} \\left|\\int_{\\beta}\\eta-h\\right| + \\frac{1}{N}\\pi||d\\eta||_{\\infty}||b||_G + \\pi L ||d\\eta||_{\\infty}|\\alpha| \\\\\n&\\leq TL(2B_0\\vol(M)+E)\\sqrt{\\vol(M)}|\\alpha| \\frac{1}{|\\beta|} \\left|\\int_{\\beta}\\eta-h\\right| \\\\\n&~~~~+ \\pi||d\\eta||_{\\infty} TL\\sqrt{\\vol(M)}|\\alpha | + \\pi L ||d\\eta||_{\\infty}|\\alpha|\n \\\\\n&= TL\\sqrt{\\vol(M)}|\\alpha|\\left((2B_0\\vol(M) + E)\\frac{1}{|\\beta|}\\left|\\int_{\\beta}\\eta-h\\right|  + \\pi||d\\eta||_{\\infty}\\right) \\\\\n&~~~~+ \\pi L ||d\\eta||_{\\infty}|\\alpha|.\n\\end{align*}\n\nSince the geodesic $\\beta$ is nullhomologous, Lemma \\ref{lem:5.4} implies $$\\left|\\int_{\\beta}(\\eta-h) \\right|\\leq 2\\pi||d\\eta||_{\\infty}\\scl(\\beta).$$ Replacing the integral term with this estimate and using that $\\frac{\\scl(\\beta)}{|\\beta|}\\leq\\rho(M)^{-1}$ gives\n\n\\begin{align*}\n\\left|\\int_{\\alpha}\\eta-h\\right| \\leq |\\alpha| TL\\sqrt{\\vol(M)} ||d\\eta||_{\\infty}\\left(2\\pi(B_0\\vol(M) + E)\\rho(M)^{-1}\\right)+ \\pi) \\\\\n+ \\pi L ||d \\eta||_{\\infty}|\\alpha|.\n\\end{align*}\n\nAgain using the existence of a minimal volume hyperbolic $n$-manifold, one can replace $B_0$ with the constant $B_1 = 2B_0 + E\/\\vol(W)$ since $B_1\\vol(M) > 2B_0\\vol(M) + E$. Then, after combining constants in the first summand (and using that $2\\pi > \\pi$ to pull out the terms containing $\\pi$) into a single constant $L_1$, one obtains:\n\n$$\\left|\\int_{\\alpha}(\\eta - h)\\right|\\leq |\\alpha| L_1\\vol(M)^{3\/2} ||d\\eta||_{\\infty}\\left(\\rho(M)^{-1} + 1 \\right) + \\pi L_1 ||d\\eta||_{\\infty}|\\alpha|. $$\n\nSet $L_0 =2\\max\\{L_1,\\frac{\\pi L}{\\vol(W)^{3\/2}}\\}$ and multiply the second summand by $\\vol(M)^{3\/2}$ to obtain the claim.\n\n\\end{proof}\n\n\\begin{lem}  \\label{lem:5.6} Let $M$ have deeply embedded triangulation $K$ and let $\\tilde K$ be the pullback of this triangulation to $\\H^n$.\nThen there is a fundamental domain $\\mathcal D\\subset \\H^n$ for $M$ that is a union of simplices from $\\tilde K$ such that the diameter of $\\mathcal D$ satisfies $\\diam(\\mathcal D) \\leq 3\\diam(M).$ \\end{lem}\n\\begin{proof}\nFix a top dimensional simplex $\\sigma_0\\in K^{(n)}$ and let $\\tilde \\sigma_0$ be a lifted copy in $\\H^n$. Let $\\tilde x_0$ be the barycenter of $\\tilde \\sigma_0$. For every other top dimensional simplex $\\sigma$ in $K^{(n)}$ there is a lift \t$\\tilde \\sigma$ whose barycenter $\\tilde x_{\\sigma}$ is within $\\diam(M)$ of $\\tilde x_0$. Choose one such lift for every $\\sigma$ in such a way the resulting fundamental domain $\\mathcal D$ is connected. Then the diameter of the fundamental domain satisfies $\\diam(\\mathcal D)\\leq \\diam(M) + 2e$, where $e$ is the maximum distance from the barycenter of a simplex in a deeply embedded triangulation to its boundary. Clearly $e<\\diam(M)$, so the lemma immediately follows.\n\\end{proof}\n\n\nNow, assume $M$ has a fixed deeply embedded triangulation and let $\\mathcal D$ be a fundamental domain as in Lemma \\ref{lem:5.6}. Let $\\gamma_i$ be the geodesics in the free homotopy class of the side pairing transformations of the fundamental domain $\\mathcal D$, and notice by construction $|\\gamma_i|\\leq 3\\diam(M)$. With this, we modify the estimate in Proposition \\ref{prop:5.1}to obtain the following.\n\n\\begin{prop}\\label{prop:5.7}\nLet $\\eta$ be a coclosed 1-form on $M$. Let $h$ be the harmonic form of Proposition \\ref{prop:5.5} associated to $\\eta$. Then for a constant $A_0 = A_0(\\varepsilon)>0$, the following holds:\n\\begin{align*}\n||\\eta-h||_2^2 \\leq A_0\\vol(M)||\\eta-h||_{\\infty}\\left(3\\pi||d\\eta||_{\\infty} + 3L_0B_0\\vol(M)^{5\/2} ||d\\eta||_{\\infty} \\left(\\rho(M)^{-1} + 1\\right)\\right) \\\\ + \\frac{1}{2}||d\\eta||_{\\infty}||\\eta-h||_2\\sqrt{\\vol(M)}.\n\\end{align*}\n\n\\end{prop}\n\\begin{proof}\n\nLet $\\gamma_i$ realize the maximum among the integrals $\\int_{\\gamma_i}\\eta$.\nSubstitute the estimate of Proposition \\ref{prop:5.5} for the integral term in Proposition \\ref{prop:5.1} applied to the coclosed form $\\eta-h$ and the fundamental domain $\\mathcal D$ to obtain\n\\begin{align*}\n    ||\\eta-h||_2^2 \\leq \\text{Area}(\\d \\mathcal D)||\\eta-h||_{\\infty}\n    \\left(3\\pi||d\\eta||_{\\infty} +L_0\\vol(M)^{3\/2} ||d\\eta||_{\\infty} |\\gamma_i| \\left(\\rho(M)^{-1} +1 \\right)\\right) \\\\ + \\frac{1}{2}||d\\eta||_{\\infty}||\\eta-h||_2\\sqrt{\\vol(M)}.\n\\end{align*}\n\n\nThen, replace $|\\gamma_i|$ with $3B_0\\vol(M)$, using Lemma \\ref{lem:5.6} and Lemma \\ref{lem:5.3}.\nLastly, since there is an upper bound on the area of a face of any simplex in $\\mathcal G_\\varepsilon$ (a consequence of the bounds on the dihedral angles), the total area of the boundary of a complex made from no more than $T\\vol(M)$ simplices from $\\mathcal G_\\varepsilon$ is bounded by $A_0\\vol(M)$ for a constant $A_0$ depending on $\\varepsilon$. Substituting this estimate for the $\\text{Area}(\\d \\mathcal D)$ term completes the proof.\n\\end{proof}\n\n\\begin{prop}\\label{prop:5.8} (Proposition 2.2 of \\cite{LS}) Let $M$ be a closed hyperbolic n-manifold with $\\inj(M)>\\varepsilon$. Assume the first positive eigenvalue $\\lambda$ of the Laplacian acting on coexact 1-forms is less than some fixed constant $H>0$. Then there is a constant $C(H,\\varepsilon)>0$ such that for a coexact $\\lambda$-eigenform $\\omega$, one has $$||\\omega||_{\\infty}\\leq C(H,\\varepsilon)||\\omega||_2.$$\n\\end{prop}\n\n\\begin{prop}\\label{prop:5.9}\nLet $M$ be a closed hyperbolic n-manifold with $\\inj(M)> \\varepsilon$. Let $\\lambda < H$ be the first positive eigenvalue for the Hodge Laplacian acting on coexact 1-cochains. Then the following holds:\n\\begin{align*}\\frac{1}{\\sqrt{\\lambda}} \\leq A_0\\vol(M)C(H,\\varepsilon)^2\n    \\left(3\\pi+ 3L_0B_0\\vol(M)^{5\/2}\\left(\\rho(M)^{-1} + 1\\right)\\right) \\\\\n    + \\frac{C(H,\\varepsilon)}{2}\\sqrt{\\vol(M)}.\n\\end{align*}\n\\end{prop}\n\n\\begin{proof}Let $\\eta$ be a $\\lambda$ coexact eigenform.\nApplying the Sobolev type estimate of Proposition \\ref{prop:5.8} to each instance of the sup norm in Proposition \\ref{prop:5.7} and using that $||d\\eta||_2=\\sqrt{\\lambda}||\\eta||_2 \\leq \\sqrt{\\lambda}||\\eta - h||_2$, where the inequality follows from the orthogonality of the Hodge decomposition, gives\n\n\\begin{align*}||\\eta-h||_2^2 \\leq A_0\\vol(M) C(H,\\varepsilon) ^2 \\sqrt{\\lambda} ||\\eta-h||_{2}^2 \\left(3\\pi + 3L_0B_0\\vol(M)^{5\/2}\\left(\\rho(M)^{-1}\n    + 1\\right)\\right) \\\\ + \\frac{C(H,\\varepsilon)}{2}\\sqrt{\\lambda} ||\\eta-h||_2^2\\sqrt{\\vol(M)}.\n\\end{align*}\n\nDividing both sides by $\\sqrt{\\lambda} ||\\eta-h||_2^2$ then gives \\begin{align*} \\frac{1}{\\sqrt{\\lambda}}\\leq A_0\\vol(M) C(H,\\varepsilon) ^2\\left(3\\pi+ 3L_0B_0\\vol(M)^{5\/2}\\left(\\rho(M)^{-1} + 1\\right)\\right) \\\\ + \\frac{C(H,\\varepsilon)}{2}\\sqrt{\\vol(M)}.\\end{align*}\\end{proof}\n\nRearanging the terms and combining constants (which again requires the existence of a minimal volume hyperbolic $n$-manifold) in the previous proposition and applying a geometric estimate of Calegari and a systolic inequality due to Sabourau leads to the main theorem of this section.\n\n\\begin{mainthm}\\label{thm:B}\nLet $M$ be a closed hyperbolic $n$-manifold with $\\inj(M) > \\varepsilon$. Let $\\lambda$ be the first positive eigenvalue for the Laplacian acting on coexact 1-forms and let $H > \\lambda$. Then there is a constant $P(H,\\varepsilon)>0$ such that $$ \\frac{P\\rho(M)}{\\vol(M)^{7\/2+1\/n}}\\leq\\sqrt{\\lambda}.$$\n\\end{mainthm}\n\\begin{proof}\nFirst we rearrange the previous proposition and combine constants into one constant $P$ to get the estimate $$\\frac{P\\rho(M)}{(1+\\rho(M))\\vol(M)^{7\/2}}\\leq\\sqrt{\\lambda}.$$ We need an estimate of Calegari's (see the proof of Theorem 3.9 in \\cite{Calegari}, the estimate at the bottom of page 58) which gives that for a genus $g$ surface $S$ with boundary $\\d S = \\gamma^m$,\none has $$\\frac{m|\\gamma|}{12g-6}\\leq 4\\mu + \\frac{2\\pi}{3\\mu} + 2|\\gamma|,$$ where $\\mu$ depends only on the dimension $n$. Since $\\chi_-(S)\\geq2g-1,$ we get $$\\frac{2m|\\gamma|}{\\chi_-(S)}\\leq 24\\left(4\\mu + \\frac{2\\pi}{3\\mu} + 2|\\gamma|\\right).$$\nSince this is true for any surface $S$ bounding a power of $\\gamma$, we obtain $$\\frac{|\\gamma|}{\\scl(\\gamma)}\\leq 24\\left(4\\mu + \\frac{2\\pi}{3\\mu} + 2|\\gamma|\\right).$$\nWe also have the commutator systolic inequality of Sabourau from Theorem 1.4 in \\cite{Sab}, which bounds the shortest nontrivial integrally nullhomologous loop $\\gamma\\in \\Gamma'$ by $$|\\gamma|\\leq c\\vol(M)^{1\/n},$$ for a dimensional constant $c$.\n\nBoth the inequality of Calegari and the systolic inequality involve a dimensional constant; let $\\mu$ be the maximum of these constants in dimension $n$ and write Calegari's inequality as $\\frac{|\\gamma|}{\\scl(\\gamma)} \\leq \\mu(1+|\\gamma|)$. Then we get $$ \\rho(M)\\leq \\frac{|\\gamma|}{\\scl(\\gamma)}\\leq \\mu(1 + |\\gamma|) \\leq \\mu (1+ \\mu\\vol(M)^{1\/n}).$$\nInserting this upper bound into the denominator of the above rearranged estimate above gives $$\\frac{P\\rho(M)}{(1 + \\mu(1+\\mu\\vol(M)^{1\/n})\\vol(M)^{7\/2}}\\leq \\frac{P\\rho(M)}{(1+\\rho(M))\\vol(M)^{7\/2}} \\leq \\sqrt{\\lambda}.$$ We can then increase $P$ to allow us to pull out the volume term and absorb $\\mu$, thereby obtaining the desired estimate.\n\\end{proof}\n\n\\section{Norm estimates}\n\\label{sec:3}\n\nIn this section, we use deeply embedded triangulations and the Whitney maps described in the previous section to compare various geometric and combinatorial norms on forms and cochains. Throughout, let be $M$ a closed hyperbolic manifold of dimension $n>2$ with injectivity radius bounded below by $\\varepsilon>0$ and a fixed deeply embedded triangulation $K$. Let $\\beta$ be the smooth barycentric partition of unity associated to $K$.\n\nWe require various comparisons of the following norms on cochain and chain complexes associated to $M$ and $K$. The relevant norms are:\n\n\n\\begin{enumerate}\n    \\item The combinatorial Gromov norm $||\\cdot||_G$ on any chain or cochain complex given by $||\\sum a_i\\sigma_i||_G = \\sum|a_i|$ and $||\\sum a_i\\delta_{\\sigma_i}||_G = \\sum|a_i|$.\n    \\item The combinatorial Euclidean norm $||\\cdot||_E$, which is the usual $\\ell^2$ norm on chains and cochains given by $ ||\\sum a_i\\sigma_i||_E = \\sqrt{\\sum|a_i|^2}$ and  $ ||\\sum a_i\\delta_{\\sigma_i}||_E = \\sqrt{\\sum|a_i|^2}$.\n     \\item The combinatorial max norm $||\\cdot||_{\\max}$ on any chain or cochain complex given by $||\\sum a_i\\sigma_i||_{\\max} = \\max|a_i|$.\n\n    \\item The Whitney induced $L^2$-norm $||~\\cdot~||_2$ on the cochain complex $C^{\\bullet}(K)$, given by $||f||_2 = \\sqrt{\\int_MW_{\\beta}(f)\\wedge\\star W_{\\beta}(f)}$.\n\n    \\item The Whitney induced $L^{\\infty}$-norm $||\\cdot||_{\\infty}$ on the cochain complex $C^{\\bullet}(K)$, given by taking the essential supremum of the pointwise Riemannian metric operator norms $||f||_{\\infty} = \\esssup\\limits_{p\\in M} ||W_{\\beta}(f)_p||_{\\infty}.$\n\\end{enumerate}\n\nGiven a norm $||\\cdot||$ on the cochain complex $C^{\\bullet}(K)$, let $||\\cdot||^*$ denote the dual norm on the linear dual chain complex $C_{\\bullet}(K)$ induced by the integration pairing: $$||a||^* = \\sup\\limits_{\\substack{||f||=1\\\\ f\\in C^{\\bullet}(K)}} \\int_a W_{\\beta}(f).$$\n\n\\begin{prop} \\label{prop: 3.1} There is a constant $B =  B(\\varepsilon)>0$ such that the norms $||\\cdot||_G$ and $||\\cdot||_2$ on $C^{\\bullet}(K)$ satisfy $$||\\cdot||_G\\leq B\\sqrt{\\vol(M)}||\\cdot||_2,$$ and the norms $||\\cdot||_G$ and $||\\cdot||_2^*$ on $C_{\\bullet}(K)$ satisfy  $$||\\cdot||_G\\leq B \\sqrt{\\vol(M)}||\\cdot||_2^{*}.$$\n\\end{prop}\n\\begin{proof}\n\n Let $f = \\sum\\limits_Fa_F\\delta_F$ be a cochain.\n Then for any $n$-simplex $\\sigma$, $f|_{\\sigma} = \\sum\\limits_{F\\subset \\sigma}a_F\\delta_F$ and $$||f||^2_2 = \\sum\\limits_{\\sigma \\in K^{(n)}} || W_{\\beta}(f|_{\\sigma})|_{\\sigma}||_2^2 =\\sum\\limits_{\\sigma \\in K^{(n)}} ||W_{\\beta}(f)|_{\\sigma}||_2^2 .$$\nApply Proposition \\ref{prop: smooth comparison} to obtain $D = \\mathcal B(\\varepsilon,||\\cdot||_G)$. This gives $||f|_{\\sigma}||_G\\leq D ||f|_{\\sigma}||_{2,\\sigma},$ where $||\\cdot||_{2,\\sigma}$ is the $L^2$-norm on the simplex $\\sigma$ associated to the smooth barycentric coordinate $\\beta$.\nThen, by applying the Euclidean $\\ell^1$-$\\ell^2$-comparison to the cochain complex and using the fact there is a constant $T$ such that the number of $n$-simplices in $K$ is less than $T\\vol(M)$, we find\n\\begin{align*}\n||f||_G  &\\leq \\sum\\limits_{\\sigma \\in K^{(n)}} ||f|_{\\sigma}||_G\\\\\n\t\t&\\leq \\sum\\limits_{\\sigma \\in K^{(n)}}  D||f|_{\\sigma}||_{2,\\sigma}\\\\\n\t   &\\leq D\\sqrt{T\\vol(M)\\sum\\limits_{\\sigma \\in K^{(n)}}||W_{\\beta}(f)|_{\\sigma}||_2^2}\\\\\n\t   &\\leq D\\sqrt{T\\vol(M)}||f||_2.\n\\end{align*}\nFor the second inequality, notice $||\\cdot||_G$ is the usual $\\ell^1$-norm on a finite dimensional vector space, so its dual norm is the max norm $||\\cdot||_{\\max}$.\n\\color{black}\n\nApply Proposition \\ref{prop: smooth comparison} and set $ D' =\\mathcal B(\\varepsilon, ||\\cdot||_{\\max}),$ so that if $\\sigma$ is the simplex in which $||f||_{\\infty}$ is realized, then $$||f||_{\\infty} = ||f|_{\\sigma}||_{\\infty}\\leq  D'||f|_{\\sigma}||_{\\max}\\leq D'||f||_{\\max}.$$\nThen, $$||f||_2 \\leq\\sqrt{\\vol(M)} ||f||_{\\infty} \\leq D' \\sqrt{\\vol(M)}||f||_{\\max}.$$\nDualizing gives $$  ||\\cdot||_G = ||\\cdot||_{\\max}^*\\leq  D' \\sqrt{\\vol(M)}||\\cdot||_2^*,$$\nsince\n\\begin{align*}\n\t||a||_G &= ||a||_{\\max}^*\\\\\n\t\t   &= \\sup\\limits_{||f||_{\\max} \\leq 1}\\int_a W_{\\beta}(f)\\\\\n\t\t   &=\\sup\\limits_{|| D'\\sqrt{\\vol(M)}f||_{\\max} \\leq 1}\\int_{a} D'\\sqrt{\\vol(M)} W_{\\beta} (f)\\\\\n\t\t   &\\leq \\sup\\limits_{||f||_2\\leq1}\\int_{a} D'\\sqrt{\\vol(M)} W_{\\beta} (f)\\\\\n\t\t   &=  D'\\sqrt{\\vol(M)} \\sup\\limits_{||f||_2\\leq 1} \\int_a W_{\\beta} (f)\\\\\n\t\t   &= D'\\sqrt{\\vol(M)}||a||_2^*.\n\\end{align*}\nSet $B = \\max\\{D\\sqrt{T}, D'\\}$ to obtain the claim.\n \\end{proof}\n\nRecall from Section 2 that there is a polyhedral celluation $K^*$ dual to $K$ that can be canonically subdivided into a triangulation $\\tau(K)$. Equipping these dual complexes with the Gromov norm, we have the following two propositions relating these norms by the Poincar\\'e duality and subdivision maps.\n\n\\begin{prop} \\label{prop: 3.2} \nThere is a constant $D = D(\\varepsilon)$ such that for any $\\bullet$-cochain $f\\in C^{\\bullet}(K)$ one has $||f||_2 \\leq D||f||_G$.\n\\end{prop}\n\n\\begin{proof} Let $f = \\sum a_i\\delta_{\\sigma_i}$ be a $\\bullet$-cochain. Then $||\\omega||_G = \\sum|a_i|$ and $||f||_2 \\leq \\sum |a_i| ||\\delta_{\\sigma_i}||_2.$ Then, for any fixed $\\bullet$-simplex $\\sigma$ that is a face of an $n$-simplex from $\\mathcal G_e$, using the $L^2$-change of variables formula and Proposition \\ref{prop: smooth L2 comparison} we can take $D = A\\mathcal L^{n\/2}||W(\\delta_{\\sigma})||_2$, so that $||\\delta_{\\sigma_i}||_2 \\leq D.$\n\nThe comparison \\[||f||_2\\leq\\sum |a_i| ||\\delta_{\\sigma_i}||_2 \\leq D\\sum |a_i| = D||f||_G \\] then follows.\n\\end{proof}\n\n\n\\begin{prop} \\label{prop: 3.3} \nThe Poincar\\'e duality map $\\Phi:C^{\\bullet}(K)\\to C_{n-\\bullet}(K^*)$ preserves the Gromov norm $$||f||_G= ||\\Phi(f)||_G.$$\n\\end{prop}\n\n\\begin{proof}\n  Let $f = \\sum a_i\\delta_{\\sigma_i}$, then $\\Phi(f) = \\sum a_i (\\sigma_i)^*$.\n\\end{proof}\n\n\\begin{prop} \\label{prop: 3.4} \nLet $N$ be the constant from Proposition \\ref{prop: star bound}, which bounds the number of simplices in the star of a simplex in a deeply embedded triangulation. Then the subdivision map $\\tau: C_{2}(K^*)\\to C_{2}(T)$ satisfies $$ || \\tau(c)||_G \\leq N ||c||_G.$$\n\\end{prop}\n\n\\begin{proof}\n  The number of sides of a $2$-cell in $K^*$ dual to a $(n-2)$-simplex $\\sigma$ in $K$ corresponds to the number of $n$-simplices in $K$ that contain $\\sigma$. The number of such simplices is bounded by $N$.\n\\end{proof}\n\nThe following estimates are essential in comparing the first eigenvalue of the Whitney Laplacian to the genuine first eigenvalue. For this, we need to work with various Sobolev spaces to control the orthogonal projection of a Whitney form onto its coexact part. This discussion is the reason we use the smoothed Whitney forms in place of the standard ones.\n\nWe will require the following version of the Gaffney inequality, which follows from Lemma 2.4.10 in \\cite{schwarz}. To simplify the following discussion, for a smooth manifold $Y$ possibly with boundary, we introduce an alternative Sobolev norm on $H^{k+1}_{\\nabla}\\Omega^{\\bullet}(Y)$: $$||\\omega||_{A^{k}(Y)}:= ||\\omega||_{H^k_{\\nabla}(Y)} + ||d\\omega||_{H^k_{\\nabla}(Y)} + ||d^*\\omega||_{H^k_{\\nabla}(Y)}.$$\n\nSince $d$ and $d^*$ are bounded operators $H^{k+1}\\Omega^{\\bullet}_{\\nabla}(Y)\\to H^{k}\\Omega^{\\bullet\\pm 1}_{\\nabla}(Y)$, we immediately have that there is a constant $C$ such that $||\\omega||_{A^{k}(Y)}\\leq  C ||\\omega||_{H^{k+1}_{\\nabla}(Y)}.$\n\nRecall that the marking $\\mathring H$ denotes the subspace of given Sobolev space that is the closure of smooth functions supported in the interior.\n\n\\begin{lem}  \\label{lem: 3.5} (Gaffney inequality) Let $Y$ be a smooth manifold with boundary. Let $\\omega\\in \\mathring{H}^k_{\\nabla} \\Omega^{1}(Y)$. Then there is a constant $C = C(Y)>0$ such that $||\\omega||_{\\mathring{H}^k_{\\nabla}(Y)}\\leq C ||\\omega||_{A^{k-1}(Y)}$.\n\\end{lem}\n\n\\begin{lem} \\label{lem: 3.6} \nLet $B_0 = B_r(p)$ and $B_1= B_{r+\\delta}(p)$ be a pair of concentric balls in $\\H^n$ and let $\\phi$ be a smooth bump function that is identically 1 on $B_0$ and vanishes in a neighborhood of $\\d B_1$. There is a constant $C = C(\\phi,k)$ that depends only on the norm of the covariant derivatives of $\\phi$ up to order $k+1$ such that if $\\omega \\in \\Omega^{1}(\\H^n)$, then $$||\\phi\\omega||_{A^k(B_1)}\\leq C ||\\omega||_{A^k(B_1)}.$$\n\\end{lem}\n\n\\begin{proof}\nNotice that $d\\phi\\omega = d\\phi\\wedge\\omega + \\phi d \\omega$ and $d^*\\phi\\omega = \\phi d^*\\omega + g(\\nabla \\phi, X_{\\omega}),$ where $X_{\\omega}$ is the vector field dual to $\\omega$.\nAs a result, the triangle inequality yields \\begin{align*}\n||\\phi\\omega||_{A^k(B_1)} &\\leq ||\\phi d\\omega||_{H^k_{\\nabla}(B_1)} + ||\\phi d^*\\omega||_{H^k_{\\nabla}(B_1)}+ ||\\phi\\omega||_{H^k_{\\nabla}(B_1)} \\\\ &+ ||d\\phi\\wedge\\omega||_{H^{k}_{\\nabla}(B_1)} + ||g(X_{\\omega},\\nabla\\phi)||_{H^k_{\\nabla}(B_1)}.\n\\end{align*}\n\nThe estimate $||\\alpha\\wedge\\beta||_2\\leq ||\\alpha||_{\\infty}||\\beta||_2$ implies $$||d\\phi\\wedge\\omega||_{H^k_{\\nabla}(B_1)} \\leq C||\\omega||_{H^k_{\\nabla}(B_1)},$$ where the constant $C$ is given by the sum of the $||\\cdot||_{\\infty}$-norms of the covariant derivatives of the bump function $\\phi$.\nThe same argument gives for any form $\\xi$ that $||\\phi\\wedge\\xi||_{H^k_{\\nabla}(B_1)}\\leq C ||\\xi||_{H^k_{\\nabla}(B_1)}$. Applying this estimate to $\\phi d\\omega, \\phi d^*\\omega$, and $\\phi \\omega$ handles all terms in the comparison save for $||g(X_{\\omega},\\nabla\\phi)||_{H^{k}_{\\nabla}(B_1)}$.\nFor this term, notice that $\\nabla g(X_{\\omega},\\nabla\\phi) = g(\\nabla X_{\\omega},\\nabla \\phi) + g( X_{\\omega}, \\nabla^2\\phi).$ We therefore have the pointwise estimate\n\n\\begin{align*}\n|\\nabla g(X_{\\omega},\\nabla\\phi) | &\\leq |g(\\nabla X_{\\omega},\\nabla \\phi)| + |g( X_{\\omega}, \\nabla^2\\phi)| \\text{ by the triangle inequality,} \\\\\n&\\leq |\\nabla \\omega|^2|\\nabla\\phi|^2 + |\\omega|^2|\\nabla^2\\phi|^2,\n\\end{align*}\n\nby applying Cauchy-Schwarz and the musical isomorphism.\nIntegrating then gives the corresponding inequality for the Sobolev norm $H^1_{\\nabla}$. Repeating this calculation for higher order covariant derivatives completes the proof.\n\\end{proof}\n\nThe two previous lemmas combine to give the following statement.\n\\begin{prop} \\label{prop: 3.7} \nLet $B_0 = B_r(p)$ and $B_1= B_{r+\\delta}(p)$ be a pair of concentric balls in $\\H^n$ Then there is a constant $C = C(r,\\delta)$ such that for any $\\omega \\in H^k_{\\nabla}(B_1)$ one has $$||\\omega||_{H^k_{\\nabla}(B_0)}\\leq C||\\omega||_{A^{k-1}(B_1)}.$$\n\\end{prop}\n\\begin{proof}\nLet $\\sqrt{C}$ be the maximum of the constants from Gaffney's inequality and Lemma 3.6 with bump function $\\phi$. Then for a form $\\omega \\in H^k_{\\nabla}(B_1)$, one has $$||\\omega||_{H^k_{\\nabla}(B_0)}\\leq ||\\phi\\omega||_{H^k_{\\nabla}(B_1)},$$ since $\\phi\\omega|_{B_0} = \\omega.$\nSince $\\phi\\omega$ vanishes on $\\d B_1$, Gaffney's inequality and Lemma 3.6 give $$||\\phi\\omega||_{H^k_{\\nabla}(B_1)} = ||\\phi\\omega||_{\\mathring{H}^k_{\\nabla}(B_1)}\\leq \\sqrt{C} ||\\phi\\omega||_{A^{k-1}(B_1)}\\leq C||\\omega||_{A^{k-1}(B_1)}.$$ Combining these two estimates gives the proposition.\n\\end{proof}\n\n\\begin{prop}\\label{prop: 3.8} \nLet $M$ be a hyperbolic $n$-manifold with injectivity radius greater than $\\varepsilon>0$. There is a constant $H(\\varepsilon) = H >0$ depending only on $\\varepsilon$ such that if the $L^2$-Hodge decomposition of a smooth 1-form $\\omega\\in \\Omega^1$ has coexact part $\\alpha$, then for any point $p\\in M$ there is a ball $B\\subset M$ centered at $p$ of radius determined by $\\varepsilon$ such that for $k\\geq n$ one has $$|\n\\nabla \\alpha(p)|\\leq H\\left(||\\omega||_{H^k_{\\nabla}(B)} + ||\\alpha||_{2,B}\\right)$$ and consequently $$|\n\\nabla \\alpha(p)|\\leq H||\\omega||_{H^k_{\\nabla}(M)}.$$\n\\end{prop}\n\n\\begin{proof}\n\nThe $L^2$-Hodge decomposition of $M$ determines an orthogonal decomposition of $\\omega= \\alpha + \\eta + h$, where $\\alpha = d^*A$, $\\eta = db$, and $h$ harmonic. Cantor's estimate (Theorem \\ref{thm: Cantor Sobolev} implies at any point $p$ of $M$, $$|\\nabla \\alpha (p)| \\leq C(r)||\\alpha||_{H^k_{\\nabla}(B_r(p))}$$ so long as $r<\\inj(M)$ and $k > n\/2 + 1$. Since we assume dimension $n>2$, any $k\\geq n$ suffices.\nTake a concentric family of balls $B_i = B_{r + i\\delta}(p)$ where $r = 6\\varepsilon$, and $k\\delta + r < 10\\varepsilon < \\inj(M)$; note that $B_0$ contains the 0-star of $\\sigma$. Let $\\phi_i$ be a bump function that is identically one on $B_i$ and vanishes on $\\d B_{i+1}$.\n\nLetting $C_i$ be the maximum of the constant from Cantor's estimate on the ball $B_i$ and the constant from Proposition \\ref{prop: 3.7} for the balls $B_i\\subset B_{i+1}$, we have\n$$|\\nabla \\alpha (p)| \\leq C_i||\\alpha||_{H^k_{\\nabla}(B_i)}\n   \\leq  C_i^2||\\alpha||_{A^{k-1}(B_{i+1})}.$$\nSince $\\alpha$ is coexact, $$||\\alpha||_{A^{k-1}(B_{i+1})} = ||\\alpha||_{H^{k-1}_{\\nabla}(B_{i+1})} + ||d\\alpha||_{H^{k-1}_{\\nabla}(B_{i+1})}.$$ Notice that $d\\alpha = d\\omega$, and that $d:H^k_{\\nabla}(B_i)\\to H^{k-1}_{\\nabla}(B_i)$ is a bounded operator, say with operator norm $C$. Thus, we have $$||d\\alpha||_{H^{k-1}_{\\nabla}(B_{i+1})} \\leq C||\\omega||_{H^k_{\\nabla}}.$$ As a result, we get the estimate $$||\\alpha||_{A^{k-1}(B_{i+1})} \\leq ||\\alpha||_{H^{k-1}_{\\nabla}(B_{i+1})} + C||\\omega||_{H^{k}_{\\nabla}(B_{i+1})}$$\nCombining this with the estimate of $\\nabla \\alpha$ above gives $$|\\nabla \\alpha(p)|\\leq C_i\\left(||\\omega||_{H^{k}_{\\nabla}(B_{i+1})}+||\\alpha||_{H^{k-1}_{\\nabla}(B_{i+1})}\\right).$$\nWe can repeat argument this using the family of balls $B_i$ to reduce the order of the Sobolev norm of $\\alpha$ on the right-hand side until we obtain $$|\\nabla \\alpha(p)|\\leq H\\left(||\\omega||_{H^k_{\\nabla}(B_n)} + ||\\alpha||_{2,B_n}\\right),$$ where $H$ is obtained by combining all the constants appearing in the iterated calculation. Orthogonality of the Hodge decomposition implies $||\\alpha||_2\\leq ||\\omega||_2$, and we clearly have $||\\omega||_{2, B_n}\\leq ||\\omega||_{H^k_{\\nabla}(M)},$ so we are done after increasing $H$ by 1.\n\\end{proof}\n\nWhen the form in the previous proposition is a Whitney form, we can make the following refinement.\n\n\\begin{prop}\\label{prop: 3.9} \nLet $M$ be a hyperbolic $n$-dimensional manifold with a deeply embedded triangulation $K$ and let $f\\in C^1(K)$. There is a constant $H(\\varepsilon) = H >0$ depending only on $\\varepsilon$ such that if the $L^2$-Hodge decomposition of the generalized Whitney form $W_{\\beta}(f)$ has coexact part $\\alpha$,  then  $$\n||\\nabla \\alpha||_{\\infty}\\leq H||W_{\\beta}(f)||_2.$$\n\\end{prop}\n\\begin{proof}\nLet $\\omega = W_{\\beta}(f)$ be a smooth Whitney form. We will apply Proposition \\ref{prop: 3.8} at a point $p$ at which $||\\nabla\\alpha||_{\\infty}$ is realized. Proposition \\ref{prop: 3.8} gives a ball $B$ about $p$ of radius depending only on $\\varepsilon$ such that $$|\n\\nabla \\alpha(p)|\\leq H\\left(||\\omega||_{H^k_{\\nabla}(B)} + ||\\alpha||_{2,B}\\right),$$ for a constant $H$ depending only on $\\varepsilon$.\nThe ball $B$ intersects some uniformly bounded collection of $n$-simplices $\\sigma'$ from $K$ where the constant depends only on $\\varepsilon$; let $T = T(\\varepsilon)$ be this bound. We can therefore estimate the norm of the Whitney form term by $$||\\omega||_{H^k_{\\nabla}(B)}\\leq \\sum\\limits_{\\sigma'\\cap B \\neq \\emptyset} ||\\omega|_{\\sigma'}||_{H^k_{\\nabla}(\\sigma')}.$$\nApplying Lemma 2.13 to each summand in the previous estimate then gives\n$$ ||\\omega||_{H^k_{\\nabla}(B)} \\leq \\sum\\limits_{\\sigma'\\cap B_i \\neq \\emptyset} ||\\omega|_{\\sigma'}||_{H^k_{\\nabla}(\\sigma')}\\leq R\\sum\\limits_{\\sigma'\\cap B \\neq \\emptyset} ||\\omega|_{\\sigma'}||_2\\leq R\\sqrt{T}||\\omega||_2.$$\nCombining the above then gives that $$|\\nabla \\alpha(p)|\\leq HR\\sqrt{T}||\\omega||_2 + H||\\alpha||_{2,B}.$$ Clearly $$H||\\alpha||_{2,B}\\leq H||\\alpha||_2 \\leq H||\\omega||_2,$$ so that after increasing $H$ to absorb the $R\\sqrt{T}$ term we are done.\n\\end{proof}\n\n\\begin{prop}\\label{prop: 3.10} \nLet $M$ be a hyperbolic $n$-manifold. Let $\\omega\\in \\Omega^1(M).$ Assume there exists a constant $H$ such that $|\\nabla \\omega| \\leq H$. Then there is a constant $C(H,\\varepsilon)$ such that $||\\omega||_{\\infty}\\leq C(H,\\varepsilon) ||\\omega||_2.$\n\\end{prop}\n\n\\begin{proof} Assume $||\\omega||_{\\infty} = 1$ and is realized at the point $p$. By Kato's inequality and the hypothesis, away from the zeros of $\\omega$ one has $|\\nabla|\\omega||\\leq |\\nabla \\omega| \\leq H$. Fix a normal coordinate frame $x_0,\\dots,x_{n-1}$ at $p$ of radius $2\\varepsilon$. Define the function $\\phi$ on this normal coordinate neighborhood by $\\phi(x) = 1- Hd(x,p)$ for $ d(x,p) < 1\/H$ and extend by zero.\nThen $||\\phi||_{\\infty} = ||\\omega||_{\\infty}$ and $||\\phi||_2 \\leq ||\\omega||_2$.\nThe claim then holds for $C(H,\\varepsilon) = 1\/||\\phi||_2$ by scaling the unit norm case.\n\\end{proof}\n\\begin{prop}\\label{prop: 3.11} \nThere is a constant $C = C(\\varepsilon)$ such that if $f \\in C^1(K)$ and $\\omega = W_{\\beta}(f)  = \\alpha + \\eta$ where $\\alpha$ is $L^2$-coexact and $\\eta$ is closed, then $$||\\alpha||_{\\infty} \\leq C ||\\alpha||_2.$$\n\\end{prop}\n\n\\begin{proof}\nAssume that $||f||_2 = 1$. By Proposition \\ref{prop: 3.9}, $||\\nabla \\alpha ||_{\\infty} \\leq H||f||_2 = H$.  Proposition \\ref{prop: 3.10} gives a constant $C = C(H(\\varepsilon))$ (so this really just depends on $\\varepsilon$) such that $||\\alpha||_{\\infty}\\leq C||\\alpha||_2$. If $f$ does not have unit $L^2$-norm, then either $f=0$, in which case the result is trivial, or else $f = \\lambda f'$ for some unit $L^2$-norm cochain $f'$ and positive number $\\lambda$.\nThe coexact part of $W_{\\beta}(f')$ is $\\alpha ' = \\alpha\/\\lambda$. Hence, $||\\alpha'||_{\\infty}\\leq C||\\alpha'||_2$, and the result follows. \\end{proof}\n\n\\section*{Acknowledgments}  Countless thanks are due to my advisor Nathan Dunfield for suggesting the questions considered in this paper and for invaluable support. Thanks to Michael Lipnowski and Joel Hass for answering some questions. Thanks to the anonymous referee for helpful comments that improved this paper. This work was partially supported by US NSF grant DMS-1811156.\n\n\\input{triangulations}\n\\input{norms}\n\\input{upperbound}\n\\input{lowerbound}\n\\input{example}\n\n\\bibliographystyle{alpha}\n\n\n\\section{Triangulations and Whitney forms}\n\\label{sec:2}\n\nThe purpose of this section is to outline the basic properties of the triangulations we use in this paper and how the Whitney map relates these triangulations to the de Rham complex.\n\n\\subsection{Deeply embedded triangulations}\n\nIn this section we study certain triangulations, called deeply embedded triangulations, of hyperbolic manifolds with injectivity radius bounded below that enjoy useful combinatorial and geometric properties that will facilitate the estimates in sections 3 through 5. While we focus on the hyperbolic setting, we give an account motivated by potential generalizations to the variable negative curvature setting.\n\nThe triangulations we use are generally obtained via Delaunay complexes associated to collections of points. To obtain a Delaunay complex in a Riemannian manifold $M$, take a finite collection of points $P\\subset M$ and consider the Voronoi celluation consisting of cells $$V_p = \\{x\\in M~:~ d(x,p) \\leq d(x,q)~\\text{for all } p \\neq q \\in P\\}$$ for $p\\in P.$ Dual to the Voronoi celluation is the Delaunay complex. The cells of the Delauney complex are the convex hulls of tuples of points whose corresponding Voronoi cells have nonempty intersections. In \\cite{Bois}, it is shown that if the collection of points $P$ satisfies certain density and separation conditions, then there is a quantifiably small perturbation of the point set $P$ whose Delaunay complex is a triangulation. The simplices of this triangulation are geodesic. The precise conditions are as follows.\n\nLet $M$ be a closed Riemannian manifold with distance function $d$. Given a pair $1\\geq\\mu>0,~\\varepsilon>0$, a $(\\mu,\\varepsilon)$-net is a collection $P$ of points in $M$ for which the following hold:\n\n\\begin{enumerate}\n    \\item ($P$ is $\\varepsilon$-dense) For all $x\\in M$, there is a $p\\in P$ such that $d(x,p)<\\varepsilon$.\n\n    \\item ($P$ is $\\mu$-separated) All distinct $p,q\\in P$ satisfy $d(p,q) \\geq \\mu\\varepsilon$.\n\n\\end{enumerate}\n\nTheorem 3 of \\cite{Bois} says that if $\\mu$ and $\\varepsilon$ satisfy several inequalities relating to the injectivity radius and sectional curvatures, a $(\\mu,\\varepsilon)$-net can be perturbed to $(\\mu',\\varepsilon')$-net such that the resulting Delaunay complex is indeed a triangulation. This theorem, specialized to closed hyperbolic $n$-manifolds, becomes:\n\\begin{thm}\n\n\\cite{Bois} Let $M$ be a closed hyperbolic $n$-manifold and $P$ a $(\\mu,\\varepsilon)$-net such that \\[\\varepsilon \\leq \\min\\left\\{\\frac{\\inj(M)}{4},~\\Psi(\\mu)\\right\\},\\] where $\\inj(M)$ is the injectivity radius of $M$ and $\\Psi$ is a function of the net parameter $\\mu$. The function $\\Psi$ is described in \\cite{Bois} and is independent of the manifold $M$. Then there is a point set $P'$ that is a $(\\mu',\\varepsilon')$-net with resulting Delaunay complex a triangulation. Moreover, $\\mu'$ and $\\varepsilon'$ satisfy the following (see equation (2) in \\cite{Bois}):\n: \\begin{align*}\n     \\varepsilon &\\leq \\varepsilon' \\leq \\frac{5}{4}\\varepsilon, \\\\\n    \\frac{2}{5}\\mu &\\leq \\mu' \\leq \\mu.\n\\end{align*}\n\\end{thm}\n\nThe separation and density conditions of a $(\\mu',\\varepsilon')$-net ensure that the resulting Delaunay triangulation has edge lengths lying in the closed interval $[\\frac{2}{5}\\mu \\varepsilon ,2\\varepsilon]$. We now specialize to the case $\\mu=1$. Note that when $n=3$, $\\Psi(1) = 2\\times 3^{121.5}\\times 5^{-81}$, which is roughly 45.15.\n\nSet $\\epsilon_0 = \\min\\{\\varepsilon\/10,\\Psi(1)\\}$ and define $\\mathcal{G}_{\\varepsilon}$ to be the space of hyperbolic $n$-simplices with edge lengths in the interval $[\\frac{2\\epsilon_0}{5} ,2\\epsilon_0]$. Because the space of all hyperbolic $n$-simplices is parametrized by edge lengths, the space $\\mathcal{G}_{\\varepsilon}$ is compact. Proposition \\ref{prop: existence} below establishes that every closed hyperbolic manifold $M$ with injectivity radius bounded below by $\\varepsilon$ admits a triangulation whose simplices are isometric to those in $\\mathcal{G}_\\varepsilon$.\n\nIn a triangulation $K$, the combinatorial 1-neighborhood of a simplex is the union of the stars of the vertices of that simplex. A triangulation $K$ of a hyperbolic manifold $M$ is an \\textbf{$\\varepsilon$-deeply embedded triangulation} if:\n\n\\begin{enumerate}\n    \\item Every simplex is geodesic and contained in $\\mathcal G_\\varepsilon$,\n    \\item The combinatorial 1-neighborhood of every simplex lifts isometrically to $\\H^n$.\n\\end{enumerate}\n\nThroughout, when referring to deeply embedded triangulations, we generally suppress reference to some fixed $\\varepsilon$.\n\n\\begin{prop} \\label{prop: existence}\nLet $M$ be a closed hyperbolic $n$-manifold and let $0<\\varepsilon < \\inj(M)$. Then there is a deeply embedded triangulation $K$ of $M$. That is, there is a geodesic triangulation $K$ of $M$ whose simplices come from $\\mathcal G_{\\varepsilon}$ such that the combinatorial 1-neighborhood of every simplex isometrically embeds in $\\H^n$.\n\\end{prop}\n\n\\begin{proof} Set $\\epsilon_0 = \\min\\{\\varepsilon\/10,\\Psi(1)\\}$.\n Take a maximal collection of points $P\\subset M$ such that the balls $B_{\\epsilon_0\/2}(p)$ for $p\\in P$ are all disjoint. By maximality, the $B_{\\epsilon_0}(p)$ balls then cover $M$.\n Since the $B_{\\epsilon_0\/2}(p)$ balls are all disjoint, we have that for all $p,q\\in P$, $d(p,q) \\geq \\epsilon_0 = \\mu {\\epsilon_0} $, so $P$ is $\\mu$-separated. Since the $B_{\\epsilon_0}(p)$ balls cover, every point $x\\in M$ is ${\\epsilon_0} $-close to some point $p\\in P$, so $P$ is ${\\epsilon} $-dense.\n The collection $P$ therefore is a $(\\mu, {\\epsilon_0})$-net, with $\\mu = 1$ and ${\\epsilon_0} $ satisfying the hypotheses of Theorem 2.1. Consequently, there is a perturbation of $P$ that is a $(\\mu',{\\epsilon_0}')$-net whose Delaunay complex is a triangulation. Since, as remarked above, the edge lengths of simplices in this Delaunay triangulation lies in the interval $[\\frac{2}{5} {\\epsilon_0} ,2 {\\epsilon_0}]$, the simplices come from $\\mathcal G_{\\varepsilon} $.\n The edge length bound along with the fact ${\\epsilon_0} < \\inj(M)\/10$ ensures the diameter of any vertex star (which will be less than 3 times the length of the longest edge of a simplex) will be less than $2 {\\epsilon_0}$. Thus, the star of every vertex embeds isometrically in $\\H^n$ via the local inverse of the exponential map. \\end{proof}\n\nWe will also use cell complexes that are dual to deeply embedded triangulations. Every simplicial triangulation $K$ of a closed Riemannian manifold admits a dual celluation $K^*$ comprised of cells $\\sigma^*$ dual to the simplices $\\sigma$ of the triangulation $K$ in the following sense (for a reference, see \\cite{bredon} chapter VI.6): Take the first barycentric subdivision $\\tau(K)$ of $K$, then the $n$-cells of the dual celluation $K^*$ are the closed stars of the vertices of the original triangulation $K$ in the barycentric subdivision. This celluation is naturally triangulated by the the barycentric subdivision triangulation $\\tau(K)$. Like $K$, the dual celluation can be uniformly controlled.\n\nThe controlled geometry of deeply embedded triangulations and their dual celluations primarily manifests in the compactness of $\\mathcal G_\\varepsilon$ and the following local bound.\n\n\\begin{prop} \\label{prop: star bound}Let $M$ be a closed hyperbolic $n$-manifold of injectivity radius $\\inj(M)>\\varepsilon$ with a deeply embedded triangulation $K$. Then there is a positive constant $N = N(\\varepsilon)$ such that the number of $k$-simplices in the star of a $j$-simplex is less than $N$.\n\\end{prop}\n\n\\begin{proof}\n\nThe edge length bounds provide a lower bound on the angle between two edges meeting at a vertex that span a face via the hyperbolic law of cosines. This implies that the number of $n$-simplices meeting at a vertex $v$ is bounded uniformly, since for any ball around the vertex, there is a uniform lower bound on the volume of the intersection of an $n$-simplex containing the vertex $v$ and the ball. It follows that there is an $N$ such that the number of $k$-simplices in the star of a vertex is less than $N$ for $k = 0,\\dots, n$.\\end{proof}\n\n\nProposition \\ref{prop: star bound} and the compactness of the space $\\mathcal G_\\varepsilon$ of simplices together imply that the geometry and combinatorics of the 1-neighborhood of a simplex in a deeply embedded triangulation is uniformly controlled.\nIn particular, let $K$ be a deeply embedded triangulation of $M$. By Proposition \\ref{prop: star bound}, any vertex in $K$ is contained in at most $N$ simplices. Therefore, there are finitely many possible finite simplicial complexes that appear as the combinatorial 1-neighborhood of a simplex in a deeply embedded triangulation. Let $\\mathcal C$ be the finite set of such possible complexes.\n\nFor any complex $a \\in \\mathcal C$, say with $|a|$ many $n$-simplices, a hyperbolic structure on $a$ is given by identifying each $n$-simplex in $a$ with a model simplex in $\\mathcal G_{\\varepsilon}$ so that the face gluing maps are isometries. The possible geometric structures on $a$ are parametrized by a subspace $\\mathcal S_\\varepsilon(a)$ of $\\mathcal G_{\\varepsilon}^{|a|}$. Because the gluing conditions are closed, and because $\\mathcal G_\\varepsilon$ is compact, the space $\\mathcal S_\\varepsilon(a)$ is compact.\nBy taking the disjoint union over the finite list of possible complexes $a\\in\\mathcal C$, there is a compact space $\\mathcal S_{\\varepsilon}$ that parametrizes the geometry of the combinatorial 1-neighborhood of a simplex in a deeply embedded triangulation.\n\nWe now turn to relating closed geodesics in $M$ to cellular paths in $K$.\n\n\\begin{prop} \\label{prop: loop multiplicity}\nLet $M$ be a closed hyperbolic $n$-manifold with injectivity radius $\\inj(M)>\\varepsilon$ with a deeply embedded triangulation $K$. Let $\\gamma$ be a closed geodesic curve in $M$. Then there is a constant $J = J(\\varepsilon)$ such that the number $v$ of cells in the dual cell complex $K^*$ that $\\gamma$ intersects (counted with multiplicity) satisfies \\[v\\leq J|\\gamma|.\\] \\end{prop}\n\n\\begin{proof}\n\nSuppose $\\gamma$ moves from an $n$-cell $\\sigma$ to an $n$-cell $\\sigma'$, intersecting the ($n-1)$-skeleton of $K^*$ at a point $p\\in \\sigma\\cap \\sigma'$. Consider the closed radius $\\varepsilon$-ball at the point $p$, $V = \\bar B_{\\varepsilon}(p).$ Let $x$ be the point at which $\\gamma$ enters $V$ and let $y$ be the point at which it exits $V$.\nThen the geodesic subarc of $\\gamma$ running from $x$ to $y$ has length $2\\varepsilon$. Since the triangulation $K$ has simplices from $\\mathcal G_{\\varepsilon}$, the restrained combinatorics of the dual celluation ensures that the ball $V$ intersects a universally bounded number of dual cells. Let $R(\\varepsilon)$ denote this bound.\n\nConsider the sequence $x_n,y_n$ of points such that $x_1 = x$ and $y_1 = y$ from above for the first simplex crossing, and $x_n$ is obtained by taking the simplex crossing that happens after $y_n$. Then each pair $x_n,y_n$ corresponds to a geodesic sub arc of $\\gamma$ that intersects at most $R(\\varepsilon)$ simplices. Thus, $\\nu \\leq (\\frac{|\\gamma|}{2\\varepsilon} +1)R(\\varepsilon)$.\nIt therefore follows that $\\frac{v}{R(\\varepsilon)} 2\\varepsilon \\leq |\\gamma| + 2\\varepsilon.$ Since $\\varepsilon \\leq |\\gamma|$, we have $|\\gamma|+2\\varepsilon\\leq 3|\\gamma|$, and the stated linear bound follows with $J = \\frac{3R(\\varepsilon)}{2\\varepsilon}$. \\end{proof}\n\nThe next result compares the lengths of closed geodesics in $M$ to approximating paths in the 1-skeleton of dual celluation $K^*$. To measure the complexity of paths in $K^*$, let $||\\cdot||_G$ be the $\\ell^1$-norm on chains and $\\len(\\cdot)$ the word length of the cellular path. For a cellular path $c$, let $||c||_G$ be the $\\ell^1$-norm of the corresponding chain.\n\n\\begin{prop} \\label{prop: length comparison} There is a constant $L = L(\\varepsilon)>0$ such that for any closed geodesic curve $\\gamma$ in $M$, there is a cellular path $c$ in $K^*$ homotopic to $\\gamma$ such that $||c||_G\\leq \\len(c) \\leq L|\\gamma|$.\n\\end{prop}\n\n\\begin{proof}\n\nFix a base point and orientation for $\\gamma$ such that the base point lies on a face of a top dimensional cell. The curve $\\gamma$ can be replaced by a homotopic curve whose length is bounded by a constant times the geodesic length of $\\gamma$ and which intersects the boundary of every simplex at vertices. This follows from Proposition \\ref{prop: loop multiplicity} which gives that there is a bound on the number of simplices $\\gamma$ intersects (counting these intersections with multiplicity) that depends linearly on the length of $\\gamma$ and the fact the simplices of the triangulation have bounded diameter. Using the orientation and basepoint, we obtain a sequence of vertices with line segments between them that lie entirely in a cell. We can further modify $\\gamma$ by replacing these curve segments with curves that lie in the 1-skeleton by traversing the 1-simplex joining the two boundary vertices. Since the edges in the celluation $K^*$ have bounded length, this again adds bounded length to the curve. Let $c$ denote the cellular path we have constructed. By the previous considerations, there is a constant $L$ depending only on $\\varepsilon$ giving the comparison $\\len(c)\\leq L|\\gamma|$. The inequality $||c||_G\\leq \\len(c)$ is trivial.\n\n\\end{proof}\n\n\nThe geometry of geodesic simplices in hyperbolic manifolds can also be understood using barycentric coordinates via Thurston's straightening map (see \\cite{thurstonbook} page 124). Identify $\\H^n$ with the upper sheet of the hyperboloid in Minkowski space $\\mathbb R^{n,1}$ with quadratic form $Q = x_0^2 + x_1^2 + \\cdots + x_{n-1}^2 - x_n^2$ and consider a singular simplex $\\sigma:\\Delta\\to \\H^n$.\nLet $b_0, \\dots, b_n:\\Delta\\to [0,1]$ be the barycentric coordinates on the standard Euclidean simplex $\\Delta$ with vertices $e_0,\\dots, e_n$. Then for $v\\in \\Delta$, write $v = \\sum b_i(v)e_i$. The straightening of $\\sigma$ is the singular simplex in $\\H^n$ given by centrally projecting the affine simplex $\\sum b_i(v)\\sigma(v_i)$ from the origin to the upper sheet of the hyperboloid. This process endows each geodesic simplex with a natural barycentric coordinate.\n\nIf $\\pi:\\H^n\\to M$ is the projection map and $\\sigma$ is a singular simplex in $M$, let $\\text{st}(\\sigma):\\Delta\\to M$ be the composition of the straightening of $\\sigma$ applied to some lift of $\\sigma$ and the projection map.\nThis is well-defined and independent of the lift because the isometry group of $\\H^n$ acts linearly on $\\mathbb R^{n,1}$ preserving the quadratic form $Q$.\n\nFor a geodesic simplex $\\sigma$ in $\\H^n$, let $V_{\\sigma}:\\sigma \\to \\Delta$ be the map from $\\sigma$ to the standard simplex given by the barycentric coordinates induced by straightening. Geodesic simplices $\\sigma,\\sigma'$ can be compared using the composition $V_{\\sigma'}^{-1}\\circ V_{\\sigma}$.\nUsing the straightening construction and the barycentric coordinates, one sees that the maps $V_{\\sigma}$ depend continuously on $\\sigma$ in the sense that if $\\sigma$ is a straight simplex in $\\H^n$ and $\\sigma'$ is obtained by perturbing the vertices of $\\sigma$, then the composition map $V_{\\sigma'}^{-1}\\circ V_{\\sigma}$ is almost an isometry, where the failure to be an isometry is controlled by the size of the vertex perturbation.\n\n\n\\begin{prop}\\label{prop: simplices biLipschitz} Let $\\sigma$ and $\\sigma'$ be geodesic simplices from $\\mathcal G_\\varepsilon$ embedded in $\\H^n$. Then the map $V_{\\sigma'}^{-1}\\circ V_{\\sigma}$ is $\\kappa$-biLipschitz for some $\\kappa = \\kappa(\\varepsilon)>0$ that does not depend on $\\sigma$ and $\\sigma'$.\n\\end{prop}\n\\begin{proof}\nThe biLipschitz constant for the comparison map between any given simplex and the Euclidean simplex depends continuously on the simplex. The result then follows from the compactness of $\\mathcal G_\\varepsilon$.\n\\end{proof}\n\n\nWe also have uniform control over the geometry of the combinatorial 1-neighborhood of a simplex.\n\n\\begin{prop}\\label{prop: stars biLipschitz} There is a constant $\\mathcal L = \\mathcal L(\\varepsilon)$ such that  if $s$ and $s'$ are two complexes of the same combinatorial type in $\\mathcal S_\\varepsilon$, then $s$ and $s'$ are $\\mathcal L$-biLipschitz equivalent.\n\\end{prop}\n\\begin{proof}\n    Define maps from the combinatorial 1-neighborhood $s$ of a simplex $\\sigma$ to an abstract Euclidean model by gluing the maps $V_{\\sigma'}$ together according to the combinatorics of $s$. Since the gluing maps are isometries, this is well defined. Since the map restricted to each simplex $\\sigma'$ is uniformly biLipschitz equivalent to the model simplex, and since there are a uniformly bounded number of simplices in $s$, it follows that $s$ is uniformly biLipschitz equivalent to the Euclidean model.\n\\end{proof}\n\nLastly, we note that the lower bound on volumes of simplices in deeply embedded triangulations implies the following.\n\n\\begin{prop} \\label{prop: volume}There is a constant $T=T(\\varepsilon)$ such that if $M$ is a closed hyperbolic $n$-manifold with $\\inj(M)>\\varepsilon$ and $K$ is a deeply embedded triangulation of $M$, then the number $V_K$ of simplices in $K$ satisfies $$V_K\\leq T\\vol(M).$$\n\\end{prop}\n\n\n\n\\subsection{Whitney forms}\n\nThe combinatorial geometry of the triangulation $K$ is related to the Riemannian geometry of $M$ by way of the Whitney form map $W:C^{\\bullet}(K)\\to L^2\\Omega^{\\bullet}(M)$ relating the cochain complex $C^{\\bullet}(K)$, with $\\mathbb R$ coefficients, to the $L^2$-de Rham complex. It will be useful to view $C^{\\bullet}(K)$ as a subcomplex of the singular cochain complex. With this in mind, we often identify singular simplices with their images.\n\nThe Whitney map is readily defined using the basis for $C^{\\bullet}(K)$ dual to the basis of simplices. This basis consists of the cochains $\\delta_{\\sigma}$ that take the value 1 on the simplex $\\sigma$ and zero on all other simplices.\nThe Whitney form $W(\\delta_{\\sigma})$ associated to the cochain $\\delta_{\\sigma}$ dual to an  oriented simplex $\\sigma = [v_0,\\dots,v_q]$ is given by\n$$W(\\delta_{\\sigma}) = q! \\sum_{k=0}^q(-1)^kb_{k}db_0\\wedge \\cdots \\wedge db_{k-1}\\wedge db_{k+1}\\wedge \\cdots \\wedge d b_q,$$\nwhere $b_k:M\\to [0,1]$ is the barycentric coordinate associated to the vertex $v_k$. See \\cite{Dodziuk} for more details.\n\nAn $L^2$-form in the image of $W$  is called a Whitney form. The support of a Whitney form $W(\\delta_{\\sigma})$ is contained in the closed star of the simplex $\\sigma$. The barycentric coordinates used to define the Whitney forms are not smooth, however, they are smooth in the compliment of the $(n-1)$-skeleton of $K$. One can define the exterior derivative of a Whitney form in a weak sense, which yields a differential that is well defined as an $L^2$-form. With this exterior derivative, the Whitney map becomes a chain map. For any cochain $f$ and simplex $\\sigma$, the restriction of the Whitney cochain $\\omega= W(f)$ to $\\sigma$, denoted $\\omega|_{\\sigma}$, can be uniquely extended to a smooth form on the boundary of $\\sigma$. This extension however is not unique when $\\sigma$ lies in the boundary of multiple simplices. In addition to the restriction of Whitney forms, we have the restriction for cochains.\nIf $f= \\sum a_i \\delta_{\\sigma_i}$ is a cochain and $\\sigma$ is a simplex, then $f|_{\\sigma} = \\sum\\limits_{\\sigma_i\\subset\\sigma}a_i\\delta_{\\sigma_i}.$ This cochain restriction satisfies $\\omega|_{\\sigma} = W(f|_{\\sigma})|_{\\sigma}$.\n\nA Whitney form associated to a geodesic simplex $\\sigma$ in $M$ is the corresponding Whitney form on the standard Euclidean simplex pulled back to $\\sigma$ via the map $V_\\sigma$. Geometric norms on cochains determined by the Whitney map can be compared with various combinatorial norms. This is done using the $L^p$-change of variables formula for $k$-forms, see \\cite{stern}.\n\n\\begin{prop} \\label{prop: L2 comparisons}\nLet $s\\in\\mathcal S_\\varepsilon\\cup\\mathcal G_\\varepsilon$. Let $W(f)$ be the Whitney form associated to a cochain $f\\in C^{k}(s;\\mathbb R)$. Let $||\\cdot||$ be some fixed norm on the real vector space $C^{k}(s;\\mathbb R)$ and let $||\\cdot||_{p,s}$ be the $p$-norm associated to $s$ on  $\\Omega^k(s)$, where $p = {\\infty}$ or $p=2$.\nThen there is a constant $\\mathcal A = \\mathcal A (\\varepsilon,||\\cdot||)>0$ such that $$\\mathcal A^{-1}||W(f)||_{p,{s}}\\leq  \\mathcal ||f|| \\leq \\mathcal A||W(f)||_{p,s}.$$ The constant $\\mathcal A$ only depends on the chosen norm and combinatorial type of $s,$ not on the geometric structure of $s$.\n\\end{prop}\n\\begin{proof}\nBy Proposition \\ref{prop: stars biLipschitz}, there is a constant $\\mathcal L$ (note $\\kappa\\leq \\mathcal L$, so that if we're working with simplices, $\\mathcal L$ works as Lipschitz constant) such that any pair $s,s'\\in \\mathcal S_\\varepsilon\\cup\\mathcal G_\\varepsilon$ in the same combinatorial type are $\\mathcal L$-biLipschitz equivalent.\nFor each combinatorial type, fix a model $s_a\\in \\mathcal S_\\varepsilon\\cup\\mathcal G_\\varepsilon$ and let $\\mu:s\\to s_a$ be the $\\mathcal L$-biLipschitz comparison map described above. Then, because Whitney forms are obtained by pulling back the standard Whitney forms via the model map, we can apply the $L^2$-change of variables formula for $k$-forms and use the biLipschitz comparison to get $$\\frac{||W(f)||_{2,{s}}}{||f||}\\leq \\mathcal L^{n\/2}\\frac{||W(f)||_{2,s_a}}{||f||}.$$ Similarly, applying the $L^{\\infty}$-change of variables formula for $k$-forms, gives\n$$\\frac{||W(f)||_{{\\infty},s}}{||f||}\\leq \\mathcal L^{k}\\frac{||W(f)||_{\\infty,s_a}}{||f||}.$$\n\nFor $p=2$, set $$\\mathcal A_2 =  \\mathcal L^{n\/2}\\max\\limits_a \\sup\\limits_{g\\in C^{k}(s_a)} \\frac{||W(g)||_{2,s_a}}{||g||}$$ and for $p = \\infty$, set $$\\mathcal A_{\\infty} = \\mathcal L^{n} \\max\\limits_a\\sup\\limits_{g\\in C^{k}(s_a)} \\frac{||W(g)||_{2,s_a}}{||g||},$$\nwhere the maximum runs over all combinatorial types $a$.\nThen, $A = \\max\\{\\mathcal A_2,\\mathcal A_{\\infty}\\}$ gives the first inequality.\nThe second inequality is obtained from an identical argument via the lower bound in biLipschitz comparison. Let $\\mathcal A$ be the maximum of these two constants.\n\\end{proof}\n\nThis estimate is enough to reproduce the statements in \\cite{LS} for deeply embedded triangulations. However, to obtain the cleaner discrete-smooth eigenvalue comparison in Proposition \\ref{prop:4.2}, we need to study a smooth analogue of the Whitney map. The smooth Whitney map was introduced by Dodzuik in \\cite{dodzuik2}. The map is defined by replacing the barycentric coordinates with a smooth partition of unity indexed by the vertices of a triangulation. The particular partition of unity is provided by the following proposition.\n\n\\begin{prop} \\label{prop: partition existence} (\\cite{dodzuik2}, Lemma 2.11)\nIf $M$ admits a deeply embedded triangulation $K$, then there exists a $C^{\\infty}$ partition of unity $\\beta_i$ indexed by the vertices of $K$ and subordinate to the covering of $M$ by open stars of vertices of $K$ (indeed, compactly supported in each open star). Moreover, each $\\beta_i$ has covariant derivatives satisfying the pointwise bound $|\\nabla^k\\beta_i|< C$ for some constant $C = C(\\varepsilon)$, for $k\\leq n$.\n\\end{prop}\n\n\\begin{proof}\nLet $s\\in\\mathcal S_\\varepsilon$ be the combinatorial 1-neighborhood of a simplex. Denote the vertices of $s$ by $v_0,\\dots, v_n$ and let $b_i$ be the standard barycentric coordinates associated to the vertex $v_i$. Define\n\n\\[\\bar b_i(x) =  \\begin{cases}\n       0 & b_i(x)\\leq 1\/(n+2), \\\\\n       \\frac{(n+2)b_i(x)-1}{n+1} & b_i(x) \\geq 1\/(n+2). \\\\\n\n   \\end{cases}\n\\] Observe that $\\sum\\limits_i \\bar b_i(y) \\geq \\frac{1}{(n+2)}$. Define $$\\delta(s) = \\inf\\limits_{\\substack{x\\in\\supp(\\bar b_i)\\\\y\\in \\d \\s(v_i)}}d(x,y),$$ where $d$ is the distance function induced by the Riemannian metric.  Set $\\delta = \\frac{1}{2}\\inf\\limits_{s\\in \\mathcal S_\\varepsilon} \\delta(s)$ and notice $\\delta>0$. For any point $x\\notin \\s(v_i)$, the ball $B_{\\delta}(x)$ is disjoint from the support of $\\bar b_i$.\nLet $\\eta$ be a smooth cutoff function such that $\\eta(r) = 1$ when $|r|<\\delta\/2$ and $\\eta(r) = 0$ when $|r|>3\\delta\/4$. The function $\\eta(d(x,y))$ is smooth on the open star of a vertex, so the operator given by integrating against $\\eta(d(x,y))$ is smoothing. Therefore, if we define $$\\tilde b_i(x) = \\int_{B_{3\\delta\/4}(x)}\\eta(d(x,y))\\bar b_i(y)dy,$$ the result is a smooth function.  Notice $\\tilde b_i$ is supported in the interior of the star of $v_i$ by virtue of our choice of $\\delta$. We now define smoothed barycentric partitions of unity for a smooth manifold with deeply embedded triangulation $K$ by normalizing the functions $\\tilde b_i$ associated to the vertices of $K$:\n$$\\beta_i(x) = \\left(\\sum\\limits_j \\tilde b_j (x)\\right)^{-1} \\tilde b_i (x).$$ Notice that if $\\tilde b_i(x)\\neq 0$, then this normalizing sum really just runs over the vertices of $\\s(v_i)$. This normalizing constant can be bounded from below:\n\\begin{align*}\n    \\sum\\limits_j \\tilde b_j(x) &\\geq \\sum\\limits_j \\int_{B_{\\delta\/2}(x)}\\bar b_j(y)dy \\\\\n    &= \\int_{B_{\\delta\/2}(x)} \\sum\\limits_j\\bar b_j(y)dy \\\\\n    &\\geq \\vol(B_{\\delta\/2}(x))\\frac{1}{(n+2)(n+1)}.\n\\end{align*}\n\nThe covariant derivative bound follows from repeated application of the quotient rule and the corresponding bounds for $\\bar b_i$, which  depends only on the derivatives of cutoff function $\\eta$ and the covariant derivatives of the metric. The choice of $\\delta$ ensures each function $\\beta_i$ is compactly supported in the star of $v_i$.\n\\end{proof}\n\nOur aim now is to establish a version of Proposition \\ref{prop: L2 comparisons} for these partitions of unity that will allow us to relate the geometric norms induced by the smooth Whitney map to combinatorial norms.\n\n\nLet $M$ be a hyperbolic $n$-manifold with a deeply embedded triangulation $K$. Denote by $\\s_0(\\sigma)$ the combinatorial 1-neighborhood of an $n$-simplex $\\sigma$ in $K$.\nFix some $n$-simplex $\\sigma$ in $K$ and set $s = \\s_0(\\sigma)$. Because $K$ is deeply embedded, for any point $p\\in s$, the ball $B = B_\\varepsilon(p)$ contains $s$ and lifts isometrically to $\\H^n$. Identify $s$ with such a lift. The functions $\\beta_i$ associated to the vertices of $\\sigma$ are supported in $s$ and their value in any simplex $\\sigma$ depends only on the geometry of $s$. This enables us to isolate the local properties of the barycentric partition of unity functions.\nBy perturbing the vertices of $s$ in $\\H^n$ and modifying the various simplices making up the complex $s$ accordingly, we can then see how these functions relate to the geometry of combinatorial 1-neighborhoods as encoded by the space $\\mathcal S_\\varepsilon$.\n\n\\begin{lem} \\label{lem: partition of unity continuous}\nLet $\\sigma$ be an $n$-simplex with vertices $v_0,\\dots,v_n$ from $\\mathcal G_\\varepsilon$ contained in combinatorial 1-neighborhood $s = \\s_0(\\sigma) \\in \\mathcal S_\\varepsilon$ with additional vertices $v_{n+1},\\dots,v_m$.\nThe functions $\\beta_i$ associated to the vertices of $\\sigma$ constructed in Proposition \\ref{prop: partition existence} and their covariant derivatives $\\nabla \\beta_i$ vary continuously in $L^2(\\H^n)$ when the perturbation of $s$ in $\\mathcal S_\\varepsilon$ is realized as above in $\\H^n$.\n\\end{lem}\n\\begin{proof}\n\nAs above, we can embed the combinatorial 1-neighborhood $s$ in $\\H^n$ and the functions $\\tilde b_i$ for each vertex $v_i$ of $\\sigma$ described in Proposition \\ref{prop: partition existence} are well defined smooth functions on $\\H^n$ supported on $s$. To define the barycentric partition of unity, we also need the functions $\\tilde b_k$ for vertices $v_k$ in $s$ that are not in $\\sigma$ to be well defined on the support of the functions $\\tilde b_i$ for vertices $v_i$ of $\\sigma$.\nFor a vertex $v_k$ in $s$  that is not part of $\\sigma$ and a point $x$ in the support of $\\tilde b_i$ for $v_i$ a vertex of $\\sigma$, we have that the ball $B_{3\\delta\/4}$ used in the definition of $\\tilde b_k$ is contained in $s$, so $\\tilde b_k(x)$ only depends on $s$.\n\nFrom the definition, one sees that each function $\\tilde b_i$ varies continuously in $L^2(\\H^n)$ as the complex $s$ in $\\H^n$ is varied by perturbing the vertices and modifying the various simplices making up the complex $s$ accordingly. One similarly can see from the definition of $\\tilde b_i(x)$ that $\\nabla \\tilde b_i(x)$ varies continuously as $s$ is varied as above.\nThe functions $\\beta_i$ in the barycentric partition of unity are defined by normalizing the functions $\\tilde b_i$: $$\\beta_i(x) = \\left(\\sum\\limits_{v_k\\in s^{(0)}} \\tilde b_k (x)\\right)^{-1} \\tilde b_i (x) .$$ By the remark above, $\\tilde b_k$ is well defined on the support of $\\tilde b_i$ for every vertex $v_k$ of $s$. Since each $\\tilde b_j$ and $\\nabla \\tilde b_j$ varies continuously with $s$, the same holds for $\\beta_j$ and $\\nabla \\beta_j$.\n\\end{proof}\n\nTo a partition of unity indexed by the vertices of a triangulation and subordinate to the covering by open stars of vertices, one can define a generalized Whitney mapping, given by the same formula as the standard Whitney map but with the smooth barycentric partitions of unity in place of the standard barycentric coordinates. Let $\\beta = (\\beta_i)$ be the barycentric partition of unity. The Whitney form $W_{\\beta}(\\delta_{\\sigma})$ associated to the cochain $\\delta_{\\sigma}$ dual to an  oriented simplex $\\sigma = [v_0,\\dots,v_q]$ is given by $$W_{\\beta}(\\delta_{\\sigma}) = q! \\sum_{k=0}^q(-1)^k\\beta_{k}d\\beta_0\\wedge \\cdots \\wedge d \\beta_{k-1}\\wedge d \\beta_{k+1}\\wedge \\cdots \\wedge d \\beta_q,$$ Like the standard Whitney map, these generalized Whitney maps satisfy:\n\\begin{enumerate}\n\t\\item For a chain $a$ and cochain $f$ of the same degree, $\\int_aW_{\\beta}(f) = f(a).$\n\t\\item For any cochain $f$, $dW_{\\beta}(f) = W_{\\beta}(df)$.\n\t\\item If $p$ is contained in the interior of an  $n$-simplex $\\sigma$, and any cochain $f$, $W_{\\beta}(f)_p = W_{\\beta}(f|_{\\sigma})_p$.\n\\end{enumerate}\n\n\\begin{lem} \\label{lem: forms continuous} Fix a simplex $\\sigma\\in\\mathcal G_\\varepsilon$ contained in its combinatorial 1-neighborhood $s\\in \\mathcal S_\\varepsilon$.\nThe Whitney map $W_{\\beta}: C^{\\bullet}(\\sigma)\\to L^2\\Omega^{\\bullet}(\\H^n)$ varies continuously as the geometry of the star $s$ varies in $\\H^n$ as described above. Consequently, $||W_{\\beta}(f)||_{2,s}$ and $|| W_{\\beta}(f) ||_{2,\\sigma}$ vary continuously with the geometric structure on $s$ in $\\mathcal S_\\varepsilon$.\n \\end{lem}\n\\begin{proof} We work with the combinatorial 1-neighborhood $s$ embedded in a ball $B\\subset \\H^n$ as above. Because $\\beta_i$ and $\\nabla \\beta_i$ vary continuously with $s$ and since $||\\nabla\\beta_i||_{2,B} $ is comparable to $||d\\beta_i||_{2,B}$, it suffices to show that exterior products of $d\\beta_i$ vary continuously in the $L^2$-norm. The degree 0 case and  degree 1 case are immediate from the continuity in Lemma \\ref{lem: partition of unity continuous}. We treat only the degree 2 case as the other higher degree cases are handled similarly. Assume $||\\beta_i - \\beta_i'||_{2,B} < \\epsilon$ and $||d\\beta_i - d\\beta_i'||_{2,B}<\\varepsilon$. Then we have \\begin{align*}\n||d\\beta_0\\wedge d\\beta_1 - d\\beta'_0\\wedge d\\beta_1'||_{2,B} &=||d\\beta_0\\wedge d\\beta_1 - d\\beta'_0\\wedge d\\beta_1' + d\\beta_0 \\wedge d\\beta_1' - d\\beta_0 \\wedge d\\beta_1'||_{2,B} \\\\\n&\\leq ||d\\beta_0\\wedge d\\beta_1  - d\\beta_0 \\wedge d\\beta_1'||_{2,B} + || d\\beta_0' \\wedge d\\beta_1' - d\\beta_0\\wedge d\\beta_1'||_{2,B} \\\\\n&= ||d\\beta_0\\wedge d(\\beta_1 - \\beta_1')||_{2,B} + ||d(\\beta_1' - \\beta_1)\\wedge d\\beta_1'||_{2,B}\\\\\n&\\leq C||d\\beta_0||_{2,B} ||d(\\beta_1-\\beta_1')||_{2,B} + C||d\\beta_1'\n||_{2,B} ||d(\\beta_1'-\\beta_1)||_{2,B},\\\\\n&\\leq 2C\\epsilon, \\text{~after increasing $C$.}\n\\end{align*}\nThis immediately gives that $||W_{\\beta}(f)||_{2,s}$ varies continously with $s$. Let $\\xi_{\\sigma}$ be the characteristic function of $\\sigma$ in $\\H^n$. Then $\\xi_{\\sigma}$ varies continuously in $L^2$ when $\\sigma$ is varied by perturbing its vertices. The norm $||W_{\\beta}(f)||_{2,\\sigma} = ||W_{\\beta}(f)\\xi_{\\sigma}||_{2,B}$ therefore also varies continuously when $s$ is varied in $\\mathcal S_\\varepsilon$.\n\\end{proof}\n\n\\begin{prop} \\label{prop: smooth L2 comparison} There is a constant $A=A(\\varepsilon)>0$ such that if $\\sigma\\in \\mathcal G_\\varepsilon$ has combinatorial 1-neighborhood $s\\in\\mathcal S_\\varepsilon$ and $f\\in C^{\\bullet}(\\sigma)$ is any cochain, then there are comparisons $$ A^{-1}||W(f)||_{2,\\sigma}\\leq ||W_{\\beta}(f)||_{2,\\sigma}\\leq A||W(f)||_{2,\\sigma}.$$\n\\end{prop}\n\\begin{proof}\n    We embed $s$ in a ball $B$ in $\\H^n$ as above. The $L^2$-norm induced by $\\beta$ for a fixed geometric structure on $\\sigma$ is continuous on the vector space of cochains $C^{\\bullet}(\\sigma)$. For any cochain $f\\in C^{\\bullet}(\\sigma)$, the form $W_{\\beta}(f)$ varies continuously in $L^2\\Omega^{\\bullet}(B)$ as the combinatorial 1-neighborhood $s$ varies in $\\mathcal S_\\varepsilon$.\n    It follows that $||W_{\\beta}(f)||_{2,\\sigma}$ is continuous as a function on the component of $\\mathcal S_\\varepsilon \\times C^{\\bullet}(\\sigma)$ corresponding to the combinatorial type of the combinatorial 1-neighborhood $s$.\n\n    Since $W_{\\beta}$ sends nonzero cochains to nonzero forms, for any $f\\neq 0$, one has $0<||W_{\\beta}(f)||_{2,\\sigma}$. Let $||\\cdot||_{E}$ be the usual $\\ell^2$ norm on $C^{\\bullet}(\\sigma)$.\n    The norm $\\norm{\\cdot}_E$ is fixed as the geometric structure on $\\sigma$ varies. For $s'\\in \\mathcal S_\\varepsilon$, let $\\beta'$ be the corresponding  barycentric partition of unity defined by the combinatorial 1-neighborhood $s'$. By definition, each $s'\\in\\mathcal S_\\varepsilon$ is the combinatorial 1-neighborhood of some simplex, let $\\sigma'$ be this simplex.\n\n    Using the continuity in Lemma \\ref{lem: partition of unity continuous} and the compactness of $\\mathcal S_\\varepsilon$, we conclude the constants\n    $$A_{\\bullet} = \\inf\\limits_{s'\\in \\mathcal S_\\varepsilon}\\inf\\limits _{\\substack{f\\in C^{\\bullet}(\\sigma')\\\\ ||f||_{E} = 1}} ||W_{\\beta'}(f)||_{2, \\sigma'}$$\n    and\n    $$B_{\\bullet}= \\sup\\limits_{s'\\in \\mathcal S_\\varepsilon}\\sup\\limits_{\\substack{f\\in C^{\\bullet}(\\sigma')\\\\ ||f||_{E} = 1}} ||W_{\\beta'}(f)||_{2, \\sigma'},$$\n    are strictly positive real numbers independent of the particular geometric structure on $s$ or $\\sigma$ giving the desired comparison between $||f||_E$ and $||W_{\\beta}(f)||_{2,\\sigma}$.\n\n    We can then compare $||W(f)||_{2,\\sigma}$ and $||f||_E$ using Proposition \\ref{prop: L2 comparisons}. For $f\\in C^{\\bullet}(\\sigma)$, Proposition \\ref{prop: L2 comparisons} gives a constant $\\mathcal A$ such that $\\mathcal A^{-1}||W(f)||_{2,\\sigma} \\leq ||f||_{E}\\leq \\mathcal A||W(f)||_{2,\\sigma}$.\n    Combining these comparisons with the comparison of $||W_{\\beta}(f)||_{2,\\sigma} $ and $||f||_{E}$ above gives the claimed result for $||W_{\\beta}(f)||_{2,\\sigma} $.\n\\end{proof}\n\nThe upshot of this is that the smooth barycentric partition of unity induces a norm on the cochain complex of a simplex that locally is uniformly comparable to the $L^2$-norm induced by the standard barycentric partition of unity.  This gives an analogue of Proposition \\ref{prop: L2 comparisons} for the $L^2$-norm induced by the barycentric partition of unity. We also require such a comparison for the $L^{\\infty}$-norm. The upgraded version of Proposition \\ref{prop: L2 comparisons} appears below as Proposition \\ref{prop: smooth comparison}.\n\nFor a smooth manifold $Y$, possibly with boundary, we denote the Sobolev spaces of differential $\\bullet$-forms by $H^k_{\\nabla}\\Omega^{\\bullet}(Y)$ and their norms by $||\\cdot||_{H^k_{\\nabla}(Y)}$, where $$||\\omega||_{H^k_{\\nabla}} = \\sum\\limits_{i=0}^k||\\nabla^i \\omega||_2.$$ When $\\bullet = 0$, we drop $\\Omega^0$ from this notation.\nWhen $Y$ has boundary, the marking $\\mathring{H}^k_{\\nabla}$ denotes the subspace of forms that are approximated by smooth forms supported in the interior of $Y$.\n\n\\begin{lem} \\label{lem: constant R}\n    There is a constant $R(\\varepsilon)>0$ such that for any $n$-simplex $\\sigma\\in \\mathcal G_\\varepsilon$ with combinatorial 1-neighborhood $s\\in\\mathcal S_\\varepsilon$, the map $W_{\\beta}:C^{\\bullet}(\\sigma)\\to H^{n}_{\\nabla}\\Omega^{\\bullet}(B)$ satisfies\n    $$||W_{\\beta}(f)|_{\\sigma}||_{H^n_{\\nabla}(\\sigma)} \\leq ||W_{\\beta}(f)||_{H^n_{\\nabla}(B)} \\leq R||W_{\\beta}(f)||_{2,\\sigma},$$ where we have identified the combinatorial 1-neighborhood $s$ isometrically with a domain $D$ in $\\H^n$ and $B$ is a ball of radius $\\varepsilon$ based at $p\\in \\sigma$.\n\\end{lem}\n\n\\begin{proof}\nThe first inequality follows from the definition of the Sobolov norm.\n\nWe now observe that the covariant derivative bounds for a smooth barycentric partition of unity imply that $||W_{\\beta}(f)||_{H^n_{\\nabla}(B)}$ is bounded by some constant times $||f||_{G,\\sigma}$, where $||\\cdot||_{G,\\sigma}$ is the $\\ell^1$-norm on $C^{\\bullet}(\\sigma)$. For a cochain $f = \\sum a_i \\delta_{\\sigma_i}$, let $\\omega_i = W_{\\beta}(\\delta_{\\sigma_i})$ so that $W_{\\beta}(f) = \\sum a_i \\omega_i$. We can then compute,\n\\begin{align*}\n\t||W_{\\beta}(f)||_{H^k_{\\nabla}(B)} &= ||\\sum a_i\\omega_i||_{H^k_{\\nabla}(B)}\\\\\n\t&\\leq \\sum_j||\\nabla^j \\sum_i  a_i\\omega_i||_{2,B}\\\\\n\t&\\leq  \\sum_j\\sum_i|a_i|||\\nabla^j \\omega_i||_{2,B}.\n\\end{align*}\nEach summand above satisfies $||\\nabla^j\\omega_i||_{2,B}<C$ for a constant $C$ depending on the covariant derivative bounds of the barycentric partition of unity. There is a constant $T$ such that the number of $\\bullet$-faces of an $n$-simplex is less than $T$.\nThus, \\begin{align*}\n ||W_{\\beta}(f)||_{H^k_{\\nabla}(B)} &\\leq \\sum_j\\sum_i|a_i|||\\nabla^j \\omega_i||_{2,B} \\\\\n&\\leq \\sum_jC\\sum_i|a_i|\\\\\n&\\leq TC\\sum_i|a_i| \\\\\n&= TC||f||_{G,\\sigma}.\n\\end{align*}\n For a single simplex, this combinatorial $\\ell^1$-norm is comparable to the $\\beta$-induced $L^2$-norm by Proposition \\ref{prop: L2 comparisons} and Proposition \\ref{prop: smooth L2 comparison}. Thus, $$||W_{\\beta}(f)||_{H^n_{\\nabla}(B)}\\leq R||W_{\\beta}(f)||_{2,\\sigma}.$$\n\\end{proof}\n\nThe following version of the Sobolev inequality, a consequence of Theorem 1 in \\cite{cantor}, ensures that control over certain Sobolev norms implies pointwise norm control. Let $|\\cdot|$ be the pointwise norm induced by the Riemannian metric.\n\n\\begin{thm} \\label{thm: Cantor Sobolev}(Cantor-Sobolev)\nSuppose $M$ is a hyperbolic $n$-manifold with injectivity radius bounded below by $\\varepsilon$. Let $r<\\varepsilon$, and $l\\geq 0,~k\\geq 0$ be such that $l + n\/2 < k$. Then if $\\omega$ is in the Sobolev space $H^k_{\\nabla} \\Omega^{\\bullet}(M)$, there is a constant $C = C(r)$ such that for every $p\\in M$, $$|\\nabla^l\\omega(p)|\\leq C||\\omega||_{H_{\\nabla}^k(B_r(p))}.$$\n\\end{thm}\n\n\nThis discussion provides us with the following upgraded version of Proposition \\ref{prop: L2 comparisons}.\n\n\\begin{prop} \\label{prop: smooth comparison}\nLet $s\\in\\mathcal S_\\varepsilon$ be the combinatorial 1-neighborhood of an $n$-simplex $\\sigma \\in\\mathcal G_\\varepsilon$. Let $\\beta$ be the smooth barycentric partition of unity in Proposition \\ref{prop: smooth L2 comparison} or the standard barycentric coordinate on $s$. Let $W_{\\beta}(f)$ be the resulting generalized Whitney form associated to a cochain $f\\in C^{\\bullet}(\\sigma;\\mathbb R)$. Let $||\\cdot||$ be some fixed norm on the real vector space $C^{\\bullet}(\\sigma;\\mathbb R)$ and let $||\\cdot||_{p,s}$ be the $p$-norm associated to $s$ on $\\Omega^{\\bullet}(s)$ and likewise for $||\\cdot||_{p,\\sigma}$, where $p = {\\infty}$ or $p=2$.\nThen there is a constant $\\mathcal B = \\mathcal B (\\varepsilon,||\\cdot||)>0$ (independent of $\\sigma$ and $s$) such that $$\\mathcal B ^{-1}|| W_{\\beta}(f)||_{2,{\\sigma}}\\leq   ||f||\\leq \\mathcal B|| W_{\\beta}(f)||_{2,\\sigma}$$ and\n$$\\mathcal B^{-1}|| W_{\\beta}(f) ||_{\\infty, \\sigma}\\leq \\mathcal B^{-1} || W_{\\beta}(f)||_{\\infty,s} \\leq ||f||\\leq \\mathcal B|| W_{\\beta}(f)||_{\\infty,s}.$$\n\\end{prop}\n\n\\begin{proof}\n    Identify $s$ with a domain $D$ in $\\H^n$ and let $B$ be a ball of fixed radius $r = r(\\varepsilon)$ that contains $s$ as in Lemma 2.11 and assume the basepoint of the ball is the point at which $||W_{\\beta}(f)||_{\\infty, s}$ is realized.\n\n    From the Sobolev inequality, Propositions \\ref{prop: L2 comparisons} and \\ref{prop: smooth L2 comparison}, and Lemma \\ref{lem: constant R}, for any cochain $f$,\n        \\begin{align*}\n        ||W_{\\beta}(f)||_{\\infty,\\sigma}&\\leq ||W_{\\beta}(f)||_{\\infty,s}\\\\\n        &\\leq C||W_{\\beta}(f)||_{H^n_{\\nabla}(B)}\\\\\n        &\\leq CR||f||_{G,\\sigma} \\\\\n        &\\leq \\mathcal ACR||W_{\\beta}(f)||_{2,\\sigma}.\n        \\end{align*}\n\n    Then the comparison $$||W_{\\beta}(f)||_{2,\\sigma}\\leq ||W_{\\beta}(f)||_{2,s}\\leq\\sqrt{\\vol(s)} ||W_{\\beta}(f)||_{\\infty,s}$$ implies the smooth barycentric partition of unity induced $L^2$-norm $||W_{\\beta}(f)||_{2,\\sigma} $ and $L^{\\infty}$-norm $||W_{\\beta}(f)||_{\\infty,s}$ are comparable for any simplex $\\sigma\\in\\mathcal G_\\varepsilon$ with combinatorial 1-neighborhood\n    $s\\in \\mathcal S_\\varepsilon$.\n\n    The only constant appearing in the comparison depending on $s$ is $\\sqrt{\\vol(s)}$, which can be uniformly bounded by a constant depending only on $\\varepsilon$ by compactness of $\\mathcal S_\\varepsilon$. As remarked after the proof of Proposition \\ref{prop: smooth L2 comparison}, the $L^2$-norm version of the desired estimate follows from Proposition \\ref{prop: L2 comparisons} and Proposition \\ref{prop: smooth L2 comparison}. The above comparisons of the $L^2$-norm and the $L^{\\infty}$-norm imply the claim.\n\\end{proof}\n\n\\section{The upper bound}\n\\label{sec:4}\nIn this section we prove Theorem A, which states that in a closed hyperbolic 3-manifold $M$ the first positive eigenvalue of the Hodge Laplacian acting on coexact 1-forms is bounded above by a multiple of the stable isoperimetric ratio $\\rho(M)$. The background results of this section all hold in any dimension greater than 2, however the proof of Theorem A makes use of Poincar\\'e duality to relate 1-forms and surfaces, this forces us to restrict Theorem A to the 3-dimensional setting.\n\n\nThe cochain results of the previous section are connected to spectral geometry via the inner product induced by the Whitney map associated to a triangulation and barycentric coordinate: $$\\langle f, g\\rangle = \\int_M W_{\\beta}(f)\\wedge\\star W_{\\beta}(g),$$ which along with the corresponding norm $||\\cdot||_2$, determine a Hodge theory for the cochain complex $C^{\\bullet}(K)$. This inner product determines a codifferential $$d^*_W:C^{\\bullet}(K)\\to C^{\\bullet-1}(K)$$ which, as the adjoint of the standard differential, satisfies $\\langle d f, g\\rangle = \\langle f , d_W^* g \\rangle.$\nThe corresponding Whitney Laplacian $\\Delta_W:C^{\\bullet}(K)\\to C^{\\bullet}(K)$ is then given by the standard formula $\\Delta_W = dd_W^*+d_W^*d.$ This inner product was introduced using the standard barycentric coordinates in \\cite{Dodziuk}.\n\nThis Laplacian decomposes the space $C^{\\bullet}(K)$ into harmonic, exact, and coexact components: $$C^{\\bullet}(K)\\cong H^{\\bullet}(M) \\oplus dC^{\\bullet-1}(K) \\oplus d_W^* C^{\\bullet+1}(K).$$ This combinatorial Hodge decomposition serves as a good approximation of the $L^2$-Hodge decomposition of $M$, though it does not capture the $L^2$-Hodge decomposition exactly. In particular, the Whitney coexact chains may not be $L^2$-coexact.\n\nWe begin by relating the Whitney and the Riemannian coexact eigenvalues.\n\\begin{lem} \\label{lem:4.1}\nLet $M$ be a closed Riemannian $n$-manifold with triangulation $K$ and an associated barycentric partition of unity $\\beta$. Give the cochain complex the Whitney $L^2$-norm induced by the Whitney map determined by $\\beta.$ Likewise, give the chain complex the dual norm $||\\cdot||_2^*$ determined by the integration pairing. Then for every coexact cochain $f\\in d_W^*C^2(K)$, there is an exact chain $a\\in \\d C_2(K)$ of unit norm such that $||f||_2 = \\int_aW_{\\beta}(f).$\n\\end{lem}\n\\begin{proof}\n\nThe cochain Hodge decomposition from the Whitney inner product gives the orthogonal decomposition $$C^1(K) = H^1(M) \\oplus d_W^*C^2(K) \\oplus dC_0(K).$$ Let $Z^1(K) = H^1(M)\\oplus d C_0(K)$. Identify $C_1(K)$ with $C^1(K)^*$ via the integration pairing. The composition of the inclusion and quotient map determines an isomorphism $d_W^*C^2(K) \\to C^1(K)\/Z^1(K) $ that allows us to identify these spaces. If $\\text{Ann}$ assigns to a subspace its annihalator, then there is also an isomorphism $(C^1(K)\/Z^1(K))^*\\to \\text{Ann}(Z^1(K)).$ By Stokes' theorem and dimension counting, $\\text{Ann}(Z^1(K)) = \\d C_2(K)$.\nThus, the dual of $d_W^*C^2(K)$ is exactly $\\d C_2(K)$. The dual norm of an element $a \\in \\d C_2(K)$ is given by $$||a||_2^* = \\sup\\limits_{\\substack{f\\in C^1(K)\\\\||f||_2\\leq 1}}\\int_a W_{\\beta}(f).$$\nIf $f$ has unit $L^2$-norm and $f = g + h$ where $g\\in d_W^*C^2(K)$ and $h\\in Z^1(K)$, then orthogonality implies $||g||_2\\leq 1.$ Whence,\n$$||a||_2^* = \\sup\\limits_{\\substack{f = g + h\\in C^1(K)\\\\||f||_2\\leq 1}}\\int_a W_{\\beta}(g) = \\sup\\limits_{\\substack{g\\in d^*_WC^2(K)\\\\||g||_2\\leq 1}}\\int_a W_{\\beta}(g).$$ The isometric identification of $(d^*_WC^2(K),||\\cdot||_2)$ with its double dual therefore implies we can compute the norm of an element $f\\in d^*_WC^2(K)$ via the integration pairing integrating only against chains in $\\d C_2(K)$:\n$$||f||_2= \\sup\\limits_{\\substack{a \\in \\d C_2(K)\\\\||a||_2^* = 1}}\\int_aW_{\\beta}(f).$$ In particular, for any coexact cochain $f\\in d^*_WC^1(K)$, there exists an exact chain $a$ with $||a||^*_2=1$ and $$\\int_aW_{\\beta}(f) = ||f||_2.$$\n\\end{proof}\n\n\n\n\\begin{prop} \\label{prop:4.2}\nLet $\\lambda$ denote the first eigenvalue for the Hodge Laplacian acting on coexact $1$-forms and let $\\lambda_W$ denote the first eigenvalue for the Whitney Laplacian acting on coexact 1-cochains associated to a deeply embedded triangulation $K$ and barycentric partition of unity $\\beta$. There is a constant $G = G(\\varepsilon)$ such that $$\\lambda \\leq G \\vol(M)\\lambda_W.$$\n\\end{prop}\n\\begin{proof}\n\nThe main issue here is that a Whitney coexact cochain will not generally map to an $L^2$-coexact form. This potentially adds a closed term to the denominator in the Whitney Rayleigh quotient, causing the Whitney Rayleigh quotient to be smaller than the Riemannian Rayleigh quotient. However, this failure can be controlled.\n\nLet $f$ be a coexact eigen-cochain with eigenvalue $\\lambda_W$. Set $\\omega = W_{\\beta}(f)\\in\\Omega^1(M)$, so that $\\frac{||d\\omega||_2^2}{||\\omega||_2^2} = \\lambda_W$. Let $p:\\Omega^1(M)\\to \\Omega^1(M)$ be the $L^2$-orthogonal projection onto coexact forms. Let $a\\in C_1(K)$ be the unit norm exact chain that realizes the norm of $f$ by integration given by Lemma 4.1. Then using that $d\\omega = d(p(\\omega))$ and the fact $a$ is exact, we obtain $$||f||_2 = ||\\omega||_2 = \\int_{a}\\omega = \\int_{a}p(\\omega).$$\n\nUsing 3.11, we have $||p(\\omega)||_{\\infty}\\leq C||f||_2.$ Hence, $||f||_2\\leq C\\text{len}(a)||p(\\omega)||_2.$ We can therefore obtain a lower bound on $||p(\\omega)||_2$ by bounding $\\text{len}(a)$ from above. Applying Proposition \\ref{prop: 3.1} to the chain $a$ above gives $$\\mathcal ||a||_G\\leq B \\sqrt{\\vol(M)} ||a||_2^* =  B \\sqrt{\\vol(M)}.$$ Since the lengths of the edges in the triangulation are bounded, we conclude the length of the support of $a$ is bounded.\nTake $E$ to be the length of the largest edge possible in a deeply embedded triangulation, so that $\\text{len}(a)\\leq BE\\sqrt{\\vol(M)}$\nThen we have obtained $$||\\omega||_2 = \\int_{a} \\omega = \\int_{a}p(\\omega) \\leq ||p(\\omega)||_2  BCE\\sqrt{\\vol(M)}.$$ Setting $G = ( BCE)^2$ and using that $\\omega$ is a Whitney eigenform, we obtain the result by the following short computation:\n\\[\n \\lambda \\leq \\frac{||d\\omega||_2^2}{||p(\\omega)||_2^2}\n \\leq G\\vol(M)\\frac{||d\\omega||_2^2}{||\\omega||_2^2}\n = G\\vol(M)\\lambda_W.\n\\]\n\\end{proof}\n\n\\begin{remark}\nNote that the above estimate in fact holds for the first positive eigenvalue since the first positive eigenvalue $\\lambda$ is the minimium of the first  eigenvalue of the Laplacian acting on functions and the first eigenvalue of the Laplacian acting on coexact 1-forms. The first eigenvalue $\\lambda_f$ for the Laplacian acting on functions automatically satisfies the comparison $\\lambda_f \\leq \\lambda_W$, as can be seen by studying the Rayleigh quotient and noticing that the estimate above controlling the projection in the denominator is immaterial in the function case.\n\\end{remark}\n\n\nWe are now ready to introduce stable commutator length, a thorough reference for which is \\cite{Calegari}. For a group  $\\Gamma$, let  $\\Gamma'$ denote the commutator subgroup and define the rational commutator subgroup to be $$\\Gamma_{{\\mathbb Q}}' = \\text{Ker}(\\Gamma\\to \\Gamma^{ab}\\otimes {\\mathbb Q}).$$ Note that when $\\Gamma$ is the fundamental group of a manifold, these subgroups correspond to the integrally nullhomologous and rationally nullhomologous loops respectively. The commutator length of an element $\\gamma\\in \\Gamma'$, denoted ${\\tt{cl}}(\\gamma)$ is the shortest word length of $\\gamma$ with respect to the generating set of all commutators.\nThe stable commutator length for $\\gamma\\in \\Gamma'_{{\\mathbb Q}}$ is then defined to be \\[\\scl(\\gamma) = \\inf\\limits_{m\\geq 1}\\frac{{\\tt{cl}}(\\gamma^m)}{m}.\\] Topologically, stable commutator length corresponds to the stable complexity of a surface bounding a nullhomologous curve. In particular, for $\\gamma\\in \\Gamma'_{{\\mathbb Q}}$, one has $$\\scl(\\gamma) = \\inf\\left\\{\\frac{\\chi_-(S)}{2m}~:~S \\text{ with }\\d S = \\gamma^m \\text{ and $S$ with no closed components}\\right\\},$$\nwhere for a connected surface $S$ we define $\\chi_-(S) = \\max\\{0,-\\chi(S)\\}$, and extend this additively to disconnected surfaces.\nThere is another natural complexity measure for loops in $\\Gamma'_{\\mathbb Q}$, the Gersten filling norm. For a loop $\\gamma\\in \\Gamma'_{\\mathbb Q}$, $\\fill(\\gamma)$ is the infimum of the Gromov norm $\\frac{||A||_G}{m}$ for all singular 2-chains $A$ bounding a 1-cycle representing a singular fundamental class of $\\gamma^m$. A fundamental theorem of Bavard relates the filling norm to the stable commutator length.\n\n\\begin{thm} \\label{thm:4.3} (\\cite{Bavard}) For any group element $\\gamma$, there is an equality: \\[\\scl(\\gamma) = 4\\fill(\\gamma).\\]\n\\end{thm}\nFor proof, see for instance Lemma 2.69 in \\cite{Calegari}.\n\n\\begin{remark} Let $B(\\Gamma)$ be the $\\mathbb R$-vector space of 1-boundaries. Then stable commutator length can be extended to a psuedo-norm on $B(\\Gamma)$. After identifying chains with vanishing psuedo-norm, Bavard duality, which relates the filling norm to quasimorphisms and their defect norm, becomes a genuine functional analytic duality theorem. One could define the stable isoperimetric ratio in this chain setting, and the results of this paper would go through for that (smaller) ratio as well.\n\\end{remark}\n\nWe can now prove the main theorem of this section.\n\n\\begin{mainthm}\\label{thm:A} Let $M$ be a closed hyperbolic 3-manifold  with injectivity radius bound below by $\\varepsilon$.  There is a constant $A = A(\\varepsilon)$ that only depends on $\\varepsilon$ such that for any nontrivial boundary $\\gamma \\in \\Gamma'_{{\\mathbb Q}}$, one has\n$$\\sqrt{\\lambda} \\leq A \\vol(M)\\frac{|\\gamma|}{\\scl(\\gamma)},$$\n where $\\lambda$ is the first coexact eigenvalue of the Hodge Laplacian on $\\Omega^1(M)$.\n\\end{mainthm}\n\n\n\\begin{proof}\nFirst note that since stable commutator length and geodesic length are both multiplicative under powers, it suffices to show the claim for an integrally nullhomologous loop $\\gamma$.\n\nFix a deeply embedded triangulation $K$ of $M$ and denote by $\\lambda_W$ the first eigenvalue of the Whitney Laplacian $\\Delta_W$ acting on $d^*_WC^2(K)$ associated to a smooth barycentric partition of unity. Notice that the Hodge decomposition ensures that zero is not an eigenvalue of this operator. Let $c:S^1\\to M$ be a cellular path in the 1-skeleton of $K^*$ representing the loop $\\gamma$, constructed as in Proposition  \\ref{prop: length comparison}.\nLet $T$ be a triangulation of $K^*$. Let $a\\in C_1(K^*)$ be the fundamental cycle for $\\gamma$ corresponding to the path $c$ in $C_1(K^*)\\subset C_1(T)\\subset C_1^{\\text{sing}}(M)$.\nIf $\\Phi:C^2(K)\\to C_1(K^*)$ is the Poincar\\'e duality map, then $\\Phi^{-1}(a)$ is an exact 2-cochain. We can therefore choose $\\omega\\in d^*_WC^2(K)$ with $d\\omega = \\Phi^{-1}(a)$. Setting $A = \\Phi(\\omega)$ in $C_2(K^*)$, we have $\\d A = a$ and $||A||_G = ||\\omega||_G.$\nSince $\\lambda_W$ is nonzero, we have $||\\omega||_2 \\leq \\frac{||d\\omega||_2}{\\sqrt{\\lambda_W}}$.  Proposition \\ref{prop:4.2} implies  \\[||\\omega||_2\\leq \\frac{\\sqrt G\\sqrt{\\vol(M)} ||d\\omega||_2}{\\sqrt{\\lambda}}.\\] By Bavard's theorem relating the filling norm to stable commutator length (Theorem 4.3), our choice of $A$, and Proposition \\ref{prop: 3.4} we find that\n$$\\scl(\\gamma) = 4\\fill(\\gamma) \\leq 4||\\tau(A)||_G \\leq 4N||A||_G = 4N||\\omega||_G,$$ where, as in Proposition \\ref{prop: 3.4}, $\\tau$ is the triangulation map relating the cellular chain $A$ to the subdivided simplicial chain in $C_2(T)$. Consequently,\n\\begin{align*}\n                  \\scl(\\gamma) &\\leq 4N ||\\omega||_G\\\\\n                 &\\leq 4NB\\sqrt{\\vol(M)}||\\omega||_2 ~\\text{by Proposition \\ref{prop: 3.1},}\\\\\n                 &\\leq  4NB\\sqrt G \\vol(M)\\frac{||d\\omega||_2}{\\sqrt{\\lambda}} ~\\text{by above computation,}\\\\\n                 &\\leq 4NB\\sqrt G \\vol(M)\\frac{D||d\\omega||_G}{\\sqrt{\\lambda}}~\\text{by Proposition  \\ref{prop: 3.2},} \\\\\n                 &=4NB\\sqrt G \\vol(M)\\frac{D||\\d  A||_G}{\\sqrt{\\lambda}}~\\text{by Proposition  \\ref{prop: 3.3},} \\\\\n                 &=  4NB\\sqrt G \\vol(M)\\frac{D||c||_G}{\\sqrt{\\lambda}} \\text{ by construction of $\\d A$,} \\\\\n                 &\\leq 4NB\\sqrt G \\vol(M)\\frac{DL|\\gamma|}{\\sqrt{\\lambda}}~\\text{by Proposition  \\ref{prop: length comparison},} \\\\\n                 &= 4NB\\sqrt G DL\\vol(M)\\frac{|\\gamma|}{\\sqrt{\\lambda}}.\n\\end{align*}\nSetting $A=4NB\\sqrt G DL$ and rearranging, we are done.\n\\end{proof}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nHigh performance codes are becoming increasingly difficult to program,\ndespite a proliferation of successful (but incremental) efforts to\nincrease programmability and productivity for high performance\ncomputing (HPC) systems. The reasons for this range over several\nlayers, beginning with the need for large, international\ncollaborations to combine expertise from many different fields of\nscience,\nto the need to address a wide variety of systems and hardware\narchitectures to ensure efficiency and performance.\n\nAs heterogeneous and hybrid systems are becoming common in HPC\nsystems, additional levels of parallelism need to be addressed, and\nthe bar for attaining efficiency is being raised. Three out of ten,\nand 62 of the top 500 of the fastest computers in the world use\naccelerators of some kind to achieve their performance~\\cite{top500}.\nMore large heterogeneous systems are scheduled to be set up, especially\nincluding new Intel Xeon Phi and Nvidia K20x co-processors.\n\\todo{ES: Should we update this statement? Stampede at TACC (with Xeon\n  Phis) is open to the public, and Blue Waters (with Nvidia GK110\n  coprocessors) is open for testing.}\n\nIn this paper we present \\emph{Chemora\\xspace}, using an integrated approach\naddressing programmability and performance at all levels, from\nenabling large-scale collaborations, to separating physics, numerical\nanalysis, and computer science portions, to\ndisentangling kernel implementations from performance optimization\nannotations. Chemora\\xspace is based on the \\emph{Cactus}\nframework~\\cite{Goodale02a, cactusweb}, a well-known tool used in\nseveral scientific communities for developing HPC applications. Cactus\nis a component-based framework providing key abstractions to \nsignificantly simplify parallel programming for a large class of problems, in\nparticular solving systems of partial differential equations (PDEs) on\nblock-structured grids -- i.e.\\ adaptive mesh refinement (AMR) and\nmulti-block systems (see section \\ref{sec:cactus} below).\n\nChemora\\xspace enables existing Cactus-based applications to continue scaling\ntheir scientific codes and make efficient use of new hybrid systems,\nwithout requiring costly re-writes of application kernels or adopting\nnew programming paradigms. At the same time, it also provides a\nhigh-level path for newly developed applications to efficiently employ\ncutting-edge hardware architectures, without having to target a\nspecific architecture.\n\nWe wish to emphasize that the present work is merely the next step in\nthe currently fifteen year-long history of the Cactus framework. While\nfinding ways to exploit the power of accelerators is perhaps the\nlargest current challenge to increased code performance, it is really\nonly the latest advance in an ever-changing evolution of computer\narchitectures. Suport for new architectures is typically added to the\nlower-level components of frameworks (such as Cactus) by the framework\ndevelopers, allowing the application scientist to take advantage of\nthem without having to significantly rewrite code.\n\nTo create the Chemora\\xspace framework, we have built on top of a number of\nexisting modules that have not been written specifically for this\nproject, as well as creating new modules and abstractions. The main\nresearch and development effort has been the integration of these\nmodules, especially as regards accelerator interfaces, their\nadaptation for production codes as well as automatic optimizations to\nhandle complicated Numerical Relativity codes. The result is that this\nframework allows the use of accelerator hardware in a transparent and\nefficient manner, fully integrated with the existing Cactus framework,\nwhere this was not possible before.  The full contribution to the\ndescribed research work has been described in the section\n\\ref{sec:contribution}. The framework, along with introductory\ndocumentation, will be made publicly available \\cite{chemoracode}.\n\n\\subsection{Scientific Motivation}\n\\label{sec:science}\n\nPartial differential equations are ubiquitous throughout the fields of\nscience and engineering, and their numerical solution is a challenge\nat the forefront of modern computational science.  In particular, our\napplication is that of \\emph{relativistic astrophysics}. Some of the\nmost extreme physics in the universe is characterised by small regions\nof space containing a large amount of mass, and Newton's theory of\ngravity is no longer sufficient; Einstein's theory of General\nRelativity (GR) is required.  For example, black holes, neutron stars,\nand supernovae are fundamentally relativistic objects, and\nunderstanding these objects is essential to our understanding of the\nmodern universe. Their\naccurate description is only possible using GR\\@. The solution of\nEinstein's equations of GR using computational techniques is known as\n\\emph{numerical relativity} (NR\\@). See \\cite{Pfeiffer:2012pc} for a recent\nreview, and see \\cite{Loffler:2011ay} for a detailed description of an open-source\nframework for performing NR simulations.\n\nOne of the most challenging applications of NR is the inspiral and\nmerger of a pair of orbiting black holes.  GR predicts the existence\nof gravitational waves: ripples in spacetime that propagate away from\nheavy, fast-moving objects.  Although there is indirect evidence,\nthese waves have not yet been directly detected due to their low\nsignal strength. The strongest expected sources of gravitational waves are\nbinary black hole and neutron star mergers, and supernova explosions--\nprecisely those objects for which GR is required for accurate\nmodeling.  Several gravitational wave detectors \\cite{Fritschel:2003qw} are\npresently under construction and they are expected to see a signal within\nthe next few years.  The detection of gravitational waves will lead\nto an entirely new view of the universe, complementary to existing\nelectromagnetic and particle observations.  The existence and\nproperties of expected gravitational wave sources will dramatically\nextend our knowledge of astronomy and astrophysics.\n\nNR models the orbits of the black holes, the waveforms they produce,\nand their interaction with these waves\nusing the Einstein equations. Typically, these equations are split\ninto a 3+1 form, breaking the four dimensional character of the\nequations and enabling the problem to be expressed as a time evolution\nof gravitational fields in three spatial dimensions.\nThe Einstein equations in the BSSN formulation~\\cite{Nakamura:1987zz,\n  Shibata:1995we, Baumgarte:1998te}\nare a set of coupled nonlinear\npartial differential equations with 25 variables~\\cite{Alcubierre99d, Alcubierre02a},\nusually written for compactness in abstract index form.\nWhen fully expanded, they contain thousands of terms, and the right\nhand side requires about 7900\nfloating point operations per grid point to evaluate once, if using\neigth order finite differences.\n\nThe simulations are characterised by the black hole mass, $M$,\na length, $G M\/c^2$, and a time, $G M\/c^3$. Usually one \nuses units in which $G = c = 1$, allowing both time and distance to be\nmeasured by $M$. Typical simulations of the type listed above\nhave gravitational waves of\nsize $\\sim 10 M$, and the domain to be simulated is $\\sim\n100$--$1000 M$ in radius.  For this reason, Adaptive Mesh Refinement\n(AMR) or multi-block methods are required to perform long-term BBH\nsimulations.\n\nOver 30 years of research in NR culminated in a major breakthrough in\n2005~\\cite{pretorius2005evolution,Baker:2005vv,Campanelli:2005dd},\nwhen the first successful long-term stable binary black hole\nevolutions were performed.  Since then, the NR community has refined\nand optimized their codes and techniques, and now routinely runs\nbinary black hole simulations, each employing hundreds or thousands of\nCPU cores simultaneously of\nthe world's fastest supercomputers.\nPerformance of the codes is a critical issue, as the\nscientific need for long waveforms with high accuracy is compelling.\nOne of the motivations of the Chemora\\xspace project was taking the NR\ncodes into the era of computing with the use of accelerators (in particular\nGPUs) and improving their performance by an order of magnitude, thus enabling\nnew science.\n\n\\subsection{Related Work}\n\nTo achieve sustained performance on hybrid supercomputers and reduce\nprogramming cost, various programming frameworks and tools have been developed, e.g.,\nMerge~\\cite{Linderman:2008:MPM:1353536.1346318} (a library based framework\nfor heterogeneous multi-core systems), Zippy~\\cite{CGF:CGF1131} (a framework\nfor parallel execution of codes on multiple GPUs),\nBSGP~\\cite{Hou:2008:BBG:1360612.1360618} (a new programming language for\ngeneral purpose computation on the GPU), and\nCUDA-lite~\\cite{springerlink:10.1007\/978-3-540-89740-8_1} (an enhancement to\nCUDA that transforms code based on annotations).\nEfforts are also underway to improve compiler tools\nfor automatic parallelization and optimization of affine loop\nnests for GPUs~\\cite{Baskaran:2008:CFO:1375527.1375562} and for automatic\ntranslation of OpenMP parallelized codes to\nCUDA~\\cite{Lee:2009:OGC:1594835.1504194}.\nFinally, OpenACC is slated to provide OpenMP-like annotations for C and\nFortran code.\n\nStencil computations form the kernel of many scientific applications that \nuse structured grids to solve partial differential equations.\nThis numerical problem can be characterised as the {\\em structured grids} \"Berkeley\nDwarf\" \\cite{berkeleydwarfs2006}, one of a set of algorithmic patterns identified as important\nfor current and near-future computation.\nIn particular, stencil computations parallelized using hybrid architectures\n(especially multi-GPU) are\nof particular interest to many researchers who want to leverage the emerging hybrid\nsystems to speed up scientific discoveries.\nMicik~\\cite{Micik2009} proposed an optimal 3D finite difference\ndiscretization of the wave equation in a CUDA environment, and\nalso proposed a way to minimize the latency of inter-node communication\nby overlapping slow PCI-Express (interconnecting the GPU with the\nhost) data exchange with computations. This may be achieved by\ndividing the computational domain along the slowest varying dimension.\nThibault \\cite{Thibault2009} followed the idea of a domain division pattern and implemented\na 3D CFD model based on finite-difference discretization of the Navier-Stokes equations parallelized\non a single computational node with 4 GPUs.\n\nJacobsen \\cite{Jacobsen2010} extended this model by adding inter-node communication via\nMPI\\@. They followed the approach described in Micik~\\cite{Micik2009} and overlapped the communication with\ncomputations as well as GPU-host with host-host data exchange. However, they did not take\nadvantage of the full-duplex nature of the PCI-Express bus, which would have decreased the\ntime spent for communication. Their computational model also divides the domain along the slowest\nvarying dimension only, and this approach is not suitable for all numerical problems. For example, for large computational\ndomains, the size of the ghost zone becomes noticeable in comparison to the computed part\nof the domain, and the communication cost becomes larger than the computational cost, which can\nbe observed in the non-linear scaling of their model.\n\nNotable work on an example stencil application was selected as a finalist of the Gordon Bell Prize in \nSC 2011 as the first peta-scale result \\cite{Shimokawabe2011}. Shimokawabe et al.\\ demonstrated very high \nperformance of 1.017 PFlop\/s in single precision using 4,000 GPUs along with 16,000 CPU cores on TSUBAME 2.0. \nNevertheless, a set of new and more advanced optimization techniques introduced in the Chemora\\xspace framework as \nwell as its capabilities to generate highly efficient multi-GPU stencil computing codes from a high-level \nproblem description make this framework even more attractive for users of large-scale hybrid systems.\n\nPhysis \\cite{Physis} addresses the problem of dividing the domain in\nall dimensions, and is these days\nseen as one of the most efficient frameworks for stencil\ncomputations over regular multidimensional Cartesian\ngrids in distributed memory environments.\nThe framework in its current state, however, does not divide\nthe domain automatically; this has to be done manually\nat launch time.\nNevertheless, Physis achieves very good scaling by taking\nadvantage of memory transfers overlapped with computations.\nStencil computations are defined in the form of C-based functions (or \\emph{kernels})\nwith the addition of a few special macros that allow accessing values at grid points.\nThe framework also uses CUDA streams that allow for parallel execution\nof multiple kernels at the same time; e.g.\\ regular and boundary kernels\nmay be executed in parallel.\nData dependencies between stencil points are resolved statically,\nhence must be known beforehand, at compile time.\nThe authors put a special emphasis on ease of use, and\nindeed the time needed to write an application in Physis is relatively short.\nThis framework was evaluated using three benchmark programs running on \nthe TSUBAME~2.0 supercomputer, and proved\nto generate scalable code for up to 256 GPUs.\nBelow, we compare Chemora\\xspace with its dynamic compilation and \nauto-tuning methods to Physis, and show that Chemora\\xspace outperforms Physis\nin the area of automatically generated\ncode for GPU clusters.\n\n\\subsection{Contributions}\n\\label{sec:contribution}\nThis paper makes the following contributions:\n\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item An overview of the Chemora\\xspace framework for generating hybrid\n  CPU\\slash GPU cluster code from PDE descriptions is presented and\n  its performance is characterized.\n\n\\item A language for expressing differential equation models of\n  physical systems suitable for generating hybrid cluster simulation\n  code (based on the existing \\term{Kranc} code-generation package), was developed.\n\n\\item Model-based GPU tile\\slash thread configuration optimization\n  techniques were developed, enabling the exploration of a large\n  search space and the use of dynamic compilation (performed once on\n  the chosen configuration).\n\n\\item Automatic hybrid execution GPU\\slash CPU data staging techniques\n  were developed (the \\term{accelerator} module).\n\n\\item GPU tuning techniques were developed for large kernel codes,\n  such as register-pressure sensitive configuration.\n\n\\item The first demonstration binary black hole simulations using GPUs in full GR\n  were presented.  Since Chemora has not yet been applied to the\n  Carpet AMR driver, these are not suitable for production physics,\n  but prove that existing codes used in numerical relativity can be\n  adapted to Chemora.\n\\end{itemize}\n\n\\section{Chemora\\xspace Framework}\n\nChemora\\xspace takes a physics model described in a high level \n\\emph{Equation Description Language} (EDL) and generates highly optimized code suitable \nfor parallel execution on heterogeneous systems.\nThere are three major components in Chemora\\xspace:\nthe Cactus-Carpet computational infrastructure, \nCaKernel programming abstractions, and the Kranc\ncode generator. Chemora\\xspace is portable to many\noperating systems, and adopts widely-used parallel programming \nstandards (MPI, OpenMP and OpenCL) and models (vectorization and CUDA\\@).\nAn architectural view of the Chemora\\xspace framework is shown in\nFigure~\\ref{fig:chemora_arch}. We describe the individual components below.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.5\\textwidth]{figs\/Chemora}\n\\caption{An architectural view of Chemora\\xspace. Chemora\\xspace consists of three major\ncomponents: The Cactus-Carpet computational infrastructure, CaKernel\nprogramming abstractions, and the Kranc code generator. Chemora\\xspace takes a physics\nmodel described in a high level Equation Description Language and \nproduces highly optimized code suitable for parallel execution on \nheterogeneous systems.}\n\\label{fig:chemora_arch}\n\\end{figure}\n\n\\subsection{Cactus-Carpet Computational Infrastructure}\n\\label{sec:cactus}\n\nThe Cactus computational framework is the foundation of Chemora\\xspace.\nCactus~\\cite{Goodale02a, cactusweb} is an open-source, modular,\nhighly-portable programming environment for collaborative research\nusing high-performance computing. Cactus is distributed with a generic\ncomputational toolkit providing parallelization, domain decomposition,\ncoordinates, boundary conditions, interpolators, reduction operators,\nand efficient I\/O in different data formats. More than 30\ngroups worldwide are using Cactus for their research work in cosmology,\nastrophysics, computational fluid dynamics, coastal modeling, quantum\ngravity, etc. The Cactus framework is a vital part of the Einstein\nToolkit~\\cite{Loffler:2011ay, EinsteinToolkit:web},\nan NSF-funded collaboration enabling a\nlarge part of the world-wide research in numerical relativity by\nproviding necessary core computational tools as well as a common\nplatform for exchanging physics modules. Cactus is part of the\nsoftware development effort for Blue Waters, and in particular the\nCactus team is working with NCSA to produce development interfaces and\nparadigms for large scale simulation development.\n\nOne of the features of Cactus relevant in this context is that it\nexternalizes parallelism and memory management into a module (called\na \\emph{driver}) instead of providing it itself,\nallowing application modules (called \\emph{thorns}) to function mostly\nindependently of the system architecture. Here we employ the\n\\emph{Carpet} driver~\\cite{Schnetter-etal-03b, Schnetter06a,\n  carpetweb} for MPI-based parallelism via spatial domain\ndecomposition. Carpet provides adaptive mesh refinement (AMR) and\nmulti-block capabilities\\footnote{We do not use these capabilities in\n  the examples below.}, and has been shown to scale to more than\n16,000 cores on current NERSC and XSEDE systems.\n\nIn the typical Cactus programming style for application modules, these\nmodules consist either of \\emph{global} routines (e.g.\\ reduction or\ninterpolation routines), or \\emph{local} routines (e.g.\\ finite\ndifferencing kernels). Local routines are provided in the form of\nkernels that are mapped by the driver onto the available resources.\nAt run time, a schedule is constructed, where Cactus orchestrates the\nexecution of routines as well as the necessary data movement\n(e.g.\\ between different MPI processes).\nThis execution model is both easy to understand for application\nscientists, and can lead to highly efficient simulations on large\nsystems. Below, we refine this model to include accelerators\n(e.g.\\ GPUs) with separate execution cores and memory systems.\n\n\\subsection{CaKernel Programming Abstractions}\n\\label{sec:cakernel}\nThe Chemora\\xspace programming framework uses the CaKernel\n\\cite{parco11, ppopp11, sciprog11}, a set of high level programming\nabstractions, and the corresponding implementations.\nBased on the Cactus-Carpet computational infrastructure,\nCaKernel provides two major sets of programming abstractions:\n(1) \\emph{Grid Abstractions} that represent the dynamically \ndistributed adaptive grid\nhierarchy and help to separate the application development from the\ndistributed computational domain;\n(2) \\emph{Kernel Abstractions} that enable automatic generation of numerical\nkernels from a set of highly optimized templates and help to separate the\ndevelopment, scheduling, and execution of numerical kernels.\n\n\\subsubsection{Grid Abstractions}\nThe Cactus flesh and the Cactus computational toolkit contain a collection\nof data structures and functions that \ncan be categorized into the following three grid abstractions, which commonly appear\nin high level programming frameworks for parallel block-structured\napplications~\\cite{parabrow96}:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item The \\emph{Grid Hierarchy (GH)} represents the distributed adaptive GH\\@.\nThe abstraction enables application developers to create, operate and destroy\nhierarchical grid structures. The regridding and partitioning operations on a grid\nstructure are done automatically whenever necessary. In Cactus, grid operations\nare handled by a driver thorn which is a special module in Cactus.\n\\item A \\emph{Grid Function (GF)} represents a distributed data structure\ncontaining one of the variables in an application. Storage, synchronization, arithmetic,\nand reduction operations are implemented for the GF by standard thorns. The\napplication developers are responsible for providing routines for\ninitialization, boundary updates, etc.\n\\item The \\emph{Grid Geometry (GG)} represents the coordinates, bounding boxes,\nand bounding box lists of the computational domain. Operations on the GG, such\nas union, intersection, refine, and coarsen are usually implemented in a driver\nthorn as well.\n\\end{itemize}\n\n\\subsubsection{Kernel Abstractions}\nThe kernel abstractions enable automatic code generation with a set of highly optimized\ntemplates to simplify code construction. The definition of a kernel requires\nthe following three components:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item A \\emph{CaKernel Descriptor} describes one or more numerical\n  kernels,\n \n  dependencies, such as grid functions and parameters\n  required by the kernel, and grid point relations with its neighbors.\n  the information provided in the descriptor is then used to generate\n  a kernel frame (macros) that performs automatic data fetching,\n  caching and synchronization with the host.\n\\item A \\emph{Numerical Kernel} uses kernel-specific auto-generated\n  macros. The function may be generated via other packages (such as\n  Kranc), and operates point-wise.\n\\item The \\emph{CaKernel Scheduler} schedules CaKernel launchers and\n  other CaKernel functions in exactly the same way as other Cactus\n  functions. Data dependencies are evaluated and an optimal strategy\n  for transferring data and performing computation is selected\n  automatically.\n\\end{itemize}\nThese kernel abstractions not only enable a simple way to write and execute\nnumerical kernels in a heterogeneous environment, but also enable lower-level\noptimizations without modifying the kernel code itself.\n\n\\subsubsection{Hardware Abstraction}\nCaKernel provides an abstraction of the hardware architecture, and\nChemora code is generated on top of this abstraction. The high level\nproblem specification in the Chemora framework may thus remain\nindependent of the architecture. The support for new architectures is\nthe responsibility of the Chemora developers, and thus it is\ntransparent to the end-user, who should not need to significantly\nmodify their code once the underlying CaKernel implementation has been\nmodified.\n\n\\subsection{Describing a Physics Model}\nProgramming languages such as C or Fortran offer a very low level of\nabstraction compared to the usual mathematical notation. Instead of\nrequiring physicists to write equations describing PDEs at this level,\nwe introduce EDL, a\ndomain-specific language for specifying systems of PDEs as well as\nrelated information (initial and boundary conditions, constraints,\nanalysis quantities, etc.) EDL allows equations to be specified independent\nof their discretization, allows abstract index notation to be used as a\ncompact way to write vectors and tensors, and does not limit\nthe options for memory layout or looping order. For Chemora\\xspace, we designed EDL\nfrom scratch instead of piggybacking it onto an existing language\nsuch as Mathematica, Haskell, or C++ so that we could choose a syntax\nthat is easily understood by domain scientists, i.e.\\ physicists and\nengineers.\n\nEDL has a very simple syntax, similar to C, but extended with a\nLaTeX-like syntax for abstract index notation for vectors and tensors.\nSample \\ref{fig:edl} shows as an example the main part of specifying the\nscalar wave equation in a fully first order form (assuming, for\nsimplicity, the propagation speed is $1$.) In addition to specifying\nthe equations themselves, EDL supports constants, parameters,\ncoordinates, auxiliary fields, and conditional expressions.\n\n\\begin{lstlisting}[escapechar=!, caption={Example showing (part of) \nthe scalar wave equation written in \\emph{EDL}, a language designed to describe PDEs. A LaTeX-like\n  syntax allows a compact notation for vectors and tensors. Additional\n  annotations (not shown here) are needed to complete the\n  description.}, label=fig:edl]\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{Init}!\n  u   = !\\color{cyan}{0}!\n  rho = A exp(!\\color{cyan}{-1\/2}! (r\/W)**!\\color{cyan}{2}!)\n  v_i = !\\color{cyan}{0}!\n!\\color{ForestGreen}{end calculation}!\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{RHS}!\n  D_t u   = rho\n  D_t rho = delta^ij D_i v_j\n  D_t v_i = D_i rho\n!\\color{ForestGreen}{end calculation}!\n\n!\\color{ForestGreen}{begin calculation}! !\\color{blue}{Energy}!\n  eps = !\\color{cyan}{1\/2}! (rho**!\\color{cyan}{2}! + delta^ij v_i v_j)\n!\\color{ForestGreen}{end calculation}!\n...\n\\end{lstlisting}\n\nIn addition to describing the system of equations, EDL makes it possible\nto specify a particular discretization by specifying sets of finite\ndifferencing stencils.\nThese\nstencil definitions remain independent of the equations themselves.\n\nThe \\emph{Kranc} code-generation package (see section\n\\ref{sec:kranc}), written in Mathematica and\ndescribed below, has been enhanced in Chemora\\xspace to accept EDL as its input\nlanguage.  Via a J\/Link interface to the Piraha PEG \\cite{brandt2010piraha} Java\nparsing library, the EDL is parsed into Mathematica expressions\nequivalent to those traditionally used as input to Kranc.  The formal\ngrammar which defines the syntax of the language is available as part\nof the Kranc distribution, should other tools need to parse EDL files.\n\nIn spite of its apparent simplicity, the high-level description in EDL\ncaptures everything that is needed to create a complete Cactus module.\nMetadata such as variable declarations, schedule items, and parameter\ndefinitions are extracted from EDL, and implementation choices such as\nmemory layout and loop traversal order are made automatically or even\ndynamically at run time (see below).\n\nKranc is written in Mathematica, and prior to Chemora\\xspace was used by\nwriting a script in the Mathematica language to set up data structures\ncontaining equations and then call Kranc Mathematica functions to\ngenerate the Cactus module.  This allowed great flexibility, but at\nthe same time required users to know the Mathematica language, which\nin several ways is idiosyncratic and is unfamiliar to many users.\nAdditionally, the use of an imperative language meant that Kranc was\nunable to reason about the input script in any useful manner (for\nexample for the purpose of reporting line numbers where errors were\nfound).  A new, simple, declarative domain-specific language was\ntherefore created which allowed a concise expression of exactly the\ninformation needed by Kranc.  Existing languages familiar to the\nmajority of scientists (C, Fortran, Perl, Python) introduce a wide\nvariety of features and semantics unnecessary for our application, and\nnone of these are suitable for expressing equations in a convenient\nmanner.  The block structure of EDL was inspired by Fortran, the\nexpression syntax by C, and the index notation for tensors by LaTeX.\nWe feel that the language is simple enough that it can be learned very\nquickly by reference to examples alone, and that there is not a steep\nlearning curve.\n\nBy providing a high-level abstraction for an application scientist,\nthe use of EDL substantially reduce the time-to-solution, which includes:\nlearning the software syntax, development time from a given system of\nequations to machine code, its parallelization on a heterogeneous\narchitecture, and finally its deployment on production clusters.  It\nalso eliminates many potential sources of errors introduced by low\nlevel language properties, and thus reduces testing time. For further\ninformation about the total time-to-solution, see\n\\cite{hochstein2005parallel}.  \n\n\\subsection{Automated Code Generation with Kranc}\n\\label{sec:kranc}\n\nTranslating equations from a high-level mathematical notation into C\nor Fortran and discretizing them manually is a tedious, error-prone\ntask. While it is straightforward to do for simple algorithms, this\nbecomes prohibitively expensive for complex systems.\nWe identify two levels\nof abstraction.  The first is between the continuum equations and the\napproximate numerical algorithm (discretization), and the second is\nbetween the numerical algorithm and the computational implementation.\n\nWe employ \\emph{Kranc}~\\cite{Husa:2004ip, Lechner:2004cs, Kranc:web}\nas a code-generation package which implements these abstractions. The\nuser of Kranc provides a \\emph{Kranc script} containing a section\ndescribing the partial differential equations to solve, and a section\ndescribing the numerical algorithm to use. Kranc translates this\nhigh-level description into a complete Cactus module, including C++ code\nimplementing the equations using the specified numerical method, as\nwell as code and metadata for integrating this into the Cactus\nframework.\n\nBy separating mathematical, numerical, and computational aspects,\nKranc allows users to focus on each of these aspects separately\naccording to their specialization. Although users can write Kranc\nscripts directly in Mathematica, making use of the EDL \nshields them from\nthe (sometimes arcane) Mathematica syntax (because they are required to follow a\nstrict pattern for specifying PDEs) and provides them with much more\ninformative (high-level) error messages.  Either the traditional Mathematica language, or the new EDL language, can be used\nwith Chemora for GPU code generation.\n\nKranc is able to:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item accept input with equations in abstract index notation;\n\\item generate customized finite differencing operators;\n\\item generate codes compatible with advanced Cactus features such as\n  adaptive mesh refinement or multi-block systems;\n\\item check the consistency with non-Kranc generated parts of the\n  user's simulation;\n\\item apply coordinate transformations, in particular of derivative\n  operators, suitable for multi-block systems\n  (e.g.~\\cite{Pollney:2009yz});\n\\item use symbolic algebra based on the high-level description of the\n  physics system to perform optimizations that are inaccessible to the\n  compiler of a low-level language;\n\\item implement transparent OpenMP parallelization;\n\\item explicitly vectorize loops for SIMD architectures (using\n  compiler-specific syntaxes);\n\\item generate OpenCL code (even independent of the CaKernel framework\n  described below);\n\\item apply various transformations and optimizations (e.g.\\ loop\n  blocking, loop fission, multi-threading, loop unrolling) as\n  necessary for the target architecture.\n\\end{itemize}\n\n\\subsubsection{Optimization}\n\nIt is important to note that Kranc does not simply generate the source code for\na specific architecture that\ncorresponds $1:1$ to its input. Kranc has many of the features of a traditional compiler, including\na front-end, optimizer, and code generator, but the code generated is C++\/CaKernel\/CUDA rather than\nmachine code.\n\nThe high-level optimizations currently implemented act on discretized\nsystems of equations, and include the following:\n\\begin{itemize}\n\\setlength{\\itemsep}{-2pt}\n\\item Removing unused variables and expressions;\n\\item Transforming expressions to a normal form according to\n  mathematical equivalences and performing \\emph{constant folding};\n \n \n\\item\n \n   \n    Introducing temporaries to perform \\emph{common subexpression elimination};\n\\item Splitting calculations into several independent calculations\n \n \n \n  to reduce the instruction cache footprint and data cache pressure\n  \\emph{(loop fission)};\n\\item Splitting calculations into two, the first evaluating all\n  derivative operators (using stencils) storing the result into\n  arrays, the second evaluating the actual RHS terms but not using any\n  stencils. This allows different loop optimizations to be applied to\n  each calculation, but requires more memory bandwidth \\emph{(loop\n    fission)}.\n\\end{itemize}\n\nNote in the above that a\n\\emph{calculation} is applied to all grid points, and thus either\nloops over or uses multiple threads to traverse all grid points.\nAlso note that both the high-level and the low-level optimizations could in principle\nalso be performed by an optimizing compiler. However, none of the\ncurrently available compilers for HPC systems are able to do so,\nexcept for very simple kernels. We surmise that the reason for this is\nthat it is very difficult for a compiler to abstract out sufficient\nhigh-level information from code written in low-level languages\nto prove that these\ntransformations are allowed by the language standard. A programmer is\nforced to make many (ad-hoc) decisions when implementing a system of\nequations in a low-level language such as C or C++, and the compiler\nis then unable to revert these decisions and fails to optimize the\ncode.\n\nIt is surprising to see that these optimizations -- which are in\nprinciple standard transformations among compiler builders -- are (1)\nable to significantly improve performance, are (2) nevertheless not\napplied by current optimizing compilers, and are yet (3) so easily\nimplemented in Mathematica's language, often requiring less than a\nhundred lines of code.\n\nKranc is a developed and mature package. Since its conception in 2002,\nit has been continually developed to adapt to changing computational\nparadigms.\nKranc is not just a theoretical tool. In the Einstein\nToolkit~\\cite{Loffler:2011ay},\nKranc is used to generate a highly efficient\nopen-source implementation of the Einstein equations as well as\nseveral analysis modules.\nAll of the above features are used heavily by users of the Toolkit,\nand hence have been well-tested on many production architectures,\nincluding most systems at NERSC or in XSEDE\\@.\n\n\\subsubsection{Debugging the Numerical Code}\nIt is also important to note that Chemora significantly reduces the time\nrequired to debug the application.  The recommended approach for development\nusing Chemora is that the user's Kranc script is considered the canonical\nsource, and only this should be modified during development. The generated code\nshould not be modified, as it will be completely regenerated each time Kranc is\nrun, so any hand-modifications of the generated code will be lost. Unlike when\nwriting a C++ program, every successfully-compiled Kranc script should lead to\ncorrect computational (though not necessarily physical) code. Hence the errors\nare limited to the application domain, for example an incorrect equation is solved.\nSimilarly, use of a source-code level debugger is\nnot typical when working with Kranc, as the ``debugging'' happens at the level of\nthe scientific results (e.g. convergence tests and visualisation) rather than\nat the level of programmatic bugs in the generated code. \nAs such, Kranc is treated as a black box by the application scientist,\nmuch as a compiler would be.\n\n\\subsubsection{Code Generation for CaKernel}\n\nIn order to use Kranc as a component of Chemora\\xspace, the code-generation\nbackend was modified, and\nCaKernel (see section \\ref{sec:cakernel}) was added as an output target. This\nchange is essentially invisible to the application developer; there is merely\nan additional option to generate CaKernel code rather than C++ or OpenCL code.\nEach calculation is then annotated with whether\nit runs on the host (CPU) or the device (GPU\\@).  Kranc\nalso creates all metadata required by CaKernel.\nAdditionally, the new EDL language frontend was added to Kranc.\n\n\\subsubsection{Hybrid Codes}\n\nSince GPU accelerators have to be governed by CPU(s), it is natural to attempt\nto exploit them by employing \\emph{hybrid codes}. In this case, Kranc,\ngenerates both CPU and CaKernel codes from the same script. At\nrun time, each MPI process checks whether to attach itself to a GPU\nand perform its calculations there, or whether to use the CPU for\ncalculations. \n\nThis mechanism works in principle; however, as the Cactus driver\ncurrently assigns the same amount of work to each\nMPI process (uniform load balancing), the large performance disparity between\nCPU and GPU has led to only minimal performance gains so far. We expect this issue\nto be resolved soon.\n\n\\subsection{CaKernel GPU Code Optimization}\n\n\\bitbucket{\nChemora\\xspace optimizes at multiple levels of abstraction, starting with the\nKranc scripts, through domain decomposition and data staging,\nGPU code generation, down to execution configuration and cache\nsettings. Optimizations are performed at the appropriate level, but\ntake advantage of information provided by higher levels.}\n\n\\bitbucket{\nOne goal of Chemora\\xspace is to generate efficient code starting from a\nhigh-level description and without requiring the user to tune for\nefficient execution. CaKernel achieves this by using\nKranc-provided and run-time information to set such important\nparameters as tile shape and GPU cache settings.}\n\nThe CaKernel code generated by Kranc consists of\n\\term{numerical kernels}, routines that operate on a single grid\npoint. The CaKernel parts of Chemora\\xspace use Kranc-provided and\nrun time information to generate efficient GPU executables\nfrom the numerical kernels, without requiring the user to set tuning\nparameters. At build time, numerical kernels are wrapped with\n\\term{kernel frames}, code that implements data staging and iteration,\nproducing a source code package that is compressed and compiled into\nthe Cactus executable. At run time, CaKernel makes use of\ninformation about the kernels provided by Kranc as well as user\nparameters and information on the problem size to choose tiling, etc.\nWith this information, the code package is extracted,\nedited, compiled, loaded to the GPU, and run. This dynamic process\nresults in lightweight GPU code that makes efficient use of GPU\nresources, including caches.\nCaKernel uses several techniques to generate efficient GPU code which\nwe shall elaborate in the following subsections.\n\n\\bitbucket{\n\\term{Numerical kernels} are routines that update a single grid point,\nthey reference their own and neighboring grid points through\n\\term{indexing functions} which are restricted to purely relative\naccesses. To achieve maximum portability, numerical kernels should be\nwritten in a compatible subset of CUDA C, OpenCL, and C++, however\nthis is not enforced and so a kernel intended for CUDA execution can\nuse CUDA-specific features. The routines have Cactus parameters and\nvariables describing grid point location defined in their\nnamespaces. The numerical kernel will ultimately run as a thread on an\naccelerator. Assignment of grid points to threads is a hierarchical\nprocess. At the top, it is the responsibility of a Cactus thorn to\nassign a \\term{local section} of the grid to an accelerator device\n(actually to an MPI processes associated with the accelerator). It is\nthe important responsibility of the kernel frame to assign local grid\npoints to threads, such an assignment will be referred to as a\n\\term{tile selection}.\n\nData staging and iteration over the local grid are performed by the\n\\term{kernel frame} code, which wraps the numerical kernel. The\nparticular type of kernel frame and its options are specified in the\nCaKernel descriptor. There are separate kernel frame types for the\nboundary region and interior points, and for static and dynamic\ncompilation. For the non-boundary types, the descriptor indicates the\nextent of the stencil and tile size and caching hints. The descriptor\nalso identifies the grid functions and other data needed by the\nrespective kernel.}\n\n\\bitbucket{\n\\subsection{Optimization Challenges}\n\nChemora\\xspace was designed to generate efficient hybrid system code, without\nthe need for user tuning, from problems described in terms of\ndifferential equations, in particular those of the complexity of the\nEinstein equations. These are characterized by evolution expressions\nhaving thousands of terms, sometimes rendering standard performance\ntuning guidelines useless. Several innovations needed to achieve\nperformance are described below. They include dynamic tile size\nselection, lightweight kernel generation via dynamic compilation, the\nuse of integrated GPU performance monitoring, and source-level code\ntransformations.}\n\n\\bitbucket{\n\\subsection{Kernel Frame Types}\n\nCaKernel provides manual and automatic means of tile selection; manual\nselection is described briefly below, automatic tile selection, an\nimportant factor in achieving high performance, is described in a\nfollowing section.\n\n\\def\\tile#1,#2,#3;{\\left<#1,#2,#3\\right>}\n\nFor some of the kernel frames tile selection is specified manually by\na three-component \\param{tile} parameter. Tile $\\tile x,y,z;$\nindicates that the thread block should consist of $xy$ threads and\nshould operate on a $x\\times y\\times z$ section of the grid, where\n$x$, $y$, and $z$ are integers. Each thread operates on $z$ grid\npoints. The kernel frame code on the host will launch a kernel\nconsisting of enough such blocks to cover the local section of the\ngrid. The kernel frame code on the device, which wraps the numerical\nkernel, iterates over $z$ and computes the global indices of the grid\nfunction corresponding to the thread at each iteration.\n\nThe CaKernel descriptor is also used to specify which of the grid\nfunctions should be staged in shared memory. These choices do not\naffect the code in the numerical kernel, in particular the same\nindexing function is used whether or not the variable is in shared\nmemory. The kernel frame will load the selected grid functions into\nshared memory and update them as threads iterate. Some kernel frame\ntypes use registers rather than shared memory when stencil patterns\nallow.\n}\n\n\\subsubsection{Stencils and Dynamic Tile Selection}\n\n\\bitbucket{\nClassic CPU loop tiling involves choosing an iteration strategy to maximize\ncache reuse, see for example \\cite{rivera00}. Cache reuse is important\nfor GPU tiling too, however because of latency hiding with\nmultithreading, a primary goal can be minimization of the total data\nrequest size. Many tiling strategies for stencil computations on GPUs\nhave been reported~\\cite{renganarayana07,datta08,meng09,unat11}, the\ncommon goal being to make best use of the limited amount of high-speed\nmemory by taking advantage of the repeated access to data elements.}\n\nCPU and GPU tiling has been extensively studied, though often limited\nto specific stencils,\n\\cite{renganarayana07,datta08,meng09,unat11}. The goal for CaKernel\nwas to develop an automatic tile selection scheme that would work well\nnot just for a few specific stencils, but any stencil pattern the user\nrequested. The tile selection is based not just on the stencil shape\nbut also on the number of grid variables and on the shape of the local\ngrid. The resulting tile makes best use of the cache and potentially\nregisters for minimizing data access. The discussion below provides\nhighlights of the scheme; details will be more fully reported\nelsewhere.\n\nThe following discussion uses CUDA terminology, see \\cite{cuda40,cudatune40}\nfor background. The term \\term{tile} will be used here to mean the portion of\nthe grid assigned to a CUDA block. In GPUs, higher \\term{warp} occupancy means\nbetter latency hiding introduced by common memory access. That can\nbe achieved with multiple blocks, but to maximize L1 cache reuse\nCaKernel will favor a single large block, the maximum block size\ndetermined by a trial compilation of a numerical kernel. Within that\nblock size limit a set of candidate tile shapes are generated using\nsimple heuristics, for example, by dividing the $x$ dimension of the\nlocal grid evenly, (by 1, 2, $3,\\ldots$) and then for each tile $x$\nlength find all pairs of $t_y$ and $t_z$ lengths that fit within the\nblock limit, where $t_x$, $t_y$, and $t_z$ are the tile shape in units\nof grid points.\n\nGiven a candidate tile shape, the number of cache lines requested\nduring the execution of the kernel is computed. Such a \\term{request\n  size} is computed under the \\term{ordering assumption} that memory\naccesses are grouped by grid function and dimension (for stencil\naccesses). As an illustration, if the assumption holds a possible\naccess pattern for grid functions $g$ and $h$ is $g_{0,1,0}$,\n$g_{0,2,0}$, $g_{1,0,0}$, $h_{0,0,0}$, while the pattern $g_{0,1,0}$,\n$h_{0,0,0}$, $g_{1,0,0}$, $g_{0,2,0}$ violates the assumption because\n$h$ is between $g$'s accesses and for $g$ a dimension-$x$ stencil\naccess interrupts dimension-$y$ accesses.\n\nRequest sizes are computed under different cache line \\term{survival\n  assumptions}, and the one or two that most closely match the cache\nare averaged. One survival assumption is that all lines survive (no\nline is evicted) during an iteration in which case the request size is\nthe number of distinct lines the kernel will touch, after accounting\nfor many special cases such as alignment. Another survival assumption\nis that data accessed using stencils along one dimension (say, $x$)\nwill not survive until another dimension access (say, $y$)\n  (e.g., common lines might be evicted). The particular assumption to\nuse is based on the size of the tile and cache.\n\nSkipping details, let $r$ denote the overall request size. An\n\\term{estimated cost} is computed by first normalizing $r$ to the number of\ngrid points, $r\/It_xt_yt_z$, where $I$ is the number of iterations\nperformed by threads in the tile. To account for the lower execution\nefficiency with smaller tiles, a factor determined empirically as\n$1\/(1+256\/t_xt_yt_z)$ is used. The complete expression for the\nestimated cost\nis $\\sigma=(r\/It_xt_yt_z)\/(1+256\/t_xt_yt_z)$.\nThe tile with the lowest estimated cost is selected.\n\nTiles chosen using this method are often much longer in the $x$\ndirection than other dimensions, because the request size includes the\neffect of partially used cache lines. If a stencil extends in all three\ndimensions and there are many grid functions, the tile chosen will be\n``blocky''. If there are fewer grid functions, the tile will be\nplate-shaped, since the request size accounts for cache lines that survive\niterations in the axis orthogonal to the plate. The tile optimization\nis performed for the tile shape, but not for the number of iterations\nwhich so far is chosen empirically.\n\n\\subsubsection{Lightweight Kernel Generation}\n\nA number of techniques are employed to minimize the size of the GPU\nkernels. Dynamic compilation using program parameters and tile shape,\nseen by the compiler as constants, was very effective. Another\nparticularly useful optimization given the large size of the numerical\nkernels is \\term{fixed-offset loads}, in which a single base address\nis used for all grid functions. Normally, the compiler reserves two\n32-bit registers for the base address of each grid function, and uses\ntwo additional registers when performing index arithmetic since the\noverhead for indexing is significant. Fortunately, the Fermi memory\ninstructions have a particularly large offset, at least 26 bits based\non an inspection of Fermi machine code (which is still not well\ndocumented). (An offset is a constant stored in a memory instruction,\nit is added to a base address to compute the memory access address.)\nWith such generous offsets, it is possible to treat all grid\nfunctions (of the same data type) as belonging to one large array.\n\n\\subsubsection{Fat Kernel Detection}\n\nSome numerical kernels are extremely large, and perform very poorly\nusing standard techniques, primarily due to very frequent register\nspill\/reload accesses. \nCaKernel identifies and provides\nspecial treatment for such kernels. The kernels can be automatically\nidentified using CaKernel's integrated performance monitoring code by\nexamining the number of local cache misses. (Currently, they are\nautomatically identified by examining factors such as the number\nof grid functions.)\nSuch fat kernels are handled using two techniques: they are launched\nin small blocks of 128 threads, and source-level code restructuring\ntechniques are applied. Launching in small blocks relieves some\npressure on the L1 cache. (A dummy shared memory request prevents\nother blocks from sharing the multiprocessor.) The source code\nrestructuring rearranges source lines to minimize the number of live\nvariables; it also assigns certain variables to shared memory.\n\n\\subsubsection{Integrated Performance Monitoring}\n\nCaKernel provides performance monitoring using GPU event counters,\nread using the NVIDIA Cupti API\\@. If this option is selected, a\nreport on each kernel is printed at the end of the run. The report\nshows the standard tuning information, such as warp occupancy and\nexecution time, and also cache performance data. To provide some insight\nfor how well the code is performing, the percentage of potential\ninstruction execution and memory bandwidth used by the kernel is\noutput. For example, a 90\\% instruction execution potential would\nindicate that the kernel is close to being instruction bound. We plan\nto use these data for automatic tuning, e.g.\\ to\nbetter identify fat kernels.\n\n\\subsubsection{Effectiveness of Low-Level Optimizations}\n\nMost of the optimizations are highly effective, including dynamic\ncompilation and fixed-offset loads.\nThere are two areas where some\npotential has been left unexploited: tile shape, and the handling of\nfat kernels. \n\nAutomatic tile size selection greatly improves performance over\nmanually chosen tile sizes, however kernels are still running at just\n20\\% of execution utilization while exceeding 50\\% of available memory\nbandwidth, suffering L1 cache miss ratios well above what was\nexpected. The primary weakness in tile selection is assuming an\nordering of memory accesses that does not match what the compiler\nactually generates. (The compiler used was NVIDIA \\code{ptxas} release\n4.1 V0.2.1221.) For example, for a kernel with a $5\\times5\\times5$\nstencil and a $102\\times3\\times3$ tile, the compiler interleaves $n$\naccesses along the $y$ and $z$ axes. The cache can hold all grid\npoints along one axis (273 cache lines would be needed in this\nexample) but not along two (483 cache lines). Several solutions have\nbeen identified, including modifying the model to match compiler\nbehavior, waiting for a better compiler, restructuring the code to\nobtain a\nbetter layout, or rescheduling the loads at the object-file level.\n\nOne of the kernels performing the last step in the time evolution\nhas over 800 floating point\ninstructions in straight-line code. This executes at only 14\\%\ninstruction utilization, suffering primarily from L1 cache misses on\nregister spill\/reload accesses. We address this via fixed offsets and\nother dynamic compilation techniques that reduce register pressure. A\ncombination of source-level scheduling and shared memory use yielded\nfrom 5\\% to 10\\% better performance, and there seems to be a large\npotential for further improvement.\n\n\\subsection{Accelerator Framework}\n\\label{sec:accelerator}\n\nIn large, complex applications based on component frameworks such as\nCactus, GPUs and other accelerators are only useful to those\ncomponents which perform highly parallel arithmetic computations. As\nsuch, it is neither necessary nor useful to port the entire framework\nto run on GPUs -- in fact, much of the code in Cactus-based\napplications is not numerical, but provides support in the form of\norganizing the numerical data.\n\nOne approach to porting a component to run on a GPU is to identify the\nentry and exit points of that component, copy all required data to the\nGPU beforehand, and copy it back after the GPU computation.\nUnfortunately, such data transfer is prohibitively slow, and \nthe performance of this approach is not acceptable.\n\nInstead, we track which data (and which parts of the data) is read and\nwritten by a particular routine, and where this routine executes (host\nor GPU). Data is copied only when necessary, and then only those\nportions that are needed. Note that data is not only accessed for computations,\nbut also by inter-process synchronization and I\/O.\n\nThe metadata available for each Cactus component (or thorn) already contains\nsufficient information in its schedule description for such tracking,\nand during Chemora\\xspace we refined the respective declarations\nto further increase performance. This metadata needs to be provided\nmanually for hand-written thorns, but can be deduced automatically\ne.g.\\ by Kranc in auto-generated thorns.\n\nIn keeping with the Cactus spirit, it is a Cactus component (thorn\n\\term{Accelerator}) that tracks which parts of what grid functions are\nvalid where, and which triggers the necessary host--device copy\noperations that are provided by other, architecture-specific thorns.\n\n\\section{Case studies}\n\n\\subsection{Computing Resources}\nWe tested our framework on different computational systems.\nUnfortunately, clusters available to us at the time this paper was written were\ninsufficient for the purpose of challenging scaling tests.\n\\todo{Steve: Maybe we can we add Tienhe-1 and\/or Super Mike?}\n\n\\subsubsection{Cane}\n\\label{sec:canedesc}\n\n\\emph{Cane} is a heterogeneous cluster located at the Pozna\\'{n} Supercomputing\nand Networking Center.  Although it consists of 334 nodes, at the time we\nperformed the tests only 40 of them were available as\nthe cluster was still being set up.\nEach node is equipped with two AMD Opteron\u2122 6234 2.7GHz processors (with two\nNUMA nodes each; 12 cores per CPU), 64GB of main memory,\nand one NVIDIA M2050 GPU with 3GB of RAM\\@. The computational nodes\nare interconnected by InfiniBand QDR network with the fat-tree topology\n(32Gbit\/s bandwidth). CUDA 4.1 and gcc 4.4.5 were used for GPU and CPU code\ncompilation, respectively.\n\n\\subsubsection{Datura}\n\\label{sec:daturadesc}\n\n\\emph{Datura} is an CPU-only cluster at the Albert-Einstein-Institute in\nPotsdam, Germany. Datura has 200 nodes, each consisting of two Intel\nWestmere 2.666GHz processors with 6 cores and 24GB of memory.\nThe nodes are connected via QDR InfiniBand (40Gbit\/s bandwidth).\nWe used the Intel compilers version 11.1.0.72.\n\n\\subsection{CFD with Chemora\\xspace and Physis}\n\nWe employed a simple CFD (Computational Fluid Dynamics) benchmark\napplication to compare the performance of Chemora\\xspace and Physis.\nThis code solves the Navier-Stokes equations;\nfor details about the problem and\nits discretization see~\\cite{VOF_Hirt79,NASA_VOF2D_Torrey}, and for its\nimplementation in Cactus and CaKernel see~\\cite{parco11,sciprog11,ppopp11}.\nThe setup consists of three stencil kernels:\none that explicitly updates velocity values, one that iteratively solves \nthe conservation of mass (updating velocity and pressure), and one that\nupdates the boundary conditions.\nFor simplicity, we ran 4 iterations of the mass conservation kernel,\nand applied the boundary condition after each iteration. Although the CFD code\nwas written directly in CaKernel native language and its performance was\nalready reported along with our previous\nwork~\\cite{parco11,sciprog11,ppopp11}, we used\nCaKernel's new optimization facilities in this work.\nThese allowed us to obtain improved\nperformance compared to our previous results as well as compared to\nsimilar, publicly\navailable frameworks (e.g.\\ Physis).\n\nTo obtain statistically stable performance results, as many as 1000 iterations\nwere executed in each run. The CFD benchmark uses single-precision\nfloating-point data, which provides sufficient accuracy for this test case.\nBoth frameworks use the GPUs only for computation, and use CPUs only for\ndata transfer and management.\n\nFigure~\\ref{fig:sc12bench_cfd} compares the scalability of the \nframeworks in this CFD benchmark. The problem size of the weak scaling test\nfor each GPU was fixed at $256^3$, and \nthe performance was evaluated using 1 to 36 GPUs with two-dimensional\ndomain decompositions\nalong the $y$ and $z$ directions. We present results for\nthe best domain decompositions for each framework. The performance of both implementations increases\nsignificantly with increasing number of the GPU nodes. Numerous optimizations in Chemora\\xspace such as dynamic \ncompilation and auto-tuning allowed us to find the best GPU block size for\nthe domain size, and execute on the correct number of warps to limit the number of L1 cache \nmisses. As a result, for a single GPU, Chemora\\xspace obtained 90.5\nGFlop\/s, whereas Physis only obtained 43 GFlop\/s.\nThis gap may be also due to the fact that Physis does not make any use of\nshared memory on the GPUs.\n\nFigure~\\ref{fig:sc12bench_cfd} also compares\nthe performance of the two frameworks in a strong scaling test.\nThe problem size for this test was fixed at $656^3$. Both implementations\nscale up very well;\nChemora\\xspace achieved 270 GFlop\/s and 1055 GFlop\/s for 4 and 36 GPUs, respectively, \nwhereas Physis achieved 170 GFlop\/s and 965 GFlop\/s in the same configurations.\nThe parallel efficiency (when increasing the number of GPUs from \n4 to 36) is 43\\% and 63\\% for Chemora\\xspace and Physis, respectively.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{benchmark\/physis\/results\/both-cane-cfd}\n\\caption{Weak and strong scaling test comparing Chemora\\xspace and Physis\n  running on multiple nodes for the same CFD application.\n  Smaller numbers are better, and ideal scaling correspoonds to a\n  horizontal line.\n  Chemora\\xspace\n  achieves a higher per-GPU performance, whereas Physis shows a higher\n  strong scalability. Details in the main text.}\n\\label{fig:sc12bench_cfd}\n\\end{figure*}\n\n\\subsection{Binary Black Hole Simulations with Chemora\\xspace}\n\nWe demonstrate the integration of Chemora\\xspace technologies into our\nproduction-level codes by performing a  Numerical Relativity (NR)\nsimulation. This simulation of a binary black\nhole (BBH) merger event shows that our GPU-accelerated main evolution\ncode can be seamlessly integrated into the pre-existing CPU framework, and that\nit is not necessary to port the entire framework to the GPU\\@.  It also\ndemonstrates the use of the data management aspect of Chemora\\xspace, showing\nhow data is copied between the host and the device on demand.\nAnalysis modules\nrunning on the CPU can make use of data generated on the GPU\nwithout significant modification.\n\nOur production simulations differ from this demonstration only in\ntheir use of adaptive mesh refinement (AMR), which allows a much larger\ncomputational domain for a given computational cost.  This allows the\nsimulation of black hole binaries with larger separations, many more\norbits before merger, and hence longer waveforms when AMR is used.\n\nThe initial condition consists of two black holes on a quasi-circular\norbit about their common center of mass (``QC-0'' configuration). This\nis a benchmark configuration; in a production simulation, the black\nholes would have a much larger separation.\nThis configuration performs approximately one orbit before the energy\nloss due to gravitational wave emission cause the black holes to\nplunge together and form a single, highly-spinning black hole.\n\nGravitational waves are emitted from the orbiting and merging system.\nThese are evaluated on a sphere and decomposed into spherical\nharmonics.\nIt is this waveform which is used in gravitational wave detection.\n\nWe use a 3D Cartesian numerical grid $x^i \\in [-6.75, 6.75]^3$ with\n$270^3$ evolved grid points. To ensure a balanced domain decomposition\nwe run on 27 processes, corresponding to $90^3$ evolved points per\nprocess. This is the largest grid that fits in the 3 GB of GPU memory\non Cane, given the large number of grid variables required.\nAll calculations are performed in double precision.\nWe evolve the system using the \\code{McLachlan} code (see section\n\\ref{sec:science} above), using 8th order finite differencing and a\n3rd order Runge-Kutta time integrator.\n\nAny production Cactus simulation makes use of a large number of\ncoupled thorns; e.g.\\ this simulation contains 42 thorns. Most of\nthese do not need to be aware of the GPU, CaKernel, or the Accelerator\ninfrastructure. In our case, only \\code{McLachlan} and the\n\\code{WeylScal4} gravitational wave extraction thorns were running on a\nGPU\\@. Additional thorns, e.g.\\ tracking the location or shape of the\nblack holes, were run on the CPU\\@.\n\nWe use 27 nodes of the Cane cluster (see section~\\ref{sec:canedesc})\nwith one GPU per node. We do not run any CPU-only processes.\n\nFig.~\\ref{fig:bbh} shows the numerical simulation domain.  On the\n$x-y$ plane we project the $\\Psi_4$ variable which represents\ngravitational waves.  The black hole trajectories are shown as black\ncurves near the center of the grid; they end when the black holes\nmerge into a single black hole located at the center.\nThe sphere on which\nthe multipolar decomposition of the gravitational waves is performed\nis also shown.  In the insets, we show (a) the time evolution of the\n(dominant) $\\ell = 2, m = 2$ mode\nof the gravitational radiation computed on the sphere at $r = 4 M$,\nand (b) the (highly distorted) shape of the common apparent horizon\nformed when\nthe two individual black holes merge.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.50\\textwidth]{figs\/bbhsim}\n\\caption{Visualization of a binary black hole system}\n\\label{fig:bbh}\n\\end{figure}\n\nTable~\\ref{tbl:bbhtimers} shows a break-down of the total run time of\nthe BBH simulation. The routines labeled in bold face run on the\nGPU\\@. The times measured are averaged across all processes. The\n\\emph{Wait} timer measures the time processes wait on each other\nbefore an interprocessor synchronization. This encapsulates the\nvariance across processes for the non-communicating routines.\n\n\\begin{table}\n\\centering\n\\begin{tabular}{lr}\n        & \\hspace{-0.5cm}Percentage of \\\\\n  Timer & \\hspace{-0.5cm}total evolution time \\\\\n\\hline\n \\text{Interprocess synchronization} & 39\\%\\\\\n \\textbf{RHS advection} & 13\\%\\\\\n \\textbf{RHS evaluations} & 12\\%\\\\\n \\text{Wait} & 11\\%\\\\\n \\textbf{RHS derivatives} & 6\\%\\\\\n \\textbf{Compute Psi4} & 5\\%\\\\\n \\text{Multipolar decomposition} & 3\\%\\\\\n \\text{File output} & 3\\%\\\\\n \\text{BH tracking} & 3\\%\\\\\n \\text{Time integrator data copy} & 2\\%\\\\\n \\text{Horizon search} & 2\\%\\\\\n \\textbf{Boundary condition} & 1\\%\\\\\n \\text{BH tracking (data copy)} & 1\\%\\\\\n\\end{tabular}\n\\caption{Timer breakdown for the binary black hole\n  simulation.  Routines in bold face (48\\%) are executed on the GPU\\@.}\n\\label{tbl:bbhtimers}\n\\end{table}\n\nWe see that the interprocess synchronization is a significant portion\n(38\\%) of the total run time on this cluster. One reason for this is\nthat the large number of ghost zones (5) needed for partially-upwinded 8th order stencils\nrequire transmitting a large amount of data. This could likely be\nimproved by using a cluster with more than one GPU or more GPU memory\nper node, as this would reduce the relative cost of inter-process\ncommunication relative to computation.\n\n\\subsection{McLachlan Benchmark}\n\nWe used part of the binary black hole simulation as a weak-scaling\nperformance benchmark. We chose a local problem size that fitted into\nthe GPU memory of Cane (see section~\\ref{sec:canedesc}), corresponding\nto $100^3$ evolved points plus boundary and ghost zones. We ran\nthe benchmark on Cane (on GPUs) and Datura (on CPUs; see\nSec.~\\ref{sec:daturadesc}), using between 1 and 48 nodes.\nFigure~\\ref{fig:s12bench_ml} shows results comparing several\nconfigurations, demonstrating good parallel scalability for these core\ncounts. One of Cane's GPUs achieved about twice the performance\nof one of its CPUs, counting each NUMA node as a single\nCPU\\@.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{figs\/sc12bench_ml_weak_cane}\n\\caption{Weak-scaling test for McLachlan code performed on the Cane and\n  Datura clusters. ($n$)p(($m$)t) stands for $n$ processes per node\n  using $m$ threads each. (no) x-split stands for (not) dividing\n  domain along the $x$ axis. Smaller numbers are better, and ideal\n  weak scaling corresponds to a horizontal line. The benchmark scales\n  well on these platforms.}\n\\label{fig:s12bench_ml}\n\\end{figure*}\n\nAs a measurement unit we use time per grid point update per GPU (or\nCPU\\@). The best performance was achieved for a single GPU: 25\nGFlop\/s, which is 5\\% of the M2050 GPU's peak performance of 515\nGFlop\/s. On 40 nodes, we observed 50\\% scaling efficiency due to\nsynchronization overhead, and achieved a total performance of 500\nGFlop\/s.\n\nCPU performance tests were performed on both Cane and Datura. The\ntotal performance of the parallel OpenMP code, properly vectorized, was\nsimilar to the performance of a single GPU, with similar scaling factor.\n\nWe note that our floating point operation counts consider only those\noperations strictly needed in a sequential physics code, and e.g.~do\nnot include index calculations or redundant computations introduced by\nour parallelization. \\todo{Steve: I'm not quite sure what this\nparagraph is trying to say, it sounds like we're saying our code\nhas inefficiencies and we artificially removed their effects in\nour results.}\n\n\\section{Conclusion}\n\nWe have presented the Chemora\\xspace project, a component-based approach to \nmaking efficient use of current and future accelerator\narchitectures for high-performance scientific codes. \nAlthough the examples we present run on the GPU and use CUDA, \nour work is general and will be applied e.g.\\ to OpenCL and other approaches in\nfuture work. Using Chemora\\xspace, a scientist can \ndescribe a problem in terms of a system of PDEs in our Equation\nDescription Language.\nA module for the Cactus framework is then\ngenerated automatically by Kranc for one or more\ntarget architectures.\nKranc applies many optimizations \nat code-generation time, making use of symbolic algebra,\nand the resulting source code can then be compiled on a\ndiverse range of machines (taking advantage of the established\nportability of Cactus and the availability of CUDA as a uniform GPU\nprogramming environment).  At run-time, the CUDA code is recompiled\ndynamically to enable a range of runtime optimizations.\n\nWe have presented two case studies. The first is a \nComputational Fluid Dynamics (CFD) code, and we demonstrated weak scaling\nusing our infrastructure running on GPUs.  We also used\nthe Physis framework for this same problem and compared the scaling.\nChemora\\xspace has comparable\nor higher performance, a result we attribute to the dynamic\noptimizations that we employ.  The second case study is a Numerical\nRelativity simulation based on the\nMcLachlan code, a part of the freely available open-source (GPL)\nEinstein Toolkit (ET\\@).  McLachlan solves a\nsignificantly more complex set of equations, and integrates with many\nother components of the ET\\@.  We performed a simulation of a binary\nblack hole coalescence using the same codes and techniques as we would\ncurrently use in production CPU simulations, with the omission of\nAdaptive Mesh Refinement (AMR), which is not yet adapted to Chemora\\xspace.\n\nWe plan to implement AMR and multi-block methods next.\nAMR and multi-block are implemented in Cactus in a\nway which is transparent to the application programmer, hence we\nexpect that including AMR in Chemora\\xspace will be straightforward using\nthe Accelerator architecture developed in this work\n(which maintains knowledge of which variables are valid on the host\n(CPU) and which on the device (GPU)).  As with the time integration, we\nwill implement only the basic low-level interpolation operators\nrequired for mesh refinement on the GPU, and the existing AMR code\nCarpet will marshal the required operations to the device.\n\nWith AMR and\/or multi-block methods, Chemora\\xspace will be an even more\ncompelling option for implementing scientific codes, and fields of\nscience (such as Numerical Relativity) requiring the solution of\ncomplex systems of PDEs will be able to reach a new level of\nperformance.\nShould the specifics of accelerator devices change in the future, the\nChemora\\xspace framework, much of which is general, should be easily adaptable to\nthe new technology, and codes built with Chemora\\xspace will have a head start in\nadvancing computational science on the new platform.\n\n\\section*{Acknowledgments}\n\nThe authors would like to thank Gabrielle Allen and Joel E. Tohline at\nthe CCT\nand Krzysztof Kurowski at PSNC for their vision, encouragement, and \ncontinuous support to this project.\n\nThis work was supported by the UCoMS project under award number MNiSW\n(Polish Ministry of Science and Higher Education) Nr 469~1~N~-\nUSA\/2009 in close collaboration with U.S. research institutions\ninvolved in the U.S. Department of Energy (DOE) funded grant under\naward number DE-FG02-04ER46136 and the Board of Regents, State of\nLouisiana, under contract no.\\ DOE\/LEQSF(2004-07)\nand LEQSF(2009-10)-ENH-TR-14.\nThis work was also supported by NSF award 0725070 \\emph{Blue Waters},\nNFS awards 0905046 and 0941653\n\\emph{PetaCactus}, NSF award 0904015 \\emph{CIGR}, \nand NSF award 1010640 \\emph{NG-CHC} to Louisiana State University, and\nby the DFG grant SFB\/Transregio~7 ``Gravitational-Wave Astronomy''.\n\nThis work was performed using computational resources of XSEDE\n(TG-CCR110029, TG-ASC120003), LONI (loni\\_cactus), LSU, and PSNC,\nand on the Datura cluster at the AEI\\@.\n\n\\bibliographystyle{unsrt}\n{\\footnotesize","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nThe real analytic Eisenstein series is a special function\nthat has been studied classically. It is used in the representation theory of $SL(2,\\mathbb{R})$,\nand in analytic number theory (e.g., cf.\\,\\cite{Kub}).\nIts generalization to the case of many variables was initiated by\nSiegel and later studied more extensively by Langlands \\cite{Lang} and Shimura \\cite{S2}.\n\nLet\n$$\nE_k^{(m)}(z,s)=\\text{det}(y)^s\n                         \\sum_{\\{ c,d\\}}\\text{det}(cz+d)^{-k}\\,|\\text{det}(cz+d)|^{-2s}\n$$\nbe the Eisenstein series of degree $m$ (for a precise definition, see $\\S$ \\ref{sec3-1}).\n\nShimura \\cite{S2} studied the analytic properties of the Eisenstein series, including\nthis type. He reveals the holomorphy of $E_k^{(m)}(z,s)$ in $s$ at $s=0$ by\nanalyzing the Fourier coefficients. The Fourier coefficient essentially consists of\ntwo parts. One is the confluent hypergeometric function, and the other is the Siegl series.\nTherefore, the analytic properties of Fourier coefficients, and the Eisenstein\nseries results in the study of the analytic properties of these two parts. In \\cite{S1},\nShimura established the analytic theory of confluent hypergeometric functions\non tube domains and then applied them to analysis of Eisenstein series.\nThe results of holomorphy of $E_k^{(m)}(z,s)$ studied and extended by Weissauer\n\\cite{W}.\n\nIn Shimura's paper \\cite{S2}, apart from the holomorphy, the residue of Eisenstein\nseries is mentioned. His statement is as follows:\n\\vspace{2mm}\n\\\\\nThe residue of the Eisenstein series $E_{(m-1)\/2}^{(m)}(z,s)$ at $s=1$ can be expressed \nas the product of $\\pi^{-m}$ and a holomorphic modular form of weight $(m-1)\/2$,\nwith rational Fourier coefficients.\n\\vspace{2mm}\n\\\\\n(It is known that the holomorphic modular form stated above is\na (rational) constant multiple of Eisenstein series $E_{(m-1)\/2}^{(m)}(z,0)$.)\n\\vspace{2mm}\n\\\\\nOther than his work, few papers mention concrete forms of the residue for the Eisenstein series,\nexcept for the classical work by Kaufhold \\cite{Kau} (see $\\S$ \\ref{other}).\n\nThis study aims to provide concrete forms of residue $E_0^{(m)}(z,s)$\nat $s=m\/2$. Our results strongly depend on Mizumoto's work \\cite{Mi}, especially his work on\nthe Fourier expansion of $E_k^{(m)}(z,s)$, which is a refinement of Maass's result.\n\\vspace{4mm}\n\\\\\n\\textbf{Theorem} \n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n&=\\mathbb{A}^{(m)}(y)+\\mathbb{B}^{(m)}(y)\n                \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n                \\boldsymbol{e}(\\sigma(hx)),\n\\end{align*}\n{\\it where}\n\\begin{align*}\n\\mathbb{A}^{(m)}(y) &=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n                                                    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\\\\n                         &+\\frac{1}{8}\\,\\beta'_m(y,m\/2),\\\\\n\\mathbb{B}^{(m)}(y) &=2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1}\\\\\n                          &\\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\,\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\n \\end{align*}     \n\\vspace{2mm}\n\\\\\nHere, $C_{m-1}^{(m)}$ is the constant term of the completed Koecher--Maass zeta function\n$\\xi_{m-1}^{(m)}(2y,s)$ at $s=m\/2$ (see (\\ref{DefC})), and $\\alpha_m(y,s)$ and $\\beta_m(y,s)$\nare defined in (\\ref{alpham}) and (\\ref{betam}), respectively,  which are essentially products of the gamma functions\nand zeta functions.\n\\vspace{2mm}\n\\\\\nIn the degree 2 case, the constant $C_{1}^{(2)}(y)$ can be calculated explicitly from the first\nKronecker limit formula.\n\\vspace{2mm}\n\\\\\n\\textbf{Corollary}\n\\\\\n\\begin{align}\n\\underset{s=1}{{\\rm Res}}\\,E_0^{(2)}(z,s) \n& =\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right)\n\\nonumber \\\\\n& \\quad +\\frac{36\\,\\text{det}(y)}{\\pi^2}\\sum_{h\\in\\Lambda_2^{(1)}}\\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;1,1)\n             \\boldsymbol{e}(\\sigma(hx)). \\label{A}\n\\end{align}\n(for the notation, see $\\S$ \\ref{Degree2}.)\nIn \\cite{Na}, the author provided a formula for $E_2^{(2)}(z,0)$ (Siegel Eisenstein series of degrees 2 and 2):\n\\begin{align}\nE_2^{(2)}(z,0)=& 1-\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(1+\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right) \\nonumber \\\\\n         &-\\frac{72}{\\pi^3}\\sum_{\\substack{0\\ne h\\in\\Lambda_2\\\\ \\text{discr}(h)=\\square}}\\varepsilon_h\n           \\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;2,0)\\boldsymbol{e}(\\sigma(hx))\\nonumber \\\\\n         &+288\\sum_{0\\ne h\\in \\Lambda_2}\\sum_{d\\mid\\text{cont}(h)}d\\,H\\left(\\frac{|\\text{discr}(h)|}{d^2}\\right)\n              \\boldsymbol{e}(\\sigma(hz)). \\label{B}\n\\end{align}\nIt is interesting that the same term appears in each Fourier coefficient in (\\ref{A}) and\n(\\ref{B}).\n\n\n\n\n\\section{Notation}\n\\label{notation}\n$1^\\circ$\\quad\nIf $a$ is an $m\\times m$ matrix, we write it as $a^{(m,n)}$, and as $a^{(m)}$ if $m=n$,\n${}^ta$ denotes the transpose of $a$, and $a_{ij}$ denotes the $(i,j)$-entry of $a$. For a\nmatrix $a$, we write $\\sigma (a)$ as the trace of $a$. If the right-hand side is defined as\nthe identity matrix (resp. zero matrix) of size $m$ and is denoted by $1_m$ (resp. $0_m$).\nFor a commutative ring $R$ with $1$, we denote $R^{(m,n)}$ by the $R$-module of\nall $m\\times n$ matrices with entries $R$. We set $R^{(m)}:=R^{(m,m)}$ and $R^m:=R^{(1,m)}$.\n\\vspace{2mm}\n\\\\\n$2^\\circ$\\quad\nWe put\n\\vspace{1mm}\n\\\\\n$\\bullet$\\quad $\\displaystyle\\kappa (\\nu)=\\frac{\\nu+1}{2}$ for $\\nu\\in\\mathbb{Z}_{\\geqq 0}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $\\boldsymbol{e}(z)=\\text{exp}(2\\pi iz)$ for $z\\in\\mathbb{C}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $H_m:=\\{\\,z\\in\\mathbb{C}^{(m)}\\,\\mid\\, {}^tz=z,\\;\\text{Im}(z)>0\\,\\}$\\,:\\;upper\nhalf space.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m=\\{\\,x\\in \\mathbb{R}^{(m)}\\,\\mid\\, {}^tx=x\\,\\}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m(\\mathbb{C})=V_m\\otimes_{\\mathbb{R}}\\mathbb{C}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $P_m:=\\{\\, x\\in V_m\\,\\mid\\,x>0\\,\\}$.\n\\vspace{2mm}\n\\\\\n$\\bullet$\\quad $V_m(p,q,r)$: subset of $V_m$ consisting of the elements with $p$ positive,\\\\\n             \\qquad\\qquad\\qquad\\quad $q$ negative, $r$ zero eigenvalues.\n\\vspace{2mm}\n\\\\\n$3^\\circ$\\quad\nThe function $\\Gamma_m(s)$ is defined by\n$$\n\\Gamma_m(s)=\\pi^{\\frac{m(m-1)}{4}}\\prod_{\\nu=0}^{m-1}\\Gamma\\left(s-\\frac{\\nu}{2}\\right)\n$$\nfor $m>0$, and $\\Gamma_0(s):=1$. \n\\vspace{2mm}\n\\\\\n$4^\\circ$\\quad The set of symmetric half-integral matrices of size $m$ is denoted by\n$\\Lambda_m$. We place\n$$\n\\Lambda_m^{(\\nu)}:=\\{\\,h\\in\\Lambda_m\\,\\mid\\,\\text{rank}(h)=\\nu\\,\\}.\n$$\nFor $\\nu\\in\\mathbb{Z}$ with $1\\leqq \\nu\\leqq m$,\n$$\n\\mathbb{Z}_{\\text{prim}}^{(m,\\nu)}=\\{\\, a\\in \\mathbb{Z}^{(m,\\nu)}\\,\\mid\\,\n                                             a\\; {\\rm is\\; primitive} \\,\\}.\n$$\n$5^\\circ$\\quad Throughout the paper, we understand that the product (resp. sum)\nover an empty set is equal to $1$ (resp. $0$).\n\\section{Preliminary}\n\\label{Pre}\n\\subsection{Eisenstein series}\n\\label{sec3-1}\nFor $m\\in\\mathbb{Z}_{>0}$ and $k\\in 2\\mathbb{Z}_{\\geq 0}$, let\n\\begin{equation}\n\\label{DefEis}\nE_k^{(m)}(z,s)=\\text{det}(y)^s\n                         \\sum_{\\{ c,d\\}}\\text{det}(cz+d)^{-k}\\,|\\text{det}(cz+d)|^{-2s}\n\\end{equation}\nbe the Eisenstein series for $\\Gamma_m=Sp_m(\\mathbb{Z})$ (Siegel modular\ngroups of degrees $m$). Here, $z=x+iy$ is a variable on $H_m$, $s$ is a complex\nvariable, and $\\{ c,d\\}$ runs over a complete system of representatives $\\binom{\\,*\\,\\;*\\,}{c\\;d}$\nof $\\left\\{ \\binom{\\,*\\,*\\,}{\\,0\\;*\\,}\\in \\Gamma_m \\right\\}\\backslash \\Gamma_m$.\nThe right-hand side of (\\ref{DefEis}) converges absolutely, locally, and uniformly on the\n$$\n\\{\\; (z,s)\\in H_m\\times\\mathbb{C}\\;\\mid\\; \\text{Re}(s)>(m+1-k)\/2\\;\\}.\n$$\nAs is well known, the Eisenstein series $E_k^{(m)}(Z,s)$ has a meromorphic continuation\nto the whole $s$-plane (Langlands \\cite{Lang}, Mizumoto \\cite{Mi}).\n\\subsection{Confluent hypergeometric functions}\n\\label{sec3-2}\nShimura studied the confluent hypergeometric functions on the tube domains (\\cite{S1})\nand applied his results to develop the theory of the Eisenstein series (\\cite{S2}). In this section,\nwe summarize some results on the confluent hypergeometric functions that will be used later.\n\nFor $g\\in P_m$, $h\\in V_m$, and $(\\alpha,\\beta)\\in\\mathbb{C}^2$,\n\\begin{equation}\n\\label{xi}\n\\xi_m(g,h;\\alpha,\\beta)=\\int_{V_m}\\boldsymbol{e}^{-\\sigma(hx)}\\text{det}(x+ig)^{-\\alpha}\n                                     \\text{det}(x-ig)^{-\\beta}dx,\n\\end{equation}\nwith $dx=\\prod_{i\\leqq j}dx_{ij}$, which is convergent for $\\text{Re}(\\alpha+\\beta)>m$;\n\\begin{equation}\n\\label{eta}\n\\eta_m(g,h;\\alpha,\\beta)= \\int_{\\substack{V_m\\\\ x\\pm h>0}}\ne^{-\\sigma(gx)}\\text{det}(x+h)^{\\alpha-\\kappa(m)}\n                                     \\text{det}(x-h)^{\\beta-\\kappa(m)}dx,\\\\\n\\end{equation}\nwhich is convergent for $\\text{Re}(\\alpha)>\\kappa (m)-1$, $\\text{Re}(\\beta)>m$.\nWe also use\n\\begin{equation*}\n\\label{etastar}\n\\eta^*_m(g,h;\\alpha,\\beta)=\\text{det}(g)^{\\alpha+\\beta-\\kappa(m)}\\eta_m(g,h;\\alpha,\\beta),\n\\end{equation*}\nwhich satisfies the property\n\\begin{equation*}\n\\eta_m^*(g[a],h[{}^ta^{-1}];\\alpha,\\beta)\n=\\eta^*_m(g,h;\\alpha,\\beta)\n\\quad\n\\text{for all}\n\\quad a\\in GL_m(\\mathbb{R}).\n\\end{equation*}\nBy \\cite{S1}, (1.29),\n\\begin{equation}\n\\label{xieta}\n\\xi_m(g,h;\\alpha,\\beta)= i^{m(\\beta-\\alpha)}\\cdot 2^m\\pi^{m\\kappa(m)}\n                                          \\Gamma_n(\\alpha)^{-1}\\Gamma_n(\\beta)^{-1}\n                                        \\eta_m(2g,\\pi h;\\alpha,\\beta).\n\\end{equation}\nfor $\\text{Re}(\\alpha)>\\kappa(m)-1$, $\\text{Re}(\\beta)>m$.\nWhen $h=0_m$, the following identity holds:\n\\begin{Prop}\n(Shimura \\cite{S1}, (1.31)) {\\it If ${\\rm Re}(\\alpha+\\beta)>2\\kappa(m)-1$, then}\n\\begin{align}\n\\xi_m(g,0_m;\\alpha,\\beta) &= i^{m\\beta-m\\alpha}\\cdot 2^{m(1-\\kappa(m))}(2\\pi)^{m\\kappa(m)}\\nonumber \\\\\n                     & \\cdot \\Gamma_m(\\alpha)^{-1}\\Gamma_m(\\beta)^{-1}\\Gamma_m(\\alpha+\\beta-\\kappa(m)) \n                     \\nonumber \\\\\n                     & \\cdot \\text{det}(2g)^{\\kappa(m)-\\alpha-\\beta}.\n                     \\label{xi0}\n\\end{align}\n\\end{Prop}\nFor $g\\in P_m$, $h\\in V_m(p,q,r)$ with $p+q+r=m$, we put\n\\begin{align*}\n& \\delta_+(hg):=\\text{the product of all positive eigenvalues of} \\;\\;g^{\\frac{1}{2}}hg^{\\frac{1}{2}},\\\\\n& \\delta_{-}(hg):=\\delta_+((-h)g).\n\\end{align*}\nWe then put\n\\begin{align}\n\\omega_m(g,h;\\alpha,\\beta) :=&  2^{-p\\alpha-q\\beta}\\Gamma_p\\left(\\beta-(m-p)\/2\\right)^{-1}\n                                                               \\Gamma_q\\left(\\alpha-(m-q)\/2\\right)^{-1}\\nonumber \\\\\n                                   & \\cdot \\Gamma_r\\left(\\alpha+\\beta-\\kappa(m) \\right)^{-1} \\nonumber\\\\\n                                   & \\cdot \\delta_{+}(hg)^{\\kappa(m)-\\alpha-q\/4}\n                                     \\delta{-}(hg)^{\\kappa(m)-\\beta-p\/4}\\,\n                                     \\eta_n^*(g,h;\\alpha,\\beta), \\label{omegaeta}\n\\end{align}\nOne of the main results in \\cite{S1} is as follows:\n\\begin{Thm}\n\\label{ShimuraMain}\n(Shimura \\cite{S1}, Theorem 4.2)\\;{\\it \nFunction $\\omega_m$ can be continued as a holomorphic function in $(\\alpha,\\beta)$ to\nthe whole $\\mathbb{C}^2$ and satisfies\n$$\n\\omega_m(g,h;\\alpha,\\beta)=\\omega_m\\left(g,h;\\kappa(m)+(r\/2)-\\beta,\\kappa(m)+(r\/2)-\\alpha\\right).\n$$\n}\n\\end{Thm}\n\\subsection{Fourier expansion}\n\\label{FourierEx}\nFor $m\\in\\mathbb{Z}_{>0}$ and $k\\in 2\\mathbb{Z}_{\\geqq 0}$, let $s$ be a complex variable,\nwhere $\\text{Re}(s)>\\kappa (m)$, and let $z=x+iy$ be a variable on $H_m$ with $x\\in V_m$.\nand $y\\in P_m$. Maass (\\cite{Ma}) provided a formula for the Fourier expansion of the\nEisenstein series $E_k^{(m)}(z,s)$:\n\\begin{align}\nE_k^{(m)}(z,s) &= \\text{det}(y)^s+\\text{det}(y)^s\\sum_{\\nu=1}^m\\sum_{h\\in\\Lambda_\\nu}\n                                         \\sum_{q\\in\\mathbb{Z}_{\\text{prim}}^{(m,\\nu)}\/GL_\\nu(\\mathbb{Z})}\\nonumber\\\\\n                 & S_\\nu(h,2s+k)\\xi_{\\nu}(y[q],h;s+k,s)\\boldsymbol{e}(\\sigma(h[{}^tq]x)),  \\label{ExMaass}                 \n\\end{align}\nwhere\n\\begin{equation}\nS_\\nu(h,s)=\\sum_{r\\in V_\\nu\\cap\\, \\mathbb{Q}^\\nu \\text{mod}\\, 1}n(r)^{-s}\\boldsymbol{e}(\\sigma (hr))\n\\end{equation}\nis the singular series (Siegel series), where $n(r)$ is the product of the reduced positive denominators\nof the elementary divisors of $r$, and $\\xi_\\nu$ is the confluent hypergeometric function defined in  (\\ref{xi}).\n\nFrom \\cite{Mi}, Lemma 1.1, we have\n\\begin{Lem}\n{\\it For $\\nu\\in\\mathbb{Z}_{>0}$, each $h\\in\\Lambda_\\nu$ of {\\rm rank} $\\lambda>0$ (that is,\n$h\\in \\Lambda_\\nu^{(\\lambda)}$) is expressed uniquely as\n$$\nh=h_0[{}^tw]\n$$\nwith $h_0\\in\\Lambda_\\lambda^{(\\lambda)}$ and \n$w\\in \\mathbb{Z}_{{\\rm prim}}^{(\\nu,\\lambda)}\/GL_\\lambda(\\mathbb{Z})$.}\n\\end{Lem}\nMizumoto provided a reduced formula for $\\xi_\\nu$ (\\cite{Mi}, Lemma 1.4):\n\\begin{Prop}\n\\label{xi-eta}\n{\\it Let $h=h_0[{}^tw]$ be, as in the above lemma. Suppose that ${\\rm Re}(s)>\\nu$.\nThen, in {\\rm (\\ref{ExMaass})}, we have}\n\\begin{align}\n& \\xi_{\\nu}(y[q],h;s+k,s) \\nonumber \\\\\n& = (-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)+\\lambda(\\nu-\\lambda)\/2)}\n                                \\cdot \\Gamma_{\\nu-\\lambda}(2s+k-\\kappa(\\nu))\n                                \\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1} \\nonumber \\\\\n                &\\cdot {\\rm det}(2y[q])^{\\kappa(\\nu)-k-2s}\n                \\eta_{\\lambda}^*(2y[qw],\\pi h_0;s+k+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2).\n\\end{align}\n\\end{Prop}\nLet $m,\\,\\lambda\\in\\mathbb{Z}$ with $m\\geqq \\lambda\\geqq 1$. We define the subgroup\n$\\Delta_\\lambda^{(m)}$ of $GL_m(\\mathbb{Z})$ by\n$$\n\\Delta_\\lambda^{(m)}:=\\left\\{ \\begin{pmatrix} * & * \\\\ 0^{(m-\\lambda,\\lambda)} & * \\end{pmatrix}\\in\nGL_m(\\mathbb{Z}) \\;\\right\\}.\n$$\nFor $r\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}$, $u_r$ is an element of $GL_m(\\mathbb{Z})$\ncorresponding to $r$ under a bijection\n\\begin{align*}\n\\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/&GL_\\lambda(\\mathbb{Z})\n \\qquad\n\\longleftrightarrow\n\\qquad\nGL_m(\\mathbb{Z})\/\\Delta_\\lambda^{(m)} \\\\\n& r\\qquad\\qquad\\quad  \\longmapsto\\qquad\\qquad u_r\n\\end{align*}\nwhich is determined up to the right action of $\\Delta_\\lambda^{(m)}$.\n\nFor $y\\in P_m$, we write the Jacobi decomposition of $y[u_r]$ as\n\\begin{equation*}\ny[u_r]={\\rm diag}(y[r],g(y,u_r))\\begin{bmatrix} 1_\\lambda & b \\\\ 0 & 1_{m-\\lambda} \\end{bmatrix}.\n\\end{equation*}\nExplicitly, we place $u_r=(r\\,r_1)$ and then\n\\begin{equation}\n\\label{gyur}\ng(y,u_r)=y[r_1]-(y[r])^{-1}[{}^tryr_1].\n\\end{equation}\n\\vspace{3mm}\n\\\\\n\\quad Next, we provide a definition of Koecherer--Maass zeta functions.\nFor $1\\leqq \\nu\\leqq m$ and $g\\in P_m$, we define\n\\begin{equation}\n\\label{KM}\n\\zeta_\\nu^{(m)}(g,s):=\\sum_{a\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\nu)}\/GL_\\nu(\\mathbb{Z})}\n\\text{det}(g[a])^{-s}\n\\end{equation}\nwhich is convergent for $\\text{Re}(s)>m\/2$. By definition,\n$$\n\\zeta_m^{(m)}(g,s)=\\text{det}(g)^{-s}.\n$$\nFor later purposes, we put\n$$\n\\zeta_0^{(m)}(*,s):=1\\qquad\n\\text{for all}\n\\quad m\\in\\mathbb{Z}_{\\geqq 0}.\n$$\nMizumoto's refinement of Maass' expression is as follows:\n\\begin{Thm} (Mizumoto \\cite{Mi}, Theorem 1.8) \n\\label{FourierMi}\n{\\it\nFor $m\\in\\mathbb{Z}_{>0}$, $k\\in 2\\mathbb{Z}_{\\geqq 0}$, and ${\\rm Re}(s)>m$,\nthe Eisenstein series $E_k^{(m)}(z,s)$ has the following expression:\n\\begin{equation}\n\\label{MiEx}\nE_k^{(m)}(z,s)=\\sum_{\\nu=0}^m\\sum_{\\lambda=0}^\\nu F_{k,\\nu,\\lambda}^{(m)}(z,s)\n\\end{equation}\nwhere\n\\begin{align}\n\\label{F0}\n& F_{k,\\nu,0}^{(m)}(z,s) \\nonumber \\\\\n& =(-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)}\\Gamma_\\nu(2s+k-\\kappa(\\nu))\n            \\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1} \\nonumber \\\\\n& \\cdot S_\\nu(0_\\nu,2s+k)\\,{\\rm det}(y)^s\\,\\zeta_\\nu^{(m)}(2y,2s+k-\\kappa(\\nu)),\n\\end{align}\nfor $0\\leqq \\nu\\leqq m$, and\n\\begin{equation}\n\\label{Fb}\nF_{k,\\nu,\\lambda}^{(m)}(z,s)\n=\n\\sum_{h\\in\\Lambda_{\\lambda}^{(\\lambda)}}\\sum_{r\\in \\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/GL_\\lambda(\\mathbb{Z})}\nb_{k,\\nu,\\lambda}^{(m)}(h[{}^tr],y,s)\\boldsymbol{e}(\\sigma(h[{}^tr]x))\n\\end{equation}\nfor $1\\leqq \\lambda\\leqq \\nu\\leqq m$ with\n\\begin{align}\n& b_{k,\\nu,\\lambda}^{(m)}(h[{}^tr],y,s) \\nonumber \\\\\n             &\\quad := (-1)^{k\\nu\/2}2^\\nu\\pi^{\\nu\\kappa(\\nu)+\\lambda(\\nu-\\lambda)\/2}\n              \\Gamma_{\\nu-\\lambda}(2s+k-\\kappa(\\nu))\\Gamma_\\nu(s)^{-1}\\Gamma_\\nu(s+k)^{-1}\n              \\nonumber \\\\\n              &\\qquad \\cdot S_\\nu({\\rm diag}(h,0_{\\nu-\\lambda}),2s+k)){\\rm det}(y)^s{\\rm det}(2y[r])^{\\kappa(\\nu)-k-2s}\n              \\nonumber \\\\\n              &\\qquad \\cdot \\eta_\\lambda^*(2y[r],\\pi h;s+k+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)\\nonumber \\\\\n             &\\qquad  \\cdot \\zeta_{\\nu-\\lambda}^{(m-\\lambda)}(2g(y,u_r),2s+k-\\kappa(\\nu)) \\label{Fb1}.\n\\end{align}\nHere, $\\zeta_\\nu^{(m)}(g,s)$ for $0\\leqq \\nu\\leqq m$ is the Koecher--Maass zeta function\ndefined in {\\rm (\\ref{KM})}, and $g(y,u_r)$ is defined by {\\rm (\\ref{gyur})}. Matrix $h[{}^tr]$ runs\nover the set $\\Lambda_m^{(\\lambda)}$ exactly once if $h$ runs over $\\Lambda_\\lambda^{(\\lambda)}$\nand $r$ runs over a complete set of representatives of \n$\\mathbb{Z}_{{\\rm prim}}^{(m,\\lambda)}\/GL_\\lambda(\\mathbb{Z})$.}\n\\end{Thm}\n\\subsection{Siegel series}\n\\label{sec3-4}\nIn this section, we summarize the results of the Siegel series $S_\\nu(h,s)$ that appear in\nthe Fourier expansions (\\ref{ExMaass}) and (\\ref{Fb1}).\n\nFor $h\\in\\Lambda_\\lambda^{(\\lambda)}$, we set\n$$\nd(h):=(-1)^{[\\lambda\/2]}2^{-\\delta((\\lambda-1)\/2)}\\text{det}(2h)\n$$\nwhere\n$$\n\\delta (x):=\\begin{cases} 1  &  x\\in\\mathbb{Z},\\\\\n                                0  &   x\\notin \\mathbb{Z}\n                                \\end{cases}\n$$\nfor $x\\in\\mathbb{Q}$. By \\cite{Mi}, (5.1), \n\\begin{align}\n\\label{Siegelreduce}\nS_\\nu(\\text{diag}(h,0_{\\nu-\\lambda}),s) &=\\zeta(s+\\lambda-\\nu)\\zeta(s)^{-1}\\nonumber\\\\\n                                                  & \\cdot \\prod_{j=1}^{\\nu-\\lambda}(\\zeta(2s-\\nu-j)\\,\\zeta(2s-2j)^{-1})\n                                                  \\nonumber\\\\\n                                                   & \\cdot S_\\lambda(h,s-\\nu+\\lambda)\n\\end{align}\nand\n\\begin{align*}\n& S_\\lambda(h,s)=\\sum_{d\\in A(h)}(\\text{det}(d))^{\\lambda+1-2s}\\widehat{S}_{\\lambda}(h[d^{-1}],s),\\\\\n& \\widehat{S}_{\\lambda}(h,s)=\\zeta(s)^{-1}\\prod_{j=1}^{[\\lambda\/2]}\\zeta(2s-2j)^{-1}\n             L\\left(s-\\lambda\/2,\\left(\\frac{d(h)}{*} \\right)\\right)^{\\delta(\\lambda\/2)}\\prod_p a_p(h,s)\n\\end{align*}\nwhere $L\\left(s,\\left(\\frac{d(h)}{*}\\right)  \\right)$ is Dirichlet $L$-function associated to the quadratic\ncharacter $\\left(\\frac{d(h)}{*}\\right)$, the product of $p$ runs over the prime divisors of $d(h)$,\n$$\nA(h):=GL_\\lambda(\\mathbb{Z})\\backslash\n     \\{\\,d\\in\\mathbb{Z}^{(\\lambda)}\\,\\mid\\, \\text{det}(d)\\ne 0\\;\\;\\text{and}\\;\\; h[d^{-1}]\\in\\Lambda_\\lambda\\,\\},\n$$\nand from \\cite{Bo}, we have\n\\begin{align*}\n& a_p(h,s)=\\\\\n&\n\\begin{cases}\n\\prod_{j=1}^{r\/2}(1-p^{2j-1+\\lambda-2s}) & (\\lambda,r) \\equiv (1,0) \\pmod{2},\\\\\n(1+\\lambda_p(h)p^{(\\lambda+r)\/2-s})\\prod_{j=1}^{(r-1)\/2}(1-p^{2j-1+\\lambda-2s}) \n                                                    & (\\lambda,r) \\equiv (1,1) \\pmod{2},\\\\   \n\\prod_{j=1}^{(r-1)\/2}(1-p^{2j+\\lambda-2s})\n                                                   & (\\lambda,r) \\equiv (0,1) \\pmod{2},\\\\ \n(1+\\lambda_p(h)p^{(\\lambda+r)\/2-s})\\prod_{j=1}^{r\/2-1}(1-p^{2j+\\lambda-2s}) \n                                                  & (\\lambda,r) \\equiv (0,0) \\pmod{2}.\n\\end{cases}\n\\end{align*}\nHere, $r:=r(p)$ is the maximal number, which is the condition \n$h[u] \\equiv \\begin{pmatrix}h^* & 0 \\\\ 0 & 0_r \\end{pmatrix} \\pmod{p}$\nfor some $u\\in\\mathbb{Z}^{(\\lambda)}$ and $\\lambda_p(h):=\\left(\\frac{d(h^*)}{p} \\right)$.\n\\vspace{2mm}\n\\\\\n\\begin{Rem}\n(1)\\quad We understand that $S_0(*,s)=1$.  Therefore, from (\\ref{Siegelreduce}), we obtain the following.\nFormula for $S_\\nu(0_\\nu,s)$:\n\\begin{align}\n\\label{S0}\nS_\\nu(0_\\nu,s) &=\\zeta(s-\\nu)\\zeta(s)^{-1}\\prod_{\\nu=1}^\\nu(\\zeta(2s-\\nu-j)\\,\\zeta(2s-2j)^{-1}) \\nonumber\\\\\n               &= \\zeta(s-\\nu)\\zeta(s)^{-1}\\prod_{\\nu=1}^{[\\nu\/2]}(\\zeta(2s-2\\nu-1+2j)\\,\\zeta(2s-2j)^{-1}).\n\\end{align}\n(2)\\quad In the following discussion, the concrete form of $a_p(h,s)$ is not needed, only its\nholomorphy in $s$.\n\\end{Rem}\n\\subsection{Koecher--Maass zeta function}\n\\label{sec:3-5}\nThe Koecher--Maass zeta function $\\zeta_\\nu^{(m)}(g,s) $ in $ \\S$ \\ref{FourierEx}.\nAnalytic properties of this function are important for the analysis of the Fourier coefficient\n$F_{k,\\nu,\\lambda}^{(m)}(z,s)$. In this section, we recall Arakawa's results for the Koecher--Maass\nzeta function.\n\nFor $1\\leqq \\nu\\leqq m$ and $g\\in P_m$, we define the completed Koecher--Maass zeta\nfunction by\n\\begin{equation}\n\\label{CompKM}\n\\xi_{\\nu}^{(m)}(g,s):=\\prod_{i=0}^{\\nu-1}\\xi(2s-i)\\,\\zeta_\\nu^{(m)}(g,s)\n\\end{equation}\nwhere\n$$\n\\xi(s):=\\pi^{-s\/2}\\Gamma(s\/2)\\,\\zeta(s)\n$$\nand we understand\n$$\n\\xi_0^{(m)}(g,s):=1.\n$$\nThe following result is due to Arakawa, which plays an important role in our investigation.\n\\begin{Prop}\\;(Arakawa \\cite{Ara})\\quad \n\\label{Ara}\n{\\it (1)\\; Suppose $m\\geqq 2\\nu-1$.\nThe function $\\xi_{\\nu}^{(m)}(g,s)$ has simple poles at $s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{\\nu-1}{2}$\nand $s=\\frac{m-\\nu+1}{2},\\,\\cdots\\,,\\frac{m}{2}$. For $0\\leqq \\mu\\leqq\\nu-1$, the residues of\n$\\xi_{\\nu}^{(m)}(g,s)$ at $s=\\frac{\\mu}{2}$ and $s=\\frac{m-\\mu}{2}$ are given by\n\\begin{align}\n& \n\\underset{s=\\mu\/2}{\\rm Res}\\xi_{\\nu}^{(m)}(g,s)=-\\frac{1}{2}v(\\nu-\\mu)\\xi_{\\mu}^{(m)}(g,\\tfrac{\\nu}{2}),\n\\label{Res1} \\\\\n&\n\\underset{s=(m-\\mu)\/2}{\\rm Res}\\xi_{\\nu}^{(m)}(g,s)=\\frac{1}{2}v(\\nu-\\mu)\n                                               {\\rm det}(g)^{-\\frac{\\nu}{2}}\\xi_{\\mu}^{(m)}(g^{-1},\\tfrac{\\nu}{2}),\n\\label{Res2}\n\\end{align}\nwhere\n$$\nv(\\nu)=\n\\begin{cases}\n\\displaystyle \\prod_{i=2}^\\nu \\xi(i)   &  (\\nu\\geqq 2),\\\\\n1                                             & (\\nu=1).\n\\end{cases}\n$$\n(2)\\; Suppose $\\nu\\leqq m\\leqq 2\\nu-2$. The function \n$\\xi_{\\nu}^{(m)}(g,s)$ has poles at $s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{m}{2}$ of which\n$s=0,\\,\\frac{1}{2},\\,\\cdots\\,,\\frac{m-\\nu}{2}$ and $s=\\frac{\\nu}{2},\\,\\frac{\\nu+1}{2},\\,\\cdots\\,,\\frac{m}{2}$\nare simples poles. The poles at $s=\\frac{m-\\nu+1}{2},\\,\\frac{m-\\nu+2}{2},\\,\\cdots\\,,\\frac{\\nu-1}{2}$ are\ndouble poles. For $0\\leqq \\mu\\leqq m-\\nu$, the residues of $\\xi_{\\nu}^{(m)}(g,s)$ at $s=\\frac{\\mu}{2}$\nand $s=\\frac{m-\\mu}{2}$ are given by {\\rm (\\ref{Res1}) and (\\ref{Res2})}, respectively.\n}\n\\end{Prop}\n\\begin{Rem}\nWhen $m=2$ and $\\nu=1$, the function $\\zeta_1^{(2)}(g,s)$ appears as a simple\nfactor of Epstein's zeta function for $g$. Therefore, the residue and constant term at\n$s=1$ is explicitly expressed by the Kronecker limit formula (see $\\S$ \\ref{Degree2}).\n\\end{Rem}\n\\section{Residue of Eisenstein series}\n\\label{Main}\nIn the rest of this paper, we assume that $m\\geqq 2$.\nIn this section, we provide an explicit formula for\n$$\n\\underset{s=m\/2}{\\text{Res}}E_0^{(m)}(z,s)\n$$\nwhich is the main result of this paper.\n\n\\subsection{Fourier coefficient of $\\boldsymbol{E_0^{(m)}(z,s)}$}\n\\label{sec:4-1}\nWe recall the Fourier expansion\n$$\nE_0^{(m)}(z,s)=\\sum_{\\nu=0}^m\\sum_{\\lambda=0}^\\nu F_{0,\\nu,\\lambda}^{(m)}(z,s)\n$$\nin Theorem \\ref{FourierMi} and study the analytic property, particularly the singularity\nof $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$.\nFor this purpose, we use the results introduced in $\\S$ \\ref{Pre} and consider\nthem dividing into several cases.\n\\vspace{2mm}\n\\\\\n$1^\\circ$\\quad $(\\nu,\\lambda)=(0,0)$:\n$$\nF_{0,0,0}^{(m)}(z,s)=\\text{det}(y)^s.\n$$\n$2^\\circ$\\quad $(\\nu,\\lambda)=(\\nu,0)$,\\;$(0<\\nu<m)$:\n\\begin{align*}\nF_{0,\\nu,0}^{(m)}(z,s) &= c_{\\nu,0}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s-\\nu)\\zeta(2s)^{-1}\\\\\n                           &\\quad\\cdot \\prod_{j=1}^\\nu\\zeta(4s-2j)^{-1}\\,\\xi_\\nu^{(m)}(2y,2s-\\kappa(\\nu)).\n\\end{align*}\n$3^\\circ$\\quad $(\\nu,\\lambda)=(m,0)$:\n\\begin{align*}\nF_{0,m,0}^{(m)}(z,s) &= c_{m,0}(s)\\Gamma_m(2s-\\kappa(m))\\Gamma_m(s)^{-2}\n\\zeta(2s-m)\\zeta(2s)^{-1}\\\\\n                           &\\quad\\cdot \\prod_{j=1}^m(\\zeta(4s-m-2j)\\zeta(4s-2j)^{-1}).\n\\end{align*}\n$4^\\circ$\\quad $(\\nu,\\lambda)=(\\nu,\\nu)$,\\;$(0<\\nu\\leqq m)$:\n\\begin{align*}\nb_{0,\\nu,\\nu}^{(m)}(*,y,s) &= c_{\\nu,\\nu}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s)^{-1}\n                                    \\prod_{j=1}^{[\\nu\/2]}\\zeta(4s-2j)^{-1}\n                                    \\cdot \\eta_\\nu(*,*;s,s).\n\\end{align*}\n$5^\\circ$\\quad $(\\nu,\\lambda)$,\\;$(0<\\lambda<\\nu<m)$:\n\\begin{align*}\nb_{0,\\nu,\\lambda}^{(m)}(*,y,s) &= c_{\\nu,\\lambda}(s)\\Gamma_\\nu(s)^{-2}\\zeta(2s)^{-1}\n                                           \\prod_{j=1}^{\\nu-\\lambda}\\zeta(4s-2j)^{-1}\\\\\n                                 & \\cdot \\prod_{j=1}^{[\\lambda\/2]}\\zeta(4s-2\\nu+2\\lambda-2j)^{-1}\\\\\n                                   &\\cdot  \\eta_\\lambda(*,*;s+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)\n                                    \\cdot \\xi_{\\nu-\\lambda}^{(m-\\lambda)}(*,2s-\\kappa(\\nu)).\n\\end{align*}\n$6^\\circ$\\quad $(\\nu,\\lambda)=(m,\\lambda)$,\\;$(0<\\lambda<m)$:\n\\begin{align*}\nb_{0,m,\\lambda}^{(m)}(*,y,s) &= c_{m,\\lambda}(s)\\Gamma_{m-\\lambda}(2s-\\kappa(m))\\Gamma_\\nu(s)^{-2}\\\\\n                 &\\cdot \\prod_{j=1}^{m-\\lambda}\\zeta(4s-m-j)\\zeta(4s-2j)^{-1}\n                    \\prod_{j=1}^{[\\lambda\/2]}\\zeta(4s-2m+2\\lambda-2j)^{-1}\\\\\n                 &\\cdot \\eta_{\\lambda}(*,*;s+(\\lambda-m)\/2.s+(\\lambda-m)\/2).\n\\end{align*}\nwhere $c_{\\nu,\\lambda}(s)$ are holomorphic function in $s$.\n\\subsection{Analytic property of Fourier coefficients}\n\\label{sec:4-2}\nWe investigate the analytic property of $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$\nat $s=m\/2$, based on the description in $\\S$ \\ref{sec:4-1}.\n\\begin{Prop}\n\\label{Triang}\n{\\it Functions $F_{0,\\nu,\\lambda}^{(m)}(z,s)$ and $b_{0,\\nu,\\lambda}^{(m)}(*,y,s)$ are holomorphic in $s$ at\n$s=m\/2$, except for the following three cases:\n$$\n{\\rm (i)}\\; \\nu=m-1,\\;\\lambda=0\\qquad\n{\\rm (ii)}\\; \\nu=m,\\;\\lambda=0\\qquad\n{\\rm (iii)}\\; \\nu=m, \\;\\lambda=m-1.\n$$\n}\n\\end{Prop}\n\\begin{proof}\nWe use the expressions $1^\\circ-6^\\circ$ given in the previous section.\n\nThe holomorphy for the case $1^\\circ$ is\ntrivial.\n\nFirst, we consider the case $5^\\circ$. The $\\Gamma$-factor $\\Gamma_\\nu(s)^{-2}$,\nand $\\zeta$-factors $\\zeta(2s)^{-1},\\cdots$ are all holomorphic at $s=m\/2$.\nFrom Theorem \\ref{ShimuraMain}, the holomorphy of \n$\\eta_\\lambda(*,*;s+(\\lambda-\\nu)\/2,s+(\\lambda-\\nu)\/2)$ is reduced to that of\n$$\n\\Gamma_p(s+(\\lambda-\\nu)\/2-q\/2)\\quad\n\\Gamma_q(s+(\\lambda-\\nu)\/2-p\/2)\\qquad (p+q=\\lambda),\n$$\nand are both holomorphic at $s=m\/2$. (Note that the factor $\\Gamma_r(*)^{-1}$\ndoes not appear in $\\eta_\\lambda$.)\n\nFunction $\\xi_{\\nu-\\lambda}^{(m-\\lambda)}(*,2s-\\kappa(\\nu))$ is holomorphic at $s=m\/2$\nbecause $\\text{Re}(2s-\\kappa(\\nu))>(m-\\lambda)\/2$.\nConsequently, the functions in the case $5^\\circ$ are holomorphic at $s=m\/2$.\n\nBy a similar argument, we observe that the functions in the case of $4^\\circ$ are holomorphic at $s=m\/2$.\nThe cases we must consider are the cases of $2^\\circ$ and $6^\\circ$.\n\nIn the case of $2^\\circ$, only $F_{0,m-1,0}^{(m)}(z,s)$ and $F_{0,m,0}^{(m)}(z,s)$ are non-holomorphic\nat $s=m\/2$ because $F_{0,m-1,0}^{(m)}(z,s)$ has a factor $\\zeta(2s-m+1)\\xi_{m-1}^{(m)}(*,2s-m\/2)$,\nand $F_{0,m,0}^{(m)}(z,s)$ have the factors $\\Gamma(2s-m)\\zeta(4s-2m+1)$, respectively. In fact,\nthe factors above have double poles at $s=m\/2$.\n\nIn the case $6^\\circ$, only the function $b_{0,m,m-1}^{(m)}(*,y,s)$ is nonholomorphic at $s=m\/2$,\nbecause it contains the factor $\\zeta(4s-2m+1)$, which has a simple pole at $s=m\/2$.\n\nThese facts complete the proof.\n\\end{proof}\n\\begin{Rem}\nThe explicit formulas for $F_{0,m-1,0}^{(m)}(z,s)$, $F_{0,m,0}^{(m)}(z,s)$, \\\\\nand $F_{0,m,1}^{(m)}(z,s)$ will be given in the next sections.\n\\end{Rem}\nHere, we arrange functions $F_{0,\\nu,\\lambda}^{(m)}$ as follows:\n\n{\\small\n\\begin{center}\n\\begin{tabular}{llllll}\n$F_{0,0,0}^{(m)}$  &    {}                    &     {}         &    {}    &     {}    &    {}       \n\\vspace{2mm}\n\\\\\n$F_{0,1,0}^{(m)}$  & $F_{0,1,1}^{(m)}$ &    {}          &   {}     &     {}    &    {}      \n\\vspace{2mm}\n\\\\\n$\\vdots$          &  $\\vdots$         &    $\\ddots$ &  {}    &     {}     &   {},        \n\\vspace{2mm}\n\\\\\n$F_{0,m-2,0}^{(m)}$ &$F_{0,m-2,1}^{(m)}$&  $\\ldots$ & $\\ddots$   &    {}     &    {}       \n\\vspace{2mm}\n\\\\\n$\\boldsymbol{F_{0,m-1,0}^{(m)}}$ &$F_{0,m-1,1}^{(m)}$&$F_{0,m-1,2}^{(m)}$&$ \\ldots$&$F_{0,m-1,m-1}^{(m)}$ &{} \n\\vspace{2mm}\n\\\\\n$\\boldsymbol{F_{0,m,0}^{(m)}}$ &$\\boldsymbol{F_{0,m,1}^{(m)}}$&$F_{0,m,2}^{(m)}$& $\\ldots$&$F_{0,m,m-1}^{(m)}$ &$F_{0,m,m}^{(m)}$\n\n\\end{tabular}\n\\end{center}\n}\nThe proposition asserts that only functions $F_{0,\\nu,\\lambda}^{(m)}$ printed in bold are non-holomorphic\nat $s=m\/2$.\n\\subsection{Residue of the constant term}\n\\label{sec:4-3}\nWe investigate the analytic property of the constant term\n$$\n\\sum_{\\nu=0}^mF_{0,\\nu,0}^{(m)}(z,s)\n$$\nat $s=m\/2$. More specifically, we show that the constant term has a simple pole at $s=m\/2$\nand calculate the residue.\n\nBy Proposition \\ref{Triang}, it is sufficient to investigate only $F_{0,m-1,0}^{(m)}(z,s)$ and\\\\\n$F_{0,m,0}^{(m)}(z,s)$ as far as considering the residue.\n\\vspace{2mm}\n\\\\\n\\textbf{Analysis of} $\\boldsymbol{F_{0,m-1,0}^{(m)}(z,s)}:$\n\\vspace{2mm}\n\\\\\nFrom the definition of $F_{0,\\nu,0}^{(m)}(z,s)$ (see (\\ref{F0})), we have\n\\begin{align}\n\\label{Fm-1}\n F_{0,m-1,0}^{(m)}(z,s) \n & =2^{m-1}\\pi^{2(m-1)s}\\text{det}(y)^s\\Gamma_{m-1}(s)^{-2}\\zeta(2s-m+1)\\zeta(2s)^{-2}\n         \\nonumber \\\\\n&\\quad \\cdot\\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}\\cdot\\xi_{m-1}^{(m)}(2y,2s-m\/2). \n\\end{align}\n(We rewrote (\\ref{F0}) with the complete Koecher--Maass zeta function $\\xi_{m-1}^{(m)}$.)\n\nWe separate $F_{0,m-1,0}^{(m)}(z,s)$ into holomorphic and non-holomorphic parts.\nWe define the function $\\alpha_m(y,s)$ by\n$$\nF_{0,m-1,0}^{(m)}(z,s)=\\zeta(2s-m+1)\\xi_{m-1}^{(m)}(2y,2s-m\/2)\\cdot \\alpha_m(y,s).\n$$\nExplicitly,\n\\begin{equation}\n\\label{alpham}\n\\alpha_m(y,s):=2^{m-1}\\pi^{2(m-1)s}\\text{det}(y)^s\\Gamma_{m-1}(s)^{-2}\\zeta(2s)^{-1}\n                  \\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}.\n\\end{equation}\nFunctions $\\zeta(2s-m+1)$ and $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$ have a simple pole at\n$s=m\/2$ (for $\\xi_{m-1}^{(m)}$, see Proposition \\ref{Ara}), and $\\alpha_m(y,s)$\nis holomorphic at $s=m\/2$. These facts imply that $F_{0,m-1,0}^{(m)}(z,s)$ has a double pole at\n$s=m\/2$.\n\nWe set\n$$\nF_{0,m-1,0}^{(m)}(z,s)=\\sum_{l=-2}^\\infty A_l^{(m)}(y)(s-m\/2)^l\n\\quad (\\text{Laurent expansion at}\\; s=m\/2)\n$$\nand calculate $A_{-2}^{(m)}(y)$ and $A_{-1}^{(m)}(y)$.\n\nAs a preparation, we investigate the analytic behavior of $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$\nat $s=m\/2$. We consider the completed Koecher--Maass zeta function $\\xi_{m-1}^{(m)}(2y,s)$.\nAccording to Arakawa's result (Proposition \\ref{Ara}), this function has a simple pole with\nresidue\n\\begin{equation}\n\\label{Resm-1}\n\\underset{s=m\/2}{\\text{Res}}\\xi_{m-1}^{(m)}(2y,s)=\\frac{1}{2}v(m-1)\\text{det}(2y)^{-(m-1)\/2}\n\\end{equation}\n\\begin{Def}\n\\label{DefC}\nDefine a constant $C_{m-1}^{(m)}(y)$ by\n$$\nC_{m-1}^{(m)}(y):=\n\\lim_{s\\to m\/2}\\left(\\xi_{m-1}^{(m)}(2y,s)-\\underset{s=m\/2}{\\text{Res}}\\xi_{m-1}^{(m)}(2y,s)(s-m\/2)^{-1}\\right).\n$$\n\\end{Def}\nThat is, $C_{m-1}^{(m)}(y)$ is the constant term of the Laurent expansion of $\\xi_{m-1}^{(m)}(2y,s)$\nat $s=m\/2$.\n\\begin{Rem}\n(1)\\; It should be noted that the constant $C_{m-1}^{(m)}(y)$ is defined from $\\xi_{m-1}^{(m)}(2y,s)$\nnot $\\xi_{m-1}^{(m)}(y,s)$, and the constant term of $\\xi_{m-1}^{(m)}(2y,2s-m\/2)$ at $s=m\/2$\nis equal to that of $\\xi_{m-1}^{(m)}(2y,s)$.\n\\\\\n(2)\\; In the case $m=2$, the constant $C_1^{(2)}(y)$ is explicitly expressed by the Kronecker\nlimit formula (see $\\S$ \\ref{Degree2}).\n\\end{Rem}\n\\begin{Prop} \n{\\it Explicit forms of $A_{-2}(y)$ and $A_{-1}(y)$ are given as follows:\n\\begin{align}\nA_{-2}^{(m)}(y) &= \\frac{1}{8}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\alpha_m(y,m\/2),\n\\label{EA-2}\n\\\\\nA_{-1}^{(m)}(y) \n&=\\alpha_m(y,m\/2)\\left(\\frac{1}{2}\\,C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\right)\n        \\nonumber \\\\\n&\\qquad +\\frac{1}{8}\\,v(m-1){\\rm det}(2y)^{-(m-1)\/2}\\cdot \\alpha'_m(y,m\/2),\n\\label{EA-1}\n\\end{align}\nwhere $\\gamma$ is the Euler constant and \n$\\alpha'_m(y,m\/2)=\\left.\\frac{d}{ds}\\alpha_m(y,s)\\right|_{s=m\/2}$.}\n\\end{Prop}\n\\begin{proof}\nThe formulas are derived from the expression\n\\begin{align*}\n\\zeta(2s-m+1)&\\xi_{m-1}^{(m)}(2y,2s-m\/2) \\\\\n                  &= \\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}(s-m\/2)^{-2}\\\\\n             &+\\left(\\frac{1}{2}\\,C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2} \\right)(s-m\/2)^{-1}\\\\\n             &+(\\text{a holomorphic function at}\\; s=m\/2). \n\\end{align*}\n\\end{proof}\n\\noindent\n\\textbf{Analysis of} $\\boldsymbol{F_{0,m,0}^{(m)}(z,s)}:$\n\\vspace{2mm}\n\\\\\nBy definition (\\ref{Fb}),\n\\begin{align}\n\\label{Fm}\n F_{0,m,0}^{(m)}(z,s) \n & =2^{-2ms+m(m+3)\/2}\\pi^{m(m+1)\/2}\\text{det}(y)^{-s+(m+1)\/2}\n\\Gamma_m(2s-\\kappa(m)) \\nonumber \\\\\n& \\cdot \\Gamma_{m}(s)^{-2}\\zeta(2s-m)\\zeta(2s)^{-1}\n         \\cdot\\prod_{j=1}^{m}(\\zeta(4s-m-j)\\zeta(4s-2j)^{-1}).\n\\end{align}\nSimilar to that in case $F_{0,m,-1,0}^{(m)}(z,s)$, we define the function $\\beta_m(y,s)$ as\n$\\alpha_m(y,s)$:\n$$\nF_{0,m,0}^{(m)}(z,s)=\\Gamma(2s-m)\\zeta(4s-2m+1)\\,\\beta_m(y,s).\n$$\nExplicitly,\n\\begin{align}\n\\label{betam}\n\\beta_m(y,s):&=2^{-2ms+m(m+3)\/2}\\pi^{(m^2+2m-1)\/2}\\text{det}(y)^{-s+(m+1)\/2}\\nonumber \\\\\n                            & \\cdot \\Gamma_{m-1}(2s-\\kappa(m))\\Gamma_{m}(s)^{-2}\\zeta(2s)^{-1} \\nonumber \\\\\n                            &\\cdot \\prod_{j=1}^{m-2}\\zeta(4s-m-j)\\cdot\\prod_{j=1}^{m-1}\\zeta(4s-2j)^{-1}.\n\\end{align}\nFunctions $\\Gamma(2s-m)$ and $\\zeta(4s-2m+1)$ have a simple pole at $s=m\/2$, respectively,\nand $\\beta_m(y,s)$ is holomorphic at $s=m\/2$. Consequently, we observe that $F_{0,m,0}^{(m)}(z,s)$,\nhas a double pole at $s=m\/2$, as in the previous case.\n\nWe set\n$$\nF_{0,m,0}^{(m)}(z,s)=\\sum_{l=-2}^\\infty B_l^{(m)}(y)(s-m\/2)^l\n$$\nand calculate $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$.\n\\begin{Prop} \n\\label{B-2-1}\n{\\it The explicit forms of $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$ are as follows:\n\\begin{align}\n& B_{-2}^{(m)}(y) = \\frac{1}{8}\\,\\beta_m(y,m\/2),\n\\label{EB-2}\n\\\\\n& B_{-1}^{(m)}(y) \n=\\frac{\\gamma}{4}\\,\\beta_m(y,m\/2)+\\frac{1}{8}\\,\\beta'_m(y,m\/2)\n\\label{EB-1}\n\\end{align}\nwhere\n$\\beta'_m(y,m\/2)=\\left.\\frac{d}{ds}\\beta_m(y,s)\\right|_{s=m\/2}$.}\n\\end{Prop}\n\\begin{proof}\nFunction $\\Gamma(2s-m)\\zeta(4s-2m+1)$ has the Laurent expansion as\n\\begin{align*}\n &\\Gamma(2s-m)\\zeta(4s-2m+1) \\\\\n & = \\frac{1}{8}\\,(s-m\/2)^{-2}+\\frac{\\gamma}{4}\\,(s-m\/2)^{-1}+(\\text{a holomorphic function at}\\; s=m\/2). \n\\end{align*}\nThe formulas for $B_{-2}^{(m)}(y)$ and $B_{-1}^{(m)}(y)$ are obtained from this expression.\n\\end{proof}\nAn important point is the following relationship between $A_{-2}^{(m)}(y)$ and\\\\\n $B_{-2}^{(m)}(y)$.\n\\begin{Prop}\n{\\it The following identity holds.}\n\\begin{equation}\n\\label{A=-B}\nA_{-2}^{(m)}(y)=-B_{-2}^{(m)}(y).\n\\end{equation}\n\\end{Prop}\n\\begin{proof}\nA direct calculation shows that\n\\begin{align}\nA_{-2}^{(m)}(y) &= 2^{(-m^2+4m-8)\/2}\\pi^{(m^3+3m-2)\/4}\\text{det}(2y)^{1\/2}\\zeta(m)^{-1} \\nonumber \\\\\n        & \\cdot \\prod_{i=2}^{m-1}\\Gamma(i\/2)\\prod_{j=0}^{m-2}\\Gamma((m-j)\/2)^{-2}\\prod_{i=2}^{m-1}\\zeta(i)\n        \\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\\label{ExplicitA-2}\n\\end{align}\nMeanwhile,\n\\begin{align}\nB_{-2}^{(m)}(y) &= -2^{(-m^2+4m-8)\/2}\\pi^{(m^3+3m)\/4}\\text{det}(2y)^{1\/2}\\zeta(m)^{-1} \\nonumber \\\\\n        & \\cdot \\prod_{j=0}^{m-2}\\Gamma((m-1-j)\/2)\\prod_{j=0}^{m-1}\\Gamma((m-j)\/2)^{-2}\\nonumber \\\\\n        & \\cdot  \\prod_{j=1}^{m-2}\\zeta(m-j)  \\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\\label{ExplicitB-2}\n\\end{align}\nNoting that\n$$\n\\prod_{j=0}^{m-2}\\Gamma((m-1-j)\/2)=\\Gamma(1\/2)\\cdot\\prod_{i=2}^{m-1}\\Gamma(i\/2),\n$$\nwe conclude that $A_{-2}^{(m)}(y)=-B_{-2}^{(m)}(y)$.\n\\end{proof}\nFrom this proposition, we observe that the singularity of function\n$$\nF_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s)\n$$\nat $s=m\/2$ is a simple pole.\n\\begin{Thm}\n{\\it The residue of the constant term is as follows: }\n\\begin{align}\n& \\underset{s=m\/2}{{\\rm Res}}\\sum_{\\nu=0}^{m}F_{0,\\nu,0}^{(m)}(z,s)\n  =\\underset{s=m\/2}{{\\rm Res}}(F_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s)) \\nonumber \\\\\n&=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\nonumber \\\\\n   &\\quad +\\frac{1}{8\\,}\\beta'_m(y,m\/2).\n    \\label{ResConst}\n\\end{align}\n\\end{Thm}\n\\begin{proof}\nWe have\n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}(F_{0,m-1,0}^{(m)}(z,s)+F_{0,m,0}^{(m)}(z,s))\n   =A_{-1}^{(m)}(y)+B_{-1}^{(m)}(y)\\\\\n&=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)+\\frac{\\gamma}{4}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha_m(y,m\/2)\\\\\n& \\quad +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)+\\frac{1}{8}\\,\\beta'_m(y,m\/2)+\n      \\frac{\\gamma}{4}\\,\\beta_m(y,m\/2).\n\\end{align*}\nBy the identity (\\ref{A=-B}), the sum of the second and fifth terms in the last formula is equal to zero.\nThis implies (\\ref{ResConst}).\n\\end{proof}\n\\subsection{Calculation of $\\boldsymbol{F_{0,m,1}^{(m)}(z,s)}$}\n\\label{sec:4-4}\nWe have one more non-holomorphic term, that is, $F_{0,m,1}^{(m)}(z,s)$.\n\\begin{Prop}\n{\\it Function $F_{0,m,1}^{(m)}(z,s)$ has the following expression:}\n\\begin{align}\n& F_{0,m,1}^{(m)}(z,s) \\nonumber \\\\\n&  =2^m\\pi^{m\\kappa(m)}\\text{det}(y)^s\\Gamma_m(s)^{-2}\\zeta(2s)^{-1}\n      \\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\cdot\\zeta(4s-2j)^{-1}) \\nonumber \\\\\n&\\cdot \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_{m-2s}(\\text{cont}(h))\\cdot \\eta_m(2y,\\pi h;s,s)\n          \\boldsymbol{e}(\\sigma (hx)),\n\\end{align}\n{\\it where $\\sigma_s(a)=\\sum_{0<d|a}d^s$ and for $0\\ne h\\in\\Lambda_m$, \n${\\rm cont}(h):={\\rm max}\\{l\\in\\mathbb{N}\\,\\mid\\,l^{-1}h\\in\\Lambda_m\\}$.}\n\\end{Prop}\n\\begin{proof}\nBy definition (\\ref{Fb1}),\n\\begin{align}\n& F_{0,m,1}^{(m)}(z,s) \\nonumber \\\\\n& =2^m\\pi^{m\\kappa(m)+(m-1)\/2}\\Gamma_{m-1}(2s-\\kappa(m))\\Gamma_m(s)^{-2}\n\\text{det}(y)^s\\text{det}(2y)^{\\kappa(m)-2s} \\nonumber \\\\\n&\\quad \\cdot\\sum_{0\\ne h\\in\\mathbb{Z}}\\sum_{w\\in\\mathbb{Z}_{\\text{prim}}^{(m,1)}\/\\{\\pm 1\\}}\n                   S_m(\\text{diag}(h,0_{m-1}),2s) \\nonumber \\\\\n& \\qquad\\qquad \\cdot (2y[w])^{2s-m}\\eta_1(2y[w],\\pi h;s-(m-1)\/2,s-(m-1)\/2)\\boldsymbol{e}(\\sigma(h[{}^tw]x).\n\\label{DefF0m0}\n\\end{align}\nIt follows from (\\ref{Siegelreduce}) that\n\\begin{align}\n& S_m(\\text{diag}(h,0_{m-1}),2s)\\nonumber \\\\\n& =\\zeta(2s-m+1)\\zeta(2s)^{-1}\\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\,\\zeta(4s-2j)^{-1})\n        S_1(h,2s-m+1) \\nonumber \\\\\n        &=\\sigma_{m-2s}(h)\\zeta(2s)^{-1}\\prod_{j=1}^{m-1}(\\zeta(4s-m-j)\\,\\zeta(4s-2j)^{-1}).\n        \\label{SiegelS}\n\\end{align}\nIn the above, we used the formula $S_1(h,s)=\\zeta(s)^{-1}\\sigma_{1-s}(h)$ for $h\\in\\mathbb{Z}_{>0}$.\n\nFrom Proposition \\ref{xi-eta}, function $\\eta_1$ is expressed as\n\\begin{align}\n& \\eta_1(2y[w],\\pi h;s-(m-1)\/2,s-(m-1)\/2) \\nonumber \\\\\n& =\\pi^{-(m-1)\/2}\\Gamma_{m-1}(2s-\\kappa(m))^{-1}\\text{det}(2y)^{2s-\\kappa(m)}(2y[w])^{-2s+m} \\nonumber \\\\\n&\\quad \\cdot \\eta_m(2y,\\pi h[{}^tw],s,s) . \\label{eta1}\n\\end{align}\nSubstituting (\\ref{SiegelS}) and (\\ref{eta1}) into (\\ref{DefF0m0}), we obtain the following expression:\n\\end{proof}\nFrom the above proposition, we obtain the following result:\n\\begin{Thm}\n{\\it Function $F_{0,m,1}^{(m)}(z,s)$ has a simple pole at $s=m\/2$. }\n\\begin{align}\n \\underset{s=m\/2}{{\\rm Res}}F_{0,m,1}^{(m)}(z,s)  &=\n2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1} \\nonumber \\\\\n& \\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1} \\nonumber \\\\\n& \\cdot \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n\\boldsymbol{e}(\\sigma(hx)). \\label{Main2}\n\\end{align}\n\\end{Thm}\n\\begin{proof}\nIn the expression of $F_{0,m,1}^{(m)}(z,s)$, only the last factor $\\zeta(4s-2m+1)$ in the product\n$\\prod_{j=1}^{m-1}\\zeta(4s-m-j)$ has a simple pole at $s=m\/2$ with residue $1\/4$.\nFrom this fact, we obtain (\\ref{Main2}).\n\\end{proof}\n\\subsection{Conclusion}\n\\label{sec:4-5}\nWe summarize our results in the previous sections.\n\nThe following is a main result of this study.\n\\begin{Thm}\n\\label{Conclusion}\n\\begin{align*}\n& \\underset{s=m\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n&=\\mathbb{A}^{(m)}(y)+\\mathbb{B}^{(m)}(y)\n                \\sum_{h\\in\\Lambda_m^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_m(2y,\\pi h;m\/2,m\/2)\n                \\boldsymbol{e}(\\sigma(hx)),\n\\end{align*}\n{\\it where}\n\\begin{align*}\n\\mathbb{A}^{(m)}(y) &=\\frac{1}{2}\\,\\alpha_m(y,m\/2)\\cdot C_{m-1}^{(m)}(y)\n                                                    +\\frac{1}{8}\\,v(m-1)\\text{det}(2y)^{-(m-1)\/2}\\alpha'_m(y,m\/2)\\\\\n                         &+\\frac{1}{8}\\,\\beta'_m(y,m\/2),\\\\\n\\mathbb{B}^{(m)}(y) &=2^{m-2}\\pi^{m\\kappa(m)}\\text{det}(y)^{m\/2}\\Gamma_m(m\/2)^{-1}\\zeta(m)^{-1}\\\\\n                          &\\cdot \\prod_{j=1}^{m-2}\\zeta(m-j)\\,\\prod_{j=1}^{m-1}\\zeta(2m-2j)^{-1}.\n \\end{align*}                                                                          \n\\end{Thm}\n\\section{Remarks}\n\\subsection{Low degree cases}\nIn this section, we provide more explicit formulas for $\\text{Res}_{s=m\/2}E_0^{(m)}(z,s)$.\n$m=2,\\,3$. We used the notations in the previous sections as they are.\n\\subsubsection{Case $\\boldsymbol{m=2}$}\n\\label{Degree2}\nIn this case, the constant $C_1^{(2)}(y)$ appearing in the term $\\mathbb{A}^{(2)}(y)$\ncan be expressed more explicitly because we can apply the Kronecker limit formula.\nFor $g\\in P_2$, we consider the Epstein zeta function\n$$\n\\zeta_g(s):=\\sum_{\\boldsymbol{0}\\ne a\\in\\mathbb{Z}^{(2,1)}\/\\{\\pm 1\\}}g[a]^{-s},\n\\qquad\\quad \\text{Re}(s)>1.\n$$\nThe first Kronecker limit formula asserts that $\\zeta_g(s)$ has the following expression:\n\\begin{align}\n& \\zeta_g(s)=\\frac{1}{2}(4\\text{det}(g))^{-s\/2}\\left[\\frac{2\\pi}{s-1}+4\\pi\\beta(g)+O(s-1) \\right]\n\\label{Epstein}\\\\\n& \\beta(g)=\\gamma+\\frac{1}{2}\\log\\frac{v'}{2\\sqrt{\\text{det}(g)}}-\\log|\\eta(W_g)|^2.\n\\end{align}\nHere, for $g=\\begin{pmatrix} v' & w \\\\ w & v \\end{pmatrix}\\in P_2$,\n$$\nW_g:=\\frac{w+i\\sqrt{\\text{det}(g)}}{v'}\\in H_1,\n$$\nand $\\eta(z)$ is the Dedekind eta function:\n$$\n\\eta(z)=\\boldsymbol{e}(z\/24)\\prod_{n=1}^\\infty (1-\\boldsymbol{e}^n(z)),\\qquad z\\in H_1.\n$$\nThe relationship between the complete zeta function $\\xi_1^{(2)}(g,s)$ and $\\zeta_g(s)$\nis expressed as follows:\n$$\n\\xi_1^{(2)}(g,s)=\\xi(2s)\\zeta_1^{(2)}(g,s)=\\pi^{-s}\\Gamma(s)\\zeta_g(s).\n$$\nTherefore, the constant $C_1^{(2)}(y)$, which is the constant term of $\\xi_1^{(2)}(2y,s)$, \ncan be expressed as\n$$\nC_1^{(2)}(y)=\\frac{1}{2}(4\\text{det}(y))^{-1\/2}\\left(\\gamma+\\log\\frac{v'}{8\\pi}-\\log(\\text{det}(y))\n                -2\\log|\\eta(W_g)|^2\\right).\n$$\nConcerning $\\alpha_m(y,m\/2)$, $\\alpha'_m(y,m\/2)$, $\\cdots$ appearing in $\\mathbb{A}^{(m)}(y)$, \nwe can calculate them explicitly as\n\\begin{align*}\n\\alpha_2(y,1) &=72\\pi^{-2}\\text{det}(y)\\,\\zeta(2)^{-2},\\\\\n\\alpha'_2(y,1) &=2\\pi^2\\text{det}(y)\\,\\zeta(2)^{-2}(2\\log\\pi+\\log(\\text{det}(y))+2\\gamma-6\\zeta'(2)\\,\\zeta(2)^{-1}),\\\\\n\\beta_2(y,1) &=-\\pi^2\\text{det}(y)^{1\/2}\\,\\zeta(2)^{-2},\\\\\n\\beta'_2(y,1) &= 2\\pi^2\\text{det}^{1\/2}\\,\\zeta(2)^{-2}\n                     \\left(2\\log 2+\\tfrac{1}{2}\\log(\\text{det}(y))-\\gamma+2\\zeta'(0)+3\\zeta'(2)\\zeta(2)^{-1}\\right).\n\\end{align*}\nFrom these formula,\n\\begin{align*}\n& \\mathbb{A}^{(2)}(y)=\\underset{s=1}{{\\rm Res}}(F_{0,1,0}^{(2)}(z,s)+F_{0,2,0}^{(2)}(z,s))\\\\\n& =\\frac{1}{2}\\,\\alpha_2(y,1)\\cdot C_1^{(2)}(y)+\\frac{1}{8}\\,v(1)\\cdot\\text{det}(y)^{-1\/2}\\alpha'_2(y,1)+\\frac{1}{8}\\,\\beta'_2(y,1)\\\\\n& =\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right).\n\\end{align*}\nCombining with $\\text{Res}_{s=1}F_{0,2,1}^{(3)}(z,s)$, we obtain\n\\begin{Prop}\n\\begin{align}\n \\underset{s=1}{{\\rm Res}}\\,E_0^{(2)}(z,s) &\n=\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right)\\nonumber\n\\\\\n& \\quad +\\frac{36\\text{det}(y)}{\\pi^2}\\sum_{h\\in\\Lambda_2^{(1)}}\\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;1,1)\n             \\boldsymbol{e}(\\sigma(hx)). \\label{Degrre2Main}\n\\end{align}\n\\end{Prop}\n\\begin{Rem}\nIn \\cite{Na}, the author provided a formula for $E_2^{(2)}(z,0)$ (Siegel Eisenstein series of degree 2 and weight 2):\n\\begin{align}\nE_2^{(2)}(z,0)=& 1-\\frac{18}{\\pi^2\\sqrt{\\text{det}(y)}}\\left(1+\\frac{1}{2}\\gamma+\\frac{1}{2}\\log\\frac{v'}{4\\pi}-\\log|\\eta(W_g)|^2 \\right) \\nonumber\\\\\n         &-\\frac{72}{\\pi^3}\\sum_{\\substack{0\\ne h\\in\\Lambda_2\\\\ \\text{discr}(h)=\\square}}\\varepsilon_h\n           \\sigma_0(\\text{cont}(h))\\eta_2(2y,\\pi h;2,0)\\boldsymbol{e}(\\sigma(hx))\\nonumber \\\\\n         &+288\\sum_{0\\ne h\\in \\Lambda_2}\\sum_{d\\mid\\text{cont}(h)}d\\,H\\left(\\frac{|\\text{discr}(h)|}{d^2}\\right)\n              \\boldsymbol{e}(\\sigma(hz)).\\label{Degree2Weight2}\n\\end{align}\nHere, $H(N)$ is the Kronecker--Hurwitz class number and $\\varepsilon_h=1\/2$ if $\\text{rank}(h)=1;$\n$=1$ if $\\text{rank}(h)=2$.\n\nIt is interesting that the same term appears in each Fourier coefficient in (\\ref{Degrre2Main}) and\n(\\ref{Degree2Weight2}).\n\\end{Rem}\n\\subsubsection{Case $\\boldsymbol{m=3}$}\n\\label{Degree3}\nFrom Theorem \\ref{Conclusion}, we can write \n\\begin{align*}\n& \\underset{s=3\/2}{{\\rm Res}}E_0^{(3)}(z,s)\\\\\n&=\\mathbb{A}^{(3)}(y)+\\mathbb{B}^{(3)}(y)\n                \\sum_{h\\in\\Lambda_3^{(1)}}\\sigma_0({\\rm cont}(h))\\eta_3(2y,\\pi h;3\/2,3\/2)\n                \\boldsymbol{e}(\\sigma(hx)).\n\\end{align*}\nThe quantities $\\mathbb{A}^{(3)}(y)$ and $\\mathbb{B}^{(3)}(y)$ are given as follows:\n\\begin{align*}\n& \\mathbb{A}^{(3)}(y) \\\\\n& = 2^3\\pi^4\\text{det}(y)^{3\/2}\\zeta(2)^{-1}\\zeta(3)^{-1}\\zeta(4)^{-1}\\cdot C_2^{(3)}(y)\n                         +2^{-2}\\pi^3\\text{det}(y)^{1\/2}\\zeta(3)^{-1}\\zeta(4)^{-1}\n\\\\\n& \\quad\\cdot (-2\\Gamma'(1)-4\\zeta'(2)\\zeta(2)^{-1}+4\\zeta'(0)+2\\log(\\text{det}(y))+4\\log\\pi+6\\log 2),\n\\vspace{2mm}\n\\\\\n& \\mathbb{B}^{(3)}(y)\\\\ \n& = 2^2\\pi^{7\/2}\\text{det}(y)^{3\/2}\\zeta(3)^{-1}\\zeta(4)^{-1}.\n\\end{align*}\n\\begin{Rem}\nIn the above formulas, we may substitute\n$$\n\\zeta(2)=\\pi^2\/6,\\quad \\zeta(4)=\\pi^4\/90,\\quad \\zeta'(0)=(-\\log 2\\pi)\/2,\\quad \\Gamma'(1)=-\\gamma.\n$$\n\\end{Rem}\n\\subsection{Residue at the other point}\n\\label{other}\nThe residue we considered above was to $s=m\/2$, and it is represented as a Fourier series.\nThe case $\\text{Res}_{s=(m+1)\/2}E_0^{(m)}(z,s)$ is easier than in the above case. In fact,\nit becomes a constant, explicitly\n\\begin{align*}\n& \\underset{s=(m+1)\/2}{{\\rm Res}}E_0^{(m)}(z,s)\\\\\n& =\\underset{s=(m+1)\/2}{{\\rm Res}}\\xi(2s-m)\\xi(2s)^{-1}\\prod_{j=1}^{[m\/2]}(\\xi(4s-2m-1+2j)\\,\\xi(4s-2j)^{-1}).\n\\end{align*}\n\\begin{Rem}\nKaufhold \\cite{Kau} noted that the residue of\n$$\n\\varPhi_0(s):=E_0^{(2)}(z,s\/2)\n$$\nat $s=3$ is $90\\pi^{-2}$. This is a special case of the above formula because\n$$\n \\underset{s=3\/2}{{\\rm Res}}E_0^{(2)}(z,s)= \\underset{s=3\/2}{{\\rm Res}}\\xi(2s-2)\\xi(2s)^{-1}\\xi(4s-3)\\xi(4s-2)^{-1}\n=\\frac{45}{\\pi^2}.\n$$\n\\end{Rem}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n{\\sc Gist}\\\/ (Game solver from IST) is a tool for (a)~qualitative analysis of \\emph{turn-based probabilistic\ngames ($2\\slopefrac{1}{2}$-player games)} with $\\omega$-regular objectives, and \n(b)~computing environment assumptions for synthesis of unrealizable \nspecifications. \nThe class of $2\\slopefrac{1}{2}$-player games arise in several important applications \nrelated to verification and synthesis of reactive systems. Some key \napplications are: (a) synthesis of stochastic reactive systems; \n(b) verification of probabilistic systems; and (c) synthesis of unrealizable\nspecifications. We believe that our tool will be useful for the above\napplications.\n\n\n\\smallskip\\noindent{\\bf $2\\slopefrac{1}{2}$-player games.} \n$2\\slopefrac{1}{2}$-player games are played on a graph by two players along with \nprobabilistic transitions.\nWe consider $\\omega$-regular objectives over infinite paths specified by \nparity, Rabin and Streett (strong fairness) conditions that can express \nall $\\omega$-regular properties such as safety, reachability, liveness, \nfairness, and most properties commonly used in verification. \nGiven a game and an objective, our tool determines\nwhether the first player has a strategy to ensure that the objective \nis satisfied with probability~1, and if so, it constructs \nsuch a witness strategy. Our tool provides the first implementation of \nqualitative analysis (probability~1 winning) of $2\\slopefrac{1}{2}$-player games \nwith $\\omega$-regular objectives.\n\n\\smallskip\\noindent{\\bf Synthesis of environment assumptions.}\nThe synthesis problem asks to construct a finite-state reactive \nsystem from an $\\omega$-regular specification. In practice, initial specifications \nare often unrealizable, which means that there is no system that implements \nthe specification. A common reason for unrealizability is that assumptions \non the environment of the system are incomplete. The problem of \ncorrecting an unrealizable specification $\\Psi$ by computing an environment \nassumption $\\Phi$ such that the new specification $\\Phi \\to \\Psi$ \nis realizable was studied in~\\cite{CHJ08}.\nThe work~\\cite{CHJ08} constructs an assumption $\\Phi$ that constrains only \nthe environment and is as weak as possible. \nOur tool implements the algorithms of~\\cite{CHJ08}.\nWe believe our implementation will be useful in analysis \nof realizability of specifications and computation of \nassumptions for unrealizable specifications.\n\n\n\n\\section{Definitions}\\label{section:definition}\nWe first present the basic definitions of games and objectives.\n\n\\smallskip\\noindent{\\bf Game graphs.} \nA \\emph{turn-based probabilistic game graph} (\\emph{$2\\slopefrac{1}{2}$-player game graph})\n$G =((S, E), (S_0,S_1,S_{P}),\\delta)$ consists of a directed graph \n$(S,E)$, a partition $(S_0$, $S_1$,$S_{P})$ of the finite set $S$ of states, \nand a probabilistic transition function $\\delta$: $S_{P} \\rightarrow {\\cal D}(S)$, \nwhere ${\\cal D}(S)$ denotes the set of probability distributions over the \nstate space~$S$. \nThe states in $S_0$ are the {\\em player-$0$\\\/} states, where player~$0$\ndecides the successor state; the states in $S_1$ are the {\\em \nplayer-$1$\\\/} states, where player~$1$ decides the successor state; \nand the states in $S_{P}$ are the {\\em probabilistic\\\/} states, where\nthe successor state is chosen according to the probabilistic transition\nfunction~$\\delta$. \nWe assume that for $s \\in S_{P}$ and $t \\in S$, we have $(s,t) \\in E$ \niff $\\delta(s)(t) > 0$.\nThe {\\em turn-based deterministic game graphs} (\\emph{2-player game graphs})\nare the special case of the $2\\slopefrac{1}{2}$-player game graphs with $S_{P} = \\emptyset$.\n\n\\smallskip\\noindent{\\bf Objectives.} We consider the three canonical\nforms of $\\omega$-regular objectives: Streett and its dual Rabin\nobjectives; and parity objectives.\nThe Streett objective consists of $d$ request-response pairs\n$\\set{(Q_1,R_1),(Q_2,R_2),\\ldots, (Q_d,R_d)}$ where $Q_i$ denotes a\nrequest and\n$R_i$ denotes the corresponding response (each $Q_i$ and $R_i$ are subsets of \nthe state space). The objective requires that if a request $Q_i$ happens \ninfinitely often, then the corresponding response must happen infinitely often.\nThe Rabin objective is its dual. \nThe parity (or Rabin-chain objective) is the special case of Streett objectives\nwhen the set of request-responses \n$Q_1 \\subset R_1 \\subset Q_2 \\subset R_2 \\subset  Q_3 \\subset \\cdots \\subset \nQ_d \\subset R_d$ form a chain.\n\n\\smallskip\\noindent{\\bf Qualitative analysis.} The qualitative analysis for \n$2\\slopefrac{1}{2}$-player games is as follows: the input is a \n$2\\slopefrac{1}{2}$-player game graph, and an objective $\\Phi$ (Streett, Rabin or parity \nobjective), and the output is the set of states such that player~0 \ncan ensure $\\Phi$ with probability~1.\nFor detailed description of game graphs, plays, strategies, objectives and \nnotion of winning see~\\cite{KrishThesis}.\nWe focus on qualitative analysis because:\na)~In applications like synthesis the relevant analysis is qualitative\nanalysis: the goal is to synthesize a system that behaves correctly\nwith probability~1; (b)~Qualitative analysis for probabilistic games is independent of \nthe precise probabilities, and thus robust with imprecision in the \nexact probabilities (hence resilient to probabilistic modeling errors). \nThe qualitative analysis can be done with discrete graph theoretic \nalgorithms.\nThus qualitative analysis is more robust and efficient, and our \ntools implements these efficient algorithms.\n\n\\section{Tool Implementation} \nOur tool presents a solution of the following two problems.\n\n\\smallskip\\noindent{\\bf Qualitative analysis of $2\\slopefrac{1}{2}$-player games.} \nOur tool presents the first implementation for the qualitative \nanalysis of $2\\slopefrac{1}{2}$-player games with Streett, Rabin and parity objectives.\nWe have implemented the linear-time reduction for qualitative analysis of \n$2\\slopefrac{1}{2}$-player Rabin and Streett games to $2$-player Rabin and Streett games \nof~\\cite{CdAH05}, and the linear-time reduction for $2\\slopefrac{1}{2}$-player \nparity games to $2$-player parity games of~\\cite{CJH04}.\nThe $2$-player Rabin and Streett games are solved by reducing them to the\n$2$-player parity games using the LAR (latest appearance records) \nconstruction~\\cite{GH82}. The $2$-player parity games are solved using the\ntool PGSolver~\\cite{Lange09}. \n\n\\smallskip\\noindent{\\bf Environment assumptions for synthesis.} \nOur tool implements a two-step algorithm for computing the environment assumptions\nas presented in~\\cite{CHJ08}. \nThe algorithm operates on the game graph that is used to answer the \nrealizability question. \nFirst, a safety assumption that removes a minimal set of \nenvironment edges from the graph is computed. \nSecond, a fairness assumption that puts fairness conditions on some of the\nremaining environment edges is computed. \nThe problem of finding a minimal set of fair edges is \ncomputationally hard~\\cite{CHJ08}, and a reduction to $2\\slopefrac{1}{2}$-player games \nwas presented in~\\cite{CHJ08} to compute a locally minimal fairness assumption.\nThe details of the implementation are as follows: given an LTL formula $\\phi$,\nthe conversion to an equivalent deterministic\nparity automaton is achieved through GOAL~\\cite{Goal}. Our tool then converts\nthe parity automaton into a $2$-player parity game by splitting the states and\ntransitions based on input and output symbols. Our tool then computes the safety\nassumption by solving a safety model-checking problem. \nThe computation of the fairness assumption is achieved in the following steps:\n\\begin{compactitem}\n\\item Convert the parity game with fairness assumption\ninto a $2\\slopefrac{1}{2}$-player game.\n\\item Solve the $2 \\slopefrac{1}{2}$-player game (using our tool) to check whether the\nassumption is sufficient (if so, go to the previous step with a weaker fairness assumption).\n\\end{compactitem}\nThe synthesized system is obtained from a witness strategy of the parity\ngame. \nThe flow is illustrated in Figure~\\ref{fig:example}.\n\n\\begin{figure}\n\\begin{center}\n\\begin{pspicture}[showgrid=false](-1.5,0)(10,2.7)\n\\begin{psmatrix}[rowsep=1.0,colsep=1.5]\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{LTL Formula} & \n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Det. Parity \\\\ Aut. \\endtabular} & \n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Synthesis \\\\ Game \\endtabular} \\\\\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{Synthesized System}  &\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} $2\\slopefrac{1}{2}$-player \\\\ game \\endtabular} &\n\\psframebox[linearc=0.2,cornersize=absolute,framesep=2pt]{\\tabular{c} Safe \\\\ Synthesis \\\\ Game \\endtabular} \\\\\n\\end{psmatrix}\n\\ncline[linestyle=dashed]{->}{1,1}{1,2}\\naput{GOAL} \\ncline{->}{1,2}{1,3}\n\\ncline{->}{1,3}{2,3} \\ncline{->}{2,3}{2,2} \\ncline{->}{2,2}{2,1}\n\\ncloop[angleA=180,angleB=-270,loopsize=-1,linestyle=dashed]{->}{2,2}{2,2}\\naput{\\tiny{Assumption not locally minimal}}\n\\end{pspicture}\n\\end{center}\n\\caption{An example illustrating the flow of the tool}\n\\label{fig:example}\n\\end{figure}\n\n\\begin{figure}[h]\n\\subfigure[Deterministic Parity Automaton]{\n\\begin{picture}(35,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(5,15){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5](1)(20,25){1}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](2)(20,5){2}\n\t\\drawloop[loopdiam=2,ELdist=0.5](0){$\\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,1){$c \\wedge g$}\n\t\\drawedge[curvedepth=1,ELdist=0](0,2){$\\neg c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](1){$T$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,0){$\\neg g$}\n\t\\drawedge[curvedepth=-2,ELdist=0.5,ELside=r](2,1){$c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](2){$\\neg c \\wedge g$}\n\\end{picture}\n\\label{fig:parity_automaton}\n}\n\\subfigure[$2$-player Parity Game]{\n\\begin{picture}(40,30)(0,0)\n\t\\rpnode[Nmarks=i,fangle=45](0)(5,15)(4,3){0}\n\t\\rpnode[fangle=45,Nmarks=r](1)(25,5)(4,3){1}\n\t\\rpnode[fangle=45](2)(25,25)(4,3){2}\n\t\\rpnode[polyangle=45](3)(15,25)(4,3){3}\n\t\\rpnode[polyangle=45](4)(15,5)(4,3){4}\n\t\\rpnode[polyangle=45](5)(35,25)(4,3){5}\n\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,3){$c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](3,0){$\\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0,4){$\\neg c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,0){$\\neg g$}\n\t\\drawedge(3,2){$g$}\n\t\\drawedge(1,3){$c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,1){$g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](1,4){$\\neg c$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](5,2){$\\mbox{T}$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,5){$\\mbox{T}$}\n\\end{picture}\n\\label{fig:synthesis_game}\n}\n\\subfigure[A $2\\frac{1}{2}$-player game obtained for the fairness\n\tassumption which contains only $(0,4)$]{\n\\begin{picture}(50,30)(0,0)\n\t\\rpnode[Nmarks=i,fangle=45](0)(5,25)(4,3){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5](0d)(5,5){$\\overline{0}$}\n\t\\rpnode[fangle=45,Nmarks=r](1)(25,5)(4,3){1}\n\t\\rpnode[fangle=45](2)(25,25)(4,3){2}\n\t\\rpnode[polyangle=45](3)(15,25)(4,3){3}\n\t\\rpnode[polyangle=45](4)(15,5)(4,3){4}\n\t\\rpnode[polyangle=45](5)(35,25)(4,3){5}\n\n\t\\drawedge(0,3){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](0d,4){}\n\t\\drawedge(0d,0){}\n\t\\drawedge(3,0d){}\n\t\\drawedge(0,4){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,0d){}\n\t\\drawedge(3,2){}\n\t\\drawedge(1,3){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](4,1){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](1,4){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](5,2){}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,5){}\n\\end{picture}\n\\label{fig:probabilistic_game}\n}\\hfill\n\\subfigure[Environment Assumption]{\n\\begin{picture}(40,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(5,15){0}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](1)(20,25){1}\n\t\\node[Nw=5,Nh=5,Nmr=2.5,Nmarks=r](2)(20,5){2}\n\t\\drawloop[loopdiam=2,ELdist=0.5](0){$c \\wedge \\neg g$}\n\t\\drawedge[curvedepth=1,ELdist=0.5,ELside=r](0,1){$c \\wedge g$}\n\t\\drawedge[curvedepth=1,ELdist=0](0,2){$\\neg c$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](1){$T$}\n\t\\drawedge[curvedepth=1,ELdist=0.5](2,0){$c \\wedge \\neg g$}\n\t\\drawedge[curvedepth=-2,ELdist=0.5,ELside=r](2,1){$c \\wedge g$}\n\t\\drawloop[loopdiam=2,ELdist=0.5,loopangle=0](2){$\\neg c$}\n\\end{picture}\n\\label{fig:environment_assumption}\n}\\hfill\n\\subfigure[Transducer System]{\n\\begin{picture}(20,30)(0,0)\n\t\\node[Nmarks=i,Nw=5,Nh=5,Nmr=2.5](0)(10,15){0}\n\t\\drawloop[loopdiam=2.5,ELdist=0.5](0){$\\neg c \/ g$}\n\t\\drawloop[loopdiam=2.5,ELdist=0.5,loopangle=0](0){$c \/ \\neg g$}\n\\end{picture}\n\\label{fig:transducer_system}\n}\n\\caption{An example that illustrates the tool flow}\n\\label{fig:worked_example}\n\\end{figure}\n\nWe illustrate the working of our tool on a simple example shown in\nFigure~\\ref{fig:worked_example}\nConsider an LTL formula $\\Phi=GF \\mathtt{grant} \\wedge G(\\mathtt{cancel} \\to \\neg \n\\mathtt{grant})$, where $G$ and $F$ denote globally and eventually, respectively.\nThe propositions \\texttt{grant} and \\texttt{cancel} are abbreviated as \\texttt{g} \nand \\texttt{c}, respectively. \nFrom $\\Phi$ our tool obtains a deterministic parity automaton (Figure~\\ref{fig:parity_automaton})\nthat accepts exactly the words that satisfies $\\Phi$. \nThe parity automaton is then converted into a parity game. In Figure~\\ref{fig:synthesis_game},\n$\\Box$ represents player~0 states and $\\Diamond$ represents player~1 states. \nIt can be shown that in this game no safety assumption required.\nWe illustrate how to compute a locally minimal fairness assumption. \nGiven an fairness assumption on edges, our tool reduces the game with the assumption to a $2\\slopefrac{1}{2}$-player parity game\n(see details in~\\cite{CHJ08}). If the initial state in the $2\\slopefrac{1}{2}$-player game is in winning with probability~1 \nfor player~0, then the assumption is sufficient. Figure~\\ref{fig:probabilistic_game} illustrates \nthe $2\\slopefrac{1}{2}$-player game obtained with the fairness assumption on the edge\n$(0,4)$. The $\\bigcirc$ state is \nthe probabilistic state with uniform distribution over its successors.\nThe assumption on this edge is the minimal fairness assumption for the\nexample. Our tool then converts this game back into an automaton to\nobtain the environment assumption as an\nautomaton(Figure~\\ref{fig:environment_assumption}). This assumption\nis equivalent to the formula $G(\\neg (\\mathtt{cancel \\wedge grant})) \\implies GF(\\mathtt{\\neg cancel})$. From a witness\nstrategy in Figure~\\ref{fig:probabilistic_game}\nour tool obtains the system that implements the specification with the\nassumption (Figure~\\ref{fig:transducer_system}).\n\n\\smallskip\\noindent{\\bf Performance of {\\sc Gist}.} Our implementation of reduction \nof $2\\slopefrac{1}{2}$-player games to $2$-player games is linear time and efficient, and\nthe computationally expensive step is solving $2$-player games. \nFor qualitative analysis of $2\\slopefrac{1}{2}$-player games, {\\sc Gist}\\ can handle game \ngraphs of size that can be typically handled by tools solving $2$-player games.\nTypical run-times for qualitative analysis of $2\\slopefrac{1}{2}$-player parity\ngames of various sizes are summarized in Table~\\ref{table:runtimes}. The\ngames used were generated using the benchmark tools of PGSolver and then\nconverting one-tenth of the states into probabilistic states. \n\n\\begin{table}[ht]\n\\begin{center}\n\\label{table:runtimes}\n\\begin{tabular}{|c|c||c|c|c|}\n\\hline\nStates & Edges & \\multicolumn{3}{|c|}{Runtime (sec.)} \\\\\n\\cline{3-5}\n&& Avg. & Best & Worst \\\\\n\\hline\n1000 & 5000 & 1.17 & 0.63 & 1.59 \\\\\n5000 & 25000 & 15.94 & 11.10 & 19.46 \\\\\n10000 & 50000 & 51.43 & 39.38 & 62.61 \\\\\n20000 & 100000 & 282.24 & 267.40 & 310.11 \\\\\n50000 & 250000 & 2513.18 & 2063.40 & 2711.23 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Runtimes for solving $2\\slopefrac{1}{2}$-player parity games}\n\\end{table}\nIn the case of synthesis of environment assumptions, the expensive step is the\nreduction of LTL formula to deterministic parity automata. Our tool can \nhandle formulas that are handled by classical tools for translation of LTL \nformula to deterministic parity automata.\n\n\n\n\n\\smallskip\\noindent{\\bf Other features of {\\sc Gist}.}\nOur tool is compatible with several other game solving and synthesis tools: \n{\\sc Gist}\\\/  is compatible with PGSolver and GOAL. Our tool provides a graphical \ninterface to describe games and automata, and thus can also be used as a \nfront-end to PGSolver to graphically describe games.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nCosmological observations, and in particular Cosmic Microwave\nBackground (CMB) measurements \\cite{COBE,CMB,WMAP}, are\nconsistent with a nearly gaussian and practically scale invariant\nspectrum of primordial perturbations, as predicted by the inflationary\nmodels \\cite{inflation, Linde:1983gd}. An early period of inflation also\naccounts for the inferred flatness of the Universe, and provides a\nsolution to the horizon problem. This makes inflation a robust\ncandidate to account for the early evolution of our\nuniverse. Inflationary predictions are characterized by the spectral\nindex of the primordial spectrum, its tensor contribution, and the\nlevel of non-gaussianity. Present CMB data however sets at\nmost an upper limit on the level of the tensor contribution and\nnon-gaussianity \\cite{WMAP}, and it is not yet able\nto discriminate among the different implementations and models of \ninflation. This situation is expected to change with the next\ngeneration of CMB experiments, like ESA's Planck surveyor satellite\n\\cite{planck}, which will further improve our knowledge of the cosmological\nparameters.      \n\nInflation in brief is no more than an early period of accelerated\nexpansion. In the standard picture of inflation, denoted as cold\ninflation, the universe rapidly \nsupercools, and inflation should be followed by a  \nreheating period, during which the inflationary vacuum energy is\nconverted into radiation. For reheating to take place, the inflaton field has to\ncouple to other degree of freedom, such that it can decay into light,\nrelativistic degrees of freedom that thermalize and provide the\nradiation bath \\cite{reheating}. During cold inflation one assumes that those couplings\nplay no role during the accelerated expansion. The alternative, called\nwarm inflation \\cite{bf1,wi}\n(for earlier related work see \\cite{prewarm}), assumes on the contrary that\nthose coupling can lead to non-negligible dissipative effects, and radiation\nproduction can occur concurrently with the inflationary expansion. \nBoth background evolution and inflaton fluctuations are modified with\nthe inclusion of an extra friction term $\\Upsilon \\dot \\phi$\naccounting for the transfer of energy between the inflation and the radiation. \nThe dynamics of the fluctuations are now governed by a\nLangevin equation including  a noise force term from the influence\nof the radiation fluctuations into the inflaton field \n\\cite{bf1,langevin,warmdeltap,BR1,langevin2}. Thermal fluctuations in\nthe radiation are transfered to the inflaton and become the main\nsource of primordial fluctuations \\cite{bf1,warmdeltap,warmpert}.  \n\nThe dissipative coefficient can be computed from first principles in\nquantum field theory, within an adiabatic approximation. The two-stage\ninteraction configuration proposed in \\cite{BR1} has been\nshown to lead to a large enough dissipative coefficient while keeping\nthe corrections to the inflationary potential under control and\nallowing a period of slow-roll inflation\n\\cite{br05,Hall:2004zr,Moss:2008yb}. \nFor the two-stage mechanism, the inflaton field \ncouples to a heavy catalyst field, and the latter in\nturn couples to light degrees of freedom. During the motion of the\nbackground inflaton, it excites the catalyst fields which then decay\ninto light fields \\cite{Berera:2008ar}. \nUsing this mechanism, the first calculation of the\ndissipative coefficient within the close-to-equilibrium approximation\nwas done in \\cite{mossxiong}, leading to a temperature dependent\ndissipative coefficient. In the low-temperature regime, when the mass\nof the heavy catalyst field is much larger than $T$, one has $\\Upsilon \\propto\nT^3$, while in the high-temperature regime,\nwhen $T$ is above the heavy catalyst field\nmass, depending on type of interaction the  dissipative coefficient\nbecomes linear with $T$ \\cite{BasteroGil:2010pb}\nor goes goes like the inverse of $T$ \\cite{BGR}. All high-$T$ models\nsuffer in general from very large thermal corrections which spoil the\nflatness of the potential \\cite{BGR,Yokoyama:1998ju},\nwith only a few exceptions \\cite{warmdeltap,Berera:1998px}.\nHowever, viable\nmodels of warm inflation  have been studied in the low-$T$ regime \n\\cite{BasteroGil:2009ec,Zhang:2009ge,joao}.  \n\nThe temperature dependence of the dissipative coefficient induces the\ncoupling of the field and radiation fluctuations as shown in\n\\cite{mossgraham}. Previous studies of the primordial spectrum of\nperturbations in warm inflation \\cite{warmpert} did take into account\nthe influence of the thermal fluctuations on the field through the\nnoise term, but  not the coupling through the dissipative term\nitself. In \\cite{mossgraham} it was shown that for positive power of $T$\nin $\\Upsilon$, and when $\\Upsilon$ dominates over the Hubble expansion\nrate, this coupling induces a growing mode in the fluctuations before\nhorizon crossing, enhancing by several order of magnitude the\namplitude of the primordial perturbations with respect to previous\ncalculations. In the calculations of the primordial spectrum\ntypically the radiation bath is treated as a perfect fluid, with an\nequilibrium pressure $p_r \\simeq \\rho_r\/3$, where $\\rho_r$ is the radiation\nenergy density, an approximation valid in the close-to-equilibrium\nregime required for the calculation of the dissipative coefficient to\nhold. However, even in that regime,  the radiation bath is expected to\ndepart from  an ideal fluid as a consequence of the constant\nproduction of radiation particles from  the background field\ndissipation.  Imperfect fluids have dissipative\neffects that can be parameterized in terms of shear and bulk viscosity\ncoefficients, and a heat flow coefficient.  Heat flow happens as a \nconsequence of changes of conserved charges other than the\nstress-energy tensor, but we do not consider this possibility here and\nfocus on the effects of the temperature. Bulk viscous effects, which\ncan be interpreted as a  consequence of the decay of particles within\nthe fluid, have been considered for warm inflation in\n\\cite{delCampo:2010by}, where \nthey studied either a constant bulk viscous pressure or one proportional\nto the radiation energy density.  The bulk\npressure appears at both the background and the perturbation level,\nand being a negative pressure, it will favor warm inflation. For the\namplitude of the spectrum, for the phenomenological model considered\nin \\cite{delCampo:2010by} they found that it could induce a variation in\nthe amplitude of the order of 4\\%. On the other hand, the shear\nviscosity is related to changes in momentum of the  particles of the\nfluid, \nand appears only at the level of the perturbations.     \nShear and bulk viscosity coefficients due to light field have been extensively\ncompute in the literature \\cite{jeon,shear}, leading to power-law\ndependences on $T$ for these coefficients. In addition, the bulk\nviscous coefficient typically ends being much smaller than the shear\nviscosity. Therefore, we will concentrate on the effect of the shear\nviscosity on the spectrum, and do not consider those of the bulk\nviscosity. The shear viscosity, being related to dissipation, appears\nin the radiation fluid equation as a friction term which tends to damp\nthe growth of the fluctuations \\cite{mossgraham}. Eventually, the\ndamping effect will dominate \nover  the enhancement induced by the dissipative source term. The aim\nwill be therefore to quantify, in a model independent way, this effect\non the spectrum, and when it will render the system field-radiation\neffectively decoupled.     \n\nThe outline of the paper is as follow. In section II we review the\nbasic of warm inflation at the background level. In section III we\npresent the equations for the fluctuations of the coupled\nfield-radiation system, when the radiation is taken as an imperfect\nfluid. The numerical solutions for the fluctuations are presented in\nsection IV, with and without the shear viscosity. We  also study\nthe inflationary model dependence of the results on the\nspectrum by considering two generic model of inflation: a chaotic\nmodel with general power $p$, and a standard quadratic hybrid model. \nIn section V we summarize our findings.\n\n\n\n\\section{Warm inflation: background equations}\n\\label{sect2}\n\nIn any particle physics realization of the inflationary framework, the\ninflaton is not an isolated part of the model but it interacts with\nother fields.  These interactions  may \nlead to the dissipation of the inflaton energy into other degrees of\nfreedom, such that a small percent of the inflaton vacuum energy is\ntransferred into other kinds of energy. In the two-stage mechanism for\nwarm inflation, dissipation leads to particle production of light\ndegrees of freedom. When those relativistic particles thermalize fast\nenough, say in less than a Hubble time in an expanding universe, we\ncan model their contribution as that of radiation:\n\\begin{equation}\n\\rho_r \\simeq \\frac{\\pi^2}{30}g_* T^4 \\,,\n\\end{equation}\nwhere $T$ is the temperature of the thermal bath, and $g_*$ the\neffective number of light degrees of freedom\\footnote{If not otherwise\nspecified, we will take $g_* = 228.75$, the effective no. of degrees of\nfreedom for the Minimal Supersymmetric Standard Model, when presenting\nnumerical results.}.  \n\nThe dissipative term appears as an\nextra friction term in the evolution equation for the inflaton field $\\phi$,\n\\begin{equation}\n\\ddot \\phi + ( 3 H + \\Upsilon ) \\dot \\phi + V_\\phi =0\n\\,,\\label{eominf}\n\\end{equation}\n$\\Upsilon$ being the dissipative coefficient, $H=\\dot a\/a$ is the\nHubble rate of expansion, and\n$a$ the scale factor of the Friedmann-Robertson-Walker background\nmetric:\n\\begin{equation}\nds^2 = - dt^2 + a(t)^2 \\delta_{ij}dx^i dx^j \\,.\n\\end{equation} \nEq. (\\ref{eominf})\nis equivalent to the evolution equation for the inflaton energy\ndensity $\\rho_\\phi$:\n\\begin{equation}\n\\dot \\rho_\\phi + 3 H ( \\rho_\\phi + p_\\phi) = - \\Upsilon ( \\rho_\\phi +\np_\\phi) \\,,\n\\label{rhoinf}\n\\end{equation}\nwith pressure $p_\\phi = \\dot \\phi^2\/2 - V(\\phi)$, and $\\rho_\\phi\n+ p_\\phi= \\dot \\phi^2$. Energy conservation then demands that the\nenergy lost of the inflaton field must be gained by the radiation fluid\n$\\rho_r$, with the  RHS of Eq. (\\ref{rhoinf}) acting as\nthe source term:\n\\begin{equation}\n\\dot \\rho_r+ 3 H ( \\rho_r + p_r) = \\Upsilon ( \\rho_\\phi +\np_\\phi) \\,. \\label{eomrad}\n\\end{equation}\nIn warm inflation, radiation  is not\nredshifted away during inflation,\nbecause it is continuously fed by the inflaton \nthrough the dissipation \\cite{wi}. \nInflation happens when $\\rho_R \\ll \\rho_\\phi$, but even if small when\ncompared to the inflaton energy density it can be larger than the\nexpansion rate with $\\rho_R^{1\/4} > H$. Assuming thermalization, this\ntranslates roughly  into $T > H$. Otherwise, when $T < H$ (or\nsimilarly when $\\rho_R^{1\/4} < H$), one just recovers the standard cold\ninflation scenario, where dissipation can be neglected.\n\nDuring warm inflation the motion of the inflaton field has to be\noverdamped in order to have the accelerated expansion, but now this\ncan be achieved due to \nthe extra friction term $\\Upsilon$ instead of that of the Hubble\nrate. And once $\\phi$, $H$, and also $\\Upsilon$, are in this slow-roll\nregime, the\nsame will happen with the radiation energy density, the source term\nnow compensating for the Hubble dilution. In the slow-roll regime, the\nequations of motion reduce to:\n\\begin{align}\n3 H ( 1 + Q ) \\dot \\phi &\\simeq  -V_\\phi    \\,,\\label{eominfsl} \\\\\n4 \\rho_R  &\\simeq 3 Q\\dot \\phi^2\\,, \\label{eomradsl}\n\\end{align}\nwhere we have introduced the dissipative ratio $Q=\\Upsilon\/(3 H)$. \nNotice that\n $Q$ is not necessarily constant. The\ncoefficient  $\\Upsilon$ will depend on  $\\phi$ and $T$, and \ntherefore depending on the model the ratio $Q$ may increase or\ndecrease during inflation \\cite{BasteroGil:2009ec}.\nThe slow-roll conditions are given by \\cite{Moss:2008yb}: \n\\begin{align}\n\\epsilon &= \\frac{m_P^2}{2} \\left ( \\frac{V_{\\phi}} {V}\n\\right)^2 \\frac{1}{1+Q}\\ll 1\\,, \\label{eps}\\\\\n\\eta &= m_P^2 \\left ( \\frac{V_{\\phi \\phi}} {V}\\right) \\frac{1}{1+Q}\n\\ll 1 \\,, \\label{eta} \\\\\n\\beta_\\Upsilon &= m_P^2 \\left ( \\frac{\\Upsilon_\\phi V_\\phi }\n     {\\Upsilon V}\\right) \\frac{1}{1+Q}\\ll 1 \\,, \\label{beta} \\\\\n\\delta &= \\frac{T V_{T\\phi}}{V_\\phi} < 1  \\label{delta}\\,,\n\\end{align}\nwhere the slow-roll parameter $\\beta_\\Upsilon$  takes into account the\nvariation of $\\Upsilon$ with respect to $\\phi$, and \nthe last condition ensures that  thermal corrections to\nthe inflation potential are negligible. Similarly, taking also into\naccount the dependence on $T$ of $\\Upsilon$, one has: \n\\begin{equation}\n\\left|\\frac{d \\ln \\Upsilon}{d \\ln T}\\right| < 4 \\,,\n\\end{equation}\nwhich reflects the fact that radiation has to be produced at a rate\nlarger than the redshift due to the expansion of the universe.   \nThe slow-roll regime ends when any of the above conditions\nEqs. (\\ref{eps})-(\\ref{delta}) is no longer satisfied, such that  either the\nmotion is no longer overdamped and slow-roll ends, or the radiation\nbecomes comparable to the inflaton energy density. Either way,\ninflation will  end shortly  afterwards. \n\nFor warm inflation the first requirement is to have $T>H$, but the\nratio $Q$ can be larger or smaller than unity. In the former case we\nare in the strong dissipative regime (SDR), whereas the latter is\ncalled weak dissipative regime (WDR). In the weak dissipative regime\nthe extra friction added by $\\Upsilon$ is not enough to substantially\nmodify the background inflaton evolution, and it will resemble that\nof cold inflation; still the thermal fluctuations of the\nradiation energy density will modify the field fluctuations, and\naffect the primordial spectrum of perturbations. \n\nThe $T$ and $\\phi$ dependent dissipative coefficient has been computed\nin \\cite{mossxiong}, using the near-equilibrium approximation\nproposed in \\cite{BGR}. The specific field theory models \nconsidered for the inflaton interactions leading to dissipation \nall follow from the two-stage mechanism \\cite{BR1,br,br05}. \nIn this mechanism,  \nthe inflaton field  $\\phi$ \nis coupled to heavy catalyst fields  $\\chi$, which decay into light\nfields  $\\sigma_i$. Consistency of the approximations then demands the\nmicrophysical dynamics determining $\\Upsilon$ to be faster than that\nof the macroscopic motion of the background inflaton and the\nexpansion: \n\\begin{equation}\n\\Gamma_\\chi > \\left|\\frac{\\dot \\phi}{\\phi}\\right|,\\, H, \n\\end{equation}\nwhere $\\Gamma_\\chi$ is the decay width of the heavy fields. \nIn addition, the condition $T \\gg H$ allows  to neglect the expansion of \nthe universe  when computing $\\Upsilon$. In the low $T$ regime, when \n the mass of the catalyst field $m_\\chi$ is larger than $T$, one has:\n\\begin{equation}\n\\Upsilon(\\phi,T) \\propto \\frac{T^3}{m_\\chi^2} \\propto\n\\frac{T^3}{\\phi^2}\\,, \n\\end{equation}\nwhile in the high $T$ regime, where the thermal corrections to the\ncatalyst field mass start to be important,\n\\begin{equation}\n\\Upsilon(\\phi,T) \\propto T \\,.\n\\end{equation}\nAnd in the very high $T$ regime, the dissipative coefficient goes like\nthe inverse of $T$. These are the cases of study we are going to\nconsider in the next section when studying the fluctuations during\nwarm inflation. In general, we will parametrize the dissipative coefficient as:\n\\begin{equation}\n\\Upsilon = C_\\phi \\frac{T^c}{\\phi^{m}} \\,,\n\\end{equation}\nwith $c-m=1$. We will work with $c=$3,1,-1, and also $c=0$, the case\nof no $T$ dependence for the dissipative coefficient. \n\n\n\\section{Fluctuations at linear order: Primordial spectrum}\n\nDuring warm inflation we have a multicomponent\nfluid, a mixture of a \nscalar inflaton field  $\\Phi$ interacting with the radiation fluid.\nBoth components exchange energy and momentum through the dissipative\nterm $\\Upsilon$. Dissipative effects also imply small departures from\nequilibrium, and that the \nradiation fluid will not behave exactly like a perfect fluid during\ninflation. In relativistic \ntheory, these effect can be parametrized\nin terms of a shear viscous tensor $\\pi_{ab}$, an energy flux vector $q_a$\nand a bulk viscous pressure $\\pi_b$, in the stress-energy tensor for the\nradiation fluid~\\cite{weinberg,maartens},\n\\begin{equation} \nT_{ab}^{(r)}= (\\bar \\rho_r + \\bar p_r + \\pi_b) u_a^{(r)}u_b^{(r)} + (\\bar p_r +\n\\pi_b) g_{ab} + q_a^{(r)}u_b^{(r)}+q_b^{(r)}u_a^{(r)}+ \\pi_{ab} \\,,\n\\end{equation}\nwhere $\\bar \\rho_r$ is the energy density, $\\bar p_r$ the adiabatic pressure,\n$u_a^{(r)}$ the four velocity of the radiation fluid, $g_{ab}$ the\nfour-dimensional metric, and  $u_a^{(r)}\n\\pi^{ab}=0=g_{ab}\\pi^{ab}$, $u_a^{(r)} q^{a}=0$. There would be heat \nflow for example in the presence of conserved charges in the system\nother than the stress-energy tensor, but we do not consider such\npossibility in this work, and then set $q_a=0$. The shear viscous\ntensor vanishes in an homogeneous and isotropic background geometry,\nbut at linear order it is given by \\cite{maartens}:\n\\begin{equation}\n\\pi_{ab} \\simeq -2 \\zeta_s \\sigma_{ab} \\,, \\label{shear}\n\\end{equation}\nwhere $\\zeta_s$ is the shear viscosity coefficient and $\\sigma_{ab}$\nthe shear of the radiation fluid:\n\\begin{equation}\n\\sigma_{ab} = \\nabla_{(a}u_{b)} + u_{(a} u^c\\nabla_c u_{b)} -\n\\frac{h_{ab}}{3}\\nabla^c u_c \\,,\n\\end{equation}\n$\\nabla_a$ being the covariant derivative of the metric $g_{ab}$. \nThe bulk viscous pressure can be seen as a non-adiabatic pressure\ncontribution, already present at the background level. Nevertheless,\nthe contribution from the light fields is expected to be small with\nrespect to $p_r$.  Thus, we will also set $\\pi_b=0$ and  focus on studying the\nconsequences of the shear viscosity on the primordial perturbations\nduring warm inflation.   \n\nIn order to study the system of perturbations at linear order, field,\nradiation energy density  and radiation pressure \nare expanded around their background values in a\n{}Friedman-Robertson-Walker metric: \n\\begin{align}\n\\Phi(x,t) &= \\phi(t) + \\delta \\phi(t,x) \\,, \\\\\n\\bar \\rho_r (x,t)&= \\rho_r(t) + \\delta \\rho_r(t,x) \\,, \\\\\n\\bar p_r(x,t) &= p_r(t) + \\delta p_r(t,x) \\,,\n\\end{align}\nand similarly for the dissipative coefficient: $\\bar \\Upsilon(x,t)=\n\\Upsilon(t) + \\delta \\Upsilon(t,x)$.  The  perturbed FRW metric,\nincluding only scalar perturbations,  is given by\\footnote{Latin\n  indexes $i,\\,j,\\,k,\\ldots$ are used for the spatial components, and either\n  $a,\\,b,\\,c,\\ldots$ or Greek letters for space-time indexes. }:  \n\\begin{equation}\nds^2= -(1+2 \\alpha) dt^2 - 2 a \\partial_i \\beta dx^i dt \n     + a^2 [ \\delta_{ij} (1 +2 \\varphi) + 2 \\partial_i \\partial_j \\gamma] dx^i\ndx^j \\,, \\label{metric}\n\\end{equation}\nwhere $\\varphi$ is the intrinsic curvature of a\nconstant time hypersurface. For later use, we introduce the combinations:\n\\begin{align}\n\\chi & = a ( \\beta + a \\dot \\gamma) \\,, \\\\\n\\kappa&= 3 (H \\alpha - \\dot \\varphi) + \\partial_k \\partial^k \\chi \\,, \n\\end{align}\nwhere $\\chi$ is the shear and $-\\kappa$ the\nperturbed expansion scalar of the comoving frame. \n \nThe evolution equations follow from the\nconservation of the energy-momentum tensor: \n\\begin{equation}\n\\nabla^a T_{ab}^{(\\alpha)}= Q_b^{(\\alpha)} \\,,\\;\\;\\; \\sum_\\alpha\nQ_b^{(\\alpha)}=0\\,,\n\\label{DTab}\n\\end{equation}\nwhere $Q_b$ is the four-vector source term  accounting for \nthe exchange of energy and momentum:\n\\begin{equation}\n-Q_b^{(\\phi)}=Q_b^{(r)}=\\Upsilon u_{\\phi}^a \\nabla_a \\Phi \\nabla_b \\Phi \\,, \n\\end{equation}\n$u_\\phi^a$ is now the four-velocity of inflaton fluid:  \n\\begin{equation}\nu_\\phi^a=-\\frac{\\nabla^a \\Phi}{\\sqrt{\\rho_\\phi + p_\\phi}} \\,,\n\\end{equation}\nand then:\n\\begin{equation}\nQ_b^{(r)}=\\Upsilon (\\bar \\rho_\\phi + \\bar p_\\phi)^{1\/2} \\nabla_b \\Phi \\,. \n\\end{equation}\nThe projection of the four-vector source term along the direction of the fluid \ngives the energy density source term, \n\\begin{equation}\nQ^{(r)}=-u_{\\phi}^a Q_a^{(\\phi)}\\,,\n\\end{equation}\nwhich at linear order is given by:\n\\begin{align} \nQ^{(r)}&= Q_r + \\delta Q_r \\,, \\\\\nQ_r & = \\Upsilon \\dot \\phi^2 \\,, \\\\\n\\delta Q_r &= \\delta \\Upsilon \\dot \\phi^2 + 2 \\Upsilon \\dot \\phi \\delta\n\\dot \\phi - 2 \\alpha \\Upsilon \\dot \\phi^2 \\,. \\label{Qr}\n\\end{align}\nThe momentum source term $J_a$ is orthogonal to the fluid velocity:\n\\begin{equation}\nQ^{(\\phi)}_a= Q^{(\\phi)} u^{(\\phi)}_a + J^{(\\phi)}_a\\,, \\;\\;\\;\\;\nu^{(\\phi)\\,a} J_a^{(\\phi)}=0 \\,,\n\\end{equation}\nand vanishes in the FRW geometry; at linear order it reads:\n\\begin{align}\nJ_i^{(r)} &= \\partial_i {\\bf J}_r \\,, \\\\ \n{\\bf J}_r& = - \\Upsilon \\dot \\phi \\delta \\phi \\,.\\label{Jr}\n\\end{align}\nTo complete the specification of the source, we need $\\delta\n\\Upsilon$, which for a general temperature $T$ and field $\\phi$\ndependent dissipative coefficient,   \n$\\Upsilon =  C_\\phi T^c\/\\phi^m$, with $c-m=1$, is given by:  \n\\begin{equation}\n\\delta \\Upsilon =  \\Upsilon \\left(c \\frac{\\delta T}{T} - m\n\\frac{\\delta \\phi}{\\phi} \\right) \n\\,. \\label{dupsilon} \n\\end{equation}\nAlthough dissipation implies departures from thermal equilibrium in\nthe radiation fluid, the system has to be close-to-equilibrium for the\ncalculation of the dissipative coefficient to hold, therefore $p_r \\simeq\n\\rho_r\/3$, $\\rho_r \\propto T^4$ and \n\\begin{equation}\n4 \\frac{\\delta T}{T} \\simeq \\frac{\\delta \\rho_r}{ \\rho_r}\\,.\n\\end{equation}\n\n{}Finally, the evolution equations for the radiation fluctuations are obtained\nexpanding at linear order Eq. (\\ref{DTab}) \\cite{kodama,\n  hwang,hwangnoh,malik}. Working in momentum space, defining the Fourier\ntransform with respect to the comoving coordinates, the equation of\nmotion for the fluctuations with comoving wavenumber $k$ are given\nby\\footnote{For simplicity, we keep the same notation  for the\n  fluctuations $\\delta f({\\bf x},t)$ and their Fourier transform \n$\\delta f({\\bf k},t)$.}:    \n\\begin{align}\n\\delta \\dot \\rho_r + 3 H (\\delta \\rho_r + \\delta\np_r) &= -3 (\\rho_r + p_r) \\dot \\varphi + \\frac{k^2}{a^2} \\left[\n\\Psi_r +(\\rho_r + p_r) \\chi\\right] + \\delta Q_r +\nQ_r \\alpha \\,, \\label{energyalpha}\\\\\n\\dot \\Psi_r + 3 H \\Psi_r &=- (\\rho_r +\np_r)\\alpha  - \\delta p_r+\n\\frac{2 k^2}{3 a^2} \\sigma_r + {\\bf J}_r\n\\,, \\label{momentumalpha} \n\\end{align}\nwhere a ``dot'' denotes the derivative with respect to the metric time\n``$t$'', $\\Psi_r$ is the radiation momentum perturbation, $T^{0\\,(r)}_j=-\n\\partial_j \\Psi_r\/a$, and $\\sigma_r$ the shear viscous pressure at\nlinear order:\n\\begin{equation}\n\\sigma_r \\simeq - 2 \\zeta_s \\left(\\frac{\\Psi_r}{\\rho_r + p_r} + \\chi\\right)\n\\,. \n\\end{equation}\n\nOn the other hand, field fluctuations are described by a stochastic\nsystem whose evolution is determined by a Langevin equation\n\\cite{calzetta,mossgraham}:\n\n\\begin{align}\n\\delta \\ddot \\phi + (3 H + \\Upsilon) \\delta \\dot \\phi +\n\\left(\\frac{k^2}{a^2} + V_{\\phi\\phi}\\right) \\delta \\phi &= \n\\left[2(\\Upsilon + H) T\\right]^{1\/2}\na^{-3\/2}\\xi_k  \\nonumber - \\delta \\Upsilon \\dot \\phi \\\\\n& + \\dot \\phi ( \\kappa + \\dot \\alpha) + (2 \\ddot\n\\phi + 3 H \\dot \\phi) \\alpha \n-\\Upsilon ( \\delta \\dot \\phi - \\alpha \\dot \\phi) \\label{field}\\,,\n\\end{align}\nwhere the stochastic source $\\xi_k$ can be approximated by a\nlocalized gaussian distribution with correlation function:\n\\begin{equation}\n\\langle \\xi(t,x) \\xi(t^\\prime,x^\\prime)\\rangle = \\delta(t -t^\\prime)\n\\delta^{(3)}(x-x^\\prime) \\,. \n\\end{equation}\n\nSo far, the equations for the perturbations at linear\norder are written in a ``gauge ready'' form, without specifying any\nparticular gauge, but the equations can also be written in\nterms of gauge invariant (GI) variables. For any scalar quantity $f$, at\nlinear order one can define a gauge invariant perturbation\n\\cite{kodama,hwang}: \n\\begin{equation}\n\\delta f^{GI} = \\delta f - \\frac{\\dot f}{H} \\varphi \\,, \n\\end{equation}\nwhile similarly the gauge invariant momentum perturbation reads:\n\\begin{equation}\n\\Psi^{GI}= \\Psi - \\frac{\\rho + p}{H} \\varphi \\,,\n\\end{equation}\nand for the metric perturbations one has the gauge invariant combinations:\n\\begin{align}\n{\\cal A} &= \\alpha - \\left[ \\frac{\\varphi}{H} \\right]^\\cdot \\,,\\\\\n\\Phi & = \\varphi - H \\chi \\,.\n\\end{align}\nThe evolution equations then read:\n\\begin{align}\n\\delta \\ddot \\phi^{GI} + 3 H \\delta \\dot \\phi^{GI} + \n\\left(\\frac{k^2}{a^2} + V_{\\phi \\phi}\\right) \\delta \\phi^{GI} &= \n\\left[2 (\\Upsilon+H) T\\right]^{1\/2}a^{-3\/2}  \\xi \n-\\dot \\phi \\delta \\Upsilon^{GI}  -\\Upsilon \\delta \\dot \\phi^{GI}\n\\nonumber  \\\\  \n&+\\Upsilon \\dot \\phi {\\cal A} + \\dot \\phi \\dot {\\cal A}+ 2\n(\\ddot \\phi + 3 H \\dot \\phi) {\\cal A} -\\frac{k^2}{a^2} \\dot \\phi\n\\frac{\\Phi}{H}, \\label{fieldGI} \\\\\n\\delta \\dot \\rho_r^{GI} + 3 H (1 + w_r) \\delta \\rho_r^{GI}  &=\n\\frac{k^2}{a^2} \\Psi_r^{GI}+ \\delta Q_r^{GI} + \\dot \\rho_r {\\cal A} \n\\,, \\label{energyrGI}\\\\\n\\dot \\Psi_r^{GI} + 3 H \\left( 1 + \\frac{k^2}{a^2 H^2}\\bar \\zeta_s\\right) \n\\Psi_r^{GI} &=- w_r \\delta \\rho_r^{GI} -\n\\Upsilon \\dot \\phi \\delta \\phi^{GI} - (\\rho_r + p_r) {\\cal A}\n-\\frac{3 k^2}{ a^2} (\\rho_r+ p_r) \\bar \\zeta_s \\frac{\\Phi}{H} \n\\label{momentumrGI} \\,,\n\\end{align}\nwhere in Eq. (\\ref{momentumrGI}) we have defined: \n\\begin{equation}\n\\bar \\zeta_s= \\frac{4}{9} \\frac{\\zeta_s H}{\\rho_r + p_r} \\label{zetas}\\,.\n\\end{equation} \n{}Finally, from the Einstein equations at linear order, the gauge invariant\nmetric perturbations are given by \\cite{kodama, hwang}:\n\\begin{align}\n{\\cal A}&= -\\frac{\\dot H}{H^2} {\\cal R} \\,, \\label{metricA}\\\\\n\\frac{k^2}{a^2 H^2}\\Phi&= 3 {\\cal A} + \\frac{3}{2} \\frac{\\delta\n  \\rho_T^{GI}}{\\rho_T}  \\label{metricPhi}\\,,\n\\end{align}\nwhere ${\\cal R}$ is the total comoving curvature perturbation,\n\\begin{eqnarray}\n{\\cal R}&=& \\varphi - \\frac{H}{\\rho_T + p_T} \\Psi_T \n\\nonumber \\\\\n&=& -\\frac{H}{\\rho_T\n  +p_T} \\Psi_T^{GI} \\,.\n\\end{eqnarray}\n{}For a multicomponent fluid, the total momentum perturbation is given\nby the sum of the individual components, and ${\\cal R}$ can be written\nas the weighted sum of the individual contributions:\n\\begin{align}\n{\\cal R} &= \n \\sum_{\\alpha=\\phi,\\,r} \\frac{h_\\alpha}{h_T} {\\cal R}_{\\alpha} \\,, \\label{RT}\\\\\n{\\cal R}_\\alpha & = -\\frac{H}{h_\\alpha} \\Psi_\\alpha^{GI}\\,,\n\\end{align}\nwhere we have defined $h_\\alpha= \\rho_\\alpha + p_\\alpha$. \nIn particular, for a scalar field $\\Psi_\\phi^{GI}= - \\dot \\phi \\delta\n\\phi^{GI}$, $h_\\phi = \\dot \\phi^2$,  and\n\\begin{equation}\n{\\cal R}_{\\phi}= \\frac{H}{\\dot \\phi} \\delta \\phi^{GI} \\,, \\label{Rphi}\n\\end{equation}\nfor which the power spectrum would be the form for single\nfield cold inflation. For warm inflation, we shall use the total\ncomoving curvature perturbation to evaluate the primordial spectrum,  \n\\begin{equation}\nP_{\\cal R}(k)= \\frac{k^3}{2 \\pi^2} \\langle |{\\cal R}_k|^2 \\rangle\\,, \\label{PR}\n\\end{equation}\nwhere ``$\\langle \\cdots \\rangle$'' means average over different\nrealizations of the noise term in Eq. (\\ref{field}). In all numerical\nresults shown in this work we have performed averages over 1000 runs. \nThis was found to be more than enough to get convergent numerical results.\".\n \n\nThe largest observable scale in the CMB corresponds to a comoving\nscale $k$ crossing the horizon $N_e$ e-folds before the end of\ninflation, denoted by $k=a_* H_*$. The value of $N_e$ can vary in\ngeneral from 40 to 70 depending on the inflationary model and details on the\nsubsequent reheating process \\cite{reheating}. Although we shall\nconsider different inflationary potentials, we will not consider the\ndetails of reheating period, and simply fix the horizon crossing at\n$N_e\\simeq 60$. As we will see, soon after horizon crossing, the amplitude of\nthe individual curvature perturbations ${\\cal R}_\\phi$ and ${\\cal\n  R}_r$  freezes out, and so does that of the total\ncurvature. This allows to compute the primordial spectrum by\nevaluating Eq. (\\ref{PR}) at horizon crossing, mainly when getting\nanalytical approximations. Nevertheless, when\nshowing numerical results we shall integrate Eqs. (\\ref{fieldGI})-\n(\\ref{momentumrGI}) and evaluate the amplitude of\nthe spectrum say 10 e-folds after horizon crossing. \n\n\n\\section{Equations in the zero-order slow-roll approximation and beyond}\n\nIn order to gain some insight on the evolution of the perturbations,\nwe follow \\cite{mossgraham} and first study the equations for the\nperturbations at zero-order in the slow-roll parameters. That is, \nexpanding the background variables around their slow-roll values and\nneglecting all terms in the equations proportional to slow-roll\nparameters. This eliminates in the equations for the fluctuations the\ndependence on the inflationary potential, and the details on the\nevolution of the dissipative coefficient. For example, the metric\nperturbation  ${\\cal A}$ given in \nEq. (\\ref{metricA}) is proportional to the slow-roll coefficient\n$\\epsilon$, Eq. (\\ref{eps}),    \n$\\epsilon= -\\dot H\/H^2$,\nand can be neglected; \nand the same for the terms proportional to $H \\dot \\phi$, \n{\\it i.e.}, those\nproportional to $\\Phi$, and $\\dot \\phi\/H\\phi$. Defining the\ndimensionless variables:  \n\\begin{align}\ny_k &= \\frac{k^{3\/2}\\delta \\phi^{GI}} {\\left[2 (\\Upsilon + H) T\\right]^{1\/2}} \\,,\\\\\nw_k &= \\frac{k^{3\/2}\\delta \\rho_r^{GI}}{ \\left[2 (\\Upsilon + H) T\\right]^{1\/2} \n(\\Upsilon \\dot \\phi)} \\,, \\\\\nu_k &= \\frac{k^{3\/2}\\Psi_r^{GI}}{ \\left[2 (\\Upsilon + H) T\\right]^{1\/2} \\dot \\phi} \\,,\n\\end{align}\nand using the slow-roll background equations (\\ref{eominfsl}),\n(\\ref{eomradsl}), \nwe have the system:\n\\begin{align}\n\\ddot y_k + 3 H ( 1 +Q) \\dot y_k + H^2 \\left[ z^2 + 3 \\eta (1+Q) - 3m Q\n\\frac{\\dot \\phi}{H\\phi} \n\\right] y_k \n&= \\left(\\frac{k}{a}\\right)^{3\/2}\\xi_k- 3 Q c H^2 w_k \\,, \\label{EOMyk} \\\\\n\\dot w_k + H (4-c) w_k &= \\frac{H}{3 Q}z^2 u_k + 2 \\dot y_k \n\\,, \\label{EOMwk}\\\\ \n\\dot u_k + 3 H(1 + z^2 \\bar \\zeta_s) u_k &= -3 Q H \\left(\\frac{w_k}{3}\n+y_k\\right) \n\\,, \\label{EOMuk}  \n\\end{align}\nwhere $z=k\/(aH)$. Combining Eqs. (\\ref{EOMwk}) and (\\ref{EOMuk}) into\na second order differential equation, we have:\n\\begin{align}\n\\ddot w_k + H ( 9 -c + 3 z^2 \\bar \\zeta_s ) \\dot w_k + H^2 \\left[ 20 - 5 c +\n6 Q c + \\frac{z^2}{3} + 3 z^2 (4-c) \\bar \\zeta_s\\right] w_k \n&=\n2 \\left(\\frac{k}{a}\\right)^{3\/2}\\xi \\nonumber \\\\\n& \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n+ H ( 4 - 6 Q + 6 z^2 \\bar \\zeta_s) \\dot y_k \n- H^2 \\left[3 z^2 + 6 \\eta (1+Q)- 6 Q m \\frac{\\dot \\phi}{H \\phi}\\right]\ny_k  \\label{fullwk}\\,.  \n\\end{align}\n\n\\begin{figure}\n\\vspace{0.5cm}\n \\includegraphics[width=0.55\\textwidth,\n  angle=0]{spectra_chaot_kk10000_Q100_yy_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot1a} Evolution of the power\n  spectrum of the field \n  $\\langle y_k^2\\rangle$, radiation energy density $\\langle\n  w_k^2\\rangle$, and radiation momentum $\\langle u_k^2\\rangle\/Q^2$,\n  for $Q=100$, and wavenumber \n  $k=10^4 H$. The vertical thin dotted line sets the value of the freeze\n  out scale $k_F$ in warm inflation.     \n  The results are shown for different power dependence on\n  $T$ of the dissipative coefficient: \n  $c=3$ (solid lines), $c=1$ (dashed lines), $c=-1$ (dash-dotted\n  lines), and $c=0$ (dotted lines).\n} \n\\end{figure}\n\n\n\\begin{figure}\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth, angle=0]{spectra_chaot_kk10000_Q100_Pr_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot1b} Evolution  of the total curvature perturbation\n  spectrum $P_{\\cal R}^{1\/2}$ (black lines), the \n  radiation $P_{{\\cal R}_r}^{1\/2}$ (red lines), and the field\n  $P_{{\\cal R}_\\phi}^{1\/2}$ (green lines) curvature perturbation spectrum.\n  The results are shown for different power dependence on\n  $T$ of the dissipative coefficient: \n  $c=3$ (solid lines), $c=1$ (dashed lines), $c=-1$ (dash-dotted\n  lines), and $c=0$ (dotted lines).\n} \n\\end{figure}\n\nBefore including shear effects, we study the evolution of the\nperturbations setting $\\bar \\zeta_s=0$. \nThe evolution of the power spectrum of the field $\\langle y_k^2\n\\rangle$, the radiation energy  density $\\langle w_k^2 \\rangle$, and\nthe radiation momentum $\\langle u_k^2 \\rangle$, are shown \nin {} Fig. \\ref{plot1a} as a function of $z^{-1}=aH\/k$. We have\ntaken $Q=100$,  and started the\nintegration at $z_i=10^4$. As mentioned above, quantities as $\\langle\ny_k^2 \\rangle$ denotes the average over 1000 realizations of the\ngaussian noise term,  and by $y_k^2$, $w_k^2$, $u_k^2$ we mean the\nmodulus of the complex variable. We have set the initial conditions\nfor field fluctuations in the vacuum,  while $w_k$ and $u_k$ initially\nvanish, for simplicity. Starting the evolution early enough before horizon\ncrossing, the system is always controlled by the stochastic source\nterm, and the dependence on the initial conditions is quickly erased.  \n\n\nWe have considered different powers of $T$ for the dissipative\ncoefficient $\\Upsilon$, $c=3,\\,1\\,,-1$, and included the\ncase of a constant or \nfield dependent $\\Upsilon$ ( $c=0$, dotted lines) for comparison. \nThe radiation fluctuation $w_k$ acts as a source term for the field,\nbut at early times $z^{-1} \\ll 1$, for subhorizon perturbations, the\nfield evolution is dominated by the stochastic source term, and both\nradiation and field fluctuations evolve like in the case $c=0$. \nIn the latter\ncase, the freeze out of the perturbations takes place before horizon\ncrossing, due to the extra friction term in Eq. (\\ref{EOMyk}), at\naround $k_F\/(aH)\\simeq 3\\sqrt{Q\/2}$ (vertical thin dotted line)\n\\cite{warmdeltap,warmpert, mossgraham}, and\nsoon after field and radiation spectrum level off. The field spectrum\nfor a $T$ independent $\\Upsilon$  can be computed analytically and is\ngiven by \\cite{warmdeltap}:    \n\\begin{equation}\n\\langle y_k^2 \\rangle_0 \\simeq \\frac{\\sqrt{3\\pi}}{4} \n\\frac{\\sqrt{1+Q}}{(1+3 Q)}\\,,  \\label{yy0}\n\\end{equation}\nwhere the subindex ``0'' denotes the value for $c=0$. On the other\nhand, when $c > 0$, field and radiation fluctuations get\neffectively coupled before freeze out at around $z_c^2 \\approx 18 Q c$,\nand both start growing at similar\nrates. Numerically, we get: \n\\begin{equation}\n\\langle y_k^2 \\rangle \\approx \\langle y_k^2 \\rangle_{z_c} \\left( \\frac{z_c}{z}\n\\right)^{5 c} \\,. \\label{yyz} \n\\end{equation}\nThe field spectrum when the fluctuations are still subhorizon and $z >\nz_c$, which is independent of the radiation fluctuation,  can be\nfound in \\cite{mossgraham}: \n\\begin{equation}\n\\langle y_k^2 \\rangle_{z_c} \\approx \\frac{3c}{z_c} \\,, \\label{yyzc}\n\\end{equation}\nand therefore, at horizon crossing: \n\\begin{equation}\n\\langle y_k^2 \\rangle_* \\propto z_c^{5c-1} \\propto Q^{(5c-1)\/2} \\,. \\label{yystar}\n\\end{equation}\nWhen $c< 0$, the effect is the opposite, and effectively the freeze\nout is delayed until practically horizon crossing, which makes the\namplitude of the field spectrum smaller than in the $c=0$ case. \n\n{}Fig. \\ref{plot1b} shows the evolution\nof the comoving curvature power spectrum, $P_{\\cal R}^{1\/2}$, given by\nthe sum of the radiation and the field contributions as in\nEq. (\\ref{RT}), for different values of $c$. Also shown are \nthe power spectra of\nthe radiation, $P_{{\\cal R}_r}^{1\/2}$, \nand the field $P_{{\\cal R}_\\phi}^{1\/2}$. After  \nhorizon crossing they all converge to the same amplitude. During\nslow-roll one has that $h_r = \\rho_r +p_r \\simeq Q h_\\phi$, and from\nEq. (\\ref{EOMuk}) when $z\\ll 1$ the radiation momentum\nbecomes proportional to the field fluctuation:   \n\\begin{equation}    \n\\Psi_r \\simeq Q \\dot \\phi \\delta \\phi \\,,\n\\end{equation}\nand therefore: \n\\begin{equation}\nP_{{\\cal R}_r} = \\frac{H}{h_r} P_{\\Psi_r} \\simeq P_{{\\cal R}_\\phi} \\,.\n\\end{equation}\nOwing to the fact that $h_\\phi \\ll h_r \\simeq h_T$, the main\ncontribution to the primordial spectrum in Eq. (\\ref{PR}) comes from\nthe radiation, before and after horizon crossing. The\nprimordial spectrum is always dominated by the thermal fluctuations. \nBut after horizon crossing one simply has:\n\\begin{equation}\nP_{\\cal R} \\simeq P_{{\\cal R}_r}\\simeq P_{{\\cal R}_\\phi} \\,.\n\\end{equation}  \n Therefore, the amplitude of the primordial spectrum can be written as\n usual in terms of that of the inflaton field:\n\\begin{equation}\nP_{\\cal R} \\simeq \\left( \\frac{H}{\\dot \\phi} \\right)^2 \\frac{(H +\n  \\Upsilon)T}{\\pi^2} \\langle y_k^2 \\rangle_*   \\label{PRyy}\\,,\n\\end{equation}\nevaluated at horizon crossing. \n\n\n\\begin{figure}\n\\vspace{0.55cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_gralT_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot2} The spectrum of the field at horizon crossing \n  $\\langle y_k^2\\rangle_*$ as a function of the\n  dissipative parameter $Q$, at zero order in the slow-roll\n  parameters, for different values of $c$, and no shear $\\bar\n  \\zeta_s=0$. Solid lines are obtained integrating\n  Eqs. (\\ref{EOMyk})-(\\ref{EOMuk}), while dashed lines were obtained\n  with the approximation used in \\cite{mossgraham}.     \n} \n\\end{figure}\n\nIn {}Fig. \\ref{plot2} we have compared the power spectrum of the\nfield, $\\langle y_k^2 \\rangle$ at horizon crossing, as a function of the\ndissipative parameter $Q$ for different values of $c$. The equation\nfor the fluctuations has been integrated keeping the background values\nconstant (solid lines), Eqs. (\\ref{EOMyk})-(\\ref{EOMuk}).  The larger\nthe power $c>0$, the larger the enhancement with $Q$, as the\nfluctuations get coupled earlier. For $c <0$, as mentioned \nbefore, we have the opposite effect, and the spectrum diminished with\nrespect to the case $c=0$. For positive $c$, the curves can be fitted by\na function:\n\\begin{equation}\n\\langle y_k^2 \\rangle_* |_{c>0}\\simeq \\langle y_k^2 \\rangle_0 \n( A_c Q^\\alpha + B_c Q^\\beta) \\,.  \\label{yyapprox}\n\\end{equation}\nbut when $c=-1$, we have found that the curve can be best fitted by: \n\\begin{equation}\n\\langle y_k^2 \\rangle_*|_{c=-1} \\simeq \\frac{ 1+ A_{-1} Q^\\alpha}{ 1+ B_{-1}\n  Q^\\beta} \\,. \\label{yyapproxcn1}\n\\end{equation}\nThe coefficients are given in Table \\ref{table1}. \nFor $c=3,\\,1$, the approximation works well for $Q>50$, while for\n$c=-1$ it is valid for any $Q$. \n\n\n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|c|}   \n \\hline\n$c$ & $\\alpha$ &  $\\beta$ & $A_c$ & $B_c$ \\\\\n\\hline\n~~~3~~~ & ~~~7.5~~~ & ~~~7.0~~~ & ~~~$1.9\\times 10^{-8}$~~~ & ~~~$3.4 \\times 10^{-6}$~~~ \\\\\n~~~1~~~ & ~~~2.5~~~ & ~~~2.0~~~ & ~~~$2.8 \\times 10^{-2}$~~~ & ~~~$6.8 \\times 10^{-5}$~~~ \\\\\n~~~-1~~~ & ~~~0.2~~~ & ~~~1.4~~~ & ~~~0.78~~~ & ~~~0.088~~~ \\\\\n\\hline\n\\end{tabular} \n\\caption{\\label{table1} Coefficients for the numerical fit of the\n  spectrum, Eq. (\\ref{yyapprox}) and Eq. (\\ref{yyapproxcn1}).}\n\\end{table}\n\n\nWe have also included in {}Fig. \\ref{plot2} the spectrum of the field\nobtained with the approximation used in \\cite{mossgraham} (dashed\nlines) for comparison. We have confirmed the main results about the\npower spectrum obtained in \\cite{mossgraham}; {\\it i.e.}, that for $c >0$,\nthe amplitude of the primordial spectrum in warm inflation is enhanced\nby a factor $\\simeq Q^{\\alpha -1\/2}$. However, while in\n\\cite{mossgraham} they found $\\alpha = 3c$, we have a smaller power\n$\\alpha=5c\/2$, which for $c=3$ can   \nmean a difference of two or 3 orders of magnitude in the amplitude of\nthe spectrum for $Q\\simeq 50 - 100$. The differences can be traced\nback to how the radiation source term is treated. In \\cite{mossgraham}, it\nwas approximated by:\n\\begin{equation}\n\\delta Q_r \\simeq Q_r \\frac{\\delta \\Upsilon}{\\Upsilon} \\simeq c Q_r\n\\frac{\\delta \\rho_r}{4\\rho_r} \\,, \\label{Qrian} \n\\end{equation} \nwhile we have kept the dependence on the field:  \n\\begin{equation}\n\\delta Q_r \\simeq Q_r \\left( c \\frac{\\delta \\rho_r}{4\\rho_r} + 2\n\\frac{\\delta \\dot \\phi}{\\dot \\phi}\\right) \\,. \\label{Qrphi}  \n\\end{equation} \nWith no shear viscosity included, the second order differential\nequation Eq. (\\ref{fullwk}) reads:  \n\\begin{align}\n\\ddot w_k + H ( 9 -c ) \\dot w_k + H^2 \\left( 20 - 5 c  \\frac{z^2}{3}\\right) w_k \n+  H^2 z^2 y_k  \n&= 2 \\left(\\frac{k}{a}\\right)^{3\/2}\\xi  \\nonumber \\\\\n& \n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\n- H^2 6 Q c w_k +  H ( 4 - 6 Q ) \\dot y_k \n- H^2 \\left[2 z^2 + 6 \\eta (1+Q) - 6 Q m \\frac{\\dot \\phi}{H \\phi}\\right]\ny_k  \\label{fullwknoshear}\\,,  \n\\end{align}\nwhere we have written the equation such that on the RHS we have the terms\ninduced by the field dependence in Eq. (\\ref{Qrphi}). Thus, setting\nthe RHS to zero one recovers the equation derived with the source term\nas given in Eq. (\\ref{Qrian}). While the term\nproportional to the field perturbation $y_k$ acts as a source term on\nthe radiation that tends to enhance the fluctuation, the extra terms \n$6 H^2 Qc w_k$ and $H(4 -6Q) \\dot y_k$ have the opposite effect, i.e.,\nthat of damping the growth. Although these contributions are not enough\nto avoid the growth of the fluctuations, they have a sizable effect on\ntheir power-law behavior, mainly when $c=3$, as seen in Fig. \\ref{plot2}. \nWhen $c =-1$ the radiation fluctuations do not grow before horizon\ncrossing, so that the effect of the field dependent terms in\nEq.~(\\ref{Qrphi}) is negligible. \n\n\\begin{figure}\n\\vspace{0.55cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{ianeq_c3_k100_z1_all_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot3} The spectrum of the field \n  $\\langle y_k^2\\rangle$ at horizon crossing as a function of the\n  dissipative parameter $Q_*$, for different inflationary models, $c=3$\n  and no shear $\\bar \\zeta_s=0$. The solid line is the result at zero\n  order in slow-roll; dashed lines are for a chaotic model with\n  $p=$6,4,2, from top to bottom; the dash-dotted line is a quadratic\n  hybrid model. \n} \n\\end{figure}\n\nNeglecting the evolution of the background variables and working at\nzero order in the slow roll parameters,  is a good approximation when\n$Q$ is large enough, and the background parameters hardly vary during\nthe last 60 e-folds of inflation. Indeed the spectrum depends mainly\non the value of the parameters in a smaller interval, 5-6 e-folds\naround horizon crossing, where one may expect the approximation of\nkeeping them constant to be a fairly good one. Still, this is a model\ndependent question. This can be seen in {}Fig. \\ref{plot3}, where we\nshow the field spectrum for some generic inflationary models, and\n$c=3$, $\\bar \\zeta_s=0$. The value of  \n$\\langle y_k^2 \\rangle_*$ has been obtained integrating\nEqs. (\\ref{field})-(\\ref{momentumrGI}), evaluating the amplitude of\nthe comoving curvature spectrum at $N_e=20$ efolds after horizon crossing, and\nusing Eq. (\\ref{PRyy}). We have considered the inflationary models:\n\\begin{align}\nV& = \\frac{V_0}{p} \\left( \\frac{\\phi}{m_P}\\right)^p\\,, \\;\\;\\; & {\\rm\n  (chaotic)} \\,, \\label{chaotic}\\\\ \nV& = V_0 \\left[ 1 + \\frac{\\eta_\\phi}{2} \\left( \\frac{\\phi}{m_P}\\right)^2\n\\right] \\,, \\;\\;\\; & {\\rm (hybrid)}  \\label{hybrid}\\,. \n\\end{align} \n{}For the chaotic model we have run different powers $p=$ 6, 4, 2, and\nset $V_0= 10^{-14} m_P^4$, while for the hybrid $V_0= 10^{-8}m_P^4$\nand $\\eta_\\phi=3$. For each model, the initial value of the inflation\nfield is chosen such that we can have $N_e \\simeq 64$, and from the\nbackground slow-roll equations one derives the initial values of the field\nderivative, $\\rho_r$ and $Q$. We have chosen $k= 100 H_i$, $H_i$ being\nthe initial value of the Hubble parameter. Therefore, horizon crossing\n$k = a_* H_*$ takes places at around 60 e-folds before the end of\ninflation. \n\n\nIn all the examples considered above the dissipative coefficient\nincreases during inflation \\cite{BasteroGil:2009ec}, and the larger the\npower in the potential, the slower the evolution of the background\nvalues.  In the plot, the solid line is the result obtained with constant\n background variables. For a quartic chaotic model or larger power, the\napproximation at zero order in the slow-roll works fine, while for a\nquadratic chaotic it tends to overestimate the spectrum by at least an order of\nmagnitude for $Q \\gtrsim 50 $. {}For the hybrid model, the model\ndependence shows up when $Q \\lesssim 100$. \n\n\nShear effects will further damp the growth of the fluctuations.  \nIn Eq. (\\ref{fullwk}) the shear acts as an extra friction when the\nfluctuations are still subhorizon, suppressing the amplitude of the\nradiation fluctuation before the \nradiation-field system becomes effectively coupled. Whenever the shear\nis large enough, this suppression indeed can prevent altogether the\ngrowth of the field perturbations, as the amplitude of the radiation\nfluctuation is not enough to affect that of the field before horizon\ncrossing. This will happen when $\\bar \\zeta_s= \\zeta_s H\/(3 \\rho_r)\n\\gtrsim 1$ at horizon crossing. During warm inflation, we have the\ncatalyst field \ncoupled to the inflaton field, and to the light degrees of freedom\ngiving rise to the thermal bath. The shear viscosity for light fields\n(light with respect to the temperature $T$ of the thermal bath)\ntypically behaves \nas $\\zeta_s \\propto T^3$ \\cite{shear,Moore:2007ib}, although, depending on the\npattern of interactions, other powers could be\npossible and cannot be excluded. Nevertheless, the damping  is fully\ncontrolled by the value of the dimensionless parameter $\\bar \\zeta_s$ at\nhorizon crossing, and {\\it is independent of  the functional form of the shear\nwith the temperature}, as can be seen in\n{}Fig. \\ref{plot4}. We have integrated the full EOM without\napproximations Eqs. (\\ref{field})-(\\ref{momentumrGI}), for a quartic\nchaotic model, Eq. (\\ref{chaotic}) with $p=4$, and set the initial\nbackground values such that $Q_*\\simeq 40$. We show in the plot  \nthe dependence of the field spectrum with $\\bar \\zeta_s$ at horizon\ncrossing, for different values of $c$, and two \nexamples of the shear viscosity: one proportional to $T^3$ (solid lines), and\nanother linear in $T$ (dashed lines), however the curves lay on top of each\nother. We have checked that this is independent of the inflationary\nmodel considered. We have normalized the field spectrum with the value obtained\nwhen $c=0$. As the shear\nincreases, it does damp the radiation fluctuation enough for the\nfield spectrum to approach the $c=0$ value. Asymptotically, when $\\bar\n\\zeta_s \\gg 1$, for a linear dissipative coefficient with $T$ one\npractically recovers the\n$c=0$ case, for a cubic one the field spectrum is $\\simeq 2 \\langle\ny_k^2\\rangle_0\/5$, while the inverse $T$ case is slightly above $\\simeq\n5 \\langle y_k^2 \\rangle_0\/2$. In all cases, the field spectrum is well\nfitted by a function:\n\\begin{equation}\n\\log_{10} \\frac{\\langle y_k^2 \\rangle}{\\langle y_k^2\\rangle_0 } \\simeq \nA_s - B_s\\left[1+ \\tanh \\left( \\log_{10}\\bar \\zeta_s + \\Delta_s\\right)\\right] \n\\,, \\label{fittingyys}\n\\end{equation}\nwhich interpolates between the result with no shear $\\sim 10^{A_s}$ and \n the  $c=0$ case, modulo a normalization constant $\\sim\n 10^{A_s- 2 B_s}$. As an example, the coefficients $A_s$, $B_s$ and\n $\\Delta_s$ for the  potential  shown in Fig. \\ref{plot4} with\n $Q_*=40$ are given in Table\n   \\ref{table2}.  \n\n\\begin{table}\n\\begin{tabular}{|c|c|c|c|}   \n \\hline\n$c$ & $A_s$ & $B_s$ & $\\Delta_s$ \\\\\n\\hline\n~~~3~~~ & ~~~6.35~~~ & ~~~3.4~~~ & ~~~1.36~~~ \\\\\n~~~1~~~ & ~~~1.9~~~ & ~~~1.2~~~ & ~~~1.33~~~ \\\\\n~~~-1~~~ & ~~~-0.95~~~ & ~~~-0.7~~~ & ~~~-0.66~~~ \\\\\n\\hline\n\\end{tabular} \n\\caption{\\label{table2} Coefficients for the numerical fit of the\n  spectrum of the field with the shear viscosity,\n  Eq. (\\ref{fittingyys}).}\n\\end{table}\n\n\\begin{figure}[t]\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_cshear_T_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot4} Field spectrum normalized by the value with\n  $c=0$, as a function of the shear parameter $\\bar \\zeta_s$, for\n  different values of $c$, and $Q=40$. From top to bottom, $c=3,\\,1,\\,-1$. For\n  each curve, we have also considered two different $T$ dependence on\n  the shear viscosity, as indicated in the legend, but both gives the\n  same field spectrum. } \n\\end{figure}\n\nFinally, combining Eq. (\\ref{fittingyys}) with Eq. (\\ref{yyapprox}),\nthe field spectrum when $c>0$ reads:\n\\begin{equation}\n\\langle y_k^2 \\rangle_* \\simeq \\langle y_k^2 \\rangle_0 F_Q[Q]^{F_s[\\bar \\zeta_s]} \\,,\n\\end{equation}\nwhere: \n\\begin{align}\nF_Q[Q] &\\simeq \\left( A_c Q^\\alpha + B_c Q^\\beta \\right) \\,, \\\\\nF_s[\\bar \\zeta_s]& \\simeq \\frac{1}{2}\\left[1- \\tanh \\left( \\log_{10}\\bar \\zeta_s + \n\\Delta_s\\right)\\right] \\,. \n\\end{align}\nTherefore, in the strong dissipative regime when $Q>1$, the primordial\nspectrum of the curvature perturbation can be written as that obtained\nfor a $T$ independent dissipative coefficient, times an enhancement\nfunction $F_Q[Q]$ depending on the dissipative ratio $Q$, but\ncontrolled by a function depending on the shear $F_s[\\bar \\zeta_s]$ :\n\\begin{equation}\nP_{\\cal R} \\simeq \\left( \\frac{H^2}{\\dot \\phi} \\right)^2\n\\frac{\\sqrt{3\\pi}}{4 \\pi^2}\\left[\\frac{(1 +\n  Q)^{3\/2}}{1+3Q}\\right]  \\left(\\frac{T}{H}\\right) \\times F_Q[Q]^{F_s[\\bar \\zeta_s]} \\,, \n\\end{equation}\nand whenever $\\bar \\zeta_s > 1$ one recovers the amplitude of the\nprimordial spectrum obtained when $c=0$. The latter is of a magnitude\ncompatible with the observational value, for model parameters values\ncommon in inflationary model building \\cite{BasteroGil:2009ec,Zhang:2009ge}. For\nexample, without the enhancement, a quartic warm chaotic\nmodel gives rise to the right level of perturbations with a coupling\nconstant $\\lambda \\simeq 10^{-13}-10^{-14}$. From the point of view of\nmodel building, the enhancement produced\nby the backreaction of the radiation fluctuations onto the fields, if\nnot avoided by shear effects, can be compensated by lowering the\nheight of the potential, i.e., by lowering couplings and\nmasses. However, it will also have an impact on the prediction for the\nspectral index: for models with $Q$ increasing at the time of horizon\ncrossing it will render the spectrum too blue-tilted, and the other\nway round, for $Q$ decreasing the spectrum may become too\nred-tilted.  Shear viscosity damps the growth of the fluctuations, and\ntherefore will also affect the spectral index. \n\nEven though the results we have obtained are fairly model independent\nand shown not to depend on the specific dependence on the temperature\nof shear viscosity, it is useful to estimate the\nmagnitude of the ratio $\\zeta_s H\/(3 \\rho_r)$ for typical warm\ninflation models. In kinetic theory, the shear viscosity for\nrelativistic fluids can be expressed parametrically as proportional to\nthe mean free path of quasiparticles in the fluid. Considering as an\nexample a radiation fluid made of relativistic scalar particles\n$\\sigma$, with mass $m_\\sigma\/T \\ll 1$ and self-interaction potential\n$\\lambda_\\sigma \\sigma^4\/4!$, the mean free path of quasiparticles is\ndetermined by the inverse of the thermal width, which is ${\\cal\n  O}(\\lambda_\\sigma^2)$, and the computed value for the shear\nviscosity is \\cite{jeon} $\\zeta_s \\simeq 3 \\times 10^3\nT^3\/\\lambda_\\sigma^2$.  The condition $\\zeta_s H\/(3 \\rho_r) \\gtrsim 1$\ncan then be expressed, for example, as a condition on the magnitude of\nthe radiation bath self-interaction, $\\lambda_\\sigma \\lesssim 55\n\\sqrt{H\/T}$. Since warm inflation requires $T > H$ and in general we\ntypically work with values $T \\gg H$, we find that weakly interacting\nradiation fluids can easily have shear viscosities of sufficient\nmagnitude to counterbalance and suppress completely the growth of\nfluctuations caused by the coupling of the inflaton's fluctuations with\nthose from the radiation.\n\n\nIn supersymmetric warm inflation models however the radiation bath\nself coupling is the same as the coupling to the catalyst field, which\nenters in the calculation of the dissipative coefficient. In general,\nlarge multiplicities of the fields (catalyst $\\chi$ and radiation $\\sigma$)\nare required in order to have enough dissipation. Considering\na model with ${\\cal N}_\\chi$, ${\\cal N}_\\sigma$ copies of the complex fields,\nwith common coupling $h$ among the $\\chi$'s and $\\sigma$'s, the\nself-interaction potential is given by $h^2 \\sum_i |\\sigma|^2\/4$. \nIn the low-$T$ regime,  the dissipative and shear coefficients can be\nwritten as \\cite{BasteroGil:2010pb}:\n\\begin{align}\n\\Upsilon & \\simeq 0.1 h^4 {\\cal N}_\\chi{\\cal N}_\\sigma^2 \\frac{T^3}{\\phi^2}\n\\,, \\label{upslowT}\\\\\n\\zeta_s &\\simeq 127 N_\\sigma (1 + 0.3 h)\\frac{T^3}{h^4} \\,, \\label{zetaslf}\n\\end{align}\nwhere we have included the next-to-leading order correction in the\nshear viscosity \\cite{Moore:2007ib}.  \nThe condition $\\bar \\zeta_s \\geq 1$ then reads:\n\\begin{equation}\nh^4 \\lesssim 128 \\frac{{\\cal N}_\\sigma}{g_*} \\frac{H}{T} \\simeq \n69 \\frac{H}{T} \\,,\n\\end{equation}\nwhere $g_*\\simeq 15 {\\cal N}_\\sigma\/8$. Although \nin a bath of weakly interacting radiation particles, $h \\ll 1$,   \nthe shear viscosity would be rather large, it may not give rise to  enough\ndissipation to sustain a period of warm inflation, unless the weakness of\nthe coupling is compensated by having a large multiplicity for the fields.   \nThis therefore becomes a model dependent question depending on the\nparameters ${\\cal N}_\\chi$, ${\\cal N}_\\sigma$ and the coupling\n$h$. In Fig. \\ref{plot5} we show an example for a quartic chaotic\nmodel, where we compare the value of the field spectrum $\\langle\ny^2_k\\rangle_*$ as a function\nof $Q_*$ for different values of the Yukawa coupling $h$.  \nWe have kept the value ${\\cal\n  N}_\\sigma=10^3$ fixed, and vary the value of ${\\cal N}_\\chi$ to get\ndifferent values of $Q_*$. From top to bottom the value of $h$\ndecreases, starting with the largest possible one $h=\\sqrt{4\\pi}$\n(solid line) for which there is a negligible shear effect. In this case,  \na value $Q_*\\simeq 10$ requires ${\\cal N}_\\chi=35 $, while for $Q_*\\simeq\n100$ we need ${\\cal N}_\\chi \\simeq 110$. As the value of $h$\ndecreases, the multiplicity of the field to get the same value of\n$Q_*$ increases by a factor $(4\\pi\/h^2)^2$.  For $h=1.12$ and $Q_*=100$,\nwe would need ${\\cal N}_\\chi \\simeq 10^4$. \nAs the value of $Q_*$ increases, so it does the\nratio $T\/H$, and therefore the parameter $\\bar \\zeta_s$\ndecreases along the curves. Although  in this example \nviscous effects are not \nenough to completely avoid the growth of the perturbations, \nthey bring  it down to $\\langle y^2_k\\rangle_* \\propto Q^{2.5}$ for\n$h=1,12$ and $\\langle y^2_k\\rangle_* \\propto Q^{4.5}$ for $h=1.67$,\ninstead of $\\langle y^2_k\\rangle_* \\propto Q^7$.      \n\n\\begin{figure}[t]\n\\vspace{0.5cm}\n\\includegraphics[width=0.55\\textwidth,angle=0]{spectra_chaot_Qiter_kk100_pp4_shear_p.eps}\n\\vspace{0.25cm}\n\\caption{\\label{plot5} Field spectrum as a function of $Q_*$, for a\n  quartic chaotic model, with a cubic dissipative coefficient\n  (Eq. \\ref{upslowT}) and $\\zeta_s$ given by Eq. (\\ref{zetaslf}). We\n  have taken ${\\cal N}_\\sigma=10^3$ and $h=\\sqrt{4\\pi}$ (solid line),\n  $h=1.67$ (dashed line) and $h=1.12$ (dot-dashed line). \nWe include for comparison the result with $c=0$. \n} \n\\end{figure}\n\n\n\\section{Conclusions}\n\nDensity perturbations in warm inflation are seeded by thermal\nfluctuations of the inflaton. In warm inflation the inflaton decays into\nradiation through a dissipation term in the inflaton's equation of\nmotion and that originates from the microphysical interactions of the\ninflaton field with other degrees of freedom of the\nmicroscopic Lagrangian describing the complete system. The origin of\nthe dissipation term and its quantum field theoretical treatment has\nbeen extensively discussed in the literature (for a recent review, see\ne.g. Ref. \\cite{Berera:2008ar}).  However, as radiation is produced\nduring the inflaton's evolution, the full treatment of the spectrum of\nperturbations no longer involves only that of the inflaton's\nperturbations but must also account for radiation perturbations.  This\nmakes the treatment of density perturbations in warm inflation \nsimilar to\na multifluid system. Since the larger is the dissipation the larger is\nexpected to be the rate of radiation production, it becomes important\nthe study of how the produced radiation and its perturbations\nbackreacts on the inflaton's evolution and respective\nperturbations. In Ref. \\cite{mossgraham} it was shown that as\na consequence of this backreaction, the infaton's perturbations can\ngrow as the dissipation of the inflaton increases. This increasing of\nthe inflaton's perturbations with increasing dissipation can,\ntherefore, severely constrain the model parameters in warm inflation\nso as to cope with the known measured results for the CMB\nradiation. This backreaction of the produced radiation on the\ninflaton's perturbations is larger the stronger is the coupling\nbetween the radiation and the inflaton; in particular\nthe larger the power of $T$ in the dissipative coefficient,\nthe stronger the $T$ dependence on the perturbations.\nThis is exemplified by the results shown in\n{}Fig. \\ref{plot2}.\n \nIn this paper we have studied how this backreaction of the produced\nradiation, that can lead to this growth mode in the inflaton's\nperturbations, can be counterbalanced by the dissipative effects within\nthe radiation fluid. Dissipative effects in the radiation fluid\nitself are described by viscosity terms.  This is expected when the\nradiation fluid departs from equilibrium, which is the case in any\ndissipative system, where the produced radiation from the system does\nnot immediately equilibrate in the radiation bath and its approach to\nequilibrium is controlled by viscosity coefficients, like the shear\nviscosity, the bulk viscosity and heat transport coefficients.  We\nhave focused on the dissipation effect as coming dominantly from a\nshear viscosity term in the fluctuation equations. We have then shown\nthat the shear viscosity can effectively damp the radiation\nfluctuations so as to avoid altogether the appearance of the growth mode\nin the resulting perturbations. The results we have obtained are model\nindependent and we have shown that the overall effect of compensation\nof the growth mode and its control is determined by the ratio $\\zeta_s\nH\/(3 \\rho_r)$, where $\\zeta_s$ is the shear viscosity coefficient, $H$\nis the Hubble parameter and $\\rho_r$ the radiation energy density.\nWhen the ratio is ${\\cal O}(1)$ or larger, the growth mode disappears\ncompletely.\n\n\nIn this work we have only considered the coupling between the\nfluctuations in the inflaton field with those in \nthe radiation through the\n(temperature dependence on the) dissipation coefficient in the\ninflaton's dynamics. But in a thermal bath, the parameters of the\ninflaton's potential can also adquire temperature corrections. Even\nthough these thermal corrections can be kept under control and small\nin Supersymmetry model building realizations for warm inflation\n\\cite{BR1,Berera:2008ar,BasteroGil:2009ec}, they can still be large enough\nto provide extra sources of couplings between inflaton and radiation\nfluctuations and it should be interesting to analyze their effects in a\nfuture work. Likewise, there can be additional sources of dissipation\nin the radiation fluid, for example as coming from bulk viscosity,\nthat can further help to damp any leftover growing modes as resulting from\nthese additional couplings. In this work we have neglected the effects\nof the bulk viscosity on the grounds that it is in general much\nsmaller than the shear viscosity. {}For example, for the\nself-interacting scalar field radiation discussed in section IV, the ratio of\nthe bulk viscosity, $\\zeta_b$, with the shear viscosity for a high\ntemperature radiation fluid is $\\zeta_b\/\\zeta_s\\sim 10^{-9}\n\\lambda_\\sigma^3$, thus, it is negligible for a weakly interacting\nradiation bath. But there may be other interactions and energy\nregimes for the radiation fluid in which the bulk can be sizable and lead\nto effects in the density perturbation evolution (see\ne.g. Ref. \\cite{giovaninni}). \n\n\nAll these effects, starting with the shear viscosity, will also impact\nthe second order evolution of the perturbations and thus the\ncalculation of the non-gaussinity. Forthcoming cosmological data are\nexpected very soon to set the level of non-gaussinity of the\nprimordial spectrum, which clearly will help to discriminate among inflationary\nmodels. Warm inflation, being a type of \nmulti-fluid model, falls into the category of models with a\nnon-negligible value of the non-linearity parameter $f_{NL}$ for\nnon-gaussinity. This parameter has been computed for a $T$ independent\ndissipative coefficient \\cite{nongauss}, and recently the $T$\ndependence of $\\Upsilon$ has been included \\cite{Moss:2011qc}, which\nprovides an extra \nnon-linear source in the field second order equation. However, if the\ncoupling between field and radiation perturbations at first order is\nsuppressed by viscous effects, qualitatively we expect the same to\nhappen at second order. The question then is whether one simply\nrecover the prediction for a constant dissipative coefficient, or\nnon-linearities are further suppressed by viscous effects. \nThese and other effects mentioned above will be studied\nelsewhere.\n\n\\acknowledgements\n\n\nA.B. acknowledges support from the STFC. R.O.R\nis partially supported by Conselho Nacional de\nDesenvolvimento Cient\\'{\\i}fico e Tecnol\\'ogico (CNPq - Brasil).\nM.B.G. would like to thank the hospitality of the School of \nPhysics and Astronomy at the\nUniversity of Edinburgh which during her visit this work has started.\nMBG is partially supported by MICINN (FIS2010-17395) and ``Junta de\nAndaluc\\'ia\" (FQM101), and by SUPA during the realization of this work\nin the UK.    \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction }\n\nOne of the predictions of general theory of relativity is the existence of gravitational waves. The sources of  generation of these waves  vary  from   the dynamics of early universe to  massive  astrophysical objects  such as  neutron star binaries and black hole mergers etc. Thus the    waves have a  wide spectrum   of frequencies vary from very low to high ${\\cal{O}}($10$^{-19}$Hz -10$^{10}$Hz). It is possible to discriminates the  relic waves from other sources on the observational point of view also.   The relic gravitational waves are   paramount importance in cosmology because it provides valuable information on the conditions of very early universe.   It is believed  that the relic gravitational waves are mainly generated during inflationary epoch.  And the waves that amplified  during the inflation are low frequency only.  Since the higher frequency  waves are  outside the  ``barrier'' (horizon)\\cite{gh}  \\footnote{ the terminology ``barrier\" is adopted from \\cite{gh}.} the corresponding  amplitudes decreased during the evolution of the universe. \nThe features of  relic gravitational  waves  of  very high frequency range  is interesting   though  the energy scale of conventional inflationary models  are not favoring for it. \nHowever  if  extra dimensions exist (for a review, see \\cite{rex} and motivation for extra dimensions \\cite{ex}) the graviton background  can have a thermal spectrum \\cite{fry}. According to the  extra dimensional models  the  thermal gravitons with very high frequency range also be observed with  a specific peak temperature  today  \\cite{fry} and  \n therefore  the detection of very high  frequency thermal gravitational waves is  an interesting test to   see the possibility of existence of extra dimensions as well. \nThese  thermal gravitational waves   can contribute to the  higher frequency range of the spectrum.    \nThe \nexistence of  thermal graviton background\nwith  the black body type spectrum  is   also discussed in \\cite{24},\\cite{25}. \nIf  the inflation was preceded by a radiation era, then there would be  thermal gravitational waves at the time of inflation \\cite{25}. The  generation of tensor perturbations during inflation  by  the stimulated emission   process leads  to the existence of  thermal  gravitational waves \\cite{29}.  \nDirect detection of the thermal gravitons is challenging\nbut may be possible in the near future with the 21-cm\nemission line of atomic hydrogen. \n  \n  \n  The inflationary scenario \\cite{23} predicts a\nstochastic cosmic background of gravitational waves\n(CGWB) \\cite{24}.\nThe spectrum of  these relic gravitational waves  depends\nnot only on the details  of expansion  during the  inflationary era  but also \n the subsequent stages, including the current  epoch of the universe. \nComputation of  the  spectrum of the  waves   for   matter dominated   universe  is usually  done in  decelerated expanding model \\cite{1}-\\cite{6}. The resulting spectrum is used  for putting constrain on the detection of  gravitational waves  originated from  sources other than  early universe epoch.\nThe result of  astronomical\nobservations on SN Ia \\cite{11}-\\cite{12} shows that the universe is currently under going  accelerated\nexpansion  indicating a non-zero cosmological \nconstant.  According to the $\\Lambda$CDM\nconcordance model, the observed acceleration of the present universe is supposed to be driven by the dark energy. \n Effect of  the current acceleration  on the nature  of  the  spectrum  and  spectral energy density of the relic\ngravitational waves  is studied  \\cite{2},\\cite{15}.  And shown that  the current acceleration phase of the universe  does change  the shape, amplitude and spectrum  of the  waves  \\cite{15}.\n\n\nIn the present work, we   consider   contribution of  very higher frequency  relic  thermal gravitational waves  to its  spectrum and spectral  energy density  for the decelerated as well as  accelerated universe.  The focus of the present work is on the spectrum of  the higher frequency range of the  waves due to extra dimensional effects.\nThe  normalization  of the spectrum is being done with the measured  CMB  anisotropy spectrum of the WMAP.   The inclusion of the higher frequency relic thermal gravitational waves    leads to  enhancement of the spectrum.   This  enhancement  leads to  modification of the amplitude of the spectrum  in the frequency range (10$^{-16}$ -10 $^{8}$ Hz)   as an additional  feature  and is  possible to  compare these with the sensitivity of Advanced.LIGO (Adv.LIGO), Einstein Telescope  (ET) and LISA missions.  The corresponding spectral energy density can be compared with estimated upper bound of various studies.\n Also can check whether the  inclusion of  higher frequency thermal gravitational waves   in  the  total spectral energy density  exceed  the  upper  bound   of  primordial nucleosynthesis rate or not.\n In the present work, we use the unit $c=\\hbar = k_{B} =1$.\n \n  \n \\section{Gravitational waves spectrum in expanding universe}\nThe perturbed metric for a homogeneous  isotropic  flat Friedmann-Robertson-Walker (FRW) universe  can be written   as\n\\begin{equation}\nd s^{2}= S^{2}(\\eta)(d\\eta^{2}-(\\delta_{ij}+h_{ij})dx^{i}dx^{j}),\n\\end{equation}\nwhere $S(\\eta)$ is the cosmological scale factor,  $\\eta$ is the conformal time and $\\delta_{ij}$ is the Kronecker delta  symbol.  The $h_{ij} $ are  metric perturbations  field contain only the pure gravitational waves and is  transverse-traceless i.e; $\\nabla_i h^{ij} =0, \\delta^{ij} h_{ij}=0$.\n\nThe present study mainly deals  with amplitude and spectral energy density of  the  relic gravitational waves   generated by the expanding spacetime  background. Thus  the perturbed matter source is therefore not taken into account. The \n gravitational waves are described with  the  \n linearized field equation given by\n \\begin{equation}\\label{weq}\n \\nabla_{\\mu} \\left( \\sqrt{-g} \\, \\nabla^{\\mu} h_{ij}(\\bf{x}, \\eta)\\right)=0.\n \\end{equation} \nThe tensor perturbations have two independent physical degrees of freedom  and are denotes as $h^{+}$ and $h^{\\times}$, called polarization modes. To compute the spectrum of gravitational waves $h(\\bf{x},\\eta)$ in the thermal states, we express $h^{+}$ and $h^{\\times}$ in terms\nof the creation ($a^{\\dagger}$) and annihilation ($a$) operators,\n\\begin{eqnarray}\\label{1}\n\\nonumber  h_{ij}({\\bf x},\\eta)=\\frac{\\sqrt{16\\pi} l_{pl}}{S(\\eta)} \\sum_{\\bf{p}} \\int\\frac{d^{3}k}{(2\\pi)^{3\/2}} {\\epsilon}_{ij} ^{\\bf {p}}(\\bf {k}) \\\\\n \\times  \\frac{1}{\\sqrt{2 k}} \\Big[a_{\\bf{k}}^{\\bf {p}}h_{\\bf {k}}^{\\bf {p}}(\\eta) e^{i \\bf {k}.\\bf {x}} +a^{\\dagger}_{\\bf {k}} {^{\\bf {p}}} h^{*}_{\\bf {k}}{^{\\bf {p}}} (\\eta)e^{-i\\bf{k}.\\bf{x}}\\Big],\n\\end{eqnarray}\nwhere  $\\bf{k}$ is the comoving wave\nnumber, $k=|\\bf {k}|$, $l_{pl}= \\sqrt{G}$ is the\nPlanck's length and $\\bf{ p}= +, \\times$ are polarization modes. The polarization tensor \n$\\epsilon_{ij} ^{{\\bf p}}({\\bf k})$ is symmetric and transverse-traceless  $ k^{i} \\epsilon_{ij} ^{{\\bf p}}({\\bf k})=0, \\delta^{ij} \\epsilon_{ij} ^{{\\bf p}}({\\bf k})=0$ and \nsatisfy  the conditions $\\epsilon^{ij {\\bf p}}({\\bf k})   \\epsilon_{ij}^{{\\bf p}^{\\prime}}({\\bf k})= 2  \\delta_{ {\\bf p}{{\\bf p}}^{\\prime}} $ and $ \\epsilon^{{\\bf p}}_{ij} ({\\bf -k}) = \\epsilon^{{\\bf p}}_{ij} ({\\bf k}) $, the creation and annihilation operators  satisfy\n$[a_{{\\bf k}}^{{\\bf p}},a^{\\dagger}_{{\\bf k} ^{\\prime}} {{^{{\\bf p}}}^{\\prime}}]= \\delta_{{{\\bf p}} {\\bf {p}}^{\\prime} }\\delta^{3}({\\bf k}-{{\\bf k}}^{\\prime})$, the initial vacuum state is defined  as\n\\begin{equation}\na_{\\bf{k}}^{\\bf{p}}|0\\rangle = 0,\n\\end{equation}\nfor each $\\bf {k}$ and $\\bf {p}$. The energy density of the gravitational waves in vacuum state is $ t_{00}= \\frac{1}{32 \\pi l^2_{pl}} h_{ij,0} h^{ij}_{,0}$.\n\n For a fixed  wave number $\\bf{k} $ and a fixed polarization state $\\bf{p}$ the linearized wave equation (\\ref{weq}) gives\n \\begin{equation}\\label{zz1}\nh^{\\prime \\prime}_{k}+2\\frac{S^{\\prime}}{S}h^{\\prime}_{k}+k^{2}{h}_{k}=0,\n\\end{equation}\nwhere   prime means derivative with respect to the conformal time. Since the  polarization states are  same,   we here onwards denote  $h_{k}(\\eta)$ without the polarization  index.  \n\nNext, we  rescale the filed $h_{k}(\\eta)$ by taking\n$h_{k}(\\eta)=f_{k}(\\eta)\/S(\\eta)$, where the mode functions $f_{k}(\\eta)$ obey the minimally coupled Klein-Gordon equation\n\\begin{equation}\\label{zz}\nf^{\\prime \\prime}_{k}+\\Big(k^{2}-\\frac{S^{\\prime \\prime}}{S} \\Big)f_{k}=0.\n\\end{equation}\n The  general solution of the above equation  is a linear combination of  the Hankel function with a  generic power-law for the scale factor $S= \\eta^{q}$  given by\n\\begin{equation}\nf_k (\\eta)= A_k \\sqrt{ k \\eta} H^{(1)} _{(q-\\frac{1}{2})} (k \\eta)+  B_k \\sqrt{ k \\eta} H^{(2)}_{(q-\\frac{1}{2})} (k \\eta).\\end{equation}\nFor a given  model of the expansion of universe, consisting of a sequence of successive  scale factor with different $q$, we can obtain an exact solution  $f_k (\\eta)$ by matching its value and  derivative at the joining points.\n\n The  approximate computation of the spectrum of  gravitational  waves  is usually performed in two limiting cases depending up on the waves that are within  or outside of the  barrier. For the  gravitational  waves  outside barrier ($k^{2}\\gg S^{\\prime \\prime}\/S$, short wave   approximation)  the corresponding   amplitude  decrease as $h_k \\propto 1\/S(\\eta) $ while for the  waves inside the barrier ($k^{2} \\ll S^{\\prime \\prime}\/S$, long wave  approximation),  $h_k = C_k $ simply a constant. Thus these results can be used to estimate the spectrum for the present epoch of  universe.\n\n\nThe history of  overall expansion of the universe can be modeled as  following sequence\nof successive epochs of power-law expansion.\n\nThe initial stage (inflationary)\n\\begin{equation}\nS(\\eta)=l_{0}|\\eta |^{1+\\beta},\\;\\;\\;\\;\\;\\;-\\infty <\\eta\\leq \\eta_{1},\n\\end{equation}\nwhere $1+\\beta <0$, $\\eta<0$ and $l_{0}$ is a constant.\n\nThe z-stage\n\\begin{equation}\nS(\\eta)=S_{z}(\\eta - \\eta_{p})^{1+\\beta_{s}},\\;\\;\\;\\;\\;\\;\\eta_{1}<\\eta\\leq \\eta_{s},\n\\end{equation}\nwhere $\\beta_{s}+1>0$. Towards the end of inflation, during the reheating, the equation of state of\nenergy in the universe can be quite complicated and is rather model-dependent \\cite{qa}. Hence this\nz-stage is introduced to allow a general reheating epoch, see for details \\cite{3}.\n\nThe radiation-dominated stage\n\\begin{equation}\nS(\\eta)=S_{e}(\\eta-\\eta_{e}),\\;\\;\\;\\;\\;\\;\\eta_{s}\\leq \\eta \\leq \\eta_{2},\n\\end{equation}\n\nThe matter-dominated stage\n\\begin{equation}\nS(\\eta)=S_{m}(\\eta-\\eta_{m})^{2},\\;\\;\\;\\;\\;\\;\\eta_{2}\\leq \\eta \\leq \\eta_{E},\n\\end{equation}\nwhere $\\eta_{E}$ is the time when the dark energy density $\\rho_{\\Lambda}$ is equal to the matter energy density $\\rho_{m}$. Before the discovery of  accelerating expansion of the universe, the current expansion is\nused to take  as decelerating one  because of  the matter-dominated stage. Thus, following\nthe matter-dominated stage,  it reasonable to add an epoch of accelerating stage, which is probably driven\nby either the cosmological constant, or the quintessence, or some other kind of condensate \\cite{sa}. The value of  redshift $z_{E}$ at  $\\eta_{E}$ is  $(1+z_{E})=S(\\eta_{0})\/S(\\eta_{E})$, where $\\eta_{0}$ is the present time.  Since $\\rho_{\\Lambda}$ is constant and $\\rho_{m}(\\eta) \\propto S^{-3}(\\eta)$, we get\n\\begin{equation}\n\\frac{\\rho_{\\Lambda}}{\\rho_{m}(\\eta_{E})}=\\frac{\\rho_{\\Lambda}}{\\rho_{m}(\\eta_{0})(1+z_{E})^3}=1.\n\\end{equation}\nIf  the current value of $\\Omega_{\\Lambda}\\sim0.7$ and $\\Omega_{m}\\sim0.3$, then it follows that\n\\begin{equation}\n1+z_{E}=\\Big(\\frac{\\Omega_{\\Lambda}}{\\Omega_{m}}\\Big)^{1\/3}\\sim 1.33.\n\\end{equation}\n\nThe accelerating stage (up to the present)\n\\begin{equation}\\label{1w}\nS(\\eta)=\\ell_{0}|\\eta- \\eta_{a} |^{-1},\\;\\;\\;\\;\\;\\;\\eta_{E}\\leq \\eta \\leq\\eta_{0}.\n\\end{equation}\nThis stage describes the accelerating expansion of the universe.  And  is a  new feature and hence  its  influence  on  the spectrum of relic gravitational waves is of interesting to study. It is be  noted  that the actual scale factor function $S(\\eta)$ differs from equation (\\ref{1w}), since\nthe matter component exists in the current universe. However, the dark energy is dominant,\ntherefore (\\ref{1w}) is an approximation to the current expansion behaviour.\n\nGiven $S(\\eta)$ for the various epochs, the derivative $S^{\\prime}=dS\/d\\eta$ and  ratio $S^{\\prime}\/S$ follow\nimmediately. Except for  $\\beta_{s}$ which is imposed upon as the model parameter, there are ten\nconstants in the  expressions of $S(\\eta)$. By the continuity conditions of $S(\\eta)$ and $S^{\\prime}(\\eta)$ at\n four given joining points $\\eta_{1}, \\eta_{s}, \\eta_{2},$ and $\\eta_{E}$, one can fix only eight constants. The remaining\ntwo constants can be fixed by the overall normalization of $S$ and  the observed Hubble\nconstant as the expansion rate. Specifically, we put $|\\eta_{0}-\\eta_{a}|=1$ for the normalization of $S$, which fixes the  $\\eta_{a}$, and the constant $\\ell_{0}$ is fixed by the following calculation,\n\\begin{equation}\n\\frac{1}{H}\\equiv \\Big(\\frac{S^{2}}{S^{\\prime}}\\Big)_{\\eta_{0}}=\\ell_{0}.\n\\end{equation}\nwhere $\\ell_{0}$ is  the Hubble radius at present. \n\n In the expanding Friedmann-\nRobertson-Walker spacetime the physical wavelength is related to the comoving wave\nnumber  as\n$\\lambda \\equiv \\frac{2\\pi S(\\eta)}{k},$\nand the wave number $k_{0}$ corresponding to the present Hubble radius is\n$k_{0}=\\frac{2\\pi S(\\eta_{0})}{\\ell_{0}}=2\\pi.$ And\nthere is another wave number\n$k_{E}=\\frac{2\\pi S(\\eta_{E})}{1\/H}=\\frac{k_{0}}{1+z_{E}},$\nwhose corresponding wavelength at the time $\\eta_{E}$ is the Hubble radius $1\/H$.\n\nBy matching $S$ and $S^{\\prime}\/S$ at the joint points, one gets\n\\begin{equation}\\label{kk}\nl_{0}=\\ell_{0}b\\zeta_{E}^{-(2+\\beta)}\\zeta_{2}^{\\frac{\\beta-1}{2}}\\zeta_{s}^{\\beta}\\zeta_{1}^{\\frac{\\beta-\\beta_{s}}{1-\\beta_{s}}},\n\\end{equation}\nwhere $b\\equiv|1+\\beta|^{-(2+\\beta)}$, which is defined differently from  \\cite{p}, $\\zeta_{E}\\equiv\\frac{S(\\eta_{0})}{S(\\eta_{E})}$, $\\zeta_{2}\\equiv\\frac{S(\\eta_{E})}{S(\\eta_{2})}$, $\\zeta_{s}\\equiv\\frac{S(\\eta_{2})}{S(\\eta_{s})}$, and $\\zeta_{1}\\equiv\\frac{S(\\eta_{s})}{S(\\eta_{1})}$. With these specifications, the functions $S(\\eta)$ and $S^{\\prime}(\\eta)\/S(\\eta)$ are fully determined. In particular, $S^{\\prime}(\\eta)\/S(\\eta)$ rises up during the accelerating\nstage, instead of decreasing as in the matter-dominated stage. This causes the modifications to the spectrum of relic gravitational waves.\n\n\\section{Gravitational waves spectrum in thermal vacuum state}\nThe power spectrum  of   gravitational waves is defined  as\n\\begin{equation}\\label{pow}\n\\int_0 ^\\infty h^2 (k,\\eta) \\frac{dk} {k} = \\langle 0 | h^{ij}({\\bf x},\\eta) h_{ij}({\\bf x},\\eta) |0 \\rangle,\n\\end{equation}\n Substituting equation  (\\ref{1}) in (\\ref{pow}) and taking the contribution from each polarization is same, we get\n \\begin{equation}\\label{pp}\n h(k,\\eta)= \\frac{4 l_{pl}}{\\sqrt{\\pi}} k \\mid h(\\eta) \\mid.\n \\end{equation} \nThus once the mode function $h(\\eta)$ is known, the spectrum $h(k,\\eta)$ follows.\n\nThe spectrum at the present time $ h(k,\\eta_0)$ can be obtained, provided the initial  spectrum is specified. The initial condition is taken to be the during the inflationary stage. Thus the initial amplitude of the spectrum is given by\n\\begin{equation}\\label{bet}\nh(k,\\eta_i)= A{\\left(\\frac {k}{k_0}\\right)}^{2+\\beta},\n\\end{equation}\nwhere $A=8\\sqrt{ \\pi} \\frac{l_{pl}}{l_0} $ is a constant. The  power spectrum for the primordial perturbation of energy density  is $P(k)\\propto {\\mid  h(k,\\eta_0)\\mid}^2$ and in terms of  initial spectral index $n$, it is defined as $ P(k) \\propto k^{n-1}$.\nThus the scale invariant spectral index  $n=1$ for the pure de Sitter expansion can be obtained   with the relation $n= 2 \\beta +5 $  for $\\beta $= - 2. \n \n\nAn effective approach to deals with the thermal  vacuum state is the thermo-field dynamics (TFD)\\cite{34}. In this approach  a tilde space  is needed besides the usual\nHilbert space, and the direct product space is made up of the these two spaces. Every operator and state in the Hilbert space has the corresponding counter part in the tilde\nspace \\cite{34}. Therefore a  thermal  vacuum state ($Tv$) can be  defined as\n\\begin{equation}\\label{16}\n|Tv\\rangle ={\\cal T }(\\theta_{k})|0\\; \\tilde{0}\\rangle,\n\\end{equation}\nwhere \n \\begin{equation}\\label{333}\n{\\cal T }(\\theta_{k})=\\mathrm{exp} [-\\theta_{k} (a_{\\bf {k}}\\tilde{a}_{\\bf {k}}-a_{\\bf{k}}^{\\dagger}\\tilde{a}_{\\bf {k}}^{\\dagger})],\n\\end{equation}\nis the thermal operator and $|0\\; \\tilde{0}\\rangle$ is the two mode vacuum state at zero temperature. The  quantity $\\theta_{k}$ is  related to   the average number of the thermal particle, $\\bar{n}_{k}=\\mathrm{sinh}^{2}\\theta_{k}$. The $\\bar{n}_{k}$  for given  temperature T is\nprovided   by the Bose-Einstein distribution \n$\\bar{n}_{k}=[\\mathrm{exp}( k \/T)-1]^{-1}$, \nwhere $\\omega_{k}$ is the resonance frequency of the field. The $a_{{\\bf k}}$, $a_{{\\bf k}}^{\\dagger}$  and  $\\tilde{a}_{{\\bf  k}}$, $\\tilde{a}_{{\\bf k}}^{\\dagger}$, are respectively the annihilation and creation operators in Hilbert and tilde space, satisfy the  usual commutation relations,  \n$[a_{{\\bf k}},a^{\\dagger}_{{\\bf k}^{\\prime}}]=\n[\\tilde{a}_{{\\bf k}},\\tilde{a}^{\\dagger}_{{\\bf k}^{\\prime}}]=\\delta^{3}({\\bf k}-{{\\bf k}^{\\prime}}) \\; $. And all other commutation relations   of these operators are zero.\n By the appropriate action of the operator (\\ref{333})  on $a_{{\\bf k}}$ and $a_{{\\bf k}}^{\\dagger}$, we  get \\cite{35}\n\\begin{eqnarray} \\label{tt}\n\\nonumber {\\cal T }^{\\dagger}a_{\\bf {k}}{\\cal T }=a_{\\bf {k}}\\;\\mathrm{cosh}\\; \\theta_{k} +\\tilde{a}_{\\bf {k}}^{\\dagger}\\; \\mathrm{sinh} \\; \\theta_{k},\\\\\n{\\cal T }^{\\dagger}a^{\\dagger}_{\\bf {k}}{\\cal T }=a^{\\dagger}_{\\bf {k}}\\; \\mathrm{cosh}\\; \\theta_{k} +\\tilde{a}_{\\bf{k}}\\;\\mathrm{sinh}\\; \\theta_{k}.\n\\end{eqnarray} \nHence  the  occupation number   in  thermal vacuum state\n can be written as\n\\begin{equation}\\label{w}\n\\langle a^{\\dagger}_{\\bf {k}} a_{\\bf {k} ^{\\prime}}\\rangle = \\left( \\frac{1}{e^{k\/T} -1} \\right)\\delta^{3}(\\bf {k}-\\bf {k}^{\\prime}).\n\\end{equation}\n Thus, using Eq.(\\ref{1}) and Eqs.(\\ref{16}-\\ref{w}) in Eq.(\\ref{pow}) the power spectrum in thermal vacuum state is obtained as\n  \\begin{equation}\n h^2 _T(k,\\eta)=  \\frac{16 l^2 _{pl} }{\\pi} k^2 {\\mid h(\\eta)\\mid}^2  \\mathrm {coth}\\Big[\\frac{k}{2T}\\Big],\n   \\end{equation}\n\nThus in comparison with Eq.(\\ref{bet}), the spectrum  is expressed as\n\\begin{equation}\nh(k,\\eta_i)= A{\\left(\\frac {k}{k_0}\\right)}^{2+\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}.\n\\end{equation}\n   The  last  term becomes significant when the ratio $k\/(2T)$ is less than unity. The  wave number $k$ and  temperature $T$ are comoving quantity  which are \n    related  to the physical parameters at the time of inflation, see for details \\cite{25}. Thus it is expected an enhancement of the spectrum by a factor $\\mathrm {coth}^{1\/2}[{k}\/{2T}] =\\mathrm {coth}^{1\/2}[{H S_i}\/{2T_i}]$ .\n   \n   It is convenient to consider the amplitude of waves   in different  range of wave numbers \\cite{15}.  Thus the amplitude  of the spectrum in thermal vacuum state  for  different ranges are given by\n   \n       (i) when $k\\leq k_{E}$, the  corresponding wavelength is   greater the present Hubble radius. Thus the amplitude remain as the initial one and can be written as \n\n\\begin{equation}\\label{y}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{2+\\beta}\\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]},\n\\end{equation}\n\n(ii)   the amplitude remains  approximately same as long as the wave inside the barrier   but begin to  decrease when it leaves the barrier by a factor $1\/S(\\eta)$, depending the value of scale factor at that time. This  process  continue until the barrier becomes higher than $k$ at a time  $\\eta$ earlier than $\\eta_0$, so the  amplitude has decreased by  the ratio of the scale factor at the time of  leaving the barrier $S_b$ to  its value at $\\eta$, $S(\\eta)$. This   is   in the   range  $k_{E}\\leq k\\leq k_{0}$.  \n   \n\\begin{equation}\\label{ke}\nh_{T}(k ,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta-1} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\nNote  that this range is a new feature   on account of the  current acceleration of the universe  which is absent in the decelerating model as pointed out in \\cite{15}.  The amplitude of     the waves  that left  the barrier at $S_b$ with waves numbers $k > k_0$  has  been decreased  up to the present time by a factor $ S_b \/ S( \\eta _0)$. This affect the amplitude of the  present spectrum and is obtained as\n\\begin{equation}\\label{sq}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{2+\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]} \\frac{S_b}{S(\\eta_0)}.\n\\end{equation}\nThis result  can be  used to obtain  the spectrum of the waves in the remaining range of wave numbers.\n\n(iii) the  wave number that does not hit the barrier in the range $ k_{0}\\leq k\\leq k_{2} $ gives the amplitude as follows\n\\begin{equation}\\label{l}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}},\n\\end{equation}\n the spectrum  in this   interval  is differ from that of the matter dominated case by a the factor $\\frac{1}{(1+z_{E})^{3}}$. The  wave lengths of the spectrum  in the range  are  long but smaller than the present Hubble radius.\n \n (iv) in  the range of wave number $ k _{2}\\leq  k \\leq k _{s}$ , the amplitude is\n\\begin{equation}\\label{o}\nh(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta}\\Big(\\frac{k_{0}}{k_{2}}\\Big) \\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\n This is the interesting range on the observational point of view of Adv.LIGO, ET and LISA. Note that the temperature dependent factor in this  range  is negligible hence  the term  is dropped out  because of the low temperature nature of the  relic waves. \n \n (v) for the wave number  range $k _{s}\\leq k \\leq k _{1}$  which is in  the high frequency  case and   gives the corresponding amplitude as\n\\begin{equation*}\nh_{T}(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta-\\beta_{s}}\\Big(\\frac{k _{s}}{k _{0}}\\Big)^{\\beta_{s}}\\Big(\\frac{k _{0}}{k _{2}}\\Big) \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation*}\n\\begin{equation}\\label{oo}\n\\end{equation}\n \n In the usual case the temperature dependent term can also be  neglected however the extra dimensional scenario  predicts  higher temperature for the  thermal gravitational waves, hence the term again becomes significant. Therefore the contribution from the thermal relic gravitational waves is  expected increase the amplitude of spectrum  particularly in the higher frequency range also. \n \n \n It is to be noted that in  (iv)   the  thermal contribution in $ k _{2}\\leq  k \\leq k _{s}$  range is  negligible   \ndue to  the temperature dependent  term.  Similarly  the thermal effect is  insignificant  in the range $k _{s}\\leq k \\leq k _{1}$ also. However\nby taking into account  the extra dimensional effect,  the spectrum of  relic waves  is   peaked with a  temperature   $T_{*}$=1.19 $\\times$ 10$^{25}$\\;{Mpc}$^{-1}$ \\cite{fry} (See, appendix  A for a brief discussion on $T_{*}$ from extra dimensional scenario.). Therefore it is expected  enhancement  for the amplitude of spectrum (orange lines, Figs.[\\ref{ff1}] and [\\ref{ff2}]) in the range  $ k _{s}\\leq  k \\leq k _{1}$ compared   to $T= 0$ case for the accelerated as well as  decelerated universe. But at the same time, ignoring  the  thermal contribution  to  the amplitude of  spectrum  in the range $ k _{2}\\leq  k \\leq k _{s}$ leads to a discontinuity  at $ k_{s}$, see Fig.[\\ref{ff1}]. \nThis   is evaded  by  fitting a new  line  in the range  $ k _{2}\\leq  k \\leq k _{s}$\nfor  the amplitude $h$  of Eq.(\\ref{o})  as follows. \n \n Let the amplitude of the wave in the range  $k _{0}\\leq  k \\leq k _{2}$ is given by  (\\ref{l}) and can be rewritten as\n\\begin{equation}\\label{1q}\nh_{1T}(k,\\eta_{0})=A\\Big(\\frac{k}{k_{0}}\\Big)^{\\beta} \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T}\\Big]}\\frac{1}{(1+z_{E})^{3}},\n\\end{equation}\nand  the amplitude in the $k _{s}\\leq  k \\leq k _{1}$ is given by (\\ref{oo}) also rewritten as\n\n\\begin{equation}\n\\label{3q}\nh_{2T}(k,\\eta_{0})=A\\Big(\\frac{k}{k _{0}}\\Big)^{1+\\beta-\\beta_{s}}\\Big(\\frac{k _{s}}{k _{0}}\\Big)^{\\beta_{s}}\\Big(\\frac{k _{0}}{k _{2}}\\Big) \\mathrm {{coth}^{1\/2}\\Big[\\frac{k}{2T_{*}}\\Big]}\\frac{1}{(1+z_{E})^{3}}.\n\\end{equation}\nThus the new slope for  Eq.(\\ref{o}), in the range $ k _{2}\\leq  k \\leq k _{s}$, can be obtained \nby taking  $y\\equiv \\log_{10}(h)$ and $x\\equiv \\log_{10}(k)$, then \n\\begin{equation}\\label{poo}\n\\log_{10}(h)-\\log_{10}(h)_{i}=\\frac{\\log_{10}(h)_{f}-\\log_{10}(h)_{i}}{\\log_{10}(k_{f})-\\log_{10}(k_{i})}(\\log_{10}(k)-\\log_{10}(k_{i})),\n\\end{equation}\nwhere the subscribes  $i$ and $f$  are respestively indicating  the first and last points of the straight line.\nBy putting $k_{i}\\equiv k_{2}$ from Eq.(\\ref{1q}) and $k_{f}\\equiv k_{s}$ from Eq.(\\ref{3q}) in   Eq.(\\ref{poo}), we get \\footnote{ here, $\\mathrm {{coth}^{1\/2}\\Big[\\frac{k_{2}}{2T}\\Big]}=1. $}\n\\begin{equation}\\label{p1}\nh=(h_{1T})_{k_{2}}g(k),\n\\end{equation}\nwhere\n\\begin{equation}\ng(k)=\\Big(  \\frac{k}{k_{2}}\\Big)^{\\gamma},\n\\end{equation}\nand\n\\begin{equation}\\label{ss}\n\\gamma=\\frac{\\log_{10}(h_{2T})_{k_{s}}-\\log_{10}(h_{1T})_{k_{2}}}{\\log_{10}(k_{s})-\\log_{10}(k_{2})}=\\frac{\\log_{10}\\Big ( \\Big(\\frac{k_{s}}{k_{2}} \\Big)^{1+\\beta}  \\mathrm {{coth}^{1\/2}\\Big[\\frac{k_{s}}{2T_{*}}\\Big]}\\Big)}{\\log_{10}\\Big( \\frac{k_{s}}{k_{2}}\\Big)},\n\\end{equation}\nis the slope of the line and thus  we find the amplitude, for convenience we  call it as `modified amplitude', given by\n\\begin{equation}\\label{ppp}\nh(k,\\eta_{0})=A\\Big (\\frac{k_{2}}{k_{0}}\\Big)^{\\beta} \\frac{1}{(1+z_{E})^{3}} \\Big( \\frac{k}{k_{2}} \\Big)^{\\gamma}.\n\\end{equation}\nWhen $T_{*}$  becomes  zero    Eq.(\\ref{ss}) leads to $\\gamma=1+\\beta$, and  hence   (\\ref{o}) is recovered  from   (\\ref{ppp})  in the range $k _{2}\\leq  k \\leq k _{s}$.\n\n\n The overall multiplication factor $A$  in all the spectra is determined in  absence of  the temperature dependent  term  with the CMB data of  WMAP  \\cite{15}.  This is  based on the  assumption that the contribution from  gravitational waves  and  the density perturbations  are the same order of magnitude  or if  the CMB anisotropies at low multipole are induced by the gravitational waves,   therefore \n it is possible to write  $\\Delta T \/ T \\simeq h(k,\\eta_{0})$.  The observed CMB anisotropies \\cite{u} at lower multipoles is $\\Delta T \/ T \\simeq0.37\\times10^{-5}$ at $l\\sim2$ which corresponds to the largest\nscale anisotropies that have observed so far. Thus taking this to be the perturbations at the Hubble\nradius  gives\n\\begin{equation}\\label{k}\nh(k_{0},\\eta_{0})=A\\frac{1}{(1+z_{E})^{3}}=0.37 \\times 10^{-5}.\n\\end{equation}\nHowever, there is a subtlety  in the interpretation of $ \\Delta T \/ T$ at low multipoles, whose\ncorresponding scale is very large $ \\sim \\ell_{0}$. At present the Hubble radius is $\\ell_{0}$, and the Hubble\ndiameter is $2\\ell_{0}$. On the other hand, the smallest characteristic wave number is $k_{E}$, \nwhose\ncorresponding physical wave length at present is $2\\pi S (\\eta_{0} )\/k_{E}=\\ell_{0}(1+z_{E})\\simeq1.32 \\ell_{0}$, which\nis within the Hubble diameter $2 \\ell_{0}$, and is theoretically observable. So, instead of Eq.(\\ref{k}), if $ \\Delta T \/ T \\simeq 0.37 \\times 10^{-5}$ at $l\\sim2$ were taken as the amplitude of the spectrum at $\\nu_{E}$, one would\nhave $h_T(k_{E},\\eta_{0})=A\/(1+z_{E})^{2+\\beta}=0.37\\times10^{-5}$, yielding a smaller $A$ than that in Eq. (\\ref{k}) by a factor $(1+z_{}E)^{1-\\beta}\\sim2.3$ \\cite{15}. The allowed values of  $\\beta$ and $\\beta_s$ are obtained and are  respectively give by $\\beta=-1.9$, and $\\beta_{s}=-0.552$ \\cite{15}.\n\n  \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{m1.eps}}\n \\caption{ The  amplitude of  the gravitational waves  for the accelerated (solid lines) and decelerated (dashed lines) universe. }\\label{ff1}\n \\end{figure} \n \n \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{LIG.eps}}\n \\caption{  Comparison  of  the  modified  amplitude of the spectrum for the accelerated (solid black and green lines) and decelerated (dashed black and green lines) universe with  the  sensitivity curves of Adv.LIGO \\cite{alg}   and ET \\cite{et}.  }\\label{f3}\n\\end{figure}    \n\n \\begin{figure}[t]\n {\\includegraphics[scale=0.52]{LIS.eps}}\n \\caption{Comparison  of  the  modified  amplitude of the spectrum for the accelerated (solid black and green lines) and decelerated (dashed black and green lines) universe with   the   LISA sensitivity curve. }\\label{f4}\n\\end{figure}    \n \n Next, we obtain the spectrum in the thermal vacuum state with the following  parameters. By taking   $k=2\\pi\\nu$, $\\nu_{E}=1.5\\times10^{-18}$ Hz, $\\nu_{0}=2\\times10^{-18}$ Hz, $\\nu_{2}=117\\times10^{-18}$ Hz, $\\nu_{s}=10^{8}$ Hz, $\\nu_{1}=3\\times10^{10}$ Hz ( the value of  $\\nu_{1}$  is taken such a way that spectral energy density does not exceed the level of $10^{-6}$ , as required by the rate of  primordial nucleosynthesis). \n The range of frequency is  chosen  in accordance with  generation of gravitational waves  that vary from early universe to  various astrophysical sources. And hence the range  is matching with  the interest   of CMB, Adv.LIGO, ET and LISA  operations for detection of  the gravitational waves.\nThe  spectrum is computed  in the thermal vacuum state  with the chosen values of the parameters  for  the  accelerated as well as decelerated model with  $T=0.001$Mpc$^{-1}$  in the low  range $k < k_{2}$. This temperature is    considered in the context of B mode of  CMB spectrum  in   thermal  state  \\cite{25}. And  $T_{*}$  =1.19$\\times$10$^{25}$Mpc$^{-1}$  \\footnote{here,  $T_{*}$ = 0.905 K=  1.19$\\times$10$^{25}$Mpc$^{-1}.$ } for the high  range $k_{s}\\leqslant k\\leqslant k_{1}$ which is from the   extra dimensional scenario \\cite{fry}.  Since we use the natural unit, the wave number and temperature that appear in the temperature dependent term of the spectrum is computed  numerically in the Mpc$^{-1}$ unit.\nThe obtained spectra are normalized   of  the CMB  anisotropy spectrum of  WMAP data. \n The  amplitude of the spectrum of  the thermal gravitational  waves is  enhanced  compared to its zero temperature case (vacuum case). It is  observed that the  spectrum for  $T=0.001$Mpc$^{-1}$ get maximum enhancement   $\\sim 1.51$  times  than  the  vacuum case, at  $k=k_E$,\nand   it is $\\sim 20$ times for  $T_{*}$=1.19$\\times$10$^{25}$Mpc$^{-1}$ at $k=k_s$. \n\n\n The  plots  for the  amplitude  of   spectrum $h_{T}(k,\\eta_{0})$ versus the frequency $\\nu$ for $\\beta=-1.9$ and $\\beta_{s}=-0.552$  are  given in Fig.[\\ref{ff1}]. The amplitude of the  spectrum get enhanced  in the  frequency ranges, 10$^{-19}$ Hz$ \\leq \\nu <1.49 \\times 10 ^{-17}$Hz, and $\\nu_{s}\\leq \\nu\\leq \\nu_{1}$ ( the lower   value  of this range is  selected  such way that  thermal enhancement of the spectral density does not exceed the upper bound of nucleosynthesis rate.) due to the thermal effect of gravitational waves but  for the  range $ 1.49 \\times 10 ^{-17} $Hz $\\leq \\nu < \\nu_{s}$     there is a suppression because of  the $\\coth^{1\/2}[k\/2T]$   term.  For  comparison, the amplitude of the spectra  are plotted for the decelerated and accelerated  universe, see  Fig.[\\ref{ff1}].\n\n  \n The new enhancement of the gravitational waves spectrum   due to the extra dimensional effect (the  modified amplitude, see  Fig.[\\ref{ff1}], light green lines) can be compared with  the sensitivity of  Adv.LIGO, ET and LISA. An analytical expressions for the Adv.LIGO and ET  interferometers are  discussed in \\cite{bs}. For  Adv.LIGO and ET cases, consider the root mean square amplitude per root Hz which equal to\n \\begin{equation}\n \\frac{h(\\nu)}{\\sqrt{\\nu}}.\n \\end{equation}\n The  comparison of the sensitivity  (10 Hz - 10$^4$ Hz) curve of the ground based interferometer  Adv.LIGO \\cite{alg} with the  gravitational wave spectra  of  $\\beta =-1.9$   for the accelerated and decelerated universe are given in Fig.[\\ref{f3}].  Thus it shows that the Adv. LIGO  is unlikely to   detect  the enhancement   of the spectrum  from the  extra dimensional effect with its current stands but  be possible with  the sensitivity of  ET.\n  \n  \n  \n Next, we compare the enhancement of the spectrum with the sensitivity  (10$^{-7}$ Hz - 10$^0$ Hz) of  space based detector LISA \\cite{ss}. It is assumed that LISA  has one year observation time which corresponds to frequency bin $\\Delta \\nu$ = 3 $\\times$ 10 $^{-8}$Hz ( one cycle\/year) around each frequency.  Hence to make a comparison with the sensitivity curve,  a rescaling  of the spectrum $h(\\nu)$ is required in Eq.(\\ref{pp}) into the root mean square spectrum $h(\\nu, \\Delta \\nu)$ in the band $\\Delta \\nu$, given by\n \\begin{equation}\n h(\\nu, \\Delta \\nu)=  h(\\nu) \\sqrt{\\frac{\\Delta \\nu}{\\nu}}.\n   \\end{equation}\n The plots of the LISA sensitivity  with the  modified amplitude of the spectrum are  given in Fig.[\\ref{f4}] for the accelerated and decelerated universe. This show that the LISA is unlikely to detect the spectrum with the new enhancement feature of the gravitational waves. \n\n\n\nThe spectral energy density\nparameter $\\Omega_{g}(\\nu)$ of gravitational waves is defined through the relation $\\rho_{g}\/\\rho_{c}=\\int\\Omega_{g}(\\nu)\\frac{d\\nu}{\\nu}$, where $\\rho_{g}$ is the energy density of the gravitational waves and $\\rho_{c}$ is the critical energy density.\nThus\n\\begin{equation}\\label{ka}\n\\Omega_{g}(\\nu)=\\frac{\\pi^{2}}{3}h^2_{T}(k,\\eta_{0})\\Big(\\frac{\\nu}{\\nu_{0}}\\Big)^{2}.\n\\end{equation}\n Since the spacetime is assumed to be spatially flat \n$K=0$ with $\\Omega=1$, the fraction density of relic gravitational waves must be less than unity, $\\rho_{g}\/ \\rho_{c}<1$. After  \n  normalization of  the spectrum by using Eq.(\\ref{k}), we  integrate $\\int\\Omega_{g}(\\nu)d\\nu\/ \\nu$ from  $\\nu_{*}=10^{-19}$ Hz up to  $\\nu_{1}=3\\times10^{10}$ Hz,  with $\\beta=-1.9$ and $\\beta_{s}=-0.552$. \nThe integral is  evaluated   for the thermal case  and  zero temperature case, the  obtained results for the accelerated universe are  \n\n(a) $\\nu_{*}\\leq\\nu\\leq\\nu_{E}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&5.8\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&8.8\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(b) $\\nu_{E}\\leq\\nu\\leq\\nu_{H}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.3\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&3.5\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(c) $\\nu_{H}\\leq\\nu\\leq\\nu_{2}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.4\\times 10^{-11},\\;\\;\\;\\;T=0,\\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&3.7\\times 10^{-11},\\;\\;\\;\\;T=0.001\\;{Mpc}^{-1},\n\\end{eqnarray*}\n\n(d) $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&8.97\\times 10^{-9},\\;\\;\\;\\;T=0,\\\\\n\\end{eqnarray*}\n\n(e) $\\nu_{s}\\leq\\nu\\leq\\nu_{1}$,\n\\begin{eqnarray*}\n\\frac{\\rho_{g}}{\\rho_{c}}&=&2.7\\times 10^{-6},\\;\\;\\;\\;T=0. \\\\\n\\frac{\\rho_{g}}{\\rho_{c}}&=&6.67\\times 10^{-6},\\;\\;\\;\\;T_{*}=1.19 \\times10 ^{25}Mpc^{-1}.\n\\end{eqnarray*}\n\n\\begin{figure*}[t]\n{\\includegraphics[scale=0.52]{m2.eps}}\n\\caption{ The  spectral  energy density of the gravitational waves  for the accelerated (solid lines) and decelerated (dashed lines) universe.}\\label{ff2}\n\\end{figure*}\n\nIt is to be noted that in  (d)  the  thermal case are not shown  because  the  thermal contribution in this frequency range is  negligible   \ndue to  the temperature dependent  term. However\nby taking into the extra dimensional effect, the  upper limit of temperature of the relic waves  is to be $T_{*}$=1.19$\\times$10$^{25}$Mpc$^{-1}$. Thus it is expected an enhancement of the  spectral energy density in range  $\\nu_{s}\\leq\\nu\\leq\\nu_{1}$ compared   to $T= 0$ case for the accelerated as well as  decelerated universe. But at the same time ignoring the thermal contribution  on the spectral density in the range  $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$ leads to a discontinuity  at $ \\nu_{s}$, see Fig.[\\ref{ff2}]. This problem  is solved  by  fitting a new  line  as discussed  in the context  of  estimation of the amplitude of the spectrum and  hence recomputed the spectral density in the range  $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$ which gives the new value $\\frac{\\rho_{g}}{\\rho_{c}}$ = 8.21$\\times 10^{-7}$. This \n changes the slope  indicating enhancement of the spectral energy density of the gravitational waves in the range    $\\nu_{2}\\leq\\nu\\leq\\nu_{s}$, green lines, Fig.[\\ref{ff2}].\n \nThe enhancement spectral energy density $\\Omega_{g}(\\nu)$ in (d) can be  compared with  the estimated upper bound of various  studies and  are given  in Tab.[\\ref{t1}].   Thus $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g}(\\nu)$   are less than the upper bound of  the estimated values  of   the respective frequency range.\n\\begin{table}\n\\caption{\\label{t1}Comparison of  the estimated upper bound  of spectral energy density of  various studies  with   the present work.\nHere $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g} $ are   respectively the   spectral energy density  of the relic gravitational waves in  the decelerated and accelerated universe  of the present study and $\\Omega^{(est)}_{g}$ is the  estimated upper bound of  various studies.\n} \n\n\\footnotesize\\rm\n\\begin{tabular*}{\\textwidth}{@{}l*{15}{@{\\extracolsep{0pt plus12pt}}l}}\n\\br\nFrequency($\\nu$) Hz &$ \\Omega^{(dec)}_{g}(\\nu)$ & $\\Omega^{(acc)}_{g}(\\nu)$&  $\\Omega^{(est)}_{g}(\\nu)$\\\\\n\\mr\n$10^{-9}-10^{-7} $ & $ 4.98 \\times10^{-9}$ & $  1.03 \\times10^{-9}$ & $2 \\times 10^{-8} \\;\\;\\;\\cite{z}$ \\\\\n$69-156$ &  $  34.84 \\times10^{-8}$ & $  7.2 \\times10^{-8}$ & $8.4\\times 10^{-4}\\cite{zz}$ \\\\\n$41.5-169.25$  & $  4.93 \\times10^{-7}$ & $ 1.02 \\times10^{-7}$ & $6.9\\times 10^{-6}\n \\cite{zzz}$\\\\\n\\br\n\\end{tabular*}\n\\end{table}\n\n\n \nFurther  see that  the  contribution  to  $\\rho_{g}\/\\rho_{c}$  from  the low frequency range  is $ {\\cal{O}} (10^{-11}-10^{-10})$ while  from the higher frequency range it is $ {\\cal{O}}(10^{-6})$.  Since the order of  contribution  to   the total $\\rho_{g}\/\\rho_{c}$ from the lower frequency side is   very small  in  contrast with higher frequency side,  we  get  for the accelerated universe as\n \\begin{equation}\n \\rho_{g}\/\\rho_{c}\\simeq 6.67\\times10^{-6}   \\; \\; \\;  \\nu_{*}\\leq\\nu\\leq\\nu_{1},\n\\end{equation}\nand is  the same order as that of the zero temperature case.\n However    $ \\rho_{g}\/\\rho_{c}$  of the gravitational waves with $T\\neq 0$ is higher  than  the zero temperature  case at lower frequency range $\\nu_{*} \\leq\\nu\\leq\\nu_{2}$. Therefore  it is expected an enhancement for   the spectral energy density  in the thermal  vacuum state  in the  frequency range  $\\nu_{*} \\leq\\nu\\leq\\nu_{2}$ and actually  it   is the range  of interest on the observational point of view of the  relic gravitational waves.   The total   estimated value of $ \\rho_{g}\/\\rho_{c}$  by including    the thermal relic   gravitational waves in the very high frequency   does not alter the  upper bound of the nucleosynthesis rate. Thus the relic thermal  gravitons  with very high frequency range  are not ruled out and is   testable with  the upcoming  data of various missions  for detecting gravitational waves.\n \\section{Discussion and conclusion}\n \nGravitational waves are  one of the classical predictions of  Einstein's general theory of relativity. The gravitational waves are generated  during the very early  evolution stages  of the universe as well as from the various astrophysical objects.  Therefore  frequency of the waves are  varying  very widely.  There are many on going  experiments  to detect these waves and the  interested range of frequency is from 10$^{-19}$ Hz to 10$^{10}$Hz.  The existence of gravitational waves with  very high  frequency range is  not  favoring by the energy scale of the conventional  inflationary scenario. However the  very high frequency range gravitational waves are interesting   candidates  in the models with extra dimensions.  The extra dimensional theories predict the existence of thermal gravitons with black body type spectrum.  These relic thermal gravitational waves can also add to the spectrum of the waves thus the corresponding amplitude  also  get enhanced. \nThe nature of spectrum of the waves   to be observed today is   dependents on the  evolution history of the universe. Before the result of SN Ia observations,  the current evolution of the universe  is used to consider as  matter dominated  with decelerated expansion.  But, according  to the $\\Lambda$CDM concordance model the present universe is supposed to be driven by dark energy  resulting an accelerated phase. If this  is the case then the spectral property of the waves  to be studied by taking into account of the current acceleration of the universe.    In the present work we  mainly considered the  very high frequency range  (The low frequency range thermal gravitational case  is carried out   by us without including the  very high frequency thermal waves  that comes from extra dimensional scenario and the   work is under  preparation. The enhancement of the lower frequency range is shown with red lines, see Figs.[\\ref{ff1},\\ref{ff2}]) of relic gravitational waves in the thermal vacuum state and obtained the spectrum for the accelerated as well as decelerated models.  The obtained spectra are normalized with the WMAP data. It is observed that  the inclusion of the very high relic thermal gravitational waves  leads to a discontinuity  in  the  amplitude of the spectrum at  $\\nu_s$ (see Fig.[\\ref{ff1}]). This is  \ndue to the fact that the temperature dependent term is insignificant in the  higher  frequency side of  the  range $\\nu_2 \\leq \\nu \\leq \\nu_s$.  To evade this problem  a new  equation of  line  is derived and thus the amplitude get  enhanced in the range  $\\nu_2  \\leq \\nu \\leq \\nu_s$. This is the new feature of the spectrum and we designates  it as the `modified amplitude' of the spectrum.\n The  modified amplitude of the spectrum can be compared with the sensitivity of  the Adv.LIGO, ET and LISA missions. The comparison of the Adv.LIGO  sensitivity shows that the modified amplitude   is unlikely to   detect    with its current stands of  LISA or   the improved sensitivity of Adv.LIGO. Where  as the  proposed sensitivity of  the ET  is promising to verify the modified amplitude  with its upcoming mission data.\n \n \nThe spectral energy density of the gravitational waves is  estimated   in thermal vacuum state for  the accelerated and decelerated universe.  It is observed that the spectral energy density get enhanced   in  the lower frequency range  $ {\\cal{O}} (10^{-11}-10^{-10})$ and   from the higher frequency range it is $ {\\cal{O}}(10^{-6})$. A comparison of  the estimated upper bound  of spectral energy density of  various studies  with   the present work is done. It shows that\n $\\Omega^{(dec)}_{g}$ and $\\Omega^{(acc)}_{g} $   are less than  the  estimated upper bound of  various studies.  The total   estimated value of $ \\rho_{g}\/\\rho_{c}$  by including    the   very high frequency  thermal relic gravitational waves  does not alter the  upper bound of the nucleosynthesis rate. Thus the relic thermal  gravitons  with very high frequency range  are not ruled out and is   testable with  the upcoming  data of various missions  for detecting gravitational waves.\n\n\\section*{Appendix A}\n \\section*{Extra dimensional Scenario and Thermal Gravitons} \n \n Cosmology with extra dimensions have been motivated since Kaluza  and Klein (KK) showed that classical electromagnetism and general relativity could be combined in a five-dimensional framework. The modern scenarios involving extra dimensions are being explored in particle physics, with most models possessing either a large volume  or a large curvature. Although there exist  different  models of extra dimensions, there are some general features and signals common to all of them.\n\n \n In presence of  $D$ extra spatial dimensions, the 3+D+1- dimensional  action for gravity  for can be written as \n \\begin{equation}\n S = \\int d^4 x \\left [ \\int d^D y \\sqrt{- g_{D}} \\frac{R_{D}}{16 \\pi G_{D}}  + \\sqrt{-g} L_{m} \\right],\n \\end{equation}\n where \n \\begin{equation}\n G_D = G_N \\frac{m_{pl} ^2}{m_D ^{2+D}},\n \\end{equation}\n and $g$ is the four dimensional metric, $G_N$  is  Newton's constant, $g_D, G_N$ and $R_D$ denote the higher dimensional counter parts of the metric, Newton's constant, and the Ricci scalar, respectively. $m_D $ is the fundamental scale of the extra dimensional theory. \n \n    Since the gravitational interactions are not strong enough to produce a thermal gravitons at temperatures below the Planck scale ($m_{pl}  \\sim1.22 \\times 10^{19}$ GeV), the standard  inflationary cosmology  predicted the existence the cosmic gravitational waves background  which are non-thermal in nature. However\n if the universe contains extra dimensions that can generate the thermal gravitational waves and its shape and amplitude of the CGWB may change significantly.  This can happen \n when energies in the universe are higher than the fundamental scale $m_D$, the gravitational coupling strength increases significantly, as the gravitational field spreads out into the full spatial volume. Instead of freezing out at $\\sim { \\cal O}(m_{pl})$, as in 3+1 dimensions, gravitational interactions freeze-out at $\\sim {\\cal O}(m_D)$.  If the gravitational interactions become strong at an energy scale below the reheat temperature ($m_D  < T_{RH}$), gravitons get the opportunity to thermalize, creating a thermal CGWB. The qualitative result, the creation of a thermal CGWB if $m_D < T_{RH}$, is unchanged by the type of extra dimensions chosen \\cite{fry}. \n \n Thus, if extra dimensions do exist, and the fundamental scale of those dimensions is below the reheat temperature, a relic thermal CGWB ought to exist today. Compared to the relic thermal photon background (CMB), a thermal CGWB would have the same shape, statistics, and high degree of isotropy and homogeneity. The energy density ($\\rho_g$) and fractional energy density ($\\Omega_g$ )\tof\ta\tthermal\tCGWB\t are\n \\begin{eqnarray}\n \\rho_g = \\frac{\\pi^2}{15} \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{4\/3} T_{CMB}^4, \\\\\n \\Omega_g = \\frac{\\rho_g}{\\rho_c} \\simeq 3.1 \\times 10 ^{-4} (g_{\\star})^{4\/3},\n \\end{eqnarray}\n where $\\rho_c $ is the critical energy density today, $T_{CMB}$ is the present temperature of the CMB, and $g_{\\star}$ is the number of relativistic degrees of freedom at the scale of $m_D$.  $g_{\\star}$ is dependent on the particle content of the universe, i.e. whether (and at what scale) the universe is supersymmetric, has a KK tower, etc. Other quantities, such as the temperature (T), peak frequency ($\\nu$), number density ($n$), and entropy density ($s$) of the thermal CGWB can be derived from the CMB if $g_{\\star}$ is known, as \n \\begin{eqnarray}\n n_g = n_{CMB} \\left( { \\frac{3.91}{g_{\\star}}} \\right ),\\,\\, \\, \\, \\, \\,  s_g= s_{CMB}  \\left( { \\frac{3.91}{g_{\\star}}} \\right )\\\\\n  T_g = T_{CMB} \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{1\/3},\\,\\, \\, \\, \\, \\,  \\nu_g= \\nu_{CMB}  \\left( { \\frac{3.91}{g_{\\star}}} \\right )^{1\/3}.\n    \\end{eqnarray}\n  These quantities are not dependent on the number of extra dimensions, as the large discrepancy in size between the three large spatial dimensions and the $D$ extra dimensions suppresses those corrections by at least a factor of  $ \\sim 10^{ -29}$. If $m_D$ is just barely above the scale of the standard model, then $ g_{\\star }$= 106.75. The thermal CGWB then has a temperature of 0.905 Kelvin, a peak frequency of 19 GHz \\cite{fry}. \n  \n  \\section*{Acknowledgement}\nAuthors would like to thank S.Hild  for providing the ET sensitivity  data and  also Adv.LIGO and LISA web.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{introduction}\n\nExcitations in quantum spin liquids can be viewed as strongly\ninteracting bosonic quasiparticles. This circumstance enables\nexperimental studies of the physics of Bose liquids in prototypical\nquantum magnetic\nmaterials\\cite{GiamarchiPRB99,Nikuni00,GiamarchiNaturePhys}. Such\nexperiments are often possible under conditions that can not be\nrealized in more conventional models, such as\n$^4$He\\cite{London,Reppy} and ultracold trapped\nions\\cite{MHanderson,Wynar}. One recent topic of interest is teh\nbehavior of bosonic quasiparticles in the presence of disorder.\nExotic new phases such as the Random Singlet state\\cite{Ma79}, Bose\nand Mott glasses\\cite{Fisher89} have been predicted for systems with\nquenched disorder. In real prototype materials one usually tries to\ncreate such disorder by chemical doping\\cite{ManakaPRL,OosawaPRB}.\nIn the present work we demonstrate an alternative approach: a random\nmagnetic field created by disordered (paramagnetic) ions. We show\nthat such a random field acting on a simple dimer-based quantum spin\nliquid dramatically alters the excitation spectrum.\n\nLet us consider the effect of different types of magnetic fields on\nan isolated $S=1\/2$ dimer, as shown in Fig.~\\ref{fig5}. In a uniform\nfield the excited triplet is split into three levels. Eventually, at\nhigh field, $\\lvert S=1, S^z=0\\rangle $ will cross the singlet\nground state. In the presence of inter-dimer interactions, BEC of\nmagnon will occur. If local fields applied to each dimer spin are\nantiparallel to each other (referred to as ``staggered field''\nhereafter) the triplet is split into a singlet and a doublet.  The\nsinglet ground state becomes mixed with $\\lvert S=1, S^z=0\\rangle $\nand the total spin is no longer a good quantum number. Even in an\ninfinitesimal staggered field the ground state becomes ploarized.\nNow, if the field direction is spatially randomized, each dimer will\nexperience both a staggered and uniform component. The corresponding\nenergy levels can be calculated numerically. The resulting density\nof state (DOS) for excitations in a set of $N$ dimers is plotted in\nthe right panel in Fig.~\\ref{fig5}. The DOS  lower and higher\nboundaries of the DOS spread coincide with the levels of $\\lvert\nS_{z}=1\\rangle $ and $\\lvert S_{z}=-1\\rangle $ in the uniform field.\n\nThe quantum ferrimagnet \\CuFeGeO\\ \\cite{Masuda03} is a rare\npotential realization  of this random field effect. The compound\nincludes $S$ = 1\/2 Cu$^{2+}$ dimers coupled to classical Fe$^{3+}$\nchains \\cite{Masuda05}. At low temperature the cooperative ordered\nstate with classical spin and quantum spin is stabilized by a weak\ninter-subsystem coupling. In the adiabatic approximation, the\nquantum spins are effectively under the internal field from the much\nslower fluctuating classical spins. In this compound, the staggered\nnature of the exchange field is due to the magnetic structure. The\nstaggered magnetization curves of dimers in \\CuFeGeO\n~\\cite{Masuda04a} were experimentally obtained by measuring the\ntemperature dependence of sublattice moments in neutron diffraction.\nAt high temperature, in the paramagnetic phase, the classical spins\nare thermally disordered and the effective field on the quantum\nspins is randomly oriented. Then the system can be considered as the\nensemble of $N$ dimers in a random quasi-static field. As shown in\nFig.~\\ref{fig5} the effect of this random field is to broaden the\ndimer excitations at $T > T_{\\rm N}$.\n\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.5cm]{fig1.eps}\n\\end{center}\n\\caption{ Schematic diagrams of triplet excitations in $S=1\/2$\ndimers in different types of locally applied magnetic field. }\n\\label{fig5}\n\\end{figure}\n\nIn the previous inelastic neutron scattering study it was shown that\nthe energy scales of excitations in the Fe chains and Cu dimers are\nwell separated~\\cite{Masuda05,Masuda07a}. The lower energy\nexcitations up to 10 meV are Fe-based spin waves. Preliminary powder\nexperiments~\\cite{Masuda05} and comparative studies in an\nisostructural compound \\CuScGeO ~\\cite{Masuda06} associated the\ndispersionless excitations at 24 meV  with Cu-dimers. However the\neffect of a staggered and\/or fluctuating field could not been\nidentified in powders samples. In the present paper we study the\ndimer excitations by single crystal inelastic neutron scattering. By\nadopting a high resolution setup, we identify the split peaks due to\nthe staggered exchange field. Furthermore, we observe a drastic\nbroadening of the peak profile at $T > T_{\\rm N}$ that can be\nascribed to randomly oriented field from thermally fluctuated Fe\nmoments.\n\nHigh quality single crystals were grown by floating zone method. The\ncrystal (monoclinic $P2_1\/m$) were found to be twinned, so that both\nmicroscopic domains share $a^*$ - $b^*$ plane. To avoid\ncomplications due to twinning, we restrict the measurements to the\n$a^*$ - $b^*$ plane. In the setups Ia and Ib PG (002) were used for\nboth monochromator and analyzer. The Soller collimations were 48' -\n60' - 60' - 120' and open - 80' - 80' - open for Ia and Ib,\nrespectively. In setup II, to achieve high energy resolution, PG\n(004) for monochromator and PG (002) for analyzer with 30' - 20' -\n40' - 120' were used. The setups Ia and II were performed on HB1\nspectrometer in HFIR, ORNL. The setup Ib was performed on TAS1\nspectrometer in JRR-3M, JAEA. In all setups final energy of the\nneutron was fixed at $E_f$ = 14.7meV and PG filter was installed\nafter the sample to eliminate higher order contamination. A closed\ncycle He refrigerator was used to achieve low temperatures.\n\nIn a series of energy scans in a wide range of $(h~k~0)$ space shown\nin Fig.\\ref{fig1}(a) two dispersionless peaks are readily\nidentified: a pronounced one at $\\hbar \\omega \\sim 24$ meV and a\nweaker feature at $\\hbar \\omega \\sim 31$ meV. The experiments were\nperformed in setups Ia and Ib. The former is consistent with the\nCu-centered magnetic excitation in previous\nstudies~\\cite{Masuda05,Masuda06}. Constant energy scan at $\\hbar\n\\omega = 24$ meV and its temperature dependence are shown in\nFig.\\ref{fig1}(b). The observed sinusoidal intensity modulation is\ncharacteristics of dimer excitations and is observed in a wide\ntemperature range. In fig.~\\ref{fig1}(c) the temperature dependence\nof the peak intensity is shown. The intensity at ${\\bm q} =\n(0~2.5~0)$ was measured at each temperature and then subtracted as\nbackground. The decrease of the intensity at high temperature is\ncommon behavior for magnetic excitations in local spin clusters. The\nsmaller peak at $\\hbar \\omega \\sim 31$ meV was identified as a\nFe-centered excitation, as will be discussed below.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig2.eps}\n\\end{center}\n\\caption{ Inelastic neutron scattering using experimental setup Ia\nand Ib. (a) Typical energy scans at $(h~k~0)$. Dispersionless\nexcitations are observed at $\\hbar \\omega = 24$ and 31 meV. Two\npeaks are separately fit by Gaussians (dotted curves). (b) $h$ scans\nat $\\hbar \\omega =$ 24 meV at various temperatures. Sinusoidal\nintensity modulations are fitted to the dimer structure factor\ncalculated for zero field, plus a constant background (solid\ncurves). (c) Temperature dependence of the peak intensity at ${\\bm\nq} = (h~2.5~0)$ and $\\hbar \\omega =$ 24 meV. } \\label{fig1}\n\\end{figure}\n\nTo obtain a more detailed profile, we performed an energy scan using\nsetup II at $T$ = 2.0 K. As shown in Fig.~\\ref{fig3} it is revealed\nthat the primary peak at $\\hbar \\omega = 24$ meV actually has a\nshoulder structure. The main peak is located at 23.5 meV, and a\nsmaller bump is centered around 25.0 meV. This splitting is\nattributed to the staggered exchange field from the adjacent Fe\nmoments. The main peak corresponds to the excitation doublet and the\nsmall one to the singlet.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=7.5cm]{fig3.eps}\n\\end{center}\n\\caption{ Energy scan collected using the high resolution setup II\nat $T$ = 2.0 K. The shoulder structure is reproduced by the doublet\n(shaded with white background) and singlet (shaded with gray\nbackground) dimer excitations, split by a staggered exchange field.\n} \\label{fig3}\n\\end{figure}\n\nEnergy scans collected at several temperatures are shown\nFig.~\\ref{fig4}(a). The small peak at $\\hbar \\omega \\sim 31$ meV in\nFig.~\\ref{fig1} is temperature independent and has been  subtracted\nfrom the data. Well-defined peaks are observed at all temperatures.\nWhile at low temperature the peak profile is sharp and the width is\nwithin resolution limit, at $T \\gtrsim T_N$ the peak becomes\ndrastically broadened. This qualitative behavior is consistent with\nthe effect a random exchange field should have on the dimer\nexcitation triplet. The data were analyzed using Gaussian fits. The\nestimated peak positions, widths, and the integrated intensities are\nplotted as functions of temperature in Figs.~\\ref{fig4}(b)-(d). With\nincreasing temperature the peak energy decreases at $T \\sim T_{\\rm\nN}$,  and stays constant beyond. The peak width drastically\nincreases at $T \\sim T_{\\rm N}$, but also remains constant at higher\ntemperature. The integrated intensity decreases by 10 $\\sim $ 20\\%.\nIt is noted that in the previous powder experiment the peak cannot\nbe distinguished at $T \\ge 41$ K \\cite{Masuda05}. This is because\nthe powder integration in wide ${\\bm q}$ space collects phonon\nexcitations and accidental suprious peaks, masking magnetic\nexcitations at higher temperatures.\n\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=8.7cm]{fig4.eps}\n\\end{center}\n\\caption{ (a) Temperature dependence of the peak profile at ${\\bm q}\n= (2~3~0)$. Experimental resolution is indicated by the gray area.\nSmall peaks due to Fe-centered excitation at 31 meV are separately\nfitted and subtracted. Profiles at $T \\ge 54$ K are reproduced by\ndimers in a randomly oriented field model (solid curves). The\ntemperature dependence of peak positions (b), widths (c) and\nintegrated intensities (d), as estimated from Gaussin fits. }\n\\label{fig4}\n\\end{figure}\n\n\n\nFor $T < T_{\\rm N}$ we will consider the following effective\nHamiltonian:\n\\begin{equation}\nH = J_{\\rm Cu}{\\bm S_1}\\cdot {\\bm S_2} +g\\mu _{\\rm B}{\\bm S_1}\\cdot {\\bm h_1} + g\\mu _{\\rm B}{\\bm S_2}\\cdot {\\bm h_2},\n\\label{dimerHamiltonian}\n\\end{equation}\nwhere ${\\bm h_1}=(0~0~h)$ and ${\\bm h_2}=(0~0~-h)$. Here  the\n$z$-axis is chosen along the ordered Cu moment. The ground state\nenergy $E_{\\rm G}$ decreases with the field, $E_{\\rm G} =-J_{\\rm\nCu}\/4-\\sqrt {(2g\\mu _{\\rm B} h)^2+J_{\\rm Cu}^2}\/2$, and the\nexcitation triplet splits into a singlet and doublet. The\ncorresponding energy levels are giben by\n\\begin{eqnarray}\n\\Delta_{\\rm s}&=&\\sqrt {(2g\\mu _{\\rm B} h)^2+J_{\\rm Cu}^2}\n\\label{singlet} \\\\\n\\Delta_{\\rm d}&=&J_{\\rm Cu}\/2+\\sqrt{(2g\\mu _{\\rm B} h)^2+\nJ_{\\rm Cu}^2}\/2, \\label{doublet}\n\\end{eqnarray}\nand plotted vs. $h$ in Fig.~\\ref{fig5}. For $g\\mu _{\\rm B}  h\\ll\n\\Delta$,  the neutron cross section is approximately given by:\n\\begin{eqnarray}\n\\frac{d^2\\sigma}{d\\Omega dE}\\sim N(\\gamma r_0)^2\\frac{k'}{k}\n\\sin ^2(\\bm {q.d})\\left( f(q)\\right) ^2P(T) \\nonumber \\\\\n\\times \\{A(h)(1+\\cos ^2 \\theta)\\delta (\\hbar \\omega-\\Delta_{\\rm d})+\nB(h)\\sin ^2 \\theta\n\\delta(\\hbar \\omega -\\Delta _{\\rm s})\\}. \\label{crosssection}\n\\end{eqnarray}\nThe doublet and singlet terms correspond to transverse and\nlongitudinal spin fluctuation, respectively. $A(h)$ and $B(h)$ are\n$h$ dependent parameters with $A(h) \\le 1, B(h) \\le 1,$ and $A(0) =\nB(0) = 1$. Since the staggered field stabilizes the polarized spin\nconfiguration and suppresses longitudinal fluctuation, $B(h)$\ndecrease with $h$. Meanwhile $A(h)$ is almost constant in the low\nfield. $P(T)$ is a temperature factor, $P(T)=1\/\\{ 1+2\\exp (-\\beta\n\\Delta _{\\rm d}) + \\exp (-\\beta \\Delta _{\\rm s}\\}$. $\\bm q$ is the\nscattering vector, $\\bm d$ is the spin separation in each dimer, and\n$\\theta$ is the angle between $\\bm q$ and the moment of Cu. We used\n${\\bm m}_{\\rm Cu}$ = (-0.227, 0.035, -0.301) $\\mu _{\\rm\nB}$\\cite{Masuda04a} to calculate $\\theta$. Two types of domains,\nnamely antiferromagntic and crystallographic ones due to twinning,\nare considered.\n\nThe peak profile in Fig.~\\ref{fig3} is reasonably well reproduced by\nthe cross section convoluted by experimental resolution function\nwith $\\Delta _{\\rm s}$ = 25.0 meV and $\\Delta _{\\rm d}$ = 23.5 meV.\nFrom eqs.(\\ref{singlet}) and (\\ref{doublet}), $J_{\\rm Cu}=22.0$ meV\nand $h$ = 51 T are obtained. Let us check the consistency of $h$\nwith the previous study~\\cite{Masuda04a}. From the staggered\nmagnetization curve by neutron diffraction $J_{\\rm Cu-Fe}\/J_{\\rm\nCu}=0.105$ was obtained. Here $J_{\\rm Cu-Fe}$ is the interaction\nbetween Cu and Fe spins. Using the molecular field relation\n$h=m_{\\rm Fe}J_{\\rm Cu-Fe}\/(g\\mu _{\\rm B})^2$ and previously\nobtained parameters, $h \\sim 40$ T is estimated. Thus statically\nestimated value is consistent with that obtained in the present\ndynamic measurement.\n\nThe energy splitting between the singlet and doublet states is about\n1.5 meV. This value is small compared energy resolution in the\ntypical experimental setup. In setup Ia and Ib at $T < T_{\\rm N}$,\ntherefore, staggered field effect is smeared and two terms in\neq.(\\ref{crosssection}) are integrated. Then the cross section is\napproximately equivalent to that at $h$ = 0. Indeed, the constant\nenergy scan at $T$ = 3.3 K in Fig.~\\ref{fig1}(b) is reasonably\nfitted by dimers cross section in zero field shown by the thick\ncurve.\n\n\nAt $T>T_{\\rm N}$ effective field on the Cu dimers is randomly\noriented. We consider an ensemble of $N$ dimers in random field. The\nrandomly oriented field ${\\bm h}$ is assumed to have a constant\nmagnitude in eq.(\\ref{dimerHamiltonian}). The resulting DOS of the\nexcited states is then calculated numerically. The neutron cross\nsection is assumed to be approximately proportional to the DOS,\n\\begin{equation}\n\\frac{d^2\\sigma}{d\\Omega dE}=(\\gamma r_0)^2\\frac{k'}{k}\\sin ^2(\\bm {q.d})f(q)^2P_{\\rm rand}(T) D(\\hbar \\omega)  \\label{crosssection2}\n\\end{equation}\nwith $\\int D(\\epsilon )d\\epsilon = 3N$ and $P_{\\rm rand}(T) =\nN\/(N+\\int D(\\epsilon )e^{-\\frac{\\epsilon }{k_{\\rm B}T}}d\\epsilon )$.\nThe data collected at $T \\ge 54$ K are well reproduced by this cross\nsection convoluted by experimental resolution function, as indicated\nby solid curves in Fig.~\\ref{fig4}(a). The obtained fit parameters\nare $J_{\\rm Cu}$ = 22.3(4) meV and $h$ = 41.(8) T. The values are\nreasonably consistent with those obtained at $T \\le T_{\\rm N}$. The\n${\\bm q}$ dependence of the cross section is the same as for zero\nfield, and is given by dimer structure factor $\\sin ^2 (\\bm {q.d})$.\nIndeed, the ${\\bm q}$ scans at 80 K and 300 K in Fig.~\\ref{fig1}(b)\nare reproduced by this model. The temperature dependence in\nFig.~\\ref{fig1}(c) is well accounted for by the temperature factor\n$P_{\\rm rand}(T)$.\n\nWe shall now discuss the small decrease of the intensity at $T <\nT_{\\rm N}$ in Fig.~\\ref{fig4}(d). At $T > T_{\\rm N}$ dimer spins are\nfluctuated equally in all directions and the dynamical spin\ncorrelation is fully detected by neutron. In the ordered state the\npolarized magnetic ordering suppresses the longitudinal fluctuation\nof Cu spins. To estimate the reduction of the longitudinal\nexcitation we calculate $B(h = 51 {\\rm T})=0.77$. The reduction of\n$B(h)$ is about 20\\% that is consistent with the experiment. This\nmeans that 51 T is rather modest compared with the intradimer\ninteraction $J_{\\rm Cu}$ = 22 meV. If the effective field was large\nand the moment were fully polarized, the suppression would be more\ndrastic. Such a situation is in fact realized in Haldane spin chains\ncoupled to rare earth moment in Pr$_2$BaNiO$_5$ with fully saturated\nNi$^{2+}$ moment at $T < T_{\\rm N}$~\\cite{Zheludev96c}. The\nHaldane-gap mode lost half of its intensity at $T < T_{\\rm N}$ and\nit was ascribed to the total suppression of longitudinal mode.\n\nFinally we will mention the temperature independent small peak at\n$\\hbar \\omega \\sim$ 31 meV in Fig.~\\ref{fig1}(a). If the Fe $S=5\/2$\nchains were perfectly isolated from the Cu subsystem, the Fe\nexcitation spectrum  would be dominated by one-magnon excitation at\n$\\hbar \\omega \\le 5J_{\\rm Fe}$. However, a recent theory predicts\nthat the introduction of Cu dimer enhances the multi-magnon\nexcitation of Fe spins at $\\hbar \\omega$ = $10J_{\\rm Fe}$, $15J_{\\rm\nFe}$, $20J_{\\rm Fe}$ and $25J_{\\rm Fe}$. According to the Bond\noperator method~\\cite{Matsumoto04,Sachdev90} the excitation at\n$\\hbar \\omega$ = $20J_{\\rm Fe}$ is the particularly\nenhanced~\\cite{Matsumoto09}. Since $J_{\\rm Fe}=1.6$\nmeV~\\cite{Masuda07a}, the observed small peak at $\\hbar \\omega$ = 31\nmeV could be ascribed to the Fe centered longitudinal excitation.\nFurther details will be published somewhere else.\n\nTo conclude, we have experimentally investigated the dynamics of $S\n= 1\/2$ dimers in staggered and random fields in \\CuFeGeO . The\nstaggered field is realized at $T < T_{\\rm N}$ and produces a\nsplitting of the excitation triplet.  At $T > T_{\\rm N}$ a random\nexchange field produces a drastic broadening of these modes. In teh\nfuture, polarized neutron experiments may be useful to separate the\nlongitudinal and transverse excitations. Recently\nCu$_2$CdB$_2$O$_6$~\\cite{Hase05} and\nCu$_3$Mo$_2$O$_9$~\\cite{Hamasaki08} identified as new realizationsof\nthe coupled dimers and chains models. Particularly in the latter\ncompound, the dimer energy is close to that of the  chains, and more\ncomplex physics is expected.\n\nProf. M. Matsumoto is greatly appreciated for fruitful discussion.\nThis work was partly supported by Yamada Science Foundation, Asahi\nglass foundation, and Grant-in-Aid for Scientific Research (No.s\n19740215 and 19052004) of Ministry of Education, Culture, Sports,\nScience and Technology of Japan.\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nLet $G$ be a finite group of automorphisms of an abelian variety $A\/\\bbc$.  It is a classical result \\cite[II. \\S1]{La} that $A$ itself cannot contain a rational curve.\nFor $|G| > 1$, there may or may not be rational curves on $A\/G$. For general abelian varieties, $\\Aut(A)= \\pm1$, and Pirola proved \\cite{P} that for $A$ sufficiently general and of dimension at least three, $A\/\\pm1$ has no rational curves. At the other extreme, regarding $A=E^n$ as the set of $n+1$-tuples of points on the elliptic curve $E$ which sum to $0$, $A\/S_{n+1}$ can be interpreted as the set of effective divisors linearly equivalent to $(n+1)[0]$ and, as such, is just $\\bbp^n$. More generally, Looijenga has shown  \\cite{L} that the quotient of $E^n$ by the Weyl group of a root system of rank $n$ is a weighted projective space.\n\nRational curves on $A\/G$ over a field $K$ are potentially a source of rational points over $G$-extensions of $K$.  For instance, the method \\cite{Larsen} for finding pairs $a,b\\in \\bbq^\\times$ such that the quadratic twists\n$E_a$, $E_b$, and $E_{ab}$ all have positive rank amounts to finding a rational curve on $E^3\/(\\bbz\/2\\bbz)^2$.   Likewise, the theorem of Looijenga cited above gives for each elliptic curve $E$ over a number field $K$ and for each Weyl group $W$, a source of $W$-extensions $L_i$ of $K$ such that the representation of $W$ on each $E(L_i)\\otimes\\bbq$ contains the reflection representation.  On the other hand, the result of Pirola cited above dims the hope of using geometric methods to show that every abelian variety over a number field $K$ gains rank over infinitely many quadratic extensions of $K$.  Thus, it is desirable from the viewpoint of arithmetic to understand when $A\/G$ can be expected to have a rational curve over a given field $K$, and to begin with, one would like to know when $A\/G$ has a rational curve over $\\bbc$.\n\nAny automorphism $g$ of an abelian variety $A$ defines an invertible linear transformation (also denoted $g$) on $\\Lie(A)$.\nIf $g$ is of finite order, there exists a unique sequence of rationals $0\\le x_1\\le x_2\\le \\cdots\\le x_n < 1$ such that\nthe eigenvalues of $g$ are $e(x_1),\\ldots,e(x_n)$, where $e(x) := e^{2\\pi i x}$.  We say $g$ is of \\emph{type} $(x_1,\\ldots,x_n)$. Following\nKoll\\'ar and Larsen \\cite{KL}, we write $\\age(g) = x_1+\\cdots+x_n$.\nFor instance, $\\age(g) = 1\/2$ for every reflection $g$.  The main result of \\cite{KL} asserts that\n$A\/G$ is uniruled if and only if $0<\\age(g)<1$ for some $g\\in G$.  In this paper, we prove that to find a single rational curve in $A\/G$, it sufices that $\\age(g) \\le 1$.\n\nSince we need only consider the case $\\age(g) = 1$, we first classify all types of weight $1$.  This requires a combinatorial analysis, which we carried out using a computer algebra system to minimize the risk of an oversight.\nThere are thirty-five cases (see Table 2 below), and our strategy for finding rational curves depends on case analysis.  Abelian surfaces play a special role, since here we can use known results on K3 surfaces.\nThe other key  idea is to find a non-singular projective curve $X$ on which $G$ acts with quotient\n$\\bbp^1$ and a $G$-equivariant map from $X$ to $A$, or, equivalently, a $G$-homomorphism from the Jacobian variety of $X$ to $A$.\n\nWe would like to thank Yuri Tschinkel and Alessio Corti for helpful comments on earlier versions of this paper.\n\n\\section{Classifying types}\n\nIf $A = V\/\\Lambda$, then the Hodge decomposition $\\Lambda\\otimes\\bbc \\cong V\\oplus \\bar V$ respects the action of $\\Aut(A)$.\nTherefore, if $g$ is of finite order with eigenvalues $e(x_1),\\ldots,e(x_n)$, then the multiset\n\\begin{equation}\n\\tag{$\\ast$}\\{e(x_1),\\ldots,e(x_n),e(-x_1),\\ldots,e(-x_n)\\}\n\\end{equation}\nis $\\Aut(\\bbc)$-stable.\nBy a \\emph{type}, we mean a multiset $\\{x_1,\\ldots,x_n\\}$ with $x_i\\in [0,1)$ such that the multiset ($\\ast$) is $\\Aut(\\bbc)$-stable.\nEquivalently, a type can be identified with a finitely supported function $f\\colon \\bbq\/\\bbz\\to \\bbn$ such that $f(x)+f(-x)$ depends only on the order of $x$ in $\\bbq\/\\bbz$.\nBy the \\emph{weight} of $\\{x_1,\\ldots,x_n\\}$, we mean the sum $x_1+\\cdots+x_n$, so that $\\age(g)$ is the weight of the type of $g$.\n\nA type is \\emph{reduced} if $0$ does not appear, and the reduced type of a given type is obtained by discarding all copies of $0$.\nThe \\emph{sum} of types is the union in the sense of multisets; at the level of associated functions on $\\bbq\/\\bbz$ it is the usual sum.\nA type which is not the sum of non-zero types is \\emph{primitive}.\nAll the elements of a primitive type appear with multiplicity one, and they all have the same denominator.\nEvery type can be realized (not necessarily uniquely) as a sum of primitive types; if the weight of the type is $1$, each of the primitive types has weight $\\le 1$,\nso our first task is to classify primitive types with weight $\\le 1$.\n\nA primitive type $X$ of denominator $n\\ge 2$ consists of fractions $a_i\/n$ where $0<a_i<n$, and $(a_i,n)=1$.\nMoveover, if $n\\ge 3$, for each positive integer $a<n$ prime to $n$, exactly one of $a\/n$ and $1-a\/n$ belongs to $X$.  If $a<b<n\/2$ and $1-a\/n\\in X$,\nthen the weight of $X$ exceeds $1$ since either $b\/n$ or $1-b\/n$ belongs to $X$.\nThus, if $0<a<n\/2$ and $1-a\/n\\in X$, then $a$ must be the largest integer in $(0,n\/2)$ prime to $n$.\n\n\\begin{lem}\\label{primineq} If $\\phi(n) > 24$, then  $$\\sum_{x\\in S_n} \\min(x,n-x) > 2n,$$ where $S_n$ is the set of  positive integers $< n$ and prime to $n$. Moreover, the largest integer $n$ such that $\\phi(n)\\le 24$ is $90$.\n\\end{lem}\n\n\\begin{proof} Note that\n$$\\min(x,n-x) > \\frac{x(n-x)}{n} = \\frac{n^2-x^2-(n-x)^2}{2n}.$$\nIn order to prove the first statement, we want to prove if $\\phi(n)>24$, then\n$$\\sum_{x\\in S_n} \\Big(n^2-x^2-(n-x)^2\\Big) > 4n^2,$$ or equivalently,\n$$\\phi(n)n^2  - 2\\sum_{x\\in S_n} x^2-4n^2>0.$$\nBy  M\\\"{o}bius inversion, one can prove that\n$$\\sum_{x\\in S_n} x^2 = \\frac{\\phi(n) n^2}{3} +(-1)^{d_n}\\frac{\\phi(f(n)) n}6,$$\nwhere $f(n)$ denotes the larges squarefree divisor of $n$ and $d_n $ is the number of distinct prime divisors of $n$.  Thus, if $\\phi(n) > 24$, then $\\phi(n)>24\\geq 12 \\frac{n}{n-1}$, so $(n-1)\\phi(n)-12n >0$ and since $\\phi(f(n))\\leq \\phi(n)$,\n\\begin{align*}\n\\phi(n)n^2 - 2\\sum_{x\\in S_n} x^2-4n^2 &\\geq \\Big(\\frac{\\phi(n)}{3}-4\\Big)n^2-\\frac{\\phi(n)n}{3} \\\\\n&=\\frac{n((n-1)\\phi(n)-12n)}{3}>0,\n\\end{align*}\nwhich is the desired inequality.\n\nFor the second statement, if $\\phi(n)\\le 24$ and $p$ is a prime factor of $n$, then $\\phi(p)=p-1\\leq\\phi(n)\\leq 24$. Hence $p\\le 23$.  Writing\n$$n=2^{n_2}3^{n_3}5^{n_5}7^{n_7}11^{n_{11}}13^{n_{13}}17^{n_{17}}19^{n_{19}}23^{n_{23}},$$\nwe have \n$$0\\leq n_2\\leq 5,0\\leq n_3\\leq 3,0\\leq n_5\\leq 2,$$\nand $0\\leq n_i\\leq 1$ for $7\\le i\\le 23$.\nCase analysis now shows $n\\leq 90$.\n\n\\end{proof}\n\n\\begin{prop}\\label{primlist} There are $28$  primitive types with weight $ \\leq 1$:\n\\end{prop}\n\\begin{center}\n\\begin{tabular}{|l|c|c|c|}\n\\hline\n\\#& n & \\text{primitive types} &  \\text{weight}\\\\\\hline\n1&  2 & 1\/2 & 1\/2\\\\\\hline\n2&  3 & 1\/3 & 1\/3\\\\\n3&    & 2\/3 & 2\/3\\\\\\hline\n4&  4 & 1\/4 & 1\/4 \\\\\n5&    & 3\/4 & 3\/4\\\\\\hline\n6&  5 & 1\/5, 2\/5 & 3\/5 \\\\\n7&    & 1\/5, 3\/5 & 4\/5 \\\\\\hline\n8&  6 & 1\/6 &1\/6\\\\\n9&    & 5\/6 & 5\/6\\\\\\hline\n10&  7 & 1\/7, 2\/7, 3\/7 & 6\/7\\\\\n11&    & 1\/7, 2\/7, 4\/7 & 1\\\\\\hline\n12&  8 & 1\/8, 3\/8 & 1\/2 \\\\\n13&    & 1\/8, 5\/8 & 3\/4 \\\\\\hline\n14&  9 & 1\/9, 2\/9, 4\/9 & 7\/9 \\\\\n15&    & 1\/9, 2\/9, 5\/9 & 8\/9\\\\\\hline\n16&  10 & 1\/10, 3\/10 & 2\/5 \\\\\n17&    & 1\/10, 7\/10 & 4\/5 \\\\\\hline\n18&  12 & 1\/12, 5\/12 & 1\/2\\\\\n19&    & 1\/12, 7\/12 & 2\/3\\\\\\hline\n20&  14 & 1\/14, 3\/14, 5\/14 & 9\/14\\\\\n21&    & 1\/14, 3\/14, 9\/14 & 13\/14\\\\\\hline\n22&  15 & 1\/15, 2\/15, 4\/15, 7\/15 & 14\/15\\\\\n23&    &1\/15,2\/15, 4\/15, 8\/15 & 1\\\\\\hline\n24&  16 & 1\/16, 3\/16, 5\/16, 7\/16 & 1\\\\\\hline\n25&  18 & 1\/18, 5\/18, 7\/18 & 13\/18\\\\\n26&    & 1\/18, 5\/18, 11\/18 & 17\/18 \\\\\\hline\n27&  20 & 1\/20, 3\/20, 7\/20, 9\/20 & 1\\\\\\hline\n28&  24 & 1\/24, 5\/24, 7\/24, 11\/24 & 1\\\\\n  \\hline\n\\end{tabular}\n\\vskip 4pt\nTable 1\n\\end{center}\n\\begin{proof} For $n\\ge 3$, the weight of a primitive type of denominator $n$ is at least\n$$\\sum_{\\{x\\in S_n\\mid x<n\/2\\}}\\frac xn \\ge \\frac 1{2n}\\sum_{x\\in S_n} \\min(x,n-x).$$\nBy Lemma~\\ref{primineq}, it suffices to carry out an exhaustive search  up to $n = 90$.\n\\end{proof}\n\\begin{lem} There are $35$ types with $\\age ~~1$ of automorphisms given in Table 2 below.\n\\end{lem}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n\\#&  n & \\text{types} &  \\text{notes}\\\\\\hline\n1&  2 & 1\/2, 1\/2 & Prop.~\\ref{kummer} \\\\\\hline\n2&  3 & 1\/3, 1\/3, 1\/3 &  Th.~\\ref{small-factor}\\\\\n3&   & 1\/3, 2\/3 &  Prop.~\\ref{cube}\\\\\\hline\n4&  4 & 1\/4, 1\/4, 1\/4, 1\/4 &    Th.~\\ref{small-factor}\\\\\n5&   & 1\/4, 1\/4, 2\/4 & $g^2\\to\\#1$ \\\\\n6&   & 1\/4, 3\/4 & $g^2\\to\\#1$  \\\\\\hline\n7&  6 & 1\/6, 1\/6, 1\/6, 1\/6, 1\/6, 1\/6 &    Th.~\\ref{small-factor}\\\\\n8&  & 1\/6, 1\/6, 1\/6, 1\/6, 2\/6  & Th.~\\ref{small-factor} \\\\\n9&   & 1\/6, 1\/6, 1\/6, 3\/6  &  $g^2\\to\\#2$ \\\\\n10&  & 1\/6, 1\/6,  4\/6  &  $g^3\\to\\#1$ \\\\\n11&    & 1\/6, 5\/6  &  $g^3\\to\\#1$ \\\\\n12&   & 1\/6, 2\/6, 3\/6  &  $g^3\\to\\#1$  \\\\\n13&    & 1\/6, 1\/6,  2\/6, 2\/6  &  $g^3\\to\\#1$  \\\\\\hline\n14&   7 & 1\/7, 2\/7, 4\/7 & Cor.~\\ref{seven}\\\\\\hline\n15&   8 & 1\/8, 2\/8, 5\/8  &   $g^4\\to\\#1$\\\\\n16&   &  1\/8, 3\/8, 4\/8 & $g^4\\to\\#1$ \\\\\n17&   & 1\/8,  1\/8, 3\/8, 3\/8 & Th.~\\ref{small-factor} \\\\\n18&    & 1\/8, 2\/8, 2\/8, 3\/8 & $g^4\\to\\#1$  \\\\\\hline\n19&   10 &  1\/10, 2\/10, 3\/10, 4\/10 & $g^5\\to\\#1$ \\\\\\hline\n20&   12 & 4\/12, 2\/12, 1\/12, 5\/12& $g^6\\to\\#1$\\\\\n21&    & 4\/12, 3\/12, 3\/12, 2\/12 &$g^6\\to\\#1$\\\\\n22&    & 6\/12, 1\/12, 5\/12 &$g^6\\to\\#1$\\\\\n23&    & 3\/12, 3\/12, 1\/12, 5\/12 &$g^4\\to \\#3$\\\\\n24&    & 2\/12, 2\/12, 2\/12, 1\/12, 5\/12 & $g^6\\to\\#1$\\\\\n25&    & 1\/12, 1\/12, 5\/12, 5\/12 & Th.~\\ref{small-factor} \\\\\n26&    & 4\/12, 1\/12, 7\/12 & $g^6\\to\\#1$ \\\\\n27&    & 2\/12, 2\/12, 1\/12, 7\/12 & $g^6\\to\\#1$ \\\\\n28&    & 3\/12, 3\/12, 2\/12, 2\/12, 2\/12 & $g^6\\to\\#1$\\\\\\hline\n29&   15 & 1\/15, 2\/15, 4\/15, 8\/15 & Cor.~\\ref{fifteen}\\\\\\hline\n30&   16 & 1\/16, 3\/16, 5\/16, 7\/16 & Cor.~\\ref{mult-one}\\\\ \\hline\n31&   20 & 1\/20, 3\/20, 7\/20, 9\/20 & Cor.~\\ref{mult-one}\\\\\\hline\n32&   24 & 1\/24, 5\/24, 7\/24, 11\/24 & Cor.~\\ref{mult-one}\\\\\n33&    & 8\/24, 4\/24, 3\/24, 9\/24 & $g^{12}\\to\\#1$\\\\\n34&    & 3\/24, 9\/24, 2\/24, 10\/24 & $g^{12}\\to\\#1$\\\\\n35&    & 4\/24, 4\/24, 4\/24, 3\/24, 9\/24 & $g^{12}\\to\\#1$ \\\\\n  \\hline\n\\end{tabular}\n\\vskip 4pt\nTable 2\n\\end{center}\n\n\\begin{proof} Let $[a_i]$ be a formal variable representing the $i$th primitive type in Table $1$,  and let  $w_i$ denote the weight of the $i$th type.\nA monomial $\\prod a_i^{m_i}$ stands for a sum of primitive types in which the $i$th type appears $m_i$ times.\nThe g.c.d. of the denominators of the $w_i$  is $5040$. Let $y=x^{1\/5040}$, so $(1-[a_i]x^{w_i})^{-1}$ is a power series in $y$ for every $i$.\nBy \\textsc{Maple 13},   the coefficient of $y^{5040}$ in the product\n$$\\prod_{i=1}^{28} (1-[a_i]x^{w_i})^{-1}=\\prod_{i=1}^{28} (1-[a_i]y^{w_i\\cdot 5040})^{-1},$$\nis\n${{\\it a1}}^{2}+{{\\it a2}}^{3}+{\\it a2}\\,{\\it a3}+{\\it a4}\\,{\\it a5}+{\n\\it a8}\\,{\\it a9}+{\\it a4}\\,{\\it a13}+{\\it a6}\\,{\\it a16}+{{\\it a4}}^{\n2}{\\it a1}+{\\it a28}+{\\it a27}+{\\it a23}+{\\it a24}+{\\it a11}+{\\it a19}\n\\,{\\it a2}+{\\it a19}\\,{{\\it a8}}^{2}+{\\it a18}\\,{\\it a1}+{\\it a18}\\,{{\n\\it a4}}^{2}+{\\it a18}\\,{{\\it a8}}^{3}+{\\it a18}\\,{\\it a12}+{{\\it a18}\n}^{2}+{\\it a12}\\,{\\it a1}+{\\it a12}\\,{{\\it a4}}^{2}+{\\it a12}\\,{{\\it\na8}}^{3}+{{\\it a12}}^{2}+{{\\it a8}}^{2}{{\\it a2}}^{2}+{{\\it a8}}^{2}{\n\\it a3}+{{\\it a8}}^{3}{\\it a1}+{{\\it a8}}^{3}{{\\it a4}}^{2}+{{\\it a8}}\n^{4}{\\it a2}+{{\\it a8}}^{6}+{{\\it a4}}^{4}+{\\it a18}\\,{\\it a8}\\,{\\it\na2}+{\\it a12}\\,{\\it a8}\\,{\\it a2}+{\\it a8}\\,{\\it a1}\\,{\\it a2}+{\\it a8\n}\\,{\\it a2}\\,{{\\it a4}}^{2}$.\n\nEach monomial in this sum corresponds to an entry in Table 2.\n\\end{proof}\n\n\\section{Rational curves in $A\/\\langle g\\rangle$}\n\nIn this section we explain how to find rational curves on $A\/\\langle g\\rangle$ in each case in Table 2.\n\n\\begin{lem}\nIf $A\/\\langle g^n\\rangle$ has a rational curve for some positive integer $n$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{lem}\n\n\\begin{proof}\nThe morphism $A\/\\langle g^n\\rangle\\to A\/\\langle g\\rangle$ is finite, so the image of a rational curve is again a rational curve.\n\\end{proof}\n\n\\begin{prop}\n\\label{reduce}\nLet $A$ be an abelian variety and $g$ an automorphism of finite order.  Suppose that for every abelian variety $B$ and finite-order automorphism $h\\in \\Aut(B)$\nwhose type is the reduced type of $g$, $B\/\\langle h\\rangle$ has a rational curve.  Then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{prop}\n\n\\begin{proof}\nLet $B$ denote the image of $1-g$ acting on $A$.  Then $B$ is an abelian subvariety of $A$, and $g$ restricts to an automorphism $h$ of $B$\nwhose type is the reduced type of $g$.  As $B\/\\langle h\\rangle \\subset A\/\\langle g\\rangle$ has a rational curve, the same is true of $A\/\\langle g\\rangle$.\n\\end{proof}\n\n\nThe following proposition is well known.\n\n\\begin{prop}\\label{kummer}\nIf $A$ is an abelian surface, then $A\/\\pm 1$ has a rational curve.\n\\end{prop}\n\n\\begin{proof}\nResolving the $16$ singularities of $A\/\\pm 1$, we obtain a K3 surface with Picard number $\\ge 16\\ge 5$.  By work of Bogomolov and Tschinkel \\cite{BT},  any such surface is either elliptic or has infinite automorphism\ngroup and in either case has infinitely many rational curves, all but finitely many of which lie on $A\/\\pm 1$.\n\\end{proof}\n\nNote that Proposition~\\ref{kummer} covers not only case \\#1 in Table 2 but twenty other cases as well, namely those (indicated in the ``notes'' column) for which the reduced type of some power of $g$ is $(1\/2,1\/2)$.\n\n\\begin{prop}\\label{cube} Let $A\\cong V\/\\Lambda$ be an abelian surface with an automorphism $g$ of type $(1\/3,2\/3)$.  Then $A\/\\langle g\\rangle$ contains a rational curve.\n\\end{prop}\n\n\\begin{proof}\nLet $G=\\langle g\\rangle$ and $X=A\/G$. Regarding  $1-g$  as an isogeny of $A$, the number of fixed points of $g$ is\n$$\\deg(1-g)=\\#\\ker(1-g)=\\det(1-g|\\Lambda )=3^2=9.$$\nThese are singularities of  type $A_2$, since under $(x,y)\\mapsto(\\omega x, \\omega^2 y)$ where $\\omega$ is a cube root of unity, the invariants are generated by $X=x^3, Y=y^3, Z=xy$, and so\n    $$\\bbc[[x,y]]^G = \\bbc[[X,Y,Z]]\/(XY-Z^3).$$\nThis is isomorphic to $\\bbc[[x, y, z]]\/(x^2 + y^2 + z^3)$,\n    which has a Du Val singularity of type $A_2$ (see \\cite[Ch.4,~4.2]{R}).\n\nConsider the  minimal resolution $f: Y\\rightarrow X$, for which the $9$ exceptional divisors $Y_i$ each consists of two projective lines $D_{i,1}$ and $D_{i,2}$ intersecting at one point.\nThe canonical divisor of $Y$ is $K_Y=f^*K_X=0$.  Hence $Y$ is a K$3$-surface of Picard number $\\ge 18$, and again by \\cite{BT}, we deduce that $X$ has infinitely many rational curves..\n\\end{proof}\n\n\\begin{thm}\n\\label{disjoint}\nLet $B$ be an abelian variety and $h\\in \\Aut(B)$ an automorphism of finite order such that $h$ and $h^{-1}$ have disjoint types.\nLet $A$ be an abelian variety with a finite order automorphism $g$ whose type is contained in that of $h$.\nIf there is a rational curve on $B\/\\langle h\\rangle$ then there is a rational curve on $A\/\\langle g\\rangle$.\n\\end{thm}\n\n\\begin{proof}\nIt suffices to prove that there exists a surjective homomorphism $p\\colon B\\to A$ such that the diagram\n$$\\xymatrix{B\\ar[r]^p\\ar[d]_h&A\\ar[d]^g \\\\ B\\ar[r]^p&A}$$\ncommutes.  Writing $B = \\Lie(B)\/\\Lambda_B$ and $A = \\Lie(A)\/\\Lambda_A$, the goal is to find a surjective $\\bbc[t]$-linear map $\\phi\\colon \\Lie(B)\\to \\Lie(A)$\n(where $t$ acts as $g$ on $\\Lie(A)$ and as $h$ on $\\Lie(B)$)\nsuch that $\\phi(\\Lambda_B)\\subset \\Lambda_A$.  If $\\psi$ is a surjective $\\bbc[ t]$-linear map $\\Lie(B)\\to \\Lie(A)$\nsuch that $\\psi(\\Lambda_B\\otimes\\bbq) = \\Lambda_A\\otimes  \\bbq$, then we can define $\\phi := n\\psi$ for $n$ a sufficiently divisible positive integer.  It suffices, therefore, to find $\\psi$\nwith the desired properties.\n\nAs the type of $A$ is a subset of the type of $B$, there exists a surjective $\\bbq[t]$-linear map $T\\colon\\Lambda_B\\otimes\\bbq\\to \\Lambda_A\\otimes  \\bbq$.\nExtending scalars to $\\bbc$, $T\\otimes 1$ maps $\\Lambda_B\\otimes\\bbc = \\Lie(B) \\oplus \\overline{\\Lie(B)}$ to $\\Lambda_A\\otimes \\bbc = \\Lie(A) \\oplus \\overline{\\Lie(A)}$.\nThe type of $g$ acting on $\\overline{\\Lie(A)}$ is the same as the type of $g^{-1}$ acting on  $\\Lie(A)$ and therefore disjoint from the type of $g$ acting on $\\Lie(A)$, and the same is true for the type of $h$ acting on $\\Lie(B)$.\nAs $\\Lie(B)$ and $\\Lie(A)$ are direct sums of certain $t$-eigenspaces of $\\Lambda_B\\otimes\\bbc$ and $\\Lambda_A\\otimes\\bbc$ respectively and\nas the spectrum of $t$ acting on $\\Lie(A)$ is the intersection of the spectra of $t$ acting on $\\Lie(B)$ and on $\\Lambda_A\\otimes\\bbc$, it follows that\n$T\\otimes 1$ maps $\\Lie(B)$ to $\\Lie(A)$.  The restriction of $\\psi$ to $\\Lie(B)$ is therefore the desired map.\n\n\\end{proof}\n\n\\begin{cor}\n\\label{mult-one}\nLet $n\\ge 3$ be a positive integer, and let $m = \\lceil n\/2\\rceil - 1$.  If $A$ is an abelian variety and $g\\in \\Aut(A)$ is an automorphism of order $n$\nwhose type is contained in $\\{1\/n,2\/n,\\ldots,m\/n\\}$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\nLet $X$ denote the non-singular projective hyperelliptic curve of genus $m$ which contains the affine curve $y^2 = x^n-1$.\nThe order-$n$ automorphism $h(x,y) = (e(1\/n)x,y)$ extends to an automorphism of $X$ and therefore defines an automorphism of $B := \\Jac(X)$.\nThe Lie algebra $\\Lie(B)$ can be identified with the space $H^0(X,\\Omega_X)$ of holomorphic differential forms on $X$, which has a basis\n$$\\Bigl\\{\\frac {dx}y, \\frac{x\\,dx}y,\\ldots,\\frac{x^{m-1}\\,dx}y \\Bigr\\}.$$\nTherefore, the type of $h$ acting on $B$ is $\\{1\/n,2\/n,\\ldots,m\/n\\}$, which is disjoint from the type of $h^{-1}$.\nOn the other hand, $B\/\\langle h\\rangle$ contains the rational curve $X\/\\langle h\\rangle$.\nThus, Theorem~\\ref{disjoint} applies.\n\\end{proof}\n\n\\begin{cor}\n\\label{seven}\nIf $A$ is an abelian $3$-fold and $g$ is an automorphism of $A$ of type $(1\/7,2\/7,4\/7)$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\n\nLet $X$ denote the Klein quartic:\n$$X : x^3y+y^3z+z^3x=0$$\nand $B$ the Jacobian of $X$.\nThe self-map $h(x,y,z)=(\\zeta_7x, \\zeta_7^4y, \\zeta_7^2z)$ of $X$ belongs to the automorphism group\n$\\PSL_2(\\bbf_7)$ of $X$ which acts non-trivially on the Jacobian variety $B$ and therefore on $\\Lie(B) = H^0(X,\\Omega_X)$.\nConjugating $h$ by the cyclic permutations of $(x,y,z)$, we see that $h$ is conjugate to $h^2$ and $h^4$\nin $\\Aut(X)$, and therefore the type of $h$ is invariant under multiplication by $2$ (mod $1$).\nIt is therefore $(1\/7,2\/7,4\/7)$ or $(3\/7,5\/7,6\/7)$, and\nreplacing $h$ by $h^{-1}$ if necessary, we may assume that it is the former.\n\\end{proof}\n\nWe remark that $B\/\\langle h\\rangle$ in the proof of Corollary~\\ref{seven} has appeared\nin the literature; it is known to have  a Calabi-Yau resolution  \\cite[Example~6.3]{BT2}.\n\n\n\n\\begin{cor}\\label{fifteen}\nIf $A$ is an abelian variety and $g$ an automorphism such that the type of $A$ is contained in $(1\/15,2\/15,3\/15,4\/15,8\/15,9\/15)$,\nthen $A\/\\langle g\\rangle$ has a rational curve.\n\\end{cor}\n\n\\begin{proof}\nLet $X$ be the non-singular projective curve which has a (singular, affine) model $X'\\colon y^{15} = x^2(x-1)$.  This is singular only at $(0,0)$, and the inverse image of this singularity\nunder the normalization map $X\\setminus \\{P_\\infty\\}\\to X'$ is a single point $P_0\\in X$.\nThe automorphism $h\\colon (x,y)\\mapsto (x,e(1\/15)y)$ of the affine curve induces an automorphism of $X$ of order $15$.\nAs $15y^{14}dy = (3x^2-2x)dx$, any differential form $\\frac {x^m y^n  dy}{3x^2-2x}$, $m,n\\ge 0$, is holomorphic except possibly at $P_0$ and $P_\\infty$.\nOne checks that\n$$\\frac{dy}{3x-2}, \\frac{ydy}{3x-2}, \\frac{y^2dy}{3x-2}, \\frac{y^3dy}{3x-2},  \\frac{y^7dy}{3x^2-2x},   \\frac{y^8dy}{3x^2-2x}$$\nare all holomorphic, and their eigenvalues under $h$ are $e(1\/15)$, $e(2\/15)$, $e(3\/15)$, $e(4\/15)$, $e(8\/15)$, $e(9\/15)$ respectively.\nApplying the Riemann-Hurwitz theorem to the map $X\\to \\bbp^1$ given by $y$, we see that $X$ is of genus $6$, and therefore, that these differential\nforms form a basis of $\\Lie(\\Jac(B)) = H^0(X,\\Omega_X)$.\n\\end{proof}\n\n\\begin{thm}\n\\label{small-factor}\nIf $A$ is an abelian variety and $g$ is an automorphism of finite order such that $g$ and $g^{-1}$ have disjoint types and the type of $g$ is a sum\nof primitive types at least one of which has $weight$ less than $1$, then $A\/\\langle g\\rangle$ has a rational curve.\n\\end{thm}\n\n\\begin{proof}\nFor every primitive type, there exists an abelian variety $B_i$ with complex multiplication and an automorphism $h_i$ of $B_i$ with the given type.\nIndeed, the primitive types of denominator $n$ are in natural correspondence with CM-types on $\\bbq(\\zeta_n) = \\bbq(e(1\/n))$.  Any CM-type $\\Phi$ on $\\bbq(\\zeta_n)$, $n\\ge 3$\ndefines an embedding $\\bbq(\\zeta_n) \\to \\bbc^{\\phi(n)\/2}$.  The image of $\\bbz[\\zeta_n]$ defines a lattice $\\Lambda\\subset \\bbc^{\\phi(n)\/2}$, and the quotient $\\bbc^{\\phi(n)\/2}\/\\Lambda$\nis a complex torus with a natural action of $\\bbz\/n\\bbz$ of the type associated with $\\Phi$.  The quotient $\\bbc^{\\phi(n)\/2}\/\\Lambda$ admits a polarization \\cite[II~6~Theorem 4]{S}, so there\nexists a pair $(B_i,h_i)$ as claimed.  If $\\age(h_1) < 1$, then by \\cite{KL}, $B_1\/\\langle g_1\\rangle$ has a rational curve.\nIf $A = A_1\\times\\cdots\\times A_m$, and $g = (g_1,\\ldots,g_m)$ is a finite order automorphism of $A$ which stabilizes each factor, then $A_1\/\\langle g_1\\rangle \\subset A\/\\langle g\\rangle$,\nso $A\/\\langle g\\rangle$ has a rational curve.  The theorem now follows from Theorem~\\ref{disjoint}.\n\\end{proof}\n\n\\\n\nTo summarize, we have the following theorem:\n\n\\begin{thm}\\label{main2}\nLet $A$ be an abelian variety with a nontrivial automorphism $g$ of finite order such that $\\age(g)\\le 1$.  Then $A\/\\langle g\\rangle$ contains a rational curve.\n\\end{thm}\n\n\n\\begin{cor}\\label{hyper} Let $A$ be an abelian variety of dimension $n$ with a nontrivial automorphism $g$ of finite order. If $\\dim(\\ker(1-g))\\geq n-2$ (i.e. the codimension of the fixed subspace of $A$ under $g$ is less than or equal to $2$), then the quotient $A\/\\langle g\\rangle$ contains a uniruled hypersurface.\n\\end{cor}\n\n\\begin{proof} Since $\\dim(\\ker(1-g))\\geq n-2$, $B:=\\mathrm{im}(1-g)$ is an abelian variety of dimension $n-\\dim (\\ker (1-g))\\le 2$.\nLet $h$ denote the restriction of $g$ to $B$.  As $\\age(h)+\\age(h^{-1}) \\ge 2$, we may assume without loss of generality that $\\age(h)\\le 1$,\nso $B\/\\langle h\\rangle$ has a rational curve $Z$ by Theorem~\\ref{main2}.\n  Let $C$ denote the identity component of $\\ker (1-g)$, which is an abelian subvariety of dimension $\\dim \\ker(1-g)$ on which $g$ acts trivially.  The addition morphism $B\\times C\\to A$ is an isogeny and respects the action of  $\\langle g\\rangle$.  We therefore obtain a finite morphism from\n$$Z\\times C \\subset (B\/\\langle g\\rangle)\\times C \\cong (B\\times C)\/\\langle g\\rangle$$\nto $A\/\\langle g\\rangle$.  The image of an $n-1$-dimensional ruled variety under a finite morphism is a uniruled hypersurface.\n\\end{proof}\n\n\\begin{cor} Let $E$ be an elliptic curve. If $W$ is a Weyl group of simple roots of rank $n\\geq 3$ acting on $E^n$ and $W^+$ is an index $2$-subgroup of $W$, then the quotient $E^n\/W^+$ contains a rational curve.\n\\end{cor}\n\\begin{proof} $W$ is generated by reflections $s_j$ of simple roots. Since $W^+$ is an index $2$-subgroup of $W$, there exist  two reflections $s_1$ and $s_2$ such that $s_1s_2\\in W^+$. Then for each $i$, $\\ker(1-s_i)$ has codimension $1$ and their intersection has codimension $\\leq 2$. Since $\\ker(1-s_1s_2)$ contains the intersection of $\\ker(1-s_1)$ and $\\ker(1-s_2)$,  this follows from Corollary~\\ref{hyper}.\n\\end{proof}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe main aim of this work is to present a combined theory of the term structure of interest rates and of price impact, with applications to optimal execution. This objective entails the inclusion of a specific type of cross price impact that is specific to fixed income. \n\nTerm structure modeling with a view to derivatives valuation and hedging has been developed over several decades. For the purposes of this work and what we might call the classic theory we refer to the monographs by Bjork \\cite{bjork1997interest}, Brigo and Mercurio \\cite{brigo2007interest} and Filipovic \\cite{filipovic2009term}. After the crisis that started in 2007, the gap between two of the rates that were used as benchmarks for risk free rates, namely interbank and overnight rates, widened considerably, peaking in October 2008, following the defaults of several financial institutions in the space of one month \\cite{brigopallatorre}. This highlighted the fact that interbank rates could no longer be used as benchmarks for risk free rates and neither could they be used to derive zero-coupon curves that were not contaminated by credit and liquidity risk. This led to the necessity to model multiple interest rate curves, treating interbank rates as risky rates affected by credit and liquidity risk and adopting overnight based rates as new risk free rates. \n\nThis multiple interest rate curve academic literature was initiated by practitioners, see in particular the monograph by Henrard \\cite{henrard2014interest}. Substantial contributions were made later by numerous academics, where we refer to a monograph by Grbac and Runggaldier \\cite{grbac2015interest}, Crepey et al. \\cite{crepey2015levy}, Grbac et al. \\cite{grbac2015affine}, Cuchiero et al. \\cite{cuchiero2019affine,cuchiero2016general}, Nguyen and Seifried \\cite{nguyen2015multi} and finally to Bormetti et al. \\cite{bormetti2018}, for multiple curves in conjunction with valuation adjustments and credit risk.\n\nFurther recent developments include the presence of negative interest rates in many currencies, see for example the BIS report  \\cite{negativeratesbis}, and the ongoing project of eliminating current interbank rates like the London Interbank Offer Rate (LIBOR), replacing them with new types of risk free rates inspired by overnight rates. This puts the multiple-curve area in a state of uncertainty, while negative rates prompted the mainstream resurgence of Gaussian models that were previously justified only in very special economies exhibiting negative rates, such as for example Switzerland in the seventies. \n\nGiven the state of market uncertainty on benchmark interest rates, products and markets, we will not consider these recent developments in this work, except for allowing for negative rates in our formulation. We are interested in developing a combination of term structure modeling and price impact in the classic theory of interest rates. We are confident that multiple curves and further discussion of negative rates, if still present in the market after reforms and updated central bank policies, can be incorporated in further work after the classic theory has been developed.\n\nDespite the fact that the bond market size is considerably larger than the equity market size, relevance of liquidity risk in the context of bonds was pointed out in several papers. A report from the Federal Reserve Bank of New York from 2003 \\cite{chordia2005empirical}, for instance, showed that bond and stock markets have common factors driving their liquidity. A strong relationship between liquidity in the Treasury bond market and in the stock market was later highlighted also by Goyenko and Ukhov \\cite{goyenko2009stock}. Another significant contribution appears in a recent paper by Schneider and Lillo \\cite{schneider2019cross}, which studied cross price impact among sovereign bonds.\n\nThe financial crisis of 2008-2009 have also raised concerns about the inventories kept by intermediaries.\nRegulators and policy makers took advantage of two main regulatory changes (Reg NMS in the US and MiFID in Europe) and enforced more transparency on the transactions and hence on market participants positions, which pushed the trading processes toward electronic platforms \\cite{citeulike:12047995}.\nSimultaneously, consumers and producers of financial products asked for less complexity and more transparency.\n\nThis tremendous pressure on the business habits of the financial system, shifted it from a customized and high margins industry, in which intermediaries could keep large (and potentially risky) inventories, to a mass market industry where logistics have a central role. As a result, investment banks nowadays unwind their risks as fast as possible.\n In the context of small margins and high velocity of position changes, trading costs  are of paramount importance. A major factor of the trading costs is the price impact: the faster the trading rate, the more the buying or selling pressure will move the price in a detrimental way. \n \nAcademic efforts to quantify and reduce the transaction costs of large trades trace back to the seminal papers of Almgren and Chriss \\cite{OPTEXECAC00} and Bertsimas and Lo \\cite{BLA98}. In both models one large market participant (for instance an asset manager or a bank) would like to buy or sell a large amount of shares or contracts during a specified duration.\nThe cost minimization problem turned out to be quite involved, due to multiple constraints on the trading strategies.\nOn one hand, the price impact demands to trade slowly, or at least at a pace which takes into account the available liquidity (see \\cite{citeulike:13497373} and references therein).\nOn the other hand, traders have an incentive to trade rapidly, because they do not want to carry the risk of an adverse price move far away from their decision price. The importance of optimal trading in the industry generated a lot of variations for the initial mean-variance minimization of the trading costs (see \\cite{citeulike:12047995,cartea15book,olivier16book} for details). These type of problems are usually formulated as optimisation problems in the context of stochastic control where the agent tries to minimize the transaction costs which result by the price impact and to reduce the risk associated with holding the assets for too long (see e.g. \\cite{citeulike:5282787,GLFT, citeulike:13675369}). \n\nIn this spirit, we will follow the term structure theory as developed by Bjork \\cite{bjork1997interest}, modifying it to allow for the inclusion of price impact.  We will start from the simplest possible dynamics, namely one factor short rate models, and extend it later to instantaneous forward rate models. We will introduce price impact formulated on zero-coupon bonds, since they are a possible choice of building blocks for the term structure. We will later connect this with impact on coupon bearing bonds that are more commonly traded. \n\nWe assume that an agent who is executing a large order of bonds is creating two types of price impact which are extensively used in the literature. The first one is an instantaneous (or temporary) price impact, which affects the asset  price only while trading, and fades away immediately after. This type of price impact occurs due to the fact that a large buy (sell) trade consumes the liquidity which is available in the market by ``walking through'' the first few price levels of the limit order book (see e.g. \\cite{OPTEXECAC00}  and Chapter 6.3 of \\cite{cartea15book}). However, empirical studies have shown that price impact also has a \\emph{transient} effect. A short of liquidity due to a large trade creates an imbalance between supply and demand, which in turn pushes the price in a detrimental direction. This effect decays within a short time period after each trade (see \\cite{citeulike:13497373} and \\cite{Ob-Wan2005}). Execution in presence of transient price impact was studied extensively in the context of optimal control problems (see e.g.  \\cite{GSS, GSS2, AFS2,neuman2020optimal,stat-adp18}).  In this work we incorporate these two types of price impact models into a bonds trading framework, as was done in \\cite{neuman2020optimal} for equities. \n\nApplying these price impact models to the term structure will be challenging. At every point in time the term structure of interest rates is a high dimensional object, or even an infinite dimensional one when considering all possible maturities for interest rates or zero-coupon bonds at a given time. This is a unique feature of term structure modeling, where differently from FX or equity modeling for example, we model a whole curve dynamics rather than a point dynamics. We can expect that trading a bond with a specific maturity may impact the price of bonds with different maturities {\\emph{on the same currency curve}}. In this sense, the cross price impact is endogenous to the same underlying asset, differently from what happens in other markets.  \nWe will investigate how price impact interacts with no-arbitrage dynamics, and we will encapsulate the effect of price impact in the definition of a new no-arbitrage pricing measure embedding impact itself.  This will be done by extending the market price of risk to an impacted version embedding the bond price impact speed. The impacted zero-coupon bond dynamics will then be written as the unimpacted bond dynamics but under a different measure. We will also introduce an impacted physical measure that could be useful for risk management and risk analysis. Finally, we will illustrate our theory by proposing an application to optimal execution. \n\nWe will not limit ourselves to short rate models. We will also see how in the Heath-Jarrow-Morton model the no-arbitrage drift condition for instantaneous forward rate dynamics can be maintained under price impact by resorting to the modified pricing measure.\n\n\nThe paper is structured as follow. \nIn section \\ref{sec:road} we propose a roadmap for readers who are not familiar with at least one of the two areas of the paper, namely term structure modeling and market impact.  \nIn Section \\ref{sec:main} we introduce the short rate models setup and the main theoretical results. In particular, we introduce price impact for zero-coupon bonds, and we look at the impacted market price of risk, absence of arbitrage and the impacted risk neutral measure. We define the impacted yield curve and extend impact to coupon bearing bonds. We further show how to use the impact setup in a HJM framework.  \nSection \\ref{sec:examples} features a few examples including valuation of impacted Eurodollar futures with the Hull and White model.\nSection \\ref{sec:numerical_results}\t\npresents some numerical results illustrating how the yield curve behaves under impact.\nFinally, we introduce a result on optimal execution with impacted bonds in Section \\ref{sec:optexec}.\nSections \\ref{sec:proofs} and \\ref{sec:proof_linear_feedback} include proofs that have not been included in the main text. \n\n\n\n\\section{Roadmap: how to read this paper}\\label{sec:road}\n\nThis paper combines areas that are typically disjointed in the literature, such as term structure modeling and price impact\/optimal execution. It is possible that most readers will be familiar with one area but not the other one. We expect to find few readers who may read the whole paper without effort. We therefore provide a roadmap for the reader who wishes to have a quick understanding of what the paper deals with and how it is organized, with a particular focus on where finding what and what needs to be read more carefully depending on the reader's background. \n\n{\\bf Main section of the paper.} \nSection \\ref{sec:main} is the section where both the term structure model is introduced and  the impact model is postulated and developed in detail, linking the developments to no-arbitrage pricing. The reader should start ideally from here. \n\n\\medskip\n\n{\\bf Interest rate and bond dynamics under different measures.} In Section \n\\ref{subsec:def} we introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section \\ref{subsec:HJM} but the paper arguments and innovations are best appreciated in a short-rate setting. We start postulating a diffusion dynamics for the short rate $r$ under the physical measure, giving then also the dynamics of the bond price $P$ under the same measure. We introduce the market price of risk leading to the risk neutral measure dynamics for the zero-coupon bond and show that, as classically done in Bjork with no-arbitrage assumptions related to self-financing and locally risk free portfolios, it does not depend on the bond maturity. With this market price of risk we then derive the dynamics of the short rate $r$ under the risk neutral measure. All this is standard theory that the term structure expert will find immediate and may want to read quickly. For readers who are more into price impact but relatively unfamiliar with term structure modeling, we recommend a careful reading of this introductory part. \n\n\\medskip\n\n{\\bf Inventory and impact.} We define the trader's inventory $X$, the execution speed $v$ and the instantaneous and transient impacts, leading to the impacted bond price $\\tilde{P}$ in Section \\ref{subsec:def}. This part will be easier to the market microstructure expert but, given that applications of price impact to full term structure modeling are not known, we recommend to read carefully this part  given the nuances involved in developing an impact theory under term structure.\nWe then show how it is possible to incorporate the effect of price impact in a new definition of impacted market price of risk, leading to an impacted risk neutral measure.  \n \n{\\bf Impacted risk neutral and physical probability measures.} \nThis measure is introduced in Section \\ref{subsec:rnm}. Under this measure, the impacted zero-coupon bond dynamics is the same as the risk neutral dynamics of the unimpacted bond provided one replaces the Brownian motion under the original risk neutral measure with a Brownian motion under the new impacted measure. The impacted risk neutral measure allows us to define an impacted physical measure as well, and again the bond price dynamics under the impacted physical measure is the same as the bond price dynamics under the original physical measure, provided that the Brownian motion is replaced with a Brownian motion under the new measure. In this setting, we can show that the impacted zero-coupon bond is the impacted risk neutral expectation of the stochastic discount factor. In a way, the impacted risk neutral measure behaves indeed as much as possible as a risk neutral measure in presence of impact. \n\nOne of the main purposes of introducing the risk neutral measure in classic derivatives pricing is the idea that the existence of this measure implies absence of arbitrage. The reader might then wonder whether we can say the same of the impacted risk measure. This is dealt with in the Section \\ref{subsec:pricing}.\n\n\\medskip\n\n{\\bf Derivatives pricing and absence of arbitrage.} First, we show that the existence of an impacted equivalent pricing measure implies that the impacted model is arbitrage free. It then follows that we can extend our result for zero-coupon bonds, namely we can price interest-rate derivatives under impact by taking the impacted risk neutral expectation of their discounted cash flows. We present an example involving Eurodollar futures. \n\n\\medskip\n\nAn important part of the theory of price impact for interest rates is what we might call {\\bf endogenous cross-impact}. With this we point to the impact of a zero-coupon bond with a given maturity on the price of a zero-coupon bond with a different maturity {\\emph{on the same currency curve}}. Exogenous cross impact would be the impact of a bond in one curve on the bond on another curve, but this is not what we deal with here. The theory of endogenous cross impact is developed in Section \\ref{subsec:yield}. This has to be read very carefully by all readers, as it represents a very peculiar aspect of any theory of price impact applied to a term structure. In term structure modeling the underlying of derivatives is a whole curve, so price impact has to be postulated also for one part of the curve over other parts. This is not a problem in markets where assets are points rather than curves, like stocks or FX rates. \n\n\\medskip\n\n{\\bf Coupon bonds.} The theory developed up to this point of the paper deals with zero-coupon bonds. However, most traded bonds are coupon bonds, so it is necessary to develop a theory of price impact for coupon bearing bonds. This is done in Section \\ref{subsec:coupon}. First, by looking at prices as expectations of discounted cash flows, we deduce that the price of impacted coupon bearing bond is simply obtained by the unimpacted one, by replacing all unimpacted zero-coupon bonds in its expression with impacted ones. \nWe find a way to connect the impact on the coupon bearing bond to the impact of single zero-coupon bonds, and find assumptions under which the impacted dynamics may even be the same.\n\n\\medskip\n\nFinally, as explained earlier, Section \\ref{subsec:HJM} goes into quite some detail to show that the theory developed so far can be equally formulated in the {\\bf Heath-Jarrow-Morton (HJM) framework}, using instantaneous forward rates to model the term structure. It is well known to interest rate experts that the no-arbitrage condition in the HJM framework is expressed by requiring that the drift in the forward rate dynamics is a specific transformation of the dynamics volatility. We show that the same relationship between drift and volatility holds for impacted instantaneous forward rates under the impact-adjusted risk neutral measure. The classic HJM theory then carries over to the impacted case. This part may be helpful to readers who wish to see our framework work outside the short rate models setting, but if the reader is not too interested in the specific interest rate model used, she can skip this part and still get all the ideas on impacted term structure modeling. \n\n\n\\bigskip\n\n{\\bf Pricing examples.} These are presented in Section \\ref{sec:examples}, including pricing impacted Eurodollar Futures with Gaussian short rate models. This section is interesting for readers who wish to see how the model can be effectively applied to price interest rate derivatives under price impact. \n\n\n\\bigskip\n\n{\\bf Numerical results.} Section \\ref{sec:numerical_results} presents numerical examples of impacted yield curves, showing how the term structure changes once price impact is included. We analyze the interplay between the bond pull-to-par and cross-price impact parameters. This can be a good section for readers who wish to develop some intuition on how price impact may change the shape of the term structure.  \n\n\\bigskip\n\n\n{\\bf Optimal execution with impacted bonds.} For the reader wondering how this theory of price impact can be included in classic problems such as optimal execution, we solve the optimal execution problem in a specific setting. This is presented in Section \\ref{sec:optexec}. The example is rather technical and the solution involves a system of forward-backward SDEs, showing that optimal execution with term structure models can be technically demanding, and this is recommended only for readers who are interested in optimal execution. \n\n\\bigskip\n\nConcluding, the best way to read the paper is sequentially, skipping the parts that are not of interest and looking more carefully at areas outside the specific expertise of the reader. We tried to provide a guide to what can be skipped and on the different emphasis for different areas of expertise above. \n\n\n\\section{Model setup and main results}\\label{sec:main}\n\n\\subsection{Impacted market price of risk, impacted risk neutral measure and absence of arbitrage} \\label{subsec:def}\nWe introduce our initial assumption on the interest rate dynamics, assuming it is a one-dimensional short rate model dynamics. This will be relaxed later with a Heath-Jarrow-Morton setting in Section \\ref{subsec:HJM}.\n\nLet us fix a maturity $T>0$ and let $(\\Omega, \\mathcal{F}, (\\mathcal{F}_t)_{t \\geq 0}, \\mathbb{P})$ be a filtered probability space satisfying the usual conditions, on which there is a standard $(\\mathcal{F}_t)_{t \\in [0,T]}$-Brownian motion $W^\\mathbb{P}$. We consider the following dynamics of the short rate under the real world measure $\\mathbb{P}$\n\\begin{equation} \\label{r-sde} \nd r(t) = \\mu(t,r(t)) dt + \\sigma(t,r(t)) d W^\\mathbb{P}(t),\n\\end{equation}\nwhere $\\mu(t,r), \\sigma(t,r)$ are given real valued functions, assumed to be regular enough to ensure the SDE has a unique strong solution. For example one can assume that both $\\mu$ and $\n\\sigma$ are Lipschitz continuous in the $r$ coordinate, and has at most linear growth in $r$ uniformly in $t\\in [0,T]$. \nWe moreover assume that $\\sigma(t,r(t))$ is $\\mathbb{P}$-a.s.  strictly positive for any $t>0$.  \n\nAssume that the dynamics of a zero-coupon bond with maturity at time $T$, under the real world measure, is given by\n\\begin{equation} \\label{class-b}\nd P(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) d W^\\mathbb{P}(t),\n\\end{equation}\nwith $\\mu_T,\\sigma_T$ depending on the maturity $T$ and regular enough as in \\eqref{r-sde}. Typically, one might assume the price process of the $T$-bond to be of the form $P(t,T) = F(t,r(t);T)$ for some function $F$ smooth in three variables. Then, under suitable assumptions (see Assumption 3.2 in Chapter 3.2 of \\cite{bjork1997interest}) one can define for any finite maturity $T>0$ the stochastic process\n\\begin{equation}\n\\lambda(t) = \\frac{\\mu_T(t,r(t)) - r(t) P(t,T)}{\\sigma_T(t,r(t))},\n\\label{market_price_of_risk}\n\\end{equation}\nand show that $\\lambda$ may depend on $r$ but it does not actually depend on $T$. Such process is called \\emph{market price of risk}. Provided the Novikov condition holds, this process can be used to define a change of measure from the real world measure $\\mathbb{P}$ to the risk neutral measure $\\mathbb{Q}$:\n\\begin{equation}\n\\frac{d \\mathbb{Q}}{d \\mathbb{P}} = \\exp \\left( \\int_0^t \\lambda(s) d W^\\mathbb{P}(t)- \\frac{1}{2} \\int_0^t \\lambda^2(s) ds\\right).\n\\label{risk_neutral_measure_Q}\n\\end{equation}\nThe dynamics of the short rate under $\\mathbb{Q}$ becomes\n\\begin{equation*}\nd r(t) = [\\mu(t,r(t)) - \\lambda(t) \\sigma(t,r(t))] dt + \\sigma(t,r(t)) d W^\\mathbb{Q}(t).\n\\end{equation*}\nThe model will be fully specified once the stochastic process $\\lambda$ is defined.\nOur strategy for establishing a mathematical framework that encompasses both risk neutral pricing and price impact in the context of interest rates derivatives consists, first of all, in specifying the dynamics for an impacted bond with maturity $T$ under the real world measure $\\mathbb{P}$. \n\nWe consider a trader with an initial position of $x_T>0$ zero-coupon bonds with maturity $T$. Let $0< \\tau \\leq T$ denote some finite deterministic time horizon. In an optimal execution problem, the objective of the trader would be to complete her transaction by time $\\tau$, starting from the $x_T$ position at time $0$. In this sense, we should avoid confusion between $T$, which is the traded bond maturity, and $\\tau$, which is the trading horizon of the $T$-maturity bond. The number of bonds the trader holds at time $t \\in [0,\\tau]$ is given by\n\\begin{equation}  \\label{inv} \nX_T(t) = x_T- \\int_{0}^t v_T(s) ds.\n\\end{equation} \nwhere the function $v_T$ denotes the trader's selling rate, which takes negative values in case of a buy strategy. In what follows we assume that $v_T= \\{v_T(t)\\}_{0\\leq t\\leq \\tau}$ is progressively measurable and has a $\\mathbb P$-a.s. bounded derivative (in the $t$-variable), that is, there exists $M>0$ such that \n \\begin{equation} \\label{v-bnd} \n \\sup_{0\\leq t\\leq \\tau^{+}} |\\partial_{t}  v_{T}(t)|  < M, \\quad \\mathbb P-\\rm{a.s.}, \n\\end{equation} \nwhere $0\\leq t\\leq \\tau$. After the trading stops, we assume that $v_{T}(t)= 0$.\nWe denote the class of such trading speeds as $\\mathcal{A}_{T}$.  \n \nThe main idea behind the assumption of the differentiability of $v_{T}$ is that the overall impact we add to the zero-coupon bond should affect the drift only (see \\eqref{impacted_bond_differential_form}). Moreover due to price impact effect, we allow  bond price which are large than $1$ for some time intervals but we do need to control their upper bound. \n\nWe consider a price impact model with both transient and instantaneous impact, which is a slight generalization of the model which was considered in \\cite{neuman2020optimal}. The impacted bond price is therefore given by \n\\begin{equation} \\label{impacted_bond}\n\\tilde{P}(t,T) = P(t,T) - l(t,T)  v_T(t) - K(t,T) \\Upsilon_T^v(t). \n\\end{equation}\nHere $\\Upsilon_T^v$ represents the transient impact effect and it has the form \n\\begin{equation}\\label{def:transient_impact}\n\\Upsilon_T^v(t):= y e^{-\\rho t} + \\gamma \\int_0^t e^{-\\rho(t-s)} v_T(s) ds,\n\\end{equation}\nwhere $y,\\rho$ and $\\gamma$ are positive constants. The term $v_T(t)$ in \\eqref{impacted_bond} represents the instantaneous price impact, where we absorb in the function $l$ any constants that should factor it. \nLastly, $l$ and $K$ are differentiable functions with respect to both variables $(t,T)$ which take positive values on $0\\leq t <T$ and for any $0\\leq \\tau <T$ we have\n\\begin{equation}  \\label{l-pos}\n\\inf_{0\\leq t \\leq \\tau} l(t,T) >0.\n\\end{equation} \nMoreover we assume that   \n\\begin{equation}  \\label{k-assump} \n\\begin{aligned} \n&\\sup_{0\\leq t\\leq \\tau} |\\partial_{t}l(t,T)| <\\infty, \\quad \\lim_{t \\rightarrow T} l(t,T) =0,  \\\\\n&    \\sup_{0\\leq t\\leq \\tau} |\\partial_{t}K(t,T)| <\\infty, \\quad \\lim_{t \\rightarrow T} K(t,T) =0.\n\\end{aligned} \n\\end{equation}  \nWhile the assumption on boundedness of the derivatives of functions $K$ and $l$ arise from technical reasons which has similar motivation as the reason for \\eqref{v-bnd}, the assumptions on the behaviour at expiration is meant to enforce the boundary condition on the price of the impacted bond at expiration, which is $\\tilde{P}(T,T) =1$. Note that $K$ and $l$ are time-dependent versions of the parameters $\\lambda,k$ in \\cite{neuman2020optimal}. A prominent example of such functions is \n\\[\nl(t,T) = \\kappa \\left(1-\\frac{t}{T} \\right)^\\alpha, \\quad K(t,T) = \\left(1-\\frac{t}{T} \\right)^\\beta, \n\\]\nfor some constants $\\alpha , \\beta \\geq1$ and $\\kappa>0$. \n\nWe define for convenience the overall price impact:\n\\begin{equation}\\label{def:overall_price_impact}\nI_T(t) := l(t,T) v_T(t) + K(t,T) \\Upsilon_T^v(t).\n\\end{equation}\nThen, since $v_{T}$ is in $\\mathcal A_{T}$ we can rewrite \\eqref{impacted_bond} as follows: \n\\begin{equation} \\label{impacted_bond_differential_form}\nd \\tilde{P}(t,T) = d P(t,T) - J_T(t) dt, \\quad \\tilde P(T,T) = 1,\n\\end{equation}\nwith\n\\begin{equation} \\label{impact_density}\n\\begin{aligned}\nJ_T(t) & := \\partial_t I_T(t) \\\\\n& = \\partial_t l(t,T) v_T(t) + l(t,T) \\partial_t v_T(t) + \\partial_t K(t,T) \\Upsilon_T^v(t) + K(t,T) [-\\rho \\Upsilon_T^v(t) + v_T(t)].\n\\end{aligned}\n\\end{equation} \n\nOur model so far describes how trading a $T$-bond affects its price. Next, we show the existence of an \\emph{impacted market price of risk process} which will be a generalization of \\eqref{market_price_of_risk}. Using this process we will define an equivalent martingale measure, under which bonds and derivatives prices can be computed. Such a measure will be called an \\emph{impacted risk neutral measure}. It is important to remark that, as in the classic case, this change of measure will be unique for all bond maturities. \n\nBefore stating the main theorem of this section, let us first introduce a few important definitions.\n\n\\begin{definition}[Impacted portfolio]\\label{def:impacted_portfolio}\nLet $\\hat{T}<+\\infty$ be some finite time horizon. An \\emph{impacted portfolio} is a $(n+1)-$dimensional, bounded progressively measurable process $\\tilde{h} = (\\tilde{h}_t)_{t \\in [0, \\hat T]}$ with $\\tilde{h}_t = (\\tilde{h}_t^0,\\tilde{h}_t^1,\\dots,\\tilde{h}_t^n)$, where $\\tilde{h}_t^i$ represents the number of shares in the impacted bond $\\tilde{P}(t,T_i)$ held in the portfolio at time $t$. The value at time $t$ of such a portfolio $\\tilde{h}$ is defined as\n\\begin{equation*}\n\\tilde{V}(t) \\equiv \\tilde{V}(t,\\tilde{h}) := \\sum_{i=0}^n \\tilde{h}^i(t) \\tilde{P}(t,T_i).\n\\end{equation*}\n\\end{definition}\n\n\\begin{definition}[Self-financing]\\label{def:self_financing}\nLet $\\hat{T}<+\\infty$ be some finite time horizon. and let $\\tilde{h}$ be an impacted portfolio as in Definition \\ref{def:impacted_portfolio}. We say that $\\tilde{h}$ is \\emph{self-financing} if its value $\\tilde{V}$ is such that\n\\begin{equation} \\label{imp-port} \nd \\tilde{V}(t,\\tilde{h}) = \\sum_{i=0}^n \\tilde{h}^i(t) d  \\tilde P(t,T_i), \\quad \\textrm{for all } 0\\leq t \\leq \\hat T. \n\\end{equation}\n\\end{definition}\n\n\\begin{definition}[Locally risk free]\\label{def:locally_risk_free}\nLet $\\tilde{h}$ be an impacted portfolio as in Definition \\ref{def:impacted_portfolio} and let $\\tilde{V}$ be its value. Let also $\\alpha=(\\alpha_t)_{t \\in [0,\\hat T]}$ be an adapted process. We say that $\\tilde{h}$ is \\emph{locally risk-free} if, for almost all $t$,\n\\begin{equation*}\nd \\tilde{V}(t) = \\alpha(t) \\tilde{V}(t) \\implies \\alpha(t) = r(t),\n\\end{equation*}\nwhere $r(t)$ is the risk-free interest rate introduced in \\eqref{r-sde}.\n\\end{definition}\n\n\nHere is the main result of this section.\n\n\\begin{theorem}[Impacted market price of risk]\\label{thm_impacted_market_price}\nLet $\\hat{T}<+\\infty$ be some finite time horizon and let $\\mathbb T:=(0,\\hat{T}]$.\nLet $J_T$ be the impact density defined in \\eqref{impact_density}. Given an impacted portfolio $\\tilde{h}$ as in \\eqref{imp-port}, we assume that it is self-financing and locally risk-free, as in Definitions \\eqref{def:self_financing} and \\eqref{def:locally_risk_free}, respectively. Then, there exists a progressively measurable stochastic process $\\tilde{\\lambda}(t)$ such that \n\\begin{equation} \\label{def:lambda_tilde}\n\\tilde{\\lambda}(t) = \\frac{\\mu_{T_i}(t,r(t)) - r(t) \\tilde{P}(t,T_i) - J_{T_i}(t)}{\\sigma_{T_i}(t,r(t))}, \\quad t\\geq 0, \n\\end{equation}\nfor each maturity $T_i $, $i=1,..,n$, with $\\tilde{\\lambda}$  depending on the short rate $r$ but not on $T_i$.\n \\end{theorem}\n The proof of Theorem \\ref{thm_impacted_market_price} is given in Section \\ref{sec:proofs}.\n\n \n\n \\begin{remark}[Self-financing in presence of price impact]\nIn presence of price impact it is of course not obvious that the self-financing condition should hold. Adjusted self-financing conditions have been proposed, for instance, by Carmona and Webster \\cite{carmona2013self}. We notice that, in their work, the adjustment consists of two parts: the covariation between the inventory and the price process, and the bid-ask spread. In our work we will assume the inventory is a finite variation process and that the bid ask spread is negligible, thereby obtaining the classic self-financing condition.\n\\end{remark}\n\n\\begin{remark} [Intrinsic price impact] \nFrom Theorem \\ref{thm_impacted_market_price} it follows that \\emph{endogenous cross price impact} naturally emerges in our framework. Indeed, once an agent trades a bond with maturity $T_1$, the process $\\tilde{\\lambda}$ is uniquely determined. Note that $\\tilde \\lambda$ does not depend on the maturity. For any bond with maturity $T_2 \\in \\mathbb{T}$, which is not traded, we have $J_{T_2}\\equiv 0$ but by \\eqref{def:lambda_tilde}, the price $\\tilde P(t,T_2)$ will be affected by the trade on the bond with maturity $T_1$. We remark that, by \\emph{endogenous}, we mean that the bonds with different maturities $T_1$ and $T_2$ are thought of as belonging to the \\emph{same currency curve}. If we were to discuss multiple interest rate curves, then exogenous cross price impact should be taken into account as well.\n\\end{remark} \n\n\\subsection{Impacted risk-neutral measure}\\label{subsec:rnm}\nWe previously introduced two measures: the real world measure $\\mathbb{P}$ and the classic risk neutral measure $\\mathbb{Q}$, as defined in \\eqref{risk_neutral_measure_Q}. Now we use the result of Theorem \\ref{thm_impacted_market_price} to define a third measure, which we call \\emph{impacted risk neutral measure} and denote by $\\tilde{\\mathbb{Q}}$. This is defined as follows:\n\\begin{equation}   \\label{radon_nikodym_der_Q_tilde}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} = \\exp \\left\\{ \\int_0^t \\tilde{\\lambda}(s) d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\lambda}^2(s) ds \\right\\}.\n\\end{equation}\nThe well posedness of $\\tilde{\\mathbb{Q}}$ can be checked via the Novikov condition. It might be useful to recall that the usual approach does not consist in determining the conditions on $\\mu_T,\\sigma_T$ under which the Novikov condition is fulfilled. Rather, one chooses a specific short rate model to begin with. Then, one can specify the market price of risk process, exploiting the fact that it depends on $t$ and $r$, but not on $T$. For example, in the case of Vasicek model, the market price of risk is assumed to be $\\lambda(t) = \\lambda r(t)$, for some constant $\\lambda$. At this point, Novikov condition can be checked much more easily. Since we proved that $\\tilde{\\lambda}$ depends on $t$ and $r$ only, we can assume the two processes to have the same structure and follow the same idea. In the case of Vasicek model, for example, we can assume $\\tilde{\\lambda}(t) = \\tilde{\\lambda} r(t)$, for some constant $\\tilde{\\lambda}$ incorporating the impact. Consequently, determining the existence and well-posedness of $\\tilde{\\mathbb{Q}}$ is fundamentally equivalent to determining the existence and well-posedness of $\\mathbb{Q}$.\n\nThe Girsanov change of measure from the classic risk neutral measure to the impacted one is given by\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} \\frac{d \\mathbb{P}}{d \\mathbb{Q}}.\n\\end{equation*}\nwith\n\\begin{equation*}\n\\frac{d \\mathbb{P}}{d \\mathbb{Q}} = \\exp \\left\\{- \\int_0^t \\lambda(s) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\lambda^2(s) ds \\right\\}.\n\\end{equation*}\nwhere $\\lambda$ was defined in \\eqref{market_price_of_risk}.\nHence,\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\exp \\left\\{ \\int_0^t \\tilde{\\lambda}(s) d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\lambda}^2(s)ds + \\int_0^t \\lambda(s) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\lambda^2(s)ds \\right\\}.\n\\end{equation*}\nSince $W^\\mathbb{P}(t) := W^\\mathbb{Q}(t) + \\int_0^t \\lambda(s) ds$ is a Brownian motion under the measure $\\mathbb{P}$, we have\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} = \\exp \\left\\{ \\int_0^t (\\tilde{\\lambda}(s) - \\lambda(s)) d W^\\mathbb{Q}(s) - \\frac{1}{2} \\int_0^t \\left(\\lambda^2(s) + \\tilde{\\lambda}^2(s) - 2 \\lambda(s) \\tilde{\\lambda}(s) \\right) ds \\right\\}.\n\\end{equation*}\nIn other words,\n\\begin{equation*}\nW^{\\tilde{\\mathbb{Q}}}(t) := W^\\mathbb{Q}(t) - \\int_0^t (\\tilde{\\lambda}(s) - \\lambda(s)) ds,\n\\end{equation*}\nis a Brownian motion under the measure $\\tilde{\\mathbb{Q}}$. It is then straightforward to notice that the impacted zero-coupon bond under the impacted measure $\\tilde{\\mathbb{Q}}$ will be described by the dynamics\n\\begin{equation} \\label{qt-p}\nd \\tilde{P}(t,T) = r(t) \\tilde{P}(t,T) dt + \\sigma_T(t,r(t)) dW^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation}\nWe further remark that, in principle, we could start by defining a new measure $\\tilde{\\mathbb{P}}$ to get rid of the additional drift due to impact. Just rewrite the dynamics of the impacted zero-coupon bond as\n\\begin{equation*}\nd \\tilde{P}(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) \\left( \\frac{J_T(t)}{\\sigma_T(t,r(t))} dt  + d W^{\\mathbb{P}}(t) \\right).\n\\end{equation*}\nThis suggests to define\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{P}}}{d \\mathbb{P}} = \\exp \\left\\{ \\int_0^t \\frac{J_T(s)}{\\sigma_T(s,r(s))} d W^\\mathbb{P}(s) - \\frac{1}{2} \\int_0^t \\left(\\frac{J_T(s)}{\\sigma_T(s,r(s))} \\right)^2 ds \\right\\}.\n\\end{equation*}\nThe impacted bond under this measure would follow the dynamics\n\\begin{equation}\nd \\tilde{P}(t,T) = \\mu_T(t,r(t)) dt + \\sigma_T(t,r(t)) dW^{\\tilde{\\mathbb{P}}}(t).\n\\label{impacted_bond_under_impacted_P}\n\\end{equation}\nAt this point, $\\tilde{\\mathbb{Q}}$ can be defined from $\\tilde{\\mathbb{P}}$ by using the classic market price of risk $\\lambda(t)$. In other words,\n\\begin{equation*}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{Q}} \\frac{d \\mathbb{Q}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\tilde{\\mathbb{P}}}{d \\mathbb{P}} \\frac{d \\mathbb{Q}}{d \\tilde{\\mathbb{P}}} = \\frac{d \\mathbb{Q}}{d \\mathbb{P}}.\n\\end{equation*}\nPutting everything together, we have the following commuting diagram\n\\[\n\\begin{tikzcd}\n\\mathbb{P} \\arrow{r}{\\lambda}  \\arrow[swap]{dr}{\\tilde{\\lambda}} \\arrow[swap]{d}{\\tilde{\\lambda}-\\lambda} & \\mathbb{Q} \\arrow{d}{\\tilde{\\lambda} - \\lambda} \\\\\n\\tilde{\\mathbb{P}} \\arrow{r}{\\lambda} & \\tilde{\\mathbb{Q}}\n\\end{tikzcd}\n\\]\nBy \\eqref{qt-p} and usual arguments it follows that discounted impacted traded prices, that is $\\{\\tilde P(\\cdot,T)\/B(t)\\}_{t\\geq 0}$, are martingales for any $0\\leq T \\leq \\hat T$ under $\\tilde{\\mathbb{Q}}$. Here $B$ is the usual money market account at time $t$ given by \n\\begin{equation}\\label{def:bank_account}\nB(t) = e^{\\int_0^t r(s) ds}.\n\\end{equation}\nWe therefore have \n\\begin{equation} \\label{discounted_impacted_ZC_bond}\n\\frac{\\tilde{P}(t,T)}{B(t)} = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{\\tilde{P}(T,T)}{B(T)} \\Bigg| \\mathcal{F}_t \\right].\n\\end{equation}\nMultiplying both sides by $B(t)$ and exploiting the boundary condition $\\tilde{P}(T,T)=1$, we obtain the fundamental equation\n\\begin{equation}\n\\tilde{P}(t,T) = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ e^{-\\int_t^T r(s) ds} \\big | \\mathcal{F}_t \\right].\n\\label{impacted_ZCB_expectation}\n\\end{equation}\n\n\\begin{remark}[Interpretation impacted real world measure]\nFrom \\eqref{impacted_bond_under_impacted_P} we observe that, financially speaking, under the impacted real world measure $\\tilde{\\mathbb{P}}$, impacted bond dynamics $\\tilde P(\\cdot,T)$ has the same dynamics as the classic bond (without price impact modeling) in \\eqref{class-b} under $\\mathbb{P}$. In particular we have that $\\tilde P(T,T)=1$ for all maturities $T$.\n\\end{remark} \n\n\\subsection{Applications to pricing of interest rate derivatives} \\label{subsec:pricing}\n\nWe start this section by remarking that the notion of arbitrage we use in our work is the classic one (see e.g. Harrison and Kreps \\cite{harrison1979martingales} or Harrison and Pliska \\cite{harrison1981martingales}), adjusted with impacted portfolios.\n\\begin{definition}[Arbitrage portfolio]\nAn \\emph{arbitrage portfolio} is an impacted self-financing portfolio $\\tilde{h}$ such that its corresponding value process $\\tilde{V}$ satisfies\n\\begin{enumerate}\n\\item $\\tilde{V}(0) = 0$ . \n\\item $\\tilde{V}(T)  \\geq 0$ $\\mathbb{P}$-a.s.\n\\item $\\mathbb{P}(\\tilde{V}(T)>0)>0$\n\\end{enumerate}\n\\end{definition}\nUsing this definition of arbitrage we show that the first fundamental theorem of asset pricing holds in our setting.\n\n\\begin{theorem}[Absence of arbitrage]\\label{thm_absence_arbitrage}\nAssume that there exists an impacted equivalent martingale measure $\\tilde{\\mathbb{Q}}$ as in \\eqref{radon_nikodym_der_Q_tilde}. Then, our impacted model is arbitrage  free.\n\\end{theorem}\nThe proof of Theorem \\ref{thm_absence_arbitrage} is given in Section \\ref{sec:proofs}.\n\nA key consequence Theorem \\ref{thm_absence_arbitrage} is that our term structure model with price impact is indeed free of arbitrage. This allows to price interest rate derivatives by taking the expectation of discounted payoffs under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$. As a benchmark example, we consider the price of an impacted Eurodollar future. In the classic context, a Eurodollar-futures contract provides its owner with the payoff (see Chapter 13.12 of \\cite{brigo2007interest})\n\\begin{equation*}\nN (1-L(S,T)),\n\\end{equation*}\nwhere $N$ denotes the notional and $L(S,T)$ is the LIBOR rate, defined as (see Chapter 1 of \\cite{brigo2007interest}, Definition 1.2.4) \n\\begin{equation}\nL(S,T) := \\frac{1 - P(S,T)}{\\tau(S,T) P(S,T)},\n\\label{libor_rate}\n\\end{equation}\nwith $\\tau(S,T)$ denoting the year fraction between $S$ and $T$. Motivated by this, we introduce the impacted counterpart of the LIBOR rate in \\eqref{libor_rate}, i.e.\n\\begin{equation} \\label{l-t}\n\\tilde{L}(S,T) := \\frac{1 - \\tilde{P}(S,T)}{\\tau(S,T) \\tilde{P}(S,T)},\n\\end{equation}\nwith $\\tau$ defined as above and the impacted zero-coupon bond in place of the classic one. This new rate $\\tilde{L}$ is interpreted as the simply-compounded rate which is consistent with the impacted bond. This corresponds to the classic LIBOR rate which is the constant rate at which one needs to invest $P(t,T)$ units of currency at time $t$ in order to get an amount of one unit of currency at maturity $T$. Then, the fair price of an impacted Eurodollar future at time $t$ is (see \\cite{brigo2007interest}, Chapter 13, eq. (13.19))\n\\begin{equation}\\label{fair_price_impacted_Eurodollar}\n\\begin{split}\n\\tilde{C}_t & = \\mathbb{E}_t^{\\tilde{\\mathbb{Q}}} [N (1-\\tilde{L}(S,T))], \\\\\n& = N \\left(1 + \\frac{1}{\\tau(S,T)} - \\frac{1}{\\tau(S,T)} \\mathbb{E}_t^{\\tilde{\\mathbb{Q}}} \\left[\\frac{1}{\\tilde{P}(S,T)} \\right] \\right),\n\\end{split}\n\\end{equation}\nwhere the discounting was left out due to continuous rebalancing (see again Chapter 13.12 of \\cite{brigo2007interest}). We will demonstrate in Section \\ref{sec:examples} how such expectation can be computed analytically provided the short rate model is simple enough as in Vasicek and Hull-White models. \n\n\\begin{remark}[Linear and nonlinear pricing equations]\nOur success in retaining analytical tractability and linearity in the pricing equation may look surprising at first. In the context of equities, pricing derivatives in presence of price impact typically leads to nonlinear PDEs. This, in turn, motivated the study of super-replicating strategies and the so-called gamma constrained strategies. Several works provide also necessary and sufficient conditions ensuring the parabolicity of the pricing equation, hence the existence and uniqueness of a self-financing, perfectly replicating strategy. We refer, for example, to Abergel and Loeper \\cite{abergel2013pricing}, Bourchard, Loeper et al. \\cite{bouchard2016almost,bouchard2017hedging} and Loeper \\cite{loeper2018option}. The point we would like to stress here is that the nonlinearity of the pricing equation is a consequence of the trading strategy having a diffusion term, or a consequence of the presence of transaction costs. In other words, under the assumption that trading strategies have bounded variation and no transaction costs are present, the pricing PDE becomes linear again. Hence, our work is actually in agreement to what can be found in the context of equities.\n\\end{remark}\n\n\\subsection{Cross price impact and impacted yield curve} \\label{subsec:yield}\nIn this section we discuss how trading a bond $P(t,T)$ impacts the yield curve. For the sake of analytical tractability, we will consider affine short-rate models, that is, those models where bond prices are of the form\n\\begin{equation}\nP(t,T) = A(t,T) e^{-B(t,T) r(t)}, \\quad 0\\leq t \\leq T, \n\\label{affine:bond_price}\n\\end{equation}\nfor some deterministic, smooth functions $A$ and $B$ and $r$ is given by \\eqref{r-sde}. The remarkable property of these models is that they can be completely characterized as in the following theorem (see, e.g., Filipovic \\cite{filipovic2009term}, Section 5.3, Brigo and Mercurio \\cite{brigo2007interest}, Section 3.2.4, Bjork \\cite{bjork1997interest} Section 3.4 and references therein). \n\n\\begin{lemma}[Characterization affine short-rate models]\nThe short rate model \\eqref{r-sde} is affine if and only if there exist deterministic, continuous functions $a,\\alpha,b,\\beta$ such that the diffusion and the drift terms in \\eqref{r-sde} are of the form\n\\begin{align*}\n\\begin{split}\n\\sigma^2(t,r) & = a(t) + \\alpha(t) r, \\\\\n\\mu(t,r) & = b(t) + \\beta(t) r,\n\\end{split}\n\\end{align*}\nand the functions $A,B$ satisfy the following system of ODEs\n\\begin{align*}\n\\begin{split}\n- \\frac{\\partial}{\\partial t}  \\ln A(t,T) & = \\frac{1}{2} a(t) B^2(t,T) - b(t) B(t,T), \\ \\ \\ A(T,T) = 1, \\\\\n\\frac{\\partial}{\\partial t} B(t,T) & = \\frac{1}{2} \\alpha(t) B^2(t,T) - \\beta(t) B(t,T) - 1, \\ \\ \\ B(T,T) = 0, \\\\\n\\end{split}\n\\end{align*}\nfor all $t \\leq T$.\n\\end{lemma}\nAs explained in \\cite{filipovic2009term}, the functions $a,\\alpha,b,\\beta$ can be further specified by observing that any non-degenerate short rate affine model, that is a model with $\\sigma(t,r) \\ne 0$ for all $t>0$, can be transformed, by means of an affine transformation, in two cases only, depending on whether the state space of the short rate $r$ is the whole real line $\\mathbb{R}$ or only the positive part $\\mathbb{R}_+$. In the first case, it must hold $\\alpha(t)=0$ and $a(t) \\geq 0$, with $b,\\beta$ arbitrary. In the second case, it must hold $a(t)=0, \\alpha(t), b(t) \\geq 0$ and $\\beta$ arbitrary.\n\nLet $\\hat{T}<+\\infty$ be some finite time horizon. The yield curve at a pre-trading time $t_0$ (i.e. before price impact effects kick in) according to classic theory of interest rates is defined by \n\\begin{equation}\nY(t,T) := P(t,T)^{-1\/T} - 1, \\quad 0\\leq t \\leq t_0,\n\\label{def:classic_yield}\n\\end{equation}\nfor all maturities $0 \\leq T \\leq \\hat T$. Next, we consider the impacted bond dynamics \n\\begin{equation*}\nd \\tilde{P}(t,T) = d P(t,T) - J_T(t) dt, \n\\end{equation*}\nwhere $J_T$ was defined in \\eqref{impact_density}. Recall that the dynamics of $r(t)$ is given in \\eqref{r-sde}. Applying Ito's formula on $P(t,T)$ in  \\eqref{affine:bond_price} we get\n\\begin{multline*}\nd P(t,T) = e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2(t,r(t)) + \\\\\n- A(t,T) B(t,T) \\mu(t,r(t)) \\bigg] dt - \\sigma(t,r(t)) B(t,T) A(t,T) e^{-B(t,T) r(t)} d W^{\\mathbb{P}}(t). \\\\\n\\end{multline*}\nFrom this equation, we readily extract the drift and the diffusion of the zero-coupon bond with maturity $T$: \n\\begin{align} \\label{affine:drift_vol_zero_coupon_bond}\n\\begin{split}\n\\mu_T(t,r(t)) &:= e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2(t,r(t))  \\\\\n& \\quad - A(t,T) B(t,T) \\mu(t,r(t)) \\bigg], \\\\\n\\sigma_T(t,r(t)) & := - \\sigma(t,r) A(t,T) B(t,T) e^{-B(t,T) r(t)}.\n\\end{split}\n\\end{align}\nNext, we consider the effect of an agent trading on the bond with maturity $T$ on a bond which is not traded by the agent with maturity $S$. We call this effect the \\emph{endogenous cross-impact} on the bond with maturity $S$. Recall that in this case the dynamics of the $S$-bond is given by\n\\begin{equation} \\label{sde:impacted_bond_S}\nd \\tilde{P}(t,S) = \\mu_S(t,r(t)) dt + \\sigma_S(t,r(t)) d W^{\\mathbb{P}}(t), \n\\end{equation}\nwhere the coefficients $\\mu_S$ and $\\sigma_S$ are given by analogous formulas to \\eqref{affine:drift_vol_zero_coupon_bond}. Since we are trading the $T$-bond only, $J_S$ in \\eqref{impact_density} will be identically equal to zero. Hence, the definition of the impacted market price of risk \\eqref{def:lambda_tilde} implies the following relationship\n\\begin{equation*}\n\\frac{\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t)}{\\sigma_T(t,r(t))} = \\frac{\\mu_S(t,r(t)) - r(t) \\tilde{P}(t,S)}{\\sigma_S(t,r(t))}.\n\\end{equation*}\nThis equation tells us how the drift the $S$-bond has to change in order to avoid arbitrage. That is, this equation describes the \\emph{cross-price impact}. Specifically we have\n\\begin{equation*}\n\\mu_S(t,r(t)) = \\frac{\\sigma_S(t,r(t))}{\\sigma_T(t,r(t))} \\left[\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t)\\right] + r(t) \\tilde{P}(t,S).\n\\end{equation*}\nSubstituting this drift in \\eqref{sde:impacted_bond_S} we get \n\\begin{align}\\label{sde:cross_impacted_bond_S}\n\\begin{split}\nd \\tilde{P}(t,S) &= r(t) \\tilde{P}(t,S) dt + \\frac{\\sigma_S(t,r(t))}{\\sigma_T(t,r(t))} \\left[\\mu_T(t,r(t)) - r(t) \\tilde{P}(t,T) - J_T(t) \\right] dt \\\\\n& \\quad + \\sigma_S(t,r(t)) d W^{\\mathbb{P}}(t). \\\\\n\\end{split}\n\\end{align}\nFinally, we define the impacted yield curve for all $t_0 \\leq T \\leq \\hat T$ as follows: \n\\begin{equation} \\label{def:impacted_yield}\n\\tilde{Y}(t,T) := \\tilde{P}(t, T)^{-1\/T} - 1.\n\\end{equation}\n\n\\begin{remark}[Cross impacted bonds at maturity]\\label{rem:cross_impacted_bonds_at_maturity}\nWe have shown in \\eqref{impacted_bond_differential_form} that according to our model $\\tilde P(T,T) =1$. However, we should also ensure that all cross-impacted bonds with maturity $S \\not = T$ reach value $1$ at their  maturities. This of course, would make the model much more involved and we may lose tractability. \n\\end{remark}\n\n\\subsection{Coupon bonds}\\label{subsec:coupon}\nIt is worth recalling that the zero-coupon bond $P(t,T)$ is rarely traded. In practice, its price is derived using some bootstrapping procedure applied, for instance, to coupon bonds. In the classic theory, coupon bonds are defined as\n\\begin{equation*}\nB(t,T) = \\sum_{i=1}^n c_i P(t,T_i) + N P(t,T_n),\n\\end{equation*}\nwhere $N$ denotes the reimbursement notional, $(c_i,T_i)_{i=1}^n$ are the coupons and the maturities at which the coupons are paid, respectively. In order to determine an expression for the impacted coupon bond, we start from its cash flow\n\\begin{equation*}\nC(t) := \\sum_{i=1}^n c_i D(t,T_i) + N D(t,T_n),\n\\end{equation*}\nwhere $D(t,T)$ is the stochastic discount factor defined by\n\\begin{equation*}\nD(t,T) := e^{- \\int_t^T r(s) ds},\n\\end{equation*}\nwhere $r$ is given by \\eqref{r-sde}.\nThen, we define the impacted coupon bond as the expectation of this cash flow under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$ (see \\eqref{radon_nikodym_der_Q_tilde}):\n\\begin{equation*}\n\\tilde{B}(t,T) := \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[C(t) \\right].\n\\end{equation*}\nSubstituting the expression of $C$ immediately yields\n\\begin{align}\\label{impacted_coupon_bond_linear_combination}\n\\begin{split}\n\\tilde{B}(t,T) & = \\sum_{i=1}^n c_i \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ D(t,T_i) \\right] + N \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ D(t,T_n) \\right] \\\\\n& = \\sum_{i=1}^n c_i \\tilde{P}(t,T_i) + N \\tilde{P}(t,T_n),\n\\end{split}\n\\end{align}\nwhere $\\tilde{P}(\\cdot, T_i)$ is the (directly) impacted price of a zero-coupon bond as defined in \\eqref{impacted_bond}. Note that \\eqref{impacted_coupon_bond_linear_combination} gives the price of the impacted coupon bond in terms of impacted zero-coupon bonds. Since zero-coupon bonds are not always traded, we would like to get a direct pricing formula for impacted coupon bonds. Let $\\{v_{T_i}\\}_{i=1}^n$ be admissible trading speeds on zero-coupon bonds with maturities $\\{T_i\\}_{i=1}^n$ as defined in Section \\ref{sec:main}, that is $v_{T_i} \\in \\mathcal A_{T_i}$ for any $i=1,...n$.\nFrom \\eqref{impacted_bond} and \\eqref{impacted_coupon_bond_linear_combination} we get\n\\begin{subequations}\n\\begin{align*}\n\\tilde{B}(t,T) & = \\sum_{i=1}^n c_i \\tilde{P}(t,T_i) + N \\tilde{P}(t,T_n) \\\\\n& = B(t,T) - \\sum_{i=1}^n c_i l(t,T_i) v_{T_i}(t) - N l(t,T_n) v_{T_n}(t)  \\\\\n& \\quad - \\sum_{i=1}^n c_i K(t,T_i) y e^{-\\rho t} - N K(t,T_n) ye^{-\\rho t} \\\\\n& \\quad - \\gamma \\int_0^t e^{-\\rho (t-s)} \\left(\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right) ds.\n\\end{align*}\n\\end{subequations}\nLet us now assume that $l=\\kappa K$ at all times and for all maturities, where $\\kappa>0$ is a constant. Then, the impacted coupon bond dynamics can be written as\n\\begin{align}\n\\begin{split}\n\\tilde{B}(t,T) & = B(t,T) - y e^{-\\rho t} \\left[\\sum_{i=1}^n c_i K(t,T_i) + N K(t,T_n) \\right] \\\\\n& \\quad - \\int_0^t e^{-\\rho (t-s)} \\kappa\\delta(s-t) \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right] ds \\\\\n& \\quad - \\gamma \\int_0^t e^{-\\rho (t-s)} \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right] ds,\n\\end{split}\n\\end{align}\nwhere $\\delta$ denotes the Dirac delta. Notice that under this assumption the impacted zero-coupon bond dynamics defined in \\eqref{impacted_bond} boils down to\n\\begin{equation}\\label{impacted_bond_simplified}\n\\tilde{P}(t,T) = P(t,T) - K(t,T) \\left[y e^{-\\rho t} + \\int_0^t e^{-\\rho (t-s)} v_T(s) \\left(\\gamma + \\kappa\\delta(s-t)\\right) ds \\right].\n\\end{equation}\nThis suggest we can define\n\\begin{equation*}\nK^B(t,T) := \\sum_{i=1}^n c_i K(t,T_i) + N K(t,T_n).\n\\end{equation*}\nand the trading speed relative to the coupon bond as\n\\begin{equation}\nv^B(t,s) := \\frac{1}{K^B(t,T)} \\left[\\sum_{i=1}^n c_i K(t,T_i) v_{T_i}(s) + N K(t,T_n) v_{T_n}(s) \\right].\n\\label{trading_speed_coupon_bond}\n\\end{equation}\nTherefore, we obtain the following price impact model for the coupon bond: \n\\begin{equation}\\label{impacted_coupon_bond_simplified}\n\\tilde{B}(t,T) = B(t,T) - K^B(t,T) \\left[y e^{-\\rho t} + \\int_0^t e^{-\\rho (t-s)} v^B(t,s) \\left(\\gamma +  \\kappa \\delta(s-t)\\right) ds \\right].\n\\end{equation}\nInterestingly, under the simplifying assumption that the functions $l$ and $K$ are equal up to some constant, we observe that the impacted zero-coupon bond $\\tilde{P}(t,T)$ in \\eqref{impacted_bond_simplified} and the impacted coupon bond $\\tilde{B}(t,T)$ in \\eqref{impacted_coupon_bond_simplified} are described by the same kind of dynamics.\n\nThis is particularly useful because, provided enough data on traded coupon bonds are available, one might attempt to use \\eqref{trading_speed_coupon_bond} and \\eqref{impacted_coupon_bond_simplified} to bootstrap the trading speeds $v_{T_i}$ relative to the zero-coupon bonds. Using the price impact model \\eqref{impacted_bond}, it would be then possible to price impacted zero-coupon bonds consistently with market data. Finally, using these impacted zero-coupon bonds as building blocks, it would be possible to price, consistently with market data, more complicated and less liquid impacted interest rate derivatives, as discussed in Section \\ref{subsec:pricing}.  \n\n\\subsection{HJM framework}\\label{subsec:HJM}\nIn this section we turn our discussion to incorporating price impact into the Heath, Jarrow and Morton framework \\cite{heath1992bond}, in order to model the forward curve. Notice that this approach, although it may look different, has some common aspects to the framework developed in Section \\ref{subsec:def}. Namely, we start by adding artificially a price impact term to the forward rate dynamics. This corresponds to adding price impact to zero-coupon bonds in Section \\ref{subsec:def}. The important difference is that, here, we are creating an impacted interest rate, which was not done in Section \\ref{subsec:def}. Then we will develop the connection between the price impact of zero-coupon bonds and the price impact term incorporated into the forward rate, in order to reveal the financial interpretation of the latter. Note that both the zero-coupon bonds and the forward rate can be used as building blocks for the whole interest rates theory. We are therefore interested in showing the connection between the two in the presence of price impact. For a thorough discussion on the HJM framework in the classic interest rate theory, we refer to Chapter 6 of the book by Filipovic \\cite{filipovic2009term}. \n\nGiven an integrable initial forward curve $T \\to \\tilde{f}(0,T)$, we assume that the impacted forward rate process $\\tilde{f}(\\cdot,T)$ is given by \n\\begin{equation}\n\\tilde{f}(t,T) = \\tilde{f}(0,T) + \\int_0^t\\left( \\alpha(s,T) + J^f(s,T)\\right) ds + \\int_0^t \\sigma(s,T) d W^{\\mathbb{P}}(s),\n\\label{impacted_forward_rate}\n\\end{equation}\nfor any $0\\leq t \\leq T$ and each maturity $T>0$. Here $W^{\\mathbb{P}}$ is a Brownian motion under the measure $\\mathbb{P}$ and $\\alpha(\\cdot,T)$, $J^f(\\cdot,T)$ and $\\sigma(\\cdot,T)$ are assumed to be progressively measurable processes and satisfy for  any $T>0$\n\\begin{align*}\n\\begin{split}\n\\int_0^T \\int_0^T (|\\alpha(s,t)| + |J^f(s,t)| )ds dt & < \\infty, \\\\\n\\sup_{s,t \\leq T} |\\sigma(s,t) | & < \\infty.\n\\end{split}\n\\end{align*}\nWhile the roles of $\\alpha(\\cdot,T)$ and $\\sigma(\\cdot,T)$ above are as in standard HJM model, the stochastic process $J^f$ represents the impact density relative to the forward rate and accounts for the fact that the forward curve is affected by the trading activity. From a modelling perspective, it plays a completely analogous role as the quantity $J_T$ defined in \\eqref{impact_density} for the impacted zero-coupon bond. In fact, in Proposition \\eqref{prop:relationship_Jf_Jp} we will determine the mathematical relationship linking these two quantities. Such relationship will allow us to understand how the forward curve is impacted by trading zero-coupon bonds. \n\nIn this framework, the impacted short rate model is given by\n\\begin{equation}\n\\tilde{r}(t) := \\tilde{f}(t,t) = \\tilde{f}(0,t) + \\int_0^t\\left( \\alpha(s,t) + J^f(s,t)\\right)  ds + \\int_0^t \\sigma(s,t) d W^{\\mathbb{P}}(s),\n\\label{impacted_short_rate}\n\\end{equation}\nand the impacted zero-coupon bond is defined as follows \n\\begin{equation}\n\\tilde{P}(t,T) = e^{-\\int_t^T \\tilde{f}(t,u) du}.\n\\label{impacted_bond_HJM}\n\\end{equation}\n\nNext we derive the explicit dynamics of $\\{\\tilde{P}(t,T)\\}_{0\\leq t\\leq T}$. The following corollary is an impacted version of Lemma 6.1 in \\cite{filipovic2009term}.\n\n\\begin{corollary}[Impacted zero-coupon bond in HJM framework]\\label{cor:impacted_zcb_HJM} \nFor every maturity $T$ the impacted zero-coupon bond defined in \\eqref{impacted_bond_HJM} follows the dynamics\n\\begin{equation}\n\\tilde{P}(t,T) = \\tilde{P}(0,T) + \\int_0^t \\tilde{P}(s,T) \\left(\\tilde{r}(s) + \\tilde{b}(s,T) \\right) ds + \\int_0^t \\tilde{P}(s,T) \\nu(s,T) d W^{\\mathbb{P}}(s),\\ \\ \\ t \\leq T,\n\\label{impaced_ZCB_HJM}\n\\end{equation}\nwhere $\\tilde{r}$ is the impacted short rate defined in \\eqref{impacted_short_rate} and \n\\begin{equation} \\label{b-v-rel}\n\\begin{aligned}\n\\nu(s,T) & := - \\int_s^T \\sigma(s,u) du, \\\\\n\\tilde{b}(s,T) & := - \\int_s^T \\alpha(s,u) du - \\int_s^T J^f(s,u) du + \\frac{1}{2} \\nu^2(s,T).\n\\end{aligned}\n\\end{equation} \n\\end{corollary}\n\nWe now show that the impact $J^f$ can be expressed in terms of the impact relative to the zero-coupon bond, and vice versa. In order to show this correspondence in terms of agent's trading speed, we need to make an additional assumption on the trading speeds on zero-coupon bonds. We assume that the price impact in the forward curve is a result of trading by one or many agents over a continuum of zero-coupon bonds with maturities $T$ and trading speeds $\\{T \\geq 0 \\, : \\,  v_T \\in \\mathcal A_T\\}$ so that \n\\begin{equation} \\label{sp-der}  \n|\\partial_T v_T(t)| <\\infty, \\quad \\textrm{for all }  0\\leq t \\leq T,  \\quad \\mathbb{P}-\\textrm{a.s.} \n\\end{equation} \nNote that this assumption in fact makes sense in bond trading, which has discrete maturities, as it claims that when there is a highly traded $T_i$-bond, you would find that also the neighbouring $T_{i-1}$, $T_{i+1}$ are liquid. Assumption \\eqref{sp-der} implies that $\\partial_T I_T(t)$ is well defined as needed in the following Proposition.  We recall that $f$ represents the unimpacted forward rate which is given by setting $J^{f} \\equiv 0$ in \\eqref{impacted_forward_rate}.\n \n\\begin{proposition}[Forward rate and zero-coupon bond price impact relation] \\label{prop:relationship_Jf_Jp}\nLet $I_T(t)$ be the overall impact defined in \\eqref{def:overall_price_impact} and $\\tilde{P}(\\cdot,T)$ the impacted zero-coupon bond price in \\eqref{impacted_bond_HJM}. Assume $\\tilde{f}(0,t) =  f(0,t)$, meaning that the initial value of the forward curve is not affected by trading. Then, the forward rate impact $J^f$ introduced in \\eqref{impacted_forward_rate} is given by\n\\begin{equation}\\label{relationship_Jf_Jp}\nJ^f(t,T) = - \\frac{\\partial}{\\partial T} \\log \\left( 1- \\frac{I_T(t)}{P(t,T)} \\right) , \\quad \\textrm{for all } 0\\leq t \\leq T \\ \\textrm{such that }\\tilde{P}(t,T) >0.  \n\\end{equation}\n\\end{proposition}\n\nThe proof of Proposition \\ref{prop:relationship_Jf_Jp} is given in Section \\ref{sec:proofs}.\n\n\\begin{remark} Note that the requirement that $\\tilde{P}(t,T)>0$ ensures that the logarithm on the right-hand side of \\eqref{relationship_Jf_Jp} is well defined, as \\eqref{bla} in the proof suggests. The proof also gives another relation between $J^{f}(\\cdot, T)$ and $I_{T}$ which always holds but is perhaps not as direct. \n\\end{remark} \n\nA well known feature of the classic HJM framework is that, under the risk neutral measure, the drift of the forward rate is completely specified by the volatility through the so called \\emph{HJM condition}. In order to understand how this condition is affected by the introduction of price impact, we will follow Theorem 6.1 of \\cite{filipovic2009term}. In particular, we have the following key result.\n\n\\begin{theorem}[HJM condition with price impact]\\label{hjm_condition_market_impact}\nLet $\\mathbb{P}$ be the real world measure under which the impacted forward rate as in \\eqref{impacted_forward_rate}. Let $\\tilde{\\mathbb{Q}} \\sim \\mathbb{P}$ be an equivalent probability measure of the form\n\\begin{equation}\\label{def:Q_tilde_rnd_HJM}\n\\frac{d \\tilde{\\mathbb{Q}}}{d \\mathbb{P}} = \\exp \\left\\{\\int_0^t \\tilde{\\gamma}(s) d W^{\\mathbb{P}}(s) - \\frac{1}{2} \\int_0^t \\tilde{\\gamma}^2(s) ds \\right\\},\n\\end{equation}\nfor some progressively measurable stochastic process $\\tilde{\\gamma}= \\{\\tilde \\gamma(t) \\}_{t\\geq 0}$ such that $\\int_0^t \\tilde{\\gamma}^2(s) ds < \\infty$, for all $t>0$, $\\mathbb P$-a.s. Then, $\\tilde{\\mathbb{Q}}$ is an equivalent (local) martingale measure if and only if \n\\begin{equation} \n\\tilde{b}(t,T) = - \\nu(t,T) \\tilde{\\gamma}(t), \\quad \\textrm{for all } 0\\leq t \\leq T. \n\\label{HJM_condition}\n\\end{equation}\nwith $\\tilde{b}(\\cdot,T)$ and $\\nu(\\cdot,T)$ defined as in \\eqref{b-v-rel}. In this case, the dynamics of the impacted forward rate under the measure $\\tilde{\\mathbb{Q}}$ is given by \n\\begin{equation}\n\\tilde{f}(t,T) = \\tilde{f}(0,T) + \\int_0^t \\left( \\sigma(s,T) \\int_s^T \\sigma(s,u) du \\right) ds + \\int_0^t \\sigma(s,T) d W^{\\tilde{\\mathbb{Q}}}(s).\n\\label{impacted_forward_rate_Q_tilde}\n\\end{equation}\nMoreover, the prices of impacted zero-coupon bonds are\n\\begin{equation}\n\\tilde{P}(t,T) = \\tilde{P}(0,T) + \\int_0^t \\tilde{P}(s,T) \\tilde{r}(s) ds + \\int_0^t \\tilde{P}(s,T) \\nu(s,T) d W^{\\tilde{\\mathbb{Q}}}(s).\n\\label{impacted_zc_bond_Q_tilde}\n\\end{equation}\n\\end{theorem}\n\nThe proof of Theorem \\ref{hjm_condition_market_impact} is given in Section \\ref{sec:proofs}.\n\nIn our context such a measure $\\tilde{\\mathbb{Q}}$ would be clearly interpreted as an impacted risk-neutral measure, completely analogous to the measure defined in \\eqref{radon_nikodym_der_Q_tilde}. In fact, the stochastic process $\\tilde{\\gamma}$ in the HJM condition \\eqref{HJM_condition} is the counterpart in the HJM framework, of the impacted market price of risk $\\tilde{\\lambda}$ defined in Section \\ref{sec:main}. Indeed, combining equations \\eqref{market_price_of_risk} and \\eqref{impaced_ZCB_HJM} we obtain for $0\\leq t\\leq T$,\n\\begin{equation*}\n\\tilde{\\lambda}(t) = \\frac{\\tilde{P}(t,T) \\left(r(t) - \\tilde{b}(t,T) \\right) - r(t) \\tilde{P}(t,T)}{\\tilde{P}(t,T) \\nu(t,T)} = - \\frac{-\\tilde{b}(t,T)}{\\nu(t,T)} = \\tilde{\\gamma}(t).\n\\end{equation*}\n\nThe HJM framework adjusted with price impact discussed in this section is therefore perfectly consistent with the price impact model for zero-coupon bonds introduced in Section \\ref{subsec:def}.\n\nWe remark once again that, in the classic theory of interest rates, the meaning of the HJM condition lies in the fact that the drift of the forward rate is constrained under the risk neutral measure. Similarly, looking at the market price of risk, we notice that a constraint is present for the drift of the zero-coupon bond. In particular, its drift, under the risk neutral measure, has to be precisely the risk free interest rate. The interesting point we would like to make here is that, once we incorporate price impact, the same kind of constraints holds, only under the newly defined impacted measure $\\tilde{\\mathbb{Q}}$.\n\nWe conclude this section by making two remarks. We first address the question of when the measure defined in Theorem \\ref{hjm_condition_market_impact} is an equivalent martingale measure, instead of just local martingale measure. The second remark concerns the Markov property of the impacted short rate. In both cases, we see that the classic results carry over to the price impact framework, thanks to the key fact that the impact component affects only the drift of the forward rate.\n\n\\begin{remark}[Impacted risk neutral measure is an EMM]\nLet $\\nu(t,T)$ be defined as in \\eqref{b-v-rel}. From Corollary 6.2 of \\cite{filipovic2009term} it follows that the measure $\\tilde{\\mathbb{Q}}$ defined in Theorem \\ref{hjm_condition_market_impact} is an equivalent martingale measure if either\n\\begin{equation*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ e^{\\frac{1}{2} \\int_0^T \\nu^2(t,T) dt }\\right] < \\infty, \\quad \\text{for all} \\ T \\geq 0, \n\\end{equation*}\nor \n\\begin{equation*}\nf(t,T) \\geq 0, \\quad \\textrm{for all } 0\\leq t\\leq T. \n\\end{equation*}\n\\end{remark}\n\n\\begin{remark}[Markov property of the short rate]\nAs pointed out in Chapter 5 of \\cite{brigo2007interest}, one of the main drawbacks of HJM theory is that the implied short rate dynamics is usually not Markovian. Here we simply remark that, since the volatility of the forward rate $\\sigma(t,T)$ is not affected by price impact, if the Markov property of the short rate $r(t)$ is ensured under the measure $\\mathbb{Q}$ when there is no trading, hence no price impact, then it is also preserved in the presence of price impact under the $\\tilde{\\mathbb{Q}}$.\n\\end{remark}\n\n\n\\section{Examples}{\\label{sec:examples}}\t\n\n\\subsection{Pricing impacted Eurodollar futures with Vasicek model}{\\label{subsec:vasicek_example}}\nIn this section we illustrate the argument outlined in Section \\ref{subsec:pricing} by computing the explicit price of a Eurodollar-futures contract when the underlying short rate follows an Ornstein-Uhlenbeck process \\cite{vasicek1977equilibrium}. The dynamics under the risk neutral measure $\\mathbb{Q}$ is given by\n\\begin{equation}\nd r(t) = k (\\theta - r(t)) dt + \\sigma d W^\\mathbb{Q}(t),\n\\label{sde:vasicek}\n\\end{equation}\nwith $k,\\theta,\\sigma$ positive parameters. The dynamics of the short rate under the real world measure $\\mathbb{P}$ can be expressed as\n\\begin{equation}\nd r(t) = k (\\theta - r(t)) dt + \\sigma (d W^\\mathbb{P}(t) - \\lambda(t) dt),\n\\label{short_rate_P_lambda}\n\\end{equation}\nwhere we highlight the classic market price of risk process $\\lambda$ defined in \\eqref{market_price_of_risk}. Another representation for $r(t)$ under $\\mathbb{P}$ is \n\\begin{equation}\nd r(t) = \\tilde{k} (\\tilde{\\theta} - r(t)) dt + \\sigma (d W^\\mathbb{P}(t) - \\tilde{\\lambda}(t) dt),\n\\label{short_rate_P_lambda_tilde}\n\\end{equation}\nwhere $\\tilde{\\lambda}$ is the impacted market price of risk defined in \\eqref{def:lambda_tilde} and $\\tilde k, \\tilde \\theta$ are positive constants. Combining the two equivalent representations \\eqref{short_rate_P_lambda} and \\eqref{short_rate_P_lambda_tilde}, we see that the following holds for any $t\\geq 0$\n\\begin{equation}\nk \\theta - k r(t) - \\sigma \\lambda(t) = \\tilde{k} \\tilde{\\theta} - \\tilde{k} r(t) - \\sigma \\tilde{\\lambda}(t).\n\\label{relationship_vasicek_pars}\n\\end{equation}\nSimilarly to what is done in the standard theory (see Brigo and Mercurio \\cite{brigo2007interest}, section 3.2.1), we assume the short rate $r(t)$ has the same kind of dynamics under the measures $\\mathbb{P}$, $\\mathbb{Q}$ and $\\tilde{\\mathbb{Q}}$, that is \n\\begin{equation}\n\\lambda(t) = \\lambda r(t), \\ \\ \\ \\tilde{\\lambda}(t) = \\tilde{\\lambda} r (t),\n\\label{lambda_lambda_t_vasicek}\n\\end{equation}\nwith $\\lambda,\\tilde{\\lambda}$ constants. The whole impact is then encapsulated in the constant $\\tilde{\\lambda}$. By plugging \\eqref{lambda_lambda_t_vasicek} into \\eqref{relationship_vasicek_pars}, we deduce\n\\begin{align}\\label{tilde_pars}\n\\begin{split}\n\\tilde{k} & = k - \\sigma (\\tilde{\\lambda} - \\lambda), \\\\ \n\\tilde{\\theta} & = \\frac{k \\theta}{k - \\sigma (\\tilde{\\lambda} - \\lambda)}. \n\\end{split}\n\\end{align}\nClearly, in order to ensure all parameters are positive, we must require\n\\begin{equation*}\nk > \\sigma (\\tilde{\\lambda} - \\lambda).\n\\end{equation*}\n\nIn this way, the short rate $r(t)$ is normally distributed under all three measures. In particular, plugging the Girsanov transformation from the measure $\\mathbb{P}$ to the measure $\\mathbb{\\tilde{Q}}$, defined in \\eqref{radon_nikodym_der_Q_tilde}, into equation \\eqref{short_rate_P_lambda_tilde}, the short rate dynamics under $\\tilde{\\mathbb{Q}}$ can be conveniently rewritten as\n\\begin{equation} \\label{vasicek_under_Q_tilde}\nd r(t) = \\tilde{k}(\\tilde{\\theta} - r(t)) dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation}\n\nSince the short rate under $\\tilde{\\mathbb{Q}}$ is Gaussian, $\\{\\int_t^T r(s) ds\\}_{t\\geq 0}$ is also a Gaussian process. At the same time, we recall the well known fact that if $X$ is a normal random variable with mean $\\mu_X$ and variance $\\sigma^2_X$, then $\\mathbb{E}(\\exp(X)) = \\exp (\\mu_X + \\frac{1}{2} \\sigma_X^2)$. Following the same argument as in (Brigo and Mercurio \\cite{brigo2007interest}, Chapters 3.2.1, 3.3.2 and Chapter 4), we can use \\eqref{vasicek_under_Q_tilde} in order to  express the impacted zero-coupon bond price as follows\n\\begin{equation*}\n\\tilde{P}(t,T) = A(t,T) e^{-B(t,T) r(t)}\n\\end{equation*}\nwhere\n\\begin{align}\\label{A_B_coeff_vasicek}\n\\begin{split}\nA(t,T) & = \\exp \\left\\{ \\left(\\tilde{\\theta} - \\frac{\\sigma^2}{2 \\tilde{k}^2} \\right) [B(t,T) - T + t] - \\frac{\\sigma^2}{4 \\tilde{k}} B^2(t,T) \\right\\}, \\\\\nB(t,T) & = \\frac{1}{\\tilde{k}} \\left(1 - e^{-\\tilde{k} (T-t)} \\right). \n\\end{split}\n\\end{align}\nHence, the key expectation needed to compute the impacted Eurodollar future fair price in equation \\eqref{fair_price_impacted_Eurodollar} is equal to\n\\begin{equation} \\label{eu-d}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{1}{\\tilde{P}(t,T)} \\right] = \\frac{1}{A(t,T)} \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[e^{B(t,T) r(t)} \\right].\n\\end{equation}\nSince $r(t)$ is normally distributed, $B(t,T) r(t)$ will be normally distributed as well with mean and variance respectively equal to (see Brigo and Mercurio \\cite{brigo2007interest}, Eq. (3.7))\n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}}[B(t,T) r(t)] & = B(t,T) \\left[r(0) e^{-\\tilde{k}t} + \\theta (1-e^{-\\tilde{k}t}) \\right], \\\\\n\\text{Var}^{\\tilde{\\mathbb{Q}}}[B(t,T) r(t)] & = B^2(t,T) \\left[ \\frac{\\sigma^2}{2 \\tilde{k}} (1 - e^{-2 \\tilde{k} t}) \\right].\n\\end{align*}\n\\end{subequations}\nTherefore in order to get the impacted price of a Eurodollar-future  contract we need to compute the expectation in the right hand side of \\eqref{eu-d} which can be written explicitly as\n\\begin{multline*}\n\\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[ \\frac{1}{\\tilde{P}(t,T)} \\right] = \\frac{1}{A(t,T)} \\times \\\\\n\\times \\exp \\left\\{ B(t,T) [r(0) e^{- \\tilde{k} t} + \\theta (1-e^{- \\tilde{k}t})] + \\frac{1}{2} B^2(t,T) \\left[ \\frac{\\sigma^2}{2\\tilde{k}} (1-e^{-2 \\tilde{k} t}) \\right] \\right\\}.\n\\end{multline*}\nThe main conclusion here is that defining the short rate under the impacted risk neutral measure preserves analytical tractability of interest rate derivatives precises. \n\n\\subsection{Pricing impacted Eurodollar futures with Hull White model}{\\label{subsec:hullwhite_example}}\nIn this section we compute the explicit price of a Eurodollar-futures contract when the underlying short rate follows a Hull White model \\cite{hull1990pricing}. We start with the classic framework where there is not price impact. In this case the short rate is given by \n\\begin{equation*}\nd r(t) = \\left[ \\theta(t) - a r(t) \\right] dt + \\sigma d W^{\\mathbb{Q}}(t),\n\\end{equation*}\nwhere $a$ and $\\sigma$ are positive constants and the function $\\theta$ is chosen in order to fit exactly the term structure of interest rates being currently observed in the market. Denoting by $P^M(0,T)$ the unimpacted market discount factor for the maturity $T$ and defining the (unimpacted) market instantaneous forward rate at time $0$ for the maturity $T$\n\\begin{equation*}\nf^M(0,T) := - \\frac{\\partial}{\\partial T} \\ln P^M(0,T),\n\\end{equation*}\nthe function $\\theta$ is given by (see e.g. Brigo and Mercurio \\cite{brigo2007interest}, Chapter 3, Eq. (3.34))\n\\begin{equation*}\n\\theta(t) = \\frac{\\partial f^M(0,t)}{\\partial T} + a f^M(0,t) + \\frac{\\sigma^2}{2 a} \\left(1-e^{-2 a t} \\right),\n\\end{equation*}\nwhere $\\frac{\\partial f^M(0,t)}{\\partial T}$ denotes the partial derivative of $f^M$ with respect to its second variable. We start by computing the price under the classic risk neutral measure $\\mathbb{Q}$. According to eq. (3.36)--(3.37) in Chapter 3 of \\cite{brigo2007interest}, the short rate is normally distributed with mean and variance respectively equal to \n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^\\mathbb{Q}[r(t) | \\mathcal{F}_s] & = r(s) e^{-a (t-s)} + \\alpha(t) - \\alpha(s) e^{-a (t-s)} \\\\\n\\text{Var}^\\mathbb{Q}[r(t) | \\mathcal{F}_s] & = \\frac{\\sigma^2}{2 a} \\left[1 - e^{-2 a (t-s)} \\right],\n\\end{align*}\n\\end{subequations}\nwhere \n\\[\n\\alpha(t) := f^M(0,t) + \\frac{\\sigma^2}{2 a^2} (1-e^{-a t})^2.\n\\]\nAs before, we notice that the integral of the short rate will be normally distributed as well, hence the price of a zero-coupon bond under the classic risk neutral measure is given by (see eq. (3.39) in Chapter 3 of \\cite{brigo2007interest}), \n\\begin{equation*}\nP(t,T) = A(t,T) e^{-B(t,T) r(t)},\n\\end{equation*}\nwhere\n\\begin{align*}\nA(t,T) &= \\frac{P^M(0,T)}{P^M(0,t)} \\exp \\left\\{B(t,T) f^M(0,t) - \\frac{\\sigma^2}{4 a} (1-e^{-2 a t}) B^2(t,T)\\right\\},\\\\\nB(t,T) &= \\frac{1}{a} \\left[1-e^{-a (T-t)} \\right].\n\\end{align*}\nMoreover, the term $B(t,T) r(t)$ is still normally distributed and we immediately have\n\\begin{subequations}\n\\begin{align*}\n\\mathbb{E}^\\mathbb{Q}[ B(t,T) r(t) | \\mathcal{F}_s] & = B(t,T) \\left(r(s) e^{-a (t-s)} + \\alpha(t) - \\alpha(s) e^{-a (t-s)} \\right), \\\\\n\\text{Var}^\\mathbb{Q}[ B(t,T) r(t) | \\mathcal{F}_s] & = B^2(t,T) \\frac{\\sigma^2}{2 a} \\left[1 - e^{-2 a (t-s)} \\right].\n\\end{align*}\n\\end{subequations}\nThis implies that the expectation we are interested in, under the classic risk neutral measure $\\mathbb{Q}$, can be written explicitly as (see Section 13.12.1 in \\cite{brigo2007interest})\n\\begin{multline*}\n\\mathbb{E}^\\mathbb{Q} \\left[ \\frac{1}{P(t,T)} \\right] = \\frac{1}{A(t,T)} \\exp \\left\\{B(t,T) \\mathbb{E}^\\mathbb{Q}[r(t)] + \\frac{1}{2} B^2(t,T) \\text{Var}^\\mathbb{Q}[r(t)] \\right\\}.\n\\end{multline*}\nNext we derive the corresponding expression under the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$ in \\eqref{radon_nikodym_der_Q_tilde}. We assume as in Section \\ref{subsec:vasicek_example} that the market price of risk and impacted market price of risk are given by\n\\[\n\\lambda(t) = \\lambda r(t), \\ \\ \\ \\tilde{\\lambda}(t) = \\tilde{\\lambda} r(t),\n\\]\nfor some constants $\\lambda,\\tilde{\\lambda}$. Using the Girsanov change of measure from $\\mathbb{Q}$ to $\\tilde{\\mathbb{Q}}$ defined in Section \\ref{subsec:rnm}, it follows that the short rate under $\\tilde{\\mathbb{Q}}$ is given by  \n\\begin{subequations}\n\\begin{align*}\nd r(t) & = \\left[ \\theta(t) - a r(t) \\right] dt + \\sigma d W^{\\mathbb{Q}}(t) \\\\\n& = \\left[\\theta(t)- a r(t) \\right] dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t) + \\sigma(\\tilde{\\lambda} - \\lambda) r(t) dt \\\\\n& = \\left[ \\theta(t) - (a - \\sigma (\\tilde{\\lambda} - \\lambda)) r(t) \\right] dt + \\sigma d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{align*}\n\\end{subequations}\nHence, we can define the impacted parameter\n\\begin{equation*}\n\\tilde{a} := a - \\sigma (\\tilde{\\lambda} - \\lambda).\n\\end{equation*}\nThe pricing formula for $\\mathbb{E}^{\\tilde {\\mathbb{Q}}}\\left[ \\frac{1}{P(t,T)} \\right]$ is then derived by following the same steps as in the classic case. Similarly to the Vasicek model, analytical tractability is preserved.\n\n\\section{Numerical results}{\\label{sec:numerical_results}}\nIn this section we give a few numerical examples for the behaviour of the yield curve under price impact in the framework of short-rate affine models, which was described in Section \\ref{subsec:yield}. In order to compute the cross price impact, we need the drift and the volatility of the zero-coupon bond. For the sake of simplicity, we assume the short rate is described by a Vasicek model \\eqref{sde:vasicek}\n\\begin{equation*}\nd r(t) = k (\\theta - r(t)) dt + \\sigma d W^\\mathbb{Q}(t),\n\\end{equation*}\nwith $k,\\theta,\\sigma$ positive parameters. Then, the drift and the diffusion coefficients of the unimpacted zero-coupon bond are given by \\eqref{affine:drift_vol_zero_coupon_bond}:\n\\begin{align*}\n\\mu_T(t,r(t)) &= e^{-B(t,T) r(t)} \\bigg[\\frac{\\partial A}{\\partial t} - A(t,T) \\frac{\\partial B}{\\partial t} r(t) + \\frac{1}{2} A(t,T) B^2(t,T) \\sigma^2& \\\\\n&\\quad - A(t,T) B(t,T) k (\\theta -r(t)) \\bigg], \\\\\n\\sigma_T(t,r(t))& = - \\sigma B(t,T) A(t,T) e^{-B(t,T) r(t)},\n\\end{align*}\nwhere the functions $A,B$ are given as in \\eqref{A_B_coeff_vasicek} and their derivatives are given by\n\\begin{equation*}\n\\frac{\\partial B}{\\partial t} = - e^{-k (T-t)}, \\ \\ \\ \\\n\\frac{\\partial A}{\\partial t} = A(t,T) \\left[ \\left(\\theta - \\frac{\\sigma^2}{2 k^2} \\right) \\left(\\frac{\\partial B}{\\partial t} + 1 \\right) - \\frac{\\sigma^2}{2 k} B(t,T) \\frac{\\partial B}{\\partial t} \\right].\n\\end{equation*}\nWe can then plug all these quantities in equation \\eqref{sde:cross_impacted_bond_S} to determine the dynamics of the cross-impacted zero-coupon bond and therefore the corresponding impacted yield. We set the following values for the parameters in \\eqref{sde:vasicek}:\n\\begin{equation*}\nk=0.20, \\quad \\theta = 0.10, \\quad \\sigma = 0.05, \\quad r_0 = 0.01. \n\\end{equation*}\nWe consider zero-coupon bonds with maturities $ \\mathbb{T} := \\left\\{1,2,5,10,15 \\right\\}$ years and assume that an agent is trading on the bond with maturity $T=5$ years. All the other zero-coupon bonds experience cross price impact during the trading period. We fix the execution time horizon to be $\\tau = 10$ days. All bonds are simulated over the time interval $[0, 9\\ \\text{months}]$, discretized in $N=365$ subintervals with time step $\\Delta t = 1\/365$. The short rate $r$ defined in \\eqref{sde:vasicek} is simulated via Euler-Maruyama scheme. Since we are going to describe the average behaviour of the yield curve under market impact, we also set the number of Monte Carlo simulations to $M=10.000$. As we shall see below in the detailed algorithm, for each realization of the short rate, we will have a corresponding impacted yield curve. The idea is then to plot the average of such curves.\n\nFor the sake of simplicity, we discuss the benchmark trading strategy\n\\begin{equation}\\label{def:benchmark_strategy}\nv_T(s):=\n\\begin{cases} c, & \\mbox{if } s \\leq \\tau \\\\ 0, & \\mbox{otherwise }\n\\end{cases}\n\\end{equation}\nwith $c$ some positive constant if we buy, negative if we sell. In our simulations we choose $c=2$. The transient impact defined in \\eqref{def:transient_impact} reads as\n\\begin{equation}\n\\Upsilon_T^v(t) = y e^{-\\rho t} + \\gamma e^{-\\rho t} \\int_0^t e^{\\rho s} c \\mathbbm{1}_{s \\leq \\tau} ds,\n\\label{benchmark_transient_impact}\n\\end{equation}\nwhere the  parameters are set to\n\\begin{equation*}\n\\rho = 2, \\ \\ \\ \\gamma = 1, \\ \\ \\ y = 0.01.\n\\end{equation*}\nThe functions $l,K$ introduced in \\eqref{impacted_bond} are assumed to be of the form\n\\begin{equation*}\nl(t,T) = \\kappa \\left(1-\\frac{t}{T}\\right)^\\alpha, \\ \\ \\ K(t,T) =  \\left(1-\\frac{t}{T}\\right)^\\beta\n\\end{equation*}\nwith $\\kappa \\geq 0, \\alpha,\\beta \\geq 1$. In particular, we choose\n\\begin{equation*}\n\\alpha = 1, \\ \\ \\beta=1, \\ \\ \\kappa =0.01. \n\\end{equation*}\nFollowing \\eqref{impacted_bond_differential_form}, the price of the impacted bond in $T=5$y is \n\\begin{equation*}\n\\tilde{P}(t,T) = P(t,T) + \\int_0^t J_T(s) ds,\n\\end{equation*}\nwhere $J_T$, which was defined in \\eqref{impact_density}, is specified to be\n\\begin{equation*}\nJ_T(t) = - \\frac{\\kappa}{T} v_T(t) + \\left(1-\\frac{t}{T} \\right) \\left[-\\rho \\Upsilon_T^v + v_T(t) \\right] - \\Upsilon_T^v(t)\n\\end{equation*}\nThe algorithm we implemented to simulate the impacted yield curve consists of the following steps.\n\\begin{steps}\n\\item Simulate a path of the short rate $r(t)$ given in \\eqref{sde:vasicek} for $t \\in [0,9\\ \\text{months}]$. \n\\item Compute the unimpacted zero-coupon bond price $P(t,T)$ for the trading maturity  $T=5$ years using equation \\eqref{affine:bond_price} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the unimpacted yield $Y(t,T)$ by plugging $P(t,T)$ in \\eqref{def:classic_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the (directly) impacted zero-coupon bond $\\tilde{P}(t,T)$ using \\eqref{impacted_bond_differential_form} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the (directly) impacted yield $\\tilde{Y}(t,T)$ by plugging $\\tilde{P}(t,T)$ into \\eqref{def:impacted_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item For all other maturities $S=1,2, 10, 15$ years, compute the cross impacted zero-coupon bond price $\\tilde{P}(t,S)$ using equation \\eqref{sde:cross_impacted_bond_S} for $t \\in [0,9\\ \\text{months}]$.\n\\item Compute the cross impacted yield $\\tilde{Y}(t,S)$ by plugging $\\tilde{P}(t,S)$ into \\eqref{def:impacted_yield} for $t \\in [0,9\\ \\text{months}]$.\n\\item Repeat these steps $M=10.000$ times and compare the average of $Y(t,T)$ with the average of $\\tilde{Y}(t,T)$.\n\\end{steps}\n\nIn Figure \\ref{fig:imp_unimp_yield_average} we visualize for all maturities the average classic yield $\\mathbb{E}[Y(t,T)]$ versus the average impacted yield $\\mathbb{E}[\\tilde{Y}(t,T)]$ at times $t=5$ days (middle of trading), $t=11$ days (right after trading is ended) and $t=270$ days (after $9$ months). \n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{imp_unimp_yield_average_intermediate.png}\n\\caption{Trading zero-coupon bond with maturity $T=5$ years. Average unimpacted yield curve and average impacted yield curve in the middle of trading (top left panel), right after trading is concluded (top right panel) and after nine months (bottom panel).}\n\\label{fig:imp_unimp_yield_average}\n\\end{figure}\nIn the top panels we see that the yield has decreased over all maturities as result of trading. This is consistent with the fact that a buy strategy of bonds pushes their prices up due to price impact, hence the yield decreases. Clearly, the almost parallel shift of the yield curve is a just a consequence of the very simple (constant) trading strategy we defined in equation \\eqref{def:benchmark_strategy}. We expect to observe much more complicated behaviours when implementing more sophisticated strategies. In the bottom panel, instead, we observe that, roughly nine months after performing the trades, the two yield curves pretty much coincide. This is due to the transient component in the price impact model, which induces impacted yield curve to converge to its classic counterpart as time goes by. When analysing price impact due to zero-coupon bond trading, one aspect that certainly can't be ignored is the special nature of the assets we are trading. Unlike what happens with stocks, the time evolution of zero-coupon bonds is constrained, specifically by the fact that they must reach value $1$ at maturity. It therefore appears that two fundamental forces are in play: the intrinsic \\emph{pull to par} effect, which makes both the impacted and unimpacted bond price go to $1$, hence the corresponding yields to $0$, and the \\emph{price impact} effect, which induces the bond price to first increase (if we buy) or decrease (if we sell), and then revert back to its unimpacted value. Interestingly, when trading stocks, it will take the impacted asset forever to converge to the unimpacted counterpart, as the transient impact converges to $0$ as time $t$ goes to infinity. When trading bonds, though, this convergence occurs in finite time. In order to better understand the role played by price impact, in Figure \\ref{fig:imp_classic_bonds} we compare directly the behaviour of the impacted bond $\\tilde{P}(t,T)$ and of the classic bond $P(t,T)$ for different maturities $T$.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{imp_classic_bonds.png}\n\\caption{Trading bond with maturity $T=5$ years. Averaged impacted zero-coupon bond vs averaged classic zero-coupon bond for maturities $1$ year (top left panel), $2$ years (top right panel), $5$ years (middle left panel), $10$ years (middle right panel), $15$ (bottom panel). All curves are observed over the interval $[0, 1\\ \\text{year}]$.}\n\\label{fig:imp_classic_bonds}\n\\end{figure}\nWe observe that, over one year, the pull-to-par effect is somehow stronger than the transient impact effect in bonds with short maturity $(T=1,2)$. By this, we mean that the unimpacted and impacted bonds meet at, or very close to, maturity. \\footnote{It can be observed that the price of the cross-impacted zero-coupon bond with maturity $S=1$ year is not $1$ at expiration, as it should be, but slightly higher (top left panel). This is not a numerical error, but rather a consequence of our model not being able to ensure the cross-impacted bonds reach value precisely $1$ at their respective maturities. See Remark \\ref{rem:cross_impacted_bonds_at_maturity}.} For bonds with long maturity ($T=5,10,15$), instead, the transient effect is prominent. This causes the impacted bond curve and the unimpacted bond curve to cross each other significantly before their maturity. In fact, we can numerically compute the first instant the two curves meet and we observe that the longer the maturity, the sooner this happens. This is illustrated in Figure \\ref{fig:first_hitting_time}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{first_hitting_time.png}\n\\caption{Trading bond with maturity $T=5$ years. First instant (in days) impacted bond curve and unimpacted bond curve cross for maturities $5,10,15$ years. All curves are observed over the interval $[0, 1\\ \\text{year}]$.}\n\\label{fig:first_hitting_time}\n\\end{figure}\n\nThe interplay between the cross price impact effect, averaged over $10.000$ realizations, and the pull to par effect is demonstrated in Figure \\ref{fig:average_imp_bond_1_rho} for the price of a zero-coupon bond with maturity $S=1$ year when trading a bond with maturity $T=5$ years. Trading takes place on the first 10 days of the year, while the time scale in the graph is of one year. We illustrate this effect for various values of the transient impact parameter $\\rho$ in equation \\eqref{benchmark_transient_impact}.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{average_imp_bond_1_rho.png}\n\\caption{Averaged cross price impact effect vs. pull to par effect over $10.000$ realizations is demonstrated for the price of a bond with maturity $S=1$ year when trading a bond with maturity $T=5$ years, for various values of $\\rho$. Trading takes place on the first 10 days of the year, while the time scale in the graph is of one year.}\n\\label{fig:average_imp_bond_1_rho}\n\\end{figure}\nIt can be observed that the higher $\\rho$, the more aggressively the price is \"pulled down\" close to the original price before the trades. At the beginning, far from maturity, the transient impact component dominates and the price decreases. After some time, though, the bond intrinsic nature takes over and the price starts to increase.\n\nAnother phenomenon which is revealed in our framework is the interplay between the mean reversion of the short rate model and the price impact. Recall that in Section \\ref{subsec:vasicek_example} we found that the mean reversion speed $k$ under the measure $\\mathbb{Q}$ and the mean reversion $\\tilde{k}$ under the price-impacted measure $\\tilde{\\mathbb{Q}}$ are linked by \\eqref{relationship_vasicek_pars} as follows\n\\[\n\\tilde{k} = k - \\sigma(\\tilde{\\lambda}-\\lambda),\n\\]\nwith $\\tilde{\\lambda}, \\lambda$ representing the impacted market price of risk and the classic market price of risk respectively. We stress that the higher $k$, the faster the short rate $r$ under $\\mathbb{Q}$ and its counterpart under $\\tilde{\\mathbb{Q}}$ converge to their respective stationary distributions. At the same time, since the variance of the stationary distribution is $\\sigma^2\/(2k)$, large values of $k$ reduce the overall variance of the model, thereby making the two types of rates that we consider closer to each other. This, in turn, implies that after a long time ($T=10,15$ years) the zero-coupon bond $P$ and impacted zero-coupon bond $\\tilde{P}$, hence their yields, will be closer to each other. Conversely, if $k$ is small, the two short rates are quite far from each other and the overall variance of the model is large. Furthermore, looking again at \\eqref{relationship_vasicek_pars}, we notice that the larger $k$, the less significant the impact component $-\\sigma (\\tilde{\\lambda} - \\lambda)$, and vice versa. In a way, the speed of mean reversion works in an opposite direction to the price impact. We demonstrate this in Figure \\ref{fig:yield_curve_mean_reversion} for $k=0.01$ (top panel) and for $k=0.20$ (bottom panel). As above, we trade the zero-coupon bond with maturity $T=5$ years, trading occurs for the first $10$ days and the yields are observed after $9$ months. The difference in behaviour is evident for long maturities $(T=5,10,15)$. While in the bottom panel unimpacted yield and impacted yield are really close to each other (as in Figure \\eqref{fig:imp_unimp_yield_average}, right panel), in the top panel the distance between the two yields is rather significant.\n\\begin{figure}[H]\n\\centering\n\\includegraphics[scale=.50]{yield_curve_mean_reversion.png}\n\\caption{Impacted and unimpacted yield curves for $k=0.01$ (top panel) and for $k=0.20$ (bottom panel) when trading zero-coupon bond with maturity $T=5$ years. Trading occurs during the first $10$ days. Yield curves are observed after nine months.}\n\\label{fig:yield_curve_mean_reversion}\n\\end{figure}\n\n\\section{Optimal execution of bonds in presence of price impact }\\label{sec:optexec}\nIn this section we consider a problem of an agent who tries to liquidate a large inventory of $T$-bonds within a finite time horizon $[0,\\tau]$ where $\\tau < T$. We assume that the agent's transactions create both temporary and transient price impact and that the performance of the agent is measured by a revenue-cost functional that captures the transaction costs which result by price impact, and the risk of holding inventory for long time periods.  Our optimal execution framework is closely related to the framework which was proposed for execution of equities in Section \\ref{subsec:def} of \\cite{neuman2020optimal}. The main difference between the two frameworks is that in our framework the price impact has to vanish at the bond's maturity in order to satisfy the boundary condition $\\tilde P(T,T)=1$. \n\nLet $T>0$ denote the bond's maturity. We assume that the unimpacted bond price $P(\\cdot,T)$ is given by \\eqref{class-b} and we consider the canonical decomposition $P(\\cdot,T)= A(\\cdot,T) + \\bar{M}(\\cdot,T)$, where \n\\begin{equation*}\nA(t,T) := \\int_0^t \\mu_T(s,r(s)) ds, \\quad 0\\leq t\\leq T, \n\\end{equation*}\nis a predictable finite-variation process and  \n\\begin{equation*}\n\\bar{M}(t,T) := \\int_0^t \\sigma_T(s,r(s)) d W^{\\mathbb{P}}(s) , \\quad 0\\leq t\\leq T, \n\\end{equation*}\nlocal martingale. We assume that the coefficients $\\sigma_{T},\\mu_{T}$ in \\eqref{class-b} are such that we have \n\\begin{equation} \\label{def: mathcal_h_squared} \n\\mathbb{E}[\\langle \\bar{M}(\\cdot,T) \\rangle_\\tau] + \\mathbb{E} \\left[ \\left( \\int_0^\\tau |d A (\\cdot,T)| \\right)^2 \\right] < \\infty.\n\\end{equation} \nIn this case we say that a bond price $\\{P(t,T)\\}_{t \\in [0,T]}$ is in $\\mathcal{H}^2$.\n\nThe initial position of the agent's inventory is denoted by $x>0$ and the number of shares the agent holds at time $t\\in [0,\\tau]$ is given by \n\\begin{equation} \\label{def:X}\nX^{v_T}(t)\\triangleq x-\\int_0^t v_T(s)ds\n\\end{equation}\nwhere $\\{v_T(t)\\}_{t \\in [0,\\tau]}$ denotes the trading speed. We say that the trading speed is admissible if it belongs to the following class of admissible strategies \n\\begin{equation} \\label{def:admissset} \n\\mathcal A \\triangleq \\left\\{ v_T \\, : \\, v_T \\textrm{ progressively measurable s.t. } \\mathbb E\\left[ \\int_0^\\tau v^2_T(s) ds \\right] <\\infty \\right\\}.\n\\end{equation} \nWe assume that the trader's trading activity causes price impact on the bond's price as described by $\\{\\tilde P(t,T)\\}_{t \\in [0,T]}$ in \\eqref{impacted_bond}. \n\nAs in Section 2 of \\cite{neuman2020optimal}, we now suppose that the trader's optimal trading objective is to unwind her initial position $x>0$ in the presence of temporary and transient price impact through maximizing the following performance functional \n\\begin{equation}  \\label{j-fun}\n\\begin{aligned}\n\\mathcal{J}(v) := \\mathbb{E} \\bigg [\\int_0^\\tau \\bigg(P(t,T) - K(t,T) \\Upsilon_T^v(t) \\bigg) v_T(t) dt - \\int_0^\\tau l(t,T) v_T^2(t) dt + X_T^v(\\tau) P(\\tau,T)  \\\\\n- \\phi \\int_0^\\tau (X_T^v(t))^2 dt - \\varrho (X_T^v(\\tau))^2 \\bigg].\n\\end{aligned}\n\\end{equation} \nThe first, second and third terms in $\\mathcal J$ represent the trader's terminal wealth, meaning the final cash position including the accrued trading costs induced by temporary and transient price impact, as well as the remaining final risky asset position's book value. The fourth and fifth terms, instead, account for the penalties $\\phi,\\varrho>0$ on the trader's running penalty (i.e. the risk aversion term) and the penalty of holding any terminal inventory, respectively. \n\nSince $T$ is fixed, for the sake of readability we will omit the subscripts $T$ for the rest of this section. \nThe main result of this section is the derivation of the unique optimal admissible strategy, namely \n\\begin{equation}\n\\mathcal{J}(v) \\to \\max_{v \\in \\mathcal{A}}.\n\\label{opt_stoch_control}\n\\end{equation}\nand exhibiting an explicit expression for the optimal trading strategy. We define\n\\begin{equation}\nA(t) :=\n\\left(\n\\begin{matrix}\n0 & 0 & -1 & 0 \\\\\n0 & -\\rho & \\gamma & 0 \\\\\n-2 \\phi \\Lambda(t) & \\begin{matrix} \\rho K(t,T) \\Lambda(t)\\\\ \\hfill{} - \\Lambda'(t) K(t,T) - \\Lambda(t) \\partial_t K(t,T) \\end{matrix}  & 0 & \\Lambda'(t) + \\rho \\Lambda(t) \\\\\n0 & 0 & K(t,T) \\gamma & \\rho\n\\end{matrix}\n\\right),\n\\label{def:time_dep_matrix_A}\n\\end{equation}\nwhere\n\\begin{equation}\n\\Lambda(t) := \\frac{1}{2 l(t,T)}. \n\\label{def:Lambda_optimal_exe}\n\\end{equation}\nNote that $\\Lambda(t)$ is well defined for $0\\leq t \\leq \\tau$ since $l(t,T)>0$ on this interval by \\eqref{l-pos}. Let $\\Phi$ be the fundamental solution of the matrix-valued ordinary differential equation\n\\begin{align}\\label{matrix_ODE}\n\\begin{split}\n\\frac{d}{dt} \\Phi(t) & = A(t) \\Phi(t), \\\\\n\\Phi(0) & = \\text{Id}.\n\\end{split}\n\\end{align}\nLet us define the matrix\n\\begin{equation}\\label{def:psi_matrix}\n\\Psi(t,\\tau) := \\Phi^{-1}(\\tau) \\Phi(t). \n\\end{equation}\nWe also define the vector $G$:\n\\begin{align}\\label{def:G_components}\n\\begin{split}\nG^1(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{11}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Psi^{21}(t,\\tau) - \\Psi^{31}(t,\\tau) ,\\\\\nG^2(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{12}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Psi^{22}(t,\\tau) - \\Psi^{32}(t,\\tau)  ,\\\\\nG^3(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{13}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)}\\Psi^{23}(t,\\tau) - \\Psi^{33}(t,\\tau) , \\\\\nG^4(t,\\tau) & := \\frac{\\varrho}{l(\\tau,T)} \\Psi^{14}(t,\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)}\\Psi^{24}(t,\\tau) - \\Psi^{34}(t,\\tau) . \n\\end{split}\n\\end{align}\nNext, we define the process\n\\begin{equation}\n\\Gamma^{\\hat{v}}(t) := \\frac{\\Lambda'(t)}{\\Lambda(t)} \\left(P(t,T) + \\tilde{M}(t) - 2 \\phi \\int_0^t X^{\\hat{v}}(u) du \\right),\n\\label{def:Gamma_quantity}\n\\end{equation}\nwhere $\\tilde{M}$ is the square integrable martingale \n\\begin{equation}\\label{def:square_int_mart_M_tilde}\n\\tilde{M}(s) := \\mathbb{E}_s \\bigg[2 \\phi \\int_0^\\tau X^{\\hat{v}}(u) du + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg], \n\\end{equation}\nand $\\mathbb{E}_t$ denotes the expectation conditioned on the filtration $\\mathcal{F}_t$ for all $t \\in [0,\\tau]$. Finally we define the following functions on $0\\leq t \\leq \\tau$,\n\\begin{align}\\label{def:v_terms}\n\\begin{split}\nv_0(t,\\tau) & :=  \\left(1-\\frac{G^4(t,\\tau) \\Psi^{43}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} \\right)^{-1}, \\\\\nv_1(t,\\tau) & := \\left(\\frac{G^4(t,\\tau) \\Psi^{41}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} \\right), \\\\\nv_2(t,\\tau) & := \\left(\\frac{G^4(t,\\tau) \\Psi^{42}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\right) ,\\\\\nv_3(t,\\tau) & := \\frac{G^4(t,\\tau)}{G^3(t,\\tau)}.\n\\end{split}\n\\end{align}\nIn order for the optimal strategy to be well defined, we will need additional assumptions. Note that if $l, K$ are positive constants these assumptions translate to Assumption 3.1 and Lemma 5.5 in \\cite{neuman2020optimal}. \n\\begin{assumption} \\label{ass-opt}  \n\\begin{enumerate}[label=(A.\\arabic*), ref=A.\\arabic*] We assume that the following hold:\n\\item \\label{A.1}\n\\[\n\\inf_{0\\leq t \\leq \\tau} |G^4(t,\\tau) \\Psi^{43}(t,\\tau) - G^3(t,\\tau) \\Psi^{44}(t,\\tau)| > 0,\n\\] \n\\item \\label{A.2}\n\\[\n\\sup_{0 \\leq t \\leq \\tau} |\\Psi^{4j}(t,\\tau)| < \\infty, \\ \\ \\sup_{0 \\leq t \\leq \\tau} |G^j(t,\\tau)| < \\infty, \\ \\ \\ j \\in \\left\\{1,2,3,4\\right\\}\n\\]\n\\item \\label{A.3}\n\\[\n\\inf_{0\\leq t\\leq \\tau}  |\\Psi^{44}(t,\\tau)| >0, \\quad  \\inf_{0\\leq t\\leq \\tau} | G^3(t,\\tau)|>0.\n\\]\n\\end{enumerate}\n\\end{assumption}\n\\begin{remark}\nAt this point we stress the fact that the conditions in Assumption \\ref{ass-opt} are not very general, however the purpose of this section is to show how to incorporate optimal execution into the impacted bonds framework. Future work may improve the theoretical results on this topic. \n\\end{remark} \nNext we present the main result of this section, which derives the unique optimal trading speed.\n\n\\begin{theorem}[Optimal trading strategy]\n\\label{linear_feedback_form}\nUnder Assumption \\ref{ass-opt}, there exists a unique optimal strategy $\\hat{v} \\in \\mathcal{A}$ which maximises \\eqref{opt_stoch_control} and it is given by the following feedback form \n\\begin{align}\\label{optimal_strategy_linear_feedback}\n\\begin{split}\nv(t) & = v_0(t,\\tau) \\Bigg( v_1(t,\\tau) X^v(t) + v_2(t,\\tau) \\Upsilon^v(t) \\\\\n& \\qquad \\qquad + v_3(t,\\tau) \\mathbb{E}_t \\left[ \\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{v}(s)) ds \\right] \\\\\n&  \\qquad \\qquad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{v}(s)) ds \\right] \\Bigg), \n\\end{split}\n\\end{align}\nfor all $t \\in (0,\\tau)$.\n\\end{theorem}\nThe proof Theorem \\ref{linear_feedback_form} is given in Section \\ref{sec:proof_linear_feedback}.\n\n\\section{Proofs of the results from Section \\ref{sec:main}}{\\label{sec:proofs}}\n\n\\begin{proof} [Proof of Theorem \\ref{thm_impacted_market_price}]\n\nWe adapt the argument by Bjork in Section 3.2 of \\cite{bjork1997interest} to our case. We fix two maturities $T$ and $S$, and we consider a portfolio $V$ consisting of $S$-bonds and $T$-bonds. \nWe further assume that both bonds are traded with admissible trading speeds $v_T$ and $v_S$ which correspond by \\eqref{impact_density} to impact densities $J_T$ and $J_S$.\n\nFrom \\eqref{class-b} and \\eqref{impacted_bond_differential_form} we can write the dynamics of the impacted bonds as follows: \n\\begin{equation}\\label{dyn_imp_bonds}\n\\begin{aligned}\nd \\tilde{P}(t,T) = & \\mu_T(t,r(t)) dt + J_T(t) dt + \\sigma_T(t,r(t)) d W^\\mathbb{P}(t), \\\\\nd \\tilde{P}(t,S) = & \\mu_S(t,r(t)) dt + J_S(t) dt + \\sigma_S(t,r(t)) d W^\\mathbb{P}(t).\n\\end{aligned}\n\\end{equation}\nLet $\\tilde{h}_T,\\tilde{h}_S$ by locally bounded predictable processes representing the weights of the $T$ and $S$ bonds, respectively. We denote by $\\tilde{V}(t)$ the portfolio value process, i.e.\n\\begin{equation*}\n\\tilde{V}(t) \\equiv \\tilde{V}(t;\\tilde{h}) := \\tilde{h}_T(t) \\tilde{P}(t,T) + \\tilde{h}_S(t) \\tilde{P}(t,S).\n\\end{equation*}\nSince, by assumption, the impacted-portfolio is self-financing, it holds at any time $t$ (see Definition \\ref{def:self_financing})\n\\begin{equation*}\nd \\tilde{V}(t;\\tilde{h}) = \\tilde{h}_T(t) d \\tilde{P}(t,T) + \\tilde{h}_S(t) d \\tilde{P}(t,S).\n\\end{equation*}\nIt is convenient to define the relative (impacted) weights\n\\begin{equation*}\n\\alpha_{T_i}(t) := \\frac{\\tilde{h}_{T_i}(t) \\tilde{P}(t,T_i)}{\\tilde{V}(t;\\tilde{h})},\\ \\ T_i \\in \\left\\{T,S\\right\\},\n\\end{equation*}\nWe conclude that if the impacted portfolio is self financing, then\n\\begin{equation}\n\\frac{d \\tilde{V}(t)}{\\tilde{V}(t)} = \\alpha_T(t) \\frac{d \\tilde{P}(t,T)}{\\tilde{P}(t,T)} + \\alpha_S(t) \\frac{d \\tilde{P}(t,S)}{\\tilde{P}(t,S)}.\n\\label{self_fin_imp_portfolio_relative}\n\\end{equation}\nIn order to ease the notation, we suppress the dependence on $r(t)$ in the drift and volatility. Substituting the dynamics \\eqref{dyn_imp_bonds} into \\eqref{self_fin_imp_portfolio_relative}, we have\n\\begin{multline}\n\\frac{d \\tilde{V}(t)}{\\tilde{V}(t)} = \\frac{\\alpha_T(t)}{\\tilde{P}(t,T)} (\\mu_T(t) - J_T(t)) dt + \\frac{\\alpha_S(t)}{\\tilde{P}(t,S)} (\\mu_S(t) - J_S(t)) dt + \\\\\n+ \\left( \\alpha_S(t) \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} + \\alpha_T(t) \\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} \\right) d W^\\mathbb{P}(t).\n\\label{relative_dynamics}\n\\end{multline}\nAt this point, we choose the relative weights so that the diffusive part of the equation above vanishes, that is, \n\\begin{equation}\\label{sys1}\n\\begin{aligned}\n\\alpha_T(t) + \\alpha_S(t) & = 1, \\\\\n\\alpha_T(t) \\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} + \\alpha_S(t) \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} & = 0.\n\\end{aligned}\n\\end{equation}\nSolving this system gives \n\\begin{equation}\\label{expression_relative_weights}\n\\begin{aligned}\n\\alpha_S(t) & = \\frac{\\sigma_T(t)\/\\tilde{P}(t,T)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)}, \\\\\n\\alpha_T(t) & = - \\frac{\\sigma_S(t)\/\\tilde{P}(t,S)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)}.\n\\end{aligned}\n\\end{equation}\nNotice that the above expressions are well defined. Indeed, if the denominator was approaching zero, then the sum of the two weights would be zero and this would contradict \\eqref{sys1}. Next, we substitute \\eqref{expression_relative_weights} into \\eqref{relative_dynamics}. Following again Bjork's argument, we use the fact that our impacted portfolio is locally risk-free (as in Definition \\ref{def:locally_risk_free}) by assumption and deduce the following relationship must hold:\n\\begin{multline*}\n\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} \\left(- \\frac{\\sigma_S(t)\/\\tilde{P}(t,S)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)} \\right) + \\\\\n+ \\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} \\left( \\frac{\\sigma_T(t)\/\\tilde{P}(t,T)}{\\sigma_T(t)\/\\tilde{P}(t,T) - \\sigma_S(t)\/\\tilde{P}(t,S)} \\right) = r(t).\n\\end{multline*}\nMultiplying both sides by the term\n\\begin{equation*}\n\\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} - \\frac{\\sigma_S(t)}{\\tilde{P}(t,S)},\n\\end{equation*}\nwe obtain\n\\begin{equation*}\n\\left(\\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} - r(t) \\right) \\left(\\frac{\\sigma_T(t)}{\\tilde{P}(t,T)} \\right) = \\left(\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} - r(t) \\right) \\left(\\frac{\\sigma_S(t)}{\\tilde{P}(t,S)} \\right).\n\\end{equation*}\nIt follows that,\n\\begin{equation*}\n\\left( \\frac{\\mu_S(t) - J_S(t)}{\\tilde{P}(t,S)} - r(t) \\right) \\left( \\frac{\\tilde{P}(t,S)}{\\sigma_S(t)}\\right) = \\left(\\frac{\\mu_T(t) - J_T(t)}{\\tilde{P}(t,T)} - r(t) \\right) \\left( \\frac{\\tilde{P}(t,T)}{\\sigma_T(t)}\\right),\n\\end{equation*}\nand rearranging we deduce\n\\begin{equation} \\label{eq:lambda_tilde_independent_of_mat}\n\\frac{\\mu_S(t) - J_S(t) - r(t) \\tilde{P}(t,S)}{\\sigma_S(t)} = \\frac{\\mu_T(t) - J_T(t) - r(t) \\tilde{P}(t,T)}{\\sigma_T(t)}.\n\\end{equation}\nNotice that the left hand side of \\eqref{eq:lambda_tilde_independent_of_mat} depends on $S$ but not on $T$, while the right hand side of \\eqref{eq:lambda_tilde_independent_of_mat} depends on $T$, but not on $S$. Since $S$ and $T$ are arbitrary, we conclude that both sides of \\eqref{eq:lambda_tilde_independent_of_mat} depend only on $t$ and $r(t)$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm_absence_arbitrage}]\nThe proof is similar to the proof of Proposition 1.1 in Chapter 1.2 of \\cite{bjork1997interest} (see also Harrison and Kreps \\cite{harrison1979martingales} Theorem 2 and relative Corollary in Section 3 and Harrison and Pliska \\cite{harrison1981martingales}, Theorem 2.7, Section 2). For the sake of completeness, we give the proof here, translated in our price impact environment. Let $T<+\\infty$ be some finite maturity. Let $\\tilde{h}$ be an arbitrage portfolio and $\\tilde{V}$ the corresponding value process. Then, given the positivity of the discount factor (bank account) defined in \\eqref{def:bank_account} and the equivalence between the real world measure $\\mathbb{P}$ and the impacted risk neutral measure $\\tilde{\\mathbb{Q}}$, we immediately deduce\n\\begin{equation}\n\\tilde{\\mathbb{Q}} \\left( \\frac{\\tilde{V}(T)}{B(T)} \\geq 0 \\right) = 1, \\ \\ \\ \\tilde{\\mathbb{Q}} \\left( \\frac{\\tilde{V}(T)}{B(T)} >0 \\right) > 0.\n\\label{Q_tilde_probs}\n\\end{equation}\nMoreover we have\n\\begin{equation*}\n0 = \\tilde{V}(0) = \\frac{\\tilde{V}(0)}{B(0)} = \\mathbb{E}^{\\tilde{\\mathbb{Q}}} \\left[\\frac{\\tilde{V}(T)}{B(T)} \\right] > 0,\n\\end{equation*}\nwhere the first equality comes from the definition of arbitrage, the second from the fact that $B(0)=1$ and the third from the fact that $\\tilde{V}(t)\/B(t)$ is a martingale under $\\tilde{\\mathbb{Q}}$. Finally, the positivity of the expectation is a consequence of \\eqref{Q_tilde_probs}. We get a contradiction so we conclude that absence of arbitrage must hold.\n\\end{proof}\n\n\\begin{proof}[Proof of Proposition \\ref{prop:relationship_Jf_Jp}]\nWe start by writing the impacted forward rate defined in \\eqref{impacted_forward_rate} as\n\\begin{equation*}\n\\tilde{f}(t,T) = f(t,T) + \\int_0^t J^f(s,T) ds, \n\\end{equation*}\nwhere $f$ represents the unimpacted forward rate (see e.g. Chapter 6, of \\cite{filipovic2009term}) and we used the assumption $\\tilde{f}(0,t) = f(0,t)$. Then, using \\eqref{impacted_bond_HJM}, we deduce\n\\begin{align}\\label{eq:dummy}\n\\begin{split}\n\\tilde{P}(t,T) & = \\exp \\left\\{-\\int_t^T \\tilde{f}(t,u) du\\right\\} \\\\\n& = \\exp \\left\\{-\\int_t^T f(t,u) du - \\int_t^T J^f(s,u) du \\right\\} \\\\\n& = P(t,T) \\exp \\left\\{-\\int_t^T J^f(s,u) du \\right\\},\n\\end{split}\n\\end{align}\nwhere $P$ denotes the unimpacted zero-coupon bond and we used the well known relation between $P(t,T)$ and $f(t,T)$. From \\eqref{impacted_bond} and \\eqref{def:overall_price_impact} we have \n\\begin{equation} \\label{bla} \n\\tilde{P}(t,T) = P(t,T) - I_T(t).\n\\end{equation}\nSubstituting this last expression into \\eqref{eq:dummy} and rearranging, we obtain\n\\begin{equation*}\n\\exp \\left\\{-\\int_t^T J^f(s,u) du \\right\\} = \\frac{\\tilde{P}(t,T)}{\\tilde{P}(t,T) + I_T(t)}.\n\\end{equation*}\nBy taking logarithms on both sides yields and using \\eqref{bla} we get \n\\begin{equation*}\n\\begin{aligned} \n\\int_t^T J^f(s,u) du &= - \\log \\left( \\frac{\\tilde{P}(t,T)}{\\tilde{P}(t,T) + I_T(t)} \\right) \\\\\n&= - \\log \\left( 1- \\frac{I_T(t)}{P(t,T)} \\right).\n\\end{aligned}  \n\\end{equation*}\nDifferentiating with respect to maturity, we get \\eqref{relationship_Jf_Jp}.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{hjm_condition_market_impact} ]\nLet $B(t)$ be the bank account defined in \\eqref{def:bank_account} and let the impacted zero-coupon bond $\\tilde{P}$ follow the dynamics \\eqref{impaced_ZCB_HJM}. By applying Ito's formula to the discounted impacted zero-coupon bond price, we immediately find\n\\begin{equation*}\nd \\frac{\\tilde{P}(t,T)}{B(t)} = \\frac{\\tilde{P}(t,T)}{B(t)} \\tilde{b}(t,T) dt + \\frac{\\tilde{P}(t,T)}{B(t)} \\nu(t,T) d W^{\\mathbb{P}}(t),\n\\end{equation*}\nwith $\\tilde{b}$ and $\\nu$ defined as in \\eqref{b-v-rel}. Changing measure form the real world $\\mathbb{P}$ to the impacted risk neutral $\\tilde{\\mathbb{Q}}$ as in \\eqref{def:Q_tilde_rnd_HJM} implies \n\\begin{equation*}\nd \\frac{\\tilde{P}(t,T)}{B(t)} = \\frac{\\tilde{P}(t,T)}{B(t)} \\left(\\tilde{b}(s,T) + \\nu(t,T) \\tilde{\\gamma}(t) \\right) dt + \\frac{\\tilde{P}(t,T)}{B(t)} \\nu(t,T) d W^{\\tilde{\\mathbb{Q}}}(t).\n\\end{equation*}\nTherefore, we clearly see that\n\\begin{equation*}\n\\frac{\\tilde{P}(t,T)}{B(t)} \\ \\ \\text{local martingale under}\\ \\ \\tilde{\\mathbb{Q}} \\iff \\tilde{b}(s,T) = - \\nu(t,T) \\tilde{\\gamma}(t).\n\\end{equation*}\nThis is our new HJM condition. Notice also that since both functions $\\nu$ and $\\tilde{b}$ are continuous with respect to $T$, this condition is equivalent to saying that the impacted measure $\\tilde{\\mathbb{Q}}$ is an equivalent local martingale measure. Following Theorem  6.1 in \\cite{filipovic2009term}, Chapter 6, we can plug in the explicit expression for $\\tilde b$ in \\eqref{b-v-rel} and write the HJM condition \\eqref{HJM_condition} as\n\\begin{equation}\\label{gh0} \n-\\int_s^T \\alpha(s,u) du - \\int_s^T J^f(s,u) du + \\frac{1}{2} \\nu^2(s,T)  = - \\nu(t,T) \\tilde{\\gamma}(t).\n\\end{equation} \nDifferentiating both sides with respect to the maturity $T$ yields the equation \n\\begin{equation*}\n- \\alpha(t,T) + \\sigma(t,T) \\int_t^T \\sigma(t,u) du - J^f(t,T) = \\sigma(t,T) \\tilde{\\gamma}(t),\n\\end{equation*}\nthat is\n\\begin{equation} \\label{gh1} \n\\alpha(t,T) + J^f(t,T) = \\sigma(t,T) \\int_t^T \\sigma(t,u)du - \\sigma(t,T) \\tilde{\\gamma}(t).\n\\end{equation}\nSubstituting \\eqref{gh1} in the dynamics of the forward rate \\eqref{impacted_forward_rate} and using Girsanov yields \\eqref{impacted_forward_rate_Q_tilde}. Using \\eqref{HJM_condition} along with \\eqref{impaced_ZCB_HJM} and Girsanov gives \\eqref{impacted_zc_bond_Q_tilde}.\n\\end{proof}\n\n\\section{Proof of Theorem \\ref{linear_feedback_form}}{\\label{sec:proof_linear_feedback}}\nThe uniqueness of the optimal strategy follows by a standard convexity argument for the performance functional \\eqref{j-fun}. Hence we only need to derive the optimal strategy. \n  \nWe start by deriving a system of coupled forward-backward stochastic differential equations (FBSDEs) which is satisfied by the solution to the stochastic control problem. \n\n\\begin{lemma}[FBSDE system]\n\\label{FBSDE_system}\nA control $\\hat{v} \\in \\mathcal{A}$ solves the optimization problem \\eqref{opt_stoch_control} if and only if the processes $(X^{\\hat{v}},\\Upsilon^{\\hat{v}},\\hat{v},Z^{\\hat{v}})$ satisfy the coupled forward-backward stochastic differential equations  \n\\begin{equation}\n\\begin{cases}\nd X^{\\hat{v}}(t) &= - \\hat{v}(t) dt, \\ \\ \\ X^{\\hat{v}}(0) = x, \\\\\nd \\Upsilon^{\\hat{v}}(t)& = - \\rho \\Upsilon^{\\hat{v}}(t) dt + \\gamma \\hat{v}(t) dt, \\ \\ \\ \\Upsilon^{\\hat{v}}(0) = y, \\\\\nd \\hat{v}(t) &= \\Lambda(t) d P(t,T) - 2 \\Lambda(t) \\phi X^{\\hat{v}}(t) dt \\\\\n& \\quad + \\Upsilon^{\\hat{v}}(t) \\left[-\\Lambda'(t) K(t,T) - \\Lambda(t) \\partial_t K(t,T) + \\rho K(t,T) \\Lambda(t) \\right] dt \\\\\n& \\quad + Z^{\\hat{v}}(t) \\left[\\Lambda'(t) + \\rho \\Lambda(t) \\right] dt + \\Lambda(t) \\Gamma^{\\hat{v}}(t) dt + d M(t), \\\\\n &\\qquad \\qquad \\qquad \\qquad \\qquad\\qquad\\qquad \\hat{v}(\\tau) = \\frac{\\varrho}{l(\\tau,T)} X^{\\hat{v}}(\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Upsilon^{\\hat{v}}(\\tau), \\\\\nd Z^{\\hat{v}}(t)& = \\left(\\rho Z^{\\hat{v}}(t) + K(t,T) \\gamma \\hat{v}(t) \\right) dt + d N(t), \\ \\ Z^{\\hat{v}}(\\tau) = 0,\n\\end{cases}\n\\end{equation}\nfor two suitable square integrable martingales $M=(M(\\cdot,T))_{0 \\leq t \\leq \\tau}$ and $N=(N(\\cdot,T))_{0 \\leq t \\leq \\tau}$, where the $\\Lambda, \\Gamma^{\\hat{v}}$ and $\\tilde{M}$ are defined in \\eqref{def:Lambda_optimal_exe}, \\eqref{def:Gamma_quantity} and \\eqref{def:square_int_mart_M_tilde} respectively.\n\\begin{proof}\nThe proof follows the same lines as Lemmas 5.1 and 5.2 in \\cite{neuman2020optimal}. Since for all $v \\in \\mathcal{A}$ the map $v \\to \\mathcal{J}(v)$ is strictly concave, we can study the unique critical point at which the Gateaux derivative of $\\mathcal{J}$, which is defined as\n\\begin{equation*}\n\\langle \\mathcal{J}'(v), \\alpha \\rangle := \\lim_{\\epsilon \\to 0} \\frac{\\mathcal{J}(v+\\epsilon \\alpha) - \\mathcal{J}(v)}{\\epsilon},\n\\end{equation*}\nis equal to $0$ for any $\\alpha \\in \\mathcal{A}$. This derivative can be computed analytically as follows. Let $\\epsilon>0$ and $v,\\alpha \\in \\mathcal{A}$. Since for all $t \\in [0,\\tau]$,\n\\begin{align} \\label{bb1}\n\\begin{split}\nX^{v + \\epsilon \\alpha}(t) & = x - \\int_0^t (v(s) + \\epsilon \\alpha(s) )ds = X^v(t) - \\epsilon \\int_0^t \\alpha(s) ds \\\\\n\\Upsilon^{v+\\epsilon \\alpha}(t) & = \\Upsilon^v(t) + \\epsilon \\gamma \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds,\n\\end{split}\n\\end{align}\nFrom \\eqref{j-fun} and \\eqref{bb1} we have\n\\begin{align*}\n&\\mathcal{J}(v+\\epsilon \\alpha) = \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\left( P(t,T) - K(t,T) \\Upsilon^v(t) - K(t,T) \\epsilon \\gamma \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds \\right) \\left(v(t) + \\epsilon \\alpha(t) \\right) dt  \\\\\n&\\qquad - \\int_0^\\tau l(t,T) v^2(t) + \\epsilon^2 l(t,T) \\alpha_t^2 + 2 l(t,T) v(t) \\epsilon \\alpha(t) dt + X^v(\\tau) P(\\tau,T) - \\epsilon P(\\tau,T) \\int_0^\\tau \\alpha(s) ds  \\\\\n&\\qquad - \\phi \\int_0^\\tau (X^v(t))^2 + \\epsilon \\left(\\int_0^t \\alpha(s) ds \\right)^2 - 2 X^v(t) \\epsilon \\int_0^t \\alpha(s) ds dt  \\\\\n&\\qquad - \\varrho \\left( (X^v(\\tau))^2 + \\epsilon^2 \\left(\\int_0^\\tau \\alpha(s) ds \\right)^2 - 2 X^v(\\tau) \\epsilon \\int_0^\\tau \\alpha(s) ds \\right) \\Bigg]. \n\\end{align*}\nIt follows that \n\\begin{align*}\n\\mathcal{J}(v+\\epsilon \\alpha) - \\mathcal{J}(v) &= \\epsilon \\mathbb{E} \\Bigg[ \\int_0^\\tau \\left( P(\\tau,T) - K(t,T) \\Upsilon^v(t) \\right) \\alpha(t) dt \\\\\n&\\qquad - \\int_0^\\tau K(t,T) v(t) \\int_0^t \\gamma e^{-\\rho (t-s)} \\alpha(s) ds dt - 2 \\int_0^\\tau l(t,T) v(t) \\alpha(t) dt \\\\\n&\\qquad + 2 \\phi \\int_0^\\tau X^v(t) \\int_0^t \\alpha(s) ds dt + 2 \\varrho X^v(\\tau) \\int_0^\\tau \\alpha(s) ds\n- P(\\tau,T) \\int_0^\\tau \\alpha(s) ds \\Bigg]  \\\\\n&\\qquad+ \\epsilon^2 \\mathbb{E} \\Bigg[ \\gamma \\int_0^\\tau K(t,T) \\alpha(t) \\int_0^t e^{-\\rho (t-s)} \\alpha(s) ds dt - \\int_0^\\tau l^2(t,T) \\alpha^2(t) dt  \\\\\n&\\qquad - \\phi \\int_0^\\tau \\left( \\int_0^t \\alpha(s) ds \\right)^2 dt - \\varrho \\left(\\int_0^\\tau \\alpha(s) ds \\right)^2 \\Bigg].\n\\end{align*}\nNote that all the terms above are finite since $\\ell$ and $K$ are bounded functions and since $\\alpha, v \\in \\mathcal A$. Applying Fubini's theorem twice, we obtain\n\\begin{multline*}\n\\langle \\mathcal{J}'(v), \\alpha \\rangle = \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\Bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\int_s^\\tau K(t,T) e^{-\\rho(t-s)} \\gamma v(t) dt + \\\\\n- 2 l(s,T) v(s) + 2 \\phi \\int_s^\\tau X^v(t) dt + 2 \\varrho X^v(\\tau) - P(\\tau,T) \\Bigg) ds \\Bigg],\n\\end{multline*}\nfor any $\\alpha \\in \\mathcal{A}$. We get the following condition on the optimal strategy\n\\begin{equation}\n\\begin{aligned}\n\\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\Bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\int_s^\\tau K(t,T) e^{-\\rho(t-s)} \\gamma v(t) dt  \\\\\n- 2 l(s,T) v(s) + 2 \\phi \\int_s^\\tau X^v(t) dt + 2 \\varrho X^v(\\tau) - P(\\tau,T) \\Bigg) ds \\Bigg] = 0.\n\\label{first_order_condition}\n\\end{aligned}\n\\end{equation}\nNext we show that given the optimal strategy $\\hat{v} \\in \\mathcal{A}$, the vector $(X^{\\hat{v}},\\Upsilon^{\\hat{v}})$ satisfies the first order condition \\eqref{first_order_condition} if and only if the vector $(X^{\\hat{v}},\\Upsilon^{\\hat{v}},\\hat{v},Z^{\\hat{v}})$ solves a FBSDE system, for some auxiliary process $Z$.\n\nFor any $s>0$ we denote by $\\mathbb{E}_s$ the conditional expectation with respect to the filtration $\\mathcal F_s$. Assume $\\hat{v} \\in \\mathcal{A}$ maximizes the functional $\\mathcal{J}$. Applying the optional projection theorem we obtain  \n\\begin{align*}\n\\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg( P(s,T) - K(s,T) \\Upsilon^v(s) - \\mathbb{E}_s \\bigg[\\int_s^\\tau K(t,T) e^{-\\rho (t-s)} \\gamma \\hat{v}(t) dt \\bigg] - 2 l(s,T) \\hat{v}(s)  \\\\\n+ \\mathbb{E}_s \\bigg[2 \\phi \\int_s^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] \\bigg) ds \\Bigg] = 0,\n\\end{align*}\nfor all $\\alpha \\in \\mathcal{A}$. This implies\n\\begin{equation}  \\label{explicit_sol_BSDE_v_hat} \n\\begin{aligned}\n&P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) - e^{\\rho s} \\mathbb{E}_s \\bigg[\\int_s^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\bigg] - 2 l(s,T) \\hat{v}(s) \\\\\n&\\qquad+ \\mathbb{E}_s \\bigg[2 \\phi \\int_s^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] \\\\\n&= P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) \\\\\n&\\qquad - e^{\\rho s} \\left( \\mathbb{E}_s \\bigg[ \\int_0^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\bigg] - \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) \\\\\n&\\qquad - 2 l(s,T) \\hat{v}(s) + \\mathbb{E}_s \\bigg[2 \\phi \\int_0^\\tau X^{\\hat{v}}(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg] - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt \\\\\n&= 0, \\ \\ \\ d \\mathbb{P} \\otimes ds\\ \\ \\text{a.e. on} \\ \\ \\Omega \\times [0,\\tau]. \\\\\n\\end{aligned}\n\\end{equation} \nNext, we define the square-integrable martingale\n\\begin{equation}\\label{def:square_int_mart_N_tilde}\n\\tilde{N}(s) := \\mathbb{E}_s \\left[\\int_0^{\\tau} K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right]\n\\end{equation}\nand the auxiliary square-integrable process\n\\begin{equation*}\nZ^{\\hat{v}}(s) := e^{\\rho s} \\bigg( \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt - \\tilde{N}(s) \\bigg),\n\\end{equation*}\nfor all $s \\in [0,\\tau]$. Note that since both $l$ and $K$ are assumed to be uniformly bounded and $v \\in \\mathcal A$, we have that $P(\\tau,T) \\in L^2(\\Omega,\\mathcal{F}_\\tau,\\mathbb{P})$. Therefore, we obtain\n\\begin{equation} \\label{fgr}\nP(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) - 2 l(s,T) \\hat{v}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt = 0\n\\end{equation}\nalmost everywhere on $\\Omega \\times [0,\\tau]$, where $\\tilde{M}$ is the square-integrable martingale defined in \\eqref{def:square_int_mart_M_tilde}, and we immediately see that the process $Z^{\\hat{v}}$ satisfies the BSDE\n\\begin{equation*}\nd Z^{\\hat{v}}(t) = \\left(\\rho Z^{\\hat{v}}(t) + K(t,T) \\gamma \\hat{v}(t) \\right) dt - e^{\\rho t} d \\tilde{N}(t), \\quad Z^{\\hat{v}}(\\tau) = 0.\n\\end{equation*}\nFrom \\eqref{def:transient_impact} we get that $\\Upsilon^{\\hat{v}}$ satisfies   \n\\begin{equation*}\nd \\Upsilon^{\\hat{v}}(t) = - \\rho \\Upsilon^{\\hat{v}}(t) dt + \\gamma \\hat{v}(t) dt, \\quad \\Upsilon^{\\hat{v}}(0) = y.\n\\end{equation*}\nRecall that $\\Lambda$ was defined in \\eqref{def:Lambda_optimal_exe}. From \\eqref{fgr} it follows that $\\hat{v}$ satisfies the backward stochastic differential equation \n\\begin{align*}\n\\begin{split}\nd \\hat{v}(s) & = \\Lambda'(s) \\left(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\right)ds \\\\\n& \\quad + \\Lambda(s) \\bigg(d P(s,T) - \\partial_s K(s,T) \\Upsilon^{\\hat{v}}(s) ds - K(s,T) d \\Upsilon^{\\hat{v}}(s) + d Z^{\\hat{v}}(s) \\\\\n& \\quad \\quad + d \\tilde{M}(s) - 2 \\phi X^{\\hat{v}}(s) ds \\bigg) \\\\\n& = \\Lambda'(s) \\left(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(s) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\right)ds \\\\\n& \\quad + \\Lambda(s) d P(s,T) - \\Lambda(s) \\partial_s K(s,T) \\Upsilon^{\\hat{v}}(s) ds + \\rho K(s,T) \\Upsilon^{\\hat{v}}(s) \\Lambda(s) ds  \\\\\n& \\quad   + \\Lambda(s) \\rho Z^{\\hat{v}}(s) ds  - 2 \\Lambda(s) \\phi X^{\\hat{v}}(s) ds + \\Lambda(s) d \\tilde{M}(s) - \\Lambda(s) e^{\\rho s} d \\tilde{N}(s)  \\\\\n\\hat{v}(\\tau) & = \\frac{\\varrho}{l(\\tau,T)} X^{\\hat{v}}(\\tau) - \\frac{K(\\tau,T)}{2 l(\\tau,T)} \\Upsilon^{\\hat{v}}(\\tau),\n\\end{split}\n\\end{align*}\nPutting these equations together with \\eqref{inv}, we obtain the FBSDE system \\eqref{FBSDE_system} with $M,N$ square-integrable martingales defined as\n\\begin{align*}\n\\begin{split}\nM(t) & := \\int_0^t \\Lambda(s) d \\tilde{M}(s) - \\int_0^t \\Lambda(s) e^{\\rho s} d \\tilde{N}(s) \\\\\nN(t) & := - \\int_0^t e^{\\rho s} d \\tilde{N}(s).\n\\end{split}\n\\end{align*}\nIn order to check the integrability of $M$, recall that $\\Lambda$ was defined in \\eqref{def:Lambda_optimal_exe}. Since $l$ is bounded away from $0$ on $[0,\\tau] $ (see \\eqref{l-pos}) we have \n\\[\n\\sup_{0 \\leq t \\leq \\tau} | \\Lambda(t) | < \\infty.\n\\]\nThen, it holds\n\\begin{align*}\n\\begin{split}\n\\mathbb{E}[M^2(t)] & \\leq \\mathbb{E} \\left[ \\int_0^t \\Lambda^2(s) d [\\tilde M]_s \\right] + \\mathbb{E} \\left[ \\int_0^t\\Lambda^2(s) e^{2\\rho s} d[\\tilde N]_s \\right] \\\\\n& \\leq C_1 \\mathbb{E} [\\tilde M]_T + C_2 \\mathbb{E} [ \\tilde N]_T \\\\\n&< \\infty\n\\end{split}\n\\end{align*}\nfor some constants $C_1,C_2$, where in the last inequality we used the fact that both $\\tilde{M}$ and $\\tilde{N}$ are square integrable martingales.\n\nNext, assume that  $(\\hat{v},X^{\\hat{v}},\\Upsilon^{\\hat{v}},Z^{\\hat v})$ is a solution to the FBSDE system \\eqref{FBSDE_system} and $\\hat{v} \\in \\mathcal{A}$. We will show that $\\hat{v}$ satisfies the first order condition \\eqref{first_order_condition}, hence it maximizes the cost functional \\eqref{j-fun}. First, note that the BSDE for $\\hat{v}$ can be solved explicitly and the solution is indeed given in \\eqref{explicit_sol_BSDE_v_hat}\n\\begin{align*}\n\\begin{split}\n\\hat{v}(s) & = \\frac{1}{2 l(s,T)} \\Bigg(P(s,T)- K(s,T) \\Upsilon^{\\hat{v}}(s) + Z^{\\hat{v}}(t) + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(t) dt \\Bigg) \\\\\n& = \\frac{1}{2l(s,T)} \\Bigg(P(s,T) - K(s,T) \\Upsilon^{\\hat{v}}(s) - e^{\\rho s} \\left(\\tilde{N}(s) - \\int_0^s K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) \\\\\n& \\quad + \\tilde{M}(s) - 2 \\phi \\int_0^s X^{\\hat{v}}(u) du \\Bigg),\n\\end{split}\n\\end{align*}\nwith $\\tilde{N},\\tilde{M}$ defined in \\eqref{def:square_int_mart_N_tilde} and \\eqref{def:square_int_mart_M_tilde}, respectively. Plugging this into the first order condition \\eqref{first_order_condition} yields\n\\begin{align*}\n&\\mathbb{E} \\Bigg[\\int_0^\\tau \\bigg(e^{\\rho s} \\left(\\tilde{N}(s) - \\int_0^\\tau K(t,T) e^{-\\rho t} \\gamma \\hat{v}(t) dt \\right) - \\tilde{M}(s) \\\\\n&\\qquad + 2 \\phi \\int_0^\\tau X(t) dt + 2 \\varrho X^{\\hat{v}}(\\tau) - P(\\tau,T) \\bigg)ds \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg( e^{\\rho s} (\\tilde{N}(s) - \\tilde{N}(\\tau)) - \\tilde{M}(s) + \\tilde{M}(\\tau) \\bigg) ds \\Bigg] \\\\\n&= \\mathbb{E} \\Bigg[ \\int_0^\\tau \\alpha(s) \\bigg(e^{\\rho s} (\\tilde{N}(s) - \\mathbb{E}_s[\\tilde{N}(\\tau)]) - \\tilde{M}(s) + \\mathbb{E}_s[\\tilde{M}(\\tau)] \\bigg)ds \\Bigg] \\\\\n& = 0,\n\\end{align*}\nfor all $\\alpha \\in \\mathcal{A}$. Since $\\tilde{N},\\tilde{M}$ are martingales, hence the first order condition \\eqref{first_order_condition} is satisfied and $\\hat{v} \\in \\mathcal{A}$ is the optimal strategy.\n\\end{proof}\n\\end{lemma}\n\nBefore giving the proof of our main theorem, we will need the following Lemma, which will help us to show the optimal strategy in \\eqref{linear_feedback_form} is indeed admissible.\n\n\\begin{lemma}\\label{lem:Gamma_quantity}\nLet $\\Gamma^{\\hat{v}}$ be defined as in \\eqref{def:Gamma_quantity}. Then, there exist constants $C_1, C_2 > 0$ such that \n\\begin{equation*}\n\\mathbb{E} \\left[  \\int_0^\\tau \\big(\\Gamma^{\\hat{v}}(s) \\big)^{2}  ds \\right] \\leq C_1 + C_2\\mathbb{E} \\left[\\int_0^\\tau v^2(s) ds \\right].\n\\end{equation*}\n\\begin{proof}\nFirstly, by the assumptions on $l$ (see \\eqref{l-pos} and \\eqref{k-assump}) it follows that\n\\begin{equation}\\label{bounded_ratio_lambda_lambda_prime}\n\\sup_{0 \\leq t \\leq \\tau} \\bigg | \\frac{\\Lambda'(t)}{\\Lambda(t)} \\bigg| = \\sup_{0 \\leq t \\leq \\tau} \\bigg| \\frac{\\partial_t l(t,T)}{l(t,T)} \\bigg| < \\infty,\n\\end{equation}\nwhere $\\Lambda$ is given in \\eqref{def:Lambda_optimal_exe}. Therefore, from \\eqref{bounded_ratio_lambda_lambda_prime},  \\eqref{def:Gamma_quantity} and Jensen's inequality we get that there exist constants $C_1,C_2>0$ such that\n\\begin{align*} \n \\mathbb{E} \\left[  \\int_0^\\tau \\Gamma^{\\hat{v}}(s)^2  ds \\right] \n&\\leq \\mathbb{E} \\left[  \\int_0^\\tau \\Big(\\frac{\\Lambda'(s)}{\\Lambda(s)}\\Big)^2 \\left(P^2(s,T) + \\tilde{M}^2(s) + 4 \\phi^2 \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 \\right) ds \\right] \\\\ \n&\\leq C_1 \\mathbb{E} \\left[ \\int_0^\\tau \\left(  P^2(s,T) + \\tilde{M}^2(s) +  4 \\phi^2  \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 \\right)ds \\right] \\\\ \n&\\leq C_2 +  4 \\phi^2  \\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 ds \\right],\n\\end{align*}\nwhere we used \\eqref{def: mathcal_h_squared} and the fact that the martingale $\\tilde{M}$ defined in \\eqref{def:square_int_mart_M_tilde} is square-integrable. \n\nNext, using the definition of $X^{\\hat{v}}$ in \\eqref{def:X} and Jensen's inequality twice, we deduce\n\\begin{align*} \n\\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s X^{\\hat{v}}(u) du \\Big)^2 ds\\right] & = \n\\mathbb{E} \\left[ \\int_0^\\tau \\Big(\\int_0^s \\big(x-\\int_0^u  \\hat v(y) dy  \\big)du\\Big)^2 ds \\right] \\\\\n&\\leq C_1 + C_2\\mathbb{E} \\left[\\int_0^\\tau \\int_0^s \\int_0^u \\hat v^2(y) dy du ds \\right] \\\\\n&\\leq C_1 + C_2 \\mathbb{E} \\left[ \\int_0^\\tau \\int_0^\\tau \\int_0^\\tau \\hat v^2(y) dy du ds  \\right] \\\\\n&\\leq C_1 + C_2 \\mathbb{E} \\left[\\int_0^\\tau  \\hat v^2(y) ds \\right], \n\\end{align*} \nfor some constants $C_1,C_2$, and we are done. \n\\end{proof}\n\\end{lemma}\nWe are now ready to prove Theorem \\ref{linear_feedback_form}. \n\n\\begin{proof}[Proof of Theorem \\ref{linear_feedback_form}] \nDefine\n\\begin{equation*}\n\\mathbf{X}^{{v}}(t) := \n\\left(\n\\begin{matrix}\nX^{\\hat v}(t) \\\\\n\\Upsilon^{\\hat v}(t) \\\\\n\\hat v(t) \\\\\nZ^{\\hat v}(t)\n\\end{matrix}\n\\right), \\ \\ \\ \n\\mathbf{M}(t) :=\n\\left(\n\\begin{matrix}\n0 \\\\\n0 \\\\\nP(t,T) + \\int_0^t \\Gamma^{\\hat v}(s) ds + \\int_0^t \\Lambda^{-1}(s) d M(s) \\\\\n\\int_0^t \\Lambda^{-1}(s) d N(s)\n\\end{matrix}\n\\right),\n\\end{equation*}\nwhere $\\Lambda$ and $\\Gamma^{\\hat v}$ are defined in \\eqref{def:Lambda_optimal_exe} and \\eqref{def:Gamma_quantity} respectively. The FBSDE system \\eqref{FBSDE_system} can be written as\n\\begin{equation*}\nd \\mathbf{X}_t^{\\hat v}= A(t) \\mathbf{X}_t^{\\hat v} dt + \\Lambda(t) d \\mathbf{M}(t), \\ \\ \\ 0 \\leq t \\leq \\tau,\n\\end{equation*}\nwhere the matrix $A(t)$ is defined in \\eqref{def:time_dep_matrix_A}, with initial conditions\n\\begin{equation*}\n\\mathbf{X}^{{\\hat v},1}(0) = x, \\ \\ \\ \\mathbf{X}^{{\\hat v},2}(0) = y,\n\\end{equation*}\nand terminal conditions\n\\begin{equation} \\label{term}\n\\left(\\frac{\\varrho}{l(\\tau,T)}, - \\frac{K(\\tau,T)}{2 l(\\tau,T)}, -1, 0 \\right) \\mathbf{X}^{\\hat v}(\\tau) = 0, \\ \\ \\ (0,0,0,1) \\mathbf{X}^{\\hat v}(\\tau) = 0.\n\\end{equation}\nExploiting linearity, the unique solution can be expressed as\n\\begin{equation*}\n\\mathbf{X}^{\\hat v}(\\tau) = \\Phi(\\tau) \\Phi^{-1}(t) \\mathbf{X}^{\\hat v}(t) + \\int_t^\\tau \\Phi(\\tau) \\Phi^{-1}(s) \\Lambda(s) d \\mathbf{M}(s),\n\\end{equation*}\nwhere $\\Phi$ solves the ODE \\eqref{matrix_ODE}. Moreover, it can be immediately seen that the first terminal condition in \\eqref{term} yields\n\\begin{align*}\n\\begin{split}\n0 & = G^1(t,\\tau) X^{\\hat v}(t) + G^2(t,\\tau) \\Upsilon^{\\hat v}(t) + G^3(t,\\tau) \\hat v(t) + G^4(t,\\tau) Z^{\\hat v}(t) \\\\\n& \\quad + \\int_t^\\tau \\Lambda(s) \\left(G^3(s,\\tau) \\left(d P(s,T) + \\Gamma^{\\hat v}(s) ds + \\Lambda^{-1}(s) d M(s) \\right) + G^4(s,\\tau) \\Lambda^{-1}(s) d N(s) \\right)\n\\end{split}\n\\end{align*}\nwith $G = (G^1,G^2,G^3,G^4)$ defined in \\eqref{def:G_components}. Solving for the trading speed $v$, taking expectations and using that $P \\in \\mathcal{H}^2$, together with the fact that both $M$ and $N$ are square integrable martingales, implies\n\\begin{align}\\label{trading_speed_u_with_z}\n\\begin{split}\n\\hat v(t) & = - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} X^{\\hat v}(t) - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\Upsilon^{\\hat v}(t) - \\frac{G^4(t,\\tau)}{G^3(t,\\tau)} Z^{\\hat v}(t) \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)}(\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right].\n\\end{split}\n\\end{align}\nRecall that $\\Psi$ was defined in \\eqref{def:psi_matrix}. Then the second terminal condition in \\eqref{term} implies\n\\begin{align*}\n\\begin{split}\n0 & = (0,0,0,1) \\Psi(t,\\tau) \\mathbf{X}^{\\hat v}(t) + (0,0,0,1) \\int_t^\\tau \\Psi(s,\\tau) \\Lambda(s) d \\mathbf{M}(s) \\\\\n& = \\Psi^{41}(t,\\tau) X^{\\hat v}(t) + \\Psi^{42}(t,\\tau) \\Upsilon^{\\hat v}(t) + \\Psi^{43}(t,\\tau) \\hat v(t) + \\Psi^{44}(t,\\tau) Z^{\\hat v}(t) \\\\\n& \\quad + \\int_t^\\tau \\Lambda(s) \\left(\\Psi^{43}(s,\\tau) \\left(d P(s,T) + \\Gamma^{\\hat v}(s) ds + \\Lambda^{-1}(s) d M(s) \\right) + \\Psi^{44}(s,\\tau) \\Lambda^{-1}(s) d N(s) \\right).\n\\end{split}\n\\end{align*}\nHence, taking expectation and solving for $Z^u$ yields\n\\begin{align}\\label{aux_process_Z}\n\\begin{split}\nZ^{\\hat v}(t) & = - \\frac{\\Psi^{41}(t,\\tau)}{\\Psi^{44}(t,\\tau)} X^{\\hat v}(t) - \\frac{\\Psi^{42}(t,\\tau)}{\\Psi^{44}(t,\\tau)} \\Upsilon^{\\hat v}(t) - \\frac{\\Psi^{43}(t,\\tau)}{\\Psi^{44}(t,\\tau)} {\\hat v}(t) \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right].\n\\end{split}\n\\end{align}\nTherefore, plugging \\eqref{aux_process_Z} into \\eqref{trading_speed_u_with_z} gives\n\\begin{align*}\n\\begin{split}\n\\hat v(t) & = - \\frac{G^1(t,\\tau)}{G^3(t,\\tau)} X^{\\hat v}(t) - \\frac{G^2(t,\\tau)}{G^3(t,\\tau)} \\Upsilon^{\\hat v}(t) + \\frac{G^4(t,\\tau) \\Psi^{41}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} X^{\\hat v}(t) \\\\\n& \\quad + \\frac{G^4(t,\\tau) \\Psi^{42}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} \\Upsilon^{\\hat v}(t) + \\frac{G^4(t,\\tau) \\Psi^{43}(t,\\tau)}{G^3(t,\\tau) \\Psi^{44}(t,\\tau)} {\\hat v}(t) \\\\\n& \\quad + \\frac{G^4(t,\\tau)}{G^3(t,\\tau)} \\mathbb{E}_t \\left[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right] \\\\\n& \\quad - \\mathbb{E}_t \\left[\\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\right]. \n\\end{split}\n\\end{align*}\nRearranging and using the Definitions \\ref{def:v_terms}, we obtain the linear feedback form \\eqref{optimal_strategy_linear_feedback}. Finally, we prove that the optimal trading strategy is admissible, that is, $\\hat{v} \\in \\mathcal{A}$, as defined in \\eqref{def:admissset}. Thanks to assumptions  \\eqref{A.1} and \\eqref{A.2}, we immediately see that\n\\begin{equation*}\n\\sup_{0 \\leq t \\leq \\tau} |v_0(t,\\tau)| < \\infty.\n\\end{equation*}\nSimilarly, from assumptions \\eqref{A.1}-\\eqref{A.3} we deduce that $v_1$ and $v_2$ are both bounded on $[0,\\tau]$. Exploiting again assumptions \\eqref{A.1}-\\eqref{A.3}, together with \\eqref{def: mathcal_h_squared} we get that  \n\\begin{align*}\n& \\sup_{0 \\leq t \\leq \\tau} \\bigg| \\mathbb{E}_t \\bigg[\\int_t^\\tau \\frac{\\Lambda(s) \\Psi^{43}(s,\\tau)}{\\Psi^{44}(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \n - \\mathbb{E}_t \\int_t^\\tau \\Lambda(s) \\frac{G^3(s,\\tau)}{G^3(t,\\tau)} (\\mu(s) + \\Gamma^{\\hat v}(s)) ds \\bigg] \\bigg|\\\\\n& \\leq C \\mathbb{E} \\left[\\int_0^\\tau( |\\mu(s) |+ |\\Gamma^{\\hat v}(s)| )ds \\right] \\\\\n& \\leq \\tilde{C}_1 + \\tilde{C}_2\\left( \\mathbb{E} \\left[\\int_0^\\tau \\Gamma^{\\hat v}(s)^{2}ds \\right] \\right)^{1\/2} \\\\\n& \\leq \\tilde{C}_1 +\\tilde C_2\\left(\\mathbb{E} \\left[\\int_0^\\tau \\hat{v}^2(s)ds \\right] \\right)^{1\/2}, \n\\end{align*}\nwhere we have used Jensen's inequality and Lemma \\ref{lem:Gamma_quantity} in the last two inequalities. \nUsing the above bound, together with  equations \\eqref{def:X} and \\eqref{def:transient_impact} we get from  \\eqref{optimal_strategy_linear_feedback} that\n\\begin{equation*}\n\\mathbb{E}[\\hat{v}^2(t)] \\leq C_1 + C_2 \\int_0^\\tau \\mathbb{E}[\\hat{v}^2(s)] ds, \\ \\ \\ 0 \\leq t \\leq \\tau,\n\\end{equation*}\nfor some positive constants $C_1,C_2$, where we used again Jensen's inequality. Thanks to Gronwall's lemma, we get that\n\\begin{equation*}\n\\sup_{0 \\leq t \\leq \\tau} \\mathbb{E}[\\hat{v}^2(t)] < \\infty,\n\\end{equation*}\nwhich implies\n\\begin{equation*}\n\\int_0^\\tau \\mathbb{E}[\\hat v^2(s)] ds < \\infty.\n\\end{equation*}\nHence Fubini's theorem, we conclude that $\\hat v\\in \\mathcal A$. \n\\end{proof}\n\t\n\\section*{Acknowledgements}\nWe are very grateful to an anonymous referees for  careful reading of the manuscript,\nand for a number of useful comments and suggestions that significantly improved this paper.\n\n\n\\newpage\n \n\t\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nWhen Tunka-Rex started its operation in 2012 with 18 antennas it had two main goals. \nFirst, a cross-calibration with the Tunka-133 photomultiplier array using the established non-imaging air-Cherenkov technique for air showers \\cite{Tunka133_NIM2014}. \nBy this the precision of the radio measurements for the energy and the depth of the shower maximum, $X_\\mathrm{max}$, could be determined experimentally \\cite{TunkaRex_Xmax2016}. \nThis goal is complementary to the comparison of radio to air-fluorescence measurements at the Pierre Auger Observatory \\cite{Holt_AERA_ICRC2017}. \nIt also gives additional confidence to the measurements of radio arrays whose analysis is based mainly on Monte Carlo simulations m\\cite{BuitinkLOFAR_Xmax2014}. \nSecond, the practical demonstration that the radio technique is cost-effective -- at least when the antennas are attached as extension to other detectors in order to improve the total accuracy. \n\nMeanwhile both goals have been achieved and the new main goal of Tunka-Rex is a mass-sensitive measurement of the energy spectrum between $10^{17}$ and $10^{18}\\,$eV, since in this energy range the transition from galactic to extra-galactic cosmic-rays is assumed, but not yet understood. \nFor this purpose the Tunka-Rex array has been extended to 63 antennas that measure in coincidence with particles and air-Cherenkov detectors (the latter only during clear nights).  \nIn combination with other arrays at different locations of the Earth, Tunka-Rex can search for anisotropies of different mass components and consequent differenced of the energy spectrum between the northern and southern hemispheres. \nA precise measurement of the energy spectrum can be used to study whether the second knee really exists at a few $100\\,$PeV and whether it is a distinct feature from the heavy knee discovered by KASCADE-Grande \\cite{KGheavyKnee2011}. \nFinally, Tunka-Rex remains a testbed for future developments of the radio technique. \nWhile the physics of the radio emission by air showers is understood sufficiently well \\cite{HuegeReview2016, SchroederReview2016}, significant technical developments have to be done for sparse and large arrays of the next-generation, such as GRAND \\cite{GRAND_ICRC2017}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.7\\linewidth]{TunkaRexMap2017.pdf}\n  \\caption{Map of the cosmic-ray detectors of TAIGA: the Tunka-133 air-Cherenkov array consisting of 25 clusters of seven photomultipliers each and a local data acquisition (DAQ) at each cluster center, the Tunka-Grande particle-detector array with one detector station at each of the 19 inner cluster centers, and the Tunka-Rex radio array, with 3 antenna stations at each of the 19 inner clusters triggered by both, Tunka-133 and Tunka-Grande, and one antenna stations at each outer cluster triggered by Tunka-133.}\n  \\label{fig_tunkaRexMap}\n\\end{figure*}\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{TunkaRexCluster5.jpg}\n  \\caption{Photo of one cluster center with three antenna stations, a particle-detector station with surface scintillators in the gray shelter and underground scintillators in a concrete tunnel below the longish hump attached to the shelter, and a photomultiplier detector (silver cylinder) next to the local data-acquisition of this cluster (white box).}\n  \\label{fig_tunkaPhoto}\n\\end{figure*}\n\n\\section{Status of Tunka-Rex}\nSince October 2016 Tunka-Rex consists of 63 antenna stations distributed over a total area of about $3\\,$km$^2$ (figure \\ref{fig_tunkaRexMap}) \\cite{Schroeder_TunkaRex_ARENA2016}. \nEach of the 19 inner clusters of the cosmic-ray detectors of TAIGA (Tunka Advanced Instrument for cosmic ray physics and Gamma Astronomy) features 3 antenna stations (Tunka-Rex), 7 photomultiplier detectors for the air-Cherenkov light (Tunka-133), and 1 scintillator station with surface and underground particle detectors for electron and muon measurements (Tunka-Grande). \nThe 6 outer clusters feature only 1 antenna station and 7 photomultiplier detectors, each.\nDuring clear nights, when the air-Cherenkov array Tunka-133 operates it triggers all radio antennas, and during the remaining time the particle-detector array Tunka-Grande triggers all inner antennas. \nThis means that for every air shower all 57 inner antennas are read out, and additionally the 6 outer antennas exclusively when the air-Cherenkov detector operates. \n\nEach radio station consists of a set of two perpendicularly aligned SALLAs with low-noise amplifiers integrated directly in the antenna \\cite{AERAantennaPaper2012}. \nAfter passing a another filter-amplifier, the radio signal is digitized by the same digital data-acquisition used for the other cosmic-ray detectors of TAIGA (figure~\\ref{fig_tunkaPhoto}) \\cite{TAIGA_2014}. \nUsing the core-position of the denser Tunka-133 as input, the radio data are analyzed by a special version of the Offline software frame work developed by the Pierre Auger Collaboration \\cite{RadioOffline2011}, and afterwards compared to the Tunka-133 measurements. \nA combined analysis with the particle signals of Tunka-Grande is planned, too. \n\nTunka-Rex is calibrated on an absolute scale using the same external reference source that was used first by LOPES and later by LOFAR \\cite{TunkaRex_NIM_2015, 2015ApelLOPES_improvedCalibration, NellesLOFAR_calibration2015}. \nHowever, the antennas and the analog electronics of the subsequent signal chain are from three different production series, i.e., each of the inner clusters has one antenna of each series (figure~\\ref{fig_exampleEvent}). \nWhile all three series use the same electronics scheme, the gain and phase response varies slightly from series to series, and the inter-calibration is not yet completed. \nThus, at the moment most of the analyses are still based on almost 200 events detected by Tunka-Rex during the first two years of data taking with only one antenna per cluster triggered by the Tunka-133 air-Cherenkov detector. \n\n\n\n\\section{Event Reconstruction}\nThe reconstruction of the air-shower parameters consists of several steps described in more detail in the given references. \nFirst, the detector response is deconvoluted from the raw data using the phase and gain obtained by calibration measurements\\footnote{ \nLately we found that the correction for the phase responses was insufficient, i.e., delays were calculated too small and the distortion of the radio pulses due to dispersion was not accounted for properly. \nNevertheless, the pulse distortion by the signal chain is only a second-order effect of about $2\\,\\%$ on the maximum pulse amplitude we use in our standard analyses. \nThe necessary corrections on our previous results have been partially applied already: the figures presented here are still preliminary and we expect further changes that are small compared to other uncertainties, though. \nTogether with other minor improvements, e.g., in fitting procedures, this is the main reason why the results in this proceedings slightly differ from the ones published earlier. \n}.\nThen, we search for pulses with a signal-to-noise ratio $SNR > 10$ (in power) in a signal windows depending on the trigger time by Tunka-133 \/ Tunka-Grande \\cite{TunkaRex_NIM_2015}, and correct the pulse amplitude for the average bias due to noise \\cite{TunkaRex_Xmax2016}. \nThe threshold is tuned such that pure noise has a pass-chance of slightly below $5\\,\\%$. \nPhase and gain of the antenna response depend on the arrival direction of the radio signal and are deconvoluted in an iterative process while reconstructing the direction of the air shower from the pulse arrival times in the individual antenna stations. \nAt the moment we still use a plane-wave approximation for this purpose, since we need the arrival direction mainly as quality cut, where the expected difference to the more accurate hyperbolic wavefront of the order of $1^\\circ$ is unimportant \\cite{2014ApelLOPES_wavefront}.\nNonetheless, this simple method yields an accuracy of the arrival direction of about $1^\\circ$ for showers with zenith angles $\\theta < 50^\\circ$, and all events for which the Tunka-Rex direction disagrees by more than $5^\\circ$ from Tunka-133 are rejected as false-positive. \nThis removes almost all events passing the SNR cut by chance, and the potentially remaining ones have not deteriorated our main analysis results, since we do not observe any obvious outliers. \n\nEnergy and $X_\\mathrm{max}$ are reconstructed in a subsequent step from the lateral-distribution of the radio amplitude \\cite{KostuninTheory2015}. \nIn order to fit a simple, one dimensional lateral distribution function (LDF), we correct the amplitude in each station for the azimuthal asymmetry of the radio footprint. \nFor the correction we simply assume that the strength of the Askaryan effect accounts for a constant $8.5\\,\\%$ of the maximum geomagnetic amplitude (i.e, more for showers that are not perpendicular to the geomagnetic field), since more complicated correction formulas did not significantly improved the subsequent accuracy for the reconstructed energy and $X_\\mathrm{max}$.\nFinally we fit a Gaussian LDF and use the amplitude at $120\\,$m as estimator for the shower energy and the slope at $180\\,$m for the reconstruction of $X_\\mathrm{max}$ (see figure \\ref{fig_exampleEvent} for an example event). \nAll parameters, i.e., the proportionality coefficient for the energy and the function used for $X_\\mathrm{max}$, have been tuned against CoREAS simulations and not to the Tunka-133 measurements we compare to. \n\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.99\\linewidth]{exampleEventCombined.pdf}\n  \\caption{Example event after correction of the amplitudes in individual antennas for the geomagnetic angle and for the azimuthal asymmetry caused by the interference of the Askaryan and the geomagnetic effects. \n  Gray crosses are antenna below threshold, the small outer crosses are clusters without particle detectors that were not operating when the event was recorded.}\n  \\label{fig_exampleEvent}\n\\end{figure*}\n\n\\section{Results}\n\\label{section_results}\nUsing the standard method described above we can determine the energy of every event. \nFor $X_\\mathrm{max}$ the uncertainties are usually too large for events with only three or four antennas above threshold which is easy to understand: \nWhile the axis distance of $120\\,$m used for energy estimation is close to the maximum of the LDF, the amplitude is lower and closer to the noise level around the axis distance of $180\\,$m used for $X_\\mathrm{max}$ determination. \nConsequently, the threshold for $X_\\mathrm{max}$ is about $0.2$ higher in $\\lg E$ and less than one third of our events have sufficient quality to measure $X_\\mathrm{max}$. \nFigure \\ref{fig_energyXmax} shows the correlation of the radio and air-Cherenkov measurements of the same showers. \n\nAs cross-check that the correlation is real and not due to any unwanted implicit tuning, we kept the Tunka-133 reconstruction of half of the events blind to the persons working on the Tunka-Rex analysis. \nThe collaboration members responsible for the Tunka-133 reconstruction at first revealed only the shower axis and kept the energy and $X_\\mathrm{max}$ secret until they had received the corresponding Tunka-Rex values. \nThus, we consider the observed correlations an experimental proof that radio measurements can be used to measure not only the energy, but also $X_\\mathrm{max}$, as indicated earlier by LOPES \\cite{2012ApelLOPES_MTD}.\nThe Tunka-Rex precisions we have derived from the deviations to the Tunka-133 values are better than $15\\,\\%$ for the energy and about $40\\,$g\/cm$^2$ for $X_\\mathrm{max}$. \n\nAs next step, we aim at a lower threshold for $X_\\mathrm{max}$ by further developing the computing-intensive analysis method used by LOFAR \\cite{BuitinkLOFAR_Xmax2014, Kostunin_TopDownXmax_ICRC2017}.\nWe have already shown that the threshold can be lowered for the energy.\nThe energy can be reconstructed with slightly worse precision even for events with only a single antenna station above threshold when using the shower axis measured by Tunka-133 or Tunka-Grande.\nThis is possible because shower-to-shower fluctuations have only a small impact on the amplitude at the reference distance of $120\\,$m so that we can reconstruct the energy by using the average shape of the lateral distribution \\cite{Hiller_ARENA2016}. \nMoreover, we will check whether we can further improve the accuracy of our energy reconstruction by implementing ideas used by AERA \\cite{AERAenergyPRL_PRD_combined2016}, and we work on a calculation of the time-, energy- and direction-dependent efficiency as basis for an energy spectrum \\cite{Fedorov_TunkaRexEfficiency_ICRC2017}.\n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.48\\linewidth]{TunkaRexEnergyCorrelation.pdf}\n  \\hfill\n  \\includegraphics[width=0.50\\linewidth]{TunkaRexXmaxCorrelation.pdf}\n  \\caption{Correlation of the energy (left) and the distance to the shower maximum (right) between Tunka-Rex radio measurements and the Tunka-133 air-Cherenkov measurements of the same air-showers \\cite{TunkaRex_Xmax2016}.}\n  \\label{fig_energyXmax}\n\\end{figure*}\n\n\nMotivated by the high accuracy of the energy reconstruction we have applied our radio measurements on an important open issue in cosmic-ray physics: the absolute energy scale of different experiments \\cite{TunkaRexScale2016}. \nWhile the absolute scale accuracy of Tunka-Rex of $20\\,\\%$ is not better than the one of the host experiment Tunka-133, we were able to make a relative comparison to KASCADE-Grande with a higher accuracy of about $10\\,\\%$, since Tunka-Rex was calibrated by exactly the same reference source as LOPES, the radio extension of KASCADE-Grande. \nThe comparison of the energy scales has been done in two different ways, both using the primary energy reconstructed by the host experiments and the radio amplitudes measured by Tunka-Rex and LOPES. \n\nThe first method uses the ratio between the energy reconstructed by the host experiments and the radio amplitude at a common reference distance after correction for the local strengths of the geomagnetic field. \nNeglecting smaller effects, e.g., due to different observation levels and various peculiarities of the two radio arrays, different ratios between the radio amplitude and the energy directly translate into corresponding differences of the absolute energy scales of the host experiments. \nThe second method is based on CoREAS simulations using the energy reconstructed by the host experiments as true input. \nThus, the ratio between the simulated and observed radio amplitudes corresponds to the ratio of the energy scales of the host experiments. \nSince the real composition is unknown, we simulated both, a pure proton and a pure iron composition, as extreme cases with almost identical results. \nThe general advantage of the second method is that it takes the specific differences between the LOPES and Tunka-Rex arrays into account because the detector simulation is included, while the advantage of the first method is that it does not depend on the theoretical models implicit in the Monte Carlo simulations. \n\nBoth methods yield the same final result within their (correlated) systematic uncertainties. \nThe energy scales of Tunka-133 and KASCADE-Grande are equal within a systematic uncertainty of about $10\\,\\%$ (figure \\ref{fig_scaleComparison}).\nThe small and not significant difference of the energy scales measured by the radio detectors is the same as the one obtained directly from the energy spectra of the host experiments \\cite{2012ApelKGenergySpectrum, Tunka133_NIM2014}.\nThis is remarkable since Tunka-133 and KASCADE-Grande use completely different detection techniques, namely the measurement of air-Cherenkov light and the detection of secondary particles at ground, respectively. \n\n\\begin{figure*}[t]\n  \\centering\n  \\includegraphics[width=0.39\\linewidth]{TunkaRex_vs_CoREAS.pdf}\n  \\hfill\n  \\includegraphics[width=0.59\\linewidth]{energyScaleResult.pdf}\n  \\caption{Left: Comparison of the radio amplitudes in individual antenna stations measured by Tunka-Rex and simulated by CoREAS using the Tunka-133 energy as input. \n  They agree within uncertainties for both extreme cases of a pure proton or a pure iron composition. \n  Right: Ratio of the Tunka-133 and KASCADE-Grande energy scales determined with different methods via their radio extension Tunka-Rex and LOPES; within a total uncertainty of about $10\\,\\%$ both experiments have the same absolute scale \\cite{TunkaRexScale2016}.}\n  \\label{fig_scaleComparison}\n\\end{figure*}\n\n\\section{Conclusion}\nTunka-Rex has shown that radio arrays can provide a cost-effective enhancement for existing air-shower arrays in order to increase their total accuracy. \nThe radio measurements can compete with the established optical air-fluorescence and air-Cherenkov techniques regarding the absolute accuracy of the energy, which enables several science cases, e.g., the comparison of the absolute energy scales of different experiments. \nMoreover, radio measurements provide an measurement of $X_\\mathrm{max}$ during day time. \nThe $X_\\mathrm{max}$ resolution of the first Tunka-Rex measurements, however, cannot yet compete with the optical techniques. \nSince these first results are based on only one antenna station per cluster, we will soon study how much the resolution improves with the current configuration of three antenna stations per cluster. \nFinally, the combination of the radio measurements with the muon measurements by Tunka-Grande will provide an additional sensitivity to the mass composition.\n\n\\clearpage\n\n\\section*{Acknowledgements}\nThe construction of Tunka-Rex was funded by the German Helmholtz association and the Russian Foundation for Basic Research (grant HRJRG-303).\nThis work has been supported by the Helmholtz Alliance for Astroparticle Physics (HAP),\nby Deutsche Forschungsgemeinschaft (DFG grant SCHR 1480\/1-1),\nby the Russian Federation Ministry of Education and Science (projects 14.B25.31.0010, 2017-14-595-0001-003, No3.9678.2017\/8.9, No3.904.2017\/4.6, 3.6787.2017\/7.8, 1.6790.2017\/7.8), \nby the Russian Foundation for Basic Research (grants 16-02-00738, 16-32-00329, 17-02-00905),\nand by grant 15-12-20022 of the Russian Science Foundation (section~\\ref{section_results}).\n\n\\bibliographystyle{JHEP}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLet $\\Sigma_{g,p}$ denote a connected orientable surface of genus-$g$ with $p$-punctures, when $p=0$ we write  $\\Sigma_{g}$. \nThe \\textit{mapping class group} of $\\Sigma_{g,p}$ is the group of isotopy classes of orientation-preserving homeomorphisms of $\\Sigma_{g,p}$ preserving the set of punctures. \n\nHere is a brief history of generating sets for ${\\rm Mod}(\\Sigma_{g,p})$:\nDehn~\\cite{de} showed that  ${\\rm Mod}(\\Sigma_{g})$ can be generated by $2g(g-1)$ Dehn twists. About a quarter century later, Lickorish~\\cite{l3} gave a generating set  consisting of $3g-1$ Dehn twists. \nLater, Humphries~\\cite{H} reduced the number of Dehn twist generators to $2g+1$. He also proved that  the number $2g+1$ is minimal for $g\\geq 2$. \nJohnson~\\cite{j} proved that  the same set of Dehn twists also generates ${\\rm Mod}(\\Sigma_{g,1})$. \nIn the presence of multiple punctures, Gervais~\\cite{G} proved that ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by $2g+p$ Dehn twists for $p\\geq1$.\n\nIf it is not required that the generators are Dehn twists, then it is possible to obtain smaller generating sets for ${\\rm Mod}(\\Sigma_{g,p})$:   \nFor $g\\geq 3$ and $p=0$, Lu~\\cite[Theorem~$1.3$]{lu} proved that  \n${\\rm Mod}(\\Sigma_{g})$ can be generated by three elements\nFor $g\\geq 1$ and $p=0$ or $1$, \na minimal (since the group is not abelian) generating set of two elements, a product of two Dehn twists  and a product of $2g$ Dehn twists,   \nwas first given by Wajnryb~\\cite{w}.  Korkmaz~\\cite[Theorem~$5$]{mk2} improved this result by showing that one of these two generators can be taken as a Dehn twist. \nHe also showed that this group is generated by two elements of finite order ~\\cite[Theorem~$14$]{mk2}.  \nFor $g\\geq 3$, Kassabov obtained a generating set of  involution elements where the number of generators depends on $g$ and the parity of $p$  (see ~\\cite[Theorem~1]{ka}).\nLater, Monden~\\cite{m1} removed the parity condition on $p$ for $g=7$ and $g=5$. For $g\\geq1$ and $p\\geq 2$, Monden~\\cite{m2} also gave a generating set for ${\\rm Mod}(\\Sigma_{g,p})$ \nconsisting of three elements.  Recently, he~\\cite{m3} gave a minimal generating set for ${\\rm Mod}(\\Sigma_{g,p})$ containing two elements for $g\\geq3$.\n\n\nNote that any infinite group generated by two involutions must be isomorphic to the infinite dihedral group whose subgroups are either cyclic or dihedral.  Since \n${\\rm Mod}(\\Sigma_{g,p})$ contains nonabelian free groups, it cannot be generated by two involutions.  In this paper, we obtain the following result (cf. ~\\cite[Remark~5]{ka}):\n\n\\begin{thma}\\label{thma}\nFor every even integer $p\\geq 8$ and $g\\geq 14$, ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by three involutions.\nMoreover, for every even integer $p\\geq 4$ and for $g=3, 4, 5$ or $6$,  ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by four involutions.\n\\end{thma}\n\nAt the end of the paper, we also show that Theorem~A also \nholds for the cases $p=2$ or $p=3$.  For surfaces with odd number of punctures, we have the following result:\n\n\\begin{thmb}\\label{thmb}\nFor every odd integer $p\\geq 9$ and $g\\geq 13$, ${\\rm Mod}(\\Sigma_{g, p})$ is generated by four involutions.\nMoreover, for every odd integer $p\\geq 5$ and for $g=3, 4, 5$ or $6$,  ${\\rm Mod}(\\Sigma_{g,p})$ can be generated by five involutions.\n\\end{thmb}\n\nThe paper is organized as follows: In Section~\\ref{S2}, we quickly provide the necessary background on mapping class groups. \nThe proofs of Theorem~A  and Theorem~B  are given in Section~\\ref{S3}.  \n\n\n\\medskip\n\n\\noindent\n{\\bf Acknowledgements.}\nThis work was supported by the Scientific and Technological Research Council of Turkey (T\\\"{U}B\\.{I}TAK)[grant number 120F118].\n\n\n\\par  \n\\section{Background and Results on Mapping Class Groups} \\label{S2}\n\n Let $\\Sigma_{g,p}$ denote a connected orientable surface of genus $g$ with $p$ punctures specified by the set $P=\\lbrace z_1,z_2,\\ldots,z_p\\rbrace$ of $p$ distinguished points. If $p$ is zero then we omit it from the notation and write $\\Sigma_{g}$. {\\textit{The mapping class group}} \n ${\\rm Mod}(\\Sigma_{g,p})$ of the surface $\\Sigma_{g,p}$ is defined to be the group of the isotopy classes of orientation preserving\n self-diffeomorphisms of $\\Sigma_{g,p}$ which fix the set $P$. {\\textit{The mapping class group}} ${\\rm Mod}(\\Sigma_{g,p})$ of the surface $\\Sigma_{g,p}$ is defined to be the group of isotopy classes of all orientation preserving self-diffeomorphisms of $\\Sigma_{g,p}$ which fix the set $P$. Let ${\\rm Mod}_{0}(\\Sigma_{g,p})$ denote the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ consisting of elements which fix the set $P$ pointwise. It is obviuos that we have the following exact sequence:\n \\[\n1\\longrightarrow {\\rm Mod}_{0}(\\Sigma_{g,p}) \\longrightarrow {\\rm Mod}(\\Sigma_{g,p}) \\longrightarrow Sym_{p}\\longrightarrow 1,\n\\]\nwhere $Sym_p$ denotes the symmetric group on the set $\\lbrace1,2,\\ldots,p\\rbrace$ and the last projection is given by the restriction of the isotopy class of a diffeomorphism to its action on the punctures. \\par\nLet $\\beta_{i,j}$ be an embedded arc that joins two punctures $z_i$ and $z_j$ and does not intersect $\\delta$ on $\\Sigma_{g,p}$. \nLet $D_{i,j}$ denote a closed regular neighbourhood of $\\beta_{i,j}$, which is a disk with two punctures. \nThere exists a diffeomorphism $H_{i,j}: D_{i,j} \\to D_{i,j}$, which interchanges the punctures such that $H_{i,j}^{2}$ is equal to the right handed Dehn twist about $\\partial D_{i,j}$ and is the identity on the complement of the interior of $D_{i,j}$. Such a diffeomorphism is said to be \\textit{the (right handed) half twist} about $\\beta_{i,j}$. It can be extended to a diffeomorphism of ${\\rm Mod}(\\Sigma_{g,p})$. Throughout the paper we do not distinguish a \n diffeomorphism from its isotopy class. For the composition of two diffeomorphisms, we\nuse the functional notation; if $f$ and $g$ are two diffeomorphisms, then\nthe composition $fg$ means that $g$ is applied first and then $f$.\\\\\n\\indent\n For a simple closed \ncurve $a$ on $\\Sigma_{g,p}$, following ~\\cite{apy,mk1}, we denote the right-handed \nDehn twist $t_a$ about $a$ by the corresponding capital letter $A$.\nLet us also remind the following basic facts of Dehn twists that we use frequently throughout  the paper: Let $a$ and $b$ be \nsimple closed curves on $\\Sigma_{g,p}$ and $f\\in {\\rm Mod}(\\Sigma_{g,p})$.\n\\begin{itemize}\n\\item  If $a$ and $b$ are disjoint, then $AB=BA$ (\\textit{Commutativity}).\n\\item If $f(a)=b$, then $fAf^{-1}=B$ (\\textit{Conjugation}).\n\\end{itemize}\nLet us finish this section by noting that we  denote the conjugation relation $fgf^{-1}$ by $f^{g}$ for any $f,g \\in {\\rm Mod}(\\Sigma_{g,p})$.\n\n\n\n\n\\section{Involution generators for ${\\rm Mod}(\\Sigma_{g,p})$}\\label{S3}\nLet us start this section by recalling the following set of generators given by Korkmaz~\\cite[Theorem~$5$]{mk1}.\n\n\\begin{theorem}\\label{thm1}\nIf $g\\geq3$, then the mapping class group ${\\rm Mod}(\\Sigma_g)$ can be generated by the four elements $R$, $A_1A_{2}^{-1}$, $B_1B_{2}^{-1}$, $C_1C_{2}^{-1}$.\n\\end{theorem}\n\nLet us also recall the following well known result from algebra.\n\\begin{lemma}\\label{lemma1}\nLet $G$ and $K$ be groups. Suppose that the following short exact sequence holds,\n\\[\n1 \\longrightarrow N \\overset{i}{\\longrightarrow}G \\overset{\\pi}{\\longrightarrow} K\\longrightarrow 1.\n\\]\nThen the subgroup $\\Gamma$ contains $i(N)$ and has a surjection to $K$ if and only if $\\Gamma=G$.\n\\end{lemma}\n\\par\n\nIn our case where $G={\\rm Mod}(\\Sigma_{g,p})$ and $N={\\rm Mod}_{0}(\\Sigma_{g,p})$,\nwe have the following short exact sequence:\n\\[\n1\\longrightarrow {\\rm Mod}_{0}(\\Sigma_{g,p})\\longrightarrow {\\rm Mod}(\\Sigma_{g,p}) \\longrightarrow Sym_{p}\\longrightarrow 1.\n\\]\nTherefore, we obtain the following useful result which follows immediately from Lemma~\\ref{lemma1}. Let $\\Gamma$ be a subgroup of ${\\rm Mod}(\\Sigma_{g,p})$. If the subgroup $\\Gamma$ contains ${\\rm Mod}_{0}(\\Sigma_{g,p})$ and has a surjection to $Sym_p$ then $\\Gamma={\\rm Mod}(\\Sigma_{g,p})$.\n\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.3}{\\includegraphics{gevenpeven.png}}\n\\caption{The involutions $\\rho_1$ and $\\rho_2$ if $g=2k$ and $p=2b$.}\n\\label{EE}\n\\end{center}\n\\end{figure}\n\n\nThroughout the paper, we consider the embeddings of $\\Sigma_{g,p}$ into $\\mathbb{R}^{3}$ in such a way that it is invariant under the rotations \n$\\rho_1$ and $\\rho_2$.  Here, $\\rho_1$ and $\\rho_2$ are the rotations by $\\pi$ about the $z$-axis  (see Figures~\\ref{EE} and~\\ref{OE}). \nNote that ${\\rm Mod}(\\Sigma_{g,p})$ contains the element $R=\\rho_1\\rho_2$ which satisfies the following:\n\\begin{itemize}\n\\item [(i)]   $R(a_i)=a_{i+1}$, $R(b_i)=b_{i+1}$ for $i=1,\\ldots,g-1$ and $R(b_g)=b_{1}$,\n\\item [(ii)]  $R(c_i)=c_{i+1}$ for $i=1,\\ldots,g-2$,\n\\item [(iii)] $R(z_1)=z_p$ and $R(z_i)=z_{i-1}$ for $i=2,\\ldots,p$.\n\\end{itemize}\n\n\n\n\nWe want to note here that, in the following lemmata, where we present generating sets for surfaces with even number of punctures, we mainly follow the proof \nof \\cite[Theorem~$5$]{mk1}.   We use them in the proof of Theorem A and then for surfaces with odd number of punctures we explain how our arguments are modified.\n\n\\begin{lemma}\\label{lemeven}\nFor every even integer $g=2k\\geq14$ and every even integer $p=2b\\geq 10$, the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n\\[\n\\rho_1, \\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}\\]\ncontains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\n\\begin{proof}\n\n\nConsider the models of $\\Sigma_{g,p}$ depicted in Figure~\\ref{EE}. Let $F_1:=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}$ and let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n$\\rho_1$, $\\rho_2$ and $\\rho_1F_1$. One can see that the elements $R=\\rho_1\\rho_2$ and $F_1=\\rho_1 \\rho_1 F_1$ are contained in\nthe subgroup $\\Gamma$.\nLet $F_2$ be the element obtained by the conjugation of $F_1$ by $R^{-3}$. Since \n\\[\nR^{-3}(c_{k-3}, b_{k-1}, a_k, a_{k+2}, b_{k+3}, c_{k+4}) = (c_{k-6}, b_{k-4}, a_{k-3}, a_{k-1}, b_{k}, c_{k+1})\n\\]\nand\n\\[\nR^{-3}(z_{b-1},z_{b},z_{b+1})=(z_{b+2},z_{b+3},z_{b+4}),\n\\]\n\\[F_2=F_{1}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-4}A_{k-3}A_{k-1}^{-1}B_{k}^{-1}C_{k+1}^{-1}\\in \\Gamma. \n\\]\n\\noindent \nLet $F_3$ be the element $F_1^{F_1F_2^{-1}}$, that is\n$\nF_3=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1} \\in \\Gamma.\n$\n\nSince we use repeatedly similar calculations in the remaining parts of the paper, let us provide  some details here. It can be shown that the diffeomorphism $F_1F_2^{-1}$ maps the curves $\\lbrace c_{k-3},b_{k-1},a_k,a_{k+2},b_{k+3},c_{k+4} \\rbrace$ to the curves $\\lbrace c_{k-3},b_{k-1},a_k,b_{k+2},a_{k+3},c_{k+4}  \\rbrace$, respectively. Also it follows from the factorizations of half twists $H_{b-1,b}H_{b+1,b}^{-1}$ and $H_{b-4,b-3}H_{b-2,b-3}^{-1}$ commute and  we get\n\\begin{eqnarray*}\nF_3&=&F_1^{F_1F_2^{-1}}\\\\\n&=&(F_1F_2^{-1})(H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1})(F_1F_2^{-1})^{-1}\\\\\n&=&H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}.\n\\end{eqnarray*}\n The subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_4&=&F_{3}^{R^{-3}}=H_{b+2,b+3}H_{b+4, b+3}^{-1}C_{k-6}A_{k-4}B_{k-3}A_{k-1}^{-1}B_{k}^{-1}C_{k+1}^{-1},\\\\\nF_5&=&F_{3}^{F_3F_4}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}.\n\\end{eqnarray*} \n From these, we obtain the element \n $F_5F_3^{-1}=B_{k-3}C_{k-3}^{-1}$, which is contained in $\\Gamma$. By conjugating this element with powers of $R$, we conclude that  \n \\[\n B_{i}C_{i}^{-1}\\in \\Gamma \\  \\textrm{for}  \\ i=1, \\ldots, g-1.\n \\]\nThe subgroup $\\Gamma$ also contains the element $F_1F_3^{-1}=B_{k-1}A_{k}B_{k}^{-1}A_{k-1}^{-1}$.\nAfter conjugating with $R^3$ and considering the inverse, we have $A_{k+2}B_{k+3}A_{k+3}^{-1}B_{k+1}^{-1} \\in \\Gamma$.\nThis in turn  implies that for $i=1,\\ldots,g-1$, the elements\n\\[\nA_iB_{i+1}A_{i+1}^{-1}B_i^{-1}\\in \\Gamma.\n\\]\nWe also have the following elements in $\\Gamma$:\n\\begin{eqnarray*}\nF_6&=&F_1 (A_{k+2}B_{k+3}A_{k+3}^{-1}B_{k+2}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_{k}\nB_{k+2}^{-1}A_{k+3}^{-1}C_{k+4}^{-1},\\\\\nF_7&=&F_{6}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-4}A_{k-3}\nB_{k-1}^{-1}A_{k}^{-1}C_{k+1}^{-1} \\textrm{ and }\\\\\nF_8&=&F_{6}^{F_6F_7}=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nC_{k+1}^{-1}A_{k+3}^{-1}C_{k+4}^{-1},\n\\end{eqnarray*}\nHence, we can conclude that $F_8^{-1}F_6=C_{k+1}B_{k+2}^{-1}\\in \\Gamma$. Again conjugating with $R$  implies that\n\\[\nC_iB_{i+1}^{-1}\\in \\Gamma, \\ \\textrm{for all} \\  i=1,\\ldots,g-1. \n\\]\nFurthermore, we can see that  $\\Gamma$  contains the following elements: \n\\begin{eqnarray*}\nF_9&=&(B_{k-3}C_{k-3}^{-1})F_3(C_{k+4}B_{k+5}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_kA_{k+2}^{-1}B_{k+3}^{-1}B_{k+5}^{-1},\\\\\nF_{10}&=&F_{9}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-6}A_{k-4}B_{k-3}A_{k-1}^{-1}B_{k}^{-1}B_{k+2}^{-1},\\\\\nF_{11}&=&F_{9}^{F_9F_{10}}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}A_{k-1}B_kB_{k+2}^{-1}B_{k+3}^{-1}B_{k+5}^{-1}.\n\\end{eqnarray*}\nFrom these, we obtain $F_{11}F_{9}^{-1}=B_{k+2}^{-1} A_{k+2} \\in \\Gamma$. By the action of $R$, we can conclude that\n\\[\nA_iB_i^{-1}\\in \\Gamma \\ \\textrm{for all}  \\ i=1, \\ldots, g. \n\\]\nThis completes the proof by  Theorem~\\ref{thm1} since the subgroup $\\Gamma$ contains the elements\n\\begin{eqnarray*}\nA_1A_{2}^{-1}&=&(A_1B_1^{-1})(B_1C_1^{-1})(C_1B_{2}^{-1})(B_{2}A_{2}^{-1}),\\\\\nB_1B_{2}^{-1}&=&(B_1C_1^{-1})(C_1B_{2}^{-1}) \\textrm{ and }\\\\\nC_1C_{2}^{-1}&=&(C_1B_{2}^{-1})(B_{2}C_{2}^{-1}).\n\\end{eqnarray*}\n\\end{proof}\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.2}{\\includegraphics{goddpeven.png}}\n\\caption{The involutions $\\rho_1$ and $\\rho_2$ for $g=2k+1$ and $p=2b$.}\n\\label{OE}\n\\end{center}\n\\end{figure}\n\nIf $g$ is  odd and $p$ is even, we have the following result:\n\\begin{lemma}\\label{lemodd}\nFor every odd integer $g=2k+1 \\geq15$ and even integer $p=2b \\geq 10$, the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$  generated by the elements\n\\[\n\\rho_1,\\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\n\\begin{proof}\nConsider the models for $\\Sigma_{g,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ denote the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$ \ngenerated by the elements $\\rho_1$, $\\rho_2$ and $\\rho_1G_1$, where $G_1=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}$. \nThe elements $R=\\rho_1\\rho_2$ and $G_1=\\rho_1 \\rho_1 G_1$ belong to the subgroup $\\Gamma$. \nLet $G_2$ denote the conjugation of $G_1$ by $R^{-3}$,\n\\[\nG_2=G_{1}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}C_{k+2}^{-1}.\n\\]\nIt is easy to verify that the element\n\\begin{eqnarray*}\nG_3&=&G_{1}^{G_1G_2}\\\\\n&=&H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}B_kA_{k+1}A_{k+2}^{-1}C_{k+2}^{-1}C_{k+5}^{-1}\n\\end{eqnarray*}\nis contained in $\\Gamma$.\nLet\n\\[\nG_4=G_{3}^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}C_{k-1}^{-1}C_{k+2}^{-1}.\n\\]\nThus we get the element\n\\begin{eqnarray*}\nG_5=G_3^{G_3G_4^{-1}}\n=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}C_{k-1}A_{k+1}A_{k+2}^{-1}C_{k+2}^{-1}C_{k+5}^{-1},\n\\end{eqnarray*}\nwhich is contained in $G$. This implies that $G_3G_5^{-1}=B_kC_{k-1}^{-1}\\in \\Gamma$. By conjugating $B_kC_{k-1}^{-1}$ with powers of $R$, we see that\n\\[\nB_{i+1}C_{i}^{-1}\\in \\Gamma,\n\\]\nfor all $i=1,\\ldots,g-1$. In particular, the element $C_{k+5}B_{k+6}^{-1}\\in \\Gamma$. Hence, the subgroup $\\Gamma$ contains the following element:\n\\[\nG_6=G_1(C_{k+5}B_{k+6}^{-1})=H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}.\n\\]\nThen, we see that the elements\n\\begin{eqnarray*}\nG_7&=&G_6^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}C_{k-6}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}B_{k+3}^{-1} \\textrm{ and}\\\\\nG_8&=&G_6^{G_6G_7}=H_{b-1,b}H_{b+1,b}^{-1}B_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}\n\\end{eqnarray*}\nare contained in $\\Gamma$, which implies that the subgroup $\\Gamma$ contains the element $G_6G_8^{-1}=C_{k-3}B_{k-3}^{-1}$. By the action of $R$ we see that\n\\[\nC_iB_i^{-1} \\in \\Gamma\n\\]\nfor all $i=1,\\ldots, g-1$.  Moreover, we get \n\\begin{eqnarray*}\nG_9&=&(B_{k-2}C_{k-3}^{-1})G_6=H_{b-1,b}H_{b+1,b}^{-1}B_{k-2}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1} \\in \\Gamma,\\\\\nG_{10}&=&G_9^{R^{-3}}=H_{b+2,b+3}H_{b+4,b+3}^{-1}B_{k-5}B_{k-3}A_{k-2}A_{k-1}^{-1}B_{k}^{-1}B_{k+3}^{-1}\\in \\Gamma \\textrm{ and}\\\\\nG_{11}&=&G_9^{G_9G_{10}}=H_{b-1,b}H_{b+1,b}^{-1}A_{k-2}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}B_{k+6}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nFrom these, we have $G_9G_{11}^{-1}=B_{k-2}A_{k-2}^{-1}\\in \\Gamma$ so that \n\\[\nB_iA_i^{-1}\\in \\Gamma,\n\\]\nfor $i=1,\\ldots,g$, by the action of $R$. The remaining part of the proof can be completed as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\nIn the following four lemmata, we give generating sets for smaller genera\n\\begin{lemma}\\label{lem6}\nFor $g=6$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_2B_2A_3A_4^{-1}B_5^{-1} \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{6,p}$ as shown in Figure~\\ref{EE}. Let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{6,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_2F_1$ and $\\rho_1E_1$ where $F_1=B_2A_3A_4^{-1}B_5^{-1}$ and $E_1=H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}$. Hence the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_2 \\rho_2 F_1$ and $E_1=\\rho_1\\rho_1E_1$ are contained in the subgroup $\\Gamma$. \n\nThe subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_2&=&F_1^{R}=B_3A_4A_5^{-1}B_6^{-1} \\in \\Gamma,\\\\\nF_{3}&=&F_{1}^{F_1F_2}=B_2B_3A_4^{-1}A_5^{-1}\\in \\Gamma\\\\\nF_4&=&F_3^{R}=B_3B_4A_5^{-1}A_6^{-1}\\in \\Gamma\n\\textrm{ and}\\\\\nF_{5}&=&F_3^{F_3F_{4}^{-1}}=B_2B_3B_4^{-1}A_5^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nHence we get the element $F_5^{-1}F_3=B_4A_4^{-1}\\in \\Gamma$. By the action of $R$, for all $i=1,\\ldots,6$,\n\\[\nA_iB_i^{-1}\\in \\Gamma.\n\\]\nMoreover, we have\n\\begin{eqnarray*}\nF_6&=&E_1^{E_1F_3}=H_{b-1,b}H_{b+1,b}^{-1}B_{3}C_{4}^{-1} \\in \\Gamma\n\\textrm{ and}\\\\\nF_{7}&=&E_1^{E_1F_{1}}=H_{b-1,b}H_{b+1,b}^{-1}C_{3}B_{5}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nThis implies that $F_6E_1^{-1}=B_3C_3^{-1}\\in \\Gamma$ and $F_7^{-1}E_1=B_5C_4^{-1}\\in \\Gamma$ and so\n\\[\nB_iC_i^{-1}\\in \\Gamma \\textrm{ and } B_{i+1}C_{i}^{-1}\\in \\Gamma,\n\\]\nfor all $i=1,\\ldots,5$, by conjugating these elements with powers of $R$. The proof can be completed as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem5}\nFor $g=5$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_1H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1} \\textrm{ and }\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1} \n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{5,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ denote the subgroup of ${\\rm Mod}(\\Sigma_{5,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_1F_1$ and $\\rho_2E_1$ where $F_1=H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1}$ and $E_1=A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1}$. Thus the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_1 \\rho_1 F_1$ and $E_1=\\rho_2\\rho_2E_1$ are in the subgroup $\\Gamma$. \n\nOne can obtain the following elements:\n\\begin{eqnarray*}\nF_2&=&F_1^{R^{-1}}=H_{b,b+1}H_{b+2,b+1}^{-1}A_{2}A_{3}^{-1} \n\\\\\nF_{3}&=&F_2^{E_1}=H_{b,b+1}H_{b+2,b+1}^{-1}B_{2}A_{3}^{-1} \\textrm{ and}\\\\\nF_4&=&F_3^{E_1}=H_{b,b+1}H_{b+2,b+1}^{-1}C_{2}A_{3}^{-1},\n\\end{eqnarray*}\nwhich are contained in $\\Gamma$. Thus we get that $F_2F_3^{-1}=A_2B_2^{-1}\\in \\Gamma$ and $F_3F_4^{-1}=B_2C_2^{-1}\\in \\Gamma$. By conjugating these elements with powers of $R$, we see that \n\\[\nA_iB_i^{-1}\\in \\Gamma \\textrm{ and }B_jC_j^{-1}\\in \\Gamma,\n\\]\nwhich also implies that $A_iC_i^{-1}\\in \\Gamma$ for all $i=1,\\ldots, 5$ and $j=1,\\ldots, 4$. \nFinally, it can be verified that \n\\[\nE_1(a_3,c_3)=(a_3,b_4)\n\\]\nso that the group $\\Gamma$ contains the element\n\\[\n(A_3C_3^{-1})^{E_1}=A_3B_4^{-1}.\n\\]\nHence $A_iB_{i+1}^{-1}\\in \\Gamma$ for all $i=1,\\ldots,5$ by the action of $R$. The rest of the proof is similar to that of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\n\n\\begin{lemma}\\label{lem4}\nFor $g=4$ and every even integer $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_2B_1A_2A_3^{-1}B_4^{-1} \\textrm{ and }\\rho_1H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1} \n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,\\ldots,g$.\n\\end{lemma}\n\\begin{proof}\nLet us consider the models for $\\Sigma_{4,p}$ as shown in Figure~\\ref{EE} and let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{4,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_2F_1$ and $\\rho_1E_1$ where $F_1=B_1A_2A_3^{-1}B_4^{-1}$ and $E_1=H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1}$. Thus it is clear that the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_2 \\rho_2 F_1$ and $E_1=\\rho_1\\rho_1E_1$ belong to the subgroup $\\Gamma$.   We have the element\n\\[\nF_2=E_1^{E_1F_1}=H_{b-1,b} H_{b+1,b}^{-1}C_{2}B_{4}^{-1}\\in \\Gamma.\n\\]\nThus the subgroup $\\Gamma$ contains the elements $F{_2}^{-1} E_1=B_4C_3^{-1}$ and $\\rho_1(B_4C_3^{-1})\\rho_1=B_2C_2^{-1}$. By conjugating these elements with powers of $R$, we get\n\\[\nB_{i+1}C_i^{-1}\\in \\Gamma \\textrm{ and } B_iC_i^{-1}\\in \\Gamma\n\\]\nfor all $i=1,2,3$.  One can also obtain that the subgroup $\\Gamma$ contains the following elements:\n\\begin{eqnarray*}\nF_3&=&(C_1B_1^{-1})F_1=C_1A_2A_3^{-1}B_4^{-1},\\\\\nF_4&=&F_3^{R}(B_1C_1^{-1})=C_2A_3A_4^{-1}B_1^{-1}(B_1C_1^{-1})=C_2A_3A_4^{-1}C_1^{-1}\\textrm{ and}\\\\\nF_5&=&F_3^{F_3F_4}=C_1A_2A_3^{-1}A_4^{-1}.\n\\end{eqnarray*}\nThus we obtain that $F_5F_3^{-1}=A_4B_4^{-1}\\in \\Gamma$. By the action of $R$, $A_iB_i^{-1}\\in \\Gamma$ for all $i=1,2,3,4$.\nThe remaining part of the proof is very similar to that of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{lemma}\\label{lem3}\nFor $g=3$ and every even $p\\geq4$, the group generated by the elements\n\\[\n\\rho_1,\\rho_2, \\rho_1H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1} \n\\textrm{ and }\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}\n\\]\n contains the Dehn twists $A_i$, $B_i$ and $C_i$  for $i=1,2,3$.\n\\end{lemma}\n\\begin{proof}\nConsider the models for $\\Sigma_{3,p}$ as shown in Figure~\\ref{OE}. Let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{3,p})$ generated by the elements $\\rho_1$, $\\rho_2$, $\\rho_1F_1$ and $\\rho_2E_1$ where $F_1=H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1}$ and $E_1=A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}$. Thus the elements $R=\\rho_1\\rho_2$, $F_1=\\rho_1 \\rho_1 F_1$ and $E_1=\\rho_2\\rho_2E_1$ are contained in the subgroup $\\Gamma$. We get the elements\n\\begin{eqnarray*}\nF_2&=&F_1^{R^{-1}}=H_{b,b+1} H_{b+2,b+1}^{-1}A_{1}A_{2}^{-1}\\in \\Gamma,\\\\\nF_3&=&F_2^{E_1}=H_{b,b+1} H_{b+2,b+1}^{-1}B_{1}A_{2}^{-1}\\in \\Gamma \\textrm{ and}\\\\\nF_4&=&F_3^{E_1}=H_{b,b+1} H_{b+2,b+1}^{-1}C_{1}A_{2}^{-1}\\in \\Gamma.\n\\end{eqnarray*}\nFrom these, the subgroup $\\Gamma$ contains the elements $F_2F_3^{-1}=A_1B_1^{-1}$ and $F_3F_4^{-1}=B_1C_1^{-1}$, which implies that $A_1C_1^{-1}\\in \\Gamma$. Hence\n\\[\nA_iB_i^{-1}\\in \\Gamma, B_jC_j^{-1}\\in \\Gamma \\textrm{ and } A_jC_j^{-1}\\in \\Gamma\n\\]\nfor all $i=1,2,3$ and $j=1,2$, by the action of $R$.  We also have the following element\n\\[\n(A_2C_2^{-1})^{E_1}=A_2B_3^{-1},\n\\]\nwhich is contained in $\\Gamma$. This implies that \n\\[\nA_iB_{i+1}^{-1}\\in \\Gamma\n\\]\nfor $i=1,2$ by the action of $R$. One can complete the proof as in the proof of Lemma~\\ref{lemeven}.\n\\end{proof}\n\n\\begin{remark}\\label{podd}\nOur results in lemmata~\\ref{lemeven}--\\ref{lem3}  are also valid for surfaces with odd number of punctures.  To see that our proofs also work \nfor such surfaces we refer the reader to Figures~$3$ and $5$ in \\cite{apy1}.\n\\end{remark}\n\n\n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.3}{\\includegraphics{curves.png}}\n\\caption{The curves $e_{i,j}$ and $\\gamma_i$ on the surface $\\Sigma_{g,p}$.}\n\\label{C}\n\\end{center}\n\\end{figure}\n\n\n\nNow, in the remainder of the paper let $\\Gamma$ be the subgroup of ${\\rm Mod}(\\Sigma_{g,p})$\ngenerated by the elements given explicitly in lemmata~\\ref{lemeven}--\\ref{lem3} with the conditions mentioned in these lemmata.\nThe proof of the following lemma is similar to that of ~ \\cite[Lemma~$4.6$]{apy1}, nevertheless we give a proof for the sake of completeness of the paper.\n\n\\begin{lemma}\\label{lemma4}\nThe group ${\\rm Mod}_{0}(\\Sigma_{g,p})$ is contained in the group $\\Gamma$.\n\\end{lemma}\n\n\\begin{proof}\nIt follows from the subgroup $\\Gamma$ contains the elements $A_i$, $B_i$ and $C_j$ for all $i=1,\\ldots,g$ and $j=1,\\ldots,g-1$ by lemmata~\\ref{lemeven}--\\ref{lem3} that\nit is sufficient to prove that $\\Gamma$ contains the Dehn twists $E_{i.j}$ for some fixed $i$ ($j=1,2,\\ldots,p-1$). Let us first note that $\\Gamma$ contains $A_{g}$ and $R=\\rho_1\\rho_2$. Consider the models for $\\Sigma_{g,p}$ as shown in Figures~\\ref{EE} and~\\ref{OE}. By the fact that the diffeomorphism $R$ maps $a_{g}$ to $e_{1,p-1}$, we get\n\\[\nRA_{g}R^{-1}=E_{1,p-1} \\in \\Gamma.\n\\]\nThe diffeomorphism $\\phi_{i}=B_{i+1}\\Gamma_i^{-1}C_iB_i$ which maps each $e_{i,j}$ to $e_{i+1,j}$ for $j=1,2,\\ldots,p-1$ (see Figure~\\ref{C}). \nBy the proof of~ \\cite[Lemma~$4.5$]{apy1}, the group $\\Gamma$ contains the element $\\phi_{g}$. Thus we have\n\\[\n\\phi_{g-1}\\cdots \\phi_2\\phi_1E_{1,p-1}(\\phi_{g-1}\\cdots \\phi_2\\phi_1)^{-1}=E_{g,p-1}\\in H.\n\\]\nLikewise, the diffeomorphism $R$ sends $e_{g,p-1}$ to $e_{1,p-2}$. Then we obtain\n\\[\nRE_{g,p-1}R^{-1}=E_{1,p-2}\\in \\Gamma.\n\\]\nIt follows from \n\\[\n\\phi_{g-1}\\cdots \\phi_2\\phi_1E_{1,p-2}(\\phi_{g-1}\\cdots \\phi_2\\phi_1)^{-1}=E_{g,p-2}\\in \\Gamma\n\\]\n that\n \\[\n R(E_{g,p-2})R^{-1}=E_{1,p-3}\\in \\Gamma\n \\]\n Continuing in this way, we conclude that the elements $E_{1,1},E_{1,2},$ $\\ldots,E_{1,p-1}$ belong to $\\Gamma$, which completes the proof.\n\\end{proof}\n\n\\begin{proofa}\nConsider the surface $\\Sigma_{g,p}$ as in Figures~\\ref{EE} and~\\ref{OE}.\n\n\\underline{ If $g=2k\\geq 14$ and $p=2b\\geq 10$}: In this case, consider the surface $\\Sigma_{g,p}$ as in Figure~\\ref{EE}. Since\n\\[\n\\rho_1(c_{k-3})=c_{k+4}, \\rho_1(b_{k-1})=b_{k+3} \\textrm{ and }\\rho_1(a_{k})=a_{k\n+2},\n\\]\n we get\n\\[\n\\rho_1C_{k-3}\\rho_1=C_{k+4},\n\\rho_1B_{k-1}\\rho_1=B_{k+3} \\textrm{ and }\n\\rho_1A_{k}\\rho_1=A_{k+2}.\n\\]\nAlso, since  $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$,\nit is easy to see that $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}$ is an involution. Therefore, the generators of the subgroup $\\Gamma$ given in Lemma~\\ref{lemeven} are involutions.\n\n\n\\underline{ If $g=2k+1\\geq 13$ and $p=2b\\geq10$}: In this case, consider the surface $\\Sigma_{g,p}$ as in Figure~\\ref{OE}. It follows from\n\\[\n\\rho_1(c_{k-3})=c_{k+5}, \\rho_1(b_{k})=b_{k+3} \\textrm{ and }\\rho_1(a_{k+1})=a_{k\n+2},\n\\]\n that we have\n\\[\n\\rho_1C_{k-3}\\rho_1=C_{k+5},\n\\rho_1B_{k}\\rho_1=B_{k+3} \\textrm{ and }\n\\rho_1A_{k+1}\\rho_1=A_{k+2}.\n\\]\nAlso, by the fact that $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$,\nit is easy to see that the element $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1}$ is an involution. \nWe conclude that the generators of the subgroup $\\Gamma$ given in Lemma~\\ref{lemodd} are involutions.\n\n\n\n\\underline{ If $g=3,4,5$ or $g=6$ and $p=2b\\geq4$}:\nIt follows from\n\\begin{itemize}\n\\item $\\rho_2(b_2)=b_5$,  $\\rho_2(a_3)=a_4$ and $\\rho_1(c_3)=c_4$ if $g=6$,\n\\item $\\rho_1(a_3)=a_4$,  $\\rho_2(a_2)=a_4,\\rho_2(b_2)=b_4$ and $\\rho_2(c_2)=c_3$ if $g=5$,\n\\item $\\rho_2(b_1)=b_4$,  $\\rho_2(a_2)=a_3$ and $\\rho_1(c_2)=c_3$ if $g=4$ \n\\item $\\rho_1(a_2)=a_3$,  $\\rho_2(a_1)=a_3,\\rho_2(b_1)=b_3$ and $\\rho_2(c_1)=c_2$ if $g=3$ and\n\\item $\\rho_1H_{b-1,b}\\rho_1=H_{b+1,b}$ if $g=3,4,5$ or $g=6$\n\\end{itemize}\nthat the following elements:\n\\begin{itemize}\n\\item  $\\rho_2B_2A_3A_4^{-1}B_5^{-1}$ and $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{3}C_{4}^{-1}$ if $g=6$,\n\\item $\\rho_1H_{b-1,b}H_{b+1,b}^{-1}A_{3}A_{4}^{-1}$ and $\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1}$  if $g=5$,\n\\item $\\rho_2B_1A_2A_3^{-1}B_4^{-1}$ and $\\rho_1H_{b-1,b} H_{b+1,b}^{-1}C_{2}C_{3}^{-1}$ if $g=4$ and\n\\item $\\rho_1H_{b-1,b} H_{b+1,b}^{-1}A_{2}A_{3}^{-1}$\n and $\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1}$ if $g=3$ \n\\end{itemize}\ngiven in lemmata~\\ref{lem6}--\\ref{lem3} are involutions.\n\nNext, we show that the subgroup $\\Gamma$ is equal to the mapping class group ${\\rm Mod}(\\Sigma_{g,p})$.\nBy Lemma~\\ref{lemma4}, the group ${\\rm Mod}_{0}(\\Sigma_{g,p})$ is contained in the group $\\Gamma$. Hence, by Lemma~\\ref{lemma1}, we need to prove that \n$\\Gamma$ is mapped surjectively onto $Sym_p$. The element $\\rho_1\\rho_2 \\in \\Gamma$ has the image $(1,2,\\ldots,p)\\in Sym_p$. \n\nAs proven above,  the Dehn twists $A_i$, $B_i$ and $C_i$ belong to the subgroup $\\Gamma$. Thus, it can be easily observed that the factorization of half twists $H_{b-1,b}H_{b+1,b}^{-1}$ are contained in subgroup $\\Gamma$. Therefore, the group $\\Gamma$ also contains the following element:\n\\[\nR^{b-2}(H_{b-1,b}H_{b+1,b}^{-1})R^{2-b}=H_{1,2}H_{3,2}^{-1},\n\\]\nwhich has the image $(1,2,3)\\in Sym_p$. This completes the proof since the elements $(1,2,\\ldots,p)$ and $(1,2,3)$ of $Sym_p$ generate the whole group $Sym_p$ if $p$ is even~\\cite[Theorem B]{iz}.\n\n\\end{proofa}\n\nWhen the number of punctures is odd, we introduce an additional involution $\\rho_3$ (depicted in Figure~\\ref{rho3})  to our generating set.  The main reason behind adding an extra involution is for generating \nthe symmetric group $Sym_p$.  We want to point out that aside from generating $Sym_p$, all of our proofs in the case of even number of punctures work for odd number of punctures.\nFor $\\rho_1$ and $\\rho_2$, we distribute punctures as in Figures $3$ and $5$ in \\cite{apy1} (see also Remark~\\ref{podd}).\n \n\\begin{figure}[hbt!]\n\\begin{center}\n\\scalebox{0.35}{\\includegraphics{rho3.png}}\n\\caption{The involution $\\rho_3$  on the surface $\\Sigma_{g,p}$ for $p=2b+1$.}\n\\label{rho3}\n\\end{center}\n\\end{figure}\n\n\n\\begin{proofb}\nFor the first part of the proof we show that  \n\\begin{itemize}\n\\item [(i)] For every even integer $g=2k\\geq14$ and every odd integer $p=2b+1\\geq 9$, the subgroup ${\\rm Mod}_{0}(\\Sigma_{g,p})$ of ${\\rm Mod}(\\Sigma_{g,p})$ generated by the elements \n\\[\n\\rho_1, \\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1}, \\rho_3. \n\\]\n\n\\item[(ii)] For every odd integer $g=2k+1 \\geq15$ and odd integer $p=2b+1 \\geq 9$, the subgroup ${\\rm Mod}_{0}(\\Sigma_{g,p})$ of ${\\rm Mod}(\\Sigma_{g,p})$  generated by the elements\n\\[\n\\rho_1,\\rho_2 \\textrm{ and }\\rho_1H_{b-1,b}H_{b+1,b}^{-1}C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1},  \\rho_3.\n\\]\n\\end{itemize}\nNote that, it is enough to prove that the subgroup generated by the elements above mapped surjectively onto $Sym_p$.  \nFor this, consider the images of the elements $\\rho_1$, $\\rho_2$ and $\\rho_3$ \n\\begin{align*}\n&(1, p-1) (2, p-2) \\ldots (b, b+1),\\\\\n&(1,p) (2, p-1) \\ldots (b, b+2), \\\\\n&(2, p-1) (3, p-2) \\ldots (b, b+2).\n\\end{align*}\nThis finishes the proof for the first part,since these elements generate $Sym_p$, see \\cite[Lemma 6]{m1}.  For the second part of the theorem, note that adding $\\rho_3$ to the corresponding \ngenerating set given in Theorem A, finishes the proof.\n\\end{proofb}\n\n\nAs a last observation, one can prove that Theorem~A also holds  for the cases $p=2$ or $p=3$.\nIn theses cases, the generating set of $\\Gamma$ can be chosen as \n\\[\n\\begin{array}{lll}\n\\lbrace \\rho_1,\\rho_2,\\rho_1C_{k-3}B_{k-1}A_k\nA_{k+2}^{-1}B_{k+3}^{-1}C_{k+4}^{-1} \\rbrace & \\textrm{if} & g=2k\\geq14,\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1C_{k-3}B_kA_{k+1}A_{k+2}^{-1}B_{k+3}^{-1}C_{k+5}^{-1} \\rbrace & \\textrm{if} & g=2k+1\\geq13.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_2B_2A_3A_4^{-1}B_5^{-1},\\rho_1C_{3}C_{4}^{-1} \\rbrace & \\textrm{if} & g=6.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1A_{3}A_{4}^{-1},\\rho_2A_2B_2C_2C_3^{-1}B_4^{-1}A_4^{-1} \\rbrace & \\textrm{if} & g=5.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_2B_1A_2A_3^{-1}B_4^{-1},\\rho_1C_{2}C_{3}^{-1} \\rbrace & \\textrm{if} & g=4.\\\\\n\\lbrace \\rho_1,\\rho_2,\\rho_1A_{2}A_{3}^{-1},\\rho_2A_1B_1C_1C_2^{-1}B_3^{-1}A_3^{-1} \\rbrace & \\textrm{if} & g=3.\\\\\n\\end{array}.\n\\]\nIt can be easily proven that the group $\\Gamma$ contains ${\\rm Mod}_{0}(\\Sigma_{g,p})$ by the similar arguments in the proofs of  lemmata~\\ref{lemeven}--\\ref{lem3}. \nThe element $\\rho_1\\rho_2 \\in \\Gamma$ has the image $(1,2,\\ldots,p)\\in Sym_p$. Hence, this element generates $Sym_p$ for $p=2$. If $p=3$, we distribute the punctures \nas in ~ \\cite[Figure~$1$]{ka}. Then the element $\\rho_1$ has the image $(1,3)$, which generate $Sym_p$ together with the element $(1,2,3)$. Therefore, the group $\\Gamma$ \nis mapped surjectively onto $Sym_p$ for $p=2,3$. One can conclude that the group $\\Gamma$ is equal to ${\\rm Mod}(\\Sigma_{g,p})$.\n \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\n\nComputing integrals is a core challenge in machine learning and numerical methods play a central role in this area. \nThis can be problematic when a numerical integration routine is repeatedly called, maybe millions of times, within a larger computational pipeline.\nIn such situations, the cumulative impact of numerical errors can be unclear, especially in cases where the error has a non-trivial structural component.\nOne solution is to model the numerical error statistically and to propagate this source of uncertainty through subsequent computations.\nConversely, an understanding of how errors arise and propagate can enable the efficient focusing of computational resources upon the most challenging numerical integrals in a pipeline.\n\nClassical numerical integration schemes do not account for prior information on the integrand and, as a consequence, can require an excessive number of function evaluations to obtain a prescribed level of accuracy \\cite{OHagan1984}.\nAlternatives such as Quasi-Monte Carlo (QMC) can exploit knowledge on the smoothness of the integrand to obtain optimal convergence rates \\cite{Dick2010}.\nHowever these optimal rates can only hold on sub-sequences of sample sizes $n$, a consequence of the fact that all function evaluations are weighted equally in the estimator \\cite{Owen2014}.\nA modern approach that avoids this problem is to consider arbitrarily weighted combinations of function values; the so-called \\textit{quadrature rules} (also called cubature rules). \nWhilst quadrature rules with non-equal weights have received comparatively little theoretical attention, it is known that the extra flexibility given by arbitrary weights can lead to extremely accurate approximations in many settings (see applications to image de-noising \\cite{Chen2015} and mental simulation in psychology \\cite{Hamrick2013mental}).\n\n\nProbabilistic numerics, introduced in the seminal paper of \\cite{Diaconis1988}, aims at re-interpreting numerical tasks as inference tasks that are amenable to statistical analysis.\\footnote{A detailed discussion on probabilistic numerics and an extensive up-to-date bibliography can be found at \\url{http:\/\/www.probabilistic-numerics.org}.}\nRecent developments include probabilistic solvers for linear systems \\cite{Hennig2015solvers} and differential equations \\cite{Conrad2015,Schober2014}. For the task of computing integrals, Bayesian Quadrature (BQ) \\cite{OHagan1991} and more recent work by \\cite{Oates2015} provide probabilistic numerics methods that produce a full posterior distribution on the output of numerical schemes. \nOne advantage of this approach is that we can propagate uncertainty through all subsequent computations to explicitly model the impact of numerical error \\cite{Hennig2015ProbNum}. \nContrast this with chaining together classical error bounds; the result in such cases will typically be a weak bound that provides no insight into the error structure.\nAt present, a significant shortcoming of these methods is the absence of theoretical results relating to rates of posterior contraction. This is unsatisfying and has likely hindered the adoption of probabilistic approaches to integration, since it is not clear that the induced posteriors represent a sensible quantification of the numerical error (by classical, frequentist standards).\n\nThis paper establishes convergence rates for a new  probabilistic approach to integration.\nOur results thus overcome a key perceived weakness associated with probabilistic numerics in the quadrature setting.\nOur starting point is recent work by \\cite{Bach2012}, who cast the design of quadrature rules as a problem in convex optimisation that can be solved using the Frank-Wolfe (FW) algorithm.\nWe propose a hybrid approach of \\cite{Bach2012} with BQ, taking the form of a quadrature rule, that (i) carries a full probabilistic interpretation, (ii) is amenable to rigorous theoretical analysis, and (iii) converges orders-of-magnitude faster, empirically, compared with the original approaches in \\cite{Bach2012}.\nIn particular, we prove that super-exponential rates hold for posterior contraction (concentration of the posterior probability mass on the true value of the integral), showing that the posterior distribution provides a sensible and effective quantification of the uncertainty arising from numerical error. \nThe methodology is explored in simulations and also applied to a challenging model selection problem from cellular biology, where numerical error could lead to mis-allocation of expensive resources.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Background} \\label{section:sigmapoint}\n\n\\subsection{Quadrature and Cubature Methods}\n\nLet $\\mathcal{X} \\subseteq \\mathbb{R}^d$ be a measurable space such that $d \\in \\mathbb{N}_{+}$ and consider a probability density $p(x)$ defined with respect to the Lebesgue measure on $\\mathcal{X}$.\nThis paper focuses on computing integrals of the form $\\int f(x) p(x) \\mathrm{d}x$ for a test function $f:\\mathcal{X} \\rightarrow \\mathbb{R}$ where, for simplicity, we assume $f$ is square-integrable with respect to $p(x)$.\nA \\textit{quadrature rule} approximates such integrals as a weighted sum of function values at some design points $\\{x_i\\}_{i=1}^n \\subset \\mathcal{X}$:\n\\begin{equation}\n\\int_\\mathcal{X} f(x) p(x) \\mathrm{d}x \\approx \\sum_{i=1}^n w_i f(x_i).\n\\end{equation}\nViewing integrals as projections, we write $p[f]$ for the left-hand side and $\\hat{p}[f]$ for the right-hand side, where $\\hat{p} = \\sum_{i=1}^n w_i \\delta(x_i)$ and $\\delta(x_i)$ is a Dirac measure at $x_i$. \nNote that $\\hat{p}$ may not be a probability distribution; in fact, weights $\\{w_i\\}_{i=1}^n$ do not have to sum to one or be non-negative.\nQuadrature rules can be extended to multivariate functions $f:\\mathcal{X} \\rightarrow \\mathbb{R}^d$ by taking each component in turn.\n\nThere are many ways of choosing combinations $\\{x_i,w_i\\}_{i=1}^n$ in the literature. For example, taking weights to be $w_i = 1\/n$ with points $\\{x_i\\}_{i=1}^n$ drawn independently from the probability distribution $p(x)$ recovers basic Monte Carlo integration. \nThe case with weights $w_i= 1\/n$, but with points chosen with respect to some specific (possibly deterministic) schemes includes kernel herding \\cite{Chen2010} and Quasi-Monte Carlo (QMC) \\cite{Dick2010}. In Bayesian Quadrature, the points $\\{x_i\\}_{i=1}^n$ are chosen to minimise a posterior variance, with weights $\\{w_i\\}_{i=1}^n$ arising from a posterior probability distribution.\n\nClassical error analysis for quadrature rules is naturally couched in terms of minimising the worst-case estimation error. Let $\\mathcal{H}$ be a Hilbert space of functions $f: \\mathcal{X}\\rightarrow \\mathbb{R}$, equipped with the inner product $\\langle \\cdot,\\cdot \\rangle_{\\mathcal{H}}$ and associated norm $\\|\\cdot\\|_{\\mathcal{H}}$. \nWe define the \\textit{maximum mean discrepancy} (MMD) as:\n\\begin{equation}\n\\text{MMD}\\big(\\{x_i,w_i\\}_{i=1}^n \\big) \\coloneqq  \\sup_{\\substack{f \\in \\mathcal{H}: \\|f\\|_{\\mathcal{H}}=1}} \\big|p[f] - \\hat{p}[f] \\big|.\n\\end{equation}\nThe reader can refer to \\cite{Sriperumbudur2009} for conditions on $\\mathcal{H}$ that are needed for the existence of the MMD. The rate at which the MMD decreases with the number of samples $n$ is referred to as the `convergence rate' of the quadrature rule. \nFor Monte Carlo, the MMD decreases with the slow rate of $\\mathcal{O}_P(n^{-1\/2})$ (where the subscript $P$ specifies that the convergence is in probability). Let $\\mathcal{H}$ be a RKHS with reproducing kernel $k: \\mathcal{X}\\times \\mathcal{X} \\rightarrow \\mathbb{R}$ and denote the corresponding canonical feature map by $\\Phi(x) = k(\\cdot,x)$, so that the mean element is given by $\\mu_p(x) = p[\\Phi(x)] \\in \\mathcal{H}$. \nThen, following \\cite{Sriperumbudur2009}\n\\begin{equation}\n\\text{MMD}\\big(\\{x_i,w_i\\}_{i=1}^n \\big) =  \\| \\mu_p - \\mu_{\\hat{p}} \\|_{\\mathcal{H}}.\n\\end{equation}\nThis shows that to obtain low integration error in the RKHS $\\mathcal{H}$, one only needs to obtain a good approximation of its mean element $\\mu_p$ (as $\\forall f \\in \\mathcal{H}$: $p[f] = \\langle f , \\mu_p \\rangle_{\\mathcal{H}}$). \nEstablishing theoretical results for such quadrature rules is an active area of research \\cite{Bach2015}.\n\n\n\n\n\n\n\n\\subsection{Bayesian Quadrature} \\label{subsec:BQ}\n\n\n\n\n\nBayesian Quadrature (BQ) was originally introduced in \\cite{OHagan1991} and later revisited by \\cite{Rasmussen2003,Gunter2014} and \\cite{Osborne2012}. \nThe main idea is to place a functional prior on the integrand $f$, then update this prior through Bayes' theorem by conditioning on both samples $\\{x_i\\}_{i=1}^n$ and function evaluations at those sample points $\\{\\mathrm{f}_i\\}_{i=1}^n$ where $\\mathrm{f}_i = f(x_i)$. \nThis induces a full posterior distribution over functions $f$ and hence over the value of the integral $p[f]$. \nThe most common implementation assumes a Gaussian Process (GP) prior $f \\sim \\mathcal{GP}(0,k)$. A useful property motivating the use of GPs is that linear projection preserves normality, so that the posterior distribution for the integral $p[f]$ is also a Gaussian, characterised by its mean and covariance. \nA natural estimate of the integral $p[f]$ is given by the mean of this posterior distribution, which can be compactly written as\n\\begin{equation}\n\\hat{p}_{\\text{BQ}}[f] = \\mathrm{z}^T K^{-1} \\mathrm{f}.\n\\label{BQmeaneq}\n\\end{equation}\nwhere $\\mathrm{z}_i = \\mu_p(x_i)$ and $K_{ij} = k(x_i,x_j)$. Notice that this estimator takes the form of a quadrature rule with weights $\\mathrm{w}^{\\text{BQ}} =\\mathrm{z}^T K^{-1}$. Recently, \\cite{Sarkka2015} showed how specific choices of kernel and design points for BQ can recover classical quadrature rules. This begs the question of how to select design points $\\{x_i\\}_{i=1}^n$.\nA particularly natural approach aims to minimise the posterior uncertainty over the integral $p[f]$, which was shown in \\cite[Prop. 1]{Huszar2012} to equal:\n\\begin{equation}\nv_{\\text{BQ}}\\big(\\{x_i\\}_{i=1}^n \\big) \\; = \\; p [\\mu_p] - \\mathrm{z}^T K^{-1} \\mathrm{z} \\; = \\; \\text{MMD}^2\\big(\\{x_i,w_i^{\\text{BQ}}\\}_{i=1}^n \\big).\n\\label{eq:variance}\n\\end{equation}\nThus, in the RKHS setting, minimising the posterior variance corresponds to minimising the worst case error of the quadrature rule.\nBelow we refer to Optimal BQ (OBQ) as BQ coupled with design points $\\{x_i^\\text{OBQ}\\}_{i=1}^n$ chosen to globally minimise \\eqref{eq:variance}. \nWe also call Sequential BQ (SBQ) the algorithm that greedily selects design points to give the greatest decrease in posterior variance at each iteration. \nOBQ will give improved results over SBQ, but cannot be implemented in general, whereas SBQ is comparatively straight-forward to implement. There are currently no theoretical results establishing the convergence of either BQ, OBQ or SBQ.\n\n\n\n\n{\\it Remark:} \\eqref{eq:variance} is independent of observed function values $\\mathrm{f}$. \nAs such, no active learning is possible in SBQ (i.e. surprising function values never cause a revision of a planned sampling schedule). \nThis is not always the case:\nFor example \\cite{Gunter2014} approximately encodes non-negativity of $f$ into BQ which leads to a dependence on $\\mathrm{f}$ in the posterior variance.\nIn this case sequential selection becomes an {\\it active} strategy that outperforms batch selection in general.\n\n\n\n\n\n\n\n\n\n\\subsection{Deriving Quadrature Rules via the Frank-Wolfe Algorithm} \\label{section:introFW}\n\n\n\nDespite the elegance of BQ, its convergence rates have not yet been rigorously established.\nIn brief, this is because $\\hat{p}_{\\text{BQ}}[f]$ is an orthogonal projection of $f$ onto the {\\it affine} hull of $\\{\\Phi(x_i)\\}_{i=1}^n$, rather than e.g. the {\\it convex} hull. \nStandard results from the optimisation literature apply to bounded domains, but the affine hull is not bounded (i.e. the BQ weights can be arbitrarily large and possibly negative).\nBelow we describe a solution to the problem of computing integrals recently proposed by \\cite{Bach2012}, based on the FW algorithm, that restricts attention to the (bounded) convex hull of $\\{\\Phi(x_i)\\}_{i=1}^n$.\n\nThe Frank-Wolfe (FW) algorithm (Alg. \\ref{alg:FWalgorithm}), also called the conditional gradient algorithm, is a convex optimization method introduced in \\cite{Frank1956}. It considers problems of the form $\\min_{g \\in \\mathcal{G}} J(g)$ where the function $J:\\mathcal{G}\\rightarrow\\mathbb{R}$ is convex and continuously differentiable. \nA particular case of interest in this paper will be when the domain $\\mathcal{G}$ is a compact and convex space of functions, as recently investigated in \\cite{Jaggi2013}. These assumptions imply the existence of a solution to the optimization problem.\n\nAt each iteration $i$, the FW algorithm computes a linearisation of the objective function $J$ at the previous state $g_{i-1} \\in \\mathcal{G}$ along its gradient $(DJ)(g_{i-1})$ and selects an `atom' $\\bar{g}_i \\in \\mathcal{G}$ that minimises the inner product a state $g$ and $(DJ)(g_{i-1})$. \nThe new state $g_{i} \\in \\mathcal{G}$ is then a convex combination of the previous state $g_{i-1}$ and of the atom $\\bar{g}_{i}$. This convex combination depends on a step-size $\\rho_i$ which is pre-determined and different versions of the algorithm may have different step-size sequences. \n\n\\begin{algorithm}[t]\n\\caption{The Frank-Wolfe (FW) and Frank-Wolfe with Line-Search (FWLS) Algorithms.}\n\\label{alg:FWalgorithm}\n\\begin{algorithmic}[1]\n\\Require function $J$, initial state $g_1=\\bar{g}_1 \\in \\mathcal{G}$ (and, for FW only: step-size sequence $\\{\\rho_i\\}_{i=1}^n$).\n\\For{ $i = 2,\\ldots,n$ }\n\\State Compute $\\bar{g}_i = \\text{argmin}_{g \\in \\mathcal{G}} \\big\\langle g , (DJ)(g_{i-1}) \\big\\rangle_{\\times} $\n\\State [For FWLS only, line search: $\\rho_i = \\text{argmin}_{\\rho \\in [0,1]} J \\bigl((1-\\rho) g_{i-1} + \\rho\\, \\bar{g}_i \\bigr)$]\n\\State Update $g_{i} = (1 - \\rho_i) g_{i-1} + \\rho_i \\bar{g}_i$ \n\\EndFor\n\\end{algorithmic}\n\\end{algorithm}\n\n\nOur goal in quadrature is to approximate the mean element $\\mu_p$.\nRecently \\cite{Bach2012} proposed to frame integration as a FW optimisation problem. Here, the domain $\\mathcal{G} \\subseteq \\mathcal{H}$ is a space of functions and taking the objective function to be:\n\\begin{equation}\nJ(g) = \\frac{1}{2}\\big\\| g - \\mu_p \\big\\|^2_{\\mathcal{H}}.\n\\end{equation}\nThis gives an approximation of the mean element and $J$ takes the form of half the posterior variance (or the MMD$^2$). In this functional approximation setting, minimisation of $J$ is carried out over $\\mathcal{G} = \\mathcal{M}$, the marginal polytope of the RKHS $\\mathcal{H}$. \nThe marginal polytope $\\mathcal{M}$ is defined as the closure of the convex hull of $\\Phi(\\mathcal{X})$, so that in particular $\\mu_p \\in \\mathcal{M}$.\nAssuming as in \\cite{Lacoste-Julien2015} that $\\Phi(x)$ is uniformly bounded in feature space (i.e. $\\exists R>0: \\forall x \\in \\mathcal{X}$, $\\|\\Phi(x)\\|_{\\mathcal{H}} \\leq R$), then $\\mathcal{M}$ is a closed and bounded set and can be optimised over. \n\nIn order to define the algorithm rigorously in this case, we introduce the Fr\\'{e}chet derivative of $J$, denoted $DJ$, such that for $\\mathcal{H}^*$ being the dual space of $\\mathcal{H}$, we have the unique map $DJ:\\mathcal{H} \\rightarrow \\mathcal{H}^*$ such that for each $g \\in \\mathcal{H}$, $(DJ)(g)$ is the function mapping $h \\in \\mathcal{H}$ to $(DJ)(g)(h) = \\big\\langle g - \\mu, h \\big\\rangle_\\mathcal{H}$. We also introduce the bilinear map $\\langle \\cdot, \\cdot \\rangle_{\\times}: \\mathcal{H} \\times \\mathcal{H}^* \\rightarrow \\mathbb{R}$ which, for $F \\in \\mathcal{H}^*$ given by $F(g) = \\langle g, f \\rangle_\\mathcal{H}$, is the rule giving $\\langle h, F \\rangle_{\\times} = \\langle h, f \\rangle_{\\mathcal{H}}$. \n\n\nA particular advantage of this method is that it leads to `sparse' solutions which are linear combinations of the atoms $\\{\\bar{g}_i\\}_{i=1}^n$ \\cite{Bach2012}.\nIn particular this provides a weighted estimate for the mean element:\n\\begin{equation}\\label{eq:FWsparse}\n\\hat{\\mu}_{\\text{FW}} \\coloneqq g_n = \\sum_{i=1}^n \\Big( \\prod_{j=i+1}^{n} \\big( 1 - \\rho_{j-1} \\big) \\rho_{i-1} \\Big) \\bar{g}_i  \\coloneqq \\sum_{i=1}^n w_i^{\\text{FW}} \\bar{g}_i ,\n\\end{equation}\nwhere by default $\\rho_0 = 1$ which leads to all $w_i^{\\text{FW}} \\in [0,1]$ when $\\rho_i = 1\/(i+1)$.\nA typical sequence of approximations to the mean element is shown in Fig. \\ref{fig:designpoints} (left), demonstrating that the approximation quickly converges to the ground truth (in black).\nSince minimisation of a linear function can be restricted to extreme points of the domain, the atoms will be of the form $\\bar{g}_i = \\Phi(x_i^{\\text{FW}}) = k(\\cdot ,x_i^{\\text{FW}})$ for some $x_i^{\\text{FW}} \\in \\mathcal{X}$. The minimisation in $g$ over $\\mathcal{G}$ from step 2 in Algorithm \\ref{alg:FWalgorithm} therefore becomes a minimisation in $x$ over $\\mathcal{X}$ and this algorithm therefore provides us design points. In practice, at each iteration $i$, the FW algorithm hence selects a design point $x_i^{\\text{FW}} \\in \\mathcal{X}$ which induces an atom $\\bar{g}_i$ and gives us an approximation of the mean element $\\mu_p$.\nWe denote by $\\hat{\\mu}_{\\text{FW}}$ this approximation after $n$ iterations. Using the reproducing property, we can show that the FW estimate is a quadrature rule:\n\\begin{equation}\n\\hat{p}_{\\text{FW}}[f]\n\\coloneqq \\big\\langle f,\\hat{\\mu}_{\\text{FW}} \\big\\rangle_{\\mathcal{H}}\n= \\Big\\langle f , \\sum_{i=1}^n w_i^{\\text{FW}} \\bar{g}_i \\Big\\rangle_{\\mathcal{H}}\n= \\sum_{i=1}^n  w_i^{\\text{FW}} \\big\\langle f , k(\\cdot,x_i^{\\text{FW}}) \\big\\rangle_{\\mathcal{H}}\n= \\sum_{i=1}^n w_i^{\\text{FW}} f(x_i^{\\text{FW}}).\n\\end{equation}\nThe total computational cost for FW is $\\mathcal{O}(n^2)$. \nAn extension known as FW with Line Search (FWLS) uses a line-search method to find the optimal step size $\\rho_i$ at each iteration (see Alg. \\ref{alg:FWalgorithm}). \nOnce again, the approximation obtained by FWLS has a sparse expression as a convex combination of all the previously visited states and we obtain an associated quadrature rule.\nFWLS has theoretical convergence rates that can be stronger than standard versions of FW but has computational cost in $\\mathcal{O}(n^3)$.\nThe authors in \\cite{Garber2015} provide a survey of FW-based algorithms and their convergence rates under different regularity conditions on the objective function and domain of optimisation.\n\n{\\it Remark:} The FW design points $\\{x_i^{\\text{FW}}\\}_{i=1}^n$ are generally not available in closed-form.\nWe follow mainstream literature by selecting, at each iteration, the point that minimises the MMD over a finite collection of $M$ points, drawn i.i.d from $p(x)$.\nThe authors in \\cite{Lacoste-Julien2015} proved that this approximation adds a $\\mathcal{O}(M^{-1\/4})$ term to the MMD, so that theoretical results on FW convergence continue to apply provided that $M(n) \\rightarrow \\infty$ sufficiently quickly.\nAppendix A provides full details. \nIn practice, one may also make use of a numerical optimisation scheme in order to select the points.\n\n\\begin{figure}[t]\n  \\centering\n    \\includegraphics[width=0.49\\textwidth,clip,trim = 0.3cm 0.2cm 0.8cm 0.7cm]{muapproxtest2.pdf}\n   \n    \\includegraphics[width=0.49\\textwidth,clip,trim = 1cm 0.9cm 1.5cm 1cm]{designlocations.pdf}\n  \\caption{\\textit{Left}: Approximations of the mean element $\\mu_p$ using the FWLS algorithm, based on $n = 1, 2, 5, 10, 50$ design points (purple, blue, green, red and orange respectively). It is not possible to distinguish between approximation and ground truth when $n=50$. \\textit{Right}: Density of a mixture of 20 Gaussian distributions, displaying the first $n=25$ design points chosen by FW (red), FWLS (orange) and SBQ (green). \n  Each method provides well-spaced design points in high-density regions. Most FW and FWLS design points overlap, partly explaining their similar performance in this case.}\n\\label{fig:designpoints}\n\\end{figure}\n\n\n\n\\section{A Hybrid Approach: Frank-Wolfe Bayesian Quadrature} \\label{section:BayesianFW}\n\n\nTo combine the advantages of a probabilistic integrator with a formal convergence theory, we propose Frank-Wolfe Bayesian Quadrature (FWBQ).\nIn FWBQ, we first select design points $\\{x_i^{\\text{FW}}\\}_{i=1}^n$ using the FW algorithm.\nHowever, when computing the quadrature approximation, instead of using the usual FW weights $\\{w_i^{\\text{FW}}\\}_{i=1}^n$ we use instead the weights $\\{w_i^{\\text{BQ}}\\}_{i=1}^n$ provided by BQ.\nWe denote this quadrature rule by $\\hat{p}_{\\text{FWBQ}}$ and also consider $\\hat{p}_{\\text{FWLSBQ}}$, which uses FWLS in place of FW.\nAs we show below, these hybrid estimators (i) carry the Bayesian interpretation of Sec. \\ref{subsec:BQ}, (ii) permit a rigorous theoretical analysis, and (iii) out-perform existing FW quadrature rules by orders of magnitude in simulations. \nFWBQ is hence ideally suited to probabilistic numerics applications.\n\nFor these theoretical results we assume that $f$ belongs to a finite-dimensional RKHS $\\mathcal{H}$, in line with recent literature \\cite{Bach2012,Garber2015,Jaggi2013,Lacoste-Julien2015}.\nWe further assume that $\\mathcal{X}$ is a compact subset of $\\mathbb{R}^d$, that $p(x) > 0$ $\\forall x \\in \\mathcal{X}$ and that $k$ is continuous on $\\mathcal{X} \\times \\mathcal{X}$.\nUnder these hypotheses, Theorem \\ref{theorem:posteriormean} establishes consistency of the posterior mean, while Theorem \\ref{theo1} establishes contraction for the posterior distribution. \n\n\n\\begin{theorem}[Consistency] \\label{theorem:posteriormean}\nThe posterior mean $\\hat{p}_{\\emph{FWBQ}}[f]$ converges to the true integral $p[f]$ at the following rates:\n\\begin{equation}\n\\Big|p[f] - \\hat{p}_{\\emph{FWBQ}}[f]\\Big| \\leq \\text{MMD}\\big(\\{x_i,w_i\\}_{i=1}^n\\big) \\leq \\left\\{ \\begin{array}{cl} \\frac{2 D^2}{R}  n^{-1} & \\text{for FWBQ}\\\\\n \\sqrt{2}D \\exp(-\\frac{R^2}{2 D^2} n) & \\text{for FWLSBQ}  \\end{array} \\right.\n\\end{equation}\nwhere the FWBQ uses step-size $\\rho_i = 1\/(i+1)$, $D \\in (0,\\infty)$ is the diameter of the marginal polytope $\\mathcal{M}$ and $R \\in (0,\\infty)$ gives the radius of the smallest ball of center $\\mu_p$ included in $\\mathcal{M}$.\n\\end{theorem}\n\n\nNote that all the proofs of this paper can be found in Appendix B. An immediate corollary of Theorem \\ref{theorem:posteriormean} is that FWLSBQ has an asymptotic error which is exponential in $n$ and is therefore superior to that of any QMC estimator \\cite{Dick2010}.\nThis is not a contradiction - recall that QMC restricts attention to uniform weights, while FWLSBQ is able to propose arbitrary weightings.\nIn addition we highlight a robustness property:\nEven when the assumptions of this section do not hold, one still obtains atleast a rate $\\mathcal{O}_P (n^{-1\/2})$ for the posterior mean using either FWBQ or FWLSBQ \\cite{Dunn1980}.\n\n\n{\\it Remark}: The choice of kernel affects the convergence of the FWBQ method \\cite{Hennig2015ProbNum}. Clearly, we expect faster convergence if the function we are integrating is `close' to the space of functions induced by our kernel. \nIndeed, the kernel specifies the geometry of the marginal polytope $\\mathcal{M}$, that in turn directly influences the rate constant $R$ and $D$ associated with FW convex optimisation.\n\n\nConsistency is only a stepping stone towards our main contribution which establishes posterior contraction rates for FWBQ.\nPosterior contraction is important as these results justify, for the first time, the probabilistic numerics approach to integration; that is, we show that the {\\it full} posterior distribution is a sensible quantification (at least asymptotically) of numerical error in the integration routine:\n\n\\begin{theorem}[Contraction] \\label{theo1}\nLet $S \\subseteq \\mathbb{R}$ be an open neighbourhood of the true integral $p[f]$ and let $\\gamma = \\inf_{r \\in S^C} | r - p[f]| >0$.\nThen the posterior probability mass on $S^c = \\mathbb{R} \\setminus S$ vanishes at a rate:\n\\begin{equation}\n\\emph{prob}(S^c) \\leq \\left\\{ \\begin{array}{cl} \\frac{2\\sqrt{2}D^2}{\\sqrt{\\pi}R \\gamma} n^{-1} \\exp \\Big(- \\frac{\\gamma^2 R^2}{8 D^4} n^2 \\Big) \n& \\text{for FWBQ} \\\\ \n\\frac{2 D}{\\sqrt{\\pi} \\gamma} \\exp\\Big( - \\frac{R^2}{2 D^2}  n - \\frac{\\gamma^2}{2\\sqrt{2}D} \\exp\\big( \\frac{R^2}{2D^2} n\\big)\\Big)\n& \\text{for FWLSBQ} \\end{array} \\right.\n\\end{equation}\nwhere the FWBQ uses step-size $\\rho_i = 1\/(i+1)$, $D \\in (0,\\infty)$ is the diameter of the marginal polytope $\\mathcal{M}$ and $R \\in (0,\\infty)$ gives the radius of the smallest ball of center $\\mu_p$ included in $\\mathcal{M}$.\n\\end{theorem}\n\n\n\nThe contraction rates are exponential for FWBQ and super-exponential for FWLBQ, and thus the two algorithms enjoy both a probabilistic interpretation and rigorous theoretical guarantees. \nA notable corollary is that OBQ enjoys the same rates as FWLSBQ, resolving a conjecture by Tony O'Hagan that OBQ converges exponentially [personal communication]:\n\n\\begin{corollary*}\\label{theorem:OBQ_rates}\nThe consistency and contraction rates obtained for FWLSBQ apply also to OBQ.\n\\end{corollary*}\n\n\n\n\n\n\n\n\n\n\\section{Experimental Results}\\label{section:Experimental_Results}\n\n\\subsection{Simulation Study}\n\n\nTo facilitate the experiments in this paper we followed \\cite{Bach2015,Bach2012,Rasmussen2003,Lacoste-Julien2015} and employed an exponentiated-quadratic (EQ) kernel $k(x, x') \\coloneqq \\lambda^{2} \\exp ( -\\nicefrac{1}{2 \\sigma^2} \\|x-x'\\|^2_2)$.\nThis corresponds to an infinite-dimensional RKHS, not covered by our theory; nevertheless, we note that all simulations are practically finite-dimensional due to rounding at machine precision. See Appendix E for a finite-dimensional approximation using random Fourier features.\nEQ kernels are popular in the BQ literature as, when $p$ is a mixture of Gaussians, the mean element $\\mu_p$ is analytically tractable (see Appendix C). Some other $(p,k)$ pairs that produce analytic mean elements are discussed in \\cite{Bach2015}.\n\nFor this simulation study, we took $p(x)$ to be a 20-component mixture of 2D-Gaussian distributions. \nMonte Carlo (MC) is often used for such distributions but has a slow convergence rate in $\\mathcal{O}_P(n^{-1\/2})$.\nFW and FWLS are known to converge more quickly and are in this sense preferable to MC \\cite{Bach2012}. \nIn our simulations (Fig. \\ref{fig:sim study}, left), both our novel methods FWBQ and FWLSBQ decreased the MMD much faster than the FW\/FWLS methods of \\cite{Bach2012}. Here, the same kernel hyper-parameters $(\\lambda,\\sigma) = (1,0.8)$ were employed for all methods to have a fair comparison.\nThis suggests that the best quadrature rules correspond to elements {\\it outside} the convex hull of $\\{\\Phi(x_i)\\}_{i=1}^n$.\nExamples of those, including BQ, often assign negative weights to features (Fig. S1 right, Appendix D).\n\n\nThe principle advantage of our proposed methods is that they reconcile theoretical tractability with a fully probabilistic interpretation.\nFor illustration, Fig. \\ref{fig:sim study} (right) plots the posterior uncertainty due to numerical error for a typical integration problem based on this $p(x)$.\nIn-depth empirical studies of such posteriors exist already in the literature and the reader is referred to \\cite{Chen2015,Hamrick2013mental,OHagan1991} for details.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width = 0.48\\textwidth]{MMD2decreasefinal.pdf}\n\\includegraphics[width = 0.48\\textwidth]{posteriordensitytmp.pdf}\n\\caption{Simulation study. \n{\\it Left}: Plot of the worst-case integration error squared (MMD$^2$). \nBoth FWBQ and FWLSBQ are seen to outperform FW and FWLS, with SBQ performing best overall. \n{\\it Right}: Integral estimates for FWLS and FWLSBQ for a function $f \\in \\mathcal{H}$. FWLS converges more slowly and provides only a point estimate for a given number of design points. In contrast, FWLSBQ converges faster and provides a full probability distribution over numerical error shown shaded in orange (68\\% and 95\\% credible intervals). Ground truth corresponds to the dotted black line.}\n\\label{fig:sim study}\n\\end{figure}\n\n\nBeyond these theoretically tractable integrators, SBQ seems to give even better performance as $n$ increases. \nAn intuitive explanation is that SBQ picks $\\{x_i\\}_{i=1}^n$ to minimise the MMD whereas FWBQ and FWLSBQ only minimise an approximation of the MMD (its linearisation along $DJ$). \nIn addition, the SBQ weights are optimal at each iteration, which is not true for FWBQ and FWLSBQ. \nWe conjecture that Theorem \\ref{theorem:posteriormean} and \\ref{theo1} provide upper bounds on the rates of SBQ. \nThis conjecture is partly supported by Fig. \\ref{fig:designpoints} (right), which shows that SBQ selects similar design points to FW\/FWLS (but weights them optimally). Note also that both FWBQ and FWLSBQ give very similar result. This is not surprising as FWLS has no guarantees over FW in infinite-dimensional RKHS \\cite{Jaggi2013}.\n\n\n\n\n\\subsection{Quantifying Numerical Error in a Proteomic Model Selection Problem}\n\nA topical bioinformatics application that extends recent work by \\cite{Oates2014} is presented.\nThe objective is to select among a set of candidate models $\\{M_i\\}_{i=1}^m$ for protein regulation. This choice is based on a dataset $\\mathcal{D}$ of protein expression levels, in order to determine a `most plausible' biological hypothesis for further experimental investigation.\nEach $M_i$ is specified by a vector of kinetic parameters $\\theta_i$ (full details in Appendix D).\nBayesian model selection requires that these parameters are integrated out against a prior $p(\\theta_i)$ to obtain marginal likelihood terms $L(M_i) = \\int p(\\mathcal{D}|\\theta_i) p(\\theta_i) \\mathrm{d}\\theta_i$.\nOur focus here is on obtaining the {\\it maximum a posteriori} (MAP) model $M_j$, defined as the maximiser of the posterior model probability $L(M_j) \/ \\sum_{i=1}^m L(M_i)$ (where we have assumed a uniform prior over model space).\nNumerical error in the computation of each term $L(M_i)$, if unaccounted for, could cause us to return a model $M_k$ that is different from the true MAP estimate $M_j$ and lead to the mis-allocation of valuable experimental resources.\n\nThe problem is quickly exaggerated when the number $m$ of models increases, as there are more opportunities for one of the $L(M_i)$ terms to be `too large' due to numerical error.\nIn \\cite{Oates2014}, the number $m$ of models was combinatorial in the number of protein kinases measured in a high-throughput assay (currently $\\sim 10^2$ but in principle up to $\\sim 10^4$).\nThis led \\cite{Oates2014} to deploy substantial computing resources to ensure that numerical error in each estimate of $L(M_i)$ was individually controlled.\nProbabilistic numerics provides a more elegant and efficient solution:\nAt any given stage, we have a fully probabilistic quantification of our uncertainty in each of the integrals $L(M_i)$, shown to be sensible both theoretically and empirically.\nThis induces a full posterior distribution over numerical uncertainty in the location of the MAP estimate (i.e. `Bayes all the way down').\nAs such we can determine, on-line, the precise point in the computational pipeline when numerical uncertainty near the MAP estimate becomes acceptably small, and cease further computation.\n\nThe FWBQ methodology was applied to one of the model selection tasks in \\cite{Oates2014}.\nIn Fig. \\ref{fig:model posteriors} (left) we display posterior model probabilities for each of $m = 352$ candidates models, where a low number ($n = 10$) of samples were used for each integral.\n(For display clarity only the first 50 models are shown.)\nIn this low-$n$ regime, numerical error introduces a second level of uncertainty that we quantify by combining the FWBQ error models for all integrals in the computational pipeline; this is summarised by a box plot (rather than a single point) for each of the models (obtained by sampling - details in Appendix D).\nThese box plots reveal that our estimated posterior model probabilities are completely dominated by numerical error.\nIn contrast, when $n$ is increased through 50, 100 and 200 (Fig. \\ref{fig:model posteriors}, right and Fig. S2), the uncertainty due to numerical error becomes negligible. At $n = 200$ we can conclude that model $26$ is the true MAP estimate and further computations can be halted.\nCorrectness of this result was confirmed using the more computationally intensive methods in \\cite{Oates2014}.\n\nIn Appendix D we compared the relative performance of FWBQ, FWLSBQ and SBQ on this problem.\nFig. S1 shows that the BQ weights reduced the MMD by orders of magnitude relative to FW and FWLS and that SBQ converged more quickly than both FWBQ and FWLSBQ.\n\n\n\n\n\n\\begin{figure}[t]\n\\centering\n\\includegraphics[width = 0.48\\textwidth]{n10.pdf}\n\\includegraphics[width = 0.48\\textwidth]{n100.pdf}\n\\caption{Quantifying numerical error in a model selection problem. FWBQ was used to model the numerical error of each integral $L(M_i)$ explicitly.\nFor integration based on $n=10$ design points, FWBQ tells us that the computational estimate of the model posterior will be dominated by numerical error (left). \nWhen instead $n=100$ design points are used (right), uncertainty due to numerical error becomes much smaller (but not yet small enough to determine the MAP estimate).}\n\\label{fig:model posteriors}\n\\end{figure}\n\n\n\n\\section{Conclusions}\n\nThis paper provides the first theoretical results for probabilistic integration, in the form of posterior contraction rates for FWBQ and FWLSBQ.\nThis is an important step in the probabilistic numerics research programme \\cite{Hennig2015ProbNum} as it establishes a theoretical justification for using the posterior distribution as a model for the numerical integration error (which was previously assumed \\cite[e.g.]{Rasmussen2003,Gunter2014,Oates2015,Osborne2012,Sarkka2015}).\nThe practical advantages conferred by a fully probabilistic error model were demonstrated on a model selection problem from proteomics, where sensitivity of an evaluation of the MAP estimate was modelled in terms of the error arising from repeated numerical integration.\n\nThe strengths and weaknesses of BQ (notably, including scalability in the dimension $d$ of $\\mathcal{X}$) are well-known and are inherited by our FWBQ methodology.\nWe do not review these here but refer the reader to \\cite{OHagan1991} for an extended discussion.\nConvergence, in the classical sense, was proven here to occur exponentially quickly for FWLSBQ, which partially explains the excellent performance of BQ and related methods seen in applications \\cite{Gunter2014,Osborne2012}, as well as resolving an open conjecture.\nAs a bonus, the hybrid quadrature rules that we developed turned out to converge much faster in simulations than those in \\cite{Bach2012}, which originally motivated our work.\n\nA key open problem for kernel methods in probabilistic numerics is to establish protocols for the practical elicitation of kernel hyper-parameters.\nThis is important as hyper-parameters directly affect the scale of the posterior over numerical error that we ultimately aim to interpret.\nNote that this problem applies equally to BQ, as well as related quadrature methods \\cite{Bach2012,Rasmussen2003,Gunter2014,Oates2015} and more generally in probabilistic numerics \\cite{Schober2014}.\nPrevious work, such as \\cite{Hamrick2013mental}, optimised hyper-parameters on a per-application basis.\nOur ongoing research seeks automatic and general methods for hyper-parameter elicitation that provide good frequentist coverage properties for posterior credible intervals, but we reserve the details for a future publication.\n\n\n\n\n\n\n\n\\subsubsection*{Acknowledgments}\n\nThe authors are grateful for discussions with Simon Lacoste-Julien, Simo S{\\\"a}rkk{\\\"a}, Arno Solin, Dino Sejdinovic, Tom Gunter and Mathias Cronj{\\\"a}ger. FXB was supported by EPSRC [EP\/L016710\/1]. CJO was supported by EPSRC [EP\/D002060\/1]. MG was supported by EPSRC [EP\/J016934\/1], an EPSRC Established Career Fellowship, the EU grant [EU\/259348] and a Royal Society Wolfson Research Merit Award. \n\n\\newpage\n\n{\\small \n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{PLASMA EQUIPARTITION TIMES}\n\nThe common assumption that all particles in a plasma have the same local\ntemperature may not be true if the electron-ion equilibration\ntimescale is longer than heating timescales\\cite{Takizawa1999,Kawazura2019}. \nThis timescale is \nfundamental for such processes as accretion onto black holes and \nX-ray emission from the intergalactic medium. It can be \ndirectly measured using cluster shocks. \n\nAt a low-Mach shock, ions are\ndissipatively heated to a temperature $T_i$, while electrons are adiabatically compressed to a\nlower $T_e$. The two species then equilibrate to the\nmean post-shock temperature\\cite{Zeldovich1966} (Fig.~\\ref{fig:tei}). \nFrom the X-ray brightness and spectra, we can measure the plasma density\nand $T_e$ across the shock (this requires only a modest spectral resolution). \nFor the typical low sonic Mach numbers in clusters \n($M=2-3$), the mean\npost-shock temperature can be accurately predicted from\nthe shock density jump. \nIf the equilibration is via Coulomb collisions,\nthe region over which the electron temperature $T_e$ increases \nis tens of kpc wide --- resolvable with a \\textit{Chandra}-like telescope at distances of $z<2$.  \nThis direct test is unique to cluster shocks because of the\nfortuitous combination of the linear scales and relatively low Mach numbers;\nit cannot be done for the solar wind or SNR shocks.\n\nA \\textit{Chandra}{} measurement for the \nBullet cluster shock (Fig.~\\ref{fig:tei}) suggests that \n$T_e-T_i$ equilibration is quicker than \nCoulomb\\cite{Markevitch2006}, although with a systematic uncertainty that arises from the assumption of \nsymmetry and requires averaging over a sample of shocks.\nWith \\textit{Chandra}, this measurement is limited to only three shocks, and the results\nare contradictory\\cite{Markevitch2006,Russell2012,Wang2018}.\nA more sensitive imager is needed to find many more \nshocks (most of them in the cluster outskirts), select a sample of \nsuitable ones, and robustly determine this basic plasma property.\n\n\\begin{figure*}\n    \\centering\n   \n    \\includegraphics[width=\\textwidth]{strip_visc_opt1.pdf}\n    \\caption{Plasma viscosity determines how the gas is stripped from the infalling \n    groups and galaxies. \n    {\\em Left}: \n    If viscosity is not strongly suppressed, galaxies falling into clusters \n    should exhibit prominent tails of stripped gas\\cite{Roediger2015}. {\\em Middle, right}: An infalling galaxy (NGC1404), which appears not to have \n    such a tail\\cite{Su2017}, and a much larger infalling group in the outskirts of a cluster\\cite{Eckert2014}, which \n    does.}\n    \\label{fig:tails}\n\\end{figure*}\n\n\\section*{HEAT CONDUCTIVITY}\n\nHeat conduction erases temperature gradients and competes with \nradiative cooling, and is of utmost \nimportance for galaxy and cluster formation. The effective heat\nconductivity in a plasma with tangled magnetic fields is unknown, \nwith a large uncertainty for the component parallel to the \nfield, which recent theoretical works predict to be\nreduced\\cite{Schekochihin2008,Kunz2014,Komarov2016,Komarov2018,RobergClark2018}. \nThe existence of cold fronts in clusters confirms that\nconduction across the field lines is very\nlow\\cite{Ettori2000,Vikhlinin2001,Wang2016}, but constraints\nfor the average or parallel conductivity are\npoor\\cite{Markevitch2003,Wang2016}. \nShock fronts are locations where the parallel component can be \nconstrained, because the field lines should connect the post-shock \nand pre-shock regions (unlike for the magnetically-insulated \ncold fronts), though the field structure in the narrow shock layer \ncan be chaotic. Electron-dominated conduction may result in an \nobservable $T_e$ precursor (Fig.~\\ref{fig:tei}).\n\nThe magnetic field can be stretched and untangled in a predictable way in the cluster sloshing\ncool cores. The characteristic spiral temperature\nstructure that forms there\\cite{ZuHone2013} can also be used to constrain parallel\nconductivity. A telescope with a bigger mirror than\n\\textit{Chandra}'s could look for temperature precursors in shocks \nand obtain detailed maps of temperature gradients along the field filaments in many \ncluster cores to measure the conductivity.\n\n\\section*{VISCOSITY}\n\nPlasma viscosity is a fundamental quantity that governs damping of turbulence and sound\nwaves, suppression of hydrodynamic instabilities and mixing of different gas phases, and \nthus relevant to such important processes as heating the gas, spreading metals\nejected from galaxies, and amplification of magnetic fields. At present it is largely unknown. Isotropic viscosity can be determined from the dissipation scale of the \npower spectrum of turbulence. {\\it\\small XRISM}\\ and \\textit{Athena}\\ will pursue that via the velocity measurements in the ICM, though it is\nunclear if the dissipation scale will be reachable\\cite{ZuHone2016}. The turbulence spectrum can also be constrained by observing the gas density fluctuations\\cite{Schuecker2004,Zhuravleva2015}. However, the plasma viscosity should be anisotropic and may affect turbulence and other phenomena differently. It is thus useful to approach it from several angles. Two subtle phenomena in galaxy cluster images can help us probe the viscosity through its effect on gasdynamic instabilities.\n\n\\subsubsection*{~~Galaxy stripping tails.}\n\nFigure~\\ref{fig:tails}{\\em a}\\\/ shows\na striking difference in the simulated X-ray appearance of the tail of the\ncool stripped gas behind a galaxy as it flies through the \nICM\\cite{Roediger2015}. In an inviscid plasma, the gas\npromptly mixes with the ambient ICM, but a modest viscosity suppresses the\nmixing and makes the long tail visible. Deep \\textit{Chandra}{} images of such infalling galaxies \nNGC1404 (Fig.\\ \\ref{fig:tails}{\\em b}) and M89 favor efficient mixing and a reduced \nviscosity\\cite{Su2017,Kraft2017}. Other infalling groups in the cluster periphery\ndo exhibit unmixed tails (e.g., Fig.\\ \\ref{fig:tails}{\\em c}). This points to a \npossibility of a systematic study to constrain effective viscosity --- and directly \nobserve its effect on gas mixing --- in various ICM regimes. However, a \nmore sensitive instrument with lower background is required to study these subtle, \nlow-contrast extended features, most of which will be found in the \nlow-brightness cluster outskirts.\n\n\\begin{figure*}\n    \\centering\n   \n    \\includegraphics[width=\\textwidth]{zuhone_b_visc_opt.pdf}\n    \\caption{MHD simulation of a sloshing cluster core with viscosity (isotropic) \n    and magnetic field\\cite{Bellomi2019}. X-ray brightness gradients are shown.\n    Initial $\\beta$ values are given; sloshing amplifies the magnetic field and produces lower \n    $\\beta$, which result in plasma depletion regions. \n    The appearance of cold fronts can be used to constrain the effective plasma \n    viscosity and magnetic field strength.}\n    \\label{fig:cf_visc}\n\\end{figure*}\n\n\n\\subsubsection*{~~Instabilities in cold fronts.}\n\nCold fronts --- contact discontinuities in the ICM that separate regions of\ndifferent density and temperature in pressure \nequilibrium\\cite{Markevitch2007} --- are\nubiquitous in merging subclusters, where they are seen as sharp X-ray brightness \nedges (e.g., the \n``bullet'' boundary in the Bullet  cluster, Fig.\\ \\ref{fig:tei}{\\em a}). \nThey are also found in most cool cores, where \nthey emerge as the dense gas of the core ``sloshes'' in\nthe cluster gravitational well\\cite{Ascasibar2006}. Sloshing produces velocity\nshear across the cold front, which should generate Kelvin-Helmholtz instabilities \n(Fig.\\ \\ref{fig:cf_visc}{\\em a}). If the ICM is viscous, K-H \ninstabilities are suppressed\\cite{Churazov2004,Roediger2013b,Zuhone2015} \n(Fig.\\ \\ref{fig:cf_visc}{\\em b}). \\textit{Chandra}\\\nhas discovered K-H instabilities in a few cold fronts and placed an \nupper limit on the effective isotropic viscosity of $\\sim0.1$ \nSpitzer\\cite{Roediger2013a,Ichinohe2017,Wang2018b} \n(or, equivalently in this context, a full \nBraginskii anisotropic viscosity\\cite{Zuhone2015}). \nTo constrain the viscosity from below requires finding \ninstabilities for a range of \ndensity contrasts. These subtle wiggles can be seen only\nwith high resolution and lots of photons, and a systematic study \nrequires a larger-area telescope.\n\n\n\\section*{PLASMA DEPLETION LAYERS}\n\nThe velocity shear at cold fronts (and elsewhere in the cluster) should stretch and \namplify the magnetic fields, forming magnetic layers parallel to the front. \nSuch layers can suppress the instabilities even without the \nviscosity\\cite{Vikhlinin2001a}, although a certain \ninitial field strength is required (compare Figs.\\ \\ref{fig:cf_visc}{\\em a,c}). \nA distinguishing feature between these two suppression\nmechanisms is seen in Fig.\\ \\ref{fig:cf_visc}{\\em c}.  Wherever the \nfield is amplified, thermal plasma is squeezed out, forming \nplasma depletion layers (PDL, like the ones in the solar wind around \nplanets\\cite{Oieroset2004}) that can become visible in the X-ray\nimage\\cite{Markevitch2007}. \n\n\\begin{figure*}\n    \\centering\n   \n    \\includegraphics[width=\\textwidth]{slosh_pdl_per_opt.pdf}\n    \\caption{Plasma depletion layers in a cluster core. ({\\em a}) MHD simulation of a sloshing core\\cite{ZuHone2011}; color shows the field strength. As the gas swirls in the core, it forms filaments of stretched and amplified field. ({\\em b}) Pressure profiles across two $\\beta\\sim10$ filaments, extracted along the line in panel {\\em a}. While total pressure is monotonic, thermal pressure shows dips (both the density and the temperature dip). ({\\em c}) Possible observation of such ``feathery'' structure in the Perseus core\\cite{Ichinohe2019}. Subtle X-ray ``channels,'' possibly of similar origin, have also been seen by \\textit{Chandra}\\ in A520\\cite{Wang2016} and A2142\\cite{Wang2018b}.}\n    \\label{fig:pdl}\n\\end{figure*}\n\nIn Fig.\\ \\ref{fig:pdl}, we show how PDL can form in a \nsloshing core. \\textit{Chandra}\\ has reported hints of this new phenomenon --- \nlow-contrast ``channels'' in A520 and A2142\\cite{Wang2016,Wang2018b} and \n``feathery'' structures in Virgo and Perseus\\cite{Werner2016,Ichinohe2019}, \nFig.\\ \\ref{fig:pdl}c). Apart from disentangling the effects of viscosity and \nmagnetic fields on cold front stability, observing PDL in clusters would have a \nmore general significance --- it allows us literally to see the structure \nof the intracluster magnetic field. Combined with the radio\nimages, this can map the distribution of cosmic ray electrons in the ICM.\nObserving these subtle image features requires many more photons than \\textit{Chandra}\\ \ncan collect for most clusters. A future imager with a much bigger mirror can \ngive us this novel tool for cluster plasma studies.\n\n\n\\section*{COSMIC RAY ACCELERATION}\n\nAcross the universe, shocks accelerate particles to very high energies\nvia the first-order Fermi mechanism\\cite{Blandford1987}. Microscopic \ndetails of this fundamental process remain poorly\nknown for astrophysical plasmas, and particle-in-cell simulations are still\nfar from covering realistic plasma parameters.\n\nMany galaxy clusters exhibit striking ``radio relics'' in their \noutskirts\\cite{vanWeeren2010}. These Mpc-long, arc-like \nstructures are synchrotron signatures of\nultrarelativistic ($\\gamma \\sim 10^4$) electrons. Some relics, as well as sharp \nedges of giant radio halos, coincide with ICM shocks\\cite{Giacintucci2008,Shimwell2015,Wang2018}, \nsuggesting that shocks have something to do with those electrons. However, the shock \nMach numbers are low ($M=1.5-3$) and it is unclear how they reach the acceleration\nefficiency needed to produce the relics\\cite{Macario2011,Brunetti2014}. \nOther puzzles include similar-Mach shocks that produce very different radio \nfeatures\\cite{Shimwell2015} and a relic for which the shock is ruled out\\cite{Markevitch2019}. \nParticle acceleration in the ICM appears more complex than a classical Fermi picture. Proposed\nsolutions involve re-acceleration of aged relativistic particles\\cite{Brunetti2014} \nas well as modifications to the Fermi mechanism in a magnetized plasma. To gain insight \ninto these universal processes, we need a systematic comparison of shocks in \nthe X-ray and radio. However, most radio relics are found far in the cluster \noutskirts, where the X-ray emission is too dim for \\textit{Chandra}. A low-background, high-area, high-resolution X-ray imager is needed to discover and study shocks there.\n\n\n\\section*{FINDING MOST POWERFUL AGN OUTBURSTS}\n\nAGN that reside in many cluster cores eject copious amounts of energy \ninto the ICM, preventing runaway radiative cooling of the gas at the \ncluster centers\\cite{Mcnamara2007}. \nThey inflate X-ray cavities in the ICM; radio observations show \nthat these cavities are filled with relativistic plasma. A recent discovery of \na giant ghost bubble outside the core in Ophiuchus\\cite{Giacintucci2019} \nsuggest that the AGN effects may extend far beyond the cluster cool cores, and that AGN \ncan produce far more powerful outbursts than we infer from the energetics \nof the cavities in the cluster cores\\cite{Mcnamara2005}. \nIf this phenomenon is widespread, as hinted at by recent \nlow-freqency radio surveys by {\\it\\small LOFAR}\\ and {\\it\\small MWA}, \nclusters can be affected more strongly by the AGN feedback than previously thought. \nForensic evidence for that can be provided by large, low-contrast ghost cavities outside \ncluster cores\\cite{Sanders2009}. Their detection requires a low-background, high-area X-ray imager.\n\n\\section*{WHAT KIND OF INSTRUMENT WE NEED}\n\nAll the above studies require a\nmuch greater collecting area and much lower background than the current X-ray \ninstruments can provide. \nCritically, they also require high angular resolution --- at least \\textit{Chandra}-like --- both \nto resolve the sharp spatial features and to remove the faint point sources of the Cosmic \nX-ray Background that dominate the flux in the cluster outskirts, \nwhere most of those features \nwill be found. {\\it\\small AXIS}, a proposed Probe, and the imaging detector \nof \\textit{Lynx}, a proposed Flagship, \nwill have the requisite resolution and photon-collecting capabilities. They will also enable \nunsurpassed low-background imaging for $E>1$ keV (where the soft diffuse Galactic \nbackground becomes insignificant), as shown in the accompanying \nwhite paper\\cite{Walker2019}.\n\n\\clearpage\n\\section*{REFERENCES}\n\\small\n\\parskip=0mm\n\\vspace*{-4mm}\n\n\\bibliographystyle{naturemag}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nQuantum mechanics allows remote parties to be entangled in a way that is beyond classical physics. Bell tests, being an elegant illustration of this phenomenon, perform projective measurements on entangled parties who cannot signal to each other, for instance, enforced by space-like separation or a shield between the remote parties, to generate correlations of measurement outcomes that are impossible for the classical theory (i.e., local realism) \\cite{bell}. By exploiting Bell tests, Popescu and Rohrlich generalize the theories beyond quantum mechanics, which are still restricted by the no-signaling condition \\cite{prbox}. This immediately raises several interesting questions. One is to understand the restriction on quantum mechanics beyond the no-signaling theory, which inspires a line of works on quantum foundations \\cite{Hardy2001quant}. Another is whether there exists a theory more nonlocal than the quantum theory in nature, such as the one only limited by the no-signaling condition (later referred to as \\emph{the maximally nonlocal theory}). Posing constraints in addition to the no-signaling condition, such as the uncertainty principle, induces various no-signaling theories, that are less nonlocal than the maximally nonlocal theory. Though such theories have not been experimentally found, researchers postulate that they may exist in ultra-high-density objects such as black holes \\cite{Preskill1992Do} or when the scale is out of the quantum regime, for example, smaller than a Planck length. Independently, this research has spurred interest for device-independent quantum information processing where the adversary is only limited to the no-signaling condition \\cite{mayers1998quantum,QKDNoSignal05}.\n\nOne of the major obstacles of Bell tests\nis the detection efficiency loophole. In fact, loophole-free Bell tests have only been very recently demonstrated \\cite{Hensen2015Loophole, LoopholeFree:Zeilinger, LoopholeFree:NIST}. They are all based on Bell inequalities that have two measurement settings for two parties and thus need a detection efficiency bound of at least $2\/3$ \\cite{PhysRevA.68.062109}. It was known that $2\/3$ is tight for two measurement settings \\cite{Eberhard93}. However, beyond that, the efficiency bounds of no-signaling theories for more measurement settings and\/or parties are largely unknown, even for the most-studied quantum theory. The first result in this direction is due to Larsson and Semitecolos \\cite{larsson2001strict}, who show that, for $N$ parties and two measurement settings, the detector efficiency requirement is no larger than $N\/(2N+1)$. This bound is later shown to be tight by Massar and Pironio \\cite{PhysRevA.68.062109}. In the case of two parties, the best upper bound with four measurement settings is $61.8\\%$ \\cite{vertesi2010closing} and with a large number of measurement settings, the upper bound can approach zero \\cite{massar2002nonlocality}. In the tripartite case, the best upper bound for eight measurement settings is $0.501$ \\cite{pal2015closing}. For an infinite number of parties,\na series of works\n\\cite{PhysRevLett.91.047903,PhysRevA.86.062111,PhysRevA.92.052104}\nhave led to the best upper bound $2\/(2+M)$ for $M$ measurement settings.\n\nIn practice, to lower the detection efficiency requirement, Bell tests with more than two measurement settings are more important than the ones with two measurement settings because they potentially have a lower efficiency requirement. Consequently they could make experimental realizations easier, especially for optical systems where the loss is normally high. The minimum detector efficiency to violate the local realism can also be viewed as a measure to characterize nonlocal correlations and thus is an important operational quantity.\n\n\\section{Preliminaries}\nBefore continuing, we first formalize the problem of Bell tests.\nIn Bell tests, there are two space-separated parties, $A$ and $B$. In each turn, $A$ ($B$) will be given a number $x \\in [1,M_A]$ ($y \\in [1,M_B]$)  randomly. Then $A$ and $B$ will output the outcomes $a$ and $b$ respectively according to the inputs and their shared resources.\n\nSince the two parties cannot signal to each other in a Bell test, $A$ $(B)$ is unaware of the input to $B$ $(A)$. Thus a no-signaling probability $p^{NS}$ satisfies\n\\begin{equation}\n\\begin{aligned}\n\\forall x,y \\quad p^{NS}(a|x,y)= p^{NS}(a|x), \\\\\np^{NS}(b|x,y) = p^{NS}(b|y),\\\\\n\\end{aligned}\n\\end{equation}\n which represents that the probability of one party $A$ $(B)$ outputting an outcome $a$ $(b)$ is independent of the input of the other party $B$ $(A)$ and only depends on its own input $x$ ($y$).\n\nDenote the detector efficiency as $\\eta$, which is independent of the input. The probability under such inefficiency of detectors, denoted as $p^{NS}_{\\eta}$, is related to the ideal no-signaling probability $p^{NS}$ by\n\\begin{equation}\n\\begin{aligned}\np^{NS}_\\eta(a,b|x,y) =& \\eta^2p^{NS}(a,b|x,y),\n\\\\p^{NS}_\\eta(\\Phi,\\Phi) = &(1-\\eta)^2,\n\\\\\np^{NS}_\\eta(a,\\Phi|x) =& \\eta(1-\\eta)p^{NS}(a|x),\\\\\np^{NS}_\\eta(\\Phi,b|y) =& \\eta(1-\\eta)p^{NS}(b|y),\n\\end{aligned}\n\\end{equation}\nwhere $\\Phi$ denotes the empty output corresponding to a failed detection. The derivation of these relations is straightforward. For example, for the first relation, the outcomes $a$ and $b$ would be obtained only when both detectors respond. Hence the probability shrinks by a factor $\\eta^2$ because the detector of each party responds with a probability $\\eta$ and the detectors of different parties are independent.\n\nA local hidden variable  (LHV) model assumes that $A$ and $B$ share a random variable $\\lambda$ but cannot communicate. The strategy of $A$ ($B$) can then be characterized by the probability $p_A(a|x,\\lambda)$ ($p_B(b|y, \\lambda)$), which uses $\\lambda$ and $x$ ($y$) to determine the probability of outputting $a$ ($b$). If $p^{NS}_{\\eta}$ can be  simulated by a LHV model, there exists a local strategy such that for any outcomes $a$ and $b$ (including the empty output $\\Phi$),\n\\begin{equation}\np^{NS}_{\\eta} (a,b|x,y) = \\int_{\\lambda}p(\\lambda)p(a|x,\\lambda)p(b|y,\\lambda)d\\lambda.\n\\end{equation}\nSince the efficiency $\\eta$ and the simulation capability of a LHV model are monotonic (as shown in  the Appendix \\ref{app}), one can define the minimum value of detector efficiency $\\eta^*_{NS}$ for showing the nonlocality of the maximally nonlocal theory.\n\n\n\\section{Results}\nWe are now ready to state our main results.\n\n\\begin{theorem}\nIn Bell tests with two parties,\n\\begin{equation}\n\\eta_{NS}^{*}\\ge \\frac{M_A+M_B-2}{M_AM_B-1},\n\\end{equation}\n where $M_A,M_B$ are the numbers of inputs of $A$ and $B$ respectively.\n\\end{theorem}\n\\begin{proof}\nTheorem 1 in Ref.~\\cite{PhysRevA.68.062109} designs a LHV model to simulate any quantum strategy if the efficiency is lower than or equals $(M_A+M_B-2)\/(M_AM_B-1)$. Since this design only leverages the no-signaling condition, the efficiency bound also holds in the maximally nonlocal theory which completes the proof.\n\\end{proof}\n\nWhen $M_A=M_B=2$, Theorem 1 gives a well-known detector efficiency bound of $2\/3$ which is tight for the quantum theory\t\\cite{larsson2001strict}. We now show that, for arbitrary  $M_A$ and $M_B$, the efficiency bound $(M_A+M_B-2)\/(M_AM_B-1)$ is tight for the maximally nonlocal theory. The same statement is open for the quantum theory.\n\n\\begin{theorem}\nIn two-party Bell tests with $M_A$ and $M_B$ inputs respectively,\n\\begin{equation}\n\\label{eq:twoupper}\n\\eta_{NS}^{*}\\le \\frac{M_A+M_B-2}{M_AM_B-1}.\n\\end{equation}\n\\end{theorem}\n\n\\begin{proof}\nDenote $P$ as the minimal prime number such that $P\\geq \\max(M_A,M_B) $. We construct a Bell inequality with $P$ outputs excluding $\\Phi$,\n\\begin{equation}\n\\begin{aligned}\n&I_{M_AM_B(P+1)(P+1)} = \\sum_{a,b,x,y} f(a,b,x,y)p(a,b|x,y) \\\\\n&\\quad\\quad+ \\sum_{a,x} g(a,x)p(a|x) + \\sum_{b,y} h(b,y)p(b|y),\\\\\n&f(a,b,x,y) =\n\\begin{cases}\n1 &  ((x>1) \\vee (y>1)) \\wedge (a+b \\equiv xy)\\\\\n0 &  (x=1) \\wedge(y=1) \\wedge (a+b \\equiv xy) \\\\\n-P^4 & \\text{otherwise}\n\\end{cases}\n\\\\\n&g(a,x) =\n\\begin{cases}\n-1 & x > 1\\\\\n0 & x = 1\n\\end{cases},\n\\quad h(b,y) =\n\\begin{cases}\n-1 & y > 1\\\\\n0 & y = 1\n\\end{cases},\n\\end{aligned}\n\\end{equation}\nwhere $1\\leq x\\leq M_A, 1\\leq y \\leq M_B, 0\\leq a,b<P$ and the modulo is over $P$. An illustration of this Bell inequality is shown in Fig.~\\ref{fig:2Bell}(a).\n\\begin{figure}\n\\centering\n    \\subfigure[]{%\n      \\includegraphics[width=4.2cm]{2Bell.eps}\n    }\n    \\subfigure[]{%\n      \\includegraphics[width=4cm]{nBell.eps}\n    }\n\\caption{Geometric interpretation of the Bell inequalities $I_{M_AM_B(P+1)(P+1)}$ and $I^{\\text{multi}}_{P+1}$. (a) In $I_{M_AM_B(P+1)(P+1)}$, which is a linear combination of $p(a,b|x,y)$, whose coefficients are first divided into blocks according to $x,y$, and then divided into entries according to $a,b$. The red, green, white entries are 1, $-P^4$ and 0 respectively. Each blue arrow pointing to a column or a row is a term $-p(a|x)$ or $-p(b|y)$ in $I_{M_AM_B(P+1)(P+1)}$.\n(b) $I^{\\text{multi}}_{P+1}$ is similarly divided into higher-dimensional blocks and entries. The red entries still maintain the property that there is one and only one red entry on every line within a block according to the construction.}\n\\label{fig:2Bell}\n\\end{figure}\n\nFirst, we prove $I_{M_AM_B(P+1)(P+1)} \\leq 0$ with LHV strategies. Since LHV strategies can be regarded as the linear combination of deterministic strategies, the LHV strategy can be assumed deterministic,\n\\begin{equation}\nF(x) = a, G(y) = b.\n\\end{equation}\nIn other words, the input will determine output by the functions $F$ and $G$.\n\nCase 1: If $F(x)  = \\Phi \\ \\forall x$, then $I_{M_AM_B(P+1)(P+1)} \\le 0$. Similarly, if $G(y) = \\Phi \\ \\forall y$, then $I_{M_AM_B(P+1)(P+1)} \\le 0$.\n\nCase 2:\nIf $x_1\\neq x_2, y_1\\neq y_2$, $a_1 = F(x_1), a_2 = F(x_2), b_1 = G(y_1), b_2 = G(y_2)$ and none of $a_1,a_2,b_1,b_2$ equals $\\Phi$, then one of $f(a_1,b_1,x_1,y_1)$, $f(a_1,b_2,x_1,y_2)$,\n $f(a_2,b_1,x_2,y_1)$, $f(a_2,b_2,x_2,y_2)$ must equal $-P^{4}$. Otherwise\n\\begin{eqnarray}\na_1 + b_1 &\\equiv& x_1y_1 \\mod{P}, \\label{12.1}\\\\\na_1 + b_2 &\\equiv& x_1y_2 \\mod{P},  \\label{12.2}\\\\\na_2 + b_1 &\\equiv& x_2y_1 \\mod{P}, \\label{12.3}\\\\\na_2 + b_2 &\\equiv& x_2y_2 \\mod{P}.\\label{12.4}\n\\end{eqnarray}\nBy applying \\eqref{12.1}+\\eqref{12.4}$-$\\eqref{12.2}$-$\\eqref{12.3}, we get $(x_1-x_2)(y_1-y_2) \\equiv 0 \\mod{P}$. Since $P$ is a prime number, and $1\\leq x_1 \\neq x_2 \\le P,1\\leq y_1\\neq y_2 \\le P$, there is a contradiction. Since $P^{4}$ is bigger than the sum of positive coefficients in the Bell inequality, $I_{M_AM_B(P+1)(P+1)}\\leq 0$ follows.\n\nCase 3:\nIf there is only one $x$ such that $F(x) = a \\neq \\Phi$, the Bell inequality is simplified to\n\\begin{equation}\n\\begin{aligned}\n&\\sum_{y} f(a, G(y), x, y) + \\sum_{y} h(G(y),y) + g(a, x)\\\\\n=&\\sum_{y\\neq 1} (f(a,G(y),x,y) - 1) + f(a,G(1),x,1) +\ng(a, x)\\\\\n\\leq& f(a,G(1),x,1) + g(a,x)\\leq 0.\n\\end{aligned}\n\\end{equation}\nIf there is only one $y$ such that $G(y) = b \\neq \\Phi$, similarly $I_{M_AM_B(P+1)(P+1)} \\leq 0$.\n\nNext, we construct a no-signaling strategy\n\\begin{equation}\np^{NS}(a,b|x,y) =\n\\begin{cases}\n\\frac{1}{P} & a+b\\equiv xy\\\\\n0 & \\text{otherwise}\n\\end{cases}.\n\\end{equation}\nThis probability distribution satisfies the no-signaling condition because when $x,y,a$ are determined, there is a unique $b$ such that $p^{NS}(a,b|x,y) = \\frac{1}{P}$ and therefore\n\\begin{equation}\n\\begin{aligned}\n\\forall y,\\quad p^{NS}(a|x,y) = \\sum_{b}p^{NS}(a,b|x,y) = \\frac{1}{P};\n\\end{aligned}\n\\end{equation}\nsimilarly $\\forall x,\\  p^{NS}(b|x,y)=1\/P$.\nIts Bell value is\n\\begin{equation}\n\\begin{aligned}\n&I_{M_AM_B(P+1)(P+1)} \\\\\n=& \\sum_{a,b,x,y}f(a,b,x,y)p^{NS}_{\\eta}(a,b|x,y) \\\\\n&\\quad+ \\sum_{a,x}g(a,x)p^{NS}_{\\eta}(a|x) + \\sum_{b,y}h(b,y)p^{NS}_{\\eta}(b|y)\\\\\n=&\\sum_{x>1\\vee y>1} \\eta^{2} - \\sum_{x>1} \\eta - \\sum_{y>1} \\eta\\\\\n=& \\eta^{2}(M_AM_B-1) - \\eta(M_A+M_B-2).\n\\end{aligned}\n\\end{equation}\nRecall that  $\\eta > \\eta^*_{NS}$ is necessary to violate the Bell inequality, i.e., $I_{M_AM_B(P+1)(P+1)}>0$.\nHence Eq.~\\eqref{eq:twoupper} holds.\n\\end{proof}\n\n Through proving Theorem 2, we design a useful Bell inequality $I_{M_AM_B(P+1)(P+1)}$. It has a similar condition with the CHSH inequality: $a\\oplus b \\equiv x\\cdot y$ \\cite{clauser1969proposed}, but generalizes module $2$ to module $P$. Compared to another generalization of the CHSH inequality \\cite{collins2004relevant}, our Bell inequality is more advantageous at persisting nonlocality when detector inefficiency occurs. There are relatively few quantum efficiency upper bound results. The quantum efficiency upper bound is $61.8\\%$ when $M_A=M_B=4$, showing the $2\/3$ bound can be beaten with relatively few inputs \\cite{vertesi2010closing}.\n  With more inputs $M_A=M_B=2^{d}$, the quantum efficiency bound can be as low as $\\eta = d^{1\/2}2^{-0.0035d}$  \\cite{massar2002nonlocality}, which is however hard to be met in practice, requiring $10^{285}$ inputs when the efficiency is $1\/10$. Our Bell inequality suggests much fewer inputs might suffice.\n\nWe next generalize the efficiency bound $\\eta_{NS}^{*}$ to $N$ parties and have the following lower bound.\n\\begin{theorem}\nIn Bell tests with $N\\le 500$ space-separated parties, the efficiency bound satisfies\n\\begin{equation}\n\\eta_{NS}^{*}\\ge  \\frac{N}{M(N-1)+1},\n\\end{equation}\n where all $N$ parties have $M$ inputs.\n\\end{theorem}\n\\begin{proof}\nFrom Ref.~\\cite{PhysRevA.68.062109} Conjecture 2, a multipartite LHV model is designed to simulate any quantum strategy when the efficiency is no larger than $M\/[M(N-1)+1]$ and $N\\leq 500$. Since the only condition used in that proof is the no-signaling condition, this finishes the proof. Ref.~\\cite{PhysRevA.68.062109} also conjectures that this bound holds for any value of $N$.\n\\end{proof}\n\nWhen $N\\rightarrow \\infty$, this theorem suggests $1\/M$ is the detector efficiency lower bound  for showing nonlocality. We also have the following asymptotically matched upper bound.\n\n\\begin{theorem}\nConsider $N$-party Bell tests with $M_1, M_2, ..., M_N$ inputs respectively,\n\\begin{equation}\n\\label{multiupper}\n\\eta_{NS}^{*}\\le \\left(\\frac{M_1+M_2+\\cdots+M_N-N}{M_1M_2\\cdots M_N-1}\\right)^{\\frac{1}{N-1}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nDenote $P$ as the minimal prime number such that $P> \\max(M_1,M_2,\\cdots,M_N)$.\nWe construct a general Bell inequality with $P$ outputs excluding $\\Phi$,\n\\begin{equation}\n\\begin{aligned}\nI_{P+1}^{\\text{multi}}\n&= \\sum_{\\vec{a},\\vec{x}} f(\\vec{a},\\vec{x})p(\\vec{a}|\\vec{x})\n+ \\sum_i\\sum_{a_i,x_i} g_i(a_i,x_i)p(a_i|x_i) \\\\\nf(\\vec{a},\\vec{x}) &=\n\\begin{cases}\n1 & (\\exists t, x_t \\neq 1)\\wedge  (\\sum_{j}{a_j} \\equiv \\prod_{i} x_i)\\\\\n0  &\t(\\forall t, x_t = 1)\\wedge (\\sum_{i}{a_i} \\equiv \\prod_{i} x_i)\\\\\n-P^{2N} & \\text{otherwise}\\\\\n\\end{cases}\n\\\\\ng_i(a_i,x_i) &=\n\\begin{cases}\n-1 & x_i > 1\\\\\n0 & x_i = 1\n\\end{cases},\n\\end{aligned}\n\\end{equation}\nwhere $\\vec{a} =\\{a_1, ..., a_N\\}, \\vec{x} = \\{x_1, ..., x_N\\}$, $1\\leq x_i\\leq M_i, 0\\leq a_i<P$ and the modulo is over $P$. An illustration of this Bell inequality is shown in Fig.~\\ref{fig:2Bell}(b).\n\nFirst, we prove $I_{(P+1)}^{\\text{multi}} \\leq 0$ for any LHV strategy. Since a LHV strategy can always be regarded as a linear combination of deterministic strategies, we consider a deterministic strategy $\nF_i(x_i) = a_i\\ \\forall i$.\n\nCase 1:\nIf there exists some $i$ such that $F_i(x_i) = \\Phi \\ \\forall 1\\leq x_i \\le M_i$, then $I_{P+1}^{\\text{multi}} \\le 0$.\n\n\nCase 2:\nThere exist $i,j,x_i',x_i'', x_j', x_j''$, where $x_i',x_i''$ are different inputs of the $i$-th party, $x_j', x_j''$ are different inputs of the $j$-th party,  such that none of the corresponding outputs $a_i'=F_i(x_i'), a_i''=F_i(x_i''), a_j'=F_j(x_j'), a_j''=F_j(x_j'')$ equals $\\Phi$.\nAlso for any $k$, there exists $x_k$ such that $F_k(x_k) = a_k \\neq \\Phi$ because otherwise Case 1 arises. Denote $\\vec{a_1} = \\{a_k\\}$, $\\vec{x_1} = \\{x_k\\}$ and $\\vec{a_1}_{a_i', a_j'}$ as the vector which replaces the $i$-th and the $j$-th items in $\\vec{a_1}$ with $a_i', a_j'$ respectively and similarly for $\\vec{x_1}_{x_i',x_j'}$. Consider the following four Bell coefficients:\n\\begin{align*}\nf(\\vec{a_1}_{a_i',a_j'},\\vec{x_1}_{x_i',x_j'}),f(\\vec{a_1}_{a_i'',a_j'},\\vec{x_1}_{x_i'',x_j'}),\\\\\nf(\\vec{a_1}_{a_i',a_j''},\\vec{x_1}_{x_i',x_j''}),f(\\vec{a_1}_{a_i'',a_j''},\\vec{x_1}_{x_i'',x_j''}).\n\\end{align*}\nDenote $c = \\sum_{k\\neq i, k\\neq j} a_k, z = \\prod_{k\\neq i, k\\neq j} x_k$ and consider\n \\begin{eqnarray}\na_i' + a_j' +c&\\equiv x_i'x_j' z \\mod{P}  \\label{(1)},\\\\\na_i' + a_j''  +c&\\equiv x_i'x_j'' z \\mod{P}\\label{(2)},\\\\\na_i'' + a_j'  +c&\\equiv x_i''x_j' z \\mod{P}\\label{(3)},\\\\\na_i'' + a_j''  +c&\\equiv x_i''x_j'' z \\mod{P}\\label{(4)}.\n\\end{eqnarray}\nIf one of the above equations does not hold, then $-P^{2N}$ will contribute to the Bell inequality which induces $I^{\\text{multi}}_{P+1} \\leq 0$, because the sum of the positive coefficients is smaller than $P^{2N}$.\nIf all equations hold, $\\eqref{(1)}+\\eqref{(4)}-\\eqref{(2)}-\\eqref{(3)}$ yields $(x_i'-x_i'')(x_j'-x_j'')z \\equiv 0$, which contradicts with that $P$ is a prime number and $1\\leq x_i',x_i'',x_j',x_j'',x_k< P$.\n\nCase 3:\nWithout loss of generality, we assume for $1\\leq i\\leq N-1$, there is only one $x_i'$ such that $F_i(x_i') = a_i' \\neq \\Phi$. Denote $\\vec{a'} = \\{a_i'\\}_{1 \\leq i \\leq N-1}$, $\\vec{x'} = \\{x_i'\\}_{1 \\leq i \\leq N-1}$. Then $I^{\\text{multi}}_{P+1}=\\sum_{x_n} f(\\vec{a'},F_n(x_n),\\vec{x'},x_n)+\\sum_{i=1}^{N-1} g_i(a_i',x_i') + \\sum_{x_n} g_n(F_n(x_n), x_n) =\\sum_{x_n\\neq 1} (f(\\vec{a'},F_n(x_n),\\vec{x'},x_n) - 1)\n+ \\sum_{i=1}^{N-1} g_i(a_i',x_i') + f(\\vec{a'}, F_n(1), \\vec{x'}, 1)\n\\leq \\sum_{i=1}^{N-1} g_i(a_i',x_i') + f(\\vec{a'}, F_n(1), \\vec{x'}, 1)\n\\leq 0$. The last inequality holds because if $f(\\vec{a'}, F_n(1), \\vec{x'}, 1)=1$, then there exists $1\\le j<n$ such that $x_j'>1$ and consequently $g_j(a_j',x_j')=-1$.\n\nNext, we construct a no-signaling strategy\n\\begin{equation}\n\\begin{aligned}\np^{NS}(\\vec{a}|\\vec{x})= \\begin{cases}\n\\frac{1}{P^{N-1}} & \\sum_{i=1}^{N}a_i\\equiv \\prod_{i=1}^{N}x_i\\\\\n0 & \\text{otherwise}\n\\end{cases},\n\\end{aligned}\n\\end{equation}\nwhich satisfies the no-signaling condition\n\\begin{equation}\n\\begin{aligned}\n\\forall \\vec{a},\\vec{x},1\\le k\\le N \\quad  p^{NS}(a_k|\\vec{x}) = p^{NS}(a_k|x_k),\n\\end{aligned}\n\\end{equation}\nbecause for any value of $a_k$,\n\\begin{equation}\n\\begin{aligned}\np^{NS}(a_k|\\vec{x}) =\\sum_{\\vec{a'}:a'_k = a_k, \\sum_{i}a'_i \\equiv \\prod_{i}x_i} \\frac{1}{P^{N-1}} =\\frac{1}{P}.\n\\end{aligned}\n\\end{equation}\nThe last equality holds because there are only $N-2$ free variables.\n\nThe Bell value of the no-signaling strategy $p^{NS}_{\\eta}$ is\n\\begin{equation}\n\\begin{aligned}\nI_{P+1}^{\\text{multi}}=& \\sum_{\\vec{a},\\vec{x}} f(\\vec{a},\\vec{x})p(\\vec{a}|\\vec{x}) +\\sum_i \\sum_{a_i,x_i}g_i(a_i, x_i)\\\\\n=&\\sum_{\\vec{x}:\\exists t, x_t \\neq 1} \\eta^{N} - \\sum_i \\sum_{x_i>1} \\eta \\\\\n=&\\eta^N(\\prod_{i=1}^nM_i-1)- \\eta(\\sum_{i=1}^nM_i-N).\n\\end{aligned}\n\\end{equation}\nCombined with the definition of $\\eta^*_{NS}$ that $\\eta > \\eta^*_{NS}$ is necessary to violate the Bell inequality, i.e., $I_{P+1}^{\\text{multi}}>0$, this leads to Eq.~\\eqref{multiupper}.\n\\end{proof}\n\nWe compare the efficiency lower bound and upper bound in Theorem 3 and Theorem 4 for different numbers of parties $N$ in Fig.~\\ref{fig:upperlower}. In the comparison, the number of inputs for each party is taken to be the same value $M$. It can be seen that when $N=2$, the two bounds coincides. Also when $N$ becomes large, the two bounds converge to the same value. This is consistent with the analytic upper and lower bound formulas, which both converge to $1\/M$ when $N$ goes to infinity.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=9cm]{many_party_efficiency_bound.eps}\n\\caption{(a)-(d) varies $M$ from 4 to 256 and for each $M$, the efficiency upper bound and lower bound are shown for $N=2,3,\\cdots,200$. When $M$ increases, the ratio between the efficiency bound of $N=2$ and $N=200$ gradually increases but is always smaller than 2.}\n\\label{fig:upperlower}\n\\end{figure}\n\nIt is also instructive to fix $M$ and compare the efficiency bound for different $N$. It can be seen that with increasing $N$, the efficiency bound is lowered. However, even when $N$ goes to infinity, the efficiency bound is lowered by at most half compared to $N=2$. On the other hand, increasing the number of input settings $M$ greatly reduces the efficiency requirement. For example, when $M=256$ and $N=2$, an  efficiency  of $8\\%$ is enough to demonstrate nonlocality. Thus in this respect, increasing the number of input settings is more efficient than increasing the number of parties.\n\nCompared to quantum efficiency bounds, there exist significant gaps. When $N=3$ and $M=8$, the best bound of the quantum theory is $0.501$ \\cite{pal2015closing} while the best bound of the maximally nonlocal theory is $0.20$. For $N\\to \\infty$, the best bound of the quantum theory $2\/(2+M)$ \\cite{PhysRevA.92.052104} is approximately twice of the best bound of the maximally nonlocal theory $1\/M$. Therefore the advantage brought by our no-signaling strategies may provide inspiration to design more robust quantum strategies.\n\n\\section{Discussion}\nIn summary, we have investigated the efficiency requirement for the violation of Bell inequalities in the maximally nonlocal theory. Our result implies that, for showing the maximally nonlocal theory, the detector efficiency requirement can be arbitrarily low with enough input settings. Our work opens a few interesting avenues for future investigations. First, though our work essentially closes the gap between the upper bound and the lower bound of the detector efficiency for the maximally nonlocal theory, the corresponding question in the quantum theory is still wide open. It would be interesting to apply our techniques to solve the analog question in the quantum theory.\nSecond, there is a small gap between the detector efficiency bounds of the maximally nonlocal theory when $2<N<\\infty$. Closing this gap may require some more delicate estimates on both the upper bound and  the lower bound.\n\n\\section*{Acknowledgements}The authors acknowledge insightful discussions with X. Ma. This work was supported by the 1000 Youth Fellowship program in China.\n\nZ. C. and T. P. contributed equally to this work.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\nThere is a rapidly expanding use of low temperature platforms ($T < 1$~K), for fundamental science, sensitive instrumentation, and new technologies of potentially significant commercial impact. The availability of commercial ``push button'' cryogen-free dilution refrigerators is a key game-changing recent development, in which European companies have played the leading role. The availability of cryogen-free commercial platforms to 1~mK is a future prospect \\cite{Casey2013}.\n\nPrecise measurement of the thermodynamic temperature of these low temperature platforms is crucial for their operation. However, despite the sophisticated nature of the cooling technology, reliable and convenient thermometers are unavailable.\n\nIn the temperature range below 1~K the dissemination of the kelvin is usually realised using secondary thermometers and superconducting reference points \\cite{LounasmaaB,EnssB,PobellB}. Examples include: resistance thermometers; paramagnetic susceptibility; vibrating wire\/tuning fork viscometry. These thermometers do not reliably cover the temperature range 1~K to 10~mK (a fortiori to 1~mK). The calibration of these thermometers against the PLTS-2000 \\cite{PLTS2000,Rusby2002} is impractically time consuming and expensive. Resistance thermometers are intrinsically subject to shifts in calibration and prone to self heating and pick-up of electromagnetic interference. Paramagnetic-salt susceptibility thermometers are subject to degradation with time, and viscometric methods measure temperature inside the mixing chamber and not on the typical sample mounting platform.\n \nThe dependence of $^{3}$He melting pressure on temperature has been used as the laboratory standard in specialist laboratories for many years. The PLTS-2000 scale is based on the melting pressure of $^{3}$He but this is viewed as difficult to implement for the rapidly expanding community of users of ultra-low temperature platforms. $^{60}$Co nuclear orientation thermometry is a primary technique that relies on the known anisotropy of the decay of radioactive nuclei as a function of temperature but is only effective in a narrow temperature range, and is slow and user unfriendly. Nuclear magnetic susceptibility thermometry is likewise insensitive above 20~mK, and user unfriendly. Reliable and traceable thermometry below 1~K thus offers demanding challenges even to the specialist, and, despite the sophisticated low temperature platforms now readily available with ``push-button'' operation, a solution which is reliable, traceable and user-friendly has yet to be developed. \n\nA quote attributed to Prof O V Lounasmaa, founder of the Low Temperature Laboratory at Helsinki University of Technology (now Aalto University) in the context of the T \\textless 1 K regime reads ``A person with one thermometer knows what the temperature is; a person with two will forever have doubt''. The challenge is to rectify this state of affairs.\n\nThe need to have a thermometer that measures thermodynamic temperature outside of international standards laboratories has led to the development of thermometers with a simple relationship between the measured parameter and the Boltzmann constant, $k_{\\mathrm{B}}$, for example thermometers that couple noise to a SQUID either directly \\cite{Lusher2001} or indirectly via the magnetic Johnson noise \\cite{Varpula1993,Beyer2007,Engert2009,Enss2013} and Coulomb Blockade thermometers \\cite{Pekola1994} based on electric conductance characteristics of tunnel junction arrays.\n\nIn this paper we describe recent advances in current sensing noise thermometer, exploiting low noise and high bandwidth dc SQUID amplifiers, for the high precision and rapid measurement of absolute temperature, covering the entire temperature range of interest from 4K to below 100 microK.\n\n\\section{Current Sensing Noise Thermometry}\n\\label{sec:1a}\n\nThe current sensing noise thermometer was pioneered by Giffard et al.\\cite{Giffard1972,Webb1973} and developed by Webb and Washburn \\cite{Webb1983}. The principle is to measure the thermal Johnson noise \\cite{Johnson1928} in a resistor to determine its temperature. The performance of the thermometers developed in \\cite{Giffard1972,Webb1973,Webb1983} was limited by the noise of the RF SQUID amplifiers; in \\cite{Webb1983} the typical device noise temperature was a few mK. Using a low transition temperature dc SQUID as the front end amplifier the current sensing technique becomes a practical approach for measuring absolute temperature from 4.2 K down to sub mK temperatures, as shown in \\cite{Lusher2001}; this work used commercial dc SQUIDs. The improved sensitivity of dc SQUIDs fabricated at PTB \\cite{Drung2007} allows the use of sensor resistors with much larger resistance, thereby increasing the speed of the thermometer. In this paper we demonstrate experimentally an improvement in speed by orders of magnitude and we demonstrate that the percentage precision is independent of temperature.\n\nThe principle of operation of a current sensing noise thermometer is discussed in detail in our earlier paper \\cite{Lusher2001} but is summarised below. The mean square open circuit noise voltage per unit bandwidth across the terminals of a resistor, $R$, at temperature $T$ is given by the Nyquist expression:\n\\begin{equation}\n\\label{eq:1}\n\\left\\langle V^{2}_{N}\\right\\rangle=4k_{\\mathrm{B}}TR\n\\end{equation}\nThis is a white noise spectrum over the frequency range of interest. In our current sensing noise thermometer, the resistor whose absolute temperature is to be measured is connected in series with the superconducting input coil of the readout SQUID through a superconducting twisted pair. The mean square noise current flowing in the SQUID input coil per unit bandwidth arising from the thermal noise in the resistor is then given by:\n\\begin{equation}\n\\label{eq:2}\n\\left\\langle I^{2}_{N}\\right\\rangle=\\frac{4k_{\\mathrm{B}}T}{R}\\left(\\frac{1}{1+\\omega^{2}\\tau^{2}}\\right)\n\\end{equation}\nwhere $\\omega = 2 \\pi f$. The time constant $\\tau=L\/R$, where the total inductance in the input circuit $L$ includes the SQUID input coil inductance $L_{i}$ and any additional stray inductance including that of the twisted pairs. By increasing the resistance, it is possible to increase the bandwidth of the noise spectrum. However, this results in a reduced noise power from the resistor at $\\omega \\tau < 1$, which must be kept much greater than the SQUID noise in order to ensure a reasonable device noise temperature, $T\\mathrm{_{N}}$, much less than the minimum temperature we wish to measure . The SQUID is operated in flux-locked loop (FLL) mode where the gain depends only on fixed, known parameters. Thus, if the value of the resistive sensor and the SQUID gain can be measured, a current noise spectrum can be used to determine absolute temperature using Equation \\ref{eq:2}.\n\\section{Experimental Details}\n\\label{sec:1}\n\\begin{figure}\n  \\includegraphics{Fig1.eps}\n\\caption{Schematic diagram of the current sensing noise thermometer. The dc SQUID amplifies the thermal noise current of the resistor with the output captured by a digitiser. $T_{1}$ can be at a different temperature to $T_{2}$, the temperature of the SQUID.}\n\\label{fig:1}      \n\\end{figure}\nFigure \\ref{fig:1} shows a schematic of the measurement set-up. The SQUID sensor used in this work is a PTB two-stage SQUID \\cite{Drung2007} with an input coil inductance of $L_{i}=1.05$~$\\mu$H and a mutual inductance $M_{i}$ between the SQUID and the input coil equivalent to $1\/M_{i}=0.29$~$\\mathrm{\\mu}$A\/$\\phi_{0}$, where $\\phi_{0}$ is the flux quantum. When operated with a 10~k$\\mathrm{\\Omega}$ feedback resistor, the FLL gain is 0.42~V\/$\\phi_{0}$. The SQUID is located remotely from the sensor resistor, allowing the resistor to be placed in environments hostile to SQUID operation. In this case the SQUID is mounted on the continuous heat-exchanger plate of the dilution fridge at approximately 100~mK. At this temperature, the white flux noise floor of the SQUID is $0.58~\\mathrm{\\mu}\\phi_{0}$Hz$^{-1\/2}$, equivalent to a coupled energy sensitivity, $\\epsilon_{c}=22 h$ in units of Planck's constant, where:\n\\begin{equation}\n\\label{eq:3}\n\\epsilon_{c}=\\frac{1}{2}L_{i}\\frac{\\left\\langle \\phi^{2}_{N}\\right\\rangle}{M^{2}_{i}}\n\\end{equation}\nand  $\\left\\langle \\phi^{2}_{N}\\right\\rangle$ is the flux noise power. This can be compared to  $\\epsilon_{c}=500 h$ for the commercial Quantum Design \\cite{QuantDesign} SQUID operated at 4.2~K used in our previous work \\cite{Lusher2001,Lusher2003}. This lower noise floor corresponds to a lower amplifier noise temperature  $T\\mathrm{_{N}}$. Thus through choice of resistor the user can minimise the noise temperature for absolute precision or increase the bandwidth of the measurement for increased speed. The primary focus of this work was to establish the optimum choice of resistance value for operation of the thermometers on a dilution refrigerator (with a typical base temperature of 5~mK). In addition in order to investigate the $^{3}$He melting curve fixed points on the provisional low temperature scale PLTS-2000 we need a thermometer that can be reliably operated below 1~mK. For these reason we chose sensor resistance values of 0.2~m$\\mathrm{\\Omega}$, 0.14~$\\mathrm{\\Omega}$ and 1.290~$\\mathrm{\\Omega}$.\n\n\\subsection{Construction of noise sensors}\n\\label{sec:2}\n\\begin{figure}\n  \\includegraphics[width=0.5\\textwidth]{Fig2.eps}\n\\caption{Schematic of a PtW noise thermometer mounted onto a copper sample plate. The plate has either a thermal link to the mixing chamber of the dilution fridge or the sensor is mounted directly onto the copper nuclear demagnetisation stage.}\n\\label{fig:2}      \n\\end{figure}\nThe 0.2~m$\\mathrm{\\Omega}$ resistor was fabricated with a copper foil as described in \\cite{Lusher2001}. Using this material it is not practical to make a sensor with a resistance greater than $\\approx$~10~m$\\mathrm{\\Omega}$. A higher resistivity material was needed for the 0.14~$\\mathrm{\\Omega}$ and 1.290~$\\mathrm{\\Omega}$ sensors. For this work 50~$\\mu$m diameter platinum tungsten wire (Pt92\\%\/W8\\%) \\cite{Goodfellow} was used. PtW was chosen as it has a weakly temperature dependant resistivity as well as having a very small magneto-resistance \\cite{Cieloszyk1975}. Figure \\ref{fig:2} is a schematic diagram of the sensor resistor and the sample plate. The required length of PtW wire is achieved by hard soldering it to two pieces of 100~$\\mu$m thick copper foil separated by a well-defined gap, using Johnson Matthey EasyFlo2 solder. One foil is electrically grounded and consequently heat sunk to a large annealed copper holder with a copper screw contact. The other foil is electrically isolated from the holder. The sensor resistor is then connected to a pair of niobium screw terminals via short sections of CuNi clad NbTi wire. The screw terminals are connected to the remote SQUID input coil through a shielded NbTi superconducting twisted pair. The resistor and copper former are enclosed in a niobium can for shielding and heat sunk to a copper sample plate with an M4 screw and cone joint as depicted in Figure \\ref{fig:2}. For the tests of the  1.290~$\\mathrm{\\Omega}$ sensor the sample plate was mounted at the mixing chamber plate of the dilution fridge, the  0.2~m$\\mathrm{\\Omega}$ and 0.14~$\\mathrm{\\Omega}$ sensors were mounted directly on the top plate of a copper nuclear demagnetisation stage. \n\\subsection{Characterisation of the sensors}\n\\label{sec:3}\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig3.eps}\n\\caption{Current noise spectra at 500~mK for the three noise sensors evaluated. Solid lines are fits using equation \\ref{eq:2}.}\n\\label{fig:3}      \n\\end{figure}\nIn principle, the current sensing noise thermometer is a primary thermometer; if the resistance of the sensor and the transfer function of the SQUID are known accurately. In practice however, it is more convenient to stabilise at a known reference temperature and determine the parameters $R$ and $L$ from a fit of the output noise power. For this single point calibration of our thermometers, the sample plate was held at 500~mK, measured by the melting curve thermometer. The temperature stabilisation was realised with a digital PID temperature controller to within 5~$\\mu$K. The SQUID output voltage is read out using room temperature Magnicon XXF electronics \\cite{Magnicon} and is captured with a PXI digitiser card \\cite{NationalInstruments}. Time traces were captured with sampling frequencies of 50~kS\/s, 500~kS\/s, 2.5~MS\/s for 0.2~m$\\mathrm{\\Omega}$, 0.14~$\\mathrm{\\Omega}$ and 1.290~$\\mathrm{\\Omega}$ sensors respectively, with a record length of 2$^{20}$ points. 1000 time domain traces were captured for each sensor.  Each captured time trace is individually Fourier transformed and the power spectral density is averaged in the frequency domain for improved precision. Before fitting, it was necessary to remove the noise power contribution of the SQUID, which was measured at high frequencies and assumed to be frequency independent over the range used for the fits.\nThe current noise spectra obtained in this way are shown in figure \\ref{fig:3}. Fits were performed on the resulting current noise spectrum using Equation \\ref{eq:2}, with $T=500$~mK. The resulting values for the resistance of the noise sensor $R$ and total inductance of the circuit $L$ are summarised in table \\ref{tab:1}. These parameters were constrained in the fitting routine to determine the temperature of subsequent noise traces.\n\nAn alternative method to obtain the single fixed point calibration is to integrate a superconducting fixed point directly into the input circuit of the thermometer. An additional coil can be added to the superconducting flux transformer,wound around a sample with a known superconducting transition temperature; thermalized at the same point as the resistive sensor. The total inductance, $L$, of the input circuit is then sensitive to the state of the superconducting fixed point. Figure \\ref{fig:4} shows the temperature dependence of the inductance measured in a current sensing noise thermometer in which we have included an in-situ tantalum \\cite{Milne1961} fixed point device ( $T\\mathrm{_{c}}$ = 4.45 K). Here 12 turns of NbTi wire are wound around a tantalum bobbin forming part of the superconducting flux transformer. The sharp transition of the tantalum is clearly visible in the noise thermometer measurement.\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig4.eps}\n\\caption{Total inductance, $L$, of the noise thermometer input circuit as a function of temperature measured on the calibrated Germanium resistance thermometer. A sharp decrease in inductance is observed when the noise thermometer cools through the superconducting transition of the integrated tantalum fixed point device.}\n\\label{fig:4}       \n\\end{figure}\n\\begin{table}\n\\caption{Summary of the parameters of the noise thermometers evaluated,  $T\\mathrm{_{N}}$ is measured from the high frequency noise.}\n\\label{tab:1}      \n\\begin{tabular}{llll}\n\\hline\\noalign{\\smallskip}\nResistance & Total Inductance &  $T\\mathrm{_{N}}$ & Bandwidth (3dB) \\\\\n\\noalign{\\smallskip}\\hline\\noalign{\\smallskip}\n0.20436~m$\\mathrm{\\Omega}$  & 1.5016~$\\mu$H & $<$1~$\\mu$K & 22 Hz\\\\\n0.13781~$\\mathrm{\\Omega}$ & 1.5483~$\\mu$H  & 40~$\\mu$K & 14.5 kHz\\\\\n1.290~$\\mathrm{\\Omega}$ & 1.315~$\\mu$H & 400~$\\mu$K & 156 kHz \\\\\n\\noalign{\\smallskip}\\hline\n\\end{tabular}\n\\end{table}\n\n\\section{Performance}\n\\label{sec:4}\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig5.eps}\n\\caption{Comparison of temperature obtained from the current sensing noise thermometers with that of a $^{3}$He melting curve thermometer, a calibrated germanium resistance thermometer and a pulsed platinum NMR thermometer over a wide temperature range.  The linear fit (solid line) to high temperature data is also shown. The decoupling of the thermometer at the lowest temperatures is modelled by fits that assume a metallic thermal link to the nuclear stage with a constant heat leak into the noise thermometer (dotted lines).}\n\\label{fig:5}       \n\\end{figure}\nThe current sensing noise thermometers were operated over a wide temperature range from 4.2~K down to 200~$\\mathrm{\\mu}$K. Over this range there is not a single thermometer against which a comparison can be made. In this work we used a $^3$He melting curve thermometer to determine the Provisional Low Temperature Scale, PLTS-2000 (between 1 and 700~mK); a calibrated germanium thermometer from 50~mK to 4.2~K and a pulsed NMR platinum thermometer, calibrated by the $^3$He melting curve thermometer, was used below 1~mK. Figure \\ref{fig:5} shows this comparison for all three thermometers. At each temperature above 6~mK the temperature was stabilised for 1 hour. Excellent agreement between the thermometers was observed over a wide temperature range. Deviations from the reference thermometer are observable up to progressively higher temperature for higher noise sensor resistance. This can be accounted for by a heat leak to the resistor resulting in a temperature gradient across the resistor. Reducing the size of the heat leak through improving the heat sinking and filtering of the wires to the `floating' side of the sensor resistor will be the subject of future development.\n\nFigure \\ref{fig:6} shows the percentage deviation of the comparison between the 0.14~$\\mathrm{\\Omega}$ sensor and the melting curve thermometer. Here an agreement to better than 1~\\% is achieved down to the lowest temperatures attainable on a commercial dilution refrigerator.\n\nIt is clear that below 1~mK a relatively low sensor resistance is desirable to ensure good thermalization. The non-contact measurements, which form the basis of the magnetic field fluctuation thermometer \\cite{Enss2013} are an alternative method for $T<100\\mathrm{\\mu}$K. At the lowest temperature obtained, 165~$\\mu$K on the platinum thermometer, the 0.2~m$\\mathrm{\\Omega}$ sensor cooled to 180~$\\mu$K.\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig6.eps}\n\\caption{Comparison of temperature obtained from the 0.14~$\\mathrm{\\Omega}$ current sensing noise thermometers with the $^{3}$He melting curve thermometer, down to the lowest temperatures typically reached using a dilution refrigerator. Plotted as a percentage of $\\Delta T\/T$.}\n\\label{fig:6}       \n\\end{figure}\n\nAnother figure of merit for a thermometer is the measurement time required to obtain a given precision of temperature measurement. If we consider capturing a time trace of $N$ points with a time $\\Delta t$ between each point, for each point to be an independent measure of the square of the noise current,  $\\Delta t$ must be of order the correlation time of the noise or greater (which for this system is the time constant $\\tau = L \/ R$). Therefore in a measuring time $t_{meas}$ one can make approximately $N = t_{meas} \/ \\tau$ independent measurements. A rough estimate for the precision in a given measuring time is thus given by:\n\\begin{equation}\n\\label{eq:4}\n\\frac{\\Delta T}{T}\\approx \\left(\\frac{2}{N}\\right)^{1\/2}=\\left(\\frac{2 \\tau}{t_{meas}}\\right)^{1\/2}\n\\end{equation}\n\nThe percentage precision in a given measuring time is thus independent of temperature and determined by the noise bandwidth. The noise bandwidth for the sensors used is given in table \\ref{tab:1}, and can be seen in figure \\ref{fig:3}. Increasing the resistance of the sensor results in an improved precision in a given time, which was the motive behind using PtW resistors with a larger resistance value.\n\nTo experimentally determine the precision of the 1.29~$\\mathrm{\\Omega}$ PtW thermometer, the sample plate was stabilised at a known fixed temperature. Single shot noise traces were taken consecutively. Approximately 1000 individual spectra were captured for accurate statistics. A sampling frequency of 2.5 MS\/s with 2$^{20}$ points was used, giving a measurement time of 419 ms per trace. Figure \\ref{fig:7}A shows data taken with the PtW thermometer stabilised at 493~mK using the germanium thermometer. This follows a normal distribution as shown by Fig.\\ref{fig:7}B, where a Gaussian fit to a histogram of temperatures is shown. The precision is calculated using $\\sigma \/ T$, where $\\sigma$ is the standard deviation of the data and $T$ is the mean temperature. In a measurement time of 419 ms, a precision of 0.50 \\% was obtained. This precision measurement was repeated at 1.5 K, 100 mK and at 13 mK; the percentage precision was confirmed to be independent of temperature. Reducing the measurement time to 105 ms, resulted in a temperature independent precision of 1.15~\\%. We contrast this with the pioneering work on current sensing noise thermometers \\cite{Webb1973}, which required a measurement time of 2500~s to achieve a similar performance. The 1.29~$\\mathrm{\\Omega}$ PtW thermometer is about two orders of magnitude faster than our previous noise thermometers \\cite{Lusher2003}, magnetic field fluctuation thermometers \\cite{Beyer2007} and on-chip current sensing noise thermometers \\cite{Engert2009}, which all have achieved 1 \\% precision in times of the order of 10 s. Ultra-low temperature measurements using a contact free cross-correlation noise thermometer \\cite{Enss2013} have been reported down to 45~$\\mu$K, here using two SQUIDs the cross-correlation technique allows them to reduce the noise contribution from SQUID amplifiers, and the reported measurement time was 400 seconds.\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig7.eps}\n\\caption{(colour online) A: The scatter of temperature measurements using the 1.29~$\\mathrm{\\Omega}$ PtW current sensing noise thermometer, stabilised with a germanium thermometer at 493~mK. The x axis shows the real time taken to capture the traces. Each temperature reading was acquired in a measurement time of 419~ms. The horizontal line of best fit is the mean temperature. B: A histogram of temperatures showing the scatter follows a normal distribution.}\n\\label{fig:7}       \n\\end{figure}\n\nFigure \\ref{fig:8} illustrates the dependence of the precision obtained from all three of the current sensing noise thermometers as a function of the measurement time at a temperature of 100~mK. To vary the measurement time, the sampling frequency was kept constant at 50~kS\/s, 500~kS\/s, 2.5~MS\/s for 0.2~m$\\mathrm{\\Omega}$, 0.14~$\\mathrm{\\Omega}$ and 1.290~$\\mathrm{\\Omega}$ respectively; 2$^{20}$ points were captured for each trace and the number of averages was changed to give total measurement times of between 0.1 and 250 seconds. The best fit lines shown are proportional to the inverse of the square root of the measurement time, as in Equation \\ref{eq:4}.  The measured precision of 0.03~\\% obtained with the 1.29$\\mathrm{\\Omega}$, sensor in a measurement time of 100~s, is approaching the limit of our ability to stabilise the temperature at 100~mK.\n\\begin{figure}\n  \\includegraphics[width=0.75\\textwidth]{Fig8.eps}\n\\caption{Measured precision as a function of measurement time for each of the three thermometers tested, a best fit line proportional to the inverse of the square of the measurement time is shown for the 1.29 and 0.14~$\\mathrm{\\Omega}$ sensors. }\n\\label{fig:8}       \n\\end{figure}\n\n\\section{Conclusions}\n\\label{sec:5}\nIn conclusion, a series of fast, precise current sensing noise thermometers based on a range of resistive sensors has been tested. Using a single temperature calibration, the noise thermometers can be used to accurately measure a wide range of temperatures. We have presented data on a current sensing noise thermometer that has been optimised for speed, taking advantage of the improvements in SQUID noise and bandwidth. This is two orders of magnitude faster than previous noise thermometers, achieving 1.15~\\% precision in just 0.1~seconds. The 0.14~$\\mathrm{\\Omega}$ sensor tested is in agreement with the $^{3}$He melting curve thermometer to within better than 1~\\% over the operating temperature range of a dilution refrigerator, and in this case less than 2 seconds of measurement time is required for 1~\\% precision. Finally the lowest value of resistance chosen, 0.2~m$\\mathrm{\\Omega}$ resulted in a thermometer with a noise temperature of 1~$\\mu$K that showed reasonable agreement with a pulsed platinum NMR thermometer to below 200~$\\mu$K.\n\nThe current sensing noise thermometers described here constitute practical devices for disseminating thermodynamic temperature. The system is user-friendly and could be installed on any commercial ultra-low temperature platform, to provide a reliable and traceable measure of thermodynamic temperature, with high precision (0.1~\\%) and high accuracy (better than~1\\%) in a reasonable measurement time ($<$~10~s).The potential provision of absolute temperature to all users promises to be a revolutionary technical development in a temperature range expected to be of growing technological and commercial significance.\n\n\n\\begin{acknowledgements}\nThis work was supported by the European Metrology Research Program (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. Additional support was through the MICROKELVIN project. We acknowledge the support of the European Community - Research Infrastructures under the FP7 Capacities Specific Programme, MICROKELVIN project number 228464.\n\\end{acknowledgements}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAs quantum computing is becoming practical~\\cite{google2020hartree, jurcevic2020demonstration, ebadi2021quantum, gong2021quantum}, there has been a growing interest in employing near-term quantum algorithms to help solve problems in quantum chemistry~\\cite{moll2018quantum}, quantum machine learning~\\cite{benedetti2019parameterized}, and combinatorial optimization~\\cite{farhi2014quantum}. \nAny such near-term algorithm must consider the primary restrictions of Noisy Intermediate Scalable Quantum (NISQ) devices; e.g., the number of qubits, decoherence etc.\nVariational Quantum Algorithms (VQAs) such as the Quantum Approximation Optimization Algorithm (QAOA)~\\cite{farhi2014quantum} were developed as a potential approach to achieve a quantum advantage in practical applications keeping in mind these design restrictions.\n\nFor a user-specified input depth \\(l\\), QAOA consists of a quantum circuit with \\(2l\\) variational parameters. In the limit of infinite depth, for optimal parameters, the solution of QAOA converges to the optimum for a given combinatorial optimization problem~\\cite{farhi2014quantum}.\nHowever, a significant body of research has produced negative results~\\cite{hastings2019classical,marwaha2021local, barak2021classical, bravyi2020obstacles, farhi2020mis, farhi2020mc, chou2021limitations, marwaha2021bounds} for QAOA limited to logarithmic depth (in the number of qubits), exploiting the notion of \\emph{locality} or \\emph{symmetry} in QAOA.\nThis motivates the study of techniques that circumvent the restriction of locality or symmetry in QAOA, which exploit the information-processing capabilities of low-depth quantum circuits by employing non-local pre-and post-processing steps\\footnote{The time complexity of these auxiliary steps should be polynomial in input size for the algorithm to remain practically viable.}.\n\n\nOne such proposal is the recursive QAOA (RQAOA), a non-local variant of QAOA, which uses QAOA iteratively, and at every iteration, the size of the problem (usually expressed in terms of a graph) is reduced by one.\nThe authors in~\\cite{bravyi2020obstacles, bravyi2020hybrid, bravyi2021classical} empirically show that RQAOA always performs better than QAOA and is competitive to best known classical algorithms based on rounding of a semidefinite programming relaxation for Ising and graph colouring problems. \nHowever, given that these problems are \\(\\mathsf{NP}\\)-hard, there must also exist instances that RQAOA fails to solve, unless \\(\\mathsf{NP} \\subseteq \\mathsf{\\mathsf{BQP}}\\).\nHence, to further push the boundaries of combinatorial optimization on NISQ devices (and beyond), it is helpful to determine when RQAOA fails, as this can aid in developing better variants of RQAOA.\n\nIn this work, we study extensions of RQAOA, which perform better than RQAOA for the Ising problem (or equivalently, the weighted Max-Cut problem, where the external field is zero, refer to Sec.~\\ref{subsec:classical_sim}).\nWe do this by identifying cases where RQAOA fails (i.e., find small-scale instances with approximation ratio \\(\\leq 0.95\\)).\nThen, we analyze the reasons for this failure and, based on this, employ reinforcement learning (RL) to enhance the performance of RQAOA in a new algorithm that we call RL-RQAOA. \nIn particular, the proposed hybrid algorithm provides a suitable test-bed for assessing the potential benefit of RL: we perform simulations of (depth-1) RQAOA, and RL-RQAOA on an ensemble of randomly generated weighted $d$-regular graphs and show that RL-RQAOA consistently outperforms its counterparts. \nIn the proposed algorithm, the RL component itself plays an integral role in finding the solution, so this raises the question of the actual role of the QAOA circuit and thus potential quantum advantages.\nTo show that the QAOA circuits have a non-trivial contribution to the advantage, we compare RL-RQAOA to an entirely classical RL agent (which, given exponential time, imitates a brute force algorithm) and show that RL-RQAOA converges both faster and to better solutions than the simple classical RL agents.\n\n\nWe present our results as follows: Sec.~\\ref{sec:background} introduces QAOA, recursive QAOA (RQAOA), and fundamental concepts behind policy gradient methods in RL.\nSec.~\\ref{sec:rqaoa_lim} describes the limitations of RQAOA, and we illustrate their validity by performing numerical simulations.\nIn Sec.~\\ref{sec:rl_rqaoa}, we provide a sketch of the policies of RL-RQAOA (quantum-classical) and RL-RONE (classical, introduced to characterize the role of quantum aspects of the algorithm) and their learning algorithms. \nSec.~\\ref{sec:numerics} presents our computational results for the comparison between classical and hybrid algorithms (RQAOA, RL-RQAOA, and RL-RONE) on an ensemble of Ising instances.\nFinally, we conclude with a discussion in Sec. \\ref{sec:disc}.\n\n\n\\section{Related Work}\\label{sec:related_work}\n\n\nIn the context of RL, two works~\\cite{yao2020policy,sung2020using} developed optimizers based on policy gradient methods for VQA optimization, highlighting the robustness of RL-based techniques against off-the-shelf optimizers in the presence of noise.\nAs opposed to our work, both these works use an external RL policy to choose the angles of QAOA in a one-step Markov Decision Process (MDP) environment, and otherwise rely on the basic QAOA algorithm.\nA series of works~\\cite{yao2021reinforcement, yao2021rl} have also used RL-based optimization to generalize the approach of QAOA for preparing the ground state of quantum many-body problems.\nIn~\\cite{yao2021reinforcement}, an agent uses an auto-regression mechanism to sample the gate unitaries in a one-step MDP and employs an off-the-shelf optimizer to optimize angles to prepare a generalized QAOA ansatz. \nThe same set of authors then unify their previous works~\\cite{yao2020policy, yao2021reinforcement} with both the use of a generalized autoregressive architecture that incorporates the parameters of the continuous policy and an extended variant of Proximal Policy Optimization (PPO) applicable to hybrid continuous-discrete policies~\\cite{yao2020noise}. \nWe note that for all the works~\\cite{yao2020policy, yao2020noise, yao2021reinforcement, yao2021rl}, the quantum circuit (QAOA-type ansatz) is a part of an environment.\nIn our case, we focus on employing reinforcement learning to enhance the performance of the RQAOA, inspired by a recent work~\\cite{jerbi2021parametrized} on using quantum circuits to design RL policies. \nIn contrast to the approaches discussed above, we design an RL policy based on QAOA ansatz in a multi-step MDP environment where the quantum circuit (QAOA ansatz) is \\emph{not} a part of the environment. \nOther works have used Q-learning to formulate QAOA into an RL framework to solve difficult combinatorial problems~\\cite{wauters2020reinforcement} and in the context of digital quantum simulation~\\cite{khairy2020learning}. \n\nIn the context of employing non-local post-processing methods in quantum optimization algorithms akin to classical iterated rounding, there have been a few proposals to modify RQAOA. \nThe main idea behind RQAOA is to use QAOA iteratively to compute correlations and then, at every iteration, employ a rounding (variable elimination) procedure to reduce the size of the problem by one. \nThe variants of RQAOA proposed in the literature primarily differ in how the correlations are computed and how the variables are eliminated. \nFor instance, in~\\cite{bravyi2020obstacles, bravyi2020hybrid}, variable elimination scheme of RQAOA is deterministic and relies on correlations between qubits (qudits). On the other hand, the authors in~\\cite[Sec. \\RomanNumeralCaps{5}.A.]{mcclean2021low} propose a modified RQAOA where the rounding procedure is stochastic (controlled by a fixed hyper-parameter \\(\\beta\\)), and a variable is eliminated based on individual spin polarizations. \nIn contrast, our proposal of RL-RQAOA trains analogous parameter(s) \\(\\vec{\\beta}\\) via RL (See Appendix~\\ref{app:implementation}) and uses correlations between qubits to perform variable elimination.\n\n\\section{Background}\\label{sec:background}\n\nIn this section, we first provide a brief overview of QAOA (Sec.~\\ref{subsec:qaoa}) and its classical simulatability for the Ising problem (Sec.~\\ref{subsec:classical_sim}). Later, we introduce recursive QAOA (RQAOA) (Sec.~\\ref{subsec:rqaoa}) upon which we base our proposal for RL-enhanced RQAOA and introductory concepts behind policy gradient in RL (Sec.~\\ref{subsec:rl_primer}). \nThese notions will give us tools to develop policies based on the QAOA ansatz and their learning algorithms in the upcoming sections.\n\n\n\n\n\\subsection{Quantum Approximate Optimization Algorithm}\\label{subsec:qaoa}\n\nQAOA seeks to approximate the maximum of the binary cost function $\\mathcal{C}: \\{0,1\\}^n \\rightarrow \\mathbb{R}$ encoded into a Hamiltonian as $H_n = \\sum_{x \\in \\{0,1\\}^n} \\mathcal{C}(x) \\ket{x}\\bra{x}$. \nStarting from an initial state $\\ket{s} = \\ket{+^n}$ (uniform superposition state), QAOA alternates between two unitary evolution operators $U_p(\\gamma) = \\exp(-i \\gamma H_n)$ (phase operator) and $U_m(\\alpha) = \\exp(-i \\alpha H_b)$ (mixer operator) respectively, where $H_b = \\sum_{j=1}^{n} X_j$. \nHereafter, $X, Y, Z$ are standard Pauli operators and $P_j$ is a Pauli operator acting on qubit $j$ for $P \\in \\{X, Y, Z\\}$. The phase and mixer operator are typically applied a total of $l$ times, generating the quantum state,\n\\begin{eqnarray}\n        \\ket{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})} &=& \\prod_{k=1}^l \\exp(-i \\alpha_k H_b) \\exp(-i \\gamma_k H_n) \\ket{s},\n    \\label{eq:variationalstate}\n\\end{eqnarray}\nwhere the variational parameters \\{\\(\\vec{\\alpha}, \\vec{\\gamma}\\} \\in [0,2\\pi]^{2l}\\) and the integer $l$ is called the QAOA \\emph{depth}. The depth $l$ controls the non-locality of the QAOA circuit. \nDuring the operation of QAOA, these parameters are tuned to optimize the expected value of $H_n := \\bra{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})} H_n \\ket{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})}$. \nThe preparation of the state (\\ref{eq:variationalstate}) is followed by a measurement in the computational basis, which outputs a bitstring \\(x\\) corresponding to a candidate solution of the cost function \\(\\mathcal{C}\\). \nThe probability \\(\\mathbb{P}_l(x)\\) of obtaining a bitstring \\(x \\in \\{0,1\\}^n\\) is given by Born's rule,\n\\begin{eqnarray}\\label{eqn:prob_bit}\n  \\mathbb{P}_l(x) =  \\left| \\langle x|\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})\\rangle \\right|^2.\n    \\label{eq:prob_bitstring}\n\\end{eqnarray}\n\n\nA candidate bitstring \\(x^*\\) is called an \\(r\\)-approximation solution to a given instance, for \\(0 \\leq r \\leq 1\\) if,\n\\begin{eqnarray}\n    \\mathcal{C}(x^*) \\geq r \\cdot \\max_x \\mathcal{C}(x).\n    \\label{eq:apprx}\n\\end{eqnarray}\nAn algorithm is said to achieve an approximation ratio of \\(r\\) for a cost function \\(\\mathcal{C}\\) if it returns an \\(r\\)-approximation or better for every problem instance in the class (i.e., in the worst case).\n\n\nWe say that depth-$l$ QAOA achieves an approximation ratio of $r$ for a problem instance of a cost function $\\mathcal{C}$ if there exists parameters $\\{\\vec{\\alpha}, \\vec{\\gamma}\\}$ such that\n\\begin{eqnarray}\n    \\langle H_n \\rangle_l := \\bra{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})} H_n \\ket{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})} \\geq r \\cdot \\max_x \\mathcal{C}(x)\n    \\label{eq:approx_ratio}\n\\end{eqnarray}\n\nWe note that repeating a sequence of state preparations and measurements approximates the distribution of \\(x\\) given by (\\ref{eqn:prob_bit}) and that (\\ref{eq:approx_ratio}) is the mean of this distribution. \nThe candidate bitstring \\(x^*\\) may then be selected to yield the maximum approximation ratio \\(r\\).\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=2\\columnwidth]{figure_1_rl_rqaoa_big_picture.pdf}\n    \\caption{\\textbf{Training QAOA-based policies for reinforcement learning.} We consider an RL-enhanced recursive QAOA (RL-RQAOA)  scenario where a hybrid quantum-classical agent learns by interacting with an environment which we represent as a \\emph{search tree} induced by the recursive framework of RQAOA. \n    The agent samples the next action \\(a\\) (corresponding to selecting an edge and its sign) from its policy \\(\\pi_{\\theta}(a|s)\\) and receives feedback in the form of a reward \\(r\\), where each state corresponds to a graph (the state space is characterized by a search tree of weighted graphs, where each node of the tree corresponds to a graph). \n    The nodes at each level of the search tree correspond to the candidate states for an agent to perceive by taking action. \n    For our hybrid agents, the policy \\(\\pi_{\\theta}\\) of RL-RQAOA (see Def.~\\ref{def:softmax_qaoa}) along with the gradient estimate \\(\\nabla_{{\\theta}} \\log \\pi_{{\\theta}}\\) is evaluated on a CPU as we are in the regime where depth \\(l=1\\). \n    However, the policy can also be evaluated on a quantum processing unit (QPU) for higher depths, when classical simulations can only be performed efficiently for graphs of small size. The training of the policy is performed by a classical algorithm such as \\(\\mathsf{REINFORCE}\\) (see Alg.~\\ref{alg:reinforce}), which uses sample interactions and policy gradients to update parameters.}\n    \\label{fig:big_picture}\n\\end{figure*}\n\n\\subsection{Classical Simulatability of QAOA for the Ising problem}\\label{subsec:classical_sim}\n\nNext, we review the classical simulatability of a paradigmatic case of QAOA for the Ising problem. \nThis is a core building block for simulating both (depth-1) RQAOA and RL-RQAOA. It enables their efficient classical simulation at depth-1 for arbitrary graphs.  \nGiven a graph \\(G_n = (V,E)\\) with \\(n\\) vertices \\(V = [n]\\) (where \\([n] = \\{1, 2,\\ldots, n\\}\\)) and edges \\(E \\subset V \\times V\\), as well as an external field \\(h_u \\in \\mathbb{R}\\) and a coupling coefficient (edge weight) \\(J_{uv} \\in \\mathbb{R}\\) associated with each vertex and edge respectively, then the Ising problem aims to find a spin configuration \\(s \\in\\{-1, +1\\}^n\\) maximizing the cost Hamiltonian\\footnote{The textbook Ising problem definition has a negative sign \\((-)\\), and the goal is to minimize the Hamiltonian.},\n\\begin{eqnarray}\n    H_n = \\sum_{u \\in V} h_u Z_u + \\sum_{(u,v) \\in E} J_{uv} Z_u Z_v.\n    \\label{eq:ising_func}\n\\end{eqnarray}\nThe Ising problem without any external field is equivalent to the weighted Max-Cut problem, where the goal is to find a bi-partition of vertices such that the total weight of the edges between them is maximized.\nThe expected value of each Pauli operator \\(Z_u\\) and \\(Z_uZ_v\\) on depth-1 QAOA can be computed classically in \\(O(n)\\) time using analytical results stated in Theorem~\\ref{thm:analytical_form} in Appendix~\\ref{app:classical_sim_qaoa}. \nSince the cost function has \\(O(n^2)\\) many terms in the worst case, computing the final expected value of (\\ref{eq:ising_func}) hence takes a total time in \\(O(n^3)\\) given the variational parameters.\n\n\\subsection{Recursive QAOA}\\label{subsec:rqaoa}\nIn this subsection, we outline the RQAOA algorithm of Bravyi et al.~\\cite{bravyi2020obstacles} for the Ising problem as defined in (\\ref{eq:ising_func}) with no external fields \\((h_u = 0, \\forall u \\in V)\\). \nThis will serve as a base for our proposal of RL-enhanced RQAOA.\nThe RQAOA algorithm aims to approximate the maximum expected value\\footnote{The bitstring \\(\\{0,1\\}^n\\) is analogous to the spin configuration \\(\\{-1,+1\\}^n\\) where \\(0\\) corresponds to \\(-1\\) and +1 to \\(1\\). Hereafter, we will use both of them interchangeably.}  \\(\\max_x \\bra{x} H_n \\ket{x}\\), where \\(x \\in \\{0, 1\\}^n\\).\nIt consists of the following steps. \nFirst, a standard depth-\\(l\\) QAOA is executed to find the quantum state \\(\\ket{\\Psi^*_l(\\vec{\\alpha}, \\vec{\\gamma})}\\) (with optimal variational parameters) as in (\\ref{eq:variationalstate}) that maximizes the expectation value of \\(H_n\\). \n For each edge \\((u, v) \\in E\\), the two-correlation \\(M_{u, v} = \\bra{\\Psi^*_l(\\vec{\\alpha}, \\vec{\\gamma})} Z_uZ_v \\ket{\\Psi^*_l(\\vec{\\alpha}, \\vec{\\gamma})}\\) is computed. \n A variable \\(Z_u\\) with largest \\(|M_{u, v}|\\) is then eliminated (breaking ties arbitrarily) by imposing the constraint \n \\begin{eqnarray}\n    Z_u = \\mathrm{sign}(M_{u, v})Z_v\n    \\label{eq:constraint_rqaoa}\n \\end{eqnarray}\n which yields a new Ising Hamiltonian \\(H_{n-1}\\) with \\(n-1\\) variables. The resulting Hamiltonian is processed iteratively, following the same steps. Finally, this iterative process stops once the number of variables is below a predefined threshold \\(n_c\\). \n The remaining Hamiltonian with \\(n_c\\) variables can then be solved using a classical algorithm (e.g., brute force method). \n The final solution can then be obtained iteratively by reconstructing eliminated variables using (\\ref{eq:constraint_rqaoa}). \n\n We note that the variable elimination scheme in RQAOA is analogous to rounding solutions obtained by solving continuous relaxations of combinatorial optimization problems. \n We refer the interested reader to~\\cite[Sec. \\RomanNumeralCaps{5}.A.]{mcclean2021low} for a detailed discussion on the connection between quantum optimization algorithms and classical approximation algorithms. \n Recall that the final expected value of \\(H_n\\) as in (\\ref{eq:ising_func}) can be computed in \\(O(n^3)\\) time. Since we can choose \\(n_c\\) such that \\(n_c \\approx O(1)\\), RQAOA runs for approximately \\(n\\) iterations, so that the total running time is \\(O(n^4)\\) (neglecting the running time needed to select the variational parameters).\n \n\n\n\n\\subsection{Reinforcement Learning Primer}\\label{subsec:rl_primer}\n\nAs our proposal to improve upon RQAOA is based on reinforcement learning, we introduce basic concepts behind RL and the policy gradient method in this subsection. \n\nIn RL, the agent learns an optimal policy by interacting with its environment using a trial-and-error approach~\\cite{sutton2018reinforcement}. \nFormally, RL can be modeled as a Markov Decision Process (MDP) defined by the tuple \\((\\mathcal{S}, \\mathcal{A}, p, R)\\), where \\(\\mathcal{S}\\) and \\(\\mathcal{A}\\) represent the state and action spaces (both can be continuous and discrete), the function \\(p: \\mathcal{S} \\times \\mathcal{S} \\times \\mathcal{A} \\rightarrow [0,1]\\) defines the transition dynamics, and \\(R: \\mathcal{S} \\times \\mathcal{A} \\rightarrow \\mathbb{R}\\) describes the reward function of the environment.\nAn agent's behaviour is governed by a stochastic policy \\(\\pi_{\\theta}(a|s) : \\mathcal{S} \\times \\mathcal{A} \\rightarrow [0, 1]\\), for \\(a \\in \\mathcal{A}\\) and \\(s \\in \\mathcal{S}\\).\nHighly expressive function approximators, such as deep neural networks (DNN), can be used to parametrize a policy \\(\\pi_{\\theta}\\) using tunable parameters \\(\\theta \\in \\mathbb{R}^d\\).\nAn agent's interaction governed by a policy \\(\\pi_{\\theta}(a|s)\\) in the environment can be viewed as sampling a trajectory \\(\\tau \\sim \\mathbb{P}_E(\\cdot)\\) from the MDP, where \\(\\mathbb{P}_E(\\tau) = p_0(s) \\pi_{\\theta}(a|s) p(s_1|s,a) \\cdots \\pi_{\\theta}(a_{H-1}|s_{H-1}) p(s_H|s_{H-1},a_{H-1})\\) is the probability of the trajectory \\(\\tau\\) of length \\(H\\) to occur, where \\(p_0\\) is a distribution of initial state \\(s\\).\nAn example of a trajectory is \\(\\tau = (s, a, s_1, a_1, \\ldots, s_{H-1}, a_{H-1}, s_{H})\\). An agent collects a sequence of rewards based on its interactions with the environment. \nThe metric that assesses an agent's performance is called the value function \\(V_{\\pi_{\\theta}}\\) and takes the form of a discounted sum as follows,\n\\begin{eqnarray}\\label{eqn:value_func}\nV_{\\pi_{\\theta}}(s) = \\mathbb{E}_{\\pi_{\\theta}, \\mathbb{P}_E} \\left(\\sum_{t=0}^{H-1} \\gamma^t r_t \\right)  = \\mathbb{E}_{\\pi_{\\theta}, \\mathbb{P}_E} \\left(R(\\tau)\\right)\n\\end{eqnarray}\nwhere \\(s\\) is an initial state of an agent's trajectory \\(\\tau\\) within an environment, \\(\\mathbb{P}_E\\) describes the environment dynamics (i.e., in the form of an MDP)), and \\(r_t\\) is the reward at time step \\(t\\) during the interaction. \nEvery trajectory has a horizon (length) \\(H \\in \\mathbb{N} \\cup \\{\\infty\\}\\) and the expected return involves a discounting factor \\(\\gamma \\in [0,1]\\). Most often one chooses \\(\\gamma < 1\\) to avoid unwanted diverging value functions for a horizon \\(H = \\infty\\). \nFinally, the goal of an RL algorithm is to learn an \\emph{optimal policy} \\(\\pi^*_{\\theta}\\) such that the value function is maximized for each state. \nOne way of finding a good policy is through the policy gradient method, i.e., finding an optimal set of parameters \\(\\theta\\) which maximize the value function of the policy (by evaluating its gradient).\nFor the sake of brevity, we defer the explanation of the policy gradient method to Appendix~\\ref{app:pgm}.\n\n\\section{Limitations of RQAOA}\\label{sec:rqaoa_lim}\n\nThis section highlights some algorithmic limitations of RQAOA by introducing an alternative perspective on it. Then, based on this perspective, we provide insights into when RQAOA might fail and why.\nIt is obvious that (depth-1) RQAOA must fail on some instances, since we assume \\(\\mathsf{BPP} \\subsetneq \\mathsf{NP}\\)\\footnote{Here, we use the complexity-theoretic assumption of \\(\\mathsf{BPP} \\subsetneq \\mathsf{NP}\\) because depth-1 RQAOA can be simulated classically.}, but these instances may be quite big a priori.\nWe primarily focus on finding small-size instances since we need a data set of small instances to be able to computationally efficiently compare the performance of (depth-1) RQAOA and RL-RQAOA.\n\nFirst, let us motivate the use of QAOA as a subroutine in RQAOA. In other words, why would one optimize depth-\\(l\\) QAOA (i.e., find energy-optimal parameters for a Hamiltonian) and then use it in a completely different way (i.e., perform variable elimination by computing two-correlation coefficients \\(M_{u,v}\\)). \nIntuitively, using QAOA in such a fashion makes sense because as depth \\(l \\rightarrow \\infty\\), the output of QAOA converges to the quantum state which is the uniform superposition over all optimal solutions, and hence, for each pair \\((u,v) \\in E\\), computing the coefficient \\(M_{u,v}\\) \\emph{exactly} predicts if the edge is correlated (vertices with the same sign; i.e. lie in the same partition) or anti-correlated (vertices with the different sign; i.e. lie in the different partition) in an optimal cut. \nThe next piece of intuition, which is not any kind of a formal argument, is that low-depth QAOA prepares a superposition state where low-energy states are more likely to have high probability amplitudes.\nThen the RQAOA selects the edge which is most correlated or anti-correlated in these low-energy states. \nFurthermore, assuming that an ensemble of reasonable solutions often agree on which edges to keep and which ones to cut, RQAOA will select good edges to cut or keep (from the Max-Cut perspective).\nHowever, we also expect RQAOA to fail sometimes, for instance, when the intuition mentioned above is wrong, or it assigns a wrong edge-correlation sign to an edge for other reasons. \nHence, as RQAOA fails, this raises the question of whether there are better angles to select an edge and its correct edge-correlation sign at every iteration than those which coincide with energy-optimal angles (see Fig.~\\ref{fig:couterexample}).\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=2\\columnwidth]{figure_2_rqaoa_counterexample.pdf}\n    \\caption{\\textbf{Illustration of a counterexample where the heuristic of using the energy-optimal QAOA angles in RQAOA fails.} Here, we show that for the weighted graph (9 vertices and 24 edges) depicted in (a), RQAOA makes a mistake even in its strongest regime, so at the very first iteration (i.e., \\(n_c = 8\\)). The two-correlation coefficients for each edge (at energy-optimal angles) are shown in the form of a horizontal bar plot in (b), where the edge \\((0,2)\\) has the maximal correlation coefficient. For the graph in (a), RQAOA with energy-optimal angles assigns a wrong edge-correlation (sign) to this edge which is precisely highlighted by a bold star in (c) and (d). Both (c) and (d) characterize the sets of good and bad QAOA angles where RQAOA makes a correct and a wrong choice, respectively.\n    This example is counter-intuitive: as the edge \\((0,2)\\) has the highest weight in the graph, intuitively, the variables should be correlated (same sign) as to maximize the energy. However, this leads to a sub-optimal solution which RQAOA achieves with energy-optimal angles. Yet, for different settings of QAOA angles which do not maximize the overall energy, this edge will still have the largest magnitude of correlation, but in this case, anti-correlation, which leads to the true optimum (see sub-figure (c)). \n    }\n    \\label{fig:couterexample}\n\\end{figure*}\n\n\nRQAOA can alternatively be visualized as performing a tree search to find the most probable spin configuration close to the ground state of the Ising problem. \nIn particular, at the \\(k^{th}\\) level of the tree, nodes correspond to graphs with \\(n-k\\) vertices, each having different edge sets. \nSuppose that a node has \\(n-k\\) vertices with \\(e\\) edges, then it will have \\(e\\) many children where each child corresponds to a graph with \\(n-k-1\\) vertices having different edge sets following the edge contraction rules by imposing (\\ref{eq:constraint_rqaoa}). \nThe original RQAOA proposal~\\cite{bravyi2020obstacles} is a randomized algorithm (in the sense that ties between maximal two-correlation coefficients are broken uniformly at random) on this tree exploring only a single path during one run and terminating at the \\((n-n_c)^{th}\\) level. \nThe decision of choosing an appropriate branch is performed based on the largest magnitude of the absolute value of two-correlation coefficients \\(M_{u, v}\\) computed via a depth-$l$ QAOA using \\(H_n\\)-energy optimal parameters. \nWhile exploring level-by-level, RQAOA assigns the edge correlations \\((-1~\\text{or}~+1)\\)\nwhere a vertex is eliminated according to the constraint (\\ref{eq:constraint_rqaoa}). \nWe note that in the case of ties between maximal two-correlation coefficients, independent runs of RQAOA might not necessarily induce the same search tree.\n\n\n\nThis alternative perspective described above provides some insights regarding the limitations of RQAOA: (i) when there are ties and branching occurs, it could be that only one path within a set of induced search tree leads to a good approximate solution; and (ii) it may be the case that even when there are no ties (i.e., one path and no branching), selecting edges to contract according to the maximal correlation coefficient stemming from energy-optimal parameters of QAOA is an incorrect choice to attain a good solution. \nA priori, it is not obvious if any of the above mentioned two possibilities can occur under the choice of energy-optimal angles. \nHowever, note that one of (i), (ii), or a combination of both must happen; otherwise, RQAOA is an efficient polynomial-time algorithm for the Ising problem. \nHence, in the case that RQAOA makes an incorrect choice, RQAOA lacks the ability to explore the search tree to find better approximate solutions. \nKeeping these considerations in mind, we will show later that both phenomena (i) and (ii) occur by performing an empirical analysis of RQAOA.\nWe now describe both the limitations mentioned above in detail below.\n\n\\begin{itemize}\n    \\item[(i)]\\label{item:ties} It may be the case that eliminating a variable by taking the \n    \\emph{argmax} of the absolute value of two-correlation coefficients is always a correct choice, but there can be more than one choice at every iteration.\n    Moreover, it is possible to construct instances with a small number of optimal solutions, where for the majority of \\(n-n_c\\) iterations (corresponding to the level of the tree) there is at least one tie (here, \\(m\\) ties corresponds to \\(m+1\\) pairs \\((u_1, v_1), \\ldots, (u_{m+1}, v_{m+1})\\) with the same two-correlation coefficient).\n    In other words, the number of times RQAOA needs to traverse the search tree in the worse case to reach the ground state (optimum) may be exponentially large; i.e., every argmax tie break leads to a new branching of the potential choices of RQAOA, and this happens at each level of the tree.\n    We showcase this phenomenon in our empirical analysis for one such family of instances (see Fig.~\\ref{fig:cage_ties}).\n    One may imagine perturbing the edge weights to avoid ties while preserving the ground states of the Hamiltonian, but no such perturbation is generally known. \n    \n    \\item[(ii)]\\label{item:rqaoa_mistake} It may be the case that the path to reach the ground state requires the selection of a pair \\((u,v)\\) (and its correlation sign) for which the two-correlation coefficient is \\emph{not} maximal according to QAOA at energy-optimal parameters (see Fig.~\\ref{fig:couterexample}).\n    This implies that RQAOA might be prematurely locking out on optimal solutions. \n    \n\\end{itemize}\n\n\\begin{figure}[htb]\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figure_3_cage_ties.pdf}\n    \\caption{Number of ties per iteration of RQAOA (average over 200 runs) for \\((3,8)\\)-cage graph (\\(30\\) vertices, \\(45\\) edges, edge weights \\((\\{-1, +1\\})\\). We chose \\(n_c=8\\) in our simulations where RQAOA achieved a mean approximation ratio of \\(0.955 \\pm 0.036\\) and the probability to reach the ground state was \\(33.5\\%\\). The y-axis (Number of Ties) is log-scaled.\n    The black crosses depict the mean values, with the error bar showing the 95\\% confidence interval of 200 independent runs.\n    The figure illustrates that one would invariably encounter a constant fraction of ties between maximal two-correlations no matter whatever path is chosen in the search tree, implying an exponential blow-up in the size of the search tree to be explored by RQAOA.}\n    \\label{fig:cage_ties}\n\\end{figure}\n\nWe provide examples of graphs to prove the validity of the observations above.\nIn the regime where there are ties between maximal correlation coefficients [(i)], we performed 200 independent RQAOA runs for the family of weighted \\((d,g)\\)-cage graphs\\footnote{A \\((d,g)\\)-cage graph \\((d \\geq 3, g \\geq 5)\\) is a \\(d\\)-regular graph of girth \\(g\\) (length of a shortest cycle contained in the graph)  consisting of the smallest number of vertices possible.}\n\\((3 \\leq d \\leq 7; 5 \\leq g \\leq 12;~\\text{edge weights}~\\{-1, +1\\})\\) where ties are broken uniformly at random for the \\(n-n_c\\) iterations (levels of the tree). \nWe work with these graphs because the subgraphs that (depth-1) QAOA sees are regular trees (for most edges at every iteration of RQAOA, QAOA will see a \\((d-1)\\)-ary tree, as cage graphs are \\(d\\)-regular graphs, which creates the situation of ties between correlation coefficients).\nHere, by \\emph{seeing} we refer to the fact that the output of depth-\\(l\\) QAOA for a qubit (vertex) only depends on the neighbourhood of qubits that are within \\(l\\) distance to the given qubit~\\cite{farhi2020mc}.\nFor these graphs, we found that in \\(86.4 \\pm 9.63\\%\\) of the \\(n-n_c\\) iterations, the variable to eliminate was chosen from the ties between maximal correlation coefficients (see Fig.~\\ref{fig:cage_ties}). \n\n\n\n\nTo investigate the scenario of [(ii)], we focus on a particular case where there are no ties (or comparatively less ties) and find instances such that taking the maximal two-correlation coefficient does not reach the optimum solution in the tree. For this, we performed a random search over an ensemble of \\(10600\\) weighted random \\(d\\)-regular graphs and found several small-size instances (\\#nodes \\(\\leq 30\\)) for which RQAOA did not attain the optimum.\n\n\nUsing both the theoretical and numerical observations discussed above, we create a dataset of graphs (containing both hard and random instances for RQAOA) for our later analysis. \nIn the next section, we develop our new algorithm (RL-RQAOA) and compare its performance to RQAOA to assess the benefit of employing reinforcement learning in the context of recursive quantum optimization specifically for hard instances. \nFinally, we give the relevant details about the data set of the graph ensemble considered in Sec.~\\ref{subsec:hard_inst}.\n\n\n\\section{Reinforcement Learning Enhanced RQAOA \\& Classical Brute Force Policy}\\label{sec:rl_rqaoa}\n\n\n\nHaving introduced the background of policy gradient methods and the limitations of RQAOA, we develop a QAOA-inspired policy which selects a branch in the search tree (eliminate a variable) at every iteration of RL-RQAOA. \nRecall that, even though selecting an edge to contract according to the maximal two-correlation coefficient is often a good choice, it is not always an optimal one, and also often, there is no single best option, but more (for instance, see Fig.~\\ref{fig:couterexample}).\nOur basic idea is to train an RL method to learn how to select the edges to contract (along with its edge-correlation sign) correctly while using the information generated by QAOA.\nAdditionally, to investigate the power of the quantum circuit within the quantum-classical arrangement of RL-RQAOA, we design a classical analogue of RL-RQAOA called reinforcement-learning recursive ONE (RL-RONE) and compare it with RL-RQAOA. \n\nTo overcome the limitations of RQAOA, one needs to carefully tweak (a) RQAOA's variable elimination subroutine and (b) the use of QAOA as a subroutine; i.e., instead of finding energy-optimal parameters, we learn the parameters of QAOA. For (a), we apply the non-linear activation function \\(\\mathsf{softmax}_{\\vec{\\beta}}\\) (see Def.~\\ref{def:softmax_qaoa}) on the absolute value of two-correlation coefficients \\(|M_{u,v}|\\) measured on \\(\\ket{\\Psi_l(\\vec{\\alpha}, \\vec{\\gamma})}\\). \nBy doing this, the process of selecting a variable to eliminate (and its sign) is represented by a smooth approximation of \\emph{argmax} that is controlled by a vector of \\emph{trainable} inverse temperature parameters \\(\\vec{\\beta}\\) (one \\(\\beta\\) per edge).\nThe parameters \\(\\vec{\\beta}\\) (initialized at low values) are then trained such that the probability of selecting an edge (or a branch at every iteration) with the highest expected reward tends to 1.\nIn the case of (b), we train the variational angles of QAOA in the course of learning rather than using the ones that give optimal energy. \nWe do this because of the following two reasons: (i) to avoid costly optimization\\footnote{We train the QAOA angles at depth-1 even though we can optimize them efficiently (see Appendix~\\ref{app:var_opt}). However, the optimization becomes non-trivial with an increase in depth.}; (ii) different angle choices can help the algorithm sometimes to choose optimal paths in the search tree that are not possible otherwise (see Fig.~\\ref{fig:couterexample}).\nWe note that the entire learning happens on one instance of the Ising problem.\nEven though it is conceivable to train the algorithm over an ensemble of instances by introducing suitable generalization mechanisms such that \\(\\vec{\\beta}\\) are dependent on instances, we solely focus on learning parameters of the policy of RL-RQAOA on one instance so that it eventually performs better than RQAOA. \\hfill \\break\n\n\\begin{algorithm*}[H]\\label{alg:reinforce}\n\\DontPrintSemicolon\n\n  \\SetAlFnt{\\small\\sffamily}\n\n  \\SetNlSkip{1em}\n  \n  \\KwInput{The policy of RL-RQAOA (Def.~\\ref{def:softmax_qaoa}) or RL-RONE (Def.~\\ref{def:softmax_one})}\n    \\BlankLine\nInitialize the policy parameters \\(\\theta\\).\\;\n\\While{True}\n{\n     Generate \\(N\\) episodes \\(\\left\\{(s_0, a_0, r_1, \\ldots, s_{H-1}, a_{H-1}, r_H)\\right\\}_i\\) following \\(\\pi_{\\theta}\\).\\;\n     \n     \\For{episode i in batch}{\n        Compute the returns \\(G_{i,t} \\leftarrow \\sum_{t'=1}^{H-t}\\gamma^{t'} r_{t+t'}^{(i)}\\).\\;\n        \n        Compute the gradients \\(\\nabla_{{\\theta}} \\log \\pi_{{\\theta}}(a_{t}^{(i)}| s_{t}^{(i)})\\).\\;\n     }\n    \n    Compute \\(\n\\Delta{\\theta}=\\frac{1}{N} \\sum_{i=1}^{N} \\sum_{t=0}^{H-1} \\nabla_{{\\theta}} \\log \\pi_{{\\theta}}(a_{t}^{(i)}| s_{t}^{(i)}) \\ G_{i, t}\\).   \\;   \n\n    Update \\(\\theta \\leftarrow \\theta + \\delta \\Delta\\theta\\).\n\n}\n       \n\\caption{\\(\\mathsf{REINFORCE}\\) algorithm for the policies of RL-RQAOA and RL-RONE}\n\\end{algorithm*} \\hfill \\break\n\n\nTo provide further details on the effective Markov Decision Process (MDP) that the above described policy will be exploring, note that the RQAOA method can be interpreted as a multi-step (also called a \\(n\\)-step) MDP environment (with a delayed reward and a non-trainable policy), where at every iteration, a variable is eliminated based on the information generated by QAOA. Let us now cast the learning problem of variable elimination in the RL framework, inspired by recent work~\\cite{jerbi2021parametrized} on using quantum circuits to design RL policies.\nFor every step of the episode\\footnote{Here, one episode corresponds to one complete run of (RL)-RQAOA.}, our RL agent is required to choose one action out of the discrete space equivalent to an edge set of the underlying graph; i.e., in the worse case, selects one edge from \\(n \\choose 2\\), on which it imposes a constraint of the form (\\ref{eq:constraint_rqaoa}).\nHence, the state space \\(\\mathcal{S}\\) consists of weighted graphs (which we could encounter during an RQAOA run) and the action space \\(\\mathcal{A}\\) consists of edges (and $\\pm 1$ edge-correlations to impose on them).\nThe actions are selected using a parameterized policy \\(\\pi_{\\theta}(a|s)\\) which is based on the QAOA ansatz.\nSince, we use the expectation value of the Hamiltonian \\(H_n\\) of the Ising problem as an objective function, the reward space is \\(\\mathcal{R} = [0, \\max_{x \\in \\{0,1\\}^n} \\bra{x}H_n\\ket{x}]\\). \n \nNext, we formally define the policy of RL-RQAOA and its learning algorithm, which is a crucial part of RL-RQAOA.  \n\n\\begin{defn}[Policy of RL-RQAOA]\\label{def:softmax_qaoa}\nGiven a depth-l QAOA ansatz acting on \\(n\\) qubits, defined by a Hamiltonian \\(H_n\\) (with an underlying graph \\(G_n = (V,E)\\)) and variational parameters \\(\\{{\\vec{\\alpha}}, \\vec{{\\gamma}}\\} \\in [0, 2\\pi]^{2l}\\), let \\(M_{u, v} = \\bra{\\Psi_l(\\vec{{\\alpha}}, \\vec{{\\gamma}})} Z_u Z_v \\ket{\\Psi_l(\\vec{{\\alpha}}, \\vec{{\\gamma}})}\\) be the two-correlations that it generates. We define the policy of RL-RQAOA as\n\\begin{equation}\\label{eqn:softmax_policy}\n\\pi_{\\theta}(a = (u, v)|s = G_{n})=\\frac{\\exp(\\beta_{u,v} \\cdot |M_{u,v}|)}{\\sum\\limits_{(u,v) \\in E} \\exp(\\beta_{u,v} \\cdot |M_{u,v}|)}\n\\end{equation}\nwhere actions \\(a\\) correspond to edges \\((u,v) \\in E(G_n)\\), states \\(s\\) to graphs \\(G_n\\) and \\(\\beta_{u,v} \\in \\mathbb{R}\\) (exists for every possible edge) is an inverse temperature parameter. Here, \\(\\theta = (\\vec{\\alpha}, \\vec{\\gamma}, \\vec{\\beta})\\) constitutes all trainable parameters, where \\(\\vec{\\beta} \\in \\mathbb{R}^{(n^2 - n)\/2}\\).\n\\end{defn}\n\n\n\n\nThe reader is referred to Alg.~\\ref{alg:rlrqaoa} for the pseudo-code of RL-RQAOA (for one episode), where the addition of RL components are highlighted in the shade of green. \nFurthermore, we note that RL-RQAOA is a generalized version of RQAOA because the former is exactly equivalent to the latter when the energy-optimal parameters \\(\\{\\vec{\\alpha}, \\vec{\\gamma}\\}\\) are specified by QAOA on \\(H_n\\), and for all \\((u, v) \\in E,~\\beta_{u,v} \\in \\vec{\\beta},~\\text{where}~\\beta_{u,v} = \\infty\\). \n\nSince the vector \\(\\vec{\\beta}\\) is edge specific and as we learn \\(\\beta_{u,v} \\in \\vec{\\beta}~\\text{for}~\\{u,v\\} \\in E\\) separately for every instance, we develop a fully classical RL algorithm, namely RL-RONE, to simply learn \\(\\beta_{u,v}\\) for all edges directly in spite of where the two-correlation coefficients \\(M_{u,v}\\) are generated from.\nIt is natural to consider this because it might be the case that in the hybrid quantum-classical arrangement of RL-RQAOA, the classical part (learning of \\(\\beta_{u,v}~\\text{for}~\\{u,v\\} \\in E\\)) is more powerful than the quantum part (computing two-correlations \\(M_{u, v}~\\text{for}~\\{u,v\\} \\in E\\) from the QAOA ansatz at given variational angles \\(\\{{\\vec{\\alpha}}, \\vec{{\\gamma}}\\}\\)). \nHence, in order to assess the contribution of the quantum circuit in RL-RQAOA, we define the policy of RL-RONE\nsuch that for each edge, we fix the two-correlation \\(M_{u, v} = 1\\); i.e., we do not use any output from the quantum circuit. \nAlthough simply using \\(M_{u, v} = 1\\) in Def.~\\ref{def:softmax_qaoa}, the policy will select an edge and always assign it to be correlated, rendering it to be less expressive. \nA solution to this problem is to simultaneously learn the parameters \\(\\beta_{u,v}^{+1}~\\text{(correlated edge)}~\\text{and}~\\beta_{u,v}^{-1}~\\text{(anti-correlated edge)}\\) for each edge. \nThen the resulting RL-RONE algorithm is expressive enough to reach the optimum solution. Moreover, it has trainable inverse temperature parameters \\(\\vec{\\beta}\\) where \\(|\\vec{\\beta}| = n^2 - n\\) for \\(n\\) the number of nodes of the graph \\(G_n\\).\nThe notion of an action slightly differs from the  RL-RQAOA policy as the action here corresponds to selecting an edge along with its sign ($+1$ and $-1$ for correlated and anti-correlated edges, respectively), while in RL-RQAOA, the two-correlation coefficient implicitly selects this sign.\nWe formally define the policy of RL-RONE below.\n\n\\begin{defn}[Policy of RL-RONE]\\label{def:softmax_one}\nGiven a Hamiltonian \\(H_n\\) (with an underlying graph \\(G_n = (V,E)\\)), we define the policy of RL-RONE as \n\\begin{equation}\\label{eqn:softmax_one}\n\\pi_{\\theta}(a = ((u, v), b)|s = G_{n})=\\frac{\\exp(\\beta^b_{u,v})}{\\sum\\limits_{b \\in \\{\\pm 1\\}}\\sum\\limits_{(u,v) \\in E} \\exp(\\beta^b_{u,v})}\n\\end{equation}\nwhere actions \\(a\\) correspond to edges \\((u,v) \\in E(G_n)\\) along with an edge correlation \\(b \\in \\{\\pm 1\\}\\), states \\(s\\) correspond to graphs \\(G_n\\) and \\(\\beta_{u,v}^{\\pm1} \\in \\mathbb{R}\\) (exists for every edge) are inverse temperature parameters. Here, \\(\\theta = (\\vec{\\beta}^{+1}, \\vec{\\beta}^{-1})\\) constitutes all trainable parameters, where \\(\\vec{\\beta}^{\\pm1} \\in \\mathbb{R}^{(n^2 - n)\/2}\\). \n\n\\end{defn}\n\n\nThe classical analogue RL-RONE can then be simulated by performing the following modifications to Alg. \\ref{alg:rlrqaoa}: (i) modify the parameters \\(\\theta\\) of the policy of RL-RONE by \\(\\theta = (\\vec{\\beta}^{+1}, \\vec{\\beta}^{-1})\\), (ii) delete \\(\\mathsf{Lines~4}\\) and \\(\\mathsf{5}\\), (iii) update \\(\\mathsf{Line~6}\\) by incorporating the policy of RL-RONE and the constraint (\\ref{eq:constraint_rqaoa}) in \\(\\mathsf{Line~7}\\) is imposed by feeding the correlation sign of the edge from the output (\\(b \\in \\{\\pm1\\}\\)) of the policy of RL-RONE. \\hfill \\break\n\n\n \\begin{algorithm*}[H]\\label{alg:rlrqaoa}\n\\DontPrintSemicolon\n\n  \\SetAlFnt{\\small\\sffamily}\n\n  \n  \\KwInput{A graph $G_n = (V, E)$ with $|V| = n$; a weighted adjacency matrix $J_n$ defined by Ising Hamiltonian $H_n$; the policy of RL-RQAOA (Def.~\\ref{def:softmax_qaoa}) or RL-RONE (Def.~\\ref{def:softmax_one}); $n_c$}\n  \\KwOutput{A binary bitstring $x \\in \\{\\pm 1\\}^n$}\n    \\BlankLine\n\n{\\color{black}Initialize the parameters \\(\\theta = (\\vec{\\alpha}, \\vec{\\gamma}, \\vec{\\beta})\\) of the policy; \\(G_{\\mathrm{current}} \\gets G_n\\); \\(H_{\\mathrm{current}} \\gets H_n\\); \\(J_{\\mathrm{current}} \\gets J_n\\).}\\;\n{\\color{black}Let \\(\\kappa^{(0)} : \\{\\pm 1\\}^n \\rightarrow \\{\\pm 1\\}^n\\) be an identity map.}\\;\n\\For{\\(i = 1\\) \\text{\\textbf{to}} \\(n-n_c\\)}\n{   \n    \n    {\\color{black} Run $\\mathsf{QAOA}_l(H_{\\mathrm{current}})$ to find a state $\\ket{\\Psi_l}$ for \n     angles \\(\\{{\\vec{\\alpha}}, \\vec{{\\gamma}}\\} \\in [0, 2\\pi]^{2l}\\).}\\; \n    \n    {\\color{black}Compute \\(M_{u, v} = \\bra{\\Psi_l} Z_uZ_v \\ket{\\Psi_l}, \\forall (u,v) \\in E\\).}\\;\n\n    {\\color{green!55!blue} Select \\((u,v) = \\textsf{softmax}_{(u,v) \\in E}(|M_{u,v}|, \\beta_{u,v})\\) as defined in (\\ref{eqn:softmax_policy}).}\\;\n    \n    {\\color{black} Imposing a constraint \\(Z_v = \\textsf{sign}(M_{u,v})Z_u\\), eliminate \\(Z_v\\) from \\(H_{\\text{current}}\\) resulting into a Hamiltonian \\(H_{n-i}\\) with \\(n-i\\) variables.}\\;\n    \n    {\\color{black}Define a function \\(\\kappa : \\{\\pm1\\}^{| V(G_{n-i})|}\n    \\rightarrow \\{\\pm1\\}^{|V(G_{\\text{current}})|}\\) \n    by updating for all \\(w \\in V(G_{\\text{current}})\\)\n        $ \\kappa(x)_w =\n    \\begin{cases}\n                    \\textsf{sign}(M_{u,v})x_u, & \\text{if } w = v.\\\\\n                    x_w, & \\text{otherwise}\\\\\n    \\end{cases}$}\\;\n    \n    $\\kappa^{(i)} \\leftarrow \\kappa^{(i-1)} \\circ \\kappa$\\;\n        \n    {\\color{black} Update \\(G_{\\mathrm{current}} \\gets G_{n-i}\\); \\(H_{\\mathrm{current}} \\gets H_{n-i}\\); and \\(J_{\\mathrm{current}} \\gets J_{n-i}\\).}\\;\n       }\n    {\\color{black} Let \\(G_{n_c} = (V,E)\\) be the final graph with \\(|V| = n_c\\) vertices.}\\;\n    Find \\(x^* = \\argmax_{x\\in \\{-1, 1\\}^{n_c}} \\bra{x}H_{n_c}\\ket{x}.\\)\\;\n    {\\color{green!55!blue} Update parameters \\(\\theta\\) using \\(\\mathsf{REINFORCE}\\) (Alg.~\\ref{alg:reinforce}).}\\;\n    \\Return \\(\\kappa^{(n-n_c)}(x^*)\\)\\;\n\n\n\\caption{RL-RQAOA \\((G_n= (V,E), J_n, n_c)\\) for one episode}\n\\end{algorithm*} \n\n\n\nWe train both the policies of RL-RQAOA and RL-RONE using the Monte Carlo policy gradient algorithm \\(\\mathsf{REINFORCE}\\), as explained in Appendix~\\ref{app:pgm}. \nAlso, refer to Alg.~\\ref{alg:reinforce} for the pseudo-code. \nThe horizon (length) of an episode is \\(n-n_c\\). \nThe value function is defined as \\(V_{\\pi_{\\theta}}(H_n) = \\mathbb{E}_{{\\pi_{\\theta}}}\\left[\\gamma^{n-n_c} \\cdot \\bra{x}H_n\\ket{x} \\right]\\), where \\(\\gamma \\in [0,1]\\), \\(H_n\\) is the Hamiltonian defined on \\(n\\) variables for a problem instance and \\(x\\) is a binary bitstring as defined in \\(\\mathsf{Line~14}\\) of Alg.~\\ref{alg:rlrqaoa}. \n\nIn this work, we only focus on simulations of depth-1 RQAOA and RL-RQAOA. Indeed, the particular case of depth-1 quantum circuits and Ising Hamiltonian RQAOA can be simulated efficiently classically; see Sec.~\\ref{subsec:classical_sim} and Appendix~\\ref{app:classical_sim_qaoa}. \nHowever, classical simulatibility is not known for Ising cost functions at depth larger than 2~\\cite{bravyi2021classical}, and more general cost Hamiltonians even at depth-1 (e.g., Max-\\(k\\)-XOR on arbitrary hypergraphs), leaving room for both quantum and RL-enhanced quantum advantage.\n\n\\section{Numerical Advantage of RL-RQAOA over RQAOA}\\label{sec:numerics}\n\nIn the previous section, we have introduced both our quantum (inspired) policy of RL-RQAOA and an entirely classical policy of RL-RONE, and their design choices, and based on these; we propose an RL-enhanced RQAOA and its classical analogue RL-RONE.\nAlthough we gave justifications for these choices, it is natural to evaluate their influence on the performance of RL-RQAOA and RL-RONE.\nIn this section, we first describe how we found hard instances for RQAOA and discuss their properties. \nWe then describe the results of our numerical simulations, where we consider both hard instances and random instances to benchmark the performance of (depth-1) RQAOA, RL-RQAOA, and RL-RONE.\nThe reader is referred to Appendix~\\ref{app:implementation} for implementation details for the above algorithms.\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=2\\columnwidth]{figure_4_probability_rqaoa_vs_rl_rqaoa.pdf}\n    \\caption{\\textbf{Comparison of success probability of RL-RQAOA and RQAOA on cage graphs.} The x-axis depicts the properties of cage graph(s), for instance, d3-g6 denotes that the instance is \\(3\\)-regular with girth (length of the shortest cycle) being 6. \n    The error-bars appear only for few instances (specifically for d3-g9, d3-g10 and d5-g5) because of the existence of multiple graph instances with the same properties (degree and girth). The evaluation of RL-RQAOA was done by evaluating the average learning performance over 15 independent runs. While, for RQAOA, the best energy is taken when given a fixed budget of 1400 runs. The probability for RL-RQAOA-max is computed by taking the maximum energy attained by the agent over all 15 independent runs for a particular episode. One the other hand, the probability for RL-RQAOA-vote (statistically more significant) is computed by aggregating the maximum energy attained for a particular episode only if more than 50\\% of the runs agree. We chose \\(n_c=8\\) for instances with nodes \\(\\leq 50\\) and \\(n_c=10\\) otherwise. The parameters \\(\\theta = (\\alpha, \\gamma, \\vec{\\beta})\\) of the RL-RQAOA policy were initialized by setting \\(\\vec{\\beta} = \\{25\\}^{{(n^2-n)}\/2}\\) and the angles \\(\\{\\alpha, \\gamma\\}\\) (at every iteration) to energy-optimal angles (i.e., by following one run of RQAOA). All agents were trained using \\(\\mathsf{REINFORCE}\\) (Alg. \\ref{alg:reinforce}).} \n    \\label{fig:rl_rqaoa_vs_rqaoa_random}\n\\end{figure*}\n\n\n\n\n\\subsection{Hard Instances for RQAOA}\\label{subsec:hard_inst}\n\n\nHere, our focus is on finding \\emph{small-size hard instances} (with approximation ratio as a metric) for the Ising problem where RQAOA fails. Note that, we assume it must fail eventually as if it does not, then \\(\\mathsf{NP} \\subseteq \\mathsf{BQP}\\) as the Ising problem is \\(\\mathsf{NP}\\)-hard in general.\nAs we lack techniques to analyze the performance guarantees of RQAOA at arbitrary depth \\(l\\) apart from special cases like ``ring of disagrees\" at depth-1~\\cite{bravyi2020obstacles}, it is a non-trivial task to find hard instances for RQAOA. \nIn this spirit, we generate an ensemble \\(\\mathcal{G}[n, d, w]\\) of \\emph{weighted random d-regular instances} with \\(n\\) vertices and edge weight distribution \\(w: E \\rightarrow \\mathbb{R}\\).\nWe then perform a random search over \\(\\mathcal{G}[n, d, w]\\) to find hard instances.\nConcretely, we construct a graph ensemble \\(\\mathcal{G}[n,d,w]\\) as follows: for each tuple of parameters \\((n, d, w) \\in \\{14, 15, \\ldots, 30\\} \\times \\{3, 4, \\ldots, 29\\} \\times \\{\\mathrm{Gaussian}, \\mathrm{bimodal}\\}\\), we generate 25 graphs whenever possible\\footnote{For generating \\emph{d}-regular graphs with \\(n\\) vertices, \\(1 \\leq d \\leq n-1\\) and further if \\(d\\) is odd, \\(n\\) must be even.} \nyielding \\(10600\\) graphs in total, where Gaussian \\((\\mathcal{N}(0,1))\\) and bimodal \\((\\{\\pm1\\})\\) are edge weight distributions.\nIntuitively, the instances with bimodal edge weights would have a huge level of degeneracy within the ground states, which is confirmed by our simulations. Moreover, for the instances with bimodal edge weights, where ties between two-correlation coefficients were encountered, the final approximation ratio was computed based on the best energy attained by running RQAOA for a maximum of 1400 independent runs.\nOn the other hand, for the instances with Gaussian edge weights \\(\\mathcal{N}(0,1)\\), we found that all instances had unique ground states. Hence, we ran RQAOA only once to get the best approximation ratio for instances with Gaussian edge weights.\n\n\n\n\n\nWe filter out \\(1027\\) (857 with bimodal weights and 170 with Gaussian weights) instances for which RQAOA's approximation ratio is less than \\(0.95\\). \nNote that RQAOA can only be closer to optimal the larger \\(n_c\\) is. \nIn other words, it monotonically improves the quality of the solution with an increase in \\(n_c\\).\nSince we want to improve upon RQAOA in its strongest regime, we choose \\(n_c = 8\\) (unless specified otherwise) for our numerical simulations.\nHowever, interestingly for the \\(1027\\) hard instances found above, even with \\(n_c = 4\\), we only found 26 instances (5 with bimodal weights and 21 with Gaussian weights) for which the approximation ratio decreased (for the rest, the approximation ratio remained the same). We chose \\(n_c=4\\) for the previously mentioned experiment because, for some instances, the edge weights cancelled out after an edge contraction subroutine, and as a consequence, the intermediate graph ended up being an empty graph (a graph with zero edge weights) for \\(1 \\leq n_c < 4\\).\n\n \\begin{figure}\n    \\centering\n    \\includegraphics[width=\\columnwidth]{figure_5_box_plot_rqaoa_vs_rl_rqaoa.pdf}\n    \\caption{\\textbf{Numerical evidence of the advantage of RL-RQAOA over RQAOA in terms of approximation ratio on hard instances}. The box plot is generated by taking the mean of the best approximation ratio over 15 independent runs of 1400 episodes for RL-RQAOA. The RL-RQAOA clearly outperforms RQAOA in terms of approximation ratio for the instances considered (these are exactly the instances where RQAOA's approx. ratio \\(\\leq 0.95)\\). We chose \\(n_c=8\\) in our simulations and the parameters \\(\\theta = (\\alpha, \\gamma, \\vec{\\beta})\\) of the RL-RQAOA policy were initialized by setting \\(\\vec{\\beta} = \\{25\\}^{{(n^2-n)}\/2}\\) and the angles \\(\\{\\alpha, \\gamma\\}\\) (at every iteration) were initialized randomly. All agents were trained using \\(\\mathsf{REINFORCE}\\) (Alg. \\ref{alg:reinforce}).}\n    \\label{fig:rl_rqaoa_vs_rqaoa_hard}\n\\end{figure}\n\n\n\n\n\\begin{figure*}\n    \\centering\n    \\includegraphics[width=2\\columnwidth]{figure_6_rl-rqaoa_rl-rone.pdf}\n    \\caption{\\textbf{Numerical evidence of relevance of the quantum circuit in RL-RQAOA and separation between RL-RQAOA and RL-RONE.} The above plot illustrates the separation between learning curves of RL-RQAOA and RL-RONE agents averaged across 15 bimodal weighted random 3-regular graphs with 100 (left) and 200 (right) nodes each. We chose \\(n_c=10\\) and \\(n_c=18\\) for 100 and 200 nodes, respectively in our simulations and the parameters \\(\\theta = (\\alpha, \\gamma, \\vec{\\beta})\\) of the RL-RQAOA policy were initialized by setting \\(\\vec{\\beta} = \\{25\\}^{{(n^2-n)}\/2}\\) and the angles \\(\\{\\alpha, \\gamma\\}\\) (at every iteration) to energy-optimal angles (i.e., by following one run of RQAOA). All agents were trained using \\(\\mathsf{REINFORCE}\\) (Alg. \\ref{alg:reinforce}).}\n    \\label{fig:rl_rqaoa_vs_rl_rone}\n\\end{figure*}\n\n\n\n\\subsection{Benchmarking}\\label{subsec:benchmarking}\n\n\\subsubsection{RQAOA vs RL-RQAOA on Cage Graphs}\nIn our first set of experiments, illustrated in Fig.~\\ref{fig:rl_rqaoa_vs_rqaoa_random}, we compare the performance of RL-RQAOA with RQAOA on random Ising instances derived from \\((d,g)\\)-cage graphs \\((3 \\leq d \\leq 7; 5 \\leq g \\leq 12;~\\text{edge weights}~\\{-1, +1\\})\\). \nThe aim of this experiment is twofold: first, to show that RL-RQAOA does not perform much worse than RQAOA on instances where the latter performs quite well;\nsecond, to test the advantage of RL-RQAOA over RQAOA in terms of the probability of attaining the optimal solution when there are many ties between two-correlation coefficients \\(M_{u,v}\\) at every iteration. \nNotably, we already demonstrated earlier (see Fig.~\\ref{fig:cage_ties}) that for cage graphs, RQAOA has a constant number of ties between maximal two-correlation coefficients for the majority of the \\(n-n_c\\) iterations. \nFor assessing our hypotheses, we evaluate the average learning performance over \\(15\\) independent RL-RQAOA runs over 1400 episodes.\nIn order to fairly compare RL-RQAOA with RQAOA, we run RQAOA independently for 1400 runs and choose the best solution from the result these runs.\nNote that, this is a more powerful heuristic than the vanilla-RQAOA (which outputs the first solution it finds) where the hyperparameter (the number of independent runs) controls the solution quality. \nBoth RL-RQAOA (vote variant) and RQAOA fail to reach the optimum for \\((3, 12)\\)-cage graph within the given budget (see Fig.~\\ref{fig:rl_rqaoa_vs_rqaoa_random}).\nHowever, by evaluating the resulting learning curves of RL-RQAOA, both our hypotheses can be confirmed for most of the instances.\n\n\n\n\n\\subsubsection{RQAOA vs RL-RQAOA on hard instances}\n For the next set of experiments, presented in Fig.~\\ref{fig:rl_rqaoa_vs_rqaoa_hard}, the flavour here is similar to the previous experiment but with the aim to show separation between RL-RQAOA and RQAOA for \\emph{hard} instances found in Sec.~\\ref{subsec:hard_inst}. \n More specifically, we show that RL-RQAOA always performs better than RQAOA on these instances in terms of the best approximation ratio achieved. \n We do this by evaluating average learning performance over \\(15\\) independent RL-RQAOA runs to assess this claim. Interestingly, RL-RQAOA outperformed RQAOA even when the angles of the QAOA circuit were initialized randomly.\n\n\n\n \n \\subsubsection{RL-RQAOA vs RL-RONE}\n \n However, the results in the previous two subsections do not indicate the importance of the quantum part in the quantum-classical arrangement. \n To address this, we performed a third set of experiments, presented in Fig.~\\ref{fig:rl_rqaoa_vs_rl_rone} where both algorithms were tested on random 3-regular graphs of 100 and 200 nodes. \n By comparing the performance of RL-RONE with RL-RQAOA, we can see a clear separation between learning curves of the agents of these algorithms, highlighting the effectiveness of the quantum circuits in solving the Ising problem. \n\n\n\\section{Discussion and Conclusion}\\label{sec:disc}\n\nIn this work, we analyzed the bottlenecks of a non-local variant of QAOA, namely recursive QAOA (RQAOA), and based on this, propose a novel algorithm that uses reinforcement learning (RL) to enhance the performance of the RQAOA (RL-RQAOA). \nIn the process of analyzing the bottlenecks of RQAOA for the Ising problem, we find small-size \\(hard\\) Ising instances from a graph ensemble of random weighted \\(d\\)-regular graphs. \nTo avoid missing out on better optimal solutions at every iteration, we cast the variable elimination problem within the RQAOA as a reinforcement learning framework; we introduce a quantum (inspired) policy of RL-RQAOA, which controls the task of switching between exploitative or exploratory behaviour of RL-RQAOA. \nWe demonstrate via numerical simulations that formulating RQAOA into the RL framework boosts the performance and performs as well as RQAOA on random instances and beats RQAOA on all hard instances we have identified.\nFinally, we note that all the numerical simulations for RQAOA (depth-1) and the proposed hybrid algorithm RL-RQAOA (depth-1) were performed classically, and no quantum advantage is to be expected unless we simulate both of them at higher depths.\nAn interesting follow-up to this work would be to assess the performance of both RQAOA and RL-RQAOA at higher depths on an actual quantum processing unit (QPU) in both noise and noise-free regimes.\n\n\\section*{Acknowledgements}\n\nYJP would like to thank Simon Marshall and Charles Moussa for useful discussions.\nSJ would like to thank Hans Briegel for useful discussions in the early phases of this project.  \nThe authors thank Adri{\\'a}n P{\\'e}rez-Salinas, and Andrea Skolik for useful comments on an earlier version of this manuscript and Casper Gyurik for reading the final version of this manuscript. \nTB, VD, and YJP acknowledge support from TotalEnergies. \nSJ acknowledges support from the Austrian Science Fund (FWF) through the projects DK-ALM:W1259-N27 and SFB BeyondC F7102. SJ also acknowledges the Austrian Academy of Sciences as a recipient of the DOC Fellowship. The computational results presented here have been achieved in part using the LEO HPC infrastructure of the University of Innsbruck and DSlab infrastructure of the Leiden Institute of Advanced Computer Science (LIACS) at Leiden University.\nThis work was in part supported by the Dutch Research Council (NWO\/OCW), as part of the Quantum Software Consortium programme (project number 024.003.037). \nVD acknowledges the support by the project NEASQC funded from the European Union's Horizon 2020 research and innovation programme (grant agreement No 951821).\nVD also acknowledges support through an unrestricted gift from Google Quantum AI.\n\n\n\\bibliographystyle{apalike}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nThe ability estimate position is key to most, if not all, robotics application involving mobility. In this paper we focus on the problem of visual odometry (VO), i.e. relative motion estimation based on visual information. This is the corner stone in vision-based SLAM systems, which will be the setting we will demonstrate our work in. As in our previous work, \\cite{GCN}, we estimate the motion using only an RGB-D sensor, and our target platform is a drone operating in an indoor environment. The RGB-D sensor makes scale directly observable without the need for visual-inertial integration or the computational cost of inferring depth using a neural network as in \\cite{Left-right,Single-stereo,SFMLearner}. This increases robustness, which is a key property when used on a drone and in particular for indoor environments where the margin for error is small and the there is typically less obvious textures than in outdoor environments. Our method is designed to be applicable to any system simply by adding an RGB-D sensor, without the need for complicated calibration and synchronisation routines with additional sensors, such as cameras or IMUs. Fusion can instead take place at a lower rate and with less need for precise timing, which, for example, makes integration with the flight control system of a drone simpler.\n\n\t\\begin{figure}\n\t    \\centering\n\t    \\resizebox{\\hsize}{!}{\n\t\t\\includegraphics[width=0.5\\textwidth]{uav.pdf}\n\t\t}\n\t\t\\caption{The top figure shows our drone preforming position hold using GCN-SLAM. The figures below show the intermediate output for comparison of binary features, ORB and GCNv2, in ORB-SLAM2 and GCN-SLAM respectively. GCNv2 (left) tends to predict more repeatable and evenly distributed features compared with ORB (right.)}\n\t\t\\label{fig:digest}\n\t\\end{figure}\n\nLike many other areas of research, there is a trend in SLAM to investigate deep learning-based methods. In \\cite{SuperPoint} a keypoint detector and descriptor called SuperPoint is presented. Experimental results show that this CNN-based method has more distinctive descriptors than classical descriptors such as SIFT, and a detector on par with them. However, evaluation on homography estimation in \\cite{SuperPoint} shows that it is only working on par with other keypoint extractors, classical or learning-based. In our previous work~\\cite{GCN}, presented before SuperPoint, we introduced the Geometric Correspondence Network, GCN, specifically tailored for producing keypoints for camera motion estimation, achieving better accuracy than classical methods. However, due to the computation and structure limitation of GCN, it is difficult to achieve real-time performance in a fully-operational system, e.g. on board a drone. Both keypoint extraction and matching are computationally too expensive. Furthermore, due to the multi-frame matching setup, integrating GCN into existing SLAM systems becomes non-trivial.\n\nIn this paper we introduce GCNv2, based on the conclusions from~\\cite{GCN}, to improve computational efficiency while still maintaining the high precision of GCN. Our contributions are:\n\\begin{itemize}\n\\item GCNv2 maintains the accuracy of GCN, providing significant improvements in motion estimation in comparison to related deep learning-based feature extraction methods as well as classical methods. \n\\item The inference of GCNv2 can run on embedded low-power hardware, such as a Jetson TX2, compared to GCN which requires a desktop GPU for real-time inference.\n\\item We design GCNv2 to have the same descriptor format as the ORB feature so that it can be drop-in substituted as the keypoint extractor in SLAM systems like ORB-SLAM2~\\cite{ORBSLAM2-TRO17} or SVO2~\\cite{SVO-TRO17}.\n\\item We demonstrate the effectiveness and robustness of our work by using GCN-SLAM\\footnote{Built on ORB-SLAM2 with ORB substituted for GCNv2} on a real drone for control, and show that it handles situations where ORB-SLAM2 fails.\n\\end{itemize}\n\n\\section{Related Work}\n\\label{sec:related}\nIn this section we cover related work in two areas. First VO and SLAM methods are covered and then we focus specifically on work on deep learning-based methods for image correspondence.\n\n\\subsection{VO and SLAM}\nIn direct methods for VO and SLAM, motion is estimated by aligning frames based directly on the pixel intensities, with~\\cite{Comport07} being an early example. DVO (Direct Visual Odometry), presented in~\\cite{DVO-Kerl13}, adds a pose graph to reduce the error. DSO~\\cite{DSO} is a direct and sparse method that adds joint optimisation of all model parameters. An alternative to the frame-to-frame matching is to match each new frame to a volumetric representation as in KinectFusion~\\cite{KinectFusion}, Kintinous~\\cite{Kintinous-IJRR14} and ElasticFusion~\\cite{ElasticFusion}. \n\nIn indirect methods, the first step in a typical pipeline is to extract keypoints, which are then matched to previous frames to estimate the motion. The matching is based on the keypoint descriptors and geometric constraints. The state-of-the-art in this category is still defined by ORB-SLAM2~\\cite{ORBSLAM-TRO15, ORBSLAM2-TRO17}. The ORB descriptor is a binary vector allowing high-performance matching. \n\nSomewhere between direct and indirect methods we find the semi-direct approaches. SVO2~\\cite{SVO-TRO17} is a sparse method in this category, and can run at hundreds of Hertz. There are also semi-dense methods, in which category LSD-SLAM~\\cite{LSD-SLAM} was one of the first. RGBDTAM~\\cite{RGBDTAM} combines both semi-dense photometric and dense geometric errors for pose estimation.\n\nThere are a number of recent deep learning-based mapping systems like \\cite{CNN-SLAM, DVSO}. The focus in these methods is deep learning-based single view depth estimation to reduce the scale drift inherent in monocular systems. CNN-SLAM~\\cite{CNN-SLAM} feeds the depth into LSD-SLAM. In DVSO~\\cite{DeepVO}, depth is predicted in a similar way to \\cite{Left-right}, using a virtual stereo view. CodeSLAM \\cite{code-slam} learns an optimizable representation from conditioned auto-encoding for 3D reconstruction. In S2D~\\cite{S2D}, we build on DSO~\\cite{DSO} and exploit both depth and normals predicted by a jointly optimised CNN. Some work on end-to-end training for motion estimation also exist. Image reconstruction loss is used for unsupervised learning in~\\cite{SFMLearner, Undeepvo}. However, geometry-based optimization methods still outperform end-to-end systems as shown in~\\cite{DeepVO}.\n\n\\subsection{Deep Correspondence Matching}\nThere is a an abundance of recent works that deploy variants of metric learning for training deep features for finding image correspondences~\\cite{feat_ref1, feat_ref2, feat_ref3, feat_ref4, feat_ref5, feat_ref6, LIFT, UCN, SuperPoint}.\nWorks in~\\cite{cd_ref1, cd_ref2} focus on improving learning-based detection with better invariances. \nAimed at a different aspect, \\cite{cd_ref3, cd_ref4, cd_ref5} use synthetic samples generated in a self-supervised manner to improve general feature matching.\n\nAmong the aforementioned methods, LIFT~\\cite{LIFT} in particular uses a patch-based method to perform both keypoint detection and descriptor extraction. SuperPoint~\\cite{SuperPoint} predicts the keypoints and descriptors using a single network together with the self-supervised strategy in~\\cite{cd_ref5}. Notably, \\cite{SuperPoint} shows that \\cite{SuperPoint, LIFT, UCN} work on par with classical methods like SIFT for motion estimation.\n\nIn GCN~\\cite{GCN}, we show that by learning keypoints and descriptors specifically targeting motion estimation, performance is improved -- contrary to what is reported for other more general deep learning-based keypoint extractor systems~\\cite{SuperPoint,UCN}. In this paper we introduce an extension to GCN, GCNv2. We demonstrate the applicability of these keypoints for SLAM and build on ORB-SLAM2 as it offers a comprehensive multi-threaded state-of-the-art indirect SLAM system with support for monocular as well as RGB-D cameras. ORB-SLAM2 complements the tracking front-end with a back-end that does both pose graph optimisation using g2o~\\cite{g2o} and loop closure detection using a binary bag of words representation~\\cite{Galves-BoW-TRO12}. To simplify this integration, we design the GCNv2 descriptor to have the same format as that of ORB.\n\n\\section{Geometric Correspondence Network}\n\nIn this section, we present the design of GCNv2, aimed at making it suitable for real-time SLAM applications running on embedded hardware rather than a powerful desktop computer. We first introduce the modifications to the network structure. Then, we present the training scheme for the binarized feature descriptor and keypoint detector.\n\n\\subsection{Network Structure}\nThe original GCN structure, proposed in~\\cite{GCN}, consists of two major parts: an FCN~\\cite{FCN} structure with a ResNet-50 backbone and a bidirectional convolutional network. The FCN is adopted for dense feature extraction and the bidirectional network is used for temporal keypoint prediction. Although impressive tracking performance has been achieved using GCN compared with existing methods, GCN has practical limitations when it comes to the use in a real-time SLAM system with limited hardware. \nThe network architecture requires relatively powerful computational hardware which renders it unable to run in real-time on board e.g. the Jetson TX2 used on our drone.\nFurthermore, the bidirectional structure requires matching between two or more frames at the same time. This significantly increases computational complexity for a window-based SLAM method, since keyframes then are updated dynamically based on the current camera position.\n\nInspired by SuperPoint~\\cite{SuperPoint}, which uses a simple structure to perform detection using a single frame,\nwe deploy a network with even fewer parameters and working on a lower scale than SuperPoint. Intuitively, the network performs an individual prediction for each grid cell of size $16\\times16$ pixels in the original image. \nIn GCNv2 all the pooling layers are replaced by convolutions with kernel size $4\\times4$, stride ${2}$ and padding $1$. As in SuperPoint, the network takes $320\\times240$ images as input. This is also the image size we used later for SLAM. Further details on the GCNv2 network specifics can be found in our publicly available source code\\footnote{\\url{https:\/\/github.com\/jiexiong2016\/GCNv2_SLAM}}. \n\nGCN-SLAM with GCNv2 runs at 20 Hz on Jetson TX2 and runs at around 80 Hz on a laptop with Intel i7-7700HQ and mobile version NVIDIA 1070. To achieve even higher frame rates we introduce a smaller version of GCNv2, called GCNv2-tiny, where we reduce the number of feature maps by half from the second layer and onward. GCN-SLAM with GCNv2-tiny runs at 40 Hz on a TX2 and is therefore well-suited for deployment on a drone.\n\n\n\\subsection{Feature Extractor}\n\n\\textbf{Binarized Descriptor}\nWe trained the features of GCNv2 to be binary for accelerating the matching procedure and to match those of ORB. To binarize the features, we add a binary activation layer on top the final output. It is essentially a hard sign function and is therefore not differentiable. The challenge is how to back-propagate the loss properly through this layer of the network. We used the method proposed in~\\cite{Binarization}. The binary activation layer can be written as follows:\n\\begin{equation} \\label{eq:bin_activation}\n\\begin{split}\n\\text{Forward: }\\,\\bm{b}(\\bm{x}) &= \\operatorname{sign}(\\bm{f}(\\bm{x})) = \n\\left\\{\n\\begin{array}{c l}\t\n     +1 & \\bm{f}(\\bm{x}) \\ge 0\\\\\n     -1 & \\text{otherwise}\n\\end{array}\\right. \\\\\n\\text{Backward: }\\,\\frac{\\partial \\bm{b}}{\\partial \\bm{f}} &=  \\frac{\\partial L_{metric}}{\\partial \\bm{b}} \\bm{1}_{|\\bm{f}| \\le 1}\n\\end{split}\n\\end{equation}\nwhere $\\bm{x} = (u, v)$ is the 2D coordinates in the image and $\\bm{f}(\\cdot)$ is the feature vector at a given location. $L_{metric}$ is the loss for metric learning. $\\bm{1}$ is the indicator function.\nWe found that it is more efficient to train the network with the above method rather than forcing the network to directly predict a binary output by minimizing quantification loss as in \\cite{Lin-DeepBit-CVPR16}. One possible reason is that forcing the value to be clustered around $\n\\{+1,-1\\}$ conflicts with the metric learning that follows, which uses distance as margin making the training unstable.\n\nThe number of feature maps $\\bm{f}$ is set to $256$ to make the descriptor have the same bit width as ORB features so that the descriptor can be directly incorporated into existing ORB-based visual tracking systems.\n\n\\textbf{Nested Metric Learning} The pixel-wise metric learning is used for training the descriptor in a nearest-neighbour manner. The triplet loss for binarized features is as follows:\n\\begin{equation} \\label{eq:triplet}\n\\begin{split}\nL_{metric} &= \\sum_{i} \\operatorname{max} (0, d_{\\bm{x}_i, \\bm{x}_{i,+}} - d_{\\bm{x}_i, \\bm{x}_{i,-}} + m) \\\\\nd_{\\bm{x}_1, \\bm{x}_2} &= (\\operatorname{sign}(\\bm{f}_1 ( {\\bm{x}_1} ) ) - \\operatorname{sign}(\\bm{f}_2 ( {\\bm{x}_2} ) )  )^2\n\\end{split}\n\\end{equation}\nwhere $m$ is the distance margin for the truncation. $d$ is equivalent to the squared Hamming distance for the 32-byte descriptor. We use the squared distance since we found it leads to faster and better convergence for training. $(\\bm{x}_i, \\bm{x}_{i,+})$ is a matching pair obtained using the ground truth camera poses from the training data as follows:\n\\begin{equation} \\label{eq:cor_gt}\n\\bm{x}_{i,+} =  \\pi^{-1}(\\mathbf{R}_{gt} \\cdot \\pi(\\bm{x}_i, d_i) + \\bm{t}_{gt})\n\\end{equation}\nwhere $\\mathbf{R}\\in \\mathbb{R}^{3 \\times 3}$ is the rotation matrix and $\\bm{t} \\in \\mathbb{R}^3$ is the translation vector. $(\\bm{x}_i, \\bm{x}_{i,-})$ is a non-matching pair retrieved by negative sample mining. The mining procedure is described in \\cref{alg:ENSM}. The exhaustive search will further penalize the already matched features with the relaxed criteria described in~\\cite{GCN}. The relaxed criteria is used to increase the tolerance to potentially noisy data.\n\n\\addtolength{\\topmargin}{2mm}\n\\begin{algorithm}[!t]\n\\caption{Exhaustive Negative Sample Mining}\n\\label{alg:ENSM}\n\\SetKwInOut{Input}{Input}\n\\SetKwInOut{Output}{Output}\n\\Input{Feature  in 1st frame: $   \\bm{f}_1 ( \\bm{x}_{i} )$, \\\\ \n       Features in 2nd frame: $ \\{\\bm{f}_2 ( \\bm{x}_{j} ) \\, | \\, j\\in[0,N] \\} $, \\\\ \n       Correspondence Ground Truth: $\\bm{x}_{i, +}$, \\\\\n       Relaxed Criteria: $\\bm{c} = (u_c, v_c)$}\n\\Output{Negative sample $\\bm{f}_2 ( \\bm{x}_{i,-} ) $ \\textbf{or} None}\n\\textbf{k-NN Matching}: Find top $k$ matched $\\bm{x}_j$ using Hamming distance\\\\\n\\For{$n=1;n<=k;n\\text{\\scshape++}$}{\n    Query the $n$th nearest distance $\\bm{x}_{j}^{n}$ \\\\\n    \\If{$\\operatorname{abs}(  \\bm{x}_{j}^{n}- \\bm{x}_{i}) > c $}{\n        \\Return $\\bm{f}_2 ( \\bm{x}_{j}^{n} )$\n    }\n}\n\\Return None\n\\end{algorithm}\n \n\\subsection{Distributed Keypoint Detector}\nThe training loss for keypoint detection is the same as in the original GCN. It is computed using two consecutive frames as follows:\t\n\\begin{equation} \\label{eq:NLL}\n\\begin{gathered}\n\t\\resizebox{0.55\\hsize}{!}{$L_{mask} = L_{ce}(\\bm{o}_1, \\bm{x}) + L_{ce}(\\bm{o}_2, \\bm{x}_{+})$} \\\\\n\t\\resizebox{1.0\\hsize}{!}{\n    $L_{ce}(\\bm{o}, \\bm{x}) = - \\sum_{i}  ( \\alpha_1 c_{\\bm{x}_i} \\mathrm{log}(\\bm{o}({\\bm{x}_i})) +\n\t\\alpha_2 (1-c_{\\bm{x}_i}) \\mathrm{log}(1-\\bm{o}({\\bm{x}_i})) $}\n\t\\end{gathered}\n\t\\end{equation}\n\nA notable difference from SuperPoint is that GCNv2 specifically targets motion estimation. SuperPoint tries to accomplish SIFT-like corner detection and its performance gain can chiefly be attributed to its superior descriptor. However, a CNN is capable of generating more representative features with a larger receptive field than classical methods. We therefore generate the ground truth by detecting Shi-Tomasi corners in a $16x16$ grid and warp them to the next frame using \\cref{eq:cor_gt}.  This leads to better distribution of keypoints and the objective function directly reflects the ability to track the keypoints based on texture.\n\n\\subsection{Training Details}\n\nThe triplet loss in \\cref{eq:triplet} and cross entropy in \\cref{eq:NLL} are weighted by ${100, 1}$ during training to provide a coarse normalization of the two terms. The learning rate for the adaptive gradient descent method, ADAM~\\cite{ADAM}, used for training is started from $10^{-4}$ and halved every 40 epoch for a total of 100 training epochs. The weights of GCNv2 are randomly initialized. \nWe mapped the squared Hamming distance to the $L_2$ unit sphere to perform the fast nearest neighbour matching as in \\cite{SuperPoint}. The margin for the triplet loss is set to $1$.\nThe weights $[\\alpha_1, \\alpha_2]$ of the weighted cross entropy is set to $[0.1, 1.0]$.\n\n\\section{GCN-SLAM}\n\n\\begin{figure}\n    \\centering\n    \n    \\includegraphics{orbvsgcnpipe.pdf}\n    }\n    \\caption{Illustration of the original ORB-SLAM2 keypoint extraction process on the left side, and our method on the right. The keypoint extraction in GCN-SLAM is comparatively simple, in large part because it relies on 2D convolutions and matrix multiplication which is off-loaded to the GPU.}\n    \\label{fig:piplines}\n\\end{figure}\n\nOne of the most important design choices for a keypoint-based SLAM system is the choice of keypoint extractor. The keypoints are often re-used at multiple stages in such systems. ORB features \\cite{ORB}, the namesake of ORB-SLAM2, are a particularly well-suited candidate as ORBs are invariant to rotation and scale, cheap to compute compared to other keypoint detectors with equivalent properties such SIFT or SURF, and have a binary feature vector to cater for fast matching. \n\nAs previously shown~\\cite{GCN}, GCN used in a naive motion estimation pipeline performs better than or on par with ORB-SLAM2~\\cite{ORBSLAM2-TRO17}. Notably, this is without higher-order SLAM functionality such as pose graph optimization, global bundle adjustment, or loop detection. Incorporating GCN into a system with such functionality would therefore be likely to yield better results. However, as mentioned, GCN is prohibitively expensive for real-time use on embedded hardware which we target in this work. In what follows we show how we modify ORB-SLAM2 to incorporate GCNv2, in a system we call GCN-SLAM.\n\n\n\nORB-SLAM2's motion estimation is based on frame-to-frame keypoint tracking and feature-based bundle adjustment. We will briefly describe the detection and description of these features. ORB-SLAM2 employs a \\textit{scale pyramid} where the input image is iteratively scaled down to enable multi-scale feature detection by running single-scale algorithms on the multiple rescaled images. For each scale level, the FAST corner detector is applied in a $30\\times{}30$ grid. If no detections are found in a cell, FAST is run again with a decreased threshold. After all detections have been gathered from all cells at a given level in the scale pyramid, a space partitioning algorithm is used to cull the keypoints first by their image coordinates, then by detection score. Finally, once typically 1000 keypoints have been selected in total, the viewing angle of each keypoint is computed, then each pyramid scale level is filtered with Gaussian blur, and the 256-bit ORB descriptor for each keypoint at each level is computed from the blurred image.\n\nOur method computes both keypoint locations and descriptors simultaneously in a single forward-pass of the network, and as stated before, its end result is designed to be a drop-in replacement for the ORB feature extractor outlined above. In GCNv2, we input a single grey image frame to the network which outputs two matrices: a $1\\times{}320\\times{}240$ keypoint mask, and a $256\\times{}320\\times{}240$ feature descriptor matrix. The keypoint mask is thresholded to obtain a set of keypoint locations, their confidences, and their corresponding 256-bit feature descriptors. As in \\cite{SuperPoint}, we apply non-maximum suppression with a grid size of $8\\times{}8$. As it is not possible to know the orientation of the detected features, we set the angle to zero. The two keypoint methods are illustrated in \\cref{fig:piplines}.\n\nOnce keypoints and their respective descriptors are found, ORB-SLAM2 relies primarily on two methods for frame-to-frame tracking: first, by assuming constant velocity and projecting the previous frame's keypoints into the current frame, and if that fails, by matching the keypoints of the current frame to the last-created keyframe using bag-of-words similarity. We have disabled the former so as to use only use the latter keypoint-based reference frame tracking. We have also replaced the matching algorithm with a standard nearest-neighbor search in our experiments. These modifications are made to examine the performance of our keypoint extraction method, rather than that of ORB-SLAM2's other tracking heuristics. \n\nFinally, we have left ORB-SLAM2's loop closure and pose graph optimization intact, apart from having regenerated the bag-of-words vocabulary to suit GCNv2 feature descriptors by computing them on the training dataset presented in \\cref{sec:trainingdata}.\n\n\n\n\\section{Experimental Results}\n\n\\addtolength{\\topmargin}{2mm}\n\\begin{table*}[!ht]\n\\caption{ATE USING FRAME TO FRAME TRACKING}\\label{tab:ate_open}\n\\centering\n\\begin{tabular}{ l | c | c | c | c || c | c | c | c}\n\\hline\nDataset (200 Frames) & GCN & ORB & SIFT & SURF & SuperPoint & GCNv2 & GCNv2-tiny & GCNv2-large\\\\ \n\\hline \\hline\nfr1\\textunderscore floor & \\textbf{0.015m} & 0.080m  &  0.073m  & 0.074m & -  & -  & -& -\\\\  \\hline\nfr1\\textunderscore desk & \\textbf{0.037m} & 0.151m  & 0.144m & 0.148m & 0.166m & 0.049m & 0.084m & 0.038m \\\\  \\hline\nfr1\\textunderscore 360 & \\textbf{0.059m} & 0.278m & 0.305m  & 0.279m  & - & - & - & 0.097m \\\\  \\hline\nfr3\\textunderscore long\\textunderscore office& 0.061m & 0.090m  &  0.076m & 0.070m & 0.105m & \\textbf{0.046m} & 0.085m & 0.067m\\\\ \\hline\nfr3\\textunderscore large\\textunderscore cabinet & 0.073m & 0.097m &  0.091m & 0.143m & 0.195m & 0.064m & 0.067m & \\textbf{0.056m} \\\\ \\hline\nfr3\\textunderscore nst & 0.020m &  0.061m &  0.036m & 0.030m & 0.055m & \\textbf{0.018m} & 0.024m & 0.021m\\\\ \\hline\nfr3\\textunderscore nnf & \\textbf{0.221m} & -  &  - & - & - & - & - & -\\\\ \\hline\n\\end{tabular}\n\\end{table*}\n\nIn this section, we present experimental results to justify our conclusions regarding the performance of our keypoint extraction method, and its embodiment in the GCN-SLAM system. We first introduce our training dataset, then four datasets on which we examined our method's performance and compare to some related methods, and finally we outline the quantitative and qualitative conclusions of these results. \n\nNote that our aim in this section is not to show that GCN-SLAM is better that ORB-SLAM2 but to show that GCNv2 is: i) better suited for accurate motion estimation, ii) computationally efficient, and iii) providing robustness for a SLAM system.\n\nIn the results below, evaluations on datasets were performed on a laptop with an Intel i7-7700HQ and a mobile version of NVIDIA 1070. For real-world experiments we used an NVIDIA Jetson TX2 embedded computer for processing and an Intel RealSense D435 RGB-D camera sensor on our drone (see \\cref{fig:digest}.)\n\n\n\\subsection{Training Data}\n\\label{sec:trainingdata}\nThe original GCN was trained using the TUM dataset from sensor fr2. It provides accurate pose through a motion capturing system. In GCNv2, we trained the network using a subset of the SUN-3D~\\cite{SUN-3D} dataset we created in our recent work~\\cite{S2D}. SUN-3D contains millions of real-world recorded RGB-D images in various typical indoor environments. A total $44,624$ frames were extracted by roughly one frame per second from the videos. It is very large and can potentially produce a more generalized network. However, the ground truth poses provided by SUN-3D are estimated by visual tracking with loop closure and so are relatively accurate globally, but have misalignments at frame level. To account for this local error, we extract SIFT features and use the provided poses as initial guesses for bundle adjustment to update the relative pose of each frame pair. In this sense, the training of GCNv2 is using self-annotated data with only vision information. \n\n\\subsection{Quantitative Results}\nFor comparison with the original GCN, we select the same sequences of the TUM datasets as in~\\cite{GCN} and evaluating tracking performance with an open and a closed loop system. We use the Absolute trajectory error~(ATE) as the metric. \n\nSince we trained GCNv2 on a different dataset than the original GCN~\\cite{GCN}, we also show results using the original recurrent structure for comparison. We have therefore also created GCNv2-large, with ResNet-18 as the backbone and deconvolutional up-sampling for the feature maps. The bidirectional feature detector is moved to the lowest scale as the other two versions of GCNv2.\n\n\\begin{table}[ht]\n\\centering\n\\caption{ATE USING CLOSED LOOP SYSTEM}\\label{tab:ate_closed}\n\\resizebox{\\hsize}{!}{\\begin{tabular}{ l | c | c | c | c || c } \n\\hline\n\\multirow{2}{*}{Dataset} & \\multirow{2}{*}{GCN} & ORB       & Elastic        & RGBD    & GCN     \\\\\n                         &                      &     SLAM2 &         Fusion &     TAM &     SLAM\\\\\n\\hline \\hline\nfr1\\textunderscore floor & 0.038m & 0.036m  &  - &  - & \\textbf{0.021m} \\\\ \\hline\nfr1\\textunderscore desk & 0.029m & \\textbf{0.016m}  & 0.020m & 0.027m & 0.031m \\\\ \\hline\nfr1\\textunderscore 360 & \\textbf{0.069m} & 0.213m & 0.108m  & 0.101m & 0.155m \\\\ \\hline\nfr3\\textunderscore long\\textunderscore office & 0.040m & \\textbf{0.010m}  &  0.017m & 0.027m & 0.021m \\\\ \\hline\nfr3\\textunderscore large\\textunderscore cabinet & 0.097m &  - &  0.099m & \\textbf{0.070m} & \\textbf{0.070m} \\\\ \\hline\nfr3\\textunderscore nst & 0.020m &  0.019m &  0.016m & \\textbf{0.010m} & 0.014m\\\\ \\hline\nfr3\\textunderscore nnf & \\textbf{0.064m} & -  &  -   &  - & 0.086m \\\\ \\hline\n\\end{tabular}}\n\\end{table}\n\nFrame-to-frame tracking results are shown in \\cref{tab:ate_open}. The first columns, before SuperPoint, are from~\\cite{GCN} where 640x480 images were used and GCN was trained on TUM fr2 data only. All versions of GCNv2 use the same image resolution as SuperPoint, i.e. 320x240. The results are consistent with the results reported in \\cite{SuperPoint}, SuperPoint performs on par with classical method like SIFT. GCNv2 has a performance close to GCN, and like GCN, significantly better than both SuperPoint and classical keypoints. GCNv2 performance is on par with GCN, and even slightly better in two cases -- likely due to using a much larger dataset for training. The exceptions are fr1\\textunderscore floor and fr1\\textunderscore 360. These sequences require fine details, and the detection and descriptor extraction in GCNv2 is performed with a lower scale feature map for efficiency, though GCNv2-large handles one of these sequences. The smaller version of GCNv2, GCNv2-tiny, is only slightly less accurate than GCNv2. \n\nIn \\cref{tab:ate_closed}, we compare the closed loop performance of GCN-SLAM with our previous work, as well as ORB-SLAM2, Elastic Fusion, and RGBDTAM. GCN-SLAM successfully tracks the position in all sequences with an error similar to that of GCN, whereas ORB-SLAM2 fails on two sequences. GCNv2 has significantly reduced drift error compared to ORB-SLAM2 in the fast rotations of fr1\\textunderscore360. It is also noteworthy that for this particular sequence, the original GCN does significantly better than both ORB-SLAM2 and GCN-SLAM. ORB-SLAM2 is tracking well in all other sequences, and the errors of both GCN-SLAM and ORB-SLAM2 are small.\n\n\\subsection{Qualitative Results}\n\n\\begin{figure*}[hp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{cor_traj}\n\\caption{\\textit{Corridor}: indoor, handheld.}\n\\label{fig:trajs_cor}\n\\end{subfigure\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{park_traj}\n\\caption{\\textit{Parking lot}: outdoor, handheld.}\n\\label{fig:trajs_park}\n\\end{subfigure}\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{atrium_traj.pdf}\n\\caption{\\textit{Alcove}: indoor, flying with primitive optical flow sensor.}\n\\label{fig:trajs_atrium}\n\\end{subfigure\n\\hfill\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{kitchen_traj.pdf}\n\\caption{\\textit{Kitchen}: indoor, flying with GCN-SLAM for positioning.}\n\\label{fig:trajs_kitchen}\n\\end{subfigure\n\\caption{Estimated trajectory on each of the four datasets using the GCN-SLAM pipeline and illustrating the difference between GCNv2 and ORB features. Note that these trajectories do not have any ground truth but we mark track lost with a cross and we see that this happened in all four cases when using ORB features.}\n\\label{fig:trajs}\n\\end{figure*}\n\n\\begin{figure*}[hp]\n\\centering\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{cor}\n\\end{subfigure\n\\begin{subfigure}{0.45\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{park}\n\\end{subfigure}\n\\caption{Keypoint extractor performance in terms of tracking shown for the \\textit{Corridor} (left) and \\textit{Kitchen} datasets (right.) Lines indicate the total number of keypoints detected per frame. The filled-in area under the lines shows the fraction of keypoints that were successfully used for local map tracking, i.e. contribute to tracking and is plotted against the right-hand axis.}\n\\label{fig:inliers}\n\\end{figure*}\n\nTo further verify the robustness of using GCNv2 in a real-world SLAM system, we show results on datasets collected in our environment under different conditions: a) going up a corridor, turning 180 degrees and walking back with a handheld camera, b) walking in a circle on an outdoor parking lot with a handheld sensor in daylight, c) flying in an alcove with windows and turning 180 degrees, and d) flying in a kitchen and turning 360 degrees while using GCN-SLAM for positioning.\n\n\\begin{figure*}[ht]\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{room_crop.png}\n\\end{subfigure\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\includegraphics[width=\\textwidth]{rpl_crop.png}\n\\end{subfigure}\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\caption{\\textit{Lab room}: handheld.}\n\\end{subfigure\n\\centering\n\\begin{subfigure}[c]{0.49\\textwidth}\n\\centering\n\\caption{\\textit{Lab corridor}: handheld.}\n\\end{subfigure}\n\\caption{Mesh reconstruction results using GCN-SLAM as input to TSDF volume integration from Open3D. Loop closure detection in GCN-SLAM was disabled to demonstrate the accuracy in the tracking alone.}\n\\label{fig:mesh}\n\\end{figure*}\n\nSince they are without ground truth, the results are only qualitative.\nThese datasets were chosen to show that our method handles difficult scenarios, is robust, and can be used for positioning of a real drone. \\Cref{fig:trajs} shows the estimated trajectory of GCN-SLAM using ORB versus GCNv2 as keypoints. Note that both features are evaluated in exactly the same tracking pipeline for fair comparison, i.e. GCNv2 or ORB features is the \\textit{only} differences. Refer to the source-code for further details.\nusing GCN-SLAM as a basis for drone control improves performance as can be seen by comparing \\cref{fig:trajs_atrium,fig:trajs_kitchen}. In \\cref{fig:trajs_atrium}, the position is estimated using only an optical flow sensor whereas \\cref{fig:trajs_kitchen} uses GCN-SLAM as a source of position, and it is clear that the drone is able to hold its position better as there is significantly less jitter in this trajectory. In all four datasets, tracking is maintained with GCNv2, but lost with ORB. We used a remote control to send setpoints to the flight control unit on the drone for control, using the built-in position holding mode.\n\nIn \\cref{fig:inliers} we compare the performance of our keypoint extractor to the original ORB keypoint extractor. We plot the number of inliers during tracking of the local map for our adapted SLAM system, first with ORB keypoints, and then with GCNv2 keypoints. As the figure illustrates, while there are many more ORB features, our method has a higher percentage of inliers. In addition, as shown in \\cref{fig:digest}, GCNv2 results in better distributed features compared with ORB.\n\n\\Cref{fig:mesh} shows the mesh reconstruction using GCN-SLAM output poses from two additional sequences. The left sequence was from an office and the right was acquired walking between two floors using stairs. TSDF volume integration from Open3D\\footnote{\\url{http:\/\/www.open3d.org}} was used to create the mesh. To show the accuracy of our method, the loop closure detection of GCN-SLAM is disabled. \n\n\\section{Conclusions}\n \nIn our previous work~\\cite{GCN}, we showed that our method, GCN, achieves better performance in visual tracking compared with existing deep learning and classical methods. However, it cannot be directly deployed into a real-time SLAM system in an efficient way due to its deep recurrent structure. In this paper, we addressed these issues by proposing a smaller, more efficient version of GCN, called GCNv2, that is readily adaptable to existing SLAM systems. We showed that GCNv2 can be effectively used in a modern feature-based SLAM system to achieve state-of-the-art tracking performance. The robustness and performance of the method was verified by incorporating GCNv2 into GCN-SLAM and using it on-board for positioning on our drone.\n\n\\textbf{Limitations}\nGCNv2 is trained mainly for projective geometry and not generic feature matching.\nAs always with learning-based methods generalization is an important factor. GCNv2 works relatively well for outdoor scenes, as demonstrated in the experiments (Cf. \\cref{fig:trajs_park}). However, since no outdoor data was used for training, further improvements can likely be made. Our target here is an indoor setting and we did not investigate this further.\n\n\\textbf{Future work}\nIn this paper, we mainly improved the efficiency of GCN and achieve stable tracking perform on our platform, a drone with NVIDIA Jetson TX2. However, since the original recurrent structure is removed, there is trade-off between the accuracy and running speed. In the future, we would like to further investigate to use the self-supervised learning to improve our system.\n\n\n\n\n\n    \\bibliographystyle{IEEEtran}\n\t","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nFor any given vector $\\bf{c} = [c_0, c_1, \\dots, c_{n-1}]^T$, the circulant matrix $C$ with the first row $\\bf{c}$ is defined as follows (\\cite{golub2013matrix,gray2006toeplitz,karner2003spectral}):\n\\begin{equation}\n\\label{Eq:C}\nC = \\left(\\begin{matrix}\nc_0      & c_{n-1} & \\cdots  & c_2     & c_1     \\\\\nc_1      & c_0     & c_{n-1} &         & c_2     \\\\\n\\vdots   & c_1     & c_0     & \\ddots  & \\vdots  \\\\\nc_{n-2}  &         & \\ddots  & \\ddots  & c_{n-1} \\\\\nc_{n-1}  & c_{n-2} & \\cdots  & c_1     & c_0     \\\\\n\\end{matrix}\\right)\n\\end{equation}\nThe eigenspace of the circulant matrices are formed by Fourier matrices; therefore, the eigienvectors and the eigenvalues of $C$ are defined analytically in the following form of pairs:\nA $j$th ($j = 0, \\ldots, n-1$) eigenvector is given by\n\\begin{equation}\nv_j=\\frac{1}{\\sqrt{n}} \\left(1, \\omega^j, \\omega^{2j}, \\ldots, \\omega^{(n-1)j}\\right),\n\\end{equation}\nwhere $\\omega=\\exp \\left(\\tfrac{2\\pi i}{n}\\right)$.\nAnd its corresponding eigenvalue is\n\\begin{equation}\n\\label{Eq:EigenC}\n\\lambda_j = c_0+c_{n-1} \\omega^j + c_{n-2} \\omega^{2j} + \\dots + c_{1} \\omega^{(n-1)j}.\n\\end{equation}\nAs a result, the eigendecomposition of any circulant matrix can be written as $C = F\\Lambda F^{-1}$ where $F$ represents the normalized matrix for the discrete Fourier transform and $\\Lambda = diag({F \\bf c})$, which is a diagonal matrix representing the eigenvalues. \n\n\\subsection{Direct Implementation as a quantum circuit}\n\nOne can implement $C$ as a quantum circuit by using its eigendecomposition $C = F\\Lambda F^{-1}$: It is well known that $F$ can be implemented in $O(\\log(n)^2)$ time as the quantum Fourier transform \\cite{nielsen2002quantum}. \nIn addition, since any vector of dimension $n$ can be implemented in $O(n)$ time, we can also implement $\\Lambda$ in linear time. Therefore, $C$ can be implemented as a quantum circuit by using $O(n)$ two and single qubit quantum gates, which is less than the classical matrix size $O(n^2)$. \n\n\\subsection{Circuit implementation through permutations}\n$C$ can be represented as a function of the following permutation matrix, which is also circulant:\n\\begin{equation}\n\\label{EqMatrixP}\nP = \\left( \\begin{matrix}\n 0&0&\\cdots&0&1\\\\\n 1&0&\\cdots&0&0\\\\\n 0&\\ddots&&\\vdots&\\vdots\\\\\n \\vdots&\\ddots&\\ddots&0&0\\\\\n 0&\\cdots&0&1&0\n\\end{matrix} \\right)\n\\end{equation}\nThen, we can redefine $C$ as a polynomial function of $P$ with the coefficients given by $\\bf{c}$: \n\\begin{equation}\nC = c_0 I + c_1 P + c_2 P^2 + \\dots + c_{n-1} P^{n-1} = \\sum_{j=0}^{n-1}c_jP^j.\n\\end{equation}\nSince $P$ is an orthogonal matrix, we can construct its exact quantum circuit decomposition. \nMoreover, it is known that sum of unitary matrices can be implemented as a subpart of a larger quantum  system with the help of ancilla registers (e.g.,  \\cite{childs2012hamiltonian,daskin2012universal,low2019hamiltonian}).  \nHere, we shall show that $C$ can be constructed as a vector by using $V_P\\times \\ket{\\psi}$ where \\ket{\\psi} represents a combinations of the vector $\\bf{c}$ and $V_P$ includes the powers of $P$ used in the above polynomial. \nLater in the paper, we shall show that this way of implementation eases the way to solve some string problems. \n\n\\subsubsection{Circuit for $P$}\nSince $P$ is a circulant matrix, we can define its any $j$th eigenvalue by using Eq.\\eqref{Eq:EigenC}:\n\\begin{equation}\n\\lambda_j = w^j.\n\\end{equation}\nTherefore, its eigendecomposition can be written as:\n\\begin{equation}\n\\label{EqLambdaP}\nF\\times \n\\left(\\begin{matrix}\n1&&&&\\\\\n&w^1&&\\\\\n&&w^2&\\\\\n&&&\\ddots&\\\\\n&&&&w^{n-1}\n\\end{matrix}\\right)\\times F^{-1} =F \\Lambda_P F^{-1} .\n\\end{equation}\nAs a quantum circuit, we can implement $\\Lambda_P$ as follows:\n First, observe that the power $j$ in $w^j$ is equal to the row index which can be found from the binary expansion of $j = (b_{n-1}\\dots b_0)_2$. Therefore, if a qubit is in \\ket{1}, we apply a control phase gate to the first qubit. The phase of these gates are determined from the decimal index of the qubit in the binary expansion.\nWe define the phase gate as follows:\n\\begin{equation}\n\\label{Eq:Rw}\nR_w(k) = \\left(\\begin{matrix}\n  1 & 0\\\\\n  0 & w^{2^k}\n\\end{matrix}\\right),\n\\end{equation}\nwhere $k$ is the order of the control qubit in the binary expansion $(b_{n-1}\\dots b_0)_2$.\nThe resulting circuit is drawn in Fig.\\ref{FigCircuitLambdaP} for four qubits. The number of quantum gates for this implementation is equivalent to the number of qubits which is $(\\log n)$. \n\\begin{figure}[t]\n\t\\begin{center}\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\qw &\\qw &  \\qw &\\ctrl{3} & \\qw & \\\\\n\\lstick{} &\\qw &\\qw& \\ctrl{2}  & \\qw &  \\qw &   \\\\\n\\lstick{} & \\qw &\\ctrl{1}&\\qw & \\qw &  \\qw &   \\\\\n\\lstick{}&\\gate{R_w(0)} &\\gate{R_w(1)}  &\\gate{R_w(2)}  & \\gate{R_w(3)}&\\qw   \\\\\n\t\t\t}\n}\n\t\\end{center}\n\t\\caption{\\label{FigCircuitLambdaP} Quantum circuit for $\\Lambda_P$}\n\\end{figure}\n\n\n\\subsubsection{Circuit for $V_P$}\nConsider the following matrix:\n\\begin{equation}\nV_P = \n\\left(\\begin{matrix}\nI&&&&\\\\\n&P^1&&\\\\\n&&P^2&\\\\\n&&&\\ddots&\\\\\n&&&&P^{n-1}\n\\end{matrix}\\right),\n\\end{equation}\nwhich highly resembles $\\Lambda_P$ given in Eq.\\eqref{EqLambdaP}. Therefore, we can implement the same way as we implemented $\\Lambda_P$: In this case,  we only need to somehow replace $R_w({k})$ with the power of $P$: \n\\begin{equation}\nP^{2^k}=F\\Lambda_P^{2^k}F^{-1}.\n\\end{equation} \nSince all the powers of $P$s have the same eigenvectors, we apply $F$ and $F^{-1}$ only once at the beginning and at the end of the circuit. We use the following quantum operation in the circuit:\n\\begin{equation}\nU_P(k) = \\left(\\begin{matrix}\n  I & 0\\\\\n  0 & \\Lambda_P^{2^k}\n\\end{matrix}\\right).\n\\end{equation}\nHere,  when we look at the circuit for $\\Lambda_P$ in Fig.\\ref{FigCircuitLambdaP}, we can see that the power can be distributed to the quantum gates inside the circuit since the order of the gates is not important. \nThat means, we can obtain any power of $\\Lambda_P$ by simply adjusting the angle values of the quantum gates. \nTherefore, for power of $\\lambda_P^{2^k}$, we adjust the parameter of the rotation gates in Fig.\\ref{FigCircuitLambdaP} as: $R_w(0\\times k), R_w(1\\times k), \\dots$, and so on. The final circuit for $V_P$ is just composed of the controlled versions of these quantum gates which is shown in Fig.\\ref{FigCircuitV} and \\ref{FigUpk}.\n\n\\begin{figure}[t]\n\t\\begin{center}\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\qw&\\qw &\\qw &  \\qw &\\ctrl{3} & \\qw & \\qw\\\\\n\\lstick{} &\\qw&\\qw &\\qw& \\ctrl{2}  & \\qw &  \\qw & \\qw  \\\\\n\\lstick{} &\\qw& \\qw &\\ctrl{1}&\\qw & \\qw &  \\qw & \\qw  \\\\\n\\lstick{}&\\gate{F^{-1}}&\\gate{U_P(0)} &\\gate{U_P(1)}  &\\gate{U_P(2)}  & \\gate{U_P(3)}&\\gate{F}&\\qw   \\\\\n}}\n\t\\end{center}\n\t\\caption{\\label{FigCircuitV} The final quantum circuit of $V_P$ illustrated for $n=4$. The explicit circuit for $U_P(k)$ is given in Fig.\\ref{FigUpk} below.}\n\\end{figure}\n\n\\begin{figure}[t]\n\\colorbox{green!10}{\n\t\t\\Qcircuit @C=0.7em @R=1em {\n\\lstick{} &\\ctrl{4}&\\ctrl{4} &\\ctrl{4} &  \\ctrl{4}  & \\qw & \\\\\n\\lstick{} &\\qw &\\qw &  \\qw &\\ctrl{3} & \\qw & \\\\\n\\lstick{} &\\qw &\\qw& \\ctrl{2}  & \\qw &  \\qw &   \\\\\n\\lstick{} & \\qw &\\ctrl{1}&\\qw & \\qw &  \\qw &   \\\\\n\\lstick{}&\\gate{R_w(0\\times k)} &\\gate{R_w(1\\times k)}  &\\gate{R_w(2\\times k)}  & \\gate{R_w(3\\times k)}&\\qw    \\\\\n}}\n\t\\caption{\\label{FigUpk} The explicit circuit implementation of $U_P(k)$  which implements the $2^k$th power of $\\Lambda_P$ given in Eq.\\eqref{EqLambdaP}.}\n\\end{figure}\n\n\\subsection{The overall complexity}\nAs shown in Fig.\\ref{FigCircuitV} and \\ref{FigUpk} the number of quantum gates are limited to the number of qubits used in the main system. The size of the permutation matrix $P$ is $n$ by $n$, therefore the main system is described by $\\log n$ qubits. Then, an additional $\\log n$ qubit is used to construct $V_P$. Therefore, in total the circuit requires $(2\\times \\log n)$ qubits.\nThe quantum gates are at most controlled by two qubits, therefore the total number of required single and two qubit gates are bounded by $O(poly(\\log n))$ in general. More specifically, if we assume the general implementation of quantum Fourier transform requires $O((\\log n)^2)$ gates (with an optimization, it may require $O((\\log n)\\log\\log n)$ ) and the decomposition of each Toffoli gate requires less than $\\log n$ gates \\cite{nielsen2002quantum}, then the total complexity becomes bounded by $O((\\log n)^2)$.\nThis shows that we can form $V_P$ on quantum computers very efficiently since it requires   a number of quantum gates which is a poly-logarithm of the dimension $n$.\n\n\n\\section{Applications}\n\\subsection{Suffix trees and arrays}\nMany string problems can be solved easier if they are stored by using well-known data structures such as suffix tries, trees, and arrays \\cite{manber1993suffix}. Below, we will follow the lecture notes in Ref.\\cite{langmeadburrows} to give a brief introduction of these data structures and explain how to implement them on quantum computers with the help of the circuits introduced in the previous section.\n\nFrom a given string, a suffix can be chosen by determining a starting position. For instance, we can choose the following suffixes from string \\textbf{``banana\"}:\n\\begin{equation}\n\\begin{bmatrix}\nb&a&n&a&n&a&\\$\\\\\na&n&a&n&a&\\$\\\\\nn&a&n&a&\\$\\\\\na&n&a&\\$\\\\\nn&a&\\$\\\\\na&\\$\\\\\n\\$\n    \\end{bmatrix}\n\\end{equation}\nHere, ``\\$\" is used to indicate the end of the string and   considered less than any letter of the alphabet in lexicographical order.\n\nIn suffix arrays, these strings are put in the buckets based on their orders and sorted. \nIn tries, the prefixes are put in the rooted tree in which every node has a child for each letter of the alphabet used in the string. \nIn the suffix tries, the suffixes are put in a rooted tree in which every node has a child for each letter of the alphabet used in the string. \nTherefore, by following the paths from the root of the tree, it is possible to determine if a letter follows another and if the prefix exists or not.\nIn the suffix trees, the tries are built for the suffixes and the number of nodes in the tries are optimized by compressing the nodes in the straight paths ( i.e. the paths that do not go more than one way in any nodes on the path.). \nEach path in the suffix tree ends with the letter ``\\$\".\n\nWhile constructing suffix arrays and trees  can be done in $O(n\\log n)$ time \\cite{ukkonen1995line}, they can be used to solve many string problems more efficiently such as string searching and matching and sequence alignment \\cite{mielczarek2016review}. \n\n\\subsection{Burrows Wheeler Transform\\cite{burrows1994block}}\n\nThe main idea in Burrows Wheeler transform (BWT) is to first  generate a circulant matrix by using the given string and then sort this matrix column by column. In the sorted matrix, the last column is used as the equivalent transformed string. As an illustration, consider the following example string \"banana\", which is used in most textbooks: \n\\begin{equation}\n    \\begin{bmatrix}\nb&a&n&a&n&a&\\$\\\\\na&n&a&n&a&\\$&b\\\\\nn&a&n&a&\\$&b&a\\\\\na&n&a&\\$&b&a&n\\\\\nn&a&\\$&b&a&n&a\\\\\na&\\$&b&a&n&a&n\\\\\n\\$&b&a&n&a&n&a\n    \\end{bmatrix} \\xrightarrow{sort}\n        \\begin{bmatrix}\n\\$&b&a&n&a&n&\\bf a\\\\\na&\\$&b&a&n&a&\\bf n\\\\\na&n&a&\\$&b&a&\\bf n\\\\\na&n&a&n&a&\\$&\\bf b\\\\\nb&a&n&a&n&a&\\bf \\$\\\\\nn&a&n&a&\\$&b&\\bf a\\\\\nn&a&\\$&b&a&n&\\bf a\\\\\n    \\end{bmatrix} \n\\end{equation}\nFrom the above, BWT(``banana\") = ``annb\\$aa\". In the above matrix, if one considers ``\\$\" as 1s and the rest of the letters as 0s, then BWT of any string can be represented as a permutation matrix. \n\\subsection{Quantum implementation of suffixes}\nIn the previous section we show how to construct $V_P$. \nHere, consider that we are given the text \\ket{c} including also the ``\\$\" sign.\nWe apply \\ket{c} in the main register and $H^{\\otimes \\log n}\\ket{0}$ in the ancilla register. This generates the following vector:\n\\begin{equation}\n(H^{\\otimes \\log n}\\ket{0}) \\ket{c} = \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{0\\ \\ \\ }\\\\\n\\vdots\\ \\ \\ \\ \\\\\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{n-1}\\\\\n\\end{matrix}\\right).\n\\end{equation}\nIf we multiply this vector by $V_P$, we get $C$ in Eq.\\eqref{Eq:C} in vector form:\n\\begin{equation}\nV_P (H^{\\otimes \\log n}\\ket{0}) \\ket{c} = \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\n\\begin{bmatrix}\n c_0\\\\c_1\\\\ \\vdots \\\\c_{n-1}\\\\\n\\end{bmatrix}_{0\\ \\ \\ }\\\\\n\\vdots\\ \\ \\ \\ \\\\\n\\begin{bmatrix}\n c_{n-1}\\\\c_{n-2}\\\\ \\vdots \\\\c_{0}\\\\\n\\end{bmatrix}_{n-1}\\\\\n\\end{matrix}\\right) \n=  \n\\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\nS_{0}\\\\\n\\vdots \\\\\nS_{n-1}\\\\\n\\end{matrix}\\right),\n\\end{equation}\nwhere each $S_j$ represents a circularly permuted string.\nThe above vector provides the unsorted suffixes whose end indicated by some end of file character \"\\$\". \nSince the order of this character provides  information about the suffixes, the amplitude of this char may be adjusted higher so that from the measurements, the order can be obtained in an efficient way.\n\nBy using the indices of the end of file characters, we can collapse the quantum state onto any desired direction. \nIn addition, we can draw some conclusion from the measurements on the collapsed state: e.g., . Therefore, the measurement statistics can be used to draw conclusions about the most common prefixes in the suffixes  in the collapsed state. \n\nSince in most string algorithms sorted structures allow us to develop more efficient algorithms, the same sorting can be also done on this vector.\n\n\\section{Sorting}\nSorting problem simply can be described as finding an ordered list of items given as an unordered list.\nIt is known that comparison based sorting algorithms such as merge sort and quicksort have running time $\\Omega(n log n)$ for a list of $n$ items. \nThis lower bound can be broken by using  bucket sorting (counting sort) where each item is considered as a direct pointer to the bucket; therefore, the sorting is done in $O(n)$ time. This simple approach is further improved to give $O(nd)$ time, where $d$ is the number of bits used to represent each item and $O(nlog log n)$ time. \nBecause of the memory constraints and requirements,  in practice comparison sorts are used  more often in practice than these kinds of sorting algorithms. \\cite{hagerup1998sorting}\n\nOn quantum computers, the sorting problem is considered based on registers $\\ket{x_1} \\dots \\ket{x_n}$. Here, each register represents an item in the unsorted list. \nTherefore, the algorithms try to prepare the registers in the output to encode the natural ordered list of the given items: i.e., $\\ket{q_1} \\geq \\dots \\geq \\ket{q_n}$.\nBased on this representation, it is shown that quantum algorithms based on comparison models have similar complexity bounds for the sorting problem: i.e. $\\Omega (\\sqrt{n}logn)$ \\cite{hoyer2002quantum,ambainis2002quantum}.\n\nAs done in the classical sorting algorithms, bucket sort with an order preserving hash function can be used to sort the items stored in memory as:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{\\bf 0}\\ket{i}\\ket{x_i},\n\\end{equation}\nwhere $i$ is the index of the item $x_i$ in the given unordered list.\nThe sorting problem becomes constructing the following quantum state:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{o_i}\\ket{i}\\ket{x_i}\n\\end{equation}\nwhere \\ket{o_i} represents the index of the item in the sorted list.\nWe can rewrite this in terms of a hash function $h(x_i)$ that maps an item $x_i$ to the index $o_i$:\n\\begin{equation}\n\\sum_{i=1}^n\\ket{h(x_i)}\\ket{i}\\ket{x_i}.\n\\end{equation}\n$h$ can be as simple as a direct map or more general hash function. \nIn particular consider\n$h$ as a partial order preserving hash function: i.e., if $x_i$ and $x_j$, their real ordered locations at some distance $d$ from each other so that $o_i + d < o_j$, then $h(x_i) < h(x_j)$.  \nThen, since we can apply an operator $h$ simultaneously to all items, we can generate their orders in $O(1)$ time.\nSometimes knowing the elements' rough order in the array may be considered enough. \nIn those cases,  the hash function need not be perfect; therefore,\none can use similar ideas to classical sorting algorithms such as bucket sort or the shell-sort \\footnote{Shell sort is a generalization of insertion sort algorithm, where items at certain distances are compared and if necessary swapped to have a k-sorted array: i.e. an array where the numbers are grouped into regions based on their orders. } to generate some k-sorted array in which buckets are sorted, however, the items in the same buckets are not sorted. \n\nAs mentioned above, if we use comparison based sorting algorithms, then the sorting is almost the same as classical sorting algorithm:  \nLet us consider the following vector whose construction is given in the previous section:\n\\begin{equation}\n    \\frac{1}{\\sqrt{n}}\\left(\\begin{matrix}\nS_{0}\\\\\n\\vdots \\\\\nS_{n-1}\\\\\n\\end{matrix}\\right),\n\\end{equation}\nIn Burrows Wheeler transform, the items are sorted by columns.\nWe can do the same sorting on this vector: A particular direct sorting may be as follows: \n\n\\begin{itemize}\n    \\item \nFirst, we compare each element with its left neighbor, then if it is necessary we swap them.  \\begin{itemize}\n    \\item This step can be done in parallel, if the swap and comparison can be implemented in $O(poly(\\log n))$ number of quantum gates, then it takes $O(poly(\\log n))$ time \\footnote{Here, since the comparison and swap operations depend on the number of qubits, it may require controlled gates whose decomposition requires number of gates that are polynomial in the system size $O(\\log n)$. If this step can be done in $O(1)$, then sorting can be done more efficiently.}.\n    \\end{itemize} \n\\item In the second step we do the same thing for the right neighbors (we group two elements together and compare them.).\n \\begin{itemize}\n    \\item This step also requires $O(poly(\\log n))$ number of quantum gates.\n\\end{itemize} \n\\item If we repeat the above steps for O(n) time, we basically get a simple $O(n\\log n)$ time sorting algorithm.\n\\end{itemize}\n\n\n\\section{Conclusion}\nIn this paper, we describe how to generate suffix structures efficiently as a vector by using quantum circuits for circulant matrices.\nWe discuss how the generated vector can be used in the string algorithms and sorted if necessary.\nAs a future direction, we will apply this circuit to the sequence alignment and pattern matching problems.\nSince circulant matrices are used in convolutions, it can be also applied to problems in different areas such as convolution neural network, time series analysis \\cite{pollock2002circulant, daskin2022walk}.\n\n\n\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nColloidal metal chalcogenide NPLs offer well defined advantages over their quantum dot \nand rod counterparts as semiconductor building blocks for optical applications.\\cite{LhuillierACR,MinNR,DirollJMCc,SharmaIEEE}\nSome of the most distinctive features are order-of-magnitude shorter radiative lifetimes,\nwhich result from the strong exciton binding energies in quasi-2D systems\n(Giant Oscillator Strength effect),\\cite{FeldmannPRL,PlanellesACSph}\nand precisely controlled thickness of the nanostructure,\\cite{IthurriaNM,RiedingerNM,ChristodoulouNL,BhandariCM}\nwhich suppresses the emission broadening due to size dispersion usually observed in dots.\nThese properties give rise to bright and narrow emission lines, which\nis of interest for displays, lighting and lasers.\\cite{DirollJMCc,SharmaIEEE}\n\nUnfortunately, ligand passivation of surface dangling bonds is usually incomplete because of\n labile binding and steric hindrance between ligands. This can translate into significant \nnon-radiative losses.\\cite{TessierACS} To overcome this problem, core-only NPLs are sometimes\nreplaced by sandwich-like core\/shell heterostructures, where top and bottom facets of the \ncore material are coated with a higher band gap inorganic material. \nTypical core\/shell combinations are CdSe\/CdS,\\cite{TessierNL,AchtsteinACS} \nCdSe\/ZnS\\cite{PolovitsynCM,SaidzhonovJL} and their alloys.\\cite{RossinelliCM,KelestemurACS}\nThese heterostructures succeed in isolating the photogenerated carriers, \nwhich remain in and around the core, from top and bottom surfaces,\nthus translating into enhanced fluorescence quantum efficiency and \nphotostability.\\cite{LhuillierACR,SharmaIEEE,YadavJPCc}\nThe shell growth has however a negative side effect, namely the systematic broadening of\nthe emission line width, e.g. from $\\sim 35-40$ meV in CdSe NPLs to $\\sim 60-80$ meV in CdSe\/CdS NPLs.\\cite{TessierNL,RabouwNL}\n Linewidth broadening in core\/shell NPLs was initially ascribed to the presence of traps\ninduced upon shell coating.\\cite{TessierNL} Graded interface composition was then shown to \nnarrow the line width down to $\\sim 55$ meV,\\cite{RossinelliCM,KelestemurACS} \nbut this figure is still larger than in core-only NPLs, which suggests that interface \ndefects are not the only source of broadening.\n\nTo shed light into this problem, Antolinez and co-workers recently investigated the origin \nof the fluorescence line width broadening in CdSe\/CdS NPLs by means of single-particle spectroscopy.\\cite{AntolinezNL} \nThey observed that individual NPLs present a series of 2 to 4 narrow peaks split from each other by $\\sim 10$ meV.\nAltogether, the peaks fit well the asymmetric lineshape of ensemble NPLs at cryogenic temperatures.\\cite{TessierNL}\nA similar feature was soon after reported in core-only CdSe NPLs.\\cite{AntolinezNL2}\nThe nature of these peaks was tentatively ascribed to SU processes of negative trions (X$^-$).\nThese are partly radiative Auger processes, whereby an electron-hole pair recombines radiatively but \ntransfers part of its energy to the remaining electron by exciting it into a higher single-electron level\n(in-plane excitation).\n They have been previously reported in epitaxial quantum wells\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB} \nand self-assembled quantum dots\\cite{PaskovPE} under the magnetic fields, corresponding \nto inter-Landau level excitations of the excess carrier.\\cite{HawrylakPRB}\n Clarifying the role of SU processes in the emission of colloidal NPLs is then a desirable step\nto fully understand and control the emission line width, which would be advantageous for optical \napplications.\n\nIn this work, we analyze the possible occurence of SU processes in colloidal CdSe-based NPLs\nfrom a theoretical perspective. The goal is to determine which physical conditions enable \nthese processes.\nTo this end we use effective mass models and full CI simulations,\nwhich provide an intuitive description of the underlying physics.\n We shall confirm that at least one intense SU replica can be expected for $X^-$\nupon electron-hole recombination, in both core-only and core\/shell NPLs, \ncorresponding to the excitation of the remaining electron into a higher orbital \nwith the same symmetry as the ground state.\nFor this to take place, the trion must be weakly bound to an off-centered acceptor impurity.\nThe role of the impurity is to lower the system symmetry, thus relaxing selection rules, \nand to stimulate electron-electron repulsion (quench electron-hole attraction) in the \nground orbital.\n By doing so, SU peaks can reach intensities exceeding 10\\% of the fundamental (band edge, fully radiative) transition. \nThis is one order of magnitude higher than in epitaxial quantum wells, which can be rationalized\nfrom the stronger Coulomb interactions, which result from the pronounced dielectric confinement,\nand the presence of lateral sidewalls, which are prone to surface traps.\n We discuss connections with experiments in the literature and propose potential \n strategies to suppress these processes.\n \n\n\\section{Results}\n\nWe analyze the emission spectra of trions in core-only and core\/shell NPLs. \nNegative trions are studied unless otherwise noted, as it is the\nmost frequently reported species in these structures, but the conclusions do not\ndepend on the sign of the charged exciton (see Fig.~S2 in the supporting information, SI).\nOnce the general behavior of SU processes in these systems is understood, \nwe discuss how our conclusions fit the interpretation of different experimental \nobservations and the practical implications of our findings.\n\n\n\\subsection{Core-only NPLs}\n\nWe start by studying core-only CdSe NPLs. The NPLs are chosen to have $4.5$ monolayer (ML) thickness \nand a lateral size of $20 \\times 20$ nm$^2$, for similarity with the core dimensions \nof Ref.~\\cite{AntolinezNL}. They have a pronounced dielectric mismatch with the organic \nenvironment, which we model with $\\epsilon_{in}=6$ and $\\epsilon_{out}=2$ as dielectric \nconstants inside and outside the NPL, unless otherwise stated.\\cite{Sadao_book,AchtsteinNL}\n The presence of few-meV spectral jumps in photoluminescence experiments\\cite{AntolinezNL} \nsuggests that the trion is subject to the influence of carriers temporarily trapped on \nthe surface.\\cite{RabouwNL,BeylerPRL} \nTo model this phenomenon, a fractional point charge is placed on the surface, \nwith charge $Q=e \\, Q_X$ ($|Q_X| \\leq 1$ and $e$ the full electron charge).\nThe fractional value of $Q_X$ accounts for the screening of trapped charged (e.g. hole)\nby the trap defect itself (e.g. surface dangling bond).\\cite{CalifanoNL}\nTwo scenarios are considered: a charge centered on the top facet ($Q_{top}$) \nand an off-centered charge, located along the edge of a lateral facet ($Q_{edge}$). \nThe latter setup is suggested by studies showing that edge and vertex atoms in CdSe \nstructures have weaker binding to oleate ligands.\\cite{DrijversCM}\nThe two systems are represented in Figure \\ref{fig1}a and \\ref{fig1}b.\nThe corresponding emission spectra are shown in Fig.~\\ref{fig1}c and \\ref{fig1}d. \nThe figure reveals a number of important observations.\n(i) In the absence of surface charge ($Q_X=0$, thick lines),\nonly the fundamental transition shows up, with no sizable SU replica. \n(ii) Charges on the top facet induce SU peaks (see arrow in Fig.~\\ref{fig1}{c}),\nbut their strength is two orders of magnitude smaller than that of the \nfundamental transition (main line). This is similar to the case of epitaxial quantum wells.\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB}\n(iii) Stronger SU replica are however obtained for charges located on the \nlateral sidewall, provided the charge is attractive (acceptor impurity) \nand binding to the trion is moderately weak, see Fig.~\\ref{fig1}d.\nFor $Q_{edge}=0.4$ (marked with a star in the figure),\nthe SU peak reaches $\\sim 25$\\% of the main peak height.\nThis ratio is about 20 times higher than in epitaxial quantum wells,\nand it holds despite the Giant Oscillator Strength enhancing the band edge \nrecombination,\\cite{FeldmannPRL,PlanellesACSph,IthurriaNM,AchtsteinNL}\nwhich suggests that SU satellites also benefit from this phenomenon.\nFor $Q_{edge}>0.4$, however, the SU peak intensity is lowered again\nand the energy splitting (redshift) with respect to the main line increases.\nSecond and third SU lines are built for strong surface charges \n(see inset in Fig.~\\ref{fig1}d at $Q_{edge}=0.7$), \nbut their magnitude is negligible.\nWe have also explored different locations of the charge, obtaining intermediate\nresults between those shown in Fig.~\\ref{fig1} (see Fig.~S3 in SI). \nThese results point out the potentially significant role of lateral sidewalls, \nwhich are characteristic feature of colloidal quantum wells as compared to epitaxial ones, \nin obtaining high SU peaks.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{fig1.eps}\n\\caption{\n(a,b) Schematics of core-only NPLs with different location of the surface charge. \n(c,d) Corresponding X$^-$ emission spectrum for charge strength $Q=Q_X\\,e$. \nThe arrows point at the SU satellites \n(dotted lines are guides to the eyes). The highest SU peak is observed \nfor off-centered acceptor charges weakly bound to the trion ($Q_{edge} = 0.4$, \nmarked with a star in (d)). \nThe spectra are normalized to the intensity of the fundamental transition\nat $Q_X=0$, and offset vertically for clarity.\nThe insets for $Q_{edge}=0.7$ in (d) show amplified SU peaks.\n}\\label{fig1}\n\\end{figure}\n\nTo gain understanding on the origin of strong SU peaks when trions bind\nto lateral surface acceptors, beyond the full numerical calculation\nof Fig.~\\ref{fig1}, in Fig.~\\ref{fig2}a and \\ref{fig2}b we \ncompare sketches of the SU processes,\nin the absence and presence of an attractive edge charge.\nWithin effective mass theory, the conduction band \nand valence band energy levels of (non-interacting) \nelectrons and holes can be described as particle-in-the-box states, \nwith quantum numbers $(n_x,n_y,n_z)$. \nIt is useful however to label the states by their symmetry \n(irreducible representation).\nWhen $Q_{edge}=0$, because the NPL has squared shape, the \npoint group is $D_{4h}$. When $Q_{edge}\\neq 0$, the electrostatic\npotential yields a symmetry descent to $C_s$.\nAs a consequence, degeneracies are lifted and additional\nstates with the same symmetry as the ground orbital ($A'$)\nare obtained. This is important because after electron-hole\nrecombination, the excess electron can only be excited to\nan orbital with the same symmetry as the initial one\n(vertical arrows in Fig.~\\ref{fig2}a and \\ref{fig2}b).\nTherefore, lowering the system symmetry opens new channels\nfor SU processes. Furthermore, these can involve low-energy \norbitals, which have fewer nodes and will then have larger\noverlap with the trion ground state, as we shall see below. \nBoth the number and the intensity of the SU processes \nare in principle enhanced. \nBy contrast, a centered charge on the top surface barely affects \nthe system symmetry, which remains high ($C_{4v}$), and\nSU processes are only slightly stronger than in the $Q_{edge}=0$ case.\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=13cm]{fig2.eps}\n\\caption{\n(a,b) Sketch of SU processes in NPLs with (a) and without (b) an edge charge.\nLabels on the left are $(n_x, n_y, n_z)$ quantum numbers\nfor the (independent particle) energy levels.\nLabels on the right are the corresponding irreducible representation. \nThe surface charge lowers the point group symmetry, \nfrom $D_{4h}$ to $C_s$, lifting degeneracies and enabling\nnew channels for SU transitions (vertical arrows).\n(c,d) Two main configurations $|m_{X^-}\\rangle$ in the CI expansion of $|GS_{X^-}\\rangle$,\nwith and without edge charge. Thin (thick) arrowsheads denote electron (hole) spin.\nOnly when $Q_{edge}\\neq 0$ a SU process is expected.\n(e) Energy splitting between $|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$ at an \nindependent particle level.\n(f) average value of electron-electron repulsion and (g) electron-hole\nattraction in configurations $|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$.}\\label{fig2}\n\\end{figure}\nThe qualitative reasoning above can be substantiated with a \nCI formalism on the basis of independent particle (non-interacting) \nelectron and hole states, which has the additional advantage of giving \nintuitive insight on how Coulomb interactions affect the likelihood of SU processes.\nWe consider that the transition rate from the trion ground state \n$|GS_{X^-}\\rangle$ to an electron spin-orbital $|f_e\\rangle$, \nis proportional to:\\cite{Pawel_book}\n\\begin{equation}\nP_{GS \\rightarrow f} = \\left| \\langle f_e | \\,{\\hat P}\\, | GS_{X^-} \\rangle \\right|^2 .\n\\label{eq:trans}\n\\end{equation}\n\\noindent ${\\hat P}$ is the dipolar transition operator,\n${\\hat P} = \\sum_{i_e,i_h} \\, \\langle i_e | i_h \\rangle \\, e_{i_e}\\,h_{i_h}$,\nwhere $e_{i_e}$ and $h_{i_h}$ are annihilation operators for independent electron\nand hole spin-orbitals $|i_e\\rangle$ and $|i_h\\rangle$, respectively.\nWe describe the trion ground state with a CI expansion,\n\\begin{equation}\n|GS_{X^-} \\rangle  = \\sum_m c_m \\, | m_{X^-} \\rangle, \n\\end{equation}\n\\noindent where $|m_{X^-}\\rangle$ is a trion configuration:\n$|m_{X^-} \\rangle  = e_{r_e}^\\dagger e_{s_e}^\\dagger |0 \\rangle_e \\, h_{t_h}^\\dagger |0 \\rangle_h$,\n with $e_{r_e}^\\dagger$ and $h_{t_h}^\\dagger$ creator operators,\n $|0\\rangle_e$ and $|0\\rangle_h$ the vacuum occupation vectors of electron and hole, \nand $c_m$ the coefficient in the expansion.\n Inserting ${\\hat P}$ and $|GS_{X^-}\\rangle$ into Equation (\\ref{eq:trans}), \n one obtains:\n\\begin{equation}\nP_{GS \\rightarrow f} = \n\\left| \\sum_m  c_m \\, \\left( \\langle r_e | t_h \\rangle\\, \\delta_{f_e\\, s_e} \n- \\langle s_e | t_h \\rangle \\delta_{f_e\\, r_e} \\right) \\right|^2.\n\\label{eq:trans2}\n\\end{equation}\n\\noindent In SU processes, $|f_e\\rangle$ is an excited spin-orbital. \nIt then follows from Equation (\\ref{eq:trans2}) that such a transition will \nonly take place if $|GS_{X^-}\\rangle$ contains at least one configuration $|m_{X^-}\\rangle$\nin the CI expansion where one electron is in the excited spin-orbital \nand the other electron has finite overlap with the hole ground state \n($|s_e\\rangle = |f_e\\rangle$ and $\\langle r_e | t_h \\rangle \\neq 0$\nor $|r_e\\rangle = |f_e\\rangle$ and $\\langle s_e | t_h \\rangle \\neq 0$). \nThe larger the weight of this configuration, $|c_m|^2$, the more likely the SU process.\nIt is worth noting that in the strong confinement limit, the trion ground state is well\ndescribed by a single configuration where all carriers are in the lowest-energy spin-orbitals\n(configuration $|1_{X^-}\\rangle$ in Fig.~\\ref{fig2}c and \\ref{fig2}d).\nThat is, $c_1 \\approx 1$ and $c_m \\approx 0$ for $m > 1$. \nSU transitions are then forbidden, which is why SU peaks are rarely reported \nin quantum dots.\nOn the contrary, in systems where Coulomb interaction energies exceed quantum\nconfinement energies, the CI expansion contains mono- and biexcitations of electrons.\nSU processes are then enabled.\nColloidal NPLs constitute an ideal system at this regard, because they combine weak\nconfinement in the lateral direction with strong Coulomb \ninteractions.\\cite{RajadellPRB,RichterPRM}\nHereafter, we refer to this condition ($c_m \\neq 0$ for $m > 1$) as Coulomb admixture.\n\nThe role of Coulomb correlation and symmetry breaking in activating SU processes\ncan be illustrated, in the simplest approximation, by considering the two lowest-energy \nconfigurations of the\ntrion ground state, \n\\begin{equation}\n|GS_{X^-}\\rangle \\approx c_1 |1_{X^-}\\rangle + c_2 |2_{X^-}\\rangle. \n\\end{equation}\nIn Fig.~\\ref{fig2}c and \\ref{fig2}d we depict such configurations\nin the absence and presence of an edge charge, respectively.\nThese can be expected to be the two most important configurations in the full CI expansion.\nNotice that the two configurations must have the same symmetry, \nfor Coulomb interaction to couple them.\nBecause the lowest-energy configuration, $|1_{X^-}\\rangle$, is always totally symmetric,\nso must be $|2_{X^-}\\rangle$.\nThus, when $Q_{edge}=0$ ($D_{4h}$ group), the electronic configuration of $|1_{X^-}\\rangle$ is\n$[A_{1g}^2]_e \\, [A_{1g}]_h$, and that of $|2_{X^-}\\rangle$ is $[E_{u}^2]_e \\, [A_{1g}]_h$. \nThe recombination of the $E_u$ electrons with the hole, which stays in a $A_{1g}$ orbital, is then \nsymmetry forbidden ($\\langle r_e | t_h \\rangle = \\langle s_e | t_h \\rangle = 0$ in Eq.~(\\ref{eq:trans2})).\nBy contrast, when $Q_{edge} \\neq 0$ ($C_s$ group), $|2_{X^-}\\rangle$ is formed by a monoexcitation where\none electron is placed in the $(n_x,n_y,n_z)=(2,1,1)$ orbital, which also has $A'$ symmetry, resulting\nin an electronic configuration $[A'\\,A']_e \\, [A']_h$ (see Fig.\\ref{fig2}d).\n\n The hole can then recombine with the ground orbital electron, as both have $A'$ symmetry \n ($\\langle r_e | t_h \\rangle \\neq 0$ or $\\langle s_e | t_h \\rangle \\neq 0$ in Eq.~(\\ref{eq:trans2}))\nand leave the excited electron as the final state. \nThis constitutes a SU process. \n Because both SU and fundamental transition rely on the recombination of the same electron-hole pair\n (same overlap integral, e.g. $\\langle r_e | t_h \\rangle$), the ratio between SU and fundamental\n radiative rates can be approximated as:\n %\n \\begin{equation}\n\t \\frac{ P_{GS \\rightarrow (2,1,1)_e} }{ P_{GS \\rightarrow (1,1,1)_e} } \\approx \n\t \\frac{|c_2|^2}{|c_1|^2}.\n \\end{equation}\n %\n \\noindent i.e. it is set exclusively by the degree of Coulomb admixture.\n\nOne can guess the requirements that maximize $|c_2|^2$ by looking which\nconditions favor energetically $|2_{X^-}\\rangle$ over $|1_{X^-}\\rangle$. \nThese include: (i) small energy splitting between the two configurations, \nat an independent particle level, $\\Delta_{sp}$ in Fig.~\\ref{fig2}d, \n(ii) weaker electron-electron repulsion ($V_{ee}$) and \n(iii) stronger electron-hole attraction ($V_{eh}$) in $|2_{X^-}\\rangle$ as compared to $|1_{X^-}\\rangle$.\nFigures ~\\ref{fig2}e-g show that these conditions are met for moderately attractive (positive) \ncharges ($Q_{edge} \\sim 0.3-0.4$).\nWhen the off-centered charge is switched on, $\\Delta_{sp}$ rapidly decreases (see Fig.~\\ref{fig2}e) because \nthe symmetry descent turns one of the $E_u$ ($p$-like) electron orbitals into a $A'$ ($s$-like) one.\nHowever, the surface charge brings about electrostatic confinement and hence $\\Delta_{sp}$ increases again soon after.\nAs for inter-electron repulsion, $\\langle 1_{X^-} | V_{ee} | 1_{X^-} \\rangle$ \nincreases more rapidly than $\\langle 2_{X^-} | V_{ee} | 2_{X^-} \\rangle$\n(see Fig.~\\ref{fig2}f) because the former involves placing the two electrons in \nidentical orbitals, while the latter does not.\n\nLast, $\\langle 1_{X^-} | V_{eh} | 1_{X^-} \\rangle$ is rapidly quenched (see Fig.~\\ref{fig2}g) \nbecause it involves the ground orbitals of electron and hole --$(1,1,1)_e$ and $(1,1,1)_h$--, \nwhich dissociate rapidly under an external charge.\n$\\langle 2_{X^-} | V_{eh} | 2_{X^-} \\rangle$ stays strong up to $Q_{edge} \\sim 0.3$ because it involves the $(2,1,1)_e$ orbital,\nwhich is spatially more extended and then keeps significant overlap with the $(1,1,1)_h$ hole. \nFigs.~\\ref{fig2}e-f further evidence that $Q_{edge} > 0.3-0.4$ is inconvenient for SU processes, \nbecause the electrostatic potential increases lateral \nquantum confinement ($\\Delta_{sp}$ increases) and because electrons and hole in configuration $|2_{X^-}\\rangle$ are\neventually dissociated as well ($\\langle 2_{X^-} | V_{eh} | 2_{X^-} \\rangle$ is quenched in Fig.~\\ref{fig2}g).\n\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{fig3.eps}\n\\caption{In-plane charge density of the two electrons and hole in the X$^-$ ground state (top rows), \nand wave functions of the two lowest electron orbitals with $A'$ symmetry (bottom rows), \nas a function of the edge charge magnitude. The edge charge is located on the top edge, in this view. \nThe strongest SU peak corresponds to $Q_{edge}\\approx 0.4$,\nwhen the X$^-$ electron charge density reveals a clear contribution from $(2,1,1)_e$, and the hole\nis not yet fully dissociated from electrons.\n\t}\\label{fig3}\n\\end{figure}\n\nMuch of the above observations can be visualized by analyzing the evolution\nof charge densities and wave functions under $Q_{edge}$. \nIn Figure \\ref{fig3} we show the two-electron (first row)\nand one-hole (second row) charge densities of $|GS_{X^-}\\rangle$, \nobtained from the CI calculations of Fig.~\\ref{fig1}.\nThe wave functions of the two lowest electron orbitals which can constitute \nconfiguration $|2_{X^-}\\rangle$, --$(n_x,n_y,n_z)_e=(1,1,1)_e$ and $(2,1,1)_e$-- \nare also plotted (bottom rows).\nAt $Q_{edge}\\approx 0$, the two orbitals are quasi-orthogonal. \nAs a result, Coulomb interaction cannot couple configurations \n$|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$.\n and $c_2 \\approx 0$. This is why the two-electron charge density \nclosely resembles the $(1,1,1)_e$ orbital. SU processes are not expected in this case.\n\nAt $Q_{edge} \\approx 0.4$, symmetry lowering and energetic considerations \nenable efficient Coulomb coupling. \nThe oval shape of the two-electron charge density reflects a significant contribution \nfrom $(2,1,1)_e$ to $|GS_{X^-}\\rangle$ (i.e. $|c_2| > 0$). \nAt the same time, the electron $(1,1,1)_e$ orbital and the hole ground state have \nsizable overlap.  This is an optimal situation for the appearance for\nthe transition $P_{GS \\rightarrow (2,1,1)_e}$ to show up as a SU process,\naccording to Equation (\\ref{eq:trans2}).\nFurther increasing $Q_{edge}$ separates the $(2,1,1)_e$ electron orbital from\nthe hole. Coulomb attraction is then weaker, making $c_2$ and consequently \n$P_{GS \\rightarrow (2,1,1)_e}$ small again.\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=6cm]{fig4.eps}\n\\caption{Normalized X$^-$ emission as a function of the environment dielectric constant.\nWith increasing dielectric contrast, the SU peak increases and becomes more redshifted.\nFor every value of $\\epsilon_{out}$, the value of $Q_{edge}$ that maximizes SU transitions\nis shown. In all cases, $\\epsilon_{in}=6$.}\\label{fig4}\n\\end{figure}\n \nWe have argued above that strong Coulomb admixture of configurations facilitates the appearance of SU processes.\nA distinct feature of colloidal NPLs when compared to epitaxial quantum wells is the presence\nof a prounounced dielectric contrast with the organic ligands surrounding the NPL,\nwhich enhances Coulomb interactions by effectively reducing the system dielectric screening.\\cite{AchtsteinNL,RajadellPRB,BenchamekhPRB}\nTo study the influence of this phenomenon over SU transitions, in Figure \\ref{fig4}\nwe compare the trion emission spectrum for different values of the environment\ndielectric constant $\\epsilon_{out}$, while fixing that of the NPL to the high-frequency\nCdSe value, $\\epsilon_{in}=6$. For the sake of comparison, the emission spectrum is \nnormalized so that the band edge peak has the same intensity in all cases. \nAlso, we have selected the value of $Q_{edge}$ that maximizes the relative size of the SU peak in each case.\nBecause $\\epsilon_{out}$ screens the surface charge electrostatic field, \nlarger $Q_{edge}$ values are needed when $\\epsilon_{out}$ increases.\nThe figure evidences that lowering $\\epsilon_{out}$ increases the SU peak height \nand energetic redshift.\nFor typical ligands of CdSe NPLs (e.g. oleic acid), $\\epsilon_{out} \\sim 2$.\\cite{AchtsteinNL,EvenPCCP}\nWe then conclude that dielectric confinement makes SU processes in colloidal NPLs\nmore conspicuous.\n\n\\subsection{Core\/shell NPLs}\n\nWe next consider heterostructured core\/shell NPLs. The first case under study are CdSe\/CdS NPLs.\\cite{TessierNL,AchtsteinACS,PolovitsynCM,LlusarJPCc} \n The NPLs have the same CdSe core as in the previous section and 6 ML thick CdS shells\non top and bottom  (see inset in Figure \\ref{fig5}a).\nIn general, the behavior of SU replicas is found to be analogous to that of core-only NPLs. \nAn off-centered acceptor impurity is needed to yield sizable SU replicas,\nwith an optimal value of $Q_{edge}$ maximizing the relative size of the SU peak. \n\nFigure \\ref{fig5}a shows the emission spectrum of X$^-$ for the optimal $Q_{edge}$ value,\nin CdSe\/CdS NPLs (green line) against CdSe core-only NPLs (black, dashed line).\nOne can see that the SU replica of the CdSe\/CdS structure is again significant\n (11\\% of the main transition), but less pronounced than in the core-only structure \n (26\\%). \n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=13cm]{fig5.eps}\n\\caption{(a) Normalized X$^-$ emission spectrum a CdSe\/CdS NPL with 6 ML-thick shell (solid line), \ncompared to that of a core-only CdSe NPL (dotted line). \nThe spectra are centered at the energy of band edge transition.\n$Q_{edge}=0.6$ ($0.4$) is used for the CdSe\/CdS NPL (core-only NPL), to maximize the relative height of SU lines.\nThe SU peak for the core\/shell system is smaller than for core-only NPLs. \n\n(b,c) Average Coulomb integrals of $|GS_{X^-}\\rangle$ configurations \n$|1_{X^-}\\rangle$ and $|2_{X^-}\\rangle$: (b) electron-electron repulsion, (c) electron-hole attraction.\nSolid (dotted) lines are used for core\/shell (core-only) NPLs. The interactions\nare weaker in the core\/shell structure.\n(d) Charge densities of\nelectrons (left) and hole (right) for the trion ground state in the CdSe\/CdS NPL at $Q_{edge}=0.6$.  \nThe electron stays in the vicinity of the core, despite the shallow band offset.}\\label{fig5}\n\\end{figure}\nThe smaller SU replica in the core\/shell structures is a robust result, which holds\nfor different shell thickness and surface charge locations. It is a consequence of the\nweaker Coulomb interactions. The electron leakage into the CdS shell reduces \nelectron-electron repulsions and electron-hole attractions. \n The quenching of dielectric confinement by the CdS shell, which pushes organic ligands far from\nthe core, further contributes to the weakening.\nThis observation is reflected by Figs.~\\ref{fig5}b and \\ref{fig5}c, \nwhich show that Coulomb interactions (especially $V_{ee}$) are weakened in core\/shell NPLs \n(solid lines) as compared to core-only NPLs (dotted lines). \nConfiguration $|2_{X^-}\\rangle$ is then less stabilized with respect to $|1_{X^-}\\rangle$,\nwhich implies smaller $|c_2|$ coefficient in the CI expansion.\n\nFigure \\ref{fig5}d compares the charge density of the two electrons (left) \nand hole (right) in $|GS_{X^-}\\rangle$.\n The trion electrons are found to stay in the vicinity of the core, rather than delocalizing all over the structure, \n to benefit from interaction with the hole. \n This is consistent with the observed behavior of CdSe\/CdS NPLs being similar to that\n of core-only structures, albeit with weakened Coulomb interactions due to the lessened confinement.\n\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6cm]{fig6.eps}\n\\caption{(a) Normalized X$^+$ emission spectrum in a CdSe\/CdTe NPL\nwith 6 ML thick shells, as a function of the lateral charge strength.\nDotted line is a guide to the eye. SU peaks and fundamental transition\nhave comparable intensities.\n(b) Two main $|GS_{X^+}\\rangle$ configurations in the CI expansion\nin the presence of a charge. The weight of $|2_{X^+}\\rangle$ is comparable\nto that of $|1_{X^+}\\rangle$ in this system, which explains the high SU peaks in (a).\n(c) Wave function of $(1,1,1)_h$ and $(1,1,2)_h$ hole orbitals under $Q_{edge}=-0.5$.\nThe states have the same symmetry but localize on opposite sides of the core\nto stay orthogonal.}\\label{fig6}\n\\end{figure}\n\nUnderstanding the conditions which promote SU processes allows us to devise\nstructures where their impact would be maximal. In Fig.~\\ref{fig6} we consider \na core\/shell NPL with the same dimensions as before, but CdSe\/CdTe composition.\nThe NPL is chosen to be charged with a positive trion (X$^+$).\nBecause of the type-II band alignment, the electron stays in the CdSe\ncore and the holes in the CdTe region, as observed in related core\/crown structures.\\cite{AntanovichNS,KelestemurJPCc}\nIn the absence of external charges, the two first hole orbitals are $(1,1,1)_h$ and $(1,1,2)_h$, \ni.e. the symmetric ($A_{1g}$) and antisymmetric ($A_{1u}$) solutions of the double well potential, respectively,\nwhich are almost degenerate because tunneling across the core is negligible\n(i.e. $\\Delta_{sp} \\rightarrow 0$). Switching on a negative surface charge, \n$Q_{edge} < 0$, lifts the inversion symmetry so that both orbitals acquire $A'$ symmetry\nand can be Coulomb coupled. \nThe admixture between configurations $|1_{X^+}\\rangle$ and $|2_{X^+}\\rangle$, \ndepicted in Fig.~\\ref{fig6}b, is then very strong.\n In the presence of the charge, the two hole orbitals tend to localize \non opposite shell sides to remain orthogonal, as shown in Fig.~\\ref{fig6}c. \nThis implies that configuration $|1_{X^+}\\rangle$, which has two holes\nin the same orbital, has much stronger repulsion than configuration $|2_{X^+}\\rangle$, \nwhich distributes the two electrons on opposite sides of the core. This makes\n$\\langle 1_{X^+} | V_{hh} | 1_{X^+} \\rangle \\gg \\langle 2_{X^+} | V_{hh} | 2_{X^+}\\rangle$. \nAltogether, the small $\\Delta_{sp}$ value and the large difference in hole-hole repulsion\nexplain the strong admixture between configurations $|1_{X^+}\\rangle$ and $|2_{X^+}\\rangle$.\nAs shown in Fig.~\\ref{fig6}a, this gives rise to SU peaks whose magnitude is almost\nas large as that of the fundamental transition ($72\\%$ for $Q_{edge}=-0.5$).\n\n\n\n\\section{Discussion}\n\nOur simulations show that SU processes can be expected for trions in core-only \nand core\/shell NPLs, if off-centered impurities are present.  %\nWe discuss here the potential relationship of this finding with experiments\nand practical implications.\n\n\\subsection{Relationship with experiments}\n\nIn core-only CdSe NPLs, the low temperature photoluminescence is thought to arise \nfrom subpopulations of excitons and negative trions.\\cite{ShornikovaNL,YuAMI,AntolinezNL2} \nVery recently, Antolinez and co-workers have reported that the X$^-$ \nemission shows a distinct peak or a shoulder (depending on the film thickness)\nredshifted from the trion band edge transition. The redshift is $\\sim 19$ meV \nand the relative height $15-25\\%$ that of the main peak.\\cite{AntolinezNL2} \nThey speculated that the origin could be a SU process of the kind we study.\nOur calculations support the feasibility of this interpretation.\nFigure \\ref{fig1}a shows excellent agreement with the experimental measurements,\nboth in energy and relative intensity of the SU peak, \nassuming a lateral charge with $Q_{edge}=0.3-0.4$,\nwhich gives a redshift of $19-25$ meV and a relative height of $15-23$ \\%.\n\nThe presence of acceptor impurities in CdSe NPLs likely originates when\nthe hole of a photoexcited electron-hole pair is trapped by a surface\ndefect. The next electron-hole pair generated in the NPL joins the\nresidual electron to form X$^-$, while the trapped hole exerts \na screened electrostatic potential.\\cite{RabouwNL,FengNL,CalifanoNL}\nThe coexistence of X$^-$ and trapped surface charges in CdSe NPLs \nis backed up by studies reporting correlation between surface-to-volume ratio, \nlaser irradiation time and trion emission intensity.\\cite{YuAMI}\nA plausible location for surface charges are the\nlateral sidewalls of the NPL (as in Fig.~\\ref{fig1}b).\nThis possibility is suggested by studies showing that Z-type \nligand desorption --and hence surface traps-- in CdSe NPLs is more \nfrequent on these facets,\\cite{LeemansJPCL}\nand by the fact that CdSe\/CdS core\/crown NPLs generally improve the\nphotoluminescence quantum yield as compared to core-only structures,\ndespite having larger surfaces on top and bottom.\\cite{TessierNL2}\nBecause off-centered charges are needed to originate SU peaks, \nlateral charges are candidates to trigger such processes.\\\\\n\n\n\nIn core\/shell CdSe\/CdS NPLs, SU processes have been also proposed \nas the origin of multi-peaked fluorescence emission \n--and hence broadened line width--.\\cite{AntolinezNL}\nOur simulations in Fig.~\\ref{fig5}a confirm one can indeed expect \na sizable SU peak in such structures.\nWe note that earlier experimental studies had so far interpreted the\nline width broadening as a result of either SU processes\\cite{AntolinezNL}\nor of surface defects.\\cite{TessierNL} By showing that the second effect\nis a prerequesite for the first one, our study helps to reconcile both\ninterpretations.\nNonetheless, two remarkable disagreements are observed between our simulations\nand Ref.~\\cite{AntolinezNL} measurements. \nFirst, the experiments show from 2 to 4 emission peaks, \nwhich are interpreted as the X$^-$ fundamental transition plus up to three redshifted, SU peaks.  \nIn our calculations, however, we fail to see more than one significant SU replica.\nSecond, the highest-energy peak in the experiment is never the brightest one. \nThis is inconsistent with our results and with earlier studies on epitaxial\nquantum wells and dots, where the higher-energy peak corresponds to the fundamental\ntransition, which is the most likely recombination channel.\\cite{NashPRL,FinkelsteinPRB,BryjaPRB,DzyubenkoPRB,PaskovPE}\n\nTentatively, one may suspect that a large number of SU peaks in core\/shell CdSe\/CdS NPLs \ncould be connected with the thick CdS shell (12 ML in Ref.~\\cite{AntolinezNL}), which\nmakes surface defects more likely than in core-only structures. \nA significant presence of defects in these structures has been hinted \nby studies showing that the long radiative lifetime is not due to electron delocalization \nbut to the influence of impurities.\\cite{AchtsteinACS}\nHowever, Coulomb interactions are weaker than in core-only structures\n(Fig.~\\ref{fig5}c,d), where only one SU peak has been measured.\\cite{AntolinezNL2} \nIt is then not surprising that, despite investigating different charge locations\n(Figs. S3, S6 and S7 in SI), conduction band-offset values (Fig. S4) and\nshell thicknesses (Fig.S5), we see at most one significant SU satellite.\n \nRegarding the relative intensity of the peaks,\n as mentioned in the previous section, the highest-energy one (fundamental transition) \n is proportional to the weight of configuration $|1_{X^-}\\rangle$ in the CI expansion, $|c_1|^2$, \nwhile subsequent (SU) peaks would be proportional to $|c_2|^2$, $|c_3|^2$, \\ldots\nConfiguration $|1_{X^-}\\rangle$ (all carriers in the ground orbital, Fig.~\\ref{fig2}c) \nis nodeless and hence naturally expected to be the dominant one, \nso the highest-energy peak is also the brightest one.\nWe have not observed SU peaks exceeding the fundamental transition height\n despite considering different charge locations and shell thicknesses (see SI). \n Even in CdSe\/CdTe NPLs, which constitute a limit case, \n SU peaks never exceed the height of the main transition, see Fig.~\\ref{fig6}a.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=6cm]{fig7.eps}\n\\caption{(a) Normalized X$^-$ emission spectrum in a CdSe\/CdS NPL,\n as a function of the lateral charge strength.\nAn electron spin relaxation bottleneck is imposed, so that emission comes from \nthe lowest singlet ($S_e=0$, ground state) and triplet ($S_e=1$) states.\nDotted lines are guides to the eyes.\n(b-c) Sketches showing the relevant electron-hole recombination channels\nof singlet and triplet states. (b) The singlet can give rise to one fully radiative \n($s$-$R1$) plus one SU transition ($s$-$SU$). \n(c) The triplet can give rise to two fully radiative transitions, $t$-$R1$\nand $t$-$R2$.}\\label{fig7}\n\\end{figure}\n\nAs an alternative interpretation for the experiments, a multi-peaked emission spectrum could \nresult from stacking of colloidal NPLs,\\cite{DirollNL} which leads to electronic \ncoupling through dielectric confinement.\\cite{MovillaJPCL}\nHowever, the time-dependent spectral shifts observed by Antolinez \\emph{et al.} \nsuggest that all peaks arise from the same NPL, and significant stacking was not expected\nin the experiment samples.\\cite{AntolinezNL}\nWe thus propose a different interpretation. Namely, simultaneous emission from the X$^-$ \nground state, with singlet electron spin ($S_e=0$), and a metastable excited state with triplet\nelectron spin ($S_e=1$). The decay from the triplet to the singlet state is slowed down \nby spin selection rules, as phonons are spinless. This could allow simultaneous occupation\nof the two states even if the energy splitting exceeds thermal energy.\n \nTo illustrate this point, in Figure \\ref{fig7}a we show the calculated emission of X$^-$ \nassuming equipopulation of $S_e=0$ and $S_e=1$ trion states. One can see that the number \nof sizable peaks in the spectrum ranges from two to four, depending on the \nstrength of surface charge, $Q_{edge}$.\n The origin of these peaks is summarized in the sketches of Fig.~\\ref{fig7}b and \\ref{fig7}c.\nThe singlet (Fig.~\\ref{fig7}b) can give rise to a fully radiative transition ($s$-$R1$) \nand a SU transition ($s$-$SU$), as described in the previous sections. \nIn turn, the triplet (Fig.~\\ref{fig7}c) can give rise to two fully radiative transitions \n($t$-$R1$ and $t$-$R2$), depending on which electron recombines with the hole.\n$t$-$R2$ is readily visible at $Q_{edge}=0$, but $t$-$R1$ requires recombining the hole \nwith an excited electron, a process which is again activated when the surface charge \nlifts symmetry restrictions.\nHowever, unlike in SU processes, the two triplet transitions come from the main configuration \nof the trion CI expansion. Therefore, their intensity can be comparable to that of the band\nedge transition, $s$-$SU$, even if Coulomb admixture is weak.\nThe triplet transitions are built on both sides of $s$-$SU$, with inter-peak energy \nsplittings up to few tens of meV.\nThe relative sizes of the peaks will be further modulated in realistic situations\nby a finite triplet-singlet decay rate. This relaxation channel would possibly reduce the relative \npopulation of $S_e=1$, and hence the intensity of $t$-$R1$.  \n\nAltogether, the number of peaks, the magnitude of the energy splitting between the peaks\nand the flexible intensities provide a framework to explain the \nmulti-peaked photoluminescence of Ref.~\\cite{AntolinezNL}.\nSeveral other aspects of this proposal are consistent with the experiments.\nFor example, because all peaks in Fig.~\\ref{fig7}a arise from the same NPL, \nthey will experience simultaneous spectral shifts when surface impurities migrate.\\cite{AntolinezNL} \nAlso, the hot trion emission is expected to vanish when the impurities are removed, \nas $t$-$R1$ becomes deactivated and $t$-$R1$ almost merges with the singlet emission, $s$-$R1$,\nsee Fig.~\\ref{fig7}a for $Q_{edge}=0$. This fits the transition from asymmetric to symmetric band shape\nas temperature increases.\\cite{TessierNL}\n\nThe fact that triplet emission is observed in CdSe\/CdS NPLs, but not in CdSe ones,\nmay be explained from the strong spin-spin interaction of resident carriers and \nsurface dangling bonds in the latter case,\\cite{ShornikovaNN}\nwhich should speed up spin relaxation through flip\/flop processes.\nThis mechanism is expected to be inhibited in core\/shell structures, because\nX$^-$ carriers stay far from the surface, as shown in Fig.~\\ref{fig5}d.\nOn the other hand, the triplet trion is expected to have fine structure\nthrough electron-hole exchange interaction\\cite{WarePRL}, \nwhich may not fit the mono-exponential photoluminescence decay reported in Ref.~\\cite{AntolinezNL}. \nFurther experiments are needed, e.g. on polarisation of the different peaks under external fields\\cite{ShornikovaNL,JovanovPRB},\nto confirm the different spin of the emissive states in CdSe\/CdS NPLs. \n\nThe observation of metastable triplet trion photoluminescence has been previously reported \nin epitaxial quantum wells\\cite{BryjaPRB,ShieldsPRB} and dots\\cite{JovanovPRB},\nand more recently in transition metal chalcogenide monolayers.\\cite{VaclavkovaNT\nTo our knowledge, however, its presence in colloidal nanostructures has not been confirmed.\n\n\n\\subsection{Control of SU processes}\n\nInasmuch as SU processes can be responsible for the line width broadening NPLs,\ntheir supression is desirable to improve color purity in optical applications.\nIt has been suggested that this job could be achieved by increasing quantum confinement,\nreducing either lateral dimensions or shell thickness --the latter would favor \nelectrostatic confinement.\\cite{AntolinezNL} \nBoth strategies have the drawback of introducing size dispersion in ensemble\nluminescence. \nFrom our theoretical analysis, we confirm that reducing Coulomb admixture \nwould minimize SU processes, but this can be achieved by weakening Coulomb interactions \ninstead of increasing quantum confinement. For example, reducing dielectric \nconfinement or using thinner cores to enhance the quasi-type-II character \nshould contribute to this goal.  Obviously, this approach would have the \ndrawback of reducing the band edge recombination rate as well.\n\nAlternatively, since our study shows that impurities are ultimately responsible\nfor SU processes, experimental routes to suppress SU processes could be directed \nto control of traps. Appropriate choice of surface ligands\\cite{LeemansJPCL},  electrochemical \npotentials\\cite{GallandNAT} and interface alloying\\cite{RossinelliCM,KelestemurACS} \ncould contribute to this end. \n\nBecause we find surface charges on lateral sidewalls particularly suited to induce SU processes,\nthe growth of core\/crown heterostructures is expected to reduce their influence by \nkeeping the outer rim away from the photogenerated carriers. \nThis suggestion seems to agree with experimental observations by Kelestemur and co-workers, \nindicating that core\/crown\/shell CdSe\/CdS NPLs have more symmetric emission behavior than core\/shell \nones at cryogenic temperatures,\\cite{KelestemurAFM}\nThis can be understood as a consequence of the suppression of SU processes in \nthe low-energy tail of the emission band. \nIt is also consistent with recent single-particle studies showing that the \nline width in CdTe\/CdSe core\/crown NPLs is set by LO phonon replica, \nrather than SU ones.\\cite{SteinmetzJPCc}\n\n\nShould the role of metastable triplet states be confirmed in CdSe\/CdS NPLs, \nstrategies to control the line width should rather focus on enhancing the interaction of\nconfined carriers with surface spins\\cite{ShornikovaNN} or intrinsic spin-orbit interaction\\cite{TadjinePRB}, \nto shorten their lifetime.\nReplacing trion by neutral exciton emission through thermal dissociation,\\cite{AyariNS} \nis yet another possibility to avoid SU and high spin peaks.\n\nThus, our calculations propose a wealth of experiments targeted at material design to \ntune quantum and dielectric confinement, and exciton-surface\/interface interactions, \nand set suitable temperature ranges to control SU\/triplet emission.\n\n\\section{Conclusions}\n\nWe have shown that SU processes in colloidal NPLs are enabled by severe Coulomb admixture \n--which results from strong Coulomb interactions and weak lateral confinement-- \nand the presence of off-centered electrostatic traps, which suppress the\n protection against Auger processes provided by symmetry conservation. \nSurface charges on lateral sidewalls seem particularly efficient to this end. \n\nUnder typical experimental conditions, core-only and core\/shell NPLs are susceptible of showing a SU peak\nwith oscillator strength $0.1$-$0.3$ times that of the band edge transition.\nThis is at least one order magnitude larger than in epitaxial quantum wells.\nThe SU peak is redshifted from the band edge peak by up to few tens of meV,\nthus providing a source of line width broadening.\n \nThese results are in excellent agreement with recent experimental findings in CdSe NPLs\\cite{AntolinezNL2}\nin terms of number of emission peaks, energy splitting and relative intensity,\nbut only partially so with those of core\/shell CdSe\/CdS NPLs.\\cite{AntolinezNL}\nExperiments in the latter structure are however in line with an alternative interpretation \ninvolving simultaneous participation from trion singlet and metastable triplet states.\n\nStrategies to narrow the line width of NPLs through suppression of SU processes should\naim at controlling electrostatic impurities or Coulomb admixture.\n\n\n\n\\section{Methods}\n\nCalculations are carried within k$\\cdot$p-continuum elastic theory framework.\nIndependent electron and hole states are calculated with single-band Hamiltonians\nincluding strain and self-energy potential terms. \nModel details and material parameters are given in Ref.~\\cite{LlusarJPCc}.\nPoint charge electrostatic potentials and Coulomb integrals for \nCI matrix elements, including dielectric mismatch effects, \nare calculated solving Poisson Equation with Comsol $4.2$.\nThe CI basis set is formed by all possible combinations of the first\n22 single-electron and 22 single-hole spin-orbitals. For X$^-$,\nthese are combined to form configurations $|m_{X^-}\\rangle$ \nas the Hatree product of one hole spin-orbital with a two-electron \nSlater determinant.\n\n\n\n\\begin{acknowledgement}\nThe authors acknowledge support from MICINN project CTQ2017-83781-P.\nWe are grateful to I. Moreels, A. Achtstein and F. Rabouw for useful discussions.\n\\end{acknowledgement}\n\n{\\bf Supporting Information Available:} \n\nAdditional calculations on the influence of trion charge, surface charge\nlocation and conduction band offset over the formation of SU processes are\nprovided.\n\n\n\n\n\\section{Additional calculations}\n\nWe present here additional calculations for further understanding of SU processes. \n\n\\subsection{Convergence of CI calculations}\n\nConfiguration Interaction (CI) calculations on the basis of independent particle (or Hartree-Fock) orbitals provide an excellent description of repulsions in few- and many-fermion systems\\cite{JacakSpringer,RontaniJCP}. However, large basis sets are needed to describe strong attractions,\\cite{RontaniJPB,ShumwayPRB} which are certainly present in colloidal NPLs\\cite{RajadellPRB} and are involved in a correct description of SU processes.\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{convergence_su.eps}\n\t\t\\caption{X$^-$ emission spectrum for $Q_{edge}=0.4$ (see main text). Zero energy is set for the fundamental transition with $ne=nh=22$.\n$ne$ and $nh$ are the number of single-electron and single-hole spin-orbitals, respectively, used to build the CI basis sets.}\n\n\\label{figS}\n\\end{figure}\n \n In Fig.~\\ref{figS} we compare the X$^-$ emission spectrum calculated for CdSe NPLs --same dimensions as in main text-- \nusing different basis sizes.  The basis is formed by all possible combinations of the first $ne$ ($nh$) independent particle \n spin-orbital states of electrons (holes). \n With increasing basis dimensions, the band edge transition peak redshiftes and gains intensity, which reveals an improved\n description of electron-hole correlation. The intensity of the SU peak height and its redshift with respect to the \n band edge transition are however less sensitive to the basis dimensions.\n %\n It follows from the figure that quantitative assessment on the ratio of fundamental vs SU peak heights requires \n large basis sets. In the main text we use $ne=nh=22$. By comparing with smaller values of $ne\/nh$ in the figure,\n it is clear that for this value --which involves very time-consuming computations-- the ratio is reaching saturation.\n This validates the order of ratios provided in the main text.\n For the calculations in this Supporting Information, however, we may resort to $ne=nh=12$, \n which overestimates the relative height of SU peaks, but suffices to provide qualitative assessment.\n\n\\subsection{Positive trion behaviour} \\label{positive_trion}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=8cm]{emission_X+.eps}\n\\caption{$X^{+}$ normalized spectrum emission for different charge intensities. The arrows are pointing to SU satellites (dotted lines are guides to the eyes). The highest SU peak ($Q_{edge}=-0.3$) is marked with a star. The origin of energies is set at the band edge recombination peak. The insets correspond to $Q_{edge}=0.5$ amplified SU peaks.}\n\\label{figSXXX3}\n\\end{figure}\n\n\nIn the main text, we have mostly considered the case of negative trions.\nWe show here that the same behavior holds for positive ones.\nTo illustrate this point, we choose the case of the core-only NPL with an edge charge,\nequivalent to Fig.1d of the main text.\nFigure \\ref{figSXXX3} shows that the presence of SU peaks in the emission \nspectrum is again strongly dependent on the value of the surface charge.\nFor $Q_{edge}=0$, no SU peak is observed. For repulsive ($Q_{edge}>0$) charges,\nSU are formed but very small in magnitude. The highest SU peaks are formed\nfor weakly bound donor charges ($Q_{edge}<0$), which attract the holes of X$^+$,\nmarked with a star in the figure. As in the X$^-$ case, if the attractive charge\nfurther increases it starts dissociating the trion. \nConsequently, SU peaks are quenched again.\nNotice however energy splittings for X$^+$ (Fig.~\\ref{figSXXX3}) \nare smaller than for X$^-$ (Fig.~1d in the main text).\nThis is expected from the heavier masses of holes.\n\n\n\\subsection{Effect of charge impurity location}\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{effect_charge_location.eps}\n\\caption{X$^-$ emission spectra for different locations of surface charges. \nThe spectra are normalized with respect to the energy and intensity of the $Q_{edge}$ fundamental transition.\n(a) Edge-located vs. corner-located impurity. Blue and red lines stand for edge and corner, respectively. \n(b) Edge-located vs. edge-top-located vs. corner-top-located. Blue, green and pink lines stand for edge, \ntop-corner and top-edge, respectively. $ne=nh=12$.}\n\\label{figSX}\n\\end{figure}\n\nIn the main text we present the representative cases of a surface charge centered on top of the NPL ($Q_{top}$), \nand centered and that of a charge on the edge of lateral sidewall ($Q_{edge}$). In Figure ~\\ref{figSX} we compare with\ndifferent locations. One can see that the effect of a charge located in the corner, red line in Fig.~\\ref{figSX}a, \nprovides similar SU peaks to that of the edge charge, blue line in the figure, both in energy and intensity.\nWe recall that these traps seem to be particularly likely according to recent studies on ligand desorption.\\cite{LeemansJPCL,DrijversCM}\nOff-centered charges on top and bottom surfaces are studied in Fig.~\\ref{figSX}b. They give rise to SU peaks\nof similar height to that of $Q_{edge}$, although they reach the optimal charge value sooner than $Q_{edge}$\n($Q_{top-edge} \\sim Q_{top-corner} \\approx 0.2$ versus $Q_{edge}=0.4$), \nbecause they lie closer to the center of the NPL, where photogenerated carriers tend to localize.\n\n\n\n\n\n\\subsection{Effect of conduction band offset in CdSe\/CdS NPLs} \\label{red_cbo}\n\n\nThe value of the CdSe\/CdS conduction band offset (CBO) has been a subject of debate in nanocrystal heterostructures.\\cite{AIPZunger,SteinerNL,LlusarJPCc}. We used, along our main text, an upper-bound unstrained value of $0.48$ eV\\cite{AIPZunger}, which is partly reduced by compressive strain in the core.\\cite{LlusarJPCc} Here we explore the scenario where we use a lower-bound\\cite{SteinerNL} value as well, to see the possible effect of enhancing electron delocalization over the CdS shell. Figure \\ref{figSXX} compares the two cases.\nLowering the CBO gives rise to slightly weaker electron-electron repulsion ($V_{ee}$) and electron-hole attraction ($V_{eh}$),\nhowever the differences are very small. One can then expect similar role of SU processes as in the main text.\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{CBO.eps}\n\t\\caption{(a,b) Average Coulomb integrals for $Q_{edge}=0.6$: (a) electron-electron repulsion and (b) electron-hole attraction for every CBO. $ne=nh=22$}\n\\label{figSXX}\n\\end{figure}\n\n\\newpage\n\n\\subsection{Effect of shell thickness in CdSe\/CdS NPLs} \\label{correspondence}\n\n\nAlong the main text, core\/shell NPLs under study had a shell thickness of 6ML on each side of the core. \nThe experiments of Antolinez et al.\\cite{AntolinezNL} however used thicker shells (12 ML). \nIn this section we compare qualitatively the response in the two cases using a moderate basis set ($ne=nh=12$),\nwhich permits addressing the experimental dimensions without the computational burden of the \nlarge basis set (for 12 ML thickness, the extended CI computation is beyond our current resources).\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{fig_correspondence.eps}\n\\caption{(a) Sketch of the NPLs we are comparing: 12ML shell (top) and 6ML (bottom). \nThe charge is located at same coordinates. \n(b,c) Coulomb interactions: (b) repulsions e-e and (c) attractions e-h for $Q=0.5$ and $Q=0.8$. \n(d,e) Normalized emission spectra of 6ML vs 12ML: (d) $Q=0.5$ and (e) $Q=0.8$; \n$Q=0$ is centred at band edge recombination energy for 6ML in both cases. $ne=nh=12$}\n\\label{figSXXX}\n\\end{figure}\n\nIf we focus on the charge location in both systems, Fig.~\\ref{figSXXX}a, one may expect similar behaviour. \nThe main difference, as can be seen in Fig.~\\ref{figSXXX}b (left panel) occurs for repulsive electron-electron interactions,\nwhich are slightly weaker for thick shells. This is a consequence of the larger electron delocalization,\nwhich translates into smaller $|c_2|$ coefficients in the CI expansion (see main text) and hence slightly smaller SU satellite,\nas observed in Fig.~\\ref{figSXXX}c.\n\n\n\\subsection{Effect of inserting multiple impurities in CdSe\/CdS} \\label{impurity_effect}\n\nWe consider here the possibility that two surface traps, instead of one, \nare acting as electrostatic impurities in CdSe\/CdS NPLs.\nSince there is a general preference of forming defects in the heterostructure interfaces \n-- because of lattice mismatch\\cite{LlusarJPCc,LiACS} -- and on lateral facets \n-- where ligand desorption is more likely to happen\\cite{LeemansJPCL}--, we choose the charges\nto be located as shown in Fig.~\\ref{figSXXX1}a.\nThe presence of two charges, combined with the weak in-plane confinement, easily dissociates the trion \nby driving one electron to each surface impurity. This can be seen in the charge densities of Fig.~\\ref{figSXXX1}b. The number of visible SU peaks, however, remains one (see Fig.~\\ref{figSXXX1}c).\nIn the case of strong surface charges ($Q=1.0$), the trion triplet (discussed in the main text) becomes so close in \nenergy to the singlet ground state that it shows up in the spectrum at 4 K, see right panel in Fig.~\\ref{figSXXX1}c.\n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16.5cm]{fig_bicharged.eps}\n\\caption{(a) Schematic of a CdSe\/CdS NPL with 2 charges on edges intersecting interface and sidewall facet. \n$Q_{edge(1)}=Q_{edge(2)}=Q_{edge}$. The NPL shell is 12 ML thick. \n(b) In-plane electrons and hole charge densities for the X$^{-}$ singlet ($S=0$) ground state at $Q=0.5$ and $Q=1.0$; \n(c) Normalized emission spectra at $Q=0.5$ (left) and $Q=1.0$ (right). $ne=nh=12$.}\n\\label{figSXXX1}\n\\end{figure}\n\nIf we further increase the charge $Q$ (e.g. by assuming double point charges on each side of the NPL, see Fig.~\\ref{figSXXX2}a),\nadditional peaks start showing up in the emission spectrum, which is shown in Fig.~\\ref{figSXXX2}g.\nThe sketches in Fig.~\\ref{figSXXX2}e-f assign each peak to a corresponding recombination process.\nTwo transitions come from the X$^-$ singlet ground state, namely its band edge ($s$-$R1$) and first SU ($s$-$SU$) recombinations.\nThe other transitions are fully radiative recombinations arising from the triplet state, $t$-$R1$ and $t$-$R2$.\nThe picture is analogous to that proposed in the Discussion section of the main text to explain the multi-peaked emission of\nRef.~\\cite{AntolinezNL}, but in this case the triplet state is thermally occupied at 4 K, so there is no need to assume\nslow spin relaxation. The top panel in Fig~\\ref{figSXXX2}g qualitatively resembles the clusters of four peaks often\nobserved by Antolinez and co-workers in their photoluminescence measurements\\cite{AntolinezNL}, \nalthough the inter-peak energy splittings here are one order \nof magnitude smaller. As mentioned in the main text, assuming the triplet state is metastable even if it is beyond $kT$\nfrom the singlet ground state, and varying trapped charge location, it may be possible to retrieve the experimental spectra. \n\n\n\\begin{figure}\n\t\\centering\n\t\t\\includegraphics[width=16cm]{fig_tetracharged.eps}\n\\caption{(a) Schematic of a CdSe\/CdS NPL with 2 double charges on edges intersecting interface and sidewall facet. \n(b,c) Coulomb interactions: (b) electron-electron repulsion and (c) electron-hole attraction for configurations $\\ket{1}$ and $\\ket{2}$. \n(d-f) Recombination processes involved in each transition. \n(g) Normalized emission spectrum. The energy origin is set at position of the brightest peak, $t$-$R2$. \n$Q_{edge}$ is the net charge on each edge (times the fundamental electron charge).  $ne=nh=12$. }\n\n\\label{figSXXX2}\n\\end{figure}\n\n\\newpage\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe neutrino mixing information is encapsulated in the unitary PMNS mixing matrix which, in the standard PDG parameterisation~\\cite{PDG}, is given by\n\\begin{equation}\nU_\\text{PMNS}=\\left(\\begin{matrix}\nc_{12} c_{13} &s_{12} c_{13} &s_{13}e^{-i\\delta} \\\\\n-s_{12} c_{23}-c_{12} s_{23} s_{13} e^{i\\delta} &c_{12}c_{23} -s_{12}s_{23} s_{13} e^{i\\delta} &s_{23} c_{13} \\\\\ns_{12} s_{23}-c_{12} c_{23} s_{13}e^{i\\delta} &-c_{12}s_{23}-s_{12}c_{23}s_{13}e^{i\\delta} &c_{23}c_{13}\n\\end{matrix}\\right)\\left(\\begin{matrix}\n1 &0 &0 \\\\\n0 &e^{i\\frac{\\alpha_{21}}{2}} &0 \\\\\n0 &0 &e^{i\\frac{\\alpha_{31}}{2}}\n\\end{matrix}\\right)\n\\label{eq:pmns}\n\\end{equation}\nwhere $s_{ij}=\\sin \\theta_{ij}, c_{ij}=\\cos \\theta_{ij}$. The three mixing angles $\\theta_{12}$ (solar angle), $\\theta_{23}$ (atmospheric angle) and $\\theta_{13}$ (reactor angle) along with the $CP$-violating complex phases (the Dirac phase, $\\delta$, and the two Majorana phases, $\\alpha_{21}$ and $\\alpha_{31}$) parameterise $U_{PMNS}$. In comparison to the small mixing angles observed in the quark sector, the neutrino mixing angles are found to be relatively large~\\cite{NeutrinoGlobalFit}:\n\\begin{align}\n\\sin^2 \\theta_{12} &= 0.313_{-0.012}^{+0.013}\\,,\\label{eq:anglevalues1}\\\\\n\\sin^2 \\theta_{23} &= 0.444_{-0.031}^{+0.036}\\,\\, \\text{and} \\,\\, 0.600_{-0.026}^{+0.019}\\,,\\label{eq:anglevalues2}\\\\\n\\sin^2 \\theta_{13} &= 0.0244_{-0.0019}^{+0.0020}\\,. \\label{eq:anglevalues3}\n\\end{align}\nThe values of the complex phases are unknown at present. Besides measuring the mixing angles, the neutrino oscillation experiments also proved that neutrinos are massive particles. These experiments measure the mass-squared-differences of the neutrinos and currently their values are known at about $3\\%$ precision~\\cite{NeutrinoGlobalFit},\n\\begin{gather}\n\\Delta m_{21}^2=75.0_{-1.7}^{+1.9}~\\text{meV}^2,\\label{eq:massvalues1}\\\\\n|\\Delta m_{31}^2|=2429_{-54}^{+55}~\\text{meV}^2.\\label{eq:massvalues2}\n\\end{gather}\n\nSeveral mixing ansatze with a trimaximally mixed second column for $U_\\text{PMNS}$, i.e.~$|U_{e2}|=|U_{\\mu2}|=|U_{\\tau2}|=\\frac{1}{\\sqrt{3}}$, were proposed during the early 2000s~\\cite{TM, TBM, TXPM, Xing, Demo}. Here we briefly revisit two of those, the tri-chi-maximal mixing ($\\txm$) and the tri-phi-maximal mixing ($\\tpm$),\\footnote{$TM_i$ ($TM^i$) has been proposed~\\cite{TM2a, TM2b} as a nomenclature to denote the mixing matrices that preserve various rows (columns) of the tribimaximal mixing~\\cite{TBM}. Under this notation, both $\\txm$ and $\\tpm$ fall under the category of $TM_2$. To be more specific, $TM_2$ which breaks $CP$ maximally is $\\txm$ and $TM_2$ which conserves $CP$ is $\\tpm$.} which are relevant to our model. They can be conveniently parameterised~\\cite{TXPM} as follows   \n\\begin{align}\nU_{\\txm}&=\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right),\\label{eq:txmform}\\\\\nU_{\\tpm}&=\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right)\\label{eq:tpmform}.\n\\end{align}\nBoth $\\txm$ and $\\tpm$ have one free parameter each ($\\chi$ and $\\phi$) which directly corresponds to the reactor mixing angle, $\\theta_{13}$, through the $U_{e3}$ elements of the mixing matrices. The three mixing angles and the Dirac $CP$ phase obtained by relating Eq.~(\\ref{eq:pmns}) with Eqs.~(\\ref{eq:txmform},~\\ref{eq:tpmform}) are shown in Table~\\ref{tab:anglesandphase}.\n{\\renewcommand{\\arraystretch}{1.6}\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{||c||c|c|c|c||}\n\\hline\n\\hline\n\t&$\\sin^2 \\theta_{13}$\t&$\\sin^2 \\theta_{12}$\t&$\\sin^2 \\theta_{23}$\t&$\\delta$\t\\\\\n\\hline\n\\hline\n$\\txm$\t&$\\frac{2}{3} \\sin^2 \\chi$\t&$\\frac{1}{\\left(3-2\\sin^2 \\chi\\right)}$\t&$\\frac{1}{2}$\t&$\\pm\\frac{\\pi}{2}$\t\\\\\n\\hline\n$\\tpm$\t&$\\frac{2}{3} \\sin^2 \\phi$\t&$\\frac{1}{\\left(3-2\\sin^2 \\phi\\right)}$\t&$\\frac{2 \\sin^2 \\left(\\frac{2\\pi}{3}+\\phi\\right)}{\\left(3-2\\sin^2 \\phi\\right)}$\t&$0,~\\pi$\t\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The standard PDG observables $\\theta_{13}$, $\\theta_{12}$, $\\theta_{23}$ and $\\delta$ in terms of the parameters $\\chi$ and $\\phi$. Note that the range of $\\chi$ as well as $\\phi$ is $-\\frac{\\pi}{2}$ to $+\\frac{\\pi}{2}$. In $\\txm$ ($\\tpm$), the parameter $\\chi$ ($\\phi$) being in the first and the fourth quadrant correspond to $\\delta$ equal to $+\\frac{\\pi}{2}$ ($0$) and $-\\frac{\\pi}{2}$ ($\\pi$) respectively.}\n\\label{tab:anglesandphase}\n\\end{table}}\n\\noindent In $\\txm$, since $\\delta = \\pm \\frac{\\pi}{2}$, $CP$ violation is maximal for a given set of mixing angles. The Jarlskog $CP$ violating invariant~\\cite{JCP1, JCP2, JCP3, JCP4, JCP5} in the context of $\\txm$~\\cite{TXPM} is given by \n\\begin{equation}\\label{eq:jcp}\nJ=\\frac{\\sin 2\\chi}{6\\sqrt{3}}.\n\\end{equation} \nOn the other hand, $\\tpm$ is $CP$ conserving, i.e.~$\\delta = 0,~\\pi$, and thus $J=0$. Since the reactor angle was discovered to be non-zero in 2012~\\cite{DayaBay}, there has been a resurgence of interest~\\cite{TM21, TM22, TM23, TM24, TM25, S4Paper, LIS, SteveFour, Delta16, Thomas} in $\\txm$ and $\\tpm$ and their equivalent forms\\footnote{Any $CP$-conserving ($\\delta = 0,~\\pi$) mixing matrix with non-zero $\\theta_{13}$ and trimaximally mixed $\\nu_2$ column is equivalent to $\\tpm$. Observationally they differ only with respect to the Majorana phases. Similarly any mixing matrix with $\\delta = \\pm \\frac{\\pi}{2}$, $\\theta_{13}\\neq0$ and trimaximal $\\nu_2$ column is equivalent to $\\txm$.}.  \n\nRecently~\\cite{LIS} it was shown that $\\txm_{(\\chi=\\pm \\frac{\\pi}{16})}$ as well as $\\tpm_{(\\phi=\\pm \\frac{\\pi}{16})}$ results in a reactor mixing angle, \n\\begin{equation}\n\\begin{split}\n\\sin^2 \\theta_{13} &= \\frac{2}{3} \\sin^2 \\frac{\\pi}{16} \\\\\n&= 0.025,\n\\end{split}\n\\end{equation} \nconsistent with the experimental data. The model was constructed in the Type-1 see-saw framework. Four cases of Majorana mass matrices were discussed:\n\\begin{align}\nM_\\text{Maj} & \\propto \\left(\\begin{matrix}2-\\sqrt{2} & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 0\n\\end{matrix}\\right), & M_\\text{Maj} & \\propto \\left(\\begin{matrix}0 & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 2-\\sqrt{2}\n\\end{matrix}\\right),\\label{eq:txmmat}\\\\\nM_\\text{Maj} & \\propto \\left(\\begin{matrix}i+\\frac{1-i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & -i+\\frac{1+i}{\\sqrt{2}}\n\\end{matrix}\\right), & M_\\text{Maj} & \\propto \\left(\\begin{matrix}-i+\\frac{1+i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & i+\\frac{1-i}{\\sqrt{2}}\n\\end{matrix}\\right)\\label{eq:tpmmat}\n\\end{align}\nwhere $M_\\text{Maj}$ is the coupling among the right-handed neutrino fields, i.e.~$\\overline{(\\nu_R)^c}M_\\text{Maj} \\nu_R$. In Ref.~\\cite{LIS}, the mixing matrix was modelled in the form\n\\begin{equation}\nU_\\text{PMNS} = \\mathcal{T} U_\\nu\n\\end{equation}\nwhere the $3\\times3$ trimaximal contribution,\n\\begin{equation}\n\\mathcal{T}=\\frac{1}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & 1\\\\\n1 & \\omega & \\bar{\\omega}\\\\\n1 & \\bar{\\omega} & \\omega\n\\end{matrix}\\right) \\quad \\text{with} \\quad \\omega=e^{i\\frac{2\\pi}{3}}, \\,\\,\\, \\bar{\\omega}=e^{\\text{-}i\\frac{2\\pi}{3}},\n\\end{equation}\ncame from the charged-lepton sector. $U_\\nu$, on the other hand, was the contribution from the neutrino sector. The four $U_\\nu$s vis-a-vis the four Majorana neutrino mass matrices given in Eqs.~(\\ref{eq:txmmat}) and Eqs.~(\\ref{eq:tpmmat}), gave rise to $\\txm_{(\\chi=\\pm \\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm \\frac{\\pi}{16})}$ respectively. All the four mass matrices, Eqs.~(\\ref{eq:txmmat},~\\ref{eq:tpmmat}), have the eigenvalues $\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$, $1$ and $\\frac{-1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$. Due to the see-saw mechanism, the neutrino masses become inversely proportional to the eigenvalues of the Majorana mass matrices. As a result we obtained the mass relation\n\\begin{equation}\\label{eq:numass}\nm_1:m_2:m_3=\\frac{\\left(2+\\sqrt{2}\\right)}{1+\\sqrt{2(2+\\sqrt{2})}}:1:\\frac{\\left(2+\\sqrt{2}\\right)}{-1+\\sqrt{2(2+\\sqrt{2})}}\\,\\,.\n\\end{equation}\nUsing this mass relation and given the experimentally measured mass-squared differences, we also predicted the light neutrino mass to be around $25~\\text{meV}$.\n\nIn this paper we use the discrete group $\\Sigma(72\\times3)$ to construct a flavon model that essentially reproduces the above results. Unlike the original paper~\\cite{LIS} where the neutrino mass matrix was decomposed into a symmetric bi-product, here a single representation of the flavour group is used to build the symmetric mass matrix. A brief discussion of the group $\\Sigma(72\\times3)$ and its representations is provided in Section~2. Appendix~A contains further details such as the tensor product expansions of its various irreducible representations (irreps) and the corresponding Clebsch-Gordan (C-G) coefficients. In Section~3, we describe the model with its fermion and flavon field content in relation to these irreps. The flavons are assigned specific Vacuum Expectation Values (VEVs) to obtain the required mass matrices. How we may construct suitable flavon potentials to generate the given set of VEVs is demonstrated in Appendix~B. In Section~4, we obtain the phenomenological predictions and compare them with the current experimental data along with the possibility of further validation from future experiments. Finally the results are summarised in Section~5.\n\n\\section{The Group $\\Sigma(72\\times3)$ and its Representations}\n\nDiscrete groups have been used extensively in the description of flavour symmetries. Historically, the study of discrete groups can be traced back to the study of symmetries of geometrical objects. Tetrahedran, cube, octahedran, dodecahedran and icosahedran, which are the famous Platonic solids, were known to the ancient Greeks. These objects are the only regular polyhedra with congruent regular polygonal faces. Interestingly, the symmetry groups of the platonic solids are the most studied in the context of flavour symmetries too - $A_4$ (tetrahedron), $S_4$ (cube and its dual octahedron) and $A_5$ (dodecahedron and its dual icosahedron). These polyhedra live in the three-dimensional Euclidean space. In the context of flavour physics, it might be rewarding to study similar polyhedra that live in three-dimensional complex Hilbert space. In fact, five such complex polyhedra that correspond to the five Platonic solids exist as shown by Coxeter~\\cite{Coxpoly}. They are $3\\{3\\}3\\{3\\}3$, $2\\{3\\}2\\{4\\}p$, $p\\{4\\}2\\{3\\}2$, $2\\{4\\}3\\{3\\}3$, $3\\{3\\}3\\{4\\}2$ where we have used the generalised schlafli symbols~\\cite{Coxpoly} to represent the polyhedra. The polyhedron $3\\{3\\}3\\{3\\}3$ known as the Hessian polydehron can be thought of as the tetrahedron in the complex space. Its full symmetry group has 648 elements and is called $\\Sigma(216\\times3)$. Like the other discrete groups relevant in flavour symmetry, $\\Sigma(216\\times3)$ is also a subgroup of the continuous group $SU(3)$. \n\nThe principal series of $\\Sigma(216\\times3)$~\\cite{Sigma1} is given by\n\\begin{equation}\n\\{e\\} \\triangleleft Z_3 \\triangleleft \\Delta(27) \\triangleleft \\Delta(54) \\triangleleft \\Sigma(72\\times3) \\triangleleft \\Sigma(216\\times3).\n\\end{equation}\nOur flavour symmetry group, $\\Sigma(72\\times3)$, is the maximal normal subgroup of $\\Sigma(216\\times3)$. So we get $\\Sigma(216\\times3)\/\\Sigma(72\\times3)=Z_3$. Various details about the properties of the group $\\Sigma(72\\times3)$ and its representations can be found in Refs.~\\cite{Sigma1, Sigma2, Smallgroup, SigmaHagedorn, Merle}. Note that $\\Sigma(72\\times3)$ is quite distinct from $\\Sigma(216)$ which is defined using the relation $\\Sigma(216\\times3)\/Z_3=\\Sigma(216)$. In other words, $\\Sigma(216\\times3)$ forms the triple cover of $\\Sigma(216)$. $\\Sigma(216\\times3)$ as well as $\\Sigma(216)$ is sometimes referred to as the Hessian group. In terms of the GAP~\\cite{GAP4} nomenclature, we have $\\Sigma(216\\times3)\\equiv \\text{SmallGroup(648,532)},\\,$ $\\Sigma(72\\times3)\\equiv \\text{SmallGroup(216,88)}\\,$ and $\\,\\Sigma(216)\\equiv \\text{SmallGroup(216,153)}$.\n\nWe find that, in the context of flavour physics and model building, $\\Sigma(72\\times3)$ has an appealing feature: it is the smallest group containing a complex three-dimensional representation whose tensor product with itself results in a complex six-dimensional representation, i.e.\n\\begin{equation}\\label{eq:tensor0}\n\\boldsymbol{3}\\otimes\\boldsymbol{3}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}.\n\\end{equation}\nWith a suitably chosen basis for $\\boldsymbol{6}$ we get\n\\begin{equation}\\label{eq:tensor1exp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{\\sqrt{3}}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{6}} a_1 b_1 - \\sqrt{\\frac{2}{3}} a_2 b_2 + \\frac{1}{\\sqrt{6}} a_3 b_3\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_1 - a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 + a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 + a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 + a_2 b_1\\right)\n\\end{matrix}\\right), \\quad \\quad \\boldsymbol{\\bar{3}} \\equiv\n\\left(\\begin{matrix}\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nwhere $(a_1, a_2, a_3)^T$ and $(b_1, b_2, b_3)^T$ represent the first triplet and the second triplet respectively appearing in the LHS of Eq.~(\\ref{eq:tensor0}). All the symmetric components of the tensor product together form the representation $\\boldsymbol{6}$ and the antisymmetric components form $\\boldsymbol{\\tb}$. For the $SU(3)$ group it is well known that the tensor product of two $\\boldsymbol{3}$s gives rise to a symmetric $\\boldsymbol{6}$ and an antisymmetric $\\boldsymbol{\\tb}$. $\\Sigma(72\\times3)$ being a subgroup of $SU(3)$, of course, has its $\\boldsymbol{6}$ and $\\boldsymbol{\\tb}$ embedded in the $\\boldsymbol{6}$ and $\\boldsymbol{\\tb}$ of $SU(3)$.  \n\nConsider the complex conjugation of Eq.~(\\ref{eq:tensor0}), i.e.~$\\boldsymbol{\\tb}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}$. Let the right-handed neutrinos form a triplet, $\\nu_R=(\\nu_{R1},\\nu_{R2},\\nu_{R3})^T$, which transforms as a $\\boldsymbol{\\tb}$. A symmetric (and also Lorentz invariant) combination of two such triplets leads to a sextet, $X_\\nu$, which transforms as a $\\boldsymbol{\\xb}$,\n\\begin{equation}\\label{eq:Xnu}\nX_\\nu = \\left(\\begin{matrix}\\frac{1}{\\sqrt{3}}\\left(\\nu_{R1}.\\nu_{R1} + \\nu_{R2}.\\nu_{R2} + \\nu_{R3}.\\nu_{R3}\\right)\\\\\n\t\\frac{1}{\\sqrt{6}} \\nu_{R1}.\\nu_{R1} - \\sqrt{\\frac{2}{3}} \\nu_{R2}.\\nu_{R2} + \\frac{1}{\\sqrt{6}} \\nu_{R3}.\\nu_{R3}\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R1}.\\nu_{R1} - \\nu_{R3}.\\nu_{R3}\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R2}.\\nu_{R3} + \\nu_{R3}.\\nu_{R2}\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(\\nu_{R3}.\\nu_{R1} + \\nu_{R1}.\\nu_{R3}\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(\\nu_{R1}.\\nu_{R2} + \\nu_{R2}.\\nu_{R1}\\right)\n\\end{matrix}\\right)\\equiv \\boldsymbol{\\xb}\n\\end{equation}\nwhere $\\nu_i.\\nu_j$ is the Lorentz invariant product of the right-handed neutrino Weyl spinors. We may couple $X_\\nu$ to a flavon field $\\phi=(\\phi_1,\\phi_2,\\phi_3,\\phi_4,\\phi_5,\\phi_6)^T$ which transforms as a $\\boldsymbol{6}$ to construct the invariant term\n\\begin{equation}\\label{eq:Tnu}\nX_\\nu^T \\phi = \\left(\\begin{matrix}\\nu_{R1}\\\\\n\t\\nu_{R2}\\\\\n\\nu_{R3}\n\\end{matrix}\\right)^T \\left(\\begin{matrix}\\frac{\\phi_1}{\\sqrt{3}}+\\frac{\\phi_2}{\\sqrt{6}}+\\frac{\\phi_3}{\\sqrt{2}} & \\frac{\\phi_6}{\\sqrt{2}} & \\frac{\\phi_5}{\\sqrt{2}}\\\\\n       \\frac{\\phi_6}{\\sqrt{2}} & \\frac{\\phi_1}{\\sqrt{3}}-\\frac{\\sqrt{2}\\phi_2}{\\sqrt{3}} & \\frac{\\phi_4}{\\sqrt{2}}\\\\\n       \\frac{\\phi_5}{\\sqrt{2}} & \\frac{\\phi_4}{\\sqrt{2}} & \\frac{\\phi_1}{\\sqrt{3}}+\\frac{\\phi_2}{\\sqrt{6}}-\\frac{\\phi_3}{\\sqrt{2}}\n\\end{matrix}\\right)\\left(\\begin{matrix}\\nu_{R1}\\\\\n\t\\nu_{R2}\\\\\n\\nu_{R3}\n\\end{matrix}\\right).\n\\end{equation}\nIn general, the $3\\times3$ Majorana mass matrix is symmetric and has six complex degrees of freedom. Therefore, using Eq.~(\\ref{eq:Tnu}), any required mass matrix can be obtained through a suitably chosen Vacuum Expectation Value (VEV) for the flavon field. Constructing the symmetric Majorana neutrino mass matrix with the help of flavon sextets has been attempted before, eg. scalar fields transforming as the antisextets of $SU(3)_L$ are used in Refs.~\\cite{Long1, Long2}.\n\nTo describe the representation theory of $\\Sigma(72\\times3)$ we largely follow Ref.~\\cite{Sigma1}. $\\Sigma(72\\times3)$ can be constructed using four generators, namely $C$, $E$, $V$ and $X$~\\cite{Sigma1}. For the three-dimensional representation, we have\n\\begin{equation}\\label{eq:gen3}\nC \\equiv\n\\left(\\begin{matrix}1 & 0 & 0\\\\\n       0 & \\om & 0\\\\\n       0 & 0 & \\ob\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}0 & 1 & 0\\\\\n       0 & 0 & 1\\\\\n       1 & 0 & 0\n\\end{matrix}\\right), \\quad V\\equiv\n-\\frac{i}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & 1\\\\\n       1 & \\om & \\ob\\\\\n       1 & \\ob & \\om\n\\end{matrix}\\right), \\quad X\\equiv\n-\\frac{i}{\\sqrt{3}}\\left(\\begin{matrix}1 & 1 & \\ob\\\\\n       1 & \\om & \\om\\\\       \n       \\om & 1 & \\om\n       \\end{matrix}\\right).\n\\end{equation}\nThe characters of the representations of $\\Sigma(72\\times3)$ are given in Table~\\ref{tab:charactertable}. From the character table it is easy to infer that the one-dimensional representations $\\boldsymbol{1^p}$, $\\boldsymbol{1^q}$ and $\\boldsymbol{1^r}$ involve a multiplication with $\\pm1$ only. For these representations, the generators $C$, $E$, $V$ and $X$ are given by\n\\begin{align}\n\\boldsymbol{1^p}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv -1, \\quad X \\equiv 1 \\label{eq:gen1p},\\\\ \n\\boldsymbol{1^q}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv 1, \\quad X \\equiv -1 \\label{eq:gen1q},\\\\ \n\\boldsymbol{1^r}:\\quad \\quad &C \\equiv 1, \\quad E \\equiv 1, \\quad V \\equiv -1, \\quad X \\equiv -1 \\label{eq:gen1r}.\n\\end{align}\nThe representations $\\boldsymbol{1^p}$, $\\boldsymbol{1^q}$ and $\\boldsymbol{1^r}$ along with the representation $\\boldsymbol{3}$ can be used to construct $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$:\n\\begin{equation}\\label{eq:3times1}\n\\boldsymbol{3^p} = \\boldsymbol{1^p} \\otimes \\boldsymbol{3}, \\quad \\boldsymbol{3^q} = \\boldsymbol{1^q} \\otimes \\boldsymbol{3}, \\quad \\boldsymbol{3^r} = \\boldsymbol{1^r} \\otimes \\boldsymbol{3}.\n\\end{equation}\nFor $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$, we use the basis defined using the generator matrices given in Eqs.~(\\ref{eq:gen3}) multiplied with $\\pm1$ in accordance with Eqs.~(\\ref{eq:gen1p},~\\ref{eq:gen1q},~\\ref{eq:gen1r},~\\ref{eq:3times1}). Tensor product expansions of various representations relevant to our model along with the $SU(3)$ embeddings (branching rules) are given in the Appendix~A. We have also provided the C-G coefficients and the generator matrices in the bases corresponding to those coefficients.\n\n\n\\begin{table}[H]\n\\begin{center}\n\\scriptsize\n\\begin{tabular}{||c||c c c|c|c c c|c c c|c c c|c c c||}\n\\hline\n\\hline\n$\\Sigma(72\\times3)$&$C_1$&$C_2$&$C_3$&$C_4$&$C_5$&$C_6$&$C_7$&$C_8$&$C_9$&$C_{10}$&$C_{11}$&$C_{12}$&$C_{13}$&$C_{14}$&$C_{15}$&$C_{16}$\\\\\n$\\#C_k$&$1$&$1$&$1$&$24$&$9$&$9$&$9$&$18$&$18$&$18$&$18$&$18$&$18$&$18$&$18$&$18$\\\\\n$ord(C_k)$&$1$&$3$&$3$&$3$&$2$&$6$&$6$&$4$&$12$&$12$&$4$&$12$&$12$&$4$&$12$&$12$\\\\\n\\hline\n\\hline\n$\\boldsymbol{1}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$1$&$1$&$1$\\\\\n\n\\hline\n$\\boldsymbol{1^p}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\\\\\n$\\boldsymbol{1^q}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$-1$&$-1$&$-1$\\\\\n$\\boldsymbol{1^r}$&$1$&$1$&$1$\t\t&$1$&$1$&$1$&$1$\t\t&$-1$&$-1$&$-1$\t\t&$-1$&$-1$&$-1$\t\t&$1$&$1$&$1$\\\\\n\\hline\n$\\boldsymbol{2}$&$2$&$2$&$2$\t\t&$2$&$-2$&$-2$&$-2$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n$\\boldsymbol{3}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$1$&$\\om$&$\\ob$\t&$1$&$\\om$&$\\ob$\t&$1$&$\\om$&$\\ob$\\\\\n\\hline\n$\\boldsymbol{3^p}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$-1$&$-\\om$&$-\\ob$\t&$1$&$\\om$&$\\ob$\t&$-1$&$-\\om$&$-\\ob$\\\\\n$\\boldsymbol{3^q}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$1$&$\\om$&$\\ob$\t&$-1$&$-\\om$&$-\\ob$\t&$-1$&$-\\om$&$-\\ob$\\\\\n$\\boldsymbol{3^r}$&$3$&$3\\om$&$3\\ob$\t&$0$&$-1$&$-\\om$&$-\\ob$\t\t&$-1$&$-\\om$&$-\\ob$\t&$-1$&$-\\om$&$-\\ob$\t&$1$&$\\om$&$\\ob$\\\\\n\\hline\n$\\boldsymbol{\\tb}$&$3$&$3\\ob$&$3\\om$\t&$0$&$-1$&$-\\ob$&$-\\om$\t\t&$1$&$\\ob$&$\\om$\t&$1$&$\\ob$&$\\om$\t&$1$&$\\ob$&$\\om$\\\\\n\\hline\n$\\boldsymbol{\\ab}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\t&$-1$&$-\\ob$&$-\\om$\\\\\n$\\boldsymbol{\\bb}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\\\\\n$\\boldsymbol{\\cb}$&$3$&$3\\ob$&$3\\om$&$0$&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$-1$&$-\\ob$&$-\\om$\t&$1$&$\\ob$&$\\om$\\\\\n\\hline\n$\\boldsymbol{6}$&$6$&$6\\ob$&$6\\om$&$0$&$2$&$2\\ob$&$2\\om$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n$\\boldsymbol{\\xb}$&$6$&$6\\om$&$6\\ob$\t&$0$&$2$&$2\\om$&$2\\ob$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n$\\boldsymbol{8}$&$8$&$8$&$8$\t\t&$-1$&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\t\t&$0$&$0$&$0$\\\\\n\\hline\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Character table of $\\Sigma(72\\times3)$.}\n\\label{tab:charactertable}\n\\end{table} \n\n\\section{The Model}\n\\addtocontents{toc}{\\protect\\setcounter{tocdepth}{1}}\n\nIn this paper we construct our model in the Standard Model framework with the addition of heavy right-handed neutrinos. Through the type~I see-saw mechanism, light Majorana neutrinos are produced. The fermion and flavon content of the model with the representations to which they belong is given in Table~\\ref{tab:flavourcontent}. The Standard Model Higgs field is assigned to the trivial (singlet) representation of $\\Sigma(72\\times3)$.\n\n\\begin{table}[H]\n\\begin{center}\n\\begin{tabular}{|c|c c c c c c c c c|}\n\\hline\n\t&$e_R$\t&$\\mu_R$&$\\tau_R$&$L$\t&$\\nu_R$&$\\phi_e$&$\\phi_\\mu$&$\\phi_\\tau$&$\\phi$\\\\\n\\hline\n\n$\\Sigma(72\\times3)$\t&$\\boldsymbol{1^p}$&$\\boldsymbol{1^q}$&$\\boldsymbol{1^r}$&$\\boldsymbol{\\tb}$&$\\boldsymbol{\\tb}$&$\\boldsymbol{\\ab}$&$\\boldsymbol{\\bb}$&$\\boldsymbol{\\cb}$&$\\boldsymbol{6}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The flavour structure of the model. The three families of the left-handed-weak-isospin lepton doublets form the triplet $L$ and the three right-handed heavy neutrinos form the triplet $\\nu_R$. The flavons $\\phi_e$, $\\phi_\\mu$, $\\phi_\\tau$ and $\\phi$, are scalar fields and are gauge invariants. On the other hand, they transform non-trivially under the flavour group.}\n\\label{tab:flavourcontent}\n\\end{table}\n\nFor the charged leptons, we obtain the mass term\n\\begin{equation}\\label{eq:mclept}\n\\left(y_e L^\\dagger e_R \\frac{\\phi_e}{\\Lambda}+y_\\mu L^\\dagger \\mu_R \\frac{\\phi_\\mu}{\\Lambda}+y_\\tau L^\\dagger \\tau_R \\frac{\\phi_\\tau}{\\Lambda}\\right)H+H.C.\n\\end{equation}\nwhere $H$ is the Standard Model Higgs, $\\Lambda$ is the cut-off scale and $y_i$ are the coupling constants. The VEV of the Higgs, $(0, h_o)$, breaks the weak gauge symmetry. For the flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$, we assign the vacuum alignments\\footnote{Refer to Appendix~B for the details of the flavon potential that leads to these VEVs.} \n\\begin{equation}\\label{eq:leptvev}\n\\langle\\phi_e\\rangle= \\frac{i}{\\sqrt{3}}(1,1,1),\\quad\\langle\\phi_\\mu\\rangle= \\frac{i}{\\sqrt{3}}(1,\\ob,\\om),\\quad \\langle\\phi_\\tau\\rangle= \\frac{i}{\\sqrt{3}}(1,\\om,\\ob).\n\\end{equation}\nAs a result of these vacuum alignments we get the following charged-lepton mass term\n\\begin{equation}\\label{eq:leptcontrib}\n\\left(\\begin{matrix} e_L\\\\\n\\mu_L\\\\\n\\tau_L\n\\end{matrix}\\right)^\\dagger V^\\dagger \\left(\\begin{matrix} m_e & 0 & 0\\\\\n0 & m_\\mu & 0\\\\\n0 & 0 & m_\\tau\n\\end{matrix}\\right)\\left(\\begin{matrix} e_R\\\\\n\\mu_R\\\\\n\\tau_R\n\\end{matrix}\\right)\n\\end{equation}\nwhere $m_e=\\frac{y_e h_o}{\\Lambda}$ etc. $V$ is the $3\\times3$ trimaximal matrix and is one of the generators of $\\Sigma(72\\times3)$ as given in Eqs.~(\\ref{eq:gen3}).\n\nNow, we write the Dirac mass term for the neutrinos:\n\\begin{equation}\\label{eq:mdl2dirac}\n2 y_w L^\\dagger \\nu_R \\tilde{H}+H.C.\n\\end{equation}\nwhere $\\tilde{H}$ is the conjugate Higgs and $y_w$ is the coupling constant. With the help of Eq.~(\\ref{eq:Tnu}), we also write the Majorana mass term for the neutrinos:\n\\begin{equation}\\label{eq:mnurnur}\ny_G X_\\nu^T\\frac{\\phi}{\\Lambda}\n\\end{equation}\nwhere $y_G$ is the coupling which gives rise to the heavy right-handed Majorana masses. Let $\\langle\\phi\\rangle$ be the VEV acquired by the sextet flavon $\\phi$, and let $\\boldsymbol{ \\langle\\phi\\rangle}$ be the corresponding $3\\times 3$ symmetric matrix of the form given in Eq.~(\\ref{eq:Tnu}). Combining the mass terms, Eq.~(\\ref{eq:mdl2dirac}) and Eq.~(\\ref{eq:mnurnur}), and using the VEVs of the Higgs and the flavon, we obtain the Dirac-Majorana mass matrix:\n\\begin{equation}\nM=\\left(\\begin{matrix}0 & y_w h_o I\\\\\n       y_w h_o I & \\,\\,y_G \\frac{1}{\\Lambda}\\boldsymbol{ \\langle\\phi\\rangle}\n\\end{matrix}\\right).\n\\end{equation}\nThe $6\\times6$ mass matrix $M$, forms the coupling\n\\begin{equation}\nM_{ij} \\,\\nu_i.\\nu_j \\quad \\text{with} \\quad  \\nu=\\left(\\begin{matrix}\\nu_{L}^*\\\\\n\t\\nu_{R}\n\\end{matrix}\\right)\n\\end{equation}\nwhere $\\nu_L=(\\nu_e,\\nu_\\mu,\\nu_\\tau)^T$ are the left-handed neutrino flavour eigenstates.\n\nSince $y_w h_o$ is at the weak scale and $y_G$ is at the GUT scale, small neutrino masses are generated through the see-saw mechanism. It can be shown~\\cite{Majorana} that the resulting effective see-saw mass matrix is of the form\n\\begin{equation}\\label{eq:seesaw}\nM_\\text{ss}=-\\left(y_w h_o\\right)^2\\left(y_G \\frac{{\\boldsymbol{ \\langle\\phi\\rangle}}}{\\Lambda}\\right)^{-1}.\n\\end{equation}\nFrom Eq.~(\\ref{eq:seesaw}) it is clear that the see-saw mechanism makes the light neutrino masses inversely proportional to the eigenvalues of the matrix $\\boldsymbol{ \\langle\\phi\\rangle}$. As foreseen in the Introduction, we now construct the four cases of the mass matrices, Eqs.~(\\ref{eq:txmmat},~\\ref{eq:tpmmat}), all of which resulting in the neutrino mass relation, Eq.~(\\ref{eq:numass}). To achieve this we choose suitable vacuum alignments\\footnote{Refer to Appendix~B for the details of the flavon potentials that lead to these VEVs.} for the sextet flavon $\\phi$. \n\n\\subsection{$\\txm_{(\\chi=+\\frac{\\pi}{16})}$} \n\nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtxmp}\n\\langle\\phi\\rangle = \\left(\\frac{3-\\sqrt{2}}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},-1+\\sqrt{2},0,1,0\\right).\n\\end{equation}\nUsing the symmetric matrix form of the sextet given in Eq.~(\\ref{eq:Tnu}), we obtain\n\\begin{equation}\\label{eq:vevsym1}\n\\boldsymbol{ \\langle\\phi\\rangle} =\\left(\\begin{matrix}2-\\sqrt{2} & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 0\n\\end{matrix}\\right).\n\\end{equation}\nDiagonalising the corresponing effective see-saw mass matrix $M_{ss}$, Eq.~(\\ref{eq:seesaw}), we get\n\\begin{equation}\\label{eq:nucontrib}\nU_\\nu^\\dagger M_{ss} U_\\nu^* = \\left(y_w h_o\\right)^2\\frac{\\Lambda}{y_G} \\text{Diag}\\left({\\textstyle\\frac{\\left(2+\\sqrt{2}\\right)}{1+\\sqrt{2(2+\\sqrt{2})}},1,\\frac{\\left(2+\\sqrt{2}\\right)}{-1+\\sqrt{2(2+\\sqrt{2})}}}\\right)\n\\end{equation}\nleading to the neutrino mass relation, Eq.~(\\ref{eq:numass}). The unitary matrix $U_\\nu$ is given by\n\\begin{equation}\\label{eq:unu1}\nU_\\nu = i \\left(\\begin{matrix}\\cos\\left(\\frac{3\\pi}{16}\\right) & 0 & -i\\sin\\left(\\frac{3\\pi}{16}\\right)\\\\\n0 & 1 & 0\\\\\n\\sin\\left(\\frac{3\\pi}{16}\\right) & 0 & i\\cos\\left(\\frac{3\\pi}{16}\\right)\n\\end{matrix}\\right).\n\\end{equation}\nThe product of the contribution from the charged-lepton sector i.e.~$V$ from Eqs.~(\\ref{eq:leptcontrib},~\\ref{eq:gen3}) and the contribution from the neutrino sector i.e.~$U_\\nu$ from Eqs.~(\\ref{eq:nucontrib},~\\ref{eq:unu1}) results in the $\\txm_{(\\chi=+\\frac{\\pi}{16})}$ mixing:\n\\begin{equation}\\label{eq:mix1}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & i\n\\end{matrix}\\right)\n\\end{equation}\nwith $\\chi=+\\frac{\\pi}{16}$.\n\n\\subsection{$\\txm_{(\\chi=-\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtxmm}\n\\langle\\phi\\rangle = \\left(\\frac{3-\\sqrt{2}}{\\sqrt{3}},-\\frac{1}{\\sqrt{3}},1-\\sqrt{2},0,1,0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym2}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}0 & 0 & \\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} & 0 & 2-\\sqrt{2}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\cos\\left(\\frac{5\\pi}{16}\\right) & 0 & i\\sin\\left(\\frac{5\\pi}{16}\\right)\\\\\n0 & 1 & 0\\\\\n\\sin\\left(\\frac{5\\pi}{16}\\right) & 0 & -i\\cos\\left(\\frac{5\\pi}{16}\\right)\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\chi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\chi\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}-i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\chi}{\\sqrt{6}}+i\\frac{\\sin \\chi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -i\\frac{\\cos \\chi}{\\sqrt{2}}-\\frac{\\sin \\chi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & -i\n\\end{matrix}\\right)\\label{eq:mix2}\n\\end{equation}\nwith $\\chi=-\\frac{\\pi}{16}$.\n\n\\subsection{$\\tpm_{(\\phi=+\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtpmp}\n\\langle\\phi\\rangle = \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, -i \\left(1-\\sqrt{2}\\right),0, -1+\\sqrt{2}, 0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym3}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}i+\\frac{1-i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & -i+\\frac{1+i}{\\sqrt{2}}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}} & 0 & -\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}} & 0 & \\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}}\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\\label{eq:mix3}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & i\n\\end{matrix}\\right)\n\\end{equation}\nwith $\\phi=+\\frac{\\pi}{16}$.\n\n\\subsection{$\\tpm_{(\\phi=-\\frac{\\pi}{16})}$} \nHere we assign the vacuum alignment\n\\begin{equation}\\label{eq:vevtpmm}\n\\langle\\phi\\rangle = \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, i (1-\\sqrt{2}),0, -1+\\sqrt{2}, 0\\right)\n\\end{equation}\nresulting in the symmetric matrix\n\\begin{equation}\\label{eq:vevsym4}\n\\boldsymbol{ \\langle\\phi\\rangle}=\\left(\\begin{matrix}-i+\\frac{1+i}{\\sqrt{2}} & 0 & 1-\\frac{1}{\\sqrt{2}}\\\\\n0 & 1 & 0\\\\\n1-\\frac{1}{\\sqrt{2}} & 0 & i+\\frac{1-i}{\\sqrt{2}}\n\\end{matrix}\\right).\n\\end{equation}\nIn this case, the diagonalising matrix is\n\\begin{equation}\nU_\\nu =i \\left(\\begin{matrix}\\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}} & 0 & \\frac{1}{\\sqrt{2}} e^{i\\frac{\\pi}{16}}\\\\\n0 & 1 & 0\\\\\n\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}} & 0 & -\\frac{1}{\\sqrt{2}} e^{-i\\frac{\\pi}{16}}\n\\end{matrix}\\right)\n\\end{equation}\nand the corresponding mixing matrix is\n\\begin{equation}\nU_\\text{PMNS}=V.U_\\nu=\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & \\om & 0\\\\\n0 & 0 & \\ob\n\\end{matrix}\\right).\\left(\\begin{matrix}\\sqrt{\\frac{2}{3}}\\cos \\phi & \\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\sin \\phi\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}-\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & \\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\\\\\n-\\frac{\\cos \\phi}{\\sqrt{6}}+\\frac{\\sin \\phi}{\\sqrt{2}} & \\frac{1}{\\sqrt{3}} & -\\frac{\\cos \\phi}{\\sqrt{2}}-\\frac{\\sin \\phi}{\\sqrt{6}}\n\\end{matrix}\\right).\\left(\\begin{matrix}1 & 0 & 0\\\\\n0 & 1 & 0\\\\\n0 & 0 & -i\n\\end{matrix}\\right)\\label{eq:mix4}\n\\end{equation}\nwith $\\phi=-\\frac{\\pi}{16}$.\n\nAs stated earlier, the four cases, Eqs.~(\\ref{eq:vevsym1},~\\ref{eq:vevsym2},~\\ref{eq:vevsym3},~\\ref{eq:vevsym4}), result in the same neutrino mass relation Eq.~(\\ref{eq:numass}).\n\n\\section{Predicted Observables}\n\nFor comparing our model with the neutrino oscillation experimental data, we use the global analysis done by the NuFIT group and their latest results reproduced in Eqs.~(\\ref{eq:anglevalues1}-\\ref{eq:massvalues2}). The prediction $\\sin^2 \\theta_{13} = \\frac{2}{3} \\sin^2 \\frac{\\pi}{16} = 0.025$,\\footnote{Besides in Ref.~\\cite{LIS}, this value was predicted in the context of $\\Delta(6n^2)$ symmetry group in Ref.~\\cite{Delta16} and later obtained in Ref.~\\cite{Thomas}} is within $1\\sigma$ errors. For the solar angle, using the formula given in Table~\\ref{tab:anglesandphase}, we get \n\\begin{equation}\n\\begin{split}\n\\sin^2 \\theta_{12} &= \\frac{1}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)}\\\\ \n&= 0.342\\,.\n\\end{split}\n\\end{equation}\nThis has a small tension with the experimental value, but it is still within $3\\sigma$ errors. For the atmospheric angle, $\\txm$ predicts maximal mixing:\n\\begin{equation}\n\\sin^2 \\theta_{23} = \\frac{1}{2}\\,.\n\\end{equation}\nThe NuFIT data as well as other global fits~\\cite{Global1,Global1b,Global2} are showing a preference for non-maximal atmospheric mixing. As a result there has been a recent interest in the problem of octant degeneracy of $\\theta_{23}$~\\cite{Octant1,Octant2,Octant3}. $\\tpm$ predicts this non-maximal scenario of atmospheric mixing. $\\tpm_{(\\phi=\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=-\\frac{\\pi}{16})}$ correspond to the first and the second octant solutions respectively. Using the formula for $\\theta_{23}$ given in Table~{tab:anglesandphase}, we get \n\\begin{gather}\n\\tpm_{(\\phi=+\\frac{\\pi}{16})}:\\,\n\\begin{split}\n\\quad \\sin^2 \\theta_{23} &= \\frac{2\\sin^2\\left(\\frac{2\\pi}{3}+\\frac{\\pi}{16}\\right)}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)} \\\\\n&= 0.387\\,,\\\\\n\\end{split}\\\\\n\\tpm_{(\\phi=-\\frac{\\pi}{16})}:\\,\n\\begin{split}\n\\quad \\sin^2 \\theta_{23} &= \\frac{2\\sin^2\\left(\\frac{2\\pi}{3}-\\frac{\\pi}{16}\\right)}{3-2\\sin^2\\left(\\frac{\\pi}{16}\\right)}\\\\\n&= 0.613\\,.\n\\end{split}\n\\end{gather}\nThe dirac $CP$ phase, $\\delta$, has not been measured yet. The discovery that the reactor mixing angle is not very small has raised the possibility of a relatively earlier measurement of $\\delta$~\\cite{MHCP1,MHCP2}. $\\txm$ having $\\delta=\\pm\\frac{\\pi}{2}$ should lead to large observable $CP$ violating effects. Substituting $\\chi=\\pm\\frac{\\pi}{16}$ in Eq.~(\\ref{eq:jcp}), our model gives\n\\begin{equation}\n\\begin{split}\nJ&=\\pm\\frac{\\sin \\frac{\\pi}{8}}{6\\sqrt{3}}\\\\\n&= \\pm0.0368.\n\\end{split}\n\\end{equation} \nwhich is about $40\\%$ of the maximum value of the theoretical range, $-\\frac{1}{6\\sqrt{3}}\\leq J\\leq +\\frac{1}{6\\sqrt{3}}$.\nOn the other hand, $\\tpm$, with $\\delta=0,\\,\\pi$ and $J=0$, is $CP$ conserving. \n\nFrom Figure~\\ref{fig:neutrinopredict2}, it is clear that the neutrino mass relation~Eq.~(\\ref{eq:numass}), is consistent with the measured mass-squared differences, Eqs.~(\\ref{eq:massvalues1},~\\ref{eq:massvalues2}). Here we have assumed the normal mass hierarchy. Using Eq.~(\\ref{eq:numass}) and Eqs.~(\\ref{eq:massvalues1},~\\ref{eq:massvalues2}), we predict the neutrino masses:\\footnote{The best fit values correspond to $\\chi^2_\\text{min}=1.1$ and the error ranges correspond to $\\Delta\\chi^2=1$ where $\\chi^2=\\displaystyle\\sum_{{\\displaystyle x}=\\Delta m^2_{21}, \\Delta m^2_{31}} \\left(\\frac{x_\\text{model}-x_\\text{expt}}{\\sigma_{x\\,\\text{expt}}}\\right)^2$ and $\\Delta\\chi^2=\\chi^2-\\chi^2_\\text{min}$.}\n\\begin{gather}\nm_1=24.77^{+0.20}_{-0.19}~\\text{meV},\\notag\\\\\nm_2=26.22^{+0.21}_{-0.20}~\\text{meV},\\label{eq:m123}\\\\\nm_3=55.49^{+0.45}_{-0.42}~\\text{meV}.\\notag\n\\end{gather}\nNote that the mass relation Eq.~(\\ref{eq:numass}), is incompatible with the inverted mass hierarchy. Considerable experimental studies are being conducted to determine the mass hierarchy~\\cite{MHCP2,MH1,MH2,MH3} and we may expect a solution in the not-too-distant future. Observation of inverted hierarchy will obviously rule out the model.\n\n\\begin{figure}[]\n\\begin{center}\n\\includegraphics[scale=1.0]{massplot.pdf}\n\\caption[$\\Delta m_{31}^2$ vs $\\Delta m_{21}^2$ plane]{$\\Delta m_{31}^2$ vs $\\Delta m_{21}^2$ plane. The straight line shows the neutrino mass relation Eq.~(\\ref{eq:numass}). As a parametric plot, the line can be represented as $\\Delta m_{21}^2=(r_{21}^2-1)m_1^2$ and $\\Delta m_{31}^2=(r_{31}^2-1)m_1^2$ where $r_{21}=\\frac{m_2}{m_1}=\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{\\left(2+\\sqrt{2}\\right)}$ and $r_{31}=\\frac{m_3}{m_1}=\\frac{1+\\sqrt{2(2+\\sqrt{2})}}{-1+\\sqrt{2(2+\\sqrt{2})}}$ are the mass ratios obtained from Eq.~(\\ref{eq:numass}). The parametric values of the light neutrino mass, $m_1$, (denoted by the black dots on the line) are in terms of meV. The red marking indicates the experimental best fit for $\\Delta m_{21}^2$ and $\\Delta m_{31}^2$ along with $1\\sigma$ and $3\\sigma$ errors.}\n\\label{fig:neutrinopredict2}\n\\end{center}\n\\end{figure}\n\nCosmological observations can give us limits on the sum of the neutrino masses. The strongest such limit has been set recently by the data collected using the Planck satellite~\\cite{Planck}:\n\\begin{equation}\n\\sum_i m_i < 230~\\text{meV}.\n\\end{equation}\nOur predictions Eqs.~(\\ref{eq:m123}), give a sum \n\\begin{equation}\n\\sum_i m_i = 106.48^{+0.86}_{-0.81}~\\text{meV}\n\\end{equation}\nwhich is not far below the current cosmological limit. By including the data from future projects such as Polarbear and SKA, we may be able to lower the cosmological limit to around $100~\\text{meV}$~\\cite{FutureCosmo}. Such a result may support or rule out our model.\n\nNeutrinoless double beta decay experiments seek to determine the Majorana nature of the neutrinos. These experiments have so far set limits on the effective electron neutrino mass~\\cite{DBD} $|m_{\\beta\\beta}|$, where\n\\begin{align}\n\\begin{split}\nm_{\\beta\\beta}&=m_1 U_{e1}^2+m_2 U_{e2}^2+m_3 U_{e3}^2\\\\\n&=m_1 |U_{e1}|^2+m_2 |U_{e2}|^2 e^{i\\alpha_{21}}+m_3 |U_{e3}|^2 e^{i\\alpha_{31}}\n\\end{split}\n\\end{align}\nwith $U$ representing $U_\\text{PMNS}$. In all the four mixing scenarios predicted by the model, Eqs.~(\\ref{eq:mix1},~\\ref{eq:mix2},~\\ref{eq:mix3},~\\ref{eq:mix4}), we have $|U_{e1}|=\\sqrt{\\frac{2}{3}}\\cos \\frac{\\pi}{16}$, $|U_{e2}|=\\frac{1}{\\sqrt{3}}$ and $|U_{e3}|=\\sqrt{\\frac{2}{3}}\\sin \\frac{\\pi}{16}$. Also, all of them result in the Majorana phases:\\footnote{Note that both $+i$ and $-i$ appearing in the diagonal phase matrices in Eqs.~(\\ref{eq:mix1},~\\ref{eq:mix2},~\\ref{eq:mix3},~\\ref{eq:mix4}) correspond to $\\alpha_{31}=\\pi$}\n\\begin{equation}\n\\alpha_{21}=0, \\quad \\alpha_{31}=\\pi. \n\\end{equation}\nTherefore the model predicts\n\\begin{equation}\\label{eq:modelbb}\nm_{\\beta\\beta}=\\frac{2}{3}m_1\\cos^2 \\frac{\\pi}{16}+\\frac{1}{3}m_2-\\frac{2}{3}m_3\\sin^2 \\frac{\\pi}{16}.  \n\\end{equation}\nSubstituting the neutrino masses from Eqs.~(\\ref{eq:m123}) in Eq.~(\\ref{eq:modelbb}) we get \n\\begin{equation}\nm_{\\beta\\beta}=23.22^{+0.19}_{-0.18}~\\text{meV}.  \n\\end{equation}\n\nThe most stringent upper bounds on the value of $|m_{\\beta\\beta}|$ have been set by Heidelberg-Moscow, Cuoricino, NEMO3 and GERDA experiments. These bounds are of the order of a few hundreds of meV. In future, experiments such as GERDA-2, CUORE and EXO can improve the bounds on $|m_{\\beta\\beta}|$ to a few tens~\\cite{DBD} of meV and thus may support or rule out our model.\n\n\\section{Summary}\n\nIn this paper we utilise the group $\\Sigma(72\\times3)$ to construct fully-constrained Majorana mass matrices for the neutrinos. These mass matrices reproduce the results obtained in Ref.~\\cite{LIS} i.e.~$\\txm_{(\\chi=\\pm\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm\\frac{\\pi}{16})}$ mixings along with the neutrino mass relation Eq.~(\\ref{eq:numass}). The mixing observables as well as the neutrino mass relation are shown to be consistent with the experimental data. $\\txm_{(\\chi=\\pm\\frac{\\pi}{16})}$ and $\\tpm_{(\\phi=\\pm\\frac{\\pi}{16})}$ predict the Dirac $CP$ violating effect to be maximal (at fixed $\\theta_{13}$) and null respectively. Using our neutrino mass relation in conjunction with the experimentally-observed neutrino mass-squared differences, we calculate the individual neutrino masses. We note that the neutrino mass relation Eq.~(\\ref{eq:numass}) is incompatible with the inverted mass hierarchy. We also predict the effective electron neutrino mass for the neutrinoless double beta decay, $|m_{\\beta\\beta}|$. We briefly discuss the current status and future prospects of experimentally determining the neutrino observables leading to the confirmation or the falsification of our model. In the context of model building, we carry out an in-depth analysis of the representations of $\\Sigma(72\\times3)$ and develop the necessary groundwork to construct the flavon potentials satisfying the $\\Sigma(72\\times3)$ flavour symmetry. In the charged-lepton sector, we use three triplet flavons with a suitably chosen set of VEVs which provide a $3\\times 3$ trimaximal contribution towards the PMNS mixing matrix. In the neutrino sector, we discuss four cases of Majorana mass matrices. The $\\Sigma(72\\times3)$ sextet acts as the most general placeholder for a fully constrained Majorana mass matrix. The intended mass matrices are obtained by assigning appropriate VEVs to the sextet flavon. It should be noted that we need additional symmetries to `explain' any specific texture in the mass matrix. Further research in this direction is a work in progress.\n\nI would like to thank Paul Harrison and Bill Scott for helpful discussions. This work was supported by the UK Science and Technology Facilities Council (STFC). I acknowledge the hospitality of the Centre for Fundamental Physics (CfFP) at the Rutherford Appleton Laboratory and support from the University of Warwick. \n\n\\section{Appendix A: Tensor Product Expansions of Irreps of $\\Sigma(72\\times3)$}\n\n\\begin{flalign}\\label{eq:tensor288}\n&\\text{{\\textit {\\textbf {i}}})\\,\\,\\,}\\boldsymbol{3}\\otimes\\boldsymbol{3}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}&\n\\end{flalign}\nThe generator matrices for the triplet representation were provided in Eq.~(\\ref{eq:gen3}). We define the basis for the sextet representation using Eqs.~(\\ref{eq:tensor1exp}). The resulting generator matrices are \n\\begin{align}\\label{eq:gen6}\n\\begin{split}\n&C \\equiv\n\\left(\\begin{matrix}0 & -\\frac{\\ob}{\\sqrt{2}} & \\frac{i\\ob}{\\sqrt{2}} & 0 & 0 & 0\\\\\n\t-\\frac{\\ob}{\\sqrt{2}} & \\frac{\\ob}{2} & \\frac{i\\ob}{2} & 0 & 0 & 0\\\\\n\t\\frac{i\\ob}{\\sqrt{2}} & \\frac{i\\ob}{2} & -\\frac{\\ob}{2} & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 1 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & \\ob & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & \\om\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & -\\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0\\\\\n\t0 & -\\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 1 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 1\\\\\n\t0 & 0 & 0 & 1 & 0 & 0\n\\end{matrix}\\right),\\\\\n&V\\equiv\n\\left(\\begin{matrix}-\\frac{1}{3} & -\\frac{1}{3\\sqrt{2}} & -\\frac{1}{\\sqrt{6}} & -\\sqrt{\\frac{2}{3}} & 0 & 0\\\\\n\t-\\frac{1}{3\\sqrt{2}} & \\frac{1}{3}+\\frac{\\om}{2} & -\\frac{\\om}{2\\sqrt{3}} & 0 & \\frac{\\ob}{\\sqrt{3}} & \\frac{\\om}{\\sqrt{3}}\\\\\n\t-\\frac{1}{\\sqrt{6}} & -\\frac{\\om}{2\\sqrt{3}} & -\\frac{i\\om}{2\\sqrt{3}} & 0 & -\\frac{i\\ob}{\\sqrt{3}} & \\frac{i\\om}{\\sqrt{3}}\\\\\n\t-\\sqrt{\\frac{2}{3}} & 0 & 0 & \\frac{1}{3} & \\frac{1}{3} & \\frac{1}{3}\\\\\n\t0 & \\frac{\\ob}{\\sqrt{3}} & -\\frac{i\\ob}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\ob}{3} & \\frac{\\om}{3}\\\\\n\t0 & \\frac{\\om}{\\sqrt{3}} & \\frac{i\\om}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\om}{3} & \\frac{\\ob}{3}\n\\end{matrix}\\right),\\\\\n&X\\equiv\n\\left(\\begin{matrix}\\frac{\\om}{3} & \\frac{1}{3\\sqrt{2}} & -\\frac{1}{\\sqrt{6}} & -\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\sqrt{2}}{3}\\\\\n\t\\frac{\\ob}{3\\sqrt{2}} & \\frac{1}{2}+\\frac{\\om}{6} & -\\frac{\\ob}{2\\sqrt{3}} & -\\frac{i}{3} & \\frac{i2}{3} & -\\frac{i\\ob}{3}\\\\\n\t\\frac{\\ob}{\\sqrt{6}} & \\frac{\\ob}{2\\sqrt{3}} & -\\frac{i\\ob}{2\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & -\\frac{i\\ob}{\\sqrt{3}}\\\\\n\t\\frac{i\\ob\\sqrt{2}}{3} & \\frac{i}{3} & -\\frac{i}{\\sqrt{3}} & \\frac{1}{3} & \\frac{1}{3} & \\frac{\\om}{3}\\\\\n\t\\frac{i\\om\\sqrt{2}}{3} & \\frac{i\\ob}{3} & \\frac{i\\ob}{\\sqrt{3}} & \\frac{1}{3} & \\frac{\\ob}{3} & \\frac{\\ob}{3}\\\\\n\t\\frac{i\\om\\sqrt{2}}{3} & -\\frac{i\\ob2}{3} & 0 & \\frac{\\ob}{3} & \\frac{1}{3} & \\frac{\\ob}{3}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\n\n\\newpage\n\\begin{flalign}\\label{eq:tensor3pqr}\n&\\text{{\\textit {\\textbf {ii}}})\\,\\,\\,}\\boldsymbol{3^p}\\otimes\\boldsymbol{3^p}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}, \\quad \\boldsymbol{3^q}\\otimes\\boldsymbol{3^q}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}, \\quad \\boldsymbol{3^r}\\otimes\\boldsymbol{3^r}=\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}&\n\\end{flalign}\nWe have already defined the bases of $\\boldsymbol{3}$, $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$, $\\boldsymbol{3^r}$ and $\\boldsymbol{6}$. Since the representation matrices corresponding to $\\boldsymbol{3}$, $\\boldsymbol{3^p}$, $\\boldsymbol{3^q}$ and $\\boldsymbol{3^r}$ differ only with respect to the multiplication with $\\pm1$, the C-G coefficients for the tensor product expansions in Eqs.~(\\ref{eq:tensor3pqr}) are exactly those given in Eqs.~(\\ref{eq:tensor1exp}). However that is not the case for tensor products involving different types of triplets, \n\\begin{flalign}\\label{eq:tensor3pqr2}\n&\\quad \\,\\,\\,\\boldsymbol{3^p}\\otimes\\boldsymbol{3^q}=\\boldsymbol{6}\\oplus\\boldsymbol{\\cb}, \\quad \\boldsymbol{3^q}\\otimes\\boldsymbol{3^r}=\\boldsymbol{6}\\oplus\\boldsymbol{\\ab}, \\quad \\boldsymbol{3^r}\\otimes\\boldsymbol{3^p}=\\boldsymbol{6}\\oplus\\boldsymbol{\\bb}.&\n\\end{flalign}\nThe C-G coefficients for the sextets appearing in the first, the second and the third tensor product expansions in Eqs.~(\\ref{eq:tensor3pqr2}) are given by\n\\begin{equation}\\label{eq:tensorpqrexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+\\om\\left(a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{\\om}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\om\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{\\om}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{\\om}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\ob \\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right),\n\\end{matrix}\\right),\n\\end{equation}\n\\begin{equation}\\label{eq:tensorqrexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+\\ob\\left(a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{\\ob}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\ob\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{\\ob}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{\\ob}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\om \\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nand\n\\begin{equation}\\label{eq:tensorrpexp}\n\\boldsymbol{6}\\equiv\\left(\\begin{matrix}\\frac{1}{3}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3+a_2 b_3+a_3 b_2+a_3 b_1 + a_1 b_3 +a_1 b_2 + a_2 b_1\\right)\\\\\n\t\\frac{1}{3\\sqrt{2}} a_1 b_1 - \\frac{\\sqrt{2}}{3} a_2 b_2 + \\frac{1}{3\\sqrt{2}} a_3 b_3+\\frac{1}{3\\sqrt{2}} (a_2 b_3+a_3 b_2) - \\frac{\\sqrt{2}}{3} (a_3 b_1 + a_1 b_3) + \\frac{1}{3\\sqrt{2}} (a_1 b_2 +a_2 b_1)\\\\\n\t\\frac{1}{\\sqrt{6}}\\left(a_1 b_1 - a_3 b_3\\right)+\\frac{1}{\\sqrt{6}}\\left(a_2 b_3 + a_3 b_2 - a_1 b_2 - a_2 b_1\\right)\\\\\n\t\\frac{\\sqrt{2}}{\\sqrt{3}}a_1 b_1 - \\frac{1}{\\sqrt{6}}\\left(a_2 b_3+a_3 b_2\\right)\\\\\n\t\\frac{\\sqrt{2}}{\\sqrt{3}}a_2 b_2 - \\frac{1}{\\sqrt{6}}\\left(a_3 b_1+a_1 b_3\\right)\\\\\n\\frac{\\sqrt{2}}{\\sqrt{3}}a_3 b_3 - \\frac{1}{\\sqrt{6}}\\left(a_1 b_2+a_2 b_1\\right)\n\\end{matrix}\\right)\n\\end{equation}\nrespectively. On the other hand, the C-G coefficients for $\\boldsymbol{\\cb}$, $\\boldsymbol{\\ab}$ and $\\boldsymbol{\\bb}$ (the triplets in the RHS of Eqs.~(\\ref{eq:tensor3pqr2})) are given by the same expression as the one for $\\boldsymbol{\\tb}$ in Eq.~(\\ref{eq:tensor1exp}), i.e.~\n\\begin{equation}\n\\boldsymbol{\\cb},\\, \\boldsymbol{\\ab},\\, \\boldsymbol{\\bb} \\equiv\n\\left(\\begin{matrix}\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right).\n\\end{equation}\n\n\\newpage\n\\begin{flalign}\\label{eq:tensor18}\n&\\text{{\\textit {\\textbf {iii}}})\\,\\,\\,}\\boldsymbol{3}\\otimes\\boldsymbol{\\bar{3}}=\\boldsymbol{1}\\oplus\\boldsymbol{8}&\n\\end{flalign}\n\\begin{equation}\\label{eq:tensor18exp}\n\\boldsymbol{1}\\equiv \\frac{1}{\\sqrt{3}}\\left(a_1 b_1 + a_2 b_2 + a_3 b_3\\right), \\quad \\quad \\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{1}{\\sqrt{6}} a_1 b_1 - \\sqrt{\\frac{2}{3}} a_2 b_2 + \\frac{1}{\\sqrt{6}} a_3 b_3\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_1 - a_3 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 + a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 + a_1 b_3\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 + a_2 b_1\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_2 b_3 - a_3 b_2\\right)\\\\\n\t\\frac{1}{\\sqrt{2}}\\left(a_3 b_1 - a_1 b_3\\right)\\\\\n\\frac{1}{\\sqrt{2}}\\left(a_1 b_2 - a_2 b_1\\right)\n\\end{matrix}\\right).\n\\end{equation}\nWe define the basis for the octet representation using Eqs.~(\\ref{eq:tensor18exp}). The resulting generator matrices are \n\\begin{align}\n&C \\equiv\n\\left(\\begin{matrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0\\\\\n\t0 & 0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0\\\\\n\t0 & 0 & 0 & 0 & -\\frac{1}{2} & 0 & 0 & -\\frac{i \\sqrt{3}}{2}\\\\\n\t0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2} & 0 & 0\\\\\n\t0 & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2} & 0\\\\\n\t0 & 0 & 0 & 0 & -\\frac{i \\sqrt{3}}{2} & 0 & 0 & -\\frac{1}{2}\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}-\\frac{1}{2} & \\frac{\\sqrt{3}}{2} & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t-\\frac{\\sqrt{3}}{2} & -\\frac{1}{2} & 0 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\\\\n\t0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\\\\n\t0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\\\\n\t0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\n\\end{matrix}\\right),\\notag \\\\\n&V \\equiv\n\\left(\\begin{matrix}0 & 0 & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{i}{2} & \\frac{i}{2} & \\frac{i}{2}\\\\\n\t0 & 0 & \\frac{1}{2} & \\frac{1}{2} & \\frac{1}{2} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & \\frac{2}{3} & -\\frac{1}{3} & -\\frac{1}{3} & 0 & 0 & 0\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{6} & -\\frac{i}{\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & \\frac{1}{6} & \\frac{1}{6} & \\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{i}{\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & 0 & 0\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{1}{2} & \\frac{1}{2}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{1}{2} & -\\frac{1}{2}\n\\end{matrix}\\right),\\label{eq:gen8}\\\\\n&X \\equiv\n\\left(\\begin{matrix}0 & 0 & -\\frac{1}{\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & \\frac{1}{2\\sqrt{3}} & 0 & -\\frac{i}{2} & \\frac{i}{2}\\\\\n\t0 & 0 & 0 & -\\frac{1}{2} & \\frac{1}{2} & \\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & -\\frac{1}{2} & \\frac{1}{6} & \\frac{1}{6} & \\frac{2}{3} & -\\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & 0\\\\\n\t-\\frac{1}{\\sqrt{3}} & 0 & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{6} & 0 & \\frac{i}{\\sqrt{3}} & \\frac{i}{2\\sqrt{3}}\\\\\n\t\\frac{1}{2\\sqrt{3}} & \\frac{1}{2} & -\\frac{1}{3} & -\\frac{1}{3} & \\frac{1}{6} & -\\frac{i}{\\sqrt{3}} & 0 & -\\frac{i}{2\\sqrt{3}}\\\\\n\t-\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & \\frac{i}{2\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & \\frac{1}{2} & \\frac{1}{2} & 0\\\\\n\t0 & \\frac{i}{\\sqrt{3}} & \\frac{i}{\\sqrt{3}} & 0 & \\frac{i}{2\\sqrt{3}} & 0 & 0 & -\\frac{1}{2}\\\\\n\t\\frac{i}{2} & -\\frac{i}{2\\sqrt{3}} & 0 & -\\frac{i}{\\sqrt{3}} & -\\frac{i}{2\\sqrt{3}} & 0 & 0 & -\\frac{1}{2}\n\\end{matrix}\\right).\\notag\\\\ \\notag\n\\end{align}\nNote that for the $SU(3)$ group, the tensor product of a $\\boldsymbol{3}$ and a $\\boldsymbol{\\tb}$ gives a $\\boldsymbol{1}$ and an $\\boldsymbol{8}$, i.e.~the tensor product expansion Eqs.~(\\ref{eq:tensor18},~\\ref{eq:tensor18exp}) is applicable to the $SU(3)$ group as well.\n\\begin{flalign}\\label{eq:tensor288}\n&\\text{{\\textit {\\textbf {iv}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{3}=\\boldsymbol{2}\\oplus\\boldsymbol{8}\\oplus\\boldsymbol{8}&\n\\end{flalign}\n\\begin{align}\\label{eq:tensor288exp}\n\\begin{split}\n&\\boldsymbol{2}\\equiv\\left(\\begin{matrix}\\frac{a_4 b_1}{\\sqrt{3}}+\\frac{a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3}+\\frac{a_1 b_2}{3} +\\frac{a_1 b_3}{3}+\\frac{a_2 b_1}{3 \\sqrt{2}}-\\frac{\\sqrt{2}a_2 b_2}{3}+\\frac{a_2 b_3}{3 \\sqrt{2}}+\\frac{a_3 b_1}{\\sqrt{6}} -\\frac{a_3 b_3}{\\sqrt{6}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{a_1 b_1}{\\sqrt{6}}-\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}-\\frac{a_2 b_3}{2 \\sqrt{3}}+\\frac{a_3 b_1}{2}+\\frac{a_3 b_3}{2}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{\\sqrt{2} a_1 b_2}{3}-\\frac{a_1 b_3}{3\\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{2 a_2 b_2}{3}-\\frac{a_2 b_3}{6}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}\\\\\n\t\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_2}{6}+\\frac{a_2 b_3}{3}-\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{\\sqrt{3}}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}+\\frac{a_2 b_3}{6}+\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_5 b_1}{\\sqrt{3}}-\\frac{a_5 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3}-\\frac{a_2 b_2}{6}-\\frac{a_3 b_2}{2\\sqrt{3}}-\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{3}-\\frac{a_3 b_2}{2 \\sqrt{3}}+\\frac{a_4 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{\\sqrt{3}}\\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{6}+\\frac{a_2 b_3}{6}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_5 b_1}{\\sqrt{3}}+\\frac{a_5 b_3}{\\sqrt{3}}\\\\\n\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3}+\\frac{a_2 b_2}{6}+\\frac{a_3 b_2}{2\\sqrt{3}}+\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{\\sqrt{3}}\n\\end{matrix}\\right), \\\\ &\\boldsymbol{8}\\equiv\\left(\\begin{matrix}\\frac{a_4 b_1}{\\sqrt{2}}-\\frac{a_6 b_3}{\\sqrt{2}}\\\\\n\t-\\frac{a_4 b_1}{\\sqrt{6}}+\\frac{\\sqrt{2}a_5 b_2}{\\sqrt{3}} -\\frac{a_6 b_3}{\\sqrt{6}}\\\\\n\t-\\frac{a_2 b_1}{\\sqrt{2}}+\\frac{a_3 b_1}{\\sqrt{6}}+\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t-\\frac{\\sqrt{2} a_3 b_2}{\\sqrt{3}}-\\frac{a_4 b_3}{\\sqrt{6}}+\\frac{a_6 b_1}{\\sqrt{6}}\\\\\n\t\\frac{a_2 b_3}{\\sqrt{2}}+\\frac{a_3 b_3}{\\sqrt{6}}+\\frac{a_4 b_2}{\\sqrt{6}}-\\frac{a_5 b_1}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_1}{3}-\\frac{a_2 b_1}{3 \\sqrt{2}}-\\frac{a_3 b_1}{\\sqrt{6}}-\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_2}{3}+\\frac{\\sqrt{2} a_2 b_2}{3} -\\frac{a_4 b_3}{\\sqrt{6}}-\\frac{a_6 b_1}{\\sqrt{6}}\\\\\n\t\\frac{2 a_1 b_3}{3}-\\frac{a_2 b_3}{3 \\sqrt{2}}+\\frac{a_3 b_3}{\\sqrt{6}}-\\frac{a_4 b_2}{\\sqrt{6}}-\\frac{a_5 b_1}{\\sqrt{6}}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\nWe define the basis for the doublet representation using Eqs.~(\\ref{eq:tensor288exp}). The resulting generator matrices are \n\\begin{equation}\\label{eq:gen2}\nC \\equiv\n\\left(\\begin{matrix}1 & 0\\\\\n0 & 1\n\\end{matrix}\\right), \\quad E \\equiv\n\\left(\\begin{matrix}1 & 0\\\\\n0 & 1\n\\end{matrix}\\right), \\quad V \\equiv\ni\\left(\\begin{matrix}-\\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\\\\n\\sqrt{\\frac{2}{3}} & \\frac{1}{\\sqrt{3}}\n\\end{matrix}\\right), \\quad X \\equiv\ni\\left(\\begin{matrix}-\\frac{1}{\\sqrt{3}} & \\sqrt{\\frac{2}{3}}\\om\\\\\n\\sqrt{\\frac{2}{3}}\\ob & \\frac{1}{\\sqrt{3}}\n\\end{matrix}\\right).\n\\end{equation}\nIn relation to the tensor product expansion Eq.~(\\ref{eq:tensor288}), we may embed the irreps of $\\Sigma(72\\times3)$ in the irreps of $SU(3)$:\n\\begin{equation}\n  \\label{eq:orig}\n    \\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n      \\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n        SU(3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{3}&=&\\boldsymbol{10}&\\oplus&\\boldsymbol{8}\\\\\n        \\,\\\\\n        \\Sigma(72\\times3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{3}&=&\\boldsymbol{2}&\\oplus&\\boldsymbol{8}&\\oplus&\\boldsymbol{8}.\\\\\n      };\n\\path[->] ($(foil-1-1.south)$)  edge[]     ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$)  edge[]     ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$)   edge[]    ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$)   edge[]    ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-7.south)$)   edge[]    ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-11.north)$);\n    \\end{tikzpicture}\n  \\end{equation}\n\\begin{flalign}\\label{eq:tensor315}\n&\\text{{\\textit {\\textbf {v}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{3}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}&\n\\end{flalign}\n\\begin{equation*}\n\\boldsymbol{3}\\equiv\\left(\\begin{matrix}\\frac{a_1 b_1}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_1}{2}+\\frac{a_5 b_3}{2}+\\frac{a_6 b_2}{2}\\\\\n\t\\frac{a_1 b_2}{\\sqrt{6}}-\\frac{a_2 b_2}{\\sqrt{3}}+\\frac{a_4 b_3}{2}+\\frac{a_6 b_1}{2}\\\\\n\t\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_3}{2 \\sqrt{3}}-\\frac{a_3 b_3}{2}+\\frac{a_4 b_2}{2}+\\frac{a_5 b_1}{2}\n\\end{matrix}\\right),\n\\end{equation*}\n\\newpage\n\\begin{align}\n\\begin{split}\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}-\\frac{a_4 b_2}{\\sqrt{6}}+\\frac{a_4 b_3}{\\sqrt{6}}+\\frac{a_5 b_1}{\\sqrt{6}}-\\frac{a_5 b_3}{\\sqrt{6}}-\\frac{a_6 b_1}{\\sqrt{6}}+\\frac{a_6 b_2}{\\sqrt{6}}\\\\\n\t-\\frac{a_4 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_3}{\\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_1}{\\sqrt{3}}+\\frac{a_6 b_2}{2 \\sqrt{3}}\\\\\n\t\\frac{a_4 b_2}{2}-\\frac{a_5 b_1}{2}-\\frac{a_5 b_3}{2}+\\frac{a_6 b_2}{2}\\\\\n\t-\\frac{a_1 b_2}{\\sqrt{6}}+\\frac{a_1 b_3}{\\sqrt{6}}-\\frac{a_2 b_2}{2 \\sqrt{3}}-\\frac{a_2 b_3}{\\sqrt{3}}+\\frac{a_3 b_2}{2}\\\\\n\t\\frac{a_1 b_1}{\\sqrt{6}}-\\frac{a_1 b_3}{\\sqrt{6}}+\\frac{a_2 b_1}{2 \\sqrt{3}}-\\frac{a_2 b_3}{2 \\sqrt{3}}-\\frac{a_3 b_1}{2}-\\frac{a_3 b_3}{2}\\\\\n-\\frac{a_1 b_1}{\\sqrt{6}}+\\frac{a_1 b_2}{\\sqrt{6}}+\\frac{a_2 b_1}{\\sqrt{3}}+\\frac{a_2 b_2}{2 \\sqrt{3}}+\\frac{a_3 b_2}{2}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^p}\\equiv\\left(\\begin{matrix} -\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{\\om a_1 b_2 }{3 \\sqrt{2}}-\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{\\om a_2 b_2}{6}+\\frac{\\om a_2 b_3}{3} -\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{\\om a_3 b_2}{2 \\sqrt{3}} -\\frac{\\ob a_4 b_1}{\\sqrt{3}} +\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_2}{2\\sqrt{3}} \\\\\n\t-\\frac{\\om a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{\\om a_2 b_1}{6}+\\frac{a_2 b_2}{3}-\\frac{\\om a_2 b_3}{6} +\\frac{\\om a_3 b_1}{2 \\sqrt{3}} -\\frac{\\om a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_3}{2 \\sqrt{3}}-\\frac{\\ob a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_1}{2 \\sqrt{3}}\\\\\n\t-\\frac{\\om a_1 b_1}{3 \\sqrt{2}}-\\frac{\\om a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{\\om a_2 b_1}{3} -\\frac{\\om a_2 b_2}{6}-\\frac{a_2 b_3}{6} -\\frac{\\om a_3 b_2}{2\\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_2}{2 \\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{\\ob a_6 b_3}{\\sqrt{3}} \n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^q}\\equiv\\left(\\begin{matrix}-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}-\\frac{a_2 b_2}{6}+\\frac{a_2 b_3}{3}-\\frac{a_3 b_1}{2 \\sqrt{3}}+\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{a_4 b_1}{\\sqrt{3}}+\\frac{a_5 b_3}{2 \\sqrt{3}}+\\frac{a_6 b_2}{2 \\sqrt{3}}\\\\\n\t-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{6}+\\frac{a_2 b_2}{3}-\\frac{a_2 b_3}{6}+\\frac{a_3 b_1}{2 \\sqrt{3}}-\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_3}{2 \\sqrt{3}}-\\frac{a_5 b_2}{\\sqrt{3}}+\\frac{a_6 b_1}{2 \\sqrt{3}}\\\\\n-\\frac{a_1 b_1}{3 \\sqrt{2}}-\\frac{a_1 b_2}{3 \\sqrt{2}}-\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{3}-\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{6}-\\frac{a_3 b_2}{2\\sqrt{3}}+\\frac{a_3 b_3}{2 \\sqrt{3}}+\\frac{a_4 b_2}{2 \\sqrt{3}}+\\frac{a_5 b_1}{2 \\sqrt{3}}-\\frac{a_6 b_3}{\\sqrt{3}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^r}\\equiv\\left(\\begin{matrix}\\frac{\\om a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3\\sqrt{2}}+\\frac{\\om a_2 b_1}{6}+\\frac{a_2 b_2}{6}-\\frac{a_2 b_3}{3} +\\frac{\\om a_3 b_1}{2 \\sqrt{3}} -\\frac{a_3 b_2}{2 \\sqrt{3}}+\\frac{\\ob a_4 b_1}{\\sqrt{3}}-\\frac{\\om a_5 b_3}{2\\sqrt{3}}-\\frac{\\om a_6 b_2}{2 \\sqrt{3}} \\\\\n\t\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{\\om a_1 b_2}{3 \\sqrt{2}}+\\frac{a_1 b_3}{3 \\sqrt{2}}+\\frac{a_2 b_1}{6}-\\frac{\\om a_2 b_2}{3}+\\frac{a_2 b_3}{6}  -\\frac{a_3 b_1}{2 \\sqrt{3}} +\\frac{a_3 b_3}{2 \\sqrt{3}}-\\frac{\\om a_4 b_3}{2 \\sqrt{3}}+\\frac{\\ob a_5 b_2}{\\sqrt{3}}-\\frac{\\om a_6 b_1}{2 \\sqrt{3}}\\\\\n\\frac{a_1 b_1}{3 \\sqrt{2}}+\\frac{a_1 b_2}{3 \\sqrt{2}}+\\frac{\\om a_1 b_3}{3 \\sqrt{2}}-\\frac{a_2 b_1}{3} +\\frac{a_2 b_2}{6}+\\frac{\\om a_2 b_3}{6}+\\frac{a_3 b_2}{2 \\sqrt{3}}-\\frac{\\om a_3 b_3}{2 \\sqrt{3}} -\\frac{\\om a_4 b_2}{2 \\sqrt{3}}-\\frac{\\om a_5 b_1}{2 \\sqrt{3}}+\\frac{\\ob a_6 b_3}{\\sqrt{3}}\n\\end{matrix}\\right).\n\\end{split}\n\\end{align}\nThe $SU(3)$ embedding corresponding to Eq.~(\\ref{eq:tensor315}) is \n\\begin{equation}\n  \\label{eq:orig}\n    \\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n      \\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n        SU(3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{\\tb}&=&\\boldsymbol{3}&\\oplus&\\boldsymbol{15}\\\\\n        \\,\\\\\n        \\Sigma(72\\times3)&\\,:\\,&\\boldsymbol{6}&\\otimes&\\boldsymbol{\\tb}&=&\\boldsymbol{3}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3^p}&\\oplus&\\boldsymbol{3^q}&\\oplus&\\boldsymbol{3^r}.\\\\\n      };\n\\path[->] ($(foil-1-1.south)$)  edge[]     ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$)  edge[]     ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$)   edge[]    ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$)  edge[]     ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-11.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-13.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-15.north)$);\n    \\end{tikzpicture}\n  \\end{equation}\n\\begin{flalign}\\label{eq:tensor66}\n&\\text{{\\textit {\\textbf {vi}}})\\,\\,\\,}\\boldsymbol{6}\\otimes\\boldsymbol{6}=\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}}_\\text{symm}\\oplus\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}}_\\text{antisymm}&\n\\end{flalign}\nHere the sextet, $\\boldsymbol{\\xb}$, appears more than once in the symmetric part. So there is no unique way of decomposing the product space into the sum of the irreducible sextets, i.e.~the C-G coefficients are not uniquely defined. To solve this problem, we utilise the group $\\Sigma(216\\times3)$ which has $\\Sigma(72\\times3)$ as one of its subgroups. $\\Sigma(216\\times3)$ has three distinct types of sextets~\\cite{Sigma1}, $\\boldsymbol{6^x}$, $\\boldsymbol{6^y}$, $\\boldsymbol{6^z}$. The sextet of $\\Sigma(72\\times3)$ can be embedded in any of these three sextets of $\\Sigma(216\\times3)$. A tensor product expansion for $\\Sigma(216\\times3)$, equivalent to Eq.~(\\ref{eq:tensor66}), is given by \n\\begin{equation}\\label{eq:tensor66hes}\n\\boldsymbol{6^x}\\otimes\\boldsymbol{6^x}=\\underbrace{\\boldsymbol{\\xb^x}\\oplus\\boldsymbol{\\xb^y}\\oplus\\boldsymbol{\\xb^z}\\oplus\\boldsymbol{3^x}}_\\text{symm}\\oplus\\underbrace{\\boldsymbol{\\xb^x}\\oplus\\boldsymbol{9}}_\\text{antisymm}.\n\\end{equation}\nIn Eq.~(\\ref{eq:tensor66hes}), the decomposition of the symmetric part into the irreducible sextets is unique. Hence we embed the irreps of $\\Sigma(72\\times3)$ in the irreps of $\\Sigma(216\\times3)$,\n\\begin{equation}\\label{eq:orig}\n\\begin{tikzpicture}[>=stealth,baseline=(current bounding box.center),anchor=base,inner sep=0pt]\n\\matrix (foil) [matrix of math nodes,nodes={minimum height=1em}] {\n\\Sigma(216\\times3)&\\,:\\,& \\boldsymbol{6^x}&\\otimes&\\boldsymbol{6^x}&=&\\boldsymbol{\\bar{6^x}}&\\oplus&\\boldsymbol{\\bar{6^y}}&\\oplus&\\boldsymbol{\\bar{6^z}}&\\oplus&\\boldsymbol{3^x}&\\oplus&\\boldsymbol{\\bar{6^x}}&\\oplus&\\boldsymbol{9}\\\\\n\\,\\\\\n\\Sigma(72\\times3)&\\,:\\,& \\boldsymbol{6}&\\otimes&\\boldsymbol{6}&=&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3}&\\oplus&\\boldsymbol{\\xb}&\\oplus&\\boldsymbol{3^p}&\\oplus&\\boldsymbol{3^q}&\\oplus&\\boldsymbol{3^r}.\\\\};\n\\path[->] ($(foil-1-1.south)$)  edge[]     ($(foil-3-1.north)$);\n\\path[->] ($(foil-1-3.south)$)  edge[]     ($(foil-3-3.north)$);\n\\path[->] ($(foil-1-5.south)$)   edge[]    ($(foil-3-5.north)$);\n\\path[->] ($(foil-1-7.south)$)  edge[]     ($(foil-3-7.north)$);\n\\path[->] ($(foil-1-9.south)$)   edge[]    ($(foil-3-9.north)$);\n\\path[->] ($(foil-1-11.south)$)  edge[]     ($(foil-3-11.north)$);\n\\path[->] ($(foil-1-13.south)$)   edge[]    ($(foil-3-13.north)$);\n\\path[->] ($(foil-1-15.south)$)  edge[]     ($(foil-3-15.north)$);\n\\path[->] ($(foil-1-17.south)$)   edge[]    ($(foil-3-17.north)$);\n\\path[->] ($(foil-1-17.south)$)  edge[]     ($(foil-3-19.north)$);\n\\path[->] ($(foil-1-17.south)$)   edge[]    ($(foil-3-21.north)$);\n\\end{tikzpicture}\n\\end{equation}\nto obtain a unique decomposition for the case of $\\Sigma(72\\times3)$ as well.\n\\newpage\n\n{\\small\n\\begin{align}\\label{eq:tensor66exp}\n\\begin{split}\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{2}{3}a_1 b_1-\\frac{1}{3}a_2 b_2-\\frac{1}{3}a_3 b_3-\\frac{1}{3}a_4 b_4-\\frac{1}{3}a_5 b_5-\\frac{1}{3}a_6 b_6\\\\\n\t-\\frac{\\sqrt{2}}{3}a_2 b_2  +\\frac{\\sqrt{2}}{3}a_3 b_3 -\\frac{1}{3 \\sqrt{2}}a_4 b_4+\\frac{\\sqrt{2}}{3}a_5 b_5 -\\frac{1}{3 \\sqrt{2}}a_6 b_6-\\frac{1}{3}a_{\\{1} b_{2\\}}\\\\\n\t-\\frac{1}{\\sqrt{6}}a_4 b_4+\\frac{1}{\\sqrt{6}}a_6 b_6 -\\frac{1}{3}a_{\\{1} b_{3\\}}+\\frac{\\sqrt{2}}{3}a_{\\{2} b_{3\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{4\\}}-\\frac{1}{3 \\sqrt{2}}a_{\\{2} b_{4\\}}-\\frac{1}{\\sqrt{6}}a_{\\{3} b_{4\\}}+\\frac{1}{\\sqrt{6}}a_{\\{5} b_{6\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{5\\}}+\\frac{\\sqrt{2}}{3}a_{\\{2} b_{5\\}}+\\frac{1}{\\sqrt{6}}a_{\\{4} b_{6\\}}\\\\\n\t-\\frac{1}{3}a_{\\{1} b_{6\\}}-\\frac{1}{3 \\sqrt{2}}a_{\\{2} b_{6\\}}+\\frac{1}{\\sqrt{6}}a_{\\{3} b_{6\\}}+\\frac{1}{\\sqrt{6}}a_{\\{4} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\n\\frac{1}{3}(a_1 b_1+a_2 b_2+a_3 b_3)-\\frac{\\sqrt{2}}{3\\sqrt{3}}(a_{\\{1}b_{4\\}}+a_{\\{1}b_{5\\}}+a_{\\{1}b_{6\\}}) +\\frac{1}{6\\sqrt{3}}(a_{\\{2}b_{4\\}}+a_{\\{2}b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{2}b_{5\\}}+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{6}(a_{\\{3}b_{4\\}}-a_{\\{3}b_{6\\}})\\\\\n\t\\frac{1}{3 \\sqrt{2}}(-a_2 b_2+a_3 b_3)+\\frac{1}{3} a_{\\{1} b_{2\\}}+\\frac{1}{6 \\sqrt{3}}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{1} b_{5\\}}+\\frac{\\sqrt{2}}{3\\sqrt{3}} (a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad-\\frac{1}{3 \\sqrt{6}}a_{\\{2} b_{5\\}}+\\frac{1}{3 \\sqrt{2}}(-a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})\\\\\n\t\\frac{1}{3} a_{\\{1} b_{3\\}}+\\frac{1}{6} (a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{3 \\sqrt{2}}(a_{\\{2} b_{3\\}}-a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\frac{1}{\\sqrt{6}}a_{\\{3} b_{5\\}}\\\\\n\t\\frac{\\sqrt{2}}{3\\sqrt{3}} \\left(-a_1 b_1+a_2 b_2\\right)-\\frac{\\sqrt{2}}{\\sqrt{3}} a_4 b_4+\\frac{1}{6 \\sqrt{3}}a_{\\{1} b_{2\\}}+\\frac{1}{6} a_{\\{1} b_{3\\}}-\\frac{1}{3 \\sqrt{2}} a_{\\{2} b_{3\\}}\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1 -\\frac{1}{3\\sqrt{6}} a_2 b_2+\\frac{1}{\\sqrt{6}} a_3 b_3-\\frac{\\sqrt{2}}{\\sqrt{3}} a_5 b_5-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}\\\\\n\\frac{\\sqrt{2}}{3\\sqrt{3}}(-a_1 b_1+a_2 b_2)-\\frac{\\sqrt{2}}{\\sqrt{3}} a_6 b_6+\\frac{1}{6\\sqrt{3}}a_{\\{1} b_{2\\}}-\\frac{1}{6} a_{\\{1} b_{3\\}}+\\frac{1}{3\\sqrt{2}}a_{\\{2} b_{3\\}}\n\\end{matrix}\\right),\\\\ \n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{1}{3 \\sqrt{6}}(a_{\\{1} b_{4}\\}+a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})+\\frac{1}{6 \\sqrt{3}}(a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})-\\frac{1}{3 \\sqrt{3}}a_{\\{2} b_{5\\}}+\\frac{1}{6} \\left(a_{\\{3} b_{4\\}}-a_{\\{3} b_{6\\}}\\right)+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{3} \\left(a_{\\{4} b_{5\\}}+a_{\\{4} b_{6\\}}+a_{\\{5} b_{6\\}}\\right)\\\\\n\t\\frac{1}{6 \\sqrt{3}}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{5\\}}+\\frac{1}{6 \\sqrt{6}}(a_{\\{2} b_{4\\}}+a_{\\{2} b_{6\\}})+\\frac{\\sqrt{2}}{3\\sqrt{3}}a_{\\{2} b_{5\\}}+\\frac{1}{6\\sqrt{2}}(a_{\\{3} b_{4\\}}-a_{\\{3} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\frac{1}{3\\sqrt{2}}(a_{\\{4} b_{5\\}}+a_{\\{5} b_{6\\}})-\\frac{\\sqrt{2}}{3}a_{\\{4} b_{6\\}}\\\\\n\t\\frac{1}{6}(a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{6\\sqrt{2}}(a_{\\{2} b_{4\\}}-a_{\\{2} b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})+\\frac{1}{\\sqrt{6}}(-a_{\\{4} b_{5\\}}+a_{\\{5} b_{6\\}})\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{1}{3\\sqrt{6}}a_2 b_2-\\frac{1}{\\sqrt{6}}a_3 b_3-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}-\\frac{1}{6}(a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{3}a_{\\{1} b_{3\\}}+\\frac{1}{3\\sqrt{2}}(-a_{\\{2} b_{3\\}}+a_{\\{2} b_{6\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad\\,\\,+\\frac{1}{2\\sqrt{6}}a_{\\{3} b_{5\\}}-\\frac{1}{6\\sqrt{2}}a_{\\{2} b_{5\\}}\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{2\\sqrt{2}}{3\\sqrt{3}}a_2 b_2+\\frac{2}{3\\sqrt{3}}a_{\\{1}b_{2\\}}-\\frac{1}{6}(a_{\\{1}b_{4\\}}+a_{\\{1}b_{6\\}})-\\frac{1}{6\\sqrt{2}}(a_{\\{2}b_{4\\}}+a_{\\{2}b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{3}b_{4\\}}-a_{\\{3}b_{6\\}})\\\\\n\t-\\frac{\\sqrt{2}}{3\\sqrt{3}}a_1 b_1-\\frac{1}{3\\sqrt{6}}a_2 b_2-\\frac{1}{\\sqrt{6}}a_3 b_3-\\frac{1}{3\\sqrt{3}}a_{\\{1} b_{2\\}}+\\frac{1}{3}a_{\\{1} b_{3\\}}-\\frac{1}{6}(a_{\\{1} b_{4\\}}+a_{\\{1} b_{5\\}})+\\frac{1}{3\\sqrt{2}}(a_{\\{2} b_{3\\}}+a_{\\{2} b_{4\\}})+\\\\\n\t\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\qquad\\quad-\\frac{1}{6\\sqrt{2}}a_{\\{2} b_{5\\}}-\\frac{1}{2\\sqrt{6}}a_{\\{3} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}(-a_{\\{1} b_{5\\}}+a_{\\{1} b_{6\\}})-\\frac{1}{2\\sqrt{6}}a_{\\{2} b_{5\\}}-\\frac{1}{\\sqrt{6}}a_{\\{2} b_{6\\}}+\\frac{1}{2\\sqrt{2}}a_{\\{3} b_{5\\}}\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{\\{1} b_{4\\}}-a_{\\{1} b_{6\\}})+\\frac{1}{2\\sqrt{6}}(a_{\\{2} b_{4\\}}-a_{\\{2} b_{6\\}})-\\frac{1}{2\\sqrt{2}}(a_{\\{3} b_{4\\}}+a_{\\{3} b_{6\\}})\\\\\n\t\\frac{1}{2\\sqrt{3}}(-a_{\\{1} b_{4\\}}+a_{\\{1} b_{5\\}})+\\frac{1}{\\sqrt{6}}a_{\\{2} b_{4\\}}+\\frac{1}{2\\sqrt{6}}a_{\\{2} b_{5\\}}+\\frac{1}{2\\sqrt{2}}a_{\\{3} b_{5\\}}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{\\xb}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}(a_{[6} b_{3]}+a_{[3} b_{4]})+\\frac{1}{3\\sqrt{2}}(a_{[1} b_{4]}+a_{[1} b_{5]}+a_{[1} b_{6]})+\\frac{1}{6}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{3}a_{[5} b_{2]}\\\\\n\t\\frac{1}{2\\sqrt{6}}(a_{[3} b_{4]}+a_{[6} b_{3]})+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{3}a_{[5} b_{1]}+\\frac{\\sqrt{2}}{3}a_{[2} b_{5]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[1} b_{6]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{1}{2\\sqrt{6}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{1}{2\\sqrt{2}}(a_{[3} b_{4]}+a_{[3} b_{6]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{5]}+a_{[1} b_{6]})+\\frac{1}{2\\sqrt{6}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[6} b_{2]}+\\frac{1}{2\\sqrt{2}}a_{[5} b_{3]}\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{6]}+a_{[1} b_{4]})+\\frac{1}{2\\sqrt{6}}(a_{[2} b_{4]}+a_{[2} b_{6]})+\\frac{1}{2\\sqrt{2}}(a_{[3} b_{6]}+a_{[4} b_{3]})\\\\\n\t\\frac{1}{2\\sqrt{3}}(a_{[1} b_{4]}+a_{[1} b_{5]})+\\frac{1}{2\\sqrt{6}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[4} b_{2]}+\\frac{1}{2\\sqrt{2}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^p}\\equiv\\left(\\begin{matrix}\\frac{\\om}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[6} b_{2]}+\\om a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{6]}+a_{[5} b_{1]}+\\om a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{\\ob}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{\\om}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{\\om}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{\\ob}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{\\om}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+\\om a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{5]}+a_{[4} b_{1]}+\\om a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{\\ob}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^q}\\equiv\\left(\\begin{matrix}\\frac{1}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[3} b_{2]}+a_{[6} b_{2]})+\\frac{1}{6}(a_{[1} b_{3]}+a_{[1} b_{6]}+a_{[5} b_{1]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{1}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{1}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{1}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{1}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{1}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{3]}+a_{[1} b_{5]}+a_{[4} b_{1]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{1}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right),\\\\\n&\\boldsymbol{3^r}\\equiv\\left(\\begin{matrix}\\frac{\\ob}{2\\sqrt{3}}a_{[2} b_{1]}+\\frac{1}{3\\sqrt{2}}(a_{[6} b_{2]}+\\ob a_{[3} b_{2]})+\\frac{1}{6}(a_{[5} b_{1]}+a_{[1} b_{6]}+\\ob a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[5} b_{2]}+\\frac{\\om}{\\sqrt{6}}a_{[5} b_{6]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\\\\\n\t\\frac{\\ob}{3\\sqrt{2}}a_{[3} b_{2]}+\\frac{1}{6}(a_{[1} b_{4]}+a_{[6} b_{1]})+\\frac{\\ob}{3}a_{[3} b_{1]}+\\frac{1}{6\\sqrt{2}}(a_{[2} b_{4]}+a_{[6} b_{2]})+\\frac{\\om}{\\sqrt{6}}a_{[6} b_{4]}+\\frac{1}{2\\sqrt{6}}(a_{[4} b_{3]}+a_{[6} b_{3]})\\\\\n\t\\frac{\\ob}{2\\sqrt{3}}a_{[1} b_{2]}+\\frac{1}{3\\sqrt{2}}(a_{[2} b_{4]}+\\ob a_{[3} b_{2]})+\\frac{1}{6}(a_{[1} b_{5]}+a_{[4} b_{1]}+\\ob a_{[1} b_{3]})+\\frac{1}{6\\sqrt{2}}a_{[2} b_{5]}+\\frac{\\om}{\\sqrt{6}}a_{[4} b_{5]}+\\frac{1}{2\\sqrt{6}}a_{[3} b_{5]}\n\\end{matrix}\\right).\\\\\n\\end{split}\n\\end{align}\n}\n\nIn Eqs.~(\\ref{eq:tensor66exp}) we have used the curly bracket and the square bracket to denote the symmetric sum and the antisymmetric sum respectively, i.e.~$a_{\\{i} b_{j\\}}=a_i b_j + a_j b_i$ and $a_{[i} b_{j]}=a_i b_j - a_j b_i$ . \n\n\\section{Appendix B: Flavon Potentials}\n\nHere we discuss the flavon potentials that lead to the vacuum alignments assumed in our model. The potentials we construct contain only up to the sixth-order flavon terms. It should be noted that even though our construction results in the required VEVs, we are not doing an exhaustive analysis of the most general flavon potentials involving all the possible invariant terms. However, the content we include is sufficient to realise our VEVs.\n\n\\subsection{The triplet flavons: $\\phi_e$, $\\phi_\\mu$, $\\phi_\\tau$}\\label{sec:triplets}\n\nFirst we consider the triplet flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$. Our target is to obtain the VEVs $\\langle\\phi_e\\rangle=\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\langle\\phi_\\mu\\rangle=\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $\\langle\\phi_\\tau\\rangle=\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$, Eqs.~(\\ref{eq:leptvev}). The flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ transform as $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ respectively. The $3\\times3$ maximal matrix $V$, Eqs.~(\\ref{eq:gen3}), is one of the generators of $\\boldsymbol{3}$. The corresponding generators of $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ are $-V^*$, $V^*$ and $-V^*$ respectively, Eqs.~(\\ref{eq:gen1p}-\\ref{eq:3times1}). If the potentials of $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ have minima at $(-1,0,0)^T$, $(0,1,0)^T$ and $(0,0,-1)^T$, then they have minima also at $-V^*(-1,0,0)^T=\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $V^*(0,1,0)^T=\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $-V^*(0,0,-1)^T=\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$ as required. The $3\\times3$ cyclic matrix $E$, Eqs.~(\\ref{eq:gen3}), is another generator of $\\boldsymbol{3}$ and thus of $\\boldsymbol{\\ab}$, $\\boldsymbol{\\bb}$ and $\\boldsymbol{\\cb}$ as well. Therefore, if the potential has minima at $(\\pm1,0,0)^T$, then it has minima also at $(0,\\pm1,0)^T$ and $(0,0,\\pm1)^T$. In a nutshell, for obtaining the required VEVs, Eqs.~(\\ref{eq:leptvev}), all we need to do is to construct potentials with minima at $(\\pm1,0,0)^T$.\n\nWe start with the term $\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2$ which is $SU(3)$-invariant and which leads to a continuous set of minima that corresponds to unit magnitude for $\\phi_e$, i.e.~$\\phi_e^\\dagger\\phi_e=1$. Now we add terms which are invariant under $\\Sigma(72\\times3)$, but which break $SU(3)$ and result in a discrete set of points of minima including $(\\pm1,0,0)^T$.\n\nWith two $\\phi_e$ triplets which transform as $\\boldsymbol{\\ab}$, we may construct a conjugate sextet $\\boldsymbol{\\xb}$, i.e.~$\\boldsymbol{\\bar{3^p}}\\otimes\\boldsymbol{\\bar{3^p}}=\\boldsymbol{\\bar{6}}\\oplus\\boldsymbol{3}$, Eqs.~(\\ref{eq:tensor3pqr}). Note that the antisymmetric part, $\\boldsymbol{3}$, vanishes.  With the help of Eqs.~(\\ref{eq:tensor288},~\\ref{eq:tensor288exp}), we combine the symmetric part, $\\boldsymbol{\\xb}$, with another $\\phi_e$ ($\\boldsymbol{\\bar{3^p}}$) to obtain\\footnote{Here we have omitted the distinction among $\\boldsymbol{\\tb}$, $\\boldsymbol{\\ab}$, $\\boldsymbol{\\ab}$ and $\\boldsymbol{\\ab}$. These triplets differ only with respect to a multiplication with $\\pm1$. In the potential terms constructed subsequently in Sec.~(\\ref{sec:triplets}), each type of triplet appears an even number of times and hence a sign flip does not have any impact.}\n\\begin{equation}\\label{eq:tensor288conj}\n\\boldsymbol{\\xb}\\otimes\\boldsymbol{\\tb}=\\boldsymbol{2}\\oplus\\boldsymbol{8}\\oplus\\boldsymbol{8}.\n\\end{equation} \nIn terms of $\\phi_e=(a_1,a_2,a_3)^T$, we provide the explicit expressions for the doublet,\n\\begin{equation}\\label{eq:doubletexplicit}\n\\boldsymbol{2} \\equiv \\frac{1}{\\sqrt{3}}\\left(3\\sqrt{2}a_1 a_2 a_3, a_1^3+a_2^3+a_3^3\\right)^T,\n\\end{equation}\nand the first octet,\n\\begin{equation}\\label{eq:octetexplicit}\n\\begin{split}\n\\boldsymbol{8}&\\equiv\\frac{\\sqrt{3}}{\\sqrt{2}}\\left(\\frac{\\left(a_1^3-a_3^3\\right)}{\\sqrt{3}}, \\frac{\\left(-a_1^3+2a_2^3-a_3^3\\right)}{3},a_3a_2(a_3-a_2),a_1a_3(a_1-a_3),a_2a_1(a_2-a_1),\\right.\\\\\n&\\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\qquad \\left. \\vphantom{\\frac{1}{\\sqrt{3}}}a_3a_2(a_3+a_2),a_1a_3(a_1+a_3),a_2a_1(a_2+a_1)\\right)^T.\\\\\n\\end{split}\n\\end{equation}\nIt can be shown that the second octet in Eq.~(\\ref{eq:tensor288conj}) is totally antisymmetric with respect to the permutation of the indices of the $\\phi_e$ triplets and therefore it vanishes. Using the doublets\\footnote{$\\boldsymbol{2}$ as well as $\\boldsymbol{8}$ are quaternionic representations. Their characters are real, but the use complex numbers can not be avoided in their representation matrices, eg.~Eqs.~(\\ref{eq:gen8},~\\ref{eq:gen2}). Real, complex and quaternionic representations can be identified by calculating the Frobenius-Schur indicator which gets the values $+1$, $0$ and $-1$ respectively.} from Eq.~(\\ref{eq:doubletexplicit}) we obtain the invariant term $T_{e\\boldsymbol{2}}\\equiv\\boldsymbol{2}^\\dagger \\boldsymbol{2}$. Similarly using the octets from Eq.~(\\ref{eq:octetexplicit}) we obtain the invariant term $T_{e\\boldsymbol{8}} \\equiv \\boldsymbol{8}^\\dagger \\boldsymbol{8}$.\n\nIn the previous paragraph, we obtained a conjugate sextet $\\boldsymbol{\\xb}$ from two $\\phi_e$ triplets. With the help of Eqs.~(\\ref{eq:tensor66},~\\ref{eq:tensor66exp}), we take the tensor product of two of these $\\boldsymbol{\\xb}$s to construct three $\\boldsymbol{6}$s and a $\\boldsymbol{\\tb}$:\n\\begin{equation}\\label{eq:tensor66conj}\n\\boldsymbol{\\xb}\\otimes\\boldsymbol{\\xb}=\\underbrace{\\boldsymbol{6}\\oplus\\boldsymbol{6}\\oplus\\boldsymbol{6}\\oplus\\boldsymbol{\\tb}}_{sym}\\oplus\\underbrace{\\boldsymbol{6}\\oplus\\boldsymbol{\\ab}\\oplus\\boldsymbol{\\bb}\\oplus\\boldsymbol{\\cb}}_{antisym}.\n\\end{equation}\nOf course, the antisymmetric part vanishes. It can also be shown that the first sextet in the symmetric part is antisymmetric with respect to the permutation of the indices of the original triplets ($\\phi_e$) and therefore it vanishes too. We combine the second sextet in the symmetric part with the original $\\boldsymbol{\\xb}$ in the LHS of Eq.~(\\ref{eq:tensor66conj}) to obtain an invariant term $T_{e\\boldsymbol{6}}\\equiv\\boldsymbol{6}^T\\boldsymbol{\\xb}$. $T_{e\\boldsymbol{6}}$ in terms of the components of $\\phi_e$ is given by\\footnote{The invariant of lowest degree that breaks $SU(3)\\rightarrow\\Sigma(72\\times3)$ as calculated in \\cite{Merle} agrees with our expression for $T_{e\\boldsymbol{6}}$.}:\n\\begin{equation}\nT_{e\\boldsymbol{6}}=\\frac{1}{\\sqrt{3}}\\left(a_1^6+a_2^6+a_3^6-10a_1^3a_2^3-10a_2^3a_3^3-10a_3^3a_1^3 \\right).\n\\end{equation} \nNote that $T_{e\\boldsymbol{6}}$ is complex. A similar invariant term can be constructed using the third sextet in the RHS and the $\\boldsymbol{\\xb}$ in LHS. However this term, when viewed as the tensor product of the three $\\boldsymbol{\\xb}$s, is totally antisymmetric with respect to the permutation of indices of the $\\boldsymbol{\\xb}$s and thus vanishes. Note that $T_{e\\boldsymbol{2}}$, $T_{e\\boldsymbol{8}}$ and $T_{e\\boldsymbol{8}}$ are sixth-order flavon terms. \n\nCombining all the non-vanishing invariant terms we write the potential\n\\begin{equation}\\label{eq:leptpotinit}\n\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2+k_{e1} T_{e\\boldsymbol{2}} +k_{e2} T_{e\\boldsymbol{8}} +k_{e3} \\text{Re}\\left(T_{e\\boldsymbol{6}}\\right)+k_{e4} \\text{Im}\\left(T_{e\\boldsymbol{6}}\\right). \n\\end{equation}\nBy the suitable choice of coefficients $k_{e1}$, $k_{e2}$, $k_{e3}$ and $k_{e4}$, the above potential can be made to have minima at the required points, eg.~at $(\\pm1,0,0)^T$. The first partial derivates at the points of extrema must be zero. To make sure that these points are minima, we do the second derivative test using the Hessian partial derivative matrix. Such a procedure has been followed in previous works eg. in~\\cite{Derivative}. The first partial derivatives vanishing at $(\\pm1,0,0)^T$ leads to the conditions\n\\begin{equation}\\label{eq:leptpotfirstder}\nk_{e1}+2 k_{e2} + \\sqrt{3} k_{e3} = 0, \\quad k_{e4}=0.\n\\end{equation}\nSubstituting Eqs.~(\\ref{eq:leptpotfirstder}) in Eq.~(\\ref{eq:leptpotinit}), we obtain the potential\n\\begin{equation}\\label{eq:leptpot}\n\\left(\\phi_e^\\dagger\\phi_e-1\\right)^2-\\left(2k_{e2}+\\sqrt{3} k_{e3}\\right) T_{e\\boldsymbol{2}} +k_{e2} T_{e\\boldsymbol{8}} +k_{e3} \\text{Re}\\left(T_{e\\boldsymbol{6}}\\right).\n\\end{equation}\nNow we apply the second partial derivative test which gives the constraints\n\\begin{equation}\\label{eq:leptpotcond}\nk_{e2}>0, \\quad k_{e3}<0.\n\\end{equation}\nTo sum up our discussion; the potential, Eq.~(\\ref{eq:leptpot}), with the constraints, Eq.~(\\ref{eq:leptpotcond}), has minima at $(\\pm1,0,0)^T$, $(0,\\pm1,0)^T$, $(0,0,\\pm1)^T$ and also at $\\pm\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\pm\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$, $\\pm\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$.\n\nPotentials similar to Eq.~(\\ref{eq:leptpot}) can be written for the flavons $\\phi_\\mu$ and $\\phi_\\tau$ also, i.e.\n\\begin{gather}\n\\left(\\phi_\\mu^\\dagger\\phi_\\mu-1\\right)^2-\\left(2k_{\\mu2}+\\sqrt{3} k_{\\mu3}\\right) T_{\\mu\\boldsymbol{2}} +k_{\\mu2} T_{\\mu\\boldsymbol{8}} +k_{\\mu3} \\text{Re}\\left(T_{\\mu\\boldsymbol{6}}\\right)\\label{eq:potleptmu}\\\\\n\\left(\\phi_\\tau^\\dagger\\phi_\\tau-1\\right)^2-\\left(2k_{\\tau2}+\\sqrt{3} k_{\\tau3}\\right) T_{\\tau\\boldsymbol{2}} +k_{\\tau2} T_{\\tau\\boldsymbol{8}} +k_{\\tau3} \\text{Re}\\left(T_{\\tau\\boldsymbol{6}}\\right).\\label{eq:potlepttau}\n\\end{gather}\nWe need to ensure that the vacuum alignments of $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ are orthogonal to each other, Eqs.~(\\ref{eq:leptvev}). For that purpose, we construct the cross term\n\\begin{equation}\\label{eq:leptcross}\nk_{e \\mu}|\\phi_e^\\dagger \\phi_\\mu|^2 + k_{\\mu \\tau} |\\phi_\\mu^\\dagger \\phi_\\tau|^2 + k_{\\tau e} |\\phi_\\tau^\\dagger \\phi_e|^2,\n\\end{equation}\nwhere $k_{e \\mu}$, $k_{\\mu \\tau}$ and $k_{\\tau e}$ are positive constants. Therefore, the complete potential for the triplet flavons is the sum of Eq.~(\\ref{eq:leptpot}), Eq.~(\\ref{eq:potleptmu}), Eq.~(\\ref{eq:potlepttau}) and Eq.~(\\ref{eq:leptcross}). Such a potential has minima at $\\phi_e=(\\pm1,0,0)^T$, $\\phi_\\mu=(0,\\pm1,0)^T$ and $\\phi_\\tau=(0,0,\\pm1)^T$, and also at $\\phi_e=\\pm\\frac{i}{\\sqrt{3}}(1,1,1)^T$, $\\phi_\\mu=\\pm\\frac{i}{\\sqrt{3}}(1,\\ob,\\om)^T$ and $\\phi_\\tau=\\pm\\frac{i}{\\sqrt{3}}(1,\\om,\\ob)^T$ as originally proposed.\n\n\\subsection{The sextet flavon: $\\phi$}\n\nNow we turn our attention towards constructing the potentials for the sextet flavon $\\phi$. We discussed four different cases of VEVs and here the construction of potentials corresponding to all these cases are done in a rather similar framework. Let us list all the four VEVs:\\footnote{Note that compared to Eqs.~(\\ref{eq:vevtxmp},~\\ref{eq:vevtxmm}), there is an extra negative sign in Eqs.~(\\ref{eq:vevlisttxmp}, \\ref{eq:vevlisttxmm}). The reason for this becomes apparent later in our discussion, but it is clear that such a choice does not have any observable consequence.}.\n\\begin{align}\n\\txm_{(\\chi=+\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{-3+\\sqrt{2}}{\\sqrt{3}},\\frac{1}{\\sqrt{3}},1-\\sqrt{2},0,-1,0\\right),\\label{eq:vevlisttxmp}\\\\\n\\txm_{(\\chi=-\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{-3+\\sqrt{2}}{\\sqrt{3}},\\frac{1}{\\sqrt{3}},-1+\\sqrt{2},0,-1,0\\right),\\label{eq:vevlisttxmm}\\\\\n\\txm_{(\\phi=+\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, -i \\left(1-\\sqrt{2}\\right),0, -1+\\sqrt{2}, 0\\right),\\label{eq:vevlisttpmp}\\\\\n\\txm_{(\\phi=-\\frac{\\pi}{16})}:\\quad \\langle\\phi\\rangle &= \\left(\\frac{1+\\sqrt{2}}{\\sqrt{3}},\\frac{1-\\sqrt{2}}{\\sqrt{3}}, i (1-\\sqrt{2}),0, -1+\\sqrt{2}, 0\\right).\\label{eq:vevlisttpmm}\n\\end{align}\nThese VEVs have the same magnitude, i.e.~$\\langle\\phi\\rangle^\\dagger\\langle\\phi\\rangle=8-4\\sqrt{2}$ for every given $\\langle\\phi\\rangle$ in Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). Therefore we may write an $SU(3)$-invariant term $|\\phi^\\dagger\\phi-(8-4\\sqrt{2})|^2$. Just as we did in the case of the triplet flavons, here also we add terms that break $SU(3)$ but respect the $\\Sigma(72\\times3)$ symmetry to obtain a discrete set of minima. \n\nWith the help of Eqs.~(\\ref{eq:tensor66},~\\ref{eq:tensor66exp}), We analyse the tensor product of three $\\phi$s ($\\boldsymbol{6}$s):\n\\begin{align}\n\\boldsymbol{6}\\otimes\\boldsymbol{6}\\otimes\\boldsymbol{6}&=\\left(\\underbrace{\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{\\xb}\\oplus\\boldsymbol{3}}_{sym}\\oplus\\underbrace{\\xcancel{\\boldsymbol{\\xb}\\oplus\\boldsymbol{3^p}\\oplus\\boldsymbol{3^q}\\oplus\\boldsymbol{3^r}}}_{antisym}\\right)\\otimes\\boldsymbol{6}\\label{eq:tensor666a}\\\\\n&=(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})\\oplus(\\boldsymbol{3}\\otimes\\boldsymbol{6})\\label{eq:tensor666b}.\n\\end{align}\nEach tensor product $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) contains the invariant term $\\boldsymbol{\\xb}^T\\boldsymbol{6}$. Such invariants constructed from the first $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ and the second $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) are named $T_{\\boldsymbol{6}}$ and $T'_{\\boldsymbol{6}}$ respectively. In terms of the components of the sextet flavon, $\\phi=(a_1,a_2,a_3,a_4,a_5,a_6)^T$, they are given by\n\n\\begin{align}\\label{eq:t6invar}\n\\begin{split}\nT_{\\boldsymbol{6}} &=\\frac{2}{3}a_1^3-\\frac{\\sqrt{2}}{3}a_2^3-a_1\\left(a_2^2+a_3^2+a_4^2+a_5^2+a_6^2\\right)\\\\\n&\\qquad\\qquad+\\sqrt{2}a_2\\left(a_3^2+a_5^2\\right)-\\frac{1}{\\sqrt{2}}a_2\\left(a_4^2+a_6^2\\right)-\\frac{\\sqrt{3}}{\\sqrt{2}}a_3\\left(a_4^2+a_6^2\\right)+\\sqrt{6} a_4 a_5 a_6,\n\\end{split}\\\\\n\\begin{split}\nT'_{\\boldsymbol{6}}&=\\frac{1}{3}a_1^3 -\\frac{1}{3\\sqrt{2}}a_2^3-\\frac{\\sqrt{2}}{\\sqrt{3}}\\left(a_4^3+a_5^3+a_6^3\\right)+a_1\\left(a_2^2+a_3^2\\right)+\\frac{1}{\\sqrt{2}}a_2 a_3^2\\\\\n&\\qquad\\qquad-\\frac{\\sqrt{2}}{\\sqrt{3}}a_1^2\\left(a_4-a_5+a_6\\right)+\\frac{\\sqrt{2}}{\\sqrt{3}}a_2^2\\left(a_4+a_6\\right)-\\frac{1}{\\sqrt{6}}a_2^2a_5+\\frac{\\sqrt{3}}{\\sqrt{2}}a_3^2 a_5\\\\\n&\\qquad\\qquad+a_1 a_3 \\left(a_4-a_6\\right) + \\sqrt{2}a_2 a_3\\left(a_6-a_4\\right) + \\frac{1}{\\sqrt{3}}a_1 a_2\\left(a_6+a_4\\right) +\\frac{2}{\\sqrt{3}} a_1 a_2 a_5.\n\\end{split}\n\\end{align}\n\nThe invariant term constructed from the third $(\\boldsymbol{\\xb}\\otimes\\boldsymbol{6})$ in Eq.~(\\ref{eq:tensor666b}) is totally antisymmetric under the permutation of the indices of the three $\\boldsymbol{6}$s in the LHS of Eq.~(\\ref{eq:tensor666a}) and therefore it vanishes. \n\nConsider the tensor product space $\\phi\\otimes\\phi\\otimes\\phi$. Any specific alignment of $\\phi$, eg.~the VEV $\\langle\\phi\\rangle$, has a corresponding alignment in the tensor product space, eg.~$\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle$. We have shown that the tensor product space $\\phi\\otimes\\phi\\otimes\\phi$ contains two non-vanishing invariant directions, the ones that correspond to $T_{\\boldsymbol{6}}$ and $T'_{\\boldsymbol{6}}$. We take the projection of the alignment $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle$ along these directions and call them $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}$ and $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}$ respectively. It can be shown that, for every VEV given in Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm})\\footnote{The reason for changing the signs of Eqs.~(\\ref{eq:vevlisttxmp},~\\ref{eq:vevlisttxmm}) compared to Eqs.~(\\ref{eq:vevtxmp},~\\ref{eq:vevtxmm}) was to ensure that for all the VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), the values of $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}$ (and $\\langle\\phi\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}$) have the same sign.}, we get $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T_{\\boldsymbol{6}}}=\\sqrt{3}$ and $\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle\\otimes\\langle\\phi\\rangle_{T'_{\\boldsymbol{6}}}=\\sqrt{3}\\left(5\\sqrt{2}-7\\right)$. In other words $|T_{\\boldsymbol{6}}-\\sqrt{3}|^2$ and $|T'_{\\boldsymbol{6}}-\\sqrt{3}\\left(5\\sqrt{2}-7\\right)|^2$ have the minimum value zero when the flavon field acquires any of the four VEVs. Thus we construct the flavon potential\n\\begin{equation}\\label{eq:commonpot}\n\\left|\\phi^\\dagger\\phi-(8-4\\sqrt{2})\\right|^2 + k_{\\nu1} \\left|T_{\\boldsymbol{6}}-\\sqrt{3}\\right|^2 + k_{\\nu2} \\left|T'_{\\boldsymbol{6}}-\\sqrt{3}\\left(5\\sqrt{2}-7\\right)\\right|^2\n\\end{equation}\nwhere $k_{\\nu 1}$ and $k_{\\nu 2}$ are positive constants. This potential, Eq.(\\ref{eq:commonpot}), has minima at the required VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), as required. Further analysis shows that the points of minima are not discrete, but rather they form a continuous set. In order to remove this ambiguity and ensure a discrete set of minima we add more invariant terms to the potential. We construct such terms by coupling the triplet flavons $\\phi_e$, $\\phi_\\mu$ and $\\phi_\\tau$ with sextet flavon $\\phi$.\n\nFirst we construct sextets ($\\boldsymbol{\\xb}$s) by combining two triplets ($\\boldsymbol{\\tb}$s) using the conjugate forms of Eqs.~(\\ref{eq:tensor3pqr},~\\ref{eq:tensor3pqr2}). The various possiblilities are $\\phi_e\\otimes \\phi_e$, $\\phi_\\mu\\otimes \\phi_\\mu$, $\\phi_\\tau\\otimes \\phi_\\tau$, $\\phi_\\mu\\otimes \\phi_\\tau$, $\\phi_\\tau\\otimes \\phi_e$ and $\\phi_e\\otimes \\phi_\\mu$. The sextets so constructed are combined with the sextet flavon $\\phi$ to obtain invariants, namely $T_{ee}$, $T_{\\mu\\mu}$, $T_{\\tau\\tau}$, $T_{\\mu\\tau}$, $T_{\\tau e}$ and $T_{e\\mu}$. In the tensor product space $\\phi_\\alpha\\otimes \\phi_\\beta \\otimes \\phi$ where $\\alpha, \\beta = e, \\mu, \\tau$, we consider the specific alignment $\\langle\\phi_\\alpha\\rangle\\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle$ which corresponds to the required VEVs of the flavons given in Eqs.~(\\ref{eq:leptvev},~\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). As was done previously, we take the projection of this alignment along the direction of the invariant $T_{\\alpha\\beta}$, i.e.~$\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta}}$. Finally we construct the potential term\n\\begin{equation}\\label{eq:uniquepot} \n\\displaystyle \\sum_{\\alpha,\\beta} k'_{\\alpha\\beta} \\left|T_{\\alpha\\beta}-\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta\\nu}}\\right|^2\n\\end{equation}\nwhere $k'_{\\alpha\\beta}$ are positive constants and the summation is over $(\\alpha,\\beta)=$ $(e,e)$, $(\\mu,\\mu)$, $(\\tau,\\tau)$, $(\\mu,\\tau)$, $(\\tau,e)$ and $(e,\\mu)$. The values of $\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta}}$ corresponding to the VEVs, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}), are given in Table~\\ref{tab:vev}. \n\n\\begin{table}[]\n\\begin{center}\n\\begin{tabular}{||c||c|c|c|c||}\n    \\hline \\hline\n    \\multirow{2}{*}{} & \\multicolumn{4}{c||}{$\\langle\\phi_\\alpha\\rangle \\otimes \\langle\\phi_\\beta\\rangle \\otimes \\langle\\phi\\rangle_{T_{\\alpha\\beta\\nu}}$} \\\\\n    \t\t\t\t    \\cline{2-5}\n    $(\\alpha,\\beta)$&$\\txm_{(\\chi=+\\frac{\\pi}{16})}$& $\\txm_{(\\chi=-\\frac{\\pi}{16})}$& $\\tpm_{(\\phi=+\\frac{\\pi}{16})}$& $\\tpm_{(\\phi=-\\frac{\\pi}{16})}$\\\\\n    &({\\small Eq.~(\\ref{eq:vevlisttxmp})}) & ({\\small Eq.~(\\ref{eq:vevlisttxmm})}) & ({\\small Eq.~(\\ref{eq:vevlisttpmp})}) & ({\\small Eq.~(\\ref{eq:vevlisttpmm})}) \\\\\n    \\hline \\hline\n    $(e,e)$\t\t&$1$ \t       & $1$         & $-1$         & $-1$           \\\\\n    \\hline \n    $(\\mu,\\mu)$\t\t&$\\frac{1-\\sqrt{2}}{2}+i\\frac{1+\\sqrt{2}}{2\\sqrt{3}}$  \t& $-\\frac{1}{2}+i\\frac{-1+2\\sqrt{2}}{2\\sqrt{3}}$          & $\\frac{(-1+\\sqrt{2})(1+\\sqrt{3})(1-i\\sqrt{3})}{2\\sqrt{6}}$          & $\\frac{(-1+\\sqrt{2})(-1+\\sqrt{3})(1-i\\sqrt{3})}{2\\sqrt{6}}$           \\\\\n    \\hline \n    $(\\tau,\\tau)$\t& $\\frac{1-\\sqrt{2}}{2}-i\\frac{1+\\sqrt{2}}{2\\sqrt{3}}$\t&   $-\\frac{1}{2}-i\\frac{-1+2\\sqrt{2}}{2\\sqrt{3}}$         &  $\\frac{(-1+\\sqrt{2})(-1+\\sqrt{3})(1+i\\sqrt{3})}{2\\sqrt{6}}$    & $\\frac{(-1+\\sqrt{2})(1+\\sqrt{3})(1+i\\sqrt{3})}{2\\sqrt{6}}$           \\\\\n    \\hline \n    $(\\mu,\\tau)$\t&$\\frac{3-\\sqrt{2}}{2\\sqrt{3}}-i\\frac{1}{2}$ \t       \t& $\\frac{3-\\sqrt{2}}{2\\sqrt{3}}-i\\frac{1}{2}$          & $-\\frac{1+\\sqrt{2}}{2\\sqrt{3}}+i\\frac{1}{2}$          & $-\\frac{1+\\sqrt{2}}{2\\sqrt{3}}+i\\frac{1}{2}$           \\\\\n    \\hline\n    $(\\tau,e)$\t\t&$-\\frac{1}{2\\sqrt{6}}+i\\frac{1}{2\\sqrt{2}}$ \t       \t& $-\\frac{1}{2\\sqrt{6}}+i\\frac{1}{2\\sqrt{2}}$          & $\\frac{-1+\\sqrt{2}}{2\\sqrt{6}}-i\\frac{-1+\\sqrt{2}}{2\\sqrt{2}}$          & $\\frac{-1+\\sqrt{2}}{2\\sqrt{6}}-i\\frac{-1+\\sqrt{2}}{2\\sqrt{2}}$           \\\\\n    \\hline \n    $(e,\\mu)$\t\t&$\\frac{-1+2\\sqrt{2}}{2\\sqrt{6}}-i\\frac{1}{2\\sqrt{2}}$ \t& $\\frac{3-2\\sqrt{2}}{2\\sqrt{6}}-i\\frac{1}{2\\sqrt{2}}$          & $\\frac{1-\\sqrt{2}}{2\\sqrt{6}}+i\\frac{(-1+\\sqrt{2})(3-2\\sqrt{3})}{6\\sqrt{2}}$   & $\\frac{1-\\sqrt{2}}{2\\sqrt{6}}+i\\frac{(-1+\\sqrt{2})(3+2\\sqrt{3})}{6\\sqrt{2}}$           \\\\\n    \\hline \\hline\n  \\end{tabular}\n\\end{center}\n\\caption{The projection of the tensor products of VEVs along the corresponding invariant directions.}\n\\label{tab:vev}\n\\end{table}\n\nThe terms given in Eq.~(\\ref{eq:commonpot}) and Eq.~(\\ref{eq:uniquepot}) form a potential with a discrete set of minima which includes the requied VEV, Eqs.~(\\ref{eq:vevlisttxmp}-\\ref{eq:vevlisttpmm}). Note that, we use highly specific constants in our potential, eg. $(8-4\\sqrt{2})$, $\\sqrt{3}$ and $\\sqrt{3}\\left(5\\sqrt{2}-7\\right)$ in Eq.~(\\ref{eq:commonpot}). Put another way, we may be able to construct any given mass matrix by suitably tweaking such constants. This situation can be made a lot less arbitrary by imposing additional symmetries on top of $\\Sigma(72\\times3)$. In this context, we can not help noticing the appearance of $\\frac{\\pi}{16}$ and the factor $-1+\\sqrt{2}$ ($=\\tan \\frac{\\pi}{8}$) throughout this paper. They give hints towards the presence of additional symmetries like $Z_{16}$. Such topics are beyond the scope of this paper, but will be discussed in a future publication.\n\n\\providecommand{\\href}[2]{#2}\\begingroup\\raggedright","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nWhile the standard model proves an excellent framework for fundamental interactions at energy scales up to at least the sub TeV range, it nevertheless leaves a number of fundamental problems. Theoretically, one of the most outstanding puzzles centers on the origin and mechanism of electroweak symmetry breaking, and the quantum mechanical stability of the hierarchy generated between the electroweak scale and the Planck scale. In addition, recent astrophysical observations \\cite{DarkMatter} concord with $0.094 < \\Omega_{CDM}h^2 < 0.129$, indicating the presence of cold non-baryonic dark matter as the principle form of matter in the Universe, of which the standard model provides no explanation. The most popular candidate for dark matter assumes a non-standard model, stable, electrically neutral, and weakly interacting particle -- the WIMP hypothesis. Clearly, from both a theoretical and phenomenological perspective, the standard model requires extension. In this letter we wish to explore the consequences of the universal extra dimension (UED) \\cite{Appelquist:2000nn} extension of the standard model.\n\nIn the UED scenario, all standard model particles can freely propagate in the bulk of one or more extra dimensions and thus each standard model particle is associated with a Kaluza-Klein (KK) tower of states. Each state in the KK tower has the same spin as its standard model counterpart. An important consequence of UED models concerns the existence of a conserved discrete symmetry, KK-parity, which guarantees the stability of the lightest KK particle (LKP) and thus provides a dark matter candidate. Suitable thermal relic dark matter candidates that have been studied extensively \\cite{Servant:2002aq} include the first KK-excitations of the hypercharge boson, the photon, and the neutrino {\\it i.e. } $B^{(1)}$, $\\gamma^{(1)}$, or $\\nu^{(1)}$.\n\nThe tree-level mass spectrum of the KK-excitations of UED models reveals a nearly degenerate spectrum. As an example, a UED model with one extra-dimension compactified on an $S_1\/Z_2$ orbifold of radius $R$, leads to the tree level mass relation \n\\begin{equation}\\label{mass} m^{(n)} = \\sqrt{(n\/R)^2 + (m^{(0)})^2} \n\\end{equation} \nfor the n-th KK mode, where $m^{(0)}$ constitutes the zero-mode mass ({\\it i.e. } the standard model particle value). Quantum corrections typically dominate over zero mode level contributions and therefore the resulting mass spectrum depends crucially on radiative effects. In general, a moderately split UED mass spectrum \\cite{Cheng:2002ej} develops. In the minimal UED model (MUED) \\cite{Cheng:2002ej}, one-loop calculations suggest that the LKP is well approximated by the KK hypercharge boson, $B^{(1)}$, and numerous studies have examined the thermal production and prospects of direct and indirect detection of $B^{(1)}$ and $\\gamma^{(1)}$ LKP dark matter \\cite{Servant:2002aq,KKDMcollection}. \n\nHowever, the non-renormalizability of UED models imply the existence of an ultraviolet cut-off, typically of the order of a few tens of TeV, at which point the model requires UV completion. As such, UED models must be regarded as an effective theory. The presence of incalculable boundary terms arising from the UV complete theory can potentially change the mass spectrum, resulting in different LKP candidates. Recent studies of the relic density of $B^{(1)}$ LKP dark matter with the full MUED spectrum \\cite{coannmat,coannkribs} reveal substantial observational tension with constraints from electroweak precision data \\cite{Flacke:2005hb}. Thus, non-minimal models with brane-localized terms appear as a likely alternative if UED models are to provide a successful phenomenology. Furthermore, model independent studies \\cite{Servant:2002aq} show that $B^{(1)}$, $\\gamma^{(1)}$, and $\\nu^{(1)}$ can all be thermally produced with abundances sufficient to provide the dark matter. Constraints on minimal UED models from limits on weak neutral current nucleon-$\\nu^{(1)}$ elastic scattering in direct searches, together with thermal dark matter production mechanisms, disfavour $\\nu^{(1)}$ dark matter \\cite{Servant:2002hb}. However, given the need for possible new non-minimal interactions, we consider further consequences of KK-neutrino dark matter model-independently.\n\nWhile there exists compelling evidence for dark matter in the form of WIMPS, there also exists strong constraints on possible electromagnetic interactions of dark matter, even in the limit of complete charge neutrality. A neutral Dirac fermion can posses both a permanent magnetic dipole moment, $\\mu$, and a permanent electric dipole moment, $d$, arising from the dimension-five operator, \n\\begin{equation} \n\\mathcal{L}_{D} = \\frac{i}{2} \\bar f\n\\sigma_{\\mu\\nu}\\left(\\mu +\\gamma_5 d \\right)f F^{\\mu\\nu}. \n\\end{equation}\nWhile the presence of the magnetic dipole moment does not violate any discrete symmetries, the electric dipole moment requires the violation of parity and CP. Severe constraints exist on the dipole moments of $\\sim 1$ TeV dark matter WIMPS \\cite{Sigurdson:2004zp}. (We are aware that the authors of \\cite{Sigurdson:2004zp} are currently revising their estimates and, as a result, the strength of the constraints in \\cite{Sigurdson:2004zp} may change substantially \\cite{private}.) Thus, if a model predicts Dirac fermionic dark matter, it is important to determine the strength of the induced dipole moment.\n\nSince the KK-neutrino of UED models is a Dirac fermion from the 4-dimensional perspective, UED models that assume KK-neutrino dark matter are constrained by the strength of the induced dipole operator.\n\nIn this letter we derive model independent bounds on KK-neutrino dark matter by examining the induced dipole moment and comparing the predictions with the current observational bound. Section II provides a brief review on the properties of KK-fermions in UED models along with a discussion on dipole moments relevant to the calculation presented in section III. Finally, in section IV we present our conclusions.\n\n\\section{KK-neutrino LKP and induced dipole moments}\n\nIn UED models, standard model fields become identified with the zero modes of KK towers of states once the extra dimensions are integrated out of the theory. For concreteness, we will restrict our discussion to five dimensional UED models compactified on an $S_1\/Z_2$ orbifold. Five dimensional theories do not posses a chirality condition since $\\gamma_5$ becomes part of the five dimensional Clifford Algebra. Thus, in order to arrive at a chiral theory at the zero mode level, we require one five dimensional \\emph{Dirac} spinor for every \\emph{Weyl} spinor of the standard model. By use of the orbifold boundary conditions half the number of states project out of the spectrum leaving a zero mode level chiral theory. In the case of the lepton doublet, the decomposition of the corresponding five dimensional Dirac spinor reads ({\\it c.f. } {\\it e.g. }\n\\cite{Appelquist:2000nn}),\n\\begin{equation} \\label{fermiondecomp}\n\\begin{split}\n\\hat{\\mathcal{L}}(x^{\\mu},y) = \\frac{1}{\\sqrt{\\pi R}}\nP_L\\mathcal{L}(x^\\mu) +  \\sqrt{\\frac{2}{\\pi\nR}}\\sum_{n=1}\\left[P_L\\mathcal{L}(x^\\mu)\\cos(\\frac{ny}{R})\\right.&\\\\\n  \\left.+P_R\\mathcal{L}(x^\\mu)\\sin(\\frac{ny}{R})\\right]&\\\\\n\\end{split}\n\\end{equation}\nwhere $\\hat{\\mathcal{L}}$ denotes the five dimensional Dirac spinor, $\\mathcal{L}$ denotes a four dimensional \\emph{4\ncomponent} spinor with $P_{L,R} = (1\\pm\\gamma_5)\/2$, and $\\hat{\\mathcal{L}}$ is associated with the (4D left-handed) lepton doublet via the identification $P_L\\mathcal{L}(x^\\mu)\\leftrightarrow (\\nu_L,e_L)$. We see from eq(\\ref{fermiondecomp}) that $\\hat{\\mathcal{L}}$ contains a purely left-handed zero mode while all higher KK modes contain both chiralities. Therefore, the non-zero mode KK-level fermions appear as Dirac spinors from the 4-dimensional perspective. Specifically, the KK-neutrino charged under the weak $SU(2)$ appears with both chiralities.\n\nTranslational invariance in the fifth direction implies (once the extra dimension becomes integrated out) that a UED model compactified solely on $S_1$ conserves KK-excitation-number in every vertex. The orbifold reduces the conserved KK-number to a discrete $Z_2$ symmetry, called KK-parity. The conservation of KK-parity in every vertex implies the stability of the lightest KK-particle.\n\nSince the fermions receive mass through Yukawa couplings after electroweak symmetry breaking and through the KK-level expansion itself, the theory, in general, requires a unitary transformation to connect mass and gauge eigenstates. For example, in the lepton sector at the $j$th KK-level we have,\n\\begin{equation} \n\\left(\\begin{array}{c} \\mathcal{E}^j \\\\ \\mathcal{L}^j \\end{array} \\right) =\n\\left(\\begin{array}{cc}-\\gamma_5\\cos\\alpha_j & \\sin\\alpha_j\\\\\n\\gamma_5\\sin\\alpha_j & \\cos\\alpha_j\n\\end{array}\\right)\\left(\\begin{array}{c} \\mathcal{E}^{\\prime j} \\\\\n\\mathcal{L}^{\\prime j} \\end{array} \\right) \n\\end{equation} \nwhere $\\mathcal{E}^j$ and $\\mathcal{L}^j$ denote the $j^{\\mbox{th}}$ KK mode of the lepton $SU(2)$ singlet and doublet respectively, and where \\begin{equation} \\tan (2\\alpha_j) = \\frac{m_l^{(0)}}{j\/R}. \\end{equation} As the lepton masses are small compared to $j\/R$, we ignore the effects of the mixing matrix for the remainder of this letter. Furthermore, we ignore the effects of lepton flavour violation and neutrino mixing.\n\nA Dirac fermion can posses a dipole moment, derived from the transition amplitude ({\\it c.f. } {\\it e.g. } \\cite{neutrinobook}), \n\\begin{equation} \nT =\n-i\\epsilon^\\mu q^\\nu \\bar f (p^\\prime) \\sigma_{\\mu\\nu} \\left(F_2 + G_2\n\\gamma_5\\right) f(p) \n\\end{equation} \nwhere $q=p^\\prime -p$. The magnetic moment is defined by $\\mu = F_2(0)$ while the electric dipole is defined by $ d = G_2(0)$. At low energies compared to the mass of the particle, the photon does not distinguish between $\\mu$ or $d$ provided that one ignores other time-reversal violating observables. We will focus on the limits established on $\\mu$ throughout. On dimensional grounds, we naively expect the induced magnetic dipole moment to scale as, \n\\begin{equation} \n\\mu \\lesssim e \\frac{M_{\\nu^{(1)}}}{R^{-2}} \\simeq \\frac{e}{M_{\\nu^{(1)}}} = 1.022\\times\n10^{-6}\\mu_B\\left(\\frac{\\mathrm{TeV}}{M_{\\nu^{(1)}}}\\right) \n\\end{equation}\nwhere $M_{\\nu^{(1)}}$ denotes the mass of KK-neutrino, $\\nu^{(1)}$, and $R$ indicates the radius of compactification.\n\n\\section{Calculation}\n\nThe KK-neutrino develops a magnetic dipole moment through the diagrams tabulated in figure \\ref{Figgraphs}.\n\\begin{figure}[h]\n     \\begin{center}\n     \t\\vspace{5pt}\n \t\\hspace{-100pt}{\\includegraphics[height=120pt,width=120pt]{testgraph.ps}}\n     \t\\vspace{45pt}\n    \t\\caption{Magnetic dipole moment inducing one loop Feynman graphs for\n             the first KK-excitation of the neutrino contained in the $\\mathrm{SU(2)}_L$ doublet in UED models.\n             KK-number assignments are denoted in graph a), to be understood analogously in graphs b) - f).\n             Graphs with dashed lines denote the Goldstone modes along their\n             KK-excitations, and the KK-excitation of the Higgs scalar. The Feynman rules are taken from \\cite{Buras:2002ej}.}\n\t\\label{Figgraphs}     \n     \\end{center}\n\\end{figure}\nIn general, the entire Kaluza-Klein tower of states participate, however we estimate the leading order effect by considering only the first level KK excitations. Furthermore, we restrict our calculation to KK-number conserving graphs since KK-number violation proceeds with volume suppression. For simplicity, we ignore flavour violation in the lepton sector and we assume that the lightest KK-neutrino is $\\nu_e^{(1)}$. Relaxing these assumptions will not significantly alter our conclusions.\n\nAs we make no assumptions on the exact UED spectrum, we consider the mass of the KK-$W$ ($W^{(1)}$) and the KK-electron ($e^{(1)}$) as free parameters. In our numerical calculations we do not consider a KK-electron\/KK-neutrino mass difference in excess of 5\\% as any substantial splitting will lead to unacceptably large contributions to the $T$ parameter.\n\nThe relevant UED Feynman rules are listed in \\cite{Buras:2002ej} and we calculate in the Feynman-'t Hooft gauge. In the limit of exact $M_{\\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy and where the effects of Yukawa couplings are ignored, we arrive at the semi-analytic result,\n\\begin{equation} \n\\label{dipole_result} \n\\begin{split}\n\\mu =   & \\frac{eg^2}{(4\\pi)^2}\\frac{1}{M_{\\nu^{(1)}}} \\times\\\\\n\t&\\left\\{\\frac{3}{2}\\ln (\\epsilon)+r+\\frac{7}{2}+\\frac{1}{2r}-\\frac{5}{2}(r-1)\\ln \\left(\\frac{r}{r-1}\\right)\\right.\\\\\n\t&\\left. -(r-1)^2\\ln \\left(\\frac{r}{r-1}\\right) + \\mathcal{O}(\\sqrt{\\epsilon})\\right\\}\t\n\\end{split}\n\\end{equation} \nwith the approximation $\\epsilon\\equiv M_{W^{(0)}}^2\/M_{\\nu^{(1)}}^2\\ll 1$ and $r\\equiv M_{W^{(1)}}^2\/M_{\\nu^{(1)}}^2\\simeq 1$.\nNumerically, we find agreement with our semi-analytical result as seen in figure \\ref{plot_1}.\n\\begin{figure}[ht!]\n   \\newlength{\\picwidtha}\n   \\setlength{\\picwidtha}{3.5in}\n   \\begin{center}\n       \\resizebox{\\picwidtha}{!}{\\includegraphics{plot_v2.eps}}\n   \\end{center}\n   \\caption{Magnetic moment of the KK-neutrino $\\nu^{(1)}$ vs.\n            KK dark matter mass $M_{\\nu^{(1)}}$ for varying $M_{e^{(1)}}$, and $M_{W^{(1)}}$.\n            Upper line (solid): $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ degeneracy with tree-level $M_{W^{(1)}}$  ({\\it c.f. } analytical result eq(\\ref{dipole_result})).\n            Middle line (dashed-dotted): 5\\% splitting of $M_{e^{(1)}} \/ M_{\\nu^{(1)}}$ with with tree-level $M_{W^{(1)}}$.\n            Lower line (dashed): $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ degeneracy with 5\\% splitting of $M_{W^{(1)}} \/ M_{\\nu^{(1)}}$}\n\t    \\label{plot_1}\n\\end{figure}\n\n  \nIn figure \\ref{plot_1} we display the result of the dipole moment as a function of the KK-neutrino mass. The upper curve illustrates the magnetic dipole moment with exact degeneracy $M_{e^{(1)}}$-$M_{\\nu^{(1)}}$ while holding the KK-$W$ mass at its tree level value. The middle curve plots the effect of a KK-electron 5\\% heavier than the KK-neutrino. This has an $\\mathcal{O}(1)$ effect on the dipole moment. The predicted value of the dipole moment exceeds the current upper bound for a KK-neutrino mass $M_{\\nu^{(1)}}\\sim 1 \\hspace{1mm}\\mathrm{TeV}$ by more than five orders of magnitude \\cite{Sigurdson:2004zp}. The lower curve displays the effect of maintaining $M_{\\nu^{(1)}}$-$M_{e^{(1)}}$ degeneracy while varying the KK-$W$ mass. Allowing the mass difference, $M_{W^{(1)}}$-$M_{\\nu^{(1)}}$, to vary by up to 5\\% has at most an $\\mathcal{O}(10)$ effect.\n\nWe should note that the calculation presented above determines only the radiatively induced part of the KK-neutrino magnetic dipole moment. The presence of boundary terms or effects arising from the UV complete theory may also contribute a non-renormalizable dimension-five dipole operator which, a priori, may be of the same order as the radiative part itself.\n\n\\section{Conclusion}\n\nUED models have attracted attention as a possible extension to the standard model. A particular appealing feature of the model class centers on the existence of plausible dark matter candidates as the result of KK-parity conservation. While the minimal UED model suggests $B^{(1)}$ dark matter \\cite{Cheng:2002ej}, detailed studies of the relic abundance in the minimal UED model \\cite{coannmat,coannkribs, Matsumoto:2005uh} in combination with electroweak precision constraints \\cite{Flacke:2005hb} show strong observational tension, and thus provide motivation for new possible UED model building avenues. The need for non-minimality has been reported \\cite{Hewett:2004py}. Extensions of the MUED scenario by incalculable boundary terms arising from the UV completion of the model can lead to a different LKP and therefore different possible dark matter candidates. We have taken a model independent approach, following \\cite{Servant:2002aq}, and examined the consequences of UED KK-neutrino dark matter.\n\nAs the KK-neutrino is a Dirac fermion, UED models predict an induced KK-neutrino dipole moment. We find that the induced dipole moment, typically $\\mu \\lesssim 10^{-7} \\mu_B$, strongly conflicts -- by over five orders of magnitude -- with the current observational bounds stated in \\cite{Sigurdson:2004zp} for TeV scale dipole dark matter. We reiterate that the constraints provided by \\cite{Sigurdson:2004zp} are currently under revision, and the strength of the stated bounds are expected to change \\cite{private}. The constraints on magnetic dipole moments given in \\cite{Sigurdson:2004zp} would provide the strongest limits on KK-neutrino dark matter. Even in the absence of the strong limits provided by \\cite{Sigurdson:2004zp}, the bounds on dipole moments remain an important constraint on future model building. Not only will new, non-minimal models that predict KK-neutrino LKP need to circumvent the constaints provided by \\cite{Servant:2002hb}, but will also have to evade the constraints provided by radiatively induced magnetic dipole moments, which are generically at least as large as the current experimental bounds. \n\nWhile we have restricted our discussion to five-dimensional models compactified on $S_1\/Z_2$, we expect that the qualitative features carry over to UED models with multiple extra dimensions. We have also assumed the absence of any fine-tuning between the radiatively induced magnetic dipole moment and possible non-renormalizable dimension-five dipole operators arising from the UV complete theory or boundary terms.\n\nOur results indicate that observational limits on dipole dark matter can place significant constraints on UED scenarios where the KK-neutrino is the LKP dark matter candidate.\n\n\n\n\\section{Acknowledgments} \n\nWe would like to thank J.~March-Russell, G.~Starkman, B.~A.~Campbell, and K.~Sigurdson for useful discussions. DM wishes to acknowledge the support of the Natural Science and Engineering Research Council of Canada and the Canada-United Kingdom Millennium Research Fellowship. The work of TF is supported by ``Evangelisches Studienwerk Villigst e.V.\" and PPARC Grant No. PPA\/S\/S\/2002\/03540A. This work was also supported by the ``Quest for Unification\" network, MRTN 2004-503369.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nPower system operations aim to optimally utilize available electricity generation resources to satisfy projected demand, at minimal cost, subject to various physical transmission and operational security constraints. Traditionally, such operations involve numerous sub-tasks, including short-term load forecasting, unit commitment, economic dispatch, voltage and frequency control, and interchange scheduling between distinct operators. Most recently, renewable generation units in the form of geographically distributed wind and solar farms have imposed the additional requirement to consider uncertain generation output, increasingly in conjunction with the deployment of advanced storage technologies such as pumped hydro. Growth in system size and the introduction of significant generation output uncertainty contribute to increased concerns regarding system vulnerability. Large-scale blackouts, such as the Northeast blackout of 2003 in North America and, more recently, the blackout of July 2012 in India, impact millions of people and result in significant economic costs. Similarly, failure to accurately account for renewables output uncertainty can lead to large-scale forced outages, as in the case of ERCOT on February 26, 2008. Such events have led to an increased focus on power systems reliability, with the goal of mitigating against failures due to both natural causes and intelligent adversaries.\n\nOptimization methods have been applied to power system operational problems for several decades; Wood and Wollenberg \\cite{Wood1996} provide a brief overview. The coupling of state-of-the-art implementations of core optimization algorithms (including simplex, barrier, and mixed-integer branch-and-cut algorithms) and current computing capabilities (e.g., inexpensive multi-core processors) enable optimal decision-making in real power systems. One notable example involves the unit commitment problem, which is used to determine the day-ahead schedule for all generators in a given operating region of an electricity grid. A solution to the unit commitment problem specifies, for each hour in the scheduling horizon (typically 24 hours), both the set of generators that are to be operating and their corresponding projected power output levels. The solution must satisfy a large number of generator (e.g., ramp rates, minimum up and down times, and capacity limits) and transmission (e.g., power flow and thermal limit) constraints, achieving a minimal total production cost while satisfying forecasted demand. The unit commitment problem has been widely studied, for over three decades. For a review of the relevant literature, we refer to \\cite{Hobbs2001} and the more recent \\cite{Padhy2004}. Many heuristic (e.g., genetic algorithms, tabu search, and simulated annealing) and mathematical optimization (e.g., integer programming, dynamic programming, and Lagrangian relaxation) methods have been introduced to solve the unit commitment problem. Until the early 2000s, Lagrangian relaxation methods were the dominant approach used in practice. However, mixed-integer programming implementations are either currently in use or will soon be adopted by all Independent System Operators (ISOs) in the United States to solve the unit commitment problem \\cite{fercreport}.\n\nSecurity constraints (i.e., which ensure system performance is sustained when certain components fail) in the context of unit commitment are now a required regulatory element of power systems operations. The North American Electric Reliability Corporation (NERC) develops and enforces standards to ensure power systems reliability in North America. Of  strongest relevance to security constraints for unit commitment is the NERC Transmission Planning Standard (TPL-001-0.1, TPL-002-0b, TPL-003-0b, TPL-004-0a, \\cite{NERC2011}). The TPL specifies 4 categories of operating states, labeled A through D. Category A represents the baseline ``normal\" state, during which there are no system component failures. Category B represents so called $N$-$1$ contingency states, in which a single system component has failed (out of a total of $N$ components, including generators and transmission lines). NERC requires no loss-of-load in both categories A and B, which collectively represent the vast majority of observed operational states. Categories C and D of the TPL represent more extreme states, in which multiple system components fail (near) simultaneously. Large-scale blackouts, typically caused by cascading failures, are Category D events. Such failure states are known as $N$-$k$ contingencies in the power system literature, where $k$ ($k \\geq 2$) denotes the number of component failures. In contrast to categories A and B, the regulatory requirements for categories C and D are vaguely specified, e.g., ``some\" loss of load is allowable, and it is permissible to exceed normal transmission line capacities by unspecified amounts for brief time periods.\n\nThe computational difficulty of security-constrained unit commitment is well-known, and is further a function of the specific TPL category that is being considered. The unit commitment problem subject to $N$-$1$ reliability constraints is, given the specific regulatory requirements imposed for category B events of the TPL, addressed by system operators worldwide. However, we observe that it is often solved approximately in practice, specifically in the context of large-scale (ISO-scale) systems \\cite{personalcommunication}. For example, a subset of contingencies based on a careful engineering analysis is often used to obtain a computationally tractable unit commitment problem. Alternatively, the unit commitment problem can be solved without considering contingencies, and the solution can be subsequently ``screened\" for validity under a subset of contingencies (again identified by engineering analysis). Additional constraints can then be added to the master unit commitment problem, which is then resolved; the process repeats until there is no loss-of-load in the contingency states. We raise this issue primarily to point out that even the full $N$-$1$ problem is not considered a ``solved\" problem in practice, such that advances (including those introduced in this paper) in the solution of unit commitment problems subject to general $N$-$k$ reliability constraints can potentially impact the practical solution of the simpler $N$-$1$ variant.\n\nNumerous researchers have introduced algorithms for solving both the security-constrained unit commitment problem and the simpler, related problem of security-constrained optimal power flow. In the latter, the analysis is restricted to a single time period, and binary variables relating to generation unit statuses are fixed based on a pre-computed unit commitment schedule. \\cite{Capitanescu2011} provides a recent review of the literature on security-constrained optimal power flow. Of specific relevance to our work is the literature on security-constrained optimal power flow in situations where large numbers of system components fail.  This literature is mostly based on worst-case network interdiction analysis and  includes solution methods based on bi-level and mixed-integer programming (see \\cite{Salmeron2004,Salmeron2009,Arroyo2010,Fan2011,Zeng2013,Zeng2014}) and graph algorithms (see \\cite{Pinar2010,Bienstock2010,Fan2011,LeRoDoPi06,LePiRo08}).\n\nFollowing the Northeast US blackout of 2003, significant attention was focused on developing improved solution methods for the security-constrained unit commitment problem. In particular, various researchers introduced mixed-integer programming and decomposition-based methods for more efficiently enforcing $N$-$1$ reliability, e.g., see \\cite{Fu2005,Fu2006,Wang2008,Lotfjou2010,Hedman2010,ONeill2010}. However, due to its computational complexity, security-constrained unit commitment considering the full spectrum of NERC reliability standards has not attracted a comparable level of attention until very recently. Specifically, \\cite{Street2011a} and \\cite{Wang2012} consider the case of security-constrained unit commitment under the more general $N$-$k$ reliability criterion. Similarly, \\cite{Street2011a,Street2011b} and \\cite{Wang2012} use robust optimization methods for identifying worst-case $N$-$k$ contingencies.\n\nIn this paper, we extend the $N$-$k$ reliability criterion to yield the more general $N$-$k$-$\\boldsymbol \\varepsilon$ criterion. This new criterion dictates that for all contingencies of size $j$, $j \\in \\{1,\\cdots, k\\}$, at least $(1-\\varepsilon_ j)$ fraction of the total demand must be met, with $\\varepsilon_j \\in [0,1]$ and $\\varepsilon_1 \\le \\varepsilon_2 \\le \\cdots \\le \\varepsilon_k$. The primary motivation for introducing this metric is that it provides a practical  and quantifiable bound on system performance under Categories C and D TPL contingencies, and can easily be expressed in mathematical optimization models. We refer to the security-constrained unit commitment problem subject to $N$-$k$-$\\boldsymbol \\varepsilon$ reliability as the contingency-constrained unit commitment (CCUC) problem. In the CCUC, all contingencies with $k$ or fewer element failures (generation units or transmission lines) are implicitly considered when checking for the feasibility of post-contingency corrective recourse. The CCUC is formulated as a large-scale mixed-integer linear program (MILP). To solve the CCUC, we develop two decomposition strategies: one based on a Benders decomposition \\cite{Benders1962}, and another based on cutting planes derived from the solution of power system inhibition problems  \\cite{Chen2012a,Chen2014}.  We then show the computational effectiveness of our algorithms on a range of benchmark instances.\n\nOur specific contributions, as detailed in this paper, include:\n\\begin{itemize}\n  \\item We ensure the existence of a feasible post-contingency corrective recourse, taking into consideration generator ramping constraints and the no-contingency (nominal) state economic dispatch;\n  \\item We consider the loss of both generation units \\emph{and} transmission lines;\n  \\item We propose novel decomposition methods to solve the contingency-constrained CCUC efficiently, and show that models and methods proposed by \\cite{Hedman2010}, \\cite{ONeill2010}, \\cite{Street2011a}, \\cite{Street2011b}, and \\cite{Wang2012} are all special cases of our general approach.\n\\end{itemize}\n\nThe remainder of this paper is organized as follows. In Section \\ref{sec:model}, we formulate the MILP model for the contingency-constrained unit commitment problem under the $N$-$k$-$\\boldsymbol \\varepsilon$ reliability criterion. In Section \\ref{sec:solution}, two approaches based on decomposition methods are presented for solving this large-scale MILP. In Section \\ref{sec:experiments}, we test our algorithms on several IEEE test systems and a simplified model of the Western interconnection. Finally, we conclude in Section \\ref{sec:conclusions} with a summary of our results and directions for future research.\n\n\\section{Problem Formulation}\n\\label{sec:model}\nThis section presents our mixed-integer linear programming model for the contingency-constrained unit commitment (CCUC) problem. In Table~\\ref{tab:nom}, we introduce the core sets, parameters, and decision variables of the model. The baseline unit commitment formulation, without contingency constraints, is described in Section~\\ref{sec:model-standarduc}. We discuss key concepts involving $N$-$k$-$\\boldsymbol \\varepsilon$ contingency analysis in Section~\\ref{sec:model-contingencies}, which are subsequently illustrated on an example in Section~\\ref{ieee6_example}. Finally, we combine the baseline unit commitment model with $N$-$k$-$\\boldsymbol \\varepsilon$ contingency analysis in Section~\\ref{sec:model-ccuc}, for our contingency-constrained unit commitment model.\n\n\\begin{table}\n\\caption{Nomenclature}\\label{tab:nom}\n\\begin{tabular}[t]{|c|l|}\n\\hline\n\\multicolumn{2}{|c|}{ \\bf Sets and Indices} \\\\\n\\hline\n $\\mathcal I$&  Set of buses. Indexed by $i$ for individual buses, $i$ and $j$ for pairs of buses. \\\\\n      $I$ &   Number of buses. $I = |\\mathcal I|$. \\\\\n$\\mathcal G$ &  Set of generation units. Indexed by $g$. \\\\\n $G$ &  Number of generation units. $G = |\\mathcal G|$. \\\\\n$\\mathcal G_i$ &  Set of generation units located at bus $i \\in \\mathcal I$. \\\\\n$\\mathcal E$ &  Set of directed transmission lines connecting pairs of buses. Indexed by $e$. \\\\\n$E$ &  Number of transmission lines. $E = |\\mathcal E|$. \\\\\n$\\mathcal E_{.i}$ &  Set of transmission lines oriented into bus $i \\in \\mathcal I$. \\\\\n$\\mathcal E_{i.}$ &  Set of transmission lines oriented out of bus $i \\in \\mathcal I$. \\\\\n$i_e,j_e$ &  Tail bus $i_e$ and head bus $j_e$ of transmission line $e \\in \\mathcal E$. \\\\\n$\\mathcal T$ &  Set of time periods in the planning horizon. Indexed by $t \\in \\{1,2,\\cdots,T\\}$. \\\\\n$( \\mathcal I, \\mathcal G, \\mathcal E)$ & triple that defines a power system. \\\\\n\\hline\\hline\n\\multicolumn{2}{|c|}{ \\bf Parameters}\\\\\n\\hline\n$B_e$, $F_e$ &    Susceptance and power flow (i.e., thermal) limit of transmission line $e$.  \\\\\n$D_i^{t}$ &   Demand (load) at bus $i \\in  \\mathcal I$ at time $t $.  \\\\\n$D^{t} =  \\displaystyle \\sum_{i \\in \\mathcal I} D_{i}^{t}$ &  Total demand, summed across all buses, in period $t $.  \\\\\n$P^{\\min}_g$, $P^{\\max}_g$ &    Lower\/upper limits on power output for generation unit $g $.  \\\\\n$T_g^{d0}$, $T_g^{u0}$ &  Minimum time period generation unit $g\\in  \\mathcal G$ must be initially offline\/online.  \\\\\n $T_g^d$, $T_g^u$  &   Minimum time period generation unit $g\\in  \\mathcal G$ must remain offline\/online once  the unit \\\\& is shut down\/started up. \\\\\n$R_g^d,R_g^u$  & Maximum ramp-down and ramp-up rate for generation unit $g \\in  \\mathcal G$ between  adjacent \\\\& time periods.  \\\\\n$\\tilde{R}_g^d,\\tilde{R}_g^u$  & Maximum shutdown\/startup ramp rates for generation unit $g \\in  \\mathcal G$ for the time period \\\\&  in which $g$ is turned off\/on. \\\\\n $C_g^u,C_g^d$ &   Fixed startup\/shutdown cost for generation unit $g $.  \\\\\n $C_g^p(\\cdot)$ &   Production cost function for generation unit $g$. \\\\\n\\hline\\hline\n\\multicolumn{2}{|c|}{ \\bf Variables}\\\\\n\\hline\n  $x_g^t$ &  Binary variable indicating if a generation unit $g \\in  \\mathcal G$ is committed ($x_g^t=1$) or not \\\\& ($x_g^t=0$) at time $t $.\\\\\n$\\mathbf x^t$ &  Unit commitment decision vector for all generation units at  time $t$.\\\\\n$\\mathbf x$ &    $G\\times T$ unit commitment decision vector for all generation units over all time periods. \\\\\n $c_g^{ut},c_g^{dt}$ & Incurred startup\/shutdown cost for a generation unit $g \\in  \\mathcal G$ at time  $t $  (if unit $g$ is started \\\\& up or shut down at time $t$, the respective costs are $C_g^u$ and $C_g^d$. Otherwise, $C_g^u=C_g^d=0$.) \\\\\n$\\tilde p_g^{t}$ & No-contingency state power output by generation unit $g $  at time $t $. \\\\\n$\\tilde f_e^{t}$ & No-contingency state power  flow on transmission line $e $ at time  $t $. \\\\\n$\\tilde \\theta_i^{t}$ & No-contingency state phase angle at bus $i $ at time  $t $. \\\\\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\subsection{The Baseline Unit Commitment Model}\n\\label{sec:model-standarduc}\n\nWe now present our baseline unit commitment (BUC) formulation, without contingency constraints. Our formulation is based on the mixed-integer linear programming UC formulations introduced by \\cite{Carrion2006,Wu2010,Zheng2013}. We extend these formulations to capture network transmission constraints, in the form of a DC power flow model. Our BUC model is intended to reflect steady-state operational conditions, such that the system is in a no-contingency state. Consequently, we require that the demand at each bus $i \\in  \\mathcal I$ must be fully satisfied, i.e., no loss-of-load is allowed.\n\nOur BUC formulation for a power system $( \\mathcal I, \\mathcal G, \\mathcal E)$ is given as follows:\n\\begin{subequations}\\label{suc}\n\\begin{align}\n\\min_{\\mathbf x}\\quad &\\sum_{t \\in \\mathcal T} \\sum_{g \\in \\mathcal G}(c_g^{ut}+c_g^{dt})+ \\mathcal Q(\\mathbf x) \\label{suc-obj}& \\\\\n\\text{s.t.} \\quad & \\sum_{t=1}^{T_g^{u0}} (1-x_g^t) = 0 & \\forall g \\in \\mathcal G  \\label{suc-init-on} \\\\\n        & \\sum_{t=1}^{T_g^{d0}} x_g^t = 0  & \\forall g \\in  \\mathcal G      \\label{suc-init-off}\\\\\n\t&\\sum_{t'=t}^{t+T_g^u -1} x_g^{t'}\\geq T_g^u (x_g^t- x_g^{t-1}) & \\forall g \\in \\mathcal G, t = T_g^{u0}+1, \\cdots, T-T_g^u+1 \\label{suc-on}\\\\\n\t&\\sum_{t'=t}^T \\big (x_g^{t'}-(x_g^t- x_g^{t-1}) \\big)\\geq 0  & \\forall g \\in \\mathcal G, t=T-T_g^u+2, \\cdots, T \\label{suc-on-end}\\\\\n\t&\\sum_{t'=t}^{t+T_g^d-1} (1-x_g^{t'})\\geq T_g^d (x_g^{t-1} - x_g^t)  & \\forall g \\in \\mathcal G, t=T_g^{d0}+1, \\cdots, T-T_g^d+1\\label{suc-off}\\\\\n\t&\\sum_{t'=t}^T \\big ((1-x_g^{t'})-(x_g^{t-1} - x_g^t) \\big) \\geq 0  & \\forall g \\in \\mathcal G, t = T-T_g^d+2, \\cdots, T \\label{suc-off-end}\\\\                    \n\t&c_g^{ut} \\geq C_g^u (x_g^t - x_g^{t-1})  & \\forall g \\in  \\mathcal G,t \\in \\mathcal  T \\label{suc-c-up}\\\\\n\t&c_g^{dt} \\geq C_g^d (x_g^{t-1} - x_g^{t}) & \\forall g \\in  \\mathcal G,t \\in  \\mathcal T \\label{suc-c-down}\\\\\n\t&c_g^{ut}, c_g^{dt} \\geq 0 & \\forall g \\in \\mathcal  G,t \\in \\mathcal  T  \\label{buc_cost_vars}\\\\\n\t&x_g^t \\in \\{0,1\\} & \\forall g \\in  \\mathcal  G,t \\in  \\mathcal  T \\label{buc_binary_const}\n\\end{align}\n\\end{subequations}\n\nThe optimization objective \\eqref{suc-obj} is to minimize the sum of the startup costs $c_g^{ut}$, shutdown costs $c_g^{dt}$, and generation cost $\\mathcal Q(\\mathbf x)$.  Constraints \\eqref{suc-init-on} - \\eqref{buc_binary_const} include (in order): initial online requirements for generating units \\eqref{suc-init-on}; initial offline requirements for generating units \\eqref{suc-init-off}; minimum online constraints in nominal time periods for generating units \\eqref{suc-on}; minimum online constraints for the last $T_g^u$ time periods \\eqref{suc-on-end}; minimum offline constraints in nominal time periods for generating units \\eqref{suc-off}; minimum offline constraints for the last $T_g^d$ time periods \\eqref{suc-off-end}; startup cost computation \\eqref{suc-c-up}; shutdown cost computation \\eqref{suc-c-down}; non-negativity for startup\/shutdown costs \\eqref{buc_cost_vars}; and binary constraints for the on\/off status of generating units \\eqref{buc_binary_const}. For clarity of exposition and conciseness, we define the set $\\mathcal X$ given by $\\mathcal X = \\{\\mathbf x \\in \\{0,1\\}^{G\\times T} | \\textmd{ constraints }\\eqref{suc-init-on} - \\eqref{buc_binary_const}\\}$.\n\nThe minimum generation cost $\\mathcal Q(\\mathbf x)$, given a unit commitment $\\mathbf x$ is constrained by a combination of DC power flow constraints and unit ramping constraints, as follows:\n\n\\begin{subequations}\\label{suc-sp}\n\\begin{align}\n\\mathcal Q(\\mathbf x)=\\min_{{\\tilde {\\textbf f}}, {\\tilde {\\textbf p}}, {\\tilde {\\boldsymbol \\theta}}} \\quad &\\sum_{g \\in \\mathcal G}\\sum_{t \\in \\mathcal T} C_g^p(\\tilde p_g^t)& \\label{sp-obj} \\\\%+ \\sum_{t= 1}^T\\sum_{c \\in \\mathcal C(k)} Q(x^t,p^t, c)\n\\text{s.t.} \\quad & \\sum_{g \\in   \\mathcal G_i} \\tilde p_g^t + \\sum_{e \\in   \\mathcal E_{.i}}\\tilde  f_{e}^t - \\sum_{e \\in   \\mathcal E_{i.}}\\tilde  f_{e}^t = D_i^t  &  \\forall i \\in \\mathcal I, \\forall t \\in  \\mathcal T                          \\label{sp-bus-bal} \\\\\n\t& B_e (\\tilde \\theta_{i_e}^t - \\tilde \\theta_{j_e})^t - \\tilde f_e^t =  0 \\quad \\forall e \\in  \\mathcal E &  \\forall t \\in \\mathcal  T                \\label{sp-flow}                                \\\\     \n\t& -F_e \\leq \\tilde f_e^t \\leq F_e  &  \\forall e \\in \\mathcal  E,  \\forall t \\in \\mathcal  T    \\label{sp-flow-cap}                  \\\\     \n\t& P^{\\min}_g x_g^t\\leq \\tilde p_g^t \\leq P^{\\max}_g x_g^t  &   \\forall g \\in \\mathcal  G, \\forall t \\in \\mathcal  T    \\label{sp-p-bounds}   \\\\  \n\t& \\tilde p_g^{t} - \\tilde p_g^{t-1} \\leq R_g^u x_g^{t-1} + \\tilde{R}_g^u (x_g^t-x_g^{t-1}) + P^{\\max}_g ( 1-x_g^t) &  \\forall g \\in \\mathcal  G, \\forall t \\in \\mathcal  T \\label{sp-ramp-up} \\\\   \n\t& \\tilde p_g^{t-1}- \\tilde p_g^t\\leq R_g^d x_g^{t}+\\tilde{R}_g^d (x_g^{t-1}-x_g^t)+P^{\\max}_g(1-x_g^{t-1}) &  \\forall g \\in  \\mathcal G, \\forall t  \\in  \\mathcal T\\label{sp-ramp-down}\n\n\\end{align}\n\\end{subequations}\n\nThe optimization objective \\eqref{sp-obj} is to minimize generation cost given a unit commitment $\\mathbf x$, where $C_g^p(\\tilde p_g^t)$ is a linear approximation of  generation cost for thermal units, as is commonly employed. We discuss this linearization further below. Constraints \\eqref{sp-bus-bal}-\\eqref{sp-ramp-down} constitute an optimal power flow formulation, and include (in order): power balance at each bus \\eqref{sp-bus-bal}; power flow on a line, proportional to the difference in voltage phase angles at the terminal buses  \\eqref{sp-flow}; transmission line capacity limits \\eqref{sp-flow-cap}; lower and upper bounds for committed generation unit output levels \\eqref{sp-p-bounds}; and generation ramp-up\/ramp-down constraints for pairs of consecutive time periods \\eqref{sp-ramp-up}  and \\eqref{sp-ramp-down}.\n\nBy linearizing the generation cost functions, \\eqref{suc}-\\eqref{suc-sp} provides a mixed-integer linear programming (MILP) formulation of the unit commitment problem with transmission constraints, but without contingency constraints. A solution to the BUC provides an on\/off schedule for all generation units, over all time periods in the horizon $T$. In practice, committed generation units are adjusted on an hourly or sub-hourly basis, by ramping up or down specific units in order to satisfy realized demand. Further, additional fast-reaction (i.e., ``peaker\") units can be brought online if necessary. However, this process occurs on a different time scale than the BUC, i.e., one or two hours prior to real-time execution.\n\n\\remark {Our BUC model most closely represents the reliability unit commitment problem, which ISOs and vertically integrated utilities solve  every night. In contrast, the day-ahead unit commitment problem is executed earlier in the day, and is used to clear the market and set nodal electricity prices. While there are differences between the two problem variants, specifically in terms of the inputs (e.g., bids driving aggregate demand, in contrast to ISO-forecasted load), the basic BUC model can be easily re-cast into either variant.}\n\n\\remark {The number of time periods that unit $g$ has been online\/offline prior to $t=1$ should satisfy $T_g^{u0} \\times T_g^{d0}=0$. That is, if a unit $g$ is online prior to time period $1$, $T_g^{u0} \\ge 0$ and $T_g^{d0} = 0$. Similarly, if unit $g$ is offline before time period  $1$, $T_g^{u0} = 0$ and $T_g^{d0} \\ge 0$.}\n\n\\remark {The structure of the BUC solution space is known to be degenerate, due to the nature of the phase angles $\\tilde \\theta_i^t$. In particular, alternative optimal solutions can be obtained by shifting all of the $\\tilde \\theta_i^t$ of a given optimal solution by a constant factor. To mitigate this degeneracy, and following common practice in the literature, we require in our numerical experiments that the value $\\tilde \\theta_r^t$ for a pre-defined ``reference'' bus $r$ be equal to 0 for $t \\in \\mathcal{T}$.}\n\n\\remark {Generation cost curves $C_g^p (p_g^t)$ are generally specified as quadratic functions of the form: $C_g^p (p_g^t)=c_g^{p2}(p_g^{t})^2+c_g^{p1}p_g^{t}+c_g^{p0}$. However, because the $C_g^p (p_g^t)$ are non-decreasing convex functions of $p_g^{t}$, they can be easily approximated using a piecewise linear function (see \\cite{Carrion2006}). Many researchers make a further simplification by assuming a linear cost function, which corresponds to the not uncommon case in which a generator offers into the market with a single marginal cost factor. We make this assumption below in our numerical experiments, specifically that $C_g^p (p_g^{t})=c_g^p p_g^{t}$. The extension to the more general piecewise construct discussed above is straightforward, and does not impact the algorithms we introduce in Section~\\ref{sec:solution}. Practically, piecewise constructs would inflate the solve times, but not significantly.}\n\n\\remark {Variables and constraints to capture reserve requirements are common in the unit commitment literature, but are absent in our unit commitment models. As noted in \\cite{Hedman2010}[p. 1056], ``The primary purpose of spinning and non-spinning reserves is to ensure there is enough capacity online to survive a contingency''. Hedman et al. \\cite{Hedman2010} make this argument in the context of $N$-$1$ reliabiliy; the argument for the exclusion of reserve models is even stronger for $N$-$k$ contingencies. Reserves,  specifically spinning reserves, also serve as proxies for explicitly dealing with uncertainty in demand and variable generation (e.g., wind and solar plant) output. However, again following \\cite{Hedman2010}, we argue that enforcing $N$-$k$ reliability (even when $k=1$) is  likely to ensure sufficient spinning reserves are online to deal with forecast errors in both demand and variable generation. We demonstrate that this is indeed the case in Section \\ref{ieee6_example} by analyzing the CCUC for a 6-bus system.}\n\n\\subsection{$N$-$k$-$\\boldsymbol \\varepsilon$ Contingency Constraints for Reliability Requirements}\n\\label{sec:model-contingencies}\nAccording to the NERC TPL standard, in the event of a loss involving a single component (i.e., an $N$-$1$ contingency), a power system must remain stable and satisfy all demand.\n\nIn the case of  two or more  simultaneous losses (i.e., an $N$-$k$ contingency with $k \\ge 2$), the system must maintain stability; however, a pre-planned or controlled loss-of-load is allowed. Therefore, prior to analyzing the contingency-constrained unit commitment problem, we must first ensure that the BUC model can yield solutions that satisfy such requirements.\n\nWe consider the loss of elements in a power system $(\\mathcal I, \\mathcal G, \\mathcal E)$ in both the set $\\mathcal G$ of generating units and the set $\\mathcal E$ of transmission lines.\n The parameters and the variables in our formulation are defined in Table~\\ref{tab:nke}.\n\\begin{table}[thb]\n\\caption{ Variables and parameters $N$-$k$-$\\boldsymbol \\varepsilon$ contingency analysis }\n\\label{tab:nke}\n\\begin{center}\n\\begin{tabular}{|c|l|}\n\\hline\n\\multicolumn{2}{| l |}{\\bf Parameters}\\\\ \\hline\n  $k$ &  Maximum number of simultaneous element failures. \\\\\\hline\n   $\\mathcal{C}(j)$  & Set of all contingencies with \\emph{exactly} $j$ failed generation units \\\\& and\/or transmission lines  for $j \\in \\{1,\\cdots,k\\}$. Indexed by $c$.\\\\\\hline\n      $|c|$ &  Size of contingency $c$, i.e., the number of failed elements.\\\\\\hline\n  $\\mathcal C = \\displaystyle \\cup_{j=1}^{k} \\mathcal C(j)$:  & Set of all contingencies with $k$ \\emph{or fewer} failed elements (generation \\\\& units and\/or transmission lines). $|\\mathcal C| = C$. \\\\\\hline\n   $\\tilde d_g^c\\in \\{0,1\\}$  & Parameter specifying whether generation unit $g \\in  \\mathcal G$ is involved in \\\\& contingency $c \\in \\mathcal{C}$.\\\\\\hline\n   $\\tilde d_e^c\\in \\{0,1\\}$ & Parameter specifying whether transmission line $e \\in  \\mathcal E$ is involved in \\\\& contingency $c \\in \\mathcal{C}$.\\\\\\hline\n   $\\tilde{\\textbf d}^c\\in \\{0,1\\}^{G+T}$ &  Vector that is the concatenation of $d_g^c \\ \\forall g \\in \\mathcal G$ and $d_e^c \\ \\forall e \\in \\mathcal E$.\\\\\\hline\n    $\\varepsilon_ j$  & Parameter indicating the maximum fraction of total system load that \\\\& can be shed in a size $j$ contingency state, for $j=1,\\cdots,k$.\\\\\\hline\n     $\\boldsymbol \\varepsilon$  & Parameter vector indicating the maximum fraction of total  load \\\\& that can be shed for each contingency size, $\\boldsymbol \\varepsilon = (\\varepsilon_ 1,\\cdots,\\varepsilon_ k)$. \\\\\\hline\n     $\\Delta_g^j$  & Multiplicative factor applied to the ramping limits of generator $g \\in \\mathcal G$ \\\\& during a size $j \\in \\{1,\\cdots,k\\}$ contingency ($\\Delta_g^j \\ge 1$). \\\\\\hline\n $\\Delta_e^j$  & Multiplicative factor applied to the power flow limits of transmission \\\\&  line $e \\in \\mathcal E$ during a size $j \\in \\{1,\\cdots,k\\}$ contingency ($\\Delta_e^j \\ge 1$). \\\\\n \\hline\\hline\n\\multicolumn{2}{| l |}{\\bf Variables}\\\\ \\hline\n$p_g^{ct},f_e^{ct},\\theta_i^{ct}$ & corresponding values of $\\tilde p_g^{t}, \\tilde f_e^{t}, \\tilde \\theta_i^{t}$ during contingency $c \\in \\mathcal{C}$. \\\\ \\hline\n  $q_i^{ct}$ & Loss-of-load amount during contingency $c $ at bus $i$ at time  $t $. \\\\ \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\nWe express the $N$-$k$ contingency set $\\mathcal{C}$ as follows:\n\\begin{align}\n\\mathcal{C}=\\Big \\{ \\tilde {\\textbf{d}^c}  : \\left( \\sum_{g \\in  \\mathcal G} \\tilde d_g^c + \\sum_{e \\in  \\mathcal  E} \\tilde d_e^c \\right) \\le k  \\Big \\}.\n\\end{align}\n\n\\remark {\nThe number of contingencies within the set $\\mathcal{C}$ is then given by:\n$${G+E\\choose 1}+\\cdots+{G+E\\choose k}\\le (G+E+1)^k-1.$$\nPractically, the number of contingencies grows so rapidly  that explicit enumeration-based approaches are almost certain to fail even for modestly-sized  systems.\n}\n\nWe assume that a given contingency $c $ holds for all time periods $t \\in \\mathcal T$. Or alternatively, a power system must be $N$-$k$-$\\boldsymbol \\varepsilon$ compliant in all time periods $t \\in \\mathcal T$, for all contingencies $c \\in \\mathcal{C}$. We are not modeling specific issues relating to when a contingency may occur, how long it may last, and what corrective measures may be taken to restore functionality. Such issues can significantly expands the size and difficulty of the associated unit commitment problem, and is beyond the scope of this work. Further, generation costs are not optimized in post-contingency operation; following precedence in the literature, only constraints related to power flow on the non-contingency system elements must be enforced. In other words, the primary goal during a contingency state is operational feasibility and not cost minimization.  Additionally, multiple failure contingencies are extreme events with correspondingly low occurrence probabilities.  Therefore, consideration of the cost of these extreme events during operations planning is unnecessary, and may result in prohibitively expensive operations.\n\nGiven these assumptions, we formulate the \\emph{post-contingency corrective recourse} constraints (i.e., the constraints that must be satisfied as the system state is altered in response to a contingency event, starting from a given steady state) $\\mathcal R({\\textbf{x}},\\tilde{\\textbf{p}}, \\tilde {\\textbf d^c})$ for a contingency prescribed by $\\tilde {\\textbf d^c}$, under a unit commitment decision vector $\\mathbf x$ and the no-contingency state generation schedule $\\tilde{\\textbf{p}}$ as follows:\n\\begin{subequations}\\label{cont_const}\n\\begin{align}\n        \\mathcal R({\\textbf{x}},\\tilde{\\textbf{p}}, \\tilde {\\textbf d^c}): \\quad & \\sum_{g \\in  \\mathcal G_i} p_g^{ct} + \\sum_{e \\in  \\mathcal E_{.i}}  f_{e}^{ct}  - \\sum_{e \\in  \\mathcal E_{i.}}  f_{e}^{ct}  + q_i^{ct}= D_i^t \\quad &  \\forall i \\in \\mathcal I, \\forall t \\in  \\mathcal T                          \\label{csp_flow_bal} \\\\\n       \n\t& B_e (\\theta_{i_e}^{ct} - \\theta_{j_e}^{ct} )(1- \\tilde d_e^c) - f_e^{ct}=  0 \\quad & \\forall e \\in  \\mathcal E, \\forall t \\in  \\mathcal T  \\label{csp_kirchoffs} \\\\\n\t& - F_e\\Delta_e^{|c|}(1-  \\tilde d_e^c) \\leq f_e^{ct} \\leq F_e\\Delta_e^{|c|}(1- \\tilde d_e^c ) \\quad & \\forall e \\in  \\mathcal E, \\forall t \\in \\mathcal  T                   \\label{csp_trans_bounds} \\\\\n\t&  p_g^{ct} \\leq P^{\\max}_g x_g^t (1- \\tilde d_g^c)\\quad & \\forall g \\in  \\mathcal G, \\forall t \\in  \\mathcal T \\label{csp_gener_bounds} \\\\\n\t&p_g^{ct}\\leq R^u_g \\Delta_g^{|c|} + \\tilde p_g^t  \\quad  & \\forall g \\in  \\mathcal G, \\forall t \\in  \\mathcal T \\label{csp_ru}  \\\\\n\t&-p_g^{ct}\\leq (R^d_g\\Delta_g^{|c|} - \\tilde p_g^t)(1- \\tilde d_g^c) \\quad & \\forall g \\in  \\mathcal G, \\forall t \\in  \\mathcal T    \\label{csp_rd}  \\\\\n\t& q_i^{ct} \\leq  D_i^t \\quad & \\forall i \\in I, \\forall t \\in  \\mathcal T   \\label{csp_lol_i}   \\\\\n\t& \\sum_{i \\in  \\mathcal I} q_i^{ct} \\leq \\varepsilon_ {|c|} D^t \\quad & \\forall t \\in  \\mathcal T \\label{csp_lol_budget}\\\\\n\t& p_g^{ct} \\ge 0 \\quad & \\forall g \\in  \\mathcal G, \\forall t \\in  \\mathcal T\\\\\n\t& q_i^{ct} \\ge 0 \\quad & \\forall i \\in  \\mathcal I, \\forall t \\in \\mathcal  T.\n\\end{align}\n\\end{subequations}\n\nConstraints \\eqref{csp_flow_bal} enforce power balance at each bus, leveraging additional loss-of-load variables $q^{ct}$. Constraints \\eqref{csp_kirchoffs} enforce DC power flow on those lines that are not part of contingency $\\tilde {\\textbf d^c}$ . Constraints \\eqref{csp_trans_bounds} enforce transmission line capacity limits. If a line is not part of a contingency, then the power flow limit is given by $F_e\\Delta_e^{|c|}$; otherwise, the power flow is constrained to equal zero. Constraints \\eqref{csp_gener_bounds} enforce upper bounds on power output of committed (or ``on'') generation units not involved in the contingency $\\tilde {\\textbf d^c}$; otherwise, the power output is constrained to equal to zero. Constraints \\eqref{csp_ru} enforce generator ramp-up constraints.  If a generation unit is part of the contingency, then its corresponding power output level during the contingency is zero ($p_g^{ct}=0$) and the constraint is non-binding. Otherwise, a generator can only ramp-up by $R_g^u$ from its pre-contingency level. Similarly,  constraints \\eqref{csp_rd}  enforce generation ramp-down constraints.  If a generation unit is involved in a contingency, then $(1-d_g^c)=0$  and the ramp-down constraint is non-binding. Otherwise, a generator can only ramp-down by $R_g^d$ from its pre-contingency level. Constraints \\eqref{csp_lol_i} ensure that loss-of-load at each bus does not exceed the demand at that bus. Finally, constraints \\eqref{csp_lol_budget} ensure that at most $\\varepsilon_{|c|}$ fraction of the aggregate demand can be shed in a size $|c|$ contingency.\n\nObserve that in (\\ref{cont_const}), lower limits on power output for generation units not in the contingency are relaxed to ensure sufficient operational flexibility.  These lower limits can be easily incorporated for systems with sufficiently flexible generation units.  In addition, we implicitly assume that the on\/off state of generation units \\emph{not} involved in a contingency are fixed and cannot be changed via recourse variables during the contingency. For those generation units that are not involved in the contingency, the power output levels $p^{ct}$ are not allowed to deviate from the baseline (pre-contingency) power output levels $\\tilde p_g^{t}$ beyond the interval $[\\tilde p_g^t  - R_g^d, \\tilde p_g^t + R_g^u]$, given physical ramping limitations.\n\n\\subsection{A 6-Bus Illustrative Example}\n\\label{ieee6_example}\n\nWe now examine the impact of contingency constraints on optimal BUC solutions using the 6-bus test system introduced in \\cite{Fu2005}-\\cite{Fu2006}. Our goals are to concretely illustrate (a) the often significant changes in solution structure induced by the requirements to maintain $N$-$k$-$\\boldsymbol \\varepsilon$ in unit commitment, relative to the baseline and $N$-$1$ cases, and (b) the redundant nature of contingency constraints, in that satisfaction of one contingency state yields solutions that can ``cover\" many other contingency states. The original test system consists of 6 buses, 7 transmission lines, and 3 generating units.  We modified this instance for purposes of our analysis as follows. We augmented the system with three additional, fast-ramping generators G4, G5, and G6, located at buses 1, 2, and 6, respectively. This modification ensures there is sufficient generation capacity to satisfy the $N$-$k$-$\\boldsymbol \\varepsilon$ criterion during contingency states. Data for the original generator set and the three additional generators is summarized in Table \\ref{tab10}. Transmission line data is summarized in Table \\ref{tab11}.\n\n\\begin{table}[t]\n\n\\centering\n\\small{\n\\caption{Generator data for the 6-bus test system}\n\\centering\n\\begin{tabular}{ c | c | c  c  | c c c}\n\\hline\nUnit \t\t&\tBus \t\t &Prod.\t\t&Start-            \t& $P^{\\max}$\t&  $P^{\\min}$\t\t& Ramp\t\t \\\\[-0.3ex]\n\t\t&\tNo. \t\t&(\\$\/MW)\t\t&up (\\$)\t\t&(MW)\t\t& (MW)\t\t\t& (MW\/h)\t\\\\[0.2ex]\n\\hline\nG1\t\t&\t1\t\t&\t13.51\t&\t125\t\t& 220\t\t& 100\t\t\t& 55\t\t\\\\[-0.3ex]\nG2\t\t&\t2\t\t&\t32.63\t&\t249\t\t& 100\t\t& 10\t\t\t\t& 50\t\t\\\\[-0.3ex]\nG3\t\t&\t6\t\t&\t17.69\t&\t0\t\t& 100 \t\t& 10\t\t\t    \t& 20\t\t\\\\[-0.3ex]\nG4\t\t&\t1\t\t&\t42\t\t&\t50 \t\t& 100 \t\t& 0\t\t\t    \t& 50\t\t\\\\[-0.3ex]\nG5\t\t&\t2\t\t&\t42\t\t&\t50 \t\t& 100 \t\t& 0\t\t\t    \t& 50\t\t\\\\[-0.3ex]\nG6\t\t&\t6\t\t&\t42\t\t&\t50 \t\t& 100 \t\t& 0\t\t\t    \t& 50\t\t\\\\[-0.3ex]\n\\hline\n\\end{tabular}\n}\n\\label{tab10}\n\\end{table}\n\nConsistent with \\cite{Fu2005}-\\cite{Fu2006}, the unit shutdown cost is negligible and assumed to be zero in our analysis.  For illustrative purposes, we only consider the BUC with a single time period, with loads of 51.2, 102.4, and 42.8 at buses 3, 4, and 5, respectively. Runtime results for the full 24-hour instance are presented in Section \\ref{sec:experiments}.\n\n\\begin{table}[t]\n\\centering\n\\caption{Transmission line data for the 6-bus test system}\n\\centering\n\\small{\n\\begin{tabular}{ c | c | c  | c  | c c c}\n\\hline\nLine \t\t&\tFrom \t&To\t\t&$B_e$  \t&$F_e$\t\\\\[-0.3ex]\nNo.\t\t&\tBus\t\t&Bus\t&\t\t&(MW)\t\t\\\\[0.2ex]\n\\hline\nL1\t\t&\t1\t\t&2\t\t&5.88\t& 200\t\t\\\\[-0.3ex]\nL2\t\t&\t1\t\t&4\t\t&3.88\t& 100\t\t\\\\[-0.3ex]\nL3\t\t&\t2\t\t&4\t\t&5.08\t& 100 \t\t\\\\[-0.3ex]\nL4\t\t&\t5\t\t&6\t\t&7.14\t& 100 \t\t\\\\[-0.3ex]\nL5\t\t&\t3\t\t&6\t\t&55.56 \t& 100 \t\t\\\\[-0.3ex]\nL6\t\t&\t2\t\t&3\t\t&27.03 \t& 100 \t\t\\\\[-0.3ex]\nL7\t\t&\t4\t\t&5\t\t&27.03\t& 100 \t\t\\\\ [-0.3ex]\n\\hline\n\\end{tabular}\n}\n\\label{tab11}\n\\end{table}\n\nA single line diagram of the 6-bus test system is shown in Figure \\ref{figure1}(a). Generator capacity bounds, transmission line capacity bounds, and loads are shown adjacent to their corresponding system elements. When contingencies are ignored, the optimal BUC solution commits a single unit (G1 at bus 1), generating 196.4 MW to meet the total demand. The no-contingency economic dispatch is shown graphically in Figure \\ref{figure1}(b).\n\n \\begin{figure}[t]\n   \\centering\n \n    \\includegraphics[width=1\\textwidth, angle=0]{6BusExample1.pdf}\n    \\vskip -4.5cm\n    \\caption{\\footnotesize {(a) Single line diagram of the modified 6-bus test system. (b) An optimal BUC solution to the 6-bus test system, ignoring contingency constraints. Green, blue, and red text respectively refers to characteristics of generators, lines, and buses.}}\n    \\label{figure1}\n\\end{figure}\n\nIn accordance with NERC's TPL standard, loss-of-load is not permitted in single-component-failure contingency states. In order for the  6-bus test system to be fully $N$-$1$ compliant, i.e., to operate the system in such a way that there exists a post-contingency corrective recourse action for \\emph{all} possible $N$-1 contingencies, 5 generation units must be committed, as shown in Figure~\\ref{figure3}(a). Of these, two units (G1 and G3) provide generation capacity during the no-contingency state, while three units (G4, G5, and G6) function as spinning reserves.  Unlike the practical approach of explicitly setting aside spinning reserves (e.g., equivalent to the capacity of the largest unit) via constraints, our proposed CCUC model implicitly and automatically selects units to provide spinning reserves, within the context of satisfying contingency constraints. Further, in contrast to the approach of explicitly allocating spinning reserves, our proposed CCUC model guarantees that there is adequate transmission capability to dispatch the generator outputs during all contingency states.\n\nThe optimal $N$-$1$-compliant BUC solution shown in Figure \\ref{figure3}(a) represents the system in steady state operations, i.e., under no observed contingency. Figures \\ref{figure3}(b) and  \\ref{figure3}(c) illustrate feasible corrective recourse power flows for single-component contingency states corresponding to the failure of generation unit 1 and transmission line 1 (connecting buses 1 and 2), respectively. The total operating cost of the  $N$-$1$ compliant solution is approximately 6.52\\% higher than an optimal no-contingency BUC solution.\n\n \\begin{figure}[t]\n   \\centering\n \n    \\includegraphics[width=1\\textwidth, angle=0]{6BusExample3.pdf}\n    \\vskip -4.5cm\n    \\caption{\\footnotesize {(a) An optimal BUC solution to the 6-bus test system that is fully $N$-$1$ compliant. (b) A corrective recourse power flow after the failure of generation unit 1.  (c) A corrective recourse power flow after the failure of  the transmission line connecting buses 1 and 2.}}\n    \\label{figure3}\n\\end{figure}\n\nThe modified 6-bus system has 13 (7 transmission lines and 6 generators) possible single-component contingency states. We observe that it is sufficient to consider \\emph{only} the two contingency states shown in Figure \\ref{figure3}(b) and Figure \\ref{figure3}(c) in order to achieve full $N$-$1$ compliance.  In other words, accounting for those two contingencies \\emph{implicitly} yields feasible corrective recourse actions for the other $N$-$1$ contingency states. As we discuss in Section~\\ref{sec:solution}, in most practical systems only a small number of contingency states are likely to impact the optimal unit commitment solution. Consequently, we design our algorithm to screen for these critical contingencies implicitly, without the need to explicitly consider all possible combinations of system component failures -- thus avoiding the combinatorial explosion in the number of possible contingencies.\n\nIf the maximum allowable contingency size is increased to $k=2$, the optimal BUC solution for the  6-bus test system commits four generation units, as shown in Figure \\ref{figure2}. In addition to including $k=2$ contingencies in our analysis, we require that loads must be fully served in the no-contingency state and that a post-contingency corrective resource exist for all $k=1$ contingencies with zero loss-of-load, per TPL standards. For all $k=2$ contingencies, the allowable loss-of-load threshold is set to $\\varepsilon_2 = 0.27$, to ensure that there is sufficient slack  to accommodate the loss of both transmission lines connected to bus 5. If both transmission lines connected to bus 5 fail, then the load at that bus cannot be served; the factor $0.27$ corresponds to the minimal loss-of-load under this contingency. For systems with greater redundancy and flexibility, such as those presented Section \\ref{sec:experiments}, the loss-of-load threshold can be set more conservatively (i.e., lower).\n\nOf the four committed units, one unit (G1) is producing at maximum capacity and three units (4, 5, and 6) are producing at at levels below their maximum rating. Taken together, these three units can provide up to 150MW of spinning reserves. Although fewer units are committed (4 compared to 5) relative to the $N$-$1$ solution, the two least expensive units (G1 and G2) are not committed while the three most expensive units (G4, G5, and G6) are committed in the $N$-$2$-$\\boldsymbol \\varepsilon$ compliant solution.\n\n \\begin{figure}[t]\n   \\centering\n \n    \\includegraphics[width=1\\textwidth, angle=0]{6BusExample2.pdf}\n    \\vskip -5.2cm\n    \\caption{\\footnotesize {An optimal BUC solution to the  6-bus test system that is fully $N$-$2$-$\\boldsymbol \\varepsilon$ compliant with $\\varepsilon_ 2 = 0.27$ allowable loss-of-load.}}\n    \\label{figure2}\n\\end{figure}\n\nWe conclude with the obvious, yet critical, observation that contingency constraints must be considered in normal (no-contingency) unit commitment operations in order to ensure that a feasible post-contingency corrective recourse exists for all contingency states under consideration.\n\n\\subsection{Contingency-Constrained Unit Commitment Formulation}\\label{sec:model-ccuc}\n\nGiven the baseline unit commitment model (BUC) and associated contingency constraints as defined respectively in Sections~\\ref{sec:model-standarduc} and \\ref{sec:model-contingencies}, we can now describe our full contingency-constrained unit commitment (CCUC) problem:\n\n\n\\begin{align}\\label{CCUC}\n\\text{CCUC}:  \\quad  \\min_{\\mathbf x \\in \\mathcal X} \\quad & \\sum_{t \\in \\mathcal T} \\sum_{g \\in \\mathcal G}(c_g^{ut}+c_g^{dt})+ \\mathcal Q(\\mathbf x)      \\\\\n\\text{s.t.}  \\quad  & (\\mathbf f, \\mathbf p, \\mathbf q, \\boldsymbol \\theta) \\in \\mathcal R(\\mathbf x^t, {\\tilde {\\textbf p}}^t, \\mathbf d^c) \\quad \\forall c \\in \\mathcal{C},\\forall t \\in \\mathcal T \\nonumber\n\\end{align}\nThe resulting unit commitment decision vector $\\mathbf x$ represents a minimal-cost solution that satisfies (1) the non-contingency demands $D_i^t$ for each bus $i \\in \\mathcal I$ for each time period $t \\in \\mathcal T$, (2) the generation unit ramping constraints and startup\/shutdown constraints, and (3) the network security and DC power flow constraints for each contingency, subject to loss-of-load allowances $\\varepsilon_j$. We again note that generation costs incurred during a contingency are not considered in the objective function. Rather, only power system feasibility need be maintained, subject to the loss-of-load allowances $\\varepsilon_ j$, for all $j \\in \\{1, \\cdots, k\\}$.\n\n\\section{Solution Methods}\n\\label{sec:solution}\n\nThe extensive formulation (EF) \\eqref{CCUC} of the CCUC problem is a large-scale MILP, and has an extremely large number of variables and constraints.  For large power systems and\/or non-trivial contingency budgets $k$, the EF formulation will quickly become computationally intractable. For example, the number of constraints in the second stage of the CCUC (which drives the overall problem size) is approximately given as: $C \\times T(3I+2E+4G)=O\\big(T\\times (G+E)^k\\times(I+G+E)\\big)$.\n\nAlternatively, the EF formulation of the CCUC problem has a structure that is amenable to a Benders decomposition (BD) approach, which partitions the constraints in the EF formulation into (1) a BUC problem prescribing the unit commitment decisions and the corresponding economic dispatch in the no-contingency state (this is commonly referred to as the master problem in BD), and (2) a subproblem corresponding to each contingency feasibility check, for each contingency state $c \\in \\mathcal C$ and time period $t \\in \\mathcal T$.  The BD algorithm iterates between solving the master problem (BUC), to prescribe the lowest cost unit commitment and economic dispatch, and the linear subproblems, until an optimal solution with a feasible post-contingency corrective recourse for all contingency states is obtained. In the following sub-section, we describe our Benders decomposition solution method, as it is applied to CCUC.\n\n\\subsection{A Benders Decomposition Approach}\n\nWe begin by observing that given a time period $t$, a unit commitment decision $\\mathbf x^t$ and the no-contingency generation schedule $\\tilde{\\textbf{p}}^t$, feasibility under contingency state $c$, as prescribed by $\\tilde{\\textbf d^c}$, is contingent on satisfying the following DC power flow constraints.  We refer to this problem as the contingency feasibility problem {\\bf{CF}}$(\\mathbf x^t, \\tilde{\\textbf p}^t, \\tilde{\\mathbf d}^c)$. For conciseness of notation, we eliminate the superscript ``$ct$'' from the $f_e^{ct}, p_g^{ct}, q_i^{ct}$ and $\\theta_i^{ct}$ decision variables.\n\\begin{subequations}\\label{mod_cf}\n\\begin{align}\n(\\alpha) \\quad &\\sum_{g \\in \\mathcal G_i} p_g + \\sum_{e \\in \\mathcal E_{.i}}  f_{e}  - \\sum_{e \\in \\mathcal E_{i.}}  f_{e}  + q_i= D_i^t \\quad &\\forall i \\in \\mathcal  I \\label{cf_bal} \\\\\n(\\beta) \\quad & B_e (\\theta_{i_e} - \\theta_{j_e} )(1- \\tilde d_e^c) -f_e=  0 \\quad &\\forall e \\in \\mathcal E \\label{cf_kirk}  \\\\\n(\\hat \\delta) \\quad &  f_e \\leq F_e \\Delta_e^{|c|} (1- \\tilde d_e^c ) \\quad& \\forall e \\in \\mathcal E \\label{cf_f_ub}  \\\\\n(\\check \\delta) \\quad & -f_e \\leq F_e \\Delta_e^{|c|} (1- \\tilde d_e^c ) \\quad& \\forall e \\in \\mathcal E \\label{cf_f_lb} \\\\\n(\\gamma) \\quad & p_g \\leq P^{\\max}_g x_g^t (1- \\tilde d_g^c)\\quad& \\forall g \\in \\mathcal G  \\label{cf_p_ub} \\\\\n(\\hat \\lambda) \\quad &p_g \\leq R^u_g \\Delta_g^{|c|} +  \\tilde p_g^t \\quad& \\forall g \\in \\mathcal G  \\label{cf_r_ub} \\\\\n(\\check \\lambda) \\quad &-p_g \\leq (R^d_g  \\Delta_g^{|c|} -  \\tilde p_g^t)(1- \\tilde d_g^c) \\quad& \\forall g \\in \\mathcal G  \\label{cf_r_lb} \\\\\n(\\zeta) \\quad &  q_i \\leq  D_i^t &\\quad \\forall i \\in \\mathcal I  \\label{cf_q_ub}\\\\\n(\\pi) \\quad & \\sum_{i \\in \\mathcal I} q_i \\le \\varepsilon_ {|c|} D^t\t& \\label{cf_tot_q}\\\\\n&p_g,q_i\\ge 0 &\\forall g\\in \\mathcal G, i \\in \\mathcal I.\n\\end{align}\n\\end{subequations}\n\nUsing the dual variables associated with each constraint set in (\\ref{cf_bal})-(\\ref{cf_tot_q}), we have, by strong duality in linear programming, that $(\\mathbf x^t, \\tilde{\\textbf p}^t, \\tilde{\\mathbf d}^c)$ is feasible if and only if the following dual problem  {\\bf{DCF}}$(\\mathbf x^t, \\tilde{\\textbf p}^t, \\tilde{\\mathbf d}^c)$ is bounded:\n\\begin{subequations}\\label{mod_dcf}\n\\begin{align}\n\\max_{\\boldsymbol \\alpha,  \\boldsymbol \\beta,  \\hat {\\boldsymbol \\delta}, \\check {\\boldsymbol \\delta}, \\boldsymbol  \\gamma,  \\hat {\\boldsymbol \\lambda}, \\check {\\boldsymbol \\lambda}, \\boldsymbol  \\zeta, \\boldsymbol  \\pi} \\quad &\\sum_{i \\in \\mathcal I} D_i^t (\\alpha_i+\\zeta_i)+ \\sum_{e \\in \\mathcal E} F_e\\Delta_e^{|c|} (1- \\tilde d_e^c)(\\check \\delta_e + \\hat \\delta_e) + \\sum_{g \\in \\mathcal G}  P^{\\max}_g x_g^t (1- \\tilde d_g^c)\\gamma_g   &  \\\\\n& + \\sum_{g \\in \\mathcal  G} \\Big( (R^u_g \\Delta_g^{|c|} + \\tilde p_g^t)\\hat \\lambda_g + (R^d_g\\Delta_g^{|c|}  - \\tilde p_g^t )(1- \\tilde d_g^c)\\check \\lambda_g +   \\varepsilon_ {|c|} D^t \\pi \\Big) &\\nonumber \\\\\n\\textmd{s.t.} \\quad & \\alpha_{i_e} - \\alpha_{j_e} - \\beta_e - \\check \\delta_e + \\hat \\delta_e = 0  \\hspace{22ex}  \\forall e \\in \\mathcal E  \\\\\n& \\alpha_{i_g} + \\gamma_g + \\hat \\lambda_g - \\check \\lambda_g \\le 0 \\hspace{27ex} \\forall g \\in \\mathcal G \\\\\n& \\alpha_i + \\zeta_i \\le 0  \\hspace{37ex}\\forall i \\in \\mathcal  I   \\\\\n&\\sum_{e \\in \\mathcal E_{i.}} B_e(1- \\tilde d_e^c)\\beta_e - \\sum_{e \\in \\mathcal E_{.i}}B_e(1- \\tilde d_e^c)\\beta_e=0  \\hspace{10ex} \\forall i \\in  \\mathcal   I\\\\\n&\\hat {\\boldsymbol \\delta}, \\check {\\boldsymbol \\delta}, \\boldsymbol  \\gamma,  \\hat {\\boldsymbol \\lambda}, \\check {\\boldsymbol \\lambda}, \\boldsymbol  \\zeta, \\boldsymbol  \\pi\\le 0\n\\end{align}\n\\end{subequations}\n\nNote that the feasible domain for  {\\bf{DCF}}$(\\mathbf x^t, \\tilde{\\textbf p}^t, \\tilde{\\mathbf d}^c)$, is a polyhedral cone and any solution in the domain is a ray.  By Minkowski's theorem, every such ray can be expressed as a non-negative linear combination of the extreme rays of the domain. Therefore, the dual problem {\\bf{DCF}}$(\\mathbf x^t, \\tilde{\\textbf p}^t, \\tilde{\\mathbf d}^c)$ is bounded if and only if  its optimal objective value is less than or equal to zero. And this happens   if and only if\n\\begin{align}\\label{f_cut}\n& \\sum_{i \\in \\mathcal I} D_i^t (\\alpha_i+\\zeta_i)+ \\sum_{e \\in \\mathcal E} F_e\\Delta_e^{|c|} (1- \\tilde d_e^c)(\\check \\delta_e + \\hat \\delta_e) + \\sum_{g \\in \\mathcal G}  P^{\\max}_g x_g^t (1- \\tilde d_g^c)\\gamma_g     \\\\\n& + \\sum_{g \\in \\mathcal G} \\Big( (R^u_g \\Delta_g^{|c|} + \\tilde p_g^t)\\hat \\lambda_g + (R^d_g\\Delta_g^{|c|}  - \\tilde p_g^t )(1- \\tilde d_g^c)\\check \\lambda_g +   \\varepsilon_ {|c|} D^t \\pi \\Big) \\le 0. \\nonumber\n\\end{align}\n\nWe call these the Benders feasibility cuts or $f$-$cut$ for short. Below we outline the Benders decomposition algorithm as it applied to CCUC.\n\\begin{algorithm\n\\caption{\\emph{Benders Decomposition Algorithm (BD)}}\n\\begin{algorithmic}[1]\n\\State  Initialization: let $\\ell \\leftarrow 1$\n\\State Solve BUC$_\\ell$\n\\State \\textbf{if} BUC$_\\ell$ is infeasible, CCUC has no feasible solution, EXIT\n\\State  \\textbf{else}, let  $(\\mathbf{x}_\\ell, \\tilde{\\textbf{p}}_\\ell)$ be an optimal solution of BUC$_\\ell$\n\\State \\hskip 0.8cm \\textbf{for} each $c\\in \\mathcal{C}$, $t \\in \\mathcal T$,\n\\State \\hskip 1.6cm solve  {CF${(\\mathbf{x}^t_\\ell, \\tilde{\\textbf{p}}^t_\\ell,\\tilde{\\mathbf{d}}^c)}$}   and let $w^*$ be the optimal objective value\n\\State \\hskip 1.6cm \\textbf{if}  $w^*$ unbounded  add $f$-cut \\eqref{f_cut} to \\text{BUC$_\\ell$}\n\\State \\hskip 1.6cm \\textbf{end if}\n\\State \\hskip 0.8cm \\textbf{end for}\n\\State \\hskip 0.8cm \\textbf{if} $f$-cut(s) added in (7), let $\\ell \\leftarrow \\ell+1$ and return to (2)\n\\State \\hskip 0.8cm \\textbf{else}, $(\\mathbf{x}_\\ell, \\tilde{\\textbf{p}}^t_\\ell)$ is an optimal solution, EXIT\n\\State \\hskip 0.8cm  \\textbf{end if}\n\\State \\textbf{end if}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsection{A Cutting Plane Method Based on the Power System Inhibition Problem}\nEven with a BD approach CCUC  may not be tractable for practical size power systems, since  for every contingency $c\\in \\mathcal{C}$ and time period $t \\in \\mathcal T$, we need to ensure that a feasible DC power flow with limited loss-of-load exists, which is intractable in most cases.\n\n In this section, we describe a cutting plane algorithm that uses a bi-level separation problem to {\\em implicitly} identify a contingency state that would result in the worst-case loss-of-load for each contingency size $j$ , $j \\in \\{1,\\cdots,k\\}$.  If the worst-case generation shedding is non-zero and\/or loss-of-load is above the given contingency budget $\\varepsilon_j$, the current solution is infeasible, and we generate a cutting plane, corresponding to $f$-$cut$ \\eqref{f_cut} to add to the BUC to protect against this particular contingency state.\n\n\\subsubsection{The Bi-Level Power System Inhibition Problem}\nGiven a time period $t \\in \\mathcal T$ and a contingency budget $j \\in \\{1,\\cdots,k\\}$, unit commitment $\\mathbf x^t$, and the no-contingency generation levels $\\tilde{\\textbf{p}}^t$, we solve a bi-level \\emph{power system inhibition problem} (PSIP), to determine the worst-case generation\/load  shedding under a contingency with \\emph{exactly} $j$ failed elements. In the context of PSIP, the contingency vector $\\mathbf d$ is no longer a parameter but a vector of upper-level decision variables. In PSIP, the upper-level decisions $(\\mathbf d)$ correspond to binary contingency selection decisions and the lower level decisions $(\\mathbf f, \\mathbf p, \\mathbf q, \\mathbf r, s, \\boldsymbol \\theta)$ correspond to recourse generation schedule and DC power flow under the state prescribed by the the unit commitment decisions $\\mathbf x^t$, the no-contingency state economic dispatch $(\\tilde{\\textbf p}^t)$, and upper-level contingency selection variables $(\\mathbf d)$.\n\n\nBefore we introduce the power system inhibition problem (PSIP), we augment the direct current power flow constraints as follows.  We introduce three sets of non-negative, continuous variables corresponding to generation shedding $r_g$ for all $g \\in \\mathcal G$, loss-of-load at each bus $q_i$ for all $i \\in \\mathcal I$ and auxiliary variable $s$ corresponding to total system loss-of-load above the allowable threshold $\\varepsilon_ {j}D^t$. These variables in conjunction with additional constraints ensure that PSIP has a feasible recourse power flow for any unit commitment $\\mathbf x^t$, no-contingency state economic dispatch ${\\tilde{\\textbf p}^t}$ and upper-level contingency selection decisions $\\mathbf d$. We now state the power system inhibition problem.\n\\begin{subequations} \\label{bpsip}\n\\begin{align}\n\\textrm{B-PSIP}({\\textbf{x}}^t, \\tilde{\\textbf{p}}^t, j): &\\nonumber  \\\\\n\\max_{\\mathbf d} \\ \\min_{\\mathbf f,\\mathbf p, \\mathbf q, \\mathbf r, s, \\boldsymbol \\theta} \\quad &  \\sum_{g \\in \\mathcal G} r_g + s \\label{bpsip_obj}  \\\\\n\\text{s.t.}  \\quad\n        & \\sum_{e \\in\\mathcal  E} d_e + \\sum_{g \\in \\mathcal G} d_g = j\\label{bpsip_budget} \\\\\n    \t&\\sum_{g \\in \\mathcal G_i} (p_g-r_g) + \\sum_{e \\in \\mathcal E_{.i}}  f_{e} - \\sum_{e \\in \\mathcal E_{i.}}  f_{e} + q_i= D_i^t  &\\forall i \\in \\mathcal  I  \\label{bpsip_bal} \\\\\n    \t& B_e (\\theta_{i_e} - \\theta_{j_e})(1-d_e)-f_e=  0  & \\forall e \\in \\mathcal E  \\label{bpsip_vol} \\\\\n\t& -f_e \\leq F_e \\Delta_e^j (1-d_e)  & \\forall e \\in \\mathcal E  \\label{bpsip_f_lb} \\\\\n\t&  f_e \\leq F_e(1-d_e) \\Delta_e^j   & \\forall e \\in \\mathcal E  \\label{bpsip_f_ub} \\\\\n\t& p_g \\leq P^{\\max}_g   x_g^t (1-d_g)  & \\forall g \\in \\mathcal G \\label{bpsip_p_ub} \\\\\n\t&p_g \\leq R^u_g \\Delta_g^j  + \\tilde p_g^t   & \\forall g \\in \\mathcal G \\label{bpsip_ru} \\\\\n\t&-p_g \\leq R^d_g \\Delta_g^j  - \\tilde p_g^t(1-d_g)   & \\forall g \\in \\mathcal G \\label{bpsip_rd} \\\\\n\t&  q_i\\leq  D_i^t   & \\forall i \\in \\mathcal I \\label{bpsip_q} \\\\\n\t& r_g - p_g \\leq  0  & \\forall i \\in \\mathcal I \\label{bpsip_r} \\\\\n \t& \\sum_{i \\in \\mathcal I} q_i - s \\le \\varepsilon_ {j} D^t \\label{bpsip_s}\\\\\n\t& p_g \\ge 0,\\; q_i \\ge 0,\\; r_g \\ge 0,\\; s \\ge 0  & \\forall i \\in \\mathcal I, g \\in \\mathcal G \\label{bpsip_non_neg}\\\\\n\t& d_e \\in \\{0,1\\}, \\;d_g \\in \\{0,1\\} \\quad & \\forall e \\in \\mathcal E, \\forall g \\in \\mathcal G. \\label{bpsip_d}\n\\end{align}\n\\end{subequations}\n\nThe bi-level objective \\eqref{bpsip_obj} seeks to maximize, the minimum generation shedding and loss-of-load quantity above the allowable threshold. Since $r_g$ for all $g\\in \\mathcal G$ and $s$ are non-negative variables, the objective value is bounded below by zero.  If the objective value is equal to zero, the current solution $({\\textbf{x}}^t, \\tilde{\\textbf{p}}^t)$ has a feasible corrective recourse for all contingencies of size $j$.  Otherwise, the current solution cannot survive the contingency prescribed by upper-level contingency selection variables $\\mathbf d$. Given a contingency state defined by $\\mathbf d$, the objective of the power system operator (the inner minimization problem) is to find a corrective recourse power flow such that generation shedding and loss-of-load quantity above the allowable threshold is minimized. \\eqref{bpsip_budget} is a budget constraint on the number of power system elements, generation and\/or transmission, that must be in the selected contingency state. Constraints \\eqref{bpsip_bal} enforce power balance at each bus, with additional generation shedding variables $r_g \\in \\mathcal G$ for each generator located at a bus and a bus load-shedding variable $q_i \\in \\mathcal I$ to ensure system feasibility.  Constraints \\eqref{bpsip_vol}-\\eqref{bpsip_q} are as stated in (\\ref{cont_const}).  Constraints (\\ref{bpsip_r}) restrict the  amount of generation shedding to be at most the generation output for each generator $g \\in \\mathcal G$.  Constraint (\\ref{bpsip_s}) defines the amount of load shedding above the allowable threshold. If  $\\sum_{i \\in \\mathcal I} q_i > \\varepsilon_ {|c|} D^t$ then $ s = \\sum_{i \\in \\mathcal I} q_i - \\varepsilon_ {|c|} D^t$, otherwise, $s=0$.\n\n\n\\remark {Observe that  \\eqref{bpsip_vol} are nonlinear constraints with terms associated with products of binary contingency-selection variables (upper level) and continuous voltage phase angles  variables (lower level); thus, B-PSIP is a bi-level, nonlinear program. }\n\n\\remark {Observe that B-PSIP is feasible for any first-stage solution $(\\mathbf x^t, \\tilde{\\textbf{p}}^t)$ and any contingency prescribed by $\\mathbf d$; the  solution $\\mathbf f = \\mathbf 0, \\mathbf p^t = \\mathbf r^t = \\tilde{\\textbf{p}}^t , \\mathbf q^t = D^t$,  $s=(1-\\varepsilon_j)D^t$ and $\\boldsymbol \\theta = \\mathbf 0$ is feasible under any contingency state. }\n\nBi-level programs, such as (\\ref{bpsip}), cannot be solved directly. Next, we describe a reformulation strategy to derive a mixed-integer linear programming equivalent for B-PSIP. We begin by fixing the upper level variables $\\mathbf d$ and dualizing the inner minimization problem.  This results in a single-level, bilinear program with bilinear terms in the objective function.  In the resulting reformulation, there are five nonlinear terms, which are products of\nbinary contingency selection variables $(d_e, d_g)$ and continuous dual variables $(\\beta, \\hat \\delta, \\check \\delta, \\gamma, \\check \\lambda)$. Each of these non-linear terms can be linearized using the following strategy.\n\nLet $u$ and $v$ be two continuous variables and $b \\in \\{0,1\\}$.  Then the bilinear term, $(1-b)u$, can be linearized as follows. Letting $v = (1-b)u$, we introduce the following four constraints to linearize the bilinear term $(1-b)u$.\n\\begin{subequations}\n\\begin{align}\n\t  u-Ub \\;\\leq\\;& v \\;\\le\\; u + Ub \\label{lin13}\\\\\n\t -U(1-b) \\;\\leq\\; &  v \\;\\le\\; U(1-b) \\label{lin24}\n\\end{align}\n\\end{subequations}\n\n\\noindent\nHere, parameter $U$ represents an upper bound for continuous variable $u$ and satisfies $U \\ge |u|$. Assessing these  constraints for both binary values of $b$ shows that they provide a linearization.  If $b=1$, then constraint  (\\ref{lin24})  implies that $v=0$.  With $v=0$, constraints  (\\ref{lin13}) implies that $-U \\le u\\le U$, which are never binding. If $b=0$, then constraints (\\ref{lin13}) implies $u=v$ and constraint (\\ref{lin24})   implies $-U \\le v \\le U$, which are never binding.\n\n\\remark {If the bilinear term is a product of a binary variable $b$ and a non-positive variable $u$, i.e. $u \\le 0$, the lower bound in  (\\ref{lin24}) is redundant, and thus, can be eliminated. Analogously, if $u$ is a non-negative variable, i.e. $u \\ge 0$, the upper bound in  (\\ref{lin24}) is redundant, and thus, can be eliminated.}\n\nWe follow a similar strategy to linearize all five bilinear terms $(\\beta, \\hat \\delta, \\check \\delta, \\gamma, \\check \\lambda)$.  Define continuous variables $(r^1, r^1, r^3, r^4, r^5)$ and let $r_e^1 = (1-d_e) \\beta_e$, $r_e^2 = (1-d_e) \\hat \\delta_e$, $r_e^3 = (1-d_e) \\check \\delta_e$, $r_g^4 =(1-d_g) \\gamma_g$ and $ r_g^5 = (1-d_g) \\check \\lambda_g$.  Following the same linearization strategy introduced above, we now state the full mixed-integer linear PSIP formulation for completeness.\n\\begin{subequations}\\label{mod_psip_full}\n\\begin{align}\n\\!\\!\\!\\!\\! \\!\\!\\!\\!\\! \\!\\!\\!\\!\\! \\!\\!\\!\\!\\!\n\\max_{\\mathbf d, \\boldsymbol \\alpha,  \\boldsymbol \\beta,  \\hat {\\boldsymbol \\delta}, \\check {\\boldsymbol \\delta}, \\boldsymbol  \\gamma,  \\hat {\\boldsymbol \\lambda}, \\check {\\boldsymbol \\lambda}, \\boldsymbol  \\zeta, \\boldsymbol  \\pi} \\quad &\\sum_{i \\in \\mathcal I} D_i^t (\\alpha_i+\\zeta_i)+ \\sum_{e \\in \\mathcal E} F_e\\Delta_e^{j} (r_e^2 + r_e^3)  + \\sum_{g \\in  \\mathcal G}  P^{\\max}_g x_g^t r_g^4\\gamma_g \\!\\!&   \\\\\n& + \\sum_{g \\in  \\mathcal G} \\Big( (R^u_g \\Delta_g^{j} + \\tilde p_g^t)\\hat \\lambda_g + (R^d_g\\Delta_g^{j}  - \\tilde p_g^t ) r_g^5 +   \\varepsilon_ {j} D^t \\pi \\Big) &\\nonumber \\\\\n\\textmd{s.t.} \\quad & \\sum_{e\\in  \\mathcal E} d_e + \\sum_{g\\in  \\mathcal G} d_g = j & \\label{psip_full_budget} \\\\\n& \\alpha_{i_e} - \\alpha_{j_e} - \\beta_e - \\check \\delta_e + \\hat \\delta_e = 0   & \\forall e \\in \\mathcal E  \\\\\n& \\alpha_{i_g} + \\gamma_g + \\hat \\lambda_g - \\check \\lambda_g \\le 0 & \\forall g \\in \\mathcal G \\\\\n& \\alpha_i + \\zeta_i \\le 0 & \\forall i \\in \\mathcal  I   \\\\\n& -\\alpha_{i_g} + \\eta_g \\le 1 & \\forall g \\in \\mathcal G\\\\\n& -\\pi \\le 1 & \\\\\n&\\sum_{e \\in  \\mathcal E_{i.}}B_e r_e^1 - \\sum_{e \\in  \\mathcal E_{.i}}B_e r_e^1 = 0 & \\forall i \\in  \\mathcal   I \\\\\n& r_e^1  \\ge \\max\\{ \\beta_e - U d_e, - U (1-d_e)   \\} & \\forall e \\in \\mathcal E \\\\\n & r_e^1  \\le \\min \\{ \\beta_e + U d_e,  U (1-d_e)\\} & \\forall e \\in \\mathcal E \\\\\n&r_e^2  \\ge \\max \\{ \\hat \\delta_e - U d_e, - U (1-d_e)  \\} & \\forall e \\in \\mathcal E \\\\\n&r_e^2  \\le \\hat \\delta_e + U d_e& \\forall e \\in \\mathcal E \\\\\n&r_e^3  \\ge \\max\\{ \\check \\delta_e - U d_e, - U (1-d_e) \\}& \\forall e \\in \\mathcal E \\\\\n&r_e^3  \\le \\check \\delta_e + U d_e& \\forall e \\in \\mathcal E \\\\\n&r_g^4  \\ge \\max \\{ \\gamma_g - U d_g, - U (1-d_g) \\} & \\forall g \\in \\mathcal G \\\\\n&r_g^4  \\le\\gamma_g + U d_g& \\forall g \\in \\mathcal G \\\\\n&r_g^5  \\ge \\max \\{ \\check \\lambda - U d_g,  - U (1-d_g)\\} & \\forall g \\in \\mathcal G \\\\\n&r_g^5  \\le \\check \\lambda + U d_g& \\forall g \\in \\mathcal G\\\\\n& \\hat {\\boldsymbol \\delta}, \\check {\\boldsymbol \\delta}, \\boldsymbol  \\gamma,  \\hat {\\boldsymbol \\lambda}, \\check {\\boldsymbol \\lambda}, \\boldsymbol  \\zeta, \\boldsymbol  \\pi\\le 0\n\\end{align}\n\\end{subequations}\n\nNext, we outline an algorithm for \\emph{optimally} solving problem CCUC that combines a Benders decomposition with the aid of an oracle given by \\eqref{mod_psip_full}, which acts as a separation problem.  A given solution $({\\textbf{x}}^t, \\tilde{\\textbf{p}}^t)$ is feasible if the oracle cannot find a contingency of size $j$ that results in a loss-of-load above the allowable threshold. That is, if the optimal objective value is zero.  For each contingency budget $(j=1,\\cdots,k)$, we can check for the worst-case $j$-element contingencies by solving $(\\ref{mod_psip_full})$ using a failure budget of $j$ (i.e. the right-hand side of inequality (\\ref{psip_full_budget}) is set to $j$, as it is right now).  Whenever the oracle determines that the current solution is \\emph{not} $N$-$k$-$\\boldsymbol \\varepsilon$ compliant,  it returns a contingency state, prescribed by $\\mathbf d$, that results in a generation shedding and\/or  loss-of-load, above the allowable threshold $\\varepsilon_ j D^t$ for $j$-element failures.\n\n\\subsubsection{Contingency Screening Algorithms}\nWe now present a cutting plane algorithm,  referred to as the \\emph{Contingency Screening Algorithm 1 (CSA1)} to solve CCUC  implicitly by screening for the worst-case contingency, in terms of total generation and load shedding.\n\\begin{algorithm\n\\caption{\\emph{Contingency Screening Algorithm 1 (CSA1)}}\n\\begin{algorithmic}[1]\n\\State  Initialization: let $\\ell \\leftarrow 1$\n\\State Solve BUC$_\\ell$\n\\State \\textbf{if} BUC$_\\ell$ is infeasible, CCUC has no feasible solution, EXIT\n\\State  \\textbf{else}, let  $( {\\textbf{x}}_\\ell, \\tilde{\\textbf{p}}_\\ell)$ be an optimal solution of BUC$_\\ell$\n\\State \\hskip 0.8cm \\textbf{for} all $j \\in \\{1,\\cdots, k\\}$, $t \\in \\mathcal T$,\n\\State \\hskip 1.6cm solve  {PSIP${( {\\textbf{x}}^t_\\ell, \\tilde{\\textbf{p}}^t_\\ell,j)}$}   and let $w^*$ be the optimal objective value\n\\State \\hskip 1.6cm \\textbf{if}  $w^* > 0,$\n\\State \\hskip 2.4cm Add $f$-cut \\eqref{f_cut} to \\text{BUC$_\\ell$}\n\\State \\hskip 1.6cm \\textbf{end if}\n\\State \\hskip 0.8cm \\textbf{end for}\n\\State \\hskip 0.8cm \\textbf{if} $f$-cut(s) added in (7), let $\\ell \\leftarrow \\ell+1$ and return to (2)\n\\State \\hskip 0.8cm \\textbf{else}, $({\\textbf{x}}_\\ell, \\tilde{\\textbf{p}}_\\ell)$ is an optimal $N$-$k$-$\\boldsymbol \\varepsilon$ compliant solution, EXIT\n\\State \\hskip 0.8cm  \\textbf{end if}\n\\State \\textbf{end if}\n\\end{algorithmic}\n\\end{algorithm}\n\n\\subsubsection{Contingency Sharing Using a Dynamic Contingency List}\n\nIn preliminary testing using CSA1, we found that run time is significantly impacted by the need to solve a large number of PSIP instances at each master iteration of the algorithm. Specifically, we solve one instance of PSIP for each contingency-size and period pair $(j,t)$ for each master iteration.  The solution time of PSIP, as expected, is heavily impacted by the size of the power system $(\\mathcal I, \\mathcal G, \\mathcal E)$.  Figure \\ref{figure0} shows the solution time (on a logarithmic scale) of PSIP for various power system sizes and maximum contingency budgets $k$.\n\n \\begin{figure}[t]\n   \\centering\n  \\vskip -0.5cm\n    \\includegraphics[width=0.7\\textwidth, angle=0]{psip_sol_times.pdf}\n    \\vskip -1cm\n    \\caption{\\footnotesize {Average run times (sec.) of PSIP for varies power systems and contingency budgets ($k=1,2,3$).}}\n    \\label{figure0}\n\\end{figure}\n\nIn solving CCUC using CSA1 we also made three observations.  First, the majority of the the total run-time was spent solving PSIP (\\ref{mod_psip_full}) instances.  Secondly, a contingency $c$ that fails the system in one time period $t$ often fails the system in other time times as well, which suggests sharing of contingencies across time periods.  Thirdly, in the final solution only a small number of contingencies are actually identified.  That is to say, it is often prudent to consider a small number of contingencies \\emph{explicitly} in solving CCUC.\nBased on these observations, we found that it is most efficient to develop a version of the CSA algorithm that minimizes the number times we solve PSIP (\\ref{mod_psip_full}) instances and allows for sharing of contingencies across time periods.  We achieve this buy using a dynamic contingency list.\n\nWe begin with an empty contingency list $\\mathscr L$.  At each master iteration, we first screen all contingencies in the list for each time period $t \\in \\mathcal T$.  For each time period $t$, we generate feasibility cuts  \\eqref{f_cut} for each violated contingency in the list.  If none of the contingencies in the list is violated in any time period $t$, we proceed to solving PSIP instances to identify a new  violated contingency. This simple procedure ensures that each violated contingency identified by solving PSIP is never redundant.  That is to say, the new contingency is not in our existing contingency list.  When a new contingency is identified, we add it to the contingency list and check for its violation in all other time periods by solving a linear DCF problem.\nOur computational results indicate that this procedure results in the fewest total PSIP instances solved on average, which results in the fastest run\ntime. The key idea is that this procedure avoids redundant PSIP solutions to re-identify violated contingencies. This algorithm is referred to as the {\\em Contingency Screening Algorithm 2 (CSA2)}.\n\n\\begin{algorithm\n\\caption{\\emph{Contingency Screening Algorithm 2 (CSA2)}}\n\\begin{algorithmic}[1]\n\\State  Initialization: $\\ell \\leftarrow 1$, $\\mathscr L = \\emptyset$\n\\State Solve BUC$_\\ell$\n\\State \\textbf{if} BUC$_\\ell$ is infeasible, CCUC has no feasible solution, EXIT\n\\State  \\textbf{else}, let  $( {\\textbf{x}}_\\ell, \\tilde{\\textbf{p}}_\\ell)$ be an optimal solution of BUC$_\\ell$\n\\State \\hskip 0.8cm \\textbf{for} each $c\\in \\mathcal{C}$, $t \\in \\mathcal T$,\n\\State \\hskip 1.6cm solve  {CF${(\\mathbf{x}^t_\\ell, \\tilde{\\textbf{p}}^t_\\ell,\\mathbf{d}^c)}$}   and let $w^*$ be the optimal objective value\n\\State \\hskip 1.6cm \\textbf{if}  $w^* > 0,$\n\\State \\hskip 2.4cm Add $f$-cut \\eqref{f_cut} to \\text{BUC$_\\ell$}\n\\State \\hskip 1.6cm \\textbf{end if}\n\\State \\hskip 0.8cm \\textbf{end for}\n\\State \\hskip 0.8cm \\textbf{if} $f$-cut(s) added in step (7)\n\\State \\hskip 1.6cm let $\\ell \\leftarrow \\ell+1$, return to step (2)\n\\State \\hskip 0.8cm \\textbf{end if}\n\\State \\hskip 0.8cm \\textbf{for} all $j \\in \\{1,\\cdots, k\\}$, $t \\in \\mathcal T$,\n\\State \\hskip 1.6cm solve  {PSIP${( {\\textbf{x}}^t_\\ell, \\tilde{\\textbf{p}}^t_\\ell,j)}$}   and let $z^*$ be the optimal objective value\n\\State \\hskip 1.6cm \\textbf{if}  $z^* > 0,$  add $f$-cut \\eqref{f_cut} to \\text{BUC$_\\ell$}\n\\State \\hskip 2.4cm let $\\ell \\leftarrow \\ell+1$,  $\\mathscr L \\leftarrow \\mathscr L \\cup \\{c\\}$, return to step (2)\n\\State \\hskip 1.6cm \\textbf{end if}\n\\State \\hskip 0.8cm \\textbf{end for}\n\\State  \\textbf{end if}\n\\State $( \\tilde{\\textbf{x}}_\\ell, \\tilde{\\textbf{p}}_\\ell)$ is an optimal $N$-$k$-$\\boldsymbol \\varepsilon$ compliant solution, EXIT\n\\end{algorithmic}\n\\end{algorithm}\n\n\n\n\\section{Computational Experiments}\n\\label{sec:experiments}\n\nWe implemented our proposed models and algorithms in C++ using IBM's Concert Technology Library 2.9 and CPLEX 12.1 MILP solver. All experiments were performed on a workstation with two quad-core, hyper-threaded 2.93GHz Intel Xeon processors with 96GB of memory.  This yields a total of 16 threads allocated to each invocation of  CPLEX. The default behavior of CPLEX 12.1 is to allocate a number of threads equal to the number of machine cores. In the case of hyper-threaded architectures, each core is presented as a virtual dual-core -- although it is important to note that the performance is not equivalent to a true dual core. The workstation is shared by other users, such that our run-time results should be interpreted as conservative. With the exception of the optimality gap, which we set to 0.1\\%, we used the CPLEX default settings for all other parameters. All runs were allocated a maximum of $10,800$ seconds (3 hours) of wall clock time.\n\nWe executed our models and algorithms on five test systems of varying size: the 6-bus, IEEE 24-bus, RTS-96, and IEEE 118-bus test systems \\cite{IEEEtest}, and a simplified model of the US Western interconnection (WECC-240)\\cite{Price2011}. The 6-bus system described in Section \\ref{ieee6_example} is further augmented with three fast ramping generation units located at bus 1, 2, and 6, respectively, to ensure there is sufficient generation capacity for larger-size contingencies.  Generator data for these three units are identical to G4-G6 in Table \\ref{tab10}.  To ensure there is sufficient operational flexibility in the WECC-240  system, we made eight transmission lines and one generation unit immune to failures. These nine elements include serial lines, pairs of transmission lines, and generation unit and transmission line pairs, whose failure would result in islanding of subsystems (buses). Additionally, we assume that non-dispatchable generation injections into the system can be shed during contingency states. For each test system, we consider a 24 hour planning horizon and the four contingency budgets $k=0, 1, 2,$ and $3$, yielding a total of 20 instances.\n\n \\begin{table}[t]\n\\caption{Runtimes for different solution approaches to the CCUC problem}\n\\centering\n\\small{\n\\begin{tabular}{l c c c| c c c}\n\\hline\n\\multicolumn{4}{c|}{} & \\multicolumn{3}{c}{Solution time} \\\\[-0.3ex]\n\\multicolumn{4}{c|}{} & \\multicolumn{3}{c}{(exit status or feasibility gap)} \\\\[0.5ex]\nTest System \t\t&$C$ \t\t& $k$\t&  $\\varepsilon_ {k}$ \t\t& EF        \t\t\t\t & BD        \t\t& CSA2           \t \t\\\\[0.5ex]\n\\hline\n\n 6-bus\t\t\t&\t0\t\t\t&\t0\t&\t0 \t\t    \t\t\t\t\t& 0 \t\t\t\t& 0 \t\t\t & 0\t\t\t \t\\\\[-0.3ex]\n         \t\t\t\t&\t16\t\t\t&\t1\t&\t0 \t\t    \t\t\t\t\t& 3 \t\t\t\t& 2 \t\t\t & 1\t\t\t\t\\\\[-0.3ex]\n           \t\t\t&\t136\t\t\t&\t2\t&\t0.29 \t\t\t\t\t\t\t& 7 \t\t\t\t& 16 \t\t\t & 2\t\t\t\t \\\\[-0.3ex]\n        \t\t\t\t&\t696\t\t\t&\t3\t&\t0.77 \t\t\t\t\t\t\t& 134 \t\t\t& 189 \t\t & 4\t\t\t\t \\\\[-0.3ex]\n\\hline\nIEEE 24-bus     \t\t&\t0\t\t\t&\t0\t&\t0 \t\t    \t\t\t\t\t& 0 \t\t\t\t& 1 \t\t\t& 0\t\t\t\t\\\\[-0.3ex]\n         \t\t\t\t&\t70\t\t\t&\t1\t&\t0 \t\t    \t\t\t\t\t& 108 \t\t\t& 58\t\t\t& 11\t\t\t\t\\\\[-0.3ex]\n            \t\t\t&\t2,485\t\t&\t2\t&\t0.08 \t\t\t\t\t\t\t& x(LPR) \t\t\t& 3,861 \t\t& 101\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t57,225\t\t&\t3\t&\t0.21 \t\t\t\t\t\t\t& x(OM)\t\t\t& x(0.03) \t\t& 397\t\t\t \\\\[-0.3ex]\n\\hline\nRTS-96     \t\t&\t0\t\t\t&\t0\t&\t0 \t\t    \t\t\t\t\t& 1 \t\t\t\t& 2 \t\t\t & 2\t\t\t\t\\\\[-0.3ex]\n         \t\t\t\t&\t216\t\t\t&\t1\t&\t0 \t\t    \t\t\t\t\t& x(LPR)\t\t\t& 303 \t\t& 41\t\t\t\t\\\\[-0.3ex]\n            \t\t\t&\t23,436\t\t&\t2\t&\t0.05 \t\t\t\t\t\t\t& x(OM) \t\t\t& 8,139 \t \t& 4,04\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t1,679,796\t\t&\t3\t&\t0.09 \t\t\t\t\t\t\t& x(OM) \t\t\t & x(0.050) \t\t& 4,989\t\t\t \\\\[-0.3ex]\n\\hline\nIEEE 118-bus     \t\t&\t0\t\t&\t0\t&\t0 \t\t    \t\t\t\t\t& 1 \t\t\t\t& 1 \t\t\t& 1\t\t\t\t\\\\[-0.3ex]\n         \t\t\t\t&\t240\t\t\t&\t1\t&\t0.01 \t\t    \t\t\t\t\t& x(LPR) \t\t\t & 3,513 \t\t& 352\t\t\t\\\\[-0.3ex]\n            \t\t\t&\t28,920\t\t&\t2\t&\t0.12 \t\t\t\t\t\t\t& x(OM) \t\t\t& x(0.204) \t\t& 1,232\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t2,304,200\t\t&\t3\t&\t0.25 \t\t\t\t\t\t\t& x(OM) \t\t\t & x(0.249) \t\t& 8,586\t\t\t \\\\[-0.3ex]\n\\hline\nWECC-240     \t\t&\t0\t\t\t&\t0\t&\t0 \t\t    \t\t\t\t\t& 1 \t\t\t\t& 2 \t\t\t & 2\t\t\t\t\\\\[-0.3ex]\n         \t\t\t\t&\t424\t\t\t&\t1\t&\t0 \t\t    \t\t\t\t\t& x(LPR) \t\t\t & 262 \t\t& 108\t\t\t\\\\[-0.3ex]\n            \t\t\t&\t90,100\t\t&\t2\t&\t0.06 \t\t\t\t\t\t\t& x(OM) \t\t\t& x(0.004) \t\t& 2,484\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t12,704,524\t&\t3\t&\t0.15 \t\t\t\t\t\t\t& x(OM) \t\t\t& x(0.004) \t\t& x(0)\t\t\t\\\\[-0ex]\n\\hline\n\\end{tabular}}\n\\label{tab1}\n\\end{table}\n\nWe first consider the run-times for the three different algorithms for solving the CCUC problem: the extensive form MILP, Benders decomposition, and the Contingency Screening Algorithm 2 (CSA2). The results are presented in Table \\ref{tab1}. All times are reported in wall clock (elapsed) seconds. The column labeled ``$C$\" reports the number of distinct contingencies for a given budget $k$, while the column labeled ``$\\varepsilon_k$\" reports the fraction of total load (demand) that can be shed. Entries in Table \\ref{tab1} reporting ``x\" indicate that the corresponding algorithm failed to locate a $N$-$k$-$\\boldsymbol \\varepsilon$ compliant solution within the 0.1\\% optimality gap within the 3 hour time limit.  For those instances that could not be solved within the allocated time, we provide exit status or feasibility gaps, indicating the maximum fraction of total demand shed \\emph{above} the allowable threshold $\\varepsilon_k$ in the final solution. In all runs of the CSA2 algorithm, we initialize the contingency list $\\mathscr L$ to the empty list.\n\nAs expected, the extensive form approach (EF) can only solve the smallest  instances, since for each contingency, a full set of DC power flow constraints (\\ref{cont_const}) must be explicitly embedded in the formulation. As the number of contingencies grows, this formulation quickly becomes intractable.  The exit status ``LPR\" and ``OM'' represent ``solving Linear Programming Relaxation at root node'' and ``Out of Memory'',  respectively. Note that our test instances only represent small to at most moderate sized systems relative to real power systems (which can contain on the order of thousands to tens of thousands of elements), indicating that even significant advances in solver technology are unlikely to mitigate this issue. Further, even given significant algorithmic advances, the memory requirements associated with the EF will likely cause the intractability to persist.\n\nThe BD approach attempts to address the exponential, as shown in Remark 6, explosion in the number of contingencies via a Benders reformulation\/decomposition, with corresponding delayed cut generation. However, although the BD approach does not explicitly incorporate  power flow constraints (\\ref{cont_const}) for each contingency into the formulation, those power flow constraints  must still be solved to identify violated feasibility cuts (which are then added to the master problem). In summary, the BD approach mitigates the memory issues associated with the EF approach, but the cost of identifying feasibility violations for a rapidly growing number of contingencies remains prohibitive. Overall, the BD approach can solve larger instances than the EF approach, but still fails given larger $k$ and larger test instances.\n\nFinally, we consider the performance of our third approach: CSA2. Here, we see that all of our test instances, with the sole exception of the WECC-240 system with $k=3$, can be solved within the 3 hour time limit. This result is enabled by the combination of using a dynamic contingency list (significantly reducing the number of PSIP solves) and the fact that we are able to implicitly evaluate all the contingencies in order to identify a violated contingency, and then quickly find a corresponding feasibility cut by solving a single linear program (DCF). These features of the CSA2 algorithm allow it to mitigate the effects of a combinatorial number of contingencies and the associated impact on run-times and memory requirements.  Lastly, we note that although CPA2 failed to solve the WECC-240 system with $k=3$ within the allocate time, the final solution at the three hour mark is in fact a $N$-$k$-$\\boldsymbol \\varepsilon$ compliant solution.  For large power systems and\/or contingency budgets, significant computational time is required to ``prove'' feasibility.  Eliminating the three hour time limit, we observed that the WECC-240 system with $k=3$ could be solved in approximately 18 hours, with the majority of this time taken to prove feasibility of the final solution.\n\nWe next examine the run-times of our CSA2 algorithm in further detail, as reported in Table \\ref{tab2}. For each instance, we report the total number of possible contingency states $C$ and the number of contingencies for which corresponding feasibility cuts were actually generated. The latter corresponds to the final size of the dynamic contingency list, which is reported in the column labeled ``$|\\mathscr L|$\". Clearly, $|\\mathscr L|$ corresponds to a vanishingly small fraction of the possible number of contingencies, which is critical to the tractability of the approach. The remaining columns of Table \\ref{tab2} break down the total run time (in wall clock seconds) by the three main components of the algorithm -- the RMP, which identifies unit commitments; the power system inhibition problem (PSIP), which identifies a contingency that has no feasible corrective recourse power flow given the current RMP UC decisions and no-contingency economic dispatch; and the contingency feasibility subproblems (DCF), which yield the feasibility cuts.  The final column, labeled ``cuts'', reports the total number of feasibility cuts generated in solving the instance. It is clear from Table \\ref{tab2} that the computational bottleneck in the CSA2 algorithm is the solution of the PSIP, such that any improvements to that process will yield immediate reductions in CSA2 run-times.\n\n\\begin{table}[t]\n\\caption{Runtime breakout for the CSA2 algorithm}\n\\centering\n\\footnotesize{\n\\begin{tabular}{l r r r | r r r | r r r r }\n\\hline\nTest Systems \t\t&$C$ \t\t& \t$k$\t&  $\\varepsilon_ k$ \t\t&RMP        \t&PSIP       \t\t &DCF        \t\t& itr\t\t&$|\\mathscr L|$ \t&cuts \t\\\\[0.5ex]\n\\hline\n 6-bus\t\t\t&\t0\t\t\t\t&\t0\t&\t0 \t\t    \t\t& 0 \t\t    \t& 0\t\t        \t & 0\t\t\t\t& 1\t\t&0\t\t&0\t\t\t\\\\[-0.3ex]\n         \t\t\t\t&\t16\t\t\t\t&\t1\t&\t0 \t\t    \t\t& 0 \t\t\t& 1 \t\t\t & 0\t\t\t\t& 2\t\t&1\t\t&21\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t136\t\t\t\t&\t2\t&\t0.29 \t\t\t\t& 0 \t\t\t& 2 \t\t\t & 0\t\t\t    \t& 7\t\t&3\t\t&48\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t696\t\t\t\t&\t3\t&\t0.77 \t\t\t\t& 0 \t\t\t& 4 \t\t\t & 0\t\t\t    \t& 11\t\t&4\t\t&89\t\t\t\\\\ [-0.3ex]\n\\hline\nIEEE 24-bus    \t\t&\t0\t\t\t\t&\t0\t&\t0 \t\t    \t\t& 0 \t\t    \t& 0\t\t        \t & 0\t\t\t\t& 1\t\t&0\t\t&0\t\t\t\\\\[-0.3ex]\n       \t\t\t\t&\t70\t\t\t\t&\t1\t&\t0 \t\t    \t\t& 6\t\t\t& 46 \t\t\t& 1\t\t\t\t & 185\t&1\t\t&185\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t2,485\t\t\t&\t2\t&\t0.08\t\t\t\t& 26 \t\t\t& 69 \t\t\t & 6\t\t\t   \t& 857\t&3\t\t&857\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t57,225\t\t\t&\t3\t&\t0.21 \t\t\t\t& 64 \t\t\t& 324 \t\t& 9\t\t\t   \t & 928\t&4\t\t&928\t\t\t\\\\ [-0.3ex]\n\\hline\nRTS-96    \t\t\t&\t0\t\t\t\t&\t0\t&\t0 \t\t    \t\t&2 \t\t    \t& 0\t\t        \t & 0\t\t\t\t& 1\t\t&0\t\t&0\t\t\t\\\\[-0.3ex]\n       \t\t\t\t&\t216\t\t\t\t&\t1\t&\t0 \t\t    \t\t& 13\t\t\t& 25 \t\t\t& 3\t\t\t\t & 12\t\t&3\t\t&27\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t23,436\t\t\t&\t2\t&\t0.05\t\t\t\t& 17\t\t\t& 385 \t\t& 2\t\t\t   \t & 15\t\t&4\t\t&33\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t1,679,796\t\t\t&\t3\t&\t0.09 \t\t\t\t& 19 \t\t\t& 4,965 \t\t & 5\t\t\t   \t& 17\t\t&5\t\t&38\t\t\t\\\\ [-0.3ex]\n\\hline\nIEEE 118-bus    \t&\t0\t\t\t\t&\t0\t&\t0 \t\t    \t\t&1 \t\t    \t& 0\t\t        \t & 0\t\t\t\t& 1\t\t&0\t\t&0\t\t\t\\\\[-0.3ex]\n       \t\t\t\t&\t240\t\t\t\t&\t1\t&\t0.01 \t\t    \t\t& 243\t\t& 72 \t\t\t& 37\t\t\t\t & 85\t\t&5\t\t&1,305\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t28,920\t\t\t&\t2\t&\t0.12\t\t\t\t& 377 \t\t& 796 \t\t& 59\t\t\t\t & 120\t&7\t\t&1,671\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t2,304,200\t\t\t&\t3\t&\t0.25 \t\t\t\t& 405 \t\t& 8,114 \t\t & 67\t\t\t\t& 132\t&9\t\t&1,743\t\t\\\\ [-0.3ex]\n\\hline\nWECC 240-bus    \t\t&\t0\t\t\t&\t0\t&\t0 \t\t    \t\t&0 \t\t    \t& 0\t\t        \t & 0\t\t\t\t& 1\t\t&0\t\t&0\t\t\t\\\\[-0.3ex]\n       \t\t\t\t&\t424\t\t\t\t&\t1\t&\t0 \t\t    \t\t& 4\t\t\t& 102 \t\t& 2\t\t\t\t & 5\t\t&2\t\t&48\t\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t90,100\t\t\t&\t2\t&\t0.06\t\t\t\t& 3 \t\t\t&2,479 \t\t & 2\t\t\t\t& 5\t\t&2\t\t&48\t\t\\\\[-0.3ex]\n        \t\t\t\t&\t12,704,524\t\t&\t3\t&\t0.15 \t\t\t\t& x \t\t\t& x \t\t\t & x\t\t\t\t& x\t\t&x\t\t&x\t\t\t\\\\ [-0ex]\n\\hline\n\\end{tabular}}\n\\label{tab2}\n\\end{table}\n\n\\section{Conclusions}\n\\label{sec:conclusions}\n\nWe have investigated the problem of committing generation units in power system operations, and determining a corresponding no-contingency state economic dispatch, such that the resulting solution satisfies the $N$-$k$-$\\boldsymbol \\varepsilon$ reliability criterion. This reliability criterion is a generalization of the well-known $N$-$k$  criterion, and requires that at least $(1-\\varepsilon_ j)$ fraction of the total demand is met following the failure of $j$  system components, for $j \\in \\{1,\\cdots,k\\}$. We refer to this problem as the contingency-constrained unit commitment problem, or CCUC. We proposed two algorithms to solve the CCUC: one based on the Benders decomposition approach, and another based on contingency screening algorithms. The latter method avoids the combinatorial explosion in the number of contingencies by seeking vulnerabilities in the current solution, and generating valid inequalities  to exclude such infeasible solutions in the master problem.  We tested our proposed algorithms on test systems of varying sizes.  Computational results show our proposed Contingency Screening Algorithm (CSA2), which uses a bi-level separation program to implicitly consider all contingencies and a dynamic contingency list to avoid re-identification of contingencies, significantly outperforms the Benders decomposition approach.  We were able to solve all test systems, with the exception of the largest WECC-240 instance, in under 3 hours. In contrast, both the Benders decomposition algorithm and a straightforward solution of the CCUC extensive form, failed to solve all but the smallest instances within 3 hours.\n\nWe believe that  this paper will provide a significant basis for subsequent research in contingency-constrained unit commitment. For example, we are working to apply these methods to full-scale systems. While our results are promising in terms of scalability, full-scale problems pose more significant computational challenges, and consequently will require stronger formulations for the power system inhibition problem and possible adoption of high-performance computing resources.  Further, our current CCUC model assumes  all  component failures occur simultaneously. In order to reflect practical operational situations, where failures may happen consecutively, new CCUC models that consider timing between system component failures are needed. We plan to extend our CCUC models to include these cases. Finally, we worked exclusively with a deterministic CCUC model to date. However, it is ultimately essential to take uncertainty into account in unit commitment, e.g., to account for uncertain demand and renewable generation units. We believe our current cutting plane framework can be naturally extended to robust optimization and stochastic programming formulations via a nested decomposition approach.\n\n\\vskip 0.5cm\n\n{\\bf\\it Acknowledgement}. {\nSandia National Laboratories' Laboratory-Directed Research and Development Program and the U.S. Department of Energy's Office of Science (Advanced Scientific Computing Research program) funded portions of this work. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.}\n\n\\bibliographystyle{}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Supplementary Information}\n\n\nHere we provide the details of the kinetic equation approach used in the main text to describe collisions in the presence of a magnetic field. This analysis links\nthe ee scattering crosssection $\\sigma(\\theta)$ angular dependence \nto the magnetic steering response measured by a voltage probe placed at system boundary near current injector.\n\n\nAs a first step, we recall the basics of the collisionless Hamiltonian transport in magnetic field. The evolution of particle distribution in a four-dimensional phase space parameterized by particle coordinates and velocity components, $\\xi=(\\vec r,\\vec v)$, is described by\n\\begin{equation}\\label{eq:Liouville_eqn}\n(\\partial_t+\\dot \\xi\\partial_\\xi)f(\\xi,t)=J(\\xi)\n,\\quad\n\\dot\\xi=(v_1,v_2,-\\omega v_2,\\omega v_1)\n\\end{equation}\nwhere $J$ is a time-independent particle source representing the electron injector. The last two components of $\\dot\\xi$ represent the Lorentz force expressed through the  cyclotron frequency $\\omega=\\frac{e}{m}B$. For a generic point source at the origin,\n\\begin{equation}\nJ(\\vec r, \\vec v)=\\delta(\\vec r)S(\\vec v)\n,\n\\end{equation}\nEq.\\eqref{eq:Liouville_eqn}\nis satisfied by a steady-state distribution given by a sum of the contributions due to trajectories $\\eta(t)=(\\vec r(t),\\vec v(t))$ with all possible initial conditions, \n\\begin{equation}\n\\dot\\eta(t)=\\dot\\xi\n,\\quad\n\\eta(t=0)=\\eta'\n.\n\\end{equation}\nNamely, \n\\begin{equation}\\label{eq:Liouville_theorem}\nf(\\xi)=\\int_0^\\infty dt e^{-\\delta t}\\int d^4\\eta'\\delta^{(4)}(\\xi-\\eta(t))J(\\eta')\n\\end{equation}\nwhere the positive-$t$ integration domain\naccounts for the cause-effect relation in the dynamics, Eq.\\eqref{eq:Liouville_eqn}, with a small $\\delta>0$ added to assure convergence of the integral over $t$. Physically, the value $\\delta^{-1}$ represents the time scale after which the free-particle picture becomes inapplicable e.g. due to particles escaping the system through contacts or colliding with other particles, phonons, or disorder. \nIn the limit $\\delta\\to 0$, Eq.\\eqref{eq:Liouville_theorem} is in agreement with\nthe Liouville theorem which asserts that the phase-space distribution function is constant along the trajectories of the system.\n\n \\begin{figure}[t]\n \n\\begin{center}\n\\includegraphics[width=0.7\\columnwidth]{Fig3}\n \\end{center}\n\\caption{a) Trajectory of a particle injected into an electron system from a current source at $\\vec r=0$, and propagating over a cyclotron orbit to point $\\vec r$. Particle velocity direction at $\\vec r$, described by the angle $\\theta_{\\vec v}$, is determined by the radius vector $\\vec r$ and magnetic field, as given in Eq.\\eqref{eq:f(xi)_1}.\nb) Particles injected in the electron system propagate away from the injector and, after scattering off the background electrons, propagate to the probe placed at a distance $a$ from the source. Shown are the quantities used in the analysis. \n} \n \\label{fig2extra}\n\\end{figure}\n\nWe consider a point-like source, represented by a delta function in space and a broad angular distribution of velocity the injected particles\n\\begin{equation}\nJ(\\vec r,\\vec v')=\\delta(\\vec r)S(\\vec v')\n,\n\\end{equation}\nwhere without loss of generality we place the source at the origin of the coordinate system. In this case, the four-dimensional delta function $\\delta^{(4)}(\\xi-\\eta(t))$, after being integrated over $t$ and the two components of $\\vec v'$, gives a one-dimensional delta function. The latter can be represented as a delta-function  of the velocity $\\vec v$\ndeflected by magnetic field. Namely, the angle $\\theta_{\\vec v}$ is such that there exists a classical trajectory that makes it from $0$ to $\\vec r$ and passes through $\\vec r$ at that angle. Accordingly, the phase space density $f(\\xi)$, defined in  Eq.\\eqref{eq:Liouville_theorem}, equals \n\\begin{equation}\\label{eq:f(xi)_1}\nf(\\xi)=A(r)S(v)\\delta(\\theta_{\\vec v}-\\theta_{\\vec r}-\\theta_B)\n,\n\\end{equation}\nwhere\n$\\theta_B=\\arcsin(r\/2R_c)$ is the magnetic deflection angle, with $R_c=mv\/B$ the cyclotron radius. \nThe prefactor\n\\begin{equation}\nA(r)=\\frac1{vr\\cos(\\theta_B)}=\\frac1{vr\\sqrt{1-r^2\/4R_c^2}}\n.\n\\end{equation}\naccounts for the spreading of trajectories originating from the source at slightly different initial angles.\nThe divergence in the phase-space density for $r\\approx 2R_c$ (magnetic focusing) occurs when the velocity is nearly perpendicular to $\\vec r$, so that the radial velocity is tiny. The inverse square-root divergence has a simple meaning: Because of the Lorentz force  particles cannot go away from the injector by more than $2R_c$, after reaching that distance they must turn around and come back. Thus at distances $r\\approx 2R_c$ velocity is nearly perpendicular to the radius vector $\\vec r$. The small value of the radial velocity causes a divergence in the phase-space density $f(\\xi)$. \n\nThis scheme can now be extended to describe an interacting system in which particles propagate along cyclotron orbits between collisions, with the collisions causing abrupt switching between different orbits. To account for collisions which transfer particles between cyclotron orbits, we add \nin Eq.\\eqref{eq:Liouville_eqn} a collision term as\n\\begin{equation}\\label{eq:Liouville_eqn_Iee}\n(\\partial_t+\\dot \\xi\\partial_\\xi -I_{\\rm ee})f(\\xi,t)=J(\\xi)\n,\\quad\n\\dot\\xi=(v_1,v_2,-\\omega v_2,\\omega v_1)\n\\end{equation}\nwhere $I_{\\rm ee}$ is a linearized collision integral (for details, see Ref.\\cite{shytov2018}). \nWe will be interested in the \nsignal measured by a probe placed at the boundary at a distance $a$ from the source (see Fig.\\ref{fig2extra}). By combining the above treatment of the free-particle problem with the perturbation expansion of the phase-space density $f(\\xi)$ in powers of $I_{\\rm ee}$, developed in Ref.\\cite{shytov2018}, yields a closed-form expression for \nthe particle flux into the probe.\n\nWe will focus on the contribution first-order in $I_{\\rm ee}$, which gives the measured voltage signal\n\\begin{equation}\\label{eq:delta_V_general}\n\\delta V=evR \\int d^2r e^{-\\gamma(t_1+t_2)}\\sin(\\theta_{+}')A(r_1)A(r_2)\\sigma_\\pi(\\theta-\\tilde\\theta_B\n)S(v)\n\\end{equation}\nwhere $\\sigma_\\pi(\\theta)$ is a shorthand for $\\sigma(\\theta-\\pi)$ less the delta function term $-\\gamma\\delta(\\theta-\\pi)$,\n$\\theta_{+}'$ is the incidence angle at the probe shown in Fig.\\ref{fig2extra}, $R$ is the probe inner resistance, the times $t_{1(2)}$ are given by $\\omega t_{1(2)}=\\arcsin{(r_{1(2)}\/2R_c)}$, and $\\gamma$ is the scattering rate, defined implicitly via Eq.\\eqref{eq:crossection_forward} and Eq.\\eqref{eq:sum_rules}. \n\nIn the above derivation, the delta function contribution to $\\sigma(\\theta)$ of the form $-\\gamma\\delta(\\theta)$ has been moved from $I_{\\rm ee}$ to the transport operator. After combining it with the $\\vec v\\nabla$ term, the perturbation expansion in $I_{\\rm ee}$ can be carried out in a straightforward manner. In doing so, the decay rate $\\gamma$ will replace the fictitious decay rate $\\delta$ in Eq.\\eqref{eq:Liouville_theorem}, generating the exponential decay factor in Eq.\\eqref{eq:delta_V_general}.\n\n\nThe scattering angle, at which the crosssection\nis evaluated, is shifted by a sum of two deflection angles\n\\begin{equation}\n\\tilde\\theta_B=\\frac{\\omega t_1}2+\\frac{\\omega t_2}2\n=\\arcsin\\left(\\frac{r_1}{2R_c}\\right)+\\arcsin\\left(\\frac{r_2}{2R_c}\\right)\n.\n\\end{equation}\nThe two terms account \nfor the change of the particle velocity orientation due to cyclotron motion before and after scattering, as illustrated in Fig.\\ref{fig2extra}.\nEq.\\eqref{eq:delta_V_general} describes the magnetic steering response in a wide range of parameters. It was used to generate the dependence of the steering signal plotted in Fig.2 of the main text. \n\n\\end{document}\n----------------------------\n\nThe expression in Eq.\\eqref{eq:delta_V_general} can be used to evaluate the response in a wide range of parameters. We are interested in the limit of weak fields where the essential physics can be distilled directly by taking the lowest order in $B$. To that end, we will assume that the cyclotron radius $R_c$ is greater than the ee collision mean free path. In this case, at first order in $B$, we can suppress the terms $r^2\/R_c^2$ in the expressions for $A(r_{1(2)})$ and approximate $\\tilde\\theta_B$ by $\\omega(r_1+r_2)\/2v$. We also ignore the field dependence in $\\theta'$, approximating it with a $B=0$ value. Finally, we assume an isotropic source, so $S(v)=S_0\\sin{\\phi}$, where $\\phi$ is the angle at which a particle leaves the injector. The right side of Eq. \\eqref{eq:delta_V_general} then becomes \n\\begin{equation}\\label{eq:delta_V_apprx}\nS_0Rev\\int d^2r e^{-\\gamma(r_1+r_2)\/v}\\frac{\\sin\\theta'\\sin\\phi}{v^2r_1r_2} \\sigma_\\pi\\left( \\theta-\\frac{\\omega(r_1+r_2)}{2v}\\right)\n\\end{equation}\nDue to the $1\/r_{1, 2}$ factors imposed by particle number conservation, the integral Eq.\\eqref{eq:delta_V_apprx} gets significant contributions from points with small $r_1$ or $r_2$. These contributions are not very sensitive to changes in $B$, because they sample the cross-section $\\sigma_{\\pi}$ at large $\\theta$, where it is small and slowly varying. Contributions at large $r_{1, 2}$, however, sample $\\sigma_{\\pi}$ at small values of $\\theta$, near the backscattering peak. As a result, contributions with large $r_{1, 2}$ can still substantially affect the integral Eq.\\eqref{eq:delta_V_apprx}, and can also be strongly modulated by small changes in $B$, via $\\tilde\\theta_B$. We will compute the voltage response due to points with $r_{1, 2}\\gg a$, so that we may approximate $r_1\\approx r_2\\approx r$, $\\theta\\approx\\sin{\\theta}=(a\/r)\\sin{\\phi}$, and $\\theta'\\approx\\phi$. We then can write\n\\begin{equation}\\label{eq:delV1}\n\\delta V_{r\\gg a}=\\frac{S_0Re}{v}\\int d^2r e^{-2\\gamma r\/v}\\frac{\\sin^2{\\phi}}{r^2}\\sigma_\\pi \\left(\\frac{a}{r}\\sin{\\phi}-\\frac{\\omega r}{v}\\right).\n\\end{equation}\nSince this approximation is only good for $r\\gg a$, the integral Eq.\\eqref{eq:delV1} is not a good approximation to the full voltage response $\\delta V$ in Eq.\\eqref{eq:delta_V_general}). However, since the $B$ field dependence of $\\delta V$ is mostly due to points with large $r$, $\\delta V_{r\\gg a}$ should be able to capture the \\textit{change} in $\\delta V$ due to a changing $B$ field in some suitable range of parameters. That is, \n\\begin{equation}\\label{eq:change_approx}\n\\delta V\\big|_0^B=\\delta V_{r\\gg a}\\big|_0^B\n\\end{equation}\nfor some choice of field and probe-source distance. Specifically, we expect this condition to hold if the integrand in Eq.\\eqref{eq:delV1} is maximized with $a\\ll r\\ll R_c$ (recall that we assumed $r\\ll R_c$ past Eq.\\eqref{eq:delta_V_apprx}). If this is the case for the approximate expression Eq.\\eqref{eq:delV1}, then the exact expression in Eq.\\eqref{eq:delta_V_general} will receive its large-$r$ (and hence magnetic field-sensitive) contributions from $r\\gg a$. Analyzing the integrand of Eq.\\eqref{eq:delV1} shows that this condition is satisfied for\n\\begin{equation}\n\\frac{a\\omega}{v}\\lesssim\\theta_T\\ll 1.\n\\end{equation}\nIn this regime Eq.\\eqref{eq:delV1} may be manipulated to yield\n\\begin{equation}\\label{eq:voltage_diff}\n\\delta V\\big|_0^B =\\frac{2S_0Re}{v}\\int_{-\\pi}^{\\pi}d\\alpha \\sigma_{\\pi}(\\alpha)F(\\omega', \\alpha)\\big|_{\\omega'=0}^{\\omega'=\\omega}\n\\end{equation}\nfor\n\\begin{equation}\\label{eq:F_func_pos}\nF(\\omega, \\alpha)=\\int_0^1 du \\frac{u^2}{\\sqrt{1-u^2}}\\frac{\\exp{\\left(\\frac{\\gamma}{\\omega}(\\alpha-\\sqrt{\\alpha^2+\\frac{4a\\omega}{v}u})\\right)}}{\\sqrt{\\alpha^2+\\frac{4a\\omega}{v}u}}\n\\end{equation}\nfor $\\omega>0$ and\n\\begin{align}\\label{eq:F_func_neg}\nF(\\omega, \\alpha)=\\int_0^{\\min{(1, \\frac{v\\alpha^2}{4a|\\omega|})}} &du \\frac{\\Theta(\\alpha)u^2}{\\sqrt{1-u^2}} \\times \\\\\n& \\frac{e^{\\frac{2\\gamma}{\\omega}\\alpha}\\cosh\\left(\\frac{\\gamma}{\\omega}\\sqrt{\\alpha^2+\\frac{4a\\omega}{v}u}\\right)}{\\sqrt{\\alpha^2+\\frac{4a\\omega}{v}u}}\n\\end{align}\nfor $\\omega<0$.\nPhysically, the difference between positive and negative $\\omega$ is due to $\\tilde\\theta_B$ increasing with $r$ for negative $\\omega$ and decreasing for positive $\\omega$, while $\\theta$ always decreases with increasing $r$. This causes the positive-angle values of $\\sigma$ to be sampled twice, and the negative-angle values not at all, in a negative applied magnetic field, while with positive field both positive- and negative-angle values are sampled once. From Fig. \\ref{fig2}, it is clear that negative-angle parts of the cross-section can never be sampled with negative field.\n\n\\begin{figure}[t]\n\\begin{center}\n\\\n\\includegraphics[width=0.48\\columnwidth]{Fig4a.png}\n\\\n\\includegraphics[width=0.48\\columnwidth]{Fig4b.png}\n\\end{center}\n\\caption{Shows the exact form Eq.\\eqref{eq:delta_V_general} (solid lines) and the approximation Eq.\\eqref{eq:voltage_diff} (dashed lines) with the parameters $a=0.1$, $\\theta_T=0.02$, $l_{ee}=4$, for a test cross-section $\\sigma$. Left panel shows the full response, while right panel shows the symmetrized and antisymmetrized functions $U_{\\pm}(B)$. Note the sharp, single peak at $B=0$, in contrast to Fig. \\ref{fig5}. Here $\\sigma$ has a $1\/\\sqrt{\\delta\\theta^2+\\theta_T^2}$ dependence about $\\theta=0$ and $\\pi$, with the $\\pm 1$ harmonics projected out. This dependence was chosen to mimic the cross section predicted in Eq.\\eqref{eq:crossection_backward} and shown in Fig. \\ref{fig1}. Uses $a=0.1$, $\\theta_T=0.05,$ and $l_{\\rm ee}=4$. The scale factor $B_a$ on the $B$ field is $mv\/ea$.} \n\\label{fig4}\n\\end{figure}\n\n\\begin{figure}[t]\n\\begin{center}\n\\\n\\includegraphics[width=0.48\\columnwidth]{Fig5a.png}\n\\\n\\includegraphics[width=0.48\\columnwidth]{Fig5b.png}\n\\end{center}\n\\caption{Shows Eq.\\eqref{eq:delta_V_general} (solid lines) and Eq.\\eqref{eq:voltage_diff} (dashed lines) with the parameters $a=0.1$, $\\theta_T=0.01$, $l_{ee}=1.7$, and cross-section $\\sigma$ with the same functional form as Fig. \\ref{fig4}. Left panel shows the full response, while right panel shows the symmetrized and antisymmetrized functions $U_{\\pm}(B)$. Note the distinctive two-peak feature that may appear in the symmetrized voltage response when the mean free path $l_{ee}$ is not so much larger than the source-probe distance $a$.}\\label{fig5}\n\\end{figure}\n\n\n\nFig. \\ref{fig4} shows the approximation Eq.\\eqref{eq:voltage_diff} along with the exact voltage difference predicted by Eq.\\eqref{eq:delta_V_general}, for a test cross section $\\sigma_{\\pi}$. It is evident that the approximation is quite accurate within a wide range of $B$ field values.\n\nFig. \\ref{fig5} shows the voltage response predicted by Eq.\\eqref{eq:delta_V_general} for a different set of parameters, most significantly with a shorter mean free path. In this regime, the voltage response symmetrized with respect to $B$ field displays a characteristic double-peak structure about $B=0$, which should be detectable in experiment.\n\n\n\n\\end{document}\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Assigning errors to operations}\\label{exampleSection}\nWe now illustrate the gauge freedom with a simple, experimentally relevant example, namely, amplitude damping.\nA gate-set is a mathematical description of the possible actions executable in an experiment, typically consisting of models for initial states ($\\mathds{S}$), gate operations ($\\mathds{G}$), and measurements ($\\mathds{M}$).\nIf an experimentalist with an ideal quantum system could initialize a qubit in the state $\\ket{0}$, apply an arbitrary unitary gate, and measure the expectation value of $Z$, then their control can be represented by the gate-set\n\\begin{equation}\n\\Phi = \\left\\{ \\mathds{S}_\\Phi = \\begin{pmatrix}\n1 & 0\\\\ 0 & 0\n\\end{pmatrix},\\ \\mathds{G}_\\Phi = \\rm{SU}(2) ,\\ \\mathds{M}_\\Phi = Z\\right\\}.\n\\end{equation}\nNow suppose that the experimentalist prepares a mixed initial $Z$ state with polarization $\\epsilon_1$ and performs a measurement with signal-to-noise ratio $\\epsilon_2$.\nSuppose further that before each gate is applied, the system undergoes amplitude damping with strength $\\gamma$ but that the target Hamiltonian is still implemented perfectly.\nThe Pauli-Liouville representation (see Appendix or e.g., \\cite{greenbaum2015introduction}) of the noisy gate-set is then\n\\begin{equation}\\label{gateSetPauli}\n\\Theta = \\left\\{\\mathds{S}_\\Theta = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n1 \\\\ 0 \\\\ 0 \\\\ \\epsilon_1\n\\end{pmatrix},\\ \n\\mathds{G}_\\Theta = \\{\\mc{UA}_\\gamma: U\\in\\rm{SU}(2)\\},\\ \n\\mathds{M}_\\Theta =\\begin{pmatrix}\n0 & 0 & \\frac{\\epsilon_2}{\\sqrt{2}} & 0\n\\end{pmatrix} \\right\\}\n\\end{equation}\nwhere $\\mathcal{U} = U\\rho U^\\dagger$ denotes the unitary channel acting via conjugation and\n\\begin{align}\n\\mc{A}_\\gamma = \\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n0 & \\sqrt{1 - \\gamma} & 0 & 0\\\\\n0 & 0 & \\sqrt{1 - \\gamma} & 0\\\\\n\\gamma & 0 & 0 & 1 - \\gamma\n\\end{pmatrix}.\n\\end{align}\n\nThe expectation value of an operator $M$ given an input state $\\rho$ is the vector inner product between the Pauli-Liouville representations of the state and measurement operators, \n\\begin{equation}\n\\text{prob} = \\dbraket{M}{\\rho}.\n\\end{equation}\nIf $m$ gates $\\mathcal{G}_1,\\ldots,\\mathcal{G}_m\\in\\mathds{G}$ are applied to the state in chronological order before the measurement takes place, the expectation value becomes\n\\begin{equation}\n\\text{prob} = \\dbra{M} \\mathcal{G}_{m:1}  \\dket{\\rho}\n\\end{equation}\nwhere we use the shorthand notation \n\\begin{equation}\n\t\\mathcal{G}_{b:a} \\coloneqq \\begin{cases}\n\t\\mathcal{G}_b \\mathcal{G}_{b-1}...\\mathcal{G}_a & \\text{if}\\ b \\geq a \\\\\n\t\\mathcal{I} & \\text{otherwise.}\n\t\\end{cases} \n\\end{equation} \n\nThe above probabilities are preserved under the family of gate-set transformations\n\\begin{equation}\\label{GaugeEqn}\n\\dket{\\rho} \\rightarrow B \\dket{\\rho},\\ \\dbra{M} \\rightarrow \\dbra{M}B^{-1},\\ \\mathds{G}_\\Phi \\rightarrow B\\mathds{G}_\\Phi B^{-1}\n\\end{equation}\nfor some invertible matrix $B$.\nBecause these probabilities are the only experimentally accessible quantities, the same experimental results can be predicted equally well by these two gate-sets.\nThis is the gauge freedom inherent in mathematically representing quantum experiments, in analogy with concepts in thermodynamics and electromagnetism \\cite{jackson2017nonholonomic}, with $B$ being called a gauge transformation matrix.\nThe analogy arises from the fact that changing the gauge does not result in observable effects in an experiment, just as changing the electromagnetic gauge would not result in any difference in the measurable electric or magnetic fields.\n\nGenerally, a gate-set is considered valid if all quantum states can be represented as density matrices, measurements as expectation values of Hermitian operators, and quantum gates as completely-positive, trace-preserving (CPTP) maps as these conditions ensure that probabilities for arbitrary experiments are positive.\nGauge transformations do not generally preserve these \\textit{canonical constraints}, although the resulting gate-set is nevertheless an equally valid mathematical description of the same experiment.\n\nWe now present a simple, physically motivated, gauge transformation that yields a gate-set that suggests a different physical interpretation of the experimental system.\nApplying the gauge transformation matrix\n\\begin{equation}\\label{gauge}\nB=\\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n0 & q & 0 & 0\\\\\n0 & 0 & q & 0\\\\\n0 & 0 & 0 & q\n\\end{pmatrix}\n\\end{equation}\nfor any $q\\in[-1,1]$ to the noisy gate-set in \\cref{gateSetPauli} yields the equivalent gate-set\n\\begin{equation}\\label{gateSetPauli2}\n\\Theta_q = \\left\\{\\mathds{S}_{\\Theta_q} = \\frac{1}{\\sqrt{2}}\\begin{pmatrix}\n1 \\\\ 0 \\\\ 0 \\\\ q \\epsilon_1\n\\end{pmatrix},\\ \n\\mathds{G}_{\\Theta_q} = \\{\\mc{U} \\mc{A}_{\\gamma,q}: U\\in\\rm{SU}(2)\\},\\\n\\mathds{M}_{\\Theta_q} =\\begin{pmatrix}\n0 & 0 & 0 & \\frac{\\epsilon_2}{q\\sqrt{2}}\n\\end{pmatrix} \\right\\},\n\\end{equation}\nwhere\n\\begin{align}\n    \\mc{A}_{\\gamma,q} = \\begin{pmatrix}\n1 & 0 & 0 & 0\\\\\n0 & \\sqrt{1 - \\gamma} & 0 & 0\\\\\n0 & 0 & \\sqrt{1 - \\gamma} & 0\\\\\nq\\gamma & 0 & 0 & 1 - \\gamma\n\\end{pmatrix}\n\\end{align}\nand we have used the fact that $\\mc{U}$ commutes with $B$ for any $U\\in\\rm{SU}(2)$.\nThe gauge transformation results in equivalent statistics  but suggests a different noise model, namely, relaxation to a mixed state rather than a pure state (corresponding to a different effective temperature).\nAs long as $|q|\\in[|\\epsilon_2|,1]$, the states, measurements, and transformations all satisfy the canonical constraints for gate-set elements.\n\nNote that this gauge freedom does not change the average gate fidelity as $\\tr \\mc{A}_{\\gamma,q}$ is independent of $q$~\\cite[Eq. 2.5]{Kimmel2014}.\nHowever, the diamond distance from the identity depends on $q$, with $\\|\\mc{A}_{\\gamma,1} - \\mc{I}\\|_\\diamond \\approx 2 \\|\\mc{A}_{\\gamma,0} - \\mc{I}\\|_\\diamond $ for $\\gamma\\in[0,1]$~\\cite{Kueng2016}.\nMoreover, this example illustrates that noise can be artificially reassigned to different objects, as the state in \\cref{gateSetPauli} is closer to pure than the one in \\cref{gateSetPauli2}.\nNote that the range of gauge transformations is constrained by $\\epsilon_2$ and so cannot significantly change the effective temperature for systems with high quality readout.\nWe could have added larger errors by considering a non-unital (e.g., $\\mc{A}_{\\gamma,q}B$) or unitary gauge transformation at the cost of making the errors gate-dependent and consequently giving a more complicated example.\nWe did not do this as our intent is to clarify that the gauge freedom is more than a basis mis-match~\\cite{carignan2018randomized} and is distinct from the issue of gate-dependent noise~\\cite{proctor2017randomized, wallman2018randomized, carignan2018randomized}.\nIn particular, we note that the full effect of the gauge freedom for states and measurements is unknown.\n\n\\section{Gauge and Representation of Quantum States}\\label{gaugeSection}\nWe have seen that under realistic circumstances, the same experiment can be described by distinct gate-sets that suggest different physical noise models due to a gauge freedom. \nIn this section we illustrate how representations of quantum states are related to the concepts of gauges and gauge transformations.\nFor clarity we focus on the representation for quantum states, but similar arguments can be made about gate and measurement operations.\n\nFrom the point of view of scientific realism, the apparatus (e.g., a qubit) has a physical existence and properties (which may be relative to the environment) independent of our representation.\nWe describe the abstract state of this physical object as a \\textit{noumenal state} following the terminology in \\cite{brassard2017equivalence}, denoted as $\\mathds{N}$ in Figure \\cref{VennDiag}.\nHere we slightly change their definition to include in $\\mathds{N}$ both physically allowed (denoted as $P$) and forbidden states, such that the set $\\mathds{N}$ contains both states that the system can be in, and ones that it cannot be in based on the physics.\nQuantum mechanics allows us to assign to each noumenal state a mathematical \\textit{representation} which is an element of a Hilbert space $\\mathcal{H}^d$: for example, one can associate the system with a matrix that summarizes its properties, and the set of all $d \\times d$ matrices is called $\\mathds{R}$ in the same figure.\nSuch an association is what we call a \\textit{gauge} $\\Gamma$, which is a bijective map from $\\mathds{N}$ to $\\mathds{R}$: the bijectivity of the map should be clear from our inclusion of physically-forbidden states in $\\mathds{N}$, which allows assigning ``some state'' to every $d \\times d$ matrix.\nDifferent choices of $\\Gamma$ thus correspond to different mathematical descriptions of the noumenal states.\n\n\\begin{figure}[ht]\n\\centering\n\\includegraphics[width=0.7\n\\textwidth]{NewVenn.png}\n\\caption{Diagram illustrating the distinction between the set of noumenal states $\\mathds{N}$ and their mathematical representations $\\mathds{R}$. \nFor a single qubit, $\\mathds{R}$ is the set of $2 \\times 2$ matrices.\nA gauge is a bijective map $\\Gamma:N\\to R$.\nLet $\\mathds{P}$ be the set of all noumenal states that can possibly be prepared physically.\nIn general, $\\mathds{P}$ is unknown, but $\\Gamma(\\mathds{P})$ is assumed to satisfy the canonical constraints, that is, to be the set of density matrices.\nIn our example, $\\Gamma_1(\\mathds{P})$ is the set of density operators in the corresponding Hilbert space and $\\mathfrak{B}_{12}$ is a gauge transformation from $\\Gamma_1$ to $\\Gamma_2$.\nWhether an object in $\\mathds{R}$ directly corresponds to objects in $\\mathds{P}$ or not depends on the particular gauge under which the object is represented.\nFor example, $r_2^1 = \\Gamma_1(n_2)$ is a density operator while $r_2^2 = \\Gamma_2(n_2)$ is not, despite both being the image of the same physical state.}  \n\\label{VennDiag}\n\\end{figure}\n\nThe common formulation of quantum mechanics says that every state of a quantum object can be described by a density operator \\cite{hardy2001quantum}, which belongs to a subset of the \\textit{canonical constraints} defined at the beginning of \\cref{exampleSection}.\nThis means that there exists a \\textit{canonical gauge} \\begin{equation}\n\\Gamma_1:\\ N \\rightarrow R,\\ \\Gamma_1(\\mathds{P}) = \\mathds{D}_d\n\\end{equation}\nwhere $\\mathds{D}_d$ is the set of $d \\times d$ density operators. In fact, there exists a family of canonical gauges that are all related to $\\Gamma_1$ through unitary gauge transformations which preserves the shape of $\\mathds{D}_d$.\nSatisfying the canonical constraint implies that we should work in one of these canonical gauges.\nNow, consider another gauge $\\Gamma_2$ which can be converted from $\\Gamma_1$ with a \\textit{gauge transformation} $\\mathfrak{B}_{12}$, defined by\n\\begin{equation}\n\\mathfrak{B}_{12} \\coloneqq \\Gamma_2 (\\Gamma_1^{-1}),\\ \\mathfrak{B}_{12}(r_{*}^{1}) = r_*^2\n\\end{equation}\nwhere the $r$'s are members in $\\mathds{R}$ and the superscript denotes the gauge in which they are represented.\nIn the light of \\cref{GaugeEqn}, this transformation can be represented in Pauli-Liouville representation as\n\\begin{equation}\\label{gauge B12}\n\\dket{\\mathfrak{B}_{12} (\\rho)} \\coloneqq B_{12} \\dket{\\rho},\\ \\dbra{\\mathfrak{B}_{12}(M)} \\coloneqq \\dbra{M}B_{12}^{-1},\\ \\mathds{G}_{\\mathfrak{B}_{12}(\\Phi)} \\coloneqq  B_{12}\\mathds{G}_{\\Phi} B_{12}^{-1} \n\\end{equation}\nAs a subset of $\\mathds{R}$, $\\Gamma_1(\\mathds{P})$ is generally not invariant under an arbitrary gauge transformation: consider a general trace-preserving transformation given by the following transformation matrix\n\\begin{equation}\nB_{12} = \\begin{pmatrix}\n1 & 0\\\\ \\vec{x} & y\n\\end{pmatrix}\n\\end{equation}\nwhere $\\vec{x}$ is a $(d^2-1) $ by $ 1$ real vector and $y$ is a $(d^2-1) $ by $ (d^2-1)$ real matrix: the image of this affine transformation of $\\Gamma_1(\\mathds{P})$ is a different subset of $\\mathds{R}$. \nSuch a gauge is perfectly valid in principle, provided that \\textit{all} the gates and measurement operators are transformed according to \\cref{gauge B12} as well, even though $\\Gamma_2(\\mathds{P})$ is no longer the set of density operators. \n\nThe existence of a non-canonical gauge implies, for example, that a physical state may or may not be represented by a density operator: as illustrated in \\cref{VennDiag}, $r_1^2 \\in \\Gamma_1(\\mathds{P})$ whereas $r_2^2 \\notin \\Gamma_1(\\mathds{P})$.\nSimilarly, a density operator in a non-canonical gauge does not necessarily correspond to a physical state, as $r_3^2 \\in \\Gamma_1(\\mathds{P})$ but $n_3 \\notin \\mathds{P}$.\nOne example for the state $n_3$ is a qubit state represented as $\\frac{1}{2} (I+\\frac{10}{9} \\sigma_z)$ in a canonical gauge. \nIt is not positive semidefinite and thus lies outside $I_1$, representing an abstract state the qubit cannot be in. \nNow, using $B_{12} = B$ from \\cref{gauge} with $q = \\frac{9}{10}$, the image of $n_3$ under $\\Gamma_2$ becomes $\\frac{1}{2} (I+ \\sigma_z)$, which \\textit{is} a density operator, but only as a consequence of this non-canonical gauge.\nWe conclude that if the gauge is unknown, the mathematical representation does not imply the noumenal state is physically possible.\nRepresentations satisfying the canonical constraints are easier to work with, so it is often implicitly assumed that all gate-set elements (obtained from a tomography experiment, for example) are expressed in a canonical gauge. \nHowever, this assumption can only be verified by performing perfect experiments, which are axiomatically the operations specified by the canonical constraints (up to a unitary change of basis).\n\n\\section{Operational interpretations of figures of merit}\\label{implicationSection}\nThe existence of this gauge freedom has direct implications for figures of merit used to evaluate quantum operations.\nThe main problem is that there is no way to know whether an experimentally-determined gate-set element is expressed in a canonical gauge.\nWe have already seen in \\cref{exampleSection} that by changing the gauge, the states can appear as having different expressions; the same holds true for gates and measurement operators.\n\nFrom quantum information theory, we have successfully attached some operational meanings to various distance metrics: an important example is the interpretation for the diamond norm distance between two channels $\\mc{A}$ and $\\mc{B}$ as the maximum distinguishability between output states under a fixed input~\\cite{watrous2018theory}.\nMathematically,\n\\begin{equation}\\label{optimalInputOutput}\n\\frac{1}{2} \\norm{\\mc{A} - \\mc{B}}_\\diamond =  \\max_{M\\in\\Gamma(\\mathds{M}), \\rho\\in\\Gamma(\\mathds{P})} \\dbra{M_1} (\\mc{A} - \\mc{B}) \\otimes I) \\dket{\\rho}\n\\end{equation}\nwhere $\\mathds{P}$ and $\\mathds{M}$ are the set of physically possible states and measurements respectively. \nThis operational meaning is gauge invariant, provided one consistently transforms $\\mc{A}$, $\\mc{B}$, $\\Gamma(\\mathds{P})$, and $\\Gamma(\\mathds{M})$.\nHowever, when $\\mc{A}$ is an experimentally reconstructed gate and $\\mc{B}$ is its ideal target, $\\Gamma(\\mathds{P})$ and $\\Gamma(\\mathds{M})$ are unknown and so the above maximization that leads to its operational meaning cannot be performed.\nTo obtain concrete numbers, people calculate\n\\begin{equation}\n\\frac{1}{2} \\norm{\\mc{A} - \\mc{B}}_\\diamond =  \\max_{M\\in \\mu(\\mathds{D}_{d}), \\rho\\in\\mathds{D}_{d}} \\dbra{M_1} (\\mc{A} - \\mc{B}) \\otimes I) \\dket{\\rho}\n\\end{equation}\nwhere $\\mu(\\mathds{D}_{d})$ is the set of all POVMs.\nHowever, this assumes that the reconstructed $\\mc{A}$ and the ideal target $\\mc{B}$ are expressed in a canonical gauge.\nWhile $\\mc{B}$ is an ideal gate, its representation may not be unitary in a non-canonical (and unknown) gauge.\nOther works have reported that the quantity $\\frac{1}{2} \\norm{\\mc{A} - \\mc{B}}_\\diamond$ can be changed by changing the gauge and used this to minimize reported error rates~\\cite{blume2017demonstration}, however, such changes are obtained by implicitly changing the set of physically allowed states and measurements.\nNote that even in one special case of interest where $\\mc{B} = \\mc{I}$, which is gauge invariant, $\\mc{A}$ is still reconstructed in an unknown gauge.\n\nWe briefly discuss several common practices related to this gauge freedom in quantum tomography.\nThe process known as ``gauge optimization'' is commonly adopted in GST experiments whereby the gauge transformation matrix $B$ is varied to minimize the distance from the target gate-set according to a (non-gauge-invariant) weighted distance measure \\cite{blume2017demonstration}.\nHowever, this optimized gauge is just as arbitrary as any other gauge, and one still cannot know whether the resultant gate-set is a faithful representation of the apparatus, in particular, whether the states and measurements that satisfy the canonical constraints are actually the images of the set of physically possible states and measurements respectively.\nMoreover, such optimization undermines a common use of tomography, namely assessing the performance of a system against some external threshold (e.g., a fault-tolerance threshold). \nAltering the gauge to make the tested channel similar to its target will artificially reduce the distance between the two.\nAdditionally, assigning different weights on SPAM and gates during gauge optimization will result in a difference in the output, and such weights are only based on rough initial guess about the relative quality of these components.\nAnother common approach is Maximum Likelihood Estimation (MLE), which takes the  estimated gate-set to be the one that maximizes the likelihood function of obtaining the experimental data, while restricting the gate-set elements to satisfy the canonical physicality constraints \\cite{medford2013self,brida2012quantum}.\nMLE does not resolve the gauge ambiguity either, because all gauge-equivalent gate-sets are equally likely to produce the data by definition.\nIn the process of optimization, one will find that the likelihood function profile has the same value wherever two points are related by a gauge-transformation, and the actual output is largely a matter of the optimization algorithm and the initial parameters \\cite{blume2015turbocharging}.\n\n\\section{A Gauge-Invariant Measure for Gate-Sets}\\label{newMeasureSection}\nThe gauge freedom prevents one from using conventional distance measures to faithfully evaluate the quality of individual quantum operations.\nNote that our discussion is carried out in the absence of any additional errors such as finite-counting, and in a real experiment the situation becomes even more complicated.\nFundamentally, this problem is due to the limited information that can be gained from experimental probabilities.\nA gauge-transformation re-assigns state, gate, and measurement ``errors'' by adjusting their relative appearance in different representations, while keeping the experimental measurables unchanged, although some degrees of freedom can be fixed by convention (e.g., that the state preparation is diagonal in the $Z$ eigenbasis).\n\nWe now propose a gauge-invariant figure of merit for a \\textit{gate-set}.\nAs far as we know, this is the first fully gauge-invariant measure, addressing a problem raised in Ref.~\\cite{blume2015turbocharging}.\nLet $\\Phi$ denote the gate-set $\\{\\mathds{S}, \\mathds{G}, \\mathds{M}\\}$ and $C$ denote a particular experiment with input state $\\rho \\in \\mathds{S}$, measurement $M \\in \\mathds{M}$, and a set of $m$ gates $\\mc{G}_1...\\mc{G}_m$ each selected from $\\mathds{G}$.\nThe only observable properties of the experiment $C$ is the probability distribution over outcomes.\nWe can quantify the error of the experiment by the total variation distance between the observed and ideal distributions over outcomes,\n\\begin{equation}\n\\delta d (C,\\tilde{C}) \\coloneqq \\frac{1}{2} \\sum_i \\abs{\\Tr[\\tilde{M}_i^\\dagger \\tilde{G}_{m:1} (\\tilde{\\rho})] - \\Tr[M_i^\\dagger G_{m:1} (\\rho)]} \n\\end{equation}\nwhere the tilde represents real versions of the operations.\nThe total variation distance is a metric over probability distributions.\nDenoting the set of all experiments with $m$ gates by $\\mathds{A}_{m}$, we further define the \\textit{Mean Variation Error} (MVE) over $\\mathds{A}_{m}$ with the underlying gate-set $\\Phi$ as\n\\begin{equation}\\label{MVE definition}\nx(\\Phi, m) \\coloneqq \\frac{1}{|\\mathds{A}_{m}|}\\sum_{C\\in\\mathds{A}_m} [\\delta d (C, \\tilde{C})]\n\\end{equation}\nNote that although in each $C$ only one state and one measurement is allowed (in order for the final outcome to be a valid probability distribution), there is no constraint on how many are included in the gate-set.\nThe size of $\\mathds{A}_{m}$ is thus given by $\\abs{\\mathds{A}_{m}} = \\abs{\\mathds{S}} \\abs{\\mathds{G}}^m \\abs{\\mathds{M}}$.\n\n\\begin{figure}[ht]\n\\centering\n\\begin{tikzpicture}\n\\begin{axis}[xmin=0,ymin=0,width = 0.5\\linewidth,xlabel={$m$},scaled y ticks=base 10:2 \n\/pgf\/number format\/sci subscript, ylabel={$x(\\Phi,m)$}, title={Depolarizing error}]\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, y error=std, only marks, col sep=comma] {depolarizing_identity.csv};\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, y error=std, only marks, col sep=comma] {depolarizing_general.csv};\n\\end{axis}\n\\end{tikzpicture}\n\\begin{tikzpicture}\n\\begin{axis}[xmin=0,ymin=0,width = 0.5\\linewidth,xlabel={$m$}, title={Unitary error},scaled y ticks=base 10:2\n\/pgf\/number format\/sci subscript]\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, y error=std, only marks, col sep=comma] {unitary_identity.csv};\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, y error=std, only marks, col sep=comma] {unitary_general.csv};\n\\end{axis}\n\\begin{semilogyaxis}[at={(rel axis cs:0,1)},anchor={outer north west},tiny,width = 0.215\\linewidth]\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, only marks, col sep=comma] {shortunitary_identity.csv};\n\\addplot+[error bars\/.cd,y dir=both,y explicit]\ntable [x=m, y=mean, only marks, col sep=comma] {shortunitary_general.csv};\n\\end{semilogyaxis}\n\\end{tikzpicture}\n\\caption{Simulated mean variation error from \\cref{MVE definition} under two error models for a gate-set with $\\rho = M_1 = \\ketbra{0}{0}$, and $\\mathds{G} = Cl_1 $ being the 1-qubit Clifford group. The depolarizing error channel is $\\mathcal{E}_D (\\rho,r) = (1-2r)\\rho + r I $, whereas the unitary error is $\\mathcal{E}_U (\\rho,\\theta) = e^{-i \\theta Z} $ with $\\theta = \\arccos(\\sqrt{1-3r\/2})$, such that the error channel on every gate has an averaged infidelity of  $r = 10^{-4}$. Blue circles indicate self-inverting (identity) circuits whereas red squares indicate random circuits. Each point is generated from averaging 200 random circuits with length $m$; error bars are standard error in the mean and data shows significant spread for unitary error. MVE may have different behaviors under different error types ($m$ or $\\sqrt{m}$) for a random circuit, as compared to the linear behavior for a self-inverting circuit. The inset in the second plot is a zoom-in view for small $m$, showing the significant underestimation of MVE by restricting to self-inverting circuits.}\n\\label{New-metric-plot}\n\\end{figure}\n\nThe MVE quantifies how well the apparatus performs an average experiment from the gate-set.  \nIn the case where the measurement is a projective measurement in the basis of the initial state (i.e., $\\rho$ = $M_i$ for some $i$) and the gate sequence is self-inverting, $\\delta d(C,\\tilde{C})$ can be simplified as\n\\begin{equation}\n\\begin{aligned}\n\\delta d (C, \\tilde{C}) &= \\frac{1}{2} \\left(\\abs{\\Tr[\\tilde{M}_i^\\dagger \\tilde{G}_{m:1} (\\tilde{\\rho})] - 1} + \\sum_{j \\neq i} \\abs{\\Tr[(I - \\tilde{M}_{j})^\\dagger \\tilde{G}_{m:1} (\\tilde{\\rho})] - 0} \\right)\\\\\n&= 1 - \\Tr[\\tilde{M}_i^\\dagger \\tilde{G}_{m:1} (\\tilde{\\rho})]\n\\end{aligned}\n\\end{equation}\nwhose average over $\\mathds{A}_m$ is just 1 minus the ``survival probability'' plotted in a conventional randomized benchmarking experiment.\nWhen $\\mathds{G}$ is a unitary 2-design, the MVE restricted to self-inverting gate sequences is well-approximated by a linear relation to first order in the average error rate \\cite{wallman2018randomized,proctor2017randomized}.\n\nHowever, for generic gate sequences, the MVE behaves differently depending on the underlying error model.\nThis behavior provides additional information about the underlying error mechanism compared to a conventional randomized benchmarking experiment~\\cite{wallman2015bounding}.\nTo illustrate this, we simulated random circuits of varying length $m$ sampled from the gate-set $\\Phi = \\{\\mathds{S} = \\ketbra{0}, \\mathds{G} = \\textit{Cl}_1, \\mathds{M} = \\ketbra{0}\\}$ (with $\\textit{Cl}_1$ denoting the set of 1-qubit Clifford gates), where erroneous gates are represented as $\\tilde{\\mathcal{G}} =\\mathcal{E} \\mathcal{G}$ for a fixed error channel $\\mathcal{E}$.\nWe simulated two types of random circuits: circuits from the entire set of possible experiments allowed by the gate-set, and circuits restricted to self-inverting gate sequences. \nIn both simulations, the state and measurements are assumed to be error-free.\nThe results are shown in \\cref{New-metric-plot}. \nWhen the error is a depolarizing channel, the MVE scales linearly with the gate sequence length $m$ for both random and self-inverting circuits, with the slope for random circuits being $\\sim 1\/3$ the slope for self-inverting circuits.  \nThis is because when the state is transformed onto the xy-plane of the Bloch sphere right before measurement (which happens about 2\/3 of the time), the depolarizing channel does not affect the outcome probability of a z-axis measurement, resulting in an MVE of 0 for those circuit sequences. \nAdditionally, there is no statistical error present for the self-inverting circuit under this error model because all circuits of the same sequence length have exactly the same overall error, as the error channel commutes with all the gates in $Cl_1$.\nIn contrast, for a gate-independent unitary error, the scaling remains linear for the self-inverting circuits but exhibits a $\\sqrt{m}$ scaling for generic circuits.  \nThis occurs because when the state system is in the xy-plane before measurement, each error contributes a random sign to the probability of each outcome, whereas when the system is on the $z$ axis each error has to contribute a systematic sign \\cite{wallman2015bounding}. \nAs shown in \\cref{New-metric-plot}, this implies that restricting to self-inverting circuits can underestimate the MVE by over an order of magnitude in the small-$m$ regime, which is relevant for near-term quantum computer applications.\n\nUnlike other distance measures where an improvement in quality can be caused by a bias in choosing a gauge, a decrease in MVE is unequivocally an improvement due to its gauge-invariance and because, by definition, the output probability distribution gets closer to the ideal distribution.\nFurthermore, the MVE captures the relevant behavior for generic circuits, rather than just self-inverting circuits which, by design, perform no useful computation.\n  \nA protocol for estimating the MVE of a gate-set $\\Phi = \\{\\mathds{S}, \\mathds{G}, \\mathds{M}\\}$ is as follows:\n\\begin{enumerate}\n\t\\item Select $N_m$ random experiments $C\\in\\mathds{A}_m$, for some $N_m$ large enough to accurately approximate the average.\n\t\\item Repeat each experiment $C$ $K_m$ times to estimate $\\langle \\tilde{M}_i, \\tilde{G}_{m:1}(\\tilde{\\rho}) \\rangle$ for each $C$. \n\t\\item Compute the ideal probabilities $\\langle M_i, G_{m:1}(\\rho) \\rangle$ for each observed outcome of $\\tilde{C}$. \n\t\\item Calculate $\\delta d(C, \\tilde{C})$ for each experiment $C$, average over them to estimate $x(\\Phi, m)$.\n\t\\item Repeat step 1--4 for different values of $m$ to measure the scaling behaviour of MVE.\n\\end{enumerate}\nNote that if $\\mathds{G}$ is a unitary 2-design and the states and measurements are chosen appropriately, the applied operations are identical to those used to estimate the unitarity \\cite{Wallman2015}.\nThe primary difference in the protocol is that it is more scalable, more general, and has different post-processing.\n\nThe scalability of the above protocol is affected by the number of experiments $N_m$, the number of repetitions for each experiment $K_m$, and the complexity of calculating the ideal probabilities.\nThe number of experiments determine the accuracy of the MVE, and can be estimated using Hoeffding's inequality independently of the number of qubits \\cite{hoeffding1963probability}.\nThe complexity in the protocol is determined by the complexity of calculating probabilities and by the number of repetitions required to estimate $\\delta d(C,\\tilde{C})$ to a fixed precision.\nThe number of repetitions required to estimate $\\delta d(C,\\tilde{C})$ to a fixed precision is polynomial in the number of outcomes \\cite{chan2014}.\nTo efficiently characterize multi-qubit gate-sets (where the number of raw outcomes grows exponentially with the number of qubits), we can coarse-grain the measurements over sets of outcomes.\nThe computational complexity of calculating each probability will depend on the gate-set in question.\nThe ideal probabilities can be efficiently computed if $\\mathds{G}$ is the $N$-qubit Clifford group \\cite{gottesman1998heisenberg,nest2008classical}.\nFor gate-sets containing only one- and two-qubit gates and product states and measurements, the MVE can be computed for small values of $m$.\nHowever, for a generic gate-set, each probability will be hard to compute.\nOf course, for small systems with a few qubits, this procedure can nonetheless be performed quickly on a classical computer.\n\nAn experimentalist can perform a feedback loop whereby they update the control parameters, rerun the MVE evaluation experiment (potentially for some fixed value of $m$) and compare to the previous result to see if the error has decreased.\nProtocols that use feedback from experimental outcomes to improve control over quantum devices have been proposed before, such as in \\cite{kelly2014optimal} where control parameters were optimized by maximizing the randomized benchmarking survival probability for a fixed sequence length.\nOptimizing the MVE instead of the randomized benchmarking survival probability corresponds to minimizing the effect of errors on generic quantum circuits, rather than minimizing the effect of errors on self-inverting circuits.\nAs demonstrated in \\cref{New-metric-plot}, errors in self-inverting circuits may be substantially smaller than those in generic circuits because such circuits suppress coherent errors and implement a form of randomized dynamical decoupling \\cite{viola2005}.\n\n\\begin{acknowledgments}\nJJW acknowledges helpful discussions with Robin Blume-Kohout.\nThis research was supported by the U.S. Army Research Office through grant W911NF-14-1-0103, the Government of Ontario, and the Government of Canada through CFREF, NSERC and Industry Canada.\n\\end{acknowledgments}\n\n\n\\clearpage\n\\newpage\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{intro}\nIn India R\\&D projects on various gaseous detectors such as Gas Electron Multiplier (GEM), Resistive Plate Chamber (RPC), Single Wire Proportional Chamber (SWPC), Multi Wire Proportional Chamber (MWPC) and scintillator detectors are being carried out now a days for various low energy nuclear physics and high energy physics (HEP) experiments \\cite{RNP, KKM, RG, SR14, APN}. During the characterisation of any detector, the analog input signals from the detector are fed into discriminator and the corresponding digital signals are counted. Different scalers or counters are commercially available for this purpose. The available commercial scalers are quite expensive. An effort is going on in our collaborative work to build a cheap in house scaler. In that spirit  a rising edge triggered 4-channel  TTL (Transistor Transistor Logic) scaler has been developed to record the number of pulses in a given interval of time. The four channels are independent and each channel is capable of capturing maximum 4,294,967,295~(2$^{32}$-1) number of pulses i.e. each channel can count maximum 4,294,967,295~(2$^{32}$-1) number of signals. In this report the details of the design, fabrication and calibration of the scaler is presented.\n\n\\section{Design principle}\\label{construct}\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.35]{front.pdf}\n\\caption{\\label{Front} Front side view of the scaler.}\\label{Front}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.35]{back.pdf}\n\\caption{\\label{Back} Back side view of the scaler.}\\label{Back}\n\\end{center}\n\\end{figure}\nOne standalone 4-channel edge triggered TTL scaler is designed here to count the pulses from the detector \\cite{SS}. The scaler is powered directly from the 220~V AC line. The front and back side view of the scaler are shown in Fig.~\\ref{Front} and Fig.~\\ref{Back} respectively. The power switch of the scaler is placed at the rare side of the device as shown in Fig.~\\ref{Back}. There is a red LED which will glow when the scaler is switched on. There are four inputs to the scaler. All the inputs are fed with BNC connector as shown in Fig.~\\ref{Back}. The analogue signals from the detector is first discriminated from a discriminator module and the TTL signal is generated either from the SCA or from a NIM-TTL converter. The TTL signal is fed to one of the inputs of the scaler. The scaler can accept user command by external knob (potentiometer) for setting up of sampling time. The maximum sampling time can be set to 120 minutes. The black knob is marked as \"Time Regulator\" in  Fig.~\\ref{Front}. After setting up of sampling time one pulse switch (KEY) is pressed to start the count. This switch is marked as the \"Set Time\" in Fig.~\\ref{Front}. The sampling time is set in minutes as shown in Fig.~\\ref{Display}. The elapsed time during counting are displayed on the LCD in seconds as shown in Fig.~\\ref{Display2}. At the end of counting the number is displayed on the LCD as shown in Fig.~\\ref{Display3}. Each channel has 10 digit display as seen in Fig.~\\ref{Display3}. The scaler can be controlled by computer and the counted numbers can automatically be stored in the computer.\n\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.25]{display.pdf}\n\\caption{\\label{Display} Display unit.}\\label{Display}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.25]{display_running.pdf}\n\\caption{\\label{Display2} Display during run time.}\\label{Display2}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.25]{count_end.pdf}\n\\caption{\\label{Display3} Display at the end of counting.}\\label{Display3}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.35]{scaler_block_diagram.pdf}\n\\caption{\\label{block} The block diagram of the scaler.}\\label{block}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.6]{flowchart_master.pdf}\n\\caption{\\label{master} The flowchart of master controller.}\\label{master}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.6]{flowchart_slave.pdf}\n\\caption{\\label{slave} The flowchart of slave controller.}\\label{slave}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb!]\n\\begin{center}\n\\includegraphics[scale=0.5]{160317_scaler_calibration_curve.pdf}\n\\caption{\\label{calib} The calibration curve.}\\label{calib}\n\\end{center}\n\\end{figure}\n\nThe scaler has been designed to avoid multiple counting with larger pulse width. So, the edge trigger with fast response has been designed i.e. the scaler counts only when the digital signal changes its state from 0 to 1. A rate of maximum 140 kHz can be counted without any delay independently in four channels. The time is calculated in millisecond accuracy so a high precession counting can be achieved. The entire scaler is designed using independent Atmega328 microcontrollers. There are  five numbers of  microcontrollers out of them one is MASTER and other four are SLAVES and the MASTER-SLAVE communicates with (Inter-Integrated Circuit)~I$^2$C protocol. The block diagram of the scaler is shown in Fig.~\\ref{block}. The functions of the Master and Slave controllers are given as flow chart in Fig.~\\ref{master} and Fig.~\\ref{slave} respectively.\n\n\n\\section{Calibration of the TTL scaler}\\label{calibration}\nThe TTL scaler is calibrated using a function generator. The function generator can provide different kind of signals of different frequencies. In this work square wave signal is considered. The amplitude of the square wave signal is kept constant and the frequency is varied from a few Hz to a few kHz. The frequency of the output signals from the function generator can be obtained from the function generator itself. That square wave signal of known frequency is fed to the input of the scaler. The number of signals are counted for 2 minutes for each settings and from it the rate is calculated. This calibration is done for all four channels individually and the calibration curve is drawn. The calibration curve for the count rate of one channel is shown in Fig.~\\ref{calib}. The curve is found to be a straight line up to 140 kHz, with a slope of 0.99 and intercept $\\sim$~0.16. Same parameters are obtained for other three channels also. \n\n\\section{Summary and outlooks}\nOne 4-channel TTL scaler has been fabricated to count the signals from any detector. This is a part of our regular detector R\\&D programme. The scaler has the following features. (a) The scaler has 4 channels, (b) each channel has 10 digit display, (c) the scaler can accept TTL input, (d) it can accept the maximum count rate of 140 kHz, (e) the maximum preset time can be set to 120 minutes and (f) count is displayed once the counting is stopped. The count rate of the TTL scaler is calibrated with a function generator. The calibration curve is found to be a straight line with a calibration factor of 0.99 and intercept $\\sim$~0.16. So this scalar is a low priced one and suitable for the counting of signals from a detector.\n\nIn future we are planning to build a scaler in standard NIM format and will be communicated at a later stage.\n\n\\section{Acknowledgements}\nWe would like to acknowledge Prof. S.~Raha, Prof. S.~K.~Ghosh, Dr. S.~Das, Dr. R.~Ray of Bose Institute, Kolkata, Prof. S.~Panda of IOP, Bhubanswar and Dr. S.~Chattopadhyay, Dr. T.~K.~Nayak of VECC, Kolkata for  their support during course of this work. We would also like to thank the RD51 collaboration for helping us and giving valuable suggestions in the course of our different works.\n\n\\noindent\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n\nThe aim of this survey is to report on new connections between the\nrepresentation theory of finite groups and the theory of cluster algebras.\n\nBlocks of group algebras form an important class of indecomposable, symmetric\nfinite-dimensional algebras. Blocks of finite representation type are Morita\nequivalent to Brauer tree algebras and are well understood.\nIn order to understand blocks of tame representation type,\nErdmann~\\cite{Erdmann90} introduced\nthe classes of algebras of dihedral, semi-dihedral and quaternion type,\nwhich are defined by properties of their Auslander-Reiten quiver,\nproved that blocks with dihedral, semi-dihedral or generalized quaternion\ndefect group belong to the respective class of algebras and classified\nthe possible quivers with relations these algebras may have.\n\nOne of these classes consists of the algebras of quaternion type, which are\nindecomposable, symmetric algebras of tame representation type having\nnon-singular Cartan matrix, with the property that\nany indecomposable non-projective module is $\\Omega$-periodic of period\ndividing $4$, where $\\Omega$ is Heller's syzygy functor. This class\nof algebras is closed under derived equivalences~\\cite{Holm99}.\n\n2-Calabi-Yau triangulated categories with cluster-tilting objects arise in the\nadditive categorification of cluster algebras with skew-symmetric exchange\nmatrices~\\cite{Amiot09,BMRRT06,Keller10,Keller11}.\nThe role of the clusters in the cluster algebra is played by \nthe cluster-tilting objects, whose endomorphism algebras, called\n\\emph{2-CY-tilted algebras}, have remarkable representation theoretic and\nhomological properties~\\cite{BMR07,KellerReiten07}.\n\nWe show that symmetric algebras $\\Lambda$ that are in addition 2-CY-tilted have\ninteresting structural properties analogous to those of the algebras of\nquaternion type; firstly, the functor $\\Omega^4$ is\nisomorphic to the identity functor on the stable module category\n$\\stmod \\Lambda$ (Proposition~\\ref{p:period});\nsecondly, such algebras tend to come in derived equivalence classes\n(Proposition~\\ref{p:dereq}). More precisely, if $\\Lambda=\\operatorname{End}_{\\mathcal{C}}(T)$ \nfor a cluster-tilting object $T$ in a 2-Calabi-Yau triangulated category $\\mathcal{C}$,\nthen the 2-CY-tilted algebra $\\Lambda'=\\operatorname{End}_{\\mathcal{C}}(T')$ is derived equivalent\nto $\\Lambda$ for any other cluster-tilting object $T'$ obtained from $T$\nby a finite sequence of Iyama-Yoshino~\\cite{IyamaYoshino08} mutations.\n\nMotivated by this analogy one is naturally led to ask whether the algebras\nof quaternion type can be realized as 2-CY-tilted algebras, and even more\ngenerally, what are the symmetric algebras that are also 2-CY-tilted?\n\nIn this survey we provide an affirmative answer to the first question\nand attempt to answer the second question, first by \nclassifying the symmetric, 2-CY-tilted algebras of finite representation type\nand then by constructing a new class of\nsymmetric, 2-CY-tilted algebras of tame representation type.\nNote that there are also many wild symmetric, 2-CY-tilted algebras, but we\nwill not discuss them here.\nLet us describe the main results along with the structure of this survey.\n\n\nIn Section~\\ref{sec:motivation} we review some basic notions including\nblocks, stable categories, symmetric algebras, periodic modules and the\ndefinition of algebras of quaternion type.\nWe also introduce the algebras of quasi-quaternion type, which are defined\nsimilarly to the algebras of quaternion type,\nthe only difference being the omission of the condition\nthat the Cartan matrix is non-singular.\n\nIn Section~\\ref{sec:sym2CY} we investigate symmetric 2-CY-tilted algebras.\nWe start by recalling the definition and basic properties of 2-CY-tilted\nalgebras. Since many of them arise as Jacobian algebras of quivers\nwith potentials~\\cite{Amiot09,DWZ08,Keller11},\nwe review this notion as well, and introduce the\nnotion of hyperpotential~\\cite{Ladkani14a} which is useful over ground fields\nof positive characteristic.\nThen we present results concerning the periodicity of\nmodules and derived equivalences for these algebras.\n\nA classification of symmetric, 2-CY-tilted, indecomposable algebras of finite\nrepresentation type which are not simple is presented in\nSection~\\ref{sec:sym2CYfin}.\nWe show that these algebras are precisely the Brauer tree algebras\nwith at most two simple modules (Theorem~\\ref{t:sym2CYfin}).\n\nThen, we construct a large class of symmetric, 2-CY-tilted algebras of tame\nrepresentation type (Theorem~\\ref{t:quasi}).\nOur construction is based on the combinatorial notion of \\emph{triangulation\nquivers}, which are quivers with the property that for each vertex\nthe set of incoming arrows and that of outgoing arrows have cardinality\n$2$, together with bijections between these sets that combine to yield\na permutation on the set of all arrows which is of order dividing $3$.\nTriangulation quivers can be built from ideal triangulations of surfaces\nwith marked points in a way which is analogous to, but different than the\nconstruction of the adjacency quivers of Fomin, Shapiro and\nThurston~\\cite{FST08} arising in their work on cluster algebras from\nsurfaces.\n\nThe ingredients behind our construction are presented in\nSections~\\ref{sec:quivers}, \\ref{sec:surface}, \\ref{sec:algebras}\nand~\\ref{sec:triangquasi}.\nSection~\\ref{sec:quivers} forms the combinatorial heart of this survey.\nWe introduce ribbon quivers and the dual notion of ribbon graphs,\ndefine the subclass of triangulation quivers, and present a block decomposition\nof the latter into three basic building blocks.\nSection~\\ref{sec:surface} explains how triangulations of marked surfaces\ngive rise to triangulation quivers. We discuss the\ndifferences and similarities to adjacency quivers and provide a dimer model\nperspective on these constructions.\n\nIn Section~\\ref{sec:algebras} we introduce two classes of algebras which turn\nout to be important for our study, one consists of the well known\nBrauer graph algebras~\\cite{Alperin86,Benson98,Kauer98}, while the other\nis the newly defined \\emph{triangulation algebras}. Roughly speaking,\na Brauer graph algebra arises from any ribbon quiver and\nauxiliary data given in the form of scalars and positive integer\nmultiplicities, whereas\na triangulation algebra arises from any triangulation quiver with similar\nauxiliary data.\n\nIn Section~\\ref{sec:triangquasi} we investigate triangulation algebras in\nmore detail and prove that they are finite-dimensional, tame, symmetric,\n2-CY-tilted algebras and hence of quasi-quaternion type.\nBy using Iyama-Yoshino mutations of\ncluster-tilting objects~\\cite{IyamaYoshino08} we are able to construct even\nmore, derived equivalent, algebras with the same properties. The\nfinite-dimensionality of the triangulation algebras relies on computations\ninside complete path algebras of quivers, whereas the proof of their\nrepresentation type uses the observation that apart from a few exceptions,\nthe triangulation algebras are deformations of the corresponding\nBrauer graph algebras~(Proposition~\\ref{p:degen}).\n\nOur construction yields new symmetric 2-CY-tilted algebras in addition to the\nones constructed by Burban, Iyama, Keller and Reiten~\\cite{BIKR08} arising from\nthe stable categories of maximal Cohen-Macaulay modules over odd-dimensional\nisolated hypersurface singularities.\nMoreover, it provides new insights on the important problem of classifying\nthe self-injective algebras with periodic module categories, as\nthe algebras we construct are instances of\nnew tame symmetric algebras with periodic modules which seem not to appear in\nthe classification announced by Erdmann and\nSkowro\\'{n}ski~\\cite[Theorem~6.2]{ES08}.\n\nIn Section~\\ref{sec:known} we prove that \nour newly constructed class of algebras\ncontains two known classes of algebras as subclasses.\nFirstly, it contains all the members in Erdmann's lists of\nalgebras of quaternion type (Theorem~\\ref{t:quat2CY}).\nSince these algebras turn out to be 2-CY-tilted, this gives a new proof of\nthe fact that they are indeed of quaternion type, which was first shown\nin~\\cite{ES06} by constructing bimodule resolutions.\nAs a consequence, we are able to characterize all the blocks of group algebras\nthat are 2-CY-tilted algebras (Proposition~\\ref{p:block2CY}).\n\nIn order to illustrate the advantage of\nthis new point of view on the algebras of quaternion\ntype, we discover new algebras of quaternion type which seem not to appear\nin the existing lists (Proposition~\\ref{p:newquat}).\n\nSecondly,\nour newly constructed class of algebras contains also all the Jacobian algebras\nof the quivers with potentials associated by\nLabardini-Fragoso~\\cite{Labardini09} to triangulations of closed surfaces with\npunctures. As a consequence, we deduce that the latter algebras are\nfinite-dimensional of quasi-quaternion type and their derived equivalence\nclass depends only on the surface and not on the particular triangulation\n(Corollary~\\ref{c:Jaclosed}, see also~\\cite{Ladkani12}).\n\nOur newly constructed class contains also all the symmetric algebras of\ntubular type $(2,2,2,2)$ and their socle deformations classified in~\\cite{BS03,BS04}\n(Proposition~\\ref{p:sympoly}). \n\nIn Section~\\ref{sec:mut} we introduce a notion of mutation on triangulation\nquivers and compare it to various other notions of mutations existing in the\nliterature, including flips of triangulations, Kauer's elementary\nmoves~\\cite{Kauer98} for Brauer graph algebras and mutations of quivers with\npotentials~\\cite{DWZ08}. \nWe observe that the Brauer graph algebras arising from different\ntriangulations of the\nsame marked surface are derived equivalent (Corollary~\\ref{c:BGAsurface}),\na result which has also been obtained by Marsh and\nSchroll~\\cite{MarshSchroll14}, however the algebras they consider in the case\nof surfaces with non-empty boundary are different.\nAnalogously, under mild conditions the triangulation algebras of\ntriangulation quivers related by a mutation are derived equivalent\n(Proposition~\\ref{p:mutriang}).\n\nFinally we outline an application to the theory of quivers with potentials.\nNon-degenerate potentials are important in various approaches to the\ncategorification of cluster algebras~\\cite{DWZ10,Plamondon11}.\nIt was proved by Derksen, Weyman and Zelevinsky~\\cite{DWZ08} that over an\nuncountable field, any quiver without loops or 2-cycles has at least one\nnon-degenerate potential. For certain classes of quivers, a non-degenerate\npotential is unique~\\cite{GLS16,Ladkani13}.\nOn the other hand, we construct infinitely many families of quivers, each\nhaving infinitely many non-degenerate potentials with pairwise non-isomorphic\nJacobian algebras (Corollary~\\ref{c:infpot}).\n\n\\subsection*{Acknowledgements}\nI discussed various aspects of this work with Thorsten Holm, Maxim Kontsevich,\nRobert Marsh and Andrzej Skowro\\'{n}ski. I thank them for their interest.\nI would like to thank Peter Littelmann for his encouragement while writing\nthis survey.\nI thank also the anonymous referee for many valuable comments and suggestions.\n\nParts of the material were presented in various talks that I gave in Germany\nduring Summer 2013 and in Israel\nduring Winter 2014\/15, as well as at the workshop on Cluster algebras and\nfinite-dimensional algebras that was held in June 2015 at Leicester, UK.\nThey were also scheduled to be presented at the ARTA conference that was held\nin September 2013 at Torun, Poland, and at the final meeting of the\npriority program ``Representation Theory''\nthat took place at Bad Honnef, Germany in March 2015.\nI thank the organizers of these meetings for their invitations.\n\nMany of the results presented in this survey were obtained during my stay at\nthe University of Bonn which was supported by my DFG grant LA~2732\/1-1 within\nthe framework of the priority program SPP 1388 ``Representation Theory''.\n\nA report containing these results~\\cite{Ladkani14b} was\nwritten during my visit to the IHES at Bures-sur-Yvette in the spring of 2014.\nIn a subsequent visit during Spring 2015\nsome aspects of the theory were refined.\nI would like to thank the IHES for the hospitality and the\ninspiring atmosphere.\n\nWhile writing this survey I have been supported by the\nCenter for Absorption in Science, Ministry of Aliyah and Immigrant Absorption,\nState of Israel.\n\n\n\n\\section{Motivation: blocks of tame representation type}\n\\label{sec:motivation}\n\n\\subsection{Group algebras}\n\\label{ssec:group}\n\nLet $G$ be a finite group and $K$ be a field. The group algebra\n$KG$ can be written as a direct product of indecomposable rings,\nwhich are called \\emph{blocks}.\nBy Maschke's theorem, if the characteristic of $K$ does not divide \nthe order of $G$,\nthen $KG$, and hence each block, is semi-simple. In particular, when $K$\nis also algebraically closed, each block is isomorphic to a matrix ring\nover~$K$.\n\nHowever, when the characteristic of $K$, denoted here and throughout the\npaper by $\\operatorname{char} K$, divides the order of $G$, a block may not be\nsemi-simple anymore. The \\emph{defect group} of a block $B$\nmeasures how far it is from being semi-simple. It may be defined\nas a minimal subgroup $D$ of $G$ such that any $B$-module\nis $D$-projective (i.e.\\ it is isomorphic to a direct summand of\n$W \\otimes_{KD} KG$ for some $KD$-module $W$).\nA defect group is a $p$-subgroup of $G$ (where $p=\\operatorname{char} K$),\ndetermined up to $G$-conjugacy. A defect group of the principal block\n(the block which the trivial $KG$-module $K$ belongs to) is\na $p$-Sylow subgroup of $G$, and\na block is semi-simple if and only if its defect group is trivial.\nWe refer to the survey article~\\cite{Linckelmann11} for further details.\n\nMany aspects of the representation theory of a block are controlled by its\ndefect group. One such important aspect is the representation type.\nIndeed, if $B$ is a block with defect group $D$ over an algebraically closed\nfield of\ncharacteristic $p$, then $B$ is of finite representation type if and only if\n$D$ is cyclic~\\cite{Higman54},\nwhile $B$ is of tame (but not finite) representation type if and only if\n$p=2$ and $D$ is either dihedral, semi-dihedral of generalized quaternion\ngroup~\\cite{BD75}.\nIn all other cases, $B$ is of wild representation type.\n\nBlocks of finite representation type, that it, blocks with cyclic defect group,\nare Morita equivalent to Brauer tree algebras~\\cite{Dade66,Janusz69} and hence\nare well understood.\nIn order to understand blocks of tame representation type (over algebraically\nclosed fields), Erdmann introduced families of symmetric algebras defined by\nproperties of their Auslander-Reiten quivers. These are the algebras\nof dihedral, semi-dihedral and quaternion type.\nShe showed that a block with dihedral (respectively, semi-dihedral, generalized\nquaternion) defect group is an algebra of the corresponding type and moreover\nshe classified the quivers with relations these algebras may possibly\nhave~\\cite{Erdmann90}.\n\nIn this section we focus on the algebras of quaternion type and start by\nreviewing the relevant notions.\n\n\\subsection{Stable categories and periodicity}\nLet $A$ be a finite-dimensional algebra over a field $K$. Denote by \n$\\operatorname{mod} A$ the category of finitely generated right $A$-modules,\nand by $\\mathcal{D}^b(A) = \\mathcal{D}^b(\\operatorname{mod} A)$ its bounded derived category.\nThe latter contains as triangulated subcategory the category\n$\\operatorname{per} A$ of perfect complexes whose objects are bounded complexes\nof finitely generated projective $A$-modules.\nThe Verdier quotient $\\mathcal{D}^b(A)\/\\operatorname{per} A$ is known as the\n\\emph{singularity category} of $A$, see~\\cite{Orlov09}. \nIts name comes from the fact that it vanishes precisely when $A$ has finite\nglobal dimension (i.e.\\ $A$ is ``smooth'')~\\cite[Remark~1.9]{Orlov09},\nas in this case any $A$-module has a finite projective resolution,\nthus any object in $\\mathcal{D}^b(A)$ is isomorphic to a perfect complex.\n\nAssume that the algebra $A$ is \\emph{self-injective}, i.e.\\ $A$ is injective\nas left and right module over itself, and\nconsider the \\emph{stable module category} $\\stmod A$\nwhose objects are the same as those of $\\operatorname{mod} A$ and the space of morphisms\nbetween any two objects $M, N \\in \\stmod A$ is given by\n\\[\n\\underline{\\operatorname{Hom}}_A(M,N) = \\operatorname{Hom}_A(M,N)\/\\mathcal{P}(M,N)\n\\]\nwhere $\\mathcal{P}(M,N)$ consists of all the morphisms $M \\to N$ in $\\operatorname{mod} A$\nwhich factor through some projective module over $A$.\n\nBy a result of Happel~\\cite[Theorem~I.2.6]{Happel88}, the additive category\n$\\stmod A$ is triangulated. Moreover, by a theorem of\nRickard~\\cite[Theorem~2.1]{Rickard89}, $\\stmod A$ can be identified with the\nsingularity category of $A$.\n\nLet $M \\in \\stmod A$ and consider a projective cover $P_M$ of $M$. Define\na module $\\Omega M$ by the exact sequence (in $\\operatorname{mod} A$)\n\\[\n0 \\to \\Omega M \\to P_M \\to M \\to 0 .\n\\]\nThe \\emph{syzygy} $\\Omega M$ is well defined in the category $\\stmod A$ and\ngives rise to \\emph{Heller's syzygy functor}\n$\\Omega \\colon \\stmod A \\to \\stmod A$, see~\\cite{Heller61}.\nSometimes, when we want to stress the role of the\nalgebra $A$, we will denote the syzygy functor by $\\Omega_A$ instead of $\\Omega$.\n\nSimilarly, by taking an injective envelope $I_M$ of $M$ and the exact sequence\n\\[\n0 \\to M \\to I_M \\to \\Omega^{-1} M  \\to 0\n\\]\none can define the cosyzygy functor $\\Omega^{-1} \\colon \\stmod A \\to \\stmod A$,\nwhich is an inverse of $\\Omega$.\nThe suspension functor of the triangulated category $\\stmod A$ is\ngiven by $\\Omega^{-1}$.\n\nAn important class of self-injective algebras is formed by the\nsymmetric algebras, which we now define. First, observe that\nfor a finite-dimensional algebra $A$ over $K$, the vector space\n$DA = \\operatorname{Hom}_K(A,K)$ is an $A$-$A$-bimodule.\n\n\\begin{definition}\nA finite-dimensional $K$-algebra $A$ is \\emph{symmetric} if\n$A \\simeq DA$ as $A$-$A$-bimodules. Here and throughout the paper,\nthe symbol $\\simeq$ denotes isomorphism.\n\\end{definition}\n\nWe recall a few alternative characterizations of symmetric algebras.\nIn order to formulate them, we need the notion of a Calabi-Yau\ntriangulated category which is given below.\n\n\\begin{definition}\nLet $d \\in \\mathbb{Z}$.\nA $K$-linear triangulated category $\\mathcal{T}$ with suspension $\\Sigma$\nand finite-dimensional morphism spaces is \\emph{$d$-Calabi-Yau} if\nthere exist functorial isomorphisms\n\\[\n\\operatorname{Hom}_{\\mathcal{T}}(X, Y) \\simeq D \\operatorname{Hom}_{\\mathcal{T}}(Y, \\Sigma^d X)\n\\]\nfor all $X, Y \\in \\mathcal{T}$.\n\\end{definition}\n\n\\begin{proposition} \\label{p:symmetric}\nThe following conditions are equivalent for a finite-dimensional $K$-algebra\n$A$.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:sym}\n$A$ is symmetric;\n\n\\item \\label{it:p:symform}\nThere exists a symmetrizing form on $A$, that is,\na $K$-linear map $\\lambda \\colon A \\to K$\nwhose kernel does not contain any non-trivial left ideal of $A$\nand moreover $\\lambda(xy)=\\lambda(yx)$ for any $x,y \\in A$;\n\n\\item \\label{it:p:symper}\nThe triangulated category $\\operatorname{per} A$ is $0$-Calabi-Yau.\n\n\\item \\label{it:p:sym0CY}\n$A$ is isomorphic to the endomorphism algebra of an object in a triangulated\n$0$-Calabi-Yau category.\n\\end{enumerate}\n\\end{proposition}\n\\begin{proof}\nThe equivalence of~\\eqref{it:p:sym} and~\\eqref{it:p:symform} is standard, see\ne.g.~\\cite[Theorem~IV.2.2]{SY11}.\nThe implication~\\eqref{it:p:sym}$\\Rightarrow$\\eqref{it:p:symper} follows from\nthe fact that for any finite-dimensional algebra $A$ one has\n\\[\n\\operatorname{Hom}_{\\mathcal{D}^b(A)}(X, Y) \\simeq D\\operatorname{Hom}_{\\mathcal{D}^b(A)}(Y,\nX \\stackrel{\\mathbf{L}}{\\otimes}_A DA)\n\\]\nfor any $X \\in \\operatorname{per} A$ and $Y \\in \\mathcal{D}^b(A)$, see the proof\nof~\\cite[Theorem~I.4.6]{Happel88}.\nFor the implication~\\eqref{it:p:symper}$\\Rightarrow$\\eqref{it:p:sym0CY},\nobserve that $A \\simeq \\operatorname{End}_{\\operatorname{per} A}(A)$.\nFor~\\eqref{it:p:sym0CY}$\\Rightarrow$\\eqref{it:p:sym}, note that if $X$ is an\nobject in a $0$-Calabi-Yau triangulated category $\\mathcal{T}$ and $A = \\operatorname{End}_{\\mathcal{T}}(X)$,\nthen the functorial isomorphism $\\operatorname{Hom}_{\\mathcal{T}}(X,X) \\simeq D\\operatorname{Hom}_{\\mathcal{T}}(X,X)$\nimplies that $A \\simeq DA$ as $A$-$A$-bimodules.\n\\end{proof}\n\n\\begin{corollary}[\\protect{\\cite[Theorem~IV.4.1]{SY11}}]\n\\label{c:sym:idempotent}\nLet $A$ be a symmetric algebra and $e \\in A$ an idempotent. Then the algebra\n$eAe$ is also symmetric.\n\\end{corollary}\n\\begin{proof}\nOne has $D(eAe) \\simeq e(DA)e$; alternatively,\nuse Proposition~\\ref{p:symmetric} for the algebra\n$eAe \\simeq \\operatorname{End}_{\\operatorname{per} A}(eA)$.\n\\end{proof}\n\n\\begin{example}\nAny group algebra $KG$ is symmetric. Indeed, a symmetrizing form on $KG$ is\ngiven by\n\\[\n\\lambda(\\sum_{g \\in G} a_g g) = a_1 .\n\\]\nIt follows from Corollary~\\ref{c:sym:idempotent} that any block of a group\nalgebra is also symmetric.\n\\end{example}\n\n\\begin{example}\nIf $A$ is any finite-dimensional $K$-algebra, the bimodule structure\non $DA$ allows to define a symmetric algebra whose underlying vector space is\n$A \\oplus DA$ called the \\emph{trivial extension algebra} of $A$ and\ndenoted by $T(A)$. The elements of $T(A)$ are\npairs $(a,\\mu)$ where $a \\in A$ and $\\mu \\in DA$. Addition and\nmultiplication are given by the formulae\n\\begin{align*}\n(a, \\mu) + (a', \\mu') = (a+a', \\mu + \\mu') \\\\\n(a,\\mu) \\cdot (a',\\mu') = (aa', a \\mu'+\\mu a')\n\\end{align*}\nfor $a,a' \\in A$ and $\\mu, \\mu' \\in DA$.\nThe symmetrizing form on $T(A)$ is given by $\\lambda(a,\\mu) = \\mu(1)$.\n\\end{example}\n\n\\begin{remark} \\label{rem:CYminus1}\nThe stable category of a symmetric algebra $A$ is\n$(-1)$-Calabi-Yau, i.e.\\\n\\[\n\\stHom_A (M, N) \\simeq D \\stHom_A (N, \\Omega M)\n\\]\nfor $M, N \\in \\stmod A$, see for example~\\cite[Proposition~1.2]{ES06}\nand the end of~\\cite[\\S1]{Amiot09}.\n\\end{remark}\n\n\n\\begin{definition}\nA module $M \\in \\stmod A$ is \\emph{$\\Omega$-periodic} if\n$\\Omega^r M \\simeq M$ for some integer $r>0$.\n\\end{definition}\n\nThe category $\\stmod A$ has Auslander-Reiten sequences, and \nwhen $A$ is symmetric there is a\nclose connection between the Auslander-Reiten translation $\\tau$ on\n$\\stmod A$ and the syzygy $\\Omega$, namely $\\tau = \\Omega^2$.\nIn particular, a module is $\\Omega$-periodic if and only if it\nis $\\tau$-periodic.\n\n\\begin{example} \\label{ex:Kxn}\nLet $n \\geq 1$ and consider the algebra $A = K[x]\/(x^n)$.\nIt is a commutative, local, symmetric algebra over $K$ of finite\nrepresentation type whose indecomposable modules are given by $M_i = x^i A$\nfor $0 \\leq i < n$. The module $M_0 = A$ is projective, and the exact\nsequence\n\\[\n0 \\to x^{n-i}A \\to A \\xrightarrow{x^i \\cdot -} x^i A \\to 0\n\\]\nshows that $\\Omega(M_i) = M_{n-i}$ for any $0 < i < n$.\nHence $\\Omega^2 M \\simeq M$ for any $M \\in \\stmod A$.\nNote that if $\\operatorname{char} K = p$ and $n=p^e$ for some $e \\geq 1$,\nthen $A \\simeq KG$ for $G=\\mathbb{Z}\/p^e\\mathbb{Z}$ and its defect group\nequals $G$.\n\\end{example}\n\n\n\\subsection{Algebras of quaternion type}\nIn this section we assume that the ground field $K$ is algebraically\nclosed. The algebras of quaternion type were introduced by Erdmann, and\nwe refer to the articles~\\cite{Erdmann88} and the monograph~\\cite{Erdmann90}\nfor a detailed presentation.\n\n\\begin{definition}[\\cite{Erdmann88}]\nA finite-dimensional algebra $A$ is of \\emph{quaternion type} if\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item\n$A$ is symmetric, indecomposable as a ring;\n\\item\n$A$ has tame (but not finite) representation type;\n\\item\n$\\Omega^4 M \\simeq M$ for any $M \\in \\stmod A$;\n\\item \\label{it:q:cartan}\n$\\det C_A \\neq 0$, where $C_A$ denotes the Cartan matrix of $A$. \n\\end{enumerate}\n\\end{definition}\n\nRecall that the \\emph{Cartan matrix} of a basic algebra $A$ is the\n$n$-by-$n$ matrix with integer entries given by\n$(C_A)_{i,j} = \\dim_K e_i A e_j$, where $e_1, e_2, \\dots, e_n$ form a complete\nset of primitive orthogonal idempotents in $A$.\nThe motivation behind condition~\\eqref{it:q:cartan} lies in the fact that\nif $B$ is a block over a field of characteristic $p$, then the determinant\nof its Cartan matrix is a power of $p$.\n\nLet $n \\geq 3$. The \\emph{generalized quaternion group} $Q_{2^n}$ is\ngiven by generators and relations as follows:\n\\[\nQ_{2^n} = \n\\langle\nx,y \\mid x^{2^{n-2}}=y^2 \\,,\\, y^4=1 \\,,\\, y^{-1}xy=x^{-1}\n\\rangle .\n\\]\nIn particular, for $n=3$ we recover the usual quaternion group with\n$8$ elements.\n\nErdmann proved the following facts:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nBlocks of group algebras with generalized quaternion defect groups are of\nquaternion type.\n\n\\item \\label{it:q12lists}\nAn algebra of quaternion type is Morita equivalent to an algebra in 12\nfamilies of symmetric algebras given by quivers with relations.\nIn particular, an algebra of quaternion type has at most three isomorphism\nclasses of simple modules.\n\\end{enumerate}\n\nThe lists of the quivers with relations of point~\\eqref{it:q12lists}\ncan be found in~\\cite[pp.\\ 303--306]{Erdmann90}\nor in the survey articles~\\cite[Theorem~5.5]{ES08}\nand~\\cite[Theorem~8.4]{Skowronski06}.\nLater, Holm~\\cite{Holm99}\npresented a derived equivalence classification of the algebras appearing in\nthese lists and proved that these algebras are indeed tame. Finally,\nin~\\cite{ES06} Erdmann and Skowro\\'{n}ski showed that the algebras in these lists\nhave the required periodicity property and hence they are indeed of quaternion\ntype.\n\n\\begin{example}\nOne of the families in Erdmann's list consists of local algebras whose\nquiver is\n\\[\n\\xymatrix@=1pc{\n{\\bullet} \\ar@(dl,ul)[]^{\\alpha} \\ar@(dr,ur)[]_{\\beta}\n}\n\\]\nwith the relations\n\\begin{align*}\n\\alpha^2 = (\\beta \\alpha)^{m-1} \\beta &,&\n\\beta^2 = (\\alpha \\beta)^{m-1} \\alpha &,&\n\\alpha \\beta^2 = \\alpha^2 \\beta = \\beta \\alpha^2 = \\beta^2 \\alpha = 0\n\\end{align*}\ndepending on an integer parameter $m \\geq 2$. \nWhen the ground field $K$ is algebraically closed, $\\operatorname{char} K = 2$ and $m=2^{n-2}$\nfor some $n \\geq 3$, this algebra is the group algebra $K Q_{2^n}$ of the\ngeneralized quaternion group $Q_{2^n}$.\n\\end{example}\n\n\\subsection{Algebras of quasi-quaternion type}\n\nIt seems natural to lift the restriction on the Cartan determinant in the\ndefinition of algebras of quaternion type and consider a wider\nclass of algebras, which we call algebras of quasi-quaternion type.\n\n\\begin{definition} \\label{def:quasi}\nA finite-dimensional algebra $A$ is of \\emph{quasi-quaternion type} if:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item\n$A$ is symmetric, indecomposable as a ring;\n\\item\n$A$ has tame (but not finite) representation type;\n\\item \\label{it:qq:period4}\n$\\Omega^4 M \\simeq M$ for any $M \\in \\stmod A$;\n\\end{enumerate}\n\\end{definition}\n\n\\begin{remark}\nSince $\\tau=\\Omega^2$, the stable Auslander-Reiten quiver of an algebra of\nquasi-quaternion type consists of tubes of ranks $1$ and $2$.\n\\end{remark}\n\nIn analogy with Erdmann's description of the algebras of quaternion type,\nthe following problem arises naturally.\n\\begin{problem}\nDescribe the algebras of quasi-quaternion type.\n\\end{problem}\n\nAlgebras of quasi-quaternion type are in particular tame symmetric algebras\nwith periodic modules. A classification of the latter algebras has been\nannounced in~\\cite[Theorem~6.2]{ES08},\nsee also~\\cite[Theorem~8.7]{Skowronski06}. However, many of the algebras\nof quasi-quaternion type to be constructed in Section~\\ref{sec:triangquasi}\nseem to be missing from the aforementioned classification.\n\nSince the derived equivalence of self-injective algebras implies their stable \nequivalence~\\cite[Corollary~2.2]{Rickard89}\nand stable equivalence preserves representation type~\\cite{Krause97},\nan argument as in Prop.~2.1 and Prop.~2.2 of~\\cite{Holm99} yields the\nfollowing observation.\n\\begin{proposition} \\label{p:derquasi}\nAny algebra which is derived equivalent to an algebra of quasi-quaternion\ntype is also of quasi-quaternion type.\n\\end{proposition}\n\nOne approach to guarantee the condition~\\eqref{it:qq:period4} in the\ndefinition of algebras of quasi-quaternion type\nis to show that the algebra $A$ is periodic as $A$-$A$-bimodule with\nperiod dividing $4$, that is, \n$\\Omega_{A^e}^4(A) \\simeq A$, where $A^e = A^{op} \\otimes_K A$.\nThis is usually done using a projective resolution of $A$ as\na bimodule over itself. In fact, such strategy is used in~\\cite{ES06}\nto prove that the algebras in Erdmann's list are of quaternion type.\n\nWe suggest an alternative approach using 2-Calabi-Yau categories.\nIt turns out that symmetric algebras that are also the endomorphism\nalgebras of cluster-tilting objects in such categories always satisfy the \nperiodicity condition~\\eqref{it:qq:period4}. We explain this in the\nnext section.\n\n\\section{Symmetric 2-CY-tilted algebras}\n\\label{sec:sym2CY}\n\nIn this section we study properties of symmetric algebras that are also\n2-CY-tilted, i.e.\\ being isomorphic to the endomorphism algebras of\ncluster-tilting objects in 2-Calabi-Yau triangulated categories. \n\nWe start by recalling the definition and basic properties of 2-CY-tilted\nalgebras. Since many of them arise as Jacobian algebras of quivers with\npotentials, we review this notion as well, and introduce the notion of\nhyperpotential which is useful over ground fields of positive characteristic.\nThen we present two new results whose details will appear elsewhere;\nthe first concerns the periodicity of modules over symmetric 2-CY-tilted\nalgebras (Proposition~\\ref{p:period}), and the second concerns derived\nequivalences of neighboring 2-CY-tilted algebras (Proposition~\\ref{p:dereq}). \n\nAs a consequence of the first result, we deduce that indecomposable, tame,\nsymmetric, 2-CY-tilted algebras are of quasi-quaternion type. For more\nbackground on 2-CY-tilted algebras, we refer the reader to the survey\narticle~\\cite{Reiten10}.\n\n\\subsection{2-CY-tilted algebras}\n\nLet $\\mathcal{C}$ be a $K$-linear triangulated category with\nsuspension $\\Sigma$. We assume:\n\\begin{itemize}\n\\item\n$\\mathcal{C}$ has finite-dimensional morphism spaces.\n\n\\item\n$\\mathcal{C}$ is Krull Schmidt (i.e.\\ any object has a decomposition\ninto a finite direct sum of indecomposables which is unique up to\nisomorphism and change of order).\n\n\\item\n$\\mathcal{C}$ is $2$-Calabi-Yau.\n\\end{itemize}\n\nSuch triangulated categories $\\mathcal{C}$ arise in the additive categorification\nof cluster algebras, see the survey~\\cite{Keller10}.\nThe role of the clusters in a cluster algebra\nis played by cluster-tilting objects in the category $\\mathcal{C}$.\n\n\\begin{definition}\nAn object $T \\in \\mathcal{C}$ is \\emph{cluster-tilting} if:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item\n$\\operatorname{Hom}_{\\mathcal{C}}(T, \\Sigma T) = 0$;\n\n\\item\nFor any $X \\in \\mathcal{C}$ with $\\operatorname{Hom}_{\\mathcal{C}}(T, \\Sigma X) = 0$,\nwe have that $X \\in \\operatorname{add} T$, where \n$\\operatorname{add} T$ denotes the full subcategory of $\\mathcal{C}$ consisting of\nthe objects isomorphic to finite direct sums of summands of $T$.\n\\end{enumerate}\n\\end{definition}\n\n\\begin{definition}\nAn algebra is called \\emph{2-CY-tilted} if it is isomorphic to\nan algebra of the form $\\operatorname{End}_{\\mathcal{C}}(T)$ with $\\mathcal{C}$ as above and\n$T$ a cluster-tilting object in $\\mathcal{C}$.\n\\end{definition}\n\nThe cluster categories associated to quivers without oriented cycles\n\\cite{BMRRT06} were the first instances of triangulated 2-Calabi-Yau\ncategories with cluster-tilting object.\nThey are constructed as orbit categories of the\nbounded derived category of the path algebra of the quiver\nwith respect to a suitable auto-equivalence~\\cite{Keller05}.\nThe corresponding endomorphism algebras of cluster-tilting objects\nare called cluster-tilted algebras~\\cite{BMR07}.\nSelf-injective cluster-tilted algebras were classified by\nRingel~\\cite{Ringel08}; there are very few such algebras\nas all of them are of\nfinite representation type and up to Morita equivalence there are at most two\nsuch algebras having a given number of non-isomorphic simple modules.\nIn particular, except for the quiver $A_1$ with\none vertex whose cluster-tilted algebra equals the ground field,\ncluster-tilted algebras are never symmetric.\n\nMore generally, 2-CY-tilted algebras were investigated by Keller and\nReiten~\\cite{KellerReiten07}. The next proposition records the\nrelevant properties we need.\n\n\\begin{proposition} \\label{p:2CYtilted}\nLet $\\Lambda$ be a 2-CY-tilted algebra. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\n\\emph{\\cite[Prop.~2.1]{KellerReiten07}}\n$\\Lambda$ is Gorenstein of dimension at most 1, i.e.\\ the\nprojective dimension of any injective module and the injective dimension of\nany projective module are at most 1;\n\n\\item\n\\emph{\\cite[Theorem~3.3]{KellerReiten07}}\nThe singularity category of $\\Lambda$ is $3$-Calabi-Yau.\n\\end{enumerate}\n\\end{proposition}\n\nGiven a 2-CY-tilted algebra $\\Lambda$, there is a procedure to construct new\n2-CY-tilted algebras from idempotents of $\\Lambda$.\nThe corresponding statement for cluster-tilted algebras has been shown\nin~\\cite[Theorem~2.13]{BMR08}, see also~\\cite[Theorem~5]{CalderoKeller06}.\nThe general case follows from Calabi-Yau reduction~\\cite{IyamaYoshino08},\nsee also~\\cite[\\S{II.2}]{BIRS09}. For the convenience of the reader, we give\nthe short proof.\n\\begin{proposition}\nLet $\\Lambda$ be a 2-CY-tilted algebra and let $e \\in \\Lambda$ be an idempotent.\nThen the algebra $\\Lambda\/\\Lambda e \\Lambda$ is 2-CY-tilted.\n\\end{proposition}\n\\begin{proof}\nLet $\\Lambda=\\operatorname{End}_{\\mathcal{C}}(T)$ where $\\mathcal{C}$ is a triangulated 2-Calabi-Yau category\nand $T$ is a cluster-tilting object in $\\mathcal{C}$. Let $T'$ be the summand of $T$ \ncorresponding to the idempotent $e$. The category\n$\\mathcal{C}' = \\{X \\in \\mathcal{C} : \\operatorname{Hom}_{\\mathcal{C}}(X,\\Sigma T')=0\\}\/(\\operatorname{add} T')$\nis a triangulated 2-Calabi-Yau category by~\\cite[Theorem~4.7]{IyamaYoshino08}\nand $T$ is a cluster-tilting object in $\\mathcal{C}'$\nby~\\cite[Theorem~4.9]{IyamaYoshino08}.\nFinally, $\\operatorname{End}_{\\mathcal{C}'}(T) \\simeq \\Lambda\/\\Lambda e \\Lambda$.\n\\end{proof}\n\n\\subsection{Quivers with potentials}\n\\label{ssec:QP}\n\nThanks to the works of Amiot~\\cite{Amiot09} and Keller~\\cite{Keller11},\na rich source of 2-Calabi-Yau triangulated categories with cluster-tilting\nobject is provided \nby quivers with potentials whose Jacobian algebras are finite-dimensional.\nQuivers with potentials and their Jacobian algebras were defined and\nstudied by Derksen, Weyman and Zelevinsky~\\cite{DWZ08}.\n\nA \\emph{quiver} is a finite directed graph. Formally, it is a quadruple\n$Q=(Q_0,Q_1,s,t)$ where $Q_0$ and $Q_1$ are finite sets (of \\emph{vertices}\nand \\emph{arrows}, respectively) and $s,t \\colon Q_1 \\to Q_0$ are functions\nspecifying for each arrow its starting and terminating\nvertex, respectively.\n\nThe \\emph{path algebra} $KQ$ has the set of paths of $Q$ as a basis, with the\nproduct of two paths being their concatenation, if defined, and zero otherwise.\nThe \\emph{complete path algebra} $\\widehat{KQ}$ is the completion of\n$KQ$ with respect to the ideal generated by all the arrows of $Q$.\nIt is a topological algebra, with a topological basis given by\nthe paths of $Q$. Thus, an element in $\\widehat{KQ}$ is a possibly\ninfinite linear combination of paths. We denote by $\\bar{I}$ the closure of an\nideal $I$ in $\\widehat{KQ}$.\n\n\\begin{example}\nThe path algebra of the quiver with one vertex and one loop at that vertex\nis the ring $K[x]$ of polynomials in one variable, \nwhereas the complete path algebra is the ring $K[[x]]$ of power series \nin one variable.\n\\end{example}\n\nA \\emph{cycle} in $Q$ is a path that starts and ends at the same vertex.\nOne can consider the equivalence relation on the set of cycles given by\nrotations, i.e.\\\n\\[\n\\alpha_1 \\alpha_2 \\dots \\alpha_n \\sim\n\\alpha_i \\dots \\alpha_n \\alpha_1 \\dots \\alpha_{i-1}\n\\]\nfor a cycle $\\alpha_1 \\alpha_2 \\dots \\alpha_n$ and $1 \\leq i \\leq n$.\n\nThe \\emph{zeroth continuous Hochschild homology} $\\operatorname{HH}_0(\\widehat{KQ})$ is $\\widehat{KQ}\/\\overline{[\\widehat{KQ},\\widehat{KQ}]}$, i.e.\\ the\nquotient of $\\widehat{KQ}$ by the closure of the subspace spanned by all the\ncommutators of elements in $\\widehat{KQ}$. It has a topological\nbasis given by the equivalence classes of cycles of $Q$ modulo rotation.\n\n\\begin{definition}[\\protect{\\cite[Definition~3.1]{DWZ08}}]\nA \\emph{potential} on $Q$ is an element in $\\operatorname{HH}_0(\\widehat{KQ})$.\nIn explicit terms, a potential is a (possibly infinite) linear\ncombination of cycles in $Q$, considered up to rotations.\n\nA pair $(Q,W)$ where $Q$ is a quiver and $W$ is a potential on $Q$\nis called a \\emph{quiver with potential}.\n\\end{definition}\n\nFor any arrow $\\alpha$ of $Q$, there is a \\emph{cyclic derivative} map\n$\\partial_\\alpha \\colon \\operatorname{HH}_0(\\widehat{KQ}) \\to \\widehat{KQ}$ which is\nthe unique continuous linear map whose value on each\ncycle $\\alpha_1 \\alpha_2 \\dots \\alpha_n$ is given by\n\\[\n\\partial_\\alpha(\\alpha_1 \\alpha_2 \\dots \\alpha_n) =\n\\sum_{i \\,:\\, \\alpha_i = \\alpha}\n\\alpha_{i+1} \\dots \\alpha_n \\alpha_1 \\dots \\alpha_{i-1}\n\\]\nwhere the sum goes over all indices $1 \\leq i \\leq n$ such that\n$\\alpha_i=\\alpha$.\n\n\\begin{definition}[\\cite{DWZ08}]\nLet $(Q,W)$ be a quiver with potential. Its \\emph{Jacobian algebra}\n$\\mathcal{P}(Q,W)$ is the quotient of the complete path algebra $\\widehat{KQ}$\nby the closure of its ideal generated by the\ncyclic derivatives $\\partial_\\alpha W$ with respect to the arrows\n$\\alpha$ of $Q$,\n\\[\n\\mathcal{P}(Q,W) = \\widehat{KQ} \/ \\overline{(\\partial_\\alpha W : \\alpha \\in Q_1)} .\n\\]\n\\end{definition}\n\n\\begin{remark}\nWhen the potential $W$ is a finite linear combination of cycles, one\ncan also consider a non-complete version of the Jacobian algebra, \nnamely, the quotient of the path algebra $KQ$ by its ideal generated\nby the cyclic derivatives of $W$. While in many cases this variation\ngives the same result,\nthe next example shows that in general these two notions differ.\n\\end{remark}\n\n\\begin{example} \\label{ex:QP}\nLet $Q$ be the quiver\n\\[\n\\xymatrix@=0.5pc{\n& {\\bullet_3} \\ar[ddl]_{\\gamma} \\\\ \\\\\n{\\bullet_1} \\ar[rr]_{\\alpha} && {\\bullet_2} \\ar[uul]_{\\beta}\n}\n\\]\nwith the potential $W = \\alpha \\beta \\gamma - \\alpha \\beta \\gamma\n\\alpha \\beta \\gamma$. Let $\\mathcal{J}$ be the closure of the ideal generated\nby the cyclic derivatives of $W$, so that $\\mathcal{P}(Q,W) = \\widehat{KQ}\/\\mathcal{J}$.\n\nComputing the cyclic derivative with respect to the arrow $\\gamma$,\nwe get\n\\[\n\\partial_\\gamma W = \\alpha \\beta - \\alpha \\beta \\gamma \\alpha \\beta\n- \\alpha \\beta \\gamma \\alpha \\beta\n= \\alpha \\beta - 2 \\alpha \\beta \\gamma \\alpha \\beta ,\n\\]\nhence $\\alpha \\beta - 2 \\alpha \\beta \\gamma \\alpha \\beta \\in \\mathcal{J}$.\nTherefore, for any $n \\geq 1$,\n\\[\n\\alpha \\beta - (2 \\alpha \\beta \\gamma)^n \\alpha \\beta\n= \\sum_{i=0}^{n-1}\n(2 \\alpha \\beta \\gamma)^i \n(\\alpha \\beta - 2 \\alpha \\beta \\gamma \\alpha \\beta) \\in \\mathcal{J} .\n\\]\n\nSince $\\mathcal{J}$ is closed, this implies that $\\alpha \\beta \\in \\mathcal{J}$.\nMoreover, one can verify that $\\mathcal{P}(Q,W) \\simeq KQ\/(\\alpha \\beta,\n\\beta \\gamma, \\gamma \\alpha)$. In particular, we see that in the\npresentation of the Jacobian algebra as quiver with relations,\nthe relations are not necessarily the cyclic derivatives of the potential.\n\nConsider now the non-complete Jacobian algebra $A$ and assume that\n$\\operatorname{char} K \\neq 2$. Since $\\alpha \\beta = 2 \\alpha \\beta \\gamma \\alpha \\beta$\nin $A$, one has\n\\begin{align*}\n2 \\alpha \\beta \\gamma &= 4 \\alpha \\beta \\gamma \\alpha \\beta \\gamma\n= (2 \\alpha \\beta \\gamma)^2 ,\\\\\n(e_1 - 2 \\alpha \\beta \\gamma)^2\n&= e_1^2 - 4 \\alpha \\beta \\gamma + 4 \\alpha \\beta \\gamma \\alpha \\beta \\gamma\n= e_1 - 2 \\alpha \\beta \\gamma ,\\\\\n2 \\alpha \\beta \\gamma (e_1 - 2 \\alpha \\beta \\gamma) &= \n(e_1 - 2 \\alpha \\beta \\gamma) 2 \\alpha \\beta \\gamma = 0 ,\n\\end{align*}\nhence the idempotents in $A$ corresponding to the paths of length zero are\nno longer primitive; for example, $e_1$ can be written as a sum\n$e_1 = (e_1 - 2 \\alpha \\beta \\gamma) + 2 \\alpha \\beta \\gamma$\nof two orthogonal idempotents. Using these idempotents one can\nverify that the algebra $A$ decomposes into a direct\nsum of $\\mathcal{P}(Q,W)$ and the matrix ring $M_3(K)$.\n\\end{example}\n\nFor a quiver with potential $(Q,W)$, Ginzburg~\\cite[\\S4.2]{Ginzburg06}\nhas defined a dg-algebra $\\Gamma(Q,W)$ which is concentrated in\nnon-positive degrees and its zeroth cohomology is isomorphic to the\nJacobian algebra, i.e.\\ $\\operatorname{H}^0(\\Gamma(Q,W)) \\simeq \\mathcal{P}(Q,W)$.\nIn~\\cite[Theorem~6.3]{Keller11}, Keller shows that the Ginzburg dg-algebra\n$\\Gamma=\\Gamma(Q,W)$ is homologically smooth and bimodule 3-Calabi-Yau,\nthat is, $\\RHom_{\\Gamma^e}(\\Gamma, \\Gamma^e) \\simeq \\Gamma[-3]$ in\n$\\mathcal{D}(\\Gamma^e)$, where $\\Gamma^e = \\Gamma^{op} \\otimes_K \\Gamma$.\n\nGiven a dg-algebra $\\Gamma$ which is concentrated in non-positive degrees,\nhomologically smooth, bimodule 3-Calabi-Yau and whose zeroth cohomology\n$\\operatorname{H}^0(\\Gamma)$ is finite-dimensional, \nAmiot constructs in~\\cite[\\S2]{Amiot09} a triangulated \n2-Calabi-Yau category with a cluster-tilting object\nwhose endomorphism algebra is $\\operatorname{H}^0(\\Gamma)$. She then\napplies this construction to $\\Gamma(Q,W)$ for quivers with potentials\n$(Q,W)$ whose Jacobian algebra is finite-dimensional to obtain\nthe \\emph{generalized cluster category} associated with $(Q,W)$\n\\cite[Theorem~3.5]{Amiot09}.\n\n\\begin{proposition}[\\protect{\\cite[Corollary~3.6]{Amiot09}}]\nAny finite-dimensional Jacobian algebra of a quiver with potential is\n2-CY-tilted.\n\\end{proposition}\n\nA notion of equivalence of quivers with potentials was introduced by Derksen,\nWeyman and Zelevinsky~\\cite{DWZ08}. Let $Q$ be a quiver. Any continuous\nalgebra automorphism $\\varphi \\colon \\widehat{KQ} \\to \\widehat{KQ}$ induces a continuous\nlinear\nautomorphism, denoted $\\overline{\\varphi}$, of the topological vector space\n$\\operatorname{HH}_0(\\widehat{KQ})=\\widehat{KQ}\/\\overline{[\\widehat{KQ},\\widehat{KQ}]}$.\nFor a vertex $i$ of $Q$, denote by $e_i$ the path of length zero at $i$.\nIt is an idempotent of the algebra $\\widehat{KQ}$.\n\n\\begin{definition}[\\protect{\\cite[Definition~4.2]{DWZ08}}]\nTwo potentials $W$ and $W'$ on $Q$ are \\emph{right equivalent} if there exists\na continuous algebra automorphism $\\varphi \\colon \\widehat{KQ} \\to \\widehat{KQ}$ satisfying\n$\\varphi(e_i)=e_i$ for each $i \\in Q_0$ and $W'=\\overline{\\varphi}(W)$ in\n$\\operatorname{HH}_0(\\widehat{KQ})$.\n\\end{definition}\n\nThe Ginzburg dg-algebras of right equivalent potentials are isomorphic\nand hence also their Jacobian algebras~\\cite[Lemmas~2.8 and~2.9]{KellerYang11}.\nIf the latter are finite-dimensional, then the associated 2-Calabi-Yau\ncategories are equivalent as triangulated categories, since they depend only on\nthe corresponding Ginzburg dg-algebras.\n\n\\begin{example}\nConsider the quiver with potential $(Q,W)$ of Example~\\ref{ex:QP} and let\n$W'=\\alpha \\beta \\gamma$ be another potential on $Q$.\nA continuous algebra automorphism of $\\widehat{KQ}$ fixing each $e_i$ is determined\nby its value on the arrows. The endomorphism $\\varphi$ whose value on the arrows\nis given by\n\\begin{align*}\n\\varphi(\\alpha) = \\alpha - \\alpha \\beta \\gamma \\alpha &,&\n\\varphi(\\beta)  = \\beta &,&\n\\varphi(\\gamma) = \\gamma\n\\end{align*}\nis an automorphism of $\\widehat{KQ}$; indeed,\n\\[\n\\varphi^{-1}(\\alpha) = \\alpha + \\alpha \\beta \\gamma \\alpha\n+ 2 (\\alpha \\beta \\gamma)^2 \\alpha + 5 (\\alpha \\beta \\gamma)^3 \\alpha\n+ 14 (\\alpha \\beta \\gamma)^4 \\alpha + \\dots\n\\]\n(where the coefficients are the Catalan numbers).\nMoreover, $\\varphi(W')=W$, hence the potentials $W$ and $W'$ are right\nequivalent.\n\\end{example}\n\n\\subsection{Hyperpotentials}\nThe following extension of the notion of a potential, introduced \nin~\\cite{Ladkani14a}, allows to prove that certain algebras defined over\nground fields of positive characteristic are 2-CY-tilted. This will\nbe particularly important when considering blocks of group algebras.\n\n\\begin{definition}[\\cite{Ladkani14a}]\nA \\emph{hyperpotential} on $Q$ is an element in $\\operatorname{HH}_1(\\widehat{KQ})$.\nIn explicit terms, it is a collection of elements\n$(\\rho_\\alpha)_{\\alpha \\in Q_1}$ in $\\widehat{KQ}$ indexed by the arrows of $Q$ satisfying the\nfollowing conditions:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item\nIf $\\alpha \\colon i \\to j$ then $\\rho_\\alpha \\in e_j \\widehat{KQ} e_i$.\nIn other words, $\\rho_\\alpha$ is a (possibly infinite) linear combination\nof paths starting at $j$ and ending at $i$.\n\n\\item \\label{it:comm}\n$\\sum_{\\alpha \\in Q_1} \\alpha \\rho_\\alpha =\n\\sum_{\\alpha \\in Q_1} \\rho_\\alpha \\alpha$ in $\\widehat{KQ}$.\n\\end{enumerate}\n\nThe \\emph{Jacobian algebra} of $(\\rho_\\alpha)_{\\alpha \\in Q_1}$ is the\nquotient of $\\widehat{KQ}$ by the closure of the ideal generated by\nthe elements $\\rho_\\alpha$,\n\\[\n\\mathcal{P}(Q, (\\rho_\\alpha)_{\\alpha \\in Q_1}) = \\widehat{KQ}\/\n\\overline{(\\rho_\\alpha : \\alpha \\in Q_1)} .\n\\]\n\\end{definition}\n\nAny potential $W$ gives rise to a hyperpotential by taking its cyclic\nderivatives $(\\partial_\\alpha W)_{\\alpha \\in Q_1}$. This is essentially\nConnes' map $B$ from $\\operatorname{HC}_0(\\widehat{KQ})$ to $\\operatorname{HH}_1(\\widehat{KQ})$.\nConversely, when $\\operatorname{char} K = 0$, any hyperpotential arises in this way, see\nthe discussion at the end of~\\cite[\\S6.1]{Keller11}.\n\nIt is possible to define a Ginzburg dg-algebra for a hyperpotential and\nfollow Keller's proof to show that it has the same homological properties\nas in the case of potentials, see~\\cite{Ladkani14a}.\nTherefore Amiot's construction applies and we deduce the following.\n\n\\begin{proposition} \\label{p:hyperCY}\nAny finite-dimensional Jacobian algebra of a quiver with hyperpotential is\n2-CY-tilted.\n\\end{proposition}\n\n\\begin{example} \\label{ex:KxnCY}\nConsider the algebra $A=K[x]\/(x^n)$ of Example~\\ref{ex:Kxn}\nover a field $K$ with characteristic $p \\geq 0$, and\nconsider the quiver $Q$ consisting of one vertex and one loop, denoted $x$,\nat that vertex.\nIf $p$ does not divide $n+1$, then for any $c \\in K^{\\times}$, the algebra\n$A$ is the Jacobian algebra of the potential $W=cx^{n+1}$ on $Q$.\nHowever, if $p$ divides $n+1$, then $A$ is not a Jacobian algebra of a\npotential on $Q$.\nNevertheless, the sequence consisting of the single element $x^n$ is always\na hyperpotential on $Q$,\nhence $A$ is 2-CY-tilted regardless of the characteristic of $K$.\n\\end{example}\n\n\\subsection{Periodicity}\n\nA large class of symmetric 2-CY-tilted algebras has been constructed by\nBurban, Iyama, Keller and Reiten~\\cite{BIKR08}. In their construction, the\nambient 2-Calabi-Yau triangulated categories are the stable categories of\nmaximal Cohen-Macaulay\nmodules over odd dimensional isolated hypersurface singularities.\nThese categories are also $0$-Calabi-Yau since the square of the suspension\nfunctor is isomorphic to the identity. Therefore, the endomorphism algebra of\nany object is symmetric (cf.\\ Proposition~\\ref{p:symmetric}).\n\nThe next proposition provides a partial converse. We start with one\ncluster-tilting object in a 2-Calabi-Yau category $\\mathcal{C}$ whose endomorphism\nalgebra $\\Lambda$ is symmetric and study the implications this has on the structure\nof $\\mathcal{C}$ and $\\stmod \\Lambda$.\n\n\\begin{proposition} \\label{p:period}\nLet $\\Lambda$ be a finite-dimensional symmetric algebra that is also\n2-CY-tilted, i.e.\\ $\\Lambda = \\operatorname{End}_{\\mathcal{C}}(T)$ for some cluster-tilting object $T$\nwithin a triangulated 2-Calabi-Yau category $\\mathcal{C}$ with suspension functor\n$\\Sigma$.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:omega4}\nThe functor $\\Omega^4$ on the stable module category $\\stmod \\Lambda$\nis isomorphic to the identity, hence all non-projective\n$\\Lambda$-modules are $\\Omega$-periodic with period dividing $4$.\n\n\\item \\label{it:p:sigma2}\nThe functor $\\Sigma^2$ acts as the identity on the objects of $\\mathcal{C}$.\n\n\\item \\label{it:p:nonrigid}\nAssume that $\\Lambda$ is a Jacobian algebra of a hyperpotential.\nThen this hyperpotential is rigid if and only if $\\Lambda$ is semi-simple.\n\\end{enumerate}\n\\end{proposition}\n\nFor part~\\eqref{it:p:nonrigid}, note that rigid quivers with potentials have\nbeen defined in~\\cite[Definitions~3.4 and~6.10]{DWZ08} in terms of vanishing\nof the deformation space of their Jacobian algebras. This definition carries\nover without any modification to hyperpotentials. In particular,\na hyperpotential with finite-dimensional Jacobian algebra $\\Lambda$ is\n\\emph{rigid} if and only if $\\operatorname{HH}_0(\\Lambda) = \\Lambda\/[\\Lambda,\\Lambda]$ is spanned by the\nimages of the primitive idempotents corresponding to the vertices.\n\nLet us give the short proof of part~\\eqref{it:p:omega4}. We note that\nparts~\\eqref{it:p:omega4} and~\\eqref{it:p:sigma2} of the proposition have\nalso been recently observed by Valdivieso-Diaz~\\cite{Valdivieso13}.\n\n\\begin{proof}[Proof of part~(a)]\nOn the one hand, $\\Lambda$ is symmetric, hence $\\stmod \\Lambda$ is $(-1)$-Calabi-Yau\n(Remark~\\ref{rem:CYminus1}). On the other hand, $\\Lambda$ is $2$-CY-tilted, hence\n$\\stmod \\Lambda$ is $3$-Calabi-Yau (Prop.~\\ref{p:2CYtilted}). The uniqueness of\nthe Serre functor implies that the fourth power of the suspension on\n$\\stmod \\Lambda$ is isomorphic to the identity functor, and since the suspension\nis $\\Omega^{-1}$, we get the result.\n\\end{proof}\n\n\\begin{example}\nLet $n \\geq 1$ and consider the algebra $A = K[x]\/(x^n)$. It is symmetric\nand 2-CY-tilted (Example~\\ref{ex:KxnCY}). By Proposition~\\ref{p:period},\n$\\Omega^4_A M \\simeq M$ for any $M \\in \\stmod A$.\nIndeed, in this case even $\\Omega^2_A M \\simeq M$, see Example~\\ref{ex:Kxn}.\n\\end{example}\n\nAs a direct consequence of Proposition~\\ref{p:period}\\eqref{it:p:omega4} and\nDefinition~\\ref{def:quasi}, we obtain the next statement.\n\n\\begin{corollary}\nAn indecomposable, symmetric, 2-CY-tilted algebra of tame representation type\nis of quasi-quaternion type.\n\\end{corollary}\n\n\\subsection{Derived equivalences}\n\\label{ssec:dereq}\n\nIn this section all cluster-tilting objects are assumed to be\n\\emph{basic}, i.e.\\ they decompose into a direct sum of\nnon-isomorphic indecomposable objects.\nIyama and Yoshino~\\cite{IyamaYoshino08} have shown that there is\na well-defined notion of mutation of (basic) cluster-tilting objects in a\ntriangulated 2-Calabi-Yau category $\\mathcal{C}$.\n\n\\begin{proposition}[\\protect{\\cite[Theorem~5.3]{IyamaYoshino08}}]\nLet $T$ be a cluster-tilting object in $\\mathcal{C}$, let $X$ be an \nindecomposable summand of $T$ and write $T = \\bar{T} \\oplus X$.\nThen there exists a unique indecomposable object $X'$ of $\\mathcal{C}$ which\nis not isomorphic to $X$ such that $T' = \\bar{T} \\oplus X'$ is \na cluster-tilting object in $\\mathcal{C}$.\n\\end{proposition}\n\nThe cluster-tilting object $T'$ in the proposition is called the\n\\emph{Iyama-Yoshino mutation} of $T$ at $X$. The algebras\n$\\Lambda=\\operatorname{End}_{\\mathcal{C}}(T)$ and $\\Lambda'=\\operatorname{End}_{\\mathcal{C}}(T')$ are said to be\n\\emph{neighboring} 2-CY-tilted algebras.\n\nLet $(Q,W)$ be a quiver with potential and let $k$ be a vertex in $Q$\nsuch that no 2-cycle (i.e.\\ a cycle of length 2) passes through $k$.\nDerksen, Weyman and Zelevinsky have defined in~\\cite[\\S5]{DWZ08}\nthe \\emph{mutation} of $(Q,W)$ at $k$, which is a quiver with\npotential denoted $\\mu_k(Q,W)$.\nBuan, Iyama, Reiten and Smith have shown in~\\cite{BIRS11} that \nunder some mild conditions the notions of Iyama-Yoshino\nmutation and mutation of quivers with potentials are compatible.\nThis is expressed in the next proposition.\n\\begin{proposition}[\\protect{\\cite[Theorem~5.2]{BIRS11}}] \\label{p:QPmutIY}\nLet $T$ be a cluster-tilting object in $\\mathcal{C}$. Assume that\n$\\operatorname{End}_{\\mathcal{C}}(T) \\simeq \\mathcal{P}(Q,W)$ for some quiver with potential $(Q,W)$\nand that $\\operatorname{End}_{\\mathcal{C}}(T)$ satisfies the vanishing condition.\nLet $k$ be a vertex of $Q$ such that no $2$-cycle passes through $k$,\nlet $X$ be the corresponding indecomposable summand of $T$ and let\n$T'$ be the Iyama-Yoshino mutation of $T$ at $X$. Then\n$\\operatorname{End}_{\\mathcal{C}}(T') \\simeq \\mathcal{P}(\\mu_k(Q,W))$.\n\\end{proposition}\nFor the precise formulation of the vanishing condition we refer the\nreader to~\\cite{BIRS11}, but for our purposes it is sufficient to note\nthat this condition holds when the algebra $\\operatorname{End}_{\\mathcal{C}}(T)$ is\nself-injective, and in particular when it is symmetric.\n\nNeighboring 2-CY-tilted algebras are nearly Morita equivalent in the\nsense of Ringel~\\cite{Ringel07}, that is, there is an equivalence of categories\n\\[\n\\operatorname{mod} \\Lambda\/\\operatorname{add} S \\simeq \\operatorname{mod} \\Lambda'\/\\operatorname{add} S'\n\\]\nwhere $S$ (respectively, $S'$) is the simple module which is the top of the\nindecomposable projective $\\Lambda$-module (respectively, $\\Lambda'$-module)\ncorresponding to the summand $X$ of $T$ (respectively, $X'$ of $T'$),\nprovided there are ``no loops'', i.e.\\ any non-isomorphism $X \\to X$\n(or $X' \\to X'$) factors through $\\operatorname{add} \\bar{T}$,\nsee~\\cite[Proposition~2.2]{KellerReiten07}.\nHowever, neighboring 2-CY-tilted algebras are not necessarily derived\nequivalent, see for example~\\cite[Example~5.2]{Ladkani10}.\n\nThe next statement concerns the derived equivalence of neighboring\n2-CY-tilted algebras. \nIt is an improvement of~\\cite[Theorem~5.3]{Ladkani10} which has\nturned out to be a very useful tool in derived equivalence\nclassifications of various cluster-tilted algebras and Jacobian\nalgebras~\\cite{BHL13,BHL14,Ladkani11a}.\nThe derived equivalences are instances of\n(refined version of) good mutations introduced in our previous\nwork~\\cite{Ladkani10}.\nBefore formulating the result, we recall some relevant notions.\n\nLet $\\Lambda$ be a basic algebra and $P$ an indecomposable projective\n$\\Lambda$-module and write $\\Lambda = P \\oplus Q$. Consider the silting\nmutations in the sense of Aihara and Iyama~\\cite{AiharaIyama12}\nof $\\Lambda$ at $P$ within the triangulated category $\\operatorname{per} \\Lambda$\nof perfect complexes, which are the following two-term complexes\n\\begin{align} \\label{e:silt}\nU^-_P(\\Lambda) = (P \\to Q') \\oplus Q &,&\nU^+_P(\\Lambda) = (Q'' \\to P) \\oplus Q ,\n\\end{align}\nwhere $Q', Q'' \\in \\operatorname{add} Q$, the maps are left (resp., right)\n$(\\operatorname{add} Q)$-approximations and $Q, Q', Q''$ are in degree 0.\nThese two-term complexes of projective modules are known also as\nOkuyama-Rickard complexes. In~\\cite{Ladkani10} we considered these\ncomplexes in relation with our definition of mutations of algebras.\n\nAn algebra is \\emph{weakly symmetric} if for any simple module, its\nprojective cover is isomorphic to its injective envelope. Symmetric\nalgebras are weakly symmetric and if $\\Lambda$ is weakly symmetric, then\nthe complexes $U^-_P(\\Lambda)$ and $U_P^+(\\Lambda)$ are tilting complexes.\n\n\\begin{proposition} \\label{p:dereq}\nLet $T$ be a cluster-tilting object in a triangulated 2-Calabi-Yau category\n$\\mathcal{C}$, let $X$ be an indecomposable summand of $T$ and let $T'$ be the\nIyama-Yoshino mutation of $T$ at $X$.\nConsider the algebras $\\Lambda = \\operatorname{End}_{\\mathcal{C}}(T)$ and $\\Lambda'= \\operatorname{End}_{\\mathcal{C}}(T')$.\nLet $P$ be the indecomposable projective $\\Lambda$-module corresponding to $X$ and\nlet $P'$ be the indecomposable projective $\\Lambda'$-module corresponding to $X'$.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nIf $U^-_P(\\Lambda)$ and $U^+_{P'}(\\Lambda')$ are tilting complexes\n(over $\\Lambda$ and $\\Lambda'$, respectively), then\n\\[\n\\operatorname{End}_{\\cD^b(\\gL)} U^-_P(\\Lambda) \\simeq \\Lambda' \\qquad \\text{and} \\qquad\n\\operatorname{End}_{\\mathcal{D}^b(\\Lambda')} U^+_{P'}(\\Lambda') \\simeq \\Lambda .\n\\]\n\n\\item\nIf $U^+_P(\\Lambda)$ and $U^-_{P'}(\\Lambda')$ are tilting complexes\n(over $\\Lambda$ and $\\Lambda'$, respectively), then\n\\[\n\\operatorname{End}_{\\cD^b(\\gL)} U^+_P(\\Lambda) \\simeq \\Lambda' \\qquad \\text{and} \\qquad\n\\operatorname{End}_{\\mathcal{D}^b(\\Lambda')} U^-_{P'}(\\Lambda') \\simeq \\Lambda .\n\\]\n\n\\item \\label{it:algmut}\nIf $\\Lambda$ is weakly symmetric, then $\\Lambda'$ is also weakly symmetric\nby~\\cite[\\S4.2]{HerschendIyama11}, hence all\nthe complexes $U^-_P(\\Lambda)$, $U^+_P(\\Lambda)$, $U^-_{P'}(\\Lambda')$\nand $U^+_{P'}(\\Lambda')$ are tilting complexes and\n\\[\n\\operatorname{End}_{\\cD^b(\\gL)} U^-_P(\\Lambda) \\simeq \\Lambda' \\simeq\n\\operatorname{End}_{\\cD^b(\\gL)} U^+_P(\\Lambda) .\n\\]\nIn particular, $\\Lambda$ and $\\Lambda'$ are derived equivalent.\n\n\\item\nIf $\\Lambda$ is symmetric then $\\Lambda'$ is symmetric.\n\\end{enumerate}\n\\end{proposition}\nWe note that there are related works by Dugas~\\cite{Dugas15}\nconcerning derived equivalences of symmetric algebras and by\nMizuno~\\cite{Mizuno15} concerning derived equivalences of\nself-injective quivers with potential.\n\nAs the category of perfect complexes over a symmetric algebra is 0-Calabi-Yau,\nthe derived equivalences in part~\\eqref{it:algmut} can be\nconsidered as 0-CY analogs of the derived equivalences of\nIyama-Reiten~\\cite{IyamaReiten08} and\nKeller-Yang~\\cite[Theorem~6.2]{KellerYang11}\nfor 3-CY-algebras.\n\n\\begin{definition}\nLet $T$ be a cluster-tilting object in a triangulated 2-Calabi-Yau\ncategory $\\mathcal{C}$.\nA cluster-tilting object $T'$ in $\\mathcal{C}$ is \\emph{reachable} from $T$\nif it can be obtained from $T$ by finitely many Iyama-Yoshino mutations at\nindecomposable summands.\n\\end{definition}\n\n\\begin{corollary} \\label{c:dereq}\nLet $T$ be a cluster-tilting object in a triangulated 2-Calabi-Yau \ncategory $\\mathcal{C}$ and assume that $\\Lambda=\\operatorname{End}_{\\mathcal{C}}(T)$ is (weakly) symmetric.\nThen for any cluster-tilting object $T'$ in $\\mathcal{C}$ that is reachable from $T$,\nthe algebra $\\Lambda'=\\operatorname{End}_{\\mathcal{C}}(T')$ is (weakly) symmetric and derived\nequivalent to $\\Lambda$.\n\\end{corollary}\n\n\\begin{remark}\nThere are examples of triangulated 2-Calabi-Yau categories $\\mathcal{C}$ with a\ncluster-tilting object $T$ such that $\\Sigma T$ is not reachable from\n$T$, see~\\cite[\\S3]{Ladkani13} and~\\cite[Example~4.3]{Plamondon13}.\nInterestingly, in all of these examples the algebra $\\operatorname{End}_{\\mathcal{C}}(T)$ is\nsymmetric. Note, however, that $\\operatorname{End}_{\\mathcal{C}}(\\Sigma T) \\simeq \\operatorname{End}_{\\mathcal{C}}(T)$ and\nin particular these algebras are derived equivalent.\n\\end{remark}\n\nWe can rephrase part~\\eqref{it:algmut} of Proposition~\\ref{p:dereq} as\nfollows.\n\\begin{corollary} \\label{c:sym2CYmut}\nLet $\\Lambda$ be a weakly symmetric 2-CY-tilted algebra and let $P$ be\nan indecomposable projective $\\Lambda$-module.\nThen the two algebras $\\operatorname{End}_{\\cD^b(\\gL)} U^-_P(\\Lambda)$\nand $\\operatorname{End}_{\\cD^b(\\gL)} U^+_P(\\Lambda)$ are isomorphic,\n2-CY-tilted and derived equivalent to~$\\Lambda$.\n\\end{corollary}\n\nWe see that derived equivalences of a particular kind preserve the property\nof an algebra being symmetric 2-CY-tilted. One may ask whether this is still\ntrue for arbitrary derived equivalences.\n\n\\begin{question} \\label{q:sym2CYder}\nLet $\\Lambda$ be a symmetric 2-CY-tilted algebra and let $\\Lambda'$ be an\nalgebra derived equivalent to $\\Lambda$. Is $\\Lambda'$ also 2-CY-tilted?\n\\end{question}\n\nOne may also ask if a converse to Proposition~\\ref{p:period}\\eqref{it:p:omega4}\nholds.\n\n\\begin{question} \\label{q:sym42CY}\nLet $\\Lambda$ be a symmetric algebra such that $\\Omega_\\Lambda^4 M \\simeq M$ for\nany $M \\in \\stmod \\Lambda$. Is $\\Lambda$ then 2-CY-tilted?\n\\end{question}\n\nObserve that by Proposition~\\ref{p:derquasi} and Proposition~\\ref{p:period},\nan affirmative answer to Question~\\ref{q:sym42CY} will yield\nan affirmative answer to Question~\\ref{q:sym2CYder}.\nWe note that the answer to Question~\\ref{q:sym42CY}\n(and hence Question~\\ref{q:sym2CYder}) is positive in the following cases:\n$\\Lambda$ is of finite representation type (Theorem~\\ref{t:sym2CYfin});\n$\\Lambda$ is tame with non-singular Cartan matrix (Theorem~\\ref{t:quat2CY}); or\n$\\Lambda$ is tame of polynomial growth (Proposition~\\ref{p:sympoly}).\n\n\\section{Ribbon quivers and triangulation quivers}\n\\label{sec:quivers}\n\nIn this section we develop a theory of ribbon quivers and ribbon\ngraphs, with an emphasis on a particular class of ribbon quivers called\ntriangulation quivers. The connections to ideal triangulations of marked\nsurfaces and dimer models will be explained in Section~\\ref{sec:surface}.\nRibbon quivers and triangulation quivers are the combinatorial ingredients\nunderlying the definition of Brauer graph algebras and triangulation algebras\nwhich will be introduced in Section~\\ref{sec:algebras} and studied later in\nthis survey.\nThe combinatorial statements in this section will be stated without proofs,\nand the details will appear elsewhere.\n\n\\subsection{Ribbon quivers}\n\\label{ssec:ribbon}\n\nRecall from Section~\\ref{ssec:QP} that a quiver $Q$ is quadruple\n$Q=(Q_0,Q_1,s,t)$ where $Q_0,Q_1$ are finite sets and\n$s,t \\colon Q_1 \\to Q_0$.\n\n\\begin{definition} \\label{def:ribbon}\nA \\emph{ribbon quiver} is a pair $(Q,f)$ consisting of a quiver\n$Q$ and a permutation $f \\colon Q_1 \\to Q_1$ on its set of arrows\nsatisfying the following conditions:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item \\label{it:ribbon:deg2}\nAt each vertex $i \\in Q_0$ there are exactly two arrows starting at $i$ and\ntwo arrows ending at $i$;\n\n\\item \\label{it:ribbon:f}\nFor each arrow $\\alpha \\in Q_1$, the arrow $f(\\alpha)$ starts where $\\alpha$\nends.\n\\end{enumerate}\nNote that loops are allowed in $Q$. A loop at a vertex is counted both\nas an incoming and outgoing arrow at that vertex.\n\\end{definition}\n\n\\begin{example} \\label{ex:ribbon1}\nConsider a ribbon quiver $(Q,f)$ with one vertex.\nCondition~\\eqref{it:ribbon:deg2} implies that $Q$ must have two loops\nas in the following picture\n\\[\n\\xymatrix@=1pc{\n{\\bullet} \\ar@(dl,ul)[]^{\\alpha} \\ar@(dr,ur)[]_{\\beta}\n}\n\\]\nand condition~\\eqref{it:ribbon:f} is empty in this case, so that $f$ equals\none of the two permutations $f_1$ or $f_2$ on $Q_1$ given in cycle form \nby $f_1 = (\\alpha)(\\beta)$ and $f_2 = (\\alpha \\, \\beta)$.\nIn particular we see that the underlying quiver does not determine the\nribbon quiver structure.\n\\end{example}\n\nLet $(Q,f)$ be a ribbon quiver.\nSince at each vertex of $Q$ there are exactly two outgoing arrows,\nthere is an involution $\\alpha \\mapsto \\bar{\\alpha}$ on $Q_1$ mapping\neach arrow $\\alpha$ to the other arrow starting at the vertex $s(\\alpha)$.\nComposing it with $f$ gives rise to the permutation\n$g \\colon Q_1 \\to Q_1$ given by $g(\\alpha) = \\overline{f(\\alpha)}$\nso that for each arrow $\\alpha$,\nthe set $\\{f(\\alpha),g(\\alpha)\\}$ consists of\nthe two arrows starting at the vertex which $\\alpha$ ends at.\n\nDenote by $Q_1^f$ and $Q_1^g$ the subsets of arrows fixed by $f$ and $g$,\nrespectively, i.e.\\ $Q_1^f = \\{ \\alpha \\in Q_1 : f(\\alpha)=\\alpha \\}$ and\n$Q_1^g = \\{ \\alpha \\in Q_1 : g(\\alpha)=\\alpha \\}$. The set of loops in $Q$\nthus decomposes as a disjoint union $Q_1^f \\cup Q_1^g$.\n\nGiven a quiver $Q$ satisfying condition~\\eqref{it:ribbon:deg2} in the\ndefinition, the data of the permutation $f$ is equivalent to the data of the\npermutation $g$. Thus from now on when considering a ribbon quiver $(Q,f)$\nwe will freely refer to the involution $\\alpha \\mapsto \\bar{\\alpha}$ and\nthe permutation $g$ as defined above.\n\n\\begin{lemma} \\label{l:gf2fg2}\nLet $\\alpha \\in Q_1$. Then $f^{-1}(\\alpha)=g^{-1}(\\bar{\\alpha})$\nand $gf^{-2}(\\alpha) = fg^{-2}(\\bar{\\alpha})$.\n\\end{lemma}\n\n\\begin{definition}\nLet $(Q,f)$ be a ribbon quiver and define $g \\colon Q_1 \\to Q_1$ by\n$g(\\alpha) = \\overline{f(\\alpha)}$.\nThe \\emph{dual} of $(Q,f)$ is the ribbon quiver $(Q,g)$.\n\\end{definition}\n\n\\begin{example}\nIn Example~\\ref{ex:ribbon1}, $\\bar{\\alpha} = \\beta$ and $\\bar{\\beta} = \\alpha$,\nso in cycle form $g_1 = (\\alpha \\, \\beta) = f_2$ and\n$g_2 = (\\alpha)(\\beta) = f_1$.\nHence $(Q,f_1)$ and $(Q,f_2)$ are dual to each other.\n\\end{example}\n\n\\begin{definition}\nLet $(Q,f)$ and $(Q',f')$ be ribbon quivers with $Q=(Q_0, Q_1, s, t)$,\n$Q'=(Q'_0, Q'_1, s', t')$. Recall that a pair of bijections\n$\\varphi_0 \\colon Q_0 \\xrightarrow{\\sim} Q'_0$ and\n$\\varphi_1 \\colon Q_1 \\xrightarrow{\\sim} Q'_1$ is an \\emph{isomorphism} between\nthe quivers $Q$ and $Q'$ if $\\varphi_0 s = s' \\varphi_1$ and\n$\\varphi_0 t = t' \\varphi_1$.\nIf, in addition, $\\varphi_1 f = f' \\varphi_1$ and $\\varphi_1(\\bar{\\alpha})\n= \\overline{\\varphi_1(\\alpha)}$ for any $\\alpha \\in Q_1$, \nwe say that $(\\varphi_0, \\varphi_1)$ is \\emph{isomorphism} between the ribbon\nquivers $(Q,f)$ and $(Q',f')$.\n\\end{definition}\n\nRibbon quivers are closely related to ribbon graphs.\nTo avoid confusion, we shall use the term ``node'' for the graph\nin order to distinguish it from a vertex in the quiver.\nInformally speaking, a ribbon graph is a graph consisting of nodes and\nedges together with a cyclic ordering of the edges around each node.\nThis can be made more formal in the next definition.\n\n\\begin{definition}\nA \\emph{ribbon graph} is a triple $(H, \\iota, \\sigma)$ where $H$ is a finite\nset, $\\iota$ is an involution on $H$ without fixed points and\n$\\sigma$ is a permutation on $H$.\n\\end{definition}\n\nThe elements of $H$ are called \\emph{half-edges}.\nA ribbon graph gives rise to a graph $(V,E)$ (possibly with loops and\nmultiple edges between nodes) as follows.\nThe set $V$ of nodes consists of the cycles of $\\sigma$ and\nthe set $E$ of edges consists of the cycles of $\\iota$.\nAn edge $e \\in E$ can be written as $(h \\, \\iota(h))$ for some $h \\in H$.\nThe $\\sigma$-cycles that $h$ and $\\iota(h)$ belong to are the nodes that\n$e$ is incident to.\nMoreover, $\\sigma$ induces a cyclic ordering of the edges around each node.\n\nConversely, given a graph $(V,E)$ with a cyclic ordering of the edges around\neach node, we think of each edge $e \\in E$ incident to the nodes $v', v'' \\in V$\n(which may coincide) as composed of two half-edges $e'$ and $e''$, with $e'$\nincident to $v'$ and $e''$ incident to $v''$. This yields a ribbon graph\n$(H,\\iota,\\sigma)$ where $H$ is the set of all half-edges,\n$\\iota = \\prod_{e \\in E} (e' \\, e'')$ is the product of all the transpositions\n$(e' \\, e'')$ for $e \\in E$, and\nfor any half-edge $h$ incident to a node $v$, the half-edge $\\sigma(h)$\nis the one following $h$ in the cyclic order around $v$.\n\n\\begin{example} \\label{ex:ribbon1g}\nConsider a ribbon graph with one edge. In this case the set $H$ of half-edges\nconsists of two elements, which we denote by $\\alpha$ and $\\beta$, and the\ninvolution $\\iota$ can be written as $\\iota = (\\alpha \\, \\beta)$ in cycle form.\nThe permutation $\\sigma$ equals one of the two permutations $\\sigma_1$ or\n$\\sigma_2$ given in cycle form by $\\sigma_1 = (\\alpha \\, \\beta)$ and\n$\\sigma_2 = (\\alpha)(\\beta)$.\n\nThe corresponding graphs, with their half-edges labeled, are shown in the\npicture below. Since $\\sigma_1$ has one cycle, the graph of\n$(H,\\iota,\\sigma_1)$, shown to the left, has one node.\nSimilarly, since $\\sigma_2$ has two cycles, the graph of $(H,\\iota,\\sigma_2)$,\nshown to the right, has two nodes.\n\\begin{align*}\n\\begin{array}{cc}\n\\xymatrix{{\\circ} \\ar@{-}@(ul,dl)_(0.15)\\alpha_(0.85)\\beta} &\n\\sigma_1 = (\\alpha \\, \\beta)\n\\end{array}\n&& &&\n\\begin{array}{cc}\n\\xymatrix{{\\circ} \\ar@{-}[r]^(0.25)\\alpha_(0.75)\\beta & {\\circ}} &\n\\sigma_2 = (\\alpha)(\\beta)\n\\end{array}\n\\end{align*}\n\\end{example}\n\n\\begin{definition}\nLet $(H,\\iota,\\sigma)$ and $(H',\\iota',\\sigma')$ be ribbon graphs.\nAn \\emph{isomorphism} between $H$ and $H'$ is a bijection\n$\\varphi \\colon H \\to H'$ satisfying $\\iota' \\varphi = \\varphi \\iota$\nand $\\sigma' \\varphi = \\varphi \\sigma$.\n\\end{definition}\n\nAny ribbon quiver $(Q,f)$ gives rise to a ribbon graph\n$(H,\\iota,\\sigma)$ by taking $H=Q_1$ and defining $\\iota(\\alpha)=\\bar{\\alpha}$\nand $\\sigma(\\alpha) = \\overline{f(\\alpha)}$ for each $\\alpha \\in Q_1$.\n\nConversely, a ribbon graph $(H,\\iota,\\sigma)$ gives rise to a ribbon quiver\n$(Q,f)$ as follows.\nSet $Q_1=H$ and take $Q_0$ to be the set of cycles of $\\iota$. Define\nthe maps $s,t \\colon Q_1 \\to Q_0$ and the permutation $f \\colon Q_1 \\to Q_1$\nby letting, for each $h \\in H$, $s(h)$ to be the $\\iota$-cycle that $h$ belongs\nto and setting $t = s \\sigma$ and $f = \\iota \\sigma$.\n\nNote that these two constructions are inverses of each other, hence we deduce\nthe following.\n\n\\begin{proposition} \\label{p:ribbon}\nThere is a bijection between the set of\nisomorphism classes of ribbon quivers and the set of isomorphism classes\nof ribbon graphs,\n\\[\n\\left(\n\\{\\text{ribbon quivers}\\}\/\\simeq\n\\right) \\longleftrightarrow\n\\left(\n\\{\\text{ribbon graphs} \\}\/\\simeq\n\\right) .\n\\]\n\\end{proposition}\n\nUnder this bijection, the various notions concerning ribbon quivers and\nribbon graphs are related as in the dictionary given in Table~\\ref{tab:ribbon}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{ccc}\n$(Q,f)$ & $(H,\\iota,\\sigma)$ & $(V,E)$ \\\\ \\hline\nvertex & cycle of $\\iota$ & edge \\\\\narrow & element of $H$ & half-edge \\\\\n$f$ & $\\iota \\sigma$ \\\\\n$g$ & $\\sigma$ & cyclic ordering \\\\\ncycle of $g$ & cycle of $\\sigma$ & node\n\\end{tabular}\n\\end{center}\n\\caption{Dictionary between ribbon quivers and ribbon graphs.}\n\\label{tab:ribbon}\n\\end{table}\n\n\\begin{example} \\label{ex:ribbonG1}\nWe illustrate the bijection between the ribbon quivers with one vertex\ndiscussed in Example~\\ref{ex:ribbon1} and the ribbon graphs with one edge\ndiscussed in Example~\\ref{ex:ribbon1g}.\nWe denote the set of half-edges by $\\{\\alpha, \\beta\\}$ and let\n$\\iota = (\\alpha \\, \\beta)$. The underlying quiver $Q$ is always\n\\[\n\\xymatrix@=1pc{\n{\\bullet} \\ar@(dl,ul)[]^{\\alpha} \\ar@(dr,ur)[]_{\\beta}\n}\n\\]\nand the corresponding graphs are shown in the right column below.\n\\[\n\\begin{array}{lclcc}\nf_1 = (\\alpha)(\\beta) & &\n\\sigma_1 = g_1 = (\\alpha \\, \\beta) & &\n\\xymatrix{{\\circ} \\ar@{-}@(ul,dl)}\n\\\\[3pt]\nf_2 = (\\alpha \\, \\beta) & &\n\\sigma_2 = g_2 = (\\alpha)(\\beta) & &\n\\xymatrix{{\\circ} \\ar@{-}[r] & {\\circ}}\n\\end{array}\n\\]\n\\end{example}\n\n\\medskip\n\nThe data of a graph can be encoded in matrix form in the following way.\nLet $(V,E)$ be a graph. For a node $v \\in V$, define a vector\n$\\chi_v \\in \\mathbb{Z}^E$ by\n\\[\n\\chi_v(e) = \\begin{cases}\n2 & \\text{$e$ is a loop incident to $v$,} \\\\\n1 & \\text{$e$ is incident to $v$ but is not a loop,} \\\\\n0 & \\text{$e$ is not incident to $v$}\n\\end{cases}\n\\]\nand think of it as a row vector. Obviously, $\\chi_v(e) \\geq 0$ and\n$\\sum_{v \\in V} \\chi_v(e) = 2$ for any $e \\in E$, so by arranging the vectors\n$\\chi_v$ as a $V \\times E$ matrix, one gets an integer matrix with non-negative\nentries whose sum of rows equals the constant vector $(2,2,\\dots,2)$.\nConversely, any such matrix $\\chi$ gives rise to a graph whose nodes are\nindexed by the rows of $\\chi$, its edges are indexed by the columns of $\\chi$\nand the incidence relations are read from the entries $\\chi_v(e)$.\n\nNow let $(Q,f)$ be a ribbon quiver. In the underlying graph $(V,E)$ of the\nribbon graph corresponding to $(Q,f)$ under the bijection of\nProposition~\\ref{p:ribbon}, the set $V$ corresponds to the set $\\Omega_g$ of\nthe cycles of the permutation $g$, the set $E$ corresponds to the set $Q_0$\nof vertices of $Q$ and the entries of the matrix $\\chi$ are given by\n$\\chi_{\\omega}(i) = |\\{\\alpha \\in \\omega : s(\\alpha)=i\\}|$\nfor any $g$-cycle $\\omega \\in \\Omega_g$ and vertex $i \\in Q_0$.\n\n\n\\subsection{Triangulation quivers}\n\n\\begin{definition}\nA \\emph{triangulation quiver} is a ribbon quiver $(Q,f)$ such that\n$f^3$ is the identity on the set of arrows.\n\\end{definition}\n\n\\begin{example} \\label{ex:triang1}\nConsidering the ribbon quivers with one vertex of Example~\\ref{ex:ribbon1},\nwe see that $(Q,f_1)$ is a triangulation quiver whereas $(Q,f_2)$ is not.\n\\end{example}\n\n\\begin{remark}\nGiven any quiver $Q$ satisfying condition~\\eqref{it:ribbon:deg2} of\nDefinition~\\ref{def:ribbon}, there is always at least one \n(and in general, many) permutation(s) $f$ on\nthe arrows making $(Q,f)$ a ribbon quiver. Indeed, for each $i \\in Q_0$ \nlabel by $\\alpha, \\beta$ the arrows ending at $i$ and by $\\gamma, \\delta$\nthe arrows starting at $i$ and set, for instance,\n$f(\\alpha)=\\gamma$ and $f(\\beta)=\\delta$.\n\nHowever, as the next example demonstrates, there may not exist\na permutation~$f$ making $(Q,f)$ a triangulation quiver.\nIn other words, the existence of a triangulation\nquiver $(Q,f)$ imposes some restrictions on the shape of a quiver $Q$.\n\\end{remark}\n\n\\begin{example}\nUp to isomorphism, there are two ribbon quivers whose underlying quiver is the\none given below,\n\\[\n\\xymatrix{\n{\\bullet_1} \\ar@<1.5ex>[r] \\ar@<-0.5ex>[r] &\n{\\bullet_2} \\ar@<1.5ex>[l] \\ar@<-0.5ex>[l]\n}\n\\]\nNamely, denoting the arrows from $1$ to $2$ by $\\alpha$, $\\gamma$ and those\nfrom $2$ to $1$ by $\\beta$, $\\delta$, the ribbon quivers are given by the\npermutations\n$(\\alpha \\beta)(\\gamma \\delta)$ and $(\\alpha \\beta \\gamma \\delta)$.\nNone of them is a triangulation quiver.\n\\end{example}\n\nWe have seen that not every quiver satisfying condition~\\eqref{it:ribbon:deg2}\nof Definition~\\ref{def:ribbon} is an underlying quiver of a triangulation quiver.\nThe next proposition tells us that if such triangulation quiver exists,\nthen it is unique up to isomorphism.\n\n\\begin{proposition} \\label{p:triauniq}\nLet $(Q,f)$ and $(Q',f')$ be two triangulation quivers. If the\nquivers $Q$ and $Q'$ are isomorphic, then $(Q,f)$ and $(Q',f')$\nare isomorphic as ribbon quivers.\n\\end{proposition}\n\nSince the number of triangulation quivers with a given number of vertices\nis finite, they can be enumerated on a computer.\nTable~\\ref{tab:quivers} lists (up to isomorphism)\nthe connected triangulation quivers with at most\nthree vertices and their corresponding ribbon graphs.\nNote that the ribbon graph of quiver $2$ could have also been drawn as\n\\[\n\\xymatrix{\n{\\circ} \\ar@{-}[r]^1 & \\circ \\ar@{-}@(ur,dr)[]^{2}\n}\n\\]\nbut the drawing in Table~\\ref{tab:quivers} emphasizes the relation of this\nquiver to the punctured monogon, as we shall see in~Section~\\ref{sec:surface}.\n\n\\begin{table}\n\\[\n\\begin{array}{c|cc}\n& \\text{\\textbf{Triangulation quiver}}\n& \\text{\\textbf{Ribbon graph}}\n\\\\ \\hline & & \\\\\n1 &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n{\\bullet_1} \\ar@(dl,ul)[]^{\\alpha} \\ar@(dr,ur)[]_{\\beta}\n}\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix{\n\\circ \\ar@{-}@(ul,dl)_{1}\n}\n\\end{array}\n\\\\ & (\\alpha) (\\beta)\n\\\\ \\hline\n2 &\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet_1} \\ar@(ul,dl)[]_{\\alpha} \\ar@<-0.5ex>[r]_{\\beta}\n& {\\bullet_2} \\ar@(dr,ur)[]_{\\eta} \\ar@<-0.5ex>[l]_{\\gamma}\n}\n\\\\ \n(\\alpha \\beta \\gamma) (\\eta)\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n{_2} & {\\circ} \\ar@{-}[rr]^1\n& & {\\circ} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n}\n\\end{array}\n\\\\ \\hline\n3a &\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet_1} \\ar@(ul,dl)[]_{\\alpha} \\ar@<-0.5ex>[r]_{\\beta}\n& {\\bullet_2} \\ar@<-0.5ex>[r]_{\\delta} \\ar@<-0.5ex>[l]_{\\gamma}\n& {\\bullet_3} \\ar@(dr,ur)[]_{\\xi} \\ar@<-0.5ex>[l]_{\\eta}\n}\n\\\\\n(\\alpha \\beta \\gamma) (\\delta \\xi \\eta)\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n{_2} & {\\circ} \\ar@{-}[rr]^1\n& & {\\circ} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n\\ar@{-}[rr]^3\n& & {\\circ}\n}\n\\end{array}\n\\\\ \\hline\n3b &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& {\\bullet_3} \\ar@<-0.5ex>[ddl]_{\\alpha_3} \\ar@<-0.5ex>[ddr]_{\\beta_2} \\\\ \\\\\n{\\bullet_1} \\ar@<-0.5ex>[rr]_{\\alpha_1} \\ar@<-0.5ex>[uur]_{\\beta_3}\n&& {\\bullet_2} \\ar@<-0.5ex>[ll]_{\\beta_1} \\ar@<-0.5ex>[uul]_{\\alpha_2}\n}\n\\\\\n(\\alpha_1 \\alpha_2 \\alpha_3) (\\beta_3 \\beta_2 \\beta_1)\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix@=1pc{\n{\\circ} \\ar@{-}[rr]^3 && {\\circ} \\ar@{-}[ddl]^2 \\\\ \\\\\n& {\\circ} \\ar@{-}[uul]^1\n}\n\\end{array}\n\\\\ \\hline\n3' &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& {\\bullet_3} \\ar@(ur,ul)[]_{\\alpha_3} \\ar[ddl]_{\\beta_3} \\\\ \\\\\n{\\bullet_1} \\ar@(ul,dl)[]_{\\alpha_1} \\ar[rr]_{\\beta_1}\n&& {\\bullet_2} \\ar@(dr,ur)[]_{\\alpha_2} \\ar[uul]_{\\beta_2}\n}\n\\\\\n(\\alpha_1) (\\alpha_2) (\\alpha_3) (\\beta_1 \\beta_2 \\beta_3)\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix{\n{\\circ} \\ar@{-}@(ur,ul)[]_3 \\ar@{-}@(l,dl)[]_1 \\ar@{-}@(r,dr)[]^2\n}\n\\end{array}\n\\\\ \\hline\n3'' &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& {\\bullet_3} \\ar@<-0.5ex>[ddl]_{\\alpha_2} \\ar@<0.5ex>[ddl]^(.4){\\alpha_5} \\\\ \\\\\n{\\bullet_1} \\ar@<-0.5ex>[rr]_{\\alpha_0} \\ar@<0.5ex>[rr]^(.4){\\alpha_3} &&\n{\\bullet_2} \\ar@<-0.5ex>[uul]_{\\alpha_1} \\ar@<0.5ex>[uul]^(.4){\\alpha_4}\n}\n\\\\\n(\\alpha_4 \\alpha_2 \\alpha_0) (\\alpha_5 \\alpha_3 \\alpha_1)\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n&& {_2} \\ar@{-}@(r,r)[ddd] \\\\ \\\\ \\\\\n&& {\\circ} \\ar@{-}@(l,l)[uuu] \\ar@{-}@(ul,ul)[ddll] \\ar@{-}@(dl,dl)[ddrr]\n\\\\ \\\\\n{_1} \\ar@{-}@(dr,dr)[uurr]\n&& && {_3} \\ar@{-}@(ur,ur)[uull] \\\\\n{}\n}\n\\end{array}\n\\\\ \\hline\n\\end{array}\n\\]\n\\caption{The connected triangulation quivers with at most 3 vertices.\nWe list the triangulation quivers and the corresponding ribbon graphs,\nwhere we write the permutation~$f$ in cycle form below each quiver.}\n\\label{tab:quivers}\n\\end{table}\n\n\\begin{remark}\nAs the entries in rows $3'$ and $3''$ of Table~\\ref{tab:quivers} demonstrate,\ntwo different ribbon graphs can have the same underlying graph (in this case\na node with three loops).\n\\end{remark}\n\n\\begin{remark}\nIn a triangulation quiver $(Q,f)$,\nthe permutations $\\alpha \\mapsto \\bar{\\alpha}$ and $\\alpha \\mapsto f(\\alpha)$\nare of orders $2$ and $3$, respectively, hence the group\n$\\operatorname{PSL}_2(\\mathbb{Z})$, which is the free product of the cyclic groups\n$\\mathbb{Z}\/2\\mathbb{Z}$ and $\\mathbb{Z}\/3\\mathbb{Z}$, acts on the set of arrows $Q_1$.\nThis action is transitive when $Q$ is connected.\n\\end{remark}\n\nThe dual of a triangulation quiver $(Q,f)$ need not be a triangulation quiver.\nHowever, when it is, then by Proposition~\\ref{p:triauniq}, it must be isomorphic\nto $(Q,f)$, hence $(Q,f)$ is \\emph{self dual}. The next proposition shows that\nthere are only two connected self dual triangulation quivers.\n\n\\begin{proposition}\nA connected triangulation quiver whose dual is also a triangulation quiver\nis isomorphic to one of the two triangulation quivers shown in\nFigure~\\ref{fig:selfdual}.\n\\end{proposition}\n\n\\begin{figure}\n\\[\n\\begin{array}{ccc}\n \\text{\\textbf{Triangulation quiver}}\n&& \\text{\\textbf{Ribbon graph}} \\\\ \\\\\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet} \\ar@(ul,dl)[] \\ar@<-0.5ex>[r]\n& {\\bullet} \\ar@(dr,ur)[] \\ar@<-0.5ex>[l]\n}\n\\end{array}\n& \\qquad &\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n&& {\\circ} \\ar@{-}[rr]\n&& {\\circ} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n}\n\\end{array}\n\\\\ \\\\ \\\\\n\\begin{array}{c}\n\\xymatrix@=0.5pc{\n& {\\bullet} \\ar[rr] \\ar@\/^\/[dddd]\n&& {\\bullet} \\ar[ddr] \\ar@\/_3pc\/[ddlll] \\ar@{<-}@\/^3pc\/[dddd] \\\\ \\\\\n{\\bullet} \\ar[uur] & && &\n{\\bullet} \\ar[ddl] \\ar@\/^\/[uulll] \\ar@{<-}@\/_\/[ddlll] \\\\ \\\\\n& {\\bullet} \\ar[uul]\n&& {\\bullet} \\ar[ll] \\ar@{<-}@\/^3pc\/[uulll]\n}\n\\end{array}\n&&\n\\begin{array}{c}\n\\xymatrix@=1pc{\n&& {\\circ} \\ar@{-}[dddll] \\ar@{-}[dddrr] \\ar@{-}[dd] \\\\ \\\\\n&& {\\circ} \\ar@{-}[dll] \\ar@{-}[drr] \\\\\n{\\circ} \\ar@{-}[rrrr] &&&& {\\circ}\n}\n\\end{array}\n\\end{array}\n\\]\n\\caption{The connected self dual triangulation quivers and the corresponding\nribbon graphs, a punctured monogon (top) and a tetrahedron (bottom).}\n\\label{fig:selfdual}\n\\end{figure}\n\nWe call the ribbon graph with two nodes appearing in Figure~\\ref{fig:selfdual}\na \\emph{punctured monogon}, for reasons that will become apparent in\nSection~\\ref{sec:surface}. Similarly, we call the ribbon graph with\nfour nodes appearing in Figure~\\ref{fig:selfdual} a \\emph{tetrahedron}.\nIn the triangulation quiver corresponding to the tetrahedron there are four\n$f$-cycles and four $g$-cycles, each of length $3$, and for any arrow $\\alpha$,\neach of the arrows $\\alpha, f(\\alpha), \\bar{\\alpha}, f(\\bar{\\alpha})$ belongs to a\ndifferent $g$-cycle.\n\n\n\\subsection{Block decomposition of triangulation quivers}\n\\label{ssec:blocks}\n\nIn this section we analyze the structure of triangulation quivers in terms\nof three types of building blocks. This is similar in spirit to the\nblock decomposition of~\\cite[\\S13]{FST08}, however the number of blocks in\nour case is smaller and only full matchings are used.\n\n\\begin{definition}\nA \\emph{block} is one of the three pairs, each consisting of a quiver\nand a permutation on its set of arrows, shown in Figure~\\ref{fig:blocks}.\nA vertex of a block marked with white circle ($\\circ$) is called\nan \\emph{outlet}.\n\\end{definition}\n\n\\begin{figure}\n\\[\n\\begin{array}{ccccc}\n\\begin{array}{c}\n\\xymatrix{\n{\\circ} \\ar@(ul,dl)[]_{\\alpha}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet} \\ar@(ul,dl)[]_{\\alpha}\n\\ar@<-0.5ex>[r]_{\\beta} & \n{\\circ} \\ar@<-0.5ex>[l]_{\\gamma}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& {\\circ} \\ar[ddl]_{\\gamma} \\\\ \\\\\n{\\circ} \\ar[rr]_{\\alpha} &&\n{\\circ} \\ar[uul]_{\\beta}\n}\n\\end{array}\n\\\\\n(\\alpha)\n& &\n(\\alpha \\, \\beta \\, \\gamma)\n& &\n(\\alpha \\, \\beta \\, \\gamma)\n\\\\ \\\\\n\\text{Type A}\n& &\n\\text{Type B}\n& &\n\\text{Type C}\n\\end{array}\n\\]\n\\caption{Blocks for triangulation quivers. The permutation is given in\ncycle form below each quiver.}\n\\label{fig:blocks}\n\\end{figure}\n\nLet $B_1, B_2, \\dots, B_s$ be a collection of blocks.\nDenote by $V_1, V_2, \\dots, V_s$ their corresponding sets of outlets\nand let $V = \\bigsqcup_{i=1}^s V_i$ be their disjoint union.\nA \\emph{matching} on $V$ is an involution $\\theta \\colon V \\to V$ without\nfixed points such that $\\theta(V_i) \\cap V_i$ is empty for each\n$1 \\leq i \\leq s$\n(in other words, an outlet cannot be matched to an outlet in the same block).\n\nGiven a collection of blocks and a matching $\\theta$ on their outlets,\nconstruct a quiver $Q$ and a permutation $f$ on its set of arrows as follows;\ntake the disjoint union of the blocks and identify each outlet\n$v \\in V$ with the outlet $\\theta(v)$ to obtain $Q$.\nThe permutation $f$ on the set of arrows of $Q$ is induced by the permutations\non each of the blocks.\n\n\\begin{definition}\nA pair $(Q,f)$ consisting of a quiver $Q$ and a permutation $f$ on its set of\narrows is \\emph{block-decomposable} if it can be obtained by the above\nprocedure.\n\\end{definition}\n\n\\begin{proposition}\nA block-decomposable pair $(Q,f)$ is a triangulation quiver.\nConversely, any triangulation quiver is block-decomposable.\n\\end{proposition}\n\n\\begin{example}\nSince each of the blocks of types A and B has only one outlet, there is\nonly one way to match a pair consisting of two such blocks. In contrast,\nthere are two different ways to completely match two blocks of type C,\nyielding the triangulation quivers $3b$ and $3''$ of Table~\\ref{tab:quivers}.\nThe block decompositions of the triangulation quivers with at most three\nvertices are given in Table~\\ref{tab:block}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cc}\n\\textbf{Quiver} & \\textbf{Block decomposition} \\\\\n\\hline\n$1$ & A, A \\\\\n$2$ & B, A \\\\\n$3a$ & B, B \\\\\n$3b$ & C, C \\\\\n$3'$ & C, A, A, A \\\\\n$3''$ & C, C\n\\end{tabular}\n\\end{center}\n\\caption{Block decompositions of the triangulation quivers with at most three\nvertices. The numbers of the quivers refer to Table~\\ref{tab:quivers}.}\n\\label{tab:block}\n\\end{table}\n\\end{example}\n\n\\begin{remark}\nIn a block decomposition of a triangulation quiver $(Q,f)$, the blocks of type\nA are in bijection with the elements of $Q_1^f$, whereas those of type B are\nin bijection with the elements of $Q_1^g$.\n\\end{remark}\n\nIn the theory of cluster algebras, quivers without loops (i.e.\\ cycles of\nlength~$1$) and $2$-cycles (cycles of length~$2$) play an important role. The\nblock decomposition allows to quickly characterize those triangulation\nquivers without loops and $2$-cycles. Indeed, a loop can only arise from a\nblock of types A or B, whereas a $2$-cycle arises either from a block\nof type B or from gluing two blocks of type C, identifying two pairs of\nvertices at opposing directions of the arrows. This can be rephrased as\nfollows.\n\n\\begin{proposition} \\label{p:tri2cyc}\nLet $(Q,f)$ be a triangulation quiver. Then the length of any non-trivial\ncycle in $Q$ is at least 3 if and only if the following conditions hold:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\roman{enumi}}\n\\item\nThere are no arrows fixed by the permutation $f$; and\n\n\\item\nthe length of any cycle of the permutation $g$ is at least $3$.\n\\end{enumerate}\n\\end{proposition}\n\nThe block decomposition is also useful in proving the next statement.\n\n\\begin{proposition} \\label{p:gcycles}\nLet $(Q,f)$ be a triangulation quiver. Then the number of cycles of the\npermutation $g$ does not exceed the number of vertices of $Q$, and equality\nholds if and only if $(Q,f)$ is a disjoint union of any of the triangulation\nquivers $1$, $2$, $3a$ or $3b$ of Table~\\ref{tab:quivers}.\n\\end{proposition}\n\n\\section{Triangulations of marked surfaces and their quivers}\n\\label{sec:surface}\n\nIn this section we explain how (ideal) triangulations of marked\nsurfaces give rise to triangulation quivers. Marked surfaces\nwere considered by Fomin, Shapiro and Thurston~\\cite{FST08} in their\nwork on cluster algebras from surfaces. Let us recall the setup and\ndefinitions.\n\nA \\emph{marked surface} is a pair $(S,M)$ consisting of a compact,\nconnected, oriented, Riemann surface $S$ (possibly with boundary\n$\\partial S$) and a finite non-empty set $M$ of points in $S$,\ncalled \\emph{marked points}, such that each connected component of\n$\\partial S$ contains at least one point from $M$. The points in $M$\nwhich are not on $\\partial S$ are called \\emph{punctures}.\nWe exclude the following surfaces:\n\\begin{itemize}\n\\item\na sphere with one or two punctures;\n\n\\item\nan unpunctured digon;\n\\end{itemize}\n(a \\emph{sphere} is a surface of genus $0$ with empty boundary, a \\emph{disc}\nis a surface of genus $0$ with one boundary component, an \\emph{$m$-gon} is a\ndisc with $m$ marked points on its boundary, and for $m=1,2,3$ an $m$-gon is\ncalled \\emph{monogon}, \\emph{digon} and \\emph{triangle}, respectively).\n\nUp to homeomorphism, $(S,M)$ is determined by the following discrete data:\n\\begin{itemize}\n\\item\nthe genus $g$ of $S$;\n\n\\item\nthe number $b \\geq 0$ of boundary components;\n\n\\item\nthe sequence $(n_1,n_2,\\dots,n_b)$ where\n$n_i \\geq 1$ is the number of marked points on the $i$-th boundary component,\nconsidered as a multiset;\n\n\\item\nthe number $p$ of punctures.\n\\end{itemize}\n\n\\subsection{Triangulation quivers from triangulations}\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& & & {\\circ} \\ar@{-}@\/^\/[dd]^k \\\\\n{\\circ} \\ar@{-}@\/_\/[drrr]_i \\ar@{-}@\/^\/[urrr]^j \\\\\n& & & {\\circ}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@=0.5pc{\n{\\bullet_j} \\ar[drr]^{f(\\alpha)} \\\\\n&& {\\bullet_k} \\ar[dll] \\\\\n{\\bullet_i} \\ar[uu]^{\\alpha}\n}\n\\end{array}\n\\end{align*}\n\\caption{A triangle (left) and the corresponding block in the triangulation\nquiver (right).}\n\\label{fig:triquiver}\n\\end{figure}\n\nLet $(S,M)$ be a marked surface. An \\emph{arc} $\\gamma$ in $(S,M)$\nis a curve in $S$ satisfying the following:\n\\begin{itemize}\n\\item\nthe endpoints of $\\gamma$ are in $M$;\n\\item\n$\\gamma$ does not intersect itself, except that its endpoints may\ncoincide;\n\\item\nthe relative interior of $\\gamma$ is disjoint from $M \\cup \\partial S$;\n\\item\n$\\gamma$ does not cut out an unpunctured monogon or an unpunctured digon.\n\\end{itemize}\nArcs are considered up to isotopy.\nTwo arcs are \\emph{compatible} if there are curves in their respective\nisotopy classes whose relative interiors do not intersect.\nA \\emph{triangulation} of $(S,M)$ is a maximal collection of pairwise\ncompatible arcs. The arcs of a triangulation cut the surface $S$ into\nideal triangles. The three sides of an ideal triangle need not be\ndistinct. Sides on the boundary of $S$ are called \\emph{boundary\nsegments}.\n\n\\begin{definition}\nLet $\\tau$ be a triangulation of a marked surface $(S,M)$\nwhich is not an unpunctured monogon. Construct a quiver\n$Q_\\tau$ and $f_\\tau \\colon (Q_\\tau)_1 \\to (Q_\\tau)_1$ as follows:\n\\begin{itemize}\n\\item\nThe vertices of $Q_\\tau$ are the arcs of $\\tau$ together with\nthe boundary segments.\n\\item\nAt each vertex corresponding to a boundary segment add a loop\n$\\delta$ and set $f(\\delta)=\\delta$.\n\n\\item\nFor each ideal triangle in $\\tau$ with sides $i, j, k$\n(which may be arcs or boundary segments)\narranged in a clockwise order induced by the orientation of $S$,\nadd three arrows $i \\xrightarrow{\\alpha} j$, $j \\xrightarrow{\\beta} k$,\n$k \\xrightarrow{\\gamma} i$ and set\n$f(\\alpha)=\\beta$, $f(\\beta)=\\gamma$, $f(\\gamma)=\\alpha$ as in\nFigure~\\ref{fig:triquiver}.\n\\end{itemize}\n\\end{definition}\n\nThe next statement is immediate from the definitions, observing that in any\ntriangulation $\\tau$, an arc $\\gamma$ of $\\tau$ is either the side of two\ndistinct triangles or there exists a triangle $\\Delta$ such that two of its\nsides are $\\gamma$. In the latter case we say that the triangle $\\Delta$ is\n\\emph{self-folded} and $\\gamma$ is its \\emph{inner side}.\n\n\\begin{lemma}\n$(Q_\\tau,f_\\tau)$ is a triangulation quiver.\n\\end{lemma}\n\n\\begin{remark}\nWhen $(S,M)$ is an unpunctured monogon, a triangulation is empty,\nthere is one boundary segment, and we agree that the associated\ntriangulation quiver is the one with one vertex shown in the top\nrow of Table~\\ref{tab:quivers}.\n\\end{remark}\n\n\\begin{example}\nFigure~\\ref{fig:square} shows a triangulation of the square and the\ncorresponding triangulation quiver. There are four boundary segments and for\neach $1 \\leq i \\leq 4$ the loop  $\\delta_i$ corresponds to the boundary segment \nlabeled $i$. The permutation $f$ on the arrows is given in cycle form by\n$(\\alpha_1 \\, \\alpha_2 \\, \\alpha_3)(\\beta_1 \\, \\beta_2 \\, \\beta_3)\n(\\delta_1)(\\delta_2)(\\delta_3)(\\delta_4)$.\n\\end{example}\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix{\n& {\\circ} \\ar@{-}[dr]^3 \\ar@{-}[dd] \\\\\n{\\circ} \\ar@{-}[ur]^2 && {\\circ} \\ar@{-}[dl]^4 \\\\\n& {\\circ} \\ar@{-}[ul]^1\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n{\\bullet} \\ar@(l,ur)[]^{\\delta_2} \\ar[drr]^{\\alpha_3}\n&& && {\\bullet} \\ar@(ul,r)[]^{\\delta_3} \\ar[dd]^{\\beta_2} \\\\\n&& {\\bullet} \\ar[dll]^{\\alpha_1} \\ar[urr]^{\\beta_1} \\\\\n{\\bullet} \\ar@(dr,l)[]^{\\delta_1} \\ar[uu]^{\\alpha_2}\n&& && {\\bullet} \\ar@(r,dl)[]^{\\delta_4} \\ar[ull]^{\\beta_3}\n}\n\\end{array}\n\\end{align*}\n\\caption{A triangulation of a square (left) and the corresponding triangulation\nquiver (right).}\n\\label{fig:square}\n\\end{figure}\n\n\\begin{remark} \\label{rem:nvertex}\nBy using Euler characteristic considerations one sees that\nif $(S,M)$ is not an unpunctured monogon, then\nthe number of vertices of the triangulation quiver associated to\nany of its triangulations is\n\\begin{equation} \\label{e:nvertex}\n6(g-1) + 3(p+b) + 2(n_1 + n_2 + \\dots + n_b) ,\n\\end{equation}\ncompare~\\cite[Proposition~2.10]{FST08}.\n\\end{remark}\n\n\\begin{remark} \\label{rem:triblocks}\nIn terms of the block decomposition of triangulation quivers described\nin Section~\\ref{ssec:blocks}, \nthere is a natural block decomposition of $(Q_\\tau, f_\\tau)$ induced\nby the triangulation $\\tau$ with bijections\n\\begin{align*}\n&\\text{blocks of type A}\n\\longleftrightarrow \\text{boundary segments}, \\\\\n&\\text{blocks of type B}\n\\longleftrightarrow \\text{self-folded triangles in $\\tau$}, \\\\\n&\\text{blocks of type C}\n\\longleftrightarrow \\text{the other triangles in $\\tau$}.\n\\end{align*}\nIn addition, there are also bijections\n\\begin{align*}\n\\text{cycles of $f_\\tau$ of length $1$}\n&\\longleftrightarrow \\text{boundary segments}, \\\\\n\\text{cycles of $f_\\tau$ of length $3$}\n&\\longleftrightarrow \\text{triangles in $\\tau$}, \\\\\n\\text{cycles of $g_\\tau$ of length $1$}\n&\\longleftrightarrow \\text{self-folded triangles in $\\tau$}, \\\\\n\\text{cycles of $g_\\tau$}\n&\\longleftrightarrow \\text{punctures and boundary components}.\n\\end{align*}\n\\end{remark}\n\nWe can also obtain the triangulation quiver via a ribbon graph naturally\nassociated to the triangulation. Informally speaking, one thinks of the\ntriangulation as the graph, but some modifications are needed at the\nboundary components, as in the next definition.\n\n\\begin{definition} \\label{def:trigraph}\nLet $\\tau$ be a triangulation of a marked surface $(S,M)$.\nAssociate to $\\tau$ a ribbon graph defined as a graph $(V,E_\\tau)$ with cyclic\nordering of the edges around each node as follows:\n\\begin{itemize}\n\\item\nthe set $V$ of nodes consists of the punctures in $M$ and the connected\ncomponents of $\\partial S$,\n\\item\nthe set $E_\\tau$ of edges consists of the arcs of $\\tau$ and the boundary\nsegments.\n\\end{itemize}\n\nDenote by $\\pi \\colon M \\twoheadrightarrow V$ the map taking each puncture to\nitself and each marked point on $\\partial S$ to the boundary component it\nbelongs to. In the graph $(V,E_\\tau)$, each edge is incident to the nodes\nwhich are the images under $\\pi$ of its endpoints.\n\nThe cyclic ordering is determined as follows.\nIf $v \\in V$ is a puncture, then the edges incident to $v$ are arcs of $\\tau$\nand the cyclic ordering of them is the counterclockwise ordering induced by\nthe orientation of $S$.\n\nIf $v \\in V$ is a boundary component, we arrange the set $\\pi^{-1}(v)$ of\nmarked points on $v$ in a counterclockwise order $\\{q_0,q_1,\\dots,q_{n-1}\\}$\nsuch that for each $0 \\leq i < n$ there is a boundary segment $\\varepsilon_i$ whose\nendpoints are $q_i,q_{i+1}$ (where indices are taken modulo $n$).\nThe set of edges incident to $v$ thus consists of the boundary segments\n$\\varepsilon_i$, which become loops in the graph (see Figure~\\ref{fig:node}), and\nthe arcs incident to any of the marked points $q_i$. Their cyclic ordering is\nobtained by taking the arcs incident to $q_0$ in the counterclockwise order\ninduced by the orientation of $S$, then $\\varepsilon_0$, then the arcs incident to\n$q_1$ in a counterclockwise order, then $\\varepsilon_1$, etc.\n\n\\begin{figure}\n\\[\n\\xymatrix{\n{\\circ} \\ar@{-}@(ur,ul)[] \\ar@{-}@(ul,dl)[]\n\\ar@{-}@(dl,dr)[] \\ar@{-}@(dr,ur)[]\n}\n\\]\n\\caption{A boundary component with $4$ marked points becomes a node with $4$\nloops. The arcs incident to the marked points are to be placed between these\nloops.}\n\\label{fig:node}\n\\end{figure}\n\\end{definition}\n\nThe next statement is a consequence of the definitions.\n\n\\begin{proposition}\nFor any triangulation $\\tau$ of a marked surface $(S,M)$,\nthe ribbon quiver corresponding under the bijection of\nProposition~\\ref{p:ribbon} to the ribbon graph\nconstructed in Definition~\\ref{def:trigraph} \nis the triangulation quiver $(Q_\\tau, f_\\tau)$.\n\\end{proposition}\n\n\\begin{example} \\label{ex:surface3}\nTable~\\ref{tab:surface} lists the marked surfaces whose\ntriangulation quivers have at most three vertices. For each surface,\nwe list the corresponding triangulation quivers (and ribbon graphs)\nappearing in Table~\\ref{tab:quivers}.\n\nNote that the unpunctured monogon and unpunctured triangle have only\nempty triangulations, so for each of these surfaces there is only one quiver.\nSimilarly, a punctured monogon has only one triangulation, consisting of one\narc. A sphere with three punctures has two topologically inequivalent\ntriangulations and hence two triangulation quivers.\n\\end{example}\n\n\n\\subsection{Triangulation vs.\\ adjacency quivers}\n\\label{ssec:triangadj}\n\nThe construction of the triangulation quiver of an ideal triangulation\nresembles that of the adjacency quiver defined in~\\cite[Definition~4.1]{FST08},\nhowever there are several differences:\n\\begin{enumerate}\n\\renewcommand{\\labelenumi}{\\theenumi.}\n\\item\nIn the triangulation quiver there are vertices corresponding to the\nboundary segments and not only to the arcs, as in the adjacency\nquiver.\n\n\\item\nOur treatment of self-folded triangles is different; in the triangulation\nquiver there is a loop at each vertex corresponding to the inner side\nof a self-folded triangle.\n\n\\item\nWe do not delete 2-cycles that arise in the quiver (e.g.\\ when there\nare precisely two arcs incident to a puncture).\n\\end{enumerate}\n\n\\begin{example}\nConsider the triangulation of the square shown in Figure~\\ref{fig:square}.\nIts triangulation quiver consists of $5$ vertices whereas its adjacency\nquiver is the Dynkin quiver $A_1$ (one vertex, no arrows).\n\\end{example}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}[c]{cl}\n\\textbf{Quiver} & \\textbf{Marked surface} \\\\\n\\hline\n$1$ & monogon, unpunctured \\\\\n$2$ & monogon, one puncture \\\\\n$3a$, $3b$ & sphere, three punctures \\\\\n$3'$ & triangle, unpunctured \\\\\n$3''$ & torus, one puncture\n\\end{tabular}\n\\end{center}\n\\caption{The marked surfaces whose triangulation quivers have at most three\nvertices. The numbers of the quivers refer to Table~\\ref{tab:quivers}.}\n\\label{tab:surface}\n\\end{table}\n\nAs Example~\\ref{ex:surface3} demonstrates,\nthese differences allow to attach triangulation quivers to \nmarked surfaces that do not admit adjacency quivers, such as\na monogon, a triangle or a sphere with three punctures.\nOn the other hand, there are situations where\nthe triangulation quiver and the adjacency quiver of a triangulation\ncoincide. By abuse of notation, in the next statements by referring\nto a triangulation quiver we actually mean the underlying quiver $Q$ of\nthe pair $(Q,f)$.\nThis is not ambiguous in view of Proposition~\\ref{p:triauniq}\n\nRecall that a surface $S$ is \\emph{closed} if $\\partial S$ is empty.\nIf $(S,M)$ is a marked surface and $S$ is closed, then all marked\npoints are punctures. The next statement is a reformulation of our result\nin~\\cite[\\S2]{Ladkani12}.\n\n\\begin{lemma} \\label{l:adjacency}\nLet $(S,M)$ be a closed marked surface which is not a sphere with\nless than four punctures.\nThen for any triangulation $\\tau$ of $(S,M)$ with at least three\narcs incident to each puncture, the triangulation quiver\nand adjacency quiver associated to $\\tau$ coincide.\n\\end{lemma}\nThe condition on $\\tau$ in the lemma was called (T3) in~\\cite{Ladkani12}.\nIn particular, we get the following corollary (cf.\\\n\\cite[Lemma~5.3]{Ladkani12}).\n\n\\begin{corollary} \\label{c:onep}\nLet $(S,M)$ be a closed surface with exactly one puncture, i.e.\\ $|M|=1$.\nThen for any triangulation $\\tau$ of $(S,M)$, the triangulation quiver\nand the adjacency quiver associated to $\\tau$ coincide.\n\\end{corollary}\n\n\\begin{example}\nFor a torus with empty boundary and one puncture, the adjacency quiver of\nany triangulation is known as the Markov quiver and is given by the\nquiver~$3''$ in the last row of Table~\\ref{tab:quivers}.\n\\end{example}\n\nAnother difference between triangulation quivers and adjacency quivers\nconcerns the possibility to recover the topology of the underlying marked\nsurface. It is known~\\cite[\\S12]{FST08} that a quiver may arise \nas adjacency quiver of two triangulations of topologically inequivalent marked\nsurfaces. On the other hand, \nif $(Q_\\tau, f_\\tau)$ is the triangulation quiver\ncorresponding to a triangulation $\\tau$ of a marked surface $(S,M)$,\nthen the topology of\n$(S,M)$ can be completely recovered from $(Q_\\tau, f_\\tau)$.\nIndeed, the cycles of the permutation $g$ on $(Q_\\tau)_1$ are in bijection\nwith the punctures and boundary components of $(S,M)$.\nFor each such cycle $\\omega$ set\n$m_\\omega = \\left| \\{ \\alpha \\in \\omega : f(\\alpha)=\\alpha \\}\\right|$.\nIf $m_\\omega = 0$, then $\\omega$ corresponds to a puncture, otherwise\nit corresponds to a boundary component with $m_\\omega$ marked points on it.\nIn this way we recovered the parameters $p$, $b$ and the numbers\n$n_1, \\dots, n_b$. Once these are known, the genus of $S$ can be recovered\nusing Eq.~\\eqref{e:nvertex}.\n\n\\subsection{A dimer model perspective}\n\\label{ssec:dimer}\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix@=0.5pc{\n& {\\circ} \\ar@{-}[rr] && {\\bullet} \\ar@{-}[ddr] \\\\ \\\\\n{\\bullet} \\ar@{-}[uur]^{g(e)} &&&& {\\circ} \\ar@{-}[ddl] \\\\ \\\\\n& {\\circ} \\ar@{-}[uul]^{e} && {\\bullet} \\ar@{-}[ll]^{f(e)}\n}\n\\end{array}\n&&\n\\begin{array}{c}\n\\xymatrix@=1pc{\n&& \\ar[dd] \\\\ \n&&&& {} \\\\\n&& {\\bullet} \\ar[ull]_{g(\\alpha)} \\ar[urr] \\ar[dd]^{f(\\alpha)} \\\\\n\\ar[urr]^{\\alpha} &&&& \\ar[ull] \\\\\n&& {} \n}\n\\end{array}\n\\end{align*}\n\\caption{A $2$-cell in a dimer model (left) and the corresponding vertex\nwith incident arrows (right).}\n\\label{fig:dimer}\n\\end{figure}\n\nDimer models on a torus have been used to construct non-commutative\ncrepant resolutions of toric Gorenstein\nsingularities~\\cite{Bocklandt13,Broomhead12}.\nSuch resolution is a 3-Calabi-Yau algebra which is a (non-complete)\nJacobian algebra of a quiver with potential constructed from the dimer\nmodel. In this section we explain how triangulations of closed surfaces\ngive rise to a very special kind of dimer models, yet the corresponding\n(complete) Jacobian algebras (which are triangulation algebras to be\ndefined in Section~\\ref{ssec:trialg}) have completely different properties,\nas we shall see in Section~\\ref{sec:triangquasi}.\n\nA \\emph{dimer model} on a closed, compact, connected, oriented surface $S$\nis a bipartite graph\non $S$ whose complement is homeomorphic to a disjoint union of discs.\nThe set of nodes of this graph can thus be written as a disjoint union\n$V^+ \\cup V^-$. We call the elements of $V^+$ \\emph{white nodes} and\nthose of $V^-$ \\emph{black nodes}.\nDenote by $E$ the set of edges. An edge $e \\in E$ defines a pair\n$(v^+_e,v^-_e) \\in V^+ \\times V^-$ consisting of the nodes incident to $e$.\nEach connected component of the complement defines a $2$-cell, and an\nedge is incident to exactly two $2$-cells. \n\nDefine two permutations $f,g \\colon E \\to E$ on the set of edges as follows.\nFor an edge $e \\in E$, let $f(e)$ be the edge following $e$ when going\nclockwise around the node $v^+_e$ and let $g(e)$ be the edge following $e$\nwhen going counterclockwise around the node $v^-_e$, see the left drawing\nin Figure~\\ref{fig:dimer}.\n\nA dimer model gives rise to a quiver $Q$ by taking the graph dual to the\ngraph $(V^+ \\cup V^-, E)$. The vertices of $Q$ are thus the $2$-cells,\nand the arrows are in bijection with the edges. \nLet $\\alpha$ be an arrow corresponding to an edge $e \\in E$.\nThe endpoints of $\\alpha$ are the two $2$-cells that $e$ is incident to,\nand $\\alpha$ is oriented in such a way that when going forward in the\ndirection of the arrow, the white node $v^+_e$ is seen to the right while\nthe black node $v^-_e$ is to the left, see Figure~\\ref{fig:dimer}.\n\nThe permutations $f, g$ on $E$ induce permutations (denoted by the same\nletters) on the set of arrows $Q_1$. For any vertex $i \\in Q_0$, each of the\npermutations $f$ and $g$ induces a bijection between the sets of\narrows starting at $i$ and those ending at $i$.\n\nNow we restrict attention to dimer models whose $2$-cells are \\emph{quadrilaterals}, i.e.\\ consist of exactly four edges.\nIn this case, for each vertex $i$ of the quiver $Q$ there are exactly two \narrows starting at $i$ and two arrows ending at $i$, and $(Q,f)$ thus becomes\na ribbon quiver.\nLet us construct the corresponding ribbon graph $(V',E')$ explicitly in terms\nof the dimer model. We have $V' = V^-$, that is, the nodes of the ribbon graph\nare the black nodes of the dimer model, and the edges $E'$ of the ribbon graph\nare in bijection with the quadrilaterals.\nThere are exactly two black nodes incident to each quadrilateral\nand the corresponding edge $e'$ in the ribbon graph connects these nodes,\nsee Figure~\\ref{fig:dimeribbon}.\nThe cyclic ordering around each node of $V'$ is induced by the embedding into\nthe oriented surface $S$.\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix@=0.5pc{\n& {\\circ} \\ar@{-}[ddr] \\\\ \\\\\n{\\bullet} \\ar@{-}[uur] && {\\bullet} \\ar@{-}[ddl] \\\\ \\\\\n& {\\circ} \\ar@{-}[uul]\n}\n\\end{array}\n&&\n\\begin{array}{c}\n\\xymatrix@=0.5pc{\n{\\bullet} \\ar@{-}[rr] && {\\bullet}\n}\n\\end{array}\n\\end{align*}\n\\caption{A quadrilateral in a dimer model (left) corresponds to an edge\nin the ribbon graph (right).}\n\\label{fig:dimeribbon}\n\\end{figure}\n\nFinally we further restrict to dimer models whose $2$-cells are quadrilaterals\nand moreover their white nodes are \\emph{trivalent}, i.e.\\ each $v^+ \\in V^+$\nis incident to exactly three edges. In this case the associated ribbon graph\n$(V',E')$ is a triangulation of the marked surface $(S,V')$ and the\nribbon quiver $(Q,f)$ is a triangulation quiver.\n\nConversely, given a set $M$ of punctures, any triangulation $\\tau$ of\n$(S,M)$ without self-folded triangles gives rise to a dimer model\nwhose white nodes $V^+_\\tau$ are the triangles of $\\tau$, its black nodes\n$V^-_\\tau$ are the punctures $M$, and there is an edge connecting\n$\\Delta \\in V^+_\\tau$ with $v \\in V^-_\\tau$\nif and only if $v$ is incident to $\\Delta$ in $\\tau$. The $2$-cells of\nthis dimer model are quadrilaterals (corresponding bijectively to the\narcs of $\\tau$) and any $\\Delta \\in V^+_\\tau$ is trivalent.\nFor example, Figure~\\ref{fig:dimertetra} shows two dimer models; one for\nthe triangulation of a sphere with three punctures corresponding to the\ntriangulation quiver $3b$ of Table~\\ref{tab:quivers}; and the other for\nthe tetrahedron of Figure~\\ref{fig:selfdual},\nwhich is a triangulation of a sphere with four punctures.\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix{\n& {\\circ} \\ar@{-}[dl] \\ar@{-}[d] \\ar@{-}[dr] \\\\\n{\\bullet} & {\\bullet} & {\\bullet} \\\\\n& {\\circ} \\ar@{-}[ul] \\ar@{-}[u] \\ar@{-}[ur]\n}\n\\end{array}\n&&\n\\begin{array}{c}\n\\xymatrix@=1pc{\n&& {\\bullet} \\ar@{-}[d] \\ar@{-}[ddrr] \\ar@{-}[ddll] \\\\\n&& {\\circ} \\ar@{-}[dr] \\ar@{-}[dl] \\\\\n{\\circ} \\ar@{-}[r] & {\\bullet} && {\\bullet} \\ar@{-}[r] & {\\circ} \\\\\n&& {\\circ} \\ar@{-}[ur] \\ar@{-}[ul] \\\\\n&& {\\bullet} \\ar@{-}[u] \\ar@{-}[uurr] \\ar@{-}[uull]\n}\n\\end{array}\n\\end{align*}\n\\caption{Dimer models on a sphere corresponding to the triangulation quiver\n$3b$ of Table~\\ref{tab:quivers} (left) and that of the tetrahedron (right).\nSince they arise from triangulations,\nall $2$-cells are quadrilaterals and all white nodes are trivalent.}\n\\label{fig:dimertetra}\n\\end{figure}\n\nThe various notions concerning dimer models, ribbon graphs, ribbon quivers and\ntriangulations are related as in the dictionary given in Table~\\ref{tab:dimer}.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{cccc}\nDimer model & Ribbon graph & Ribbon quiver & Triangulation \\\\ \\hline\n$V^+$ &      & cycles of $f$ & triangles \\\\\n$V^-$ & $V'$ & cycles of $g$ & punctures \\\\\n$2$-cells & $E'$ & $Q_0$ & arcs \\\\\n$E$   &      & $Q_1$\n\\end{tabular}\n\\end{center}\n\\caption{Dictionary between dimer models, ribbon graphs, ribbon quivers\nand triangulations of a closed surface.\nA ribbon graph\/quiver arises when all the $2$-cells in the dimer model are\nquadrilaterals, and a triangulation arises when, in addition, all the white\nnodes $V^+$ are trivalent.}\n\\label{tab:dimer}\n\\end{table}\n\n\\section{Brauer graph algebras and triangulation algebras}\n\\label{sec:algebras}\n\nIn this section we introduce two classes of algebras which turn out to be\nimportant for our study, one consists of the well known Brauer graph algebras\nand the other consists of the newly defined triangulation algebras.\nRoughly speaking, a Brauer graph algebra arises from any ribbon quiver\nand auxiliary data given in the form of scalars and positive integer\nmultiplicities, whereas a triangulation algebra arises from any triangulation\nquiver with similar auxiliary data.\n\n\\begin{definition}\nLet $(Q,f)$ be a ribbon quiver. \nRecall from Section~\\ref{sec:quivers} the permutation $g \\colon Q_1 \\to Q_1$\ndefined by $g(\\alpha) = \\overline{f(\\alpha)}$ for any $\\alpha \\in Q_1$.\nGiven a function $\\nu$ on the set $Q_1$ of arrows, we write\n$\\nu_\\alpha$ instead of $\\nu(\\alpha)$.\nWe say that $\\nu$ is \\emph{$g$-invariant} if $\\nu_{g(\\alpha)} = \\nu_\\alpha$\nfor any $\\alpha \\in Q_1$. Similarly, we say that $\\nu$ is\n\\emph{$f$-invariant} if $\\nu_{f(\\alpha)} = \\nu_\\alpha$ for any\n$\\alpha \\in Q_1$.\n\nSince the $g$-cycles are in bijection with the nodes of the\ncorresponding ribbon graph, \na $g$-invariant function can thus be regarded as a function on the\nnodes of that ribbon graph.\n\\end{definition}\n\nLet $(Q,f)$ be a ribbon quiver. For an arrow $\\alpha \\in Q_1$, set\n\\begin{align*}\nn_\\alpha &= \\min \\{n>0 \\,:\\, g^n(\\alpha)=\\alpha\\} \\\\\n\\omega_\\alpha &= \\alpha \\cdot g(\\alpha) \\cdot \\ldots \\cdot g^{n_\\alpha-1}(\\alpha) \\\\\n\\omega_\\alpha' &= \\alpha \\cdot g(\\alpha) \\cdot \\ldots \\cdot g^{n_\\alpha-2}(\\alpha)\n\\end{align*}\n\nThe function $\\alpha \\mapsto n_\\alpha$ is obviously $g$-invariant, telling the\nlength of the $g$-cycle $\\omega_\\alpha$ starting at $\\alpha$. The path $\\omega_\\alpha'$ is\n``almost'' a cycle; when $n_\\alpha=1$ the arrow $\\alpha$ is a loop at some vertex\n$i$ and $\\omega_\\alpha'$ is understood to be the path of length zero starting at $i$.\nSimilarly, for an arrow $\\alpha \\in Q_1$, set\n\\begin{align*}\nk_\\alpha &= \\min \\{k>0 \\,:\\, f^k(\\alpha)=\\alpha\\} \\\\\n\\xi_\\alpha &=\n\\alpha \\cdot f(\\alpha) \\cdot \\ldots \\cdot f^{k_\\alpha-1}(\\alpha) \\\\\n\\xi'_\\alpha &=\n\\alpha \\cdot f(\\alpha) \\cdot \\ldots \\cdot f^{k_\\alpha-2}(\\alpha)\n\\end{align*}\n\nThe function $\\alpha \\mapsto k_\\alpha$ is obviously $f$-invariant, telling the\nlength of the $f$-cycle $\\xi_\\alpha$ starting at $\\alpha$. The path\n$\\xi'_\\alpha$ is ``almost'' a cycle; when $k_\\alpha=1$ the arrow $\\alpha$ is a\nloop at some vertex $i$ and $\\xi'_\\alpha$ is understood to be the path of\nlength zero starting at $i$.\n\n\\begin{lemma}\nFor any $\\alpha \\in Q_1$, the paths $\\omega_\\alpha'$ and $\\xi'_{\\bar{\\alpha}}$ are parallel,\ni.e.\\ they both start at the same vertex and end at the same vertex.\n\\end{lemma}\n\n\\subsection{Brauer graph algebras}\n\nIn this section we fix a field $K$.\nBrauer graph algebras form a generalization of Brauer tree algebras.\nThey are algebras defined from combinatorial data consisting of\na ribbon graph together with multiplicities and scalars associated to its\nnodes, see~\\cite{Alperin86,Benson98,Kauer98}. Many authors start\nwith the ribbon graph and construct the quiver with relations of the\ncorresponding Brauer graph algebra, see for\nexample~\\cite{GSS14,MarshSchroll14}.\nWe prefer to give the definition directly in terms of the associated ribbon\nquiver.\n\n\\begin{definition} \\label{def:BGA}\nLet $(Q,f)$ be a ribbon quiver, and let $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nand $c \\colon Q_1 \\to K^{\\times}$ be $g$-invariant functions of\n\\emph{multiplicities} and \\emph{scalars}, respectively.\nThe \\emph{Brauer graph algebra} \n$\\Gamma(Q,f,m,c)$\nassociated to these data is the\nquotient of the path algebra $KQ$ by the ideal generated by two\ntypes of elements; the elements of the first type are the paths\n$\\alpha \\cdot f(\\alpha)$ for each $\\alpha \\in Q_1$ (``zero-relations'')\nand the elements of the second type are the differences\n$c_\\alpha \\omega_\\alpha^{m_\\alpha} - c_{\\bar{\\alpha}} \\omega_{\\balpha}^{m_{\\bar{\\alpha}}}$\n(``commutativity-relations'').\nIn other words, \n\\[\n\\Gamma(Q,f,m,c) = KQ \/\n( \\alpha \\cdot f(\\alpha) \\,,\\,\nc_\\alpha \\omega_\\alpha^{m_\\alpha} - c_{\\bar{\\alpha}} \\omega_{\\balpha}^{m_{\\bar{\\alpha}}}\n)_{\\alpha \\in Q_1}.\n\\]\n(It is clearly enough to take one commutativity-relation for each\npair of arrows $\\alpha$ and $\\bar{\\alpha}$, so these relations can be\nseen as indexed by the vertices of $Q$).\n\\end{definition}\n\nThe next proposition is well known. Special biserial algebras have been defined\nin~\\cite[\\S1]{SW83} and a classification of the indecomposable modules over\nthese algebras, implying that they are of tame representation type, is given\nin~\\cite[\\S2]{WW85}.\n\n\\begin{proposition} \\label{p:BGA}\nA Brauer graph algebra is finite-dimensional, symmetric, special biserial and\nhence of tame representation type.\n\\end{proposition}\n\nMoreover, it has been recently shown that over an algebraically closed field\nthe classes of symmetric special biserial algebras and that of Brauer graph\nalgebras coincide~\\cite[Theorem~1.1]{Schroll15}.\n\nWe present a few examples of Brauer graph algebras related to group algebras.\n\n\\begin{example} \\label{ex:BGAf1}\nConsider the ribbon quiver $(Q,f_1)$ of Example~\\ref{ex:ribbonG1}.\nIn this case there is only one $g$-cycle and hence the auxiliary data\nconsists of one multiplicity $m \\geq 1$ and one scalar $c \\in K^{\\times}$.\nThe path algebra $KQ$ is the free algebra $K\\langle \\alpha, \\beta \\rangle$\non the generators $\\alpha$ and $\\beta$, and\nthe Brauer graph algebra is\n\\[\nK\\langle \\alpha, \\beta \\rangle \/\n( \\alpha^2 , \\beta^2 , c(\\alpha \\beta)^m - c(\\beta \\alpha)^m ) ,\n\\]\nhence, up to isomorphism, we may set $c=1$. \nWhen $m=1$ and $\\operatorname{char} K = 2$, this algebra is isomorphic the group algebra of\nKlein's four-group.\n\\end{example}\n\n\\begin{example} \\label{ex:BGAf2}\nConsider now the ribbon quiver $(Q,f_2)$ of Example~\\ref{ex:ribbonG1}.\nIn this case there are two $g$-cycles and hence the auxiliary data\nconsists of two multiplicities $m, m' \\geq 1$ and two scalars\n$c, c' \\in K^{\\times}$.\nThe Brauer graph algebra is given by\n\\[\nK\\langle \\alpha, \\beta \\rangle \/\n(\\alpha \\beta, \\beta \\alpha, c \\alpha^m - c' \\beta^{m'}) .\n\\]\nWhen $m' = 1$, the arrow $\\beta$ can be eliminated so the relations in the\nabove presentations are no longer minimal and the algebra becomes isomorphic\nto the algebra $K[\\alpha]\/(\\alpha^{m+1})$ considered in Example~\\ref{ex:Kxn}.\n\\end{example}\n\n\\begin{example}\nLet's describe as group algebras some Brauer graph algebras for a few\ntriangulation quivers appearing in Table~\\ref{tab:quivers},\nunder the assumption that $\\operatorname{char} K = 2$.\n\nThe Brauer graph algebra of the triangulation quiver number $1$ with\nmultiplicity~$1$ was discussed in Example~\\ref{ex:BGAf1}; it is the group\nalgebra of Klein's four group.\n\nAssume now that $K$ contains a primitive third root of unity.\nThen the Brauer graph algebra of the\ntriangulation quiver number $2$ with multiplicities $m_\\alpha=2$ and\n$m_\\beta=m_\\gamma=m_\\eta=1$ is Morita equivalent to the group algebra\nof the symmetric group $S_4$ \\cite[V.2.5.1]{Erdmann90},\nwhereas that of the triangulation quiver number $3b$ with all multiplicities\nset to $1$ is isomorphic to the group algebra of the alternating group\n$A_4$ \\cite[V.2.4.1]{Erdmann90}\n(in both cases the scalars take the constant value~$1$).\n\\end{example}\n\nWe list a few remarks concerning Brauer graph algebras.\n\n\\begin{remark} \\label{rem:BGAelim}\nAs Example~\\ref{ex:BGAf2} shows, if $\\alpha$ is an arrow such that $n_\\alpha=1$\nand $m_\\alpha=1$, the corresponding commutativity-relation becomes\n$c_\\alpha \\alpha - c_{\\bar{\\alpha}} \\omega_{\\balpha}^{m_{\\bar{\\alpha}}}$ and in the\npresentation of the Brauer graph algebra as quiver with relations the\narrow $\\alpha$ can be eliminated at the expense of adding zero-relations\nof a third kind, namely $\\omega_\\beta^{m_\\beta} \\beta$\nfor $\\beta \\in \\{\\bar{\\alpha}, g^{-1}(\\bar{\\alpha})\\}$.\nHowever, in order to keep the presentation unified, we will not eliminate\narrows and add the corresponding new relations.\n\\end{remark}\n\n\\begin{remark}\nIf $K$ is algebraically closed, or more generally, if $K$ contains an\n$m_\\alpha$-th root of $c_\\alpha$ for each $\\alpha \\in Q_1$, then by\nconsidering the automorphism of $KQ$ defined by choosing from each $g$-cycle\none arrow $\\alpha$, sending it to $c_\\alpha^{1\/m_\\alpha} \\alpha$ and keeping\nall other arrows intact, we see that\n$\\Gamma(Q,f,m,c) \\simeq \\Gamma(Q,f,m,\\mathbf{1})$ where $\\mathbf{1}$\nis the constant function $\\mathbf{1}_\\alpha = 1$.\n\\end{remark}\n\nIn the next statements we explicitly compute the Cartan matrix of a Brauer\ngraph algebra and show that it depends only on the multiplicities and\nthe underlying graph of the ribbon graph corresponding to its defining ribbon\nquiver.\n\nThroughout, we fix a ribbon quiver $(Q,f)$ together with $g$-invariant\nfunctions $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ and $c \\colon Q_1 \\to K^{\\times}$\nof multiplicities and scalars, and consider the Brauer graph algebra\n$\\Gamma=\\Gamma(Q,f,m,c)$.\nFor any $i \\in Q_0$, let $\\alpha, \\bar{\\alpha}$ be the two arrows starting at $i$.\nBy definition, the images of the paths $c_\\alpha \\omega_\\alpha^{m_\\alpha}$ and\n$c_{\\bar{\\alpha}} \\omega_{\\bar{\\alpha}}^{m_{\\bar{\\alpha}}}$ in $\\Gamma$ are equal, and we\ndenote their common value by $z_i \\in \\Gamma$.\nThe next statement is a consequence of the definition.\n\\begin{lemma} \\label{l:basisBGA}\nA basis of $\\Gamma(Q,f,m,c)$ is given by the images of the paths\n\\[\n\\{e_i\\}_{i \\in Q_0} \\cup\n\\left\\{ \\alpha \\cdot g(\\alpha) \\cdot \\ldots \\cdot g^r(\\alpha)\n\\right\\}_{\\alpha \\in Q_1, 0 \\leq r < m_\\alpha n_\\alpha -1} \\cup\n\\{z_i\\}_{i \\in Q_0} .\n\\]\n\\end{lemma}\n\nGiven the basis of Lemma~\\ref{l:basisBGA}, an argument as\nin~\\cite[\\S4.4]{Ladkani12} allows to compute the Cartan matrix and to draw some\nconclusions.\nFor a $g$-cycle $\\omega$ in $Q_1$, define a row vector\n$\\chi_\\omega \\in \\mathbb{Z}^{Q_0}$ by\n$\\chi_\\omega(i) = |\\{\\alpha \\in \\omega : s(\\alpha)=i\\}|$ for $i \\in Q_0$.\nDenote by $\\Omega_g$ the set of $g$-cycles in $Q_1$.\nRecall from Section~\\ref{ssec:ribbon} that the matrix \n$(\\chi_\\omega(i))_{\\omega \\in \\Omega_g, i \\in Q_0}$ encodes the underlying\ngraph of the ribbon graph corresponding to $(Q,f)$ and hence depends only on\nthat graph.\nFor any $\\omega \\in \\Omega_g$, the square matrix $\\chi_\\omega^T \\chi_\\omega$ is\nsymmetric of rank $1$ whose $(i,j)$-entry is $\\chi_\\omega(i) \\chi_\\omega(j)$\nfor $i,j \\in Q_0$. The $g$-invariant function $m$ on $Q_1$ induces a\nfunction on $\\Omega_g$ which will be denoted by the same letter.\n\n\\begin{proposition} \\label{p:CartanBGA}\nLet $C_\\Gamma$ be the Cartan matrix of $\\Gamma(Q,f,m,c)$. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\n$C_\\Gamma = \\sum_{\\omega \\in \\Omega_g} m_\\omega \\chi_\\omega^T \\chi_\\omega$.\n\n\\item\n$\\dim_K \\Gamma = \\sum_{\\omega \\in \\Omega_g} m_\\omega |\\omega|^2$.\n\n\\item\nThe quadratic form $q_{C_\\Gamma} \\colon \\mathbb{Z}^{Q_0} \\to \\mathbb{Z}$ defined by\n$q_{C_\\Gamma}(x) = x C_{\\Gamma} x^T$ takes non-negative even values;\nin particular it is non-negative definite.\n\n\\item\n$\\operatorname{rank} C_\\Gamma \\leq \\min(|Q_0|, |\\Omega_g|)$.\n\\end{enumerate}\n\\end{proposition}\n\nThe next remark will not be used in the sequel. Nevertheless, we list it here\nfor completeness.\n\n\\begin{definition}\nGiven a non-zero power series\n$p(x) = \\sum_{i=0}^{\\infty} a_i x^i \\in K[[x]] \\setminus \\{0\\}$,\nlet $m = \\min \\{i \\geq 0 : a_i \\neq 0\\}$. The \\emph{least order term} of $p(x)$\nis $c_m x^m$.\n\\end{definition}\n\nFor any non-trivial cycle $\\omega$ in $Q$, the evaluation map\n$\\operatorname{ev}_\\omega \\colon K[[x]] \\to \\widehat{KQ}$ sending $p \\in K[[x]]$ to $p(\\omega)$\nis a continuous ring homomorphism.\n\n\\begin{remark} \\label{rem:genBGA}\nIn analogy with the triangulation algebras to be defined in the next section\nas quotients of complete path algebras by closed ideals,\none could consider an apparently more general, continuous, version of\na Brauer graph algebra defined by taking a $g$-invariant function\n$p \\colon Q_1 \\to xK[[x]] \\setminus \\{0\\}$ of non-zero power series without\nconstant term and forming the quotient of $\\widehat{KQ}$ by the closure of the\nideal generated by zero-relations and commutativity-relations\n\\[\n\\widehat{KQ} \/\n\\overline{(\\alpha \\cdot f(\\alpha) \\,,\\,\np_\\alpha(\\omega_\\alpha) - p_{\\bar{\\alpha}}(\\omega_{\\balpha}))}_{\\alpha \\in Q_1}.\n\\]\n\nHowever, it turns out that this algebra is isomorphic to the Brauer graph\nalgebra $\\Gamma(Q,f,m,c)$ where the multiplicities $m_\\alpha$ and scalars\n$c_\\alpha$ are such that each $c_\\alpha x^{m_\\alpha}$ is the least order\nterm of $p_\\alpha(x)$.\n\\end{remark}\n\n\n\\subsection{Triangulation algebras}\n\\label{ssec:trialg}\n\nIn this section we define, for any triangulation quiver together with some\nauxiliary data, a new algebra called triangulation algebra.\nThroughout, we fix a field $K$.\nWe start with a construction of hyperpotentials on ribbon quivers described\nin the following somewhat technical statement whose proof is omitted.\n\n\\begin{proposition} \\label{p:hyperib}\nLet $(Q,f)$ be a ribbon quiver.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:hyperib}\nLet $p \\colon Q_1 \\to K[[x]]$ be $f$-invariant and\nlet $q \\colon Q_1 \\to K[[x]]$ be $g$-invariant.\nThen the collection $(\\rho_\\alpha)_{\\alpha \\in Q_1}$ given by\n\\[\n\\rho_\\alpha = \np_\\alpha(\\xi_{f(\\alpha)}) \\cdot \\xi'_{f(\\alpha)} -\nq_{\\alpha}(\\omega_{g(\\alpha)}) \\cdot \\omega'_{g(\\alpha)}\n\\]\nis a hyperpotential on $Q$.\n\n\\item \\label{it:potrib}\nLet $P \\colon Q_1 \\to K[[x]]$ be $f$-invariant and\nlet $R \\colon Q_1 \\to K[[x]]$ be $g$-invariant.\nConsider\n\\[\nW = \\sum_{\\alpha} P_\\alpha(\\xi_\\alpha) - \\sum_{\\beta} R_\\beta(\\omega_\\beta)\n\\]\nwhere the left sum runs over representatives $\\alpha$ of the $f$-cycles\nin $Q$ and the right sum runs over representatives $\\beta$ of the\n$g$-cycles. Then $W$ is a potential on $Q$ and\n$\\partial_\\alpha W = \\rho_\\alpha$ for each $\\alpha \\in Q_1$, where\n$\\rho_\\alpha$ are defined as in part~\\eqref{it:hyperib} for the \nfunctions $p,q \\colon Q_1 \\to K[[x]]$ given by\n$p_\\alpha(x) = P'_\\alpha(x)$ and $q_\\alpha(x) = R'_\\alpha(x)$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}\nIf $p_\\alpha(x)$ and $q_\\alpha(x)$ are monomials, that is,\nthere exist $f$-invariant function $\\ell \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nand $g$-invariant function $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ such that\n$p_\\alpha(x)=x^{\\ell_\\alpha-1}$ and $q_\\alpha(x) = x^{m_\\alpha-1}$\nfor any $\\alpha \\in Q_1$, then the (non-complete) Jacobian algebra of the\nhyperpotential in Proposition~\\ref{p:hyperib}\\eqref{it:hyperib} is the one\nassociated by Bocklandt to a weighted quiver polyherdon~\\cite{Bocklandt13}.\nIn particular, if $P_\\alpha(x)=Q_\\alpha(x)=x$ for any $\\alpha \\in Q_1$,\nthen the potential in Proposition~\\ref{p:hyperib}\\eqref{it:potrib} is the\npotential arising from the dimer model corresponding to $(Q,f)$, see\nSection~\\ref{ssec:dimer}.\n\\end{remark}\n\nWe are now ready to define what a triangulation algebra is.\n\n\\begin{definition} \\label{def:triang}\nLet $(Q,f)$ be a triangulation quiver. Let $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nand $c \\colon Q_1 \\to K^{\\times}$ be $g$-invariant functions of\n\\emph{multiplicities} and \\emph{scalars}, respectively, and assume\nthat $m_\\alpha n_\\alpha \\geq 2$ for any $\\alpha \\in Q_1$.\nLet $\\lambda \\colon Q_1^f \\to K$, i.e.\\ $\\lambda$ is an assignment of a scalar\n$\\lambda_\\alpha \\in K$ for each $\\alpha \\in Q_1$ such that\n$f(\\alpha)=\\alpha$.\n\nThe \\emph{triangulation algebra} $\\Lambda(Q,f,m,c,\\lambda)$\nassociated to these data is the quotient\n$\\Lambda(Q,f,m,c,\\lambda) = \\widehat{KQ}\/\\overline{\\mathcal{J}}$\nof the complete path algebra $\\widehat{KQ}$ by the closure of the ideal\n$\\mathcal{J}$ generated by the commutativity-relations\n\\begin{equation} \\label{e:triangJ}\n\\begin{split}\n\\mathcal{J} = \\Bigl(&\n\\left\\{\\bar{\\alpha} \\cdot f(\\bar{\\alpha}) - c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n\\right\\}_{\\alpha \\in Q_1 \\,:\\, f(\\bar{\\alpha}) \\neq \\bar{\\alpha}},  \\\\\n& \\left\\{\\bar{\\alpha}^2 - \\lambda_{\\bar{\\alpha}} \\bar{\\alpha}^3\n- c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n\\right\\}_{\\alpha \\in Q_1 \\,:\\, f(\\bar{\\alpha}) = \\bar{\\alpha}}\n\\Bigr)\n\\end{split}\n\\end{equation}\n(when the set $Q_1^f$ is empty then evidently $\\lambda$ does not play any role\nin the definition).\n\\end{definition}\n\nThe data defining a triangulation algebra can be used to define an\n$f$-invariant function $p \\colon Q_1 \\to K[[x]]$ and a $g$-invariant function\n$q \\colon Q_1 \\to K[[x]]$ as follows; set $p_\\alpha(x)=x^2-\\lambda_\\alpha x^3$\nif $\\alpha \\in Q_1^f$ and $p_\\alpha(x)=1$ otherwise. Similarly, set\n$q_\\alpha(x) = c_\\alpha x^{m_\\alpha-1}$ for any $\\alpha \\in Q_1$.\nWe observe that the commutativity-relation in~\\eqref{e:triangJ} corresponding\nto an arrow $\\alpha \\in Q_1$ equals the element $\\rho_{g^{-1}(\\alpha)}$ of the\nhyperpotential considered in Proposition~\\ref{p:hyperib} arising from the\nfunctions $p$ and $q$. This yields the following basic property of\ntriangulation algebras.\n\n\\begin{proposition} \\label{p:potential}\nLet $(Q,f)$ be a triangulation quiver. Let $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nand $c \\colon Q_1 \\to K^{\\times}$ be $g$-invariant functions of\nmultiplicities and scalars, respectively and let $\\lambda \\colon Q_1^f \\to K$.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nThe triangulation algebra $\\Lambda(Q,f,m,c,\\lambda)$ is always a Jacobian algebra\nof a hyperpotential on $Q$.\n\n\\item\nLet $\\mu_g = \\operatorname{lcm}(\\{m_\\alpha\\}_{\\alpha \\in Q_1})$ and let\n\\[\n\\mu_f = \n\\begin{cases}\n6 & \\text{if $Q_1^f$ is non-empty and $\\lambda_\\alpha \\neq 0$ for some\n$\\alpha \\in Q_1^f$,} \\\\\n3 & \\text{if $Q_1^f$ is non-empty and $\\lambda_\\alpha = 0$ for any\n$\\alpha \\in Q_1^f$,} \\\\\n1 & \\text{if $Q_1^f$ is empty.}\n\\end{cases}\n\\]\nIf $\\operatorname{char} K$ does not divide $\\mu_f \\mu_g$, then  $\\Lambda(Q,f,m,c,\\lambda)$\nis a Jacobian algebra of a potential on $Q$.\n\\end{enumerate}\n\\end{proposition}\n\n\\sloppy\nAdditional properties of triangulation algebras will be presented in\nSection~\\ref{sec:triangquasi}. Since these algebras are given as quotients by\nclosure of ideals generated by commutativity-relations, \\emph{a-priori} it is\nnot even clear from the outset if they are finite-dimensional or not.\nHowever, it turns out that under some mild conditions on the auxiliary data,\nthis is indeed the case as the closure $\\overline{\\mathcal{J}}$ contains sufficiently\nmany paths (that is, zero-relations), see Section~\\ref{ssec:findim}.\n\n\\fussy\nIt turns out that the concept of triangulation algebra is versatile enough to\ncapture two seemingly unrelated classes of algebras occurring in the literature.\nIndeed,\n\\begin{itemize}\n\\item\nmany algebras of quaternion type are in fact triangulation algebras\n(see Section~\\ref{ssec:quat}); and\n\n\\item\nfor many triangulations of closed surfaces with punctures, the Jacobian\nalgebras of the quivers with potentials associated by\nLabardini-Fragoso~\\cite{Labardini09} are triangulation algebras (Section~\\ref{ssec:Jaclosed}).\n\\end{itemize}\n\nLet us quickly discuss the triangulation algebras on the triangulation quivers\nwith small number of vertices shown in Table~\\ref{tab:quivers}\nand refer to the relevant statements in the sequel. Under some mild\nconditions on the auxiliary data, the triangulation algebras on the quivers\n$1$, $2$, $3a$ and $3b$ are algebras of quaternion type\n(Remark~\\ref{rem:quat}); triangulation algebras on the quiver $1$ are\nfurther discussed in Section~\\ref{ssec:triang1}, whereas those on the quivers\n$2$ and $3b$ are considered in Lemma~\\ref{l:quatriang}.\nA triangulation algebra on the quiver $3''$ with all multiplicities set to\n$1$ coincides with the Jacobian algebra of the quiver with potential\nassociated with a triangulation of a torus with one puncture\n(a special case of Proposition~\\ref{p:QPtriang}); this algebra\nhas been considered in~\\cite[Example~8.2]{Labardini09b}\nand~\\cite[Example~4.3]{Plamondon13}.\n\nIn order to complete the picture and also to provide some concrete examples,\nthe triangulation algebras on the quivers $3a$ and $3'$ are given in the next\ntwo examples.\n\n\\begin{example} \\label{ex:triang3a}\nLet $(Q,f)$ be the triangulation quiver $3a$ of Table~\\ref{tab:quivers} shown\nin the picture below\n\\[\n\\xymatrix{\n{\\bullet_1} \\ar@(ul,dl)[]_{\\alpha} \\ar@<-0.5ex>[r]_{\\beta}\n& {\\bullet_2} \\ar@<-0.5ex>[r]_{\\delta} \\ar@<-0.5ex>[l]_{\\gamma}\n& {\\bullet_3} \\ar@(dr,ur)[]_{\\xi} \\ar@<-0.5ex>[l]_{\\eta}\n}\n\\]\nwith $f=(\\alpha \\beta \\gamma)(\\xi \\eta \\delta)$.\nThen $g=(\\alpha)(\\beta \\delta \\eta \\gamma)(\\xi)$ has three cycles and\nany $g$-invariant function $\\nu$ on $Q_1$ satisfies\n$\\nu_\\beta = \\nu_\\gamma = \\nu_\\delta = \\nu_\\eta$, hence it depends on\nthree values which by abuse of notation will be denoted by $\\nu_1, \\nu_2, \\nu_3$\nwhere $\\nu_1=\\nu_\\alpha$, $\\nu_2 = \\nu_\\beta$ and $\\nu_3=\\nu_\\xi$.\n\nThe auxiliary data needed to define a triangulation algebra on $(Q,f)$\nthus consists of three positive integer multiplicities $m_1, m_2, m_3$\nsatisfying $m_1, m_3 \\geq 2$ and three scalars $c_1, c_2, c_3 \\in K^{\\times}$\n(the function $\\lambda$ has empty domain and hence can be ignored).\nThe triangulation algebra  $\\Lambda(Q,f,m,c,\\lambda)$ is the quotient of the\ncomplete path algebra of $Q$ by the closure of the ideal generated by the six\nelements\n\\begin{align*}\n&\\beta \\gamma - c_1 \\alpha^{m_1-1}, &&\n\\alpha \\beta - c_2 (\\beta \\delta \\eta \\gamma)^{m_2-1} \\beta \\delta \\eta, &&\n\\delta \\xi - c_2 (\\gamma \\beta \\delta \\eta)^{m_2-1} \\gamma \\beta \\delta, \\\\\n&\\eta \\delta - c_3 \\xi^{m_3-1}, &&\n\\xi \\eta - c_2 (\\eta \\gamma \\beta \\delta)^{m_2-1} \\eta \\gamma \\beta, &&\n\\gamma \\alpha - c_2 (\\delta \\eta \\gamma \\beta)^{m_2-1} \\delta \\eta \\gamma.\n\\end{align*}\n\\end{example}\n\n\\begin{example}\nLet $(Q,f)$ be the triangulation quiver $3'$ of Table~\\ref{tab:quivers} shown\nin the picture below\n\\[\n\\xymatrix@=1pc{\n& {\\bullet_3} \\ar@(ur,ul)[]_{\\alpha_3} \\ar[ddl]_{\\beta_3} \\\\ \\\\\n{\\bullet_1} \\ar@(ul,dl)[]_{\\alpha_1} \\ar[rr]_{\\beta_1}\n&& {\\bullet_2} \\ar@(dr,ur)[]_{\\alpha_2} \\ar[uul]_{\\beta_2}\n}\n\\]\nwith $f=(\\alpha_1)(\\alpha_2)(\\alpha_3)(\\beta_1 \\beta_2 \\beta_3)$.\nThen $g=(\\alpha_1 \\beta_1 \\alpha_2 \\beta_2 \\alpha_3 \\beta_3)$ has one cycle,\nthe set $Q_1^f$ consists of the arrows $\\alpha_1, \\alpha_2, \\alpha_3$ and the\nauxiliary data needed to define a triangulation algebra on $(Q,f)$ consists of\na multiplicity $m \\geq 1$, one scalar $c \\in K^{\\times}$ and three scalars\n$\\lambda_1, \\lambda_2, \\lambda_3 \\in K$.\n\nThe triangulation algebra  $\\Lambda(Q,f,m,c,\\lambda)$ is the quotient of the\ncomplete path algebra of $Q$ by the closure of the ideal generated by the six\nelements\n\\begin{align*}\n&\\beta_i \\beta_{i+1} -\nc (\\alpha_i \\beta_i \\alpha_{i+1} \\beta_{i+1} \\alpha_{i-1} \\beta_{i-1})^{m-1}\n\\alpha_i \\beta_i \\alpha_{i+1} \\beta_{i+1} \\alpha_{i-1} \n&& (1 \\leq i \\leq 3) \\\\\n&\\alpha_i^2 - \\lambda_i \\alpha_i^3 - \nc (\\beta_i \\alpha_{i+1} \\beta_{i+1} \\alpha_{i-1} \\beta_{i-1} \\alpha_i)^{m-1}\n\\beta_i \\alpha_{i+1} \\beta_{i+1} \\alpha_{i-1} \\beta_{i-1}\n&& (1 \\leq i \\leq 3)\n\\end{align*}\nwhere index arithmetic is taken modulo $3$ (i.e.\\ $3+1=1$ and $1-1=3$).\n\\end{example}\n\n\\begin{definition}\nWe say that a $g$-invariant multiplicity function\n$m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ is \\emph{admissible}\nif $m_\\alpha n_\\alpha \\geq 3$ for any arrow $\\alpha \\in Q_1$.\n\\end{definition}\n\nOne needs to check the condition in the definition only for the arrows $\\alpha$\nwith $n_\\alpha \\leq 2$. In particular, these arrows occur as loops or as\npart of $2$-cycles. The admissibility condition thus reads as follows:\nif $n_\\alpha=1$ then $m_\\alpha \\geq 3$, while\nif $n_\\alpha=2$ then $m_\\alpha \\geq 2$.\nNote that when the pair $(n_\\alpha,m_\\alpha)$ equals $(1,2)$ or $(2,1)$ the\ntriangulation algebra is defined but the multiplicity is not admissible,\nand when it equals $(1,1)$ the triangulation algebra is not even defined.\n\n\\begin{example}\nFor the triangulation quiver $3a$ of Table~\\ref{tab:quivers} considered in\nExample~\\ref{ex:triang3a}, one has $n_\\alpha = n_\\xi = 1$ and $n_\\beta = 4$.\nHence the multiplicity function $m$ is admissible if and only if\n$m_\\alpha \\geq 3$ and $m_\\xi \\geq 3$.\n\\end{example}\n\nWe conclude this section by a series of remarks concerning the definition of\ntriangulation algebras and possible extensions thereof. The reader might skip\nthese remarks on first reading.\n\n\\begin{remark}\nSince the path $\\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'$ is of length\n$m_\\alpha n_\\alpha - 1$, the definition of a triangulation algebra makes\nperfect sense when $m_\\alpha n_\\alpha=2$ for some arrow $\\alpha$, but in this\ncase the right hand side of the corresponding commutativity-relation is just\n$c_\\alpha \\alpha$, so the arrow $\\alpha$ could be eliminated from $Q$\ncomplicating somewhat the remaining relations.\nThe admissibility condition ensures that the generating relations lie\nin the square of the ideal generated by all arrows of $Q$ so that no arrows\nhave to be deleted, compare with Remark~\\ref{rem:BGAelim} for Brauer graph\nalgebras.\n\\end{remark}\n\n\\begin{remark}\nWhen $\\operatorname{char} K \\neq 2$, the scalars $\\lambda_{\\alpha}$ occurring in the\ndefinition of a triangulation algebra do not play any role, \ni.e.\\ $\\Lambda(Q,f,m,c,\\lambda) \\simeq \\Lambda(Q,f,m,c,\\mathbf{0})$.\nThis can be shown by considering the automorphism of\n$\\widehat{KQ}$ sending each arrow $\\alpha$ with $f(\\alpha)=\\alpha$ \nto $\\alpha - (\\lambda_\\alpha\/2) \\alpha^2$ and keeping the other arrows\nunchanged. For the proof ones needs to know the additional zero relations\nthat hold in a triangulation algebra given in Proposition~\\ref{p:qrel}.\n\\end{remark}\n\n\\begin{remark}\nEven if $\\lambda_\\alpha=0$ for all the arrows $\\alpha \\in Q_1^f$,\nthere may be different $g$-invariant functions of scalars\n$c,c' \\colon Q_1 \\to K^{\\times}$ yielding isomorphic triangulation algebras,\nthat is,\n\\[\n\\Lambda(Q,f,m,c,\\mathbf{0}) \\simeq \\Lambda(Q,f,m,c',\\mathbf{0}),\n\\]\nbut in this survey we will not pursue a systematic study of this equivalence\nrelation on $g$-invariant functions of scalars.\n\\end{remark}\n\n\\begin{remark} \\label{rem:trianga}\nIt is possible to slightly generalize Definition~\\ref{def:triang} by\nconsidering\nalso an $f$-invariant function $a \\colon Q_1 \\to K^{\\times}$ of scalars\nand setting\n\\begin{align*}\n\\mathcal{J} = \\Bigl(&\n\\left\\{a_{\\bar{\\alpha}} \\bar{\\alpha} \\cdot f(\\bar{\\alpha}) - c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n\\right\\}_{\\alpha : f(\\bar{\\alpha}) \\neq \\bar{\\alpha}},  \\\\\n& \\left\\{a_{\\bar{\\alpha}} \\bar{\\alpha}^2 - \\lambda_{\\bar{\\alpha}} \\bar{\\alpha}^3\n- c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n\\right\\}_{\\alpha : f(\\bar{\\alpha}) = \\bar{\\alpha}}\n\\Bigr) \n\\end{align*}\n(the current definition uses the constant function $a=\\mathbf{1}$).\n\nAll the results of Section~\\ref{sec:triangquasi} are valid also in this\nmore general setting, but for simplicity, we chose to present the material\nwithout these extra scalars, since in many cases this apparent generalization\ndoes not yield any new algebras.\nIndeed, by using scalar transformation of the arrows\nand replacing the scalar function $c$ by another $g$-invariant function\n$c' \\colon Q_1 \\to K^{\\times}$, we may always assume\nthat $a_{\\alpha}=1$ for any arrow with $f(\\alpha) \\neq \\alpha$, and if\n$\\alpha$ is an arrow such that $f(\\alpha) = \\alpha$ and the\nground field $K$ contains a third root of $a_\\alpha$, we may assume\nthat $a_{\\alpha}=1$ as well. This holds in particular when $K$ is\nalgebraically closed.\n\\end{remark}\n\n\\begin{remark} \\label{rem:gentrian}\nOne could define an even more general version of a triangulation algebra by\nutilizing the full power of Proposition~\\ref{p:hyperib},\ntaking an $f$-invariant function $p \\colon Q_1 \\to K[[x]]^{\\times}$ of\ninvertible power series, a $g$-invariant function $q \\colon Q_1 \\to\nK[[x]] \\setminus \\{0\\}$ such that the least order term\n$c_\\alpha x^{m_\\alpha-1}$ of\neach $q_\\alpha(x)$ satisfies $m_\\alpha n_\\alpha \\geq 2$,\nand forming the quotient of the complete path algebra $\\widehat{KQ}$ by the\nclosure of the ideal $\\mathcal{J}$ given by\n\\[\n\\mathcal{J} = \\left(\n\\bigl\\{\np_{\\bar{\\alpha}}(\\xi_{\\bar{\\alpha}}) \\cdot \\bar{\\alpha} \\cdot f(\\bar{\\alpha}) -\nq_{\\alpha}(\\omega_\\alpha) \\cdot \\omega_\\alpha' \\bigr\\}_{\\alpha \\in Q_1}\n\\right).\n\\]\n\nHowever, it turns out by using techniques similar to that in the proof of\nTheorem~\\ref{t:quasi} that if the induced multiplicity function\n$m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ is admissible and $((Q,f),m)$ is not exceptional\n(see Section~\\ref{ssec:exceptional} below)\nthen the algebra $\\widehat{KQ}\/\\bar{\\mathcal{J}}$ already occurs as an algebra of the form\ndiscussed in Remark~\\ref{rem:trianga} above,\ncompare with Remark~\\ref{rem:genBGA} for Brauer graph algebras.\n\nNevertheless, for some triangulation quivers with non-admissible multiplicities\nthis generalized version does yield new algebras, see for example\nProposition~\\ref{p:newquat} describing some new algebras of quaternion type not\nappearing in the known lists.\n\\end{remark}\n\n\n\\subsection{Example -- triangulation algebras with one vertex}\n\\label{ssec:triang1}\n\nIn this section we work out in some detail the case of triangulation algebras\nwith one vertex. Already in this rather special case, one is able to\ndemonstrate many of the ideas and techniques that apply also in the general\ncase to be treated in Section~\\ref{ssec:findim}.\n\nRecall that the only triangulation quiver $(Q,f)$ with one\nvertex has two loops $\\alpha$ and $\\beta$ with $f$ being the identity function\n(Example~\\ref{ex:triang1}).\nHence $\\bar{\\alpha} = \\beta$, $\\bar{\\beta}=\\alpha$\nand $\\omega_\\alpha = \\alpha \\beta$, $\\omega_\\alpha' = \\alpha$,\n$\\omega_\\beta = \\beta \\alpha$, $\\omega'_\\beta = \\beta$.\nSince there is only one $g$-cycle, the multiplicities and scalars\nare given by an integer $m \\geq 1$ and some $c \\in K^{\\times}$.\nIn addition, there are parameters $\\lambda_\\alpha, \\lambda_\\beta \\in K$\ncorresponding to the fixed points of $f$.\n\nThe triangulation algebra is the quotient\n$\\Lambda$ = $\\widehat{K \\langle \\alpha, \\beta \\rangle} \/ \\bar{\\mathcal{J}}$\n(see the notation in Example~\\ref{ex:BGAf1}),\nwhere the generators of the ideal $\\mathcal{J}$ are given by\n\\begin{align*}\n\\alpha^2 - \\lambda_\\alpha \\alpha^3 - c (\\beta \\alpha)^{m-1} \\beta &,&\n\\beta^2 - \\lambda_\\beta \\beta^3 - c (\\alpha \\beta)^{m-1} \\alpha .\n\\end{align*}\n\nIf $m=1$, the multiplicity is not admissible and\n$\\beta \\in (\\alpha^2, \\mathcal{J})$, $\\alpha \\in (\\beta^2, \\mathcal{J})$, so by\ninduction we get $\\alpha \\in (\\alpha^{4n}, \\mathcal{J})$ for any\n$n \\geq 1$, therefore\n$\\alpha \\in \\bar{\\mathcal{J}}$ and similarly for $\\beta$.\nHence the image of the arrows $\\alpha$, $\\beta$ in $\\Lambda$ vanishes\nand $\\Lambda = K$.\n\nIf $m \\geq 2$, the multiplicity is admissible. Define elements\n$E_\\alpha, E_\\beta \\in \\widehat{K \\langle \\alpha, \\beta \\rangle}$ by\n\\begin{align*}\nE_\\alpha &= (1 - \\lambda_\\alpha \\alpha)^{-1} = \n1 + \\lambda_\\alpha \\alpha + \\lambda_\\alpha^2 \\alpha^2 + \\dots \\\\\nE_\\beta &= (1 - \\lambda_\\beta \\beta)^{-1} = \n1 + \\lambda_\\beta \\beta + \\lambda_\\beta^2 \\beta^2 + \\dots\n\\end{align*}\nThen $\\alpha^2 = E_\\alpha (\\alpha^2 - \\lambda_\\alpha \\alpha^3)$\nand $\\beta^2 = (\\beta^2 - \\lambda_\\beta \\beta^3) E_\\beta$, hence\n\\begin{align*}\n\\alpha^2 \\beta - c E_\\alpha (\\beta \\alpha)^{m-2} \\beta \\alpha \\beta^2\n&= E_\\alpha \\left(\n\\alpha^2 - \\lambda_\\alpha \\alpha^3\n- c (\\beta \\alpha)^{m-1} \\beta \\right) \\beta \\in \\mathcal{J},\n\\\\\n\\alpha \\beta^2 - c \\alpha^2 \\beta (\\alpha \\beta)^{m-2} \\alpha E_\\beta\n&= \\alpha \\left( (\\beta^2 - \\lambda_\\beta \\beta^3)\n- c (\\alpha \\beta)^{m-1} \\alpha \\right) E_\\beta \\in \\mathcal{J},\n\\end{align*}\nso\n$\\alpha^2 \\beta - u \\alpha \\beta^2 \\in \\mathcal{J}$ and\n$\\alpha \\beta^2 - \\alpha^2 \\beta v \\in \\mathcal{J}$\nfor some (infinite) linear combinations $u$ and $v$ of paths of positive\nlengths. Therefore\n\\[\n\\alpha^2 \\beta - u \\alpha^2 \\beta v \n= (\\alpha^2 \\beta - u \\alpha \\beta^2) + u(\\alpha \\beta^2 - \\alpha^2 \\beta v)\n\\in \\mathcal{J} .\n\\]\n\nIt follows that $\\alpha^2 \\beta - u^n \\alpha^2 \\beta v^n\n=\\sum_{i=0}^{n-1} u^i ( \\alpha^2 \\beta - u \\alpha^2 \\beta v ) v^i \\in \\mathcal{J}$\nfor any $n \\geq 1$, hence $\\alpha^2 \\beta \\in \\bar{\\mathcal{J}}$. Similarly,\n$\\alpha \\beta^2, \\beta^2 \\alpha, \\beta \\alpha^2 \\in \\bar{\\mathcal{J}}$ so their\nimage in the quotient $\\Lambda$ is zero.\nTherefore, in $\\Lambda$ we have\n\\[\n\\alpha^4 = E_\\alpha (\\alpha^2 - \\lambda_\\alpha \\alpha^3) \\alpha^2 =\nE_\\alpha c (\\beta \\alpha)^{m-1} \\beta \\alpha^2 = 0\n\\]\nand similarly $\\beta^4=0$. We deduce that\n\\begin{align*}\n(\\alpha \\beta)^m &= (\\alpha \\beta)^{m-1} \\alpha \\beta = \nc^{-1}(\\beta^2 - \\lambda_\\beta \\beta^3) \\beta = c^{-1} \\beta^3 =\nc^{-1} \\beta (\\beta^2 - \\lambda_\\beta \\beta^3) \\\\\n&= \\beta (\\alpha \\beta)^{m-1} \\alpha = (\\beta \\alpha)^m\n= (\\beta \\alpha)^{m-1} \\beta \\alpha =\nc^{-1}(\\alpha^2 - \\lambda_\\alpha \\alpha^3) \\alpha = c^{-1} \\alpha^3\n\\end{align*}\n(compare with Remark~\\ref{rem:dimercyc})\nand a basis for $\\Lambda$ is given by the $4m$ elements\n\\[\n\\{1\\} \\cup \\left\\{(\\alpha \\beta)^i, (\\beta \\alpha)^i \\right\\}_{0 < i < m}\n\\cup \\left\\{(\\alpha \\beta)^i \\alpha, (\\beta \\alpha)^i \\beta\n\\right\\}_{0 \\leq i < m} \\cup \\{(\\alpha \\beta)^m = (\\beta \\alpha)^m\\}\n\\]\n(compare with Proposition~\\ref{p:basis}).\n\nThe discussion above shows parts~\\eqref{it:p:rel1} and~\\eqref{it:p:dim4m}\nof the next statement.\nFor part~\\eqref{it:p:rel2} one uses similar considerations whose details\nwill appear elsewhere.\n\n\\begin{proposition}\nLet $\\Lambda$ be a triangulation algebra with one vertex.\nThen there exist parameters $m \\geq 2$, $c \\in K^{\\times}$\nand $\\lambda_\\alpha, \\lambda_\\beta \\in K$ such that the following hold.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:rel1}\n$\\Lambda = KQ\/I$ where $Q$ is the quiver\n\\[\n\\xymatrix@=1pc{\n{\\bullet} \\ar@(dl,ul)[]^{\\alpha} \\ar@(dr,ur)[]_{\\beta}\n}\n\\]\nand $I$ is the ideal of $KQ$ generated by the three elements\n\\begin{align}\n\\label{e:I1gen}\n& \\alpha^2 - c (\\beta \\alpha)^{m-1} \\beta -\nc \\lambda_\\alpha (\\beta \\alpha)^m, \\, \n\\beta^2 -  c (\\alpha \\beta)^{m-1} \\alpha - c \\lambda_\\beta (\\alpha \\beta)^m \\\\\n& \\alpha^2 \\beta .\n\\end{align}\n\n\\item \\label{it:p:rel2}\nThe ideal $I$ is generated by the elements in~\\eqref{e:I1gen} together with\nthe elements\n\\[\n(\\alpha \\beta)^m - (\\beta \\alpha)^m ,\\, \n(\\alpha \\beta)^m \\alpha ,\\, (\\beta \\alpha)^m \\beta .\n\\]\n\n\\item \\label{it:p:dim4m}\nThe algebra $\\Lambda$ is finite-dimensional and $\\dim_K \\Lambda = 4m$.\n\\end{enumerate}\n\\end{proposition}\n\nComparing the description of $\\Lambda$ with that of the local algebras of\nquaternion type in~\\cite[\\S III]{Erdmann90}, we deduce:\n\n\\begin{corollary} \\label{c:triang1}\nAn algebra of quaternion type with one simple module is a triangulation\nalgebra.\n\\end{corollary}\n\n\\subsection{Exceptional triangulation quivers with multiplicities}\n\\label{ssec:exceptional}\n\nIn proving the results on triangulation algebras one needs to distinguish\ntwo exceptional cases where the triangulation quiver is self dual and\nthe admissible multiplicities are the minimal possible.\n\n\\begin{definition}\nA pair $((Q,f),m)$ \nconsisting of a connected triangulation quiver $(Q,f)$ and\na $g$-invariant multiplicity function $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nis \\emph{exceptional}\nif $(Q,f)$ is one of the two self-dual triangulation quivers shown\nin Figure~\\ref{fig:selfdual} and the function $m$ is the following:\n\\begin{itemize}\n\\item\n$m_\\alpha=3$ and $m_\\beta = m_\\gamma = m_\\eta = 1$ in the punctured\nmonogon case (for the labeling of the arrows see row $2$ of\nTable~\\ref{tab:quivers});\n\n\\item\n$m_\\alpha=1$ for any arrow $\\alpha$ in the tetrahedron case.\n\\end{itemize}\n\\end{definition}\n\nThe next statement characterizes the exceptional triangulation quivers with\nmultiplicities among all triangulation quivers with admissible multiplicities.\n\n\\begin{proposition} \\label{p:except}\nLet $(Q,f)$ be a connected triangulation quiver and\n$m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ an admissible $g$-invariant multiplicity function.\nThen the following conditions are equivalent:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:except}\n$((Q,f),m)$ is exceptional;\n\n\\item \\label{it:p:mn3}\n$m_\\alpha n_\\alpha = 3$ for all $\\alpha \\in Q_1$;\n\n\\item \\label{it:p:mn1}\n$(m_\\alpha n_\\alpha)^{-1} + (m_{f(\\alpha)}n_{f(\\alpha)})^{-1} + \n(m_{f^2(\\alpha)} n_{f^2(\\alpha)})^{-1} = 1$ for some $\\alpha \\in Q_1$.\n\\end{enumerate}\n\\end{proposition}\n\nThe implications \\eqref{it:p:except}$\\Rightarrow$\\eqref{it:p:mn3}\nand~\\eqref{it:p:mn3}$\\Rightarrow$\\eqref{it:p:mn1} are trivial.\nAs with the other combinatorial statements of Section~\\ref{sec:quivers},\nthe proof of the implication~\\eqref{it:p:mn1}$\\Rightarrow$\\eqref{it:p:except}\nwill be given elsewhere.\n\n\n\\section{Representation-finite symmetric 2-CY-tilted algebras}\n\\label{sec:sym2CYfin}\n\nIn this section we classify all the symmetric 2-CY-tilted\nalgebras of finite representation type over an algebraically closed field.\n\nAssume that $K$ is algebraically closed and consider first Brauer graph\nalgebras. By invoking the results of Erdmann and\nSkowro\\'{n}ski on the structure of the stable Auslander-Reiten\nquiver of a self-injective special biserial\nalgebra~\\cite[Theorems~2.1 and~2.2]{ES92}, recalling\nthat $\\tau$-periodicity and $\\Omega$-periodicity are equivalent for\nsymmetric algebras, we obtain:\n\n\\begin{proposition} \\label{p:BGAperiodic}\nThe following conditions are equivalent for a Brauer graph algebra $\\Gamma$.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nAny indecomposable non-projective $\\Gamma$-module is $\\Omega_{\\Gamma}$-periodic;\n\n\\item\n$\\Gamma$ is of finite representation type.\n\\end{enumerate}\n\\end{proposition}\n\nThe class of Brauer graph algebras of finite representation type coincides\nwith that of the Brauer tree algebras.\nA \\emph{Brauer tree algebra} is a Brauer graph algebra whose underlying ribbon\ngraph is a tree and at most one node has multiplicity greater than 1\n(this node is called \\emph{exceptional}).\n\nOne of the first applications of Rickard's Morita theory for derived\ncategories was the derived equivalence classification of the Brauer\ntree algebras~\\cite[Theorem~4.2]{Rickard89}.\nRickard proved that any Brauer tree algebra is derived equivalent to a\nBrauer tree algebra whose graph has a special shape, called a\n\\emph{Brauer star}, with the same number of edges and same\nmultiplicity of the exceptional node.\n\n\\begin{figure}\n\\begin{align*}\n\\xymatrix@=1pc{\n& & {\\circ} \\\\\n{\\circ} && && {\\circ} \\\\\n{\\vdots} && {\\circ_m} \\ar@{-}[ull] \\ar@{-}[uu] \\ar@{-}[urr] && {\\vdots}\n}\n& &\n\\xymatrix@=0.5pc{\n&& {\\bullet} \\ar@(ur,ul)[]_{\\beta} \\ar[dll]_{\\alpha} \\\\\n{\\bullet} \\ar[dd]_{\\alpha} \\ar@(u,l)[]_{\\beta} && &&\n{\\bullet} \\ar[ull]_{\\alpha} \\ar@(r,u)[]_{\\beta} \\\\ \\\\\n\\ar@{.}@\/_1pc\/[rrrr] && && \\ar[uu]_{\\alpha}\n}\n\\end{align*}\n\\caption{A Brauer star and the corresponding ribbon quiver.}\n\\label{fig:star}\n\\end{figure}\n\nFigure~\\ref{fig:star} shows a Brauer star with $n$ edges and multiplicity\n$m$ of the exceptional node. The corresponding ribbon quiver is shown to\nthe right. The Brauer tree algebra has commutativity-relations of the form\n$\\beta - \\alpha^{nm}$ and zero-relations\n$\\alpha \\beta$ and $\\beta \\alpha$, hence the arrows $\\beta$ can be\neliminated and one gets a symmetric Nakayama algebra with zero-relations\n$\\alpha^{nm+1}$. Some properties of this algebra are reviewed\nin~\\cite[\\S4]{KessarLinckelmann10}. For the next statement, see \nparagraphs 4.11 and 4.12 there.\n\n\\begin{lemma} \\label{l:Bstar}\nLet $\\Gamma$ be the Brauer star algebra with $n$ simple modules and\nmultiplicity $m$ of the exceptional node.\nLet $S$ be a simple $\\Gamma$-module.\nIf $m=n=1$ then $\\Omega_\\Gamma S \\simeq S$, otherwise $\\Omega^{2n}_\\Gamma S\n\\simeq S$ but $\\Omega^i_\\Gamma S \\not \\simeq S$ for any $0 < i < 2n$.\n\\end{lemma}\n\nWe are now ready to state the classification of symmetric, 2-CY-tilted\nalgebras of finite representation type.\n\n\\begin{theorem} \\label{t:sym2CYfin}\nThe following conditions are equivalent for an indecomposable, basic,\nfinite-dimensional algebra $\\Gamma$ which is not simple.\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:t:sym2CYfin}\n$\\Gamma$ is symmetric, 2-CY-tilted of finite representation type;\n\n\\item \\label{it:t:sym4fin}\n$\\Gamma$ is symmetric of finite representation type and\n$\\Omega_\\Gamma^4 M \\simeq M$ for any $M \\in \\stmod \\Gamma$;\n\n\\item \\label{it:t:BGA2CY}\n$\\Gamma$ is a 2-CY-tilted Brauer graph algebra;\n\n\\item \\label{it:t:BGA4}\n$\\Gamma$ is a Brauer graph algebra and $\\Omega_\\Gamma^4 M \\simeq M$ for any\n$M \\in \\stmod \\Gamma$;\n\n\\item \\label{it:t:BTA2}\n$\\Gamma$ is a Brauer tree algebra with at most two simple modules;\n\n\\item \\label{it:t:BTAfig}\n$\\Gamma$ belongs to one of the three families of Brauer tree algebras shown in\nFigure~\\ref{fig:BGA2CY}.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{figure}\n\\[\n\\begin{array}{ccccc}\n\\xymatrix{\n{\\circ_m} \\ar@{-}[r] & {\\circ_1}\n}\n& \\quad &\n\\xymatrix{\n{\\circ_1} \\ar@{-}[r] & {\\circ_m} \\ar@{-}[r] & {\\circ_1}\n}\n& \\quad &\n\\xymatrix{\n{\\circ_m} \\ar@{-}[r] & {\\circ_1} \\ar@{-}[r] & {\\circ_1}\n}\n\\\\ \\\\\n\\xymatrix{\n{\\bullet} \\ar@(ul,dl)[]_{\\alpha}\n}\n& \\quad &\n\\xymatrix{\n{\\bullet} \\ar@<-0.5ex>[r]_{\\beta} & {\\bullet} \\ar@<-0.5ex>[l]_{\\gamma}\n}\n& \\quad &\n\\xymatrix{\n{\\bullet} \\ar@(ul,dl)[]_{\\alpha} \\ar@<-0.5ex>[r]_{\\beta}\n& {\\bullet} \\ar@<-0.5ex>[l]_{\\gamma}\n} \\\\\n(\\alpha^{m+1})\n& \\quad &\n\\left((\\gamma \\beta)^m \\gamma, (\\beta \\gamma)^m \\beta\\right)\n& \\quad &\n\\left(\\beta \\gamma - \\alpha^m, \\gamma \\alpha, \\alpha \\beta \\right)\n\\\\\n_{m \\geq 1}\n& \\quad &\n_{m \\geq 1}\n& \\quad &\n_{m \\geq 2}\n\\end{array}\n\\]\n\\caption{The indecomposable 2-CY-tilted symmetric algebras of\nfinite representation type that are not simple.\nThese algebras are Brauer tree algebras, and\nfor each family we show the ribbon graph with multiplicities (top);\nthe quiver, where we eliminated arrows (middle); and the corresponding\nhyperpotential (bottom).\nBy allowing the value $m=0$ in the leftmost family one includes also the\nsimple, symmetric, 2-CY-tilted algebra $K$.}\n\\label{fig:BGA2CY}\n\\end{figure}\n\nThe implications \\eqref{it:t:sym2CYfin}$\\Rightarrow$\\eqref{it:t:sym4fin}\nand~\\eqref{it:t:BGA2CY}$\\Rightarrow$\\eqref{it:t:BGA4} follow from\nProposition~\\ref{p:period}.\nThe implication~\\eqref{it:t:BGA4}$\\Rightarrow$\\eqref{it:t:BTA2} is\na consequence of Proposition~\\ref{p:BGAperiodic} and Lemma~\\ref{l:Bstar}.\nThe equivalence~\\eqref{it:t:BTA2}$\\Leftrightarrow$\\eqref{it:t:BTAfig} is\nclear. As each of the algebras shown in Figure~\\ref{fig:BGA2CY} is\na Jacobian algebra of a hyperpotential and hence 2-CY-tilted by\nProposition~\\ref{p:hyperCY}, this proves the implications\n\\eqref{it:t:BTAfig}$\\Rightarrow$\\eqref{it:t:sym2CYfin}\nand~\\eqref{it:t:BTAfig}$\\Rightarrow$\\eqref{it:t:BGA2CY}.\n\nIt remains to show that~\\eqref{it:t:sym4fin}$\\Rightarrow$\\eqref{it:t:BTA2}.\nBy Riedtmann~\\cite{Riedtmann80}, the stable Auslander-Reiten quiver of a\nself-injective algebra of finite representation type has the form\n$\\mathbb{Z}\\Delta\/\\langle \\phi \\tau^{-r} \\rangle$,\nwhere $\\Delta$ is a Dynkin graph $A_n$ ($n \\geq 1$),\n$D_n$ ($n \\geq 4$) or $E_n$ ($n=6,7,8$), $\\tau$ is the\ntranslation of $\\mathbb{Z}\\Delta$, $\\phi$ is an automorphism of $\\mathbb{Z}\\Delta$ with a\nfixed vertex and $r \\geq 1$.\nFollowing Asashiba~\\cite{Asashiba99}, these data are encoded in the\n\\emph{type} $(\\Delta, r\/(h_\\Delta-1), t)$, where $h_\\Delta$ is the Coxeter\nnumber of $\\Delta$ and $t$ is the order of $\\phi$.\n\nAsashiba~\\cite{Asashiba99} classified the self-injective algebras up to\nderived equivalence and described the possible types that\ncan occur. If $\\Gamma$ is symmetric of finite representation type,\nthen our assumption in~\\eqref{it:t:sym4fin} implies that $\\tau_{\\Gamma}^2$\nacts as the identity on the vertices of the stable Auslander-Reiten quiver\nof $\\Gamma$ and hence either $r=2$ and $t=1$ or $r=1$ and $t \\leq 2$.\nComparing this with the list of possible types in~\\cite{Asashiba99},\none gets that the type of $\\Gamma$ must be $(A_n, r\/n, 1)$\nfor some $n \\geq 1$ and $r \\leq 2$ dividing $n$.\nIn particular, $\\Gamma$ is derived equivalent to a symmetric Nakayama\nalgebra with $r \\leq 2$ simple modules.\nBy~\\cite{GabrielRiedtmann79,Rickard89}, $\\Gamma$ is a Brauer tree algebra.\n\n\\begin{remark}\nAs a consequence of Theorem~\\ref{t:sym2CYfin}, we see that the answer to\nQuestion~\\ref{q:sym2CYder} and Question~\\ref{q:sym42CY}\nis affirmative in the representation-finite case.\n\\end{remark}\n\n\\begin{remark}\nThe algebras listed in Figure~\\ref{fig:BGA2CY} occur also as the endomorphism\nalgebras of cluster-tilting objects in the 2-Calabi-Yau\nstable categories of maximal Cohen-Macaulay\nmodules over one dimensional simple hypersurface singularities\nof types $A_{2m+1}$ and $D_{2m+2}$, see Proposition~2.4 and\nProposition~2.6 in~\\cite{BIKR08}.\n\\end{remark}\n\n\n\\section{Triangulation algebras are of quasi-quaternion type}\n\\label{sec:triangquasi}\n\nLet $K$ be a field.\nConsider a connected triangulation quiver $(Q,f)$ together with\nthe following auxiliary data:\n\\begin{itemize}\n\\item\n$g$-invariant function $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ of multiplicities;\n\n\\item\n$g$-invariant function $c \\colon Q_1 \\to K^{\\times}$ of scalars;\n\n\\item\na function $\\lambda \\colon Q_1^f \\to K$, i.e.\\ a scalar $\\lambda_\\alpha \\in K$\nfor each arrow $\\alpha$ with $f(\\alpha)=\\alpha$.\n\\end{itemize}\nAssume that the following conditions hold:\n\\begin{itemize}\n\\item\n$m$ is admissible, i.e.\\ $m_\\alpha n_\\alpha \\geq 3$ for each arrow $\\alpha$;\n\n\\item\n$((Q,f),m)$ is not exceptional; or \n\n\\item\n$((Q,f),m)$ is exceptional and the scalars $c \\colon Q_1 \\to K^{\\times}$\nsatisfy\n\\begin{itemize}\n\\item\n$\\prod_{\\alpha \\in Q_1} c_\\alpha \\neq 1$ in the punctured monogon case; or\n\\item\n$c_\\alpha c_{\\bar{\\alpha}} c_{f(\\alpha)} c_{f(\\bar{\\alpha})} \\neq 1$ for some $\\alpha \\in Q_1$\nin the tetrahedron case.\n\\end{itemize}\n\\end{itemize}\n\nDenote by $\\Lambda = \\Lambda(Q,f,m,c,\\lambda)$ the triangulation algebra\n(Definition~\\ref{def:triang}) associated with the above data and by\n$\\Gamma = \\Gamma(Q,f,m,c)$ the Brauer graph algebra\n(Definition~\\ref{def:BGA}).\n\n\\begin{theorem} \\label{t:quasi}\nUnder the above conditions, we have:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:t:finite}\n$\\Lambda$ is finite-dimensional.\n\n\\item \\label{it:t:symmetric}\n$\\Lambda$ is symmetric.\n\n\\item \\label{it:t:tame}\n$\\Lambda$ is of tame representation type. Moreover, if $((Q,f),m)$ is not\nexceptional then the Brauer graph algebra $\\Gamma$ is a degeneration of $\\Lambda$.\n\n\\item \\label{it:t:potential}\n$\\Lambda$ is a Jacobian algebra of a hyperpotential\nand therefore it is 2-CY-tilted, i.e.\\ there is a 2-Calabi-Yau\ntriangulated category $\\mathcal{C}$ and a cluster-tilting object $T$ in $\\mathcal{C}$\nsuch that $\\Lambda \\simeq \\operatorname{End}_{\\mathcal{C}}(T)$.\n\n\\item \\label{it:t:quasi}\n$\\Lambda$ is of quasi-quaternion type.\n\n\\item \\label{it:t:quasimut}\nMore generally, for any cluster-tilting object $T'$ in $\\mathcal{C}$\nwhich is reachable from $T$, the 2-CY-tilted algebra $\\operatorname{End}_{\\mathcal{C}}(T')$\nis derived equivalent to $\\Lambda$ and of quasi-quaternion type.\n\\end{enumerate}\n\\end{theorem}\n\n\\begin{remark}\nThe theorem holds also when the multiplicities are not admissible\n(but still $m_\\alpha n_\\alpha \\geq 2$ for any arrow $\\alpha$),\nbut then more exceptional cases are needed to be taken care of.\n\\end{remark}\n\n\\begin{remark}\nRemark~\\ref{rem:nvertex} implies that for any positive integer $n \\geq 1$\nthere exists a triangulation quiver with $n$ vertices and hence\na triangulation algebra with $n$ simple modules.\nThe algebras of part~\\eqref{it:t:quasimut} of the theorem thus provide\nmany instances of tame, symmetric, indecomposable algebras with periodic\nmodules which seem\nto be missing from the classification announced\nin~\\cite[Theorem~6.2]{ES08} and~\\cite[Theorem~8.7]{Skowronski06}. Moreover,\nthey provide counterexamples to~\\cite[Corollary~8.8(3)]{Skowronski06} which\nclaims to bound the number of simple modules of such algebras \nof infinite representation type by $10$.\n\\end{remark}\n\nPart~\\eqref{it:t:potential} of the theorem follows from\npart~\\eqref{it:t:finite}, Proposition~\\ref{p:potential}\nand Proposition~\\ref{p:hyperCY}.\nPart~\\eqref{it:t:quasi} is a consequence of parts~\\eqref{it:t:symmetric},\n\\eqref{it:t:tame}, \\eqref{it:t:potential} and Proposition~\\ref{p:period}.\nFinally, part~\\eqref{it:t:quasimut} is a consequence of\nparts~\\eqref{it:t:symmetric}, \\eqref{it:t:potential},\nCorollary~\\ref{c:dereq}, part~\\eqref{it:t:quasi}\nand Proposition~\\ref{p:derquasi}.\nThe ideas behind the proof of parts~\\eqref{it:t:finite} and~\\eqref{it:t:tame}\nare explained in the next sections.\n\n\\subsection{Remarks on finite-dimensionality}\n\\label{ssec:findim}\n\nWe keep the notations as in the preceding section.\n\nMotivated by the dimer model perspective of Section~\\ref{ssec:dimer},\na path $\\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha)$ may be called \n\\emph{zig-zag} path. A crucial point in proving part~\\eqref{it:t:finite}\nof Theorem~\\ref{t:quasi} is the vanishing of the images of these zig-zag paths\nin $\\Lambda$, whose proof we sketch below.\nLet $\\mathcal{J}$ be the ideal defining the triangulation algebra, so that\n$\\Lambda = \\widehat{KQ} \/ \\bar{\\mathcal{J}}$, and let $\\alpha \\in Q_1$ be any arrow.\nLemma~\\ref{l:gf2fg2} implies that $gf(\\beta) = fg^{-2}(\\beta)$ for any\narrow $\\beta$, so we can repeatedly use the commutativity-relations\ndefining $\\mathcal{J}$ to deduce that\n\\[\n\\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) -\nu \\cdot \\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) \\cdot v \\in \\mathcal{J}\n\\]\nfor some $u, v \\in \\widehat{KQ}$.\n\nIf $((Q,f),m)$ is not exceptional, then one shows using\nProposition~\\ref{p:except} that $u$ and $v$ are\nlinear combinations of paths of positive length, and therefore\ndeduces that $\\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) \\in \\bar{\\mathcal{J}}$\nas done in the case discussed in Section~\\ref{ssec:triang1}.\n\nIf $((Q,f),m)$ is exceptional, then one shows that $u = C_u + u'$ and\n$v = C_v + v'$ where $u',v'$ are linear combinations of paths of positive\nlength (in the tetrahedron case even $u'=v'=0$) and\n$C_u, C_v \\in K^{\\times}$ are scalars satisfying $C_u C_v \\neq 1$ by our\nadditional assumption on the function $c \\colon Q_1 \\to K^{\\times}$. Hence\n$\\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) \\in \\bar{\\mathcal{J}}$ as well.\n\nThe next proposition provides a more refined version of part~\\eqref{it:t:finite}\nof Theorem~\\ref{t:quasi} by giving a presentation of the triangulation algebra\nas quiver with relations. \n\\begin{proposition} \\label{p:qrel}\nUnder the hypotheses of Theorem~\\ref{t:quasi}, the triangulation algebra\n$\\Lambda(Q,f,m,c,\\lambda)$ is the quotient of the path algebra $KQ$ by the\nideal generated by the elements\n\\begin{align}\n\\label{e:comm3}\n&\\bar{\\alpha} \\cdot f(\\bar{\\alpha}) - c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n&& \\alpha \\in Q_1 \\text{ and } f(\\bar{\\alpha}) \\neq \\bar{\\alpha}, \\\\\n\\label{e:comm1}\n&\\bar{\\alpha}^2 - c_\\alpha \\omega_\\alpha^{m_\\alpha-1} \\cdot \\omega_\\alpha'\n- c_\\alpha \\lambda_{\\bar{\\alpha}} \\omega_\\alpha^{m_\\alpha}\n&& \\alpha \\in Q_1 \\text{ and } f(\\bar{\\alpha}) = \\bar{\\alpha}, \\\\\n\\label{e:zigzeroF}\n&\\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) && \\alpha \\in Q_1.\n\\end{align}\n\\end{proposition}\n\n\\begin{remark}\nThe ideal of relations in Proposition~\\ref{p:qrel} is not changed if \nthe relations of type~\\eqref{e:zigzeroF} are replaced by\n\\begin{align}\n\\label{e:zigzeroG}\n&\\alpha \\cdot g(\\alpha) \\cdot fg(\\alpha) && \\alpha \\in Q_1.\n\\end{align}\n\nMoreover, it turns out that it is enough to specify a zero-relation as\nin~\\eqref{e:zigzeroF} or~\\eqref{e:zigzeroG} for just one arrow\n$\\alpha \\in Q_1$, as the relations for the other arrows would then\nfollow from the commutativity-relations~\\eqref{e:comm3} and~\\eqref{e:comm1}.\n\nIn the cases where $((Q,f),m)$ is exceptional, \nour assumption on the scalars~$c$ implies that all the\nzero-relations~\\eqref{e:zigzeroF} and~\\eqref{e:zigzeroG} already follow from\nthe relations~\\eqref{e:comm3} and~\\eqref{e:comm1},\nprovided that in the punctured monogon case the scalar $\\lambda_\\eta$\nassociated to the loop $\\eta \\in Q_1^f$ vanishes\n(no additional assumption is needed in the tetrahedron case).\n\\end{remark}\n\n\\begin{remark} \\label{rem:dimercyc}\nLet $i \\in Q_0$ and let $\\alpha, \\bar{\\alpha}$ be the two arrows starting at $i$.\nThen the images in $\\Lambda$ of the following cycles starting at $i$ are equal:\n\\[\n\\alpha \\cdot f(\\alpha) \\cdot f^2(\\alpha) =\nc_{\\bar{\\alpha}} \\omega_{\\bar{\\alpha}}^{m_{\\bar{\\alpha}}} =\nc_{\\alpha} \\omega_\\alpha^{m_\\alpha} = \\bar{\\alpha} \\cdot f(\\bar{\\alpha}) \\cdot f^2(\\bar{\\alpha}).\n\\]\nDenote this common value by $z_i \\in \\Lambda$.\n\\end{remark}\n\nThe next statement is a generalization of our result\nin~\\cite[\\S4.1]{Ladkani12} and its proof is similar.\n\n\\begin{proposition} \\label{p:basis}\nA basis of the triangulation algebra $\\Lambda(Q,f,m,c,\\lambda)$ is given\nby the images of the paths\n\\[\n\\{e_i\\}_{i \\in Q_0} \\cup\n\\left\\{ \\alpha \\cdot g(\\alpha) \\cdot \\ldots \\cdot g^r(\\alpha)\n\\right\\}_{\\alpha \\in Q_1, 0 \\leq r < m_\\alpha n_\\alpha -1} \\cup\n\\{z_i\\}_{i \\in Q_0} .\n\\]\n\\end{proposition}\n\n\\begin{example}\nIn the case of the triangulation quiver with one vertex, the same basis for the\ntriangulation algebra has been constructed in Section~\\ref{ssec:triang1}.\n\\end{example}\n\nUsing the basis of Proposition~\\ref{p:basis}, one can explicitly compute the\nCartan matrix of a triangulation algebra in terms of its defining combinatorial\ndata. As the details of the computation are similar to those given\nin~\\cite[\\S4.4]{Ladkani12}, we will state the result without proof.\n\nFor a triangulation quiver $(Q,f)$, recall that $\\Omega_g$ denotes the set\nof $g$-cycles in $Q_1$. The vectors $\\chi_\\omega \\in \\mathbb{Z}^{Q_0}$ for\n$\\omega \\in \\Omega_g$ have been defined at the end of Section~\\ref{ssec:ribbon},\nsee also the paragraph preceding Proposition~\\ref{p:CartanBGA}.\n\n\\begin{proposition} \\label{p:Cartan}\nConsider $\\Lambda=\\Lambda(Q,f,m,c,\\lambda)$ and $\\Gamma=\\Gamma(Q,f,m,c)$.\nUnder the hypotheses of Theorem~\\ref{t:quasi}, we have:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:Cartan}\n$C_\\Lambda = C_{\\Gamma} =\n\\sum_{\\omega \\in \\Omega_g} m_\\omega \\chi_\\omega^T \\chi_\\omega$.\n\n\\item \\label{it:p:dim}\n$\\dim_K \\Lambda = \\dim_K \\Gamma = \n\\sum_{\\omega \\in \\Omega_g} m_\\omega |\\omega|^2$.\n\n\\item \\label{it:p:nonneg}\nThe quadratic form $q_{C_\\Lambda} \\colon \\mathbb{Z}^{Q_0} \\to \\mathbb{Z}$ defined by\n$q_{C_\\Lambda}(x) = x C_{\\Lambda} x^T$ takes non-negative even values;\nin particular it is non-negative definite.\n\n\\item \\label{it:p:rank}\n$\\operatorname{rank} C_\\Lambda \\leq |\\Omega_g| \\leq |Q_0|$.\n\n\\item \\label{it:p:det0}\n$\\det C_\\Lambda \\neq 0$ if and only if $(Q,f)$ is one of the\ntriangulation quivers $1$, $2$, $3a$ or $3b$ of Table~\\ref{tab:quivers}.\nIn this case, $\\det C_{\\Lambda} = 4 \\cdot \\prod_{\\omega \\in \\Omega_g} m_\\omega$.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark} \nFrom part~\\eqref{it:p:det0} we see that any triangulation algebra with\n$n>10$ simple modules provides a counterexample\nto~\\cite[Corollary~8.8(1)]{Skowronski06} which states that a tame, symmetric\nalgebra of infinite representation type with periodic modules and singular\nCartan matrix must be isomorphic to the trivial extension of a tubular algebra\n(and hence has at most $10$ simple modules).\n\\end{remark}\n\n\\begin{remark} \\label{rem:quat}\nCombining part~\\eqref{it:p:det0} of Proposition~\\ref{p:Cartan} with \npart~\\eqref{it:t:quasi} of Theorem~\\ref{t:quasi} we deduce that the\ntriangulation algebra $\\Lambda(Q,f,m,c,\\lambda)$ is of quaternion type\n(and not just of quasi-quaternion type)\nif and only if $(Q,f)$ is any of the triangulation quivers $1$, $2$, $3a$ or\n$3b$. Further details will be given in Section~\\ref{ssec:quat}.\n\\end{remark}\n\nParts~\\eqref{it:p:Cartan}, \\eqref{it:p:dim}, \\eqref{it:p:nonneg} follow from\nthe corresponding statements of Proposition~\\ref{p:CartanBGA}, observing that\nthe basis constructed in Proposition~\\ref{p:basis} for $\\Lambda$ and\nthat constructed in Lemma~\\ref{l:basisBGA} for $\\Gamma$ consist of images (in\nthe respective algebras) of the same set of paths.\nThe inequality $|\\Omega_g| \\leq |Q_0|$ in part~\\eqref{it:p:rank} is a\nconsequence of Proposition~\\ref{p:gcycles}, which also implies the ``only if''\ndirection of part~\\eqref{it:p:det0}. The ``if'' direction follows from explicit\ncalculations which are presented in the next example for the purpose of\nillustration.\n\n\\begin{example}\nFor each of the triangulation quivers $1$, $2$, $3a$ and $3b$ in\nTable~\\ref{tab:quivers} we compute the Cartan matrix of a triangulation algebra\non that quiver. Recall that for a $g$-cycle $\\omega$, the quantity $m_\\omega$\nused in Proposition~\\ref{p:Cartan} equals any of the values $m_\\alpha$ for\n$\\alpha \\in \\omega$.\n\n\\begin{enumerate}\n\\item\nThere is one $g$-cycle $(\\alpha \\beta)$ with $\\chi_{(\\alpha \\beta)} = (2)$.\nThe Cartan matrix is $(4m_\\alpha)$.\n\n\\item\nThere are two $g$-cycles $(\\alpha)$ and $(\\eta \\gamma \\beta)$ with\n$\\chi_{(\\alpha)}=(1,0)$ and $\\chi_{(\\eta \\gamma \\beta)} = (1,2)$, the Cartan\nmatrix is\n\\[\nm_\\alpha \\begin{pmatrix} 1 & 0 \\\\ 0 & 1 \\end{pmatrix}\n+ m_\\eta \\begin{pmatrix} 1 & 2 \\\\ 2 & 4 \\end{pmatrix}\n\\]\nand its determinant is $4 m_\\alpha m_\\eta$.\n\n\\item[($3a$)]\nThere are three $g$-cycles $(\\alpha)$, $(\\beta \\delta \\eta \\gamma)$ and $(\\xi)$\nwith\n$\\chi_{(\\alpha)} = (1,0,0)$,\n$\\chi_{(\\beta \\delta \\eta \\gamma)} = (1,2,1)$ and\n$\\chi_{(\\xi)} = (0,0,1)$,\nthe Cartan matrix is\n\\[\nm_\\alpha \\begin{pmatrix} 1 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} +\nm_\\beta \\begin{pmatrix} 1 & 2 & 1 \\\\ 2 & 4 & 2 \\\\ 1 & 2 & 1 \\end{pmatrix} +\nm_\\xi \\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 0 & 0 \\\\ 0 & 0 & 1 \\end{pmatrix}\n\\]\nand its determinant is\n$4 m_\\alpha m_\\beta m_\\xi$.\n\n\\item[($3b$)]\nThere are three $g$-cycles $(\\alpha_1 \\beta_1)$, $(\\alpha_2 \\beta_2)$ and\n$(\\alpha_3 \\beta_3)$ with\n$\\chi_{(\\alpha_1 \\beta_1)} = (1,1,0)$,\n$\\chi_{(\\alpha_2 \\beta_2)} = (0,1,1)$ and\n$\\chi_{(\\alpha_3 \\beta_3)} = (1,0,1)$,\nthe Cartan matrix is\n\\[\nm_{\\alpha_1}\n\\begin{pmatrix} 1 & 1 & 0 \\\\ 1 & 1 & 0 \\\\ 0 & 0 & 0 \\end{pmatrix} +\nm_{\\alpha_2}\n\\begin{pmatrix} 0 & 0 & 0 \\\\ 0 & 1 & 1 \\\\ 0 & 1 & 1 \\end{pmatrix} +\nm_{\\alpha_3}\n\\begin{pmatrix} 1 & 0 & 1 \\\\ 0 & 0 & 0 \\\\ 1 & 0 & 1 \\end{pmatrix}\n\\]\nand its determinant is\n$4 m_{\\alpha_1} m_{\\alpha_2} m_{\\alpha_3}$.\n\\end{enumerate}\n\\end{example}\n\n\\subsection{Remarks on tameness}\n\nIn this section we sketch the proof of part~\\eqref{it:t:tame} of\nTheorem~\\ref{t:quasi}. We keep the notations as in the preceding sections.\n\nFirst, we recall the notion of degeneration of algebras appearing in the\nstatement of part~\\eqref{it:t:tame}.\nFor a positive integer $d$, denote by $\\operatorname{alg}_d(K)$ the affine variety of\nassociative algebra structures with unit on the vector space $K^d$. The\ngroup $\\operatorname{GL}_d(K)$ acts on $\\operatorname{alg}_d(K)$ by transport of structure and its\norbits correspond bijectively to the isomorphism classes of\n$d$-dimensional $K$-algebras. Given two $d$-dimensional algebras\n$\\Gamma$ and $\\Lambda$ viewed as points in $\\operatorname{alg}_d(K)$, we say that $\\Gamma$ is a\n\\emph{degeneration} of $\\Lambda$ if $\\Gamma$ lies in the closure of the \n$\\operatorname{GL}_d(K)$-orbit of $\\Lambda$ in the Zariski topology of $\\operatorname{alg}_d(K)$.\n\nAssume that $((Q,f),m)$ is not exceptional and\nlet $N=\\operatorname{lcm}(m_\\alpha n_\\alpha)_{\\alpha \\in Q_1}$, so that\n$N\/(m_\\alpha n_\\alpha)$ is a positive integer for any $\\alpha \\in Q_1$.\nFor each $\\alpha \\in Q_1$,\nset\n\\[\ne_\\alpha = 1 - \\left(\n(m_\\alpha n_\\alpha)^{-1} + (m_{f(\\alpha)} n_{f(\\alpha)})^{-1}\n+ (m_{f^2(\\alpha)} n_{f^2(\\alpha)})^{-1} \\right) .\n\\]\nIf $f(\\alpha)=\\alpha$, set also $e'_\\alpha = 1 - 2 (m_\\alpha n_\\alpha)^{-1}$.\nNote that $e_\\alpha$ and $e'_\\alpha$ are rational numbers and moreover\n$N e_\\alpha$, $N e'_\\alpha$ are integers by construction.\n\nSince $((Q,f),m)$ is not exceptional, Proposition~\\ref{p:except}\nimplies that $e_\\alpha > 0$ for any arrow $\\alpha$ (hence also\n$e'_\\alpha > 0$ for any $\\alpha \\in Q_1^f$).\nFor $t \\in K$, let $I_t$ be the ideal of the path algebra $KQ$ generated by\nthe elements\n\\begin{align*}\n&\\bar{\\alpha} \\cdot f(\\bar{\\alpha})\n- c_\\alpha t^{N e_{\\bar{\\alpha}}} \\omega_\\alpha^{m_\\alpha-1} \\omega_\\alpha'\n&& \\alpha \\in Q_1 \\text{ and } f(\\bar{\\alpha}) \\neq \\bar{\\alpha}, \\\\\n&\\bar{\\alpha}^2\n- c_\\alpha t^{N e_{\\bar{\\alpha}}} \\omega_\\alpha^{m_\\alpha-1} \\omega_\\alpha'\n- c_\\alpha \\lambda_{\\bar{\\alpha}} t^{N e'_{\\bar{\\alpha}}} \\omega_\\alpha^{m_\\alpha}\n&& \\alpha \\in Q_1 \\text{ and } f(\\bar{\\alpha}) = \\bar{\\alpha}, \\\\\n& \\alpha \\cdot f(\\alpha) \\cdot gf(\\alpha) && \\alpha \\in Q_1, \\\\\n& c_\\alpha \\omega_\\alpha^{m_\\alpha} - c_{\\bar{\\alpha}} \\omega_{\\balpha}^{m_{\\bar{\\alpha}}} && \\alpha \\in Q_1,\n\\end{align*}\nand let $\\Lambda_t = KQ\/I_t$.\n\n\\begin{proposition} \\label{p:degen}\nAssume that $((Q,f),m)$ is not exceptional.\n\n\\noindent\nLet $\\Gamma=\\Gamma(Q,f,m,c)$ be the Brauer graph algebra\nand let $\\Lambda=\\Lambda(Q,f,m,c,\\lambda)$ be the triangulation algebra\nas in Theorem~\\ref{t:quasi}. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\n$\\Lambda_0 \\simeq \\Gamma$.\n\n\\item\n$\\Lambda_1 \\simeq \\Lambda$.\n\n\\item\nFor any $t \\in K^{\\times}$, the automorphism of $KQ$ defined by sending each\narrow $\\alpha$ to $t^{N\/(m_\\alpha n_\\alpha)} \\alpha$ maps $I_1$ onto $I_t$,\nhence $\\Lambda_t \\simeq \\Lambda$.\n\n\\item\n$\\Gamma$ is a degeneration of $\\Lambda$.\n\\end{enumerate}\n\\end{proposition}\n\nWe constructed a one-parameter family of algebras $\\{\\Lambda_t\\}$ for $t \\in K$\nsuch that $\\Lambda_t \\simeq \\Lambda$ for $t \\neq 0$ and $\\Lambda_0 \\simeq \\Gamma$.\nSince $\\Gamma$ is tame (Proposition~\\ref{p:BGA}), a degeneration\ntheorem of Geiss~\\cite{Geiss95} implies that $\\Lambda$ is also tame.\n\n\\begin{remark}\nIn~\\cite[\\S6]{Holm99} Holm establishes the tameness of the algebras of\nquaternion type with $2$ or $3$ simple modules\nby showing that some of them degenerate to algebras of dihedral type\nand then applying the result in~\\cite{Geiss95}. Proposition~\\ref{p:degen}\ncan be seen as a generalization of this statement to arbitrary triangulation\nquivers.\n\\end{remark}\n\n\\begin{example}\nThe algebras of quaternion type with one simple module are precisely the\nalgebras listed as items (5) and (5') in the paper~\\cite{Ringel74} by Ringel\ndealing with the representation type of local algebras, but their representation\ntype was not determined in that paper.\nTheir tameness was later established by Erdmann~\\cite[III.1.2]{Erdmann90}\nas a consequence of the result of~\\cite{BD75} mentioned in\nSection~\\ref{ssec:group}.\n\nSince these algebras are triangulation algebras (see Corollary~\\ref{c:triang1}),\nProposition~\\ref{p:degen} thus yields an alternative proof of their tameness.\nFor the purpose of illustration, let us carry out the explicit calculations.\n\nRecall from Section~\\ref{ssec:triang1} that a triangulation quiver with one\nvertex has two loops $\\alpha$ and $\\beta$ with the function $f$ being the\nidentity. Hence there is one $g$-cycle and the auxiliary algebraic data\nis given by a positive integer multiplicity $m$, which is admissible if\n$m \\geq 2$, and scalars $c \\in K^{\\times}$ and $\\lambda_\\alpha, \\lambda_\\beta\n\\in K$.\n\nTherefore $n_\\alpha = n_\\beta = 2$ and $m_\\alpha = m_\\beta = m$, hence\n$N=2m$ and\n\\begin{align*}\ne_\\alpha = e_\\beta = 1- \\frac{3}{2m} &,&\nN e_\\alpha = N e_\\beta = 2m-3, \\\\\ne'_\\alpha = e'_\\beta = 1 - \\frac{2}{2m} &,&\nN e'_\\alpha = N e'_\\beta = 2m-2,\n\\end{align*}\nso the defining relations of the algebra $\\Lambda_t$ (for any $t \\in K$)\nare given by\n\\begin{align*}\n&\\alpha^2 - c t^{2m-3} (\\beta \\alpha)^{m-1} \\beta\n- c \\lambda_\\alpha t^{2m-2} (\\beta \\alpha)^m,\\, \\beta^2 \\alpha,\\\\\n&\\beta^2 - c t^{2m-3} (\\alpha \\beta)^{m-1} \\alpha\n- c \\lambda_\\beta t^{2m-2} (\\alpha \\beta)^m,\\, \\alpha^2 \\beta,\\,\n(\\alpha \\beta)^m - (\\beta \\alpha)^m .\n\\end{align*}\n\nIf $t \\neq 0$, the linear map defined by sending $\\alpha$ to $t \\alpha$\nand $\\beta$ to $t \\beta$ induces an isomorphism between the algebras $\\Lambda_1$ \nand $\\Lambda_t$, the former being equal to the triangulation algebra associated\nwith the auxiliary data as described in Section~\\ref{ssec:triang1}.\nTherefore the algebra $\\Lambda_0$, which is precisely the Brauer graph algebra\nassociated with these data (see Example~\\ref{ex:BGAf1}), is a degeneration of\nthe corresponding triangulation algebra. It follows that the latter algebra is\nalso tame.\n\\end{example}\n\n\\begin{remark}\nWhen $((Q,f),m)$ is exceptional, Proposition~\\ref{p:degen} does not apply but\nthe triangulation algebra is still tame since it is of tubular type~\\cite{BS03},\nsee also Section~\\ref{ssec:tubular} below.\n\\end{remark}\n\n\\section{Known families of algebras as triangulation algebras}\n\\label{sec:known}\n\n\\subsection{Algebras of quaternion type}\n\\label{ssec:quat}\n\nIn~\\cite[pp.~303-306]{Erdmann90}, Erdmann gave a list of the possible quivers\nwith relations of the algebras of quaternion type, and asked whether any such\nalgebra is indeed of quaternion type~\\cite[VII.9]{Erdmann90}.\nLater, Holm~\\cite[\\S6]{Holm99}\nproved that the algebras in this list are of tame representation type.\nErdmann and Skowro\\'{n}ski~\\cite[Theorem~5.9]{ES06} proved that these algebras\nare periodic\nof period dividing $4$ by constructing projective bimodule resolutions for them\nand deduced that they are indeed of quaternion type.\nIn this section we give an alternative proof of the periodicity \nof modules for these algebras by showing that all the algebras in Erdmann's\nlist are 2-CY-tilted.\n\nAssume that the ground field is algebraically closed.\nWe say that an algebra is of\n\\emph{possibly quaternion type} if it appears in Erdmann's list.\nConsider two families of algebras in Erdmann's list whose quivers with\nrelations are shown in Figure~\\ref{fig:quat}, where \nfor the convenience of the reader we tried to keep the\nnotations as close as possible to the original ones.\nThe first family $\\mathcal{Q}(2\\mathcal{B})_1^{k,s}(a,c)$ depends on integer parameters\n$k \\geq 1$, $s \\geq 2$ such that $k+s \\geq 4$\nand scalars $a \\in K^{\\times}$ and $c \\in K$.\nIf $(k,s)=(1,3)$, one should assume that $a \\neq 1$, otherwise one could set\n$a=1$.\nThe second family $\\mathcal{Q}(3\\mathcal{K})^{a,b,c}$\ndepends on three integers $1 \\leq a \\leq b \\leq c$ such that at most one\nof them equals $1$. The scalar $d \\in K^{\\times}$ should be set to $1$,\nunless $(a,b,c)=(1,2,2)$ and then $d \\neq 1$.\n\n\\begin{figure}\n\\[\n\\begin{array}{lcl}\n\\begin{array}{l}\n\\mathcal{Q}(2\\mathcal{B})_1^{k,s}(a, c) \\\\\n_{k \\geq 1,\\, s \\geq 2,\\, k+s \\geq 4} \\\\\n_{a \\in K^{\\times}\\!, \\, c \\in K}\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet_0} \\ar@(ul,dl)[]_{\\alpha} \\ar@<-0.5ex>[r]_{\\beta}\n& {\\bullet_1} \\ar@(dr,ur)[]_{\\eta} \\ar@<-0.5ex>[l]_{\\gamma}\n}\n\\end{array}\n&\n\\begin{array}{l}\n\\alpha^2 - a (\\beta \\gamma \\alpha)^{k-1} \\beta \\gamma\n- c (\\beta \\gamma \\alpha)^k \\\\\n\\beta \\eta - (\\alpha \\beta \\gamma)^{k-1} \\alpha \\beta \\\\\n\\eta \\gamma - (\\gamma \\alpha \\beta)^{k-1} \\gamma \\alpha \\\\\n\\gamma \\beta - \\eta^{s-1} \\\\\n\\alpha^2 \\beta \\,,\\, \\gamma \\alpha^2\n\\end{array}\n\\\\ \\\\\n\\begin{array}{l}\n\\mathcal{Q}(3\\mathcal{K})^{a,b,c} \\\\\n_{1 \\leq a, \\, \\max(2,a) \\leq b \\leq c} \\\\\n_{d \\in K^{\\times}}\n\\end{array}\n&\n\\begin{array}{c}\n\\xymatrix@=1pc{\n& {\\bullet_2} \\ar@<-0.5ex>[ddl]_{\\lambda} \\ar@<-0.5ex>[ddr]_{\\eta} \\\\ \\\\\n{\\bullet_0} \\ar@<-0.5ex>[rr]_{\\beta} \\ar@<-0.5ex>[uur]_{\\kappa}\n&& {\\bullet_1} \\ar@<-0.5ex>[ll]_{\\gamma} \\ar@<-0.5ex>[uul]_{\\delta}\n}\n\\end{array}\n&\n\\begin{array}{l}\n\\beta \\delta - (\\kappa \\lambda)^{a-1} \\kappa \\\\\n\\eta \\gamma - (\\lambda \\kappa)^{a-1} \\lambda \\\\\n\\delta \\lambda - (\\gamma \\beta)^{b-1} \\gamma \\\\\n\\kappa \\eta - (\\beta \\gamma)^{b-1} \\beta \\\\\n\\lambda \\beta - d (\\eta \\delta)^{c-1} \\eta \\\\\n\\gamma \\kappa - d (\\delta \\eta)^{c-1} \\delta \\\\\n\\lambda \\beta \\gamma \\,,\\, \\kappa \\eta \\delta\n\\end{array}\n\\end{array}\n\\]\n\\caption{Quivers with relations of some algebras of possibly quaternion type.}\n\\label{fig:quat}\n\\end{figure}\n\nOur presentation slightly deviates from the lists in the existing literature.\nThe next two remarks explain the differences and the motivation behind them.\n\n\\begin{remark}\nIn the literature, the family $\\mathcal{Q}(2\\mathcal{B})^{k,s}_1(a,c)$ is defined only for\n$s \\geq 3$ and $k \\geq 1$. We extended the definition to include the case\nwhere $s=2$ and $k \\geq 2$. In this case the arrow $\\eta$ can be\neliminated from the quiver and one actually gets the algebras in another\nfamily $\\mathcal{Q}(2\\mathcal{A})^k(c)$ for $k \\geq 2$ and $c \\in K$.\nThe reason for including these algebras is to have a complete list of the\nderived equivalence classes of the algebras of (possibly) quaternion type,\nneeded in the proof of Theorem~\\ref{t:quat2CY} below. \nThus one has to modify the statements of~\\cite[Proposition~5.8]{ES06}, \\cite[Theorem~5.7]{ES08}, \\cite[Theorem~5.1]{Holm99}\nand~\\cite[Theorem~8.6]{Skowronski06} accordingly,\notherwise the derived equivalence classes of the algebras $\\mathcal{Q}(2\\mathcal{A})^2(c)$\nwould be missing.\n\\end{remark}\n\n\\begin{remark}\nThe parameter $d$ for the family $\\mathcal{Q}(3\\mathcal{K})^{a,b,c}$ does not appear in the\nliterature. In fact, in the original tables of~\\cite{Erdmann90}, the\nparameters were assumed to satisfy $2 \\leq a \\leq b \\leq c$.\nOnly in~\\cite{Holm99} one value of $1$ was allowed. Note that if $a=1$\nthen the two arrows $\\kappa$ and $\\lambda$ can be eliminated from the\nquiver and one actually gets the algebras in another family\n$\\mathcal{Q}(3\\mathcal{A})_1^{b,c}(d)$ for $b,c \\geq 2$. \nFor this family, if $(b,c)=(2,2)$, one should assume that $d \\neq 1$,\notherwise one could set $d=1$.\n\nWe also slightly modified the presentation of the zero-relations.\nIf $a \\geq 2$, then as noted in~\\cite[Theorem~VII.8.8]{Erdmann90},\nall the twelve zig-zag paths\n\\begin{align*}\n\\beta \\gamma \\kappa &,& \\beta \\delta \\eta &,&\n\\gamma \\beta \\delta &,& \\gamma \\kappa \\lambda &,&\n\\delta \\eta \\gamma &,& \\delta \\lambda \\kappa &,&\n\\eta \\delta \\lambda &,& \\eta \\gamma \\beta &,&\n\\lambda \\kappa \\eta &,& \\lambda \\beta \\gamma &,&\n\\kappa \\lambda \\beta &,& \\kappa \\eta \\delta\n\\end{align*}\nvanish, and it suffices to specify one such zero-relation.\nHowever, if $a=1$, there are zig-zag paths that do not vanish; only those\npaths containing $\\kappa$ or $\\lambda$ do vanish, and it suffices to\nspecify two zero-relations as in Figure~\\ref{fig:quat}. These relations\ncorrespond to the two zero-relations occurring in the definition of the\nfamily $\\mathcal{Q}(3\\mathcal{A})_1^{b,c}(d)$.\n\\end{remark}\n\n\\begin{figure}\n\\[\n\\begin{array}{ccc}\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n{_0} & {{^1}\\circ_{s}} \\ar@{-}[rr]^(0.4)1\n&& {{^{a^{k\/3}}}\\circ_{k}} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n}\n\\end{array}\n& \\qquad &\n\\begin{array}{c}\n\\xymatrix@=1pc{\n{{^1}\\circ_a} \\ar@{-}[rr]^2 && {{^d}\\circ_c} \\ar@{-}[ddl]^1 \\\\ \\\\\n& {{^1}\\circ_b} \\ar@{-}[uul]^0\n}\n\\end{array}\n\\\\\n\\mathcal{Q}(2\\mathcal{B})_1^{k,s}(a,c)\n& \\qquad &\n\\mathcal{Q}(3\\mathcal{K})^{a,b,c}\n\\end{array}\n\\]\n\\caption{Description of some families of algebras of (possibly) quaternion\ntype~\\cite{Erdmann90} as triangulation algebras.\nThe labeling of the edges\ncorresponds to that of the vertices in Figure~\\ref{fig:quat}.\nThe subscript at each node indicates the corresponding multiplicity whereas\nthe superscript indicates the scalar.}\n\\label{fig:quatriang}\n\\end{figure}\n\n\\begin{lemma} \\label{l:quatriang}\nAn algebra in the family $\\mathcal{Q}(2\\mathcal{B})_1^{k,s}(a,c)$ or\n$\\mathcal{Q}(3\\mathcal{K})^{a,b,c}$ is a triangulation algebra.\n\\end{lemma}\n\\begin{proof}[Proof (sketch)]\nFirst, rescaling the arrow $\\alpha$ by a factor of $a^{1\/3}$\nwe slightly change the presentation of the algebra $\\mathcal{Q}(2\\mathcal{B})_1^{k,s}(a,c)$\nand get the relations\n\\begin{align*}\n&\\alpha^2 - a^{k\/3}(\\beta \\gamma \\alpha)^{k-1} \\beta \\gamma\n- \\lambda (\\beta \\gamma \\alpha)^k\n&& \\gamma \\beta - \\eta^{s-1} \\\\\n&\\beta \\eta - a^{k\/3}(\\alpha \\beta \\gamma)^{k-1} \\alpha \\beta\n&& \\alpha^2 \\beta \\\\\n&\\eta \\gamma - a^{k\/3}(\\gamma \\alpha \\beta)^{k-1} \\gamma \\alpha\n&& \\gamma \\alpha^2\n\\end{align*}\nfor some scalar $\\lambda \\in K$.\n\nThis algebra is isomorphic to the triangulation algebra \n$\\Lambda(Q,f,m,c,\\lambda)$ with the following data:\n\\begin{itemize}\n\\item\nThe triangulation quiver $(Q,f)$ is isomorphic to quiver $2$ in \nTable~\\ref{tab:quivers}, with the permutation $f$ written in cycle form as\n$(\\alpha)(\\eta \\, \\gamma \\, \\beta)$, so that the permutation $g$ is\n$(\\alpha \\, \\beta \\, \\gamma)(\\eta)$;\n\n\\item\nIn terms of the corresponding ribbon graph, the $g$-invariant functions $m$\nand~$c$ are shown in Figure~\\ref{fig:quatriang}.\nThe multiplicities are admissible when $s \\geq 3$. The triangulation quiver\nwith multiplicities is exceptional precisely when $(k,s)=(1,3)$, and the\nassumption that $a \\neq 1$ in this case ensures that Theorem~\\ref{t:quasi}\nholds;\n\n\\item\nThe scalar $\\lambda_\\alpha$ for the loop $\\alpha$ with $f(\\alpha)=\\alpha$ \nis $\\lambda$.\n\\end{itemize}\n\nSimilarly, the algebra $\\mathcal{Q}(3\\mathcal{K})^{a,b,c}$ is a triangulation algebra\nfor the following data:\n\\begin{itemize}\n\\item\nthe triangulation quiver is isomorphic to quiver $3b$ in\nTable~\\ref{tab:quivers}, with the permutation $f$ written in cycle form as\n$(\\beta \\, \\delta \\, \\lambda)(\\kappa \\, \\eta \\, \\gamma)$,\nso that the permutation $g$ is\n$(\\beta \\, \\gamma)(\\delta \\, \\eta)(\\kappa \\, \\lambda)$;\n\n\\item\nIn terms of the corresponding ribbon graph, the $g$-invariant multiplicity\nand scalar functions are shown in Figure~\\ref{fig:quatriang}.\nThe multiplicities are admissible when $a \\geq 2$.\n\\end{itemize}\n\nIn both cases Theorem~\\ref{t:quasi} holds (even when the multiplicities are\nnot admissible) and one deduces the extra zero-relations as in\nSection~\\ref{ssec:findim}.\n\\end{proof}\n\n\\begin{theorem} \\label{t:quat2CY}\nThe following assertions are true:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nAn algebra of possibly quaternion type is 2-CY-tilted.\n\n\\item \\label{it:t:posquat}\nAn algebra of possibly quaternion type is actually of quaternion type.\n\n\\item \\label{it:t:quat2CY}\nAn algebra of quaternion type is 2-CY-tilted.\n\\end{enumerate}\n\\end{theorem}\n\\begin{proof}[Proof (sketch)]\nAn algebra of possibly quaternion type has at most three simple modules.\nThe case of one simple module was considered in Corollary~\\ref{c:triang1},\nso let $\\Lambda$ be such algebra with two or three simple modules.\nA careful look at the derived equivalences constructed by Holm~\\cite{Holm99}\nfor algebras of (possibly) quaternion type shows that there exist algebras\n$\\Lambda_0, \\Lambda_1, \\dots, \\Lambda_n$ such that:\n\\begin{itemize}\n\\item\n$\\Lambda_0$ is one of the algebras of possibly quaternion type appearing in\nFigure~\\ref{fig:quat};\n\n\\item\nFor each $0 \\leq i < n$, there exists an indecomposable projective\n$\\Lambda_i$-module $P_i$ such that $\\Lambda_{i+1} \\simeq \\operatorname{End} U^+_{P_i}(\\Lambda_i)$\nor $\\Lambda_{i+1} \\simeq \\operatorname{End} U^-_{P_i}(\\Lambda_i)$ (cf.\\ Section~\\ref{ssec:dereq});\n\n\\item\n$\\Lambda_n \\simeq \\Lambda$.\n\\end{itemize}\n\nThe algebra $\\Lambda_0$ is a triangulation algebra by Lemma~\\ref{l:quatriang},\nhence it is 2-CY-tilted by Theorem~\\ref{t:quasi}\\eqref{it:t:potential}.\nBy repeatedly applying Corollary~\\ref{c:sym2CYmut} we see that\nsince $\\Lambda_i$ is symmetric and 2-CY-tilted, so is $\\Lambda_{i+1}$.\nTherefore $\\Lambda \\simeq \\Lambda_n$ is 2-CY-tilted.\n\nPart~\\eqref{it:t:posquat} now follows from Proposition~\\ref{p:period} and\nthe tameness of the algebras of possibly quaternion type established by\nHolm~\\cite[\\S6]{Holm99}. Part~\\eqref{it:t:quat2CY} is a consequence of\nErdmann's classification, but see the caveat in Proposition~\\ref{p:newquat}\nbelow.\n\\end{proof}\n\nThe above proof also shows that all the algebras of quaternion type arise\nas algebras of the form given in Theorem~\\ref{t:quasi}\\eqref{it:t:quasimut}.\n\n\\begin{corollary} \\label{c:quat2CY}\nBlocks of finite groups with generalized quaternion defect group are\n2-CY-tilted.\n\\end{corollary}\n\n\\begin{remark}\nIt is actually possible to present all the algebras of (possibly) quaternion\ntype as Jacobian algebras of hyperpotentials and thus deduce an alternative,\ndirect proof of Theorem~\\ref{t:quat2CY}.\n\nIt is also possible to present more families in the list of algebras of\nquaternion type as triangulation algebras.\nHowever, not all the algebras of quaternion type are triangulation algebras.\nFor example, the algebras in the family $\\mathcal{Q}(3\\mathcal{C})^{k,s}$ have the quiver\n\\[\n\\xymatrix{\n{\\bullet} \\ar@<-0.5ex>[r] &\n{\\bullet} \\ar@<-0.5ex>[r] \\ar@<-0.5ex>[l] \\ar@(ur,ul)[]^{} &\n{\\bullet} \\ar@<-0.5ex>[l]\n}\n\\]\nwhich is not a triangulation quiver or obtained from one by\ndeleting arrows.\n\\end{remark}\n\n\\begin{remark}\nWhen the ground field is of characteristic zero,\nBurban, Iyama, Keller and Reiten have shown in~\\cite[\\S7]{BIKR08}\nthat certain algebras of quaternion\ntype occur as endomorphism algebras of cluster-tilting objects in\nthe 2-Calabi-Yau stable categories of maximal Cohen-Macaulay modules over\nminimally elliptic curve singularities, and hence they are 2-CY-tilted.\nMoreover, they described these algebras as quotients of the complete path\nalgebra by closed ideals. \n\nThese algebras are organized in two families, denoted\n$A_q(\\lambda)$, where $q \\geq 2$ and $\\lambda \\in K^{\\times}$, \nand $B_{p,q}(\\lambda)$, where $p,q \\geq 1$ and $\\lambda \\in K^{\\times}$.\nThe scalar $\\lambda$ could be set to~$1$ except for the algebras\n$A_2(\\lambda)$ and $B_{1,1}(\\lambda)$ corresponding to the simply elliptic\nsingularities, where one should assume $\\lambda \\neq 1$.\n\nComparing their definition in~\\cite[\\S7]{BIKR08} with\nDefinition~\\ref{def:triang}, we see that\nthe algebras $A_q(\\lambda)$ and $B_{p,q}(\\lambda)$ are triangulation algebras\nwith the triangulation quivers numbered $2$ and $3a$ of\nTable~\\ref{tab:quivers}, respectively. The corresponding\nmultiplicities and scalars are shown in Figure~\\ref{fig:elliptic}.\n\\end{remark}\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n{} & {{^{\\lambda^{-1}}}\\circ_{q+1}} \\ar@{-}[rr]^{}\n& & {{^1}\\circ_1} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@=0.75pc{\n{} & {{^1}\\circ_{p+1}} \\ar@{-}[rr]\n& & {{^1}\\circ_1} \\ar@{-}@(ul,u)[lll]_{} \\ar@{-}@(dl,d)[lll]_{}\n\\ar@{-}[rr]\n& & {{^\\lambda}\\circ_{q+1}}\n}\n\\end{array}\n\\\\\nA_q(\\lambda) \n\\text{ \\scriptsize{($q \\geq 2$, $\\lambda \\in K^{\\times}$)}}\n& &\nB_{p,q}(\\lambda)\n\\text{ \\scriptsize{($p,q \\geq 1$, $\\lambda \\in K^{\\times}$)}}\n\\end{align*}\n\\caption{Description of 2-CY-tilted algebras arising from minimally elliptic\ncurve singularities~\\cite{BIKR08} as triangulation algebras.\nThe subscript at each node indicates the corresponding multiplicity whereas\nthe superscript indicates the scalar.}\n\\label{fig:elliptic}\n\\end{figure}\n\nAs the next proposition shows, by using triangulation quivers and power series\nas in Remark~\\ref{rem:gentrian}, we are able to find algebras of quaternion\ntype which seem not to appear in the known lists. Consider a new family of\nalgebras $\\mathcal{Q}(3\\mathcal{A})_3^k$ defined for the integers $k>2$\nby the quivers with relations given in Figure~\\ref{fig:newquat}.\nBy computing their Cartan matrices, one verifies that these algebras do\nnot belong to any of the families in Erdmann's list.\n\n\\begin{figure}\n\\begin{align*}\n\\begin{array}{l}\n\\mathcal{Q}(3\\mathcal{A})_3^k \\\\\n_{k>2}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix{\n{\\bullet_1} \\ar@<-0.5ex>[r]_{\\beta} &\n{\\bullet_0} \\ar@<-0.5ex>[r]_{\\delta} \\ar@<-0.5ex>[l]_{\\gamma} &\n{\\bullet_2} \\ar@<-0.5ex>[l]_{\\eta}\n}\n\\end{array}\n& &\n\\begin{array}{l}\n\\beta \\delta \\eta - \\beta \\gamma \\beta  \\\\\n\\delta \\eta \\gamma - \\gamma \\beta \\gamma \\\\\n\\eta \\gamma \\beta - \\eta \\delta \\eta + (\\eta \\delta)^{k-1} \\eta \\\\\n\\gamma \\beta \\delta - \\delta \\eta \\delta + (\\delta \\eta)^{k-1} \\delta \\\\\n(\\delta \\eta)^k \\delta \\,,\\, (\\eta \\delta)^k \\eta\n\\end{array}\n\\end{align*}\n\\caption{A new family of algebras of quaternion type.}\n\\label{fig:newquat}\n\\end{figure}\n\n\\begin{proposition} \\label{p:newquat}\nThe algebras in the family $\\mathcal{Q}(3\\mathcal{A})_3^k$ are 2-CY-tilted and of quaternion\ntype.\n\\end{proposition}\n\\begin{proof}\nLet $\\Lambda$ be the algebra $\\mathcal{Q}(3\\mathcal{A})_3^k$ for some $k>2$.\nBy slightly modifying the proof of Lemma~5.12 in~\\cite{Holm99}, one shows that\n$U^-_{P_2}(\\Lambda)$ is a tilting complex over $\\Lambda$ whose endomorphism algebra\nis isomorphic to an algebra of the form $\\mathcal{Q}(3\\mathcal{A})_1^{k,2}$ in Erdmann's list.\nThe latter algebra has also the form $\\mathcal{Q}(3\\mathcal{K})^{1,2,k}$.\nHence, by~\\cite[Proposition~2.1]{Holm99} and Theorem~\\ref{t:quat2CY},\nthe algebra $\\Lambda$ is of quaternion type.\n\nCorollary~\\ref{c:sym2CYmut} would then imply that $\\Lambda$ is 2-CY-tilted, but\nlet us give a direct proof of this fact. Indeed, the algebra $\\Lambda$ is\na generalized version of a triangulation algebra, as considered in \nRemark~\\ref{rem:gentrian}, for the triangulation quiver $3b$ \nof Table~\\ref{tab:quivers} with the\n$f$-invariant invertible power series $p_\\alpha(x)$ all set to $1$\nand the $g$-invariant power series given on the nodes of the corresponding\nribbon graph by\n\\[\n\\xymatrix@=1pc{\n{\\circ_{1}} \\ar@{-}[rr]^2 && {\\circ_{q(x)}} \\ar@{-}[ddl]^1 \\\\ \\\\\n& {\\circ_{x}} \\ar@{-}[uul]^0\n}\n\\]\nwhere $q(x)$ is any power series such that the least order term of $q(x)-x$\nhas degree $k-1$.\n\\end{proof}\n\nAll the algebras of quasi-quaternion type constructed so far are 2-CY-tilted.\nIn view of Theorem~\\ref{t:sym2CYfin} and Theorem~\\ref{t:quat2CY},\nthe following question, which is a reformulation of\nQuestion~\\ref{q:sym42CY} in the tame case, arises naturally.\n\\begin{question}\nLet $\\Lambda$ be an algebra of quasi-quaternion type. Is $\\Lambda$ 2-CY-tilted?\n\\end{question}\n\n\\subsection{2-CY-tilted blocks}\n\nBy using results on the stable Auslander-Reiten quivers of\ntame blocks~\\cite{ES92} and wild blocks~\\cite{Erdmann95},\nErdmann and Skowro\\'{n}ski have characterized the blocks of group algebras\nwhose non-projective modules are periodic~\\cite{ES15},\nsee also~\\cite[Theorem~5.3]{ES08}.\nAs a consequence, by invoking Proposition~\\ref{p:period},\nTheorem~\\ref{t:sym2CYfin} and Corollary~\\ref{c:quat2CY}\nwe obtain the following characterization of 2-CY-tilted blocks.\n\n\\begin{proposition} \\label{p:block2CY}\nLet $B$ be a block of a group algebra over an algebraically\nclosed  field of characteristic~$p$ with defect group $D$.\nThen $B$ is a 2-CY-tilted algebra if and only if either:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\n$D$ is cyclic and $B$ has at most two simple modules; or\n\n\\item\n$p=2$ and $D$ is a generalized quaternion group.\n\\end{enumerate}\n\\end{proposition}\n\n\\subsection{Symmetric algebras of tubular type $(2,2,2,2)$}\n\\label{ssec:tubular}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{clllc}\n\\textbf{No.\\ of} & \\textbf{Algebra} &\n\\textbf{Alternative} &\n\\textbf{Marked surface} & \n\\textbf{Multiplicities} \\\\\n\\textbf{simples} & & \\textbf{description} \\\\\n\\hline\n\\rule{0pt}{2.6ex}\n2 & $A_2(\\lambda)$ & $\\mathcal{Q}(2\\mathcal{B})_1^{1,3}(\\lambda,0)$ &\npunctured monogon & $(1,3)$ \\\\\n  & $\\Lambda_3(\\lambda)$ & $\\mathcal{Q}(2\\mathcal{B})_1^{1,3}(\\lambda,\\lambda)$ \\\\\n\\hline \n\\rule{0pt}{2.6ex}\n3 & $A_1(\\lambda)$ & $\\mathcal{Q}(3\\mathcal{K})^{1,2,2}(\\lambda)$ &\nsphere, 3 punctures & $(1,2,2)$ \\\\\n\\hline\n\\rule{0pt}{2.6ex}\n6 & $T(B_i(\\lambda))$ & \\cite[Fig.\\ 1]{GKO13},\n& sphere, 4 punctures & $(1,1,1,1)$ \\\\\n& $^{1 \\leq i \\leq 4}$ & \\cite[Fig.\\ 1.6]{Jasso15} \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The symmetric algebras of tubular type $(2,2,2,2)$ and their socle\ndeformations. \nEach family depends on a parameter $\\lambda \\in K \\setminus \\{0,1\\}$.}\n\\label{tab:tubular}\n\\end{table}\n\nIn this section we show that the class of algebras considered in\nTheorem~\\ref{t:quasi} contains all the symmetric algebras of tubular type\n$(2,2,2,2)$ and their socle deformations.\nAs a consequence, Question~\\ref{q:sym42CY} has a positive\nanswer for the tame symmetric algebras of polynomial growth. For the\ndefinitions of the terms in the next proposition we refer the reader to\nthe classification of tame symmetric algebras of polynomial growth by\nSkowro\\'{n}ski~\\cite{Skowronski89} and to the surveys~\\cite{ES08,Skowronski06}.\nRecall that two self-injective algebras $\\Lambda$ and $\\Lambda'$ are \\emph{socle\nequivalent} if the factor algebras $\\Lambda\/\\operatorname{soc} \\Lambda$ and $\\Lambda'\/\\operatorname{soc} \\Lambda'$\nare isomorphic.\n\n\\begin{proposition} \\label{p:sympoly}\nLet $\\Lambda$ be a basic, indecomposable, representation-infinite tame symmetric\nalgebra of polynomial growth.\nThen the following conditions are equivalent:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:sympoly4}\n$\\Omega_{\\Lambda}^4 M \\simeq M$ for any $M \\in \\stmod \\Lambda$;\n\n\\item \\label{it:p:tubular2222}\n$\\Lambda$ is socle equivalent to a symmetric algebra of tubular type $(2,2,2,2)$;\n\n\\item \\label{it:p:sympoly2CY}\n$\\Lambda$ is a 2-CY-tilted algebra.\n\\end{enumerate}\n\\end{proposition}\n\nThe implication \\eqref{it:p:sympoly4}$\\Rightarrow$\\eqref{it:p:tubular2222}\nfollows from known results in the literature, we refer\nto~\\cite[Proposition~6.2]{BES15}, \\cite{BS02} or~\\cite[Theorem~6.1]{ES08}.\nThe implication \\eqref{it:p:sympoly2CY}$\\Rightarrow$\\eqref{it:p:sympoly4} is\na consequence of Proposition~\\ref{p:period}. We prove the implication\n\\eqref{it:p:tubular2222}$\\Rightarrow$\\eqref{it:p:sympoly2CY} by using the\nclassification of the tame symmetric algebras of tubular type and their\nsocle deformations in~\\cite{BS03,BS04}, keeping the notation introduced in\nthese papers.\n\nLet $\\Lambda$ be socle equivalent to a symmetric algebra of tubular type\n$(2,2,2,2)$. Then $\\Lambda$ may have 2, 3 or 6 simple modules.\n\nIn the case of 2 simple modules, the algebra $\\Lambda$ is either $A_2(\\lambda)$\nof~\\cite{BS03} or the non-standard $\\Lambda_3(\\lambda)$ of~\\cite{BS04}, where\n$\\lambda \\in K \\setminus \\{0,1\\}$. We observe that $A_2(\\lambda)$ is\nisomorphic to the algebra $\\mathcal{Q}(2\\mathcal{B})_1^{1,3}(\\lambda,0)$ whereas\n$\\Lambda_3(\\lambda)$ is isomorphic to the algebra\n$\\mathcal{Q}(2\\mathcal{B})_1^{1,3}(\\lambda,\\lambda)$, hence both algebras are triangulation\nalgebras by Lemma~\\ref{l:quatriang}.\n\nIn the case of 3 simple modules, the algebra $\\Lambda$ is $A_1(\\lambda)$\nof~\\cite{BS03},\nwhich is isomorphic to $\\mathcal{Q}(3\\mathcal{K})^{1,2,2}(\\lambda)$, so again\nLemma~\\ref{l:quatriang} gives that $\\Lambda$ is a triangulation algebra.\n\nIn the case of 6 simple modules, by~\\cite[Proposition~5.2]{BHS03}\nthe algebra $\\Lambda$ is the trivial extension algebra of a tubular algebra of\ntype $(2,2,2,2)$, and there are exactly four such algebras, denoted\nby $T(B_i(\\lambda))$ for $1 \\leq i \\leq 4$,\nsee~\\cite[\\S3.3]{Skowronski89} or~\\cite[\\S4]{BS02}.\nIt is instructive to compare the description of these trivial extension\nalgebras as quivers with relations with the lists of quivers with potentials\ngiven in~\\cite[Figure~1]{GKO13} or in~\\cite[Figure~1.6]{Jasso15} describing\nthe endomorphism algebras of the cluster-tilting objects within the cluster\ncategory associated to a weighted projective line with weights $(2,2,2,2)$,\nand to see that these are identical.\n\nMoreover, we observe that $T(B_4(\\lambda))$ is a triangulation algebra for the\ntriangulation quiver whose ribbon graph is the tetrahedron with all\nmultiplicities set to~$1$. The marked surfaces realizing the\nsymmetric algebras of tubular type $(2,2,2,2)$ and their socle deformations\nare summarized in Table~\\ref{tab:tubular}.\n\n\n\\subsection{Jacobian algebras from closed surfaces}\n\\label{ssec:Jaclosed}\n\nIn this section we explain how Theorem~\\ref{t:quasi}\nimplies that the Jacobian algebras of the quivers with potentials associated by\nLabardini-Fragoso to triangulations of closed surfaces with punctures are of\nquasi-quaternion type.\n\nIn~\\cite{Labardini09}, Labardini-Fragoso constructed potentials on the\nadjacency quivers of triangulations of marked surfaces and proved that flips\nof triangulations result in mutations of their associated quivers with\npotentials.\nDenote by $Q'_\\tau$ the adjacency quiver of a triangulation $\\tau$ of\na marked surface $(S,M)$ as defined by Fomin, Shapiro and\nThurston~\\cite[Definition~4.1]{FST08}\n(we use the notation $Q'_\\tau$ to distinguish it\nfrom the underlying quiver $Q_\\tau$ of the triangulation quiver associated\nto $\\tau$, see Section~\\ref{ssec:triangadj}) and let $W_\\tau$ be the\nassociated potential on $Q'_\\tau$.\nThe notion of flip occurring in the next proposition is explained later in \nSection~\\ref{ssec:flip}.\n\n\\begin{proposition}[\\protect{\\cite[Theorem~30]{Labardini09}}] \\label{p:QPflip}\nIf a triangulation $\\tau'$ of $(S,M)$ is obtained from $\\tau$ by flipping an\narc $\\gamma$, then the quiver with potential $(Q'_{\\tau'},W_{\\tau'})$ is\nright equivalent to the mutation\nof $(Q'_\\tau,W_\\tau)$ at the vertex of $Q'_\\tau$ corresponding to $\\gamma$.\n\\end{proposition}\n\nWe now assume that the surface $S$ is closed. In this case the potentials\ndepend on scalars attached to the punctures of $S$.\nFor ``nice'' triangulations of $(S,M)$, an equivalent\ndescription of the quivers with potentials was given in~\\cite{Ladkani12},\nwhere we also showed that their Jacobian algebras are finite-dimensional and\nsymmetric.\nIn particular, the scalars can be encoded as a $g$-invariant function\n$c \\colon Q_1 \\to K^{\\times}$ and the Jacobian\nalgebra of the associated potential is a triangulation algebra, where all the\nmultiplicities are set to $1$.\n\n\\begin{proposition}[\\protect{\\cite[\\S2]{Ladkani12}}] \\label{p:QPtriang}\nLet $\\tau$ be a triangulation of a closed surface which is not a sphere with\nless than four punctures, and assume that\nat each puncture there are at least three incident arcs. Then \n$Q'_\\tau=Q_\\tau$, the constant multiplicity function $\\mathbf{1}$ is\nadmissible, and the Jacobian\nalgebra $\\mathcal{P}(Q_\\tau, W_\\tau)$ is isomorphic to the triangulation algebra\n$\\Lambda(Q_\\tau, f_\\tau, \\mathbf{1}, c)$.\n\\end{proposition}\n\nLet $(S,M)$ be a closed surface which is not a sphere with less than four\npunctures. In~\\cite[\\S5]{Ladkani12} we proved the existence of a triangulation\n$\\tau$ of $(S,M)$ satisfying the condition in Proposition~\\ref{p:QPtriang}.\nTherefore Theorem~\\ref{t:quasi} applies for the triangulation algebra\n$\\mathcal{P}(Q_\\tau, W_\\tau)$.\nWe note that in the case of a sphere with exactly four\npunctures the ribbon graph of $\\tau$ is a tetrahedron and\nthe corresponding assumption on the scalars attached to the\npunctures has to be made.\n\nLet $\\mathcal{C}$ be the triangulated 2-Calabi-Yau category of \nTheorem~\\ref{t:quasi}\\eqref{it:t:potential} such that\n$\\operatorname{End}_{\\mathcal{C}}(T) \\simeq \\mathcal{P}(Q_\\tau,W_\\tau)$ for some cluster-tilting object\n$T$ of $\\mathcal{C}$.\nIt is well known that any other triangulation $\\tau'$ of $(S,M)$ can be\nobtained from $\\tau$ by a sequence of flips. Let $T'$ be the\ncluster-tilting object of $\\mathcal{C}$ obtained from $T$ by the corresponding\nsequence of Iyama-Yoshino mutations. Repeated application of\nProposition~\\ref{p:QPmutIY} and Proposition~\\ref{p:QPflip} shows that\n$\\operatorname{End}_{\\mathcal{C}}(T') \\simeq \\mathcal{P}(Q'_{\\tau'}, W_{\\tau'})$,\nhence part~\\eqref{it:t:quasimut} of Theorem~\\ref{t:quasi} applies and we get\nthe following result.\n\n\\begin{corollary} \\label{c:Jaclosed}\nLet $(S,M)$ be a closed surface which is not a sphere with less than\nfour punctures. Then the Jacobian algebras of the quivers with potentials\nassociated to the ideal triangulations of $(S,M)$ are finite-dimensional of\nquasi-quaternion type and they are all derived equivalent to each other.\nMoreover, each of these algebras arises as an algebra in\npart~\\eqref{it:t:quasimut} of Theorem~\\ref{t:quasi} for a suitable\ntriangulation quiver.\n\\end{corollary}\n\n\\begin{remark}\nThe tameness of the algebras $\\mathcal{P}(Q'_\\tau, W_\\tau)$ has also been proved \nin~\\cite{GLS16} using a different degeneration argument.\n\\end{remark}\n\n\\begin{remark}\nLabardini-Fragoso showed also that the potentials $W_\\tau$ are\nnon-degenerate~\\cite{Labardini16}, but this fact is not needed in order to\nestablish Corollary~\\ref{c:Jaclosed}.\n\\end{remark}\n\n\n\\section{Mutations}\n\\label{sec:mut}\n\nMany of the algebras occurring in part~\\eqref{it:t:quasimut} of\nTheorem~\\ref{t:quasi} are themselves triangulation algebras. In this\nsection we introduce a notion of mutation for triangulation quivers\nand study its relations to other notions of mutation in the literature\nincluding flips of triangulations, Kauer's elementary moves for Brauer graph\nalgebras~\\cite{Kauer98}, mutations of quivers with potentials~\\cite{DWZ08}\nand Iyama-Yoshino mutations~\\cite{IyamaYoshino08} within the triangulated\n2-Calabi-Yau categories appearing in Theorem~\\ref{t:quasi}.\n\n\\subsection{Mutation of triangulation quivers}\n\\label{ssec:mut}\n\nA mutation of a triangulation quiver at some vertex is a new triangulation\nquiver. We first give the definition in the case the vertex we mutate at\nhas no loops.\n\n\\begin{definition} \\label{def:mut}\nLet $(Q,f)$ be a triangulation quiver and let $k$ be a vertex of $Q$\nwithout loops. Denote by $\\alpha$, $\\bar{\\alpha}$ the two arrows that start at $k$\nand observe that our assumption on $k$ implies that there are six distinct arrows\n\\begin{align*}\n\\alpha_1 = \\alpha &,&\n\\beta_1 = f(\\alpha) &,&\n\\gamma_1 = f^2(\\alpha) &,&\n\\alpha_2 = \\bar{\\alpha} &,&\n\\beta_2 = f(\\bar{\\alpha}) &,&\n\\gamma_2 = f^2(\\bar{\\alpha})\n\\end{align*}\nwhich form two cycles of the permutation $f$.\n\nThe \\emph{mutation} of $(Q,f)$ at $k$, denoted $\\mu_k(Q,f)$,\nis the triangulation quiver $(Q',f')$\nobtained from $(Q,f)$ by performing the following steps:\n\\begin{enumerate}\n\\item\nRemove the two arrows $\\beta_1$ and $\\beta_2$;\n\n\\item\nReplace the four arrows $\\alpha_1$, $\\alpha_2$, $\\gamma_1$ and $\\gamma_2$ \nwith arrows in the opposite direction $\\alpha^*_1$, $\\alpha^*_2$,\n$\\gamma^*_1$ and $\\gamma^*_2$;\n\n\\item\nAdd new arrows $\\delta_{12}$ and $\\delta_{21}$ with\n\\begin{align*}\ns(\\delta_{12}) = s(\\gamma_1) &,&\nt(\\delta_{12}) = t(\\alpha_2) &,&\ns(\\delta_{21}) = s(\\gamma_2) &,&\nt(\\delta_{21}) = t(\\alpha_1) ,\n\\end{align*}\nsee Figure~\\ref{fig:mut}(a).\n\n\\item\nDefine the permutation $f'$ on the new set of arrows $Q'_1$ \nby $f'(\\varepsilon)=f(\\varepsilon)$ if $\\varepsilon$ is an arrow of $Q$ which has not been\nchanged, and by\n\\begin{align*}\nf'(\\alpha^*_1) = \\gamma^*_2 &,&\nf'(\\gamma^*_2) = \\delta_{21} &,&\nf'(\\delta_{21}) = \\alpha^*_1 \\\\\nf'(\\alpha^*_2) = \\gamma^*_1 &,&\nf'(\\gamma^*_1) = \\delta_{12} &,&\nf'(\\delta_{12}) = \\alpha^*_2\n\\end{align*}\nfor the other arrows.\n\\end{enumerate}\n\\end{definition}\nAt the level of the underlying quivers, this is similar to Fomin-Zelevinsky\nmutation~\\cite{FZ02}.\nNote, however, that the quivers $Q$ and $Q'$ may have $2$-cycles.\n \n\\begin{figure}\n\\[\n\\begin{array}{ccccc}\n(a) & &\n\\begin{array}{c}\n\\xymatrix@R=1.5pc{\n{\\bullet} \\ar[dr]^{\\gamma_2} & & {\\bullet} \\ar[dd]^{\\beta_1}  \\\\\n& {\\circ} \\ar[ur]^{\\alpha_1} \\ar[dl]^{\\alpha_2} \\\\\n{\\bullet} \\ar[uu]^{\\beta_2}  & & {\\bullet} \\ar[ul]^{\\gamma_1}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@R=1.5pc{\n{\\bullet} \\ar[rr]^{\\delta_{21}} && {\\bullet} \\ar[dl]^{\\alpha^*_1} \\\\\n& {\\circ} \\ar[dr]^{\\gamma^*_1} \\ar[ul]^{\\gamma^*_2} \\\\\n{\\bullet} \\ar[ur]^{\\alpha^*_2}  && {\\bullet} \\ar[ll]^{\\delta_{12}}\n}\n\\end{array}\n\\\\\n(b) & &\n\\begin{array}{c}\n\\xymatrix@R=1.5pc{\n&& {\\bullet} \\ar[dd]^{\\beta} \\\\\n& {\\circ} \\ar[ur]^{\\alpha} \\ar@(dl,ul)[]^{\\delta} \\\\\n&& {\\bullet} \\ar[ul]^{\\gamma}\n}\n\\end{array}\n& &\n\\begin{array}{c}\n\\xymatrix@R=1.5pc{\n&& {\\bullet} \\ar[dl]_{\\alpha^*} \\\\\n& {\\circ} \\ar[dr]_{\\gamma^*} \\ar@(ul,dl)[]_{\\delta^*} \\\\\n&& {\\bullet} \\ar[uu]_{\\beta^*}\n}\n\\end{array}\n\\end{array}\n\\]\n\\caption{Mutation of triangulation quivers at the middle vertex $\\circ$;\n(a) without loops; (b) with a loop fixed by the permutation $f$.\nSome of the other vertices may coincide, and only the arrows that change are\nshown.}\n\\label{fig:mut}\n\\end{figure}\n\nNext, we define mutation at a vertex with loop.\n\\begin{definition} \\label{def:mutloop}\nLet $(Q,f)$ be a triangulation quiver and let $k$ be a vertex of $Q$ with\na loop. Denote by $\\alpha$, $\\bar{\\alpha}$ the two arrows that start at $k$ and\nassume that $\\bar{\\alpha}$ is a loop.\nThe \\emph{mutation} of $(Q,f)$ at $k$, denoted $\\mu_k(Q,f)$,\nis the triangulation quiver $(Q',f')$\nobtained from $(Q,f)$ by performing the following steps:\n\\begin{enumerate}\n\\setcounter{enumi}{-1}\n\\item\nIf $g(\\bar{\\alpha})=\\bar{\\alpha}$, or if $\\alpha$ is also a loop, then set\n$(Q',f')=(Q,f)$.\n\nOtherwise, there are four distinct arrows\n\\begin{align*}\n\\alpha &,& \\beta=f(\\alpha) &,& \\gamma=f^2(\\alpha) &,& \\delta=\\bar{\\alpha}=f(\\bar{\\alpha})\n\\end{align*}\nwhich form two cycles of the permutation $f$.\n\n\\item\nReplace the four arrows $\\alpha$, $\\beta$, $\\gamma$ and $\\delta$\nby arrows in the opposite direction $\\alpha^*$, $\\beta^*$, $\\gamma^*$\nand $\\delta^*$, see Figure~\\ref{fig:mut}(b);\n\n\\item\nDefine the permutation $f'$ on the new set of arrows $Q'_1$ \nby $f'(\\varepsilon)=f(\\varepsilon)$ if $\\varepsilon$ is an arrow of $Q$ which has not been\nchanged, and by\n\\begin{align*}\nf'(\\alpha^*) = \\gamma^* &,&\nf'(\\beta^*) = \\alpha^* &,&\nf'(\\gamma^*) = \\beta^* &,&\nf'(\\delta^*) = \\delta^*\n\\end{align*}\nfor the other arrows.\n\\end{enumerate}\nNote that the arrow $\\delta^*$ is also a loop at $k$ so we could have\navoided the reversal of $\\delta$. This reversal is done in order to stress the\nanalogy to the general case of Definition~\\ref{def:mut}.\n\\end{definition}\n\n\\begin{example}\nWe describe all the mutations of the triangulation quivers appearing in\nTable~\\ref{tab:quivers}.\nFor each of the triangulation quivers $1$, $2$, $3'$ and $3''$, a mutation at\nany vertex gives an isomorphic triangulation quiver.\nFor the triangulation quiver $3b$, a mutation at any vertex is isomorphic to\nthe triangulation quiver $3a$.\nFor the triangulation quiver $3a$, a mutation at the vertex $2$ is\nisomorphic to the triangulation quiver $3b$, whereas a mutation at any of the\nother vertices gives the triangulation quiver $3a$.\n\\end{example}\n\n\\begin{remark}\nAs can be seen from Figure~\\ref{fig:mut}, mutation is an involution.\nIn other words, if $(Q,f)$ is a triangulation quiver and $k$ is a vertex of\n$Q$, then the triangulation quiver $\\mu_k(\\mu_k(Q,f))$ is isomorphic to\n$(Q,f)$.\n\\end{remark}\n\nThe permutation $f'$ on $Q'_1$ defines the permutation $g'$ by\n$g'(\\alpha')=\\overline{f'(\\alpha')}$ for any $\\alpha' \\in Q'_1$.\nThe next statement is a consequence of the definitions.\n\n\\begin{lemma} \\label{l:mutcycle}\nLet $(Q',f')$ be a mutation of the triangulation quiver $(Q,f)$\nat some vertex. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:l:fcyc}\nThe permutations $f$ and $f'$ have the same cycle structure.\n\n\\item \\label{it:l:gnum}\nThe permutations $g$ and $g'$ have the same number of cycles.\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{example} \\label{ex:mutg}\nAlthough $g$ and $g'$ have the same number of cycles, the lengths of the\ncycles may change. For example, for the triangulation quiver 3b of\nTable~\\ref{tab:quivers} the lengths are $2$, $2$, $2$ whereas for\nthe mutated triangulation quiver 3a they are $1$, $4$, $1$.\n\\end{example}\n\nLemma~\\ref{l:mutcycle}\\eqref{it:l:gnum} implies that\nany $g$-invariant function $\\nu$ gives rise to a $g'$-invariant function\n$\\nu'$ on $Q'_1$ with the same image. Explicitly, this is done by setting\n$\\nu'_\\varepsilon = \\nu_\\varepsilon$ for the arrows in $Q'_1$ that are also in $Q_1$ and\n\\begin{align*}\n\\nu'_{\\alpha^*_1} = \\nu'_{\\gamma^*_1} = \\nu_{\\beta_1} &,&\n\\nu'_{\\alpha^*_2} = \\nu'_{\\gamma^*_2} = \\nu_{\\beta_2} &,&\n\\nu'_{\\delta_{12}} = \\nu_{\\gamma_1} (= \\nu_{\\alpha_2}) &,&\n\\nu'_{\\delta_{21}} = \\nu_{\\gamma_2} (= \\nu_{\\alpha_1})\n\\end{align*}\nfor the other arrows in the case of Definition~\\ref{def:mut} and\n\\begin{align*}\n\\nu'_{\\alpha^*} = \\nu'_{\\gamma^*} = \\nu'_{\\delta^*} = \\nu_\\beta &,&\n\\nu'_{\\beta^*} = \\nu_{\\gamma} (=\\nu_{\\delta} = \\nu_{\\alpha})\n\\end{align*}\nin the case of Definition~\\ref{def:mutloop}.\nIn particular, any two $g$-invariant functions $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ and\n$c \\colon Q_1 \\to K^{\\times}$ of multiplicities and scalars on $(Q,f)$\ngive rise to $g'$-invariant functions of multiplicities\n$m' \\colon Q'_1 \\to \\mathbb{Z}_{>0}$ and scalars $c' \\colon Q'_1 \\to K^{\\times}$\non $(Q',f')$.\n\n\\begin{remark}\nSince the lengths of the cycles of $g$ may change under mutation, even\nif a multiplicity function $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ \non $(Q,f)$ was admissible, the multiplicity function $m'$ on\n$(Q',f')$ may not be admissible anymore.\n\\end{remark}\n\n\\begin{example}\nContinuing Example~\\ref{ex:mutg}, if $m$ is the multiplicity function for\nthe triangulation quiver $3b$ taking the constant value $2$,\nthen $m'$ takes the constant value $2$ on the arrows of the triangulation\nquiver $3a$. Hence $m$ is admissible while $m'$ is not.\n\\end{example}\n\nSimilarly, Lemma~\\ref{l:mutcycle}\\eqref{it:l:fcyc} implies that any function\n$\\theta$ on the set $Q_1^f$ of fixed points of $f$ gives rise to a function\n$\\theta'$ on the set $(Q'_1)^{f'}$ of fixed points of $f'$.\nExplicitly, in the case of\nDefinition~\\ref{def:mut} we have $\\theta'=\\theta$, whereas in the case of\nDefinition~\\ref{def:mutloop} we have\n$\\theta'_{\\delta^*} = \\theta_\\delta$\nand $\\theta'_\\varepsilon=\\theta_\\varepsilon$ for any\narrow $\\varepsilon \\neq \\delta$ with $f(\\varepsilon)=\\varepsilon$.\n\n\\subsection{Mutations and flips}\n\\label{ssec:flip}\n\nFomin, Shapiro and Thurston have shown in~\\cite[Proposition~4.8]{FST08} that\nif two triangulations\nare related by flipping an arc, then their adjacency quivers are related by a\nFomin-Zelevinksy mutation at the vertex corresponding to that arc. In this\nsection we discuss an analogous statement for triangulation quivers.\n\nLet $\\tau$ be a triangulation of a marked surface $(S,M)$.\nIf $\\gamma$ is an arc of $\\tau$ which is not the inner side of a self-folded\ntriangle, then it is possible to replace $\\gamma$ by another arc $\\gamma'$\nto obtain a triangulation $\\tau' = \\tau \\setminus \\{\\gamma\\} \\cup \\{\\gamma'\\}$\nwhich is not topologically equivalent to $\\tau$, see Figure~\\ref{fig:flip}.\nThe triangulation $\\tau'$ is called the \\emph{flip} of $\\tau$ at $\\gamma$.\n\n\\begin{figure}\n\\begin{align*}\n\\xymatrix{\n& {\\circ} \\ar@{-}[dr] \\ar@{-}[dd]_{\\gamma} \\\\\n{\\circ} \\ar@{-}[ur] \\ar@{-}[dr] && {\\circ} \\\\\n& {\\circ} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n&&\n\\xymatrix{\n& {\\circ} \\ar@{-}[dr] \\\\\n{\\circ} \\ar@{-}[ur] \\ar@{-}[rr]^{\\gamma'} \\ar@{-}[dr] && {\\circ} \\\\\n& {\\circ} \\ar@{-}[ul] \\ar@{-}[ur]\n}\n\\end{align*}\n\\caption{Flip of a triangulation at the arc $\\gamma$.\nThe sides of the quadrilateral may be arcs or boundary segments.}\n\\label{fig:flip}\n\\end{figure}\n\n\\begin{lemma}\nThe triangulation quivers of two triangulations related by a flip at some arc\nare related by a mutation at the vertex corresponding to that arc.\n\\end{lemma}\n\\begin{proof}\nFirst we verify that a vertex corresponding to a flippable arc cannot\nhave loops. Indeed, for a loop $\\alpha$ at some vertex $k$ we have that either\n$f(\\alpha)=\\alpha$ or $g(\\alpha)=\\alpha$. In the former case $k$ corresponds\nto a boundary segment, whereas in the latter case it corresponds to an arc\nwhich is the inner side of a self-folded triangle.\n\nNow the claim follows by comparing\nFigure~\\ref{fig:flip} and Figure~\\ref{fig:mut}(a)\nusing the construction of triangulation quiver visualized in \nFigure~\\ref{fig:triquiver}.\n\\end{proof}\n\nConsider now a mutation of a triangulation quiver $(Q,f)$\nat a vertex with a loop fixed by the permutation $f$. The change of the\nassociated ribbon graphs is illustrated in Figure~\\ref{fig:ribmut}.\nIn particular, if $\\tau$ is a triangulation of a marked surface $(S,M)$\nand $k$ is a vertex corresponding to a boundary segment of $(S,M)$, then\na mutation of $(Q_\\tau, f_\\tau)$ at $k$ is \na triangulation quiver $(Q_{\\tau'}, f_{\\tau'})$\nof a triangulation $\\tau'$ of a new marked\nsurface $(S',M')$ which is obtained from $(S,M)$ \nas follows: remove the boundary segment corresponding to $k$\nfrom the boundary component containing it represented by the left node of\nthe ribbon graph in Figure~\\ref{fig:ribmut}, and\nadd it to the component (or puncture) represented by the right node.\nThe arcs of $\\tau'$ are identical to those of $\\tau$.\n\n\\begin{figure}\n\\begin{align*}\n\\xymatrix@=1pc{\n{\\circ} \\ar@{-}@(dr,ur) \\ar@{-}@\/_2pc\/[rrr] \\ar@{-}@\/^2pc\/[rrr]\n&&& {\\circ}\n}\n&&\n\\xymatrix@=1pc{\n{\\circ} \\ar@{-}@\/_2pc\/[rrr] \\ar@{-}@\/^2pc\/[rrr]\n&&& {\\circ} \\ar@{-}@(dl,ul)\n}\n\\end{align*}\n\\caption{The mutation of Figure~\\ref{fig:mut}(b) in terms of ribbon graphs.}\n\\label{fig:ribmut}\n\\end{figure}\n\nAdding or removing a boundary segment is equivalent to adding or removing\none marked point.\nHere, it makes sense to consider punctures as boundary components with zero\nmarked points. So, when we remove a boundary segment from a component with just\none marked point we get a puncture, and conversely, when we add\na boundary segment to a puncture we get a boundary component with one marked\npoint.\n\nThis point of view can be made more systematic by using the notion of\norbifolds and their triangulations as introduced by Felikson, Shapiro and \nTumarkin~\\cite[\\S4]{FST12}. The precise details are outside the scope of this\nsurvey, but let us just mention that any marked surface $(S,M)$ gives rise to\na closed orbifold $\\mathcal{O}$ by replacing each boundary component of\n$(S,M)$ containing $n$ marked points by a puncture and $n$\norbifold points, each connected to that puncture by a so-called pending arc.\nAny triangulation of $(S,M)$ yields a triangulation of the orbifold\n$\\mathcal{O}$.\n\nThe transitivity of flips on triangulations of orbifolds implies the next\nproposition, which provides a partial converse to Lemma~\\ref{l:mutcycle}.\n\n\\begin{proposition} \\label{p:mutflip}\nLet $\\tau$ be a triangulation of a marked surface $(S,M)$ with\n$p$ punctures and $b$ boundary components, and let $\\tau'$ be a triangulation\nof a marked surface $(S',M')$ with $p'$ punctures and $b'$ boundary\ncomponents.\nThen the following conditions are equivalent:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nThe triangulation quiver $(Q_{\\tau'}, f_{\\tau'})$ can be obtained from\n$(Q_\\tau, f_\\tau)$ by a finite sequence of mutations;\n\n\\item\nThe topological parameters of the marked surfaces $(S,M)$ and $(S',M')$\nsatisfy\n\\begin{align} \\label{e:topomut}\n\\operatorname{genus}(S)=\\operatorname{genus}(S') &,& p+b=p'+b' &,&\n|M|-p=|M'|-p';\n\\end{align}\n\n\\item\nThe permutations $f_\\tau$ and $f_{\\tau'}$ have the same cycle structure\nand the permutations $g_\\tau$ and $g_{\\tau'}$ have the same number of cycles.\n\\end{enumerate}\n\\end{proposition}\n\n\\begin{remark}\nTwo closed surfaces $(S,M)$ and $(S',M')$ satisfy \\eqref{e:topomut}\nif and only if they are homeomorphic (i.e.\\ they have the same genus and\nthe same number of punctures).\n\\end{remark}\n\n\\subsection{Mutations and Kauer moves}\nRickard~\\cite[Theorem~4.2]{Rickard89} proved that a Brauer tree algebra is\nderived\nequivalent to a Brauer star algebra by constructing a tilting complex over the\nformer whose endomorphism algebra is isomorphic to the latter.\nLater, K\\\"{o}nig and Zimmermann~\\cite{KZ97} have shown that a Brauer tree can be\ntransformed to a Brauer star by applying a sequence of small changes, replacing\none edge at a time. In each such replacement, the Brauer tree algebras of the\ntwo trees are related by a tilting complex of length 2 which is of the form\ngiven in~\\eqref{e:silt}, so in particular they are derived equivalent.\n\nIn~\\cite{Kauer98}, Kauer considered more generally Brauer graph algebras\nand defined similar moves, which he called \\emph{elementary moves}.\nFor each edge $e$ of a Brauer graph he defined a new graph \nobtained by replacing $e$ (i.e.\\ taking it out and putting it back in a \ndifferent place) such that\nif $\\Gamma$ is the Brauer graph algebra corresponding to the original graph\nand $P$ is the indecomposable projective $\\Gamma$-module corresponding to\nthe edge $e$, then $\\operatorname{End}_{\\mathcal{D}^b(\\Gamma)} U^+_P(\\Gamma)$ is the Brauer graph\nalgebra corresponding to the new graph.\n\nThere are three kinds of elementary moves; the first involves edges that are\nleaves in the graph (i.e.\\ they are incident to nodes without any additional\nincident edges); the second involves edges that are loops whose two half-edges\nare successive in the cyclic ordering around their common node; and the third\ninvolves the other edges. In terms of the ribbon quiver, the first case\ncorresponds to vertices with a loop $\\alpha$ such that $g(\\alpha)=\\alpha$; the\nsecond to vertices with a loop $\\alpha$ such that $f(\\alpha)=\\alpha$; and\nthe third to vertices without loop.\n\n\\begin{proposition} \\label{p:mutKauer}\nLet $(Q,f)$ be a triangulation quiver, let $k$ be a vertex of $Q$ and\nlet $(Q',f')$ be the mutation of $(Q,f)$ at $k$. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nThe ribbon graphs of $(Q,f)$ and $(Q',f')$ are related by an elementary move\nat the edge corresponding to the vertex $k$.\n\n\\item\nLet $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$ and $c \\colon Q_1 \\to K^{\\times}$ be\n$g$-invariant functions of multiplicities and scalars, respectively, and let \n$m' \\colon Q'_1 \\to \\mathbb{Z}_{>0}$ and $c' \\colon Q'_1 \\to K^{\\times}$ be the\n$g'$-invariant functions induced from $m$ and $c$.\nThen the Brauer graph algebras\n$\\Gamma = \\Gamma(Q,f,m,c)$ and $\\Gamma'= \\Gamma(Q',f',m',c')$\nsatisfy\n\\begin{align} \\label{e:BGAmut}\n\\operatorname{End}_{\\mathcal{D}^b(\\Gamma)} U^-_{P_k}(\\Gamma) \\simeq \\Gamma' \\simeq\n\\operatorname{End}_{\\mathcal{D}^b(\\Gamma)} U^+_{P_k}(\\Gamma)\n\\end{align}\nand in particular they are derived equivalent.\n\\end{enumerate}\n\\end{proposition}\n\nIf $(Q_\\tau, f_\\tau)$ is a triangulation quiver arising from a triangulation\n$\\tau$ of a marked surface $(S,M)$, then by Remark~\\ref{rem:triblocks} we can\nthink\nof the multiplicities and scalars as quantities attached to each puncture\nand boundary component of $(S,M)$. By combining Proposition~\\ref{p:mutflip}\nand Proposition~\\ref{p:mutKauer} we deduce the next corollary which implies\nin particular that the derived equivalence class of a Brauer graph algebra\nfrom a triangulation quiver may depend only on the surface and not on the\nparticular triangulation.\n\n\\begin{corollary} \\label{c:BGAsurface}\nLet $(S,M)$ and $(S',M')$ be two marked surfaces whose topological parameters\nsatisfy Eq.~\\eqref{e:topomut}.\nLet $\\tau$ be any triangulation of $(S,M)$ and let $\\tau'$ be\nany triangulation of $(S',M')$. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nThe triangulation quiver $(Q_{\\tau'}, f_{\\tau'})$ can be\nobtained from $(Q_\\tau, f_\\tau)$ by a sequence of mutations, hence\nany $g$-invariant function $\\nu$ on $(Q_\\tau)_1$ yields a\n$g$-invariant function $\\nu'$ on $(Q_{\\tau'})_1$.\n\n\\item\nThe Brauer graph algebras \n$\\Gamma(Q_\\tau, f_\\tau, m, c)$ and $\\Gamma(Q_{\\tau'}, f_{\\tau'}, m', c')$\nare derived equivalent for any $g$-invariant function of multiplicities\n$m \\colon (Q_\\tau)_1 \\to \\mathbb{Z}_{>0}$ and scalars\n$c \\colon (Q_\\tau)_1 \\to K^{\\times}$.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{remark}\nIt has also been observed by Marsh and Schroll~\\cite{MarshSchroll14} that\nby viewing triangulations of marked surfaces as ribbon graphs, flips of\ntriangulations become elementary moves of Brauer graphs and hence a marked\nsurface gives rise to a collection of derived equivalent Brauer graph algebras.\nNote that in the\ncase of surfaces with non-empty boundary, the Brauer graph algebras they\nconsider are somewhat different than the algebras considered here.\n\\end{remark}\n\n\\begin{remark}\nRecently, a description of Kauer's elementary moves in terms of the ribbon\nquivers has been given in~\\cite{Aihara15}.\n\\end{remark}\n\n\\subsection{Mutations and quivers with potentials}\nLet $(Q,f)$ be a triangulation quiver and let $(Q',f')$ be a\nmutation of $(Q,f)$ at a fixed vertex $k$ of $Q$.\n\nLet $R \\colon Q_1 \\to K[[x]]$ be a $g$-invariant function and let\n$P \\colon Q_1^f \\to K[[x]]$ be a function whose values are power series\n(i.e.\\ $P_\\alpha(x)$ is a power series for each $\\alpha \\in Q_1$ such that \n$f(\\alpha)=\\alpha$).\nConsider the potential on $Q$ defined by\n\\begin{equation} \\label{e:potR}\nW = \\sum_{\\alpha \\,:\\, f(\\alpha) = \\alpha} P_\\alpha(\\alpha) +\n\\sum_{\\alpha \\,:\\, f(\\alpha) \\neq \\alpha}\n\\alpha \\cdot f(\\alpha) \\cdot f^2(\\alpha) -\n\\sum_\\beta R_{\\beta}(\\omega_\\beta) ,\n\\end{equation}\nwhere the first sum runs over the fixed points of $f$, the second\nruns over representatives $\\alpha$ of the $f$-cycles of length $3$\nthe third runs over representatives $\\beta$ of the $g$-cycles in $Q_1$.\nThis is a special case of a potential considered in\nProposition~\\ref{p:hyperib}\\eqref{it:potrib}, as $P$ can be extended to\nan $f$-invariant function on all the arrows by setting $P_\\alpha(x)=x$ for\nany arrow $\\alpha$ with $f(\\alpha) \\neq \\alpha$.\n\nBy the discussion in Section~\\ref{ssec:mut}, the function $R$ gives rise to a\n$g'$-invariant function $R'$ and the function $P$ gives rise to a function $P'$\non the set $(Q'_1)^{f'}$ of fixed points of $f'$, hence to the potential on\n$Q'$ given by\n\\begin{equation} \\label{e:potR2}\nW' = \\sum_{\\alpha' \\,:\\, f'(\\alpha')=\\alpha'} P'_{\\alpha'}(\\alpha') + \n\\sum_{\\alpha' \\,:\\, f'(\\alpha') \\neq \\alpha'}\n\\alpha' \\cdot f'(\\alpha') \\cdot f'^2(\\alpha') -\n\\sum_{\\beta'} R'_{\\beta'}(\\omega_{\\beta'}) ,\n\\end{equation}\nwhere the sums run over  fixed points $\\alpha'$ of $f'$, representatives\n$\\alpha'$ of the $f'$-cycles of length $3$ and representatives $\\beta'$ of\nthe $g'$-cycles in $Q'_1$.\n\nThe next proposition compares $(Q',W')$ with the mutation of the quiver\nwith potential $(Q,W)$ at the vertex $k$ as defined in~\\cite[\\S5]{DWZ08}.\n\n\\begin{proposition} \\label{p:QPmut}\nAssume that there are no $2$-cycles in $Q$ passing through the vertex~$k$.\nThen $(Q',W')$ is right equivalent to the mutation of $(Q,W)$ at $k$.\n\\end{proposition}\n\nThe assumption in the proposition implies that $k$ has no loops and\ntherefore the mutation is governed by Definition~\\ref{def:mut}.\nIn the notations of that definition, the condition in the proposition\nis equivalent to the conditions that\n$n_{\\alpha_1} > 2$, $n_{\\gamma_1} > 2$, $n_{\\beta_1} > 1$ and\n$n_{\\beta_2} > 1$.\n\n\\subsection{Mutations and triangulation algebras}\n\\sloppy\nLet $(Q,f)$ be a triangulation quiver and let $k$ be a vertex of $Q$.\nLet $m \\colon Q_1 \\to \\mathbb{Z}_{>0}$\nand $c \\colon Q_1 \\to K^{\\times}$ be $g$-invariant functions of multiplicities\nand scalars, respectively and let $\\lambda \\colon Q_1^f \\to K$. Assume that:\n\\begin{itemize}\n\\item\n$m$ is admissible;\n\n\\item\nif $((Q,f),m)$ is exceptional, the scalars $c \\colon Q_1 \\to K^{\\times}$\nsatisfy the conditions stated before Theorem~\\ref{t:quasi};\n\n\\item\n$\\operatorname{char} K$ does not divide $\\mu_f \\mu_g$ (see Proposition~\\ref{p:potential} for\nthe definition);\n\n\\item\nthere are no $2$-cycles in $Q$ passing through the vertex $k$.\n\n\\end{itemize}\n\\fussy\n\nConsider the triangulation algebra $\\Lambda=\\Lambda(Q,f,m,c,\\lambda)$. Our first\ntwo assumptions imply that Theorem~\\ref{t:quasi} holds for $\\Lambda$ and that in\nparticular, $\\Lambda$ is symmetric and there is a triangulated 2-Calabi-Yau category\n$\\mathcal{C}$ with a cluster-tilting object $T$ such that $\\Lambda \\simeq \\operatorname{End}_{\\mathcal{C}}(T)$.\n\nLet $(Q',f')$ be the mutation of $(Q,f)$ at $k$, let\n$m' \\colon Q'_1 \\to \\mathbb{Z}_{>0}$ and  $c' \\colon Q'_1 \\to K^{\\times}$ be the\n$g'$-invariant functions induced from $m$ and $c$, and\nlet $\\lambda' \\colon (Q'_1)^{f'} \\to K$ be the function on the arrows fixed by\n$f'$ induced from $\\lambda$. Our last assumption implies that the triangulation\nalgebra $\\Lambda'=\\Lambda(Q',f',m',c',\\lambda')$ is well defined\n(i.e.\\ $m'_{\\alpha'} n_{\\alpha'} \\geq 2$ for any $\\alpha' \\in Q'_1$),\nbut $m'$ is not necessarily admissible.\n\n\\begin{proposition} \\label{p:mutriang}\nUnder the above assumptions, the following assertions hold true:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item \\label{it:p:potR}\n$\\Lambda \\simeq \\mathcal{P}(Q,W)$, where the potential $W$ takes the form\nin~\\eqref{e:potR} for suitable $g$-invariant function\n$R \\colon Q_1 \\to K[[x]]$ and function $P \\colon Q_1^f \\to K[[x]]$.\n\n\n\\item \\label{it:p:potR2}\n$\\Lambda' \\simeq \\mathcal{P}(Q',W')$ for the potential $W'$ given in~\\eqref{e:potR2}\nwith the functions $R'$ and $P'$ corresponding to the functions $R$ and $P$\nof part~\\eqref{it:p:potR}.\n\n\\item \\label{it:p:QP}\nThe quiver with potential $(Q',W')$ is right equivalent to the mutation\nof the quiver with potential $(Q,W)$ at the vertex $k$.\n\n\\item \\label{it:p:IY}\n$\\Lambda' \\simeq \\operatorname{End}_{\\mathcal{C}}(T')$, where $T'$ is the Iyama-Yoshino mutation of $T$\nwith respect to the indecomposable summand corresponding to the vertex $k$.\n\n\\item \\label{it:p:mutriang}\n$\\Lambda'$ is derived equivalent to $\\Lambda$ and is of quasi-quaternion type.\nMore precisely, we have isomorphisms\n\\begin{equation} \\label{e:mutriang}\n\\operatorname{End}_{\\cD^b(\\gL)} U^-_{P_k}(\\Lambda) \\simeq \\Lambda' \\simeq\n\\operatorname{End}_{\\cD^b(\\gL)} U^+_{P_k}(\\Lambda)\n\\end{equation}\nwhere $P_k$ is the indecomposable projective $\\Lambda$-module corresponding to\nthe vertex $k$.\n\\end{enumerate}\n\\end{proposition}\n\nClaim~\\eqref{it:p:potR} follows by our assumption on $\\mu_f \\mu_g$. Since\n$\\mu_{f'}=\\mu_f$ and $\\mu_{g'}=\\mu_g$, claim~\\eqref{it:p:potR2} follows in a\nsimilar way. Claim~\\eqref{it:p:QP} is a consequence of the previous claims\ntogether with our assumption on the vertex $k$ and Proposition~\\ref{p:QPmut}.\nClaim~\\eqref{it:p:IY} follows from~\\eqref{it:p:QP}\nand Proposition~\\ref{p:QPmutIY}. Finally, claim~\\eqref{it:p:mutriang} is\na consequence of~\\eqref{it:p:IY} and Proposition~\\ref{p:dereq}.\n\n\\begin{remark}\nOur assumptions on the characteristic of $K$ and the vertex $k$ are needed\nin order to use the theory of mutations of quivers with potentials.\nIt seems very likely that the statements in parts~\\eqref{it:p:IY}\nand~\\eqref{it:p:mutriang}\nof Proposition~\\ref{p:mutriang} are still true even if we drop the assumption\non the characteristic of $K$ and weaken the assumption on the vertex $k$,\nrequiring only that the triangulation algebra $\\Lambda'$ is defined.\n\\end{remark}\n\n\\subsection{Construction of infinitely many non-degenerate potentials}\n\nWe conclude by presenting an application of the preceding results to\nthe theory of quivers with potentials.\n\nFor a mutation $(Q',W')$ of a quiver with potential $(Q,W)$, the underlying\nquiver $Q'$ may have $2$-cycles even if the quiver $Q$ did not have such.\nThus, $(Q',W')$ could not be further mutated at the vertices lying on these\n2-cycles.\n\nA quiver with potential $(Q,W)$ is \\emph{non-degenerate} if, for any sequence\nof mutations of quivers with potentials, the underlying quiver does not contain\nany $2$-cycles~\\cite[Definition~7.2]{DWZ08}.\nThe existence of non-degenerate potentials is crucial to\nseveral approaches to solve various conjectures on cluster algebras,\neither via the representations of Jacobian algebras and their mutations\nas in~\\cite{DWZ10}, or via the generalized cluster categories~\\cite{Plamondon11}.\n\nDerksen, Weyman and Zelevinsky proved~\\cite[Corollary~7.4]{DWZ08} that if\nthe ground field is\nuncountable, then over any quiver without loops and $2$-cycles there is at\nleast one non-degenerate potential. It is interesting to know when such\nnon-degenerate potential is unique (up to right equivalence). For instance,\non quivers without oriented cycles there is only one potential, namely the\nzero potential. In~\\cite[Theorem~1.4]{GLS16},\nGeiss, Labardini-Fragoso and Schr\\\"{o}er\nproved that apart from one exception, the adjacency quiver of\na triangulation of a marked surface with non-empty boundary has only one\nnon-degenerate potential.\nMore generally, we proved in~\\cite[\\S4]{Ladkani13} that a non-degenerate\npotential\nis unique on any quiver belonging to the class $\\mathcal{P}$ of Kontsevich and\nSoibelman~\\cite[\\S8.4]{KS08}, and that this class of quivers actually contains\nthe previous two instances.\n\nIn this section we consider the other extremity, namely, we apply the previous\nresults to construct quivers which have infinitely many non-degenerate\npotentials whose Jacobian algebras are pairwise non-isomorphic.\n\nConsider a triangulation quiver $(Q,f)$ such that:\n\\begin{equation} \\label{e:g1f3}\n\\tag{$\\star$}\n\\text{The permutation $g$ has one cycle and\nall the cycles of $f$ are of length $3$.}\n\\end{equation}\nThese assumptions imply that the quiver $Q$ does not have loops or 2-cycles\n(see Proposition~\\ref{p:tri2cyc}).\nMoreover, a potential as in Eq.~\\eqref{e:potR} is controlled by one power series\n$R(x) \\in K[[x]]$, and all the cycles $\\omega_\\alpha$ (where $\\alpha$ runs over the arrows\nof~$Q$) are rotationally equivalent. Denote by $\\omega$ one of these cycles.\n\nIf $(Q',f')$ is a mutation of $(Q,f)$, then by Lemma~\\ref{l:mutcycle} it\nalso satisfies~\\eqref{e:g1f3} and hence Proposition~\\ref{p:QPmut} can be\napplied indefinitely to yield the following.\n\n\\begin{proposition}\nLet $(Q,f)$ be a triangulation quiver satisfying condition~\\eqref{e:g1f3}.\nThen for any power series $R(x) \\in xK[[x]]$, the potential $W_R$ on $Q$ given\nby\n\\begin{equation} \\label{e:potRg1f3}\nW_R = -R(\\omega) + \\sum_\\alpha \\alpha \\cdot f(\\alpha) \\cdot f^2(\\alpha)\n\\end{equation}\n(where the sum runs over representatives $\\alpha$ of the $f$-cycles)\nis non-degenerate.\n\\end{proposition}\n\nConsider now a triangulation $\\tau$ of a closed surface with exactly one\npuncture. Then its triangulation quiver $(Q_\\tau, f_\\tau)$ satisfies\ncondition~\\eqref{e:g1f3} by Remark~\\ref{rem:triblocks}.\nMoreover, the adjacency quiver of $\\tau$ is $Q_\\tau$ by Corollary~\\ref{c:onep}.\n\n\\begin{corollary} \\label{c:infpot}\nLet $Q$ be the adjacency quiver of a triangulation of a closed surface with\nexactly one puncture, and view it as triangulation quiver $(Q,f)$. Then:\n\\begin{enumerate}\n\\renewcommand{\\theenumi}{\\alph{enumi}}\n\\item\nFor any power series $R(x) \\in xK[[x]]$, the potential $W_R$ on $Q$ defined by\nEq.~\\eqref{e:potRg1f3} is non-degenerate.\n\n\\item\nLet $R_0(x)=0$ and $R_m(x)=x^m$ for $m \\geq 1$. Then\n\\[ \n\\{W_{R_0} \\} \\cup \\{W_{R_m} : \\text{$m \\geq 1$ is not divisible by $\\operatorname{char} K$}\\}\n\\]\nis an infinite set of non-degenerate potentials on $Q$ whose Jacobian algebras\nare pairwise non-isomorphic.\n\\end{enumerate}\n\\end{corollary}\n\n\\begin{remark}\nFor a quiver as in Corollary~\\ref{c:infpot}, it was known that there are\nat least two inequivalent non-degenerate potentials, denoted in our notation\nby $W_0$ and $W_x$, see~\\cite[\\S4.3]{Ladkani11b}, \\cite[\\S3]{Ladkani13}\nand~\\cite[Proposition~9.13]{GLS16}. Note that\nthe Jacobian algebra of $W_0$ is infinite dimensional whereas that\nof any $W_{x^m}$ with $m \\geq 1$ not divisible by $\\operatorname{char} K$ is a triangulation\nalgebra and hence finite-dimensional of quasi-quaternion type.\nThe latter Jacobian algebras are pairwise non-isomorphic because their\ndimensions are all different; indeed, if the surface has genus $g \\geq 1$ then\n$Q$ has $6g-3$ vertices (Remark~\\ref{rem:nvertex}) and hence the Jacobian\nalgebra of $W_{x^m}$ has dimension $m(12g-6)^2=36m(2g-1)^2$\n(Proposition~\\ref{p:Cartan}).\n\\end{remark}\n\n\\bibliographystyle{amsplain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":"\\section{Introduction}\nIt is well-known since the mid-1970's that $\\Lambda$ hyperons produced\nin unpolarized $p\\, p$ collisions are to a large degree polarized\ntransversely to the production plane \\cite{Lesnik:1975my}.  There have\nbeen many experimental and theoretical investigations aimed at\nunderstanding this striking polarization phenomenon, but no consensus\nhas been reached about its origin. One of the difficulties in\ninterpreting the available (mostly fixed target) data is that they are\nnot or only partly in a region where a factorized description of the\ncross section is expected to be applicable. For a comprehensive review\nof these relatively low energy data ($\\sqrt{s} \\leq 62$ GeV) see Ref.\\ \n\\cite{Panagiotou:1989sv}. New high-energy hadron collider data would\nbe very welcome, for instance from RHIC, Tevatron, or LHC, but there\nthe capabilities to measure the $\\Lambda$ polarization $P_\\Lambda$ via\nthe self-analyzing parity violating decay $\\Lambda \\to p \\, \\pi^-$ are\ntypically restricted to the midrapidity region ($\\eta \\approx 0$), \nwhere protons can be \nidentified\\footnote{An alternative may be to use neutral decays \n$\\Lambda \\to n\\, \\pi^0$ (50\\% less frequent than $p\n\\pi^-$) \\cite{Cork:1960zz}.}. For symmetry reasons $P_\\Lambda=0$ at\n$\\eta=0$ in $p\\, p$ collisions in the center of mass frame, hence \nthe degree of transverse polarization $P_\\Lambda$ around midrapidity\nis expected to be very small. As an\nalternative to $p + p \\rightarrow \\Lambda^{\\uparrow} + X$, it has been\nsuggested \\cite{Boer:2007nh} to perform $\\Lambda$ polarization studies\nin the process $p + p \\to (\\Lambda^\\uparrow \\text{jet}) + \\text{jet} +\nX$, where the $\\Lambda$ and jets can be measured in the midrapidity region\nwithout paying a suppression penalty. It is especially of interest at\nLHC, where the factorized description is expected to apply and certain\nsimplifications may arise due to the large center of mass energy\n$\\sqrt{s}$. \n\n\\section{Spontaneous $\\Lambda$ polarization in a factorized approach}\nThe magnitude of the $\\Lambda$ polarization in $p + p \\rightarrow\n\\Lambda^{\\uparrow} + X$ at large $x_F$ remains large up to the highest measured\ntransverse momentum $p_T$ of the $\\Lambda$: $p_T \\sim 4$ GeV\/$c$. For\nsufficiently high $p_T^{}$, perturbative QCD and collinear\nfactorization should become applicable. \nConsider for example the $q g \\to q g$ subprocess contribution to\nthe $p + p \\to \\Lambda + X$ cross section in collinear\nfactorization. It is of the form:\n$\\sigma \\sim {q(x_1)} \\otimes {g(x_2)}  \\otimes\n\\hat{\\sigma}_{q g \\to q g} \\otimes {D_{\\Lambda\/q}(z)}$, where\n${q(x_1)}$ is the quark density in proton 1, ${g(x_2)}$ is the gluon\ndensity in proton 2, and \n${D_{\\Lambda\/q}(z)}$ is the $q \\to \\Lambda$ fragmentation function\n(FF). In a\nsimilar way the transverse polarization should be of the form:\n$P_{\\Lambda} \\sim q(x_1)  \\otimes g(x_2) \\otimes\n\\hat{\\sigma}_{q g \\to q g} \\otimes {?}$, involving the unpolarized\nparton densities and the {\\it unpolarized} hard partonic subprocess \nbecause $\\Lambda$ polarization created\nin the hard partonic scattering is very small, \n$P_\\Lambda \\sim \\alpha_s m_q\/\\sqrt{\\hat s}$ \\cite{Kane:1978nd}.\nThe question mark indicates\nthat for symmetry reasons at leading twist \nthere is no collinear fragmentation function\ndescribing $q \\to \\Lambda^\\uparrow X$. In\ncollinear factorization $P_{\\Lambda}$ is thus necessarily power\nsuppressed. \nDropping the demand of {\\em collinear\\\/} factorization, \ndoes allow for a leading twist solution: the transverse momentum\ndependent (TMD) fragmentation function \n$D_{1T}^\\perp(z,{\\mathbf{k}_T})$ \\cite{Mulders:1995dh} for\n$\\Lambda$'s. It describes a nonperturbative \n{$\\mathbf{k}_T \\times \\mathbf{S}_T$} dependence in the fragmentation\nprocess, which is allowed by the symmetries (parity and time\nreversal). As the $\\Lambda$ polarization arises in the fragmentation of \nan {{\\em unpolarized}} {quark}, it carries the descriptive name \n``polarizing fragmentation function'' \\cite{Anselmino:2000vs}. \n${D_{1T}^\\perp}$ has been extracted \\cite{Anselmino:2000vs} \nfrom the low energy \n$p + p\/Be \\to \\Lambda^{\\uparrow}\/\\bar\\Lambda^{\\uparrow} + X$ data, \nwhere gluon FFs ($g \\to \\Lambda \\, X$) are expected to be hardly\nrelevant. Reasonable valence quark FFs are obtained: \n$D_{1T}^\\perp$ has opposite signs for $u\/d$ versus \n$s$ quarks, and the latter is larger. This is responsible for the \ncancellations that ensure $P_{\\; \\bar \\Lambda} \\approx 0$.\nThe extraction has been done under the restriction of $p_{T} > 1$\n  GeV\/$c$, in order to exclude the soft regime but to retain\nsufficient data to make a fit to. Whether this restriction \nis sufficiently strict is a matter of concern, due to the large \n$K$ factors required to obtain a cross section description. \nData at higher $\\sqrt{s}$ and $p_T$ would be much safer to ensure the validity\nof the factorized description. As pointed out, these\ndo not necessarily require large $x_F$, if one goes beyond \n$p + p \\rightarrow \\Lambda^{\\uparrow} + X$. \nIf the origin of the transverse $\\Lambda$ polarization\nis indeed due to polarizing fragmentation, then another, related asymmetry\ncould be observed that does not need to vanish at $\\eta_\\Lambda=0$, \nnamely in the process $p + p \\to (\\Lambda^\\uparrow \\text{jet}) +\n  \\text{jet} + X$ \\cite{Boer:2007nh} and actually also in  \n$p + p \\to (\\Lambda^\\uparrow \\text{jet}) + X$.\n\n\\section{Jet-$\\Lambda^\\uparrow$~production}\n\nThe suggestion of \\cite{Boer:2007nh} \nis to select two-jet events and to measure\nthe jet momenta $K_j^{}$ and $K_{j'}^{}$ (with $K_{j}^{}\\cdot\nK_{j'}^{} = {\\cal O}(\\hat{s})$), in addition to the momentum\n$K_\\Lambda^{}$ and polarization $S_\\Lambda^{}$ of the $\\Lambda$ that\nis part of either of the two jets. A single spin asymmetry proportional to\n$\\epsilon_{\\mu\\nu\\alpha\\beta} K_j^{\\mu} K_{j'}^{\\nu}\nK_\\Lambda^{\\alpha} S_\\Lambda^\\beta$ can then arise, which is neither\npower suppressed, nor needs to be zero at midrapidity. \nIn the center of mass (c.o.m.) \nframe of the two jets the asymmetry is of the form:\n\\begin{equation}\n\\text{SSA}\n=\\frac{d\\sigma({+}\\mathbf{S}_\\Lambda)\\,\n{-}\\,d\\sigma({-}\\mathbf{S}_\\Lambda)}\n{d\\sigma({+}\\mathbf{S}_\\Lambda)\\,\n{+}\\,d\\sigma({-}\\mathbf{S}_\\Lambda)}\n=\\frac{\\hat{\\mathbf{K}}{}_j{\\cdot}\n({\\mathbf{K}_\\Lambda{\\times}\\mathbf{S}_\\Lambda})}{\nz\\,M_\\Lambda}\\,{\\frac{d\\sigma_T}{d\\sigma_U}}\\ \n\\label{SSA}\n\\end{equation}\nThe analyzing power {$d\\sigma_T\/d\\sigma_U$} of the asymmetry \ndepends on $D_{1T}^\\perp$. This new\n$\\Lambda$+jets observable could allow for a more trustworthy extraction\nof $D_{1T}^\\perp$ (for both quarks and gluons) and subsequent\npredictions, for instance for\nsemi-inclusive DIS \\cite{Anselmino:2001js}.\n\nAt RHIC and LHC this process $p \\, p \\to \\left(\n\\Lambda^\\uparrow \\text{jet}\\right) \\, \\text{jet} \\, X$ can be studied.\nFor instance, the ALICE experiment has \nexcellent PID capabilities which allow measurements of \n$\\Lambda$'s over a wide $p_T$ range.\nThe ALICE rapidity coverage is ${-}0.9\\,{\\leq}\\,\\eta\\,{\\leq}\\,{+}0.9$.\nIf the jet rapidities ($\\eta_{j,j'}$) are in this range \nand if gluon fragmentation is at least\nas important as quark fragmentation for both unpolarized and polarized\n$\\Lambda$ production, then the process\nis dominated by gluon-gluon ($gg{\\rightarrow}gg$) scattering\\footnote{Unlike in Ref.\\ \\cite{Boer:2007nh}, here it will be assumed \nthat universality of $D_{1T}^\\perp$ holds throughout \\cite{Metz:2002iz}.},\ni.e.\\ $d\\sigma_T\/d\\sigma_U \\approx\nD_{1T}^{\\perp\\,g}(z{,}K_{\\Lambda\\,T}^2)\/D_1^g(z{,}K_{\\Lambda\\,T}^2)$.\nBecause no model or fit for $D_{1T}^{\\perp\\,g}$ is available yet, no\npredictions can be made in this case. \nIf it happens that {$D_{1T}^{\\perp\\, g}\\,{\\ll}\\,D_{1T}^{\\perp\\, q}$}, then\none can use the extracted $D_{1T}^{\\perp\\,q}$ to obtain an\nestimate. Taking into account the $qg\\to qg$ subprocess, one finds \nfor $\\eta_{j'}{\\approx}\\,{-}\\eta_j \\approx 0$ \n\\begin{equation}\n\\frac{d\\sigma_T}{d\\sigma_U}\n\\approx \\frac{\\sum_q f_1^q(x)D_{1T}^{\\perp\\,q}(z{,}K_{\\Lambda\\,T}^2)}\n{\\sum_q f_1^q(x)(D_1^q(z{,}K_{\\Lambda\\,T}^2)\n+D_1^g(z{,}K_{\\Lambda\\,T}^2))+f_1^g(x) D_1^g(z{,}K_{\\Lambda\\,T}^2)\/0.4},\n\\label{SigTSigU}\n\\end{equation}\nwhere $x\\,{\\approx}\\,2|\\mathbf{K}_{j\\perp}|\/\\sqrt{s}$. The factor\n$0.4$ comes from the ratio $d\\hat\\sigma_{qg\\rightarrow\n  qg}\/d\\hat\\sigma_{gg\\rightarrow gg}$ at midrapidity; other partonic channels \ncan be neglected \\cite{Boer:2007nh}.\n\\begin{figure}[htb]\n\\includegraphics[width=0.5\\textwidth]{LambdaJetAsymmDSVDIS2010-punt4.eps}\n\\includegraphics[width=0.5\\textwidth]{LambdaJetAsymmIMRDIS2010-punt4.eps}\n\\caption{The asymmetry $d\\sigma_T\/d\\sigma_U$ for $\\eta_{j},\\eta_{j'} =\n  0$, $|K_{j\\, \\perp}|=|K_{j'\\, \\perp}| = 70 $ GeV, $\\sqrt{s}=14$ TeV,\n  using DSV (left) and IMR (right) FFs.}\n\\label{Asymm}\n\\end{figure}\nFor estimates we will use the DSV \\cite{deFlorian:1997zj}\nand IMR \\cite{Indumathi:1998am} unpolarized FFs sets, and their accompanying\n$D_{1T}^\\perp$ parametrizations from Ref.\\ \\cite{Anselmino:2001js}. \nBoth DSV and IMR satisfy $D_1^g \\ll D_1^q$ at larger $z$, as both\nhave been obtained from fits to $e^+ e^- \\to \\Lambda \\, X$ data only, \nwhich is not sensitive to the fragmentation process $g \\to \\Lambda \\, X$. \nThe resulting asymmetry (\\ref{SigTSigU}) evaluations \nare given in Fig.\\ \\ref{Asymm} for three different values of the\n$\\Lambda$ momentum component transverse to the jet direction (see\n\\cite{dis2009} for additional comments).\nVery small asymmetries are obtained at smaller $z$, due to \n$D_{1T}^\\perp$ being fitted to low energy\ndata where mostly valence quarks matter. \nThis behavior need not be realistic at high energies. \nALICE would have most data in the region $z <0.5$ and could therefore \nprovide valuable information on sea quark and gluonic contributions.  \n\nThe asymmetry is quite sensitive to the\ncancellation between $u\/d$ and $s$ contributions, like in SIDIS\n\\cite{Anselmino:2001js}, and can even flip its overall sign depending \non the amount of $SU(3)$ breaking in the unpolarized fragmentation \nfunctions: IMR includes $SU(3)$ breaking, whereas DSV does not. \nThis aspect represents a large uncertainty, but more reliable estimates are \nnot possible at this stage. ALICE may help to determine the importance \nof $SU(3)$ breaking and flavor cancellations. \n\nUnpolarized $\\Lambda$ FFs have also been obtained \ntaking into account midrapidity \nhadronic $\\Lambda$ production data from STAR at RHIC: the AKK\n\\cite{Albino:2005mv} and AKK08 \\cite{Albino:2007ns} sets. These have\nvery different characteristics compared to fits to $e^+ e^-$ data only. \nFig.~\\ref{Ratios} shows the ratio $D_1^g\/D_1^{q+\\bar{q}}$ for\n$q=u,d,s$, respectively, for DSV, AKK and AKK08. The curves\ndiffer widely and represent yet another uncertainty in the\npredictions of (\\ref{SigTSigU}). \nNo $D_{1T}^\\perp$ extraction has been performed with AKK or AKK08. \nA problem with that is that the most recent fit AKK08 considerably\nundershoots the cross section of \n$p \\, p \\to \\Lambda\/\\overline{\\Lambda} \\, X$ at midrapidity, even at \n$\\sqrt{s}=200$ GeV and transverse momenta in the range \n$2 \\leq \\mathbf{p}_T \\leq 5$ GeV\/$c$. In\n\\cite{Albino:2007ns} it was therefore concluded that there is\n``a possible inconsistency between the $p\\, p$ and $e^+e^-$ reaction\ndata for $\\Lambda\/\\overline\\Lambda$ production'' (cf.\\ Fig.~5 of\n\\cite{Albino:2007ns}). This issue ought to be clarified first.\n\n\\begin{figure}[htb]\n\\includegraphics[width=0.32\\textwidth]{Lambda-FF-FSVAKKupd-c1.eps}\n\\includegraphics[width=0.32\\textwidth]{Lambda-FFd-FSVAKKupd-c1.eps}\n\\includegraphics[width=0.32\\textwidth]{Lambda-FFs-FSVAKKupd-c1.eps}\n\\caption{The ratio $D_1^g\/D_1^{q+\\bar{q}}$ as function of $z$ \nfor $q=u,d,s,$ respectively, for the leading order DSV, AKK and AKK08 sets, \nat the scale $Q=10$ GeV.} \n\\label{Ratios}\n\\end{figure}\n\nThe suggestion of \\cite{Boer:2007nh} requires the opposite side jet\n$j'$ to be measured, such that the jet-jet or equivalently the\npartonic c.o.m.\\ frame can be selected. \nHowever, if universality of $D_{1T}^\\perp$ and gauge\ninvariance of the factorized expression are taken to hold, \nthen one can actually consider \n$p \\, p \\to \\left(\\Lambda^\\uparrow \\text{jet}\\right) \\, X$,\nincreasing the statistics considerably. The asymmetry \nis also given as in Eq.\\ (\\ref{SSA}), but now in the\n{\\it hadronic} c.o.m.\\ frame, which\ncan be shown by considering the Sudakov decomposition of the fragmenting\nquark momentum $k=K_\\Lambda\/z+\\sigma_\\Lambda n_\\Lambda+k_T$ \n(but not of the other \nmomenta), with the uncommon choice $n_\\Lambda=P_1+P_2$. Due to the large\n$\\sqrt{s}$ at the LHC, $\\sigma_\\Lambda$ is still suitably \nsuppressed (at least by a factor $E_j\/\\sqrt{s}$), \ndespite $n_\\Lambda$ being non-lightlike.  \nThe asymmetry is now $\\propto \\epsilon_{\\mu\\nu\\alpha\\beta} K_j^{\\mu}\n(P_1+P_2)^{\\nu} K_\\Lambda^{\\alpha} S_\\Lambda^\\beta$, which in the\nhadronic c.o.m.\\ frame indeed becomes $\\propto \\mathbf{K}_j{\\cdot}\n(\\mathbf{K}_\\Lambda{\\times}\\mathbf{S}_\\Lambda)$. Eq.\\ (\\ref{SigTSigU})\nis now replaced by \n\\begin{equation}\n\\frac{d\\sigma_T}{d\\sigma_U} \n\\approx \n\\frac{\n\\int \\frac{dy}{y}\\sum_q \\left(f^{qg}+f^{gq}\\right) \nd\\hat{\\sigma}_{qg} D_{1T}^{\\perp\\,q}}{\\int \\frac{dy}{y}  \n\\left[ \\sum_q \\left(f^{qg}+f^{gq}\\right)\nd\\hat{\\sigma}_{qg} \\left(D_1^q + D_1^g\\right) \n+ f^{gg}d\\hat{\\sigma}_{gg} D_1^g \\right]} , \n\\label{SigTSigUnew}\n\\end{equation}\nwhere the arguments of the fragmentation functions have been\nsuppressed and we introduced the notation: $f^{ab}\\equiv x_1 f_1^a(x_1) \nx_2 f_1^b(x_2)$, $d\\hat\\sigma_{ab}\\equiv d\\hat\\sigma_{ab\\to ab}(y)+\nd\\hat\\sigma_{ab\\to ab}(1-y)$ with $y=-\\hat t \/\\hat s$. \nThe momentum fractions are fixed to be \n$x_1=x_\\perp e^\\eta\/(2(1-y))$ and $x_2=x_\\perp e^{-\\eta}\/(2y)$ for\ngiven observed $x_\\perp=2|\\mathbf{K}_{j\\perp}|\/\\sqrt{s}$ and $\\eta=\\eta_{j}$.\nThe $y$ integration is between \n$y_{\\min}=x_\\perp e^{-\\eta}\/2$ and $y_{\\max}=1- x_\\perp e^\\eta\/2$. The\nexpressions for the partonic cross sections can be found in Ref.\\\n\\cite{Boer:2007nh}. For $D_1^g \\gg D_1^q$ the last term in the denominator\nis the dominant one: for $x_\\perp=0.01, \\eta=0$ it is\napproximately a factor of 3 larger than the other term proportional to $D_1^g$.\n\n\\section*{Acknowledgments}\n\nI thank Markus Diehl, Dae Sung Hwang, Piet Mulders, Pasquale Di Nezza,\nWerner Vogelsang, and Feng Yuan for useful discussions. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nLooking at the large-scale distributions of galaxies, one may notice\nthat along with clusters and groups of galaxies there are also\nconspicuously oblong concentrations of galaxies: filaments. One may also\nget an impression that filaments are connected to each other forming a\nsingle network spanning through the entire sample.\nThe two-point\ncorrelation function (the most common way of describing the distribution\nof galaxies) obviously is not sensitive to the geometry and topology of the\ngalaxy distribution.\nThe three- and many- point correlation functions generally speaking\nare sensitive to the shapes and probably the topology\nbut become cumbersome very quickly.\nCurrently popular averaged moments are easy to interpret, but they loose\nsensitivity to the geometry after averaging over the volume \\cite{Bern95}.\nThere have been suggested various statistics to characterize the geometry\nand topology of the large-scale structure.\nIn this talk I briefly review the studies of the topology\nof the large-scale galaxy distribution.\n\nThe first mention of topology in the context of the large-scale\nstructure problem (I am familiar with) was in the 1970 paper by Doroshkevich\n\\cite{Dor70}.\nStudying the formation of the large-scale structure in the pancake\nscenario Doroshkevich calculated the Euler characteristic\nof the isodensity surfaces of the initial\n{\\it Gaussian} density field.\nBoth the Euler characteristic and a common topological measure used\nin cosmology, genus, are\ndetermined by the mean Gaussian curvature of the surface of a\nconstant density.\n\nIn 1982 Zel'dovich noticed that the percolation properties of the\n{\\it nonlinear} density distribution in the HDM\n(Hot Dark Matter) model are very different from the initial Gaussian field.\nHe also suggested characterizing the topology of the nonlinear density\ndistribution by the percolation thresholds \\cite{Z82}.\nPercolation theory deals with the number and properties of the ``clusters'',\nwhich are defined as connected regions bounded by the surfaces of a\nconstant density. Following Zel'dovich's\nidea the author of this talk suggested to use percolation properties\nof {\\it the galaxy distribution} as an objective quantitative measure of the\ntopology of the large-scale structure and also as a discriminator\nbetween cosmological models \\cite{Sh83}, \\cite{Sh-Z83}.\n\nThe percolation technique was utilized in the study of the CfA I catalog\n\\cite{E-K-S-Sh84}. It was found that the large-scale\ndistribution\nof galaxies had a network structure. Theoretical studies of the models\nwith the power law initial spectra showed that the $n=-1$ model clearly\npercolated better than the $n=0$ model and also in the  $\\Omega=1$ universe\nthe $n=-1$ model was in an agreement with the observations \\cite{B-B83}.\nThe percolation\nmethod showed that the CDM (Cold Dark Matter) model appeared filamentary\nrather than hierarchical \\cite{M-etal83}, \\cite{D-etal85}.\nIt was also pointed out that the major disadvantage of percolation\ntechnique was the dependence\nof the percolation thresholds on the mean density of the sample \\cite{D-W85}\nwhich made it difficult to apply to sparse samples. Similarly, we note that at\npresent some believe that sparse samples can be reliably used for\nthe estimation of the two-point correlation function only \\cite{Bouch95}.\n\nIn 1986 Bardeen et al \\cite{Bard-etal86} and\nGott, Melott and Dickinson rediscovered Doroshkevich's\nidea of utilizing the Euler characteristic and expanded it\nto the {\\it nonlinear} distributions\nas well as {\\it galaxy} catalogs \\cite{G-M-D86}.\n(Both percolation and genus techniques assumes some kind of smoothing when\napplied to galaxy distributions.)\nHowever, instead of\nthe mean density of the Euler characteristic $\\chi$ used by Doroshkevich\nthey introduced  the mean density of genus, $g$, which is\nproportional to the Euler characteristic: $g=-\\chi\/2$.\n(For a general review of this method see e.g. \\cite{Mel90}.)\nTomita \\cite{Tom86} gave a very elegant analytic expression for the mean Euler\ncharacteristic for a D-dimensional Gaussian field which in three-dimensional\nspace yields the familiar equation for the mean genus density\n\n\\begin{equation}\ng = {1 \\over 4\\pi^2}\\Bigl({<k^2> \\over 3}\\Bigr)^{3\/2}(1-\\nu^2)\\exp(-\\nu^2\/2),\n\\label{eq:genus} \\end{equation}\n\nwhere $\\nu=\\delta\/\\sigma_{\\delta}$ is the number of standard\ndeviations by which the threshold density\ndeparts from the mean density,\n$\\delta \\equiv {(\\rho -\\bar{\\rho})}\/\\bar{\\rho}$,\n$<k^2> = \\int k^2P(k)d^3k\/\\int P(k)d^3k$,\nand $P(k)$ is the power spectrum (see e.g. \\cite{Vog-etal94}).\nIn a Gaussian field the genus curve has a maximum\nat $\\nu =0$ with the amplitude determined by\nthe characteristic scale of the density field $<k^2>$.\nSince it depends on the\nslope of the spectrum it is often used as a measure of the\n``effective'' slope of\nthe spectrum (see e.g. \\cite{Moore-etal92});\nhowever, it is worth mentioning that $<k^2>$ has a stronger\ndependence on the smoothing scale than on the spectral index.\nTherefore even weak nonlinearity on the smoothing scale may influence\nthe estimate of the spectral index.\nThe genus curve changes sign two times at $\\nu =\\pm 1$ which signifies\nthe qualitative change of the topology of the surface separating\nhigh ($\\delta>\\delta_c$) and low ($\\delta<\\delta_c$) density regions where\n$\\delta_c$ is a chosen density threshold.\nAs suggested by  Eq.\\ref{eq:genus} there are only two types of\nqualitatively different topologies: 1) one phase percolates and the other\ndoes not (negative genus)\nand 2) both phases percolate (positive genus).\nIn the range\n$-1< \\nu <1$ both phases percolate through the whole region and this is\noften reffered to as sponge topology. At $\\nu <1$ the low density regions\ndo not percolate and at $\\nu >1$ the high density regions do not percolate.\nThe topology of the separating surfaces is obviously the same\nat $\\nu =\\pm\\nu_c$ as measured by Eq.\\ref{eq:genus}.\nBut in cosmological literature it is labeled either as a meatball or bubble\ntopology depending on whether the high or\nlow density phase does not percolate. The network structure obviously\nhas a sponge topology however the term emphasizes a geometrical aspect\nof a non-Gaussian density field: the high density regions at the percolation\nthreshold occupy a smaller volume than that of a Gaussian field.\n\nThe change of the genus sign is believed to coincide with the percolation\nthresholds however there is no theorem proving that. Intuitively, it is\nplausible for distributions resulting from Gaussian fields\ndue to gravitational instability\nif one believes in a common interpretation of the genus\nas the mean density of the number of holes minus the number of the isolated\nregions and that clusters with holes inside do not form. Under these\nconditions, we\nassume that percolation thresholds coincide with the changes of the genus\nsign at least approximately.\n\nOne advantage of the genus method is the existence of the analytic expression\nfor Gaussian random fields (Eq.\\ref{eq:genus}). Recently there has also been\nan analytic expression obtained in the weakly nonlinear regime \\cite{Mats94}.\nHowever, one should not forget that the mean genus is a statistical\nmeasure and therefore an estimate of errors is needed before it becomes\nmeaningful. The errors for finite samples having finite resolution\ncan be estimated only from numerical simulations. Percolation parameters\nare also calculated numerically, but if one can estimate the errors he\nalmost certainly can estimate the mean with similar accuracy.\n\nIt has been claimed that the percolation thresholds are the most sensitive\ndiscriminators of the models \\cite{Sh83}. The recent study of the\nCfA II catalog using the genus method \\cite{Vog-etal94}\nseems to support that suggestion.\nThe authors reduced the\ninformation of the genus curve to three numbers one of which was\nthe genus peak width $W_{\\nu}=\\nu_+ -\\nu_-$ where $\\nu_+$ and\n$\\nu_-$ are the levels at which the genus changes the sign.\nFig. 12 through 14 in \\cite{Vog-etal94} clearly\ndemonstrate that $W_{\\nu}$ has the highest discriminating power.\nHowever,\nwe still believe that the percolation thresholds, $\\nu_+$ and $\\nu_-$,\nshould be interpreted separately because\nthey carry independent information about the topology of the structure.\n\n\\section{Largest ``cluster'' and largest ``void''}\nPercolation theory deals with the number and properties of the ``clusters''.\nIn the absence of a better term we label as ``clusters'' the regions bounded\nby the surfaces of chosen constant density. In order to avoid confusion\nwith clusters of galaxies, we will use quotation marks when talking about\n``clusters'' with this non-astronomical meaning. The density threshold\n$\\delta_c$ separating\nhigh ($\\delta>\\delta_c$) and low ($\\delta<\\delta_c$) density regions is\nassumed to be a free parameter $\\delta_c >-1$. Analyzing discrete distributions\n(e.g. galaxy distributions) we assume a smoothing procedure creating\na continuous density distribution.\n\nAt every density threshold all ``clusters'' and ``voids'' are identified and\nvarious types of analysis can be performed \\cite{K-Sh93}.\nHowever here we present only\nthe results of the study of the largest ``cluster'' and the largest ``void'' as\nfunctions of a density threshold.\n(The largest ``void'' is defined as the largest ``cluster'' in the low density\nphase.) The full analysis will be presented elsewhere \\cite{Y-Sh95},\n\\cite{Y-Sh-F95}.\nThe choice of the largest structure is determined by the fact that at the\npercolation threshold the largest structure become infinite which signifies\nthe change of topology.\n\nThe density threshold is not a convenient parameter\nif linear (Gaussian) and nonlinear density distributions are to be compared.\nInstead we utilize the filling factor to parameterize the density threshold\n\\cite{dL-G-H91}.\nThe filling factor is the fraction of the volume occupied by the given phase.\nIn this case one can easily compare the properties of ``clusters'' with that\nof ``voids'' and also linear and nonlinear density distributions. It is similar\nto comparing different patterns provided that the same amount of paint was\nused to make each pattern. The filling factor as a function of the density\nthreshold is obviously the cumulative distribution function.\n\nThe largest ``cluster'' and ``void'' are measured as a fraction of\nthe corresponding filling factor. Thus if the largest ``cluster'' is $0.9$ at\nfilling factor of $0.2$ it means that the density is higher than the chosen\nthreshold in $20\\%$ of the volume and almost all of that volume\n($90\\%$) comprised of only one connected region.\n\nThe top two panels of Fig.1 show the largest ``cluster''\nand the largest ``void''\nfor the density distributions obtained in the N-body simulation of the\npower law model with $n=-1$ at two stages of evolution: $\\lambda_{nl}=1\/8$\nand $1\/4 L_{box}$. The simulations have been done with $128^3$\nparticles on the equivalent mesh \\cite{M-Sh93} but for this analysis\nthe mesh has been reduced to $64^3$.\nError bars show $1\\sigma$ deviations from the mean obtained in four\ndifferent realizations of the model.\n\nThe qualitative behavior of both of the largest structures is universal:\nat small filling\nfactors the largest structure is negligible then at some filling factor\nit quickly grows and becomes the only significant structure in the\ncorresponding phase. This is the percolation transition and also the\nindication of the change of topology.\nIn Gaussian fields there is no statistical\ndifference between ``clusters'' and ``voids'' and the transition happens at\na filling factor of about $16\\%$ corresponding to $\\nu =\\pm1$.\nHowever, a finite size of the sample\nas well as finite resolution biases the transition.\nIn order to avoid these effects, we obtain\nthe ``Gaussian'' distribution\nby mixing the phases of the Fourier transform of the nonlinear density\ndistributions in question. This automatically includes all finite\ngrid effects in the reference Gaussian field keeping the Fourier amplitudes\nexactly the same.\nThis allows for the generation of as many Gaussian realizations with identical\namplitudes as needed to estimate the dispersion.\nThe Gaussian largest structure is shown as a dotted\nline in Fig.1\n(hidden by the shade of the error bars) lying between the solid and dashed\nlines.\n\nThe major feature of the nonlinear distribution is that the largest\n``cluster'' percolates easier and the largest\n``void'' harder than in the Gaussian case.\nThe significance of this conclusion for the largest cluster is at the\nmany-$\\sigma$ level (see Fig.1). Qualitatively this remains true for\nall models we have\nstudied ($n=1,0,-1,-2,-3$, CDM, and C+HDM \\cite{K-Sh93}), but quantitatively\nthe transitions are different. The high density\nregions form a connected network spanning through the whole region when the\nfilling factor is relatively small (smaller than in the Gaussian case) and\ntherefore this transition can be labeled as a shift\ntoward the network structure.\nOn the contrary the low density regions do not form a percolating\nvoid (remain isolated)\neven when the filling factor of the low density phase\nis greater than that in the\nGaussian field. This type of transition can be labeled as a shift toward\nthe bubble structure.\nThe range of the sponge topology  is typically (but not necessarily)\nincreased compared\nto the Gaussian case. Thus the above changes also can be labeled\nas a shift to a sponge topology.\nHowever, the major point is not how to label a structure but rather\nto show that in a general case the two shifts are independent of each other\nand carry independent information about the structure. Therefore combining\nthem into one parameter (like $W_{\\nu}=\\nu_+ - \\nu_-$ mentioned above)\nresults in lost information.\n\nAt small filling factors the largest structure must be negligible in\nsufficiently large samples. The actual (finite) sizes of the largest structures\ncan be used as an internal characteristic of the fairness of the sample.\n\nIn the past the percolation technique has been successfully applied to\nvolume limited samples \\cite{E-K-S-Sh84}.\nAnalyzing the statistically homogeneous distributions is very easy and the\nlargest structures clearly distinguish between the models \\cite{Y-Sh95}.\nHowever, the analysis is much more difficult if the distribution\nhas a radial gradients, like the IRAS $1.2 Jy$ redshift catalog.\n\n\\section{The IRAS $1.2 Jy$ catalog}\nWe analyze the whole-sky galaxy distribution in real space\nreconstructed from the redshift IRAS $1.2 Jy$ catalog  using a Wiener\nfilter and an expansion in spherical harmonics \\cite{Lah-etal94}. The Wiener\nfilter effectively uses a variable window size which is about $500 km\/s$ at\n$2000 km\/s$ and increases to $1800 km\/s$ at $10000 km\/s$. The resulting\nsmoothed galaxy distribution is not statistically homogeneous.\nThis is the major\nchallenge for applying the percolation technique.\n\nIn order to test the effect of the reconstruction we generated two density\ndistributions from the N-body simulation ($n=-1$, $k_{nl}=8$): one with the\nWiener filtering and the other without it. The percolation statistics of\nthese distributions are shown in the middle and bottom  panels of Fig.1.\nThe panels on the\nleft hand side show the largest structures in the reconstructed density field\nwithout the Wiener filter and on the right hand side with the Wiener filter\napplied. The panels in the middle row show the results for the sphere with\nthe radius of $30$ mesh units and the bottom panels show the results for\nsmaller sphere with the radius of $24$ mesh units.\n\nThe middle and bottom panels show only one realization which demonstrates\na substantial change in the topological properties due to the reconstruction\nprocedure. We plot here only one realization for better visual comparison with\nthe data.\n\\begin{figure}[t]\n     \\caption{The N-body simulations of the $n=-1$ model.\nThe top row show two stages of evolution $\\lambda_{nl} =1\/8 $ and\n$1\/4 L_{box}$ before reconstruction.\nThe largest ``cluster'' (solid line) and largest ``void''\n(dashed line) is shown as a function of filling factor;\nthe Gaussian realization\nwith the identical Fourier amplitudes is shown in between (shaded by the error\nbars). The middle and bottom panels are the reconstructions without the Wiener\nfilter (on the left hand side) and with the Wiener filter (on the right hand\nside). Two different radii of the sphere has been used: $R=30$ and $24$ mesh\nunits.}\n\\end{figure}\nFig.2 shows the largest structures in the filtered galaxy distribution\nassuming two different $\\beta =\\Omega^{0.6}\/b$. This preliminary result\nis in a very general agreement with the $n=-1$ model. The models with\n$\\beta=0.1$ and $\\beta=1$ do not look much different. Unfortunately the\nmethod of estimating the dispersion described above does not work in the\ninhomogeneous case. We are working now on the improved version of estimating\nthe dispersion.\n\nThe largest structures in the galaxy distribution are quite large at small\nfilling factors: $30 - 40 \\%$ of the corresponding filling factor. It may mean\nthat the sample is not large enogh for this kind of analysis.\n\n\\begin{figure}[t]\n     \\caption{The Wiener reconstruction of the IRAS $1.2 Jy$ redshift\ncatalog in spherical harmonics. Solid lines show the largest ``cluster'' and\ndashed lines show the largest ``void'' at two values of\n$\\beta =\\Omega^{0.6}\/b$  and at two radii of the spherical region.}\n\\end{figure}\n\n\\section{Summary}\nThe quantitative topology proved to be a useful technique\nfor studying the large-scale structure in the universe. There have been\nsuggested two different methods for quantitative characterization of the\ntopology. One widely used method measures the mean density of genus of\na constant density surface as a function of the density threshold.\nIt actually measures the mean Gaussian curvature of the surface.\nThe other more geometrical method is based on percolation theory.\nHere the number and properties of the ``clusters'' are studied.\nIn particular the volume of the largest ``cluster'' measured\nas a function of a density threshold plays an important role.\n\nIn general the questions addressed by the two methods are similar but not\nidentical. In the case where the questions coincide the two approaches\nuse different methods of solving them. In particular, they treat the\nboundaries differently and are probably affected differently by noise.\n\nThe genus curve method is more developed and has been applied to many\ncatalogs of galaxies (both two-dimensional and  redshift surveys).\nThe major results indicate that.\nnon-Gaussian behavior has been\ndetected in both types of the catalogs (see e.g. \\cite{Col-etal93},\n\\cite{Vog-etal94}, \\cite{Moore-etal92} and references therein). The\ncharacterization of the structure is somewhat conflicting. Almost all\npossible labels have been assigned to the galaxy distributions: meatball,\nsponge, network, bubble topology. However, it is likely that the structure\nseen in different catalogs looks different. \\\nThe estimate of effective slope is in agreement with $n=-1$\n\\cite{Moore-etal92}. \\\n\nThe percolation method has been tested in various theoretical models and\ndeveloped to the level when it can be applied to the galaxy catalogs.\nFor the first time we try to apply it to the magnitude limited sample\nand find the preliminary result encouraging.\n\n{I am grateful to Capp Yess and Karl Fisher\nfor allowing me to\nreport the preliminary results of a common unfinished work and\nFrancis Bernardeau for useful discussions during the meeting.\nI acknowledge the AAS travel grant, NSF grant AST-9021414, and\nUniversity of Kansas GRF-94 grant.}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section*{Introduction}\n Harmonic maps into (semi-simple) Riemannian symmetric spaces\n can be studied by infinite dimensional Lie group theory (\\textit{loop\n groups}).\n The dressing action of the loop group enables one to construct\n ``non-trivial solutions'' from a trivial solution and, meromorphically, \n new solutions from old\/known ones. This is based on a\n Weierstrass type representation for harmonic maps which\n  was established by the first named author of the present paper, \n F.~Pedit and H.~Wu \\cite{DPW} in terms of loop group decompositions.\n\n As an application of the loop group method, constant mean curvature\n surfaces in Euclidean  $3$-space $\\mathbb{E}^3$, pseudo-spherical\n surfaces in $\\mathbb{E}^3$ and their indefinite analogues (spacelike or\n timelike surfaces in Minkowski $3$-space) are studied extensively\n \\cite{DIT}, \\cite{I}. The loop group method to construct such\n \\textit{integrable surfaces} is frequently called \\textit{DPW method} or \n \\textit{generalized Weierstrass type representation} for surface\n geometry.\n\n In Euclidean geometry, the starting point of the DPW method is that\n integrable surfaces are naturally associated with harmonic maps into the\n $2$-sphere $\\mathbb{S}^2$  with respect to appropriate conformal\n structures. More precisely, a surface in $\\mathbb{E}^3$ is of constant\n mean curvature if and only if its Gauss map is harmonic. This\n characterization of constancy of mean curvature is referred to as the\n \\textit{Ruh-Vilms property}. On the other hand, let $f:M\\to\n \\mathbb{E}^3$ be a surface in $\\mathbb{E}^3$ with negative or positive\n Gaussian curvature. Then the Gaussian curvature is constant if\n and only if its Gauss map is harmonic with respect to the conformal\n structure determined by the second fundamental form. \n This is another instance of the Ruh-Vilms property.\n Analogously, in Minkowski 3-space the corresponding integrable\n surfaces are associated with harmonic maps into hyperbolic $2$-space\n $\\mathbb{H}^2$ or de Sitter $2$-space $\\mathbb{S}^{1, 1}$.\n\n In M{\\\"o}bius geometry, a conformally immersed surface in\n M{\\\"o}bius $3$-space $\\mathcal{M}^3$ is M{\\\"o}bius minimal (Willmore)\n if and only if its conformal Gauss map (central sphere congruence)\n is a harmonic map into the\n moduli space of oriented $2$-spheres. Note that via the projective\n light-cone model of the M{\\\"obius} $3$-space, the\n moduli space of all oriented $2$-spheres (including point spheres)\n is identified with the de Sitter $4$-space $\\mathbb{S}^{1, 3}$, a\n semi-simple Lorentzian symmetric space \\cite{Bryant:Willmore}.\n\n F.E.~Burstall and U.~Hertrich-Jeromin \\cite{BHJ} proved the Ruh-Vilms\n property for certain surfaces in Lie sphere geometry as well as\n projective differential geometry. More precisely,  let $f:M \\to\n \\mathcal{M}^3$ be a surface in M{\\\"o}bius space. Then $M$ is said to be\n a \\textit{Lie minimal surface} if $M$ is a critical point of the\n Lie-area functional. Note that the Lie-area functional is invariant \n under the M{\\\"o}bius group  as well as the Lie sphere transformation group.\n On the other hand, for every surface in real projective $3$-space\n $\\mathbb{R}P^3$, a projectively invariant area functional (projective\n area) has been introduced. A critical point of the projective area is\n called a \\textit{projective minimal surface}.\n Both the Lie-minimality and projective minimality are characterized by\n the harmonicity of  appropriate Gauss maps taking values in a certain \n non-compact Grassmann manifold equipped with an invariant indefinite\n semi-Riemannian metric.\n\n In addition, E.~Musso and L.~Nicolodi \\cite{MN} showed that in Laguerre\n geometry, Laguerre-minimality of surfaces is characterized by the\n harmonicity of the Laguerre Gauss map (\\textit{middle sphere\n congruence}). \n Thus all these  surfaces classes are defined by the Ruh-Vilms property.\n\n Here we should remark that all the target manifolds of \\textit{harmonic\n Gauss maps} in Lie-sphere, M\\\"obius, projective and Laguerre\n geometries are semi-simple semi-Riemannian \n \\textit{symmetric spaces} equipped with invariant indefinite metrics.\n\n It would be interesting to characterize all surface geometries to\n which the DPW method is applicable. So far only partial results are\n known: See \\textit{e.g.}, the survey \\cite{D} by the first named\n author and the recent classification, carried out by the third named\n author, of surfaces in 3-dimensional space forms\n which are \\textit{real forms} of complex constant mean curvature\n surfaces  \\cite{Kobayashi}.\n \n In this article, we give a \\textit{new} example of such a surface\n geometry. More precisely we shall show that the DPW method is\n also applicable to surfaces in hyperbolic $3$-space\n $\\mathbb{H}^3$ of constant mean curvature $0 \\leq H <1$.\n\n The geometry of a CMC surface in $\\mathbb{H}^3$ depends on\n the range of the mean curvature.\n In case $H>1$, there exist compact CMC surfaces  in $\\mathbb{H}^3$.\n It is well known that there exist natural, locally bijective correspondences \n (so-called \\textit{Lawson correspondences}) between CMC surfaces in \n 3-dimensional space forms.\n In particular, CMC surfaces in $\\mathbb{H}^3$ with $H\\geq 1$ have \n corresponding CMC or minimal surfaces (\\textit{Lawson correspondents}) \n in Euclidean 3-space or the $3$-sphere. Based on the Lawson correspondences,\n M.~Kilian, W.~Rossman, N.~Schmitt and the third named author \\cite{KKRS}\n studied CMC surfaces in $\\mathbb{H}^3$ with $H>1$ via the DPW method. \n In case $H=1$, the Lawson correspondent in $\\mathbb E^3$ \n is a minimal surface, and  R.~Bryant \\cite{Br} \n gave the Weierstrass type representation  \n in terms of a  holomorphic differential equation.\n\n On the other hand, by the maximum principle, there are no compact \n CMC surfaces in $\\mathbb{H}^3$ (without boundary) such that \n $0\\leq H \\leq1$. Moreover, \n CMC surfaces in $\\mathbb{H}^3$ with $0\\leq H<1$ have \n neither Lawson correspondents in Euclidean 3-space $\\mathbb{E}^3$ nor in\n the $3$-sphere  $\\mathbb{S}^3$. \n This means that CMC surface geometry with $0\\leq H<1$\n in hyperbolic space has special and unusual features and hence \n it is of interest. \n\n There is another motivation for the study of CMC surfaces with $0<H<1$. \n D.~Chopp and J.~Velling \\cite{CV} found numerical evidence that \n $\\mathbb{H}^3$ can be foliated by CMC discs with $0<H<1$ that share a common \n Jordan curve boundary in the ideal boundary of $\\mathbb{H}^3$. \n\n In contrast to the case $H\\geq 1$, there are relatively few papers on \n CMC surfaces in $\\mathbb{H}^3$ with  $0\\leq H<1$.\n Initial studies on the existence and regularity of CMC surfaces in \n $\\mathbb{H}^3$ with $0\\leq H<1$ can be found in  \n M.~Anderson \\cite{Anderson}, \n K.~Uhlenbeck \\cite{Uhlenbeck}, \n K.~Polthier \\cite{Polthier}, \n Y.~Tonegawa \\cite{Tonegawa}\n and J.~Velling \\cite{Velling}.\n\n In \\cite{BaBo}, M.~Babich and A.~Bobenko considered minimal surfaces \n of finite type in $\\mathbb H^3$ and described them explicitly in terms \n of theta functions.\n\n One of the remarkable points of the present study is that, --as in the\n already known cases--\n we associate a certain kind of \\textit{Gauss map} to constant mean\n curvature surfaces in $\\mathbb H^3$. And the constancy of the mean\n curvature is equivalent to the harmonicity of the Gauss map. \n Thus the Ruh-Vilms property holds.\n However, unlike all surface geometries discussed by the DPW method\n so far, our target space is \\textit{not} a symmetric space but a \n real $5$-dimensional $4$-symmetric space.\n\n In general, the loop group approach can not be applied to harmonic maps\n into non-symmetric homogeneous spaces. \n Thus it is not obvious how  to apply the loop group method to our case.  \n More precisely, the loop group method has two main \n ingredients. One is a zero curvature representation and the other one \n is a loop group decomposition.\n The zero curvature representation is equivalent with the existence of a \n flat connections and this representation enables us to use \n loop groups. A loop group decomposition is at the heart \n of the DPW method and recovers surfaces from holomorphic potentials.\n\n In this paper, the Gauss map of CMC surfaces in $\\mathbb H^3$\n has as target space a $4$-symmetric \n space. Thus the DPW method is, a priori, not applicable.\n However it turns out that these Gauss maps are Legendre maps. \n In the situation under consideration this suffices to obtain a \n zero curvature representation for all Gauss maps of CMC\n surfaces in $\\mathbb H^3$ (and in particular for mean curvature \n $H$ satisfying $0 \\leq H <1$).\n\n Applying the DPW method in this setting, we obtain a potential\n in a $4$-graded Lie algebra. However, not every potential in this \n $4$-graded Lie algebra will admit the application of the DPW method, \n since the admissibility condition\n will not be satisfied in general \\cite{BP}.\n\n It is a fortunate accident that the Maurer-Cartan form of \n the moving frame of every CMC immersion into $\\mathbb H^3$\n with $0 \\leq H <1$ is also contained in same real form of \n $\\Lambda \\mathrm{SL}_2 \\mathbb C_{\\sigma} \\times \\Lambda \\mathrm{SL}_2 \\mathbb C_{\\sigma}$\n and this is part of some ``symmetric space setting'' \n (which actually is not induced from any finite dimensional symmetric space).\n This enables us to apply the DPW method via standard way and therefore \n two main featured of the loop group method can be established and \n the loop group method is applicable.\n\n The reason for the applicability of the DPW method in our case\n also can be explained by using the Lawson correspondence which \n transforms our original harmonic Gauss map into a {\\it primitive} \n map, to which the DPW method is applicable.\n By the Lawson correspondence for a CMC surface \n with mean curvature $0 \\leq H <1$ in $\\mathbb H^3$, \n the CMC surface can be considered \n as a minimal surface in $\\mathbb H^3 (c)$ for some $c <0$, where  \n $\\mathbb H^3 (c)$ denotes the hyperbolic $3$-space of sectional \n curvature $c$, \\cite{AA:CMCH3}.\n At this point it is important to note that \n the Gauss map of a CMC surface in $\\mathbb H^3(c)$ is primitive\n if and only if the surface is minimal. \n Finally we recall that primitive maps \n into $k$-symmetric ($k>2$) spaces\n form a special class of harmonic maps and it is known that the DPW method \n is applicable to primitive maps \\cite{DMPW}. Moreover, we interpret\n our original Gauss map of a CMC surface in $\\mathbb H^3$,\n which is only a Legendre harmonic map, as a primitive \n Gauss map of some CMC surface in $\\mathbb H^3(c)$, \n and thus the loop group method becomes applicable.\n\n We would like to point out that the DPW method for CMC surfaces \n in $\\mathbb{H}^3$ presented in \\cite{KKRS} is based on the \n Lawson correspondence, so the authors of \n \\cite{KKRS} can treat CMC surfaces in $\\mathbb{H}^3$ with mean curvature $H>1$.\n In the present study, we give a unified approach to both\n cases: $H>1$ and $0 \\leq H  <1$. The case $H =1$ has been studied\n extensively already, \\textit{e.g.}, \\cite{Br}, \\cite{UY},\n and will not be considered in  this paper.\n\n This paper is organized as follows.\n After establishing the requisite knowledge on harmonic maps into normal \n semi-Riemannian homogeneous spaces and homogeneous geometry of\n hyperbolic space in sections \\ref{sc:Harmonic}--\\ref{sc:Hyperbolic},\n we shall devote  sections \\ref{sc:SurfacesH3}--\\ref{sc:2by2} to \n surface geometry in $\\mathbb{H}^3$ in terms of \n $\\mathrm{SL}_{2}\\mathbb{C}$-valued functions. \n In section \\ref{sc:CMCH>1}, \n we shall  give a loop group formulation of CMC surfaces with $H>1$. \n This formulation is different from the one used in \\cite{KKRS}. \n We shall clarify the Lawson correspondences between CMC surfaces \n in $\\mathbb{H}^3$ with $H>1$ and  CMC surfaces in $\\mathbb{E}^3$ \n in terms of loop groups.\n\n In the next section \\ref{sc:CMCH<1}, we shall give a loop group formulation\n for CMC surfaces with  $0\\leq H<1$. The two cases discussed in section \n \\ref{sc:CMCH>1} and section  \\ref{sc:CMCH<1} are distinguished \n by the automorphism of the Kac-Moody Lie algebra which characterizes\n the Lax equations of corresponding types of CMC surfaces.  \n \n The key tool of the present study is a ``contact geometric characterization'' \n of CMC surfaces. \n For every (oriented) surface $f:M\\to \\mathbb{H}^3$, there exists \n a smooth map $F$ into the unit tangent sphere bundle $\\mathrm{U}\\mathbb{H}^3$ \n of $\\mathbb{H}^3$. \n The map $F$ is referred to as  the \\textit{Gauss map} of $M$.\n One can see that the Gauss map $F$ of a surface $M$ is a Legendre map\n in the sense of V.I.~Arnold \\cite{Arnold}, that is, \n it is tangent to the canonical contact structure. \n Moreover, the constancy of the mean curvature is equivalent \n to the harmonicity of that Gauss map. \n Thus the Ruh-Vilms property holds.\n  \n In section \\ref{sc:4symUnittangent}, we shall characterize harmonic Gauss maps\n in a way which is different from the Ruh-Vilms property.\n The Legendre property of the Gauss map will be characterized in terms of \n $4$-symmetric structure of $\\mathrm{U}\\mathbb{H}^3$. \n These characterizations in terms of contact geometry and \n $4$-symmetric structure yield a zero-curvature representation for\n Legendre harmonic maps.  Based on these results, in sections \n \\ref{sc:Potential}--\\ref{sc:DPW}, \n we shall give a DPW method for Legendre harmonic maps \n (and hence for CMC surfaces with $0\\leq H<1$). \n In the final section, we shall exhibit some examples of CMC surfaces \n with $0\\leq H<1$ via the DPW method established in this paper.\n\n In the surface geometry of $\\mathbb{H}^3$, several notions of \n \\textit{Gauss map}  have been introduced.  \n For the convenience of the reader, we collect in the appendix \n several notions of Gauss map that have been used by \n different authors. We shall explain how these other \n Gauss maps can be derived  from our Gauss map.  \n As a side result we obtain that only the Gauss map considered in \n this paper is suitable for a DPW method of CMC surfaces in $\\mathbb H^3$.\n\n\\textbf{Acknowledgements:} We would like to thank Idrisse Khemar \n for helpful comments on a preliminary \n version of this paper. Part of this\n work was carried out during the workshop ``Surface Theory: Research in\n Pairs'' at Kloster Sch\\\"{o}ntal, March 2008, funded by DFG Grant\n DO776. This work was started when the second named author visited the\n University of Kansas in 2000. He would like to express his sincere thanks\n to the Department of Mathematics. \n Some of the results of this article were \n reported at  the workshop ``Progress in Surface Theory'' held at\n Mathematisches Forschungsinstitut Oberwolfach, May,  2010.\n\\section{Harmonic maps into normal semi-Riemannian homogeneous spaces}\n\\label{sc:Harmonic}\n\\subsection{}\n Let $G\/H$ be a reductive homogeneous space with semi-simple Lie group\n $G$. We equip $G\/H$ with a $G$-invariant semi-Riemannian metric which is\n derived from (a constant multiple of) the Killing form of $G$.\n Assume that the Lie algebra $\\mathfrak{h}$ of $H$ is non-degenerate with \n respect to the induced scalar product.  \n Then the orthogonal complement $\\mathfrak{p}$ of $\\mathfrak{h}$ is \n non-degenerate and can be identified with the tangent space of $G\/H$ \n at the origin $o=H$.  \n The Lie algebra $\\mathfrak{g}$ is decomposed into the direct sum:\n$$\n\\mathfrak{g}=\\mathfrak{h}\\oplus \\mathfrak{p}\n$$\n of linear subspaces.  \n The resulting homogeneous semi-Riemannian manifold is a \n \\textit{normal semi-Riemannian homogeneous space} \\cite{CE}.\n\n\n\\subsection{}\nA smooth map $\\psi:M \\to N$ of a Riemann surface $M$ into a semi-Riemannian \nmanifold $N$ is said to be a \\textit{harmonic map} if its tension field  \n$\\mathrm{tr}\\>(\\nabla \\mathrm{d}\\psi)$ vanishes \\cite{Ur}. \nWhen the target space $N$ is a normal semi-Riemannian homogeneous space $G\/H$,\nthe harmonic map equation for $\\psi$ has a particularly simple form.\n\nNow let $\\psi:\\mathbb{D} \\to G\/H$ be a smooth map from a simply connected\n domain $\\mathbb D \\subset \\mathbb C $ into a normal semi-Riemannian homogeneous space. \nTake a frame $\\Psi:\\mathbb{D} \\to G$ of\n $\\psi$ and put $\\alpha:=\\Psi^{-1}\\mathrm{d}\\Psi$. Then we have the\n identity (\\textit{Maurer-Cartan equation}):\n$$\n\\mathrm{d}\\alpha+\\frac{1}{2}[\\alpha \\wedge \\alpha]=0.\n$$\n Decompose $\\alpha$ along the Lie algebra decomposition\n $\\mathfrak{g}=\\mathfrak{h}\\oplus\\mathfrak{p}$ as\n$$\n\\alpha=\\alpha_{\\mathfrak{h}}+\\alpha_{\\mathfrak{p}},\n\\ \\\n\\alpha_{\\mathfrak{h}}\\in\n\\mathfrak{h},\n\\ \\\n\\alpha_{\\mathfrak{p}}\n\\in\n\\mathfrak{p}.\n$$\nWe decompose $\\alpha_{\\mathfrak{p}}$ \nwith respect to the conformal\nstructure of $\\mathbb{D}$ as\n$$\n\\alpha_{\\mathfrak p}=\n\\alpha_{\\mathfrak p}^{\\prime}+\n\\alpha_{\\mathfrak p}^{\\prime \\prime}.\n$$\nHere $\\alpha_{\\mathfrak{p}}^{\\prime}$ and \n$\\alpha_{\\mathfrak p}^{\\prime \\prime}$\nare the $(1,0)$ and $(0,1)$ part of \n$\\alpha_{\\mathfrak p}$, respectively.\n\n\nThe harmonicity of $\\psi$ is equivalent to\n\\begin{equation}\\label{harmonicity}\n\\mathrm{d}(*\\alpha_{\\mathfrak p})+\n[\\alpha \\wedge *\\alpha_{\\mathfrak p}]\n=0.\n\\end{equation}\nHere $*$ denotes the Hodge star operator of $\\mathbb{D}$.\nThe Maurer-Cartan equation is split into its $\\mathfrak{h}$-component and \n$\\mathfrak{p}$-component:\n\\begin{equation}\\label{MC-h}\n\\mathrm{d}\\alpha_{\\mathfrak{h}}+\\frac{1}{2}\n[\\alpha_{\\mathfrak{h}}\\wedge \\alpha_{\\mathfrak{h}}]\n+[\\alpha_{\\mathfrak{p}}^{\\prime}\\wedge \n\\alpha_{\\mathfrak{p}}^{\\prime\\prime}]_{\\mathfrak{h}}=0,\n\\end{equation}\n\\begin{equation}\\label{MC-p}\n\\mathrm{d}\\alpha\n_{\\mathfrak{p}}^{\\prime}\n+\n[\n\\alpha_{\\mathfrak{h}}\n\\wedge \n\\alpha_{\\mathfrak{p}}^{\\prime}\n]\n+\n\\mathrm{d}\\alpha_{\\mathfrak{p}}^{\\prime\\prime}\n+\n[\n\\alpha_{\\mathfrak{h}}\n\\wedge \n\\alpha_{\\mathfrak{p}}^{\\prime\\prime}\n]\n+\n[\n\\alpha_{\\mathfrak{p}}^{\\prime}\n\\wedge \n\\alpha_{\\mathfrak{p}}^{\\prime\\prime}\n]_{\\mathfrak{p}}\n=0.\n\\end{equation}\nHence for a harmonic map $\\psi:\\mathbb{D}\\to G\/H$ with a framing $\\Psi$, \nthe pull-back $1$-form $\\alpha=\\Psi^{-1}\\mathrm{d}\\Psi$ satisfies\n(\\ref{harmonicity}), (\\ref{MC-h}) and (\\ref{MC-p}). \nCombining (\\ref{harmonicity}) and (\\ref{MC-p}), we have\n\\begin{equation}\\label{harm+MC}\n\\mathrm{d}\\alpha^{\\prime}_{\\mathfrak p}+\n[\\alpha_{\\mathfrak h} \\wedge \n\\alpha_{\\mathfrak p}^{\\prime}]=\n-\\frac{1}{2}[\\alpha_{\\mathfrak p}^{\\prime}\n\\wedge \n\\alpha_{\\mathfrak p}^{\\prime \\prime}\n]_{\\mathfrak p}.\n\\end{equation}\nOne can easily check that \nthe harmonic map equation for $\\psi$ combined with\nthe Maurer-Cartan equation is equivalent to the system (\\ref{MC-h}) and (\\ref{harm+MC}).\n\nAssume that\n\\begin{equation}\\label{admissible}\n[\\alpha_{\\mathfrak p}^{\\prime}\n\\wedge \n\\alpha_{\\mathfrak p}^{\\prime \\prime}\n]_{\\mathfrak p}=0.\n\\end{equation}\nThen the harmonic map equation together with the Maurer-Cartan equation\nis reduced to the system of equations:\n$$\n\\mathrm{d}\\alpha_{\\mathfrak p}^{\\prime}+\n[\\alpha_{\\mathfrak h} \\wedge \n\\alpha_{\\mathfrak p}^{\\prime}]=0,\n$$\n$$\n\\mathrm{d}\\alpha_{\\mathfrak h}+\n\\frac{1}{2}\n[\\alpha_{\\mathfrak h} \\wedge\n\\alpha_{\\mathfrak h}]\n+[\\alpha_{\\mathfrak p}^{\\prime}\n\\wedge\n\\alpha_{\\mathfrak p}^{\\prime \\prime}\n]=0.\n$$\nThis system of equations is equivalent to the\nfollowing \\textit{zero-curvature representation}:\n$$\n\\mathrm{d}\\alpha_{\\lambda}+\n\\frac{1}{2}[\\alpha_{\\lambda}\n\\wedge \\alpha_{\\lambda}]\n=0,\n$$\nwhere $\\alpha_{\\lambda}:=\n\\alpha_{\\mathfrak h}\n+\\lambda^{-1}\\alpha_{\\mathfrak p}^{\\prime}+\n\\lambda\\> \\alpha_{\\mathfrak p}^{\\prime \\prime}$\nwith $\\lambda \\in S^1$. \n\n\\begin{proposition}\n Let $\\mathbb D$ be a region in  $\\mathbb C$ and \n $\\psi:\\mathbb{D} \\to G\/H$ a harmonic map\n which satisfies the admissibile condition {\\rm (\\ref{admissible})}.\n Then the loop of connections $\\mathrm{d}+\\alpha_{\\lambda}$ is flat for\n all $\\lambda$. Namely:\n\\begin{equation}\\label{eq:flatconnections}\n\\mathrm{d}\\alpha_{\\lambda}+\\frac{1}{2}\n[\\alpha_{\\lambda}\\wedge\n\\alpha_{\\lambda}]=0\n\\end{equation}\n for all $\\lambda$.\n Conversely assume that $\\mathbb{D}$ is simply connected. Let\n $\\alpha_{\\lambda}=\\alpha_{\\mathfrak h}+\\lambda^{-1} \\alpha_{\\mathfrak\n p}^{\\prime}+\\lambda\\alpha_{\\mathfrak  p}^{\\prime\\prime}$ be an\n $S^1$-family of $\\mathfrak{g}$-valued $1$-forms \n satisfying \\eqref{eq:flatconnections} for all $\\lambda \\in S^1$.\nThen there exists a $1$-parameter family of maps\n$\\Psi_{\\lambda} :\\mathbb{D} \\to G$ such that\n$$\n\\Psi_{\\lambda}^{-1}{\\mathrm d}\\Psi_{\\lambda}=\\alpha_{\\lambda}\n\\;\\;\\;\\mbox{and}\\;\\;\\;\n\\psi_{\\lambda}=\\Psi_{\\lambda}\\;{\\mbox{\\rm mod}}\\; H:\\mathbb{D} \\to G\/H\n$$\nis harmonic for all $\\lambda$.\n\\end{proposition}\n\nWhen the target space $G\/H$ is a semi-Riemannian symmetric space, then the\nadmissible condition \nis fulfilled automatically for any $\\psi$, since\n$[\\mathfrak{p},\\mathfrak{p}]\\subset \\mathfrak{h}$.\n\n\n\\subsection{Primitive maps}\nLet $G$ be a semi-simple Lie group \nwith automorphism $\\tau$ of order $k> 2$.\nA normal semi-Riemannian \nhomogeneous space $G\/H$ is said to be a \n(regular)  \\textit{semi-Riemannian $k$-symmetric space}\nif $G_{\\tau}^{\\circ}\\subset H \\subset G_{\\tau}$.\nHere $G_{\\tau}$ is the Lie subgroup of all fixed points of \n$\\tau$ and \n$G_{\\tau}^\\circ$ \nthe identity component of it.\n\nWe denote the induced Lie algebra automorphism of \n$\\mathfrak{g}$ by the same letter $\\tau$. \nNow we have the eigenspace decomposition of \nthe complexified Lie algebra \n$\\mathfrak{g}^{\\mathbb{C}}$;\n$$\n\\mathfrak{g}^{\\mathbb{C}}=\\sum_{j\\in \\mathbb{Z}_k}\n\\mathfrak{g}_{j}^{\\mathbb{C}}.\n$$ \nHere $\\mathfrak{g}_j^{\\mathbb{C}}$ is the eigenspace of $\\tau$ with\neigenvalue $\\zeta^j$. Here $\\zeta$ is the primitive $k$-th root of unity.\nIn particular, $\\mathfrak{g}_0^{\\mathbb{C}}=\\mathfrak{h}$ and $\\overline{\\mathfrak{g}_{-1}^{\\mathbb{C}}}=\\mathfrak{g}_1^{\\mathbb{C}}$.\n\\begin{definition}[\\cite{BP}]\n{\\rm\nLet $\\psi:M\\to G\/H$ be a smooth map of a Riemann surface into a regular \nsemi-Riemannian $k$-symmetric space. Then $\\psi$ is said to be a\n\\textit{primitive map} if any frame $\\Psi$ has\n$\\alpha_{\\mathfrak{p}}^{\\prime}$  taking value in $\\mathfrak{g}_{-1}^{\\mathbb{C}}$.\n}\n\\end{definition} \nOne can see that every primitive map satisfies the admissible condition \n(\\ref{admissible}). In fact,\n$$\n[\\alpha_{\\mathfrak{p}}^{\\prime}\n\\wedge\n\\alpha_{\\mathfrak{p}}^{\\prime\\prime}]\n=[\\alpha_{-1}\\wedge\\alpha_{1}]\n\\in \\mathfrak{g}_0^{\\mathbb{C}}=\n\\mathfrak{h}.\n$$ \nFor more information on primitive maps, we refer to \\cite{BP} and\n\\cite{DMPW}.\n\n\\section{Hyperbolic space}\\label{sc:Hyperbolic}\n\\subsection{}\n Let $\\mathbb{E}^{1,n}$ be the $n+1$-dimensional Minkowski space\n with scalar product.\n$$\n\\langle \\Vec{x},\\Vec{y}\\rangle=-x_{0}y_{0}+x_{1}y_{1}+\\cdots+x_{n}y_{n}.\n$$\n We denote by $\\Vec{e}_0=(1, 0, \\dots, 0), \\Vec{e}_1=(0, 1, \\dots, 0), \\dots,\n \\Vec{e}_n=(0, 0, \\dots, 1)$ the basis of $\\mathbb{E}^{1,n}$.\n We will identify Euclidean $n$-space $\\mathbb E^{n} = \\mathbb\n E^{0,n} = \\{ \\Vec{x} \\in \\mathbb{E}^{1,n} \\;| \\;x_0 =0\\}$. \n Let $\\epsilon$ denote the \\textit{signature matrix} defined by\n $\\epsilon=\\operatorname{diag} (-1,1,\\cdots,1)$. Then we define the\n \\textit{Lorentz group} $\\mathrm{O}_{1,n}$ by\n$$\n \\mathrm{O}_{1,n}=\\{A\\in\\mathrm{GL}_{n+1}\\mathbb{R}\n \\\n \\vert\n \\\n A^{t}\\epsilon A=\\epsilon\\}.\n$$\n Note that $\\mathrm{O}_{1,n}$ acts isometrically on $\\mathbb{E}^{1,n}$,\n and has four connected components. We denote by $\\mathrm{SO}^{+}_{1,n}$\n the identity component of $\\mathrm{O}_{1,n}$. The Lie algebra\n $\\mathfrak{so}_{1,n}$ of $\\mathrm{SO}^+_{1,n}$ is given by\n$$\n \\mathfrak{so}_{1,n} =\n \\left\n \\{\n X=\n \\left(\n \\begin{array}{cc}\n 0 & \\Vec{x}^{t}\\\\\n \\Vec{x} & b\n \\end{array}\n \\right)\n \\\n \\biggr \n \\vert\n \\\n b\\in\n \\mathfrak{o}_{n},\n \\\n \\Vec{x}\\in \\mathbb{R}^{n}\n \\\n \\right\\}.\n$$\n We equip the Lie algebra $\\mathfrak{so}_{1,n}$ with an invariant\n scalar product $\\langle \\cdot,\\cdot\\rangle$. \n\\begin{equation}\\label{scalarproduct}\n \\langle X,Y\\rangle=\\frac{1}{2}\\mathrm{tr}\\>(XY),\\ \\\n X,Y \\in \\mathfrak{so}_{1,n}.\n\\end{equation}\n Since $\\mathrm{O}_{1,n}$ is non-compact, this scalar product\n is indefinite. More precisely it has signature $(n(n-1)\/2,n)$.\n\\subsection{}\n The Lie group $\\mathrm{SO}^{+}_{1,n}$ acts transitively and \n isometrically on the hyperbolic $n$-space\n$$\n \\mathbb{H}^{n}=\\{\\Vec{x}\\in \\mathbb{E}^{1,n}\n \\\n \\vert\n \\ \\langle \\Vec{x},\\Vec{x}\\rangle=-1,\\\n x_{0}>0\\\n \\}.\n$$\n The isotropy subgroup of \n $\\mathrm{SO}^{+}_{1,n}$ at $\\Vec{e}_{0}$ is\n$$\n\\left\\{\n\\left(\n\\begin{array}{cc}\n1 & 0\n\\\\\n0 & a \n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\\na \\in \\mathrm{SO}_{n}\n\\right\\}.\n$$\nHence $\\mathbb{H}^{n} \\cong \\mathrm{SO}^{+}_{1,n}\/ \\mathrm{SO}_{n}$.\nThe tangent space $T_{\\Vec{e}_0}\\mathbb{H}^n$ of $\\mathbb{H}^n$ at $\\Vec{e}_0$ \nis identified with the following linear subspace of $\\mathfrak{so}_{1,n}$;\n$$\n\\left\\{\n\\left(\n\\begin{array}{cc}\n0 & \\Vec{x}^{t}\\\\\n\\Vec{x} & 0\n\\end{array}\n\\right)\n\\\n\\biggr \n\\vert\n\\\n\\Vec{x}\\in \\mathbb{R}^{n}\n\\\n\\right\\}.\n$$\n On $T_{\\Vec{e}_0}\\mathbb{H}^{n}$, the scalar product\n (\\ref{scalarproduct}) is positive\n definite. Moreover the Riemannian metric on\n $\\mathrm{SO}^{+}_{1,n}\/\\mathrm{SO}_{n}$ induced from\n (\\ref{scalarproduct}) is of constant curvature $-1$.\n\n Next we define an involution $\\sigma_{H}$ of $\\mathrm{SO}^{+}_{1,n}$ by\n $\\sigma_{H}=\\mathrm{Ad}(\\epsilon)$. Then\n $(\\mathrm{SO}^{+}_{1,n},\\sigma_{H})$ is a symmetric pair.  \n\\begin{remark}\n{\\rm\nThe Killing form $\\varphi$ of $\\mathfrak{so}_{1,n}$\nis \n$$\n\\varphi(X,Y)=-(n-1)\\mathrm{tr}\\>(XY),\n\\ \\\nX,Y \n\\in \\mathfrak{so}_{1,n}.\n$$\n We equip the tangent space $T_{\\Vec{e}_0}\\mathbb{H}^{n} \\subset\n \\mathfrak{so}_{1,n}$ with the inner product $\\langle\\cdot,\\cdot\\rangle$\n given  by\n$$\n\\langle\n X,Y\\rangle=-\\frac{1}{2(n-1)}\\varphi(X,Y)=\\frac{1}{2}\\mathrm{tr}\\>(XY).\n$$\n}\n\\end{remark}\n\n\\subsection{}\\label{sc:unit tangent sphere bundle}\nLet us denote by $\\mathrm{U}\\mathbb{H}^n$ the \n\\textit{unit tangent sphere bundle} of $\\mathbb{H}^n$.\nNamely, $\\mathrm{U}\\mathbb{H}^n$ is the manifold of all unit tangent \nvectors of $\\mathbb{H}^n$.\nThen $\\mathrm{U}\\mathbb{H}^n$ is identified with the submanifold\n$$\n\\left\\{\n(\\Vec{x},\\Vec{v})\\\n\\vert\n\\\n\\langle \\Vec{x},\\Vec{x}\\rangle=-1,\\\n\\langle \\Vec{v},\\Vec{v}\\rangle=1,\\\n\\langle \\Vec{x},\\Vec{v}\\rangle=0,\\\nx_{0}>0\n\\right\\}\n$$\n of $\\mathbb{E}^{1,n}\\times \\mathbb{E}^{1,n}$. The tangent space\n $T_{(\\Vec{x},\\Vec{v})}\\mathrm{U}\\mathbb{H}^{n}$ at a point\n $(\\Vec{x},\\Vec{v})$ is expressed as\n$$\nT_{(\\Vec{x},\\Vec{v})}\\mathrm{U}\\mathbb{H}^{n}=\n\\{(X,V)\\in \\mathbb{E}^{1,n}\\times \n\\mathbb{E}^{1,n}\n\\\n\\vert\n\\\n\\langle \\Vec{x},X\\rangle=0,\\\n\\langle \\Vec{v},V\\rangle=0,\\ \n\\langle\\Vec{x}, V \\rangle + \\langle \\Vec{v},\nX\\rangle =0 \n\\}.\n$$\n Define a $1$-form $\\omega$ on $\\mathrm{U}\\mathbb{H}^{n}$ by\n$$\n\\omega_{(\\Vec{x},\\Vec{v})}(X,V)=\\langle X,\\Vec{v}\\rangle=-\\langle\n\\Vec{x},V\\rangle.\n$$\n Then one can see that $\\omega$ is a \\textit{contact form} on\n $\\mathrm{U}\\mathbb{H}^{n}$, \\textit{i.e.},\n $(\\mathrm{d}\\omega)^{n-1}\\wedge \\omega\\not=0$. The distribution \n$$\n\\mathcal{D}_{(\\Vec{x},\\Vec{v})}:=\n\\{(X,V)\\in T_{(\\Vec{x},\\Vec{v})}\\mathrm{U}\n\\mathbb{H}^n\n\\\n\\vert\n\\\n\\omega_{(\\Vec{x},\\Vec{v})}(X,V)=0\n\\}\n$$\n is called the \\textit{canonical contact structure} of\n $\\mathrm{U}\\mathbb{H}^{n}$, see section \\ref{sc:contactmfd}.\n\n\n The Lorentz group $\\mathrm{SO}^+_{1,n}$ acts on $\\mathrm{U}\\mathbb H^n$ via\n $A \\cdot (\\Vec{x}, \\Vec{v}) = (A \\Vec{x},A\\Vec{v})$. It is easy to see that\n under this action the unit tangent sphere bundle\n $\\mathrm{U}\\mathbb{H}^{n}$ is a homogeneous space \n of $\\mathrm{SO}^{+}_{1,n}$. The isotropy subgroup at\n $(\\Vec{e}_{0},\\Vec{e}_1)$ is\n$$\n\\left\\{\n\\left(\n\\begin{array}{cc}\n\\mathbf{1} & 0\\\\\n0 & b \n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\\nb \\in \\mathrm{SO}_{n-1}\n\\right\\}.\n$$\n Hence $\\mathrm{U}\\mathbb{H}^n \\cong\n \\mathrm{SO}^{+}_{1,n}\/\\mathrm{SO}_{n-1}$. \n The Lie algebra of this isotropy subgroup is\n$$\n\\left\\{\n\\left(\n\\begin{array}{cc}\n0 & 0 \\\\\n0 & t\n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\\nt\\in \\mathfrak{so}_{n-1}\n\\right\\}.\n$$\n The tangent space $T_{(\\Vec{e}_0,\\Vec{e}_1)}(\\mathrm{U}\\mathbb{H}^{n})$\n is identified with\n$$\n\\left\n\\{\nX=\n\\left(\n\\begin{array}{lll}\n0 & x_1 &\\Vec{x}^t \\\\ \nx_1 &0 & \\Vec{y}^t \\\\ \n\\Vec{x} &- \\Vec{y} &\\Vec{0}\n\\end{array}\n\\right)\n\\\n\\biggr \n\\vert\n\\\nx_1\\in \\mathbb R, \\Vec{x}, \\Vec{y} \\in \\mathbb R^{n-1}\n\\\n\\right\\}\\subset \\mathfrak{so}_{1,n}.\n$$\n The semi-Riemannian metric induced on the homogeneous space\n $\\mathrm{U}\\mathbb{H}^n \\cong \\mathrm{SO}^{+}_{1,n}\/\\mathrm{SO}_{n-1}$\n via the scalar product (\\ref{scalarproduct}) on $\\mathfrak{so}_{1,n}$\n has signature $(n-1,n)$. One can see that $\\mathrm{U}\\mathbb{H}^{n}$ is\n a normal semi-Riemannian homogeneous space (hence it is naturally\n reductive) but not a semi-Riemannian symmetric space.\n\n\\subsection{}\n Let us denote by $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$ the\n Grassmann manifold\n of all oriented \\textit{timelike} planes in\n $\\mathbb{E}^{1,n}$. There exists a natural projection\n $\\pi_1:\\mathrm{U}\\mathbb{H}^{n}\\to\n \\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$:\n$$\n\\pi_1 (\\Vec{x},\\Vec{v})= \\Vec{x} \\wedge \\Vec{v}. \n$$\n The Grassmann manifold\n is a homogeneous space of\n $\\mathrm{SO}^{+}_{1,n}$. In fact, $\\mathrm{SO}^+_{1,n}$ acts\n isometrically and transitively on $\\mathrm{Gr}_{1, 1} (\\mathbb\n E^{1,n})$ by \n $$\n A \\cdot (\\Vec{x} \\wedge \\Vec{v}) = (A \\Vec{x}) \\wedge ( A \\Vec{v}).\n $$\n The isotropy subgroup at $\\Vec{e}_{0}\\wedge\n \\Vec{e}_{1}$ is $\\mathrm{SO}_{1,1}\\times \\mathrm{SO}_{n-1}$.  The \n semi-Riemannian metric induced from the Killing form of $\\mathrm{SO}^+_{1,n}$ \n has signature $(n-1,n-1)$.\n\n The contact form $\\omega$ on $\\mathrm{U}\\mathbb{H}^n$ induces \n a symplectic form $\\Omega$ on the Grassmann manifold \n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$ so that \n $\\pi_{1}^{*}\\Omega=\\mathrm{d}\\omega$.\n\n The Grassmann manifold\n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$ admits an invariant product\n structure $\\mathrm{P}$, \\textit{i.e.}, an endomorphism field \n $\\mathrm{P}$ satisfying $\\mathrm{P}^2=\\mathrm{Id}$, \n compatible with the metric. Moreover, $\\mathrm{P}$ is parallel\n with respect to the Levi-Civita connection. The resulting homogeneous\n space $\\mathrm{SO}^{+}_{1,n}\/\\mathrm{SO}_{1,1}\\times \\mathrm{SO}_{n-1}$ is\n an indefinite para-K\\\"ahler symmetric space \\cite{KK}. \n The symplectic form $\\Omega$ is related to the para-K\\\"ahler \n structure by \n $$\n \\Omega(X, Y) = 2 \\langle X, \\mathrm{P}Y \\rangle\n $$\n for all vector fields $X$ and $Y$, see \\cite{Honda}.\n\n\\subsection{}\n From now on we will concentrate on the case $n=3$. Then \n$$\n\\mathfrak{so}_{1,3}=\n\\left\\{X=\n\\left(\n\\begin{array}{cccc}\n0 & x_1 & x_2 & x_3\\\\\nx_1 & 0 & x_{12} & x_{13}\\\\\nx_1 & -x_{12} & 0 & x_{23}\\\\\nx_{3} & -x_{13} & -x_{23} & 0\n\\end{array}\n\\right)\n=\n\\left(\n\\begin{array}{cc}\n0 & \\Vec{x}^{t}\\\\\n\\Vec{x} & b^{x}\n\\end{array}\n\\right)\n\\right\\}.\n$$\n Here we put \n$$\n\\Vec{x}=(x_1,x_2,x_3)^{t} \\in \\mathbb R^3 \\subset \\mathbb\nE^{1,3}\\ \\ \\mbox{and} \\ \\\nb^{x}=\\left(\n\\begin{array}{ccc}\n0 & x_{12} & x_{13}\\\\\n -x_{12} & 0 & x_{23}\\\\\n -x_{13} & -x_{23} & 0\n\\end{array}\n\\right).\n$$\n Then one can check that\n$$\nXY=\\left(\n\\begin{array}{cc}\n\\Vec{x}^{t}\\Vec{y} & \\Vec{x}^{t}b^{y}\n\\\\\nb^{x}\\Vec{y} & b^{x}b^{y}+\\Vec{x}\\Vec{y}^{t}\n\\end{array}\n\\right)\n$$\n and\n\\begin{equation}\\label{eq:innerproduct}\n\\frac{1}{2} \\mathrm{tr}\\>(XY)=\\frac{1}{2}\\mathrm{tr}\\>(b^xb^y) +\\Vec{x}^{t}\\Vec{y}\n= - (x_{12}y_{12}+x_{13}y_{13}+x_{23}y_{23}) + \\langle \\Vec{x},\\Vec{y}\\rangle\n\\end{equation}\n for \n$$\nX=\\left(\n\\begin{array}{cc}\n0 & \\Vec{x}^{t}\\\\\n\\Vec{x} & b^{x}\n\\end{array}\n\\right),\n\\ \\\nY=\n\\left(\n\\begin{array}{cc}\n0 & \\Vec{y}^{t}\\\\\n\\Vec{y} & b^{y}\n\\end{array}\n\\right).\n$$\n Now we identify $\\mathfrak{so}_{1,3}$ with $\\mathbb{R}^{6}$ by\n\\begin{equation}\\label{eq:identifyso}\n\\left(\n\\begin{array}{cccc}\n0 & x_1 & x_2 & x_3\\\\\nx_1 & 0 & x_{12} & x_{13}\\\\\nx_{2} & -x_{12} & 0 & x_{23}\\\\\nx_{3} & -x_{13} & -x_{23} & 0\n\\end{array}\n\\right)\n\\longleftrightarrow \n(x_{12},x_{13},x_{23}, x_1,x_2,x_3).\n\\end{equation}\n Then $\\mathfrak{so}_{1,3}$ is identified with the semi-Euclidean\n $6$-space $\\mathbb{E}^{3,3}=\\left(\\mathbb{R}^{6},\\langle\n \\cdot,\\cdot\\rangle\\right)$ with scalar product \n$$\n \\langle \\cdot,\\cdot\\rangle = \n- \\mathrm{d}x_{12}^2\n- \\mathrm{d}x_{13}^2\n- \\mathrm{d}x_{23}^2\n+ \\mathrm{d}x_{1}^2\n+ \\mathrm{d}x_{2}^2\n+ \\mathrm{d}x_{3}^2\n\\;.\n$$ \n The isotropy subgroup of $\\mathrm{SO}^{+}_{1,3}$ acting on\n $\\mathrm{U}\\mathbb{H}^{3}$ at $(\\Vec{e}_0,\\Vec{e}_1)$ is \n$$\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n1 & 0 & 0 & 0 \\\\\n0 & 1 & 0 & 0\\\\\n0 & 0 & \\cos \\theta  & -\\sin \\theta \\\\\n0 & 0 & \\sin \\theta & \\cos \\theta\n\\end{array}\n\\right)\n\\right\\}\n\\cong \\mathrm{SO}_2\n$$\nwith Lie algebra\n$$\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 0 \\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & -t \\\\\n0 & 0 & t & 0\n\\end{array}\n\\right)\n\\\n\\biggr \n\\vert\n\\\nt \\in \\mathbb{R}\n\\right\\}\\cong \\mathfrak{so}_2.\n$$\n The tangent space $T_{(\\Vec{e}_0,\\Vec{e}_1)} \\mathrm{U}\\mathbb{H}^3$ is\n naturally  identified with the complement\n$$\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & x_1 & x_2 & x_3\\\\\nx_1& 0 & x_{12} & x_{13}\\\\\nx_2 & -x_{12} & 0 & 0 \\\\\nx_{3} & -x_{13} & 0 & 0\n\\end{array}\n\\right)\n\\right\\}\n\\cong \\mathbb{R}^{5}\n$$\n of the isotropy algebra in $\\mathfrak{so}_{1,3}$. The scalar product \n $\\langle X,Y\\rangle$ of\n $X$, $Y\\in T_{(\\Vec{e}_0,\\Vec{e}_1)}\\mathrm{U}\\mathbb{H}^3$ is computed\n as\n\\begin{equation}\\label{eq:scalarUH3}\n\\langle X,Y\\rangle=-(x_{12}y_{12}+x_{13}y_{13})+x_{1}y_{1}\n+x_{2}y_{2}+x_{3}y_{3}.\n\\end{equation}\n Let $\\pi_{2}:\\mathrm{U}\\mathbb{H}^{3}\\to \\mathbb{H}^3$ denote the\n natural projection. Then the vertical subspace $\\mathcal{V}_2$  of\n $T_{(\\Vec{e}_0,\\Vec{e}_1)}\\mathrm{U}\\mathbb{H}^3$ with\n respect to $\\pi_2$  at $(\\Vec{e}_0,\\Vec{e}_1)$ is\n$$\n\\mathcal{V}_2 (\\Vec{e}_0,\\Vec{e}_1) =\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & 0 & 0 & 0\\\\\n0 & 0 & x_{12} & x_{13}\\\\\n0 & -x_{12} & 0 & 0\\\\\n0 & -x_{13} & 0 & 0\n\\end{array}\n\\right)\n\\right\\}\n\\cong \\mathbb{E}^{2,0}.\n$$\n Here $\\mathbb E^{2,0}$ is $\\mathbb R^2$ with scalar\n product $-\\mathrm{d}x_{12}^2-\\mathrm{d}x_{13}^2$.\n Since the restriction of the scalar product \\eqref{eq:scalarUH3} to\n $\\mathcal{V}_2$ is non-degenerate, the horizontal\n subspace $\\mathcal{H}_2$ can be defined by\n $\\mathcal{H}_2 =\\mathcal{V}_2^{\\perp}$. \n The submersion $\\pi_2$ satisfies\n $$\n \\langle \\mathrm{d}\\pi_2(X), \\mathrm{d}\\pi_2(Y)\\rangle = 4 \\langle X, Y\\rangle\n $$\n for any vector fields $X$ and $Y$ on $\\mathrm{U}\\mathbb H^3$.\n\n Next we consider the Grassmann manifold $\\mathrm{Gr}_{1,1} (\\mathbb\n E^{1,3})$.\n The isotropy subgroup of $\\mathrm{SO}^{+}_{1,3}$ acting on\n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$ at $\\Vec{e}_{0}\\wedge\n \\Vec{e}_1$ is $\\mathrm{SO}_{1,1}\\times \\mathrm{SO}_2$ with Lie algebra\n$$\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & s & 0 & 0\\\\\ns & 0 & 0 & 0\\\\\n0 & 0 & 0 & t\\\\\n0 & 0 & -t & 0\n\\end{array}\n\\right)\n\\right\\}.\n$$\n Therefore, the tangent space $T_{\\Vec{e}_{0}\\wedge\n \\Vec{e}_1}\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$\n  can be identified with a subspace of\n $\\mathfrak{so}_{1,3}$ \n$$\nT_{\\Vec{e}_{0}\\wedge\n \\Vec{e}_1}\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})\n\\cong\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & 0 & x_2 & x_3\\\\\n0 & 0 & x_{12} & x_{13}\\\\\nx_{2} & -x_{12} & 0 & 0\\\\\nx_{3} & -x_{13} & 0 & 0\n\\end{array}\n\\right)\\right\\}\n\\cong\n\\mathbb{E}^{2,2},\n$$\n which is complementary to the isotropy algebra.\n Here $\\mathbb E^{2,2}$ is a semi-Euclidean $4$-space with \n scalar product $-\\mathrm{d}x_{12}^2-\\mathrm{d}x_{13}^2\n +\\mathrm{d}x_{2}^2 +\\mathrm{d}x_{3}^2$.\n Let $\\pi_{1}:\\mathrm{U}\\mathbb{H}^3 \\to\n \\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$ denote the natural\n projection. Then the vertical subspace $\\mathcal V_1$ \n of $T_{(\\Vec{e}_0,\\Vec{e}_1)} \\mathrm{U}\\mathbb{H}^3$ with respect to\n $\\pi_1$ at $(\\Vec{e}_0,\\Vec{e}_1)$ is\n$$\n\\mathcal{V}_1 (\\Vec{e}_0,\\Vec{e}_1) =\n\\left\\{\n\\left(\n\\begin{array}{cccc}\n0 & x_1 & 0 & 0\\\\\nx_1 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\\\\\n0 & 0 & 0 & 0\n\\end{array}\n\\right)\n\\right\\}\\cong \\mathbb{E}^{1}.\n$$\n Since the restriction of the scalar product \\eqref{eq:scalarUH3} to\n $\\mathcal{V}_1$ is non-degenerate, the horizontal\n subspace of $\\mathcal{H}_1$ can be defined by\n $\\mathcal{H}_1 =\\mathcal{V}_1^{\\perp}$. \n Moreover, it is easy to see \n that $\\mathrm{d} \\pi_1$ restricted to $\\mathcal {H}_1$ preserves the scalar\n product. Thus $\\pi_1$ is a semi-Riemannian submersion, see appendix\n \\ref{sc:submersion}.\n\\subsection{}\\label{sc:Geo}\n Next we consider $\\mathrm{Geo}(\\mathbb{H}^3)$ the space of all oriented\n geodesics in $\\mathbb{H}^3$. Take a point $\\gamma\\in\n \\mathrm{Geo}(\\mathbb{H}^3)$, then  $\\gamma$ is given by the\n intersection of $\\mathbb{H}^3$ with a timelike plane $W\\in\n \\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$. By identifying $\\gamma$ with $W$,\n the space $\\mathrm{Geo}(\\mathbb{H}^3)$ is identified with the \n Grassmann manifold\n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$.\n In contrast to the general case of $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$,\n in the case $n=3$, the Grassmann\n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,n})$ admits an invariant \n complex structure $J$ compatible with the metric such that \n the resulting homogeneous\n space $\\mathrm{SO}^{+}_{1,3}\/\\mathrm{SO}_{1,1}\\times \\mathrm{SO}_{2}$ is\n an indefinite K\\\"ahler symmetric space.  \n The K\\\"ahler structure is related to the symplectic form $\\Omega$ by \n $$\n \\Omega(X, Y) = -2 \\langle X, JY\\rangle\n $$\n for any vector fields $X$ and $Y$, see \\cite{Honda}. \n\\subsection{}\\label{sc:fibering}\n In addition to the fibrations of $\\mathrm{U}\\mathbb H^3$ mentioned\n above there also exists a fibration onto the de Sitter $3$-space\n $\\mathbb{S}^{1, 2}$: $\\pi_3: \\mathrm{U}\\mathbb{H}^3\n \\to \\mathbb{S}^{1, 2}, \\pi_3 (\\Vec{x}, \\Vec{v}) = \\Vec{v}\\in \\mathbb\n {S}^{1, 2} \\subset\n \\mathbb{E}^{1,3}$, where we consider $\\mathrm{U}\\mathbb H^3$ again as \n a subspace of $\\mathbb E^{1,3} \\times \\mathbb E^{1,3}$ (see also appendix\n \\ref{App:Obata}). Altogether the unit tangent sphere bundle\n $\\mathrm{U}\\mathbb{H}^3$ has the fibrations:\n\\begin{align*}\n&\\pi_{1}:\\mathrm{U}\\mathbb{H}^3\\to \\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3});\n\\ \\ \\pi_{1}(\\Vec{x},\\Vec{v})=\\Vec{x} \\wedge \\Vec{v}, \\\\\n&\\pi_{2}:\\mathrm{U}\\mathbb{H}^3\\to \\mathbb{H}^{3};\n\\ \\\n\\pi_{2}(\\Vec{x},\\Vec{v})=\\Vec{x}\\in \\mathbb{H}^3\\subset\n\\mathbb{E}^{1,3}, \\\\\n&\\pi_{3}:\\mathrm{U}\\mathbb{H}^3\\to \\mathbb{S}^{1, 2};\n\\ \\\n\\pi_{3}(\\Vec{x},\\Vec{v})=\\Vec{v}\\in \\mathbb{S}^{1, 2}\\subset\n\\mathbb{E}^{1,3}.\n\\end{align*}\nThese fibrations are realized as homogeneous projections:\n\\begin{align*}\n&\\pi_{1}:\\mathrm{SO}^{+}_{1,3}\/\\mathrm{SO}_2\\to\n \\mathrm{SO}^{+}_{1,3}\/\\mathrm{SO}_{1,1}\\times \\mathrm{SO}_2, \\\\\n&\\pi_{2}:\\mathrm{SO}^{+}_{1,3}\/\\mathrm{SO}_2\\to \\mathrm{SO}^{+}_{1,3}\/\n\\mathrm{SO}_3,\\\\\n&\\pi_{3}:\\mathrm{SO}^{+}_{1,3}\/\\mathrm{SO}_2\\to \\mathrm{SO}^{+}_{1,3}\/\n\\mathrm{SO}_{1,2}.\n\\end{align*}\n\n\\section{Surfaces in $\\mathbb{H}^3$}\\label{sc:SurfacesH3}\n\n\\subsection{}\n Let $f:M\\to \\mathbb{H}^3\\subset \\mathbb{E}^{1,3}$ be a conformal\n immersion of a Riemann surface with unit normal vector field $n$.\n  Clearly, by replacing, if necessary, the unit normal $n$ of an\n immersion by $-n$, we can assume that the mean curvature satisfies\n $H\\geq 0$.\n Let $\\mathbb D$ denote the universal cover of $M$. Since $\\mathbb D\n \\cong S^2$ can only occur for totally umbilic CMC immersions with $H>1$\n and cannot occur for CMC immersions with $0\\leq H \\leq 1$, \\cite{Spivak}, \n we can assume that $\\mathbb D \\subset \\mathbb C$ being an open. \n (Usually we will assume $\\mathbb D = \\mathbb C$ or $\\mathbb D=$ open unit disk.) \n Hence, without loss of generality,\n $\\mathbb D$ is open in $\\mathbb C$, and the first\n fundamental form $\\mathrm{I}$ is written as\n\\begin{equation}\n\\mathrm{I}=e^{u}\\mathrm{d}z\\mathrm{d}\\bar{z}.\n\\end{equation}\n The \\textit{Hopf differential} of $(M,f)$ is a quadratic differential on \n $M$ defined by \n$$\nQ\\>\\mathrm{d}z^{2},\n\\ \\\nQ=\\langle f_{zz},n\\rangle.\n$$\n The \\textit{Gauss-Codazzi equations} of $(M,f)$ are given by\n\\begin{equation}\nu_{z\\bar{z}}+\\frac{1}{2}(H^2-1)e^{u}-2|Q|^{2}e^{-u}=0, \\;\\;\nQ_{\\bar z}=\\frac{1}{2}H_{z}e^{u}.\n\\end{equation}\n Here $H$ denotes the \\textit{mean curvature} of $(M,f)$, \n which is explicitly given by \n $H = 2 e^{-u} \\langle f_{z \\bar z}, n\\rangle$.\n The constancy of the mean curvature is characterized as follows.\n\n\\begin{proposition}\n Let $M$ be a Riemann surface. \n A conformal immersion $f:M\\to \\mathbb{H}^3$ is of constant mean curvature\n if and only if its Hopf differential is holomorphic.\n\\end{proposition}\n Assume that $H$ is constant, then the Gauss-Codazzi equations\n are invariant under the deformation\n$$\nQ\\longmapsto \\lambda^{-1}Q,\\ \\ \n\\lambda\\in S^1.\n$$ \n Hence, on the region $\\mathbb{D}$, there exists a $1$-parameter \n deformation family of conformal constant mean curvature immersions \n $\\{f_{\\lambda}\\}$ through $f_1=f$. All these immersions have the \n same induced metric and mean curvature. The family $\\{f_\\lambda\\}$ \n is referred to as the \n \\textit{associated family} of the original immersion $f$. \n\n\\subsection{}\\label{sc:tangentGauss}\n Let again $f:M\\to \\mathbb{H}^3$ be a conformal immersion with unit\n normal $n$. For the purposes of this paper it will be important to\n consider the map\n\\begin{equation}\\label{eq:mapsinUH3}\nF:=(f,n):M\\to \\mathrm{U}\\mathbb{H}^3.\n\\end{equation}\n In our paper, the smooth map $F$ will be  called the \\textit{Gauss\n map} of $f$. The Gauss map satisfies  $\\langle\n \\mathrm{d}f,n\\rangle=0$. In section \\ref{sc:unit tangent sphere bundle}\n we have introduced the canonical contact form\n $\\omega$ of $\\mathrm{U}\\mathbb{H}^3$. By the definition of $\\omega$, we\n have\n$$\nF^{*}\\omega=\\langle \\mathrm{d}f,n\\rangle.\n$$\n Hence the Gauss map $F$ satisfies the \\textit{Legendre condition}:\n$$\n F^{*}\\omega=0.\n$$\n Note that the Gauss map $F$ is also called the \\textit{Legendre\n lift} of $f$ as in \\cite{Arnold}.\n The Legendre property will be discussed \n in section \\ref{sc:contactmfd}. The\n following result which is very important for this paper is due to\n T.~Ishihara \\cite{Ishihara}. \n\\begin{proposition}{\\rm (T.~Ishihara)}\\label{prop:Ishihara}\n Let $M$ be a Riemann surface. \n A conformal immersion $f:M \\to \\mathbb{H}^3$ has constant mean curvature  \n if and only if its Gauss map is harmonic with respect to the metric \n induced from \\eqref{scalarproduct}. \n\\end{proposition} \n In section \\ref{sc:Harmonicity}, we will give a proof of this result in\n terms of frames and the Sym formula.\n\n\\section{The $2 \\times 2$-matrix model for immersions into $\\mathbb H^3$}\\label{sc:2by2}\n\\subsection{}\\label{sc:Minkowski}\n The Minkowski 4-space $\\mathbb{E}^{1,3}$ is identified with the space\n $\\mathrm{Her}_2 \\mathbb C$ of all complex Hermitian $2\\times\n 2$-matrices: \n\\begin{equation}\\label{eq:basis}\n\\mathbb E^{1,3} \\cong \\mathrm{Her}_2 \\mathbb C=\\left \\{\n\\xi=\\left (\n\\begin{array}{cc}\n\\xi_0+\\xi_1 &\n\\xi_3-i\\xi_2 \\\\\n\\xi_3+i\\xi_2 &\n\\xi_0-\\xi_1 \n\\end{array}\n\\right)\\ \n\\Biggr \\vert\n\\\n\\xi_0,\n\\xi_1,\\xi_2,\\xi_3\n\\in \\mathbb{R}\\\n\\right \\}.\n\\end{equation}\n The space $\\mathrm{Her}_2 \\mathbb C$ is spanned by the orthonormal basis\n$$\n\\Vec{e}_{0}=\\left(\n\\begin{array}{cc}\n1 & 0\\\\\n0 & 1\n\\end{array}\n\\right),\n\\ \\\n\\Vec{e}_{1}=\\left(\n\\begin{array}{cc}\n1 & 0\\\\\n0 & -1\n\\end{array}\n\\right),\n\\\n\\Vec{e}_{2}=\\left(\n\\begin{array}{cc}\n0 & -i\\\\\ni & 0\n\\end{array}\n\\right),\n\\\n\\Vec{e}_{3}=\\left(\n\\begin{array}{cc}\n0 & 1\\\\\n1 & 0\n\\end{array}\n\\right).\n$$\n It is easy to see that $-\\det \\xi=-\\xi_0^2+\\xi_1^2+\\xi_2^2+ \\xi_3^2$ \n for $\\xi \\in \\mathrm{Her}_2 \\mathbb C$.  Thus the Lorentzian metric\n of $\\mathrm{Her}_2 \\mathbb C$ is described as \n$$\n\\langle \\xi, \\eta\\rangle=-\\frac{1}{2} {\\mathrm{tr}}( \\xi \\Vec{e}_2 \\eta^t \\Vec{e}_2), \\ \\ \\xi, \\eta \\in \\mathrm{Her}_2 \\mathbb C.\n$$\n In particular, we have\n$$\n \\langle \\xi, \\xi\\rangle = -\\det \\xi, \\ \\ \\xi \\in \\mathrm{Her}_2 \\mathbb C.\n$$\n Thus  we have the identification: \n$$\n\\mathbb{H}^3=\\{ \\xi \\in \\mathrm{Her}_2 \\mathbb C \\ \\vert \\ \\det \\xi=1,\\\n \\mathrm{tr}\\: \\xi>0\\}.\n$$\n The special linear group $G=\\mathrm{SL}_{2}\\mathbb C$ acts isometrically and\n transitively on the hyperbolic $3$-space via the action:\n\\begin{equation}\\label{eq:actionA}\n\\mathrm{A}:\\mathrm{SL}_{2}\\mathbb C\\times \\mathbb{H}^3\\to \n\\mathbb{H}^3,\\\n(g,\\xi) \\mapsto g\\> \\xi \\>g^{*},\n\\end{equation}\n where $g^*$  denotes $\\bar g^t$.\n The isotropy subgroup of this action at $\\Vec{e}_{0}$ \n is the special unitary group $\\mathrm{SU}_2$. Hence\n $\\mathbb{H}^3$ is represented by\n $\\mathbb{H}^3=G\/K=\\mathrm{SL}_{2}\\mathbb C\/\\mathrm{SU}_2$ as a \n \\textit{Riemannian symmetric space}. The natural projection $\\pi:G\\to\n \\mathbb{H}^3$ is given explicitly by $\\pi(g)=gg^{*},\\ g\\in G$. In other\n words, $\\mathbb{H}^3$ is represented as\n$$\n\\mathbb{H}^{3}=\\{gg^{*}\\\n\\vert\n\\\ng \\in G\\}.\n$$\n\\begin{remark}\n{\\rm \n It is important to note that in this context the simple Lie group\n $\\mathrm{SL}_{2}\\mathbb C$ is regarded as a simple real Lie group \n and as a double covering of the special Lorentz group\n $\\mathrm{SO}^{+}_{1,3}$. The real Lie algebra $\\mathfrak{sl}_{2}\\mathbb C$\n is spanned by the basis \n\\begin{equation}\\label{orthonormalbasis}\n\\{i\\Vec{e}_{1},i\\Vec{e}_{2},i\\Vec{e}_{3}, \\Vec{e}_{1},\\Vec{e}_{2},\n \\Vec{e}_{3}\\}.\n\\end{equation}\n The bi-invariant semi-Riemannian metric $\\langle\\cdot,\\cdot \\rangle$ on\n $G$ corresponding to the scalar product\n (\\ref{eq:innerproduct}) via the \n isomorphism \\eqref{eq:basis} has the signature $(-,-,-,+,+,+)$. The\n tangent space $\\mathfrak{m}=T_{\\Vec{e}_0}\\mathbb{H}^3$ is given by\n$$\n\\mathfrak{m}=\\mathfrak{sl}_{2}\\mathbb C\n\\cap \\mathrm{Her}_2 \\mathbb C=\n\\left\\{\n\\left(\n\\begin{array}{cc}\na & \\bar{b}\n\\\\\nb & -a\n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\ a \\in \\mathbb{R},\\\nb\\in \\mathbb C\n\\right\\}=\n\\mathbb{R}\\Vec{e}_{1}\\oplus \\mathbb{R}\\Vec{e}_{2}\n\\oplus \\mathbb{R}\\Vec{e}_3.\n$$\n}\n\\end{remark}\n\n\\subsection{}\\label{sc:tangentspherebundle}\n The unit tangent sphere bundle $\\mathrm{U}\\mathbb{H}^3$ is represented\n as\n$$\n\\mathrm{U}\\mathbb{H}^{3}=\n\\{ (\\Vec{x},\\Vec{v})\\in \\mathrm{Her}_2 \\mathbb C\\times \n\\mathrm{Her}_2 \\mathbb C\n\\\n\\vert\n\\\n\\det \\Vec{x}=1, \\mathrm{tr}\\>\\Vec{x}>0,\\>\n\\det \\Vec{v}=-1, \\> \\langle \\Vec{x},\\Vec{v}\\rangle =0\\}. \n$$\n The special liner group $G=\\mathrm{SL}_{2}\\mathbb C$ acts\n isometrically and transitively on $\\mathrm{U}\\mathbb{H}^{3}$ via the\n action:\n$$\ng\\cdot (\\Vec{x},\\Vec{v})=(g \\Vec{x} g^{*}, g \\Vec{v} g^{*}).\n$$\n The isotropy subgroup of $G$ at $(\\Vec{e}_{0},\\Vec{e}_{1})$ is \n$$\nH=\\left\\{\n\\left(\n\\begin{array}{cc}\ne^{i\\theta} & 0\\\\\n0 & e^{-i\\theta}\n\\end{array}\n\\right)\n\\right\\}=\\mathrm{U}_1.\n$$\n Thus the unit tangent sphere bundle $\\mathrm{U}\\mathbb H^3$ is as \n homogeneous space $G\/H = \\mathrm{SL}_2 \\mathbb C\/\\mathrm{U}_1$.\n The Lie algebra $\\mathfrak{h}$ of $H$ is\n\\begin{equation}\\label{eq:h}\n\\mathfrak{h}=\\left\\{\n\\left(\n\\begin{array}{cc}\nia_{2} & 0\\\\\n0 & -ia_{2}\n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\\na_{2}\\in \\mathbb{R}\\>\n\\right\\}=\\mathfrak{u}_1=\\mathbb{R}(i\\Vec{e}_{1}).\n\\end{equation}\n The tangent space $\\mathfrak{p}:=T_{(\\Vec{e}_0,\\Vec{e}_1)}\n \\mathrm{U}\\mathbb{H}^3$ is given by\n\\begin{equation}\\label{eq:p}\n\\mathfrak{p}=\n\\left\\{\n\\left(\n\\begin{array}{cc}\na_{1} & b_{1}+ib_{2}\\\\\nc_{1}+ic_{2} & -a_{1}\n\\end{array}\n\\right)\n\\\n\\biggr\n\\vert\n\\\na_{1},b_{1},b_{2},\nc_{1},c_{2}\n\\in \\mathbb{R}\n\\>\n\\right\\}.\n\\end{equation}\n Note that\n\\begin{eqnarray*}\n\\mathfrak{p}&=&\n\\{\\mathbb{R}\\Vec{e}_1\\oplus\n\\mathbb{R}\\Vec{e}_2\n\\oplus\n\\mathbb{R}\\Vec{e}_3\\}\n\\oplus\n\\{\n\\mathbb{R}(i\\Vec{e}_2)\\oplus\n\\mathbb{R}(i\\Vec{e}_3)\n\\}\\\\\n&=& \\mathfrak{m}\\oplus \n\\{\n\\mathbb{R}(i\\Vec{e}_2)\\oplus\n\\mathbb{R}(i\\Vec{e}_3)\n\\}.\n\\end{eqnarray*}\n This equation shows that the horizontal distribution $\\mathcal{H}_{2}$\n and the vertical distribution $\\mathcal{V}_{2}$ with respect to the fibering\n $\\pi_2 : \\mathrm{U}\\mathbb{H}^{3}\\to \\mathbb{H}^3$ are generated via\n the identifications \\eqref{eq:identifyso} and \\eqref{eq:basis} by\n$$\n\\mathcal{H}_{2}=\\mathfrak{m},\\ \\\n\\mathcal{V}_{2}=\\mathbb{R}(i\\Vec{e}_2)\\oplus\n\\mathbb{R}(i\\Vec{e}_3).\n$$\n\n\\subsection{}\n Next we consider the fibering $\\pi_{1}:\\mathrm{U}\n \\mathbb{H}^3\\to \\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$.\n From section \\ref{sc:fibering}, the Grassmann manifold\n $\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3})$ is a homogeneous space of $G = {\\rm\n SL}_2 \\mathbb C$. The isotropy subgroup of $G$ at $\\Vec{e}_{0}\\wedge \\Vec{e}_{1}$\n is \n$$\nD=\\mathrm{GL}_{1}\\mathbb{C}=\\left\\{\\left(\n\\begin{array}{cc}\nw &0\\\\\n0 & 1\/w\n\\end{array}\n\\right)\n\\\n\\biggr \n\\vert\n\\\nw\\in \\mathbb C^{\\times}\n\\right\\}\n\\cong \\mathbb C^{\\times}.\n$$ \n The Lie algebra $\\mathfrak{d}$ of $D$ is\n\\begin{equation}\\label{eq:d}\n\\mathfrak{d}=\n\\left\\{\\left(\n\\begin{array}{cc}\nv &0\\\\\n0 & -v\n\\end{array}\n\\right)\n\\\n\\biggr \n\\vert\n\\\nv\\in \\mathbb C\n\\right\\}=\n\\mathbb{R}\\Vec{e}_{1}\\oplus \\mathbb{R}(i\\Vec{e}_1).\n\\end{equation}\n The tangent space $\\mathfrak{q}:=T_{\\Vec{e}_{0}\\wedge\n \\Vec{e}_1}\\mathrm{Gr}_{1,1} (\\mathbb{E}^{1,3})$ of $\\mathrm{Gr}_{1,1}\n (\\mathbb{E}^{1,3})$ at $\\Vec{e}_{0}\\wedge \\Vec{e}_1$ is\n\\begin{equation}\\label{eq:q}\n \\mathfrak{q}=\\mathbb{R}\\Vec{e}_{2}\\oplus \\mathbb{R}(i\\Vec{e}_{2})\n \\oplus \\mathbb{R}\\Vec{e}_{3}\\oplus \\mathbb{R}(i\\Vec{e}_{3}).\n\\end{equation}\n The horizontal and vertical distributions with respect to $\\pi_1$ are\n generated by\n$$\n\\mathcal{H}_{1}=\\mathfrak{q},\n\\ \\\n\\mathcal{V}_{1}=\\mathbb{R}\\Vec{e}_{1}.\n$$\n\n\\subsection{}\n Next we consider the fibering $\\pi_{3}:\\mathrm{U}\n \\mathbb{H}^3\\to \\mathbb S^{1,2}$.\n From section \\ref{sc:fibering}, the de Sitter $3$ space $\\mathbb\n S^{1,2}$ is a homogeneous space of $G = {\\rm\n SL}_2 \\mathbb C$. The isotropy subgroup of $G$ at $\\Vec{e}_{1}$ is \n $\\mathrm{SU}_{1,1}$. Hence\n $\\mathbb{S}^{1, 2}$ is represented by\n $\\mathbb{S}^{1, 2}=G\/K=\\mathrm{SL}_{2}\\mathbb C\/\\mathrm{SU}_{1,1}$ as a \n \\textit{Lorentzian symmetric space}. The natural projection \n $\\pi:G\\to \\mathbb{S}^{1, 2}$ is given explicitly by \n $\\pi(g)=g\\Vec{e}_1g^{*},\\ g\\in G$. In other\n words, $\\mathbb{S}^{1, 2}$ is represented as\n$$\n\\mathbb{S}^{1, 2}=\\{g\\Vec{e}_1g^{*}\\\n\\vert\n\\\ng \\in G\\}.\n$$\n The horizontal and vertical distributions with respect to \n $\\pi_3$ are generated by\n$$\n\\mathcal{H}_{3}=\\mathbb{R}\\Vec{e}_1 \\oplus \\mathbb{R}(i \\Vec{e}_2)\\oplus \n \\mathbb{R}(i\\Vec{e}_3),\n\\ \\\n\\mathcal{V}_{3}=\\mathbb R\\Vec{e}_2\\oplus \\mathbb R\\Vec{e}_3.\n$$\n\\begin{remark}\n{\\rm \n Precisely speaking, to represent the de Sitter 3-space\n $\\mathbb{S}^{1,2}$ as a Lorentzian symmetric space, we need to equip the \n scalar product $-\\langle \\cdot,\\cdot\\rangle$ on the linear space\n $\\mathbb{R}\\Vec{e}_1 \\oplus \\mathbb{R}(i \\Vec{e}_2)\\oplus \n \\mathbb{R}(i\\Vec{e}_3)$.\n}\n\\end{remark}\n\n\\subsection{}\n Now let again  $f:M\\to \\mathbb{H}^3$ be a CMC surface with unit normal\n $n$. Take a simply connected complex coordinate region \n $\\mathbb D \\subset \\mathbb C$ as before. Denote by\n $(x,y)$ the associated isothermal coordinates, \\textit{i.e.},\n $z=x+iy$, where the induced metric is expressed as\n $\\mathrm{I}=e^{u} (\\mathrm{d}x^2+\\mathrm{d}y^2)$. \n\n The \\textit{coordinate frame} $\\Psi$ of $f$ with respect to\n $(x,y)$ is a map from $\\mathbb{D}$ into the Lorentz group\n $\\mathrm{SO}^{+}_{1,3}$ defined by\n$$\n\\Psi=(f,n,e^{-u\/2}f_{y},e^{-u\/2}f_{x}).\n$$\n As mentioned above, the action \\eqref{eq:actionA} induces a double covering\n $\\tilde \\pi:\\mathrm{SL}_{2}\\mathbb C\\to \\mathrm{SO}^{+}_{1,3}$. Since\n $\\mathbb D$ is simply connected, the lift $\\hat{\\Phi}$ of the\n coordinate frame $\\Psi$ to $\\mathrm{SL}_{2}\\mathbb C$ is\n determined uniquely by\n\\begin{equation}\\label{eq:hatframe}\n\\Vec{e}_j \\longmapsto \\hat \\Phi \\Vec{e}_j \\hat \\Phi^*\n\\end{equation}\n up to sign. The lift $\\hat{\\Phi}$ satisfies the following\n Gauss-Weingarten formulas (see appendix\n \\ref{sc:Gauss-Codazzi}): \n\\begin{equation}\\label{eq:hatMau1}\n\\hat{\\Phi}^{-1}\\hat{\\Phi}_z=\n\\left(\n\\begin{array}{cc}\nu_{z}\/4 & \\frac{1}{2}(H+1)e^{u\/2} \\\\\n-Q e^{-u\/2} & -u_{z}\/4\n\\end{array}\n\\right), \\;\\;\n\\hat{\\Phi}^{-1}\\hat{\\Phi}_{\\bar z}=\n\\left(\n\\begin{array}{cc}\n-u_{\\bar z}\/4 &\n{\\bar Q}e^{-u\/2} \\\\\n-\\frac{1}{2}(H-1)e^{u\/2} &\nu_{\\bar z}\/4\n\\end{array}\n\\right).\n\\end{equation}\n Let us consider the associated family $\\{f_\\lambda\\}_{\\lambda\\in S^1}$\n of $f=f_1$. The immersion $f_{\\lambda}$ has the Hopf differential\n $\\lambda^{-1}Q\\>\\mathrm{d}z^2$. The corresponding\n $\\mathrm{SL}_{2}\\mathbb C$-valued frame is denoted by\n $\\hat{\\Phi}_{\\lambda}$. For our purposes, it will be useful to perform\n the following gauge transformation\n\\begin{equation}\\label{eq:tildephi}\n\\tilde{\\Phi}:=\\hat{\\Phi}_{\\lambda^2}\n\\left(\n\\begin{array}{cc}\n\\sqrt{\\lambda} & 0\\\\\n0 & 1\/\\sqrt{\\lambda}\n\\end{array}\n\\right).\n\\end{equation}\n Then we obtain $\\tilde U = \\tilde{\\Phi}^{-1}\\tilde{\\Phi}_{z}$\n and $\\tilde V = \\tilde{\\Phi}^{-1}\\tilde{\\Phi}_{\\bar z}$ with\n\\begin{equation}\\label{Lax}\n{\\tilde U}=\n\\left(\n\\begin{array}{cc}\nu_{z}\/4 & \\frac{\\lambda^{-1}}\n{2}(H+1)e^{u\/2} \\\\\n-\\lambda^{-1}Q e^{-u\/2} & -u_{z}\/4\n\\end{array}\n\\right), \\;\\;\n{\\tilde V}=\n\\left(\n\\begin{array}{cc}\n-u_{\\bar z}\/4 &\n\\lambda{\\bar Q}e^{-u\/2} \\\\\n-\\frac{\\lambda}{2}(H-1)e^{u\/2} &\nu_{\\bar z}\/4\n\\end{array}\n\\right).\n\\end{equation}\n From \\eqref{Lax}, $\\tilde U$ and $\\tilde V$\n are elements  of the \\textit{loop algebra} of $\\mathfrak{sl}_2 \\mathbb C$:\n$$\n \\Lambda \\mathfrak{sl}_2 \\mathbb C_{\\sigma} = \\left\\{\n g:S^1\\to \\mathfrak{sl}_{2}\\mathbb C\\ \\vert \\\n g(-\\lambda)=\\sigma g(\\lambda)\\>\\right\\},\n$$\n where \n\\begin{equation}\\label{eq:sigma}\n \\sigma = \\mathrm{Ad}(\\Vec{e}_1).\n\\end{equation}\n It will turn out to be useful to consider the analytic loops in\n $\\Lambda \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$. Those loops will\n be denoted by $\\tilde \\Lambda \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$. \n We will use a similar notation for loop groups.\n It is easy to see that \n $\\tilde U, \\tilde V \\in \\tilde \\Lambda \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$ \n and $\\tilde \\Phi \\in \\tilde \\Lambda \\mathrm{SL}_2 \\mathbb C_{\\sigma}$.\n Then we obtain the {\\it Sym formula} for CMC surfaces in $\\mathbb H^3$\n which is easily seen to be equivalent to the corresponding formula \n used in \\cite{Bob:Russ},  \\cite{BaBo}. \n\\begin{proposition}\\label{prop:Sym}\n Let $\\tilde{\\Phi}$ be a solution to \\eqref{Lax}. Then\n$$\n{\\tilde f}_{\\lambda}={\\tilde \\Phi}{\\tilde \\Phi}^{*}\n$$\n is a loop of immersions of constant mean curvature $H$ with unit normal\n vector field \n$$\n {\\tilde n}_{\\lambda}={\\tilde \\Phi} \\Vec{e}_{1}\\>\n{\\tilde \\Phi}^{*}.\n$$\n For $\\lambda =1$ we obtain $\\tilde f_{\\lambda =1} = f$.\n\\end{proposition}\n{\\bf Proof}.\n A direct computation shows\n\\begin{eqnarray*}\n\\tilde{f}_{z}=\n\\frac{\\lambda^{-1}}{2}e^{u\/2}{\\tilde\\Phi}\n(\\Vec{e}_{3}+i \\Vec{e}_{2})\\tilde \\Phi^*,\\;\\;\n\\tilde{f}_{\\bar z}=\n\\frac{\\lambda}{2}e^{u\/2}{\\tilde \\Phi}\n(\\Vec{e}_{3}-i \\Vec{e}_{2}){\\tilde \\Phi}^*.\n\\end{eqnarray*}\n From these equations we obtain\n$$\n\\mathrm{I}_{\\lambda}=e^{u}\\mathrm{d}z\\mathrm{d}{\\bar z}.\n$$ \n Hence $\\tilde{f}_{\\lambda}$ is an immersion for all $\\lambda \\in\n S^1$. The unit normal for $\\tilde f_{\\lambda}$ is given by ${\\tilde\n n}_{\\lambda}=\\tilde{\\Phi}\\Vec{e}_1\\tilde \\Phi^*$. Thus \n$$\n{\\tilde n}_{z}=\n{\\tilde \\Phi}(U\\Vec{e}_{1}+\\Vec{e}_{1}V^{*}){\\tilde \\Phi}^*=\n-\\frac{\\lambda^{-1}}{2}\\left\\{ He^{u\/2}\\tilde{\\Phi}(\\Vec{e}_{3}+\\Vec{e}_{2})\n\\tilde{\\Phi}^*+2 Q e^{-u\/2}\\tilde{\\Phi}\n(\\Vec{e}_{3}-\\Vec{e}_{2})\\tilde{\\Phi}^*\\right\\}.\n$$\n From these equations one can check that each $\\tilde{f}_{\\lambda}$ has\n constant mean curvature $H$ and Hopf differential $\\lambda^{-2}Qdz^2$.\n $\\Box$\n\\begin{remark}\n{\\rm\n The construction above shows that all one needs for the construction\n of constant mean curvature surfaces are a real number $H$, a\n holomorphic function $Q$ and a real valued function $u$ such that \n with $\\tilde U$ and $\\tilde V$ as in \\eqref{Lax} the one \n form $\\tilde \\alpha = \\tilde U dz + \\tilde V d\\bar z$ is integrable.\n}\n\\end{remark}\n\\subsection{}\\label{sc:Harmonicity}\n In this section, we discuss the harmonicity of the Gauss map \n associated with $\\tilde f_{\\lambda}$. Let $f:M\\to \\mathbb{H}^3$ be a\n conformal immersion as above.\n Let $\\hat{\\Phi}$ denote the $\\mathrm{SL}_{2}\\mathbb C$-valued frame\n which is a lift of the coordinate frame $\\Psi$\n and put $\\hat{\\alpha}:=\\hat{\\Phi}^{-1}\\mathrm{d}\\hat{\\Phi}$.\n Then we decompose $\\hat{\\alpha}$ as\n$$\n\\hat{\\alpha}=\\hat{\\alpha}_{\\mathfrak{h}}+\n\\hat{\\alpha}_{\\mathfrak{p}}^{\\prime}+\n\\hat{\\alpha}_{\\mathfrak{p}}^{\\prime\\prime}\n$$\n according to the Lie algebra decomposition\n $\\mathfrak{sl}_2 \\mathbb C=\\mathfrak{h}\\oplus \\mathfrak{p}$ as in\n \\eqref{eq:h} and \\eqref{eq:p} respectively. A direct computation shows\n$$\n[\\hat{\\alpha}^{\\prime}_{\\mathfrak{p}}\\wedge\n\\hat{\\alpha}^{\\prime\\prime}_{\\mathfrak{p}}]\n=-\\frac{1}{4}\n\\left\\{(H^2-1)e^{u}-4|Q|^{2}e^{-u}\n\\right\\}\\Vec{e}_{1}\\>\\mathrm{d}z\\wedge \\mathrm{d}\\bar{z}.\n$$\n This is contained in $\\mathfrak h$. Therefore the $\\mathfrak p$-part  of \n $[\\hat \\alpha_{\\mathfrak p}^{\\prime} \\wedge \n \\hat \\alpha_{\\mathfrak p}^{\\prime \\prime}]$ vanishes\n and $\\hat{\\alpha}$ satisfies the admissibility condition\n (\\ref{admissible}). Moreover, it is easy to check that \n$$\n\\mathrm{d}(*\\hat{\\alpha}_{\\mathfrak{p}})\n+\n[\\hat{\\alpha}^{\\prime}\\wedge\\>*\n\\hat{\\alpha}_{\\mathfrak{p}}]\n=-ie^{u\/2}\n\\left(\n\\begin{array}{cc}\n0 & H_{\\bar z}\\\\\n-H_{z} & 0\n\\end{array}\n\\right)\\mathrm{d}z\\wedge \\mathrm{d}\\bar{z}\n$$\n holds. Since \\eqref{harmonicity} describes the harmonicity of the \n Gauss map, this formula implies Proposition \\ref{prop:Ishihara}.\n\n\\begin{remark}\n{\\rm \n On the unit tangent sphere bundle $\\mathrm{U}\\mathbb{H}^3$, we can\n define  Riemannian metrics so that the natural projection $\\pi_2$ is a\n Riemannian submersion. One of such metrics is the \\textit{Sasaki\n lift metric}. It is not difficult to see that the Gauss map $F$ of a\n non-minimal CMC\n surface is \\textit{never} harmonic  with respect to the Sasaki lift\n metric. See \\cite[p.~271, proof of Corollary]{JR}.\n\n In 3-dimensional homogeneous Riemannian spaces of\n non-constant curvature, the harmonicity of the Gauss map with respect to the \n Sasaki lift metric is a very\n strong restriction for CMC surfaces. In fact, the\n only CMC surfaces with harmonic Gauss map in a 3-dimensional\n homogeneous Riemannian space with 4-dimensional isometry group are inverse\n images of geodesics under the Hopf-fibration or totally geodesic\n leaves. The latter case only occurs if the ambient space is a\n direct product space \\cite{Sanini}, \\cite{Tamura}. \n\n CMC surfaces with harmonic Gauss maps in \n 3-dimensional homogeneous Riemannian spaces with 3-dimensional isometry\n group have been classified by J.~Van der Veken and the second named\n author \\cite{IV}.\n}\n\\end{remark}\n\\section{CMC surfaces with $H>1$}\\label{sc:CMCH>1}\n As pointed out in the introduction, the case $H =1$ is\n special and has been investigated already intensively.\n Therefore this case will not be considered in this paper.\n In this section, we study CMC surfaces with mean curvature $H$ such\n that $H>1$. In this case, we may write $H=\\mathrm{coth}\\>q,\\; q \\in\n \\mathbb R_{>0}$.\n\n We perform a gauge transformation:\n\\begin{equation}\\label{eq:extendedH>1}\n\\Phi:=\\tilde{\\Phi}\n\\left(\n\\begin{array}{cc}\ne^{q\/4} & 0\\\\\n0 & e^{-q\/4}\n\\end{array}\n\\right).\n\\end{equation}\n We call $\\Phi$ the \\textit{extended frame} of a CMC immersion $f$ with\n $H>1$.  Moreover we put\n$$\n\\mathscr{H}=e^{-q}(H+1) \\in \\mathbb R,\n\\ \\\n\\nu=-e^{-q\/2}\\lambda.\n$$\n Then $H = \\coth q$ implies $\\mathscr H =e^{-q}(H+1) = e^q (H-1)$. \n Moreover, the Lax pair\n$$\n U=\\Phi^{-1}\\Phi_{z},\\  \\\n V=\\Phi^{-1}\\Phi_{\\bar z}\n$$\n is given by \n\\begin{equation}\\label{LaxH>1}\nU=\\left(\n\\begin{array}{cc}\nu_{z}\/4 & -\\frac{1}{2}\\nu^{-1}\\mathscr{H} e^{u\/2}\n\\\\\n\\nu^{-1}Q e^{-u\/2} & -u_{z}\/4\n\\end{array}\n\\right),\n\\ \\\nV=\n\\left(\n\\begin{array}{cc}\n-u_{\\bar z}\/4 & \n-\\nu \\bar{Q}e^{-u\/2}\n\\\\\n\\frac{1}{2}\\nu \\mathscr{H}e^{u\/2}\n&\nu_{\\bar z}\/4\n\\end{array}\n\\right).\n\\end{equation}\n Clearly, the matrices $U$ and $V$ are holomorphic in the parameter $\\nu\n \\in \\mathbb C^{\\times}$. \n In particular, the gauged frame $\\Phi$ defined in\n \\eqref{eq:extendedH>1} above can be considered to be a holomorphic\n function in $\\nu$, where $\\nu$ is restricted to the circle of radius $ r\n = e^{-q\/2}$.\n Noting that everything is holomorphic in $\\nu \\in \\mathbb C^{\\times}$, \n it is straightforward to  check that the $1$-form \n $\\alpha = U \\mathrm{d} z + V \\mathrm{d} \\bar z$ \n is fixed by the following loop algebra automorphism:\n\\begin{equation}\\label{eq:invc3}\n \\tau_3:g(\\nu)\\longmapsto -g^{*}(1\/\\bar{\\nu}).\n\\end{equation}\n This automorphism is said to be of type $C_3$ (almost compact\n automorphism of the third kind), \\cite{Kobayashi}. The Maurer-Cartan form\n $\\alpha=\\Phi^{-1}\\mathrm{d}\\Phi$ has the decomposition\n\\begin{equation}\\label{eq:decalpha}\n\\alpha=\\nu^{-1}\\alpha_{-1}+\\alpha_{0}+\\nu\\alpha_{1}.\n\\end{equation}\n $\\tau_3 (\\alpha) = \\alpha$ translates into \n$$\n\\overline{\\alpha_{0}}=-\\alpha_{0},\n\\ \\\n\\alpha_{-1}= - \\alpha_{1}^{*}.\n$$\n The mapping $\\Phi$ takes values in the twisted loop group\n$$\n\\Lambda \\mathrm{SL}_{2}\\mathbb C_{\\sigma,\\tau_3}:=\n\\left\\{\ng: S^1\\to \\mathrm{SL}_{2}\\mathbb C\n\\\n\\vert\n\\\ng(-\\nu)=\\sigma g(\\nu),\n\\ \\\n\\tau_3 (g)(\\nu)=g(\\nu)\\>\n\\right\\},\n$$\n where $\\sigma$ is defined in \\eqref{eq:sigma} and \n $\\tau_3(g)(\\nu) = g(1\/\\bar{\\nu})^{*-1}$.\n\n Moreover, the first formula in Proposition \\ref{prop:Sym} can now be\n reinterpreted as\n\\begin{equation}\\label{eq:SymH>1}\nf_{\\nu}:=\\Phi \n\\left(\n\\begin{array}{cc}\ne^{-q\/2} & 0\\\\\n0 & e^{q\/2}\n\\end{array}\n\\right)\n\\Phi^{*}.\n\\end{equation}\n This CMC immersion into $\\mathbb H^3$ has mean curvature $H = \\coth q$\n and the unit normal\n\\begin{equation}\\label{eq:normalH>1}\nn_{\\nu}=\n\\Phi\n\\left(\n\\begin{array}{cc}\ne^{-q\/2} & 0\\\\\n0 & -e^{q\/2}\n\\end{array}\n\\right)\n\\Phi^{*}.\n\\end{equation}\n Note that the matrix $\\operatorname{diag} (e^{-q\/2}, e^{q\/2})$ corresponds under\n the isomorphism \\eqref{eq:basis} to the point\n $(\\cosh(q\/2),-\\sinh(q\/2),0,0) \\in \\mathbb H^3$.\n Conversely, the following result holds:\n\\begin{proposition}\\label{prop:fundamentalH>1}\n Let $\\mathscr H$ be a positive real number, $u$ a real valued function and\n $Q$ a holomorphic function on the simply connected domain \n $\\mathbb D \\subset \\mathbb C$. \n Let $\\nu$ be a complex parameter.\n Assume that the differential $1$-form \n $\\alpha = U\\mathrm{d}z + V \\mathrm{d}\\bar z$ is\n integrable and let $\\Phi$ denote a solution to $ \\Phi^{-1} \\mathrm{d} \\Phi =\n \\alpha$. Then \\eqref{eq:SymH>1} defines for $\\nu$ of absolute value\n $e^{-q\/2}$ a CMC immersion into $\\mathbb H^3$ with mean\n curvature $ H = \\coth q$ and the unit normal is defined in \n \\eqref{eq:normalH>1}.\n\\end{proposition}\n\n The Lawson correspondence (see appendix \\ref{sc:Lawsoncorrespond} for\n more details) between CMC surfaces in\n $\\mathbb H^3$ with mean curvature $H>1$ and CMC surfaces in $\\mathbb E^3$ \n with mean curvature $\\mathscr H$ has now in our setting the\n following simple explanation:\n Consider a CMC surface in $\\mathbb H^3$ and let $U$ and $V$ denote the\n associated matrices, but now consider these  matrices as functions of\n $\\nu \\in S^1$. \n Then it is straightforward to check that the $1$-form  $\\alpha = U\\mathrm{d}z +\n V \\mathrm{d}\\bar z$ has the decomposition as in \\eqref{eq:decalpha} and\n satisfies all the conditions for being the Maurer-Cartan form of the\n extended frame of\n some CMC surface in $\\mathbb E^3$ with Hopf differential $Q$ and mean\n curvature $\\mathscr H$.\n\n The converse construction, starting from some CMC surface in $\\mathbb E^3$ \n and ending up with some CMC surface in $\\mathbb H^3$ works out analogously.\n\n Thus for the case $H>1$, constructions of CMC surfaces in\n $\\mathbb{H}^3$ are reduced to those for CMC surfaces in Euclidean\n 3-space \\cite{DPW}. In other words, we can construct CMC surfaces in\n $\\mathbb{H}^3$ with $H>1$ via the generalized Weierstrass\n type representation (DPW method) for CMC surfaces in Euclidean 3-space.  \n See appendix \\ref{sc:Lawsoncorrespond} for an other explanation for this\n fact. \n\n \\begin{remark}\n{\\rm \n One can make this relation into a 1-1 relation by fixing the\n initial conditions of the extended frames at some base point.\n}\n\\end{remark}\n\nLet us  identify \n$\\mathfrak{su}_2$ with Euclidean $3$-space via the \ncorrespondence\n$$\nx_{1}(i\\Vec{e}_1)+x_{2}(i\\Vec{e}_2)+\nx_{3}(i\\Vec{e}_3)\n\\longleftrightarrow\n(x_1,x_2,x_3).\n$$\nThe Euclidean inner product \n$\\mathrm{d}x_{1}^{2}+\\mathrm{d}x_{2}^{2}+\\mathrm{d}x_{3}^{2}$ corresponds to \nthe inner product\n$$\n\\langle X,Y \\rangle=-\\frac{1}{2}\\mathrm{tr}\\>(XY),\n\\ \\  X, Y \\in \\mathfrak{su}_2.\n$$\nThen one can see that \n$$\n\\varphi_{\\nu}=\\mathrm{Ad}(\\Phi)(i\\Vec{e}_{1}):\\mathbb{D}\\times C_r\n\\to \\mathbb{S}^{2}\\subset \\mathfrak{su}_2\n$$ is a loop of harmonic maps, where $C_r$ is a radius $r$ circle.\n\n\\section{CMC surfaces with $0 \\leq H<1$}\\label{sc:CMCH<1}\n\\subsection{}\n Now we start our study of CMC surfaces with mean curvature $H$ such\n that $0\\leq H <1$. In this case, we may write $H=\\mathrm{tanh}\\>q, \\;\n q \\in \\mathbb R_{\\geq 0}$.\n\n We perform the gauge transformation:\n\\begin{equation}\\label{eq:extendedH<1}\n\\Phi:=\\tilde{\\Phi}\n\\left(\n\\begin{array}{cc}\ne^{(q+\\pi{i})\/4} & 0\\\\\n0 & e^{-(q+\\pi{i})\/4}\n\\end{array}\n\\right).\n\\end{equation}\n We call $\\Phi$ the \\textit{extended frame} of a CMC immersion $f$ with\n $0 \\leq H <1$. Moreover we put\n$$\n\\mathscr{H}=ie^{-q}(H+1) \\in i \\mathbb R,\n\\ \\\n\\nu=e^{-q\/2}\\lambda,\n\\ \\\n\\mathscr{Q}=-iQ.\n$$\n Note that $H = \\tanh q$ implies, $\\mathscr H = i e^{-q} (H+1) = -i e^q\n (H-1)$, and the Lax pair\n$$\nU=\\Phi^{-1}\\Phi_{z},\\  \\\nV=\\Phi^{-1}\\Phi_{\\bar z}\n$$\n is given explicitly  by the matrices\n\\begin{equation}\\label{LaxH<1}\nU=\\left(\n\\begin{array}{cc}\nu_{z}\/4 & -\\frac{1}{2}\\nu^{-1}\\mathscr{H}e^{u\/2}\n\\\\\n\\nu^{-1}\\mathscr{Q} e^{-u\/2} & -u_{z}\/4\n\\end{array}\n\\right),\n\\ \\\nV=\n\\left(\n\\begin{array}{cc}\n-u_{\\bar z}\/4 & \n-\\nu \\overline{\\mathscr{Q}}e^{-u\/2}\n\\\\\n\\frac{1}{2}\\nu \\mathscr{H}e^{u\/2}\n&\nu_{\\bar z}\/4\n\\end{array}\n\\right).\n\\end{equation}\n Considering, as in section \\ref{sc:CMCH>1}, everything as holomorphic \n expressions in $\\nu \\in \\mathbb C^{\\times}$ it is straightforward to \n check that the $1$-form $\\alpha = U \\mathrm{d} z + V \\mathrm{d} \\bar z$ is fixed by \n the following automorphism of the loop algebra:\n\\begin{equation}\\label{eq:invc4}\n\\tau_4:g(\\nu)\\longmapsto -\\mathrm{Ad}\n(\\mathscr{R})\n\\{g(i\/\\bar{\\nu})\\}^{*},\\ \\ \n\\mathscr{R}=\\left(\n\\begin{array}{cc}\n1\/\\sqrt{i} & 0\\\\\n0 & \\sqrt{i}\n\\end{array}\n\\right).\n\\end{equation}\n This automorphism is said to be of type $C_4$ (almost compact\n automorphism of the fourth kind), \\cite{Kobayashi}. The Maurer-Cartan form\n $\\alpha=\\Phi^{-1}\\mathrm{d}\\Phi$ has the decomposition\n\\begin{equation}\\label{eq:Maurernu}\n\\alpha=\\nu^{-1}\\alpha_{-1}+\\alpha_{0}+\\nu\\alpha_{1}.\n\\end{equation}\n Moreover, $\\tau_4 (\\alpha) = \\alpha$ translates into \n$$\n\\overline{\\alpha_{0}}=-\\alpha_{0},\n\\ \\\n\\alpha_{-1}=\ni\\mathrm{Ad}(\\mathscr{R})(\\alpha_{1})^{*}.\n$$\n The mapping $\\Phi$ takes values in the twisted loop group\n$$\n\\Lambda \\mathrm{SL}_{2}\\mathbb C_{\\sigma,\\tau_4}:=\n\\left\\{\ng:S^1\\to \\mathrm{SL}_{2}\\mathbb C\n\\\n\\vert\n\\\ng(-\\nu)=\\sigma g(\\nu),\n\\ \\\n\\tau_4 (g)(\\nu)=g(\\nu)\\>\n\\right\\}, \n$$\n where $\\sigma$ is defined in \\eqref{eq:sigma} and \n \\begin{equation}\\label{eq:tau4}\n \\tau_4 (g)(\\nu) = \\mathrm{Ad} (\\mathscr{R}) \\{g(i\/\\bar{\\nu})\\}^{*-1}.\n \\end{equation}\n Moreover, the first formula in Proposition \\ref{prop:Sym} can be\n reinterpreted as \n\\begin{equation}\\label{eq:SymH<1}\nf_{\\nu}:=\\Phi \\left(\n\\begin{array}{cc}\ne^{-q\/2} & 0\\\\\n0 & e^{q\/2}\n\\end{array}\n\\right)\n\\Phi^{*},\n\\end{equation}\n thus reproducing the given CMC immersion in $\\mathbb H^3$ with $0\\leq H\n = \\tanh q<1$. Its unit normal can be written in the form\n\\begin{equation}\\label{eq:normalH<1}\nn_{\\nu}=\n\\Phi\n\\left(\n\\begin{array}{cc}\ne^{-q\/2} & 0\\\\\n0 & -e^{q\/2}\n\\end{array}\n\\right)\n\\Phi^{*}.\n\\end{equation}\nConversely, the following result holds:\n\\begin{proposition}\\label{prop:fundamentalH<1}\n Let $\\mathscr H$ be a purely imaginary constant, $u$ a real function and\n $Q$ a holomorphic function on the simply connected domain $\\mathbb D \\subset \\mathbb C$. \n Let $\\nu$ be a complex parameter.\n Assume that the differential $1$-form $\\alpha = U\\mathrm{d}z + V \\mathrm{d}\\bar z$ is\n integrable and let $\\Phi$ denote a solution to $ \\Phi^{-1} \\mathrm{d} \\Phi =\n \\alpha$. Then \\eqref{eq:SymH<1} defines for $\\nu$ of absolute value\n $e^{-q\/2}$ a CMC immersion into $\\mathbb H^3$ with mean\n curvature $ H = \\tanh q$ and normal as defined in \\eqref{eq:normalH<1}.\n\\end{proposition}\n\\begin{remark}{\\rm \n As pointed out above,\n the Maurer-Cartan form\n $\\alpha=\\lambda^{-1}\\alpha_{-1}+\\alpha_{0}+\\lambda\\alpha_{1}$ is a type\n $C_4$ real form of the complex CMC surface equation \\cite{Kobayashi}.\n But it does not correspond naturally to a CMC surface in $\\mathbb E^3$, \n since $\\mathscr H$ is not real. \n}\n\\end{remark}\n\n\\subsection{}\n As indicated in the introduction\n the extended frame \\eqref{eq:extendedH<1} \n of a CMC surface $f$ with mean curvature $H=\\tanh q$ \n can also be considered as the extended frame of a minimal surface in \n the hyperbolic $3$-space of sectional curvature $-1\/\\cosh^2 q$. \n For simplicity, this paper primarily considers surfaces in $\\mathbb H^3(-1)$.\n However, the formalism can easily be adjusted to fit surfaces in \n $\\mathbb H^3(c)$ with $c<0$.\n It suffices to ``scale'' a given surface, and thus $\\mathbb H^3(-1)$ inside \n $\\mathrm{Her}_2 \\mathbb C$, by the factor $1\/\\sqrt{|c|}$. The radial \n deformation of the loop parameter $\\lambda$ on unit circle\n to a radius $r$ circle changes the mean curvature for a CMC surface, \n which is given by conjugation of a diagonal matrix to the extended frame.\n Combining the scaling and the radial deformation, \n the Lawson correspondence for CMC surfaces \n in $\\mathbb H^3(c)$ is obtained.\n\n More precisely, let $\\tilde{\\Phi}=\\tilde{\\Phi}_{\\lambda}$ be the\n $\\Lambda\\mathrm{SL}_{2}\\mathbb{C}_{\\sigma}$-valued map\n defined by (\\ref{eq:tildephi}) which frames the associated family of\n a CMC surface $f:\\mathbb{D}\\to \\mathbb{H}^3(-1)$ with mean curvature $H$. \n Then for any real number $q$,\n \\begin{equation}\\label{defomedimmersion}\n f_{\\lambda} = \\left. \\frac{1}{\\cosh q - H \\sinh q} \n \\tilde \\Phi_{\\lambda}\n \\begin{pmatrix}\n e^{q\/2} & 0 \\\\ \n 0 & e^{-q\/2} \n \\end{pmatrix}\n \\tilde \\Phi_{\\lambda}^*\\right|_{\\lambda=e^{q\/2}}\n \\end{equation}\n defines a CMC surface of mean curvature $H_{\\lambda} = H \\cosh q -\\sinh q$\n in the hyperbolic space $\\mathbb{H}^{3}(K_{\\lambda})$ of sectional curvature \n $K_{\\lambda} = - (\\cosh q - H \\sinh q)^2$. \n The surface $f_{\\lambda}$ has the same metric and \n the same Hopf differential as $f$. \n By definition, $H_{\\lambda}^2+K_{\\lambda} = H^2-1$. \n Thus $f_\\lambda$ is a Lawson correspondent of \n $f:\\mathbb{D}\\to \\mathbb{H}^{3}(-1)$ in $\\mathbb{H}^{3}(K_\\lambda)$.\n\n\n Now we consider a CMC surface $f$ with mean curvature \n $H=\\tanh q$, then the Lawson correspondent $f_\\lambda$  has the mean \n curvature $H_{\\lambda} =0$ and thus is a minimal surface.\n Note that the sectional curvature of \n the ambient space is $K_\\lambda=-1\/\\cosh^{2}q$ and the extended \n frame of $f_{\\lambda}$ is given as in \\eqref{eq:extendedH<1}.\n In section \\ref{subsc:primitiveminimal} we will show that \n the minimality of the surface in $\\mathbb H^3(c), c<0$ and primitivity of the \n Gauss map of the surface are equivalent.\n\\section{$4$-symmetric structure of the unit tangent sphere bundle}\\label{sc:4symUnittangent}\n\n\\subsection{}\\label{sc:4symmetric}\n As we have seen in the preceding section, every CMC surface with\n $0 \\leq H <1$ admits a loop group valued map $\\Phi$ which is fixed under the\n type $C_4$ automorphism $\\tau_4$.\n\n In this section we study the  automorphism $\\tau$ of $\\mathfrak{sl}_2 \\mathbb C$ \n which is obtained by first extending $\\tau_4$ to the untwisted \n loop algebra $\\Lambda \\mathfrak{sl}_2 \\mathbb C$ using the formula \n \\eqref{eq:invc4} and then\n restricting it to $\\mathfrak{sl}_2 \\mathbb C \\subset \\Lambda\n \\mathfrak{sl}_2 \\mathbb C$.\n The automorphism $\\tau$ is given on\n $\\mathfrak{g}=\\mathfrak{sl}_{2}\\mathbb C$ by the formula\n\\begin{equation}\\label{6.1}\n\\tau(X)=-\\mathrm{Ad}(\\mathscr{R})X^{*}, \n\\end{equation} \n where $\\mathscr{R}$ is defined in \\eqref{eq:invc4}.\n More explicitly, \n\\begin{equation}\\label{eq:tau}\n\\tau\n\\left(\n\\begin{array}{cc}\na & b\\\\\nc & -a\n\\end{array}\n\\right)=\n\\left(\n\\begin{array}{cc}\n-\\bar{a} & i\\bar{c}\\\\\n-i\\bar{b} & \\bar{a}\n\\end{array}\n\\right).\n\\end{equation}\n\n It is easy to see that $\\tau$ is of order $4$. The eigenspace\n decomposition of  the \\textit{complexified Lie algebra}\n $\\mathfrak{g}^{\\mathbb C}$ with respect to $\\tau$ is given by\n$$\n\\mathfrak{g}^{\\mathbb C}=\\mathfrak{g}_{0}^{\\mathbb C}\\oplus\n\\mathfrak{g}_{1}^{\\mathbb C}\\oplus\n\\mathfrak{g}_{2}^{\\mathbb C}\\oplus\n\\mathfrak{g}_{3}^{\\mathbb C},\n$$\n where $\\mathfrak{g}_{k}^{\\mathbb C}$ is the eigenspace corresponding to the\n eigenvalue $i^k$. \n Note that the complexified Lie algebra \n $\\mathfrak{g}^{\\mathbb C}=(\\mathfrak{sl}_{2}{\\mathbb C})^{\\mathbb C}$ is realized as\n $\\mathfrak{g}\\times\\mathfrak{g}$. This construction can be described\n as follows: \n \n Consider the map \n$$\n \\iota: X \\in \\mathfrak{sl}_2 \\mathbb C  \\longmapsto (X,  \\; \\bar X) \\in \\mathfrak{sl}_2 \\mathbb C \\times\n \\mathfrak{sl}_2 \\mathbb C\\;. \n$$\n This is an injective homomorphism of the real Lie algebra\n $\\mathfrak{sl}_2 \\mathbb C$ into $\\mathfrak{sl}_2 \\mathbb C \\times\n \\mathfrak{sl}_2 \\mathbb C$. Since  $\\iota(\\mathfrak{sl}_2 \\mathbb C) \\cap \n i \\iota(\\mathfrak{sl}_2 \\mathbb C) = \\{0\\}$,\n the image $\\iota(\\mathfrak{sl}_2 \\mathbb C)$ is a real form of \n $\\mathfrak{sl}_2 \\mathbb C \\times \\mathfrak{sl}_2 \\mathbb C$.\n The latter Lie algebra carries the natural complex structure \n given by multiplying a complex number to each of the two factors.\n By transporting $\\tau$ via $\\iota$ we obtain\n\\begin{enumerate}  \n\\item[(i)] $\\tau$ acts on $\\iota(\\mathfrak{sl}_2 \\mathbb C)$ as $\\tau(X, \\bar\n     X) = (\\tau(X), \\;  \\overline{\\tau(X)})$.\n\\end{enumerate}\n We now define the complex linear extension $\\hat \\tau$  to\n $\\mathfrak{sl}_2 \\mathbb C \\times \\mathfrak{sl}_2 \\mathbb C$ as\n follows:\n\\begin{enumerate}\n\\item[(ii)] $\\hat \\tau(X,Y) = ( \\tau(\\bar Y),\\; \\overline{\\tau(X)})$.\n\\end{enumerate}\n It is straightforward to show that, indeed, $\\hat \\tau$ is complex linear\n relative to $i$ acting on the first and the second factor equally by\n multiplication.\n \n Moreover, on $\\iota (X)$ the new $\\hat \\tau$ acts like {\\rm (i)}.\n Thus {\\rm (ii)} is the complex linear extension of the original\n $\\tau$ on $\\mathfrak{sl}_2 \\mathbb C$ to the product \n $\\mathfrak{sl}_2 \\mathbb C \\times \\mathfrak{sl}_2 \\mathbb C$.\n \n Since $\\tau$ is an automorphism of order $4$, also $\\hat \\tau$\n is an automorphism of order $4$ and we have\n\\begin{lemma}\\label{grading}\n$$\n[\\mathfrak{g}_{k}^{\\mathbb C},\\mathfrak{g}_{l}^{\\mathbb C}] \\subset\n\\mathfrak{g}_{k+l}^{\\mathbb C}\\ \\ (\\mathrm{mod} \\>4).\n$$\n\\end{lemma}\n\n The automorphism has two \\textit{real eigenvalues} $i^{0}=1$ and\n $i^{2}=-1$. The corresponding eigenspaces $\\mathfrak g_0^{\\mathbb C}$\n and $\\mathfrak g_2^{\\mathbb C}$ of\n $\\mathfrak g^{\\mathbb C} \\cong \\mathfrak g \\times \\mathfrak g$ are\n computed explicitly as\n\\begin{align*}\n\\mathfrak{g}_0^{\\mathbb C} = \n\\left\\{\n\\left(\n\\begin{pmatrix}\nx_1 & 0 \\\\ 0 & -x_1\n\\end{pmatrix},\n\\begin{pmatrix}\n-x_1 & 0 \\\\ 0 & x_1\n\\end{pmatrix}\n\\right)\n\\right\\},\\;\\;\\;\n\\mathfrak{g}_2^{\\mathbb C} = \n\\left\\{\n\\left(\n\\begin{pmatrix}\nx_1 & 0 \\\\ 0 & -x_1\n\\end{pmatrix},\n\\begin{pmatrix}\nx_1 & 0 \\\\ 0 & -x_1\n\\end{pmatrix}\n\\right)\n\\right\\},\n\\end{align*}\n with $x_1 \\in \\mathbb C$.\n The corresponding real subspaces of $\\mathfrak{sl}_2 \\mathbb C$ are\n $$\n \\mathfrak g \\cap \\mathfrak {g}_0^{\\mathbb C} =\n \\mathfrak h = \\mathbb R  (i \\Vec{e}_1)\n \\; \\mbox{and}\\;\\;  \\mathfrak g \\cap \\mathfrak {g}_2^{\\mathbb C} =\n \\mathbb R (\\Vec{e}_1).\n $$\n Moreover, the eigenspaces $\\mathfrak g_1^{\\mathbb C}$ and $\\mathfrak g_3^{\\mathbb C}$ \n for eigenvalues $i$ and $-i$ are computed explicitly as\n\\begin{align*}\n\\mathfrak{g}_1^{\\mathbb C} =\n\\left\\{\n\\left(\n\\begin{pmatrix}\n0 & x_2 \\\\ x_3 & 0\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & -x_3 \\\\ x_2 & 0\n\\end{pmatrix}\n\\right)\n\\right\\}, \\;\\;\\;\n\\mathfrak{g}_3^{\\mathbb C} =\n\\left\\{\n\\left(\n\\begin{pmatrix}\n0 & x_2 \\\\ x_3 & 0\n\\end{pmatrix},\n\\begin{pmatrix}\n0 & x_3 \\\\ -x_2 & 0\n\\end{pmatrix}\n\\right)\n\\right\\},\n\\end{align*}\n with $x_2, x_3 \\in \\mathbb C$.\n The automorphism $\\tau$ defines the semi-Riemannian $4$-symmetric\n space $\\mathrm{SL}_2 \\mathbb C\/\\mathrm{U}_1$. The space is isomorphic with\n the unit tangent sphere bundle $\\mathrm{U}\\mathbb H^3$ as shown in\n section \\ref{sc:tangentspherebundle}. The complexified tangent space\n $\\mathfrak{p}^{\\mathbb C}=\n (T_{(\\Vec{e}_0,\\Vec{e}_1)}\\mathrm{U}\\mathbb{H}^3)^{\\mathbb C}$ is given\n by\n$$\n\\mathfrak{p}^{\\mathbb C}=\n\\mathfrak{g}_{1}^{\\mathbb C}\\oplus\\mathfrak{g}_{2}^{\\mathbb C}\n\\oplus\\mathfrak{g}_{3}^{\\mathbb C}.\n$$\n Comparing this with the fibration $\\pi_{1}: \\mathrm{U}\\mathbb{H}^3\\to\n \\mathrm{Geo}(\\mathbb{H}^3)$ discussed in section\n \\ref{sc:tangentspherebundle}, we have\n$$\n\\mathcal{H}_{1}^{\\mathbb C}=\\mathfrak{g}_{1}^{\\mathbb C}\\oplus\\mathfrak{g}_{3}^{\\mathbb C},\n\\ \\\n\\mathcal{V}_{1}^{\\mathbb C}=\\mathfrak{g}_{2}^{\\mathbb C}.\n$$\n\n\n\n\\subsection{}\\label{sc:contactmfd}\n Here we recall the notion of a contact manifold.\n\\begin{definition}{\\rm\n A 1-form $\\omega$ on a manifold $L$ of dimension $2n-1$ is said to be a\n \\textit{contact form} if $(\\mathrm{d}\\omega)^{n-1} \\wedge \\omega \\not=0$ on\n $M$. A hyperplane field $\\mathcal{D}\\subset TL$ on $L$ is called a\n \\textit{contact structure} if for any point $p\\in L$, there exists a\n contact form $\\omega$ defined on a neighborhood $U_p$ of $p$ such that\n $\\mathrm{Ker}\\> \\omega=\\mathcal D$ on $U_p$. \n}\n\\end{definition}\n A $(2n-1)$-manifold $L$ with a contact structure  $\\mathcal{D}$ is\n called a \\textit{contact manifold}. If a contact manifold\n $(L,\\mathcal{D})$ admits a globally defined contact form $\\omega$ which\n annihilates $\\mathcal{D}$,  \\textit{i.e.},\n $\\mathrm{Ker}\\>\\omega=\\mathcal{D}$, then $(L,\\mathcal{D})$ is said to\n be a contact manifold in the \\textit{strict sense}.\n\\begin{definition}\n{\\rm\nLet $M^n$ be an $n$-manifold and $F:M \\to L$ a smooth map into\na contact $(2n-1)$-manifold. Then $F$ is said to be \\textit{Legendre} if \n$\\mathrm{d} F(TM)\\subset \\mathcal{D}$.\n}\n\\end{definition}\n In particular, if $L$ admits a global contact form $\\omega$, \n then $F$ is Legendre if and only if  $F^{*}\\omega=0$.\n\n Now let $(N^n,g)$ be a Riemannian $n$-manifold. Then its unit tangent\n sphere bundle $\\mathrm{U}N$ admits a canonical contact structure. In case\n $N=\\mathbb{H}^3$, one can check that the canonical contact structure of\n $\\mathbb{H}^3$ is given by\n$$\n \\mathcal{D}=\\mathcal{H}_{1}=(\\mathfrak{g}_{1}^{\\mathbb C}\n \\oplus\\mathfrak{g}_{3}^{\\mathbb C}) \\cap \\mathfrak g.\n$$\n By using this fact and results of section \\ref{sc:tangentGauss} we\n obtain:\n\n\\begin{proposition}\\label{prop:Legendre}\n Let $F=(f, n):\\mathbb{D}\\to\\mathrm{U}\\mathbb{H}^3$ be a smooth map with\n frame $\\Phi:\\mathbb{D}\\to \\mathrm{SL}_{2}\\mathbb C$,\n \\textit{i.e.}, $\\Phi$ is a map satisfying \n$$\nF=(\\Phi \\Phi^*, \\Phi \\Vec{e}_1\\Phi^{*}).\n$$\n Denote by $\\alpha=\\Phi^{-1}\\mathrm{d}\\Phi$\n the pull-back of the Maurer-Cartan form by $\\Phi$ and decompose\n $(\\alpha,\\overline{\\alpha})$ \nas\n$$\n(\\alpha,\\overline{\\alpha})=\n\\alpha_{0}^\\mathbb{C}\n+\n\\alpha_{1}^\\mathbb{C}\n+\n\\alpha_{2}^\\mathbb{C}\n+\n\\alpha_{3}^\\mathbb{C}\n$$\naccording to the eigenspace decomposition with respect to \n$\\tau$. Then $\\alpha_{2}^{\\mathbb{C}}$ is given by\n$\\alpha_{2}^{\\mathbb{C}}=(\\alpha_2,\\overline{\\alpha_2})$ with\n$$\n\\alpha_{2}=\\frac{1}{2}F^{*}\\omega\\>\\Vec{e}_{1}.\n$$\n Thus $F$ is Legendre if and only if $\\alpha_2=0$.\n In particular, if $F=(f,n)$ is the Gauss map of a conformal\n immersion $f:\\mathbb{D}\\to\\mathbb{H}^3$, then \n$$\n\\alpha_{2}=\\frac{1}{2}F^{*}\\omega\\>\\Vec{e}_{1} = \\frac{1}{2}\\langle \\mathrm{d}f,n\\rangle \\>\\Vec{e}_{1}=0.\n$$\n\\end{proposition}\n\\subsection{} \n Let $F: \\mathbb{D}\\to G\/H = \\mathrm{SL}_2 \\mathbb C\/\\mathrm{U}_1$ be a\n Legendre map with frame $\\Phi$. \n Then we have the eigenspace decomposition\n of $\\alpha=\\Phi^{-1}\\mathrm{d}\\Phi$.\n$$\n(\\alpha,\\overline{\\alpha})=\n\\alpha_{0}^{\\mathbb{C}}+\n\\alpha_{1}^{\\mathbb{C}}+\n\\alpha_{3}^{\\mathbb{C}},\n\\ \\ \n\\alpha_{0}^{\\mathbb{C}}=\n(\\alpha_0,\\overline{\\alpha_0}).\n$$\n On the other hand, we have the decomposition\n $\\alpha=\\alpha_{\\mathfrak{h}}+\n \\alpha_{\\mathfrak{p}}$. \n We denote the first component of $\\alpha_j^{\\mathbb C}$ by $\\alpha_j$.\n Comparing these decompositions, we get\n\\begin{equation}\\label{eq:alphaLegendre}\n\\alpha=\\alpha_{\\mathfrak{h}}+\n\\alpha_{\\mathfrak{p}},\n\\ \\\n\\alpha_{\\mathfrak{h}} = \\alpha_0, \\\n\\alpha_{\\mathfrak{p}} =\n\\alpha_{\\mathfrak{p}}^{\\prime}+\n\\alpha_{\\mathfrak{p}}^{\\prime\\prime}=\n\\alpha_{1}+\\alpha_{3}. \n\\end{equation}\n We express the type-decompositions of $\\alpha_{1}$ and $\\alpha_{3}$ \n with respect to the conformal structure of $\\mathbb{D}$ as\n$$\n\\alpha_{1}=\n\\alpha_{1}^{\\prime}+\n\\alpha_{1}^{\\prime \\prime},\n\\ \\\n\\alpha_{3}=\n\\alpha_{3}^{\\prime}+\n\\alpha_{3}^{\\prime \\prime}.\n$$\nThen from Lemma \\ref{grading} and the integrability of $\\alpha$, we derive \n$$\n[\\alpha^{\\prime}_{\\mathfrak{p}}\n\\wedge \n\\alpha^{\\prime\\prime}_{\\mathfrak{p}}\n]_{\\mathfrak{p}}=\n[\n(\\alpha_{1}^{\\prime}\n+\n\\alpha_{3}^{\\prime})\n\\wedge \n(\n\\alpha_{1}^{\\prime\\prime}\n+\n\\alpha_{3}^{\\prime \\prime})\n]\n_{\\mathfrak{p}}=\n[\\alpha_{1}^{\\prime}\n\\wedge \n\\alpha_{1}^{\\prime\\prime}]\n+\n[\\alpha_{3}^{\\prime}\n\\wedge \n\\alpha_{3}^{\\prime\\prime}]\n$$ \n and \n$$\n[\\alpha_1 \\wedge \\alpha_1]+ \n[\\alpha_3 \\wedge \\alpha_3]= 0.\n$$\n Noting \n $[\\alpha_1 \\wedge \\alpha_1] = 2 [\\alpha^{\\prime \\prime}_1 \\wedge \\alpha^{\\prime}_1]$  and \n $[\\alpha_3 \\wedge \\alpha_3] =2 [\\alpha^{\\prime\\prime}_3 \\wedge \\alpha^{\\prime}_3]$, \n we conclude that\n$$\n[\\alpha^{\\prime}_{\\mathfrak{p}}\n\\wedge \n\\alpha^{\\prime\\prime}_{\\mathfrak{p}}]_{\\mathfrak{p}}\n=\n[\\alpha_{1}^{\\prime}\n\\wedge \n\\alpha_{1}^{\\prime\\prime}]\n+\n[\\alpha_{3}^{\\prime}\n\\wedge \n\\alpha_{3}^{\\prime\\prime}]\n=\n\\frac{1}{2} [\\alpha_{1}\n\\wedge \n\\alpha_{1}]\n+\n\\frac{1}{2} [\\alpha_{3}\n\\wedge \n\\alpha_{3}]\n=\n0.\n$$\n Now we arrive at the following \n \\textit{zero curvature representation} for Legendre harmonic maps.\n\\begin{proposition}\\label{ZCR-Legendre}\n Let $F:\\mathbb{D}\\to \\mathrm{SL}_2 \\mathbb C\/\\mathrm{U}_1$ be a\n Legendre harmonic map with frame\n $\\Phi:\\mathbb{D}\\to \\mathrm{SL}_2 \\mathbb C$.\n Then $\\alpha_{\\lambda}= \\Phi^{-1}\\mathrm{d}\\Phi= \n \\alpha_{\\mathfrak{h}}+ \\lambda^{-1} \\alpha_{\\mathfrak{p}}^{\\prime} + \\lambda\n \\alpha_{\\mathfrak{p}}^{\\prime \\prime} $\nsatisfies\n$$\n\\mathrm{d}\\alpha_{\\lambda}+\\frac{1}{2}[\\alpha_\\lambda\n \\wedge \\alpha_\\lambda]=0\n$$\n for all $\\lambda \\in \\mathbb C^{\\times}$.\n Here $\\alpha_{\\mathfrak{h}}$ and $\\alpha_{\\mathfrak{p}}$ are\n defined in \\eqref{eq:alphaLegendre} and $\n \\alpha_{\\mathfrak{p}}^{\\prime}$ (resp. $\n \\alpha_{\\mathfrak{p}}^{\\prime \\prime}$) is the $(1, 0)$-part\n (resp. $(0,1)$-part)  of  $\\alpha_{\\mathfrak{p}}$.\n\\end{proposition}\n\n\\subsection{} \n The square $\\tau^{2}$ of $\\tau$ is an involutive \n automorphism of  $\\mathfrak g$. The $1$-eigenspace and\n $(-1)$-eigenspace of $\\tau^2$ on $\\mathfrak g$\n are\n$$\n\\mathfrak{g} \\cap (\\mathfrak{g}_0^{\\mathbb C} \\oplus\n\\mathfrak{g}_2^{\\mathbb C}) = \\mathfrak{d}\n\\ \\mbox{and}\\\n\\mathfrak{g} \\cap (\\mathfrak{g}_{1}^{\\mathbb C}\\oplus\\mathfrak{g}_{3}^{\\mathbb C})\n =\\mathfrak{q},\n$$\n where $\\mathfrak{d}$ and $\\mathfrak{q}$ are defined in \\eqref{eq:d} and\n \\eqref{eq:q}, respectively. Hence $(G,\\tau^2)$ defines the\n semi-Riemannian symmetric space \n$$\n G\/D=\\mathrm{SL}_{2}\\mathbb C\/\\mathbb C^{\\times}\n    =\\mathrm{Gr}_{1,1}(\\mathbb{E}^{1,3}).\n$$\n The $4$-symmetric space $G\/H = \\mathrm{SL}_2 \\mathbb C \/ \\mathrm{U}_1$\n is a fiber bundle over $G\/D$ with standard fiber\n$$\nD\/H=\\mathbb C^{\\times}\/\\mathrm{U}_1=\\mathbb{R}_{>0}.\n$$\n Now let $F:\\mathbb{D}\\to G\/H$ be a Legendre harmonic map with frame\n $\\Phi: \\mathbb D \\to G$. Since $F$ is Legendre,\n Integrating $\\Phi_{\\lambda}^{-1}\n \\mathrm{d}\\Phi_{\\lambda}=\\alpha_{\\lambda}$, \n we get the associated\n family $\\{F_{\\lambda}\\}$ of $F$.\n\n Decompose $\\alpha$ according to the Lie algebra decomposition\n $\\mathfrak{g}=\\mathfrak{d}\\oplus \\mathfrak{q}$:\n$$\n\\alpha_{\\mathfrak{d}}=\n\\alpha_{0}+\\alpha_{2} = \\alpha_0,\n\\ \\\n\\alpha_{\\mathfrak{q}}=\\alpha_{1}+\\alpha_{3}.\n$$\n Then the decomposition above can be rephrased as \n$$\n\\alpha_{\\lambda}\n=\\alpha_{\\mathfrak{d}}\n+\\lambda^{-1}\n\\alpha^{\\prime}_{\\mathfrak q}\n+\\lambda\n\\alpha^{\\prime\\prime}_{\\mathfrak q}.\n$$\n This formula implies that the projected map $\\mathcal{G}=\\pi_{1}\\circ\n F$ is harmonic and \n$$\n \\mathcal{G}_{\\lambda}:=\\pi_{1}\\circ F_{\\lambda}\n$$\n gives the associated family $\\{\\mathcal{G}_\\lambda\\}$ of $\\mathcal{G}$,\n where $\\pi_1$ is the natural projection $\\pi_1 : G\/H \\to G\/D$.\n\n\\begin{remark}\n {\\rm Proposition \\ref{ZCR-Legendre} is valid for any horizontal harmonic maps \n into semi-Riemannian $4$-symmetric spaces. More precisely, let $(G\/H,\\tau)$ \n be a semi-Riemannian $4$-symmetric space with semi-Riemannian homogeneous \n projection $\\pi_1:G\/H\\to G\/D$ onto the semi-Riemannian symmetric space \n $(G\/D,\\tau^2)$. \n Take a map $F:\\mathbb{D}\\to G\/H$ which is horizontal with respect to $\\pi_1$ \n and let $\\Phi:\\mathbb{D}\\to G$ be its frame.\n Decompose $\\alpha=\\Phi^{-1}\\mathrm{d}\\Phi$ as $\\alpha=\\sum_{j=0}^{3}\\alpha_j$ \n according to the eigenspace decomposition of $\\tau$. Then one can see that \n $F$ is horizontal if and only if $\\alpha_2=0$. In addition, the harmonicity \n of $F$ is equivalent to the flatness of the connections \n $\\mathrm{d}+\\alpha_\\lambda$, where $\\alpha_\\lambda$ is defined \n as in Proposition \\ref{ZCR-Legendre}.\n}\n\\end{remark}\n\\subsection{}\\label{subsc:primitiveminimal}\n In this section, we prove the following characterization of minimal surfaces.\n\\begin{proposition}\\label{prop:prm}\n  Let $\\mathbb H^3(c)$ be the hyperbolic $3$-space of sectional curvature $c<0$.\n Then the unit tangent sphere bundle \n $\\mathrm{U} \\mathbb H^3(c)$ is a $4$-symmetric space.\n Moreover, a surface $f: \\mathbb D \\to \\mathbb H^3(c)$ is minimal \n if and only if its Gauss map is a primitive map with \n respect to the $4$-symmetric structure of $\\mathrm{U}\\mathbb H^3(c)$.\n\\end{proposition}\n {\\bf Proof}. It is clear that $\\mathrm{U} \\mathbb H^3(c)$ is \n a $4$-symmetric space with respect to $\\tau$ defined in \\eqref{eq:tau}. \n Let $f:M\\to \\mathbb{H}^3(c)$ be a conformal \n immersion with unit normal $n$.\n Take a simply connected coordinate domain $(\\mathbb{D},z)\\subset M$ and denote \n by $\\Psi$ the coordinate frame defined on $\\mathbb{D}$. Let $\\hat{\\Phi}$ be \n a lift of $\\Psi$ to $\\mathrm{SL}_{2}\\mathbb{C}$ as in (\\ref{eq:hatframe}).\n Then the Maurer-Cartan form $\\hat{\\alpha}=\\hat{\\Phi}^{-1}\n \\mathrm{d}\\hat{\\Phi}$ is given by  (\\ref{eq:hatMau1}) \n with $H+1$ and $H-1$ replaced by $H+ \\sqrt{|c|}$ and $H-\\sqrt{|c|}$,  \n respectively.\n Now we decompose $(\\hat \\alpha, \\overline{\\hat{\\alpha}})$\n according to the eigenspace decomposition of $\\tau$. Then we have\n$$\n(\\hat \\alpha, \\overline{\\hat{\\alpha}})=\n\\hat{\\alpha}^{\\mathbb{C}}_{0}+\n\\hat{\\alpha}^{\\mathbb{C}}_{1}+\n\\hat{\\alpha}^{\\mathbb{C}}_{2}+\n\\hat{\\alpha}^{\\mathbb{C}}_{3},\n$$\nwhere\n\\begin{align*}\n\\hat{\\alpha}^{\\mathbb{C}}_{0}\n&=\n\\left(\n\\begin{pmatrix}x_1 & 0 \\\\ 0 & -x_1 \\end{pmatrix}, \n\\begin{pmatrix}-x_1 & 0 \\\\ 0 & x_1 \\end{pmatrix}\n\\right), \\;\\;\n\\hat{\\alpha}^{\\mathbb{C}}_{2}\n=(\\Vec{0},\\Vec{0}), \\\\\n\\hat{\\alpha}^{\\mathbb{C}}_{1}\n&=\n\\left(\n\\begin{pmatrix}0 & x_2 \\\\ x_3 & 0 \\end{pmatrix}, \n\\begin{pmatrix}0  & -x_3 \\\\ x_2 & 0 \\end{pmatrix}\n\\right), \\;\\;\n\\hat{\\alpha}^{\\mathbb{C}}_{3}\n=\n\\left(\n\\begin{pmatrix}0 & x_4 \\\\ x_5 & 0 \\end{pmatrix}, \n\\begin{pmatrix}0  & x_5 \\\\ -x_4 & 0 \\end{pmatrix}\n\\right)\n\\end{align*}\n with $x_1 = \\frac{1}{4}(u_{z}\\mathrm{d}z-u_{\\bar z}\\mathrm{d}\\bar{z})$, \n $x_2 =  \\frac{\\sqrt{|c|}}{2}e^{u\/2}\\mathrm{d}z$, \n $x_3 =-Qe^{-u\/2}\\mathrm{d}z-\\frac{H}{2}e^{u\/2}\\mathrm{d}\\bar{z}$, \n $x_4 = \\frac{H}{2}e^{u\/2}\\mathrm{d}z+\\bar{Q}e^{-u\/2}\\mathrm{d}\\bar{z}$ \n and $x_5 = \\frac{\\sqrt{|c|}}{2}e^{u\/2}\\mathrm{d}\\bar{z}$.\n From these equations, we deduce that $f$ is minimal if and only if its \n Gauss map $F$ is primitive. $\\Box$\n\n\\begin{remark}\n\\mbox{}\n {\\rm\n\\begin{enumerate}\n\\item The unit tangent sphere bundle $\\mathrm{U}\\mathbb{H}^3$ is \n the twistor $CR$-manifold of $\\mathbb{H}^3$ in the sense of \\cite{LeBrun}. \n There exist two standard $f$-structures $J_1$ and $J_2$ on $\\mathrm{U}\\mathbb{H}^3$, \n \\textit{i.e.}, endomorphism fields $J$ on  $\\mathrm{U}\\mathbb{H}^3$ \n such that $J^3+J=0$.\n One can see that a map $F:M\\to \\mathrm{U}\\mathbb{H}^3$\n from a Riemann surface to $\\mathrm{U}\\mathbb{H}^3$ \n is $J_2$-holomorphic if and only if $F$ is a primitive map. \n On the other hand, \n for a conformal immersion $f:M\\to \\mathbb{H}^3$,\n its Gauss map is $J_1$-holomorphic if and only if \n $f$ is totally umbilical (see \\cite[Theorem 7.1]{Salamon}). \n\n\\item  Setting $\\tau (g) = {\\rm Ad} \\mathscr R (g^*)^{-1}$ we \n obtain an automorphism of $G = {\\rm SL}_2 \\mathbb C$, the differential \n of which coincides with $\\tau$ as given in \\eqref{6.1}. \n By abuse of language we will\n use the same notation for the group level and for the Lie algebra level\n as well as for the corresponding complexified objects.\n\n \\item In \\cite{DMPW},  the original loop group approach \\cite{DPW} \n was extended to include primitive harmonic maps into compact $k$-symmetric \n spaces. In our case the symmetric space under consideration is non-compact.\n This has far-reaching consequences. We have therefore included in sections \n \\ref{sc:Potential} and \\ref{sc:DPW}\n a brief description of the corresponding technical details.\n In particular, the Iwasawa decomposition has not only one, but two open cells\n (the union of which is dense).\n Implications of this can already be seen in the examples presented in section \n \\ref{sc:Ex}.\n\\end{enumerate}\n}\n\\end{remark}\n\n\\section{Potentials}\\label{sc:Potential}\n\\subsection{}\n We recall loop groups and the Birkhoff decomposition.\n The twisted loop group is defined as\n \\begin{equation}\\label{eq:loopgroups}\n \\Lambda{\\rm SL}_2 \\mathbb C_\\sigma = \\left\\{ g:S^1  \\to {\\rm\n SL}_2 \\mathbb C\\; \\left| \\right. \\; g \\;\\mbox{is continuous and}\\;\n g(-\\lambda) =  \\sigma g(\\lambda)  \\; \\right \\} \\; ,\n \\end{equation}\n where $\\sigma$ is defined in \\eqref{eq:sigma}.\n More strictly, we assume that the coefficients of all $g \\in\n \\Lambda{\\rm SL}_2 \\mathbb C_\\sigma$ are in the {\\it Wiener algebra}\n $\\mathcal A = \\left\\{ f(\\lambda) = \\sum_{n \\in \\mathbb Z} f_n\n \\lambda^n\\;  |\\;  S^1 \\to \\mathbb C \\;\\; ; \\;\\; \\sum_{n\n \\in \\mathbb Z}|f_n| < \\infty \\right\\}$. \n  The Wiener algebra is a Banach algebra relative to the norm \n $\\| f\\| = \\sum |f_n|$, and $\\mathcal A$\n consists of continuous functions. Thus \n $\\Lambda{\\rm SL}_2 \\mathbb C_\\sigma$ is a Banach Lie group. \n We denote the Lie algebra of\n $\\Lambda{\\rm SL}_2 \\mathbb C_\\sigma$ by $\\Lambda\\mathfrak{sl}_2\n \\mathbb C_\\sigma$, which consists of maps  $g : S^1 \\to\n \\mathfrak{sl}_2 \\mathbb C$.\n\n We will need to consider two subgroups of $\\Lambda{\\rm SL}_2 \\mathbb C_{\\sigma}$, \n the {\\it twisted plus loop group} and the {\\it minus loop group} \n as follows:\n Let $\\boldsymbol B$ a subgroup of ${\\rm SL}_2 \\mathbb C$.\n Let $ \\Lambda_{B}^+ {\\rm SL}_2 \\mathbb C_\\sigma$\n be the group of maps into $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}$\n which can be extended holomorphically to $D = \\{\\lambda \\in \\mathbb C\n \\;|\\;  |\\lambda| < 1\\}$ and which take values\n in $\\boldsymbol B$ at $\\lambda =0$. Similarly, let $ \\Lambda_{B}^-\n {\\rm SL}_2 \\mathbb C_\\sigma$ be the group of maps into $\\Lambda {\\rm\n SL}_2 \\mathbb C_{\\sigma}$ which can be extended holomorphically to\n $E =\\{\\lambda \\in \\mathbb C  \\;|\\; 1 <\n |\\lambda| \\} \\cup \\{\\infty\\}$ and take values\n in $\\boldsymbol B$ at $\\lambda =\\infty$. If $\\boldsymbol B = \\{\\rm Id\\}$ \n we write the subscript $*$  instead of $\\boldsymbol B$, if $\\boldsymbol B = \n {\\rm SL}_2 \\mathbb C$ we  abbreviate  $\\Lambda_{B}^+ {\\rm SL}_2 \\mathbb C_\\sigma$ and \n $\\Lambda_{B}^- {\\rm SL}_2 \\mathbb C_\\sigma$ by\n $\\Lambda^+ {\\rm SL}_2 \\mathbb C_\\sigma $ and \n $\\Lambda^- {\\rm SL}_2 \\mathbb C_\\sigma$, respectively. \n\n It is clear that the loop groups \n $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j}, \\;j = 3, 4,$\n defined in sections \\ref{sc:CMCH>1} and \\ref{sc:CMCH<1} are also \n subgroups of $\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$.  \n The Lie algebras of  $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j}$\n are denoted by $\\Lambda {\\mathfrak{sl}}_2 \\mathbb C_{\\sigma, \\tau_j}, \\;j = 3, 4,$ \n and those are subalgebras of $\\Lambda {\\mathfrak{sl}}_2 \\mathbb C_{\\sigma}$.\n \\begin{theorem}[Birkhoff decomposition\n  \\cite{LoopGroup}]\\label{thm:Birkhoff}\n The maps\n\\begin{equation*}\n \\Lambda_{*}^- {\\rm SL}_2 \\mathbb C_\\sigma \\times \\Lambda^+\n {\\rm SL}_2 \\mathbb C_\\sigma \\to \\Lambda {\\rm SL}_2 \\mathbb C_\\sigma\n \\;\\; \\mbox{and}\\;\\;\\Lambda_{*}^+ {\\rm SL}_2 \\mathbb C_\\sigma \\times \\Lambda^-\n {\\rm SL}_2 \\mathbb C_\\sigma \\to \\Lambda {\\rm SL}_2 \\mathbb C_\\sigma \\\\\n\\end{equation*}\n are analytic diffeomorphisms onto the open dense subsets \n $\\Lambda_{*}^- {\\rm SL}_2 \\mathbb C_\\sigma \\cdot \\Lambda^+\n {\\rm SL}_2 \\mathbb C_\\sigma$ and $\\Lambda_{*}^+ {\\rm SL}_2 \\mathbb C_\\sigma \n \\cdot \\Lambda^- {\\rm SL}_2 \\mathbb C_\\sigma$ of \n $\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$ respectively.\n The open dense subsets will be called the {\\rm left big cell} and \n the {\\rm right big cell} respectively.\n\\end{theorem}\n\\begin{remark}\n {\\rm In this paper the big cell always means the left big cell.}\n\\end{remark}\n\n\\subsection{}\\label{subsc:holopot}\n The holomorphic potential for a CMC surface $f$ in $\\mathbb H^3$\n is a $\\Lambda \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$-valued holomorphic $1$-form\n determined from the extended frame of $f$, which is an analogue of \n Weierstrass data for a minimal surface in $\\mathbb R^3$. \n The holomorphic potential reproduces the CMC surface $f$ \n using the generalized Weierstrass\n type representation recalled in section \\ref{sc:DPW}.\n \n Let $\\mathbb D$ be a simply connected domain in $\\mathbb C$, and \n let $\\Phi :\\mathbb D\\to \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j}, j = 3, 4,$ \n the extended frame of a CMC surface $f$ in $\\mathbb H^3$ with $H \\neq 1$ \n as defined in \\eqref{eq:extendedH>1} or \\eqref{eq:extendedH<1}, \n respectively.\n\\begin{proposition}\\label{prop:delbar}\n There exists a loop $g : \\mathbb D \\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$, \n holomorphic for $\\lambda \\in \\mathbb C$, such that \n $\\Phi g : \\mathbb D \\to \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}$ \n is holomorphic in $\\lambda \\in \\mathbb C$ and $z \\in \\mathbb D$.\n\\end{proposition}\n {\\bf Proof}.\n The holomorphicity of $\\Phi g$ is equivalent to \n that \n \\begin{equation}\\label{eq:delbar}\n g_{\\bar z} + Vg =0\\;,\n \\end{equation}\n where $V$ is defined in \\eqref{LaxH>1} or \\eqref{LaxH<1} respectively.\n Since $V$ is real analytic in $z$, one can extend $V$ holomorphically to \n $\\mathbb D_{\\epsilon} (p_0) \\times \\mathbb D_{\\epsilon} (\\bar p_0)$, \n \\textit{i.e.}, \n there exists, for sufficiently small $\\epsilon > 0$, a holomorphic matrix \n function  \n $\\tilde V(z, w) :\\mathbb D_{\\epsilon} (p_0) \\times \\mathbb\n D_{\\epsilon} (\\bar p_0) \\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$\n such that  $\\tilde  V|_{(z, \\bar z)} = V$, where $\\mathbb D_{\\epsilon}\n (p_0)$  (resp. $\\mathbb D_{\\epsilon} (\\bar p_0)$) denotes the\n $\\epsilon$-disk around $p_0 \\in \\mathbb D$ (resp. $\\bar p_0 \\in \n \\bar{\\mathbb D}$).\n Let us consider the ordinary differential equation:\n  \\begin{equation}\n  \\tilde g_{w} + \\tilde V \\tilde g =0\\;, \\;\\;\\tilde g(z, \\bar p_0) ={\\rm\n   Id}.\n  \\end{equation}\n This equation has, for every fixed $z$,  a unique solution \n $\\tilde g = \\tilde g (z, w)$. Setting $g =  \\tilde  g(z, \\bar z)$, we obtain \n  \\begin{equation}\n  g_{\\bar z} + V g =0\\;, \\;\\; g(p_0, \\bar p_0) ={\\rm Id}.\n  \\end{equation}\n Therefore, on every $U_{\\alpha}$ of an open cover $(U_{\\alpha})$ \n of $\\mathbb D$, there exists \n a real analytic solution $g_{\\alpha}$. On $U_{\\alpha} \\cap U_{\\beta}$, \n we define \n  \\begin{equation}\n  h_{\\alpha \\beta} = g_{\\alpha}^{-1} g_{\\beta} : U_{\\alpha} \\cap\n   U_{\\beta} \\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}.\n  \\end{equation}\n It is easy to verify that $h_{\\alpha \\beta}$ is holomorphic in $z$. \n Moreover, the $h_{\\alpha \\beta}$ satisfy the co-cycle condition \n $h_{\\alpha  \\beta} h_{\\beta \\gamma} = h_{\\alpha \\gamma}$. Thus \n this data defines a holomorphic $\\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$-principal \n fiber bundle $P \\to \\mathbb D$. Since $\\mathbb D$ is simply connected,\n by a generalization of Grauert's theorem \\cite{Bungart}, the\n holomorphic bundle $P$ is trivial. Thus $h_{\\alpha \\beta}$ splits, \n $h_{\\alpha \\beta} = h_{\\alpha} h_{\\beta}^{-1}$ on $U_{\\alpha} \\cap U_{\\beta}$,  where $h_{\\gamma}: U_{\\gamma }\\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$ is \n holomorphic. \n Therefore $g=  g_{\\alpha} h_{\\alpha}$  is well-defined on $\\mathbb D$\n and \\eqref{eq:delbar} holds. This completes the proof.  $\\Box$\n\\begin{corollary}[Existence of a holomorphic potential]\n Let $\\Phi :\\mathbb D\\to \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j} \n (j = 3\\;\\mbox{or}\\;4)$\n be the extended frame for some CMC surface $f$ in $\\mathbb H^3$ with $H \\neq 1$ \n as defined in \\eqref{eq:extendedH>1} or \\eqref{eq:extendedH<1}.\n Then there exist $g: \\mathbb D \\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$\n and a $\\Lambda \\mathfrak{sl}_2 \\mathbb C$-valued $1$-form $\\eta$\n such that\n \\begin{equation}\\label{eq:holopot}\n  \\eta = (\\Phi g)^{-1} \\mathrm{d} (\\Phi g) \n       = \\sum_{j= -1}^{\\infty} \\lambda^j \\eta_j\\;,\n  \\end{equation} \n where the $\\eta_j$ are $\\lambda$-independent diagonal (resp. off-diagonal)\n holomorphic $1$-forms if $j$ is even (resp. odd). In particular, \n the $1$-form $\\lambda \\eta$ is holomorphic in $z \\in \\mathbb D$ and \n $\\lambda \\in \\mathbb C$ and the upper right entry of $\\eta_{-1}$ does not vanish \n on $\\mathbb D$.\n\\end{corollary}\n {\\bf Proof}. Let $g$ be the loop defined in Proposition \\ref{prop:delbar} \n and set $\\eta = (\\Phi g)^{-1} \\mathrm{d} (\\Phi g)$. Then $\\eta$\n defines a $\\Lambda  \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$-valued holomorphic $1$-form\n and can be computed as \n \\begin{equation}\\label{eq:holopot2}\n \\eta = (g^{-1} U g + g^{-1} g_{z}) \\; \\mathrm{d}z\\;,\n \\end{equation}\n where $U$ is defined in \\eqref{LaxH>1} and \\eqref{LaxH<1}, respectively. \n Thus $\\eta$\n has the form  $\\eta = \\sum_{j= -1}^{\\infty} \\lambda^j \\eta_j$.\n Moreover, since $\\eta$ satisfies the twisting condition, $\\eta_j$ is\n diagonal  (resp. off-diagonal) if $j$ is even  (resp. odd). $\\Box$\n \\begin{remark}\n\\mbox{}\n\n{\\rm \n \\begin{enumerate}\n \\item The holomorphic $1$-form $\\eta$ in \\eqref{eq:holopot} \n       will be called  the {\\it holomorphic potential} \n       of the immersion $f$.\n\n \\item Holomorphic potentials are not unique, since the right \n       multiplication of $C$ by some holomorphic loop $C_+ : \\mathbb D \\to\n       \\Lambda^+       \n       {\\rm SL}_2  \\mathbb C_{\\sigma}$ gives  another holomorphic\n       potential, \\textit{i.e.}, $\\tilde \\eta = \\tilde C^{-1} \\mathrm{d} \\tilde C$\n       is a new holomorphic potential, where $\\tilde  C = C C_+$, $C_+$ holomorphic.\n\n \\item The generalization of Grauert's theorem has so far only been \n       proven for non-compact Riemann surfaces. Therefore, at this point \n       we can only infer that  holomorphic potentials \n       exist for all CMC immersions  defined on a non-compact \n       Riemann surface. We would expect that the analogous result holds \n       for a compact Riemann surface. But in this case the potential will \n       probably only be meromorphic.\n \\end{enumerate}\n}\n \\end{remark}\n\n \\subsection{}\n In section \\ref{subsc:holopot}, the holomorphic potential was \n derived from a CMC surface $f$.\n In this section, we give \n the normalized potential for $f$ which is the \n $\\Lambda \\mathfrak{sl}_2 \\mathbb C$-valued meromorphic $1$-form \n determined from the extended frame of $f$. Unlike the holomorphic \n potential, the normalized potential is not holomorphic, \n however, the Fourier expansion of it has only one coefficient.\n \\begin{proposition}[Existence of a normalized potential]\\label{prop:norpot}\n  Let $\\Phi :\\mathbb D\\to \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j}, \n  j = 3, 4$\n  be the extended frame for some CMC surface $f$ in $\\mathbb H^3$ with $H \\neq 1$ \n  as defined in \\eqref{eq:extendedH>1} or \\eqref{eq:extendedH<1}.\n  Then there exist $\\Phi_+: \\mathbb D \\to \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$\n  and a $\\Lambda \\mathfrak{sl}_2 \\mathbb C$-valued $1$-form $\\xi$\n  such that\n  \\begin{equation}\\label{eq:normalizedpot}\n  \\xi = (\\Phi \\Phi_+)^{-1} \\mathrm{d} (\\Phi \\Phi_+) = \\lambda^{-1} \\xi_{-1},\n  \\end{equation}\n   where $\\xi_{-1}$ is an off-diagonal $1$-form which is  meromorphic on\n  $\\mathbb D$. \n \\end{proposition}\n {\\bf Proof}.\n Let us consider the Birkhoff decomposition of $\\Phi$:\n \\begin{equation}\n \\Phi = \\Phi_{-} \\Phi_{+}, \\;\\; \\Phi_{-} \\in \\Lambda_{*}^{-} {\\rm SL}_2\n  \\mathbb C_{\\sigma} \\;\\mbox{and}\\;\\Phi_{+} \\in\\Lambda^{+} {\\rm SL}_2 \\mathbb C_{\\sigma}.\n \\end{equation}\n It can be shown as in \\cite{DPW} that this decomposition holds in $\\mathbb D\n \\setminus S$, where $S$ is a discrete subset of $\\mathbb D$. \n Moreover, this decomposition can be extended meromorphically across $S$.\n Differentiating $\\Phi$ with respect to $\\bar z$, one obtains \n \\begin{equation}\\label{eq:Phiz}\n \\Phi_{-}^{-1} \\Phi_{-, \\bar z} = \\Phi_{+} V \\Phi_{+}^{-1} - \\Phi_{+, \\bar\n  z} \\Phi_{+}^{-1}\\;.\n \\end{equation}\n Clearly, $\\Phi^{-1}_{-} \\Phi_{-, z} \\in\n \\Lambda^{-}\\mathfrak{sl}_2 \\mathbb C_{\\sigma}$, and in view of \n $\\Phi_{-} \\to {\\rm Id}$ as $\\lambda \\to \\infty$, the coefficient matrix \n at $\\lambda^{0}$ vanishes.\n Hence the left side of \\eqref{eq:Phiz} contains only powers \n $\\lambda^{k}$ with $k <0$.\n Moreover, since $V$ only contains $\\lambda^{0}$ and $\\lambda^{1}$, \n the right side of \\eqref{eq:Phiz} contains only powers \n $\\lambda^{k}$, $k \\geq 0$. \n Therefore both sides of \\eqref{eq:Phiz} vanishes.\n Thus $\\Phi_{-, \\bar  z} =0$, \\textit{i.e.}, $\\Phi_{-}$ is holomorphic \n in $z$, where it is non-singular. Set $\\xi = \\Phi_{-}^{-1} \\mathrm{d}\n  \\Phi_{-}$. Differentiating $\\Phi$ with respect to $z$, one obtains\n similar to \\eqref{eq:Phiz}\n \\begin{equation}\n \\xi = (\\Phi_{+} U \\Phi_{+}^{-1} - \\Phi_{+, z} \\Phi_{+}^{-1})\\mathrm{d}z.\n \\end{equation}\n Since $U$ only contains $\\lambda^{-1}$ and $\\lambda^{0}$, \n the right hand side does not have $\\lambda^{j}$ with $j \\leq  -2$. \n Thus $\\xi$ has the form $\\xi = \\lambda^{-1} \\xi_{-1}$,\n and the twisting condition implies that $\\xi_{-1}$ is off-diagonal.\n $\\Box$\n\\begin{remark}\n{\\rm \n  The meromorphic $1$-form $\\xi$ in \\eqref{eq:normalizedpot}\n  will be called  the {\\it normalized potential} of the immersion $f$.\n}\n\\end{remark}\n\\section{Generalized Weierstrass type representation}\\label{sc:DPW}\n\\subsection{}\n In Proposition \\ref{prop:delbar} and \\ref{prop:norpot} we have\n considered objects which are\n holomorphic in $\\lambda \\in \\mathbb C^{\\times}$. For the construction of\n  CMC surfaces we need to obtain frames.\n Therefore we consider the double loop groups $\\mathcal H = \\Lambda\n {\\rm SL}_2 \\mathbb C_\\sigma \\times \\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$.\n Then the subgroups $\\mathcal H_+$ and $\\mathcal\n H_-$ of $\\mathcal H$ are defined as follows:  \n\\begin{align*}\n\\mathcal H_+ = \\Lambda^+ {\\rm SL}_2  \\mathbb C_\\sigma\n \\times \\Lambda^- {\\rm SL}_2 \\mathbb C_\\sigma,\\;\\;\n \\mathcal H_- = \n \\left\\{ (g_1,\\;\\;g_2) \\in \\mathcal H \\;\n \\left| \n \\;\\mbox{$g_1= g_2$}\n \\right. \n \\right\\}.\n\\end{align*}\n We quote Theorem 2.6 in \\cite{DW:CMC-loop}.\n\\begin{theorem}[Generalized Iwasawa decomposition]\\label{thm:generalIwasawa}\n The map $\\mathcal H_- \\times \\mathcal H_+ \\rightarrow \\mathcal H_-\n \\mathcal H_+$ is an analytic diffeomorphism. The image is open and\n dense in $\\mathcal H$. Moreover,\n $$\n \\mathcal H = \\bigcup_{n=0}^{\\infty} \\mathcal H_- w_n \\mathcal H_+\\;\\;,\n $$\n where $ w_n =   \\left({\\rm Id},\\;\\;\\left(\\begin{smallmatrix}\\lambda^{n} & \n 0\\\\ 0 & \\lambda^{-n}\\end{smallmatrix}\\right)\\right) $ if $n=2k$ and \n $  \\left( {\\rm Id},\\;\\; \\left( \\begin{smallmatrix} 0 & \n \\lambda^{n} \\\\ -\\lambda^{-n} & 0 \\end{smallmatrix}\\right)\\right)$ if\n $n=2k+1$.\n\\end{theorem}\n\n \\subsection{}\n We recall that $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_j},\\; j = 3, 4$, \n defined in sections \\ref{sc:CMCH>1} and \\ref{sc:CMCH<1}, \n are real forms of $\\Lambda  {\\rm SL}_2 \\mathbb C_{\\sigma}$, \\cite{Kobayashi}.\n These real forms naturally induce Iwasawa decompositions of \n $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}$:\n \\begin{theorem}[Iwasawa decomposition for $\\tau_3$, \\cite{LoopGroup}]\n \\label{thm:Iwasawa}\n The map \n \\begin{equation}\n \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_3} \\times \\Lambda^+ {\\rm\n  SL}_2 \\mathbb C_{\\sigma}  \\to  \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}\n \\end{equation}\n is an analytic diffeomorphism. \n \\end{theorem}\n  The automorphism $\\tau_3$ in the theorem above is induced from some \n  automorphism of ${\\rm SL}_2 \\mathbb C$. And this is used essentially in \n the proof of this theorem. For the automorphism $\\tau_4$ the situation \n is completely different, since $\\tau_4$ is not induced by some automorphism \n of the (finite dimensional) Lie group ${\\rm SL}_2 \\mathbb C$.\n\\begin{theorem}[Iwasawa decomposition for $\\tau_4$]\\label{thm:Iwasawa2}\n Let $\\omega_0 =  \\left(\\begin{smallmatrix} 0 & \\lambda^{-1} \\\\\n  - \\lambda & 0\\end{smallmatrix}\\right)$. Then the map\n \\begin{equation}\n \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4} \\times\n \\left\\{{\\rm Id}, \\omega_0\\right\\} \\times \\Lambda^+ {\\rm\n  SL}_2 \\mathbb C_{\\sigma}  \\to  \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma},\\;\\;\n (g, \\delta, w_+) \\mapsto g \\delta w_+,\n \\end{equation}\n is an analytic diffeomorphism onto an open dense subset\n of $\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$. The open dense subset \n will be called  the {\\rm Iwasawa core} with two open cells.\n\\end{theorem}\n{\\bf Proof}.\n Consider the map $g \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma} \\mapsto\n (g, \\tau_4 (g))  \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma} \\times \\Lambda {\\rm\n SL}_2 \\mathbb C_{\\sigma}$, where $\\tau_4$ is defined in \\eqref{eq:tau4}.\n Applying the generalized Iwasawa decomposition of \n Theorem \\ref{thm:generalIwasawa}, we have\n \\begin{equation*}\n (g, \\tau_4(g)) = (\\Phi, \\Phi) ({\\rm Id}, W) (V_+,  V_-),\n \\end{equation*}\n Since $\\Phi = g V_+^{-1}$ and $\\Phi = \\tau_4(g) V_{-}^{-1}W^{-1}$, \n we obtain\n \\begin{equation}\\label{eq:splitggsharp}\n g^{-1}\\tau_4(g) =V_+^{-1} W V_-\\;\\;\\mbox{on $S^1$.}\n \\end{equation}\n \n Since \n $g^{-1}\\tau_4(g)$ is in general not positive definite, the middle term \n $W$ is not the identity element for the Birkhoff  decomposition \n \\eqref{eq:splitggsharp} in general.\n Consider the injective real analytic group homomorphism into \n $$\n \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}  \\to \\mathrm{Gr}(H), \n \\;\\;g \\mapsto A_{g} \\cdot H_+,\n $$\n where $H = L^2 (S^1,\\mathbb C)$ with polarization $H = H_+ \\oplus H_-$, \n \\cite[Chap. 2]{DPW}. \n Let $s$ be the non-trivial holomorphic section of the dual of the \n determinant bundle over $\\mathrm{Gr}(H)$. \n We now set $\\ell (g) =\n s (A_{g^{-1} \\tau_4(g)} \\cdot H_{+})$. It is known that \n $\\ell(g) \\neq 0$ if and only if $g^{-1} \\tau_4(g)$ is \n in the big cell, \\textit{i.e.}, $W = {\\rm Id}$, \n \\cite[Corollary 2.5]{DPW}. \n The set  given by \n $\\ell(g) =0$  in $\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$\n can not contain any open subset and is closed, \n since $\\ell (g)$ is  a real analytic function and $\\ell ({\\rm Id}) \\neq 0$.\n Thus the set given by\n $\\ell(g) \\neq 0$ in $\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma$\n is an open dense subset.\n Let's assume $\\ell(g)\\neq 0$ and $\\ell({\\rm Id}) \\neq 0$, \n \\textit{i.e.}, $W ={\\rm Id}$. \n Then taking $\\tau_4$ on both sides of \\eqref{eq:splitggsharp}, we obtain\n $\\tau_4(V_{+}^{-1} V_{-}) = (V_{+}^{-1} V_{-})^{-1}$\n and hence\n \\begin{equation*}\\label{eq:V+V-}\n \\tau_4(V_{+}) = k^{-1} V_{-}\\;\\mbox{and}\\;\\tau_4(V_{-}) = k^{-1}\n  V_{+},\n \\end{equation*}\n where $k$ is a $\\lambda$-independent diagonal matrix with entries\n $k_0, k_0^{-1} \\in \\mathbb R$.  Then from these symmetries, $\\Phi$ \n acquires the symmetry  $\\tau_4(\\Phi) = \\Phi k$.\n If $k_0 >0$, we let $\\tilde k$ be a real matrix such that \n $\\tilde k^2 =k$ holds. Then with \n $\\tilde \\Phi = \\Phi \\tilde k$ and $\\tilde V_+ = \\tilde k^{-1} V_+$, \n we obtain\n \\begin{equation*}\n g = \\tilde \\Phi \\tilde V_+, \\;\\;\\tilde \\Phi \\in \n \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}, \\;\\; \\tilde V_+ \n \\in \\Lambda^+ {\\rm  SL}_2 \\mathbb C_{\\sigma}.\n \\end{equation*}\n If $k_0 < 0$, we set \n  $\\omega_0 =  \\left(\\begin{smallmatrix} 0 & \\lambda^{-1}  \n \\\\  - \\lambda & 0\\end{smallmatrix}\\right)$, and choose\n a real matrix  $\\check k$  such that \n $\\check k^2 = - k$ holds.\n Then with $\\check \\Phi = \\Phi \\check k \\omega_0$ and \n $\\check V_+ = - \\check k^{-1} V_+$, we have\n \\begin{equation*}\n g = \\check \\Phi\\omega_0 \\check V_+, \\;\\;\\check \\Phi \\in \n \\Lambda{\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}, \\;\\; \\check V_+ \n \\in \\Lambda^+ {\\rm  SL}_2 \\mathbb C_{\\sigma}.\n \\end{equation*}\n To show that the sets $ \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4} \\cdot \n \\Lambda^+ {\\rm SL}_2 \\mathbb C_\\sigma $ and\n $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}\n  \\omega_0\\Lambda^+ {\\rm SL}_2 \\mathbb C_\\sigma$ are open in \n $\\Lambda{\\rm SL}_2 \\mathbb C_\\sigma$, it suffices to show \n that \n \\begin{equation}\\label{eq:openset1}\n {\\rm Lie}(\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma)  = \n {\\rm Lie}(\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4})  \\oplus \n {\\rm Lie}(\\Lambda^+ {\\rm SL}_2 \\mathbb C_\\sigma)\n \\end{equation}\n and \n \\begin{equation}\\label{eq:openset2}\n {\\rm Lie}(\\Lambda {\\rm SL}_2 \\mathbb C_\\sigma)  = \n \\omega_0^{-1} {\\rm Lie}(\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4})\\omega_0 \\oplus \n {\\rm Lie}(\\Lambda^+ {\\rm SL}_2 \\mathbb C_\\sigma),\n \\end{equation}\n where ${\\rm Lie} (G)$ denotes the Lie algebra of \n $G$ \\cite[Chap. II, Lemma 2.4]{Hel:Diff}. \n But these two equations can be proven by a straightforward computation.\n This completes the proof. $\\Box$\n\\begin{remark}\n{\\rm \n The Iwasawa decomposition for $\\tau_4$ in Theorem \\ref{thm:Iwasawa2} can \n be rephrased as follows: Let $\\hat \\tau$ be an extension of $\\tau$ \n in \\eqref{6.1} to $G^{\\mathbb C} = {\\rm SL}_2 \\mathbb C \\times  {\\rm SL}_2 \\mathbb C$ \n in section \\ref{sc:4symmetric} and \n\\begin{align*}\n \\Lambda G^{\\mathbb C}&= \n \\Lambda {\\rm SL}_2 \\mathbb C \\times \\Lambda {\\rm SL}_2 \\mathbb C, \\\\\n \\Lambda G^{\\mathbb C}_{\\hat \\tau} &= \n \\{(g, h) \\in  \\Lambda G^{\\mathbb C} \\;|\\; \n \\hat{\\tau}(g(\\lambda),h(\\lambda))=(g(-i \\lambda),h(-i \\lambda))\\},  \\\\\n \\Lambda G_{\\hat \\tau} &= \n \\left\\{(g, h) \\in  \\Lambda G^{\\mathbb C}_{\\hat \\tau} \\;|\\; \n h(\\lambda) = \\overline{ g(1\/\\bar \\lambda)} \\right\\}, \\\\\n \\Lambda^+ G^{\\mathbb C}_{\\hat \\tau}  &= \n \\{(g, h) \\in  \\Lambda G^{\\mathbb C}_{\\hat \\tau} \\;|\\;  \n g \\in  \\Lambda^+ {\\rm SL}_2 \\mathbb C,\\;\\; h \\in  \\Lambda^- {\\rm SL}_2 \\mathbb C \\}.\n\\end{align*}\n Since $\\hat \\tau^2$ is the involution $\\hat \\sigma = (\\sigma, \\sigma)$ \n with $\\sigma$ defined in \\eqref{eq:sigma}, \n we have \n $$\n \\Lambda G^{\\mathbb C}_{\\hat \\tau} \\subset \\Lambda G^{\\mathbb C}_{\\hat \\sigma} = \n \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma} \\times \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}.\n $$\n Then the Iwasawa decomposition Theorem \\ref{thm:Iwasawa2}\n can be rephrased as follows:\n The multiplication map \n \\begin{equation}\n \\Lambda G_{\\hat \\tau} \\times\n \\left\\{({\\rm Id}, {\\rm Id}), (\\omega_0, \\omega_0)\\right\\} \\times \n \\Lambda^+ G^{\\mathbb C}_{\\hat \\tau} \n \\to   \\Lambda G^{\\mathbb C}_{\\hat \\tau}, \\;\\; \n (g, \\delta, w_+) \\mapsto g \\delta w_+\n \\end{equation}\n is an analytic diffeomorphism onto an open dense subset of \n $\\Lambda G^{\\mathbb C}_{\\hat \\tau}$. \n}\n\\end{remark}\n\n \\subsection{}\n In section \\ref{sc:Potential}, we discussed potentials associated with CMC\n surfaces  in $\\mathbb H^3$ with $H \\neq 1$. In this section,\n we give  conversely a  construction of CMC surfaces from potentials, the\n  {\\it Generalized Weierstrass type representation}:\n\n{\\bf The case $H>1$:}\n\n\\noindent{\\bf Step 1:} Take a holomorphic potential $\\eta$ as defined \n in \\eqref{eq:holopot}. \n\n\\noindent{\\bf Step 2:} Solve the ordinary differential equation \n $\\mathrm{d} C = C \\eta$ with $C(z_*, \\lambda) = {\\rm Id}$, where $z_*$ \n is some base point.\n                      \n\\noindent{\\bf Step 3:} \n Perform the Iwasawa decomposition for $C = \\Phi V_{+}$ by \n Theorem \\ref{thm:Iwasawa}, where \n $\\Phi \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_3}$ \n and $V_{+} \\in \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$.\n\n\\begin{theorem}\\label{thm:ExtFrame}\n Up to a diagonal gauge $D \\in {\\mathrm U}_1$\n and a change of coordinates, \n the matrix  $\\Phi$ obtained in\n Step 3 is  the extended frame of some CMC\n surface with $H>1$.\n\\end{theorem}\n\n\\noindent{\\bf Step 4:} Inserting $\\Phi$ obtained in Step 3\n into the Sym formula in  \\eqref{eq:SymH>1} we obtain a CMC surface \n in $\\mathbb H^3$ with $H = \\coth q >1$.\n\n {\\bf Proof of Theorem \\ref{thm:ExtFrame}}.\n  Let us compute the Maurer-Cartan form of $\\Phi = C V_{+}^{-1}$, where\n $\\Phi \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_3}$. \n A direct computation shows that \n \\begin{equation}\n \\alpha = \\Phi^{-1} \\mathrm{d} \\Phi = V_{+} C^{-1} \\mathrm{d} \n C V_{+}^{-1} - \\mathrm{d} V_{+} V_{+}^{-1}.\n \\end{equation}\n Thus $\\alpha$ has the form $\\alpha = \\sum_{j = -1} ^{\\infty}\n \\lambda^{j} \\alpha_{j}$. Since $\\Phi$ is an element in\n $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_3}$, $\\alpha$ satisfies \n that $\\tau_3 (\\alpha) = \\alpha$, where $\\tau_3$ \n is defined in \\eqref{eq:invc3}. \n Therefore \n \\begin{equation}\n \\alpha = \\lambda^{-1} \\alpha_{-1} + \\alpha_0 + \\lambda \\alpha_1\\;,\n \\end{equation}\n where $\\overline{\\alpha_0} = - \\alpha_0$ and $\\alpha_{-1} =\n {\\alpha^*}_1$.\n It is easy to check that $\\alpha$ has the form \\eqref{LaxH>1}\n up to a diagonal gauge \n $D =\\operatorname{diag} (e^{i \\theta}, e^{-i \\theta})$ and a change of coordinates\n \\cite[Section A.8]{DH:Mero}. \n Since $D \\in {\\mathrm U}_{1}$, $C = \\Phi D \\cdot D^{-1} V_+$ is\n also an Iwasawa decomposition. Thus, by Proposition\n \\ref{prop:fundamentalH>1}, $\\Phi D$ is\n the extended frame of some CMC surface  with $H>1$. \n This completes the proof. $\\Box$\n\n{\\bf The case $0 \\leq H<1$:}\n\n\\noindent{\\bf Step 1:} Take a holomorphic potential $\\eta$ as defined \n in \\eqref{eq:holopot}. \n\n\\noindent{\\bf Step 2:} Solve the ordinary differential equation \n $\\mathrm{d} C = C \\eta$ with $C(z_*, \\lambda) = {\\rm Id}$, where $z_*$ \n is some base point.\n                      \n\\noindent{\\bf Step 3:} \n Perform the Iwasawa decomposition for $C$, \\textit{i.e.}, \n $C = \\Phi V_{+}$ or \n $C = \\Phi \\omega_0 V_{+}$\n for all $z$ in $\\mathbb D$ such that $C(z, \\lambda)$\n is in $\\mathbb D_1 \\subset \\mathbb D$ \n or $\\mathbb D_2 \\subset \\mathbb D$ respectively, where \n $\\Phi \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$, \n $\\omega_0 = \\left(\\begin{smallmatrix} 0 & \\lambda^{-1} \n \\\\ -\\lambda & 0\\end{smallmatrix}\\right)$, \n $V_{+} \\in \\Lambda^+ {\\rm SL}_2 \\mathbb C_{\\sigma}$, and\n $\\mathbb D_1$ and $\\mathbb D_2$ are open subsets in $\\mathbb D$ \n such that the real valued functions $k_0$ defined in \n Theorem \\ref{thm:Iwasawa2} are positive and negative on \n $z \\in \\mathbb D_1$ and $z \\in \\mathbb D_2$ respectively.\n\n\\begin{theorem}\\label{thm:ExtFrame2}\n Up to a diagonal gauge $D \\in {\\mathrm U}_1$ and a change of coordinates, \n the matrices $\\Phi$ and $\\Phi \\omega_0$ obtained in\n Step 3 are the extended frames of some CMC\n surface with $0 \\leq H <1$. \n\\end{theorem}\n\n\\noindent{\\bf Step 4:} Inserting $\\Phi$ or $\\Phi\\omega_0$ \n obtained in Step 3 into the Sym formula in \\eqref{eq:SymH<1}, \n we obtain a CMC surface in $\\mathbb H^3$ with $0\\leq H <1$.\n\n{\\bf Proof of Theorem \\ref{thm:ExtFrame2}}.\n Let us compute the Maurer-Cartan form of $\\Phi$ if \n  $C = \\Phi V_+ $ and for $\\Phi\\omega_0$ if $C = \\Phi \\omega_0 V_+$ with \n $\\Phi  \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$ and $V_+ \n \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$. \n Since $\\Phi$ is an element in $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$, \n $\\alpha$ satisfies that $\\tau_4(\\alpha) = \\alpha$, where \n $\\tau_4$ is defined in \\eqref{eq:invc4}.\n Similar to the proof of Theorem \\ref{thm:ExtFrame},  \n the Maurer-Cartan form of $\\Phi$ and $\\Phi \\omega_0$ respectively\n has the form\n \\begin{equation}\n \\alpha = \\lambda^{-1} \\alpha_{-1} + \\alpha_0 + \\lambda \\alpha_1\\;,\n \\end{equation}\n where $\\overline{\\alpha_0} = - \\alpha_0$ and $\\alpha_{-1} =\n i {\\rm Ad}(\\mathscr{R})(\\alpha_1)^*$.\n It is easy to check that \n up to a diagonal gauge \n $D =\\operatorname{diag} (e^{i \\theta}, e^{-i \\theta})$ \n and a change of coordinates \\cite[Section A.8]{DH:Mero},\n $\\alpha$ has the form \\eqref{LaxH<1}.\n Since $D \\in {\\mathrm U}_{1}$ and $D \\omega_0 D = \\omega_0$,\n $C = \\Phi D \\cdot D^{-1} V_+$ and \n $C = \\Phi \\omega_0 D \\cdot D^{-1} V_+ = \n \\Phi D^{-1} \\omega_0 \\cdot D^{-1} V_+$ are \n also Iwasawa decompositions. Thus, by Proposition\n \\ref{prop:fundamentalH<1}, $\\Phi D$ and $\\Phi D^{-1} \\omega_0 $ are\n the extended frame of some CMC surface  with $0\\leq  H<1$. \n$\\Box$\n\n\\subsection{}\n Let $\\eta$ be a potential, defined on $\\mathbb D$, for a CMC immersion\n into $\\mathbb H^3$ with mean curvature $0\\leq H <1$. Let $C$ be defined \n by $\\mathrm{d} C = C\\eta$ and $C(z_*, \\lambda) ={\\rm Id}$, \n where $z_* \\in \\mathbb D$\n is some base point. In the proof of Theorem \\ref{thm:Iwasawa2}, \n we have seen that \n there exists a real analytic function $k_0 : \\mathbb D \\to \\mathbb R$\n such that the ``singular set'' $S_0 =\\{ z \\in \\mathbb D \\;\\vert\\; \n k_0 (z, \\bar z) =0\\}$ divides $\\mathbb D$ into two open subsets \n $\\mathbb D_1$ and $\\mathbb D_2$, where $C(z)$ is in the open Iwasawa cell \n containing ${\\rm Id}$ for $z \\in \\mathbb D_1$ and in the other open Iwasawa \n cell for $z \\in \\mathbb D_2$. Let $C  = \\Phi V_+$ be an Iwasawa decomposition \n on the open Iwasawa cell containing ${\\rm Id}$ for $z \\in \\mathbb D_1$. \n Examples like the one given in \n section \\ref{subsec:sphere} show that \n the frame $\\Phi$ associated with $C$ will be generically singular along $S_0$.\n Hence it does not seem to make sense to extend the immersion $f$ associated \n with $\\eta$ in $\\mathbb D_1$ across $S_0$ to $\\mathbb D_2$. \n However, in view of \n \\cite[Theorem 3.2]{DK:cyl},  one can extend $\\Phi l $ meromorphically to \n $\\mathbb D \\times \\bar{\\mathbb D}$,\n if $l$ is an appropriately chosen diagonal matrix independent of $\\lambda$.\n Actually, with $\\tilde k$ as in the proof of Theorem \\ref{thm:Iwasawa2} \n the matrices, \n $\\hat \\Phi = \\Phi \\tilde k^{-1}$ and $\\tilde k^2$ have meromorphic \n extension to $(z, w) \\in \\mathbb D \\times \\bar{\\mathbb D}$. As a consequence, \n consider the CMC immersion of mean curvature $H = \\tanh q$ given by the \n Sym formula $f = \\Phi D_0 \\Phi^* $ for $z\\in \\mathbb D_1$, where \n $D_0 = \\operatorname{diag} (e^{-q\/2}, e^{q\/2})$.\n Then we rephrase the Sym formula as  \n\\begin{eqnarray*}\n f &=& \\Phi D_0 \\Phi^*=(\\Phi \\tilde k^{-1}) \\tilde k^{2}D_0 \n       (\\Phi \\tilde k^{-1})^*.\n\\end{eqnarray*}\n Then by the argument above, $\\Phi \\tilde k^{-1}$ and  $\\tilde k^2$ \n can be extended meromorphically  to \n $(z, w) \\in  \\mathbb D \\times \\bar{\\mathbb D}$.\n Let's put $\\hat \\Phi = \\Phi \\tilde k^{-1}$  and $k=\\tilde k^2$. \n Then $\\hat f = \\hat \\Phi k D_0 \\hat \\Phi^*$\n has a meromorphic extension to $(z, w) \\in \\mathbb D \\times \\bar {\\mathbb D}$.\n We note that, on $w = \\bar z$, \n $\\tau_4(\\hat \\Phi) = \\hat \\Phi k$ and $k$ has real diagonal entries,\n where $\\tau_4$ is defined in \\eqref{eq:tau4}.\n  We can now consider the \n analytic continuation from the one open Iwasawa cell for\n $z \\in \\mathbb D_1$, to  the other open Iwasawa cell for $z \\in \\mathbb D_2$, \n \\textit{i.e.}, \n $C = \\tilde \\Phi \\omega_0 \\tilde V_+$ is the Iwasawa \n decomposition for $z \\in \\mathbb D_2$.\n Then, it is easy to check that\n the entries of $k$ are negative on $z \\in \\mathbb D_2$. \n Thus we can rephrase $k$ as $k = - \\check k^2$, \n where $\\check k$ is the diagonal matrix with positive entries \n which are independent of $\\lambda$ defined as in Theorem \\ref{thm:Iwasawa2}. \n Moreover\n $C =(\\hat \\Phi \\check k \\omega_0) \\omega_0 (- \\check k^{-1} \\hat V_+)$\n is the Iwasawa decomposition for $z \\in \\mathbb D_2$,  \n where $\\hat V_+$ is the meromorphic extension\n  of $\\tilde k V_+$. Using an obvious abbreviation\n $\\tilde \\Phi =\\hat \\Phi \\check k \\omega_0$, \n we can rephrase again the Sym formula as  \n\\begin{eqnarray*}\n \\hat f &=& \\tilde \\Phi  \\omega_0 \n \\check k^{-2}  k D_0 \\omega_0^*  \\tilde \\Phi^*= -(\\tilde \\Phi \\omega_0) D_0 \n     (\\tilde \\Phi\\omega_0)^*.\n\\end{eqnarray*}\n Therefore, it is natural to use for $z$ in $\\mathbb D_2$ \n the negative of the Sym formula \\eqref{eq:SymH<1} and to \n use this formula for $\\tilde\\Phi \\omega_0$.\n Thus in the second open Iwasawa cell actually $\\tilde\\Phi \\omega_0$ \n is the ``frame'' to use.\n\n\\subsection{}\n We have seen in section \\ref{sc:CMCH<1} that coefficient matrices of\n the form \\eqref{LaxH<1} define CMC surfaces in $\\mathbb H^3$ with mean\n curvature $0 \\leq H < 1$.\n\n In section \\ref{sc:DPW} we have shown that all such matrices can be\n obtained via a generalized Weierstrass type representation procedure from\n ``potentials''. Thus we have a loop group procedure to construct all CMC\n surfaces in $\\mathbb H^3$ with mean curvature $0 \\leq H < 1$.\n\n On the other hand, a simple minded approach would follow \\cite{DPW} more\n closely, since we know (Ishihara, Proposition \\ref{prop:Ishihara}) that\n the surfaces under consideration are characterized by the fact that\n their ``Gauss map'' \\eqref{eq:mapsinUH3} is a harmonic map into\n $\\mathrm{U}\\mathbb{H}^3 = \\mathrm{SL}_2 \\mathbb C\/\\mathrm{U}_1$. Since\n $\\mathrm{U}\\mathbb{H}^3$ is not a symmetric space, only a $4$-symmetric\n space, \\cite{DPW} or \\cite{BD1} cannot be applied directly. \n\n In section\n \\ref{sc:4symUnittangent} we have shown that\n one can indeed introduce a loop parameter into the Maurer-Cartan form\n $\\alpha$ as in \\cite[section 3.2]{BP},\n since the Gauss map is Legendre:\n$$\n \\alpha = \\alpha_{\\mathfrak h} + \\nu^{-1} \\alpha^{\\prime}_{\\mathfrak p}\n + \\nu \\alpha^{\\prime \\prime}_{\\mathfrak p}, \\;\\; \\nu \\in \\mathbb C^{\\times},\n$$\n where $\\alpha_{\\mathfrak h}$, $\\alpha_{\\mathfrak p}$\n are defined in \\eqref{eq:alphaLegendre}. \n Integration of $\\alpha$ yields an extended frame $\\Phi = \\Phi(z,\\bar z,\\nu)$.\n\n There is no loop group method known for general harmonic maps into\n $k$-symmetric spaces if $k>2$. However, since we were able to introduce\n a loop parameter one could decompose $\\Phi$ anyway, say \\'a la Birkhoff,\n see Theorem \\ref{thm:Birkhoff}:\n$$\n\\Phi = \\Phi_{1-} \\Phi_{2+} = \\Phi_{1+} \\Phi_{2-}, \\;\\;\\Phi_{1\\pm} \\in\n\\Lambda^{\\pm}_{*} \\mathrm{SL}_2 \\mathbb C_{\\sigma}, \\;\\Phi_{2\\pm} \\in\n\\Lambda^{\\pm} \\mathrm{SL}_2 \\mathbb C_{\\sigma}.\n$$\n This would yield a potential $\\xi = (\\xi_-, \\xi_+) = (\\Phi_{1-}^{-1}\n \\mathrm{d} \\Phi_{1-}, \\Phi_{1+}^{-1}\\mathrm{d} \\Phi_{1+})$.\n\n A priori, the only information we have about $\\xi_\\pm$ is that they are \n elements in $\\Lambda^{\\pm} \\mathfrak{sl}_2 \\mathbb C_{\\sigma}$ with\n simple loop behavior. \n \n If one would want to construct, conversely, extended frames from\n potentials of this type, one would run irreparably into the problem of\n violating the admissibility condition \\eqref{admissible}.\n The observation that $\\alpha$ is fixed under $\\tau_4$, however, permits to\n fix this problem. Since $\\tau_4 (\\Phi_{1-}) = \\Phi_{1+}$, we have \n$$\n\\tau_4(\\xi_-)=\\xi_+\\;.\n$$\n Similar to \\cite{Kobayashi}, one can now show that for such potentials\n one can obtain an extended frame which is fixed under $\\tau_4$.\n \n It is not difficult to see that following \\cite{Kobayashi}\n corresponds exactly to the approach (using an Iwasawa decomposition)\n explained in section \\ref{sc:CMCH<1} and section \\ref{sc:DPW}.\n\n\\subsection{} \\label{subsec:finitetype}\n Among the classes of surfaces that can be constructed by integrable \n system methods the surfaces of ``finite type'' have received special\n attention. This is due to the fact that these surfaces can be constructed\n fairly explicitly. Moreover, the algebro-geometric methods used for these \n surfaces are classical and beautiful. These methods have been highly \n successful for the construction of tori. On the other hand, since until\n now only surfaces without umbilical points can be constructed by \n algebro-geometric methods, it is necessary to develop more general\n methods which also work when umbilical points are present.\n\n In the context of this paper the property of being of finite type \n can be expressed on all levels of our construction scheme. \n We sketch  these constructions, but will not present any proofs.\n\n The starting point for the construction of a CMC surface in $\\mathbb H^3$\n with $0 \\leq H <1$ in this paper are ``potentials''. In the spirit\n of \\cite{BP} and \\cite{DoSter}, we call a potential $\\eta$ to be of\n ``finite type'' if there exists a Laurent polynomial \n ${\\xi}^o(\\lambda)$\n satisfying $\\tau_4 \\xi^o = \\xi^o$ such that \n $\\eta = \\lambda^{d-1} \\xi^o(\\lambda)\n \\mathrm{d}z$, where $d$ is odd and the maximal degree of $\\lambda$ \n occurring in $\\xi^o$. In our construction scheme one obtains from \n $\\eta$ a CMC surface in $\\mathbb H^3$ with $0 \\leq H<1$. \n Surfaces constructed from some  $\\xi^o$ as above are usually \n called to be of {\\it Symes finite type} in the sense of \\cite{DoSter}. \n\n Our construction scheme produces from $\\eta$ an ODE solution \n $\\mathrm{d} C = C \\eta$\n satisfying $C(z_*, \\lambda) = {\\rm Id}$ and from $C$ via Iwasawa decomposition \n $C = \\Phi V_+$, $\\Phi(z_*, \\bar z_*, \\lambda) ={\\rm Id}$, \n the frame $\\Phi$ of the CMC  surface constructed from $\\eta$. \n As in \\cite{BP}  (see also \\cite{DoSter}) one can show \n the following theorem:\n\\begin{theorem}\n The Maurer-Cartan form $\\alpha = \\Phi^{-1} \\mathrm{d} \\Phi$ can be represented \n in the form \n\\begin{equation}\n \\alpha = \\alpha_{\\xi} = \n (\\lambda^{-1}\\xi_{-d}+\\frac{1}{2} \\xi_{-d+1})\\mathrm{d}z +\n \\tau_4 (\\lambda^{-1}\\xi_{-d}+\\frac{1}{2} \\xi_{-d+1})\\mathrm{d}\\bar z, \n\\end{equation}\n where $\\xi = \\sum_{n \\in \\mathbb Z} \\xi_n$ satisfies $\\tau_4 \\xi = \\xi$ \n and the Lax equation \n \\begin{equation}\\label{eq:phixi}\n d \\xi =[\\xi, \\alpha_{\\xi}], \\;\\;\\xi(z_*) = \\xi^o.\n \\end{equation}\n Moreover the $\\xi$ is polynomial in \n $\\lambda$ and satisfies\n\\begin{equation}\n \\xi = \\Phi^{-1} \\xi^o \\Phi.\n\\end{equation}\n Such a  $\\xi$ is called the \n {\\rm polynomial Killing field}.\n\\end{theorem}\n\\begin{remark}\n{\\rm \n In view of $\\Phi = C V_+^{-1}$ one obtains that the degree of $\\xi$ \n is equal to the degree of $\\xi^o$.\n Again following \\cite{BP} (see also \\cite{DoSter}) one calls a frame $\\Phi$ \n to be of {\\it finite type} if there exists some $\\xi^o$ such that the \n solution to \\eqref{eq:phixi} satisfies $\\Phi^{-1} \\mathrm{d} \\Phi = \\alpha_{\\xi}$. \n Moreover, a CMC surface in $\\mathbb H^3$ with $0 \\leq H <1$ is \n called of {\\it finite type} if its frame is of finite type. With \n this notation the theorem above states that finite type surfaces \n are equivalent to Symes finite type surfaces. \n}\n\\end{remark}\n There are basically two ways to treat finite type surfaces \n algebro-geometrically. One way is to consider the spectral curve \n $Y =\\{ (\\lambda, \\mu) \\in \\mathbb C^{\\times} \\times \\mathbb C \\;\\vert \\;\n \\mathrm{det}(\\mu \\cdot {\\rm Id}- \\xi(z, \\lambda)) =0\\}$ and the eigenline \n bundle $\\mathscr L$ of $\\xi$ over $Y$. \n This approach was carried out by I.~McIntosh for \n finite type harmonic maps into $\\mathbb C P^n$ \\cite{Mc:CPn} \n and it seems that his method should also apply to our case. \n The second approach is due to A.~Bobenko, it produces algebro-geometric\n solutions to the complex sine-Gordon equation from line bundles over \n hyperelliptic curves via the corresponding theta functions. For the \n surfaces considered in this paper this was carried out in \\cite{BaBo}. \n \n In the case of CMC surfaces in $\\mathbb R^3$, the relation between \n the loop group method as used in this paper and the approach of Bobenko\n was established in \\cite{DoH}. It would be interesting to see whether also for \n CMC surfaces of finite type in $\\mathbb H^3$ a relation with Bobenko's \n approach can be realized in a similar fashion.\n\n\\section{\\bf Examples of CMC immersions with $0 \\leq H <1$ in  $\\mathbb H^3$}\n \\label{sc:Ex}\n In \\cite{KKRS}, examples  of {\\sc CMC}-immersions with $H>1$ have\n been constructed from  holomorphic potentials. In this section, we\n present some examples of {\\sc\n CMC}-immersions with $0 \\leq H <1$ starting from holomorphic potentials.\n From now on we assume always $q \\in \\mathbb R_{\\geq 0}$ and use the symbol\n $\\lambda$ instead of $\\nu$. We recall that the resulting immersion by the\n Sym formula $f_{\\lambda = e^{-q\/2}}$ in \\eqref{eq:SymH<1} has\n the mean curvature $0 \\leq H = \\tanh q<1$. \n \n All the examples given below have the feature that there exist some curves \n $\\mathcal C$ along which are immersions which tend to infinity. \n Each of the connected components of the immersion from $M \\setminus \\mathcal C$\n is well defined  and takes values in the forward $\\mathbb H^3$ or \n the backward  $\\mathbb H^3$ (in the light cone picture of hyperbolic space). \n We hope to study this behavior in detail in a separate publication.\n\n\n\\subsection{Umbilical surfaces (Figure \\ref{fig:one})} \\label{subsec:sphere}\n Assume an CMC immersion $f$ with $0 \\leq H<1$ to be totally umbilical.\n It is easy to see that the Hopf differential $Q \\mathrm{d}z^2$ of an\n umbilic CMC immersion with $0 \\leq H < 1$ vanishes, \\textit{i.e.}, $Q =0$. \n Then the corresponding normalized potential $\\xi$ as in\n \\eqref{eq:normalizedpot} has the form \n $$\n \\xi = \\lambda^{-1} \\begin{pmatrix}0 & h(z) \\\\ 0 & 0 \\end{pmatrix} \\mathrm{d}z,\n $$\n where $h(z)$ is a non-vanishing meromorphic\n function. Using the coordinate change\n  $w = \\int h \\mathrm{d}z$, $\\xi$ has the form defined in\n \\begin{equation}\\label{eq:umbiliceta}\n \\eta =\n \\begin{pmatrix}\n 0 & \\lambda^{-1} \\\\\n 0 & 0\n \\end{pmatrix} \\mathrm{d}z. \n \\end{equation}\n Let us compute totally umbilic surfaces explicitly from holomorphic\n potentials $\\eta$  defined in \\eqref{eq:umbiliceta}. \n A solution $C$ to the holomorphic differential equation \n $\\mathrm{d} C = C \\eta$ with the above $\\eta$ \n and the Iwasawa decomposition for $C = \\Phi V_+$\n can be computed explicitly as follows:\n \\begin{align*}\n C =\n \\begin{pmatrix} 1 & \\lambda^{-1} z \\\\ 0 &1 \\end{pmatrix}\n   = \\Phi V_{+} = \n  \\begin{pmatrix}\n \\frac{1}{\\sqrt{1-|z|^2}}  & \\frac{\\lambda^{-1} z}{\\sqrt{1-|z|^2}} \\\\\n  \\frac{\\lambda \\bar z }{\\sqrt{1-|z|^2}} & \\frac{1}{\\sqrt{1-|z|^2}}\n \\end{pmatrix}\n \\begin{pmatrix}\n  \\frac{1}{\\sqrt{1-|z|^2}}   & 0 \\\\\n \\frac{-\\lambda \\bar z}{\\sqrt{1-|z|^2}} & \\sqrt{1-|z|^2}\n \\end{pmatrix}\\;. \n\\end{align*}\n The Sym formula \\eqref{eq:SymH<1} for the above $\\Phi$ with $\\lambda =\n e^{-q\/2}$ gives  \n\\begin{equation*}\nf_{\\lambda} = \\frac{1}{1-|z|^2}\\begin{pmatrix}\ne^{-q\/2} + e^{3q\/2 }|z|^2 & (e^{-q} + e^{q}) z\\\\ (e^{-q}+e^{q}) \\bar z&\ne^{q\/2}+ e^{-3q\/2 } |z|^2\\end{pmatrix}\n\\;.\n\\end{equation*}\n Thus the resulting immersion is a totally umbilic surface, which is \n called an {\\it equidistance surface} for $0 < H <1$ and a {\\it\n totally geodesic surface} for \n $H=0$  \\cite[Theorem 29, p.~77]{Spivak}.\n\n\\begin{figure}[h]\n\\begin{center}\n \\includegraphics[width=0.4\\textwidth]{sphere}\\hspace{2cm}\n \\includegraphics[width=0.4\\textwidth]{sphere-out}\n \\caption{\n An equidistance surface with $H = \\tanh (1)$, $|z|<1$ (left) and \n an equidistance surface with $H = \\tanh (1)$, $|z|>1$  (right).\n Surfaces are shown in the Poincar{\\'e} ball model and the outside \n of the Poincar{\\'e} ball model respectively.\n }\\label{fig:one}\n\\end{center}\n\\end{figure}\n\n \\subsection{Surfaces of revolution (Figure \\ref{fig:two})} \\label{subsec:Equivariant}\n We set\n %\n \\begin{equation}\\label{eq:Delpotential}\n \\eta = A \\mathrm{d}z =\n  \\begin{pmatrix}\n   0 & \\lambda^{-1} a + \\lambda b \\\\\n  \\lambda^{-1} b - \\lambda a& 0\n \\end{pmatrix} \\mathrm{d}z\\;, \n\\end{equation}\n where $a, b \\in \\mathbb R$ and $b^2 -a ^2 + a b(e^{q} - e^{-q}) =1\/4$.\n A solution to the holomorphic differential equation $\\mathrm{d} C = C \\eta$ with \n the above potential $\\eta$ is \n\\begin{equation}\\label{eq:Delsol}\n C = \\exp \\left(A z \\right)\\;.\n\\end{equation}\n Let $\\gamma : z \\mapsto z + p$ be an automorphism of $\\mathbb C$ with $p  \\in \\mathbb C^{\\times}$. \n Noting that $\\tau_4 A = - A$, it is easy to check that \n $\\gamma^* C = C (p, \\lambda) C$ and $C(p, \\lambda) \\in\n  \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$ for all \n $p \\in i \\mathbb R$ and $|\\lambda| = e^{-q\/2}$. \n  \n Let $p$ be purely imaginary and let $C = \\Phi  V_+$ denote the\n Iwasawa decomposition of $C$. \n Then the Iwasawa decomposition for $\\gamma^* C =  \\hat \\Phi \\hat V_+$\n can be  computed as $\\hat \\Phi= C (p, \\lambda) \\Phi$ and  $\\hat V_+ = V_+$. \n Thus $C_{q}(p) = C(p, \\lambda=e^{-q\/2})$ acts on the resulting\n immersion $f_{\\lambda}$  at $\\lambda = e^{-q\/2}$, which is denoted by\n $f_q$, as a $1$-parameter group of isometries with group\n parameter $p \\in i  \\mathbb R$:\n \\begin{equation}\\label{eq:oneparameter}\n  \\gamma^* f_{q} = C_{q}(p) f_{q} C_{q}(p)^*.\n \\end{equation}\n Moreover, $C_q(p)$ has the eigenvalues\n $$\n \\exp (\\pm p \\sqrt{b^2 -a^2 + a b(\\lambda^{-2} - \\lambda^2)})|_{\\lambda\n =  e^{-q\/2}} = \\exp (\\pm  p\/2).\n $$ \n Since  $C_q(p) = \\pm {\\rm  Id}$ for $p = 2k \\pi i$ with \n $k \\in \\mathbb Z$, the resulting immersion closes up for $p= 2k \\pi i$, \n \\textit{i.e.}, $\\gamma^* f_{q} = f_{q}$. \n The resulting immersion defines a surface\n of revolution with respect to the $1$-parameter group\n \\eqref{eq:oneparameter}. As its profile curve one can choose:\n  \\begin{equation*}\n  g (x) = \\Phi \\begin{pmatrix} e^{-q\/2} & 0\\\\ 0& e^{q\/2}\\end{pmatrix}\n          {\\Phi}^* |_{z = x, \\lambda = e^{-q\/2}},\n  \\end{equation*}\n  where $x \\in \\mathbb R$. For the axis of the surface of revolution we obtain\n \\begin{equation*}\n  \\ell =\\{ X \\in {\\rm Her}_2 \\mathbb C\\;|\\; C_q(p)X -\n   XC_q(p)^{*-1} = 0, \\; p \\in i\\mathbb R\\} \\cap \\mathbb H^3.\n \\end{equation*}\n Even though the axis $\\ell$ could be an empty set, it is natural to call\n the resulting immersion a surface of revolution \\cite{Mori}.\n\n\\begin{figure}[h]\n\\begin{center}\n \\includegraphics[width=0.4\\textwidth]{Del-a-03-q-03}\\hspace{2cm}\n \\includegraphics[width=0.4\\textwidth]{Del-a-02-q-0}\n \\caption{\n A portion of a surface of revolution with $H = \\tanh (0.3)$ \n and $a =0.3$ (left) and \n a portion of a minimal surface of revolution with $a =0.2$ (right).\n Surfaces are shown in the Poincar{\\'e} ball model and the outside \n of the Poincar{\\'e} ball model.\n }\\label{fig:two}\n\\end{center}\n\\end{figure}\n\n\\subsection{Radially symmetric surfaces}\n\\label{subsec:radiall}\n We set \n\\begin{equation*}\n \\eta = \\begin{pmatrix} 0 &  \\lambda^{-1}\\\\\n        z^k \\lambda^{-1} & 0\\end{pmatrix} \\mathrm{d}z\\;,\n\\end{equation*}\n where $k \\in \\mathbb N$. Set $\\epsilon : (z, \\lambda) \\mapsto (a z,\n a^{\\frac{k+2}{2}}\\lambda)$. Then $\\epsilon^* \\eta  = T \\eta T^{-1}$, \n where $T= \\operatorname{diag} (a^{\\frac{-k}{4}}, a^{\\frac{k}{4}})$. Moreover,\n we assume that $a \\in S^1$. Let $C$ denote the solution to $\\mathrm{d} C = C\n \\eta$, $C(0, \\lambda) ={\\rm Id}$. Let $C = \\Phi V_+$ be the Iwasawa\n decomposition of $C$ and assume that $\\Phi$ is in the first open Iwasawa\n cell. Then the Iwasawa decomposition for $\\epsilon^* C$ is \n $(\\epsilon^* \\Phi) (\\epsilon^* V_+)$, thus $\\epsilon^* \\Phi$ is \n again in the first open Iwasawa cell. Moreover $\\epsilon^* \\Phi$ and \n $\\epsilon^* V_+$ can be computed\n as $\\epsilon^* \\Phi = T \\Phi T^{-1}$ and\n $\\epsilon^* V_+ = T V_+ T^{-1}$.  Since $V_+(\\lambda =0)$ is diagonal,\n the last relation shows that $V_{+}(\\lambda =0)$ only depends on $|z|$.\n In view of $C = \\Phi V_+$, comparison of the potential considered in\n this section with \\eqref{LaxH<1} shows that the metric only depends on\n $|z|$. The resulting surface is a radially symmetric surface. The minimal \n surface of this type has been investigated in \\cite{Novokshenov}.\n\n\\subsection{Cylinders}\\label{subsec:cylinders}\n We set \n\\begin{equation*}\n \\eta = S \\mathrm{d}z = \\begin{pmatrix} 0 &  (a \\lambda^{-1} + b \\lambda) h(z)\\\\\n\t       (- b \\lambda^{-1} + a \\lambda) \\overline{h(\\bar z)}&\n\t       0\\end{pmatrix} \\mathrm{d}z\\;, \n\\end{equation*}\n where $h(z)$ is a periodic holomorphic function on $\\mathbb C$ with \n period $p \\in \\mathbb R^{\\times}$ and $a, b \\in \\mathbb R^{\\times}$. Moreover we\n assume $ b = a e^{-q}$. It is easy to check that $S$ has the following two\n properties: $\\tau_4 S = S$ and\n $S ( z, \\lambda = e^{-q\/2})$ is an upper triangular matrix.\n \n Let $C(z, \\lambda)$ denote the solution to $\\mathrm{d} C = C \\eta$, $C(z=0,\n \\lambda) ={\\rm Id}$. By the Picard-Lindel\\\"{o}f iteration, \n $C$ can be computed as\n \\begin{equation*}\n C = {\\rm Id} + \\int_0^z C \\eta \\mathrm{d}t+ \n     \\int_0^z \\left(\\int_0^t \\eta \\mathrm{d}t_1\n     \\right) \\eta  \\mathrm{d}t+ \\cdots\\;.\n \\end{equation*}\n Set $\\gamma : z \\mapsto z +p$ with $p \\in \\mathbb R^*$. \n Then $\\eta$ is invariant under $\\gamma$, \n and we obtain $\\gamma^* C = C(p) C$.\n From the properties of $S(z, \\lambda)$ stated above, \n we obtain $C(p) \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$\n and \n \\begin{equation*}\n C(p)|_{\\lambda = e^{-q\/2}} = \\begin{pmatrix} 1 & \\int_0^p h(t)\n\t\t\t       \\mathrm{d}t \\\\ 0 & 1\n\t\t       \\end{pmatrix}\\;.\n \\end{equation*}\n Let $C = \\Phi V_+$ be the Iwasawa decomposition of $C$. Then \n $\\gamma^* C$ has the Iwasawa decomposition $\\gamma^* C = \\hat \\Phi \\hat\n V_+$, where $\\hat \\Phi = C(p) F$ and $\\hat V_+ = V_+$, since \n $C(p) \\in \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4}$. Therefore the \n immersion $f_q$ obtained by the Sym formula\n closes up, \\textit{i.e.}, $\\gamma^* f_q = f_q$, if and  only if\n $\\int_0^p h(t)  \\mathrm{d}t =0$. It is not difficult to find \n periodic holomorphic functions $h(z)$ with period $p$ satisfying this\n condition.\n\n\\subsection{Totally symmetric surfaces}\n \\label{subsec:trinoids}\n We set \n\\begin{equation}\\label{eq:trinoidspot}\n \\eta = \n - \\begin{pmatrix}\n    0 & \\lambda^{-1} \\\\ \\lambda X^2 &0  \n   \\end{pmatrix} \n   \\frac{\\mathrm{d}z}{z}  + \n   \\begin{pmatrix}\n    0 & \\lambda^{-1} \\\\  \\lambda X^2 &0  \n   \\end{pmatrix} \n   \\frac{\\mathrm{d}z}{z-1}  +  \n   \\begin{pmatrix}  \n    0 & 0 \\\\ \\lambda (X^2-1\/4) &0 \n   \\end{pmatrix} \n   \\mathrm{d}z,\n\\end{equation}\n where $a, b \\in \\mathbb R$, $X = \\sqrt{ b^2 -a ^2 + a b (\\lambda^{-2} -\n \\lambda^2)}$ with  $b^2 -a ^2 + a b(e^{q} - e^{-q}) =1\/4$. We assume \n that \n \\begin{equation}\\label{eq:trinoidcondition}  \n  b^2\\not\\equiv a^2 \\;(\\mbox{mod}\\;\\boldsymbol S) \\;\\;\\mbox{and}\\;\\;\n  -1+ 4 \\sin^2 (\\pi X) \\neq 0 \\;\\;\\mbox{on}\\;\\;\\lambda \\in S^1,\n \\end{equation}\n where $\\boldsymbol S = \\{y \\in \\mathbb Z \\;|\\; \\mbox{\n $x^2 = y$ for some $x \\in \\mathbb Z$}\\}$.\n It is known that one can construct CMC immersions from $\\mathbb C P^1\n \\setminus \\{0, 1, \\infty\\}$ into $\\mathbb E^3$ from potentials $\\eta$\n which have the form \\eqref{eq:trinoidspot}. However, in this case $a$\n and $b$ need to be chosen differently, see \\cite{DW-n-noids} and\n \\cite{KKRS}. It turns out that \n the conditions \\eqref{eq:trinoidcondition} are equivalent to that\n monodromy matrices can be simultaneously  conjugated into \n $\\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma, \\tau_4} \n \\subset \\Lambda {\\rm SL}_2 \\mathbb C_{\\sigma}$.\n In our case, following an analogous procedure, \n it can be shown that the resulting\n CMC immersion with $H = \\tanh q$ in $\\mathbb H^3$  given by the holomorphic\n potential in \\eqref{eq:trinoidspot} is well-defined on $\\mathbb C P^1\n \\setminus \\{0, 1, \\infty\\}$. The detailed computation has been discussed in \n \\cite{Kobayashi:tri}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nIt is common knowledge that the scalar potential  $\\Phi(\\mathbf{r})$ of any \ncharge distribution  $\\rho(\\mathbf{r'})$ can be calculated (using the Gaussian \nabsolute system of units) by the relation (see, e.g., \\cite{1}, Chapter 2)\n\\begin{equation}\n\\label{1}\n\\Phi(\\mathbf{r})=\\int\\frac{\\rho(\\mathbf{r'})dV'}{|\\mathbf{r}-\\mathbf{r'}|}.\n\\end{equation}  \n\nIn the spherical coordinate system  $r, \\vartheta, \\varphi$ the  \n$\\Phi(\\mathbf{r})$ potential \ncan be conveniently found in many cases by expansion in terms of\nspherical harmonics $Y_{lm}$ (see, e.g., \\cite{2}). The potential at a point\nis a sum of actions of all the charges involved. Therefore, if\nwe are interested in the potential at a point inside a charge \ndistribution, we will have to split the region of integration\nin Eq. (\\ref{1}) into two parts by a sphere of radius $r$ centered at \nthe pole $O$. As a result, we come to equalities of two types, \nfor $r>r'$ and for  $r<r'$. Here  $r$ refers to the point with the\npotential we are interested in, and  $r'$ is the point over which \nintegration will be performed. Accordingly, the part of the \npotential formed by charges with coordinates $r'<r$ we shall \ndenote with $\\Phi_Q(\\mathbf{r})$, and the other part, produced by charges\nwith coordinates  $r'>r$, with $\\Phi_G(\\mathbf{r})$. The expressions for \n$\\Phi_Q(\\mathbf{r})$\nand $\\Phi_G(\\mathbf{r})$ written in the form appropriate for us here can be \nfound, for instance, in  \\cite{2} and are reproduced below:\n\n\\begin{equation}\n\\label{2}\n\\Phi_Q(\\mathbf{r})=\\sum_{l=0}^\\infty\\sum_{m=-l}^l\\sqrt{\\frac{4\\pi}{2l+1}}\\cdot\n\\frac{Q_{lm}(r)Y_{lm}(\\vartheta, \\varphi)}{r^{l+1}}  \\qquad (r>r'),\n\\end{equation}\nwhere $Q_{lm}(r)$ is a multipole moment of order $l, m$:\n\n\\begin{equation}\n\\label{3}\nQ_{lm}(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_0^r\\int\\int\\rho(\\mathbf{r'})r'^lY_{lm}^*\n(\\vartheta',\\varphi')dV',\n\\end{equation}\n\n\\begin{equation}\n\\label{4}\n\\Phi_G(\\mathbf{r})=\\sum_{l=0}^\\infty\\sum_{m=-l}^l\\sqrt{\\frac{4\\pi}{2l+1}}\nr^l{G_{lm}(r)Y_{lm}(\\vartheta, \\varphi)}  \\qquad (r<r'),\n\\end{equation}\nwhere $G_{lm}(r)$: \n\n\\begin{equation}\n\\label{5}\nG_{lm}(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_r^\\infty\\int\\int\\frac{\\rho(\\mathbf{r'})}\n{r'^{l+1}}Y_{lm}^*(\\vartheta',\\varphi')dV'.\n\\end{equation}\n\nThe functions $Q_{lm}(r)$ and $G_{lm}(r)$ depend on $r$ on the upper and \nlower limits of integration as on a parameter.\n\nThe potential at a given point $\\mathbf{r}$ is a sum of the potentials \ncalculated for $r>r'$ and $r<r'$: \n$\\Phi(\\mathbf{r})=\\Phi_Q(\\mathbf{r})+\\Phi_G(\\mathbf{r})$.\n\nThe spherical functions $Y_{lm}$ can be written as (see, e.g., \\cite{2}):\n\n\\begin{equation}\n\\label{6}\nY_{lm}(\\vartheta,\\varphi)=\\delta_m\\sqrt{\\frac{2l+1}{4\\pi}\\cdot\\frac{(l-|m|)!}\n{(l+|m|)!}}P_{lm}(\\cos\\vartheta)e^{im\\varphi}.\n\\end{equation}\n\nIn this expression, the integers  $l,m$ satisfy the conditions \n$l\\ge 0$, $-l\\le m\\le l $; $\\delta_m=(-1)^m$ for $m\\ge 0$, $\\delta_m=1$\nfor $m<0$.\n$P_{lm}$ denotes here the associated Legendre polynomial. \n\nThe vector potential $\\mathbf{A}(\\mathbf{r})$ of any current distribution \n$\\mathbf{j}(\\mathbf{r'})$ in vacuum can be calculated from the relation \n(see, e.g., \\cite{1}, Chapter 2)\n\n\\begin{equation}\n\\label{7}\n\\mathbf{A}(\\mathbf{r})=\\frac{1}{c}\\int\\frac{\\mathbf{j}(\\mathbf{r'})dV'}\n{|\\mathbf{r}-\\mathbf{r'}|},\n\\end{equation}\nwhere $c$ is the electrodynamic constant.\n\nRelation (\\ref{7}) resembles in its form equality (\\ref{1}), the difference \nbeing that integration is performed here not over the scalar but \nrather over the vector function $\\mathbf{j}(\\mathbf{r'})$. If one wanted\nto employ the above method of potential calculation with the use of spherical\nharmonics, one would have to apply all the calculations to each \ncomponent separately, with subsequent vector addition of the results\nthus obtained. This approach may meet with considerable difficulties.\n\nOn the other hand, if the currents have axial symmetry, one can suggest \na fairly simple and straightforward method of calculation of the vector \npotential which reduces essentially to calculating an integral of one\nscalar function only.\n\n\n\n\\section{Method of calculation of the vector potential of axially \nsymmetric currents}\n\nAssume a system of currents with a volume density $\\mathbf{j}(\\mathbf{r'})$\nwhich have axial symmetry. Denote the axis of symmetry of \nthe system with  $Z$.  Introduce a spherical coordinate system \n$r, \\vartheta, \\varphi$ such that the  $\\vartheta=0$ axis coincides with \nthe symmetry axis $Z$. Denote the unit vectors along the \n$r, \\vartheta, \\varphi$ axes by  $\\mathbf{n}_r, \\mathbf{n}_\\vartheta, \n\\mathbf{n}_\\varphi$. \nThe $\\vartheta=\\pi\/2$ plane will be called the equatorial plane. \n\nThe system being symmetric, the currents do not depend on the $\\varphi$\ncoordinate and have only one component along it: \n $\\mathbf{j}(\\mathbf{r'})=j(r',\\vartheta')\\mathbf{n}_{\\varphi'}$.\nThis means that all $\\mathbf{j}(\\mathbf{r'})$ are circular currents, and \nthat the vector of any element of the current lies in the plane parallel\nto the equatorial plane (we shall call it the $S'$ plane). The symmetry \nof the current system suggests that the vector potential\n$\\mathbf{A}(\\mathbf{r})$ is also axially symmetric, i.e., the magnitude of the \nvector potential $A(r,\\vartheta)$ does not depend on coordinate $\\varphi$. \nBut this means that the magnitude of the vector potential $A(r,\\vartheta)$ \ncan be determined on any plane passing through the symmetry axis $Z$.\nThe simplest way appears to determine the potential on the plane \npassing through the $Z$ axis and the coordinate $\\varphi=0$, and it is this\nwhat we are going to do. Call this plane the $M$ plane.\n\nCalculate the vector potential at a point $\\mathbf{r}$ lying on the $M$ plane.\nThe coordinates of this point are  $(r,\\vartheta,0)$. It is convenient for \nthe purpose of this analysis to pass through this point a plane $S$\nparallel to the equatorial plane. \n\nConsider our situation in more detail. Vector potential at a point \n$\\mathbf{r}$ (lying on plane $M$) is produced by currents flowing throughout the \nspace of interest here. Consider the potentials created by the current \nelements $d\\mathbf{j}(\\mathbf{r'})$. Figure 1 illustrates our case.\n\n\\begin{figure}\n\\includegraphics{Figure1.eps}\n\\caption{}\n\\end{figure}\n\n\nIn Fig. 1, the origin of the spherical system of coordinates is at point $O$,\nthe $Z$ axis points directly at us, and the dashed line visualizes one of \nthe circular current trajectories. This trajectory lies in the $S'$ plane.\nThe $S$ and $S'$ planes are parallel but do not coincide. The direction of\nthe $\\varphi$ axis at point $(r,\\vartheta,0)$ is specified in the figure. We are \nlooking for the potential at this point. The $\\varphi$ axis at point \n$(r,\\vartheta,0)$ is perpendicular to the $M$ plane. \nThe $r$ axis lies in the $M$ plane. \n\nThe dashed line $L$ on the $S'$ plane is perpendicular to plane $M$,\ni.e., parallel to the $\\varphi$ axis at point $(r,\\vartheta,0)$. \n\nFigure 1 shows two elements of the circular current with densities \n$d\\mathbf{j}_1(\\mathbf{r'}_1)$ and $d\\mathbf{j}_2(\\mathbf{r'}_2)$. \nThe current element $d\\mathbf{j}_1(\\mathbf{r'}_1)$ produces at point \n$(r,\\vartheta,0)$ on plane $M$ the potential $d\\mathbf{A}_1(r,\\vartheta,0)$.  \nAs seen from vector equality (\\ref{7}), this potential is \ndirected in space in the same way as the current element\n$d\\mathbf{j}_1(\\mathbf{r'}_1)$. This means that the angles between a direction \nin space (for instance, straight lines perpendicular to plane $M$)\nand vectors $d\\mathbf{j}_1(\\mathbf{r'}_1)$ and $d\\mathbf{A}_1(\\mathbf{r})$\nshould be equal (in Fig. 1, these are the $\\varphi'$ angles).\n\nThe vector of the current element $d\\mathbf{j}_1(\\mathbf{r'}_1)$ is confined to\nplane $S'$, which is parallel to the equatorial plane. As seen from\nexpression (\\ref{7}), this means that the element of the vector potential\n$d\\mathbf{A}_1(\\mathbf{r})$ generated by the current element \n$d\\mathbf{j}_1(\\mathbf{r'}_1)$ \nshould likewise lie fully in the plane parallel to \nthe equatorial plane, i.e., in plane $S$ (the $S$ and $S'$\nplanes do not coincide). Therefore, the $d\\mathbf{A}_1(\\mathbf{r})$ potential\ncan be resolved into two components, $dA_{1,\\perp}$ and $dA_{1,\\varphi}$\nin the $S$ plane.  These components are shown in Fig. 1: \n $dA_{1,\\perp}=dA_1\\sin\\varphi'$ and $dA_{1,\\varphi}=dA_1\\cos\\varphi'$,\n where the angle $\\varphi'$\nis the coordinate of the current element $d\\mathbf{j}_1(\\mathbf{r'}_1)$.\n\nSince, however, the current is axially symmetric, there will always be \nan element of current $d\\mathbf{j}_2$ equal in magnitude to the element \n$d\\mathbf{j}_1$ and located at the same distance from point $(r,\\vartheta,0)$\nas the  $d\\mathbf{j}_1$ element. The coordinates of this current element are \n$(r',\\vartheta',-\\varphi')$. It produces at the same point  $(r,\\vartheta,0)$\nthe potential $d\\mathbf{A}_2(r,\\vartheta,0)$. Resolving potential \n$d\\mathbf{A}_2(r,\\vartheta,0)$ in the  $S$ plane into two components, \n$dA_{2,\\perp}$ and $dA_{2,\\varphi}$, we see immediately that the components \n$dA_{1,\\perp}$ and  $dA_{2,\\perp}$ cancel, while $dA_{1,\\varphi}$ and\n$dA_{2,\\varphi}$ add (see Fig. 1). Thus, after integration over all elements\nof current $d\\mathbf{j}(\\mathbf{r'})$ the potential $\\mathbf{A}(\\mathbf{r})$\nwill have only one component, the one along the $\\varphi$ axis, left. \nBut this implies that in determination of the $\\mathbf{A}(\\mathbf{r})$\npotential one may, rather than taking into account the whole potential \n$d\\mathbf{A}(\\mathbf{r})$ produced by the current element \n$d\\mathbf{j}(\\mathbf{r'})$, restrict oneself to the projection \n$dA_\\varphi$ of this potential onto the $\\varphi$ axis: \n$dA_\\varphi=dA\\cos\\varphi'$, where $\\varphi'$ is the coordinate of the\nelement of current $d\\mathbf{j}(\\mathbf{r'})$. Said otherwise, the \npotential $dA_\\varphi$ is generated not by the current element\n$d\\mathbf{j}(\\mathbf{r'})$ as a whole but by the projection of this \nelement on the line $L$ parallel to vector $\\varphi$ at point $(r,\\vartheta,0)$.\n\nWe can now, on replacing vector $\\mathbf{j}(\\mathbf{r'})$ in Eq. (\\ref{7})\nwith a scalar \n$J(\\mathbf{r'})=J(r',\\vartheta',\\varphi')=j(r',\\vartheta')cos\\varphi'$,\n calculate the magnitude of the vector potential\n\n\\begin{equation}\n\\label{8}\nA(r,\\vartheta,0)=\\frac{1}{c}\\int\\frac{J(\\mathbf{r'})dV'}\n{|\\mathbf{r}-\\mathbf{r'}|}=\\frac{1}{c}\\int\\frac{j(r',\\vartheta')\\cos\\varphi'dV'}\n{|\\mathbf{r}-\\mathbf{r'}|}.\n\\end{equation}\nThe potential at any point $\\mathbf{r}$ can now be written as\n\n\\begin{equation}\n\\label{9}\n\\mathbf{A}(\\mathbf{r})=A(r,\\vartheta,0)\\mathbf{n}_\\varphi.\n\\end{equation}\n\nBecause expression (\\ref{8}) contains scalar quantities, in calculation \nof the integral one may resort to expanding the integrand in spherical\nharmonics. Note, however, that because expression (\\ref{8}) contains \n$\\cos\\varphi'$ as a factor, one may conveniently use in expressions \nfor $Y_{lm}$ in place of exponential functions employed customarily, \ntrigonometric ones (see. e.g., \\cite{3}, Chapter 21). In this case \none will have to replace expression (\\ref{6}) with the following relations:\n\n\\begin{equation}\n\\label{10}\nY_{lm}^c(\\vartheta,\\varphi)=\\sqrt{\\frac{2l+1}{2\\pi}\\cdot\\frac{(l-m)!}\n{(l+m)!}}P_{lm}(\\cos\\vartheta)\\cos{m\\varphi},\n\\end{equation}\n\n\\begin{equation}\n\\label{11}\nY_{lm}^s(\\vartheta,\\varphi)=\\sqrt{\\frac{2l+1}{2\\pi}\\cdot\\frac{(l-m)!}\n{(l+m)!}}P_{lm}(\\cos\\vartheta)\\sin{m\\varphi},\n\\end{equation}\n$$\n(l=0, 1, 2, ...; m=0, 1, 2, ..., l).\n$$\nThis involves a minor change in the form of expansion of the function.\n\n\\begin{flushleft}\nFor $r>r'$:\n\\end{flushleft}\n\n\\begin{equation}\n\\label{12}\nA_Q(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=0}^\\infty\\sqrt{\\frac{4\\pi}{2l+1}}\\left\n(\\frac{1}{2}\n\\frac{Q_{l0}^cY_{l0}^c}{r^{l+1}}+\\sum_{m=1}^l\\frac{Q_{lm}^cY_{lm}^c}{r^{l+1}}+\n\\sum_{m=1}^l\\frac{Q_{lm}^sY_{lm}^s}{r^{l+1}}\\right),\n\\end{equation}\nwhere the multipole moments $Q_{lm}^c$ and $Q_{lm}^s$ have the form\n\n\\begin{equation}\n\\label{13}\nQ_{lm}^c(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_0^r\\int\\int J(\\mathbf{r'})r'^lY_{lm}^c\n(\\vartheta',\\varphi')dV',\n\\end{equation}\n\n\\begin{equation}\n\\label{14}\nQ_{lm}^s(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_0^r\\int\\int J(\\mathbf{r'})r'^lY_{lm}^s\n(\\vartheta',\\varphi')dV'.\n\\end{equation}\nFor $r<r'$:\n\n\\begin{equation}\n\\label{15}\nA_G(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=0}^\\infty\\sqrt{\\frac{4\\pi}{2l+1}}\n\\left(\\frac{1}{2}r^l\nG_{l0}^cY_{l0}^c+\\sum_{m=1}^lr^lG_{lm}^cY_{lm}^c+\n\\sum_{m=1}^lr^lG_{lm}^sY_{lm}^s\\right),\n\\end{equation}\nwith the multipole moments $G_{lm}^c$ and $G_{lm}^s$\n\n\\begin{equation}\n\\label{16}\nG_{lm}^c(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_r^\\infty\\int\\int\\frac\n{J(\\mathbf{r'})}{r'^{l+1}}Y_{lm}^c(\\vartheta',\\varphi')dV',\n\\end{equation}\n\n\\begin{equation}\n\\label{17}\nG_{lm}^s(r)=\\sqrt{\\frac{4\\pi}{2l+1}}\\int_r^\\infty\\int\\int\\frac\n{J(\\mathbf{r'})}{r'^{l+1}}Y_{lm}^s(\\vartheta',\\varphi')dV'.\n\\end{equation}\n\nThe function $\\cos\\varphi'$ entering the expression for $J(\\mathbf{r'})$\nalso belongs to the system of functions $Y_{lm}^c$ (see expression (\\ref{10})).\nTherefore, because of the functions $\\cos{m\\varphi'}$ and $\\sin{m\\varphi'}$ \nbeing orthogonal, part of the multipole moments will vanish in the course \nof integration of expressions (\\ref{13}), (\\ref{14}) and (\\ref{16}), (\\ref{17})\n(orthogonality in index $m$). The only terms left will be those with $m=1$,\ni.e., the terms $Q_{l1}^c(r)$ and $G_{l1}^c(r)$. As a result, the double \nseries (\\ref{12}) and (\\ref{15}) will become single series and look as\n\n\\begin{equation}\n\\label{18}\nA_Q(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=1}^\\infty\\sqrt{\\frac{4\\pi}{2l+1}}\n\\frac{Q_{l1}^cY_{l1}^c}{r^{l+1}}\n\\qquad \\mbox{for } r>r', \n\\end{equation}\n\n\\begin{equation}\n\\label{19}\nA_G(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=1}^\\infty\\sqrt{\\frac{4\\pi}{2l+1}}\nG_{l1}^cY_{l1}^cr^l\n\\qquad \\mbox{for } r<r'. \n\\end{equation}\n\nThe next step appears to be substitution in these series of the expression\nfor $Y_{l1}^c$ from formula (\\ref{10}). We have to remember, however, \nthat we are looking for the potential on the $\\varphi=0$ plane only.\nFor all other values of the $\\varphi$ coordinate, the relations we have\nderived will yield wrong values of the potential. Therefore, prior to \nsubstituting the expression for $Y_{l1}^c$ from relation (\\ref{10}), \nwe will have to zero the coordinate $\\varphi$. We obtain naturally\n$\\cos\\varphi=1$. Now the potentials take on the form\n\n\\begin{equation}\n\\label{20}\nA_Q(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=1}^\\infty\\frac{2\\pi}{l(l+1)}\n\\frac{Q^A_{l1}(r)P_{l1}(\\cos\\vartheta)}{r^{l+1}}\n\\qquad \\mbox{for } r>r',\n\\end{equation}\nwhere the multipole moments $Q^A_{l1}$ are\n\n\\begin{equation}\n\\label{21}\nQ^A_{l1}(r)=\n\\int_0^r\\int_0^\\pi j(r',\\vartheta')r'^lP_{l1}(\\cos\\vartheta')r'^2\\sin\\vartheta' \nd\\vartheta' dr',\n\\end{equation}\n\n\\begin{equation}\n\\label{22}\nA_G(r,\\vartheta,0)=\\frac{1}{c}\\sum_{l=1}^\\infty\\frac{2\\pi}{l(l+1)}\nr^lG^A_{l1}(r)P_{l1}(\\cos\\vartheta)\n\\qquad \\mbox{for } r<r',\n\\end{equation}\nwith the multipole moments $G^A_{l1}$\n\n\\begin{equation}\n\\label{23}\nG^A_{l1}(r)=\n\\int_r^\\infty\\int_0^\\pi \\frac{j(r',\\vartheta')}{r'^{l+1}}P_{l1}(\\cos\\vartheta')\nr'^2\\sin\\vartheta' d\\vartheta' dr'.\n\\end{equation}\n\nIn these formulas, $j(r',\\vartheta')$ is the circular current density,\nand $Q^A_{l1}(r)$ and $G^A_{l1}(r)$  are the parts of the multipole moments \nleft after integration over the $\\varphi$ coordinate (truncated multipole \nmoments). The vector potential at any point in space can be written as\n\n\\begin{equation}\n\\label{24}\n\\mathbf{A}(\\mathbf{r})=[A_Q(r,\\vartheta,0)+A_G(r,\\vartheta,0)]\n\\mathbf{n}_\\varphi,\n\\end{equation}\nwhence we finally come to\n\\medskip\n\n\\fbox{\n\\parbox{12.8cm}{\n\\begin{equation}\n\\label{25}\n\\mathbf{A}(\\mathbf{r})=\\frac{1}{c}\\sum_{l=1}^\\infty\\frac{2\\pi}{l(l+1)}\n\\left(\\frac{Q^A_{l1}(r)}{r^{l+1}}+r^lG^A_{l1}(r)\\right)\nP_{l1}(\\cos\\vartheta)\\mathbf{n}_\\varphi.\n\\end{equation}\n}}\n\\medskip\n\nThis is the final expression for the vector potential $\\mathbf{A}(\\mathbf{r})$.\nAs evident from Eq. (\\ref{25}), vector potential $\\mathbf{A}(\\mathbf{r})$\nhas only one component, $A_\\varphi$.\n\nNow we impose an additional constraint, namely, that the currents are \nsymmetrical with respect to the equatorial plane (which quite often is \nthe case). In this case, expression (\\ref{25}) may contain only functions \n$P_{l1}$ symmetric about the equator, i.e., $P_{2n+1,1}$ functions. \nNow expression (\\ref{25}) for the vector potential has to be replaced by\n\n\\begin{equation}\n\\label{26}\n\\mathbf{A}(\\mathbf{r})=\\frac{1}{c}\\sum_{n=0}^\\infty\\frac{\\pi}{(n+1)(2n+1)}\n\\left(\\frac{Q^A_{2n+1,1}(r)}{r^{(2n+1)+1}}+r^{2n+1}G^A_{2n+1,1}(r)\\right)\nP_{2n+1,1}(\\cos\\vartheta)\\mathbf{n}_\\varphi.\n\\end{equation}\n\nIt may be of interest to consider a particular relevant case. Consider\naxially symmetric currents with a  $\\sin\\vartheta$ dependence on the \n$\\vartheta$ angle. This case becomes realized, for instance, if the currents \nare generated by rotation of charges, with all the charges in a layer \nof thickness $dr$ distributed uniformly over the angle  $\\vartheta$ \nand rotating with the same angular velocity.\n\nThe $\\sin\\vartheta$ function belongs to the system of associated Legendre \nfunctions. Therefore, the $P_{l1}$ functions being orthogonal with respect \nto index $l$, the larger part of the terms in the $Q^A_{l1}(r)$ and\n$G^A_{l1}(r)$ expansions (see expressions (\\ref{21}) and (\\ref{23})) \nwill vanish in calculation, with only $Q^A_{11}(r)$ and $G^A_{11}(r)$\nterms with index $l=1$ retained. In this case, the potential will come \nout as a sum of two terms only:\n\n\\begin{equation}\n\\label{27}\n\\mathbf{A}(\\mathbf{r})=\\frac{\\pi}{c}\\left\n(\\frac{Q^A_{11}(r)}{r^2}+rG^A_{11}(r)\\right)\\sin\\vartheta\\mathbf{n}_\\varphi.\n\\end{equation} \n\n\\section{Magnetic fields of axially symmetric currents}\n\nMagnetic field $\\mathbf{H}(\\mathbf{r})$ can be calculated with the general\nexpression $\\mathbf{H}(\\mathbf{r})=\\nabla\\times\\mathbf{A}(\\mathbf{r})$. \nThe vector potential of axially symmetric currents has only one component,\n$A_\\varphi$:\n$\\mathbf{A}(\\mathbf{r})=A(r,\\vartheta)\\mathbf{n}_\\varphi$.\nWith this in mind, and using the general expression for\n$\\nabla\\times\\mathbf{A}(\\mathbf{r})$ in the spherical coordinate system \n(see, for instance, \\cite{3}, Chapter 6)\n\n$$\n\\nabla\\times\\mathbf{A}=\\left[\\frac{1}{r\\sin\\vartheta} \\left(\\frac{\\partial}\n{\\partial\\vartheta}(\\sin\\vartheta A_\\varphi)-\\frac{\\partial A_\\vartheta}\n{\\partial\\varphi}\\right)\\right]\\mathbf{n}_r+\n$$\n$$\n\\left[\\frac{1}{r\\sin\\vartheta}\\frac{\\partial A_r}{\\partial\\varphi}-\n\\frac{1}{r}\\left(\\frac{\\partial}{\\partial r}(rA_\\varphi)\\right)\\right]\n\\mathbf{n}_\\vartheta+\n\\left[\\frac{1}{r}\\frac{\\partial(rA_\\vartheta)}{\\partial r}-\\frac{1}{r}\n\\frac{\\partial A_r}{\\partial\\vartheta}\\right]\\mathbf{n}_\\varphi,\n$$\nwe finally come to \n\n\\begin{equation}\n\\label{28}\n\\mathbf{H}=\\nabla\\times\\mathbf{A}=\\left(\\frac{1}{r}\\frac{\\cos\\vartheta}\n{\\sin\\vartheta}A_\\varphi+\\frac{1}{r}\\frac{\\partial}{\\partial\\vartheta}\nA_\\varphi\\right)\\mathbf{n}_r-\n\\left(\\frac{1}{r}A_\\varphi+\\frac{\\partial}{\\partial r}A_\\varphi\\right)\n\\mathbf{n}_\\vartheta.\n\\end{equation} \nwith  $A_\\varphi$ expressed by formula (\\ref{25}). \n\nConsider the expression $\\frac{\\partial}{\\partial r}A_\\varphi$ in any of \nthe terms of expansion (\\ref{25}):\n\n\\begin{equation}\n\\label{29}\n\\frac{\\partial}{\\partial r}A_\\varphi=\\frac{1}{c}\\frac{2\\pi}{l(l+1)}\n\\left(\\frac{-(l+1)}{r^{l+2}}Q^A_{l1}+lr^{l-1}G^A_{l1}+\n\\frac{1}{r^{l+1}}\\frac{\\partial}{\\partial r}Q^A_{l1}+r^l\\frac{\\partial}\n{\\partial r}G^A_{l1}\\right)P_{l1}(\\cos\\vartheta).\n\\end{equation} \nBy the theorem of Leibniz--Newton (see, e.g., \\cite{3}, Chapter 4, item 4.6-5):\n\n$$\n\\frac{d}{dx}\\int_a^xf(t)dt=f(x),\n$$\nthe last two terms in expression (\\ref{29}) cancel, with the final\nexpression taking on the form\n\\newpage\n\n$$\n\\mathbf{H}(\\mathbf{r})=\\frac{1}{c}\\sum_{l=1}^\\infty\\frac{2\\pi}{l(l+1)}\n\\left\\{\\left[\\left(\\frac{Q^A_{l1}(r)}{r^{l+2}}+r^{l-1}G^A_{l1}(r)\\right)\n\\right.\\right.\\cdot\n$$\n\n\\begin{equation}\n\\label{30}\n\\left.\\left(\\frac{\\cos\\vartheta}{\\sin\\vartheta}P_{l1}(\\cos\\vartheta)+\n\\frac{\\partial P_{l1}(cos\\vartheta)}\n{\\partial\\vartheta}\\right)\\right]\\mathbf{n}_r+\n\\end{equation}\n$$\n\\left.\\left[\\left(\\frac{lQ^A_{l1}(r)}{r^{l+2}}-(l+1)r^{l-1}G^A_{l1}(r)\\right)\nP_{l1}(\\cos\\vartheta)\\right]\\mathbf{n}_\\vartheta\\right\\}.\n$$\n\nIf the currents are symmetric about the equator, expression (\\ref{30}), \nsimilar to formula (\\ref{26}), will contain only equatorially symmetric \nfunctions $P_{2n+1,1}$, and summation over $n$ should be performed from \n$0$ to $\\infty$.\n\nConsider a few particular cases. In the case covered by potential (\\ref{27}),\nthe magnetic field should be written as\n\n\\begin{equation}\n\\label{31}\n\\mathbf{H}=\\frac{\\pi}{c}\\left[\\left(\\frac{Q^A_{11}}{r^3}+\nG^A_{11}\\right)2\\cos\\vartheta\\mathbf{n}_r+\n\\left(\\frac{Q^A_{11}}{r^3}-2G^A_{11}\\right)\\sin\\vartheta\\mathbf{n}_\n\\vartheta\\right].\n\\end{equation} \n\nTo illustrate the use of this expression, consider the following simple \nproblem: rotation of a charged sphere about one of the diameters. Let the \nsphere of radius $R$, with charge  $q$ distributed uniformly over its surface,\nrotate about the $Z$ axis with an angular velocity  $\\mathbf{w}$. Calculate \nthe magnetic field and the vector potential inside and outside the sphere.\n\nThe density of the surface current produced by rotation of the sphere can \nbe expressed in the following way:\n\n\\begin{equation}\n\\label{32}\n\\mathbf{j}=\\frac{qw}{4\\pi R}\\delta(r-R)\\sin\\vartheta\\mathbf{n}_\\varphi.\n\\end{equation} \n\nSubstituting the expression for current density (\\ref{32}) in relations \n(\\ref{16}) and (\\ref{17}), we come to the following expressions for the \n(truncated) multipole moments\n\n$$\n Q^A_{11}=0, \\qquad G^A_{11}=\\frac{qw}{3\\pi R} \\qquad \\mbox{for } r<R,\n$$ \n\n$$\n Q^A_{11}=\\frac{qwR^2}{3\\pi}, \\qquad G^A_{11}=0 \n \\qquad \\mbox{for } r>R.\n$$\n\nNow the magnetic field and the vector potential acquire their final form\n\n\\begin{equation}\n\\label{33}\n\\mathbf{H}=\\frac{2qw}{3cR}(\\cos\\vartheta\\mathbf{n}_r-\\sin\\vartheta\\mathbf{n}\n_\\vartheta), \\qquad  \n\\mathbf{A}=\\frac{qwr}{3cR}\\sin\\vartheta\\mathbf{n}_\\varphi\n\\qquad \\mbox{for } r<R,\n\\end{equation} \n\n\\begin{equation}\n\\label{34}\n\\mathbf{H}=\\frac{2qwR^2}{3cr^3}(\\cos\\vartheta\\mathbf{n}_r+\\frac{1}{2}\n\\sin\\vartheta\\mathbf{n}_\\vartheta), \\qquad\n\\mathbf{A}=\\frac{qwR^2}{3cr^2}\\sin\\vartheta\\mathbf{n}_\\varphi\n\\qquad \\mbox{for } r>R.\n\\end{equation}\n\nSignificantly, this technique permits one to determine the magnetic field\nwithout prior calculation of the vector potential.\n\nThe solution to this problem can be found, for instance, in monographs \n\\cite{1} (problem 2.85) and \\cite{2} (problem 253):\n\n$$\n\\mathbf{H}=\\frac{2q\\mathbf{w}}{3cR} \\qquad \\mbox{for }r<R, \\qquad\n\\mathbf{H}=\\frac{3\\mathbf{r}(\\mathbf{m}\\cdot\\mathbf{r})}{r^5}-\\frac{\\mathbf{m}}\n{r^3} \\qquad \\mbox{for }r>R,\n$$\nwhere $\\mathbf{m}=\\frac{qR^2}{3c}\\mathbf{w}$ is the magnetic moment of \nthe system. One can readily verify that these expressions are identical. \nIn monographs  \\cite{1} and \\cite{2} it is proposed to solve this problem\nby differential calculus. Our approach, by contrast, has yielded the answers \nto it by straightforward calculation.\n\n\\addcontentsline{toc}{chapter}{\u041b\u0438\u0442\u0435\u0440\u0430\u0442\u0443\u0440\u0430}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:intro}\n\nThe game of darts has experienced a surge in popularity in recent years and now has a substantial presence in many countries including the U.K., Germany, the Netherlands, and Australia, among others. Indeed, a recent headline in \\cite{Economist2020} explains that darts has moved from pastime to prime time and how it is quite common today to have tens of thousands of fans attend darts tournaments. The article also notes how in recent years darts has become the second-most-watched sport over the Christmas period on Sky Sports in the U.K., coming second only to football. Further growth is expected and professional tournaments now take place in locations such as Shanghai and Madison Square Garden in New York - locations that are far removed from the U.K. and the Netherlands, the traditional hot-beds of darts.\n The game is also becoming increasingly attractive to women as evidenced by the exploits of Fallon Sherrock who in 2019 became the first woman to win a match at the PDC World Darts Championship and in fact made\\footnote{Her other successes include making it to the quarter-final of the Grand Slam of Darts in 2021 and reaching the final of the 2021 Nordic Darts Masters in Copenhagen.} it to the third round of that tournament.\n\nIn this paper we conduct an exploratory data analysis of a darts data-set based on 16 professional players' dart throws in the 2019 season. Our ultimate goal is to build skill models that accurately reflect the players' skill levels and that can be used as inputs for zero-sum games (ZSGs) that model real-world darts matches. There are two major issues with the data-set, however. First, the data is too coarse and we do not have enough data for some player \/ target-region combinations. Second, while we know the target area for each dart in the data-set, we do not know the precise location in the target area at which it was aimed. Both of these issues lead to several problems with identifying a player's skill model. These problems include poor estimation of the skill model in certain regions of the dartboard and difficulty in inferring whether the mean outcome of a dart differs from the intended target, i.e. whether or not there is a {\\em bias}.\n\nWe propose to partially resolve these issues via an empirical Bayes approach that utilizes the Dirichlet-Multinomial (DM) distribution to borrow strength from all players when fitting each player's skill model. Our overall approach to building skill models consists of two stages. The first stage utilizes the DM model to convert the raw data-counts into pseudo-counts. This ensures there is sufficient data-coverage for all players in the important target areas of the board. The second stage then feeds the pseudo-counts into a bivariate normal skill model that allows us to consider targeting any part of the dartboard in a ZSG setting. Our bivariate normal skill model extends \\cite{Tibshirani} (hereafter TPT) by (i) partitioning the dartboard by target area to reflect the fact that a professional player's skill level varies considerably by target area and (ii) allowing for a bias in the players' skill models.\n\nOur DM approach resolves several issues but bias and correlation identification issues remain. We finesse these issues in the context of ZSGs by considering so-called ``single-action'' and ``multi-action'' players. The single-action player is only allowed to aim at a single point in each possible target region, (e.g. treble 20), whereas the multi-action player can aim at any point in the target area. The single-action player only needs the skill model given directly by the pseudo-counts in the first stage above whereas the multi-action player needs the bivariate normal skill model from the second stage. We show the single-action player is only at a very small disadvantage to the multi-action player in real-world darts matches and in fact even this small disadvantage may be overstated if we factor in bias and model estimation errors. We therefore conclude the first-stage skill model (based on pseudo-counts arising from the DM empirical Bayes model) is a viable alternative to the more general two-stage skill model that allows us to target any precise location on the dart-board.\n\nWe also show how the DM-based skill models can be used to analyze some specific situations that have arisen in real-world dart matches where a player's decisions were deemed surprising by some pundits. We consider two examples where our skill models allow us to argue that the player's decisions were in fact not surprising and suggest that the players themselves often have a better understanding of their own skills than the pundits \/ commentators.\n\nWe are certainly not the first to study the game of darts. \\cite{Stern-CHANCE} and \\cite{Percy99} consider the problem of where on the dartboard to aim in order to maximize the score of an individual dart. More recently, the aforementioned work of TPT proposed several models based on the Gaussian and skew Gaussian bivariate distributions to model the throwing skills of players. They develop an EM algorithm together with importance sampling to fit these distributions to dart-throwing data. Another recent development is the work by \\citet{HotHands_RSSA} who find some evidence for a weak-form of the so-called ``hot-hand'' phenomenon in darts. In particular, they find strong evidence for it during the three dart throws {\\em within} a turn\\footnote{We clarify what we mean by a ``turn'' in Section \\ref{sec:Rules}.}, but they find little evidence for it persisting {\\em across} turns. \n\\cite{Significance-RSS-2015} consider whether the standard layout of a darts board disadvantages left-handers and propose an alternative layout to counter any such disadvantage. \\cite{Otting_2020_Pressure} use darts as a vehicle for understanding and predicting how individuals will perform in high pressure situations. They find no evidence in favor of either choking or excelling under pressure.\n\nAnother stream of research is concerned with finding optimal {\\em strategies} for playing darts. \\cite{Kohler} uses a dynamic programming (DP) formulation to minimize the expected number of darts throws to {\\em check out}, i.e. to reach a score of zero with the last throw being a double. More recently \\citet{Baird2020} considers the slightly more general DP formulation where the goal is to minimize the expected number of turns until checking out. Both of these papers ignore their opponent's score when constructing a strategy and therefore produce sub-optimal policies. More recently, \\cite{HaughWang-Darts-2021} formulate the game of darts as a ZSG and show how to solve the ZSG efficiently using DP methods. Employing the same data-set considered in this paper, they use their ZSG solutions to argue that the importance of playing strategically, i.e. taking an opponent's score into account, could improve a player's win-probability by approximately 2\\% - 3\\% in real-world matches. In contrast to this paper, however, their focus was on formulating and solving the ZSGs. In particular, they did not consider at all the various problems that can arise when using their data-set to estimate the players' skill models which is the focus of this work.\n\nThe remainder of the paper is organized as follows. In Section \\ref{sec:Rules} we describe the rules of darts and we describe our data in Section \\ref{sec:Data}. We present the bivariate normal skill model in Section \\ref{sec:SkillModel} while in Section \\ref{sec:Problems} we describe some of the main problems that arise in fitting the skill model to our data. We propose our empirical Bayes approach, i.e. the DM model, in Section \\ref{sec:Empirical-Bayes} as a partial solution to some of these problems and discuss the fit of this model in Section \\ref{sec:Fitted-Models}. In Section \\ref{sec:Empirical Bayes-ZSG} we consider the DM-based skill models in the context of ZSGs and we conclude in Section \\ref{sec:Conclusions}. A {\\em Supplementary Material} file contains additional material including the EM algorithm for fitting our extension of the bivariate skill model as well as complete versions, i.e. for all 16 players, of all figures and tables that appear in the main text.\nFinally, we note that some sections, e.g. Sections \\ref{sec:Rules} and \\ref{sec:Data}, closely follow analogous sections in \\citet{HaughWang-Darts-2021}.\n\n\n\\section{The Rules of Darts}\n\\label{sec:Rules}\n\nThe score obtained by a dart is determined by where it lands on the board. The small concentric circles in the middle of the board define the ``double bulls-eye'' (DB) and the ``single bulls-eye'' (SB) regions. If a dart lands in the DB region (the small red circle) then it scores 50 points while a dart landing in the SB region (the green annulus surrounding the DB region) scores 25. See Figure \\ref{fig:DartB1}. Beyond the SB region, the dartboard is divided into 20 segments of equal area corresponding to the numbers 1 to 20. \\\\\n\n\\noindent\n\\begin{minipage}[b]{0.55\\linewidth}\nIf a dart lands in the ``20'' segment, for example, then it will land in the treble twenty (T20) region, the double twenty region (D20), or the single twenty region (S20) for scores of 60, 40 or 20, respectively. The double region is the region between the two outermost circles on the dartboard whilst the treble region is the region between the two circles beyond the SB region. The single region is then the union of the two disjoint regions between the SB and treble region and between the treble and double regions. If a dart lands beyond the double region then it scores zero.\n\\end{minipage} \\hspace{.5cm}\n\\begin{minipage}[t]{0.4\\linewidth} \\vspace{-6.4cm}\n\\centering\n\\includegraphics[width=.9\\linewidth]{Annotated_DartBoard_20.png}\n\\captionof{figure}{A standard dartboard}\n\\label{fig:DartB1}\n\\end{minipage}\n\\vspace{.05cm}\n\\noindent\n\n\nThe rules of darts that we describe here\\footnote{The rules for 501 and other less common forms of the game are described at \\url{https:\/\/www.mastersofgames.com\/rules\/darts-rules.htm}. Furthermore the geometry of the dartboard is described in Appendix \\ref{sec:geom} of the {\\em Supplementary Materials} file.} are for the most commonly played form of the game, namely ``501''. In 501, each player starts on a score of 501 and takes turns in throwing three darts. These three darts constitute a ``turn''. After a player's turn, his scores on the three darts are added together and subtracted from his score at the beginning of the turn. The first player to reach a score of {\\em exactly} zero wins the game as long as his final dart is a double, i.e. a dart that lands on D1, D2, ..., D20, or DB in which case we say the player has {\\em checked out}. If a player's turn would result in an updated score of 1, 0 (but not achieved via a double on his final dart) or a negative score, then the turn is invalidated, the player remains on the score he had prior to the turn and his opponent then takes his turn. It's possible for a player to win without having to throw all three darts in his turn. For example, suppose a player has a score of 16 just prior to his turn. If he then scores D8 with the first dart of his turn he wins. Alternatively, if he scores S8 with his first dart, then he could still win by throwing a D4 with his second dart.\n\nThe game we have just described is known as a ``leg'' and in practice darts matches are typically played over many legs with the winner being the first to win some fixed number of legs. Alternatively, some tournaments have a legs and ``sets'' structure whereby the winner of the match is the first to win a fixed number of sets and the winner of a set is the first to win a fixed number of legs.\nFinally, because the player that throws first has a considerable advantage, players alternate in starting legs (and sets).\n\n\n\\section{The Data}\n\\label{sec:Data}\n\nOur data-set relates to matches that were played by the top 16 professional players in the world during the 2019 season.\nFor each of the 16 players we have data of the form (TR, $z$, $n$) where $n$ is the number of darts that were aimed at the {\\em target region} (TR) and achieved a score of $z$. There are a total of 62 possible target regions, namely the single regions S1, ..., S20, the double regions D1, ..., D20, the treble regions T1, ..., T20, and the single and double bulls-eye, i.e. SB and DB.\nBecause many regions of the dartboard are very rarely targeted, the target regions that appear in our data-set are the treble regions T20, T19, T18, and T17 together with all the double regions D1, ..., D20, and the double bulls-eye DB.\n\nThe possible realized value of the score $z$ depends on the target region TR since a professional darts player will only very rarely miss his target region TR by more than a small distance.\nFor each treble region, there are therefore 6 possible $z$ scores. In the case of TR = T20, for example, the possible values are $z \\in \\{\\text{T20, S20, T5, S5, T1, S1} \\}$ because the 5 and 1 segments of the dartboard are adjacent to the 20 segment; see Figure \\ref{fig:DartB1}. Because we only have data for 4 treble regions, this means a total of $6 \\times 4 = 24$ data-vectors (TR, $z$, $n$) for each player where TR is a treble region.\nFor each double target region TR = D$x$ for $1 \\leq x \\leq 20$, we have 7 possible $z$ values represented in the data-set. For example, if TR = D16, then the corresponding possible values of $z$ are $z\\in \\{\\text{D16, S16, D8, S8, D7, S7, M}\\}$ because 8 and 7 are adjacent to 16 on the dartboard and where M denotes a ``miss'', i.e. a dart that fails to score because it landed outside the double region; again, see Figure \\ref{fig:DartB1}.\nWhen TR = DB, we have 22 possible $z$'s corresponding to \\{DB, SB, S1, S2, ..., S20\\}.\nBecause targeting DB and some of the double regions is relatively rare, we note that for some of the (TR, $z$, $n$) combinations we often have $n=0$.\n\nWe provide the treble data in Appendix \\ref{app:Data} of the {\\em Supplementary Materials} file and the entire data-set is available at \\url{https:\/\/github.com\/wangchunsem\/OptimalDarts}. We now provide some summary statistics of the data. Aggregated across the 16 players, there were  117,600, 27,709, 7,717, and 2,461 attempts at T20, T19, T18, and T17, respectively, and 41.2\\%, 41.7\\%, 36.9\\%, and 33.5\\% of these attempts, respectively, were successful. However, the success rates at the individual level vary considerably more than the aggregate numbers. For example, several players have a success rate of less than 30\\% when targeting T18 and over 40\\% when targeting T20. There were a total of 16,777 attempts at the 21 double regions with D20 (4,399), D16 (2,338), and D10 (2,064) together accounting for over 50\\% of this total. In contrast, there were only 1,866 attempts in total at the 10 odd doubles. These numbers reflect the preference for even doubles and, in particular, doubles that have powers\\footnote{If a player needs say D20 to check out on the second dart of his turn, then if he hits an M and or S10 (the most likely fails) he can still check out with his third and final throw in the turn.} of 2 in their factorizations. The average success rate at doubles across the 16 players was 40.2\\%. Note this is smaller than the success rates for T20 and T19, despite the fact that the double beds are approximately 50\\% larger than the treble areas or ``beds''. There are also substantial variations in success rates both across regions for a given player and across players. Some of this variation can certainly be explained by small number effects but even for the most popular double, i.e. D20, we see success rates vary from as low as 27\\% to as high as 47\\%.\n\n\n\\subsubsection*{Limitations of the Data-Set}\nUnfortunately, the data-set has two important natural limitations. The first is that while we always know the realized value of the score $z$, we never know the precise point where each dart lands. This was also true of TPT's data-set and led to their development of EM algorithms for estimating skill models. The second natural limitation is that when a player targets a particular region, e.g. T20, we do not know the exact point in the region at which he was aiming. This was not true of TPT since they used self-generated data where they aimed at the center of the DB. As we shall see, this limitation in our data-set leads to some difficulty in interpreting the skill models.\n\nThere are two other limitations of our data-set that we can assume away. The first one concerns so-called ``bounce-outs'' which are dart throws that fail to land in the dartboard. A ``bounce-out'' occurs because of a poor throw or because the tip of the dart strikes the wire that defines the boundaries between the different regions. With a sufficiently sharp dart tip, however, this rarely happens and in fact only approximately 3 in every 1,000 darts thrown by top players were bounce-outs in the 2019 season. We will assume there is no possibility of a bounce-out occurring when a dart is thrown.\n\nThe second limitation is that the data-set is not sufficiently granular to estimate a different skill model for each throw in a turn. It is known (see, for example, \\citealp{HotHands_RSSA}) that the success rate of the first dart in a turn is generally smaller than the success rate of the second and third throws of the turn. This can be explained by the need of the dart-thrower to ``re-calibrate'' at the beginning of each turn.\nIf we had sufficiently granular data, then for each player we could fit one skill model for the first throw in a turn and a separate skill model for the combined second and third throws. As we do not have such data, however, we will assume that each player has the same skill model for each throw in his turn. That said, it should be clear the issues we raise in Section \\ref{sec:Problems} would continue to hold even with more granular data and that our DM model could also be applied immediately to such data.\n\n\n\n\n\\section{A Skill Model for Dart Throwing}\n\\label{sec:SkillModel}\n\nLet $\\bvartheta \\in \\mathbb{R}^2$ and $(x,y) \\in \\mathbb{R}^2$ denote the intended target and outcome, respectively, of a dart throw. We use $z:=g(x,y)$ to denote the function $g$ that maps $(x,y) \\in \\mathbb{R}^2$ to the dart score $z$, e.g. D16, SB, T20, S7, etc. We will also let $h(z) \\in \\mathbb{N}$ denote the actual numerical score achieved by the dart, so for example, $h(\\text{D16}) = 32$, $h(\\text{T20})=60$, $h(\\text{SB}) = 25$, $h(\\text{S7}) = 7$, etc. Our goal then is to construct a model for $p(x,y;\\, \\bvartheta)$, the distribution of $(x,y)$ given $\\bvartheta$, that can be easily estimated with the available data.\nTPT proposed several such models including the model $[x \\ y]^\\top  \\sim  \\mbox{N}_2(\\bvartheta,\\sigma^2 \\bI)$, i.e. a bivariate Gaussian distribution with mean $\\bvartheta$ and covariance matrix $\\sigma^2 \\bI$ where $\\bI$ is the $2 \\times 2$ identity matrix. They also proposed $ [x \\ y]^\\top  \\sim \\mbox{N}_2(\\bvartheta, \\bSigma)$\nwhere $\\bSigma$ is an arbitrary covariance matrix. They developed an EM algorithm based on importance-sampling for estimating $\\bSigma$. They assumed the intended target $\\bvartheta$ was known for each data-point and that only the result $z=g(x,y)$ was observed rather than the realized location $(x,y)$. That only the $z$'s were observed is a very reasonable assumption given the difficulty of measuring the $(x,y)$-coordinates of dart throws and, as mentioned earlier, this is also true for the data-set we consider in this paper.\n\nIn their models TPT implicitly assumed the mean outcome $\\bmu$, say, and target  $\\bvartheta$ were one and the same. While this may be a reasonable assumption for professional players, we suspect it is unlikely to hold for amateurs. Even for professionals, it is quite possible there may be some {\\em bias}, i.e. discrepancy between the target $\\bvartheta$ and the mean $\\bmu$, especially for less frequently targeted parts of the dartboard. A further complication is that in our data-set we do not know $\\bvartheta$ even though we always know the target region. So for example, we will know if a dart was aimed at T20 but we will not know precisely where in the T20 bed the dart was aimed. We shall often (but not always) make the following assumption.\n\n\\begin{assum} \\label{ass:Center}\nGiven a target region TR, the specific target $\\bvartheta$ that a player aims for is the center of the target region with the center being defined as the midpoint of the polar-coordinates defining the region.\n\\end{assum}\nWhile this assumption may not always be true, absent any other information it seems like the most natural assumption to make. Indeed it is not difficult to imagine a player deviating from this assumption on occasion. For example, suppose a player has two darts remaining in his turn and needs a D5 to check out, i.e. win the game. Rather than aiming at the center of D5, he may prefer to aim a little closer to the outer circular boundary of the D5 region on the basis that if he is going to miss D5 he would prefer to miss the scoring region entirely rather than hit S5 which would leave him unable to check out on his final dart of the turn. This argument does not apply to even doubles such as D20, D18, D16, etc. because if the corresponding single is hit then the player can still exit on the next dart in his turn.\n\n\nEven if we invoke Assumption \\ref{ass:Center}, however, there remains the possibility of bias. It therefore makes sense to decompose the mean outcome $\\bmu$ according to $\\bmu = \\bvartheta +\\btheta$ where $\\btheta$ denotes a bias term. Given sufficient data, we can infer $\\bmu$ but if we do not know the target $\\bvartheta$ then we cannot infer the bias $\\btheta = \\bmu-\\bvartheta$. TPT assumed\\footnote{Throughout the paper we take the center of the intended target region, e.g. T20, D16, DB, etc., to be the origin. Hence a zero bias corresponds to $\\btheta = {\\bf 0}$.} $\\btheta = {\\bf 0}$ and since they knew $\\bvartheta$, they therefore also knew $\\bmu=\\bvartheta$.\nRegardless then of whether we invoke Assumption \\ref{ass:Center}, $\\bmu$ will be unknown and therefore must be estimated in our skill models. From a model estimation point of view, our decomposition of $\\bmu$ into $\\bvartheta +\\btheta$ has no bearing: we simply fit\n\\begin{eqnarray}  \\label{eq:DM2}\n[x \\ y]^\\top & \\sim & \\mbox{N}_2(\\bmu, \\bSigma).\n\\end{eqnarray}\nBut the decomposition of $\\bmu$ has significant\\footnote{To see this, consider two cases: (a) $\\btheta = {\\bf 0}$ so $\\bmu = \\bvartheta$ and (b)  $\\btheta$ is ``large'' in magnitude. In case (b) the player is presumably unaware of this large bias since if he was aware of it then he could make adjustments to counteract it. Therefore, if employing the skill model in a game of darts we would have to assume the darts player under interpretation (a) would be considerably more skillful than the same dart player under interpretation (b).} ramifications for using the skill model in a ZSG.\n\nRegardless of how we interpret $\\bmu$,\na simple model such as (\\ref{eq:DM2}) will not be sufficiently rich for modeling the skills of professional darts players. This is because these players tend to focus on (and practice throwing at) specific parts of the darts board, e.g. T20, T19, DB, etc. This means that a player's skill level, as determined by his $\\bSigma$ (and bias $\\btheta$), is likely to be a function of $\\bvartheta$. Indeed this is what we observe in the data. For example, the Dutch player Michael van Gerwen was successful 45.3\\% of the time when targeting T20 but only 30.2\\% of the time when targeting T17. This difference is statistically significant\\footnote{Moreover the approximate 99.9\\% confidence intervals (CIs) ($\\hat{p} \\pm 2.578 \\sqrt{\\hat{p} (1-\\hat{p} )\/n}$) for Van Gerwen's success percentages when targeting T20 and T17 do not overlap. Nor is this an outlier case. Taken together, the 16 players are successful 41.2\\% of the time when targeting T20 but only  33.5\\% of the time when targeting T17. Again, approx. 99.9\\% CIs for these proportions do not overlap.} and, more importantly, {\\em practically significant}. To see this we ran the following simple experiment. (We omit some of the less relevant details.) We assumed both players have no bias so their skill levels are determined by $\\bSigma$. We assumed player A's skill level was constant throughout the board and such that, when aiming at any treble region, had a success rate of $(45.3+ 30.2)\/2 = 42.75\\%$. Player B's skill level was identical to A's throughout the board except on T20 and T17 where his success rates were $45.3\\%$ and $30.2\\%$, respectively. Both players were therefore equally skillful when {\\em averaged} across the entire board and identical on all target areas except T17 and T20.  Over the course of a best-of-35 leg dart match, we found that player B would win approximately\\footnote{The precise number depends on which player starts the first leg.} 70\\% of the time. This is because T20 is far more important than T17 and so even if both players are equally skillful on average, the skill level on T20 is much more important than the skill level on T17. Hence, assuming the same skill model throughout the board can result in a drastic underestimation of a player's skill level and should be avoided. Finally, van Gerwen is in no way exceptional. As mentioned in Section \\ref{sec:Data}, on average the players are significantly more skillful at targeting T20 and T19 than T18 or T17.\n\n\n\n\n\\subsection{A Conditional Gaussian Skill Model}\n\\label{sec:CondGaussian}\n\nIn light of the data-set that we do have and the heterogeneity of success rates across target regions for each player, it seems appropriate to assume a separate skill model for different target regions. We therefore generalize (\\ref{eq:DM2}) to\n\\begin{eqnarray}\n \\label{eq:DM4}\n[x \\ y]^\\top & \\sim & \\mbox{N}_2(\\bmu_i, \\bSigma_i), \\ \\ \\mbox{ for } \\bvartheta_i \\in R_i.\n\\end{eqnarray}\nwhere $R_i$ denotes the $i^{th}$ possible target region in our data-set. That is, we will infer a separate skill model (\\ref{eq:DM2}) for each player and each target region (T20, T19, DB, D20, etc) in our data-set.\nThere are several reasons for using a separate Gaussian skill model for each target region. First, it is not at all clear how different target regions should be combined and, as we noted earlier when discussing van Gerwen's success rates at T20 and T17, a naive approach could lead to severe inaccuracies in estimating the win-probabilities of players in real-world matches. Second, as we shall see in Section \\ref{sec:Problems}, using a separate skill model for each target area allows us to highlight the natural limitations of the data-set.\n\nIt is worth noting that using a model such as (\\ref{eq:DM4}) (or indeed (\\ref{eq:DM2})) allows us to consider any potential target $\\bvartheta$ on the dartboard when solving ZSG's. We could not do this if we just used the players' empirical distributions even if we had data for each player on all (TR, $z$) combinations which (as discussed earlier) is not the case.\nWe note that (\\ref{eq:DM4}) is as easy to estimate as (\\ref{eq:DM2}) since we can partition our data-set by target regions. For each player we can therefore fit the skill model (\\ref{eq:DM4})  using a simple extension of the EM algorithm developed by TPT. This extension allows for the case where the mean $\\bmu$ is also unknown and therefore needs to be inferred. The EM algorithm is described in Appendix \\ref{app:EM} of the {\\em Supplementary Materials} file.\n\n\n\n\n\n\\section{Problems with the Conditional Gaussian Skill Model}\n\\label{sec:Problems}\n\nWe consider two possible specifications of the skill model (\\ref{eq:DM4}):\n\\begin{enumerate}\n\\item The first invokes Assumption \\ref{ass:Center} and assumes a zero bias, i.e. $\\btheta = {\\bf 0}$, so that  $\\bmu_i=\\bvartheta_i$ is {\\em known} for each $i$ and is equal to the center (in polar coordinates) of the $i^{th}$ target region. We refer to this specification as the {\\bf unbiased} model.\n\\item The second does not invoke Assumption \\ref{ass:Center} and assumes $\\bmu$ is {\\em unknown} and therefore must be inferred. We refer to his specification as the {\\bf inferred-$\\bmu$} model. This specification provides a better fit to the data as it increases the number of free parameters by two. It brings with it, however, the issue of how to decompose $\\bmu$ into the target $\\bvartheta$ and bias $\\btheta$ if and when the model is used in a ZSG setting.\n\\end{enumerate}\nWe begin with the skill model for DB. This is perhaps the most natural starting point and is also what TPT considered. Indeed for professional players the DB data is capable of providing the most information regarding the skill model. Specifically, and as discussed in Section \\ref{sec:Data}, a professional darts player is quite likely to hit any of the 22 regions when aiming at the DB. In contrast, when aiming at a treble or double region, a professional is only likely to hit 6 and 7 regions, respectively. We will use the term {\\em coverage} to refer to the number of regions with non-zero observations. So the maximum coverage for DB is 22 while the maximum coverage for trebles and doubles is 6 and 7, respectively.\n\n\nFigure \\ref{figure:FittedEllipseRealDataDB} displays 95\\% confidence ellipses\\footnote{Rather than present numerical values for $(\\bmu_i, \\bSigma_i)$, it is more informative to display results visually via CEs on the appropriate section of the dartboard. Due to space considerations, we only display results for 4 players here but Appendix \\ref{app:AdditionalResults} of the {\\em Supplementary Materials} file displays results for all 16 players.} (CEs) for 4 of the 16 players. The unbiased model fits looks reasonable although it is interesting to note the non-zero-correlations, i.e. off-diagonal term in $\\bSigma_i$, that are manifested by the non-alignment of the ellipses' axes with the x- and y- axes. (Our prior was that correlations should be relatively small and we will return to the issue of (possibly spurious) correlations in Section \\ref{sec:CorrelIdent}.)\nThe inferred-$\\bmu$ model fits are more interesting. Although they provide better fits to the data than the unbiased model, there are clear problems with the fits of Anderson, Lewis, and Whitlock due to insufficient coverage. Besides DB and SB, the coverage of Anderson (S14, S9, S5), Lewis (S14, S13, S9), and  Whitlock (S14, S2) in particular is insufficient to accurately estimate a model with 5 parameters. (The problem with Whitlock hints at the major problems to come when we consider fits to the double target regions.)\nIn contrast, the coverage of van Gerwen was 13 which results in a reasonable model fit and this was also true of most of the other 12 players in the data-set.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_B50_all_fourplayers.png}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) DB models. }\n\\label{figure:FittedEllipseRealDataDB}\n\\end{figure}\n\nThere are three major issues that arise when fitting the model separately on each of the main target regions, i.e. the high trebles and all of the doubles: (i) coverage and quantity of data, (ii) correlation identification, and (iii) interpreting $\\bmu$. We will discuss these in the following subsections.\n\n\\subsection{Coverage and Quantity of Data}\n\\label{sec:coverage}\n\nSevere problems can occur when fitting the skill model to double regions, including even D20, which accounts for over 25\\% of all doubles in the data-set. Fits for D20 and D14 are displayed in Figure \\ref{figure:Doubles} for 4 of the 16 players. In the case of D20, for example, while all 4 players have observations in D20, S20, and M, these are the only regions covered by Cross, while Cullen (S1), van Gerwen (D5), and Gurney (D5) only have a coverage of 4 when targeting D20. With such limited coverage the likelihood function will be quite flat in large neighborhoods and terminating at a reasonable (local) maximum is quite unlikely although it can happen. For example, the fits for both unbiased and inferred-$\\bmu$ models for van Gerwen and Gurney on D20 look reasonable despite only having a coverage of 4.  The quality of fits deteriorate as we consider doubles with even less data. There were only 312 darts targeted at D14, for example, which averages out to approximately 19.5 darts per player. It is no surprise then that the D14 coverages were low for all players including Cross (4), Cullen (3), van Gerwen (4), and Gurney (3) whose fits are displayed in the lower row of Figure \\ref{figure:Doubles}. Matters are even worse for the odd doubles with D19 (the best of them) having only 154 darts targeting it for an average of 9.6 per player and all but 1 player having a coverage of just 2 or 3.\n\nIt is worth emphasizing that in general we would not recommend using any of the fitted skill models for the double regions, even when the coverage is sufficiently high and the fits seem reasonable. This is because even when the coverage is good, the actual quantity of data may still be quite low and result in a reasonable-looking fit that nonetheless may be far from the true skill model. Our empirical Bayes approach will also help to address this quality issue.\n\n\n\n\n\n\n\n\n\\vspace{-2mm}\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_D20D14_all_fourplayers.png}\n\\end{center}\n\\vspace{-0.5cm}\n\\small\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) D20 and D14 models.}\n\\label{figure:Doubles}\n\\end{figure}\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\vspace{-8mm}\n\\subsection{Correlation Identification}\n\\label{sec:CorrelIdent}\n\nThe unbiased and inferred-$\\bmu$ models both fit the data well when there is sufficient coverage. This is demonstrated for T20, the target region with most data, in Tables \\ref{table:FittedProb_RealData_T20_fixedmu_fourplayers} and \\ref{table:FittedProb_RealData_T20_nonfixedmu_fourplayers} for the unbiased and inferred-$\\bmu$ models, respectively. The bottom row (``Fitted Error\") of these tables shows the mean (across the 4 players) absolute difference between the observed and fitted scoring percentages for each of the 6 possible outcomes when T20 is targeted. In the unbiased model, the average fitted error is approximately 0.3 percentage points. Not surprisingly, the inferred-$\\bmu$ model does even better with an average fitted error of less than 0.2 percentage points. We obtain similarly good fits\\footnote{It is worth emphasizing that we would obtain similarly low errors even for the doubles but, as is clear from Figure \\ref{figure:Doubles}, the resulting skill models are generally not credible and would not extrapolate well. For example, the D14 fitted skill models for van Gerwen in Figure \\ref{figure:Doubles} would suggest a probability of hitting S9 that is far too small.} for all 4 of the trebles.\n\\begin{table}[h]\n\\begin{center}\n\\captionsetup{justification=centering}\n\n\\renewcommand{\\tabcolsep}{1.0mm}\n\\caption{Observed vs fitted scoring percentages \\\\ The unbiased model for T20 is fitted using the raw data.}\n\\vspace{-1mm}\n\\label{table:FittedProb_RealData_T20_fixedmu_fourplayers}\n\\small\n\\begin{tabular}{lc rrr rrr rrr rrr rrr rrr } \n\\toprule\n\\multirow{2}{*}{Player} &~& \\multicolumn{2}{c}{T20} &~& \\multicolumn{2}{c}{S20} &~& \\multicolumn{2}{c}{T5} &~& \\multicolumn{2}{c}{S5} &~& \\multicolumn{2}{c}{T1} &~& \\multicolumn{2}{c}{S1}  \\\\\n\\cline{3-4} \\cline{6-7} \\cline{9-10} \\cline{12-13} \\cline{15-16} \\cline{18-19}  \\noalign{\\smallskip}\n~ &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit.  \\\\\n\\midrule\nCross&~&42.3&42.3&~&51.6&51.8&~&1.0&0.8&~&1.4&1.6&~&1.1&1.3&~&2.6&2.2   \\\\\nCullen&~&38.0&38.2&~&52.6&52.3&~&1.9&2.0&~&2.6&3.3&~&1.3&1.4&~&3.5&2.7    \\\\\nvan Gerwen&~&45.3&45.6&~&48.2&48.0&~&2.0&1.6&~&2.1&2.0&~&0.9&1.0&~&1.5&1.7 \\\\\nGurney&~&40.4&40.9&~&52.8&52.4&~&2.2&1.4&~&2.7&2.6&~&0.5&0.8&~&1.4&1.9    \\\\\n\\midrule\nFitted Error&~&~&0.3&~&~&0.3&~&~&0.4&~&~&0.3&~&~&0.2&~&~&0.5  \\\\\n\\bottomrule\n\\end{tabular}}~\\\\\n\\end{center}\n\\end{table}\n\n\n\n\\begin{table}[h]\n\\begin{center}\n\\captionsetup{justification=centering}\n\n\\renewcommand{\\tabcolsep}{1.0mm}\n\\caption{Observed vs fitted scoring percentages \\\\ The inferred-$\\bmu$ model for T20 is fitted using the raw data. }\n\\vspace{-1mm}\n\\label{table:FittedProb_RealData_T20_nonfixedmu_fourplayers}\n\\small\n\\begin{tabular}{lc rrr rrr rrr rrr rrr rrr } \n\\toprule\n\\multirow{2}{*}{Player} &~& \\multicolumn{2}{c}{T20} &~& \\multicolumn{2}{c}{S20} &~& \\multicolumn{2}{c}{T5} &~& \\multicolumn{2}{c}{S5} &~& \\multicolumn{2}{c}{T1} &~& \\multicolumn{2}{c}{S1} \\\\\n\\cline{3-4} \\cline{6-7} \\cline{9-10} \\cline{12-13} \\cline{15-16} \\cline{18-19}  \\noalign{\\smallskip}\n~ &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. &~& Obs. & Fit. \\\\\n\\midrule\nCross&~&42.3&42.3&~&51.6&51.6&~&1.0&0.7&~&1.4&1.6&~&1.1&1.2&~&2.6&2.5    \\\\\nCullen&~&38.0&38.0&~&52.6&52.8&~&1.9&2.0&~&2.6&2.3&~&1.3&1.1&~&3.5&3.8   \\\\\nvan Gerwen&~&45.3&45.3&~&48.2&48.3&~&2.0&2.0&~&2.1&1.9&~&0.9&0.7&~&1.5&1.7   \\\\\nGurney&~&40.4&40.6&~&52.8&52.8&~&2.2&2.2&~&2.7&2.4&~&0.5&0.3&~&1.4&1.7   \\\\\n\\midrule\nFitted Error&~&~&0.1&~&~&0.1&~&~&0.1&~&~&0.3&~&~&0.2&~&~&0.2  \\\\\n\\bottomrule\n\\end{tabular}}~\\\\\n\\end{center}\n\\end{table}\n\n\n\n\n\n\nFigure \\ref{figure:Trebles} displays 95\\% CEs for the two models for\\footnote{The 95\\% CEs for T19 are displayed in Appendix \\ref{app:AdditionalResults} of the {\\em Supplementary Materials} file. We omit them here since our observations for T20 also apply to T19.} T20, T18, and T17. For T20, we are pleased to see there is very little discrepancy between the unbiased and inferred-$\\bmu$ fits. T20 (and T19) are very popular targets and we would expect players to target the center of their treble beds with little bias. That said, there are some discrepancies as measured by the distance between the blue and red crosses in the sub-figures. In the case of Gurney and T20, for example, we see the inferred value of $\\bmu$ (for the inferred-$\\bmu$ model) lies close to the boundary between the T20 and S20 beds. We observe similar behavior for T18 (with the unbiased and inferred-$\\bmu$ models quite close), but we begin to see some divergence for T17 where, due to the smaller number of darts targeted at it, we begin to see smaller coverage values. Cross and Cullen, for example,  only have a coverage of 4 while van Gerwen and Gurney have a coverage of 6.\n\nA noticeable feature of the CEs in Figure \\ref{figure:Trebles} is that the correlations generally appear to be quite different from zero. However, we believe these correlations to be spurious and a function of the form of the data-set. In particular, we do not know the exact $(x,y)$-coordinates of where a dart lands. Instead we only know the region, e.g. S20, T20, etc., and this means identifying the correlation parameter is very difficult. To see this, consider Figure \\ref{figure:DataExample_T20} which displays three synthetic data-sets each consisting of 100 darts aimed at T20. It is clear the skill-models generating these data-sets have very different correlations with the first, second, and third data-sets displaying positive, negative, and (approximately) zero correlations, respectively. However, all three data-sets have identical outcomes, i.e. 37, 54, 6, 1, and 2 darts hitting T20, S20, S5, T1, and S1, respectively. As such, the three data-sets would have identical likelihood functions and there is no way for us to identify the correct sign (never mind value) of the correlation. While we have no darts hitting T5 in these artificial data-sets, it should be clear that not having full coverage\\footnote{It is true, however, that in the limit of infinite data eventually some darts would land outside the 20, 5, and 1 segments and allow for the identification of the correlation. But with professional darts players, we would be waiting a very long time for that to happen.} is not the cause of this problem. Moreover it is easy to see that this problem also applies to double regions.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_T20T18T17_all_fourplayers.png}\n\\end{center}\n\\small\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) T20, T18, and T17 models.}\n\\label{figure:Trebles}\n\\end{figure}\n\n\n\n\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=.8\\linewidth]{data_symmetry_example_T20.png}\n\\end{center}\n\\vspace{-0.5cm}\n\\small\n\\caption{Three synthetic T20 data-sets with identical outcomes but different correlations}\n\\label{figure:DataExample_T20}\n\\end{figure}\n\n\n\n\n\n\n\\subsection[Interpreting Mu]{Interpreting $\\bmu$}\n\\label{sec:mu}\n\nAs discussed in Section \\ref{sec:SkillModel}, if we do not invoke Assumption \\ref{ass:Center} then the question arises as to how interpret the inferred $\\bmu$ in terms of the unknown target $\\bvartheta$ and bias $\\btheta$. Moreover this interpretation has a significant impact when employing the skill model in a ZSG setting if we assume (as seems reasonable) that a player is unaware of his bias so that a larger bias implies less skill and poorer performance in a ZSG.\nIn this section, we briefly describe two points of evidence to suggest that there is a significant non-zero bias.\n\nFirst, if we assumed $\\bmu$ obtained in the inferred-$\\bmu$ model was indeed the intended target then that would imply players are intentionally lowering their chances of hitting the intended target. Consider, for example, Gurney when he is targeting T20. As may be seen from the corresponding sub-figure in the top row of Figure \\ref{figure:Trebles},  the inferred $\\bmu$ for Gurney is very close to the boundary separating T20 from the upper S20 bed. As may be seen from Table \\ref{table:FittedProb_RealData_T20_nonfixedmu_fourplayers}, his estimated probability of hitting T20 in his fitted inferred-$\\bmu$ model is 40.6\\%. However, if the fitted $\\bmu$ was indeed his intended target then by moving his target to the center of T20 he could increase this success probability\nto 44.5\\%. This would be a significant increase and it is difficult to imagine a player is intentionally behaving as sub-optimally as this.\n\nSecond, consider the average distance between the inferred $\\bmu$ (the red cross) and the center of the target region (the blue cross in the sub-figures). We can see from Figure \\ref{figure:Trebles} that this distance is increasing as we move successively through T20, T18, and T17. Because players throw considerably more often at the higher trebles, we would expect the magnitude of any bias to also increase as we move successively through these trebles and so our data is consistent with this observation. That said, coverage and data quantity issues certainly muddy this story particularly in the case of T17 and, to a lesser extent, T18.\n\n\n\n\n\n\n\n\n\\subsection{Potential Solutions to these Problems}\n\\label{sec:PotentialSolns}\n\nThere are various approaches we could take to mitigate the problems of Sections \\ref{sec:coverage} to \\ref{sec:mu}. For example, we could address the\ncoverage issue by adding (say) 0.5 observations to each neighboring region (of the target) with 0 observations. This would not address the data quantity issue, however. (We essentially do this more systematically in our DM model.) We could also consider combining regions but which regions should be combined? As mentioned just before Section \\ref{sec:CondGaussian}, a careless combination of regions could lead to a significant practical impact on predicted match outcomes. We could cluster based on success rates but this seems ad-hoc and would result in different partitions of the regions for different players. The issue of correlation identification could be finessed by simply setting correlations to zero and then using the inferred-$\\bmu$ model.  In that case, however, the inferred $\\bmu$ can move even further away from the center and, in some circumstances, even lie outside the target area. The question of bias becomes even greater then and assuming there is no bias makes little sense for the reasons outlined in Section \\ref{sec:mu}. Another possibility is to assume zero correlation and to then use the unbiased model. The fit to the data deteriorates and when we do this we are essentially assuming there is no bias despite the evidence outlined in Section \\ref{sec:mu}. This approach is therefore not entirely satisfactory although it is perhaps the most plausible of the approaches we have mentioned thus far.\n\nOur preferred approach is an empirical Bayesian one that relies on the DM distribution. As we shall see, this approach largely resolves the data coverage and quantity problems of Section \\ref{sec:coverage}. It will also help clarify the bias story of Section \\ref{sec:mu}. Finally, as we shall see in Section \\ref{sec:Empirical Bayes-ZSG}, we can use the empirical Bayesian approach to finesse the correlation identification and bias issues by considering ZSGs with only a single action per target area.\n\n\n\n\\section{An Empirical Bayes Approach: the DM Model}\n\\label{sec:Empirical-Bayes}\n\nThe main idea behind our empirical Bayesian approach is to use a hierarchical Bayesian model that leverages information across all players to fit a skill model for each target region. Model parameters are chosen\\footnote{We use the {\\em DirichletMultinomial} package \\citet{DM-R_Package} in \\texttt{R} for this.} via maximum likelihood estimation (MLE) rather than via the specification of a hyper-prior, and this results in updated ``raw'' data-sets which can then be used for fitting the conditional Gaussian skill models (\\ref{eq:DM4}) for each player to each target region. Our specific model, the DM distribution, is a well-known generalization of the Beta-Binomial distribution and has been applied extensively in baseball analytics for example; see \\citet{Jiang-Zhang2010} and \\citet{Robinson_Emp-Bayes}. More generally, the DM distribution is often very useful for modelling proportional data and \\citet{Minka2000} provides an excellent review of the model as well as various approaches to estimating it via MLE. Several chapters of \\citet{EfronHastie} provide an overview of empirical Bayes as well as recent developments and applications of the methodology.\n\n\n\\subsection{The DM Model}\n\\label{sec:DM-Model}\n\nThe Dirichlet distribution is a distribution on the $(K-1)$-dimensional simplex ${\\cal S}_K :=\\{{\\bf p} \\in \\mathbb{R}^K_+ \\, : \\, \\sum_{k=1}^K p_k = 1 \\}$ and is therefore a distribution over probability vectors in $\\mathbb{R}^K_+$. The density $f$ at a point $\\bp = (p_1, \\ldots , p_K) \\in {\\cal S}_K$ satisfies\n\\[\nf(\\bp) \\sim {\\cal D}({\\bsym{\\alpha}}) := \\frac{\\Gamma \\left( \\sum_k \\alpha_k \\right)}{\\prod_k \\Gamma \\left( \\alpha_k \\right)} \\prod_k p_k^{\\alpha_k - 1},\n\\]\nwhere ${\\bsym{\\alpha}} := (\\alpha_1, \\ldots , \\alpha_K) \\in \\mathbb{R}^K_+$. It is well known that $\\Ex[p_k] = \\alpha_k\/\\sum_k \\alpha_k$ and that $\\Var(p_k)=\\Ex[p_k](1-\\Ex[p_k])\/\\left(1+ \\sum_k \\alpha_k\\right)$. The Dirichlet distribution is conjugate to the Multinomial distribution which makes the Dirichlet distribution very convenient for modelling proportions.\n\nIn our darts setting, we will assume that the skill model of the $j^{th}$ darts player\\footnote{Recall that there are 16 professional darts players in our data-set.} targeting a particular region, e.g. D20, is represented by a draw $\\bp_j \\sim {\\cal D}({\\bsym{\\alpha}})$. In this case $K=7$ because there are $7$ possible outcomes of a dart aimed at D20. Conditional on $\\bp_j$, the dart scores of this player are $\\mbox{Multinomial}(\\bp_j)$. That is\n\\begin{eqnarray}\n\\bp_j & \\sim & {\\cal D}({\\bsym{\\alpha}}), \\label{eq:Post-DM} \\\\\n\\bx_j & \\sim & \\mbox{Mult}(n_j,\\bp_j), \\label{eq:Post-DM0}\n\\end{eqnarray}\nwhere $\\bx_j := (x_{j1}, \\ldots , x_{jK})$ is the $K \\times 1$ vector of dart scores and $n_j = \\sum_k x_{jk}$ is the total number of darts thrown by the $j^{th}$ player at the target region. In our example with D20 as the target, the possible outcomes are D20, S20, D5, S5, D1, S1, and M. In this case, $x_{jk}$ is the number of times the $k^{th}$ possible score, e.g. D20, S20, etc., was achieved. It is easily seen that the posterior distribution is Dirichlet, i.e.\n\\begin{equation}  \\label{eq:Post-DM1}\nf(\\bp_j \\mid \\bx_j) \\sim {\\cal D}({\\bsym{\\alpha}} + \\bx_j).\n\\end{equation}\nUnfortunately, we do not know ${\\bsym{\\alpha}}$ but we can easily estimate\\footnote{The alternative is to adopt a full Bayesian approach which would require us to specify a hyper-prior distribution for ${\\bsym{\\alpha}}$.} it via an empirical Bayes approach.\nTowards this end, it is easily shown that, unconditional on $\\bp$, the dart scores have a DM distribution with PMF\n\\[\nf(\\bx_j \\, ; {\\bsym{\\alpha}}) = \\frac{\\Gamma \\left(\\sum_k \\alpha_k \\right)}{\\Gamma \\left(\\sum_k x_{jk}+\\alpha_k \\right)} \\, \\prod_k \\frac{\\Gamma \\left(x_{jk} + \\alpha_k \\right)}{\\Gamma \\left(\\alpha_k \\right)}.\n\\]\nLet $D := \\{\\bx_1, \\ldots , \\bx_{16} \\}$ denote the data from the 16 professionals when targeting the region in question. The likelihood is then given by\n\\begin{equation}  \\label{eq:Post-DM2}\nL({\\bsym{\\alpha}} \\, ; D) = \\prod_{j=1}^{16} f(\\bx_j \\, ; {\\bsym{\\alpha}}).\n\\end{equation}\nWe can maximize (\\ref{eq:Post-DM2}) to obtain the MLE $\\widehat{{\\bsym{\\alpha}}}:= (\\widehat{\\alpha}_1, \\ldots , \\widehat{\\alpha}_K)$,\nand then substitute $\\widehat{{\\bsym{\\alpha}}}$ into (\\ref{eq:Post-DM1}) to take this as our posterior for player $j$. That is, we assume\n\\begin{equation}  \\label{eq:Post-DM3}\nf(\\bp_j \\mid \\bx_j) \\sim {\\cal D}(\\widehat{{\\bsym{\\alpha}}} + \\bx_j).\n\\end{equation}\nUnder this posterior model, the fraction of new throws from the $j^{th}$ player that would have the $k^{th}$ outcome is $(\\widehat{\\alpha}_k + x_{jk})\/\\sum_{l=1}^K(\\widehat{\\alpha}_l + x_{jl})$. We can therefore use these fractions in fitting the skill model (\\ref{eq:DM4}) for the $j^{th}$ player via the EM algorithm as described in Appendix \\ref{app:EM} of the {\\em Supplementary Materials} file. The  $(\\widehat{\\alpha}_k + x_{jk})$ quantities are often called {\\em pseudo-counts} and the {\\em pseudo-fractions}  $(\\widehat{\\alpha}_k + x_{jk})\/\\sum_{l=1}^K(\\widehat{\\alpha}_l + x_{jl})$ are shrinkage estimators that shrink the raw estimators  $x_{jk}\/\\sum_{l=1}^K x_{jl}$ towards their population means. It is easy to see that the amount of shrinkage that takes place for the $j^{th}$ player decreases in the amount of data $\\sum_{l=1}^K x_{jl}$ that we have for that player. This of course is a desirable property of these estimators.\n\n\\begin{rem}\nInstead of leveraging information across players for a given target region, an interesting alternative might be to leverage across regions for a given player. Unfortunately the DM model does not immediately adapt to this setting because the target regions do not all share the same value of $K$. We could circumvent this problem by using one DM model for the doubles (with $K=7$ and treating DB separately) and another for the trebles (with $K=6$). Even then an issue arises, however. Consider the targets T20 and T19, for example. Should a dart targeted at T20 that hits T1 be mapped to misses at T19 that hit T7 or T3? We have no way of answering this and so it is not all clear how to apply the DM model across regions for a given player.\n\\end{rem}\n\n\n\n\\section{The Fitted DM Models}\n\\label{sec:Fitted-Models}\n\nIn this section we display the same figures of Section \\ref{sec:Problems} except now the skill models (\\ref{eq:DM4}) are estimated using the pseudo-fractions $(\\widehat{\\alpha}_k + x_{jk})\/\\sum_{l=1}^K(\\widehat{\\alpha}_l + x_{jl})$ obtained from the fitted DM distributions. Figure \\ref{figure:FittedEllipse_DM_DB} is the analog to Figure \\ref{figure:FittedEllipseRealDataDB} in Section \\ref{sec:Problems} and displays the updated fitted skill models for the DB target area. We see the problems with the skill models in Figure \\ref{figure:FittedEllipseRealDataDB} have been resolved and that the correlations look much closer to zero.\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_DM_B50_all_fourplayers.png}\n\\end{center}\n\\vspace{-0.5cm}\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) DB models using DM pseudo-counts.}\n\\label{figure:FittedEllipse_DM_DB}\n\\end{figure}\nBecause there is ample data for players targeting T20, the pseudo-fractions for T20 are almost identical to the raw fractions and so minimal shrinkage takes place with the T20 data. Indeed the analogs for Tables \\ref{table:FittedProb_RealData_T20_fixedmu_fourplayers} and \\ref{table:FittedProb_RealData_T20_nonfixedmu_fourplayers} as well as the top row of Figure \\ref{figure:Trebles} (all for T20)  in Section \\ref{sec:Problems} are almost entirely unchanged when we use the DM data, i.e. the pseudo-fraction data, and so we do not display them here.\n\nThe amount of shrinkage increases as we move progressively through T20, T19, T18, and T17 but is still relatively small even for T18 and T17. Nonetheless, for players with relatively little data or coverage, we see the fitted skill models based on the pseudo-fractions help resolve the issues that arose when we used the raw data.  Figure \\ref{figure:FittedEllipse_DM_T18T17}, for example, displays the updated fitted skill models for T18 and T17 and should be compared to the bottom two rows of Figure \\ref{figure:Trebles} in Section \\ref{sec:Problems}. The problem with Cullen's skill models on T17 has been resolved, and we can also see closer agreement between the unbiased and inferred-$\\bmu$ models. We will return to this issue below.\n\nFigure \\ref{figure:DoublesDM} displays the DM analogs to Figure \\ref{figure:Doubles}. The shrinkage supplied by the DM model means that the coverage issues have been resolved and the updated skill models of Figure \\ref{figure:DoublesDM} look much more reasonable. It is worth noting that the DM approach does not work for some of the rarely targeted doubles. In particular, D11, D13, D15, and D17 are so rarely targeted\\footnote{There were only 232 darts in total thrown by the 16 players at these 4 target areas in the data-set. Darts targeting D11, D13, D15, and D17 therefore represent less than .15\\% of all dart throws in the data-set.} that even when we aggregate the data for these targets across all players, the coverages are still too low to allow a reasonable estimation of (\\ref{eq:DM4}).\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_DM_T18T17_all_fourplayers.png}\n\\end{center}\n\\small\n\\vspace{-0.5cm}\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) T18 and T17 models using DM pseudo-counts.}\n\\label{figure:FittedEllipse_DM_T18T17}\n\\end{figure}\n\n\n\\begin{figure}[h]\n\\begin{center}\n\\includegraphics[width=1\\linewidth]{ellipse_ModelFit_DM_D20D14_all_fourplayers.png}\n\\end{center}\n\\vspace{-0.5cm}\n\\small\n\\caption{95\\% CEs for the unbiased (blue) and inferred-$\\bmu$ (red) D20 and D14 models using DM pseudo-counts.}\n\\label{figure:DoublesDM}\n\\end{figure}\n\nIn Section \\ref{sec:mu}, we discussed the interpretation of $\\bmu$ as a bias term and noted that the average distance between the inferred $\\bmu$ and the center of the target region was increasing as we moved successively through T20, T18, and T17. We argued that because players throw considerably more often at the higher trebles, we would expect the magnitude of any bias to also increase as we move successively through these trebles and that our data in Section \\ref{sec:mu} was consistent with this observation. We now return to that issue. Table \\ref{table:distance_change} displays the absolute distance (in mm's) between the inferred $\\bmu$ and the center of the target region (denoted as {\\bf 0}) averaged across the 16 players for the four treble regions when (i) the raw data is used and (ii) when the DM \/ pseudo-fraction data is used. In both cases we see this distance increase as we move progressively from the most popular treble (T20) to the least popular treble (T17). As expected, the DM \/ pseudo-fraction data reduces these average distances (particularly for T17) but the effect persists nonetheless and in our opinion provides evidence for at least some bias being present, i.e. a non-zero $\\btheta$, in our interpretation of $\\bmu$.\n\\begin{table}[h]\n   \n    \\begin{center}\n    \\captionsetup{justification=centering}\n    \n    \\renewcommand{\\tabcolsep}{1.0mm}\n    \\caption{Average absolute distance (in mm's) between the inferred $\\bmu$ and ${\\bf 0}$}\n    \\vspace{-1mm}\n    \\label{table:distance_change}\n    \\small\n    \\begin{tabular}{lc rrr rrr rrr rrr rrr rrr r}\n    \\toprule\n    Model Fitted &~& T20 & T19 & T18 & T17\\\\\n    \\midrule\n    Raw Data  &~& 1.50 & 1.62 & 3.56 & 5.23 \\\\\n    DM \/ Shrunk Data        &~& 1.47 & 1.48 & 3.42 & 3.88 \\\\\n    \\bottomrule\n    \\end{tabular}}~\\\\\n    \\end{center}\n    \\end{table}\n\nUnfortunately, the superior fits provided by the DM approach bring to light a new problem. In particular, we see from Figure \\ref{figure:DoublesDM} that the main axis of the CEs tends to be aligned with the principal direction of the double region in question. For example, in the case of D14 we see the CEs tend to be oriented in a north-south direction which is also the general orientation of the D14 bed. Similarly, we see the CEs for D20 are oriented east-west which is also the orientation for the D20 bed. Indeed, as may be seen from {\\em Supplementary Material} file, this general behavior also persists for the other doubles as well. We believe this to be a shortcoming of the bivariate normal skill model in (\\ref{eq:DM4}) as it stretches the variance along the main access in order to fit the fractions of darts that landed in the neighboring doubles regions. A bivariate distribution with fatter tails might resolve this problem but owing to space considerations and the issues with correlation and interpreting $\\bmu$ (which would not be handled by a fatter-tailed distribution) we leave this for future research.\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{Using the Empirical Bayes Model in Zero-Sum Game Settings}\n\\label{sec:Empirical Bayes-ZSG}\n\nWhile the skill models of the professional players are of interest in their own right, they also play a vital role in the solution of the ZSG that models a game of darts between players. \\citet{HaughWang-Darts-2021} showed how a leg of darts could be formulated as a ZSG and used dynamic programming (DP) methods to find the unique Nash equilibrium of the game. In Section \\ref{sec:SingleActionPerTarget} we explain how the correlation identification issue of Section \\ref{sec:CorrelIdent} is relatively insignificant in the context of real-world darts matches among top players. Then in Section \\ref{sec:RealWorld} we show via real-world examples how a player's skill model in conjunction with a player's optimal strategy can explain observed behavior that might otherwise appear surprising. But first we provide a very brief review of ZSG concepts from \\citet{HaughWang-Darts-2021}.\n\nThe {\\em state} of a game or leg is denoted by $\\bs = (\\sA,\\sB,t,i,u)$ where $\\sA, \\sB \\in \\{0,2,3,\\ldots,501\\}$ are the scores of players A and B, respectively, at the beginning of the turn, $t \\in \\{\\mbox{A},\\mbox{B}\\}$ denotes whose turn it is,\n$i \\in \\{1,2,3\\}$ denotes how many throws are left in player $t$'s turn and $u \\in \\{0,1,\\ldots,(3-i)\\times60\\}$ is player $t$'s cumulative score thus far within the current turn. We use ${\\cal S}_{\\mbox{\\tiny ZSG}}$ to denote the state-space for the ZSG and it contains all possible values of the state $\\bs$ that can arise during the leg. The action-space ${\\cal A}$ is the set of possible targets on the dartboard so, for example, if we assume each square millimeter on the board is a feasible target then it is easily seen that there are approximately 90,000 feasible targets or actions.\n\nA strategy $\\piA$ ($\\piB$) for player A (B) is a rule telling him where in ${\\cal A}$ he should target his next dart as a function of the current state $\\bs$. Let $\\JpiApib (\\bs)$ be the probability that player A wins the leg when A and B play strategies $\\piA$ and $\\piB$, respectively, given the current state of the leg is $\\bs$. The min-max and max-min values of the game are then defined to be\n\\begin{eqnarray*}\n\\underline{J}(\\bs) &:=& \\min_{\\piB } \\max_{\\piA } \\JpiApib (\\bs), \\\\\n\\overline{J}(\\bs) &:=& \\max_{\\piA} \\min_{\\piB }  \\JpiApib (\\bs).\n\\end{eqnarray*}\nIt can be shown that $\\underline{J}(\\bs) = \\overline{J}(\\bs) =: J^*(\\bs)$ and that there exist strategies $\\piA^*$ and $\\piB^*$  satisfying\n\\begin{equation}\\label{eq:EquilZSG1}\n\\JpisApibs(\\bs) = \\max_{\\piA} \\JpiApibs (\\bs) =  \\min_{\\piB} \\JpisApib (\\bs) = \\underline{J}(\\bs) = \\overline{J}(\\bs) = J^*(\\bs).\n\\end{equation}\nIt follows from (\\ref{eq:EquilZSG1}) that $\\piA^*$ and $\\piB^*$ are optimal or Nash equilibrium strategies for A and B, respectively,\nand we say $J^*(\\bs)$ is the equilibrium value of the game starting from state $\\bs$.\nLet $\\VA(\\bs)$ denote player A's best-response (BR) value function to player B's strategy $\\piB$ for arbitrary states $\\bs \\in {\\cal S}_{\\mbox{\\tiny ZSG}}$. Then, player A's BR problem can be formulated as\n\\begin{equation} \\label{eq:BR-PRobForm1}\n\\VA(\\bs_1) = \\max_{\\piA } \\JpiApib (\\bs_1)\n\\end{equation}\nwhere $\\bs_1 := (501,501,\\text{A},3,0)$ is the initial\\footnote{For ease of exposition, we assume A is first to throw in the leg.} state of the game. \nProblem (\\ref{eq:BR-PRobForm1}) is a so-called stochastic shortest path problem and standard DP techniques can be used to show that it has a unique solution.\nThe optimal solution \/ Nash equilibrium for the game can be found by repeatedly solving A's and B's BR problems until convergence occurs. Convergence is guaranteed and typically occurs within just 2 to 3 BR iterations if one player's initial strategy is the optimal strategy\\footnote{This initial strategy takes no account of the opponent's score and is easy to compute via DP methods.} that minimizes the number of turns required to check out.\n\nGiven the optimal strategies $\\piA^*$ and $\\piB^*$, and A's equilibrium win probability $J^*(\\bs_1)$, it is straightforward to compute $\\PA(N)$, the equilibrium probability that A wins an $N$-leg match assuming A starts the first leg. Specifically, let $V$ denote the number of legs that A wins out of $N$ legs in total where A starts the first leg and the players then alternate in starting legs. If we let $\\VVA$ and $\\VVB$ denote the number of legs won by A that were started by A and B, respectively, then we have $V = \\VVA+\\VVB$. As a typical darts match has an odd number of legs, we can assume $N=2K+1$ so that a player needs to win at least $K+1$ legs to win the match. A simple conditioning argument then implies\n\\begin{equation}\n\\PA(N) = \\sum_{j=1}^{K+1} \\Pb(\\VVB \\geq K+1-j  ) \\Pb(\\VVA = j),  \\label{eq:Adv2}\n\\end{equation}\nwhich is easily calculated since $\\VVA \\sim \\mbox{Bin}(K+1,\\pA^*)$ and $\\VVB \\sim \\mbox{Bin}(K,\\pB^*)$ where $\\pA^* = J^*(\\bs_1)$ and  $\\pB^*$ is the equilibrium probability of A winning a leg given that B starts it.\n\n\n\\subsection{Limiting the Action Space to a Single Target Per Target Region}\n\\label{sec:SingleActionPerTarget}\n\nIn this section, we compare the performances of players having access to two different action sets. The first action set ${\\cal A}_{\\text{m}}$ is one where the player (the ``multi-action player'') can target every square millimeter on the dartboard and the second action set ${\\cal A}_{\\text{s}}$ is one where the player (the ``single-action player'') can only target the center of each possible target region. ${\\cal A}_{\\text{m}}$ and ${\\cal A}_{\\text{s}}$ therefore have approximately 90,000 and 61 actions, respectively. This comparison is of interest because the multi-action player must estimate the skill model (\\ref{eq:DM4}) for each target region whereas the single-action player's skill model relies only on the pseudo-counts \/ pseudo-fractions for each target region and therefore he has no use for the skill model (\\ref{eq:DM4}). The single-action player will inevitably be inferior to the corresponding multi-action player in a ZSG since he has a much smaller action set. If we find the degree of inferiority to be mild, however, then this would suggest that we lose little in the ZSG setting by restricting players to the action set ${\\cal A}_{\\text{s}}$ thereby finessing the issues of interpreting $\\bmu$ and identifying the correlation that we discussed earlier.\n\nTo make the comparison between the two players fair, we need to ensure that regardless of the target region, the success probabilities are equalized for the single-action and multi-action players when the latter targets the center of the target region. We do this as follows. Using the DM data, we fit the skill model (\\ref{eq:DM4}) to all\\footnote{As noted in Section \\ref{sec:Fitted-Models}, we do not have enough data to estimate for D11, D13, D15, and D17 so for these doubles we simply assume the estimated skill model for DB applies.} target regions. We then take the single-action skill model to be the multi-action skill model when the target is the center of the target region. We also assume both players have full information and therefore know their own skill models and their opponent's skill models. Finally we assume the multi-action player has zero bias, i.e the $\\btheta$ of $\\bmu$ is zero. This obviously favors the multi-action player in light of our discussion in Section \\ref{sec:Fitted-Models} where we argued there is some evidence supporting a non-zero $\\btheta$.\n\nIn Table \\ref{table:winprob_modelcompare} we report match-win probabilities (computed using (\\ref{eq:Adv2})) for single-leg, 21-leg, and 35-leg matches for 4 of the 16 players. In order to isolate the impact of using ${\\cal A}_{\\text{m}}$ versus ${\\cal A}_{\\text{s}}$ we only consider players playing against themselves, so for example, the first row of the table reports results for multi-action Cross versus single-action Cross, etc. Because starting the first leg provides a considerable advantage (especially for smaller values of $N$), we report separate win-probabilities for when the multi-action and single-action players start the first leg. The reported win-probabilities in the table are for the payer starting the first leg.  The ``gap'' columns in the table show the difference in win-probabilities and represent the edge that the multi-action player has over the single-action player.\n\n\\begin{table}[H]\n\\begin{center}\n\\captionsetup{justification=centering}\n\n\\renewcommand{\\tabcolsep}{1.0mm}\n\\caption{Win-Probabilities in $N$-leg Matches for Multi-Action vs Single-Action Players}\n\\vspace{-1mm}\n\\label{table:winprob_modelcompare}\n\\small\n\\begin{tabular}{lc cccc cccc ccc}\n\\toprule\n\\multirow{2}{*}{Player} &~& \\multicolumn{3}{c}{1-leg} &~& \\multicolumn{3}{c}{21-leg} &~& \\multicolumn{3}{c}{35-leg} \\\\\n\\cline{3-5} \\cline{7-9} \\cline{11-13} \\noalign{\\smallskip}\n~ &~& ${\\cal A}_{\\text{m}}$\/${\\cal A}_{\\text{s}}$ & ${\\cal A}_{\\text{s}}$\/${\\cal A}_{\\text{m}}$ & gap &~& ${\\cal A}_{\\text{m}}$\/${\\cal A}_{\\text{s}}$ & ${\\cal A}_{\\text{s}}$\/${\\cal A}_{\\text{m}}$ & gap &~& ${\\cal A}_{\\text{m}}$\/${\\cal A}_{\\text{s}}$ & ${\\cal A}_{\\text{s}}$\/${\\cal A}_{\\text{m}}$ & gap \\\\\n\\midrule\nCross     &~&       65.4&    64.9&     0.4&~&     53.6&      52.0&     1.7&~&     53.2&    51.1&     2.2\\\\\nCullen    &~&       64.5&    64.0&     0.5&~&     53.6&      51.7&     1.9&~&     53.3&    50.8&     2.5\\\\\nvan Gerwen&~&       65.8&    65.4&     0.4&~&     53.6&      52.2&     1.5&~&     53.2&    51.3&     1.9\\\\\nGurney    &~&       65.1&    64.8&     0.3&~&     53.3&      52.3&     1.0&~&     52.8&    51.5&     1.3\\\\\n\\midrule\nMean      &~&           &        &     0.4&~&         &          &     1.5&~&         &        &     2.0\\\\\n\\bottomrule\n\\end{tabular}}~\\\\\n\\end{center}\n\\justify\n\\small\n{\\em Notes.} Numbers are in percentages. ${\\cal A}_{\\text{m}}$ and ${\\cal A}_{\\text{s}}$ denote the multi-action and single-action players, respectively, with the player before the ``\/'' throwing first in the first leg of the match.\n\\end{table}\n\nOur first observation is that the gap numbers are very small, ranging from 0.3\\% - 0.5\\% in the single-leg case to 1.3\\% - 2.5\\% in the N=35 leg case. For any given player we see the gap increase in $N$ but this must be the case since any edge in win-probability over a single leg will be magnified\\footnote{Indeed the ${\\cal A}_{\\text{m}}$\/${\\cal A}_{\\text{s}}$ and ${\\cal A}_{\\text{s}}$\/${\\cal A}_{\\text{m}}$ numbers will converge to 1 and 0, respectively, as $N \\to \\infty$ and so the gap will also converge to 1 as $N \\to \\infty$.} as we increase the number of legs $N$. That the gap is relatively small even in the best of 35-leg case (a typical number of legs in finals of major competitions), suggests that the simpler action set ${\\cal A}_{\\text{s}}$ is more than adequate in ZSG settings. Indeed, these gaps numbers are almost surely overstated since (i) we assumed the bias $\\bmu$ was zero for each player and (ii) we assumed the skill model (\\ref{eq:DM4}) was an accurate representation of the multi-action player's skill model. At best, these assumptions will only be approximately true.\n\n\n\n\n\n\n\n\\subsection{Analysis of Some Real-World Game Situations}\n\\label{sec:RealWorld}\n\nIn this section we consider two specific real-world match-play situations each of which featured a player's decision that was viewed by pundits and fans as being at least somewhat surprising. We show how our DM-multi-action models can be used to analyze these decisions and in fact argue that the decisions were not surprising at all. To be clear, these situations are really only intended to be indicative of the types of analysis we could conduct given sufficient data, and for several reasons should not be viewed as definitive. For example, our models were fit using data from the 2019 season whereas our first example is from a match during the 2018 season. It is quite possible that the form (and therefore skill levels) of the players in 2018 were quite different from their average form in the 2019 season. This is also true for the second example even though it is from the 2019 season. It is possible that at the time of these matches the players' forms were different from their average 2019 form which is essentially what the skill models aim to capture. Moreover, and as discussed in Section \\ref{sec:Data}, we have ignored the possibility of bounce-outs and assumed the same skill model applies to each of the throws within a turn. All of this reinforces that our fitted skill models are only approximations to reality and more granular data would be required to improve them.\n\n\n\\subsubsection{Price vs Wright in 2018 Shanghai Masters Quarter-Final}\n\nThe first example we consider is a quarter-final match between Gerwyn Price (player A) and Peter Wright (player B) in the 2018 Shanghai Masters. This was a best of 15 legs match and the situation occurred in the $14^{th}$ leg with Wright leading by 7 legs to 6. In the $14^{th}$ leg Wright was on a score of 18 and it was his turn to shoot with Price on 20. An obvious exit strategy for Wright was to aim at D9. Instead, however, he (successfully) aimed his first throw of the turn at S2 leaving him two throws to exit on D8. However, he missed both of these throws and Price went\\footnote{Wright did win the final deciding leg, however, and therefore won the match. The incident in question and indeed the entire match may be seen via \\cite{WrightPrice} on \\texttt{YouTube}.} on to win the leg in his next turn.\n\nAfter losing this leg there was some questioning of Wright's decision to go for the S2 (and then D8) rather than having three throws at D9. The analysis using our fitted skill model for Price may be seen in Figure \\ref{figure:PriceWrightExample1_DM}.\nWe see that, according to the fitted DM skill model, the optimal decision for Wright was indeed to aim for S2 with his first throw (and to then check out via D8) rather than D9. While the difference in winning probabilities (71.4\\% for S2-D8 vs 70.5\\% for D9) is small, if anything the evidence points to his having made the correct decision.\nIt is interesting to consider what other players should have done had they been in the same situation as Wright.\nFor example, as displayed in Figure \\ref{figure:PriceWrightExample2_DM}, we find that Michael Smith should have targeted a D9 exit rather than the S2-D8 strategy of Wright. In Smith's case the difference in winning probabilities (77.8\\% for D9 vs 70.3\\% for S2-D8) is substantial. The difference in optimal strategies for the two players is explained by their fitted skill models (estimated using the DM pseudo-fractions) when targeting D8 and D9. In particular, Smith succeeds in targeting D8 and D9 with probabilities 41.5\\% and 44.7\\%, respectively, while Wright succeeds in targeting D8 and D9 with probabilities 43.0\\% and 36.4\\%, respectively.\n\n\\vspace{-2mm}\n\\begin{figure}[h]\n  \\begin{center}\n   \n    \\subfigure[Leg 14 of Price (player A) vs. Wright (player B) in QF of 2018 Shanghai Masters.]\n    {\\label{figure:PriceWrightExample1_DM}\n    \\includegraphics[width=0.48\\linewidth]{example_PriceWright_DM_fixmu_fig1.png}}\n    \\hspace{1mm}\n    \\subfigure[A hypothetical situation of Price (player A) vs. Smith (player B).]\n    {\\label{figure:PriceWrightExample2_DM}\n    \\includegraphics[width=0.48\\linewidth]{example_PriceSmith_DM_fixmu_fig1}}\n  \\end{center}\n  \\vspace{-3mm}\n  \\small\n  \\emph{Note:}\n  The heat-maps display win-probabilities as a function of where player B targets the first throw in his turn assuming both players play optimally thereafter. The left-hand heat-map displays the situation at the beginning of Wright's (player B's) turn.\n  In the first throw, Wright wins the leg with probability 71.4\\% if he targets S2 (the green ``+'') followed (if successful) by D8, or 70.5\\% if he aims at D9 (the blue ``$\\times$'') instead.\n  The right-hand heat-map displays the same situation but where player B is now Michael Smith.\n  In this case the optimal decision for Smith is to target D9 (the green ``+''), with a corresponding win-probability of 77.8\\%,\n  comparing to 70.3\\% if he aims at S2 (the blue ``$\\times$'').\n  \\caption{Comparing S2-D8 and D9 exit strategies. }\n  \\label{figure:PriceWrightExample_DM}\n\\end{figure}\n\n\n\\vspace{-0.5cm}\n\\subsubsection{Anderson vs Price in 2019 Grand Slam of Darts Final}\n\nOur second example comes from a match between Gary Anderson (player A) and Gerwyn Price (player B) in the quarter-final of the 2019 Grand Slam of Darts. This was a best of 31 legs match. In one of the legs Anderson was on a score of 80 and it was his turn while Price was on a score of 20. Anderson tried to exit via D20-D20 but missed outside (for a score of zero) with his first throw, failed to hit D20 with his second throw and therefore failed to exit on his turn. The most common approach to exiting from a score of 80 is to target T20-D10, however, and so we consider this situation using Anderson's fitted skill model.\n\nIn Figure \\ref{figure:AndersonPriceExample_DM} we display heat-maps for the various scenarios. Figure \\ref{figure:AndersonPriceExample1_DM} considers the first throw of the turn and in fact it suggests that the optimal action for Anderson was to target S20. It is interesting to note, however, that the optimal target (as indicated by the green cross) is located on the boundary of the D20 and S20 regions. This suggests the real goal was a score of D20 but that the strategy wanted to ``hedge its bets'' so that if Anderson missed D20, it was important to miss and score S20 rather than miss outside and score nothing.\nThis highlights one of the advantages of building a skill model which can analyze any target within a region. In Figure \\ref{figure:AndersonPriceExample2_DM} it is clearly optimal for Anderson to target D20 on his second throw if he had been successful in hitting D20 on his first throw. It is interesting to note that the green cross in Figure \\ref{figure:AndersonPriceExample2_DM} is now located in the center of the D20 region rather than on the boundary as in Figure \\ref{figure:AndersonPriceExample1_DM}.\n\\begin{figure}[h]\n  \\begin{center}\n    \\subfigure[Optimal to aim at S20 with first throw.]\n    {\\label{figure:AndersonPriceExample1_DM}\\includegraphics[width=0.48\\linewidth]{example_AndersonPrice_DM_fixmu_fig1.png}}\n    \\subfigure[Optimal to aim at S20 with second throw if first throw hits D20.]\n    {\\label{figure:AndersonPriceExample2_DM}\\includegraphics[width=0.48\\linewidth]{example_AndersonPrice_DM_fixmu_fig2.png}} \\\\\n  \\end{center}\n  \\small\n  \\emph{Note:} The main takeaway is that on the first throw of his turn, Anderson should hedge his bets by targeting the boundary between the S20 and D20 regions.\n  \\caption{Anderson (player A) vs. Price (player B) in QF of 2019 Grand Slam of Darts}\n  \\label{figure:AndersonPriceExample_DM}\n\\end{figure}\nWe note that Anderson's success rates (based on his fitted DM models) are 39.1\\% and 45.5\\% when targeting D10 and D20, respectively. In contrast, James Wade, for example, has success rates (again based on his fitted DM models) of 48.3\\% and 39.4\\% when targeting D10 and D20, respectively. It is therefore not surprising that he would have preferred\\footnote{And we confirmed this using our ZSG analysis.} the more typical T20-D10 checkout attempt than the D20-D20 preference of Anderson in the scenario above.\n\n\n\n\n\n\\section{Conclusions and Further Research}\n\\label{sec:Conclusions}\n\nWe performed an exploratory data analysis on a data-set for the top 16 professional players from the 2019 season. We identified several problems that arose due to natural limitations in the data and proposed an empirical Bayesian approach based on the DM distribution to overcome some of these problems. We also explained how the remaining problems could be finessed in the context of ZSGs by using the DM model with a restricted action set. There are several directions for future research. First, by interviewing players we could resolve the issue regarding the decomposition of  $\\bmu$ into the target $\\bvartheta$ and the bias $\\btheta$. Second, the technology to capture the precise landing location of a dart now exists and we anticipate that at some point this technology will be used in professional competitions. If this occurs and the resulting data is made available, then fitting skill models will no longer be a missing-data problem requiring the use of the EM algorithm. We could then certainly resolve the correlation identification issue in that event and develop skill models beyond (\\ref{eq:DM2}) that might more accurately reflect the players' skill levels.\n\n\n\n\n\\section*{Acknowledgements}\nWe are very grateful to Christopher Kempf (\\texttt{Twitter}  @ochepedia), Statistical Analyst for the PDC  for providing us with the data underlying this work and for some very insightful conversations. All errors are our own.\n\n\\bibliographystyle{ormsv080}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
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+{"text":" \nSUBURGATORY\nSUBURGATORY\n\nTwisted Tales from Darkest Suburbia\n\nLINDA ERIN KEENAN\n\nGuilford, Connecticut   \nAn imprint of Globe Pequot Press\nCopyright \u00a9 2012 by Linda Keenan\n\nALL RIGHTS RESERVED. No part of this book may be reproduced or transmitted in any form by any means, electronic or mechanical, including photocopying and recording, or by any information storage and retrieval system, except as may be expressly permitted in writing from the publisher. Requests for permission should be addressed to Globe Pequot Press, Attn: Rights and Permissions Department, P.O. Box 480, Guilford, CT 06437.\n\nskirt! is an imprint of Globe Pequot Press.\n\nText design: Sheryl P. Kober\n\nEditor: Lara Asher\n\nProject editor: Meredith Dias\n\nLayout: Joanna Beyer\n\nLibrary of Congress Cataloging-in-Publication Data is available on file.\n\nPrinted in the United States of America\nThis book is dedicated to the best man I could have   \never chosen to have a son with, Steve Mendes, and  \nthe now boy-man who made life in Suburgatory a   \ndaily joy amid the madness, Frank Keenan Mendes.   \nAlso to my late Daddy, Joe Keenan, and my beloved,  \nirreplaceable Mommy, Marie Thibodeau Keenan.   \nIf only you were here to see this all, Mommy.\nThe individuals and situations described in this book are composites based in part on the author's experiences and in part on her imagination. In some instances, names and other information have been changed to protect the privacy of the people involved. No resemblance to specific individuals is intended or should be inferred, unless specifically stated otherwise. This book is not and should not be construed as associated with or endorsed by any of the individuals herein named.\nIntroduction\n\nBehold, in graphic detail, the career suicide note of one Linda Erin Keenan. Each year since having my son and having the luxury to stay home with him, the Social Security Administration has quite graciously let me know, with this, my lifetime earnings statement, that after a decade of steady raises at work, I have gone from Hero in 2003, to pregnant and disabled with severe nausea in 2004, to Big Fat Zero in 2005 and beyond. Well, I prefer the term, \"Unpaid Mommy, Raising America's Future,\" you woman-hating bureaucrats!\n\nEvery year when I find this hateful scrap of paper in the mailbox, I once again see the clearest evidence I have of how I tumbled so hard and fast from a trash-talking, urban CNN news writer in New York City to an unemployed, depressed, suddenly suburban, and still trash-talking stay-at-home mom. I do have an unstoppable potty mouth as you'll see in this book, and I'm not talking about diaper chat here.\n\nThis Career Suicide Social Security Earnings document, in a way, was my ticket into an utterly foreign place I now call Suburgatory, where potty mouths (and minds) like mine are about as rare as black people\u2014in the 1 percent range, I'd say. Actually there are probably more black people than potty mouths, now that I think about it. In this strange land, I had a new baby, no friends, and not much more than a prescription for Zoloft to keep myself afloat. Apparently, the ticket was one-way, because I'm sure as hell still stuck in Suburgatory.\n\nThe original proposal for this book was picked by Warner Brothers in 2010, and you can see their imagining of Suburgatory on the ABC show of the same title, which debuted in Fall 2011. But while the TV series focuses on a transplanted teen from New York City, my book offers a vision of suburbia and contemporary American life that I witnessed when I myself was transplanted from the city after having my son. It is satirical local news that skewers mostly upper-middle-class American pieties and parenting obsessions (not least, my own). I also target racism, sexism, mommy wars, body self-hatred, sublimated suburban sexuality, and class warfare; willful ignorance to the broader world along with an America in decline; and the all-around bad behavior that I have seen raging underneath the surface of those obsessively tended suburban lawns and bikini lines.\n\nDo those \"isms\" make this sound like that annoying late 1980s sociology class you skipped, or worse, a women's studies course?! Well, fear not, there's lots of swear words and dirty talk all over Suburgatory. (See later, my combination of both obsessions with the lawns and the bikini lines, in my fake ad for Suburgatory's new, hot landscaping service: \"The Lawnzilian.\")\n\nTessa, the teenage character created by ABC's Suburgatory, was forced by the man in her life\u2014her dad\u2014to leave her beloved city life for this supposed suburban utopia, which the show creator says was inspired by her own experience as a teen. As I wrote about in my book proposal, I followed the same trajectory\u2014with my husband, Steve. Steve couldn't imagine raising our son in New York, which had become too unimaginably scary to him once he gazed upon his new, miraculous baby boy clone. I did not feel this way. But I couldn't imagine handling my scary-intense job and being a mom at the same time. (Several mom friends could handle it\u2014though not most\u2014it must be said. Sorry, bra-burning, second-wave feminists disgusted with their weak, pathetic daughters! You can ask my weary shrink and \u00adpharmaceutical-battered liver: I simply couldn't cut it.)\n\nOK, here's the point where you might be thinking of flinging your book or iPad or Kindle across the room, and saying to yourself: Oh my God, I've bought another whiny white mommy book. What was I thinking? Here she goes, complain, complain, annoying, spoiled whiny white mommy. I'm toning her! Well, let me explain precisely what this whiny white mommy is complaining about.\n\nI fully acknowledge that I am among the luckiest of women in the luckiest place anywhere on the planet. I chose to stay at home and that's not a choice most can make. That Big Fat Zero on my Career Suicide Note did not bankrupt me. I made good money for a handful of years, and have a husband with a good salary, and we are also both pathologically cheap.\n\nHow cheap are we? So cheap that we could swing living in modest homes in expensive suburbs with great schools. But full disclosure: This luxury was subsidized by money I inherited when both my parents died (by the time I turned twenty-nine). Considering their modest salaries as teacher and career guidance expert in my hometown of Albany, New York (black\/hispanic population near 40 percent\u2014take that, whitey-town suburbia!), the fact that they amassed any savings at all after sending three daughters to pricey colleges amazes me. Clearly they were cheap, too\u2014or, to use my technical term, \"Super Crazy Mega Cheap.\"\n\nMy problem, and of course it is a problem only in the upper-middle-class sense of the word, is that abruptly leaving my career for suburban mommyhood made me a foreigner in a place where conformity was king, subversion seemed policed, and where I often felt like I had been taken hostage by an adult Girl Scout troop.\n\nNo surprise that my first friends in suburbia were actual foreigners. Naoko and Yuki were my treasured Japanese lady friends who fit in far better than I did, even though they weren't part of the rich \"Power Asian\" set, a significant demographic in my new land. \"Rinda, this my home now!\" Naoko would say, with several kids to take care of and a husband deployed in Iraq (and now, sigh, he's in Afghanistan. Semper Fi, Kevin Conway.) Still, she somehow found time to scrapbook, and, oh, work overnights at a place where she championed and cared for the severely disabled. What she lacked in language skills, she more than made up for with her indomitable spirit and trays of homemade sushi rolls. I wasn't the least bit surprised at the strength we all saw after the Japanese earthquake. Not me, not after Naoko and Yuki, who's a gorgeous beam of steel herself.\n\nBut what made me a foreigner? Really, it was my love of the transgressive and the unspeakable, spoken out loud. I've always been this way. As a seven-year-old in 1977, I drove my very conservative, Depression-era mother insane after I read the book William's Doll and began my own tiny fag hag crusade on behalf of \"sissies\" everywhere.\n\nNo surprise, I was the all-purpose outcast of my Catholic school\u2014St. Bully's of the All Sadists\u2014where my only outlet from constant harassment was to furtively read Judy Blume's Deenie (the one where Deenie touches her \"special place\" with a washcloth! God, I'm still aroused thirty years later).\n\nNo surprise that my sole childhood friend was the only Jew my Irish Catholic family really knew\u2014Sheryl Olinsky\u2014who set me up for a lifetime of Heeb-lovin' and Chinese-food eatin', and whose Barry Manilow and purple-powered Bat Mitzvah was the defining social event of my childhood. Pretty much the only social event, now that I think about it.\n\nAnd no surprise I ended up in New York, a glorious Jewzapalooza and Homo Heaven rolled into one, working in a place where having an eye for the deranged and twisted was not just tolerated, but a job requirement. For me, my truest home was a network TV newsroom.\n\nEarnest moments are rare for me, and here comes one of them: I felt genuinely grateful, especially after 9\/11 at CNN, that in my own tiny deskbound way, writing hermit-like on my bits of anchor copy, I got to call bullshit on asshole thugs on a global scale, on a daily basis. \"Oh, I see, crazy Taliban nut-jobs, you plan to tip over a brick wall on top of an 'accused homosexual?' Now you're stoning a thirteen-year-old who was raped because she 'asked for it'? Allow me to help tell the world how sick and horrible you are!\"\n\nAnd beyond that ample satisfaction, I had the social benefits. I had found a family of trash-talkers like me, and we all relished incidents of guests gone wild. To get a control room full of satellite jockeys, camera guys, and nail-gnawing producers to laugh, you need to bring it hard, like the night when one guest, comedian Bill Maher, discussed basketball player Kobe Bryant \"trying to get some stanky on his hang-down. Oh, can you say that on CNN?\" Anchor Aaron Brown smoothly said, \"I didn't understand most of those words,\" and for a few seconds neither did we as we speed-translated in our heads what the \"hang-down\" and \"skanky\" were. (For you nice, innocent types, \"getting stanky on your hang-down\" means getting a girl to have sex with you, as in \"Linda was over last night? Did you get some stanky on your hang-down?\") Once those seconds were up, we began bellowing at the fact that Maher had slipped that tender bit of sweet talk onto CNN air, with no apparent FCC fine either.\n\nAnother favorite was no-holds-barred sex columnist Dan Savage. When we booked him as a guest, I remember thinking in the back of my mind, This guy could go X-rated, nasty-nuclear on live air. He was picked to appear after a U.N. weapons inspector and Marine was reported as being an open and active member of the sadomasochistic \"community.\" In the span of, um, four minutes? Savage hit the following topics: S&M (\"sort of cops and robbers for grown-ups with your pants off, and it usually ends in masturbation\"), vaginal and anal intercourse, balloon fetishes, smoking fetishes, and plushophiles (folks turned on by stuffed animals, and\/or who dress up like stuffed animals. Savage's advice: \"I hope there's a lot of Scotchguarded fabric on it.\"). He closed the interview by saying \"Personally, I haven't spanked a Marine, but I would make an exception for this man if I could see him first.\"\n\nNo one went ballistic about Maher's \"stanky hang-down,\" but I think Savage's bravura performance almost got a few of us fired.\n\nThere were also moments of sublime retribution: When we, the lowly staff, saw our own private Ron Burgundys\u2014the most imperious of anchors\u2014humbled. Some were famous, or, more annoying still, wannabe famous. These are things that never ended up on Page Six in the New York Post, but damn well should have. (I should add that I have worked with or around dozens of anchors and reporters at three different news organizations. Good luck trying to figure out who I'm talking about. To tease you further on this, see the \"Toddler or Anchor\" piece later in the book, satire that's actually all true. And to the many nice, normal anchors or reporters? Know that your staff worships you for your sanity.)\n\nA favorite was when one particularly annoying gasbag anchor left his wallet in the bathroom. Inside was a topless photo of his socially prominent, flat-chested girlfriend. (Is there any socially prominent woman who isn't flat-chested? And what about the booty? No booty either? Is there any preppy person with a booty?) That image got photocopied, by the way. Lucky for them this was back in the day, well before bare, semifamous A-cup titties would go viral within minutes. In any case, Sorry socially prominent, flat-chested, booty-free girlfriend. That's what you get for dating a pompous ass who can be nasty to his staff. A bumbling pompous ass who loses his wallet. He's booty-free too, by the way.\n\nAnd there are few things sweeter than seeing the anchor who just humiliated your beloved work friend having his bald spot spray-covered with Hair-In-A-Can after the previous application of \"hair\" melted off in the rain. Or hearing a know-it-all anchor mispronounce a word everyone with a pulse should know, leaving the tech guys busting a rib laughing.\n\nSo the humor was on the jagged edge and the pace was intense. There were a handful of times when I was writing copy for Anderson Cooper about forty seconds before it was to come out of his mouth. Lucky for me Anderson can edit on the fly, on air, while these never-before-seen words popped from my fingers onto the prompter.\n\nOne time with Anderson, I blasted in, with literally seconds to spare, something thoughtlessly inappropriate about kidnapping victim Elizabeth Smart, in some misguided attempt to be \"edgy.\" I watched Anderson process the words live and reject them, forcing him to vamp and instantaneously come up with something new and tasteful. Forced anchor vamping = massive fuckup for any copywriter. Crashing that hard, as we call it in news, left my hands sweating and my heart racing in one of the control rooms nicknamed the \"Screamatorium.\"\n\nBy 2003 (the peak \"Hero\" moment on my Career Suicide Note), it was in my head that my life needed radical change. The seed had been planted two years before on 9\/11. I heard the first plane crash from my apartment, just like millions of other New Yorkers, while Steve watched it fly in overhead. We were by no means affected in a truly personal way by 9\/11, that is, having a direct family member or friend murdered that day, as so many other friends actually did. (Yeah, you heard me, murdered. I'm one of those liberal pukes who felt zero ambivalence shouting out \"Fuck, YEAH\"\u2014and then crying\u2014when Bin Laden got it. Might have been a small fist pump, too. \"About fucking time\" is how this liberal reacted.)\n\nBut despite not having a family member or friend murdered, 9\/11 had a cumulative effect on me: watching people cry in the streets holding \"Missing\" signs; seeing half my local firehouse wiped out; waking up to the smell of burning rubble for months after; being evacuated after a bomb threat with my boss screaming, \"Everyone Get Out Now!\"; writing endless stories about it at work; watching field producers come back from Ground Zero looking stricken; having guys in puffy hazmat suits walk around the office while my only protection from possible anthrax was an old, ugly Ann Taylor outfit. It added up, as I'm sure it did with countless other New Yorkers.\n\nI vowed\u2014quite uncharacteristically for a cynical sort like me\u2014to take a chance on life, clean up my act, and have a baby. Steve said, \"Sure, what the hell.\" I think 9\/11 affected him too, though in a different way. His foul-mouthed daddy lion voice spoke to him saying, \"There's no fucking way I'm raising my kid in this crime-ridden terror trap. Roar!\" Not that we even had to debate the question. I knew I couldn't be a mom, keep my sanity, and continue in the Screamatorium; and we couldn't afford to stay in Manhattan on one salary. So off we went to the first of three suburbs over the next four years.\n\nWithin months of shvitzing in the Screamatorium, I found myself marooned in suburbia. Now, years later, with a happy son; a small house overrun with his brothers from different mothers; a select group of trash-talking friends; and Zoloft pills a-poppin', I see it as the birthplace of my new life as a thirty-something mom. I have a new set of satisfactions and, of course, a bigger ass. But at first suburbia seemed like nothing more than the graveyard for my twenty-something dreams. (Sooo overdramatic, spoiled whiny white mommy! Trying to score the Pulitzer Prize for Overwriting? Good effort!)\n\nI had gone from 80 miles an hour to zero, and I had vastly underestimated the crash this staggering deceleration might cause. My landscape now included Wal-Mart on one end of the class scale and Whole Foods on the other; and crossover between the two looked nonexistent. Nearly everyone was white (see Whole Foods), except nannies, cleaning ladies, and yard workers (see Wal-Mart), and all the dads and the working moms I never met vanished on the 6:04 a.m. train.\n\nSuburbanites seemed to me to have one-track minds, and mighty clean ones, too. They were about one business, and one business alone: baby and then child-raising. On paper, I didn't look so different from the other stay-at-home moms. They were mostly former professionals, too, some from Wall Street trading floors, which are easily as rough and tumble as any newsroom. But whatever edge these type-A women might have had now seemed gone, replaced by a version of hyper-vigilant parenting that was, to me, brutally boring and faintly absurd. They blathered on and on about how people without kids \"just didn't get it.\" They seemed busy busy busy and, most important to me, appeared very short on laughs. One day I spotted some obscene playground graffiti and I was the only one even willing to acknowledge it, as in, \"Look, that slide says VAGINA on it!\" I wished I could share my glee with my potty-mouth ex-coworkers, since my new \"coworker\" moms had no interest. But they were back in the newsroom slamming a show together, while I was soon to establish myself as that weirdo mommy at Gymboree.\n\nI knew very quickly that moving to the suburbs was a mistake when I realized that I missed my New York City doorman I'll call Rob, who was such a clich\u00e9 of the portly, klutzy, up-in-your-business doorman that no self-respecting comedy writer would ever dream him up. (And in fact, one of the world's most successful comedy writers lived in my building as he, too, became a parent and eventually left for the suburbs: Adam McKay. How do I know this, since I've never met Adam McKay? Because Rob told me all about him, what a great guy he was, his nice wife and his adorable baby and his latest fantastic SNL sketch and. . . .)\n\nEven at 11:30 p.m.\u2014no, especially at 11:30 p.m.\u2014when I was coming home to the East Village from work at CNN, exhausted, Rob would start in on an obscure Civil War fact, or deliver the results of the game he listened to on his handheld radio, or describe the latest outing he had with girlfriend \"Gladys\" (her real name, like Gladys, is straight out of an old-school sitcom, just like Rob himself). Rob could and would talk about anything, and he would still be talking as the elevator doors closed. Years later, he's surely still talking. If Rob had suddenly found himself in my suburb, an army of concerned mommies would have dragged his fifty-year-old ass in for a special-ed evaluation, because Rob might well have had Asperger's.\n\nSo it was while pushing a baby stroller through a suburban mall (my new \"town square\") that I started missing not just my urban friends and job, but especially Rob, my go-to conversation machine, and all the other random faces I would bump into, sometimes literally, going about my city life. I had gone from living vertically with dozens of couples or single people in the same building (using the same elevators, clogging the same trash room), to living horizontally, families cut off from other families in their own cocoons: self-imposed segregation in a most concrete way. Feeling so cut off surely magnified a nasty case of postpartum depression and the crushing loneliness that came with it. That's when I turned into, well, the stay-at-home-mommy version of Rob.\n\nI began talking to everyone, anywhere, anytime, all the time. Were people's facial cues telling me to back the fuck off, you crazy mommy? I didn't care. I followed a circuit of library story times with the devotion of a Dead Head (story time for tiny infants, mind you, who still don't know the difference between you and their own hand). I ate at the same diner every morning, ordering the same two-dollar egg sandwich until the waitress busting her ass recognized me. Yeah, she recognized me alright, as the spoiled mommy bitch who didn't have to work and wouldn't stop smiling at her, attempting pleasantries. I became an avid student of nanny culture and racially profiled them to find the most talkative ones. My inappropriate, sweeping generalization is that Caribbean nannies seemed the chattiest, and often cattiest, and therefore most desirable to me. I swear one favorite nanny showed such contempt for the parents of the boy-prince she was caring for that I thought her eyes were going to roll out and drop on to the park bench.\n\nI attended a ragtag sing-along at a bookstore that usually attracted just a few passersby and me, the sad-sack regular with her quarrelsome baby. The sing-along leader was Jean, a sixty-something sweetheart, who was so scattershot I thought she was either drunk or mis-medicated. She would sort of punt on her kiddie playlist after just a few highly awkward songs and one day even said, \"Linda, you take over.\" She seemed more unhinged than me, and that's saying a lot. (Jean didn't have a car, I learned, when I saw her blowing around in the rain waiting for a bus on a hugely busy, dispiriting commercial thoroughfare. I picked her up. A suburbanite without a car: the ultimate outcast.)\n\nAnd, as I mentioned, I joined a Gymboree class. That's where I met Bridget, Luv, as I called her in my head. Bridget, Luv, of course, was Irish; an older woman and a local legend among the baby-raising set for her savant-like knowledge of newborns. In my haze of postpartum depression, I had five of the worst seconds of my life in her class, when I actually forgot my beloved son's name, Frank. After class, I was so distraught I didn't want to leave Gymboree or Bridget, Luv, who said to me, \"You have the bad baby blues, luv. I seen it a-tousand times.\"\n\nAll my pent-up loneliness plus my suppressed impropriety had to go somewhere, and that somewhere was online. Google my name and in a few clickety-clicks you'll find a sorry list of intimate grotesqueries I cataloged about myself with abandon when the meds finally kicked in, I got my writing act together, and I started submitting to the Boston Globe and the Huffington Post. I was determined to entertain myself, even if it meant looking like a self-obsessed exhibitionist begging for laughs.\n\nMuch of the indignity happened on Facebook, which is just vastly more diverse than my real life in suburbia. (Please friend me, Linda Erin Keenan, on Facebook if you're so inclined. The crazier you are, the better. I genuinely love it.) Like other lonely souls out there, I fell into that vortex of making Facebook my real, not-real community. How could snow-white suburbia compete with this picture? I realized it could be my own massively Awkward Facebook Family Photo: the homeless artist, the Pakistani mariner, the military fetishist, the Renaissance faire\u2013loving transsexual lesbian massage therapist, the evangelical Republicans, my fashionista Mormon, the very sweet Sikh, the homeschooling pagan, two home-birthing doulas, the Texas BBQ restaurateur who promotes \"burnt end sandwiches\" right next to the hard-core vegan telling us that, say, my beloved Hot Pockets are killing me. All there and much much more in my wonderful Facebook nuthouse. Oh, and Buddy the Elf. He's there, too. He works for Santa. Says he's a real bastard.\n\nIn private, I pushed my boundaries further and began writing fake news satire, because eventually I went from bored to fascinated with the habits and fixations of upper-middle-class suburban life and parenting culture\u2014like the bubble-wrapping of the affluent child\u2014and what that says about America. Why do so many of the world's luckiest people seem so damn anxious?\n\nI'm fascinated by the way some women mercilessly judge other women's choices, and what motivates the harshest proponents of the \"pure\" and \"natural,\" especially in terms of breast-feeding. I see a lot of gory, competitive masochism in this area, like, say, \"My nipples bled more than your nipples.\" \"No, MY nipples bled more AND I got mastitis and then septic-shock!\" Well, at the risk of having frozen bags of breast milk pelted at my door, I really don't get why people are so passionately interested in how I feed my child or how I use or view my own breasts. As I recall, they are my breasts, and that baby is my baby, and it's actually quite an intimate act to press on others with such vigor. I also don't understand the many women and men who vocally trash those who breast-feed their kids publicly, or for years and years. None of your business.)\n\nI do have friends who believe strongly in breast-feeding, but they are lovely, advocate for all women, including the poor here and around the world, where breast-feeding can be a life-and-death choice for a baby if water is dirty. These activists are not toxically judgmental like, say, the \"Breast-feeding Nazi Really a Nazi\" I write about in the book. But vicious invective from others can be found all over the Internet. And passive-aggressive, thoughtless comments on breast-feeding, C-sections, epidurals, circumcision, staying at home versus working, and organic-food eating can have ugly impacts on fragile moms who choose to do things differently, or who might not even have a choice. It sure did on me, and on innumerable other friends.\n\nSadly, I think some of these movements inadvertently add to the yawning divide between rich and poor, or educated versus less educated. Not because the underlying goals of breast-feeding or eating organic food are unworthy, but because we are simply not set up in this country for subsidizing healthy food or fully supporting working mothers who breast-feed, poor or otherwise. Really, how does a woman working at low pay afford a souped-up Medela breast pump? How many women have jobs that will allow time to pump? Breast-feeding is only free if you don't put value on a woman's time.\n\nMichelle Obama can exercise her ass off all over the country (and God love her for it), but until we end, say, distorting food industry subsidies, the poor and middle class have every incentive imaginable to eat cheap crap, and many of the \"well meaning\" make the less fortunate feel guilty about it. Eating like a locovore is all good and great, but it's often expensive, time-consuming, and simply impossible for people of average means.\n\nSo the poor and middle class seem to be getting less and less healthy, while affluent suburbia (and \"urbia,\" too, for that matter) plies itself with every high-priced age-defying product, time-consuming betterment program or Whole Foods supplement, and basically jogs itself straight into its hale and hearty future. These days, Fat = Poor = Shame, as seen in \"Woman Shops at Wal-Mart to Feel 'Pretty, Thin,'\" among other pieces. Class, race, and religious collisions in mostly monochrome suburbia really interest me.\n\nIn a broader way, I also began thinking about America's diminished place in the world and how it might translate to the everyday business of raising a family; this whole idea of \"living the dream\" and maintaining a kind of phantom affluence in the so-called Great Recession.\n\nFor instance, why did Amy Chua's tough-parenting Tiger Mom book strike such a chord? Do we fear she may be right, that we Americans are coddling our kids into mediocrity? I wrote \"Indian Child Taunted as 'New Jew' at Middle School\" several years ago and thought Chua's \"shaming\" as an overdemanding mommy-shrew was quite similar to what happens to my overperforming immigrant Chaudry family.\n\nAmidst this anxiety, I feel like in wealthy suburbia we not only Bubble-Wrap our kids from the broader world, but also ourselves, even as wars are being fought in our name by our less fortunate, rural, urban, and not comfortably suburban countrymen.\n\nMeanwhile, I have seen the insistent creep of anti-Muslim, anti-\"other,\" anti-teacher, and anti-union resentment that has been percolating since 9\/11, but really seemed to explode as the economy collapsed and when President Obama was elected.\n\nI was actually asked if I would prefer another doctor because the one I selected, a Sikh, wore a turban. You'll see that story, a Sikh gyno's desperate bid to keep patients, in an op-ed titled \"I Am Certified Not Muslim . . . And I Love Your Feminine Area!\" I labeled these op-eds Shout Outs.\n\nOn a more personal level, I'm interested in the way we transmit our biases and neuroses (especially my own), like female body self-hatred, to our kids. If you see pieces attacking women, just know that, very often, I am attacking myself. Not to be hopelessly clich\u00e9, but I love how a child's unspoiled view of the world challenges our own jaded beliefs and often leaves us flummoxed. If I used to call bullshit on the Taliban, my own son now calls bullshit on me; and if you're a parent, you surely get what I'm talking about. And I see the way we rewrite our sometimes sketchy pasts once we get to suburbia, because You're a parent now, and that old life is over\u2014especially the dirty, sexy parts. I am fascinated and frankly sad that Suburgatory seems to be where sex goes to die, or at least gets suppressed. Well, it comes pouring out in this book, so get your raincoat!\n\nI have also included history's worst advice columnist\u2014Dr. Drama\u2014who gets earnest questions and wants nothing more than to stir the shit out of your already messed-up life. This was inspired by the often riotous and toxic comment sections of real website advice columns, where anonymity lets people project and splay their crazy any which way.\n\nAnd it should go without saying that anything labeled Paid Advertising Content is not a real ad. Believe me, I wish those were real. That would mean more money for me and my amazing agent, editors, and publisher. Also, as I mentioned above, I have lived in three suburbs in two states and have gotten ideas and themes from friends who live in a dozen more, mostly white, mostly affluent suburbs.\n\nSuburgatory is not the town where I live now. I wouldn't stay here if it was, because some of the fictitious people you're about to meet are truly awful\u2014and, hopefully, awfully hilarious. Any of the real people mentioned are, of course, used solely for parody purposes; Cynthia Nixon, a great actress and public school advocate, did not suddenly move to suburbia with her partner. Blogger Perez Hilton did not take a job as suburbia's zaniest new \"Manny.\" New York Times columnist Tom Friedman did not threaten an all-powerful high-school guidance counselor with a nasty column. And surely, Wolf Blitzer did not really report \"Live from the Lactation Room,\" though I'd give my left nut to see that actually happen for real. It's all satire and I'd hope Wolf, an anchor I long admired from afar at CNN, would get the joke.\n\nSo do I think of myself as a Big Fat Zero, like the Social Security Administration thinks I am, a no-paycheck parasite? No, because I neglected to mention that the Big Fat Zero did include a Plus One, and that would be the love of my life, my son, Frank Keenan Mendes. I only hope he never reads this book and realizes how sick and twisted his \"Best Mom Ever\" really is. Maybe after my funeral! And so I begin with a piece that's in large part true (though not the baptism part) about the near-year I spent as a secret atheist surrounded by simply wonderful Baptist believers. It is no exaggeration to say that these ladies helped save me from the abyss of postpartum depression. But sadly for them, they did not save me from hell.\nAtheist Mom So Lonely   \nShe Accepts Christ\n\nSuburgatory, USA\u2014Overwhelmed by the isolation of being a newly suburban stay-at-home mom in a town \"not cool enough\" for her, a local atheist has accepted Christ so she has someone to talk to.\n\nNonbeliever Mara Scully says her path to Christ was paved by the relentlessly friendly Baptists she met at Redemption Hill Church soon after moving here from Manhattan's East Village with her young son and husband. \"They were so friendly and cool in a really weirdo kind of way and not at all like those plastic mommy-drones in my neighborhood. And I was so lonely. When I heard there were other moms all gathered in one place, I didn't care if it was a Baptist church or a crack house. The crack house would have been edgier and more my speed, but, you know, whatever.\"\n\nScully enjoyed the weekly church play zone, despite what she described as \"a lot of crazy Christ stuff on the walls.\" She stresses that she is open-minded. \"Just because I'm an atheist doesn't mean I'm bigoted. Is bigoted the right word?\" It was nothing like what Scully recalled from her Catholic girlhood. \"No 'scary Jesus,' no gory wounds at all! Their Jesus is sooo happy, and you know what? So are they!\"\n\n\"They\" are the members of Redemption Hill, who are puzzled by their new recruit. \"She's not very pretty or turned out or 'New York,' is she?\" whispered Pastor Kevin Barnett's wife, Karen. \"And we've tried every polite way we can think of to tell her to stop swearing. Can you tell her?\"\n\nAfter attending the church twice weekly, enjoying potluck dishes like Tater Tot Pie, and developing warm if casual friendships, Scully said there were some contradictory feelings. \"Sure, I find their views on abortion, gay rights, and a woman's place in the home repugnant and all. But after a while they began talking more and more about living a Godly life and I knew where the train was going. What was I going to do? Lose my only friends out here in the middle of Bumblefuck Nowheresville?\"\n\nScully said she met with the pastor, talked about the Glory Christ can bring to a young family, the eternal paradise awaiting her after the Rapture, and then proceeded with the ritual Baptism. Is she now a believer? Scully laughed heartily. \"Not me. I'm still an atheist even though I did 'technically' accept Christ,\" she said, using air quotes.\n\n\"Of course I feel a little bad. But who's really getting hurt here? My Baptist buddies are thrilled that I'm saved. It would have hung over every conversation if I hadn't accepted Him, like when I wouldn't get a tattoo with my old girlfriends in Syosset. So awkward. Now I know how to talk the talk and we can keep having these nice coffees and playdates and, seriously, you gotta try that Tater Tot pie. Can't find that in the East Village! And since nothing happens after you die, no one's ever going to find out, right?\"\n\n\"Breast-feeding Nazi\" Really a Nazi\n\nSuburgatory, USA\u2014A \"Breast-feeding Nazi\" is an actual Nazi, combining her fierce lactivist advocacy with her membership in the American Nazi Party.\n\n\"You know who is really behind the formula industry? The Jews. Why don't you hear this on the news? The Jews. Why can I only buy my organic meat here at Whole Foods rather than hunt for it myself, as my White ancestors were able to? This one is a trick question. No, just kidding! It's the Jews again!\"\n\nJanie Tipton is a young mother with four children under the age of seven who believes that \"Aryan Americans\" need to repopulate America. This is one of two key tenets of National Socialism: the struggle for Aryan racial survival. \"We believe in the Fourteen Words,\" said Tipton, while searching the Whole Foods aisle for sprouted grain bread. \"The Fourteen Words are this: We must secure the existence of our people and a future for White children. And I personally would add six more: Breast-feed your babies, lazy White bitches.\"\n\nHow did her involvement in the American Nazi Party lead to her breast-feeding activism? \"Oh, you've got that backwards. The breast-feeding came first. After I had my first son, I became possessed with the power of natural birth, and saw that there was only one true, pure method of feeding your child: at your breast, and nowhere else. Anything less is a corruption, a defilement. I became frustrated with my lactivist sisters who were too accommodating, too easy on those weak and inferior bottle addicts. And the more I investigated who was enabling the addicts, I finally discovered the real enemy of breast-feeding: the corrupt Judeo-Capitalist system.\"\n\nDoes her husband believe as she does? \"Oh yeah, he is just trying to make enough as a money manager for a few more years and then we'll move back to the land to unchain ourselves from this horrible suburb, the teeming savages right across the town lines, and from the Jew-gamed agribusiness industry poisoning our bodies and babies.\"\n\nTipton is disturbed by the casual use of the term \"Breast-feeding Nazi\" to refer to anyone with a harshly judgmental attitude toward bottle feeding. \"Some of them think they're all badass for saying 'You should need a prescription to feed your baby formula!' You know what's badass? Telling bottle addicts to turn over their White babies to people who actually care about their health and future and freedom from allergies, their strength and purity and intelligence, which is what we will base our educational system on in our National Socialist future. If you don't believe that, stop patting yourself on the back by calling yourself a Breast-feeding Nazi, get out of the way, and let a real White woman handle this.\"\n\nThe other tenet of the American Nazi Party is social justice for the White working class. \"Of course, I believe that the White working man is now nothing more than a wage-slave, tax-cow, and cannon fodder, with their White babies forced to suck off the teat of the Jew-controlled formula-industrial complex,\" said Tipton.\n\n\"The only problem with this one . . .\" she whispered, \"is that I grew up in [the posh Connecticut suburb] Darien, so it's a little hard for me to talk the talk on the working man stuff, you know, The People of the Folk, and all that. And not to be a total bitch, but I have such nice teeth compared to my White brothers and sisters who believe as I do. So, yes, I really am the 'Aryan from Darien!' But you know, we hated Jews there, too.\"\n\nSHOUT OUT\n\nI Am Certified Not Muslim . . .   \nAnd I Love Your Feminine Area!\n\nDr. Vijay Singh is a Harvard-trained gynecologist who practices at the Marley Street office building.\n\nGreetings, gentle townswomen! I am passionate about your genital and reproductive health and have been trained at the finest institutions, including the Harvard Medical School. Despite my proven commitment to ladies' health, there seems to be some confusion about just who I am.\n\nI am a doctor first, a Sikh second, and certifiably not a Muslim. Sikhs are not Muslims. Trust me, Muslims are as strange to me as they are to you! I do have brown skin and I wear a turban, but I am not Osama bin Laden coming at you with a speculum, like some of you seem to think I am! And remember he is dead anyway. A turban is not a message that says, \"I'm about to kill you, infidel American.\" It's just part of my religion and identity. I don't look at all those baseball caps everywhere and think Red Sox Nation is coming to get me, even if it sure seems that way sometimes.\n\nNow to be fair we Sikhs have our terrorists, too, like the Muslims\u2014one of them killed Gandhi's daughter! That probably didn't sound very good. But really, don't you ladies know that we all have crazies in our shared genetic pool? I don't see the British throwing cans of spotted dick at every Irish person they see. So yes Sikhs have terrorists, too, but none of them has ever hurt an American that I know of. Those were just a teeny handful of Muslim crazies that killed Americans, and I just happen to look like them! I am part of the 99.9 percent of nice, boring, not-Muslim Sikhs out there.\n\nIn light of the many recent incidents involving ladies seeing my turban and immediately walking out, I am forced to change my cancellation policy. Now that my not-Muslimness and nice, boring qualities are on the public record, you will be charged a fifty-dollar cancellation fee if you decide I am too scary to do your pap smear. For those of you who are fair-minded and can see that I am simply a Sikh who just happens to look exactly like Mohammed Atta and wants nothing more than to keep your insides pink and shiny and healthy, I hope to see you in stirrups very soon.\n\nPAID ADVERTISER CONTENT\n\nBar Mitzvahs by Shiksas\n\nSo! You married a Jew! Maybe fifteen years ago or so? It was your Irish-American mother's dream come true. \"Don't marry some Irish stumblebum, find a Jew. They make wonderful husbands. They never cheat. Just avoid the ones named Spitzer, Weiner, and Madoff.\" Gosh, how did Mom know that? Because she's Mom, of course, she knows everything!\n\nBut your little not-really Jewish son is almost a man. And that shiksa in you wants a little representin' at his upcoming bar mitzvah. That's where Bar Mitzvahs by Shiksas comes in! Founded by goy goddess extraordinaire Erin Goodwin-\u00adGotbaum, our team of experienced shiksas will show you how to slip your cultural touchstones into the event with only the barest ripple of, \"Oh, that's the shiksa wife at it again.\" Well, it's your not-really Jewish child too, right?\n\nAt Bar Mitzvahs by Shiksas we can make sure that \"Danny Boy\" and \"When Irish Eyes Are Smiling\" just, you know, accidentally pop up on the DJ list. And of course we'll have the bar fully stocked. What is with these Jews and their constitutional inability to get down to business and drink? Quite the cross for them to bear, it seems.\n\nFor the Italian American, rest assured that Old Nonna Carnivale's gravy with meatballs\u2014pork, beef, and cheese meatballs, of course (are there any other kind?)\u2014will suddenly appear on the catering tables. Oh, those kosher guests will never know the difference. Or if they do, they'll think: Boy, these Italians might be a bunch of thugs but they sure know how to make a meatball. Nonna Carnivale's meatballs will put that horrible, bland kreplach to shame! And don't be alarmed, Jewish friends, when never-before-seen paisans\u2014local guys from the shiksa's own corner of the Old Country\u2014just show up. Because the meatballs are that good. And like Jews, paisans stick together.\n\nOr what if your heritage is just a bit . . . trashy? Now for you, no event is complete without Pigs in a Blanket, but your adopted Jewish community might find that a bit . . . d\u00e9class\u00e9, and your Pigs sure as hell aren't kosher either. Well, as we say quite often at Bar Mitzvahs by Shiksas: Tough titties! You, as the shiksa mom, let your child go unbaptized and now he's probably going straight to hell after death\u2014Jesus, Mary, and Joseph. The least these people can do is eat your Pigs in a Blanket! And with our help, very strong encouragement, and well-toned Shiksa biceps, they will. Oh yes, that's a promise.\n\nSo call us at Bar Mitzvahs by Shiksas! You can take the girl away from the goys, but you can't take the goy out of the girl. Embrace it, shiksas! And of course, mazel tov to your wonderful bar mitzvah boy and his loving, attentive, totally faithful, moral, and stone-cold sober Jewish dad.\n\nTown's Sole Goth Couple   \nWins Over Hearts, Minds\n\nSuburgatory, USA\u2014The only teenage goth couple in town, once considered an oddity or even a menace, has won over local citizens with the intensity of their devotion to each other and their lifestyle.\n\n\"Boy, that's a lot of velvet. They're like one tangled up unit\u2014oh my God! Look, you can't see their feet! It's like they're floating. Floating weirdo Siamese twins,\" said postwoman Julie Serra. She had just delivered mail to resident Frieda Graber. \"I didn't even know there still were goths. I remember way back, that guy with the black hair from that gloomy rock band but then when I saw he got fat, I thought, Well, that's over. You can't love food and hate life, right? Hypocrite. But these kids, I think they're for real.\"\n\nThe couple, who go by the names \"Thanatos\" and \"Sylvrefyre,\" first came to be known at Wagner High School by their refusal to separate during the school day.\n\n\"These two were a couple of losers before they found each other. They were sad plus scary, to be totally honest. Like, maybe not Columbine scary, but . . . you know, like small- to medium-size time bombs. And now look at 'em.\"\n\nPrincipal Gary Briscoe gestured to the couple, who were sitting silently on the basketball court, tracing invisible tear lines down each other's faces. \"Seriously, have you ever seen love like that? So yeah, I made some accommodations for them. I let them stay together and let 'em out of gym. Violates their 'beliefs' or whatever. And look what they gave me!\"\n\nHe fumbled under paperwork and produced an ornate pendant. \"They told me it's . . . where's that Amazon slip . . . here it is . . . It's a . . . 'Vladeptus Black Rose Gothic Pendant,' a 'stealthy bat who guards the rose noir, whose perfume reeks of death.' $14.95, on sale. Not bad. Now, I thought that was really thoughtful of them.\"\n\n\"I bet the sex is out-of-this-world great, too,\" the principal said quietly, apparently not realizing he was on the record. \"Wait, do goths have sex?\"\n\nParents who thought it was a phase that would end with the school year changed their minds during the summer heat. \"I saw them walking all the way to Dunkin' Donuts . . . in August. All cloaked up and crazy and all. I mean, a goth in August? That's commitment,\" said Seena Murray. \"It's a little sad because I remember Ashley\u2014sorry I mean Sylvrefyre\u2014when she was little and she was so pretty. I can still see that face, though, no matter how much of that insane makeup she puts on.\"\n\nThe couple tries to speak as little as possible, but did issue a written statement: \"We are thankful that the doomed, beautiful, and terrible people of this town have embraced us, and in return, we will honor their life essence long after their corpses begin their spectacular, eternal rot.\"\n\nThe one citizen not won over by Thanatos and Sylvrefyre is thirty-five-year-old Gina Hartnett, a former goth herself, who serves the couple at Dunkin' Donuts. \"Oh, please. I hated life before those brats were even born,\" said Hartnett. \"They'll be at one of those fancy weirdo colleges like [nearby] Hampshire College before you know it.\" Hartnett traded her goth getup for a Dunkin' Donuts uniform several years ago, after running out of tuition money for Green Valley Community College when her father was incarcerated for meth production. \"You want to really understand the excruciatingly awful pain of being alive? Spend eight hours making Coolattas. And go home with donut smell that won't wash off. Try that for a few weeks, posers.\"\n\n\"Funny Racist Lady\" Enchants Prominent Black Townsman\n\nSuburgatory, USA\u2014A woman couldn't contain her racist statements when encountering a black dad in town today, but rather than finding her offensive, the dad found her to be delightfully funny.\n\nKellie Alda is a kindhearted and irrepressible mother of two who is so disturbed by racism that when she actually interacts with a person of another race\u2014which is rare in this community\u2014she can't stop herself from injecting her darkest racial preoccupations into the conversation.\n\nShe first saw Deshaun Watson and his daughter Amahlia while standing next to them at the annual marathon.\n\n\"Oh hi! Good morning!\" she said, holding the hands of her twins, Peter and Emma.\n\n\"So . . . What's a black guy like you doing in a place like this?\" she asked, laughing nervously.\n\nWatson stared at her quizzically. \"Just showing my daughter the marathon.\"\n\nFive-year-old Peter looked up at Watson very gravely and said, \"Are you a jigaboo?\"\n\nAlda's hand went to her mouth. \"Oh my God! Peter! How mortifying, I'm so so sorry. I've been trying to teach Peter and Emma about the history and legacy of racism, which is a hugely important issue to me, so I was telling him all the nasty names for brown people that they should never use: sambos, coons, coloreds, negroes, blackies, jigaboos, jungle bunny, macaca, and you know, the big one, the N-one.\"\n\n\"Yeah, Peter, you might want to forget those other words and just stick with 'black,'\" said Watson.\n\nAfter a few moments, Alda leaned over and said, \"I hope seeing this doesn't bother you.\"\n\n\"Seeing a marathon?\" asked Watson.\n\nAlda said, \"Well, maybe I'm just really sensitive to race, but it's like a white power rally to me. There's a few black people being chased by an army of white people. I mean, I know it's a marathon and all but doesn't it look a little weird to you? Like they're out to run down and lynch those poor Kenyans? Not that these Kenyans are poor. I'm sure they are rich in Kenya\u2014I've seen them running on National Geographic\u2014I mean\u2014oh my God\u2014I mean, on ESPN. They don't wear shoes, but it's by choice\u2014better for running I guess! It's not that they can't afford it, hahahaha.\"\n\nAlda never asked Watson what he did for a living, because, \"I would just never want to ask a black gentleman what he does for a living. I mean, you don't want to make them uncomfortable if they aren't working, or doing something, you know, well you know, something else. This guy did seem kind of like a Mr. Mom. Which is great because, you know, black guys aren't always so great on the dad thing let's be honest. . . . What a fine man.\"\n\nLater on, she saw Watson again at the park with Amahlia. Their familiar greeting attracted the interest of other park-goers. \"Those moms are whispering and trying to hide their pointing! How disgusting, how utterly disgusting,\" said Alda, convinced the other park-goers were racists. \"A white mom and a black dad can't talk to each other without thinking about, you know, interracial porn? No, even worse, I bet they are thinking about Civil War slave porn, which is the sickest thing I've ever seen. It was so dirty and wrong and I just can't ever get it out of my head . . . and that slave's upper body, wow, just wow. . .\"\n\nWatson beamed at her in sheer amazement. \"Wanna come back to my house? We're going to get takeout,\" he said.\n\n\"Gee, well, hmmmm,\" Alda thought. \"Of course!\" She whispered to this reporter, \"How could I say 'No'? He'd think I was scared of him, but I wasn't, of course!\"\n\nAs she put the address in her car's GPS, the system began guiding her away from the park and her own relatively modest neighborhood, and slowly but surely the houses got bigger and bigger until they pulled up at a gated house\u2013complex of no less than twenty-thousand square feet, in the exclusive Westgate community.\n\n\"Oh wait,\" Alda said. \"Is this guy a manny [male nanny]? But the kid is black, too. Could she be an adopted child of a white family? How many white families choose black babies . . . not that many . . . isn't that awful? What horrible people there are in this world. Wait, could his bosses be . . . gay men? Hmmm.\"\n\nBut as Alda walked in, she slowly passed through a hallway lined with dozens of pictures of Watson in his NFL uniform, a picture here with Bill Clinton, there with Bono and Nelson Mandela. \"Ohhhh. So that's why the people in the park were looking at us? Not because you're black but because you're famous?\" Alda said.\n\n\"Well, probably a little of both,\" said Watson.\n\nBy this reporter's count, Alda had said a dozen moderate to appallingly racist things. Did it bother Watson?\n\n\"No! She cracks me up. Though not sure about my wife. Kellie's the first person in town who's even said the word 'black' to me. She just says what the rest of them are thinking and you can tell she's a sweetheart. Nicest racist white lady I've met in a long long time. Who clearly doesn't know shit about football, but that's gonna change. She's getting season tickets.\"\n\nMom Gives Up Pubic Hair for Lent\n\nSuburgatory, USA\u2014An area mom is giving up pubic hair for Lent and can't understand why others don't see this as perhaps the most appropriate choice to honor the suffering and death of the Lord Savior Jesus Christ.\n\n\"I mean, I've never done it, but my husband just mentioned it to me offhand. I know it will involve the flaying of my most sensitive flesh and then very itchy stubble and ingrown hairs. Now if that suffering doesn't bring me closer to knowing what Christ went through on the cross, well, I don't know what will,\" said Polly Tanner.\n\nWhen Tanner told her \"small group\" of Women of the Word at church of her decision, most seemed stunned and suggested other possibilities such as giving up Starbucks or gossip. \"Ha! Not giving up that last one!\" Tanner winked. \"I really pushed back on them and said [husband] James fully supported my decision. In fact, James even said this to me, 'I would love you even more than I do now, if I saw how much strength you had in giving up pubic hair to honor Christ.'\"\n\nRegina Clark, known as the most cynical of the church group, said, \"Riiiight. Your husband wasn't actually pushing you toward this idea?\"\n\nTanner said, \"Of course not! I've heard ladies take it all off and I've thought about doing it and giving up my pubic hair in the past few years. But this year, I really felt God nudge me on my shoulder.\"\n\nClark said, \"You know I love you, Poll, but you're gonna get a 'nudge' the likes of which you will not believe, and it's not on your shoulder.\"\n\nTanner replied, \"Yeah and isn't that horrific pain what Lent's all about?\"\n\nAs she made her way to the appointment, Tanner talked about how she has always liked to take good care of herself to honor her Creator, whether it was through maintaining her hair, nails, or teeth. She thought the time had come to add her pudendum to the list.\n\n\"It's disgusting down there, to be totally honest. Like a hairy smelly swamp monster,\" Tanner said.\n\nDoesn't God accept her this way?\n\n\"Well, smooth, hairless and clean, that's how I was born right? It was only after I turned into a totally out-of-\u00adcontrol trashy sin-crazy teenager that the hairy swamp monster appeared. I'm returning myself to God's original innocent, perfect vision.\"\n\nTanner chose the salon where she normally gets her hair cut. She had requested that Cristina, who emigrated from Guatemala, do the waxing. \"When they don't speak English, it's like they're not even here,\" Tanner whispered. \"I just thought it would make things more, you know, socially comfortable.\"\n\nCristina warmed up the wax and asked Tanner what exactly she wanted.\n\nCristina: Brazilian?\n\nTanner: What? I'm American.\n\nCristina: Clean? You want clean?\n\nTanner: Oh yes, clean.\n\nAs Christina began to apply the wax and started ripping, Tanner let the hot pain wash over her. \"Wow. Regina sure wasn't kidding.\" The waxing continued in speedy fashion and tears began to collect in Tanner's eyes. She said, \"This was the right thing to do. I really understand suffering, better than I ever have.\"\n\nOnce she was mostly finished, Cristina asked Tanner, \"The back? You want the back, too? Backdoor?\" Once again, Tanner was confused. Cristina struggled to explain that she wanted Tanner to get on all-fours to touch up the buttocks area. \"Your husband, he like the butt?\"\n\nTanner said, \"What? I'm doing this for God.\"\n\nCristina just shook her head and muttered \"Pinche gringo pendejo,\" which translates to \"fucking American idiot.\"\n\nWhen Polly walked out, James Tanner was waiting for her. \"Oh my God!\" said Polly. \"James never comes to pick me up at the salon! Look, he's so proud of me.\"\n\nPAID ADVERTISER CONTENT\n\nPetty Crimes Private Investigator\n\nEver wonder why that girl you used to work with and had no apparent problem with wouldn't friend you back on Facebook? Or why your child wasn't chosen for the status-team in your town's soccer league? Oh, and what about when your idea for the PTO fund-raiser was ripped out from underneath you? What the fuck was that all about? If it all sounds petty, but you just can't rest till you get an answer, then it sounds like a job for us at Petty Crimes Private Investigator.\n\nDon't call us at Petty Crimes Private Investigator if you've been involved in a crime, a domestic dispute, or anything that might end up on a permanent record. That's not what we do. We tackle those maddening mysteries you think about maybe a few times a year, but that consistently bug you when you do think about them.\n\nOur investigative techniques include light interrogation, Facebook and Twitter \"stalking,\" and perhaps some subtle playground or playing field observation. We can charge you less, because our private investigators are not those fancy \"prestige\" gumshoe PIs. If they were, they'd be working on real crimes! But you probably won't even come close to real crime in your lifetime anyway. It's the little things that matter to you, and at Petty Crimes Private Investigator, we sweat the little things right along with you. And don't think our PIs are pushovers just because they don't have the \"prestige.\" Many come straight from war zones after exciting and previously litigated stints with the private security firm Blackwater! They'll make quite the impression on the playground, you can be sure of that.\n\nSo the next time you ask yourself that bedeviling question, \"Why did that bitch from spin class pretend she didn't know me when she cut me in line at Starbucks last week? What the fuck was that all about?\" call or e-mail us at Petty Crimes Private Investigator.\n\nLesbian Hamsters \"Just Grew Apart\"\n\nSuburgatory, USA\u2014A pair of lesbian hamsters has apparently separated, a development that has united a family once divided over the same-sex couple in their home.\n\n\"We are all crushed. First, Ellen and Anne, then Rosie and Kelli, then Melissa and Tammy, and now Trixie and Fuff, members of our own family,\" said Flora Greenbaum, a longtime supporter of gay rights. She added, \"Lesbian break-ups hit me hard. I mean, if lesbians can't make it, who can?\"\n\nNot everyone in the Greenbaum household was pleased, at first, to see the hamsters growing close. \"My home is a judgment-free zone\u2014you know, like Planet Fitness. All shapes, sizes, and backgrounds welcome at Chez Greenbaum!\" said Flora.\n\nBut her husband David was not so welcoming. \"Two days in, Hannah came running in and said 'Look, Daddy, Fuff and Trixie are hugging like crazy!' And it didn't stop. They were glued to each other. Now, I'm a reasonable guy, I voted Obama. I'm not some homophobe, but no, I did not think my daughter needed to see that kind of thing at her age. And no, I was not going to have a 'conversation' about it either. And did I want the neighbors to come in and see two girl hamsters dry-humping each other? NO.\"\n\nFlora Greenbaum vehemently disagreed, and returned the hamster cage to the house after David tried to put it in the garage. \"I thought this was a great teachable moment for Hannah and I told David he can kiss my ass. I told Hannah that if a girl hamster falls in love with another girl hamster, they should have the same rights and freedoms and opportunities that all hamsters have,\" Flora said, as Fuff frantically clawed the sides of the cage.\n\nDavid tried to blunt this advocacy by challenging the very idea that Fuff and Trixie are actually gay. \"Think about it this way. Fuff and Trixie are actually in hamster prison. They have no one but each other. Do you think all those guys on that show Oz were gay? No! They had no choice but each other.\" David thought about that for a moment and then added, \"Wow, you don't think Fuff was raping Trixie, do you?\"\n\nDespite David's reluctance to embrace the hamster's sexuality, he was slowly won over by their fierce devotion to Thor, Fuff's son from a previous relationship. \"Don't even go near Thor\u2014they'll totally turn Seal Team Six on you, I'm serious. I mean, they're better parents than those loser slackers we never see at Hannah's soccer games. You know, if hamsters had soccer you can bet those lesbians would be at every single game, cheering every single goal without fail. So yeah, I'll admit, I was a little sad when Trixie and Fuff started drifting apart.\"\n\nThat happened, the Greenbaums say, a few weeks ago. Fuff started burrowing in the cedar bedding of the cage's right corner, while Trixie shuffled around listlessly and just leaned next to her wheel for hours on end. The only time their energy level seemed to increase was when they cared for Thor, separately, apparently trying to keep things as normal as possible for him.\n\nToday the Greenbaums finally decided to take them to the vet. \"I heard they give cats Prozac, so who knows what advancements there are these days for hamsters and mental health,\" said Flora Greenbaum. They explained the situation to Dr. Phoebe Macul. Dr. Macul looked a bit perplexed and said, \"I think you might be a bit confused. Hamsters are usually sold from the same litter. Trixie and Fuff are surely sisters. And Thor isn't Fuff's son\u2014Thor's a girl and also their sister, but she obviously was born small and didn't grow any hair, poor little thing; they shouldn't have sold her to you. Actually, I'm shocked their mom didn't just eat Thor. Anyway, I'm happy to hear Trixie and Fuff were getting along, a lot of times hamsters tear each other apart. Even though they are drifting now, you've actually lucked out.\"\n\nThe Greenbaums nodded silently and after a pause, David said quietly to Flora, \"Sisters? So this was . . . incest?\" Flora responded, \"Oh my God, like Flowers in the Attic. It's sort of sad and sick but sweet. . . .\" When Hannah Greenbaum asked, \"What's incest?\" Flora began to respond, until David raised his hand and said, \"Don't. Even.\"\n\nToddler News Junkie Thinks   \nGlenn Beck Is Kids' Show\n\nSuburgatory, USA\u2014A four-year-old news junkie recently helped Glenn Beck deal with the Mad That He Feels, something he learned from the late legendary child educator Mr. Rogers.\n\nJamie Baker watches various news programs with his mother and believes the newscasters are his friends. He especially likes Glenn Beck's show, which\u2014because of its occasional use of props, cartoons, and puppets\u2014he believes to be a kids' program. With Beck's \"cozy\" sweaters, as Jamie puts it, he even seems like Mr. Rogers. \"But Mr. Rogers doesn't get mad and then cry like Glenn does,\" said Jamie.\n\n\"Yeah, I sort of think it's a kids' program, too,\" said his mother Tricia Baker, who is a Cultural Studies Professor specializing in Marxist and post-Marxist Teddyry. \"I watch everything, including Fox News. No, especially Fox News. That's how I got pretty obsessed with Glenn Beck. I can't believe I am actually paying money to him now that he's off Fox and has this web-only thing. But it's nuts, totally crackers, and I just can't look away from that train wreck.\"\n\n\"Glenn! Glenn!\" said Jamie, as his mom began playing his nightly web program on her iPad.\n\nBut the child's face darkened as Beck began to discuss \"the people who hate America.\"\n\nJamie: What's wrong with Glenn tonight, Mommy?\n\nTricia: He's mad again, Jamie. You know when Mr. Rogers sings about \"What Do You Do with the Mad that You Feel?\"\n\nJamie: Yes! \"Do you punch a bag? Do you pound some clay or some dough? Do you round friends for a game of tag?\"\n\nTricia: Well, Glenn is working on using his words like you do at school, to get out the Mad that He Feels. He has a lot\u2014a lot\u2014of Mad he needs to get out.\n\nJamie: Mommy, he is talking about \"fascists.\" What's a \"fascist\"?\n\nTricia: Glenn is, honey.\n\nJamie: But he doesn't like fascists! How could he be one?\n\nJamie petted Beck's head on the screen. \"It's OK, Glenn. It will be OK. Everything will be fine!\" Beck's commentary continued to escalate in intensity, until he abruptly shifted tone and began to tear up and cry. Tricia leaned out of Jamie's earshot and said, \"You see? You couldn't make this shit up if you tried!\"\n\nJamie was troubled. \"Don't cry, Glenn! Ask your mommy for snack! Maybe you should go tinkle. You'll feel better!\"\n\nAfter a moment or two, Jamie said, \"Oh look, he's OK. He's calming his body. Yay! 'Adventures of Spooky Dude!'\" Adventures of Spooky Dude is a cartoon that features financier George Soros, to which Jamie asked, \"Why doesn't Glenn like Spooky Dude, Mommy?\" Tricia said, \"Because Spooky Dude gave a billion dollars away but not to Glenn.\" Jamie took this in. \"Oh. I like it when he gets out his blackboard or sits on his desk. He's just like you, Mommy. A mafessor!\" At that, Tricia snorted. \"Riiiight. Just like me. Professor Goebbels.\"\n\nSo if she views Beck as a fascist, why does Tricia let her young son watch him? \"Yeah it's a little nuts, I know. In my house, Jamie watches either news or kids' programming produced before 1980. And only kids' programming produced before 1980. Have you ever seen Dora the Explorer? As in actually sat through it? Talk about a fascist.\"\n\nSHOUT OUT\n\nCasey Anthony Was Always Guilty   \nby Reason of Eyebrows\n\nMaureen Bentley is a mom and part-time accountant who lives on Stratford Street.\n\nLike every mom in America, I was disgusted and appalled by that \"Not Guilty\" verdict in the recent Casey Anthony let's-dump-my-toddler-who's-getting-in-my-big-slutty-way trial. Now I'm not some fancy lawyer or anything, but I've seen more than my share of Law and Order episodes, and I am mystified that the jury showed a complete disregard for what I saw as the clearest evidence of her guilt. And that was her intense devotion to eyebrow hygiene.\n\nI'm really not a snap judgment person at all, and in fact I have served with integrity on several juries, including one where we sent a glue-huffing scumball to prison for burglary.\n\nBut the first time I saw that mug shot of Casey Anthony, I didn't need to see any more \"evidence\" beyond those over-coiffed eyebrows. I knew she was guilty as sin. No normal mother would have a daughter go missing and think to herself, \"Hmmm, gee, my eyebrows are looking a little shaggy, better touch them up in front of my lighted mirror!\"\n\nNot to say that her eyebrows looked good. My eye-shaping genius Irina shuddered when she saw that mug shot because of how that \"mom\" butchered those eyebrows! Then the whole eyebrow thing got even crazier during the trial where she seemed to go on some wild, uneven plucking orgy. Here she is, still worried about her brow arch, while sitting in jail not potty training little Caylee and not waking up in the middle of the night, exhausted and pissed off like the rest of us worn-out mothers who don't kill their babies?\n\nCome on, jury! A woman still worried about those brows while \"grieving\" a dead toddler? If that was me, my eyebrows would be as big as Borat's mustache. Frida Kahlo would have nothing on me! Why? Because I love my children and I'm not some piece-of-trash whore and neither are any of my friends. There have been times now and again when I've let myself go, you know why? Because I was too busy running myself ragged being a great mom.\n\nRule of Law and Reasonable Doubt my ass. Get your act together, American Justice System, we're supposed to be a beacon of sanity around the world. Next time, look to the eyebrows.\n\n\"Intactivist\" Mom Celebrates   \nArea Foreskins\n\nSuburgatory, USA\u2014A local \"intactivist\" mom has formed an unlikely alliance with a gay man who fetishizes the \"uncut\" penis.\n\n\"Circumcision is barbaric, an all-out human rights atrocity that we are inflicting on the most helpless among us,\" said activist for \"genital integrity\" Tara Cote. \"At first I was just plain angry. I put a homemade sticker on my car with a slogan: People Who Circumcise Aren't Playing with a FULL DICK. But a lot of people honked and gave me the finger.\"\n\nCote said that experience made her change her tack. \"Instead of guilting parents, I'm trying to show how natural and beautiful the foreskin really is. I'm still judging them, of course. I just don't tell them.\" Isn't that hypocritical since Cote admits she had her first son circumcised? \"No! I was uneducated, uninformed. There was no movement when I deformed my baby. Well, now there's no one that can say, 'I didn't know.'\" Cote's \"reeducation,\" as she calls it, came after her involvement with an influential Facebook campaign called \"Fuck You, Keep Your Babies Whole.\"\n\nWhile trying to dispel the notion that foreskins \"are stinky,\" \"collect cheese,\" and \"look sad,\" Cote stumbled upon websites that glorified the foreskin, for example \"Uncut, Uncensored\" and \"Hooded Heartthrobs.\" From this, a fruitful political partnership was formed. \"I found some fringe groups, but after I started chatting with them, they were talking about how vicious the Jews are and how they are controlling the world, it was really scary. So I'm so glad I found Libearache, which is what he likes to be called in the\u2014what did he call it?\u2014the 'bears, cubs, and chubs community.' Don't call him Donald. He gets really mad, but that's his real name,\" she whispered.\n\nLibearache explained his advocacy. \"You know when I'm in my comfort zone with my cubs and chubs, I talk about how succulent and fragrant a ripe, unwashed foreskin can be. But out here when I find these moms and moms-to-be, I just talk about how foreskins are God's creation and all that bullshit. And how circumcision is cruel and horrible and dangerous. Which it is.\"\n\nWhen some mothers asked Cote about Libearache's fetish-wear, which Libearache toned down considerably for his suburban audience, Cote replied, \"Um, he's just . . . a motorcycle enthusiast.\" Some moms were also disturbed that he was approaching pregnant women, saying, \"May the foreskin be with you!\" But others found Libearache an engaging presence, with Cote adding, \"He can be a real chatty Patty when he wants to be and loves looking at the latest gossip on TMZ. He's fun!\"\n\nCote is confident she and Libearache can convince moms of the horrors of circumcision even if they, like her, have already chosen it for their first child. \"We can save the next one,\" she said. What about the growing body of research suggesting circumcision reduces rates of STD transmission? \"Oh, yuck! Look, gays are the only ones who really have to worry about that stuff and my kid's sure as hell not gonna be gay.\"\n\nPAID ADVERTISER CONTENT\n\nSwagger-Speak International\n\nWhen your teenager learns a second language, you envision him or her taking a school trip to another country. Do you want your child to come across as a timid naif ready to be taken advantage of? Or as a proud American with all the swagger the world has come to expect and fear? Of course you want the swagger, you love your country!\n\nAnd you can bet your bottom dollar \"Monsieur Fitzgerald\" at school won't have the balls to risk all that politically correct censure and teach the swagger-speak your child really needs. That's where Swagger-Speak International comes in! Our swagger-speak is taught by a veritable United Nations of tough-as-nails nannies and au pairs from around the world. And trust me, they love hurling these phrases at your bright American child; we barely have to pay them! No matter where your children choose to do their overseas college application-builders, they'll be armed with phrases like these, when visiting a pub or jostling for position in a train ticket \"queue,\" or on that African eco-safari!\n\nIn Polish:\n\nOn jest zasrany skurwysyn.\n\nHe is a shit-covered bastard.\n\nIn Mandarin:\n\nCiao Ni Zu Zong!\n\nFuck your mom!\n\nIn Swahili:\n\nMshenzi we!\n\nYou savage!\n\nIn Japanese:\n\nOmae no Kaasan Sakana kusa.\n\nYour mom has a bad fish smell.\n\nIn Russian:\n\nChto b ty provalilsia, mudak dolbanyi!\n\nMay you fall through hollow cunt!\n\nIn Swedish:\n\nUrs\u00e4kta mig, men din fitta syns.\n\nExcuse me, but I can see your pussy.\n\nIn Spanish:\n\nMe Cago En Su Puta Madre.\n\nI shit on your whore mother.\n\nIn German:\n\nDu Saftsack.\n\nYou stupid bastard.\n\nIn Tamil:\n\nKandaraOli - Cun-daara-Olee.\n\nSlut . . . who sleeps with anyone.\n\nIn Arabic:\n\nAir il'e yoshmotak.\n\nMay you be struck by a dick.\n\nNow, as a concerned parent, you might fear that these phrases will get your child in a bit of trouble. And yet we've only had two \"international incidents\" . . . and consider the outcomes! One of our students got to meet Secretary of State Hillary Clinton, and the other now has his own Wikipedia page! Think how impressive that will look for college admission. And isn't the small risk of a modest scuffle worth the pride that your child will feel as an American who stood up to an uppity foreigner on his own turf and using his own language? It's like the bluster of George W. Bush plus the globe-trotting know-how of Barack Obama. Call us at Swagger-Speak International. And remember, if you ever meet a real America-hating fuckwad in Bucharest, repeat after me: Sa-mi bagi mana-n cur si sa-mi faci laba la cacat. That will say to him, \"Stick your hand in my ass and jerk off with my shit.\" Now that'll show him who's boss!\n\nWoman Shops at Wal-Mart   \nto Feel \"Pretty, Thin\"\n\nSuburgatory, USA\u2014An affluent local woman chooses to shop at Wal-Mart, not because of the low, low prices, but because shopping there makes her feel \"pretty\" and \"thin.\"\n\n\"Just get a look at this place! I'm like a total rock star here!\" said Terry Gotlieb, who often feels inferior in her regular social circle at both the Temple Beth-El Sisterhood Social Group and the Junior League, because of what she calls her \"fat giraffe body\" and unmanageable \"Jew-fro.\"\n\n\"I just feel so relaxed and energized when I come in here. I can let it all hang out and still feel like a million bucks. \"Hey, Fred!\" Fred Upton, one of the store's many senior citizen employees, is Gotlieb's favorite Wal-Mart greeter. \"That guy, he's the best. He knows all my kids' names. Always says, 'Hi Sunshine! You look beautiful today.' You know, Fred has to take two buses to come work in this shithole?\"\n\nUpton, taking his break sitting in the store's blood-\u00adpressure testing stall, was asked for his impression of Gotlieb. \"Fat? She thinks she's fat? In my day, Terry'd be called a 'tall drink of water.' I think she has a fine figure. Don't know where she gets 'fat' from. She's the most beautiful Jew I've ever seen. But she's a sad lady, too. Her values are a little cuckoo in the head. You know, when I talk to her, I always feel so much better about my own life. So I love it when she comes in.\"\n\nGotlieb believes that the diversity in the store lends her an advantage no matter the looks of the other shoppers. \"Even the best-looking Puerto Rican girl is no match for an average white lady like me. I'm coming out on top every time.\" Gotlieb paused. \"Except for J-Lo. But I don't have to worry about seeing J-Lo at Wal-Mart, do I?\"\n\nGotlieb marvels at the number of children some of the patrons of Wal-Mart have, and the impossibility that they are saving for the future. \"Oh God, if these people don't start saving, their kids will never have what I have. They'll have to shop here. Not like me, just here for fun.\"\n\nGotlieb particularly likes the pharmacy section, where she feels \"the thinnest and the fittest.\" Here, Gotlieb observes, \"I see all these gigantic obese people with diabetes buying crap for 'wound care' and oh grosssssss. I am so lucky.\"\n\nGotlieb uses the Wal-Mart pharmacy, not because it's cheap, but because, as she puts it, \"I don't have those cunts from the Temple hanging over me seeing me pick up my Zoloft and Ativan.\" Do her friends know she shops at Wal-Mart? \"Are you out of your fucking mind? They'd think we were being foreclosed on. Or they'd bitch I was an 'enemy of the people' or something for shopping here. Well, unlike them, I actually know people of different colors and different backgrounds, like Fred, because I go to Wal-Mart.\"\n\nGotlieb had never heard of the popular website Shoppers of Wal-Mart, where contributors surreptitiously photograph Wal-Mart shoppers for the amusement value. On this reporter's iPad, Gotlieb began clicking through and laughing. \"This is great! Oh my God, look at that,\" referring to a photo of an overweight African-\u00adAmerican woman's loose back fat.\n\n\"I've totally seen that, like, a million times.\" But her mood darkened when she arrived on a picture entitled \"The Old & The Frizzy-full,\" which showed Gotlieb on an especially bad hair day, slack-jawed, speaking to Fred Upton.\n\nGotlieb abruptly ended the interview and began walking out. A checkout girl said, \"Terry, you haven't paid yet.\" Without turning around, Gotlieb thrust her hand behind her and said, \"I don't have to pay for anything here.\" Upton said goodbye as Gotlieb left the store unchallenged, but got no response. \"There goes my sunshine,\" he said.\n\nHome-Schooled Girl Excels in Competitive Spelling, Blow Jobs\n\nSuburgatory, USA\u2014A home-schooled girl excels in championship-level spelling and blow jobs, with both skills giving her local renown. \"Thanks to my amazing parents, I have the confidence and commitment to excellence that I would have never had if I spent years as a slave to an educational system that creates kids who can't think for themselves, who have no real love for learning, and know nothing about the life of the mind,\" said sixteen-year-old Catherine Busby.\n\nAt a very young age, Busby says, she knew exactly what she wanted to accomplish. \"Memorize words to win or at least place in the biggest contests and get on the news, and, separately, to get boys to like me. So I dug in, and dug in hard.\"\n\n\"Did she ever!\" said Paul Minnow, seventeen, one of Busby's early boyfriends\u2014a starter project in her quest to perfect and master her blow-job skills. \"I can't believe she dumped me. If I was her 'training subject,' I can't even believe how good Catherine is now. Those guys seriously must faint.\"\n\nAnother boy expressed his pride at her success at age fifteen, coming in third in the Scripps National Spelling Bee. \"I mean, appoggiatura? Autochthonous? How does she know this shit? But I have to tell you, when she is up there and thinking really hard and then starts that slow spell, letter by letter, the excitement building and building, all I can think about is my cock in her mouth. Then when she gets it right? Oh my God, it's explosive. Even her dumpy outfits turn me on now.\"\n\nBusby's family members describe themselves as \"neo-pagan\" and believe they honor The Goddess by purchasing only secondhand clothing. Why didn't they give their daughter a more traditional pagan name? \"Oh, we were psycho yuppies in the '80s and '90s.\"\n\nBusby's mother, Sophie, is very proud of her daughter's spelling accomplishments, and has little problem with the fact that she has earned the nickname \"Blow Job Babe\" by local boys. \"First of all, I trust Catherine's judgment without reservation. She spent years on my breast, and years more learning right at my knee. I'm convinced we gave her the security to express her deepest desire, coupled with the wisdom to do it responsibly and with integrity. Who am I to say that achievement in sexual ability is worthless? It's a skill she can utilize and enjoy her whole life!\"\n\nWhile she seems equally skilled in both spelling and blow jobs, the spelling holds a more sentimental place in Busby's heart.\n\nSays current boyfriend Jonah Klein:\n\n\"I tried to get her to spell out one of her crazy words on me with her tongue, but she wouldn't do it. She said 'Spelling is sacred.' I had to settle for 'Heartbeat of America,' where she squeezes me, like thump-thump.\" Klein demonstrated a pulsing grip. \"Thump-thump. Thump-thump. And that was fine, too, of course. I really respect her!\"\n\nSo where did Busby learn all these techniques? \"Oh God, the Christian home-school girls of course,\" referring to the kids she meets with regularly for home-schooling field trips and other enrichment activities. There is often tension between the neo-pagan and the Christian home-schoolers, but Busby made fast friends. \"Oh, they're lovely! And they are like blow-job ninjas. But, well, not to be mean, but their spelling is for shit.\"\n\nDr. Drama\n\n\"When life hands you a problem, let's make it more interesting!\"\n\nDear Dr. Drama:\n\nOne of my best friends, I'll call her \"Meg,\" is being emotionally abused by her husband, I'll call him \"Brad.\" He berates her, controls her, and has made her a prisoner in her own home. She's so under siege she can't even imagine leaving. I'm desperate to find a solution and to force her to get help. I'm at my wit's end!\n\n\u2014Hopeless in Suburgatory\n\nDear Hopeless:\n\nIn my many years of online training to become a clinical \"psychologist,\" while not slaving away in that awful call center, I learned the technical term for men like \"Brad.\" We old pros in the biz call them Asshole Dickwads. Now, Dr. Drama doesn't want to be a Debbie Downer here, but the fact is research shows that the recovery rate for Asshole Dickwads is extremely low. You know what's even lower? The chances that doormat \"Meg,\" living under siege with an untreatable Asshole Dickwad, will do something about it. So that leaves you, and if this is a hopeless situation, I always say, why don't you have a little fun with it? Someone should!\n\nI like to give clients action items to achieve their goals. So here goes: First, sabotage him at work. You wouldn't believe how easy this is, I know firsthand! Find even one coworker and start feeding him shit about your Asshole Dickwad. In the age of social networking, we'll have Asshole Dickwad the talk of Twitter before you know it.\n\nSecond, contact your friend's dad. Just because \"Meg\" says, \"He'll kill Brad if he finds out what he's doing!\" Hey, that's no skin off your back or hers. What do you care if Asshole Dickwad gets the shit beat out of him? I have four words for you: It's about fucking time.\n\nAnd finally, start rocking that passive aggression you always wanted to use, face-to-face, with Asshole Dickwad. Trust me, your beaten down friend can't do it, so it's up to you. Maybe you can demean his career choice: \"Oh, I've heard that field is the least competitive of the fields you could have chosen, and there's a lot less money, and your peers are really not the brightest, but I'm sure the quality of life you have is so worth being in that really not-competitive field.\" Or \"Yes, the kids are a bit . . . large. But you must love that they're so like you!\" He will smile, but inside his rage will burn with the intensity of a thousand suns. Enjoy!\n\nMom Befriends, Infuriates   \nMormon Missionaries\n\nSuburgatory, USA\u2014A local mother has based her entire social life around a pair of young missionaries from the Church of Jesus Christ of Latter-Day Saints, but she has unwittingly alienated them with her ignorance.\n\n\"If she offers me Starbucks one more time, Oh. My. Heck.\" said Buck Berkeley, age twenty, of Salt Lake City, Utah. Berkeley is spreading the message of the Book of Mormon with his companion or \"comp,\" Cason Mabry, twenty-two, of Harris Landing, Idaho, in what is known as \"tracting,\" or door-to-door proselytizing of selected suburban neighborhoods.\n\n\"Case, how many times have we told her about the coffee? Six times? She couldn't give a scrud about our eternal damnation. I mean, has she ever heard of Wikipedia? It's not that hard to figure out that we can't even get near her gross coffee.\"\n\nBerkeley was referring to Kim Ballante, forty-one, the mom who first greeted the young men at her door two months ago. \"Aren't they just the cutest?\" she said.\n\nBallante says she has learned much from the pair about the tenets and history of the Mormon faith, including a description of the angel that adherents believe visited church founder Joseph Smith in the late 1800s. \"Moroni was the last Nephite prophet, whatever the hell that means, and then his angel wrote the Mormon Bible on golden plates?! Can you be-leeve this shit? Mo-RONE-eye. Buck said it's spelled like moron with an 'i.'\"\n\nBerkeley laughed when this was mentioned to him. \"She had no clue I was calling her a moron. She just kept going 'uh-huh, uh-huh,' and 'wow, that's so interesting!', which is all she says when we actually try to give her the Word of Wisdom. One day she kept calling Joseph Smith, Robert Smith. Remember that, comp?\" Cason said, \"Yeah, because she loves that old creepy band The Cure. Man she is so old.\" Buck agreed. \"So so SOOOO old.\"\n\nOn this day, Ballante welcomed the boys in and said, \"You're just in time for 'Hot Topics!'\" referring to the topical portion of the show The View, which they frequently watch with her. \"I thought they would like Elisabeth [Hasselbeck]. She's all religious like them. But they don't seem to like her. She wears a cross, and they don't like crosses, these people. Like, like vampires or something, they are seriously scared of them.\"\n\nAt the end of The View, Ballante said, \"Ethan [Ballante's two-year-old] is still sleeping! You know what that means . . . Appletini Time!\" which apparently meant she would mix up some drinks for herself, even though the young men are forbidden to drink alcohol.\n\n\"Sex and the City,\" Berkeley said quietly to Mabry when Ballante went to prepare her Appletinis. \"Do you think she got Appletini Time from Sex and the City?\" Mabry asked, \"What's Sex and the City again?\" Berkeley said, \"You know, that show from when we were, like, twelve, and those movies? Those trashy old bags running around New York drinking and having sad sex and all? The women in this town pretend it's, like, real or something. It's kinda tragic. . . . They just need real love from their husbands. What is wrong with these men?\"\n\nAfter Ballante consumed several drinks, she lost her inhibitions with the young men, whom she believed secretly found her attractive. \"Come on, you know you want more than one wife. Admit it! I'd totally be a sister wife if I could have Bill Paxton! But not that creep from Sister Wives.\" Ballante was referring to the polygamist husbands on the HBO show Big Love and the TLC program Sister Wives. She gets most of her knowledge about the Latter-Day Saints from the shows, and more recently from what she's read about the Broadway send-up, Book of Mormon.\n\nBerkeley, as the more senior missionary of the two, handled the polygamy question. \"Ma'am, our Church disavowed polygamy more than a century ago. While there are some fringe groups who continue to practice polygamy, they do not represent the people or beliefs of our Church in any way, shape, or form. I will have one wife, and we will be together with our children forever.\"\n\n\"Holy shit, you are so adorable,\" Ballante said. \"Alright, alright, alright. But you need to at least give me something,\" Ballante said, leaning in closely to the young men, who were visibly apprehensive.\n\n\"Show me your magic underwear. We read this book in book club about crazy Mormons and they were talking about magic underwear.\"\n\nMabry cleared his throat and said, \"Ma'am, they are temple garments that we wear at all times to remind us of our sacred covenants. Other religions like Judaism also include special garments in their faith.\"\n\n\"Jeez-us, you guys are all God and no fun. It's gonna be a long long life for you two if you don't loosen up. Trust me on this,\" Ballante said.\n\nAfter leaving the home for the day, Berkeley said, \"You know, the thing that kills me the most is that she is so pathetic that she sits around day after day with a couple of twenty-year-olds and she thinks WE are the freaks.\" Mabry nodded in agreement. Ballante confirmed this impression. \"Yeah I love those boys, but of course they're freaks. They're Mormons knocking on doors! If that's not a freak, I don't know what is.\"\n\nSo why do Mabry and Berkeley keep going back if they feel both offended and hopeless at their chances of getting Ballante baptized? \"Oh, I don't know, maybe because I'm-With-Stupid?\" Berkeley said, gesturing to Mabry. \"Case, like an idiot, let it slip to our Zone Lord [the mission supervisor] that this woman loved us and now they are convinced they have a golden.\" \"Golden,\" they say, refers to an easy conversion target.\n\nMabry added: \"I don't even want her at this point even though we look like rock stars if we get her dunked [baptized]. I don't give a fudge about her or her soul\u2014but if she becomes LDS and has a celestial marriage, then that poor guy she's married to and that kid she totally, completely ignores, are going to be stuck with her for all Eternities.\"\n\nIce Cream Man Assaulted   \nBecause He's the Ice Cream Man,   \n\"Not Because He's Muslim\"\n\nSuburgatory, USA\u2014The dads who admit to the harassment and second-degree assault last month of Egyptian-born ice cream man Suleiman Rahman insist they did not attack Rahman because of his faith, but because he's the ice cream man. They also say he is a \"possible perv,\" a \"rolling extortionist,\" and a \"kiddie-poison pusher.\" But they claim they have no problem whatsoever that Rahman is a Muslim, and argue this incident, which involved pushing and yelling, does not fit the definition of a federal hate crime.\n\n\"That's straight-up slander!\" said Mark Watson, one of the accused parents. \"We aggressively suggested the ice cream man leave the school parking lot last month because we are fed up with him and all the problems he causes, following families all over town, ruining every nice event!\"\n\nJanet Maroney said Rahman's truck arrives just as the kids begin playing soccer, making it impossible to keep the kids on the field and forcing parents to bring cash to every game. \"I feel like every time I see that stick-up truck headed for me, I can kiss five bucks goodbye unless I want an epic meltdown. Then if a friend forgets her cash, I say buh-bye to ten bucks.\"\n\nJodi Keyes wishes she could say no when the ice cream man arrives but doesn't want to look like a \"joy-sucking cheapskate\" or one of those \"granola moms.\" Peggy Davies is proud to call herself a \"granola mom\" and even she can't resist buying her kids what she calls the \"frozen death on a stick\" with \"neon gumball eyes.\" \"Yes, I'm granola, but I hate a tantrum just like the rest of you,\" Davies said.\n\nOthers alluded to what they see as a corruption of this classically American institution. Some who asked to remain anonymous thought Rahman was \"a Gypsy or something\" and mourned the days when ice cream trucks were manned by \"wholesome teenagers\" and not \"old possible pervs who don't even live in town.\" One wondered, \"Does he sleep in there?\"\n\nParent Roger Jackson asked, \"Have you heard that toy piano tune his truck plays? It will seriously haunt your soul. It's like Satan on four wheels.\" And these feelings have nothing to do with the fact that he's a Muslim? \"I said Satan, not Osama, didn't you hear me?\"\n\nSo what does Rahman say about the incident? \"In 2002 I was held for three months by the Egyptian secret police. You think these homosexual-looking men scare me?\" After some cajoling, Rahman admitted that he didn't think town residents hated him because he's a Muslim, even though his lawyer is pursuing a hate crime charge. \"I don't think they even knew what I was before this happened. I think they just thought I was the poor brown stranger taking their money and annoying them with my tempting and delicious ice cream. They blame me because they can't say no to their spoiled-rotten children. So no, I don't really think they hate me because I'm Muslim. But I couldn't really blame them if they did. Because I definitely hate them because they're American.\"\n\nWolf Blitzer\u2014  \nLive From the Lactation Room\n\nSuburgatory, USA\u2014This is Wolf Blitzer. And you're in . . . the Lactation Room. We have a situation developing in the Lactation Room at the Unum Provident office building today, as two mothers battle over their degree of virtue and commitment to pumping breast milk for their babies.\n\nJill Branson is trying to convince Susan Markle to \"tough it out,\" \"don't be a quitter,\" and \"remember all the troubles formula-fed babies have,\" as Susan struggles to produce enough breast milk. For those of you unfamiliar with the process\u2014as I was before I discovered this oasis of feminine splendor\u2014working mothers use electric pumps and then store their breast milk for their babies to drink later. We go to the fight playing out live.\n\nJill: Breast-feeding for me has been excruciating, bloody, and by far, the most important and life-fulfilling job I've ever had. I know it's a gift that I'm able to do this. Susan's wavering commitment is an insult to those who can't breast-feed at all, even though most women can, if they try hard enough, and care enough.\n\nSusan: Like Wolf? He can't breast-feed. What if he was a gay man? He and his husband couldn't breast-feed.\n\nWolf: Ladies and gentlemen watching out there at home, to clarify, I am not gay.\n\nJill: Well, gay Wolf and his partner could get donated milk. I'd donate to them! But really, I'm most interested in women and what women are capable of doing.\n\nSusan: Of course you are! Wait, you think people would just take your milk, no questions asked? Isn't that like letting your kid have unprotected sex with a stranger? Hey, why is Wolf Blitzer in the Lactation Room with us?\n\nWolf: Because I'm a breast man. I told you I wasn't gay.\n\nJill: Wolf, my breasts are for my baby.\n\nWolf: Sorry, Jill. I'm not made of stone.\n\nJill: Well, anyway, I nursed through multiple nipple cracks and nipple psoriasis, thrush, a kidney infection, an abscess, a shattered elbow, major surgery with my first son, and a ruptured appendix. I don't think anything can stop me from breast-feeding. I willed every last damn drop out of my body.\n\nSusan: Why don't you take up another kind of competition that will pit you against other women, like a marathon? Women's roller derby? Probably get some gore there to brag about, too.\n\nJill: I'm not competing with you! I just want what's best for your child!\n\nSusan: You don't even know my last name! THAT'S IT! I'm going up to the roof to SMOKE. Did you hear that, Jill? Going to report me to protective services? Michelle Obama?\n\nWolf: While the First Lady is a vocal supporter of breast-feeding, I should note that the President himself struggles with smoking.\n\nSusan: WhatEVER Wolf, what are you really adding here?\n\nLadies, I will add that, like the First Lady, I'm also a big supporter of breast-feeding whenever and wherever a beautiful, luscious, ripe new mother chooses to do so. And if you're done pumping, my work here is done. I'm Wolf Blitzer, and you've just been in . . . the Lactation Room.\n\nMom Unaware of Two American Wars\n\nSuburgatory, USA\u2014An area mom is unaware of two American wars fought over the last ten years.\n\n\"Huh? What are you talking about?\" asked Carol Stewart. \"And I can't talk for long, I've got Tommy's soccer pickup at 3:30 p.m., have to swing back and pick up Sarah from Mindy's house, then get all of us to the store. Jesus, that sounds like absolute hell, doesn't it?\"\n\nThis reporter explained to her that 9\/11 had led to a \"war on terror\" that still has US servicemen and women in harm's way in both Iraq and Afghanistan. Had she heard of 9\/11? \"Now that is extremely insulting. In fact, 9\/11 touched me close to home. I actually saw it happen. Live. They pre-empted Regis and Kelly for it. So just back off with the 'hey-ignorant-Mommy' line of questioning.\" Arriving at the soccer field, she said, \"Get in the van, Tommy. No. You've had enough ice cream. No. Now.\"\n\nBut is she aware that the United States went on to fight two wars after that?\n\n\"Here's what I know. I know that we went into Afghanistan, found the smoking gun mushroom cloud WMDs, thank God, did that amazing rescue of that adorable blond soldier girl\u2014poor little thing. Then Seal Team Six got al-Qaeda's top guys\u2014Saddam, bin Laden, Qaddafi. Then George W. Bush did that whole thing on the boat with the big We Won! sign, and Axles of Evil were finished. See? I know a little something about something besides Mommying.\" She looked satisfied as she retrieved Sarah from her playdate. \"Thanks, Mindy. Did she behave nicely, I hope?\"\n\nAs she headed to the store, this reporter told her that major combat did not end with the Mission Accomplished sign, and in fact had gone on for years, costing a trillion dollars. She was still quite dubious.\n\n\"I haven't seen Ken Burns do anything about it. I guess you're going to tell me you know more than Ken Burns? You, some small-town suburban reporter? And if there were really two wars, wouldn't I know someone who had gone? Hellooo! And wouldn't every house have a flag out?\" Not necessarily. Since the military is all volunteers these days, it attracts mostly lower- or middle-class recruits, and those who have no connection to those socioeconomic groups would be almost fully insulated from the impact of two wars.\n\n\"So I guess you're asking, 'Where've I been?' Oh, I know, raising the future of America.\"\n\nDid she want to know the number of American servicemen and women and civilians killed in the two wars?\n\n\"You know what?\" she said, exasperated with the questions and all three of her kids yelling or throwing things across the minivan. \"I've got my own army to worry about. Sorry. Priorities.\"\n\nSHOUT OUT\n\nToddler or Anchor: I Report, You Decide\n\nLinda Mendes is a former TV news producer turned stay-at-home mother who lives on Rice Street.\n\nI take to the Shout Out today because I know a lot of moms leave their jobs and think, \"All I know how to do these days is wipe a baby's ass and listen to a baby scream!\" Well, I'm here to tell you that if you can handle that, you can handle a challenging career in TV news! Getting both crapped on and screamed at by anchor babies is what it's all about.\n\nOK, I exaggerate. An anchor never did literally crap on me, as my son did many times. But metaphorically, yes, my friends and I were crapped on with alarming regularity.\n\nSo here's a ten-part puzzler for those moms out there who don't think they have what it takes to work in TV news. Trust me, after dealing with a child, you probably have the chops, as you will soon see. So, am I describing my toddler or one of the many anchors (or reporters) I've worked for or worked around or heard about from colleagues over the years? I report, you decide.\n\nQuestion #1. This person broke wind, frequently, loudly, shamelessly. This person also, how might I put this delicately, often mined for the mother lode, usually at the same time as the aforementioned wind-breaking. Toddler or Anchor?\n\nAnswer: Anchor. At least my tiny boy had some shame. Frank would tell the lunch crowd at Applebee's, \"Excuse me, I farted.\" This particular anchor just said \"Welcome back to the show!\" And my toddler, mercifully, lacked the fine motor skills to flagrantly pick his nose. This two-fer of anchor farting and picking would happen during commercial breaks, eliciting a chorus of \"Oh gross!\" \"He's at it again!\" and \"Is he done yet?\" in the control room.\n\nQuestion #2. This person was nicknamed \"Cranky Pants\" by his caretakers. Toddler or Anchor?\n\nAnswer: Anchor. As in, \"Watch out, Cranky Pants hit traffic, his BlackBerry crashed, he just saw that we tanked in the last quarter-hour ratings, and he's on the fucking warpath.\"\n\nQuestion #3. This person was a huge admirer of the sober, thoughtful reporting of NewsHour with Jim Lehrer and watched it religiously. Toddler or Anchor?\n\nAnswer: Toddler. My toddler loved when Jim \"Ware\" used to \"wee-cap\" the news at the end. When the show would list servicemen and women killed in Iraq and Afghanistan, Frank would say \"Jim Ware is sad now.\" He also loved substitute anchor Gwin Eye-full, and analysis by David Bwooks from the New York Times. He's really mad Jim Ware retired. Crazy-mad.\n\nQuestion #4. This person's caretaker had to remove carrot shreds from his lunch because he \"hates orange food\u2014no orange food!\" Toddler or Anchor?\n\nAnswer: Anchor. Though, to be fair, Frank didn't like orange food either.\n\nQuestion #5. This person was inconsolable when told he couldn't have a monkey as a pet. Toddler or Anchor?\n\nAnswer: Toddler. OK, you guessed it; that was my three-year-old. Most anchors care a tad more about their crucial demographics than they do any living creature, other than themselves.\n\nQuestion #6. This person struggled mightily with language development. Toddler or Anchor?\n\nAnswer: Anchor(s). One of my proudest moments while driving around suburbia was when Frank heard a public radio anchor say rapprochement, and he repeated it flawlessly. But with reporters and anchors? Some have left us writers and producers awestruck at their ignorance. It's also a very delicate dance as a writer, whether to spell out a word phonetically for the \"talent,\" because if it's a word or name they know, that's you effectively telling them, \"Hey, boss, I think you're a moron!\" But if you didn't put the \"prono\" in, and then they prove themselves to be morons on live television, then you, the writer, get ripped a new one, and a big new one at that. I polled TV friends for favorite prono mistakes. Here's a sampling: Remember the Alamo: \"Remember the a-LAMB-o.\" Fidel Castro: \"Feye-dell (like Fido) Castro.\" Mao: \"Mayo.\" Pneumonia: \"Puh-numonia.\"\n\nQuestion #7. This person turned a very angry red, balled up his fists, and screamed when he was read something he didn't like. Toddler or Anchor?\n\nAnswer: Anchor. My toddler had things he didn't like to read, but at least he didn't crumple up the offending material and throw it at me. He also didn't know how to say \"Who the FUCK wrote this?\"\n\nQuestion #8. This person has trouble looking into people's eyes. Toddler or Anchor?\n\nAnswer: Anchor(s). When the camera is on? No problem. With the staff? Not so great on the eye contact. Thank goodness this wasn't my toddler, because I definitely would have worried about autism.\n\nQuestion #9. This person whipped out his penis any chance he got. Toddler or Anchor?\n\nAnswer: Toddler and Anchor(s). With apologies to my sweet boy for violating his privacy, my toddler son loved showing us his \"nudie rudy.\" (Blame entirely his baby-talking mama. When he was streaking around the house, I started calling him a \"nudie rudy.\" He eventually decided that his penis was actually called a \"nudie rudy.\") As for the anchors, well, their nudy rudies were deployed in far less innocent circumstances, and just like the maids and the butlers in a secretly steamy English manor, we, the news-servants of the all-powerful, knew a lot more than our masters ever suspected.\n\nQuestion #10. This person just up and ran away from his caretakers, leaving them terrified and heartbroken. Toddler or Anchor?\n\nAnswer: Anchor. Actually, the above isn't completely true. The anchor did just disappear without warning, not showing up or calling in for days, weeks, never to return. But we weren't terrified and heartbroken. In fact, we had a lot of fun in the newsroom during that time and some nice relaxing lunches.\n\nAs I review my puzzler, I realize I better snuggle on down in this crazy suburb I now call home and get cozy, because I'll probably never eat a nice lunch, or produce TV, in that town again. But that's OK. I'm happy with the less-than-relaxing lunches I now have at, yes, Applebee's, with my little ball of energy who tells me he loves me ten times a day. I never got that with an anchor. Frank can crap on me all he wants.\n\nMom Crushed to Learn   \nthat Facebook Isn't Job\n\nSuburgatory, USA\u2014A local stay-at-home mom who calls herself the \"Facebook Queen\" was crushed to find out, after nine months of avid social networking, that Facebook is not a job and she won't be getting paid.\n\nMolly Brooks, thirty-eight, has become an instant legend among her 732 Facebook friends for seeking and accepting friend requests from people she doesn't know, or barely knows.\n\nIn a very short period of time, Brooks has made herself a fixture, by being the first on anyone's Wall to offer birthday greetings and by invariably being the first person to \"like\" a status update. Her response time leaves her friends awestruck and a bit concerned.\n\n\"Seriously, my update will be fourteen seconds old and she's already lunging for it,\" said Patrick Mulleavy, who knows Brooks through a one-time fraternity party hand job back in college. \"It was great,\" he said. The hand job? \"No, no, my fraternity. I miss that life so much. Thank God for Facebook. Reconnecting with Molly brought me a little closer to those memories.\"\n\nHe considers Brooks's Facebook prowess similar to her approach to a hand job. \"A little too eager, a little forced, but total A for effort,\" said Mulleavy. \"She was a really nice girl in college, and from what little I can tell from Facebook, she's turned into a really nice lady and mom. With maybe not a whole lot going on in her life.\"\n\nOther friends worry that her Facebook vigilance means she is neglecting her family responsibilities. \"How does she keep up that totally insane pace with the liking and the commenting, the 'Get better soon!' and 'How fun!' and still watch her kid? He's only two years old!\" said Maura Tanner, a childhood friend.\n\n\"Molly's like a sister to me and I've known her in real life for decades, which is more than I can say for these other 'friends' she has. 'Shahgan Vatan'? His Facebook page says he's a Pakistani mariner. Did you hear that? A PAKISTANI MARINER,\" said Tanner, disturbed that Diehl would be wasting her time with what she called \"some rando across the world.\"\n\nTanner explains, \"Molly met him in . . . the Jewish Maritime Historical Society page on Facebook. I mean, what the FUCK. I'm not sure what's weirder\u2014a Pakistani who joins the Jewish Maritime Historical Society or a stay-at-home who's a) not Jewish, b) barely ever been on a boat, and c) gets history from, like, going to see The Help. Molly watches that one stupid movie and she thinks she marched on Washington arm in arm with MLK or something.\"\n\nBrooks says she joined the Jewish Maritime Historical Society \"because it looked cool and my friend Herb asked me and he just had a baby, Henry. Isn't he soooo cute?\" She thumbs her iPhone to find the pictures. Does Brooks feel, as some of her friends do, that she's wasting her time and life on Facebook? \"Well no, of course not. I do get paid, you know. Though Facebook is apparently total crap with their payroll, since I've been on nine months and still haven't gotten paid. Gotta get on them for that.\"\n\nThis reporter then explained to Brooks that she must be confused, Facebook is not a job. As she let this information sink in, Brooks looked as if she was going to cry. \"I can't speak for a little bit,\" she said. Instead, she updated her status on Facebook. \"Molly Brooks is the world's biggest moron. All this time I thought Facebook was a job and was going to pay me. Feeling like the most massive loser. :(\"\n\nBut before Brooks could even finish typing, her update was flooded with comments. \"Molly = The Awesome.\" \"MB, don't go changin'!\" \"Molly\u2014I may not 'know you' IRL, but you were the ONLY person to say something when Aunt Lucille died. XOXO.\" And from Shahgan Vatan in Pakistan, where he was apparently on Facebook at 2:00 in the morning his time: \"Molly, when I tell my friends that they are wrong, when I tell them that the US does have good and kind people, I say, let me show you Mrs. Molly Brooks of America.\"\n\nWith that, Brooks did indeed begin to cry and said, \"I guess Facebook taught me what really matters in life. It's not about the money. It's about real people and real community. . . . OK, I need to direct message Shahgan\u2014his third wife was having some huge fibroids removed yesterday.\"\n\nTeacher Says What She Actually Thinks,  \nInfuriates Parents\n\nSuburgatory, USA\u2014A local teacher, fed up with the media and the public attacking her profession and union, has stopped sugarcoating things, infuriating parents with her brutal honesty.\n\n\"These parents think teachers have it sooo easy? That our union is not much better than a bunch of public-sector gangsters?\" says fifty-six-year-old teacher Lily Peterson. \"Well, I have two words for them and they start with an F and a U. It's high time they hear exactly what I think of some of them and their precious babies.\"\n\nAnd so the weekly PTO meeting was replete with stories of the teacher's disturbing candor. Instead of being told to schedule more outdoor activities for her son, Missy Carter was asked, \"Are you really going to let Blake eat himself into type 2 diabetes?\"\n\nSarah Martin admits that her son has had a few behavioral incidents, but offers that \"the cat healed up just fine.\"\n\nShe was flabbergasted when, instead of offering strategies for encouraging appropriate behavior, Peterson simply told her, \"Hide your knives, and lock your bedroom door at night.\"\n\nAlexandra Petit, whose son is an outstanding student, but also a loner and an obsessive \"deep thinker,\" also got some troubling honesty. Peterson told Petit, \"Your kid's either Bill Gates or the Unabomber. It could really go either way.\" Petit says, \"I'm really clinging to the Bill Gates thing and trying to forget the rest, so can we just not talk about it?\"\n\nHer comments were also a mixed bag for Tracy Heffernan and her daughter Erin, as described by Peterson.\n\n\"Yeah, I told Heffer that Erin is an annoying tight-ass, but she's super bright and bossy and I wouldn't be shocked if she was another Hillary Clinton. I'd probably vote for her but she's still kind of a stuck-up tool.\"\n\nHeather Lee says her daughter Bella is a lovely child and that she was never told anything of note other than that Bella is \"very anxious to please, no matter the consequences.\" At the latest teacher's conference, Peterson clarified what she meant: \"She's going to be the girl giving group blow jobs at frat parties if you're not careful.\"\n\nBut Peterson is quick to point out that she loves many, if not most, of the kids, especially Vincent Stone. Peterson says Vincent is \"unusually mature.\" When his mother\u2014whom Peterson describes as a \"no-good slacker\"\u2014said thank you, Peterson shot back, \"You deserve exactly no credit for it.\"\n\nAs is the case with Stone's mother, Peterson notes that it's largely the parents she has the problem with, not the kids.\n\nRobin McIntyre\u2014another parent irate about the teacher's decision to be honest\u2014wouldn't reveal what was said to her, though Peterson was happy to oblige. \"Oh, I'll tell you exactly what I said to her. I said, 'I know you're hungover and tired from banging that sleazy boyfriend of yours, but get your kid's ass in here on time.'\"\n\nPeterson sits back, looking both satisfied and energized. \"Maybe I shouldn't retire after all.\"\n\nFamily's A-Ha Moment: All Uniquely   \nDevastated by Oprah's Departure\n\nSuburgatory, USA\u2014Each member of the Houlihan household, in their own unique way, is devastated by the departure of American talk show treasure, Oprah Winfrey.\n\nThese days, at 4:00 p.m. two-year-old \"Little\" Cathy Houlihan walks around the house, listlessly looking for Oprah, and banging the TV. \"Black Mommy? Black Mommy? Where's Black Mommy? When Black Mommy coming? Is she coming back?\"\n\n\"I know it's awful to watch, isn't it?\" said Cathy's mom, \"Big\" Cathy Houlihan. \"But at least Little Cathy has hope that Oprah's coming back. I have to live with the knowledge that she's not!\"\n\nBig Cathy explains Oprah's impact on her life. \"Oprah taught me everything I know about being a white lady! I mean, that didn't come out right at all. She taught me what it meant to be a mom in America. A mom who lives in her own truth and has a sassy black friend to give her the kind of down-home Southern advice she needs. And sometimes that advice meant accepting things I wasn't raised to approve of. I did what Oprah told me, and I'm a better person for it,\" Cathy said.\n\n\"And thank God, I mean, thank Oprah for that,\" said Houlihan's now openly gay teenage son, George, who's also in mourning for Winfrey.\n\n\"I might have killed myself had Oprah not made my mom believe that it was OK for me to be gay. FUCK those people who pick on Oprah. FUCK. THEM.\"\n\nHis dad, Ian Houlihan, has his own reason for bemoaning Oprah's farewell, but at this point it's a secret. \"I'm gay, too, and I was hoping she would do a few more of those shows on closeted gay husbands to, you know, help ease the blow to Big Cathy. \"The Wrath of Cath,\" we call it, it's a scary scary thing. So, no hope for coming out anytime soon. Thank god for gay porn. I'd probably kill myself if it wasn't for gay porn.\"\n\nIan's father, Walter, who lives with the family, is apparently the only straight male in the household, but the ladies he prefers are the black ladies. \"I can't believe my Ebony Goddess is gone.\"\n\nDid he have a problem with her weight? Walter was nonplussed. \"In my day, youngster, women were a lot . . . roomier, like Oprah. Boy, I used to love watching her, imagining all the wonderful zestful steam baths I might have had with her, her dark black gorgeous flesh glistening with sweat. . . . that smile lighting up every afternoon with her wit and beauty. . . .\"\n\nEven the children's live-in nanny is missing Winfrey. \"Just to be clear,\" said Elvira Martinez. \"I don't like the blacks. Even the rich blacks. Especially the rich blacks. But Oprah gave me one whole hour every day of peace and quiet without this whole house of babies needing something.\"\n\nSHOUT OUT\n\nAct Like Grown-ups at the Drugstore\n\nEliot Dubin is a concerned pharmacist at the Bartlett Pharmacy on Cabot Street.\n\nI had hoped I would not have to take to the public Shout Out forum to address certain behaviors we are witnessing at the Bartlett Pharmacy, but the health of our patrons and the dignity of our workers must take top priority.\n\nFirst, calling in prescriptions you don't pick up.\n\nNow I know that Maria Osnos really can't predict when her nervous collapse will occur and when it will ease. I know that Bill Sanford sometimes looks down at his painful toes and thinks, \"Well, as the youngest person ever to have gout, maybe I should do something about both my lifestyle and this excruciating pain.\" But then Maria's nervous collapse goes away after a half-dozen glasses of wine, or Bill decides he just can't rouse his fat ass out of the house to pick up his meds. But all of you should know that when our pharmacist rushes to fill orders that you never pick up, he is putting aside the needs of others. I can only assume that John Maron was able to get the erection he sought without the help of the medication that he ordered because he never picked it up. I give a hearty guy-to-guy mazel tov to that, but please, be considerate of our time as professionals.\n\nSecond, cutting ahead in line.\n\nMelissa Henry, I understand your urgency in getting your medication last week. I know better than anyone, except perhaps a prostitute, that a vaginal yeast infection requiring prescription medication is quite \"a situation\" indeed. But to stride so aggressively to the counter that you knocked over Millie Wexler's walker? It is true that Millie uses the pharmacy counter as a place to socialize now that those worthless piece of shit kids, Bob and Sheila, have decided to forget they have a mother and leave her to us. But that means Millie is practically a member of our family now, and no yeast infection, no matter how itchy, steamy, or smelly is worth disrespecting her. At Bartlett Pharmacy, we simply won't tolerate it.\n\nThird, yelling at us because your doctor didn't refill the prescription.\n\nWhen you are hopelessly addicted to painkillers, like Daniel Chelmsford, it is hard to keep your head about you. When things go wrong, you blame the wrong people. You need to remember, Daniel, that it is the doctor, not us, who has control over your stash. You want your meds? Knock over the store after hours like the rest of you criminal drug seekers, but don't scream at me during my shift.\n\nFourth, spying on people's medication.\n\nNow this one I really am the go-to expert on. Of course you're spying on what other people are picking up, Jessie Borden. It's completely natural to sit there and use your iPhone to Google the name on the bottle Greg Silver is holding, only to discover that he is picking up medication to block his unstoppable cravings for alcohol. It's human nature! But it is the divine right of pharmacists to know this information and no one else. And we don't even need to Google the stuff, it's all right in our noggins! We dream in medication names!\n\nI hope this clears up some of the poor conduct we've been seeing at Bartlett. We're all grown-ups here. Let's start acting like it.\n\nChild Convinced Being Disabled Rocks\n\nSuburgatory USA\u2014An area child is convinced being disabled \"rocks,\" despite his mother's best efforts to explain the struggles that disabled people face.\n\n\"Why can't we park right in front of Starbucks, Mommy?\" said five-year-old Mikey Purcell.\n\n\"Because that's only for people who have disabilities\u2014who can't use their legs, let's say, as well as we can. It's so they can use the store just like we can,\" said his mother, Sandy Purcell.\n\n\"Hmmmm. That sounds pretty good,\" Mikey said as he and his mother entered the Starbucks and asked to use the bathroom. The barista told them the regular one was occupied, so he gave them the key for the handicapped restroom. Mikey had never seen one. \"Mommy, this bathroom's huge! And so clean. It smells like, like, rainbows and miracles and fruit punch.\" Sandy began telling him this was because the handicapped restrooms aren't used as much as regular ones.\n\nAs Mikey thought about this, he said, \"I wish I was disabled.\"\n\n\"No, honey, you don't. Disabled people are people just like you and me, and they matter a lot, just like you and me, but life can be pretty hard for them. It's no walk in the park.\"\n\n\"I know that. You just told me they can't use their legs,\" said Mikey. The boy and his mom returned to the car and made their way to the Edgewood Elementary School.\n\n\"What I'm trying to tell you, sweetie, is that it's not an easy life. If you were stuck in a wheelchair, you could never walk to school with your friends,\" said Sandy. \"But I never do that now,\" Mikey pointed out. \"We go to Starbucks, get your iced chai latte, you play with your phone, and then we ride to school. Hey, if I was disabled could I ride that super cool bus with all those ramps and buttons that go 'zzzhoooop'? The floors move up and down! Those kids are so lucky. Being disabled rocks!\"\n\nSandy Purcell grew somewhat exasperated. \"I don't think you're getting it, dear. It's not a cool thing. It's not like being Justin Bieber. I mean, there's nothing wrong with being different and we all have struggles, but it's not something to wish for. You might not be able to talk, like we're doing right now . . . \"\n\nHe thought about that. \"So you could sit criss-cross applesauce on the carpet and not say anything when Miss Barrett says something to you and that would be fine. . . .\"\n\nHis mother added he might need to be fed, like he was still a baby. \"Great!\" said the boy.\n\nPurcell finally exploded. \"Michael, be happy you're not disabled! I don't want to make it sound awful but sometimes people can't have regular lives at all, like getting married or wiping their own bottoms!\"\n\nMikey went over to his mother right before lining up to go inside for school. He cupped her face in his hands. \"But Mommy, you know I'm marrying you and you already love me so much AND you still help me wipe my bottom sometimes when I can't reach.\"\n\n\"So,\" he said quietly to himself, \"how do I get disabled? Would it hurt?\"\n\nVegetarian Mom Vexed   \nby Son's Meat-Lust\n\nSuburgatory, USA\u2014A vegetarian mother is vexed by her son's newfound passion for meat and his complete lack of empathy for the animals that, as she puts it, \"suffered horribly and died so you can eat a disgusting chicken leg.\"\n\nLiz Stakun blames a friend of hers who, without her permission, provided her son Max with a fried chicken leg, which he was eating as Stakun arrived at her friend's home to pick up her child from a playdate. \"OK\u2014Anne has been undermining me since college. First she horned in on my major, then surprise, surprise, she rushed my sorority. So passive aggressive.\"\n\n\"Hey Anne! Hugs!\" Stakun stiffly embraced her friend and tried to hide her horror, as she saw her son tearing into the chicken leg with abandon. \"Um, Anne, you know I've been a vegetarian. Since 1994. After the U2 concert where you blacked out. Remember?\" said Liz, trying to contain her irritation. \"Oh, Liz! Of course I know you're a vegetarian but I never in a million years thought you'd impose your beliefs on a growing boy! And look how much he loves it. How can that be so wrong?\"\n\nIndeed, Max was enjoying it so much that Liz had trouble getting his attention away from the greasy chicken leg, which he was examining in a methodical way that he sees on his favorite forensic crime program, Bones.\n\nLiz: Max, do you remember the nice mommy chicken we saw at Landsakes Farm, the one who loved her chicks sooooo much, like Mommy loves you?\n\nMax: Oh yeah, they were sooooo cute, Mommy. What a good mommy that hen was.\n\nLiz: Yes, but honey, I just want you to understand that what you saw at the farm and what you are eating on this plate are one and the same.\n\nMax: But Mommy, they're dead now and they're delicious!\n\nAnd with that Max returned to dismembering the leg, telling his mother, \"Mom, this is the best part; you need to try it.\" He pulled the chicken meat off the bone. \"You're gonna die when you taste this, the skin's the best!\" he said as he pulled the skin away from the meat. Liz sighed and shot Anne a perturbed look but said nothing more. \"Max, we'll talk about this more at home. By the way, I realize we found a pair of bunnies in the backyard.\"\n\n\"Awesome!\" said Max, as he wiped the chicken grease off his face. \"Are they dead, too?\"\n\nPAID ADVERTISER CONTENT\n\nSurrogates of a Certain Age\n\nAttention women of a certain age! We know that, for years, in your swinging twenties, you probably thought to yourself with pleasure, Well, if I ever really get down on my luck, I can always sell my bountiful eggs for a tidy sum!\n\nAnd when those wonderful glory days came and went, you probably remember reading ads about couples seeking eggs for in vitro and realizing, \"What, I'm too old to donate?! No one wants my eggs anymore? My eggs are worthless?\" This moment can be especially cruel if you were one of those gals who attended a prestige school. You spent years secretly relishing the idea that you harbored golden Faberge status-eggs that could be harvested for top dollar at any time (practically worth auctioning off at Sotheby's, for fuck's sake) and sold off to a couple who wanted their own little Ivy League bundle of joy.\n\nWell, part of growing up is learning a simple but hard fact: Your lady parts lose value. But at Surrogates of a Certain Age, we can ease that hard fact with a compelling one of our own: Some parts don't age as fast as others. Your warm, still moist, roomy and inviting uterus is fully usable until age forty!\n\nYes, you can carry someone else's child long after your eggs have shriveled into oblivion. Talk about grapes of wrath! Well, forget those little old rejects. Our match team will place you with a couple who will pay a little less, considering your advancing age, but who will never make you feel less of a woman, even if you really are.\n\nSo why don't you consider doing something wonderful for another couple, but more important, something wonderful for yourself.\n\nE-mail us for a promotional package at Surrogates of a Certain Age! Proof of age required.\n\nMother Discovers Russian   \nAu Pair Is Richer than She Is\n\nSuburgatory, USA\u2014An area mom has discovered, much to her chagrin, that her Russian au pair is far wealthier than her in Russia.\n\n\"Actually, what's worse is that she's not just richer than me in crappy, depressing Russia. She's richer than me here, too!\" said thirty-nine-year-old Jen Barofsky.\n\nBarofsky says she chose the eighteen-year-old au pair mostly because of her English skills and her unattractiveness. \"Oh please, I'm not stupid. 'eighteen-year-old Russian au pair'? I don't need my husband sitting back and watching a hot Russian teen romping around here. So yeah, I picked the frumpy one who'd probably be a workhorse and eager to please.\"\n\nBut when Galina Popov first arrived, she spoke little, though her English was near flawless. Popov barely acknowledged the children she was supposed to care for. And she walked around slowly, making small \"snorts,\" as she surveyed the 5,000-square-foot home and the property.\n\n\"I mean, at first I just thought it was a Russian thing. They're not exactly the most uplifting people, you know? Maybe that's why they hated us Jews\u2014we're loud and proud, I like to say. Life's not one long pity party, Russia!\" said Barofsky. Her husband Jeff started calling Popov the \"Crabby Cossack.\"\n\nWhen Popov's attitude failed to improve, Barofsky considered that maybe the girl actually was an anti-Semite. But in fact it was not anti-Semitism at all. \"One day Galina left her Facebook page open, and holy fucking shit. That girl and her life make me look like trailer trash. Who knew there even were rich people in Russia?\"\n\nBarofsky got to see just what Popov left behind in her native country. \"There were pictures of her houses. Houses, plural. I counted at least three. There were hundreds of nightlife shots of her all over the place dancing and drinking, and there were all these dressing-room pictures of her visits to some really swank mall. Still pretty ugly but definitely not frumpy. Nothing like her au pair video at all.\"\n\nBarofsky decided to ask Popov about it.\n\nGalina: Well, yes, you do live quite modestly compared to me.\n\nJen: But why didn't you tell me?\n\nGalina: I was supposed to tell you that you are far less fortunate than I am?\n\nJen: No, I mean, of course, it's OK that you are blessed with a lot of money. I'm just curious as to why you chose to become an au pair.\n\nGalina: Chose this? [Snort] No. I'm more suited for [French resort area] Cap Ferrat. But my father is a . . . ran into trouble with an ally of an oligarch, and he thought it best for me to leave my country for a while.\n\nJen: What's an oligarch?\n\nGalina: The smartest, fastest people in Russia who were in the right place at the right time when Communism fell.\n\nJen: Oh. Is that why you left all your fancy clothes at home and didn't wear them in your au pair video? You're in hiding or something?\n\nGalina: No, my mom told me to do that after we looked you up on Facebook and saw your . . . house. She told me an American woman wouldn't want an au pair who looked richer than her, or looked like she might try to steal her husband.\n\nJen: [Nervous laugh] Hahaha! Now that's just silly.\n\nGalina: Well, my mother is grateful you took me and I know she would want me to ask you if you would benefit from some of our . . . what do you call it in English . . . hand-me-downs? We have so many.\n\nJen: Uh, no. Nyet. Thank you.\n\nSo did Barofsky keep Popov? \"What, and have my dead Bubby rise from the grave and pelt me with boiling matzoh balls? She'd be mad enough that I had the Crabby Cossack living with me, but a rich one? It's a shande.\"\n\nDr. Drama\n\n\"When life hands you a problem, let's make it more interesting!\"\n\nDear Dr. Drama:\n\nWe are having a battle in town over whether or not to remove a house close to mine on Shelton Street, to make room for a new Wal-Mart. Many of my neighbors are thrilled that they're going to get rid of that house, because a Level 3 sex offender lives there. Now, I'm no fan of sex offenders, but as a person of conscience I would rather have him in town than Wal-Mart, which I believe does things far worse to our community\u2014collectively\u2014than any one man. I have told no one this, just you, and I look forward to getting some of your sage advice.\n\n\u2014Wal-Not in Suburgatory\n\nDear Wal-Not:\n\nSooooooo you would rather have a neighbor who has, say, ass-raped a child than a store that offers rock-bottom prices? Because low prices are bad for the poor? Have you ever been to a Wal-Mart? No? That's because you're not poor!\n\nI know your type, Wal-Not. You just have to have your special mom-and-pop bullshit coffee and ironic old-time-y hardware store that's highway robbery, because you can afford it, and you think Wal-Mart is the Evil Empire. Well, here's my advice for you, since you've told no one but Dr. Drama about your troubled \"conscience.\" Start walking around telling people your views. And send the medical bills to the ass-raper.\n\nKid's Book Used to Explain:   \nDon't Cockblock Your Gay Uncle\n\nSuburgatory, USA\u2014A gay uncle is using the heartwarming book Uncle Bobby's Wedding to urge his overly attached niece to stop \"cockblocking him\" in his search for love and passion.\n\n\"I love her to bits, I really do,\" said \"Gunkle\" Rob Marino, referring to his niece Ella. \"But just like her mom, she thinks I'm her funny gay play toy. But you know what? I'm a human being like everyone else who likes to come home from work, watch Netflix instant play, and get fucked as much as possible by my boyfriend. So when I saw this book, Uncle Bobby's Wedding, I lunged for it. Yeah, the book is a great celebration of committed gay love, blah blah blah, but to be honest, really I just saw it and thought, Oh my god, this might just get it through Ella's adorably thick head that I need my life back. And I need my own committed gay love. Like fucking Gunkle Bobby did.\"\n\nUncle Bobby's Wedding tells the story of a gay guinea pig named Bobby, whose niece Chloe is troubled when her favorite \"special\" uncle finds love. Slowly Chloe has to accept that Uncle Bobby is going to marry another special guinea pig, Jamie.\n\nMarino describes how his life as \"Gunkle\" evolved.\n\n\"Oh, this is all her mother's fault\u2014my sister Angie\u2014who I also love to bits but you know, her idea of a gay man is someone to 'be there for you.' You know, cuz I'm gay. I know it's hard for her, a single mom in suburbia? That's almost as bad as being a single gay dude in suburbia. But seriously, it takes me an hour to get out of the city to see them, and it's like I'm on call 24-7, paging Rob to the rescue! Because I'm supposed to be really good at wiping your tears and going to the mall. I do more than that idiot shrink of hers. She should be giving me those co-pays she hands over to that woman.\"\n\nAnd then Angie had Ella, and gay uncle Rob became the go-to \"Gunkle.\"\n\n\"You know, they all assume I have no life because I'm gay, which is hilarious because if I was a whore, and I'm not, I could be out every night fucking five different guys if I wanted to. If that's not a life, I don't know what is. My sister always assumes I can babysit or can come over when Ella's loser dad doesn't step up. If she hadn't had Ella, I'd be so out of here. Ella really is the best thing that ever happened to me. I just need more space.\"\n\nSo Rob and new boyfriend Jayson took Ella out for Afternoon Tea at American Girl Doll Place, and sat down with Ella's best-loved \"Molly\" doll, described by the company as a \"lively, lovable schemer and dreamer growing up in 1944.\" And Rob had his own \"Kanani\" doll.\n\n\"Ella insisted I have one, and Angie chose it because she thought it was really kitschy for me. Why does she think I'm some screaming queen\u2014I hate kitsch! Why does my own sister think every gay man is a screaming queen?\"\n\nRob brought Uncle Bobby's Wedding with him to tea and began to explain why nieces need to stop cock-blocking their Gunkles.\n\nRob: Now Ella, see, Bobby found Jamie after many many years of being alone.\n\nElla: But you haven't been alone, Gunkle Rob, you've had mommy and me.\n\nRob: Of course, honey, and you know I love taking you to American Girl and talking to you every night before bed and having frozen hot chocolates and everything we do all the time, but at some point, everyone wants to have their own family.\n\nElla: You're my Gunkle! You are my family!\n\nElla began crying but Jayson, who'd been quietly assessing Ella, whispered to Rob, \"She's playing you. Let's throw money at this.\"\n\nJayson said, \"Ella, is there something maybe we can get you, something you've been wanting, so that Rob and I can carve some time alone together and it won't be too sad for you?\"\n\nElla carefully composed herself and said, \"The new Josefina doll from 1824 on the Santa Fe trail. The whole six-book series. And extra moccasins and shawls.\"\n\nJayson said, \"Deal!\"\n\nAs they left tea to go purchase Josefina, Jayson looked down at the first American Girl doll Rob ever bought for Ella, \"the lovable schemer,\" and said, \"Wow, Ella's a real 'Molly,' huh?\"\n\nPAID ADVERTISER CONTENT\n\nC-Secrets, Inc.\n\nAre you sick and tired of women telling you that you simply don't understand what it's like to have a child because you had a C-section? Don't you just want to kick them in the pussy? Well nothing's going to change their hippie-addled minds, so C-Secrets gives you a way to avoid the issue completely. Our scar-mitigation technique makes you bikini-ready for beach time. In fact, the only person who will even see your scar now is your husband, and he's sure as hell not complaining about your C-section! Your vagina is still as tight as a twenty-year-old cheerleader's! Thank you, C-section!\n\nBut the real test of C-Secrets is in the carefully constructed, well-rehearsed birthing back-story that we offer you. We provide a practice session with an actor portraying a typical judgmental mom who will blather on about the C-section industrial complex denying women the only \"true\" route to motherhood, through vaginal birth. And you will be able to offer her a fully believable birth story of your own, including \"I nearly died when I started bleeding out but I got through it!\" and \"I bit the tip of my husband's finger off because I refused the epidural, but the pain was really part of the ecstatic beauty!\"\n\nSo consider C-Secrets. Your secret's safe with us, and after using our service, your secret will be safe from every judgy mommy you'll ever meet (and silently despise)!\n\nMom Buys Muscle Massager   \nto Really Massage Muscles\n\nSuburgatory, USA\u2014A local woman was unable to convince the men at the Eaton pharmacy that she was buying a muscle massager to really massage her muscles.\n\nMary Pickering was first noticed in the orthopedic care aisle where she was purchasing supplies for her elderly mother. Tom Carrothers and Joey Marti, on their break from the construction site down on Milford Street, noticed her and thought she fit the sexual clich\u00e9 that has run rampant among men of their age: \"MILF.\"\n\n\"Hey, Tommy she's pretty MILF-y, huh? Tits are a little droopy, but me still likey,\" said Marti.\n\nCarrothers said, \"Yeah, that yummy mummy's aged pretty well.\" Pickering ignored them.\n\nBut soon Pickering was in the personal massage area and Marti and Carrothers were electrified. As she picked up the Ergonomic two-speed Handheld Massager and began heading to the checkout, they decided they were done shopping and would check out along with her. While standing in line, she overheard Tom Carrothers say, \"You know, I have just the thing to 'massage those muscles' of hers. Right, Joey?\" Joey said, \"Yeah, you know if I had a pretty lady like that at home, she'd be getting her muscles massaged every night, real hard, too. No batteries required!\" They laughed and elbow-poked one another.\n\nPickering turned to the men and said, \"Look, I genuinely need this device for my shoulders. My physical therapist told me to buy it.\" Both men were silent, smiling and slowly nodding with their mouths a bit ajar. As she arrived at the checkout, the clerk, who looked to be nineteen, had the same expression. Pickering said, slowly and forcefully, \"I have sore shoulders.\" \"Of course! That will work wonders on shoulders. You know, we are also having a special on RePhresh vaginal gel,\" the clerk said.\n\nAt this point, she said, \"Little boy, I don't have a dry vagina.\"\n\n\"Yeah, definitely not, no dry vaginas if I was in the sack with her, right?\" said Carrothers to Marti, who held up his hand and said, \"Don't leave me hangin'!\" Marti high-fived him.\n\nPickering finally lost her composure. \"Do I look like a fool to you? Do you really think I would trust my orgasms to a piece of junk I buy at a drugstore? I have a whole goddamn family of vibrators at home. And a husband who fucks me into oblivion whenever I want. Oh, and could you be any more clich\u00e9? 'Don't leave me hanging,' all that MILF and Yummy Mummy shit? Does anyone even still use those stupid terms?\"\n\nMarti, looking chastened, said, \"In porno they do.\"\n\n\"What are you even buying anyway?\" asked Pickering, accusingly. The men dutifully extended their hands: Marti was holding Lamisil anti-fungal cream. Carrothers had Preparation H, and a copy of Martha Stewart Living magazine.\n\nMercedes-Driving Dad Dreams of   \nEasier Life for His Children\n\nSuburgatory, USA\u2014A local dad spoke about the current economic hardship and his hopes for an \"easier life\" for his children, while getting his Mercedes detailed at the Hooper Street Car Wash & Detailing.\n\nEric Sellers' S-class sedan had a bumper sticker that read Had Enough?, which referred to what he called President Obama's \"war on the little guy.\"\n\n\"I mean, I am not one of those Tea Party crazies who thinks Obama is a socialist or a secret Kenyan terrorist or anything. I think he's dangerous just because he's an old-fashioned tax-and-spend liberal. The tax rates in this country are criminal,\" he said, looking for agreement from the mostly Latino men using small brushes on the crevices of the car.\n\n\"You know, I'm just a regular Joe, a suburban small-\u00adbusiness man. We are the backbone of this economy! And yet I'm expected to pay more than 40 percent of my income to the government. Money that should be going to my kids, who are suffering in this terrible recession. Thank God I have a whiz-bang accountant who can get that 40 percent down to about 10 percent, but that's still highway robbery.\"\n\nAt a 40 percent tax rate, this would put Sellers in the top 1 percent of America's wealthiest individuals. Sellers scoffs at this notion. \"Oh please, do I look like Donald Trump? I may be 'technically' in the top 1 percent, but with prices these days I'm really just average like everyone else. Right, guys?\" He gestured to the men working, who murmured their approval.\n\n\"You know, America used to be great. Everyone had a fair shot. But I don't know anymore, I just don't know. I just hope and pray life is easier when my kids are grown up than it's been for me these last few years. And for them.\"\n\nHe went on to describe some of their recent hardships. \"This year, I told my wife 'No more Nordstrom. It's Nordstrom Rack or nothing.' To see her face, oh God, as a man, that was quite a blow. That really hit my dignity.\"\n\nSellers has also had to curtail the travel and leisure budget.\n\n\"I had to sit them all down as a family and tell them we couldn't go to Vail this year. Our ski-week would have to happen in Vermont. I thought the kids would be sad, but really, they looked terrified. Seemed like a perfect time to teach them about the tyranny of marginal tax rates.\"\n\nWhile some give too much, Sellers says, other \"protected classes\" enjoy outrageous benefits.\n\n\"These teachers and their extortionist unions get the whole fucking summer off. Wouldn't you like to have the summer off too, guys?\" he said, addressing the detailing crew. \"I know I would!\"\n\nWith his car finished, Sellers checked his iPhone for the time and said, \"Guys, not to be a dick here but I dropped the car off at noon and it's 2:06 p.m. Two-hour guarantee?\"\n\nThe workers looked at him and without saying a word rang up his 25 percent discount.\n\n\"Hey, every penny counts, am I right?\" Sellers said.\n\nPAID ADVERTISER CONTENT\n\nPrecious Toddler PR\n\nIf you're a good parent, and we know you are, you are convinced that your little Aidan or sweet Isabella is the single most extraordinary, precious child ever to walk this fine earth. But be assured: You are the only one who thinks this. Well, maybe your husband and the grandparents do, too, but that's it.\n\nAt Precious Toddler PR, we utilize third-party confirmation to boost your incorrect assessment of your child's extraordinary nature. It could be the well-placed comment on Twitter from a seemingly objective observer. Or a preschool \"teacher\" no one's ever seen before appearing at afternoon pickup to note your child's amazing attributes. Precious Toddler PR will create the buzz you believe\u2014again, incorrectly\u2014that your child deserves.\n\nThink of Precious Toddler PR as an investment in your child's future. Word of mouth will open doors when he has to go to college. Why not start when he's still in the sandbox? Give us a buzz and we'll give it back, at Precious Toddler PR.\n\nPTO Stunner: New President   \n\"Not a Power-Mad Psycho\"\n\nSuburgatory, USA\u2014Longtime PTO participants at the Burns Elementary School are reeling at news that their new president is neither crazy, nor annoying, nor a \"power-mad psycho.\"\n\n\"Oh my God, I'm afraid to say it and end up jinxing it but she seemed completely normal!\" said Joellen Joyce. \"And. And. She told a dirty joke. In front of her kid. And then said, 'Don't tell on me!'\"\n\nJoyce is referring to Laura Beazley, who was a dark-horse candidate to succeed Emily Fahdin, the previous PTO head, who was nicknamed \"Bin Fahdin.\" \"Yeah, it's stupid, but we laughed so hard after we came up with that one,\" said Joyce.\n\nAs for Beazley, Joyce said, \"No one knew anything about Laura so we were convinced that she was a 'Bin Fahdin embed,' that is, an acolyte and pawn of Fahdin, who has moved on to a bigger PTO stage at the Lexington Middle School.\n\nFahdin was loathed and feared for fomenting dissent, or in PTO parlance, \"shit-stirring\"; for launching bizarre vanity projects like commissioning a video crew to document her activities for, in her words, \"posterity\"; for alienating the working parents by calling them \"blood-sucking PTO parasites\"; and for demanding total control.\n\nKatie Mulheren described one run-in with Fahdin, after Mulheren floated a new, and apparently unwanted, idea for improving one aspect of the school's annual Rodeo Fun Fair. \"She trapped me in the spaghetti aisle at the store and got her face right up to mine. I mean, an inch away from my face, and said, 'The Fun Fair has always been done this way, and always will. People in this town don't like change. You need to be very very careful and think really hard about what you are doing.' I really can't do it justice, it was so insane.\"\n\nFahdin also policed members for insufficient spirit or irreverence, unless it was directed at working moms, whom she seemed to openly loathe. \"Oh you don't need Freud for that one! It's because she didn't make partner at her law firm, and had to plow her type-A crazy into something. Lucky us! It was the PTO,\" said Mulheren.\n\nSo when Beazley scheduled her first PTO meeting, members were prepared for the worst. Here's some of what she said:\n\n\"Thanks for coming out at night, I know that's a change for many of you but we need to try to include working parents in the PTO, so they do not feel like PTO pariahs.\n\nMy PTO philosophy can be summed up in four words: No Bullshit, No Drama. I believe PTO is for making extra money for things the school can't provide, and offering a reasonable\u2014and I stress the word reasonable\u2014number of projects to get families involved in the community or enrich education. That's it. If you, as a working parent, want to kick in a little money, that counts as volunteering, and you will be thanked alongside those fortunate enough to have time to lend their labor. There'll be no guilting under my watch. And while I welcome any and all ideas, I do not welcome cliques or backstabbing. I don't know about you, but I remember high school and have no desire to relive it.\"\n\nAs Beazley finished, it was apparent several mothers were tearing up with relief at this unexpected sanity. Mulheren: \"I know it's ridiculous, especially since we had that silly name for her, but this is how I felt when I heard we got rid of Bin Laden. The real Bin Laden.\"\n\nPrincipal Replaces Pledge of Allegiance with \"Eye of the Tiger\"\n\nSuburgatory, USA\u2014 A principal has been placed on paid administrative leave after having children recite lyrics from \"Eye of the Tiger\" rather than the Pledge of Allegiance.\n\nParents learned of the principal's \"lesson\" when kids began using phrases from \"Eye of the Tiger.\" At first, they assumed the kids had learned them from Wii's \"Guitar Hero\" game. But they weren't singing the words, rather, they were reciting them, and one parent contacted the media. \"Yeah, that principal was so far out of line, but I gotta say, I do love that song,\" said parent Richard Dunn, who began humming it himself and lightly pumping his fist.\n\nThe principal in question is Jon Bohrman.\n\n\"They're making a big deal out of nothing! I was trying, very creatively, I might add, to challenge the kids at Maginn Elementary to pursue their dreams, question assumptions, and think for themselves. That's why I forced them to learn 'Eye of the Tiger.'\"\n\nCritics of Bohrman say that was not his sole motivation. They say the answer can be found on his Facebook page, which shows he has \"liked\" several groups: Ayn Rand Greatest Philosopher That Ever Lived, Ayn Rand Rocks, and Ayn Rand's Breasts. Bohrman flinched a bit at the mention of that last group, but readily admits that that he is an adherent of Rand's Objectivist philosophy, which argues that morality is about the pursuit of one's own happiness or self-interest. Objectivists believe only unfettered capitalism can allow for this to flourish, and in general, believers oppose many or most forms of collective action.\n\n\"Yes, I'm a believer. Not in God, of course, but in Objectivism. I got a copy of Atlas Shrugged back in college from the really smart guy in my frat, a genius, hedge-fund god now. This is my favorite quote: 'The world you desire can be won. It exists . . . it is real . . . it is possible . . . it's yours.'\"\n\nBohrman thinks the time has come for Objectivism to flourish. \"Let's face it, America is in decline. Shouldn't we emphasize to our children the importance of personal drive, of having the motivation to face down detractors and fight another day? Now, that's what \"Eye of the Tiger\" teaches, not the Pledge. Listen. . .\" As he played the song, he spoke the lyrics that the kids themselves recited.\n\n'Risin' up, back on the street.\n\nDid my time, took my chances.\n\nWent the distance, now I'm back on my feet.\n\nJust a man and his will to survive.\n\nHe muted the song. \"Now compare that to the Pledge. All the Pledge does is force kids to mouth empty, robotic promises to be loyal to a nation and a god. Nothing about shooting for the moon. What's that all about? I'm sure if Ayn Rand was alive, she would be a fan of the band Survivor.\"\n\nWhen this reporter corrected Bohrman's pronunciation of \"Ayn\" (he was calling her \"Ann\" rather than \"Ein\"), Bohrman paused for a moment and said, \"The liberal elitist media is the reason I'm on leave in the first place, instead of back in school where I belong. But I take solace and get my drive from Rand's own words: 'The question isn't who is going to let me; it's who is going to stop me.'\"\n\n\"I'll tell you who's going to stop him. You're lookin' at him,\" said Superintendent Phil Troutman. \"But of course, the union will try to save him.\"\n\nDoesn't Bohrman see the irony that a union\u2014a collective organization loathed by many Rand believers\u2014is making sure he is still getting paid and will try to fight his dismissal?\n\n\"Oh, not at all. Being in the union is in my rational self-interest.\"\n\nPAID ADVERTISER CONTENT\n\nThe Totally, Completely Not-Creepy Chimney Sweep\n\nNow, ladies, we all know when you get that loud knock on the door, you're thinking, Oh no! Is that the chimney sweep? I forgot that I made that appointment and now I'm barely dressed! Where, oh where, could my clothes beeeee? Well, if you enlisted our fine professionals at The Totally, Completely Not-Creepy Chimney Sweep, you have absolutely nothing to fear!\n\nAt Totally, Completely Not-Creepy Chimney Sweep, our serviceman will not size you up and down and think about all the dirty things he wants to do to you. The Totally, Completely Not-Creepy Chimney Sweep will never ask to use the bathroom just to look in your medicine cabinet in hopes of learning more about you.\n\nThe Totally, Completely Not-Creepy Chimney Sweep will not find an excuse to \"check something with the chimney upstairs,\" slip into your bedroom, slowly open the dresser drawer next to your bed, and look for something naughty that will give him days of material for self-pleasuring. He will not go to his buddies back at Totally, Completely Not-Creepy Chimney Sweep headquarters and say, \"You need to get a load of the rack on that lady down on Milford Street.\" And it should go without saying that your Totally, Completely Not-Creepy Chimney Sweep will not then look you up on the Internet and try to friend you on Facebook or follow you on Twitter.\n\nThat will never happen. Why? Because we are not men. We are professionals. Wear that thong all you want, we're not looking! Trust in us, and we'll treat your chimney like the Queen you are. Call us today!\n\nDad Confirms Child's   \nWorst Fears in Life\n\nSuburgatory, USA\u2014A dad who works as an actuary can't resist ghoulishly confirming his child's worst fears in life, in what's been described as perhaps the worst bedtime tuck-in talk ever captured on video.\n\nJoe Gardner's wife Cherie surreptitiously recorded the incident with the couple's ten-year-old son, Andrew, and is about to post it on YouTube.\n\n\"Why would I put this horror show with the two people I love most in the world up for all to see? I'll tell you why. Joe is the greatest, most wonderful dad, but he doesn't believe me when I tell him that he is surely scaring the living shit out of our child, not to mention destroying his dreams and ambitions. And I need the rest of the world to tell him he's out of his mind and needs to stop,\" said Cherie.\n\nJoe tried to defend himself. \"What can I say? I've always had a weird talent for doom even as a child. Lady Gaga's right: I was born this way. I'm not ashamed! Then when I kicked ass in math, well, doom + math = boom! I'm an actuary! I love my job and yes, I do want my kid to know what disasters he really has to fear in life and what he doesn't.\"\n\n\"But honey, you're also a dad, and you should be saying to him, these terrible things will never happen,\" Cherie said.\n\n\"Actuaries never say never. You should know that by now, Cherie,\" Joe responded.\n\nExasperated, Cherie loaded the YouTube clip and played it. Then she pointed her finger at this reporter. \"You tell me how crazy this is.\"\n\nJoe: Bedtime, Polar Bear [Andrew's nickname]! How was your day?\n\nAndrew: Great, Dad! We did a unit on space in school. It was soooo cool; I think I want to be an astronaut. My teacher thought that was great.\n\nJoe: Well, it's a really cool field that's for sure, but you should know the odds of becoming an astronaut are 13,000,200 to 1. The odds are better at becoming President. But you know, that's a real long shot, too. That's 10 million to 1! Probably better to think a little more practically about things to do with your life.\n\nAndrew: Oh.\n\nJoe: Hey, Mom said you looked a little freaked when that thunderstorm came through this afternoon.\n\nAndrew: Well . . . a little.\n\nJoe: Listen, Polar Bear. Daddy knows more than anyone about whether you should worry\u2014it's what I do all day long at work. I'll tell you whether you really have to worry a lot about lightning. Compared to other stuff that can kill you, the answer is no. I mean, you have a way way higher chance of dying in the car on the way to school! Or in the bathtub right there [pointing to the bathroom]! And not by drowning, but by falling; so try to keep your feet steady in there, buddy! Ha ha ha. Heck, even the lawnmower's way more deadly than lightning. You know what's weird?\n\nAndrew: What, Dad?\n\nJoe: You'd think your risk of dying by chainsaw would be higher than dying by the lawnmower but, well, you'd be wrong\u2014lawnmower wins! Maybe we shouldn't talk about all this.\n\nAndrew: No, tell me, Daddy.\n\nJoe: Well you always hear that the most dangerous thing you do is get in the car every day and that's true, no doubt about it. But while the chances are very high that you will be in an accident at some point, the chances you'll die in that accident are pretty low. Lower, in fact, than the chances of being murdered! Lower even than the chances that a catastrophic asteroid will hit the Earth! Which sounds crazy, but boy when you crunch the numbers, it doesn't start looking so unlikely. Those big scary Hollywood movies? Well, they may not be so far off.\n\nWithin just a half hour of the YouTube clip being up, comments began to appear: \"Douchebag.\" \"You are one sick and scary bastard.\" \"So, I can't get pregnant and this psycho gets to be a parent?\" \"Paging protective services!\"\n\nWith that last comment, Cherie quickly took the clip down, but not before saying to Joe, accusingly, \"You see?????\" Joe, looking deflated, said he would go check in on Andrew to see if he was OK.\n\nJoe: Bear, I'm so sorry for our little talk tonight. So, so sorry.\n\nAndrew: Why, Daddy?\n\nJoe: Well I guess I sort of told you to forget about being an astronaut and then I talked about all that awful stuff that can happen.\n\nAndrew: It's OK, Dad! I wasn't scared at all. I thought it was so cool! I can't wait to tell the boys all that cool stuff at school! Especially that death by chainsaw part!\n\nJoe: Well, don't scare them too much. And you can be an astronaut if you want to be. You're my best boy. You can be anything you want to be.\n\nAndrew: I don't want to be an astronaut anymore. Dad?\n\nJoe: Yes, Andrew?\n\nAndrew: What are my chances of becoming an actuary?\n\nIn a flash, Joe went from looking chastened to elated. He gave Andrew a bear hug and said, \"Chances are sky-high.\"\n\nSHOUT OUT\n\nMessage to Husband:   \nStop Getting Boners When I Cook\n\nBeatrice Mathers is a mom and corporate lawyer who lives on Thomas Street.\n\nI take to the Shout Out today, as a proud feminist married to a husband who I thought was a proud feminist, too. Except now the only time\u2014and I'm not exaggerating\u2014the only time he gets a boner for me is when I'm in the kitchen.\n\nWhen I was a girl, I remember watching the classic '70s horror movie The Stepford Wives, and being disgusted that the men were massively turned on by the fantastic cooking of the sexy domestic robots they had created. \"She cooks as good as she looks\" is the line I remember best.\n\nBut I don't have one of those suburban body-snatcher husbands, right? Or do I? My husband Bill is my own private Dennis Kucinich. Most of his friends are women, half of which are lesbians, and there is the guy from high school who got a sex change but still likes women so she's now a lesbian, too. All are welcomed in the warm embrace of our anything-goes, super-liberal home. We've been together for nearly twenty years but didn't get married until I was eight months pregnant, and when we did it was at the Justice of the Peace and I wore a stained chambray maternity jumper (gay men everywhere are weeping at this). My husband got really sad when feminist Bella Abzug died, cried during the Harvey Milk movie, and once told me he didn't care if I ever shaved.\n\nAnd yet as soon as our son was safely swaddled in our first suburban home, he seemed to forget that I used a microwave almost exclusively when we lived together for the three or so meals a week I ate at home. Back then, \"cooking\" meant using the stove top to boil water. Now, I quickly saw that it was here, in the kitchen, that he was most attracted to me. Those times when he remembered when I was a real professional kicking ass on the job? Oh, that's nice, honey. But put me in front of the stove endlessly stirring some chocolate pudding? Watch out, Daddy's zooming in like a predator drone to sex me up! \"Woman, why don't you go in the kitchen and make me a sandwich.\" That was now my life, not a joke. (OK, apology break here to Bill for serving up my own steaming pile of hyperbole. He would never, ever say that to me, and not because I would lacerate his genitals, which I would, but because he's not a caveman. That said, you can damn well bet that he'd still love that sandwich, and love me a little more because I made it for him.)\n\nThe problem for wives, of course, is that a woman who achieves at work can be an absolute flop at homemaking. Not me, of course (cough, splutter). And we are often blindsided, left to wonder if we really knew our husbands and whether they ever valued us for what we are truly good at: Working in a goddamn office, dumbass! What do you think I went to an Ivy League school for? To flip your fucking pancakes? The fact that it isn't intentional makes it even more insidious, as if even an enlightened man like my own can't resist the siren song of sexist expectations. Bill said it feels completely natural and instinctual that he has this attraction to my domestic side. Yeah, well, my talents in the domestic arts feel about as natural to me as a pair of silicone double-Ds. Oops, gotta go, almost dinnertime. Now, let's see, was that bean burrito supposed to be nuked for a minute thirty or two minutes?\n\nAsperger's Dad Unlikely   \nSex Symbol at School Pickup\n\nSuburgatory, USA\u2014A dad with undiagnosed Asperger's syndrome has become an unlikely sex symbol for the moms in the Walker School pickup line, because of his candor, weight-lifting regimen, and \"special interest\" that is unusually appropriate for socializing with other moms.\n\n\"How do we know he has Asperger's?\" said Lindsay Cooper. \"Gee, I don't know, maybe because these days every one of us either knows a kid with Asperger's or has one ourselves? Trust me. We know it when we see it. And with Mark, we like what we see.\"\n\nMark Toomey works from home as a computer programmer and so is more available than his wife to pick up their children from school.\n\nBeth Barton describes him this way:\n\n\"He's hot, he's blunt, he's emotionally unavailable, and, well that's like a triple whammy turn-on. Oh, and you heard about his awesome 'special interest,' right? And don't forget those rock-hard abs,\" said Barton.\n\nPart of Toomey's never-changing routine is intensive weight-lifting. \"Ladies, guess who was at the pond last week. Oh. My. God,\" said Melissa Bandar to the other moms, while smiling and nodding. They joined her, smiling and nodding.\n\nBut the ladies say his looks come second to his \"thrilling\" and \"provocative\" honesty. Bandar said, \"One time he walked right up to me, while I was talking to someone else, and he just blurted out, 'You have the most beautiful breasts.' Then he immediately looked worried, and I wanted to touch him on the arm, but you know, I know he flinches a bit, so I held back. He was so cute. He said, 'I'm sorry, I don't have much of a filter, do other people tell you that you have beautiful breasts?' I said, 'Not enough of them, honey!'\"\n\nGeri McGovern had her own sizzling encounter.\n\n\"One time I was sitting next to him while waiting for Carter to get out of physical therapy. We were talking about our pasts and he just abruptly launched into this little speech about puberty: 'Do you remember when you were a teenager and you just had that surge of sexual hormones, and you could just feel that sexual drive coursing through your veins, pump pump pump, like you didn't really know what you needed but whatever it was you really really needed it?' By the time he was done, oh God, I was seriously breathless.\"\n\nHis candor and unusual talent for detail and memory also make him an incorrigible gossip, which the moms love. Said Bandar, \"He'll always remember which parent was shit-faced at the school benefit or the time he noticed that Gina's skirt had changed between drop-off and pickup and the whiff of cologne on her, which he said he knew was Karl Wagner's [not Gina's husband]. He's like the Sherlock Holmes of gossip!\"\n\nBut McGovern doesn't think he's a gossip, exactly. \"You know I was thinking about it and he's not really a gossip because to be a gossip you have to know you're not supposed to say something, and say it anyway. But Mark, he doesn't know he's supposed to zip a lip. So he's totally blameless shoveling all that great dirt at us. And we lap it up.\"\n\nBut perhaps Toomey's most unlikely selling point involves his \"special interest,\" which is an obsession with a single topic: Oprah Winfrey. While Winfrey's show has ended, she is still a subject of fascination to the moms at pickup, who grew up watching her.\n\n\"He's like a walking Oprah bible. He can tell you what the audience got at the \"Favorite Things\" episode\u2014every single year they did it. He knows every detail behind the Tom Cruise couch-jumping thing. But the best is that he has a spreadsheet that tracks public records and the fates of everyone who has ever appeared on the show. So you want to know whatever happened to that horrible crackhead mother or the Klansman who came out as gay, he might just have it. Oprah shoulda hired that guy,\" said McGovern.\n\nWhile Toomey will tell you anything you want about Oprah, he is not forthcoming emotionally, which is typical for many with Asperger's. This hasn't deterred the moms. \"Oh, no way! He maintains the mystery,\" Bandar said. \"You know,\" she added, \"he has a lot of trouble, no surprise, with eye contact. But when he does make eye contact? It's like a few seconds of pure magic.\"\n\nAnd what does Toomey say about all this? He shifted a bit uncomfortably and didn't look this reporter in the eyes, but said, \"I haven't had it so easy my whole life making friends. My wife saved my life but next to her, these moms are the best things that ever happened to me. Well, them and Oprah.\"\n\nWoman with Eating Disorder   \nConsiders Meth\n\nSuburgatory, USA\u2014A church drug-awareness program has backfired in spectacular fashion for a mom who has long battled an eating disorder.\n\nJanet Gosling attended a session on drug addiction in the community, which included a harrowing anti-drug photo collection called \"Faces of Meth.\" It shows in frightening detail the physical toll that meth abuse takes on a body, tracking mug shots from habitual users over time.\n\n\"Now, ladies and gentlemen,\" said session leader and police officer Bill Barry, clicking through the befores and afters. \"As you can see, meth simply ravages people who once had bright futures and leaves them diminished and often deformed.\" There were audible gasps in the room.\n\nGosling was one of those gasping, but not for the reasons Barry intended. Gosling has long wrestled with an eating disorder and body image dysmorphia, which she actually considers a \"gift\" from what she calls \"thin God.\" Janet sat with her friend Debbie Flander.\n\nDebbie: Oh my God. Nasty.\n\nJanet: Right . . . yeah . . . wow. Um . . . but . . . look how thin that one got.\n\nDebbie: But Janet, she has sores covering her face. Oozing sores.\n\nJanet: Oh yeah, I guess that's bad but you know, there's always a product for that. Try laxatives and see what that does to you\u2014that stuff is poison. Works like you wouldn't believe, but you end up dribbling poop uncontrollably. This cop should be talking about that stuff.\n\nBarry clicked on the \"before\" image of another abuser.\n\nJanet: Look at that fatty! She musta been 135 pounds. No wonder she felt the need to abuse drugs.\n\nDebbie put her hand to her forehead as Barry clicked on the \"after.\"\n\nJanet: Now look at that. See, she's totally normal now.\n\nDebbie: Janet, she looks like a zombie ghoul about to come and scoop out your brains for breakfast.\n\nJanet: Oh, I think these people are so amazingly high that they don't even remember what breakfast is. Imagine that.\n\nBarry described the woman's fate. \"This abuser lit her trailer on fire when her home meth lab exploded. She lost everything.\"\n\nJanet: Well, at least she can go buy some new \"thin\" clothes! Debbie, stop it, I'm just kidding! I mean, sort of?\n\nBarry chose a final woman to illustrate meth's degrading effects. She now suffers from so-called \"Meth Mouth,\" in which the constant use of meth causes catastrophic effects to the oral cavity.\n\nJanet: Look at how sculpted her face is now! She started out looking like Rosie O'Donnell and now she looks like a thinner Maria Shriver! So dramatic.\n\nDebbie: Yeah Janet, a toothless, wild-eyed, sore-scratching Maria Shriver.\n\nJanet: You know, bulimia rots your teeth, too. It gives you horrific breath and burns up your GI lining. We all pick our battles in this life.\n\nAs the presentation ended, Janet went up to thank Officer Barry for his outreach and said, \"Which parts of town are known for selling meth? I . . . I . . . really want to make sure I keep my kid away from those places.\"\n\nPAID ADVERTISER CONTENT\n\nStatus Wrappers, Inc.\n\nEver felt the shame of being caught carrying . . . a Wal-Mart shopping bag? Or being spotted feeding your beloved children non-organic crap that they love, and you love as well? Does that cute new sweater really seem that cute when people see it sitting there in a bag from Kohl's?\n\nAt Status Wrappers we provide you with the cover you need to tote around your purchases without embarrassment. Whether it's the spanking new Nordstrom bag or the well-worn, but current issue of the New Yorker, which of course you read cover to cover (Wink, Wink!), or the wide selection of Annie's Organic boxes, we will provide you with the perfect receptacle to house or hide the crap products you really love, without the slightly raised eyebrow or snippy behind-the-back comment.\n\nHere's a testimonial from one satisfied customer:\n\n\"One day I had a stack, and I'm not lying, a stack of flattened McDonald's Happy Meal boxes sitting in my passenger's seat when I saw that totally snotty bitch from my daughter's ballet class walking toward my car. I was immediately able to cover the stack with both a copy of Vanity Fair and a Whole Foods reusable tote with three boxes of Earth's Best Elmo's Pasta with Sauce with Carrots and Broccoli. Thank you, thank you, Status Wrappers!\"\n\nYou know what your parents always told you? Don't judge a book by its cover? Well, you know now, that was a load of shit! Everyone judges a book by its cover. Welcome to adulthood! So what's that cover going to be? Let Status Wrappers be your cover, in a judge-or-be-judged world!\n\nDad Pretends IKEA Is   \nChild Cultural Enrichment\n\nSuburgatory, USA\u2014A local dad was outed for pretending IKEA was a culturally enriching outing for his son.\n\nPeter Marello lost his job eight months ago and is now the primary care provider for his four-year-old son James Patrick. Marello admits to being \"kinda lost\" and one day, on a quest to find a garlic press, he made his first-ever visit to IKEA. He found a lot more than he bargained for. \"They have free childcare, food that doesn't break the five-dollar mark, clean bathrooms, and nice people. And of course, modern Swedish design at quality prices. Life's just a lot brighter and shinier at IKEA.\"\n\nHow did his regular IKEA pilgrimages begin?\n\n\"It sort of evolved . . . how I started going so much.\" Marello said sheepishly. \"At first, I made a list of those little odds and ends you need but never buy\u2014the new silverware tray, new knobs for the bathroom, new curtain rod\u2014but then I just bagged the list and said, 'Fuck it.'\"\n\nNow they go a few times a week, with Marello dropping James Patrick in \"Smaland,\" IKEA's free child-care drop-off and what James Patrick actually thinks IKEA is called. Then Marello finds a comfortable spot in the store's vast cafeteria and sits down with his bottomless coffee and iPad. If no children are there, he switches the TV channel to ESPN to catch up on the scores; a few IKEA associates usually join him for that.\n\nWhen James Patrick's child-care hour is up, Marello retrieves him and they visit parts of the store that James Patrick has named: the \"Land of a Thousand Bedrooms\" and \"Magical Forest,\" where they play hide and seek among the artificial plants. They finish their visit with a cinnamon bun and a quick chat with Stephen Marsden in Returns and Exchanges, who especially loves James Patrick because his own grandchild lives hours away. \"Sometimes if my wife works late we go for Wednesday Rib Night\u2014it is off the wall cheap and James Patrick loves the Swedish meatballs,\" said Marello.\n\nBut he doesn't tell his wife, Jill, a busy corporate executive, where they go for ribs. Instead of admitting that he and James Patrick have unofficially joined the IKEA family, he tries to avoid detailing their outings, or, when pressed, says that they were attending a \"museum of Scandinavian design and culture.\" Then James Patrick adds, \"It's called Smaland.\"\n\nMarello was outed, however, while at the home of friends Marisa and Joe Mucha, empty nesters Marello used to work with. While catching up, they asked Marello what he and James Patrick do together during the day and James Patrick immediately said \"Smaaalannd!\" Marello quickly described Smaland as a \"small museum with a focus on Scandinavian design and culture,\" and hoped that would be the end of it. \"Wow, Scandinavian design and culture? That sounds totally random, but these days I guess there's a museum for everything,\" said Marisa.\n\nJames Patrick said, \"It's not small, Daddy, it's huge. You should see the furniture pickup area!\"\n\nJoe asked, \"What did you say, James Patrick?\"\n\nMarello said, \"Oh he's just talking about a part of the museum where you can touch the furniture. So what else is going on with you guys?\"\n\nMarisa, already looking suspicious asked James Patrick, \"Sooooo, what else do you do at the museum of Scandinavian design and culture?\"\n\n\"Oh, eat cinnamon buns and jump into the huge ball pit and throw balls all over and go to the Land of a Thousand Bedrooms and we talk to Stephen and . . . hey,\" James Patrick said, running his hands up and down the living room chair. \"This is the Ektorp Jennylund!\"\n\nAt this point, the jig was up, and while Marello looked mortified, Joe said, \"James Patrick, does the museum of Scandinavian design and culture have a big sign with the letters I K E A?\"\n\n\"Yes! Does that spell Smaaaland?\"\n\nMarello's friends erupted in riotous laughter and asked why he didn't just say he was going to IKEA. \"It sounds pathetic! And don't you dare tell Jill, I don't know how, but she still hasn't figured it out. She's totally turned into a man now, and barely even asks about our day. But she'd still kill me if she found out I wasn't providing James Patrick with 'enrichment.'\"\n\nJoe Mucha said, \"Oh, don't beat yourself up. When you're a parent you gotta do what you gotta do to just get through it. Don't make yourself crazy. We know all about that, right Maris?\"\n\nMarisa agreed. \"James Patrick enjoys it, you enjoy it. End of story.\"\n\nNot quite. James Patrick emerged from the kitchen with a melon baller.\n\n\"This toy is miscontinued.\"\n\n\"Discontinued, James Patrick,\" Marello said.\n\n\"But you can bring it back and get store credit\u2014ask for Stephen and tell him James Patrick said Hi!\"\n\nMom's Thrill Dashed by   \nSex and the City's Miranda\n\nSuburgatory, USA\u2014A mom tried to appear blas\u00e9\u2014and failed\u2014about news that actress Cynthia Nixon has not only moved to town, but has also enrolled her child at the local school.\n\n\"Oh yeah, it's a great school that we're zoned for. We've really lucked out. Oh and you know, I think Cynthia Nixon's new in town and she's zoned for the school, too,\" said Joyce Birney, in an intentionally offhanded way en route to the first day of classes for her nine-year-old, Aaron.\n\nArriving at the school, Birney said, \"Yeah I mean we could have easily afforded to send Aaron to Briarcliff, but we just thought, we have a great school right here, people love it, and\u2014Holy shit, there's Miranda! I mean, um, there's Cynthia Nixon.\"\n\nBirney's husband, Robert, turned away quickly. \"I don't know how I'm going to talk to her when I know what her boobs look like the whole time. They're little and super white, not my bag. I mean, not bad or anything, any boobs are better than no boobs at all, but, I don't know, it's going to be distracting to me.\"\n\nNixon had arrived with her partner, public education activist Christine Marinoni. As Birney and her husband attempted to hide their furtive glances, Birney discussed the values that she believes she shares with Cynthia Nixon. \"You know, Miranda, she realized you have to give up your 'city dream' eventually. Leave Manhattan. Try Brooklyn. Now she's here. Just like me! Can't send your kids to those crap schools in the city. She was always the most sensible of the four girls. Oh, and she's a lawyer, too, who didn't stop working after Brady came along. Oh, there's Aaron's teacher. We need to talk to her, I guess.\"\n\nThe teacher was already speaking with Marinoni, as Birney waited her turn. \"Cynthia must really respect her. She never respected Steve. Now this relationship must be built on something real and lasting. That woman must worship Miranda!\"\n\nAs Marinoni finished, Birney whispered to her husband, \"Let's blow off the teacher,\" and decided it was \"too much of a coincidence not to say hello\" to Nixon, the coincidence being \"She's just a mom like I am.\"\n\nBirney introduced herself to Nixon and said, \"We're thrilled that you've moved to town! It'll take some getting used to life outside the big city and all but it's worth it for the kids, right?!\"\n\nNixon very cordially said, \"It's lovely to meet you! But we are only in town for six months.\"\n\nBirney looked crestfallen. \"So you didn't move here\u2013move here?\" Nixon said no. Birney then brightened, \"Wait, you're not shooting the next Sex and the City movie out here, are you????\"\n\n\"Oh no no! I'm doing the play Angels in America at the McClaren.\" Nixon said.\n\n\"Wow, that must be such a drag for you after shooting with all that glamour and fun costumes in Abu Dhabi!\" said Birney, in what her husband Robert calls her \"annoying girl-talk tone.\"\n\n\"Actually that was shot in Morocco. And I'm not knocking Sex and the City at all, but really theater is my first love, so it's a thrill for me to be here, even though we are heartbroken to take the kids out of P.S. 34 back in the city\u2014it's . . . um . . .\" Nixon looked around. \"It's a little more diverse. And in a way, this is really a lot more like what you'd find in private school. Christine and I really believe in diverse public education. I'm sure it's great here though!\" said Nixon, seemingly trying to soften the blow.\n\n\"Ah well, great to meet you,\" Birney said. As they walked away, Birney looked deflated, and whispered to Robert, \"Such a boring snotty lesbian.\"\n\nPAID ADVERTISER CONTENT\n\nThe Calorie-Schmalorie Pizza Kitchen\n\nBoy, there's no bigger buzzkill than eating that enormous bacon cheese hamburger with fried onions and fried egg and then stumbling upon that nasty nutritional information. We know what it feels like: Oh My Fucking God. I just ate in one meal what a whole African village might get in a day!\n\nWell, at The Calorie-Schmalorie Pizza Kitchen we will never, ever reveal the calories in our delectable dishes. To anyone. What if we're forced by the arm of the law to post our calories? Then we'll just pack it in; that's how important our commitment to ignorance in eating is.\n\nNot only will we avoid public posting and online customizable calculators, but we will resist all direct questions from our patrons. Let's say you're thinking of eating our wonderful, decadent Death by Chocolate Molten Lava Cake, and you think, \"Gee, I should ask the waiter the nutritional content of that Lava Cake even though I really, really don't want to.\" Finally you cave and ask. Our waiters will simply tell you, \"You look so thin! Do you work out? You must really deserve our wonderful cake, because you are really taking care of yourself.\" Don't try to persist. Our waiters are trained to withstand al-Qaeda terrorist interrogation\u2014they're not going to spill our secrets to little ole pesky (and thin) you!\n\nWorry about your ass somewhere else. Eat, enjoy, and don't look back. The Calorie-Schmalorie Pizza Kitchen.\n\nFeared Room Mommies   \nDivide Iraqi Oil Revenue\n\nSuburgatory, USA\u2014The Room Mommies, infamous for their take-no-prisoners negotiating tactics, have been dispatched to settle an issue that's dogged Iraq in the years since the 2003 US invasion: the division of oil revenue among rival factions.\n\nMommies Christina Hohn, Rachel Tovacs, and Lisa Epstein traveled with military escort to Baghdad, at the behest of Iraqi president Jalal Talabani. Hohn spoke from inside the still heavily fortified Green Zone.\n\n\"Listen, last year when we had to manage the holiday gifts at the preschool and divide them between the teachers, it was just a major, massive shit storm. Every teacher is supposed to get the same amount, the float teachers get a little less, Bernardo the facilities guy, he gets some too. And what happens? We have to drag these mothers kicking and screaming out of Starbucks to cough up the cash. And do they accept our authority?\"\n\nEpstein, as is custom among the Room Mommies, finished Hohn's thought for her.\n\n\". . . NO! One parent went renegade and snuck in some big Neiman card on the sly. One bought organic food baskets with kombucha or some bullshit granola crap. Some gave nothing, but still wanted their kid's handprint on the group card. The Bible thumper gave her own card with a creepy hologram Jesus, you know, like, right out of Carrie, and then gives it to Miss Levine. LUH-vine! One actually gave swag from work. . . .\"\n\nMommy Tovacs grabbed the microphone from Epstein and tried to shout over the din of mortar fire.\n\n\". . . then the teacher who got swag is pissed at the teacher who got the Neiman gift card. The teacher who got kombucha had no clue what it was. Then Miss Levine filed a complaint about the Jesus card. But you know what? We got them to take their junk back and give us cash, talked Levine off the ledge, and handed out perfectly apportioned Amex gift cards, all before holiday break. And we're going to get the dirty work done here in Baghdad, too. The Kurds, the Shia, and the Sunnis, they know our rep. We're not here to make friends. We're gonna knock some heads before bath and bedtime, and everyone will get their fair share of those oil dollars.\"\n\nAfter the morning's first meeting, which featured a nutrient-dense snack and water sippy cups, Mommy Tovacs was ebullient at their progress. \"Now, look, with 59 percent of Iraq's oil reserves concentrated in southern Basra we pointed out to the Shia that poor Dahook, Anbar, Babel, and Dewaniya have nothing! Is that fair to Dahook, Anbar, Babel, and Dewaniya? No! And the Shia said politely, 'No, Mommy Tovacs.'\"\n\nEpstein said this to the Kurds. \"I know, I know the Sunnis did the gassing and all, but being good friends means forgiving. And sharing. And with 12 percent of total reserves held in Kurd-controlled Kirkuk, you have to share. You must share.\"\n\nThe Kurdish representative, however, was visibly shaken by what happened behind closed doors. \"Saddam . . . we praise Allah and the George Bush for saving us from that pure evil set upon our land. But these ladies, uhhhh, Saddam never met these ladies. Saddam, filthy dog, he terrified me. And the Sunnis, those puppies of the filthy dog, they scare me. But not like this. Now I see that I need to work on my sharing. I will work on my sharing.\"\n\nThe Shia cleric, who demanded that the Room Mommies wear the hijab head cover during the talks, was similarly rattled by the morning's brinkmanship. \"I thought, well, I mean, I thought the hijab would soften the Room Mommies, but it didn't work. These ladies . . . well . . . I've faced fedayeen [torture squads] who'll give you cigarettes and sweets. And mercy.\"\n\nThe Sunni negotiator, a former Baath party member and accused war criminal, was more blunt. \"We'll do whatever the Room Mommies tell us and, inshallah, they'll go home, they'll take those terrible devices\u2014the nasal aspirator and the anal thermometer\u2014away, and we can, what did that Jew lady call it? 'Go nappy.'\"\n\nWaitress Wages Anti-Foodie   \nJihad on Chowhound\n\nSuburgatory, USA\u2014A local woman, fed up with the high-end restaurant where she waitresses and the people who eat there, has launched an anonymous online jihad on the foodie website Chowhound.\n\n\"OK, [head chef] Graydon would be horror-fied that you are calling it 'high-end.' Because that sounds fancy and contrived, which, of course, Ploughshare isn't at all. It's just farm-to-table pure authenticity on a plate! This shit will set you back two hundred bucks for one dinner; the most expensive food in suburbia within a hundred miles, but it doesn't matter. It's still 'rustic comfort food,' isn't it? Whatever that means.\"\n\nThe woman spoke to this reporter at the Elm Street Applebee's. She asked to remain anonymous, and would like to be referred to by the Internet name she uses while terrorizing the unsuspecting foodies on the section of Chowhound devoted to the region. Her Internet name is EatMyShit.\n\nEatMyShit feels like it is her responsibility to puncture the illusions and pretensions of the foodies who make her job torture.\n\n\"So go to Ploughshare and look at the communal tables with that tiny hint of dust. That is not naturally occurring dust. It's artfully dusted every morning. Do you know that the maple used for those tables is recovered wood from a 1950s bowling alley? Because you know what foodies also like when they're not eating food that's farm-to-table? Irony! Mmmmm mmmm yummy yummy, gobble gobble, gimme my lobster gruyere mac and cheese and a Pabst Blue Ribbon, please!\"\n\nEatMyShit realized she could take out her many frustrations on the foodie website Chowhound, and away she went.\n\n\"Loco-More\" asked if anyone knew where he could find regionally sourced wild ramps. EatMyShit responded: \"You mean you need a bag of onions? Yes, Wal-Mart has started selling produce. Local enough for you? And does every fucking ingredient have to have a zip code attached to it?\"\n\n\"Chowdah-hound\" was in search of the perfect Tunisian Mahdjouba Djazairia sandwich with \"round, flat griddled bread.\" EatMyShit wrote, \"Did you hear that a poor Tunisian man selling vegetables from a pushcart set himself on fire and touched off a revolution that swept the Middle East? No? Oh right, you're too busy chasing down your super-special-ethnic-I'm-the-coolest-sandwich.\"\n\nEatMyShit's favorite guerrilla tactic is searching for people who use the words \"famished\" or \"starved\" or \"dying\" for something like, say, white truffle oil or nettle soup, and then posting pictures of emaciated Somali children in response.\n\nJust where did this seemingly bottomless pit of anger come from? \"You know, it's not really that fun serving food you yourself can't afford. I'd almost rather work in a place that actually screams out that it's high-end, instead of pretending to be so simple and virtuous. Then you see these people trying to seem so casual snapping pictures of their precious dinners and putting it on Facebook like they just won the fucking Nobel Peace Prize for Eating.\"\n\nSo when \"Gordough\" asked on Chowhound about the ambiance at Ploughshare, EatMyShit was eager to respond: \"Douchebag with a side of Hipster. Oh, and you know the only thing worse than a hipster? An old, gray-haired suburban hipster. Give it the fuck up already.\"\n\n\"Yeah, so I guess I have a really strong position on this topic and . . . and . . .\" EatMyShit unexpectedly started tearing up. \"Well, to be really honest, my boyfriend Graydon\u2014he's the chef at Ploughshare. He broke up with me last month. Maybe it's made me a little crazy. . . . Love sucks, doesn't it? Miss?\" she said to the Applebee's waitress. \"Can I get the Bloomin' Onion?\"\n\nSHOUT OUT\n\nThe House that Ate My Husband\n\nCarla Baker is a wife and mother who lives on Linden Street.\n\nI take to the Shout Out section today to deliver a cautionary tale to my fellow wives out there.\n\nIt was July 16, 2007. Yes I remember the exact date, how could I not? How can a loving wife and mother erase from her mind that horrible day when her family was ripped away from her without warning? The day I signed my name on all those dotted lines, page after page, thinking I was forging a bright future for my son, my husband, and me, too, in our new suburban town. But that wasn't how it all worked out. No, no it did not.\n\nIt was the moment Steve was stolen from me by a mistress who consumes his heart, his time, his very soul. She is unrelenting in her demands, and her steely grip on him is complete. To add insult to injury, while I can still rock a size 8 on a thin day, she is built like a brick house. Because she is a brick house. Our house. And I curse the day I let that fat bitch into my life.\n\nAs soon as he saw her, when that old-hag broker pimped her out to us, he had to have her. His hand ran gently across her mantle. He traced the curves of her countertops. It was only then that I realized that I hadn't seen Steve look like this in years: He looked happy. I could tell that this was it, it was her or nothing. I thought she was borderline white trash, I mean, her kitchen? Those tacky cabinets? Steve insisted we could class her up. She just needed our help. Steve really meant his help, his tender, knowing touch. And within weeks of that day we closed, as the tools piled up and the projects got their own Excel spreadsheet, I knew he was hooked. He was already out the door emotionally, and he was taking my preschool son with him.\n\nShe makes Steve feel needed in ways I never could. \"Steeeve, my gutters are so clogged. Can you clean them out? Pleeeeze?\" And he's out there in a flash. Sometimes he'll try to duck out and I'll catch him and demand to know where he's going. He can't look me in the eye or say it, but I know where he's headed: Home Depot. Because she loves sending him out on a whim, hoping he'll come back with some bauble to make her prettier. Whore-red paint for her shutters. A gaudy spotlight to show off her shapely front door. Anything to tart her up, to keep him coming back for more. And when he's not hanging off some part of her, I can see him drift, get that glazed look. I know he's thinking about her, what he wants to do with her next and how fast he can start. Then there was that time she called during dinner.\n\n\"Steeeve, the acid rain is falling on me. I'm burning! Can you come and power wash me? It's stinging me! Ow!\" I told him if she ever called during dinner again, I was out of there. He shot back at me, \"You know, every guy in town is just like me. You act like I'm some criminal or something!\" I said, \"They are not just like you. They bring in plumbers, landscapers, and handymen so they don't get too attached. But you couldn't resist, could you?\"\n\nThere's always drama with her. I've always been sensible, reliable, predictable, easy-peasy; now I see what Steve has always wanted: a train wreck. He can't get enough of the excitement, the challenge. \"Oh, no! How did this happen? Steve, helllp! My basement just flooded and you need to clean me up now! The mold, it's coming!\" Then, \"I don't know how this could have happened but my furnace shut down and I'm getting soooo cold, Steve!\"\n\nYou know, I think I could accept this betrayal from Steve; I get it, relationships change, mature, grow old, grow tired. We're both adults. But she's sucked our little boy into her sickening web, too. He follows Steve around with his little play toolbox, anxious to see what Dad's all hot and bothered about. When I ask him to make muffins with me, Jackson will say \"No Mommy, I have to help Daddy wee-gwout the tiles in the baff-woom!\" And I see Steve's example imprinting itself on my little boy. It's what Steve's dad did to him. And I look ahead at Jackson's future and think, it's what Jackson will do to his wife, too. We all know it's a cycle.\n\nShe's taken everything dear from me. But I'm trapped. I can't leave because then that cunt would win. And we're underwater on our mortgage. Because of her. That filthy, good-for nothing homewrecker.\n\nParents Called \"Bad Jews\" for   \nRejecting Sleepaway Camp\n\nSuburgatory, USA\u2014An area Jewish family has been harassed online and in person by those in their community who are flabbergasted by the parents' decision not to send their seven-year-old son to sleepaway camp.\n\n\"I want my boy home this summer. These people will have to pry him from my cold, dead hands,\" said Lori Metzner.\n\n\"How do you like that,\" said Bari Weiss, whose daughter attends Hebrew school with Metzner's son, Josh.\n\n\"There she is, quoting Charlton Heston. People thought he was a Jew, too. Well, he wasn't. I'm starting to think Lori is less of a Jew than he was! Ben Hur would have sent his kid to sleepaway camp, you can be damn sure of that.\"\n\nAt first, friends and acquaintances of Lori and Jeremy Metzner were gentle with the couple, as they tried to process the idea of a Jewish child just aimlessly kicking a ball around at home all summer, completely bereft of other young Jews. Some asked them, delicately, \"Is there something wrong with Josh\u2014is he sick?\"\n\nBut once word got out that Josh was not sick, the gloves came off. It was decided among the Highland Street Jews that an intervention was needed. Two parents, Roni Sussman and Lisa Scher, banged hard on the door without warning one night and barreled in, giving the Metzners no chance to keep them out.\n\nRoni: Lori, we are really, really concerned about Josh.\n\nLori: Why?\n\nLisa: How is he going to learn about his Jewish identity if he doesn't go to sleepaway camp?\n\nJeremy: Considering I never see either of you at temple even on high holidays, I'm starting to think your Jewish identity is sleepaway camp.\n\nRoni: Think of our terrible past. Our people died in the Holocaust and would have wanted our kids to go to sleepaway camp.\n\nJeremy: They had sleepaway camp in Nazi Germany?\n\nLori: Wait, are you saying you think Holocaust victims would want me to put my child on a bus to be sent away to a camp out in the woods a hundred miles away?\n\nRoni: Lori, that's not funny.\n\nJeremy: Good one, Lor!\n\nLisa: You two are letting Hitler win!\n\nUndeterred, Lisa and Roni put up a Facebook page called \"Save Joshy's Summer\" in hopes of putting pressure on the Metzners. The page encouraged people to post their favorite camp stories and it attracted a few thousand camp-crazed Jewish adults from all over the world.\n\nIt remained generally positive, that is, until the Metzners decided to have some fun with it. First, Lori posted this. \"All I learned at Camp Shalom was how to give a blow job.\" Then Jeremy said they had changed their minds and decided to send Josh to camp, which got dozens of \"Likes\" within minutes. Then he posted which camp\u2014it was Sunnyvale\u2014a well-known high-end camp that caters exclusively to WASPs. A few minutes later, as those dozens of people \"Unliked\" the post, Jeremy added \"Psych!\" One response to the Metzner's ruse was this: \"Why don't you send him to the Gaza Strip Hamas training camp, because that's the only place that'd want you.\"\n\nDid these attacks upset the Metzners? Jeremy Metzner snorted, and said, \"No. We're tough Jews.\" And is Josh going to miss being around his Jewish brethren this summer? He said, \"I'm going to be with my best and favorite Jews in the whole world,\" pointing to his mom and dad.\n\nDr. Drama\n\n\"When life hands you a problem, let's make it more interesting!\"\n\nDear Dr. Drama:\n\nI know this is going to sound really awful but I recently dealt with a painful breakup with my husband, at the same time that my single mom friend says she found the love of her life. \"He's great with the kids, he's great in bed, he's got a great job, you name it, he's The One.\" Meanwhile I'm stuck in this suburb that has only about five single mothers, tops, and I'm suddenly the saddest loser around. I can't deal with the resentment, and I feel like a terrible person for even feeling this way. Any advice?\n\n\u2014Jealous in Suburgatory\n\nDear Jealous:\n\nWait, you feel bad? Your friend is the one who should feel bad, because if there's anything I've learned in my many decades on this horrible rock we call Earth, it's this: Happy couples need to shut the fuck up. Now if they are teenagers, I give them a pass. If they're so unattractive that this is their first, crazy-making burst of love, fine. But any average suburban person over the age of thirty? If they haven't figured out how much pain their joy causes 99 percent of the rest of the world, well, they are about two baby steps away from sociopath.\n\nDon't feel bad, Jealous. Just be patient. That happy will be gone by the time the snow flies and you and your friend will be back together saying \"shut the fuck up\" to the next clueless couple shmushing their eternal love in your face. Because we all know eternal love has a shelf life shorter than the box of your kids' Go-Gurts in the fridge.\n\nNew Atheist Bigger Asshole   \nthan Old Catholic\n\nSuburgatory, USA\u2014In a stunning development, our community's so-called New Atheist has out-assholed the Old Catholic.\n\n\"Yes, I definitely didn't think this was possible,\" said Brian Marooney, who judged this afternoon's clash of assholes at the Community Comes Alive! town event.\n\n\"Those Old Catholics are the worst. They try to explain away pedophile priests. They defend that psycho scumbag Mel Gibson. But still, those New Atheists, man, what else do they do other than think up new dorky ways to call five billion people around the world morons?\" said Marooney.\n\n\"New Atheists\" are a relatively recent addition to town and the broader community of non-believers. They strike a militant tone compared to \"old\" atheists, on what they say are the evils of organized religion and belief in general. But in doing so, many, though not all, have adopted a sarcastic attack-dog method, with slogans like \"WWJD = We Won. Jesus Died.\" and \"Too Stupid to Understand Science? Try Religion.\"\n\nAnd so the clash of assholes at Community Comes Alive! began with the New Atheist and the Old Catholic sitting side-by-side at adjacent booths, though not communicating.\n\n\"That boy needs a good haircut and a draft card,\" said Old Catholic Gerry O'Connor. \"Dad, they don't have draft cards anymore,\" said son Bob O'Connor, who insists he's not an Old Catholic, just keeping an eye on his dad, \"who wanders off sometimes.\" What does Bob O'Connor believe in? \"I attend the Church of Don't-Give-a-Shit,\" he said.\n\nGerry O'Connor festooned his booth with photographs of mangled fetuses and featured pamphlet material on why radical homosexuality, not criminal pedophilia, was the cause of the priest abuse scandal. He began a discussion about how \"a few poofs and queers and that Lady Gaga should go back into the closet and slam the door.\" O'Connor continued, \"Not seen and not heard. Those poofs and queers made our wonderful priests look like monsters.\"\n\nJudge Brian Marooney said, \"Wow. That was really bad. The New Atheist is going to have to bring it.\"\n\nEthan Barthold, who has tattooed a New Atheist \"A\" on his arm, was ready for the challenge. \"People who don't want their beliefs laughed at shouldn't have such funny beliefs. We feel sorry for the theists, but my patience with their stupidity has come to an end. If they were capable of rational thought\u2014and the jury's out on that one\u2014then maybe they would see the evil that their feeble-minded delusions cause.\"\n\nBarthold's booth had a banner touting \"National Idiot Outreach Day,\" and it featured various ironic and sarcastic attacks on people of faith such as Atheists\u2014Winning Since 33 AD; No Gods. No Mullets; JESUS SAVES . . . You from Thinking for Yourself.\n\n\"Whoa, this is a really tough call,\" said Judge Marooney. \"I mean, that Old Catholic really is hateful, but he is sort of sad, like, stuck in an Archie Bunker time warp. But that New Atheist, I mean, he's just a complete asshat. So I'm going to say, after careful thought, that the New Atheist has out-assholed the Old Catholic.\"\n\nMarooney did say to take his judgment with a grain of salt. \"I'm an atheist and I think people like Ethan are hurting the cause. If they spent half as much time doing charity outreach in the name of atheism as they did thinking up those ridiculous insults, and gotchas, then maybe we'd be getting somewhere. How do they think evangelicals took over half of Latin America? Charity. Part of me wants to be wrong about atheism just so I can see God smack that smirk off Barthold's face. Watching that dude be wrong for an eternity? I'm there. Oh my God, look!\"\n\nMarooney pointed at Barthold's sign, Athiests\u2014\u00adWinning Since 33 AD, but realized Barthold had misspelled \"Atheists\" as \"Athiests.\"\n\nMarooney laughed hard and said, \"A is for Awesome. And Asshole.\"\n\nPAID ADVERTISER CONTENT\n\nFaith for a Day Seat Fillers\n\nThere's nothing more disheartening on your sacred day of worship than walking into your church or temple and finding the room where you commune with your savior nearly empty. God's in the room, no doubt, but where is everyone else? If you're thinking Dunkin' Donuts, you're probably right. They better hope there's Dunkin' Donuts in Hell!\n\nThe empty seats, the unmistakable whiff of dying faith. It's a pox on the soul of the congregation, and a self-fulfilling prophecy: Empty seats beget empty seats until one day, the temple board or the diocese calls and Father Murphy or Rabbi Moshe is knocking hard on the door of God's welfare office. So what's a priest, reverend, or rabbi to do?\n\nIntroducing Faith for a Day Seat Fillers, where for a small per-person fee, young and vibrant church- and temple-goers will enthusiastically attend your services. They will be trained in the specific faith, but cannot directly be asked any questions beyond what is found in Wikipedia.\n\nSo take action. Take just a little out of that dwindling pass-the-basket pocket change from those Grandmas and Grandpas still turning out week after week, God bless 'em. Turn that mother out, and pack that place to the rafters, with Faith for a Day Seat Fillers. You know The Boss upstairs is watching and he's one tough bastard. Don't let God decide you're just not worth His effort.\n\nMom Discovers   \nThe Sociopath Next Door\n\nSuburgatory, USA\u2014After reading the book The Sociopath Next Door, an area mom identified the sociopath, who is next door.\n\n\"It's Griffin! It's him in a nutshell, it's uncanny! I stayed up all night Googling everything you'd ever want to know about sociopaths!\" said Mary Thibodeau.\n\nThe Sociopath Next Door by Martha Stout takes the idea of a sociopath\u2014usually thought of as a violent criminal\u2014and expands it to include the everyday deviant who might be \"next door\"\u2014your coworker, college roommate, or in this case, an actual next-door neighbor, Griffin Driscoll. The popular book spawned an army of armchair psychologists diagnosing those around them, and Thibodeau is just the latest.\n\n\"So here's more sociopath, psychopath, and narcissist stuff I found on Google: 'Pecking order is extremely important to the sociopath. His outward appearance may be the picture of success with all the trappings of status aggressively and elaborately displayed. But his inner life is empty,'\" read Mary from her iPad.\n\n\"Well, that would explain 'The Rev,'\" she said. That's what Mary and her husband Jim call Driscoll's habit of loudly revving his Porsche convertible each day in his driveway, a sound the Thibodeaus can only imagine is designed to attract attention to the expensive car.\n\n\"Yeah, well you say that annoys you, but you run out every time he does it just to look at that car,\" said Jim.\n\n\"Not true! It's because I can't believe how rude it is!\" Mary said.\n\n\"So this stuff was on a support group for wives of everyday psychos,\" Mary continued. \"A sociopath will show little or no empathy and may lie to cover up his lack of feeling\u2014like the time that little girl from down the street got hurt on his property, and he did nothing and pretended he didn't see her even though I saw him strutting around half-dressed like a peacock on the deck.\"\n\n\"Yep,\" said Jim.\n\n\"The sociopath, while perhaps not violent to people, may use animals to satisfy his thirst for causing pain. Oh my God. The squirrel. Do you really think?\" said Mary.\n\nMary Thibodeau was referring to a squirrel found on their property border that had been mauled to death in a way that, to Jim, looked highly unnatural.\n\n\"OK let me finish this paragraph from the support group: A sociopath is often highly sexually appealing to . . .\" Mary abruptly stopped and turned red. Jim looked at her, grabbed the iPad, and finished \". . . women!\" adding emphasis and drawing it out. \"The sociopath has a surface charm. And that can often be an aphrodisiac for women. Even to those who claim to find his behavior abhorrent.\"\n\nJim put the iPad down triumphantly. \"You're hot for the Sociopath Next Door. Nice! Maybe there are some cute serial killers in prison that you can start sending letters to! God, this is just like when you and every other thirty-something housewife was obsessed with that psychotic killer Tony Soprano. What is wrong with you women? I mean, yeah, we men like big boobs and young girls but some of you freaks are attracted to men who might hurt you?\"\n\n\"Leave my Tony out of it. He would never hurt an animal,\" Mary said.\n\nNew Black Resident   \nWorst Racist in Town\n\nSuburgatory, USA\u2014A new black father who has joined our overwhelmingly white community is being described as the \"worst racist in town.\"\n\nBradford Johnson described his family's move here very openly as a classic case of \"white flight\" from the section of the city where the population is 50 percent black or Hispanic and growing. This reporter thought using the term \"white flight\" was an unusual choice for a black person to use.\n\n\"Just because I'm black doesn't mean I give a rat's ass about other blacks. I want the same things white people do. To be far, far away from black people. To suggest that I love blacks just because I'm black, well, that's just racist.\"\n\nJohnson has been wholeheartedly welcomed into his new affluent Westgate neighborhood, with residents appreciating the chance to have the appearance of diversity, even though Johnson's background makes him far more similar culturally to his neighbors than to the black or Hispanic server at the local Dunkin' Donuts. \"The diversity I provide is that I am the first person of any color on our street to attend MIT.\"\n\nPeggy Marist was thrilled at the idea of what she called \"our own, personal Obamas!\" Lowering her voice, Marist said, \"They didn't seem black-black at all, and not even Michelle Obama\u2013black. They're Barack-black all the way. Some of the kids bused in from the city are really black-black and kind of wild, so I was really excited for my Madison to have her own sweet 'Malia' to be friends with on the bus! We already have a gay, a bunch of Chinese all in one house, and even one Mexican. So the Johnsons seemed like the perfect addition. But . . . well, Bradford has a little . . . anger. I think he's, like, mad at America or something. I mean, not mad at America but other black people.\" She had loudly whispered the words \"black people.\"\n\nThe white residents had a very clear idea of how their interaction would be with Johnson: They would have light, friendly contact in which his race would go unmentioned, at all costs. But Johnson didn't comply. People would frequently say, \"You must have really wanted your kids to enjoy the great schools here!\" He would be honest and reply, \"Yes, we wanted them away from those black people. Believe me, you don't want your kids around them. If it's more than 10 percent black, well, I'd never send my kids there. They just bring you down.\"\n\nJohnson found himself very frustrated, and alienated people with his blunt race-talk. \"I thought they would want to talk about black people as much as I do. Isn't that why I moved here? Thank goodness I found Old Bill.\"\n\n\"Old Bill\" Jesper, also known by his harsher critics as \"the Starbucks Klansman,\" now meets Bradford for coffee a few days a week to discuss some of their favorite areas of attack, like unwed mothers in the black community or the evils of the drug trade. \"They call me a racist? If I was a black guy and got stuck in an alley, it's Bradford I'd worry about, not poor old falling-apart Bill Jesper. I just like to rile people up to pass the time. Johnson wants a race war. Him versus the rest of 'hims.'\"\n\nInterestingly, Bradford has faced at least one moment of true racism, but blames the incident on the failings of black people. One time a new neighbor assumed that Bradford was actually NFL player Deshaun Watson, one of the few other prominent black residents in the area. The neighbor believed that the only way a black man could afford to live in the area was if he was a pro athlete. But rather than attack the neighbor for that assumption, Bradford blamed \"the blacks.\" \"That assumption is not racist. It's incontrovertible fact. There are not hardly enough rich black professionals and we have no one to blame but themselves.\"\n\nSo what does he make of the idea that he is viewed as the town's Obama? \"I don't get that man. Obama had the perfect life, completely free from blacks, and then he picks up and moves to Chicago to be with all those, those people? But notice he didn't decide to stay in blackie-town. He had a plan to get out. And notice also,\" he said triumphantly, \"did he send his girls to public school in D.C. to be with all those no-good blacks? Nope. Because he is a proud black man who wants the best for his black daughters, just like me.\"\n\nPAID ADVERTISER CONTENT\n\nMacho Pottery Camp\u2014  \nWhere Boys Become Men   \nWho Become Potters\n\nSo your son wants to take up pottery. You're not worried he'll become gay. You're a modern man; gay is completely fine with you. What's not fine is the idea that you might be fostering a sensitive straight man. No one likes them!\n\nAt Macho Pottery Camp, we offer the full array of potter skills; but you won't walk in and find a gray-haired lady with a brocade vest and comfort clogs as your son's teacher. Oh no. Your pottery instructor is Marine Corps all the way, and the majority of them were honorably discharged. OOH-RAH! They will whip that clay into shape, just like they will your son. He'll \"Semper Fire\" a misshapen cup for Father's Day that says, \"I love you Dad . . . for making me a man at Macho Pottery Camp.\"\n\nAlso consider for your son our brother facility, Macho Mime Camp. That Invisible Box won't know what hit it when your badass son crushes it in a single imaginary blow! Call us now!\n\nPoor by Choice Meets Just Plain Poor\n\nSuburgatory, USA\u2014A woman who is poor by choice gave some less than helpful advice to a woman who is just plain poor, as the two of them were checking out groceries at the Bay Street Stop-n-Shop.\n\nKristin Perry lives in town and shops at the store regularly, as does Callie Bennett, who works there as a cashier and lives right outside the town line in the Edgemont trailer park. When Perry saw that Bennett had a food stamp swipe card, like she has, she thought she had found a kindred spirit. \"Not too many poor people around here, we have to stick together!\"\n\nPerry attended Cornell University and then the Wharton Business School; her husband attended Dartmouth. Both grew up in an upper-\u00admiddle-class suburb, and after college they both stepped on the upwardly-mobile work treadmill along with most of their peers. That is, until their first child was born.\n\n\"We just decided we wanted more out of life, and being poor was the answer. All that stuff we were accumulating and the time spent earning money to buy it was just making us emptier. Not everyone can handle the impoverished life, but for us it was the challenge of a lifetime, and we were eager to conquer it. With a small stipend from Mike's dad's stock dividends, I was able to leave my job and Mike was able to fulfill his dream of perfecting his craft.\" Which is? \"His craft? I don't know, we keep our passions separate. You should ask him!\"\n\nWhy did they choose to live in such an expensive town? \"Of course, education is number one to us, so we were really lucky Mike's father supported our decision to be poor and bought us the house, which helped him on his taxes, too. Win-win!\"\n\nStore clerk Callie Bennett says she's just plain poor and comes from a long line of just plain poor people who came before her. With no chance to attend college, she feels lucky, though hardly pleased, to have her job at Stop-n-Shop. But, as it goes with the just plain poor, she is only just getting by, and has a new baby to boot. Bennett described her encounter with Perry.\n\n\"When I pulled out my EBT [food stamp] card, the scary hippie lady behind me sort of waved hers at me, with a small smile, like it was sign language for 'I'm poor, too, let's be friends!' Ugh I was so zonked from work, I just wanted to go the fuck home.\"\n\nPerry noticed Bennett's several boxes of diapers. \"Knowing that many other poor women don't know about the value of cloth-diapering to both the environment and their budgets, I thought maybe I would do some outreach.\" So she said, \"Have you considered cloth diapers for your baby? It's soft on their bottoms, good for the globe, and it's practically free, and free's good for poor women like us, right?\"\n\nBennett for a second just stared. Then she responded: \"It's only free if you don't put any value on my time, and my time is worth fifteen bucks an hour, and I need every dollar from every hour and every minute that I work. You think I want to add disgusting 'cloth' diapers to my insane laundry load?\"\n\nPerry seemed chastened and said, \"Well, it's really not that hard. You boil them, using tongs, about six times for about fifteen minutes to make them more absorbent and then you . . . \"\n\nBennett looked at her, slack-jawed. \"Why don't you worry about your own life and your own baby's bottom and your own boiled shit and I'll worry about mine? I have to think about my future and I don't plan to be poor forever.\" Bennett assumed, correctly, that Perry doesn't work. \"Maybe if you got a job, you could dream bigger, too.\"\n\nAs Bennett walked out, Perry said, \"See, she's still on that terrible treadmill we were on. She hasn't discovered the freedom that being poor can give you.\" Perry shook her head and said, \"I so hope it happens for her someday.\"\n\nPAID ADVERTISER CONTENT\n\nScrappers for Slackers\n\nMoms, have you ever felt the shame when another mother asks, \"Gee, where are all your scrapbooks?\" and you are forced to hide the unthinkable: You have failed to document each moment of your child's life. 'Fess up, sister: You're a slacker. You always were a slacker, muddling your way through college and the brief \"job\" you had, and motherhood didn't change a damn thing. But now you don't have to look like one, with Scrappers for Slackers. Scrappers for Slackers will hunt through every stray computer file, every tagged photo at your cousin's graduation party, every dusty side closet and desk drawer for some evidence of your child's life. Then with the magic of Photoshop, we'll construct a picture-perfect, expertly bound scrapbook that screams, \"This child was in no way, shape, or form a big drunken mistake.\" Give us a call. You may be a slacker, but you're no Casey Anthony. Let Scrappers for Slackers prove it.\n\nChild Precocious in Sarcasm\n\nSuburgatory, USA\u2014A local child has both impressed and alarmed his teachers with his precocious fluency in sarcasm.\n\nTex Holter attends the Mason Elementary School. His teacher, Jared Bauer, is bowled over by his abilities and describes the first time he became aware of Tex's sarcastic talents.\n\n\"He's six years old. And yet he overheard me saying Principal Massey accidentally sent out an embarrassing personal e-mail to the whole staff and, without missing a beat, said\u2014I shit you not\u2014'Awkward!' and walked away.\"\n\nAt first, Bauer and others thought it was a fluke, but it was not.\n\n\"One day a specialist came in to help one of the disabled kids in the class. This specialist is just a bumbling idiot. Of course, what do the kids know? Well, one of them knows; Tex Holter knows. He sees her and says, 'Oh great, that one.'\"\n\nBut Holter's strange fluency in sarcasm has a serious downside. It sets him well apart from his classmates, according to Bauer. \"These are six-year-olds. Their idea of a joke is to put the word 'poop' at the end of every sentence and then say, 'Get it? Poop. Get it? Poop. Get it? Poop.' Again and again. That's normal\u2014and annoying after a few minutes, by the way\u2014but the kids love it and laugh and laugh. So what does a kid like Tex do with that? They keep yelling 'poop!' in his face waiting for him to laugh and he just says, 'Talk to the hand.' So yeah, he's become a total outsider and we're worried about him. And I shouldn't have said 'a kid like Tex.' I've never met another one like him. I've seen more albinos than I've seen this.\"\n\nThe school took action to help protect Tex, allowing him to join the teacher's lunchroom. Now he has become fully conversant in school gossip and has become such a welcome source of humor among the teachers that they would be sad to see him go. \"We know, we know, he needs to negotiate how to get along with his peers, but when I told him once that he was my own personal Jay Leno, he just gave me this stare and said, 'I'm with Coco,' meaning I had insulted him by choosing Leno instead of Conan, who I guess he thinks is far superior. Oh God, I laughed so hard.\"\n\nThe school felt it was crucial to pull Holter's parents into the mix to try to understand what's going on with their son. So Ben and Teresa Holter came in for a talk with teacher Jared Bauer.\n\nJared: So I just want to say that I love having Tex in my class and he's a great kid, but he seems to have this unusually, um, developed way with humor, a kind of sarcasm that is very atypical for his age.\n\nTeresa [rolling her eyes]: Right, we wouldn't want him to excel in anything, just try to make him mediocre like the rest of them.\n\nBen: Zing!\n\nJared: Ummmm, OK . . . it's just that his strange, I mean, extraordinary ability is causing him social problems.\n\nBen: Oh, like Bill Gates had problems? Yeah, he turned out just awful.\n\nJared: Sorry? Not quite understanding either of you.\n\nBen: Are you new here?\n\nJared: Huh?\n\nBen: Great, you're going to force me to speak in your language with all its ugly directness. We as a family are fluent in sarcasm, it's our primary language, our culture, our mother tongue. Tex is just doing what he was born to do. So what do you suggest be done?\n\nJared: Wow. Weird, never heard of this one. Well, I think we need to stop accommodating for Tex's \"different ability\" and force him back with his peers, or he'll never learn to speak and interact well with kids his age.\n\nTeresa: Yeah, it's always been my dream that Tex think up the wittiest poop joke; yours too, Ben?\n\nBen: Harvard weights quality of poop jokes right alongside SAT scores.\n\nJared: Listen you two, Harvard's a long long way off. Right now I just want to make sure Tex makes some friends and finds a supportive group to thrive in.\n\nAnd with that, Teresa began a \"slow clap.\"\n\nAs the meeting ended, Jared Bauer said, \"Wow, that was brutal. What's cute and funny on a six-year-old is pure D-Bag on a forty-year-old. I'm going to help Tex with every fiber of my being to knock some of that smarm out of his system. 'Culture,' my ass. Poor kid!\"\n\nBauer has his work cut out for him. When he told Tex that he could no longer have lunch with the teachers in the faculty lounge, he added, \"Tex, we just think it's best for you to be with kids your age, to try to fit in and be with the regular boys.\" Tex stood there, in his ironic Nobody Puts Baby in a Corner T-shirt, and said, \"Epic FAIL.\"\n\nMom \"Never Yells\" at Kids,   \nUses Scorn Instead\n\nSuburgatory, USA\u2014A local mom prides herself on never yelling, preferring to use pointed scorn instead.\n\n\"Those people who yell, they should be hauled off to protective services. Don't they know what they're doing to their kids? I never even raise my voice when my\u2014\" Gina Burke's four-year-old daughter interrupted. \"Mama, can I have a cup of water? I'm, I'm thwirsty now.\"\n\nGina took a very big breath, as if winding up, clenched her jaw and slowly articulated each word. \"Right, Anna. I would just love. To get up. And get you something. Right after I sat down. Right after I asked you. If you needed something. It would be so so much fun for me. To get up. Again.\"\n\nShe got up in a huff and continued speaking about the lasting legacy yelling can have on children. \"I grew up in a yelling household and I flinch every time I hear a raised voice. I simply never do it, in any context, no matter my frustration level. In fact, I won't let my kids even go to a house where I know the parent yells. My kids would just shut down, crumble.\"\n\nIn fact, what Gina doesn't know is that she is a mom whispered about by other parents for her \"terrifying\" discipline style. \"I just don't have any reference point for it\u2014I find it astonishing! It's so carefully thought out, and designed for maximum damage, like she's tossing an emotional shit bomb in her kids' faces,\" said friend Madeleine Golden.\n\nSaid another mom, \"In every other way she is Miz Model-Mommy, I mean, she acts like if they were to eat a single non-organic blueberry, they'd keel over and die. But then she talks to them like that? I'd rather send my kid for a playdate with the creepy single guy next door to her who looks like Dog the Bounty Hunter, that's how bad I think she is.\"\n\nBack at the Burke's home, Kenneth had just asked his mother if she could buy him a new Lego Star Wars kit.\n\nGina breathed deep again, and clenched. She picked up Anakin and Obi Wan and quietly but intensely started to playact with them. \"'What do you think, Anakin?' 'Well, Obi Wan, I think Kenneth is pretty selfish when he knows we don't have the money to buy another Lego set.' 'Yeah, Anakin, and does Kenneth ever pick up the ones he already has?' 'No, Obi Wan, he sure is a spoiled rotten Sith Lord.'\"\n\nKenneth just stared at his mother as Gina got up to make lunch. While Anna fussed, Gina said, \"This must be why I spent eight years getting a PhD, so that I could cut happy faces out of sandwiches. Dreams do come true!\"\n\nAt this point, Anna is too young to understand what her mother is getting at, and Kenneth just thinks \"Mama is mad.\" But Gina's oldest, fourteen-year-old Kendra, fully understands what's behind her mother's \"technique.\" She stated her take on this: \"My mother. Is mad. That we. Ruined. Her professional. Ambitions. We didn't. She did that. Herself. And she. Can bite. My. Ass.\"\n\nSHOUT OUT\n\nJoin Our Weirdo Junior League!\n\nJenny Jorgenson is a mom and self-described \"freeganista\" who lives on Blanco Street.\n\nWe, as \"freeganistas,\" take to the Shout Out today not just to scorn your throw-away culture, with your constant visits to big box stores and relentless focus on regular bathing. We want to win you over, too.\n\n(Just to clarify. We are not \"frugalistas,\" a term that has been overused to the point of becoming a pathetic Great Recession clich\u00e9. Frugalistas, put simply, are pussies. Only when you scavenge for completely free items can you truly disconnect from America's nauseating consumer culture. We are freeganistas. Not trite at all, right?)\n\nThough I do enjoy scorning this culture of waste, I wanted to show the human side of dumpster diving and scouring this great town for free items. My freeganista adventures have brought me far more than untold savings and one admittedly nasty case of intestinal worms. Freegan living has brought me firmly into the fold of what I call our town's Weirdo Junior League. Trust me, they're the only people worth knowing within a hundred miles.\n\nThe Weirdo Junior League is centered at the town dump, which is in essence a free garage sale. It's stuff that the \u00adcorporate-controlled shopper-slaves drop off\u2014perfectly good items\u2014so they can resume their cycle of shop and dump and shop and dump in hopes of forgetting the futility of their lives and the inevitability of death. Meanwhile, my Weirdo Junior League members are eating their lunch, sometimes literally!\n\nMy Disabled Home-Boy\n\nAn emotionally disabled man who regularly visits the dump with his aide taught me that the band Journey is a great unifier; I was afraid he was going to wrestle that Frontiers cassette out of my hand (being a freeganista means, of course, that your decade-old car still has a cassette player).\n\nThe Vagitarian\n\nHer bumper sticker says, I'm a Vagitarian. Even better would have been Pussy. It's What's for Dinner. I saw her once at Dunkin' Donuts, so she is apparently a Donutarian, too, a proclivity to which I could relate. Did you know that Dunkin' Donuts server Mariela will give you all their leftovers at 6:00 p.m.? And the donuts freeze beautifully!\n\nNow if only we could get the gay guy in town who has the bumper sticker, Rock Out with Your Cock Out, and our little Weirdo Junior League would be homosexually complete. I should be honest, I have never had the balls to talk to the Vagitarian, because she looks a little mad all the time. Sometimes I'm a real bottom that way. Maybe she needs more veg. I mean, Vag!\n\nHaunted House Guy\n\nEvery town's got at least one, right? This guy's \"home\" is packed floor to ceiling with, well, what in the Sam Hill is in there? Oh yeah, crap from the dump. All I know is it's busting out the windows, and no wonder both of his neighbors have their houses on the market. He's one of those crazy people who is sweet and exasperating in equal parts\u2014also grimy, which I love, and buoyant. And he has a real touch with kids, finding magic everywhere in the unexpected. If he wasn't so clearly deranged, I'd have him babysit.\n\nBeryl the Yenta\n\nAs a daily attendee at the dump, Beryl is like the Elder Stateswoman of town secrets. One man's trash is another man's treasure? Actually one man's trash is another lady's gossip. What do you think happens when an old wedding album tragically appears at the dump? Trust me, word spreads fast\u2014like, Twitter-fast\u2014and she doesn't own a single texting device! If you think you're hiding anything when you dump your crap, just ask Beryl. She could give you a profile of your life like what you might see on the show Criminal Minds. Seriously, it will make your hair stand on end. Shred before you dump. Beryl's on to you.\n\nFreecyclers\n\nThis is a website where you post items you can offer to others to come pick up, bits and bobs you'd end up chucking in a landfill. Through freecycling, I've met a saintly foster mom and a few delightful junkster shut-ins. Here are some of my favorite offerings from freecyclers.\n\nOffer: Ovaltine. We have promised this twice and it is still here. Please, for the LOVE come get the Ovaltine. It is starting to develop a complex. It's a really nice 12 oz. jar. I hope someone out there can give it a good home.\n\n\u2014Miserable (Ovaltine) in Marlborough\n\nOffer: Gynecologist examining chair from maybe the 1940s. Your grandma might have been examined in this!\n\nOffer: Extra progesterone vaginal suppositories for hormone replacement.\n\nDo you have any idea how much those suppositories cost retail? I hope someone snagged them.\n\nAnd sometimes being Super-Crazy-Mega-Cheap brings friends closer together.\n\nOne day recently, a friend, PTO goddess Laura Beazley, who is not in the Weirdo Junior League (not yet, anyway) looked over at me, and, knowing I had just gone on a thrifting adventure that week, started laughing uncontrollably. \"What?\" I asked. She pointed at my outfit and said, \"That's my shirt! The shirt I left at the dump! You found my shirt!\"\n\nIt's a freeganista miracle!\n\nFive-Year-Old Loves, But No Longer   \n\"In Love with,\" Mommy\n\nSuburgatory, USA\u2014A five-year-old boy \"loves\" but is no longer \"in love\" with his mommy, and thinks she has grown \"needy and possessive.\"\n\nEvan Morton was in a reflective mood about his situation while sitting in his Batman Underoos at the kitchen island, nursing the last of his Horizon Organic Chocolate Milk Box. \"See?\" he said, pointing at the label certifying the milk as antibiotic-, pesticide-, and hormone-free. \"See, what good care she takes of me? God, this is hard. So, so hard.\"\n\nHe gestured in a defeated way to his dad, who was at the refrigerator. \"Dad, can I get another one of these?\"\n\n\"Comin' right up, Ev,\" his dad replied.\n\nAccording to Morton, he and Mommy have been together for five years. \"Let's be honest. Early on I was in it just for the boobs. That first year, it was all boobs, all the time, all I wanted and needed. I didn't really look at her as a person. I know I sound awful for saying that, but it's true. She was more like some . . . thing . . . attached to those wonderful boobs.\" He sucked down the last of the second chocolate milk.\n\n\"Listen, Dad.\" He belched. \"Good one, right?\"\n\nHis dad said, \"Good one, buddy!\"\n\nMorton went on. \"But eventually she was more than boobs. In years two, three, and four, the relationship deepened. She gave me solid food, and we really connected as human beings. We were really communicating. Seeing her face lit me up like nothing else. I'll never forget our first visit to Bugaboo Creek together\u2014my choice, of course\u2014to see the robot moose. We laughed together so hard. Sounds silly now. Sad too,\" he said.\n\n\"You OK, Ev?\" his dad asked.\n\n\"Yeah, I'm alright. Anyway, yes, I was in love with her. Me and her and no one else. Well, at that point there was also Bob the Builder and Thomas the Tank Engine and Diego, and, I'm embarrassed to admit, The Wiggles, but at that time if I had to make the choice between Bob and Tom and Diego and The Wiggles and her, I would have chosen her.\"\n\nMorton moved out to the living room and flipped on the TV. Power Rangers: Samurai was on. \"Yes!\" He did the same frenetic dance he always does when the opening sequence of Power Rangers comes on, which involves cartwheels, handstands, fist thrusts, and running around in a circle.\n\nAs he sacked out to watch Power Rangers, Morton started to describe how things have been going downhill for the two of them. \"She's so needy and possessive. You know, after school I want to play the Pip Penguin Club with my boys. We're space penguins who kill zombies who are trying eat our space brains. It's really important to me. It's my thing, it's what I do. She demeans it. And she is always embarrassing me in front of my buds and dragging me home. Then she's pissed and doesn't get it when she asks, 'Why don't your friends like meeeeee?'\"\n\nAlso, Morton believes her nagging is taking a real toll. \"She's always asking me, 'Are you going to wear that Clone Wars T-shirt again?' 'Yes I am, Mommy, until freedom is restored to the galaxy, and General Grievous and Count Dooku are taken down, YES I AM. What does it matter to you?' Or with the food, always the food. 'Eat your gummy Vites. Just try these eggie-eggs, just once.' I mean, do I look like the kind of guy who wants to eat tofu? I ask her, 'How does this even affect you?' Then she starts crying and telling me it's because she loves me more than life itself, and it's just awful, and, hey, I'm not made of stone.\"\n\nAt that point, Morton heard his Mommy come in with groceries. She walked in and said, \"Hey, handsome!\" Morton looked at her and said, \"hey.\" She came over to give him at least ten kisses and to ruffle his hair, while Morton squirmed away. \"God, that was awkward, wasn't it? Sorry you had to see that.\" Morton said, cringing.\n\n\"What do you want to do tonight?\" she asked.\n\n\"Cats and Dogs: The Revenge of Kitty Galore is on,\" Morton replied.\n\n\"OK, well, we are going to Bertucci's, so maybe it will be on when we get back. You've been in those Underoos all day. Go change!\" Mommy said.\n\nMorton groaned and stomped upstairs. \"Yeah, I'm annoyed. But I still know, when all is said and done, that she's a wonderful woman. Who I still love. Who probably deserves better,\" he said, pulling out his Clone Wars T-shirt from the hamper. \"NOT the Clone Wars T-Shirt again, Evan!\" she yelled from downstairs.\n\nDad: Guppies Represent   \n\"Everything that's Wrong with America\"\n\nSuburgatory, USA\u2014A dad is telling his daughter that the guppies in their home represent \"everything that's wrong with America.\"\n\nGreg Mazur, forty-nine, recently lost his job as sales manager at the Piermont Insurance Company and now is spending more time at home with his ten-year-old daughter, Ava.\n\n\"Time to feed the freeloaders!\" Mazur said, grabbing the fish food to shake into the aquarium.\n\n\"God, it's like Sodom and Gomorrah in there. Ughhh . . . disgusting,\" said Mazur. Ava has been noticing that her dad now gets agitated every time he has to feed the guppies, a community of dozens that grew from a single guppy brought home from school last year. \"Yeah, that first guppy slut must have been knocked up when Ava brought her home,\" said Mazur quietly.\n\nAva: Daddy, why are you so mad at the guppies?\n\nMazur: Well, sweetie, because I look in there and see everything that is wrong with America. You know, when Mommy and Daddy decided to have you and your brother, we planned it out and made sure we could swing it, money-wise. But look at these guppies, do you think they plan anything? They just have guppy after guppy after goddamn guppy, I mean do they think they have any chance of paying for, I don't know, college?\n\nAva: Daddy, guppies don't have college.\n\nMazur: But if they did, all these little babies, they'd be out of luck wouldn't they?\n\nAva: What does it matter?\n\nMazur: What does it matter? What matters is that they are relying on us to feed them, money out of our pockets, stuck paying for their bad life choices. [muttering] Welfare queens. . . .\n\nAva: What's a welfare queen?\n\nMazur [muttering]: They're guppies who can't keep their legs together. See there's no respect for life in there. They swim around in their own poop and pee. Diseases all over the place\u2014white spot disease, gold dust disease, fish lice, dropsy. Those are lifestyle diseases, Ava. You choose to get them because you don't take care of yourself.\n\nAva: Daddy, if it's dirty in there, that's our fault.\n\nMazur: Right, it's always our fault. I repeat\u2014no respect for life. These people eat their own. Once they shoot them out they don't even bother with taking care of them.\n\nAva: But Daddy, that's what Miss Dalton said they're supposed to do; this is nature.\n\nMazur: That's fine for Miss Dalton, but we don't have to like it, or celebrate it. [muttering] Typical liberal bullshit they feed my kids. That's why they hand out these guppies. Start trainin' 'em early to hand over their hard-earned cash to a bunch of lazy thugs.\n\nAva [defiantly]: I love my guppy family.\n\nMazur [muttering]: Family. Like the Manson Family maybe or some filthy commune. Seriously, Ava, does that look like any family you've ever seen? How many are in there? Do they know who their fathers are? Who are the moms?\n\nAva: The moms are the fat ones.\n\nMazur [triumphantly]: Bingo.\n\nAt that point, Mazur's wife Emily came home, walked in, and kissed Ava and said, \"Oh no, has Daddy been yelling at the guppies again? Greg, ease up on the poor guppies! They didn't lay you off from your job, you know. Did you put in for unemployment today? Or just yell at the guppies again?\"\n\n\"Ummmm,\" Mazur said, looking dejected.\n\n\"Greg. Honey,\" she said, hands in the air.\n\n\"OK, right. I'll do it. We'll be OK,\" he said, shaking more fish food into the aquarium.\n\nDr. Drama\n\n\"When life hands you a problem, let's make it more interesting!\"\n\nDear Dr. Drama:\n\nI'm afraid my husband might be gay. He doesn't seem to have much interest in me, you know, that way, and he just seems a lot more, um, fixated on the dads when we go to school events or soccer games. Also, and I know I shouldn't have done this, but I looked at his search history on the computer and found gay porn! And then I found a strange number on his cell phone that came up a lot, I called it, and it was a man. Do you think he might be gay? He's my best friend, I don't want to lose him!\n\n\u2014Paranoid in Suburgatory\n\nDear Paranoid:\n\nYour subconscious is screaming at you, and your conscious is covering its ears and yelling \"La la la. I can't see the big fag sleeping right next to me!\", so I'll say it loud and clear for you: Your husband is gay. You can pretend all you want that just being a little curious about gay porn doesn't mean anything, but take it from another sucker like me: Where there's gay porn and a mystery man, there's a late night circle jerk or early morning gym tug fest not far behind. Then he'll settle down, find that special guy, and have a beautiful gay wedding you won't be invited to.\n\nNow if this was Oprah hell-bent on offering a happy ending in that final ten minutes of the show, she'd be telling you to \"get some therapy, figure out what's really going on, maybe sex isn't the most important thing in the world if this is your best friend.\" But Dr. Drama is Old Testament all the way. Retribution, not redemption, that's my bible. He stole your most potent sexual years! So here's what you do. Tell your \"best friend\" that he can still be your best gay friend, but pack his bags right now and tell him to get the hell out. Don't worry about the kids, it might be hard at first, but gay dads make the greatest dads, once they're getting it up the ass, which is what they've been dreaming about the whole time. So eventually, they'll be fine. And while he's packing, you're going to put on your best tramp outfit, and you're going out, driving into the big city, getting hammered, and getting fucked by someone who loves vagina. Your vagina, and all vaginas. This is your moment. It was stolen from you. Steal it back.\n\n\"America the So-So\" Campaign   \nMars Fourth of July Celebration\n\nSuburgatory, USA\u2014A group promoting the slogan \"America the So-So\" caused a ruckus at the annual Fourth of July celebration on the town green, which attracts a more diverse crowd from several different towns.\n\nDave Sheehan runs the bipartisan advocacy group, American Realists for a Real America. \"We get accused a lot of being unpatriotic, which just . . . ugh . . . makes me crazy. And I'm a Republican! So we thought putting 'America' in twice might help.\"\n\nSheehan's group is committed to puncturing some of the illusions Americans might have about just how \"great\" America really is, and he feels he was too mild for the Tea Party's hard-edge. \"I won't yell at people or name-call, but I am determined to tell it like it is. A true patriot looks himself in the eye and says, 'You can't change what you don't acknowledge!'\"\n\nSheehan is referring to Life Law Number 4 as expounded by the inspiration behind American Realists for a Real America\u2014Dr. Phil. Sheehan, who's been out of full-time work for eight months, took Dr. Phil's \"get real\" message to heart, and began to see that the true enemy of America was self-\u00addelusion. That's why Sheehan chose the Fourth of July to roll out the group's slogan: \"America the So-So.\"\n\n\"I wanted something a lot stronger, but I figured I'd pull more people in, then boom! Rock 'em, sock 'em with my pamphlets,\" he said.\n\nSheehan explains the trouble with America. \"Math skills, life expectancy, roads and bridges, our debt rating, bungled wars, obesity, you name it, when you consider how rich we are, we're in a death spiral. America the Great? It's just not true. And yet the thing we come in Number 1 on over and over again? Self-regard.\"\n\nWas there anything he could think of that America does well?\n\nSheehan watched a man bite into a giant sausage-and-peppers hoagie, while his son pressed a sugar-coated fried dough to his face.\n\n\"Eat well? And look how great that's going!\"\n\nSheehan had set up his booth with his \"America the So-So\" sign, handing out pamphlets he had prepared, a veritable library of doom. He had \"Nation of the Living Dead\u2014America's Demographic Timebomb,\" \"Rotting Stump: The Sugaring of America's Life-Blood,\" and \"War and the Military Meat-Grinder,\" among others.\n\nThe same man who had just finished his hoagie looked at the sign and the pamphlets, and said, \"What the fuck is this shit? You know it's the Fourth of July, right? Are you a fucking Communist? You know, I am a veteran of the Iraq war and I have diabetes.\"\n\nSheehan said, \"Sir, 'America the So-So' is my own patriotic way of saying America needs to . . .\" He looked at the man's stomach. \". . . shape up. That's 'getting real.' That's loving America.\"\n\nThe man was fuming. \"Asswipe. It's America the Beautiful. Put your hand over your heart or go the fuck home. Or better yet, get a one way ticket to . . . Kenya.\"\n\nThe fireworks began. Sheehan looked up at the patriotic display and said, \"The cost of every one of these colorful little explosions could have fed a hungry orphan in Kenya for months. But, well, I still love you, America, you batty old broad! Happy Birthday!\"\n\nAs he packed up his booth, he said, \"Well, I guess that only went so-so, right?\" He laughed ruefully at his own attempt at a joke. \"Still if I can open only one person's mind, it's worth it.\" But was he offering any solutions to these problems, beyond getting real?\n\n\"Actually I haven't had a chance to get beyond Dr. Phil's Life Law Number 4, but we'll have more time now that our big debut is over.\"\n\nIn the spirit of puncturing self-delusions, this reporter was a bit suspicious and curious as to who the \"we\" was in American Realists for a Real America, since Sheehan was very much alone all day. It turns out that his only outlet, on Facebook, has just three Facebook fans: Sheehan, his wife, and one man with no picture named Gene Juluca. When presented with this news, Sheehan, rather than being embarrassed, said, \"You just might have what it takes to be an American Realist for a Real America!\"\n\nAnd who is Gene Juluca? \"Oh, that's the Facebook page for my kid's stuffed monkey.\"\n\nPAID ADVERTISER CONTENT\n\nThe Following Is a Paid Political Announcement\n\nVote Billie Carson for Mayor\n\nAs a longstanding exercise bulimic, I know your community better than most. Whether it's the dangerous rocks that need rearranging on the Brook Path or dismantling that deadly Rotary on Atwood Road, I don't need to get up to speed on the issues facing our town. Oh, I'm up to speed\u2014on high speed, a speed like you wouldn't believe possible by a menopausal woman.\n\nI am also one of the best-known, and surely one of the best-loved faces in this great little patch of America. In fact, one time, I even heard a boy shout out of his car, \"Look, Mommy, there's the flying skeleton with the big head!\" Well, son, that comment meant the world to this flying skeleton. My name is Billie Carson, and I'm asking for your mom and dad's vote for Mayor November 3rd. Come to the police station this Saturday, where we'll have a wonderful lunch of bread-free lettuce and mustard sandwiches and pickles. I'll even take a break to stationary jog, all to hear my constituents' most pressing concerns!\n\nSo Vote Billie Carson. I simply won't stop pounding the pavement on your behalf.\n\nDad and Hot Nanny   \nReally Just Good Friends\n\nSuburgatory, USA\u2014A local dad and a hot nanny are \"really just good friends.\"\n\n\"Hi, Mr. H!\" said vivacious and buxom Mandy Mistrall, eighteen, a nanny wearing daisy dukes and high-heeled sneakers and licking a large lollipop.\n\n\"Mandy! I know you're planning to wash the car, but it's so hot out; why don't we get you out of that shirt?\" said Rock Hardt, a father of two who hired Mistrall days after she turned eighteen.\n\n\"Mandy's had a bit of a rocky road in her path to becoming a nanny. Her father walked out on her and now she has what I think they call 'daddy issues.' Good thing I found her. Now she has someone strong and nurturing attending to all her needs.\"\n\nWas she experienced? \"No, she was a completely fresh, unspoiled virgin to the job at hand. We decided to overlook some trouble she had fallen into at the Reform School for Wayward Girls. Let's just say our Mandy is innocent, but a bit of a vixen. We know now after much more experience with the issue that Mandy was just getting in touch with her emerging bisexuality. The tickle fights in the girls' shower area at the school got a little out of control. That's how she ended up on the side of the road that fateful night.\"\n\nMistrall, sudsing up the car with long methodical strokes while sprawled out on the hood of the car, describes meeting Mr. H. \"It's a really funny story. It was a stormy night and I was stranded on the side of the road. I was soaking wet. Good thing Mr. H had an extra shirt with him. It was really big and that worked out well, of course, because he didn't have extra pants. We were stuck in the car for many hours and really had some special intimate time getting to know each other better.\"\n\nNow Mistrall is part of the family and, as Rock Hardt put it, \"up for anything,\" which is really important in the freewheeling Hardt household.\n\n\"With my wife now confined to a wheelchair, Mandy is so nice to oil up my sore muscles when I need it, which is to say, often,\" said Hardt.\n\nMistrall finished the car and came in to change. She emerged an hour later in thigh-high boots and a micro-mini. \"Don't you look just good enough to eat, Mandy!\" said Hardt.\n\nTwo other similarly attired and similarly vivacious and buxom girls arrived. \"Enjoy your three-way!\" Hardt said.\n\nThree-way? \"Three-way date. What did you think I meant?\"\n\nThis reporter wondered if having such an attractive nanny, along with an infirm wife, presented Hardt with perhaps too much temptation.\n\nRock Hardt was aghast. \"First of all this is a barely legal girl you are talking about. And second, that is such a silly clich\u00e9 from, well, I think you must be watching pornography! It appears that someone here, not me, has a very dirty mind. What kind of journalism school did you go to, anyway?\"\n\nHeartwarming Herpes Tale   \nBrings a Family Together\n\nSuburgatory, USA\u2014In a heartwarming tale of first-lust, untreatable sores, and eventual redemption, a four-year-old has discovered how the two people he calls \"Mommy\" and \"Daddy\" became a family. And in the telling, they all learned what really matters in life.\n\nIt began when Devon Corrie spotted the so-called \"tramp stamp\" tattoo on \"Mommy\"\u2014Eve Corrie\u2014which became visible when Corrie was fumbling with the attachments on the vacuum.\n\nAs she bent over, Devon saw a strange picture on Corrie's lower back. \"Mommy, what is that? There's a picture of a naked lady on your back! She's in a garden and there's, there's, there's . . . a snake! Mommy, it has the letters E-V-E.\" It was a tattoo of a sexed-up Eve with metalhead hair in the Garden of Eden.\n\n\"Yeah, Mommy's pretty hot, isn't she?\" said Rich Corrie.\n\n\"Rich, stop! You want him running around saying \"Mommy's hot, Mommy's hot?\"\n\n\"So Mommy, what does E-V-E spell?\" asked Devon. Eve Corrie gave her son that look that says, \"Isn't my child the most adorable moron?\"\n\nCorrie said, \"Eve's my name, sweetie! You have a name, Devon. And Mommy's name is Eve.\"\n\nDevon looked at her, utterly confused. \"But you are Mommy. Mommy Corrie.\"\n\n\"Well of course I'm Mommy, but before you were born, Mommy wasn't a mommy yet. Back then I was just Eve. I had a life before you were born.\"\n\n\"Oh did she ever!\" Rich Corrie couldn't resist interrupting.\n\n\"Rich. Your son is confused and I'm trying to explain. Mommy existed. Mommy was a person. Mommy had, well, I had a lot of, what should we call it, um, bad fun before you came into my life and made me Mommy. But you can forget about Eve, that's not me anymore. I'll never set foot on that boardwalk again,\" said Eve, actually starting to tear up.\n\n\"Why are you crying, Mommy? Don't cry!\" Devon ran up to her and started to console her.\n\n\"Eve, it's OK, honey! Let me give it a try,\" Rich said, turning to Devon. \"Mommy is sad because when Mommy was Eve she, um, had some bad fun when she went out for a playdate one night on the boardwalk. She made a friend who liked bad fun, too, and who left Mommy with a nasty bug she can never, ever get rid of.\"\n\nEve Corrie, muttering, said, \"Oh my God you just told our child that his mom is a dirty herpes whore.\"\n\n\"But you know what Daddy thinks?\" said Rich Corrie. \"Daddy thought Eve was the most fun, most funny, most beautiful, and most wonderful girl he'd ever met. And Mommy is exactly the same, except now we have you, and it's more wonderful than ever, except for the times now and then when the itchy painful sores come out.\" Eve began tearing up again.\n\nDevon still looked confused, and said, \"How did Mommy get those nasty itchy sores?\"\n\nRich and Eve looked at each other. \"Well this is probably not going to make much sense but the friend she met who loved the bad fun? Well, that was your Daddy. I was 'Rich' back then and I had just met Mommy that night on the boardwalk and I gave the nasty bug to her. I have it, too.\"\n\n\"You two didn't wash your hands, did you!\" Devon said, thrilled to have caught his parents having bad fun.\n\n\"You're right, we didn't protect ourselves from that nasty bug. But at the same time Daddy gave me the nasty bug, he also gave me you. He became Daddy that night of bad fun,\" Eve said.\n\n\"So if that was a mistake, Devon, thank God, because it was the best mistake I ever made. What really matters in life is love. And family,\" she said.\n\n\"And Valtrex,\" Rich said. \"Herpes may be forever, but family is, too,\" as they gathered in a warm three-way embrace.\n\nBoy Loves Steve Jobs   \nMore than Parents\n\nSuburgatory, USA\u2014A twelve-year-old boy loves Steve Jobs \"way more\" than his parents, a development that's been years in the making and showing no signs of ebbing away.\n\n\"That smug, know-it-all motherfucker!\" said Phil Macon of Apple founder Steve Jobs. \"That guy has singlehandedly ruined my relationship with my son!\"\n\nJust as Phil finished his rant, \"Steve Jobs,\" as his son Benno will now only answer to, walked into the room, holding his Steve Jobs plush toy.\n\n\"Hello, Father,\" he said coolly, with a somewhat dreamy countenance.\n\n\"Hello, Steve,\" Phil said, choking out the name with hate in his voice.\n\nMacon dates his son's fandom back at least six years, when he realized that his son was calling himself \"Steve Jobs,\" choosing to wear a black turtleneck and jeans and even sleeping in the outfit.\n\n\"All I did was want a stupid iPod. And yes, I took Benno to the Apple Store with me. It was like he walked in and thought 'This is my real and true family.' Like a very very clean religious cult where there are no moms and no dads, just a gang of self-satisfied little fucks. That Apple Store is now like, like his sacred temple and his God is a charisma-bot\u2014named Steve Jobs.\"\n\nPhil Macon said that the cost of acquiring the latest Apple gear is bankrupting the family, but it's what has happened to Benno's personality that's the hardest pill to swallow. \"He's become . . . I can't believe I'm saying this about my own kid, but he's become a dick. He'd throw his own parents overboard in a heartbeat if it meant saving Steve Jobs from whatever disease it is he has.\"\n\nPhil Macon says his own appearance doesn't help matters either. \"Yeah, look at me. Think about it. Who do I look like? A little schlubby, dirty-blond hair, wearing my 'lame' glasses and 'lame' pleated khakis? Yep\u2014dead ringer for the PC Guy from those unbelievably obnoxious Mac-PC Guy ads. Benno even printed out a picture for me that said, 'How to Dress Like a PC Guy,' the dude was all decked out in clothes from Sears. Benno handed it to me, without a word, like he was quietly slipping me a giant shit sandwich. It's not even that he thinks I'm the enemy, his 'PC guy' dad. He thinks I'm pathetic.\"\n\nThis reporter spoke with \"Little Steve Jobs,\" who was a bit worn out after watching the World Wide Developers' Conference the night before\u2014the must-see event for Jobs' fanatics where they not only worship the new products and features discussed, but also obsessively scrutinize Jobs's appearance for signs of health and vitality.\n\nAs is his custom, the day after WWDC, Benno was ritually rewatching previous WWDC presentations. What does he love so much about Steve Jobs? He laughed smoothly. \"You mean what do I love about myself? That's immodest . . . don't you think? Really my role in the world is quite simple, spare, and elegant. I want to put a 'ding in the universe' and I think I'm achieving that. Don't you?\" he said, gesturing to the array of devices on his desk.\n\nDoes it bother him that his parents feel shoved aside by his Steve Jobs fandom? \"I really do care for those people, but I simply won't engage in petty jealousies when so many exciting discoveries are yet to be made. Really, if you look at, say, Father, he should perhaps start thinking about his own health and appearance. As you can see, he is looking a bit . . . dated.\"\n\nWhen told of Benno's statement, Phil Macon exploded. \"You see?\"\n\n\"Little Steve Jobs\" admitted that there have been longtime strains with his \"parents,\" using air quotes while saying \"parents.\"\n\n\"Of course, I do appreciate them taking me in when I needed a family, but I suspect my birth parents might have been a bit more . . . visionary. More Silicon Valley campus than suburban office park.\"\n\nConfused, this reporter consulted with the real Steve Jobs's Wikipedia page and saw that he was born in 1955 and given up for adoption.\n\nGo the F*ck to Sleep?   \nMeet Get a F*cking Life\n\nSuburgatory, USA\u2014Annoyed by the barely sublimated parental rage found in the smash bestseller, Go the F*ck to Sleep by Adam Mansbach, a child sensation has penned an acidic rejoinder called, Get a F*cking Life.\n\n\"We as kids just thought we needed our own potty-mouth satirical send-up that expresses our frustrations\u2014and believe me, we have many,\" said nine-year-old phenom author Patrick Bryson. \"As Go the F*ck to Sleep so ably demonstrates, the child-parent relationship is fraught with complexities. We hope Get a F*cking Life honors those complexities while also giving everyone a five- or even seven-minute chuckle.\"\n\nBryson is considered an up-and-comer, named by the New Yorker as one of its \"10 under 10\" young writers to watch. \"We expect Get a F*cking Life to be the gift for Father's Day, Mother's Day, any time you want to give Mom and Dad a little zing,\" said Bryson.\n\nThis reporter was fortunate enough to get a sneak preview of Get a F*cking Life, which is already zooming up the bestseller list months before publication. And Bryson allowed us to excerpt it here.\n\nWhen your day is bleak . . . and you need some peace . . . and you find it in a box of wine . . . I come to you, Mommy, lift you from the floor, and say: Get a F*cking Life.\n\nWhen I sit in your lap . . . at the day's end . . . see the porno on the iPad screen . . . I cup your face, Daddy, with my little hands and say: Get a F*cking Life.\n\nOur weekends unspool . . . like a cat pulling yarn . . . you telling Dad, \"You ruined my life.\" And Dad saying, \"You ruined mine, too, you f*cking shrew!\" I curl at your feet and say: Get a F*cking Life.\n\nCome Sunday night . . . I don't have the blues . . . I count the minutes till I can return to school . . . where I feel safe with Miss Kenney . . . far away from you. And I say, one last time, to Mommy and Dad: Get a F*cking Life.\n\nWhere did Bryson come up with his material? \"As with any writer, I have mined my personal experience. Oh, and the family court documents after my parents' divorce, too. That thing was huge! Turns out, I had blocked out a lot. Authenticity is key.\"\n\nJust as Go the F*ck to Sleep enlisted a celebrity, Samuel L. Jackson to voice some of the story, Bryson brought on Gilbert Gottfried, former voice of Aflac, who was let go from the insurance company after making inappropriate jokes after the Japanese earthquake. \"We kids love him because he's the voice of Digit on Cyberchase. And his voice will drive my parents, I mean, all parents, out of their f*cking minds. Get it? I said 'fucking'! Adding \"fuck\" to anything is hilarious!\"\n\nPAID ADVERTISER CONTENT\n\nBRIARCLIFF ACADEMY\u2014  \nEducating the Stupid Rich Since 1903\n\nA message from Briarcliff Academy   \nHeadmaster Mason Siegel\n\nFor the high net-worth individual who has attended the very best schools, there can be nothing more challenging than discovering that your child is stupid. That's why, for more than a century, Briarcliff Academy has catered exclusively to the needs of this overlooked and underserved population. At Briarcliff, our mission is clear: We endeavor to insulate your child from his or her own inadequacies, and insulate you from the harsh realities those inadequacies create.\n\nIn a world that is growing more complex by the minute, your child simply won't be able to keep up. We will arm you and your child with the skills needed to hide his or her stupidity with elegance and aplomb. Briarcliff's commitment to no academic standards and no testing means neither you nor your child will face the tyranny of the bad report card. Our emphasis on nontraditional learning and out-of-the-box thinking ensures that your child can accomplish something that requires little actual ability, but has all the hallmarks of real creative achievement.\n\nHow do we maintain the exclusivity that someone of your stature has come to expect? Besides our lush grounds and state-of-the-art facilities, the answer can be found on your first bill. With tuition priced twice as high as conventional private schools, you can be sure that your child will be among only the most elite of his stupid brethren. Scholarship children are not accepted because we cannot serve the needs of smart, poor children. But of course we welcome all races and religions as long as your child is stupid and rich. And as your child gets close to that exciting time to apply to college, our expert placement team will guide your child to the stupid rich college of her choice, where she will find a similarly select group of dim-witted, wealthy peers, without those taxing standards.\n\n\"The rich are different.\" The stupid rich don't know where that phrase comes from. And that's just fine with us. Briarcliff Academy.\n\nPlayground Vagina, Loved   \nand Loathed Town Landmark\n\nSuburgatory, USA\u2014The fight over the so-called Playground Vagina has come to a head as both sides war over the fate of a landmark either loved or loathed among squabbling townspeople.\n\nIt's unclear when exactly Playground Vagina came to be. \"I mean there was this enclosed slide that the dads always thought looked like, you know, a vagina. We would joke about it,\" said Brad Silver.\n\n\"Yeah, we would look at all the kids rolling around in it and say, 'Hey that vag is seeing more action than Lindsay Lohan's! Or Paris Hilton's! Or Kim Kardashian's! Or whoever the fun slut of the moment is, you know?\" said Harry Manwald.\n\n\"Then one day,\" said Silver, \"it was like the smut gods smiled on us bored-out-of-our mind dads or something, and we arrived at the playground to see that someone, probably some teenage squirt, had written Vagina right at the mouth of the slide. We laughed our asses off. That was a great day, wasn't it?\" Manwald agreed. \"Totally.\"\n\nCarey Manheim didn't see it that way. She immediately assembled a group of moms to scrub the Vagina off. \"We, as concerned moms, did not think that when we take our children to the playground anyone should be thinking about their genitals. That is the only purpose of Playground Vagina\u2014to stimulate talk about the genitals and stimulate the genitals themselves.\"\n\nThe dads grumbled, and each side, the dads and the concerned moms, thought that was the end of that. But they were very wrong.\n\nAngered by the removal of the first Vagina, the playground prankster stepped it up a notch. The next phrase to appear was Pink Taco. \"Oh God, we were high-fiving when we saw it, we loved it so much. And we could never have dreamed it would get even funnier,\" said Silver. And yes, over a series of six months, the following names appeared on the top of Playground Vagina, unleashing a cycle of removals and reappearances:\n\nFurburger. Then Juicy Box. Then Mrs. Fluffy. Then Spasm Chasm. And then, finally, the one that was both the dads' favorite and the final straw for the concerned moms, Cooz McSlimy.\n\nAt that point, the majority of moms thought the only solution was to remove the actual piece of equipment. Manheim, the first mom to rail against Playground Vagina was surprisingly against the idea, arguing that the children shouldn't have to suffer from the actions of some filthy teenage boy.\n\nThe mystery was solved one night after a park supervisor realized he'd left his toolbox and returned to find Manheim, with spray can in hand. After she was fined by the police, Manheim was asked to explain her bizarre actions. \"I'm bored out of my mind. These dads think they're bored out of their minds? They're only here a few times a week, not a few times a day.\" But why did she advocate removing the original Vagina? \"Oh, you know when you get away with something you get hungry for more? Like Anthony Weiner? That's me,\" said Manheim.\n\nThe dads, when they first saw Manheim again, were so in awe of her they could barely speak. The bravest among them said, \"We didn't think girls . . . knew all those names.\" \"You're forgetting I actually have a vagina!\" And with that Manheim joined the brotherhood, having already been cast out by her old sister-moms.\n\nLittle Loman's Lemonade Stand\n\nSuburgatory, USA\u2014A pint-sized Willy Loman is selling lemonade, Nilla wafers, and despair over on the corner of Cartwright Street and Elm.\n\n\"We're out there, baking in the sun, dreaming of closing a few measly sales, and what do we get? Dust in our face from the Caddies just whizzing by without a care in the world,\" declared sad-sack eight-year-old Jonah Miller.\n\n\"What's a caddie?\" asked seven-year-old Abby Green, who'd do anything, anything in the world to save Miller from the terminal gloom that's descended on him these past two days of selling no more than two lemonades and one Nilla wafer. But mostly she has no idea what he's talking about.\n\n\"It's what the great man drives, doll, not you and not me,\" said Jonah.\n\nMiller has spent much of the weekend manning the stand. And what a weekend it has been. It began with the exhilarating promise of little-boy riches and is ending with the dying dreams on the hard streets of a suburban town on the edge of a haunted future.\n\n\"I look around . . . I see these other lemonade stands . . . every one of them grabbing for just one tiny crumb off the delicious cake that is America. But what are we really in this for, this rat-race that takes a boy who gives his blood, sweat, and guts and eats him alive? Why Abby? Why?\" Jonah implored.\n\n\"Because we wanted money to get tokens at Chuck E. Cheese, remember?\" responded Abby.\n\n\"It isn't right, what kind of life is this, in the greatest town in the loudest country in the world?\" Jonah questioned, putting his arms and head down on the stand.\n\nAbby ran to get her mother. \"You know, I could kick myself for even letting Abby and her weird friend Jonah set up that little dread-factory. I should have known that they'd be out there, just asking to get their hearts broken,\" said Peggy Green. \"Everyone drives in suburbia. You know, his cousins Ben, Josh, and Daniel in New York set up a stand at 82nd and Madison\u2014\"Lempops\"\u2014and made, I'm not kidding, two hundred bucks in two hours? It wasn't even real lemonade! But here, the only people on the roads or sidewalks are these psychotic runner-mommies, who wouldn't ever think of stopping to give these kids one moment of dignity. 'What, you want us to eat your fake lemonade and cardboard carb-laden poison cookies?' Well if it was your kid, you'd stop. I'm sure if it was a Botox stand they'd be lined up half a block long.\"\n\nGreen immediately put out the message\u2014\"Attention Must Be Paid\" to locals on Facebook and Twitter, telling them if they didn't go out and buy a Dixie cup and a Nilla wafer now, they risked her unfriending and unfollowing. \"And public backstabbing, too,\" she said.\n\nWhen people finally started arriving, Jonah picked up his head slowly and seemed at least relieved, but by no means redeemed. Would he now ever consider a career in sales? Jonah took a deep, defeated breath. \"No. There's only one place where a beaten soul can hold his head high and shoot for honor and esteem in a boy-eat-boy world, and that's behind the toy counter at Chuck E. Cheese.\"\n\nPAID ADVERTISER CONTENT\n\nMcCaskill Garden Design Presents . . .   \nThe Lawnzilian\n\nDo you want your lawn to look like an old French whore? No? Then consider the very latest in cutting-edge landscape design: At McCaskill Garden Design, we're proud to present: The Lawnzilian.\n\nOur patented Lawnzilian technique gives your lawn the closest, cleanest, and freshest cut you've ever seen.\n\nYour grass is the first thing a stranger sees when entering your inner sanctum, your most special, private place. What will that first impression be? Will your guests and neighbors think, \"This is someone who lets herself go. Who doesn't care about having an unkempt, unruly mess?\" Or will they think, \"This is someone who takes care of business.\"\n\nWith The Lawnzilian, no thickets will tangle up your smooth, dewy sheen. No tufts will poke up from your concrete cracks. You can run, stray grass blades, but you can't hide from The Lawnzilian! A little nature is God's work. But too much nature is just plain disgusting. Act now, and receive a special offer, The Fluffer-nator, the soon-to-be patented technique to freshen up and re-energize even the most bone-dry of lawns. Call McCaskill Garden Design now!\n\nAnti-Vaxxer Barbie Doll Unveiled\n\nSuburgatory, USA\u2014Anti-vaccination activists have unveiled their latest ammunition in the battle to raise awareness of the harm they say that vaccines can cause: Anti-Vaxxer Barbie. The blond, amply proportioned anti-vaccination crusader spouts a number of slogans written by Charlotte Burger, head of the advocacy group Vaxxer Zappers, including:\n\nMeasles schmeasles.\n\nVax are whack.\n\nProtect me, don't inject me.\n\nYour gut knows, your pediatrician doesn't.\n\nThink for yourself. Just say no.\n\nBefore you poke, Google it.\n\nPox or vax? I choose pox..\n\nBurger demonstrated the doll, a retrofitted 1992 Teen Talk Barbie, with matted hair and a disheveled outfit. \"I know she looks like, well, she's looking a little used, and the clothes are so dated. But we hope people focus on the important message that Anti Vaxxer Barbie is delivering.\" She began playing the doll, pushing the button on the back.\n\nVaccines\u2014pushing poison.\n\nWill I ever have enough clothes?\n\n\"Oh no,\" said Burger. \"That last one isn't supposed to be in there. I thought we had gotten all the old Teen Talk Barbie phrases cleared out. Let's try again,\" she said.\n\nNo to the needle! No to the needle!\n\nWanna have a pizza party?\n\n\"Goddamn it!\" Burger exploded, fumbling with the doll. \"One more time.\"\n\nMeet me at the mall!\n\nScientists don't know everything.\n\nBig pharma, big bullies.\n\nMath is hard!\n\nThe alternating phrases were made even more jarring because the original perky Barbie voice clashed dramatically with the harshly strident voice of Vaxxer Zapper Charlotte Burger.\n\n\"Ugh. We have a dad who tinkers with this kind of stuff, taking out the old computer chip and futzing around with it. We're really into 'do-it-yourself,' 'think-for-yourself,' but maybe we should have asked a professional or something,\" said Burger. She tried it one last time.\n\nYou vaccinate? Go fuck yourself, sheeple.\n\nThis reporter waited for an explanation, but Burger simply said, \"Oh that one is supposed to be in there!\"\n\nWhen vaccine proponents got wind of Anti-Vaxxer Barbie, they commissioned a toy maker found on the homemade crafters' website Etsy to begin work on their own doll, a girl confined to a wheelchair in a world where childhood diseases are once again running rampant. They plan to call her \"Polio Polly.\" Their doll will say, \"I wish my legs worked!\" and \"Why did hippies let polio come back?\"\n\nWhen told of this development, Burger just threw up her hands and said, \"You see? So overdramatic.\"\n\nPAID ADVERTISER CONTENT\n\nKeller Piano Academy for the Low to Moderately Talented\n\nIs your child showing a slight interest in music, but not necessarily the talent to take him to the heights of a career in piano? Then we might be just the school for you! At the Keller Piano Academy for the Low to Moderately Talented, we will teach your child to shoot, but only shoot just far enough. Gone is the tension created when your piano teacher believes her student will someday exceed her in ability or stature. It happens all the time, and it's not pretty.\n\nAs the parent of a child with low to moderate talents in the musical realm, you probably don't even want to get your kid lessons but are feeling the pressure. What would those bitches at school say if you don't? You're not going to be one of those moms who doesn't give your child every chance in life to excel, are you?\n\nWell, at the Keller Piano Academy for the Low to Moderately Talented, we will \"teach\" your child at cut-rate prices, and you never have to worry about us secretly hating your beloved.\n\nBecause we'll let you in on a dirty little secret about piano teachers: Most of us sit there during your child's lesson, dreaming of performing and wishing we didn't have that fucking mortgage to pay and maybe if we didn't have these bills and these kids and that useless husband, we'd be prepping for Carnegie Hall right now. When a superstar lands in our midst, burning up the keyboard, all our hopes and dreams tickle past us like a tragic arpeggio across our lives.\n\nWe get angry but can't show it, of course, because that would be crazy, and so we engage in passive-aggressive sabotage. So if you have the next Lang Lang at home, the Keller Music Academy for the Low to Moderately Talented is probably not for you. But if you have the thoroughly average child, our teachers might be the perfect fit to take him far in his musical odyssey\u2014but, well, not too far! We love seeing his little fingers stumbling helplessly across the keyboard, never to amount to much of anything beyond a robotic rendition of \"Fur Elise,\" which is probably all you want anyway. Call or friend us on Facebook today! Keller Music Academy for the Low to Moderately Talented. P.S. We even discount for the least talented kids!\n\nMcDonald's a Very Bad   \nSetting to Explain Slavery\n\nSuburgatory, USA\u2014A local mom is kicking herself for choosing McDonald's to explain slavery and its legacy to her six-year-old son. \"The legacy of this stupid decision, well, I'm going to live with it 'til at least the seventh grade when Teddy finally actually understands slavery. That is, unless we are ripped apart in the giant race riot he causes before then,\" said Jan Maxwell.\n\nJan and Teddy were heading to lunch at the Boone Street McDonald's; Jan was listening to Fresh Air with Terry Gross on public radio. As part of Black History Month, Gross was interviewing an author about the history of the slave trade. \"Terry Fucking Gross. Another white liberal moron just like me,\" Jan said, berating herself.\n\nAs they arrived at the McDonald's, Teddy and Jan ordered their food and sat down. It was then that Teddy asked his mom, \"What's slavery?\"\n\nJan: Well, Tadpole [Teddy's nickname], this is hard. Slaves were people with darker skin who were forced to serve other people\u2014lighter people like me and you. They had terrible jobs, doing the same boring or tiring thing over and over and over . . .\n\nTeddy processed this, looked at the entirely black or Hispanic staff of McDonald's sweating, stone-faced and slinging fries with lightning speed, and then looked at what was, on that day, the entirely white clientele, mostly moms and kids, eating contentedly.\n\nTeddy: You mean, like them? [pointing to the counter] They are slaves?\n\nJan: No, no, not at all, Tadpole! Slavery's over. They get paid. Slaves didn't get paid. They got food and a place to sleep, that's it.\n\nTeddy: Oh, that's good. I'm sure these brown people get a lot of money now after all that slavery stuff.\n\nJan: Well. . . .\n\nAs a mother, Jan knew that being honest about salaries at McDonald's might complicate matters immensely. But as a good liberal, \"like Terry Fucking Gross,\" Jan made the regrettable mistake of being honest.\n\nJan: Well, actually, I'd be lying to you if I said they do. They don't make much. Very little in fact. They make enough to just get by, but not much more than that. It's a hard life.\n\nTeddy: Do they at least get to eat the food?\n\nJan: Um, no. They take their small amount of money from McDonald's and buy food, probably somewhere else, like we do, at the store.\n\nTeddy: So slaves got paid in food and these brown people get a teeny bit of money and buy their own food?\n\nJan: Yes.\n\nTeddy: Then what's the difference? They are slaves!\n\nJan: No sweetie, they're not. No one owns them.\n\nTeddy: If McDonald's pays them their tiny money and but doesn't feed them, then doesn't McDonald's own them?\n\nJan: No . . . look, Teddy, they don't make much money but they aren't slaves. They have their families\u2014slaves mostly didn't have their kids with them. Moms lost their babies.\n\nTeddy: What about those moms? [gesturing to the women working behind the counter] I don't see any brown babies here. Where are their brown babies?\n\nJan: They put them in daycare\u2014you know, the place to keep them so the workers can do their work for McDonald's.\n\nTeddy: So McDonald's takes them away!\n\nTeddy started crying and said, \"I don't want my Happy Meal if a brown mommy gets her brown baby taken from her!\" And then, before Jan could grab him, he ran up to the counter and said, \"I'm so sorry you are slaves and I'm a slavemaker! You shouldn't be slaves!\"\n\nJan said, \"Teddy, no!\"\n\nThen he told white people in line that they were \"slavemakers,\" too.\n\nMaxwell shudders even recalling it. \"Basically I've turned my son into a little Malcolm X. A Malcolm X who still wets his bed.\"\n\nWill she go to McDonald's again? \"No. From now on we're getting served by mostly white people only. Teddy X better start liking Starbucks.\"\n\nDog Fed Better than Scholarship Child, Says School Nurse\n\nSuburgatory, USA\u2014A concerned school nurse asked for a meeting with the mother of Tom Mason, a scholarship student at Bundy Academy, upon deciding that her dog is fed better than the student.\n\n\"A dog is a child, my child, and of course I feed Roxie only hormone-free grass-fed beef, real cheeses and yogurts, pureed vegetables, whole wheat pasta, brown rice, and flax seed,\" said Jenny Maurice, who added that on special nights he gets \"prepared dinners,\" including one described this way on the menu at the high-end Delicious Ruff Doggie Bistro: \"Ground Shoulder of Farm-Raised Beef served over Couscous and Oven-Roasted Leeks. Served with a saut\u00e9 of Fresh Pan-Wilted Kale, Fresh Garbanzo Beans, Roasted Polenta, and Hint of Garlic. Then drizzled with High Oleic Kosher Olive Oil.\" Maurice said, \"I know, I know, it sounds a tiny bit excessive, but it really keeps Roxie's coat shiny.\"\n\nSo with all her attention to Roxie's diet, \"and my own,\" she added, Maurice was disturbed when she saw Tom's monthly diet diary, which is required of all the students. \"You're damn right I requested a meeting with his mother!\" said Maurice.\n\nThis reporter asked whether she also requested that the boy's father attend.\n\n\"Oh come on,\" she said, exasperated. \"Father? What father? What planet are you on? Even if there was a father, and I doubt it, I wouldn't call him. They're pretty much all useless, no matter where they come from.\"\n\nMaurice decided that Mason's mother wasn't going to be receptive after it took a week to set up the meeting. \"Oh, because I'm a bad mother? Is that what she thinks?\" said mother Terry Quillan. \"You know, he is with his dad half the time but I'm guessing she didn't call him in for a meeting. Which took me a week to set up because we live twenty miles from school and I work. At Clucky's Chicken. They don't give you a ton of 'me-time' at Clucky's Chicken.\"\n\nAs Maurice and Quillan sat down together, Maurice placed a sheet down showing the new USDA \"plate\" with its nutritional recommendations. \"Ms. Maurice, I'm fully aware of what's healthy and what's not,\" said Quillan.\n\n\"Well, not to overstep . . .\" said Maurice hesitantly, \"but Tom's diet diary had a lot of carbs and not much high-quality protein like, say, wild salmon, and no real 'rainbow' of fruits and vegetables. It looks mostly frozen or from cans and certainly not organic or locally sourced. Now since I care so much about this issue, I took it upon myself to talk with the folks at Whole Foods to put together a possible meal plan for you!\"\n\nThe list included Pineapple-Chicken Kabobs with Quinoa, Fruit, and Hemp Seed Muesli, and Lebanese-Style Grass-Fed Ground Beef Kabobs.\n\n\"I thought the chicken and grass-fed ground beef would be more affordable for you. Things like wild sea scallops can really add up. I know firsthand!\" said Maurice, hoping to be helpful.\n\nQuillan looked at the list in enraged wonder. \"This food would wipe out a week's worth of food stamps in two meals. Tell me, is my son doing poorly in school? Is he overweight? No, he is neither. He eats what I can afford and what he'll actually eat. And what he eats at his father's house half of the time. This meeting's over.\"\n\nAs Quillan stormed out, Maurice lovingly fingered the frame on her desk showing Roxie's photo. \"Oh Roxie. You are so lucky to have me.\"\n\nDiscount Doula \"A Really Bad Choice\"\n\nSuburgatory, USA\u2014A couple admits that hiring a \"discount doula\" was a \"really bad choice,\" making the delivery of their first child unforgettably awful. But their nightmare had an unexpectedly happy ending.\n\n\"Yes, looking back, trying to save money on a doula was a big mistake,\" said Alysia Verderese. \"But I just thought that anyone who calls themselves a doula is probably a caring, thoughtful person. Anyone who even knows what a doula is is probably a caring, thoughtful person, right? \"\n\n\"My mom never heard of doulas,\" said husband John.\n\n\"I rest my case.\"\n\nThe couple was hoping for as natural a birth as possible and thought a doula labor-and-delivery coach could help, especially because Alysia's mother was dead, and she would have had no women in the room with her. But with money tight, the Verdereses found a Craigslist ad that read \"Doula For Less Moola.\" John seemed embarrassed upon being reminded of this. \"Yeah, I guess a doula with a corny, rhyming name should have been a tip-off. And the University of Phoenix reference.\"\n\nBut Alysia appreciated the honesty of discount doula Maggie Brown and got a \"warm vibe\" from her ad:\n\nI am offering my services at far less than the going rate, because I admit to being new to the delivery room. But I do have certification from the University of Phoenix and proven experience in very stressful human situations. I love meeting new people! And I look forward to making your birth experience an unforgettable one!\n\nAlysia Verderese prided herself on taking chances on people. \"I've always tried to help the little guy, and every doula has to have a first time, right?\"\n\nOnce labor began, the couple arrived at the hospital and met Maggie Brown there. \"Ummmm, I expected a kind-looking, natural older woman,\" said Alysia. \"Translation: an old lady, no bra, with an ass-length gray braid,\" said John.\n\nInstead, Maggie Brown \"looked like Joan Jett. Only meaner,\" said John. \"Or The Girl with the Dragon Tattoo,\" said Alysia. \"I thought she was going to man-snap all of a sudden and rip John's face off.\"\n\nBut Alysia did notice that the tattoos, scary at first glance, were actually half the cast of Fraggle Rock all over both her arms. \"I mean, how mean could she really be?\" asked Alysia.\n\nBrown came in with supplies, which Alysia in her doula daydreams imagined might be aromatic oils, candles, and washcloths in an earth-toned, organic fabric bag. Instead it was a plastic Walgreen's bag with Cheetos, Capri Suns, the latest issue of US Magazine, tampons\u2014\"I was out,\" explained Maggie\u2014and a package of diapers. Alysia took the diapers and said, \"Maggie, we're in a hospital. They have diapers here. Also, these are for eight-month-old babies.\"\n\n\"Oh, does that really matter?\" said Maggie. She threw herself down in the chair and started reading her US Magazine and eating Cheetos. \"You want some Cheetos?\" she asked Alysia, who replied, \"No, Maggie, I can't eat during labor.\"\n\n\"Oh! Sucks to be you!\" said Maggie. \"Must have missed learning that one at ITT Technical.\"\n\n\"Your ad said it was University of Phoenix,\" John said.\n\n\"What's the difference?\" Maggie said.\n\n\"She got me there,\" John admitted.\n\nWhen Alysia started contracting, Maggie stood up and silently watched her, staring directly at her face. \"Wow, this looks like it really hurts. Hurts like a total motherfucker. You poor, poor lady.\"\n\nThe nurse came in to see if she wanted her epidural. \"Oh no no, no epidural. I have my doula here with me\u2014\" Alysia smiled tentatively \"\u2014to help me get through it.\" The nurse gave Maggie a slow up-and-down look and said, \"You look like Lisbeth Salander, the 'Girl With the Dragon Tattoo,'\" \"Oh my God, THANK YOU!\" Maggie said, momentarily elated. \"Wait, is the epidural the pain stuff? Dude, don't be crazy! You have to get it!\"\n\nAlysia: Maggie, I told you I wanted a natural birth. Do you even know what a doula is?\n\nMaggie: Well, the lady whose dog I walk\u2014she's kind of rich like you\u2014she had one and it sounded so, so, caring and thoughtful.\n\nAlysia: Yes! See, John? Caring and thoughtful. [John rolls his eyes.] Exactly. But I'm not rich. I mean, maybe to you I am.\n\nMaggie: Well, anyway, dude it's totes your body and for whatever reason you seem to want to suffer, and I really really don't like to watch people in pain or suffer! You wouldn't say 'no drugs' at the dentist's office getting something pulled out of your mouth, would you? [She starts to tear up.]\n\nAlysia: Maggie, what did you mean in your ad when you said that you had 'proven experience in very stressful human situations'?\n\nMaggie: Oh . . . well, I was in Iraq. One day our supply convoy got hit. I was OK, but, it was bad. So now I just want to do happy things like help babies and mommies and shit.\n\nJohn said, \"Great, so now I had no chance of throwing her out of the room. First, she's Lisbeth Salander, weirdo punk doula, and now she's Lisbeth Salander, American fucking hero.\"\n\nAs Alysia's contractions got stronger and stronger, Maggie covered her face and kept repeating \"I can't watch! I can't watch!\"\n\nAlysia, panting, said, \"It's OK, Maggie, you'll be OK!\"\n\nWhen the Verdereses' son finally emerged, and Maggie saw the blood, John said, \"I swear to God, she got the thousand-yard stare.\"\n\nBut then, once little Cory was cleaned up, Maggie slowly walked toward him and, like Alysia and John, looked utterly transformed. \"Hi, little man! Hi, little man! I'd take such great care of you if I could just figure out what to do and get my act together a little bit and let this piercing heal up a little better, and I'm really good with dogs,\" she said, softly touching his fingers and toes and nose. \"Do you guys maybe need a babysitter sometime?\"\n\nReader, they hired her.\n\nMommy War Combatants Embrace Mutually Assured Destruction\n\nSuburgatory, USA\u2014Combatants in the Mommy Wars have achieved a carefully calibrated detente, with each side amassing large stockpiles of vastly destructive Guilt, Pity, and Scorn bombs.\n\nBoth the stay-at-home moms and the working moms have absorbed the lessons of Mutually Assured Destruction, which helped keep the peace during the Cold War. That means the combatants make it clear the weaponry they possess, so that each side knows an attack by one would result in total annihilation of both.\n\nValerie Snow, accused war criminal and guerrilla leader of the stay-at-home moms (SAHMs), described her side's weaponry while cutting decorative radishes for the evening's PTO volunteer appreciation dinner. \"If they so much as hint that we might have something better to do with our time than making decorative radishes, by God, they will live to regret it.\"\n\nSnow describes the catastrophic impact that would result from deployment of their Guilt, Pity, and Scorn (GPS) bombs against the working moms.\n\n\"Well, our carefully engineered GPS bombs would unleash some devastating, shattering intelligence. They would inform the working moms each time their children said, 'Why can't my mommy volunteer all the time like you do?' (Which happens all the time, just sayin'.) They would disclose the intimate details the child shared with a stay-at-home mom because, where the hell is their own mother this time? Oh yeah, working, because of her pathetic priorities. They would catalog all the joys the working mothers had missed while on the job\u2014the first soccer goal, first crush, weekday summer visits to the local pond. And they would throw in some anecdotes from empty nesters dying to have their kids back and wishing they had done things differently, like be with their children.\n\nAmber Ostroff, who leads the working moms (WMs), betrayed no fear of the other side's lethal capacity. \"Oh please, you think that's gonna scare us?\" said Ostroff, speaking from the Dallas-Fort Worth airport where she was en route to a client meeting. \"We are corporate lawyers and doctors and college professors. Remember college, ladies??? What was that even for again? Right, you got an Ivy League degree in making decorative radishes.\" Snow flinched upon hearing the statement and the radish crack, but maintained control.\n\nThe WMs' own GPS weapon, designed by a working mom engineer at a leading defense contractor, would strafe the SAHMs with the charge that they insult poor and working women by suggesting there's any \"choice\" in staying home for most women. It would inform them that they are weak role models and an \"insult to every feminist who fought to get them in the workplace in the first place,\" said Ostroff. It would force stay-at-home moms to talk about \"anything other than their kids.\" Says Ostroff, \"Good luck with that one, ladies!\"\n\nTheir GPS would also feature new research suggesting that overparenting is hurting, not helping kids, and statistics showing that stay-at-home moms who give up careers take an enormous financial risk if their husbands decide to divorce them and snap up a younger model. When that happens, says Ostroff, \"The SAHMs turn from being parasites in their own home to parasites on the whole nation.\" It would also question whether they had remained sexually appealing to their husbands. \"Ever see a working mom in a hideous pair of mom jeans? No. We have some self-respect.\"\n\nWith these equally fearsome weapons aimed at each other, both sides have been able to manage a very fragile peace. \"While your leaders fight the public battle, we suggest that when our SAHMs see a WM, that they just be polite and ask one token 'How is work?' question, and try to leave it at that. If the WM starts droning on about how challenging her latest project is or bragging about a big raise, we'll advise the SAHM to just excuse herself and go back to collecting box tops.\"\n\n\"Agreed,\" says Amber Ostroff. \"Keep it all on the surface, smile and nod, smile and nod, and as long as they don't start passive-aggressively guilting me about these stupid school events I've missed, we might all make it out of this alive.\"\n\nSHOUT OUT\n\nWalk All Over Me, I Don't Care,   \nbut Don't Call Me a Perv\n\nThe Halden Street sidewalk, known to local police as the \"Sidewalk Perv,\" was installed over an eight-month period two years ago.\n\nListen up, there's a reason this is called Shout Out, townspeople. I am the Halden Street Sidewalk, not the Sidewalk Perv. I might be a little rough around the edges and have a few chips and cracks. I am made of gravel and rock, after all, and yes I do come from that industrial park across town lines where only the trucks go and the Oxy addicts squat. Excuse me for not being classy enough for you, Halden Street.\n\nNow I'm not saying I'm easy to like. I do hate a lot of you. I hate fat people pounding on me, but I hate those crazy joggers even more; scary bitches, faces all scrunched up in terror like they're being chased by the big bad old-age monster. Guess what? He's coming for you anyway! Also, I hate people in wheelchairs barreling over me all day long without a thought in the world. I hate snow and drunks who puke on me and those mouth-breathers who spit their gum on me, and most of all, of course, every damn dog on the street shitting in my face.\n\nBut what I like, I like a lot. And that's the ladies. Is that a crime? I like all types of ladies, except the fat ones and the crazy joggers. And, come summer time, pull up a chair, because I got the best view in town. Yeah, when I saw those pink french-tie hip hugger panties Cindy Kramer was wearing yesterday, yeah, you could say it made my day go a little faster. And if you ladies go all Lindsay Lohan and ditch the french-ties, the booty shorts, the boy leg briefs, g-string thong, or the classic high-waist bikini altogether, giving me a show of a lifetime, that's my fault? It seems like you're the Perv here, Miss Eileen Marple, not me.\n\nBut I want to make one thing absolutely clear. I do not ever look at an underage girl. Who do you think I am, Roman Polanski? When a young lady in a Catholic school uniform walks over me, I turn my eyes deep into the bowels of the earth, and trust me, what's down there is not pretty. I am no pedophile. I am a completely passive lover of women, not a menace. So stop walking all over me, curb your dogs somewhere else, and stop shitting on my good name.\n\nMom Plans School Auction   \nDuring Dreary Sex\n\nSuburgatory, USA\u2014A local mom who is a skilled multitasker used her once a week love-making session with her husband to plan her school's much-anticipated school auction.\n\nIt's hugely, massively important to the PTO's budget, thought Sheryl Winnick as her husband Dave ran his finger up her calf and asked, \"Did you shave?\" That was a typical signal to begin their lovemaking, along with the comment, \"Hmmmm, you smell good,\" and a playful head nuzzle.\n\n\"Yes, I shaved,\" she said. This was his cue to perform mostly perfunctory oral sex on his wife. \"Oh no, it's not fun at all but I so appreciate his effort. It's just a nice courtesy.\" Now . . . I was thinking a good theme would be Arabian Nights, she thought, as he continued attempting and failing to pleasure her.\n\nArabian Nights . . . would that sound too . . . Muslim? Well, screw those bigots, that's so ignorant. Anyway, maybe the PTO moms could wear harem outfits. \"That's it, give it to me, honey . . .\" This was her way of telling Dave it was time to perform her own perfunctory and generally substandard oral sex on her husband. \"Yeah it's lame,\" he admits. \"She's bad at it, and tonight I know she is planning the auction while blowing me, but you know, it's so important to her. I'll take what I can get.\"\n\nAt this point, says Dave, it was \"go time,\" and he entered Sheryl from the side, which was typically a two-minute prelude to the finale: climax for him and perhaps her, occasionally, through the tried-and-true missionary position. We'll get the VFW Hall and have a preview the night before, thought Sheryl. We can do a spa package, haircut certificates from Snippety Crickets, what else? A grown-ups Wii party . . . that actually did great last year . . . ski house weekend. . . \"OH YES!\"\n\nAt that point, Dave finished and Sheryl gave out a hearty, but secretly forced, gasp. Dave was satisfied despite the fact that Sheryl's mind was clearly elsewhere, and pretended to have an orgasm. \"You can see how devoted she is to the school and the kids. Marrying her was the best thing I ever did.\"\n\n\"Honey, should we watch the Daily Show now?\"\n\n\"Sure!\" Sheryl said, snuggling up to her husband. I wonder if Jon Stewart would ever sign a book for us to auction off at Arabian Nights\u2014he's always defending teachers. Hmmmm, now how would I get in touch with him?\n\nGuidance Counselor Feared, Loathed   \nas World's Sole Remaining Hyperpower\n\nSuburgatory, USA\u2014Area guidance counselor Mavis Goodstone has been declared the world's undisputed hyperpower by some of our region's most prominent residents, who were discussing Goodstone's massive, destabilizing power while attending the World Economic Forum in Davos, Switzerland.\n\nGoodstone is renowned locally as the high school guidance counselor with the \"golden touch\" in terms of college admissions. But few parents have been able to decipher her motivations or her delphic comments, and many are left enraged by the capricious and arbitrary ways that she decides who to help and who to spurn.\n\n\"That frigid bitch used her power to keep my kid out of Harvard, and I teach there,\" said historian of empires past and present, Denis Ferguson. \"Since then, Goodstone has amassed unparalleled military and economic might. A unipolar world is a dangerous world, and Goodstone is a very dangerous and, I should add, very unpleasant woman.\"\n\nHedge fund manager Gareth Faber agreed. \"Yeah, you know people hated Bush. But at least that decider was pretty personable. This decider won't even make eye contact.\" Faber, whose son is applying to Brown, still has no idea if his child is on Goodstone's anointed list.\n\n\"Goodstone's currency is just murdering the dollar and the Euro and you know, my wife Jenny just texted me to say Golden Goblin [Goodstone's nickname] appeared a little aloof this morning when she went in to discuss Tristan's applications, and now she is fuh-reaking out. Total disregard for human rights.\"\n\nSome predicted, and hoped, that Goodstone's overweening power and hubris would lead to her eventual downfall as a hyperpower. \"She's pushing the envelope. Her preemptive strike on Syria, her runaway deficit spending on arming herself, and her tactical blunders\u2014snubbing Anna Wintour's daughter?! Who snubs Anna Wintour? Oh I know. Mavis Fucking Goodstone. In her JC Penney pantsuits,\" said foreign policy expert Hassein Makaria. \"All these bold but shortsighted moves could signal the end of a long, and, as history will prove, ugly reign. Gareth . . . um . . . did Jenny mention how Goblin acted toward [Makaria's wife] Valerie? Her appointment was right before Jenny's.\"\n\nFaber became agitated. \"Where did she even go to college? Have you heard her do assembly at that open house? Can she even speak proper English? How did that woman get to be the global hegemon? I just don't get it.\"\n\nThe participants at Davos tried to get their underlings to negotiate with Goodstone while in attendance at the World Economic Forum but their diplomatic entreaties were harshly rebuffed.\n\n\"I don't know what I'm supposed to do. Write a column for the Times that just says, 'Bitch, Call Me Back'? Like that would do it, I'm sure she doesn't read the Times. She's USA Today all the way,\" said New York Times oracle Thomas Friedman.\n\nFriedman went on, \"Folks, pull up a chair and listen. Humiliation is the most underrated geopolitical force, and if I don't get a callback soon, she's catching hell, and I'm calling for sanctions. That piano-legged shrew. She will be her own undoing.\"\n\nDr. Drama\n\n\"When life hands you a problem, let's make it more interesting!\"\n\nDear Dr. Drama:\n\nI think my husband, I'll call him \"Tom,\" is flirting with other women on Facebook. I should say, I know he is flirting with other women on Facebook. It is making me really crazy! What should I do?\n\n\u2014Face-Freaked in Suburgatory\n\nDear Face-Freaked:\n\nSo let me guess. The pictures \"Tom\" has up on Facebook are the best pictures ever taken of him to the point that you barely recognize him? And for some reason you don't seem to be in any of them? And aside from the flirting, does Tom post updates like \"Didn't ever think I could bench-press that much!\" or \"Margaritas chilling in the fridge, marinated steak on the grill, relaxing on the deck. LIFE IS GOOD.\" Am I right?\n\nWell, you could ask Tom to stop flirting, sit down like grown-ups, and ask, \"What's missing here that you need to take to Facebook for stimulation?\" and boring boring blah blah. But I have a much more satisfying idea.\n\nYou're going to reveal the real Tom on Facebook. You surely have photos lying around of Tom with his crack showing or his flabby stomach sagging or stuffing his cake-hole with Funyuns, say. Simply post them, preferably at times he's not on Facebook, and tag him. He won't be around to untag and there he is, in all his real Tom disgustingness, for all his little pathetic nubile groupies to see. You could also pepper in Wall comments like \"Honey, the bank keeps calling about the foreclosure, can you call them back?\" or \"Are your hemorrhoids feeling better? I'm so sorry they kept you up last night!\"\n\nWith Facebook customization, you can even hide them from everyone you know! (Thanks, Zuckerberg!) The only ones who'll see it are strangers, and, of course, those cunts makin' time with the fake, attractive Tom. That way, you can get back to having the true Tom, the repulsive slob you love more than anything in the world, all to yourself.\n\nDad Invents Label for Unruly Child\n\nSuburgatory, USA\u2014Frustrated with his failed attempts to get his unruly child labeled, a local dad has taken matters into his own hands, promoting his own made-up label as a genuine disability to other parents and his child's teachers.\n\n\"Oh, yeah, Kilkenny has FG-NOS: Franticocious Gravidarum\u2014Not Otherwise Specified. It combines rare academic talent with a frenetic, sometimes uncontrolled pace. Kilkenny is compelled by his expansive mind to explore boundaries, so that explains the inappropriate touching incident. It's something, you know, the schools just have trouble keeping up with. But we're going to change that,\" said Kilkenny's dad, Jerry Lipton.\n\nWhen the school district intervention team was asked about FG-NOS, it had no knowledge of the disorder. Lipton was undeterred when presented with this information.\n\n\"FG-NOS is cutting edge. Fifteen years ago, who had ever heard of Asperger's? People who were troubled geniuses were considered retarded, warehoused. I refuse to give up on my seriously disabled child, and neither will this school district, not while I'm paying taxes here.\"\n\nKilkenny's teacher, who chose to remain anonymous, was nonplussed by Lipton's assertions. \"I don't know what the hell he is talking about with that FG-NOS or whatever that was he was spitting at me at the teacher conference. All I know is Kilkenny is just a seven-year-old jerk, pure and simple. He won't listen. He throws stuff. He can't spell his own name. He groped little Madison! The other kids hate him. And I can't say I blame them. 'Rare academic talent?' Where the fuck did he get that from? Not from my teacher conference, I can tell you that. I can't wait to tell Principal Harris. She's going to laugh her ass off that he talked to you about his 'disorder.'\"\n\nLipton asserts that FG-NOS will soon be part of the roster of well-established developmental maladies that require classroom assistance and government-supported occupational therapy. He expects it will be added to the next Diagnostic and Statistical Manual of Mental Disorders, or DSM.\n\n\"You know I bet forty years from now we'll look back and realize that some of the giants of yesteryear\u2014David Hasselhoff, Michele Bachmann, all the Kardashians\u2014they had undiagnosed FG-NOS and were misunderstood as the quirky child geniuses that they were. And my Kilkenny will be able to speak out as one of the first to say 'I had it, I'm hugely successful now, and you can make it out the other side, too.'\"\n\nIndian Child Taunted as   \n\"New Jew\" at Middle School\n\nSuburgatory, USA\u2014The District Attorney is investigating a possible hate crime at the Lexington Middle School, after an Indian child reported being taunted as a \"New Jew\" by several classmates.\n\nThe student, Nikhil Chaudry, thirteen, said that four children, over the span of approximately one month, referred to him as the \"New Jew\" after Chaudry captured the high score in the regional televised teen trivia show, Answers Please.\n\nChaudry's family arrived from India four years ago and quickly developed the successful Currying Flavor chain of restaurants. They have been seeking, and gaining, political influence with large donations to politicians on both sides of the aisle.\n\nThe Chaudry child is described by his teachers variously as \"a superstar,\" \"a relentless worker,\" and \"a joy to have in the classroom.\" One said, \"if only I had twenty more like him.\"\n\nThey said Chaudry's academic success and the family's emerging prominence has apparently engendered resentment among the other children.\n\nThe Chaudry child gave a statement to reporters. \"One kid said I was the 'New Jew.' He said we just hoard all our money. Another kid chanted 'New Jew, New Jew' and said 'You people want to take over the world. You took our jobs over there and now you want them here and I'm going to end up going to Green Valley Community College.' Another one said 'We don't need you making us look bad every day to Mrs. Kelley. Don't you have a Wii? And where's your iPhone? What do you do after school? Sample SATs?' Someone else said, 'You people can make all the money you want but my dad said you are still going to smell like curry no matter what [expletive] Ivy League school you get into.'\"\n\nChaudry added, \"I didn't actually think it was that bad. I thought they were going to call me a terrorist or something. I didn't even know what they meant by 'New Jew.' It was my mom and dad who got really mad, after I told them.\"\n\nParents of two of the accused bullies spoke on condition of anonymity.\n\n\"Nikhil doesn't play any sports. It's just school, school, school, school. Now how are our kids supposed to compete academically against that? I think activities should be weighted along with academics. I mean, does our country really need all these single-minded brainiacs? Is that the message we want to give the kids, that soccer is meaningless and that the teamwork they learn in lacrosse doesn't matter? That's not my America.\"\n\nAnother visibly angry parent said this: \"By the way, why do you think his parents are pushing this case? To raise his profile even further? Of course that's why. It's good free advertising for Currying Flavor, like they need any more, and it'll make Nikhil stand out even more on those college applications.\"\n\nNo matter the outcome of the case, Nikhil Chaudry believes he'll have the last laugh. \"They can call me 'New Jew' all they want. All I care about is getting into MIT or Cal Tech and making tons of cash. And in twenty years they'll be pumping gas into my Escalade, at one of the gas stations my cousin Naresh owns. And I'll be married to Miranda Cosgrove. I'll be bigger than Zuckerberg.\"\n\nSHOUT OUT\n\nIn Fifteen Years, Awkward Girl,   \nYou'll Be Totally Hot\n\nNathan Brodie is a novelist who lives on Mareau Street.\n\nAs a college professor I do quite a bit of work at home, as my desk points out the window of the red house with yellow shutters. Each day I see the young, fresh teenagers trundling past me from the high school. In a strange way I feel as if I have gotten to know them, in my fantasies, in an utterly nonsexual and nonthreatening way.\n\nSo know this, if you ever read this, Awkward Teen Girl with the red windbreaker and funky retro purple glasses: I am not stalking you. (You hear that too, Awkward Teen Girl's father? No need to come a-knocking on this door! Nothing but pure intentions on my part!) This is merely a paternal word of advice. Awkward Teen Girl, you'll be totally hot in fifteen years.\n\nNow, I know, you don't believe it. The popular boys surely have no use for you, and you've already gotten used to that. But what you're realizing now is that even the barely cute geeky ones, like I used to be, don't like you either! Well, they don't know what they'll be missing in fifteen years. The Pretty, Not-Awkward Girls have figured out their look at age fifteen and it's all downhill from there. Trust me, I know all too well. Having no originality, because it is not required of the Pretty, Not-Awkward Girls, they rest on their aesthetic laurels and cling to that one trashy look that worked so well for them in high school. Well, by thirty, that ain't looking so great anymore.\n\nYou, on the other hand, Awkward Teen Girl, are already struggling to distinguish yourself from the Pretty, Not-\u00adAwkward Girl and this is making you stronger! By twenty-five you'll have given up your adorably funky glasses for something more professionally appropriate for your challenging and exciting career. You'll be getting your hair, nails, and toes done and feel great inside and out. Look to Tina Fey\u2014and see your bright, Not Awkward, Totally Hot future.\n\nBelieve me I wish I could shake those boys who spurn you and say, \"Don't make the mistake I made!\" But that's how life works, as you probably know already, because precocious wisdom is surely one of your signature gifts. Sadly, that's just one of those talents that alienates or threatens teenage boys, but is utterly charming to a forty-year-old man. So I just want to assure you that in fifteen years, I'll probably still be here, rooting for you in a nonsexual way. Now you know where I live, if you want to wave, please do! And it's true what the gays say, sweetie, it really does get better!\n\nSanta Asks Child for   \nNew Liver for Christmas\n\nSuburgatory, USA\u2014Santa Claus, the giver of joy and delight to children the world over, today asked a four-year-old local boy to wish for a new liver for Santa, and gave the back-story on his cirrhosis diagnosis, along with other maladies.\n\n\"Hi, Santa!\" said Nick Gardner, as he leaped onto Santa's lap at the town's high-end Northway Mall. Gardner lives with his parents in the affluent Westgate neighborhood.\n\nSanta: Why, hello boy. Merry Christmas! What's the name. Oh, Nick? Hey, don't bounce too much, boy, got some stones I'm passin'. Wha-chu want for Christmas, little man?\n\nNick: Um um um I want the Power Rangers Operation Overdrive Mega Mission Helmet . . . and the . . . and the . . .\n\nSanta: You know what I want, Nick? I want to kick this infernal cirrhosis. Let me tell you a little story, Nick. Back in the '80s, my woman, Mrs. Claus, had a little problem with a very dangerous game, let's just call it \"smack.\"\n\nNick: Smack, smack, smack!\n\nSanta: My lady and I, we were travelin' 'round this great big country\u2014you know, when we're not at the North Pole\u2014parking our RV at whatever KOA we could find, takin' in whatever life brought our way. KOA is Campgrounds of America in case you don't know. . . .\n\nNick: I've never been camping! I swim in the hot time at the Club! You know what else I want?\n\nSanta: . . . and Mrs. Claus couldn't stop playin' that smack game. And there was a whole passel of people at the campgrounds who played that game, too. And then Santa started playin' that game, too. Again and again and again.\n\nNick: That musta been fun, Santa!\n\nSanta: Well, I'm not gonna lie, Nick, Santa loved the game and so did Santa's lady. Those were some crazy times for your ole Santa. Matter a fact, I think about playin' that game all the dang time. But now I got a little something called cirrhosis. And now somethin' real important inside Santa\u2014Santa's liver\u2014well, it's shot.\n\nNick: Oh gross! I hate liver! Mommy tried to give me that because she said, she said, she said it's got a lotta good iron in it! It makes me grow important like Daddy to make big money! It grows my brain! [Nick put both hands to his head.]\n\nSanta: Well, Nick, you wouldn't like my liver at all. It's all scarred up and spent and hard like the coal we give bad little boys, not like you, the bad ones I'm talking 'bout. Santa's got his own list you know, a list for a newfangled liver, all fresh and shiny and red and tied up with a bow. But, well, it's in God's hands now.\n\nNick: But aren't you, aren't you, like God, Santa? Could you call him on your Santa phone?\n\nSanta: Ho! Ho! Ho! That'd be a real kick in Santa's big red pants if he could! Santa would get a brand new liver for Christmas and maybe even get rid of that sweet blood, too. I shouldn't make it sound all that bad, Nick. That darn cirrhosis is why my tummy is so puffed up and that's how I got to be Santa here at the mall. And this year, Santa needed work. Santa needed it real real bad.\n\nNick: You're a silly Santa, Santa! See you next year!\n\nSanta: Well if you are a very good little boy, and wish for Santa to get that new liver he was tellin' you about, you just might see me next year and I'll have lots more silly stories 'bout me and Mrs. Claus to tell you.\n\nNick: Bye Santa! Merry Christmas!\n\nAnd with that, Nick and his mom, who was on her iPhone during the whole discussion, left for last-minute shopping at Bloomingdale's.\n\nDad Forcibly Removed from   \nMall Massage Chair\n\nSuburgatory, USA\u2014An unemployed, reluctant stay-at-home dad has become an unwelcome regular at the Brookstone within the struggling Atrium Mall. Yesterday he was forcibly removed from the Ultimate Robotic Human Touch Massage Chair.\n\nIan Brown was unrepentant. \"I don't know what the big deal is. Don't you think seeing someone in ecstasy sitting in that chair for a little while is good advertising for Brookstone?\" Brookstone associates offer a starkly different portrayal. They say Brown has been frequenting this particular chair for months, several times a week, and stays for at least forty-five minutes each time.\n\n\"First of all, we've been trying to cut back on people parking there, so we put a sign on it saying 'Please ask an associate before you use the chair.' He just takes the sign and throws it on the floor.\"\n\nAnother associate says Brown has a technique for ignoring the store employees. \"He just sits down and closes his eyes immediately. But I can see him do these tiny squints. He's watching us. And he's obviously awake\u2014he just parks his kid's stroller in front of him and lets her scream. And when the auto program stops at the end, he just picks up the remote and hits it again. With his eyes still closed.\"\n\nThe supervisor is worried about the physical condition of the chair and also the safety of the child. \"Sometimes to shut the kid up he drags her onto the chair with him and all that machinery, God, I don't even want to think about what could happen to that little body. And then there is the 'groove.'\"\n\nThe \"groove,\" as he calls it, is the indentation Brown is leaving in the chair. The supervisor said, \"If corporate comes in here, and sees that groove on a three-grand chair, I'm going to get my ass handed to me.\"\n\nThis is what led to yesterday's forcible removal. First they tapped Brown, and then said, \"Sir, we know you are awake and your child is crying.\" When he was roused, Brown grew combative, screaming, \"Don't call me sir, motherfucker! I'm just an out-of-work dad who needs a little break. You college flunky dipshits get a break. What about me?\"\n\nSources say after some tense discussions, he came to an agreement with security that he would take his child to Build-a-Bear, and then leave the mall immediately.\n\nPAID ADVERTISER CONTENT\n\nContractor Distractors\n\nBored out of your mind in your marriage? Look over at your mate and think, \"I wish she'd buy some new clothes\" or \"I wish he would pluck that disgusting ear hair?\" Always wanted that sundeck you never knew how to pull off? Well, at Contractor Distractors, we build your dream deck and rekindle that flame at the same time!\n\nWith a maddeningly complex design plan, multiple changes, and constant no-shows you'll forget all about those freakish bunions. With the budget running over and the total tally mounting, you'll be sitting amid exposed two-by-fours and dangling electrical fixtures for months on end. You'll be attending cocktail parties where the \"nightmare\" is all you'll be talking about. You'll have one common enemy and it won't be each other: It will be us. That's a guarantee. And by the time we're done, which won't be soon, your marriage will be as sunny as a July afternoon of Mojitos on your new sundeck. So when home improvement meets marital malaise, think Contractor Distractors! We are currently offering a midlife crisis special: The kitchen of your dreams with two full years of hassle . . . and hugs.\n\nPedophile Quietly Mourned   \nas Amazing Coach\n\nSuburgatory, USA\u2014A convicted pedophile is being quietly mourned by local dads as an \"amazing\" and \"outstanding\" girls' varsity soccer coach.\n\n\"That disgusting pervert. It took months to put that psycho away,\" said parent Kevin Mainer about Tom Miller, who was charged with statutory rape against several minor girls in February and convicted last week.\n\n\"But,\" added Mainer, a bit sheepishly, \". . . well . . . I mean, uhh, now we're shit outta luck if we think we'll kick Benchley School's ass ever again.\"\n\nOther parents agreed. Said Paul Hofstetter: \"Oh yeah, if it was my daughter? He'd be fucking dead right now. I'd have his ball sack run up the flagpole. I gotta give it to him, though, anyone who saw that upset he pulled off last year against those amazons from Holston, I mean, whoever takes over is gonna have big shoes to fill, that's all I have to say.\"\n\nThe men went on to debate the gravity of Miller's crime. \"Come on, just a few decades ago, this wouldn't have even been a crime. Those girls were marrying-age back then. What if we were in Utah?\" said Nick Natola.\n\nOne mom joined the men, speculating about whether the crimes might be considered consensual. \"Yeah, I know one of their parents; the girl says she's 'totally in love' with the pervert, so we know she wanted it. And I heard the other charges were trumped up. It was just backrubs with their shirts off. Believe me, I remember being that age. I knew exactly what I was doing.\"\n\nSeveral dads wondered if some special accommodations could be made. Mainer had this to say: \"Couldn't they just give Coach Miller one of those ankle bracelets? I mean, for this to happen mid-season? Did you see the looks on the girls' faces?\"\n\nOne dad, Donald McPherson, a professor of cultural and gender studies at Carlson University, termed the arrest a \"classically American witch hunt, a puritanical sex panic that you would never see in Europe. . . . And good God, what a coach.\"\n\nAs a Level 2 sex offender, Miller will now have to register with local police if he returns to the community. When confronted with that possibility, no parent was ready to accept Miller back as a neighbor. But as a coach, they feel like they could work around it.\n\n\"You know, the parents could take turns monitoring him every minute he's with the girls,\" said Natola. \"Let's face it. The guy's a douche. But he's got a gift. One of those we'll-never-see-the-likes-of-him-again-once-in-a-\u00adgeneration kind of gift. To throw that all away, jeez, I don't know. It just doesn't seem right.\"\n\nDad Loves Carving Ducks, Parenthood\n\nSuburgatory, USA\u2014A local dad says parenthood \"is the best thing that ever happened to him,\" as he made his way to his basement shop to spend the evening carving ducks.\n\n\"Hi, kids!\" Steve Anthony patted five-year-old Parker and three-year-old David as they watched the PBS show Arthur.\n\nOnce in Anthony's shop, this reporter saw twenty or so carved ducks in various states of completion.\n\nAnthony said this about his role as a dad. \"I never thought that I could have this much love, this much patience for my kids.\"\n\nHe fired up his Dremel Multi-Max power drill, which, Anthony says, is when the kids usually come down to say hello. Sitting at his table with the Dremel motor whirring and his kids standing silently at the door, Anthony yelled over the sound of the power tool: \"I love that I'm able to share this beloved hobby with my children and that they'll be able to share it with their children.\"\n\nThe kids then returned to Curious George and turned up the volume due to the noise of the Dremel. \"Mallards, buffleheads, egrets, you name it. . . .\" Anthony said, as his wife arrived in the shop with his dinner, which he takes nightly during carving. \"The kids know all the ducks' names by now, what kind of feathers they have, what makes them distinct from one another. I'm so lucky to have them and the kids.\"\n\nSeveral hours later, Anthony had completed a mallard. \"The boys are just such a dream. Every moment with them is just a new way to see the world, through their eyes.\" The boys arrived at the shop to say goodnight to their father. \"Goodnight, boys! I love you!\"\n\nParker said, \"I love you, too, Dad.\" David said, \"Bufflehead Bufflehead Bufflehead!\"\n\n\"Aren't they just little miracles?\" Anthony said, smoothing his hand over several of his finished ducks.\n\nSHOUT OUT\n\nLet's Do that Key Party Right the Next Time\n\nDavid Dowd is a polyamorous swinger who lives on Larrabie Street, the house with the purple door.\n\nTo say I was disappointed about the conduct at and disintegration of my long anticipated key party last Saturday night would be a colossal understatement. This is not a \"hobby\" for me, or a one-time lark. So I take to the Shout Out in hopes of making sure the next key party is done right.\n\nSome ground rules:\n\nFirst, just because I am overweight, a bit hairy, and missing that one tooth (which I'm getting replaced, by the way), you cannot put the key you chose back in the bowl hoping to draw someone else. Number one, you really hurt my dignity. Number two, you denied yourself a banging the likes of which would erase every sexual experience you've ever had. Your loss. But if you don't want to lose out again, take my words seriously.\n\nSecond, my parties are not for bored moms who watch Cougar Town or Desperate Housewives and want to do something naughty. Or say they did something naughty at the PTO meeting. Because when they actually get to my home, even when dressed like the tackiest forty-year-old prostitutes I've ever seen, they are not willing to get down to action at all. This violates the spirit of swinging. I don't care that you're hot. If you're not going to let me or another of your neighbors give it to you, and give it to you hard, you could be Angelina Fucking Jolie herself and I'd still kick you out of my house.\n\nThird, it's a party for swingers, not exhibitionists. To that couple who just came in to fuck in front of us, we say, take your sick pathology elsewhere! You need to hand your partner over to a complete stranger for sex, and she will do the same with you. That's swinging with integrity.\n\nFourth, we don't have daycare for your kids. Please, get a goddamn babysitter. There's only so long we can stick kids upstairs to watch Fanboy and Chum Chum. Now this party was a big bust, but most parties are not and they are quite loud. You want your kids to hear that, dirty bastards?\n\nNow, I'm left with shattered expectations and more Fritos and leftover seven-layer dip than I would ever eat in my life. So next time, please, come committed, come with condoms, or don't come at all.\n\nPurchased Breast Milk   \nTainted by McDonald's\n\nSuburgatory, USA\u2014Breast-feeding activists are up in arms after one working mother purchased breast milk later found to be contaminated by McDonald's.\n\nThe mother, Tinsley Maher, corporate lawyer for Proskauer Rose, purchased what she called \"premium\" milk at four dollars per ounce, after her own milk supply dried up. \"Good God, I wasn't going to give Maeve formula. I work eighty hours a week. I couldn't pump. So I found some on ThanksfortheMammaries.com. And then this happens.\"\n\nMaher felt like her daughter \"just wasn't her unique, singular self when I was reading a bedtime story to her one night on Skype. Maeve's essence just wasn't there.\" She suspected the breast milk might be to blame and had the Board of Health test the samples in her freezer. The Board subsequently found traces of the Angus Beef Third Pounder, Sausage Biscuit, and, most troubling to Maher, the McRib. \"McRib. Nanny Elvie has been giving my baby girl liquid McRib. In a bottle.\"\n\nMaher is particularly troubled because the source of the milk came from within the well-known local breast-feeding collective and progressive social group called the Titty Tribe.\n\n\"I mean, if this came from a disadvantaged person, someone for whom McDonald's is part of their culture, what their people do, well, then I would understand. Even if maybe she was a poor single mother or something, and was desperate for money and sold her milk. I understand that deep socioeconomic disparities can drive people to do sickening, crazy things, like eat McDonald's,\" Maher said. She noted that she was a sociology major at Smith and had read Nickel & Dimed by Barbara Ehrenreich on the flight to a client meeting. \"Proskauer Rose has a great program every year where we paint festive murals [in poor urban neighborhoods] and clean up their filthy playgrounds,\" Maher said.\n\n\"But this was 'premium milk'! The mom is part of the Titty Tribe, for Chrissake! She eats at McDonald's, and then sells her revolting, corporate, pesticide and growth hormone\u2013filled milk to me?\" Authorities say they would prosecute if they could, but there are no laws on the books that pertain to McDonald's-tainted breast milk. \"There will be soon enough,\" said Maher. \"I didn't go to Harvard Law School for nothing.\"\n\nThe mother who sold the milk refused to give her name. She has weaned her own child but is still lactating and thought she was, in her words, \"paying it forward by selling my precious mothers' milk.\" She says in her defense that she does not normally eat at McDonald's, but that pumping breast milk makes her \"lose her mind\" and \"eat like an animal.\" \"I only went through the drive-thru a few times,\" she said. \"I never thought it would show up in a test. I just hope I didn't do Maeve any lasting damage. I'll never do it again.\"\n\nPAID ADVERTISER CONTENT\n\nSober Sully's\n\nThe First Bar Specifically Designed   \nfor the Recovering Alcoholic\n\nWe at Sober Sully's applaud your decision to take control of your life and quit drinking. But we also know what helped get you in that situation in the first place: your insatiable, truly unquenchable appetite for a little drama. And at Sober Sully's, we provide the drama you crave without the alcohol and resulting DUIs, marital heartbreak, and end-stage liver failure.\n\nStarting at 8:00 p.m. each night, our \"bad behavior\" is professionally facilitated by actors who will turn Sober Sully's into that veritable crazy train of dysfunction you loved so much. Loved too much! Whether it's Rolling Thunder Night, Toxic Gays Night, or Saturday Sluttytown, Sober Sully's will engineer, say, the unexpected biker brawl or the sloppy kiss with the age-inappropriate stranger or the bathroom line bitch-fight. Hair will fly and we can guarantee there will be a nipple slip or two. But what happens in Sober Sully's stays in Sober Sully's and absolutely no alcohol is involved in any way. No law enforcement needed! Other than maybe a few actors who look like the police, because we know how much you aging suburbanite drunks love and miss the rush of a good bust.\n\nSober Sully's\u2014\"One day at a time\" just got a lot more interesting. Join us, and bring your sponsor, too!\n\nMom of Eight Amazingly Taut,   \nExcept for Vagina\n\nSuburgatory, USA\u2014An energetic mother of eight is a local legend at her fitness club for her \"amazingly taut\" body, but she has revealed exclusively to this reporter that her vagina is not included in this characterization.\n\n\"Oh no no, are you kidding? God has allowed me to birth eight beautiful babies the way He intended! I have to wear a cup to keep that thing from flip-flapping around at the gym!\"\n\nLaurie Bishop is well-known in the community for being pregnant for much of the last fourteen years, during which she has maintained her lithe body until the very end of the last trimester. She is also known as an active member of the Junior League, a resident of the exclusive Westgate community, and an avid churchgoer who abstains from alcohol.\n\n\"We've been so blessed, so blessed. God sent Sofia, Fernando and Paco, Sylvia, and Oksana to us,\" Bishop said, referring to the household staff. \"And Beeta. Ugh I always forget Beeta. Well, whatever, they are my superstars. I could never manage Team Bishop without them!\"\n\nDoes her husband mind the slackness of her vagina? \"My husband? You mean that useless piece of skin attached to my credit card? Ha! No, I shouldn't say that. Bryan is the leader of our family and our relationship comes before everything, including the kids, like the Bible says it should blah blah blah. I just let him do his business and it may take a little longer but hey I can always go over the kids' day plans in my head so it's no big deal at all, really! An hour and he's done. And then, if God thinks it's right, we get another precious baby.\"\n\nBishop could afford vaginal reconstructive surgery, and indeed has received enhancements to her face and breasts. But she has no interest in vaginoplasty because she views her gaping birth canal as her \"gift to God,\" and also because her husband doesn't complain. \"Complain? Bryan? You're confusing him with a guy who has balls.\"\n\nWhile at the pharmacy, Bishop said, \"And anyway, this is what my Lord God Jesus Christ put me on Earth for. Oh yes, I'm picking up for Bishop? Yes \"Klon-o-pin.\" The stuff that knocks you out?\" With that, Bishop muttered \"fucking morons\" before brightening. \"No, I don't have any questions, thanks. God loves my vagina for what it's done, its joyous bounty. Would I like the flubbering and the occasional whistle to stop? Of course I would. I wouldn't be human if I didn't. But other than that, I wouldn't change a thing.\"\n\nFour-Year-Old Gets   \nPerez Hilton as \"Manny\"\n\nSuburgatory, USA\u2014An old-school couple unwittingly hired celebrity blogger Perez Hilton as a \"manny\" for their four-year-old son, who in turn schooled them all in his special brand of queeny humor.\n\n\"Well, we know him as Mario. I just thought Mario had a lot of energy and fun hair and I enjoyed his whimsical outfits,\" said mother Susan Topping Huntington, who was born into one of the area's oldest and most prominent families, and married into another. Mario Lavandeira is Perez Hilton's real name.\n\n\"He was like a performer I'd seen who did gorgeous origami at the holiday benefit for the kids at the library. My salon guy, Jeffrey, really recommended him. And I loved that he was a Mario who spoke such perfect English,\" said Huntington, unaware that Hilton's main job is reveling in Hollywood gossip on his heavily trafficked website, and until relatively recently, drawing cocaine and semen drawings across the faces of targeted celebrities.\n\n\"He did seem a bit, well, a bit of a dandy. But he is a rather tall man, and these days aren't like the old days. We really have, how should I put this delicately, quite a lot of financial interests that make it crucial for Robert to have a man guard him rather than the traditional Scottish nannies we grew up with.\"\n\nInvestment banker Peter Huntington soon noticed some odd behavior from Robert. \"He kept talking about 'Perez' and 'Perezers.' One day he ran into the great room and asked me who was more\u2014what was the word he used?\u2014'fierce,' I think. 'Who was more fierce, Miz Dita Von Teese or Miz Kylie Minogue?'\" Both parents assumed that these were teachers or specialists at the exclusive preschool Robert attends. They are actually a burlesque star and an aging pop singer, respectively, both lionized as divas by Hilton.\n\nIt was Peter Huntington who finally discovered Hilton's true identity while attending a benefit for at-risk youth. Featured was Hilton's anti-bullying testimonial for \"It Gets Better.\" He also learned that Hilton had recently toned down his site and had stopped drawing those apparent cocaine and semen scribblings.\n\nWhen confronted by the Huntingtons, Hilton told them, \"You caught me! Took you hella long to figure it out!\" And when asked why he decided to become a manny, Hilton said it was part of his reinvention from viciously catty to more mainstream. \"I got tired of drawing cum and coke on people's faces. No more calling Jennifer Aniston 'Maniston.' I was tired of being the cattiest 'Queen of all Media' and wanted something real, authentic. And Robert's a great kid. He totally brought the ferocity on some of La Lohan's captions. He loves RiRi [Rihanna] and Selenita [Selena Gomez] as much as I do. And I just got a cameo on Glee!I realized pretty fast, dahlings, this bitch can't quit.\"\n\nPAID ADVERTISER CONTENT\n\nFrom the Offices of \"Dr.\" Victor Brown\n\nDo you want a psychotherapist who will maintain appropriate boundaries, YAWN? Or do you want your sessions to feature the crackling sexual tension you might have seen between Paul and Laura in the HBO series In Treatment? If you're like most bored, mildly depressed housewives, I'm guessing it's the latter.\n\nAnd I, \"Dr.\" Victor Brown will take your \"erotic transference\" and run with it! My practices have run me afoul of the American Psychological Association, and thus I am no longer a certified therapist, nor can I travel beyond the town line without notifying authorities. But I can assure you that you will leave our sessions tingling and exhilarated. Now, I may be a little more Abe Vigoda than Gabriel Byrne, but I can still daddy you up with the best of them. Call the doc, and let this \"Paul\" take care of all your needs, \"Laura.\" One hour will never fly by so fast.\n\nChild with \"Mullet\" Pressured   \nto Leave School\n\nSuburgatory, USA\u2014A six-year-old child has been pressured to leave his private school because his \"mullet\" hair style has been deemed \"disruptive\" and \"disturbing\" by school officials.\n\nOfficials at Hamilton Knoll Academy refused to comment on the matter, but several parents, who all requested anonymity, described the controversy. \"No one here has one of those trashy mullets! My kid kept coming home and asking me about it. 'Mommy, [name withheld to protect the child] has this really short hair in the front and long in back! What is that, Mommy?' I don't want my child to have to know what a mullet is.\"\n\nThis reporter met with several concerned mothers at a local Stitch-n-Bitch meet-up on Central Street. \"Look, I just got done explaining what Diwali is to my Eamon. It was so confusing for him. And for me, frankly. Now I have to explain what a mullet is? I seriously didn't think anyone still had those!\"\n\nAnother parent tried to explain to her daughter that this child came from \"another culture.\" The girl asked, \"What culture, Mom? Is he Native American?\" The parent remarked, \"What was I supposed to say? The kid's white trash? My daughter knows culture. She's been to Venice!\"\n\nThe attendants of the Stitch-n-Bitch gathering were asked to describe the parents of the child in question, who were unavailable for interview. In the words of one mom: \"You know, it's really weird. They can afford the tuition, but they have a dumpy car with a public radio sticker, and so for a while I thought, 'Oh, they must be lesbians! That's why the kid has a mullet!' But they weren't lesbians. I saw the dad at Open House.\"\n\nThe mothers were nearly unanimous in agreeing that it would have been much easier if the parents had, in fact, been lesbians, as this culture has already been well examined in the typical kindergarten curriculum.\n\nAnother mother who attended that open house said: \"Yeah, I was thinking before I saw him, 'Maybe he's a famous ex-hockey player? And has a family history of hockey hair mullets?' But he was a total schlub.\"\n\nOne mother disputed the notion that the family was white trash. \"No way. Not with that public radio sticker. I heard they don't even have cable. And their last name sounds Jewish. Jews aren't white trash. But a Jew with a mullet? I mean, I'm at a complete loss with that one. Could they be hipsters? Is it an ironic mullet?\"\n\nShe agreed that regardless the mysterious cultural background of the parents, the child's mullet was an unwelcome distraction for her child. \"McDermott could not stop staring at it. He's got ADHD. It's hard enough for him to keep it together in the classroom. I'm just glad the head of the lower school agreed to do something about it.\"\n\nPAID ADVERTISER CONTENT\n\nFriend of Dorothy Salon\n\nIn times such as these, in the age of Glee and Lady Gaga, we at Friend of Dorothy Salon know that there are no greater accessories for the modern woman than having that one fabulous gay friend, and that perfect, stylish haircut. At Friend of Dorothy we give you both for the same competitive price.\n\nWe at Friend of Dorothy know that in life, it's not always easy to find a fabulous gay friend on your own, someone to reference in conversation to your straight friends. At Friend of Dorothy, we will guarantee that one of our fantastic stylists can be that friend.\n\nHaven't you always wanted to wow those drab straight friends with lines like this: \"Well, my fabulous stylist friend Dave was hooked on meth for years\u2014he calls it 'Tina'\u2014he said you could have sex for days on end! But after a lot of rehab, and the love of a good boyfriend, he's OK now . . . \" or \"My gay friend Scott, he's so fabulous, he just saw my hair and said 'girlfriend, you call that shaggy animal hair?! That looks like overgrown pubes on your head! Let's fix that bad boy for you!'\"\n\nYour fabulous gay stylist friend will also joke about how hot your husband is, but actually he's not kidding about that part. Every six weeks, you'll get your one-hour dose of fab, and the hottest hair design available within fifty miles. Why do we know it will be the hottest design? Because your Friend is gay! And very, very gay, we might add.\n\nSo call us now and get your own fabulous friend at Friend of Dorothy.\n\n*Note: Friend of Dorothy does not discriminate, but we don't employ lesbian stylists, not because of their orientation, but because the modern straight woman simply doesn't consider them catty or bitchy enough. This bias does not reflect the views of Friend of Dorothy Salon. We know plenty of bitchy lesbians.\n\nChild Can't Convince Mom She's Beautiful Inside and Out\n\nSuburgatory, USA\u2014A local mom insists she is \"fat, hairy, and hideous,\" despite the protestations of her son, who tells her she is beautiful \"just the way she is.\"\n\n\"Mommy, why are you crying?\" said six-year-old Brian Gardner to his mother, Judi Gardner, who was hand-\u00adtweezing some stray lip hairs after a wax appointment earlier in the day. \"I'm just getting rid of the hair, honey, no big deal. It's just . . . just gross!\"\n\nBrian was troubled by this. \"But I like you a little fuzzy, Mommy! Like here!\" Brian touched his mother's arm. \"Ugh! See, even you noticed my freak arms. Mrs. Sasquatch. I have to do something about that. Maybe laser.\"\n\n\"Sabers? Like Star Wars?! Mommy, that sounds scary,\" said Brian.\n\n\"No honey, LASER. It's not scary. It just burns all the hair off with a scorchy light and you just have to wear special glasses and make sure to never, ever look at the light. Not scary at all!\" Gardner said, as she boarded her home elliptical machine and Brian settled into the chair and snack tray he sets up during her lengthy sessions.\n\n\"But Mommy, I thought you said that God loves all of us just the way we are!\" said Brian. \"Yeah right, God's never seen this giant flat ass of mine,\" Judi muttered.\n\n\"What, Mommy?\" asked Brian.\n\n\"Nothing honey, you're right. God does love us just the way we are. But other people aren't nice like God is,\" she said breathlessly, as she increased the resistance on the elliptical. \"You know when you see those other mommies at school, when they're guest reader or volunteering at the library, how good they look?\"\n\n\"Mommies are beautiful, all mommies, but you're the most beautifullest in the whole world,\" said Brian.\n\n\"Well, I will be, after I get something called dermabrasion,\" Judi said, panting.\n\n\"What's that, Mommy?\" Brian said.\n\n\"Oh, it's where they sand your yucky face off.\" Brian looked terrified. \"No, Mommy! Your face is my most favorite face!\"\n\n\"Honey, it's OK, I'll still have my face. It will just be as soft as a baby's bottom.\"\n\nBrian started to cry. \"I don't want your face to look like a butt! That's where poop comes from.\" Judi tried calming him down but continued on the elliptical, saying, in short bursts, \"Sweetie . . . I meant that . . . it will be super soft . . . you'll love it.\"\n\nBrian got up from his chair and tried to play with his mother's stomach while she was exercising.\n\n\"Soft like your fluffy tummy, Mommy? It's so warm and squidqy.\" Judi paused the elliptical, got off, and appeared triumphant.\n\n\"You see? Out of the mouths of babes. I knew I was fat. Kids don't lie to you about that stuff\u2014they tell it just like it is.\" She ruffled Brian's hair. \"I'm so glad I have him around to keep me honest. Brian, who's the best little man in the world? You are! Someday I know you'll meet a wonderful, beautiful girl who'll love you exactly like you are, just like your Mommy does.\"\n\nMom Literally Dragged   \nBack to Suburbs\n\nSuburgatory, USA\u2014After two days of traipsing around Manhattan reliving her carefree single-girl days, an area mother had to be literally dragged back to the suburbs.\n\n\"Ma'am. It's time,\" said New York City police officer Peter Clark. \"Noooo!!!!!!! NOOOOOOO!!! Get your hands off me!\" wailed a belligerent Trink Giroux as Officer Clark heaved her into the back of the car. \"Watch your head getting into the cruiser, ma'am! Did you just bite me?\"\n\nGiroux was reported missing by her husband one night ago. She joins hundreds of mothers in recent months who have fled their homes and responsibilities, leaving their husbands utterly unprepared to handle their children's hygiene, homework, and food requirements.\n\n\"Fucking God. That was like right out of The Sopranos when they took Adriana for a 'drive' in the country,\" said Officer Clark, shaking his head in contempt and looking for teeth marks in his hand. \"She was fighting it hard, all right, real hard.\"\n\nGiroux was picked up while trying to pose as a twenty-something hipster at a Lower East Side club, which she thought was Tonic. But Tonic has closed since she lived there. A new club is in its place, and after noting her dowdy attire, dated haircut, and weary, medicated expression, the actual twenty-something hipster manning the door immediately notified police that there was an escaped mom inside the establishment. \"What are you saying to me?\" she kept asking the young man. \"What? I don't get what you're saying.\" It appeared that he was speaking mostly in the unaffected style of Mumblecore movies, a trend Giroux has utterly missed because of parenthood. She found the hipster's speech patterns incomprehensible, which enraged her further.\n\nFrom the back of the police cruiser, Giroux tried to explain leaving her family and home to Officer Clark. \"I just wanted to turn back the clock, just for a day or two. No whining kids, no clueless husband, no 'accident reports' from the preschool. NO goddamned OLIVE GARDEN. NO PLAYDATES.\"\n\n\"Do you realize what you've done, Mrs. Giroux? You left your little kids in the hands of their father. The school said one of your kids came in with a single mitten and another with no scarf. No juice boxes either. Your husband thought Fruit-by-the-Foot was real fruit. Their field trip forms, remember those? No, you don't, do you? And you don't care. You're disgusting. You are disgusting to me.\"\n\n\"You're right, you're right, I am disgusting,\" Giroux said, weeping as they made the inevitable slog back to town through the commuter traffic. \"I just wanted to go back. But you can't go back, can you? Not even for a visit.\"\nBack-stories\n\nOkay, I didn't need to be literally dragged back to the suburbs, like \"escaped\" mom \"Trink Giroux,\" but let's just say that a lot of the material in this book was based on personal experience. I wish I could dish even more dirt I've heard over the years, but I have to be careful in giving you the back-stories, at the advice of counsel. (I've always wanted to say that! It sounds like I've arrived!) So if this stuff sounds very much like \"me me, more about me!\" the reason is simple: I don't have to worry about suing myself.\n\nThe \"escaped mom\" piece and the one that led the book, \"Atheist Mom So Lonely She Accepts Christ,\" were among the very first I wrote back in 2007\u2014when I was still praying to Brooke Shields, the patron saint of deeply depressed mommies. I was then and still am an atheist. Sort of like, \"God? Come again? Oh yeah, THAT dude!\" But loneliness is a powerful motivator, and it was a mom friend, Liz, who first suggested going to a Baptist church playgroup\/\"parenting classes\" by saying, \"I'm a Jew from Jersey, and I love it!\" At first, this idea seemed\u2014hmm, what's the formal term\u2014'ass-stupid'? Still, I began going, and sure enough, this battle-scarred ex-Catholic got an unexpected religious education. The Christ I knew from childhood seemed like He was saying, \"Look what a scary mess I am because of you, sinner! Guilty now? Good!\" But at this Church, their Christ looked serene, as if He were fresh from yoga class.\n\nI was constantly stepping in shit, yet they always politely looked away. I asked one of the ladies, \"Oh, what's Ryan going to be for Halloween?\" And she said, haltingly, \"Well, we don't celebrate Halloween.\" I looked over at my son, who was wearing a snaggle-toothed pagan pumpkin, and my friend Naoko's son, sporting a merry white skeleton, and I thought, \"Great, we just paraded our boys around their church in Satan-shirts.\"\n\nI was bowled over by their hospitality to me. But I still felt like a fraud, hiding my lack of faith and boundless social liberalism, and I wondered if at, say, a gay rights rally or pro-choice rally, would my new friends be on the other side of the line? Did I care?\n\nWell, I do care very much, as a long-standing fag hag, about gay rights. In \"Lesbian Hamsters 'Just Grew Apart'\" I am definitely the annoying \"Flora\" who foists my homophilia on my child. We have a really old game of Life (free from the town dump) with the classic blue and pink pegs, and I actually did tell my son that \"a blue peg should be able to marry a blue peg if they love each other.\" Now my son asks his friends when playing, \"Do you want to marry a blue peg or pink peg?\"\n\nYou might notice, ahem, a little anger in places over breast-feeding. When I told an (ex) pediatrician that breastfeeding wasn't working for me, while looking disheveled and ready to careen off a bridge, she looked at me as if I had served her a turd on a plate. I wanted to say, \"You know, a near-suicidal mom is quite the problem, too, lady, and breast-feeding is making it worse.\" Just a note, ex-ped: My kid has no allergies, so here's a heartfelt \"fuck you\" to you.\n\nI have spoken to at least a dozen women about the guilting they've gotten over breast-feeding, and for a while I got sucked into a real online breast-feeding collective, which I called the Titty Tribe in \"Purchased Breast Milk Tainted by McDonald's.\" I found it fascinating how obsessed they seemed to be with breasts\u2014their own and other women's. They seemed to be mostly highly-educated, third-wave feminists. I was flabbergasted that a few seemed willing to take even untested bodily fluids from perfect strangers rather than bottle-feed, as I mention in \"Wolf Blitzer: Live from the Lactation Room.\"\n\nClosely allied to the breast-feeding guilters are the natural birthing guilters, the target of my ad for \"C-Secrets,\" a business that will give you a believably \"natural\" birth story to throw off finger-waggers. This is in tribute to friends Colleen, Kate, and all my many, many other C-section moms who've been told the only \"true\" way to have a baby is through your vagina. Sure, C-section rates do seem excessive. (Call that the Ricki Lake Business of Being Born concession.) But that doesn't mean women should demean one another about such a private experience that the vast majority probably had little choice in making.\n\nYou might notice Wal-Mart gets a starring role a few times in the book. I'm intrigued by Wal-Mart's role in affluent suburbia as a class divider. Few friends will admit to going there. \"Terry Gotlieb\" in \"Woman Shops at Walmart to Feel 'Pretty, Thin'\" was actually based on a woman who one day described Wal-Mart shoppers as if they were sub-human. Wal-Mart is the single most diverse place I visit in suburbia. Chuck E. Cheese comes in second.\n\nI also get my cheap on at the town dump, described in \"Join My Weirdo Junior League!\"\u2014which is almost entirely true, except for Dumpster-diving and the very end, when my friend Laura notices I'm wearing her discarded shirt. That actually did happen, but I got the shirt at her school thrift store, where I've gotten literally $1,200 worth of clothes in one bag for, oh, thirty bucks.\n\nLaura is the unexpectedly sane leader in \"PTO Stunner: New President 'Not a Power-Mad Psycho.'\" Now, the PTO leaders at my son's school have been fantastic\u2014nothing at all like the evil PTO terrorist Emily \"Bin Fahdin.\" But that supermarket ambush by a crazed PTO honcho actually happened to a friend in another corner of Suburgatory. This piece was also inspired by working parent friends who feel shut out of PTO. I see a lot of lip service given to the idea of \"Can't we all just get along, mommies?\" But the fact is, I see a whole lot of judgin' going on everywhere. That's what inspired \"Mommy War Combatants Embrace Mutually Assured Destruction.\" I so hate it when women tear each other apart.\n\nOh, who am I kidding? I love it!\n\nReally, though, I wished I could have worked more into the book for working moms, but I'm a stay-at-home mom. It's what I know, it's pretty much who I know, and many pieces reflect my efforts to combat whiny white mommy malaise. Besides the church, I did become an IKEA regular and a mall semi-regular who bought nothing but took massage chair breaks, not unlike the \"Dad Forcibly Removed from Mall Massage Chair.\" And I am the pathetically eager, unstoppable Facebook queen lambasted in \"Mom Crushed to Learn that Facebook Isn't Job.\" If you can believe it, not one but two apparently single Pakistani mariners did friend me through the Jewish Maritime Historical Society. But they are not learning how marvelous the US is: All they're learning about American women, from me at least, is that we dress and speak like whores.\n\nYou might notice race comes up quite a lot, and race is certainly whispered about in the very white towns I've lived in. The playground encounter described in \"'Funny Racist Lady' Enchants Prominent Black Townsman\" actually happened to a friend. She is in no way racist, and I doubt the famous black athlete would be \"enchanted\" if she was. But my friend did think the other park-goers were being racist for subtly pointing at them. And he did invite her home for take-out, which is when she figured out he was a superstar and apparently a very nice one at that.\n\nOther moments I can cop to include an acquaintance referring to Indians as the \"New Jews\" because of their fierce determination to succeed. It had that strain of admiration plus disgust I see in anti-Semitism. The Ice Cream Man really is universally hated by my parent-friends, no matter his religion or color. But I did indeed meet someone who talked about one of them\u2014a \"brown\" man of indeterminate ethnicity\u2014as if he was a gypsy at best or a terrorist at worst, saying, \"Just who ARE these people?\"\n\nOne of the fun things about writing this book was getting back the very insightful copyeditor notes from the whip-smart Imee Curiel. In a couple spots, she said, \"Come on, this is just far-fetched.\" But in the grand cliche of fact being stranger than fiction, these instances were actually real. There is indeed a high-end car in town with a bumper sticker that says Had Enough?, which is what inspired \"Mercedes- Driving Dad Dreams of Easier Life for His Children.\" It was a different fancy make of car I couldn't place. (Being Super Crazy Mega Cheap, I don't know a thing about new cars; my own cars are old enough to start cramming for their PSATs.)\n\nI wanted to tailgate this guy, to ask, \"Had enough of what?\" Inherited wealth? Profound luck? Because I can say without hesitation that the vast majority of people I've met who live in affluent suburbia got here by growing up affluent, marrying someone affluent, or getting themselves advanced degrees through hard work but also because they won the IQ lottery\u2014better known as luck. That's how I got here: luck. So, yes, rich dad apparently fed up with your enviable life, I've had quite enough. Enough of you and other rich people complaining endlessly about their taxes.\n\nOne person who endlessly complains to me is a delightfully inappropriate mom friend determined to snag an invite to a supposed swinger party held each year on Halloween. I really did think that key parties were suburban folklore, but I've since been convinced that, while surely a teeny-tiny subculture, they actually do exist. I have no doubt that if we did go, we would be the moms who are all talk and no walk that swinger \"David Dowd\" complains about in his Shout Out. Though I would be all over his seven-layer dip.\n\nI do love my trashy food, and at some point, obsessive \"foodie culture\" began to both annoy and alarm me. It alarmed me to think that the healthiest food seemed to be becoming the sole province of the affluent, which is what inspired \"Dog Fed Better than Scholarship Child, Says School Nurse.\" And like the character \"EatMyShit\" in \"Waitress Wages Anti-Foodie Jihad on Chowhound,\" I became irked by constant Facebook pictures of everyone's spectacular, one-of-a-kind dinners. My own response to this is on Facebook was a \"Moms Against Food Porn\" picture series I did of really gross crap food sitting around my kitchen.\n\nAnd there's quite a lot in here about women aging and loathing their own bodies. I go through phases when I become obsessed with one topic, and for a while it was that photo-collage \"Faces of Meth\" described in \"Woman with Eating Disorder Considers Meth.\" I showed it to everyone I knew, and no fewer than three mom friends looked at those ghoulish faces, paused, and then said, essentially, \"Wow, meth really makes you lose a lot of weight, huh?\" The fact is, I thought the same thing and hated myself for even thinking that.\n\nAnd I leave you with one more self-loathing incident that I didn't explain fully in \"Child Can't Convince Mom She's Beautiful Inside and Out,\" because I thought no one would believe it. I did go through a midlife crisis a few years ago, the cheapness was very briefly tossed aside, and I actually bought laser hair removal\u2014bikini line\u2014on an impulse buy. It came with a special bonus: micro-dermabrasion! The day I went to redeem my \"bonus,\" I brought my son. (Too cheap for baby-sitting, but not for lasering? Hypocrite!) When I got there, they said, \"So sorry, we have you in for lasering your bikini line, and you'll be charged if you cancel.\" So, yes, dear reader, my son sat in the corner, oblivious, wearing oversized protective glasses that kept slipping off his face, while his mommy sat on a table, legs spread, getting her bush lasered off. At least, I thought he was oblivious, until I heard him say, \"Mommy, why are your pants on the floor?\" \nAcknowledgments\n\nI had this fear that some of my work might seem, at times, woman-hating, so I'm happy to see how many actual women I have to thank here.\n\nTo my sisters Joellen, and especially on this project, Terry (plus Benno and Ron), who pushed me to get a backbone.\n\nTo the amazing Laura Yorke and Carol Mann at the Carol Mann Agency in New York, to Steve Fisher at APA in Los Angeles, and to attorney Eric Rayman. And thank you to Willow Bay, for connecting me to Laura Yorke. Laura saw just a few writing samples in 2009 and said, \"You must write something.\"\n\nLaura placed this book with another superstar, Lara Asher. Besides being a terrific, turbo-charged editor, Lara got the material instantly and is just a lovely, spunky force. And thanks to project editor extraordinaire Meredith Dias for her expert execution.\n\nTo beloved college friend, architect David Tabenken. David named this book, and I am only sad his wonderful mom, Helen, along with my own Mommy, aren't here to see David's title appear on national television. To my Aunt Nancy, Uncle Matt and J. Spellman, among the few people left who knew and loved my Mommy and Daddy.\n\nTo my favorite Reuters anchor, Jen Rogers, who fought hard alongside my sister in making this project happen. I consider Jen a sister in everything but blood. The same goes for my dearest college friends\u2014Rachel Laiserin, Lisa Epstein Jay, Tina Hohn Schissel, Geri Clark, and Ashley Gravelle Morse\u2014as well as CNN's finest, Rachel Brown (who brought me sisters-in-crime Peg Rettino and Joanna Joplin) and all-around producer\/ball-buster Amber Briscoe. And to my honorary brothers: scary-smart journalist Chris Nolter, my first friend at CNN and someone I still treasure all these years later, and the brilliant Dr. Scott Schissel.\n\nTo bosses and boss-friend hybrids who accepted my considerable idiosyncrasies over the years: Steve Rosenbaum\/Pam Yoder, Kathleen Campion, Kathy O'Hearn, David Bohrman, Jen Zeidman Bloch and Gene Bloch, the late Jeff Gralnick, Andy Breslau, Warren Kozack, Terry Baker, the irrepressible star-producer Kara Kasarjian, and especially Jenny Harris, a force of nature I was blessed to sit next to and learn from for years. Jenny makes that Holly Hunter character in Broadcast News look like a slacker.\n\nTo Hillary St. Pierre of Baldie's Blog, whose courage against cancer leaves me slack-jawed daily, and to her son Xander, who's even braver than she is.\n\nTo Arianna Huffington, who, at a dark moment confidence-wise, responded to an email seeking help in, oh, forty-five minutes? Thank you.\n\nTo some of the real parents of Suburgatory, the friends I'm lucky to have in town: Adam, Alex, Alexandra, Carl, Dan, Doug, Glenn, Glynis, Jen, Jeremy, Jim, Julie, Laura, Liz and Liz, Mark, Michele and Michele, Molly, Nancy, Naoko, Sandra, Sarah and my book club ladies, Suzanne, Valerie, and Yuki. I left out last names, in case I get run out of the state and some of you get run out with me.\n\nFinally, the two groups people who made Suburgatory bearable besides my actual family and helped save me from postpartum depression: my Facebook family (those I know IRL, and those I don't). Wish I could name you all. You deserve your own sitcom. I'm working on it.\n\nAnd, a diaspora of comrades who have fed me ideas and friendship and I needed a lot of both for this book: Adam, Alda\/Bo, Alex, Alexandra, Alicia, Allison, Amanda, Ami, Amy, Andre, Andrea, Angela\/Paul, Anne, Annemarie, Antonella, Antonetta, Bruce, Caitlin, Caleb, Cheryl, Chris, Christine, my \"Church Ladies,\" Claire, Clint, Colby, Colleen, Dan, Dana, Daniel, Dave, Debbie\/Ed, Derek\/Manon, Diane, Dianne, Eddie, Elaine, Elizabeth\/Kati, Ellen\/Tom, Eric, Erin, Evelyn, Evelyn\/Andy, F & E, Fitz, Flip\/Tom, Frankie, Fred, George, Gretchen, Gudveig, Harold, Hasmeena, Iain, Ildi, Inger, Jackie, Jagger, all the Jennifers, Jess\/Dan, Jessica\/John, Jill, Jim\/Geoff, Jodi, Johanna, Jonathan, JTL, Judi, Judy, Julia, Julie, Karen, Kate, Katey, Kelli, Kellie\/Jens, Kerry, KimChi, Kirsten, Kirstin, Kitty, Laura, Leanne, Leslie, Leyea, Lindsey, Lisa, Liz, Lobo, Luis, Lynne, Marc, Marcia, Marcy, Marina, Mary Anne, Melissa, Merideth, Merlynda, Michelle, Nancy, Nicole, Pat, Penelope, Peter, Rebekka, Rev, Ripu, Rob, Robin, Roo, Robin\/Lee, Roo, Russell, Ruth, Sally, Sarah, Sarav, Scott, Shane, Sharon, Sherman, Steve\/Kyle, Sue, Susan, Susswein, Suzanne, Tanya, Tarun, Theresa, the Thibodeau and Morlock cousins, Tina, Toby, Tod, Todd, Tom, Tommy, Tony, Trader Guy, Tricia, Vanessa, Varman\/Sarwat, and Wendy. Love to all! \nAbout the Author\n\nLinda Erin Keenan spent seven years as a CNN senior producer\/head writer for top anchors including Anderson Cooper, Aaron Brown, Willow Bay, Stuart Varney, and Lou Dobbs during 9\/11, the Afghanistan and Iraq Wars, the Dot Com rise and fall, several key national elections, and the New York City Blackout. Before that, she was at Bloomberg TV. It was 9\/11 that pushed her to re-evaluate her 24-7 lifestyle, eventually trading it all in for baby and suburbia. Her first-person essays have appeared in the Boston Globe Magazine and the Huffington Post. She invites readers, especially the craziest ones with lots of problems, to friend her on Facebook.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Al Haines.\n\n\n\n\n[Illustration: Cover]\n\n\n\n[Illustration: Carolina Lee]\n\n\n\n                              CAROLINA LEE\n\n\n                                   By\n\n                              LILIAN BELL\n\n\n                    Author of \"Hope Loring,\" \"Abroad\n                  with the Jimmies,\" \"At Home with the\n                            Jardines,\" etc.\n\n\n\n                    With a frontispiece in colour by\n\n                           DORA WHEELER KEITH\n\n\n\n                                NEW YORK\n                           A. WESSELS COMPANY\n                                  1907\n\n\n\n\n                           _Copyright, 1906_\n                        BY L. C. PAGE & COMPANY\n                             (INCORPORATED)\n\n                         _All rights reserved_\n\n\n\n\n                          I DEDICATE THIS BOOK\n                              TO MY FRIEND\n\n                           Ella Berry Rideing\n\n                     AS AN AFFECTIONATE RECOGNITION\n                 OF THE EVIDENCES OF HER BEAUTIFUL WORK\n                        AND LOVE FOR ME AND MINE\n\n\n\n\n                                CONTENTS\n\n\nCHAPTER\n\n      I. Captain Winchester Lee\n     II. The First Grief\n    III. The Danger of Wishing\n     IV. The Turn of the Wheel\n      V. Brother and Sister\n     VI. The Stranger\n    VII. Mortal Mind\n   VIII. Man's Extremity\n     IX. The Trial of Faith\n      X. Cross Purposes\n     XI. In Which Truth Holds Her Own\n    XII. Whitehall\n   XIII. Guildford\n    XIV. Kinfolk\n     XV. The Blind Baby\n    XVI. A Letter from Carolina\n   XVII. In the Barnwells' Carryall\n  XVIII. A Letter from Kate\n    XIX. The Fear\n     XX. Moultrie\n    XXI. The Light Breaks\n   XXII. In The Voodoo's Cave\n  XXIII. Loose Threads\n   XXIV. The House-party Arrives\n    XXV. Bob Fitzhugh\n\n\n\n\n                              CAROLINA LEE\n\n\n\n                               CHAPTER I.\n\n                         CAPTAIN WINCHESTER LEE\n\n\nHaving been born in Paris, Carolina tried to make the best of it, but\nbeing a very ardent little American girl, she always felt that her\nforeign birth was something which must be lived down, so when people\nasked her where she was born, her reply was likely to be:\n\n\"Well, I was born in Paris, but I am named for an American State!\"\n\nThen if, in a bantering manner, her interlocutor said:\n\n\"Then, are you a Southerner, Carolina?\" the child always replied:\n\n\"My father says we are Americans first and Southerners second!\"\n\nColonel Yancey, himself from Savannah, upon hearing Carolina make this\nreply commented upon it with unusual breadth of mind for a Southern man,\nwith:\n\n\"I wish more of my people felt as you do, little missy.  Most of my\nkinfolk call themselves Southerners first and Americans second and are\nprouder of their State than of their country.\"\n\n\"I don't see how they can be,\" said the child with a puzzled frown\nbetween her great blue eyes. \"It would be just as if I liked one hand\nbetter than my whole body!\"\n\nWhereat the colonel slapped his leg and roared in huge enjoyment, and\nwent to Henry's to drink Carolina's health and to tell the Americans\nassembled there that he knew a little American girl that would be heard\nfrom some day.\n\nAll this took place in Paris, when General Ravenel Lee, Carolina's\ngrandfather, was ambassador to France, and when her father, Captain\nWinchester Lee, was his first secretary.\n\nMany brilliant personages surrounded the child and influenced her more\nor less, according to the fancy she took to them, for she was a magnetic\npersonality herself, and accepted or rejected an influence according to\nsome unknown inner guide.\n\nHer mother was a woman of refinement and breeding, and to her the child\nowed much of her good taste and charmingly modest demeanour. But it was\nher father who captured her imagination.\n\nOne of her earliest recollections was of her father's voice and manner\nwhen she looked up from her novel and asked him why he did not spell his\nname Leigh as men in books spelled theirs.\n\nShe had not known her father very well, so she was totally unprepared\nfor his reply.  Although she had been but a little child, she could see\nhis face and hear his voice as distinctly to-day as she did when he\nwhirled around on the hearth-rug and looked down at her as she sat on a\nlow stool with a book on her knees.\n\n\"Spell my name Leigh?\" he had said, in a tone she never had heard him\nuse before.  \"Child, you little know what blood flows in your veins, or\nyou would thank God every night in your prayers that you inherit the\nname of Lee, spelled in its simplest way.  Honest men, Carolina, pure\nwomen, heroes in every sense of the word; statesmen, warriors, brave,\nwith the bravery which risks more than life itself, are your ancestors.\nThey date back to the Crusaders, and down the long line are men of title\nin the old world, distinguished in ways you are too young to understand.\nBooks, did you say? Your name appears in many a book, child, which\nrecords heroic deeds.  On both your dear Northern mother's side and\nmine, you come of blood which is your proudest heritage.  Were you poor\nand forced to earn your daily bread, you would still be rich in that\nwhich the world can never take away--good blood and a proud name.  And\nremember this, too, little daughter, although your life has been spent\nin foreign lands, I loved America so well that I gave you the name of my\nnative State, and my dearest wish is to restore Guildford and to pass\nthe remainder of my life there.\"\n\nIt was a long, long speech for a little girl to remember, but it burned\nitself into her memory and kindled her pride to such a degree that she\ncould hardly wait to tell some one of her newly discovered treasure.\n\nFortunately her first auditor happened to be her governess, and\nfortunately, also, her father chanced to overhear her as she translated\nhis remarks into shrill French.  He immediately stopped her, and these\nwords also were seared into her memory through poignant mortification.\n\n\"I was wrong to tell you that, little daughter. I see that you are too\nyoung to have understood it properly.  I can only undo the mischief by\nreminding you never to boast of your old family to any one.  If we\nSoutherners have one fault more than another, it is our tendency to\nmention the antiquity of our families--as if that counted where breeding\nwere absent.  You will observe that your dear mother never mentions\nhers, though she is a De Clifford.  Let others boast if they will.\nSpeak you of their family and name and be silent concerning your own.\nIt is sufficient to feed your pride in secret by the inward knowledge of\nwho you are. Will you try to remember that, little daughter, and forgive\nme for putting notions into that head of yours?\"\n\nShe flew into his arms, and in that moment was born the passionate love\nand understanding which ever afterward existed between them.\n\n\"Oh, father!\" she cried.  \"Don't be sorry you told me!  I am not too\nyoung.  I will show you that I am not.  I will never speak of it again,\nand only in my heart I will always be proud that I am Carolina Lee!\"\n\nIn after years, Carolina dated her life--her most poignant happiness and\nher dearest anguish--from the moment when her father thus opened his\nheart to her and she found how intensely they were akin.  He became her\nidol, and she worshipped him not only with the abandonment of youth, but\nwith all the passion of her tempestuous nature.  She set herself to be\nworthy of his love and companionship with such ardour that she\nunwittingly broke the first commandment every day of her life.\n\nHer father realized it, perhaps because of his answering passion, for he\noften sighed as he looked at her.  He knew, as did no one else, what an\ninheritance was hers.  He felt in his own bosom all the ardour and\npassion and furious love of home which as yet his child only suspected\nin herself. As long as he could remain at her side he felt that he could\ncontrol it in both, but his heart sometimes stood still at the thought\nof what could happen were Carolina left defenceless.  How could the\nchild battle with her own nature?  He shook his head with his fine smile\nas he realized how more than competent she was to fight her own battles\nwith an alien.\n\nThey saw a good deal of Colonel Yancey in those days.  He had some\nbusiness with the French government which kept him abroad or going back\nand forth, and because of his companionable qualities, his sympathy as\nwell as his brilliance, Captain Lee discussed his most intimate plans\nwith him.\n\nCarolina always made it a point to be present when her father and\nColonel Yancey smoked their cigars in the library after dinner, for\nthere it was that conversations took place concerning the South and\nGuildford, of so breathless an interest that not one word would she\nwillingly have missed.\n\nShe had a confused feeling concerning Colonel Yancey which she was too\nyoung to analyze.  He was only a little past forty, and had won his\ntitle of colonel in the Spanish war.  She knew that her father, like\nmost Southern men, trusted Colonel Yancey, simply because he also was a\nSouthern man, when he would have been cautious with a Northerner.  He\nspoke freely of the most intimate plans and dearest hopes of his life,\nwith all the hearty, generous, open freedom of a great nature. Yet the\nwatchful child saw something in Colonel Yancey's eyes, especially when\nher father spoke of Guildford, and his passionate hope of the part it\nwould play in Carolina's future, which reminded the little girl of the\nlook in the gray cat's eyes when she pretended to fall asleep by the\nhole of a mouse.\n\nThis feeling was too intangible for her to realize at first, but as\nyears passed by, and Colonel Yancey's business brought him to Paris\nevery season while General Lee was ambassador, and when her father was\ntransferred to the Court of St. James, even oftener, she grew better\nable to understand her childish fears.\n\nOne day in London, when Carolina was about fifteen, Colonel Yancey made\nhis appearance, dressed in deep mourning.  Carolina did not hear the\nexplanation made of his loss, but she resented vaguely yet consciously\nthe glances he cast at her during dinner, and when her father whispered\nto her that the colonel had lost his wife and no questions were to be\nasked, her lip curled and her delicate nostrils dilated.  She listened\nwith more than her usual attention to the conversation which followed,\nand in after years it often came to her mind, and never without giving\nher some help.\n\nColonel Yancey opened the conversation with an inexplicable remark.\n\n\"When I hear you talk, captain, I always feel sorry for you.\"\n\nCarolina lifted her head with instant hauteur, but her father only\nsmiled and knocked the ashes from his cigar.\n\n\"Yes, an enthusiast of my type is always to be pitied,\" he said, gently.\n\n\"Not entirely that,\" responded Colonel Yancey. \"In some strong\ncharacters, their enthusiasms only indicate their weak points, but it is\nnot so in your case.  It is rather that you have idealized your\nhomesickness.\"\n\n\"I am homesick,\" said Captain Lee, \"for what I never had.\"\n\n\"Exactly.  Now you left Guildford when you were a mere lad, so it is\nlargely your father's opinion of the South--your father's love for the\nold place that you have inherited and made your own, just as, in Miss\nCarolina's case, it is wholly vicarious.  Have you any idea of the\ndeterioration your own little town of Enterprise has suffered?\"\n\n\"I suppose you are right,\" said Captain Lee.\n\n\"I hope, then,\" said Colonel Yancey, slowly, \"that you will never go\nback South to live, especially to Enterprise.\"\n\nCarolina's sensitive face flushed, but she was too well bred to\ninterrupt.\n\n\"You mean,\" said Captain Lee, with a keen glance at his friend, \"that I\nwould find the South a disappointment?\"\n\n\"It would break your heart!  It hurts me, tough as I am and little as I\ncare compared to an enthusiast like yourself.  It would wound you,\nbut\"--and here he turned his magnetic glance on the young girl--\"for an\nidealist like missy here, it would be death itself!\"\n\nCaptain Lee reached out and laid his hand, on his daughter's head.\n\n\"I am afraid so!  I am afraid so!\" he said, with a sigh.\n\n\"You understand me?\" questioned Colonel Yancey.  It was a pleasure,\nwhich Colonel Yancey seldom experienced, to converse with so\ncomprehending a man as Captain Lee.  He was accustomed to dazzling\npeople by his own brilliancy, but he seldom dived into the depths of his\npenetrating mind for the edification of men, simply for the reason that\nthe ordinary run of men seldom care to be edified.  But in diplomatic\ncircles, Colonel Yancey was a welcome guest.  He possessed an instinct\nso keen that it amounted almost to intuition in his understanding of\nmen, a business ability amounting almost to genius, and a philosophic\nturn of mind which permitted him to apply his knowledge with almost\nunerring judgment.  As a promoter, he had served governments with marked\nability, and had the reputation of having amassed fortunes for those of\nhis friends who had followed his lead and advice.\n\nAll this Carolina knew and yet--\n\nHowever, she had the good taste to listen further, without attempting to\ndraw a hasty conclusion.\n\n\"The South,\" said Colonel Yancey, with a sigh of regret, \"is like a\nbeautiful woman asleep--no, not asleep, but standing in the glorious\nsunlight of God, with her eyes deliberately shut.  Shut to opportunity!\nShut to advancement!  Shut to progress!  Her ears are closed also.\nClosed to advice! Closed to warning!  Closed to truth!  Her mind is\nlocked.  Locked against common sense!  Locked against the bitter lesson\ntaught by a jolly good licking.  And the key which thus locks her mind\nis a key which no one but God Almighty could turn, and that is\nprejudice!  Blind, bitter, unreasoning, stupid prejudice!  That is why\nher case is hopeless!  That is why fifty or a hundred years from now the\nSouth will still be ignorant, stagnant, and indigent!\"\n\n\"But why?  Why?\" cried Carolina, carried quite out of herself by her\nexcitement.\n\n\"I beg your pardon!\" she added, flushing.\n\nColonel Yancey whirled upon her, delighted to have moved her so that she\nspoke without thinking.\n\n\"Why?  My dear young lady--why?  Because she spends half her days and\nall her evenings fighting over the lost battles of the Lost Cause.\nBecause she still glories in her mistakes of judgment! Because, almost\nto a man, the South to-day believes in the days of '61!\"\n\n\"Do they still talk about it?\" asked Captain Lee.\n\n\"Talk about it?\" cried Colonel Yancey.  \"Talk about it?  They talk of\nlittle else!  They dream about it!  They absorb it in the food they eat\nand the air they breathe!  Every anniversary which gives them the ghost\nof an excuse they get up on platforms and spout glorious nonsense, which\nis so out-of-date--so prehistoric that it would be laughable, if it were\nnot pitiable--as pitiable as a beautiful woman would be who paraded\nherself on Fifth Avenue in hoop-skirts and a cashmere shawl.  You lose\nsight of even great beauty if it is clad in garments so old-fashioned\nthat they are ludicrous.\"\n\nAs Colonel Yancey paused, Captain Lee said, with a quiet smile:\n\n\"And yet, Wayne, haven't I heard you breathe fire and brimstone against\nthe 'damned Yankees,' and when they come South to invest their capital,\ndon't you feel that they are legitimate prey?\"\n\nColonel Yancey rose to his feet and strode around the room for a few\nmoments before replying.\n\n\"Well, Savannah has had her fill of them, I think.  Perhaps I do\nconsider the most of them damned Yankees, but believe me, captain, in\nthe first place, we Southerners fully believe that they deserve that\ntitle, and in the second place, we don't want them!  No, nor their money\neither!  Let them stay where they are wanted!\"\n\n\"Ah-h!\" breathed Winchester Lee.  \"Who now has been talking beautiful\nnonsense which he didn't in the least subscribe to?\"\n\n\"There!  There!\" said Colonel Yancey.  \"It is a temptation to me to\nfollow the dictates of my brain, but my heart, Winchester, is as\nunreconstructed as ever!  After all, I am no better than the rest of\nthem!\"\n\n\"But why do they--do you all feel that way?\" asked Captain Lee.  \"I\nassure you from my soul that I do not.\"\n\n\"I know you don't.  But you have had strong meat to feed your brain upon\nduring all these years. The rest of us have had nothing to feed our\nintelligence upon except the daily papers--and you know what they are.\nOur intellects are ingrowing, and have been for years.\n\n\"It is difficult for you to believe this, captain, and almost impossible\nfor missy.  But let me explain a bit further.  For nearly forty years\nthe South has been poor, with a poverty you cannot understand, nor even\nimagine.  There has been no money to buy books--scarcely enough to buy\nfood and clothes.  The libraries are wholly inadequate.  Consequently\ncurrent fiction--that ephemeral mass of part-rubbish, part-trash, which\nmany of us despise, but which, nevertheless, mirrors, with more or less\nfidelity, modern times, its business, politics, fashions, and trend of\nthought--is wholly unknown to the great mass of Southern people. The few\nwho can afford it keep up, in a desultory sort of way, with the names of\nmodern novelists and a book or two of each.  But compared to the\nomnivorous reading of the Northern public, the South reads nothing.\nTherefore, in most private libraries to-day, you find the novels which\nwere current before the war.\n\n\"Now take forty years out of a people's mind, and what do you find?  You\nfind a mental energy which must be utilized in some manner.  Therefore,\nafter a cursory knowledge of whatever of the classics their grandfathers\nhad collected, and which the fortunes of war spared, you find a\ncommunity, like the Indians, forced to confine themselves to narratives\nhanded down from mouth to mouth.  It creates an appalling lack in their\nmental pabulum.\"\n\n\"Are they conscious of this?\" asked Captain Lee.  He had been following\nColonel Yancey with the closeness of a man accustomed to learn of all\nwho spoke.  Carolina had hardly breathed.\n\n\"In a way--yes!  In a manner--no!  The comparative few who are able to\ntravel see it when they return, but years of parental training have bred\na blind loyalty to the mistakes of the South which paralyzes all outside\nknowledge.  Even those who see, dare not express it.  They know they\nwould simply brand themselves as traitors.\"\n\nCarolina opened her lips to speak, then closed them again.  She had been\ntrained as a child to have her opinions asked for before she ventured\nthem. Her father, who always saw her with his inner eye, whether he was\nlooking at her or not, said:\n\n\"You were going to say something, little daughter?\"\n\n\"I was only going to ask Colonel Yancey if they would not welcome\nsuggestions from one of themselves?\"\n\n\"Welcome suggestions, missy?  They would welcome them with a shotgun!\nTake myself, for instance.  I have travelled.  I am supposed to have\nlearned something.  I and my family have been Georgians ever since\nGeorgia was a State.  Yet when I notice things which my fellow citizens\nhave become accustomed to, and suggest remedying them, what do I get?\nAbuse from the press!  Abuse from the pulpit!  Abuse from friends and\nenemies alike!\"\n\n\"What did you say, colonel?\" asked Captain Lee, smiling.\n\n\"Why, I noticed the shabbiness of my little city--and a well-to-do\nlittle city she is.  Yet half the residences in town need paint.\nSouthern people let their property run down so, not from poverty, but\nfrom shiftlessness.  _You_ know, captain!  It is the Spanish word\n'_manana_' with them.  The slats of a front blind break off.  They stay\noff!  Paint peels off the brickwork.  It hangs there.  A window-pane\ncracks.  They paste paper over it.  A board rots in the front porch.\nThey leave it, or if they replace it, they don't paint it, and the new\nboard hits you in the eye every time you look at it.  They decide to put\non an electric door-bell.  In taking the old one off they leave the hole\nand never think of the wildness of painting the door over!  They just\nleave the hall-mark of untidiness, of shiftlessness, over everything\nthey own.  And if you tell them of it?  Well!\"\n\n\"I see,\" said Captain Lee.  \"I have often wondered why Northerners\nalways spoke of the South as such a shabby place.  They must have meant\nwhat you have just described--a lack of attention to detail.\"\n\n\"You have noticed it yourself?\" asked Colonel Yancey, eagerly.\n\n\"You must remember that I have not been south of Washington for thirty\nyears.\"\n\n\"Ah, yes, I remember.  You had the luck to be in the Civil War.\"\n\n\"I was in it only the last two years before the surrender.  I enlisted\nwhen I was fourteen, was a captain at sixteen, and was wounded in my\nlast engagement.\"\n\n\"And you've never been back since?\"\n\n\"Never!\"\n\nColonel Yancey leaned back and sighed.\n\n\"Never go, then!\" he said.  \"Take my advice and never go.  Remember your\nbeautiful unspoiled South as you see her in your dreams!\"\n\n\"The South is like a petted woman who openly declares that she would\nrather be lied to agreeably than be told the truth to, objectionably,\"\nsaid Captain Lee, with a regretful smile.  Then he added, with a\nmischievous glance at Carolina, \"Do the ladies still--er--gossip,\nColonel Yancey?\"\n\nThe colonel simply flung up his hands.\n\n\"Gossip?  My God!\"\n\nIt was Carolina who rebuked him.  Her voice was grave, but her eyes\nflashed fire.\n\n\"Do Southern ladies gossip more than Parisian or London ladies?\"\n\n\"Fairly hit, colonel!\" said Captain Lee.  \"To answer that truthfully,\nyou must admit that they do not, for nothing can equal the malice of\nParis and London drawing-rooms.\"\n\n\"Quite right, captain.  No, missy,\" he answered, \"it is only because we\nexpect so much more of Southern ladies that their gossip sounds more\nmalicious by way of contrast.\"\n\nCarolina smiled, well pleased by the brilliant tact with which he always\nextricated himself from a dilemma.\n\nWhen Colonel Yancey had gone, Captain Lee put one arm around Carolina's\nshoulder, and with the other hand tilted the girl's flowerlike face up\nto his, with a remark which, if he had made it to his son, would have\nchanged the whole current of the girl's life.  He said:\n\n\"Ah, little daughter, the colonel is like all the rest of the\nSoutherners.  He can see the truth and can spout gloriously about her,\nbut in a money transaction between himself and a Northern man, he would\nforget it all, and would consider it no more than honest to 'skin the\ndamned Yankee,' to quote his own language.\"\n\nAnd with that the subject was dropped.\n\nThe Lee household at that time consisted of Captain and Mrs. Lee, the\ntwo children, Sherman and Carolina, and the widow of a cousin of Captain\nLee, Rhett Winchester, whom they called Cousin Lois.\n\nMrs. Winchester had abundant means of her own, which were all in the\nhands of the Lee family agents, and she was distinguished by her\nidolatry of Carolina.  No temptation of travel, no wooing of elderly\nfortune hunters, had power to move her.  All the love which in her early\nlife had been given to her husband, relations, and friends, she now\npoured out on the child of her husband's cousin. She had been denied\nchildren of her own, which, perhaps, was just as well, as she would have\nruined them with indulgence.  Mrs. Winchester was a born aunt or\ngrandmother.  She took up the spoiling just where a mother's firmness\nceased.\n\nShe cared very little for Sherman, who was three years older than\nCarolina, and who resembled his Northern mother as closely as Carolina\nmodelled herself upon her father, except that Sherman was weak, whereas\nMrs. Lee, as a De Clifford of England, inherited great strength of\ncharacter as well as a calm judgment and a governable quality, which\nmade her an admirable helpmeet for the fiery, if controlled, nature of\nher Southern husband.\n\nNever was there a happiness so complete as Carolina's seemed to be.  She\ngrew from a beautiful child into a still more beautiful young girl.  She\nabsorbed her education without effort, learning languages from much\ntravel and from hearing them constantly spoken, and breathing in the\ntruest culture from her daily surroundings.  How could an intelligent\ngirl be ignorant of art and science and literature and diplomacy when\nshe heard them discussed by some of the greatest minds of the day as\ncommonly as most children hear continual conversations about the\nshortcomings of the servants? She did not realize that she was unusually\nequipped because it had been absorbed as unconsciously as the air she\nbreathed, but other American girls who came into contact with her felt\nand resented it or admired it, according to their calibre.\n\nIn religion Carolina was outwardly orthodox and conventional, but many\nwere the discussions she and her father held on the subject, in strict\nprivacy, and many were the questions she put to him which he could not\nanswer.  He often ended these interrogations by gathering her up in his\narms and saying: \"My little girl will need a new religion, made\nespecially for her, if she continues to trouble her head about things\nwhich no man knoweth!\"\n\n\"But why don't they know, dearest?  And why does the Bible contradict\nitself so?  And how can God be a 'father' if he sends pain and sickness\nand death?  Is He any worse than a real father would be?  And why does\nHe not answer prayers when He promises to?  And when did the healing\nJesus taught His disciples disappear?  Did He only let them possess the\npower for a few years?  Why are we commanded to be 'perfect' when God\nknows we can't be?  And how can you believe in a God who punishes you\nand sends all manner of evil on you while calling Himself a God of\nLove?\"\n\n\"Carolina!  Carolina!  You make my head swim with your heresies!  I\ndon't know, child!  I don't know the answer to a single one of your\nquestions. Such things do not trouble me.  I believe in God, and that\nsatisfies me.\"\n\n\"No, it doesn't, daddy!\" cried the girl, astutely, \"but you try to make\nyourself believe that it does.\"\n\n\"Then try to make yourself believe it, dear.  It has done me very well\nfor nearly forty years.\"\n\nAnd as usual, such footless discussion ended in nothingness and a burst\nof human love which effectually put out of mind all gropings after\nDivine Love!\n\n\n\n\n                              CHAPTER II.\n\n                            THE FIRST GRIEF\n\n\nThen, with no illness to prepare her for so awful a blow, with nothing\nbut a stopping of the heart-beats, Carolina's father fell into his last,\nlong sleep, and before she could fairly realize her loss, her mother\nfollowed him.\n\nWithin six weeks, the girl found herself orphaned and mistress of the\ngreat Lee fortune, but utterly alone in the world, for her grandfather\nhad died the year previous and Sherman had just married and gone back to\nAmerica.\n\nThat Carolina felt her mother's loss no one could doubt, but the change\nin the young girl wrought by her father's death was something awful to\nbehold.  She had not dreamed that he could die.  He was so young, so\nstrong, so noble, so upright, such an honour to his country and to his\nrace!  Why should perfection cease to exist and the ignorant, wicked,\nand common live on?  Carolina resisted the thought with tigerish\nfierceness, and openly blasphemed the God who created her.\n\n\"God my father?\" she stormed at Cousin Lois, who listened with blanched\nface and trembling fear of further vengeance on the part of outraged\nDeity. \"Why, would my own precious father send me a moment of such\nsuffering as I have passed through ever since they took him away from\nme?  He would have given his life to save me from one heart-pang, and\nyou ask me to believe that God is a father, when He sends such awful\nanguish into this world?\"\n\n\"He sends it for your good, Carolina, dear,\" pleaded Cousin Lois.\n\n\"Oh, He does, does He?  He thinks it will do me good to suffer?  _Daddy_\nthought so, didn't he? Daddy _liked_ to make me unhappy, didn't he?  He\ndidn't realize how blissful heavenly love could be, so he only loved me\nin a poor, blind, earthly fashion, which made every day a joy and every\nhour we spent together a song!  Poor daddy!  To be so ignorant of the\nreal way to love his children!\"\n\n\"Oh, Carolina!\" moaned Mrs. Winchester.\n\n\"God hates me, Cousin Lois,\" said the girl, dropping her impassioned\nmanner and speaking with bitter calmness.\n\n\"I have been recognizing it for some time.  I have felt that He was\njealous of my happiness. You know it says: 'For I the Lord thy God am a\njealous God.'  He admits it Himself.  So He took vengeance on me through\nHis power and killed my parents just to show me that He could! But if He\nthinks that I am going to kneel down and thank Him for murder, and love\nHim for ruining my life--\"\n\nA steel blue light seemed to blaze from the girl's eyes as she thus\nraised her tiny hand and shook it at her Creator.\n\nCousin Lois burst into tears.  Carolina viewed her without sympathy.\n\n\"I am so little,\" she said, suddenly.  \"It is a brave thing for God to\npit His great strength against mine, isn't it?  Listen to me, Cousin\nLois, I am done with religion from now on.  I will never say another\nprayer as long as I live.  The worst has happened to me which could\nhappen. Nothing more counts.\"\n\nIt was while she was in this terrible state of mind that Mrs. Winchester\ntook charge of her.\n\nSherman and his wife came over for the funeral of their father, and\nbefore they could so arrange their affairs as to be able to leave for\nhome, they were called upon to bury, instead of try to console, their\nmother.\n\nNeither Carolina nor Mrs. Winchester liked Adelaide, Sherman's wife.\nShe was selfish and ignorant, but, with true loyalty to their own, they\nnever expressed themselves on the subject, even to each other.  After\nthe period of mourning was over, they accepted her invitation to visit\nher, and spent a month in New Work.  Then, with no explanation whatever,\nMrs. Winchester and Carolina went abroad and travelled--travelled now\nfuriously, now in a desultory way; now stopping for one month or six;\nnow hurrying away from a spot as if plague-stricken--all at Carolina's\nwhim.\n\nIt was a strange life for an ardent young American to lead, but Noel St.\nQuentin and Kate Howard, who knew Carolina best, shook their heads, and\nfancied that the two travellers found in Mrs. Sherman Lee their\nincentive to remain away from America so long and so persistently.\n\nMrs. Winchester and Carolina were an oddly assorted pair, but their very\ndissimilarity made them congenial.\n\nMrs. Winchester was a woman who merited the attention she always\nreceived.\n\nAt first sight she did not invariably attract, being stout, asthmatic,\nvague of manner, and of middle age.  She had her figure well in hand,\nhowever, large though she was.  Her waist-line, she was fond of saying,\nhad remained the same for twenty years, though the rest of her had\noutgrown all recollection of the trim young girl she doubtless had been.\nBut it was her complexion of which she was most proud.  It was still a\nblending of cream and roses, and her blush was famous.\n\n\"Carolina, child,\" she used to say, \"don't let me be ridiculous, just\nbecause I am large.  Promise me that you will never leave crumbs on my\nbreast, even if they fall there and I can't see them.  If you only knew\nhow I suffered from not knowing where all of me is.  Why, with my\nfigure, it is just like the women we used to see in Russia with little\ntables on each hip and a tray around their necks.  Don't laugh, child.\nIt's dreadful, my dear.\"\n\n\"Well, but Cousin Lois, it wouldn't be so bad if you wouldn't pinch your\nwaist in so.  Just let that out and you will find yourself falling into\nplace, so to speak.\"\n\n\"What!\" cried Mrs. Winchester.  \"Lose the only--the only thing I have\nleft to be proud of, except my complexion?  Carolina, you are crazy. I'd\nrather never draw another comfortable breath than to add one inch to my\nwaist-line.  No, Carolina.  Don't advise me.  Just watch for the crumbs.\nFor I will not be guilty of the inelegance of tucking a napkin under my\nchin if I ruin a dress at each meal.\"\n\nThus it will be seen that Mrs. Winchester was quite determined in spite\nof the gentlest manner of putting her ultimatum into words.\n\nShe carefully cultivated her asthma, as, without affording her too much\ndiscomfort, it was always an excuse to travel.\n\n\"Asthma is the most respectable disease I know of,\" she often said to\nCarolina.  \"Gout is more aristocratic, but so uncomfortable.  Asthma is\nrefined and thoroughly convenient, besides always forming a safe topic\nof conversation, especially with strangers.\"\n\n\"That makes it almost indispensable for persistent travellers like us,\ndoesn't it?\" said Carolina.\n\n\"Well, you may get tired of hearing about it, but with me it is always a\ntest of a person's manners.  When a stranger says to me 'How do you do,\nMrs. Winchester?'  I don't consider him polite if he makes that merely a\nform of salutation.  I want him to stand still and listen while I answer\nhis question and tell him just how I feel!\"\n\nShe also had a slight cast in her eye, which added to this gentleness\nand likewise led the casual observer to suspect her of vagueness of\npurpose, but her intimates made no such mistake.  The mere fact that one\nof her light gray eyes was not quite in line with the other rather added\nto her attractions, for if her features and manner had carried out the\nsuggestions of her figure, she would have been a formidable addition to\nsociety instead of the charming one she really proved.\n\nShe habitually wore light mourning for the two excellent reasons she\nherself gave, although General Winchester had been dead these twelve\nyears.\n\n\"In the first place,\" she always said, when Carolina tried to coax her\nto leave off her veil at least in warm weather, \"mourning is so\ndignified, especially in the chaperoning of a young and charming girl.\nIn the second place, age shows first of all in a woman's neck, try as\nshe may to conceal it.  In the third place, a large woman ought always\nto wear black if she knows what she is about, and as to my bonnet always\nbeing a trifle crooked, as you say it is, well, Carolina, little as I\nlike to say it, I really think that is your fault.  It would be so easy\nfor you to keep your eye on it and give me a hint.  I only ask these two\nthings of you.\"\n\n\"I'll try, Cousin Lois,\" Carolina always hastened to say, \"though really\na crooked bonnet on you does not look as bad as it would on some women.\nIf you can understand me, it really seems to become you--it looks so\nnatural and so comfortable.\"\n\n\"Now, Carolina, that is only your dear way of trying to set me _a mon\naise_!  As if a crooked bonnet ever could look nice!\"\n\nYet she cast a glance into the mirror as she spoke, and seeing that her\nbonnet was even then a point off the compass she forebore to change it.\nSuch graceful yielding to flattery was in itself a charm.  But the thing\nabout Mrs. Winchester, which proved a never-failing source of amusement\nto the laughter-loving, was her amusing habit of miscalling words.  She\nhabitually interpolated into her sentences words beginning with the same\nletter as the term she had intended, as if her brain had been switched\noff before completing its thought and her tongue did the best it could,\nleft without a guide.\n\n\"Carolina,\" she would say, \"come and look up Zurich on the map for me; I\ncan't see without my gloves.\"\n\nIn her hours of greatest depression this trait never failed to amuse\nCarolina, and when, on one occasion, Cousin Lois took the tissue-paper\nfrom around a new bonnet, folded the paper carefully and put it in the\nhat-box and threw the bonnet in the waste-basket, Carolina laughed\nherself into hysterics.\n\nCarolina was genuinely fond of Cousin Lois, but it must be confessed\nthat one great secret of her attractiveness for the girl was because\nmuch of Cousin Lois's early childhood had been spent at Guildford, when\nshe had been a ward of General Lee's, and thus had met his nephew, Rhett\nWinchester, whom she afterward married.\n\nThus, while not related to their immediate family, Cousin Lois was\ninextricably mixed up with their history and knew all the traditions\nwhich Carolina so prized.\n\nAlthough Mrs. Winchester deplored Carolina's persistence in so dwelling\nupon the past and brooding over her loss, nothing ever really interested\nthis girl except to talk about her father or the golden days of\nGuildford.\n\nShe cared nothing for her wealth.  She shifted the burden of investing\nit upon Sherman's shoulders, and refused even to read his reports upon\nits earnings.\n\nAdmirers failed to interest her for the reason that she was unable to\nbelieve that they sought her for herself alone.  Her fortune had the\neffect upon her of keeping her modest concerning her own great beauty.\n\nBut grief and a rooted discontent with everything life has to offer will\nmar the rarest beauty and undermine the most robust health, and the\nchange struck Colonel Yancey with such force when he met them in Rome\nthat he became almost explosive to Mrs. Winchester.\n\n\"The girl is losing her beauty, madam!\" he said.  \"Look at the healthful\nglow of your complexion and then look at her pale face!  Her eyes used\nto dance!  Her lips were all smiles!  Her cheeks were like two roses!\nAnd what do I find now?  A sneer on that perfect mouth!  Coldness,\ncruelty, if you like, in those eyes!  Why, madam, it is a sin for so\nbeautiful a creature as Miss Carolina to destroy herself in this way.\nShe might as well shoot herself and be done with it!  What does she\nwant?\"\n\n\"She wants what she can never have, Colonel Yancey,\" said Mrs.\nWinchester, sadly.  \"Carolina wants her father to come back.\"\n\n\"We all want that, madam!\" said the colonel, gravely.  \"I no less than\nthe others.  His loss never grows less.\"\n\nWhen Cousin Lois repeated this conversation to Carolina, she laughed at\nwhat he said about her beauty, but flushed with gratitude at his praise\nof her father, and was so kind to the colonel for two days afterward\nthat he proposed to her again and so fell from grace, as he persisted in\ndoing with somewhat annoying regularity.\n\nThey travelled for another year, and Carolina grew no better.  She\nseldom complained, but her lack of interest in everything, added to her\nrestless love of change, preyed upon Mrs. Winchester.\n\nThey were in Bombay when this restlessness got beyond control.\n\n\"I am not happy!\" she cried, passionately, \"and knowing I ought to be is\nwhat makes me even more miserable!\"\n\n\"What you need is a good dose of America,\" said Cousin Lois, decidedly.\n\"You are homesick!\"\n\n\"I believe I am!\" she answered, with brightening eyes.  \"I am homesick,\nthough, for something in America which I've never found there.\"\n\n\"You are homesick for South Carolina,\" said Cousin Lois, with timid\ndaring.\n\nAt these words a look came into Carolina's eyes which half-frightened\nMrs. Winchester, for Carolina had suddenly recalled her father's words.\n\n\"My dearest wish is to restore Guildford, and pass the remainder of my\ndays in the old place.\"\n\nInstantly her life-work spread itself out before her.  Here was the\nsolution to all her restlessness, the answer to all her questionings of\nFate, the link which could bind her closer to her beloved father! If he\ncould have spoken, she knew that he would have urged her to give her\nlife, if need be, to the restoration of Guildford.\n\nHer interest in existence returned with a gush. A new light gleamed in\nher eyes.  A new smile wreathed her too scornful lips.  Her face was\nirradiated by the first look of love which Cousin Lois had seen upon it\nsince her father's death.\n\nThey began to pack in an hour.\n\n\n\n\n                              CHAPTER III.\n\n                         THE DANGER OF WISHING\n\n\nThe Lees' dinner-table was round, and about it were gathered six\npeople--Sherman and his wife, Carolina, Mrs. Winchester, Noel St.\nQuentin, and Kate Howard, Carolina's most intimate girl friend. It was\nthe first time they had all met since the return of the travellers from\nIndia.  Later they were going to hear Melba in \"Faust,\" but there was no\nhurry.  It was only nine o'clock.\n\n\"Carolina, if you could have the dearest wish of your heart, what would\nit be?\" asked Noel St. Quentin.\n\n\"If I should tell, it might not come true,\" Carolina answered.  \"And I\nwant it so much!\"\n\n\"I never saw such a girl as Carolina in all my life,\" complained her\nsister-in-law.  \"Her mind is always made up.  She keeps her ideas as\norderly as an old maid's bureau-drawer.  No odds and ends anywhere.  You\nmay ask her any sort of a question, and she has her answer ready.  She\nknows just what box in her brain it is in.  Just fancy having thought\nout what your wish would be, and having it at your tongue's end to tell\nat a dinner-party!\"\n\nMrs. Lee leaned back and fanned herself with a fatigued air.\n\n\"You almost indicate that Carolina thinks,\" said St. Quentin.\n\n\"Oh, don't accuse me of such a crime in public!\" cried the girl,\nlaughing.\n\n\"Carolina seems to me the one person on earth whose every wish had been\ngratified before it could be uttered,\" said St. Quentin, who was in some\noccult way related to the Lees.  \"I would be interested to know just\nwhat her dream in life could be.\"\n\nCarolina smiled at him gently.\n\n\"She--she's had Europe, Asia, and Africa a-all her life,\" cried Kate\nHoward, who always stuttered a little in the excitement of the moment.\nTo Carolina this slight stutter was one of Kate's greatest fascinations.\nYou found yourself expecting and rather looking forward to it.  At least\nit spelled enthusiasm.  \"She's had masters in every known\naccomplishment.  She--she can do all sorts of things.  She can speak any\nlanguage except Chinese, I do believe.  She's pretty.  She's rich in her\nown right--no waiting for dead men's shoes or trying to get along on an\nallowance--a-and what under the sun can she want--e-except a husband?\"\n\n\"Perhaps, if she's good, she may even get that,\" said St. Quentin.\n\nAgain Carolina smiled.  But her smile faded when her eyes met those of\nher sister-in-law, who viewed the girl with a thinly veiled dislike.\nThe girl's eyes flashed.  Then she spoke.\n\n\"I have wanted one thing so much that I am sure sometime I must achieve\nit,\" she said, slowly. \"I want to be so poor that I shall be forced to\nearn my own living with no help from anybody!\"\n\nShe was not looking at her brother as she spoke, or she would have seen\nhim start so violently that he upset his champagne-glass, and that his\nface had turned white.\n\n\"What did I tell you?\" murmured St. Quentin.\n\n\"Carol likes to be sensational,\" said Mrs. Lee. \"No one would dislike to\nbe poor more than she, and no one would find herself more utterly\nhelpless and dependent, if such a calamity were to overtake her.\"\n\n\"I wouldn't call it a calamity,\" said Carolina, quietly.\n\n\"Yes, you would!\" cried Kate.\n\n\"I am inclined to agree with Carol,\" said St. Quentin, deliberately,\n\"and to disagree, if I may, with Cousin Adelaide.  In my opinion, Carol\ncould go out to-morrow with only enough money to pay her first week's\nboard, and support herself.\"\n\n\"I hope she may never be obliged to try,\" said her brother, harshly.\n\"Addie, if you intend to hear any of the music, we'd better be starting.\nIt is a quarter to ten now.\"\n\nAddie raised her shoulders in a slight shrug.\n\n\"When Carolina holds the centre of the stage, it is impossible to carry\nout one's own ideas of promptness,\" she said.\n\n\"Nasty old cat,\" whispered Kate to St. Quentin, as he stooped for her\nglove and handkerchief. \"Thanks so much.  I don't know how I managed it,\nbut I held on to my fan.\"\n\nLater in the Lees' box with Melba singing Marguerite, St. Quentin turned\nto Carolina again. She had swept the house with her glass as soon as the\nparty were seated, and had noted but one old acquaintance whose face\nseemed to invite study. The girl's name was Rosemary Goddard, and among\nthe discontented faces which thronged the boxes in the horseshoe, hers\nalone was peaceful. Nay, more.  It was radiant.  Carolina remembered her\nface--a cold, aristocratic mouth, disdainful eyes, haughty brows, and a\nnose which seemed to spurn friend and foe alike.  What a\ntransfiguration! How beautiful she had grown!\n\nShe was so occupied with the enigma Rosemary presented that St. Quentin\nwas obliged to repeat his question.\n\n\"How would you go to work, Carol?\"\n\nThe girl turned with a sigh.  Sometimes it seemed to her that she never\nwould become accustomed to talking at the opera.  She almost envied a\ntall young man, who stood in the first balcony.  His evening clothes\nwere of a hopeless cut. His manner was that of a stranger in New York,\nbut in his face, one of the finest she had ever seen, was such a passion\nfor music that she watched him, even while she answered St. Quentin with\na grace which hid her unwillingness to talk.\n\n\"For what I really would love to do,\" she said over her white shoulder,\nwith her eyes on the strange young man, \"you started me off a little too\npoor.  I might have to borrow a hundred or two from you to begin with!\nI want to pioneer!  I don't mean that I want to go into a wilderness and\nbe a squatter.  I want to reclaim some abandoned farm--make over some\nugly house--make arid acres yield me money in my purse--money not given\nto me, left to me, nor found by me, but money that I, myself--Carolina\nLee--have earned! Does that amuse you?\"\n\n\"It interests me,\" said St. Quentin, quietly.\n\nTo be taken seriously was more than the girl expected.  She was only\ntelling him a half-truth, because she did not consider him privileged to\nhear the whole.  She continued to test him.\n\n\"I never see an ugly house that I do not long to go at it, hammer and\ntongs, and make it pretty. Not expensive, you understand,--I've lived in\nParis too long not to know how to get effects cheaply,--but attractive.\nOh, Noel!  The ugliness of rural America, when Nature has done so much!\"\n\n\"You ought to have been a man,\" said St. Quentin.\n\n\"I would have been more of a success,\" said the girl, quickly.  \"I\nbelieve I could have started poor and become well-to-do.\"\n\n\"How you do emphasize beginning poor and how you never mention becoming\nrich!  Don't millions appeal to you?\"\n\n\"Not at all! nor do these common men, even though they did begin poor,\nwho have acquired millions by speculation.  They but make themselves and\ntheir sycophants ridiculous.  No, I mean honest commerce--buying and\nselling real commodities at a fair profit--establishing new\nindustries--developing situations--taking advantage of Nature's\nbeginnings.  Such thoughts as these are the only things in life which\nreally thrill me.\"\n\n\"I understand you,\" said St. Quentin, \"but I fear your wish will never\ncome true.  Years ago I held similar desires.  All my plans fell\nthrough. I had too much money.  And so have you.  You'll have to go on\nbeing a millionairess, whether you will or no, and you'll marry another\nmillionaire and eat and drink more than is good for you and lose your\ncomplexion and your waist line and end your life a dowager in black\nvelvet and diamonds.\"\n\nA messenger boy entered and handed a telegram to Sherman Lee, just as\nMelba rose from her straw pallet and led the glorious finale to \"Faust.\"\n\nHer brother leaned over and touched her arm.\n\n\"You may get your infernal wish sooner than you expected,\" he said, with\na wry smile twisting his pale face.\n\nCarolina turned to St. Quentin with indifference.\n\n\"Possibly I may yet keep my waist line,\" she said, as he laid her cloak\non her shoulders.\n\nOn the way out she came face to face with the tall young man who had\nstood through the whole opera, in the balcony.\n\nHe gave back all her interest in him in the one look he cast upon her\nloveliness.  A sudden light of incredulous surprise dilated her eyes and\na swift blush stained her cheeks.  She recognized, in some intangible,\nunknown way, that he possessed kindred traits with her father and with\nherself.  He had the same look in his eyes--or rather back of them, as\nif his eyes were only a hint of what lay hid in his soul.  He was of\ntheir temperament.  He dreamed the same dreams.  He was akin to her.\n\n\"I could have told him the truth,\" she whispered. \"He would have\nunderstood that I meant Guildford all the time, and that the reason I\nwant to be poor is so that I can show that I am willing to work, to\ncarry out my father's dearest wish.  Just to spend money on it is too\nsordid and too easy.  I want it to be made hard for me, just to show\nthem what I will do!  He would have understood!\"\n\nBut with one's best friends it is as well to be on the defensive, and\nnot let them know our true aims, lest they take advantage of their\nfriendship and treat our heart's dearest secrets with mockery.\n\n\n\n\n                              CHAPTER IV.\n\n                         THE TURN OF THE WHEEL\n\n\nA week later St. Quentin dropped in at Mrs. Lee's for a cup of tea.  He\nwould have preferred to have Carol brew it, for she had not only learned\nhow in Russia, but had brought with her a brand of tea which, to St.\nQuentin's mind, was not to be ignored for mere conversation, and once\ndrunk, was not to be forgotten.  When Mrs. Lee was out, Carol dispensed\nthis tea, but when Addie was in her own house, she was mistress of it in\nmore ways than tea-drinking.\n\nSt. Quentin found several people there for whom he had little use, so he\nsat silent until they had gone and no one except Kate, Adelaide, and\nCarol were left.\n\nCarol was wearing a pale blue velvet gown trimmed with sable and a\npicture hat with a long white ostrich plume which swept her shoulder.\nBoth St. Quentin and Kate plied her with admiring comments until Addie\ncould bear it no longer, and excused herself with unnatural abruptness.\n\n\"There are more ways than one of killing a cat,\" murmured St. Quentin,\nstooping for Kate's immense ermine muff, which she had dropped for the\nthird time, \"than by choking it to death with cream.\"\n\nKate laughed delightedly.\n\nCarolina turned from the doorway.\n\n\"Don't go, either of you,\" she said.  \"I am only going for some tea.\nNoel, ring for some more hot water, will you?\"\n\n\"I wonder how it would be,\" said Kate, dreamily, \"to be born without any\nrelations at all! Could one manage to be happy, do you think?\"\n\n\"Carol couldn't.  She is very fond of Sherman.\"\n\n\"I wouldn't be fond of any brother who had lost all his own fortune and\nmine and was millions in debt besides.  One couldn't love a fool, you\nknow.\"\n\n\"I know.  But do you remember what Carol said about wanting to be poor?\"\n\n\"Of course I remember!\" said Kate, \"but I d-didn't believe her then and\nI d-don't believe her now.  Carol was s-simply lying--that's the answer\nto that!\"\n\n\"Lying about what?\" asked Carolina, reentering, with a square box in her\nhand.  The box was of old silver, heavily carved and set with turquoise.\n\n\"Lying about being g-glad Sherman has lost all your money.  Of course\nyou were lying, w-weren't you?  No-nobody but a raving maniac could be\nglad to be p-poor.\"\n\n\"Then I am a raving maniac,\" said Carolina, pouring the delicately\nbrewed tea carefully into the tall, slender glasses.  \"Lemon or rum,\nKate?\"\n\n\"W-which will I like best?  I--I've had four cups already to-day.\"\n\n\"Then you'd better have rum.  It makes you sleep when you have had too\nmuch tea.\"\n\n\"Lemon for me, please,\" said St. Quentin.\n\n\"I remembered that,\" said Carolina, smiling. \"And three lumps.\"\n\n\"P-put in some m-more rum, Carol.  I can't taste it.\"\n\n\"What a Philistine!\" cried St. Quentin.  \"To insult such tea with rum.\"\n\n\"It's quite g-good,\" murmured Kate, with her glass to her lips.  \"When\ny-you have enough of it.\"\n\n\"So you really think I can't mean it when I tell you I am glad that\nSherman has lost all our money?\" said Carolina.  \"Of course I am sorry\non Addie's account--she cares a great deal and is quite miserable over\nher future prospects.  But she has ten thousand a year from her own\nestate, so she can still educate the children and get along in some\ndegree of comfort.  But as for me\"--she leaned forward in her chair with\nthe whimsical idea of testing their calibre kindling in her eyes--\"if\nyou will believe me and will not scoff, I will tell you what my plan\nis.\"\n\n\"Promise,\" said Kate, briefly.\n\n\"If Sherman can manage it, I want,\" said Carolina, slowly, but with an\nodd gleam in her eye, \"to buy an abandoned farm in New England and raise\nchickens.\"\n\nIn spite of her promise, Kate looked at the beautiful face and figure of\nthe girl in blue velvet and sables who said this, and burst into a\nshriek of laughter, which St. Quentin, after a moment's decorous\nstruggle, joined.\n\n\"I know,\" said Carolina, leaning back, still with that curious look in\nher eyes.  \"I know it sounds absurd.  I know you are thinking of me out\nfeeding chickens in these clothes.  But oh, if you only knew how tired I\nam of--of everything that my life has held hitherto.  If you only knew\nhow unhappy I am!  If you only knew how I want a farm with pigs and\nchickens and cows and horses.  If you only knew how I long to plant\nthings and see them grow.  But above everything else in the world, if\nyou only knew how I want a dark blue print dress!  I saw a country girl\nin one once when I was a child in England, and I've never been really\nhappy since.\"\n\nShe joined in the burst of laughter which followed.\n\n\"But do things grow on farms in New England?\" asked Kate.  \"And isn't\nthat just why so many are abandoned?\"\n\n\"I suppose so,\" answered Carolina, \"but those are the only ones which\nare cheap, and chickens don't need a rich soil.  All you've got to do is\nto--\"\n\n\"I'd go South,\" interrupted Kate, \"or to California, where the c-climate\nwould help some. I've read in the papers how farmers suffer when their\ncrops fail.  I--I'd hate to think of you suffering if your turnips\ndidn't sprout properly, Carol!\"\n\n\"Laugh if you want to, but I'll get my farm in some way.\"\n\n\"How about the old Lee estate in South Carolina?\" asked St. Quentin.\n\nFor the first time in his life St. Quentin was actually conscious that\nCarolina was mocking him. The thought was startling.  Why should she\ndissemble?  Carolina's face fell, and a trace of bitterness crept into\nher voice.  This seemed so natural that he forgot his curious suspicion.\n\n\"I suppose that went, too.  I haven't questioned Sherman, but he told me\neverything was gone. That, although the house was burned during the war,\nand only the land itself remained, is the only thing I regret about our\nloss.  I did love Guildford.\"\n\n\"But you never saw it!\" exclaimed Kate.\n\nCarolina's eye flashed with enthusiasm.\n\n\"I know that!  Nevertheless, I love it as I love no spot on earth\nto-day.\"\n\nThere was a little pause, full of awkwardness for the two who had\naccidentally brought Carolina's loss home to her.  To Carolina it\nbrought home a sense of real guilt.  If she had believed that Guildford\nwas lost she would have screamed aloud and gone mad before their very\neyes.  She was almost afraid to juggle with the truth even to protect\nher sacred enthusiasm from their profane eyes.\n\nIt was St. Quentin who spoke first.\n\n\"I can understand wanting a farm or country estate in England,\" he\nbegan.  \"I myself enjoy the thought of thatched roofs and cattle\nstanding knee-deep in waving, grassy meadows; of tired farm horses; of\nmugs of ale and thick slices of bread and the sweat of honest toil--\"\n\n\"On another person's brow!\" interrupted Carolina.  \"You want your farm\nfinished.  I want to make mine.  I want to see it grow.  I almost\nbelieve when it was complete, that I would want to leave it.\"\n\n\"You'd want to leave it long before that,\" cried Kate.\n\n\"Oh, can't you understand my idea?\" cried Carolina, with sudden passion.\n\"I want to get back to Nature and sit in the lap of my mother earth!\"\n\nSt. Quentin nodded his head.\n\n\"I do understand,\" he said, \"and _apropos_ of your idea, I have a piece\nof news for you.\"\n\nCarolina looked at him distrustfully.\n\n\"You will take that look back when you hear,\" he said, with a trifle of\nreproach in his tone.  \"I know you expect no help from any of\nus--discouragements, rather--but I have only to-day heard of business\nwhich calls me to Maine, and as I expect to be obliged to wait there a\nfortnight, I will devote that time to looking up a farm for your\npurpose.\"\n\n\"You will?\" cried Carolina, in a faint voice. Her deception was already\ntripping her up.\n\nKate looked at him with undisguised amazement, mingled with a little\nreluctant contempt.\n\nSt. Quentin's eyes dilated when he saw the flash of personal interest in\nCarolina's demeanour.  Her eyes and voice and manner all underwent a\nsubtle but delightful change.  For the first time, although he was\ndistantly related to her family and had known her since childhood, she\nseemed to approach him of her own accord.  Hitherto her fine sense of\npride had kept her individuality inviolate.  She was not a girl to\npermit familiarity even from an intimate.  She seemed to hold aloof even\nfrom Kate's verbal impertinences, but this was largely due to the fact\nthat Kate's own nature was such that she never attempted to break down\nthe barriers in deeds.  There was always a dignified reserve between\nthem--a respect for each other's privacy, which was the foundation for\ntheir friendship.  One of the greatest proofs of this was that neither\nhad ever thought of suggesting that they spend the night together, with\nthe result that they had never exchanged indiscreet secrets.\n\nOf the relations in which St. Quentin stood to the two; neither had\ngiven any particular thought until that moment.  Kate surprised the look\nin St. Quentin's eyes and the response in Carolina's attitude.  Carolina\nhad never appeared to her friend \"so nearly human,\" as she expressed it\nto herself, as at that moment.  It gave her two distinct shocks of\nsurprise.  One, that Carolina was, for the first time in her life,\nreally interested in something, and therefore she was honest in wishing\nto be poor and left free to pursue her idea.  The other, and a far more\ndisquieting one, was the fact that St. Quentin's glance at Carolina had\nbrought a distinct pang to Kate's heart.\n\nShe regarded both emotions with dismay.  They threatened an upheaval in\nher life.\n\nShe dropped her muff, and, as St. Quentin did not even see it, she\nstooped hastily for it herself, murmuring:\n\n\"That let's me down hard!\"  But with characteristic energy she wasted no\ntime in repining nor even in analyzing her emotions.  She was not yet\nsure whether she was experiencing wounded vanity or the first pangs of a\nlove-affair.  She was extraordinarily healthy-minded and instinctively\nloyal.\n\nIt was this latter feeling which prompted her to leave herself out of\nthe matter, for the present, at least, and to be sure wherein lay her\nfriend's happiness before she proceeded further.\n\nAs she and St. Quentin left the house together, they met Sherman Lee\njust coming up the steps, looking pale and anxious.\n\n\"Is Carol at home?\" he inquired, eagerly, and before they could reply,\nadded, \"and alone?\"\n\n\"Yes, she is,\" answered Kate, \"and if you hurry, you will be in time to\nget a cup of tea.\"\n\nHe thanked them and ran hastily up the steps.\n\n\"How I admire a woman's tact,\" said St. Quentin, giving her a grateful\nglance.\n\n\"How do you mean?\" asked Kate to gain time, though the quick colour flew\nto her face.\n\n\"My man's first idea would have been to ask Sherman what the matter\nwas--he was plainly distraught--\"\n\n\"And to offer to help him!\" said Kate.\n\n\"Perhaps.  But your woman's quickness leaped ahead of my blundering\nintentions with the instinctive knowledge that any cognizance of his\nmanner, no matter how friendly, would be unwelcome. Therefore you sent\nhim away with the comforting assurance in his mind that we had noticed\nnothing amiss.  Thus, in an instant, you saved the feelings and kept\nintact the _amour propre_ of two men.\"\n\n\"That's what women are for!\" said Kate, bluntly.\n\n\n\n\n                               CHAPTER V.\n\n                           BROTHER AND SISTER\n\n\nCarolina had left the drawing-room before Sherman sought her there, but\non receipt of a message from him that he wished to see her immediately\nin the library, she once more descended the stairs to wait for him.\n\nAn anxious look swept over her face as she passed the door of his room,\nfor she heard Addie's voice raised in shrill accents, and to hear it\nthus was growing to be an every-day affair.  She knew her brother's\nsensitive, yet proud and gentle nature, and she knew how difficult his\nwife's loud reproaches were to endure.\n\nSuddenly the door opened and his rapid footsteps were heard running down\nthe stairs and hurrying to the library.  She rose to meet him with her\nanxiety to make up to him for his wife's conduct written in her face.\nHe saw the look and misunderstood it.\n\n\"Don't look at me like that, Carol!\" he cried, raising his hands as if\nto ward off a blow.  \"If you, too, feel the loss of the money as Addie\ndoes and you reproach me, I shall go mad.\"\n\n\"Sherman!\" cried his sister.  \"Don't insult me by the suggestion of my\nreproaching you! Haven't you lost all your money as well as mine? And\nwould you have done either if you could have helped it?\"\n\nHer brother turned uneasily.\n\n\"You don't know how it came about?\" he asked.\n\nCarolina shook her head.\n\n\"Ah,\" he breathed, \"then I must wait until you have heard before I dare\ntrust such generous statements.\"  He hesitated, then burst out.  \"But at\nleast you shall know the truth.  We are absolute beggars, you and I, and\nCousin Lois, and wholly dependent upon Adelaide's bounty until I can\npull myself together.\"\n\nCarolina recoiled as if he had struck her.  A sudden sickening fear\nclutched her heart.  Sherman said \"everything.\"  Did he include\nGuildford? She could not clear her eyes and voice sufficiently to\nmention that beloved name.  Sherman went on, not heeding her silence.\n\n\"I know what you mean, but it's the truth.  She acknowledges it as well\nas I.  Her money is intact, and she will keep it so.  She cannot spare\nany of it to start me again.  I must trust in strangers.\"\n\n\"Why strangers?\" asked Carolina.  \"Have you no friends?\"\n\n\"Friends!\" sneered her brother.  \"What do friends do for a man when he\nis down?  Give him good advice, offer to lend him a few hundreds for\nliving expenses, but trust him to make a second success after one\nfailure?  Never!  Not even St. Quentin, one of the best fellows who ever\nlived, would do that!\"\n\n\"I think you do Noel an injustice,\" said Carolina, quietly.  \"He has\noffered to help me!\"\n\nSherman looked quizzically at his sister and laughed a little.\n\n\"Has he, indeed?\" he said, with a lift of his eyebrows.\n\nCarolina noticed his manner with a slight inward start of surprise.\nWhat could he be thinking of? She had known Noel all her life, and not\nonce had the idea Sherman's tone suggested entered her mind. Noel St.\nQuentin?  She dismissed the thought with impatience.  Sherman did not\nknow what he was talking about.\n\n\"I have not yet told you,\" he broke out suddenly, \"how the money was\nlost.  Have you no idea?  You ought to know.  You warned me against the\nman, but I refused to believe you.\"\n\nCarolina leaned forward and her eyes blazed.\n\n\"Not Colonel Yancey?\" she half-whispered.\n\nHer brother nodded.\n\n\"Tell me,\" she said, with white lips.\n\n\"There is very little to tell.  The whole thing was an elaborate lie--a\nswindle from one end to the other.  I don't believe there ever was any\noil on the lands he sold us.  He swore there was, and bought outright\nthe man I sent down to Texas to investigate.  I could put him in jail, I\nsuppose, but what good would that do me?  Yancey says he has used all\nthe money in speculation and lost it, so even to prosecute him would not\nget a penny back. Now he has disappeared--Algiers, I believe they say.\nIt makes no difference where.  He was so plausible, and his enthusiasm\nwas so contagious, we kept handing over the money like born fools.  I\nwonder that he did not laugh in our faces.  But he deceived well.  He\nplanned from the ground up, and was ready with letters and witnesses of\nall sorts whenever we began to show signs of weakening.  I can see it\nall now with fatal clearness.  But then he had me thoroughly blinded by\nhis own artful proceedings.  He has wrecked two others besides myself.\nThe other three men in the syndicate suspected him and sold out to\nBrainard and me.  We continued to believe in him and he has ruined us.\"\n\nCarolina listened in silence, dreading, yet waiting, for the next blow.\n\n\"He could be the most charming man in the world when he wanted to,\"\nSherman continued. \"I will admit that I felt his spell, but all the time\nthere was something in his face which I distrusted. First I thought it\nwas his shifty eyes, and then, as if he had read my thoughts, he would\nmeet my glance with perfect candour and frankness and the craft would go\nto his lips, and when I looked again for it, I would be disarmed by the\nsincerity of his smile, so I was left to fall back on my Doctor Fell\ndislike of him, which always attacked me most strongly when I was not in\nhis magnetic presence.\"\n\nSherman looked at his sister expectantly.  He noticed for the first time\nhow pale she was.  Her own recollections of Colonel Yancey, his\nceaseless pursuit of her, his intimacy with her father in Paris, her\nfear that he knew of the Lees' great wish to restore Guildford were all\ngathering themselves together into a horrible certainty.  She was\nobliged to listen with an effort to her brother's next words.\n\n\"I've always thought that he tried to make love to you, Carol.  Did he?\"\n\n\"I believe there was something of the sort suggested,\" answered his\nsister, carelessly.  She did not choose to admit that Colonel Yancey had\nproposed to her regularly ever since his wife died, and that he had\npursued her with letters as far as India itself.\n\nA silence fell between them.  It struck Sherman Lee as most\nextraordinary that his sister should evince no more curiosity or even\ninterest in the loss of her fortune than she had hitherto expressed. He\nfelt that possibly she was only holding herself in check.\n\n\"You said a moment ago,\" she began so suddenly and in such a different\ntone that her brother nerved himself for the explosion he felt sure was\nat hand, \"that we were both--you and I--dependent upon Addie.  Just what\ndid you mean?\"\n\n\"Simply that neither of us has a dollar of ready money.\"\n\n\"That is all very well for you,\" pursued Carolina, in a low voice, \"but\nfor me to be Adelaide's guest for even a day would be intolerable.  I\nshall sell my jewels and accept Kate Howard's invitation to spend a few\nweeks with her until I find something to do.  I made Cousin Lois go to\nBoston to see her niece.  I feel that I ought to tell you how glad--how\nmore than glad I am that the money is gone.  I never wanted it!  I never\nliked it!  But Cousin Lois!  What will she do?  Oh, Sherman! If only I\nhad been a man, too!\"\n\n\"If only you had been a man instead of me,\" he cried, \"you never would\nhave lost it.  I always made money when I took your advice.  I always\nlost it when I went against you.\"\n\nCarolina's face glowed.  She felt equal now to putting the question.\n\n\"What has become of Guildford?\" she asked, in a low tone.\n\n\"Guildford?\" he repeated, to gain time.\n\nAt the mere mention of that beloved name Carolina's face was aflame.\nHer great blue eyes flashed and she seemed illumined from within.  Her\nbrother stared at her with astonishment and a growing uneasiness.\n\n\"Yes, Guildford!\" she whispered.  \"Oh, Sherman! I have been so afraid to\nask.  Tell me, is that lost, too?\"\n\nThe man's eyes fell before her accusing gaze.\n\n\"Not--not entirely,\" he stammered.  \"I--I raised money on it--I forget\njust how much--I will investigate--I had no idea you cared--it is\ndeserted--the house burned, you know--\"\n\nHe broke off, as he realized his sister's gathering anger.\n\n\"Stop!\" she said.  \"I have not uttered one complaint because you lost\nour money, nor would I complain at the loss of Guildford.  You could not\nknow how I cared for the place, because no one knew it.  I never even\ntold Cousin Lois.  But don't, if you love me, belittle the place or try\nto excuse your having mortgaged it because it had no value in your eyes!\nI know the house is gone, but the ground is there, and we Lees have\nowned it since we bought it from the Indians.  That same ground that the\nCherokees used to tread with moccasined feet has been in our family ever\nsince they owned it, and the dream of my life has been to restore the\nhouse and to live there--to marry from Guildford and to give my children\nrecollections that you and I were denied, and of which nothing can take\nthe place.  Oh, Sherman, doesn't it fairly break your heart to think\nthat we are the only generation that Guildford skipped?  Father\nremembered it and loved it beyond words to express.\"\n\n\"And you are like him,\" said her brother, gloomily.  \"I am like my\nmother.  She never cared for Guildford, and refused to let father\nrestore it.  It was she who urged him into diplomacy--\"\n\n\"Where he distinguished himself,\" cried Carolina, loyally.\n\n\"Yes, where he distinguished himself, as all the Lees have done except\nme!\" he said, bitterly.\n\n\"It's your name!\" cried Carolina, passionately. \"What could you expect\nwith those two names pulling you in opposite directions!  Why did they\never name you, a Southern man, Sherman?\"\n\n\"Father named you, and mother named me,\" answered her brother.  \"I have\nheard them say that it was all planned before either of us was born.\nThen, too, you must remember that--well, that I am not as enthusiastic\nover the traditions of the Lee family as you are.  I think that my\nleanings are all toward the de Cliffords, if anything.\"\n\n\"It's only fair,\" said Carolina, with justice, \"that you should be like\nmother and love her family best. Only--only I am glad my name is\nCarolina!\"\n\nHer brother bent down and kissed her flushed face.\n\n\"And I am glad, too, little sister, for you are a veritable Lee, and one\nto be proud of.\"\n\nCarolina felt herself grow warm in every fibre of her being over the\nfirst compliment which had ever reached her heart.\n\nSherman was still holding her hand, and she pressed his fingers\ngratefully.\n\n\"I will look up the papers to-morrow, and let you know the moment I\ndiscover anything.  I can easily guess what your plan is, but--without\nmoney?\"\n\nCarolina laughed strangely.\n\n\"Thank you, brother.  And in the meantime I shall go to stay with Kate.\"\n\nAgain the slight lift to Sherman's eyebrows.\n\n\"You will doubtless be happier there,\" he said, quietly.\n\n\n\n\n                              CHAPTER VI.\n\n                              THE STRANGER\n\n\nBut when Carolina was comfortably established in the suite of rooms\nwhich Kate had joyfully placed at her disposal, she found that she could\nneither fix her attention on the new decorations of which Kate was so\ninordinately proud, nor could she wrench her mind from the subject of\nGuildford.\n\nShe had been so stunned by the knowledge, not that the estate was\nmortgaged, but that it had been parted with so lightly, with little\nthought and less regret, that she had not been able, nor had she wished\nto express to Sherman her intense feeling in the matter.  The more she\nthought, the more she believed that some turn of the wheel would bring\nGuildford back.  If it were only mortgaged and not sold, she felt that\nher yearning was so strong she even dared to think of assuming the\nindebtedness and taking years, if need be, to free the place and restore\nthe home of her fathers.\n\nHer intimacy with her father had steeped her in the traditions of\nGuildford.  The mere fact of their having lived abroad seemed to have\naccentuated in Captain Lee's mind his love for his native State, and no\nhistorian knew better the history of South Carolina than did this little\nexpatriated American girl, Carolina Lee.  By the hour these two would\npace the long drawing-rooms and discuss this and that famous act or\nchivalric deed, Carolina's inflammable patriotism readily bursting into\nan ardent flame from a spark from her father's scintillant descriptions.\nShe fluently translated everything into French for her governess, and to\nthis day, Mademoiselle Beaupre thinks that every large city in the Union\nis situated in South Carolina, that the President lives in Charleston,\nand that Fort Sumter protects everything in America except the Pacific\nCoast.\n\nCarolina knew and named over all the great names in the State's history.\nShe could roll them out in her pretty little half-foreign English,--the\nRutledges, the Pinckneys, the Gadsdens, the Heywards, the Allstons, the\nHugers, the Legares, the Lowndes, the Guerards, the Moultries, the\nManigaults, the Dessesseurs, the Rhetts, the Mazycks, the Barnwells, the\nElliotts, the Harlestons, the Pringles, the Landgravesmiths, the\nCalhouns, the Ravenels,--she knew them all.  The Lees were related to\nmany of them.  She knew the deeds of Marion's men as well as most men\nknow of battles in which they have fought.  She knew of the treaties\nwith the Indians, those which were broken and those which were kept.\nShe had been told of some of the great families which even boasted\nIndian blood, and were proud to admit that in their veins flowed the\nblood of men who once were chiefs of tribes of savage red men.  She\nfound this difficult to believe from a purely physical prejudice, but\nher father had assured her that it was true.\n\nIn vain she tried to interest herself in Kate's plans for her amusement.\nIn vain she attempted to fix her attention on the white and silver\ndecorations of her boudoir, all done in scenes from \"Lohengrin.\"\nInstead she found herself dreaming of the ruins of an old home; of the\nchimneys, perhaps, being partially left; of a double avenue of\nlive-oaks, which led from the gate to the door and circled the house on\nall sides; of fallow fields, grown up in rank shrubbery; of palmetto and\nmagnolia trees, interspersed with neglected bushes of crepe myrtle,\nopopinax, sweet olives, and azaleas; of the mocking-birds, the\nnonpareils, and bluebirds making the air tremulous with sound; of broken\nhedges of Cherokee roses twisting in and out of the embrace of the\nhoneysuckle and yellow jessamine.  Beyond, she could picture to herself\nhow the pine-trees, left to themselves for forty years, had grown into\ngreat forests of impenetrable gloom, and she longed for their perfumed\nbreath with a great and mighty longing.  She felt, rather than knew, how\nthe cedar hedges had grown out of all their symmetry, and how raggedly\nthey rose against the sky-line.  She knew where the ground fell away on\none side into the marshes which hid the river--the river, salt as the\nocean, and with the tide of the great Atlantic to give it dignity above\nits inland fellows.  She knew of the deer, the bear even, which\nfurnished hunters with an opportunity to test their nerve in the\nwildness beyond, and of the wild turkeys, quail, terrapin, and oysters\nto be found so near that one might also say they grew on the place.  In\nher imagination the rows upon rows of <DW64> cabins were rebuilt and\nwhitewashed anew.  The smoke even curled lazily from the chimneys of the\ngreat house, as she dreamed it.  Dogs lay upon the wide verandas; songs\nand laughter resounded from among the trimmed shrubbery, and once more\nthe great estate of Guildford was owned and lived upon by the Lees.\n\nFilled so full of these ideas that she could think of nothing else, she\nsprang to her feet and decided to see Sherman without losing another\nday.  She would put ruthless questions to him and see if any power under\nHeaven could bring Guildford within her eager grasp.  What a life work\nwould lie before her, if it could be accomplished!  Europe, with all its\nhistory and glamour, faded into a thin and hazy memory before the\nliving, vital enthusiasm which filled her heart almost to the point of\nbursting.\n\nIt was, indeed, the intense longing of her ardent soul for a home.  All\nher life had been spent in a country not her own, upon which her eager\nlove could not expend itself.  It was as if she had been called upon to\nlove a stepmother, while her own mother, divorced, yet beloved, lived\nand yearned for her in a foreign land.\n\nIt was four o'clock on a crisp January day when Carolina found herself\nin the throng on Fifth Avenue.  It was the first pleasant day after a\nweek of wretched weather, and the whole world seemed to have welcomed\nit.\n\nCarolina was all in gray, with a gray chinchilla muff.  Her colour\nglowed, her eyes flashed, as she walked along with her chin tilted\nupward so that many who saw her carried in their minds for the rest of\nthe day the recollection of the girl who had formed so attractive a\npicture.\n\nSuddenly and directly in front of her, Carolina saw a young woman, arm\nin arm with a tall man, whose broad-brimmed, soft felt hat, added to a\ncertain nameless quality in his clothes and type of face, proclaimed him\nto be a Southerner.  They were laughing and chatting with the blitheness\nof two children, frankly staring at the panorama of Fifth Avenue on a\nbright day.  If the whim seized them to stop and gaze into shop windows,\nthey did it with the same disregard of appearances which induced them to\nlink arms and not to notice the attention they attracted.  No one could\npossibly mistake them for anything but what they were--bride and groom.\n\nHaving reached her brother's house, Carolina paused for a moment in an\nunpremeditated rush of interest in the young couple.  Something in the\nman's appearance stirred some vague memory, but even as she searched in\nher mind for the clue, she saw an expression of abject terror spread\nover the young bride's face, and pulling her husband madly after her by\nthe arm to which she still clung, she darted across the walk and into a\nwaiting cab.  Her husband, after a hasty glance in the direction she had\nindicated, plunged after her, and the wise cabby, scenting haste, if not\ndanger, without waiting for orders, lashed his horse, the cab lurched\nforward and was quickly swallowed up in the line of moving vehicles.\n\nThis had necessarily created a small commotion in the avenue, and a tall\nman who had also been walking south behind Carolina and who would soon\nhave met the young couple face to face, chanced to raise his head at the\ncrack of the cabman's whip, and thus caught a glimpse of the bride's\nface out of the window of the cab.\n\nInstantly, with an exclamation, he looked wildly for another cab.  None\nwas at hand, but Sherman Lee's dog-cart stood at the curb, and Carolina\nhad paused on the lowest step of the house and was looking at him.\nThere was desperate anxiety in his face.\n\n\"May I use your carriage, madam?  I promise not to injure the horse!\"\n\nIt was the strange young man who had stood in the balcony all during the\nopera of \"Faust.\"\n\nCarolina never knew why she did it, but something told her that this\nyoung man's cause was just.  In spite of the pleading beauty of the\nyoung couple, she arrayed herself instinctively on their pursuer's side.\n\n\"Yes, yes!\" she cried.  \"Follow them!\"\n\nHe sprang in, and the groom loosed the horse's head and climbed nimbly\nto his place.  A moment more and the dog-cart was lost to view.\n\nMost of the good which is done in this world is the result of impulse,\nyet so false is our training, that the first thing we do after having\nbeen betrayed into a perfectly natural action is to regret it.\n\nThe moment Carolina came to herself and realized what she had done, a\ngreat uneasiness took possession of her.  She had no excuse to offer\neven to herself.  She felt that she had done an immeasurably foolish\nthing and that she deserved to take the consequences, no matter what\nthey might be.  If the stranger injured Sherman's favourite horse, that\nwould be bad enough, but the worst result was the mortification her rash\nact had left in her own mind.  It is hard for the most humble-minded to\nadmit that one has been a fool, and to the proud it is well-nigh\nimpossible.\n\nBut Carolina admitted it with secret viciousness, directed, let it be\nsaid, entirely against herself.  In her innermost heart she realized\nthat she had yielded, without even the decent struggle prompted by\nself-respect, to the compelling influence of a strong personality.  This\nunknown man had wrested her consent from her by a power she never had\nfelt before.\n\nAt first she decided that it was her duty to tell her brother at once\nwhat she had done.  Then she realized that, in that case, they must both\nwait some little time before the dog-cart could possibly be expected to\nreturn, and Sherman would no doubt exhaust himself in an anxiety which,\nif the horse returned in safety, could be avoided.  She therefore\ncompromised on a bold expedient.\n\n\"Sherman,\" she said, when she found her brother, \"I saw the dog-cart at\nthe door; were you going out?\"\n\n\"I was, but since I came in, I have decided differently.  Ring, that's a\ngood girl, and tell Powell to see that the horse is well exercised and\nput him up.\"\n\n\"I saw Marie in the hall.  I'll just send her with the message to\nPowell,\" said Carolina.  \"There is no doubt in my mind,\" she murmured,\nas she went out, \"that the horse will be well exercised.\"\n\nShe sent word by Marie that when Powell returned he was to be told to\nsee to the condition of the horse himself by Miss Carol's express\norders, and then to report to Miss Carol herself privately.\n\nBut these precautions were taken in vain, for not ten minutes had\nelapsed before Sherman was summoned to the drawing-room, there to meet\nthe stranger, who introduced himself, told a most manly and\nstraightforward story, and, having produced an excellent impression of\nsincerity on his host, left with profuse apologies.\n\nSherman returned to his sister with a quizzical smile on his face.\n\n\"Carol,\" he said, \"what have you been doing?\"\n\nCarolina's reply was prompt and to the point.\n\n\"I own to being reckless, of trying to conceal my recklessness, under a\nmistaken sense that I was clever enough to cover my tracks.  I vainly\nendeavoured to spare you an hour's anxiety, and I feel that I am a fool\nfor my pains.\"\n\nHer brother laughed.\n\n\"The man is unmistakably a gentleman.  He is in deep trouble over a\nyoung woman, not his sister, who has run away, presumably with a man.\nHe tried to trace them and failed.\"\n\n\"Failed?\"\n\n\"Failed.  If she is his wife, may God help her when he catches her, for\nthere was danger in that man's eye.  But his pride forbade him to give\nme more than the bare facts necessary to explain his extraordinary\naction in surprising you into lending him my horse.\"\n\n\"Was that the way he put it?\" asked Carolina.\n\n\"It was.\"\n\n\"He is a gentleman!\"\n\nShe waited a moment, hesitated, and then said:\n\n\"Did he say anything else, anything about--\"\n\n\"About the woman in the case?  Not a word about anything more than I\nhave told you.  He seemed to take it for granted, however, that you were\nmy wife.\"\n\n\"And didn't you deny it?\" demanded Carolina, with such spirit that she\nsurprised herself.  She felt her cheeks grow hot.\n\n\"He didn't give me time.\"\n\n\"And you let him go, still thinking it?\"\n\n\"I didn't let him do anything.  He mastered the situation, and carried\nit off with such ease that I almost felt grateful to him for borrowing\nthe dogcart.\"\n\nCarolina opened her lips to say something, then changed her mind.\n\n\"It is of no importance,\" she said lightly.  But there was an odd\nsinking at her heart which belied her words.  She had never believed in\nlove at first sight, yet she had watched this stranger at a distance all\none evening, and at their first meeting in the throng leaving the opera,\nshe had not been mistaken in the look of--well, of welcome, she had\nfelt.  Their second meeting had been equally striking, and Carolina\ncalmly said to herself that she would meet this man again, and the third\ntime it would be even more strange.  She was so sure of this that she\nwould not allow her mind to be disturbed by the two blundering\nconclusions Sherman had forced--one that the man was in pursuit of a\nrunaway wife or love and the other that she was the wife of the master\nof the horse.  She was so sure of her own premises that she overlooked\nthe possibility that the stranger might have put the supposition\ntentatively to Sherman and had been misled by her brother's lack of\ndenial.\n\nIn fact, Carolina at this time was a very self-centred young woman.  It\nwas so of necessity.  She had never been taught self-denial, nor\npermitted to be unselfish.  Her father and mother, in yielding to every\nwhim, had quite overlooked the fact that the pretty child's character\nneeded discipline, so that Carolina was selfish without knowing it.\nQuite unconsciously she placed her own wishes before those of any other,\nand regarded the carrying of her point as the proper end to strive for.\nNo one had ever taught her differently.  Cousin Lois had pampered her\neven more than her parents had done, and when she became dissatisfied\nwith life, offered, as a remedy, change of scene.\n\nNow the girl possessed an inherently unselfish nature, and for this\nreason--that she never had been called upon to sacrifice her own\nwill--she was not happy.  Although she possessed much that young girls\nenvied in wealth and the freedom to travel, the two things which would\nhave made her happiest, a permanent home and some one--father or mother\nor lover--upon whom to lavish her heart's best love, were lacking.  Not\nbeing of an analytical turn of mind, she had never realized her lack,\nuntil suddenly she had been given a glimpse of both, and then both had\nbeen snatched away.\n\nOpposition always made the girl more spirited. Guildford lost was more\nto be desired than Guildford idle and only waiting for her to reclaim\nand restore it.  This dominant stranger interested in another\nwoman--Carolina lifted her chin.  It was her way.\n\nHer brother saw it and smiled.  It was a pretty trick she had inherited\nfrom the Lees.  It was a gage of battle.  It betokened unusual interest.\nIt meant that their blood was fired and their pride roused.  He mistook\nthe cause, that was all.  He was so engrossed in his own thoughts and so\npleased by his efforts to gain something which his sister actually\ndesired, that he had forgotten the episode of the strange visitor.  So\nthat when he said:\n\n\"So that is the way you feel, is it?\" Carolina started violently and\nblushed.  She was diplomatic enough to make no reply, so that Sherman's\nnext remark saved her from further embarrassment.\n\n\"Do you really care for Guildford so much?\"\n\n\"How do you know I am thinking of Guildford?\" asked Carolina, quickly.\n\"I have not spoken of it.\"\n\n\"Ah,\" said her brother, lifting his hand, \"I can read your thoughts.  I\nnotice that you only have that look on your face when you are thinking\nof something you love.  But I wouldn't waste such a blush on a measure\nof cold earth, even if they are your ancestral acres.\"\n\n\"My ancestral acres!\" repeated Carolina, softly. \"How beautiful that\nsounds!  Oh, Sherman, tell me if we can save them!\"\n\nSherman hesitated a moment and knit his brow. Then he lifted his head\nand looked Carolina in the eyes.\n\n\"I will do what I can,\" he said.  \"You may be sure of that.\"\n\nCarolina had all a strong woman's belief in the power of a man to do\nanything he chose.  His words were not particularly reassuring, but his\nmanner, as she afterwards thought it over, was vaguely comforting.\n\nIt was the more comforting, because, deep down in her heart, she\nintended to supplement his efforts, weak or strong, and win victory even\nfrom defeat.\n\nGuildford?\n\nShe _would_ have it!\n\n\n\n\n                              CHAPTER VII.\n\n                              MORTAL MIND\n\n\nTherefore, when the blow fell and Sherman had written her a letter, not\ndaring to see her, telling her as gently as he could, but with an air of\nfinality which there was no mistaking, that the mortgage on Guildford\nhad been bought and foreclosed by Colonel Yancey, and therefore, in his\nopinion, it was lost to the Lees for ever, Carolina realized for the\nfirst time how tenacious had been her hold on the hope of possessing it.\nIn an instant, with her woman's instinct, she saw what it had taken\nyears for Sherman to discover.  Colonel Yancey had, as Carolina found,\nlearned that it was Captain Lee's and Carolina's dearest wish to restore\nGuildford. The two men had talked intimately.  Both were Southern,\nalthough Colonel Yancey was a Georgian, but with the confidence in each\nother's integrity, which is typical of most Southern men, and which has\nled to the ruin of many an honest man, Captain Lee confided his hopes to\nColonel Yancey, who profited by them to secure Guildford for himself,\nand thus gain a hold over Carolina.\n\nIt was so easy to do this, in the most ordinary business manner, with\nSherman both unsuspicious of him and his sister's love for the place,\nthat at times Colonel Yancey almost had the grace to be ashamed of\nhimself.\n\nCarolina saw the whole vile plot, and the shock and disappointment put\nher fairly beside herself. She was so sure that she had got at the root\nof the matter that she at once disbelieved that part of Sherman's story\nwhich said that Colonel Yancey was a fugitive from justice.  If he had\ncheated this syndicate, he had done it in such a manner that it left no\nillegal entanglements, and she was sure that he was free to return to\nthis country whenever he chose.  If not, her whole theory fell to the\nground, for she knew that Colonel Yancey would not dare to offer her a\nreputation which the law had power to smirch.\n\nIt never was Carolina's way to wax confidential, but one day Kate\nsurprised her in a particularly desperate mood.  Carolina was in her\nhabit, waiting for her horse to be brought around, and when Kate\nentered, she was walking up and down the peaceful blue and silver\nboudoir like an outraged lioness.\n\n\"It's no use, Kate!\" she cried, when her friend began to remonstrate.\n\"I have come to the end of my rope.  You don't know the truth because I\nhave been afraid to tell you.  You couldn't have understood if I had\ntold you.  Even if I should sit down now and spend a whole day trying to\nexplain why I adored Guildford and why I am so upset over its loss, at\nthe end of the time you would only shake your head and say, 'Poor\nCarolina,' without in the least understanding me.  No one ever did\nunderstand about Guildford except dear Daddy, and since he died, I've\nbeen afraid to let even God know how much I wanted it, because I knew if\nHe did, He would take it away from me! He takes everything away from me\nthat I love! That is His way of showing His vaunted kindness. He is\nindeed a God of vengeance!  He punishes His children as no earthly\nfather would be mean enough to do.  Oh, I won't hush!  But the end has\ncome, Kate, to even God's power to hurt me.  I have nothing left for Him\nto take.  Let Him be satisfied with His revenge.  I wouldn't care if He\ntook my life now, so He is practically powerless!  He has reached His\nlimit!\"\n\n\"Oh, Carolina!\" almost screamed Kate.  \"Do be careful how you blaspheme!\nGoodness knows I am not religious, but I am a member of the Church and I\nam not wicked!\"\n\n\"You have never suffered, Kate, or you could bear, not only to hear, but\nto say worse things than I am saying.  If you only knew how much worse\nmy thoughts are!\"\n\n\"But you will be punished for them, Carolina! I--I don't like to preach,\nbut God always sends afflictions to those who defy Him!\"\n\n\"I wouldn't care if He killed me!\" cried Carolina, furiously.  \"I have\nnothing left to live for. I hope I shall never come back alive from this\nride!\"\n\nWhen she had rushed from the room, leaving that terrible wish in Kate's\nmemory, Kate shivered with apprehensions.\n\n\"Something awful will happen to Carolina!\" she muttered.  \"I never knew\nit to fail!\"  But her eyes filled with tears.  \"What if I had to bear\nwhat she has!\" she thought.  \"Loss of father, mother, home, and fortune!\nPoor girl!  Poor girl!\"\n\nShe had intended to go out, but some inner voice told her to wait.\nCarolina's dreadful mood and reckless words haunted her.  She went\nrestlessly from room to room, and anxiously listened for sounds of her\nreturn.  And so keenly was she expecting a misfortune that when the\ntelephone-bell rang sharply, it calmed her at once.\n\n\"It has happened!\" she said to herself, as she flew to answer.\n\nThe message was that Carolina had been thrown from her horse and\ndragged.  They were bringing her home.\n\n\"I knew it!\" said Kate.  \"She was in too awful a mood to wear spurs with\nAstra.  I ought to have made her take them off.\"\n\nCarolina was still unconscious when they brought her in.  Kate caught a\nglimpse of her still, white face as they carried her up-stairs.  She\nwaited with feverish impatience for the doctor's verdict, with her mind\nfull of Carolina's awful words.  \"I knew it!\" she kept whispering to\nherself through a rain of tears.  \"God always gets even with people who\ndare Him to do His worst!\"\n\nIt seemed hours before Doctor Colfax finally came out, with his refined\nface full of pain.\n\n\"Is she dead?\" whispered Kate, catching at his arm.  He shook his head.\n\n\"Disfigured?\" continued Kate, with growing anxiety.\n\n\"Worse!\" said the doctor.  \"She has broken her hip badly.  Even if she\nrecovers, she will be lamed for life!\"\n\nKate covered her mouth to repress a scream.\n\nBeautiful Carolina lamed for life!\n\n\"Crutches?\" whispered Kate.\n\n\"I am afraid so!\" said the doctor, with a deep sigh.  \"I am going to\nhave a consultation.  We will do everything we can to preserve her\nhealth--and her beauty, poor child!\"\n\nKate turned away in a passion of tears, well knowing that to Carolina's\nproud spirit dependence would be far worse than death.\n\nBad news travels on the wings of the wind, and before the day was over\nCarolina's accident was on everybody's tongue.\n\nHer sister-in-law was indignant, in a sense outraged by Carolina's\nbehaviour.  She blamed her first of all for existing in her radiant\nyouth and beauty and so far outshining her own modest charms.  She\nblamed her secondly for permitting Sherman to lose her money and thus\nmake it Addie's duty to offer her a home.  She blamed her thirdly, and\nmost bitterly of all, for injuring herself so hopelessly that she could\nnever marry, thus placing herself upon Addie to support for life.  Was\never a more unkind fate invented?  Addie's temper, never of the best,\nburst all bounds as this situation became plain to her, and she\nexpressed herself fluently to Sherman, who felt himself included in her\nmisfortunes as part author of them.\n\nIt was an unhappy time for all concerned, for Carolina's bitter\ndenunciations of her fate and her grief over her dependence could hardly\nbe checked even in the presence of Kate and her family, whose\nhospitality and friendship, so generously offered, put the girl under at\nleast civilized bonds of restraint.  There were times, however, when she\nwas alone, that she relapsed into such a savage state that she tore her\nhair and bit her own tender flesh.\n\nThe sight of such rebellion reduced even Kate's mutinous nature to peace\nand quiet by contrast, and Kate was developed into a gentle friend of\nChristian sentiments by Carolina's great need.\n\nThe conversations they held with each other were long and intimate.\nKate tried to put faith in the series of doctors who succeeded each\nother like chapters in a book, but the sufferer's clear eyes saw not\nonly through Kate's kind intentions, but through the great surgeon's\nhopeless hopes, and from the first she knew the worst.  Knew that her\nbright youth was for ever gone; that her usefulness was ended; that\nnever again could she expect even to ornament a social function,\ncrippled as she was and disfigured by ungainly crutches.  Her one hope\nwas to die.  Thus she made no effort to recover, and her strength,\ninstead of aiding her, gradually faded away until her accident, though\nnot at first of a fatal nature, began to be looked on as her death-blow.\n\nAt this juncture, Addie, struck with remorse, came and offered Carolina\na home, but Carolina shook her head.\n\n\"Thank you, Addie, but when I move from here it will be to rest for\never.  I want to die here with Kate.  She loves me!\"\n\nIt was a bitter thrust, and Addie felt it to the verge of tears.\nIndeed, she was so moved by pity for the frail shadow that Carolina had\nbecome, that she forgave the girl for having been so beautiful and began\nto be fond of her, as one is fond of a crippled child, who had been\nobnoxious in health.\n\nTrouble develops people.\n\nMrs. Winchester was detained in Boston by the dangerous illness of the\nniece she had gone to visit, and although greatly fretting at being kept\naway from Carolina, was fairly obliged to stay.\n\nCarolina felt that she was welcome at the Howards, for not only Kate's\nmother but her father often came to sit with her and cheer her and to\nurge upon her how glad they were to be able to help her when she needed\nhelp.\n\nCarolina was grateful, the more so because she felt that she had not\nlong to live.  She had been in bed several months, and while the\nsurgeons said the broken bones had knit, yet it was agony for her to\nmove.  She almost fainted with pain when they were obliged to lift her\nfrom one position to another.\n\nKate spent hours in trying to interest her in the life around her.  She\nfelt frightened when she discovered the depth of Carolina's\nlistlessness.  Her weakness took a stubborn form.\n\n\"I am only one of the crowd now, Kate dear,\" she said one day after a\nlong argument from her friend.  \"There is no use in wasting so much\nenergy over me.  Go and forget me and enjoy yourself.  I used to be of\nthe exclusive few who got their own ways always.  Now I belong to the\ngreat mob of malcontents--the anarchists of the social world.  I shall\nnot want to blow up kings and presidents, but I would like to throw a\nbomb at every happy face I see.\"\n\nHer voice trailed off to a weak whisper.\n\n\"Y-you wouldn't need many bombs, then,\" said Kate, \"for I never s-see\nany really happy faces. Did you ever in all your life--either at balls\nabroad or the opera here, see a perfectly happy face?\"\n\nCarolina shook her head and closed her eyes wearily.\n\nSuddenly she opened them again.\n\n\"Yes,\" she said, \"I have seen one--the night of 'Faust.'  It was\nRosemary Goddard!\"\n\nKate gave a little scream.\n\n\"Well, I'd rather follow you to the grave you seem so bent on f-falling\ninto,\" she stammered, \"than to get happiness from such a source.  My\ndear, Rosemary Goddard is a C-Christian Scientist!\"\n\nKate's tone indicated that Rosemary had contracted a loathsome disease.\n\nCarolina fixed her eyes on Kate.  She was not of a contrary disposition,\nyet the difference between Kate Howard's tone and Rosemary Goddard's\nface made her stop to think.\n\n\"I should like to talk to Rosemary,\" she said at last.  To her surprise\nand consternation, Kate burst into tears.\n\n\"If you g-go and turn into one of those n-nasty things,\" she sobbed, \"it\nwill end everything.  I'd rather you died!\"\n\n\"Then never mind,\" said Carolina, wearily.  \"I don't want to vex\nanybody.  Perhaps I shall die.\"\n\nKate jumped up.  The momentary colour faded from Carolina's face and the\nstrength from her voice.  Kate recognized the change.\n\n\"I'll go and f-fetch her,\" she said, with her old-time change of front.\n\"She may do you good.\"\n\nWhen she came back with Rosemary, she saw what Carolina had seen in\nRosemary's face--an illumination which no one could understand.  It\ntransfigured her.\n\nKate left the two girls together, and walked the floor in tempestuous\nanger all during Rosemary's stay in the house.  Something in Carolina's\neyes as they first met Rosemary's told Kate that the poison was already\nat work, and that Carolina was ripe for the hated new religion.\n\n\n\n\n                             CHAPTER VIII.\n\n                            MAN'S EXTREMITY\n\n\nRosemary approached the bed wherein lay the wreck of the girl she had\noften, when in the grasp of mortal mind, envied.  A great wave of\nsympathy, not pity, swept over her, as she noted the weary eyes and the\nlines of dissatisfaction and despair around Carolina's mouth.  With an\nimpulse of love, she knelt at the bedside and took Carolina's little\nthin hand in both of hers.\n\n\"Oh, my dear Carol,\" she said, \"I am so glad to see you.  I heard of\nyour accident while I was in California.  I only got back yesterday.\"\n\n\"Would you have come to see me if I had not sent for you?\" asked\nCarolina, childishly.\n\n\"I was coming to-day.  Mother suggested it, and I was only too happy to\nput off everything of less importance and come at once.\"\n\n\"Your mother!\" said Carolina, involuntarily. Then, as she saw Rosemary's\nface flush, she hastened to cover her awkward exclamation.  \"I did not\nknow your mother knew me well enough to--to care!\"\n\n\"Mother is very much changed since you knew her,\" said Rosemary, gently.\n\"She has been healed.\"\n\nCarolina did not know the nature of Mrs. Goddard's infirmity, so she\nforbore to ask of what. She only knew, as all the smart world knew, that\nMrs. Goddard did something dreadful, and did it to excess.  It was\nwhispered that it was a case of drugs, but there were those, less kind,\nwho hinted at a more vulgar excess, either of which would explain the\ndreadful scenes Mrs. Goddard had occasioned in public.  Her intimates\nasserted that a terrible malady was at the bottom of her habits,\nwhatever they were.  At any rate, a somewhat scandalous mystery hung\nover Mrs. Goddard's name, although she had been at the forefront of\nevery mad scene of pleasure the fashionable world could invent to kill\ntime.\n\n\"You are changed, too,\" said Carolina, wonderingly, more and more\nsurprised to see Rosemary Goddard--of all girls!--kneeling at her\nbedside, holding her hand in a warm grasp, pressing it now and then to\nemphasize an affection she felt shy of expressing, and talking in a\ngentle, altogether unknown tone of voice.  In Carolina's uncompromising\nvocabulary she had privately stigmatized Rosemary as a snob, and rather\nridiculed her exaggeration of aristocracy.  But the coldness, the tired\nexpression, the aloofness, were all gone.  The weary eyes shone.  The\nbored eyebrows were lowered. The curved lips smiled.  The withdrawn\nhands were reached out to help.  The whole attitude was radiant of\nsympathy and love.\n\nRosemary could not forbear to smile at Carolina's unconscious scrutiny.\n\n\"What has done it?\" asked Carolina, abruptly.\n\n\"Christian Science,\" said Rosemary, frankly.\n\nCarolina was disappointed that she did not rush on and explain.  She had\nheard that Scientists thrust their views upon you and were instant in\nseason, out of season.  She was piqued that Rosemary did not give her\nthe opportunity to argue and refute.  Carolina wanted to be coaxed.\n\n\"The change in you is wonderful,\" she said at last.  \"I think it is\nalways a little insulting to tell a woman how she has improved, so I\nwill not harp on it.  But I don't think I care to investigate Christian\nScience.  It has always bored me when people have tried to explain it to\nme.\"\n\n\"You have a perfect right to leave it alone, then,\" said Rosemary.\n\"Christian Science does not need you in the least.\"\n\nAlthough her tone was perfectly sweet and kind, it was dignified, and\nCarolina's quickness at once comprehended the almost unbearable\npriggishness of her remark.\n\n\"I did not intend to be rude,\" she said, hurriedly. Then she hesitated\nas another thought struck her, and in a more timid voice she said:\n\n\"Did you mean that Christian Science does not need me as much as I need\nChristian Science?\"\n\nRosemary pressed her hand as her only reply.\n\n\"Can it help me?\" cried Carolina, with sudden fervour.  \"I am a wreck,\nphysically and mentally. I have lost parents, fortune, home, health, and\nambition.  I long to die!  I have even lost my God!\"\n\n\"Christian Science will give you back your God,\" said Rosemary.\n\n\"I hate God!\" said Carolina, calmly.\n\n\"I used to hate Him, too,\" said Rosemary.  \"In the old thought there was\nnothing else to do, for a just mind, than to hate Him.  We had made an\nimage of hate and vengeance and set it up to worship and called it God.\"\n\n\"We?  Did we do it?\"\n\n\"Of course!  Who else?\"\n\n\"Then it is all our fault?\"\n\n\"It certainly is not God's fault,\" said Rosemary. \"He has declared\nHimself to be Love Incarnate. If we have been stupid enough to endow Him\nwith human attributes of our own distorted imagination, is He to blame?\"\n\n\"He never answered a prayer of mine in all my life!\" cried Carolina,\npassionately, looking at the ceiling as if to make sure that God heard\nher accusation, and as if she hoped to irritate Him into hearing future\nprayers.\n\n\"Nor of mine, either, until I learned how to pray.\"\n\n\"Who discovered the new way?  That Eddy woman?\"\n\n\"Mrs. Eddy did.\"\n\n\"How, I should like to know?  Why was all this given to her to know and\nnot to some man?\"\n\n\"By the way,\" said Rosemary, as if changing the subject, \"I hear that\nyou speak both Japanese and Russian and that you did some important\ninterpreting at a banquet on board the Kaiser's yacht at Cowes, last\nspring.  Did you?\"\n\n\"I believe so,\" said Carolina, wearily.\n\n\"However did you manage to master two such awfully difficult languages?\"\n\n\"I studied years to do it.\"\n\n\"How strange that my brother was not called upon to do that\ninterpreting,\" said Rosemary, in a musing tone.  \"He was at that\nbanquet, and he is a man.\"\n\nCarolina opened her lips to make an incautious reply, but caught herself\njust in time.  A gleam in Rosemary's eyes warned her.\n\n\"I see,\" she said, reddening.  \"But I must say you baited the hook\nskilfully.\"\n\n\"I had to, in order to catch you,\" said Rosemary.\n\nCarolina turned her head on her pillow restlessly.\n\n\"Tell me about how you came to accept it,\" she said, pleadingly.\n\n\"Well, I was so abnormally miserable!  I had everything in the world I\nwanted--apparently, yet my home was full of discord.  I had only a big,\nbeautiful house.  I wanted the love of a certain man.  He held aloof\nwhile all the others were at my feet.  I prayed wildly to my God for\nhelp, and He mocked me.  Then I grew bitter and vengeful. I vowed that I\nwould have all that life held without God, for it seemed to me, in my\nvicious interpretation of Him, that every time He saw me poke my head\nout of my hole, He hit it--\"\n\n\"Just to show that He could!\" cried Carolina, almost with a scream of\ncomprehension.\n\n\"Exactly--just to show that He could.  Well, then I plunged into a\nmadness I called gaiety, and grew more and more unhappy because I saw\nthat each day I was putting myself further and further from the man I\nloved.  Then, as if to fill my already full cup to overflowing, mamma\ngrew very much worse, so much so that I wanted her to die. I really felt\nthat she had exhausted all that _materia medica_ could do for her, and\nthat death was the only way to end it, both for her and for us.  Then I\nheard of a Christian Science practitioner, named Mrs. Seixas.  I went to\nsee her, and, impossible as it may sound, in the first fifteen minutes,\nI had told her the whole truth, mortifying as it was.  But she seemed\nnot only to inspire confidence, but to radiate help.  I felt that,\nalthough I was a perfect stranger to her, yet she wanted to help\nme--that she would go out of her way to do it, and that the reason she\nwould do it was because she loved much.  I took her to mamma that same\nday, and mamma's complete healing is so great a marvel that we never can\nget used to it.  Our happiness is almost too much to bear.\"\n\nRosemary's eyes filled with tears which rolled down her cheeks.\nCarolina viewed her with an astonishment that she could ill conceal.\nRosemary Goddard to be talking, nay, more, feeling like that! A question\nwas so unmistakably in Carolina's eyes, which her tongue could not gain\npermission to utter, that Rosemary found herself answering it.\n\n\"Then, when God had made me worthy of a good man's love, the desire of\nmy heart came to me, in so sweet and natural a way that it broke down\nthe last barrier of pride and left me humbly at the foot of the cross,\nmarvelling at God's goodness!\"\n\nCarolina drew Rosemary's face down to hers and laid her cheek against\nit.\n\nThere was a long silence between them.  Then Carolina said, fearfully:\n\n\"My hip is broken.  Can that be cured?\"\n\n\"God can do anything.\"\n\n\"So that I needn't use crutches?\"\n\n\"Most certainly.  You won't even limp.  You will be made perfectly\nwhole!\"\n\n\"Just as I was before?\"\n\n\"Just as you were before--except these bonds.\"\n\nCarolina thought a moment.\n\n\"But what do I want to get well for?  I have lost Guildford!\"\n\n\"Nothing can be lost in Truth!\"\n\nRosemary felt her two hands grasped firmly, and without thinking\nCarolina raised herself to a sitting posture in bed without pain.\n\n\"Do you mean to tell me that there is the--that Christian Science\nteaches that there is any remote possibility of my getting Guildford\nback?\"\n\n\"Guildford belongs to you, and has never been lost.  It is only error\nwhich makes such a law for you.  Truth emancipates everybody and\neverything.\"\n\n\"I don't believe it!\" said Carolina.  \"I can't! It's too good to be\ntrue!  I don't understand it!\"\n\n\"You do understand it!\" said Rosemary.\n\n\"What makes you think so?\"\n\n\"Because you are sitting up in bed, and you raised yourself without\npain.  That is because, for a moment, your soul accepted God as Love and\nthe source of all supply.  Unconsciously your mind looked into His mind,\nand you saw the truth.\"\n\n\"I believe that I could get up!\" said Carolina, in a sort of ecstasy.\n\n\"I know that you can!  Give me your hand.\"\n\nRosemary helped Carolina to dress, and in half an hour Carolina was\nsitting, for the first time in months, in a chair by the window, with\nRosemary reading and marking for her the passages in \"Science and\nHealth\" which bore immediately upon her case.  Carolina's mind opened\nunder it like a flower.\n\n\"Oh, I need so much teaching!\" cried Carolina. \"Who will help me?\"\n\n\"Did you know that my mother is a practitioner and holds classes?\" asked\nRosemary.\n\nCarolina almost felt her new-found rock melting beneath her feet at this\nintelligence.\n\n\"No, I did not.  Will she take me?  And will you help?\"\n\n\"We will both do all we can for you with the greatest joy.\"\n\nWhen Rosemary left, Kate came in and Carolina explained everything to\nher.\n\nKate called Noel St. Quentin by telephone and told him that Carolina had\ngone insane.\n\nThe next morning Carolina awakened with the happy consciousness that\nsomething pleasant had happened.  Hitherto she had gone to sleep, glad\nof the respite of a few hours of unconsciousness. Simply not to\nknow--simply not to be awake and to realize her load of pain and\ndisappointment, had been her prayer.  With her definite aim in life\nswept away, she felt rudderless, forlorn, despairing.\n\nBut suddenly everything was changed.  Her weakness vanished as if by\nmagic.  Instead of dreading to open her eyes and clarify her brain for\nthought her mind leaped to a lucid clearness without effort.  The glow\nof happiness which pervaded her she could liken to nothing so much as\nthe awakening in her hated school-days to the knowledge that to-day was\nSaturday!\n\nAnd what had brought her healing?  Only a few hours' talk from Rosemary\nGoddard which seemed to untangle all the knots of her existence and to\nwipe the mists from the window-panes, out of which she had been vainly\ntrying to get a clear view of her life, its reason for being, and its\nduties.  Always the question with Carolina had been \"To what end?\"  And\nall the answers had been vague and unsatisfactory, until suddenly she\nhad stumbled by reason of her infirmity upon one who could answer her\nvehement questions clearly and lucidly.\n\nEmerson must have been largely of the thought when he wrote: \"Put fear\nunder thy feet!\"  Carolina, with her sensitive, mystic nature had been,\nin common with all imaginative persons, literally a slave to her fear.\nWhat could it mean, this sudden freedom, except that she had found the\nonly true way out of bondage?\n\nWith a little assistance, she was able to dress herself and sit in a\nchair to wait for the promised visit of Rosemary's mother.\n\nShe had known of Mrs. Goddard for years, although she seldom appeared in\npublic.  No one spoke the name of her malady, but everyone knew of her\nintense suffering and of the days she spent unconscious from the effects\nof quieting drugs.  Secretly every one expected to hear at any time of\nMrs. Goddard's madness or death, and Carolina had heard no news of her\nexcept what Rosemary had said until Mrs. Goddard was announced and found\nher, dressed and sitting up to meet her guest, with outstretched hand\nand happy, smiling face.  As usual Carolina's expressive countenance\nbetrayed her.\n\n\"No wonder you look surprised, my dear,\" said Mrs. Goddard, kissing the\ngirl on the cheek with warmth.  \"Rosemary evidently did not have time\nyesterday to tell you what brought us both into Science.  I was cured of\ncancer in its worst form.  Did you never know?\"\n\n\"I knew you were very, very ill and suffered horribly,\" said Carolina,\n\"but--\"\n\n\"I know.  My friends were very kind.  They never gave it a name.  But\nthat was it.\"\n\n\"Oh, how wonderful!\" cried Carolina, with shining eyes.\n\n\"Not half as wonderful as what it did for me mentally,\" said Mrs.\nGoddard.  \"I used to feel that I had brought my malady on myself by my\nway of life.  I was the gayest of the gay in my youth, and in middle\nlife I found that stimulants had such a hold on me that I was not myself\nunless I was drugged.  I ran the gauntlet of those until I came to\nmorphine.  There I stayed, and whether the morphine came of the cancer\nor the cancer of the morphine I never knew.  But the horror of my life I\ncan readily recall.  It came to a point when the best physicians and\nsurgeons in New York said that there must be an operation and frankly\nadded that no one could tell whether I would come out of it or not.\nPleasant, wasn't it?\"\n\nCarolina only clasped her hands together, and Mrs. Goddard proceeded:\n\n\"Then Rosemary heard of Christian Science, and without saying a word to\nme, she looked up the names of one or two practitioners and called. The\nfirst one she did not care for and came away discouraged.  But something\ntold her to try again, and her second attempt led her to the door of the\nangel of healing who, under God, worked this cure, Mrs. Seixas.\nRosemary had not talked with her ten minutes before she knew that she\nhad been led aright.  She wanted Mrs. Seixas to get into the brougham\nand come at once, but according to Science practice she insisted upon\nRosemary's coming home and getting my consent.\n\n\"You can imagine that I was not slow to accept the hope it offered, and\nthat same afternoon I had my first treatment.  Carolina, inside of an\nhour the pain all left me!  Child, you have suffered, so you know, you\ncan fathom as many cannot, what that means!  I promised when the pain\nreturned to call her by telephone, instead of taking the morphine, but\nit never did come back!  She gave me treatments from her office every\nhour for the rest of the day and came back after dinner that night and\ngave me another.  That was three years ago.  To-day I am a well woman.\nI eat whatever I please and not once has the old craving for stimulants\nattacked me.  I am a free woman and a very happy one!\"\n\n\"Oh, Mrs. Goddard,\" cried Carolina, \"thank you so much for telling me.\nIt helps me to know that I am being cured!\"\n\n\"That you are cured.\"\n\n\"Yes, I must believe that.\"\n\n\"Pardon me--not so much believe it, as you must understand it and\nunderstand why it is so. Every orthodox Christian is ready to state\nglibly that God is All, but they never act as if they believed it and\nthat is the chief difference between members of churches and Christian\nScientists.\"\n\n\"Why does every one hate Christian Science so before they understand\nit?\"\n\n\"Christian Science is like a large crystal bowl full of the pure water\nof life.  Left alone it simply sparkles in the sunlight of God's smile.\nBut if you bring to it the alkali of ignorance and the acid of\nprejudice, this clear water becomes the vehicle of a most energetic\nboiling and fizzing.  But when it has assimilated the two foreign\ningredients the residue sinks to the bottom harmlessly, the water\nclarifies itself by its reflected power, and the crystal bowl resumes\nits placid, sparkling aspect.\"\n\n\"I understand,\" said Carolina, \"that I must have caused that commotion\nrather often, for I used to hate Christian Science so vigorously and I\nhated Mrs. Eddy so intensely that I used to rejoice at every adverse\ncriticism of her or her work, and I used to go to the trouble (when I\nnever would have bothered to make a scrap-book) of cutting things out of\nthe papers, and mailing them to my friends.  I deliberately put myself\nout in order to hate it more adequately!\"\n\n\"I know,\" said Mrs. Goddard.  \"Isn't it strange, when you look back on\nit in the light of your new understanding and your healing?\"\n\n\"Ye-es,\" said Carolina, dubiously, \"but to be quite truthful, I am\nafraid I am not cured of all my prejudice yet!\"\n\n\"Let it go,\" said Mrs. Goddard.  \"It will pass of itself.  Don't fret\nabout it.  Now tell me about yourself.  You know we do not dwell upon\nour ailments, mental or physical, but if you state them to me, as your\nphysician I can work more intelligently.\"\n\n\"Oh,\" sighed Carolina, \"what is there not the matter with me!  Where\nshall I begin?\"\n\n\"Let it console you to know in advance that there is a remedy in Divine\nScience for everything. 'Not a sparrow falleth'--you remember!  The\ntable of comfort for every woe is spread before you in the presence of\nyour enemies.  Fear neither them nor to partake freely of God's gifts.\nThe more eagerly you come and the more you partake of the feast Divine\nLove spreads, the more generously God will pour out His blessings upon\nyou.\"\n\nThus encouraged Carolina told her suspicions of the fate of Guildford\nand of Colonel Yancey, without, however, mentioning him by name, until,\nled on by Mrs. Goddard's sympathetic manner, she threw her whole soul\ninto the recital of her own and Mrs. Winchester's loss, and of how she\nhad hoped to restore Guildford.\n\nOccasionally Mrs. Goddard interrupted her to ask a pertinent question.\nIt gave Carolina a feeling of comfort to realize her new friend's\nmentality. Carolina, was so accustomed to knowing people of capacity and\nbrilliant intelligence that her mind reached after such naturally.\n\n\"Guildford is not lost to you,\" said Mrs. Goddard, just as Rosemary had.\n\n\"It will be restored to you, and you will be able to make good Mrs.\nWinchester's loss.  You must have harmony in your life.  That is your\nright--your God-bestowed right.  You are an heir of God's boundless\naffluence.  It is a crime for one of God's little ones to be poor, or\nneglected, or sick, or forsaken.  Not to believe this is to doubt His\npromises, which are sure, and to limit His power, which is limitless.\n\n\"We do not know the way, nor must we make laws nor dictate means.  But\nGod is even now preparing the broad highway which shall lead your feet\nstraight to the gates of Guildford.  Let Him find you humble, grateful,\nand ready for the blessing.  Don't fret.  Don't worry.  Don't be\nanxious.  'Be still, and know that I am God!'\"\n\nFor her only reply Carolina bowed her face upon her hands, and burst\ninto an uncontrollable fit of weeping.\n\nMrs. Goddard made no effort to check or comfort her, except by thought.\nWhen she had finished, Mrs. Goddard nodded her head, saying:\n\n\"That did you good.  Now for your physical self!  Was the hip broken?\"\n\n\"Yes, and set by six of the best surgeons in New York.  Doctor Colfax is\nthe most hopeful, but even he says that if ever I grow strong enough to\nleave off crutches, I shall limp all my life.\"\n\nMrs. Goddard smiled.\n\n\"Doctor Colfax is one of the best men I ever knew.  His left hand knows\nnot what his right hand does in the way of charity, and his whole life,\ninstead of being devoted to amassing a fortune, is given up to the\nhealing of mankind.\"\n\n\"Why, I thought Scientists did not like doctors!\" cried Carolina.\n\n\"We admire their intentions.  Who could fail to?  Among them are some of\nthe noblest characters I have ever known in any walk of life.\"\n\n\"But,\" cried Carolina, alarmed by this praise, \"you don't believe that\nwhat he says is true?  Why, Rosemary assured me--\"\n\n\"And I assure you no less than Rosemary,\" said Mrs. Goddard, \"that God\nis able and willing to heal all such as repent of their sins and come to\nHim with an humble and contrite heart.  You are the best judge of\nwhether your heart is right toward your enemies.  Can you bring yourself\nto love this man who has defrauded you of your inheritance?  If not, you\nhave no right to expect God to restore it to you.  Now think this over\nwhile I give you a treatment.\"\n\nCarolina watched her in so great a surprise that she forgot to think\nover her grievance against Colonel Yancey.  Mrs. Goddard leaned her\nelbow on the arm of her chair, and pressed the tips of her fingers\nlightly against her closed eyes as if in silent prayer.  Her lovely face\nframed in large ripples of iron-gray hair, her gown of silvery gray, her\nfigure still youthful in its curves, her slender, spiritual hands, her\nearnest voice, and tender, helpful manner, formed so beautiful an image\nin Carolina's mind, and she longed so ardently to model herself upon the\nspirit she represented, that tears welled to her eyes when she\ncontrasted her own attitude with Mrs. Goddard's, and when she recalled\nherself with a start, to the subject of Colonel Yancey, she found to her\nsurprise that his importance had so diminished that he had receded into\nthe background of her thought, and the thing she most ardently desired\nwas not Guildford, but to put herself right with God, her Father!\n\nAt the moment that this thought formulated in her mind, a flood of\ndivine peace poured over her whole spirit, and for the first time the\npain of her bereavement lessened, and then gently passed into\nnothingness.\n\nGod her Father!  A God of infinite tenderness and love!  One who loved\nher even as her own dear father had loved!  One who was not responsible\nfor all the evil which had descended upon her!  One who owed her only\nlove and protection, and a tenderness such as she had received in its\nhighest earthly form from her father.\n\nIn vain Carolina struggled to deify God above her earthly father.  She\nhad loved him in so large and deep and broad a manner that she could\nonly realize her new God by comparing Him to her father. And Divine\nScience had sent this new interpretation of God to her to take the place\nin her sore heart of the ever-present aching sense of her great loss.\n\nWhen Mrs. Goddard ended her treatment and opened her eyes, she sat for a\nmoment in silent contemplation of the transfigured face before her.\nCarolina's beauty, as she thus, for the first time, beheld the face of\nher Father, was almost unearthly. It was as that of the angels in\nheaven.\n\nA wave of generous thanksgiving and rejoicing swept over the soul of her\npractitioner, for she knew that she had been permitted to be the\ninstrument in God's hands of healing a soul which had been sick unto\ndeath.  Carolina's bodily healing took second place in her thought, yet\nher confidence was sound that that was even now being accomplished.\n\nWhen Carolina met her eyes, she smiled.  She had found peace.\n\n\"Now, dear child, I want to leave with you the ninety-first Psalm.  Read\nit with your new thought in mind, and you will realize that you never\nhave even apprehended it before.  Remember, too, that you are not alone\nany more.  You are cradled in Divine Love, for God is both Mother and\nFather to His children.  'The eternal God is thy refuge, and underneath\nare the everlasting arms!'\"\n\nMrs. Goddard bent and kissed the girl, and Carolina, usually so\nreserved, laid her flowerlike face against the older woman's cheek in a\nsilence too deep for words.\n\n\"Remember, dear, to call on me by day or night exactly as if I were\nDoctor Colfax, for I am your physician now.  But deny your error as soon\nas it makes its appearance and you won't need to send for me.  I will\ncome of my own accord every day and help you in your studies.  Now I\nmust go. Rosemary and I love you already.  Both Divine and human love\nare pouring in upon you in such a manner that you shall not be able to\nreceive it. Good-bye and God bless you, my dear!\"\n\n\n\n\n                              CHAPTER IX.\n\n                           THE TRIAL OF FAITH\n\n\nTo understand Carolina's complete and instant acceptance of the\ndoctrines of Christian Science in addition to her healing, it is\nnecessary to take a more intimate view of her character.\n\nA person of little or no understanding, or of little or no depth, would\nnaturally have accepted the boon of restored health, whether she ever\nwent any further in the doctrine or not.  But Carolina was different.\nTo her the blessing was in a change of thought.  Marvellous as she felt\nher healing to be, her greatest gain was in the peace and happiness\nwhich descended upon her like a garment.\n\nTo be sure she had been in a desperate plight, both physically and\nspiritually, when this wonderful hand was stretched out to her in her\ndarkness and despair, yet many to whom it reaches out refuse its grasp\nsimply from a blind prejudice. Having ears, they hear not, nor will they\nwhen they might.  It argues a particularly lovely spirit to be able to\naccept so freely and gladly.  Carolina was not free from prejudice.  Far\nfrom it.  But she was not stupid.  Aside from a clear, spiritual\nunderstanding, to be able to accept Christian Science demonstrates no\nsmall degree of mentality, clearness of perception, and a capacity for\nhigher education.  The Science of Metaphysics does not appeal to fools,\nand only wise men pursue it. Christian Science is the only religion\nwhich calls in any dignified way upon a man's brain.  All the others\nstuff one's intelligence with cotton wool, bidding the questioner not to\nquestion but believe. Believe what his ordinary human intelligence\nrepudiates.  \"If you don't understand all of me,\" says popular religion,\n\"skip what you don't understand and go on to the next.  If you keep on\nlong enough you will find something that you can believe without any\ntrouble.  Let that satisfy you. Forget the rest.\"\n\nBut when a metaphysical interpretation of the Scriptures comes along\nsaying: \"Ask any question you will and I will give you an answer that\nwill satisfy the best brains and highest order of intelligence among\nyou, for the day of blind belief is past, and the day of understanding\nis at hand,\" then the highest compliment which can be paid to the\nmentality of the most brilliant man and woman, is to say: \"They are\nChristian Scientists.\"\n\nThere may be--there are, many erratic minds attracted by Christian\nScience, but there are no complete and utter fools among its followers,\nfor the mere fact that a man has sense enough to grope after the very\nbest, instead of being satisfied with that which never completely\nsatisfied the mentality of any man or woman of real intelligence, is an\nevidence that some degree of wit must be entangled in the meshes of his\nfoolishness.  While on the other hand it is doubtful if there ever was a\nforty-year old sect in the knowledge of man which numbered the multitude\nof brilliant minds which are within the annals of Christian Science.\n\nCarolina, all her life, had been, not only surrounded by, but familiar\nwith the best.  Her father's and mother's brilliance and good taste had\ndrawn around them many of the finest minds in Europe, so that the girl's\nmentality was as ripe for the highest form of religion as it was of\nliterature or art.\n\nShe plunged into the study of it with all the ardour of an enthusiastic\nintelligence, and heaved a sigh of relief when she realized that at last\nshe had found a dignified religion, free from every form of\nsuperstition, from all material symbols, and, above all, one which made\nit possible intelligently to obey the command, \"Be ready always to give\nan answer to every man that asketh you a reason of the hope that is in\nyou\" (1 Peter iii. 15).\n\nHer greatest fear was that she would be unable to curb the hot temper\nwhich mortal mind had made into the law that it was a Lee inheritance.\n\nShe particularly dreaded her first interview with Noel St. Quentin,\nKate, and Cousin Lois.  She had yet, also, to face Doctor Colfax.  She\nhad not seen him since, by Mrs. Goddard's advice, she wrote him a frank\nlittle note, saying that her healing had been marvellously hastened by\nChristian Science, and that she had so much faith in it that she felt\ncompelled to relinquish all claim on materia medica, but that, in doing\nso, she wished to acknowledge most gratefully all that his skill had\naccomplished in her case.\n\nIt was a hard note to write, for Kate's assertion, which at first\nCarolina had indignantly repudiated, that Doctor Colfax was falling in\nlove with her, had proved true, and Carolina knew that this dismissal of\nhim as her physician would indicate that he need expect nothing more of\nher in any other capacity, either.\n\nHe wrote her a polite but stiff letter of acknowledgment, and soon\nafterward went away for a brief vacation.\n\nCarolina realized how much antagonism she had aroused among her own\nimmediate friends, and she spent many hours consulting Mrs. Goddard how\nto conduct herself with tact.\n\nWhen Mrs. Winchester returned from Boston, Carolina experienced her\nfirst battle with error. She possessed a high spirit, and to see Cousin\nLois sit and look at her in silent despair, with tears rolling unchecked\ndown her cheeks, irritated Carolina almost to the verge of madness, so\nthat instead of waving aloft the glorious banner of a new religion,\nCarolina found herself longing to box Cousin Lois's ears.  Anything,\nanything to stop those maddening tears!\n\nShe could only control herself by a violent effort. Mrs. Winchester,\nlike Kate Howard, was an ardent churchwoman, and to both these women\nCarolina's acceptance of Christian Science was the greatest blow which\ncould have fallen on them, short of her eloping with the coachman.  They\nfelt ashamed, and in no small degree degraded.\n\n\"Whatever can you see in it?\" demanded Mrs. Winchester, plaintively, one\nSunday morning just after she returned from church.  \"Why need you go to\ntheir church?  Why can't you continue in the church you were baptized\ninto as a baby?  I don't care what you believe, just so you go to the\nEpiscopal church!  It is so respectable to be an Episcopalian!  Oh,\nCarolina, as I sat there listening to that sermon to-morrow--oh,\nCarolina, how can you laugh when I am so serious!\"\n\n\"Do forgive me, Cousin Lois, but you couldn't be any funnier if you said\nyou had seen something week after next!\"\n\n\"I am glad to know that a Christian Scientist can laugh,\" sighed Mrs.\nWinchester, whose mild persistency in investing the new thought with\nevery attribute that she particularly disliked was, to say the least,\ndiverting.\n\n\"Am I improved or not since I began to study with Mrs. Goddard?\"\ndemanded Carolina, with recaptured good humour.\n\n\"I don't see any improvement, my dear.  To me you were always as nearly\nperfect as a mortal could be!\"\n\n\"Dear loyal Cousin Lois!\" said Carolina.\n\nShe seldom kissed any one, but she kissed Mrs. Winchester, who blushed\nwith pleasure under the unusual caress.\n\n\"Perhaps,\" she added, cautiously, \"you are a trifle more demonstrative,\nbut I always thought your apparent coldness was aristocratic.\"\n\n\"It wasn't,\" said Carolina, decidedly.  \"It was because I didn't care.\"\n\n\"And now?\" questioned Mrs. Winchester, wistfully.\n\n\"Now,\" cried Carolina, \"I care vitally for everything good!\"\n\n\"You always did, I think,\" said Mrs. Winchester. \"Even as a child you\nalways gravitated toward the highest of everything.  You are too\nremarkable a girl, Carolina, to throw yourself away at this late day on\na fad which will die a natural death of its own accord.\"\n\n\"May I be there to see when Christian Science dies!\" cried Carolina,\nbrightly.  She felt ashamed that she had ever lost patience with any one\nwho loved her as idolatrously as Cousin Lois.\n\n\"Doctor Colfax--I forgot to tell you that I met him on the train, and\nthat he asked fifty questions about you that I couldn't answer--Doctor\nColfax will certainly be nonplussed when he sees you walking with only\nthat cane.  He told me he never expected to see you walk without two\ncrutches.\"\n\n\"Then you do give Christian Science credit for that much, do you?\" asked\nCarolina.\n\n\"Oh, yes.  It must have some wonderful power. I simply don't understand\nit, that's all.  And Carolina, it seems so--excuse me, but so\ndisreputable!\"\n\n\"Does it?  I hadn't thought of it in that light.\"\n\n\"And so unsexing!  Don't you have women in the pulpit?\"\n\n\"Yes.  Christian Science recognizes woman as the spiritual equal, if not\nthe spiritual superior, of man.\"\n\n\"There!\" said Mrs. Winchester, triumphantly, as if having scored a point\nagainst the new religion. \"Yet woman caused man's fall!\"\n\n\"No, she didn't, Cousin Lois.  Christian Science doesn't take that\nallegory as history.\"\n\n\"Oh, Carolina!  Carolina!  You are indeed in a sad way when you forsake\nthe faith of your ancestors!  Such disloyalty cannot fail to have a\ndepressing effect upon your character!\"\n\n\"On the contrary,\" said Carolina, \"it is as exhilarating to kick down\nall one's old, stale beliefs as a game of football.\"\n\nAt this Mrs. Winchester's asthma returned. There was nothing left for\nher to do, in her state of mind, but to choke or to swoon.\n\nA few evenings later Doctor Colfax telephoned to Kate that he would drop\nin for a few minutes after dinner.\n\n\"H-he can't stand it for another minute, Carolina!\" cried Kate.  \"I am\ncrazy to see his face when you walk in without your crutches!  C-Carol,\ncouldn't you take an extra treatment or so, and come in without even\nyour c-cane?\"\n\nCarolina's eyes blazed with joy at this unconscious admission on Kate's\npart that she believed even that little in the new faith.\n\nFor reply Carolina rose by means of the arms of her chair, and without\nany material aid whatsoever took half a dozen steps.\n\n\"Oh, Carol!  Carol!\" shrieked Kate, bursting into tears.  \"Y-you never\neven limped!  Oh, it's l-like the d-days when Christ was on earth to\ns-see a m-miracle like that!\"\n\nShe seized her friend in her arms and almost lifted her from her feet.\n\n\"D-do it to-night, Carolina, and we'll knock their eye out!  I'll get\nthe whole family together, a-a-and you j-just walk in like that!  Will\nyou?\"\n\n\"Yes, if you will go away and let me work over it this afternoon.  And\ndon't tell anybody!\"\n\n\"Oh, certainly not!  That would spoil the surprise.\"\n\n\"I don't mean for that reason.  I mean that outsiders' adverse thought\nwould hinder my work. Mortal mind makes false laws.\"\n\n\"C-could you just as well t-talk United States when you are heaving your\nideas at me?\" pleaded Kate.  \"Y-you know I'm not on to the new jargon,\nand I fail to connect more than half the time.\"\n\nAs Carolina laughed, Kate nodded her head with great satisfaction.\n\n\"I am glad to see that Christian Science has not destroyed your royal\nsense of humour,\" she said.  \"Now I'm off to let you w-work!\"\n\nBut when the door closed behind Kate, a prolonged sense of\ndiscouragement seized Carolina. She looked forward to the evening with\ndread. Kate made fun of it, Doctor Colfax was coming purposely to scoff,\nand she knew that she was to be made conspicuous because of her\nreligion.\n\nShe tried to walk without her cane, but her knee bent under her and she\nfell to the floor.  Her first impulse was to burst into tears, but, as\nshe lay there alone, too far from the bell to summon help, apparently\nwithout human aid, she fancied she heard the voice of Mrs. Goddard\nrepeating: \"For He shall give His angels charge over thee, to keep thee\nin all thy ways.  They shall bear thee up in their hands, lest thou dash\nthy foot against a stone.\"\n\nShe said this over and over to herself, and it comforted her.  Then the\nface of Mrs. Goddard came before her mental vision, and the lovely\nearnestness of her voice sounded in Carolina's ear. She remembered her\nlast words, which now came back to her with strange and timely\nsignificance:\n\n\"The way will not always be smooth beneath your feet.  Error in the\nguise of fear, selfish or vainglorious thoughts, revenge, self-pity, or\ndesire to shine before others will sometimes cause you to stumble and\nfall.  But at such times, remember to blame, not circumstances nor\nothers, but your own faulty thought.  Be severe with yourself.  Then\nturn your thought instantly to the Source of your supply.  No one can\nhelp you, Carolina, but God, your Father, Divine Love, the All in All of\nyour existence, your very Reason for being.  Realize that God is all\nthere is.  Beyond Him there is nothing and nothingness.  Breathe His\nspirit.  Drink in His divine power.  Make yourself one with Him, and you\nwill instantly find that the mists which covered the surface of your\nspiritual reflection of His image will disappear, and you will begin to\nreflect His government clearly.  At that same moment, you will be healed\nof your infirmity.\"\n\nAs she repeated these last few words aloud, a feeling of complete\nsecurity took possession of her, and she rose, first to her knees, then\nto her feet, and walked confidently to her chair by the window.\n\nIn great thankfulness she took her Bible and read the fifth chapter of\nLuke, and, when she came to the twenty-fourth and twenty-fifth verses,\nshe read them three times, with a heart full of gratitude.\n\nStill she was not satisfied.  She was groping after a sign, and she read\non until she came to the words, \"And when they bring you unto the\nsynagogues, and unto the magistrates and powers, take ye no thought how\nor what thing ye shall answer, or what ye shall say.  For the Holy Ghost\nshall teach you in that hour what ye ought to say.\"\n\n\"The Holy Ghost!\" thought Carolina.  \"I wonder what that really is.\nThat is one of the things I never could understand in the old thought.\"\n\nShe turned to the Glossary in \"Science and Health,\" and there the first\ndefinition of Holy Ghost was \"Divine Science.\"\n\n\"I am answered,\" she said, with a sigh of complete satisfaction.  \"For\nthe first time in my life I begin to understand the fourteenth chapter\nof John.\"\n\nShe leaned her head against the window-pane to watch the postman come\ndown the street.  Then she heard his whistle, and presently the maid\nbrought her a letter.  She asked the maid to turn on the electric light,\nand, when she had done so and left the room, Carolina read the following\nletter:\n\n\n\"LONDON, May 6, 19--\n\n\"MY DEAR MISS CAROLINA:--You have rejected my suit so often, when I had\nno inducement to offer you except a heart which beats for you alone,\nwhich seems to be no temptation to you, that I shall not pay you the\npoor compliment of offering myself to you again when, as you must have\nheard, I have become the owner of Guildford.\n\n\"But, having heard of your great misfortune and of your change of\nreligion, and knowing that you love the old home so ardently that its\natmosphere might effect a cure when all else failed, I beg you to accept\nGuildford as it stands, as a gift from your father's old friend,\n\n\"WAYNE YANCEY.\"\n\n\nCarolina's first impulse, having read the letter twice, was one of the\ncold fury she used to feel when a child, and she turned pale with a rage\nwhich was unspeakable in its violence.\n\nToo well she saw through the malice of the whole affair.  Colonel Yancey\nknew that, after her first impact of anger had passed, her next thought\nwould be to wish she could buy the estate back, and these terms he\nintended to make prohibitive. Carolina wondered if he expected to wear\nout her patience, and so force her to marry him, or what? She could not\nhope to follow with accuracy the tortuous windings of a mind as\nintricate as Colonel Yancey's, and she despaired of ever realizing that\nthe labyrinth could untwist into the straight and narrow way to which\nshe was accustomed.  But, so far from crushing her, this letter simply\nroused in her the valiant spirit of the Lees.  So far from feeling\ndownhearted, she began to sing.\n\nBut it was not a worldly courage which was sustaining her.  It was the\nspirit which had grown out of her afternoon of work.\n\nShe deliberately took her cane with her as she went down to dinner,\nalthough she felt that she could walk without it.  She knew that Kate\nwanted the surprise to be complete.\n\nWith this end in view, she sat at the table until the footman announced\nDoctor Colfax, and then she allowed all the others to precede her.\n\n\"N-now wait until we have all had time to shake hands, and a-ask him how\nhe enjoyed himself, and give him a chance to be disappointed or\ng-gloating, just as he feels, because y-you aren't down.  Then y-you\nskate in and w-watch him drop!  We'll have him a Christian Science\npractitioner b-before we are done with him!\"\n\nCarolina obeyed.\n\nThey were all there,--Mr. and Mrs. Howard, Kate, Cousin Lois, Doctor\nColfax, and Noel St. Quentin, and all were under the impression that\nCarolina would never be able to walk without some slight support.  So\nthat, when she walked slowly through the door, taking her steps with\ngreat care, that she might more gloriously reflect the Light, a hush\nfell upon them all.  They did not greet her. They rose to their feet and\nstood watching her in perfect silence, and it was not until Kate sobbed\nin her excitement that the spell was broken.\n\nNoel St. Quentin bit his lips, and Doctor Colfax's face went from red to\nwhite in an emotion which no one could fathom.  Was he chagrined to see\nthe woman he loved cured?  Did he grudge her healing at other hands than\nhis?\n\nThey all began to speak at once.  Only Mr. Howard, Kate's father, sat\nback and watched and listened.\n\nRoscoe Howard was a remarkable man in many ways.  He possessed a\ncritical mind, large wealth, great depth of character, and a sureness\nand quickness of perception, which had all contributed to his success in\nlife.  He was a student, above all, of human nature, and he had insisted\nupon Kate's willing hospitality to her friend, partly from affection to\nthe daughter of his old friend, Winchester Lee, and partly to see what\neffect such an avalanche of misfortunes would have upon the proud spirit\nand high-strung nature of Carolina.  When he heard of her embrace of\nChristian Science, he became still more interested.  He had once gone in\nto sit with her when her arm was bandaged from wounds from her own teeth\nin one of her fits of despairing rage.\n\nTherefore, when he learned from his daughter that this was to be the\ngirl's first appearance before her old friends, he could imagine the\nordeal it would prove to her, and in his own mind he said: \"Carolina\nwill show us to-night whether she is The Lady or The Tiger!\"\n\nAt first they all tried to be polite and remember that they were\ncivilized, but soon that curious unable-to-let-it-alone spirit which\nChristian Science invariably stirs in mortal mind began to manifest\nitself in hints and covert remarks and side glances and meaning\nsilences, until Carolina calmly looked them in the eyes and said, in her\ngentlest manner: \"I am perfectly willing to talk about it.\"\n\nKate clutched her mother's arm.\n\n\"I-isn't Carolina a d-dandy?\" she whispered. \"Takes every hurdle without\neven stopping to measure it with her eye!\"\n\n\"Well, doctor, since Carolina has given us permission to discuss it,\nwhat have you to say about it?\" asked Mrs. Howard.\n\n\"I can simply say this,\" said Doctor Colfax.  \"I don't understand it.\nBut, then,\" he added frankly, \"I don't understand the Bible, either.\"\n\n\"Then that is why you don't understand my cure, doctor,\" said Carolina,\nquietly, \"for it is founded on the promises which Christ explicitly made\nto His disciples.\"\n\n\"To His disciples,--yes,\" replied Doctor Colfax, quickly, \"but not to\nus.  We are not His disciples.\"\n\n\"If you are a thorough Bible student,\" said Carolina, \"please tell me\nthe exact words of His promise.\"\n\n\"I am not.  You have me there, Miss Lee.\"\n\n\"Well,\" persisted Carolina, \"where did He limit the power He gave, and\nwhich you admit existed at one time, to His disciples?  Did He ever say,\n'I will give it to you and to no other?' or 'I will give it to you\nduring my lifetime, but after my ascension it will return unto me,\nbecause you will no longer have need of it?'\"\n\n\"No, I can't remember any such passages,\" admitted Doctor Colfax.\n\n\"W-well, He never s-said anything of the kind,\" put in Kate.  \"I don't\nknow much, but I know that!\"\n\n\"What did He say, Carolina?\" asked St. Quentin. \"Do you remember the\nexact words?\"\n\n\"Yes, I do.  In one place He said: 'He that believeth on me, the works\nthat I do shall he do also.  And greater works than these shall he do\nbecause I go unto my father.'  And at another time He said: 'Heal the\nsick, cleanse the lepers, raise the dead, cast out devils.  Freely ye\nhave received. Freely give.'  Now when did the time limit to those\ncommands end?\"\n\n\"Oh, nonsense, Carolina!\" said Mrs. Howard, with the amused toleration\nof the already saved. \"How can you bring up such absurd speculations?\nAll those questions have been settled for us by the heads of the\nChurches years and years before we were born.\"\n\n\"They were settled, dear Mrs. Howard, for all who choose to accept such\ndecisions, but how about those of us who have questioned all our lives\nand never found an answer which satisfied?  I can remember, as a little\ngirl in Paris, I used to come home from the convent and ply my father\nwith this very question: 'Why can't priests and preachers heal in these\ndays the way Jesus commanded?'\"\n\n\"Well, does Mrs. Eddy have the nerve to assert that she rediscovered the\nway to perform Christ's miracles?\" asked Doctor Colfax.\n\n\"Mrs. Eddy asserts that in 1866 she discovered the Christ Science, or\nthe power of healing disease as Jesus healed it, by a mental process\nwhich is so simple that to all Christian Scientists Christ's so-called\nmiracles are not miracles at all, but as simple and natural as any other\nmental phenomenon which has become common by reason of its frequency.\"\n\n\"That sounds like sacrilege,\" said St. Quentin.\n\n\"It sounds like tommy-rot!\" said Kate.\n\n\"And yet,\" put in Mr. Howard, \"we must all admit that Carolina has been\nmiraculously healed. Do you not admit that, doctor?\"\n\nDoctor Colfax's face became suffused.  He bit his lip, then said, with\nquiet distinctness:\n\n\"If I had cut off a man's leg with my own hands, and Mrs. Eddy, under my\nvery eyes, caused a new leg to grow in the place of the old one, I would\nnot believe in her or in anything she taught!\"\n\nExpressions of varying emotions swept over the faces of his listeners at\nthis sincere statement of unbelief,--some were triumphant, some\nincredulous, some surprised, and one contemptuous.\n\n\"But, doctor, when you see Christian Science enrolling the names of the\nmost brilliant minds; when you see the loveliest women forsaking a life\nof ease and pleasure and becoming practitioners,--Christian Science\ndoctors just as selfless and single-minded as you--\"\n\n\"If you are referring to that depraved woman who claims to have cured\nyou, Miss Lee, that morphine fiend, that drunkard, that reformed\ncharacter, I beg that you will not name her as a physician in any sense\nof the word.  The medical profession is too noble to be degraded in such\na manner!\"\n\n\"Oh, doctor,\" cried Carolina, reproachfully, \"if you could only hear the\nbeautiful way in which she speaks of you!\"\n\n\"Oh, doctor, aren't you a little severe?\" asked Mrs. Winchester.\n\nNoel St. Quentin smothered an amused laugh.\n\n\"Pooh!\" cried Kate.  \"Why pay any attention to him?  He's o-only a man,\nand men are always wrong!  H-he's talking through his h-hat, that's\nw-what he's doing.  He's jealous.\"\n\nShe was sitting near St. Quentin, and, turning to him under cover of the\nconversation, she murmured:\n\n\"What are you laughing at behind your hand?\"\n\n\"I was simply remarking a phenomenon that I have often remarked before,\nand that is, that Christian Science seems to possess a peculiar power--\"\n\n\"Oh, oh! are you going over to the enemy?\" asked Kate.\n\n\"You didn't let me finish.  I was going to say that it possesses a\npeculiar power of making well-bred people forget what is due a civilized\ncommunity.  I have never, I think, heard so much rudeness, such rank\ninelegance, such brutal prejudice expressed on any subject which polite\nsociety discusses.  It takes Christian Science every time to make people\nabsolutely insulting to their best friends.\"\n\n\"Funny, isn't it?  I don't mind it so much since Carolina got into it;\nshe is so honest and so brave about answering it, b-but I used to hate\nit so it c-cankered the roof of my mouth j-just to speak the name of\nit.\"\n\n\"Another curious thing I have noticed,\" said St. Quentin, speaking for\nKate's ear only, \"is that those who hate it most violently at first\ngenerally end by adopting it, so look out!\"\n\n\"You don't mean it!\" cried Kate, in such a horror-stricken voice that\nevery one heard her. \"D-don't ask me what we are t-talking about,\nbecause it is not f-fit for you to hear,\" she cried.\n\n\"Carolina,\" said Mr. Howard, tactfully, \"please tell us what you have\nfound in Christian Science. I have always had a great respect for your\nintelligence, and I am not prepared to find it befogged in this\ninstance, or that you have been deceived.\"\n\nHe never forgot the luminous gratitude of her look.\n\n\"Thank you, dear Mr. Howard.  Let me see if I can tell you what it is\nand what it has done for me.  It is the theory of mind over matter, put\nin practice and lived up to.  It teaches us to understand before we are\ncalled upon to believe.  It is the study of Christian metaphysics, or\nmetaphysics spiritualized.  It takes all the impossible out of the\nScriptures, and makes them understandable, not to a fool, but to the\nwise man,--the man capable of understanding a great matter.  Having done\nthis for the brain, it teaches so absolutely a God of Love, a God who is\nboth father and mother in the love and yearning tenderness of His\nthought toward us, that it eliminates all fear from our lives.  All\nfear! Can you take that in at once?  It makes the ninety-first psalm a\npersonal talk between a father and his dearly loved child.  To me it\nsounds just as if daddy were talking to me from the Beyond.  That would\nbe just his attitude toward me if he possessed God's power.  And if you\nbelieve it,--if you can once let yourself believe it, it makes this\nearth instantly into heaven.\"\n\n\"Yes, yes, I can see that it would,\" said Mr. Howard.  \"But do not\nScientists believe that it also prospers you in a worldly sense?\"\n\n\"Are you giving Kate everything that heart could wish now, and are you\ngoing to leave her all your money when you die?\" asked Carolina.\n\n\"That knocked his eye out,\" murmured Kate, in an aside to St. Quentin,\nbut he observed that she looked singularly pleased when Carolina scored\na point.\n\nMr. Howard waved his hand in a slightly deprecatory way.\n\n\"Ah, that is just it!\" cried Carolina.  \"You are thinking, 'Oh, but,\nCarolina, I am Kate's own father, and God is just God!'  Heavenly Father\ndoesn't mean a thing to most Christians.  Christian Scientists can't\nshirk their beliefs.  If they do, they are just as they were\nbefore,--pretending or rather trying to believe what they feel that they\nought to believe, but getting no satisfaction and no comfort from it.  A\nScientist who does not put his belief into practice can neither heal his\nown body nor others.  So he is literally forced to be honest.\"\n\n\"Well,\" said St. Quentin, \"I can easily see where the supreme and\nslightly irritating happiness of Christian Scientists comes in.  I could\nbe supremely happy myself if I could believe in it.\"\n\n\"So could I,\" declared Kate.  \"A-and I suppose it is sheer envy on my\npart, when I see their Cheshire-cat grins, to want to slap their faces\nfor being happier than I am!\"\n\n\"But what makes them so happy?\" asked Mrs. Winchester, plaintively.\n\"Why should they be any happier than we are?  We both have the same\nBible, and I flatter myself that I am just as capable of understanding\nit as any self-styled priestess of a new religion.\"\n\n\"But _do_ you understand it, Cousin Lois?\" asked Carolina, gently.\n\n\"I understand all that is good for me, dear child. I understand all that\nour Lord wants me to, or He would have made me Mrs. Eddy and made Mrs.\nEddy, Mrs. Winchester.  We are fulfilling God's will.\"\n\n\"I d-don't believe that, either,\" whispered Kate to St. Quentin.  \"I--I\nhave to admit that Carolina's God is a more consistent Being than Mrs.\nWinchester's.\"\n\n\"But you have not answered my question, Carolina,\" said Cousin Lois.\n\n\"What makes us so happy?  Well, I wonder if I can tell you.  In the\nfirst place, it is the relief of dropping all anxiety.  We don't have to\nworry about a single solitary thing.  We put all responsibility off on\nGod.  You know it says 'Cast thy burdens on the Lord!'\"\n\n\"But how can you?\" cried Kate.  \"I--I'm sure I'd like to, but I c-can't\nget my own consent.\"\n\n\"That's exactly it.  Well, we do it.  Then, having put all fear out of\nour lives, what is there left to make one unhappy?  If you are no longer\nafraid of losing your health or your money or of dying or of being\nmaimed or injured in accidents by land or sea, or of old age or any\nmisfortune coming to any of your dear ones, so that it leaves you\nperfectly free to come and go as you please, to eat at all hours things\nwhich used to produce indigestion, to eat lobster and ice-cream\ntogether, drink strong coffee late at night and drop off to sleep like a\nbaby, and, if it eliminates all dread of the unseen and the unknowable,\nwhat more is there left to fret about, I'd like to know?\"\n\n\"How about waking up in the middle of the night to worry about your\ndebts?\" asked St. Quentin.\n\n\"The answer to that is that, at first you begin by remembering that as\nGod is the Source of all supply, if you are consistent, the way will be\nopened to pay your debts.  And, after you once master that comforting\nfact, it is easy to see that the next thing will be that you won't wake\nup in the night to worry or even to think.\"\n\n\"Carolina!\" exclaimed Mrs. Winchester, \"do you mean to tell me that you,\nwho used to lie awake hours and hours every night of your life, can\nsleep through till morning?\"\n\n\"I do, Cousin Lois.  Often actually without turning over.  And with no\nbad dreams.  Can you believe me?\"\n\nDoctor Colfax rose abruptly, as if he could bear no more, and when, with\na little more leave-taking, St. Quentin had offered to drive Mrs.\nWinchester back to Sherman's in his new motor-car, and the Howards and\nCarolina were left alone, Mr. Howard turned to Carolina and said:\n\n\"Carol, I have heard a great deal, here and there, about your interest\nin Guildford and your wish to restore the place.  Would you mind telling\nme your plans?\"\n\n\"Not in the least, Mr. Howard.  The place has been sold under its\nmortgage, as you doubtless know, but it is of no more value to its\npresent owner than any of the land surrounding it, which is equally\narable.  Its only value to us was because it was our ancestral estate.\nIt has a water-front, and, having been left intact for over two hundred\nyears, its timber is enormously valuable.  If I owned it, and had a\nlittle working capital, I could pay off the mortgage and restore the\nhouse with the timber alone.\"\n\n\"Why, how is that, Carolina?  Is it so extensive as all that?\"\n\n\"It is only about two thousand acres,--a mere handful of land to a\nNorthern millionaire, who buys land along the Hudson and in the\nCatskills and Adirondacks of ten times that amount, but that is a very\ndecent size for a Southern plantation.  But the value is in the kind of\ntimber.  It is long-leaf yellow pine, which produces turpentine and\nrosin first, by the orchard process, then what is left is suitable for\nthe lumber men, and the fallen trees and stumps for the new process of\nmaking turpentine. My plan was to sell the turpentine rights to the\norchard people for, say, three years, then sell the timber, and\nafterward sell the stumpage and refuse to the patent people, or perhaps\nerect a plant myself.  There is a tremendous profit in turpentine and a\nconstant and ready market.\"\n\nMr. Howard sat in a large armchair, with his finger-tips together and\nhis head bent forward, looking at the girl from under his heavy\neyebrows. He was amazed at her statement of Guildford's possibilities.\nHitherto he had regarded her unknown plan as probably only a woman's\nsentimental idea, and doubtless wild and impracticable.\n\n\"You say that the timber has been untouched for two hundred years?\"\n\n\"Practically untouched.  We had it examined four years ago, and I have\nheard of nothing since.\"\n\n\"Is any of this land suitable for cotton?\"\n\n\"Yes, for both cotton and rice, and I should raise both.  There is no\nreason to my mind why a Southerner should not be as thrifty with every\nacre of ground as the Northerner is, nor why every inch should not be\nmade to yield in America as it does in France.\"\n\n\"Right! right!  And the Southerners will accept such incendiary\nsentiments from you, because you are one of them, but, when I ventured\nsomething on the same order, but much more mild, I was called 'a damned\nYankee,' who wanted to 'make truck-farmers out of gentlemen.'\"\n\n\"Oh, oh!\" laughed Carolina, merrily.  \"How like them that sounds!  You\nknow, dear Mr. Howard, they think we have no gentlemen in the North.\"\n\n\"T-they aren't far from it,\" cried Kate.  \"There are f-few gentlemen\nanywhere in the world, according to m-my definition of one.\"\n\n\"You say Guildford is sold?\" said Mr. Howard.\n\n\"Yes, Sherman was obliged to mortgage it, but he did so without knowing\nhow dearly I loved it. Then some one bought the mortgage and foreclosed\nit.\"\n\n\"Why, who could have done such a thing? There must have been a motive.\nHas coal been discovered on any of the surrounding property?\"\n\n\"Not that I know of,\" said Carolina, in a guarded tone.\n\n\"Then there must have been some motive in the mind of the purchaser,\"\nsaid Mr. Howard, decisively.\n\nCarolina was silent.\n\n\"Can you throw any light on the subject, Carol?\" he persisted, but his\nmanner was so kindly that Carolina could not take offence.\n\nHer reticence arose from two causes.  One, her natural wish not to bruit\nher private affairs abroad, and the other that Mrs. Goddard had enjoined\nstrict silence on her.  \"Nothing can be lost in Truth,\" Mrs. Goddard had\nsaid, \"nor are the channels of God's affluence ever clogged, but mortal\nmind makes laws which we are obliged to overcome. Therefore, the fewer\npeople who know about it, the easier our work will be.\"\n\nHowever, something in Mr. Howard's manner led Carolina to suspect that\nhe was not seeking to be informed out of idle curiosity, and her heart\ngave a bound at the thought that perhaps Divine Love might be using him\nas a channel.\n\nNoticing her momentary hesitation, he said:\n\n\"You need not fear to confide in me, Carol. Perhaps I can be of some\nhelp to you.\"\n\nAgain she hesitated.  She knew that the Howard family knew of Colonel\nYancey's attentions to her. Still she felt that she must venture.\n\n\"The present owner of Guildford is Colonel Yancey,\" she said, in a low\nvoice.\n\n\"Colonel Yancey!\"\n\n\"Colonel Yancey!\"\n\n\"Colonel Yancey!\"\n\nAnd so occupied was each listener with his own thoughts and mental\nprocesses that each regarded that exclamation as an original remark.\n\nCarolina looked from one to the other of them anxiously, in the short\nsilence which followed.\n\n\"I understand,\" said Mr. Howard, slowly.  \"I think--I--understand!\"\n\n\"And this afternoon,\" Carolina went on, \"I received a most extraordinary\nletter from him, dated at London, making me a present of Guildford.\"\n\n\"Making you a p-present of it!\" cried Kate. \"What g-gigantic impudence!\"\n\n\"He did it to irritate her into taking some notice of him!\" declared\nMrs. Howard.\n\n\"H-he did it to show her how h-helpless she is!\" cried Kate.  \"He knows\nshe has n-no money.  But I think I see him hanging around until he wears\nCarolina out.  That is his g-game!  A n-nice step-m-mother you w-would\nmake to those two children of his,--and the l-little one a <DW36>!\"\n\n\"Children!\" cried Carolina, turning white.  \"I never knew that there\nwere any!  He never mentioned them.\"\n\n\"Oh, h-he didn't want to d-discourage you t-too much,\" cried Kate.\n\n\"And one of them--the little one--a <DW36>, did you say?\"\n\nThe eager pity in Carolina's voice frightened Kate.  She looked at\nCarolina in wonder.  The girl was leaning forward in her chair, her lips\nparted, her eyes shining, her cheeks blazing.  Kate felt physically sick\nas the thought flashed through her mind that perhaps this altruistic\npity might rush her friend into the marriage with Colonel Yancey, which\neven Guildford had been unable to do.\n\n\"Where is the child?\" asked Carolina.\n\n\"She is at the Exmoor Hospital.  Her aunt, Sue Yancey, brought here\nthere last week for an examination.  They are trying to gain Colonel\nYancey's consent to an operation.\"\n\n\"How do you know all this?\" asked Kate's mother.\n\n\"I went there to take some flowers to-day, and I saw this child,--she is\na little beauty,--and I asked Doctor Shourds who she was and he told me.\nThe trouble is with her ankles.  Her feet are perfectly formed, but they\nturn in and she can't bear her weight upon them, nor walk a step.\"\n\n\"She _can_ walk!\" said Carolina, in a low, earnest voice.  \"God, in His\nDivine Love, never made a crippled baby!\"\n\nSomething smarted in Mr. Howard's eyes.  He, was no believer in\nChristian Science, but he loved little children, and Carolina's tone of\ndeep and quiet conviction wrenched his heart.\n\n\"Carol, Carol!\" wailed Kate, wringing her nose and mopping her eyes,\nwith utter disregard of their redness, \"you do make me howl so!\"\n\n\"Carolina,\" said Mr. Howard, suddenly, \"you know that I do not\npersonally subscribe to the teachings of your new religion, but I am an\nobserver of human nature, and I know the hall-marks of real\nChristianity.  I have seen you to-night keep your temper under trying\ncircumstances, defend your faith with spirit, and exemplify the command\nto love your enemies, and I want to tell you that if there is anything I\ncan do toward financing a plan to buy Guildford from Colonel Yancey, and\ninstalling you there to pursue your life-work, you can count on me.\"\n\nCarolina made an attempt to speak, but her eyes swam in tears, and she\nburied her face in her arm.\n\n\"Oh, daddy! daddy!  D-dear old daddy!\" cried Kate, dancing up and down\nin her excitement. \"I knew y-you were up to something!  Y-you may not\ncare for C-Christian Science, b-but, when you s-see a good thing, you\nknow enough to p-push it along!\"\n\n\n\n\n                               CHAPTER X.\n\n                             CROSS PURPOSES\n\n\n\"Noel must take me for a f-fool if he thinks I don't see through him!\"\nsaid Kate, angrily, to her own image in the glass.\n\nIt was about three months after Mr. Howard had offered to help Carolina\nto regain Guildford.\n\n\"H-he wants to p-pump me,\" she went on, adjusting her motor veil.  \"I\nd-don't mind trying his automobile, b-but I hate to t-think he takes me\nfor a s-sucker!\"\n\nShe rummaged viciously in her top drawer for her goggles.\n\n\"I wonder if he th-thinks I don't know he asked Carol first.  Men are\ns-such fools!  But j-just wait! He wants m-me to tell him things.\nM-maybe I won't g-give him a run for his money!\"\n\nBut, as she ran down the steps and jumped into the powerful new racing\nmachine, all outward trace of vexation was gone, and St. Quentin was\nquite as excusable as most men who believe they can outwit a clever\nwoman.\n\nNot that St. Quentin was particularly noticeable for his conceit.  He\nseemed like the majority of men, who are merely self-absorbed.  Yet in\nmany respects he was quite different.\n\nFor example, he was interested in other things besides his motor-cars.\nHe read, thought even, and was somewhat interested in other people's\nmental processes,--a thing which Kate quite overlooked in her flash of\njealousy, for Kate had been obliged to admit to herself that, if the\nsigns spoke truly and Noel were really in love with Carolina, it would\nbe a melancholy thing for her to face.\n\n\"But I'm game!\" she often said to herself.  \"I won't give up the fight\nuntil I have to.  Then, if I get left, I won't howl.\"\n\nThere were several things in Kate's favour. First, Carolina showed no\nsymptoms of being in love with Noel, although she must know that she\ncould have him if she wanted him.  Second, but this thought gave her\nalmost the same discomfort as if Carolina should fancy St. Quentin,\nCarolina was in a fair way to become violently interested in another\nman,--Colonel Yancey.\n\nThe thought of how this news would stir Noel brought such a colour into\nKate's cheeks that Noel, turning his eyes for the fraction of a second\nfrom the wheel, said:\n\n\"Motoring becomes you, Kate.\"\n\n\"I-it's more than I can s-say for y-you, then,\" she answered.  \"You look\nlike a burglar in that mask.\"\n\n\"Now sit tight,\" said St. Quentin, \"I'm going to let her out a little\nhere.\"\n\nNoel's idea of letting her out a little was more than Kate's nerves\ncould stand.  She touched Noel's arm imploringly and he obediently\nslowed up. Kate could hardly get her breath.\n\n\"Wasn't that fine?\" asked St. Quentin.\n\n\"It was s-simply devilish.  I'd rather travel in a wheelbarrow.  It\ng-gives you more time for the scenery.\"\n\n\"You are just like Carolina.  She hates racing. She likes to jog along\nabout like this.\"\n\nKate leaned over and looked at the speedometer. They were going at the\nrate of thirty miles an hour.\n\n\"P-poor Carolina!\" said Kate, mockingly. \"How old-fashioned we both\nare!\"\n\nNoel laughed and slowed up a little more.\n\n\"There, is that better?\" he asked, with the toleration a man shows when\nhe is fond of a woman.\n\n\"Yes, now I can tell the trees from the telegraph-poles. A m-moment ago\nI thought the r-road was fenced.\"\n\n\"What is Carolina up to these days?  I haven't seen her for over a\nfortnight,\" said St. Quentin.\n\nKate reluctantly admired him for being so honest about it.  Most men\nwould have tried to come at it from around the corner.  Nevertheless,\nshe wanted to carry out her original purpose.\n\n\"She goes to the hospital every day.\"\n\n\"The hospital?  What for?\"\n\n\"Oh, haven't you heard?  Then I have some news for you.\"\n\nKate smiled with wicked enjoyment.  Noel was now about to receive a dose\nof his own medicine, and she was to administer it.  She viciously hoped\nit was in her power to make him as uncomfortable over Colonel Yancey as\nhe made her about Carolina.\n\n\"Well, soon after--why, it was the very night you were at our\nhouse--after you and Doctor Colfax had gone, we still kept on talking,\na-and it came out that Colonel Yancey had never told Carolina that he\nhad children, whereas he has t-two,--the dearest little\ncreatures,--b-but the little one, Gladys, is a hopeless <DW36>.\"\n\nSt. Quentin turned with a start.\n\n\"Yes, that's just the way it struck me.  Of course you g-get the vista.\nCarolina instantly investigated her c-case, and she and Mrs. Goddard got\nit out of the doctors that there was only about one chance in ten of the\noperation being successful, whereas--well, N-Noel, I am not sentimental,\nbut I thank God I--I am human, and when I s-saw the frightened look in\nthe b-blue eyes of that l-little child--that b-baby--she's only\nsix--when she found out th-they were going to cut her, I c-could have\nscreamed.  As it w-was, I c-called them criminals and b-burst out\ncrying, and I b-begged Carol to c-cable Colonel Yancey for p-permission\nto try Christian Science.\"\n\n\"You did just right,\" said St. Quentin.  \"It seems to me that the\nlegitimate and proper place for Christian Science is in a desperate case\nlike that, when doctors agree that they are practically powerless.\"\n\n\"I--I think so, too.  And especially when time cuts no i-ice,--not like\na fever, you know, which must b-be checked at once.  Well, Carol cabled,\nand Colonel Yancey answered in these very words, 'Have no faith, but\nmust respect your intelligence. Do as you think best.'\"\n\n\"By Jove!\"\n\n\"You see?  Oh, Noel, it's s-such a comfort to t-talk to you.  Y-you're\nso clever.  Most men are f-fools.  But do you s-see the diabolical\nflattery of the cablegram?  Do you also see that it puts Carolina in the\np-place of the c-child's mother?  Oh, when I saw the c-colour come into\nher face, as she read that cablegram, and that s-sort of d-dewy\nmother-look she s-sometimes gets in her eyes, I--I could have s-slapped\nColonel Yancey's face for him!\"\n\n\"I know,\" said Noel, in a low, strained tone which woke Kate from her\nenthusiasm to a sense of her own folly.  Her face flamed.\n\n\"Well, I'll be switched!\" she said to herself. \"If N-Noel took me for a\ns-sucker, he didn't half state the case.\"\n\n\"Why don't you go on?\" asked St. Quentin. He looked at her flushed face\nand quivering lips in surprise.  \"Why, I didn't think she had it in her\nto show such feeling!\" he said to himself.\n\n\"I am the m-more afraid,\" she went on, looking straight before her,\n\"b-because Carol doesn't care for any other m-man, so she is f-free to\nfall in l-love with Colonel Yancey, if she wants to.  He is only a\nlittle over forty, is quite the most fascinating man I ever m-met, and\nhe owns Guildford.\"\n\nIf Kate expected St. Quentin to betray any violent emotion on hearing\nthese statements, she was doomed to disappointment.  However, she seemed\nsatisfied at Noel's utter silence.  A smile quivered at the corners of\nher mouth.\n\n\"Well?\" said St. Quentin at last.\n\n\"C-can't you picture the rest?  Can't you see Carol and Mrs. Goddard\ngoing there d-day after day, until Mrs. Goddard got permission to move\nGladys to her house?  I b-believe they were to t-take her there this\nmorning.\"\n\n\"Is there any improvement in the child?\" asked St. Quentin.\n\n\"A little.  She is old enough to understand and help herself, and she\nknows she is g-going to get well, or as she puts it, 'I know that I am\nwell.'  Her ankles have become flexible and her little feet can b-be put\nstraight with the hand, b-but, as yet, they don't stay straight.  S-she\nhas not gained c-control over them.\"\n\n\"Can she stand at all?\"\n\n\"J-just barely.  But she s-sinks right down.\"\n\n\"Do you believe she will be cured?\"\n\n\"I s-suppose you will think I am f-foolish, but I do.\"\n\n\"Not at all, Kate.  I am not sure but that I believe it myself.\"\n\n\"Why, Noel S-St. Quentin!  And you a Roman Catholic!\"\n\n\"Well, why not?  Wouldn't I be an acceptable convert if I should decide\nto join their ranks?\"\n\n\"I-indeed you would not!\" cried Kate, delighted to be able to administer\na stinging rebuff.  \"I have an idea that they would refuse even to\ninstruct you without a w-written permission from your priest.  Ah, ha!\nCan't you j-just see your confessor g-giving up a l-little white\nw-woolly lamb like you?  Y-ye are of more value than many s-sparrows.\"\n\nSt. Quentin accelerated the speed of the machine so suddenly that the\nmotor seemed to leap into the air.\n\n\"Oh, Lord, Noel!  D-don't do that again!  The m-machine can't feel it!\nN-now if you had struck your horse--\"\n\nSt. Quentin turned on her savagely, but said nothing.\n\n\"T-that's right, Noel.  D-don't speak.  There's a good deal in being a\ng-gentleman, after all.  If you h-hadn't been, you would have said,\n'S-shut up, Kate!'\"\n\n\"If your husband,\" said St. Quentin, slowly, \"ever goes to jail for\nwife-beating, I shall bail him out.\"\n\n\"I-it's strange how men agree with one another,\" said Kate, pensively.\n\"M-my cousin has always said that a g-good beating with a bed-slat would\nabout fit my c-case.\"\n\n\"Bright boy!\" said St. Quentin.  \"He ought to get on in the world.\"\n\n\"Hadn't we better turn back, Noel?  I have an engagement at five.\"\n\n\"Do you have to go home to dress, or shall I drop you anywhere?\"\n\n\"I was just going to see Gladys for half an hour. You may drop me at\nMrs. Goddard's if you will.\"\n\n\"Will Carolina be there?\" asked St. Quentin.\n\n\"Yes, I think so.  Do you want to see her?\" asked Kate, innocently.\n\n\"Well, I'd rather like to see her with the child. Will you let me come\nin with you?\"\n\n\"By all means.  I should be delighted.\"\n\n\"Then I can bring you home afterward.\"\n\n\"Most thoughtful of you,\" murmured Kate.\n\n\"I say, Kate,\" said St. Quentin, after a pause, \"keep your eye open for\na toy shop, will you? One oughtn't to call on a child without some\nlittle present, ought one?\"\n\n\"You won't find one up in this part of the country, such as you want,\"\nsaid Kate.  \"Let her out a little and we will have time to go down to\nTwenty-third Street.\"\n\nWhen they came out of the shop, even Kate, extravagant as she was, was\naghast.\n\n\"Noel, it's w-wicked to spend money like that. Why, that child is only a\nb-baby.  She can't appreciate all those hand-made clothes for that doll.\nAnd real lace!  It's absurd!\"\n\n\"Kate,\" said St. Quentin, slowly, \"if you were that crippled baby, I'd\nhave bought you everything in that whole shop!\"\n\nA lump came into Kate's throat so suddenly that it choked her.\n\nWhen they arrived at Mrs. Goddard's, there was no need to ask the butler\nif the ladies were at home, for, instead of the formal household Mrs.\nGoddard used to boast, the house seemed now to have become a home.  Even\nthe butler looked human, as laughter and childish screams of delight\nfloated down the hall from the second floor.\n\n\"Perkins, what is it?\" asked Kate, pausing suddenly.\n\n\"Little Miss Gladys finds that she can stand alone, Miss Howard, and we\nare so delighted none of the servants can be got to do their work.  They\njust stand around and gape at her and clap their hands.\"\n\nBut Perkins himself was smiling as Kate rushed past him up the stairs.\n\n\"Here, Perkins, my man,\" said St. Quentin, \"lend a hand with this, will\nyou, and send a footman out to the motor for the rest of those parcels.\"\n\nThe sight which met the eye was enough to make any one's heart leap, as\nKate flung open the door and joined the group.\n\nThere were Mrs. Goddard, Rosemary, Miss Sue Yancey, Carolina, and the\ntwo children, Emmeline and Gladys.  Gladys was standing in the corner,\npartly supporting herself by leaning in the angle of the walls, but\nstanding, nevertheless, bearing her entire weight upon her slender,\nbeautiful little feet, which never before had been of any use to her,\nnor, in their distorted position, even sightly. Now they were in a\nnormal position and actually bearing her weight, and so excited was\neverybody that no one turned even to give the newcomers a greeting.\nRosemary and Carolina were kneeling on the floor in front of the child,\nwhile Mrs. Goddard was audibly affirming that Gladys could walk. Gladys\nalone looked up at Kate and St. Quentin, and smiled a welcome.\n\n\"Thee, Katie!\" she lisped, \"Gladyth can thtand alone!\"\n\n\"Gladys can walk,\" affirmed Mrs. Goddard, and, as they saw the child\ncautiously begin to remove her hands from the supporting walls and\nevidently intend to attempt a step, Kate snatched the huge box from\nNoel's hands, and, hastily unfastening it, silently held up before her a\ngorgeously beautiful French doll, in a long baby dress, frilled and\ntrimmed with cobweb lace, and calculated not only to set a child crazy,\nbut to turn the heads of the grown-ups, for such a doll is not often\nseen.\n\nNo one saw it at first.  Then Gladys, looking up for encouragement,\nglanced at Kate, and, as her eyes rested on the baby doll, with one\ndelighted mother-cry of \"Baby, baby!\" she started forward and fluttered\nacross the floor, light as any thistle-down, until she clasped the doll\nin her arms, and Kate seized her little swaying body to keep her from\nfalling.\n\n\"See what Divine Love has wrought!\" exclaimed Mrs. Goddard, in a voice\nso filled with gratitude and a reverent exultation that it sounded like\na prayer.\n\nThere were tense exclamations, excited laughter which ended in sudden\ntears, quivering smiles and murmurs of thanksgiving, until Carolina,\nturning to Noel, said:\n\n\"Noel, I am sure that doll was your doing,\" when error again claimed\nKate for its own, for the look of gratitude Noel sent in return.\n\n\"Lord, but this Christian Science does make me t-tired,\" murmured Kate\nto herself, as she released Gladys, and the two children, in a fever of\nexcitement, sat down on the floor to undress the doll.  \"F-first we go\nup, up, up, and th-then we go down, down, down!  J-just as surely as I\nhave an up feeling, I g-get it in the neck inside of the next thirty\nseconds.  A-at any rate, there's no m-monotony about it.  It k-keeps you\nguessing where it will hit you n-next.\"\n\nKate unconsciously made such a wry face as she murmured these words\nunder her breath that Rosemary leaned over and whispered:\n\n\"What's the matter, Kate?\"\n\n\"I th-think I've got an attack of what you call Error, but it cramps me\nmost cruel.  Or d-do you think I could have caught cholera infantum from\nholding that d-doll baby?\"\n\n\"Kate, you are so funny!\" laughed Rosemary.\n\n\"I s-spend a good deal of v-valuable time amusing m-myself,\" said Kate.\n\"I sorta have to, in a way.  Everybody else seems o-occupied.\"\n\nAs Kate made this indiscreet remark about error, Rosemary looked back at\nthe other groups in the room, and surprised Noel looking at Carolina\nwith an expression in his eyes he gave to no other, and again a spasm of\npain crossed Kate's face.  At once Rosemary understood, and Kate saw\nthat she did. Kate's face flamed.  She pushed Rosemary into the\nwindow-seat, thrust her violently down, and pulled the thick crimson\ncurtains together, shutting them in.\n\n\"It's n-not so!\" she whispered, excitedly.  \"I know w-what you think,\nb-but it's not true.  He loves C-Carolina, and in time, no doubt, she'll\nl-love him.  I d-don't see how she can help it.  I d-don't care.\"\n\n\"Oh, Kate, that is not true!  I certainly hope Carolina will not fall in\nlove with him.  He is not suited to her, she doesn't want him, and he is\nsuited to you.  You can't deny it.\"\n\n\"I do d-deny it!\" cried Kate, but the look that swept over her face at\nRosemary's remark belied her words.  \"And you are to t-think no more\nabout it.  And Rosemary Goddard, if you go to t-treating the situation,\nas if N-Noel and I were a couple of hunchbacks or yellow fevers or\ns-snake-bites, I'll h-half kill you!  I--I'm no subject for p-prayer,\nlet me tell you that now.\"\n\n\"Kate, I wouldn't think of such a thing!\" cried Rosemary, biting her\nlips.  \"Now go on.  There's Noel calling for you to go home!\"\n\n\"As if she could mislead me,\" said Rosemary to herself.  \"She wouldn't\neven try if she could have seen her own face when I said, on purpose to\ntry her, 'There's Noel calling you to go home.'  Well, bless her dear\nheart!  I hope her love-affair will turn out as luckily as mine has, and\nwithout all my misery.  Good-bye, all!\"\n\n\n\n\n                              CHAPTER XI.\n\n                      IN WHICH TRUTH HOLDS HER OWN\n\n\nPerhaps, as a student of human nature, Roscoe Howard rather looked\nforward with enjoyment to his encounter with Colonel Yancey in the\nmatter of the purchase of Guildford.  With the promptness and decision\nwhich gave the fundamental strength to his character, he at once\ninvestigated the whole transaction, beginning with the private history\nof the syndicate, which, in his bitterness, Sherman Lee was only too\nready to give him.  He drew from Carolina, by adroit conversations, much\nof the story of Colonel Yancey's connection with the Lee family abroad,\nand, to a man with an imagination, he soon was able to formulate, though\nby a somewhat elliptical process, a theory concerning Colonel Yancey's\ndesigns on Carolina, which fitted the case as it stood, but which needed\na personal interview with the colonel to enable Mr. Howard to decide\nwhether the man was anxious to marry Carolina from love of herself alone\nor with the ulterior motive of having discovered some unsuspected source\nof wealth on the Guildford estate.\n\n\"This man is a very accomplished rascal!\" he said to himself, as he\nfollowed the winding clues in the labyrinth of the colonel's\ntransactions.  \"I feel sure that Sherman's money is done for.  He will\nnever get any of that back.  Yet Yancey, rascal as he is, is too shrewd\nto put himself in the clutches of the law.  However, he is also clever\nenough to be willing to have Sherman think him a fool for failing.  At\nthe same time, I believe that Yancey has made a fortune.  The question\nis, where is it?\"\n\nHe fell to musing on the man's extraordinary career.  Serving\ngovernments with honesty for years, waiting, studying, learning, biding\nhis time until he could make a grand haul without fear of detection,\nwith his honourable career to throw suspicion off the scent, and finding\nhis quarry at last in wrecking the orphaned children of his best friend.\n\nIt was a curious type of character,--a curious code of honour,--but not\nphenomenal.  It simply showed the effect of climate on a man's\ndefinition of honesty.  Doubtless Colonel Yancey considered the\nsyndicate of New Yorkers \"damned Yankees,\" and therefore his legitimate\nprey.  Did not the carpet-baggers rob the South?  And, as to getting\npossession of Guildford, even if only in order to force Carolina to\naccept him with it--all's fair in love and war.  Doubtless Colonel\nYancey was an honourable man in his own eyes, and ready to defend his\nhonour to the death if necessary. Mr. Howard had spent several years in\nthe South, and did not underestimate his personal danger in the coming\ninterview should he impinge on what the colonel was pleased to call his\n\"honour.\"  Mr. Howard felt that he must fortify himself with\nserpent-wisdom and dove-harmlessness.\n\nFor Colonel Yancey was coming home, and Mr. Howard had arranged for a\nmeeting with him without stating his errand.\n\nHe was prepared for a confident, even a dignified, bearing in the\ncolonel, but let it be said that he had not looked for the jaunty air\nwith which Colonel Yancey met him when Mr. Howard called at his office\nat the time appointed.  Considering that Colonel Yancey must be aware\nthat Mr. Howard knew of the crookedness of the whole transaction in oil,\nhis audacity was, to say the least, extraordinary when he rose, held out\nhis hand to the older man, and said, genially:\n\n\"Well, sir, what can I do for you?\"\n\nThe impertinence of the remark, to say nothing of its bad taste under\nthe circumstances, for a moment staggered even the Northerner's good\nbreeding, and, for one brief breathing spell, Mr. Howard felt impelled\nto imperil the whole situation by the trenchant reply:\n\n\"Not a damned thing, sir!\"\n\nBut his self-control came to his rescue, and with it a determination to\nmaster the natural and inevitable irritation which many Northern men\nfeel at being called upon to transact business with a Southern man, and\nwhich all Southern men feel when doing business with Northern men.  The\nwhole code is different and all the conditions misunderstood. Nor will\nthere be harmony until each endeavours to obtain and comprehend the\nother's point of view.\n\nIt was only by detaining the conversation upon strictly neutral grounds\nfor a few moments that Mr. Howard was able to see that the fault lay\nlargely with himself.  Perhaps Colonel Yancey was unaware that his\nvisitor knew anything of his private history or was at all interested in\nthe Lees.  It was only Mr. Howard's smarting under the real injuries\nColonel Yancey had inflicted on Winchester Lee's children which caused\nhim to resent Colonel Yancey's assumption of the role which he essayed\non all occasions and inevitably with strangers.  At first, he was the\nbland, suave, genial, open-hearted Southerner.  But at the first hint of\nMr. Howard's errand, the openness snapped shut.  The thin lips were\ncompressed, the crafty eyes narrowed, and Colonel Wayne Yancey, like a\npirate craft, \"prepared to repel boarders.\"\n\n\"Now, Mr. Howard,\" he said, \"in broaching the subject of the purchase of\nGuildford, may I ask whom you are representing?\"\n\n\"Why should you imagine that I am representing any one?\" inquired Mr.\nHoward.  \"Why not imagine that I want Guildford for my own use? It is a\ngood property.  It has a water-front.  It is picturesque.  Why not\nsuppose that I merely want to acquire a winter home in South Carolina?\"\n\n\"Then why not look at property just as good, nearer to the town of\nEnterprise than Guildford lies, and with a good stone house already on\nit? For instance, my sister's late husband's place, Whitehall, is for\nsale.\"\n\n\"Thank you for mentioning it,\" said Mr. Howard, \"but I especially want\nGuildford.\"\n\n\"Then--pardon me for saying so--you must have some ulterior motive for\nwanting it, for the place is worth no more than the adjoining property\nof Sunnymede or half a dozen other contiguous estates.\"\n\n\"That is exactly the thought which came to me, if you will pardon me for\nmentioning it, when I heard that you had bought and foreclosed the\nmortgage on Guildford!\"\n\nMr. Howard laid his finger-tips together, with a quiet satisfaction in\nthus having trapped his antagonist.  But he little knew Wayne Yancey.\n\nWith an assumption of honesty, which fairly took the Northern man's\nbreath away, Colonel Yancey looked first out of the window, as if to\nconsider, and then said:\n\n\"You are right, Mr. Howard, and to a man of honour like yourself, I will\ntell you the real reason why I bought the mortgage on Guildford, why I\nforeclosed it in order to own the place, and why I hope you will drop\nthe idea of purchasing it, for I tell you frankly at the outset that, if\nyou press the matter, I shall simply put a prohibitive price upon the\nproperty, and you have no legal recourse by which you can compel me to\npart with it.  Please bear this in mind.  And for explanation of this\nunalterable decision--here it is.  I love Carolina Lee.  I told her\nfather so when she was only a girl of sixteen in London.  He gave me his\nblessing, and told me he would rather leave her to me than to any other\nman in the world.  He was my dearest friend.  I was the unhappy means of\nbringing a loss on Sherman, which it shall be my life-work to make good.\nIf Winchester Lee can hear me in the place where he has gone, he knows\nthat I mean well by both of his children.  I adore Carolina, but she has\nrefused to marry me, and, knowing her love for her old home, I obtained\npossession of it in order to restore it to her.  If you do not believe\nthat I mean this, ask her if I did not offer her Guildford as a free\ngift.\"\n\n\"You are a clever man, Colonel Yancey, and you knew then, as well as you\nknow now, that to offer a girl of Carolina's spirit a valuable gift like\nthat was to insult the Lee pride.  What did you hope to gain by it?\"\n\n\"The girl herself!  I confess it without shame, sir.  I would move\nheaven and earth in order to have that girl for my wife!  You do not\nknow Wayne Yancey, Mr. Howard, or you would know that that means more\nthan appears on the surface.\"\n\n\"I may not know you completely, Colonel Yancey, but I know you well\nenough to believe that part of your statement implicitly.  But you will\nnever win her either by force or by coercion of any kind.  Give her a\nfree hand and let her come to you of her own accord, or she will not\ncome at all.\"\n\nBy the expression which flitted across the colonel's slightly cruel face\nat Mr. Howard's words, he was convinced of one thing, and that was that\nthe man was honestly and deeply in love with Carolina.  This fact\nilluminated the matter somewhat.\n\n\"It would be quite true with horses,\" mused Colonel Yancey.  \"And a\nblooded horse and a spirited woman have many points in common.\"\n\n\"I freely confess to you that I wish to purchase Guildford in order to\nlet Carolina go down there and work her will with the place.  The girl\nhas courage, good business ideas; she is a friend of my daughter's, and\nI am interested in the development of her character.  I would just as\nsoon leave you to make the same arrangement with her which I propose to\nmake, if she would consent to have money transactions with you, but she\nwill not.  For what reason you and she probably know.  I confess that I\ndo not, but what you have just been good enough to tell me concerning\nyour feelings toward her would seem to throw light upon the situation.\nNow, may I make a suggestion?\"\n\n\"A thousand, if you will!\"\n\n\"Thank you.  Now, possibly an outsider may be able to give you a new\npoint of view.  Suppose you yield to Carolina's wishes, sell me the\nplace, and thus give her the opportunity to carry out her dead father's\nplans.  You thus provide her with a cherished life-work.  You know the\nLees.  They are proud and grateful.  To whom would her heart naturally\nturn?  To an old married man like me, through her friendship for my\ndaughter, or to a comparatively young man like yourself, in whose\nchildren she is as vitally interested as she must have been to heal your\nbaby girl?\"\n\nNow Mr. Howard was deliberately playing upon the man's feelings, but he\nwas not prepared for the change in Colonel Yancey's face.\n\n\"Did she do that?\" he said, in a hoarse voice, \"Did she do it?\"\n\n\"Certainly she did.  Who else?\"\n\n\"They told me that Mrs. Goddard did it--Sister Sue told me.\"\n\n\"No, it is considered by the Christian Scientists--this new sect which\nyou may have heard that Carolina has joined--that Gladys is her first\ncase of healing.  Carolina is Mrs. Goddard's pupil, and doubtless Mrs.\nGoddard helped her,--in the curious way they have, for I overheard\nCarolina telephoning Mrs. Goddard to treat her--Carolina--for fear, in\nyour little daughter's case.  I believe they heal by confidence in God's\npromises and the theory that mind controls matter.  Wonderful, isn't\nit?\"\n\n\"Wonderful, indeed, but the  most wonderful part of it to me is that\nMiss Carolina was induced to render me this inestimable benefit when\nshe--well, she used to hate me, to be quite frank.  If you knew the\nrebuffs I have taken at her hands!\"\n\n\"Well, that is one of the results of this new religion of hers.  It is\nfounded on love, and they are obliged to live it, or they fail to\nreceive any benefits.  It is a self-acting religion, and is its own\ndetective.  They regard hatred, for example, as a disease, and naturally\nCarolina could not, in their code, be healed herself or heal others as\nlong as she hated you.  Thus, in healing your little girl, she was\nworking out her own salvation.\"\n\n\"Mr. Howard,\" said Colonel Yancey, with his face working painfully, \"you\ndon't know what it is to have a crippled child.  You don't know the\nagony I have endured, looking at her beautifully formed little body and\ninto her dear face, with its intelligent eyes, broad brow, and sweet\nmouth, and then realizing that all her life she must be helpless, unable\nto walk or even to stand, a burden to herself and others.  Her feet, as\nperhaps you know, were perfect in shape and form.  They were simply\nturned inward.  I have gone through Gethsemane itself wondering when her\ntender little heart would learn its first taste of bitterness against\nthe parents who brought her into the world to suffer so.  And then to\nhave all this load of grief lifted, to see my baby walk about and play\nwith her little sister, and frolic as other children do, and suddenly to\nlearn that I owe it to the woman who is my all in life--I assure you,\nsir, it is almost more than my heart can bear.  Take Guildford on your\nown terms, sir! It is a small return!\"\n\nMr. Howard held out his hand, and Colonel Yancey grasped it.\n\n\"The human heart is a curious thing, Mr. Howard. I was as determined\nfive minutes ago as ever a man was on earth to let you plead until you\nlost your breath, yet I would never part with my hold on Miss Carolina\nthrough owning Guildford.  Now, in the twinkling of an eye, I am ready\nto let you have it.  I can't give it to you quickly enough. What price\nare you willing to pay?\"\n\n\"Suppose we say the face of the mortgage,--just what it cost you?\"\n\n\"Ten thousand dollars less, if you say so, Mr. Howard.\"\n\n\"No, I prefer to let you show your gratitude to her in some other way.\nI will pay what you paid.\"\n\n\"Good!  I will have the deed made out to-day. But lose no time in\ntelling her that Guildford is hers.  She has won it for herself.\"\n\n\"If I tell her that, do you know what she will say?\" asked Mr. Howard.\n\n\"No, what?\"\n\n\"She will give all the credit to her new thought. She told me before I\nstarted that I would be successful.  As she puts it, 'Nothing is ever\nlost in Truth.'\"\n\n\"Then she considers, even though Guildford has been in my power for\nseveral years, that it was never really lost to her?\"\n\n\"In her new conception of the truth, that is the way she argues.\"\n\n\"By Jove, Mr. Howard, I'm going to join them! I wonder if she would let\nme go to church with her next Sunday?\"\n\n\"I'm sure she would.\"\n\nBut, as he turned away, Mr. Howard shook his head and said to himself:\n\"Carolina will have to tell him what she told Noel,--of the futility of\nattempting to be a Scientist for the sake of the loaves and fishes.\"\n\nBut, indeed, Carolina had not only believed it, but, with her Bible and\n\"Science and Health\" on her knees, during the hour of the interview she\nhad made her demonstration, so that she knew it without words.  She felt\nit by the uplift in her own heart and the nearness of her own soul to\nthe Infinite, so that, when Mr. Howard appeared with a beaming face to\ntell her, the radiance on Carolina's admonished him that she knew\nalready.\n\n\"But you don't know all, young lady!  After I had left his office, the\ncolonel came post-haste after me to say that his sister and the children\nare to leave to-morrow for Whitehall, his brother-in-law's estate, which\nlies some twelve miles from Guildford, but northeast from Enterprise,\nthe little station, where you leave the railroad, and Miss Yancey is\ngoing to call on you and Mrs. Winchester this evening, to invite you to\nmake Whitehall your headquarters until you can establish yourself\nelsewhere.\"\n\n\"Oh, how kind of them!\" said Carolina.\n\n\"Then y-you will accept?\" demanded Kate, in old-thought surprise.\n\n\"Why, what could possibly be better?\" asked Carolina, in new-thought\nsimplicity and gratitude.\n\n\"T-ten to one on Colonel Yancey!\" murmured Kate in her father's ear as\nthey turned away.\n\n\"W-was it a d-difficult job, d-daddy?\" she asked, tucking her arm into\nhis.\n\n\"Kate, child, it was an absolute triumph for Carolina's new religion.  I\ndeserve no credit.  The man set his jaws and looked as hard as nails,\nuntil I mentioned that Carolina had healed his baby. He had been\ncarefully led--probably by Carolina's instructions--to believe that Mrs.\nGoddard did it--\"\n\n\"Y-yes, Miss Yancey believes it, too.\"\n\n\"Well, they forgot to coach me, so I told him it was Carolina.  My dear,\n_voila tout_!\"\n\n\"C-Christian Science p-plays ball every time, doesn't it?\" observed\nKate, thoughtfully.\n\n\n\n\n                              CHAPTER XII.\n\n                               WHITEHALL\n\n\n\"Well,\" said Mrs. Winchester, looking out of the car-window as the train\napproached Enterprise, \"if any man had told me that two years from the\nday we left Bombay I should find myself going back to Guildford to live,\nI should have said he was a thousand dollars from the truth.  What are\nyou laughing at, Carolina?\"\n\n\"And if any man had told me that I could ever have brought myself to\naccept an invitation from Miss Sue Yancey to visit them at Whitehall\nuntil we could establish ourselves comfortably, when I used to dislike\nher brother so much, I should have said the same,\" said Carolina, \"but\nlove works many miracles in the human heart.\"\n\nMrs. Winchester looked sharply at the young girl, but Carolina's\nexpression was so innocent Cousin Lois decided that she was not\nreferring to Colonel Yancey.  Then, with one of her rare caresses, which\nMrs. Winchester prized above gold, Carolina laid her hand on Mrs.\nWinchester's arm and said:\n\n\"And, dear Cousin Lois, no mother could have been sweeter and more\nunselfish about the loss of her money than you have been, or more\nself-sacrificing to come down here with me.\"\n\n\"Nonsense, my dear!\" said Mrs. Winchester, colouring like a girl of\neighteen.  Her blush was still beautiful and was her only comfort,\nexcept her waist-line.  \"You know that I love to be where you are.  In\nfact, Carolina, if you knew how I suffered, actually suffered, child,\nlast winter in Boston, when I was separated from you, you would believe\nme when I say that I cannot live without you.  I must be with you.  You\nare all I have in the world,--and the money,--what is money good for\nexcept to buy things with?  Haven't I everything I want?\"\n\nCarolina listened with a beating heart.\n\n\"Yet, you are even going to have the money back!\" she said, with another\npressure of Cousin Lois's hand.\n\n\"Yes, I really believe I am.  That new religion of yours seems to be a\nsort of magic carpet, to take you anywhere you want to go and to get you\neverything you want to have.\"\n\n\"It brings perfect harmony into your life,\" said Carolina.\n\n\"Well, harmony is heaven!\" said Mrs. Winchester, emphatically.\n\n\"Oh, what bliss to be coming home!\" breathed Carolina, fervently.  \"I\nwonder if any shipwrecked sailor or prodigal son or homesick child ever\nyearned as cruelly for his father's house as I yearn for my first sight\nof Guildford!\"\n\nMrs. Winchester turned, a little frightened at the passion in the girl's\ntone.  She felt that Carolina was unconsciously preparing herself for a\nbitter disappointment.\n\n\"How dear those little <DW54>s are!\" she cried. \"But, oh, did you see\nwhat that woman did?  She knocked that little boy sprawling!  She\nknocked that child down!  Did you ever hear of such cruelty? Do you\nsuppose she could possibly have been his own mother, Cousin Lois?\"\n\n\"Sit down, Carolina, and don't get so excited. Of course she was his\nmother.  That's the way <DW52> women do.  It saves talking,--which\nseems to do no good.  I've seen old Aunt 'Polyte, in your father's time\nat Guildford, come creeping around the corner of her cabin to see if her\nchildren were obeying her, and, if she found that they were not, I've\nseen her knock all ten of them down,--some fully six feet away.  And\nsuch yells!\"\n\n\"Did grandfather allow it?\" demanded Carolina, with blazing eyes.\n\n\"I can fairly see him now, sitting his horse Splendour, draw rein and\nshake with silent laughter, till he had to take his pipe out of his\nmouth.  It was too common a sight to make a fuss about. Besides, they\nneeded it.  Of all the mischievous, obstinate, thick-headed little\ndonkeys you ever saw, commend me to a raft of black children,--Aunt\n'Polyte's in particular.  <DW52> women are nearly always inhuman on\nthe surface to their own children.\"\n\n\"Wasn't Aunt 'Polyte my father's black mammy? Wasn't she kind to the\nwhite children in her charge?\"\n\n\"Ah, that was a different matter.  Kind?  'Polyte would have let all her\nown children die to save your father one ache.  I remember when her\nchildren got the measles, she locked them all in the cabin, and sent her\nsister to feed them at night, while she stayed in the big house and kept\nher white children from contagion.  Fortunately, none of her own died,\nbut, if they had, it wouldn't have changed her idea of her duty.\"\n\n\"What was there queer about Aunt 'Polyte?  I remember that daddy told me\nonce, but I have forgotten.\"\n\n\"She had one blue eye and one biack one, and not one of her children\ninherited her peculiarity except her youngest child,--a boy,--born when\nshe was what would be called an old woman.  I know she thought it was a\nbad omen to have a child after she was fifty, and, when she saw his blue\neye, she said he was marked for bad luck.\"\n\n\"Oh, how dreadful!\" cried Carolina.  \"Cousin Lois, you know enough about\nChristian Science to know that she made a law for that child which may\nhave ruined him for life.\"\n\n\"Yes, I suppose she did.  But, Carolina, dear, don't get your hopes of\nthe South up too high.  I am afraid it won't come up to your\nexpectations.\"\n\nCarolina smiled, sighed, and shook her head.\n\n\"I can't modify my anticipations, Cousin Lois. Don't try to help me.  If\nI am to be disillusioned, let it come with an awful bump.  Nothing short\nof being knocked down with a broadside like that little <DW64> boy can do\nmy case any good.  I'm hopeless.\"\n\n\"I believe you are.  Well, we shall see.  We must be nearly there.  The\nlast time the train stopped,--was it to shoo a cow off the track or to\nrepair the telegraph wires?--the conductor said we were only five hours\nlate.  But that was six hours ago.  I wonder what we are stopping at\nthis little shed for?  Oh, hurry, Carolina!  He is calling Enterprise\nand beckoning to us.\"\n\n\"No hurry, ma'am,\" said the conductor.  \"The train will wait until you\nall get off in comfort, or I'll shoot the engineer with my own hand!\"\n\nCarolina stepped from the train to the platform and looked around.  Then\nshe bit her lip until it bled.  Cousin Lois was counting the\nhand-luggage and purposely refrained from looking at her.\n\nThere was a platform baking in the torrid heat of a September afternoon.\nFrom a shed at one end came the clicking of a telegraph instrument.\nThat, then, must be the station.  Six or eight <DW64> boys and men, who\nhad been asleep in the shade of a dusty palmetto, roused up at the\narrival of the train and came lazily forward to see what was going on.\nThere were some dogs who did not take even that amount of trouble.  A\nwide street with six inches of dust led straight away from the station\nplatform.  There was a blacksmith shop on one side and a row of huts on\nthe other.  Farther along, Carolina could see the word \"Hotel\" in front\nof a one-story cottage.  The town fairly quivered with the heat.\n\n\"Was you-all expectin' any one to meet you?\" inquired the conductor.\n\n\"Why, yes,\" answered Mrs. Winchester.  \"Miss Yancey said she would send\nfor us.\"\n\n\"Miss Yancey?  Miss Sue or Miss Sallie Yancey?  Fat lady with snappin'\nbrown eyes?\"\n\n\"Yes, that describes her.\"\n\n\"The one that's just been to New York with the colonel's children?\"\n\n\"Yes.\"\n\n\"Oh, well, that's Miss Sue.  She'll send all right, but likely's not\nyou've got to wait awn her.  She's so fat she can't move fast.  Have you\never heard how the colonel's little girl was kyored?  She went to one of\nthese here spiritualists and was kyored in a trance, they tell me.\"\n\n\"Ah, is that what they say?\" said Mrs. Winchester, in a tone of deep\nvexation.  She felt insulted to think of so dignified a belief as\nChristian Science being confounded with such a thing as spiritualism.\nBut she realized the absurdity of entering into a defence of a new\nreligion with the conductor of a waiting train.  She had, however,\nforgotten what Southern railroads are like.\n\n\"Yes'm.  They say a lady done it.  Jest waved her hands over the child,\nand Gladys hopped up and began to shout and sing and pray!\"\n\n\"My good man,\" said Mrs. Winchester, \"do start your train up.  You are\nseven hours late as it is!\"\n\n\"What's your hurry, ma'am?  Everybody expects this train to be late.  I\ncan't go till my wife's niece comes along.  She wants to go on this\ntrain, and I reckon I know better than to leave her.  She's got a tongue\nsharper'n Miss Sue Yancey's.\"\n\nMrs. Winchester turned her majestic bulk on the conductor, intending to\nannihilate him with a glance, but he shifted his quid of tobacco to the\nother cheek, spat neatly at a passing dog, lifted one foot to a\nresting-place on Carolina's steamer-trunk, and continued, pleasantly:\n\n\"Now, that there dust comin' up the road means business for these parts.\nI'd be willin' to bet a pretty that that is either Moultrie La Grange or\nMiss Sue Yancey.  But whoever it is, they are sho in a hurry.\"\n\nCarolina stood looking at the cloud of dust also. Most of the passengers\non the waiting train, with their heads out of the car-windows, were\ndoing the same.  It seemed to be the only energetic and disturbing\nelement in an otherwise peaceful landscape, and only one or two\npassengers, who were obviously from the North and therefore impatient by\ninheritance, objected in the least to this enforced period of rest.\n\n\"And from here, I'd as soon say it was Moultrie as Miss Sue.  They both\nkick up a heap of dust in one way or another, on'y Moultrie, he don't\nraise no dust talking.  If it _is_ Moultrie, he'll be mighty sore at\nbein' away when the train come in, on'y I reckon he didn't look for her\nso soon.  We was thirteen hours late yestiddy.\"\n\nHow much longer the train would have waited, no one with safety can say,\nhad not the cloud of dust resolved itself into a two-seated vehicle, in\nwhich sat two ladies, both clad in gray linen dusters, which completely\nconcealed their identity.  One of the dusters proved to be the\nconductor's niece, who took the time to be introduced to Mrs. Winchester\nand Carolina by the other duster, which turned out to be Miss Sue\nYancey.  When the conductor's niece had fully examined every item of\nCarolina's costume with a frank gaze of inventory, she stepped into the\nstation to claim her luggage, and then, after bidding everybody good-bye\nall over again, she got into the train, put her head out of the window,\ncalled out messages to be given to each of her family, and, after a few\nmoments more of monotonous bell-ringing by the engineer, in order to\ngive everybody plenty of notice that the train was going to start, it\ncreaked forward and bumped along on its deliberate journey farther\nsouth.\n\nCarolina took an agonized notice of all this.  If it had been anywhere\nelse in the world, she could have been amused; she would have listened\nin delight to the garrulous conductor, and would have laughed at the\ncrawling train.  But here at Enterprise,--that dear town which was\nnearest to the old estate of Guildford,--why, it was like being asked to\nlaugh at the drunken antics of a man whom you recognized as your own\nbrother!\n\nShe listened to Miss Yancey's apologies for being late with a stiff\nsmile on her lips.  She must have answered direct questions, if any were\nasked, because no breaks in the conversation occurred and no one looked\nquestioningly at her, but she had no recollection of anything except the\njolting of the springless carriage and the clouds of dust which rolled\nin suffocating clouds from beneath the horses' shuffling feet.\n\nThey drove about four miles, and then turned in at what was once a gate.\nIt was now two rotting pillars.  The road was rough and overgrown on\neach side with underbrush.  The house before which they stopped had been\na fine old colonial mansion. Now the stone steps were so broken that\nMiss Yancey politely warned her guests with a gay:\n\n\"And _do_ don't break your neck on those old stones, Mrs. Winchester.\nYou see, we of the old South live in a continuous state of decay.  But\nwe don't mind it now.  We have gotten used to it.  If you will believe\nme, it didn't even make me jealous to see the prosperity of those\nYankees up North.  I kept saying to myself all the time, 'But _we_ have\ngot the blood!'\"\n\nAs they entered the massive hall, cool and dim, the first thing which\nstruck the eye was a large family tree, framed in black walnut, hanging\non one side of the wall, while on the other was a highly  coat\nof arms of the Yanceys, also framed and under glass.\n\nMiss Yancey took off her duster and hung it on the hat-rack.\n\n\"Now, welcome to Whitehall!  Will you come into the parlour and rest\nawhile, or would you like to go to your rooms and lie down before\nsupper? I want you to feel perfectly at home, and do just as you\nplease.\"\n\n\"I think we will go to our rooms, please,\" said Mrs. Winchester, with\none glance into Carolina's pale, tired face.\n\n\"Here, you Jake!  Carry those satchels to Mrs. Winchester's room, and,\nLily, take these things and go help the ladies.  And mind you let me\nknow if they want anything.\"\n\nA few moments afterward, Lily, the <DW64> maid, came hurrying\ndown-stairs, her eyes rolling.\n\n\"Laws, Miss Sue!  Dey wants a bath!  Dey axed me where wuz de bathroom,\nen I sez, 'Ev'ry room is a bathroom while y'all is takin' a bath in it.'\nEn Miss Sue, Miss Calline, she busted right out laffin'.\"\n\n\"They want a bath?\" cried Miss Sue.  \"Well, go tell Angeline to heat\nsome water quick, and you fill this pitcher and take it up to them.  But\nmind that you wash it out first,--if you don't, you'll hear from\nme,--and don't be all day about it. Now, see if you can hurry, Lily.\"\n\nWhen the sun went down, the oppressive quality in the heat seemed to\ndisappear, and when Cousin Lois and Carolina came down in their cool,\nthin dresses, they found themselves in the midst of the most delightful\npart of a Southern summer day.\n\nMiss Sue was nowhere to be seen, but another lady, as thin as she was\nfat, came out of the dimness and introduced herself.\n\n\"I am Mrs. Elliott Pringle, ladies, though you will nearly always hear\nme called Miss Sallie Yancey.  Sister Sue is out in the garden.  Shall\nwe join her?  I know she wants you to see her roses.\"\n\nCarolina's spirits began to rise.  She felt ashamed of her hasty\ndisillusionment.  Where was her courage that she should be depressed by\nclouds of dust and the lack of a bathroom?\n\nIn the early evening, with the shadows lengthening on the grass and the\npitiless sun departed, the ruin everywhere apparent seemed only\npicturesque, while the warm, sweet odours from the garden were such as\nno Northern garden yields.\n\nThere were narrow paths bordered with dusty dwarf-box, with queer-shaped\nflower-beds bearing four-o'clocks, touch-me-nots, phlox, azaleas, and\nsweet-william.  Then there were beds upon beds of a flower no Northerner\never sees,--the old-fashioned pink, before gardeners, wiser than their\nMaker, attempted to graft it.  In its heavy, double beauty it always\nbursts its calyx and falls of its own weight of fragrance, to lie\nprostrate on the ground, dying of its own heavy sweetness.  Against a\ncrumbling wall were tea-roses.  In another spot grew a great pink\ncabbage rose, as flat as a plate when in full bloom, with its inner\nleaves still so tightly crinkled that its golden heart was never\nrevealed except by a child's curious investigating fingers.  And\ncuriously twisting in and out of the branches of this rose-tree was a\nhoneysuckle vine. Over one end of the porch climbed a purple clematis.\nOver the other a Cherokee rose.  But the great glory of the garden was\nover against the southern wall, where roses of every sort bloomed in\nriotous profusion.  Evidently they bloomed of their own sweet will, and\nwith little care, for the garden was almost as neglected as the rest of\nthe place.\n\nStill it was the first thing which brought back to Carolina \"a memory of\nsomething\" she \"never had seen,\" as she told Cousin Lois when she went\nin, and she made an excuse to go out alone after supper was over and the\nthree ladies were comfortably seated in rocking-chairs on the front\nporch.\n\n\"Don't sit in that chair, Mrs. Winchester,\" Carolina heard Miss Sallie's\nvoice say, as she ran down the steps into the garden.  \"That chair has\nno seat to it, and the back is broken to this one.  Sit in this chair.\nI think it won't be too damp here to wait for Moultrie.\"\n\nThe girl could smile now, for the witchery of the evening was on the\ngarden, and its perfume enthralled her senses.  She walked until she got\nbeyond the sound of voices on the front porch, and, at the head of a set\nof shallow terraces, set like grassy steps to lead down to the brook\nwhich babbled through the lower meadow, she sat down to let her mind\ntake in the sudden change in her life.\n\nShe rested her chin on her hands and was quite unaware that, in her thin\nblue dress, with frills of yellow lace falling away from the arms above\nthe elbows, and with her neck rising from the transparent stuff like an\niris on its slender stem, she made anything of a picture, until she\nbecame aware that some one was standing quite still on a lower terrace\nand looking at her with so fixed an expression that she turned until her\neyes met his.  Most girls would have started with surprise, but to\nCarolina it was no surprise at all to find the stranger of the\nMetropolitan Opera and the stranger who had borrowed her brother's\ndog-cart, a part of the enchanted garden, and to feel in her own heart\nthat he was no stranger to her, nor ever had been, nor ever could be.\n\nThey looked at each other for a few moments, the man and the woman, and\nthe sound of the brook came faintly to their ears.  But the scent of the\ngarden was all about them and there was no need of speech.\n\nSlowly Carolina smiled, and he reached up his hand to hers and took it\nand said:\n\n\"You know me?\" and she said:\n\n\"Yes.\"\n\n\"And I know you,\" he said, \"for I have felt ever since that first night\nthat you would come.\"\n\n\"That first night?\" she breathed.\n\n\"At the opera,\" he said.\n\nThen he drew back strangely and looked around at the garden and frowned,\nas if it had been to blame for the words he had spoken when he had not\nmeant to speak.  But, although Carolina saw the look and the frown, she\nonly smiled and breathed a great sigh of content and looked at the\ngarden happily.\n\nThen he turned to her again and said:\n\n\"Did you know that you and I are related?\"  And he saw with a great lift\nof the heart that she turned pale before answering, so to spare her he\nwent on, hurriedly:\n\n\"I have been talking to Mrs. Winchester, and we find that the La Granges\nand Lees are kin.  You and I are about twelfth cousins, according to\nMiss Sallie Yancey.\"\n\n\"So we are of the same blood,\" said Carolina, gently.  Then she added:\n\"I am glad.\"\n\n\"And so am I,--more glad than I can say, for it will give me the\nopportunity to be of service to you--in a way I could not--perhaps--if\nwe were not kin.\"\n\nCarolina looked at him inquiringly, but he had turned his head away, and\nagain a frown wrinkled his smooth, brown forehead.  Carolina looked at\nhim eagerly.  He was a man to fill any woman's eye,--tall, lean, lithe,\nand commanding, with long brown fingers which were closed nervously upon\nthe brim of his soft black hat.  His nose was straight, his lips\nsensitive yet strong, and his eyes had a way of making most women sigh\nwithout ever knowing why.  Moultrie La Grange was said to have \"a way\nwith him\" which men never understood, but which women knew, and knew to\ntheir sorrow, for everywhere it was whispered that \"Moultrie would never\nmarry, since--\" and here the whispers became nods and half-uttered words\nand mysterious signs which South Carolinians understood, but which\nmystified Mrs. Winchester, and Carolina did not happen to hear the\nsubject discussed.\n\n\"You have come down here,\" said Moultrie, \"to restore Guildford.\"\n\n\"Yes,\" said Carolina, seeing that he paused for a reply.\n\n\"I wish that I could restore Sunnymede.  Our place joins yours.\"\n\n\"It does?\" cried Carolina.  \"Then why don't you?\"\n\nHe looked at her sharply.  Was she making fun of him?\n\n\"You are a rich young lady.  I am a poor man. Can I rebuild Sunnymede\nwith these?\"  He held out two fine, strong, symmetrical hands.\n\nCarolina looked at them appreciatively before she answered.\n\n\"I am a poor young woman, but I intend to rebuild Guildford with,\nthese!\"  And she held out beside his two of the prettiest hands and\nwrists and arms that Moultrie La Grange had ever seen in his life, and\nhe at once said so.  And Carolina, instead of being bored, as was her\nwont in other days, was so frankly pleased that she blushed, and said to\nherself that the reason she believed this man meant what he said was\nbecause she was poor, and he could not possibly be paying court to a\nwealth that she had lost.  But the truth of the matter was that she\nbelieved him because she wanted to.  It gave her an exquisite and\nunknown pleasure to have this man tell her over and over, as he did,\nthat her hands were the most beautiful he had ever seen, and Carolina\nlooked at them in a childish wonder, and as if she had never seen them\nbefore.  And it was not until she had laid them in her lap again, and\nthey were partly hidden, that she could bring the conversation back to\nanything like reason.\n\n\"How do you mean?\" he questioned.  \"You can't do a thing without money.\nAnd I hear--\" he stopped in confusion, and his forehead reddened.\n\n\"You know that we have lost ours,\" supplemented Carolina.  \"Well, you\nhave heard correctly. Every dollar of my fortune is gone!\"  Her voice\ntook on so triumphant a ring that Moultrie looked up at her in surprise.\nHe did not know that part of her exultation came from the joy it gave\nher to be able to proclaim her poverty to this man out of all the world,\nand thus put herself on a level with him.\n\n\"I have only,\" she continued, \"a little laid by which came from the sale\nof my jewels.\"  Then, as she still saw the questions in his eyes which\nhe forebore to ask, she added: \"Do you want me to tell you about it\nall?\"\n\n\"More than anything in the world,\" he assured her.  And something in his\ntone shook the girl so that she paused a little before she began.\n\n\"Well, I suppose you know that when Sherman, my brother, mortgaged\nGuildford, Colonel Yancey bought the mortgage and foreclosed it.  That\nis how he got possession of Guildford.\"\n\n\"But why?\" interrupted the man.  \"What in the world did he especially\nwant Guildford for, when there are a dozen other estates he could have\nbought for less money, and some of them with houses already built?\"\n\n\"I don't know,\" said Carolina, so hurriedly that the man turned his eyes\nupon her, and, noticing the wave of colour mount to her brow under his\ngaze, he looked away and all at once he knew why. Carolina did not see\nhis hands clench and his teeth come together with a snap, as he thought\nof the Colonel Yancey that men knew.\n\n\"But Mr. Howard, the father of my dearest friend, persuaded Colonel\nYancey to sell it to him for the face value of the mortgage, so that now\nI have no fear of losing it, for Mr. Howard will give me all the time I\nwant to pay for it.\"\n\n\"But what are you going to pay for it with?\" asked the young man.\n\n\"Well, if you will go with us when we look over the estate, I can tell\nyou better than I can now. Do you happen to know anything about this new\nprocess of making turpentine?\"\n\n\"Of course I do,\" said La Grange, with a frown. \"I suppose that your\nbrother and his friends have organized a company with Northern capital\nto erect a plant which will make everybody rich.  That's what all\nNortherners tell us when they want us to invest.  Money is all Yankees\nseem to think about.\"\n\n\"My brother will have nothing to do with the affair at all!\" said\nCarolina, with some heat. \"Guildford is mine, and I'm going to make it\npay for itself.\"\n\nMoultrie said nothing, but his chin quivered with a desire to laugh, and\nCarolina saw it.  Then he turned to her.\n\n\"You have never seen the home of your ancestors? How are you going to\nhave your first view of it?  From the Barnwells' carryall?\"\n\nCarolina's eyes dilated and she bit her lip.\n\n\"How else could I go?\" she said, gently.\n\n\"If you would allow me,\" he said, eagerly, \"we would go on\nhorseback,--just you and I,--early, early in the morning.  It would be\nthe best time. Will you?\"\n\n\"Oh, will you take me?\" cried Carolina.  There was only a look from\nMoultrie La Grange's eyes for an answer.  But Carolina's flashed and\nwavered and dropped before it.\n\n\"Did you ever hear of a magnificent horse your grandfather owned, named\nSplendour?\" he asked, quietly.\n\n\"Ah, yes, indeed.\"\n\n\"Well, I own a direct descendant of the sire of that very animal.  Her\nname is Scintilla, and my friend, Barney Mazyck, owns Scintilla's full\nsister, a mare named Araby.  I'll borrow her for you.  Would you like\nthat?\"\n\n\"Oh, Mr. La Grange!\" breathed Carolina.\n\n\"Please _never_ call me that.  Do let me claim kin with you sufficiently\nto have you call me 'Moultrie.'\"\n\n\"And will you call me 'Carolina?'\" she asked, shyly.\n\n\"We never do that down here with young ladies, unless we are own\ncousins.  But I will call you 'Miss Carolina,' if I may.\"\n\n\"Then you are asking me to take more of a privilege than you will,\" said\nCarolina.\n\n\"I want you to take every privilege with me that you can permit\nyourself,\" he said, earnestly.\n\nWhen Carolina went indoors that night, the first thing she did was to\ntake two candlesticks, and, holding them at arm's length above her head,\nto study her own face in the great pier-glass which, in its carved\nmahogany frame, occupied one corner of her large bedchamber.  Whatever\nthe picture was which she saw reflected there, it seemed to give her\npleasure, for she  and smiled as her eyes met those of the girl\nin the mirror.\n\n\"I am glad _he_ thinks so!\" she whispered to herself, as she turned\naway.\n\n\n\n\n                             CHAPTER XIII.\n\n                               GUILDFORD\n\n\nCarolina never forgot that morning.  She was up at four o'clock, and, by\na previous arrangement with old Aunt Calla, the cook, she had a cup of\ncoffee at dawn.  Aunt Calla brought it into the dining-room herself.\n\n\"'Scuse me, honey, fer waiting awn you myself, but do you reckon I could\n'a' got dat no 'count fool, Lily, to git up en wait awn ennybody at dis\ntime in de mawnin'?  Not ef she knowed huh soul gwine be saved by doin'\nit.  Dese yere chillen ob mine is too fine to wuk lake dere mammy does.\"\n\n\"But how did you manage to wake up so early?\" asked Carolina.\n\n\"Lawd, honey, I'se done nussed sick chillen tell I sleeps wid one eye\nopen from habit.  En when I see what a pretty day it gwine turn out, en\nwhen I see dat en de fust five minutes you laid eyes awn him, you done\ncotched de beau what half de young ladies in Souf Calliny done set dere\ncaps for, I says to myself, 'Ole 'ooman, ef you wants to see courtin' as\nis courtin', you jes' hump doze ole rheumatiz laigs ob yours, en get dar\n'fore dey suspicion it demselves!'  Law, Mis' Calline, how you is\nblushing! Ump! ump!\"\n\n\"Here, Aunt Calla, take this for your trouble, and go and see if Mr. La\nGrange has come,\" cried Carolina.\n\n\"Why, Mis' Calline, dis yere will buy me a new bunnet!  Thank you,\n_ma'am_.  Yas'm, dah he is! I kin tell de way Mist' Moultrie rides wid\nmy eyes shut.  He rides lake one ob dese yere centipedes!\"\n\nOld Calla made it a point to see the riders mount. The sun was just\ncoming into view, sending the mists rolling upwards in silvery clouds,\nwhen Carolina stepped out of the door.  Her habit was of a bluish\nviolet, so dark that it was almost black.  It matched the colour of her\neyes.  Her hair caught the tinge of the sun and held it in its shining\nmeshes.\n\nMoultrie La Grange was waiting for her at the foot of the steps.\n\nHe held the mare Araby by the bridle, and leaned on the saddle of his\nown mare, Scintilla, shielding his eyes.\n\n\"Good morning,--Moultrie.\"\n\n\"Is that you, Miss Carolina?  The sun, or something blinds me.\"\n\nCarolina had heard it all many times before. Why, then, this difference?\nShe pretended to herself that she did not know, but she did know, and\nwas happy in the knowing.  He was so handsome! She gloried in his looks.\nShe felt as she had felt when she stood before the Hermes of Praxiteles,\nand wondered, if such glorious beauty should ever come to life, how she\ncould _bear_ it!\n\nMoultrie La Grange was not considered handsome by everybody.  His beauty\nwas too cold--too aloof--for the multitude to appreciate.  But does the\nordinary tourist go to Olympia?\n\nCarolina had rather dreaded the four miles to Enterprise, if their way\nshould lie over the dusty highway of yesterday.  But she was not\nsurprised; in fact, it seemed in keeping with what she had expected of\nhim when he struck off through the woods, and she found herself, not\nonly on the most perfect animal she had ever ridden, but in an enchanted\nforest.\n\nMoultrie led the way both in conversation and in direction, and Carolina\nfound herself glad to follow.  His sarcasm, his wit, and the poetry of\nhis nature were displayed without affectation.  She kept looking at him\neagerly, gladly, and yet expectantly.  What was she waiting for?  He\ndiscussed men but not deeds; amusements but not occupation; designs but\nnot achievements.  She wondered what he did with his time.  He was\nstrong, magnetic, gentle, charming.  His voice was melodious. His manner\nfull of the fineness of the old South.\n\nYet there was a vague lack in him somewhere.  He just failed to come up\nto her ideal of what a man should be.  Wherein lay this intangible lack?\n\nSuddenly they emerged from the woods and struck the highway, and in\nanother moment they were in Enterprise.\n\nNot a breath of life was anywhere visible. Although it was six o'clock,\nnot a wreath of smoke curled upward from any chimney.  They rode through\nthe sleeping town in silence.\n\n\"Now here,\" said Moultrie, \"is a very remarkable town.  It is, I may\nsay, the only town in the world which is completely finished.  Most\ntowns grow, but not a nail has been driven in Enterprise, to my\nknowledge, since I was born.  This town is perfectly satisfactory to its\ninhabitants _just as it is_!\"\n\nAgainst her will Carolina laughed.  His tone was irresistible.\n\n\"Ought you to make fun of your own--your home town?\" she asked.\n\n\"My more than that!  Enterprise yields me my bread--sometimes.\"\n\nCarolina looked at him.  He pointed with his whip at the shed on the\nrailroad platform.\n\n\"I am telegraph operator there six months in the year.  I teach a\ncountry school in winter.\"\n\nIf he had struck her in the face with that same riding-whip, the red\nwould not have flamed into Carolina's cheeks with more sudden fury.  She\ndug her spurless heel into Araby's side, and the mare jumped with a\nswerve which would have unseated most riders.  Moultrie looked at her in\nswift admiration, but she would not look at him.  She struck her horse,\nand, with a mighty stride, Araby got the lead and kept it for a mile,\neven from Scintilla.  Then the man overtook her and reached out and laid\na hand on Carolina's bridle hand, and looked deep into her eyes and\nsaid:\n\n\"Why did you do that?  Why did you try to escape from me?  Don't you\nknow that you _never can_?\"\n\nAnd all the time Carolina's heart was beating heavily against her side,\nand her brain was spinning out the question over and over, over and\nover:\n\n\"Oh, how can he?  How can he be satisfied with that?  How can he endure\nhimself!\"\n\nIt was not the lack of money, it was the lack of ambition in the man at\nher side, which stung her pride until it bled.\n\n\"Better go West on a cattle ranch,\" she thought, with bitter passion.\n\"Better hunt wolves for the government.  Better take the trail with the\nIndians than to lie down and rot in such a manner!  And _such_ a man!\"\n\nBut suddenly a realization came to her of how marked her resentment\nwould seem to him if he should discover its cause, and she hastened to\nplay a part.  But he was in no danger of discovering, because he did not\neven suspect.  All the young fellows he knew, no matter how aristocratic\ntheir names, were at work for mere pittances at employments no\nself-respecting men would tolerate for a moment, because they offered no\nhope of betterment or promotion.  Men with the talent to become lawyers,\nartists, bankers, and brokers were teaching school for less than Irish\nbricklayers get in large cities.  Therefore, it could not be alleged\nthat they were incapable of earning more or of occupying more dignified\npositions.  It was simply the lack of ambition--the inertia of the\nSouth--which they could not shake off.  It is the heritage of the\nSouthern-born.\n\nPresently Moultrie again pointed with his whip:\n\n\"Over yonder is Sunnymede, our place.  Poor old Sunnymede!  Mortgaged to\nits eyes, and with all its turpentine and timber gone!  Guildford is\nintact.  We just skirt the edge of Sunnymede riding to Guildford.  And\nright where you see that tall blasted pine standing by itself is where I\nmade one of my usual failures.  I'm like the man with the ugly mule, who\nalways backed.  He said if he could only hitch that mule with his head\nto the wagon, he could get there.  So, if my failures were only turned\nwrong side out, I'd be wealthy.\"\n\nCarolina tried to smile.  Moultrie continued:\n\n\"Once I thought I'd try to make some money, so I sold some timber to a\nYankee firm who wanted fine cypress, and with the money I constructed a\nterrapin crawl.  I knew how expensive terrapin are, and, if there is one\nthing I do know about, it is terrapin.  So I canned a few prize-winners,\nand sent them to New York, and got word that they would take all I could\nsend.  Well, with that I began to feel like a Jay Gould.  I could just\nsee myself drinking champagne and going to the opera every night.  So I\nimmediately raised some mo' money in the same way,--out of the\nYankees,--organized a small company, and built a canning factory.  The\nlumber company was interested with me and advanced me all the money I\nwanted.  So I got the thing well started, and left special word with the\nforeman, a cracker named Sharpe, to be sure and not can the claws, then\nI went off to New York to enjoy myself.  I stayed until all my money was\ngone and then came home, intending to enjoy the wealth my foreman had\nbuilt up in my absence. But what do you reckon that fool had done?  Why,\nhe had turned the work over to the <DW65>s, and they had canned the\nterrapin just so,--claws, eyebrows, and all!  Well, of course, the New\nYork people went back on me,--wrote me the most impudent letters I ever\ngot from anybody.  It just showed me that Yankees can never hope to be\nconsidered gentlemen.  Why, they acted as if I had cheated them!  Said\nthey had advertised largely on my samples, and had lost money and credit\nby my dishonest trickery.  Just as if _I_ were to blame! Then, of\ncourse, the Yankee lumbermen got mad, too, and foreclosed the mortgage\nand liquidated the company, and left me as poor as when I went in. I\nbelieve they even declare that I owe them money.  Did you ever hear of\nsuch a piece of impudence?\"\n\n\"Never,\" said Carolina, coolly, \"if you mean on your part!  You did\neverything that was wrong and nothing that was right.  And the worst of\nit is that you are morally blind to your share of the blame.\"\n\n\"Why, Miss Carolina, what do you mean?  I didn't go to lose their money.\nIt hit me just as hard as it did them.  I didn't make a cent.\"\n\n\"But the money that you lost wasn't yours to lose,\" cried Carolina,\nhotly.\n\n\"No, but I didn't do wrong intentionally.  You can't blame a man for a\nmistake.\"\n\n\"There is such a thing as criminal negligence,\" said the girl,\ndeliberately.  \"You had no business to trust an affair where your honour\nwas pledged to an incompetent cracker foreman, and go to New York on the\ncompany's money, even if you did think you would earn the money to pay\nit back.  How do you ever expect to pay it?\"\n\n\"I don't expect to pay it at all, and I reckon those Yankees don't\nexpect it, either.\"\n\n\"No, I don't suppose they do,\" said Carolina, bitterly.\n\n\"Well, if they are satisfied to lose it, and have forgotten all about\nit, would you bother to pay it back if you were in my place?\"\n\n\"I would pay it back if I had to pay it out of my life insurance and be\nburied in a pine coffin in the potter's field!  And as to those\nNortherners having forgotten it,--don't you believe it!  They have\nsimply laid it to what they call the to-be-expected dishonesty of the\nSouth when dealing with the North.  The South calls it 'keeping their\neyes peeled,' 'being wide-awake,' 'not being caught napping,' or catch\nphrases of that order.  But the strictly honest business man calls it\ndishonest trickery, and mentally considers all Southerners inoculated\nwith its poison.  Do you know what Southern credit is worth in the\nNorth?\"\n\nMoultrie only looked sulky, but Carolina went on, spurred by her own\ndespair and disillusionment.\n\n\"Well, you wouldn't be proud of it if you did! And just such a tolerant\nview of a thoroughly wrong transaction as you have thus divulged is\nresponsible.  Colonel Yancey was right.  The South is heart-breaking!\"\n\n\"Do you care so much?\" asked Moultrie, softly.\n\nCarolina lifted herself so proudly that the mare danced under her.  She\nsaw that she had gone too far.  She also felt that error had mocked her.\nShe had despaired of Moultrie's blind and false point of view when the\nLight of the world was at hand.  Immediately her thought flew upwards.\n\nBut with Carolina absorbed in her work, and Moultrie puzzling over the\nsudden changes in her behaviour, it could not be said that the remainder\nof the ride was proving as pleasant as each had hoped.  However, a\nperfect day, a fine animal, and the spirits of youth and enthusiasm are\nnot to be ignored for long, and presently Carolina began to feel\nGuildford in the air.  She looked inquiringly at Moultrie, and he\nanswered briefly:\n\n\"In another mile.\"  But there was a look in his eyes which made\nCarolina's heart beat, for it was the glance of comprehension which one\nsoul flings to another in passing,--sometimes never to meet again,\nsometimes which leads to mating.\n\nIn another five minutes Moultrie raised his arm.\n\n\"There!\"\n\nCarolina reined in and Araby stood, tossing her slim head, raising her\nhoofs, champing her bit, and snuffing at the breeze which came to her\nred nostrils, laden with the breath of piny woods and balsam. Moultrie,\nsitting at parade rest on Scintilla and watching Carolina catch her\nbreath almost with a sob, said to himself: \"She feels just as that horse\nacts.\"\n\nCarolina could find no words, nor did she dare trust herself.  She was\nafraid she would break down.  She lifted her gauntleted hand and the\nhorses drew together and moved forward.\n\nFor more than a mile an avenue as wide as a boulevard led in a straight\nline, lined on each side by giant live-oaks.  Ragged, unkempt shrubbery,\nthe neglect of a lifetime, destroyed the perfectness of the avenue, but\nthe majesty of those monarchs of trees could not be marred.  The sun was\nonly about an hour high, and the rays came slantingly across meadows\nwhose very grasses spoke of fertility and richness.  The glint of the\nriver occasionally flashed across their vision, and between the\nbird-notes, in the absolute stillness, came the whispering of the\ndistant tide.\n\nAt the end of the avenue lay the ruined stones of Guildford.\n\nCarolina sprang down, flung her bridle-rein to Moultrie, and ran\nforward.  She would not let him see her eyes.  But she stumbled once,\nand by that he knew that she was crying.  They were, however, tears of\njoy and thanksgiving.  Guildford!  Her foot was on its precious turf.\nThese stones had once been her father's home.  And she was free, young,\nstrong, and empowered to build it up, a monument to the memory of her\nancestors.  Every word which Mrs. Goddard had prophesied had come true,\nand Carolina's first thought was a repetition of her words:\n\n\"See what Divine Love hath wrought!\"\n\nWhen she came back, instead of a tear-stained face, Moultrie saw one of\nsuch radiance that her beauty seemed dazzling.  Where could be found\nsuch tints of colouring, such luminous depths in eyes, such tendrils of\ncurling hair, such a flash of teeth, such vivid lips, and such a\nspeaking smile? As he bent to receive her foot in his hand, he trembled\nthrough all his frame, and, as he felt her light spring to her mare's\nback, he would not have been at all surprised to discover that she had\nsimply floated upward and vanished from his earthly sight to join her\nwinged kindred.  But, as she gathered up her reins and watched him\nmount, it was a very businesslike angel who spoke to him, and one whose\nbrain, if the truth must be told, was full of turpentine.\n\n\"Now, let's explore,\" she said.  \"I have paid my respects to the shrine\nof my forefathers, now let's see what I have to sell my turpentine\nfarmers.\"\n\n\"Your what?\" asked the man, with the amused smile a man saves for the\npretty woman who talks business.\n\n\"I am going to sell the orchard turpentine rights of Guildford to get\nmoney for building,\" she said, in a matter-of-fact tone.\n\n\"And I was thinking of you in a white robe playing a harp!\" he said,\nwith a groan.\n\n\"I often wear a white robe, and I play a harp quite commendably,\nconsidering that I have studied it since I was nine years old, but when\nI am working, I don't wear my wings.  They get in my way.\"\n\nCarolina by instinct rode to an elevation which commanded a view of the\npine forests of Guildford.\n\n\"How much do I own?\" she asked.\n\n\"As far as you can see in that direction.  Over here your property runs\ninto ours just where you see that broad gap.\"\n\n\"Why don't you rebuild Sunnymede?\"\n\n\"No money!\" he said, with a shrug.\n\n\"You have plenty of fallen timber and acres of stumpage to sell to the\npatent turpentine people.\"\n\n\"I don't know.  I have never heard it discussed. We wouldn't sell to\nYankees.  We feel that we wouldn't have come to grief with the terrapin\naffair if we had been dealing with Southerners.\"\n\n\"Who are there to discuss?  Who owns it with you?\" asked Carolina,\ncalmly ignoring the absurdity of his remarks.\n\n\"My brother and sister--\"  He paused abruptly, and then said: \"You are\nsure to hear it from others, so I will tell you myself.  The La Grange\nfamily skeleton shall be shown to you by no less a hand than my own!  My\nbrother has made a very--I hardly know what to call it.  It is an\nunfortunate marriage, since no one knows who the girl is. When you saw\nme in New York, I was hoping to prevent their marriage, but it was too\nlate.  They had eloped and had been married immediately on arriving in\nNew York.  As soon as her aunt, with whom she lived, learned that Flower\nhad eloped with my brother, she sent for me.  She had been a great\ninvalid, and the excitement had upset her so that when I arrived she\nlooked as if she had not an hour to live.  She caught me by the arm and\nsaid: 'Flower must not marry a La Grange. She is not my niece nor any\nrelative of mine.  Her mother was--' and with that her speech failed.\nShe struggled as I never saw a being struggle to speak the one word\nmore,--the one word needful,--and, failing, she fell back against her\npillow--dead!\"\n\nCarolina's face showed her horror.  He felt soothed by her understanding\nand went on, in a low, pained voice.\n\n\"It ruined my life.  And it has ruined Winfield's.\"\n\n\"And the girl,\" said Carolina, in a tense voice, \"Flower!\"\n\n\"It has ruined hers.  They are the most unhappy couple I ever saw.  And\nmore so since the baby came.\"\n\n\"It will all come right,\" declared Carolina, straightening herself.\n\"You will discover that Flower is entitled to a name, and that your\nworst fears are incorrect.\"\n\n\"My worst fears--\" began Moultrie.  Then he stopped abruptly.  \"I cannot\nexplain them to you,\" he said.\n\n\"I know what you mean.  But remember that I, too, have seen Flower.  I\nsaw her that day, and I say to you that not one drop of <DW64> blood\nflows in that girl's veins, and your brother's child is safe.\"\n\n\"You think so?\" he exclaimed, moved by the earnestness of her voice and\nthe calm conviction of her manner.  Then he shook his head.\n\n\"It seems too good to be true.\"\n\n\"I can understand,\" she said, \"the terrible strain you are all under,\nbut, believe me, it will all come out right.\"\n\n\"They think the baby is bewitched,--that he has been voodooed,--if you\nknow what that means. The <DW64>s declare that an evil spirit can be\nseen moving around whatever spot the child inhabits.\"\n\n\"What utter nonsense!\" cried Carolina.  \"I hope your brother has too\nmuch sense, too much religion, to encourage such a belief.\"\n\n\"My poor brother believes that the devil has marked him for his own.\"\n\n\"Does your brother believe in a devil?\" asked Carolina.\n\n\"Why, don't you?\" asked Moultrie, in a shocked tone.\n\n\"I was not aware that any enlightened person did nowadays,\" answered\nCarolina, with a lift of her chin.\n\nThe movement irritated her companion far more than her words, just as\nCarolina had intended it to.\n\nThere are some subjects which cannot be argued. They must be obliterated\nby a contempt which bites into one's self-love.\n\nThe mare saved the situation by a soft whinny. She turned her head\nexpectantly, and, following her eyes, the riders saw the tall, lithe\nfigure of a man making his way toward them through the underbrush.\nMoultrie gave vent to an exclamation.\n\n\"What is it?\" asked Carolina.\n\n\"Oh, only a bad <DW64> who haunts places where he has no business to.  He\nis a perfect wonder with horses, and broke in that mare you are riding,\nwho will follow him anywhere without a bridle, pushing her nose under\nhis arm like any dog who thrusts a muzzle into your palm.  He is always\nup to something.  From present appearances, I should say that he had\nprobably been bleeding your trees.\"\n\nThe <DW64>, hearing voices, stopped, glanced in their direction, and\npromptly disappeared.  Carolina only had time to notice that he was very\nblack, but she followed him in thought, mentally denying dishonesty and\ndeclaring that harm could not come to her through error in any form.\n\nShe was struck, too, by the manner in which her sensitive, high-bred\nmare lifted her pretty head and looked after his retreating form, pawing\nthe earth impatiently and sending out little snuffling neighs which were\nhardly more than bleatings. Surely, if a man had the power to call forth\ndevoted love from such an animal, there must be much good in him!\n\n\"What makes you so quiet?\" asked Moultrie, breaking in on her thought.\n\nCarolina looked at him abruptly and decided her course of action.\n\n\"You have told me of the skeleton in your closet. Let me be equally\nfrank and tell you of mine.  I am a Christian Scientist.\"\n\n\"A what?\"\n\n\"A Christian Scientist!\"\n\n\"I never heard of one,\" said the young man, simply.  \"What is it?\"\n\nFor the second time the girl's face flushed with a vicarious\nmortification.\n\n\"It is a new form of religion founded on a perfect belief in the life of\nChrist and a literal following of His commandments to His disciples,\nregardless of time,\" said Carolina, slowly.\n\nMoultrie allowed a deep silence to follow her words.  Then he drew a\nlong breath.\n\n\"I think I should like that,\" he said.  \"Does it answer all your\nquestions?\"\n\n\"All!  Every one of them!\" she answered, with the almost too eager\nmanner of the young believer in Christian Science.  But an eagerness to\nimpart good news and to relieve apparent distress should be readily\nforgiven by a self-loving humanity. Curiously, however, the most blatant\nego is generally affronted by it.\n\n\"I was raised a Baptist,\" he said, reluctantly, \"but I reckon I never\nwas a very good one, for I never got any peace from it.\"\n\n\"My religion gives peace.\"\n\n\"And my prayers were never answered.\"\n\n\"My religion answers prayers.\"\n\n\"Not even when I lifted my heart to God in earnest pleading to spare my\nbrother the unhappiness I felt sure would follow his marriage.  _How_ I\nprayed to be in time to prevent it!  God never heard me!\"\n\n\"My religion holds the answer to that unanswered prayer.\"\n\n\"Not even when I prayed, lying on the floor all night, for the life of\nmy father.\"\n\n\"My religion heals the sick.\"\n\nHe turned to her eagerly.\n\n\"Do you believe so implicitly in Christ's teachings that you can\nreproduce His miracles?\" he cried.\n\n\"Christ never performed any miracles.  He healed sickness through the\nsimplest belief in the world,--or rather an understanding of His\nFather's power.  That same privilege of understanding is open to me--and\nto you.  You have the power within you at this very moment to heal any\ndisease, if you only know where to look for the understanding to show\nyou how to use it.\"\n\n\"Do you believe that?\"\n\n\"I do better than believe it.  I understand it.  I know it.\"\n\n\"Is there a book which will tell me how to find it?\"\n\n\"Yes.\"\n\n\"Will you order it for me, or tell me where to order it?\"\n\n\"It is a very expensive book,\" said Carolina, hesitatingly, thinking of\nthe telegraph-office.\n\n\"How expensive?\"\n\n\"Three dollars.\"\n\n\"Do you call that expensive for what you promise it will do?\"\n\nWhen Carolina looked at him, he saw that she was smiling, but there were\ntears in her eyes.  And he understood.\n\n\"You only said that to try me.\"\n\nAnd she nodded.  Her heart was too full of mingled emotions for her to\nspeak.  She had loved, despised, been proud of, and mortified for this\nman,--all with poignant, pungent vehemence,--during this three-hour\nride, and at the last he had humbled and rebuked her by his childlike\nreadiness to believe the greatest truth of the ages.  She sat her horse,\nbiting her lips to keep back the tears.\n\n\"Give me just one fact to go on,\" he begged.\n\n\"Do you read your Bible?\"\n\n\"I used to, till I found I was getting not to believe in it.  Then I\nstopped for my dead father's sake.  He believed in it implicitly.\"\n\n\"Then you have read the fourteenth chapter of John?\"\n\n\"I got fifty cents when I was twelve years old for learning it by\nheart.\"\n\n\"Then run it over in your own mind until you come to the twelfth verse.\nWhen you get to that, say it aloud.\"\n\n\"'Verily, verily I say unto you, He that believeth on me, the works that\nI do shall he do also; and greater works than these shall he do; because\nI go unto my Father.'\"\n\nHe did not glance her way again, which Carolina noticed with gratitude.\nIt showed that he was not accepting it for her sake.  Presently he spoke\nagain.\n\n\"Did you yourself ever heal any one?\"\n\n\"Through my understanding of Divine Love, I healed Gladys Yancey,\" she\nsaid, quietly.\n\nThe man's face flushed with his earnestness.  He lifted his hat and rode\nbareheaded.\n\n\"Do you remember what the father of the dumb child said?  'Lord, I\nbelieve!  Help thou mine unbelief!'\"\n\nWhen they rode in at the gates of Whitehall, Moultrie was astonished at\nthe radiance of the girl's countenance.  She seemed transfigured by\nlove. Moultrie's ready belief had glorified her, and for the second time\nher grateful thought ascended in the words, \"See what Divine Love hath\nwrought!\"\n\n\n\n\n                              CHAPTER XIV.\n\n                                KINFOLK\n\n\nCarolina took her writing materials out on the back porch.  There was\nnot a small table in the house whose legs did not wabble, so she propped\nthe best of them with chips from Aunt Calla's wood-pile and wrote until\nAunt Calla could stand it no longer.\n\n\"Miss Calline, honey,\" she said, \"you writes so fas' wid yo' fingahs,\nwould you min' ef I brung de aigplant out here to peel it en watch you?\nI won't make no fuss.\"\n\n\"Certainly not, Aunt Calla.  I'd be glad to have you.\"\n\n\"Hum! hum!  You sho have got pretty mannahs, Miss Calline.  Youse got de\nmannahs ob de ole ladies of de South.  You don't see 'em now'days wid de\nyoung ladies.  De young people got de po'est mannahs I ebber did\nsee,--screechin' and hollerin' to each odder 'cross de street, or from\none eend ob de house to de other.  Ole mahster would 'a' lammed his\nchillen ef dey'd cut up sech capers en his time!  But Miss\nPeachie,--she's got de La Grange mannahs.  She's Mist' Moultrie's\nsistah.  Dey calls her 'Peachie' caze she's got such pretty red in huh\ncheeks,--lake yores.  Most ladies down in dese pahts is too white to\nsuit me. I lakes 'em pinky and pretty.\"\n\n\"Thank you, Aunt Calla!\" cried Carolina.  \"I wonder if I couldn't get\nCousin Lois to give you that black grenadine you thought was so pretty\nyesterday.\"\n\nAunt Calla laid down her knife.\n\n\"Miss Calline, is you foolin' me?\"\n\n\"No, Calla, I am not.\"\n\n\"Dish yere grenadier dress I mean is lined wid black silk!\"\n\n\"I know it.\"\n\n\"En you gwine gib dat to me?\"\n\n\"I am thinking of it.\"\n\n\"Well, glory be!  Ef you does dat, Ise gwine jine de chutch all over\nag'in, en I reckon I'll jine de Babtis' dish yere time.  Dey's mo' style\nto de Babtis' den to de Meth'diss.  Ise 'bleeged to live up to dat silk\nlinin'!\"\n\nThe old woman's face took on a worried look.\n\n\"I don' keer!\" she said aloud.  \"I don' keer! Nemmine, Miss Calline!\nYou wouldn' laff so ef you knew what Ise studyin' 'bout doin'.  Ise been\nsavin' my money foh two years now to get a gravestone foh my fou'th\nhusban' what done died three yeahs ago.  He baiged me wid his las'\nbreath to bury him stylish, en I promus him I would.  He was all for\nstyle.  Do you know, Miss Calline, dat man would 'a' gone hongry rathah\ndan turn his meat ovah awn de fiah.  He was de mos' dudish man I ebber\nsee.  But I can't he'p it.  Ise gwine take dat grave-stone money and hab\ndat dress made to fit me good en stylish.  En I bet Miss Peachie will\ncharge me eve'y cent I got to do it!\"\n\n\"Who?\" demanded Carolina.\n\n\"Miss Peachie La Grange.  She does all my sewin' foh me, an' foh Lily,\ntoo.  Dat's de way she mek huh money.  Yas, _ma'am_.  Sewin' foh\nniggahs!\"\n\nAunt Calla paused with her mouth open, for Carolina, regardless of what\nanybody thought, sprang up, overturning her table, spilling her ink over\nAunt Calla's clean porch floor, and scattering her papers to the four\nwinds of heaven.\n\n\"Ump!  So dat's de way de win' blows!  Well, ef she ain't a Lee sho\nnuff.  She's got de pride of huh ole gran'dad, en mo', too.  She looked\nat me ez if she'd lake to kill me.  I wondah ef I'll evah git dat dress\nnow!\"\n\nShe sent Lily to reconnoitre.\n\n\"Jes' creep up en see what she's doin'.  De keyhole in huh room is\nbusted, en you kin see de whole room thoo it.  Jis' go en peek.  But ef\nyou let huh ketch you, she'll know who sont you, en she'll be so mad, I\nnevah will git dat dress.  Den I'll bust yo' yallah face open wid de\ni'nin' boa'd!\"\n\n\"She ain't cryin' nor nothin'!\" cried Lily, bursting into the kitchen\ntwenty minutes later.  \"She's settin' in huh rockin'-cheer, wid a open\nbook awn huh lap, en huh eyes is shut en huh lips a-movin', lake she's\nstudyin'.\"\n\n\"T'ank de Lawd!\" observed Calla.  \"Somehow er odder, Ise gwine git hole\nob a fryin' chicken foh huh.  You tell Jake I wants tuh see him dis\nevenin'. Run, Lily!  See who's dat drivin' in outen de big road!\"\n\n\"Hit's de La Granges!  De whole kit en bilin' ob 'em.  Dey's done\nborried de Barnwells' double ca'y-all.\"\n\nFortunately, there were many rocking-chairs at Whitehall, and, although\nmany of them were war veterans, all were pressed into service the day\nthe La Granges came to call.  Miss Sue and Miss Sallie Yancey glanced at\neach other expressively when they saw that even Flower, Mrs. Winfield La\nGrange, was one of the party.  It was the first time that she had ever\nbeen openly recognized by the La Grange family, except in name, and no\none knew that it was by Moultrie's express wish that Peachie had asked\nher to go with them.  Thus, indirectly, Carolina was at the bottom of\nit, after all.\n\nPeachie was pretty, but her delicate prettiness was scarcely noticeable\nwhen Carolina was in the room.  Aunt Angie La Grange, Cousin Elise La\nGrange, Cousin Rose Manigault, with her little girl Corinne, who had\ncome to play with Gladys and Emmeline Yancey,--all these insisted on\nclaiming kin with Mrs. Winchester and Carolina, and, as Aunt Angie and\nCousin Lois had known each other in their girlhood, and had spent much\ntime at Guildford and Sunnymede, it was easy for them to fall into the\nold way of claiming cousinship, even when a slender excuse was called\nupon to serve.\n\nThe conversation was very gay and kindly, but, under cover of its\nuniversality, Carolina managed to seat herself next to Flower La Grange,\nwhose pale cheeks and frightened eyes proclaimed how much of a stranger\nshe was to such scenes.  When Carolina called her \"Cousin Flower,\" the\nflush on her face and the look of passionate gratitude in her eyes gave\nCarolina ample evidence that any kindness she might choose to bestow\nhere would be appreciated beyond reason.\n\nAt first Flower was constrained and answered in monosyllables, but when\nCarolina adroitly mentioned the baby, Flower's whole manner thawed, and,\nin her eagerness, she poured forth a stream of rapturous talk which\ncaused the others to look at her in a chilling surprise.  But Flower's\nback was toward her haughty relatives, and only Carolina caught the\nglances,--Carolina, who calmly ignored them.\n\n\"You must come to see my baby!\" cried Flower, impulsively.  \"He is so\ndear!  And so smart! You can't imagine how hard it is to keep him\nasleep. He hears every sound and wants to be up all the time.\"\n\n\"I suppose he notices everything, doesn't he?\"\n\n\"No-o, I can't say that he does.  He likes things that make a noise.  He\ndoesn't care much for looks. If you hold a rattle right up before his\neyes, he won't pay any attention to it.  But, if you shake it, he smiles\nand coos and reaches out for it.  Oh, he is a regular boy for noise!\"\n\nAs Flower said this upon a moment of comparative silence, Carolina\nnoticed that Aunt Angie grew rather pale and said:\n\n\"I haven't seen your baby for several months, Flower.  May I come to see\nhim to-morrow?\"\n\n\"Oh, I should be so glad if you would, Mrs.--\"\n\n\"Call me mother, child,\" said the older woman, looking compassionately\nat her daughter-in-law.\n\nFlower flushed as delicately as a wild rose, and looked at Carolina, as\nif wondering if she had noticed this sudden access of cordiality.  But\nto Carolina, a stranger, it seemed perfectly natural, and she rather\nhurriedly resumed her conversation with Flower, because she had the\nuneasy consciousness that Miss Sue and Aunt Angie, on the other side of\nthe room, were talking about her.  Fragments of their conversation\nfloated over to her in the pauses of her talk with Flower.\n\n\"She thinks nothing of sending off ten or a dozen telegrams a day--\"\n\n\"--she'll wear herself out--\"\n\n\"--it can't last long.  Moultrie says she shows a wonderful head for--\"\n\n\"--and she never gets tired.  I never saw such power of concentration--\"\n\n\"--when I was a girl--\"\n\n\"--writes--writes--writes the longest letters, and if you could see her\nmail!\"\n\n\"--the very prettiest girl I ever saw,--a perfect beauty, Moultrie\nthinks.\"\n\nCarolina's little ears burned so scarlet that she got up and took\nPeachie and Flower out into the garden, and, as the three girls went\ndown the steps, a perfect babel of voices arose in the parlour. Plainly\nCarolina's going had loosened their tongues. They drew their chairs\naround Mrs. Winchester's, and, although the day was cool, they gave her\nthe warmest half-hour she could remember since she left Bombay.  They\ncould understand and excuse every feminine vagary, from stealing another\nwoman's lover to coaxing a man to spend more than he could afford, or\nidling away every moment of a day over novels or embroidery, but for a\nbeauty, a belle, a toast, a girl who had been presented at three courts\nbefore she was twenty, to come down to South Carolina and live on\nhorseback or in a buggy, meeting men by appointment and understanding\nlong columns of figures, sending and receiving cipher telegrams, and in\nall this aided and abetted by no less exclusive and particular a\nchaperon than Cousin Lois Winchester, Rhett Winchester's widow, herself\nrelated to the Lees,--this was a little more than they could comprehend.\nNor could Miss Sue Yancey nor Miss Sallie (Mrs. Pringle), although they\nwere in the same house with her, throw any light on the subject or help\nthem in any way.  Carolina was plainly a puzzle to the La Granges, at\nleast, and when, that same afternoon, Carolina and the two girls in the\ngarden saw another carryall and a buggy drive in at Whitehall,\ncontaining her father's relatives, the Lees, she frankly said that she\nwould stay out a little longer and give them a chance to talk her over\nbefore she went in to meet them.\n\nPeachie laughed at Carolina's high colour when she said this.\n\n\"You mustn't get mad, Cousin Carol, because you are talked about.  We\ntalk about everybody,--it's all we have to do in the country.  But you\nought to be used to it.  You are such a little beauty, you must have\nbeen talked about all your life.\"\n\n\"Nonsense, Peachie!\" cried Carolina, blushing. \"I am not half as\ngood-looking as you and Flower. But the way you all watch me here makes\nme feel as if I were a strange kind of a beetle under a powerful\nmicroscope, at the other end of which there was always a curious human\neye.\"\n\n\"Oh, Cousin Carol, you do say such quayah things!\" cried Peachie,\nlaughing.\n\n\"We ought to go in, I think,\" said Carolina. But at her words the two\ngirls, as if nerving themselves for an ordeal planned beforehand, looked\nat each other, and then Peachie, in evident embarrassment, said:\n\n\"Cousin Carol, I want to ask you something, and I don't want you to be\noffended or to think that we have no manners, but--\"\n\n\"Go on, Peachie, dear.  Ask anything you like. You won't offend me.\nRemember that we are all cousins down here.\"\n\n\"I know, you dear!  But maybe when you know what I want,--but you see,\nwe never get a chance to see any of the styles--\"\n\n\"Do you want to see my clothes?\" cried Carolina. \"You shall see every\nrag I possess, you dear children!  Don't I know how awful it must be\nnever to know what they are wearing at Church Parade. Five trunks came\nyesterday that haven't even been unpacked.  They are just as they were\npacked by a frisky little Frenchman in Paris, and, as they were sent\nafter me, they were detained in the custom-house, and, before I could\nget them out, I was hurt. While I was in bed, my brother got them out of\nthe custom-house and took them to _his_ house, where I forgot all about\nthem until I was preparing to come here.  Then I thought of clothes!\nAnd I also thought I might find some pretty girls down here among my\nrelatives who would like to see the Real Thing just as it comes from the\nhands of the Paris couturieres,--so there you are!\"\n\n\"Oh, Carolina Lee!\" shrieked Peachie, softly. \"What a sweet thing you\nare!  Just think, Flower, Paris clothes!\"\n\n\"And better still, Vienna clothes!\" said Carolina, laughing.\n\n\"You said you were hurt, Cousin Carol,\" said Flower, in her soft little\nvoice.  \"How were you injured?\"\n\n\"I was thrown from my horse, Flower, dear, and my hip was broken.  I was\nin bed for months with it.\"\n\n\"But you were cured,\" said Flower.  \"I never heard of a broken hip that\ndidn't leave a limp. There must be mighty fine doctors in New York.\"\n\n\"There are!\" said Carolina, softly.  Then she turned suddenly and led\nthe way to the house, the girls eagerly following.\n\nIt will be difficult and not at all to the point to try to learn the\nrelationship of the Lees and La Granges to Carolina and to each other.\nAunt Angie La Grange was Moultrie's, Winfield's, and Peachie's mother.\nRose Manigault was Aunt Angie's married sister, and Elise an unmarried\none.\n\nOf the Lees, there was Aunt Evelyn Lee, Carolina's own maiden aunt.\nAunt Isabel Fitzhugh, her married aunt, with her two daughters, Eppie\nand Marie.  Uncle Gordon Fitzhugh, Aunt Isabel's husband, and a bachelor\ncousin of Carolina's, De Courcey Lee, were the ones who had come in the\nbuggy with the two little Fitzhugh boys, Teddy and Bob.\n\nThe children could not be induced to leave the parlour until they had\nseen their new cousin, they had heard so much of her beauty from\nMoultrie, so that, when Carolina entered and was introduced to her\nadmiring relatives, none was more admiring than the children.  Indeed,\nBob Fitzhugh announced to his father, as they were driving home that\nevening, that he was going to marry Cousin Carol. He said that he had\nalready asked her, and that she had told him that she was ten years\nolder than he was, but that, if he still wanted her when he was\ntwenty-one and she hadn't married any one in the meantime, she would\nmarry him.\n\n\"You couldn't do better, son,\" said his father, nudging De Courcey, \"and\nI commend your promptness, for, as Carolina is the prettiest--the very\nprettiest little woman I ever saw, the other boys will doubtless get\nafter her, and it's just as well to have filed your petition\nbeforehand.\"\n\nIndeed the verdict on Carolina was universally favourable.  Her\nrelatives were familiar with her photographs, and were proud of the\naccounts which at intervals had filtered home to them through letters\nand newspapers, but the girl's beauty of colouring had so far outshone\ntheir expectations, and her exquisite modesty had so captivated them\nthat they annexed her bodily, and quoted her and praised and flattered\nher until she hardly knew where to turn.  She won the Fitzhugh hearts by\nher devotion to Teddy, the seven-year-old boy, who could not speak an\nintelligible word on account of a cleft palate.  She took him with her\non the sofa and talked to him and encouraged him to try to answer, until\nthe mother, though her soul was filled with the most passionate\ngratitude, unselfishly called the boy away, saying, in a hurried aside\nto Carolina:\n\n\"Thank you, and God bless you, my darling girl, for trying to help my\nbaby boy, but you owe your attention to the grown people, who, some of\nthem, have driven twenty miles to see your sweet face. Some day,\nCarolina, I want you to come and spend a week with us, and tell me about\nthe best doctor to send the child to.  You must know all about such\nthings, coming from New York.\"\n\nShe won the heart of her bachelor cousin, a man of nearly sixty, by\nallowing him to lead her to a sofa and question her about her father,\nhis last days in London, and of how she had inherited her love for\nGuildford.\n\n\"For it is an inheritance, Carolina, my dear. Your father loved the\nplace as not one of us do who have stayed near it.\"\n\n\"Yes, Cousin De Courcey, I think you are right. Daddy used to dream of\nit.\"\n\n\"Did he ever tell you of the loss of the family silver?\"\n\n\"Yes, he said it was lost during the war.\"\n\n\"Did he never tell you of his suspicions concerning it?\"\n\n\"No, because I don't think he had any.\"\n\n\"Pardon me for disagreeing with you, my dear, but in letters to me he\nhas stated it.  You know our family silver included many historical\npieces,--gifts from great men, who had been guests at\nGuildford,--besides all that the family had inherited on both sides for\ngenerations.  Many of these pieces were engraved and inscribed, and,\nunless they were melted at once, could have been traced.  Your\ngrandfather and your father, being the only ones fortunate enough to\nhave increased their fortunes, undertook to search the world over for\ntraces of this silver, but, as not so much as a teaspoon of it was ever\nfound, we think it is still buried somewhere near here,--possibly on the\nestate.  Aunt 'Polyte, your father's black mammy, and her husband buried\nit, and to the day of their death they swore it was not stolen by the\nYankees, for, when they missed it, there were no Federal troops within\nfifty miles.  They both declared that some one traced them in their\nfrequent pilgrimages to its hiding-place to ascertain that it was\nintact, and that the Lee family will yet come into its own.  As you seem\nto be our good angel, it will probably be you who will find it.  Doesn't\nsomething tell you that you will?\"\n\n\"Yes, something tells me that it is not lost,\" said Carolina, with grave\neyes.  \"I came into the possession of Guildford so wonderfully, perhaps\nI shall find the Lee silver by the same means.\"\n\nJust then Mrs. Pringle hurried into the room, saying hospitably:\n\n\"Now listen to me, good people.  You all don't come to Whitehall so\noften that we don't feel the honour, and now that you are here, you must\nstay to supper.  Don't say a word!  I'll tell Jake to hitch up and go\nafter Moultrie and Winfield, and there's a full moon to-night, so you\nwon't have any trouble in getting home.  Elise, if you are too big a\ncoward to drive twenty miles after dark, you can stay here all night.\nFlower, do you trust your nurse to stay with the baby?\"\n\n\"Oh, yes, indeed, thank you, Miss Sallie.  I'll just write a note to\nWinfield and send it by Jake, if I may, telling him to see that Aunt\nTempy and the baby are all right before he starts, then I won't be a bit\nuneasy.\"\n\nThe La Granges had never heard their unpopular kinswoman make so long a\nspeech before, and, as they listened to it, with critical, if not\nhostile ears, they were forced to admit that she exhibited both spirit\nand breeding, and her voice had a curious low-toned dignity which\nindicated an inherited power.\n\nWhitehall had not been famous for its hospitality since the death of\nElliott Pringle, Miss Sallie's husband.  During his lifetime they had\nkept open house, and Miss Sallie was the soul of hospitality. She would\ndearly have loved to continue his policy and the prestige of Whitehall,\nbut her sister, Sue Yancey, was, in popular parlance, called \"the\nstingiest old maid in the State of Georgia,\" and when she came to live\nwith her widowed sister she watched the expenditures at Whitehall, until\nnobody who ever dined there had enough to eat. There was a story going\naround that the reason she lost the only beau she ever had, was because\nonce when he was going on a journey she asked him to take out an\naccident insurance policy, and when he told her that he was all alone in\nthe world and that no one would be benefited by his death, she told him\nto send the ticket to her.  Rumour said that he sent the ticket, but\nthat he never came back to Sue.\n\nSue either cared nothing for the good opinion of other people or she\nmade the mistake of underestimating her friends' intelligence, for she\ncarried her thrift with a high hand.  At Sunday-school picnics it was no\nuncommon sight for the neighbours to see Miss Sue Yancey going around to\nthe different tables gathering all that was edible into her basket to\ntake home with her.  And that these scraps subsequently appeared on the\ntable at Whitehall often led to high words between the sisters; but in\nthe end it always happened that Sue conquered, because Mrs. Pringle\ndreaded her sister's bitter tongue and ungoverned temper.\n\nYet Sue often complained that she felt so alone in the world because no\none understood her.\n\n\"Don't stay,\" whispered Gordon Fitzhugh, in his wife's ear.  \"Sue never\ngives me enough sugar in my tea!\"\n\nCarolina could not help overhearing.  She looked up quickly and laughed.\n\n\"Are you getting thin?\" he whispered.  \"Does Sue give you as hash for\nsupper the beef the soup is made from?\"\n\n\"I think Miss Sallie is ordering while we are here,\" said Carolina,\nloyally.  She would not tell her Uncle Fitzhugh that one morning when\nLily was taking Cousin Lois's breakfast up to her, when her asthma was\nbad, that Sue had waylaid Lily in the hall and had taken the extra\nbutter ball off the tray and carried it back to the dining-room in\ntriumph.\n\n\"I admire economy,\" said Uncle Fitzhugh. \"Sue's ancestors were French,\nbut, in her case, French thrift has degenerated into American meanness.\"\n\n\"You stay,\" said Carolina, dimpling, \"and I'll see that you get all the\nsugar you want, if I have to ask for it myself!\"\n\n\"Then I'll stay,\" chuckled Uncle Fitzhugh, and he beckoned to De Courcey\nto come out into the garden and have a smoke--in reality to gossip.\n\nHardly were the gentlemen out of sight when Peachie said, excitedly:\n\n\"Mamma, do beg them all to excuse Cousin Carol, Flower, and me!  Carol\nhas promised to show us her Paris clothes--five trunks full of them!\"\nHer voice rose to a little shriek of ecstasy, which was echoed in\nvarious keys all over the room. Every face took on a look of intense\nexcitement and anticipation.\n\n\"Excuse you!\" cried Aunt Angie La Grange. \"We shall do no such thing.\nIf Carol thinks we old people are not just as crazy over pretty clothes\nas we were when we were girls, she doesn't know the temperament of her\nown blood and kin.  Carol, child, lead the way to those trunks\nimmediately. My fingers fairly burn to turn the keys in those locks!\"\n\n\"Really, Aunt Angie?  Why, we shall be delighted.  You should see the\ngowns Cousin Lois had made for the Durbar.  They are simply regal!\"\n\n\"Lois Winchester,\" said Aunt Angie, as they went up-stairs, \"they tell\nme that you actually rode an elephant while you were in India!\"\n\n\"I did, Cousin Angie,\" said Mrs. Winchester, imperturbably.  \"And what\nis more, I had my picture taken on one.  You can hardly tell me from the\nelephant!\"\n\nNow Cousin Lois so seldom jested that this sally met with the usual\nreception which non-jokers seem to expect, and the walls fairly reeled\nwith the peals of laughter from the delighted kinfolk.  But when they\nwere all gathered in Carolina's room and the chairs were brought from\nall the other rooms to seat the guests, a hush fell upon the assemblage\nsimilar to that which falls upon Westminster Abbey when a funeral\ncortege arrives.\n\nCarolina was unlocking her Paris trunks!\n\n\n\n\n                              CHAPTER XV.\n\n                             THE BLIND BABY\n\n\nThe same terrible suspicion which had entered Aunt Angie La Grange's\nmind when she overheard Flower's innocent words had occurred to\nCarolina, and as there seemed to be one of those sudden new-born bonds\nof sympathy between the beautiful old woman and the beautiful young\ngirl, which sometimes spring into existence without warning, yet with\ngood reason, as afterwards transpires, Carolina was not surprised to\nhave Aunt Angie draw her aside after supper and say:\n\n\"Carolina, child, what did you think when you heard what Flower said\nabout little Arthur?\"\n\n\"I thought just what you thought, Aunt Angie, at first, then--\"\n\n\"Then what?\"\n\n\"Nothing.\"\n\n\"Now, Carol, you were going to say something! What was it?  I am sure\nthe thought that I am a comparative stranger to you stopped the words on\nyour lips.\"\n\n\"I am afraid that you wouldn't understand what I was going to say, Aunt\nAngie, dear, and I don't want to antagonize you.  I like you too much.\"\n\n\"Dear child, nothing that your silver tongue could utter could\nantagonize me after your sweet generosity to my daughter this afternoon.\nOh, Carol, don't you think my mother-heart aches at not being able to\ndress my pretty girl in such fairy fabrics as you showed us?  And then\nto think of your giving her that pink silk!  Why, Peachie won't sleep a\nwink for a week, and I doubt if her mother does, either!  Now she can go\nto the Valentine German in Savannah.  You must go, too.  I will arrange\nit.  I--but my tongue is running away with me.  Tell me what you were\ngoing to say.\"\n\n\"Well,\" said Carolina, hesitatingly, \"you have heard that I am a\nChristian Scientist, haven't you?\"\n\n\"Yes, dear, I have, and I must say that I deeply regret it.  Not that I\nknow anything about it, but--\"\n\n\"That's the way every one feels who doesn't know about it,\" cried\nCarolina, earnestly; \"but that is nothing but prejudice which will wear\naway. Indeed, indeed it will, Aunt Angie.\"\n\nMrs. La Grange shook her head.\n\n\"I am a dyed-in-the-wool Presbyterian, and I've fought, bled, and died\nfor my religion in a family who believe that God created the Church of\nEngland first and then turned His attention to the creation of the\nearth, so you can't expect me to welcome a new fad, can you, my dear?\nBut I beg your pardon, Carol.  What were you going to say?\"\n\n\"It was only this,\" said Carolina, gently.  \"That even if Flower's baby\nis blind to mortal sight, he is not blind in God's eyes.  There he is\nperfect, for God, who is Incarnate Love, never created a blind or dumb\nbaby.\"\n\nTears rushed suddenly to the old woman's eyes.\n\n\"Are you thinking of poor little Teddy Fitzhugh?\" she whispered.\n\n\"Yes, I was.\"\n\n\"Oh, Carolina!  If you could have seen his mother's anguish all these\nyears!  But you would have to be a mother yourself before you could even\napprehend it.\"\n\n\"Yes, I suppose I would.\"\n\n\"And now,\" said the older woman, with that patient tightening of the\nlips with which so many Christian women prepare themselves to bear the\nheart-breaking calamities which they believe a tender Heavenly Father\ninflicts on those He loves, \"I suppose I must steel my heart to see poor\nFlower writhe under a worse agony.  Indeed, Carol, God's ways are hard\nto understand.\"\n\n\"Yes, God is such a peculiar sort of parent,\" observed Carolina.  \"He\nseems to do things with impunity, which if an earthly father did, the\nneighbours would lynch him.\"\n\nAunt Angie La Grange sat up with a spring of fright.\n\n\"Why, Carolina Lee!  What sacrilege!  You will certainly be punished by\nan avenging God for such blasphemy.  You shock me, Carolina. You really\ndo.\"\n\n\"Forgive me, Aunt Angie.  I only meant to imply that the God I believe\nin is a God of such love that He never sends anything but good to His\nchildren.\"\n\n\"Then how do you get around that saying, 'Whom the Lord loveth He\nchasteneth?'\"\n\n\"There is authority for translating that word 'chasteneth,'\n'instructeth.'  But even if you leave it 'chasteneth,' it doesn't mean a\nlife-long disfigurement or crippling of innocent babies.  Supposing\nPeachie should disobey you, or even disgrace you, would you deliberately\ninfect her with smallpox to destroy her beauty or send her into a train\nwreck to lame her or paralyze for life?\"\n\nMrs. La Grange only looked into Carolina's eyes for reply, but her hands\ngripped the arms of her chair until her nails were white.\n\n\"Yet you are only her earthly--her human--her finite mother.  How much\ngreater capacity has the Infinite Heart for love!\"\n\nMrs. La Grange stirred restlessly.\n\n\"It is beautiful,\" she breathed, \"but--disquieting. It upsets all my old\nbeliefs.\"\n\n\"'And good riddance to bad rubbish,' as we children used to say,\" said\nCarolina, smiling.  Aunt Angie smiled in answer, but a trifle dubiously.\n\n\"Carolina,\" she said, \"Moultrie told me--but of course you never said\nsuch a thing and I told him then that he must have misunderstood\nyou--that Gladys Yancey was cured by Christian Science! Now, what _did_\nyou say?\"\n\n\"I said just that.  She _was_ cured by Christian Science.\"\n\n\"I don't believe it!\" cried Aunt Angie.  \"Excuse me, dear child, for\nsaying so.  I know that you are truthful and that you believe it, but\n_I_ don't.  I'd have to see it done.\"\n\n\"If you saw Teddy Fitzhugh taught to speak plainly, would you believe?\"\n\n\"My dear, I'd leave the Presbyterian Church and join the Christian\nScientists so quickly my church letter would be torn by the way I'd\nsnatch it.\"\n\nCarolina laughed and squeezed Aunt Angie's hand, who added with a smile:\n\n\"I suppose you think I am as good as caught already, don't you?\"\n\n\"I hope you are.  You can't imagine how much peace it brings.\"\n\n\"Peace!  It's something I never have had, child.\"\n\n\"Nor I.  But I have it now.\"\n\n\"What does your religion compel you to give up? Peachie absolutely\nrefuses to join the church because it won't allow dancing, and the child\nloves to dance better than anything in the world.  They tell me, too,\nthat she dances like a fairy.\"  Aunt Angie pronounced it \"fayry.\"\n\n\"Why, that is one of the best things about Christian Science.  It\nrequires you to give up no innocent pleasure.  It only cautions one\nagainst indulging to excess in anything.  Dancing, card-playing,\ngames,--why, some of the best card-players I know are Christian\nScientists, but they don't lose their tempers when they lose a game and\nthey don't cheat to win.  In fact, one of the most graceful things I\nhave ever seen done was when two ladies tied for the prize--a beautiful\ngold vase--at a bridge party Addie gave just before she closed her\nhouse, and the lady who won had played coolly, well, and won by merit.\nThe other flung herself back in her chair with an exclamation, showing\nby her suffused face and clenched hands every sign of ill-temper.  My\nsister-in-law brought the prize to the winner, who, with the prettiest\ngrace imaginable, thanked her and then presented it, by Addie's\npermission, to the vexed lady who had lost.  You should have seen the\nrecipient's face!  Surprise, humiliation, and cupidity struggled almost\naudibly for supremacy.  She protested feebly, but ended by taking it.  A\nnumber of others gathered around, attracted by the unusual scene, and\nsuddenly the owner of the vase said to the giver of it: 'I would like to\nknow what church you go to.'  'Well, as none of you know, you may\nguess,' she answered. They guessed Baptist, Methodist, Unitarian,\nEpiscopal, and finally the recipient of the vase said: 'No, you are all\nwrong.  I believe she is a Christian Scientist, because no one but a\nChristian Scientist would give up a gold vase!'\"\n\n\"I like that,\" said Aunt Angie, promptly.  \"And I think the churches\nmake a mistake in forbidding innocent pleasures.  Oh, why don't they\ndwell on the good instead of squabbling over the bad?\"\n\n\"You have described one of the chief differences between the Christian\nScience and the other churches,\" cried Carolina.  \"Why, Aunt Angie, you\nare a ready-made Scientist!\"\n\n\"Am I?  Well, we shall see.  Now tell me when you can go to see Flower.\nWas Moultrie able to buy Araby for you?\"\n\n\"No, Mr. Mazyck refused to sell her.  But Moultrie has lent me Scintilla\nuntil he can find another good horse for me.\"\n\n\"But you especially wanted Araby, didn't you?\"\n\n\"Yes, because she is a direct descendant of the sire of my grandfather's\nfavourite saddle-horse. And she is simply perfect, Aunt Angie.\"\n\n\"I am afraid Barney Mazyck is hopeless.  If he wants a thing, he wants\nit and is going to keep it.\"\n\n\"I know; but I have not despaired of getting her yet.  Perhaps I am just\nas bent upon getting her as Mr. Barnwell Mazyck is upon keeping her.\"\n\n\"And in that case--\"\n\n\"Well, I wouldn't put any money on Mr. Mazyck!\" laughed Carolina.\n\nIn the slight pause which ensued, Carolina could see that Mrs. La Grange\nwas ill at ease.  Suddenly she turned to the girl and said:\n\n\"My dear, doubtless you think it strange that I do not know beyond a\ndoubt the state of my own little grandson's sight, but--\"\n\n\"I know,\" said Carolina, gently.  \"I have heard.\"\n\n\"Who told you?  Some stranger?\"\n\n\"No, Moultrie told me.\"\n\n\"Ah, then you have heard the truth!  It is a terrible grief to us,\nCarolina.  Think of the child! I do not know who my own grandson is\ndescended from!\"\n\n\"But you will know,\" said Carolina, earnestly. \"And soon.  I--we have a\nright to expect God's harmony in our lives.\"\n\nMrs. La Grange looked at her curiously, but only said, with a sigh:\n\n\"I am sure I hope you may be right.\"\n\nIt was arranged that Carolina was to meet Mrs. La Grange at Flower's the\nnext afternoon at three o'clock.\n\n\"Can't you go in the morning?\" asked Mrs. La Grange.\n\n\"I have an appointment with the architect from Charleston and the\nbuilders at Guildford at ten. We wouldn't get through in time, I am\nafraid, for there will be so much to discuss.\"\n\n\"Won't you be too tired?\"\n\n\"I never get tired.  There is rest in action for me.\"\n\nMrs. La Grange shook her head, but not in disapproval.\n\n\"I hope I am going to like it.  If I like all of it as well as I do the\nsample bits you have fed me with, I think, as you say, you may find that\nI have been a Scientist all my life without knowing it.\"\n\nMrs. La Grange looked into the girl's pure, beautiful face\nscrutinizingly, as if to learn her secret of happiness, and, as she did\nso, she was surprised to see it suffused by a blush which rose in\ndelicate waves to her hair.  Looking about in surprise for a cause, Mrs.\nLa Grange saw her son Moultrie approaching.  Could Carolina have\nrecognized his step without seeing him, and was that blush for Moultrie?\n\nThe question could not be answered at once, nor did she see them\ntogether the next day, for Carolina was late in keeping her appointment,\nand, by the time she arrived, the awful truth was known. Mrs. La Grange\nhad been so overcome that Moultrie was obliged to take her home.\n\nThe moment Carolina rode up to the house, she knew that something had\nhappened.  The house, a mere cabin, was ominously quiet, and no one came\nto meet her.\n\nShe dismounted hurriedly, fastened Scintilla to the fence, and ran up\nthe steps.  No one answered her knock.  She pushed open the door and\nentered.\n\nAt first she saw no one, but presently she heard heavy breathing, and,\ncrouching on the floor, in the darkest corner of the room, she saw\nFlower, holding the still form of her baby in her arms.  Her posture and\nthe glare in her eyes were tigerish.\n\nWith a low cry, Carolina sprang to her side.\n\n\"Oh, Flower, darling!  What is the matter with your baby?\"\n\n\"You may take him,\" said Flower, dully.  \"You care!  You cared\nyesterday.  I can tell.  She only cares because Arthur is a La Grange.\nYou will care just because he a helpless little blind baby. Oh! oh!\"\n\n\"Not blind, Flower!  Don't say it.  Don't think it.  Your baby sees.\"\n\n\"No, Cousin Carol.  You are good and kind, but Mrs. La Grange made me\nsee for myself.  We took a candle and held it so close to his eyes we\nnearly burned his little face--\"\n\n\"You?\" cried Carolina.  \"Were you in the room?\"\n\n\"That's what Moultrie said, but you don't either of you know.  When you\nhave a child of your own, you will both understand that a mother can't\nkeep away.  She must know the worst, and she must be there when it\nhappens.\"\n\n\"Oh, poor Flower!  Poor child!\" cried Carolina, weeping unrestrainedly.\nShe cuddled the baby's face in her neck, and Flower watched her\napathetically.  Flower's face was suffused from stormy weeping, but she\nhad wept herself out.\n\n\"And you had to bear this all alone, poor lamb!\"\n\n\"I wanted to be alone!  I wanted her to go. They meant to be kind, but\nthey don't love me, and they don't love my little baby.  I would rather\nbe alone.  Who could I send for--the priest? When he predicted it?\"\n\n\"What did he predict?\" asked Carolina, quickly.\n\n\"He was very angry because we went to New York to be married.  He lost\nfifty dollars by it. That is what he charges even poor people like me.\nAnd because I married a heretic, and because I was not married by a\npriest, he cursed me and my offspring.  Then--\" she broke off suddenly\nand cried: \"Oh, why do I tell it all?  Why do I trust even you?\"\n\n\"Because you know that I can help you,\" said Carolina, gravely.\n\n\"No one can help me--not even God!\"\n\n\"Say what you were going to,\" urged Carolina.\n\n\"Well, the child is bewitched.  Every time there is a thunder-storm, or\nif I am even left alone with the baby, like to-day, when I let Aunt\nTempy have her afternoon--there she is now!\"\n\nWith a shriek of terror she pointed to the window, and Carolina looked\njust in time to see a dark face disappear from view.  She ran to the\ndoor, but nothing could be seen.  Not a sound could be heard.\n\n\"It is the voodoo!\" whispered Flower.  \"That face always comes.  Once I\nsaw it in the room, bending over the cradle when the baby was asleep.\nBut I never can catch her.  Aunt Tempy has seen her, so has Winfield.\nShe has cast an evil spirit over my baby.\"\n\n\"Her face looked kind--it even looked worried,\" thought Carolina to\nherself, but she said nothing to Flower.  She only sat rocking the\nsleeping baby, wiping the tears which rolled down her cheeks at the\nsight of the mother's anguish.\n\n\"Flower,\" she said, suddenly, \"did you ever see Gladys Yancey before\nMiss Sue took her North?\"\n\n\"Heaps of times.\"\n\n\"Did you ever hear how she was cured?\"\n\n\"Why, Moultrie told Winfield that it was a new kind of religion that did\nit, and Winfield just hollered and laughed.\"\n\n\"Well, if I could prove to you that your baby could be made to see,\nwould you holler and laugh?\"\n\n\"I reckon I wouldn't.  I'd kiss your feet.\"\n\n\"The only trouble,\" murmured Carolina, half to herself, \"is that you are\na Roman Catholic.  We do not like to interfere with them.\"\n\n\"I am not a Roman Catholic,\" said Flower. \"The lady who brought me up,\nand whom I was taught to believe was my aunt, was a Catholic, but I\nnever was baptized.  I believe Father Hennessey knows who I am, and\nthat, if he would, he could clear up the mystery of my birth and give me\nback my happiness.  But he never will until I join his church.  He told\nme so.\"\n\n\"Is he an old man?\" asked Carolina.\n\n\"Oh, a very old man.  He must be over eighty,\"\n\nA slight pause ensued.  Then Carolina said: \"Would you like to hear of\nthis new religion?\"\n\n\"If it will give my baby eyes, Cousin Carolina, how can you even stop to\nask?\"\n\n\"Oh, my dear, it is only because we are taught to go cautiously,--to be\nsure our help is wanted before we offer.\"\n\n\"Well, offer it to me.  I want your help with all my soul!\"\n\nShe rose from her corner and came and sat at Carolina's feet.  Something\nof Carolina's sincerity, which always appealed to people, moved her to\nbelieve that Carolina could help her.  Flower's mind, too, though it may\nsound like an anomaly, had been trained by her aunt's Catholicism to\nbelieve in signs and wonders, and her superstitions had been carefully\neducated.  Therefore, when a more analytical mind might have hesitated\nto believe that material help for a supposed hopeless affliction could\ncome from religion, instead of from a knife or a drug, which even the\nmost skeptical may see and handle and thus believe, Flower, by her very\nchildishness, held up a receptive mind for the planting of the seed of\nan immortal truth.\n\nThe gravity of the situation caused Carolina a moment's wrestle with\nerror.  The burning eyes of the young mother fastened on Carolina's face\nwith such agonizing belief,--the feeble flutterings of the sleeping baby\nin her arms terrified her for a brief second.  Then she lifted her heart\nto the boundless source of supply for every human need, and in a moment\nshe felt quieted and could begin.\n\n\"Flower,\" she said, \"do you believe in God?\"\n\n\"Of course I do.\"\n\n\"Did you ever read your Bible?\"\n\n\"No.\"\n\n\"Have you one?\"\n\n\"No.\"\n\n\"Will you promise to read it if I will give you one?\"\n\n\"I will do whatever you want me to.\"\n\nCarolina hesitated a moment.\n\n\"Will your husband object to your trying Christian Science with the\nbaby?\"\n\n\"I don't know--yes, I suppose he will.  What shall we do?\"\n\n\"What will he want to do when he first learns that the baby is blind?\"\n\n\"I reckon he'll want to have Doctor Dodge see him.\"\n\n\"There is no objection to that.  Then what will he do?\"\n\n\"There isn't anything we can do just now, Cousin Carol.  We have had a\ndreadful time even to live since we were married.  And look what a\nshanty we live in!  Not fit for a <DW64>.  And Winfield a La Grange!  Of\ncourse, if the crops are better next year we might be able to take him\naway to consult some big doctor, but this winter we can't do anything at\nall.\"\n\n\"I don't know what to do,\" said Carolina. \"You ought to get your\nhusband's consent first.\"\n\n\"Well, what do you want me to do?  Does your treatment commence right\naway?\"\n\n\"It is already begun.\"\n\n\"Why, how?  You haven't done anything that I could see.  Do you pray?\"\n\n\"Not to any virgin or saint, Flower.\"\n\n\"No, I know that Protestants pray to God.  Is that what you want me to\ndo?\"\n\n\"I want you first to have a talk with Winfield and Moultrie--\"\n\n\"Moultrie will help me!\" interrupted Flower. \"I'll ask him to talk to\nWinfield.\"\n\n\"Well, do that.  Then if he says you may try it, I want you not to tell\nanother soul, especially don't let Aunt Tempy or any of the <DW64>s know\na word about it.  I want you to get up about twelve o'clock every night\nand light your candle, and put it where it shines directly in the baby's\neyes.  It can't hurt him.  Then read the whole of the New\nTestament,--just as much every night as you can for one hour, believing\nthat everything which was true of Jesus and His disciples then, can be\nand is true of His disciples on earth to-day, and that, if any one of us\ncould ever be as pure and holy as He was, that we could do the one thing\nwhich is denied us yet,--that is, raise the dead!  Will you?\"\n\n\"Indeed, I will.\"\n\n\"Then every night I will treat your baby's eyes by mind-healing, which I\nwill explain to you a little later.  In the meantime, you watch very\nclosely to see the first indication which Arthur's eyes give of the\nlight's making him stir, for that will show that his darkness is lifting\nand that he is beginning to see.\"\n\nFlower raised herself up and clung to Carolina's knees and buried her\nface in her dress, weeping bitterly.\n\n\"Oh, oh!  Don't think I am unhappy.  I am crying because I think you can\ndo it.  How long will it take?\"\n\n\"No one can say.  It may only take one treatment, or it may take years.\n'According to your faith be it unto you.'\"\n\nJust then, as Carolina rose to go, the baby wakened, and Flower reached\nfor him and pressed him to her bosom in a passion of grief and hope.\n\n\"Look!\" she whispered to Carolina, \"you can tell from the very\nexpression of his little eyes that he can't see.  I remember now that\nonce the sun was shining right into his eyes, and he kept them open, but\nI didn't notice it at the time.\"\n\n\"Remember this, Flower.  We think that he can't see.  But in God's eyes\nhe is perfect.  With Him there is no blindness nor sickness nor sin nor\nsorrow.  He will take away your grief.  He will wipe away all tears from\nyour eyes.\"\n\n\n\n\n                              CHAPTER XVI.\n\n                         A LETTER FROM CAROLINA\n\n\n\"'THE BATH,' ENTERPRISE, S.C.,\n       \"January 27, 19--\n\n\n\"MY DEAR MR. HOWARD:--If only I could drop in on you this evening and\nmake my report in person, what couldn't I tell!  You would laugh if you\nknew why we call our house The Bath.  But first, have I ever told you\nthat we have a house?  Well, Guildford is so far from even Whitehall,\nwhich is the nearest place we visited, that I lost too much time in\ncoming and going.  I must have been eight hours in the saddle some days,\nand I didn't get on fast enough to suit my leaping\nambition,--and--bathrooms are scarce in the country, so Cousin Lois and\nI decided to build a model cabin or quarters before we started the\nhouse, and live on the place. There was already a windmill, so I ordered\na porcelain tub in Charleston, and built my house around it.  Cousin\nLois preempts it most of the time, but I get my full share, and it is a\nluxury. Did you ever try going without a bathroom?  Try it.  It will\nmake you 't'ink ob yo' marcies,' as the <DW64>s say.\n\n\"Oh, we are so happy!  Every day some of the dear neighbours who knew\nGuildford in its prime ride or drive over to tell me little forgotten\nquirks of the blessed place, and to assure me that I am copying it\nfaithfully.  Cousin Lois calls it curiosity, but I think it is interest.\nBut the primitive methods in vogue in the South--well, you simply would\nnot believe me unless you saw them.  For example, at the turpentine\nplant at Schoville, which I will tell you more of later, my engineer\nfound them ladling out the crude turpentine by hand, when you know it\nought to be piped, and half the time this cheap <DW64> labour, which they\nhire to save machinery, is drunk or striking, which often shuts down the\nplant for days at a time,--ten days at Christmas always.  Machinery may\nbe expensive, but, at least, it doesn't get drunk, and by means of it a\nman may run his business, even in the South, regularly, and so build up\na reputation for reliability, which, honestly, Mr. Howard, nobody down\nhere seems to know the meaning of, as we understand it!  Any excuse\nserves.  Just make your excuse--that's all.  It not only seems to\nrelieve the conscience of the purveyor, but satisfies the consumer as\nwell.  In Georgia it is a State law not to move freight on Sunday.\nImagine that, added to the railroad service as it stands!  And in a\ncertain town in Middle Georgia, the fire-engines are drawn by oxen.  I\nenclose the kodak I took of it, for I know you won't believe me else.\nOne thing the South needs more than anything else is some of our\nNorthern Italian labour.  Then the <DW64>s will see what it really is to\nwork.\n\n\"But I am running away with myself.\n\n\"I shall skip all I can, and only tell the essentials.\n\n\"After we left Whitehall, nothing would do but we must pay a round of\nvisits among the Lees and La Granges, which we did, staying as short a\ntime as possible with each, partly because I could not properly attend\nto my work, and partly because of the heart-breaking poverty of all my\npoor dear relatives.  If you could only see their bravery, their pride,\nand their wholly absurd fury at the bare suggestion that ease and\ncomfort might come to them from admitting Northern capital!  I think if\nthey knew that my money comes through you, they would force me to starve\nwith them rather than be indebted to a ---- Yankee.  The ladies don't\nuse that word with their lips, but their eyes say it. As it is, they\nthink I am still selling my jewels. And I don't contradict them, simply\nbecause there is no use in giving them pain.  Their hatred of the North\nis something which cannot be eradicated in a day.  It is a factor in\nbusiness which blocks the path of every well-wisher of the South, and is\nan entity to be reckoned with just as palpably as credit. The man who\nignores it makes a mistake which sooner or later will bring him up with\na jerk.  I dwell upon this, because, if we form the syndicate which you\npropose, it must be managed craftily, and I know you will not disregard\nmy warning.\n\n\"As an example of it, let me tell what has befallen the plant for making\nwood turpentine at Schoville, Georgia.  It is a fine, modern, up-to-date\nplant of the steam process, backed and controlled by Judd Brothers &\nMorgan, of Brooklyn.  Their representative approached my counsel,\noffering to sell.  The Brooklyn firm own fifty-one per cent. of the\nstock, and the rest is taken by citizens of Schoville.  I sent my man,\nDonohue, down to investigate the process, intending, if I didn't buy, to\norganize a similar company and operate under their patents, as I find\ntheirs, if not the best, is at least a satisfactory process, and turns\nout a pure water-white turpentine with a specific gravity of 31.70. And\nDonohue asserts that by the use of steam he can eliminate the\nobjectionable odour.  He has been in the employ of both the Schoville\nand the Lightning companies and is a valuable man, though not strictly\nhonest.  Donohue was satisfied that there was something wrong at\nSchoville, and advised me to hold off.  He reported the plant out of\nrepair, although the books showed money in plenty supplied by the\nowners.  Donohue then visited the plant at Lightning, Georgia, and found\neverything all right.  It has since transpired that the foreman of the\nplant at Schoville, a cracker named Leakin, had deliberately shipped\ncrude turpentine, which of course was of rank odour and off colour, to\nthe factors at Savannah, who shipped it to Germany and South America\nwithout giving it a very careful examination.  As is usual with these\nmen, they were too slack to make the thorough examination before making\nshipment which the law requires, and paid over an advance of thirty-five\ncents a gallon to Leakin like innocent little lambs.  Of course, the\ninevitable occurred. Buenos Ayres and Berlin not only refused to pay,\nbut returned the consignment, and the Savannah factors now refuse to\ntouch wood turpentine at any price.\n\n\"It seems that, when the Northern owners sent their representative down\nto investigate, Leakin frankly told him that he did not intend to make\nmoney for any ---- Yankees.  They thereupon swore out a warrant for his\narrest, but he wrecked the plant at night and was hurried out of town by\nhis relatives.\n\n\"Now, so far from discouraging me, this serves my purpose well.  For\nwith sixty per cent. profit on the manufacture of wood turpentine on\npaper (as per my previous reports), which cuts to between forty and\nfifty in actual operation, it is one of the future industries of the\nSouth.  Of course the little plant I propose to build at Guildford or\nnear by will only be a mouthful.  I figure that between ten and twelve\nmillions of dollars would corner the turpentine market, and then put the\nprice of orchard turpentine so high that it would practically be off the\nmarket.  Then we could force the consumers to take wood turpentine in\nits place, and in this way show them that it will do the same work and\nbring the same results as the regular orchard turpentine. They are\nafraid of it now, so they must be reduced by compulsion to giving it a\nfair trial.  I bought ten barrels of wood turpentine made by the company\nat Lightning, and sent a small sample to every paint and varnish\nmanufacturer in the United States, with a letter giving them the\nchemical analysis and asking the recipient to give it a fair trial.\nAbout one-third replied that it seemed satisfactory, and sent me orders\nfor from five to ten barrels for a trial, but they want it at about ten\ncents per gallon less than the orchard.  It seems that no one will pay\nwithin ten cents of the regular market price. I turned these orders over\nto the Lightning company on a commission, and am making quite a neat\nlittle sum out of it, though I never thought of that end of the\nproposition when I sent out the samples.  I tried the experiment to see\nwhat sort of a market I could look for.  There is no reason why this\nwood turpentine should not be shipped and sold as regular turpentine,\nand one good strong corner on the market will bring this about.\n\n\"To continue my investigations, I want you to organize a small company,\ngiving me control.  I shall erect a twenty-cord plant between Enterprise\nand Guildford, within wagon distance of the wood-supply of the estate.\nRecollect that this process uses only the fallen trees and stumps of the\nlong-leafed pine, which are reduced to a sawdust, and this is then put\ninto the retorts.  Steam is then injected, which tries out the\nturpentine, which is then run into the refining still.\n\n\"I can arouse no interest whatever among my relatives.  They simply\nthink I am crazy.  I even suggested to my uncle, Judge Fanshaw Lee, of\nCharleston, the simple proposition of joining me in the purchase of a\nstump-puller to clear his land for rice and cotton, but he wouldn't do\nit, and continues to plant in fields dotted with old stumps. But he will\nrent it from me if _I_ buy one!  So please order immediately the most\nimproved sort, and consign it to me at Enterprise, S.C.\n\n\"Even though I am a Southerner by blood, and anxious to improve the\ncountry in general, and my relatives in particular, I work under\ninconceivable difficulties.  I sent my lawyer to one of the biggest\nfactors in Savannah, by the name of James Oldfield, to suggest a combine\nto corner turpentine, offering to raise nine million dollars, if he and\nhis friends would raise one million.  Legare reported that 'Oldfield's\nhead hit the ceiling' at the mere suggestion.  But, upon being drawn\nout, Oldfield admitted that twenty years ago he had entertained a\nsimilar idea, although, of course, at that time not for the purpose of\nintroducing wood turpentine. But his ideas were on too narrow-gauge a\nplan to admit the suggestion now.  So we shall simply be obliged to do\nit without him.\n\n\"It seems to me that, with the South in the mental attitude it now\nholds, it will need some radical means, such as a turpentine corner, to\nforce Southern landowners to reinvest money in their own property.  Many\na man is land poor with thousands of dollars' worth of stumps and fallen\ntrees on his land which are suitable for wood turpentine. In order to\nsupply the demand, the orchard people are obliged each year to find two\nmillion acres of virgin forest for their operations.  After bleeding\nthese for three years, the lumber men then enter and cut the timber,\nthus leaving millions of fallen trees and stumps, all of which are\nsuitable for our process.  Now, it would take years to educate these\nlandowners in the process of extracting turpentine from this stumpage,\nwhile a corner in orchard turpentine would, in three months, turn the\nattention of half the chemists and inventors in the United States toward\nbettering present processes and discovering new ones.  Every newspaper\nin the land would give this New Southern Industry millions of dollars'\nworth of free advertising, and inside of ten years the whole South would\nblossom as a rose.\n\n\"I have hinted at this before, but have not explained it because the\ntime was not ripe.  Now, after six months of untiring investigation by\ntrustworthy agents, and after bitter personal experience, I find that no\nhelp whatsoever can be expected from the South.  Rather they will fight\nus at every step, like children compelled to take medicine.  Did you\never see a health officer try to vaccinate a <DW64> settlement on the\noutbreak of a smallpox epidemic?\n\n\"You understand me, do you not?  Tell me if I make my point sufficiently\nclear.  I propose to corner turpentine, not for the purpose of raising\nthe price, but to take the orchard stuff completely off the market until\nwe have forced the public to give wood turpentine a trial.  It has been\ndemonstrated in every department that the patented product will do the\nwork of the orchard, not only just as well, but in some cases, as that\nof paint, it actually holds the colour better.\n\n\"If you are still interested, let me know and I will explain my\ndeveloped plan.  Meanwhile I welcome suggestions from you, or any of\nyour interested parties.\n\n\"With devoted love to all in your dear house, I am,\n\nAlways affectionately yours,\n       \"CAROLINA LEE.\"\n\n\n\n\n                             CHAPTER XVII.\n\n                       IN THE BARNWELLS' CARRYALL\n\n\nAunt Angie La Grange descended from the Barnwells' carryall in front of\nthe station platform at Enterprise, and tapped on the window of the\ntelegraph-agent's box.\n\n\"How late is the train from Savannah, Barney, son?\"\n\nMr. Mazyck sauntered out.\n\n\"Only about three hours to-day, Aunt Angie. Expecting the folks?\"\n\n\"Only Peachie.  Mrs. Winchester and Carolina went on down to\nJacksonville on business.  Did you ever see such a girl?\"\n\n\"I never did.  She scares me 'most to death.  I'd like to marry her,\nAunt Angie, but what could I--what could any man do with such a wife?\"\n\n\"She'd make any man rich.  Moultrie says she goes so far ahead of him in\nher ideas of business, he can't even keep her in sight.\"\n\n\"Oh, any man has got to make up his mind to take her dust!\" laughed\nBarnwell.\n\n\"Are you in earnest about marrying her, Barney?\"\n\n\"Of cou'se I am!  Aren't all the boys?  Isn't Moultrie?\"\n\nA shade darkened Aunt Angie's face.\n\n\"You know, son, that Moultrie will never marry unless--\"\n\n\"Exactly!  Unless!  Well, there's a heap of unlesses which may he'p him\nto change his mind. And maybe Miss Carolina is one of them.\"\n\n\"I'd be proud to have him win her, but, as you say, all the boys are in\nlove with her, here and in Charleston, and now she has been to Savannah,\nI suppose they will follow suit, and--\"\n\n\"Poor Jacksonville!\" sighed Barnwell.\n\nMrs. La Grange laughed.\n\n\"We haven't had such a belle in South Carolina in many years,\" she said.\n\"Before the war--\" and she sighed.\n\nBarney laughed unfeelingly, and Mrs. La Grange continued:\n\n\"How about Araby, son?  Are you going to sell her to Carolina?\"\n\n\"Indeed I am not, Aunt Angie.  I'd give her to Miss Carolina before I'd\nsell her to anybody else; but, to tell you the truth, I'd about die if I\nhad to part with that mare!  She's human.  Sound as a dollar and not a\ntrick of any kind.  That <DW65> horse-trainer is a magician with\nanimals.  I'm blest if I don't believe he'll teach Araby to talk before\nhe quits.  And she whinnies if she even passes him in a crowd.\"\n\n\"Carolina wants her worse than anything in the world.\"\n\n\"Well, she can just go awn wantin',\" said the usually gallant Mr.\nMazyck, ungallantly.  \"If I'd give Araby to her, I'd lose both my mare\nand my sweetheart.\"\n\n\"Somehow or other I can't help thinking that Carolina will get that\nhorse in spite of you. Barney, do go and see what time it is!  This is\nthe third time I've been down here to wait for this mean train!\"\n\n\"Yonder she comes now.  Only three hours and fifteen minutes late.\nThat's not so bad, Aunt Angie.  When she tries, she can tardy herself up\na heap mo' than that!\"\n\nMrs. La Grange anxiously scanned the shabby coaches for a sight of her\ndaughter's blooming face. Peachie jumped from the car steps and ran to\nher mother's arms.  They kissed each other like two lovers who had been\nparted for years.\n\n\"Have you had a pleasant week, darling baby?\" asked her mother.\n\nPeachie's pink cheeks paled and her face clouded over.\n\n\"No, I haven't,\" she whispered, hurriedly, \"but I don't want anybody but\nyou to know.  Don't let Barney ask me.  Let's hurry.\"\n\nMrs. La Grange led the way to the borrowed carriage with a sinking\nheart.  Aside from two visits to her aunt in Charleston, this was the\nonly time Peachie had ever been away from home.  And now to have this\ninvitation to visit Savannah, given the year before and anticipated all\nthis time, turn into the failure which Peachie's face indicated, was\nalmost as great a disappointment to Mrs. La Grange as to the girl\nherself.\n\nIn the carriage, where Old Moses could not hear them, the mother\nanxiously awaited the story.\n\n\"Begin at the beginning and don't skip a word. We've two good hours\nbefore us with nobody to interrupt.\"\n\n\"Well, you know how happy Carolina was at the prospect of taking me to a\nfine hotel like the De Soto, and how lovely my clothes were, and how\npleased Cousin Lois was at the prospect of seeing her old friends there?\nWell, people called, of course,--none of the girls, though,--and Mrs.\nGeneral Giddings, who is the leader of Savannah society, at once asked\nCousin Lois to be a chaperon at the Valentine Ball.  John Hobson invited\nme, and Jim Little asked Carolina, and, do you know, it was the first\ntime in all her life that Carolina had ever been to a ball with a man!\nShe says she always went with a chaperon and met her partners at the\ndance.  And she wanted to do that in Savannah, but Mrs. Giddings assured\nher that it was all right, and so she did.\n\n\"Oh, mother, I wish you could have seen us that night!  You know how I\nlooked, but Cousin Lois wore a black satin brocade, studded with real\nturquoises and blue ostrich feathers woven into the goods.  And, with\nall her size, she looked perfectly lovely.  Carolina wore a white Paris\nmuslin over white silk, with every flounce trimmed with real lace.  Her\nhair looked as if she only had one pin in it, it was so loose and fluffy\nand--well, artistic is the only word to describe her.  She looked like a\nfairy princess.  It began in the dressing-room.\"\n\n\"What began?\n\n\"Well--Savannah began!\" cried Peachie.  \"I never heard of such things\nhappening to our girls when they go to Atlanta and Columbus and Augusta\nand Macon, while as for Charleston!--well, I needn't defend Charleston\nmanners to _you_, mother!\n\n\"Not a soul spoke to us, although everybody knew we were strangers and\neverybody knew who we were, for of course it was in the papers,--such\ndistinguished arrivals as Mrs. Rhett Winchester and Carolina Lee!  But\nnot a girl came near.  They hollered and joked among themselves, and\nsomebody would whisper to two or three, then the whole roomful would\nscream like wild Indians, and once one of the boys came to the door and\ncalled to them to hurry up, and one girl screamed back, 'Shut yo' big\nmouth!' and the rest fairly yelled with approval.\n\n\"Then one girl was just going out with her bodice all gaping open, and\nCarolina stepped up to her as sweetly as if she had been received with\nperfect politeness and asked if she mightn't fasten it.  The hooks were\nhalf off, so Carolina took a paper of pins and fairly pinned that girl\ninto her clothes,--her waist and skirt didn't meet.  She accepted all\nthis help, thanked her, and went out, leaving us all alone.  Then our\nboys came and took us down to the ballroom, and, if you will believe it,\nmother, not a girl came near us or asked to be introduced or introduced\na single boy!  Not even the girl that Carolina had helped.  I looked at\nCarolina to see if she noticed it, but her face was as calm as it always\nis.  Her colour, however, was a little less than usual at first.\n\n\"We noticed that things sort of dragged at first, and soon we found out\nwhat it was.  An English yacht was in the river, and its owner, Sir\nHubert Wemyss, a young man only about thirty, was expected, and all the\ngirls were trying to save dances for him, and all the boys were trying\nto get the choice ones.\n\n\"The first dance I didn't watch Carolina, because I had heard that Jim\nLittle was a good dancer, but, after it was over, I saw him take her to\nthe door and she went up to the dressing-room.  I made John stop near\nhim, and I asked him what was the matter.  'Oh, I stuck my foot through\nthe lace of her dress, and she's gone to be sewed up.  Say, Miss\nPeachie, that girl can't dance!  I never saw a Yankee that could!'\n\n\"Well, mother, I could scarcely believe my ears! The conceit of that raw\nSouthern boy, who never had been outside of his own little town in the\nwhole of his life, except to go duck-shooting in the swamps, to presume\nto criticize Carolina's dancing!\"\n\n\"What did you say to him, sweetheart?\"\n\nAunt Angie's cheeks were as red as any girl's. She sat bolt upright in\nthe borrowed carriage, in her cheap print dress and cotton gloves,\nlooking like an empress.  The proudest blood in South Carolina flowed in\nher veins and she had the spirit of her State.\n\n\"I said, 'Are you sure, Mr. Little, that the fault was all hers?'  And\nhe laughed and said, 'Well, the Savannah girls never find fault with my\ndancing, Miss Peachie!'  'Oh,' I said, 'if such criterions have stamped\ntheir approval on you, Mr. Little, of course there is no more to be\nsaid!'  He didn't see the sarcasm at all,--he seems a trifle dense.  So\nwe waited for Carolina, and when she came back, I saw that her dress was\nruined, but she had managed to hide it pretty well, and her manner was\njust as sweet to that man as if he had been fanning her, and we all four\nwent back to Cousin Lois.\n\n\"The next dance we changed partners, Jim Little taking me and John\nHobson taking Carolina.  Now John is said to be the best dancer in\nSavannah, so I kept an eye on them, but they didn't do very well.\nCarolina's colour began to rise and her eyes began to grow that purplish\nblack--you remember?  Oh, she looked so beautiful!  But she wasn't\nenjoying herself, and she stopped near me to rest.  Then I heard John\nsay, 'You dance more like a Southern girl than any Yankee I ever knew!'\nThink, mother!  That was twice she had been called a Yankee before we\nhad been there an hour.  A Lee of South Carolina!  Her cheeks just grew\na little warmer and she lifted her chin a little higher, but didn't\ncorrect him--just said, 'I suppose you intend that for a compliment, Mr.\nHobson?'  'I should say I did!' he said.  'I never saw a Yankee girl who\ncould dance in all my born days!'  'How do you account for that?' asked\nCarolina, in just as sweet a tone, mother, as she always uses.  Me? I\nwas just boiling!  I was ready to cry!\"\n\nHer mother pressed her hand.  Aunt Angle's own lips were trembling with\nindignation.\n\n\"'Oh,' the fool said, 'I reckon they don't get as many chances to dance\nas our girls do!'  Well, that saved me.  I began to laugh and I laughed\nuntil I nearly went into hysterics.  I had to excuse myself and ask Jim\nto get me some water!\"\n\n\"Did Carolina laugh, too?\" asked Mrs. La Grange.\n\n\"Well, she smiled, and I knew from that, that she was only holding\nherself in.\n\n\"The next was a Lancers.  Carolina danced with Rube Bryan.  He is very\ntall and from the first he tried to get fresh with Carolina.  I was in\nthe same set dancing with John again.  And I want to say right here that\nI never saw such unladylike and ungentlemanly dancing in all my life.\nWhy, in Charleston the chaperons would have requested the whole dance to\nbe stopped. They wouldn't have permitted such hootings and yellings,\nsuch jumps and shouts.  Girls yelled at each other across the whole\nhall--just like <DW64>s.  'Go it, Virgie!'  'Shake a foot, Nell!'  In\nthe ladies' chain the boys jerked the girls so that one girl in our set\nwas thrown down and her wrist sprained.\"\n\n\"I was getting frightened and I could see that Carolina was on the verge\nof leaving the set. Then she seemed to brace herself, for Mrs.\nWinchester had left the line of chaperons and was making her way down to\nwhere we were dancing. And mother, there was rage in her whole bearing.\nShe just looked as if Carolina were being insulted by dancing with such\nrowdies.  But Carolina gave her a look and she did not interfere.  She\nstood there, however.\"\n\n\"Did anything happen, Peachie?\" asked Mrs. La Grange, unable to wait for\nthe sequel.\n\n\"Yes, mother, it did.  I believe those girls had dared him to, because\nhe waited until the very last, then he lifted Carolina off her feet\nclear up into the air, and landed her in front of Mrs. Winchester with a\ndeep bow.  Everybody laughed and screamed for a minute, then something\nin the attitude of both Mrs. Winchester and Carolina made them hush.\nCousin Lois's voice was low, but you could hear it all over the room.\n\n\"'Young man,' she said, 'your name is unknown to me, but let me say to\nyou that you are not a gentleman!'\n\n\"What happened then?\" cried Mrs. La Grange.\n\n\"Mrs. Giddings, of course.  She always says the cutting thing.  'You are\nperfectly right, Lois,' she said, 'the man is a nobody.  We expect such\nmanners from nobodies.  Not that the somebodies are any better, if this\ndance is a sample.  This is my first appearance.  Rest assured that it\nwill be my last.  We Giddings don't chaperon barn dances!'\n\n\"That, from Mrs. Giddings, seemed to sober them.  They all moved away\nleaving Rube Bryan bowing and scraping and trying to square himself.\nCousin Lois simply waved him aside as if he were a piccaninny.  She\nasked Carolina if she wanted to go home.  Carolina hesitated a minute,\nthen she lifted that chin of hers and said, 'No; a Lee cannot be driven\nfrom a ballroom by rudeness.  Just let me go and put on my truth!\"\n\n\"Bless the child!\" cried Mrs. La Grange, who was as excited as a\nspectator at his first horse-race. \"Bless her!  There is pride!  There\nis what the French call 'race'!  And to see the dear _putting on the\narmour of her religion even in a ballroom_!\"\n\n\"Mother, Carolina's religion helps her in everything. Why, she just\nstepped out of sight behind a row of palms.  She went to a window and\nreached up one arm and leaned her head against it.  With the other hand\nshe drew back the curtain and looked up at the stars.  I put my arm\naround her and she said, in a low, distinct voice. 'The eternal God is\nthy refuge and underneath are the everlasting arms.'  'And mother, it\nmade the tears come to my eyes.  To think of my beautiful Carolina, with\nnothing but love in her heart for the whole South, to come home to us\nand be treated so rudely that she had to appeal to God to help her to\nget through something which ought to have been only a pleasure to her!\"\n\n\"I know, my dear baby,\" said her mother, whose own eyes were\nsuspiciously bright, \"but I rather imagine that to a girl who has seen\nthe best society that Europe and America have to offer, a dance with a\nlot of Savannah boys and girls could not be considered in the light of\nmuch of a treat.\"\n\n\"I know it, mother.  Yet Cousin Carol's manners are so perfect that she\nnever lets you suspect that.  She enters into everything with such\nlove.\"\n\n\"That is her religion,\" said Mrs. La Grange.\n\n\"Oh, that reminds me.  She went on talking aloud as we stood there.  She\nsaid, 'I must remember that the vesture of truth is my raiment. I must\nstand sentinel at the door of my thought and not allow error to enter\nit.  And the way to keep error out, is to pour love in.  Love!  Love!\nLove!  That is the way to meet them. Father--mother--God!  Help me to\nlove mine enemies!' Oh, and mother dearest, by that time I was weeping,\nbut Carol's eyes were quite dry.  'Don't cry, little girl,' she said, 'I\ndon't any more, for I have got beyond the belief that religion is an\nemotion. It is too real--too lasting.  Emotions die out.'  And a little\nlight seemed to dawn for me--just as I have seen clouds break on a dark\nnight and a single star shine through.\"\n\n\"Then did you go back?\" asked her mother, after a pressure of the hand\nto show that she understood.  There was a singular bond between these\ntwo.\n\n\"Yes, she turned and pressed my hand just as you did then, with such\nunderstanding, and her face was fairly shining, but with such a\ndifferent radiance.  'Come, Peachie, darling! faithful little comrade.\nYou would not have been one of the disciples who slept and left their\nMaster to pray alone, would you?  Well, I have conquered my little\nmoment of error.  Now let's go back.'  'And show them how South Carolina\nfaces her foes,' I said.  'Wouldn't it be better to go back and show\nthem how South Carolina can forgive?' she asked.\"\n\n\"Bless her heart!\" murmured Mrs. La Grange. \"I know how a young girl\nfeels to be mistreated at a ball.\"\n\n\"Yes, but wait.  The grandest, glorious-est thing happened.  Just as we\ncame from behind the palms who should be bowing to the chaperons but the\nhandsomest man I ever saw in my life. Tall, dark, distinguished-looking,\nwith one white lock of hair and all the rest black as a coal.  He has a\nslight limp from a wound at Magersfontein, but it only distinguished him\nthe more and doesn't interfere with his dancing a bit.  Well, when he\nsaw Carolina, his face lighted up and he said, 'Oh, Miss Lee, how\nawfully jolly to see you again!  To tell the truth, I had half a mind\nnot to come, after all I had promised, and I wanted to get out of it the\nworst way until I heard that you were to be here.  Then I couldn't get\nhere fast enough.'  Well, mother, even if every girl there hadn't\nsuddenly found that side of the room strangely attractive, his voice has\na carrying tone, and--well, I wish you could have seen those girls.\nThey looked as though they had been slapped in the face.\"\n\n\"As they deserved!\" said Mrs. La Grange, grimly.\n\n\"Then the band struck up a two-step and he turned to Mrs. Winchester and\nasked her if she would save her first square dance for him, but she said\nshe wasn't dancing.  So then he asked Carolina.  She gave me a little\nlook which meant that I could have him next, and then!  Well, I've seen\ndancing all my life, but I never saw anybody dance as those two did.  It\nwas like the flight of swallows.  So graceful, so dignified, so\ndistinguished, and yet so spirited.  Carolina dances like a breeze.\"\n\n\"I can imagine just how she dances,\" cried Mrs. La Grange, excitedly.\n\"Go on, child!\"\n\n\"Well, the funniest sight of all was Cousin Lois.  She drew her chin in\nand waved her fan and puffed herself out for all the world like our\nturkey-hen.  I could have laughed.\"\n\n\"I know just how she felt--just how I should have felt in her place if\nyou had been treated as Carolina was.  Then did he dance with you?\"\n\n\"Yes, then he danced with me.  Then with Carolina again.  Then she said\nto him, 'Now, Sir Hubert, I want you to meet some of these pretty girls,\nbut as I don't know them myself, I shall ask Mr. Little to take you\naround and introduce you to the brightest of them, so that you will take\naway with you the best impression of our Southern girls.'\"\n\n\"Oh, Peachie!  I couldn't have done that!\"\n\n\"Nor I either, mother.  I just couldn't.  So Jim started to take him,\nbut he said, 'Just wait a moment.'  Then he came to me and took--\"\n\n\"I hope he took more than one!\" cried Mrs. La Grange, jealously.\n\n\"He took seven, mother.  And in the German he favoured me until--\"\n\n\"That was too many, Peachie.  You ought not--\"\n\n\"I know, dearest honey mother.  I ought not to do heaps of things I do\ndo, but after all, what do I care what those people think of me?  All\nthey can say is that I flirted with him--\"\n\n\"Or that he flirted with you,\" laughed her mother.\n\n\"Oh, yes, they will say that, never fear.  And yet--\"\n\n\"And yet what, my darling?  Here we are at home.\"\n\n\"And yet he took Cousin Lois and Carolina to Jacksonville on his yacht,\nand he asked me to go, but I said I had to get back to you, and he was\nwith us all the rest of the time we were there--\"\n\nHer mother turned and looked at her.\n\n\"And he is coming to see me on his way back.\"\n\nAs Mrs. La Grange stepped from the carriage with the air of a queen\ndescending from her chariot, she put her arm around her daughter's waist\nand said:\n\n\"I think I have to be proud of a dear, generous little girl whose\nloyalty caused an otherwise pleasant week to be spoiled.\"\n\nPeachie's cheeks flushed and her eyes sparkled.\n\n\"It wasn't quite spoiled, mother dear.  Oh, honey, he is the handsomest\nman and the best dancer!  Just wait till you see him!\"\n\n\n\n\n                             CHAPTER XVIII.\n\n                           A LETTER FROM KATE\n\n\n\"NEW YORK.\n\n\"DEAREST CAROLINA:--Great news!  Three pieces of it.  First, I have\nturned Christian Scientist!  Second, Rosemary Goddard is married to the\nHonourable Lionel Spencer!  Third, daddy is so tickled over all that you\nhave done, as you may have suspected from his letters lately, that he is\ngoing down.  He will take the car, and Noel and Mrs. Goddard, mother,\nand I are coming, too!  Don't bother about accommodations.  We will\nswitch the car to a siding and live in it.  We may all have to go to\nCharleston and Jacksonville, so that you and Peachie and a handy man or\ntwo had better get ready for a rip-roaring old time, for we are going to\nmake Rome howl.  Noel wants to go to Ormond for the automobile races.\nHe has entered his machine.  I named it for him,--'The White\nMoth,'--don't you think that's a dandy name?\n\n\"Now to go back to the really important thing. I've wanted to be a\nScientist ever since I found out that it wasn't a drag-net to catch all\nthe cranks in the world, as I at first supposed.  I found that out in\ntwo ways.  One, by knowing a lot of you who were not in the least\ncranks.  The other, by seeing what a lot of cranks there are left!  Yet\nall the time I was hating myself and struggling against the compelling\ninfluence.  Did you ever drag a cat across the carpet by the tail?\nWell, that is just about the easy, gliding gait I used to reach\nChristian Science!\n\n\"Still, you'll never guess who influenced me most.  Not you nor that\nheavenly Mrs. Goddard nor the wonderful cures I've seen.  Nuh!  Guess\nagain.  Old Noel!  Yes, sir.  Old skeptical Noel! Brought up for a\nCatholic, too.  Wouldn't that freeze you?  Well, think si to myself,\nthink si, 'if old Noel can see good in it, and he's the best all-round\nsport, man of the world, and gentleman I know, it's time little Katie\ngot aboard.'  So I just climbed on the raft without saying a word to\nanybody, expecting everybody to raise Cain, but, to my astonishment,\ndaddy was as pleased as Punch, and he and mother go to church with me\nevery Sunday.  What do you say to that?\n\n\"At the ball the Goddards gave for Rosemary just before she sailed, I\nwas doing a two-step with Noel, and I saw a dandy girl, whose gown\nsimply reeked of Paris, it was so delicious.  She was dancing with a\ncorking looking man, and, as we stopped near them for me to get a better\nlook at her clothes, I heard her say, 'Are you going to communion at the\nMother Church?' and he said, 'I never miss it.  It is the treat of my\nwhole year!'  I looked at Noel and he looked at me.\n\n\"'Noel,' I said, 'Did I hear aright?  They weren't betting on a\nhorse-race in cipher, were they?'  'No,' sez he, giggling, 'they were\nnot. They are Christian Scientists, and they are now talking about an\nincorporeal God.'  'In a ballroom,' murmurs I to myself.  'Noel,' I\nsaid, in a weak voice, 'Take me out and lay me softly under a pump and\nbring me to.  I am too young to go dotty without any warning.'  But,\ninstead of that, we joined them and Noel introduced us to each other,\nand we finished the two-step talking about how hard it was to change\nfrom our old idea of a God who was so much like a man that we had to\nflag Him and shout out our prayers to be sure to get His attention.  I\nused to feel as if I were on the floor of a convention, trying to catch\nthe Speaker's eye.\n\n\"But I want to ask you two things that I can't quite get up my nerve to\nask Mrs. Goddard.  What did you do about praying while changing your\nidea of a personal, corporeal God to one of spirit?  Why, Carolina, I've\nlost the combination!  I feel as though I were praying through a\nmegaphone out of an open window.  My prayers don't seem to strike\nagainst anything.  Will I get over this feeling in time?  It is only\nfair to state, however, that even this queer hit-or-miss method brings\nanswers which my most frantic screams for help and my most humble and\ndependent clinging to the robe of my personal God never did.  So you can\njust bet that I'm going to stick to the new method, whether I ever\nunderstand it or not, because it does deliver the goods.  Am I right or\nwrong?  I want to know.\n\n\"Now, I did tackle Mrs. Goddard on this point. I feel a perfect wretch\nto mention it, but the fact is, I simply cannot endure the name of Mrs.\nEddy! Every time they mention 'Science and Health' in church, they say,\n'By Mary Baker G. Eddy.'  Every time they give out a hymn that she\nwrote, they say, 'By Mary Baker G. Eddy.'  And every time they do it, my\nblood boils and my face burns and I grab my hymn-book until--well, I\nsplit a pair of gloves nearly every Sunday!\n\n\"The conceit of that woman!  Suppose she has given the world a new\nreligion,--why not let us show our gratitude spontaneously.  Why need\nshe say such conceited, sacrilegious things in her book? She throws hot\nair at herself indirectly in every chapter.  It reminds me of a page in\nRoosevelt's 'Alone in Cubia.'  I counted sixty-three I's on one page in\nthat book, until I felt like the little boy who said to his father,\nafter an evening of war experiences, 'Papa, couldn't you get any one to\nhelp you put down the rebellion?'\n\n\"I don't believe, unless my feeling changes, that I shall ever join the\nchurch while its by-laws remain as they are.  I will work for the cause,\nand be diligent and faithful and studious, but I disapprove of a church\nbeing such a close corporation and for one finite, human being to\npossess such power as Mrs. Eddy holds, and holds with such pertinacity\nand deliberate love of power.\n\n\"When I said some of this to Mrs. Goddard, she said that she never\nchemicalized over Mrs. Eddy the way great numbers did, but she said you\nhad a claim at one time, and I want to know if you are over it.  I feel\nlike a brute to have to admit it even to you, for of course I am\ngrateful and appreciative and all that.  But if you call what I feel\n'chemicalizing,' I can only say that I can hear myself sizzling like a\nbottle of Apollinaris whenever I come across the name of Eddy, and\nrealize how she holds the power of a female Pope.\n\n\"I told Noel about it, but he doesn't feel it at all.  Never did.  But\nhe understands how intensely I suffer from it, and he said if I didn't\nmind my eye, I'd blow off a tire right in church.  And once, when he\ntook me and saw me getting red in the face, he said, 'Now sit tight, old\ngirl!' and I nearly laughed aloud.\n\n\"Now let me tell you my first demonstration. I am so happy over it I am\ngoing to do something to celebrate it, and that's another thing I want\nto consult you about.\n\n\"Yesterday Noel and I were out in the White Moth, and every time I know\nI am going out in the thing I read in 'Science and Health' about\naccidents, and declare the truth, so that my mind will be filled with a\npreventive.  It comforts me a great deal and is the only thing that\nenables me to enjoy an automobile ride in New York, for, with the danger\nof blowing up and other people's bad driving and frightened horses and\nthe absolute recklessness of pedestrians, you take, if not your life, at\nleast your enjoyment of life, in your hand whenever you get into a\nmachine.\n\n\"Noel is the most careful chauffeur I ever saw, and we were just\ntrundling along out in the Bronx, when, without a word of warning, a\nlittle bit of a boy jumped from a crowd of children and stumbled right\nin front of us.  I saw him fall, and to my dying day I never shall\nforget the sight of his little white, upturned face as he disappeared\nunder the machine.  We ran right squarely over him!\n\n\"I stood up and screamed out: 'You said accidents could not happen!  You\npromised!  You promised! We have not hurt that baby!  He is alive! He is\nnot hurt!  He is not even run over!'  And by that time we had both\njumped down and run back, and a big crowd was gathering.  Talk about\ntreating audibly!  I was screeching at the top of my voice.  Yet still\nthere lay the child apparently dead. I picked him up in my arms and sat\ndown in the mud with him, still, as Noel declares, talking aloud. Oh,\nCarolina, I never shall forget the sight of his little hands!  So dirty,\nbut so _little_!  And his little limp body,--I feel as if I had it in my\nlap still. The crowd kept getting bigger, and some policemen came, and\nsuddenly, with a scream I never can forget even in my dreams, the\nchild's mother rushed up.  She raised her fist to strike me in the face,\nand I thought I was done for, when suddenly the child's eyes opened, and\nsomething made me say: 'Here is your baby, little woman.  He is not hurt\nat all!'  She fairly snatched him from me and began to feel him all\nover, but she could find no broken bones. She was crying and laughing\nand kissing him, and I still kept telling her that he was unhurt.  Just\nthen the police got through with Noel, and he insisted on putting mother\nand child and a policeman in the tonneau and taking them to the nearest\nhospital to have the child examined.  We did so, and, if you will\nbelieve it, there wasn't a scratch on him. He either fainted from fright\nor we stunned him, the doctor said.\n\n\"Two of the surgeons came out and examined the machine, and they found\nthat there is only a foot of space between the lowest part of the car\nand the ground.\n\n\"'It is the most miraculous escape I ever saw,' said one of them, 'to\nrun over a five-year-old boy and not even scratch him.  To make the\nstory quite complete you ought to claim to be Christian Scientists.\nThat is the sort of game they always play on a credulous public.'\n\n\"'We are both Christian Scientists,' said Noel, in his most polite\nmanner, 'and I am deeply impressed with your involuntary tribute to its\nefficacy in case of accident.'\n\n\"Between you and me, I don't believe that doctor got his mouth together\nagain without help.\n\n\"Well, we had the greatest time when we got back.  First, we took every\nchild on the scene--and I believe there must have been a hundred--to an\nice-cream saloon and treated them.  And while they were waiting their\nturns, Noel filled the White Moth with them and gave them a ride.  I\nnever had so much fun in my life.  I went home with the mother, with a\nquart of ice-cream in each hand, and got her to tell me the story of her\nlife. Poor soul!  She has nine children, but she loves each one as if it\nwere her all.  Noel and I are both going to do something for that child.\nHis name is Dewey Dolan.\n\n\"When it was all over, and we were sneaking along back streets to get\nhome without being seen, for we were both sights, and the Moth will have\nto be done over, I began to think of the way I had acted, and I have\nmade Noel promise never to take me out again unless I have my Amityville\ntag on, so that, if I go crazy out loud again, they will know where I\nhave escaped from.\n\n\"But Noel, dear old thing, confessed that he was declaring the truth no\nless, only in a quieter way, and we both firmly believe that our little\nknowledge of Science and our understanding, incomplete though it is, are\nwhat turned that calamity into a blessing, for a blessing I am\ndetermined to crown it.\n\n\"What do you think of my idea?  You know how I have always been carried\naway over children,--how their sufferings and deaths have almost turned\nme into an infidel,--how the carelessness of parents and nurses has\nalmost driven me insane,--well, if they can be protected by Christian\nScience thought and healed by mind, why not hasten the day by\nestablishing a Christian Science kindergarten, and, if it succeeds, by a\nseries of them? There must be plenty of kindergartners among Scientists\nwho would welcome a combination of their work, and in the crowded\ntenement districts it would be a boon.  But, oh, how carefully we must\ngo, for the poor will only allow themselves to be helped in their own\nblind way.  Tell me if you think there is any hope for the philanthropic\nend of it.  I am going to open one for the children of ready-made\nScientists in my own house,--you know I studied kindergartning, and I\nhave ten already promised.  I shall have no trouble about assistants for\nmy Fifth Avenue school.  But the other place is the one my heart is in.\nTell me what you think of that.\n\n\"Rosemary is coming back here to live.  Her husband is a Christian\nScientist, and has gone into business in New York, so I know she will\nhelp me, but, oh, Carolina, you will never know how I miss you!  New\nYork is not the same place since you left it.  You have such a way of\ndominating every spot you are in by your own personality.  Does this hot\nair sound natural from Kate Howard?\n\n\"I am crazy--fairly daffy--over your success in the turpentine, and\ndaddy goes around swelling out his chest and strutting like a turkey\ngobbler. Why, Carolina, do you realize that you will not only make\nyourself rich and anybody you choose to let into the game, but that you\nwill be opening up by force, so to speak, with your Educational\nTurpentine Corner, an industry which will revolutionize the entire\nturpentine pine country?  It is a big project, my dear, to have emanated\nfrom the brain of a woman.  But, oh, won't the papers fairly eat you\nraw!\n\n\"I will attend to all the commissions you sent and bring the stuff down\nin the car.  A good many of us want newer and finer editions of 'Science\nand Health,' and, if you utterly refuse to make presents of them for the\ngood of the cause, we will sell our old books at whatever you think your\nfriends can afford to pay.  I agree with you that it is better to make\nthem pay something for them.\n\n\"Rawlins, our butler, and two of the footmen go regularly to the\nChristian Science church, and Rawlins has been healed of intemperance\nthrough Mrs. Goddard's butler.  Perkins says he owes his conversion to\nthe day Gladys Yancey walked across the floor for Noel's doll.  So you\nsee we all had a hand in the work you started, and a little leaven is\nleavening the whole lump.\n\n\"Oh, Carolina, you know how discontented and fractious I used to be?\nWell, it is all gone,--all the fear, the dread of the unknown, the\nunhappiness, and the temper, and I am happy for the first time in my\nlife!\n\n\"But now good-bye, my dearest friend.  I am bringing some dandy glad\nrags with which to astonish the natives.  Tell Peachie that I go to\nevery sale I hear of, and that I am bringing her and Flower some of the\ndearest little inexpensive remnants they ever saw.  Bless those girls!\nIt sorta makes my old heart ache to think they haven't the clothes they\nneed to set off their good looks.\n\n\"Again good-bye.  Best love to Cousin Lois and yourself from all of us.\nAnd I am as ever your slave.  KATE.\"\n\n\n\n\n                              CHAPTER XIX.\n\n                                THE FEAR\n\n\nCarolina had not been a week among her kinsmen before they began to warn\nher of the terror of the South.  They definitely forbade her ever riding\nalone, except in broad daylight along the public highway, and even then\nsome white man of her acquaintance generally made it his business to be\ncalled in whatever direction she happened to be going.\n\nAll this Carolina saw and felt and appreciated, but with the natural\nfearlessness of her character and the total want of comprehension which\nwomen seem to feel who have never come into contact with this universal\ndread of all Southern States, Carolina often forgot her warnings, and\ntempted opportunity by striking off the highway into the pine woods to\ninspect her turpentine camps.\n\nOnce Moultrie La Grange found her unaccompanied by any white man,\ntalking to a burly <DW64> in a camp, and when he had taken her away and\nthey had gained the road where she could see distinctly, she found him\nwhite and shaking. Knowing his physical courage, this exhibition of fear\nstartled her, and for a few weeks she was more cautious.\n\nThen one afternoon she mounted Scintilla and rode into Enterprise for\nthe mail.  She received the letter from Kate which has just been quoted\nand read it as she rode along.  It contained so much food for thought\nthat Carolina forgot everything else, until, looking up, she found that\nshe was just opposite the new terrapin crawl she was having prepared\nunder Moultrie's direction.  Without thinking, she struck into the woods\nand threaded her way among the giant pine-trees toward the coast.\n\nIt was virgin forest and on her own land, a tract she intended leasing\nto some orchard turpentine factors in Jacksonville.  It was twilight in\nthe forest, but Carolina rode forward fearlessly, glancing sharply at\nthe trees for signs of their having been boxed by thieving <DW64>s.\n\nSuddenly she saw a boxed tree, and, springing down, she drew Scintilla's\nbridle over her arm and stooped to examine the suspected tree.  As she\nwas bending down, Scintilla jerked her head, and the bridle slipped from\nCarolina's arm.  She sprang to her feet, but, with a nicker of delight,\nthe handsome horse kicked up her heels and pranced away from her,\nlooking for all the world like a child ready for a romp.\n\nSo free from fear she was that Carolina laughed aloud, but the laugh\nfroze on her lips, for, without turning her head, she could see,\ncrouching down and creeping toward her, the huge form of a <DW64> man,\nwhose half-open mouth and half-closed eyes, as he stole noiselessly\ncloser and closer, instantly told her of her dire peril.\n\nThe girl's whole body became rigid with terror,--a terror so intense and\nso unspeakable that she realized how it was that women can go mad from\nthe effect of it.  In a moment, every warning, every hint, every word\nthat she had heard on the subject flashed through her brain with\nlightning quickness. An intense silence reigned in the forest, broken\nonly by Scintilla's cropping a stray tuft of spring grass and the\nfootsteps of the black creeping nearer to his white prey.\n\nCarolina never thought of screaming.  No white man was within a mile of\nher.  Oh, for Moultrie,--Moultrie, who had saved her once before!  A\nsick feeling came over her--things began to swim before her eyes--she\nswayed--and at the sight of her weakness the <DW64> stood upright.\n\nHe was no longer a crawling horror.  He was a man, and her God was at\nhand!\n\nThe girl smote her hands together.  \"His truth shall be thy shield!\"\n\"God is my all!\"  \"He is my rock and my fortress!\"  \"Thou shalt call\nupon me and I will answer!\"  \"Fear not, for I am with thee!\"  Detached\nsentences, phrases, half-sentences fell from her lips in frozen\nwhispers.  But the man stood still.  He was no longer crawling toward\nher. And they stood looking at each other.  He had queer eyes,--one blue\nand one black--where had she heard of such eyes--where had she seen this\nvery man?\n\n\"'Polyte!\" she cried.\n\nInstantly the white woman got the ascendency over the black blood of the\nman.\n\n\"'Polyte, do you know who you are?  You are the son of my father's\nnurse!  Your mother was my father's black mammy!\"\n\nThe assurance, even the confidence, left the man's manner.  His\nshoulders drooped perceptibly.  He took a backward step.  Surely she did\nnot know what he was or she would not speak to him except to scream for\nhelp.\n\n\"Do you know who I am?\"\n\n\"Yas, missis.\"\n\n\"You don't know how you frightened me, until I saw who you were.  Then I\nknew that you would catch Scintilla for me.  Mr. Moultrie has told me\nwhat a way you have with animals.\"\n\nIn an instant the man was her servant, the son of her grandfather's\nslave.  His fear of detection and punishment left him, and he was quick\nenough to know that her supposed ignorance of his intentions had saved\nhim from a horrible death.  He was a bad <DW64> partly because he was so\nintelligent.\n\n\"I'll git her for you.  Jes' watch me!\"\n\nHe turned eagerly toward the horse and snapped his fingers.  Scintilla\nraised her head and began to step gingerly toward the man.  'Polyte's\npower over animals may have been hypnotism, but to Carolina it was like\nmagic to see Scintilla's bridle in 'Polyte's hand.  The man proudly led\nthe mare to her.\n\n\"Help me to mount,\" said Carolina, her shaking knees threatening every\nminute to give way beneath her.  \"No, hold your hand, and when I put my\nfoot in it, you lift me.  There!\"\n\nOnce on her horse's back, Carolina felt her heart begin to beat with\nless noise.  It seemed as if he could see how it pounded against her\nside.\n\n\"'Polyte,\" she said, \"you are what people call a bad man.  You have been\nbleeding my trees, and I don't know what all.  Why don't you behave?\"\n\nThe man kicked at a tuft of moss.\n\n\"Nobody won't hire me, Miss Calline.  Ise done been in de chain-gang too\noften.  Nobody won' trus' me!\"\n\n\"Well, if I will trust you, for the sake of your dead mother, will you\nbe good and faithful to me?\"\n\nThe man's face lighted up.  He took a step toward her.\n\n\"Will I?  Miss Calline, on'y jes' try me!  I kin do anyt'in'!\"\n\n\"I believe you.  Well, I'm going to try you.  I want you to be my--well,\nmy body-servant.  To go everywhere I go and take care of\nme--so--I--won't--be--frightened--again.  Will you?\"\n\nThe man's eyes wavered in momentary terror. But he kept his head.\n\n\"On'y jes' try me!\"\n\n\"I'm going to.  But you must have a horse to ride.  Look out for a good\none, and one for me, too.  You must get me, 'Polyte, the best\nsaddle-horse in South Carolina!\"\n\n\"Yas'm.  I'll do my bes'.  I kin git you a hawse.\"\n\n\"I'll pay you good wages, 'Polyte.  But you mustn't drink.  If a lady\nhires you, you can never get drunk, you know.\"\n\n\"I'll tek de pledge.\"\n\n\"Take any pledge that you can keep,\" said Carolina.  She gathered up the\nreins and turned her horse.  The man took a step nearer.\n\n\"Well, 'Polyte?\"\n\n\"Miss Calline--\"\n\n\"Well?\"\n\n\"Nobody ain't ever trusted me befo'!\"\n\n\"Well?\"\n\n\"Not even my ole mammy.  She voodooed me. She said I brought her bad\nluck, an' everybody tuk up de bad word agin me--\"\n\n\"Well?\"\n\n\"Even when I was a child, dey laid ever'thin' awn to me.\"\n\n\"I know.\"\n\n\"Well, you say ''Polyte, I trus' you.  You tek care ob me.'\"\n\n\"Yes, that is what I say.\"\n\n\"Well, Miss Calline, _you gwine be teken cah ob_!\"\n\n\"I am sure of it.  Good-bye, 'Polyte.\"\n\nAs she rode away, Carolina's shoulders drooped until she seemed fairly\nto shrink in her saddle.\n\n\"If he had touched me--oh, my God!--if he had touched me, I would have\nkilled myself!\"\n\nShe bowed her face in her hands, and the bitter tears streamed through\nher fingers.\n\nShe strove to think--to quiet herself--no one must know.  Suddenly she\nheard the hoof-beats of a horse behind her.  She dashed away her tears\nand straightened herself in her saddle.  If any white man suspected the\ncause of her agitation, a human life--the life of some black man--would\npay the forfeit.  'Polyte's life was in her keeping. She began to think\nof him as her property,--a human soul given into her power until it\ncould be saved through her ministrations.  God help him to have got\naway!  God protect him!  Black or white, he was God's child!  The\ntear-stained face of a white woman,--a woman riding alone?\n\nScintilla had never felt a spur before in her life. Carolina knew it by\nher snort of fright and surprise.  But she needed her best speed to draw\naway from the avenging white man on her track.\n\nIn her stall that night, Scintilla knew that there was a sharp-toothed\nanimal which had bitten her twice in one short ride.  She had tried to\nrun away from it, but it was fastened to a woman's heel.\n\n\n\n\n                              CHAPTER XX.\n\n                                MOULTRIE\n\n\nIt was the last of March.  Spring, which comes so early in the South,\nwas already in the fulfilment of her promise, and no lovelier spot could\nbe found than that portion of South Carolina which contains the estates\nof Guildford, Sunnymede, and Whitehall.\n\nCarolina, although working hard all of every day and often far into some\nnights, was happier than she had ever been in her life.  She was free\nfrom the persecutions of Colonel Yancey at last.  Little Gladys was now\nperfectly healed and as active as other children.  Moultrie was proving\na most eager and progressive student of Christian Science, and, while\nmost of his narrowness and astonishing ignorance was still painfully in\nevidence at times when discussions of import took place, yet Carolina\nheld faithfully to the thought that perfect harmony must result in time,\nand that such a fine mind as he naturally possessed must yield to the\nenlightenment which most men inherit.  Instead of this, however,\nMoultrie La Grange inherited prejudices which had dwarfed and hampered\nhis mental and spiritual advancement, and which mere friends overlooked.\nBut to Carolina, who loved him, they were heart-breaking.  It was as\nimpossible to discuss history with most of her relatives as to expect\nthem to speak Chinese.  In the country schools they used a history which\ndescribed the Civil War as a series of rebel victories, and the outcome\nof the war was not accounted for in any way.  Carolina, in reading the\nbook at Moultrie's request, wondered if the pupils, after a study of its\nfacts, did not question the sanity of Gen. Robert E. Lee for\nsurrendering a victorious and a gloriously successful army to a\nconquered and outnumbered foe, simply because General Grant asked him\nto.  When she handed the history back to Moultrie, Carolina said, sadly:\n\n\"I wonder what you will say when I tell you that my dear father, who was\nas loyal a Southerner as ever lived, and who entered the Confederate\narmy when he was only sixteen years old, was engaged at the time of his\ndeath in an elaborate life of Abraham Lincoln, whom he regarded as the\nbest friend the South ever had, and the noblest patriot America ever\nproduced!\"\n\nThe young man's face flushed with feeling, but he was too wise to\nexpress his bitter disagreement with Carolina's views.\n\nBut she knew how he felt and that, unless he deliberately determined to\nopen his mind to the truth in every way, that she never could bring\nherself to marry him, and thus court discord in her daily life.\n\nHe did the best he could, but among his own people he passed muster as\nan unusually fine fellow, well-educated and progressive.  It was only\nwhen brought into contact with a broad-minded, cultured young woman like\nCarolina that Moultrie's intellect showed its limitations.  However, the\nfact that he was proud of his prejudices was the only alarming thing\nabout the whole situation.  Carolina saw his possibilities.  She\nrecognized his courage; she trusted in his capacity to rouse himself\nfrom his ignorance; she knew that he would some day awaken to the\nimpression he made upon cultivated minds.  And the more she yielded to\nhis charm, to his chivalrous care of her, to the attraction his almost\nideal beauty had for her, the more she was determined to save him in\nspite of himself.  She knew that she could expect no help from his\nfamily, who idealized him just as he was, and who would have regarded an\nintimation that even a Benjamin Franklin would have found him crude, as\nsacrilege. Nor could relatives or friends avail, for did not all in his\nlittle community think as he did, and were not prejudices respected?\nNo, she realized that she must save him unaided and alone.  Therefore,\nwhen, in a burst of passion which nearly swept her off her feet and left\nher shaken and trembling, he asked her to marry him, she took her\ncourage in both hands and refused.\n\nHe stared at her in a dismay so honest and unfeigned that she almost\nsmiled.  Then his face flushed, and he said, in a low, hurt tone:\n\n\"I understand.  You have urged me to believe that Flower's ancestry was\nnot the disgraceful thing I suspect, when you could not bring yourself\nto believe it.  That can--that must be your only reason, for you love\nme, Carolina.  You have shown me in a hundred ways that you liked my\ncare of you; you have permitted my attentions, you have not discouraged\nmy honest, ardent love, which every one has been a witness to.  You do\ncare for me!  You cannot deny it.\"\n\n\"Moultrie,\" said the girl, slowly, \"I do not wish to deny it.  I never\nsaid I did not love you, for I love you more dearly than you know or\nthan you ever will know.  I said I would not marry you, but not, oh, not\non Flower's account.  I believe implicitly in all I have said of her.\nIf that were all, I would marry you to-morrow.  But that is not the\nreason.\"\n\n\"Then what is?  Oh, Carolina, love, _love_!\"\n\n\"You don't know me at all, Moultrie, or you would know what I am going\nto say.\"\n\n\"I reckon I don't, dear, for I haven't an idea of the reason.\"\n\n\"Well, it is because we never could be happy together, holding such\ndifferent ideals and such different codes of honour.  Colonel Yancey\ntold my father in London that he would find the South heart-breaking,\nand it is.\"\n\nThe young man stared into her lovely face in a very genuine\nastonishment.\n\n\"Our codes of honour different, Carolina?\" he said.  \"Oh, I hope not.  I\nshould be sorry to think that your code of honour differed from mine.\"\n\n\"And, dear friend--\"\n\n\"Don't call me friend!  I am not your friend! I am your lover!\"\n\n\"No, let me call you friend, for that is all that I can call you at\npresent.  I should be sorry to hold a code of honour no higher than\nyours.\"\n\nThe slow, dark flush of pride and race rose in the man's fine face.\nCarolina was daring to say such words to a La Grange.  But Carolina\nherself was a Lee.\n\n\"I should be sorry,\" said Carolina, deliberately, not waiting for his\nreply, \"to be so narrow that I could refuse an offer to improve my land,\ndenuded and mortgaged as it is,--an offer for the only rights I had left\nto sell, and which would give me plenty of money to enable me to restore\nthe home of my ancestors,--simply because the syndicate furnishing the\nmoney was composed of Northern men, thus, for a senseless prejudice,\ncompelling my mother and sister to eke out their income by sewing for\n_negroes_!\"\n\nHad Carolina struck him in the face, he could not have turned a whiter\ncountenance upon her than he did.  Twice he opened his lips to speak and\ntwice closed them again with the futile words still unspoken.  His hands\nwere clenched at his side, his whole figure rigid with outraged pride.\nYet he continued to look his accuser in the face, and Carolina honoured\nhim for his courage even while she could see self-knowledge dawn and\nhumiliation take the place of his dethroned pride.  The first blow had\nbeen struck which was to unmask his pitiable attitude,--the attitude of\nthe typical young Southerner of to-day, proud of his worn-out\nprejudices, and unaware that his very pride in them is in rags.\n\nCarolina clasped her hands to hide their trembling.  She could have\ncried out in pity for the suffering in the face of the man she loved,\nbut she dared not speak one word of the sympathy her heart ached to\nshow, for fear of undoing her work. Blindly she steeled herself for the\nwords she feared would pour forth.  Dully she wondered if, when they\ncame, they would end everything between them, and preclude any possible\novertures on her part when the leaven should have worked.  But the\nwords, bitter or otherwise, did not come.  Still he simply stood and\nlooked at her.\n\nThen, with a gesture both graceful and dignified, he bent and took her\nhand and kissed it.\n\n\"I understand,\" he said, simply, and Carolina, turning away, albeit sick\nat heart, felt a dawning thrill of pride--her first--that she had come\nto love this man.\n\n\n\n\n                              CHAPTER XXI.\n\n                            THE LIGHT BREAKS\n\n\nOne afternoon, a few days later, there came an hour of stifling heat,\nand Carolina, sitting in her little cottage room with \"Science and\nHealth\" on her knees, heard the rise and fall of voices in earnest\ndiscussion, which seemed to come from the back porch.  When she appeared\nat the door to ascertain who it was, she found Aunt Calla, the cook at\nWhitehall, and Aunt Tempy, Flower's baby's mammy, in animated\nconversation with Rose Maud, her own cook.\n\n\"Dar she is now!\" exclaimed Calla.  \"Miss Calline, I was jes' awn my way\nover hyah to ax yoh advice as to what I shall do wid dat no 'count Lily\nob mine, when erlong come Sis Tempy in de Barnwells' cah'yall, sent by\nMiss Flower to say will you please come over to see de baby right away,\nen Sis Tempy done fetch me wid her.\"\n\n\"Is anything wrong with the baby?\" asked Carolina, quickly.\n\n\"No'm! no'm!\" cried Tempy.  \"Miss Flowah got somepin' mighty fine to\nshow you.  Miss Callina, de lill fellah kin see!\"\n\n\"Oh, Tempy, how glad I am to hear it!\"\n\n\"Well'm, I reckon you is de one what otto hyah it fust,\" said the old\nwoman, with a shrewd glance.\n\n\"Why, what do you mean?\" asked Carolina.\n\nThe three women settled themselves with such an air of having come to\nthe point that Carolina felt reasonably sure that they had been\ndiscussing the affair, and that further concealment was no longer of any\navail.  She was surprised to see that, instead of the hostility she had\nfeared, each old woman had the appearance of eager curiosity if not of\nreal interest.\n\n\"I means, Miss Callina, dat I believes--we all believes--dat you done\nkunjered\" (conjured) \"de chile en kyored him,\" said Calla.\n\n\"I ain't a-saying dat,\" put in Tempy.  \"I ain't a-saying but what you is\nraised de spell what de voodoo done put awn de chile.\"\n\n\"En I tells um, Miss Callina,\" ventured Rose Maud, Carolina's own cook,\n\"dat hit's yoh new religion what done it, en I tole em I believed dat\nyou is de Lawd Jesus come down to yearth de secon' time, wid power to\nheal de sick, to cast out debbils, en to raise de dead.\"\n\n\"Rose Maud, Jesus was a man, and you know that He will never take the\nform of a woman,\" said Carolina, \"so don't ever say such a foolish thing\nagain.  But He gave that power to His disciples, and this new religion\nof mine you are talking about gives that same power both to men and\nwomen.\"\n\n\"Miss Callina,\" cried Tempy and Calla at the same time, \"has you got dat\npower?\"\n\n\"Ask Rose Maud,\" said Carolina.\n\n\"I done tole 'em, Miss Callina,\" cried Rose Maud.  \"But dey is bofe\ndoubtin' Thomases.  Dey won't believe until dey sees.\"\n\n\"Miss Callina,\" pleaded Calla, \"I cain't believe jis' caze I _wants tuh\nso bad_.  Ef you kin mek me believe, I would fall down awn my face wid\njoy.  I ain't never been satisfied wid no religion.  Sis Tempy will tell\nyou.  Ise done jined de chutch en fell from grace mo' times den I kin\ncount.  But, missy, _even niggers_ want a trufe dat dey kin cling tuh!\"\n\n\"Dat's a fack, Miss Callina!\" broke in Aunt Tempy.  \"En ef you will jis'\nput awn yoh hat en go wid us in de Barnwells' cah'yall, en 'splain\nt'ings to us lake Jesus done when He tuk de walk to Emyus\" (Emmaus),\n\"you will be talkin' to thirsty sinners what are des a-begging of you\nfur de water ob life!\"\n\nCarolina remembered the great number of intelligent <DW52> faces which\nwere scattered through the congregations of the beautiful white marble\nchurch, with its splendour and glory of stained glass, in New York, and\nshe wondered if here, in the pleadings of these three fat old <DW52>\nwomen in the pine forest of South Carolina lay the answer to the great\nand ever burning question of the white man's burden.  As she debated\nswiftly, her heart leaped to the task.  It was not for her to refuse to\nspread the truth when it was so humbly and earnestly desired.\n\n\"Come then,\" she said, \"ask me questions, and I will tell you the\nanswers that my new religion teaches.  You may come, too, Rose Maud.\"\n\nThe Barnwells' carryall went slowly out through the great avenue of\nlive-oaks from Carolina's little cottage at Guildford into the \"big\nroad\" which led to Sunnymede.  But no one thought of the incongruity of\nthe three old <DW52> women and Jake, letting the horses drive\nthemselves, while he listened with pathetic eagerness to the clear,\nearnest tones of the white young lady, who simply and sincerely answered\nthe questions all four asked of her with such painful anxiety and eager\nunderstanding.\n\nMeanwhile the storm, which the intense heat presaged, gathered, and they\nhurried the horses in order to reach Sunnymede before it broke.\n\n\"Dat's all I ask,\" cried Aunt Tempy.  \"I don' need to ax no mo'\nquestions.  Miss Callina done fixed t'ings for old Tempy.\"\n\n\"I allus knowed dat I was a worshipper ob de unknown God,\" cried Calla.\n\"Ef I had 'a' knowed de right One, does y'all reckon He would 'a' let me\nget away?  No, suh!  De Lawd hol's awn tuh His own!\"\n\nThe storm broke just as they reached Flower's little cabin in the dreary\nstump-filled waste which had once been the handsome estate of the La\nGranges.  Flower met them at the door and welcomed them in.\n\n\"Hurry, Jake, and get the horses safe before the rain comes.  Aunt\nTempy, take Calla and Rose Maud to the kitchen and give them some\nsassafras tea.  Oh, Cousin Carolina, dearest, did Tempy tell you?  Oh,\nthe blessed, blessed news!  For two nights now, the lamb has turned over\nin his crib because the light hurt his eyes.  I didn't send for you the\nfirst time because I wanted to be sure.  I was reading the fourteenth of\nJohn, and when I came to the verse, 'And if ye shall ask anything in my\nname, I will do it,' I just threw the Bible down and fell on my face on\nthe floor and begged God for my baby's eyesight.  And, when I looked, he\nhad turned over.  Oh, Cousin Carol, Cousin Carol, I think I shall go mad\nwith joy!\"\n\n\"Let me see him,\" cried Carolina, rushing past Flower and snatching up\nthe baby.  \"Oh, yes, dearest, I can see even a different expression in\nhis eyes. And see how he blinks in the light!  Flower, your baby is\nhealed!\"\n\n\"I know it,\" said Flower, reverently.  \"And I shall thank God for it on\nmy knees every day of my life.\"\n\nA terrific flash of lightning at that moment almost blinded them.  It\nwas followed instantaneously by a clap of thunder which nearly rent the\ncabin in twain.  Flower immediately seized her baby, with a face made\nashen by fear, and looking apprehensively at windows and doors, she\nwhispered:\n\n\"The voodoo!  Watch for her!  She always comes in a thunder-storm!\"\n\nAt the same time the three old women, with Jake, and Flower's black\ncook, old Eloise Lu, stumbled into the room, crying:\n\n\"Foh de Lawd's sake, Miss Flower, honey, let us in hyah!  De Day of\nJudgment sho has come!\"\n\n\"Nonsense!\" cried Carolina, with a sternness none of them had ever\nsuspected her of possessing. \"For shame, you Tempy and Rose Maud and\nCalla!  Where is your new religion?  Where is your understanding of the\ntruth?  Is God going to punish you for coming to Him as you just told me\nyou had come?  Oh, faithless disciples!  Now see if _I_ am afraid of a\nlittle thunder and lightning!\"\n\nThey straightened up under her words, and, with rapidly clearing faces,\nthey watched her go toward the open door.  The rain was coming straight\ndown with a terrific tropical downpour, and, as Carolina stepped\nsuddenly to the open door, she saw the same figure she had seen before,\nin the act of leaving a little clump of pine-trees to come nearer to the\ncabin. The figure spied Carolina at the same time, and, lifting a hand,\nbeckoned to the girl.  Without a thought of fear, but with rather a wild\nquestioning hope in her heart, Carolina, to the amazement of the cabin\ninmates, and later on no less to her own, stepped out into the pouring\nrain and ran toward the shelter of the trees.\n\nThey all crowded into the doorway to see her go, and, when they\nrecognized the other figure, they were speechless with awe.\n\nMiss Carolina had deliberately gone to meet the voodoo and lift the\ncurse!  Then she was indeed a chosen one of God!\n\n\n\n\n                             CHAPTER XXII.\n\n                          IN THE VOODOO'S CAVE\n\n\nAs Carolina felt the rain drenching her to the skin, the thought came to\nher, \"This is the first time in all my life that I ever was thoroughly\nwet with rain, yet to how many of the less favoured ones of earth this\nmust be no unusual occurrence. How sheltered my life has been!\"\n\nAnd the thought of God's protection went with her as she approached the\nmotionless figure under the pines.\n\nAt first Carolina took the woman to be a quadroon, but, on a nearer\nview, she saw that none of the features was African.  Rather the high\ncheekbones and sombre eyes suggested the Indian.\n\nThe woman held out her hand, and, as Carolina yielded hers, the woman\nsaid, in a voice whose tones vibrated with a resemblance to Flower's:\n\n\"You must come with me.  You will not be afraid.  You are a Lee.  I have\nbeen waiting a long, long time to get speech with you, but your wet\nclothes must be dried.  Will you follow me?\"\n\n\"Willingly,\" said Carolina, gently.\n\nThe woman did not smile, but her face lighted.\n\n\"You will not be sorry,\" she said, tersely.  Then she turned and led the\nway.\n\nThe rain still came down in torrents, but, as Carolina was already wet\nthrough, she thoroughly enjoyed the novel sensation.  She remembered how\noften, as a child, she had begged to be allowed to go out and get\nsopping wet--just once!--and had been denied.\n\nSuddenly the woman paused.\n\n\"Do you know where we are?\" she said.\n\nCarolina looked around, but could see no possible place of concealment.\nThe ground was flat and somewhat rocky.  The river made a sudden bend\nhere, and in this clearing lay huge pieces of rock half-embedded in the\nsoil.  The timber had been cut, and now a second growth of scrubby trees\nhad grown up, hedging the spot in a thicket of underbrush.\n\n\"No,\" said Carolina.  \"I never was here before.\"\n\n\"But you will come many times again,\" said the woman.  \"Look!\"\n\nShe knelt in the sand and scratched away with both hands at the base of\na great rock, until she came to its edge.  Then with one hand she\npushed, and the great boulder was balanced so neatly on its fellow that\nit slid back, revealing a natural cave.\n\nThe cool, underground air came in a wave to Carolina's nostrils, laden\nwith mystery.  Only one moment she hesitated.\n\n\"You are sure we can get out?\" she said.\n\n\"I am sure.  From where I stand I can see through this underground\npassage the sail of a ship on the ocean.  But this rock will not slip.\nWatch me.\"\n\nShe was already in the cave, and she reached out, and, with apparently\nlittle effort, pulled the boulder into place, closing herself in.\nCarolina put her hand under the rock and felt its perfect balance give.\nShe herself opened the cave again.\n\n\"I will come,\" said Carolina.  \"Have you a light?\"\n\nNever could she forget the hour which followed. She sat in this cavern,\nwrapped in an Indian blanket, watching her thin clothes dry before the\nfire the woman had kindled and listening to the following story:\n\n\"I have watched you,\" said the Indian, \"ever since you came, and when I\nfound that you were the one to cause my daughter to take her rightful\nplace in the La Grange family--you start.  Flower is my own daughter.  I\nam a half-breed Indian. My name is Onteora.  Both my grandfather and his\nfather were chiefs of the Cherokee tribe.  I am a direct descendant of\nthe great chief Attakullakulla, friendly to your people, who, in 1761,\nmade peace between the Cherokees and the great war governor, Bull.  My\nfather married a white woman of good family, named Janet Christopher.\nI, too, married white blood.  I was married by Father Hennessey, the\nJesuit priest, to a Frenchman named Pierre Pellisier, who died in\nCharleston in 1889. I have the documents to prove all these things.\nHere, I will show them to you.\n\n\"I am educated beyond my class.  I speak French.  I can read and write,\nbut no one knows what I can do, because I have lived as an Indian woman\nin order to avert suspicion from my child. All my children died except\nFlower.  She was my baby,--pure white, as you see, and so pretty!  Miss\nLe Moyne, who educated Flower, knew the truth. We agreed upon terms.\nMiss Le Moyne would have gone to the poorhouse if it had not been for\nthe money I gave her every week for the care of Flower.  And yet she\nwould have betrayed the secret she swore by her crucifix to keep, if\ndeath had not struck her dumb just in time!\"\n\n\"But why,\" interrupted Carolina, \"did you not come forward after\nFlower's marriage and tell the La Granges of her honourable birth?  It\nis a proud heritage to have the blood of kings run in her veins.\"\n\nOnteora shook her head.\n\n\"The time was not ripe.  _It needed you to open their eyes_.  Now they\nwill listen because Fleur-de-lys has found a friend!  You have rescued\nher from their contempt.  You have rescued my grandson from blindness--a\nblindness I knew the moment I looked at him.  And for that reason I have\na gift for the daughter of the Lees--a gift she will not despise!\"\n\nOnteora disappeared and when she came back she held in one hand two\nsilver coasters, beautifully carved and inscribed in French, \"From the\nMarquis de La Fayette to his friend Moultrie Lee, Esquire, of Guildford,\n1784.\"  And in the other a large silver tankard engraved, \"To\nMajor-General Gadsden Lee, of Guildford, from his obliged friend, George\nWashington, 1791.\"\n\nCarolina's shining eyes were lifted from the massive silver pieces to\nOnteora's face.  The woman nodded.\n\n\"The famous Lee silver!  I have it all!  It was I who removed it and hid\nit here.  It was in 1866, before I was married.  I tracked 'Polyte and\nher husband to its hiding-place and took it away.  No one ever knew--not\neven my husband!  I never knew why I kept it secret.  I saw the rewards\noffered.  I could have been rich.  I could have dowered Fleur-de-lys so\nthat even the La Granges would have welcomed her.  But something told me\nto wait.  Wait!  Wait!  Now, I know why.  It was to give it to you in\nreturn for my child's happiness!  If I had returned it for the money,\nthat money would have gone to help ruin the La Granges, and I should\nhave come to you empty-handed!\"\n\nThe woman was barbaric in this speech.  She showed her Indian blood, her\nIndian power, her Indian patience.\n\nCarolina reached out her hand and Onteora took it in both of hers.\n\n\"What do you wish me to do?\" Carolina asked, gently.\n\n\"Take these,\" said Onteora with sudden passion, thrusting the documents\ntoward Carolina, \"and show them to the La Granges!\"\n\nShe sprang to her feet and folded her arms in a matchless pride.\n\nShe was, in truth, an Indian.\n\nThe rain had ceased and Carolina's things were dried.  Onteora helped\nher to dress, her eyes shining with delight at Carolina's beauty, but\nshe expressed nothing in words.\n\n\"Come and see your silver,\" she said.\n\nShe led Carolina to a smaller cavern, where, by the light of a candle,\nCarolina could see the black shapes of all the silver Cousin De Courcey\nhad described to her.  But so cunningly was this cavern concealed, that\neven one who discovered the cave wherein they stood would never have\nfound the cavern.\n\n\"It reminds me of Monte Cristo!\" she said to herself in the breathless\ndelight every one feels at the touch of the romantic and mysterious in a\nhumdrum daily life.\n\nThen, as she realized the boundless Source of Supply whence this\nprecious silver and thrice precious information had come, Carolina\nturned and put her arms around Onteora.\n\nAt this sign of human love, tears filled the eyes of the Indian.\n\n\n\n\n                             CHAPTER XXIII.\n\n                             LOOSE THREADS\n\n\nMrs. Goddard alone knew of Carolina's discouragements, disappointments,\nand dangers, as the summer came and went.  To all others the girl turned\na smiling face, and Mrs. La Grange often wondered at her courage.  How\ncould she know that there were times when that sorely tried courage\nebbed so low that many a cipher telegram winged its soft way to her\npractitioner for help, and that the battle with tears and disheartenment\nwas fought out alone in the silence and sanctuary of her closet?\n\nOften things went very wrong.  She was cheated by men because she was a\nwoman.  She was hated by the rural doctors because she healed diseases.\nShe was an object of suspicion among the neighbours because she was not\n\"orthodox.\"  She was accused of inciting the <DW64>s to an idea of\nsocial equality because she taught them.  Father Hennessey gave her all\nthe trouble he could, but Carolina's constant and unvarying kindness to\nthe poor in his parish finally drove him to an armed neutrality.  He\nhated her, but dared not show it too openly, because she had powerful\ninfluence back of her.  The La Granges rose to her defence _en masse_,\nand carried all their enormous relationship with them.  Carolina had\nremoved the largest blot from their escutcheon, and no price was too\ngreat to pay.  Flower became the pet of the whole family, and, in their\ngratitude, they even endeavoured to provide for Onteora, but that wise\nwoman, having seen justice meted out to her child, silently disappeared,\nand, beyond knowing that she lived and wanted for nothing, they could\ndiscover no more about her.\n\nShe was not too far away, however, to keep the unruly <DW64>s in order,\nand many a warning went out from the voodoo when Carolina's interests\nwere jeopardized.\n\n'Polyte's surveillance was something Carolina had not bargained for.  At\nfirst his devotion was engendered by gratitude for the trust she placed\nin him, and fear, for he knew that she actually held over him the power\nof life and death.  Even if she were ignorant of the true significance\nof that meeting in the woods, at what moment might not some stray\nanecdote bring home to her its meaning? 'Polyte was no fool, and there\nwere times when he writhed in a hell of fear.\n\nThen gradually Carolina's personality began to gain ascendency over him,\nas it had over Tempy and Calla and Rose Maud, and even flighty ones like\nLily and her kind, and he worshipped her as a superior being.  Carolina\nembodied to the <DW64>s the old times of prosperity and the patriarchal\nprotection of the whites.  They liked the idea of the restoration of the\nold Guildford mansion.  Aged <DW64>s, who had known the place in its\nprime, heard of its rebuilding and journeyed back many weary miles to\nsee \"old mahstah's\" granddaughter, and to test her hospitality.  Several\nof these Carolina annexed and housed in the clean and shining new\nquarters, and she was amply repaid by their real knowledge of past\nevents and their idolatry of herself as the last of the Lees.\n\n'Polyte studied her every whim, and carried it out with the zeal of a\nfetich.\n\nThe mare Araby became her property almost by magic.  'Polyte would never\nsay one word concerning it, but one day Barnwell Mazyck sent word to\nCarolina that she could have the mare on her own terms, only he felt\nobliged to warn her that Araby had turned vicious.\n\n'Polyte spoke only one sentence.\n\n\"Ef you tek her, missy, she won't trick _you_!\"\n\n\"Oh, 'Polyte!\" cried Carolina, \"what have you been doing?\"\n\n\"Not a t'ing, Miss Callina.  Honest!  Only I raised dat mah, en I knows\nhuh!\"\n\nCarolina still hesitated until Moultrie brought word that Araby had\nnipped at Barney's hand, and in a rage he had kicked her.  After that,\nthe mare would not allow him to approach, but even at the sight of him\nshe would rear, bite, and kick, so that, being quite useless to her\nowner, he proposed to sell her,--if not to Carolina, then to some one\nelse.\n\nHearing that decided the girl.  She bought Araby, and sent 'Polyte to\nfetch her.\n\nThe beautiful creature proved as gentle as a lamb, and, even on the day\nwhen 'Polyte led her up for Carolina to see, she nosed her new mistress\nlovingly.\n\n\"Why, she seems just as usual,\" said Carolina, but she did not see\n'Polyte's heaving shoulders and convulsed face.\n\nThus, for the most part, the <DW64>s were Carolina's friends.  They not\nonly stood in awe of her body-guard, 'Polyte, who knew them root and\nbranch, good and bad alike, but their childish vanity was tickled by the\nbeauty of the small white marble chapel Carolina built on the estate,\nwhich had an organ and stained-glass windows and a gallery for <DW64>s.\n\nThis had been Mr. Howard's gift to the little band of Christian\nScientists which he had found on his first trip down South, meeting\nevery Sunday on Carolina's cottage porch, which, vine-shaded and\nscreened and furnished daintily, was as large as the cottage itself.  He\ntook infinite pleasure in furnishing the finest material and in rushing\nthe work with Northern energy, and personally supervising the building.\n\nHe well knew that he could please Carolina in no better way, and, when\nRosemary Goddard's husband, the Honourable Lionel Spencer, became\npresident of the turpentine company, which was organized on the basis of\nCarolina's investigations, and confirmed by Mr. Howard's agents, and it\nbecame necessary for the Spencers to live in South Carolina, Rosemary\nwas elected first reader of the little church, and Carolina offered them\nthe use of her cottage until they could build, while she and Cousin Lois\ntook possession of the now completed Guildford mansion.\n\nThings were prospering with the La Grange family.  Peachie had become\nengaged to Sir Hubert Wemyss, who, urged by the example of his friend\nLionel Spencer, and the enormous profits of the turpentine company, had\ninvested largely, and, after taking Peachie to England to meet his\nfamily and make her bow as Lady Wemyss to the king and queen, he\npromised to return to America for half of the year.\n\nCarolina went to New York twice during the summer, and visited Sherman\nand Addie at their camp in the Adirondacks.\n\nTo her surprise, she found Colonel Yancey there. He had paid one or two\nmysterious visits to his sisters at Whitehall, and had been deeply\npleased to discover that they were both members of the little Christian\nScience church there.  He even went so far as to ask Carolina to\norganize a Sunday school, which had not then been done, and to enroll\nEmmeline and Gladys as its first members.\n\nHe also took this opportunity, let it be said, to offer himself to\nCarolina again, but promised her, if she refused him this time, after he\nhad declared himself a believer in the new thought, that he would never\ntrouble her again.\n\nMr. Howard viewed Colonel Yancey's conversion to Christian Science with\namused toleration, but Carolina, who knew why, held steadfastly to the\nthought that there can be no dishonesty in the perfect man, and so\nfirmly did she cling to this affirmation that, when Colonel Yancey, in\nthe Adirondacks, announced that the old oil wells had again begun to\nyield, and that all the money which she and Sherman had considered lost\nwas by way of being restored to them, Carolina resolutely closed her\neyes to any investigations which might unearth disagreeable discoveries,\neven opposing her best friend, Mr. Howard, in this decision, and simply\nopened her arms to her reappearing fortune and her heart in gratitude\ntherefor.\n\nNeither she nor Mrs. Goddard was even surprised.\n\n\"From the moment I knew that the man's change of heart was sincere and\nthat he was a true Christian Scientist, I knew this restoration must\ncome,\" she said, \"otherwise no blessing of peace nor untroubled night's\nsleep could come to him. Christian Science lays bare the very root of\nerror, and when error is recognized in the light of day, it must\ndisappear from the heart of an honest man.\"\n\nBut Carolina only said in the depths of her own soul:\n\n\"See what Divine Love hath wrought!\"\n\nThere were changes, too, going on in Moultrie. He had never repeated his\ndeclaration of love to Carolina, but in every unobtrusive way he made\nher feel that she was surrounded by it, while as to the lesson she had\nconveyed to him in that one stinging sentence, which was never absent\nfrom the minds of either of them, it was his mother who brought word of\nits effect.\n\n\"Carolina, child, I never saw such a change in any man in my life, as\nthere is in Moultrie.  He has subscribed for three or four Northern\nnewspapers, and as to books!  Not novels, mind you. They are histories\nand biographies and Congressional reports,--the driest things!  Peachie\nand I tried to read them, but we couldn't, and, when I asked Moultrie if\nhe were getting ready to write a book, he answered me in such a short\nway, 'No, mother.  I am only trying to educate myself for the first\ntime.'  'Oh, son!' I said, for I assure you I was hurt to hear my son,\nwho has had the best education of any of the boys around here, speak as\nif he weren't satisfied with his education.  But he only patted my head\nand said he was only studying now for a purpose.  What do you reckon it\nis?\"\n\n\"He has said nothing to me about it,\" said Carolina, but Mrs. La Grange\nnoticed her scarlet cheeks, and, thinking it might be only a\nself-conscious blush, dropped the subject.\n\nMoultrie had asked Carolina if he might write to her while she was away,\nand she had assented, though with fear and trembling, for some of the\nletters she had received on business from various people contained\nserious shocks for a fastidious and cultivated mind, but Moultrie's\nletters proved a pleasant surprise.  Not only were they correctly\nwritten and correctly spelled, but in them he had dared to let himself\ngo as he never had done in conversation, and Carolina found not only a\ndistinct literary style but an imagination which astonished her.\nAlthough he carefully avoided subjects which had been discussed between\nthem, he showed a breadth and largeness of view which could only come\nfrom a wider vision of things in general.\n\nThen came the time, after Carolina's return, when the great turpentine\ncompany was being organized, backed by unlimited capital, and destined\nto corner the market \"for educational purposes,\" as Kate put it, when\nthere arose a crying need for an honest Southern man, one who knew the\ncountry well, one who possessed the confidence of the sly, tricky\ncrackers,--those crackers so crafty that straight-forward dealing is\nimpossible,--who possess little sense of honour, who are prejudiced\nbeyond belief, narrow beyond credence, ignorant beyond imagination, who\nare only honest under compulsion, and who require the greatest tact, not\nto say craft, in handling.  These are the men who, for the most part,\nproduce the orchard turpentine, and who, for the company's purpose, had\nto be tied up by contract in long leases.  A Northern man could not have\ntouched them.  They will deal only with their own, and even then must be\n\"managed.\"\n\nFor two months the organization of the company was held up because no\none could be found capable of filling this delicate position.\n\nThen, to the relief of all, and to Carolina's secret delight, Moultrie\nLa Grange offered himself, and, upon being instantly accepted, upon Mr.\nHoward's and Carolina's advice, he leased them the stumpage rights of\nSunnymede, and then and there was born the purpose to restore the home\nof the La Granges, even as Carolina had restored Guildford--out of money\nearned by the place itself.\n\n\n\n\n                             CHAPTER XXIV.\n\n                        THE HOUSE-PARTY ARRIVES\n\n\nEver since the restoration of Guildford had been an assured fact,\nCarolina had looked forward to gathering the dearest of her friends and\nrelatives under its roof for a housewarming, and as Thanksgiving Day was\nthe first festival to occur after its completion, she issued her\ninvitations for that day, and anticipated the arrival of her guests with\na heart so full of gratitude that she walked with her head in the\nclouds.\n\nBeautiful Guildford stood upon its ancient site, more beautiful by far\nthan it ever had been before, for Carolina had allowed herself a few\nliberties, which, after seeing, even Judge Fanshaw Lee approved.\n\nFor example, the great flight of steps, as broad as an ordinary house,\nwas lengthened to raise the house to an even more commanding position,\nand to allow a better view of the ocean and river from the upper windows\nand the flat, railed-in roof.  In the midst of this great flight of\nsteps was a platform, where twenty persons might have dined at ease,\nwith a collateral flight of steps on each side, leading, as well as the\nsecond section of the central staircase, to the porch.  No one who has\nnot seen Guildford can form any idea of the imposing beauty of this\nsnowy expanse of steps leading to its veranda.  And such a veranda!\nSurely, the observer exclaimed, the whole house could be no larger! so\ngreat was the idea its size first induced. It ran around all four sides\nof the house, and was lived in for fully nine months of the year.  It\nwas fitted with screens and glass, which could be removed at will, but\nfor her house-party, so perfect was the weather, even these slight\nobstructions to the view were dispensed with.\n\nInside the house, however, Carolina had carried out the original plan,\nwith only the necessary additions of bathrooms to each suite and plenty\nof closets, which the old Guildford had never possessed.  This did not\ninterfere with the installation of the great carved wardrobes, without\nwhich no Southern house could look natural to a Southerner.\n\nThese she designed from old cuts and had made to order, preferring new\nones exactly like those which had been in the family for generations to\npurchasing old pieces which rightly belonged to other histories than\nhers.  Guildford was frankly a restoration, so she boldly reproduced the\nfurniture as well as the house.\n\nWith the papering she had some difficulty.  No one could remember the\nexact patterns, and there was more friction over diverse recollections\nof wallpaper than over any other point.  But Carolina waived all advice\nfinally, deciding that decorations were but temporary at best, and\nresting upon the absolute word of Judge Fanshaw Lee, of Charleston, that\nGuildford had been utterly redecorated in 1859.\n\nThis decision gave Carolina a free hand, and she exercised her taste to\nsuch good purpose that the new Guildford, in its decorations, maintained\nan air of age, yet so skilfully was it done that it was also essentially\nmodern.  Only patterns were used which had borne the test of time, as\none who discarded in cut glass the showier designs for the dignified\nsimpler patterns, considering them more restful to live with than those\nmore ornate and modern.\n\nIn her cut glass Carolina had been more fortunate, owing to the\npossession of a few precious pieces, preserved among the Lees, from\nwhich to design.  The largest was a huge epergne, with glittering\npendants, which rose almost to the chandelier, and was designed for\npyramids of fruit.  It was so delightfully old-fashioned that Carolina\nviewed it with clasped hands.\n\nAlthough electric light glowed unobtrusively from submerged globes in\nwalls and ceilings, Carolina used sconces for the wax tapers of her\nancestors, and the delicate light was so deftly shaded and manipulated\nthat it seemed only to aid and abet the candles.\n\nThe central staircase of the house rose from the midst of a square hall,\nturned on a broad landing, and wound, in two wings, back upon itself to\nreach the second floor.  On this landing was an enormous window,\ncushioned and comfortable, from which the view of the fallow fields and\nwinding river was quite as attractive as the front view, which gave upon\nthe distant ocean.\n\nThe main hall pierced the roof, in the centre of which was a gorgeous\nskylight of stained glass. Here, too, Carolina had departed from the\nlines of ancient Guildford, for no less a hand than that of John La\nFarge designed that graceful group, whose colours drenched the marble\nfloor beneath with all the colours of the rainbow.\n\nA high carved balustrade ran around this space on the second floor, from\nbehind which, in years gone by, the children and black mammies had\nviewed the arrival of distinguished guests, whose visits had helped to\nmake Guildford famous.\n\nFrom this square space, transverse halls ran each way, with suites of\nrooms on both sides, ending in doors which led to the upper porch, as\nlarge and commodious and more beautiful than the lower, because the view\nwas finer.\n\nThis gives an idea of the plan of Guildford, but not necessarily of\nother Southern houses, unless you go back to old New Orleans, for\nGuildford partook largely of the beauty of the Creole estates, owing to\nthe originator of the present design, who had felt the influence of many\nforeign countries in his travels.  Returning to spend the remainder of\nhis life in his native land, he had built Guildford--a mansion in those\ndays--in 1703, on the site of the first house, built originally in 1674.\nThus, the Guildford which Carolina built was the third actual house to\nbear that name.\n\nThe morning of Thanksgiving Day dawned clear, cool, and beautiful.\nCarolina was up at sunrise, full of delightful anticipations, and as\nbrimming with zeal for the pleasure of her guests as any young bride in\nher first house.\n\nMr. Howard was bringing most of his guests in his car, and only\nyesterday she had received a telegram from him saying: \"Am bringing an\nextra guest, an old friend of yours, as a surprise.  Due Enterprise nine\nA.M. to-morrow.  All Lees aboard.\"\n\nJust as he had anticipated, this threw her into a fever of curiosity.\nIt must be some one who would be congenial, yet she fancied she had\nasked everybody who seemed to belong.  Who could the newcomer be?  Man\nor woman?  Old or young?\n\n\"All Lees aboard.\"  That meant that Sherman and Addie had decided to\ncome, after all.  She wondered if they had brought the children.  All\nLees.  That _must_ mean the children, because she had invited them.  All\nLees,--that meant also the Fanshaw Lees, of Charleston, whom he had\npromised to pick up on the way.  But who could the other be?  Carolina\nalmost shook the scrap of yellow paper to make it divulge the secret.\nHow uncommunicative telegrams can be!\n\nThere was plenty of room at Guildford,--that was fortunate.  And every\nroom was in order. She would give him (?) her (?) the violet room and\nbath in the south wing.  But if she only knew!\n\nRosemary and her husband were comfortably ensconced in the cottage, and\nhad asked to have Mrs. Goddard under their own roof.  Colonel Yancey and\nhis children would, of course, be the guests of Mrs. Pringle at\nWhitehall, but Carolina expected as her very own, Mr. and Mrs. Howard,\nKate, Noel, and Sir Hubert Wemyss, Judge Fanshaw Lee and his wife and\nchildren, from Charleston, Cousin De Courcey Lee, Aunt Evelyn Lee, Aunt\nIsabel and Uncle Gordon Fitzhugh, with the children, Eppie, Marie,\nTeddy, and Bob.\n\nEvery neighbour within a radius of twenty miles was anxious to help\nCarolina entertain her guests. Moultrie had arranged a hunt, Aunt Angie\nwas to give an oyster roast on the shore, Colonel Yancey had declared\nfor an old-fashioned barbecue, whereat all the <DW64>s promptly lost\ntheir minds. Mrs. Gordon Fitzhugh, after consulting Carolina's plans,\nadvised a fishing-party and picnic, rather an oddity in November, with\neverything to be cooked on the ground, including a 'possum with sweet\npotatoes.  Carolina greeted each of these proposals with tears in her\neyes.  Never before had she been so loved!  Hitherto, she had been\nsurrounded by courtiers, flattered and admired, always, however, with a\ngenerous appreciation of favours to come.\n\nBut here, she was with her own, and her own had received her with open\narms and taken her into their inmost hearts.\n\nAs Carolina walked in her garden, after her morning canter on Araby, she\nwondered if any one on earth was so fortunate as she.\n\nA messenger came up the broad avenue, and Carolina went to meet him.  It\nwas with a note from Mrs. Barnwell, saying that she was sending the\ncarryall to the station at Enterprise, for fear Carolina, at the last\nmoment, might not have room for all her guests.\n\nThe Barnwells' carryall!  Carolina gave a laugh that was half a sob, to\nthink of the part that ancient vehicle had played in her life during the\nlast year.  The neighbours had not seen the glistening carriages and\nautomobiles which stood as impatiently as inanimate things so beautiful\nand alert can be,--inanimate things which know that they can go.  She\nturned to the messenger.\n\n\"Give my love to Mrs. Barnwell, Sam, and say that I will ride home in\nthe carryall myself, and that I thank her for her kindness.  Can you\nremember that, or shall I write a note?\"\n\n\"I kin 'member it, Miss Calline.  Thank you, ma'am!\"\n\nMrs. Barnwell subsequently got a message from Sam to the effect that\n\"Miss Calline sed she'd 'a' had to walk her own self ef Mrs. Barnwell\nhadn't 'a' sont de ca'yall.\"  Which is about as accurate as any message\ncan be after going through the brain of a <DW64>.\n\nFinally it was time to go to the train.  Carolina had no fear that the\ntrain carrying the car of a president of a Northern road would be late,\nso she hurried Rosemary and Lionel and Cousin Lois into her big blue\nFrench touring-car, and started.\n\nAs they sped down the great avenue, Carolina looked back at Guildford,\nas a mother looks back at her first-born child.  There rose the\nbeautiful house, just as the strangers would get their first glimpse of\nit; for the last time the Howards came South, only a dim idea of it\ncould have been obtained.\n\nThere was not a hint of frost as yet.  Late roses bloomed riotously in\nthe garden, which Carolina had been tending for the last eight months\nwith a view to this very day.  She had planned well.  She did not intend\nto have a rebuilt Guildford look down upon patches of brown earth,\nremains of mortar beds, and broken-down shrubbery.  Every day she had\ncautioned the workmen against destroying any of her outdoor work, and,\nas fast as she could, she had made the gardens, the lawns, and the\nhedges keep pace with the builders, so that everything might be\ncompleted practically at the same time.  A dozen black forms were\nhurrying hither and thither, bent on carrying out \"lill mistis's last\norders.\"  The quarters glistened in the sunshine, even the dogs asleep\non the steps were just as Carolina had pictured Guildford in her\nchildish dreams in Paris.\n\nIt was a very excited little group which stood on the tiny platform at\nEnterprise, waiting for the train.\n\nFinally, only half an hour late, its warning whistle sounded, and\nscarcely had the brakes squeaked, when Mr. Howard sprang from the\nforward end of the rear car, followed by--Doctor Colfax!\n\nCarolina could scarcely believe her eyes.  She did not speak.  She only\nwent with outstretched hands to meet her friends, and something in the\nway Doctor Colfax looked at her hinted at some great change.  Then Mrs.\nGoddard followed, and, even in the excitement of placing her people in\nthe proper vehicles, and in the midst of unanswered questions and\nunlistened-to replies, Carolina noticed that Doctor Colfax hovered near\nMrs. Goddard.  She wondered if he remembered the last thing he said\nabout her.  But, oh, the joy of seeing them friends!\n\nAddie was wonderfully friendly.  She kissed Carolina quite\naffectionately, and told her that Kate Howard had succeeded in curing\nher neuralgia, to which Carolina knew Addie had been a slave for years.\n\nAddie's children, Cynthia and Arthur, were wild with delight.  It was\nthe first time they ever had been South, and to leave snow in New York\non one day and see roses blooming the next was more than their young\nimaginations could stand.\n\nThey always had been fond of their Aunt Carolina, but now their comments\non her beauty were quite embarrassing.\n\nAs Kate sprang from the steps, a close observer might have seen a\ntelegraphic question flash from Carolina's eyes to hers and a quick\nnegative flash back.  No one but a woman would have known what it\nsignified.  Still Carolina seemed satisfied with Kate's radiant aspect.\n\nJudge Fanshaw Lee was pompous but plainly delighted, and ready to be\npleased with everything. Carolina was keen to see what he would think of\nher daring, for he had promptly wet-blanketed her every effort to assist\nhim in any way.  But she could see that he was impressed with the\nappearance of her automobiles, and she fairly ached to have him see\nGuildford.\n\nTo achieve this end, she gave personal instructions to each chauffeur\nand driver to go by roads which would enable her, even in the Barnwells'\ncarryall, to arrive at Guildford first.\n\n\"You aren't going in that thing?\" cried Kate. \"There's plenty of room\nhere.\"\n\n\"I'm going in it to accept the hospitality of a dear neighbour,\" said\nCarolina.\n\nKate and Noel were seated in a little electric runabout.  As they\nstarted ahead, Kate turned to Noel and said:\n\n\"Somehow, I can't listen to anything Carolina says lately without\nknowing that the bridge of my nose is going to ache before she turns me\nloose.\"\n\n\"She certainly is the most angelic creature!\" said Noel.\n\nKate looked at him out of the tail of her eye.\n\n\"Do you like angels?\"\n\n\"I do, indeed.\"\n\nA pause.\n\n\"But I could never fall in love with one.\"\n\n\"Oh!\" said Kate.\n\nNoel cleared his throat once or twice, as if trying to say something.\nFinally he said:\n\n\"Kate, won't you be hurt if I say an indiscreet thing?\"\n\n\"Certainly not.  You know you can say anything you like to me.  I'm not\na fool.\"\n\n\"Well, here goes, then.  I've been noticing lately that you don't\nstammer any more.  Are you being treated for it?\"\n\n\"No,\" cried Kate, plainly delighted.  \"I am treating myself.\"\n\n\"Then, don't!\" cried Noel.  \"Kate, I can't bear it.  Yours was the most\nattractive, the dearest little mannerism--not a bit disagreeable.  Your\nspeech, so far from being marred by it, was only made distinctive.  I--I\nfeel as if I had lost my Kate!\"\n\nHis voice sank with unmistakable tenderness at the last words, and Kate\nstiffened herself, as if prepared for a plunge into ice-water.  Finally\nshe caught her breath sufficiently to say, awkwardly:\n\n\"If you care, Noel, of course I w-won't.\"\n\n\"If I care!\" cried St. Quentin.  \"Do I care about anything or anybody\nelse in all this world except Kate Howard?  Don't talk as if you didn't\nknow it.\"\n\n\"K-know it!\" cried Kate, stammering quite honestly.  \"Indeed,\" as she\ntold Carolina later, \"after that, I'd have stammered if I'd been cured\nof it fifty times over.  A proposal is enough to make any woman\nstammer!\"\n\n\"Indeed, and I didn't.  I th-thought you were in love with C-Carolina.\"\n\n\"Carolina!\" cried Noel.  \"Carolina!  Well, you are blind!  As if she\nwould ever look at me, in the first place--\"\n\n\"Oh, so that was your reason,\" interrupted Kate.\n\n\"And in the second place,\" pursued Noel, calmly ignoring the\ninterruption, \"she is in love with--\"\n\n\"With whom?\" exploded Kate, gripping his arm.\n\n\"Why, with La Grange!  Did you never notice them together last spring,\nand then the way she speaks of him?\"\n\nKate let her own love-affair slip from her mind, while she thought\nrapidly for a few minutes.\n\n\"I believe you are right,\" she said, slowly, \"but I can tell you\nsomething more.  They are not engaged.  Something is separating them.\"\n\n\"I think so, too.  Possibly Carolina is holding off.  I've noticed that\ngirls have a way of doing that.\"\n\nKate's face crimsoned.  She afterward told Carolina that, if Noel had\ncaught her laughing, he would have known all.\n\nBut her obstinate silence left it to Noel to continue.\n\n\"Kate,\" he said, finally, \"when you get through playing with me, will\nyou begin to take me seriously?  I'm tired of your game.  Now don't\npretend that you haven't been baiting me.\"\n\n\"Honestly, Carolina,\" said Kate, afterward, \"I'm telling you this j-just\nso you'll know how d-dog funny the whole thing was.  Here I've nearly\nhad nervous prostration for a year, wondering if he ever _would_\npropose, and then he went and accused me of playing a game to hold him\noff!  Aren't men fools?\"\n\n\"I--I thought when you g-got good and ready, y-you'd speak your mind,\"\nsaid Kate to Noel.  \"I c-couldn't go down on my knees and b-beg you to\nname the day, could I?\"\n\n\"Do you mean to tell me,\" said St. Quentin, \"that you will accept\nme,--that you will marry me, Kate?\"\n\n\"T-that's just what my p-poor, feeble speech is t-trying to g-get\nthrough your th-thick head,\" said Kate.\n\nBut Noel refused to be amused.  He reached for Kate's hand, and, in\nspite of Kate's impertinence, if he had looked, he would have seen tears\nin her eyes.\n\n\n\n\n                              CHAPTER XXV.\n\n                              BOB FITZHUGH\n\n\nEven Carolina was satisfied with the expression on Judge Fanshaw Lee's\nface when he was whirled up the great avenue of live-oaks, and the new\nGuildford burst upon his view.  He had snow-white hair, a pale olive\ncomplexion, and piercing black eyes.  His eyebrows were still black, and\nhe had a ferocious way of working them back and forth very rapidly when\nhe was moved.  This was one sign by which Carolina could tell; another\nwas that the unusual colour came into his face.\n\nEven before the guests had been to see their own rooms, Carolina was\nimplored to lead the way and let them explore Guildford.  This she was\nas eager to do as a young bride, and yet, in spite of her natural pride\nin her achievement, her modesty was so sincere and delightful that Judge\nLee and Mr. Howard were obliged to ply her with questions.\n\nThe exclamations of delight were perfectly satisfactory, even to Mrs.\nWinchester, who moved with majestic mien in their midst, listening with\na jealous ear for praises of her idol, and, by her questioning eyes,\nplainly demanding more of the same kind.\n\nMrs. Goddard's eyes were dewy with gratitude, and Carolina whispered to\nher that she--Mrs. Goddard--was Guildford's fairy godmother.\n\nWhen they had all returned to the drawing-room, Mr. Howard turned to\nJudge Lee and said:\n\n\"Well, judge, what is your opinion?  Isn't this pretty good for one\nlittle girl to accomplish all by herself?\"\n\n\"Mr. Howard,\" said Judge Lee and his eyebrows, \"it is the most\nmarvellous thing I ever heard of a young girl achieving.  Why, sir, to\nus Southerners, it is nothing short of miraculous.  Here are scores of\nmy own dear friends, similarly situated,--land poor, they call\nthemselves,--yet, as I cannot doubt Carolina's word or your figures, and\nyou both assert that Guildford has paid for itself, each and every one\nof them might restore their property in a similar manner.  I had no idea\nof the value of this new turpentine company of yours.\"\n\n\"Aren't you sorry now, Cousin Fanshaw,\" said Carolina, mischievously,\n\"that you wouldn't invest when we wanted you to?\"\n\nJudge Lee cleared his throat and reddened slightly.  He did not relish\nbeing jested with.\n\n\"I think I am, Carolina,\" he said.  \"God knows I needed the money, but,\nif you will allow me, under the circumstances of your great triumph, to\nbe ungallant, I will tell you that I did not have any faith in a woman's\nhead for business.\"\n\n\"Few of us have, I think,\" said Mr. Howard, coming to his rescue.  \"At\nfirst, I did not, but Carolina was so sure that I began it as an\nexperiment which was likely to cost me dear.  I have ended by believing\nin it with all my heart.\"\n\n\"Of course I have had a great deal of help,\" said Carolina, generously.\n\"Mr. La Grange is very influential, and I am sure I could not have got\nthe telephone and electric light without him.  They were carrying\nlanterns in Enterprise when we first came down here, and I expected to\nhave to get along with acetylene, which I greatly dislike.  But he told\nme that for the last ten years the subject of electric lighting had been\nagitated, and that he believed a little new blood and ready money would\nstart the thing.  That was easily managed, but the cost of bringing the\nwires to Guildford was greater than I expected.  However, in another\nyear several other estates will need lighting, and I shall carry it for\nthem over my wires, and thus reduce my initial expense materially.\"\n\n\"Who owns the control in the electric company?\" asked Judge Lee.\n\n\"Why, Carolina does, of course!\" said Mr. Howard.  \"You don't suppose my\nlittle Napoleon of Finance would commit such an error of judgment as not\nto keep that?  Nevertheless, she put up the poles from Enterprise to\nGuildford at her own expense.  She wouldn't take any unfair advantage of\nher control.\"\n\nJudge Lee glanced at his cousin in half-way disapproval.  He greatly\ndisliked a woman who understood finance, and he privately considered\nCarolina unsexed.  If she had not been beautiful, he would have said so,\nbut her girlish loveliness saved her.\n\nJudge Lee looked around.  On every side familiar objects met his eye.\nIt was the same Guildford of his ancestors, yet enlarged, dignified,\nengrandeured.  His gaze clung affectionately to the heavy, quaint\nfurnishings, so cunningly reproduced that they might well pass as the\nancient pieces they represented.  He began to realize the enormous\namount of hard work this indicated,--of the hours and days of\nunremitting toil,--of the discouragements overcome,--the obstacles\nsurmounted,--the love this mirrored.\n\nFinally he turned to Carolina, with his keen eyes softened.\n\n\"I do not understand how you accomplished it, little cousin.  It is a\nmarvellous achievement for any one!\"\n\n\"I did not accomplish it of myself,\" said Carolina, gravely.  \"I never\nin the world could have done it if--\"\n\n\"If what?\"\n\n\"I hear that it annoys you even to hear the words,\" said Carolina.\n\"Nevertheless, I must tell you that the whole of Guildford is a\ndemonstration of Christian Science.\"\n\nA deep silence fell, and the eyes of the two men met.  Judge Lee's fell\nbefore the corroboration he met in Mr. Howard's.  A sudden softening\ntook place in his heart.\n\n\"I begin to believe that there is something in this thing, after all,\"\nhe said, slowly.\n\nA babel of voices broke in upon their conversation just here, as the\nguests trooped down from their rooms, exclaiming with admiration on\nevery hand.  Sherman and Addie were particularly delighted, but they\nlooked at Carolina wonderingly, as if uncertain whether this were the\nsame sister they had known before.\n\nCarolina bloomed like a rose under all the admiration her work received,\nbut she was too busy to drink it all in.  She had, for one thing, the\nchildren to amuse.  Emmeline Yancey, a serious-browed child with grave\neyes, was her right hand, and to Emmeline and Bob Fitzhugh she confided\nher plans.  Hardly had the children learned of the delights in store for\nthem, when the guests began to arrive.\n\nThen, such a rushing to and fro!  Such a calling for servants!  Such\nhurried dressing!  Such a gathering up of children, and a general\nhastening of duties which should have been performed before!\n\nIntroductions to the few who had not met before seemed like a meeting of\nold friends, so warm was the welcome and so well known the existing\nfriendships.\n\nCarriage after carriage rolled up the drive and deposited Fitzhughs, La\nGranges, Manigaults, Pringles, and Yanceys, until Guildford resembled\nthe palmiest days of its predecessors.\n\nPeachie and Sir Hubert Wemyss and Noel and Kate were receiving sub rosa\ncongratulations, and beaming faces were everywhere.  Moultrie's eyes\nfollowed Carolina wherever she was, and none noticed it more jealously\nthan a slim, blue-eyed boy who would not mingle with the other children,\neven when Emmeline begged him to.  He only shook his head, and continued\nto watch his divinity.\n\nThen old Israel, who had been a rascally boy in the days of Carolina's\ngrandfather, flung open the doors and the guests trooped out to the\ndining-room.\n\nEvery one stood and exclaimed with delight at the sight which met their\neyes.  The majestic dinner-table of Guildford, which would seat forty,\nstood in the centre of the room, flanked by side-tables groaning under\nthe glorious old Lee silver and glass and china, such as no\ncontemporaneous eye had seen, but so often had those gathered here heard\nits beauty described that it seemed a familiar sight.\n\nThe children had a table to themselves, and this was set across one end\nof the room.  Emmeline was to be the mother and Bob Fitzhugh the father,\nand actually carve the turkey.\n\n\"He'll spill the gravy and drop the turkey on the floor, Carolina,\"\ncried his mother.\n\n\"Let him,\" said Carolina.  \"Who cares?  But this turkey will be so good\nthat he will stay on the platter, as I shall bid him, and Bob shall\ncarve him, and Emmeline shall serve the plum pudding!\"\n\nShrieks of joy went up from the children at this daring announcement,\nand all the parents were made radiant by their babies' happiness.\n\nThe table was long and low, with chairs to match, and the children saw\nwith jealous delight that it was copied exactly from the big table, even\nto the bowls of flowers and pyramids of fruit.  They even had their tiny\nchampagne glasses, in which 'Polyte, who was their butler, poured\nfoaming ginger ale, so that they could join in the toasts which Judge\nFanshaw Lee proposed.  They wriggled with an ecstasy they never had felt\nbefore, and never, never did they have such a time as at Cousin\nCarolina's Thanksgiving dinner at Guildford.\n\nThe climax came to their awe when, at the end of everything, Mr. Howard\narose, glass in hand, and announced--what everybody knew--the engagement\nof his daughter Kate and Noel St. Quentin, and gave them his blessing,\nand everybody cried and laughed and drank their health.  The children's\nround eyes almost popped out of their heads.  To be present at a real\nbetrothal!  It was more exciting to the little Southerners than a <DW64>\nbaptism.\n\nBob Fitzhugh's face was seen to grow very red, and then suddenly he\npushed back his chair and strode to where Carolina sat, and said, in a\nsturdy voice:\n\n\"Cousin Carolina, why can't we announce our engagement?  You know you\npromised to marry me.\"\n\nHe stood crimson but dauntless under the shrieks of laughter which\nfollowed his speech.  Carolina's face was very rosy also, and she was\nseen to steal a mischievous glance at Moultrie La Grange, which somehow\nset his heart to beating with hope.\n\nShe put her arm around Bob and kissed him on the forehead before them\nall.\n\n\"Bob, dear, it is too soon,\" she whispered, consolingly.  \"You know I\nsaid if you wanted me in ten years and I was still unmarried--\"\n\n\"Oh, but Cousin Carol!\" cried the boy, \"you are so beautiful that unless\nyou promise to wait for me you are sure to be snapped up.  Father said\nso.\"\n\nAn added wave of colour flew to Carolina's face, and she hid her face in\nthe boy's shoulder, when, to her surprise, she heard the voice of Col.\nWayne Yancey saying:\n\n\"Bob, my boy, if she should promise you, you'd have to fight me, and\nfight me to the death.\"\n\nBob looked at him, and stiffened.\n\n\"Are you after her, too?\" he cried, angrily.\n\n\"I've been after her longer than you have.  And I'm not the only one.\"\n\nBob turned despairingly to his father.\n\n\"How many does that make?\" he roared.\n\nThe laughter of the grown people passed unheeded.\n\n\"Never mind, son,\" said his father.  \"Colonel Yancey's name completes\nthe list.  There isn't another bachelor or widower left in South\nCarolina. It's just the way the girls used to treat me, son, but\nafterward I met your mother and she made everything all right.\"\n\nThe boy flew to his father's side, and hid his head.\n\n\"Girls are all alike, son.  You'll have to bear it. We all have to.\nTurn around here and ask your Uncle De Courcey why he is a bachelor.\nAsk your mother how many boys she flirted with before I came along.  Be\na man.  Look there at Emmeline and Gladys and--\"\n\nBob burst away with a roar of pain.\n\n\"Emmeline is about right for Teddy!\" he exclaimed, in wrath.  \"I want a\ngrown woman.  I don't want anybody but Miss Carolina Lee. Moultrie knows\nhow it is, don't you, Moultrie?  When you've once loved a girl like\nCarolina, how would you like it to be told to take up with anybody\nelse?\"\n\n\"I just wouldn't do it, that's all!\" said Moultrie, looking squarely at\nCarolina.\n\n\"Bob,\" said Carolina, severely, \"you are embarrassing Mr. La Grange and\nme dreadfully. Won't you please go back to your place and make me feel\nthat I can depend upon you to protect me instead of exposing me to\nlaughter like this?\"\n\nThe boy's eagle glance flew from one convulsed face to another.  Then he\nshowed his blood.  He came to Carolina's side, and put his arms around\nher neck and kissed her cheek, whispering:\n\n\"I'll never speak of it again.  They can laugh if they want to, but some\nday you'll remember that I behaved when you asked me to.\"\n\nHe went back to his seat and Carolina looked at Emmeline, and both\nlittle ladies rose from the heads of their tables and led the way to the\ndrawing-room.\n\nBut Carolina was uneasy.  She could not forget the look that Moultrie La\nGrange shot at her, when Bob said, \"After you have once loved a girl\nlike Carolina, how would you like to be told to take up with anybody\nelse?\"\n\nShe knew the time was approaching when he would ask his question over\nagain, and she was not prepared yet to give an answer.  She was sure he\nwas on the right track, but she was not sure that he would persevere.\n\nThe chill of autumn always manifests itself in November days in South\nCarolina after the sun goes down, and when the guests repaired to the\nlibrary, they found a great log fire, the size of which they had never\nseen before.  For weeks Carolina's servants had scoured the woods for a\nbacklog of sufficient girth to please their mistress, but it was 'Polyte\nwho finally secured the prize.\n\nAround this glorious fire they all gathered, and something of the way\nGuilford had been restored, as well as the gentle tranquillity of the\ntwilight hour, crept into their hearts and tinged the conversation with\nan intimacy which years of ordinary social intercourse could not have\naccomplished. Christian Scientists all over the world will recognize\nthis as a fact peculiar to themselves.  If church-member meets\nchurch-member of any other denomination, they are forced to become\nacquainted as is usual in society, because there is no unanimity of\nthought, and each is bound for his or her particular goal by independent\nand widely diverse routes.  But in Christian Science instantaneous\nintimacies are possible, because it is the one religion which requires\ncomparative unanimity of thought, and all are travelling in the\nidentical path which leads to the ultimate perfection of harmony.\n\nThus, with no other light than the firelight and with no further\nintroduction to the dear people of the Southland, than that they were\neither Christian Scientists or Carolina's beloved kinfolk, no one was\nsurprised when Doctor Colfax said:\n\n\"You showed no astonishment this morning, Miss Carolina, when you saw me\namong the guests Mr. Howard was bringing to your beautiful\nhouse-warming.  And as I know the type of your mind, I know that you\nwill ask no questions.  Therefore, I owe it to you to tell you, and\nbelieve me, I am delighted to include your friends.\n\n\"You, Mrs. Winchester, remember meeting me on the train as you were\ncoming from Boston. You thought I had been to take a rest.  I had.  But\nit was a rest in a hospital from an operating-table. It was my second\noperation for cancer of the throat. My inexcusable show of anger at your\nhouse, Mrs. Howard, the night I saw the miracle of Miss Carolina's\nhealing, was induced and aggravated by the knowledge of the ordeal\nbefore me and of the futility of it.  My brutal words against Mrs.\nGoddard, this dear, dear woman, whom I have learned to revere and love\nas my best friend, were uttered because I longed to go and fling myself\nat her feet and ask her if she could cure me.  If any of you men who\nwere there that night--if you, St. Quentin, had knocked me senseless and\ntaken my unconscious body to a Christian Scientist for treatment, I\nshould have thanked you on my knees.  But none of you knew.\n\n\"Well, I went through this second operation, and it proved as futile as\nthe first had done.  Within six months I was confronted by the certainty\nof the third, and this I felt sure would be fatal.\n\n\"With the horrible fear of death before my mental vision, and no faith\nin surgery, I one day made up my mind to call on Mrs. Goddard, to tell\nher the ungentlemanly, unmanly words I had used against her in public,\nto beg her pardon, and if she forgave me, to implore her help for my\nhideous malady.\n\n\"Dear friends, you, who know her, know how she received me.  But none of\nyou know that under her treatment I was entirely cured.  Nor does she\nknow what I am about to say, for only since I came down here and lived\namong you and saw your beautiful lives, have I decided.  Mrs. Goddard, I\nowe it to you to tell you first.  I have decided to give up the practice\nof materia medica, which failed me in the hour of my greatest need, and\nI intend to study to be a Christian Science practitioner.\"\n\nA startled murmur ran through the group.  Even with all their faith,\nthis came as a surprise, for the name of Doctor Colfax stood for so much\nin the medical world.  Few men would have dared to show so much moral\ncourage.  Only Mrs. Goddard seemed to understand, for she reached out\nher hand to him, and he bent and kissed it before them all.\n\n\"I give up!\" cried Colonel Yancey, to relieve the tension.  \"Cousin\nLois, look at all these lovers holding hands, and thinking we don't see\nthem, and say whether you and I shall be left out.\"\n\n\"Wayne Yancey,\" said Mrs. Winchester, \"I'm not going to be left out of\nanything.  I have come to the point where I don't believe in the Church\nof England the way I did, and, if I decide to become a Christian\nScientist, there is no telling but that I may forget what a rascal you\nused to be in what they call 'the old thought' and decide to marry you\nin the new!\"\n\nThus Guilford began at once to take her proper place as the mystic spot\nwhere lovers' vows were plighted almost before they knew it, so replete\nit was with all that goes to make a home, and, as the dancing flames\ndied down, Carolina felt a soft hand steal into hers, and looked down\ninto the wide eyes of her niece, little Cynthia Lee.\n\n\"What is it, darling?\" she asked.\n\n\"I feel,\" whispered the child, \"that strange things are going to happen\nat Guildford, and that you and I shall always be in the midst of them!\"\n\nCarolina, instinctively realizing that this was a psychic moment for the\nimaginative child, slipped her arm around Cynthia's delicate waist,\nsaying:\n\n\"Why do you feel it, Cynthia?\"\n\n\"Listen, Aunt Carolina.  Something of all the queerness I have heard\nsince I came down here makes me feel that I shall lead a stormy life,\nand that I shall need this thing and want it and be unable to accept it\nuntil I am beaten by everything else.  Do you understand me?\"\n\n\"Only too well,\" sighed Carolina.\n\n\"Then I shall want you, and want you terribly.\"\n\n\"I shall always be here, dearest.\"\n\n\"That is what comforts me,\" said the child, the mystic light dying out\nof her eyes.  \"It is what comforts me about the whole thing.  I know it\nwill always be there when I want it.  I have talked to Emmeline about\nit.  Even little Gladys taught me her hymn.\"\n\nAnd the child and the woman looked into each other's eyes, knowing that\ntheir souls were akin, and that the witchery of the twilight hour had\nopened floodgates closed by day, but which opened when the soul felt the\nneed of speech.\n\n\"I am glad you told me, Cynthia,\" said Carolina. \"The only answer to all\nof life's puzzles, I have found in this awakened sense of mine, which\nwill surely come to you some day.  Remember it when the waters grow too\ndeep.\"\n\n\"The answer to all life's puzzles,\" echoed Cynthia.\n\n\"Sing, child,\" said Carolina.\n\nAnd Cynthia, whose voice was like the rippling water and the sounding of\nsilver bells, began to sing what Gladys called her hymn:\n\n    \"'And o'er earth's troubled, angry sea\n      I see Christ walk,\n    And come to me and tenderly,\n      Divinely talk!'\"\n\n\nAs the child sang, every feeling in every heart melted, until only love\nremained, and, when she finished, Kate cried out:\n\n\"It's all over!  I d-don't hate Mrs. Eddy any more.  I--I've been healed\nof it by Cynthia's singing.\"\n\nThe child's lovely voice had so sadly shaken Carolina's composure that,\nunder cover of the half-darkness, she rose and made her way quietly to a\nlittle hall which led to a private staircase, intending to gain her own\nroom and recover herself before her guests began to take leave.\n\nAs the voices rose and fell, she moved nearer and nearer the door, too\nintent upon her own ends to notice that Moultrie La Grange had likewise\ndetached himself from the fireside group and disappeared.\n\nAs she finally stepped behind a group of palms which concealed the door,\nshe sprang lightly into the dark passage and flung herself headlong into\nthe arms of Moultrie La Grange, who had come in that way to intercept\nher flight.\n\nHe was not slow to take advantage of the very opportunity he had come to\nseek, and, after one brief struggle, so slight that it was like the\nfluttering of a bird, she hid her face in his shoulder, with a little\nsob in which relief and joy and love were mingled.\n\nHe said nothing, only held her close and kissed her hair, until her arms\nstole upward and curled around his neck, and she whispered:\n\n\"Moultrie, dear, dear Moultrie, will you forgive me for what I said to\nyou that day?\"\n\n\"I have nothing to forgive, dear heart.  You only said it because you\nloved me.\"\n\nTears filled her eyes, and she drew closer to him, whispering:\n\n\"I knew that first night in New York at the opera--that this hour would\ncome--and just now, while Cynthia was singing, I knew that--you would\nunderstand--everything!\"\n\n\"I would not have dared to speak to you again, dearest,\" he answered,\n\"if I had not emptied my soul of self and got rid of that which\nseparated us. But--I have been working since you showed me where I stood\nwith you, and I, too, under the spell of that child's voice, have come\nto the point where I can say that, if you think I am capable of it,--and\nworthy to be the successor of such a man as your idolized father,--I\nwould be proud to complete his work on Abraham Lincoln, and, with your\nconsent, we will call it 'The Debt of the South to Lincoln.'\"\n\nFor reply, Carolina lifted his hand to her lips and kissed it.  She\ncould make no reply to such a surrender as that, but in that hour she\nlifted her hero to a pinnacle, whence he never was dislodged.\n\n\n\n\n                                THE END.\n\n\n\n\n\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n_Manage Your Time to Reduce Your Stress_\n**BY THE SAME AUTHOR**\n\n_The Procrastinator's Handbook_\n\n_The Procrastinating Child_\n\n_The Clutter-Busting Handbook_\nManage Your Time  \nto Reduce Your Stress\n\n**A HANDBOOK FOR THE OVERWORKED,  \nOVERSCHEDULED, AND OVERWHELMED **\n\nRita Emmett\n\nCopyright \u00a9 2009 by Rita Emmett\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 Walker & Company, 175 Fifth Avenue, New York, New York 10010.\n\nPublished by Walker Publishing Company, Inc., New York\n\nAll papers used by Walker & Company are natural, recyclable products made from wood grown in well-managed forests. The manufacturing processes conform to the environmental regulations of the country of origin.\n\nLIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA\n\nEmmett, Rita.\n\nManage your time to reduce your stress : a handbook for the\n\noverworked, overscheduled, and overwhelmed \/ Rita Emmett. \u20141st U.S. ed.\n\np. cm.\n\nIncludes bibliographical references.\n\neISBN: 978-0-802-71976-8\n\n1. Time management. 2. Stress management. I. Title.\n\nBF637.T5E46 2009\n\n640'.43\u2014dc22\n\n2008022110\n\nVisit Walker & Company's Web site at www.walkerbooks.com\n\nFirst U.S. edition 2009\n\n1 3 5 7 9 10 8 6 4 2\n\nDesigned by Rachel Reiss\n\nTypeset by Westchester Book Group\n\nPrinted in the United States of America by Quebecor World Fairfield\n_This book is dedicated to all you wonderful,_\n\n_intelligent, hardworking people who want to add a_\n\n_richer, more peaceful and relaxed quality to your life_\n\n_and care enough about it to actually read this book._\n\n_Here's wishing you times of love and laughter,_\n\n_the blessings of health and wealth,_\n\n_the joy of family and friends,_\n\n_and many restful, re-creating moments of serenity._\nAuthor's Note\n\nTo protect the privacy of my readers and seminar participants who have given permission for their stories to appear in this book, their names and identifying characteristics have been changed, with the exception of those professionals who have provided comments and have given permission for their names to be published. \nContents\n\n _[Acknowledgments and Appreciation _ ](chap9_manageyo_9780802719768_epub_part9.html#d41151541)\n\n_Introduction  _\n\n1. It's Time to Get Control of That Stress\n\n2. Search for and Select What Is Valuable in Your Life\n\n3. Trash Perfectionism\n\n4. Reach for Realistic Goals That Are Both Relevant and Rewarding\n\n5. Eliminate What You Can and Streamline Everything Else\n\n6. Set Boundaries at Work and at Home\n\n7. Strive to Recharge Your Battery Daily\n\n8. Now Let's Put It All Together\nAcknowledgments and Appreciation\n\nWriting a book and having it published is a dream come true, a blessing from God, a huge hunk of the luck of the Irish, and\u2014no matter what you call it\u2014it's a blast.\n\nThe author gets all the praise and honor (also criticism and bad reviews, but that's another story), but this book wouldn't be in your hands if it weren't for a glorious gang offering help, support, and encouragement on both the professional and personal levels.\n\nMy deepest gratitude goes to the three\u2014editor, agent, and publisher\u2014who helped turn my random, chaotic thoughts into a readable book and who have guided me through the writing and publishing of my three previous books as well as this one. Thank you to:\n\nJackie Johnson, who I am convinced is not only one of the world's finest editors, but also one of the world's most patient people. She works hard to polish my words, so if there is a shine or a sparkle to this book, we have Jackie to thank for it. She guides and answers, coaches and coaxes, and never makes fun of my disorganized way of thinking . . . and writing.\n\nDanielle Egan-Miller, my fabulous literary agent, who listens patiently, works tirelessly, treats me as if she has all the time in the world for me, and turns my weird ideas into something that can be published\u2014all with a delightful sense of humor and a wealth of wisdom.\n\nGeorge Gibson, publisher, who, while surrounded at Walker &Company by so many authors who are scholars and researchers, continues to welcome, support, and promote my books.\n\nIt's a delight and honor to be associated with all three as well as the whole team at Walker & Company who have enriched my life and worked hard to turn this book into a reality.\n\nAn enormous thank-you to so many others, including Mickey Forster, my niece, Web goddess, friend, assistant, and general guiding light. I don't know what I would have done without her fun and loving help.\n\nShe and the rest of my Dream Team, Cheryl Guidry, Brian Redding, and Stormi Willis, have played a vital role in supporting and encouraging this book and providing ways for Emmett Enterprises, Inc., to help people break the procrastination and clutter habits.\n\nA heartfelt thanks to Randy Davis for mentoring, teaching, coaching, and guiding me to new ways of thinking; for pushing me off the edge so I can fly; for wisdom, humor, and caring; and most of all for his friendship.\n\nTremendous gratitude to all my family and friends who cheered me on, helped in any way they could, and put up with my absentmindedness and lack of availability when I was writing, and special thanks to those of you who offered your enthusiasm and stories or were willing to read and offer guidance on this book, especially Michelle Emmett, Ruth Coleman, Jessa Forster, Carolyn (CJ) Jonasen, Tomas Dorney, JoAnne Knight, Linda Brakeall, and Curt Hansen, all of whom offered constant friendship and support. Also thank you for help and stories: Malachy McCourt, Kay Merkle, Sandra Baumgardner, Phaedra Vaughan, and Norma Maloney.\n\nThere's always a special hug and smooch for my long-suffering husband, Bruce, who did such a brilliant job at picking up the slack and keeping our lives on an even keel while I was cloistered in my \"cell where I dwell\" writing this book.\n\nThank you to all of you (and I do apologize if I accidentally left out your name). And most of all, my deepest gratitude to clients, subscribers to our Tip Sheet electronic newsletter, and readers of my books, including you.\n\nThank you for reading this book and thank you, God, for this joyful journey.\n_Manage Your Time to Reduce Your Stress_\nIntroduction\n\nTWENTY YEARS AGO, WHEN I FIRST started presenting \"Blast Away Procrastination\" seminars, I observed that people put off things they hate to do. These days, people still put off what they hate to do, but\u2014to me\u2014the biggest change in that area is that they also put off what they _love_ to do.\n\nOver and over I hear of how people used to see their friends every week, but now their life is so busy that visits occur only a few times a year. Or how they love to read novels, but they have so many other things they have to read, there is never time for \"reading the books I love.\" Or the whole family used to gather for dinner every weekend, but now they come together only on holidays. Or they tell of how much they love going to the movies or plays, walking in nature, visiting museums, entertaining, playing the tuba, working out, building models, sewing, and all sorts of activities that nurture their spirits and recharge their batteries, but . . . well, you know how it goes: so much to do and so little time.\n\nPeople are busy building the lives they _thought_ they wanted; they have become so stressed out that they have no life.\n\nAre you putting off the life you want to enjoy? Regardless of how productive, effective, and efficient you may be, if you answer yes to one of the following questions, you are procrastinating.\n\nDo you put off learning ways to manage your stress until the busy time ends?\n\nDo you put off spending time with people you love until you're less busy?\n\nAre you waiting for a busy time to end before you take steps to improve your life? _Manage Your Time to Reduce Your Stress_ will address a very common problem shared by many: the feeling that the more productive you become, the more there is to do, and there just isn't any time or energy left to do those enjoyable things you really want to do.\n\nWhere would you find the time?\n\nNo one really _finds_ time. Imagine walking along and stumbling across a big ol' bucket of extra time. It just doesn't happen. You have to _make_ time, and this book will help you figure out how to make time (guilt-free) and use it in areas of your life that you value.\n\nPeople who have read my previous books or attended my seminars tell me that they are successful in breaking the procrastination and clutter habits, but they are still outrageously stressed and have more responsibilities than they can handle. They know they need to nurture their health, but they keep putting it off because there are so many more important things to do. They are overworked, overscheduled, and overwhelmed. And what makes matters worse is they are concerned that if they start to use stress management concepts, all of their time management techniques will suffer because they will have to sit under a tree smelling the roses, accomplishing nothing and never drinking caffeine again.\n\n_Manage Your Time to Reduce Your Stress: A Handbook for the_ _Overworked, Overscheduled, and Overwhelmed_ will tackle time management _not_ in terms of becoming more productive or efficient, but in terms of how we spend our time in relation to all that is important and valuable to us in our lives, and how to continue to be productive and efficient without burning out.\n\nLife isn't time management, it is \"stuff\" management\u2014things to do, people to see, commitments and obligations to fulfill. And _mis_ managing all that stuff-to-do is what leads to stress and feeling fragmented, frantic, frazzled, and frustrated.\n\nThis book will address stress _not_ in terms of nutrition, exercise, what you drink, or whether you smoke, but will focus on the impact time management has on stress, what you can and cannot control about it, and how to make changes in your time management that will have an enormous impact on alleviating your stress. In addition, this book will help you:\n\n\u2022 Understand that learning to manage your stress can be one of the finest gifts you ever give yourself\n\n\u2022 Acknowledge that eliminating all stress is not the solution\n\n\u2022 Unearth ways to add fun to your life\n\n\u2022 Reassess that guilty feeling that might pop up when you do something to add fun to your life\n\n\u2022 Understand why it is crucial to stop being a perfectionist (and how to do that)\n\n\u2022 Find ways to simplify your life\n\n\u2022 Move from fatigue to energy\n\nAt the end of each chapter, there is an Extra Credit section of questions and exercises prompting you to record insights, lessons learned, and decisions. You can skip this section, but doing these exercises will deepen your experience and understanding of how to manage your time to lessen your stress. Also make note of the places in this book that discuss areas you want to work on. You'll read some ideas that point out the good things you're already doing and other areas that you need to work on. You will identify your strengths and your weaknesses.\n\nHighlight strategies you believe may work for you, then put them into practice. If one doesn't work, try another until you have your own personalized plan. If you find yourself backsliding and feeling close to burnout, you can review any section that applies to you. Use this book as a reference that you can refer to over and over. It will change your life. \n **CHAPTER 1**\n\n[It's Time to Get  \nControl of That Stress](chap8_manageyo_9780802719768_epub_part8.html#d41151181)\n\nJULIE TOOK GREAT PRIDE IN BEING a super-achiever in a waste management company; she loved a sense of accomplishment and success. But she felt that there was an emptiness inside her. She had way too much to do and never enough time to do it. As a result she didn't have time for the people she loved, and when she was asked if she ever had fun, she laughed, shrugged her shoulders, and replied, \"When would I have time for fun?\" But Julie assured everyone that she loved her work. Even though she was working with people all day long, Julie was lonely. She was always tired and took medication for depression and anxiety.\n\nDennis is a financial adviser who constantly races from meeting to meeting. He handles a huge list of phone calls as he drives from place to place and says he loves his fast-paced life. Yet he has no life except his work. His family has drifted away from him and he has no friends. He's a likable guy and many people would love to spend time with him, but he's always too busy to socialize. He too says he loves his work.\n\nFamily members express concern over his health but he says the stress doesn't bother him. He does plan to start an exercise program to lose the weight around his middle, and he knows that if he eats in a more healthy way he will feel better. But there just isn't time. Also, he has insomnia\u2014he wakes up around 2:00 A.M. and stays awake. As a result, he is exhausted most of the time.\n\nHe says others might need self-care, but he doesn't. His blood pressure is out of control, he is on medication for cholesterol and several other challenges, and he is always in a frantic, frazzled state. What Dennis doesn't realize is that those who say they don't need downtime, quiet time, or time for themselves are often the ones who need it the most.\n\nShay and Marcie are partners in their own network marketing business, which they run out of their home. They love what they do but have no boundaries. So they accept phone calls during family time and drop their work to take care of the kids, but then feel that they have to make up that time by working late into the night. On weekends, if a business crisis comes up or a deadline looms, they put aside time with the children to work in the office, and they often feel guilty about canceled family plans, so they make up for it by buying the latest toy or techno-gadget for the children.\n\nAs a consequence, they are deeply in debt, which makes them decide to spend more time working to catch up financially, which leaves them feeling guilty for ignoring the kids, so they buy more and more stuff for them. This vicious circle has contributed to arguments, sleep deprivation, overweight for both Shay and Marcie, and short tempers on everyone's part.\n\nSome people are so caught up in being productive, efficient, and stressed-out that they accept burnout as a normal way of life. They think the concept of enjoying life is airy-fairy New Age baloney.\n\nIn a seminar, Stan said he was tired of being swamped by all that he had to do and the busy-ness of his life. He felt he was under so much stress and there were so many demands being made on his time each day that stress management couldn't possibly work for him.\n\nOn the other hand, he wondered about several people he knew\u2014 including his boss\u2014who experienced enormous demands on their time but were able to get an amazing amount of work done and meet an awesome amount of deadlines, yet remained calm and had time for family and friends. So he decided to learn about stress management, trusting that he too could find a way to avoid burnout in the midst of his super-busy life.\n\nAs we work to be more productive and accomplish more and more, many of us are hyper-scheduled\u2014so busy that there is no time to:\n\nSleep late\u2014ever\u2014or read the Sunday newspaper or magazines\n\nMake love or cuddle\n\nRecharge batteries or refresh your soul\n\nGo to the zoo, museum, botanical garden, movies, theater, or concerts\n\nClutter-bust desks, e-mails, cars, or homes\n\nGarden or pursue other hobbies\n\nCreate a home that feels like a sanctuary of comfort and rest\n\nHelp a neighbor or work for a favorite worthy cause\n\nLoaf or think or pray or relax or journal or meditate or . . . rest\n\nThink . . . or feel . . . or enjoy life\n\nAsk yourself what the purpose of your busy-ness is. Does it bring you contentment or joy? Some people overschedule themselves or take on more work and responsibilities than they can handle reasonably because they are highly competitive and want to be perceived as the best or most powerful or strongest or most successful at whatever they do. Some overextend themselves out of a desire for the love and approval of others.\n\nFear also can be motivation, especially in a highly competitive job market where those who have jobs must work hard to keep them, sometimes increasing their workload to absorb the duties of laid-off colleagues.\n\nHow will you know when you achieve success? Do you know what you are striving for? Is it the right success for you? Are you working to make enough money? But how much is enough? When will you know you've got enough?\n\nWe become accustomed to stress, so used to it that in spite of the frenetic pace, multiple top priorities, and high demands of everyday life in the twenty-first century, it seems that managing stress has become a low priority for many people.\n\nWhat Stress Really Is\n\nDr. Jamie Kahon, a chiropractor in Illinois, explains that in stressful situations\u2014whether the stress hits you physically, intellectually, emotionally, or spiritually\u2014your body releases a hormone called cortisol. It is produced by the adrenal glands, which lie on top of the kidneys, and it helps you to adapt to stressful situations.\n\nCortisol, often referred to as the \"stress hormone,\" is intricately involved in many physiological functions, including the regulation of healthy blood sugar metabolism, maintenance of healthy blood pressure levels, establishment of healthy immune system function, and promotion of the body's natural anti-inflammatory response.\n\nBut when exposed to excessive demands, the brain sends a message to the adrenal glands to increase cortisol secretion. The body responds by providing a surge in energy, increasing mental alertness, and raising blood pressure, thereby preparing the body for the \"fight or flight\" response.\n\nWhile this response provides an effective mechanism for combating an acute stressor, increased or prolonged exposure to stress can lead to elevated cortisol levels. And heightened cortisol, in turn, can lead to negative changes in body chemistry, altering the balance of hormones and affecting the systems of the body.\n\nAs cortisol increases, it can reach a level where it becomes life-threatening as it increases the body's secretion of insulin, which can cause insulin resistance, metabolic syndrome, infertility, etc. This can lead to diabetes, which can cause heart disease, kidney failure, tingling in the hands and feet, blindness, and even amputation of limbs, among a host of other complications.\n\nWhen illness hits you, it can touch you on the physical, intellectual, emotional, or spiritual level. Physical symptoms might be headaches or stomach problems; intellectual symptoms might be absentmindedness or indecisiveness; emotional symptoms might be out of control anger or weeping. Even people who aren't religious recognize that we all have a spiritual part to ourselves, and spiritual stress can manifest in that feeling of emptiness or the feeling that we have nothing left to give. Burnout.\n\nThere are many different concepts of what stress is. For example, one definition of stress is the result of trying to do too much, for too many people, in too little time, in an environment that is too hard to deal with. When you let go of trying to control an inflexible environment and all the people in it, you stop struggling with fantasy and the pain of unrealistic expectations. You let go of stress.\n\nWe can all relate to this, but let's turn to Dr. Hans Selye, the Canadian endocrinologist and father of stress research. His definition is the one I use in my seminars. He writes, \"Stress is the nonspecific response of the body to any demand made upon it.\"\n\nThe secret of stress management is grasping the concept that we have zero control over the stressors or demands made upon us, yet we have 100 percent control over our response. That's where the management of stress must start.\n\nThe demands\/stressors in your life are neither good nor bad. It is only our reaction or response that turns us into raving lunatics. And we could have a negative stress response to negative _or_ positive demands. Getting married or divorced, hired or fired, graduating or dropping out of school\u2014all are demands upon us.\n\nFor example, vacations are supposed to be fun, positive. Have you ever taken a road trip for three days with kids in the backseat? See what I mean? Vacations are _supposed_ to be positive. But many times, people are outrageously stressed by vacations or other positive demands such as holidays or weddings.\n\nWhy We Experience Burnout\n\nTechnology and labor-saving devices are time savers, but using them may not lead to us getting more rest or relaxation. What are we doing with all the extra time we have that has been saved for us by dishwashers, cell phones, e-mail, computers, and all our time-saving appliances?\n\nWe have more choices, but we also have more expectations, noise, complications, anxiety, and depression. We have more loneliness than ever before, and have more worries about jobs, debt, family, finances, insecurity at work, and the future of the world.\n\nCarolyn, a participant in my time-management seminar, had an interesting insight. She said she often receives e-mails with articles by people longing for \"the good ol' days.\" \"But remember,\" she wrote in her blog, \"the good ol' days were kind of rough to live through\u2014there were no televisions, no phones, no cars, no indoor plumbing, no electricity, no air-conditioning, no deodorant, no computers; so many luxuries that we take for granted weren't available.\"\n\nCarolyn is right. The good ol' days were not a great time for minorities, widows, orphans, or the poor; there was little assistance or advocacy for the elderly or children, the mentally ill or disabled, or for people who were battered, shattered, scared, abused, or confused.\n\nYes, people lived slower lives, but that doesn't mean they were free of stress. Jake's grandfather told of working on the farm from sunup until sundown when a hailstorm wiped out all of his crops. His grandfather said, \"There was no insurance when that happened and also no miracles of modern medicine to cure illnesses or save lives.\" Just those two examples alone must have caused great worry, concern, and, yes, stress.\n\nIda, a woman in her late eighties, told me that her three young brothers all died the same week of the same illness, and she said that was not uncommon. She said the heartbreak from that almost killed her mother.\n\nAs long as people have lived, there have been stressful situations of one degree or another. Let's not wish for the peace and stress-free serenity of the good ol' days. That is a fantasy. But the reality is, we can improve our lives right now regardless of the demands being made upon us.\n\nWe have a different stress, that's all. Never has there been such a combination of pressures such as:\n\n\u2022 living on the edge\n\n\u2022 being available to work every minute of the day with cell phones, e-mail, text messages, PDAs, and all the other \"helpful technology\" coming our way\n\n\u2022 constantly rushing at a fast pace\n\n\u2022 trying to multitask with two, three, or four activities demanding our attention at the same time\n\n\u2022 overcommitting ourselves\n\n\u2022 needing medication for going to sleep, waking up, and staying alert or energized\n\n\u2022 fear of losing a job\n\n\u2022 being sandwiched between needy aging parents and needy adult kids\n\n\u2022 complications\n\n\u2022 lack of boundaries\n\n\u2022 insecurity\n\n\u2022 fear\n\n\u2022 lack of fulfilling relationships\n\n\u2022 complete exhaustion\n\nBurnout is often called the illness of the very caring. Someone who doesn't give a dang about other people or the world isn't as prone to burnout as you are.\n\nObviously, you are out there living life and caring. Does that mean you _will_ burn out? No. But it does mean that you could be prone to it, so you must be aware of the high price you could pay for stress and need to learn positive ways to prevent it . . . such as reading this book.\n\nWhen you manage your stress in negative or unhealthy ways, the outcome often is sickness. For example, high blood pressure and heart disease can be caused by stress. And there are many illnesses and conditions such as asthma that are not caused by stress but can be exacerbated by it.\n\nOne sign of burnout is when someone (maybe you?) says phrases such as \"I just don't care,\" \"I hate it here,\" \"I can't take this anymore,\" \"Every day I wish I could quit my job,\" and \"Stay away from me. Leave me alone.\"\n\nWhat is the solution? Should we strive for a life free of all stress? Nope, not possible and not desirable. No matter what your life circumstance, there will be stress. A life without stress would mean there's no reason to get out of bed in the morning. As glorious as that may sound, life would get pretty boring in no time, and guess what? Boredom, for most people, is more stressful than rush-hour traffic in New York.\n\nThe secret is to learn to manage stress in positive ways. One of the best gifts you can ever give yourself is to become an expert in stress management.\n\nHow to Control Your Responses to Stressors\n\nDaniel, a cable guy, told a story that was a great example of several people having widely different responses to the exact same demand. He had been stuck for fifteen minutes waiting for a train to pass. The train finally ended, two cars ahead of him crossed the tracks, and then he had to wait for another train coming from the opposite direction to pass.\n\nHe started to feel his heart pounding in his head because he was late for an important appointment. To distract himself from feeling frantic, he decided to observe people in the cars around him.\n\nA few cars behind him, a guy was in major meltdown, pounding his horn and swearing out the window at the top of his lungs. Daniel wondered what his purpose could possibly be. Did the guy think the railroad engineer would hear him, stop the train, and say, \"Oh my,my,my. Please excuse me, am I inconveniencing you? Allow me to back up and clear this track for you.\"\n\nThe woman next to him reached into her glove box, pulled out a large plastic bag filled with postcards, stamps, and a pen, and sat calmly writing notes to people. Daniel said, \"Now there's a woman who has a train-waiting strategy.\"\n\nThe fellow in front of him fell sound asleep at the wheel and when the train finally ended, Daniel had to beep to wake him up.\n\nOur level of stress has nothing to do with the demands made upon us and has everything to do with our response. It's up to us to control our responses; our feelings contribute to our responses, but our thoughts generate those feelings. What we say to ourselves (our self-talk) determines those thoughts. Sound hard? It's really not, and the rest of this book will help you develop new attitudes and new ways of talking to yourself.\n\nIn Daniel's instance, we saw examples of a whole range of self-talk. The fellow frantically beeping and yelling probably told himself that this was the worst thing that could happen to him at that moment and his whole life would be ruined if that train didn't end right then. Isn't that a common message that teens give their parents? \"If you don't let me go out tonight, my whole life will be ruined.\"\n\nThe other two drivers could have had the same time crunch as yelling-beeper guy and could have had the same amount of pressure on them, but they told themselves that no matter how frazzled or frustrated they got, it was not going to get them across those train tracks any faster. So they might as well make the best of it. (And maybe leave a little earlier next time.)\n\nThese messages of self-talk have great power over our mental and physical health. Yelling-beeper guy's blood pressure reading probably would have been soaring at that moment.\n\nAnother example of how self-talk can overcome stress would be if your absentminded neighbor always forgets to tell his wife you called and want her to call you back; your self-talk would be more important than his lack of communication. If you tell yourself he is rude and inconsiderate and you deserve to be treated better, you can work yourself into a frenzy. On the other hand, if you tell yourself, \"Well, I left a message but I know he never remembers to tell her. So I'll just pretend I didn't make the call and I'll try to reach her later,\" that self-talk will likely reduce your stress response.\n\nGreat Questions to Ask Yourself Frequently\n\nExamine your existing concept of time management in order to recognize when doing more is actually taking time away from what you really _want_ to do. Some of the questions we all need to ask ourselves occasionally include:\n\n1. Do I work excessive hours because I'm passionate about my work or because I'm scared not to?\n\n2. Right now, is my work exhausting or exhilarating?\n\n3. Do I have some source of joy in my life?\n\nIf that last question sounds naive and simplistic to you, then you desperately need this book. It is not normal and should not be acceptable to you to live with no expectation of happiness or joy.\n\nOur lives are like bank accounts. You need to make deposits in the form of doing things to care for yourself physically, intellectually, emotionally, and spiritually. If you keep making withdrawals for work and other commitments without making a deposit, you'll burn out. If you went to the doctor and she told you that you will die if you don't slow down and take care of yourself, you would make the time. Why wait till the doctor tells you to take care of yourself? Why not prevent that doctor visit from happening?\n\nBurnout often happens when your stress, work, or frustration is greater than the reward, success, or appreciation you receive. Have you ever had an experience where you worked hard and got a thank-you and a great pat on the back, and you said, \"I'll do it again. I'll work for that person [or organization] any time they want me to.\" For most people, a little appreciation goes a long way. And the opposite is true, too. Lack of appreciation usually leads to cranky people.\n\nBurnout is not limited to white-collar professionals. Anyone can suffer burnout on many levels, such as an adult son or daughter caring for an aging (or ill, or simply mean) parent. You could burn out when you need a break from your kids, a spouse, a volunteer organization, needy friends, clutter, work, or drudgery.\n\nSinking a Float\n\nFor example, Sylvia, a graphic designer, was in charge of building a rainbow float to represent the school PTA in her town's Independence Day parade. She formed a huge rainbow out of chicken wire, and then thousands of paper napkins were fluffed open and stuffed into each and every hole.\n\nMany people offered to help fluff and stuff, but truly it was Sylvia's project, and it took her every waking moment for over a month. But Sylvia kept telling everyone it was worth the time and effort, because the float would be used several times later. After the parade, it would be stored in a PTA member's storehouse to be used again for the harvest parade, the Brownie fly-up ceremony, and the Christmas pageant.\n\nWhen the July parade ended, Sylvia went to the PTA president to let her know she was going to deliver the float to the storehouse and was shocked to hear that there no longer was room for the float in there. Sylvia asked what she should do and was told that she should dump the float in the garbage.\n\nShe did.\n\nBut she was burned out from too much work and too little appreciation. She never volunteered for that PTA again. But every year she is the arts and crafts lady for the local church \"Summer Happening.\" At the end of summer, the group gives her a thank-you mug, a certificate, and applause. None of these cost much time or money, but they made Sylvia feel appreciated and that's all the reward she needs.\n\nMost of us don't work in a place or live in a world that has a system for rewards or appreciation. So guess where I think the reward or appreciation has to come from? Not from the boss\u2014 when they go to \"Boss School\" they learn delegation, leadership, and how to spell \"priorities,\" but not how to show appreciation.\n\nSo who's in charge of making you feel appreciated and rewarding you? That's right. You are. It's up to you to give yourself a reward: \"I just finished a big project at work, and I'm going to loaf this Sunday afternoon.\"\n\nTip: It's important to communicate these plans to your family. Otherwise you'll have the whole gang circling you like buzzards asking,\"Why is she relaxing when we need her to go to the store?\"\n\n**QUESTIONS TO CONTEMPLATE**\n\nIf you could wave a magic wand and change your stressful situations, what areas of your life would you change?\n\nWould you connect with people more often? Or less often?\n\nWould you stop feeling lonely?\n\nWould you stop spending time with people who make you feel miserable?\n\nWould you have more (or less) social activities? Group meetings? Business engagements?\n\nWould you spend time out in nature more? Have more quiet time or meaningful time in your life?\n\nWould you stop feeling guilty, worried, or anxious as often as you do now?\n\nWould you stop wishing that someone or something would change so that you could finally feel happy?\n\nWould you start to have fun?\n\nWould you stop finding excuses to put off \"recharging your battery\"?\n\nWould you have more control over your schedule? Be less hurried and harried?\n\nWould you stop allowing impossible standards to make you crazy, standards such as perfection or never making a mistake?\n\nWould you be more positive? Pleasant to be with? Less critical? Less judgmental?\n\nWould you be more active? More energetic? More organized? More dependable?\n\nWould you finally let go of your overwhelming stress?\n\nWould you enjoy your life more?\n\nHow would your life be different?\n\nAfter you give thought to these questions, here's one more: What is keeping you from spending your time on what is valuable to you?\n\nObviously, we don't have control over everything. You don't have much control of your time if you are caring for someone not able to care for himself, but that must not completely stop you from enjoying life, adding fun to your life, eating healthier, or whatever other changes you desire. Maybe chasing after several preschoolers all day does mean that their needs come first, and maybe working a high-pressure job does mean you don't have much time for yourself, but isn't there some way to sneak in a few minutes to take care of you?\n\nIt might take some creative problem-solving but often there is some way to squeeze little kindnesses to yourself into your daily schedule.\n\nFinal questions: Do you make everybody else more important than you are? Do you allow everybody else's needs and priorities to come before yours?\n\nHmmmm, these are questions worth pondering, aren't they? If you wish that someone would reduce your stress, let you become that someone, and start down the glorious road of experiencing the joys, rewards, and improved lifestyle that come when you give yourself the gift of stress management.\n\nTake Care of You\n\nWith the fast pace and high demands of everyday life in the twenty-first century, it seems the pressures associated with accomplishing more impact almost everyone and anyone, from soccer moms to great-grandmothers, from college grads to retirees. Keep in mind the classic example of airplane oxygen masks. The flight attendant announces that in case of emergency the oxygen mask will drop down. The attendant also warns that if you're on a plane with someone who is not able to care for himself, you should put your oxygen mask on first. If you try to put it on the other person first, you might pass out and be totally useless to help the person who depends on you.\n\nThe same is true with self-care. Take care of you first, otherwise you might burn out and not be any good to those you care for.\n\nDuring a seminar, Edwardo, a civil engineer, argued, \"But wait! You think we should be responsible for our response to demands and for giving ourselves appreciation. Isn't that just adding to our responsibilities? Adding to our stress?\"\n\nI told him no, not at all. As you start to take charge of your selftalk to control your response to the demands that are bombarding you, you will see your stress start to melt away and your view of the world change.\n\nTime management is not the only solution if you find yourself running like a hamster in a wheel, rushing to get too much done in too little time. In fact, if you are that busy, trying to do too many things _is_ causing you stress. There is hidden, surprising pain in efficiency and productivity for its own sake\u2014depression, broken relationships, financial difficulties, health problems, overwhelming schedules, and exhaustion all come under the general headings of stress and burnout.\n\nAs you start to give yourself some appreciation, rewards, and self-care, you will be amazed to find new zest, enthusiasm, and energy. With a little thought\u2014and a few changes\u2014you can learn how to continue to work in an efficient, productive manner without generating harmful stress and still make the time to engage in the activities you truly enjoy. This book will show you how to reach that balance.\n\nThe first letter of the next six chapters spell the word STRESS:\n\n**S** earch for and select what is valuable in your life\n\n**T** rash perfectionism\n\n**R** each for realistic goals that are both relevant and rewarding\n\n**E** liminate what you can and streamline everything else\n\n**S** et boundaries at work and at home\n\n**S** trive to recharge your battery daily\n\nThe questions to begin with are: Do you know what activities you enjoy? Do you know what is important or of value in your life? Many people don't. This is what we tackle next.\n\nTHOUGHTS TO PONDER\n\n_Stress is the spice of life. Without stress, life lacks_ _excitement, challenge, and a sense of adventure._\n\n\u2014HANS SELYE\n\n_No man need stay the way he is._\n\n\u2014HARRY EMERSON FOSDICK\n\n_Two men look out through the same bars:_   \n_One sees the mud, and one the stars._\n\n\u2014FREDERICK LANGBRIDGE\n\n_When you complain, all you do is broadcast, \"There's a_ _victim in the neighborhood.\"_\n\n\u2014MAYA ANGELOU\n\n_Slow down and enjoy life. It's not only the scenery you_ _miss by going too fast\u2014you also miss the sense of where_ _you are going and why._\n\n\u2014EDDIE CANTOR\n\n_Stress is an ignorant state. It believes that everything is_ _an emergency._\n\n\u2014NATALIE GOLDBERG\n\n_We either make ourselves miserable or make ourselves_ _strong, either way, the amount of work is the same._\n\n\u2014CARLOS CASTANEDA\n\n_Expecting the world to treat you fairly because you are_ _good is like expecting the bull not to charge because you_ _are a vegetarian._\n\n\u2014DENNIS WHOLEY\n\nEXTRA CREDIT\n\n1. What areas of your life cause the most challenge, difficulty, or stress for you? Identify those areas. Some ideas:\n\nwork\n\nhome\n\nrelationships\n\nvolunteer groups\n\nfinancial obligations\n\nschedules\n\nsleep\n\nhealth\n\nemotions\n\n2. How does stress hit you?\n\nPhysically:\n\nIntellectually:\n\nEmotionally:\n\nSpiritually:\n\n3. What is your biggest cause of stress?\n\n4. What is your best way to manage stress? (What do you do to help yourself when you become stressed out?)\n\n5. Occasionally ask yourself:\n\na) Do I work excessive hours because I'm passionate about my work or because I'm scared not to?\n\nb) Right now, is my work exhausting or exhilarating?\n\nc) Do I have some source of joy in my life?\n\nd) Am I trying to control something that is beyond my control?\n **CHAPTER 2**\n\n[Search for and Select What Is  \nValuable in Your Life](chap8_manageyo_9780802719768_epub_part8.html#d41151211)\n\nJOSH, A REAL ESTATE BROKER, TRULY enjoyed his career and never minded working twelve, fourteen, or sixteen hours per day. However, he battled Crohn's disease and his doctor constantly urged him to reduce the stress in his life. He didn't feel he was under stress because he enjoyed his work. But as he started experiencing more flare-ups with his disease, the doctor strongly pushed Josh to seek counseling, and he even set up the first appointment. So simply to appease the doctor, Josh decided he would go to one session.\n\nThe therapist asked him what was important in his life and he said his work. When encouraged to add anything else to the list, he quickly added, \"Of course my wife and kids, and my parents and the rest of my family. Oh yeah, and my health, and the Chicago Cubs.\"\n\n\"What about money?\"\n\nJosh replied, \"Yes, money is super important, too, and especially all the stuff money buys. Love it.\"\n\nThen the therapist asked, \"What on that list would you miss the most if you lost it?\" Josh thought a minute and told him, \"My kids and wife, then maybe my health.\"\n\n\"Where do your job and money fit in?\"\n\nJosh was surprised to hear himself say, \"They come after the others. I would be sad to lose my job or money, but I'd be devastated if I lost my kids or wife, and I don't want my Crohn's disease to get any worse. I want to become healthy again.\"\n\nThe therapist paused, then recapped: \"You value your family and your health most, after that comes your career and money and all that money buys.\n\n\"So let's take a look at your life. How much time do you spend with your family, on your health . . . and with your work?\"\n\nJosh felt as if he'd been kicked in the gut.\n\n\"I spend all my time working,\" he said. \"My wife says she feels like a single mom raising our kids. They are usually in bed by the time I get home. And over the weekend, I'm either on the phone or the computer or my BlackBerry. All my time is spent on my career and hardly any on the things that are really valuable to me.\"\n\nThe therapist leaned forward, his eyes riveted on Josh's, and said, \"When the way you live is not in sync with your values, the result is always stress, which can lead to dis-ease.\"\n\nWhen you spend more time doing what is not important to you, and little or no time on what is truly important to you, the result is usually stress, frustration, and a sense of inadequacy. Eventually, you have an empty feeling, as if there is not a drop of energy or joy left inside you.\n\nSome people work progressively longer hours, their personal lives becoming emptier; then in order to avoid the emptiness, they find it easier to just keep working. You might be super busy, yet where are the relationships, the joy, and the fun you would like in life? How is it that we all get the same number of hours each day, yet how you're spending your time feels so unsatisfying?\n\nThe True Meaning of Time Management\n\nHow do you spend your time? Time management doesn't mean running around like a nut doing twenty things at once. True time management means actually spending as much of your time as you can doing those things you want to do rather than activities you don't care about. It involves clarifying your values, deciding what is important, and working to spend your time doing that.\n\nFor example, a person who highly values financial security yet devotes a great deal of time to shopping and running up debts will usually find herself stressed, joyless, and sometimes sick. Once you clarify and identify your values, then you can brainstorm ways to adhere to them.\n\nFor Josh, this meant becoming more conscious of the time he spends with his family and what he would allow to interrupt it. He has made some changes. Now when he's home he screens his calls, taking only the absolutely necessary ones. And he checks his e-mail once a day over weekends and responds only to those needing immediate attention. Most of all, he makes an effort to be present when he is with his family instead of letting his mind focus on his business during family time. As a result, his family life is happier, and he feels much closer to his wife and kids. They in turn are delighted to \"have more of Daddy\" and there is less stress in the whole house, which has had a positive impact on Josh's stress at work.\n\nWhat our values are and how we spend our time regarding those values usually happens at an unconscious level, so it's often mystifying why you are so frazzled. To bring your values from your unconscious to your conscious mind, read this list of values and see which of them strike you as important in your life.\n\nMaking time to do the things you love will reduce your stress. Are you willing to devote time to uncovering what is important to you and figuring out ways to incorporate that into your life? This is a necessary step to prevent your time management from draining you.\n\nWhen people do not give some thought to their values, all their goal-setting, multitasking, and organizational strategies can lead them straight into feeling overwhelmed, which leads to burnout. And if you live your life in the fast lane, accomplishing huge numbers of projects, tasks, and goals, yet none of these brings you joy or gives you a sense of purpose, then frustration and stress will build up. You might work hard to get more done in less time, but is there any time left for experiencing fun in life? The satisfaction of a job well done? The close friendships and connections with people who nourish and enrich your spirit? The peace and serenity at the end of the day that brings soothing rest?\n\nDo You Procrastinate About You?\n\nDo you make time for yourself to recharge your battery, or are you too busy taking care of everyone else? People can have all the time management skills in the world, but if they believe that everyone else's priorities come first, then they will manage their time to care for everyone else, ignore themselves, be unproductive, and eventually burn out.\n\nWhy is everyone else more important than you are? Why are everyone else's priorities more important than yours?\n\nWhen I first started talking in my keynote presentations and training sessions about making time for yourself and what is important to you, a woman in the audience told me that was being selfish. She said that she was raised with the Bible telling her to love her neighbor and she wondered if I was contradicting that.\n\nI reminded her that the Bible says, \"Love your neighbor as yourself.\" That part that says \"as yourself\" tends to be left off by lots of people. When you think about it, we are called to love ourselves with a good, healthy, positive love, and _then_ we can love our neighbors in the same way.\n\nMany of us reverse that\u2014we offer much more kindness to our neighbors, friends, and family (even strangers) than we do to ourselves. One common example is the way people handle compliments. Have you ever heard someone receive a compliment such as \"Nice blouse\" and instead of saying \"thank you\" she sputters things like: \"Oh, this old rag? It's been hanging in my closet for eight years. I got it at a half-price sale.\"\n\nDo _you_ blow off compliments that way? When we do that we're treating ourselves much more poorly than we would ever treat others. Could you imagine yourself saying the same thing to others that you say to yourself? Would you consider greeting a friend with \"Hi, Mary, I see you're wearing that old rag again. Isn't that the one that's been hanging in your closet for eight years? The one you got at the half-price sale?\" Of course you wouldn't. That would be stupid and rude, yet we put ourselves down in that way all the time.\n\nTo avoid burnout, you have to start with being a little kinder to yourself and treating yourself with good, healthy, positive self-love. You have to put the oxygen mask on you first\u2014or you won't be able to help anyone.\n\nSomehow, some way, you have to first figure out what is important to you. What are your values in life? Then you have to decide what you're willing to do to include those values in your life.\n\nYou don't want to eventually end up getting sick and hearing a doctor tell you that now you _have_ to do something to take care of you.\n\nYou have to be extra vigilant if you have the type of job that makes extreme demands on you, such as:\n\n\u2022 being available 24 \/7\n\n\u2022 having a heavy travel schedule\n\n\u2022 being physically at the workplace ten hours a day or more\n\n\u2022 having responsibility for a large number of direct reports\n\n\u2022 having an unpredictable flow of work\n\n\u2022 handling a huge amount of responsibility\n\n\u2022 having many deadlines\n\n\u2022 being responsible for training and mentoring over and above your regular responsibilities\n\n\u2022 being expected to attend frequent events outside regular work hours\n\nWhen expectations and responsibilities at work are excessive and you can't do anything about it, then you need to make serious adjustments in your personal life.\n\nSometimes it takes creative problem-solving to find ways to add enjoyment to your life. Mario, a security guard, loved to sing in a chorus in high school, but after graduation, his job and working on his fixer-upper house took up all his time. Although he loved his life, one day he realized he was bored. He couldn't afford to change jobs, so he decided to search for a way to add some joy to his life outside of work.\n\nMario looked around for a chorus to join and discovered the answer was right in front of him\u2014every Sunday morning. He joined his church choir and once again is delighted to be part of the glorious harmonies of a singing group. He says his choir has lifted his spirits and given him a new zest for life.\n\nThe word \"recreation\" breaks down to \"re-create.\" Do you have any form of recreation in your life which leaves you feeling \"recreated\" physically, mentally, emotionally, or spiritually?\n\n**SEVENTEEN IDEAS FOR INCORPORATING  \nYOUR VALUES INTO YOUR LIFE**\n\nMaybe none of these will grab you, but the hope is that they will open your mind and your heart to starting a new, healthier, more balanced, more tranquil, less stressed way of life.\n\n1. Sign up for a class.\n\n2. Laugh.\n\n3. Make one choice to improve your health; ask a friend to be your \"accountability buddy.\"\n\n4. Spend time with a small child (consider borrowing one; the parents might appreciate the break).\n\n5. Call a friend long-distance.\n\n6. Ask for advice in starting a financial plan.\n\n7. Watch a sunrise or sunset.\n\n8. Make love.\n\n9. Read a book for no practical purpose but enjoyment.\n\n10. If quiet time is a value, figure out how you want to spend that quiet time (journaling? walking in nature?).\n\n11. Ask someone you care about to introduce you to an activity that they love. This even works if that person is a little child.\n\n12. Have lunch with a good friend or someone you'd like to know better.\n\n13. Give a hug.\n\n14. Receive a hug.\n\n15. Make a list of your favorite activities (you might find clues from what you loved to do in your childhood).\n\n16. Listen to your favorite music while doing nothing else for twenty minutes.\n\n17. Do something you love to do.\n\nDuring a dinner with friends, Eva shared that she remembers the delight of coloring books when she was little and it's reignited a whole new passion for her. A fellow asked her, \"What are you doing? Taking painting or sketching classes?\"\n\n\"No,\" said Eva, \"I've finished four coloring books and have started a new one.\"\n\nMaybe what you decide to add to your life might seem silly or childish to others, but why let that bother you? This is your self-care, your stress buster, your health, and your life. Forget what people might say or think, and take charge of starting to manage your stress.\n\nBut sometimes what we think is an important part of our lives turns out to be just a habit that does not reinvigorate us at all. Sam, a banker, was always tired, yet he claimed that he did take care of himself\u2014his self-care plan was watching TV every night, because it relaxed him. Yet when his wife, Sheila, asked him a few questions, it turned out that there were very few shows that \"sparked\" him, that he was enthused about.\n\nIn fact, he felt that most of the shows were dumb or boring, but TV watching was his habit and he felt it would be a hard one to break. Sheila asked him if he realized that boredom is as exhausting as digging ditches. He said, \"Now that you say that, I gotta admit, I agree.\" She asked what he loved to do as a kid or what he would love to do now.\n\nHe said, \"The answer to both is build models. But when I get home from work I'm just too tired to do that.\" However, he asked his family to buy him a ship model for his birthday, and he surprised himself by starting to work on it a few days later.\n\n\"You're right,\" he told Sheila. \"It gets my motor running again. I worked on the ship again last night and today I feel alive. And guess what. I still watched TV while I worked on it.\"\n\nLight Up Your Life\n\nPicture a continuum\u2014a line with \"burned out\" at one end and \"unlit\" at the other. At the burned-out end, you have way too much to do, too many responsibilities, too many commitments.\n\nThe opposite of burnout is being uninspired. You wake up in the morning and there's no reason to get out of bed or to get dressed. There's no meaning in your life, no purpose. And you are bored.\n\nIn a seminar, a woman talked of being exhausted all the time. \"Could you be bored?\" I asked.\n\n\"No way,\" she barked, and then she whipped out from her purse her long list of things to do that day.\n\n\"Is there anything you enjoy on there?\" I persisted. She told the whole group that no, there was nothing on that list that she liked to do. Day after day, she spent hours doing nothing but drudgery or hated chores. Well, that's boredom.\n\nIf you're at either extreme, the result is exhaustion. Bone-numbing, lean-against-the-wall-and-you'll-fall-asleep exhaustion. Either way, you are tired and don't care about much. Your mission is to find balance, the spot on that line where you're not uninspired, where there's enough going on in your life that it has meaning and purpose, but you're not so busy that you are sleep deprived or dragging around weary to the core of your being\u2014burned out.\n\nThat balance spot on the line between burned out and unlit will vary from day to day, and your center of balance will be different from other people's. What's right for one person might be wrong for you. In one of my seminars, Maya explained that her mother would look at her spot of balance and say, \"Too much going on; too many activities. I would burn out.\" Yet her daughter is the opposite. She wants more activities, her mother wants less.\n\nYou have to find your own balance spot.\n\nHidden Benefits of Volunteering\n\nJill, a corporate executive, found herself obsessing about the plight of the homeless and the poor in Africa, then eventually the plight of the world. During a conversation with Hannah, a longtime friend, Jill shared that not a day goes by that she doesn't feel anxiety and depression over these concerns.\n\nHannah commented that despite the fact that Jill is working in a corporate world that can tend to be self-centered, money grubbing, and uncaring, Jill has always valued helping others. Hannah said, \"Jill, anytime a person frets or worries about world problems, the solution is for them to do something, no matter how small, to make this a better world.\"\n\nJill countered that she didn't have time, and besides, through her business, she was making this a better world in many ways, such as aiding the economy and providing top-quality products.\n\nHannah said, \"Yes, and you're getting paid very, very well for it. I'm suggesting you volunteer some time for a worthy cause and perhaps start regularly donating money to a charity that means something to you. Honest, Jill, you'll be amazed at how your anxiety will melt away once you start giving back.\"\n\nJill brushed aside her friend's suggestions because she absolutely had no spare time and really didn't agree with Hannah's theory, so she changed the subject.\n\nThe following week Henry, a co-worker, sent a group e-mail quoting a book called _Margin_ by Richard A. Swenson, M.D.: \"A University of Michigan study followed 2,700 people for over a decade to see how their social relationships affected their health and well-being. Those who performed regular volunteer work showed dramatically increased life expectancy. Men not involved in such altruistic activity had two-and-one-half times the morbidity during the period studied than those who volunteered at least once a week.\"\n\nAt the end of this e-mail, Henry invited everyone to join him that weekend building a house with Habitat for Humanity. Jill joined the group. By Sunday night, she was aching and exhausted but she felt that she had just made a great contribution to the world. And she understood Hannah's thinking. If you are upset with the world, do something to help it.\n\nWithin months, Jill had a new routine. She can't afford to give whole weekends to building houses, so she volunteers at a homeless shelter two evenings per month, gives time to Habitat for Humanity once in a while, and joined Women for Women, through which she donates a certain amount of money every month to support a woman in Nigeria. She exchanges letters and photos with her \"foster sister\" and is learning about positive programs being offered to women in third-world countries.\n\nThe odd thing is, even though she is learning more and more about difficulties among less-fortunate people, she no longer experiences anxiety and depression about the world. She feels terrific.\n\nJill was able to include her values into her lifestyle as a busy corporate executive. What happens if _you_ can't? Sometimes circumstances simply will not allow us\u2014for now\u2014to do that. That's OK. It is still beneficial to explore, identify, and clarify your values.\n\nLike a single mom who values painting landscapes but cannot carve out the time for it, or a husband and wife who cherish time together but spend every waking moment either at work or caring for elderly parents, what is important to you is sometimes temporarily unavailable. But as long as those values are clear in their minds, the mom might find a minute or two occasionally to sketch a beautiful scene and the couple might sneak in a few magic moments together.\n\nWilliam Butler Yeats wrote a poem when he was standing on the gray pavements of London; he reflected on how much he loved and valued the lake island of Innisfree in his homeland of Ireland. He missed the simplicity and peaceful, quiet life, and although he wasn't in a position to return to Innisfree, he went there in his mind.\n\n**The Lake Isle of Innisfree**   \nBY WILLIAM BUTLER YEATS\n\n_I will arise and go now, and go to Innisfree,_\n\n_And a small cabin build there, of clay and wattles_ _made;_\n\n_Nine bean rows will I have there, a hive for the honey_ _bee,_\n\n_And live alone in the bee-loud glade._\n\n_And I shall have some peace there, for peace comes_ _dropping slow,_\n\n_Dropping from the veils of the morning to where the_ _cricket sings;_\n\n_There midnight's all a glimmer, and noon a purple_ _glow,_\n\n_And evening full of the linnet's wings._\n\n_I will arise and go now, for always night and day_\n\n_I hear lake water lapping with low sounds by the_ _shore;_\n\n_While I stand on the roadway, or on the pavements_ _gray,_\n\n_I hear it in the deep heart's core._\n\nThat's really what we are searching for in this chapter. What brings peace and serenity to your deep heart's core? Your children or grandchildren? Your pets? Music, art, learning? Something spiritual, intellectual, emotional, or physical?\n\nWe might not always be able to access what we love when we need it, but there's no reason why you can't just experience it in your mind as Yeats did. You might be surprised at how visualizing a peaceful setting can be as soothing as actually being there.\n\nTHOUGHTS TO PONDER\n\n_If you want to be successful, it's just this simple. Know_ _what you are doing. Love what you are doing. And_ _believe in what you are doing._\n\n\u2014WILL ROGERS\n\n_There is only one success\u2014to be able to spend your life_ _in your own way._\n\n\u2014CHRISTOPHER MORLEY\n\n_Life isn't a matter of milestones but of moments._\n\n\u2014ROSE FITZGERALD KENNEDY\n\n_Things that matter most must never be at the mercy of_ _things that matter least._\n\n\u2014JOHANN WOLFGANG VON GOETHE\n\n_I firmly believe the Universe dreams a bigger dream for_ _you than you can dream for yourself . . . You've got to_ _open yourself to the dream that the Universe has for_ _you. You've got to discover your true calling._\n\n\u2014OPRAH WINFREY\n\n_The only ones among us who will be truly happy are_ _those who have sought and found how to serve._\n\n\u2014ALBERT SCHWEITZER\n\n_To be happy at home is the ultimate result of all_ _ambition._\n\n\u2014SAMUEL JOHNSON\n\n_Humankind has not woven the web of life._   \n_We are but one thread within it._   \n_Whatever we do to the web, we do to ourselves._   \n_All things are bound together._   \n_All things connect._\n\n\u2014CHIEF SEATTLE\n\n_Only you can be yourself. No one else is qualified for the_ _job._\n\n\u2014ANONYMOUS\n\n_The bitterest tears shed over graves are for words left_ _unsaid and deeds left undone._\n\n\u2014HARRIET BEECHER STOWE\n\n_You cannot live a perfect day without doing something_ _for someone who will never be able to repay you._\n\n\u2014JOHN WOODEN\n\n_I have found that if you love life, life will love you back._\n\n\u2014ARTHUR RUBINSTEIN\n\n_The game of life is the game of boomerangs. Our_ _thoughts, deeds, and words return to us sooner or later,_ _with astounding accuracy._\n\n\u2014FLORENCE SCOVEL SHINN\n\n_One's philosophy is not best expressed in words; it is_ _expressed in the choices one makes . . . The process_ _never ends until we die. And the choices we make are_ _ultimately our responsibility._\n\n\u2014ELEANOR ROOSEVELT\n\n_Love people and use things_   \n_Not the other way around._\n\n\u2014ANONYMOUS\n\nEXTRA CREDIT\n\n**_Questions About Values to Ponder_ **\n\n1. Is all your time being spent on unimportant stuff? Are you racing through life chasing things you don't value or care about?\n\n2. List your top values:\n\n3. How much time are you spending on that which is most important to you? How much time are you spending on that which is least important to you?\n\n4. If you are not living according to your values, uncover what is keeping you from doing so. Is it time? Money? Energy? Are expectations holding you back? Whose expectations? (Family? Work? The world? Yours, maybe?)\n\n5. What can you do to start spending some time in an effective, positive, battery-charging manner?\n\n6. Are you putting off happiness? Relaxation? Rest?\n\n7. Are you making yourself miserable by trying to make everybody happy?\n **CHAPTER 3**\n\nTrash Perfectionism\n\nRAINA, A WEDDING PHOTOGRAPHER, WAS chatting with me after hearing one of my keynote presentations and casually mentioned, \"My house is always a mess because I'm a perfectionist.\"\n\nDoesn't that sound backward to you? Wouldn't you think that a perfectionist would have a house that is always immaculate?\n\nBut Raina went on to explain, \"My mother taught me that the right way to wash the kitchen floor is to move the table and chairs aside, and the right way to vacuum is to move every piece of furniture. So if I don't have the time or energy to do it right, I don't do it at all.\"\n\nDo you put off doing something until you have the time and energy to do it perfectly? Then you beat up on yourself and get stressed out because it's still not done? Do you expect yourself to be perfect . . . always?\n\nJulia Cameron, in her book _The Artist's Way: A Spiritual Path_ _to Higher Creativity_ , says, \"Perfectionism is not a quest for the best. It is a pursuit of the worst in ourselves, the part that tells us that nothing we do will ever be good enough.\"\n\nWhen we were in school, we were given rules for becoming a perfect student. Later, there were criteria for being the perfect worker, spouse, partner, and parent. Eventually we become convinced that perfection is a desirable goal.\n\nWell, here's the news flash. Striving for perfection in yourself will cause stress. Perfectionists focus on the flaws and feel agitated over what is not acceptable in their eyes. Not only is this not desirable, perfection is seldom attainable in yourself or others.\n\nIf you expect perfection in your family, friends, co-workers, or others, you will always be disappointed, frustrated, and stressed, because nobody is perfect. And as a by-product, you will make them crazy. When you expect people to be perfect, you cause them to feel inferior, inadequate, and sometimes angry. This not only damages their self-esteem and diminishes their performance, it perpetuates a cycle of frustration that further increases the stress level for all involved.\n\nPeople will make mistakes, let you down, act goofy. You will try to teach someone how to do something \"the right way\" (your way) and they will do it \"the wrong way\" (their way). Life is too short to spend it being stressed and frustrated because others aren't perfect.\n\nIf you are often frustrated beyond belief by the incompetence of others around you or their meanness or lack of caring, you need to release, relax, and let go. Yes, there are many people less competent and less intelligent than you. Yes, there are many who are cruel or don't care as much or about the same things as you. But isn't it also true that there are many who are more competent and more intelligent than you, and who are kinder, more caring, and more giving than you? Accepting people's different levels of accomplishment and ways of doing things can go a long way in reducing your stress level.\n\nPerfection Is Not the Measure of Success\n\nStrive for excellence instead of perfection. Excellence is achievable; perfection seldom is. A manufacturer's rep once asserted in a seminar that the product she represented was a perfect achievement. Another member of the seminar asked her if the manufacturer had a research and development department. The answer was yes, and the fellow replied, \"Well, then, that department is working to improve that product. Therefore, even they don't consider it to be perfect. Right?\"\n\nPursuing excellence will allow you to continue to strive for success (however you may define it) without stressing over what is beyond your control.\n\nYeprem, an artist, saw a wooden plaque at a craft show and bought it. He meditates on this plaque's inscription at least once a week:\n\n**WHAT IS SUCCESS?**\n\n**_Setting Goals_ **\n\n**But not in concrete**\n\n**_Staying Focused_ **\n\n**But turning aside to help someone**\n\n**_Following a Plan_ **\n\n**But remaining flexible**\n\n**_Moving Ahead_ **\n\n**But not too fast to smell the flowers**\n\n**_Taking a Bow_ **\n\n**But applauding those who had a part in your success**\n\nOn her TV talk show, Oprah Winfrey was interviewing Colin Cowie, who she called a \"master party planner.\" They were showing some of the spectacular weddings that he had put together, and Oprah asked Colin what he replies when a bride says, \"I want the perfect wedding.\"\n\nWithout hesitation, Colin said, \"You came to the wrong man. I don't do perfect. I can't do perfect but I will give you a piece of my heart, a piece of my soul, and an inch of my hairline. Your best in life is really good enough.\"\n\nAre your expectations ridiculously high for activities or events you've planned? Roger, a plant manager, kept assuring his daughter as her wedding day drew near, \"Everything's going to be perfect.\" He was generating an expectation that was a setup for disaster. On the day of the wedding, every little minor thing that went wrong turned into a major disappointment\u2014not just for his daughter, but for his whole family, because they were expecting perfection.\n\nOn the other hand\u2014in another family\u2014when Gina threw a bridal shower for her kid sister, Angelina, she asked everyone to write out their favorite wedding \"horror story\" and she assembled them all into a notebook. The room rocked with laughter as they related stories of cakes falling, fights, police being called, rain, snow, missing wedding rings, flower girls' giggle fits, and ring bearers' \"potty needs.\"\n\nOn the day of Angelina's wedding, when her flowers couldn't be found, she sighed and said, \"I haven't even walked down the aisle yet, and already I have my story to tell.\" People in the family and wedding party were slightly upset, but not nearly as stressed as the family of Roger.\n\nYou may have memories of a perfect holiday or party or vacation, but if you could rerun that event, you probably would see that not every moment was ideal\u2014and the event was still memorable and successful.\n\nDon't Suffer in Perfect Silence\n\nWe sometimes become overwhelmed because we are too concerned about being thought of as the perfect employee or volunteer when something _less_ self-sacrificing would be completely acceptable. Here's a charming tale about St. Patrick and a king named Angus that can illustrate this point.\n\nAt the Rock of Cashel in County Tipperary, Ireland, they say that Patrick was the bishop officiating at the coronation ceremony of Angus, around the year 440 A.D. People had traveled for days, coming from miles away, to sit on the grass and experience this important day for their new king, who they were so fond of.\n\nThe opening of the ceremony called for Bishop Patrick to lift his huge shepherd's staff and thump it into the ground three times.\n\n_Thump! Thump! Thump!_\n\nWith the third thump, the bishop's staff accidentally pierced right through the foot and sandal of Angus. Bishop Patrick didn't notice it, and Angus never let out a sound, but he was suffering pain like he had never imagined. His foot was pinned to the ground, and every nerve in his body throbbed in agony.\n\nAt the close of the ceremony\u2014 _three and a half hours later_ \u2014 poor Angus was supposed to kneel, kiss the bishop's ring, and then turn to greet his followers for the first time officially as their king. In excruciating torment and wracked with pain, Angus leaned forward and whispered, \"I can't kneel; I can't turn around.\"\n\nShocked, the bishop asked, \"Why not?\"\n\nAngus looked down at his foot, which was now swollen and had turned black, yellow, and blue. Horrified at seeing the pool of blood and the wounded, discolored foot of the king, Patrick asked, \"Why didn't you tell me?\"\n\nAngus softly replied, \"I thought it was part of the ceremony.\"\n\nYou may be silently enduring difficult working conditions, abusive relationships, or an excessive workload because you \"thought it was part of the ceremony.\" In some instances, your supervisor (or the head of your committee) might not be aware of how overburdened you are. She might say, \"I wish I knew what you were going through. Why didn't you say something?\"\n\nIf you're suffering because you have too much to do at work, at home, or in volunteer activities, speak up. Don't suffer in silence, telling yourself, \"I thought this was how it was supposed to be.\"\n\nKeeping Up the Appearance of Busyness\n\nHow many hours per day do you put in at work? Are you exhausted and frazzled because you can never break away from the phone, e-mail, texting, and BlackBerry?\n\nThe problem is not your lack of time, it is how you use that time. What matters is not how much you work but what you accomplish. And sometimes our expectation of ourselves isn't to be productive, it is to look busy. This can be true even when nobody sees it but you.\n\nShift your focus to being more productive. Start by prioritizing your work. Be selective about the phone calls you'll accept when you are away from work and about how much work you'll do at home. Business guru Peter Drucker wrote, \"Nothing is less productive than to make more efficient what should not be done at all.\"\n\nSometimes people knock themselves out working extra hours because they truly believe that their boss or the company expects it of them. But nobody expects you to work 24\/7. You are simply living up to your own expectations\u2014trying to be perfect.\n\nSue, a credit union supervisor, said in a stress management class that she took great pride in not having taken a lunch break in the eight years she had been in her position. When the group questioned her, she explained she felt the company probably expected that of her. After further questions, she finally admitted that the other supervisors did take lunch breaks and had given up inviting Sue to join them.\n\nAfter a brief class discussion about stress and expectations, Sue made a resolution that before our next class meeting, she would tear herself away from her desk at lunchtime. The following week she reported back her \"shocking news\" that she went out to lunch and no one cared that she was away from her desk. Nobody commented or questioned her about it. She sheepishly admitted that she \"sort of\" had an image that her staff and the board of directors all admired her and appreciated her dedication. Sue said, \"All these years of not taking breaks\u2014no appreciation, no compensation. Nobody even noticed. And when I asked myself why I made that decision for eight years, I honestly didn't have an answer. I was just doing what I thought a perfect supervisor was supposed to do. But I don't know where I thought that 'rule' came from. I wonder if\u2014deep down, subconsciously\u2014I thought the place would fall apart without me if I left during lunchtime.\"\n\nDo you relate? Have you ever gone to work despite being sick as a dog (and watched the smiles of appreciation as you shared your germs with everyone)? Or missed an important social or family event because work interfered?\n\nDo you believe deep down that your place of business will fall apart if you stop doing what you are doing? Or that maybe your family will fall apart, or your religious or civic association cannot manage unless you plan and work tirelessly for it?\n\nUnrealistic expectations can keep you awake at night thinking about what you didn't do today and still have to do tomorrow. They can push you to exhaustion. Expectations of yourself can cause so much stress that you get sick or depressed.\n\n**WHAT HAS CONTROL OVER YOUR LIFE  \nTHAT MIGHT CAUSE YOU STRESS?**\n\npriorities of others\n\ndemands of others\n\ne-mails a\n\nmanipulative relative\n\njob\n\nboss\n\nlimiting beliefs\n\nfears\n\nprocrastination\n\naddictions\n\nThe Control Freak and the Fervent Fixer\n\nOne of the biggest causes of stress and burnout is trying to control an inflexible environment or the people in it. When you try to control something or someone that you really don't have _any_ control over, the result is always stress for you and sometimes stress for others. It's called being a Control Freak. When you finally realize, understand, and admit to yourself that you have no control over that person, place, or thing, you let go. You let go of control. You let go of the pain of unrealistic expectations of perfection. You let go of stress.\n\nA variation on the Control Freak is the Fervent Fixer, the person who sees a problem and is determined to fix it\u2014whether people want it fixed or not. It is common for a woman who is describing a problem at work to her boyfriend or husband to become frustrated if her mate keeps trying to offer solutions. The woman wants to discuss the process of what is going on, but the man is a Fervent Fixer and wants to solve the problem. Often an argument (or stress) results.\n\nOf course, when Fervent Fixers see a problem they can solve, life is good. But if they expect perfection of themselves and want to fix everything, it's inevitable that sometimes they'll come upon an unfixable problem.\n\nThe Fizzle Factor\n\nMichael, a twenty-seven-year old financial adviser, was helping to set up for an all-day seminar he was going to co-lead with his father. Suddenly his dad realized he was missing some material that accounted for two hours of the seven-hour program. He started a frantic search and Michael, looking concerned, walked over and asked if he could help.\n\nHis dad said no and Michael calmly walked away. I asked him if any of us could offer to help and Michael said, \"No, wait till he fizzles.\"\n\nEveryone in the room stopped what they were doing and all eyes were on Michael.\n\n\"Fizzles?\" a woman asked.\n\nHe said, \"Yeah, you know. Don't you have times where you are so frantic looking for something that if someone gets in your way, you practically knock them over? Or if they say something to you and interrupt your thoughts, you practically bite their head off? Well, at those times, I just keep out of the way and wait till they fizzle. Heck, I don't want to be knocked over or have my head chewed off.\"\n\nOne of the men commented, \"You know, I think that's what we need to do when our kids throw a tantrum or get all wound up over something. I always try to be the perfect dad and step in to fix things\u2014you know, calm them down. And usually I just make matters worse, but could never think of what else to do.\n\n\"I should just get out of the way till they fizzle.\"\n\nInstead of rushing in and trying to make everything perfect by fixing things with someone who is frantic or upset, simply offer help, let them know you are available if they need your help or support, and then get out of the way until they're ready to accept your assistance.\n\nDon't Fall for the Myth of \"Fairy-Tale People\"\n\nDid you ever think someone had a perfect relationship, job, partner, home, situation, life? Or have you ever thought, \"If I had what that person has, my life would be perfect and I would never have any problems, worries, or sadness\"? Then, when you got to know her better, you discovered that she has the same aches and pains, ups and downs, joys and frustrations, griefs, struggles, and trials as you and everyone else?\n\nFairy tales program our minds to believe that if certain things happen, then people get to live happily ever after. But no matter what we do or how hard we work, we never achieve the perfection that we believe others have.\n\nPeople who are obsessed with how they are perceived want the world to think their life circumstances are perfect and they work hard to create that illusion. They are likely to respond to situations that could tarnish their image by saying things like, \"I hope your grandmother never finds out you have a tattoo\" or \"How could you call me that in front of people?\" (as if it's just fine to call them that when you're alone) or \"Don't ever tell anyone this happened, OK?\" or \"You told your teacher _what_!!?\"\n\nBarb, an options broker, described how she and two friends were discussing some holiday letters she had received from people who listed all the activities their children were involved with. During the conversation, one of her friends blurted out, \"They're Fairy-Tale People! They're people who are always trying to convince you\u2014in holiday letters or not\u2014that they have perfect children, a perfect marriage or family, or are perfect parents. They're trying to convince you that they are living happily ever after\u2014 like they're in a fairy tale!\"\n\nBarb added, \"Maybe we're all that way. On one vacation we had several arguments, fights, disagreements, whatever you want to call them. But when people asked how was our vacation, I always said it was great. Why couldn't I just be honest?\"\n\nThe three women decided together that in addition to those who _claim_ to be perfect, those who _strive_ to be perfect or to have a perfect life also are Fairy-Tale People. They are trying to live happily ever after in an imperfect world\u2014they're headed for disappointment and heartache. After that discussion, Barb made some major changes. She started by changing her expectations of herself and her family.\n\nAre you starting to see the power that perfectionism has over us? How much of an impact your expectations of perfection have on creating stress in your life? And the fairy tales don't stop with you or those around you. You might find yourself becoming stressed when you convince yourself that this house or apartment is going to be perfect, or this job, boss, location, situation\u2014anything.\n\nIt's fairy-tale thinking that creates the \"honeymoon's over\" type of stress. You know that feeling when you start to see the flaw in something for the first time? When that new wonderful coworker or friend betrays you. When those sensational new shoes become furiously uncomfortable after twenty minutes of wearing them. When your perfect new software acts like a cranky kid with a mind of its own and simply refuses to do something you really really want it to do. When your dream vacation is rained out day after day. When you stumble across a flaw in anything, you need to remind yourself that there is an upside and a downside to every single thing and situation in life. And most things are not perfect.\n\n**NINETEEN WAYS TO STRESS OUT A PERFECTIONIST**   \n**TO THE POINT OF MADNESS**\n\n1. Put empty cartons and bottles back into the fridge.\n\n2. If someone is telling a joke and you've heard it before, wait until they're nearly finished and then shout out the punch line.\n\n3. Take every opportunity to give the perfectionist advice. Especially on subjects you know nothing about.\n\n4. Squeeze the toothpaste tube from the middle. Never replace the cap.\n\n5. Always be late.\n\n6. Go to the movies with your perfectionist and just keep chatting.\n\n7. When people gather at your perfectionist's place, constantly bring the conversation around to sex, politics, or religion. Preferably all three.\n\n8. If your perfectionist is single, frequently ask with a big smile, \"So when are you getting married?\" And, of course, if they are newly married, you've got to ask, \"So when is the baby coming?\"\n\n9. If your perfectionist does have a child, give a sweet little puppy as a birthday gift . . . or a kitten.\n\n10. Borrow money often from your perfectionist and forget to pay it back.\n\n11. Recognize your limitations. Then ignore them.\n\n12. Recognize your perfectionist's limitations. Then tell him or her what they are.\n\n13. Never return anything you borrow.\n\n14. Give your perfectionist a double espresso just before bedtime.\n\n15. Switch around the decaffeinated and caffeinated coffee every chance you get.\n\n16. Turn on and turn up all the appliances in any room you are in. Leave them on.\n\n17. Keep a photo of your past lover where your current lover is bound to see it.\n\n18. Constantly raise the hopes of your perfectionist that you are going to mend your ways and improve. Then take every opportunity to dash those hopes.\n\n19. Always make sure you have the last word.\n\nWhere Do Your Expectations Come From?\n\nThere isn't one clear source for where your expectations of perfection come from. These \"rules\" that are rattling around in your brain could come from your childhood, a comment you overheard at work, your best friend's pet peeve, or even television.\n\nPay attention to how holidays are portrayed on TV. Everyone is so very, very happy, the decorations are . . . well . . . perfect, and the time spent together is so very, very enjoyable. We are smart enough to know that is unrealistic, but on the other hand, isn't that a principle of brainwashing? Keep repeating something over and over until people start to believe it, and often the result is we expect family, meals, decorations, weather\u2014even the poor turkey\u2014to be perfect.\n\nSo when your holiday is less than perfect\u2014Auntie had a bit too much to drink, your uncle is arguing with everyone, the kids are running through the house screaming at the top of their lungs with your pitted olives poked on their fingertips, something navy blue is all over the carpet, the marshmallows on the sweet potatoes are golden black\u2014do you find yourself wondering \"Where did I go wrong?\"\n\nReal life seldom reflects what we've been watching on TV or those fantasy memories of celebrations from our childhood.\n\nTHOUGHTS TO PONDER\n\n_Perfectionism is the voice of the oppressor, the enemy of_ _the people. It will keep you cramped and insane your_ _whole life._\n\n\u2014ANNE LAMOTT\n\n_If everything's under control, you're going too slow._\n\n\u2014MARIO ANDRETTI\n\n_I will tell you that there have been no failures in my life._   \n_I don't want to sound like some metaphysical queen, but  \nthere have been no failures. There have been some tremendous lessons._\n\n\u2014OPRAH WINFREY\n\n_I wish someone would tell us from the beginning that_ _we are dying. Then we wouldn't be afraid to really live._\n\n\u2014MICHAEL LANDON\n\n_Life's journey is not to arrive at the grave safely in a_ _well-preserved body, but rather to skid in sideways,_ _totally worn out, shouting, \"Woo hoo . . . what a ride!\"_\n\n\u2014ANONYMOUS\n\n_Unless you're willing to have a go, fail miserably, and_ _have another go, success won't happen._\n\n\u2014PHILLIP ADAMS\n\n_The thing that is really hard, and really amazing, is_ _giving up on being perfect and beginning the work of_ _becoming yourself._\n\n\u2014ANNA QUINDLEN\n\n_Being happy doesn't mean everything's perfect. It means_ _you've decided to see beyond the imperfections!_\n\n\u2014HELEN KELLER\n\nEXTRA CREDIT\n\n1. Who's in control in your life?\n\n2. What are you trying to control that's uncontrollable?\n\n3. What is within your control?\n\n4. What controls you (phone, schedule, computer, demands of others, e-mails)?\n\n5. What impossible standards do you strive for (perfection, never making a mistake)?\n\n6. Think of a few times you've been very upset over a person or event. List them.\n\n7. Were you upset because you expected perfection?\n **CHAPTER 4**\n\nReach for Realistic Goals That Are Both Relevant and Rewarding\n\nWHEN THE DOCTOR TOLD EILEEN that her mother's Alzheimer's disease had progressed to the point where she had to be put in a nursing home, Eileen had a clear goal in mind. She was going to take a leave of absence and give her mother the very best care possible, and prove to the professionals that her mother did not need to go into a nursing home. After all, if her mother could manage to raise and care for five kids, certainly one of those kids would be able to care for her. Eileen even wrote out her plan for giving care to her mom. It was a carefully thought-out goal\u2014one that she could carry out on her own, without help from anyone.\n\nThe problem was, it was an unrealistic goal. Five weeks later, Eileen's blood pressure was soaring, she was battling acid reflux, and then she landed in bed with a serious case of the flu. The doctor said the reason it was taking her so long to recover was that stress and fatigue had weakened her immune system. And he suspected that her anger with her four siblings also contributed to her illness. Eileen was furious with them because all they ever did was squeeze in a few quick visits per week.\n\nEileen shared with the doctor her goal of being the sole, superior caregiver for her mom and explained it wasn't her nature to ask for help. But the doctor pointed out that if Eileen didn't start to take care of her stress, she might get seriously ill and not be at all available to her mother. He strongly urged her to talk to her siblings and ask them to join in with giving care to their mother.\n\nAfter much internal struggle, Eileen set new\u2014more realistic\u2014 goals, including talking to her brothers and sisters about how difficult life had been with their mother and how she was seeking help. All four were shocked. They each described how they thought Eileen was so totally self-sufficient that they didn't think she needed anything. Her big brother said, \"Eileen, how are we supposed to know that you need help if you don't tell us? I'm there for you but you have to spell out what _kind_ of help you need. Then I'm glad to be part of taking care of Mom.\"\n\nThey all wanted to help, and once each of them figured out what they could do\u2014everything from cleaning Eileen's house and yard to staying with Mom so Eileen could have some time off or running errands for her\u2014Eileen's life became simpler. They enrolled their mom in an adult day-care center, but over time, as their mother deteriorated, the five of them agreed\u2014although sadly and reluctantly\u2014that she would best be cared for by professionals in a long-term care facility.\n\nNow Eileen and her siblings visit every day and make certain that their mom is well cared for. They are at peace with their decision and their mother seems to be at peace in her new home.\n\nOften the goal that seems so kind and noble turns out to be unrealistic. Note that Eileen's goal of giving the best of care wasn't the unrealistic part at that stage of her mom's illness; it was trying to do it all on her own that did her in.\n\nSometimes it is your goal that causes stress; sometimes it is merely the way you plan to achieve your goal that frustrates you.\n\n**TIPS FOR CAREGIVERS**\n\n1. Take care of yourself; get enough rest and sleep. This is not a luxury, it's a necessity.\n\n2. Find someone to talk to, perhaps a support group or an understanding friend.\n\n3. Be social; don't become a recluse. Keep up with your friends.\n\n4. Learn to ask for help: www.eldercare.gov is a place to start searching for local agencies such as Meals on Wheels, home health aides, and adult day care facilities. Keep asking different people till you get the help you need.\n\n5. Stay emotionally healthy\u2014change the scene, go for a walk, get fresh air. Don't feel guilty for taking time for yourself. Staying emotionally healthy will make you a better caregiver.\n\n6. Stay physically healthy\u2014eat a balanced diet and get some exercise. Consider learning yoga for mental, physical, and spiritual health.\n\n7. Keep laughter and music in your life. When tensions run high and energy runs low, listening to uplifting music, watching a comedy, or simply seeing the absurdity of life can keep you sane.\n\n8. Find a way to express yourself: Journaling can be a great release for frustration.\n\n9. Have a life outside the caretaker role: Take time off to keep up with hobbies and interests or do something you enjoy.\n\n10. Appreciate the good moments.\n\n11. Search for helpful resources such as the Alzheimer's Association's Helpline (800-272-3900).\n\n12. Treat yourself to something occasionally. You deserve it.\n\nPositive and Negative Goals\n\nDid you ever set a goal and wonder why nothing ever came of it? For example, sometimes people make New Year's resolutions to become happier or lose weight or increase their self-confidence. Or perhaps you have more specific goals, such as starting a new business, having weekly meetings with a colleague, or getting your mother-in-law to like you, but still nothing happens. Many people think that any goal they set is better than nothing, that all goals are positive. Not true.\n\nA goal can be negative or unrealistic if . . .\n\n\u2022 the action is out of your control. You have control over reaching out to your mother-in-law or being friendlier, but you cannot control whether she likes you. Some people will never like you no matter _how_ kind you are to them. You may want to have a meeting with someone, but if she has the control over saying yes or no, you control only the option of inviting her. So more positive goals would be for you to invite your colleague to a meeting and to act in a different way with your mother-in-law.\n\n\u2022 the action is not specific or there's no time limit. If you say, \"I will start a new business,\" that's too vague. You must outline the steps you need to take and put them in sequence. Break the action down into incremental stages and then follow your plan. For example, your first positive, specific goal might be: By March 17, I will sign up for an evening accounting class and will have called a real estate agent to help me find the right location. If your goal is to lose weight, specify how many pounds by a certain date. Otherwise, the criteria for success are too vague. Losing one pound in a month is not much of a goal, but aiming for twenty pounds in two weeks is totally unrealistic.\n\n\u2022 the outcome is not measurable. If becoming more confident is your goal, how will you know when you've achieved it? If you confidently say no to a telemarketer? For \"becoming more confident\" to be a measurable, positive goal, you have to define what behavior would be an example of your having gained confidence. Do you want to call someone and ask him to attend a party with you? Do you want to be able to give a presentation to your co-workers? Do you want to go for a job interview without having major anxiety attacks or collapsing in a heap outside the interviewer's door? Add a date to that behavior and you have a positive, measurable goal. \"I will say yes to giving a talk at our annual meeting in June,\" or \"By January 15, I will volunteer to serve on the board of my association.\"\n\nOnce you learn to reach for specific and measurable goals that are within your control, you will no longer be setting negative goals. You'll be reaching for positive goals that are realistic, relevant, and rewarding.\n\nPeople tend to assign goals to most activities, but they do it unconsciously without realizing that they are actually goal setting. Reggie talked of attending a chamber of commerce meeting with the hope (goal) of meeting fellow leaders in his new community. Many at his table had the same desire (goal). But the meeting planner's agenda (goal) was to pack as much information into as many speeches as possible. As a result there was no time for networking, so the attendees tried to carry on conversations during the program, which was extremely distracting to the speakers.\n\nIn the end, everyone was stressed\u2014the attendees, who didn't get a chance to meet each other; the speakers, who felt they didn't give their best presentation due to the distracting conversations; and the meeting planner, who set a goal that contradicted the goals of the members.\n\nRecognizing that we have goals\u2014conscious or subconscious\u2014 for everything we do can help us eliminate many of our negative goals.\n\nGoal Stoppers\n\nEmmett's Law of Goal Setting: As soon as you set a goal that is important to you, equally important obstacles start popping up all over the place.\n\nHas something like this ever happened to you: You decide with all your heart that you are going to lose weight and then a loving friend gives you a box of your favorite candy or some homemade fudge? Or you're struggling to quit smoking and your pal brings you three cartons of your brand because he was in another state where they sold cigarettes dirt cheap? I hear stories like this all the time. Sometimes we wonder if others are trying to sabotage our goals, but usually it's just that a person didn't have a clue that we were determined to finally make that change.\n\nI call these crazy incidents \"goal stoppers.\" They are not the same as phony excuses and they are not caused by us. They come from outside our control.\n\nThere are many explanations and theories about why this happens, but the important thing is to recognize when you are in a goal-stopper situation and figure out how to get around it.\n\nGoal stoppers almost torpedoed me when I thought maybe I'd start writing my first book, _The Procrastinator's Handbook_. First came some family illnesses, and then I was in a serious auto accident, followed by even more painful occurrences. In a three-month period, my mother passed away, my father-in-law passed away, and my twenty-five-year marriage ended.\n\nWriting became the last thing on my mind. Maybe I could have found time to write during this period, but I didn't have the emotional energy to think straight and focus on anything.\n\nYears later, once I regained my balance and again decided to start my masterpiece, my daughter got married and I met a man in the cat food aisle at the grocery store and fell in love (with him, not the cat food). Again\u2014enormous emotions. Even though the emotions were happy, they were goal stoppers. Still no book.\n\nAll of these goal stoppers were real and extremely important to me, not mealy-mouthed excuses. But still\u2014no book. Finally, in 1994, Bruce (the cat-food-aisle guy) and I married. Life was good. More emotions, but what about the book?\n\nWith great determination, I decided to find the time to start writing. That year and the next, two of our sons got married. I finally figured out that life will _always_ have important and powerful ups and downs. So to defeat goal stoppers, I decided to carve out one hour per week to write.\n\nGreat plan, yes? The very first hour of my very first week, the phone rang. I decided to ignore it. And on the answering machine I heard a dear friend sobbing. She had never before in the life of our friendship called in tears. I answered it. As we talked, I tucked away my notes about the book . . . and forgot about them.\n\nA year later I still had not begun writing my book. Later that week, when I was floating on a caffeine high and could not fall asleep, I decided to get up in the middle of the night and put in an hour on my book. And that was the breakthrough. I put in an hour a week for the next four weeks; then I was on my way and the goal stoppers stopped stopping my book.\n\nWhen sabotaging goal stoppers assault you, the first thing to do is recognize what is happening and know that it happens to all of us many times in our lives. The next thing to do is figure out a way to take one small step toward your goal. See if you can find a way to sidestep the goal stoppers (like my writing in the middle of the night). It seems that once you take that one small step, you loosen the hold goal stoppers have on you.\n\nAt the end of this chapter, you will be asked to write down a goal you would like to achieve. Writing down goals is an important habit to form because having the goal in writing helps to remind you of it, so it doesn't fly out of your mind as soon as something else interesting flies in. Also, it helps keep you focused and moving forward with that goal. Many people report a sort of magic happens when they write goals. It's as if events and people seem to appear to help with that goal.\n\nConsider starting a goal notebook. But your written goals don't have to be anything formal or fancy. Writing a goal on a Post-it note and smacking it on your bathroom mirror works wonderfully.\n\nLife Goals Can Counterbalance Job Stress\n\nOften people feel that they have no control over the direction their personal lives will take because they are too busy at work to have enough time and energy to set and strive toward nonwork goals. It never occurs to them that they can reach for life goals that would contribute to reducing their stress.\n\nTo presume that your personal life will go where you want it to go simply by default doesn't make sense. There are times when work goals that increase productivity will start making you crazy, stressed, or even sick. Those are times that you need to focus on nonwork life goals to keep yourself in balance and prevent burnout.\n\nPam, an administrative assistant, was having health problems from the stress of her job. She decided to reach for a goal to add joy and energy in her life. She knew she was never going to find ways to make her job enjoyable or less stressful, and yet she couldn't afford to quit, so she determined to streamline her after-work activities and trade two nights a week of TV for a stress-busting activity.\n\nShe made a list of what she valued in life and decided to join a volleyball team as a temporary activity till she found something to be enthusiastic about. She had loved playing volleyball in high school and felt she needed something physical to get her moving. She also made a commitment to volunteer one night a week at a local homeless shelter. Three years later, she is still with the shelter and still with the volleyball team, and feeling enthusiastic about both. The funny thing is, the team was started by a group of women who realized that just because their jobs were boring or stressful, their whole lives didn't have to be.\n\nPam says she has never been more energetic, healthy, or at ease with herself and life. Also she has lost twenty-seven pounds. If work stresses you, setting goals for your nonwork life can combat that stress and melt it away.\n\nIf you do not reach for goals to direct where your life is going, where will you end up? If you do not have specific goals, you will not have specific results. As you start the habit of reaching for realistic goals, remember that they can apply to all levels of your life. There is no reason why you cannot be reaching for goals for your career and finances, as well as goals for activities, people, and events in your life. You might even pursue a spiritual goal. For example, I often ask people in seminars how they want to be ten years from now and 92 percent have said they want to be more peaceful and serene, yet few people ever set a life goal of including more peace and serenity.\n\n**SIMPLE, RELEVANT, AND REWARDING LIFE GOALS**\n\nLearn to speak another language\n\nHave a rose garden\n\nBecome a youth leader\n\nMeet your favorite athlete\n\nStay at a luxurious hotel or spa\n\nDonate regularly to your favorite charity\n\nJoin a spiritual group\n\nLearn to dance\n\nBuy your grandmother a flat-screen TV\n\nTake a scenic train trip or a cruise\n\nGo kayaking or snorkeling\n\nGet financial advice from a successful millionaire\n\nLearn to paint or take up another art form\n\nBecome a volunteer\n\nTreat your mother to a day at a spa\n\nRead the Bible or another holy text\n\nJoin a health club\n\nStart a business\n\nBuy books for orphaned children\n\nVolunteer with Habitat for Humanity\n\nAttend a ballet\n\nLife Goals and the Good Old-Fashioned Work Ethic\n\nPeople sometimes ask, \"Why do I need to reach for goals in my life to prevent stress? Our grandparents worked hard and they never had to worry about stress, right? We'd all be better off if we had an old-fashioned work ethic like our grandparents. Maybe that's what the expression 'working 24\/7' is bringing us back to.\"\n\nNot true. Historically speaking, an old-fashioned work ethic meant work hard at work. It didn't mean work hard every hour of every day and schedule every minute of your life to be productive.\n\nMany people can remember their grandparents or parents going out on the front porch after dinner and having neighbors and friends wander over to chat. If there was a sorrow in the family, this little community on the porch would grieve with you, and if there was a joy, they would celebrate with you.\n\nJohn, a loan appraiser, talked about his grandparents playing cards with friends every Saturday night. He reassured us that his grandparents were very hard workers, but they didn't work seven days a week, and they always had time for family and friends. In spite of their goals of working hard and accomplishing a lot, they knew how to recharge their batteries on a regular basis.\n\nLife in days of yore was lived to a different rhythm. Here's a great example of that difference. During the mid-1800s, a grandmother wrote out a list for her soon-to-be-married granddaughter, instructing her in seventeen steps how to do the laundry. This list was found in someone's attic trunk several years ago and has been reprinted in many magazines. It's a wonderful example of life's wisdom being passed from one generation to the next.\n\nStep number one described how to build a fire in the backyard and make your own soap out of lye in a large cauldron. But the last two steps are what captured my imagination. Step number sixteen told the granddaughter to put the teakettle on the stove and to drag the rocking chair onto the front porch. Step number seventeen said, \"Sip your tea, rock a little, relax and count your blessings.\"\n\nWhat a different rhythm, a different attitude this displays than our generation's feelings toward work. Grandma was saying, \"Oh, yes. Work hard. But when you're finished, part of the job is to relax.\"\n\nThat's certainly not how we approach work, is it? Our rhythm seems to be that we'll work, finish a project, hit the ground running, and jump right into another project. No time ever to \"sip your tea, rock a little, relax and count your blessings.\"\n\nGenerations past have set the example of a good work ethic. After the hard work of a harvest came a feast or some kind of celebration. After a hard week's work, everyone gathered in town for a barn dance. After a hard day's work, the family would relax or play games or tell stories in the kitchen or around the fireplace. We never read about the mother jumping up and asking everyone to \"hold that thought\" while she runs down to the river to do another load of laundry.\n\nA friend of mine has a sign on her desk that says:\n\n**When I work, I work hard**\n\n**When I play, I play hard**\n\n**And when I think, I sleep**\n\nWell, I'm not advocating that last line, but doesn't it make sense to keep work from taking over our leisure time? After doing hard work we need to relax, to \"re-create\" ourselves before we begin the next project in order to head off being steamrolled by stress.\n\nThe old-fashioned work ethic encouraged hard work, but it also nurtured the values of allowing people to rest, relax, and enjoy the company of others, and finding the solitude to read, pray, or think.\n\nWhen Allen, a truck driver, told his teenage daughter Natalie that when he was a youngster, stores were closed on Sundays, she almost keeled over. In a shocked voice she said, \"Yeah, but not the malls. The malls weren't closed, were they?\" When he told her there _were_ no malls when he was a kid, she was astounded. She told her dad she didn't know he was that old. (Allen decided to take that as a compliment.)\n\nWhen Natalie asked what people did on Sundays, he told her that as a kid, he thought Sundays were very boring\u2014the family went to church, read the paper, and relaxed, visited with friends, or had a big family dinner at Grandma's house.\n\n\"Oh, all the things people would love to do these days, but they don't have time,\" Natalie observed.\n\nYet no matter how boring Allen and other children may have thought Sundays were, people connected with friends, neighbors, or family, and they did something different from the rest of the week. Most important, they slowed down. How often have you arrived at work on Monday morning feeling exhausted from having been so busy all weekend?\n\nJulie, a massage therapist, has a goal to keep Sundays free for family time. Her friends all know of Julie's goal, and when they ask to set a date to get together, they work around it.\n\nIf this seems too extreme for you, try setting a goal to mark occasional days on your calendar that are set aside just to relax. You will find that these help you to become more productive in the long run.\n\nWhen Goals Match Your Values\n\nIf you get into the habit of including meaningful goals that are congruent with your values in all areas of your life, you will find yourself sidestepping tons of stress. For example, the family at Disney World with cranky kids whining and throwing tantrums and yelling parents who look frazzled and furious needs to evaluate its goals. Is the goal to get your money's worth and ride every single ride from early morning till they close at night? Or is the goal to have a good time with your children, which could mean taking a midday break and heading back to the motel for a refreshing dip in the pool or a nap, then perhaps returning in the late afternoon when it's a bit cooler and everyone has the energy to stay and enjoy the fireworks? Evaluating your goals can help make the difference between an enjoyable time with your family and a nightmare.\n\nHannah, who read in John Bradshaw's _Homecoming: Reclaiming_ _and Healing Your Inner Child_ the advice to \"surround yourself with nourishing people, not toxic ones,\" thought about the people in her life. Every single one was a negative, complaining, _toxic_ person. No wonder she was stressed out. She wasn't sure what to do, but she knew she had to make a plan (goal). She started with calling Mary Ann, an acquaintance she liked and admired. Hannah explained her situation and asked if they could get together occasionally to either play tennis or go to the movies.\n\nMary Ann wasn't into tennis but said she would love a pal to go to the movies with, so they set a date. Hannah said she felt as if she was in first grade asking someone if she would be her friend, but she was able to summon up the courage to make that call because she passionately wanted to change her life.\n\nA friendship did develop between Hannah and Mary Ann, and both of them often talked about how to distance yourself from toxic people and associate with nourishing people. They knew they were stuck with their families but both belonged to negative, gossipy groups that they decided to drop. Also, they allowed some connections with several individuals to dwindle, and they started a women's group of other like-minded souls who supported and encouraged one another in reaching for their goal of redefining relationships in their lives to battle stress.\n\nSetting Goals for Relationships Can  \nChange Your Whole Life\n\nSeven years ago, Jason and Tiffany were planning their wedding, and as the financial estimates mounted, Tiffany broke into tears from the stress of it all. \"How can we afford all this?\" she sobbed. \"We'll be so far in debt, it will take us a lifetime to save the down payment for a house. I'm so stressed out spending all this money. I feel like a nervous wreck.\"\n\nSo Jason and Tiffany started a series of long discussions. Since the wedding is usually something the bride has been dreaming of, Tiffany had to decide whether a huge wedding was more important to her than buying a house. She and Jason together had to clarify what the purpose of their wedding was. Was it to impress people and to spend a fortune or to provide a fun party to celebrate their commitment to each other? Tiffany recognized that their purpose was to have a day of family, friends, and celebration that they'd always remember, and impressing people and going into debt weren't part of her vision.\n\nAs a result of goal setting, they decided to cut back on their wedding not only to save money but also to make the wedding planning more manageable for Tiffany, who was working full-time and in the middle of taking her nursing board exams. Instead of spending for an extravagant cake, meals, and decor, they chose simpler ways to go, but in the meantime searched the Internet and asked friends and family about rituals and ceremonies to include in their wedding day.\n\nTo this day, everyone remembers the \"richness of joy and memories\" experienced at Jason and Tiffany's wedding: the special remembrances for the family members who had passed away; Tiffany wearing her mother's headpiece and the gown made by her two aunts; the special gifts presented to each set of parents; and the \"flock of flower girls\"\u2014all the nieces, each in her own choice of pastel dress, fluttering up the aisle. Instead of expensive mementos for everyone, each guest received a candle, and when the bride and groom danced for the first time as husband and wife, the lights dimmed and everyone lit their candle and sent blessings upon the couple. Thanks to what they call \"Tiffany's Magic Meltdown\" and the goal setting that resulted, they planned a wedding they loved, didn't overspend, had a great time themselves, and within a year and a half they had their down payment for a house.\n\nIdentifying your goals can help you in making big and little decisions. Hamid and his wife struggled with his decision to attend a conference. The cost of the conference was high, and when added to the cost of airfare, hotel, and meals, it was way more than they could afford, but he felt he would learn so much that he would earn back the conference cost within a month of returning home.\n\nSo Hamid attended the conference, met some terrific people, and attended some great presentations. When he came home, there was no change in his business practices or the success of his business. When his wife asked him what he specifically learned that would change the direction of his business, Hamid was stumped. If he had set realistic goals that were relevant (such as \"improve marketing\" or \"learn to use technology to generate cash flow\") he could have selected specific breakout sessions to gain the much-needed information and perhaps asked some of the new people he met to share their favorite marketing or technology tips.\n\nThinking about and setting goals can keep you from wasting money.\n\nAt a parents club meeting, Amber was talking with Trish about books, and Trish mentioned that her mother-in-law was a fairly successful published author. Amber said she'd love to meet her to ask her some questions about how to get published, but she didn't want to take advantage of her.\n\nTrish said that since their boys were on the same baseball team, she and Amber could sit together during the next game. Even though her mother-in-law is a lawyer with a super-busy schedule, she would be there, too, so Amber could meet her and maybe make an appointment to ask questions about getting published.\n\nAt the game, when introduced to Trish's mother-in-law, Mrs. Clancy, Amber was astonished to recognize her. She sputtered, \"Mrs. Clancy, haven't I seen you at every game this season? You practice law; you write books; how do you find the time to attend these games every week?\"\n\nMrs. Clancy told her, \"Well, I know what my values are and one of the top ones is family. So one of my goals is to make sure I spend time with them. This is a great way to do it. I get to cheer on Noah and his team, I get to sit and chat with the rest of the family, and afterward we all go out for pizza. Plus I get fresh air and sunshine. What could be a better use of my Saturday?\"\n\nTrish later said of her mother-in-law, \"I'm astonished how much she accomplishes, but I know she writes out her goals and has a clear vision of what she wants in life. And many times I've seen her say no to something that would take her focus off what's important to her.\"\n\nFootprints in the Snow\n\nOnce you clarify your values and then set goals to move toward the life you want, you need to write down your goals to help you keep your eyes on them. Mrs. Clancy often tells a story of three little kids who made a bet to see who could walk the straightest line from where they stood to a tree across a field of freshly fallen snow.\n\nThe first one kept her eyes on her feet as she put one foot cautiously in front of another. But when she got to the opposite end of the field, she looked up to see that instead of a straight line to the tree, she had walked at a slant that angled way off to the side.\n\nNext a little boy started across the field. Learning from the girl's mistake, he kept looking back at his footprints in the snow and correcting his direction. When he got to the tree, his footprints formed a zigzag line all the way across.\n\nThe third one stepped onto the snow, fixed her eyes on the tree, and confidently and quickly walked across the field. When they looked at her footprints, she had walked a perfectly straight line.\n\nWhen you have a clear vision of your goal, and you keep your eyes on it\u2014instead of looking back at past mistakes or watching each step you take for fear of making a mistake\u2014you will have a simpler, clearer path to accomplishing what you want in life.\n\nMrs. Clancy told the story to Trish's children many times, and the whole family was conscious of the value _and_ the rewards of goal setting. By the way, Noah's team won the championship that year and Noah was voted most valuable player. Did it have anything to do with goals or footprints in the snow? Hmmm, you never know.\n\nRemember to be vigilant in reevaluating your goals. If they are stress producing, they may be unrealistic goals. Sometimes a simple revision or adaptation of your goals can make the difference between stress and serenity.\n\nTHOUGHTS TO PONDER\n\n_Surround yourself with only people who are going to lift_ _you higher._\n\n\u2014OPRAH WINFREY\n\n_Do nothing, and nothing happens. Do something, and_ _something happens._\n\n\u2014BENJAMIN FRANKLIN\n\n_The question isn't who is going to let me; it's who is_ _going to stop me._\n\n\u2014AYN RAND\n\n_Wishing you a year that brings_   \n_True enjoyment in each moment,_   \n_True fulfillment in each goal,_   \n_True contentment in your heart,_   \n_And true delight within your soul_\n\n\u2014ANONYMOUS\n\n_The desire accomplished is sweet to the soul._\n\n\u2014KING SOLOMON\n\n_If you don't know where you are going, every road will_ _get you nowhere._\n\n\u2014HENRY KISSINGER\n\n_Obstacles don't have to stop you. If you run into a wall,_ _don't turn around and give up. Figure out how to climb_ _it, go through it, or work around it._\n\n\u2014MICHAEL JORDAN\n\n_Make the most of yourself, for that is all there is of you._\n\n\u2014RALPH WALDO EMERSON\n\n_When you reach for the stars you may not quite get one,_ _but you won't come up with a handful of mud either._\n\n\u2014LEO BURNETT\n\n_Obstacles are those frightful things you see when you_ _take your eyes off your goals._\n\n\u2014HENRY FORD\n\nEXTRA CREDIT\n\n1. Have you ever reached for a goal that never came true? What was it?\n\n2. Do you have any idea why that goal never came to be? Was it unrealistic? Not within your control? Not specific or measurable? Write your reason:\n\n3. What life goal or work goal would you like to achieve?\n\n4. Have you ever felt stressed due to a goal you pursued?\n\nDo you recognize how\u2014whether you realize it or not\u2014all of us reach for goals in every area of our lives, even when attending a meeting or going to an amusement park? Have you uncovered any negative goals, such as working 24\/7, trying for perfection, or allowing phones or computers to control every second of your life? \n **CHAPTER 5**\n\n[Eliminate What You Can and  \nStreamline Everything Else](chap8_manageyo_9780802719768_epub_part8.html#d41151331)\n\nWHILE ON A CRUISE TO ALASKA, Danny and Margaret went on a tour where they spotted a general store with a sign:\n\n**We have everything you need.**\n\n**If we don't have it,**\n\n**You don't need it.**\n\nWhen they told their fellow cruisers at dinner about the sign, everyone laughed. But then a discussion erupted about what people actually do need. Not just want, but need. All eight people at their table expressed an opinion.\n\nScott, a commodities broker, had been on a \"wilderness experience\" early in his career. He spent three days in the desert with only a knife and a blanket. He said it is astonishing to learn what we can do without\u2014and although he wouldn't want to live that way, he thinks the experience colors his life decisions. For example, he just doesn't relate to the competitiveness and one-upmanship at work, and he doesn't care that his car is not top-of-the-line like those of his colleagues. But he maintained that everyone has their weird wants that have become needs to them, and for Scott it was high-tech toys.\n\nDebbie asked what's wrong with having things that we might not need, but we want. She loves clothes, shoes, and jewelry and loves accumulating as much as she can. When her husband made a negative remark about Debbie's shoe collection, she countered with, \"Please explain why we have four cars and only two drivers in the family.\"\n\nTo head off their bickering, Kristen jumped in with a story about her brother. \"He is stressed out from all the clutter in his place, but every weekend, he and his wife scour flea markets searching for antiques and bargains. His house, condo, garage, and office overflow with stuff and then he gets stressed out over the debt he's in and how he can never seem to get ahead. And the noodle-noggin doesn't see the connection between how all that spending causes debt, which causes stress, and then all the stuff he spent money on causes clutter, which also causes stress. _Then_ they complain that they don't have a life because they both put in overtime hours and are so swamped with so much to do to make extra cash so they can . . . It's just a crazy circle and they don't see the connections between the cause and result.\"\n\nSam, a philosopher\/salesman, said, \"So when we're all stressed out from too much to do, we need to get to the cause\u2014the debt, the spending, and the stuff and clutter. The too-much-spending and the too-much-stuff.\"\n\nScott came back with, \"Well, who decides how much is too much and how much is enough? Who decides what is a want and what is a need? I think most people aren't at all clear about _why_ they are pushing themselves to do more and to always have more. I think most people are stressed out trying to acquire wealth. And who decides what is wealth?\"\n\nSam said, \"I don't think 'wealthy' means having a certain amount of money. What amount would that be? I think wealth is based on your need\u2014it means not needing anything else or not much more. The person who is financially broke but grateful he has everything he needs is a wealthy man. When you don't need much more in life, you are truly wealthy. Yet even when people possess more than what seventy percent of what the rest of the world doesn't have, they still feel deprived.\"\n\nAsk yourself: What is enough? Many people are overworked, overscheduled, and overwhelmed because they are always trying to do more, acquire more. They never question whether they have what they need or have enough.\n\nIn order to reduce your stress, you need to do some eliminating and streamlining of the causes of stress\u2014whether it's a packed calendar, too much debt, overwhelming clutter in your life, or whatever.\n\nIt also is valuable to question what is enough for kids. Not just in terms of stuff, but activities and other things that might be putting pressure on them. When younger children are asked what makes a family happy, they often come up with the same descriptions:\n\n\u2022 Spending time with their parents\n\n\u2022 Eating dinner as a family\n\n\u2022 Being tucked in at night\n\n\u2022 Being read to\n\nThey seldom mention computers, cell phones, iPods, money, video games, television, fancy cars, or playing on sports teams.\n\nShelley and Anita, two airline reservationists, were discussing a gathering of friends who were having a potluck. Everyone was to bring something to contribute to the meal, and Shelley was feeling discouraged. \"My calendar is so full,\" she said, \"I have enough time to cook something or come to the potluck, but I can't do both.\"\n\nAnita replied, \"Just bring a container of cottage cheese.\"\n\nShelley sputtered, \"What? What? What are you talking about?!! I can't show up with just cottage cheese.\"\n\nAnita countered, \"Shell, you know this group. Their self-worth isn't tied in to what they bring to a potluck.\"\n\nAre you overly busy because your self-worth is tied to what you wear, what you drive, or how your house is decorated? Is it tied to gifts you give or the amount of money you earn? Is it tied to what your business associates think of you or how your friends and family feel about you.? You are in charge of your self-worth\u2014 value yourself according to your own values, not someone else's.\n\nOne of the best ways to streamline your life is to eliminate letting what other people think rule your life.\n\nQuestion Everything Before You Commit to It\n\nWhere do you start in eliminating or streamlining the \"stuff\" of your life?\n\nChip, a video producer, says that he was raised in a house that had a banner hanging in the kitchen that said \"Live simply that others may simply live.\" He says that quote is embedded in his mind and is part of his philosophy of life. It has made it easier for him to ask the types of questions that follow, which cause him to pause and evaluate before plunging into something:\n\nDo I really need to buy this?\n\nDo I have a place to put this?\n\nDo I really need to do this?\n\nDo I really need to own this?\n\nDo I need to join this group?\n\nDo I need to spend my time on this?\n\nDo I really need to have this in my life?\n\nChip and his family are living a fairly streamlined life according to their values. A few examples are the family members drive an older car, give deliberate thought to purchases so there is very little impulse buying, and don't use charge cards, and everyone in the family has agreed to not pay big bucks for the \"privilege of having designer names on anything.\" In other words, if a designer something is a good value, they buy it. If it is similar to something else but the price is quadrupled because it has a designer name on it, then they don't buy it.\n\nIn exchange for a simplified lifestyle, his wife is able to work part-time instead of full-time, and Chip does not have to accept overtime hours at work. They are still able to treat themselves to fun family activities throughout the year.\n\nChip says that nobody ever said on their deathbed, \"I wish I'd spent more time at the office.\" And he and his wife are determined to not have regrets about missing out on their children's growing-up years. For their family, eliminating and streamlining means cutting back on cash investments, which in turn cuts down on the need to work overly long hours. For others, it means having a choice about how time is spent, money is earned, or other areas of their lives.\n\nSometimes people wonder whether if they start to say no to things that once were important\u2014for the sake of simplifying their lives and preventing stress\u2014they'll be messing up the life values we read about earlier. First, let's explore the difference between priorities and values. When you work on priorities, you decide \"This task comes first, next I have to do that.\"\n\nValues reflect what is important in your life, what is valuable to you. Is it health? Your relationship with God? Family? Friends? Love? Music? Justice? Art? Freedom? Helping others? Saving the whales? Combating global warming? Teaching others? Making the world a better place?\n\nOnce values are clarified, you can brainstorm ways to incorporate them in your life. When you know you value time alone or time with family or friends, you can focus on ways to simplify your schedule or calendar.\n\nIf you value travel or a vacation, you can focus on simplifying your needs and expenditures. In exchange for taking a fun family vacation every year, one family decided to forgo a DVD player, second computer, and sophisticated cell phone plan, and to put strict limits on eating fast food and buying expensive coffee. They arranged for others to take care of the dad's mother who was confined to the house. The family returned from vacation each year with renewed energy for taking care of the incapacitated grandmother.\n\nWhen you live life spending more time on what is not important to you and little or no time on what is truly important to you, the result is usually stress, frustration, and a sense of inadequacy.\n\nThe Excess of Success\n\nWhat do you think success is? What is your definition of success as a person, spouse, parent, worker, wage earner, or member of an organization, team, or church?\n\nWhen author and actor Malachy McCourt was asked his thoughts on streamlining our lives, he said, \"We are born with a wheelbarrow to help us carry through our lives whatever we need. But many people keep asking for\u2014and getting\u2014bigger and bigger wheelbarrows.\n\n\"Some of them hit a point that it will topple over or they can't push it anymore so they collapse right in the middle of the wheelbarrow road.\n\n\"Others just keep piling stuff higher and higher, and never realize they are heading for the edge of a cliff. And of course the stuff goes over the cliff with them, but you still can't take it with you. The moral of the story is you can't give or receive the gifts of life with hands on the handles of an overloaded wheelbarrow.\"\n\nUse a Time Log to Eliminate and Streamline\n\nMany people hate the idea of a time log, but that might be because it forces them to face reality. Listing in a log or a journal all the ways you spend your time might leave you feeling embarrassed to admit you waste a lot of time or you are spending most of your time on things you least value.\n\nYou don't have to keep a time log forever. One week will give you fabulous insights. Even logging your activities for just a day or two will help you see where you can start to eliminate or streamline activities.\n\nKevin, a CEO, said that his best time management tip was learning to say no. The first time he ever kept a time log (and he says every once in a while he goes back to keeping one) he realized that he was putting everyone else's priorities and urgent matters before his own. He was on \"commitment overload,\" making too many appointments, taking on too many relationships and too many responsibilities, volunteering for too many tasks, serving on too many committees. He was trying to be all things to everyone.\n\nWhen he asked himself why he was involved in a particular organization, he had to admit he was simply living up to his own expectation and there was no real reason to belong to it. It took him a long time to grasp that just because you _can_ do something doesn't mean you _have_ to.\n\nKevin cut back everywhere, and not only in those areas that caused him stress. \"What I had been doing was saying yes to everything that sounded like fun,\" he said. \"And I was surprised to discover that doing many, many fun activities all at once leaves me exhausted and stressed out. Now I've learned my best way to eliminate and streamline is to say no frequently.\"\n\nControl Clutter to Lessen Stress\n\nClutter in your life keeps you busy. You've got to find space for it, maintain it, clean it or clean around it, insure it, protect it. The fewer things you have surrounding you, the easier it is to keep your living and working space neat and clean. Also, when you have clutter in your work space or living space, you have clutter in your mind and in your heart; you can't think clearly and don't know what you feel.\n\nHave you ever noticed how clutter sucks the energy out of the marrow of your bones? Just looking at all that stuff and thinking about clearing it out can be so exhausting that you want to take a nap. Another benefit to clutter busting is that once the stuff you don't need or use is gone, you can finally find things when you search for them.\n\nOnce clutter is diminished, you will feel more energized and experience the overwhelming desire to dance the dance of joy all over town. Plus it will be easier to focus on what you need to and want to do.\n\nEliminate and Streamline Information and Paper\n\nRegarding information overload, Richard A. Swenson, M.D., wrote in _Margin: Restoring Emotional, Physical, Financial, and Time_ _Reserves to Overloaded Lives_ , \"A single edition of the _New York_ _Times_ contains more information than a seventeenth-century Britisher would encounter in a lifetime.\"\n\nIt is no longer possible to read or absorb all the information coming your way. You need new methods for going through your mail and e-mail and handling all the reading material that is gushing into your life like a never-ending waterfall.\n\nSkim through mail, e-mail, books, and newsletters. Cancel subscriptions to all those magazines stacking up in the corner that you plan to read someday but never find the time for. Recycle catalogs without looking through them. Better yet, go online to www.catalogchoice.org, a free service that lets you decline paper catalogs you no longer wish to receive. You will reduce the amount of unsolicited mail in your mailbox while helping to preserve the environment. But when catalogs you choose to receive start to pile up, just tell yourself there is nothing in there that is important enough to add to your clutter.\n\nBe highly selective when downloading or printing reams of material from Web sites. They may contain important information but nobody has time to read all of them. Skim them, pausing to carefully read or print out only those parts that seem to be new, fresh, important, or extremely interesting.\n\nCreate Time-Saving Systems for Dealing with Information and Paper\n\nDevelop systems for handling information or tasks that keep coming up again and again. If people routinely ask for certain information, set up a self-help section at work or on the server where colleagues can access it. Write out all the steps and hand them the paper or send them to a Web site, instead of wasting your time on the phone explaining the same steps to everyone.\n\nOrganize your own electronic and paper files so you can quickly find the answers to questions you are often asked. Notice what kinds of queries exasperate you because you have to run around looking for the answers\u2014this is a tip-off that you need to reorganize your files to make retrieval of information more automatic and less stressful.\n\nIf your kids aren't getting their chores done or need to be reminded of what to do to get ready for school, write out the steps and post your list where the kids can see it. For children who are too young to read, cut out pictures from magazines of a child brushing her teeth or getting dressed or eating breakfast. Let your child help with the list; the more involved they are with it, the more likely they will be to follow it.\n\nIf you are sending out repetitive letters or e-mails, make up templates or form letters. Or create FAQ (Frequently Asked Questions) sheets for co-workers, clients, customers, volunteers, patients, even family.\n\nIf your computer or PDA has a way of giving you reminders (most computers do), learn how to work them. Some computers have reminders that blink, flash, ding, and practically tap you on the shoulder and hand you a cup of coffee. Once you start using these reminders to help keep you on top of your list of things to do, you'll wonder how you were able to function without them.\n\nOutsource Certain Tasks\n\nBusinesses outsource work all the time. Is there anything in your life you can outsource or at least ask for help with?\n\nBill was in charge of mowing the lawn and often let it go for so long that the grass was above the height of his kids' knees. This made his wife, Jane, crazy and they had huge arguments over it. He claimed he only put it off because he was busy; she claimed he'd better change his priorities about what is important to do around the house.\n\nA friend gave Jane the contact information for a lawn mowing company. She was astonished to discover it cost less to pay people to cut their grass than it cost to buy dinner for the family once a week. Years later, she still refers to the lawn company as the wonderful guys who saved her marriage.\n\nMaybe you can't afford to have someone clean your house every week, but it's a great stress buster when you are under extreme pressure during the holidays, when you have overnight company, or when your mother-in-law is coming to dinner. Saving your sanity may be worth the cost of a cleaning service.\n\nWhether you consider the help frequently or on rare occasions, sometimes you can eliminate a great deal of stress by paying someone to wash your car, help with the children, cook a meal, clean up after a party, stay with your invalid father, or a hundred and one other stresses when you find yourself running on that hamster wheel.\n\nSchedule Streamlining Days\n\nMyron, executive director of a landscaping association, has an interesting system. \"I have days I streamline everything I can; those are my days devoted to catching up. My e-mail replies are one or two words, I scroll through and eliminate every e-mail I can, I set a timer for five minutes for all my phone calls (unless absolutely impossible), zip through any snail mail that has stacked up, and any magazines or catalogs I haven't read since my last streamline day get tossed into the recycle bin.\n\n\"I noticed that a momentum or a mood comes over me, so as long as I'm in my 'eliminating mode,' I check my closets, even the kitchen cabinets. You know, when you put yourself in a mood to get rid of and reduce papers at your desk, it can carry over to your whole house. On those days, I fill up dozens of bags of stuff to donate to the Vietnam vets organization.\"\n\nMost of us cannot devote a whole day to scaling back our work, but we can do any of a number of variations on Myron's ideas.\n\nDave, who works out of his home, has a \"power hour\" at least once a week where he powers through his e-mails\u2014deleting them, sending short replies, or making a decision and handling the complicated ones he put aside earlier and then forgot about.\n\nMelody is a teacher who collects all kinds of craft items and articles with ideas for her classroom. Every year, the week after school ends, she has a \"zip morning.\" She zips through all that she has accumulated and either tosses out, recycles, or files it. She used to be terrified of throwing away something she might need next year, but she finally realized that stuff comes into her life as if she was a magnet and will turn into overwhelming clutter if she doesn't clear it out once a year.\n\nAfter she gets rid of almost everything at the end of the school year, she somehow collects a whole new batch of articles and craft supplies even before the next school year begins. \"Now,\" Mel says, \"I zip through everything and zip most of it out the door.\"\n\nGayle says every spring her mother had the whole family go through the house and clear out what they no longer needed or used. Now Gayle does it with her family. She cleans out closets and cabinets, her husband goes through the garage getting rid of things, and the kids clear out their toys and clothes. It's a spring ritual that works for her family.\n\nSome people simply develop a habit of going through their clutter or stacks of paper every day and either putting things where they belong or getting rid of them. It doesn't matter whether you schedule a few minutes or a full day to streamline. If you do not regularly schedule _any_ time for it, you will find yourself feeling overwhelmed and stressed.\n\nLook for Creative Alternatives\n\nYou can radically reduce your stress by eliminating the negative attached to so many life events. It's almost expected that we get depressed over certain birthdays, hate Monday mornings because we are returning to work, and dread getting together with the family at holidays. What can you do to turn around a negative event or eliminate the negative aspects of it\u2014especially if you can't change the event?\n\nKatie and her three friends discussed that there was nothing they could do to prevent their upcoming fiftieth birthdays, so instead of being depressed over that milestone, they decided to create a positive. They all felt they needed to step out of their comfort circle and go on an adventure. They decided to drive from their home in Chicago to Colorado to go river rafting, because they felt that getting smacked in the face with icy cold water and navigating terrifying rapids would fill them with the exhilaration of life.\n\nAs they left Chicago, they discovered that for each of them, it was their first vacation without their children, so they made up some fun rules. Whoever sat in the backseat had to whine: \"Are we almost there yet?\" \"Her foot's on my side.\" \"How much longer till we get there?\" \"Make her stop looking at me!\" Then someone would count to three and they'd all whine, \"I have to go to the baaaaaaathroom.\" Another rule was whoever sat behind the driver had to kick the back of the seat for hours.\n\nAfter their rafting adventure, Katie described feeling like a returning hero as she pulled into the driveway and roared into the house, excited to tell her twenty-one-year-old son, Jeff, all about her trip. He had been home alone for the ten days of her trip, and when Katie went into the house she stopped dead in her tracks. The house was immaculately clean. (Wouldn't that make you nervous?) She looked around and said to Jeff, \"Wow, even the dishes are clean!\"\n\nJeff said, \"You know when a guy knows it's time to do the dishes?\"\n\nKatie said no, and Jeff, ever the comedian, said, \"When he's eating his Froot Loops out of a bundt cake pan with an ice cream scoop.\"\n\nSo Katie and her friends, instead of living up to the world's expectation that they would be depressed over their fiftieth, spent a year saving for and planning their high adventure, and now they have great stories to tell. If you can't change the situation itself, then see if you can do something to eliminate the negative and add a positive note to it.\n\nWhat about people who are taking care of elderly parents, young children, or chronically ill people? How can they streamline?\n\nFor some people, even a time log shows that there is nothing that can be omitted in their packed schedule. For example, a single mom working two or three jobs cannot opt to work fewer hours in order to reduce her stress\u2014or outsource chores so she has more time to relax.\n\nStill, there may be some alternatives. Whitney, a bagger at a grocery store who also cleans houses on her days off, is a single mom who found a very creative solution. The woman upstairs who also had daughters and who worked nights had become a close friend.\n\nThe two women decided to move in together, and because they worked opposite hours, they each could babysit when the other one was working. They saved the cost of child care and eased a lot of stress knowing their children were with a safe, loving, reliable woman. They were a bit crowded, but all were willing to put up with the crowded conditions because of the financial benefits, and it was wonderful having two women share the cooking and cleaning. Such a solution is not suitable for everyone, but it has worked for Whitney for several years.\n\n**A DO-NOT-DO LIST**\n\nJust as a to-do list helps you focus on what you need to do, a do-not-do list can help you stop doing what you don't need to do. Here are a few ideas to get you started:\n\n**DO NOT**\n\n\u2022 compare yourself to others\n\n\u2022 believe you have to work to the point of exhaustion to be considered a capable person\n\n\u2022 try to please everybody you know\n\n\u2022 spend your time acting on everybody else's priorities\n\n\u2022 say yes to every request\n\n\u2022 think that being busy is a standard of success\n\n\u2022 consider it necessary to take care of every single person you know in order to be liked\n\n\u2022 believe that you never have time for your priorities\n\n\u2022 hang on to angers and resentments\n\n\u2022 try to live up to the expectations of everyone in the world (including you)\n\n\u2022 strive to be so independent you never need (or are able) to ask for help from others\n\n\u2022 spend time on busywork that's not important to you\n\nAvoid Multifocusing When You  \nThink You're Multitasking\n\nIf you have read or heard anyone discuss multitasking, you know what the advice is.\n\nDo it.\n\nDon't do it.\n\nMaybe sometimes do it.\n\nUnder the right circumstances, do it.\n\nIt's helpful, not helpful, effective, ineffective, it's right, it's wrong, it's good, it's bad.\n\nThe funny thing is, when people are telling you how wrong it is to multitask, they often are telling you while opening mail, logging on to a computer, or folding laundry. The frustrating part is that there are so many definitions and styles of multitasking that when two people discuss it, they often are discussing two totally different activities.\n\nWhen you are doing two tasks at the same time, you are dividing your thinking, focus, attention, and concentration between two activities. If both tasks call for high thought or focus, neither will be done well. Typing an important e-mail while listening to an important phone call can result in your missing some key details on the phone and sending out a garbled, incorrect, or incomplete e-mail. This is not smart multitasking.\n\nMany people extend \"multitasking\" to mean working on three different projects in one hour and not completing any of them. If you are doing that because you cannot stay focused on one task for a long period of time or you were forced to stop each task for a reason, then you're doing fine. No problem. But if your idea of multitasking is simply to bounce from one project to another and kid yourself that you are doing three things at once, think again. Each time you start a new task, you have stopped the old one. You are not doing three things at once; you are multifocusing. You are starting and stopping; you lose focus and you lose the precious time it takes trying to wonder, \"Now where was I and what was I going to do next?\"\n\nMultifocusing your way through several projects at the same time ensures that you will feel pressured and stressed. Each task is not accomplished as well as it could be. The work on each is so fragmented that you lack the deep pleasure that comes with a job well done. You don't have the satisfaction of completing one task to help motivate you to do the next. And you might miss a deadline or drop the ball on important stuff while switching your focus from one project to another. Also you lose momentum when you interrupt good concentration and never allow yourself to move into \"the zone.\"\n\nWhen something is important and deserves your full attention and concentration, then develop the habit of putting everything aside and giving it the focus it deserves. Doing this one task and completing it will enable you to do it well.\n\nAnd what about those tasks that do not need your full attention or concentration? Go ahead and multitask. As long as you aren't trying to divide your focus in half, you are doing smart multitasking. When you have one task that needs very little concentration, then you are able to give plenty of thought and focus to the other task. Some examples of smart multitasking are:\n\n\u2022 planning a grocery list while loading the dishwasher\n\n\u2022 making a phone call while printing out a two-hundred-page report\n\n\u2022 exercising while you listen to marketing CDs\n\n\u2022 paying bills while doing the laundry\n\n\u2022 assembling promotional packets while brainstorming marketing ideas\n\nIt's not the multitasking that creates stress, it's the multifocusing that gets us into trouble and leaves us frazzled.\n\nRaiman was conscientious about answering his phone as soon as it rang and checking his e-mail as soon as he heard the \"you've got mail\" announcement. But he realized that putting this expectation on himself was resulting in constant interruptions. He found that by turning off his phone and checking his e-mail only once or twice a day he dramatically increased his productivity. Raiman said that he has been surprised to read that many successful CEOs and other high achievers frequently turn off phones and e-mail for long periods of time when they need uninterrupted hours for thinking or creative work.\n\nTHOUGHTS TO PONDER\n\n_Out of abundance He took abundance and still_ _abundance remains._\n\n\u2014THE UPANISHADS\n\n_Joy is what happens to us when we allow ourselves to_ _recognize how good things really are._\n\n\u2014MARIANNE WILLIAMSON\n\n_The truth is, the happy get happier because they know_ _how to be happy, and the troubled get more troubled_ _because they pour all their energy into their troubles._\n\n\u2014SUSAN PAGE\n\n_He who knows enough is enough will always have_ _enough._\n\n\u2014LAO-TZU\n\n_Until you make peace with who you are, you'll never be_ _content with what you have._\n\n\u2014DORIS MORTMAN\n\n_You will recognize your own path when you come upon_ _it, because you will suddenly have all the energy and_ _imagination you will ever need._\n\n\u2014JERRY GILLIES\n\n_Few things are necessary to make the wise man happy,_ _while no amount of material wealth would satisfy a fool._ _I am not a fool._\n\n\u2014OG MANDINO\n\n_Being rich isn't about money. Being rich is a state of_ _mind. Some of us, no matter how much money we have,_ _will never be free enough to take time to stop and eat the_ _heart of the watermelon. And some of us will be rich_ _without ever being more than a paycheck ahead of the_ _game._\n\n\u2014HARVEY MACKAY\n\n_And there is no greatness where there is not simplicity,_ _goodness and truth._\n\n\u2014LEO TOLSTOY\n\n_If your outgo exceeds your income, your upkeep will be_ _your downfall._\n\n\u2014ANONYMOUS\n\n_To earn the appreciation of honest critics and endure the_ _betrayal of false friends; to appreciate beauty; to find the_ _best in others; to leave the world a bit better, whether by_ _a healthy child, a garden patch, or a redeemed social_ _condition; to know even one life has breathed easier_ _because you lived. This is to have succeeded._\n\n\u2014RALPH WALDO EMERSON\n\n_Simplify your life. You do not need all the clutter you are_ _holding on to. Get rid of it now. Keep around you only_ _the things that give you energy._\n\n\u2014JOHN-ROGER\n\n_Year by year the complexities of this spinning world_ _grow more bewildering and so each year we need all the_ more to seek peace and comfort in the joyful simplicities.\n\n\u2014 _WOMAN'S HOME COMPANION_ , 1935\n\n_'Tis a gift to be simple,_   \n_'Tis a gift to be free,_   \n_'Tis a gift to come down_   \n_Where we ought to be_   \n_And when we find ourselves_   \n_In the place that's right_   \n_'Twill be in the valley_   \n_of love and delight._\n\n\u2014NINETEENTH-CENTURY SHAKER HYMN\n\n_He threw himself on his horse and rode off in all_ _directions._\n\n\u2014MIGUEL DE CERVANTES, _DON QUIXOTE_\n\n_Success is having what you want_   \n_Happiness is wanting what you have_\n\n\u2014ANONYMOUS\n\n_Think no other greatness but that of the soul, no other_ _riches but those of the heart._\n\n\u2014JOHN QUINCY ADAMS\n\n_Our remedies oft in ourselves do lie_\n\n__ \u2014William Shakespeare, _All's Well That Ends Well_\n\n_Success isn't permanent, and failure isn't fatal._\n\n\u2014MIKE DITKA\n\n_What I am is good enough, if I would only be it openly_\n\n__ \u2014Carl Rogers\n\nEXTRA CREDIT\n\n1. When you spot yourself feeling overly busy, overscheduled, and overwhelmed, stop and ask: Why are you so busy? If your answer is, \"That's how my life is,\" then question it even further. _Why_ is that how your life is?\n\n2. Do you believe you have to be exhausted to be successful?\n\n3. Do you think you have to be exhausted to achieve your dreams?\n\n4. What's the first thing you thought of when you read the chapter title \"Eliminate What You Can and Streamline Everything Else\"?\n\n5. After giving it some thought, where do you think you need to do some eliminating and streamlining?\n\n6. What would be your biggest obstacle to doing that eliminating and streamlining?\n\n7. What are the first steps you can take to get started?\n\n **CHAPTER 6**\n\nSet Boundaries at Work and at Home\n\nSETTING BOUNDARIES IN YOUR LIFE CAN go a long way toward helping you manage your time and reduce your stress.\n\nDuring a particularly difficult fourteen days of work, Traci realized she had only one evening off\u2014the upcoming Saturday. So when her evening off arrived, she decided to cut off all connections with the outside world and catch up on laundry, relax, pet her cat, and watch every episode of _Law & Order _she could find.\n\nJust as she was heading over to turn off her phone, it rang. Her sister called to tell her they needed a fourth person for their bridge game. She enthusiastically told Traci how much fun it would be if she joined them and what a rotten, boring evening it would be for them if Traci didn't come over.\n\nTraci explained her exhausting workload and said she really needed to stay home on this very precious evening off. Her sister didn't buy it. She whined. She begged. She pleaded. Then she told Traci how selfish she was to ruin their bridge game.\n\nNotice the guilt and manipulation her sister was using. She wanted Traci to be responsible for the happiness or sadness of the three bridge players. Why can't they find someone else? Why can't they do something else? Oh, no, it's up to Traci to come and save their happy evening.\n\nBoundaries define where things begin and end, but it seems there's always someone out there disagreeing with that boundary and wanting to either cross it or change it. Traci learned a lot from this experience. She learned that others might not respect her boundaries, but it's important that she respect them and respect herself and her own needs and priorities. She also learned to use her voice mail and not return calls on a night when she has decided to curl up and simply be alone and cozy.\n\nMaura firmly believes it is important to volunteer time for children and she loved being a Cub Scout leader. So she couldn't figure out why she was reluctant to agree to her friend Sharon's request to be treasurer of the parents' club.\n\nSometime later there was an emergency in Maura's extended family. Her sister-in-law was sick from chemotherapy, so Maura called a few family members to coordinate ways people could help out. Her aunt said, \"Please don't ask me to cook or take care of the kids, but I'd be happy to put on my sweats and come over and clean her house. I'd even enjoy doing her windows.\"\n\nMaura was relieved because she hated house cleaning and window washing but actually enjoyed taking care of her sister-in-law's children. In that instant, Maura realized why she had said no to being club treasurer. She would have been miserable doing the administrative task. From then on, Maura decided to set boundaries on her volunteer work and accept only jobs she likes and that suit her interests and abilities\u2014only jobs that light her up.\n\nEleven Tips for Setting Boundaries  \nThat Lessen Your Stress\n\nIn today's world, constant access to work\u2014phones, computers, BlackBerries, and PDAs\u2014controls our lives, even at the dinner table or in a restaurant. _We need boundaries_ for when, where, how, and for how long we will hand over control of our lives to outside demands. Here are eleven tested strategies for managing your time at work, at home, and doing volunteer work; for establishing a healthy balance between your priorities and those of others; and for achieving your goals without working yourself to death or alienating your family and friends.\n\nHaving a desk or office in the home blurs the line of demarcation between work and home life even more.\n\n_We need boundaries_ between work time and personal time.\n\nWe look at our schedule and feel exhausted by all that we have to do and all the places we have to go.\n\n_We need boundaries_ on our schedules.\n\nOur boundaries might cover:\n\n\u2022 what we do (I'll do this but not that)\n\n\u2022 where we do it (I'll do this at work but not at home)\n\n\u2022 how long we do it (I'll take over this job for a month but if you don't have a replacement by then, I have to remove myself from the committee)\n\nIf you are serious about reducing your stress, you must learn about and start to set boundaries in your life.\n\n1. KNOW WHEN TO CRY \"UNCLE\"\n\nMary Ann knows she can get so wrapped up in pursuing a goal that she can become obsessive. It's a good trait when the goal is important, but she can get obsessed over the craziest, most insignificant pursuits. Mary Ann said she really needed to learn \"when to stop beating my head against the wall.\"\n\nOnce she offered to find some information for her daughter's scout leader. She thought it would involve a phone call or two, but instead she ended up spending three days on the phone, with pages of notes and a list of things to follow up on. The stress of tracking down information, making and receiving calls, keeping everything straight, and making lists was giving her a headache that would not stop.\n\nAs we talked about boundaries, she said she hated the idea of telling the scout leader that she wasn't successful in her \"mission.\" But as she thought about it, she realized she had to become selective. Her persistence is a great habit for something important, but to obsess over every single search is not a good use of her time.\n\nNow Mary Ann sets boundaries for herself. If she takes on a quest, she decides beforehand how much time, energy, and attention she will devote to that project. And then she forces herself to stop.\n\nIf it relates to work and they need whatever it is Mary Ann is searching for, of course she will continue till she finds what she needs. But that is not where she usually gets into a high-stress frazzle. Her high-stress situations almost always involve noncritical, volunteer tasks. And she has learned that they often turn into wild-goose chases.\n\n2. KEEP WORK TIME AND FAMILY TIME SEPARATE\n\nJason, a publicist, works out of his home office and was battling several stress-related illnesses. He would answer the phone during dinner, start developing a marketing campaign, and still be working on it well past midnight.\n\nHis wife and children became so resentful of his putting work before them that they hardly spoke to him. Jason felt the chill in the family but believed there was nothing else he could do. He felt that was the only way to run a successful business.\n\nOne day he heard a ruckus in the house and came out of his office to discover a celebration going on. His daughter's softball team had won the championship. And even though she was beaming with pride holding her trophy, she walked right past him and didn't bother to show it to him.\n\nIn that instant, Jason realized he not only had forgotten about the championship games, he was totally out of touch with his whole family. He had treated them as if they didn't matter, and now he didn't matter enough to his daughter for her to even show him her trophy.\n\nFinally, he decided to set boundaries such as no interruptions during family time, and that included turning off his cell phone. This was painful at first but eventually he learned that life goes on and his business will succeed without the constant ringing of a cell phone. As he started to set boundaries and honor them, his health, his family dynamics, and his whole life began to turn around.\n\n3. BE CREATIVE WHEN WORK MUST INTERRUPT PERSONAL TIME\n\nSometimes work has to interrupt family time, such as when an obstetrician receives a call in the midst of a family celebration and is told he's needed in the delivery room. He has to head for the hospital regardless of how disappointed his children are.\n\nMarge, a neonatal nurse in Illinois, reports that some doctors have found ways to blend work and family. \"One night one of our best doctors brought her adorable four-year-old daughter, Ally, with her to the hospital to check on a patient,\" Marge said. \"The nurses were happy to see Ally and chat with her in the nurses' station and she got to be with her mom at work. The doc was only there for ten minutes, but if you count the ride in and out, the doc probably spent at least an extra hour with her child. Even the chief of obstetrics has brought her teenage daughter Colleen on the unit while she was taking care of some paperwork for a few minutes. Recently, a doctor came in wearing her soccer mom shirt to do surgery, then returned to her kid's game.\"\n\nOften it takes only a bit of creative problem-solving to change having to work on your day off into a pleasant, fun experience.\n\nMichael, a meeting planner for a major restaurant chain, is the single father of five-year-old Nadia and is very protective of his time off work. He doesn't use his cell phone away from work and takes no work-related phone calls or e-mails at home, although every so often, he might use his laptop to write a proposal after Nadia is asleep.\n\nOccasionally people for whom he has planned a meeting insist that he be there in person on a Sunday, his day off. Sundays are very important to Michael and Nadia, so when a customer insists on Michael showing up for a Sunday event, he and Nadia both dress up and go to the restaurant together. Michael stops in at the meeting he planned, then he and his daughter have lunch together.\n\nThey have a special meal out and the customer is happy. Michael picked up this way of setting a boundary from his boss, Norma, who follows the same system, except if she has to come to a meeting on her day off, she invites a friend to join her for lunch or dinner after she makes an appearance at the meeting. Both meeting planners have a clear determination to protect their personal time and they both make every effort to guard that boundary.\n\n4. SAY NO TO SOME COMMITMENTS\n\nLisa is a professional speaker whose generosity caused her great stress and lost her a huge amount of money. She had the habit of saying yes to every request for her to speak free of charge to organizations in her town, and she did a great job. She also received requests from branches of the same organizations in other towns. Lisa spoke free of charge as often as three times a week. As a result, she sometimes turned down paying jobs because she was already booked at a local church, school, or organization.\n\nWorse than that, Lisa found herself on the brink of exhaustion and totally stressed out from all the talks she was giving. Since she makes her living as a speaker, she had the additional stress of money concerns and worried that she couldn't pay her bills.\n\nFinally, Lisa learned to put a boundary on her free talks. One a month, that's it.\n\nBeing generous is wonderful, but if it leaves you sick, exhausted, and not able to pay the mortgage, then something is wrong. It's no longer generosity. It's just plain dumb. Those people in her audience don't go to work and say \"Hey, skip the paycheck this week. I'll work for free.\"Why do they expect Lisa to do it? Setting boundaries has allowed Lisa to present some talks free of charge without having to be the free speaker for every organization in her state.\n\nEddy, a contractor, says that he and his wife have been trying to cut back on all their commitments because they are both exhausted and stressed to the max with a schedule of trying to take care of everyone's needs and attend every activity they are invited to. They _know_ what they need to do. They just don't know how to do it.\n\n\"When a neighbor asks me to come and help him with a plumbing problem, I might be tired and I might need to fix something at our house, but I know how to fix this type of thing and he doesn't, so I say yes . . . because I don't know how to say no.\n\n\"And my wife is the same way. When someone asks her to do something or go somewhere, she might not want to do it, but she just doesn't know what to say.\"\n\nOne day it hit Eddy that he's not the only one who can help people. If he was out of town, his neighbors and friends would have to figure out a different way to solve their problems. He decided he will occasionally give them help (that's a boundary!) but not every time they ask.\n\nHe and his wife have come to the realization that they can't keep treating everybody's commitments as more important than their own. By figuring that out, their journey of stress busting is half-complete. The second half is learning what to say in order to set boundaries, believing you have the right to say no, and saying it with strength and conviction.\n\n**WHAT TO SAY WHEN YOU WANT TO SAY NO BUT**   \n**PEOPLE REALLY WANT TO HEAR YES**\n\n\"Usually I can work on your committee, but not this time. I've just got too much going.\"\n\n\"I'd love to do that but not at this time in my life.\"\n\n\"I would like to do that but I have way too many commitments right now and cannot possibly squeeze this in.\"\n\n\"You know, that would be fun, but I simply can't.\"\n\n\"I'd love to help with your project at work, but I too have priorities and if I do yours, I'll never get mine done.\"\n\n\"I'd really like to come over to help you with your plumbing problem, but I'm tied up tonight and I just can't. I'm sorry.\"\n\n\"Thank you for asking me. I'm honored and glad you asked; unfortunately I can't serve as president of the board this year.\"\n\n\"Dang! I've always wanted to do that, but this isn't the right time for me. Thank you for asking.\"\n\n\"Under normal circumstances, I would be more than happy to oblige you, but things are a bit crazy right now and I can't do it.\"\n\n\"I would love to do that. What day did you say it was? Sorry, I can't make it. I have plans.\" (The person asking you probably will not think your plans are as important as his request, so don't explain the plans. Just say you have 'em.)\n\n5. PLACE TIME LIMITS ON PHONE CALLS\n\nIf others do not respect the value of your time, it's up to you to set limits and enforce them.\n\nSue, the mother of two children under two years of age, said her sister-in-law Charlotte was making her crazy. When Charlotte was bored, she would pop in on Sue unexpectedly or call her for long phone conversations. Then she would talk about the same problems over and over but wasn't ever willing to do anything to solve them.\n\nOne evening, Sue complained to her husband, \"Doesn't she realize that with taking care of two babies, I don't have all that time to give her?\"\n\nHe replied, \"Well, either she doesn't realize it, or she does but doesn't care.\" In that instant, Sue decided to put boundaries on Charlotte.\n\nNow, when Charlotte calls, Sue tells her right up front, \"I'm swamped today, so we only have ten minutes to chat.\"Then she sets a timer. When it dings, Charlotte is never ready to hang up. Sometimes she manipulatively starts to talk about a problem or about feeling depressed, so Sue very assertively says, \"I'm so sorry, but the kids need me and I've gotta go now. Let's finish this another time.\"\n\nWhen Charlotte drops by unannounced Sue reminds her that she needs to let her know when she wants to come over because sometimes, \"like today,\" there are a million things to do.\n\nAs Charlotte settles down at the kitchen table to have a cup of coffee, Sue invites her to join her in the basement as she does the laundry. One time, Charlotte dropped in and Sue was all caught up on laundry, so Sue started cleaning out her cabinets.\n\nShe is determined to set boundaries on her sister-in-law. She feels proud of herself for reminding Charlotte to call before visiting and feels very satisfied that she's gotten Charlotte to join her in doing chores. And now that her stress has eased up, she finds herself starting to like Charlotte a lot more.\n\n6. SET REALISTICALLY REACHABLE GOALS\n\nIf you feel that you are working as hard as you possibly can but are never getting ahead or accomplishing what you want to, that can become one of your biggest sources of stress.\n\nHenry, a business\/technology consultant, says that much of the stress of managers in the corporate world is self-inflicted: \"The stress occurs when expectations of accomplishment exceed ability to deliver. Ironically, most of the unattainable expectations are set by the manager himself. In not clearly understanding their own goals, they fall into 'fire-fighting' mode. All unexpected challenges are treated as crises. They are each addressed immediately with all resources available.\"\n\nStaff is constantly asked to drop what they're doing to focus on the new problem, yet they never actually fix the problem. There is only time for temporary solutions before the next crisis.\n\nHe suggests that managers need to stop going from one crisis to another. Instead, when a problem arises, they should run a root cause analysis. \"An example of a root cause,\" says Henry, \"would be a software bug that eventually eats all available memory until a system crashes. To fix it, a major section of the code needs to be rewritten. Everybody knows this. But it will take time. And during that time, customers are complaining. So, rather then rewrite the code, it is decided that they will substantially increase the system's memory.\n\n\"The problem still exists and will reoccur, but not for a while. The customers no longer complain, but everyone involved knows, at the back of their minds, this issue isn't really off their plate. The manager knows they have worked long and hard, but also knows that none of these problems have been put to rest. But what can he do? He obviously doesn't have the people to address the root causes of problems at the moment. Everyone is too busy.\n\n\"The answer is not a technology fix; they must put a boundary on the time they are spending putting out fires. The manager needs to clearly state the nature of the problem to management (a memory leak), and sell them on the need for a long-term solution. Then it needs to be clearly explained to the customers that this is going to take some time to fix, but once it is, it will stay fixed.\" It's better to work on solving the problem than to continue to rely on superficial or temporary fixes. Not only will customers' satisfaction increase, there will be less overtime pay, and managers and staff will be less overworked and frustrated.\n\n7. LIMIT PHONE INTERRUPTIONS\n\nDeborah was working on an extremely important presentation she was giving the next morning to the company's newest customer. She felt that doing a great job could be important for her career.\n\nThe phone rang. It was a friend who had recently experienced a tragic, devastating loss. The friend was in tears. Deborah was painfully aware that this exact same phone conversation had occurred about ten days earlier. At that time, Deb sat listening and weeping with her friend for more than an hour. Deb was working on another presentation then also, and after the phone call, it had taken her till past midnight to wrap it up. When she woke up the next morning at five thirty, her usual time, she was sluggish and felt as if she were drugged. Her stress level soared as she walked in to do the program, and she wished she felt more prepared and therefore more confident. The client's response was OK, not great. Deborah had a seriously upset stomach for three days.\n\nNow she was being given a second chance to make another presentation. And here she was with the same friend in need on the phone. As much as she loved her friend and her heart ached for her friend's sorrow, Deborah knew she could not stay on this phone call or it might mean the end of (or at least a downturn in) her career. Besides, she couldn't take the stress of another late-night session and walking into a presentation scared and unconfident.\n\nShe said, \"You know I love you and care about you and I really want to spend time with you. But I'm distracted working on a really important presentation for tomorrow. Can I call you when I get home from work and then I can give you my full attention? I would be just so distracted right now.\"\n\nHer friend's reaction astonished her. Instead of sounding disappointed, her friend asked about the presentation. She said she'd love Deborah to call her the next day so she could hear all about how everything went and promised to pray for Deborah the next morning during her \"big gig.\" Deborah told her how grateful she was, and the conversation ended on a positive note.\n\nWhat Deborah said was perfect. After all, wouldn't anyone prefer to talk to us when we can give our full attention instead of at a time we admit to being distracted? She started to use this same phrase when her teenagers interrupted her while preparing for important meetings. She would say, \"I'm really distracted right now. Can we talk about this in about an hour when I can give you my full attention and be fully present to you?\"\n\nThe girls have even started to use that phrase themselves. Recently, her eldest daughter told her, \"Mom, I have so much on my mind right now, I know what you mean about being present or not. My body is here but my mind is a million miles away. Can we cover this later?\"\n\nSo Deborah has not only learned to set boundaries and prevent high levels of stress, she has also taught the skill to her daughters.\n\n8. CUT BACK ON SOME FAMILY TRADITIONS\n\nWhen you find yourself stressed from all that you have to do, stop and see whose expectations you are living up to. Charmaine tells the story of being behind schedule in putting up Christmas decorations one year. Instead of spreading out the work over several days as they usually did, she pushed to get everything up in one day. She was determined to make the house beautiful for her family.\n\nThey pulled out all the boxes of decorations, hauled them up from the basement, put up the tree, put the ornaments and lights on the tree (at this point, tempers were flaring and everyone was cranky), and continued putting up the garland, lights, and other decorations all over the house. Everyone was stressed and exhausted.\n\nAs her thirteen-year-old daughter stomped off to bed, she said, \"Mom, I hope you know we only did this because you wanted it. I don't give a darn about all the stupid decorations.\"\n\nCharmaine was startled and chalked it up to her daughter being in a bad mood. But as she gave more thought to it, she realized that much of her stress during the holidays came from her struggle to live up to her own expectations. That year she decided to cut way down on baking. Nobody complained. Nobody noticed or cared. And Charmaine saved hours and hours of time and work by not going into her customary baking frenzy. Now each year she limits what she expects of herself (and her family) during the holiday.\n\nAnother way to put boundaries on expectations is to first find out if you have a clear idea of what is expected of you. Otherwise you might be agreeing (or refusing to agree) to expectations that don't even exist.\n\nVictoria told a story about her grandparents. They came to the U.S. from Austria as newlyweds, and every Saturday morning Grandmother made homemade raisin coffee cake for Victoria's grandfather to give him a taste of home. After twenty-six years of marriage, Grandmother discovered that her husband hated raisins. \"So why didn't you tell me? Don't you think I have better things to do than to bake you a coffee cake that you hate?\" she asked him.\n\n\"I thought you were supposed to make it and I was expected to eat it to make you happy.\"\n\nEpilogue: She stopped baking every Saturday morning and instead joined a water aerobics class.\n\n9. ENLIST YOUR BOSS'S HELP MANAGING AN OVERWHELMING WORKLOAD\n\nIf on your job you have more than one person feeding work to your desk or if you have a boss who is fairly oblivious about how swamped you are, you have to set a boundary.\n\nMost bosses really don't know how much work you have or how long something takes. So often they will say, \"Here, can you do this? It shouldn't take you long.\" Yet a week later you are still buried under the burden of that task.\n\nThe place to start is to make a list of all your top-priority work and take it in to the boss or e-mail it. Say something like \"As you can see, this all cannot possibly be done today, and I've been staying late to get work done, but today I need to leave when everyone else does. Can you advise me on what's most important for me to complete today?\"\n\nUsually that works. Sometimes it doesn't. Pearl said something similar to her boss and he blew her off. He said, \"It's your job to get it all done. That's what you get paid for.\"\n\nShe asked if perhaps there was someone who could work with her on a few of the projects and was discouraged when he said there wasn't. But later Pearl noticed a definite improvement in her workload. Her boss was passing some of the work on to other staff members and a week later he asked Pearl if she was still having to stay past five o'clock.\n\nShe was touched by his concern and delighted that she was able to cut way back on staying late and to go home at five.\n\n10. CUT BACK ON YOUR CHILD'S SCHEDULE\n\nBrendan and his family kept a crazy schedule. They never had dinner together. Actually, the only thing they did together was drive the kids to and from activities, and usually it was him taking one child one way and his wife with another child heading off in a different direction.\n\nHe was becoming more aware of how the franticness caused stress for him and his wife, and also he was spotting signs of stress in each of their four children, including sleep disturbance, a drop in grades, and a lot of squabbling going on. They had a family meeting and Brendan pointed out that they _had_ to set some boundaries in their lifestyle. He told the children that each of them needed to eliminate and streamline some of their activities. Three of the four seemed almost relieved and were willing to drop down to one activity each. One child squawked a bit, but eventually worked her activities down to basketball and flute.\n\nSuddenly, life became manageable and the stress level in the family lowered.\n\nAs part of the boundary-setting everyone agreed that if they decided to add a new activity, they would eliminate an old one. The children didn't mind the new system; they don't realize they are learning to incorporate an important life skill of setting boundaries in their lives.\n\nBrendan talks to them frequently about the need to eliminate, streamline, or delegate when schedules get too busy. He's also working to help the family schedule into their lives some time to relax together.\n\n11. PUT YOUR FOOT DOWN AROUND PROCRASTINATORS\n\nDo you hate it when you are meeting someone and he is always late? Or when someone on your team procrastinates and holds up the whole project you're working on? If so, you need to figure out where to set boundaries.\n\nWhen you've waited for someone arriving late for a lunch meeting, you might say, \"Sorry you were tied up in traffic. I went ahead and ordered my meal because I have only an hour for lunch.\" Or you can stop setting up lunch meetings with that person. Depending on how much their being late bothers or stresses you, you can either tell them why you are no longer meeting with them or not.\n\nOf course, in many work situations, you can't always stop meeting with someone, so another option would be to have the meetings at your office. It won't matter if that person is late; you can continue working. Or if possible, meet with a different individual from that company or organization.\n\nIf you can avoid an in-person meeting, try getting together by phone. If that's not possible, can you make the appointment for the end of the day so that if he arrives late, it won't disrupt your entire day or throw off other meetings?\n\nSometimes you cannot make changes and simply have to put up with the late arrival. In those cases, do some stress busting so this doesn't become a high-anxiety experience every time. If meetings at work often start late because key people (such as the boss) keep everyone waiting, then don't schedule another appointment immediately afterward. Who needs that kind of pressure? Set it up so that if the meeting starts late and ends late, you can still have time to make it to wherever you have to be.\n\nInstead of fretting while you wait, you can try turning that time into a relaxing opportunity by either chatting with and getting to know your co-workers or bringing along some work you can do while you wait.\n\nBree used to feel nervous and resentful while waiting for a latecomer at a restaurant. Now she always brings to lunch or dinner meetings a carrying case with magazines and articles and uses it as a relaxing golden opportunity to catch up on her reading.\n\nWhen the procrastinator is on your team or committee, you might try saying, \"Since your procrastination is holding up the project, either you can write an e-mail to the bosses explaining that you caused the delay or I can write it. Which do you prefer?\" Harsh sounding? Yes. But sometimes a procrastinator is clueless about the consequences of her actions.\n\nSal's committee had way too many meetings about a procrastinator sabotaging a fund-raising event, so finally the chairman spoke to the culprit. His reply was, \"Sorry. I thought you guys were real laid-back about deadlines. I didn't know you were upset. I can bring things in on time from now on.\" Everyone on Sal's team was astonished at how easily the message was taken and how simple it was to solve the problem.\n\nWhen someone's behavior is causing your stress, speak up. You seldom have the power to set boundaries on other people's behavior (although in Sal's case, a simple discussion did accomplish that) but you can set boundaries on your part of the relationship or situation.\n\nTHOUGHTS TO PONDER\n\n_Time is the coin of your life. It is the only coin you have,_ _and only you can determine how it will be spent. Be_ _careful lest you let other people spend it for you._\n\n\u2014CARL SANDBURG\n\n_Your children need your presence more than your_ _presents._\n\n\u2014JESSE JACKSON\n\n_You can do anything, you just can't do everything._\n\n\u2014ANONYMOUS\n\n_The most desired gift of love is not diamonds or roses or_ _chocolate. It is focused attention. Love concentrates so_ _intently on another that you forget yourself at the_ _moment. Attention says, \"I value you enough to give you_ _my most precious asset\u2014my time.\"_\n\n\u2014RICK WARREN, _THE PURPOSE DRIVEN LIFE_\n\n_We must be willing to get rid of the life we've planned,_ _so as to have the life that is waiting for us._\n\n\u2014JOSEPH CAMPBELL\n\n_\"Kindly let me help you or you will drown,\" said the_ _monkey, putting the fish safely up in a tree._\n\n\u2014ALAN WATTS\n\nEXTRA CREDIT\n\n1. Do you ever let a ten-minute, fairly unimportant job take days out of your life?\n\n2. Do you give more control of your life than you would like to technology?\n\n3. Do you need to set boundaries on certain people in your life? Who are they and how will you do that?\n\n4. Do you need to set boundaries on the time you spend working? How will you do that?\n\n5. Do you need to set boundaries on projects? How will you do that?\n\n6. Do you need to set boundaries on commitments? On saying yes to requests? With whom and how will you do that?\n\n7. Do you have some negative expectations of experiences or situations that you need to set a boundary on or recast in a positive way? What are the situations and what can you do to turn them around?\n\n8. Do you need to set boundaries on your schedule? How will you do that?\n\n9. What other areas cause stress for you? Can you set any boundaries on them? How will you do that?\n\n **CHAPTER 7**\n\n[Strive to Recharge Your  \nBattery Daily](chap8_manageyo_9780802719768_epub_part8.html#d41151431)\n\nAT A MEETING, BOB RAN INTO HIS friend Ken, who is head of human resources for a large corporation. Ken was talking about how he was given this year's United Way campaign to run and all his many planned activities to try to generate 100 percent participation. Bob said, \"Holy cow, Ken, you better watch out or you'll burn out.\"\n\nKen replied, \"Naw, that'll never happen.\" When Bob asked how he could be so sure, Ken told him, \"Every night after dinner, I go for a brisk invigorating walk with the kids, and if that doesn't shake off the stress of the day, I sit down and bang away at the piano. That works for me every time.\"\n\nLater that evening, Bob was filling in his wife on how Ken was doing, and he said, \"The thing that surprises me the most is that he has a _plan_ for sidestepping burnout, you know, for recharging his battery if he gets stressed out. Who ever does that?\"\n\nHis wife replied, \"Lots of people, including me. Didn't you ever notice that if I have a rotten day at work, I come home and do yoga, and end the evening by taking a warm bath? And that nurtures my spirit and my mind, and my body.\"\n\nYou might think that in your life there are days when the work and responsibilities keep popping up, the time flies by, and at day's end, there is no time or energy left for activities to revitalize yourself. So you think that's just the way your life is. But it doesn't have to be.\n\nMany people who have high-stress jobs, extreme deadlines, or overwhelming responsibilities manage to stay healthy, calm, and serene because they know how to manage stress. Randy Davis, founder of the Strategic Millionaire, says his best way to recharge is to avoid becoming overwhelmed. \"I stay crystal clear on my purpose and passion. I always ask, 'What is the outcome of this activity?' I've become ruthlessly brutal in prioritizing. I might have a hundred things coming at me but only ten are truly important. That's what I focus on.\"\n\nIt's a fun exercise to ask people what they do to relieve stress. Some people have to think long and hard about it; others know exactly what their plan is. You often learn new information about friends you have known for years. People say things like, \"I've worked with Joe for ages and never knew he played bagpipes,\" or \"Bea has lived next door since last summer and I just discovered she has a passion for Lithuanian weaving.\"\n\nTen Tips for Reinvigorating Yourself  \nand Reducing Stress\n\nSometimes the battery recharger is a passion; sometimes it's a simple ritual or technique. Even people who seem to have no opportunity to recharge have figured out little luxuries that mean the world to them. Here are some popular, effective tips on how to replenish yourself, especially during times of stress.\n\n1. DUST OFF YOUR GREEN THUMB\n\nWhether you putter around in a garden, maintain houseplants, or nurture a windowsill herb garden, working with plants can be very relaxing. Cheryl is a wife and mother of four sons, one of whom is a fifteen-year-old with autism and attention deficit\/hyperactivity disorder. She wrote, \"I sleep well only when he's down for the night and the family is there to help if he decided to get up during the night. I cannot allow myself to sleep when Lance is awake unless his personal care attendant or family members are there to watch him.\n\n\"Yes. I do recharge my battery. I _have_ to! Planting amaryllis and poinsettias are my great stress busters.\"\n\n2. TAKE CHARGE OF TAKING CARE OF YOU\n\nIn my keynotes and training sessions, when I ask audiences, \"Who takes care of you?\" they give a vast variety of replies. People say that their spouses take care of them or their parents, kids, families, or friends.\n\nSome believe that their employer or organization or the government (oh, really?) will take care of them. Some seem to think that life in general or some vague entity out there will be their caretaker or that simply because they do all the right things, someone somewhere will care for them. Several people tell me that God takes care of them and I, too, believe that God cares for us, but I also believe that the Lord helps those who help themselves.\n\nThere's the old joke about a fellow who prayed to God to win the lottery . . . and didn't win. The next week he was on his knees again praying to win. Again, no luck. The third week he prayed with all his heart to win the lottery, and again didn't win. At this point, he was so angry that he yelled at God, \"I've prayed so hard to win the lottery. Why haven't you helped me?\" And he heard a booming voice answer, \"Meet me halfway. Buy a ticket!\"\n\nYou've got to do your part. If you're stressed, learn stress management techniques. If you're depressed, find a therapist who is a good match for you. With a little creativity, you can find forms of self-care in the most trying of situations. Even road warriors who travel constantly can create systems to care for themselves.\n\nLinda Brakeall, author, speaker, and self-proclaimed general hell-raiser, e-mailed me her self-care strategy: \"Travel is my biggest stress, so here are my 'on the road' stress busters. I force myself to get extra sleep, I carry my own pillow and comfort items like essential oils (lavender is good as a soothing agent!), I take a shower as soon as I get to my hotel room and wash off the travel vibes. Really makes a difference.\n\n\"I take trashy novels and other fiction with me as a treat because I read only work-related nonfiction at home, and call in every night to talk with my honey. I eat a high-protein, low carb, no-sugar diet when traveling and _no_ liquor.\n\n\"Every day I try to take a walk to absorb more oxygen, which is energizing, and I bring extra vitamin B, E, & C with me.\"\n\nOnce, when I asked the question \"Who takes care of you?\" a woman replied, \"I do, but I'm not doing a very good job.\" Don't leave it up to someone else\u2014you are in charge of taking care of you.\n\n3. ASK FOR WHAT YOU NEED\n\nDon't expect others to know what you need. They don't have a crystal ball.\n\nUmmi and her children laugh about the time she was fed up and told her family, \"Stay away from me. Keep out of my face. I need space.\" Then the very next day, she had a fender bender, and when she came home, she very pathetically said with a sigh, \"I'm so upset. I wish someone would just give me a hug.\"\n\nSometimes you need space; sometimes you need an embrace. Needs change all the time.\n\nEllen, a nurse, recently explained to me that in Germany, when you want to celebrate a special birthday, _you_ throw the party. That way, you have the type of party you want, as well as the people you would like to have there with you.\n\nHave you ever had someone ask you, \"What do you want for your birthday?\" What do you say? Did you ever say \"nothing\" and feel disappointed when that's exactly what you got?\n\nDon't sit around waiting and hoping for someone to come along, read your mind, and handle your stress. If you know what you need from family and friends to bolster your efforts at reducing stress, ask for what you want.\n\n4. ADD MORE FUN TO YOUR LIFE\n\nGeorgia went on a quest to find ways to eliminate her stress. Her minister told her that would be like taking a bucket full of water out of a lake. The \"hole\" will simply fill in. If you could wave a magic wand to make all your stress go away, it would be replaced with new stress. And sometimes the new demands on you are ten times worse than the ones you had before.\n\nThe secret is that in the midst of demands being made upon you, you have to find ways of adding joy and fun to your life. You must discover what you love and make time for it.\n\n**BUT WHAT IF I DON'T KNOW HOW TO HAVE FUN?**\n\n\u2022 What did you love to do as a kid? Ride a bike? Finger-paint? Whistle? Jump rope double Dutch? Do it!\n\n\u2022 Learn one new great joke and tell it all month long.\n\n\u2022 Sing or play an instrument you loved as a child. If you can't find someone to join you, start off singing, dancing, or playing music by yourself or with your children. If you are too shy to do karaoke in public, ask for your own karaoke machine or see if a friend might own one and let you \"play.\" You might even consider taking music lessons or an acting class at a local college.\n\n\u2022 If there's a live concert of your favorite music anywhere in your area, or a sports event (not at all the same as watching it on TV), make time to attend. If you wait until you get the time, it'll never happen. You need to nurture your spirit now.\n\n\u2022 What is something you've always wanted to do?\n\n\u2022 Cook a meal where the making of the meal is fun for the whole family (put a pizza together, chop stuff for tacos, create your own rollups).\n\n\u2022 Try some time spent outdoors in the fresh air, even if you think you're not an \"outdoor person.\" Walk around your block or through a park or botanical garden. Plant a few flowers or tomato plants. Try this alone or with someone. You never know what you might discover.\n\n\u2022 Get back to playing with kids. None of your own? Volunteer to work with an organization that lets you spend time with kids. They will teach you a lesson about fun.\n\n\u2022 If you have children, schedule time with them separate from working on homework or pushing them to do chores. Just have time to play or talk or walk together.\n\n5. PRACTICE MEDITATION\n\nRosita attended my seminar and said, \"I used to think meditation was just sitting and not thinking, and I couldn't do that. But I believe meditation is any time you really focus on something.\n\n\"Setting a beautiful table, even if dinner is only for yourself, can be a soothing, focused meditation. Walking or knitting or working on a hobby or listening to my favorite music\u2014or dancing to it\u2014can be a soothing, focused meditation.\n\n\"I multitask sometimes, but when I need to get balanced, to get centered, then I just purely focus on doing one thing, and doing it well, doing it beautifully. That's my soothing meditation.\"\n\nThe group was spellbound listening to Rosita.\n\nThen Paul spoke up. \"I find that walking is my meditation. I used to walk listening to CDs but now I just listen to the sounds of life and pay attention to my breathing. If thoughts come to me, I give them a bit of attention, then send them on their way and go back to paying attention to my walking or breathing.\n\n\"I used to feel guilty leaving the house and going for a walk when there's so much to do, but now I know that walk will leave me clearer thinking and energized.\"\n\nCody, a CFO, said, \"My way of meditating is I keep a gratitude journal. Every day I write down three things I'm grateful for. My attitude has changed, and I don't react with extreme stress like I used to.\"\n\n6. TAKE REFRESHING BREAKS\n\nRick, a physical therapy technician, told me, \"When things start to pile up on me, I build a break time into my schedule. Maybe a two-hour lunch or sleep an extra hour or two, or work hard for a couple of days so I can take time off to watch a ball game\u2014guilt free.\"\n\nAlison, a certified public accountant in public practice, described her \"recharge\" break to me: \"I play with my daughter who is almost two. I don't worry about the clients, laundry, dishes, filing, or paperwork because none of that matters to her! She's the 'little thing' that reminds me of why I work hard.\"\n\nHere's a step-by-step recipe for a refresher break from a single mother of five: \"I have cookies and tea while I read something (the newspaper or a novel) to take me mentally away from the tasks that are depressing or overwhelming. I give myself an hour and always put two or three cookies on a plate so I don't eat the whole box.\"\n\n7. MAKE AN APPOINTMENT WITH YOURSELF\n\nIf you're like many people, you need and want to make time for what is of value to you, but at the end of the day\u2014after pushing to be productive and effective\u2014there just is never any time or energy left for those other important life factors, so you say, \"Maybe tomorrow.\" But when you have zero time for those most important things in life, that's when you most need to step off that hamster wheel and do something to recharge your battery.\n\nMake an appointment with yourself to do something you enjoy\u2014and be as vigilant in honoring that commitment as you would if the appointment was with someone important. After all, it is.\n\nHave you ever heard someone say, \"Wow! I've been so busy today; I haven't even had time to go to the bathroom.\" Now I ask you, who do they think is in charge of taking care of _that_?\n\nIf you would just make the time to stop, take a break, go for a walk, or take a nap (or\u2014for heaven's sake\u2014go to the bathroom, if need be), you would most likely return to your job feeling energized and much more productive and effective.\n\nMike Dooley, an author and professional speaker, sends out a daily e-mail signed from \"the Universe.\" One that addressed time said:\n\nThere's so much time in a day, you could have breakfast, lunch and dinner on 3 different continents. You could outline the book you're going to write, start the screenplay adaptation, and watch \"Gone With the Wind,\" before the sun even sets. Spend a day at work, and still have 16 hours left over. Or you could just think 10,000 different thoughts as you tool all over. Hey, the record for climbing Mt. Everest is under 9 hours, leaving 15 to nap and go Yeti searching.\n\nThere's so much time in a day. So much. Especially when one stops claiming there's so little of it, huh?\n\nYou're rich,  \nThe Universe\n\n**BUT I'LL HAVE TO CATCH UP ON  \nSO MUCH WORK LATER **\n\nDo you worry that if you take some time to nurture yourself that you'll have to work like a dog to catch up on all the work you ignored? Maybe once in a while that will happen, but if you are feeling refreshed and re-created, you'll probably find that you can get that work done much quicker than when you were just plodding along, going from one task to another with no enthusiasm.\n\nTo cut down on the number of e-mails waiting for you, set your e-mail to send an automatic reply that says how long you will be away from the office. If possible, arrange for a colleague to be willing to be contacted in case of emergency. You could program a similar outgoing message for your voice mail system, saying when you will return and offering an alternate number for someone covering emergencies. When you do return to work, look at e-mail from the most important people first. Consider saving low-priority e-mail in a folder to be read later.\n\nShannon told of pushing herself from project to project, never taking a break because she was determined to prevent a work pileup. Then she caught the worst cold she's ever had and was so sick she could hardly get out of bed. She missed four days of work, and when she returned, there were stacks of paper to work on all over her desk. But she realized that she wasn't buried by that crush of work. Now she takes \"recharge breaks\" by planning her breaks and getting a lot of things done before she leaves so she doesn't have to play catch-up when she returns to work.\n\n8. BANISH GUILT\n\nIf your doctor told you, \"You must exercise a certain number of times per week,\" would you question whether you deserve to spend that much time and energy on yourself? Of course not. Yet how many people say they want to exercise but can't find the time? They usually find the time and energy to take care of others, but not themselves. They may feel they are being selfish if they go to the gym when they could be spending more time with their mate, children, or friends. That's what you focus on when the guilt strikes. You need to tell yourself\u2014and to believe\u2014that you deserve to do what's needed to recharge your battery. Then you'll see that guilt start to evaporate.\n\nPatricia said, \"If I did something just for me, I couldn't live with myself. I would feel so selfish and everyone would wonder how I could pamper myself like that.\"\n\nDo you find yourself nodding in agreement? What's the solution?\n\nIf by \"did something just for me\" you mean all day every day, that's not what I'm advocating. Yes, you might know a few people who disguise their selfishness as self-care, but they are not many and they are not you. By taking just a small amount of time regularly to do something that re-creates your spirit and refreshes your soul, you will find yourself feeling more energized and able to continue the good work that you do.\n\n9. BOOST YOUR ENERGY LEVEL\n\nThere is no denying that during especially busy periods you are so exhausted you're ready to keel over, and you can't imagine doing anything other than falling into bed. But sometimes fatigue and lack of desire to do things you normally enjoy are signs of burnout.\n\nDr. Norman Vincent Peale often said, \"Enthusiasm creates energy.\" Everyone has different ways of generating enthusiasm. For some, it's reconnecting with old friends or spending time with grandchildren. For others it's a hobby or some form of artistic expression. Have you ever experienced being so enthusiastic about a project or hobby that you wind up working on it way past your usual bedtime, but the next day you feel just fine? The energy is there.\n\nEach of us has to discover for ourselves what we will be enthusiastic about. The search is worth the effort because the energy created by enthusiasm gives us back our zest for living.\n\n**TWELVE ENERGY BOOSTERS**\n\n1. Exercise. Just thirty minutes of walking three or four times a week will dramatically increase your energy. The exercise might tire you out at first, but in a short time you'll feel your fatigue melt away.\n\n2. Decrease your intake of caffeine and sugar. Their energy bursts are only temporary. The crash they cause afterward will leave you more tired than before.\n\n3. Increase your intake of fruits and veggies. This\u2014 combined with number two\u2014will not only add to your energy but might put you in a better mood.\n\n4. Get enough sleep. The amount needed varies from person to person, so it's up to you to figure out your optimum hours of sleep. Not too much, but enough to leave you refreshed.\n\n5. Consider \"power naps\" (if possible). They can either revive you or leave you feeling more exhausted. Like the ideal amount of sleep you need, this is something only you can figure out.\n\n6. Drink plenty of water. Dehydration can make you feel slow and sluggish. If you drink any caffeinated beverages (coffee, tea, or soda) you will be more dehydrated, so make certain to drink plenty of water afterward.\n\n7. Do something that's fun.\n\n8. Take breaks. Any break can energize you\u2014a five-minute midmorning break stepping outside and breathing fresh air; eating lunch away from your desk; setting aside a weekend day as a re-creation day free of work; or a two-week trip.\n\n9. Eat a midmorning or midafternoon energizer. Instead of sugar or caffeine, try a handful of nuts (providing protein and omega-3) or an apple with low-fat yogurt or low-fat cheese.\n\n10. Get fresh air. Whether it's stepping outside for a few minutes or a two-mile walk, you will find fresh air clears your head and perks you up without fail.\n\n11. Consider learning some yoga. The deep breathing, stretching, and twisting soothe tired muscles, increase oxygen intake, improve circulation, and aid digestion, all of which will help you feel energized and give you a sense of well-being.\n\n12. Eat a variety of foods. To eat only one certain kind of food will prevent your getting the amount of vitamins and minerals you need for optimum health. For example, a diet of mostly carbohydrates tends to lack potassium and magnesium, which are energy boosters.\n\nHerman told me he didn't quite understand the idea that enthusiasm creates energy until one particularly difficult work day. He came home feeling burned out and exhausted, and announced to his family, \"I'm going to be in a state of total collapse tonight. Don't expect me to do anything, I'm going to be a horizontal vegetable. A couch potato. Got it?\"\n\nFive minutes later, the phone rang. It was one of his close friends who had moved to the other end of the country. \"Hi, Herman, it's Rudy. Guess what. I'm in town for a conference. Can you get away tonight to meet for a drink?\"\n\nHerman was astonished to find that his exhaustion was gone. Totally. He was up and dressed and out the door in minutes, and he had a great visit with his friend.\n\n10. GET MORE\u2014AND BETTER\u2014SLEEP\n\nSometimes you might be fatigued because you aren't getting enough rest when you go to sleep. This can affect your health and your stress level\u2014even how long you live\u2014much more than you think. If you have trouble sleeping, take a look at the tips below to help you rest better at night. Then combine them with a few of the energy-boosting tips, and you may find your fatigue melting away in spite of your overscheduled life.\n\n**FIFTEEN STEPS TO A GOOD NIGHT'S SLEEP**\n\n1. Make your bedroom a restful place. Clear out clutter, try scented candles, incorporate your favorite beautiful things\u2014pictures, pillows, attractive sheets, flowers, wonderful decorations. Make it a welcoming, soothing sanctuary.\n\n2. Protect your bedroom from office work, watching television, or playing video games (unless it helps you fall asleep).\n\n3. When it's time to sleep, make the bedroom dark, cool, and quiet.\n\n4. Avoid heavy eating before bedtime. If you must snack, stick to peanut butter, turkey, bananas, or other foods rich in tryptophan, which has a natural calming effect.\n\n5. Try sipping warm milk or herbal tea to help you doze off. Remember when mothers used to give kids warm milk before bedtime? It still works.\n\n6. For at least three to six hours before bedtime, no caffeine, nicotine, or alcohol. Some people need to put strict limits on caffeine, such as mornings only or none after two p.m. Pay attention to what your body tells you.\n\n7. Exercise helps you sleep, but not close to (within three hours of) bedtime.\n\n8. Nap no later than midafternoon, and not so long that it interferes with your sleep. Pay attention to how naps affect you to help you determine how long a nap you need to feel refreshed but not wide awake at bedtime.\n\n9. Keep a regular sleep\/wake schedule. Go to bed and wake up within two hours of the same time every day, even on your days off.\n\n10. Consider wearing socks to bed. Could it be cold feet that are keeping you awake?\n\n11. Keep a pen and pad of paper nearby to write down your worries or things to do, so you can free up your mind to drift off to dreamland.\n\n12. Put work aside at least two to three hours before bedtime and switch to something that relaxes you.\n\n13. If you are still tossing and turning twenty minutes after going to bed, get up and read, listen to soothing music, or do something else relaxing for a while. Don't fight sleeplessness.\n\n14. Let light help you. Dim the light for a few hours before you go to bed and light up the room or go out into sunshine soon after you wake up to help set your brain's internal clock to your sleeping and waking schedule.\n\n15. If you continue to struggle with going to sleep, consider a visit to a sleep specialist.\n\nVince bounces out of bed early during the week, so he used to sleep until noon on the weekend \"to catch up on sleep.\" But when he slept that late, he usually felt heavy and sluggish for the rest of the day. He often heard that it's important to keep a consistent wake-up schedule, but he didn't think it applied to him.\n\nOne Saturday he had to work, so he woke up at his usual workday time. He was surprised to notice that he felt great all day and even more surprised that he seemed to have more energy that Monday and the rest of the week. Vince now tries to wake up around the same time even on weekends because he is convinced this new schedule has helped to improve his general energy level.\n\nTHOUGHTS TO PONDER\n\n_Without enthusiasm you are doomed to a life of_ _mediocrity but with it you can accomplish miracles._\n\n\u2014OG MANDINO\n\n_Long ago when men cursed and beat the ground with_ _sticks, it was called witchcraft . . . Today, it's called golf._\n\n\u2014ANONYMOUS\n\n_People are lonely who build walls instead of bridges._\n\n\u2014MARTIN BUBER\n\n_When you love what you do and you believe that it_ _matters, what could be more fun?_\n\n\u2014MARTHA GRAHAM\n\n_Sometimes our light almost goes out but is blown again_ _into flames by an encounter with another human being._ _Each of us owes the deepest thanks to those who have_ _rekindled this inner light._\n\n\u2014ALBERT SCHWEITZER\n\n_Those who bring sunshine to the lives of others cannot_ _keep it from themselves._\n\n\u2014JAMES M. BARRIE\n\n_Don't ask yourself what the world needs, ask yourself_ _what makes you come alive. And then go do that. Because_ _what the world needs are people who have come alive._\n\n\u2014HAROLD WHITMAN\n\n_Be gentle with yourself._\n\n\u2014MAX EHRMANN\n\n_He who cannot rest cannot work; he who cannot let go_ _cannot hold on._\n\n\u2014HARRY EMERSON FOSDICK\n\nEXTRA CREDIT\n\n1. If you were given twenty-four hours with no agenda or responsibilities, what would you do?\n\n2. If you were given seven days with no agenda or responsibilities, what would you do?\n\n3. If you could wave a magic wand and have the time, money, and talent to do whatever you want to do, what would it be?\n\n4. What makes you laugh or feel happy?\n\n5. What are you enthusiastic about? Passionate about?\n\n6. What were you doing the last time you felt relaxed? Joyful? Recreated?\n\n7. Decide on one thing to recharge your battery that you enjoy or adds fun to your life, or something you are enthusiastic or passionate about. What re-creates you?\n\n8. Schedule some time this week for yourself (even if it's only twenty minutes). If you still struggle with reasons why you procrastinate about recharging your battery, list those reasons here.\n\n **CHAPTER 8**\n\nNow Let's Put It All Together\n\nGERRI HOPED TO MAIL THANK-YOU NOTES two weeks after her wedding, but when two months had gone by, she felt that the littleshort note cards she had purchased were no longer adequate. At that point, she felt she owed people more than one or two sentenceson a thank-you note.\n\nEvery time she thought about writing those notes, she felt her heart speeding up and she would become so stressed out thatshe'd end up with a blazing headache. She was overwhelmed at the thought of writing thank-you notes to 350 people. She wouldn'task her new husband for help because she had promised him ten days after the wedding that she'd take care of it, and she hatedto admit to him that she hadn't done them.\n\nThen a third month went by, so she felt she owed everyone a letter, maybe with some made-up excuses telling everyone why shewas so late thanking them. Her husband's mother had already commented that relatives hadn't received thank-yous yet. Gerri'shusband offered to help but still nothing was done. She could not get herself to do this dreaded job, and she was startingto feel sick about it.\n\nYou've probably seen something like this\u2014maybe even lived it\u2014a hundred times. A bit of stress hits you, so you put off a fewthings. Then one of the tasks you procrastinated about escalates, causing you even greater stress.\n\nYou started reading this book probably because you were feeling overworked, overscheduled, or overwhelmed. Maybe you wantedto learn more about time management without adding to your stress. Or maybe you wanted to ease your stress without dilutingyour productivity, effectiveness, or efficiency. This book has provided strategies, techniques, tips, and ideas for meltingaway your stress and stress-related symptoms. But none of them will help you if you put off implementing them. Here are fivesimple steps to help catapult you into action.\n\n1. PUT THE SITUATION INTO PERSPECTIVE\n\nSharon, who works at the counter in the post office, misplaced the forms for money orders. Her customer was kept waiting whileshe searched for and finally found them. As she completed his order, she repeatedly apologized for keeping him waiting. Then,with a warm smile, he calmly said, \"If this is the worst mistake you make today, you are having a great day.\"\n\nSharon said that has become her motto when she needs to change her reaction. When she was in a panic because she thought she'dlost her purse, she said to herself, \"Well, if this is the worst thing that ever happens to me then I'm having a pretty goodlife.\" She repeats that phrase when she's frantic because a long train is causing her to be late for an appointment, whenshe's upset that her teenage son got a speeding ticket, when she's disappointed because she didn't get a promotion.\n\nWhen she was feeling betrayed by a co-worker, she snapped herself out of it by repeating, \"Well, if this is the worst thinga friend ever does to me, I guess I'll survive.\" She says that odd little phrase has helped her gain perspective and shakeoff stress in dozens of potentially stress-causing situations.\n\n**SELF-TALK THAT ENABLES YOU TO CHANGE**   \n**YOUR REACTION OR RESPONSE**\n\nIf this is the worst thing that ever happens to me, then I'm leading a pretty good life.\n\nIf this is the worst mistake I ever make, then I'm leading a pretty good life.\n\nIf this is the worst thing my child ever does, then I'm leading a pretty good life.\n\nIf this goes wrong, I'll survive it.\n\nIt doesn't make sense to expect perfection from myself, others, or life. This is an imperfect world.\n\nEveryone doesn't have to do what I want them to do or do things my way, and I'm OK with that.\n\nIt's OK for me to not agree with someone or to tell someone I'm hurt.\n\nIt's OK for me to be too tired or busy to do something that is not important to me.\n\nWill this matter six months from now?\n\nWill I still be worried about this a year from now?\n\nWorry is unproductive. I'll write down my worry in a worry notebook, date it, and stop dwelling on it.\n\nI don't have to solve everyone's problems.\n\nNo matter what is going wrong in my life, I will look for something to be grateful for today.\n\nInstead of blaming someone for this, it's OK for me to take responsibility for it.\n\nTaking responsibility for something doesn't mean I will blame myself constantly for everything wrong with the world or my life.\n\nMy anger doesn't hurt the one I'm angry with but it can make me sick.\n\nIt's OK if others don't do something to make me happy. I'm in charge of becoming happy.\n\nWhy am I upset? No, really, what specifically is upsetting me? What part of this is upsetting me?\n\nSo what if they didn't like it? I like it.\n\nI made a mistake; now what can I learn from it?\n\nSo what if they don't agree? I have a right to have an opinion.\n\nIt's OK for me to ask for help and it's OK for someone to refuse me.\n\nIt's not possible to make everyone happy.\n\nThere's always someone who will dislike\/hate\/criticize me.\n\n2. DECIDE WHAT YOU'RE GOING TO DO\n\nOne of the biggest causes of procrastination is difficulty making a decision.\n\nBrad, a Realtor, has dreamed of starting his own small restaurant since college. He desperately wants to leave the stressfullife of real estate, and even though he knows he will have to work hard running a restaurant, he feels that doing what heloves will omit much of the stress involved. He knows what he needs to do but he hasn't even decided what steps to take. Bradsays his life has been so busy he doesn't have time, but the reality is that he hasn't made a clear decision to do anythingabout it.\n\nKaren has planned on starting an exercise program as a way to prevent burnout and also says that she hasn't had time to doanything about it. The reality is, she just hasn't decided to take some steps to get started.\n\nAbbey is embarrassed by all the clutter in her house, and just looking at the piles of paper and stuff leaves her feelingexhausted. Once she misplaced an electric bill and another time she lost a wedding invitation and missed the wedding. Shehas been planning to declutter her house for months, and she's aware that just as her clutter took a long time to accumulate,it won't disappear in a day. But she can't decide what to tackle first.\n\nRegardless of what's causing you stress, decide what your first step will be in relieving it or working toward your goal.It can be something simple. But commit to doing something\u2014give yourself a timetable for when you will finish doing whateverit is.\n\nAndrew was in his forties, overworked, and stressed out. He liked his job but felt unfulfilled. Since high school he had beenaware that he had a gift for writing but was too tired or unmotivated to write on his own at night or on the weekend. Awarethat doing something he was enthusiastic about would reenergize him, he decided that during the fall he would enroll in aten-week continuing education writing class at his local community college, even though he felt he didn't have one spare momentin his busy life.\n\nHe loved the class and was especially inspired by a class assignment to \"write about something you know.\" Andrew was an avidgardener and his article combined spirituality and gardening. The teacher said, \"Andrew, I think you have the beginning ofan excellent book here.\" And so, Andrew is now working on his first book.\n\n3. START SMALL\n\nYes, you may have a gazillion stress-busting things you want to do\u2014don't start with the biggest, most challenging thing. Onesmall change is a great way to start.\n\nInstead of committing to exercising every day of your life till the day you die, start with a ten-or fifteen-minute walk duringyour lunch hour or after work. Rather than trying to change your whole personality, just put up a few reminders that strivingfor excellence is achievable and perfection usually isn't.\n\nGerri, the bride at the beginning of the chapter, was totally overwhelmed at the thought of writing 350 thank-you letters.First of all, she needed to realize that many people came as couples, so she'd really be writing 175 to 200 thank-yous. Butthat is still overwhelming, so her best bet was to develop a time-saving system. She designated a spot where she gatheredup all she needed into one place\u2014the stationery, envelopes, stamps, address book, everything\u2014and then was able to leave itall together in that spot for several weeks. She also cut back on her expectations\u2014 instead of writing letters offering lengthyexcuses, she wrote notes that focused on the couple's gratitude.\n\nShe decided on a realistic and attainable goal of writing to five or ten people each evening. If she watched TV, she wrotenotes or addressed envelopes during commercials. Once she did this for several evenings, she realized it wasn't the wholejob that she dreaded, it was just getting started. She decided she could probably start doing more thank-yous each evening.And eventually, that job was done.\n\n4. USE A TIMER\n\nWhen Andrew decided to take a writing class one night a week he found that the more he enjoyed it, the easier it was to maketime for the class each week. I can hear you starting to panic. Fear not. Nobody is expecting you to give up any huge amountof time.\n\nIt's your stress. It's your time. It's your life.\n\nThe fact is, most people can carve out one hour occasionally. Not an hour every day, but almost everyone can free up one hourper week. That's where you start.\n\nIdentify what change you want to make in order to take control of the stress in your life, and give it one hour per week.It might be an hour of work streamlining your files so it's easier for you to retrieve information, or it might be one hourof an activity that recharges you, such as reading, working on a hobby, or meditating. Set a kitchen timer for sixty minutes.\n\nBill, a purchasing agent, used to say that he could time everything in his head, and he could. But then he realized that thetick-tick-tick of a timer really adds a sense of urgency to the job. Now he keeps a kitchen timer shaped like an apple onhis desk and uses it every day\u2014to remind him of a conference call, to place time limits on chatty vendor calls when he wantsto set boundaries on those conversations, and of course for his one-hour blitz of his goals. He can't remember how he wasable to stay organized before he kept a timer on his desk.\n\nDuring that hour of working on your goal, there are only two rules:\n\n1. Ignore everything else. Not forever, just while the timer is ticking. This includes phone calls. (If you weren't thereor if your phone was turned off, they would have to leave a voice mail. They can survive doing that while the timer is ticking.)\n\n2. No breaks. If you forgot to get a glass of water, cup of coffee, or that mint mocha frappuccino before you started, youhave to wait until the timer dings.\n\nIt's the interruptions and the breaks that sabotage our lives. How often have you started off with a firm commitment to gothrough your old e-mails, file what's important, and delete the rest, but you keep stopping to check your incoming messages.Then you decide to take just one little minute to make two quick phone calls and follow up on requests or questions raisedin the phone calls. Suddenly you look at the clock and you decide,\"Well, it's too late to get started on organizing my e-mailtoday. I'll have to tackle that when I get the time.\"\n\nOr you look around and see that you have started fifteen tasks and completed none of them. You say, \"Well, I was multitasking.\"You might be astonished at how much work you accomplish in one hour when you follow those two rules of no interruptions andno breaks.\n\n5. IT'S TIME TO DO THE WORST FIRST AND REWARD YOURSELF\n\nBob's job included writing up a report every Monday morning logging the phone calls received over the weekend on the emergencyline of the mental health center where he worked. The report was to be sent to the human resources department and took onlyabout twenty minutes to write. But often that report still was not completed in the afternoon.\n\nIt was as if a black cloud of dread hung over Bob's head all day long. If someone asked him to go out to lunch with them,Bob would say he couldn't go because he still had that danged report to do. He knew if he could get himself to do that reportfirst, he would feel free the rest of the day, but he just couldn't get himself going on it. Then he heard about using rewardsto motivate himself.\n\nHe really loved coffee. So his plan was to allow himself one cup at home on Monday. Then once he was at work, no coffee untilthe report was complete. Now every Monday, Bob starts work at nine A.M. and has that report to HR by nine thirty. Then hemakes a big deal out of \"going down the hall to meet with Juan Valdez and his burro for my cup of fresh roasted mountain-growncoffee.\"\n\nNothing worked to get him to do this report first thing in the morning until he came up with this coffee-reward idea. Twoyears later, Bob still turns in his report by nine thirty, then races down the hall for his beloved cup of fresh roasted mountain-growncoffee. He says, \"If it works, don't fix it.\"\n\nWhen you find the right reward, it will help you reach many goals. Just don't kid yourself and say, \"Well, the accomplishmentof the job itself is enough reward for me.\" Heck, if that's true, why did you put it off in the first place?\n\nJocelyn found herself working overtime almost every evening and usually brought home work, which took up most of her Saturday.Most of Sunday was spent doing laundry, cleaning the house, and grocery shopping, so there was never time to re-create herself.\n\nJocelyn would return to work on Monday feeling tired and dragging from all she did the previous week and weekend. Then, insteadof working at 100 percent, she often performed much of her work at 50 to 70 percent. As a result, she didn't complete whatwas expected of her, so she ended up putting in hours of overtime.\n\nShe knew she needed a change and decided to do some journaling to try to clarify her thoughts and values. Immediately sherecognized that what she hated most about all her overtime was missing time with her children. She loved doing activitieswith them like going to the zoo, museums, the beach, and the movies, and she hadn't done any of that in almost a year. A chillgripped her heart as she thought about how fast her children were growing up and what she was missing with them.\n\nJocelyn decided that in the coming week she would write out what she needed to accomplish each day. Then she would keep focusedon those goals and push to work at top speed to get everything complete. She set boundaries that she would not work past fivethirty and was not going to work more than one hour on any work she brought home. She was excited about her plan and enthusiasticabout her intended reward: a day with the kids at the zoo.\n\nJocelyn's enthusiasm enabled her to feel more energized and she completed everything she planned for that week. On Saturday,she told her sons about her plan, how she wrote out her goals and reached all of them. And now she was rewarding herself witha trip to the zoo, and they were all invited to join her. The whoops and hollers showed Jocelyn the joy the boys felt, andshe felt it too. What a magnificent lesson to teach them. Reach for goals, work hard, and then give yourself a fun reward.\n\nHaving a balance between work and play is very important. According to Neil Fiore, author of _The Now Habit: A Strategic Program_ _for Overcoming Procrastination and Enjoying Guilt-Free Play_ , recharging your battery and relaxing can make the difference between being productive and finishing a project or just taking forever and procrastinating. Dr. Fiore led a research project several years ago where he compared Ph.D. students who took less than two years to complete their doctoral program to those who took between three and nineteen years. The difference between those who took a short time and those who took longer had nothing to do with intellect or emotion; it was \"who suffered more.\"\n\nStudents who took the longest had put their lives on hold; they didn't make time to enjoy themselves. They felt guilty whenthey were having fun because they felt as if they had a cloud of dread hanging over their heads reminding them they oughtto be working. Yet when they did focus on their dissertations, they worked halfheartedly because they were so deprived ofpleasure. They had created lives that were joyless. Everything took a longer time to accomplish and there was a tremendousamount of resentment.\n\nPeople who completed dissertations in two years or less didn't have the same philosophy of life. They didn't set themselvesup to be martyrs. They worked hard but in short bursts, dividing time between work and play or relaxation, and enjoyed theirlives.\n\nCherish Today\n\nBy understanding stress and clarifying your values, eliminating and streamlining wherever you can, trashing perfectionismbut reaching for realistic goals, setting boundaries, and recharging your battery, you can make radical, healthy changes inyour life. By making just a few changes suggested in this book, you can live the life you want. You can lead a full, rich,productive life without the stress and anxiety that many people accept as part of that lifestyle.\n\nAs John Greenleaf Whittier wrote:\n\n_The saddest words of tongue or pen_\n\n_Are these four words: \"It might have been.\"_\n\nYou will never get today back. If you have $1,000 saved and you decide to spend it on something important, you can earn another $1,000 someday and replace the amount you spent. Time is different. You will never have this day again. You cannot replace it because nothing will ever be this way again. Children in your life will never be this age again\u2014for that matter, neither will your family or friends. Most important, _you_ will never be this age again.\n\nMaria, a cheerful, positive-thinking housecleaner, said that she moped around for weeks when she turned sixty. She now looksback on that time and believes she actually went into a depression over that birthday. When asked what pulled her out of it,she said it was her eighty-six-year-old father-in-law. He said, \"Listen, Maria, no matter what age you are, there are peoplesomewhere wishing they could be that young again.\" Then he grinned and said, \"Me, for instance. I envy you. I wish I couldbe sixty again.\"\n\nMaria said hearing those few words snapped her out of her deep blue funk. A few weeks later, she had a great laugh when sheheard a 106-year-old woman being interviewed on TV. The woman said, \"I'm doing fine, but I'm not as spry as those ninety-nine-year-oldchicks.\"\n\nLearn to cherish each day for what it is\u2014not something to regret because it is flying by too fast, not a chunk of time inwhich to rush around trying to accomplish a gazillion tasks, but a magic twenty-four hours to be treasured on the journeyof living life to the fullest and loving every minute of it.\n\nIf you have clarified your values, stopped shooting for perfection, set realistic goals, tried to eliminate and streamlineevery place you can, set boundaries, and figured out ways to recharge your battery daily, and none of that works because youstill are working every single minute of every single day, then I presume you simply cannot make changes at this point. Evenpeople who have high-pressure careers or are building their own business can make some small changes. For example, leavingwork an hour earlier once a month to spend time with a loved one or to get a relaxing massage may be doable, though it maymean you cannot cut back on overtime the rest of the month. Or if you are caring for a very sick or difficult person, youcan contact your local family service center or health department to see if any respite services might be available to you.\n\nMaking changes in your life can be extremely difficult, but it is not impossible.\n\nMy hope is that as you start this journey, you will rediscover and awaken that childlike part of you that you were born with\u2014that part of you that loves life, that person within you who is interested and curious and excited and enthusiastic.\n\nThe actress Diane Keaton was asked in an interview, \"Given what you know from your own particular life, what would you passon to your young daughter at this moment?\" Her answer was passionate. \"It would be great if she had a huge, huge, huge appetitefor life. I really wish that for her. I hope that she embraces it enormously\u2014every aspect of it. Headstrong. I hope that sheis strong and curious and excited and interested, because that will pull her through.\"\n\nYour Personal Action Plan\n\nIt is very human to come to the end of a book and have the greatest of intentions. All kinds of resolutions are bouncing aroundin your brain and heart that when accomplished will make your life so much more wonderful. Then you set the book down andthe phone is ringing, e-mails are pouring in, the fax machine is spewing out papers, people are pulling at you from everydirection, the world is spinning faster than it used to, and every once in a while, you have fond memories of that book butcannot remember anything you wanted to do as a result of reading it. What to do?\n\nFirst of all, writing your personal action plan will enable you to remember your goals and ideas anytime you want by simplypicking up this book again.\n\nSecond, there is great power in writing down goals, plans, and resolutions. Somehow, what you write seeps slowly into yoursubconscious and sometimes later it bubbles up and nudges you to do something even though you haven't reviewed your writtenplans.\n\nThird, if you regularly refer to and review your written answers, it will help you keep on the path you have chosen to travelas you move through life.\n\nFourth, it's fun to look at your action plan a few years down the road (so be sure to date it) and see what changes have occurredin your thinking, your behavior, and your life. List here those values that are most important in your life right now.\n\nAs you clarify what is important in your life and what you want to spend your time on, you will have to decide what stepsto take in order to achieve your goals. To avoid getting stressed out, think about whether you need to take a course or buysome software or any kind of equipment (gardening, fishing, workout, etc.). Do you need to search for something (a used harp,a comedy improv class, an organization looking for volunteers with your talents)? Do you need to omit something from yourlife in order to make room for your stress busting (a group you no longer enjoy, a hobby you no longer delight in, clutterin an area that you need to use)?\n\nList here what steps you need to take to move closer to a joyful and serene life.\n\nIs perfection still one of the standards you strive for? What message or reminder can you write here to get you back to thereality of the imperfect world we all live in?\n\nHave you discovered that some of your goals were unrealistic? Which ones?\n\nTo help you think about your goals for now, try imagining what your life would be like ten years from now.\n\nWrite the date it would be ten years from now____________\n\nNow picture yourself answering these questions not today, but ten years from now:\n\na) How old are you (ten years from now)?____________\n\nb) Where do you live?____________\n\nc) What are you doing to earn a living?\n\nd) Who are you with?_________________________\n\ne) Where have you traveled?\n\nf) Describe your proudest moment in the last ten years.\n\ng) Describe a typical Saturday.\n\nh) What sort of things do you own?\n\ni) Now here's a biggie: How have you changed as a person? (Would it surprise you to know that most people include somethingabout being more relaxed and less stressed?)\n\nNow come back to the present and name one thing you can do in the next year to bring yourself closer to each of your goalsin ten years.\n\nWhat can you do in the next week to bring yourself closer to your ideal life?\n\nNow write three goals here that you want to accomplish in the next two years. Consider possibilities for adventure. What doyou want to be when you grow up? Your goals can be social, spiritual, physical, financial, intellectual, or emotional, orhave to do with relationships, pets, your home, your work, your transportation, or where you live.\n\nHave you given thought to anything in your life that you can eliminate or streamline? List them here.\n\nAre there any expectations\u2014of someone else or you\u2014 that you are eliminating or streamlining?\n\nHow will your life be different with these streamlinings and eliminations?\n\nHave you decided to set any boundaries? List them here.\n\nList at least three things you can do to recharge your battery daily. You can list lots and lots more if you choose.\n\nNow list one thing you will do _for sure_ for yourself within the next twenty-four hours.\n\nHave you ever said, \"When such-and-such happens, _then_ I'll be happy\" or \" _then_ I can manage my stress\" or \" _then_ life will be worth living\"? List here all the things you would like to happen in order for you to feel happy or relaxed or as if life is worth living.\n\nWhat steps can you take to make any one of these things you've listed come true? List them.\n\nOK, that's what you _can_ do. Now what are you willing to do?\n\nWhen I was in high school, I fell in love with a poem called \"Ode\" by Arthur O'Shaughnessy. You've probably heard some ofthe lines. It starts off: \"We are the music-makers \/ And we are the dreamers of dreams.\"\n\nI believe every one of us touches other people's dreams in a hundred different ways. Sometimes we touch others with what wedo, either professionally, through our work, or personally, through our acts of kindness and our interactions with other people.Sometimes we touch others' dreams simply by being who we are. You never know when some friend or co-worker or even a childyou know might be saying, \"If she can make it through that illness, or if he can survive that divorce, maybe I can find thestrength to make it through this tough time in my life.\" Or people might be saying, \"If he can achieve that, maybe I can setmy goals a little higher and try for a bigger dream.\"\n\n**ODE**\n\n**_(excerpt)_ **\n\nBY ARTHUR O'SHAUGHNESSY\n\n_We are the music-makers,_\n\n_And we are the dreamers of dreams,_\n\n_Wandering by lone sea-breakers,_\n\n_And sitting by desolate streams;_\n\n_World-losers and world-forsakers,_\n\n_On whom the pale moon gleams:_\n\n_Yet we are the movers and shakers_\n\n_Of the world for ever, it seems._\n\nWho are the movers and shakers of the world? You are. And you owe it to yourself to become a stress buster so you can leada life of meaning and purpose, happiness and joy. Enjoy the journey.\n\nTHOUGHTS TO PONDER\n\n_If you want to know your past, look into your present_ _conditions. If you want to know your future, look into_ _your present actions._\n\n\u2014ANONYMOUS TIBETAN MONK\n\n_Perhaps the most valuable result of all education is the_ _ability to make yourself do the thing you have to do,_ _when it ought to be done, whether you like it or not._\n\n\u2014THOMAS HUXLEY\n\n_You don't get to choose how you're going to die, or_ _when. You can only decide how you're going to live._ _Now._\n\n\u2014JOAN BAEZ\n\n_Much of the stress that people feel doesn't come from_ _having too much to do. It comes from not finishing what_ _they started._\n\n\u2014DAVID ALLEN\n\n_A procrastinator's work is never done._\n\n\u2014ANONYMOUS\n\n_Time is always on the move, even when we are not._\n\n\u2014ANONYMOUS\n\n_Procrastination is the passive assassin of opportunity._\n\n\u2014ROY WILLIAMS\n\n_He has the deed half-done who has made a beginning._\n\n\u2014HORACE\n\n_We know what we are, but know not what we may be._\n\n\u2014WILLIAM SHAKESPEARE\n\n_The aim of life is to live, and to live means to be aware:_ _joyously, drunkenly, serenely, divinely aware._\n\n\u2014HENRY MILLER\n\n_Success is not the key to happiness. Happiness is the key_ _to success. If you love what you are doing, you will be_ _successful._\n\n\u2014HERMAN CAIN\n\n_Most people never run far enough on their first wind to_ _find out they've got a second. Give your dreams all_ _you've got and you'll be amazed at the energy that comes_ _out of you._\n\n\u2014WILLIAM JAMES\n\n_Hans Selye, a pioneer in the understanding of human_ _stress, was often asked, \"What is the most stressful_ _condition a person can face?\" His unexpected response:_ _\"Not having something to believe in.\"_\n\nA Note on the Author\n\n**RITA EMMETT** leads self-improvement and productivity workshops on a variety of topics, including clutter and procrastination. She is the author of _The Clutter-Busting Handbook_ , _The Procrastinating_ _Child_ , and _The Procrastinator's Handbook_. She has appeared on _Talk of the Town_ and NBC's _Today Show_ and has been featured in _Time_ , the _New York Times_ , the _Wall Street Journal_ , _Family Circle_ , and _Parents._\n\nRita would like to hear the successful ways you find to apply her different techniques and strategies to your life. You may send your story or obtain information about other products and services offered by Rita by contacting her at:\n\nwww.RitaEmmett.com  \ne-mail: Remmett412@aol.com\n\nor at\n\nEmmett Enterprises, Inc.  \n2331 Eastview Drive  \nDes Plaines, IL 60018\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Neil Postman  \n **Technopoly**\n\nNeil Postman is a critic, communications theorist, and Chair of the Department of Communication Arts and Sciences at New York University. In 1987 he was given the George Orwell Award for Clarity in Language by the National Council of Teachers of English. In 1989 he received the Distinguished Professor Award at New York University. In the spring of 1991 he was Laurence Lombard Visiting Professor of the Press and Public Policy at Harvard University. For ten years he was editor of _Et Cetera_ , the journal of General Semantics. His seventeen previous books include _Teaching as a Subversive Activity_ (with Charles Weingartner), _The Disappearance of Childhood, Amusing Ourselves to Death_ , and _Conscientious Objections_.\n\n# ALSO BY NEIL POSTMAN\n\n_Conscientious Objections_\n\n_Teaching as a Subversive Activity (with Charles Weingartner)_\n\n_Crazy Talk, Stupid Talk_\n\n_Teaching as a Conserving Activity_\n\n_The Disappearance of Childhood_\n\n_Amusing Ourselves to Death_\n\nFIRST VINTAGE BOOKS EDITION, APRIL 1993\n\n_Copyright \u00a9 1992 by Neil Postman_\n\nAll rights reserved under International and Pan-American Copyright Conventions. Published in the United States by Vintage Books, a division of Random House, Inc., New York, and simultaneously in Canada by Random House of Canada Limited, Toronto. Originally published in hardcover by Alfred A. Knopf, Inc., New York, in 1992.\n\nLibrary of Congress Cataloging-in-Publication Data  \nPostman, Neil.\n\nTechnopoly: the surrender of culture to technology \/ Neil Postman.  \np. cm.  \nOriginally published: 1st ed. New York: Knopf, 1992.  \neISBN: 978-0-307-79735-3  \n1. Technology\u2014Social aspects. I. Title.  \nT14.5.P667 1993  \n303.48\u20323\u2014dc20 92-50584\n\nv3.1_r2\n_For Faye and Manny_\nWhether or not it draws on new scientific research, technology is a branch of moral philosophy, not of science.\n\nPAUL GOODMAN, _New Reformation_\n\n# Contents\n\n_Cover_\n\n_About the Author_\n\n_Other Books by This Author_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\n_Epigraph_\n\n_Introduction_\n\n1. The Judgment of Thamus\n\n2. From Tools to Technocracy\n\n3. From Technocracy to Technopoly\n\n4. The Improbable World\n\n5. The Broken Defenses\n\n6. The Ideology of Machines: Medical Technology\n\n7. The Ideology of Machines: Computer Technology\n\n8. Invisible Technologies\n\n9. Scientism\n\n10. The Great Symbol Drain\n\n11. The Loving Resistance Fighter\n\n_Notes_\n\n_Bibliography_\n\n# Introduction\n\nIn 1959, Sir Charles Snow published _The Two Cultures and the Scientific Revolution_ , which was both the title and the subject of the Rede Lecture he had given earlier at Cambridge University. The lecture was intended to illuminate what Sir Charles saw as a great problem of our age\u2014the opposition of art and science, or, more precisely, the implacable hostility between literary intellectuals (sometimes called humanists) and physical scientists. The publication of the book caused a small rumble among academics (let us say, a 2.3 on the Richter Scale), not least because Snow came down so firmly on the side of the scientists, giving humanists ample reason and openings for sharp, funny, and nasty ripostes. But the controversy did not last long, and the book quickly faded from view. For good reason. Sir Charles had posed the wrong question, given the wrong argument, and therefore offered an irrelevant answer. Humanists and scientists have no quarrel, at least none that is of sufficient interest to most people.\n\nNonetheless, to Snow must go some considerable credit for noticing that there _are_ two cultures, that they are in fierce opposition to each other, and that it is necessary for a great debate to ensue about the matter. Had he been attending less to the arcane dissatisfactions of those who dwell in faculty clubs and more to the lives of those who have never been in one, he would surely have seen that the argument is not between humanists and scientists but between technology and everybody else. This is not to say that \"everybody else\" recognizes this. In fact, most people believe that technology is a staunch friend. There are two reasons for this. First, technology _is_ a friend. It makes life easier, cleaner, and longer. Can anyone ask more of a friend? Second, because of its lengthy, intimate, and inevitable relationship with culture, technology does not invite a close examination of its own consequences. It is the kind of friend that asks for trust and obedience, which most people are inclined to give because its gifts are truly bountiful. But, of course, there is a dark side to this friend. Its gifts are not without a heavy cost. Stated in the most dramatic terms, the accusation can be made that the uncontrolled growth of technology destroys the vital sources of our humanity. It creates a culture without a moral foundation. It undermines certain mental processes and social relations that make human life worth living. Technology, in sum, is both friend and enemy.\n\nThis book attempts to describe when, how, and why technology became a particularly dangerous enemy. The case has been argued many times before by authors of great learning and conviction\u2014in our own time by Lewis Mumford, Jacques Ellul, Herbert Read, Arnold Gehlen, Ivan Illich, to name a few. The argument was interrupted only briefly by Snow's irrelevancies and has continued into our own time with a sense of urgency, made even more compelling by America's spectacular display of technological pre-eminence in the Iraqi war. I do not say here that the war was unjustified or that the technology was misused, only that the American success may serve as a confirmation of the catastrophic idea that in peace as well as war technology will be our savior.\n\n# **1**\n\n# The Judgment Of Thamus\n\nYou will find in Plato's _Phaedrus_ a story about Thamus, the king of a great city of Upper Egypt. For people such as ourselves, who are inclined (in Thoreau's phrase) to be tools of our tools, few legends are more instructive than his. The story, as Socrates tells it to his friend Phaedrus, unfolds in the following way: Thamus once entertained the god Theuth, who was the inventor of many things, including number, calculation, geometry, astronomy, and writing. Theuth exhibited his inventions to King Thamus, claiming that they should be made widely known and available to Egyptians. Socrates continues:\n\nThamus inquired into the use of each of them, and as Theuth went through them expressed approval or disapproval, according as he judged Theuth's claims to be well or ill founded. It would take too long to go through all that Thamus is reported to have said for and against each of Theuth's inventions. But when it came to writing, Theuth declared, \"Here is an accomplishment, my lord the King, which will improve both the wisdom and the memory of the Egyptians. I have discovered a sure receipt for memory and wisdom.\" To this, Thamus replied, \"Theuth, my paragon of inventors, the discoverer of an art is not the best judge of the good or harm which will accrue to those who practice it. So it is in this; you, who are the father of writing, have out of fondness for your off-spring attributed to it quite the opposite of its real function. Those who acquire it will cease to exercise their memory and become forgetful; they will rely on writing to bring things to their remembrance by external signs instead of by their own internal resources. What you have discovered is a receipt for recollection, not for memory. And as for wisdom, your pupils will have the reputation for it without the reality: they will receive a quantity of information without proper instruction, and in consequence be thought very knowledgeable when they are for the most part quite ignorant. And because they are filled with the conceit of wisdom instead of real wisdom they will be a burden to society.\"\n\nI begin my book with this legend because in Thamus' response there are several sound principles from which we may begin to learn how to think with wise circumspection about a technological society. In fact, there is even one error in the judgment of Thamus, from which we may also learn something of importance. The error is not in his claim that writing will damage memory and create false wisdom. It is demonstrable that writing has had such an effect. Thamus' error is in his believing that writing will be a burden to society and _nothing but a burden_. For all his wisdom, he fails to imagine what writing's benefits might be, which, as we know, have been considerable. We may learn from this that it is a mistake to suppose that any technological innovation has a one-sided effect. Every technology is both a burden and a blessing; not either-or, but this-and-that.\n\nNothing could be more obvious, of course, especially to those who have given more than two minutes of thought to the matter. Nonetheless, we are currently surrounded by throngs of zealous Theuths, one-eyed prophets who see only what new technologies can do and are incapable of imagining what they will _undo_. We might call such people Technophiles. They gaze on technology as a lover does on his beloved, seeing it as without blemish and entertaining no apprehension for the future. They are therefore dangerous and are to be approached cautiously. On the other hand, some one-eyed prophets, such as I (or so I am accused), are inclined to speak only of burdens (in the manner of Thamus) and are silent about the opportunities that new technologies make possible. The Technophiles must speak for themselves, and do so all over the place. My defense is that a dissenting voice is sometimes needed to moderate the din made by the enthusiastic multitudes. If one is to err, it is better to err on the side of Thamusian skepticism. But it is an error nonetheless. And I might note that, with the exception of his judgment on writing, Thamus does not repeat this error. You might notice on rereading the legend that he gives arguments _for_ and _against_ each of Theuth's inventions. For it is inescapable that every culture must negotiate with technology, whether it does so intelligently or not. A bargain is struck in which technology giveth and technology taketh away. The wise know this well, and are rarely impressed by dramatic technological changes, and never overjoyed. Here, for example, is Freud on the matter, from his doleful _Civilization and Its Discontents:_\n\nOne would like to ask: is there, then, no positive gain in pleasure, no unequivocal increase in my feeling of happiness, if I can, as often as I please, hear the voice of a child of mine who is living hundreds of miles away or if I can learn in the shortest possible time after a friend has reached his destination that he has come through the long and difficult voyage unharmed? Does it mean nothing that medicine has succeeded in enormously reducing infant mortality and the danger of infection for women in childbirth, and, indeed, in considerably lengthening the average life of a civilized man?\n\nFreud knew full well that technical and scientific advances are not to be taken lightly, which is why he begins this passage by acknowledging them. But he ends it by reminding us of what they have undone:\n\nIf there had been no railway to conquer distances, my child would never have left his native town and I should need no telephone to hear his voice; if travelling across the ocean by ship had not been introduced, my friend would not have embarked on his sea-voyage and I should not need a cable to relieve my anxiety about him. What is the use of reducing infantile mortality when it is precisely that reduction which imposes the greatest restraint on us in the begetting of children, so that, taken all round, we nevertheless rear no more children than in the days before the reign of hygiene, while at the same time we have created difficult conditions for our sexual life in marriage.... And, finally, what good to us is a long life if it is difficult and barren of joys, and if it is so full of misery that we can only welcome death as a deliverer?\n\nIn tabulating the cost of technological progress, Freud takes a rather depressing line, that of a man who agrees with Thoreau's remark that our inventions are but improved means to an unimproved end. The Technophile would surely answer Freud by saying that life has always been barren of joys and full of misery but that the telephone, ocean liners, and especially the reign of hygiene have not only lengthened life but made it a more agreeable proposition. That is certainly an argument I would make (thus proving I am no one-eyed Technophobe), but it is not necessary at this point to pursue it. I have brought Freud into the conversation only to show that a wise man\u2014even one of such a woeful countenance\u2014must begin his critique of technology by acknowledging its successes. Had King Thamus been as wise as reputed, he would not have forgotten to include in his judgment a prophecy about the powers that writing would enlarge. There is a calculus of technological change that requires a measure of even-handedness.\n\nSo much for Thamus' error of omission. There is another omission worthy of note, but it is no error. Thamus simply takes for granted\u2014and therefore does not feel it necessary to say\u2014that writing is not a neutral technology whose good or harm depends on the uses made of it. He knows that the uses made of any technology are largely determined by the structure of the technology itself\u2014that is, that its functions follow from its form. This is why Thamus is concerned not with _what_ people will write; he is concerned _that_ people will write. It is absurd to imagine Thamus advising, in the manner of today's standard-brand Technophiles, that, if only writing would be used for the production of certain kinds of texts and not others (let us say, for dramatic literature but not for history or philosophy), its disruptions could be minimized. He would regard such counsel as extreme na\u00efvet\u00e9. He would allow, I imagine, that a technology may be barred entry to a culture. But we may learn from Thamus the following: once a technology is admitted, it plays out its hand; it does what it is designed to do. Our task is to understand what that design is\u2014that is to say, when we admit a new technology to the culture, we must do so with our eyes wide open.\n\nAll of this we may infer from Thamus' silence. But we may learn even more from what he does say than from what he doesn't. He points out, for example, that writing will change what is meant by the words \"memory\" and \"wisdom.\" He fears that memory will be confused with what he disdainfully calls \"recollection,\" and he worries that wisdom will become indistinguishable from mere knowledge. This judgment we must take to heart, for it is a certainty that radical technologies create new definitions of old terms, and that this process takes place without our being fully conscious of it. Thus, it is insidious and dangerous, quite different from the process whereby new technologies introduce new terms to the language. In our own time, we have consciously added to our language thousands of new words and phrases having to do with new technologies\u2014\"VCR,\" \"binary digit,\" \"software,\" \"front-wheel drive,\" \"window of opportunity,\" \"Walkman,\" etc. We are not taken by surprise at this. New things require new words. But new things also modify old words, words that have deep-rooted meanings. The telegraph and the penny press changed what we once meant by \"information.\" Television changes what we once meant by the terms \"political debate,\" \"news,\" and \"public opinion.\" The computer changes \"information\" once again. Writing changed what we once meant by \"truth\" and \"law\"; printing changed them again, and now television and the computer change them once more. Such changes occur quickly, surely, and, in a sense, silently. Lexicographers hold no plebiscites on the matter. No manuals are written to explain what is happening, and the schools are oblivious to it. The old words still look the same, are still used in the same kinds of sentences. But they do not have the same meanings; in some cases, they have opposite meanings. And this is what Thamus wishes to teach us\u2014that technology imperiously commandeers our most important terminology. It redefines \"freedom,\" \"truth,\" \"intelligence,\" \"fact,\" \"wisdom,\" \"memory,\" \"history\"\u2014all the words we live by. And it does not pause to tell us. And we do not pause to ask.\n\nThis fact about technological change requires some elaboration, and I will return to the matter in a later chapter. Here, there are several more principles to be mined from the judgment of Thamus that require mentioning because they presage all I will write about. For instance, Thamus warns that the pupils of Theuth will develop an undeserved reputation for wisdom. He means to say that those who cultivate competence in the use of a new technology become an elite group that are granted undeserved authority and prestige by those who have no such competence. There are different ways of expressing the interesting implications of this fact. Harold Innis, the father of modern communication studies, repeatedly spoke of the \"knowledge monopolies\" created by important technologies. He meant precisely what Thamus had in mind: those who have control over the workings of a particular technology accumulate power and inevitably form a kind of conspiracy against those who have no access to the specialized knowledge made available by the technology. In his book _The Bias of Communication_ , Innis provides many historical examples of how a new technology \"busted up\" a traditional knowledge monopoly and created a new one presided over by a different group. Another way of saying this is that the benefits and deficits of a new technology are not distributed equally. There are, as it were, winners and losers. It is both puzzling and poignant that on many occasions the losers, out of ignorance, have actually cheered the winners, and some still do.\n\nLet us take as an example the case of television. In the United States, where television has taken hold more deeply than anywhere else, many people find it a blessing, not least those who have achieved high-paying, gratifying careers in television as executives, technicians, newscasters, and entertainers. It should surprise no one that such people, forming as they do a new knowledge monopoly, should cheer themselves and defend and promote television technology. On the other hand and in the long run, television may bring a gradual end to the careers of schoolteachers, since school was an invention of the printing press and must stand or fall on the issue of how much importance the printed word has. For four hundred years, schoolteachers have been part of the knowledge monopoly created by printing, and they are now witnessing the breakup of that monopoly. It appears as if they can do little to prevent that breakup, but surely there is something perverse about schoolteachers' being enthusiastic about what is happening. Such enthusiasm always calls to my mind an image of some turn-of-the-century blacksmith who not only sings the praises of the automobile but also believes that his business will be enhanced by it. We know now that his business was not enhanced by it; it was rendered obsolete by it, as perhaps the clearheaded blacksmiths knew. What could they have done? Weep, if nothing else.\n\nWe have a similar situation in the development and spread of computer technology, for here too there are winners and losers. There can be no disputing that the computer has increased the power of large-scale organizations like the armed forces, or airline companies or banks or tax-collecting agencies. And it is equally clear that the computer is now indispensable to high-level researchers in physics and other natural sciences. But to what extent has computer technology been an advantage to the masses of people? To steelworkers, vegetable-store owners, teachers, garage mechanics, musicians, bricklayers, dentists, and most of the rest into whose lives the computer now intrudes? Their private matters have been made more accessible to powerful institutions. They are more easily tracked and controlled; are subjected to more examinations; are increasingly mystified by the decisions made about them; are often reduced to mere numerical objects. They are inundated by junk mail. They are easy targets for advertising agencies and political organizations. The schools teach their children to operate computerized systems instead of teaching things that are more valuable to children. In a word, almost nothing that they need happens to the losers. Which is why they are losers.\n\nIt is to be expected that the winners will encourage the losers to be enthusiastic about computer technology. That is the way of winners, and so they sometimes tell the losers that with personal computers the average person can balance a checkbook more neatly, keep better track of recipes, and make more logical shopping lists. They also tell them that their lives will be conducted more efficiently. But discreetly they neglect to say from whose point of view the efficiency is warranted or what might be its costs. Should the losers grow skeptical, the winners dazzle them with the wondrous feats of computers, almost all of which have only marginal relevance to the quality of the losers' lives but which are nonetheless impressive. Eventually, the losers succumb, in part because they believe, as Thamus prophesied, that the specialized knowledge of the masters of a new technology is a form of wisdom. The masters come to believe this as well, as Thamus also prophesied. The result is that certain questions do not arise. For example, to whom will the technology give greater power and freedom? And whose power and freedom will be reduced by it?\n\nI have perhaps made all of this sound like a well-planned conspiracy, as if the winners know all too well what is being won and what lost. But this is not quite how it happens. For one thing, in cultures that have a democratic ethos, relatively weak traditions, and a high receptivity to new technologies, everyone is inclined to be enthusiastic about technological change, believing that its benefits will eventually spread evenly among the entire population. Especially in the United States, where the lust for what is new has no bounds, do we find this childlike conviction most widely held. Indeed, in America, social change of any kind is rarely seen as resulting in winners and losers, a condition that stems in part from Americans' much-documented optimism. As for change brought on by technology, this native optimism is exploited by entrepreneurs, who work hard to infuse the population with a unity of improbable hope, for they know that it is economically unwise to reveal the price to be paid for technological change. One might say, then, that, if there is a conspiracy of any kind, it is that of a culture conspiring against itself.\n\nIn addition to this, and more important, it is not always clear, at least in the early stages of a technology's intrusion into a culture, who will gain most by it and who will lose most. This is because the changes wrought by technology are subtle if not downright mysterious, one might even say wildly unpredictable. Among the most unpredictable are those that might be labeled ideological. This is the sort of change Thamus had in mind when he warned that writers will come to rely on external signs instead of their own internal resources, and that they will receive quantities of information without proper instruction. He meant that new technologies change what we mean by \"knowing\" and \"truth\"; they alter those deeply embedded habits of thought which give to a culture its sense of what the world is like\u2014a sense of what is the natural order of things, of what is reasonable, of what is necessary, of what is inevitable, of what is real. Since such changes are expressed in changed meanings of old words, I will hold off until later discussing the massive ideological transformation now occurring in the United States. Here, I should like to give only one example of how technology creates new conceptions of what is real and, in the process, undermines older conceptions. I refer to the seemingly harmless practice of assigning marks or grades to the answers students give on examinations. This procedure seems so natural to most of us that we are hardly aware of its significance. We may even find it difficult to imagine that the number or letter is a tool or, if you will, a technology; still less that, when we use such a technology to judge someone's behavior, we have done something peculiar. In point of fact, the first instance of grading students' papers occurred at Cambridge University in 1792 at the suggestion of a tutor named William Farish. No one knows much about William Farish; not more than a handful have ever heard of him. And yet his idea that a quantitative value should be assigned to human thoughts was a major step toward constructing a mathematical concept of reality. If a number can be given to the quality of a thought, then a number can be given to the qualities of mercy, love, hate, beauty, creativity, intelligence, even sanity itself. When Galileo said that the language of nature is written in mathematics, he did not mean to include human feeling or accomplishment or insight. But most of us are now inclined to make these inclusions. Our psychologists, sociologists, and educators find it quite impossible to do their work without numbers. They believe that without numbers they cannot acquire or express authentic knowledge.\n\nI shall not argue here that this is a stupid or dangerous idea, only that it is peculiar. What is even more peculiar is that so many of us do not find the idea peculiar. To say that someone should be doing better work because he has an IQ of 134, or that someone is a 7.2 on a sensitivity scale, or that this man's essay on the rise of capitalism is an A \u2212 and that man's is a C + would have sounded like gibberish to Galileo or Shakespeare or Thomas Jefferson. If it makes sense to us, that is because our minds have been conditioned by the technology of numbers so that we see the world differently than they did. Our understanding of what is real is different. Which is another way of saying that embedded in every tool is an ideological bias, a predisposition to construct the world as one thing rather than another, to value one thing over another, to amplify one sense or skill or attitude more loudly than another.\n\nThis is what Marshall McLuhan meant by his famous aphorism \"The medium is the message.\" This is what Marx meant when he said, \"Technology discloses man's mode of dealing with nature\" and creates the \"conditions of intercourse\" by which we relate to each other. It is what Wittgenstein meant when, in referring to our most fundamental technology, he said that language is not merely a vehicle of thought but also the driver. And it is what Thamus wished the inventor Theuth to see. This is, in short, an ancient and persistent piece of wisdom, perhaps most simply expressed in the old adage that, to a man with a hammer, everything looks like a nail. Without being too literal, we may extend the truism: To a man with a pencil, everything looks like a list. To a man with a camera, everything looks like an image. To a man with a computer, everything looks like data. And to a man with a grade sheet, everything looks like a number.\n\nBut such prejudices are not always apparent at the start of a technology's journey, which is why no one can safely conspire to be a winner in technological change. Who would have imagined, for example, whose interests and what world-view would be ultimately advanced by the invention of the mechanical clock? The clock had its origin in the Benedictine monasteries of the twelfth and thirteenth centuries. The impetus behind the invention was to provide a more or less precise regularity to the routines of the monasteries, which required, among other things, seven periods of devotion during the course of the day. The bells of the monastery were to be rung to signal the canonical hours; the mechanical clock was the technology that could provide precision to these rituals of devotion. And indeed it did. But what the monks did not foresee was that the clock is a means not merely of keeping track of the hours but also of synchronizing and controlling the actions of men. And thus, by the middle of the fourteenth century, the clock had moved outside the walls of the monastery, and brought a new and precise regularity to the life of the workman and the merchant. \"The mechanical clock,\" as Lewis Mumford wrote, \"made possible the idea of regular production, regular working hours and a standardized product.\" In short, without the clock, capitalism would have been quite impossible. The paradox, the surprise, and the wonder are that the clock was invented by men who wanted to devote themselves more rigorously to God; it ended as the technology of greatest use to men who wished to devote themselves to the accumulation of money. In the eternal struggle between God and Mammon, the clock quite unpredictably favored the latter.\n\nUnforeseen consequences stand in the way of all those who think they see clearly the direction in which a new technology will take us. Not even those who invent a technology can be assumed to be reliable prophets, as Thamus warned. Gutenberg, for example, was by all accounts a devout Catholic who would have been horrified to hear that accursed heretic Luther describe printing as \"God's highest act of grace, whereby the business of the Gospel is driven forward.\" Luther understood, as Gutenberg did not, that the mass-produced book, by placing the Word of God on every kitchen table, makes each Christian his own theologian\u2014one might even say his own priest, or, better, from Luther's point of view, his own pope. In the struggle between unity and diversity of religious belief, the press favored the latter, and we can assume that this possibility never occurred to Gutenberg.\n\nThamus understood well the limitations of inventors in grasping the social and psychological\u2014that is, ideological\u2014bias of their own inventions. We can imagine him addressing Gutenberg in the following way: \"Gutenberg, my paragon of inventors, the discoverer of an art is not the best judge of the good or harm which will accrue to those who practice it. So it is in this; you, who are the father of printing, have out of fondness for your off-spring come to believe it will advance the cause of the Holy Roman See, whereas in fact it will sow discord among believers; it will damage the authenticity of your beloved Church and destroy its monopoly.\"\n\nWe can imagine that Thamus would also have pointed out to Gutenberg, as he did to Theuth, that the new invention would create a vast population of readers who \"will receive a quantity of information without proper instruction... [who will be] filled with the conceit of wisdom instead of real wisdom\"; that reading, in other words, will compete with older forms of learning. This is yet another principle of technological change we may infer from the judgment of Thamus: new technologies compete with old ones\u2014for time, for attention, for money, for prestige, but mostly for dominance of their world-view. This competition is implicit once we acknowledge that a medium contains an ideological bias. And it is a fierce competition, as only ideological competitions can be. It is not merely a matter of tool against tool\u2014the alphabet attacking ideographic writing, the printing press attacking the illuminated manuscript, the photograph attacking the art of painting, television attacking the printed word. When media make war against each other, it is a case of world-views in collision.\n\nIn the United States, we can see such collisions everywhere\u2014in politics, in religion, in commerce\u2014but we see them most clearly in the schools, where two great technologies confront each other in uncompromising aspect for the control of students' minds. On the one hand, there is the world of the printed word with its emphasis on logic, sequence, history, exposition, objectivity, detachment, and discipline. On the other, there is the world of television with its emphasis on imagery, narrative, presentness, simultaneity, intimacy, immediate gratification, and quick emotional response. Children come to school having been deeply conditioned by the biases of television. There, they encounter the world of the printed word. A sort of psychic battle takes place, and there are many casualties\u2014children who can't learn to read or won't, children who cannot organize their thought into logical structure even in a simple paragraph, children who cannot attend to lectures or oral explanations for more than a few minutes at a time. They are failures, but not because they are stupid. They are failures because there is a media war going on, and they are on the wrong side\u2014at least for the moment. Who knows what schools will be like twenty-five years from now? Or fifty? In time, the type of student who is currently a failure may be considered a success. The type who is now successful may be regarded as a handicapped learner\u2014slow to respond, far too detached, lacking in emotion, inadequate in creating mental pictures of reality. Consider: what Thamus called the \"conceit of wisdom\"\u2014the unreal knowledge acquired through the written word\u2014eventually became the pre-eminent form of knowledge valued by the schools. There is no reason to suppose that such a form of knowledge must always remain so highly valued.\n\nTo take another example: In introducing the personal computer to the classroom, we shall be breaking a four-hundred-year-old truce between the gregariousness and openness fostered by orality and the introspection and isolation fostered by the printed word. Orality stresses group learning, cooperation, and a sense of social responsibility, which is the context within which Thamus believed proper instruction and real knowledge must be communicated. Print stresses individualized learning, competition, and personal autonomy. Over four centuries, teachers, while emphasizing print, have allowed orality its place in the classroom, and have therefore achieved a kind of pedagogical peace between these two forms of learning, so that what is valuable in each can be maximized. Now comes the computer, carrying anew the banner of private learning and individual problem-solving. Will the widespread use of computers in the classroom defeat once and for all the claims of communal speech? Will the computer raise egocentrism to the status of a virtue?\n\nThese are the kinds of questions that technological change brings to mind when one grasps, as Thamus did, that technological competition ignites total war, which means it is not possible to contain the effects of a new technology to a limited sphere of human activity. If this metaphor puts the matter too brutally, we may try a gentler, kinder one: Technological change is neither additive nor subtractive. It is ecological. I mean \"ecological\" in the same sense as the word is used by environmental scientists. One significant change generates total change. If you remove the caterpillars from a given habitat, you are not left with the same environment minus caterpillars: you have a new environment, and you have reconstituted the conditions of survival; the same is true if you add caterpillars to an environment that has had none. This is how the ecology of media works as well. A new technology does not add or subtract something. It changes everything. In the year 1500, fifty years after the printing press was invented, we did not have old Europe plus the printing press. We had a different Europe. After television, the United States was not America plus television; television gave a new coloration to every political campaign, to every home, to every school, to every church, to every industry. And that is why the competition among media is so fierce. Surrounding every technology are institutions whose organization\u2014not to mention their reason for being\u2014reflects the world-view promoted by the technology. Therefore, when an old technology is assaulted by a new one, institutions are threatened. When institutions are threatened, a culture finds itself in crisis. This is serious business, which is why we learn nothing when educators ask, Will students learn mathematics better by computers than by textbooks? Or when businessmen ask, Through which medium can we sell more products? Or when preachers ask, Can we reach more people through television than through radio? Or when politicians ask, How effective are messages sent through different media? Such questions have an immediate, practical value to those who ask them, but they are diversionary. They direct our attention away from the serious social, intellectual, and institutional crises that new media foster.\n\nPerhaps an analogy here will help to underline the point. In speaking of the meaning of a poem, T. S. Eliot remarked that the chief use of the overt content of poetry is \"to satisfy one habit of the reader, to keep his mind diverted and quiet, while the poem does its work upon him: much as the imaginary burglar is always provided with a bit of nice meat for the house-dog.\" In other words, in asking their practical questions, educators, entrepreneurs, preachers, and politicians are like the house-dog munching peacefully on the meat while the house is looted. Perhaps some of them know this and do not especially care. After all, a nice piece of meat, offered graciously, does take care of the problem of where the next meal will come from. But for the rest of us, it cannot be acceptable to have the house invaded without protest or at least awareness.\n\nWhat we need to consider about the computer has nothing to do with its efficiency as a teaching tool. We need to know in what ways it is altering our conception of learning, and how, in conjunction with television, it undermines the old idea of school. Who cares how many boxes of cereal can be sold via television? We need to know if television changes our conception of reality, the relationship of the rich to the poor, the idea of happiness itself. A preacher who confines himself to considering how a medium can increase his audience will miss the significant question: In what sense do new media alter what is meant by religion, by church, even by God? And if the politician cannot think beyond the next election, then _we_ must wonder about what new media do to the idea of political organization and to the conception of citizenship.\n\nTo help us do this, we have the judgment of Thamus, who, in the way of legends, teaches us what Harold Innis, in his way, tried to. New technologies alter the structure of our interests: the things we think _about_. They alter the character of our symbols: the things we think _with_. And they alter the nature of community: the arena in which thoughts develop. As Thamus spoke to Innis across the centuries, it is essential that we listen to their conversation, join in it, revitalize it. For something has happened in America that is strange and dangerous, and there is only a dull and even stupid awareness of what it is\u2014in part because it has no name. I call it Technopoly.\n\n# **2**\n\n# From Tools to Technocracy\n\nAmong the famous aphorisms from the troublesome pen of Karl Marx is his remark in _The Poverty of Philosophy_ that the \"hand-loom gives you society with the feudal lord; the steam-mill society with the industrial capitalist.\" As far as I know, Marx did not say which technology gives us the technocrat, and I am certain his vision did not include the emergence of the Technopolist. Nonetheless, the remark is useful. Marx understood well that, apart from their economic implications, technologies create the ways in which people perceive reality, and that such ways are the key to understanding diverse forms of social and mental life. In _The German Ideology_ , he says, \"As individuals express their life, so they are,\" which sounds as much like Marshall McLuhan or, for that matter, Thamus as it is possible to sound. Indeed, toward the end of that book, Marx includes a remarkable paragraph that would be entirely at home in McLuhan's _Understanding Media_. \"Is Achilles possible,\" he asks, \"when powder and shot have been invented? And is the Iliad possible at all when the printing press and even printing machines exist? Is it not inevitable that with the emergence of the press, the singing and the telling and the muse cease; that is, the conditions for epic poetry disappear?\"\n\nBy connecting technological conditions to symbolic life and psychic habits, Marx was doing nothing unusual. Before him, scholars found it useful to invent taxonomies of culture based on the technological character of an age. And they do it still, for the practice is something of a persistent scholarly industry. We think at once of the best-known classification: the Stone Age, the Bronze Age, the Iron Age, the Steel Age. We speak easily of the Industrial Revolution, a term popularized by Arnold Toynbee, and, more recently, of the Post-Industrial Revolution, so named by Daniel Bell. Oswald Spengler wrote of the Age of Machine Technics, and C. S. Peirce called the nineteenth century the Railway Age. Lewis Mumford, looking at matters from a longer perspective, gave us the Eotechnic, the Paleotechnic, and the Neotechnic Ages. With equally telescopic perspective, Jos\u00e9 Ortega y Gasset wrote of three stages in the development of technology: the age of technology of chance, the age of technology of the artisan, the age of technology of the technician. Walter Ong has written about Oral cultures, Chirographic cultures, Typographic cultures, and Electronic cultures. McLuhan himself introduced the phrase \"the Age of Gutenberg\" (which, he believed, is now replaced by the Age of Electronic Communication).\n\nI find it necessary, for the purpose of clarifying our present situation and indicating what dangers lie ahead, to create still another taxonomy. Cultures may be classified into three types: tool-using cultures, technocracies, and technopolies. At the present time, each type may be found somewhere on the planet, although the first is rapidly disappearing: we must travel to exotic places to find a tool-using culture. If we do, it is well to go armed with the knowledge that, until the seventeenth century, all cultures were tool-users. There was, of course, considerable variation from one culture to another in the tools that were available. Some had only spears and cooking utensils. Some had water mills and coal- and horsepower. But the main characteristic of all tool-using cultures is that their tools were largely invented to do two things: to solve specific and urgent problems of physical life, such as in the use of waterpower, windmills, and the heavy-wheeled plow; or to serve the symbolic world of art, politics, myth, ritual, and religion, as in the construction of castles and cathedrals and the development of the mechanical clock. In either case, tools did not attack (or, more precisely, were not intended to attack) the dignity and integrity of the culture into which they were introduced. With some exceptions, tools did not prevent people from believing in their traditions, in their God, in their politics, in their methods of education, or in the legitimacy of their social organization. These beliefs, in fact, _directed_ the invention of tools and limited the uses to which they were put. Even in the case of military technology, spiritual ideas and social customs acted as controlling forces. It is well known, for example, that the uses of the sword by samurai warriors were meticulously governed by a set of ideals known as Bushido, or the Way of the Warrior. The rules and rituals specifying when, where, and how the warrior must use either of his two swords (the _katana_ , or long sword, and the _wakizashi_ , or short sword) were precise, tied closely to the concept of honor, and included the requirement that the warrior commit seppuku or hara-kiri should his honor be compromised. This sort of governance of military technology was not unknown in the Western world. The use of the lethal crossbow was prohibited, under threat of anathema, by Pope Innocent II in the early twelfth century. The weapon was judged to be \"hateful to God\" and therefore could not be used against Christians. That it could be used against Muslims and other infidels does not invalidate the point that in a tool-using culture technology is not seen as autonomous, and is subject to the jurisdiction of some binding social or religious system.\n\nHaving defined tool-using cultures in this manner, I must add two points so as to avoid excessive oversimplification. First, the quantity of technologies available to a tool-using culture is not its defining characteristic. Even a superficial study of the Roman Empire, for example, reveals the extent to which it relied on roads, bridges, aqueducts, tunnels, and sewers for both its economic vitality and its military conquests. Or, to take another example, we know that, between the tenth and thirteenth centuries, Europe underwent a technological boom: medieval man was surrounded by machines. One may even go as far as Lynn White, Jr., who said that the Middle Ages gave us for the first time in history \"a complex civilization which rested not on the backs of sweating slaves or coolies but primarily on non-human power.\" Tool-using cultures, in other words, may be both ingenious and productive in solving problems of the physical environment. Windmills were invented in the late twelfth century. Eyeglasses for nearsightedness appeared in Italy in 1280. The invention in the eleventh century of rigid padded collars to rest on the shoulder blades of horses solved the problem of how to increase the pulling power of horses without decreasing their ability to breathe. In fact, as early as the ninth century in Europe, horseshoes were invented, and someone figured out that, when horses are hitched, one behind the other, their pulling power is enormously amplified. Corn mills, paper mills, and fulling mills were part of medieval culture, as were bridges, castles, and cathedrals. The famous spire of Strasbourg Cathedral, built in the thirteenth century, rose to a height of 466 feet, the equivalent of a forty-story skyscraper. And, to go further back in time, one must not fail to mention the remarkable engineering achievements of Stonehenge and the Pyramids (whose construction, Lewis Mumford insisted, signifies the first example of a megamachine in action).\n\nGiven the facts, we must conclude that tool-using cultures are not necessarily impoverished technologically, and may even be surprisingly sophisticated. Of course, some tool-using cultures were (and still are) technologically primitive, and some have even displayed a contempt for crafts and machinery. The Golden Age of Greece, for example, produced no important technical inventions and could not even devise ways of using horsepower efficiently. Both Plato and Aristotle scorned the \"base mechanic arts,\" probably in the belief that nobility of mind was not enhanced by efforts to increase efficiency or productivity. Efficiency and productivity were problems for slaves, not philosophers. We find a somewhat similar view in the Bible, which is the longest and most detailed account of an ancient tool-using culture we have. In Deuteronomy, no less an authority than God Himself says, \"Cursed be the man who makes a graven or molten image, an abomination to the Lord, a thing made by the hands of a craftsman, and sets it up in secret.\"\n\nTool-using cultures, then, may have many tools or few, may be enthusiastic about tools or contemptuous. The name \"tool-using culture\" derives from the relationship in a given culture between tools and the belief system or ideology. The tools are not intruders. They are integrated into the culture in ways that do not pose significant contradictions to its world-view. If we take the European Middle Ages as an example of a tool-using culture, we find a very high degree of integration between its tools and its world-view. Medieval theologians developed an elaborate and systematic description of the relation of man to God, man to nature, man to man, and man to his tools. Their theology took as a first and last principle that all knowledge and goodness come from God, and that therefore all human enterprise must be directed toward the service of God. Theology, not technology, provided people with authorization for what to do or think. Perhaps this is why Leonardo da Vinci kept his design of a submarine secret, believing that it was too harmful a tool to unleash, that it would not gain favor in God's eyes.\n\nIn any case, theological assumptions served as a controlling ideology, and whatever tools were invented had, ultimately, to fit within that ideology. We may say, further, that all tool-using cultures\u2014from the technologically most primitive to the most sophisticated\u2014are theocratic or, if not that, unified by some metaphysical theory. Such a theology or metaphysics provides order and meaning to existence, making it almost impossible for technics to subordinate people to its own needs.\n\nThe \"almost\" is important. It leads to my second qualification. As the spirit of Thamus reminds us, tools have a way of intruding on even the most unified set of cultural beliefs. There are limits to the power of both theology and metaphysics, and technology has business to do which sometimes cannot be stayed by any force. Perhaps the most interesting example of a drastic technological disruption of a tool-using culture is in the eighth-century use of the stirrup by the Franks under the leadership of Charles Martel. Until this time, the principal use of horses in combat was to transport warriors to the scene of the battle, whereupon they dismounted to meet the foe. The stirrup made it possible to fight _on_ horseback, and this created an awesome new military technology: mounted shock combat. The new form of combat, as Lynn White, Jr., has meticulously detailed, enlarged the importance of the knightly class and changed the nature of feudal society. Landholders found it necessary to secure the services of cavalry for protection. Eventually, the knights seized control of church lands and distributed them to vassals on condition that they stay in the service of the knights. If a pun will be allowed here, the stirrup was in the saddle, and took feudal society where it would not otherwise have gone.\n\nTo take a later example: I have already alluded to the transformation of the mechanical clock in the fourteenth century from an instrument of religious observance to an instrument of commercial enterprise. That transformation is sometimes given a specific date\u20141370\u2014when King Charles V ordered all citizens of Paris to regulate their private, commercial, and industrial life by the bells of the Royal Palace clock, which struck every sixty minutes. All churches in Paris were similarly required to regulate their clocks, in disregard of the canonical hours. Thus, the church had to give material interests precedence over spiritual needs. Here is a clear example of a tool being employed to loosen the authority of the central institution of medieval life.\n\nThere are other examples of how technologies created problems for the spiritual life of medieval Europe. For example, the mills to which farmers flocked to have their grain ground became a favorite place for prostitutes to attract customers. The problem grew to such proportions that Saint Bernard, the leader of the Cistercian order in the twelfth century, tried to close down the mills. He was unsuccessful, because the mills had become too important to the economy. In other words, it is something of an oversimplification to say that tool-using cultures never had their customs and symbolic life reoriented by technology. And, just as there are examples of such cases in the medieval world, we can find queer but significant instances in technologically primitive societies of tools attacking the supremacy of custom, religion, or metaphysics. Egbert de Vries, a Dutch sociologist, has told of how the introduction of matches to an African tribe altered their sexual habits. Members of this community believed it necessary to start a new fire in the fireplace after each act of sexual intercourse. This custom meant that each act of intercourse was something of a public event, since when it was completed someone had to go to a neighboring hut to bring back a burning stick with which to start a fresh fire. Under such conditions, adultery was difficult to conceal, which is conceivably why the custom originated in the first place. The introduction of matches changed all this. It became possible to light a new fire without going to a neighbor's hut, and thus, in a flash, so to speak, a long-standing tradition was consumed. In reporting on de Vries' finding, Alvin Toffler raises several intriguing questions: Did matches result in a shift in values? Was adultery less or more frowned upon as a result? By facilitating the privacy of sex, did matches alter the valuation placed upon it? We can be sure that some changes in cultural values occurred, although they could not have been as drastic as what happened to the Ihalmiut tribe early in the twentieth century, after the introduction of the rifle. As described by Farley Mowat in _The People of the Deer_ , the replacement of bows and arrows with rifles is one of the most chilling tales on record of a technological attack on a tool-using culture. The result in this case was not the modification of a culture but its eradication.\n\nNonetheless, after one acknowledges that no taxonomy ever neatly fits the realities of a situation, and that in particular the definition of a tool-using culture lacks precision, it is still both possible and useful to distinguish a tool-using culture from a technocracy. In a technocracy, tools play a central role in the thought-world of the culture. Everything must give way, in some degree, to their development. The social and symbolic worlds become increasingly subject to the requirements of that development. Tools are not integrated into the culture; they attack the culture. They bid to _become_ the culture. As a consequence, tradition, social mores, myth, politics, ritual, and religion have to fight for their lives.\n\nThe modern technocracies of the West have their roots in the medieval European world, from which there emerged three great inventions: the mechanical clock, which provided a new conception of time; the printing press with movable type, which attacked the epistemology of the oral tradition; and the telescope, which attacked the fundamental propositions of Judeo-Christian theology. Each of these was significant in creating a new relationship between tools and culture. But since it is permissible to say that among faith, hope, and charity the last is most important, I shall venture to say that among the clock, the press, and the telescope the last is also the most important. To be more exact (since Copernicus, Tycho Brahe, and to some extent Kepler did their work without benefit of the telescope), somewhat cruder instruments of observation than the telescope allowed men to see, measure, and speculate about the heavens in ways that had not been possible before. But the refinements of the telescope made their knowledge so precise that there followed a collapse, if one may say it this way, of the moral center of gravity in the West. That moral center had allowed people to believe that the earth was the stable center of the universe and therefore that humankind was of special interest to God. After Copernicus, Kepler, and especially Galileo, the Earth became a lonely wanderer in an obscure galaxy in some hidden corner of the universe, and this left the Western world to wonder if God had any interest in us at all. Although John Milton was only an infant when Galileo's _Messenger from the Stars_ was printed in 1610, he was able, years later, to describe the psychic desolation of an unfathomable universe that Galileo's telescopic vision thrust upon an unprepared theology. In _Paradise Lost_ , Milton wrote:\n\n_Before [his] eyes in sudden view appear_  \n _The secrets of the hoary Deep_ \u2014 _a dark_  \n _Illimitable ocean, without bound_ ,  \n _Without dimension_....\n\nTruly, a paradise lost. But it was not Galileo's intention\u2014neither was it Copernicus' or Kepler's\u2014to so disarm their culture. These were medieval men who, like Gutenberg before them, had no wish to damage the spiritual foundations of their world. Copernicus, for example, was a doctor of canon law, having been elected a canon of Frauenburg Cathedral. Although he never took a medical degree, he studied medicine, was private physician to his uncle, and among many people was better known as a physician than as an astronomer. He published only one scientific work, _On the Revolutions of the Heavenly Spheres_ , the first completed copy arriving from the printer only a few hours before his death, at the age of seventy, on May 24, 1543. He had delayed publishing his heliocentric theory for thirty years, largely because he believed it to be unsound, not because he feared retribution from the church. In fact, his book was not placed on the Index until seventy-three years after it was published, and then only for a short time. (Galileo's trial did not take place until ninety years after Copernicus' death.) In 1543, scholars and philosophers had no reason to fear persecution for their ideas so long as they did not directly challenge the authority of the church, which Copernicus had no wish to do. Though the authorship of the preface to his work is in dispute, the preface clearly indicates that his ideas are to be taken as hypotheses, and that his \"hypotheses need not be true or even probable.\" We can be sure that Copernicus believed that the earth really moved, but he did not believe that either the earth or the planets moved in the manner described in his system, which he understood to consist of geometric fictions. And he did not believe that his work undermined the supremacy of theology. It is true that Martin Luther called Copernicus \"a fool who went against Holy Writ,\" but Copernicus did not think he had done so\u2014which proves, I suppose, that Luther saw more deeply than Copernicus.\n\nKepler's is a somewhat similar story. Born in 1571, he began his career by publishing astrological calendars, and ended it as court astrologer to the duke of Wallenstein. Although he was famous for his service as an astrologer, we must credit him with believing that \"Astrology can do enormous harm to a monarch if a clever astrologer exploits his human credulity.\" Kepler wished astrology to be kept out of sight of all heads of state, a precaution that in recent years has not always been taken. His mother was accused of being a witch, and although Kepler did not believe this specific charge, he would probably not have denied categorically the existence of witches. He spent a great deal of his time corresponding with scholars on questions concerning chronology in the age of Christ, and his theory that Jesus was actually born in 4 or 5 B.C. is generally accepted today. In other words, Kepler was very much a man of his time, medieval through and through. Except for one thing: He believed that theology and science should be kept separate and, in particular, that angels, spirits and the opinions of saints should be banished from cosmology. In his _New Astronomy_ , he wrote, \"Now as regards the opinions of the saints about these matters of nature, I answer in one word, that in theology the weight of authority, but in philosophy the weight of Reason alone is valid.\" After reviewing what various saints had said about the earth, Kepler concluded, \"... but to me more sacred than all these is Truth, when I, with all respect for the doctors of the Church, demonstrate from philosophy that the earth is round, circumhabited by antipodes, of a most insignificant smallness, and a swift wanderer among the stars.\"\n\nIn expressing this idea, Kepler was taking the first significant step toward the conception of a technocracy. We have here a clear call for a separation of moral and intellectual values, a separation that is one of the pillars of a technocracy\u2014a significant step but still a small one. No one before Kepler had asked why planets travel at variable rates. Kepler's answer was that it must be a force emanating from the sun. But this answer still had room in it for God. In a famous letter sent to his colleague Maestlin, Kepler wrote, \"The sun in the middle of the moving stars, himself at rest and yet the source of motion, carries the image of God the Father and Creator.... He distributes his motive force through a medium which contains the moving bodies even as the Father creates through the Holy Ghost.\"\n\nKepler was a Lutheran, and although he was eventually excommunicated from his own church, he remained a man of sincere religious conviction to the end. He was, for example, dissatisfied with his discovery of the elliptical orbits of planets, believing that an ellipse had nothing to recommend it in the eyes of God. To be sure, Kepler, building on the work of Copernicus, was creating something new in which truth was not required to gain favor in God's eyes. But it was not altogether clear to him exactly what his work would lead to. It remained for Galileo to make visible the unresolvable contradictions between science and theology, that is, between intellectual and moral points of view.\n\nGalileo did not invent the telescope, although he did not always object to the attribution. A Dutch spectacle-maker named Johann Lippershey was probably the instrument's true inventor; at any rate, he was the first to claim a license for its manufacture, in 1608. (It might also be worth remarking here that the famous experiment of dropping cannon balls from the Tower of Pisa was not only _not_ done by Galileo but actually carried out by one of his adversaries, Giorgio Coressio, who was trying to confirm, not dispute, Aristotle's opinion that larger bodies fall more quickly than smaller ones.) Nonetheless, to Galileo must go the entire credit for transforming the telescope from a toy into an instrument of science. And to Galileo must also go the credit of making astronomy a source of pain and confusion to the prevailing theology. His discovery of the four moons of Jupiter and the simplicity and accessibility of his writing style were key weapons in his arsenal. But more important was the directness with which he disputed the Scriptures. In his famous _Letter to the Grand Duchess Christina_ , he used arguments first advanced by Kepler as to why the Bible could not be interpreted literally. But he went further in saying that nothing physical that could be directly observed or which demonstrations could prove ought to be questioned merely because Biblical passages say otherwise. More clearly than Kepler had been able to do, Galileo disqualified the doctors of the church from offering opinions about nature. To allow them to do so, he charged, is pure folly. He wrote, \"This would be as if an absolute despot, being neither a physician nor an architect, but knowing himself free to command, should undertake to administer medicines and erect buildings according to his whim\u2014at grave peril of his poor patients' lives, and the speedy collapse of his edifices.\"\n\nFrom this and other audacious arguments, the doctors of the church were sent reeling. It is therefore astonishing that the church made persistent efforts to accommodate its beliefs to Galileo's observations and claims. It was willing, for example, to accept as hypotheses that the earth moves and that the sun stands still. This, on the grounds that it is the business of mathematicians to formulate interesting hypotheses. But there could be no accommodation with Galileo's claim that the movement of the earth is a fact of nature. Such a belief was definitively held to be injurious to holy faith by contradicting Scripture. Thus, the trial of Galileo for heresy was inevitable even though long delayed. The trial took place in 1633, resulting in Galileo's conviction. Among the punishments were that Galileo was to abjure Copernican opinion, serve time in a formal prison, and for three years repeat once a week seven penitential psalms. There is probably no truth to the belief that Galileo mumbled at the conclusion of his sentencing, \"But the earth moves\" or some similar expression of defiance. He had, in fact, been asked four times at his trial if he believed in the Copernican view, and each time he said he did not. Everyone knew he believed otherwise, and that it was his advanced age, infirmities, and fear of torture that dictated his compliance. In any case, Galileo did not spend a single day in prison. He was confined at first to the grand duke's villa at Trinit\u00e0 del Monte, then to the palace of Archbishop Piccolomini in Siena, and finally to his home in Florence, where he remained for the rest of his life. He died in 1642, the year Isaac Newton was born.\n\nCopernicus, Kepler, and Galileo put in place the dynamite that would blow up the theology and metaphysics of the medieval world. Newton lit the fuse. In the ensuing explosion, Aristotle's animism was destroyed, along with almost everything else in his _Physics_. Scripture lost much of its authority. Theology, once the Queen of the Sciences, was now reduced to the status of Court Jester. Worst of all, the meaning of existence itself became an open question. And how ironic it all was! Whereas men had traditionally looked to Heaven to find authority, purpose, and meaning, the Sleepwalkers (as Arthur Koestler called Copernicus, Kepler, and Galileo) looked not to Heaven but to the heavens. There they found only mathematical equations and geometric patterns. They did so with courage but not without misgivings, for they did their best to keep their faith, and they did not turn away from God. They believed in a God who had planned and designed the whole of creation, a God who was a master mathematician. Their search for the mathematical laws of nature was, fundamentally, a religious quest. Nature was God's text, and Galileo found that God's alphabet consisted of \"triangles, quadrangles, circles, spheres, cones, pyramids, and other mathematical figures.\" Kepler agreed, and even boasted that God, the author, had to wait six thousand years for His first reader\u2014Kepler himself. As for Newton, he spent most of his later years trying to compute the generations since Adam, his faith in Scripture being unshaken. Descartes, whose _Discourse on Method_ , published in 1637, provided nobility to skepticism and reason and served as a foundation of the new science, was a profoundly religious man. Although he saw the universe as mechanistic (\"Give me matter and motion,\" he wrote, \"and I will construct the world\"), he deduced his law of the immutability of motion from the immutability of God.\n\nAll of them, to the end, clung to the theology of their age. They would surely not have been indifferent to knowing when the Last Judgment would come, and they could not have imagined the world without God. Moreover, the science they created was almost wholly concerned with questions of truth, not power. Toward that end, there developed in the late sixteenth century what can only be described as a passion for exactitude: exact dates, quantities, distances, rates. It was even thought possible to determine the exact moment of the Creation, which, as it turned out, commenced at 9:00 a.m., October 23, 4004 B.C. These were men who thought of philosophy (which is what they called science) as the Greeks did, believing that the true object of investigating nature is speculative satisfaction. They were not concerned with the idea of progress, and did not believe that their speculations held the promise of any important improvements in the conditions of life. Copernicus, Kepler, Galileo, Descartes, and Newton laid the foundation for the emergence of technocracies, but they themselves were men of tool-using cultures.\n\nFrancis Bacon, born in 1561, was the first man of the technocratic age. In saying this, I may be disputing no less an authority than Immanuel Kant, who said that a Kepler or a Newton was needed to find the law of the movement of civilization. Perhaps. But it was Bacon who first saw, pure and serene, the connection between science and the improvement of the human condition. The principal aim of his work was to advance \"the happiness of mankind,\" and he continually criticized his predecessors for failing to understand that the real, legitimate, and only goal of the sciences is the \"endowment of human life with new inventions and riches.\" He brought science down from the heavens, including mathematics, which he conceived of as a humble handmaiden to invention. In this utilitarian view of knowledge, Bacon was the chief architect of a new edifice of thought in which resignation was cast out and God assigned to a special room. The name of the building was Progress and Power.\n\nIronically, Bacon was not himself a scientist, or at least not much of one. He did no pioneering work in any field of research. He did not uncover any new law of nature or generate a single fresh hypothesis. He was not even well informed about the scientific investigations of his own time. And though he prided himself on being the creator of a revolutionary advance in scientific method, posterity has not allowed him this presumption. Indeed, his most famous experiment makes its claim on our attention because Bacon died as a result of it. He and his good friend Dr. Witherborne were taking a coach ride on a wintry day when, seeing snow on the ground, Bacon wondered if flesh might not be preserved in snow, as it is in salt. The two decided to find out at once. They bought a hen, removed its innards, and stuffed the body with snow. Poor Bacon never learned the result of his experiment, because he fell immediately ill from the cold, most probably with bronchitis, and died three days later. For this, he is sometimes regarded as a martyr to experimental science.\n\nBut experimental science was not where his greatness lay. Although others of his time were impressed by the effects of practical inventions on the conditions of life, Bacon was the first to think deeply and systematically on the matter. He devoted much of his work to educating men to see the links between invention and progress. In _Novum Organum_ he wrote,\n\nIt is well to observe the force and effect and consequences of discoveries. These are to be seen nowhere more conspicuously than in those three which were unknown to the ancients, and of which the origin, though recent, is obscure; namely, printing, gunpowder, and the magnet. For these three have changed the whole face and state of things throughout the world; the first in literature, the second in warfare, the third in navigation; whence have followed innumerable changes; insomuch that no empire, no sect, no star seems to have exerted greater power and influence in human affairs than these changes.\n\nIn this passage, we can detect some of Bacon's virtues and the source of his great influence. Here is no sleepwalker. He knows full well what technology does to culture and places technological development at the center of his reader's attention. He writes with conviction and verve. He is, after all, among the world's great essayists; Bacon was a master propagandist, who knew well the history of science but saw science not as a record of speculative opinion but as the record of what those opinions had enabled man to do. And he was ceaselessly energetic in trying to convey this idea to his countrymen, if not the world. In the first two books of _Novum Organum_ , which consist of 182 aphorisms, Bacon sets out nothing less than a philosophy of science based on the axiom that \"the improvement of men's minds and the improvement of his lot are one and the same thing.\" It is in this work that he denounces the infamous four Idols, which have kept man from gaining power over nature: Idols of the Tribe, which lead us to believe our perceptions are the same as nature's facts; Idols of the Cave, which lead us to mistaken ideas derived from heredity and environment; Idols of the Market-place, which lead us to be deluded by words; and Idols of the Theater, which lead us to the misleading dogmas of the philosophers.\n\nTo read Bacon today is to be constantly surprised at his modernity. We are never far from the now familiar notion that science is a source of power and progress. In _The Advancement of Learning_ , he even outlines the foundation of a College for Inventors that sounds something like the Massachusetts Institute of Technology. Bacon would have the government provide inventors with allowances for their experiments and for traveling. He would have scholarly journals and international associations. He would encourage full cooperation among scientists, an idea that would have startled Tycho Brahe, Kepler, and Galileo, who used some of their genius to devise ways of concealing their work from one another. Bacon also believed that scientists should be paid well to give public lectures, and that informing the public of the utility of invention was as important as invention itself. In short, he conceived of the scientific enterprise as it is conceived today\u2014organized, financially secure, public, and mankind's best weapon in the struggle to improve his condition and to do so continuously.\n\nAs I have said, Bacon is the first man of technocracy, but it was some time before he was joined by the multitude. He died in 1626, and it took another 150 years for European culture to pass to the mentality of the modern world\u2014that is, to technocracy. In doing so, people came to believe that knowledge is power, that humanity is capable of progressing, that poverty is a great evil, and that the life of the average person is as meaningful as any other. It is untrue to say that along the way God died. But any conception of God's design certainly lost much of its power and meaning, and with that loss went the satisfactions of a culture in which moral and intellectual values were integrated. At the same time, we must remember that in the tool-using culture of the older European world, the vast majority of people were peasants, impoverished and powerless. If they believed their afterlife was filled with unending joy, their lives on earth were nonetheless \"nasty, brutish and short.\" As C. P. Snow remarked, the Industrial Revolution of the nineteenth century, which was the fruit of Baconian science, was the only hope for the poor. And if their \"true Deity became mechanism,\" as Thomas Carlyle said, it is probable that by then most people would not have traded their earthly existence for life in a godly, integrated tool-using culture. It didn't matter if they would, since there was little use in lamenting the past. The Western world had become a technocracy from which there could be no turning back. Addressing both those who were exhilarated by technocracy and those who were repulsed by it, Stephen Vincent Ben\u00e9t gave the only advice that made any sense. In _John Brown's Body_ he wrote:\n\n_If you at last must have a word to say_ ,  \n _Say neither, in their way_ ,  \n _\"It is a deadly magic and accursed,\"_  \n _Nor \"It is blest,\" but only \"It is here.\"_\n\n# **3**\n\n# From Technocracy to Technopoly\n\nSay only, \"It is here.\" But when did \"here\" begin? When did Bacon's ideology become a reality? When, to use Siegfried Giedion's phrase, did mechanization take command? To be cautious about it, we might locate the emergence of the first true technocracy in England in the latter half of the eighteenth century\u2014let us say with James Watt's invention of the steam engine in 1765. From that time forward, a decade did not pass without the invention of some significant machinery which, taken together, put an end to medieval \"manufacture\" (which once meant \"to make by hand\"). The practical energy and technical skills unleashed at this time changed forever the material and psychic environment of the Western world.\n\nAn equally plausible date for the beginnings of technocracy (and, for Americans, easier to remember) is 1776, when Adam Smith's _Wealth of Nations_ was published. As Bacon was no scientist, Smith was no inventor. But, like Bacon, he provided a theory that gave conceptual relevance and credibility to the direction in which human enterprise was pointed. Specifically, he justified the transformation from small-scale, personalized, skilled labor to large-scale, impersonal, mechanized production. He not only argued convincingly that money, not land, was the key to wealth, but gave us his famous principle of the self-regulating market. In a technocracy\u2014that is, a society only loosely controlled by social custom and religious tradition and driven by the impulse to invent\u2014an \"unseen hand\" will eliminate the incompetent and reward those who produce cheaply and well the goods that people want. It was not clear then, and still isn't, whose unseen mind guides the unseen hand, but it is possible (the technocratic industrialists believed) that God could have something to do with it. And if not God, then \"human nature,\" for Adam Smith had named our species \"Economic Man,\" born with an instinct to barter and acquire wealth.\n\nIn any case, toward the end of the eighteenth century, technocracy was well underway, especially after Richard Arkwright, a barber by trade, developed the factory system. In his cotton-spinning mills, Arkwright trained workers, mostly children, \"to conform to the regular celerity of the machine,\" and in doing so gave an enormous boost to the growth of modern forms of technocratic capitalism. In 1780, twenty factories were under his control, for which a grateful nation knighted him, and from which an equally grateful son inherited a fortune. Arkwright may fairly be thought of as the first\u2014even archetypal\u2014technocratic capitalist. He exemplified in every particular the type of nineteenth-century entrepreneur to come. As Siegfried Giedion has described him, Arkwright created the first mechanization of production \"[in] a hostile environment, without protectors, without government subsidy, but nourished by a relentless utilitarianism that feared no financial risk or danger.\" By the beginning of the nineteenth century, England was spawning such entrepreneurs in every major city. By 1806, the concept of the power loom, introduced by Edmund Cartwright (a clergyman no less), was revolutionizing the textile industry by eliminating, once and for all, skilled workers, replacing them with workers who merely kept the machines operating.\n\nBy 1850, the machine-tool industry was developed\u2014machines to make machines. And beginning in the 1860s, especially in America, a collective fervor for invention took hold of the masses. To quote Giedion again: \"Everyone invented, whoever owned an enterprise sought ways and means to make his goods more speedily, more perfectly, and often of improved beauty. Anonymously and inconspicuously the old tools were transformed into modern instruments.\" Because of their familiarity, it is not necessary to describe in detail all of the inventions of the nineteenth century, including those which gave substance to the phrase \"communications revolution\": the photograph and telegraph (1830s), rotary-power printing (1840s), the typewriter (1860s), the transatlantic cable (1866), the telephone (1876), motion pictures and wireless telegraphy (1895). Alfred North Whitehead summed it up best when he remarked that the greatest invention of the nineteenth century was the idea of invention itself. We had learned _how_ to invent things, and the question of _why_ we invent things receded in importance. The idea that if something could be done it should be done was born in the nineteenth century. And along with it, there developed a profound belief in all the principles through which invention succeeds: objectivity, efficiency, expertise, standardization, measurement, and progress. It also came to be believed that the engine of technological progress worked most efficiently when people are conceived of not as children of God or even as citizens but as consumers\u2014that is to say, as markets.\n\nNot everyone agreed, of course, especially with the last notion. In England, William Blake wrote of the \"dark Satanic mills\" which stripped men of their souls. Matthew Arnold warned that \"faith in machinery\" was mankind's greatest menace. Carlyle, Ruskin, and William Morris railed against the spiritual degradation brought by industrial progress. In France, Balzac, Flaubert, and Zola documented in their novels the spiritual emptiness of \"Economic man\" and the poverty of the acquisitive impulse.\n\nThe nineteenth century also saw the emergence of \"utopian\" communities, of which perhaps the most famous is Robert Owen's experimental community in Scotland called New Lanark. There, he established a model factory community, providing reduced working hours, improved living conditions, and innovative education for the children of workers. In 1824, Owen came to America and founded another utopia at New Harmony, Indiana. Although none of his or other experiments endured, dozens were tried in an effort to reduce the human costs of a technocracy.\n\nWe also must not omit mentioning the rise and fall of the much-maligned Luddite Movement. The origin of the term is obscure, some believing that it refers to the actions of a youth named Ludlum who, being told by his father to fix a weaving machine, proceeded instead to destroy it. In any case, between 1811 and 1816, there arose widespread support for workers who bitterly resented the new wage cuts, child labor, and elimination of laws and customs that had once protected skilled workers. Their discontent was expressed through the destruction of machines, mostly in the garment and fabric industry; since then the term \"Luddite\" has come to mean an almost childish and certainly na\u00efve opposition to technology. But the historical Luddites were neither childish nor na\u00efve. They were people trying desperately to preserve whatever rights, privileges, laws, and customs had given them justice in the older world-view.\n\nThey lost. So did all the other nineteenth-century nay-sayers. Copernicus, Kepler, Galileo, and Newton might well have been on their side. Perhaps Bacon as well, for it was not his intention that technology should be a blight or a destroyer. But then, Bacon's greatest deficiency had always been that he was unfamiliar with the legend of Thamus; he understood nothing of the dialectic of technological change, and said little about the negative consequences of technology. Even so, taken as a whole, the rise of technocracy would probably have pleased Bacon, for there can be no disputing that technocracy transformed the face of material civilization, and went far toward relieving what Tocqueville called \"the disease of work.\" And though it is true that technocratic capitalism created slums and alienation, it is also true that such conditions were perceived as an evil that could and should be eradicated; that is to say, technocracies brought into being an increased respect for the average person, whose potential and even convenience became a matter of compelling political interest and urgent social policy. The nineteenth century saw the extension of public education, laid the foundation of the modern labor union, and led to the rapid diffusion of literacy, especially in America, through the development of public libraries and the increased importance of the general-interest magazine. To take only one example of the last point, the list of nineteenth-century contributors to _The Saturday Evening Post_ , founded in 1821, included William Cullen Bryant, Harriet Beecher Stowe, James Fenimore Cooper, Ralph Waldo Emerson, Nathaniel Hawthorne, and Edgar Allan Poe\u2014in other words, most of the writers presently included in American Lit. 101. The technocratic culture eroded the line that had made the intellectual interests of educated people inaccessible to the working class, and we may take it as a fact, as George Steiner has remarked, that the period from the French Revolution to World War I marked an oasis of quality in which great literature reached a mass audience.\n\nSomething else reached a mass audience as well: political and religious freedom. It would be an inadmissible simplification to claim that the Age of Enlightenment originated solely because of the emerging importance of technology in the eighteenth century, but it is quite clear that the great stress placed on individuality in the economic sphere had an irresistible resonance in the political sphere. In a technocracy, inherited royalty is both irrelevant and absurd. The new royalty was reserved for men like Richard Arkwright, whose origins were low but whose intelligence and daring soared. Those possessed of such gifts could not be denied political power and were prepared to take it if it were not granted. In any case, the revolutionary nature of the new means of production and communication would have naturally generated radical ideas in every realm of human enterprise. Technocracy gave us the idea of progress, and of necessity loosened our bonds with tradition\u2014whether political or spiritual. Technocracy filled the air with the promise of new freedoms and new forms of social organization. Technocracy also speeded up the world. We could get places faster, do things faster, accomplish more in a shorter time. Time, in fact, became an adversary over which technology could triumph. And this meant that there was no time to look back or to contemplate what was being lost. There were empires to build, opportunities to exploit, exciting freedoms to enjoy, especially in America. There, on the wings of technocracy, the United States soared to unprecedented heights as a world power. That Jefferson, Adams, and Madison would have found such a place uncomfortable, perhaps even disagreeable, did not matter. Nor did it matter that there were nineteenth-century American voices\u2014Thoreau, for example\u2014who complained about what was being left behind. The first answer to the complaints was, We leave nothing behind but the chains of a tool-using culture. The second answer was more thoughtful: Technocracy will not overwhelm us. And this was true, to a degree. Technocracy did not entirely destroy the traditions of the social and symbolic worlds. Technocracy subordinated these worlds\u2014yes, even humiliated them\u2014but it did not render them totally ineffectual. In nineteenth-century America, there still existed holy men and the concept of sin. There still existed regional pride, and it was possible to conform to traditional notions of family life. It was possible to respect tradition itself and to find sustenance in ritual and myth. It was possible to believe in social responsibility and the practicality of individual action. It was even possible to believe in common sense and the wisdom of the elderly. It was not easy, but it was possible.\n\nThe technocracy that emerged, fully armed, in nineteenth-century America disdained such beliefs, because holy men and sin, grandmothers and families, regional loyalties and two-thousand-year-old traditions, are antagonistic to the technocratic way of life. They are a troublesome residue of a tool-using period, a source of criticism of technocracy. They represent a thought-world that stands apart from technocracy and rebukes it\u2014rebukes its language, its impersonality, its fragmentation, its alienation. And so technocracy disdains such a thought-world but, in America, did not and could not destroy it.\n\nWe may get a sense of the interplay between technocracy and Old World values in the work of Mark Twain, who was fascinated by the technical accomplishments of the nineteenth century. He said of it that it was \"the plainest and sturdiest and infinitely greatest and worthiest of all the centuries the world has seen,\" and he once congratulated Walt Whitman on having lived in the age that gave the world the beneficial products of coal tar. It is often claimed that he was the first writer regularly to use a typewriter, and he invested (and lost) a good deal of money in new inventions. In his _Life on the Mississippi_ , he gives lovingly detailed accounts of industrial development, such as the growth of the cotton mills in Natchez:\n\nThe Rosalie Yarn Mill of Natchez has a capacity of 6000 spindles and 160 looms, and employs 100 hands. The Natchez Cotton Mills Company began operations four years ago in a two-story building of 50 \u00d7 190 feet, with 4000 spindles and 128 looms.... The mill works 5000 bales of cotton annually and manufactures the best standard quality of brown shirtings and sheetings and drills, turning out 5,000,000 yards of these goods per year.\n\nTwain liked nothing better than to describe the giantism and ingenuity of American industry. But at the same time, the totality of his work is an affirmation of preindustrial values. Personal loyalty, regional tradition, the continuity of family life, the relevance of the tales and wisdom of the elderly are the soul of his work throughout. The story of Huckleberry Finn and Jim making their way to freedom on a raft is nothing less than a celebration of the enduring spirituality of pretechnological man.\n\nIf we ask, then, why technocracy did not destroy the world-view of a tool-using culture, we may answer that the fury of industrialism was too new and as yet too limited in scope to alter the needs of inner life or to drive away the language, memories, and social structures of the tool-using past. It was possible to contemplate the wonders of a mechanized cotton mill without believing that tradition was entirely useless. In reviewing nineteenth-century American history, one can hear the groans of religion in crisis, of mythologies under attack, of a politics and education in confusion, but the groans are not yet death-throes. They are the sounds of a culture in pain, and nothing more. The ideas of tool-using cultures were, after all, designed to address questions that still lingered in a technocracy. The citizens of a technocracy knew that science and technology did not provide philosophies by which to live, and they clung to the philosophies of their fathers. They could not convince themselves that religion, as Freud summed it up at the beginning of the twentieth century, is nothing but an obsessional neurosis. Nor could they quite believe, as the new cosmology taught, that the universe is the outcome of accidental collocations of atoms. And they continued to believe, as Mark Twain did, that, for all their dependence on machinery, tools ought still to be their servants, not their masters. They would allow their tools to be presumptuous, aggressive, audacious, impudent servants, but that tools should rise above their servile station was an appalling thought. And though technocracy found no clear place for the human soul, its citizens held to the belief that no increase in material wealth would compensate them for a culture that insulted their self-respect.\n\nAnd so two opposing world-views\u2014the technological and the traditional\u2014coexisted in uneasy tension. The technological was the stronger, of course, but the traditional was there\u2014still functional, still exerting influence, still too much alive to ignore. This is what we find documented not only in Mark Twain but in the poetry of Walt Whitman, the speeches of Abraham Lincoln, the prose of Thoreau, the philosophy of Emerson, the novels of Hawthorne and Melville, and, most vividly of all, in Alexis de Tocqueville's monumental _Democracy in America_. In a word, two distinct thought-worlds were rubbing against each other in nineteenth-century America.\n\nWith the rise of Technopoly, one of those thought-worlds disappears. Technopoly eliminates alternatives to itself in precisely the way Aldous Huxley outlined in _Brave New World_. It does not make them illegal. It does not make them immoral. It does not even make them unpopular. It makes them invisible and therefore irrelevant. And it does so by redefining what we mean by religion, by art, by family, by politics, by history, by truth, by privacy, by intelligence, so that our definitions fit its new requirements. Technopoly, in other words, is totalitarian technocracy.\n\nAs I write (in fact, it is the reason why I write), the United States is the only culture to have become a Technopoly. It is a young Technopoly, and we can assume that it wishes not merely to have been the first but to remain the most highly developed. Therefore, it watches with a careful eye Japan and several European nations that are striving to become Technopolies as well.\n\nTo give a date to the beginnings of Technopoly in America is an exercise in arbitrariness. It is somewhat like trying to say, precisely, when a coin you have flipped in the air begins its descent. You cannot see the exact moment it stops rising; you know only that it has and is going the other way. Huxley himself identified the emergence of Henry Ford's empire as the decisive moment in the shift from technocracy to Technopoly, which is why in his brave new world time is reckoned as BF (Before Ford) and AF (After Ford).\n\nBecause of its drama, I am tempted to cite, as a decisive moment, the famous Scopes \"monkey\" trial held in Dayton, Tennessee, in the summer of 1925. There, as with Galileo's heresy trial three centuries earlier, two opposing world-views faced each other, toe to toe, in unconcealed conflict. And, as in Galileo's trial, the dispute focused not only on the content of \"truth\" but also on the appropriate process by which \"truth\" was to be determined. Scopes' defenders brought forward (or, more accurately, tried to bring forward) all the assumptions and methodological ingenuity of modern science to demonstrate that religious belief can play no role in discovering and understanding the origins of life. William Jennings Bryan and his followers fought passionately to maintain the validity of a belief system that placed the question of origins in the words of their god. In the process, they made themselves appear ridiculous in the eyes of the world. Almost seventy years later, it is not inappropriate to say a word in their behalf: These \"fundamentalists\" were neither ignorant of nor indifferent to the benefits of science and technology. They had automobiles and electricity and machine-made clothing. They used telegraphy and radio, and among their number were men who could fairly be called reputable scientists. They were eager to share in the largesse of the American technocracy, which is to say they were neither Luddites nor primitives. What wounded them was the assault that science made on the ancient story from which their sense of moral order sprang. They lost, and lost badly. To say, as Bryan did, that he was more interested in the Rock of Ages than the age of rocks was clever and amusing but woefully inadequate. The battle settled the issue, once and for all: in defining truth, the great narrative of inductive science takes precedence over the great narrative of Genesis, and those who do not agree must remain in an intellectual backwater.\n\nAlthough the Scopes trial has much to recommend it as an expression of the ultimate repudiation of an older world-view, I must let it pass. The trial had more to do with science and faith than technology _as_ faith. To find an event that signaled the beginning of a technological theology, we must look to a slightly earlier and less dramatic confrontation. Not unmindful of its value as a pun, I choose what happened in the fall of 1910 as the critical symptom of the onset of Technopoly. From September through November of that year, the Interstate Commerce Commission held hearings on the application of Northeastern railroads for an increase in freight rates to compensate for the higher wages railroad workers had been awarded earlier in the year. The trade association, represented by Louis Brandeis, argued against the application by claiming that the railroads could increase their profits simply by operating more efficiently. To give substance to the argument, Brandeis brought forward witnesses\u2014mostly engineers and industrial managers\u2014who claimed that the railroads could both increase wages and lower their costs by using principles of _scientific management_. Although Frederick W. Taylor was not present at the hearings, his name was frequently invoked as the originator of scientific management, and experts assured the commission that the system developed by Taylor could solve everyone's problem. The commission ultimately ruled against the railroad's application, mostly because it judged that the railroads were making enough money as things were, not because it believed in scientific management. But many people did believe, and the hearings projected Taylor and his system onto the national scene. In the years that followed, attempts were made to apply the principles of the Taylor System in the armed forces, the legal profession, the home, the church, and education. Eventually, Taylor's name and the specifics of his system faded into obscurity, but his ideas about what culture is made of remain the scaffolding of the present-day American Technopoly.\n\nI use this event as a fitting starting point because Taylor's book _The Principles of Scientific Management_ , published in 1911, contains the first explicit and formal outline of the assumptions of the thought-world of Technopoly. These include the beliefs that the primary, if not the only, goal of human labor and thought is efficiency; that technical calculation is in all respects superior to human judgment; that in fact human judgment cannot be trusted, because it is plagued by laxity, ambiguity, and unnecessary complexity; that subjectivity is an obstacle to clear thinking; that what cannot be measured either does not exist or is of no value; and that the affairs of citizens are best guided and conducted by experts. In fairness to Taylor (who did not invent the term \"scientific management\" and who used it reluctantly), it should be noted that his system was originally devised to apply only to industrial production. His intention was to make a science of the industrial workplace, which would not only increase profits but also result in higher wages, shorter hours, and better working conditions for laborers. In his system, which included \"time and motion studies,\" the judgment of individual workers was replaced by laws, rules, and principles of the \"science\" of their job. This did mean, of course, that workers would have to abandon any traditional rules of thumb they were accustomed to using; in fact, workers were relieved of any responsibility to think at all. The system would do their thinking for them. That is crucial, because it led to the idea that technique of any kind can do our thinking for us, which is among the basic principles of Technopoly.\n\nThe assumptions that underlay the principles of scientific management did not spring, all at once, from the originality of Taylor's mind. They were incubated and nurtured in the technocracies of the eighteenth and nineteenth centuries. And a fair argument can be made that the origins of Technopoly are to be found in the thought of the famous nineteenth-century French philosopher Auguste Comte, who founded both positivism and sociology in an effort to construct a science of society. Comte's arguments for the unreality of anything that could not be seen and measured certainly laid the foundation for the future conception of human beings as objects. But in a technocracy, such ideas exist only as by-products of the increased role of technology. Technocracies are concerned to invent machinery. That people's lives are changed by machinery is taken as a matter of course, and that people must sometimes be treated as if they were machinery is considered a necessary and unfortunate condition of technological development. But in technocracies, such a condition is not held to be a philosophy of culture. Technocracy does not have as its aim a grand reductionism in which human life must find its meaning in machinery and technique. Technopoly does. In the work of Frederick Taylor we have, I believe, the first clear statement of the idea that society is best served when human beings are placed at the disposal of their techniques and technology, that human beings are, in a sense, worth less than their machinery. He and his followers described exactly what this means, and hailed their discovery as the beginnings of a brave new world.\n\nWhy did Technopoly\u2014the submission of all forms of cultural life to the sovereignty of technique and technology\u2014find fertile ground on American soil? There are four interrelated reasons for the rise of Technopoly in America, why it emerged in America first, and why it has been permitted to flourish. As it happens, all of these have been written about extensively in many contexts and are well known. The first concerns what is usually called the American character, the relevant aspect of which Tocqueville described in the early nineteenth century. \"The American lives in a land of wonders,\" he wrote; \"everything around him is in constant movement, and every movement seems an advance. Consequently, in his mind the idea of newness is closely linked with that of improvement. Nowhere does he see any limit placed by nature to human endeavor; in his eyes something that does not exist is just something that has not been tried.\"\n\nThis feature of the American ethos is plain to everyone who has studied American culture, although there are wide variations in the explanation of it. Some attribute it to the immigrant nature of the population; some to the frontier mentality; some to the abundant natural resources of a singularly blessed land and the unlimited opportunities of a new continent; some to the unprecedented political and religious freedom afforded the average person; some to all of these factors and more. It is enough to say here that the American distrust of constraints\u2014one might even say the American skepticism toward culture itself\u2014offered encouragement to radical and thoughtless technological intrusions.\n\nSecond, and inextricably related to the first, is the genius and audacity of American capitalists of the late nineteenth and early twentieth centuries, men who were quicker and more focused than those of other nations in exploiting the economic possibilities of new technologies. Among them are Samuel Morse, Alexander Graham Bell, Thomas Edison, John D. Rockefeller, John Jacob Astor, Henry Ford, Andrew Carnegie, and many others, some of whom were known as Robber Barons. What they were robbing\u2014it is clearer now than it was then\u2014was America's past, for their essential idea was that nothing is so much worth preserving that it should stand in the way of technological innovation. These were the men who created the twentieth century, and they achieved wealth, prestige, and power that would have amazed even Richard Arkwright. Their greatest achievement was in convincing their countrymen that the future need have no connection to the past.\n\nThird, the success of twentieth-century technology in providing Americans with convenience, comfort, speed, hygiene, and abundance was so obvious and promising that there seemed no reason to look for any other sources of fulfillment or creativity or purpose. To every Old World belief, habit, or tradition, there was and still is a technological alternative. To prayer, the alternative is penicillin; to family roots, the alternative is mobility; to reading, the alternative is television; to restraint, the alternative is immediate gratification; to sin, the alternative is psychotherapy; to political ideology, the alternative is popular appeal established through scientific polling. There is even an alternative to the painful riddle of death, as Freud called it. The riddle may be postponed through longer life, and then perhaps solved altogether by cryogenics. At least, no one can easily think of a reason why not.\n\nAs the spectacular triumphs of technology mounted, something else was happening: old sources of belief came under siege. Nietzsche announced that God was dead. Darwin didn't go as far but did make it clear that, if we were children of God, we had come to be so through a much longer and less dignified route than we had imagined, and that in the process we had picked up some strange and unseemly relatives. Marx argued that history had its own agenda and was taking us where it must, irrespective of our wishes. Freud taught that we had no understanding of our deepest needs and could not trust our traditional ways of reasoning to uncover them. John Watson, the founder of behaviorism, showed that free will was an illusion and that our behavior, in the end, was not unlike that of pigeons. And Einstein and his colleagues told us that there were no absolute means of judging anything in any case, that everything was relative. The thrust of a century of scholarship had the effect of making us lose confidence in our belief systems and therefore in ourselves. Amid the conceptual debris, there remained one sure thing to believe in\u2014technology. Whatever else may be denied or compromised, it is clear that airplanes do fly, antibiotics do cure, radios do speak, and, as we know now, computers do calculate and never make mistakes\u2014only faulty humans do (which is what Frederick Taylor was trying to tell us all along).\n\nFor these well-known reasons, Americans were better prepared to undertake the creation of a Technopoly than anyone else. But its full flowering depended on still another set of conditions, less visible and therefore less well known. These conditions provided the background, the context in which the American distrust of constraints, the exploitative genius of its captains of industry, the successes of technology, and the devaluation of traditional beliefs took on the exaggerated significance that pushed technocracy in America over into Technopoly. That context is explored in the following chapter, which I call \"The Improbable World.\"\n\n# **4**\n\n# The Improbable World\n\nAlthough it is clear that \"social science\" is a vigorous ally of Technopoly and must therefore be regarded with a hostile eye, I occasionally pay my respects to its bloated eminence by inflicting a small experiment on some of my colleagues. Like many other social-science experiments, this one is based on deceit and exploitation, and I must rely on the reader's sense of whimsy to allow its point to come through.\n\nThe experiment is best conducted in the morning when I see a colleague who appears not to be in possession of a copy of _The New York Times_. \"Did you read the _Times_ this morning?\" I ask. If my colleague says, \"Yes,\" there is no experiment that day. But if the answer is \"No,\" the experiment can proceed. \"You ought to check out Section C today,\" I say. \"There's a fascinating article about a study done at the University of Minnesota.\" \"Really? What's it about?\" is the usual reply. The choices at this point are almost endless, but there are two that produce rich results. The first: \"Well, they did this study to find out what foods are best to eat for losing weight, and it turns out that a normal diet supplemented by chocolate eclairs eaten three times a day is the best approach. It seems that there's some special nutrient in the eclairs\u2014encomial dyoxin\u2014that actually uses up calories at an incredible rate.\"\n\nThe second changes the theme and, from the start, the university: \"The neurophysiologists at Johns Hopkins have uncovered a connection between jogging and reduced intelligence. They tested more than twelve hundred people over a period of five years, and found that as the number of hours people jogged increased there was a statistically significant decrease in their intelligence. They don't know exactly why, but there it is.\"\n\nMy role in the experiment, of course, is to report something quite ridiculous\u2014one might say, beyond belief. If I play my role with a sense of decorum and coll\u00e9gial intimacy, I can achieve results worth reporting: about two-thirds of the victims will believe or at least not wholly _disbelieve_ what I have told them. Sometimes they say, \"Really? Is that possible?\" Sometimes they do a double-take and reply, _\"Where'd_ you say that study was done?\" And sometimes they say, \"You know, I've _heard_ something like that.\" I should add that for reasons that are probably worth exploring I get the clearest cases of credulity when I use the University of Minnesota and Johns Hopkins as my sources of authority; Stanford and MIT give only fair results.\n\nThere are several conclusions that might be drawn from these results, one of which was expressed by H. L. Mencken fifty years ago, when he said that there is no idea so stupid that you can't find a professor who will believe it. This is more an accusation than an explanation, although there is probably something to it. (I have, however, tried this experiment on nonprofessors as well, and get roughly the same results.) Another possible conclusion was expressed by George Bernard Shaw, also about fifty years ago, when he wrote that the average person today is about as credulous as was the average person in the Middle Ages. In the Middle Ages, people believed in the authority of their religion, no matter what. Today, we believe in the authority of our science, no matter what.\n\nHowever, there is still another possibility, related to Shaw's point but off at a right angle to it. It is, in any case, more relevant to understanding the sustaining power of Technopoly. I mean that the world we live in is very nearly incomprehensible to most of us. There is almost no fact, whether actual or imagined, that will surprise us for very long, since we have no comprehensive and consistent picture of the world that would make the fact appear as an unacceptable contradiction. We believe because there is no reason not to believe. And I assume that the reader does not need the evidence of my comic excursion into the suburbs of social science to recognize this. Abetted by a form of education that in itself has been emptied of any coherent world-view, Technopoly deprives us of the social, political, historical, metaphysical, logical, or spiritual bases for knowing what is beyond belief.\n\nThat is especially the case with technical facts. Since this book is filled with a variety of facts, I would hardly wish to shake confidence in them by trying my experiment on the reader. But if I informed you that the paper on which this book is printed was made by a special process which uses the skin of a pickled herring, on what grounds would you dispute me? For all you know\u2014indeed, for all _I_ know\u2014the skin of a pickled herring could have made this paper. And if the facts were confirmed by an industrial chemist who described to us some incomprehensible process by which it was done (employing, of course, encomial dyoxin), we might both believe it. Or not wholly disbelieve it, since the ways of technology, like the ways of God, are awesome and mysterious.\n\nPerhaps I can get a bit closer to the point with an analogy. If you open a brand-new deck of cards and start turning the cards over, one by one, you can get a pretty firm idea of what their order is. After you have gone from the ace of spades through to the nine of spades, you expect a ten of spades to come up next. And if the three of diamonds appears, you are surprised and wonder what kind of deck of cards this is. But if I give you a deck that had been shuffled twenty times and then ask you to turn the cards over, you do not expect any card in particular\u2014a three of diamonds would be just as likely as a ten of spades. Having no expectation of a pattern, no basis for assuming a given order, you have no reason to react with incredulity or even surprise to whatever card turns up.\n\nThe belief system of a tool-using culture is rather like a brand-new deck of cards. Whether it is a culture of technological simplicity or sophistication, there always exists a more or less comprehensive, ordered world-view, resting on a set of metaphysical or theological assumptions. Ordinary men and women might not clearly grasp how the harsh realities of their lives fit into the grand and benevolent design of the universe, but they have no doubt that there _is_ such a design, and their priests and shamans are well able, by deduction from a handful of principles, to make it, if not wholly rational, at least coherent. The medieval period was a particularly clear example of this point. How comforting it must have been to have a priest explain the meaning of the death of a loved one, of an accident, or of a piece of good fortune. To live in a world in which there were no random events\u2014in which everything was, in theory, comprehensible; in which every act of nature was infused with meaning\u2014is an irreplaceable gift of theology. The role of the church in premodern Europe was to keep the deck of cards in reasonable order, which is why Cardinal Bellarmine and other prelates tried to prevent Galileo from shuffling the deck. As we know, they could not, and with the emergence of technocracies moral and intellectual coherence began to unravel.\n\nWhat was being lost was not immediately apparent. The decline of the great narrative of the Bible, which had provided answers to both fundamental and practical questions, was accompanied by the rise of the great narrative of Progress. The faith of those who believed in Progress was based on the assumption that one could discern a purpose to the human enterprise, even without the theological scaffolding that supported the Christian edifice of belief. Science and technology were the chief instruments of Progress, and in their accumulation of reliable information about nature they would bring ignorance, superstition, and suffering to an end. As it turned out, technocracies did not disappoint Progress. In sanitation, pharmacology, transportation, production, and communication, spectacular improvements were made possible by a Niagara of information generated by just such institutions as Francis Bacon had imagined. Technocracy was fueled by information\u2014about the structure of nature as well as the structure of the human soul.\n\nBut the genie that came out of the bottle proclaiming that information was the new god of culture was a deceiver. It solved the problem of information scarcity, the disadvantages of which were obvious. But it gave no warning about the dangers of information glut, the disadvantages of which were not seen so clearly. The long-range result\u2014information chaos\u2014has produced a culture somewhat like the shuffled deck of cards I referred to. And what is strange is that so few have noticed, or if they have noticed fail to recognize the source of their distress. You need only ask yourself, What is the problem in the Middle East, or South Africa, or Northern Ireland? Is it lack of information that keeps these conflicts at fever pitch? Is it lack of information about how to grow food that keeps millions at starvation levels? Is it lack of information that brings soaring crime rates and physical decay to our cities? Is it lack of information that leads to high divorce rates and keeps the beds of mental institutions filled to overflowing?\n\nThe fact is, there are very few political, social, and especially personal problems that arise because of insufficient information. Nonetheless, as incomprehensible problems mount, as the concept of progress fades, as meaning itself becomes suspect, the Technopolist stands firm in believing that what the world needs is yet more information. It is like the joke about the man who complains that the food he is being served in a restaurant is inedible and also that the portions are too small. But, of course, what we are dealing with here is no joke. Attend any conference on telecommunications or computer technology, and you will be attending a celebration of innovative machinery that generates, stores, and distributes more information, more conveniently, at greater speeds than ever before. To the question \"What problem does the information solve?\" the answer is usually \"How to generate, store, and distribute more information, more conveniently, at greater speeds than ever before.\" This is the elevation of information to a metaphysical status: information as both the means and end of human creativity. In Technopoly, we are driven to fill our lives with the quest to \"access\" information. For what purpose or with what limitations, it is not for us to ask; and we are not accustomed to asking, since the problem is unprecedented. The world has never before been confronted with information glut and has hardly had time to reflect on its consequences.\n\nAs with so many of the features of all that is modern, the origins of information glut can be traced many centuries back. Nothing could be more misleading than the claim that computer technology introduced the age of information. The printing press began that age in the early sixteenth century. Forty years after Gutenberg converted an old wine press into a printing machine with movable type, there were presses in 110 cities in six different countries. Fifty years after the press was invented, more than eight million books had been printed, almost all of them filled with information that had previously been unavailable to the average person. There were books on law, agriculture, politics, exploration, metallurgy, botany, linguistics, pediatrics, and even good manners. There were also assorted guides and manuals; the world of commerce rapidly became a world of printed paper through the widespread use of contracts, deeds, promissory notes, and maps. (Not surprisingly, in a culture in which information was becoming standardized and repeatable, mapmakers began to exclude \"paradise\" from their charts on the grounds that its location was too uncertain.)\n\nSo much new information, of so many diverse types, was generated that printers could no longer use the scribal manuscript as their model of a book. By the mid-sixteenth century, printers began to experiment with new formats, among the most important innovations being the use of Arabic numerals to number pages. (The first known example of such pagination is Johann Froben's first edition of Erasmus' New Testament, printed in 1516.) Pagination led inevitably to more accurate indexing, annotation, and cross-referencing, which in turn was accompanied by innovations in punctuation marks, section heads, paragraphing, title-paging, and running heads. By the end of the sixteenth century, the machine-made book had a typographic form and a look comparable to books of today.\n\nAll of this is worth mentioning because innovations in the format of the machine-made book were an attempt to control the flow of information, to organize it by establishing priorities and by giving it sequence. Very early on, it was understood that the printed book had created an information crisis and that something needed to be done to maintain a measure of control. The altered form of the book was one means. Another was the modern school, which took shape in the seventeenth century. In 1480, before the information explosion, there were thirty-four schools in all of England. By 1660, there were 444, one school for every twelve square miles. There were several reasons for the rapid growth of the common school, but none was more obvious than that it was a necessary response to the anxieties and confusion aroused by information on the loose. The invention of what is called a curriculum was a logical step toward organizing, limiting, and discriminating among available sources of information. Schools became technocracy's first secular bureaucracies, structures for legitimizing some parts of the flow of information and discrediting other parts. Schools were, in short, a means of governing the ecology of information.\n\nWith the rise of technocracies, information became a more serious problem than ever, and several methods of controlling information had to be invented. For a richly detailed account of what those methods were, I refer the reader to James Beniger's _The Control Revolution_ , which is among the three or four most important books we have on the subject of the relation of information to culture. In the next chapter, I have relied to a considerable degree on _The Control Revolution_ in my discussion of the breakdown of the control mechanisms, but here I must note that most of the methods by which technocracies have hoped to keep information from running amok are now dysfunctional.\n\nIndeed, one way of defining a Technopoly is to say that its information immune system is inoperable. Technopoly is a form of cultural AIDS, which I here use as an acronym for Anti-Information Deficiency Syndrome. This is why it is possible to say almost anything without contradiction provided you begin your utterance with the words \"A study has shown...\" or \"Scientists now tell us that...\" More important, it is why in a Technopoly there can be no transcendent sense of purpose or meaning, no cultural coherence. Information is dangerous when it has no place to go, when there is no theory to which it applies, no pattern in which it fits, when there is no higher purpose that it serves. Alfred North Whitehead called such information \"inert,\" but that metaphor is too passive. Information without regulation can be lethal. It is necessary, then, to describe briefly the technological conditions that led to such a grim state of affairs.\n\nIf the telescope was the eye that gave access to a world of new facts and new methods of obtaining them, then the printing press was the larynx. The press not only created new sources of data collection but vastly increased communication among scientists on a continent-wide basis. The movement toward standardization of scientific discourse resulted, for example, in uniform mathematical symbols, including the replacement of Roman with Arabic numerals. Galileo's and Kepler's reference to mathematics as the language or alphabet of nature could be made with assurance that other scientists could speak and understand that language. Standardization largely eliminated ambiguity in texts and reduced error in diagrams, charts, and visual aids. Printing brought an end to the alchemists' secrets by making science into a public enterprise. And not only for scientists: printing led to the popularization of scientific ideas through the use of vernaculars. Although some scientists\u2014Harvey, for example\u2014insisted on writing in Latin, many others (Bacon, of course) eagerly employed the vernacular in an effort to convey the new spirit and methods of scientific philosophy. When we consider that Vesalius, Brahe, Bacon, Galileo, Kepler, Harvey, and Descartes were all born in the sixteenth century, we can begin to grasp the relationship between the growth of science and the printing press, which is to say, the press announced the advent of science, publicized it, encouraged it, and codified it.\n\nAs is known, the press did the same for what is now called Protestantism. Martin Luther's reliance on printed pamphlets and books as a means of religious propaganda is well documented, as is his own acknowledgment of the importance of print to his mission. And yet, for all of Luther's astuteness about printing, even he was surprised on occasion by the unsuspected powers of the press. \"It is a mystery to me,\" he wrote in a letter to the Pope, \"how my theses... were spread to so many places. They were meant exclusively for our academic circle here.... They were written in such a language that the common people could hardly understand them.\" What Luther overlooked was the sheer _portability_ of printed books. Although his theses were written in academic Latin, they were easily transported throughout Germany and other countries by printers who just as easily had them translated into vernaculars.\n\nWithout going any further into the details of the impact of print on medieval thought, all of which are lucidly presented in Elizabeth Eisenstein's _The Printing Press as an Agent of Change_ , I will instead merely assert the obvious point: By the beginning of the seventeenth century, an entirely new information environment had been created by print. Astronomy, anatomy, and physics were accessible to anyone who could read. New forms of literature, such as the novel and personal essays, were available. Vernacular Bibles turned the Word of God into the words of God, since God became an Englishman or a German or a Frenchman, depending on the language in which His words were revealed. Practical knowledge about machines, agriculture, and medicine was widely dispersed. Commercial documents gave new form and vigorous impetus to entrepreneurial adventures. And, of course, printing vastly enhanced the importance of individuality.\n\nVitalized by such an information explosion, Western culture set itself upon a course which made technocracies possible. And then something quite unexpected happened; in a word, nothing. From the early seventeenth century, when Western culture undertook to reorganize itself to accommodate the printing press, until the mid-nineteenth century, no significant technologies were introduced that altered the _form, volume_ , or _speed_ of information. As a consequence, Western culture had more than two hundred years to accustom itself to the new information conditions created by the press. It developed new institutions, such as the school and representative government. It developed new conceptions of knowledge and intelligence, and a heightened respect for reason and privacy. It developed new forms of economic activity, such as mechanized production and corporate capitalism, and even gave articulate expression to the possibilities of a humane socialism. New forms of public discourse came into being through newspapers, pamphlets, broadsides, and books. It is no wonder that the eighteenth century gave us our standard of excellence in the use of reason, as exemplified in the work of Goethe, Voltaire, Diderot, Kant, Hume, Adam Smith, Edmund Burke, Vico, Edward Gibbon, and, of course, Jefferson, Madison, Franklin, Adams, Hamilton, and Thomas Paine. I weight the list with America's \"Founding Fathers\" because technocratic-typographic America was the first nation ever to be _argued_ into existence _in print_. Paine's _Common Sense_ and _The Rights of Man_ , Jefferson's Declaration of Independence, and the _Federalist Papers_ were written and printed efforts to make the American experiment appear reasonable to the people, which to the eighteenth-century mind was both necessary and sufficient. To any people whose politics were the politics of the printed page, as Tocqueville said of America, reason and printing were inseparable. We need not hesitate to claim that the First Amendment to the United States Constitution stands as a monument to the ideological biases of print. It says: \"Congress shall make no law respecting the establishment of religion, or prohibiting the free exercise thereof; or abridging freedom of speech or of the press; or of the right of the people peaceably to assemble, and to petition the government for a redress of grievances.\" In these forty-five words we may find the fundamental values of the literate, reasoning mind as fostered by the print revolution: a belief in privacy, individuality, intellectual freedom, open criticism, and community action.\n\nEqually important is that the words of that amendment presume and insist on a public that not only has access to information but has control over it, a people who know how to use information in their own interests. There is not a single line written by Jefferson, Adams, Paine, Hamilton, or Franklin that does not take for granted that when information is made available to citizens they are capable of managing it. This is not to say that the Founding Fathers believed information could not be false, misleading, or irrelevant. But they believed that the marketplace of information and ideas was sufficiently ordered so that citizens could make sense of what they read and heard and, through reason, judge its usefulness to their lives. Jefferson's proposals for education, Paine's arguments for self-governance, Franklin's arrangements for community affairs assume coherent, commonly shared principles that allow us to debate such questions as: What are the responsibilities of citizens? What is the nature of education? What constitutes human progress? What are the limitations of social structures?\n\nThe presumed close connection among information, reason, and usefulness began to lose its legitimacy toward the mid-nineteenth century with the invention of the telegraph. Prior to the telegraph, information could be moved only as fast as a train could travel: about thirty-five miles per hour. Prior to the telegraph, information was sought as part of the process of understanding and solving particular problems. Prior to the telegraph, information tended to be of local interest. Telegraphy changed all of this, and instigated the second stage of the information revolution. The telegraph removed space as an inevitable constraint on the movement of information, and, for the first time, transportation and communication were disengaged from each other. In the United States, the telegraph erased state lines, collapsed regions, and, by wrapping the continent in an information grid, created the possibility of a unified nation-state. But more than this, telegraphy created the idea of context-free information\u2014that is, the idea that the value of information need not be tied to any function it might serve in social and political decision-making and action. The telegraph made information into a commodity, a \"thing\" that could be bought and sold irrespective of its uses or meaning.\n\nBut it did not do so alone. The potential of the telegraph to transform information into a commodity might never have been realized except for its partnership with the penny press, which was the first institution to grasp the significance of the annihilation of space and the saleability of irrelevant information. In fact, the first known use of the telegraph by a newspaper occurred _one day_ after Samuel Morse gave his historic demonstration of the telegraph's workability. Using the same Washington-to-Baltimore line Morse had constructed, the Baltimore _Patriot_ gave its readers information about action taken by the House of Representatives on the Oregon issue. The paper concluded its report by noting, \"... we are thus enabled to give our readers information from Washington up to two o'clock. This is indeed the annihilation of space.\" Within two years of this announcement, the fortunes of newspapers came to depend not on the quality or utility of the news they provided but on how much, from what distances, and at what speed.\n\nAnd, one must add, with how many photographs. For, as it happened, photography was invented at approximately the same time as telegraphy, and initiated the third stage of the information revolution. Daniel Boorstin has called it \"the graphic revolution,\" because the photograph and other iconographs brought on a massive intrusion of images into the symbolic environment: photographs, prints, posters, drawings, advertisements. The new imagery, with photography at its forefront, did not merely function as a supplement to language but tended to replace it as our dominant means for construing, understanding, and testing reality. By the end of the nineteenth century, advertisers and newspapermen had discovered that a picture was worth not only a thousand words but, in terms of sales, many thousands of dollars.\n\nAs the twentieth century began, the amount of information available through words and pictures grew exponentially. With telegraphy and photography leading the way, a new definition of information came into being. Here was information that rejected the necessity of interconnectedness, proceeded without context, argued for instancy against historical continuity, and offered fascination in place of complexity and coherence. And then, with Western culture gasping for breath, the fourth stage of the information revolution occurred, broadcasting. And then the fifth, computer technology. Each of these brought with it new forms of information, unprecedented amounts of it, and increased speeds (if virtual instancy can be increased).\n\nWhat is our situation today? In the United States, we have 260,000 billboards; 11,520 newspapers; 11,556 periodicals; 27,000 video outlets for renting video tapes; more than 500 million radios; and more than 100 million computers. Ninety-eight percent of American homes have a television set; more than half our homes have more than one. There are 40,000 new book titles published every year 300,000 worldwide), and every day in America 41 million photographs are taken. And if this is not enough, more than 60 billion pieces of junk mail (thanks to computer technology) find their way into our mailboxes every year.\n\nFrom millions of sources all over the globe, through every possible channel and medium\u2014light waves, airwaves, ticker tapes, computer banks, telephone wires, television cables, satellites, printing presses\u2014information pours in. Behind it, in every imaginable form of storage\u2014on paper, on video and audio tape, on discs, film, and silicon chips\u2014is an ever greater volume of information waiting to be retrieved. Like the Sorcerer's Apprentice, we are awash in information. And all the sorcerer has left us is a broom. Information has become a form of garbage, not only incapable of answering the most fundamental human questions but barely useful in providing coherent direction to the solution of even mundane problems. To say it still another way: The milieu in which Technopoly flourishes is one in which the tie between information and human purpose has been severed, i.e., information appears indiscriminately, directed at no one in particular, in enormous volume and at high speeds, and disconnected from theory, meaning, or purpose.\n\nAll of this has called into being a new world. I have referred to it elsewhere as a peek-a-boo world, where now this event, now that, pops into view for a moment, then vanishes again. It is an improbable world. It is a world in which the idea of human progress, as Bacon expressed it, has been replaced by the idea of technological progress. The aim is not to reduce ignorance, superstition, and suffering but to accommodate ourselves to the requirements of new technologies. We tell ourselves, of course, that such accommodations will lead to a better life, but that is only the rhetorical residue of a vanishing technocracy. We are a culture consuming itself with information, and many of us do not even wonder how to control the process. We proceed under the assumption that information is our friend, believing that cultures may suffer grievously from a lack of information, which, of course, they do. It is only now beginning to be understood that cultures may also suffer grievously from information glut, information without meaning, information without control mechanisms.\n\n# **5**\n\n# The Broken Defenses\n\nTechnopoly is a state of culture. It is also a state of mind. It consists in the deification of technology, which means that the culture seeks its authorization in technology, finds its satisfactions in technology, and takes its orders from technology. This requires the development of a new kind of social order, and of necessity leads to the rapid dissolution of much that is associated with traditional beliefs. Those who feel most comfortable in Technopoly are those who are convinced that technical progress is humanity's supreme achievement and the instrument by which our most profound dilemmas may be solved. They also believe that information is an unmixed blessing, which through its continued and uncontrolled production and dissemination offers increased freedom, creativity, and peace of mind. The fact that information does none of these things\u2014but quite the opposite\u2014seems to change few opinions, for such unwavering beliefs are an inevitable product of the structure of Technopoly. In particular, Technopoly flourishes when the defenses against information break down.\n\nThe relationship between information and the mechanisms for its control is fairly simple to describe: Technology increases the available supply of information. As the supply is increased, control mechanisms are strained. Additional control mechanisms are needed to cope with new information. When additional control mechanisms are themselves technical, they in turn further increase the supply of information. When the supply of information is no longer controllable, a general breakdown in psychic tranquillity and social purpose occurs. Without defenses, people have no way of finding meaning in their experiences, lose their capacity to remember, and have difficulty imagining reasonable futures.\n\nOne way of defining Technopoly, then, is to say it is what happens to society when the defenses against information glut have broken down. It is what happens when institutional life becomes inadequate to cope with too much information. It is what happens when a culture, overcome by information generated by technology, tries to employ technology itself as a means of providing clear direction and humane purpose. The effort is mostly doomed to failure. Though it is sometimes possible to use a disease as a cure for itself, this occurs only when we are fully aware of the processes by which disease is normally held in check. My purpose here is to describe the defenses that in principle are available and to suggest how they have become dysfunctional.\n\nThe dangers of information on the loose may be understood by the analogy I suggested earlier with an individual's biological immune system, which serves as a defense against the uncontrolled growth of cells. Cellular growth is, of course, a normal process without which organic life cannot survive. But without a well-functioning immune system, an organism cannot manage cellular growth. It becomes disordered and destroys the delicate interconnectedness of essential organs. An immune system, in short, destroys unwanted cells. All societies have institutions and techniques that function as does a biological immune system. Their purpose is to maintain a balance between the old and the new, between novelty and tradition, between meaning and conceptual disorder, and they do so by \"destroying\" unwanted information.\n\nI must emphasize that social institutions of all kinds function as control mechanisms. This is important to say, because most writers on the subject of social institutions (especially sociologists) do not grasp the idea that any decline in the force of institutions makes people vulnerable to information chaos. To say that life is destabilized by weakened institutions is merely to say that information loses its use and therefore becomes a source of confusion rather than coherence.\n\nSocial institutions sometimes do their work simply by denying people access to information, but principally by directing how much weight and, therefore, value one must give to information. Social institutions are concerned with the _meaning_ of information and can be quite rigorous in enforcing standards of admission. Take as a simple example a court of law. Almost all rules for the presentation of evidence and for the conduct of those who participate in a trial are designed to limit the amount of information that is allowed entry into the system. In our system, a judge disallows \"hearsay\" or personal opinion as evidence except under strictly controlled circumstances, spectators are forbidden to express their feelings, a defendant's previous convictions may not be mentioned, juries are not allowed to hear arguments over the admissibility of evidence\u2014these are instances of information control. The rules on which such control is based derive from a theory of justice that defines what information may be considered relevant and, especially, what information must be considered irrelevant. The theory may be deemed flawed in some respects\u2014lawyers, for example, may disagree over the rules governing the flow of information\u2014but no one disputes that information must be regulated in some manner. In even the simplest law case, thousands of events may have had a bearing on the dispute, and it is well understood that, if they were all permitted entry, there could be no theory of due process, trials would have no end, law itself would be reduced to meaninglessness. In short, the rule of law is concerned with the \"destruction\" of information.\n\nIt is worth mentioning here that, although legal theory has been taxed to the limit by new information from diverse sources\u2014biology, psychology, and sociology, among them\u2014the rules governing relevance have remained fairly stable. This may account for Americans' overuse of the courts as a means of finding coherence and stability. As other institutions become unusable as mechanisms for the control of wanton information, the courts stand as a final arbiter of truth. For how long, no one knows.\n\nI have previously referred to the school as a mechanism for information control. What its standards are can usually be found in a curriculum or, with even more clarity, in a course catalogue. A college catalogue lists courses, subjects, and fields of study that, taken together, amount to a certified statement of what a serious student ought to think about. More to the point, in what is omitted from a catalogue, we may learn what a serious student ought _not_ to think about. A college catalogue, in other words, is a formal description of an information management program; it defines and categorizes knowledge, and in so doing systematically excludes, demeans, labels as trivial\u2014in a word, disregards certain kinds of information. That is why it \"makes sense\" (or, more accurately, used to make sense). By what it includes\/excludes it reflects a theory of the purpose and meaning of education. In the university where I teach, you will not find courses in astrology or dianetics or creationism. There is, of course, much available information about these subjects, but the theory of education that sustains the university does not allow such information entry into the formal structure of its courses. Professors and students are denied the opportunity to focus their attention on it, and are encouraged to proceed as if it did not exist. In this way, the university gives expression to its idea of what constitutes legitimate knowledge. At the present time, some accept this idea and some do not, and the resulting controversy weakens the university's function as an information control center.\n\nThe clearest symptom of the breakdown of the curriculum is found in the concept of \"cultural literacy,\" which has been put forward as an organizing principle and has attracted the serious attention of many educators. If one is culturally literate, the idea goes, one should master a certain list of thousands of names, places, dates, and aphorisms; these are supposed to make up the content of the literate American's mind. But, as I will seek to demonstrate in the final chapter, cultural literacy is not an organizing principle at all; it represents, in fact, a case of calling the disease the cure. The point to be stressed here is that any educational institution, if it is to function well in the management of information, must have a theory about its purpose and meaning, must have the means to give clear expression to its theory, and must do so, to a large extent, by excluding information.\n\nAs another example, consider the family. As it developed in Europe in the late eighteenth century, its theory included the premise that individuals need emotional protection from a cold and competitive society. The family became, as Christopher Lasch calls it, a haven in a heartless world. Its program included (I quote Lasch here) preserving \"separatist religious traditions, alien languages and dialects, local lore and other traditions.\" To do this, the family was required to take charge of the socialization of children; the family became a structure, albeit an informal one, for the management of information. It controlled what \"secrets\" of adult life would be allowed entry and what \"secrets\" would not. There may be readers who can remember when in the presence of children adults avoided using certain words and did not discuss certain topics whose details and ramifications were considered unsuitable for children to know. A family that does not or cannot control the information environment of its children is barely a family at all, and may lay claim to the name only by virtue of the fact that its members share biological information through DNA. In fact, in many societies a family was just that\u2014a group connected by genetic information, itself controlled through the careful planning of marriages. In the West, the family as an institution for the management of nonbiological information began with the ascendance of print. As books on every conceivable subject become available, parents were forced into the roles of guardians, protectors, nurturers, and arbiters of taste and rectitude. Their function was to define what it means to be a child by excluding from the family's domain information that would undermine its purpose. That the family can no longer do this is, I believe, obvious to everyone.\n\nCourts of law, the school, and the family are only three of several control institutions that serve as part of a culture's information immune system. The political party is another. As a young man growing up in a Democratic household, I was provided with clear instructions on what value to assign to political events and commentary. The instructions did not require explicit statement. They followed logically from theory, which was, as I remember it, as follows: Because people need protection, they must align themselves with a political organization. The Democratic Party was entitled to our loyalty because it represented the social and economic interests of the working class, of which our family, relatives, and neighbors were members (except for one uncle who, though a truck driver, consistently voted Republican and was therefore thought to be either stupid or crazy). The Republican Party represented the interests of the rich, who, by definition, had no concern for us.\n\nThe theory gave clarity to our perceptions and a standard by which to judge the significance of information. The general principle was that information provided by Democrats was always to be taken seriously and, in all probability, was both true and useful (except if it came from Southern Democrats, who were helpful in electing presidents but were otherwise never to be taken seriously because of their special theory of race). Information provided by Republicans was rubbish and was useful only to the extent that it confirmed how self-serving Republicans were.\n\nI am not prepared to argue here that the theory was correct, but to the accusation that it was an oversimplification I would reply that all theories are oversimplifications, or at least lead to oversimplification. The rule of law is an oversimplification. A curriculum is an oversimplification. So is a family's conception of a child. That is the function of theories\u2014to oversimplify, and thus to assist believers in organizing, weighting, and excluding information. Therein lies the power of theories. Their weakness is that precisely because they oversimplify, they are vulnerable to attack by new information. When there is too much information to sustain _any_ theory, information becomes essentially meaningless.\n\nThe most imposing institutions for the control of information are religion and the state. They do their work in a somewhat more abstract way than do courts, schools, families, or political parties. They manage information through the creation of myths and stories that express theories about fundamental questions: why are we here, where have we come from, and where are we headed? I have already alluded to the comprehensive theological narrative of the medieval European world and how its great explanatory power contributed to a sense of well-being and coherence. Perhaps I have not stressed enough the extent to which the Bible also served as an information control mechanism, especially in the moral domain. The Bible gives manifold instructions on what one must do and must not do, as well as guidance on what language to avoid (on pain of committing blasphemy), what ideas to avoid (on pain of committing heresy), what symbols to avoid (on pain of committing idolatry). Necessarily but perhaps unfortunately, the Bible also explained how the world came into being in such literal detail that it could not accommodate new information produced by the telescope and subsequent technologies. The trials of Galileo and, three hundred years later, of Scopes were therefore about the admissibility of certain kinds of information. Both Cardinal Bellarmine and William Jennings Bryan were fighting to maintain the authority of the Bible to control information about the profane world as well as the sacred. In their defeat, more was lost than the Bible's claim to explain the origins and structure of nature. The Bible's authority in defining and categorizing moral behavior was also weakened.\n\nNonetheless, Scripture has at its core such a powerful mythology that even the residue of that mythology is still sufficient to serve as an exacting control mechanism for some people. It provides, first of all, a theory about the meaning of life and therefore rules on how one is to conduct oneself. With apologies to Rabbi Hillel, who expressed it more profoundly and in the time it takes to stand on one leg, the theory is as follows: There is one God, who created the universe and all that is in it. Although humans can never fully understand God, He has revealed Himself and His will to us throughout history, particularly through His commandments and the testament of the prophets as recorded in the Bible. The greatest of these commandments tells us that humans are to love God and express their love for Him through love, mercy, and justice to our fellow humans. At the end of time, all nations and humans will appear before God to be judged, and those who have followed His commandments will find favor in His sight. Those who have denied God and the commandments will perish utterly in the darkness that lies outside the presence of God's light.\n\nTo borrow from Hillel: That is the theory. All the rest is commentary.\n\nThose who believe in this theory\u2014particularly those who accept the Bible as the literal word of God\u2014are free to dismiss other theories about the origin and meaning of life and to give minimal weight to the facts on which other theories are based. Moreover, in observing God's laws, and the detailed requirements of their enactment, believers receive guidance about what books they should not read, about what plays and films they should not see, about what music they should not hear, about what subjects their children should not study, and so on. For strict fundamentalists of the Bible, the theory and what follows from it seal them off from unwanted information, and in that way their actions are invested with meaning, clarity, and, they believe, moral authority.\n\nThose who reject the Bible's theory and who believe, let us say, in the theory of Science are also protected from unwanted information. Their theory, for example, instructs them to disregard information about astrology, dianetics, and creationism, which they usually label as medieval superstition or subjective opinion. Their theory fails to give any guidance about moral information and, by definition, gives little weight to information that falls outside the constraints of science. Undeniably, fewer and fewer people are bound in any serious way to Biblical or other religious traditions as a source of compelling attention and authority, the result of which is that they make no moral decisions, only practical ones. This is still another way of defining Technopoly. The term is aptly used for a culture whose available theories do not offer guidance about what is acceptable information in the moral domain.\n\nI trust the reader does not conclude that I am making an argument for fundamentalism of any kind. One can hardly approve, for example, of a Muslim fundamentalism that decrees a death sentence to someone who writes what are construed as blasphemous words, or a Christian fundamentalism that once did the same or could lead to the same. I must hasten to acknowledge, in this context, that it is entirely possible to live as a Muslim, a Christian, or a Jew with a modified and temperate view of religious theory. Here, I am merely making the point that religious tradition serves as a mechanism for the regulation and valuation of information. When religion loses much or all of its binding power\u2014if it is reduced to mere rhetorical ash\u2014then confusion inevitably follows about what to attend to and how to assign it significance.\n\nIndeed, as I write, another great world narrative, Marxism, is in the process of decomposing. No doubt there are fundamentalist Marxists who will not let go of Marx's theory, and will continue to be guided by its prescriptions and constraints. The theory, after all, is sufficiently powerful to have engaged the imagination and devotion of more than a billion people. Like the Bible, the theory includes a transcendent idea, as do all great world narratives. With apologies to a century and a half of philosophical and sociological disputation, the idea is as follows: All forms of institutional misery and oppression are a result of class conflict, since the consciousness of all people is formed by their material situation. God has no interest in this, because there is no God. But there _is_ a plan, which is both knowable and beneficent. The plan unfolds in the movement of history itself, which shows unmistakably that the working class, in the end, must triumph. When it does, with or without the help of revolutionary movements, class itself will have disappeared. All will share equally in the bounties of nature and creative production, and no one will exploit the labors of another.\n\nIt is generally believed that this theory has fallen into disrepute among believers because information made available by television, films, telephone, fax machines, and other technologies has revealed that the working classes of capitalist nations are sharing quite nicely in the bounties of nature while at the same time enjoying a considerable measure of personal freedom. Their situation is so vastly superior to those of nations enacting Marxist theory that millions of people have concluded, seemingly all at once, that history may have no opinion whatever on the fate of the working class or, if it has, that it is moving toward a final chapter quite different in its point from what Marx prophesied.\n\nAll of this is said provisionally. History takes a long time, and there may yet be developments that will provide Marx's vision with fresh sources of verisimilitude. Meanwhile, the following points need to be made: Believers in the Marxist story were given quite clear guidelines on how they were to weight information and therefore to understand events. To the extent that they now reject the theory, they are threatened with conceptual confusion, which means they no longer know who to believe or what to believe. In the West, and especially in the United States, there is much rejoicing over this situation, and assurances are given that Marxism can be replaced by what is called \"liberal democracy.\" But this must be stated more as a question than an answer, for it is no longer entirely clear what sort of story liberal democracy tells.\n\nA clear and scholarly celebration of liberal democracy's triumph is found in Francis Fukuyama's essay \"The End of History?\" Using a somewhat peculiar definition of history, Fukuyama concludes that there will be no more ideological conflicts, all the competitors to modern liberalism having been defeated. In support of his conclusion, Fukuyama cites Hegel as having come to a similar position in the early nineteenth century, when the principles of liberty and equality, as expressed in the American and French revolutions, emerged triumphant. With the contemporary decline of fascism and communism, no threat now remains. But Fukuyama pays insufficient attention to the changes in meaning of liberal democracy over two centuries. Its meaning in a technocracy is quite different from its meaning in Technopoly; indeed, in Technopoly it comes much closer to what Walter Benjamin called \"commodity capitalism.\" In the case of the United States, the great eighteenth-century revolution was not indifferent to commodity capitalism but was nonetheless infused with profound moral content. The United States was not merely an experiment in a new form of governance; it was the fulfillment of God's plan. True, Adams, Jefferson, and Paine rejected the supernatural elements in the Bible, but they never doubted that their experiment had the imprimatur of Providence. People were to be free but for a purpose. Their God-given rights implied obligations and responsibilities, not only to God but to other nations, to which the new republic would be a guide and a showcase of what is possible when reason and spirituality commingle.\n\nIt is an open question whether or not \"liberal democracy\" in its present form can provide a thought-world of sufficient moral substance to sustain meaningful lives. This is precisely the question that Vaclav Havel, then newly elected as president of Czechoslovakia, posed in an address to the U.S. Congress. \"We still don't know how to put morality ahead of politics, science, and economics,\" he said. \"We are still incapable of understanding that the only genuine backbone of our actions\u2014if they are to be moral\u2014is responsibility. Responsibility to something higher than my family, my country, my firm, my success.\" What Havel is saying is that it is not enough for his nation to liberate itself from one flawed theory; it is necessary to find another, and he worries that Technopoly provides no answer. To say it in still another way: Francis Fukuyama is wrong. There _is_ another ideological conflict to be fought\u2014between \"liberal democracy\" as conceived in the eighteenth century, with all its transcendent moral underpinnings, and Technopoly, a twentieth-century thought-world that functions not only without a transcendent narrative to provide moral underpinnings but also without strong social institutions to control the flood of information produced by technology.\n\nBecause that flood has laid waste the theories on which schools, families, political parties, religion, nationhood itself are based, American Technopoly must rely, to an obsessive extent, on technical methods to control the flow of information. Three such means merit special attention. They are interrelated but for purposes of clarity may be described separately.\n\nThe first is bureaucracy, which James Beniger in _The Control Revolution_ ranks as \"foremost among all technological solutions to the crisis of control.\" Bureaucracy is not, of course, a creation of Technopoly. Its history goes back five thousand years, although the word itself did not appear in English until the nineteenth century. It is not unlikely that the ancient Egyptians found bureaucracy an irritation, but it is certain that, beginning in the nineteenth century, as bureaucracies became more important, the complaints against them became more insistent. John Stuart Mill referred to them as \"administrative tyranny.\" Carlyle called them \"the Continental nuisance.\" In a chilling paragraph, Tocqueville warned about them taking hold in the United States:\n\nI have previously made the distinction between two types of centralization, calling one governmental and the other administrative. Only the first exists in America, the second being almost unknown. If the directing power in American society had both these means of government at its disposal and combined the right to command with the faculty and habit to perform everything itself, if having established the general principles of the government, it entered into the details of their application, and having regulated the great interests of the country, it came down to consider even individual interest, then freedom would soon be banished from the New World.\n\nWriting in our own time, C. S. Lewis believed bureaucracy to be the technical embodiment of the Devil himself:\n\nI live in the Managerial Age, in a world of \"Admin.\" The greatest evil is not now done in those sordid \"dens of crime\" that Dickens loved to paint. It is not done even in concentration camps and labour camps. In those we see its final result. But it is conceived and ordered (moved, seconded, carried, and minuted) in clean, carpeted, warmed, and well-lighted offices, by quiet men with white collars and cut fingernails and smooth-shaven cheeks who do not need to raise their voices. Hence, naturally enough, my symbol for Hell is something like the bureaucracy of a police state or the office of a thoroughly nasty business concern.\n\nPutting these attacks aside for the moment, we may say that in principle a bureaucracy is simply a coordinated series of techniques for reducing the amount of information that requires processing. Beniger notes, for example, that the invention of the standardized form\u2014a staple of bureaucracy\u2014allows for the \"destruction\" of every nuance and detail of a situation. By requiring us to check boxes and fill in blanks, the standardized form admits only a limited range of formal, objective, and impersonal information, which in some cases is precisely what is needed to solve a particular problem. Bureaucracy is, as Max Weber described it, an attempt to rationalize the flow of information, to make its use efficient to the highest degree by eliminating information that diverts attention from the problem at hand. Beniger offers as a prime example of such bureaucratic rationalization the decision in 1884 to organize time, on a worldwide basis, into twenty-four time zones. Prior to this decision, towns only a mile or two apart could and did differ on what time of day it was, which made the operation of railroads and other businesses unnecessarily complex. By simply ignoring the fact that solar time differs at each node of a transportation system, bureaucracy eliminated a problem of information chaos, much to the satisfaction of most people. But not of everyone. It must be noted that the idea of \"God's own time\" (a phrase used by the novelist Marie Corelli in the early twentieth century to oppose the introduction of Summer Time) had to be considered irrelevant. This is important to say, because, in attempting to make the most rational use of information, bureaucracy ignores all information and ideas that do not contribute to efficiency. The idea of God's time made no such contribution.\n\nBureaucracy is not in principle a social institution; nor are all institutions that reduce information by excluding some kinds or sources necessarily bureaucracies. Schools may exclude dianetics and astrology; courts exclude hearsay evidence. They do so for substantive reasons having to do with the theories on which these institutions are based. But bureaucracy has no intellectual, political, or moral theory\u2014except for its implicit assumption that efficiency is the principal aim of all social institutions and that other goals are essentially less worthy, if not irrelevant. That is why John Stuart Mill thought bureaucracy a \"tyranny\" and C. S. Lewis identified it with Hell.\n\nThe transformation of bureaucracy from a set of techniques designed to serve social institutions to an autonomous meta-institution that largely serves itself came as a result of several developments in the mid- and late-nineteenth century: rapid industrial growth, improvements in transportation and communication, the extension of government into ever-larger realms of public and business affairs, the increasing centralization of governmental structures. To these were added, in the twentieth century, the information explosion and what we might call the \"bureaucracy effect\": as techniques for managing information became more necessary, extensive, and complex, the number of people and structures required to manage those techniques grew, and so did the amount of information _generated_ by bureaucratic techniques. This created the need for bureaucracies to manage and coordinate bureaucracies, then for additional structures and techniques to manage the bureaucracies that coordinated bureaucracies, and so on\u2014until bureaucracy became, to borrow again Karl Kraus's comment on psychoanalysis, the disease for which it purported to be the cure. Along the way, it ceased to be merely a servant of social institutions and became their master. Bureaucracy now not only solves problems but creates them. More important, it defines what our problems are\u2014and they are always, in the bureaucratic view, problems of efficiency. As Lewis suggests, this makes bureaucracies exceedingly dangerous, because, though they were originally designed to process only technical information, they now are commonly employed to address problems of a moral, social, and political nature. The bureaucracy of the nineteenth century was largely concerned with making transportation, industry, and the distribution of goods more efficient. Technopoly's bureaucracy has broken loose from such restrictions and now claims sovereignty over all of society's affairs.\n\nThe peril we face in trusting social, moral, and political affairs to bureaucracy may be highlighted by reminding ourselves what a bureaucrat does. As the word's history suggests, a bureaucrat is little else than a glorified counter. The French word _bureau_ first meant a cloth for covering a reckoning table, then the table itself, then the room in which the table was kept, and finally the office and staff that ran the entire counting room or house. The word \"bureaucrat\" has come to mean a person who by training, commitment, and even temperament is indifferent to both the content and the totality of a human problem. The bureaucrat considers the implications of a decision only to the extent that the decision will affect the efficient operations of the bureaucracy, and takes no responsibility for its human consequences. Thus, Adolf Eichmann becomes the basic model and metaphor for a bureaucrat in the age of Technopoly. When faced with the charge of crimes against humanity, he argued that he had no part in the formulation of Nazi political or sociological theory; he dealt only with the technical problems of moving vast numbers of people from one place to another. Why they were being moved and, especially, what would happen to them when they arrived at their destination were not relevant to his job. Although the jobs of bureaucrats in today's Technopoly have results far less horrific, Eichmann's answer is probably given five thousand times a day in America alone: I have no responsibility for the human consequences of my decisions. I am only responsible for the efficiency of my part of the bureaucracy, which must be maintained at all costs.\n\nEichmann, it must also be noted, was an expert. And expertise is a second important technical means by which Technopoly strives furiously to control information. There have, of course, always been experts, even in tool-using cultures. The pyramids, Roman roads, the Strasbourg Cathedral, could hardly have been built without experts. But the expert in Technopoly has two characteristics that distinguish him or her from experts of the past. First, Technopoly's experts tend to be ignorant about any matter not directly related to their specialized area. The average psychotherapist, for example, barely has even superficial knowledge of literature, philosophy, social history, art, religion, and biology, and is not expected to have such knowledge. Second, like bureaucracy itself (with which an expert may or may not be connected), Technopoly's experts claim dominion not only over technical matters but also over social, psychological, and moral affairs. In the United States, we have experts in how to raise children, how to educate them, how to be lovable, how to make love, how to influence people, how to make friends. There is no aspect of human relations that has not been technicalized and therefore relegated to the control of experts.\n\nThese special characteristics of the expert arose as a result of three factors. First, the growth of bureaucracies, which, in effect, produced the world's first entirely mechanistic specialists and thereby gave credence and prestige to the specialist-as-ignoramus. Second, the weakening of traditional social institutions, which led ordinary people to lose confidence in the value of tradition. Third, and underlying everything else, the torrent of information which made it impossible for anyone to possess more than a tiny fraction of the sum total of human knowledge. As a college undergraduate, I was told by an enthusiastic professor of German literature that Goethe was the last person who knew everything. I assume she meant, by this astounding remark, less to deify Goethe than to suggest that by the year of his death, 1832, it was no longer possible for even the most brilliant mind to comprehend, let alone integrate, what was known.\n\nThe role of the expert is to concentrate on one field of knowledge, sift through all that is available, eliminate that which has no bearing on a problem, and use what is left to assist in solving a problem. This process works fairly well in situations where only a technical solution is required and there is no conflict with human purposes\u2014for example, in space rocketry or the construction of a sewer system. It works less well in situations where technical requirements may conflict with human purposes, as in medicine or architecture. And it is disastrous when applied to situations that cannot be solved by technical means and where efficiency is usually irrelevant, such as in education, law, family life, and problems of personal maladjustment. I assume I do not need to convince the reader that there are no experts\u2014there can be no experts\u2014in child-rearing and lovemaking and friend-making. All of this is a figment of the Technopolist's imagination, made plausible by the use of technical machinery, without which the expert would be totally disarmed and exposed as an intruder and an ignoramus.\n\nTechnical machinery is essential to both the bureaucrat and the expert, and may be regarded as a third mechanism of information control. I do not have in mind such \"hard\" technologies as the computer\u2014which must, in any case, be treated separately, since it embodies all that Technopoly stands for. I have in mind \"softer\" technologies such as IQ tests, SATs, standardized forms, taxonomies, and opinion polls. Some of these I discuss in detail in chapter eight, \"Invisible Technologies,\" but I mention them here because their role in reducing the types and quantity of information admitted to a system often goes unnoticed, and therefore their role in redefining traditional concepts also goes unnoticed. _There is, for example, no test that can measure a person's intelligence_. Intelligence is a general term used to denote one's capacity to solve real-life problems in a variety of novel contexts. It is acknowledged by everyone except experts that each person varies greatly in such capacities, from consistently effective to consistently ineffective, depending on the kinds of problems requiring solution. If, however, we are made to believe that a test can reveal precisely the quantity of intelligence a person has, then, for all institutional purposes, a score on a test becomes his or her intelligence. The test transforms an abstract and multifaceted meaning into a technical and exact term that leaves out everything of importance. One might even say that an intelligence test is a tale told by an expert, signifying nothing. Nonetheless, the expert relies on our believing in the reality of technical machinery, which means we will reify the answers generated by the machinery. We come to believe that our score is our intelligence, or our capacity for creativity or love or pain. We come to believe that the results of opinion polls _are_ what people believe, as if our beliefs can be encapsulated in such sentences as \"I approve\" and \"I disapprove.\"\n\nWhen Catholic priests use wine, wafers, and incantations to embody spiritual ideas, they acknowledge the mystery and the metaphor being used. But experts of Technopoly acknowledge no such overtones or nuances when they use forms, standardized tests, polls, and other machinery to give technical reality to ideas about intelligence, creativity, sensitivity, emotional imbalance, social deviance, or political opinion. They would have us believe that technology can plainly reveal the true nature of some human condition or belief because the score, statistic, or taxonomy has given it technical form.\n\nThere is no denying that the technicalization of terms and problems is a serious form of information control. Institutions can make decisions on the basis of scores and statistics, and there certainly may be occasions where there is no reasonable alternative. But unless such decisions are made with profound skepticism\u2014that is, acknowledged as being made for administrative convenience\u2014they are delusionary. In Technopoly, the delusion is sanctified by our granting inordinate prestige to experts who are armed with sophisticated technical machinery. Shaw once remarked that all professions are conspiracies against the laity. I would go further: in Technopoly, all experts are invested with the charisma of priestliness. Some of our priest-experts are called psychiatrists, some psychologists, some sociologists, some statisticians. The god they serve does not speak of righteousness or goodness or mercy or grace. Their god speaks of efficiency, precision, objectivity. And that is why such concepts as sin and evil disappear in Technopoly. They come from a moral universe that is irrelevant to the theology of expertise. And so the priests of Technopoly call sin \"social deviance,\" which is a statistical concept, and they call evil \"psychopathology,\" which is a medical concept. Sin and evil disappear because they cannot be measured and objectified, and therefore cannot be dealt with by experts.\n\nAs the power of traditional social institutions to organize perceptions and judgment declines, bureaucracies, expertise, and technical machinery become the principal means by which Technopoly hopes to control information and thereby provide itself with intelligibility and order. The rest of this book tells the story of why this cannot work, and of the pain and stupidity that are the consequences.\n\n# **6**\n\n# The Ideology of Machines:  \nMedical Technology\n\nA few years ago, an enterprising company made available a machine called HAGOTH, of which it might be said, this was Technopoly's most ambitious hour. The machine cost $1,500, the bargain of the century, for it was able to reveal to its owner whether someone talking on the telephone was telling the truth. It did this by measuring the \"stress content\" of a human voice as indicated by its oscillations. You connected HAGOTH to your telephone and, in the course of conversation, asked your caller some key question, such as \"Where did you go last Saturday night?\" HAGOTH had sixteen lights\u2014eight green and eight red\u2014and when the caller replied, HAGOTH went to work. Red lights went on when there was much stress in the voice, green lights when there was little. As an advertisement for HAGOTH said, \"Green indicates no stress, hence truthfulness.\" In other words, according to HAGOTH, it is not possible to speak the truth in a quivering voice or to lie in a steady one\u2014an idea that would doubtless amuse Richard Nixon. At the very least, we must say that HAGOTH'S definition of truthfulness was peculiar, but so precise and exquisitely technical as to command any bureaucrat's admiration. The same may be said of the definition of intelligence as expressed in a standard-brand intelligence test. In fact an intelligence test works exactly like HAGOTH. YOU connect a pencil to the fingers of a young person and address some key questions to him or her; from the replies a computer can calculate exactly how much intelligence exists in the young person's brain.\n\nHAGOTH has mercifully disappeared from the market for what reason I do not know. Perhaps it was sexist or culturally biased or, worse, could not measure oscillations accurately enough. When it comes to machinery, what Technopoly insists upon most is accuracy. The idea embedded in the machine is largely ignored, no matter how peculiar.\n\nThough HAGOTH has disappeared, its idea survives\u2014for example, in the machines called \"lie detectors.\" In America, these are taken very seriously by police officers, lawyers, and corporate executives who ever more frequently insist that their employees be subjected to lie-detector tests. As for intelligence tests, they not only survive but flourish, and have been supplemented by vocational aptitude tests, creativity tests, mental-health tests, sexual-attraction tests, and even marital-compatibility tests. One would think that two people who have lived together for a number of years would have noticed for themselves whether they get along or not. But in Technopoly, these subjective forms of knowledge have no official status, and must be confirmed by tests administered by experts. Individual judgments, after all, are notoriously unreliable, filled with ambiguity and plagued by doubt, as Frederick W. Taylor warned. Tests and machines are not. Philosophers may agonize over the questions \"What is truth?\" \"What is intelligence?\" \"What is the good life?\" But in Technopoly there is no need for such intellectual struggle. Machines eliminate complexity, doubt, and ambiguity. They work swiftly, they are standardized, and they provide us with numbers that you can see and calculate with. They tell us that when eight green lights go on someone is speaking the truth. That is all there is to it. They tell us that a score of 136 means more brains than a score of 104. This is Technopoly's version of magic.\n\nWhat is significant about magic is that it directs our attention to the wrong place. And by doing so, evokes in us a sense of wonder rather than understanding. In Technopoly, we are surrounded by the wondrous effects of machines and are encouraged to ignore the ideas embedded in them. Which means we become blind to the ideological meaning of our technologies. In this chapter and the next, I should like to provide examples of how technology directs us to construe the world.\n\nIn considering here the ideological biases of medical technology, let us begin with a few relevant facts. Although the U.S. and England have equivalent life-expectancy rates, American doctors perform six times as many cardiac bypass operations per capita as English doctors do. American doctors perform more diagnostic tests than doctors do in France, Germany, or England. An American woman has two to three times the chance of having a hysterectomy as her counterpart in Europe; 60 percent of the hysterectomies performed in America are done on women under the age of forty-four. American doctors do more prostate surgery per capita than do doctors anywhere in Europe, and the United States leads the industrialized world in the rate of cesarean-section operations\u201450 to 200 percent higher than in most other countries. When American doctors decide to forgo surgery in favor of treatment by drugs, they give higher dosages than doctors elsewhere. They prescribe about twice as many antibiotics as do doctors in the United Kingdom and commonly prescribe antibiotics when bacteria are likely to be present, whereas European doctors tend to prescribe antibiotics only if they know that the infection is caused by bacteria _and_ is also serious. American doctors use far more X-rays per patient than do doctors in other countries. In one review of the extent of X-ray use, a radiologist discovered cases in which fifty to one hundred X-rays had been taken of a single patient when five would have been sufficient. Other surveys have shown that, for almost one-third of the patients, the X-ray could have been omitted or deferred on the basis of available clinical data.\n\nThe rest of this chapter could easily be filled with similar statistics and findings. Perhaps American medical practice is best summarized by the following warning, given by Dr. David E. Rogers in a presidential address to the Association of American Physicians:\n\nAs our interventions have become more searching, they have also become more costly and more hazardous. Thus, today it is not unusual to find a fragile elder who walked into the hospital, [and became] slightly confused, dehydrated, and somewhat the worse for wear on the third hospital day because his first 48 hours in the hospital were spent undergoing a staggering series of exhausting diagnostic studies in various laboratories or in the radiology suite.\n\nNone of this is surprising to anyone familiar with American medicine, which is notorious for its characteristic \"aggressiveness.\" The question is, why? There are three interrelated reasons, all relevant to the imposition of machinery. The first has to do with the American character, which I have previously discussed as being so congenial to the sovereignty of technology. In _Medicine and Culture_ , Lynn Payer describes it in the following way:\n\nThe once seemingly limitless lands gave rise to a spirit that anything was possible if only the natural environment... could be conquered. Disease could also be conquered, but only by aggressively ferreting it out diagnostically and just as aggressively treating it, preferably by taking something out rather than adding something to increase the resistance.\n\nTo add substance to this claim, Ms. Payer quotes Oliver Wendell Holmes as saying, with his customary sarcasm:\n\nHow could a people which has a revolution once in four years, which has contrived the Bowie Knife and the revolver... which insists in sending out yachts and horses and boys to outsail, outrun, outfight and checkmate all the rest of creation; how could such a people be content with any but \"heroic\" practice? What wonder that the stars and stripes wave over doses of ninety grams of sulphate of quinine and that the American eagle screams with delight to see three drachms [180 grains] of calomel given at a single mouthful?\n\nThe spirit of attack mocked here by Holmes was given impetus even before the American Revolution by Dr. Benjamin Rush, perhaps the most influential medical man of his age. Rush believed that medicine had been hindered by doctors placing \"undue reliance upon the powers of nature in curing disease,\" and specifically blamed Hippocrates and his tradition for this lapse. Rush had considerable success in curing patients of yellow fever by prescribing large quantities of mercury and performing purges and bloodletting. (His success was probably due to the fact that the patients either had mild cases of yellow fever or didn't have it at all.) In any event, Rush was particularly enthusiastic about bleeding patients, perhaps because he believed that the body contained about twenty-five pints of blood, which is more than twice the average actual amount. He advised other doctors to continue bleeding a patient until four-fifths of the body's blood was removed. Although Rush was not in attendance during George Washington's final days, Washington was bled seven times on the night he died, which, no doubt, had something to do with why he died. All of this occurred, mind you, 153 years after Harvey discovered that blood circulates throughout the body.\n\nPutting aside the question of the available medical knowledge of the day, Rush was a powerful advocate of action\u2014indeed, gave additional evidence of his aggressive nature by being one of the signers of the Declaration of Independence. He persuaded both doctors and patients that American diseases were tougher than European diseases and required tougher treatment. \"Desperate diseases require desperate remedies\" was a phrase repeated many times in American medical journals in the nineteenth century. The Americans, who considered European methods to be mild and passive\u2014one might even say effeminate\u2014met the challenge by eagerly succumbing to the influence of Rush: they accepted the imperatives to intervene, to mistrust nature, to use the most aggressive therapies available. The idea, as Ms. Payer suggests, was to conquer both a continent and the diseases its weather and poisonous flora and fauna inflicted.\n\nSo, from the outset, American medicine was attracted to new technologies. Far from being \"neutral,\" technology was to be the weapon with which disease and illness would be vanquished. The weapons were not long in coming. The most significant of the early medical technologies was the stethoscope, invented (one might almost say discovered) by the French physician Ren\u00e9-Th\u00e9ophile-Hyacinthe La\u00ebnnec in 1816. The circumstances surrounding the invention are worth mentioning.\n\nWorking at the Necker Hospital in Paris, La\u00ebnnec was examining a young woman with a puzzling heart disorder. He tried to use percussion and palpation (pressing the hand upon the body in hope of detecting internal abnormalities), but the patient's obesity made this ineffective. He next considered auscultation (placing his ear on the patient's chest to hear the heart beat), but the patient's youth and sex discouraged him. La\u00ebnnec then remembered that sound traveling through solid bodies is amplified. He rolled some sheets of paper into a cylinder, placed one end on the patient's chest and the other to his ear. _Voil\u00e0!_ The sounds he heard were clear and distinct. \"From this moment,\" he later wrote, \"I imagined that the circumstance might furnish means for enabling us to ascertain the character, not only of the action of the heart, but of every species of sound produced by the motion of all the thoracic viscera.\" La\u00ebnnec worked to improve the instrument, eventually using a rounded piece of wood, and called it a \"stethoscope,\" from the Greek words for \"chest\" and \"I view.\"\n\nFor all its simplicity, La\u00ebnnec's invention proved extraordinarily useful, particularly in the accuracy with which it helped to diagnose lung diseases like tuberculosis. Chest diseases of many kinds were no longer concealed: the physician with a stethoscope could, as it were, conduct an autopsy on the patient while the patient was still alive.\n\nBut it should not be supposed that all doctors or patients were enthusiastic about the instrument. Patients were often frightened at the sight of a stethoscope, assuming that its presence implied imminent surgery, since, at the time, only surgeons used instruments, not physicians. Doctors had several objections, ranging from the trivial to the significant. Among the trivial was the inconvenience of carrying the stethoscope, a problem some doctors solved by carrying it, crosswise, inside their top hats. This was not without its occasional embarrassments\u2014an Edinburgh medical student was accused of possessing a dangerous weapon when his stethoscope fell out of his hat during a snowball fight. A somewhat less trivial objection raised by doctors was that if they used an instrument they would be mistaken for surgeons, who were then considered mere craftsmen. The distinction between physicians and surgeons was unmistakable then, and entirely favorable to physicians, whose intellect, knowledge, and insight were profoundly admired. It is perhaps to be expected that Oliver Wendell Holmes, professor of anatomy at Harvard and always a skeptic about aggressiveness in medicine, raised objections about the overzealous use of the stethoscope; he did so, in characteristic fashion, by writing a comic ballad, \"The Stethoscope Song,\" in which a physician makes several false diagnoses because insects have nested in his stethoscope.\n\nBut a serious objection raised by physicians, and one which has resonated throughout the centuries of technological development in medicine, is that interposing an instrument between patient and doctor would transform the practice of medicine; the traditional methods of questioning patients, taking their reports seriously, and making careful observations of exterior symptoms would become increasingly irrelevant. Doctors would lose their ability to conduct skillful examinations and rely more on machinery than on their own experience and insight. In his detailed book _Medicine and the Reign of Technology_ , Stanley Joel Reiser compares the effects of the stethoscope to the effects of the printing press on Western culture. The printed book, he argues, helped to create the detached and objective thinker. Similarly, the stethoscope\n\nhelped to create the objective physician, who could move away from involvement with the patient's experiences and sensations, to a more detached relation, less with the patient but more with the sounds from within the body. Undistracted by the motives and beliefs of the patient, the auscultator [another term for the stethoscope] could make a diagnosis from sounds that he alone heard emanating from body organs, sounds that he believed to be objective, bias-free representations of the disease process.\n\nHere we have expressed two of the key _ideas_ promoted by the stethoscope: Medicine is about disease, not the patient. And, what the patient knows is untrustworthy; what the machine knows is reliable.\n\nThe stethoscope could not by itself have made such ideas stick, especially because of the resistance to them, even in America, by doctors whose training and relationship to their patients led them to oppose mechanical interpositions. But the ideas were amplified with each new instrument added to the doctor's arsenal: the ophthalmoscope (invented by Hermann von Helmholtz in 1850), which allowed doctors to see into the eye; the laryngoscope (designed by Johann Czermak, a Polish professor of physiology, in 1857), which allowed doctors to inspect the larynx and other parts of the throat, as well as the nose; and, of course, the X-ray (developed by Wilhelm Roentgen in 1895), which could penetrate most substances but not bones. \"If the hand be held before the fluorescent screen,\" Roentgen wrote, \"the shadow shows the bones darkly with only faint outlines of the surrounding tissues.\" Roentgen was able to reproduce this effect on photographic plates and make the first X-ray of a human being, his wife's hand.\n\nBy the turn of the century, medicine was well on its way to almost total reliance on technology, especially after the development of diagnostic laboratories and the discovery and use of antibiotics in the 1940s. Medical practice had entered a new stage. The first had been characterized by direct communication with the patient's experiences based on the patient's reports, and the doctor's questions and observations. The second was characterized by direct communication with patients' bodies through physical examination, including the use of carefully selected technologies. The stage we are now in is characterized by indirect communication with the patient's experience and body through technical machinery. In this stage, we see the emergence of specialists\u2014for example, pathologists and radiologists\u2014who interpret the meaning of technical information and have no connection whatsoever with the patient, only with tissue and photographs. It is to be expected that, as medical practice moved from one stage to another, doctors tended to lose the skills and insights that predominated in the previous stage. Reiser sums up what this means:\n\nSo, without realizing what has happened, the physician in the last two centuries has gradually relinquished his unsatisfactory attachment to subjective evidence\u2014what the patient says\u2014only to substitute a devotion to technological evidence\u2014what the machine says. He has thus exchanged one partial view of disease for another. As the physician makes greater use of the technology of diagnosis, he perceives his patient more and more indirectly through a screen of machines and specialists; he also relinquishes control over more and more of the diagnostic process. These circumstances tend to estrange him from his patient and from his own judgment.\n\nThere is still another reason why the modern physician is estranged from his own judgment. To put it in the words of a doctor who remains skilled in examining his patients and in evaluating their histories: \"Everyone who has a headache wants and expects a CAT scan.\" He went on to say that roughly six out of every ten CAT scans he orders are unnecessary, with no basis in the clinical evidence and the patient's reported experience and sensations. Why are they done? As a protection against malpractice suits. Which is to say, as medical practice has moved into the stage of total reliance on machine-generated information, so have the patients. Put simply, if a patient does not obtain relief from a doctor who has failed to use all the available technological resources, including drugs, the doctor is deemed vulnerable to the charge of incompetence. The situation is compounded by the fact that the personal relationship between doctor and patient now, in contrast to a century ago, has become so arid that the patient is not restrained by intimacy or empathy from appealing to the courts. Moreover, doctors are reimbursed by medical-insurance agencies on the basis of what they _do_ , not on the amount of time they spend with patients. Nontechnological medicine is time-consuming. It is more profitable to do a CAT scan on a patient with a headache than to spend time getting information about his or her experiences and sensations.\n\nWhat all this means is that even restrained and selective technological medicine becomes very difficult to do, economically undesirable, and possibly professionally catastrophic. The culture itself\u2014its courts, its bureaucracies, its insurance system, the training of doctors, patients' expectations\u2014is organized to support technological treatments. There are no longer methods of treating illness; there is only one method\u2014the technological one. Medical competence is now defined by the quantity and variety of machinery brought to bear on disease.\n\nAs I remarked, three interrelated reasons converged to create this situation. The American character was biased toward an aggressive approach and was well prepared to accommodate medical technology; the nineteenth-century technocracies, obsessed with invention and imbued with the idea of progress, initiated a series of remarkable and wondrous inventions; and the culture reoriented itself to ensure that technological aggressiveness became the basis of medical practice. The ideas promoted by this domination of technology can be summed up as follows: Nature is an implacable enemy that can be subdued only by technical means; the problems created by technological solutions (doctors call these \"side effects\") can be solved only by the further application of technology (we all know the joke about an amazing new drug that cures nothing but has interesting side effects); medical practice must focus on disease, not on the patient (which is why it is possible to say that the operation or therapy was successful but the patient died); and information coming from the patient cannot be taken as seriously as information coming from a machine, from which it follows that a doctor's judgment, based on insight and experience, is less worthwhile than the calculations of his machinery.\n\nDo these ideas lead to better medicine? In some respects, yes; in some respects, no. The answer tends to be \"yes\" when one considers how doctors now use lasers to remove cataracts quickly, painlessly, and safely; or how they can remove a gallbladder by using a small television camera (a laparoscope) inserted through an equally small puncture in the abdomen to guide the surgeon's instruments to the diseased organ through still another small puncture, thus making it unnecessary to cut open the abdomen. Of course, those who are inclined to answer \"no\" to the question will ask how many laparoscopie cholecystectomies are performed _because_ of the existence of the technology. This is a crucial point.\n\nConsider the case of cesarean sections. Close to one out of every four Americans is now born by C-section. Through modern technology, American doctors can deliver babies who would have died otherwise. As Dr. Laurence Horowitz notes in _Taking Charge of Your Medical Fate_ , \"... the proper goal of C-sections is to improve the chances of babies at risk, and that goal has been achieved.\" But C-sections are a surgical procedure, and when they are done routinely as an elective option, there is considerable and unnecessary danger; the chances of a woman's dying during a C-section delivery are two to four times greater than during a normal vaginal delivery. In other words, C-sections can and do save the lives of babies at risk, but when they are done for other reasons\u2014for example, for the convenience of doctor or mother\u2014they pose an unnecessary threat to health, and even life.\n\nTo take another example: a surgical procedure known as carotid endarterectomy is used to clean out clogged arteries, thus reducing the likelihood of stroke. In 1987, more than one hundred thousand Americans had this operation. It is now established that the risks involved in such surgery outweigh the risks of suffering a stroke. Horowitz again: \"In other words, for certain categories of patients, the operation may actually kill more people than it saves.\" To take still another example: about seventy-eight thousand people every year get cancer from medical and dental X-rays. In a single generation, it is estimated, radiation will induce 2.34 million cancers.\n\nExamples of this kind can be given with appalling ease. But in the interests of fairness the question about the value of technology in medicine is better phrased in the following way: Would American medicine be better were it not so totally reliant on the technological imperative? Here the answer is clearly, yes. We know, for example, from a Harvard Medical School study which focused on the year 1984 (no Orwellian reference intended), that in New York State alone there were thirty-six thousand cases of medical negligence, including seven thousand deaths related in some way to negligence. Although the study does not give figures on what kinds of negligence were found, the example is provided of doctors prescribing penicillin without asking the patients whether they were hypersensitive to the drug. We can assume that many of the deaths resulted not only from careless prescriptions and the doctors' ignorance of their patients' histories but also from unnecessary surgery. In other words, iatrogenics (treatment-induced illness) is now a major concern for the profession, and an even greater concern for the patient. Doctors themselves feel restricted and dominated by the requirement to use all available technology. And patients may be justifiably worried by reports that quite possibly close to 40 percent of the operations performed in America are not necessary. In _Health Shock_ , Martin Weitz cites the calculations of Professor John McKinlay that more deaths are caused by surgery each year in the United States than the annual number of deaths during the wars in Korea and Vietnam. As early as 1974, a Senate investigation into unnecessary surgery reported that American doctors had performed 2.4 million unnecessary operations, causing 11,900 deaths and costing about $3.9 billion. We also know that, in spite of advanced technology (quite possibly because of it), the infant-survival rate in the United States ranks only fourteenth in the world, and it is no exaggeration to say that American hospitals are commonly regarded as among the most dangerous places in the nation. It is also well documented that, wherever doctor strikes have occurred, the mortality rate declines.\n\nThere are, one may be sure, very few doctors who are satisfied with technology's stranglehold on medical practice. And there are far too many patients who have been its serious victims. What conclusions may we draw? First, technology is not a neutral element in the practice of medicine: doctors do not merely use technologies but are used by them. Second, technology creates its own imperatives and, at the same time, creates a wide-ranging social system to reinforce its imperatives. And third, technology changes the practice of medicine by redefining what doctors are, redirecting where they focus their attention, and reconceptualizing how they view their patients and illness.\n\nLike some well-known diseases, the problems that have arisen as a result of the reign of technology came slowly and were barely perceptible at the start. As technology grew, so did the influence of drug companies and the manufacturers of medical instruments. As the training of doctors changed, so did the expectations of patients. As the increase in surgical procedures multiplied, so did the diagnoses which made them seem necessary. Through it all, the question of what was being _undone_ had a low priority if it was asked at all. The Zeitgeist of the age placed such a question in a range somewhere between peevishness and irrelevance. In a growing Technopoly, there is no time or inclination to speak of technological debits.\n\n# **7**\n\n# The Ideology of Machines:  \nComputer Technology\n\nThat American Technopoly has now embraced the computer in the same hurried and mindless way it embraced medical technology is undeniable, was perhaps inevitable, and is certainly most unfortunate. This is not to say that the computer is a blight on the symbolic landscape; only that, like medical technology, it has usurped powers and enforced mind-sets that a fully attentive culture might have wished to deny it. Thus, an examination of the ideas embedded in computer technology is worth attempting. Others, of course, have done this, especially Joseph Weizenbaum in his great and indispensable book _Computer Power and Human Reason_. Weizenbaum, however, ran into some difficulties, as everyone else has, because of the \"universality\" of computers, meaning (a) that their uses are infinitely various, and (b) that computers are commonly integrated into the structure of other machines. It is, therefore, hard to isolate specific ideas promoted by computer technology. The computer, for example, is quite unlike the stethoscope, which has a limited function in a limited context. Except for safecrackers, who, I am told, use stethoscopes to hear the tumblers of locks click into place, stethoscopes are used only by doctors. But everyone uses or is used by computers, and for purposes that seem to know no boundaries.\n\nPutting aside such well-known functions as electronic filing, spreadsheets, and word-processing, one can make a fascinating list of the innovative, even bizarre, uses of computers. I have before me a report from _The New York Times_ that tells us how computers are enabling aquatic designers to create giant water slides that mimic roller coasters and eight-foot-high artificial waves. In my modest collection, I have another article about the uses of personal computers for making presentations at corporate board meetings. Another tells of how computer graphics help jurors to remember testimony better. Gregory Mazares, president of the graphics unit of Litigation Sciences, is quoted as saying, \"We're a switched-on, tuned-in, visually oriented society, and jurors tend to believe what they see. This technology keeps the jury's attention by simplifying the material and by giving them little bursts of information.\"  While Mr. Mazares is helping switched-on people to remember things, Morton David, chief executive officer of Franklin Computer, is helping them find any word in the Bible with lightning speed by producing electronic Bibles. (The word \"lightning,\" by the way, appears forty-two times in the New International version and eight times in the King James version. Were you so inclined, you could discover this for yourself in a matter of seconds.) This fact so dominates Mr. David's imagination that he is quoted as saying, \"Our technology may have made a change as momentous as the Gutenberg invention of movable type.\" And then there is an article that reports a computer's use to make investment decisions, which helps you, among other things, to create \"what-if\" scenarios, although with how much accuracy we are not told. In _Technology Review_ , we find a description of how computers are used to help the police locate the addresses of callers in distress; a prophecy is made that in time police officers will have so much instantly available information about any caller that they will know how seriously to regard the caller's appeal for help.\n\nOne may well wonder if Charles Babbage had any of this in mind when he announced in 1822 (only six years after the appearance of La\u00ebnnec's stethoscope) that he had invented a machine capable of performing simple arithmetical calculations. Perhaps he did, for he never finished his invention and started work on a more ambitious machine, capable of doing more complex tasks. He abandoned that as well, and in 1833 put aside his calculator project completely in favor of a programmable machine that became the forerunner of the modern computer. His first such machine, which he characteristically never finished, was to be controlled by punch cards adapted from devices French weavers used to control thread sequences in their looms.\n\nBabbage kept improving his programmable machine over the next thirty-seven years, each design being more complex than the last. At some point, he realized that the mechanization of numerical operations gave him the means to manipulate non-numerical symbols. It is not farfetched to say that Babbage's insight was comparable to the discovery by the Greeks in the third century B.C. of the principle of alphabetization\u2014that is, the realization that the symbols of the alphabet could be separated from their phonetic function and used as a system for the classification, storage, and retrieval of information. In any case, armed with his insight, Babbage was able to speculate about the possibility of designing \"intelligent\" information machinery, though the mechanical technology of his time was inadequate to allow the fulfillment of his ideas. The computer as we know it today had to await a variety of further discoveries and inventions, including the telegraph, the telephone, and the application of Boolean algebra to relay-based circuitry, resulting in Claude Shannon's creation of digital logic circuitry. Today, when the word \"computer\" is used without a modifier before it, it normally refers to some version of the machine invented by John von Neumann in the 1940s. Before that, the word \"computer\" referred to a person (similarly to the early use of the word \"typewriter\") who performed some kind of mechanical calculation. As calculation shifted from people to machines, so did the word, especially because of the power of von Neumann's machine.\n\nCertainly, after the invention of the digital computer, it was abundantly clear that the computer was capable of performing functions that could in some sense be called \"intelligent.\" In 1936, the great English mathematician Alan Turing showed that it was possible to build a machine that would, for many practical purposes, behave like a problem-solving human being. Turing claimed that he would call a machine \"intelligent\" if, through typed messages, it could exchange thoughts with a human being\u2014that is, hold up its end of a conversation. In the early days of MIT's Artificial Intelligence Laboratory, Joseph Weizenbaum wrote a program called ELIZA, which showed how easy it was to meet Turing's test for intelligence. When asked a question with a proper noun in it, ELIZA'S program could respond with \"Why are you interested in,\" followed by the proper noun and a question mark. That is, it could invert statements and seek more information about one of the nouns in the statement. Thus, ELIZA acted much like a Rogerian psychologist, or at least a friendly and inexpensive therapist. Some people who used ELIZA refused to believe that they were conversing with a mere machine. Having, in effect, created a Turing machine, Weizenbaum eventually pulled the program off the computer network and was stimulated to write _Computer Power and Human Reason_ , in which, among other things, he raised questions about the research programs of those working in artificial intelligence; the assumption that whatever a computer _can_ do, it _should_ do; and the effects of computer technology on the way people construe the world\u2014that is, the ideology of the computer, to which I now turn.\n\nThe most comprehensive idea conveyed by the computer is suggested by the title of J. David Bolter's book, _Turing's Man_. His title is a metaphor, of course, similar to what would be suggested by saying that from the sixteenth century until recently we were \"Gutenberg's Men.\" Although Bolter's main practical interest in the computer is in its function as a new kind of book, he argues that it is the dominant metaphor of our age; it defines our age by suggesting a new relationship to information, to work, to power, and to nature itself. That relationship can best be described by saying that the computer redefines humans as \"information processors\" and nature itself as information to be processed. The fundamental metaphorical message of the computer, in short, is that we are machines\u2014thinking machines, to be sure, but machines nonetheless. It is for this reason that the computer is the quintessential, incomparable, near-perfect machine for Technopoly. It subordinates the claims of our nature, our biology, our emotions, our spirituality. The computer claims sovereignty over the whole range of human experience, and supports its claim by showing that it \"thinks\" better than we can. Indeed, in his almost hysterical enthusiasm for artificial intelligence, Marvin Minsky has been quoted as saying that the thinking power of silicon \"brains\" will be so formidable that \"If we are lucky, they will keep us as pets.\" An even giddier remark, although more dangerous, was offered by John McCarthy, the inventor of the term \"artificial intelligence.\" McCarthy claims that \"even machines as simple as thermostats can be said to have beliefs.\" To the obvious question, posed by the philosopher John Searle, \"What beliefs does your thermostat have?,\" McCarthy replied, \"My thermostat has three beliefs\u2014it's too hot in here, it's too cold in here, and it's just right in here.\"\n\nWhat is significant about this response is that it has redefined the meaning of the word \"belief.\" The remark rejects the view that humans have internal states of mind that are the foundation of belief and argues instead that \"belief\" means only what someone or something does. The remark also implies that simulating an idea is synonymous with duplicating the idea. And, most important, the remark rejects the idea that mind is a biological phenomenon.\n\nIn other words, what we have here is a case of metaphor gone mad. From the proposition that humans are in some respects like machines, we move to the proposition that humans are little else but machines and, finally, that human beings _are_ machines. And then, inevitably, as McCarthy's remark suggests, to the proposition that machines are human beings. It follows that machines can be made that duplicate human intelligence, and thus research in the field known as artificial intelligence was inevitable. What is most significant about this line of thinking is the dangerous reductionism it represents. Human intelligence, as Weizenbaum has tried energetically to remind everyone, is not transferable. The plain fact is that humans have a unique, biologically rooted, intangible mental life which in some limited respects can be simulated by a machine but can never be duplicated. Machines cannot feel and, just as important, cannot _understand_. ELIZA can ask, \"Why are you worried about your mother?,\" which might be exactly the question a therapist would ask. But the machine does not know what the question means or even _that_ the question means. (Of course, there may be some therapists who do not know what the question means either, who ask it routinely, ritualistically, inattentively. In that case we may say they are acting like a machine.) It is meaning, not utterance, that makes mind unique. I use \"meaning\" here to refer to something more than the result of putting together symbols the denotations of which are commonly shared by at least two people. As I understand it, meaning also includes those things we call feelings, experiences, and sensations that do not have to be, and sometimes cannot be, put into symbols. They \"mean\" nonetheless. Without concrete symbols, a computer is merely a pile of junk. Although the quest for a machine that duplicates mind has ancient roots, and although digital logic circuitry has given that quest a scientific structure, artificial intelligence does not and cannot lead to a meaning-making, understanding, and feeling creature, which is what a human being is.\n\nAll of this may seem obvious enough, but the metaphor of the machine as human (or the human as machine) is sufficiently powerful to have made serious inroads in everyday language. People now commonly speak of \"programming\" or \"deprogramming\" themselves. They speak of their brains as a piece of \"hard wiring,\" capable of \"retrieving data,\" and it has become common to think about thinking as a mere matter of processing and decoding.\n\nPerhaps the most chilling case of how deeply our language is absorbing the \"machine as human\" metaphor began on November 4, 1988, when the computers around the ARPANET network became sluggish, filled with extraneous data, and then clogged completely. The problem spread fairly quickly to six thousand computers across the United States and overseas. The early hypothesis was that a software program had attached itself to other programs, a situation which is called (in another human-machine metaphor) a \"virus.\" As it happened, the intruder was a self-contained program explicitly designed to disable computers, which is called a \"worm.\" But the technically incorrect term \"virus\" stuck, no doubt because of its familiarity and its human connections. As Raymond Gozzi, Jr., discovered in his analysis of how the mass media described the event, newspapers noted that the computers were \"infected,\" that the virus was \"virulent\" and \"contagious,\" that attempts were made to \"quarantine\" the infected computers, that attempts were also being made to \"sterilize\" the network, and that programmers hoped to develop a \"vaccine\" so that computers could be \"inoculated\" against new attacks.\n\nThis kind of language is not merely picturesque anthropomorphism. It reflects a profound shift in perception about the relationship of computers to humans. If computers can become ill, then they can become healthy. Once healthy, they can think clearly and make decisions. The computer, it is implied, has a will, has intentions, has reasons\u2014which means that humans are relieved of responsibility for the computer's decisions. Through a curious form of grammatical alchemy, the sentence \"We use the computer to calculate\" comes to mean \"The computer calculates.\" If a computer calculates, then it may decide to miscalculate or not calculate at all. That is what bank tellers mean when they tell you that they cannot say how much money is in your checking account because \"the computers are down.\" The implication, of course, is that no person at the bank is responsible. Computers make mistakes or get tired or become ill. Why blame people? We may call this line of thinking an \"agentic shift,\" a term I borrow from Stanley Milgram to name the process whereby humans transfer responsibility for an outcome from themselves to a more abstract agent. When this happens, we have relinquished control, which in the case of the computer means that we may, without excessive remorse, pursue ill-advised or even inhuman goals because the computer can accomplish them or be imagined to accomplish them.\n\nMachines of various kinds will sometimes assume a human or, more likely, a superhuman aspect. Perhaps the most absurd case I know of is in a remark a student of mine once made on a sultry summer day in a room without air conditioning. On being told the thermometer read ninety-eight degrees Fahrenheit, he replied, \"No wonder it's so hot!\" Nature was off the hook. If only the thermometers would behave themselves, we could be comfortable. But computers are far more \"human\" than thermometers or almost any other kind of technology. Unlike most machines, computers do no work; they direct work. They are, as Norbert Wiener said, the technology of \"command and control\" and have little value without something to control. This is why they are of such importance to bureaucracies.\n\nNaturally, bureaucrats can be expected to embrace a technology that helps to create the illusion that decisions are not under their control. Because of its seeming intelligence and impartiality, a computer has an almost magical tendency to direct attention away from the people in charge of bureaucratic functions and toward itself, as if the computer were the true source of authority. A bureaucrat armed with a computer is the unacknowledged legislator of our age, and a terrible burden to bear. We cannot dismiss the possibility that, if Adolf Eichmann had been able to say that it was not he but a battery of computers that directed the Jews to the appropriate crematoria, he might never have been asked to answer for his actions.\n\nAlthough (or perhaps because) I came to \"administration\" late in my academic career, I am constantly amazed at how obediently people accept explanations that begin with the words \"The computer shows...\" or \"The computer has determined...\" It is Technopoly's equivalent of the sentence \"It is God's will,\" and the effect is roughly the same. You will not be surprised to know that I rarely resort to such humbug. But on occasion, when pressed to the wall, I have yielded. No one has as yet replied, \"Garbage in, garbage out.\" Their defenselessness has something Kafkaesque about it. In _The Trial_ , Josef K. is charged with a crime\u2014of what nature, and by whom the charge is made, he does not know. The computer turns too many of us into Josef Ks. It often functions as a kind of impersonal accuser which does not reveal, and is not required to reveal, the sources of the judgments made against us. It is apparently sufficient that the computer has pronounced. Who has put the data in, for what purpose, for whose convenience, based on what assumptions are questions left unasked.\n\nThis is the case not only in personal matters but in public decisions as well. Large institutions such as the Pentagon, the Internal Revenue Service, and multinational corporations tell us that their decisions are made on the basis of solutions generated by computers, and this is usually good enough to put our minds at ease or, rather, to sleep. In any case, it constrains us from making complaints or accusations. In part for this reason, the computer has strengthened bureaucratic institutions and suppressed the impulse toward significant social change. \"The arrival of the Computer Revolution and the founding of the Computer Age have been announced many times,\" Weizenbaum has written. \"But if the triumph of a revolution is to be measured in terms of the social revision it entrained, then there has been no computer revolution.\"\n\nIn automating the operation of political, social, and commercial enterprises, computers may or may not have made them more efficient but they have certainly diverted attention from the question whether or not such enterprises are necessary or how they might be improved. A university, a political party, a religious denomination, a judicial proceeding, even corporate board meetings are not improved by automating their operations. They are made more imposing, more technical, perhaps more authoritative, but defects in their assumptions, ideas, and theories will remain untouched. Computer technology, in other words, has not yet come close to the printing press in its power to generate radical and substantive social, political, and religious thought. If the press was, as David Riesman called it, \"the gunpowder of the mind,\" the computer, in its capacity to smooth over unsatisfactory institutions and ideas, is the talcum powder of the mind.\n\nI do not wish to go as far as Weizenbaum in saying that computers are merely ingenious devices to fulfill unimportant functions and that the computer revolution is an explosion of nonsense. Perhaps that judgment will be in need of amendment in the future, for the computer is a technology of a thousand uses\u2014the Proteus of machines, to use Seymour Papert's phrase. One must note, for example, the use of computer-generated images in the phenomenon known as Virtual Reality. Putting on a set of miniature goggle-mounted screens, one may block out the real world and move through a simulated three-dimensional world which changes its components with every movement of one's head. That Timothy Leary is an enthusiastic proponent of Virtual Reality does not suggest that there is a constructive future for this device. But who knows? Perhaps, for those who can no longer cope with the real world, Virtual Reality will provide better therapy than ELIZA.\n\nWhat is clear is that, to date, computer technology has served to strengthen Technopoly's hold, to make people believe that technological innovation is synonymous with human progress. And it has done so by advancing several interconnected ideas.\n\nIt has, as already noted, amplified beyond all reason the metaphor of machines as humans and humans as machines. I do not claim, by the way, that computer technology originated this metaphor. One can detect it in medicine, too: doctors and patients have come to believe that, like a machine, a human being is made up of parts which when defective can be replaced by mechanical parts that function as the original did without impairing or even affecting any other part of the machine. Of course, to some degree that assumption works, but since a human being is in fact not a machine but a biological organism all of whose organs are interrelated and profoundly affected by mental states, the human-as-machine metaphor has serious medical limitations and can have devastating effects. Something similar may be said of the mechanistic metaphor when applied to workers. Modern industrial techniques are made possible by the idea that a machine is made up of isolatable and interchangeable parts. But in organizing factories so that workers are also conceived of as isolatable and interchangeable parts, industry has engendered deep alienation and bitterness. This was the point of Charlie Chaplin's _Modern Times_ , in which he tried to show the psychic damage of the metaphor carried too far. But because the computer \"thinks\" rather than works, its power to energize mechanistic metaphors is unparalleled and of enormous value to Technopoly, which depends on our believing that we are at our best when acting like machines, and that in significant ways machines may be trusted to act as our surrogates. Among the implications of these beliefs is a loss of confidence in human judgment and subjectivity. We have devalued the singular human capacity to see things whole in all their psychic, emotional and moral dimensions, and we have replaced this with faith in the powers of technical calculation.\n\nBecause of what computers commonly do, they place an inordinate emphasis on the technical processes of communication and offer very little in the way of substance. With the exception of the electric light, there never has been a technology that better exemplifies Marshall McLuhan's aphorism \"The medium is the message.\" The computer is almost all process. There are, for example, no \"great computerers,\" as there are great writers, painters, or musicians. There are \"great programs\" and \"great programmers,\" but their greatness lies in their ingenuity either in simulating a human function or in creating new possibilities of calculation, speed, and volume. Of course, if J. David Bolter is right, it is possible that in the future computers will emerge as a new kind of book, expanding and enriching the tradition of writing technologies. Since printing created new forms of literature when it replaced the handwritten manuscript, it is possible that electronic writing will do the same. But for the moment, computer technology functions more as a new mode of transportation than as a new means of substantive communication. It moves information\u2014lots of it, fast, and mostly in a calculating mode. The computer, in fact, makes possible the fulfillment of Descartes' dream of the mathematization of the world. Computers make it easy to convert facts into statistics and to translate problems into equations. And whereas this can be useful (as when the process reveals a pattern that would otherwise go unnoticed), it is diversionary and dangerous when applied indiscriminately to human affairs. So is the computer's emphasis on speed and especially its capacity to generate and store unprecedented quantities of information. In specialized contexts, the value of calculation, speed, and voluminous information may go uncontested. But the \"message\" of computer technology is comprehensive and domineering. The computer argues, to put it baldly, that the most serious problems confronting us at both personal and public levels require technical solutions through fast access to information otherwise unavailable. I would argue that this is, on the face of it, nonsense. Our most serious problems are not technical, nor do they arise from inadequate information. If a nuclear catastrophe occurs, it shall not be because of inadequate information. Where people are dying of starvation, it does not occur because of inadequate information. If families break up, children are mistreated, crime terrorizes a city, education is impotent, it does not happen because of inadequate information. Mathematical equations, instantaneous communication, and vast quantities of information have nothing whatever to do with any of these problems. And the computer is useless in addressing them.\n\nAnd yet, because of its \"universality,\" the computer compels respect, even devotion, and argues for a comprehensive role in all fields of human activity. Those who insist that it is foolish to deny the computer vast sovereignty are singularly devoid of what Paul Goodman once called \"technological modesty\"\u2014that is, having a sense of the whole and not claiming or obtruding more than a particular function warrants. Norbert Wiener warned about lack of modesty when he remarked that, if digital computers had been in common use before the atomic bomb was invented, people would have said that the bomb could not have been invented without computers. But it was. And it is important to remind ourselves of how many things are quite possible to do without the use of computers.\n\nSeymour Papert, for example, wishes students to be epistemologists, to think critically, and to learn how to create knowledge. In his book _Mindstorms_ , he gives the impression that his computer program known as LOGO now makes this possible. But good teachers have been doing this for centuries without the benefit of LOGO. I do not say that LOGO, when used properly by a skilled teacher, will not help, but I doubt that it can do better than pencil and paper, or speech itself, when used properly by a skilled teacher.\n\nWhen the Dallas Cowboys were consistently winning football championships, their success was attributed to the fact that computers were used to evaluate and select team members. During the past several years, when Dallas has been hard put to win more than a few games, not much has been said about the computers, perhaps because people have realized that computers have nothing to do with winning football games, and never did. One might say the same about writing lucid, economical, stylish prose, which has nothing to do with word-processors. Although my students don't believe it, it is actually possible to write well without a processor and, I should say, to write poorly with one.\n\nTechnological immodesty is always an acute danger in Technopoly, which encourages it. Technopoly also encourages in-sensitivity to what skills may be lost in the acquisition of new ones. It is important to remember what can be done without computers, and it is also important to remind ourselves of what may be lost when we do use them.\n\nI have before me an essay by Sir Bernard Lovell, founder of Britain's Jodrell Bank Observatory, in which he claims that computers have stifled scientific creativity. After writing of his awe at the ease with which computerized operations provide amazing details of distant galaxies, Sir Bernard expresses concern that \"literal-minded, narrowly focused computerized research is proving antithetical to the free exercise of that happy faculty known as serendipity\u2014that is, the knack of achieving favorable results more or less by chance.\" He proceeds to give several examples of monumental but serendipitous discoveries, contends that there has been a dramatic cessation of such discoveries, and worries that computers are too narrow as filters of information and therefore may be antiserendipitous. He is, of course, not \"against\" computers, but is merely raising questions about their costs.\n\nDr. Clay Forishee, the chief FAA scientist for human performance issues, did the same when he wondered whether the automated operation of commercial aircraft has not disabled pilots from creatively responding when something goes wrong. Robert Buley, flight-standards manager of Northwest Airlines, goes further. He is quoted as saying, \"If we have human operators subordinated to technology then we're going to lose creativity [in emergencies].\" He is not \"against\" computers. He is worried about what we lose by using them.\n\nM. Ethan Katsch, in his book _The Electronic Media and the Transformation of Law_ , worries as well. He writes, \"The replacement of print by computerized systems is promoted to the legal profession simply as a means to increase efficiency.\" But he goes on to say that, in fact, the almost unlimited capacity of computers to store and retrieve information threatens the authority of precedent, and he adds that the threat is completely unrecognized. As he notes, \"a system of precedent is unnecessary when there are very few accessible cases, and unworkable when there are too many.\" If this is true, or even partly true, what exactly does it mean? Will lawyers become incapable of choosing relevant precedents? Will judges be in constant confusion from \"precedent overload\"?\n\nWe know that doctors who rely entirely on machinery have lost skill in making diagnoses based on observation. We may well wonder what other human skills and traditions are being lost by our immersion in a computer culture. Technopolists do not worry about such things. Those who do are called technological pessimists, Jeremiahs, and worse. I rather think they are imbued with technological modesty, like King Thamus.\n\n# **8**\n\n# Invisible Technologies\n\nIf we define ideology as a set of assumptions of which we are barely conscious but which nonetheless directs our efforts to give shape and coherence to the world, then our most powerful ideological instrument is the technology of language itself. Language is pure ideology. It instructs us not only in the names of things but, more important, in what things can be named. It divides the world into subjects and objects. It denotes what events shall be regarded as processes, and what events, things. It instructs us about time, space, and number, and forms our ideas of how we stand in relation to nature and to each other. In English grammar, for example, there are always subjects who act, and verbs which are their actions, and objects which are acted upon. It is a rather aggressive grammar, which makes it difficult for those of us who must use it to think of the world as benign. We are obliged to know the world as made up of things pushing against, and often attacking, one another.\n\nOf course, most of us, most of the time, are unaware of how language does its work. We live deep within the boundaries of our linguistic assumptions and have little sense of how the world looks to those who speak a vastly different tongue. We tend to assume that everyone sees the world in the same way, irrespective of differences in language. Only occasionally is this illusion challenged, as when the differences between linguistic ideologies become noticeable by one who has command over two languages that differ greatly in their structure and history. For example, several years ago, Susumu Tonegawa, winner of the 1987 Nobel Prize in Medicine, was quoted in the newspaper _Yomiuri_ as saying that the Japanese language does not foster clarity or effective understanding in scientific research. Addressing his countrymen from his post as a professor at MIT in Cambridge, Massachusetts, he said, \"We should consider changing our thinking process in the field of science by trying to reason in English.\" It should be noted that he was not saying that English is better than Japanese; only that English is better than Japanese for the purposes of scientific research, which is a way of saying that English (and other Western languages) have a particular ideological bias that Japanese does not. We call that ideological bias \"the scientific outlook.\" If the scientific outlook seems natural to you, as it does to me, it is because our language makes it appear so. What we think of as reasoning is determined by the character of our language. To reason in Japanese is apparently not the same thing as to reason in English or Italian or German.\n\nTo put it simply, like any important piece of machinery\u2014television or the computer, for example\u2014language has an ideological agenda that is apt to be hidden from view. In the case of language, that agenda is so deeply integrated into our personalities and world-view that a special effort and, often, special training are required to detect its presence. Unlike television or the computer, language appears to be not an extension of our powers but simply a natural expression of who and what we are. This is the great secret of language: Because it comes from inside us, we believe it to be a direct, unedited, unbiased, apolitical expression of how the world really is. A machine, on the other hand, is outside of us, clearly created by us, modifiable by us, even discardable by us; it is easier to see how a machine re-creates the world in its own image. But in many respects, a sentence functions very much like a machine, and this is nowhere more obvious than in the sentences we call questions.\n\nAs an example of what I mean, let us take a \"fill-in\" question, which I shall require you to answer exactly if you wish full credit:\n\nThomas Jefferson died in the year\u2014\u2014.\n\nSuppose we now rephrase the question in multiple-choice form:\n\nThomas Jefferson died in the year (a) 1788 (b) 1826\n\n(c) 1926 (d) 1809.\n\nWhich of these two questions is easier to answer? I assume you will agree with me that the second question is easier unless you happen to know precisely the year of Jefferson's death, in which case neither question is difficult. However, for most of us who know only roughly when Jefferson lived, Question Two has arranged matters so that our chances of \"knowing\" the answer are greatly increased. Students will always be \"smarter\" when answering a multiple-choice test than when answering a \"fill-in\" test, even when the subject matter is the same. A question, even of the simplest kind, is not and can never be unbiased. I am not, in this context, referring to the common accusation that a particular test is \"culturally biased.\" Of course questions can be culturally biased. (Why, for example, should anyone be asked about Thomas Jefferson at all, let alone when he died?) My purpose is to say that the structure of any question is as devoid of neutrality as is its content. The form of a question may ease our way or pose obstacles. Or, when even slightly altered, it may generate antithetical answers, as in the case of the two priests who, being unsure if it was permissible to smoke and pray at the same time, wrote to the Pope for a definitive answer. One priest phrased the question \"Is it permissible to smoke while praying?\" and was told it is not, since prayer should be the focus of one's whole attention; the other priest asked if it is permissible to pray while smoking and was told that it is, since it is always appropriate to pray. The form of a question may even block us from seeing solutions to problems that become visible through a different question. Consider the following story, whose authenticity is questionable but not, I think, its point:\n\nOnce upon a time, in a village in what is now Lithuania, there arose an unusual problem. A curious disease afflicted many of the townspeople. It was mostly fatal (though not always), and its onset was signaled by the victim's lapsing into a deathlike coma. Medical science not being quite so advanced as it is now, there was no definite way of knowing if the victim was actually dead when burial appeared seemly. As a result, the townspeople feared that several of their relatives had already been buried alive and that a similar fate might await them. How to overcome this uncertainty was their dilemma.\n\nOne group of people suggested that the coffins be well stocked with water and food and that a small air vent be drilled into them, just in case one of the \"dead\" happened to be alive. This was expensive to do but seemed more than worth the trouble. A second group, however, came up with a less expensive and more efficient idea. Each coffin would have a twelve-inch stake affixed to the inside of the coffin lid, exactly at the level of the heart. Then, when the coffin was closed, all uncertainty would cease.\n\nThe story does not indicate which solution was chosen, but for my purposes the choice is irrelevant. What is important to note is that different solutions were generated by different questions. The first solution was an answer to the question, How can we make sure that we do not bury people who are still alive? The second was an answer to the question, How can we make sure that everyone we bury is dead?\n\nQuestions, then, are like computers or television or stethoscopes or lie detectors, in that they are mechanisms that give direction to our thoughts, generate new ideas, venerate old ones, expose facts, or hide them. In this chapter, I wish to consider mechanisms that act like machines but are not normally thought of as part of Technopoly's repertoire. I must call attention to them precisely because they are so often overlooked. For all practical purposes, they may be considered technologies\u2014technologies in disguise, perhaps, but technologies all the same.\n\nAside from language itself, I don't suppose there is a clearer example of a technology that doesn't look like one than the mathematical sign known as zero. A brief word about it may help to illuminate later examples.\n\nThe zero made its way from India to Europe in the tenth century. By the thirteenth century, it had taken hold of Western consciousness. (It was unknown to the Romans and the classical Greeks, although analogous concepts were known to Babylonian mathematicians of the Hellenistic period.) Without the zero, you will find it difficult to perform any of the calculations that are quite simple to do with it. If you should try multiplying MMMMMM by MMDCXXVI, you will have this point confirmed. I have been told, by the way, that such a calculation _can_ be done, but the process is so laborious that the task is unlikely to be completed, a truth that did not escape the notice of medieval mathematicians. There is, in fact, no evidence that Roman numerals were ever used, or intended to be used, for calculation. For that purpose, mathematicians used an abacus, and between the tenth and thirteenth centuries, a struggle of sorts took place between abacists, who wrote Roman numerals but calculated with the abacus, and algorists, who used Hindu numerals employing the zero sign. The objection raised by the abacists was that the zero registered the _absence_ of a power of ten, which no Roman numeral did, and which struck them as philosophically and perhaps aesthetically offensive. After all, the zero is a sign that affects values of numerals wherever it occurs but has no value in itself. It is a sign about signs, whose very etymology, via \"cipher\" from the Hindu word for \"void,\" suggests the idea of \"nothingness.\" To the abacists, it was a bizarre idea to have a sign marking \"nothing,\" and I fear that I would have sided with the abacists.\n\nI speak of the zero for two reasons: First, to underscore that it is a kind of technology that makes both possible and easy certain kinds of thoughts which, without it, would remain inaccessible to the average person. If it does not exactly have an ideology, it contains, at least, an idea. I have previously alluded to the technology of using letters or numbers to grade students' papers, and to the Greek discovery of the technology of alphabetization: like the use of zero, these are examples of how symbols may function like machines in creating new mind-sets and therefore new conceptions of reality. Second, the use of the zero and, of course, the Hindu numbering system of which it was a part made possible a sophisticated mathematics which, in turn, led to one of the most powerful technologies now in use: statistics.\n\nStatistics makes possible new perceptions and realities by making visible large-scale patterns. Its uses in science are too well known to warrant notice here, except to remark that if, as the physicists tell us, the world is made up of probabilities at the level of subatomic particles, then statistics is the only means by which to describe its operations. Indeed, the uncertainty principle ensures that in the nature of things physics is unable to do more than make statistical predictions.\n\nOf course, it is possible that physicists conceive of the world as probabilistic _because_ statistics was invented. But that is not the question I wish to pursue here. A more practical question is, To what extent has statistics been allowed entry to places where it does not belong? Technopoly, by definition, grants free rein to any technology, and we would expect that no limits have been placed on the use of statistics. We would expect correctly.\n\nPerhaps the most abusive example is found in the work of Francis Galton, who was born in 1822, died in 1911, and therefore lived during the richest period of technological invention. He may be thought of as one of the Founding Fathers of Technopoly. Galton is also known as the founder of \"eugenics,\" a term he coined, which means the \"science\" of arranging marriage and family so as to produce the best possible offspring based on the hereditary characteristics of the parents. He believed that anything could be measured and that statistical procedures, in particular, were the technology that could open the pathway to real knowledge about every form of human behavior. The next time you watch a televised beauty contest in which women are ranked numerically, you should remember Francis Galton, whose pathological romance with numbers originated this form of idiocy. Being unsatisfied with vagueness about where the most \"beauty\" was to be found, he constructed a \"beauty map\" of the British Isles. As he told us, he classified \"the girls I passed in streets or elsewhere as attractive, indifferent, or repellent.\" He then proved statistically that London had the most beautiful girls, Aberdeen the ugliest; this no doubt made it awkward for Galton to spend his vacation in Scotland. If this were not enough, he also invented a method for quantifying boredom (by counting the number of fidgets) and even proposed a statistical inquiry for determining the efficacy of prayer.\n\nBut Galton's main interest was in demonstrating, statistically, the inheritance of intelligence. To that end, he established a laboratory at the International Exposition of 1884, where for threepence people could have their skulls measured and receive Galton's assessment of their intelligence. Apparently, a visitor received no extra credit for demanding his or her money back, which would surely have been a sign of intelligence. We can be sure that not many did, since Galton was considered a major intellect of his day. In fact, Lewis Terman, the man most responsible for promoting IQ tests in America, calculated that Galton's IQ was more than 200. Terman, who fancied making such estimates of the dead, ranked Charles Darwin (Galton's cousin, incidentally) at a mere 135, and poor Copernicus somewhere between 100 and 110.\n\nFor a definitive history and analysis of the malignant role played by statistics in the \"measurement\" of intelligence, I refer the reader to Stephen Jay Gould's brilliant book _The Mismeasure of Man_. Here, I will only cite three points made by Gould, which I believe are sufficient to convince anyone with a higher IQ than Copernicus of the dangers of abusing statistics.\n\nThe first problem is called reification, which means converting an abstract idea (mostly, a word) into a thing. In this context, reification works in the following way: We use the word \"intelligence\" to refer to a variety of human capabilities of which we approve. There is no such thing as \"intelligence.\" It is a word, not a thing, and a word of a very high order of abstraction. But if we believe it to be a thing like the pancreas or liver, then we will believe scientific procedures can locate it and measure it.\n\nThe second problem is ranking. Ranking requires a criterion for assigning individuals to their place in a single series. As Gould remarks, what better criterion can be used than an objective number? In the ranking of intelligence, we therefore assume that intelligence is not only a thing, but a single thing, located in the brain, and accessible to the assignment of a number. It is as if \"beauty\" were determined to inhere in the size of a woman's breasts. Then all we would have to do is measure breasts and rank each woman accordingly, and we would have an \"objective\" measure of \"beauty.\"\n\nThe third point is that in doing this, we would have formulated our question \"Who is the fairest of all?\" in a restricted and biased way. And yet this would go unnoticed, because, as Gould writes, \"The mystique of science proclaims that numbers are the ultimate test of objectivity.\" This means that the way we have defined the concept will recede from our consciousness\u2014that is, its fundamental subjectivity will become invisible, and the objective number itself will become reified. One would think that such a process would appear ridiculous on the breast of it, especially since, by believing it, we must conclude that Dolly Parton is objectively proved to be more beautiful than Audrey Hepburn. Or, in the case of intelligence, that Galton had twice as much of it as Copernicus.\n\nNonetheless, in Technopoly all this is taken very seriously, albeit not without a few protests. After a lifetime of working in the field of intelligence measurement, E. L. Thorndike observed that intelligence tests suffer from three small defects: \"Just what they measure is not known; how far it is proper to add, subtract, multiply, divide, and compute ratios with the measures obtained is not known; just what the measures signify concerning intellect is not known.\" In other words, those who administer intelligence tests quite literally do not know what they are doing. That is why David McClelland remarked, \"Psychologists should be ashamed of themselves for promoting a view of general intelligence that has engendered such a testing program.\" Joseph Weizenbaum summed it up by saying, \"Few 'scientific' concepts have so thoroughly muddled the thinking of both scientists and the general public as that of the 'intelligence quotient' or 'IQ.' The idea that intelligence can be quantitatively measured along a single linear scale has caused untold harm to our society in general, and to education in particular.\"\n\nGould has documented some of this harm, and Howard Gardner has tried to alleviate it (in his book _Frames of Mind)_. But Technopoly resists such reproaches, because it needs to believe that science is an entirely objective enterprise. Lacking a lucid set of ethics and having rejected tradition, Technopoly searches for a source of authority and finds it in the idea of statistical objectivity.\n\nThis quest is especially evident not only in our efforts to determine precisely how smart people are but also in our attempts to find out precisely how smart _groups_ of people are. Aside from the fact that the procedures used do not and _cannot_ give such an answer, one must ask, Of what earthly use is it to declare that one group of people is smarter than another? Suppose it is shown that according to objective measures Asians have more \"intelligence\" than Caucasians, or that Caucasians have more than African-Americans. Then what? Of what use is this information to, say, a teacher or an employer? Is the teacher or employer to assume that a particular Asian is smarter than a particular African-American? Or even that six Asians are smarter than six African-Americans? Obviously not. And yet who knows? We must keep in mind the story of the statistician who drowned while trying to wade across a river with an average depth of four feet. That is to say, in a culture that reveres statistics, we can never be sure what sort of nonsense will lodge in people's heads.\n\nThe only plausible answer to the question why we use statistics for such measurements is that it is done for sociopolitical reasons whose essential malignancy is disguised by the cover of \"scientific inquiry.\" If we believe that blacks are dumber than whites, and that this is not merely our opinion but is confirmed by objective measures, then we can believe we have an irreproachable authority for making decisions about the allocation of resources. This is how, in Technopoly, science is used to make democracy \"rational.\"\n\nPolling is still another way. Just as statistics has spawned a huge testing industry, it has done the same for the polling of \"public opinion.\" One may concede, at the start, that there are some uses of polling that may be said to be reliable, especially if the case involves a greatly restricted question such as, Do you plan to vote for X or Y? But to say a procedure is reliable is not to say it is useful. The question is as yet undecided whether knowledge of voter trends during a political campaign enriches or demeans the electoral process. But when polls are used to guide public policy, we have a different sort of issue altogether.\n\nI have been in the presence of a group of United States congressmen who were gathered to discuss, over a period of two days, what might be done to make the future of America more survivable and, if possible, more humane. Ten consultants were called upon to offer perspectives and advice. Eight of them were pollsters. They spoke of the \"trends\" their polling uncovered; for example, that people were no longer interested in the women's movement, did not regard environmental issues as of paramount importance, did not think the \"drug problem\" was getting worse, and so on. It was apparent, at once, that these polling results would become the basis of how the congressmen thought the future should be managed. The ideas the congressmen had (all men, by the way) receded to the background. Their own perceptions, instincts, insights, and experience paled into irrelevance. Confronted by \"social scientists,\" they were inclined to do what the \"trends\" suggested would satisfy the populace.\n\nIt is not unreasonable to argue that the polling of public opinion puts democracy on a sound and scientific footing. If our political leaders are supposed to represent us, they must have some information about what we \"believe.\" In principle, there is no problem here. The problems lie elsewhere, and there are at least four of them.\n\nThe first has to do with the forms of the questions that are put to the public. I refer the reader to the matter of whether it is proper to smoke and pray at the same time. Or, to take a more realistic example: If we ask people whether they think it acceptable for the environment to continue to be polluted, we are likely to come up with answers quite different from those generated by the question, Do you think the protection of the environment is of paramount importance? Or, Do you think safety in the streets is more important than environmental protection? The public's \"opinion\" on almost any issue will be a function of the question asked. (I might point out that in the seminar held by the congressmen, not one asked a question about the questions. They were interested in results, not in how these were obtained, and it did not seem to occur to them that the results and how they are obtained are inseparable.)\n\nTypically, pollsters ask questions that will elicit yes or no answers. Is it necessary to point out that such answers do not give a robust meaning to the phrase \"public opinion\"? Were you, for example, to answer \"No\" to the question \"Do you think the drug problem can be reduced by government programs?\" one would hardly know much of interest or value about your opinion. But allowing you to speak or write at length on the matter would, of course, rule out using statistics. The point is that the use of statistics in polling changes the meaning of \"public opinion\" as dramatically as television changes the meaning of \"political debate.\" In the American Technopoly, public opinion is a yes or no answer to an unexamined question.\n\nSecond, the technique of polling promotes the assumption that an opinion is a thing inside people that can be exactly located and extracted by the pollster's questions. But there is an alternative point of view, of which we might say, it is what Jefferson had in mind. An opinion is not a momentary thing but a process of thinking, shaped by the continuous acquisition of knowledge and the activity of questioning, discussion, and debate. A question may \"invite\" an opinion, but it also may modify and recast it; we might better say that people do not exactly \"have\" opinions but are, rather, involved in \"opinioning.\" That an opinion is conceived of as a measurable thing falsifies the process by which people, in fact, do their opinioning; and how people do their opinioning goes to the heart of the meaning of a democratic society. Polling tells us nothing about this, and tends to hide the process from our view.\n\nWhich leads to the third point. Generally, polling ignores what people know about the subjects they are queried on. In a culture that is not obsessed with measuring and ranking things, this omission would probably be regarded as bizarre. But let us imagine what we would think of opinion polls if the questions came in pairs, indicating what people \"believe\" and what they \"know\" about the subject. If I may make up some figures, let us suppose we read the following: \"The latest poll indicates that 72 percent of the American public believes we should withdraw economic aid from Nicaragua. Of those who expressed this opinion, 28 percent thought Nicaragua was in central Asia, 18 percent thought it was an island near New Zealand, and 27.4 percent believed that 'Africans should help themselves,' obviously confusing Nicaragua with Nigeria. Moreover, of those polled, 61.8 percent did not know that we give economic aid to Nicaragua, and 23 percent did not know what 'economic aid' means.\" Were pollsters inclined to provide such information, the prestige and power of polling would be considerably reduced. Perhaps even congressmen, confronted by massive ignorance, would invest their own understandings with greater trust.\n\nThe fourth problem with polling is that it shifts the locus of responsibility between political leaders and their constituents. It is true enough that congressmen are supposed to represent the interests of their constituents. But it is also true that congressmen are expected to use their own judgment about what is in the public's best interests. For this, they must consult their own experience and knowledge. Before the ascendance of polling, political leaders, though never indifferent to the opinions of their constituents, were largely judged on their capacity to make decisions based on such wisdom as they possessed; that is, political leaders were responsible for the decisions they made. With the refinement and extension of the polling process, they are under increasing pressure to forgo deciding anything for themselves and to defer to the opinions of the voters, no matter how ill-informed and shortsighted those opinions might be.\n\nWe can see this process of responsibility-shift even more clearly in the case of the statistically based ratings of television shows. The definition of a \"good\" television show has become purely and simply a matter of its having high ratings. A \"bad\" show has low ratings. The responsibility of a television writer, therefore, begins and ends with his or her ability to create a show that many millions of viewers will watch. The writer, in a word, is entirely responsible to the audience. There is no need for the writer to consult tradition, aesthetic standards, thematic plausibility, refinements of taste, or even plain comprehensibility. The iron rule of public opinion is all that matters. Television executives are fond of claiming that their medium is the most democratic institution in America: a plebiscite is held every week to determine which programs will survive. This claim is given added weight by a second claim: creative artists have never been indifferent to the preferences and opinions of their audiences. Writers, for example, write _for_ people, for their approbation and understanding. But writers also write for themselves and because they have something they want to say, not always because readers have something they want to hear. By giving constant deference to public preferences, polling changes the motivation of writers; their entire effort is to increase \"the numbers.\" Popular literature now depends more than ever on the wishes of the audience, not the creativity of the artist.\n\nBefore leaving the subject of the technology of statistics, I must call attention to the fact that statistics creates an enormous amount of completely useless information, which compounds the always difficult task of locating that which is useful to a culture. This is more than a case of \"information-overload.\" It is a matter of \"information-trivia,\" which has the effect of placing all information on an equal level. No one has expressed this misuse of a technology better than the _New Yorker_ magazine cartoonist Mankoff. Showing an attentive man watching television news, Mankoff has the newscaster saying, \"A preliminary census report indicates that for the first time in our nation's history female anthropologists outnumber male professional golfers.\" When statistics and computers are joined, volumes of garbage are generated in public discourse. Those who have watched television sports programs will know that Mankoff's cartoon is, in fact, less of a parody than a documentary. Useless, meaningless statistics flood the attention of the viewer. Sports-casters call them \"graphics\" in an effort to suggest that the information, graphically presented, is a vital supplement to the action of the game. For example: \"Since 1984, the Buffalo Bills have won only two games in which they were four points ahead with less than six minutes to play.\" Or this: \"In only 17 percent of the times he has pitched at Shea Stadium has Dwight Gooden struck out the third and fourth hitters less than three times when they came to bat with more than one runner on base.\" What is one to do with this or to make of it? And yet there seems to be a market for useless information. Those who read _USA Today_ , for example, are offered on the front page of each issue an idiotic statistic of the day that looks something like this: \"The four leading states in banana consumption from 1980 through 1989 are Kansas, North Dakota, Wyoming, and Louisiana. Oddly, Nevada, which was ninth in 1989, fell to twenty-sixth last year, which is exactly where it ranks in kiwi consumption.\"\n\nIt is surprising how frequently such blather will serve as the backbone of conversations which are essentially meaningless. I have heard New Yorkers, with a triumphant flourish, offer out-of-towners the statistic that New York City is only eighth in the nation in per-capita violent crimes and then decline to go outside because it was past 6:00 p.m.\n\nI do not say, of course, that all such statistical statements are useless. If we learn that one out of every four black males between the ages of twenty and thirty has spent some time in prison, and that the nation's expenditure for the education of black children is 23 percent less than it is for white children, we may have some statistical facts that will help us to see a cause-and-effect relationship, and thereby suggest a course of action. But statistics, like any other technology, has a tendency to run out of control, to occupy more of our mental space than it warrants, to invade realms of discourse where it can only wreak havoc. When it is out of control, statistics buries in a heap of trivia what is necessary to know.\n\nAnd there is another point, which in fact is the core of this chapter. Some technologies come in disguise. Rudyard Kipling called them \"technologies in repose.\" They do not look like technologies, and because of that they do their work, for good or ill, without much criticism or even awareness. This applies not only to IQ tests and to polls and to all systems of ranking and grading but to credit cards, accounting procedures, and achievement tests. It applies in the educational world to what are called \"academic courses,\" as well. A course is a technology for learning. I have \"taught\" about two hundred of them and do not know why each one lasts exactly fifteen weeks, or why each meeting lasts exactly one hour and fifty minutes. If the answer is that this is done for administrative convenience, then a course is a fraudulent technology. It is put forward as a desirable structure for learning when in fact it is only a structure for allocating space, for convenient record-keeping, and for control of faculty time. The point is that the origin of and raison d'\u00eatre for a course are concealed from us. We come to believe it exists for one reason when it exists for quite another. One characteristic of those who live in a Technopoly is that they are largely unaware of both the origins and the effects of their technologies.\n\nPerhaps the most interesting example of such lack of awareness is the widespread belief that modern business invented the technology of management. Management is a system of power and control designed to make maximum use of relevant knowledge, the hierarchical organization of human abilities, and the flow of information from bottom to top and back again. It is generally assumed that management was created by business enterprises as a rational response to the economic and technological demands of the Industrial Revolution. But research by Alfred Chandler, Sidney Pollard, and especially Keith Hoskin and Richard Macve reveals a quite different picture and leads to a startling conclusion: modern business did not invent management; management invented modern business.\n\nThe most likely place for management to have originated is, of course, in Great Britain in the late eighteenth and early nineteenth centuries. But there is no evidence that British industry knew anything about management as late as 1830, nor did there exist anything approximating a \"managerial class.\" Management was created in the United States \"out of the blue,\" as Hoskin and Macve say. It was not a creation of any obvious needs of American industry, which was only a marginal force in the world economy in the mid-nineteenth century. The roots of management may be traced to a new educational system, introduced in 1817 to the United States Military Academy by the academy's fourth superintendent, Sylvanus Thayer. Thayer made two innovations. The first, borrowed from the Ecole Polytechnique in Paris, was to grade examinations by giving numerical marks. As I have previously noted, the grading of student papers originated in Cambridge University toward the end of the eighteenth century, and the practice was taken up by several schools on the Continent. Thayer's use of this technology is probably the first instance of it in America. As every teacher knows, the numerical mark changes the entire experience and meaning of learning. It introduces a fierce competition among students by providing sharply differentiated symbols of success and failure. Grading provides an \"objective\" measure of human performance and creates the unshakable illusion that accurate calculations can be made of worthiness. The human being becomes, to use Michel Foucault's phrase, \"a calculable person.\"\n\nThayer's second innovation, apparently his own invention, was a line-and-staff system. He divided the academy into two divisions, each organized hierarchically. As Hoskin and Macve describe it: \"Daily, weekly and monthly reports were required, all in writing. There were continual relays of written communication and command, going from the bottom to the top of each line, before being consolidated and passed to the central 'Staff Office.' \" Thayer rejected the traditional leader's role of direct, visible command. He ruled indirectly through the medium of written reports, charts, memos, personnel files, etc., not unlike the way a modern CEO functions.\n\nWe do not know how most of the two hundred cadets at the academy reacted to Thayer's new system (which Hoskin and Macve term the \"grammatocentric principle,\" meaning that everything was organized around the use of writing). But we do know that two of them, Daniel Tyler and George Whistler, were impressed. Both were in the graduating class of 1819, and took with them their lieutenant's rank and Thayer's general approach to organizations.\n\nDaniel Tyler, working at the Springfield Armory, did a time-and-motion study in 1832 (sixty years before Frederick Taylor's \"scientific management\" got under way) and established objectively based norms of production for every job in the armory. Workers were kept under surveillance, and their actual productivity was measured against the established productivity norms. Tyler also introduced quality control and inventory accounting. The result of all these methods was a dramatic increase in productivity and decrease in costs.\n\nMeanwhile, George Whistler (incidentally, the father of James Whistler and therefore the husband of \"Whistler's Mother\"), having become the chief engineer of the Western Railroad, developed a managerial system in 1839 that would have made Sylvanus Thayer proud. He organized the railroad along hierarchical lines, beginning with a central staff office, descending to regional managers and then local managers. He employed, to great effect, the grammatocentric principle, which he had no doubt learned well at the academy when serving in the staff office as cadet staff sergeant major.\n\nThe principles of calculability and grammatocentrism are, of course, the foundation of modern systems of management. Calculability led inevitably to such ideas as detailed accounting systems, inventory control, and productivity norms. Grammatocentrism promoted the idea that the best way to run a business is to know it through reports of those lower down the line. One manages, in other words, by the \"numbers\" and by being removed from the everyday realities of production.\n\nIt is worth saying that the basic structure of business management originated in nonbusiness contexts. Still, it did not take very long for American businesses to begin to adopt the principles of Thayer, Tyler, and Whistler, and by doing so they created what we now think of as a modern corporation. Indeed, management defines what we mean by a corporation, and has led John Kenneth Galbraith to remark in _The New Industrial State:_ \"More perhaps than machinery, massive and complex business organizations are the tangible manifestation of advanced technology.\"\n\nThere are two reasons why the case of management is instructive. First, as suggested by Galbraith, management, like the zero, statistics, IQ measurement, grading papers, or polling, functions as does any technology. It is not made up of mechanical parts, of course. It is made up of procedures and rules designed to standardize behavior. We may call any such system of procedures and rules a technique; and there is nothing to fear from techniques, unless, like so much of our machinery, they become autonomous. There's the rub. In a Technopoly, we tend to believe that only through the autonomy of techniques (and machinery) can we achieve our goals. This idea is all the more dangerous because no one can reasonably object to the rational use of techniques to achieve human purposes. Indeed, I am not disputing that the technique known as management may be the best way for modern business to conduct its affairs. We are technical creatures, and through our predilection for and our ability to create techniques we achieve high levels of clarity and efficiency. As I said earlier, language itself is a kind of technique\u2014an invisible technology\u2014and through it we achieve more than clarity and efficiency. We achieve humanity\u2014or inhumanity. The question with language, as with any other technique or machine, is and always has been, Who is to be the master? Will we control it, or will it control us? The argument, in short, is not with technique. The argument is with the triumph of technique, with techniques that become sanctified and rule out the possibilities of other ones. Technique, like any other technology, tends to function independently of the system it serves. It becomes autonomous, in the manner of a robot that no longer obeys its master.\n\nSecond, management is an important example of how an \"invisible technology\" works subversively but powerfully to create a new way of doing things, a classic instance of the tail wagging the dog. It is entirely possible for business and other institutions to operate without a highly technicalized management structure, however hard for us to imagine. We have grown so accustomed to it that we are near to believing management is an aspect of the natural order of things, just as students and teachers have come to believe that education would be impossible without the structure of a college \"course.\" And politicians believe they would be adrift without the assistance of public-opinion polling. When a method of doing things becomes so deeply associated with an institution that we no longer know which came first\u2014the method or the institution\u2014then it is difficult to change the institution or even to imagine alternative methods for achieving its purposes.\n\nAnd so it is necessary to understand where our techniques come from and what they are good for; we must make them visible so that they may be restored to our sovereignty. In the next chapter, I hope to do this with the intricate and vast ensemble of techniques I call Scientism.\n\n# **9**\n\n# Scientism\n\nOn December 5, 1989, Daniel Goleman, covering the social-science beat for _The New York Times_ , gave considerable space to some \"recent research findings\" that doubtless unsettled readers who hadn't been keeping informed about the work of our scientists of the mind: Goleman reported that psychological researchers have discovered that people fear death. This insight led them to formulate \"a sweeping theory,\" to quote Goleman, \"that gives the fear of death a central and often unsuspected role in psychological life.\" To whom death's role is unsuspected we were not told, but the theory is sufficiently rich to allow the hypothesis that all cultures (to quote Goleman again) \"prescribe what people should do to lead a 'good' and 'meaningful' life and offer some hope of immortality, as in the the [sic] Christian afterlife or the Hindu notion of reincarnation into a better life.\" (The repetition of the word \"the\" in the sentence quoted above may have been a typographical error\u2014or else perhaps an excited stammer in the face of such an astounding hypothesis.) As if this were not enough, Goleman also reported the same psychologists as having discovered that how one reacts to death depends on one's moral code, and that those who value open-mindedness are more tolerant of people whose values differ from theirs\u2014which means that those who are open-minded tend to be open-minded, a fact that is not sufficiently appreciated, if known at all.\n\nOn September 11, 1990, Goleman revealed the results of new research which suggests that Asian-American students do well in school because they come from intact families that value advanced academic degrees. And on October 2, 1990, he reported that psychologists have discovered that children who are inept at social relations tend to be unpopular with other children.\n\nI cite these reports from _The New York Times_ because it is considered by many to be the \"newspaper of public record\" and may be assumed to be reporting the _best_ of social science. It is possible, of course, that Goleman is a \"mole,\" or an undercover agent, who is trying to reveal where our culture stands by ridiculing the trivialities of social science. But I doubt it. He seems to believe in social science, as so many in Technopoly do. That is, he believes that the study of human behavior, when conducted according to the rigorous principles established by the physical and biological sciences, will produce objective facts, testable theories, and profound understandings of the human condition. Perhaps even universal laws.\n\nI have previously attributed the origins of this belief to the work of Auguste Comte, which is a defensible position but something of an oversimplification. In fact, the beginning formulations of a \"science of man\" are more precisely attributed to a school than to a man. The school, founded in 1794 in Paris, was called the Ecole Polytechnique (the same school that, as I mentioned earlier, quickly adopted the practice begun at Cambridge of assigning number grades to student work). The Ecole Polytechnique gathered for its teaching staff the best scientists, mathematicians, and engineers France had produced, and became famous for its enthusiasm for the methods of the natural sciences. Lavoisier and Amp\u00e8re taught there, as did, later, Volta and Alexander von Humboldt. Their work in chemistry and physics helped to lay the foundation of modern science, and in that respect the Ecole Polytechnique is justly honored. But there were others associated with the school whose exuberance for the methods of the natural sciences led them to believe that there were no limits to the powers of the human mind, and in particular no limits to the power of scientific research. The most famous expression of what may be called \"scientific hubris\" appeared in Pierre-Simon de Laplace's _Essai philosophique sur les probabilit\u00e9s_ , published in 1814. He wrote: \"A mind that in a given instance knew all the forces by which nature is animated and the position of all the bodies of which it is composed, if it were vast enough to include all these data within his analysis, could embrace in one single formula the movements of the largest bodies of the universe and of the smallest atoms; nothing would be uncertain for him; the future and the past would be equally before his eyes.\"\n\nThere is, of course, no scientist today who takes this view seriously, and there were few enough who did in the nineteenth century. But the spirit behind this scientific ideal inspired several men to believe that the reliable and predictable knowledge that could be obtained about stars and atoms could also be obtained about human behavior. Among the best known of these early \"social scientists\" were Claude-Henri de Saint-Simon, Prosper Enfantin, and, of course, Auguste Comte. They held in common two beliefs to which Technopoly is deeply indebted: that the natural sciences provide a method to unlock the secrets of both the human heart and the direction of social life; that society can be rationally and humanely reorganized according to principles that social science will uncover. It is with these men that the idea of \"social engineering\" begins and the seeds of Scientism are planted.\n\nBy Scientism, I mean three interrelated ideas that, taken together, stand as one of the pillars of Technopoly. Two of the three have just been cited. The first and indispensable idea is, as noted, that the methods of the natural sciences can be applied to the study of human behavior. This idea is the backbone of much of psychology and sociology as practiced at least in America, and largely accounts for the fact that social science, to quote F. A. Hayek, \"has contributed scarcely anything to our understanding of social phenomena.\"\n\nThe second idea is, as also noted, that social science generates specific principles which can be used to organize society on a rational and humane basis. This implies that technical means\u2014mostly \"invisible technologies\" supervised by experts\u2014can be designed to control human behavior and set it on the proper course.\n\nThe third idea is that faith in science can serve as a comprehensive belief system that gives meaning to life, as well as a sense of well-being, morality, and even immortality.\n\nI wish here to show how these ideas spiral into each other, and how they give energy and form to Technopoly.\n\nThe term \"science,\" as it is generally used today\u2014referring to the work of those in the physical, chemical, and biological disciplines\u2014was popularized in the early nineteenth century, with significant help from the formation of the British Association for the Advancement of Science in 1831 (although Murray's _New English Dictionary_ gives 1867 as the earliest use of the term in its modern sense). By the early twentieth century, the term had been appropriated by others, and it has since become increasingly familiar as a description of what psychologists, sociologists, and even anthropologists do. It will come as no surprise that I claim this is a deceptive and confusing use of the term, in part because it blurs the distinction between processes and practices.\n\nUsing definitions proposed by the British philosopher Michael Oakeshott, we may say that \"processes\" refers to those events that occur in nature, such as the orbiting of planets or the melting of ice or the production of chlorophyll in a leaf. Such processes have nothing to do with human intelligence, are governed by immutable laws, and are, so to say, determined by the structure of nature. If one were so inclined, one might even say that processes are the creation of God. By \"practices,\" on the other hand, Oakeshott means the creations of people\u2014those events that result from human decisions and actions, such as writing or reading this book or forming a new government or conversing at dinner or falling in love. These events are a function of human intelligence interacting with environment, and although there is surely a measure of regularity in human affairs, such affairs are not determined by natural laws, immutable or otherwise. In other words, there is an irrevocable difference between a blink and a wink. A blink can be classified as a process; it has physiological causes which can be understood and explained within the context of established postulates and theories. But a wink must be classified as a practice, filled with personal and to some extent unknowable meanings and, in any case, quite impossible to explain or predict in terms of causal relations.\n\nWhat we may call science, then, is the quest to find the immutable and universal laws that govern processes, presuming that there are cause-and-effect relations among these processes. It follows that the quest to understand human behavior and feeling can in no sense except the most trivial be called science. One can, of course, point to the fact that students of both natural law and human behavior often quantify their observations, and on this common ground classify them together. A fair analogy would be to argue that, since a housepainter and an artist both use paint, they are engaged in the same enterprise and to the same end.\n\nThe scientist uses mathematics to assist in uncovering and describing the structure of nature. At best, sociologists (to take one example) use quantification merely to give some precision to their ideas. But there is nothing especially scientific in that. All sorts of people count things in order to achieve precision without claiming they are scientists. Bail bondsmen count the number of murders committed in their cities; judges count the number of divorce actions in their jurisdictions; business executives count the amount of money spent in their stores; and young children like to count their toes and fingers in order not to be vague about how many they have. Information produced by counting may sometimes be valuable in helping a person get an idea, or, even more so, in providing support for an idea. But the mere activity of counting does not make science.\n\nNor does observing things, though it is sometimes said that if one is empirical, one is scientific. To be empirical means to look at things before drawing conclusions. Everyone, therefore, is an empiricist, with the possible exception of paranoid schizophrenics. To be empirical also means to offer evidence that others can see as clearly as you. You may, for example, conclude that I like to write books, offering as evidence that I have written this one and several others besides. You may also offer as evidence a tape recording, which I can supply on request, on which I tell you that I like to write books. Such evidence may be said to be empirical, and your conclusion empirically based. But you are not therefore acting as a scientist. You are acting as a rational person, to which condition many people who are not scientists may make a just claim.\n\nScientists do strive to be empirical and where possible precise, but it is also basic to their enterprise that they maintain a high degree of objectivity, which means that they study things independently of what people think or do about them. The opinions people hold about the external world are, to scientists, always an obstacle to be overcome, and it is well known that the scientist's picture of the external world is quite different from what most people believe the world to be like. Moreover, in their quest for objectivity, scientists proceed on the assumption that the objects they study are indifferent to the fact that they are being studied. Heisenberg's uncertainty principle indicates that at subatomic levels particles do \"know\" they are being studied, at least in a special meaning of \"knowing.\" An electron, for example, changes either its momentum or its position when it is being tracked\u2014i.e., when it interacts with a photon\u2014but the electron does not, in the usual sense of the word, \"know\" or \"care\" that the interaction is taking place. Nor do objects like leaves, apples, planets, kidneys, or bridges. This fact relieves the scientist of inquiring into their values and motivations and for this reason alone separates science from what is called social science, consigning the methodology of the latter (to quote Gunnar Myrdal) to the status of the \"metaphysical and pseudo-objective.\"\n\nThe status of social-science methods is further reduced by the fact that there are almost no experiments that will reveal a social-science theory to be false. Theories in social science disappear, apparently, because they are boring, not because they are refuted. But, as Karl Popper has demonstrated, science depends on the requirement that theories must be stated in a way that permits experiments to reveal that they are false. If a theory cannot be tested for its falsity, it is not a scientific theory\u2014as, for example, Freud's theory of the Oedipus complex. Psychiatrists can provide many examples supporting the validity of the theory, but they have no answer to the question \"What evidence would prove the theory false?\" Believers in the God theory (sometimes called Creation Science) are silent on the question \"What evidence would show that there is no God?\"\n\nI do not say, incidentally, that the Oedipus complex and God do not exist. Nor do I say that to believe in them is harmful\u2014far from it. I say only that, there being no tests that could, in principle, show them to be false, they fall outside the purview of science, as do almost all theories that make up the content of \"social science.\"\n\nI shall say in a few moments what I believe social science to be, as well as why Technopoly wishes to link it to the scientific enterprise. Here, I should like to give an example of social science to amplify the reasons why it is misleading to call it science.\n\nA piece of work that is greatly admired as social science, at least from a technical if not an ethical point of view, is the set of experiments (so called) supervised by Stanley Milgram, the account of which was published under the title _Obedience to Authority_. In this notorious study, Milgram sought to entice people to give electric shocks to \"innocent victims\" who were in fact conspirators in the experiment and did not actually receive the shocks. Nonetheless, most of Milgram's subjects _believed_ that the victims were receiving the shocks, and many of them, under psychological pressure, gave shocks that, had they been real, might have killed the victims. Milgram took great care in designing the environment in which all this took place, and his book is filled with statistics that indicate how many did or did not do what the experimenters told them to do. Somewhere in the neighborhood of 65 percent of his subjects were rather more compliant than would have been good for the health of their victims. Milgram drew the following conclusion from his research: In the face of what they construe to be legitimate authority, most people will do what they are told. Or, to put it another way, the social context in which people find themselves will be a controlling factor in how they behave.\n\nNow, in the first place, this conclusion is merely a commonplace of human experience, known by just about everyone from Maimonides to your aunt and uncle. The exceptions seem to be American psychiatrists. Before he conducted his experiment, Milgram sent a questionnaire to a large group of psychiatrists from whom he solicited opinions as to how many subjects would be likely to continue giving electric shocks when ordered to do so. The psychiatrists thought the number would be very much smaller than it actually was, basing their estimates on their knowledge of human behavior (which only recently has admitted the idea that people fear death). I do not mean to imply that real scientists never produce commonplaces, but only that it is rare, and never a cause for excitement. On the other hand, commonplace conclusions are almost always a characteristic of social research pretending to be science.\n\nIn the second place, Milgram's study was not empirical in the strict sense, since it was not based on observations of people in natural life situations. I assume that no one is especially interested in how people behave in a laboratory at Yale or any other place; what matters is how people behave in situations where their behavior makes a difference to their lives. But any conclusions that can be drawn from Milgram's study must specify that they apply only to people in laboratories under the conditions Milgram arranged. And even if we assume a correspondence between laboratory behavior and more lifelike situations, no predictions can be made about _what_ lifelike situations these might be. Nor can any serious claim be made that there is a causal relationship between the acceptance of legitimate authority and doing what you are told. In fact, Milgram himself shows us that there is not, since 35 percent of his subjects told the \"authority figure\" to bug off. Moreover, Milgram had no idea _why_ some people did and some people did not tell him to bug off. For myself, I feel quite sure that if each of Milgram's subjects had been required to read Hannah Arendt's _Eichmann in Jerusalem_ before showing up at the laboratory, his numbers would have been quite different.\n\nBut let us suppose that I am wrong about that, and let us further suppose that Milgram had found that 100 percent of his subjects did what they were told, with or without Hannah Arendt. And now let us suppose that I tell you a story of a group of people who in some real situation refused to comply with the orders of a legitimate authority\u2014let us say, the Danes who in the face of Nazi occupation helped nine thousand Jews escape to Sweden. Would you say to me that this cannot be so because Milgram's study proves otherwise? Or would you say that this overturns Milgram's work? Perhaps you would say that the Danish response is not relevant, since the Danes did not regard the Nazi occupation as constituting legitimate authority. But then, how would we explain the cooperative response to Nazi authority of the French, the Poles, and the Lithuanians? I think you would say none of these things, because Milgram's experiment does not confirm or falsify any theory that might be said to postulate a law of human nature. His study\u2014which, incidentally, I find both fascinating and terrifying\u2014is not science. It is something else entirely.\n\nWhich leads me to say what sort of work I think Milgram was engaged in\u2014and what sort of work those who study human behavior and situations are engaged in. I will start by making reference to a famous correspondence between Sigmund Freud and Albert Einstein. Freud once sent a copy of one of his books to Einstein, asking for his evaluation of it. Einstein replied that he thought the book exemplary but was not qualified to judge its scientific merit. To which Freud replied somewhat testily that, if Einstein could say nothing of its scientific merit, he, Freud, could not imagine how the book could be judged exemplary: it was science or it was nothing. Well, of course, Freud was wrong. His work _is_ exemplary\u2014indeed, monumental\u2014but scarcely anyone believes today that Freud was doing science, any more than educated people believe that Marx was doing science, or Max Weber or Lewis Mumford or Bruno Bettelheim or Carl Jung or Margaret Mead or Arnold Toynbee. What these people were doing\u2014and Stanley Milgram was doing\u2014is documenting the behavior and feelings of people as they confront problems posed by their culture. Their work is a form of storytelling. Science itself is, of course, a form of storytelling too, but its assumptions and procedures are so different from those of social research that it is extremely misleading to give the same name to each. In fact, the stories of social researchers are much closer in structure and purpose to what is called imaginative literature; that is to say, both a social researcher and a novelist give unique interpretations to a set of human events and support their interpretations with examples in various forms. Their interpretations cannot be proved or disproved but will draw their appeal from the power of their language, the depth of their explanations, the relevance of their examples, and the credibility of their themes. And all of this has, in both cases, an identifiable moral purpose. The words \"true\" and \"false\" do not apply here in the sense that they are used in mathematics or science. For there is nothing universally and irrevocably true or false about these interpretations. There are no critical tests to confirm or falsify them. There are no natural laws from which they are derived. They are bound by time, by situation, and above all by the cultural prejudices of the researcher or writer.\n\nA novelist\u2014for example, D. H. Lawrence\u2014tells a story about the sexual life of a woman\u2014Lady Chatterley\u2014and from it we may learn things about the secrets of some people, and wonder if Lady Chatterley's secrets are not more common than we had thought. Lawrence did not claim to be a scientist, but he looked carefully and deeply at the people he knew and concluded that there is more hypocrisy in heaven and earth than is dreamt of in some of our philosophies. Alfred Kinsey was also interested in the sexual lives of women, and so he and his assistants interviewed thousands of them in an effort to find out what they believed their sexual conduct was like. Each woman told her story, although it was a story carefully structured by Kinsey's questions. Some of them told everything they were permitted to tell, some only a little, and some probably lied. But when all their tales were put together, a collective story emerged about a certain time and place. It was a story more abstract than D. H. Lawrence's, largely told in the language of statistics and, of course, without much psychological insight. But it was a story nonetheless. One might call it a tribal tale of one thousand and one nights, told by a thousand and one women, and its theme was not much different from Lawrence's\u2014-namely, that the sexual life of some women is a lot stranger and more active than some other stories, particularly Freud's, had led us to believe.\n\nI do not say that there is no difference between Lawrence and Kinsey. Lawrence unfolds his story in a language structure called a narrative. Kinsey's language structure is called exposition. These forms are certainly different, although not so much as we might suppose. It has been remarked about the brothers Henry and William James that Henry was the novelist who wrote like a psychologist, and William the psychologist who wrote like a novelist. Certainly, in my meaning of the word \"story,\" exposition is as capable of unfolding one as is narrative. Of course, Lawrence's story is controlled entirely by the limits of his own imagination, and he is not obliged to consult any social facts other than those he believed he knew. His story is pure personal perception, and that is why we call it fiction. Kinsey's story comes from the mouths of others, and he is limited by what they answered when he asked his questions. Kinsey's story, therefore, we may call a documentary. But, like all stories, it is infused with moral prejudice and sociological theory. It is Kinsey who made up the questions, and chose who would be interviewed, the circumstances of the interview, and how the answers would be interpreted. All of this gives shape and point to his story. Indeed, we may assume that Kinsey, like Lawrence, knew from the outset what the theme of his story would be. Otherwise, he probably wouldn't have cared to tell it.\n\nBoth the novelist and the social researcher construct their stories by the use of archetypes and metaphors. Cervantes, for example, gave us the enduring archetype of the incurable dreamer and idealist in Don Quixote. The social historian Marx gave us the archetype of the ruthless and conspiring, though nameless, capitalist. Flaubert gave us the repressed bourgeois romantic in Emma Bovary. And Margaret Mead gave us the carefree, guiltless Samoan adolescent. Kafka gave us the alienated urbanit\u00e9 driven to self-loathing. And Max Weber gave us hardworking men driven by a mythology he called the Protestant Ethic. Dostoevsky gave us the egomaniac redeemed by love and religious fervor. And B. F. Skinner gave us the automaton redeemed by a benign technology.\n\nI think it justifiable to say that, in the nineteenth century, novelists provided us with most of the powerful metaphors and images of our culture. In the twentieth century, such metaphors and images have come largely from the pens of social historians and researchers. Think of John Dewey, William James, Erik Erikson, Alfred Kinsey, Thorstein Veblen, Margaret Mead, Lewis Mumford, B. F. Skinner, Carl Rogers, Marshall McLuhan, Barbara Tuchman, Noam Chomsky, Robert Coles, even Stanley Milgram, and you must acknowledge that our ideas of what we are like and what kind of country we live in come from their stories to a far greater extent than from the stories of our most renowned novelists.\n\nI do not mean, incidentally, that the metaphors of social research are created in the same way as those of novels and plays. The writer of fiction creates metaphors by an elaborate and concrete detailing of the actions and feelings of particular human beings. Sociology is background; individual psychology is the focus. The researcher tends to do it the other way around. The focus is on a wider field, and the individual life is seen in silhouette, by inference and suggestion. Also, the novelist proceeds by showing. The researcher, using abstract social facts, proceeds by reason, by logic, by argument. That is why fiction is apt to be more entertaining. Whereas Oscar Wilde or Evelyn Waugh _shows_ us the idle and conspicuously consuming rich, Thorstein Veblen _argues_ them into existence. In the character of Sammy Glick, Budd Schulberg _shows_ us the narcissist whose origins Christopher Lasch has tried to _explain_ through sociological analysis. So there are differences among storytellers, and most of the time our novelists are more pleasurable to read. But the stories told by our social researchers are at least as compelling and, in our own times, apparently more credible.\n\nWhy do such social researchers tell their stories? Essentially for didactic and moralistic purposes. These men and women tell their stories for the same reason the Buddha, Confucius, Hillel, and Jesus told their stories (and for the same reason D. H. Lawrence told his). It is true, of course, that social researchers rarely base their claims to knowledge on the indisputability of sacred texts, and even less so on revelation. But we must not be dazzled or deluded by differences in method between preachers and scholars. Without meaning to be blasphemous, I would say that Jesus was as keen a sociologist as Veblen. Indeed, Jesus' remark about rich men, camels, and the eye of a needle is as good a summary of Veblen's _Theory of the Leisure Class_ as it is possible to make. As social researchers, Jesus and Veblen differed in that Veblen was more garrulous.\n\nUnlike science, social research never discovers anything. It only rediscovers what people once were told and need to be told again. If, indeed, the price of civilization is repressed sexuality, it was not Sigmund Freud who discovered it. If the consciousness of people is formed by their material circumstances, it was not Marx who discovered it. If the medium is the message, it was not McLuhan who discovered it. They have merely retold ancient stories in a modern style. And these stories will be told anew decades and centuries from now, with, I imagine, less effect. For it would seem that Technopoly does not want these kinds of stories but facts\u2014hard facts, scientific facts. We might even say that in Technopoly precise knowledge is preferred to truthful knowledge but that in any case Technopoly wishes to solve, once and for all, the dilemma of subjectivity. In a culture in which the machine, with its impersonal and endlessly repeatable operations, is a controlling metaphor and considered to be the instrument of progress, subjectivity becomes profoundly unacceptable. Diversity, complexity, and ambiguity of human judgment are enemies of technique. They mock statistics and polls and standardized tests and bureaucracies. In Technopoly, it is not enough for social research to rediscover ancient truths or to comment on and criticize the moral behavior of people. In Technopoly, it is an insult to call someone a \"moralizer.\" Nor is it sufficient for social research to put forward metaphors, images, and ideas that can help people live with some measure of understanding and dignity. Such a program lacks the aura of certain knowledge that only science can provide. It becomes necessary, then, to transform psychology, sociology, and anthropology into \"sciences,\" in which humanity itself becomes an object, much like plants, planets, or ice cubes.\n\nThat is why the commonplaces that people fear death and that children who come from stable families valuing scholarship will do well in school must be announced as \"discoveries\" of scientific enterprise. In this way, social researchers can see themselves, and can be seen, as scientists, researchers without bias or values, unburdened by mere opinion. In this way, social policies can be claimed to rest on objectively determined facts. In Technopoly, it is not enough to argue that the segregation of blacks and whites in schools is immoral, and it is useless to offer _Black Boy_ or _Invisible Man_ or _The Fire Next Time_ as proof. The courts must be shown that standardized academic and psychological tests reveal that blacks do less well than whites and feel demeaned when segregation exists. In Technopoly, it is not enough to say it is immoral and degrading to allow people to be homeless. You cannot get anywhere by asking a judge, a politician, or a bureaucrat to read _Les Mis\u00e9rables_ or _Nana_ or, indeed, the New Testament. You must show that statistics have produced data revealing the homeless to be unhappy and to be a drain on the economy. Neither Dostoevsky nor Freud, Dickens nor Weber, Twain nor Marx, is now a dispenser of legitimate knowledge. They are interesting; they are \"worth reading\"; they are artifacts of our past. But as for \"truth,\" we must turn to \"science.\" Which brings me to the crux of what I mean by Scientism, and why it has emerged in Technopoly.\n\nI have tried to show that science, social research, and the kind of work we call imaginative literature are three quite different kinds of enterprise. In the end, _all_ of them are forms of storytelling\u2014human attempts to account for our experience in coherent ways. But they have different aims, ask different questions, follow different procedures, and give different meanings to \"truth.\" In most of these respects, social research has little in common with science, and much in common with other forms of imaginative literature. Yet social \"scientists\" have consistently sought to identify themselves, and in more than name, with physicists, chemists, biologists, and others who inquire into the lawful regularities of the natural world. Why students of the _human_ condition should do this is not hard to explain. The great successes of modern times\u2014indeed, perhaps the only successes\u2014have come in medicine, pharmacology, biochemistry, astrophysics, and all the feats of mechanical, biological, and electronic engineering made possible by the consistent application of the aims, assumptions, and procedures of natural science. These successes have attached to the name of science an awesome measure of authority, and to those who claim the title \"scientist\" a similar measure of respect and prestige. Beyond that lies the nineteenth-century hope that the assumptions and procedures of natural science might be applied without modification to the social world, to the same end of increased predictability and control, and with the same kind of engineering success. This hope has proved both misguided and illusory. But the illusion is a powerful one, and, given the psychological, social, and material benefits that attach to the label \"scientist,\" it is not hard to see why social researchers should find it hard to give it up.\n\nIt is less easy to see why the rest of us have so willingly, even eagerly, cooperated in perpetuating the same illusion. In part, the explanation lies in a profound misunderstanding of the aims of natural and of social studies, and of the differences between the physical and social worlds. But there is more to it than that. When the new technologies and techniques and spirit of men like Galileo, Newton, and Bacon laid the foundations of natural science, they also discredited the authority of earlier accounts of the physical world, as found, for example, in the great tale of Genesis. By calling into question the truth of such accounts in one realm, science undermined the whole edifice of belief in sacred stories and ultimately swept away with it the source to which most humans had looked for _moral_ authority. It is not too much to say, I think, that the desacralized world has been searching for an alternative source of moral authority ever since. So far as I know, no responsible natural scientist, either of the Renaissance or of recent times, has claimed that the procedures of natural science or its discoveries can tell us what we _ought_ to do\u2014whether some way of dealing with our fellow humans is good or evil, right or wrong. Indeed, the very principles of natural science, with its requirement of an objective stance toward what is studied, compel the natural scientist to abjure _in his or her role as a scientist_ such moral judgments or claims. When natural scientists speak out on moral questions, on what is good or evil to do, they speak as the rest of us\u2014as concerned citizens on a threatened planet, as rational women and men, as people of conscience who must struggle no less than you must, or I, to answer for themselves where the ultimate authority for their moral judgment lies. It is the world of desperate listeners, longing for a more powerful moral authority, that begs the natural scientist to say it is _the science_ that speaks, not the woman or man. But the scientist cannot with honor consent.\n\nOur social \"scientists\" have from the beginning been less tender of conscience, or less rigorous in their views of science, or perhaps just more confused about the questions their procedures can answer and those they cannot. In any case, they have not been squeamish about imputing to their \"discoveries\" and the rigor of their procedures the power to direct us in how we ought rightly to behave. That is why social \"scientists\" are so often to be found on our television screens, and on our bestseller lists, and in the \"self-help\" sections of airport bookstands: not because they can tell us how some humans sometimes behave but because they purport to tell us how we _should;_ not because they speak to us as fellow humans who have lived longer, or experienced more of human suffering, or thought more deeply and reasoned more carefully about some set of problems, but because they consent to maintain the illusion that it is their data, their procedures, their science, and not themselves, that speak. We welcome them gladly, and the claim explicitly made or implied, because we need so desperately to find some source outside the frail and shaky judgments of mortals like ourselves to authorize our moral decisions and behavior. And outside of the authority of brute force, which can scarcely be called moral, we seem to have little left but the authority of procedures.\n\nThis, then, is what I mean by Scientism. It is not merely the misapplication of techniques such as quantification to questions where numbers have nothing to say; not merely the confusion of the material and social realms of human experience; not merely the claim of social researchers to be applying the aims and procedures of natural science to the human world. Scientism is all of these, but something profoundly more. It is the desperate hope, and wish, and ultimately the illusory belief that some standardized set of procedures called \"science\" can provide us with an unimpeachable source of moral authority, a suprahuman basis for answers to questions like \"What is life, and when, and why?\" \"Why is death, and suffering?\" \"What is right and wrong to do?\" \"What are good and evil ends?\" \"How ought we to think and feel and behave?\" It is Scientism on a personal level when one says, as President Reagan did, that he personally believes that abortion is wrong but we must leave it to science to tell us when a fetus enters life. It is Scientism on a cultural level when no scientist rises to demur, when no newspaper prints a rebuttal on its \"science\" pages, when everyone cooperates, willfully or through ignorance, in the perpetuation of such an illusion. Science can tell us when a heart begins to beat, or movement begins, or what are the statistics on the survival of neonates of different gestational ages outside the womb. But science has no more authority than you do or I do to establish such criteria as the \"true\" definition of \"life\" or of human state or of personhood. Social research can tell us how some people behave in the presence of what they believe to be legitimate authority. But it cannot tell us when authority is \"legitimate\" and when not, or how we must decide, or when it may be right or wrong to obey. To ask of science, or expect of science, or accept unchallenged from science the answers to such questions is Scientism. And it is Technopoly's grand illusion.\n\nToward the end of his life, Sigmund Freud debated with himself what he called _The Future of an Illusion_. The illusion he referred to was the belief in a supranatural and suprahuman source of being, knowledge, and moral authority: the belief in God. The question Freud debated was not whether God exists, but whether humankind could survive without the illusion of God\u2014or, rather, whether humankind would fare better psychologically, culturally, and morally without that illusion than with it. Freud states his own doubts (expressed through the device of an alter ego with whom he debates) in the strongest possible voice, but in the end it is the voice of Freud's reason (or faith in reason) that \"wins\": humankind may or may not fare better, but it must do without the illusion of God. Freud did not see that, even as he wrote, his own work was lending substance to another illusion: the illusion of a future in which the procedures of natural and social science would ultimately reveal the \"real\" truth of human behavior and provide, through the agency of objectively neutral scientists, an empirical source of moral authority. Had he foreseen the peculiar transformation that the image of ultimate authority would take in our own time\u2014from an old man in a long white beard to young men and women in long white coats\u2014-Freud might have changed the question that was the focus of his inquiry. He could not. And so I will change it here, not to provide an answer, but in the hope of stirring renewed debate: as among the illusion of God, the illusion of Scientism, and no illusion or hope at all for an ultimate source of moral authority, which is most likely to serve the human interest, and which to prove most deadly, in the Age of Technopoly?\n\n# **10**\n\n# The Great Symbol Drain\n\nIt is possible that, some day soon, an advertising man who must create a television commercial for a new California Chardonnay will have the following inspiration: Jesus is standing alone in a desert oasis. A gentle breeze flutters the leaves of the stately palms behind him. Soft Mideastern music caresses the air. Jesus holds in his hand a bottle of wine at which he gazes adoringly. Turning toward the camera, he says, \"When I transformed water into wine at Cana, _this_ is what I had in mind. Try it today. You'll become a believer.\"\n\nIf you think such a commercial is not possible in your lifetime, then consider this: As I write, there is an oft-seen commercial for Hebrew National frankfurters. It features a dapper-looking Uncle Sam in his traditional red, white, and blue outfit. While Uncle Sam assumes appropriate facial expressions, a voice-over describes the delicious and healthful frankfurters produced by Hebrew National. Toward the end of the commercial, the voice stresses that Hebrew National frankfurters surpass federal standards for such products. Why? Because, the voice says as the camera shifts our point of view upward toward heaven, \"We have to answer to a Higher Authority.\"\n\nI will leave it to the reader to decide which is more incredible\u2014Jesus being used to sell wine or God being used to sell frankfurters. Whichever you decide, you must keep in mind that neither the hypothetical commercial nor the real one is an example of blasphemy. They are much worse than that. Blasphemy is, after all, among the highest tributes that can be paid to the power of a symbol. The blasphemer takes symbols as seriously as the idolater, which is why the President of the United States (circa 1991) wishes to punish, through a constitutional amendment, desecrators of the American flag.\n\nWhat we are talking about here is not blasphemy but trivialization, against which there can be no laws. In Technopoly, the trivialization of significant cultural symbols is largely conducted by commercial enterprise. This occurs not because corporate America is greedy but because the adoration of technology pre-empts the adoration of anything else. Symbols that draw their meaning from traditional religious or national contexts must therefore be made impotent as quickly as possible\u2014that is, drained of sacred or even serious connotations. The elevation of one god requires the demotion of another. \"Thou shalt have no other gods before me\" applies as well to a technological divinity as any other.\n\nThere are two intertwined reasons that make it possible to trivialize traditional symbols. The first, as neatly expressed by the social critic Jay Rosen, is that, although symbols, especially images, are endlessly repeatable, they are not inexhaustible. Second, the more frequently a significant symbol is used, the less potent is its meaning. This is a point stressed in Daniel Boorstin's classic book _The Image_ , published thirty years ago. In it, Boorstin describes the beginnings, in the mid-nineteenth century, of a \"graphics revolution\" that allowed the easy reproduction of visual images, thus providing the masses with continuous access to the symbols and icons of their culture. Through prints, lithographs, photographs, and, later, movies and television, religious and national symbols became commonplaces, breeding indifference if not necessarily contempt. As if to answer those who believe that the emotional impact of a sacred image is always and ever the same, Boorstin reminds us that prior to the graphics revolution most people saw relatively few images. Paintings of Jesus or the Madonna, for example, would have been seen rarely outside churches. Paintings of great national leaders could be seen only in the homes of the wealthy or in government buildings. There were images to be seen in books, but books were expensive and spent most of their time on shelves. Images were not a conspicuous part of the environment, and their scarcity contributed toward their special power. When the scale of accessibility was altered, Boorstin argues, the experience of encountering an image necessarily changed; that is to say, it diminished in importance. One picture, we are told, is worth a thousand words. But a thousand pictures, especially if they are of the same object, may not be worth anything at all.\n\nWhat Boorstin and Rosen direct our attention to is a common enough psychological principle. You may demonstrate this for yourself (if you have not at some time already done so) by saying any word, even a significant one, over and over again. Sooner than you expect, you will find that the word has been transformed into a meaningless sound, as repetition drains it of its symbolic value. Any male who has served in, let us say, the United States Army or spent time in a college dormitory has had this experience with what are called obscene words, especially the notorious four-letter word which I am loath to reproduce here. Words that you have been taught not to use and that normally evoke an embarrassed or disconcerted response, when used too often, are stripped of their power to shock, to embarrass, to call attention to a special frame of mind. They become only sounds, not symbols.\n\nMoreover, the journey to meaninglessness of symbols is a function not only of the frequency with which they are invoked but of the indiscriminate contexts in which they are used. An obscenity, for example, can do its work best when it is reserved for situations that call forth anger, disgust, or hatred. When it is used as an adjective for every third noun in a sentence, irrespective of the emotional context, it is deprived of its magical effects and, indeed, of its entire point. This is what happens when Abraham Lincoln's image, or George Washington's, is used to announce linen sales on Presidents' Day, or Martin Luther King's birthday celebration is taken as an occasion for furniture discounts. It is what happens when Uncle Sam, God, or Jesus is employed as an agent of the profane world for an essentially trivial purpose.\n\nAn argument is sometimes made that the promiscuous use of sacred or serious symbols by corporate America is a form of healthy irreverence. Irreverence, after all, is an antidote to excessive or artificial piety, and is especially necessary when piety is used as a political weapon. One might say that irreverence, not blasphemy, is the ultimate answer to idolatry, which is why most cultures have established means by which irreverence may be expressed\u2014in the theater, in jokes, in song, in political rhetoric, even in holidays. The Jews, for example, use Purim as one day of the year on which they may turn a laughing face on piety itself.\n\nBut there is nothing in the commercial exploitation of traditional symbols that suggests an excess of piety is itself a vice. Business is too serious a business for that, and in any case has no objection to piety, as long as it is directed toward the idea of consumption, which is never treated as a laughing matter. In using Uncle Sam or the flag or the American Eagle or images of presidents, in employing such names as Liberty Insurance, Freedom Transmission Repair, and Lincoln Savings and Loan, business does not offer us examples of irreverence. It is merely declaring the irrelevance, in Technopoly, of distinguishing between the sacred and the profane.\n\nI am not here making a standard-brand critique of the excesses of capitalism. It is entirely possible to have a market economy that respects the seriousness of words and icons, and which disallows their use in trivial or silly contexts. In fact, during the period of greatest industrial growth in America\u2014from roughly 1830 to the end of the nineteenth century\u2014advertising did not play a major role in the economy, and such advertising as existed used straightforward language, without recourse to the exploitation of important cultural symbols. There was no such thing as an \"advertising industry\" until the early twentieth century, the ground being prepared for it by the Postal Act of March 3, 1879, which gave magazines low-cost mailing privileges. As a consequence, magazines emerged as the best available conduits for national advertising, and merchants used the opportunity to make the names of their companies important symbols of commercial excellence. When George Eastman invented the portable camera in 1888, he spent $25,000 advertising it in magazines. By 1895, \"Kodak\" and \"camera\" were synonymous, as to some extent they still are. Companies like Royal Baking Powder, Baker's Chocolate, Ivory Soap, and Gillette moved into a national market by advertising their products in magazines. Even magazines moved into a national market by advertising themselves in magazines, the most conspicuous example being _Ladies' Home Journal_ , whose publisher, Cyrus H. K. Curtis, spent half a million dollars between 1883 and 1888 advertising his magazine in other magazines. By 1909, _Ladies' Home Journal_ had a circulation of more than a million readers.\n\nCurtis' enthusiasm for advertising notwithstanding, the most significant figure in mating advertising to the magazine was Frank Munsey, who upon his death in 1925 was eulogized by William Allen White with the following words: \"Frank Munsey contributed to the journalism of his day the talent of a meat packer, the morals of a money changer and the manners of an undertaker. He and his kind have about succeeded in transforming a once-noble profession into an 8% security. May he rest in trust.\" What was the sin of the malevolent Munsey? Simply, he made two discoveries. First, a large circulation could be achieved by selling a magazine for much less than it cost to produce it; second, huge profits could be made from the high volume of advertising that a large circulation would attract. In October 1893, Munsey took out an ad in the New York _Sun_ announcing that _Munsey's Magazine_ was cutting its price from 23 cents to 10 cents, and reducing a year's subscription from $3 to $1. The first 10-cent issue claimed a circulation of forty thousand; within four months, the circulation rose to two hundred thousand; two months later, it was five hundred thousand.\n\nMunsey cannot, however, be blamed for another discovery, which for convenience's sake we may attribute to Procter and Gamble: that advertising is most effective when it is irrational. By irrational, I do not, of course, mean crazy. I mean that products could best be sold by exploiting the magical and even poetical powers of language and pictures. In 1892, Procter and Gamble invited the public to submit rhymes to advertise Ivory Soap. Four years later, H-O employed, for the first time, a picture of a baby in a high chair, the bowl of H-O cereal before him, his spoon in hand, his face ecstatic. By the turn of the century, advertisers no longer assumed that reason was the best instrument for the communication of commercial products and ideas. Advertising became one part depth psychology, one part aesthetic theory. In the process, a fundamental principle of capitalist ideology was rejected: namely, that the producer and consumer were engaged in a rational enterprise in which consumers made choices on the basis of a careful consideration of the quality of a product and their own self-interest. This, at least, is what Adam Smith had in mind. But today, the television commercial, for example, is rarely about the character of the products. It is about the character of the consumers of products. Images of movie stars and famous athletes, of serene lakes and macho fishing trips, of elegant dinners and romantic interludes, of happy families packing their station wagons for a picnic in the country\u2014these tell nothing about the products being sold. But they tell everything about the fears, fancies, and dreams of those who might buy them. What the advertiser needs to know is not what is right about the product but what is wrong about the buyer. And so the balance of business expenditures shifts from product research to market research, which means orienting business away from making products of value and toward making consumers feel valuable. The business of business becomes pseudo-therapy; the consumer, a patient reassured by psychodramas.\n\nWhat this means is that somewhere near the core of Technopoly is a vast industry with license to use all available symbols to further the interests of commerce, by devouring the psyches of consumers. Although estimates vary, a conservative guess is that the average American will have seen close to two million television commercials by age sixty-five. If we add to this the number of radio commercials, newspaper and magazine ads, and billboards, the extent of symbol overload and therefore symbol drain is unprecedented in human history. Of course, not all the images and words used have been cannibalized from serious or sacred contexts, and one must admit that as things stand at the moment it is quite unthinkable for the image of Jesus to be used to sell wine. At least not a chardonnay. On the other hand, his birthday is used as an occasion for commerce to exhaust nearly the entire repertoire of Christian symbology. The constraints are so few that we may call this a form of cultural rape, sanctioned by an ideology that gives boundless supremacy to technological progress and is indifferent to the unraveling of tradition.\n\nIn putting it this way, I mean to say that mass advertising is not the cause of the great symbol drain. Such cultural abuse could not have occurred without technologies to make it possible and a world-view to make it desirable. In the institutional form it has taken in the United States, advertising is a symptom of a world-view that sees tradition as an obstacle to its claims. There can, of course, be no functioning sense of tradition without a measure of respect for symbols. Tradition is, in fact, nothing but the acknowledgment of the authority of symbols and the relevance of the narratives that gave birth to them. With the erosion of symbols there follows a loss of narrative, which is one of the most debilitating consequences of Technopoly's power.\n\nWe may take as an example the field of education. In Technopoly, we improve the education of our youth by improving what are called \"learning technologies.\" At the moment, it is considered necessary to introduce computers to the classroom, as it once was thought necessary to bring closed-circuit television and film to the classroom. To the question \"Why should we do this?\" the answer is: \"To make learning more efficient and more interesting.\" Such an answer is considered entirely adequate, since in Technopoly efficiency and interest need no justification. It is, therefore, usually not noticed that this answer does not address the question \"What is learning for?\" \"Efficiency and interest\" is a technical answer, an answer about means, not ends; and it offers no pathway to a consideration of educational philosophy. Indeed, it blocks the way to such a consideration by beginning with the question of how we should proceed rather than with the question of why. It is probably not necessary to say that, by definition, there can be no education philosophy that does not address what learning is for. Confucius, Plato, Quintilian, Cicero, Comenius, Erasmus, Locke, Rousseau, Jefferson, Russell, Montessori, Whitehead, and Dewey\u2014each believed that there was some transcendent political, spiritual, or social idea that must be advanced through education. Confucius advocated teaching \"the Way\" because in tradition he saw the best hope for social order. As our first systematic fascist, Plato wished education to produce philosopher kings. Cicero argued that education must free the student from the tyranny of the present. Jefferson thought the purpose of education is to teach the young how to protect their liberties. Rousseau wished education to free the young from the unnatural constraints of a wicked and arbitrary social order. And among John Dewey's aims was to help the student function without certainty in a world of constant change and puzzling ambiguities.\n\nOnly in knowing something of the reasons why they advocated education can we make sense of the means they suggest. But to understand their reasons we must also understand the narratives that governed their view of the world. By narrative, I mean a story of human history that gives meaning to the past, explains the present, and provides guidance for the future. It is a story whose principles help a culture to organize its institutions, to develop ideals, and to find authority for its actions. At the risk of repetition, I must point out again that the source of the world's greatest narratives has been religion, as found, for example, in Genesis or the Bhagavad-Gita or the Koran. There are those who believe\u2014as did the great historian Arnold Toynbee\u2014that without a comprehensive religious narrative at its center a culture must decline. Perhaps. There are, after all, other sources\u2014mythology, politics, philosophy, and science, for example\u2014but it is certain that no culture can flourish without narratives of transcendent origin and power.\n\nThis does not mean that the mere existence of such a narrative ensures a culture's stability and strength. There are destructive narratives. A narrative provides meaning, not necessarily survival\u2014as, for example, the story provided by Adolf Hitler to the German nation in the 1930s. Drawing on sources in Teutonic mythology and resurrecting ancient and primitive symbolism, Hitler wove a tale of Aryan supremacy that lifted German spirits, gave point to their labors, eased their distress, and provided explicit ideals. The story glorified the past, elucidated the present, and foretold the future, which was to last a thousand years. The Third Reich lasted exactly twelve years.\n\nIt is not to my point to dwell on the reasons why the story of Aryan supremacy could not endure. The point is that cultures must have narratives and will find them where they will, even if they lead to catastrophe. The alternative is to live without meaning, the ultimate negation of life itself. It is also to the point to say that each narrative is given its form and its emotional texture through a cluster of symbols that call for respect and allegiance, even devotion. The United States Constitution, for example, is only in part a legal document, and, I should add, a small part. Democratic nations\u2014England, for one\u2014do not require a written constitution to ensure legal order and the protection of liberties. The importance of the American Constitution is largely in its function as a symbol of the story of our origins. It is our political equivalent of Genesis. To mock it, to ignore it, to circumvent it is to declare the irrelevance of the story of the United States as a moral light unto the world. In like fashion, the Statue of Liberty is the key symbol of the story of America as the natural home of the teeming masses, from anywhere, yearning to be free. There are, of course, several reasons why such stories lose their force. This book is, in fact, an attempt to describe one of them\u2014i.e., how the growth of Technopoly has overwhelmed earlier, more meaningful stories. But in all cases, the trivialization of the symbols that express, support, and dramatize the story will accompany the decline. Symbol drain is both a symptom and a cause of a loss of narrative.\n\nThe educators I referred to above based their philosophies on narratives rich in symbols which they respected and which they understood to be integral to the stories they wanted education to reveal. It is, therefore, time to ask, What story does American education wish to tell now? In a growing Technopoly, what do we believe education is for? The answers are discouraging, and one of them can be inferred from any television commercial urging the young to stay in school. The commercial will either imply or state explicitly that education will help the persevering student to get a good job. And that's it. Well, not quite. There is also the idea that we educate ourselves to compete with the Japanese or the Germans in an economic struggle to be number one. Neither of these purposes is, to say the least, grand or inspiring. The story each suggests is that the United States is not a culture but merely an economy, which is the last refuge of an exhausted philosophy of education. This belief, I might add, is precisely reflected in the President's Commission Report, _A Nation at Risk_ , where you will find a definitive expression of the idea that education is an instrument of economic policy and of very little else.\n\nWe may get a sense of the desperation of the educator's search for a more gripping story by using the \"television commercial test.\" Try to imagine what sort of appeals might be effectively made on a TV commercial to persuade parents to support schools. (Let us, to be fair, sidestep appeals that might be made directly to students themselves, since the youth of any era are disinclined to think schooling a good idea, whatever the reasons advanced for it. See the \"Seven Ages of Man\" passage in _As You Like It.)_\n\nCan you imagine, for example, what such a commercial would be like if Jefferson or John Dewey prepared it? \"Your children are citizens in a democratic society,\" the commercial might say. \"Their education will teach them how to be valuable citizens by refining their capacity for reasoned thought and strengthening their will to protect their liberties. As for their jobs and professions, that will be considered only at a 'late and convenient hour' \" (to quote John Stuart Mill, who would be pleased to associate himself with Jefferson's or Dewey's purpose). Is there anyone today who would find this a compelling motivation? Some, perhaps, but hardly enough to use it as the basis of a national program. John Locke's commercial would, I imagine, be even less appealing. \"Your children must stay in school,\" he might say, \"because there they will learn to make their bodies slaves of their minds. They will learn to control their impulses, and how to find satisfaction and even excitement in the life of the mind. Unless they accomplish this, they can be neither civilized nor literate.\" How many would applaud this mission? Indeed, whom could we use to speak such words\u2014Barbara Bush? Lee Iacocca? Donald Trump? Even the estimable Dr. Bill Cosby would hardly be convincing. The guffaws would resound from Maine to California.\n\nIn recent years, a valiant attempt has been made by some\u2014for example, E. D. Hirsch, Jr.\u2014to provide a comprehensive purpose to education. In his book _Cultural Literacy_ , Hirsch defines literacy as the capacity to understand and use the words, dates, aphorisms, and names that form the basis of communication among the educated in our culture. Toward this end, he and some of his colleagues compiled a list that contains, according to them, the references essential to a culturally literate person in America. The first edition of the book (1987) included Norman Mailer but not Philip Roth, Bernard Malamud, Arthur Miller, or Tennessee Williams. It included Ginger Rogers but not Richard Rodgers, Carl Rogers, or Buck Rogers, let alone Fred Rogers. The second greatest home-run hitter of all time, Babe Ruth, was there, but not the greatest home-run hitter, Hank Aaron. The Marx Brothers were there, but not Orson Welles, Frank Capra, John Ford, or Steven Spielberg. Sarah Bernhardt was included, but not Leonard Bernstein. Rochester, New York, was on the list. Trenton, New Jersey, one of our most historic cities, was not. Hirsch included the Battle of the Bulge, which pleased my brother, who fought in it in 1944. But my uncle who died in the Battle of the Coral Sea, in 1942, might have been disappointed to find that it didn't make the list.\n\nTo fill in the gaps, Hirsch has had to enlarge his list, so that there now exists a _Cultural Literacy Encyclopedia_. We may be sure that Hirsch will continue to expand his list until he reaches a point where a one-sentence directive will be all he needs to publish: \"See the _Encyclopedia Americana_ and _Webster's Third International.\"_\n\nIt is, of course, an expected outcome of any education that students become acquainted with the important references of their culture. Even Rousseau, who would have asked his students to read only one book, _Robinson Crusoe_ (so that they would learn how to survive in the wild), would probably have expected them to \"pick up\" the names and sayings and dates that made up the content of the educated conversation of their time. Nonetheless, Hirsch's proposal is inadequate for two reasons that reflect the inadequacies of Technopoly. The first, which I have discussed in chapter four, \"The Improbable World,\" is that the present condition of technology-generated information is so long, varied, and dynamic that it is not possible to organize it into a coherent educational program. How do you include in the curriculum Rochester, New York, or Sarah Bernhardt or Babe Ruth? Or the Marx Brothers? Where does Ginger Rogers go? Does she get included in the syllabus under a unit titled \"Fred Astaire's Dancing Partners\"? (In which case, we must include Cyd Charisse and, if I am not mistaken, Winston Churchill's daughter, Sarah.) Hirsch's encyclopedic list is not a solution but a description of the problem of information glut. It is therefore essentially incoherent. But it also confuses a consequence of education with a purpose. Hirsch attempted to answer the question \"What is an educated person?\" He left unanswered the question \"What is an education for?\" Young men, for example, will learn how to make lay-up shots when they play basketball. To be able to make them is part of the definition of what good players are. But they do not play basketball for that purpose. There is usually a broader, deeper, and more meaningful reason for wanting to play\u2014to assert their manhood, to please their fathers, to be acceptable to their peers, even for the sheer aesthetic pleasure of the game itself. What you have to do to be a success must be addressed only after you have found a reason to be successful. In Technopoly, this is very hard to do, and Hirsch simply sidestepped the question.\n\nNot so Allan Bloom. In his book _The Closing of the American Mind_ , he confronts the question by making a serious complaint against the academy. His complaint is that most American professors have lost their nerve. They have become moral relativists, incapable of providing their students with a clear understanding of what is right thought and proper behavior. Moreover, they are also intellectual relativists, refusing to defend their own culture and no longer committed to preserving and transmitting the best that has been thought and said.\n\nBloom's solution is that we go back to the basics of Western thought. He does not care if students know who Ginger Rogers and Groucho Marx are. He wants us to teach our students what Plato, Aristotle, Cicero, Saint Augustine, and other luminaries have had to say on the great ethical and epistemologica! questions. He believes that by acquainting themselves with great books our students will acquire a moral and intellectual foundation that will give meaning and texture to their lives. Though there is nothing especially original in this, Bloom is a serious education philosopher, which is to say, unlike Hirsch, he is a moralist who understands that Technopoly is a malevolent force requiring opposition. But he has not found many supporters.\n\nThose who reject Bloom's idea have offered several arguments against it. The first is that such a purpose for education is elitist: the mass of students would not find the great story of Western civilization inspiring, are too deeply alienated from the past to find it so, and would therefore have difficulty connecting the \"best that has been thought and said\" to their own struggles to find meaning in their lives. A second argument, coming from what is called a \"leftist\" perspective, is even more discouraging. In a sense, it offers a definition of what is meant by elitism. It asserts that the \"story of Western civilization\" is a partial, biased, and even oppressive one. It is not the story of blacks, American Indians, Hispanics, women, homosexuals\u2014of any people who are not white heterosexual males of Judeo-Christian heritage. This claim denies that there is or can be a national culture, a narrative of organizing power and inspiring symbols which all citizens can identify with and draw sustenance from. If this is true, it means nothing less than that our national symbols have been drained of their power to unite, and that education must become a tribal affair; that is, each subculture must find its own story and symbols, and use them as the moral basis of education.\n\nStanding somewhat apart from these arguments are, of course, religious educators, such as those in Catholic schools, who strive to maintain another traditional view\u2014that learning is done for the greater glory of God and, more particularly, to prepare the young to embrace intelligently and gracefully the moral directives of the church. Whether or not such a purpose can be achieved in Technopoly is questionable, as many religious educators will acknowledge.\n\nI will reserve for the next and final chapter my own view of the struggle to find a purpose for education in Technopoly. But here it must be said that the struggle itself is a sign that our repertoire of significant national, religious, and mythological symbols has been seriously drained of its potency. \"We are living at a time,\" Irving Howe has written, \"when all the once regnant world systems that have sustained (also distorted) Western intellectual life, from theologies to ideologies, are taken to be in severe collapse. This leads to a mood of skepticism, an agnosticism of judgment, sometimes a world-weary nihilism in which even the most conventional minds begin to question both distinctions of value and the value of distinctions.\"\n\nInto this void comes the Technopoly story, with its emphasis on progress without limits, rights without responsibilities, and technology without cost. The Technopoly story is without a moral center. It puts in its place efficiency, interest, and economic advance. It promises heaven on earth through the conveniences of technological progress. It casts aside all traditional narratives and symbols that suggest stability and orderliness, and tells, instead, of a life of skills, technical expertise, and the ecstasy of consumption. Its purpose is to produce functionaries for an ongoing Technopoly. It answers Bloom by saying that the story of Western civilization is irrelevant; it answers the political left by saying there is indeed a common culture whose name is Technopoly and whose key symbol is now the computer, toward which there must be neither irreverence nor blasphemy. It even answers Hirsch by saying that there are items on his list that, if thought about too deeply and taken too seriously, will interfere with the progress of technology.\n\nI grant that it is somewhat unfair to expect educators, by themselves, to locate stories that would reaffirm our national culture. Such narratives must come to them, to some degree, from the political sphere. If our politics is symbolically impoverished, it is difficult to imagine how teachers can provide a weighty purpose to education. I am writing this chapter during the fourth week of the war against Iraq; the rhetoric accompanying the onset of the war is still fresh in mind. It began with the President's calling Americans to arms for the sake of their \"life-style.\" This was followed by the Secretary of State's request that they fight to protect their jobs. Then came the appeal\u2014at a late and convenient hour, as it were\u2014to thwart the \"naked aggression\" of a little \"Hitler.\" I do not say here that going to war was unjustified. My point is that, with the Cold War at an end, our political leaders now struggle, as never before, to find a vital narrative and accompanying symbols that would awaken a national spirit and a sense of resolve. The citizens themselves struggle as well. Having drained many of their traditional symbols of serious meaning, they resort, somewhat pitifully, to sporting yellow ribbons as a means of symbolizing their fealty to a cause. After the war, the yellow ribbons will fade from sight, but the question of who we are and what we represent will remain. Is it possible that the only symbol left to use will be an F-15 fighter plane guided by an advanced computer system?\n\n# **11**\n\n# The Loving Resistance Fighter\n\nAnyone who practices the art of cultural criticism must endure being asked, What is the solution to the problems you describe? Critics almost never appreciate this question, since, in most cases, they are entirely satisfied with themselves for having posed the problems and, in any event, are rarely skilled in formulating practical suggestions about anything. This is why they became cultural critics.\n\nThe question comes forth nonetheless, and in three different voices. One is gentle and eager, as if to suggest that the critic knows the solutions but has merely forgotten to include them in the work itself. A second is threatening and judgmental, as if to suggest that the critic had no business bothering people in the first place unless there were some pretty good solutions at hand. And a third is wishful and encouraging, as if to suggest that it is well known that there are not always solutions to serious problems but if the critic will give it a little thought perhaps something constructive might come from the effort.\n\nIt is to this last way of posing the question that I should like to respond. I have indeed given the matter some thought, and this chapter is the result. Its simplicity will tell the reader that I am, like most other critics, armed less with solutions than with problems.\n\nAs I see it, a reasonable response (hardly a solution) to the problem of living in a developing Technopoly can be divided into two parts: what the individual can do irrespective of what the culture is doing; and what the culture can do irrespective of what any individual is doing. Beginning with the matter of individual response, I must say at once that I have no intention of providing a \"how to\" list in the manner of the \"experts\" I ridiculed in chapter five, on our \"broken defenses.\" No one is an expert on how to live a life. I can, however, offer a Talmudic-like principle that seems to me an effective guide for those who wish to defend themselves against the worst effects of the American Technopoly. It is this: You must try to be a loving resistance fighter. That is the doctrine, as Hillel might say. Here is the commentary: By \"loving,\" I mean that, in spite of the confusion, errors, and stupidities you see around you, you must always keep close to your heart the narratives and symbols that once made the United States the hope of the world and that may yet have enough vitality to do so again. You may find it helpful to remember that, when the Chinese students at Tiananmen Square gave expression to their impulse to democracy, they fashioned a papier-m\u00e2ch\u00e9 model, for the whole world to see, of the Statue of Liberty. Not a statue of Karl Marx, not the Eiffel Tower, not Buckingham Palace. The Statue of Liberty. It is impossible to say how moved Americans were by this event. But one is compelled to ask, Is there an American soul so dead that it could not generate a murmur (if not a cheer) of satisfaction for this use of a once-resonant symbol? Is there an American soul so shrouded in the cynicism and malaise created by Technopoly's emptiness that it failed to be stirred by students reading aloud from the works of Thomas Jefferson in the streets of Prague in 1989? Americans may forget, but others do not, that American dissent and protest during the Vietnam War may be the only case in history where public opinion forced a government to change its foreign policy. Americans may forget, but others do not, that Americans invented the idea of public education for all citizens and have never abandoned it. And everyone knows, including Americans, that each day, to this hour, immigrants still come to America in hopes of finding relief from one kind of deprivation or another.\n\nThere are a hundred other things to remember that may help one to warm to the United States, including the fact that it has been, and perhaps always will be, a series of experiments that the world watches with wonder. Three such experiments are of particular importance. The first, undertaken toward the end of the eighteenth century, posed the question, Can a nation allow the greatest possible degree of political and religious freedom and still retain a sense of identity and purpose? Toward the middle of the nineteenth century, a second great experiment was undertaken, posing the question, Can a nation retain a sense of cohesion and community by allowing into it people from all over the world? And now comes the third\u2014the great experiment of Technopoly\u2014which poses the question, Can a nation preserve its history, originality, and humanity by submitting itself totally to the sovereignty of a technological thought-world?\n\nObviously, I do not think the answer to this question will be as satisfactory as the answers to the first two. But if there is an awareness of and resistance to the dangers of Technopoly, there is reason to hope that the United States may yet survive its Ozymandias-like hubris and technological promiscuity. Which brings me to the \"resistance fighter\" part of my principle. Those who resist the American Technopoly are people\n\nwho pay no attention to a poll unless they know what questions were asked, and why;\n\nwho refuse to accept efficiency as the pre-eminent goal of human relations;\n\nwho have freed themselves from the belief in the magical powers of numbers, do not regard calculation as an adequate substitute for judgment, or precision as a synonym for truth;\n\nwho refuse to allow psychology or any \"social science\" to pre-empt the language and thought of common sense;\n\nwho are, at least, suspicious of the idea of progress, and who do not confuse information with understanding;\n\nwho do not regard the aged as irrelevant;\n\nwho take seriously the meaning of family loyalty and honor, and who, when they \"reach out and touch someone,\" expect that person to be in the same room;\n\nwho take the great narratives of religion seriously and who do not believe that science is the only system of thought capable of producing truth;\n\nwho know the difference between the sacred and the profane, and who do not wink at tradition for modernity's sake;\n\nwho admire technological ingenuity but do not think it represents the highest possible form of human achievement.\n\nA resistance fighter understands that technology must never be accepted as part of the natural order of things, that every technology\u2014from an IQ test to an automobile to a television set to a computer\u2014is a product of a particular economic and political context and carries with it a program, an agenda, and a philosophy that may or may not be life-enhancing and that therefore require scrutiny, criticism, and control. In short, a technological resistance fighter maintains an epistemological and psychic distance from any technology, so that it always appears somewhat strange, never inevitable, never natural.\n\nI can say no more than this, for each person must decide how to enact these ideas. But it is possible that one's education may help considerably not only in promoting the general conception of a resistance fighter but in helping the young to fashion their own ways of giving it expression. It is with education, then, that I will conclude this book. This is not to say that political action and social policy aren't useful in offering opposition to Technopoly. There are even now signs that Technopoly is understood as a problem to which laws and policies might serve as a response\u2014in the environmental movement, in the contemplation of legal restrictions on computer technology, in a developing distrust of medical technology, in reactions against widespread testing, in various efforts to restore a sense of community cohesion. But in the United States, as Lawrence Cremin once remarked, whenever we need a revolution, we get a new curriculum. And so I shall propose one. I have done this before to something less than widespread acclamation. But it is the best way I can think of for the culture to address the problem. School, to be sure, is a technology itself, but of a special kind in that, unlike most technologies, it is customarily and persistently scrutinized, criticized, and modified. It is America's principal instrument for correcting mistakes and for addressing problems that mystify and paralyze other social institutions.\n\nIn consideration of the disintegrative power of Technopoly, perhaps the most important contribution schools can make to the education of our youth is to give them a sense of coherence in their studies, a sense of purpose, meaning, and interconnectedness in what they learn. Modern secular education is failing not because it doesn't teach who Ginger Rogers, Norman Mailer, and a thousand other people are but because it has no moral, social, or intellectual center. There is no set of ideas or attitudes that permeates all parts of the curriculum. The curriculum is not, in fact, a \"course of study\" at all but a meaningless hodgepodge of subjects. It does not even put forward a clear vision of what constitutes an educated person, unless it is a person who possesses \"skills.\" In other words, a technocrat's ideal\u2014a person with no commitment and no point of view but with plenty of marketable skills.\n\nOf course, we must not overestimate the capability of schools to provide coherence in the face of a culture in which almost all coherence seems to have disappeared. In our technicalized, present-centered information environment, it is not easy to locate a rationale for education, let alone impart one convincingly. It is obvious, for example, that the schools cannot restore religion to the center of the life of learning. With the exception of a few people, perhaps, no one would take seriously the idea that learning is for the greater glory of God. It is equally obvious that the knowledge explosion has blown apart the feasibility of such limited but coordinated curriculums as, for example, a Great Books program. Some people would have us stress love of country as a unifying principle in education. Experience has shown, however, that this invariably translates into love of government, and in practice becomes indistinguishable from what still is at the center of Soviet or Chinese education.\n\nSome would put forward \"emotional health\" as the core of the curriculum. I refer here to a point of view sometimes called Rogerian, sometimes Maslovian, which values above all else the development of one's emotional life through the quest for one's \"real self.\" Such an idea, of course, renders a curriculum irrelevant, since only \"self-knowledge\"\u2014i.e., one's feelings\u2014is considered worthwhile. Carl Rogers himself once wrote that anything that can be taught is probably either trivial or harmful, thus making any discussion of the schools unnecessary. But beyond this, the culture is already so heavy with the burden of the glorification of \"self\" that it would be redundant to have the schools stress it, even if it were possible.\n\nOne obviously treads on shaky ground in suggesting a plausible theme for a diverse, secularized population. Nonetheless, with all due apprehension, I would propose as a possibility the theme that animates Jacob Bronowski's _The Ascent of Man_. It is a book, and a philosophy, filled with optimism and suffused with the transcendent belief that humanity's destiny is the discovery of knowledge. Moreover, although Bronowski's emphasis is on science, he finds ample warrant to include the arts and humanities as part of our unending quest to gain a unified understanding of nature and our place in it.\n\nThus, to chart the ascent of man, which I will here call \"the ascent of humanity,\" we must join art and science. But we must also join the past and the present, for the ascent of humanity is above all a continuous story. It is, in fact, a story of creation, although not quite the one that the fundamentalists fight so fiercely to defend. It is the story of humanity's creativeness in trying to conquer loneliness, ignorance, and disorder. And it certainly includes the development of various religious systems as a means of giving order and meaning to existence. In this context, it is inspiring to note that the Biblical version of creation, to the astonishment of everyone except possibly the fundamentalists, has turned out to be a near-perfect blend of artistic imagination and scientific intuition: the Big Bang theory of the creation of the universe, now widely accepted by cosmologists, confirms in essential details what the Bible proposes as having been the case \"in the beginning.\"\n\nIn any event, the virtues of adopting the ascent of humanity as a scaffolding on which to build a curriculum are many and various, especially in our present situation. For one thing, with a few exceptions which I shall note, it does not require that we invent new subjects or discard old ones. The structure of the subject-matter curriculum that exists in most schools at present is entirely usable. For another, it is a theme that can begin in the earliest grades and extend through college in ever-deepening and -widening dimensions. Better still, it provides students with a point of view from which to understand the meaning of subjects, for each subject can be seen as a battleground of sorts, an arena in which fierce intellectual struggle has taken place and continues to take place. Each idea within a subject marks the place where someone fell and someone rose. Thus, the ascent of humanity is an optimistic story, not without its miseries but dominated by astonishing and repeated victories. From this point of view, the curriculum itself may be seen as a celebration of human intelligence and creativity, not a meaningless collection of diploma or college requirements.\n\nBest of all, the theme of the ascent of humanity gives us a nontechnical, noncommercial definition of education. It is a definition drawn from an honorable humanistic tradition and reflects a concept of the purposes of academic life that goes counter to the biases of the technocrats. I am referring to the idea that to become educated means to become aware of the origins and growth of knowledge and knowledge systems; to be familiar with the intellectual and creative processes by which the best that has been thought and said has been produced; to learn how to participate, even if as a listener, in what Robert Maynard Hutchins once called The Great Conversation, which is merely a different metaphor for what is meant by the ascent of humanity. You will note that such a definition is not child-centered, not training-centered, not skill-centered, not even problem-centered. It is idea-centered and coherence-centered. It is also otherworldly, inasmuch as it does not assume that what one learns in school must be directly and urgently related to a problem of today. In other words, it is an education that stresses history, the scientific mode of thinking, the disciplined use of language, a wide-ranging knowledge of the arts and religion, and the continuity of human enterprise. It is education as an excellent corrective to the antihistorical, information-saturated, technology-loving character of Technopoly.\n\nLet us consider history first, for it is in some ways the central discipline in all this. It is hardly necessary for me to argue here that, as Cicero put it, \"To remain ignorant of things that happened before you were born is to remain a child.\" It is enough to say that history is our most potent intellectual means of achieving a \"raised consciousness.\" But there are some points about history and its teaching that require stressing, since they are usually ignored by our schools. The first is that history is not merely one subject among many that may be taught; _every_ subject has a history, including biology, physics, mathematics, literature, music, and art. I would propose here that every teacher must be a history teacher. To teach, for example, what we know about biology today without also teaching what we once knew, or thought we knew, is to reduce knowledge to a mere consumer product. It is to deprive students of a sense of the meaning of what we know, and of how we know. To teach about the atom without Democritus, to teach about electricity without Faraday, to teach about political science without Aristotle or Machiavelli, to teach about music without Haydn, is to refuse our students access to The Great Conversation. It is to deny them knowledge of their roots, about which no other social institution is at present concerned. For to know about your roots is not merely to know where your grandfather came from and what he had to endure. It is also to know where your ideas come from and why you happen to believe them; to know where your moral and aesthetic sensibilities come from. It is to know where your world, not just your family, comes from. To complete the presentation of Cicero's thought, begun above: \"What is a human life worth unless it is incorporated into the lives of one's ancestors and set in an historical context?\" By \"ancestors\" Cicero did not mean your mother's aunt.\n\nThus, I would recommend that every subject be taught _as_ history. In this way, children, even in the earliest grades, can begin to understand, as they now do not, that knowledge is not a fixed thing but a stage in human development, with a past and a future. To return for a moment to theories of creation, we want to be able to show how an idea conceived almost four thousand years ago has traveled not only in time but in meaning, from science to religious metaphor to science again. What a lovely and profound coherence there is in the connection between the wondrous speculations in an ancient Hebrew desert tent and the equally wondrous speculations in a modern MIT classroom! What I am trying to say is that the history of subjects teaches connections; it teaches that the world is not created anew each day, that everyone stands on someone else's shoulders.\n\nI am well aware that this approach to subjects would be difficult to use. There are, at present, few texts that would help very much, and teachers have not, in any case, been prepared to know about knowledge in this way. Moreover, there is the added difficulty of our learning how to do this for children of different ages. But that it needs to be done is, in my opinion, beyond question.\n\nThe teaching of subjects as studies in historical continuities is not intended to make history as a special subject irrelevant. If every subject is taught with a historical dimension, the history teacher will be free to teach what histories are: hypotheses and theories about why change occurs. In one sense, there is no such thing as \"history,\" for every historian from Thucydides to Toynbee has known that his stories must be told from a special point of view that will reflect his particular theory of social development. And historians also know that they write histories for some particular purpose\u2014more often than not, either to glorify or to condemn the present. There is no definitive history of anything; there are only histories, human inventions which do not give us _the_ answer, but give us only those answers called forth by the questions that have been asked.\n\nHistorians know all of this\u2014it is a commonplace idea among them. Yet it is kept a secret from our youth. Their ignorance of it prevents them from understanding how \"history\" can change and why the Russians, Chinese, American Indians, and virtually everyone else see historical events differently than the authors of history schoolbooks. The task of the history teacher, then, is to become a \"histories teacher.\" This does not mean that some particular version of the American, European, or Asian past should remain untold. A student who does not know at least one history is in no position to evaluate others. But it does mean that a histories teacher will be concerned, at all times, to show how histories are themselves products of culture; how any history is a mirror of the conceits and even metaphysical biases of the culture that produced it; how the religion, politics, geography, and economy of a people lead them to re-create their past along certain lines. The histories teacher must clarify for students the meaning of \"objectivity\" and \"events,\" must show what a \"point of view\" and a \"theory\" are, must provide some sense of how histories may be evaluated.\n\nIt will be objected that this idea\u2014history as comparative history\u2014is too abstract for students to grasp. But that is one of the several reasons why comparative history should be taught. To teach the past simply as a chronicle of indisputable, fragmented, and concrete events is to replicate the bias of Technopoly, which largely denies our youth access to concepts and theories, and to provide them only with a stream of meaningless events. That is why the controversies that develop around what events ought to be included in the \"history\" curriculum have a somewhat hollow ring to them. Some people urge, for example, that the Holocaust, or Stalin's bloodbaths, or the trail of Indian tears be taught in school. I agree that our students should know about such things, but we must still address the question, What is it that we want them to \"know\" about these events? Are they to be explained as the \"maniac\" theory of history? Are they to be understood as illustrations of the \"banality of evil\" or the \"law of survival\"? Are they manifestations of the universal force of economic greed? Are they examples of the workings of human nature?\n\nWhatever events may be included in the study of the past, the worst thing we can do is to present them devoid of the coherence that a theory or theories can provide\u2014that is to say, as meaningless. This, we can be sure, Technopoly does daily. The histories teacher must go far beyond the \"event\" level into the realm of concepts, theories, hypotheses, comparisons, deductions, evaluations. The idea is to raise the level of abstraction at which \"history\" is taught. This idea would apply to all subjects, including science.\n\nFrom the point of view of the ascent of humanity, the scientific enterprise is one of our most glorious achievements. On humanity's Judgment Day we can be expected to speak almost at once of our science. I have already stressed the importance of teaching the history of science in every science course, but this is no more important than teaching its \"philosophy.\" I mention this with some sense of despair. More than half the high schools in the United States do not even offer one course in physics. And at a rough guess, I would estimate that in 90 percent of the schools chemistry is still taught as if students were being trained to be druggists. To suggest, therefore, that science is an exercise in human imagination, that it is something quite different from technology, that there are \"philosophies\" of science, and that all of this ought to form part of a scientific education, is to step out of the mainstream. But I believe it nonetheless.\n\nWould it be an exaggeration to say that not one student in fifty knows what \"induction\" means? Or knows what a scientific theory is? Or a scientific model? Or knows what are the optimum conditions of a valid scientific experiment? Or has ever considered the question of what scientific truth is? In _The Identity of Man_ Bronowski says the following: \"This is the paradox of imagination in science, that it has for its aim the impoverishment of imagination. By that outrageous phrase, I mean that the highest flight of scientific imagination is to weed out the proliferation of new ideas. In science, the grand view is a miserly view, and a rich model of the universe is one which is as poor as possible in hypotheses.\"\n\nIs there one student in a hundred who can make any sense out of this statement? Though the phrase \"impoverishment of imagination\" may be outrageous, there is nothing startling or even unusual about the idea contained in this quotation. Every practicing scientist understands what Bronowski is saying. Yet it is kept a secret from our students. It should be revealed. In addition to having each science course include a serious historical dimension, I would propose that every school\u2014elementary through college\u2014offer and require a course in the philosophy of science. Such a course should consider the language of science, the nature of scientific proof, the source of scientific hypotheses, the role of imagination, the conditions of experimentation, and especially the value of error and disproof. If I am not mistaken, many people still believe that what makes a statement scientific is that it can be verified. In fact, exactly the opposite is the case: What separates scientific statements from nonscientific statements is that the former can be subjected to the test of falsifiability. What makes science possible is not our ability to recognize \"truth\" but our ability to recognize falsehood.\n\nWhat such a course would try to get at is the notion that science is not pharmacy or technology or magic tricks but a special way of employing human intelligence. It would be important for students to learn that one becomes scientific not by donning a white coat (which is what television teaches) but by practicing a set of canons of thought, many of which have to do with the disciplined use of language. Science involves a method of employing language that is accessible to everyone. The ascent of humanity has rested largely on that.\n\nOn the subject of the disciplined use of language, I should like to propose that, in addition to courses in the philosophy of science, every school\u2014again, from elementary school through college\u2014offer a course in semantics\u2014in the processes by which people make meaning. In this connection I must note the gloomy fact that English teachers have been consistently obtuse in their approach to this subject\u2014which is to say, they have largely ignored it. This has always been difficult for me to understand, since English teachers claim to be concerned with teaching reading and writing. But if they do not teach anything about the relationship of language to reality\u2014which is what semantics studies\u2014I cannot imagine how they expect reading and writing to improve.\n\nEvery teacher ought to be a semantics teacher, since it is not possible to separate language from what we call knowledge. Like history, semantics is an interdisciplinary subject: it is necessary to know something about it in order to understand any subject. But it would be extremely useful to the growth of their intelligence if our youth had available a special course in which fundamental principles of language were identified and explained. Such a course would deal not only with the various uses of language but with the relationship between things and words, symbols and signs, factual statements and judgments, and grammar and thought. Especially for young students, the course ought to emphasize the kinds of semantic errors that are common to all of us, and that are avoidable through awareness and discipline\u2014the use of either-or categories, misunderstanding of levels of abstraction, confusion of words with things, sloganeering, and self-reflexiveness.\n\nOf all the disciplines that might be included in the curriculum, semantics is certainly among the most \"basic.\" Because it deals with the processes by which we make and interpret meaning, it has great potential to affect the deepest levels of student intelligence. And yet semantics is rarely mentioned when \"back to the basics\" is proposed. Why? My guess is that it cuts too deep. To adapt George Orwell, many subjects are basic but some are more basic than others. Such subjects have the capability of generating critical thought and of giving students access to questions that get to the heart of the matter. This is not what \"back to the basics\" advocates usually have in mind. They want language technicians: people who can follow instructions, write reports clearly, spell correctly. There is certainly ample evidence that the study of semantics will improve the writing and reading of students. But it invariably does more. It helps students to reflect on the sense and truth of what they are writing and of what they are asked to read. It teaches them to discover the underlying assumptions of what they are told. It emphasizes the manifold ways in which language can distort reality. It assists students in becoming what Charles Weingartner and I once called \"crap-detectors.\" Students who have a firm grounding in semantics are therefore apt to find it difficult to take reading tests. A reading test does not invite one to ask whether or not what is written is true. Or, if it is true, what it has to do with anything. The study of semantics insists upon these questions. But \"back to the basics\" advocates don't require education to be _that_ basic. Which is why they usually do not include literature, music, and art as part of their agenda either. But of course, in using the ascent of humanity as a theme, we would of necessity elevate these subjects to prominence.\n\nThe most obvious reason for such prominence is that their subject matter contains the best evidence we have of the unity and continuity of human experience and feeling. And that is why I would propose that, in our teaching of the humanities, we should emphasize the enduring creations of the past. The schools should stay as far from contemporary works as possible. Because of the nature of the communications industry, our students have continuous access to the popular arts of their own times\u2014its music, rhetoric, design, literature, architecture. Their knowledge of the form and content of these arts is by no means satisfactory. But their ignorance of the form and content of the art of the past is cavernous. This is one good reason for emphasizing the art of the past. Another is that there is no subject better suited to freeing us from the tyranny of the present than the historical study of art. Painting, for example, is more than three times as old as writing, and contains in its changing styles and themes a fifteen-thousand-year-old record of the ascent of humanity.\n\nIn saying this, I do not mean to subsume art under the heading of archeology, although I should certainly recommend that the history of art forms be given a serious place in the curriculum. But art is much more than a historical artifact. To have meaning for us, it must connect with those levels of feeling that are in fact not expressible in discursive language. The question therefore arises whether it is possible for students of today to relate, through feeling, to the painting, architecture, music, sculpture, or literature of the past. The answer, I believe, is: only with the greatest difficulty. They, and many of us, have an aesthetic sensibility of a different order from what is required to be inspired, let alone entertained, by a Shakespeare sonnet, a Haydn symphony, or a Hals painting. To oversimplify the matter, a young man who believes Madonna to have reached the highest pinnacle of musical expression lacks the sensibility to distinguish between the ascent and descent of humanity. But it is not my intention here to blacken the reputation of popular culture. The point I want to make is that the products of the popular arts are amply provided by the culture itself. The schools must make available the products of classical art forms precisely because they are not so available and because they demand a different order of sensibility and response. In our present circumstances, there is no excuse for schools to sponsor rock concerts when students have not heard the music of Mozart, Beethoven, Bach, or Chopin. Or for students to have graduated from high school without having read, for example, Shakespeare, Cervantes, Milton, Keats, Dickens, Whitman, Twain, Melville, or Poe. Or for students not to have seen at least a photograph of paintings by Goya, El Greco, David. It is not to the point that many of these composers, writers, and painters were in their own times popular artists. What is to the point is that they spoke, when they did, in a language and from a point of view different from our own and yet continuous with our own. These artists are relevant not only because they established the standards with which civilized people approach the arts. They are relevant because the culture tries to mute their voices and render their standards invisible.\n\nIt is highly likely that students, immersed in today's popular arts, will find such an emphasis as I suggest tedious and even painful. This fact will, in turn, be painful to teachers, who, naturally enough, prefer to teach that which will arouse an immediate and enthusiastic response. But our youth must be shown that not all worthwhile things are instantly accessible and that there are levels of sensibility unknown to them. Above all, they must be shown humanity's artistic roots. And that task, in our own times, falls inescapably to the schools.\n\nOn the matter of roots, I want to end my proposal by including two subjects indispensable to any understanding of where we have come from. The first is the history of technology, which as much as science and art provides part of the story of humanity's confrontation with nature and indeed with our own limitations. It is important for students to be shown, for example, the connection between the invention of eyeglasses in the thirteenth century and experiments in gene-splicing in the twentieth: that in both cases we reject the proposition that anatomy is destiny, and through technology define our own destiny. In brief, we need students who will understand the relationships between our technics and our social and psychic worlds, so that they may begin informed conversations about where technology is taking us and how.\n\nThe second subject is, of course, religion, with which so much painting, music, technology, architecture, literature, and science are intertwined. Specifically, I want to propose that the curriculum include a course in comparative religion. Such a course would deal with religion as an expression of humanity's crea-tiveness, as a total, integrated response to fundamental questions about the meaning of existence. The course would be descriptive, promoting no particular religion but illuminating the metaphors, the literature, the art, the ritual of religious expression itself. I am aware of the difficulties such a course would face, not the least of which is the belief that the schools and religion must on no account touch each other. But I do not see how we can claim to be educating our youth if we do not ask them to consider how different people of different times and places have tried to achieve a sense of transcendence. No education can neglect such sacred texts as Genesis, the New Testament, the Koran, the Bhagavad-Gita. Each of them embodies a style and a world-view that tell as much about the ascent of humanity as any book ever written. To these books I would add the _Communist Manifesto_ , since I think it reasonable to classify this as a sacred text, embodying religious principles to which millions of people have so recently been devoted.\n\nTo summarize: I am proposing, as a beginning, a curriculum in which all subjects are presented as a stage in humanity's historical development; in which the philosophies of science, of history, of language, of technology, and of religion are taught; and in which there is a strong emphasis on classical forms of artistic expression. This is a curriculum that goes \"back to the basics,\" but not quite in the way the technocrats mean it. And it is most certainly in opposition to the spirit of Technopoly. I have no illusion that such an education program can bring a halt to the thrust of a technological thought-world. But perhaps it will help to begin and sustain a serious conversation that will allow us to distance ourselves from that thought-world, and then criticize and modify it. Which is the hope of my book as well.\n\n# Notes\n\n## ONE\n\n1. Plato, p. 96.\n\n2. Freud, pp. 38-39.\n\n3. This fact is documented in Keith Hoskin's \"The Examination, Disciplinary Power and Rational Schooling,\" in _History of Education_ , vol. VIII, no. 2 (1979), pp. 135-46. Professor Hoskin provides the following story about Farish: Farish was a professor of engineering at Cambridge and designed and installed a movable partition wall in his Cambridge home. The wall moved on pulleys between downstairs and upstairs. One night, while working late downstairs and feeling cold, Farish pulled down the partition. This is not much of a story, and history fails to disclose what happened next. All of which shows how little is known of William Farish.\n\n4. For a detailed exposition of Mumford's position on the impact of the mechanical clock, see his _Technics and Civilization_.\n\n## TWO\n\n1. Marx, p. 150.\n\n2. Perhaps another term for a tool-using culture is \"third-world country,\" although vast parts of China may be included as tool-using.\n\n3. For a detailed analysis of medieval technology, see Jean Gimpel's _The Medieval Machine_.\n\n4. Quoted in Muller, p. 30.\n\n5. See his _Medieval Technology and Social Change_.\n\n6. De Vries' findings are recounted by Alvin Toffler in his article \"Value Impact Forecaster: A Profession of the Future,\" in Baier and Rescher's book _Values and the Future: The Impact of Technological Change on American Values_ (New York: Free Press, 1969), p. 3.\n\n## THREE\n\n1. Giedion, p. 40.\n\n2. The best account of the history of Utopias may be found in Segal.\n\n3. See David Lin ton's \"Luddism Reconsidered\" in _Etcetera_ , Spring 1985, pp. 32-36.\n\n4. Tocqueville, p. 404.\n\n## FOUR\n\n1. For a detailed examination of the impact of the printing press on Western culture, see Eisenstein.\n\n2. See Postman's _Amusing Ourselves to Death_ for a more full-bodied treatment of the telegraph.\n\n## FIVE\n\n1. An emphatic exception among those sociologists who have written on this subject is Arnold Gehlen. See his _Man in the Age of Technology_.\n\n2. Though this term is by no means original with E. D. Hirsch, Jr., its current popularity is attributable to Hirsch's book _Cultural Literacy_.\n\n3. This poignant phrase is also the title of one of Lasch's most important books.\n\n4. James Beniger, _The Control Revolution_ , p. 13. As I have already noted, Beniger's book is the best source for an understanding of the technical means of eliminating\u2014i.e., controlling\u2014information.\n\n5. Tocqueville, p. 262.\n\n6. Lewis, p. x.\n\n7. See Arendt.\n\n## SIX\n\n1. I am not sure whether the company still exists, but by way of proving that it at least once did, here is the address of the HAGOTH Corporation as I once knew it: 85 NW Alder Place, Department C, Issaquah, Washington 98027.\n\n2. All these facts and more may be found in Payer, or in Inlander et al.\n\n3. Reiser, p. 160.\n\n4. Ibid., p. 161.\n\n5. Payer, p. 127.\n\n6. Quoted in ibid.\n\n7. For a fascinating account of Laennec's invention, see Reiser.\n\n8. Ibid., p. 38.\n\n9. Ibid., p. 230.\n\n10. Horowitz, p. 31.\n\n11. Ibid., p. 80.\n\n12. Cited in Inlander et al., p. 106.\n\n13. Cited in ibid., p. 113.\n\n## SEVEN\n\n1. _New York Times_ , August 7, 1990, sect. C, p. 1.\n\n2. _Personal Computing_ , June 29, 1990, p. 36.\n\n3. _New York Times_ , November 24, 1989.\n\n4. _Publishers Weekly_ , March 2, 1990, p. 26.\n\n5. _Bottom Line_ , July 15, 1989, p. 5.\n\n6. For a concise and readable review of the development of the computer, I would recommend Arno Penzias' _Ideas and Information: Managing in a High-Tech World_.\n\n7. Quoted in Hunt, p. 318.\n\n8. Searle, p. 30.\n\n9. See Gozzi, pp. 177-80.\n\n10. See Milgram.\n\n11. Weizenbaum, p. 32.\n\n12. The March 1991 issue of _The Sun_ reports that Lance Smith, who is two years old, is called \"the Mozart of video games,\" mainly because he gets astronomical scores on one of Nintendo's games. This is as close to approaching the artistry of Mozart as computers can get.\n\n13. See J. D. Bolter's 1991 book, _Writing Space: The Computer, Hypertext and the History of Writing_ (Hillsdale, N.J.: Lawrence Erlbaum Associates).\n\n14. _Science Digest_ , June 1984.\n\n15. Both men are quoted in the Raleigh, North Carolina, _News and Observer_ , Sunday, August 13, 1989.\n\n16. Katsch, p. 44.\n\n## EIGHT\n\n1. Cited in Gould, p. 75. I am indebted to Gould's wonderful book for providing a concise history of the search to quantify intelligence.\n\n2. _The National Elementary Principal_ March\/April 1975.\n\n3. Weizenbaum, p. 203.\n\n4. The occasion, in the spring of 1990, was a retreat outside of Washington, D.C The group of twenty-three Democratic congressmen was led by Richard Gephardt.\n\n5. I have, of course, made up these ridiculous statistics. The point is, it doesn't matter.\n\n6. See the preceding note.\n\n7. An interesting example of the tyranny of statistics is in the decision made by the College Board (on November 1, 1990) that its Scholastic Aptitude Test will not include asking students to write an essay. To determine the student's ability to write, the SAT will continue to use a multiple-choice test that measures one's ability to memorize rules of grammar, spelling, and punctuation. It would seem reasonable\u2014wouldn't it?\u2014that the best way to find out how well someone writes is to ask him or her to write something. But in Technopoly reason is a strange and wondrous thing. For a documentation of all of this, see the January 16, 1991, issue of _The Chronicle of Higher Education_.\n\n8. See Keith W. Hoskin and Richard H. Macve, \"The Genesis of Accountability: The West Point Connections,\" in _Accounting Organizations and Society_ , vol. 13, no. 1 (1988), pp. 37-73. I am especially indebted to these scholars for their account of the development of modern systems of management.\n\n## NINE\n\n1. Cited in Hayek, p. 201. I am indebted to Hayek's book for his history of the Ecole Polytechnique.\n\n2. Ibid., p. 21.\n\n3. Myrdal, p. 6.\n\n4. I have borrowed much of the material dealing with the distinctions between natural science and social research from my own essay \"Social Science as Moral Theology,\" in _Conscientious Objections_.\n\n## TEN\n\n1. Although in some ways Boorstin's book is dated, to him and his book go credit for calling early attention to the effects of an image society.\n\n2. _The New Republic_ , February 18, 1991, p. 42.\n\n## ELEVEN\n\n1. What follows is a version of a proposal I have made several times before. A somewhat fuller version appears in my _Teaching as a Conserving Activity_.\n\n# Bibliography\n\nAl-Hibri, A., and Hickman, L. (eds.). _Technology and Human Affairs_. London: The C. V. Mosby Company, 1981.\n\nArendt, H. _Eichmann in Jerusalem: A Report on the Banality of Evil_. New York: Penguin Books, 1977.\n\nBellah, R. N.; Madsen, R.; Sullivan, W. H.; Swidler, A.; and Tipton, S. M. _Habits of the Heart: Individualism and Commitment in American Life_. Berkeley: University of California Press, 1985.\n\nBeniger, J. R. _The Control Revolution: Technological and Economic Origins of the Information Society_. Cambridge, Mass., and London: Harvard University Press, 1986.\n\nBolter, J. D. _Turing's Man: Western Culture in the Computer Age_. Chapel Hill: The University of North Carolina Press, 1984.\n\nBury, J. B. _The Idea of Progress: An Inquiry into its Origin and Growth_. New York: Dover Publications, Inc., 1932.\n\nCallahan, R. E. _Education and the Cult of Efficiency: A Study of the Social Forces That Have Shaped the Administration of the Public Schools_. Chicago: The University of Chicago Press, 1962.\n\nChristians, C. G., and Van Hook, J. M. (eds.). _Jacques Ellul: Interpretive Essays_. Chicago: University of Illinois Press, 1981.\n\nEisenstein, E. _The Printing Revolution in Early Modern Europe_. Cambridge, Mass.: Cambridge University Press, 1983.\n\nEllul, J. _The Technological Society_. New York: Alfred A. Knopf, 1964.\n\nEllul, J. _The Betrayal of the West_. New York: The Seabury Press, 1978.\n\nFarrington, B. _Francis Bacon: Philosopher of Industrial Science_. New York: Henry Schuman, Inc., 1949.\n\nFreud, S. _Civilization and Its Discontents_. New York: W. W. Norton & Co., 1961.\n\nGehlen, A. _Man in the Age of Technology_. New York: Columbia University Press, 1980.\n\nGiedion, S. _Mechanization Takes Command: A Contribution to Anonymous History_. New York: W. W. Norton & Co., 1948.\n\nGimpel, J. _The Medieval Machine: The Industrial Revolution of the Middle Ages_. New York: Holt, Rinehart & Winston, 1976.\n\nGould, S. J. _The Mismeasure of Man_. New York: W. W. Norton & Co., 1981.\n\nGozzi, R., Jr. \"The Computer 'Virus' as Metaphor,\" in _Etcetera: A Review of General Semantics_ , vol. 47, no. 2 (Summer 1990).\n\nHayek, F. H. _The Counter-Revolution of Science: Studies on the Abuse of Reason_. Indianapolis: Liberty Press, 1952.\n\nHirsch, E. D., Jr. _Cultural Literacy: What Every American Needs to Know_. Boston: Houghton Mifflin Co., 1987.\n\nHodges, A. _Alan Turing: The Enigma_. New York: Simon & Schuster, 1983.\n\nHoffer, E. _The Ordeal of Change_. New York: Harper & Row, 1952.\n\nHorowitz, L. C., M.D. _Taking Charge of Your Medical Fate_. New York: Random House, 1988.\n\nHunt, M. _The Universe Within: A New Science Explores the Mind_. New York: Simon & Schuster, 1982.\n\nHutchins, R. M. _The Higher Learning in America_. New Haven: Yale University Press, 1936.\n\nInlander, C. B.; Levin, L. S.; and Weiner, E. _Medicine on Trial: The Appalling Story of Medical Ineptitude and the Arrogance that Overlooks It_. New York: Pantheon Books, 1988.\n\nKatsch, M. E. _The Electronic Media and the Transformation of Law_. New York and Oxford: Oxford University Press, 1989.\n\nKoestler, A. _The Sleepwalkers_. New York: The Macmillan Company, 1068.\n\nLasch, C. _Haven in a Heartless World: The Family Besieged_. New York: Basic Books, Inc., 1975.\n\nLewis, C. S. _The Screwtape Letters_. New York: Macmillan, 1943.\n\nLogan, R. K. _The Alphabet Effect: The Impact of the Phonetic Alphabet on the Development of Western Civilization_. New York: St. Martin's Press, 1986.\n\nLuke, C. _Pedagogy, Printing, and Protestantism_. Albany: State University of New York Press, 1989.\n\nMarx, K., and Engels, F. _The German Ideology_. New York: International Publishers, 1972.\n\nMilgram, S. _Obedience to Authority: An Experimental View_. New York: Harper & Row, 1974.\n\nMuller, H. J. _The Children of Frankenstein: A Primer on Modern Technology and Human Values_. Bloomington and London: Indiana University Press, 1970.\n\nMumford, L. _Technics and Civilization_. New York: Harcourt Brace Jovanovich, 1963.\n\nMyrdal, G. _Objectivity in Social Research_. New York: Pantheon Books, 1969.\n\nPapert, S. _Mindstorms: Children, Computers, and Powerful Ideas_. New York-Basic Books, Inc., 1980.\n\nPayer, L. _Medicine and Culture: Varieties of Treatment in the United States, England, West Germany, and France_. New York: Penguin Books, 1988.\n\nPenzias, A. _Ideas and Information: Managing in a High-Tech World_. New York and London: W. W. Norton & Co., 1989.\n\nPlato. _Phaedrus and Letters VII and VIII_. New York: Penguin Books, 1973.\n\nPostman, N. _Amusing Ourselves to Death: Public Discourse in the Age of Show Business_. New York: Penguin Books, 1985.\n\nRead, H. _To Hell with Culture and Other Essays on Art and Society_. New York: Schocken Books, 1963.\n\nReiser, S. J. _Medicine and the Reign of Technology_. Cambridge, London, New York and Melbourne: Cambridge University Press, 1978.\n\nRifkin, J. _Time Wars: The Primary Conflict in Human History_. New York: Henry Holt and Company, 1987.\n\nSchumacher, E. F. _Small Is Beautiful: Economics As If People Mattered_. New York, Hagerstown, San Francisco and London: Harper & Row.\n\nSchumacher, E. F. _A Guide for the Perplexed_. New York: Viking Penguin, Inc., 1977.\n\nSearle, J. _Minds, Brains and Science_. Cambridge, Mass.: Harvard University Press, 1984.\n\nSegal, H. P. _Technological Utopianism in American Culture_. Chicago: The University of Chicago Press, 1985.\n\nSnow, C. P. _The Two Cultures and the Scientific Revolution_. New York: Cambridge University Press, 1959.\n\nSturt, M. _Francis Bacon_. New York: William Morrow & Company, 1932.\n\nSzasz, T. _Anti-Freud: Karl Kraus's Criticism of Psychoanalysis and Psychiatry_. Syracuse: Syracuse University Press, 1976.\n\nTocqueville, A. de. _Democracy in America_. New York: Anchor Books (Doubleday & Co., Inc.), 1969.\n\nUsher, A. P. _History of Mechanical Inventions_. New York: Dover Publications, Inc., 1929.\n\nWeingartner, C. \"Educational Research: The Romance of Quantification,\" _Etcetera: A Review of General Semantics_ , vol. 39, no. 2 (Summer 1982).\n\nWeizenbaum, J. _Computer Power and Human Reason: From Judgment to Calculation_. San Francisco: W. H. Freeman and Company, 1976.\n\nWhite, L., Jr. _Medieval Technology and Social Change_. London: Oxford University Press, 1962.\n\nWhitehead, A. N. _The Aims of Education and Other Essays_. New York: The Free Press, 1929.\n\nWhitrow, G. J. _Time in History: The Evolution of Our General Awareness of Time and Temporal Perspective_. Oxford and New York: Oxford University Press, 1988.\n\n# BOOKS BY NEIL POSTMAN\n\n#\n\n\"No contemporary essayist writing about American pop culture is more fun to read and more on target.\"\u2014 _Los Angeles Times_\n\nCONSCIENTIOUS OBJECTIONS  \n _Stirring Up Trouble About Language_ ,  \n _Technology, and Education_\n\nIn this series of feisty and ultimately hopeful essays, readers will find themselves rethinking many of their bedrock assumptions: Should education transmit culture or defend us against it? Is technological innovation progress or a peculiarly American addiction?\n\nCurrent Affairs\/Science\/Education\/0-679-73421-X\n\nTHE DISAPPEARANCE OF CHILDHOOD\n\nFrom the vogue for nubile models to the explosion in the juvenile crime rate, this modern classic of social history and media traces the precipitous decline of childhood in America today\u2014and the corresponding threat to the notion of adulthood.\n\nMedia\/Current Affairs\/0-679-75166-1\n\nTHE END OF EDUCATION  \n _Redefining the Value of School_\n\nIn this provocative analysis, Neil Postman suggests that the current crisis in America's educational system derives from its failure to supply students with a unifying \"narrative\" like those that inspired earlier generations. Instead, today's schools promote the false \"gods\" of consumerism, technology, and ethnic separatism.\n\nEducation\/0-679-75031-2\n\nTECHNOPOLY  \n _The Surrender of Culture to Technology_\n\nPostman launches a trenchant warning against the tyranny of machines over man in the late twentieth century. _Technopoly_ chronicles our transformation from a society that uses technology to one that is shaped by it, as it also traces its effects upon what we mean by politics, religion, intellect, history\u2014even privacy and truth.\n\nCurrent Affairs\/Sociology\/0-679-74540-8\n\nAvailable at your local bookstore, or call toll-free to order:  \n1-800-793-2665 (credit cards only).\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Charles Bowen from page scans provided by\nGoogle Books (Oxford University)\n\n\n\n\n\n\n\n\n\n\n\nTranscriber's Notes:\n\n     1. Page scan source: Google Books\n        http:\/\/books.google.com\/books?id=GyAGAAAAQAAJ\n        (Oxford University)\n\n\n\n\n\n\n                            THE LAST CALL.\n\n\n\n\n\n\n                            THE LAST CALL.\n\n\n                              A Romance.\n\n\n\n                                  BY\n\n                           RICHARD DOWLING,\n\n       AUTHOR OF \"THE MYSTERY OF KILLARD,\" \"THE WEIRD SISTERS,\"\n                         \"SWEET INISFAIL,\" ETC.\n\n\n\n\n\n                         _IN THREE VOLUMES_.\n\n                               VOL. I.\n\n\n\n\n                               LONDON:\n             TINSLEY BROTHERS, 8, CATHERINE ST., STRAND.\n                                1884.\n                       [_All rights reserved_.]\n\n\n\n\n\n\n                      CHARLES DICKENS AND EVANS\n                        CRYSTAL PALACE PRESS.\n\n\n\n\n\n\n                            THE LAST CALL.\n\n\n                              * * * * *\n\n\n                               Part I.\n\n\n\n\n\n\n                            THE LAST CALL.\n\n\n                              CHAPTER I.\n\n\nThe sun was low behind a bank of leaden cloud which stood like a wall\nupon the western horizon. In front of a horse-shoe cove lay a placid\nbay, and to the westward, but invisible from the cove, the plains of\nthe Atlantic.\n\nIt was low water, and summer. The air of the cove was soft with\nexhalations from the weed-clad rocks stretching in green and brown\nfurrows from the ridge of blue shingle in the cove to the violet\nlevels of the sea.\n\nOn the ridge of shingle lay a young man, whose eyes rested on the sea.\nHe was of the middle height and figure. Twenty-seven or twenty-eight\nseemed to be his age. He had a neat, compact forehead, dark gray eyes,\nruddy, full cheeks, a prominent nose, full lips, and a square chin.\nThe face looked honest, good-humoured, manly. The moustaches were\nbrown; the brown hair curled under the hat. The young man wore a gray\ntweed suit and a straw hat.\n\nHe lay resting on his elbow. In the line of his sight far out in the\nbay a small dot moved almost imperceptibly. The lounger knew this dot\nwas a boat: distance prevented his seeing it contained a man and a\nwoman.\n\nDominique Lavirotte, the man in the boat, was of the middle height and\nfigure, twenty-four years of age, looking like a Greek, but French by\ndescent and birth. The eyes and skin were dark, the beard and\nmoustaches black. The men of Rathclare, a town ten miles off, declared\nhe was the handsomest man they had ever seen, and yet felt their\ncandour ill-requited when their sweethearts and wives concurred.\n\nWith Dominique Lavirotte in the boat was Ellen Creagh. She was not a\nnative of Rathclare, but of Glengowra, the small seaside and fishing\ntown situate on Glengowra Bay, over which the boat was now lazily\ngliding in the cool blue light of the afternoon.\n\nEllen Creagh was tall and slender, above the average height of women,\nand very fair. She had light golden-brown hair, bright lustrous blue\neyes, and lips of delicate red. The upper lip was short. Even in\nrepose her face always suggested a smile. One of the great charms of\nthe head was the fluent ease with which it moved. The greatest charm\nof the face was the sweet susceptibility it had to smile. It seemed,\nwhen unmoved, to wait in placid faith, the advent of pleasant things.\nDuring its moments of quiet there was no suggestion of doubt or\nanxiety in it. To it the world was fair and pleasant--and the face was\npleasant and wonderfully fair. Pleasant people are less degraded by\naffectation than solemn people. Your solemn man is generally a\nswindler of some kind, and nearly always selfish and insincere. Ellen\nCreagh looked the embodiment of good-humoured candour, and the ideal\nof health and beauty. She was as blithe and wholesome as the end of\nMay; she was a northern Hebe, a goddess of youth and joy.\n\nThe name of the young man lying on the shingles was Eugene O'Donnell.\nHe lived in the important seaport of Rathclare, where his father was\nthe richest and most respected merchant and shipowner. There had James\nO'Donnell been established in business for many years, and they now\nsaid he was not worth less than a quarter of a million sterling. Mrs.\nO'Donnell was a hale, brisk, bright-minded woman of fifty-seven, being\nthree years her husband's junior. The pair had but one child, Eugene,\nand to him in due time all the old man's money was to go. The\nO'Donnells were wealthy and popular. The father had a slow, methodical\nway, which did not win upon strangers, but among those who knew him no\none was more highly respected. Without any trace of extravagance,\nJames O'Donnell was liberal with his money. He was a good husband, a\ngood father, and a good employer.\n\nHe had only one source of permanent uneasiness--his son Eugene was not\nmarried, and showed no inclination towards marriage. The old man held\nthat every young man who could support a wife should take one. He\nhimself had married young, had prospered amazingly, and never for a\nmoment regretted his marriage. He was prepared to give his son a share\nin his business, and a thousand a year out of the interest of his\nsavings, if the young man would only settle. But although Eugene\nO'Donnell was as good-humoured and good-hearted a young fellow as the\ntown of Rathclare, or the next town to it, could show, and although\nthere was not in the whole town one girl who would be likely to refuse\nhim, and although there were plenty of handsome girls in Rathclare,\nEugene O'Donnell remained obdurate. It was lamentable, but what could\nanyone do? The young man would not make love, the father would not\ninsist upon his marrying whether he loved or no, and there being at\nRathclare little faith in leap-year, no widow or maiden of the town\nwas bold enough to ask him to wed her.\n\nWhile the young man lying on the shingle was idly watching the boat,\nthe young man in the boat was by no means idle. The sculls he was\npulling occupied none of his attention. He swung himself mechanically\nbackward and forward. His whole mind was fixed on the face and form of\nthe girl sitting in the stern.\n\n\"And so, you really must go back to Dublin?\" he said ruefully.\n\n\"Yes,\" she answered with a smile. \"I must really go back to Dublin\nwithin a fortnight.\"\n\n\"And leave all here behind,\" he said tenderly.\n\n\"All!\" she exclaimed, looking around sadly. \"There is not much to\nleave besides the sea, which I always loved, and my mother, whom I\nalways loved also.\"\n\n\"There is nothing else in the place, I suppose, Miss Creagh, you love,\nbut the sea and your mother?\"\n\n\"No,\" she answered, \"nothing. I have no relative living but my mother,\nand she and the sea are my oldest friends.\"\n\n\"But have you no new friend or friends?\"\n\nShe shook her head, and leaning over the side of the boat, drew her\nfingers slowly through the water.\n\n\"The Vernons,\" she said, \"are good to me, and I like the girls very\nmuch. But I am only their servant--a mere governess.\"\n\n\"A mere queen!\" he said. \"I have known you but a short time. That has\nbeen the happiest time of my life. _I_ at least can never forget it.\nMay you?\"\n\nSuddenly a slight change came over her. She lost a little of her\ngaiety, and gathered herself together with a shadow of reserve.\n\n\"I do not think, Mr.. Lavirotte, I shall soon forget the many pleasant\nhours we have spent together and the great kindness you have shown to\nme.\"\n\n\"And you do not think you will forget _me?_\"\n\n\"How can I remember your kindness and forget you?\" she asked gravely.\n\n\"Yes, yes,\" he said eagerly, \"but you know what I mean, and are\navoiding my meaning. Perhaps I have been too hasty. Shall I sing you a\nsong?\"\n\n\"Yes, please, if you will row towards home.\"\n\nThen he sang:\n\n\n    \"The bright stars fade, the morn is breaking,\n       The dew-drops pearl each flower and leaf,\n     When I of thee my leave am taking,\n       With bliss too brief.\n     How sinks my heart with fond alarms,\n       The tear is hiding in mine eye,\n     For time doth chase me from thine arms:\n       Good-bye, sweetheart, good-bye.\"\n\n\nThe boat was now well inshore.\n\n\"Lavirotte! Lavirotte's voice, by all the gods!\" cried Eugene\nO'Donnell, raising himself into a sitting posture. \"Doing the\npolite--doing the lover, for all I know. Why has he stopped there? He\nwill begin again in a moment.\"\n\n\"When you go, Ellen, will you give me leave to bid you adieu in these\nwords?\"\n\n\"Mr. Lavirotte,\" she said, in doubt and pain, \"I am exceedingly sorry\nthat----\"\n\n\"It is enough,\" he said. \"Say no more. I am a ruined man.\"\n\n\"He will not finish it,\" said O'Donnell. \"He is ungallant. I will\nfinish it for him.\n\n\n    \"The sun is up, the lark is soaring,\n       Loud swells the song of chanticleer;\n     The leveret bounds o'er earth's soft flooring:\n       Yet I am here.\n     For since night's gems from heaven did fade,\n       And morn to floral lips must hie,\n     I could not leave thee though I said,\n       Good-bye, sweetheart, good-bye.\"\n\n\nThe girl raised her head and listened for a moment, and then bent her\nhead in some confusion. There was to her a sense of surprise in\nfeeling that this song had, bearing its present associations, been\ncompleted by an unknown voice.\n\nLavirotte noticed the look of disquietude on the girl's face, and said\nlightly and bitterly: \"You need not be uneasy, Miss Creagh. I know the\nman who finished my song for me, when there was no use in my going on\nwith it. He and I are rival tenors. I will introduce you to him when\nwe get ashore. We are the closest friends. He is the best of good\nfellows, and reputed--ah, I envy him--to be a woman-hater.\"\n\nAt length the boat glided slowly through the green channel that led\nfrom the plain of the violet bay to the ridge of blue shingle.\n\nLavirotte handed the girl out as soon as they reached the beach, and,\nas he did so, said: \"You have no objection to know my friend?\"\n\nShe was anxious to conciliate him in any way she might. \"No,\" she\nwhispered. \"What a lovely voice he has.\"\n\n\"Better than mine?\" he asked abruptly and harshly.\n\n\"I--I,\" she hesitated, \"am but a poor judge.\"\n\n\"Which means,\" he said bitterly, \"that you are a good judge, and\ndecide against me.\"\n\nBy this time they were close to where O'Donnell was. He was standing,\nand looking out to sea.\n\n\"Comrade,\" said Lavirotte, touching him on the shoulder, \"I am\ndelighted to see you. I am in sore need of a _friend_. Miss Creagh has\nadmired your singing very much. Mr. O'Donnell--Miss Creagh.\"\n\n\"Am I dreaming,\" thought O'Donnell, \"or is this beauty real?\"\n\n\n\n\n                         CHAPTER II.\n\n\nThere was around Dominique Lavirotte an air of mystery which kept the\ngood simple folk of Glengowra at bay. Although, theoretically,\nFrenchmen have always been popular in Ireland, this applies rather to\nthe mass than to the individual.\n\nThere was nothing repulsive about Dominique Lavirotte. On the\ncontrary, he had attractive manners, and although he spoke English\nwith a broken accent, he spoke it fluently and faultlessly. He was\nagreeable in company, well-read, and possessed a shallow\nencyclop[ae]dic knowledge, by means of which he was enabled to give\ngreat brilliancy and point to his conversation.\n\nYet at certain moments he was taciturn, and if one attempted to break\nin upon his reserve he turned swiftly and snarled even at his best\nfriend.\n\nAccording to his own account, he was descended from Louis Anne\nLavirotte, medical doctor, born at Nolay, in the diocese of Autun,\nsomewhere about a hundred years ago, who was a most skilful physician,\nand one well versed in the English language. This dead doctor of a\nhundred years ago had devoted much of his attention while on earth to\nmore or less obscure forms of mental disease, and had written a\ntreatise on hydrophobia.\n\nDominique was very proud of this learned ancestor, and paid his\nrelative of the last century the compliment of devoting some of his\nown time to the consideration of abnormal mental developments. Indeed,\nsome of those who knew him best said that there was a twist in his own\nmind, and that under extreme provocation, mental or physical, the\nbrain would give way.\n\nLavirotte and O'Donnell were as close friends as it is possible for\nmen to be; and, notwithstanding the ten miles which separated their\nhomes, they saw much of one another. Each was young and enthusiastic,\neach sang tenor, and sang uncommonly well.\n\nIn the town of Rathclare, no young man was more popular than Eugene\nO'Donnell, and the people there thought it a thousand pities that he\nshould select as his favourite friend a man who was not only not a\nresident of Rathclare, but a foreigner, with mysterious ways and an\nuncertain temper. O'Donnell laughed off all their expostulations and\nwarnings, and said that in so far as his friend was a stranger and\nafflicted with a bad temper, there was all the more reason why someone\nshould do him any little kindness he could.\n\nBut the people of Rathclare shook their heads gravely at the young\nman's temerity, and prophesied that no good would come to O'Donnell of\nthis connection. They did not like this foreigner, with his strange\nways and mysterious retirements into himself. They were free and\nopen-hearted themselves, and they liked free and open-hearted souls\nlike O'Donnell. They did not like swarthy skins; and now and then in\nthe newspapers they read that men with swarthy skins drew knives and\nstruck their dearest friends; that foreigners were treacherous, and\nnot to be trusted with the lives, into the homes, or with the honour\nof law-abiding folk. They knew, it being a seaport, that foreigners\nspoke a gibberish which they affected to understand, and which was in\nreality no better than the language of Satan. Once a Greek, an\ninfamous Greek, had been hanged in their town for an intolerable crime\nof cruelty committed on board ship; and somehow, ever since then, all\nforeigners, particularly swarthy foreigners, seemed in their eyes\npeculiarly prone to atrocious cruelties.\n\nWhat a luxury it must have been for this swarthy man of uncertain\ntemper to meet and speak with Ellen Creagh, who was the very\nembodiment of all that is fair in the rich, warm sense of fairness in\nthe North; and free in the sense of all that is open and joyous, and\nfull of abounding confidence, in the North!\n\nDuring the fortnight in which he had been admitted to what he\nconsidered the infinite privilege of her society, he had fallen\nhelplessly, hopelessly, madly in love. He had drunk in the subtle\npoison of her beauty with an avidity almost intolerable to himself.\nAll the poetry and passion of his nature had gone forth ceaselessly\ntowards that girl, as only the poetry and passion of southern blood\ncan go forth. The violence of his feelings had astonished even\nhimself. These feelings had grown all the more intense by the fierce\nrepression in which he had kept them. For until that day in the boat\nhe had never seemed to take more than a passing, polite interest in\nEllen. Even then, in his dark and self-restrained nature, he had given\nno indication of the struggle within. The frenzy of his worship found\nno expression, and he took his dismissal with as much apparent\nindifference as though he had put the question to her merely out of\nregard to the wishes of others.\n\nYet when he said the words, \"I am a ruined man,\" he meant the words,\nor rather he meant that he was determined to take an active part in\nhis own destruction.\n\n\"If I die,\" he thought, \"what is death to me? The sun is dead, the\nmoon is dead, the stars are dead, earth is dead, and perdition will be\na release from this valley of phantoms. When life is not worth living,\nwhy should one live? I will not live. I have no cause against her, but\nI have cause against myself, for I am a failure.\"\n\nHe had determined to make away with himself; he had made up his mind\nthat he would not survive this terrible disappointment; he would go\nhome that night and take some painless and swift poison, and so pass\nout of this vain world to the unknown beyond; he would not declare his\nintention to anyone, least of all to O'Donnell, whose voice he\nrecognised in the second stanza of the song; he knew where he could\nget the poison--from a friendly apothecary. They would hold an inquest\non him, no doubt, and discover that he had done himself to death. Her\nname might even get mixed up in the affair, but he could not help\nthat. He meant to do her no harm; he simply could not and would not\nendure.\n\nWhen that meeting took place on the beach, whereat he introduced Ellen\nto O'Donnell, he had noticed the latter's start of amazed admiration.\n\n\"What,\" thought Lavirotte, \"is he hit too; he, the invincible! he, the\nadamantine man, who has hitherto withstood all the charms of her\nlovely sex? It would be curious to watch this. Will he too make love,\nand fail--succeed? Ah.\"\n\nWhen this thought first occurred to Lavirotte he paused in a dim,\ndazed way. Of all men living he wished best to O'Donnell, now that he\nmight regard himself as dead.\n\n\"If I am to die and she is to love, would it not be best that she\nshould love him?\"\n\nAnd while he was thinking thus, and as he was mentioning his friend's\nname to her, he saw her, too, start and seem for a moment confused. He\ncould easily understand why it was O'Donnell had started. Such beauty\nas hers appeared potent enough to infuse the Belvidere Apollo with\naction. But why should she start? Woman is not overwhelmed by the\nbeauty of man, as man is by the beauty of woman. Here it was that the\ndemon of jealousy first entered the soul of Dominique Lavirotte; here\nit was he first inhaled the mephitic breath of jealousy, destined to\npoison all his life and to embitter the last moment of his existence.\n\nAs the three turned away and left the blue shingle for the yellow\nroad, the sun fell behind them, and almost imperceptibly the gray dusk\nof twilight gathered in the east. Overhead the blue of day was\nbecoming fainter and fainter, making way for the intenser blue of\nnight.\n\nNeither of the men seemed disposed to speak. The heart of each was\nfull of new emotion--one of love, the other of jealousy; one of the\nfirst rapturous buoyancy of dearest hope, the other of degrading cark.\n\nNothing but the most ordinary commonplaces were uttered that night;\nand after the leave-taking each went a different way--she to the\nmodest lodging where she spent her brief holiday with her mother;\nLavirotte to his quiet room, and O'Donnell back to Rathclare by the\nlatest train leaving the village that night.\n\nWhen the last-mentioned got home, he astonished his father and mother\nby walking into the room where they were sitting, and saying abruptly:\n\n\"Sir, you have often advised me to marry, and I have put the matter\noff. Are you still of your former mind?\"\n\n\"God bless my soul!\" cried the father in astonishment. \"God bless my\nsoul, Eugene, what's the matter?\" He could get no further than this\nwith surprise, and the question he asked was put merely as a matter of\nform, and not from any desire to ascertain the condition of his son's\nmind.\n\nBut the mother was quicker--took in the whole situation at once,\nplunged at the heart of things, and asked breathlessly: \"Eugene, who\nis she?\"\n\nHe  slightly and drew back. His father was too slow, and his\nmother too quick for him. He preferred his mother's mode of treating\nthe matter. The word \"she\" brought back to his enchanted eyes the\nvision he had seen on the beach. He said to himself: \"My mother has no\nright to be so quick. For all I know to the contrary, she may be\nengaged to Lavirotte.\" Then aloud he said: \"Mother, I assure you,\nthere is no 'she.' I never said two civil words to any girl in all my\nlife.\"\n\n\"Eugene,\" she said, dropping into her lap the woollen stocking she\nwas knitting for him, \"no young man ever yet thought of marriage until\nthinking of some girl had put the thought into his head.\"\n\nHe felt in a way flattered and fluttered. It was pleasant even for a\nmoment to fancy that his mother, although she knew nothing of Miss\nCreagh, had suggested the notion he might marry her. He laughed and\nshook his head, and laughing and shaking his head became him.\n\nHis mother looked at him half sadly, and thought: \"No girl in all the\nworld could refuse my boy--my handsome boy, my noble boy. And now one\nof them is going to take him away from me, who reared him, and have\nknown him every hour since he was born.\"\n\n\"Eugene,\" said the father deliberately, \"do I understand that you wish\nme to give you my opinions on marriage?\"\n\nThe young man burst into a loud laugh. He had got far beyond the\ntheoretic aspect of the affair now, and his father's opinion would\nhave made very little impression indeed when compared with the\nimpression Ellen Creagh had left upon his heart.\n\nAfter this the three talked upon the subject of Eugene's possible\nmarriage, he telling them no more about the adventure on the beach\nthan that the notion of marriage had been put into his mind by the\nsight of a most estimable young lady, in every way suited to him, but\nof whom he had only the slightest knowledge up to this.\n\nThat night, when Ellen Creagh found herself in her own room, no\nthoughts of love were in her head. A feeling of pity for the fair\nyoung man she had met was uppermost in her head. It was not\nsentimental pity, but pity of a much more substantial and worldly\nkind.\n\nShe had a letter to write, and sat down to write it. It began, \"My\ndear Ruth,\" and continued to narrate certain trivial matters connected\nwith seaweed and shells. Then it went on to say: \"I have seen young\nMr. O'Donnell, son of your father's great friend, here. I was quite\nstartled when I heard the name. I was introduced to him by a friend\nwho had told me of him before.\" When she had finished her letter, she\naddressed it to Miss Vernon, Fitzwilliam Square, Dublin. She added a\npostscript, saying: \"I hope you will soon get out of Dublin. You must\nbe weary of it this lovely weather. I shall write again in a few\ndays.\"\n\nThen she stood awhile at the table, musing over the events in the\nboat. \"He could not have been serious,\" she thought. \"I daresay if I\nhad looked at his face I should have seen him smiling. Anyway, he took\nit very quietly.\"\n\nThat night Dominique Lavirotte slept little.\n\n\"Though he were my friend over and over again,\" he cried passionately,\n\"he shall not. No! Not if I were to----\" Here he covered his face with\nhis hands. \"What a horrible thought! I can see his white face now in\nthe moonlight. Why is it white? Why is it moonlight? Oh, God! was\nbeauty ever such as hers?\"\n\n\n\n\n                             CHAPTER III.\n\n\nIt was in the full height of summer, and by the bland sea, and while\ngathering a bouquet of wild flowers for a girl clad in white, and\nsitting on a mound hard by, that Eugene O'Donnell had for the first\ntime the courage to tell himself he was in love. A minute before and\nhe had stood in great fear of this said love--it had seemed silly,\nchildish, unworthy of a full-grown man in the perfect possession of\nall his faculties. And now, all at once, even while his back was\ntowards her, and he was not under the glamour of her eye, the magic of\nher touch, the mysterious fascinations of her motions, when,\napparently, nothing was going on in the bare daylight but the tranquil\nripple of the waves on the shore below, this fear left him, and all at\nonce he confessed to himself his love, and began to glory in it.\n\nOnce the flood-gate was broken down his nature knew no pause, saw no\nobstacle, appreciated no difficulty. Turning round hastily, with the\nflowers in his hand and a laugh upon his lips, such a laugh as he had\nnever laughed before, for now the whole nature of the man was stirred,\nhe cried: \"What a fool I have been, Ellen.\" It was the first time he\nhad called her by her name, and yet it seemed old and familiar to him.\n\"What a fool I have been,\" he said, \"to bother about these flowers.\"\n\nShe blushed, and looked up timidly, and looked down bashfully, and\nsmiled, and moved as though to rise, and then sat still. She was not\nfamiliar with her name upon his lips. \"Eugene,\" to her mind, seemed\nfamiliar, for from one reason or another, perhaps the love of brevity,\nshe so called him when she thought of him. But to hear him call her\nEllen was as though her secret had been penetrated, and the fact that\nshe called him Eugene laid bare.\n\n\"What a fool I have been to gather these idle flowers,\" he repeated.\n\"They are but the symbols of what I could say so much better in words.\nMay I speak?\"\n\nShe grew red, and then deadly pale, and seemed about to faint. Her\nlips opened, but no sound came.\n\n\"Whether you give me leave or not,\" he said, \"I must. Ellen,\" he went\non, \"I think there is at this moment but one thing I believe\nimpossible, and it is that I could ever go away from you. I never was\nin love before, and I don't exactly know the regular thing to say, but\nI'll tell you how I feel. If you were to get up off that mound now and\nwalk away, supposing back to Glengowra or to the world's end, I'd\nfollow you. And I'd never cease to follow you, even beyond the world's\nend, until you turned back and put your hand in mine. That's better\nthan these flowers,\" he said, tossing the bouquet from him. \"It's\nstraighter, anyway, Ellen. Will you give me your hand, dear?\"\n\nHe called her \"dear,\" and after a little while her hand was raised\nslightly from where it lay, and he took it, and she let it bide with\nhim.\n\nSo the stupid flowers lay--nowhere; and two pure hearts, sweet with\nGod's goodliest graces, were opened to the understanding of one\nanother.\n\nThen came moonlight nights to make the rich completion of the full\nday. He sang to her among the rocks, with the cool fresh sea washing\nbeneath their unwearied feet. She sat clasped to him, and glad to be\nso clasped; and he sat strong beside her, and conscious of his\nstrength. There was no worshipping on his part, no bowing down before\na golden image. He took her to his heart in the beauty of her\nwholesome girlhood, as one takes a melody or a flower, without\nquestion and without any exaggeration of dearness beyond the\nexaggeration compelled by all beautiful things.\n\nThese moonlit nights amid the rocks were the dearest things which had\nbeen, up to that, with him. There was no impediment in the course of\nhis true love; his father was affluent; he had explained the whole\nmatter at home; he had brought his sweetheart home, and there had\nshe been approved of. Her mother saw no reason why the handsome,\ngood-natured, good-humoured, well-off young man should not marry her\nbeautiful daughter; and the daughter, on her part, saw all the reasons\nbetween heaven and earth, and several others which had no existence in\nheaven or earth or the region between, why she should marry him.\n\nIt was their custom in these moonlight nights to stroll down to that\ncove where their first meeting had taken place, and where the glamour\nof her beauty had first fallen upon him. Here, of nights, were\nprivacy, the moon and the sea, and the perfections lent to the moon\nand the sea by the cliffs and the rocks and the sounds of the sea\n(that are subtler than any voice); and now and then the sounds of the\nland, which take away the aerial perspective of the sea and bring to\nthe soothed eye visions of homesteads and fallows, of sleeping woods\nand gentle useful beasts, of pious folk at rest by night and pious\nfolk at rest for ever; and, over all, the limitless quiet of night.\n\nHere on several occasions they sat for hours, from the late sunset,\nthrough the late dusk, into the dark. And once or twice, when he bade\nher good-bye at her mother's gate, he stole back again to the cove\nwhich had been the theatre of the magic drama in which he was acting.\nHe now lived in the village, and often sat at the cove until the blue\ndawn blotted out the bluer night, and the seagulls awoke, and the\nsails of the fishing-boats out in the bay were trimmed for home.\n\nAll this time, though he knew it not, a shadow dogged him, an evil\nshadow, a morally misshapen shadow, a pitiless dark shadow, that hid\nhere and there where it could, behind wall, or tree, or rock, and ever\nglared unwholesomely.\n\nThe shadow of a swarthy man, of a man that showed his teeth in the\nmoonlight and fumbled something in his pocket; a sinister stealthy\nshadow, that boded good to no one, lurked, and dodged, and followed in\nthe footsteps of the lovers like the evil genius of their career.\n\nWhen all had been settled between the lovers, Ellen had written to\nMrs. Vernon and obtained release from her duties in that household. A\nmonth had now gone by since that meeting on the shingle, and it was\narranged that in another month the wedding was to take place. The\ncourse of true love was running as smooth as the planets in their\norbits. The happiest man and woman in Ireland were Eugene O'Donnell\nand Ellen Creagh. As the days went by that cove grew dearer to his\nheart; and even now, when the moon was making moonlight for lovers\nsomewhere else, he, Eugene O'Donnell, could not keep away from it, nor\ncould he sleep.\n\nOne night he left her at her mother's gate and walked slowly down the\nroad to the cove. It was dark for a summer night. Yet still there was\nlight enough to see a large object, say the figure of a man, fifty\nyards off. He knew the ground as a farmer knows his farm. Following\nthe declivity of the road he soon arrived at the broken ground. Here\nwas a high rock on the right, high enough to conceal a man; and here,\nbehind this rock, was hidden a man with gleaming teeth, and in his\nright hand a gleaming blade.\n\nAs O'Donnell drew near the rock the man sprang forth, seized the other\nby the throat with the left hand, and, whirling up his right,\nwhispered: \"You shall never marry her.\"\n\n\"Lavirotte! Lavirotte! My God, Lavirotte, are you mad?\"\n\n\"Yes, and you are dead.\"\n\nThe hand holding the knife descended swiftly.\n\n\n\n\n                             CHAPTER IV.\n\n\nInstinctively O'Donnell shot his left hand upward and seized the\ndescending wrist. But the force in Lavirotte's arm was too great to be\novercome. The blow was diverted; but the long, keen blade tipped the\nshoulder, tore through the cloth of the coat, and buried itself in the\nflesh, just above the shoulder-blade.\n\n\"Heavens and earth, man! What's the matter?\" cried O'Donnell, rendered\nalmost powerless, more by astonishment than pain.\n\n\"Death!\" cried the infuriated man--\"your death!--that's what's the\nmatter.\" And, withdrawing the knife, lie raised his arm once more\naloft.\n\nO'Donnell now plainly saw that he was indeed dealing with a madman,\nor, at least, with a man who seriously intended taking his life.\n\nStill retaining his hold on the right wrist, he seized Lavirotte by\nthe throat and shook him violently. The pain in his shoulder was\nnothing. It was no more than if he had been touched by a piece of iron\njust uncomfortably hot. Yet he felt confused and queer in his head, as\nthough he had received the blow on his head, rather than on his\nshoulder.\n\nLavirotte now seized O'Donnell by the throat, and for a while, with\nthe two hands raised in the air--the one holding the knife, the other\nthe wrist of the hand that held it--the two men struggled fiercely.\n\nIt was a matter of life and death. O'Donnell had now lost all care for\nthe cause of the attack, and was simply engaged in a brute attempt to\ndefend his life against a brute attack. Both men were mad. Both men\nhad now lost everything but the instinct of victory. All the faculties\nof each were concentrated upon the muscles each used--upon the\nadvantages each gained--upon the chances each afforded. Each now meant\nto kill, and to kill speedily--to kill with all the force, all the\npower, all the devices of his body.\n\nOne was armed and whole; the other was unarmed and hurt. Both were\nsensible that this conflict could not last many minutes.\n\nThe two twisted and writhed and struggled abroad on the open way. Now\nthey swayed this way, now that. Now, as though one were about to fall;\nnow, as though the other. Now one strove to throw the other by the aid\nof mere weight and muscle; now the other sought to win by the force of\nstrangulation. Meanwhile, above the heads of both rose the two\nupstretched arms--one hand clasped around a wrist, one hand holding a\nbloody knife.\n\nThe two men's faces were livid. They breathed only now and then, and\nwith terrible difficulty. Their eyes were dilated and protruding, the\nnostrils wide set and quivering.\n\nFor some time, he knew not how long--he never knew how long the fight\nlasted--O'Donnell had felt something warm trickling down his back. He\nwas bleeding freely. He was half suffocated. He felt he must succumb.\nFor an instant everything was dark. Suddenly he saw once more; his\nvision, his senses were restored, but only to reveal to him the fact\nthat his powers were failing swiftly.\n\nThe two men rocked and swayed in the broad roadway leading towards the\ncove. Neither knew nor cared which way he went, so long as he might\ncling to the other. At the moment when O'Donnell's faculties returned,\nafter that instant's unconsciousness, the two men were struggling a\nfew feet from the rock behind which Lavirotte had hidden.\n\n\"Now,\" thought O'Donnell swiftly, \"for one last effort; if I fail he\nwill kill me.\"\n\nSuddenly relaxing his knees, he stooped so as to bring his head on a\nlevel with the shoulder of his antagonist; then, loosing his hold of\nLavirotte's throat, he seized him by the ankle, and, putting all his\nstrength into his right arm and back, he sought to lift and throw the\nother. But his strength was gone; his head was dizzy; his eyes grew\ndim. Finally, all was dark once more. He lurched heavily forward,\nstriking his antagonist in the chest with his head.\n\nLavirotte stumbled and fell backwards. O'Donnell struggled for a\nmoment to regain his upright position, but his strength was spent; he\nwas unconscious, and subsided in the middle of the road.\n\nNow was Lavirotte's opportunity. O'Donnell could not have resisted a\nchild. The most cowardly cut-throat that ever lifted steel need have\nno fear of him.\n\nThe darkness increased as the night went on. By this time it had grown\nso great that it was impossible to see an arm's length. The sky, for\nall the light it gave, might as well have been the solid earth. No\nsound stirred the profound silence save the mellow washing of the\nwaves upon the shore. It was sultry and suffocating. Now and then the\nair panted, beating this way and that in little hot gusts that brought\nno freshness and left no coolness behind. Although the murmuring of\nthe sea filled the night with a low plaintive music, the silence\nseemed to deepen as the minutes went by.\n\nAt length a form began to stir. For a while the man did not seem to\nknow where he was, or the circumstances which had led to his\ncondition. It was only by feeling around him he was able to know he\nwas in the open air. He felt the road, the stones, the sunbaked clay\nof the road. Then he listened intently awhile, and by his hearing\nconfirmed the notion that he was in the open air. That was the murmur\nof the sea. These little puffs of wind that beat against his face\nshowed he was not between walls.\n\nAh! Now something of it came back. There had been a struggle of some\nkind, a fight with someone. What was it exactly? This was the road to\nthe cove. Of course it was. The sea lay beyond there somewhere. To the\nright, to the left, no matter where, the sea was somewhere near. It\nwould be good to get down to the sea and lie down in its cool waters,\nfor he was aching and burning. What a fearful thirst! His tongue was\nparched, baked dry as the baked clay on which he sat. He had been\nhurt, how or why he could not recollect. There had been a fight. That\nwas all right. But why he had fought or with whom, these were the\nmysteries.\n\nOh! why did they not bring him some water? He was dying of thirst, and\nno one would come. He didn't remember going to bed. He never felt so\nsleepy in all his life before. It was a kind of deathly sleep, a sleep\nwith no mercy in it, a sleep that promised no ease, no repose, no\nalleviation of the torturing uncertainties. Such a bed, too; it was as\nhard as iron. What did they mean by giving so sleepy a man such a bed?\n\nWhat nonsense it was for his mother to sing a lullaby. He was a grown\nman, and needed no such inducement to sleep. Oh, this terrible,\ntyrannical sleep that brought no ease, no repose.\n\nHow strange that the cathedral organ should be booming away in the\ndark! If service was going on, why not have lights?\n\nLights! Was it magic? No sooner did he think of them than the whole\ncathedral blazed out for one brief moment, and then fell back into\ndarkness again. It was marvellous, incredible; and the cathedral\nseemed so vast, vaster than the reason could believe, although the eye\nhad seen it. And, then, there was the music once again. Why did the\norganist play only when the lights were out? That was the swell organ.\nIt was the loudest organ he had ever heard. What seemed most\nincredible of all was the organ was big enough to fill the church, and\ndid fill it, until it made the windows, the pillars, ay, the very\nground itself tremble.\n\nGround! Ay, surely it was the ground. How extraordinary that he should\nbe lying on the ground!\n\nWhat was this so delicious and cool? Cool and refreshing after that\nhorrible dream of fighting with someone, and then waking on a road.\n\nAnd yet there was something in that dream, for this was a road.\n\nHe sat up.\n\nIt was very extraordinary. It was the most extraordinary thing that\nhad ever happened to him in his life. Was he alive, in the old\nfamiliar sense of that word? Of course he was, for this was a road,\nand he knew it was a road, and----\n\nLightning--thunder--rain.\n\nWhat was that he had seen beside him? The rain was refreshing. It was\ncooling his head, collecting his thoughts.\n\nWhat was that he had seen beside him?\n\nMore lightning--thunder--rain.\n\nWhat was that beside him?\n\nLavirotte--dead.\n\n\n\n\n                              CHAPTER V.\n\n\nLavirotte dead! Absurd.\n\nNow he remembered how it had been.\n\nLavirotte had sprung upon him out of the shadow of that rock, and\nseized him and sought to kill him, because Lavirotte was mad with\njealousy, or with southern blood, or with something else or other, no\nmatter what--mad anyway. And there was that burning sensation in his\nshoulder, and the fever in his blood, and that--ugh!--clammy feeling\ndown his back, But Lavirotte dead?\n\nNo; the very notion was preposterous.\n\nNow he remembered the struggle.\n\nAnother flash. Another roar of thunder. Another deluge of rain.\n\nHe looked wonderfully like death in that blue light. And yet in that\nstruggle he (O'Donnell) did not remember having struck the other. It\nwas a common tussle, an irregular wrestle, with the supreme interest\nof a knife added by Lavirotte. That was all. Yet he lay there\nmotionless, and it must have been a considerable time since he fell.\n\nWith great difficulty and a sense of oppression, O'Donnell rose\npartly, and crawled towards the prostrate man.\n\n\"Dominique,\" he whispered, \"Dominique, what is the matter? Rouse up.\"\n\nThere was no response. The form of the Frenchman lay there motionless,\ninert, nerveless. O'Donnell raised an arm; it fell back again into the\nmud of the road, unsustained by any trace of vitality.\n\n\"What can it be?\" thought O'Donnell, straightening himself, as another\nflash of lightning revealed the pallid face of Lavirotte. He waited\nfor the thunder to pass, and then, putting his hands around his mouth,\nshouted with all the strength that was left in him:\n\n\"Help! Help! Help!\"\n\nThe storm had not been unnoticed in the village, and many were awake.\n\nJames Crotty, boatman, had been roused by the first peal of thunder,\nhad filled a pipe, undone the door of his cottage, and come out to see\nhow the night went. His boat was moored in the cove, but as there was\nno wind his mind was easy about her. His wife and little ones were\nsafe asleep in the cottage, and his mind was easy about them. At the\nbest of times he was a light sleeper and a great smoker, and took a\nboatman's interest in the weather, fair or foul, but had a particular\ninterest in the great conflicts of nature.\n\nWhile he was standing in the doorway he was within a few hundred yards\nof the two men below near the cove. His cottage was about half-way\ndown the road, and it was quite possible to hear an ordinary speaking\nvoice from where the men now were.\n\nWhen O'Donnell's loud cry for help rang out in the stillness, Crotty\nstarted, and then listened intently. No other sound followed. There\nwas no mistaking the nature of that cry. He had heard the word as\ndistinctly as though it were spoken in the dark room behind him. \"It\ncan't be any of the men,\" he said, meaning the fishermen of the place.\n\"It is too early for any of the boats to be back, and too late for\nthem to be going out. What can have brought anyone down there at this\nhour? I'd better go and see, anyway.\"\n\nHe went down the little garden in front of his cottage, and gained the\nroad. He turned to the left. Then he went on slowly, cautiously,\nkeeping to the middle of the road.\n\n\"Who's there?\" he called out. \"What's the matter?\"\n\n\"Here,\" cried O'Donnell faintly, \"This way. Help.\"\n\nThe rain had now ceased, and the silence was intense. Far out there in\nthe darkness was the soft washing of the wavelets on the shore. No\nother sound burdened the night.\n\nGuided by O'Donnell's voice, Crotty now walked on with decision.\n\n\"What's the matter?\" he called out again. \"Who is it?\"\n\nO'Donnell's voice answered from the darkness. \"It is I, O'Donnell.\"\n\n\"Oh, Mr. O'Donnell, is it you? What's the matter?\"\n\n\"I'm hurt, badly I think, and here is Mr. Lavirotte insensible. I know\nhow I got my hurt.\" Crotty was now close to the speaker. \"That makes\nno difference; but I don't know how Mr. Lavirotte was hurt.\"\n\n\"Maybe 'twas a fight,\" said Crotty, in a tone of interest. A fight is\nalways an interesting thing, but a fight here and on such a night as\nthis was something which Crotty did not feel himself justified in\ntreating with anything but the greatest respect.\n\n\"Never mind what has been,\" said O'Donnell feebly. \"The thing is to\nget him to the village and call a doctor. I can't be of much help. I\nam quite weak. Come now, Crotty, look sharp. Knock them up at Maher's,\ntell them to put a horse in, and be back here in no time, and let\nthere be a doctor at hand by the time we get back. Run now. Don't lose\na minute.\"\n\n\"And leave you here by yourself, hurt? Aren't you strong enough to\nwalk as far as Maher's, or my place even?\"\n\n\"No. Be off. Every second you wait is killing us.\"\n\nCrotty started at the top of his speed, and in less than half-an-hour\nreturned with a car from Maher's hotel. He had brought a lantern, and\nhe and the driver carried Lavirotte to the car, and sat him up on it.\nThen Crotty got up and held the insensible man. O'Donnell got up on\nthe other side, and thus they drove to the hotel.\n\nHere the doctor was awaiting them.\n\n\"What's this, O'Donnell?\" he said. He knew the two men thoroughly.\n\"You two have been quarrelling. What is the meaning of this? Blood on\nboth! Nasty scalp wound. Don't think the bone is broken. Clear case of\nconcussion. What did you hit him with?\"\n\n\"Nothing,\" said O'Donnell. \"Is it dangerous?\"\n\n\"Dangerous! I should think it is dangerous. Dangerous enough to mean\nmanslaughter, it may be.\"\n\n\"Good heavens!\" cried O'Donnell, faintly. \"I assure you I never struck\nhim.\"\n\n\"All right. Stick to that. It never does to make admissions. What's\nthe matter with you? Blood and mud all over. Cut off his coat. Here,\ngive me the scissors. No bleeding except here. Ugly cut.\"\n\n\"Is it much?\" said O'Donnell, very weak now.\n\n\"Yes, it's a good hit.\"\n\n\"Will it do for me?\"\n\n\"I don't think so, if you have luck. He has a much better chance of\ngoing than you. What _did_ you hit him with, O'Donnell? It was a\nterrible blow. Something blunt--a stone, or something of that kind.\nIt's a downright shame that two young fellows like you, of good\neducation, and so on, should fall to hacking and battering one another\nin this brutal way, and at midnight, too. It's more like assassination\nthan fighting. A woman in the matter, eh?\"\n\n\"For heaven's sake, hush, O'Malley.\"\n\n\"All right. I'm not a magistrate. My business is with the bruises, not\nwith the row, or the cause of the row; but I'm sure it's a woman. Men\ndon't go ripping one another open for anything else nowadays.\"\n\n\"I swear to you, O'Malley, as far as I am concerned, there was no row,\nand that I did not strike him.\"\n\n\"Who else was with you?--although I'm not in the least curious. That\nwas a tremendous blow. I can't make it out. If he had stabbed you\nfirst, I don't think you could have struck that blow. I can't make it\nout. I can't do any more for you now. You mustn't lie on it, you\nknow.\"\n\n\"O'Malley,\" said O'Donnell, \"I want you to do me a great favour.\"\n\n\"Oh, my dear fellow, you needn't be afraid that I'm going to swear an\ninformation. It's nothing to me if two fellows go hacking and slashing\nat one another. I shouldn't like to see either of you killed outright\nfor the finest woman in creation.\"\n\n\"Do stop, O'Malley, like a good fellow. I'll tell you what you must do\nfor me. I want you to break the matter to her to-morrow morning the\nfirst thing.\"\n\nSuddenly the manner of the glib doctor changed. \"My dear fellow, I\nhave been very impertinent, very thoughtless, very rude, and as soon\nas you are quite well you shall punch my head, and welcome. I had\nclean forgotten that you are going to be married. When you do punch my\nhead, I hope it won't be quite so terribly as poor Lavirotte's. I'll\ndo anything in the world I can for you. What am I to say? She's at her\nmother's, I suppose.\"\n\n\"Yes; she's at her mother's. The fact is, I don't exactly know what to\nsay. I can't tell her the truth.\"\n\n\"And you want me to tell her a lie, eh?\"\n\n\"No, no; I would not be so rude as to ask you to do anything of the\nkind. The fact of the matter is, I can tell and trust you----\"\n\n\"Stop, O'Donnell, don't. Don't tell me anything you want to keep\nquiet. If you told me now 'twould be known in China at breakfast-time.\nI'm dying to know all about it, but, as your friend, I recommend you\nnot to tell me a word of it. What shall I tell her?\"\n\n\"That I have been a little hurt.\"\n\n\"Lie No. 1. You are a good deal hurt.\"\n\n\"That I shall soon be all right.\"\n\n\"Lie No. 2. For a man who wouldn't be so rude as to ask me to tell a\nlie, you are getting on marvellously.\"\n\n\"And that you do not know how I got the hurt.\"\n\n\"Truth this time, by Jove, for a change. And most unpleasant truth,\ntoo, for I really am most curious to know.\"\n\n\"Then you shall know.\"\n\n\"No; as your friend I decline to listen. There, I promised to do the\nbest for you. I'll lie as much as ever I choose, and confound your\npoliteness for not asking me. There, now, you mustn't speak any more.\nYou must keep as quiet as possible.\" And after a few words more of\ninstruction the busy, talkative little doctor left O'Donnell.\n\nLavirotte had been put in another room. O'Malley went to him, and\nagain examined his condition, and then left the hotel.\n\nWhen O'Donnell was alone, he thought to himself: \"I suppose if\nLavirotte recovers, we may be able to hush the matter up. But if he\ndies--great heavens, what a thought!--there will be a trial, and how\nwill it go with me? I can prove nothing. I know nothing of how he came\nby this hurt. It will seem to anyone that we fought. It may seem that\nI was the aggressor. That I attacked him foully, and killed him\nruthlessly while he was trying to defend his life. This is a terrible\nthought. It will drive me mad. Why, they may bring in a verdict of\nMurder! They may hang me. Innocent men have been hanged before. Hang\nme on the very day that I was to have been married. What can I do for\nyou, Nellie? What better can I do for you, Nellie, than die here?\"\n\n\n\n\n                             CHAPTER VI.\n\n\nThe next morning after the encounter on the road, all nature seemed\nrefreshed, rehabilitated. The grass sparkled green with rain, the\ntrees glittered in the sun, the air was pure and cool and sweet. Not a\ncloud darkened the sky. The whole world seemed full of joy and lusty\nhealth. One felt that something had occurred, some burden had been\nwithdrawn from the earth, some portentous influence had retired.\n\nEarly bathers were hurrying towards the strand before Dr. O'Malley\nwas stirring. When he awoke, the events of the previous night at\nonce flashed into his mind. \"Here's a nice pickle,\" he thought.\n\"Mysterious event--two men half-killed--both deserve to be killed, no\ndoubt--eminent medical man called in--eminent medical man treats with\nthe utmost skill--no confidence beyond confidence in his professional\nability reposed in medical man--medical man entrusted with a\nMission--Mission to console Beauty--infernal nuisance!--infernal\nnuisance, Tom O'Malley! I suppose there's nothing for it but to keep\nyour word, and do half-an-hour's clever lying to this Miracle.\"\n\nBetween seven and eight o'clock the post was delivered in Glengowra.\n\n\"I'll wait till I see if there are any letters,\" said O'Malley to\nhimself. \"My appointment as Surgeon-General to the Forces may at this\nmoment be the property of Her Majesty's Postmaster-General. I suppose\nif they do offer I must accept. Oh, dear! why didn't I think of making\nlove to this Paragon? Poor girl! It's no laughing matter for her this\nmorning.\"\n\nThe post brought no letter for Dr. O'Malley, and as soon as the\ncarrier had gone by, O'Malley put on his hat and set out for the house\nwhere Mrs. Creagh lived.\n\nThe postman was still in the street, and O'Malley gradually overtook\nhim. At the rate the two men walked, allowing for time lost by the\npostman in delivering letters, the doctor would arrive at Mrs.\nCreagh's half-an-hour before the other. He found all stirring at the\nwidow's place.\n\nHe had some doubt as to whether he should tell the mother first; but,\non second consideration, he decided that Miss Creagh was entitled to\nthe earliest news. He knocked at the door and was shown in. \"When\nNellie entered the room she was dressed in white, the same dress she\nhad worn that day he threw away the flowers and used words instead. Of\nall the things looking fresh to the doctor's eyes that morning she\nseemed freshest. The bloom of perfect health was on her cheek, the\nlight of perfect health was in her eye. She wore no ornament but her\nengaged ring and a rose in her hair.\n\n\"It's a pity,\" thought the little doctor, \"that such a glorious\ncreature as that should ever be troubled or grow old. What are kings\nand princes and all the powers and vanities of the world--what are all\nyour Roman triumphs--compared to such amazing perfection?\"\n\n\"A very early call,\" he said, \"but I was up and I thought I'd look in.\nIt would be impertinence to ask you how you are. I had a little\nbusiness this way, and, as I said, I thought I'd look in.\"\n\nThe girl smiled. Her face remained unclouded.\n\n\"I know a call at this hour is not convenient or considerate, but I\nhad a little thing to say to you.\"\n\n\"Something to say to me?\" she said, with a look of gentle surprise.\nWhat could he have to say to her so early? She smiled faintly as\nthough to encourage him; for now it struck her suddenly that what he\nhad to say was not pleasant.\n\n\"The fact is, a little accident has occurred. I am a doctor, and know\nwhat I am saying. It is the merest scratch. You must not be alarmed.\nThere now, sit still.\"\n\nShe had risen. All the bloom had now left her cheeks. A little still\nlingered at her lips. \"You may tell me, Dr. O'Malley. I know he is not\ndead. I can see that by your face. Where is he?\"\n\n\"Sit down. My dear young lady, you are going too fast. Dead! Why he's\nnearly as well as ever, and will be better than ever in a short time.\"\n\n\"Tell me all,\" she said. \"May I go to him?\"\n\n\"I haven't seen him this morning yet. Better wait till after\nbreakfast.\"\n\n\"Where is he?\"\n\n\"At Maher's.\"\n\n\"Dr. O'Malley, tell me exactly what has happened.\"\n\nSomething strained and rigid in her voice warned him that he must be\nquick if he meant to be merciful. \"There was a stupid quarrel of some\nkind,\" he said, \"and he got a slight wound--I assure you not in the\nleast dangerous.\"\n\n\"With whom was the quarrel?\"\n\n\"With Mr. Lavirotte.\"\n\n\"Mr. Lavirotte--Mr. Lavirotte! Did Mr. Lavirotte _stab_ Eugene?\"\n\n\"Yes, a mere nothing, though, a pin-hole. You will be angry with me\nfor causing you any uneasiness when you know how slight it is.\"\n\n\"Why did Lavirotte stab Eugene?\"\n\n\"Because there was some foolish quarrel; I really don't know what.\nIt's ridiculous to call the thing a stab; it's a mere scratch.\"\n\n\"Is Lavirotte hurt?\"\n\n\"Yes; he is more hurt than O'Donnell. But putting the two hurts\ntogether, I assure you they're hardly worth talking of.\"\n\nThe straightforward calmness of this girl was terrifying him. He was\nbecoming fidgety, and not well able to gauge the value of the words he\nused.\n\n\"You know the cause of the quarrel?\"\n\n\"Upon my honour I do not.\"\n\n\"You know the cause of the quarrel. We need not mention it now. You\nsee how calm I am. You must tell me the truth. Are you sure _neither_\nof these men will die?\"\n\n\"I--I----\"\n\n\"Mind, _sure?_\"\n\n\"I am as sure as man can be O'Donnell will not die.\"\n\n\"But Lavirotte will?\"\n\n\"Lavirotte may. It is impossible to say. I left him unconscious. He is\nunconscious still.\"\n\n\"I will not wait till after breakfast. I will go now. Stay a moment--I\nmust tell mother, and get my hat; I will not keep you long.\"\n\nAs the girl left the room, the postman turned into that street. As she\ncame into the room again, with her hat and gloves on, the postman\nwalked up the little garden and handed in a letter. It bore the Dublin\npostmark, and was addressed to \"Miss Creagh.\" Her mother, who was in\nthe hall, took the letter into the room where the doctor and the girl\nwere standing. \"A letter for you, Nellie,\" the mother said. \"Will you\nkeep it until you come back? It's from Ruth, I think.\"\n\n\"I'll take it with me,\" said the girl, and put the letter in her\npocket. \"Ruth,\" she said, in the same calm, unmoved voice, \"is one of\nmy pupils in Dublin. Now, Dr. O'Malley, if you are ready, let us go.\"\n\n\"She will not let me go with her,\" said the mother, in a tone of\nconcern.\n\n\"I am better alone, mother,\" said the girl, and she turned and moved\nout of the room.\n\nO'Malley followed her, and in a few minutes, which were passed in\nsilence, they were at the hotel. O'Malley went upstairs to the room\nwhere O'Donnell lay.\n\n\"All going on well?\" he said briskly to the patient. He went through\nthe ordinary formalities. \"Yes,\" he said, \"all going on well. Very\nlittle fever. We shall have you all right in time for your wedding.\nYou can go away then and pick up strength, amuse yourself for a month\nor two.\"\n\n\"Have you seen her?\" asked O'Donnell. \"How did she take it?\"\n\n\"Yes, I've seen her. She took it like an angel, like a heroine. I gave\nher leave to come and see you later.\"\n\n\"When do you think she'll be here?\" asked the invalid.\n\n\"Oh, at some reasonable time. Young ladies don't visit at eight\no'clock in the morning. You'll promise to keep yourself quiet when she\ndoes come?\"\n\n\"Very quiet. Did she get a great shock?\"\n\n\"Not so much a shock as a turn. Will you promise to be very quiet if I\nlet her come soon? The fact is, O'Donnell, she will be here in a few\nminutes. There, of course, you guessed it; she is here already; she\ncame with me. Now I'll go down, and she may come up and see you, but\nyou must not talk too much.\"\n\nWhile the brisk little doctor was preparing O'Donnell for the visit of\nNellie, the latter took out her letter and began to read it. Suddenly\nher face, which had been pallid ever since she heard the bad news,\nflushed, and she uttered an exclamation of dismay. \"Such news,\" she\ncried, \"and on this morning!\" The letter ran as follows:\n\n\"My dear Nellie, I told you I would write you if there was any news.\nThere is news, and very bad news, I am sorry to say. Papa came home in\nthe middle of the day quite unexpectedly, and told mamma that all was\nover and we were ruined. I don't think it's known in town yet, but\nmother told me everyone would know it to-morrow. This is dreadful.\nMamma and papa are awfully cut up. I write you this news at once,\nbecause, of course, dear, you are greatly interested in Mr. O'Donnell,\nand his father is in some way mixed up with papa. I hope it will not\nhurt your _friend_.\" Then followed an account of some family matters,\nand the signature, \"Ruth Vernon.\"\n\n\"I must not say a word of this to Eugene now,\" she thought. \"He told\nme his father was very largely mixed up with Mr. Vernon. Of course I\ncould not tell Eugene. I feared there was something wrong there, but I\nwas bound in honour, and by my promise to Ruth, not to speak of it to\nanybody living. When I met him first on the beach, and Lavirotte\nintroduced us, I was greatly struck by the coincidence that I should\nmeet him, knowing as I did, that he might suffer greatly if anything\nhappened to Mr. Vernon.\"\n\nIn a few minutes O'Malley came down and said she might go up. \"He is\ngetting on well,\" he said cheerfully, \"and there's nothing in the\nworld to fear.\"\n\nThat day went over quietly at Glengowra. Early in the afternoon\nLavirotte recovered consciousness. The police had got scent of the\naffair, and were making inquiries.\n\nIn the afternoon news reached the village that the great banking-house\nof Vernon and Son had failed for an enormous sum. It was kept from\nO'Donnell, but Lavirotte heard it. \"I must telegraph to London,\" he\nsaid. \"Someone must write the telegram for me.\" The body of the\nmessage ran as follows:\n\n\"Vernon and Son bankrupt. See about your money at once. Am ill, and\ncannot go over.\"\n\nWhen the telegram reached London it was delivered to a young woman of\ntwenty years of age, who grew pale and flushed, and flushed and pale\nagain, upon reading it. \"What?\" she cried, \"Dominique ill. My darling\nsuffering and I not near him. I will leave to-night for Glengowra.\nStop! I must get money somewhere first. I have none, not a penny--the\nattorney told me he would have my money to-day. These people are\npressing me for the rent. They are hateful creatures. I will go\nto the solicitor at once. I can pay what I owe then, and go over by\nto-night's mail.\"\n\nShe put on her things. The landlady was waiting in the hall. The\nlandlady would feel obliged if Miss Harrington would give her the rent\nnow, before going out. She really must insist on being paid now. She\ncould not afford to give six weeks' credit, and she had had an\napplication for the rooms. There were six guineas for the rooms and\nten guineas for meat and drink, sixteen in all. Would Miss Harrington\npay or leave, please?\n\nMiss Harrington would pay upon her return from her solicitor.\n\nOh, that old story about the solicitor! People could not go on\nbelieving this old tale for ever. If Miss Harrington did not bring the\nmoney with her, she need not come back that day. Whatever she had\nupstairs would not pay half the bill, and indeed Miss Harrington ought\nnot to go out with her watch and chain and leave struggling people so\npressed for money.\n\nThe tears were now falling fast from the young girl's eyes. She was\nalone, friendless, in London. She had not a coin in her possession.\nShe took off her watch and chain and laid them silently upon the hall\ntable. She made a great effort at self-control, and said, pointing to\nthe third finger of her left hand: \"I have nothing else of value but\nthis. Shall I leave it also? It was given to me by one very dear to\nme.\"\n\n\"It would help,\" said the landlady, \"and I have my husband and\nchildren to think of.\"\n\nThen she took off the ring--his ring--the ring he had given her to\nwear until he gave her a simpler one with a holier meaning. She put\nthe ring down on the table beside the watch and chain. Then her heart\nhardened against this woman, and no more tears came, and bowing\nslightly she said good-bye and left the place, meaning never to\nreturn.\n\nShe went to her solicitor's. He was away. Would his managing clerk do?\n\nYes, anyone who could give her information about her affairs.\n\nThe managing clerk had bad news--it was terrible news indeed. They had\nnot been able to get the money from Vernon and Son. Vernon and Son\nwere bankrupts according to to-day's reports, and all her money was\ngone.\n\nWould there be none of it coming to her?\n\nNo. Owing to the way in which the money was lent there was no chance\nof getting any back.\n\nThen she left the office, homeless, friendless, penniless. She had not\neven a shilling to telegraph to him--her Dominique. Whither should she\ngo? Where should she turn?\n\nTo the river.\n\n\n\n\n                             CHAPTER VII.\n\n\nDora Harrington found herself in the Strand, in the full light of a\nsummer's day, homeless, friendless, penniless. Her last chance was\ngone. Vernon and Son, who held all the money she owned in the world,\nhad failed, and failed in such a way as to leave no prospect of her\never getting a penny out of the five thousand pounds confided to them.\n\nShe was an orphan, and had spent much of her life out of these\nkingdoms. She knew nothing of business. Mr. Kempston, her solicitor,\nhad been appointed her guardian, with full discretionary powers as to\nthe disposal of her property. She and he had not agreed too well, for\nshe had wished to marry Lavirotte, and he had opposed her desires. She\nhad wished to get control of her property, and had been denied, and\nthe relations between her guardian and herself had of late been most\nstraitened. Only for his good-humour in the matter there would have\nbeen an open rupture. He had politely, but firmly, refused to agree to\neither of her suggestions. She had impulsively, warmly protested\nagainst what she called his interference in her affairs.\n\nTwo years ago she had first met Lavirotte. She was then a young girl\nof eighteen. She met him at a concert of amateurs in London. He made\nlove to her, and she fell in love with him. He proposed, and she had\naccepted. Then he explained his position.\n\nHe was not rich enough to marry. She told him she had a little\nmoney--she thought about five thousand pounds. He laughed, and said\nthat might be enough for one, but was no good for two, adding,\nbitterly, that he did not know how he could possibly advance himself\nin the world. He was then the only photographer in the small town or\nvillage of Glengowra, and the chance of his getting into any better\nway of making money did not seem likely to him.\n\n\"You sing very well,\" she said. \"You have a good voice, and you know\nmusic. Have you never thought of music as a profession?\"\n\nHe had never thought of music as a profession until then. He was only\ntwenty-two at the time. He knew very well he could not afford to go to\nItaly or even to the Conservatoire. He had no money laid by, nor was\nthere any likelihood of his having money to lay by.\n\nThen she suggested that he should borrow some of her. To this he would\nnot listen. If he were not able to attain a competency himself, he\nwould never put it in the power of fools to say that he had climbed\ninto a profession aided by anyone, least of all by his future wife.\n\nAfter much talk and expostulation on her side, he was induced to agree\nto accept the loan of a few hundred pounds. Then it was that she went\nto her solicitor and guardian, told him she had made up her mind with\nregard to her future, and that the man of her choice was a Frenchman,\nby name Lavirotte, and by profession a photographer in the town of\nGlengowra, in Ireland.\n\nThe solicitor was considerably surprised, and said he should not be\nable to come to any decision for a few days. Mr. Kempston was a\nbachelor, and had no means of taking care of his ward beyond the\nordinary appliances of his profession. He could not invite her to his\nbachelor home, and her income was not sufficiently large to warrant\nhim in appointing a lady companion or chaperon of any kind; all he\ncould do in her interest was to find her moderately comfortable\nlodgings, and see that she regularly received the dividends on her\nshares in the banking concern of Vernon and Son.\n\nMr. Kempston was the sole surviving executor and trustee to her\nfather's will, and in the exercise of his discretion he had invested\nher five thousand pounds in shares of Vernon and Son, Unlimited. She\nknew nothing whatever of business, and Mr. Kempston's managing clerk,\nin alluding to her money as lent to the bankrupt firm, was simply\nusing popular language, and attorning to the ignorance of business\ninherent in the female mind. He knew very well that she, being a\nshareholder, had not only lost all the money she owned, but was liable\nto the very last shred of her possessions for any further demands\nwhich might be made upon her with regard to this failure. He had felt\nhimself fully justified in telling her she had lost all her fortune,\nthat she was, in fact, a pauper; but he had not felt himself called\nupon to explain that later on she would appear in the light of a\ndefaulter.\n\nDora Harrington, now an outcast from home, and fortune, and friends,\nfound herself in the great city of London absolutely without resources\nof any kind. Her money was gone, she knew. Her guardian and she were\nno more than business correspondents. Her lover's position in\nGlengowra forbade the hope he might ever be able to marry her, and she\nhad within herself no art or knowledge by which she could hope to earn\na living.\n\nWhat was now to be done? Where should she eat that evening? Where\nshould she sleep that night?\n\nNowhere!\n\nWhere was nowhere?\n\nThe river.\n\nAnd yet to be only twenty years of age, and beautiful, as she had been\ntold, and still driven to the river by the mere fact of a few pounds\nthis way or that, seemed terribly hard to one who knew she had done no\nharm.\n\nIf he were but near her! But he was poor and hurt, and it would only\nhelp his pain if he knew that she had been cruelly hurt by fortune.\nAnd yet, how could she live? Where could she go? Whither should she\nturn? The world of life seemed closed against her, and only the\nportals of death seemed fit for her escape.\n\nTo be so young, to love and be loved, and yet to have no avenue before\none but that leading to the ghastly tomb, appeared hard indeed.\n\nIt is true that of late her Dominique had seemed less eager in his\nhaste to write to her, less fervent in his expressions, less tender in\nhis regard. But this may have been owing to his sense of inability to\nface the future with her maintenance added to the charges upon his\nslender means. There was no prospect of his advancing himself to any\nsubstantial result. He had written her, saying he had devoted much of\nhis time lately to the cultivation of his voice and the art of music.\nThat, in fact, he was now leading tenor in the choir of the church.\nBut he was careful to explain to her that this meant no financial\nadvancement, and that in fact it was to him the source of some small\nlosses of time and money. Besides, there was no one in Glengowra who\nknew much of music save the two organists, and the knowledge of even\nthese was not of much use to anyone who had to think purely of voice\nculture as opposed to instrumentalism.\n\nIn the present there seemed no germ of hope. The future was a blank,\nor worse than a blank. And to-day, now, this hour, was an intolerable\nburden which could not be endured.\n\nAnd yet how was she to remove it? How was she to get from under this\ncrushing sense of ruin? It was plain to her that the ardour of his\naffection was cooling, not owing to any indifference on his part to\nherself, but owing to the fact that he recognised, even with the\nprospect of her five thousand pounds a year hence, the impossibility\nof their union.\n\nNow that five thousand pounds had vanished wholly, and the possibility\nof their marriage had been reduced to an almost certain negative.\n\nWhat should she do? What was there to be done? The answer to this\nquestion did not admit of any delay. Between this moment and the\nmoment of absolute want was but an hour, two hours, three hours, a\ncondition which must arise absolutely by sunset. She could do nothing.\nIt was possible to walk about the streets, no doubt, until death\novertook her; but why should she wait for death. If Death were coming,\nwhy should she not go and meet him half-way?\n\nStill it was hard to die. To die now in the full summer, when one was\nyoung and full of health, although bankrupt in hope, when the sun was\nbright, and the air was clear, and great London at its most beautiful.\nTo die now without even the chance of communicating with him,\nDominique?\n\nHe, too, was ill, dying perhaps. Yes, he was dying. His affection\ntowards her seemed waning. He had no worldly prospect, and her little\nfortune was wholly gone. If death would only come in some pleasant\nshape she would greet it gladly; but the notion of wooing death was\ncold and repugnant. The waters of the river were chill, and full of\nnoises and foul contagion. People had not willed themselves into life;\nwhy should they not be allowed to will themselves out of it?\n\nFor hours she walked along the crowded streets of London. Moment by\nmoment faintness and the sense of dereliction grew upon her. The\nactive troubles of the morning had passed away, and were now succeeded\nby a dull numbing sense of hopelessness. She had no longer the energy\nto protest against her fate. She moved through the crowded ways\nwithout hope, without fear, without anticipation, without\nretrospection. She had the dull, dead sense of being an impertinence\nin life, nothing more. She wished that life were done with her. Life\nwas now a tyrannical taskmaster, who obliged her to walk on endlessly,\nwith no goal in view; who compelled her to pass among this infinite\nmultitude, debarred of all sympathy with them, of all participation in\ntheir joys.\n\nAt length the sun fell, and minute by minute the busy streets grew\nstiller. The great human tide of London was ebbing to the cool and\nleafy suburbs.\n\nShe found herself in a neighbourhood which she had never before\ntrodden. She had passed St. Paul's, going east, and then turned down\nsome dark, deserted way, until she found the air growing cooler and\nthe place stiller.\n\n\"I must be near the Thames,\" she thought. \"Fate is directing my steps.\nThe future is a blank. Let the present be death.\"\n\nShe was now beginning to feel faint from physical exhaustion. She had\nsought that solitary way because she found she could no longer walk\nsteadily. She had eaten nothing that day.\n\nIt was now close to midnight. This place seemed so sequestered, so far\naway from the feet of men, that she felt she might lie down and sleep\nuntil the uprousing of the great city. But she thought:\n\n\"If I sleep here, I shall wake here, and what good will that be to me?\nIf I sleep in the river, I shall wake--Elsewhere.\"\n\nShe found herself under a square tower. She leaned against the wall,\nirresolute or faint. She moaned, but uttered no word. In a few moments\nshe placed her hand against the wall and pushed herself from it, as\nthough repelling a final entreaty. Then she staggered down the street\nand into a narrow laneway that led to the river.\n\n\n\n\n                            CHAPTER VIII.\n\n\nIt was midnight, and as silent as the grave. The quality of the\nsilence was peculiar; for although no sound stirred the air close at\nhand, there was, beyond the limits at which the ear could detect\nindividual sounds, from minute to minute a tone of deep murmur, which\nwould have been like the noises of a distant sea but that it was\npulseless.\n\nOverhead hung an impenetrable cloud of darkness. There was no moon, no\nstar, no light from the north. Looking right overhead, one saw\nnothing, absolutely nothing. The eyes of the living were, when turned\ntowards the sky, as useless as the eyes of the dead. But casting the\neyes down, one could see roofs, and towers, and spires, and domes, dim\nand ghastly in the veiled underlight, glowing upward from the streets\nof a vast city.\n\nNo wind stirred. The broad river, with its radial gleams of light\nshooting towards the lamps, moved no more than an inland lake into\nwhich no stream whispers, from which no stream hurries forth. It was\nhigh water.\n\nLooking down from the giddy height, no moving forms could be seen, a\npoliceman had passed under a little while ago, and none would pass\nagain for a little while more, except some thief on his way to plunder\nthe living, or some poor, troubled, outcast brother on his way to the\nriver to join the silent confraternity of the dead.\n\nThe leads were slippery with dew and green slime; the battlements were\nclammy and cold. To look straight down one should raise himself\nslightly on the parapet of the embrasure. Then he saw a perpendicular\nchasm, two hundred feet deep on his side, a hundred feet deep on the\nside opposite. On the four sides of the leads were four such chasms,\nand in all of them lay the dark heavy gloom of that summer night, save\nwhere once in each cleft there burned a fiery point--the gas-lamp--to\nscare the unlawful and light the harmless through the silent\nways--part of the mighty city-labyrinth lying below.\n\nOn the leads it was impossible to see anything. From parapet to\nparapet, from battlement to battlement, from embrasure to embrasure\nwas to the eye a purposeless void. It was impossible to guide the\nmovements except by the sense of touch; for although when one gazed\ndownward on the roofs below, the chequered glow hanging above the\nstreet gave the eye purpose, when one drew back from the parapet all\nwas dark, the dull reflection of the city's light did not reach upward\nfar enough to illume the open space within the four walls.\n\nYet there was life and motion on those leads, in that darkness set in\nthe solitude. A heavy, slow tread could be heard now and then, and now\nand then groans, and now and then words of protest and anger, bitter\nreproach, tremulous entreaty, fierce invective, and passionate\nlamentation. The voice was high and quavering like that of a woman\noverwrought, or a man overwrought or broken down by sorrows or by\nyears. Then these sounds would cease, the footsteps, the groans, the\nwords, and the silence of a blind cave in which no water dripped, and\nwhich harboured only the whispering and confounded echoes of a far-off\nstream, fell upon the place and filled out the measure of its\nisolation.\n\nThe slow measured tread of the policeman broke in once more upon the\nlistening ear, gained, reached its height, and was lost in the still\nocean of darkness.\n\n\"I am accursed. Nothing favours me. All is against me. No wind! No\nrain! Wind and rain are my only friends. They are the only things\nwhich can now be of service to me, and for a week there has been\nneither.\"\n\nThe querulous, complaining voice was hushed. The shuffling feet moved\nrapidly across the leads. Then all was still once more. Stop! what is\nthat? In the street below an echo to the wail above? No words can be\nheard, yet the purport of the voice is unmistakable.\n\nThe listener catches the import of those tones. He has heard similar\nsounds before.\n\n\"It is a woman,\" he says. \"Men never whine here, and at this hour,\ngoing that way! In a quarter of an hour it will be all over with her.\nA quarter of an hour! How long have I been here, slaving and toiling\nday and night, carrying away bit by bit what lies between me and\naffluence, and to think that in a quarter of an hour, from one bell of\nthe clock of St. Paul's to the next, I might find an end to all my\nhopes, and fears, and labours, and lie at peace, as far as this world\nis. Hark! Why does she pause beneath? She cannot suspect, no one can\nsuspect why I am here. All the dreary months of terror and sweat that\nI have spent here never drew from me one word, one sign which could\ngive a clue.\"\n\nThe figure of a woman in the street below could be seen dimly on the\nother side of the way. She leaned against the wall, irresolute or\nfaint. She moaned, but uttered no word. In a few moments she placed\nher hand against the wall and pushed herself from it as though\nrepelling a final entreaty. Then she staggered down the street and\ninto a narrow laneway that led to the river.\n\n\"She is gone,\" said the voice in the darkness. \"She is taking all her\ntroubles with her to the greasy Thames. Why should not I, too, take\nall my troubles thither and end my care? A quarter past! Before the\nhalf-hour strikes, I and my secret, my great secret, might be gone for\never. Has she a secret, or is it only the poor want of bread and\nshelter, or is it unkindness, a hope destroyed, love outraged,\naffection slighted? Why should I inquire?\"\n\nFrom the narrow lane into which she had struck, a moan reached the\nlistener's ears.\n\n\"She is in no great haste. This is not the despair of sudden ruin to\nlife or hopes. Her misfortunes have crawled gradually upon her, with\npalsied feet and blows that maddened because they never ceased--not\nbrave blows that drive one furious and to swift despair. _I_ am the\nvictim of this slow despair. Why should I drag out wearily, toilfully,\nin terrors that I make myself, the end of my old life?\"\n\nAgain the woman groaned.\n\n\"Curse her! Can she not go? Who minds a woman more or less in the\nworld? The world is overstocked with them. No one is here to pity her.\nWhy should she pity herself? It would be a mercy to her to take her\nand lead her to the brink and push her in. Why, it would shorten all\nher pains. Curse her, there she groans again. No rain, no wind to help\nme, and only these groans for a goad to my despair. I will not hear\nthem any longer. My own troubles are more than I can bear. Stay! That\nis a lucky thought. I'll go down and tell her that the police are\nhere, coming for her, and that she has not a moment to spare.\"\n\nAgain the woman's voice was heard.\n\n\"Forty years ago I could not take that voice so coldly, for all women\nwere then to me the sisters of one; my sweetheart then, my wife, the\nmother of my children, now the tenant of the neglected grave miles and\nmiles and miles away out there. Now all the children dwell in houses\nsuch as hers, and with her and them went out the life of me. I never\ncared to see the younger brood, for when my wife died it seemed to me\nthat all who loved me, or whom I loved, came to me but to die, and so\nI steeled my heart against the new brood and slunk into myself, shut\nmyself out from them and all the world, and took to lonely ways and\nsolitude until I came to this.\"\n\nFor a while no sound reached the ear. At last there was a sob, not a\nwoman's voice this time, but a man's.\n\n\"I hardened my heart against them, and the world seemed to have\nhardened its heart against me. I am lonely and alone. There is no\nwind. There is no rain. There has been no wind or rain for weeks. For\nweeks I have been ready for either, and either will not come. Twice a\nday the river gains its full height, asking me to go with it out of my\nloneliness and my toil. Heaven will not send rain or wind to me.\nHeaven took my wife and happiness. Heaven sent the river to me. I have\noften thought of going. I cannot leave this place and live. I cannot\nstay in this place and live. Hark! I hear the first rippling of the\nriver as it turns its footsteps towards the sea. What sound is that?\nShe! Five minutes by the clock and all will be over with her. What?\nStriking half-past? Idiot that I am! Why should I burden myself with\nthe despairs of another hour? I shall await the five minutes. For\nI should not care to be--disturbed. I should not care to hear or\nsee--anything of her. I am alone. I would go alone. I am in no humour\nfor company. I am too big with my own griefs to care for those of\nothers. I have feasted on sorrow until I have grown enormous,\ncolossal, distended beyond human shape. Let my great secret die with\nme. Let me die alone. I am a giant in the land of woes. I am Giant\nDespair. She has closed the door behind her ere this. It is time for\nme to knock. I have no farewells to take. That is lucky. Not one heart\nin all London will beat one beat more or one beat less when I am\ngone.\"\n\nThe feet trod the leads more vigorously than before. Then a step was\nheard descending the ladder.\n\n\n\n\n                             CHAPTER IX.\n\n\nSt. Prisca's Tower stands alone in Porter Street, hard by the Thames,\non the Middlesex side, and between Blackfriars Bridge and the Tower of\nLondon. It is all that now remains, all that remained on that night,\nof St. Prisca's Church. City improvements had swept away the main\nportion of the building, and on that silent summer night, when that\nman descended from the leads of the tower, this square structure rose\nup, a mighty isolated shaft, two hundred feet above the pavement of\nthe street and the three small alleys which skirted its other sides.\n\nIn a short time after the voice ceased finally on the roof, the figure\nof a man--Lionel Crawford--emerged from the gloomy darkness of the\ntower door, and stood in the light of the lamp.\n\nLionel Crawford was a man of sixty-five years of age, bent in the\nshoulders, and a little feeble in the legs. His walk was shuffling and\nuncertain, but still he seemed capable of great physical effort, if he\nchose to exert himself. His face was dark, and of a leathery colour.\nHis eyes were dark, almost black, and protruded a little. His mouth\nwas large, the lips full and heavy, the teeth still white and sound.\nThe forehead was broad and high, and strongly marked with wrinkles,\nperpendicular and horizontal, dividing the forehead into four parts.\nTwo smooth, wide, arch-shaped spaces stood up over the brows, and\nabove them, slightly retreating, two smooth convex expanses. His hands\nwere large, ill-made, knotty. In the lamp-light he took off the soft\nfelt hat he was wearing and disclosed a head bald to the apex, but\nhaving still around its lower edges and behind a thick covering of\ncurly black hair. He was dressed in clothes which had been those of a\ngentleman at one time, but were now nothing more than the meanest\ndevice for covering the body and keeping it warm.\n\nWhen Lionel Crawford had stood in the light of the lamp for a short\ntime he drew himself up to his full height, inflated his lungs, and\nlooked around defiantly. To judge by his face, defiance was an\nattitude familiar to his mind. But here was no one to see it, only the\ncallous walls, the imperturbable night.\n\nFrom the top of the tower he had marked the way taken by the woman. It\nwas a continuation of the narrow alley into which the door of the\ntower opened. It led directly to the river, and in order to reach it\nfrom where he stood it was necessary to cross Porter Street.\n\nOnce more the measured tread of the policeman was heard approaching.\nLionel Crawford drew himself back into the deep doorway of the tower,\nand waited until the footsteps had passed the end of the alley and\ndied away in the distance. Then he issued forth, turned to his left\nout of the doorway, crossed Porter Street with a brisk step, and\nplunged into the narrow way the woman had taken. Before he had gone\nten yards the place became as dark as a vault; it was impossible to\nsee a yard ahead, and only that he knew the place well, he could not\nhave proceeded without feeling his way.\n\nNo ordinary man in an ordinary state of mind would, at such an hour,\nventure into that narrow, dark, forbidding way. But Lionel Crawford\nwas an exceptional man, in an abnormal state of mind.\n\nFrom the time he left the top of the tower until he obliterated\nhimself in the darkness, his mind had been in a dull lethargic state.\nHe fully intended putting an end to his existence that night. That was\nhis only thought. He should walk down to the end of that narrow lane.\nAt the end of that narrow lane was a wharf, and from the edge of this\nwharf to the surface of the water he had only a few feet to fall. Then\nall would be as good as over, for he could not swim, and it was not\nlikely--the chance was one to a thousand--there would be anyone there\nto attempt a rescue.\n\nNotwithstanding his familiarity with the place, he abated his pace a\nlittle and walked more with his old shuffling gait than when he had\nthe light to guide him. All at once he stumbled and fell.\n\n\"What is this!\" he cried, as he tried to rise. His feet were entangled\nin something soft, which yielded this way and that, and for a while\nhindered him from rising. At last he rose, and leaning against the\nwall for breath, rubbed the sweat from his forehead. His faculties\nwere numbed, and for a few moments he scarcely knew where he was or\nwhither he had been going. The first thing he clearly recalled was\nthat he had entered Winter Lane. Then he realised the fact that in the\ndark he had tripped over something now lying at his feet.\n\n\"But,\" he thought, \"what can be here? What can be lying here at such\nan hour? I was down here to-day and the place was clear. Now I\nremember I had intended going to the river. I had calculated on no one\nbeing at hand to prevent me. Fool that I was! How could I have\nforgotten the watchman of the wharf. I dared not throw into the river\nthe stones I get up with so much labour, lest he might hear me and\nhand me over to the police.\"\n\nHe now was standing over what had tripped him. He stooped down and\nfelt carefully, slowly, around him. His hand touched a face--a smooth,\nbeardless face--the hat of a woman.\n\nWhat was this? A woman lying prostrate here, and at such an hour. He\nseized the form by the shoulders, and shook it.\n\n\"What are you doing here?\" he said. \"Wake up. What are you doing\nhere?\"\n\nThere was a slight motion in the form of the woman. She made an effort\nto rise. He helped her.\n\n\"What do you mean, woman,\" he said angrily, \"by going to sleep in such\na place at such a time, and tripping up an old man who is on his way\nto--his Friend?\"\n\nThe woman answered in a feeble voice:\n\n\"I don't remember exactly how it was. I did not go to sleep. I think I\nmust have fainted.\"\n\n\"But this is no place for you to be, woman, at this hour of night.\"\n\n\"I did not mean to stop here,\" she said. \"I meant to go to--the\nRiver.\"\n\n\"_You_ meant to go to the River--to my friend, the River? So did _I_.\nYou faint and trip me up. That may be an omen of good luck to both of\nus. Come, although there is neither rain nor wind I feel in better\nhumour now. Are you hungry?\"\n\n\"I have no friend--no money.\"\n\n\"Are you young?\"\n\n\"Twenty years of age.\"\n\n\"Too young to think of death. Come with me. It cannot have been a mere\naccident that brought us two together. Come with me, my child. I am\nold enough to be your grandfather. Stop!\" he cried, suddenly. \"What is\nthat? Did you notice anything?\"\n\n\"No,\" answered the woman feebly.\n\n\"Do you know it _rains?_\" he said. The tone of despondency at once\nleft his voice, and was succeeded by one of exultation. \"I told you,\"\nhe said, \"we did not meet for nothing. I have been praying and cursing\nfor rain. I meet you, and here the rain is. Twenty,\" he said, \"and\ntired of life! Nay, nay; that will not do. You have a sweetheart? I\nwas young myself once.\"\n\n\"Yes.\"\n\n\"And he is false?\"\n\n\"No, no. He is ill and poor.\"\n\n\"I am alone, old, childless, friendless. You have stopped me on my way\nto the river, and brought the rain. One day, at any hour, I may be\nrich. If I live to win my gold, I shall share with you and your lad.\nIt would be a piteous thing that a sweetheart of twenty should die.\nCome with me; cheer up and come with me.\" He drew her arm through his\nand led her in the direction of the tower. \"Sweetheart,\" he said, \"it\nmakes one young again to think of saving love. I cannot see your face\nor figure; but all are sweethearts at twenty. What is his name?\"\n\n\"He is French,\" said she.\n\n\"French! What is his name?\"\n\n\"Dominique Lavirotte.\"\n\n\"Dominique Lavirotte!\"\n\n\n\n\n                              CHAPTER X.\n\n\nWhen Lavirotte returned to consciousness, the day after the encounter\non the road, he seemed to have but a hazy notion of what had occurred,\nand yet to have known that caution was necessary. He found one of the\nwomen of the house seated in the room. He asked her had he been hurt,\nand how he had been hurt.\n\nShe said: \"I don't exactly know. Mr. O'Donnell and you came here\ntogether. He is hurt, too.\"\n\n\"Much?\"\n\n\"His shoulder is cut, I believe. They tell me he is not very bad.\nMaybe you know something about it?\"\n\n\"My head is hurt,\" he said, \"and I cannot remember well. There is no\ndanger he will die, is there?\"\n\n\"The doctor says no, but that he'll want good caring.\"\n\nThen for a long time Lavirotte was silent.\n\n\"What does Eugene say about it?\" he asked at length. \"Does he know how\nhe was hurt or how I was hurt?\"\n\n\"They did not tell me. I do not know.\"\n\n\"Will you take my compliments to Mr. O'Donnell, and ask him if he\nremembers what happened?\"\n\n\"I don't think I'd get much for my trouble if I did. The police have\nbeen here already trying to find out about the matter, and Mr.\nO'Donnell refused to tell them anything.\"\n\n\"Refused to tell them anything! Dear Eugene! dearest Eugene. Most\nloyal of friends! I always loved him.\"\n\nThen there was another long interval of silence.\n\n\"Who is with my dear friend Eugene?\"\n\n\"I don't know who is with him now. His father and mother were here\nearly in the day. They have bad news I am told. Some great man in\nDublin is closed.\"\n\n\"Some great man in Dublin. Did you hear his name?\"\n\n\"No; but they say it will be very bad for old Mr. O'Donnell.\"\n\n\"Will you ask Mr. Maher to come this way?\"\n\nWhen the landlord entered, he said: \"Who is the great man that has\nfailed in Dublin?\"\n\n\"Mr. Vernon.\"\n\n\"Ah, Mr. Vernon. So I guessed. This will be bad for the poor\nO'Donnells.\"\n\n\"There are other things bad for the poor O'Donnells as well,\" said the\nlandlord, bitterly.\n\n\"I am sincerely sorry for my dear friends. You know, Mr. Maher, they\nare the dearest friends I have on earth.\"\n\n\"Ah!\" cried the other sarcastically.\n\n\"I must telegraph to London. Someone must write the telegram for me.\"\n\n\"I will,\" said the landlord, grudgingly.\n\n\"You are always so kind,\" said the invalid; \"always so kind! You Irish\nare, I believe, the kindest-hearted race in all the world.\"\n\n\"And sometimes we get nice pay for our pains.\"\n\nThen the telegram to Dora Harrington was written.\n\n\"Have Mr. and Mrs. O'Donnell left, or are they with their son yet?\"\n\n\"Mr. O'Donnell is gone back to Rathclare. Mrs. O'Donnell is with Mr.\nEugene. It's a sorrowful business.\"\n\n\"And nobody else?\"\n\n\"Eh?\"\n\n\"And there is nobody else with Mr. Eugene O'Donnell?\"\n\n\"I say it's a sorrowful business.\"\n\n\"Dreadful. I am profoundly sorry.\"\n\n\"Eh?\"\n\n\"A sorrowful business, I say, about the failure of the bank.\"\n\n\"Eh?\"\n\n\"My dear Maher, you are growing deaf. You ought to see to this matter\nat once. Dr. O'Malley is a very clever man. You ought to mention the\nmatter to him.\"\n\n\"That'll do, now. You're bad, and I don't want to say anything to you.\nBut my ears are wide enough to hear what they say.\"\n\n\"Who are _they_ that _say_, and what do _they say?_\"\n\n\"_They say_ that you stabbed Mr. Eugene O'Donnell, one of the\npleasantest gentlemen that ever put a foot in Glengowra.\"\n\n\"But he himself denies it.\"\n\n\"He doesn't.\"\n\n\"When the police came he would not tell them anything.\"\n\n\"More fool he! But there, there--I won't say any more. This is against\nDr. O'Malley's orders. He said you were not to be allowed to speak, or\nexcite yourself. You may say what you like now, Mr. Lavirotte; I'll\nsay no more. I'll obey Dr. O'Malley.\"\n\n\"One more question and I have done. Is there anyone but Mrs. O'Donnell\nwith Eugene?\"\n\n\"Yes, Miss Creagh.\"\n\n\"Thanks; I am very much obliged to you. I will trouble you no more\nnow.\"\n\nWhen the servant returned to the room, he said to her: \"What a kind\nman your master is. Notwithstanding his belief that I made an attack\nupon Mr. Eugene O'Donnell, he was good enough to write a telegram for\nme, and to tell me some of the town gossip. I hear that Miss Creagh is\nin the sick room. I want you to do me a great favour, if you please.\nTake my compliments to Miss Creagh, and say I would feel greatly\nobliged if she would favour me with a few moments' conversation.\"\n\nThe attendant drew herself up. \"It's not likely,\" she said, \"Miss\nCreagh would come near you. When I was coming up, Mr. Maher told me\nyou were not to talk or excite yourself.\"\n\n\"Do as I tell you, woman,\" he said sharply, \"or I will get up out of\nthis bed and dash myself out of the window, and you will be the cause\nof my death, and have to answer for it.\"\n\nThe servant was cowed. She rose timidly and left the room. Almost\nimmediately the door reopened, and Ellen Creagh entered, followed by\nthe servant. Her pallor was now gone, and although her cheeks and lips\nhad not the depth of bloom usually on them, she looked nearly her own\nself.\n\nShe smiled faintly as she approached the bed on which Lavirotte lay.\n\"You wish to speak to me, and I have come.\"\n\n\"Yes,\" he said, \"I wish to speak to you. May it be with you alone?\" He\nlooked at the servant in the doorway.\n\nShe motioned the servant to withdraw, and then came close to the bed.\n\n\"Miss Creagh,\" he said, \"they tell me he will get better. They tell me\nhe has given no account of what took place last night to--the police.\nHas he told you what occurred?\"\n\n\"He has,\" she said; \"to me, and to me only. He said to his mother that\nthe secret was one concerning three only.\"\n\n\"He and I being two, and you the third?\"\n\n\"Yes,\" she said. \"What do you wish me to do?\"\n\n\"First of all to forgive me, if you can.\"\n\n\"I forgive you freely. He says you must have been mad.\"\n\n\"I was,\" he said, \"stark, raving mad. I was not responsible for what I\ndid. I am in the most grievous despair about the matter.\"\n\n\"He is sorry he injured you; but it was in self-defence.\"\n\n\"_He_ injure me! Not he. What put that into his mind? _I_ injured him.\nI will not pain you by telling you what I did. It was not I did it; it\nwas a maniac, a demon. You must tell him quickly he did not injure me.\nIn self-defence, in trying to guard himself against an accursed\nmadman, he sought to throw me. We both fell close to a rock at the end\nof the cove road, and my head struck the rock. You will tell him this,\nwill you not, Miss Creagh? It will relieve his mind. It will relieve\nthe mind of my dear friend, my dearest Eugene.\"\n\n\"He will be glad to hear he did not do it, but sorry to know you are\nso much hurt. He does not blame you at all. He says his great anxiety\nto be up is that he may come to you and shake your hand.\"\n\nThe tears stood in Lavirotte's eyes.\n\n\"God bless my boy,\" he cried. \"God bless my boy, Eugene. I am not\nworthy to know him. I am not worthy to know you. I am not worthy to\nlive. I am not fit to die. I am an outcast from earth, from heaven,\nand from hell.\"\n\n\"Just before I left him to come and see you\"--the young girl's colour\nheightened slightly--\"I took his hand to say good-bye to him, even for\nthis little time,\" she smiled. \"I took his hand in mine; in this\nhand,\" holding out her right. \"He said to me, 'You will tell Lavirotte\nI am sorry I cannot shake his hand.'\"\n\nShe stretched out her right hand to his right hand lying on the\ncounterpane. \"If I take your hand now, it will be the nearest thing to\ntouching his.\"\n\n\"Yes,\" said Lavirotte eagerly, \"it will be touching a hand that is\ndearer to him than his own.\" He took the warm white hand in his, and\nraised it to his lips reverentially. \"Now, the favour I have to ask of\nyou is this: it far exceeds in magnitude the one I first thought of\nasking you.\"\n\n\"What is it?\" she said, briskly. \"I am sure I shall be able to grant\nit.\"\n\n\"You will ask him to let me be his best man at your wedding.\"\n\nAgain the young girl .\n\n\"I will, if you wish it, and I am sure he will consent.\"\n\n\"Will you ask him, for then I shall have something to say to you?\"\n\nShe left the room and returned in a few minutes.\n\n\"Nothing will give him greater pleasure. He is delighted at the\nnotion. He would have asked you only----\" Here she paused.\n\n\"I understand,\" he said. \"Only for what occurred once between you and\nme. I am told there is bad news, the worst news, of Vernon and Son\nto-day. Do you believe in fate?\"\n\n\"I do not believe in fate.\"\n\n\"I do,\" he said, \"implicitly. I believe it was fated that you and I\nshould never be more than friends, and that you and he should be\neverything to one another. And now fate appears to me in a new aspect.\nThere is a chance--a very slender one, I admit--nay, a wonderful,\nfoolish chance that I may one day come into some money, not in the\nordinary way of succession, but by a romantic event. I will be\nperfectly frank with you. I will make a confession to you which I have\nmade to no one else here. It will damage me more in your opinion than\nit could in the opinion of anyone else living. When I said those words\nto you that day in the boat, I was engaged to be married to someone\nnow in London.\"\n\nThe girl started. \"You--you were not serious that day, you know. You\nonly meant to pay me a compliment.\"\n\n\"No, no,\" the wounded man cried quickly. \"I meant ten thousand times\nmore than I said. But there--let us drop that subject for ever. I am\nonly too glad to think of it no more. I offered you my hand when it\nwas not mine to give, and when you promised to give yours to another I\ntried to kill him. No man could have been baser or more unworthy than\nI. And yet there is a use in my baseness, for has it not given him an\nopportunity of forgiving me--fine-hearted gentleman as he is--and you\nof showing me that you are the noblest as well as the most beautiful\nwoman alive?\"\n\n\"You are too hard upon yourself, and too generous to--us,\" the girl\nsaid, colouring. \"I must not stay if you will talk in this fashion.\"\n\n\"Yes, stay by all means,\" he said, \"for I have not done speaking yet.\nI will say no more on that topic. I have another secret to tell you.\nIt will take some time. It is not unpleasant. It is, in fact,\nconnected with the only property I own, and the possible consequence\nof my owning it. It is situated in London. It is only the tower of an\nold church--St. Prisca's, in Porter Street, by the Thames. I own that\ntower. It was built many hundred years ago. The rest of the church has\nbeen pulled down----\"\n\n\"Here is Dr. O'Malley,\" said the girl.\n\n\"Miss Creagh,\" cried the doctor in astonishment. \"You here!\"\n\n\n\n\n                             CHAPTER XI.\n\n\nMr. William Vernon was a venerable, benevolent-looking man of seventy\nyears of age. His hair was white, his figure slightly stooped, his\nmanner gentle, kindly, plausible.\n\nUntil the crash came, everyone believed he was the most prosperous man\nin the city of Dublin. He had three fine private houses--one in\nDublin, a seaside residence at Bray, and a castle in Monaghan. His\nincome was believed to be somewhere between twenty and forty thousand\na year, and it was believed that he lived well within it. His savings\nwere said to be enormous, and the general conviction was that he could\nretire in splendour on his money, invested at home and abroad.\n\nNow all was confusion and dismay among those connected with him in\nbusiness. So great was the excitement, two policemen had to be told\noff to guard the door of the bank. Men and women, too, who were\ndepositors or shareholders, refused to believe the news, and came down\nto the bank to see with their own eyes confirmation of the report.\nThere, sure enough, were the massive oak, iron-studded doors closed in\ntheir faces, never again to be opened.\n\nAs the hours rolled on, the depth and breadth of the calamity\nincreased steadily. People who were supposed to have had nothing\nwhatever to do with the bank divulged, in the excitement of the\nmoment, the secret that they were shareholders or depositors. The\ncredit of the whole city was shaken. Who could be safe when the great\nhouse of Vernon and Son had collapsed?\n\nBefore nightfall three other large houses had suspended payment. They\nhad gone down into the vortex. Then it began to be realised that not\nonly had the shareholders lost all their money invested in shares, but\nthat every man who, as principal or trustee, held even one of these\nshares, was liable to the last shilling he had in the world.\n\nIt had over and over again been suggested by outside shareholders that\nthe business should be formed into a limited company. William Vernon\nalways shook his head at this, and said that if you limit the\nresponsibility you limit the enterprise, and so reduce the profits.\nThey were paying twelve per cent. on capital--did they want to cut\ndown the earnings to eight? He assured them it would <DW36> the whole\nconcern seriously, and he, for one, would retire from any\nresponsibility if such a course were urged upon him. It had been\nsuggested to him, in advocacy of this scheme, that limiting the\ncompany would enormously diminish the risk of the shareholders in case\ndisaster should overtake the bank. He had replied to this with a shrug\nof his shoulders, a smile of half pity, half amusement, and said: \"If\nyou have any fear, why not sell out? If you have any confidence in my\nword of honour, you need have no occasion for fear.\"\n\nMr. William Vernon had the reputation of unblemished honour. He was,\nmoreover, an exceedingly pious man, belonging to one of the most rigid\nforms of dissent. No one questioned his word; no one sold out; and now\nall were ruined.\n\nMr. Vernon had married late in life. Mrs. Vernon was twenty-five years\nhis junior. His elder daughter, Ruth, was now fifteen years of age;\nhis younger, Miriam, twelve. He had but these two children.\n\nMrs. Vernon was a large, florid, comely woman, who, twenty years ago,\nwhen she was married, had been considered a beauty. She was now no\nlonger beautiful. She was a well-favoured matron of forty-five, with\nan exaggerated notion of the importance of her husband, her children,\nand herself. He was courteous, insinuating, with a dash of\ninfallibility. She was dignified, not to say haughty, with a great\nnotion of the high position she occupied in the social world. She was\nnot harsh or cantankerous with servants, but she never for one moment\nallowed them to think they were anything but servants--that is to say,\nbeings of an immeasurably inferior order.\n\nDuring the time Miss Creagh had been in Mrs. Vernon's house as\nresident governess to her two daughters, the mistress had shown the\ngoverness respect in the form of conscious condescension. She had\nnever for a moment allowed anyone to slight Nellie, and even she\nherself had never slighted her. But, then, she never was by any means\ngenial or cordial, or anything but rigidly polite; and rigid\npoliteness is the perfection of rudeness.\n\nNellie had not, however, been unhappy in that house. She had conceived\na great respect for Mr. Vernon, and had grown to love the two\nchildren. Ruth was her favourite. The elder girl was flaxen-haired,\nblue-eyed, fair and pink, with a tendency to sentimental poetry and\nenthusiasm, and with a most excellent heart. Miriam, on the other\nhand, was a brunette, brown-haired, brown-eyed, vivacious, invincibly\nloquacious, with a thorough contempt for everything that was not\nmaterial to comfort, and with a heart which beat so fast for its own\nexcitements, that it rarely had time to concern itself with anything\nelse.\n\nMr. Vernon had that summer postponed their going to their house at\nBray a month beyond the usual time. The crash had not come upon him\nunexpectedly. He and a few others knew for some time that it could not\nbe avoided, but it might be put off. He was loath to leave Dublin; and\nas his family never went to Bray without him, he thought it better\nthey should not go now, as if they did it might cause talk. Bray is\nbut half-an-hour or so from Dublin; but he did not like to sleep so\nfar away from the bank, for now important telegrams were coming at all\nhours of the day and night, and the delay of an hour might hasten the\ndisaster.\n\nThe immediate cause of the ruin was the failure of a trader in\nBelfast, who owed the bank considerable sums of money, and had been\nencouraged by Mr. Vernon to play a risky business on the chance of\nmaking large profits. In fact, the relation between the Belfast and\nDublin houses would not bear the light of day, and the large profits\nwhich, it was said, enabled the Belfast house to pay a fancy price for\nmoney, had all been taken out of the capital lent by the bank.\n\nThe Belfast house had, some years ago, an extraordinary stroke of\nluck. It legitimately doubled its income in a year. It depended almost\nwholly on its export trade. It sent most of its goods to India and the\nColonies. During the good year it could not manufacture as quickly as\nit could sell. Then it borrowed in order to increase its manufacturing\npowers. It built and set up new machinery. It exported more than it\nhad orders for and stored abroad. This went on for some years, the\noutput being in excess of the demands of the prosperous year, the\nsales less than before the prosperous year. The result of this could\nbe seen--bankruptcy.\n\nNothing else was talked of in Dublin all that day, all that night, in\nthe clubs, in the hotels, between the acts at the theatre, in the\nprivate houses, in the tramcars, in the streets. No class seemed to be\nunaffected by the gigantic catastrophe. Widows and orphans were\nruined, trustees rendered penniless. Commercial fabrics which had cost\ngenerations to build up, were now tottering to the fall.\n\nAll this dreadful day Mr. Vernon sat in his study, a large back room\non the first floor of his Fitzwilliam Square house. He now fully\nrealised his own position. He had directly ruined hundreds, and\nindirectly, through them, thousands. For years the bank had\npractically been in a bankrupt state. For years the fact had been kept\nsecret by means of false balance-sheets. For years the pious, bland\nWilliam Vernon had been the author of a gigantic fraud.\n\nWhat was coming now to him? An indictment? Imprisonment? Were a common\nprison and common prison diet coming to him in his seventieth year?\nAll this time that he had been issuing false balance-sheets he had\nlived in splendour. He had kept his three houses, his horses, his\ndomestic servants, his gardeners, his grooms, his coachmen. He\nhad given dinners which were the talk, the admiration, the envy of\nDublin. His wines were the finest. He had a French cook; he had\nfootmen of the shapeliest forms and politest manners. Was he about to\nhave, instead of his three stately houses--the city jail? Instead of\nhis dining-room--a prison cell? Instead of his courteous footman--a\ngruff turnkey? Instead of cliquot--gruel? Instead of respect, honour,\nreverence--contumely, scorn, and curses?\n\nThe present was bad enough. The future looked much worse. He did not\nallow himself to waste any of his energies in grieving for those who\nhad lost through him. He said to himself: \"They speculated and lost.\nThey only lost money. I have lost all the money I once had, all the\nreputation, and now in my old age it is not unlikely I may lose my\nliberty. I have done the best I could. Had I reduced my establishment,\nsuspicion would have been aroused at once, and the blow would have\ncome much sooner. If I had earlier exposed the position of the bank,\nruin would have come then just as now. If after the first loss in\nBelfast I sanctioned wild, mad speculation, it was in the desperate\nhope of recovering what had already been sunken. What I did, I did for\nthe best. O'Donnell will, of course, be the heaviest sufferer, but he\nhas had his twelve per cent. for many years. I dare say he will not be\nable to save a penny out of his whole fortune. Neither shall I out of\nmine.\"\n\nJust as he came to the end of these self-justification reflections,\nthese comfortable sophisms, Mrs. Vernon entered the room, dressed for\ngoing out.\n\n\"Going out, Jane?\" he cried all in astonishment.\n\n\"Yes,\" she said. \"The house is so dull, I thought I'd take the\nbrougham and call upon the Lawlors.\"\n\n\"Take the _brougham_,\" he cried, \"and call upon the Lawlors! Don't you\nknow the Lawlors are shareholders in the bank, and that they, too, are\nruined?\"\n\n\"But,\" said Mrs. Vernon, drawing herself up, \"the Lawlors were old\nfriends of mine. I knew them before you did. We were children\ntogether. They will be glad to see me, although you have been\nunfortunate in business.\"\n\n\"Glad to see you! Woman, they would thrust you out of doors with\ncurses. When people are ruined they do not pay much heed to\nfriendship, nor are they over nice in the way they express their\nanger. As to the brougham,\" he said, \"I have been stupid not to tell\nyou, but I cannot think of everything. We could never with decency use\nthe brougham, or anything of the sort, again.\" He threw himself back\nin his chair and laughed harshly for a few seconds.\n\n\"I see nothing to laugh at in this disgrace and worry,\" said his wife,\nwho thought herself the most injured person of all. \"I am sure I am\nvery sorry for you, William, when I consider the respectable position,\nthe eminent position you held. I am sure you cannot say I was\nextravagant, or that I brought up the children extravagantly. You told\nme yesterday that my five thousand pounds are secured by the marriage\nsettlement. Why should I lose my old friends any more than the money\nmy father gave me when we were married?\"\n\n\"Because,\" he said, laughing harshly again, \"you married what the\nworld will agree to call a fraudulent scoundrel. When I laughed a\nmoment ago at the thought of the brougham, the idea which occurred to\nme was--it is rather painful. Shall I tell you?\"\n\n\"Yes, you had better tell me, I suppose. _Everything_ is painful now.\"\n\n\"Well,\" he said, \"I thought that the next member of the family likely\nto drive would be myself, and the next vehicle in which I was likely\nto drive would be a Black Maria.\"\n\n\"Black Maria, William,\" she said. \"I do not understand you.\"\n\n\"Black Maria, my dear,\" he explained, \"is slang for a prison van. What\nis the matter, Jane? You seem weak. Help, outside there, Mrs. Vernon\nhas fainted.\"\n\nThe door opened. A footman entered.\n\n\"If you please, sir, the brougham is at the door.\"\n\nThe old man started and looked up, became suddenly pallid. \"What did\nyou say, James?\"\n\n\"I said, sir, that the brougham was at the door.\"\n\n\"Ha! ha! ha! As I live, James, I thought you said the Black Maria.\nFetch Mrs. Vernon's maid instantly. The mistress has fainted.\"\n\n\n\n\n                             CHAPTER XII.\n\n\nWhen, on the night after the failure of Vernon and Son, Lionel\nCrawford heard from Dora Harrington the name of Dominique Lavirotte,\nand repeated it after her, he was filled with amazement. \"This is the\nmost extraordinary thing,\" he said, \"that ever happened to me in all\nmy life. Dominique Lavirotte,\" he repeated for the second time. \"I am\namazed!\"\n\n\"Do you know him?\" the girl asked.\n\n\"Well! Why, he owns the place I am taking you to. It isn't much of a\nplace. It is only the tower of an old church. They are always talking\nof buying it from him and taking it down. But you see it isn't big\nenough to give room for building a warehouse or store on the ground it\noccupies, and it is impossible to take in any other building with it.\nBut come, sweetheart,\" he said; \"when did you eat last?\"\n\n\"I--I had some breakfast.\"\n\n\"But breakfast is a long way since. You are young, and must be hungry.\nHere is the door of the tower.\" He took out a large key, and having\nturned the lock, thrust the door into the darkness. \"Now,\" he said,\nleading her in, \"be very careful; there is a hole here. Stand where\nyou are until I find the lantern and matches.\"\n\nHe groped about, and in a few seconds had lighted the candle in the\nlantern. Then he took the young girl by the hand, and said: \"This\nway.\"\n\nBy the light of the lantern she could see that they were walking on\ntwo planks, which together were not more than eighteen inches wide.\nBeyond the planks was a hole, the depth of which she could not guess.\n\n\"Don't be afraid,\" he said. \"Keep close to the wall and you are all\nright.\"\n\nThe girl shuddered. She, who a few minutes ago was on her way to the\nriver, now shrank from the notion of death. Had she not met someone\nwho knew her lover, someone who knew Dominique, her darling Dominique?\nThis was to get a new lease of life, a new interest in worldly things,\na fresh-filled cup from the fountain of hope.\n\nShe clung closely to the wall, and followed the old man through the\ngloom. They reached a corner, and here found a ladder.\n\n\"Up this ladder,\" he said; adding, \"What shall I call you? What is\nyour name?\"\n\n\"Dora,\" she said. \"Dora Harrington.\"\n\n\"Then, Dora, my dear child,\" he said, \"keep close to the wall on this\nladder, too, for there is no hand-rail, as you see.\"\n\nThey mounted the ladder. It ran along two sides of the tower. Then\nthey found themselves on the first loft. The head of the ladder was\nunprotected by any rail. Two other lofts they reached in a similar\nmanner, she clinging closely to the wall.\n\n\"This is my sitting-room,\" he said, with a laugh. \"It is not very wide\nor long, but it is lofty, airy, and, although there is not much\nfurniture, and the little I have is the worse of the wear, it will\nhave a great interest for you, for it belongs to him, Mr. Lavirotte.\nSit down here, now, on this couch. The spring is not so good as it\nonce was. You will have a cup of tea and some nice bread-and-butter.\nThat little table over there is my kitchen. See,\" he said, \"we do not\ntake long to light the fire, and we shall have boiling water in a few\nminutes. Boiling water,\" he said, \"and the prospect of a nice cup of\ntea is better for you, sweetheart, than the cold Thames. The prospect\nof--of--ugh! Let us forget that unpleasant folly of ours.\"\n\nHe had kindled the lamp in a small oil-stove, and set the kettle on\nthe stove. \"And now,\" he said, \"while the water is boiling you shall\ntell me as much as you please about yourself.\"\n\nShe was very tired, and for the present the mere rest was food and\ndrink to her. It was pleasant to sit there, half-tranced with fatigue,\nto sit upon this couch which belonged to him, in the presence of\nsomeone who knew him, and with the prospect of succour from a friendly\nhand.\n\nThe furniture in the loft was not, indeed, handsome. It never had\nbeen. When Lavirotte lived in London he had furnished a couple of\nrooms, and upon leaving them found that he could get little or nothing\nfor the furniture. So he carted it away to St. Prisca's Tower in\nPorter Street, and there it was when, at the request of Lionel\nCrawford, he let the tower to him.\n\nIn the loft where Dora Harrington now found herself there were three\nordinary chairs, one arm-chair, a couch, and two tables, besides the\n\"kitchen.\" The walls were rough, unplastered brick. The roof of the\nloft was unceiled. Under the table was a small piece of carpet.\n\n\"My own room,\" said the old man, \"is above this, and this shall be\nyours for to-night, and as long as you wish after, until you get a\nbetter one, or until he comes for you.\"\n\n\"How can I thank you for your kindness? May I ask your name?\"\n\n\"Lionel Crawford,\" said the old man. \"I live in the room above this,\nbecause my business requires me to be near the roof by night.\"\n\n\"Your business requires you,\" she said, \"to be near the roof by\nnight.\" By this time he had made the tea, and she had drunk a little,\nand begun to be refreshed. \"Can it be you are an astronomer?\"\n\n\"No, no,\" he said. \"I am no astronomer, and yet all the matters of\nweather interest me greatly. The rain to-night may be worth a fortune\nto me.\"\n\n\"You are a farmer, perhaps,\" she said. \"Or no, that cannot be; but you\nown land?\"\n\n\"Not a rood. Although I say I am much interested in the weather, I am\nneither interested in growing anything, nor in meteorology beyond the\nwinds and the rains. By day I get as far away from the sun as I can,\nas close to the rich centre of the earth as I may. By night I aspire,\nI seek the highest point I can reach, and there I worship the clouds\nand the winds that they may befriend me.\"\n\nThe old man was now sitting in the easy-chair, leaning forward, his\neyes fixed on vacancy. He had a weird, possessed expression. He seemed\nto be looking at things far off, and yet clearly within the power of\nhis vision. He seemed like one in a dream, and yet his words were as\nconsequential and coherent as the reasoning in Euclid. His might have\nbeen the head of an alchemist, or of some other man who dwelt with\nunascertained potentialities, with mystic symbols and orders and\nrites, with things transcending the ken of vulgar flesh, with\nsubtleties of matter known to few, rare drugs, rich spices, the\nvirtues of gems, the portents of earth and air, the mystic language of\nthe stars, the music of the spheres.\n\n\"And when it is winter,\" asked the girl, \"you wish, I suppose, for\nsunshine and calms?\"\n\n\"No,\" he said. \"Never. Always for rain and wind; wind and rain. Wind\nin the daytime, and rain by night, winter and summer; all the year\nround.\"\n\n\"And may I ask you,\" said the girl, timidly, \"what you are?\"\n\n\"When I met you this evening,\" he said, in the same tone as he had\nemployed since he became abstracted, \"I was Giant Despair.\"\n\n\"And now,\" she said, \"what are you?\"\n\n\"The rain and you have come,\" he said. \"I am now the humble Disciple\nof Hope.\"\n\n\"And, sir, may I ask, have you no friends, no relatives?\"\n\n\"None that I know of,\" he said. \"All my children are, I think, dead.\nMy wife is dead. My best friends are the dead.\"\n\n\"But surely, sir,\" she said, \"there is among the living someone in\nwhom you take an interest?\"\n\n\"No; no one. I am a client of the dead. If any good ever comes to me\nin life it will be out of the buried past. I doubt if good will ever\ncome. I am too old and spent. I was too old and spent when I began my\nlabours here. For years I had my great secret hidden in my breast. I\nnursed it, I fed it, I dreamed over it. For years I lived in this\nneighbourhood hoping some day or other to gain admission to this\ntower. I could not find out who owned it. It pays no rates or taxes.\nIt is not registered in any name that I could ever find out. I had\nbegun to think I should never get any nearer the goal, when one day as\nI was without the walls I saw a young man come up, thrust a key into\nthe lock of the great door, and try in vain to move the rusty bolt. I\nwatched him with consuming eagerness----\"\n\n\"This was some time ago?\"\n\n\"Years, two or three years. I drew up to the young man and said: 'I\nfear, sir, it is a tougher job than you bargained for.' I offered to\nget him a locksmith, and in less than an hour we got in. The young man\ntold me he had come from abroad----\"\n\n\"What was the young man's name?\" asked the girl.\n\n\"Dominique Lavirotte,\" said the old man, in the voice of a seer busy\nwith things remote.\n\n\"My Dominique,\" she whispered; \"my darling Dominique.\"\n\nThe old man went on without heeding the interruption. He had forgotten\nthe connection between the girl and the man.\n\n\"The stranger told me,\" said old Crawford, \"that although he had lived\nsome time in England, he had now been for years abroad. This was all\nthe property he had in the world, and he had never seen it before. He\nunderstood it was absolutely valueless, and he had merely come to see\nit now out of curiosity. 'For,' he said, 'is it not strange that in\nthe City of London, where the rent of land is six shillings a square\nfoot, I should own some for which I cannot get a penny the square\nyard? I wish I could get someone to buy it,' he said.\n\n\"'You must not think of selling it,' said I. 'I have been waiting here\nyears in the hope of meeting you.'\n\n\"'Why?' he cried in astonishment. 'Do you want to buy?'\n\n\"'No,' I said. 'May I speak to you a while in private?' The locksmith\nwas standing by. Then I took this handsome young man aside, and having\nmade him swear he would not reveal the matter to anyone----\"\n\n\"What?\" cried the girl, leaning forward eagerly.\n\n\"That is _my_ secret,\" said the old man.\n\n\n\n\n                            CHAPTER XIII.\n\n\nFoe a while Dora Harrington and Lionel Crawford were silent, he still\nwith the look of an enraptured visionary on his face, she perplexed,\nwondering, disturbed.\n\nWhat could this secret be which he, the man to whom she was engaged,\nnever told her? One thing appeared plain to her, it was not a secret\nin which Dominique was directly concerned. It was the old man's\nsecret, communicated by him to her lover. Yet it was not pleasant to\nthink that Dominique, who seemed so candid, so outspoken, so open,\nshould have something which he had concealed from her. The notion of a\nsecret was cold and dire. He had one: he might have many, as he had\nnever even told her that he owned this queer tower, standing all alone\nin those dark, forbidding ways by the river.\n\nOf late Dominique had not written to her as often or as affectionately\nas of old. True, he was not in good spirits about his worldly\nprospects. She had told him over and over again, when he asked her,\nthat she would marry him on anything or nothing. Who or what was this\nold man, that he should be mixed up with Dominique's affairs long ago;\nthat he should have stood between her and the Thames to-night? Was it\npossible this old man would tell her nothing more? He had excited in\nher curiosity, vague fears. Would he do nothing to allay either? Thus\nto be saved from the fate she intended for herself that night, to find\nin her protector a friend of his, and then to be confronted with a\nmystery in which Dominique had a part, were, surely, enough things to\nmake this night ever memorable.\n\n\"Mr. Crawford,\" said the girl, \"I can never forget the service and the\nkindness you have done me. Will you not do me an additional favour by\ntelling me something of this secret which affects him?\"\n\nThe girl had finished the tea and eaten some bread by this time.\n\n\"Take off your hat,\" he said. \"Lean back and rest yourself, and I will\ntell you something more.\n\n\"Ten years ago I was as lonely a man as I am now. All my family had\ndrifted away from me. Most of them were dead. Some of them had\nmarried, I know not whom. My studies always occupied me, and after the\ndeath of my wife, whom I tenderly loved, I went deeper than ever into\nmy books.\n\n\"Most of my children left me when they were young, and went abroad. I\nhad six children in all. From time to time one left me until all were\ngone, and ten years ago I had no more clue to the whereabouts of any\nthan I have to-day, except that I knew some were in the grave.\n\n\"I was then better off than I am now; but I have still enough to live\non, and to buy a book now and then. My books are all above. All my\ninterest lies in one direction, all my books treat of the same\nsubject--the history of the past, the history of the men and women and\nplaces of old times. My interest in the present closed with the death\nof my wife. But, somehow or other, since the time of which I speak,\nten years ago, I think I have grown less exclusively devoted to my\nfavourite pursuit than I was at the time of the dispersion of my\nfamily.\n\n\"I do not often speak to anyone except to those of whom I want to buy;\nbut I cannot help thinking there is a link between you and me, for are\nyou not betrothed to him who owns this tower, and has not this tower\nfor ten years been the chief object of my attention, of my solicitude?\nWas it not to him I first told the secret which I had carried with me\neight years? Is he not now the only person who knows my secret, and\nwhen the time comes for divulging that secret to a few, are not you to\nbe the first to hear it?\n\n\"Well, ten years ago I was, as I have said, as much alone in the world\nas now. I had always a notion that something was to be discovered in\nconnection with this Porter Street. Here and there in my books there\nwere vague hints, misty statements, that in this street had taken\nplace something of the greatest importance, something which might in\nthe greatest degree excite the interest of an archaeologist. But you\nsee, the street is long, a mile long, I dare say, and to search every\ninch of a street a mile long would be altogether out of the question.\n\n\"At that time I was living close by. There were certain old\nbook-shops, between Longacre and the Strand, which I visited almost\ndaily. Here, one evening, I picked up a battered old volume for a few\npence. It was dated 1625. It turned out to be of no great interest;\nbut on bringing it home, I was struck by two facts--first, that the\nbook, although battered, was complete; and, second, it contained some\nmemoranda in manuscript, one bearing these startling words: 'A great\nfire has broken out, and is spreading towards us. There is not a\nminute to be lost. What can be removed is to be removed to Kensington.\n_What cannot be removed is to be left where it now is_.'\n\n\"This memorandum was dated: 'Daybreak, 3rd September, 1666.'\n\n\"It was, of course, in the spelling of the period. Underneath this\nmemorandum appeared the words and figures: 'Speght's Chaucer, page 17,\nlines 17 to 27.'\n\n\"I have told you already that I had something like a hint of what I\nwished to find out. I am not free to tell you why the first of these\nmemoranda interested me profoundly, and shone before me like a\nrevelation. I seemed to be on the point of a great discovery, a\ndiscovery of the utmost importance to me, a discovery which had\nfascinated my imagination for years.\n\n\"I am free to tell you why the second memorandum filled me with\ndespair. It was essential that the book referred to in memorandum\nnumber two should be found. The clue in my possession was absolutely\nof no value without a copy of Chaucer. Before giving way to despair, I\nhad looked over the passage in the reference. I had read over twenty\nlines above and below without being able to find the slightest hint to\na clue. It was evident from this fact that the text of the poet threw\nno light on the subject, and that the intention of the man who had\nwritten the memorandum was that reference should be made, not only to\nthe particular edition specified, but to an individual copy of that\nedition.\n\n\"My despair was all the greater because I seemed to be half-way\ntowards success. I could not rest indoors. I wandered forth into the\nstreets without any definite object in view. To the average student of\nhistory, the discovery of this volume containing a reference to the\nGreat Fire, written at the very moment it was raging, would have been\ninestimable; but to one who was in quest of a particular object, and\nhad come within a measurable distance of it, without being able to\ntouch it, this book was a curse.\n\n\"Before I knew where I was I found myself standing in front of the\nidentical shop where I had bought the volume. I went listlessly over\nall the other books exposed for sale in front of the window. I saw\nnothing corresponding to the object of my search.\n\n\"Then suddenly a thought struck me. The book I had bought was\nvalueless. A copy of this particular edition of Chaucer would fetch\nmoney. I went inside, and asked the man if he had any other books\nbelonging to the lot among which the one I had purchased was.\n\n\"He told me he had several; that he bought the lot in an old,\ntumble-down house in Wych Street, where the books had lain for ever so\nlong, and that they were reputed to be salvage from the Great Fire.\n\n\"Imagine my excitement, my delight, when I found a copy of Speght's\nedition, and upon opening the volume, and referring to the passage\nindicated, I discovered writing on the margin. This writing was\nbriefer than that in the former volume. It was simply: 'St. Prisca's\nTower. See Mentor on Hawking, 1625.' This was the book I had bought a\nshort time previously. The chain was now complete. The area of inquiry\nwas absolutely limited to the ground upon which this tower now stands.\nIn the Great Fire of Charles's reign the church and tower of St.\nPrisca had been attacked by the flames, and the church had been\ncompletely destroyed. The lower portion of the tower, however, was\nfound by Wren to be sufficiently good for the purposes of rebuilding,\nand so, about ten feet above the ground of these walls belong to the\nold tower. Later on the modern church was pulled down; but for some\nreason, I cannot find out, the tower has never been interfered with\nsince.\n\n\"These books had evidently been carried away from the region of the\nfire to the fields where Kensington now stands; and then, when the\nfire was subdued, carried back to Wych Street, where they had remained\nuntil the bookseller who sold them to me had bought them about ten\nyears ago.\"\n\nHere the old man finished his narrative, which had been delivered in a\nmonotonous tone. His eyes were fixed, staring intently before him, and\nhe seemed to be wholly oblivious of the fact that Dora was listening\nto him. He was not, however, unmindful of her presence; for no sooner\nhad he concluded, than he looked at her directly and said: \"I have\ntold you all I can; all I may. Dominique Lavirotte and I are the only\npersons who know the rest, and you know more than anyone else in the\nworld except him and me. You must be tired now. I never told this\nstory before, and, in all likelihood, I never shall again.\"\n\nIt was now close to two o'clock in the morning. To the opening words\nof the old man Dora had given little attention. In fact the events of\nthat night, until she had begun to feel refreshed by the rest and tea,\nhad left a very weak impression on her mind, and she would have found\nit hard to say whether the occurrences had been real or figments of\nher brain. As the story advanced, she had felt a more lively interest\nin it, and towards the end she found that she was listening with\nawakened curiosity.\n\nThe old man said: \"I will bring you down a rug, and then you must try\nand get a little sleep. I shall have to work a couple of hours yet in\nthis welcome rain.\"\n\nHe brought the rug and spread it over her, and then emerged once more\nupon the roof.\n\n\n\n\n                             CHAPTER XIV.\n\n\nWhen Crawford reached the roof it was still dark. The intense darkness\nof a few hours ago had passed away, and it was possible on the roof to\nsee dimly the figure of the old man, the parapet, and the lead.\n\nTowards each of the four corners of the lead the roof sloped gently,\nand in each corner was a shoot leading to a pipe. In each of the four\ncorners, but so placed as not to obstruct the shoot wholly, and yet to\nimpinge upon it, lay a heap of something.\n\nTo each of those heaps the old man went in succession, moving the\nheaps so as to make them impinge a little more upon the gutter. When\nthis was done he put down his spade, resting it against the parapet,\nand leaned out of one of the embrasures.\n\nAll was still as death below. The darkest hour is the hour before the\ndawn; the most silent hour is the hour before the reawakening.\n\nIt was raining heavily now. The old man did not heed the rain. His\neyes were turned vacantly towards the east. He was watching for the\ndawn, not with eyes busily occupied on the dim outline of the huge\nstores and warehouses before him. His gaze was directed to the east\nsimply because he knew that in the east the sun would rise, and that\nas the light grew broad, and the top of the tower was overpeered by\nlofty buildings on higher ground, he must, soon after daylight,\nintermit his work on the roof if he would keep his secret.\n\nWhen the gray had moved up in the east, the old man went his rounds\nonce more, spade in hand. The rain still continued.\n\nWhen he had finished, he paused and leaned once more at the embrasure\nhe had formerly occupied. \"I always,\" he thought, \"take care to keep\nthe clay heaps about the same size. Rain is very good, no doubt. It\nworks off more than wind, except the wind is very high. The worst of\nthe rain is that when the clay gets soaked through and cakes, I have\nto take it down to dry the minute the weather gets fine, and bring up\nmore sieved earth, for the wind would have no effect on the hardened\nclay. At first I thought of putting all I excavated on the lofts; but\nI found them so old, and weak and shaky, that I durst not trust them\nbeyond a little each. There, I have put all the large stones too big\nto carry out and leave quietly here and there. There are tons and tons\nof stones upon the lofts, and I am afraid the floors will bear very\nlittle more. It would never do to overload the lofts and have the\nlabour of my two years all undone. The rain has stopped. It will help\nme no more. Heaven send the wind. Here is the day.\"\n\nIt was now bright enough to see that the roof of the tower was covered\nall over with a coating of thin mud, washed into streaks here and\nthere by the rain. In each corner lay a heap of clay. There were a\nbasket and a large pail also on the roof. The old man now began to\nwork energetically. He filled the pail with the mud, and in four\njourneys down to the first loft, succeeded in removing all that\nhad been on the roof. Then he carried up four large baskets of\nfinely-sifted clay, and put one basket in each corner near the shoots,\nso that those who had seen the roof of the tower from afar off the\nprevious day would notice very little, if any, difference, even with\nthe aid of a glass; for the nearest building that overlooked the tower\nwas a mile distant.\n\nIt was now broad daylight, and as the old man stood, his work\ncompleted, all round him rose the muffled murmurs of awaking day. He\nwas wet through, but he did not care for this. He was used to it. The\nrain and the wind were his great friends, and he hailed their advent\nwith delight.\n\nIt was plain what his object was. By day he worked in the base of the\ntower, at which the ground stood now twelve feet higher than at the\ntime of the Great Fire, and twelve feet below this was the foundation\nof the tower.\n\nFor two years Lionel Crawford had slaved in the daylight digging down\ntowards the foundation. He had a pickaxe and shovel and sieve. When he\nhad dug up some earth and rubbish, he sifted this on a piece of old\ncarpet and carried the sittings up to the top loft, there to dry and\nbecome friable for the purpose of being got rid of on the roof.\nEverything that would not go through the sieve, he carried out with\nhim, and dropped here and there as occasion offered, and the larger\nstones, which he never put on the sieve at all, he carried up to the\nlofts.\n\nWhen he had wind instead of rain he stood on the tower in the dark,\nand when all was quiet, threw away the sifted earth to leeward,\nhandful by handful. So that although he might thus in a night get rid\nof several hundred pounds weight of earth, no trace whatever of it\nappeared below the tower. When he was not helped by rain or wind he\ncould not dispose of more than fifty or sixty pounds weight a night,\nwithout drawing attention to his operations. This quantity he got rid\nof by throwing handful after handful out of the embrasure all round\nthe tower.\n\nWhen he found himself on the loft where he slept he took off his wet\nclothes, hung them up, and then lay down and slept.\n\nIt was late in the forenoon when he awoke. He dressed himself and went\ndown to what may be called the sitting-room. Here he found Dora awake.\n\"If it would amuse you, child,\" he said, \"you may light the fire and\nmake the tea. It may be a novelty to you, and it will surely be a\nnovelty to me if you do.\"\n\nDora arose with alacrity and busied herself about the simple\npreparations for breakfast.\n\n\"It is a long time,\" said he, \"since I had anyone--man, woman, or\nchild--at a meal with me. Sometimes I go out and have my dinner or\nsupper or breakfast in the poor eating-houses around here; but that is\nnot often. I have learned to shift for myself as well as Robinson\nCrusoe did in his time.\"\n\nWhen the breakfast was ready, Dora said: \"I am sure you will forgive\nme, but the excitement and confusion of last night have made me forget\nyour name. Yet I remember that when you mentioned it, it seemed\nfamiliar to me.\"\n\n\"Lionel Crawford, my dear; Lionel Crawford is my name.\"\n\n\"Crawford,\" she said, musingly resting her chin upon her hand. \"I do\nnot know how I could have forgotten that name, for Crawford was my\nmother's name before her marriage. It is not a very uncommon name in\nEngland, is it?\"\n\n\"Not very,\" he said. \"There are several families of the name in London\nalone.\"\n\nThey were now sitting at breakfast. No contrast could be much stronger\nthan that between the young, soft, gentle, beautiful girl and the\nleather-hued, gnarled-browed old man. The bright sunlight fell through\ntwo long, narrow windows high up in the thick walls of the tower. It\ntinged the white hand of the young girl lying listlessly on the table.\nIt lit up from behind the rich curve of her cheek. It touched with\ngleaming, grave bronze the outline of her dark hair. The old man sat\nat the other side of the small table, looking with abstracted eyes at\nthe partly illumined head of the young girl opposite.\n\n\"Ay,\" said the old man, \"Crawford is not an uncommon name. There were\nseveral of us brothers when I was young. I was the only one that\nmarried, and I believe all my children are dead by this time. Their\nmother was sickly. She was everything to me while she was alive. No,\nCrawford is not an uncommon name.\"\n\n\"We used not to consider it a common name in Canada,\" the girl said.\n\nThe sunlight was gradually encroaching upon the mass of dark hair.\n\n\"Ah,\" he said, still with the abstracted air, \"you were in Canada. One\nof my daughters when she was young, a child of fourteen or fifteen,\nwent to the United States.\"\n\n\"How strange,\" said Dora, shifting her position, and bringing all her\nhead under the influence of the summer sunlight.\n\n\"No,\" he said, \"not very strange. A great lot of people from these\nparts go to the United States, and, as I tell you, Crawford is not an\nuncommon name.\"\n\n\"What I meant,\" said the girl, with a somewhat puzzled look on her\nface, \"was that it is strange your daughter, whose name was Crawford,\nshould have gone to the United States when young. My mother went to\nthe United States when young. She married there and then moved up to\nCanada.\"\n\n\"And you tell me your name is Harrington, Dora Harrington? My girl's\nname was Dora, too, and I heard she married a man named Harrington.\nWhat was your mother's Christian name?\"\n\n\"Dora was her name,\" said the girl, rising. \"What do you think, sir,\nof all this?\" The girl was now standing, so that from crown to heel\nthe full sunlight shone upon her.\n\n\"It is extremely strange,\" said he, still in his absent-minded way,\n\"for I heard that my daughter moved up after her marriage.\" Suddenly\nthe old man's eyes fixed themselves upon the illuminated figure of the\ngirl. \"I had not a good look at you before, child, and my eyes are dim\nwith overmuch study. Yes! As heaven hears me, there is a look of my\ndead wife about you, child. Did they ever tell you you were like your\nmother? Do you remember your mother?\"\n\n\"I remember her very little, sir. I was very young when she died. They\ntold me I was not like her.\"\n\n\"Ay, ay. That is all in favour of my hopes, my child, for Dora was not\nlike my wife, and you are. Marvellously like! I seem to feel the coil\nof forty years falling away from me.\" His eyes once more took the\nabstracted, faraway look of the lions. \"Forty years ago,\" he said, \"I\nwas young and blithe, strong-limbed, and not repulsive as I am now. I\nwooed my Dora then, not in smoky London, but amid the green fields,\nand when the primroses were fresh with the early spring weather, and\nall the air was sweet with moist dews and fresh songs of birds. The\nleaves were all unsheathed, and each pulse of the wind brought a new\nperfume of the season. My Dora!\"\n\n\"And you think me like her?\" said the girl. \"Oh, if it should be,\nsir!\"\n\nSuddenly the old man lost his abstracted look. He rose and stretched\nout his arms towards her, looking keenly at her the while. \"You are\nshe,\" he cried. \"You are my Dora, my dead darling's grand-daughter.\nFor her own daughter, whose child you are, was like me, all said.\"\n\n\"Oh, sir,\" cried the girl, \"it is too much happiness for me to believe\nthis true.\"\n\n\"I want some happiness now, my child,\" said he, \"and no happiness\ngreater than this could come to me, for I am tired of loneliness. Come\nto me, Dora.\"\n\nThe illuminated figure of the girl moved, passed out of the sunlight\ninto the gloom of the room--into the gloom of the old man's arms.\n\n\n\n\n                             CHAPTER XV.\n\n\nThe police of Glengowra were very inquisitive about the affair of that\nnight. The town was exceedingly quiet, as a rule, and the fact that\ntwo well-dressed men had been engaged at midnight in a deadly\nencounter was unique and fascinating to the police mind. There was no\ndoubt in the town or village, for it was indifferently called either,\nthat the two men had fought, and that jealousy was at the bottom of\nthe encounter. But both O'Donnell and Lavirotte held impregnable\nsilence on the matter. Neither would make any statement. Lavirotte\nsaid they might ask O'Donnell, and O'Donnell said they might ask\nLavirotte; and it was known that no matter what may have occurred the\nprevious night, the friendship of the men was now re-established.\n\nThis last fact was gall and wormwood to the police. It was sheerly the\nloss to them of a golden opportunity. To think that the biggest crime\nwhich had been committed for years in the town should not be made the\nsubject of a magisterial inquiry, was heartbreaking. What was the good\nof having crimes and policemen cheek by jowl, if they were not to come\ninto contact?\n\nA policeman lounged all day about the door of Maher's hotel, affecting\nto take an interest in the cars and carts passing by, and in the warm\nbaths opposite, and to be supremely unconscious of the existence of\nMaher's. Nothing came of this.\n\nSupposing each man should say his hurt was the result of an accident,\nthere would be no evidence to prove the contrary, and the police would\nonly get into trouble and be laughed at if they stirred in the affair.\n\nA fussy and blusterous Justice of the Peace made it his business to\ncall at the hotel, see Maher, and impress upon him the absolute\nnecessity of doing something.\n\nDr. O'Malley absolutely forbade any \"justices of the peace, policemen,\nor such carrion,\" entering either of the sick rooms.\n\nHe said to the magistrate: \"Don't you bother about this affair. I\npromise you, on the word of a man of honour, to let you know if either\nof the men is in danger of death, so that a deposition may be taken;\nand I promise you my word, as a man of the world, that if anyone goes\npoking his nose into this affair, one or both of these young men will\nhave something unpleasant to say to that nose when they get about.\"\n\nThis speech made the worthy magistrate extremely wroth. He stamped and\nfumed for a while, and muttered something about puppies, and left the\nhotel in dudgeon.\n\nStill later in the day the sub-inspector of the district, who was a\nfriend of O'Malley's, and happened, by a miracle in which few will\nbelieve, to be a man of gentlemanly instincts and manners--called at\nO'Malley's house, spoke of the weather, the regatta, the price of\nbeasts at the last horse fair, the desirability of building a pier for\nthe fishing-boats in the cove, the hideous inconveniences of not being\nable to get ice in Glengowra in such roasting weather, the interesting\ncase at the last Quarter Sessions, and finally, he said: \"By-the-way,\nO'Malley, if you do know anything about what occurred last night on\nthe Cove Road, and if you can do so without any breach of good faith,\ntell me what you know?\"\n\n\"I don't know all about it,\" said O'Malley, briskly; \"and what I do\nknow I am bound to keep to myself. The part of the case about which I\nam game to speak is the medical aspect of it, and of that I am free to\ntell you there is no cause to fear either of the men will die. Now,\nthat is all you want to know, because you're a good sort of fellow;\nyou're not more than a thousand years old yourself. Boys will be boys.\nHave a cigar.\"\n\nThus the young sub-inspector left O'Malley's house scarcely any wiser\nthan he came. In the phrase, \"Boys will be boys,\" O'Malley had\nconveyed to him an unmistakable impression that the theory of the\nfight was the correct one, and at the same time he recognised the\nskilful way in which O'Malley avoided any breach of confidence.\n\nDirectly opposite Maher's hotel were the warm baths, and a little to\nthe right of these a shop, famous in the history of Glengowra, and\ncalled by the pretentious name of the Confectionery Hall.\n\nThis title was ludicrously out of proportion with the appearance of\nthe place. The \"Hall,\" that is, the place open to the general public,\nwas not more than twelve by fifteen feet. Here were displayed on a\ncounter, presided over by a thin-featured maiden lady of long ago\nascertained years, cakes of various kinds and sorts and ages,\nsweetmeats of universal dustiness and stickiness, ginger-beer,\nlemonade, and bottled Guinness and Bass. Sherry might, too, be\nobtained here in genteel quantities out of a cut-glass decanter, but\nthe inhabitants of Glengowra had a national antipathy to the spirit\nknown as sherry and when they wanted anything stronger than Bass or\nGuinness, they asked for whisky.\n\nNow, the great feature of the Confectionery Hall, as opposed in\nprinciple to a mere public-house, was that whisky could not be\nobtained at the counter. If a man wanted that form of mundane\nconsolation, he was obliged to enter an inner penetralia, where not\nonly could he have the \"wine of the country,\" but an easy-chair to sit\nin and tobacco for his perturbed mind.\n\nTowards the close of the evening of the day following the occurrence\non the Cove Road, two young men were seated in this cave of nicotine\ndiscussing the event of the day, nay, of the year. Both were out from\nRathclare for the cool evening by the sea, and in order to enjoy the\nmost perfect coolness of the sea, they had retired to this back room,\nwhich was heavier to the senses and less open to the air than the\nstuffiest back slum of Rathclare.\n\nBoth had of course heard the great Glengowra news, and the great\nDublin news of the day. It happened that one of these young men was in\nthe employment of the State--to wit, the Post Office, and the other in\nthat of a public company--to wit, the railway.\n\n\"I can't make it out,\" said the Railway, \"how it is that Lavirotte\nshould have fought O'Donnell about Nellie Creagh, because a fellow\ntold me that a good while ago--a couple of years, I think--when\nLavirotte was over in London, he had made it all right with some other\ngirl there.\"\n\n\"I don't like Lavirotte, and I never did,\" said the Post Office; \"but\nthis I am sure of--that he had some great friend in London, and that\nhis friend was not a man. Of course I don't wish this mentioned, and I\ntell you it in confidence. I remember his coming over here. We make up\nthe bags from Glengowra at Rathclare, and when he came here first, and\nI met him and knew his writing, I saw a letter from him to a Miss\nSomebody (I will not tell you her name) in London, and this letter\nwent two or three times a week.\"\n\n\"Who was she?\" inquired the others, inquisitively.\n\n\"I won't tell. I have already told you more than I should. You must\nnot mention the matter to anyone. I know you so long, old fellow, I am\nsure I may rely on you.\"\n\n\"Well,\" said the other, \"I don't want to seem prying. In all\nlikelihood I shall never see Lavirotte or O'Donnell again. I am off\nnext week.\"\n\n\"I am very sorry to lose you; but you're sure to come back to see the\nold ground shortly.\"\n\n\"I don't think I shall,\" said the other, carelessly. \"It costs a lot\nfor the mere travelling, and you know none of my people live here\nabout. Anyway, when I get to London, supposing I am curious, which I\nam not, I can find out all about it; for I know an artist there who\ntold me all about Lavirotte and the girl.\"\n\n\"How on earth did you find anything out about one man in such a big\nplace as London?\"\n\n\"My dear fellow, London is at once the biggest and the smallest place\nin the world. You have never been there?\"\n\n\"No, never.\"\n\n\"Well, you see, most of the nationalities and arts and professions\nlive in districts, chiefly inhabited by themselves; and when they do\nnot, they have clubs and other places of resort where they meet. So\nthat, in the case of Lavirotte, who was then thinking of being a\nfigure-painter, but hadn't got the talent, there was nothing unlikely\nin his meeting other men of similar ambition, and so it was he came\nacross there the artist I know, who happened to have a studio in the\nhouse I lodged in.\"\n\n\"I have often looked at the map of London and wondered how it was\nanyone ever found out where anyone else lived, even when he had the\naddress. But I cannot understand how two friends can fall across one\nanother accidentally in such a tremendously large place.\"\n\n\"You have never been in Dublin even, I believe?\" said the Railway.\n\n\"No, never,\" said the other.\n\n\"Well, then, all I can tell you is, that if you walk from the College\nof Surgeons in Stephen's Green to the Post Office in Sackville Street\nthree times a day, you will meet any stranger who may happen to be in\nthe city.\"\n\nFor a little while both men were silent. Then the Post Office said:\n\n\"Well, as there is but a week between you and finding out all about\nthis girl and Lavirotte, I may as well tell you, in strict confidence,\nthat her name is Miss Harrington. I forget her address. She changed it\noften, but it did not seem a swell address to me. At first he wrote to\nher two or three times a week; but of late his letters have not gone\nnearly so often, although some one in London, I suppose this Miss\nHarrington, wrote him twice a week regularly. Within the past two\nmonths I don't think he has written to her at all.\"\n\nWhile this conversation was going on in the back parlour of the\nConfectionery Hall, the policeman, who had during the day devoted most\nof his attention to the vehicles passing in front of Maher's hotel and\nto the warm baths opposite, was relieved, and came over to the \"Hall\"\nfor a small bottle of Guinness. It so happened that he had overheard,\nthrough the glass-door from the shop to the parlour, most of the\nconversation which had passed between the two friends.\n\nHe heard the two friends rise to leave. Before the handle of the door\nturned he was out of the shop. In a few minutes he was back in the\npolice-station.\n\n\"Well, any news?\" said the sergeant, gloomily.\n\n\"I have heard something that may be useful,\" said the constable; and\nhe detailed the conversation.\n\n\"And we have found something which may be useful,\" said the sergeant.\n\"After a long search among the stones we came upon the knife Lavirotte\nstabbed O'Donnell with. Here it is, with Lavirotte's name and\nO'Donnell's blood upon it. It will go hard with us if we can't get\nLavirotte seven years on this alone.\"\n\n\n\n\n                             CHAPTER XVI.\n\n\nIn the vast pile of buildings owned by James O'Donnell in Rathclare,\nby day several hundred men were employed, by night several score; for\nthe steam mills were kept going day and night, and got no rest from\nyear's end to year's end, save from twelve o'clock on Saturday night\nto six o'clock on Monday morning.\n\nIn the portion of the buildings devoted to milling operations most of\nthe night-men were employed. In fact, so far as active employment was\nconcerned, no men were engaged anywhere else in the place. There were,\nhowever, three watchmen for the other portions of the building. One of\nthese was outside in the yard fronting the river, another was on the\nground-floor of the granaries, and it was the duty of the third to\nwander about the upper lofts and corridors.\n\nOf late these men had been cautioned to observe greater vigilance. It\nwas well known in Rathclare that the strong-room of James O'Donnell\nalways contained a large sum of money, and sometimes a very large sum.\n\nThe man whose duty it was to examine the lofts passed along the\ncorridor leading to the private office. All was right, so far. He\nalways made it a habit to pause and listen at the door of the private\nroom; for if an attempt was to be made upon the safe it should be from\nthis place. The man went on in a leisurely way, ascended the next\nladder he met, strolled along the lofts, ascended another ladder, sat\ndown on a pile of empty sacks, and lit his pipe. Smoking was not, of\ncourse, allowed, but then there was no one to see him.\n\nWhen he had finished his pipe he ascended to the top loft and walked\nall round from one end of the building to the other, pausing now and\nthen to listen at the head of a ladder or at a trap-door, or to look\nout of a window into the deserted street below. This took a long time,\nfor there was no need of haste. It was an understood thing among the\nwatchmen that each should speak to the other two about once an hour.\nThus it would be known each hour that all was well throughout the\nbuilding.\n\nThe watchman now began to descend. He went down more rapidly than he\ncame up. It was quite dark, and the silence was unbroken save by the\nnoise of the machinery and the swirl of the river as it swept past the\nwharf and quays and ships below, and whispered among the chains and\nropes.\n\nThe three men generally met in fine weather such as this on the wharf.\nIt was pleasant to the two men, whose business lay indoors, to breathe\nfor a few moments the cool air by the river. From the wharf no portion\nof the offices could be seen. They looked into the great quadrangle\nround which the granaries were built.\n\nWhen the three men had stood and interchanged a few words they\nseparated, each of the two going in his own direction, the third man\nremaining on the wharf. The man whose duty lay on the upper floors\npassed into the large quadrangle, round which the granaries stood. At\nfirst he noticed nothing remarkable; but when his eyes fell on the\nwindows of James O'Donnell's office he started visibly, and uttered an\nexclamation of surprise under his breath.\n\nThe windows were full of light!\n\nWhat should he do? What could this mean?\n\nHe had, of course, heard of the misfortunes which had fallen upon his\nmaster's house that day, but he made no connection between that fact\nand this extraordinary appearance. The warning against possible\nburglars was uppermost in his mind. Although he was nearly sure no one\nwas then in that office for an honest purpose, still he resolved to\nproceed with the greatest caution, and give no unnecessary alarm.\n\nHe went out on the wharf and told the other man what he had seen. They\nboth agreed that it would now be useless to try and overtake their\nother comrade, and that it would be best for the two of them to go to\nthe office at once and see how matters stood there.\n\nWhen they got indoors they took off their boots and proceeded\ncautiously to the foot of the stairs leading to the offices. Each\ncarried a stout stick in his hand, and each man was physically\nqualified to take care of himself in a scuffle.\n\nThey agreed it wouldn't do to get some more of the hands from the mill\nand proceed to the office as though they were sure of finding burglars\nthere; for how could they tell that it was not the manager, or their\nemployer himself, who had been obliged to come back owing to some\nurgent business?\n\nThey crept cautiously up the stairs and found themselves in the\ncorridor, upon which the office door opened. Here all was dark and\nsilent. Here they were confronted by a difficulty they had not\nanticipated. If it should be that the manager or the proprietor had\ncome back at this unseasonable hour, the proper thing would be, of\ncourse, to knock at the door and ask if all were right.\n\nBut supposing there were burglars inside, knocking at the door would\nbe simply to put them on their guard, and enable them to take up a\ndefensive or offensive position before the others could enter:\n\nWhat was to be done?\n\nAs if by a common instinct, the two men retired to the further end of\nthe passage to hold a brief council.\n\nThere was no means of escaping from that room except by this passage\nor the window. That window was not barred, and nothing could be easier\nthan to get from it by a ladder or a rope. The first thing, therefore,\nto be ascertained was--did a ladder or a rope lead from that window to\nthe ground of the quadrangle?\n\nIt was then agreed between the two that one of them should go down and\nexamine the window from the outside, while the other waited in the\npassage here and watched, the door until his fellow came back.\n\nOne of the men descended to the ground-floor, got out into the\nquadrangle, and looked at the window, and the ground near the window.\nIt was a dark night, and one could not see small objects distinctly.\nThe man was not content with the evidence of his eyes alone. He stole\nover under the window, and placing his hand against the wall, walked\nforward and backward, ascertaining by touch that neither ladder nor\nrope connected the window with the yard. When he was satisfied on this\npoint, he stole back to his companion and communicated the fact to\nhim.\n\nSo far all was well. They had not now to think of any means of exit\nbut the one before them. Still it was not easy to know what to do. Now\nit occurred to them for the first time that it was not at all\nconsistent with the belief burglars were at work that the gas should\nbe fully ablaze. Although there never had been an attempt to rob the\nmill on a large scale, or by violence, and the watchmen had no\npersonal experience of burglars yet, it was their business to know\nsomething about how that predatory tribe carry on their operations.\nIt was not likely such men would attempt to force the door of a\nstrong-room, made on the very best principles, with the light turned\nfully up. A dark lantern and silent matches were more the manner of\nthe midnight thief than the great openness and defiance of gas.\n\nIt must surely be someone connected with the business. It was well\nthey had not made a fuss about the matter, and now it would be well\nthat they should delay no longer to prove their diligence by showing\nthey had observed the unusual fact of the gas being burning.\n\nYes, there could be no longer any doubt their manager or employer was\nbehind that door. There would be something absurd in the fact of two\nfine strapping fellows like them going up to that door in their vamps.\nIt would show they had suspected someone was there who had no right to\nbe there, and this might give offence. It would be best for them to\nput on their boots before knocking; besides, if they knocked as they\nwere now, whoever was inside might think they had been prying.\n\nWhen they reached the open air they put on their boots quickly. Then\nit occurred to them that, as they were now quite certain it was\nsomeone belonging to the business who was in the office, it would\nnever do for two of them to appear at the door simultaneously. The\nduty of one man was to be on the wharf, and of the other to be on the\nlofts or in the passages, and if they had no suspicion wrong was going\nforward, why should the wharfman desert his post?\n\nThey, therefore, agreed that the loftman alone should go back and\nprove his vigilance by knocking and saying that he had observed the\nlight.\n\nThe two parted. The loftman, starting with his usual measured tread,\ncrossed the quadrangle, entered the dark passages, ascended the\nstairs, and knocked at the door.\n\nTwo minutes after he rushed out upon the wharf, exclaiming in an\nundertone:\n\n\"Do you know who's there?\"\n\n\"No. Who?\"\n\n\"No one. Come back with me and see if I am right. I can't believe my\neyes. There isn't a soul there as far as I can see, in the office or\nin the passages.\"\n\nThe two men went back to the passage, entered the private office,\nfound the gas at full cock, and the place empty!\n\n\n\n\n                            CHAPTER XVII.\n\n\nMr. O'Donnell, towards the close of that unlucky day, found himself\nonce more in his comfortable home at Rathclare. Within twenty-four\nhours, the life of his only son, the hope of his age, had been placed\nin danger; and all the earnings of a long and laborious life had been\nscattered to the winds by one tremendous blast of ill-fortune. He was\nnot a communicative or demonstrative man. He took his pleasures\nsoberly, gravely, and with little exterior show of delight. Outside\nhis business, which was large and engrossing, he cared for little save\nhis wife, and son, and home. He had few wants, and a limited mind;\nbut, like all men with few wants and a limited mind, he must have what\nhe wanted, or life would not be worth living.\n\nHe did not sigh or burst into exclamations when the bad news reached\nhim. He was reading a newspaper at the time. He put down his\nnewspaper, and asked his managing man, who brought him the news, to\nrepeat his words. Then, merely saying, \"That is very bad news,\" he\ntook down an account-book, and, having looked at how his affairs stood\nwith the bank which had failed, put up the book in the safe, walked\nout of his office, and took the train to Glengowra, where his son lay\nhurt, and where his wife was already in attendance on the injured man.\n\nNow, he was back once more in his home alone. His wife was to stay\nthat night at Maher's hotel. In the present condition of his business\naffairs he did not feel himself justified in absenting himself from\nhead-quarters.\n\nUp to this he had very rarely been separated from his wife, even for a\nday. He seldom left his native town for more than a few hours, except\nwhen he went away for a week or so and took her with him.\n\nHe sat in the deserted dining-room all alone. He always carried in his\ncoat-pocket a small memorandum-book, in which he had jotted down the\nnet results of all his business transactions, so that at any moment,\nand in any place, he could see pretty well how he stood. He seated\nhimself in a large easy-chair, and having pulled down the gasalier,\ntook out this book, and sat silently consulting the pages for a long\nwhile.\n\nBy this time he had received full information from Dublin. He knew now\nthe case of Vernon and Son was absolutely hopeless. He was going over\nhis book, not in the hope of finding out anything cheerful about his\nown affairs, but just merely to convince himself through his sight of\nwhat he was already convinced through his reason.\n\nWhen he had reached the end of the written pages, and had made a few\nfigures with his pencil and arrived at a total, he tore out the page\non which he had made this last calculation, and then carefully and\ndelicately tore the page into little bits.\n\nHe put down his pocket-book on the table at his elbow, and then sat\nfor a long time arranging and re-arranging the fragments of the paper\ninto various figures on the table at his side.\n\nWhen it was about eleven o'clock, a servant came and asked him if he\nwanted anything.\n\nNo, he wanted nothing. They might all go to bed.\n\nWhen the servant had retired, he re-began his work with the fragments\nof paper.\n\nAt twelve o'clock he seemed to have made up his mind that there was no\ngood in trying any longer to arrange the pieces in a satisfactory way.\nHe pushed all the bits together, swept them into his hand, and placed\nthem on a tray, on which were some glasses, which he had not used.\n\nHe took up the pocket-book again, and quietly tore out all the blank\nleaves. \"These may be of use to someone else,\" he said. \"They can\nnever be of any use to me.\" He placed the blank leaves on the table,\nfar in from the edge. \"The books at the office will show how my\naffairs stand. This can interest no one. It was only on account of the\nmoney I considered myself worth, over and above my liabilities. I'll\nburn it;\" and then forgetting that it was summer time, and that there\nwas no fire, he threw the book into the grate, and rose. He felt in\nhis pocket, and found that he had his keys. Then he went into the\nhall, put on his hat, and left the house.\n\nHe took his way to his principal place of business--the vast\nstorehouse, wharf, and steam mill all combined. He opened a small\npostern in the main gate, trod a dark flagged passage, and reached the\nfoot of a flight of stairs that led to the chief offices. This he\nascended, and having reached his private room, lit the gas.\n\nFor a while he stood in the middle of the room, looking vacantly round\nhim. The office was luxuriously furnished; and in the wall opposite to\nthe table at which Mr. O'Donnell usually sat, and facing him, was the\ndoor of the strong-room.\n\nHe could hear the murmur of the water as it went by, if the engines\nhad stopped. But the engines were going on at full speed, making money\nnow--making money now for whom? That morning these twenty sets of\nstones had been whirling round, and at every rotation of each stone\nhe, James O'Donnell, was the richer.\n\nThese stones were going round still, making money still; but for whom\nnow?\n\nIt was a dismal thing to stand there realising the fact that the\nfruits of his forty years' hard work, sagacity, enterprise, thrift had\nall been squandered by someone else--had all been squandered by this\nVernon, in whom he had reposed implicit confidence; who was so pious,\nso sleek, so plausible, and yet had led him on into this horrible\nposition.\n\nHe sat down in his chair, and his eyes fell upon the door of the\nstrong-room. He had destroyed his pocket-book; his interest in his own\nprivate affairs was at an end. From what he had heard there was no\nchance of his saving a sixpence out of his large fortune. Some other\nman would work the mill no doubt, for it would be a valuable asset in\nthe affairs of Vernon and Son.\n\nIt was hard to think of this fine mill, for which he had made the\ntrade, and which he had built up from the foundation, passing away\nfrom him, now that he was too old to begin life again.\n\nIn that strong-room opposite him there were the books. They were all\nin perfect order. _They_ had never been made the slaves of a false\nbalance-sheet. They were the fair records of blameless transactions.\nEvery line in them could be verified. Every shilling of expense could\nbe accounted for.\n\nSoon, very soon, he knew not exactly when, strangers would come and\nexamine these books, and go through all the vouchers, but they should\nfind nothing in that strong-room of his except flawless records of\nhonest trade--and----\n\nThe vacant look left his eyes. All at once an intense, eager light\nburned in them. He grasped the back of the chair, and rose stealthily,\nas though to avoid the attention of someone acting as sentinel over\nhim, and who was half asleep.\n\nHe stole noiselessly in the bright gaslight across the room. With\nelaborate caution he took the keys out of his pocket and fitted one to\nthe lock.\n\nWith a dull, heavy sound the bolts fell back. He drew himself a foot\naway, as though he expected that door to be pushed open, and something\nto issue forth and seize him and do him deadly hurt.\n\nHe paused, breathing heavily. The door did not stir.\n\nHe stretched forth his arm and drew the door towards him. It yielded\nslowly and swung out into the bright, handsomely furnished office,\nuntil it stood at right angles to the wall.\n\nAgain he paused, and peered into the dark cavity before him. He seized\nthe outer edge of the door and steadied himself by it, leaned against\nit slightly so that it swayed slowly to and fro a little.\n\nHis face was now flushed and covered with sweat. His hands clutched\nthe door feverishly, frantically. His knees trembled so that he seemed\nin danger of sinking to the floor.\n\n\"It would be a fit ending to my life. My life is of no further use to\nme or to those I love, or to the business I have made, nor even would\nit be any use to those whom I shall not be able to pay. For although\nno one could work the business as well as I, if things had not come to\nthis pass, I am too old now to work for others where I have so long\nworked for myself.\"\n\nHe let go the door and stood unsupported for a while.\n\n\"If they should find in the strong-room of James O'Donnell nothing but\nthe unimpeachable records of his honest life, and his bones!\"\n\nHe seemed to gather strength from the thought. He drew himself up to\nhis full height. The look of intense excitement gradually faded from\nhis face. The tension of his hands relaxed, and he looked around with\nsomething like majesty in his gaze.\n\nHe was a lion at bay, but indifferent.\n\nHe walked up and down the room two or three times calmly,\ndeliberately, as if he were disturbed by a thought greater than the\nhourly commonplaces of a busy day.\n\nHe ran the matter carefully over in his mind. When in thinking of this\ndeed first, and saying to himself his creditors would find nothing in\nthat place but books, papers, and--he had paused at the word revolver.\nIt was occasionally necessary for some of his clerks to carry large\nsums of cash a distance from Rathclare, and when doing so the\nmessenger always took with him his revolver.\n\nThe lock by which the strong-door was finally secured could be turned\nonly from the outside, but there was a strong latch of three large\nbolts which caught and kept the door closed when it was slammed.\n\nThere were two keys to this door, but he had made it a rule never to\nentrust the second to anyone in his employment. When unable to be at\nhis office at ten o'clock in the morning, or at closing time in the\nevening, he had always given the key he now carried with him to his\nmanager, and had it left at his house the same night. The second key\nhe had hidden behind some books in a bookcase which he always kept\nlocked.\n\nBut the three bolts which kept the door fast during the working hours\nof the day could be shot back from the outside by means of a key, a\nduplicate of which the manager had.\n\nIn the strong-room that night there was a sum in cash of more than two\nthousand pounds.\n\nIf he went into that strong-room and used that revolver, the sound\nwould, in all likelihood, reach the ears of no one in the place, and\nnothing would be discovered for several days, as no one would suspect\nthe main bolts were not shot, since he had been seen to lock the safe\nthat day, and no one else could unlock it.\n\nHe made up his mind that, come what might, he would end his life where\nhis fortune had begun, and where now his ruin was complete.\n\nAnd still he could not think of bidding adieu for ever to the scene of\nhis life-long labours without one more look at the books which had\nbeen so honestly kept, and which he had hoped to hand down unblemished\nto his son Eugene.\n\nHe took up a lamp which lay on one of the side tables, lit it, stepped\ninto the strong-room, and drew the door sharply after him.\n\nThere was a loud bang. The three bolts shot into their places.\n\nHe was now in the strong-room with the records of his honest life, a\nrevolver, his power of retreat cut off, and the determination not to\nsurvive the night of ruin.\n\nHe had forgotten to put out the gas in his office.\n\n\n\n\n                            CHAPTER XVIII.\n\n\nThe strong-room was about ten feet by fifteen, and no more than eight\nfeet high. There were presses in it for the books, and an iron safe in\nwhich the cash and securities were kept. This safe, standing on\ntressels in a corner, was the one used by the house before the\nbusiness expanded to its present dimensions. Upon it the old man set\nhis lamp, and putting two deed-boxes one on the other, he placed them\nnear the safe for a seat. Then he opened the safe, and taking out some\nof the securities it contained, placed them beside him. He adjusted\nhis spectacles, and turned over the deeds and shares somewhat\nlistlessly.\n\nThe documents here represented a vast sum of money. Here were deeds on\nwhich he held mortgages, title-deeds, stocks, and shares.\n\nHe did not undo the tapes. He knew them all by sight, when and how he\nhad acquired them. This was the result of one speculation, that of\nanother. In his will this and this were left for life to his wife, and\nafterwards to his son and his son's children. This and this and this\nwere to go absolutely to his son.\n\nHe went on thus through all the documents in the safe. There was no\nhurry. It was still many hours to daylight. If all were over with him\nbefore people were stirring, all would be well. He had cut off his\nretreat. He could not now get out of that room even if he wished it.\n\nHe felt glad that he had come in here. This was a kind of antechamber\nto the other world. There was no going back now, and if he could\nderive any consolation from the contemplation of the past by the light\nof these records, he might do so without injuring anyone.\n\nAy, these were for Eugene. What would be his boy's fate? No doubt he\nwould recover from the hurt, for he was young and hearty. But then how\nwould he get a living? All his life he had been used to good things,\nand looking forward to a career of remarkable prosperity. Now he was a\nbeggar, an outcast from fortune. These properties and moneys had been\nintended for him. Now they would go to the greedy creditors of Vernon\nand Son.\n\nIt was too bad that just at the very moment his boy had made up his\nmind to marry, everything should be swept away from them. For some\nyears the only anxiety he had felt was that his boy should marry some\ngood amiable girl, and settle down in Rathclare, so that he (the\nfather) might feel that the successor to his business was at hand in\ncase anything should happen to himself.\n\nHe had not wished for money with the wife of his son. He had not\nwished for any social advancement. He was not a man who believed in\nfamily or society advancement. He wished his son to be an honest and\nprosperous trader in his native town, and when that sweet girl had\nbeen to their home a few times, he began to regard her as already his\ndaughter. He had intended making her a wedding-present independent of\nwhat he was to do for Eugene. Here was what he had intended for her.\nThese were the title-deeds of Rose Cottage, Glengowra, which would do\nthe young people for their summer home. It was a famous cottage for\nflowers, and there was grass for a cow, and there was a paddock, and a\nlittle lawn, and a large garden. Just the thing altogether for a young\ncouple in the summer time.\n\nLet him look at what the property consisted of. He read over slowly\nthe recital of all the things that went with Rose Cottage, the\nmeasurements of the land, and so on, as though he were about to buy,\nand it was necessary to be careful. Then he folded up the paper\nsoftly, and tied it with the tape, and set it by him on the ground.\n\nHe was not an imaginative man, but the few images which had visited\nhim seemed all the more brilliant, because of their rareness. And one\nof the visions which had come to him lately, and which pleased him\nmore than any other he had known for years, was that of Eugene and\nNellie living in this Rose Cottage, and he and his wife coming out in\nthe cool evening and having tea with them in the little arbour\noverlooking the sea.\n\nIt would be strange and delightful, now that the vigour of his youth\nand the strength of his manhood had passed away for ever, to be the\nguest of his own son; to hear his son say, \"Welcome, father,\" and to\nsee this tall, fair girl, who had such bright and pleasant ways,\ntending to his good-hearted, kindly old wife, Mary. To see her placing\nthe chair of honour for her, and making much of her, would be a thing\nto live for and enjoy. And then, later, there would be children who\nwould call him grandfather, and, with their fresh young voices and\ngallant spirits, take away the feeling of toil and the weariness of\nyears.\n\nWhat would Mary do? Mary, whom he had married long ago; and yet, now\nthat he had come to the end of his life, it seemed but yesterday. He\ncould see every event of their marriage-day more clearly than he could\nsee what had happened yesterday; for his eyes had grown dim since\nthen, and the magic charm of memory is that it forgets so well what it\ndoes not wish to retain.\n\nBah! It would never do to think of those times, and of his old Mary\nleft alone and poor upon the world. It would take the resolution out\nof him to think of her. It would rob him of his manhood to picture her\ndestitute in the face of unsympathetic men. No. It would rob him of\nthe last remains of vigour to fancy her standing alone and deserted,\nwithout a home or a meal. He had come into that room for the purpose\nof closing his life with his business career. Eugene was young and\nfull of spirits, and had many friends, and would soon get something to\ndo, and be able to give his mother a little, and to marry.\n\nHe must not take a gloomy view of the future for those he was leaving\nbehind. If he wanted to keep up his resolution he must think of the\nfuture he was losing in this great crash. It was of Eugene and\nEugene's wife he must think. The fact that he could be of no further\nuse to his son, or his wife, or his son's wife, was the thought to\nkeep him to his resolution. If things had gone otherwise with him he\ncould have made those young lives so happy as far as worldly gear was\nconcerned. What further use was he on earth? Let him leave all at\nonce. Why should he confront this trouble and disgrace--trouble\nunearned, disgrace unmerited?\n\nHe took up the documents from the floor and replaced them all\ncarefully in the safe. It was in this safe the money was kept. He\npulled out the drawer containing it. A week ago he would have thought\nthis a comparatively small sum. Now it seemed very large indeed. If it\nhad been only so managed that this two thousand pounds could have been\nhonestly saved from the wreck, it would have been sufficient to\nprovide, in an humble way--but there! Let the thought go. Nothing\ncould be saved--not a shilling.\n\nHe closed the drawer, and then drew out the one next to it.\n\nThis contained the revolver.\n\nThe light of the lamp so fell that when the drawer was fully out only\nthe barrel of the weapon was in the light.\n\nThe old man stood looking at that glittering barrel. It was as though\nthat barrel was a deadly snake slowly issuing from the darkness, and\nhe was powerless to move, to avoid it.\n\nOnce more all his strength forsook him. His face flushed, his limbs\ntrembled; he clasped his hands convulsively. He drew back a pace and\nalmost fell against the opposite side. He put his hand before his eyes\nand groaned.\n\n\"Has it come to this with me,\" he said, \"in my old age? Can it be\npossible, I, who never did a dishonest act, must fly from life because\nof the dishonesty of another?\"\n\nHe put his hand up to his neck and tore his shirt open. He dropped his\nhands, threw up his head and looked around him. \"Great God! what is\nthis?\"\n\nThe lamp was burning blue. His head was giddy. He was suffocating!\n\n\n\n\n                             CHAPTER XIX.\n\n\nSuffocating? Yes; there could be no doubt about it!\n\nUp to this, James O'Donnell had forgotten that the strong-room was\nalmost air-tight, and that the air required by him and the lamp was\nabout what should have been exhausted since he entered the room. For\nyears he had been familiar with the great safe, and it had never\nbefore occurred to him that to shut any man up in it for a lengthened\nperiod would be almost certainly death.\n\nWas he to die of suffocation, and under the circumstances of his\npresent position?\n\nAlready his thoughts were becoming obscured. There was the revolver\ngleaming at him from the drawer. But his thoughts had taken a\ncircuitous route; and although he knew that a short time ago the\nrevolver had formed the main portion of an important design, he now\ncould not connect it clearly or coherently with that intention. He was\naltogether occupied with the thought of suffocation, and but partially\nable to concern his mind with any other idea.\n\nHow would it be if he died here, and of the death that threatened him?\nHow would it be?\n\nHe could not answer. He did not know.\n\nHe felt a tightness across his forehead, an oppression upon his chest.\nThe tightness and the oppression were little more than uncomfortable.\nHe had scarcely a pain. In fact, he felt a pleasant languor out of\nwhich it would be a decided inconvenience to raise himself.\n\nThen for a moment it came forcibly home to him that he was dying, and\nwould die before succour of any kind could reach him.\n\nThe motives which had led him to come there at such an hour, and which\ninduced him to shut himself up and cut off all retreat, were now\nobscure. By a great effort he could dimly perceive that something was\nwrong with his business concerns.\n\nWhat was that?\n\nA noise without! A noise at the other side of the heavy iron door. Who\nor what could make a noise outside there in the private office at such\nan hour? It was within the duty of no one to be in his private office\nat this hour. No one could now be there for any honest purpose.\n\nThe propinquity of the material sounds enabled them to appeal to his\nreason more forcibly than the murmur of the mill or the river, or the\ntumultuous, distracting echoes of disaster beating through his brain.\n\nAll at once the sounds, his physical and financial position, converged\nand were focussed upon a single relic of memory. Long ago, in some\nbook he had read of a famous cave called \"The Cave of Dogs,\" somewhere\nin the south of Europe, where, when men and dogs entered together, the\ndogs were suffocated by the exhalation lying close to the ground,\nwhile the men, because of their greater stature, moved on unharmed. He\nknew at this brief moment of active memory the same substance which\nnow threatened his life proved fatal to these dogs.\n\nIf he now raised himself higher in his suffocating chamber, was there\nany likelihood of prolonging his life by seeking air as high up as\npossible in the room? It is true he had no great desire to prolong his\nlife. He had by this time forgotten he had had any desire to destroy\nit. Yes, he would see if any virtue of life lay in the air above his\nhead.\n\nHe mounted upon the deed-boxes and thrust his head up.\n\nNow he had pains and a tingling sensation, but the dimness and dulness\nof the intellect gradually diminished.\n\nThe noise was repeated without. What could it be? His mind had by this\ntime become comparatively clear. He now knew he had come to that place\nfor the purpose of destroying his life, with the intention of\nobliterating the world from his perception simultaneously with the\ndestruction of his fortune.\n\nBut what were those noises which again broke in upon his ear?\n\nNow he remembered. There was a considerable sum of money in cash in\nthe strong-room. Some thieves had got scent of this fact, and were now\nin the outer place with designs upon the gold and notes lying in the\nsafe?\n\nIf these wretches broke in when he was dead and carried off the money,\nand his dead body was found later there (his head was so stupid, that\nhe could not see exactly what the inference would be), would it not\nseem in some way or other that he had applied the two thousand pounds\nto his own purposes--given them to his wife or son, say--and then\ndestroyed himself?\n\nAlthough he felt relieved from the suffocation he had endured in the\nlower air, he knew now that this relief could not last long, and that\nthe air he now breathed would soon become as tainted as that which he\nhad lately left.\n\nWhat should he do?\n\nTo die in the midst of his commercial troubles--to die, leaving behind\nhim an unblemished reputation, and to die the seeming thief of a\npaltry two thousand pounds, were widely different things.\n\nAnd yet he did not appear to have much room for choice, for should he\ncontinue as he now was and make no sign, he would, beyond doubt, die\nof suffocation; and if he made any sign and these men had the means to\nbreak in, and did break in before assistance came, they would no doubt\nsacrifice his life rather than forego their design of plunder.\n\nHe paused for a moment in thought. Then, holding his breath, he\nstepped down, took the revolver out of the safe, and got up on the\ndeed-boxes once more.\n\n\"I shall sell my life dearly,\" he said to himself, \"if they force that\ndoor.\"\n\nStanding bolt upright on the deed-boxes, he fixed his eyes steadily on\nthe only means of ingress to that room.\n\n\"It is not likely,\" he thought, \"there are more than two or three of\nthese ruffians, and I have six shots here. But how long will this air\nlast? How long is it possible for a man to live on the eighteen inches\nmore air I have gained since I mounted these boxes? For a man and--a\nlamp? I don't want the lamp. I have seen here all I desire to see. If\nthey break in I will have no difficulty in seeing them, for my eyes\nwill be accustomed to impenetrable darkness, while they must carry a\nlight of some kind, which will enable me to make them out. I and the\nlamp. It is as though there was food in a ship for a certain time for\ntwo people. If the one dies the other will have the double share. If\nthe lamp or I die now the survivor will have the double share. In this\ncase the choice is easily made.\"\n\nHe filled his lungs and blew down the chimney of the lamp.\n\nThe darkness of the strong-room was now so intense that it was\nabsolutely impossible to see any object, however large or however\nnear. For all the purposes of sight the space enclosed by the four\nwalls was an absolute void. The old man, of course, knew he was\nstanding on two deed-cases in the strong-room of his business place;\nthat he held a revolver in his hand; that there were burglars without\nand money within, and that he was threatened with suffocation. The\nquestion now was, whether they would succeed in bursting open that\ndoor before the rising tide of poisonous gas reached his nostrils.\n\nThe lamp being now extinguished, and there being some ventilation to\nthe safe, the deadly gas, which would be sufficient to destroy life,\nwas rising at a greatly diminished rate. A little of the heavy\ncarbonic acid succeeded in exuding through the lower portion of the\nslight spaces between the door, threshold, and jambs; a little of the\npure exterior air infiltrated through the upper portion of the slight\nspaces between the door, lintel, and jambs.\n\nJames O'Donnell had no means of knowing at what rate the deadly gas\nwas now rising, or whether it had ceased to rise at all, or whether it\nwas declining. It was not impossible, nay, it was not improbable, that\nthe deadly vapour might rise no higher than it had stood when he put\nout the lamp.\n\nIt would not do for him to make the least noise, for the gas might\nstill be rising, and in case he made a noise the burglars might be\nscared away for a time, only to return when he had succumbed to the\ndeadly vapour, break open the room, and so blast his character for\never.\n\nIt was now necessary for him to stand bolt upright in that ebon\ndarkness, with his eyes fixed on what he knew to be the position\noccupied by the door. Then, as soon as anyone opened that door, it\nwould be his duty to fire, and to fire with as deadly an effect as\npossible, for he was an old man, no longer strong or active, and could\nnot hope to succeed against even one man who would undertake such an\nenterprise, and the chances were there would be more than one in this.\n\nHe had no means of computing time. In the disordered condition of his\nmind it was impossible to tell how the minutes went by. Now for some\nminutes the sounds in the outer room had ceased. Any moment they might\nbe renewed. There would, of course, be a sound of hammering, although\nthe sound would be very dull. He had once seen a burglar's hammer. It\nwas made of lead, the face of it being covered with leather soaked in\noil. The wedges used were always of wood. But no matter how muffled\nthe blow, or how little noise the progress of the wedge made, the\nsound could not escape his ear.\n\nHe took out his watch and listened to it. He counted the ticks, but\nfound they conveyed no idea of time. The very sound of the watch\nconfused his senses, and threw him into new perplexities. Holding the\nwatch to his ear, and the revolver in his right hand down by his side,\nhe stood motionless for what seemed to him a very long time.\n\nIt was strange, but still he heard no sounds of hammering. Could it be\nthat the first effect of the poisonous gas upon him had been to\ndisturb his senses, and that the noises he fancied he heard had been\nthe offspring of imagination?\n\nAh! They were beginning at last. He caught the sound of their first\nattempts. He knew it would take a considerable time to break in that\ndoor, and mentally he groaned at the notion of delay in his present\nperilous condition.\n\nSuddenly he started as though he had been shot.\n\nThe door swung open rapidly on its hinges. The full light of the\noffice sprang with dazzling effect into the darkness where he stood.\nHe was paralysed.\n\n\"Seize him!\" cried a voice from without.\n\nThen all at once, and before he had time to raise the arm in which he\nheld the weapon, he was in the clutches of two men, who dragged him\nout ruthlessly into the glare of the office, and then started back\nfrom him.\n\n\"It is the master himself!\"\n\nJames O'Donnell staggered for a moment, dazed by the gaslight and the\nperception that the men who held him were no burglars, but the\nwatchmen of the place, and that behind the door, as it now stood fully\nopen with the day-key in it, was the manager of all his business,\nCorcoran.\n\nWhen the watchmen made up their minds what to do they sent for Mr.\nCorcoran. He brought the key with him; and then all three, having\ntaken off their boots, stole into the private office, and finding no\nclue there, the manager, with little hope of discovering anything, put\nhis day-key into the lock, turned the bolt swiftly, and, to his\nastonishment, pulled open the door. His astonishment rose to perfect\namazement when he found a man inside, and when that man turned out to\nbe no less a person than James O'Donnell.\n\n\n\n                            END OF VOL. I.\n\n\n\n\n                         * * * * * * * * * *\n\n           CHARLES DICKENS AND EVANS, CRYSTAL PALACE PRESS.\n\n\n\n\n\n\n\n\nEnd of Project Gutenberg's The Last Call (Vol. 1 of 3), by Richard Dowling\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\nALSO BY DR. JASON SELK\n\n_10-Minute Toughness: The Mental-Training_\n\n_Program for Winning Before the Game Begins_\n\n_Executive Toughness: The Mental-Training_\n\n_Program to Increase Your Leadership Performance_\n\nMany of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Da Capo Press was aware of a trademark claim, the designations have been printed in initial capital letters.\n\nCopyright \u00a9 2015 by Jason Selk and Thomas Bartow\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, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. For information, address Da Capo Press, 44 Farnsworth Street, 3rd Floor, Boston, MA 02210\n\nDesigned by Trish Wilkinson\n\nSet in 11-point Adobe Caslon Pro by the Perseus Books Group\n\nLibrary of Congress Cataloging-in-Publication Data\n\nSelk, Jason, author.\n\nOrganize tomorrow today: 8 ways to retrain your mind to optimize performance at work and in life \/ Dr. Jason Selk and Tom Bartow, with Matthew Rudy.\n\npages cm\n\nIncludes bibliographical references and index.\n\nISBN 978-0-7382-1870-0 (e-book) 1. Time management. 2. Habit. 3. Performance. I. Bartow, Tom, author. II. Rudy, Matthew, author. III. Title.\n\nBF637.T5S45 2015\n\n158\u2014dc23\n\n2015025600\n\nFirst Da Capo Press edition 2015\n\nPublished by Da Capo Press\n\nA Member of the Perseus Books Group\n\nwww.dacapopress.com\n\nNote: The names and identifying details of people associated with events described in this book have been changed. Any similarity to actual persons is coincidental.\n\nDa Capo Press books are available at special discounts for bulk purchases in the U.S. by corporations, institutions, and other organizations. For more information, please contact the Special Markets Department at the Perseus Books Group, 2300 Chestnut Street, Suite 200, Philadelphia, PA, 19103, or call (800) 810-4145, ext. 5000, or email special.markets@perseusbooks.com.\n\n10 9 8 7 6 5 4 3 2 1\n_This book is dedicated to everyone out there trying to improve. No matter what the outcome, you never embarrass yourself by emphasizing improvement._\nIt is the effort to improve that is valuable. By repeating it over and over you will master it.\n\n\u2014SHUNRYU SUZUKI\nContents\n\nForeword, by Jim Wooden\n\nIntroduction: Sports Psychology Meets Wall Street, by Matthew Rudy\n\n**1** Organize Tomorrow Today\n\n**2** Choose Wisely\n\n**3** Maximize Your Time\n\n**4** Win Your Fight-Thrus\n\nHalftime\n\n**5** Evaluate Correctly\n\n**6** Learn How to Talk to Yourself\n\n**7** Learn How to Talk with Others\n\nCrunch Time\n\n**8** Become Abnormal\n\nEpilogue: Inspiration and Beyond\n\nNotes\n\nAbout the Authors\n\nIndex\nForeword\n\nWhen I speak to people about my dad, Coach John Wooden, one of the first things they usually want to know is what he was like in the \"real world,\" outside of what you saw at a basketball game.\n\nDad was a simple, consistent person, who developed and lived by a core set of principles his entire life. He wasn't a brimstone preacher\u2014either inside the locker room or at our house. He was a motivator in a different way\u2014by being a true teacher, and not \"just\" a coach.\n\nOver the years, his players and the other people he met through basketball came to understand very well that Dad's philosophies and approach weren't some sort of program that he followed some of the time, or \"put on\" to look better in the media or in recruiting.\n\nHe talked about what he believed. He followed it. And he never stopped thinking about ways to make small improvements. He was always teaching, but he was also always learning.\n\nBasketball was very important to my dad, but the core beliefs and philosophies he had\u2014many of which Jason and Tom talk about in this book\u2014aren't about basketball.\n\nThey're about life.\n\nIt always amazed Dad that he became more famous in retirement than he ever was as an active coach. But I think that fame came for exactly those reasons. People began to understand that those principles applied to far more than basketball\u2014especially in the business world.\n\nIn the last two and a half years of Dad's life, I was fortunate to be able to spend two days of every week with him. It was the best time in my life. So many of the friends he made in and outside of basketball would come by, and he stayed connected to the game.\n\nThe talks he was able to give in corporate America through people like Tom really brought him a lot of joy. It wasn't about the money he could make. It was about getting his core message through to people young in life\u2014the pyramid of success, the two sets of three, making friendship a fine art.\n\nI heard him say those same things for fifty years, but they weren't just catchphrases. They're the basic building blocks of a life well lived. I used the same principles with my own kids, and now they're using them with their own children.\n\nWhen we're shooting hoops, my grandson\u2014John Wooden's great-great grandson\u2014will use the backboard on a shot and say, \"That's what Paw-Paw wanted me to do.\"\n\nDad would have been thrilled to know he influenced Tom and Jason to go out and help so many people in the business and sports worlds.\n\n_Jim Wooden_\n\n_June 15, 2015_\nIntroduction\n\nSports Psychology Meets Wall Street\n\n\"YOU HAVE TEN MINUTES . . .\"\n\nJason Selk was standing in the St. Louis Cardinals' clubhouse at their spring training facility in Jupiter, Florida, in March 2006, getting ready for what he had originally thought was going to be a two-hour introductory presentation in his new role as the Cardinals' director of mental training.\n\nCardinals general manager Walt Jocketty had extended the invitation to Jason before the season, but when he brought him into manager Tony La Russa's office in the clubhouse, it was clear that what Jason had was an audition\u2014not a job.\n\nAfter Jason gave a brief synopsis of the two hours of material he had prepared, La Russa looked up from the paperwork on his desk.\n\n\"You have ten minutes.\"\n\nMoments later, looking out at that clubhouse full of All-Stars and future Hall-of-Famers\u2014guys like La Russa, Albert Pujols, Chris Carpenter, and Scott Rolen\u2014Jason decided to introduce his Mental Workout concept. The full version of the workout is designed for elite athletes, and it involves visualization, positive self-talk, and controlled breathing.\n\nAfter Jason worked through the first step of the Mental Workout, pitching coach Dave Duncan asked to go through the second step. Jason was up against the time limit, so he looked over to Tony to see if it was okay to continue. He nodded, and Jason shared the second step.\n\nWhen he finished, All-Star third baseman Scott Rolen chimed in and asked if he could share the third step. Again, Jason got the okay from Tony to continue.\n\nAfter Jason shared the third step, reigning Cy Young Award winner Chris Carpenter stopped him and got up. \"Everybody better pay attention,\" he said. \"This is what we need to take it to the next level.\"\n\nFrom that moment, Jason was accepted as part of the staff, and he built some incredible relationships with the players over the next six seasons. At the top of any professional sport, the physical differences between the teams are minuscule. The smallest mental edge can mean the difference between losing in the National League Division Series and holding up the World Series trophy. Those Cardinals teams won three division titles and two World Series championships, and at least a small part of that success came from the peak mental performance training the team got from Jason.\n\nWorking with the Cardinals certainly helped confirm Jason's credentials in the world of sports psychology and open a lot of doors. But he got something much more valuable than a professional credential or some references out of the experience.\n\nAs he was giving that first talk in the clubhouse in March 2006, the Cardinals staff and players didn't know much about him. They didn't know if he had something useful to offer. But he wasn't more than a few minutes into his talk when he noticed that the majority of the players and coaches were taking notes.\n\nThe Cardinals have long been considered one of the model franchises in all of professional sports, and at that moment some of the whys behind that reputation became crystal clear to Jason.\n\nFrom the top down, they established an \"obsession for improvement\" as a key part of their culture. No matter how successful they got, they were always on the lookout for new information that could help them improve\u2014and a better program for incorporating that information in an organized, efficient way.\n\n_Organize Tomorrow Today_ is that program.\n\nIt is a guide for using your most powerful (and often most underestimated) tool\u2014your mind\u2014the way it was designed to be used.\n\nOVERLOAD\n\nModern life can demand an almost overwhelming amount of attention.\n\nIt isn't any surprise that a huge \"time management\" industry has grown up around this reality. Amazon is stuffed with books\u2014and devices\u2014designed to do everything from manage your schedule to convert you to a paperless office. A thousand different calendar and schedule apps will turn your smartphone into a battery-powered personal assistant\u2014one that works twenty-four hours a day.\n\nThere's one problem.\n\nNobody fully understands the power of the mind. It is an incredibly powerful thing\u2014and one we constantly underestimate. But it wasn't designed to function that way.\n\nIn 1956, Dr. George Miller published one of the most influential papers in the history of psychology. He called it \"The Magical Number Seven, Plus or Minus Two,\" and it outlined for the first time a human's mental \"channel capacity\"\u2014the amount of information the average person could remember at a given time. It wasn't so clear-cut as the title, but Miller's basic premise is still valid\u2014so much so that it's referred to as Miller's Law. Humans can only process up to seven simple concepts at a given time. This channel-capacity law actually serves as the underpinning of the modern phone-number system. Seven numbers are the most people can easily remember.\n\nWhat happens when we flood our brains with information?\n\nOnce we start thinking about more than a handful of things at a time, our ability to execute _any_ of those things at a high level becomes compromised. And the problem is compounded by the fact that very few of the things we're asking our minds to do are simple or one-dimensional, like remembering the digit of a phone number. We're asking our minds to tackle multiple multidimensional tasks at one time\u2014and our channel capacity at that level falls short. We can't really carry in our \"working memory\" any more than _three_ things at one time and have a chance of doing any of them well.\n\nIt's something like being a beginner juggler. If you work at it, you can handle juggling three things at once. But once a fourth item gets thrown in, the system is overloaded and it all gets dropped.\n\nGeorge Miller found all of this to be true in the 1950s\u2014when nobody was carrying around high-powered miniature computers in their pockets, mail came in an envelope with a stamp on it, and \"twitter\" was just something the birds did.\n\nToday, his findings are more valid than ever. All of us are busier than ever before. It's way more than just work: We're making calls, sending texts, going to meetings (real and virtual), and \"networking\" with professional colleagues. We're wrestling to balance work and home life in a time when information doesn't operate on a \"normal\" 9-to-5 schedule.\n\nThe best time management plan in the world\u2014or the best calendar, or the best device, or the best app\u2014doesn't address the fundamental problem of channel capacity. If technology were the answer, success would be as simple as flipping a switch.\n\nFiguring out how to hack Miller's Law and find more mental bandwidth isn't going to help you live a more productive life. Fitting more pieces into the puzzle won't make you more successful.\n\nIn _Organize Tomorrow Today,_ we're going to show you how to embrace channel capacity instead of fighting against it. You're going to learn how to make decisions, establish priorities, and light your own motivating fire instead of continuing to chase the counterintuitive concept of multitasking. Most people still seem to believe that being busy is the equivalent of being important, but the highly successful have learned that being busy is a waste of time: being productive is the goal.\n\nWHY SELK AND BARTOW?\n\nDr. Jason Selk has been training world-class athletes for peak mental performance for almost two decades, including that six-year stint as the St. Louis Cardinals' mental performance coach. Those Cardinal teams won two Worlds Series titles in that span. He has worked with Olympians and professional athletes in every major sport, including the NFL, NBA, Major League Baseball, the PGA and LPGA tours, NASCAR, and the Ultimate Fighting Championships. Jason has written two other groundbreaking mental performance books\u2014 _10-Minute Toughness_ and _Executive Toughness._ His work has appeared in dozens of magazines, from _Men's Health_ to _Shape,_ and he has appeared on numerous television shows for major networks and cable channels like ESPN, NBC, and CBS.\n\nTom Bartow essentially rewrote the book on how to train financial advisors. His concepts have been used by tens of thousands of them, first at Edward Jones\u2014where Tom created that company's top-rated Advanced Training Program\u2014and then at American Funds Group. On the Friday after the September 11 tragedies in 2001, American Funds chairman Dave Short and Tom devised the company's business plan to move forward in the uncertain markets and rebuild confidence among investors. In 2003, American Funds received $65 billion in cash inflows, a new record for the mutual fund industry. One analyst compared this achievement to winning the World Series and the Super Bowl in the same year.\n\nThe seeds for Jason and Tom's partnership in the world of mental performance training were planted in October 2008, when the stock market lost more than 1,000 points\u2014trading under 8,000 for the first time in years. Many people in the financial markets\u2014and financial advisors, in particular\u2014were extremely nervous. On the day of the 1,200-point market drop, Tom got a number of calls from advisors\u2014most of them in the top 5 percent of their firms in revenue. Even with that success and experience, they had some panic in their voices. They asked him over and over, \"What do we do now?\"\n\nBefore getting into business coaching, Tom was very successful in another kind of coaching\u2014on the basketball floor. At age twenty-five, he took over a team that had twenty-one consecutive losing seasons. In his second year as coach, the team went 20\u20135. The speech Tom gave those investment advisors came directly from the lessons he had learned during those first two seasons coaching the college team\u2014that it is crucial to learn how to execute during times of adversity.\n\nA few months later, Tom was reading an article in _Men's Health_ about Jason's first book, _10-Minute Toughness._ The book was born from Jason's research and clinical experience with world-class athletes. It unlocked the \"why\" of elite mental performance\u2014the science behind the success patterns that high achievers use. Tom bought it and read it several times, studying the principles in detail. Jason's research and real-world experience with elite athletes almost perfectly dovetailed with Tom's work as a high-level business coach. The message in the book\u2014on simple steps to increasing your mental performance in business and life\u2014spoke directly to the challenges Tom's advisor clients were facing. It was as if they had been working together for twenty years.\n\nThe stock market continued to struggle through 2009, and Tom flew around the country giving speeches and seminars for investment advisors, helping them learn how to guide their clients through a historically rough time. He began recommending Jason's book during his speeches, promoting it as the premier mental playbook for handling adversity. Jason and Tom got together in 2011, and in their first few conversations, they came to realize two fascinating things about the world of high achievement.\n\nFirst, the greatest athletes and the greatest businesspeople are incredibly similar in their wiring. They think in many of the same ways, and they're built with the same kind of competitiveness. While Jason and Tom were using different terminology, they found they were really speaking the same language. The best coaches in both sports and business are expert \"practical psychologists.\" They're perceptive evaluators, and they know how to figure out what makes a person tick. They understand the best approaches to take to reach different kinds of people.\n\nSecond, Jason and Tom saw that many, many people in both the sports and business arenas were being held back because they believed in a common misconception. The greatest athletes and businesspeople _are_ different from average achievers\u2014and some of the differences are innate. Some people are just faster, taller, stronger, or smarter than the rest of us. But _most_ of the difference between the highest achievers and the average achievers is in how they think and how they prepare.\n\nYou can learn it, and you can grow it. That's the message Jason and Tom have shared in a series of training seminars around the country since late 2011. At their Organize Tomorrow Today workshops, they have been giving executives and salespeople a set of real-world techniques developed from the worlds of peak mental performance in sports and business to think better, prepare better, and achieve more.\n\nThat isn't just marketing copy.\n\nJason and Tom have coached not only professional athletes and financial advisors, but also executives of both large and small corporations, attorneys, physicians, insurance professionals, and others. No matter the industry, clients testify that the Organize Tomorrow Today program does two important things: it significantly reduces stress, and it creates more success.\n\nThat success isn't abstract or hard to define. It's measured in real-world financial results every quarter and every year. As word started to get out about the results Jason and Tom were helping people achieve, many different kinds of organizations tried to figure out how they were able to coach those teams to those kinds of gains. They couldn't, so they hired Jason and Tom, who have now used the principles behind Organize Tomorrow Today to train thousands more executives, managers, and salespeople.\n\nCharlie Munger is well known for opening the minds of people in businesses of all shapes and sizes. One of Munger's most famous concepts is one he calls the \"Lollapalooza Effect\"\u2014which is what occurs when strong forces work together to produce an exponential result much more substantial than what each force could produce individually.\n\nJason and Tom believe that's what they've found with their Organize Tomorrow Today program\u2014a way to exponentially improve the way you function in business and in life.\n\nAs a team, they've distilled their more than forty years of experience coaching athletes, executives, and salespeople at the very top of their disciplines. They've seen what works and what doesn't. Their Organize Tomorrow Today program condenses those success patterns into a streamlined set of core habit-building principles anyone can use. They are the same principles that their athletic clients have used to win World Series titles and Olympic gold medals, and that their professional clients have used to shatter revenue and sales records and create millions of dollars in new business.\n\nThe strategies are real, and you can incorporate them into your routine right away.\n\nHOW?\n\nOne of Tom's mentors during his basketball coaching career (and on into his next life as a business coach) was the legendary UCLA coach John Wooden\u2014who won ten national titles in a twelve-year stretch, in part by emphasizing the fundamentals of the game.\n\nAt the beginning of every season, Coach Wooden would begin the team's first practice the same way. No matter how much experience his players had, he started with the basics. He explained very specifically how to put on socks in a way that would prevent blisters, and how to get the laces on each shoe to sit flat to provide the most security and support.\n\nAs all great coaches know, skill mastery depends on learning through repetition, one step at a time. In basketball, you must learn to hold the ball correctly before you can even consider shooting it. The approach in _Organize Tomorrow Today_ follows the same building-block principles; it shows you how to identify the fundamentals, then slowly (and certainly) move to mastery.\n\nThis book is made up of eight simple, concrete, easy-to-understand concepts:\n\n\u2022 Organize Tomorrow Today\n\n\u2022 Choose Wisely\n\n\u2022 Maximize Your Time\n\n\u2022 Win Your Fight-Thrus\n\n\u2022 Evaluate Correctly\n\n\u2022 Learn How to Talk to Yourself\n\n\u2022 Learn How to Talk with Others\n\n\u2022 Become Abnormal\n\nThey work together as a performance improvement plan for both work and life\u2014but you don't even have to master them all to get a benefit. In fact, Jason and Tom don't want you to tackle them all.\n\nChannel capacity is the key. One of the biggest mistakes people make in business and in life is that that they try to change too many things too quickly. You see it on New Year's Day, when so many people resolve to change everything they eat and go to the gym five times a week. After a burst of early enthusiasm for the new goal, reality sets in, and it gets harder and harder to cope with all the wrenching changes. At that point, it only takes a few days of \"failure\" to get discouraged and pitch the whole plan.\n\nInstead, as you read the book, think about which of the eight concepts address some of the issues you're having in your professional or personal life. Pick the _one_ that resonates the most. Start with that, and commit to following the step-by-step guidelines in that chapter. The key to high-level success is to pick one thing to change\u2014yes, just one\u2014and master it. The title of this book, _Organize Tomorrow Today,_ comes from the concept that has been the most popular starting point among the attendees at Jason and Tom's seminars (which is also why it's the first one we cover).\n\nIf all you take from this book is a single, concrete change from one of the eight concepts, it's enough for you to make a true breakthrough to the next level of success\u2014however you define it.\n\nJason and Tom are realists. The \"law\" of human channel capacity pretty much dictates that, at most, you're going to be able to successfully incorporate three of these ideas. So think of the material below as a sort of menu of improvement strategies. We'll show you eight dishes. All of them have been prepared by master chefs, and all of them will give you the nutrition you need. Sample them all, begin by choosing the three that resonate the most with you and then from there choose your favorite to be the main entr\u00e9e, then attack. Over time, you can build on them, one concept at a time.\n\nThe rules are simple and straightforward, but this isn't some kind of \"get-rich-quick\" plan. You'll have to put in the work and change some habits to see improvement. It isn't any different from building your body in the gym, or improving a sports skill through practice. Follow steps in the chapters and hold yourself accountable for the results, and we guarantee that you can find the kind of performance gains that athletes, executives, and salespeople spend tens of thousands of dollars to achieve.\n\nIn _Organize Tomorrow Today,_ you're getting your success pattern blueprint\u2014a complete plan that builds on the little victories and establishes the long-term productive habits that will let you take control of your time and your life. That said, at the beginning and end of every seminar Jason and Tom teach, they offer one simple \"warning\" to the class, and they'll share it here, too: They are firm believers in increasing knowledge through training and through learning \"best practices.\" But knowing something doesn't change your life. _Doing_ something does. Getting through this book (or a class) is one thing, but there's a huge difference between acquiring information and _understanding_ it. And there's an even wider gap between understanding it and implementing it, or actually _doing_ it. This is why there is such an emphasis in _Organize Tomorrow Today_ to avoid trying to master all eight concepts at once. Doing so is a recipe for inaction and failure. Success comes in one dedicated and focused step at a time.\n\nThe most successful people we see are the ones who take this information and use it in real life. Every day.\n\nFollow the template, and you're getting a playbook for speeding up the process of getting from information acquisition to skill implementation.\n\nJason and Tom call it the Owner's Manual for Doers.\n\nLet's get started.\n\nORGANIZE TOMORROW TODAY\n\nCollege football is one of the most competitive arenas around\u2014in sports or business. The competition doesn't just happen on the field, during the season. Recruiting is a brutal business, and training eighteen-year-olds to succeed on both the field and in the classroom is a complicated job.\n\nThat's why what Nick Saban has done since 2000, first at LSU and now at Alabama, is so amazing. Saban's teams have won four national championships in that span. Only Bear Bryant, Alabama's coach from 1958 to 1982, won more.\n\nIt's tempting to attribute Saban's success to great recruiting and superior in-game coaching tactics. But in reality, most of the other power programs handle those parts of the process in a very similar way. And Saban's offensive and defensive schemes have always been considered pretty straightforward.\n\nWhat Saban does differently\u2014and what causes his program to be the one that other top schools try to copy\u2014is redefine \"success\" for each individual player and coach working under him. Coaches and players at Alabama don't talk about winning and losing. They talk about consistency of preparation and effort, and about consistently excelling at the few core priority tasks they have each day.\n\nCoach Saban doesn't ask his players and coaches to accomplish everything in a given day, and he doesn't require them to do everything they possibly can to \"improve.\" He knows that if players are trying to focus on \"everything,\" in essence they are focused on nothing. He teaches the fundamentals, and he helps them establish their priorities for the next day\u2014and the next week, and the next season. These priorities become known as \"the process.\"\n\nIn sports or business, the art of improvement comes from the skill of establishing those few daily priorities and benchmarks the same way Coach Saban does. It comes from doing it ahead of time, so you can establish where your attention needs to fall\u2014which gives your conscious and subconscious mind direction and calm.\n\nIn our world, we call this process \"Organizing Tomorrow Today.\"\n\nYou've probably jotted out a to-do list on a Post-It note and stuck it to the side of your computer monitor. Or maybe you've even built an elaborate \"to-do\" system on your smartphone or calendar. We're not here to tell you that your method\u2014if you have one\u2014is worthless or wrong. The goal here is to puncture the myth that the highest-achieving people in sports and business are the ones who get the most done, and to get you to stop chasing that as your goal. Coach Saban would be the first one to say that isn't what works.\n\nOne of our favorite quotes comes from Stephen Covey, author of the international bestseller _The 7 Habits of Highly Effective People._ He said, \"The noise of the urgent creates the illusion of importance.\" Even the best-intentioned people fall into that trap every day. They confront their own to-do list\u2014either the one in their head or the one they've written down somewhere\u2014and put out the fires causing the most smoke, or the ones that are easiest to pick off. There's no question it can be satisfying to check a particularly troublesome or annoying problem off the list once it's been handled\u2014or to clean up two or three simple tasks\u2014but that doesn't mean you've spent your time in the most effective way.\n\nIn our experience, those who enjoy the most success are the ones who do the best job prioritizing the day's activities and accomplishing the most important tasks\u2014not the greatest _number_ of tasks. It's a skill even the most successful people can lose track of along the way.\n\n* * *\n\nIn our experience, those who enjoy the most success are the ones who do the best job prioritizing the day's activities and accomplishing the most important tasks\u2014not the greatest _number_ of tasks.\n\n* * *\n\nOne of Tom's most successful clients was rated one of the top ten financial advisors in the country. He has hundreds of millions of dollars under advisement, and a long track record of accomplishing the important tasks in his business. But recently, he called Tom with an urgent request for help\u2014and he wasn't somebody to call very often.\n\nHere's how the conversation went:\n\n**Tom:** What's going on?\n\n**Advisor:** My business isn't where I want it to be. . . . Instead of me running it, it's running me. I need your help.\n\n**Tom:** Okay. . . . Today is Thursday. How many days this week have you gone into your office with the names of clients you were going to speak to that day written down, along with a summary of what you wanted to talk about?\n\n**Advisor:** That's it! I stopped getting organized for the next day. I forgot. . . . Thank you!\n\nThat prioritizing principle was the one adjustment Tom's client needed to get back to continuing his elite-level performance and feeling in control of his life again.\n\nAnother advisor had started her career strong, far exceeding her targets for each of her first three years. But then she hit a wall, and after two years of weak results was in danger of \"flunking out\" and leaving the business. At the suggestion of her regional leader, she attended class, where she learned to Organize Tomorrow Today.\n\nOver the first few months after the class, she became addicted to organizing her business life. Within a year, she was one of the highest producers in her region, and a few years after that she was given leadership responsibilities with her region. Within five years, she had been promoted to the home office, where she had even more responsibility.\n\nAll after changing one thing.\n\nTHE PROCESS\n\nWritten lists are as old as, well, writing, but there's solid scientific evidence that they work better than keeping a list in your head\u2014or even tapping one out on a keyboard. Researchers Pam Mueller and Dan Oppenheimer from Princeton University and UCLA found that students who wrote out notes longhand retained more and had a better conceptual understanding of the material than students who typed notes on a keyboard.\n\nWhy? The physical act of writing stimulates an area of the brain called the reticular activating system (RAS), which is responsible for filtering information into the \"instant access\" and \"deep storage\" folders in your mind. It tells your brain that the information is important and needs to be kept to the forefront. It also primes your subconscious mind to get to work\u2014which is why you hear so many stories of songwriters and comedians keeping a notepad by the bed to record lyrics or jokes that leap out in the middle of the night.\n\nStill, to get the full benefit of that fun physiological fact, you have to approach your list-making with some sophistication. You can take your pen and paper and start creating a laundry list of tasks that need to be finished, but unless you learn to impose some important filters on those tasks, you'll struggle to improve your efficiency.\n\nAsk yourself if this sounds like you. You make out a to-do list with eight or ten items on it, and you start your day with some of the easier items on the list because you want to build some momentum. You'll handle the more complicated or problematic things a little later\u2014once you've had some coffee, or maybe just after lunch. Or maybe at 3:00, when you have a clear window of time to really focus.\n\nIt's a natural tendency. The only problem is, it doesn't work. What ends up happening is that you put in a full day, checking off item after item, but you get to the end of the day without tackling the most important items. That produces stress and tension for the next day, and the feeling that you're falling behind\u2014even though you're as busy as can be.\n\nAs we alluded to before, the key point is to understand that high achievers aren't necessarily completing more tasks. They're accomplishing more of the ones _that matter most._ As busy as people are these days, it is no longer possible to get it all done. The key to success has become prioritization. Prioritization may very well be the most underrated skill of the highly successful. It is what will make the single biggest difference between being busy and being productive. Highly successful people never get it all done in any one given day\u2014but they always get the most important things done each day.\n\n* * *\n\nHighly successful people never get it all done in any one given day\u2014but they always get the most important things done each day.\n\n* * *\n\nIt doesn't matter how organized, efficient, and energized you are. You will _never_ get everything done every single day. That's just too high a bar to set. But you can resolve to _always_ get to your most important tasks and conversations.\n\nThe Organizing Tomorrow Today strategy will help you do this. It starts with getting into the habit of taking about five minutes the day before to identify your priorities for the upcoming day. But instead of creating that laundry list we talked about before, you produce a simple, curated, prioritized list.\n\nThe first part of the list is called the \"3 Most Important.\" It's just as it sounds\u2014the three most important tasks you need to complete the next day. Your goal is to build out your list of three tasks, along with the time of day you'll have each one completed.\n\nIt's important to say that the tasks on this list aren't full-blown projects that must have all of their steps completed in a single day. The tasks can and should be specific component tasks that work as a part of the whole. The key is to list important, ambitious, but realistic tasks that can reasonably be completed during the day. Small, day-to-day successes are the building blocks of achievement.\n\nThe second part of the list is called the \"1 Must.\" Once you've determined your \"3 Most Important,\" you choose the \"1 Must\" from these three items. It is the single most important task or conversation you need to have that day. Multiple studies in the subfield of quantitative behavioral analysis pioneered by B. F. Skinner in the 1930s have proven what you probably already intuitively feel day-to-day\u2014that if you start moving on a task or a project, it's easier to keep in motion on that task. It's the human brain's version of the classic physics rule of inertia: A body in motion tends to stay in motion, while a body at rest tends to stay at rest. The best way to promote action is to identify just one thing, and then attack. Picking that most important to-do item creates the momentum.\n\n* * *\n\nA body in motion tends to stay in motion, while a body at rest tends to stay at rest.\n\n* * *\n\nUnfortunately, many people build a massive list of daily expectations and insist to themselves that they will always get the most important things done. But channel capacity is quickly reached, and you get overwhelmed. Then it gets easier and easier to just cast the list aside with some vague idea that you'll try again the next day\u2014or lie to yourself and say that, since you accomplished eight or nine \"little things,\" you had a productive day, even though you didn't finish anything of true importance.\n\nTo set yourself on the right track, ask yourself those two critical questions: (1) What are the three most important things I need to get done tomorrow? and (2) What is the single most important task I _must_ get done? The questions work within your brain's \"channel capacity\" to give you direction and prioritization in manageable doses. When you start your day, you know the three most important things you need to get done by the end of the day, and you know which of those three things is the big, glow-in-the-dark priority. You'll be amazed at how much clearer your decisionmaking becomes\u2014and how much more efficiently you'll use your time\u2014just by taking this simple organizational step.\n\nOnce you've created your list, remember that it doesn't exist in a vacuum. You can have a big master list somewhere of all the things you want to get done this week, this month, or this year. The \"3 Most Important \/ 1 Must\" list is simply the priority filter that goes on top of the master list\u2014the day-to-day action plan that puts things in motion.\n\nTo use another analogy from the sports world, a big-picture item on the to-do list for a given baseball season might be to win the division. But on day one, winning the division isn't a part of an individual player's \"3 Most Important\" list. The prioritized list for the day would be something like completing a full stretching routine, taking fifty quality cuts in the batting cage, and playing fifteen minutes of long-toss.\n\nOne caveat: With all of this introspection and personal list making, it can be easy to give the human communication part of the \"3 Most Important \/ 1 Must\" less attention than it deserves. It's a common mistake\u2014both for people who don't have a lot of day-to-day interaction with people outside the office, and those whose businesses rely on things like sales calls or prospect meetings. In either case, it's tempting to close your door and burrow into your work, communicating mostly by email or text message. But we believe\u2014and strongly recommend\u2014that you reemphasize the personal element of your \"3 Most Important \/ 1 Must\" and make those connections directly, either face to face or over the phone. There's often a direct correlation between in-person communication and your level of success.\n\nPREPARED MEANS CONFIDENT\n\nIt's fair to say that all of us face some level of adversity each and every day. Our clients have recognized that being organized and prepared for adversity significantly increases confidence. To give you an idea of how this works in practice, we'll use two of our recent clients as an example.\n\nTina moved out to California with the dream of becoming an actress. She spent years working on her craft and taking side jobs to pay the bills while she waited for that big break. With her talent, natural good looks, and relentless work ethic, she scored a few smaller roles in shows you've heard of, but she wasn't able to find consistent work that would let her put down permanent roots.\n\nOne summer, a family friend who knew about Tina's situation came to her with a job proposition. He owned a health-care company, and he was always on the lookout for charismatic people to work in sales roles. Tina had never given sales much thought\u2014unless waiting tables counts as sales\u2014but her bank account told her she should think seriously about the offer. She came to us to get some sense of how to organize herself and develop the skills she needed to be successful in a sales role. Tina is a naturally confident person, and she felt like she would be a good salesperson\u2014as long as she was able to come into each of the meetings prepared.\n\nWe started her with Organize Tomorrow Today (OTT) as one of her first tools. She committed to spending a few minutes at the end of every day writing down the three most important tasks she would undertake the next day. By clearly delineating her priorities and breaking the tasks down into manageable\u2014and accountable\u2014chunks, Tina was able to take in a massive amount of new information and build a variety of new skills without feeling overwhelmed. She never got to the end of the day worried that she was unprepared for what was coming next.\n\nOne of Tina's daily lists looked something like this. Notice where Tina has placed the most important items in her day\u2014at the very beginning:\n\nOne of our Major League Baseball clients used the \"3 Most Important \/ 1 Must\" technique in the period leading up to becoming a free agent, because his goal was to sign a lucrative multiyear contract that would take him to the end of his career. He knew that the right deal would set his family up financially and carry him through the ten years of Major League service that would guarantee his full pension. He also knew that unless he dedicated all of his energy to improving his performance, it would be tough to find a team that would make a good match.\n\nMichael came to us to get some sense of how to organize himself and sharpen the skills he needed to be successful. He's a naturally confident person, and he felt like his best pitching was still ahead of him\u2014as long as he was able to come into each day prepared with a game plan for success.\n\nWith the OTT tool, Michael committed to spending a few minutes at the end of every day reviewing the three most important tasks he would undertake the next day, and the single most important task he had to complete. His \"3 Most Important\" looked like this:\n\nMichael's \"1 Must\" rotated between the tasks shown above, having conversations with his pitching coach about mechanics, and conferring with the starting catcher on scouting reports of the teams he'd be pitching against. He also reached out to several prominent retired pitchers he knew to get some insight into how they strategized against different kinds of batters. On other days, he talked to team strength coaches and nutritionists to get his body in peak condition.\n\nAs a result, Michael was clear on the process he needed to follow to be totally prepared for game day. Having a game plan gave him renewed purpose and passion for the daily work. He felt confident because he had a plan that he knew would work. His increased confidence allowed him to not get caught up in results and to stay calm during the heat of competition.\n\nMichael was able to make a massive amount of measurable improvement while keeping his confidence high. He felt comfortable every day knowing what he needed to do to _control_ his success.\n\nIdentifying daily priorities might seem like an obvious or insignificant step to take, but writing your most important tasks down the previous night turns your subconscious mind loose while you sleep and frees you from worrying about being unprepared. You'll probably find that you wake up with great ideas related to the tasks or conversations that you hadn't even considered!\n\nIt's important to pay attention to the mechanics of making the lists, because the goal is always to set up a daily habit with as few roadblocks to implementation as possible. If something is pure torture for you to do, you're probably not going to be able to keep it up over time. And this tool is one you need working for you every day from now own.\n\nDON'T WAIT UNTIL THE END OF THE DAY\n\nTo start, don't wait until the very end of the day to make your list. We've used this tool with thousands of clients and conference attendees, and the overwhelming feedback we get is that the closer you get to the end of the day\u2014whether that's the time you leave your office or before you switch over to family mode at home\u2014the less likely you are to set aside the time to actually do it.\n\nIt's worth repeating: don't wait until it's the very last thing in your day to organize for tomorrow. Do it a bit earlier in your day to give yourself a fighting chance to complete it. It's that important.\n\n* * *\n\nIt's worth repeating: don't wait until it's the very last thing in your day to organize for tomorrow. Do it a bit earlier in your day to give yourself a fighting chance to complete it. It's that important.\n\n* * *\n\nWe've found that the window between lunchtime and 3 p.m. seems to be the sweet spot for making the OTT plan. In fact, many of our clients make it a rule not to take the first bite of their lunch until they've built out their OTT plan for the next day.\n\nThe next big mistake to avoid is drilling too deeply. This tool is designed for you to pick the big, important priorities for the next day. Writing your list should take you about five minutes. If you're having trouble, keep in mind that the goal isn't to make an exhaustive list of everything you need to do tomorrow. You're developing the ability\u2014and the habit\u2014of prioritizing. If you don't do it for yourself, other people will do it for you\u2014or, more accurately, _to_ you.\n\nIt will be difficult at first, but you need to train yourself to understand that checking off everything on your big to-do list isn't the goal. Highly successful people get the most important things done every day\u2014\"3 Most Important,\" and \"1 Must\"\u2014and do their best to get everything else done in the time that's left. On most days, you're going to have things pop up that will require your attention at some point. How many times have you closed yourself into your office to really bear down on an important project, only to remember something urgent\u2014but not really important\u2014that you need to take care of?\n\nFor example, you forgot to schedule that doctor's appointment your daughter needs before the beginning of the school year. You tell yourself it will only take a minute, and you go online to look up the doctor's phone number. You make the call, but you're placed on hold, so you surf the web for a bit. You finally make the appointment, but in the meantime, you've found an interesting article and finish it up. You look up and it's forty minutes later\u2014and you've solved the appointment problem, but haven't even started any of the day's most important tasks.\n\nUse your smartphone to create a holding area for those urgent-but-not-overly-important tasks, and resolve to work your way down that list only after you've completed the items from your OTT list. If you get to the point where you don't think you'll be able to address something from that separate holding-area list, you have the chance to decide if something from that list needs to make it onto the \"3 Most Important\" for the next day.\n\nWhat you're doing is imposing intention onto what you do. Instead of operating day-to-day in reaction to events, you're setting priorities and getting out in front of things. You're training yourself to get better at prioritizing. And when you are better at prioritizing, you will be surprised how quickly you get things accomplished: we've found that, generally speaking, it takes somewhere between two and three hours of focused attention for a person to complete all three of their most important tasks for a given day.\n\nDoes this mean that you'll never have an emergency or need to put out a fire? Of course not. Life happens. But the most successful people have figured out a simple, effective technique for winning: they plan on emergencies happening every day.\n\nThat's right. The most successful people know that just about every day, surprises will happen, and plans will change. They plan on it, and they aren't surprised by it.\n\nEmergencies typically don't start happening until mid to late morning, and that is precisely why successful people get their \"3 Most Important \/ 1 Must\" tasks completed early in the day. The majority of our clients have learned the value of completing the most important activities before 9:30 or 10:00 a.m.\n\nIt may not be possible to get all three of your most important tasks done that early each day. If a task involves another person, that person may not be an early bird. But we highly suggest that you take advantage of the \"get it done early\" principle whenever humanly possible.\n\nSurprises are much, much easier to deal with when you have an effective handle on the most important things on your plate. Prioritizing and starting early will give you the energy to fight off a common temptation\u2014to push one of the items on the list off to the next day. And you will be tempted: inevitably, you'll be right in the middle of something when you realize that you're up against the time by which you said you'd complete one of your important tasks or conversations. Instead of telling yourself, \"I'll do it later\"\u2014which is code for \"I'm not going to do it\"\u2014refuse to forfeit that score for the day.\n\nDon't take a zero.\n\nIf nothing else, commit to spending one minute on the important task. It will help reinforce the prioritization skills you're developing. Taking a zero on one of your most important tasks is the equivalent of a professional athlete losing a game by forfeit. Passing altogether on your priorities completely erodes mental toughness. Commit to giving at least one minute of attention to each of your priorities, and you will find yourself, sooner rather than later, developing the mental toughness needed for winning consistently.\n\n* * *\n\nTaking a zero on one of your most important tasks is the equivalent of a professional athlete losing a game by forfeit. Passing altogether on your priorities completely erodes mental toughness.\n\n* * *\n\nHaving that kind of intentional, clear-headed approach to her day-to-day made all the difference for Tina. Within a year, she had become the most successful sales rep at her company. Last year, her bonus was bigger than she had made in any three years as an actress. When she tells the story, she laughs and says she should quit and go back to Hollywood to chase her dream now that she has so much freedom and control.\n\nMichael not only got the big contract, but he's still pitching to a high level in the major leagues to this day. He's tuned his body and mind so well that he isn't remotely considering retiring. He's having too much success\u2014and too much fun.\n\nWHY IT WORKS: TURN YOUR MIND LOOSE WHILE YOU SLEEP\n\nWe ask the folks who take our seminars (or read this book) to start with one rule when they begin the improvement process. In the seminars, the Organize Tomorrow Today rule is the one most people pick to do first. It's probably because the rule is so satisfying on a day-to-day basis, and because it has such a strong grounding in both science and common sense.\n\nWhen you go to the effort to make a prioritized list of what you need to do the next day, you're essentially opening a loop in your mind. As you sleep, your brain will automatically start preparing for the successful closing of those loops. It's known as the \"Zeigarnik Effect.\" In the 1920s, Russian psychology researcher Bluma Zeigarnik quantified the phenomenon after her professor, Kurt Lewin, noticed that waiters who hadn't been paid for an order had much more recall of the details of those orders than they did for orders that had been paid. Working from Zeigarnik's research, Lewin came up with the concept of \"task-specific tension,\" which persists in both the conscious and subconscious mind until the task is completed.\n\nIn other words, the mind doesn't like unfinished business! High-level mathematicians and successful writers have been using this technique for years as a tool for pushing their work forward. Before going to bed, they take a few minutes to read over the mathematical or literary work they did during the day\u2014especially if they've reached a plateau or feel stuck. The mind then works all night to close the loop, and they wake up in the morning with \"inspiration.\" It seems magical, but it isn't so much magical as it is the result of effective priming of the mental pump.\n\nThe OTT principle and prioritization with a list make sense. If you could eliminate\u2014or at least significantly reduce\u2014certain anxieties in your life with a simple, five-minute ritual, why wouldn't you try it?\n\nUSE MOMENTUM AS A CATALYST\n\nIf you can go into your next day feeling more prepared and less anxious, you will project that comfort and confidence in what you do. Instead of spinning your wheels early and trying to generate momentum, you'll come in with momentum already in place. And when you knock off that \"1 Must\" from the list, you'll be generating even more momentum\u2014not fooling yourself with little pretend \"victories\" in the stuff that didn't really matter in the first place.\n\nYou really will be staying in motion.\n\nOrganizing Tomorrow Today hones your prioritizing skill\u2014and prioritizing is what will make the single biggest difference between being productive and being busy. Busy people don't necessarily get much done. Productive people do.\n\n\"Busy\" isn't what gets rewarded long-term in the marketplace. \"Productive\" is.\n\n* * *\n\n\"Busy\" isn't what gets rewarded long-term in the marketplace. \"Productive\" is.\n\n* * *\n\nYou'll certainly be challenged on a day-to-day basis by the \"noise of the urgent,\" but having this tool in place will help you make the decisions that will separate you from the average.\n\n* * *\n\n**The Big Why:** The most successful people don't get everything done. They get the most important things done. By organizing and prioritizing your effort toward the core tasks you need to accomplish, you create momentum and confidence. You go into attack mode.\n\n**The Inversion Test:** One of our favorite devices to use within each rule is to apply a lesson that Charlie Munger adapted from German mathematician Carl Jacobi: to better understand a principle, invert it. It's simply looking at the opposite side\u2014in this case, what happens if you _don't_ organize tomorrow today.\n\nIf you don't organize, you begin the day on defense. You're extremely busy, but not very productive. If you want to slow your progress, don't organize tomorrow today.\n\n**Act Now:** The mechanics of how you make your lists aren't as important as actually doing it. You can use a small notebook, a pad of paper, or an app on your smartphone.\n\nSpend three to five minutes preparing your own \"3 Most Important\" tasks or conversations for tomorrow, and the \"1 Must\"\u2014or main priority\u2014out of the three. Remember, the goal is to schedule completion of these tasks as early as possible in your day.\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nCHOOSE WISELY\n\nAlbert Einstein came up with a popular (and fascinating) list of what he called the five levels of intelligence:\n\n1. Smart\n\n2. Intelligent\n\n3. Brilliant\n\n4. Genius\n\n5. Simple\n\nThe step beyond genius? To be able to see beyond the chaos and complications of everyday life and identify the most important solutions to the most important problems.\n\nSimplicity.\n\nYou don't have to have superhuman intellectual powers to take away a powerful message from Einstein's theory. The key is learning how to choose wisely.\n\nWe've talked about the scientific definition of \"channel capacity,\" and how it relates to the human mind. But a scientist determining how many numbers a person can remember is very different from crawling on your belly in the mud for miles, pushing yourself to the physical and mental breaking point and beyond. Let us tell you about Bobby Gassoff.\n\nBobby grew up the son of defenseman Bob Gassoff of the National Hockey League's St. Louis Blues. Bob was tragically killed in a motorcycle accident before Bobby was born. Bobby was a hockey natural, just like his dad, and went to the University of Michigan on a hockey scholarship. In 1994, he was an important member of the team that won the NCAA championship.\n\nAfter college, Bobby started on the traditional hockey prospect road, playing in the minor leagues with the hope of working his way up to the NHL. But not long into that process, he hung up his skates and set his sights on something much more important than hockey. He started an intense training regimen with the goal of joining the Navy SEALs\u2014one of the most elite fighting units in the world. When Bobby got to BUDS\u2014basic underwater demolition training\u2014he was in peak physical and mental condition. So was everybody else in his class. They were the \"best of the best\" from the armed forces.\n\nBut during \"hell week,\" these trainees would run the equivalent of three full marathons while carrying hundreds of pounds of gear. They would crawl through the mud and sand, get dropped miles out to sea, and have to swim back\u2014or drown. And they would do it all soaking wet, freezing cold, and on less than thirty minutes of sleep per night. This regimen would separate the toughest from the rest.\n\nBy the third day of hell week, the skin around Bobby's armpits and groin had been worn away by wet sand. He was completely exhausted, and he still had four days to go. That night, one of his teammates\u2014a Navy pentathlete and the best runner and swimmer in the group\u2014rang the bell and quit.\n\nDespite the nearly overwhelming urge to follow his teammate over to the bell and ring out, Bobby decided he would narrow his mind to a single focus point. He would concentrate on just one thing\u2014the very next step\u2014because the thought of what was coming the next day and the day after that was just too overwhelming to consider.\n\nWhen the trainees finally got a few minutes to wolf down some food in between activities, the instructors would walk among the exhausted men and call out the laundry list of brutal things that were scheduled for the rest of the day\u2014a tactic designed to root out trainees who could be distracted or disheartened.\n\nA six-mile run.\n\nTwo hours of surf training.\n\nThree hours on the obstacle course.\n\nAn eight-mile boat haul.\n\nBobby kept his \"one step\" focus, survived hell week, and graduated from SEAL training. Of the 240 men to start training, he was one of 24 to make it, and he was one of only 2 to graduate as an officer. He was\u2014and is\u2014one of the toughest people on the planet, and to this day, he still operates by the principles that got him through.\n\nIf that's the case for somebody like Bobby, what does it mean for the rest of us? As we touched on in the previous chapter, one of the most common traps to fall into in business and life is to try to focus on too much and lose focus on the really important things. People have a tendency to overcommit to others and to themselves. Doing so not only causes underperformance but also has a tremendously negative impact on confidence. When you commit to doing too much, you inevitably let other people down and unfortunately begin to send a message that you cannot be trusted.\n\nIt usually starts as something quite harmless but unfortunately snowballs quickly. It could be something as simple as telling a colleague you will help out with a task or even telling a loved one you will be home by a certain time. The next thing you know, you have become overwhelmed by your own mountain of responsibilities and you haven't left time for that colleague or loved one. You never intended to let someone else down but it just got away from you.\n\nThe problem is that this becomes habitual to the point where you know even when you are committing to certain things that there is a high likelihood you won't follow through when the time comes. This is a prime example of not choosing wisely. If you are trying to beat channel capacity, you will always lose. When you overload the system with requirements, the same thing happens to your mind that happens with an outdated computer running heavy-duty software. You start to freeze, and decisions get harder to make.\n\nAs we keep saying, you can't beat channel capacity.\n\nIn this chapter, we'll talk about learning how to choose wisely and pick that _one step_ that is the next most important one in the progression. You'll learn how to attack that one step with your full attention, and then move on to the next\u2014instead of overcommitting your way into mediocrity.\n\nDoing it all\u2014or being great at it all\u2014should never be the goal.\n\nThink about what the average office environment looks like. You're sitting at your desk, working on a document that has to get out the door by the end of the week. You hear your email chime, and you see the notification at the top of the screen, telling you that the message is from one of your colleagues. You click away from the document and scan the email, then type a quick response. Then you click back to where you left off on the document, and pick up your work again.\n\nThat might sound normal, or harmless, but every time you turn into one of those attention cul-de-sacs, you're making your mind work really hard to get up to speed on the new task, then making it work some more as you click back to where you were on the previous one. And that's a problem\u2014remember channel capacity\u2014because it's going to compromise your ability to do your best on the really important tasks.\n\nIs it the end of the world if it happens once? Of course not. But does it ever happen just once? The reality is that we're always answering phones, looking at text messages, sending emails, or dividing our attention by surfing the web while we talk on the phone to a client. We're surrounded by distractions. The fact that multitasking is difficult for our brains doesn't mean you won't be able to juggle the balls and accomplish those other tasks. It just means that, if you constantly divide your attention, trying to do more than one thing at the same time, you're going to use a lot more energy and time to get things done than you need to. You're going to compromise your attention and miss some of the finer details.\n\nYou don't have to look very far to find examples to illustrate the same point at the organizational level. We see it all the time. A company will call a big meeting\u2014often off-site\u2014to get everybody together so the top execs can reveal what the new corporate strategy is. Then, the group gets presented with a laundry list of goals for the coming quarter or financial year. There are eight or twelve \"metrics\" and \"measures,\" and it isn't usually clear which ones are most important. Because it's \"important\" and everybody is already together, the meeting is almost always scheduled for all or most of the day. But the reality is that everybody in the room has reached channel capacity by the first break.\n\nWorse yet, even if the people in the audience understand the metrics and measurements and know which ones apply to them personally, they usually don't know how to prioritize them. They don't know which ones are the most important for them personally.\n\nIt happens the same way in the sports world. \"Winning\" is the same kind of top-level goal as \"profit,\" but all of the additional responsibilities can get an athlete spinning his or her wheels. Imagine what happens to a college quarterback who gets picked in the first round of the NFL draft. He signs a large contract, and the team is expecting him to come in and (eventually) become the face of the franchise.\n\nWhen minicamps start in the summer before his first season, the quarterback has a huge amount of work to do. He has to learn the playbook and integrate himself into the team. He has to show not only that he's willing to work hard and learn, but also that he can become a leader. In addition, the team expects him to make numerous public appearances and make himself available to the media. Not to mention the increase in expectations and demands from family and friends. It's really three or four different jobs in one. And we're talking about a player who is twenty-one or twenty-two years old, who has just been handed more money than most people will make in their entire career.\n\nThere's a lot to focus on, and there are a lot of ways to go off the tracks. Is it really a surprise that the players who narrowed their focus to a few critical goals and tasks\u2014and basically operated as sports hermits for their first few seasons\u2014are the ones who have had the most success?\n\nLeading up to the 1998 NFL draft, Tennessee quarterback Peyton Manning was considered to be one of the top prospects in all of college football. Manning was considered to be intelligent and hard working, but his arm strength wasn't considered to be the strongest in the draft. Even though the Colts' scouts favored another quarterback, Colts general manager Ed Polian decided to pick Manning. Manning hit the ground running after the draft, focusing on one thing: burying himself at the Colts' team facility to learn the playbook. He was the Colts' starter from the first game of his rookie season, and he would go on to become one of the greatest quarterbacks of all time.\n\nYou are more like Peyton Manning than you think. He is not the most gifted athlete on the field, and other quarterbacks are stronger and faster and younger. Yet he is a consistent winner.\n\nHe walks up to the line of scrimmage with the play that has been called from the sidelines. Once the ball is snapped, he's going to have things coming at him from all angles. Throughout the day some things will go his way and some things won't. There may even be a time or two when he gets slammed to the turf and has to pick himself back up and go at it again.\n\nYou walk into your office and, chances are, things start immediately flying at you from all directions. Throughout your day, you'll feel ahead sometimes and behind other times. There may even be a time or two when you feel like a 350-pound lineman has just planted you into the dirt. Just like Manning, you have to find the strength to get back to your feet and re-engage.\n\nPeyton Manning has been able to do it year after year after year because he chooses wisely and does one thing exceptionally well. He is relentless with his preparation\u2014and you can be, too.\n\nChoosing wisely is difficult because it is counterintuitive. It is much easier to put a laundry list together of all the possible things you need to get done each day than it is to actually choose your one most important task and then master it.\n\n* * *\n\nChoosing wisely is difficult because it is counterintuitive. It is easier to put a list together of all the possible things you need to get done than it is to actually choose your one most important task and then master it.\n\n* * *\n\nINFORMATION ADDICTION\n\nThink about that big off-site meeting we were just talking about a few paragraphs ago. At many of those meetings, organizations schedule training seminars to get big groups of employees \"up to speed\" on new techniques and strategies. If you've worked in corporate America for any amount of time, you're very familiar with the kind of training session we're talking about.\n\nNow, you'll never hear us say that innovation is a bad thing, or that training employees for the skills you want to reward is a wasted effort. But think about the way the vast majority of those corporate training sessions work. You're handed a binder when you walk in (or directed to a URL to enter on your tablet). There's somebody standing in the front of the room with a big screen, getting ready to motor through fifty PowerPoint slides.\n\nThe information could be great. But setting the group up to be force-fed for hours at a time quickly creates an overload situation. You reach channel capacity before you even get to the first break.\n\nAnd when that break comes, what happens? Everybody surges from their seats to hit the snack table to eat, drink, stretch, and chitchat about how they can't believe there's three more hours of this to go.\n\nThe data is important. But it's just washing over people who are too overwhelmed by the amount of information to be able to absorb it. All of that wasted information isn't even the worst part of it. The dirty little secret that gets left out of the accounting? Saturating people with information actually paralyzes action. Think about it: when people are overwhelmed, they typically freeze. Self-doubt slows action.\n\n* * *\n\nSaturating people with information actually paralyzes action. Think about it: when people are overwhelmed, they typically freeze. Self-doubt slows action.\n\n* * *\n\nIf you are a leader in your organization and are spearheading these meetings, you're not just paying for the conference space, hotel rooms, and catering: you're slowing your staff from doing their jobs. It can be a gigantic waste of time and money if you aren't giving people something that they can, with confidence, use to make a positive, measurable change. By overloading people with information you are essentially causing them to overcommit. If you are working for an organization and attending the meetings, two of the crucial skills you probably aren't learning are how to prioritize the information being poured into you and how to understand the point of equilibrium when it becomes actionable.\n\nTechnology makes some things better, but when it comes to this subject, it often makes things worse. We have the physical ability to \"receive\" way more information in the literal sense\u2014thousands of emails in a folder, tens of thousands of documents on hard drive\u2014than in the old days. But the technology doesn't do much to prioritize that information or steer it where it needs to go. It's like having an enormous fire hose at your disposal\u2014one twenty times bigger than the normal size. But the hose shoots out so much water that when you try to hold onto the end you get flopped around like a fish.\n\nCHOOSE WISELY\n\nThis isn't some kind of anti-progress or anti-technology rant. Cutting-edge information is the lifeblood of business, and it is crucial to understand how to digest it.\n\nWe run dozens of corporate training seminars every year for a variety of different organizations. We go into every one of them with a customized game plan, but the overall goal is always the same. Step by step, we intersperse very controlled amounts of information relevant to the business with direct, hands-on practice in implementing that information.\n\nEven though we want to teach everything we know, we work very hard to respect channel capacity. We'll have discussions late into the evening about whether or not we should include that latest piece of information. In the end we usually finish the conversation with one of us saying, \"When in doubt . . . delete.\" It's a lesson Jason learned from one of the greatest coaches of all time.\n\nDan Gable's wrestling teams at the University of Iowa won 15 NCAA team titles from 1976 to 1997. Individually, his wrestlers won 45 national titles and 106 conference championships during that time. Before becoming a coach, Gable had one of the most dominant careers in the history of wrestling, going 117\u20131 during his time at Iowa State, and winning the Olympic gold medal in Munich in 1972 without giving up a single point. He knows something about elite performance, and what you have to do to achieve it.\n\nYears ago, Coach Gable told Jason that one of the basic fundamentals of his approach to competing\u2014both as an athlete and as a coach\u2014was \"choosing wisely.\" To him, that meant not over-committing. Gable learned the importance of identifying the right \"critical factor\" and directing his energy toward moving it.\n\nIn Dan Gable's world, not overcommitting allowed him to remain focused on what was most important, practicing every night until he was physically exhausted. It doesn't have to be so dramatic in your world, but it's important to understand the point.\n\nAt every seminar we teach, we introduce the attendees to the rules you're learning about in this book. Once we've covered what they are and how they're used, we ask everyone to choose just one thing they want to \"nail\"\u2014to execute as completely as possible. Whether we're talking about NFL players or business professionals, the ones who set the records and make All Pro are the ones who choose wisely and relentlessly focus on improvement one step at a time.\n\nDoes this have the potential to sound simplistic, or even goofy?\n\nSure.\n\nTHE POWER OF ONE\n\nWhat Dan Gable instinctively knew at a young age\u2014and what we try to establish in all of our work with athletes and businesspeople\u2014is this: focusing on one primary task makes _action_ much more realistic\u2014one simple, positive change builds momentum and primes you for the next success.\n\n* * *\n\nFocusing on one primary task makes _action_ much more realistic\u2014one simple, positive change builds momentum and primes you for the next success.\n\n* * *\n\nOne of our clients, a very successful small-business owner named Todd, came in with an ambitious plan for the coming year. He was determined to push his business to the next level financially. He had made plenty of good decisions in the past to get to where he wanted to be, but he was looking for that one push he needed to move to the next level.\n\nAfter hearing about the concepts you're learning here, Todd decided that the first step he was going to take was to commit to going through the one-hundred-second Mental Workout we teach in our seminars (which we will cover in Chapter 6). At precisely 5:30 every morning, he got up and found a quiet place to collect his thoughts\u2014usually the kitchen table\u2014before the hustle and bustle of his day started.\n\nDuring the one minute and forty seconds it took him to do the Mental Workout, he went through a series of positive affirmations that reminded him of his strengths and goals. He visualized things he had done well the previous day and the steps he needed to take in the upcoming day to achieve his \"win.\" It sounds like something out of an infomercial, but a year of focusing on one positive improvement later, Todd had lost fifty pounds and doubled his business.\n\nIt isn't magic. It is the natural byproduct of a phenomenon that social scientists have recognized for decades. Inertia is a powerful force in human behavior, but it only works when you use it in a focused way.\n\nYou need to choose, and choose wisely. Focusing on one thing promotes action. Learn to do less, but more often.\n\n* * *\n\nFocusing on one thing promotes action. Learn to do less, but more often.\n\n* * *\n\nWe've all been trained to believe that we can do vastly more at one time than we really can. This notion is instilled in us in many ways: by school, by the books we read, by what we watch on television, and even by the devices we carry around with us. We'll bet that many of you reading this book are \"available\" at least five more hours per day than you were ten years ago, simply because you have the ability to answer emails and text messages anytime\u2014whether you're sitting at breakfast with your family or getting ready to shut things down for the night.\n\nIn many ways, that accessibility has built artificial, unattainable expectations. If information moves twenty-four hours a day, you have to be ready to act on it twenty-four hours a day\u2014or at least that is how it seems. When expectations change\u2014and you hold yourself to that relentless, multitasking standard\u2014you're destined to fail, and you will likely be hard on yourself when you do.\n\nThat's the first step in the perfectionist cycle. It's a trap, and most people have found themselves stuck in it at some point. You try to do everything and be everything, then you fail at it and get discouraged. At some point, the discouragement makes you stop trying.\n\nAnother mistake\u2014besides not focusing on the one primary goal you've chosen to pursue\u2014can be choosing unrealistic goals. It's especially common when it comes to the goals that will take a while to achieve. One client came to one of our coaching sessions and told us his plan was to lose forty pounds and compete in a marathon within twelve months in addition to trying to grow his business by 30 percent. He went through all the steps the self-help books tell you to take\u2014including telling friends about his goal so he would stay accountable and setting up a $1,000 charitable donation \"penalty\" if he didn't accomplish his goal. He changed his diet and cut 30 percent of his daily calories, and he started a punishing workout routine every morning at 5 a.m. and attempted to make multiple changes to how he ran his business.\n\nWe tried to redirect him to a more sustainable, long-term goal, but his mind was made up and he refused to believe in the power of channel capacity. Within six months, he had lost the weight and had signed up for the marathon. His business was showings signs of improvement but nothing was really showing up in his bottom line.\n\n* * *\n\nWe tried to redirect him to a more sustainable, long-term goal, but his mind was made up and he refused to believe in the power of channel capacity.\n\n* * *\n\nBy the eighth month, the client had finished the marathon. He was so relieved not to have to follow the brutal diet and training routine he had established that he completely backed off working out. By the twelfth month he was two pounds heavier than he had been when he had started the whole process the year before and his production at work had remained flat throughout the entire year. Instead of choosing wisely, the client overcommitted. He worked harder than he ever had but really had nothing to show for it.\n\nWe don't tell that story to make you feel like it's pointless to set goals\u2014or even to set ambitious goals. As you'll see throughout the book, the opposite is actually the case. We want you to set goals and be ambitious. But we want you to learn to set goals that are attainable and sustainable. We want you to choose wisely. One success should prime you for the next success, not become a point where you can bail out. We want you to build the habit of winning\u2014not just have a onetime win.\n\nIn the early 1990s, Edward Jones was a much smaller operation than it is now, and it didn't have many million-dollar producers among its advisors. Tom developed an advanced training program for the company's highest-achieving young advisors that had as its centerpiece this concept of understanding a small group of principles and choosing one to attack. They learned to stay focused on that one goal, and they learned to choose the goal wisely.\n\nTom kept meticulous records of how the advisors who went through the program performed and compared them to a peer group of advisors who didn't get the training. For every dollar Edward Jones invested in training the experienced advisors, it got a return of $17 by the end of the first year\u2014strictly by choosing wisely and respecting the \"power of one.\"\n\nSAYING NO\n\nThe biggest obstacle that will block you from improvement is committing to too much and getting overwhelmed.\n\nIt's an especially dangerous\u2014and common\u2014problem as you work your way higher and higher on the food chain of success. If you're a smart, ambitious person who wants to learn and get better, your own drive and determination can work against you.\n\nIn every class we teach, we'll see an energetic, smart, high-achieving student who goes along with what we're presenting, and at the end grabs one of us for a quick conversation. \"I can't wait to get started,\" the conversation starts. \"It's so clear and simple, and I think I picked it up pretty quickly. I'm going to start on three changes instead of one and jump-start the process.\"\n\nIt doesn't matter how smart or successful you are. Even if you can read two hundred pages a night and work twelve-hour days, you will bump up against the self-limiter of channel capacity. That's where saying no comes in.\n\nAnd it's a talent many, many people either have never learned or have chosen to forget\u2014usually out of desire to please others or to avoid having an \"uncomfortable\" conversation. If you're going to commit to doing one thing at a time, it means, by definition, that there are some things you aren't going to be doing.\n\nMost people think of saying no as a \"negative\" thing, and that can certainly be true in some unpleasant cases. But for the most part, saying no to something just means you're saying yes to something else. All we want to teach you to do is make those decisions consciously\u2014and not get pushed into them by the unintended consequences of a series of other decisions.\n\nOne extremely successful client of ours is a retired athlete who still lives in the town where he was a star player. He's a gregarious guy who loves to help, and as a result, he is constantly asked to give his time to various causes\u2014everything from playing in charity golf tournaments to serving on volunteer boards and shooting public service commercials. For years, he was always game to help in any way he could\u2014to the point where he was run ragged six days a week and more burned out than he'd ever been during his active days as a player.\n\nHe made one simple change: he placed a picture of his wife and kids on his desk next to the phone.\n\nHe didn't stop making commitments to help, but every time a call came in, he stopped for a moment and thought about what he was really saying yes\u2014or no\u2014to. Many times, he would tell the person on the line that he appreciated being asked, but that he had to decline. \"If I say yes to you, I'm saying no to my wife and kids, and that's not the man I want to be,\" he said. \"I hope you understand.\"\n\nUnfortunately, most people confuse \"urgent\" with \"important\" and say yes to so many urgent things that they are in effect saying no to other things that are much more important.\n\n* * *\n\nUnfortunately, most people confuse \"urgent\" with \"important\" and say yes to so many urgent things that they are in effect saying no to other things that are much more important.\n\n* * *\n\nSometimes it takes some time for this lesson to sink in. When we come back to an area where we've recently run a seminar, we always get students who come back to attend another session. Inevitably, students will come back and take the class again, and they'll tell us that when they tried to incorporate three or even two of the rules at a time, they spun their wheels. When they went back to basics and committed to nailing just one, they saw results even they couldn't believe. They'd get more done by channeling that focus than they ever could multitasking on a variety of to-do tasks.\n\nNAILING IT\n\nWhat does \"nailing it\" mean?\n\nIf you've truly mastered one positive change, we call it \"nailing it.\" It's become a popular shorthand catchphrase with many of our students. For you to have fully integrated the improvement and the changes it requires, it means that for three consecutive months, you've been able to complete the change on a daily basis 90 percent of the time or better.\n\nWhatever improvement you choose\u2014whether it's Organizing Tomorrow Today or committing to doing the Mental Workout\u2014you need to be able to do it nine out of ten days for three months straight\u2014with no excuses. If you can't do it, it means you need to increase your discipline or commit to a smaller level of intensity. Get started by proving to yourself that you can nail it, even if it's a smaller commitment. You can always increase later on. An essential element of performance is for people to learn to trust themselves. When you prove 90 percent of the time that you can nail it, you can't help but grow your confidence and self-trust.\n\nWhy does it work? Because simplicity, accuracy, and direction are a powerful combination. They provide the key to action that everybody needs\u2014whether you're a financial advisor, an NFL wide receiver, or a road paver for the state highway department. At the core level, learning to \"nail one thing\" teaches you how to believe in yourself. Your confidence grows and your self-image aligns with the knowledge that you do what you commit to doing. You literally become a \"winner.\"\n\n* * *\n\n**The Big Why:** The great myth of multitasking is that we're getting more accomplished by dividing our time and energy among all of the tasks clamoring for our attention. In reality, allowing your attention to be diverted from the step-by-step completion of your most important task triggers overload. And when you overload any system, it quits working and loses energy.\n\nWhen you commit to not overcommitting and nailing just one task before you start the next, you're stacking the odds for success in your favor. Just remember to choose the most important task to attack first.\n\n**The Inversion Test:** Put choosing wisely to the Charlie Munger inversion test and you get a simple output. What happens if you don't choose wisely, and you try to do too much? Overcommitting is counterproductive. Confidence erodes, failure becomes acceptable, and losing becomes a habit. The same thing happens when you try to carry too many groceries into the house from the car in an effort to \"save time.\" You drop a bag and have a mess to clean up. Not only did you lose time, but you lost the eggs, too!\n\n**Act Now:** Identify your \"first step,\" in the form of one simple fundamental improvement you want to commit to for tomorrow. It doesn't have to be earth-shattering or complicated. Just something you can specifically identify and \"nail.\"\n\nChoose wisely.\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nPRO ATHLETE\n\nNo matter what, I am going to complete 30 minutes film study. My focus will be to more specifically identify one thing I can do to be more effective against the starting pitcher's most prominent pitch. This will be completed by 11:00 a.m. Getting it done early will give me a calm, confident, and aggressive mindset throughout the rest of my prep and into the game. What I will not do tomorrow is spend anytime playing video games.\n\nFINANCIAL ADVISOR\n\nThe one thing I will attack tomorrow is making my proactive calls. I will make sure my first call of the day is to my most important client\u2014NO EXCUSE. Everything after that will feel easy. I will not allow myself to spend more than 10 minutes reading headline financial news (I usually spend anywhere from 30 minutes to 60 minutes on this).\n\nPHYSICIAN\n\nI will process my most important overnight case report by 9:00 a.m. so the team has time to close it out before lunch. To do this, I will need to start on the case no later than 8:20.\n\n* * *\n\n* * *\n\nMAXIMIZE YOUR TIME\n\nNever mind not having enough time in the day. Randy Boll didn't have enough time in his year. Even after being named one of the top ten performers at his Fortune 500 company, he was gunning for more.\n\nRandy told us he was hungry to reach his ultimate goal\u2014to be No. 1 in his firm. He had always been a very organized and driven guy, and in the first few years in business, he had been able to crush his to-do list every day\u2014which showed in his results. But recently, something had changed. The bar had been set higher, and he felt like there just weren't enough hours in the day to get it all done\u2014and even if there were, he had so many more distractions these days.\n\nRandy decided to start small and incorporate one of the basic ideas we're going to talk about in this chapter. He was going to do what we call \"time maximization.\"\n\nOver the course of the day, Randy\u2014like anybody else\u2014had a few minutes between meetings, or some time on the schedule he hadn't planned on because of a cancellation or an early finish. He resolved that whenever three minutes of time came free\u2014and he found that this kind of pause occurred, on average, three times during the course of the day\u2014he would ask himself a simple question. \"What can I get done in the next three minutes?\" And he'd consult the to-do list and knock out at least one important item from that list.\n\nBy maximizing time and attacking the \"open space,\" instead of allowing himself to waste the time, Randy was essentially creating nine extra minutes per day, five days a week. That translates into forty additional hours per year. This one simple change gave Randy an extra week per year to gain an edge on the competition.\n\nIt wasn't a huge change, but it had huge results. Within a year, Randy broke the earnings milestone he had been dreaming about for all those years and was sitting in the top spot of his firm. He didn't do it with time _management._ He did it with time _maximization._\n\nIt might seem like a small distinction: management vs. maximization. But in terms of achievement, the difference is vast.\n\nTraditional time management theories teach people how to do more with the time they have. Time maximization teaches you to create _more time._\n\n* * *\n\nTraditional time management theories teach people how to do more with the time they have. Time maximization teaches you to create _more time._\n\n* * *\n\nThe ultimate limitation for all of us is time. There's only so much time in the day, and you can only focus your attention on a certain number of things. Hundreds and hundreds of self-help books have focused on the concept of time management\u2014ways to get your time organized and build your list of commitments into a manageable plan so you can be more \"productive.\"\n\nThey mean well. But most successful people already manage their time. To break into the ranks of the highly successful, learning how to manage time better isn't going to change your life. Even if you come up with the perfect way to stack all of the pieces on your chessboard, you're limited by the size of the chessboard. And, more importantly, if you focus your attention on stacking those pieces and filling up the board, you're not necessarily creating a system that helps you put the right value on each of the pieces\u2014or assign them the right amount of time.\n\nJust because you have a lot of things to do and get them all done doesn't mean that all those things were done the right way, or that you're spending your time the way you want to be spending it.\n\nTime _maximization_ is very different. It is the search for both efficiency and productivity but also the act of prioritization at the same time. When you maximize your time, you're creating _more time_ to do the important things. You're changing the size of the chessboard completely. It's working the other side of the ledger.\n\nA simple analogy would be walking down into your basement and seeing water leaking from around one of the windows. You could manage that leak exceptionally well by coming up with a system of buckets to catch it, and purchasing drying equipment to keep the carpet from getting ruined, and then making a bunch of circuits downstairs to check to see if more water was coming in. Or, you could figure out how the water was getting in and fix the leak. Both skills are important, but one is _more_ important.\n\nIn Chapter 1, we started the conversation about prioritizing your commitments for the next day through Organizing Tomorrow Today, or OTT. Really, that process is about assigning your time. You're predetermining what you will focus on the next day. If you're able to take that one simple step and be ready ahead of time, you've made a huge step toward successfully maximizing your time.\n\nThe next piece comes when you are actually executing on your organization. There, you're going to use three tools to maximize your time:\n\n1. Attack the Open Space\n\n2. Prioritize the Priorities\n\n3. Trim the Fat\n\nATTACK THE OPEN SPACE\n\nIt starts with a simple question: What unit of time do you think in right now? In basic terms, the more successful you become, the smaller the unit of time you need to think in. Let us explain what that means.\n\n* * *\n\nIn basic terms, the more successful you become, the smaller the unit of time you need to think in.\n\n* * *\n\nWhen you're trying to manage your time, you look at a bunch of puzzle pieces and put them all together. That's fine, but it doesn't fully represent what happens in the real world. What really happens is that you have a collection of meetings, calls, emails, and other work to do during a given day, and most of the time those things don't happen in nice, clean intervals. You might have a call you need to be on that starts ten minutes early, or ends ten minutes late. An appointment may cancel, or some other responsibility didn't take as long as you thought it would.\n\nWhat occurs when the \"real world\" happens and you have a soft spot of unscheduled time in between responsibilities? Most people exhale and spend a few minutes chatting with coworkers, surf the web, or make a call. Blowing off steam by stepping away from the work grind is fine, and it helps your long-term productivity.\n\nBut the success level you have\u2014or the level you want to reach\u2014should determine what unit of time you're thinking in.\n\nThink about what happens when you reach a lull in your day. Let's say you have fifteen minutes before you have to go down the hall to a meeting. You might spend the time looking for deals on Amazon, absorb the latest celebrity gossip headlines at Gawker, or wander over to your friend's workspace to talk about last night's game. Maybe you have a puzzle game installed on your smartphone, so you tap away on that for a few minutes. Or, it might be time to check in with your mother, since it's been a few days.\n\n**TOP 7 TIME WASTERS**\n\n1. Personal email\n\n2. Facebook\n\n3. ESPN online\n\n4. The Kardashians\n\n5. Online News\n\n6. Shopping sites\n\n7. Google searches\n\nThose are all \"normal\" ways to blow off some steam during the workday. They're probably so common that you almost don't even consciously think about doing them. But if you want to be one of the top producers in your firm, you need to do some things that are not \"normal.\" You need to fill those gaps with something productive.\n\nTake conscious control of those small gaps in your schedule by deciding what unit of time you're going to think in, and resolving that if you have a unit of time that size or larger, you're going to apply it to another one of the items on your to-do list.\n\nIn Jason's day-to-day practice, the two or three most important events of the day\u2014usually client meetings\u2014are time blocked, as are important phone calls, meetings, and travel. His \"open space\" unit of time is two minutes. Which means that if one minute of free time shows up in the schedule, he'll disengage and send a text message, check the baseball scores, or just close his eyes and recharge. If two minutes or longer of open space show up in his day, he goes back to his expanded to-do list and asks himself what is one thing he can get done before he has to head on to his next commitment. If after completing his one thing there are still more than two minutes of unaccounted-for time available, he goes at it again, attacking his next one thing. And so on until the open space is smaller than his two-minute expectation or the open time is used up completely.\n\nThe more successful you become, the smaller that time increment becomes. If you have a big staff or a lot of decisions on your plate, your window might be one minute. You can also create a wider increment, but you won't see much of the benefit of this tool if you expand the time segment out beyond fifteen minutes.\n\nIf you're a person who has never operated by a strict schedule (or you have trouble keeping the one you already have), I'm sure this seems like a severe or demanding way to operate. It's an understandable reaction, and we won't tell you that the transition to a more structured approach will be completely painless. Once you've tried it, however, you will be impressed by what you can accomplish with a few minutes here and there. When identifying your \"open space\" unit of time, be sure to choose wisely. If you are new to this kind of strategy, it wouldn't make sense to set your number at five minutes or less. Build some success in from the start and pick a fifteen-minute increment. If you nail that fifteen-minute increment 90 percent of the time for three months, then you can think about reducing it to something like twelve minutes.\n\nThis new approach to time will certainly feel foreign to a lot of people when they first try it. But the only reason it seems foreign is because it hasn't become a habit yet. You _can_ do this, and when you do you'll immediately see the benefits of attacking the open space.\n\nYou've certainly seen that person at your office who is always the one grinding it out late into the evening, long after everybody else has gone home. Heck, you might even _be_ that person. This is obviously no knock on working hard, but if you're able to intentionally deal with the unaccounted-for time in your schedule during the day, you'll be able to get more work done in fewer hours. You'll be way more efficient, you'll get more done that needs to be done, and you'll be able to go home feeling better about what you accomplished and spend more time with the important people in your life. You can win at the things that are more important than your job. It's a little like being the kid who totally attacks his study-hall time during the school day so that after school he has more time to play with his friends.\n\nInstead of emphasizing time in terms of how long things take, you'll be changing the paradigm to emphasizing productivity.\n\nPRIORITIZE THE PRIORITIES\n\nAs you start to build some momentum from attacking the open space, you're probably going to start wondering how to choose what goes into those open spaces. It's a different process than the OTT concept we talked about in Chapter 1, but definitely related.\n\nTo be clear, we don't want you waiting until open time shows up in your day to then start thinking about your \"3 Most Important \/ 1 Must\" items. Those things should already be accounted for on your schedule as early in the day as possible.\n\nAttacking the open space is simple. When you find yourself with a few extra moments, ask yourself what is the most _important_ thing you can get done with the available open time window. The tasks you will do when you have those extra few minutes are not from the \"3 Most Important\" list but are other important tasks that you can now fit into your day even though they were not on the primary lists.\n\nMost people in this situation will go to their to-do list and pick one of the easiest, quickest items to complete. Unfortunately, that runs counter to everything we've learned about prioritization. Remember, it isn't about getting everything done. It's about getting the most important things done. When the open space shows up, avoid working on those less important tasks and instead attack the absolute most important item you can get moving on within the time available.\n\nThe efficiency you gain by filling the unaccounted-for time is directly tied to how well you choose and prioritize which to-do items go into those spots. When you do it well, you're like an NFL football team with a good game plan and a full week of practice ahead of a big game. If you don't, it's as if you told your team that they'd be playing a game Sunday, but didn't have any details about who the opponent would be, where the game was to be held, or what time it was being played. The second team is pretty much relegated to showing up and winging it. Unfortunately, that's the level where most people play in business.\n\nWe teach the students in our seminars to come in with a much more specific plan. They learn to identify and rank their priorities on two scales\u2014the \"macro\" items and the \"micro\" ones. The macro scale is the list of categories themselves\u2014prospecting for new clients, emailing, writing proposals, or having meetings with clients.\n\nWe believe there are two categories of productive behavior\u2014short-term revenue collection (those activities that produce immediate financial results) and long-term revenue cultivation (the lead activities that eventually bring about a consistent supply of revenue-generating opportunities).\n\nLet's use an analogy. If you were an orange farmer, your short-term revenue collection would be going out and picking the fruit from the trees. The long-term cultivation of revenue would consist of taking care of the trees themselves, planting new trees when necessary, and so on.\n\nMost people in the business world pay plenty of attention to short-term revenue collection activities. However, there's a real weakness when it comes to prioritizing the long-term revenue-cultivation activities. Highly successful people have learned that short-term revenue collection _and_ long-term revenue cultivation need to be emphasized daily.\n\nThe micro-scale activities are the individual activities within the categories themselves. For example, within the prospecting category, you'd have items such as asking for referrals, or meeting with centers of influence. The key is to start with the most important. Choose your best client to ask for a referral first, then move to your second-best client, and so on. Reach out to your most influential center of influence before making contacts with some of your less-connected influencers.\n\nIt's easy to see how it works in a sales-driven job, but the tool is just as applicable in other lines of work. Let's say you're a physician. Along with short-term revenue generation (seeing patients), you might be thinking about the quality of patient interactions. The longer-term cultivation component might be concerned with engaging in ongoing training or meeting with centers of influence that could potentially refer you to new clients.\n\nNo matter what category of activity you are operating in, always begin with the most important thing you can do. If you are emailing, return emails to your most important clients or prospects first. If you are completing paperwork, always begin with the most critical documents.\n\nTRIM THE FAT\n\nWorking hand in hand with prioritizing is actively reducing the amount of time you spend on things that you need to do.\n\nSchedule bloat is one of the most common\u2014and most irritating\u2014issues facing people who want to become high achievers. In short, it is quite normal to allow certain activities in your day to last longer than necessary, but you can learn not to let this happen.\n\nSpending _more_ time on critical activities shouldn't be the main objective. High achievers have learned how to get more done in less time. You should be trying to figure out how to do what you're doing every day more efficiently\u2014and how to make the time you spend on them more in tune with the overall priorities you've been setting.\n\nThere are a few methods for making this happen. First, it is essential to set your \"game clock.\" You must predetermine how long you will commit to a certain activity ahead of time. Once you set that game clock, you need to treat it just like the game clocks you would see at a basketball or football game. When the time expires, the game is over, and you need to move on to the next thing on the schedule.\n\nKnowing that you have a time limit should be the first step in getting you motivated to improve your efficiency. The game-clock concept is extremely helpful when you're tasked with organizing a meeting. Instead of telling people in a general way that the meeting starts at 9 a.m., establish a specific time frame for it\u20149 to 9:25 a.m.\n\nWhen we set up meetings, we actually make it a point to emphasize twice how long it is scheduled to last\u2014first when the appointment is confirmed, and again right at the beginning of the meeting. It lets people know you plan to honor the clock\u2014and that you respect their time.\n\nGroup meetings are always the low hanging fruit in conversations like this. They've become an integral part of the culture and of so many organizations that it's doubtful we'll ever be able to completely get rid of them\u2014and some are obviously necessary. Unfortunately, most corporate meetings are filled with people talking about work instead of actually working.\n\nAnd virtually every organization does a bad job of really understanding the cost to the business when three, thirty, or three hundred staffers get together for a meeting. We're not saying the cost isn't worth it in some cases\u2014because it certainly is. But when you think about the equivalent hourly pay for everybody at the conference table, and the value of the work they would be doing if they weren't sitting in the meeting, that meeting's opportunity cost is certainly more than a plate of bagels and some cream cheese.\n\nIf you're in charge of a meeting, you have a lot of power to improve how that meeting is run. The first question you should be asking yourself is, \"Do I really need all the meetings I am scheduling?\" Even if there is one ongoing weekly meeting you could eliminate, that would be great progress. Maybe instead of weekly, you could meet every other week. The next step is to think about setting a game clock for each of your regular meetings, and choose one of those to shorten the clock by fifteen minutes from where you currently have it.\n\nWhen Tom was at American Funds, his rule was that no meeting or conference call could last more than seventeen minutes. He would address one or two major points, outline the necessary action, and then get everyone back to work. Keep in mind that this was well before anyone had heard of TED talks.\n\nCarve the meeting down to its three main elements\u2014an opening, the main purpose, and the closing. In the opening, get right into the meat of what you want to accomplish by asking if anything important has changed since the last meeting. Then quickly transition into the main purpose by declaring the (very) short list of things that have to be accomplished (ideally, this would be one thing, and absolutely never more than three).\n\nWhen the game clock gets down to five minutes or so, start the transition to the closing by asking, \"What are the main takeaways from this meeting, and what is the next step of action?\"\n\nMeeting adjourned.\n\nWhat if there's no action to take? Then you probably didn't need a meeting at all. You could have sent out a memo. Some quick tips for running effective conference calls:\n\n\u2022 Realize going in that \"distractions are a given.\"\n\n\u2022 Hook their attention with a quick story or pertinent fact in the first twenty seconds.\n\n\u2022 Make one or two points, and keep it simple.\n\n\u2022 Know how you are going to finish\u2014before you begin.\n\n\u2022 Limit your time to seventeen minutes; anything longer, and the distraction monster is likely to win.\n\nGroup meetings are just the most obvious opportunity area. Client meetings and sales calls are other activities in our schedule that potentially eat up large amounts of your time because of the obvious pressure to generate new business.\n\nThe goal is to earn new business and to keep existing clients completely happy with less time spent. Many of the individuals we work with have examined their sales meetings and reduced them from an average of sixty minutes down to forty minutes across the board, just by making a few adjustments in how they handle the relationships. Instead of rambling on, in an effort to build a \"relationship\" with the person across the table from you, approach the interaction the way a college coach would with a basketball player. We call it the \"2-1-1.\"\n\nIf it's a client, start the conversation by complimenting two things the client is doing well. Follow that up by teaching the client one thing (within your scope of expertise) that will cause him or her to improve. Lastly, coach the client on a single, actionable element. Your goal with every individual meeting should be to leave the other person motivated to take _action_ on something that will improve his or her life. Complimenting the client will deepen the relationship, and teaching the client will increase your credibility, thus leaving him or her motivated to listen and take action on what is offered in the coaching.\n\nThis process is very different from trying to become somebody's friend.\n\nWe're not suggesting that you run people in and out in an impersonal way\u2014and obviously, this exact format will not work for every kind of one-on-one meeting. But in our view, the \"relationship-building\" pendulum has swung too far the other way in the business world. The best way to deepen relationships is to actually do your job with the client.\n\n* * *\n\nThe best way to deepen relationships is to actually do your job with the client.\n\n* * *\n\nPeople like to do business with people they like, but they also like doing business with people who get the job done and don't waste their time. If your financial advisor, mechanic, or brain surgeon came into a meeting and was friendly but didn't make things better for you, how long would it take before you were looking for someone else to work with? As a professional, you have a responsibility to coach your clients to improve on one thing in every single meeting, or you have more than likely just wasted your time and theirs.\n\nThis lesson becomes exponentially more important the more you move up the food chain, because the people you will be dealing with will be operating under the same time and scheduling constraints that you're reading this chapter to solve. Small talk is fine at a cocktail party, but it can be irritating when it happens during working hours.\n\nAs you use these three tools, it's also useful to remember what Coach Wooden considered to be the most important key to developing a disciplined team.\n\nStart _every_ meeting on time. No matter who is meandering in late, reward those who showed up early by beginning on time. The more consistent you are with this rule, the more respect you will earn, and the more people will begin showing up on time for meetings.\n\nFIXING PROCRASTINATION NOW, NOT LATER\n\nWith any well-intentioned diet, workout routine, or organizational plan, it's easy to sit in the calm times and plot out the \"ideal\" way to do things. But, as Mike Tyson famously said about boxing, everybody has a plan until they get punched in the mouth.\n\nAny of the concepts in this book will only work for you if they account for what happens in the real world and give you strategies for overcoming the problems you're going to encounter. Time maximization is no different. Even if you have the willpower and determination of a Navy Seal, you're going to run into times when you don't have the same energy to push and \"change time.\" It's natural, and it's called procrastination. We've all been there.\n\nYou see something on your schedule you just don't want to do\u2014either because it's something unpleasant, like a conversation you don't want to have, or because it's complicated and time-consuming, and you know it will take time you don't have at the moment.\n\nAnd so you keep putting the thing off to later and later in the day, or move it to later and later in the week or month. Maybe you're hoping it will secretly disappear. The reality is that you're trading your time at the moment for time at some point in the future.\n\nIt wouldn't be such an issue if each problem we confront every day existed in a vacuum. But work and time and life go on, and when you procrastinate and push things down the road, the things getting pushed join the other things on your plate at the receiving end.\n\nAs you get more behind and have more on your plate, you're going to feel stressed, rushed, and out of control. That's when the size of the job at hand\u2014or the pure number of different jobs you have to do\u2014can overwhelm you. It happens every day, and the most common response is paralysis. People get overwhelmed, and they just stop.\n\nOne of our favorite ways to attack procrastination is to break those dreaded activities into much more manageable pieces. It's called \"ask and chop.\" You ask yourself, \"What is the most important thing I can get done next?\" and then you \"chop\" that activity down into the first step of action.\n\nFor example, if you have something to do that you've been avoiding, the first, small step of progress toward completing it will help overcome the \"human inertia\" we talked about earlier. Instead of allowing your mind to become overwhelmed with the activity in its entirety, emphasize only the first step. Rather than completing the entire monthly expense report, just get the first day noted. Attacking the first step of action creates confidence and puts you on the productive side of action.\n\nSomething about chopping the activity up into smaller pieces seems to free people from many of the stalls that keep them from making progress.\n\nOne of Jason's hockey clients leaned on this technique almost exclusively to get into peak shape during one off-season and earn a lucrative contract extension. The player had never been very motivated when it came to working out, but had always compensated for a relative lack of fitness by being very well prepared mentally and strategically. He was a great \"clubhouse guy,\" beloved by his teammates and coaches, but injuries had slowed him down for a few seasons, and his strength and conditioning had become much more important.\n\nAfter meeting with Jason one spring, the player decided that he needed to build another forty minutes into his physical training routine. But instead of letting his mind focus on all of the additional work, he would tell himself, \"I'm just going to focus on getting one set done.\" Instead of adding the forty extra minutes onto the end of his current routine, he decided to \"chop\" it up. He would do one set here and one set there. It worked, because completing just one set felt so much easier than adding forty extra minutes per day into his already packed schedule.\n\nThe player came into camp in the best physical shape of his career and was able to play more minutes per game than he ever had. After a strong regular season, he played even better in the playoffs. The team rewarded him with a three-year contract extension\u2014a payoff both in money and in the opportunity for his young son to become old enough to really understand and enjoy his dad playing professional hockey. The father-son experience was literally priceless to our client.\n\nTo beat procrastination and create inertia toward time maximization, remember to \"ask and chop.\"\n\n* * *\n\n**The Big Why:** The more successful you become, the smaller the unit of time you must think in.\n\n**The Inversion Test:** The average income in the United States is around $50,000 annually. The goal is to be much better than average. To do so, you must begin thinking in smaller units of time. Most people think in time periods of one week. Mondays are dreaded, Wednesday is hump day, and then there's TGIF. Allowing yourself to think in longer units of time will cause you to get less of the most important work completed\u2014costing you money and self-confidence. You will be busy but not very productive. Thinking in smaller units of time will enable you to maximize your time. That doesn't mean you will be \"busier,\" but you will be more productive.\n\n* * *\n\nThinking in smaller units of time will enable you to maximize your time. That doesn't mean you will be \"busier,\" but you will be more productive.\n\n* * *\n\n**Act Now:** Take a sticky note in your office and write down your Time Maximization \"Open Space\" Number. This is the number of minutes, between one and fifteen, that will become your default trigger for knowing you need to get at least one thing productive completed when unaccounted-for time appears.\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nPRO ATHLETE\n\nTime Maximization \"Open Space\" Number is 10 minutes. Anytime that 10 minutes or longer of open space shows up between 1:30 p.m. and game time, I will commit to getting at least one set of armband exercises done to keep my shoulders and elbows strong.\n\nFINANCIAL ADVISOR\n\nTime Maximization \"Open Space\" Number is 8 minutes. Anytime that 8 minutes or longer of open space shows up between 8:30 a.m. and 4:00 p.m., I will reach out to one potential prospect. This could include reaching out to a current client or client of interest and asking for a referral if I don't have an identified prospect to contact.\n\nLAWYER\n\nTime Maximization \"Open Space\" Number is 5 minutes. Anytime that 5 minutes or longer of open space shows up during office hours, I will check my most important in-progress file and note if any action needs to be taken in the next 24 hours.\n\n* * *\n\n* * *\n\nWIN YOUR FIGHT-THRUS\n\nJoe Berger, a client from the business world, came to us with a very common story.\n\nHe was producing terrific sales numbers, but the way he was going about it was producing a tremendous amount of stress. He also knew he was leaving a lot on the table every month.\n\nJoe would have some incredible, record-setting sales months, but they would inevitably be followed by months with very poor numbers. And his attitude certainly matched where he was on the sales dial. In the great months, he felt like he was on top of the world. But in the slow months, stress and anxiety would consume him to the point that it affected his home life, health, and diet.\n\nWhat had always eluded Joe was _consistent_ high achievement. Simply put, Joe was a professional rider on the roller coaster of success. He didn't have the habits that would push him through and keep him on the high plane of success and make the lows less low.\n\nAfter some consultation, Joe embraced the concept of the \"fight-thru,\" a strategy we're going to talk about in this chapter. Once he learned how to win many of the small battles within himself day to day, everything changed. His normal routine of up and then down was replaced by consistent performance and gradual improvement.\n\nIn the past two years, Joe's business has increased more than 40 percent, and he's lost 40 pounds by avoiding the binge stress eating he used to do. The changes have been so stark that the other people in his group at work named a scale of improvement after him\u2014the Berger Scale of Success, a combination of weight lost and percentage points of business increased.\n\nIt all came from learning to \"fight-thru,\" and it's something anybody can do\u2014with some practice.\n\nThe entire subject of habits\u2014how they're formed, how long it takes, how easy or hard they are to break\u2014is shrouded in misunderstandings and misrepresentations of the work of one man: Dr. Maxwell Maltz, who presented his ideas in his 1960 book _Psycho-Cybernetics_.\n\nYou've probably heard that it takes something on the order of twenty-one consecutive days of the same intentional activity to create a habit. That's something that was pulled out of Maltz's book and oversimplified to the point where it is far from the point he intended to make.\n\nMaltz wasn't even talking about habit formation when he introduced the time period of twenty-one days. He was talking about patients who had gone through facial surgery, and the amount of time it took them to change their self-images enough to be used to seeing their new faces. His point was that it took a _minimum_ of twenty-one days for an established mental image to be replaced with a new one.\n\nOf course, folks with bad habits in 1960 were just like all of us with bad habits today. They really, really wanted it to be as simple as grinding out twenty-one days without a cigarette, or following a new exercise routine or a different diet, desperately hoping that this magic number would enable them to make a lasting change. Unfortunately, it isn't quite that simple.\n\nThe reality is that our habits aren't so much formed as they are in a constant state of formation. They're either getting stronger or getting weaker, based on how much attention and reinforcement they're getting each day.\n\n* * *\n\nThe reality is that our habits aren't so much formed as they are in a constant state of formation.\n\n* * *\n\nIf you're looking for the technical answer, researchers from University College London followed one hundred people for twelve weeks as they attempted to establish one new habit\u2014which ranged from something as simple as incorporating a different food into a daily diet to something as ambitious as starting a long-distance running plan. The study revealed that it took the participants anywhere from two and a half weeks to three and a half months to establish their habits to the point where they did the new thing at least 95 percent of the time.\n\nOf course, the devil is in the details.\n\nThe factors that impacted how long the habit formation took? According to the study, it depended on how much time someone spent working at the new habit, what the habit was, and how different it was from the person's regular routine. In other words, there is no magic twenty-one-day \"cleanse\" that will spit you out the other side stronger, thinner, or with more oxygen in your lungs.\n\nBut before you get discouraged, there _is_ a process you can follow to develop a set of habits (or regular practices) that will serve you well in your professional and personal life. It's a matter of following some simple steps that work in concert with how your brain functions\u2014not against it.\n\nIn basic terms, behavior patterns or habits start in the same part of the brain as the one responsible for memories and emotions. There's an external trigger for the habit, the behavior itself, and then the emotional response to the behavior. When you respond to the trigger with the behavior, and the behavior makes you feel good, it reinforces itself into a habit loop. Over time, that loop becomes an ingrained part of your brain's \"programming,\" to the point that the habit activity happens almost without conscious thought.\n\nWhether you're trying to add a new, good habit to your daily routine or replace a bad one with something else, you're going to go over the same \"mental terrain.\" Our goal here is to give you a roadmap to that terrain, so that when you get to it, you aren't surprised by what you see and feel, and you have a guide to help you push through it.\n\nWe separate that guide into three different phases\u2014the honeymoon, the fight-thru, and second nature. The mere act of recognizing the phases of habit formation when you see them and calling them out to yourself is a huge positive step, and it will provide a major boost in energy.\n\n* * *\n\nThe mere act of recognizing the phases of habit formation when you see them and calling them out to yourself is a huge positive step, and it will provide a major boost in energy.\n\n* * *\n\nTHE HONEYMOON\n\nThis is a neighborhood we all know very well. You've decided to start a new, regular routine at gym, or keep your desk much more organized, or set up a system so that your client information always gets updated in a central place.\n\nIt usually happens because of some kind of jolt\u2014either positive or negative. Maybe you went to a conference and heard a talk that really inspired you, or you lost an extremely important piece of paper in the mess and missed an important deal. Either way, a \"triggering incident\" produced plenty of fuel to kick off the habit drive.\n\nAnd the first few days, it might even be easy. After that first vigorous workout in your new routine, you might say to yourself, \"Oh yeah, I can do this. I'm ready for the challenge.\"\n\nWe chart it by email every time we hold one of our training seminars. The students have follow-up access by email for any questions or comments they might have, and our mailboxes fill up the next week like clockwork\u2014with messages about how easy it was to make all the changes, and about how they are already way ahead of the pace we suggested in the program.\n\nBut all honeymoons eventually come to an end, and the day-to-day reality sets in.\n\nOn day three or day eight or day eighteen, you'll meet one of the constant stream of obstacles that will test your resolve. You'll be presented with many different versions of the same question. Will you take the easier route and go back to \"normal,\" or will you win your \"fight thru\"?\n\nTHE FIGHT-THRU\n\nThis is the point where \"I can do this\" turns into \"This is harder than I thought,\" or, \"Is it really going to matter if I miss a day?\" To make it through to the third phase, when the habit becomes second nature, you need to be able to win two or three of these important fight-thru battles with yourself.\n\nHere are the four techniques you will need to do just that.\n\n_1. Ritualize_\n\nThe less you leave up to chance (or procrastination), the better your chances are for success. Ritualize the new habit by scheduling it on purpose at the same time every day. If the habit is getting a thirty-minute run in every day, block it out on the calendar for the same time and make it nonnegotiable. Doing it this way takes most of the thinking out of doing it. You're almost automating the process. Over his years at UCLA, Coach Wooden built a dedicated set of rituals that put him where he needed to be each day\u2014from his 5:30 a.m. walk, to meeting with assistants immediately after practice to formulate the plan for the next practice, to finding his wife, Nellie, in the stands before a game and making eye contact.\n\n_2. Recognize_\n\nAs we mentioned above, just the act of recognizing the barrier in front of you is a huge step toward getting over it. When it gets to be day three of the exercise plan and you're lying in bed debating with yourself about whether or not you want to go out in the rain and get it done, recognize where you are. Learn to simply say to yourself, \"I've entered a fight-thru.\" Recognizing that you are in a \"fight-thru\" is like taking the blindfold off before the fight begins. Now you know what you're fighting. Remind yourself that \"It's important to win this one, today.\" Why? Because each fight-thru win makes the next fight-thru easier to beat, thanks to momentum. Momentum works the other way, too. Each fight you lose makes it easier to quit the next time.\n\n_3. Ask Two Questions_\n\nAs you push through the fight-thru, coach yourself up with two \"perspective\" questions. Ask yourself how you will feel if you win the fight-thru, and, conversely, ask yourself how you will feel if you lose it. You're taking the next step past recognizing the facts of the situation and bringing emotion into the equation\u2014which is the most valuable kind of fuel. Emotion promotes action. If you've committed yourself to a workout routine, for example, and you hit one of the inevitable fight-thrus, ask yourself the two questions. The first would be, \"How will I feel if I win this fight-thru?\" Chances are, if you find the strength to win, you will end up feeling like a world champion (even if only for a short period). The second would be \"How will I feel if I lose this fight-thru?\" If you lose the battle\u2014it will probably bring out the negative emotion that comes with underperforming. Using these positive and negative emotions can help propel you through.\n\n* * *\n\nEmotion promotes action.\n\n* * *\n\n_4. Life Projection_\n\nTake thirty seconds and think, in great detail, about where you think your life will be in five years if you're able to make this change and consistently win your fight-thrus. This is the time to be totally honest with yourself, and really let yourself feel the positives of doing that different thing. Do it in tandem, and go through the same process while thinking about your quality of life if you allow yourself to lose fight-thrus for the upcoming five years. Again, picturing as much detail as you can is really important for maximum effectiveness. The more emotion you bring into it, the better.\n\nThe amazing things that world-class athletes are able to accomplish are usually chalked up to freak physical ability\u2014and that certainly can be a factor. But a much bigger factor in those athletes reaching that level is their relentless ability to consistently win the fight-thrus.\n\nGreatness is predicated on consistently doing things others can't or won't do. Simply put, success is _not_ about being brilliant. It is about being consistent.\n\n* * *\n\nGreatness is predicated on consistently doing things others can't or won't do. Simply put, success is _not_ about being brilliant. It is about being consistent.\n\n* * *\n\nThe first time we met one of our baseball clients, he was bouncing around in the minor leagues. He was extremely talented, but in six years of pro ball, he had spent less than a full season at the major league level. He knew he was just a few good habits away from a permanent stay in the big leagues, and he came to us for help.\n\nThe first habit he needed to incorporate was to control his drinking. He was in no way out of control, but drinking represented a huge component of his social life. He'd finish a game and go somewhere to unwind with \"a few\" beers, and he'd eat dinner late and stay out socializing. He wouldn't get home until the early morning hours, and the next night he'd do it all over again.\n\nIt sounds like something your mother would tell you, but he really wasn't getting enough sleep. He woke up late and groggy, and would usually scarf down some fast food in his rush to the ballpark for a game. He would get to work feeling stressed and underprepared, and it certainly leaked into his performance on the field.\n\nBy making a basic habit change and committing to drinking no more than three beers on a night before a game, he self-limited his time out. This started a chain reaction of positive events\u2014more sleep, more time in the morning, a healthier breakfast, and a more relaxed and focused attitude at the park.\n\nNow that he had more time to prepare before games, the player decided to adopt another one of the rules we teach. He committed to getting to the field and spending specific time getting himself mentally prepared to play, using the Mental Workout we're going to talk about in Chapter 6.\n\nHe also committed to spending ten minutes before every game doing what is called \"deep practice\"\u2014training at full speed. One of the player's weaknesses was pitch recognition. Major league players have just a split second to diagnose what kind of pitch is being thrown and decide whether or not to swing. If you don't go in with some kind of game plan about the pitches you expect the pitcher to throw, and have some idea what the spin looks like for each, you're guessing\u2014and you aren't going to be very successful. The player spent ten minutes standing in with the pitchers during their warm-ups, practicing calling out what pitches they were throwing. His goal was to get three in a row correct before the ball hit the catcher's mitt. If he didn't get three in a row, he had to go another round.\n\nThe changes he was making were great, but it still took almost a full season of habit development to get them to be a real, integrated part of his routine. He would be the first one to say that it took a huge number of fight-thrus to get there.\n\nThe first week he made his commitment to limit drinking, one of his best friends on the team celebrated a birthday. After the game, they all went to the bar to celebrate, and the player finished three beers in the first hour. He was thinking about ordering a fourth, but he recognized that he was in the middle of one of those fight-thrus. He forced himself to think about what would happen if he ordered another drink. He realized his \"one more\" would probably turn into six more. He pictured getting home at three in the morning, and also imagined what he would feel like when he got to the park the next day. It was not a pretty picture.\n\nHe ordered another beer, but he made it a nonalcoholic one, and spent another hour celebrating with his buddy before going home and getting to bed before 1:00.\n\nThe reality, of course, is that you won't win every time. We're all human, and willpower is imperfect. Four months into his commitment, the player went to a teammate's wedding and completely lost control, drinking hard and deep into the night. It began a two-week cycle where he returned to the habits that had kept him in the minors in the first place. Even though it was the off-season, he still had responsibilities: he had to keep himself in great shape and be ready for spring training.\n\nThe key was that after those two weeks, the player realized that it was getting easier and easier to lose those internal willpower fights, and he didn't like where he was or where he was headed.\n\nHe made himself imagine where he was going to be in five years if he kept allowing himself to fall off the wagon. He was certain that the remainder of his career would be a series of disappointments and underperforming. The thought of never reaching his potential haunted and disturbed him. He learned to use the negative emotion to help recommit himself to his original goals, and doing so caused him to start winning fight-thrus again.\n\nWhen spring training came, the player was ready. He had his best spring, and he made the big league team. He hasn't seen a minor league field in three years\u2014he has become a full-time major league starter.\n\nSECOND NATURE\n\nAfter you've won a series of fight-thrus (both big and small), you'll start to enter that \"getting into the groove\" phase that feels so good. The new habit has become a regular part of your routine, and doesn't feel so much like something you have to intentionally remember or push yourself to do.\n\nIt's a great place to be, but you also need to keep your guard up to protect against some really common traps that can send you back to the fight-thru phase. These traps are the reason business professionals don't repeat strong results year over year, exsmokers become smokers again, and writers never get that first novel completed.\n\n_The Discouragement Monster_\n\nOne of the unfortunate realities of the world is that just because you work hard and do the right things, good things don't _automatically_ flow to you. Life can happen in streaks and slumps, and it can be incredibly discouraging if you put in the work to change an important habit and it does not immediately produce the benefits you expect. Maybe you changed a key way you interact with potential clients, and it hasn't yet produced the numbers you expected. Or you changed your diet and it didn't produce the weight loss you hoped. It is very easy to slip into the mindset of _\"Why do I bother? It doesn't matter what I do anyway.\"_ The discouragement monster, as we call it, is so dangerous because it saps your willingness to keep trying. You must remember that scoring often comes in spurts.\n\n* * *\n\nThe discouragement monster, as we call it, is so dangerous because it saps your willingness to keep trying. You must remember that scoring often comes in spurts.\n\n* * *\n\n_Disruption_\n\nIt happens every single year. Just before summer, millions of people start a new diet and exercise routine and get into a great pattern. It's going well, and November comes around. They get knocked off track by Thanksgiving, and in a couple of weeks the Christmas party circuit starts. By January, they're right back where they were in June. It can happen a million other ways as well. You might get sick, go on a long vacation, or even take a weekend and put work out of your mind. Whether it's vacations, illnesses, or holidays, any break in the routine is a potential disruption. And any of those disruptions can put a major dent in your positive habits.\n\n_Seduction of Success_\n\nMaybe the most dangerous of the three comes when you actually have great success with the new habit. You've won the fight-thrus, you've changed your pattern, and now you're really humming along. It's absolutely human nature to get to that point and think\u2014even for a second\u2014 _\"Hey, I've got this licked. I've figured it out, and I don't have to work quite as hard now that I have it down.\"_ Then you test that new theory, and lo and behold, the results still show up for a short while. You allow yourself to believe you're the special one who won't have to work as hard as the others to produce results. You know what happens next. You punch another ticket for a ride on the success roller coaster. A few years ago, we worked with a successful college baseball pitcher who was as talented as any athlete we'd ever seen. He pushed himself in training, and he was relentless in his throwing program. He aced his process goals, every day. He got drafted, and quickly worked his way up to the major leagues. After his first full season, though, he started to gradually tail off with his process goal work. Regardless of what we said, we couldn't get him to realize that he was headed for trouble. He started focusing more on his bank account and the size of his next contract. In three years, he was out of baseball completely. His response to what happened: \"The things I did early in my career worked so well, I stopped doing them.\"\n\nWe have another client who is one of the preeminent personal-injury attorneys in the United States. She didn't have any trouble in the courtroom, but at times she struggled with some of the other aspects of the job. Her firm compensated her on how many cases she brought in as well as whether or not she won them.\n\nShe had always been passionate about helping clients, but, like many attorneys, she didn't look forward to the part where she had to market herself to new clients. She needed to figure out how to be better at the daily grind of finding potential clients and signing them up.\n\nShe determined that she wanted to develop one new habit to get better at that part of her job. To do this, she committed to spending thirty minutes per day writing blog posts for her firm's website. The idea was to expand her profile and start to be recognized more frequently by potential clients.\n\nMost of this attorney's struggles came from discouragement. She told us that the hardest weeks were the ones where she devoted hours of time to making new connections but none of it seemed to bear fruit. She kept feeling like she wanted to quit putting effort into looking for new clients, even though she knew that would only make things worse.\n\nShe did have some days that went really well, but those days made her struggle in a different way. Since she didn't enjoy the marketing part of her job, it made her hope against hope that grabbing three or four new clients in a short time would put her in a position where she didn't have to work on the marketing side so hard for a few weeks or months.\n\nThe first step toward beating the enemy\u2014and avoiding going back to the fight-thru phase\u2014is to know the enemy. Our client learned to recognize the discouragement monster when it came\u2014as well as the seduction of success. She learned to recognize the beginnings of \"seduction\" as one of two conversations she found herself having with herself: \"I can't get the marketing work done today because . . . \" was one of them. The other was \"It'll be okay if I don't do it today because. . . .\"\n\nRecognizing seduction is such an important part of avoiding it. Anytime you catch yourself saying, \"I can't do my most important tasks today because . . . ,\" or, \"I don't need to do my most important activities today because . . . ,\" you know you are entering the seduction zone.\n\nRecognition is great, but what do you do next?\n\nAnytime you recognize that you're backsliding, choose to do \"a little bit more, for a little while.\" If you are becoming seduced, choose that day to do up to 10 percent more on your \"1 Must.\" Doing just a bit more even for that one day will bookmark in your mind the fact that you have the ability to win fight-thrus.\n\n* * *\n\nIf you are becoming seduced, choose that day to do up to 10 percent more on your \"1 Must.\" Doing just a bit more even for that one day will bookmark in your mind the fact that you have the ability to win fight-thrus.\n\n* * *\n\nFor example, when our client started feeling pretty good about her marketing work, to a point of thinking she might take the day off on the blog posts, she committed to three extra minutes writing that day. It isn't that writing for three extra minutes (or doing any other habit-related task for a few minutes) will have a huge impact on your results. That isn't the point of doing it. You're reminding yourself during that extra time that you're willing to put in the extra work, and that you're mentally strong. That you're different from other people. When \"normal\" people experience success, they have a tendency to respond by doing a little less. When mentally tough people experience success, they stay totally committed to what caused the success. It's amazing what that kind of thinking can do for your overall attitude!\n\n* * *\n\nWhen \"normal\" people experience success, they have a tendency to respond by doing a little less. When mentally tough people experience success, they stay totally committed to what caused the success.\n\n* * *\n\nIf you do fall into one of those traps, it isn't the end of the world. It's like going back to driver education school after you get a collection of speeding tickets. You go back to the fight-thru phase and win some more of those to earn your place back in second nature.\n\nThe overall win is the commitment to pushing past the fight-thru phase no matter how many times you have to go back to it. Remember, every time you win a fight-thru, the next one gets easier to win.\n\n* * *\n\nRemember, every time you win a fight-thru, the next one gets easier to win.\n\n* * *\n\nThat's not to say the process is going to be easy or pain-free. If you want to be extremely successful, you have to commit to doing things that many people aren't willing to do. If it was easy, everybody would do it. But just like physical training makes your body strong, perseverance and willingness to relentlessly fight thru the obstacles make you mentally strong.\n\nIt is the training ground for mental toughness. You're establishing winning as a habit.\n\n\"PUKE-A-LICIOUS\" WORKOUTS\n\nOne of our NHL clients showed us just how powerful the \"a little more for a little while\" concept can be. This player was one of the top players on his team, and the team had done so well they had qualified for the playoffs with three weeks to go in the season.\n\nWhen that happened, the coaching staff got everyone together and told them to lay off their strength and conditioning training, so they would have \"fresh legs\" once the postseason started.\n\nOur client was young and strong, and one of his \"3 Most Important\" items was to get four additional strength and conditioning workouts in every week. He wanted to listen to his coaches, so he stopped doing the workouts\u2014and promptly found himself playing the worst hockey of his career going into the final days of the regular season.\n\nAfter talking to the coaches and getting permission to get back onto his workout routine, the player decided to make one modification. During one of the workouts, he would push himself harder, for just a few minutes\u2014to the point where his mouth started watering and he was about to vomit.\n\nWhen we asked him why he did it, he quickly answered, \"It's in those puke-a-licious moments that I'm reminded how much of a badass I really am.\"\n\nIn those extra, intense workouts, the player was bookmarking in his mind that he was different. While other players were worn down, or were taking it easy at the end of the season, he was staying committed and even doing more. He was reminding himself that he was growing stronger as the season wore on.\n\nThis experience had a huge impact on his confidence, and he's now considered one of the top two-way players in the game.\n\n* * *\n\n**The Big Why:** Habit formation is more than a twenty-one-day set-and-forget process. Life is a constant state of fluctuating habits.\n\n**The Inversion Test:** Learning to win one fight-thru makes it easier to win the next fight-thru. Losing one fight-thru makes it easier to lose the next, too. If you want to stop making progress, do not put effort into winning your fight-thrus.\n\n**Act Now:** Learn to recognize what's happening. Write down on a piece of paper one fight-thru you are currently experiencing.\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nPRO ATHLETE (THIRTEEN DAYS AFTER THE SEASON ENDS)\n\nI do not want to start my off-season training. I want to relax. I had a great season last year and I am feeling unmotivated about getting things going for the upcoming year. Even though my trainer is on me and I know I need to get going, I am thinking about putting it off for another week. **Winning the Fight-Thru:** I reminded myself how I will feel on day one of OTA's (organized team activities) if I don't come in feeling at the top of my game. I have had years before where I started the season not in my best shape, and those years are not the years I want to repeat. The other side of it is when I show up in the best shape of my career: it feels great, my confidence is high, and it always causes me to have a strong start to the season. For me, a strong start is very, very important.\n\nFINANCIAL ADVISOR\n\nI just had my best month ever and I am feeling really strong. I find myself thinking about my results and thinking that maybe, just maybe, I will not have to keep pushing so hard. Even though it only takes about an hour each day to complete my \"1 Must\" of proactively calling three clients and one prospect per day, I am coming up with excuses to let myself off the hook. **Winning the Fight-Thru:** I remind myself that the single most important reason I had such a strong month last month was because I made my proactive contacts daily\u2014no excuse. I know I need to continue this and I force myself to realize that \"process guarantees results . . . good and bad.\" If I follow a bad process, I will have poor results. It's that simple.\n\nPHYSICIAN\n\nI have been staying up late watching the NBA playoffs. I played college basketball and I love watching the big guys hoop. When my alarm goes off at 5:00 I know I need to get up if I am going to have time to get my workout in. Problem is, I am just too tired, and I know I need more sleep if I am going to be sharp for my patients.\n\n**Winning the Fight-Thru:** I've lost ten to twelve pounds over the past six months and I feel really good about doing so. I know if I keep giving myself a pass on making my workouts, it will be just a matter of time before the lost weight comes back. If I am going to get up at 5:00, I know I need to get at least seven hours of sleep for my mind to be sharp. I need to begin setting a \"lights-out curfew\" for myself no later than 9:45 p.m. each night. I can record the games and watch them commercial-free the next morning while I work out. If pro-athletes set lights-out curfews, I guess I should be approaching my performance the same way.\n\n* * *\n\n* * *\nHALFTIME\n\nIn sports, the halftime break gives coaches a chance to evaluate what has happened so far and come up with a plan for the rest of the game.\n\nOur goal is for you to do much more than just read this book. We want you to _use_ it and to incorporate the things you've read into your everyday life. Remember, knowing something doesn't change your life. Doing something does.\n\nSo, just like the coaches at halftime in a game, we're going to go through what's happened so far, pull out the most important takeaways\u2014and then make a plan for what's coming up.\n\nBelow, we've made a list of what we think the three main ideas are from each of the first four chapters. It's by no means set in stone\u2014if you took something else to be the most important concept, we're not telling you you're wrong. For you, maybe that is the most important thing at this point in time.\n\nEither way, you now have a new assignment. Pick one of the ideas below\u2014or the different idea you identified to be the most important\u2014and go back to the chapter describing it. Reread the information pertaining to that idea to get it more firmly set in your mind.\n\nOnce you've done that, find a friend, colleague, or family member and explain the concept to that individual. In one of the most influential educational studies ever published, French researcher Jean-Pol Martin established that students who taught their peers the French language gained much more command of the material than students who learned conventionally from professional teachers or strictly studied on their own. The act of preparing to teach the information, presenting it, and interacting with others who were learning it both enhanced the learning process and increased learning speed.\n\nIn simple terms, when you teach something effectively to somebody else, you are effectively learning it at a much deeper level.\n\n**OUR TOP 3 TAKEAWAYS: CHAPTERS 1\u20134**\n\nCHAPTER 1: ORGANIZE TOMORROW TODAY\n\n1. If you prepare successfully, you're preparing yourself to succeed.\n\n2. Writing down your goals (3 Most Important \/ 1 Must) for the next day engages your subconscious mind in problem-solving while you sleep.\n\n3. Don't mistake \"busy\" for \"productive.\"\n\n4. ______________________________________________\n\nCHAPTER 2: CHOOSE WISELY\n\n1. Choose wisely what you commit to\u2014overcommitting is common and has a very negative impact.\n\n2. Respect channel capacity.\n\n3. Focus on \"1 step\" and you will have much more success crossing the finish line.\n\n4. ______________________________________________\n\nCHAPTER 3: MAXIMIZE YOUR TIME\n\n1. The more successful you become, the smaller the unit of time you should be thinking in.\n\n2. Attacking the open space, prioritizing the priorities, and trimming the fat are all effective methods of _creating_ more time.\n\n3. Once you've identified that thing, \"chop it\" by separating out the first action step you can take on it.\n\n4. ______________________________________________\n\nCHAPTER 4: WIN YOUR FIGHT-THRUS\n\n1. Habits are in a constant state of formation.\n\n2. The most essential step of winning a \"fight-thru\" is being able to recognize when you've entered a fight-thru.\n\n3. Every time you win a fight-thru, it makes it easier to win the next one.\n\n4. ______________________________________________\n\nEVALUATE CORRECTLY\n\nTom was a twenty-five-year-old high school basketball coach when he got his first chance to coach in college. The team he took over had been historically bad\u2014with twenty-one losing seasons in a row. In Tom's first week on campus, the football coach came by his office to meet the new guy.\n\n\"Do you have any idea what a mess you're getting yourself into?\" he asked.\n\nTom replied that he did, and set out to prove that the mess could be cleaned up. He was going to use the basics that John Wooden had established with his teams\u2014a focus on fundamentals and conditioning.\n\nTom put his team members through an intense off-season conditioning program, having them run miles and miles in distance training, and following that up with brutal wind-sprints backward and forward up the big hill on campus.\n\nWhen it came time to practice at the start of the season, the players were in the best shape of their lives. Tom and his assistant then began the exhausting process of establishing a set of common fundamentals in every player\u2014starting all the way back at the beginning, with how to hold the ball, and graduating to passing and shooting skills.\n\nWith constant coaching and review, it took more than a month to establish the necessary work ethic, and another month to get some of the new skills ingrained. The players\u2014and the coaches\u2014were excited to put their improvements to the test under game conditions. But in the first eight games of the season, the team only won one game\u2014even though they had run out to double-digit leads in six of them.\n\nAfter the seventh loss, Tom was sitting in a dingy little hotel room in the middle of nowhere, replaying every game in his head and trying to figure out the missing piece. In one game, against Central Arkansas (National Basketball Association Hall-of-Famer Scottie Pippen's alma mater), his team was up 18 points before halftime, only to let the lead slip away in the second half. And Tom had an epiphany: he realized that his players now had the ability, skills, and work ethic to execute, but when they were presented with adversity\u2014the other team making a run\u2014they folded.\n\nSo Tom made an adjustment. Every night, after practice, he had each player on the team go through a mental exercise. He asked the players to take one minute to think about one thing they did right during that day's practice or game.\n\nThe positives from that simple act of self-evaluation and self-encouragement grew almost immediately. Over the next fourteen games, the team went 6\u20138 while they integrated the new routine. Then, over the last six games of the season, they went 5\u20131. They were beginning to be considered one of the toughest teams in the conference.\n\nThe next year, with the same core group of players, the team went 20\u20135, the basketball program's first winning season in twenty-three years.\n\nThe athletes and business leaders we train in our practice and seminars all have different goals they're trying to reach and issues they want to learn to deal with, but there are a surprising number of common threads in the conversations we have. And those conversations would almost certainly be familiar to you, too.\n\nWhether you're a professional athlete, an analyst at a brokerage firm, or an IT manager at the company down the street, you're trying to work your way from where you are to the next rung on the success ladder. Performance is obviously the main driver, and evaluation is a critical tool. If you don't have an accurate picture of how you're doing\u2014and a way to use that information to move you forward\u2014you're flying blind. Asking yourself what you're doing well and what you want to improve is the first step of positive growth.\n\n* * *\n\nIf you don't have an accurate picture of how you're doing\u2014and a way to use that information to move you forward\u2014you're flying blind.\n\n* * *\n\nVirtually all of the successful people in any endeavor have the same basic \"emerging pattern.\" See if this sounds familiar.\n\nWhen you first achieve some success, you begin to have the expectation that you'll continue to be successful. If you're a salesperson and you smash your boss's goals for year one and get plenty of accolades, you're almost certainly going into year two with more confidence than you had before and a built-in expectation that you're going to do even more.\n\nAthletes are the same. When the highly touted recruit comes in and plays right away as a freshman and has success, everybody starts projecting out what will happen in the future. Will he come back and play another year? Will he turn pro? They're projecting that the player will continue to shine and improve at the same rate, and it becomes a given in the player's mind as well.\n\nThis expectation of improvement is where the seeds of the \"perfectionist mentality\" are planted. Thoughts of, \"Yeah, I did a really good job\" are quickly replaced with, \"Yeah, but I could always do better.\" When you're on that path of early success, you're building a constant elevation of perfection into your measurement of success and failure. You tend to write off your success as \"I expect that of myself\" and focus relentlessly on your shortcomings.\n\nWe're not here to criticize that effort and openness to learning and improvement. What we're concerned with is the tendency to overlook those things you are already doing well. Evaluation is the genesis of improvement, however if the evaluation isn't done correctly it will be counterproductive. Unfortunately most people learn to evaluate with the perfectionist mentality.\n\n* * *\n\nEvaluation is the genesis of improvement, however if the evaluation isn't done correctly it will be counterproductive. Unfortunately most people learn to evaluate with the perfectionist mentality.\n\n* * *\n\nThere's a big problem with the perfectionist pattern. It works great at the lower levels of achievement, when natural talent and a strong drive will let you fix your mistakes, and when you can make gains relatively easily by outperforming and outworking the competitors in your immediate circle. But what happens as you move into more competitive arenas? When you move from the minor leagues to the big leagues, or from the regional office to the national office?\n\nWhen the pool gets deeper and there is less separating you from the others in your group, you're going to have a harder time \"winning\" as often.\n\nAt the lower levels of competition, the perfectionist mentality isn't as damaging because there are so many external pats on the back available for solid performances. But as the level of competition increases it becomes more difficult to feel good, because those external pats on the back happen much less often.\n\nYou see it in the NFL all the time. A player who was able to freestyle and make plays with athleticism and field vision in college gets swallowed up when all the other players on the field are elite, too, and playing within a well-designed system. The level of play is so high that it becomes much harder to stand out.\n\nThere just isn't as much room to win on pure talent or determination anymore, and it becomes natural to experience more failure than you're used to.\n\nWhen the level changes like that, the perfectionist method of evaluation becomes completely ineffective and often unhealthy. If your personal sense of measurement and worth is based on wins and losses and _always_ doing better, what do you think happens when you aren't winning as often?\n\nIf you aren't evaluating yourself in a productive way, the losses and \"failures\" erode your self-confidence. When you aren't as confident, you can't perform as well. This causes you to fail more often\u2014starting a vicious cycle. Let's be clear: this isn't the \"everybody should get a medal\" mentality. Quite the contrary, we are not asking you to feel good about things you haven't done or achieved. Rather, we want you to learn to simply give credit where it's due. Most people spend great time and energy focusing on the things they didn't accomplish while totally overlooking all the things they did accomplish.\n\n* * *\n\nIf you aren't evaluating yourself in a productive way, the losses and \"failures\" erode your self-confidence. When you aren't as confident, you can't perform as well and you fail more often\u2014starting a vicious cycle.\n\n* * *\n\nThe problem resides in the fact that ultimately one cannot control \"winning.\" By focusing on things you cannot control, you minimize your emphasis on what you _can_ control.\n\nIt happens to even the most talented athletes and businesspeople in the world\u2014which means nobody is immune. To beat it, you have to learn the art of _performance_ evaluation. Using the techniques we're going to be talking about here, you replace those perfectionist tendencies with what we call the \"performance mentality.\"\n\nInstead of burying yourself in negative thoughts and emotions, you will learn how to make effort and improvement (not perfection) your main priority, which in turn gives you the greatest possible potential for impacting results.\n\nTHE PERFORMANCE MENTALITY\n\nOne of our hockey clients had come into the NHL as a highly touted prospect, but after three seasons he was still trying to live up to his potential. He certainly wasn't a bust\u2014he was getting regular playing time on his team and was a solid contributor at the relatively young age of twenty-five\u2014but he hadn't blossomed into the star everyone thought he would become. He showed flashes of excellence, but he needed to make some changes to become consistently great.\n\nIn one meeting, Jason asked the player to tell him one thing he believed he had done well in the previous two weeks. He didn't answer for a long minute.\n\nThen he said, \"I know we've talked about this before, but to be honest, I am not proud of myself for anything. I've been judging myself on my results. When things start going bad, I get discouraged and it just makes me play worse. And when good things do happen, I don't end up working as hard\u2014which causes me to make bad decisions off the ice.\"\n\nFrom that point, Jason worked out a plan with him where he would stop listening to what the sports talk radio shows were saying about him, and he would quit looking at his statistics after every game.\n\nHe started evaluating himself solely on his effort in that day's practice or game. He came up with his own scoring system based on how he moved his feet, how he attacked loose pucks, and whether the shots he took were high-quality opportunities. It wasn't just a matter of filling up the stat sheet in the traditional ways. After each practice, he would grade himself on a scale of 1 to 10 on effort. During games, he'd do the same thing after every shift. Anything less than a 9 was completely unacceptable.\n\nSlowly but surely, he began to be more physical and active on the ice. Something else was changing as well. He was becoming a happier, calmer person off the ice.\n\nOver the next few years, he began to get more and more attention as one of the NHL's up-and-coming players. And, better yet, he knew that he was in control of that success because he was measuring the right things.\n\nTHERE ARE NO SPECIAL PEOPLE . . . ONLY THOSE WHO DESERVE IT\n\nThis is another one of those places where we're going to tell you something right up front that might not be the sexiest solution you've ever heard.\n\nThere is no magic success pill. Success requires strong and consistent effort, and the act of evaluating yourself on that effort. Most people believe that it takes their best effort on _everything,_ and that couldn't be further from the truth. Highly successful people give tremendous attention to the most important activities daily and then do _fairly_ well with the rest. Remember from the earlier chapters: it's key to have focused attention on your \"3 Most Important\" and \"1 Must.\" It's definitely a different mindset than most people have, but once you try it, you'll discover something that will give you all the motivation you need: when you give your best effort to your top priorities, the success that comes to you will be deserved.\n\nThat may sound simple, but it's really very profound. When you define success by your effort, anything is truly achievable. And when you consistently work toward your goals\u2014and honestly evaluate that effort\u2014you will begin to deserve the success that comes. When it does, you will feel a tremendous sense of validation that doesn't just come when you make your numbers or achieve certain statistics. You will own that success, and it will become a part of your foundation. You will finally have the ability to control the scoreboard. It won't be something that just happens to you. It will be a part of who you are.\n\n* * *\n\nWhen you define success by your effort, anything is truly achievable. And when you consistently work toward your goals\u2014and honestly evaluate that effort\u2014you deserve the success that comes.\n\n* * *\n\nUSING THE SUCCESS LOG FOR SELF-EVALUATION\n\nSure, it all sounds great. But how do you actually _do_ it? How do you build a system of evaluation that not only tells you how you're doing but actually helps you build on the effort you need to be successful?\n\nIt starts with establishing the right kinds of goals.\n\nThere's no question that results are the main driver of almost everything we do or see in modern society. We're all constantly being evaluated on job performance, earnings, looks\u2014even the type of car we drive. It isn't a big mystery, then, that most people have developed a very strong focus on \"results thinking.\" Process goes out the window, and results are all we think about.\n\nBut when the focus is only on results, you aren't necessarily building the actual skills you need to be successful. You aren't really learning the _whys_ and _hows_ that produce those results, which makes it hard for you to pull yourself out of a slump\u2014and adds pressure you don't need.\n\nImagine that a sales agent in the middle of a client meeting begins thinking to herself, \"I have to close this deal if I am going to meet my quota.\" She wouldn't have the mental bandwidth to focus on listening to the customer's needs, addressing his concerns, and selling the features of the most appropriate product.\n\nThe same holds true for a batter up at the plate. If he's in the box thinking about how he needs a hit to get his average over .300, or to kick in one of the performance clauses in his contract, he's probably going to have the bat on his shoulder as the ball blows by him.\n\nIn other words, focusing on results\u2014or the end product\u2014actually makes it _harder_ to produce those results, and makes any results you do produce take longer to achieve.\n\nAnd that's the paradox. A focus on results doesn't produce results. Reformatting your thinking to emphasize the _process_ is the only way to effectively set goals that will actually produce the results you want to see.\n\nWhat are the differences between product- or results-oriented goals and process-oriented goals? Product or results goals are the ones that can be measured on an income statement or seen in your job title. If you want to earn $1 million in commissions next year, that's a product or results goal. If you want to become a senior vice president by the second quarter of next year, that's a product or results goal. Write a novel? Another product goal.\n\n_Process_ goals, on the other hand, are the daily activities that _cause_ the desired results or product goal. These will typically be your \"3 Most Important \/ 1 Must\" commitments daily. Using the same analogies, a process goal for a person who wanted to earn a certain level of commissions or a job title would be to make a certain number of contacts with high-net-worth clients per day, or to shadow somebody in the comptroller's office one day a week to learn some new things. For an aspiring novelist, it would be setting aside two hours per day of quiet time for writing.\n\nThere is no magic formula for setting either kind of goal, as long as you follow a couple of core principles.\n\nFirst, both kinds of goals need to be completely measurable. What does that mean? In the literal sense, they have to be something you can track, not some gauzy judgment call. Whether you made a call or not is concrete. Whether you were happy or not in a given day? Not concrete, and not measurable\u2014at least for the purpose of this tool.\n\nAthletes tend to get measured annually by their wins, their home runs per year, or the prize money they're awarded. Corporations tend to break goals into quarterly chunks\u2014earnings reports, performance evaluations, and so on. All of those are good examples of measurable results-oriented goals. A measurable process-oriented goal would be, for example, an athlete choosing to spend fifteen minutes per day on film study or a successful CEO spending thirty minutes researching development opportunities in emerging markets.\n\nSecond, it is of utmost importance that you choose wisely when it comes to both product and process goals. Product goals need to be realistically high\u2014not so out of sight that you can miss and have a viable excuse for failing (\"It was impossible anyway . . . \"). And process goals need to be completely within your control\u2014something you have the ability to do every day.\n\nSetting goals too high and hoping to \"get close\" is one of the most damaging things you can do to your performance. It gets you in the habit of losing. Set your process goals to a point where you can hit them daily, and you build confidence and your ability to \"win\" mental commitments in the future. It's not about what you will do on your best days but, rather, what you will be sure to do even on your worst day.\n\n* * *\n\nSetting goals too high and hoping to \"get close\" is one of the most damaging things you can do to your performance.\n\n* * *\n\nThe third core principle is where you have to learn to be abnormal. Normal people focus almost completely on product goals. And product goals are fine for spectators or stock pickers. But when you're talking about your own goals, it is the process goals that need to be on the forefront of your mind and at the top of your priority list each day. You need to be tracking process goals at least weekly, and preferably daily.\n\nThe highest performers learn to devote much more focus\u201485 percent, at least\u2014to process goals, and they evaluate themselves on how they do on that scale. The product goal\u2014making a certain number or getting a certain title\u2014is the destination. The process goal is how you get there.\n\nIn 2006, Jason started working with one of the St. Louis Cardinals players during spring training. The player was coming off an especially bad season the year before, and he told Jason that he felt like he played very \"tight.\"\n\nJason asked the player what he thought about when he stepped in the batter's box, and the answer was astounding:\n\nI'm looking at the Jumbotron and I can't believe what it says I'm hitting\u2014.253 or something. I start thinking to myself that this isn't going to work\u2014and that's when it gets in my head. I can actually feel myself start to tighten and press. It's so hard to stay in control of your emotions and thoughts when you have everybody watching and you're under the microscope.\n\nJason helped the player start focusing on his preparation and performance process instead of the external results of each at-bat.\n\nAt the end of the season, the Cardinals had clinched a play-off spot, and the player had improved his average more than 30 points. He had been a key member of the team all season. This is what he said during his \"exit interview\":\n\nI basically had to say \"screw it\" when it comes to results. I made a point to judge myself on the process. Every day, I followed my routine. Every cut. Every ground ball. No matter what the result, I made myself emphasize the process, and I held myself accountable for doing the work. I stopped looking at results and forced myself to think \"process\" was the win. In the end, the results speak for themselves. Focusing on process flat-out works.\n\nAnother client came to us after a successful career as a television broadcaster. She was transitioning into financial advising, and she wanted to make sure she was putting strong building blocks in place at the beginning of her career.\n\nOne of the first habits we helped her establish was to be relentless in attacking her two main process goals every day\u2014contacting two high-net-worth investors, and completing her Success Log evaluations. (We'll describe this tool and supply a log form in a few pages.)\n\nFocusing on the process instead of the results you're getting is a challenging transition for most people to make when they go from a salary job to one that relies on commission for compensation\u2014and our client wasn't immune to that. Early on, there were nights when she went to bed wondering if she was going to be successful at this new career, and how she was going to pay her bills in the short term.\n\nBut she forced herself to control her thoughts, to the point where all she would allow herself to focus on was her process goals for the day and the game plan for tomorrow. Did she nail her process goals? And what was she going to do in order to do it again the next day?\n\nAfter a year on the job, she came back to see us, and she said that the process orientation habit she learned was like a life vest keeping her from drowning in discouragement. As the years went by, our client kept to the same simple strategy. Over time, some of the process goals changed and grew, depending on where she was in her career, but the intention always stayed the same.\n\nNow, she is one of the top-grossing advisors in her office, and she's using her process goals for a completely different function. Focusing intensely on the process allows her to \"ignore\" the great sales results she's producing and keeps her humble and hungry. She isn't getting distracted by success; rather, she attacks her process every day to continue her winning ways. She's on track to become one of the few female million-dollar producers in her firm, and she attributes it to what she calls the secret of success\u2014process orientation.\n\nThe people in these two examples were extremely high achievers, and it still felt very strange to them to revise how they prioritized their goals. There's no question it will feel foreign to you, too. But it's the proven way to get you to where you want to go. And it all begins with effective evaluation. We have combined a series of evaluation questions into what we call Success Logs to help jump-start consistent improvement. They have been proven to help people win World Series titles, national championships, and Olympic gold medals and to significantly increase business production year over year.\n\nThe Success Log provided below will literally train your brain to focus more on your strengths, your effort, and your process. It prompts you to think about your process each day and to set goals for the next day accordingly. Take a look at the log on page 99, and then read on to learn how to use it.\n\nGIVE CREDIT WHERE IT'S DUE\n\nAs nice as it would be, life isn't like kindergarten. You don't get a sticker from a generous teacher for drawing an unrecognizable lump with your crayons on a piece of construction paper. All of us are getting judged every day. It's a part of life in the business world. It's something you're almost certainly doing yourself already, as are your peers, supervisors, and competitors.\n\nBut one aspect of that kindergarten star system still works the same way it did when you were five years old: rewards encourage you to keep doing your best. But your kindergarten teacher isn't around anymore, so you need to learn to recognize your own successes.\n\nSUCCESS LOG\n\n_Knowing something does nothing . . . doing something does. . . ._\n\nName: __________________________ Date: ___________\n\nWhat did I do well in the past 24 hours?\n\nWhat is one thing I want to improve in the next 24 hours?\n\nWhat is one thing I can do differently to help make the above-mentioned improvement?\n\nHow did I do today with my \"3 Most Important \/ 1 Must\"?\n\n1 2 3 4 5 6 7 8 9 10\n\nTeachers give out stars because it makes children feel good about an accomplishment and motivates them to work toward the next task. One of the biggest problems successful people have when they evaluate themselves is that they focus only on the negative. It's something known as \"problem-centric thinking.\" People have an innate tendency to obsess over the things they aren't doing well instead of giving themselves credit for the things they are doing well. From there, the issue is compounded because of expectancy theory: whatever you focus on expands.\n\nIn essence, it is totally normal to focus on how you are screwing up; unfortunately, by doing so, you make it more likely that you will screw up even more in the future.\n\n* * *\n\nIn essence, it is totally normal to focus on how you are screwing up; unfortunately, by doing so, you make it more likely that you will screw up even more in the future.\n\n* * *\n\nMany successful people often leave out the part of the evaluation that recognizes the good things they've done. They immediately go to the list of things they aren't happy about and hammer away at those. Instead, we want you to build an evaluation ritual that takes advantage of expectancy theory by increasing confidence and performance. This is where the Success Log comes in: it forces you to focus on what you have done well, asks you to identify one thing you want to improve, and prompts you to pick one thing you can do to improve in that one area. In addition, it trains you to form the habit of evaluating your _effort_ (on a 1\u201310 scale) on your most important tasks daily rather than your _results._ Evaluating yourself through the positive lens builds self-confidence and promotes action. Remember, whatever you focus on _expands._\n\nThat's it. It isn't a laundry list of should-haves and might-haves. The whole process should take you no more than three minutes. Do it during the day\u2014preferably at the same time each day\u2014and it will put you in the best frame of mind to attack the upcoming twenty-four hours. You're building what we call a \"performance mentality.\"\n\nWithout the log, your self-evaluations can easily go awry. If you're living in the perfectionist mentality, you might have a tough day at the office, and then sit down to try to analyze what went wrong. There's a real potential to beat yourself up there. You'll not only think about the things you did wrong, but question yourself about how you could have possibly made those kinds of mistakes. The problem with that strategy is that it can put you into a downward spiral. Your focus goes to your faults and the mistakes that led to them, instead of being focused on improvements and solutions.\n\nBy identifying your \"done-wells\" for the day with the Success Log, you're establishing a much more balanced scorecard. Another great way to keep track is to have a scratch pad on your desk and simply make a note each time during the course of the day that you're happy with something you've accomplished.\n\nIt can be something substantial, like closing a deal, but it certainly doesn't have to be. It can be as simple as reminding yourself after a call that you asked for the referral and even though you didn't get the introduction, \"asking is winning.\" You want to use enough second-level detail in your description that your Success Log gives you a specific picture of what you did right: reinforcing the positive action will influence your confidence level in the future. \"The call went great\" is not as useful to know as what specifically went well on the call: \"On the call with Roger, I listened well and used a great analogy to help him understand.\"\n\nEvaluating what you have done well sets a foundation of mental strength that you will eventually be able to build on. You're setting the building blocks for mental toughness. Building your foundation of mental toughness is just like building a house. If the foundation is weak, the home will crack at the first sign of adversity. If it's strong, the house can weather any storm.\n\nOnce you've identified three \"done-wells,\" it's time to pick the one thing you want to improve. Again, you're going to write this down with that second-level detail. The sentence you write shouldn't be designed to bring you down, or be overcritical; it should be an affirmative statement of what you want to do. This isn't about focusing on your screw-ups, but about identifying what you want to do better. For example, you might tell yourself that tomorrow you want to do a better job of making your ten proactive contacts and not stopping until all ten are completed. Then you want to spend a moment identifying what action step you can take to move you in the direction of the desired improvement. The rule might be that you don't let yourself check emails until all ten contacts are made.\n\nActive and positive statements are much more productive than passive and negative ones.\n\n* * *\n\nActive and positive statements are much more productive than passive and negative ones.\n\n* * *\n\nThe last thing you want to write down in your Success Log for the day is a 1\u201310 rating on how well you did with your three most important scheduled items. Start forming the habit of evaluating your effort each day toward the completion of your priorities. Your mind will actually become trained to prioritize what is most important.\n\nOne of Tom's favorite stories about the time he spent with his friend and mentor John Wooden is about how Wooden would evaluate his players during a game. It had nothing to do with the eventual score of the game, or the individual statistics any single player put up. Coach Wooden would watch to see how each player made his cuts from position to position on the court as UCLA ran its offense. If the players were making quick, straight-line cuts, they were doing their job. If they got lazy and made more banana-shaped cuts, they weren't.\n\n\"You mean to tell me the greatest coach of all time is watching to see if his players are running banana patterns and doesn't even care about the score?\" Tom asked in disbelief.\n\nCoach Wooden responded that winning was certainly important\u2014and he knew that making sure his players were cutting properly was the best way to control the outcome. The ability to make the disciplined and correct cut time after time is purely a question of effort. Coach Wooden figured that if his players were winning on the effort front, the results would take care of themselves.\n\nThey did. Over sixteen seasons, Coach Wooden's UCLA teams went 620\u2013147. They won ten national championships\u2014including seven in a row\u2014and produced twelve consensus All-Americans. Four of those championship teams had undefeated seasons, and the Bruins won an unprecedented seventy-five games in a row between 1971 and 1973.\n\nTURNING EVALUATION INTO ROUTINE\n\nAs you get better and better at the evaluation process, you can adapt it to give yourself a very useful running \"dashboard\" of how you're doing in a given day. You'll be able to perform \"mini-evaluations\" at regular intervals and be able to take quick action to get yourself back on track if you encounter a problem.\n\nBack when Tom was working as a financial advisor himself, he adapted some of his coaching habits to his new life working at a desk. He built out a daily chart of the calls and tasks he needed to accomplish that was very similar in execution to the practice and game plans he had devised as a basketball coach. On that matrix chart, he included the list of clients he needed to speak to that day along with the topics he needed to address. As he worked his way through the list, he would check off the tasks as they were completed.\n\nBut the matrix morphed into much more than a task manager. It became a mini-evaluation instrument. Every time Tom moved from his seat\u2014to get a drink of water, walk down the hall to talk to somebody, anything\u2014he would pause first, look at the chart, and ask himself three quick questions: What have I been doing well? What is one skill I need to improve? What is one thing I can do differently to make the improvement?\n\nTom would then mark his chart with the improvement goals in the places where he needed it, so that when he got to that step he was primed to do well. Much like Coach Wooden, Tom wasn't allowing himself to assess success with results\u2014whether or not he made the sale. By evaluating with a performance focus, Tom was forcing his mind to emphasize what he could control\u2014his process and effort\u2014and he was doing it through a positive lens rather than allowing his mind to emphasize mistakes.\n\nWhen done correctly like this, the evaluation process forces growth. The mere act of effective evaluation causes improvement. The evaluation element blends directly into the action phase, so that the two work as alternate footsteps in the same walking pattern. You're evaluating, adjusting, and taking action in real time\u2014when you can actually use the information you're gathering\u2014and you can get back on track quickly if you're off course.\n\n* * *\n\nThe mere act of effective evaluation causes improvement.\n\n* * *\n\nThat's something that a vast majority of people\u2014even successful people\u2014never accomplish. They're measuring the wrong output\u2014results instead of effort\u2014or hammering away at the negative instead of reinforcing the positive.\n\nThat's not evaluation. It's punishment.\n\n* * *\n\nThey're measuring the wrong output\u2014results instead of effort\u2014or hammering away at the negative instead of reinforcing the positive. That's not evaluation. It's punishment.\n\n* * *\n\nOne financial advisor told us a story that has become fairly common since we've been running our training program. He was doing about $500,000 in business, but he had been stuck at that level of production for several years. After hearing about effort-driven evaluation during one of our talks, he mapped out a plan for himself\u2014along with some evaluation metrics.\n\nHis goal for the next year was to achieve $700,000 in gross production, and he decided that the three things he needed to do in a given day to make that happen were to Organize Tomorrow Today (Chapter 1), make twelve proactive client contacts, and complete the Mental Workout (which we'll get to in the next chapter). The advisor also decided that he was going to stop looking at the commission summary on his screen, and instead judge his success in a given day by how well he did on those three process goals. If he completed them, the day was a win. If he didn't, it was a loss.\n\nIn the midst of all this, the advisor went through a terrible personal tragedy, losing a family member and a close friend in a fire. Reeling, the advisor's first reaction was to push work to the back burner. But after a few days, he saw the three straightforward process goals as simple, concrete steps he could take in a world that was swirling around him. It gave him a sense of control\u2014and even escape\u2014to apply his focus to something other than the terrible event.\n\nOn some days, it was all he could do to push his way through the calls and the Mental Workout, and he would leave for home by noon. But as the days went on, he started to heal. The process goals and the act of evaluating himself on his effort offered some needed day-to-day stability.\n\nSix months later, at the end of the year, the advisor knew that he had been able to gain some consistency in how he approached his work day-to-day, and he knew that he had done well on his own self-imposed metrics. Still, the numbers blew him away. He had doubled his gross production, to just over $1 million. And even more incredibly, he had done it while _reducing_ the amount of stress he felt over work and the number of total hours he put in at the office. The increased success and reduced stress obviously didn't eliminate the sadness of the loss he experienced in his personal life, but they did help with the healing process.\n\nHe was finally in control of his success.\n\nCELEBRATE . . . AND FORGET\n\nWhen you do successfully evaluate yourself, you will inevitably start to see positive results\u2014and you have to make sure to celebrate those wins. That kind of positive reinforcement helps to change your behavior in the long run.\n\nThink about the most effective ways to discipline a child. You can rule by fear and intimidation, and threaten to punish the child if he or she does something wrong. Or, you can positively reinforce the behavior you _want_ to see, by using rewards and praise. Which kind of process do you think the child is going to embrace and follow happily? And which kind is going to need constant policing, and then fall apart when the policing goes away?\n\nRewarding and reinforcing the good habits makes for a more lasting change than negativity and punishment\u2014and adults need even more of that positive reinforcement than children do! We all are much more ingrained in our habits\u2014both good and bad\u2014than a five-year-old is, and we have a lot more freedom to make bad choices.\n\nThe reinforcement is more than just anecdotal. It's chemical.\n\nIn Norman Doidge's book _The Brain That Changes Itself,_ he describes the chemical reactions that take place in the brain when you receive a reward for a solid effort. The brain releases acetylcholine and norepinephrine, neurotransmitters that sharpen the mental map for performance and significantly increase motivation.\n\nRewarding yourself for great effort creates a positive cycle of improved performance. You become smarter about how to make improvements, and you are much more motivated to search for and find those methods of increased success.\n\nWe're not suggesting that you go out and buy yourself a $200 dinner to celebrate nailing your process goals two days in a row. The accomplishment needs to be meaningful, and the celebration should be relative to the size of the win. When you do get that win, recognize it and celebrate. Relax and get your mind off business for a while.\n\nA common example of an effective reward is that you allow yourself to take a half day on Fridays each week that you totally nail your \"3 Most Important \/ 1 Must\" lists. One of Tom's favorite strategies when he reaches a certain milestone is to go \"off the grid\" for thirty-six hours. He isn't reachable by cell phone during that time, and he refuses to talk about business. It's a time to spend with family and friends and decompress from the day-to-day business race.\n\nWhen you're able to disconnect for a day or two, you're fresher when you come back, and you're ready to pick up the tools again. Come Monday, you've \"forgotten\" your success, and you're ready to build it all over again. Remember, the equation for _lasting_ success is achieve, celebrate, forget . . .\n\n* * *\n\n**The Big Why:** Self-evaluation is arguably the most effective performance tool you can use\u2014when used correctly. Unfortunately, most people evaluate things they cannot ultimately control, thus causing negativity, discouragement, and a lack of focus on priority activities.\n\n**The Inversion Test:** The mere act of evaluation, if done properly, causes improvement. Failing to evaluate promotes failure.\n\n**Act Now:** Take sixty seconds now and write down on paper three things you have done well over the past twenty-four hours. Remember, anything that promotes personal and\/or professional health (even one inch of improvement!) qualifies as a \"done-well.\"\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nPRO ATHLETE\n\n3 Done-Wells\n\n1. Spent 15 minutes on FaceTime with wife and kids while on the road.\n\n2. Did a nice job of attacking 50\/50 pucks . . . won 2 out of 3.\n\n3. Only drank 2 beers after the game.\n\nFINANCIAL ADVISOR\n\n3 Done-Wells\n\n1. Started the day by calling 2 high-net-worth clients.\n\n2. Asked one of my high-net-worth clients for a referral.\n\n3. Didn't get the referral, but reminded myself that \"asking is winning.\"\n\nINSURANCE EXECUTIVE\n\n3 Done-Wells\n\n1. Finished all 3 portfolio reviews.\n\n2. Sent my wife a nice text letting her know I was thinking of her.\n\n3. After an argument with my son, followed up 10 minutes later and apologized for yelling and told him that no matter what, I will always love him more than he will ever know\u2014he smiled when he heard that.\n\n* * *\n\n* * *\n\nLEARN HOW TO TALK TO YOURSELF\n\nTalk to marathon runners, and most of them will tell you the same story about the stretch between mile 21 and mile 26.\n\nThat's where the monsters lurk.\n\nWhen you get to mile 21, it isn't a question of whether you can physically survive the last stretch. You can. It's a question of whether the mental monsters will find you and convince you otherwise.\n\nKatie Sutton felt like one of those monsters was riding on her back at mile 24. She had come out fast and run sub-seven-minute miles for the first seventeen miles\u2014a terrific pace. She told herself that it was a perfect day, and her body felt terrific. All she had to do was bring it home, and she'd shatter her personal best.\n\nBut when she got to mile 24, she felt like she wouldn't be able to run another step. Her lungs were searing with pain, and her legs throbbed with every step. If she was going to make it to the end, she would have to get her mind right and confront the monster that had tightened its grip on her, whispering _you never should have gone out that fast,_ and _it hurts too much._ . . . _You should stop._\n\nKatie was no stranger to pain. A former competitive distance runner for Texas Christian University (TCU), she had been training for her first marathon after college when she'd started experiencing some strange physical symptoms. She went from running eighty miles per week to barely being able to complete a mile before becoming completely exhausted. After a grueling round of tests, Katie's doctor delivered the terrible news. Katie had breast cancer. But fourteen rounds of chemotherapy, thirty-two sessions of radiation, and two surgeries later, she was pronounced cancer-free.\n\nThe cancer was gone, but one of the remnants from the disease was the leftover negative chatter in her head. Katie got back into training, but when she started running the longer distances, those thought monsters would swirl\u2014 _You're never going to be the same runner again._ . . . _Why bother?_\n\nOver time, Katie beat back the monsters with a mental training routine like the one you're going to learn in this chapter. She learned a mental ritual to help her develop her mental toughness and replace the negative thoughts with thoughts emphasizing desire over fear and strength over weakness\u2014 _Today I am strong. Today I am healthy. Today I am a beast._ And she gained conviction about conquering the next challenge through visualization. During what we call her \"Mental Workout,\" Katie used segments of visualization to intensely focus on the visceral feeling of success in her next performance\u2014especially pushing hard through the most difficult part of the run.\n\nBy channeling the power of her mind this way, she set herself up to defeat the negative thoughts before they even came to her. Instead of having to beat them back and silence them, she established a wall that was difficult for them to penetrate in the first place. She went on the attack, instead of playing defense.\n\nIn that brutal moment of mile 24, her only goal was to finish those last three miles. Her mind wanted her to pay attention to the fire in her lungs and the agony in her legs, but Katie knew that the more she thought about the pain, the more the pain would grow.\n\nSo she defaulted back to her identity statement: _Today I am strong. Today I am healthy. Today I am a beast._\n\nShe started repeating it, over and over. Katie pushed through to mile 25, where she encountered a steep hill\u2014designed to weed out the weak at the end of the race. Using the visualization technique she had learned from the Mental Workout, she pictured herself charging up the hill and picking up speed as she made it to the crest. As she pictured defeating the hill, she realized that her mind had wandered for a few minutes, and she was already on the downslope, gaining speed.\n\nIn the last mile, Katie's speed was back. The pain in her legs was a dull memory, and her lungs opened up into the runner's high she loved so much. The monsters were gone, and she crossed the finish line in her personal best time.\n\nShe didn't conquer the marathon with her legs, or with her physical training. She beat it with her mind, and by learning how to talk to herself.\n\nThe concept of \"self-talk\" is an old and established one in psychology. Simply put, it's the inner monologue you have with yourself inside your own head about who you are, what you believe, and how you feel about what you're doing.\n\nUnfortunately, it is completely normal for your inner dialogue to have a negative slant. You must remember, beating yourself up verbally often does more damage than physically harming yourself.\n\n* * *\n\nYou must remember, beating yourself up verbally often does more damage than physically harming yourself.\n\n* * *\n\nSelf-talk is the voice in your head driving you to put in the extra time on an important project at work. When you're worn out at the end of the day and you want to pick up and finish tomorrow, it's self-talk that rallies you to keep going. Conversely, if you've decided you're going to go to the gym three days a week, and today is gym day, self-talk is the voice in your head you're arguing with in bed as you wrestle over whether or not you're going to get up and go.\n\nAt a deeper level, your self-talk represents your _self-image_ \u2014the way you see yourself. Ironically enough, Dr. Maxwell Maltz's _Psycho-Cybernetics_ \u2014the book that produced so much misunderstanding about the formation of habits\u2014was the same one to first touch on the very important idea of self-image, and how it governs a lot of what people are ultimately able to accomplish.\n\nMaltz's theory about self-image is a simple one. He said that a person will not be able to consistently overperform or underperform the self-image he or she has. In other words, if you fundamentally believe you're an average performer (or a terrific one, or a terrible one), you won't consistently be able to do a lot better or a lot worse than that baseline self-assessment.\n\nSince self-image is determined by what you consistently say to yourself about yourself, you have the power to direct your self-image by directing your self-talk.\n\n* * *\n\nSince self-image is determined by what you consistently say to yourself about yourself, you have the power to direct your self-image by directing your self-talk.\n\n* * *\n\nWe'll say that again, because it is very important: By directing your self-talk, you can direct your self-image.\n\nBy building a self-image that represents who you truly are and want to be, you can essentially program yourself to be ready for success. To do it, you need to master two tasks. First, you have to become aware of what you're saying to yourself. Then, you have to commit to stop giving yourself permission to use negative self-talk. Berating yourself\u2014even in a joking manner\u2014doesn't help in any way. It damages your self-image, and we want you to stop doing it, right now and from now on.\n\nTHE PCT TRAP\n\nMany people believe that they're at least somewhat at the mercy of their thoughts. Doubt and negativity just pop in from time to time, and sometimes you become obsessed with the problems on your plate. And, they say, there's nothing you can really do except try to ignore those thoughts and wait for them to pass.\n\nBut if you think like most people, you will be like most people\u2014average. Mentally tough people know nothing could be further from the truth. If you work on controlling your thoughts, you'll get better at it.\n\nYes, the human mind's \"default\" setting is to focus on the problems or weaknesses. The problem with focusing on the problem, though\u2014which is called \"problem-centric thought,\" or PCT\u2014is that most people do it at the expense of considering a solution or personal strengths.\n\nIn fact, lots of people are being _trained_ into problem-centric thought by mental health professionals or \"self-help\" guides because of the misguided (but well-meaning) idea that talking about a problem is akin to making your problems go away.\n\nUnfortunately, there's no empirical evidence to show that idea actually works. Just talking about a problem\u2014and moving your problem-centric thinking to the forefront\u2014doesn't do anything to solve that problem. Actually, it usually just makes the problem grow in size, thanks to something called \"expectancy theory.\"\n\nIt goes like this: When you focus on something, it literally occupies the forefront of your mind. Other thoughts and ideas are pushed to the side. As that thought goes, so do the feelings and behaviors that follow. That which you focus on expands. Focusing on the negative is essentially like fertilizing the weeds in your yard.\n\n* * *\n\nThat which you focus on expands. Focusing on the negative is essentially like fertilizing the weeds in your yard.\n\n* * *\n\nIt's actually the flip side of channel capacity on full display. You're focused intently on one thing, and your mind is using all of its horsepower on that one thing. In essence, what you're focusing on grows larger.\n\nSo, if you think mostly about your problems, and place all of your mental focus on them, you're growing them larger in mental terms. They soon are occupying much more mental and emotional space than they normally would or should. That's when you start to lose perspective and run the risk of making decisions out of fear or even panic. It's the equivalent of trying to win a football game without ever putting your offense on the field.\n\nWhat _does_ enhance self-image? Learning to talk to yourself about what you do well and how you want to improve. Concentrating on solutions instead of problems, and bolstering that focus on the positive with self-talk and visualizations. In the words of Dr. Don Miguel Ruiz, bestselling author of the book _The Four Agreements,_ \"the human mind is like fertile ground where seeds are continually being planted.\" Whatever seeds you plant in your mind are the ones that will grow\u2014so use this knowledge to plant seeds for the things you want to achieve.\n\n* * *\n\nWhat _does_ enhance self-image? Learning to talk to yourself about what you do well and how you want to improve.\n\n* * *\n\nThink about the last time you ran into a particularly thorny problem at work\u2014something you worked on that was in the middle of not turning out the way you hoped. How did you talk to yourself before, during, and after that situation?\n\nIs self-talk something you've ever consciously considered?\n\nAs we've been saying, the mind is more powerful than most people even comprehend. It would be a tragedy to waste that power by literally polluting it with negative and self-limiting thinking.\n\nBUILDING A MENTAL PICTURE\n\nVisualization has become a kind of catchword in sports\u2014something you hear many athletes talk about in their post-game interviews.\n\nIt certainly makes sense when you're talking about a complicated physical process, like hitting a golf ball or a fastball. The player gets a huge benefit by actually \"pre-seeing\" himself or herself going through the act of making the big shot or kicking the game-winning field goal, complete with the surrounding scene and the emotions that come with doing it successfully.\n\nThe act of visualizing the action before it happens gets the mind and body prepared to actually do it in real life when the time comes. It is priming the mental and physical pump. Players who visualize their game are calmer, better prepared, and much more likely to succeed in high-pressure situations.\n\n* * *\n\nThe act of visualizing the action before it happens gets the mind and body prepared to actually do it in real life when the time comes. It is priming the mental and physical pump.\n\n* * *\n\nVisualization is a technique that you can use to help yourself even if your job never takes you within fifty miles of a batter's box or a tee at a PGA Tour event. The act of pre-seeing the important events in your day-to-day life in an office or at home is just as valuable as it would be if you were doing it as a professional athlete. The exact same principles hold true. By seeing yourself doing the things you'll soon have to do for real, you're getting your mind and body prepared early. There will be fewer emotional surprises when you get to the real event, and you will feel infinitely more prepared.\n\nFor example, Tom coaches a prominent Fortune 500 executive who received a big promotion at his firm. With that promotion came much more public responsibility\u2014giving talks to shareholders and presentations to other large groups. The executive was terrific at the business of running a division and had a very strong technical background, but public speaking was something he had never enjoyed, and he really hadn't had to do it before as part of his work.\n\nSo Tom worked with him using the same process he had used with his basketball team years ago. They worked together to develop the executive's visualizing skill. First, they went through the next presentation the executive was scheduled to give. Tom asked him to imagine himself being introduced, and to focus on controlling his breathing. Then he imagined himself at the podium giving the beginning of the speech\u2014not watching himself, but looking out at his audience, as if he were really there. In his mind he went through the main point of the presentation, still at the podium, and eventually visualized delivering the actual finish of the speech\u2014how he was going to wrap it up.\n\nThe executive went through this process several times before the presentation, and when he walked up to the podium at the event itself, he felt like he was in familiar territory. He executed the steps just as he had been visualizing them over the past week\u2014breathing during the introduction, nailing the opening, explaining the main point, and then sailing along to the finish.\n\nTHE MENTAL WORKOUT\n\nAs we said in the introduction, when Jason had one chance to convince the St. Louis Cardinals baseball team that he could bring value to the clubhouse, he picked the Mental Workout as the piece to share with the team in his ten-minute window. It's that important, and it can produce amazing results.\n\nJason designed the version we teach to world-class professional athletes and Olympians. It is designed to put them in an ideal mental state for competition and to integrate their mental performance with their physical training.\n\nMost of us mere mortals don't have such a strong physical component in our day-to-day jobs, so the version we're going to teach you here was designed specifically for nonathletes, and is geared specifically toward mental performance and consistency in the business world. Done properly, it will take you about a hundred seconds to complete\u2014which means that if you have time to brush your teeth, you have time to strengthen your mind. It has five basic steps:\n\n1. Centering breath (breathe in for six seconds, hold for two, exhale for seven)\n\n2. Identity statement (personally tailored positive self-talk)\n\n3. Personal highlight reel (visualization of past and future success)\n\n4. Identity statement (personally tailored positive self-talk)\n\n5. Centering breath (breathe in for six seconds, hold for two, exhale for seven)\n\nWe'll go over each step in more detail below, and the steps are summarized again in the accompanying table. Your mind is a muscle just like your bicep. If you want your bicep to become stronger, you must complete bicep curls on a regular basis. The same is true for your mind. If you want to become mentally tough, you must complete _mental_ workouts consistently.\n\nMuscle deterioration begins within seventy-two hours of your last workout. Just as this is the case with your bicep, it also holds true for your brain. The goal should be to never let two days go by without some type of physical activity, nor should you go two days without completing a mental workout.\n\n_Centering Breath_\n\nTo start your Mental Workout, you will give yourself some oxygen: breathe in for six seconds, hold that breath for two seconds, and then breathe out for seven seconds. When you modulate your breathing this way, you're controlling your state of arousal and corralling your body's natural response to stress.\n\nThe biological response to pressure is an elevated heart rate. Unfortunately, when your heart rate increases, your ability to think effectively decreases. A very powerful way to control heart rate is to do a centering breath. Breathing in for six seconds, holding for two seconds, and then exhaling for seven seconds gets air into your diaphragm and slows your heart rate, thus allowing your brain to operate optimally.\n\n_Identity Statement_\n\nOnce you complete the breathing, recite your personal identity statement to yourself. Marathon runner Katie's is an awesome example: _Today I am strong. Today I am healthy. Today I am a beast._ You're free to use hers, or you can come up with one of your own\u2014as long as it fits a few basic criteria. It needs to emphasize one of your positive qualities, and it has to pinpoint something you want to become.\n\nThe identity statement is a personal mantra that, when repeated over and over, will manifest itself into reality. Written in the present tense, an identity statement includes positive adjectives that describe the characteristics of the person you want to be and the level of success you want to achieve. When it comes to identity statements, let desire guide you. Don't worry as much about your current reality but, rather, focus on who you want to become. As the research on identity statements makes clear: the further from the truth, the more impactful.\n\nExamples:\n\n_I am full of positive energy, I make $1 million per year, and I am an awesome mother and wife._\n\n_I outwork the competition every day, I am the most effective salesperson in the country, and I experience true love as a husband and a father._\n\n_Think it, see it, become it. I improve every day and I am consistently excellent as a leader, executive, and parent._\n\nRepeating your identity statement in your Mental Workouts causes you to believe in yourself and in your ability to accomplish great things. It is a proactive approach to overcoming all the normal negative stuff that goes on in between a person's ears.\n\n_Personal Highlight Reel_\n\nNext, quietly visualize your own personal highlight reel for sixty seconds. See in your mind's eye three things you did well the previous day, and mentally rehearse the three most important things you need to do to in the upcoming twenty-four hours. In many ways, your mind works the same way a dominant Major League pitcher's does. He'll spend his sixty seconds visualizing himself hitting his exact spots with his slider and fastball, and following his first-inning game plan, while you'll picture things like giving a PowerPoint presentation at a sales meeting, or having a one-on-one with your supervisor that afternoon. The more specific you are in your visualization, the better. Just like the athlete, you are preparing yourself for success.\n\nVisualizing is one of the most powerful tools in the field of performance psychology. It is safe to say that a person cannot perform at his or her potential without consistently using visualization as a pre-performance technique. To get the most out of visualizing, pay attention to the following three guidelines.\n\n**Guideline 1: Use the first-person vantage point.** Visualizing from the first-person point of view means looking at the video through your own eyes, so you see the things you would actually see while performing the task or skill. If you know you have a sales meeting with a client over lunch, then visualize exactly what you will see, say, and feel while sitting in your lunch seat looking across the table at your client. Visualizing in this way will help make the mental image a three-dimensional experience that feels real enough to increase your confidence and skill most efficiently.\n\n**Guideline 2: Emotionally feel the way you want to feel.** The video you play in your head needs to capture the emotional experience you want to have. Why? Because through visualization, you create your reality, and reality involves emotions. When you allow negative emotions, such as anger, embarrassment, or doubt, to creep into your performances, you will not deliver the performance you need to succeed. One way to banish these emotions is to consciously replace them with productive, positive emotions during visualization. The goal should be to feel and experience confidence in your visualizations.\n\n**Guideline 3: Visualize at the desired speed.** Make sure to watch your mental clip in real speed\u2014the speed you want your performance to be.\n\nYou may be wondering how you can visualize, for example, the delivery of an eighty-second script, a three-minute sales call, or a thirty-minute presentation at real speed if you have only a thirty-second block within your Mental Workout. That's a good question. You'll need to pick the most important specific moments within those events to run through in your visualization clips (for example, thinking through exactly what you will say in the first ten seconds of the presentation). Many people visualize in generalities, not knowing that it is far less effective than visualizing the details. The key is to visualize specific moments of success. Doing so allows for the success to actually generalize out to other areas that may not have been visualized. An NFL running back might see himself having a successful first run of the game, for example. Including detail and specifics when visualizing that very first run will set him up for increased success in each run thereafter as well.\n\nOnce you have a basic comfort level with visualization\u2014and we mean just that, \"comfort level,\" not expertise\u2014you're ready to use one of the most powerful tools in your arsenal to improve your mental performance.\n\nThe rest of the Mental Workout is simple. After the highlight reel, once again repeat your identity statement. Then, bring yourself out of your focus-building concentration by completing another round of centered breathing\u2014in for six seconds, hold for two, and out for seven.\n\nNow, you're ready to focus and perform.\n\nPlenty of our clients in the worlds of both sports and business have given us strange looks when we first introduced the Mental Workout as something they should try. All we can tell you is that the clients who have integrated it into their daily routine have seen huge benefits from it\u2014something that shouldn't be so surprising.\n\nAfter all, you wouldn't expect to build a lot of physical strength or endurance without training your body. Why would you expect to improve your mind without some kind of workout for it? Completing Mental Workouts develops the mental toughness necessary to control your self-talk. Having the strength between the ears allows you to replace negative thinking with thoughts that emphasize the positive.\n\nWe've been using this strategy with our clients for years, and there is no doubt, Mental Workouts will help you develop the mental toughness you see with high-level performers. You will train yourself to become the \"go-to\" player who makes the game-winning shot. When it comes to Mental Workouts, you can't afford not to try it.\n\n* * *\n\n**The Big Why:** What you think\u2014or how you talk to yourself\u2014dictates how you feel and behave, which in turn dictates if you will succeed or fail.\n\n**The Inversion Test:** If you learn to talk to yourself in a positive way, you will find yourself consistently on the attack, thus speeding up the realization of success. If you continue speaking to yourself with negative tones, you will routinely play on the defensive\u2014reducing your chances of success and increasing the time you spend stuck at the wall of underperformance.\n\n**Act Now:** Take two minutes and attempt to complete one Mental Workout right now. Don't expect perfection or anything even close. Just try to work through the five steps.\n\n* * *\n\n* * *\n\nSome examples of 100-second Mental Workouts from our clients:\n\n* * *\n\nPRO ATHLETE\n\n**Step 1** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total).\n\n**Step 2** \u2014Identity statement: I am more mentally and physically prepared than the competition. I am a _dominant_ Major League pitcher (5 seconds).\n\n**Step 3** \u2014Personal highlight reel:\n\n\u2022 Remembering 3 \"done-wells\" from the previous day (30 seconds total)\n\nSpent 5 minutes reading before bed (10 seconds)\n\nMade great nutrition choices at breakfast and lunch (10 seconds)\n\nDiscussed concerns with Yadi (10 seconds)\n\n\u2022 Imagining 3 \"done-wells\" in the upcoming day (30 seconds total)\n\nSpending 20 minutes in video room and identifying 1 weakness for each hitter I will face (10 seconds)\n\nStarting strong, staying strong, finishing strong in pre-game warm-up (10 seconds)\n\nAttacking mentality in first 3 pitches in the first inning (10 seconds)\n\n**Step 4** \u2014Identity statement: I am more mentally and physically prepared than the competition. I am a _dominant_ Major League pitcher (5 seconds).\n\n**Step 5** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total).\n\nFINANCIAL ADVISOR\n\n**Step 1** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total).\n\n**Step 2** \u2014Identity statement: I am the happiest, healthiest, and best family man, friend, leader, and financial advisor in the world. I have a relentless solution focus and I always attack\u2014no excuse (5 seconds).\n\n**Step 3** \u2014Personal highlight reel:\n\n\u2022 Remembering 3 \"done-wells\" from the previous day (30 seconds total)\n\nExpressed my concerns to the Smiths about them not keeping enough emergency money in cash in their account (10 seconds)\n\nCompleted my Success Log before leaving the office (10 seconds)\n\nHad great energy with Susan and the kids at home last night\u2014played 2-inning game of kickball even though I was tired from the day (10 seconds)\n\n\u2022 Imagining 3 \"done-wells\" in the upcoming day (30 seconds total)\n\nCalling high-net-worth client first and asking 4 feedback questions without getting defensive (10 seconds)\n\nSeeing myself Organizing Tomorrow Today at precisely 3:00 p.m.\u2014writing on paper \"3 Most Important \/ 1 Must\" (10 seconds)\n\nLeaving the office by 5:00 p.m. to be home by 5:20 (10 seconds)\n\n**Step 4** \u2014Identity statement: I am the happiest, healthiest, and best family man, friend, leader, and financial advisor in the world. I have a relentless solution focus and I always attack\u2014no excuse (5 seconds).\n\n**Step 5** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total).\n\nSOFTWARE SALESMAN\n\n**Step 1** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total.\n\n**Step 2** \u2014Identity statement: I outwork the competition every day, and I am the most effective and precise presenter in the country (5 seconds).\n\n**Step 3** \u2014Personal highlight reel:\n\n\u2022 Remembering 3 \"done-wells\" from the previous day (30 seconds total)\n\nWas up by 5:00 a.m. and completed my workout by 5:45 (10 seconds)\n\nKept my focus in two client meetings and had excellent closes on both (10 seconds)\n\nForced myself to go \"lights out\" by 9:00 p.m. in preparation for the long day coming up (10 seconds)\n\n\u2022 Imagining 3 \"done-wells\" in the upcoming day (30 seconds total)\n\nGetting up by 5:00 a.m. and working out by 5:45 (10 seconds)\n\nStarting strong, staying strong, finishing strong on product call (10 seconds)\n\n**Step 4** \u2014Identity statement: I outwork the competition every day, and I am the most effective and precise presenter in the country (5 seconds).\n\n**Step 5** \u2014Centering breath (inhale for 6 seconds, hold for 2, exhale for 7, for 15 seconds total).\n\n* * *\n\n* * *\n\nLEARN HOW TO TALK WITH OTHERS\n\nSteve Jobs was unquestionably one of the true business visionaries of the modern age. He was the creative genius who cofounded Apple\u2014and spearheaded the development of the mouse-controlled personal computer, the iPod, the iPhone, the iPad, and the iTunes music store. Known as the \"father of the digital revolution,\" Jobs transformed the way people interact with technology on the most basic level. The powerful phone\/computer\/camera\/jukebox you hold in your hand today is a direct result of his tireless work in product design and development and consumer marketing.\n\nFor all of the work he did harnessing the design and technological advancements in Apple's products, Jobs's ultimate genius was in communicating exactly _why_ a breakthrough product like the iPhone was something everyone both wanted and needed. He was able to get across both the technological superiority and the basic simplicity of Apple's devices in a way that avoided hyperbolic tech-speak. He explained how products like the iPod and iPhone would transform the average person's life in truly seamless ways.\n\nInstead of intimidating customers with the prospect of having to learn how to use a complicated piece of hardware, Jobs communicated with them in a fundamentally different way. He explained\u2014in language anybody could understand\u2014how Apple's products could solve problems big and small. He introduced reasons to own an iPhone that hadn't even occurred to most of the people who ended up buying them.\n\nJobs did it in his highly anticipated annual presentations at Apple's new product launches. He walked around the stage casually in front of simple images on a large screen, speaking seemingly extemporaneously for forty-five minutes about the new products and how they fit into a user's life. He was famous for taking viewers down a simple path\u2014introducing a big problem that hadn't been solved by the market, and then introducing the simple, elegant, impeccably designed solution.\n\nBut it wasn't a matter of a highly charismatic, natural speaker getting up and wowing the crowd in an off-the-cuff way, the way Bill Clinton notably does. Jobs was intensely private\u2014and famously prickly in meetings with his team. He wasn't afraid to confront those who didn't live up to his standard. The natural, flowing presentations that launched so many iconic products were actually highly scripted, and they were the result of practice and repetition. The main objective was to make consumers wonder how they had ever lived without the latest iPhone or iPad.\n\nTo say his efforts were successful would be one of the biggest understatements of all time. When Jobs returned to Apple as CEO in 1996, the company was losing money and contemplating bankruptcy. In 2014, more than 200 million people were using iPhones, and Apple had a profit of $40 billion on sales of $182 billion\u2014making it the most profitable publicly traded company in the world.\n\nAs Jobs would say about his exhaustive design sessions for the iPhone\u2014preparation and communication makes the difference.\n\nJobs and Coach Wooden came from very different worlds, but on that point they would have agreed wholeheartedly. After Coach Wooden retired from UCLA in 1975, he spent most of his time close to his condo in Encino, California, attending games as a spectator and giving talks to different groups around Los Angeles. In 1989, Tom arranged for Coach Wooden to speak at the Edward Jones General Partners' Meeting. Before Coach's speech, the two got together, and they ended up talking basketball, as they often did. During the conversation about Wooden's famous high-post offense, Tom wanted to show that he could still handle the X's and O's like he did back in his own coaching days.\n\n\"I haven't coached for seven years, but I can still draw up that play,\" Tom said.\n\nCoach Wooden nodded, and said he'd like to see it. Tom took a piece of paper from his briefcase and diagrammed the high-post offense. When he was done, Coach Wooden took the paper and looked it over carefully.\n\n\"How long did you say you had been out of coaching?\" he asked.\n\n\"More than seven years,\" Tom said.\n\nWooden complimented him on his work, but then slid the paper back across the table. \"That's really pretty good, but you left out one of the most important points,\" he said.\n\nTom was floored. He looked at the paper again and retraced his steps. He was convinced he had diagrammed all of the necessary movements. After a minute or two of not seeing the missing component, Tom slid the paper and pen back to Coach Wooden.\n\n\"You left out a very important detail. When the high post catches the ball, the guard must take two steps toward the goal to set up the play,\" said Wooden, adding the mark to the paper. \"Tom, little details make big differences.\"\n\nIn that moment, it became even clearer to Tom how dynasties are created\u2014by communicating the details.\n\nFifteen years after he had coached his last game, Wooden was still emphasizing the subtle detail of the guard's two-step cut\u2014something that even the dozens of college and professional teams running his influential offense to this day haven't grasped.\n\nWatch virtually any NCAA game today, and you'll see Wooden's high-post offense being run, but you won't see the guard make that critical movement to sell it to the defense.\n\nTo consistently win at the highest level, you have to be able to communicate effectively, with details.\n\nWhat does the high-post offense have to do with communication skills?\n\nEverything.\n\nBasketball plays are like a language. They comprise a common set of information the people in a group use to communicate with each other. When everybody knows the play, they can move in concert. The group of five playing together can do much more than five individuals can working alone.\n\nLikewise, you can have the best, most useful information in the world, but if you can't share it effectively, what good is it? Apple's products could have been groundbreaking, but the company's fight in the marketplace would have been much harder if Jobs hadn't done such a masterful job of communicating them for the consumer at launch. It doesn't matter what kind of business you're in\u2014sales, technology, administrative, creative\u2014the ability to communicate with the people around you is a critical skill.\n\nIf you aren't a natural speaker or communicator, you've probably watched some of the speeches people like Dr. Martin Luther King Jr. and Winston Churchill were able to give and thought to yourself, _They're doing something I couldn't possibly ever learn._ In this, you are mistaken. You might not be able to copy their natural charisma, but you can certainly develop many of the same skills.\n\nStudies have repeatedly shown that a vast majority of the impact from a presentation comes from the style of delivery\u2014not just the words themselves. If you can learn to project self-confidence, you can become a terrific speaker. You don't have to be \"supernatural\" to connect with a client or move a team of five (or fifty, or five hundred) into action. You merely need to learn how to be confident when you communicate. And believe it or not, it is actually much easier than you might think.\n\nAll you need to do is understand some communication basics. Most people who struggle to talk to a group or give a basic presentation have a very basic problem: they are terrified. And being terrified keeps them from being prepared. For that matter, not being prepared can make giving a presentation much scarier than it should have been. The fear of failure can then become a self-fulfilling prophecy. It's a vicious cycle, with fear, lack of preparation, and low confidence intensifying one another.\n\nWhether you're talking to a group of a thousand or making your pitch to an audience of a single person, the idea of speaking in front of people can simply be nerve-wracking. Even if you have a handle on what you want to get across, the thought of potentially making a mistake or otherwise looking foolish can be almost paralyzing. But turning this vicious cycle around is essential. As we've said before, the things you focus on expand\u2014both good and bad. So, if you focus on how frightened you are to get up in front of people, or you think about all the mistakes you could conceivably make, what do you think will happen to those thoughts and concerns?\n\nIn this chapter, we will show you how to be prepared\u2014which then will cause you to _feel_ prepared and ready for what you're going to face. The good news is that with these techniques, you can turn the vicious cycle into a virtuous one: you can banish the fears with preparation and practice, and all that preparation and practice will lead to success and greater confidence. And the same rules work whether you are doing a large group presentation or planning a one-on-one meeting.\n\nPreparation comes in three easy steps:\n\n1. Write it.\n\n2. Slow it.\n\n3. Triangle it.\n\nWhat does that mean? We'll explain.\n\nSTEP 1. WRITE IT\n\nWhen Tom joined American Funds\u2014the third largest mutual fund management organization in the world\u2014in 1999, he was required to do what all other employees delivering the American Funds message must do\u2014attend a two-day speech class, where they gave dozens of presentations. Every presentation was videotaped and analyzed by communications consultants and other experts.\n\nIt was a grueling exercise, and the way most tried to \"beat\" the test was to use extremely detailed notes. They tried to fill every two-minute talk with as much information as possible. They figured, _There must be SOMETHING in there somebody could use. I'll just give it all to them._\n\nIt's totally natural to not want to come off as if you don't know your subject\u2014especially in a pressure situation. So the totally natural response is to talk more and talk faster. Which is exactly wrong. Unfortunately, the more you say, the less believable you often become.\n\n* * *\n\nUnfortunately, the more you say, the less believable you often become.\n\n* * *\n\nEven people in jobs that require tons of talking and presentations usually don't get this part right. Mutual fund companies have representatives called \"wholesalers\" whose job it is to present the different fund products to investment advisors and explain the benefits and features of the particular funds. A high-level advisor can be managing tens of millions of dollars for clients, and he or she usually prefers to work within a set range of funds he or she knows extremely well.\n\nSo there's a lot of competition for that valuable real estate among the wholesalers. But when you watch some wholesalers having a conversation with an advisor, you'd think the wholesaler was getting paid by the word. They speak to clients every day, yet they continue to blanket each client with a machine-gun volley of words. Unfortunately, the more you say, the harder you are to understand\u2014the harder you are to follow. Think about it: the longer you speak, the more you must begin commenting on the less important points of your presentation. Unfortunately, people have a tendency to remember what you say last.\n\nIt's worth repeating\u2014the more you say, the less believable you often become.\n\nWe've hit on the concept of channel capacity enough times by now that you should know what we're going to say next: there's a limit to what people can process in one sitting, and a limit to what they can comprehend within a string of sentences.\n\nThe best presenters are literally ruthless in organizing what they will say. They identify only the most important information that needs to be communicated, and they get rid of everything else. How does that work? They write a script. It's only after you get your thoughts organized to the point where you can go through and delete the less important parts that you'll start to feel ready to communicate.\n\nIt can certainly feel overwhelming to sit down and script out an entire presentation. But it's so worth it. By scripting your presentation, you are preparing, and by preparing, you will be banishing the fears you have about presenting. By preparing your message, you are also ensuring that you are saying only the most pertinent things\u2014the things you want the client to really focus on and hear. You will find that your clients will listen and respond so much better than when you were winging it.\n\nStart by mapping out the first five minutes, and then attack what you have written with the delete button.\n\nDon't think for a second it won't be hard. It's really tough, but deleting is one of the most important parts of communicating effectively. You need to follow the simple rule that each presentation should have only one main point and a maximum of three subpoints. Loading more information in than that is asking for trouble. By making this part of your presentation razor-sharp, you're getting your message across in the cleanest, most efficient way.\n\nOnce you've finished the first five minutes, transition to the closing five minutes. By _finalizing_ your talk, you're providing the destination for the rest of the presentation. Everything you say will be designed as a lead-in to bringing it home this way. Again, be very liberal with your \"delete\" button.\n\nWe've all sat in on brutally long, muddled presentations that wander from point to point. Either the presenter had a laundry list of information that he or she needed to get through, or there was a set amount of time on the clock that he or she needed to fill. There is no quicker way to get people to check out than to start droning on in a presentation that doesn't seem to be building to a main point or conclusion.\n\nTo drill this home, start thinking to yourself, _What if I only had five minutes total to speak? What would I say? What about three minutes? One minute?_\n\nPractice the one-minute version of your talk, and you'll know you're getting to the core of the material.\n\nOnce you have your first five minutes and last five minutes mapped out, it's time to fill in what goes between. Remember, your goal is to include only the most important information. The ever-popular TED talks shoot for succinct presentations that touch on only the most pertinent information. TED presenters are thought to be some of the greatest speakers on the planet.\n\nOnce you figure out what content you want to include, it's time to bring the presentation to life with stories. Anecdotes and stories are what the audience will most remember. Be sure your stories have details that connect the audience to your topic and also build your credibility. We try to follow the rule that there should be one story for each point or subpoint. When your stories are in, you are ready to move to the next step.\n\nSTEP 2. SLOW IT\n\nOnce you've produced your script, it's time to learn how to deliver it.\n\nBy far the most common mistake people make is to deliver presentations too fast. It is important to remember that you've been thinking about what you plan to say for a long time\u2014and, presumably, you're an expert on it. But everyone else isn't as far down the road as you are. This will likely be the first time they hear a lot of what you are trying to tell them, and they need time to digest it as you proceed.\n\nPace is the biggest indicator of confidence, and the best way to slow your pace is to purposefully use pauses. Every pause serves as an inverse catalyst. Your pace will be slower after each pause you use. Hearing your pace quicken should be your mental cue to take another pause. Don't be afraid to take three-second, five-second, and even seven-second pauses in between bits of information. If you don't slow down and simply pause between your thoughts, much of what you are saying is going to be lost. And the more information you cram into the intermittent thirty-second bursts of attention that the average listener can offer, the bigger the chance the person won't retain _any_ of what you're saying.\n\nIt isn't a race, and it isn't a contest to prove how much you know. You can do much more to show strength and confidence\u2014and attract your audience's attention\u2014by pausing than you can by actually speaking. And when you project that confidence outward, it's actually absorbed by the people receiving the message.\n\nIt's how the greatest speakers of all time have communicated with their audiences. The idea is to figure out what the goal really is for the meeting and to serve that goal in the cleanest, most efficient way possible.\n\nPlenty of coaches at the most elite level have strategic genius. But the best game plan in the world doesn't work if the players either can't follow what you're saying or have tuned you out.\n\nOne client of ours had been an extremely successful Olympic coach for several individual athletes, and had been hired by a large university to run its program in that sport. The coach was extremely knowledgeable, but had never worked either in an academic setting or with scholarship athletes in a team setting.\n\nWhen the team assembled for the first time in training before the next season, the new coach handed all the players giant notebooks detailing every aspect of their training, along with a complete schedule for everything they needed to do every day until the first game of the season. During each practice, the coach would call players out and quiz them on the information in the book\u2014and embarrass them if they stumbled on the answer.\n\nThere's no question that a few of the highest-achieving members of the team thrived on the discipline and hard-core guidance they got from the new coach. But the rank-and-file team members quickly got lost, and the overall team results weren't what the new coach\u2014or the administration\u2014expected.\n\nOur goal was to help the coach preserve the quality of the guidance he was giving to the players, but to manage the stream of information so that the players were accountable for what they needed to know, but reassured by the basic fairness of the level of expectation. The first step was to \"delete\" almost 70 percent of the original team notebook. The second step was to help the coach learn to communicate at a slow enough pace\u2014one fundamental idea per day, often covering the same fundamental day after day to ensure full understanding. Over time, that translated into a set of core, nonnegotiable principles that became the repeated focus of every practice.\n\nMost speakers and coaches are afraid of repetition, but the best communicators have learned to _use_ repetition.\n\n* * *\n\nMost speakers and coaches are afraid of repetition, but the best communicators have learned to _use_ repetition.\n\n* * *\n\nDozens of studies have shown that when you're trying to give people information that you want them to retain, offering more than two or three discrete things is a huge mistake. This basic human reality drives how we schedule every single seminar we do. We take people through one or two facets of training, offer some guidance in real-world scenarios, then break and give the attendees a chance to digest and practice the information before we move on to the next phase.\n\nYou can't expect to offer people half a dozen pieces of information and expect them to somehow synthesize it on the spot and make a well-informed decision about it. In fact, if you _do_ put your colleagues, clients, and potential clients in that position repeatedly, you're going to find you have far fewer of all three over time. Slow down and you will find yourself a whole lot more on target with your audience.\n\nIn addition, you will find that slowing down is actually much easier for you as a speaker. You will be able to think about what you are saying as you are saying it. Doing so will make it much easier to follow your script and maintain your confidence.\n\nSTEP 3. TRIANGLE IT\n\nThe whole goal of presenting in a group, either with clients or in your office setting with colleagues, is to deliver your message in a way that gets the receiver to take action.\n\nA basketball team runs its offense through a set of plays. When those plays get added to the playbook, how do you think the players on the team learn both the plays and how to move together on the floor? Does the coach hand out the playbook, explain the play one time, and hope for the best when game time comes? Or does the coach break down the play into components and explain each player's part, and then supervise as the players repeat the play again and again in practice, so that it becomes second nature?\n\nThe answer is obvious in a basketball setting. Why does it seem so much less obvious in the business world?\n\nA very effective method for practicing communication is something called the \"success triangle\": for the three days before your big presentation or meeting, spend three separate three-minute segments per day mentally rehearsing what you want to say and how you want to say it. We advise people to ritualize the triangle training by spending the three minutes each day just prior to each meal\u2014breakfast, lunch, and dinner.\n\nIf you have more time than three days before, by all means expand the practice into as many days as you can. But look at the success triangle and the three-day window as the ideal minimum requirements. If for some reason you have less time than that\u2014maybe you've been called on to fill in tomorrow unexpectedly\u2014don't sweat it. You can still get most of the benefit of preparation. Get in as many of the three-minute segments as you can, but be sure to give yourself at least sixty minutes of break time in between.\n\nA personal injury attorney client of ours was terrified about an upcoming case. Although our client was very competent and quite prepared, it was her first time presenting a case in front of a jury. For three days before the trial opened, she spent three minutes three times a day visualizing herself in the courtroom with the judge and jury, slowly going through the first minute of her opening argument word by word.\n\nWhen the trial started, the attorney was actually surprised by her confidence level and the smoothness with which she presented. Her client eventually won the ruling, and the attorney so impressed her supervisors that she got on the fast track to become only the second female partner in the firm.\n\nThis triangle technique is the same one that professional athletes use to visualize how they will perform in an event. You're just using it in a different setting. When you visualize the actual situation you'll be in, and rehearse the exact words you're going to say\u2014and how you want to feel when you're doing it\u2014you're preparing yourself for game day. You'll be ready for any surprises.\n\nIt's important to point out that there's a big difference between \"practicing\" and mentally rehearsing with the success triangle. Practice _can_ be useful, but not necessarily as effective as mentally rehearsing. If you're trying to develop a skill you don't have\u2014like learning how to hit a golf ball\u2014structured practice at a driving range is going to help you. But if you're a good player, basic practice is often little more than exercise. You're not simulating game conditions, or putting yourself in the same frame of mind you will be in when it's time to compete.\n\nAnybody can hit it long and straight on the practice tee, when you have an unlimited bucket of balls next to you and there's no penalty for missing. Mentally rehearsing with the success triangle is different: it puts you right there, in the situation. You're \"pre-creating\" the pressure of game day, so you can feel it, experience it, and be prepared for it.\n\n* * *\n\nMentally rehearsing with the success triangle is different. You're \"pre-creating\" the pressure of game day, so you can feel it, experience it, and be prepared for it.\n\n* * *\n\nThe research is overwhelming. Using a technique like the success triangle is at least seven times more effective than rote practice alone. In other words, visualizing a presentation, using the triangle, for nine minutes per day is the equivalent of standing in front of the mirror for an hour and reciting it word for word. It works, and it saves you time.\n\nGOING ONE ON ONE\n\nYou might be reading this and thinking to yourself, \"It's great to know, but in my role, I don't do much speaking in front of other people.\" Regardless of how much \"public speaking\" you're doing, you still have to interact with people on a one-to-one basis in the business world. The skills we're taking about help build confidence, and confidence is the single most important ingredient for communication success, whether you're talking to a group of 150 people or 1 person across the table from you in a sales meeting.\n\nAffirmations and positive thinking are great, and your confidence is going to really grow when you start using the concrete tools of preparation to sharpen the points you want to make. All of the same principles we've been talking about when it comes to \"public\" speaking still apply, but when you're in a closer setting, such as a one-on-one meeting or a small group, a few more factors come into play. You need to be able to stay calm, listen, and control the energy level in your voice.\n\nLet's take those one at a time.\n\nWhen it's just you and another person on a call, or you're sitting two feet from somebody and you're trying to get and keep that person's undivided attention, you have to be tuned in to a finer level of detail than when you are doing a presentation to a larger group.\n\nIn a presentation, you're controlling the show from start to finish, and repetition and preparation are your friend. You know the route you're going to take, and the words will be familiar territory. In a conversation, that's still true to a certain extent\u2014you need to know your stuff\u2014but you're also responding to what the other person will say. It's like tennis. You can have a good scouting report and know what the other person's game is like, but you still have to run down each shot and return it.\n\nBecause you're interacting back and forth, you need to be as tuned in to what's coming from the other person as you are comfortable with what you're delivering. To do that, you need to be able to stay calm. If you have a high-stakes conversation on your plate, you simply aren't going to perform as well if the running thought through your mind, as it's happening, is, \"Just don't screw this up.\"\n\nIt isn't any different from learning to perform a physical task under competitive pressure, like athletes do. The single best method of staying calm is to control your breathing by using the centering breath we have already talked about. In the sixty seconds before the beginning of a presentation or meeting, be sure to take at least one centering breath (inhale for six seconds, hold for two, exhale for seven). While the other person is talking during the conversation, find times to take additional centering breaths. You don't need to announce what you're doing, and believe it or not, your deep breathing will be undetected by your counterparts.\n\nIt might sound strange to talk about a relaxation technique like breathing when it comes to a conversation, but it really works, and it's an extremely valuable tool to know. Not only does it help you get your heart rate under control, but it allows you to become a much more effective listener. It redirects nervous energy into a more positive, productive place.\n\nAny salesman with a few months of experience on the same job can recite the benefits of a given product off the top of his or her head. _Knowing_ the information isn't the separating factor by itself. It's _knowing_ the information that way, but also having the ability to deliver the information in a manner that causes the other person in the conversation to take notice and to interact. The fact is, you're doing more to establish a true _conversation_ when you slow down your end of it. Roughly 20 percent of communicating is speaking, while 65 percent is listening. Yet, when it comes to communication, a vast majority of instruction is aimed at how to speak more effectively. Rarely are you prompted to learn to listen better. Most of us fall into this common communication trap at one time or another. We get so caught up in planning what we're going to say next that we don't pay full attention to what the other person is actually saying.\n\nThis situation will often lead to a choppy series of interruptions, where each person thinks what he or she has to say is more important than what the other is saying. If you are interrupting someone, there is no possible way you are truly listening. Interrupting sends the distinct message that you really only care about what you are saying. The opportunity for listening and learning is getting lost. By slowing down, you also allow yourself to be a better listener. You have more time to think about what you're saying, more time to say it correctly, and more time to listen to what the other person is saying.\n\nYour voice is also an important tool when you're in a one-on-one situation. When presenting to a large group, you have the advantage of practice\u2014and maybe even a microphone. You can get your message crafted and practice delivering it at the right speed with the right voice energy. When it's a more intimate setting, you're balancing getting your message across with being responsive to the other person. That causes many people to start to speed up. They talk faster and faster\u2014or louder or softer\u2014and by the end of the conversation, the voice energy they hoped for wasn't there.\n\nYou can improve your presentation by consciously evaluating your voice and the speed of your delivery. Instead of trying to set the record for the amount of material covered in one call or conversation, change your approach to value calmness, positive energy in your voice, and the act of leaving conversational room for the other person to evaluate each thing you're saying.\n\nIf you're a fast-paced person\u2014and even if you're more of a medium-paced talker\u2014it will definitely feel strange at first to slow down. But you will start to notice some amazing things when you do. The person on the other side of the conversation will begin to subconsciously attribute more authority to you and more weight to what you're saying. When you both present meaningful content and leave openings for thoughtful responses, you're putting yourself in special company.\n\nMaybe you work from an office and have little face-to-face contact with outside clients. Even if so, you're not off the hook, because the rules still apply to other kinds of communication. Think about the conversations you have over the phone. We realize that phone calls are becoming more and more scarce, thanks to the advent of text messaging. But personal connections still need to be made. And most people spend their half of the phone conversation waiting for the other person to stop talking so they can announce what they want to say. In addition, communication gets even more jumbled in a phone conversation than in face-to-face interactions, because on the phone you can't see the other person and react to his or her body language.\n\nThe advice is the same for phone conversations as for other kinds of communication: when you're on the phone, slow down at least 20 percent more than you think you need to. Leave some air between your sentences. Let the person on the other end catch up with you. Whether you're giving a presentation to a hundred people in a conference room, talking to a potential client on the phone, or having a meeting with three colleagues across the table, you have to accept that channel capacity is going to prevent you from cramming those other people full of information. You must slow down to be effective.\n\nHere's a very simple example that illustrates that point. A financial advisor who had been through several of our training programs called looking for some advice. He had a conversation scheduled for the next day with a client who had $10 million to invest\u2014a potential account that would have tremendously impacted his bottom line for the year. It was the biggest potential client he had ever had the chance to meet, and he was feeling very nervous.\n\n\"What do I say?\" he asked us. \"This guy could potentially hand me a check for $10 million. How do I open? What do I bring to this meeting?\"\n\nWe started with the basics of scripting out the two main reasons he was the best man for the job and then moved on to practicing it. He focused on of how to control breathing and heart rate, so that his nervousness wasn't written all over his face\u2014or his voice. Then, we talked about ways to inspire the potential client to engage in the conversation, instead of just listening to a prepared \"speech.\"\n\nAfter a minute or two of small talk to open the conversation, the advisor said to the potential client, \"I know your time is valuable, and there are some questions I need to ask you.\" He then paused for a few seconds. \"But before I ask them, I want to assure you that anything we talk about will remain strictly confidential.\" He then moved on to his two main points and closed with a very strong request for the prospect's business.\n\nIt might not seem like a big thing, but the advisor said that the moment he stressed confidentiality, the potential client and his attorney both relaxed and sat back in their chairs. As soon as the advisor saw them relax this way, his confidence level soared. He knew they had engaged in the discussion, and he let his preparation and knowledge take over from there.\n\nHe earned the business.\n\n* * *\n\n**The Big Why:** In the business world, a person's skill level is largely realized based on his or her ability to communicate in a confident, efficient manner.\n\n**The Inversion Test:** Great communicators will experience great success. Poor communicators will experience great failure.\n\n**Act Now:** Write on paper the first minute, word for word, of what you will say in the next presentation you will deliver.\n\n* * *\n\n* * *\n\n**Some examples from our clients:**\n\n* * *\n\nPRO ATHLETE\n\n**(pitcher talking to his catcher before playing an upcoming opponent that the team has historically had trouble with)**\n\nThe first thing I would like to say is, thank you. I don't think I tell you enough how much I appreciate all the work you put into making me look good. I know your job is somewhat thankless, but I want to be at least one person who thanks you often for your preparation and approach (three-second pause). Let's figure out how to finally beat these guys. I want your input. What is the one thing you most think I need to be focused on if we are going to be lights-out tomorrow night?\n\nFINANCIAL ADVISOR\n\n**(coffee meeting with a prospective client)**\n\nGood to see you Jim. Last week when I met with Tom and Susan they both had great things to say about you. I know you're busy, so with your permission, I'd like to skip the small talk and get to the heart of the matter. With that said, let me ask you a question, if I may (three-second pause). What is the single biggest thing you are hoping I can help with?\n\n_(This is where it becomes very important to listen. The thrust of the conversation from here forward must be centered on meeting Jim's one biggest need.)_\n\nPHYSICIAN\n\n**(speaking to a group of twenty to twenty-five attorneys, with the goal of having them give him referral business and using him as an expert witness in medical litigation)**\n\nLet me begin by asking all of you a question (three- second pause). If your mother or your father, or one of your children, was injured, and it was possible that the injury was life threatening, would you have extreme confidence in any one medical professional? Think about that for a second (three-second pause). I would like to cover with you, in the next fifteen minutes, the three reasons why, if, God forbid, a life-threatening injury did occur to a loved one, friend, or client, you should seriously consider thinking of me and my team as the resource for providing the highest level of care.\n\n* * *\n\n* * *\nCRUNCH TIME\n\nIn a basketball game between two evenly matched teams, the outcome often isn't decided until the last two minutes of the game. The team that's behind plays with a little desperation and makes a run, and it's up to the team that's ahead to handle the surge. Crunch time, when the game is close, is what defines champions\u2014the teams and players we remember long after they're done playing.\n\nIn terms of this book, we're now in crunch time. The coach has called a time out. There's two minutes to play\u2014and one chapter to go. The goal is to take what you've learned since halftime and integrate it into an end-game plan.\n\nBefore you move on to the last chapter, \"Become Abnormal,\" go through the same process with the main discussion points of these three chapters that you did with Chapters 1 to 4. Pick the one main point that resonates with you (or choose one of your own), and pass it along as a teacher to one of your friends, relatives, or colleagues. Remember, when you teach it, you learn it.\n\nCHAPTER 5: EVALUATE CORRECTLY\n\n1. Learn to recognize the done-wells. People have a biological tendency to dwell on the things that aren't going well and that which you focus on expands.\n\n2. Focusing on your level of effort, on your commitment to a process, and on what you're doing well overall is far more effective than evaluating yourself through the perfectionist lens.\n\n3. Learn to give yourself mini-evaluations throughout the day. Status reports like this will give you time and information to make necessary adjustments.\n\n4. ______________________________________________\n\nCHAPTER 6: LEARN HOW TO TALK TO YOURSELF\n\n1. Become more aware of how you talk to yourself\u2014the goal is \"no more negative self-talk.\"\n\n2. Completing the Mental Workout regularly builds mental toughness\u2014and thought control. This will give you the mental toughness needed to become much more positive and much less negative.\n\n3. When you visualize, you prepare your mind and body for positive growth and action.\n\n4. ______________________________________________\n\nCHAPTER 7: LEARN HOW TO TALK WITH OTHERS\n\n1. Know your material by writing a script.\n\n2. Slow down and breathe.\n\n3. Nail the presentation by spending three separate three-minute practice sessions per day in the three days leading up to your event (the success triangle).\n\n4. ______________________________________________\n\nCoach Wooden and Charlie Munger have been two of the most influential thought leaders in our professional lives, and they both emphasized many of the same basic philosophies. One of their core values was the idea that nobody\u2014no matter how much they had achieved\u2014should stop learning.\n\nWhen you embrace the idea that you should always be learning, you're deciding to live a life open to information and experiences. You're not only giving yourself the chance to reach your own full potential, but also fully engaging in the world and the people around you.\n\nThink about it. When you're on a team or in a group, and one of the other people doesn't want to hear any of the information you want to share, what is your inclination toward that person? We all respond much better to people who are open to hearing and learning. That works both ways.\n\nAs you go through the process of teaching one of these key points to somebody else, don't hesitate to make some notes here or on any other page. Coach Wooden devoured books on many subjects during the off-season, always looking for information that could help him coach and mentor his players. And you could always tell which books were his, because he filled the margins with notes and reminders.\n\nIt's a great habit to get into.\n\nBECOME ABNORMAL\n\n\"Normal.\" That's a word that often signifies a quality a person is striving to achieve. How many times have you heard somebody say, \"I just want to live a normal life\"? Or, \"I can't wait until things get back to normal\"?\n\nNormal is fine. Acceptable. Average. But if you want to do great things, you need to be _abnormal._\n\nWe've covered many of the tools that can be used to take your success to the next level. But we would be ripping you off if we didn't talk about the real challenges that come when you try to make these kinds of changes.\n\nMany people can sleepwalk through their professional and personal lives and reach a sort of equilibrium. They do enough to get by where they are, but nothing more. They aren't really doing anything wrong enough to make headlines, but they aren't moving up, either.\n\nYou would not have made it this far into this book if you were one of those people. And we would be ripping you off if we let _normal_ be the aspiration you took away from these lessons. As the title of this chapter says, the goal is to be . . . _abnormal._\n\nIt can be tempting\u2014and it's becoming easier than ever with all our distractions\u2014to push things off into the future, or to give yourself blanket forgiveness every time you slide on a goal or a commitment and there's a \"reasonable\" excuse.\n\nMaybe you want to break an earnings goal at your company, but you haven't gotten around to taking the certification classes you need to get there. There'll be more time in the spring, right? (Or the summer, or the fall . . .) You definitely want to lose that extra fifteen pounds you've been carrying. And once you finish this one big project, you're going to commit the time to do it, right? And if not then, then right after the kids leave for summer camp. Then the time will be right.\n\nWe hate to say it, but the time is never _perfectly_ right. If you're waiting for the perfect moment and perfect information so you can make the perfect decision, you're going to wait forever. The truth is, life is a bumpy road. You're going to have periods of smooth sailing, and times when you're faced with adversity and stress. To be _abnormal,_ you need the ability to accept the adversity and stress that is inevitably coming your way\u2014and not only survive it, but thrive on it. That happens when you have confidence in the tools at your disposal\u2014the concepts we've been talking about\u2014and when you're familiar with the test.\n\nThe test preparation business offers a great analogy. In the test-prep world, the first order of business for any instructor is to get the student familiar with the test format. When you walk into the classroom and are handed the test booklet, your level of anxiety is greatly reduced if you know, in general, what kinds of questions are coming, and how long you have to complete each section. Just by eliminating the fear of the unknown\u2014and establishing a frame of reference for what's coming\u2014you've allowed yourself to concentrate on the actual task at hand instead of trying to corral your emotions.\n\nOf course, knowing the format is only part of the battle. If you don't know any of the material, you're still going to struggle. But a combination of knowing the material\u2014or, in terms of this book, mastering one or more of the concepts we've been discussing\u2014and being ready for the test itself is a powerful position to be in.\n\nYou know the material. Now, we want to warn you about three of the performance \"viruses\" that could cause you the most trouble in your quest for abnormal. We call them the \"The Trap of the Viable Excuse,\" \"Focusing on What You Can't Control,\" and \"Giving In to Problem-Centric Thought.\"\n\nJust knowing what these viruses are\u2014and how they hide in plain sight in your everyday life\u2014will help you establish immunity to them before they have a chance to establish themselves. And if you do start to see symptoms, you'll be able to stay calm, keep your perspective, and apply an effective antidote.\n\nVIRUS #1: THE TRAP OF THE VIABLE EXCUSE\n\nIn Major League Baseball, nothing spreads faster than a scouting report.\n\nOnce the \"book\" on a player has been established, news travels. If a guy murders inside fastballs\u2014or can't hit a curveball\u2014it doesn't take long for everybody in the league to know. Once everybody knows what a player can (and can't) do, it's up to the player to make whatever adjustments need to be made.\n\nJason had been working with one of his baseball clients for about a year and a half when the player ran into what we call the \"Trap of the Viable Excuse.\" The player was a veteran with a lot of success under his belt, but he went through a season in which his batting average dropped by more than 50 points.\n\nThe word had gotten out that the player was struggling to hit breaking balls. Of course, almost immediately, all he started seeing in each at-bat was breaking balls. His weakness had been exposed, and now it was getting attacked. Every time Jason brought up the obvious subject of learning to improve how he responded to breaking balls, the player had the same response.\n\nHe said that he had never been a good breaking ball hitter\u2014not in high school, college, or the pros\u2014but he destroyed fastballs. That's what he was paid to do, and that's what he did. For one season, he clung to that idea, but the results got even worse.\n\nAfter the season, Jason once again brought up the touchy subject of addressing the problem. The player started in on his speech about never being a good breaking ball hitter, but before he got to the second sentence, Jason respectfully stopped him. \"I know you're a professional hitter, and I know you get paid to hit fastballs,\" Jason said. \"But nobody is throwing you fastballs anymore, and if we don't figure this breaking ball thing out, we're both going to get fired.\"\n\nAt first, the player was angry about getting confronted that way. To ease the tension, Jason explained that what the player was experiencing was very common. He had fallen into the \"Trap of the Viable Excuse.\"\n\nViable excuses are so hard to overcome because they sound so reasonable. They're disguised to the point that the person doesn't even realize they're using an excuse.\n\n* * *\n\nViable excuses are so hard to overcome because they sound so reasonable. They're disguised to the point that the person doesn't even realize they're using an excuse.\n\n* * *\n\nHere's the most dangerous part. The more \"reasonable\" the excuse is, the more you're willing to accept the failure and make it your new normal.\n\n* * *\n\nThe more \"reasonable\" the excuse is, the more you're willing to accept the failure and make it your new normal.\n\n* * *\n\nThat's the \"Trap of the Viable Excuse.\" When you accept it, you're accepting a permanent lowering of your personal standards.\n\nMany, many clients have told us some variation of this line: \"I'm not making any excuses, but let me tell you why I couldn't get the work done that we talked about last time. . . . \" The reason could be anything\u2014traffic, bad weather, not feeling your best, or any of the other real issues that pop up in real life. Viable excuses all contain a significant element of truth. Of course there will be obstacles that appear in your path toward success. Highly successful people have learned that even when obstacles present themselves, they still have an obligation to find a way to get it done.\n\n* * *\n\nHighly successful people have learned that even when obstacles present themselves, they still have an obligation to find a way to get it done.\n\n* * *\n\nBut the only right answer\u2014if you're truly committed to improvement\u2014is to learn how to be completely accountable for what you do, even in the face of adversity. That means no excuses. Not even viable ones.\n\nWhen a person makes an excuse, it serves as a pacifier to the mind. The excuse itself gets the attention, rather than the reality that a commitment was not honored. Shifting the mental focus to the excuse stalls progress. Avoiding excuses allows a person to experience strong negative emotions\u2014and those negative emotions can serve as the internal motivation needed for growth and improvement.\n\nDeveloping a \"no-excuse\" mentality can often make you become \"abnormal\" in the best way. People who don't make excuses are the ones you want on your team. They are the ones you can count on when the pressure mounts. They are the all-stars, the leaders, the consistent winners in life.\n\n* * *\n\nPeople who don't make excuses are the ones you want on your team. They are the ones you can count on when the pressure mounts. They are the all-stars, the leaders, the consistent winners in life.\n\n* * *\n\nWe're not talking about making terrible choices here. There _are_ circumstances when getting something done isn't the priority. If you're choosing being at a business meeting over being with a family member who is having a medical emergency, you're not making good choices. You do want to \"defer to intelligence,\" and that means making those logical calls when needed.\n\nWhen you accept accountability, you're creating a powerful internal dynamic. Instead of being externally motivated by what is happening around you, you're becoming internally motivated.\n\n* * *\n\nWhen you accept accountability, you're creating a powerful internal dynamic. Instead of being externally motivated by what is happening around you, you're becoming internally motivated.\n\n* * *\n\nJason's client quickly started coming up with ways he could improve how he handled breaking balls. He resolved to take twenty pitches from the mechanical pitching machine every day\u2014half with no bat, just to improve his pitch recognition. His ability to find solutions almost immediately increased once he stopped making the excuse. By the end of the season, the player had doubled his success against breaking pitches, and his overall batting average was back where it had been two years before.\n\nYou probably don't have to think too hard to remember some common viable excuses from your own life. Do any of these sound familiar?\n\n\u2022 There's no way I can get to that. I'm too busy.\n\n\u2022 I'm doing my part, but the rest of the team is screwing up.\n\n\u2022 How can I do what I'm supposed to do? Our technology is so out of date.\n\n\u2022 I was exhausted.\n\n\u2022 I didn't get the breaks that other person got.\n\nIt certainly feels better in the moment to come up with a \"viable excuse\" about why something didn't happen successfully. But, as we said, it lowers the bar. And it also prevents you from taking advantage of one of the most powerful change motivators there is: negative emotion. Fear. Disgust. Disappointment. Anger. People do everything they can to avoid feeling those emotions. But the most successful people approach it a different way: they take those negative emotions and _use_ them as a fuel for improvement.\n\nWhen you use a viable excuse, you're stopping yourself from feeling bad enough that you want to make the change that will prevent you from coming up short the next time. The excuse makes it okay to underperform in the moment\u2014and, most importantly and unfortunately, in the future as well. Operating with a true \"no excuses\" mentality certainly takes some guts, and you'll probably be apologizing much more than you're used to. But just the act of setting the bar higher for yourself will improve your performance, and it will remind you that you're the one in control of your thoughts, goals, and actions.\n\nVIRUS #2: FOCUSING ON WHAT YOU CAN'T CONTROL\n\nIt's early 2009, and the stock market is in the free-fall that would see it drop more than 50 percent from its high in October 2007.\n\nTom was working with American Funds, and he traveled to different offices around the country to take the pulse of advisors and try to give them some guidance on how to help clients who had seen their portfolios crushed by the downturn. As Tom made what he called his \"world tour,\" he knew he needed to offer a straightforward strategy that would help advisors\u2014and clients\u2014fend off panic and make rational decisions with the right amount of perspective.\n\nIt started with two lists. First, Tom asked the advisors to make a list of the things they could control.\n\nEffort.\n\nKnowledge.\n\nOrganization.\n\nAttitude with clients.\n\nTone of voice.\n\nFrequency of communication.\n\nYou get the idea.\n\nThen, Tom asked them to make a list of the things they _could not_ control.\n\nThe markets.\n\nThe news.\n\nThe presidential election.\n\nWhat other advisors were saying or doing.\n\nCommodity prices.\n\nNatural disasters.\n\nWhen the lists were finished, Tom asked the advisors which list they had been spending most of their time thinking (and worrying) about. Virtually everyone\u2014ninety-five out of one hundred\u2014said they had been spending most of their time in crisis mode worrying about the things on the second list.\n\nAt that point in the presentation, Tom stopped and told each advisor to get on the phone and just concentrate on _one_ thing they could control\u2014his or her tone of voice. By taking control of one thing that was available to _be_ controlled, the advisors felt the ground under their feet. Then, they could move on to some of the other items on the list and build out from there. Instead of starting the day on defense, dreading the conversations that were going to come, they resolved to come into the day with a plan of attack and start on offense.\n\nTom's advice wasn't new. He adapted it from a lesson he had learned from Coach Wooden. Early in Wooden's career at UCLA, the basketball team practiced in a building called the B.O. Barn (named after the distinctive \"smell\"). They had to share space with the wrestling and gymnastics teams, and many times, all three teams had to practice at the same time.\n\nDriving home one night, Wooden was concerned because the players had been distracted by some commotion from the other teams during practice. But as he continued his drive, he realized that he was spending a lot of energy getting upset about something that was out of his control. He decided right then that he was only going to focus his energy on the factors he _could_ control.\n\nHe later said that decision was the fuel that took his coaching to the next level. People think that Wooden was an instant success at UCLA, but he lost the first four games he coached in the NCAA Tournament. Focusing on what they could control, the Bruins would go on to win ten national championships in a twelve-year period. The success of the 1964 team\u2014which went 30\u20130 and packed the 1,300-seat Barn\u2014led the school to build Pauley Pavilion, a bigger, better-smelling, and less distracting arena.\n\nThe lesson Coach Wooden learned\u2014and passed on to Tom in his own handwriting\u2014is that focusing on the things you can't control actually hurts your ability to focus on the things you _can_ control. This is the way Coach Wooden said it:\n\n4-17-03\n\nDear Tom,\n\nConcern yourself with what you can control.\n\nIf you get too caught up, concerned, engrossed or involved over the things over which you have no control, it will have an adverse effect on the things over which you have control.\n\nLove,\n\nJohn\n\nIt goes beyond the time suck that automatically comes with spinning your wheels dwelling on something that won't change\u2014no matter how much time you actually spend thinking about it. The bigger problem is how it impacts your ability to prioritize\u2014and how it steals your time, energy, and creativity.\n\nPeople spend so much time thinking about things on the \"Can't Control\" side because, frankly, it's easier. If you're angry about something that Congress did or fuming about the weather, you don't actually have to come up with a plan of action, and there's ultimately no accountability. The \"Can't Control\" virus is extremely contagious, and it's one that can take hold of you multiple times a day. Luckily, awareness is the best defense against it. The key is to learn to recognize when you are allowing yourself to focus on things you cannot control. Typically, frustration is the alarm that lets you know you're emphasizing things out of your control. When you feel frustrated, it is often a response to your thoughts being centered outside of your control.\n\nYour strategy should be the same one Tom used at the meetings during his world tour. Make a simple two-column chart on a piece of paper and write down a list of a few things you can control. Then move over to the other side and make the same kind of list of things you can't.\n\nCommon examples of things you cannot control:\n\n\u2022 Competing in a hostile environment\n\n\u2022 Bad officials\n\n\u2022 The media\n\n\u2022 Other people\n\n\u2022 The weather\n\nCommon examples of things you can control:\n\n\u2022 Activity\n\n\u2022 Preparation\n\n\u2022 Organization\n\n\u2022 Attitude\n\n\u2022 Effort\n\nThen ask yourself, \"Which side have I been spending most of my time on?\"\n\nFrustration is the direct result of the \"Can't Control\" virus. It may sound overly simplistic, but anytime you feel frustrated, pull out a piece of paper and create your own \"Can and Cannot\" control chart. Doing so will help jump-start you into action toward those things you can control in life. It might even be helpful to highlight one of the most common \"Can't Control\" viable excuses you tend to use, to sort of warn yourself in advance to be ready to fight it off!\n\n* * *\n\nAnytime you feel frustrated, pull out a piece of paper and create your own \"Can and Cannot\" control chart. Doing so will help jump-start you into action toward those things you can control in life.\n\n* * *\n\nVIRUS #3: GIVING IN TO PROBLEM-CENTRIC THOUGHT\n\nIn Game Six of the 2011 World Series, the St. Louis Cardinals were trailing by three runs as late into the game as it's possible to go without it being over\u2014with one strike left in the bottom of the ninth inning. But even as Cardinals fans were preparing for the Texas Rangers to clinch the title, the Cardinals players refused to give in to what was seemingly inevitable.\n\nInstead of concentrating on the negatives of the moment and the problem at hand, the players forced themselves to think about solutions. Refusing to focus on the scoreboard, the pressure, or how close to defeat they were, they channeled their thoughts toward the fundamental details of having a \"good at-bat.\" They kept to the routine that had gotten them to the World Series in the first place\u2014waiting for good pitches to hit, not trying to do too much in an at-bat.\n\nPlayer after player showed he thrived on the biggest stage. David Freese laced a triple off the wall, scoring three runs and sending the game to extra innings. Once again, the Rangers took the lead on a two-run homer by Josh Hamilton. This time, Lance Berkman was the last-strike hero for the Cardinals, tying the game again with an RBI single. In the eleventh inning, Freese would strike again, this time hitting the game-winning home run that would send the series to a Game Seven\u2014and ultimately a World Championship for the Cardinals.\n\nIt was the first time in World Series history that a team came back from two different two-run deficits in the ninth inning or later in the same game. Freese's homer was only the fourth walk-off in World Series history, and the Cardinals became the first team in Series history to score in the eighth, ninth, tenth, and eleventh innings.\n\nAfter the game, the players talked about how they refused to let their thoughts become poisoned by thinking about the possibility of a negative outcome: they thought about what they had done well up to that point in the game and what needed to happen to succeed, and that gave them confidence that they could continue.\n\nThat makes the St. Louis Cardinals abnormal. Most \"normal\" people would fall back to the \"default.\" In this case, the default is what we call \"problem-centric thinking,\" a concept we've discussed earlier in this book. If you're confronted with a problem\u2014anything from a simple flat tire to a complex medical issue\u2014your first reaction is probably going to be to obsess about the problem itself. But if you occupy all of your time and energy thinking about the problem itself, it will cause that problem to grow larger in your mind. Remember what we established in Chapter 6\u2014the concept that what you focus on expands.\n\nIt doesn't happen just in the mind, either. When you allow your mind to focus on a problem, your brain releases certain neurotransmitters that cause you to feel terrible physically, and those neurotransmitters also significantly hamper your creativity and intelligence. The more you focus on the problem, the more likely it is for you to look at the now-larger problem and start thinking about quitting. Focusing on problems has the potential to send you into a vicious negative cycle. For example, you might be a small business owner who gets most of his or her business from one huge client. That client goes through financial problems and eliminates your contract with no warning. You could get angry, wipe everything off your desk, and curse out your contact at that company\u2014which would be quintessential problem-centric thinking.\n\nThe next natural emotion after that would be a feeling of despair or hopelessness:\n\n_How will I ever replace that business?_\n\n_I might as well close the doors for good._\n\n_I knew I shouldn't have relied so heavily on that one client._\n\nAs we said with the viable excuse trap, we're not here to make any value judgment about the reasons behind what happened. It happened. Now what?\n\nStrong, resilient people have what we call a \"Relentless Solution Focus,\" or RSF. If a person with great RSF was in the same situation and lost that big client, he or she wouldn't be some kind of emotionless robot\u2014the loss would sting. But the immediate, laser-sharp focus would be on finding the solution path, and doing it in less than sixty seconds.\n\nWe say \"solution path\" because many, many problems aren't solved with one lightning strike of an idea, obviously. A solution is a process, and there are steps to that process. In RSF, your goal when presented with a problem is to identify one step within sixty seconds that you can take that will make the situation better\u2014even if only by a small increment of improvement. RSF is not about finding the \"perfect\" solution but, rather, about just identifying some kind of improvement. It's called the \"+1 solution,\" because any improvement whatsoever to the current situation is part of a solution. The +1 concept has been credited numerous times with making the previously deemed impossible actually possible.\n\nOne important key is to transfer your thinking about the problem from worry to RSF within sixty seconds, to avoid those negative neurotransmitters from being released into your bloodstream. The great news is that when you direct your thoughts from problems to solutions, your brain releases a whole new set of neurotransmitters. The neurotransmitters on the solutions side actually work to make you feel better, and they significantly improve your creativity, intelligence, and memory. It's as if you have a powerful army in your head, and all it needs is the right orders.\n\nWhen we focus on small, incremental improvements instead of perfection, the human spirit takes over, and all things become much more possible.\n\n* * *\n\nWhen we focus on small, incremental improvements instead of perfection, the human spirit takes over, and all things become much more possible.\n\n* * *\n\nIf you're solution focused, the loss of a client and subsequent business downturn gives you the opportunity to focus on prospecting for other, better clients. It isn't an excuse to slow down and become less active!\n\nThis advice is very different from what you might have heard from other people within the mental health field. \"Talk about your problems and you'll feel better\" has become so much of a clich\u00e9 that it's even been a punch line on Seinfeld.\n\nThere's one problem with that. No studies have conclusively shown that talking about or thinking about problems will naturally lead people to solutions.\n\nIt's the same as thinking about a broken leg and expecting the bone to heal itself quickly and cleanly. It isn't any more rational to do it on a figurative problem than it is to do it on a physical one. By immediately framing a problem in the solution framework and asking yourself to come up with one step you can take that can make things better\u2014and doing it within sixty seconds\u2014you're aligning your mind with powerful beneficial forces. And you will be taking useful actions before negative emotions can start to cycle you into a bigger and bigger hole.\n\nYou're literally manufacturing optimism and success.\n\n* * *\n\n**The Big Why:** It is totally normal to focus on excuses, things you cannot control, and your problems. The goal is to be _abnormal,_ and to do so you must learn to vaccinate yourself against the three performance viruses: \"The Trap of the Viable Excuse,\" \"Focusing on What You Can't Control,\" and \"Giving In to Problem-Centric Thought.\"\n\n**The Inversion Test:** If you do not vaccinate yourself against excuse making, allowing your mind to focus on what you cannot control, and problem-centric thinking, you will be significantly less happy, less healthy, and less successful.\n\n**Act Now:** Write down your three most common viable excuses. The act of recognizing the excuses you use causes your mind to be more sensitive to not using them in the future.\n\n* * *\n\n* * *\n\n**Here are some examples of the three most commonly used excuses from our clients to help get you started:**\n\n* * *\n\nPRO ATHLETE\n\n1. My competition makes a lot more money than I do\u2014of course he's better than I am.\n\n2. I'm not sure why but that ump has it out for me. I never get the calls with him.\n\n3. As the game or season wears on, I sometimes catch myself thinking that I am wearing down and getting tired, and that is my excuse for giving less than 100 percent to my training and preparation.\n\nFINANCIAL ADVISOR\n\n1. My assistant continues to make mistakes. She often misses deadlines or says the wrong things to clients, and then I have to continually clean up the mess. That's why I can't get organized daily.\n\n2. I have so much paperwork and my clients expect me to get it done. There just aren't enough hours in the day to get all of that stuff done and make my proactive contacts.\n\n3. When the market is bad, there is nothing I can do to make things better.\n\nLAWYER\n\n1. I am too busy making money to do any prospecting.\n\n2. I am too busy making money to get home on time.\n\n3. When the office calls, it is so important that I have to answer\u2014even during family dinner time.\n\nOnce you've admitted to your excuses, it's time to start thinking of solutions. Take the time to do another exercise: think of one problem you have been worried about, but quickly come up with one simple step you can take that is solution-oriented. This exercise should take no longer than sixty seconds. Then, write down exactly when you will take that step. You will be on your way to a brighter future.\n\n* * *\n\n* * *\nEpilogue\n\n_Inspiration and Beyond_\n\nInspiration is a powerful thing.\n\nCoach Wooden's father, Joshua, raised his children to live by two sets of simple rules. The first set dealt with integrity:\n\n\u2022 Never lie.\n\n\u2022 Never cheat.\n\n\u2022 Never steal.\n\nThe second set dealt with adversity.\n\n\u2022 Don't whine.\n\n\u2022 Don't complain.\n\n\u2022 Don't make excuses.\n\nAnd when Coach Wooden graduated from the eighth grade, in 1924, his father gave him a printed card. On one side was an inspirational poem by the Reverend Henry van Dyke, a Presbyterian minister who wrote many popular stories and hymns in the early twentieth century. On the other side was a seven-point creed that included advice like \"Make each day your masterpiece.\"\n\nWooden took the two sets of three rules and the advice on that card (which he carried with him in his pocket to the day he died in 2010) and used them every day in more than seventy years of coaching and mentoring.\n\nThe thousands of people Wooden touched were in turn inspired by Joshua Wooden's gesture nearly a hundred years ago. To this day, you can listen to the words of the all-time great players mentored by Wooden\u2014Kareem Abdul-Jabbar, Bill Walton, Gail Goodrich\u2014and hear Wooden's influence, down to the exact language he used with them in practice decades ago.\n\nOur goal in this book\u2014and in every seminar we teach\u2014is the same as the one Coach Wooden had for every practice he ran: we want you to leave a little better than you came in.\n\nHere, we'd like to share five real success stories from families who embraced the concepts we teach and changed not just their lives but the lives of their children. In many ways, these families may be like yours. And their transformations didn't come from some kind of dramatic meteor strike from outer space or thunderbolt from the sky. It came from methodical, incremental improvement\u2014and the determination to make those improvements stick.\n\nYou can do it, too.\n\nTHE LANGUAGE OF A LOSER\n\nMarcus Lopez hadn't yet attended one of our classes, but he received one of our audio CDs from a financial advisor friend who had. Marcus listened to the CD so much that it actually got stuck in the slot in his car radio\u2014so he ended up listening to it much more often than he had planned. He didn't realize just how often until he was driving back from a golf tournament that his ten-year-old son had played in.\n\nHis son had a tough round, and he spent the first part of the car ride talking about all of the problems with his game\u2014everything from his clubs to the condition of the course to the fact that everybody in the tournament was a better player than he was.\n\nWhen the ten-year-old took a breath, his five-year-old brother chimed in. \"That's whining, complaining, and making excuses,\" he said. \"You're speaking in the language of a loser.\"\n\nThat lesson became one that all of Marcus's family shared together on a regular basis. It led to a little game: When one of the kids heard something on television that sounded like whining, complaining, or making an excuse, they'd almost automatically point it out. Marcus would start the next sentence: \"That's the language of a . . . ,\" and the kids would say, in unison, \". . . a loser, Dad.\"\n\nCoach Wooden would have been pleased to learn that children were learning to adopt the \"no excuse\" mentality.\n\nGRAPPLING WITH THE MENTAL WORKOUT\n\nBrian Blough is a very successful business owner from Georgia who attended one of our seminars and incorporated the Mental Workout into his routine the first day after the program finished. He had so much success with it that he shared the process with a number of other colleagues in his area\u2014and with his sixteen-year-old son, who wrestles on the high school team.\n\nIn his second year on varsity, Brian's son had the ups and downs you would expect from a younger competitor. As he progressed through the season, his record hovered around .500. But as he got to the end of the season and started facing tougher competition, he began to wrestle much more conservatively.\n\nAfter a conversation before one of his matches, Brian realized that his son was letting the prospect of competing against talented wrestlers get in his head. That night, he introduced him to the Mental Workout, and took him through all of the steps he had learned in our seminar. They created an identity statement, and his son began to visualize a highlight reel of success from his previous matches, and pictured in his mind's eye what he wanted his next match to look like.\n\nIn the next match, his son took the mat looking much calmer and more confident. A few weeks later, he wrestled extremely well in the sectional tournament\u2014where he also advanced to make it into the state tournament. After a great showing in the state tournament, he's looking forward to a complete off-season to further develop his mental and physical skills\u2014and to making an even longer run in next year's tournament.\n\nBUILDING MENTAL TOUGHNESS\n\nRosey Hayett thought the principles he learned from Tom and Jason would be perfect to use in his role as a youth sports coach. Some of the most valuable material for the young athletes was on the subject of mental toughness.\n\nHayett's alpine ski racers faced some of the most daunting pressure of any youth athletes. Ski racing itself is a physically demanding sport that requires tremendous fitness and athleticism. The conditions are often brutal\u2014cold weather, unfamiliar terrain, bad snow conditions.\n\nAs hard as that all is, the mental demands might even be tougher. The difference between first and fifth place is usually measured in hundredths of a second. One tiny mistake can blow up what was otherwise a great run. To compound the performance pressure, racers often have to go to the top of the slope and wait more than an hour to go down the hill, giving them plenty of time to think\u2014and get nervous. A sound mental approach is absolutely crucial in the sport\u2014especially for young athletes.\n\nThe team Hayett coached adopted Mental Workouts for each racer\u2014the centering breaths, positive self-talk, and visualization helped to calm each of the skiers before their events and increased their confidence during runs. They also modified their practice routine to start treating training runs as if they were real, competitive ones\u2014which helped them develop more tolerance for pressure.\n\nBy any measure, the tools were a success. On Hayett's team there were skiers from many small towns around northern New Mexico, and they regularly competed against larger, better-funded teams from higher population areas across three states. With the help of the mental workout, their team won the Southern Series championship four times in five seasons and produced a dozen Junior Olympic champions.\n\nPRIORITIZING THE PROCESS\n\nJohn Mark Brown is an Edward Jones advisor who went through one of our training seminars a few years ago. He started to see some great improvement in his business in the twelve months after the seminar, to the tune of a 19 percent increase. But it was the effect the training had on John's family life that proved to be the most valuable gain of all.\n\nJohn's daughter Regan had been born four months premature and was diagnosed with cerebral palsy. She worked hard and pushed herself to lead the regular life of an elementary, junior high, and high school student, but she had her share of challenges. One day, about four months after John took the seminar, the superintendent of Regan's high school called. She asked what the family had been doing differently, because Regan's engagement and participation in her classes had gone way up.\n\nJohn was stumped for a minute, then realized that the act of concentrating on the process of his business\u2014the one rule he chose to hit hard after training\u2014was spilling over into his family life, too. He was choosing to concentrate on the process of family time. Everybody in the Mark house was more engaged and working together, focusing on the process of growing and improving without worrying so much about the outcome or achieving perfection. Regan was feeding off that positive energy\u2014with great results.\n\nRegan graduated from high school with a B average and is now going to college.\n\nFROM PERFECTION TO PERFORMANCE\n\nJeff Gayanski had always tried to motivate and push his kids the way he had been taught\u2014to relentlessly attack until you got the results you wanted. It had worked for him. Jeff was a managing director for a Fortune 500 company, and he had found great success during his professional career by focusing harder and harder on getting those results.\n\nBut Jeff's family life wasn't keeping up with his professional life. He felt burned out with the relentlessness of work pressure. It was a phase that had come and gone before, but this time it seemed to be lasting much longer. And more importantly, Jeff's daughter was struggling, and he didn't know how to help her. None of his usual motivational tools were working.\n\nJeff was noticing some dramatic changes in his seven-year-old. She was obviously distressed about something. She was very emotional over small things, bursting into tears almost instantly over little frustrations. She was dealing with her anxiety by pulling her eyebrows out. Imagine how difficult it would be for a seven-year-old to have to go to school without eyebrows!\n\nDoctors eventually diagnosed Jeff's daughter as having high anxiety, and said the family would need to learn how to deal with it the best they could. Jeff came to one of our seminars right about the time he was at his low point both professionally and personally. He was desperate, and he thought that if he could improve his work numbers, he could give himself some more breathing room at home.\n\nIt took a few days, but we showed him that he _wasn't_ his results. He was his effort.\n\nJeff resolved to stop focusing on his numbers and to start concentrating on things he could control. He trained himself to answer three simple effort-based questions every day:\n\n1. What three things did I do well today?\n\n2. What one thing do I want to improve tomorrow?\n\n3. What is one thing I can do differently right now that can help me make that improvement?\n\nIn the made-for-TV movie of Jeff's life, he would have answered those three questions on the first day, and everything would have solved itself immediately. Jeff lives in the real world, though, and it took a lot of determination\u2014and some pain\u2014to establish the new habit. Most days, he didn't want to bother to take the three minutes to answer the questions. But he doggedly did it, forcing himself to check off the box every day.\n\nJeff started noticing how much less anxiety he felt when he answered the questions. Then, he started to see an uptick in his results\u2014ironically enough, right when he committed to not worrying about them. More importantly, the benefits began to seep into his home life. Jeff wasn't coming home tired and stressed out. His wife and kids now enjoyed the time they had together each night. The more Jeff talked about the process he had learned in class, the more his wife wanted to know about it. Maybe the new process could help their family, and do something for their daughter. Maybe this new definition of success could reduce some of the frustration she was feeling. They decided to try. They started a new ritual. Every night, at dinner as a family, they decided to go around the table and identify one thing each did well that day.\n\nShockingly, Jeff's daughter started to respond. This one simple change helped her stop focusing so much on all the negative things in life and to be more aware of the good. Her stress level went way down, and her smile reappeared. In addition, she began using centering breaths to battle anxiety when she felt it coming on. A few months later, she went back to school for the first time in many years with all of her eyebrows intact.\n\nHer confidence grew to the point where she even started joining extracurricular activities. At the low point, she never would have considered playing on a team, because of the fear she wouldn't be able to handle the pressure. Last fall, she hit nine service winners in a row to win the title for her volleyball team. In more ways than one, she's a hero.\n\nKNOWING SOMETHING DOES NOTHING. DOING SOMETHING DOES\n\nThe success stories we've told in this chapter (and in this book) all have one thing in common. The central people in them are doers. If the commitment to doing isn't there, success doesn't exist.\n\nIt's up to you to change the lens through which you see the world. Instead of telling yourself you aren't special enough to get all the things you want in life, start telling yourself _I deserve it_ \u2014and then start deserving it by taking action!\n\nPICK ONE . . . AND WE MEAN IT\n\nOur main priority for this book has been to give you a variety of techniques you can use to increase your success\u2014and to give you permission to choose only one of those techniques to attack at a time.\n\nRitualizing one change will bring you much more success than if you try to bite off everything at once\u2014just like taking the entire bottle of aspirin isn't the right solution for muscle pain. Remember, done is better than perfect.\n\nWe can already read the minds of some of you looking over this paragraph right now. The whole concept of choosing just one thing doesn't apply to you, right? You're too advanced, too smart, and too special. It's great to have that kind of self-belief, and often, it's something you can use. But in this case, you're wrong.\n\nIn almost twenty years of working with some of the most successful and talented people in the world, we've seen the power and the value of focusing on one improvement at a time\u2014no matter what kind of physical and mental horsepower you bring to the table.\n\nOnce you've nailed your one improvement for three consecutive months (and 90 percent completion is how we define \"nailing it\"), you're ready to move on to the next challenge. By doing it this way, you're actually speeding up the growth process. It won't all happen overnight, but your improvement will be steady, and it'll be lasting.\n\n_Play Hard, Play Smart_\n\nEverything we've been talking about in this book has been designed to give you the information you need and the motivation to use it. Taken together, those two things produce the single most important factor in human performance\u2014confidence.\n\nIf you watch any sport on television, you've certainly heard \"confidence\" used as a catchphrase, or shorthand, for being \"in the groove.\" \"He's playing with a lot of confidence,\" or, \"She really needs to make this shot to get some confidence back,\" are things announcers say all the time. And it's certainly true that confidence is what often separates one athlete from another. Two athletes may both have world-class physical skill, but the one who has just a little more of that mental and emotional intangible is the one who has the better chance of success. So you probably won't be surprised to hear that confidence is just as big of a factor in everyday life, for everyday people, as it is in NCAA basketball games, NASCAR races, or Olympic sprints. Confidence matters just as much for you and me as it does for Lebron James, Jordan Spieth, or Ronda Rousey.\n\nThink about the times in your life when you've had a lot of confidence\u2014and when you've had less of it. And think about how easy it is to _lose_ confidence.\n\nIt's happened to all of us. You're feeling great about work, but then you run into a bad manager who puts you into a sidespin. Or things are running smoothly at work and one of your children runs into a problem that is complicated and hard to solve.\n\nIt happens. Life is how you deal with it when it does.\n\nConfidence isn't some kind of magic trick. It isn't something people are automatically born with. It's something you can learn, and something you can practice. It's what we've been teaching folks in dozens of training seminars all over the country.\n\nIf you apply even just one of the lessons from this book\u2014and choose wisely\u2014you're going to set up your own process for making your confidence grow.\n\nAs a reminder, we want to leave you with a few guidelines to help keep you on that confidence track.\n\n1. Choose wisely. Your mind is like a powerful fire hose. Don't dilute the strength by splitting the stream and aiming at a bunch of different fires. Pick one concept at a time and fully acknowledge the limits of channel capacity.\n\n2. Nail it\u201490 percent of the time. Once you've chosen, give it time to work and become habit. For the next three months, concentrate on that one concept and make it second nature.\n\n3. Improve, not perfect. If you expect perfection in anything\u2014yourself, your spouse, your kids\u2014you're setting yourself\u2014or them\u2014up for failure and disappointment. First, understand that just the act of deciding to make a change and trying to improve sets you apart from most people. Benjamin Franklin said it first, and he was paraphrased by General George S. Patton: \"Most people die at a very early age, only to be buried 40 or 50 years later.\" Congratulate yourself for taking the initiative, and keep pushing for improvement\u2014not perfection.\n\n4. Evaluate smartly. When you do improve, recognize it. Don't be critical about the size or speed of improvement. Congratulate yourself, then write down what you want to continue to improve. Making the effort is winning in and of itself.\n\n5. Repetition, repetition, repetition, repetition. . . . Mastery only comes from effort and repetition. You wouldn't expect your five-year-old to be able to tie her shoes the first time. In the words of the Zen master Suzuki, if you lose the spirit of repetition, your practice will become difficult. This was one of the absolute cornerstones of Coach Wooden's teaching.\n\n6. Be your own best cheerleader. Friends, family, loved ones, and colleagues might recognize the things you're trying to improve, and they might not. It's up to you to reinforce your own positive feelings. When you make progress, don't hesitate to celebrate it. You'll build a feedback loop that recruits your mind toward the next reward.\n\n* * *\n\n**The Big Why:** Life is hard and most people are overwhelmed. It's important to remember that you can do it. You have the power to improve. When you take action on your one thing, you will deserve the success you experience.\n\n**The Inversion Test:** People who try to take action on too many things become overwhelmed to a point of inaction. People who try to take action on one thing become energized to a point of action.\n\n**Act Now:** Pick one thing from the list of chapters to attack. Evaluate yourself weekly on your completion rate. Make it a goal to nail your one thing each week at 90 percent or better.\n\nChapter 1: Organize Tomorrow Today\n\n**ACTION TOOL:** Identify daily your \"3 Most Important \/ 1 Must.\"\n\nChapter 2: Choose Wisely\n\n**ACTION TOOL:** Every day, no matter what, take _action_ on your \"1 Must.\"\n\nChapter 3: Maximize Your Time\n\n**ACTION TOOL:** Choose one of the three time-maximization tools described in the chapter and commit to following it (\"Attack the Open Space,\" \"Prioritize the Priorities,\" \"Trim the Fat\").\n\nChapter 4: Win Your Fight-Thrus\n\n**ACTION TOOL:** Win fight-thrus by asking yourself two questions: How will I feel if I win the fight-thru? How will I feel if I lose it?\n\nChapter 5: Evaluate Correctly\n\n**ACTION TOOL:** Complete Success Logs a minimum of three times weekly.\n\nChapter 6: Learn How to Talk to Yourself\n\n**ACTION TOOL:** Complete Mental Workouts a minimum of three times weekly.\n\nChapter 7: Learn How to Talk with Others\n\n**ACTION TOOL:** Before meetings and presentations, spend three minutes three times daily for three days mentally rehearsing what you will say and how you will say it.\n\nChapter 8: Become Abnormal\n\n**ACTION TOOL:** Identify the one \"virus\" that is most affecting you, and take the action now to vaccinate yourself.\n\n* * *\n\n* * *\n\n**Here is how some of our clients tackled this assignment:**\n\n* * *\n\nPRO ATHLETE\n\nI am going to work on choosing wisely. I feel like there are actually two steps to this goal. First I am going to commit each and every day, no matter what, to complete my \"1 Must.\" If I am going to finish my career strong, this has to be done. The other side of it for me is that I am going to do a better job of choosing wisely what I commit to. I am always telling people that I will do this or that without really giving much thought to whether or not I will actually follow through. I am going to really work on only saying things that I fully intend on following through with. I will choose more wisely.\n\nFINANCIAL ADVISOR\n\nThere is no doubt what I need to choose to focus on. I need to get organized. I am going to attempt every day to identify my \"3 Most Important \/ 1 Must.\" The rule I am setting up for myself is that before I am allowed to take my first bite of lunch I will spend a few minutes getting organized for tomorrow. To help with accountability, I am going to ask my assistant to come in and remind\/ask me if I have completed my OTT before she takes her first bite of lunch each day.\n\nPHYSICIAN\n\nThe one thing I am going to commit to moving forward is beating the \"no excuse\" virus. I am keenly aware that my one \"go-to\" excuse has been \"I am too busy making money.\" This excuse has caused me for years to become stagnant. I don't read anymore, and that used to be a big part of making personal improvement. The worst part is I hide behind that excuse when it comes to being a good husband and father. I am embarrassed and heartbroken that I have allowed this to go on for so long, but I am committing to not using that excuse any longer.\n\n* * *\n\n* * *\n\nWe have one last thing to say. Be relentless about improvement; your progress has no limits!\nNotes\n\nINTRODUCTION\n\n. Dr. George Miller outlined the amount of information the human brain could process and retain at any given time in \"The Magical Number Seven, Plus or Minus Two: Some Limits on Our Ability to Process Information,\" originally published in _Psychological Review_ 63, no. 2 (1956).\n\n. \"The Lollapalooza Effect\" was coined by Charlie Munger in a June 1995 speech to the Harvard Law School.\n\nCHAPTER 1\n\n. \"The noise of the urgency creates an illusion of importance.\" Steven R. Covey, _The 7 Habits of Highly Effective People: Powerful Lessons in Personal Change_ (New York: Simon and Schuster, 2013).\n\n. \"Taking notes on laptops rather than in longhand is increasingly common. Many researchers have suggested that laptop note taking is less effective than longhand note taking for learning. Prior studies have primarily focused on students' capacity for multitasking and distraction when using laptops. The present research suggests that even when laptops are used solely to take notes, they may still be impairing learning because their use results in shallower processing. In three studies, we found that students who took notes on laptops performed worse on conceptual questions than students who took notes longhand. We show that whereas taking more notes can be beneficial, laptop note takers' tendency to transcribe lectures verbatim rather than processing information and reframing it in their own words is detrimental to learning.\" Abstract from Pam A. Mueller and Daniel M. Oppenheimer, \"The Pen Is Mightier Than the Keyboard: Advantages of Longhand over Laptop Note Taking,\" _Psychological Science,_ April 23, 2014.\n\n. \"Do Written Goals Really Make a Difference?,\" on a study by Dr. Gail Matthews, Dominican University, UGM Consulting, August 26, 2011, www.ugmconsulting.com\/Do%20written%20goals%20really%20make%20a%20difference%20UGM%20Briefing%2026%20Aug%202011.pdf.\n\n. A concise description of and previous research about behavioral inertia appears in Raymond B. Huey, Paul E. Hertz, and B. Sinervo, \"Behavioral Drive vs. Behavioral Inertia in Evolution: A Null Model Approach,\" _American Naturalist,_ March 2003. B. F. Skinner's research was called \"quantitative behavioral analysis.\"\n\n. Bluma Zeigarnik's research on \"the influence of tension in the achievement of memory\" originally appeared in a paper called \"On Finished and Unfinished Tasks,\" _Psychologische Forschung,_ 1927. Twenty years later, Kurt Lewin, one of Bluma Zeigarnik's earliest mentors, expanded on the concept of task-specific tension in his research; see his \"Concept, Method and Reality in Social Science: Social Equilibria and Social Change,\" _Frontiers in Group Dynamics_ 1, no. 1 (1947).\n\n. \"The 19th century geometrist Jacobi once said that one should always try to invert every geometrical theory. But his advice applies much more widely! Choose any class of relational frames and you can study its valid modal axiom.\" Johan van Bethem, \"Man Mass Immer Umkehren,\" Institute for Logic, Language and Computation, January 2007.\n\nCHAPTER 2\n\n. \"Einstein said there are five ascending levels of intelligence: smart, intelligent, brilliant, genius and simple.\" \"Understanding Circle of Competence and Knowing the Edge of Your Competency,\" _Forbes,_ GuruFocus, January 2, 2015.\n\nCHAPTER 3\n\n. Mike Berardino, \"Mike Tyson Explains One of His Most Famous Quotes,\" _Sun-Sentinel_ (Florida), November 9, 2012.\n\nCHAPTER 4\n\n. Dr. Maxwell Maltz's original book, _Psycho-Cybernetics,_ was published in 1960 by Prentice-Hall. It has been updated and republished many times since.\n\n. \"How Long Does It Take to Form a Habit?,\" article published by the University College London in the UCL News website about research performed by UCL's Phillippa Lally and the Cancer Research UK Health Behaviour Research Centre, August 4, 2009, <https:\/\/www.ucl.ac.uk\/news\/news-articles\/0908\/09080401>.\n\nHALFTIME\n\n. Anne Kathrin Barbeck and Thomas Neumann from West-falische Wilhelms-Universitat Munster gave an extensive description of Dr. Jean Pol-Martin's research in their paper \"Lernen durch Lehren (LdL) in Theory and Practice,\" for the seminar _Learner-Centered Approaches,_ Wintersemester 2005\/2006.\n\nCHAPTER 5\n\n. Dr. Norman Doidge, _The Brain That Changes Itself: Stories of Personal Triumph from the Frontiers of Brain Science_ (New York: James H. Silberman Books, 2007).\n\nCHAPTER 6\n\n. Maxwell Maltz, _Psycho-Cybernetics_ (Upper Saddle River, NJ: Prentice-Hall, 1960).\n\n. The quotation, in context, reads: \"The human mind is like a fertile ground where seeds are continually being planted. The seeds are opinions, ideas, and concepts. You plant a seed, a thought, and it grows. The word is like a seed, and a human mind is so fertile!\" Don Miguel Ruiz, _The Four Agreements: A Practical Guide to Personal Freedom_ (San Rafael, CA: Amber-Allen Publishing, 1997).\n\nCHAPTER 7\n\n. Steve Jobs's presenting skills were detailed extensively by Marcus Wohlsen in \"The Apple Watch Is Steve Jobs' iPod Launch All Over Again,\" _WIRED,_ September 10, 2014, and by Carmine Gallo in \"11 Presentation Lessons You Can Still Learn from Steve Jobs,\" Forbes.com, October 4, 2012.\n\nCHAPTER 8\n\n. David K. Randall, \"The Day the Stocks Bottomed Out,\" _Forbes,_ March 8, 2010, recounted the 2009 cratering of the stock market.\n\n. Gene Wojciechowski recounted the pivotal moment in the 2011 World Series in \"Game 6 Defines What Baseball History Looks Like,\" ESPN.com, October 28, 2011, <http:\/\/sports.espn.go.com\/espn\/columns\/story?page=wojciechowski-111027>.\n\n. In the _Seinfeld_ episode \"The Engagement,\" 1995, Kramer says, \"What are you thinking about, Jerry? Marriage? Family? They're prisons! Man-made prisons. You're doing time. You get up in the morning, she's there. You go to sleep at night? She's there. It's like you gotta ask permission to go to the bathroom. 'Is it all right if I go to the bathroom now?' And you can forget about watching TV while you're eating. . . . You know why? Because it's dinnertime. And you know what you do at dinner? You talk about your day.\"\n\nEPILOGUE\n\n. Joshua Wooden, \"Timeless Wisdom Creed Cards,\" reproduced for the John Wooden Course, available at woodencourse.com.\n\n. Suzuki said, \"If you lose the spirit of repetition, your practice will become quite difficult.\" Shunryu Suzuki, _Zen Mind, Beginner's Mind_ (Boston: Shambhala, 2011).\nAbout the Authors\n\n**Dr. Jason Selk** is one of the premier performance coaches in the United States. His clients include Olympians and professional athletes in Major League Baseball, the NFL, the NBA, the NHL, and NASCAR, along with Fortune 500 executives and organizations such as Northwestern Mutual and Edward Jones. As the director of mental training for Major League Baseball's St. Louis Cardinals, Dr. Selk helped the team win two World Series championships, in 2006 and 2011. He previously wrote two best-selling books, _10-Minute Toughness_ and _Executive Toughness;_ is a regular contributor to ESPN, _Inc.,_ and _Forbes;_ and has been featured in _USA Today, Men's Health, Muscle & Fitness,_ and _Self._ He lives outside St. Louis, Missouri. You can find him at JasonSelk.com.\n\n**Tom Bartow** left a successful career as a college basketball coach to become one of the top financial advisors at Edward Jones. Tom applied many of the concepts he had learned from John Wooden, the famed basketball coach at UCLA, to create and develop an advanced training program for higher-level advisors at the firm. In June 1999, the American Funds Group offered Tom a unique position: one of his responsibilities was to work with American Funds distributors across the nation to increase the skill set of the entire sales organization. Tom's insights immediately proved to be highly beneficial to investors and advisors. From 2002 to 2009, Coach Wooden and Tom delivered a one-two punch for the American Funds Advisor Forums. Their friendship was such that Tom was invited to join Coach Wooden at the White House for Coach Wooden's acceptance of the Medal of Freedom. Coach Wooden said of Tom, \"You are something else.\"\n\nTom Bartow and Dr. Jason Selk have become the best of friends and have worked together to bring peak performance techniques from the world of professional athletic competition to the corporate world.\n\n**Matthew Rudy** has authored or coauthored twenty-three golf, business, and travel books, including titles with Hank Haney, Dr. Michael Lardon, Dave Stockton, and Johnny Miller. He is a senior writer at _Golf Digest,_ where he has produced more than twenty-five cover stories since 1999. He lives in Bridgeport, Connecticut. You can find him at MatthewRudy.net, or on Twitter at @RudyWriter.\n\nWhen you have your own breakthrough with the help of one of the rules in this book, we want to hear about it. Email jason@jasonselk.com. We'll use the best stories as case studies to help motivate the students that come behind you.\n\nFor more information on Organize Tomorrow Today classes and seminars in your area, or to inquire about scheduling them, go to Jasonselk.com.\nIndex\n\nAbdul-Jabbar, Kareem,\n\nabnormal, goal of becoming, 96\u201397, 157\u2013175\n\naction tool,\n\nexamples, 174\u2013175\n\nfocusing on what you can't control and, 164\u2013168\n\ngiving in to problem-centric thought and, 168\u2013172\n\nintroduction to, 157\u2013159\n\nsummary,\n\nthriving on adversity and stress and,\n\ntrap of viable excuse and, 159\u2013164\n\naccountability, viable excuses and, ,\n\naction\n\nfocus on single primary task and, ,\n\ninformation overload and,\n\nprioritized list and taking,\n\naction tools, 188\u2013189\n\nadversity, rules for dealing with,\n\nAmerican Funds Group, xviii, , ,\n\nanecdotes, for presentations,\n\nanxiety, focus on factors one can control and alleviation of, 183\u2013184\n\n\"ask and chop\" method, to combat procrastination, 55\u201357\n\nathletes. _See_ pro athletes; _individual sports_\n\nattacking open space, 41\u201342, 44\u201347\n\nbacksliding, habit formation and danger of, 72\u201376\n\nBartow, Tom, xviii\u2013xxi, xxiii\u2013xiv\n\nbaseball players\n\nexample of habit formation, 68\u201371\n\nexample of setting process goals,\n\nexample of trap of viable excuses, 159\u2013160\n\nexample of use of Organize Tomorrow Today, 10\u201311,\n\n_See also_ St. Louis Cardinals\n\nbaseball scouting reports, 159\u2013160\n\nbasketball players\/coaches,\n\ncommunication and, 142\u2013143\n\nexample of focus on fundamentals, 85\u201386\n\nexample of self-evaluation,\n\nuse of visualization,\n\n_See also_ Wooden, John\n\nbehavior, habit formation and,\n\nBerger, Joe, 61\u201362\n\nBerkman, Lance,\n\nBlough, Brian, 179\u2013180\n\nBoll, Randy, 41\u201342\n\nbrain\n\nactivity during sleep, 16\u201317\n\nchannel capacity and, xvi\n\nchemical reactions in upon receiving reward,\n\neffect of focusing on problem on, ,\n\nhabit formation and,\n\ninformation retention and reticular activating system, 4\u20135\n\n_The Brain That Changes Itself_ (Doidge),\n\nbreathing\n\nalleviating anxiety and,\n\nMental Workout and, ,\n\none-on-one communication and control of, ,\n\nBryant, Bear,\n\nbusinesspeople, successful athletes and successful, xix\u2013xx\n\nCarpenter, Chris, xiii, xiv\n\ncelebration of progress,\n\ncentering breaths\n\nalleviating anxiety and,\n\nMental Workout and, ,\n\none-on-one communication and, ,\n\nchannel capacity, xvi\u2013xvii\n\nchoosing wisely and, 21\u201322\n\ncorporate training seminars and, ,\n\novercommitment and, 35\u201337\n\nskill mastery and, xxii, xxiii\n\nchoosing goals wisely,\n\naction tool,\n\nchoosing one goal, 29\u201331\n\nexamples, , 190\u2013191\n\ninformation addiction, 27\u201329\n\nintroduction to, 21\u201327\n\n\"nailing it,\"\n\npower of one, 31\u201334\n\nsaying no, 35\u201337\n\nsummary,\n\ntop 3 takeaways for, 82\u201383\n\nChurchill, Winston,\n\nclient meetings, improving efficiency of, 52\u201354\n\nClinton, Bill,\n\ncommitment\n\nto doing,\n\nto spending time on most important task,\n\n_See also_ Overcommitment\n\ncommunication, 131\u2013152\n\naction tool,\n\ndelivering, 139\u2013142\n\nof details, 133\u2013134\n\nexamples, 151\u2013152\n\nintroduction to, 131\u2013136\n\none-on-one, 145\u2013149\n\npreparation and, 131\u201332,\n\nsuccess and in-person,\n\nsuccess triangle and, 142\u2013144\n\nsummary,\n\ntop 3 takeaways for, 154\u2013155\n\nuse of repetition in, 141\u2013142\n\nwriting script, 136\u2013139\n\nconference calls, running effective,\n\nconfidence\n\ncommunication success and, ,\n\nguidelines to maintain, 186\u2013187\n\nperformance and, 185\u2013187\n\npreparation and, 8\u201312,\n\nprocess-oriented goals and,\n\nsuccessful habit formation and,\n\nSuccess Log and,\n\ncontrol, focusing on factors you can, 164\u2013168, 182\u2013184\n\nconversation, successful, 145\u2013149\n\ncorporate training seminars,\n\ncontrolling amount of information offered in, 29\u201330\n\nCovey, Stephen,\n\ncrunch time,\n\ndaily success log,\n\n\"deep practice,\"\n\ndelivery of presentation\/speech, 139\u2013142\n\ndetails\n\ncommunicating the, 133\u2013134\n\nvisualization and focus on,\n\ndiscouragement, habit formation and, 71\u201372\n\ndisruption, habit formation and,\n\ndistractions\n\nconference calls and,\n\ndealing with daily, 24\u201325\n\nDoidge, Norman,\n\ndoing, commitment to,\n\ndone-wells\n\nexamples, , 127\u2013130\n\nidentifying, 101\u2013102,\n\nDuncan, Dave, xiv\n\nEdward Jones, xviii, , ,\n\nefficiency\n\nprioritizing priorities and increase in,\n\ntime maximization and,\n\ntrimming fact in schedule and increase in, 50\u201354\n\nEinstein, Albert,\n\nemergencies, planning on  occurrence of,\n\nemotions\n\nhabit formation and, ,\n\nviable excuses and, 161\u2013162,\n\nvisualization and, 123\u2013124\n\nenergy level of voice, one-on-one communication and, , 146\u2013147\n\nESPN online, as time waster,\n\n_Executive Toughness_ (Selk), xvii\n\nexpectancy theory, ,\n\nexpectations, unattainable, 32\u201333\n\nFacebook, as time waster,\n\nfactors you can control\n\ncommon examples of, 167\u2013168\n\nfocusing on, 164\u2013168, 182\u2013184\n\nfamilies, using Organize Tomorrow Today concepts, 178\u2013184\n\nfeedback loop, building,\n\nfight-thru phase of habit formation, 61\u201362, 65\u201371\n\nask two questions and,\n\nlife projection and, 67\u201368\n\nrecognition and,\n\nritualization and,\n\nfinancial advisors\n\nexample of choosing goals wisely, , 190\u2013191\n\nexample of commitment to habit formation, 79\u201380\n\nexample of effective communication,\n\nexample of Mental Workout, 128\u2013129\n\nexample of self-evaluation, 105\u2013106,\n\nexample of setting process goals, 97\u201398\n\nexample of time maximization,\n\nexample of use of Organize Tomorrow Today,\n\nexamples of viable excuses,\n\nfirst-person vantage point, for visualization,\n\nfootball players\/coaches\n\nexample of choosing goals wisely, 26\u201327\n\nexample of focus on fundamental tasks, 1\u20132\n\nperfectionist pattern and,\n\n_The Four Agreements_ (Ruiz), 116\u2013117\n\nFranklin, Benjamin,\n\nFreese, David,\n\nfrustration, focus on factors out of one's control and, ,\n\nfundamentals, focus on, xxi, 1\u20132, 85\u201386\n\nGable, Dan, ,\n\ngame clock, setting, 50\u201351\n\nGassoff, Bobby, 22\u201323\n\nGayanski, Jeff, 182\u2013184\n\ngoals\n\nchoosing wisely ( _see_ choosing goals wisely)\n\nchoosing wrong, 33\u201334\n\nestablishing process-oriented, 93\u201398\n\nsetting attainable,\n\nGoodrich, Gail,\n\nGoogle searches, as time waster,\n\ngroup meetings, improving efficiency of, 51\u201352\n\nhabit, making prioritizing daily list-making a,\n\nhabit formation, 61\u201380,\n\naction tool,\n\nexamples, 79\u201380\n\nfight-thru phase, 65\u201371\n\nhoneymoon phase, 64\u201365\n\nintroduction to, 61\u201364\n\n\"puke-a-licious\" workouts, 76\u201377\n\nsecond nature phase, 71\u201376\n\nof self-evaluation, 104\u2013106\n\nsummary,\n\ntime required for, 62\u201363\n\ntop 3 takeaways for,\n\nHamilton, Josh,\n\nHayett, Rosey, 180\u2013181\n\nheadline news, as time waster,\n\nheart rate, centering breath and control of,\n\nhockey players,\n\nexample of performance mentality, 90\u201391\n\nexample of using \"ask and chop,\"\n\n\"puke-a-licious\" workouts and, 76\u201377\n\nholding-area lists, 13\u201314\n\nhoneymoon phase of habit formation, 64\u201365\n\nidentity statement, Mental Workout and, , 121\u2013122\n\nimportant, urgent _vs.,_\n\nimprovement, focus on single, 185\u2013187\n\ninformation\n\nlimiting amount communicated,\n\nwriting in longhand and retention of, 4\u20135\n\ninformation addiction, 27\u201329\n\ninsurance executive, example of self-evaluation,\n\nintegrity, rules for maintaining,\n\nintelligence, five levels of,\n\ninterruptions\n\nin conversation,\n\ndealing with routine, 24\u201325\n\ninversion tests\n\nof becoming abnormal,\n\nof choosing goals wisely,\n\nof communication,\n\nof fight-thru phase of habit formation,\n\nof Organize Tomorrow Today,\n\nof overcommitment,\n\nof self-evaluation,\n\nof self-talk,\n\nof time maximization,\n\nJacobi, Carl,\n\nJames, Lebron,\n\nJobs, Steve, 131\u2013132\n\nJocketty, Walt, xiii\n\nKardashians, as time waster,\n\nKing, Martin Luther King, Jr., 134\u2013135\n\nlanguage of a loser, 178\u2013179\n\nLa Russa, Tony, xiii\n\nlawyers\n\nexample of backsliding and habit formation, 73\u201376\n\nexample of time maximization,\n\nexample of viable excuses,\n\nlearning\n\nembracing continuous,\n\nteaching others and,\n\nLewin, Kurt,\n\nlife projection, habit formation and, 67\u201368\n\nlists\n\nholding-area, 13\u201314\n\nprioritizing items on written, 4\u20138\n\nusing open space to accomplish items on,\n\n_See also_ Organize Tomorrow Today (OTT)\n\n\"Lollapalooza Effect,\" xx\u2013xxi\n\nlong-term revenue collection activities,\n\nLopez, Marcus, 178\u2013179\n\nloser, language of, 178\u2013179\n\nmacro-scale activities,\n\n\"The Magical Number Seven, Plus or Minus Two\" (Miller), xv\u2013xvi\n\nMaltz, Maxwell, ,\n\nManning, Peyton, 26\u201327\n\nMark, John, 181\u2013182\n\nMark, Regan, 181\u2013182\n\nMartin, Jean-Pol,\n\nmeasurable goals, 94\u201395\n\nmeetings\n\nclient, 52\u201354\n\nconference calls,\n\ngroup, 51\u201352\n\nrespecting start and end times, 51\u201352\n\n_Men's Health_ (magazine), xvii, xix\n\nmental exercise, self-evaluation and self-encouragement and,\n\nmental image, replacing established,\n\nmental rehearsal, of communication, 142\u2013144\n\nmental toughness\/strength\n\nexample of building, 180\u2013181\n\nMental Workouts and increased,\n\nself-evaluation and,\n\nMental Workout, , 119\u2013125\n\nadoption by youth ski team, 180\u2013181\n\ncentering breath and, ,\n\ncommitment to, 31\u201332\n\nexample, 179\u2013180\n\nidentity statement and, , 121\u2013122\n\npersonal highlight reel and, 120\u2013121, 122\u2013124\n\nSt. Louis Cardinals and, xiii\u2013xv\n\nself-talk and, ,\n\nsteps in, 120\u2013121\n\nmessage, preparing succinct, 138\u2013139\n\nmicro-scale activities,\n\nMiller, George, xv\u2013xvi\n\nMiller's Law, xvi\n\nmomentum\n\nfocus on single primary task and creation of,\n\nhabit formation and,\n\nprioritized list and,\n\nmotivation, viable excuses and, ,\n\nMueller, Pam,\n\nmultitasking, xvii\n\nmyth of,\n\nMunger, Charlie, xx\u2013xxi, , ,\n\n\"nailing it,\" ,\n\nNavy SEALS training, 22\u201323\n\nNFL draft picks, choosing preparation tasks and, 26\u201327\n\nno-excuse mentality\n\nchildren adopting, 178\u2013179\n\ndeveloping, 161\u2013164\n\n\"normal,\"\n\n\"1 Must\" part of prioritized list, 6\u20137,\n\none-on-one communication, 145\u2013149\n\nopen space, attacking, 41\u201342, 44\u201347\n\nopen space number, ,\n\nOppenheimer, Dan,\n\norganizational-level meetings\n\nchoosing goals for,\n\ntraining seminars at, 27\u201329\n\n\"Organize Tomorrow Today\" workshops, xx\n\nOrganize Tomorrow Today (OTT), 1\u201319\n\naction tool,\n\ncommitting to routine time for, 12\u201316\n\nexamples, 8\u201312,\n\nintroduction to, 1\u20134\n\nmomentum as catalyst for,\n\nparts of list, 6\u20137\n\nprocess, 4\u20138\n\nreasons for success of, 16\u201317\n\nsummary,\n\ntop 3 takeaways for,\n\novercommitment\n\nchannel capacity and, 35\u201337\n\ncounterproductivity of,\n\neffect of, 23\u201324\n\ninaction and,\n\ninformation overload and,\n\npace\n\nof one-on-one conversation, 147\u2013148\n\nof presentation, 139\u2013142\n\nparalysis, procrastination and,\n\nPatton, George S.,\n\npauses\n\nin one-on-one conversation, 147\u2013148\n\nin presentations\/communication,\n\nPCT. _See_ Problem-centric thinking (PCT)\n\nperfectionist cycle,\n\navoiding,\n\nself-evaluation and, 88\u201390\n\nperformance\n\nconfidence and, 185\u2013187\n\nevaluation and,\n\nperformance mentality, 90\u201391\n\npersonal email, as time waster,\n\npersonal highlight reel, Mental Workout and, 120\u2013121, 122\u2013124\n\nperspective questions, on habit formation,\n\nphone conversations,\n\nphysicians\n\nexample of choosing goals wisely, ,\n\nexample of commitment to habit formation,\n\nexample of effective communication,\n\nexample of prioritizing priorities, 49\u201350\n\nexample of use of Organize Tomorrow Today,\n\nPippen, Scottie,\n\n\"+1 solution,\"\n\nPolian, Ed,\n\npositive reinforcement\n\nof progress,\n\nself-evaluation and, 107\u2013108\n\nSuccess Log and, 99\u2013103\n\npracticing, mental rehearsal _vs.,_\n\npreparation\n\ncommunication and, 131\u2013132, 136\u2013144\n\nconfidence and, 8\u201312,\n\nfor one-on-one conversation,\n\npresentations\n\ndelivery style and impact of,\n\nwriting script for, 136\u2013139\n\nprioritization\n\nfocus on factors out of one's control and impact on,\n\nof items on Success Log for improvement, 102\u2013103\n\nof priorities, 47\u201350\n\nof process-oriented goals, 181\u2013182\n\ntime maximization and,\n\nprioritizing day's activities. _See_ Organize Tomorrow Today (OTT)\n\npro athletes\n\nconfidence and,\n\nexample of choosing goals wisely, ,\n\nexample of commitment to habit formation, , 76\u201377,\n\nexample of effective communication,\n\nexample of Mental Workout, 127\u2013128\n\nexample of self-evaluation,\n\nexample of time maximization,\n\nexample of use of ask and chop method, 56\u201357\n\nexample of use of Organize Tomorrow Today,\n\nexample of viable excuses,\n\nperfectionist mentality and,\n\nperformance mentality and, 90\u201391\n\nsuccessful businesspeople and successful, xix\u2013xx\n\nvisualization and, 117\u2013118\n\n_See also individual sports_\n\nproblem-centric thinking (PCT),\n\ngiving in to, 168\u2013172\n\nself-talk and, 115\u2013117\n\nprocess-oriented goals, 93\u201398\n\nprioritizing, 181\u2013182\n\nprocrastination, 54\u201357\n\nproductive behavior, short-term _vs._ long-term revenue collection and,\n\nproductivity\n\nattacking open space and,\n\nOrganize Tomorrow Today and,\n\ntime maximization and, ,\n\nproduct-oriented goals, 93\u201395\n\n_Psycho-Cybernetics_ (Maltz), ,\n\nPujols, Albert, xiii\n\n\"puke-a-licious\" workouts, 76\u201377\n\nRAS. _See_ Reticular activating system (RAS)\n\nrecognition, of barriers to habit formation,\n\nreinforcement\n\nof progress,\n\nself-evaluation and, 107\u2013108\n\nSuccess Log and, 99\u2013103\n\nRelentless Solution Focus (RSF), 170\u2013172\n\nrepetition\n\nmastery and,\n\nuse of in communication, 141\u2013142\n\nresult-oriented goals, 93\u201395\n\nreticular activating system (RAS), 4\u20135\n\nrewards\n\nexample of effective,\n\nfor progress,\n\nself-evaluation and, 107\u2013108\n\nfor success, 99\u2013103\n\nritualizing new habits, , 185\u2013187\n\nRolen, Scott, xiii, xiv\n\nRousey, Ronda,\n\nroutinizing self-evaluation, 104\u2013106\n\nRSF. _See_ Relentless Solution Focus (RSF)\n\nRuiz, Don Miguel, 116\u2013117\n\nSaban, Nick, 1\u20132\n\nSt. Louis Cardinals,\n\nMental Workout and, xiii\u2013xv,\n\nWorld Series and rejection of problem-centric thinking, 168\u2013169\n\n_See also_ baseball players\n\nsales calls, improving efficiency of, 52\u201353\n\nsales role\n\nexample of Mental Workout, 129\u2013130\n\nexample of use of Organize Tomorrow Today, 9\u201310,\n\nschedule bloat,\n\nscouting reports, 159\u2013160\n\nsecond nature phase of habit formation, 71\u201376\n\ndiscouragement monster and, 71\u201372\n\ndisruption and,\n\nseduction of success and, 72\u201376\n\nself-confidence\n\ncommunication and projection of,\n\nperfectionist mentality and, 89\u201390\n\nself-doubt, information overload and,\n\nself-encouragement,\n\nself-evaluation, 85\u2013110,\n\naction tool,\n\ndefining success by effort and, 91\u201392\n\nexamples,\n\nintroduction to, 85\u201390\n\nperformance mentality and, 90\u201391\n\npositive reinforcement and, 107\u2013108\n\nreinforcing positive action and, 99\u2013103\n\nroutinizing, 104\u2013106\n\nSuccess Log sample, 98\u201399\n\nsummary,\n\ntop 3 takeaways for, 153\u2013154\n\nusing Success Log for, 92\u201398\n\nself-image, self-talk and, 114\u2013115, 116\u2013117\n\nself-talk, 111\u2013130\n\naction tool,\n\nexamples, 127\u2013130\n\nintroduction to, 111\u2013115\n\nMental Workout and, 119\u2013125\n\nproblem-centric thinking and, 115\u2013117\n\nself-image and, 114\u2013115, 116\u2013117\n\nsummary,\n\ntop 3 takeaways for,\n\nvisualization and, 117\u2013119\n\nSelk, Jason, xiii\u2013xv, xvii\u2013xviii, xix\u2013xxi, xxiii\u2013xiv\n\n_The Habits of Highly Effective People_ (Covey),\n\n_Shape_ (magazine), xvii\n\nshopping sites, as time waster,\n\nShort, Dave, xviii\n\nshort-term revenue collection activities,\n\nskill mastery, xxi\u2013xxii\n\nSkinner, B. F., 6\u20137\n\nsleeping, completing tasks while, 16\u201317\n\nslowing down pace\n\nof one-on-one conversation, 147\u2013148\n\nof presentation delivery, 139\u2013142\n\nsmartphone\n\nholding-area list on, 13\u201314\n\nprioritized list on,\n\nsoftware salesman, example of Mental Workout, 129\u2013130\n\nsolution, focus on finding, 170\u2013172\n\nspeed, visualizing at desired,\n\nSpieth, Jordan,\n\nsports, choosing goals wisely in, 25\u201327, . _See also_ Pro athletes; _individual sports_\n\nstories, for presentations,\n\nsuccess\n\ndefining, ,\n\neffort and evaluation of effort and,\n\nexpectation of continued, 87\u201390\n\nwith habit formation and danger of backsliding, 72\u201376\n\nin-person communication and,\n\nmeasuring, xx\n\nrewards for, 99\u2013103\n\nSuccess Log\n\nchoosing affirmative statement to improve, 102\u2013103\n\nas reinforcement tool, 100\u2013103\n\nsample, 98\u201399\n\nusing for self-evaluation, 92\u201398\n\nsuccess stories, 178\u2013184\n\nsuccess triangle, communication and, 142\u2013144\n\nSutton, Katie, 111\u2013113\n\nSuzuki,\n\ntasks\n\ncompletion of during sleep, 16\u201317\n\nholding-area list, 13\u201314\n\norganizing and prioritizing _(see_ Organize Tomorrow Today (OTT))\n\npreparation (see choosing goals wisely)\n\ntask-specific tension,\n\nTED talks, ,\n\n_10-Minute Toughness_ (Selk), xvii, xix\n\ntest preparation business, 158\u2013159\n\n\"3 Most Important\" part of prioritized list, ,\n\ntime\n\nfor accomplishing prioritized list, 14\u201315\n\nshrinking units of, 44\u201347\n\ntime management\n\nchannel capacity and, xvii\n\ntime maximization _vs.,_ 42\u201343\n\ntime management industry, xv\n\ntime maximization, 41\u201359\n\naction tool, 188\u2013189\n\nattacking open space and, 44\u201347\n\nexamples,\n\nintroduction to, 41\u201344\n\nprioritizing priorities and, 47\u201350\n\nprocrastination and, 54\u201357\n\nsummary,\n\ntime management _vs.,_ 42\u201343\n\ntop 3 takeaways for,\n\ntrimming fat and, 50\u201354\n\ntime wasters, top seven,\n\nto-do lists\n\nprioritizing items on _(see_ Organize Tomorrow Today (OTT))\n\nusing open space to accomplish,\n\ntriggering incident, to kick off habit drive,\n\n2\u20131-1 method, for client meetings,\n\nTyson, Mike,\n\nUrgent, important _vs.,_\n\nvan Dyke, Henry,\n\nviable excuses\n\ncommon,\n\nexamples of, 174\u2013175\n\ntrap of, 159\u2013164\n\nvisualization\n\nguidelines for, 123\u2013124\n\npersonal highlight reel and, 122\u2013124\n\nof presentation, 142\u2013144\n\nself-talk and, , , 117\u2013119\n\nvoice, one-on-one communication and energy level of, , 146\u2013147\n\nWalton, Bill,\n\n\"wholesalers,\"\n\nwinning, establishing as habit,\n\nWooden, John, xi\u2013xii,\n\nconditioning program and focus on fundamentals, 85\u201386\n\ncontinuous learning and,\n\nevaluating players during game,\n\nfather's advice to, 177\u2013178\n\nfocus on factors one can control, 164\u2013165\n\nfundamentals and, xxi, 85\u201386\n\nno excuse mentality,\n\npaying attention to details, 133\u2013134\n\nritualized habits and,\n\nstarting meetings on time,\n\nWooden, Joshua, 177\u2013178\n\nwriting a script, 136\u2013139\n\nwritten lists, prioritizing items on, 4\u20138\n\nZeigarnik, Bluma,\n\nZeigarnik Effect, \nTable of Contents\n\n  1. Cover\n  2. Title Page\n  3. Copyright\n  4. Dedication\n  5. Contents\n  6. Foreword\n  7. Introduction: Sports Psychology Meets Wall Street\n  8. 1. Organize Tomorrow Today\n  9. 2. Choose Wisely\n  10. 3. Maximize Your Time\n  11. 4. Win Your Fight-Thrus\n  12. Halftime\n  13. 5. Evaluate Correctly\n  14. 6. Learn How to Talk to Yourself\n  15. 7. Learn How to Talk with Others\n  16. Crunch Time\n  17. 8. Become Abnormal\n  18. Epilogue: Inspiration and Beyond\n  19. Notes\n  20. About the Authors\n  21. Index\n\n# Guide\n\n  1. Cover\n  2. Contents\n  3. Title Page\n\n  1. i\n  2. ii\n  3. iii\n  4. iv\n  5. v\n  6. vi\n  7. vii\n  8. viii\n  9. ix\n  10. x\n  11. xi\n  12. xii\n  13. xiii\n  14. xiv\n  15. xv\n  16. xvi\n  17. xvii\n  18. xviii\n  19. xix\n  20. xx\n  21. xxi\n  22. xxii\n  23. xxiii\n  24. xxiv\n  25. \n  26. \n  27. \n  28. \n  29. \n  30. \n  31. \n  32. \n  33. \n  34. \n  35. \n  36. \n  37. \n  38. \n  39. \n  40. \n  41. \n  42. \n  43. \n  44. \n  45. \n  46. \n  47. \n  48. \n  49. \n  50. \n  51. \n  52. \n  53. \n  54. \n  55. \n  56. \n  57. \n  58. \n  59. \n  60. \n  61. \n  62. \n  63. \n  64. \n  65. \n  66. \n  67. \n  68. \n  69. \n  70. \n  71. \n  72. \n  73. \n  74. \n  75. \n  76. \n  77. \n  78. \n  79. \n  80. \n  81. \n  82. \n  83. \n  84. \n  85. \n  86. \n  87. \n  88. \n  89. \n  90. \n  91. \n  92. \n  93. \n  94. \n  95. \n  96. \n  97. \n  98. \n  99. \n  100. \n  101. \n  102. \n  103. \n  104. \n  105. \n  106. \n  107. \n  108. \n  109. \n  110. \n  111. \n  112. \n  113. \n  114. \n  115. \n  116. \n  117. \n  118. \n  119. \n  120. \n  121. \n  122. \n  123. \n  124. \n  125. \n  126. \n  127. \n  128. \n  129. \n  130. \n  131. \n  132. \n  133. \n  134. \n  135. \n  136. \n  137. \n  138. \n  139. \n  140. \n  141. \n  142. \n  143. \n  144. \n  145. \n  146. \n  147. \n  148. \n  149. \n  150. \n  151. \n  152. \n  153. \n  154. \n  155. \n  156. \n  157. \n  158. \n  159. \n  160. \n  161. \n  162. \n  163. \n  164. \n  165. \n  166. \n  167. \n  168. \n  169. \n  170. \n  171. \n  172. \n  173. \n  174. \n  175. \n  176. \n  177. \n  178. \n  179. \n  180. \n  181. \n  182. \n  183. \n  184. \n  185. \n  186. \n  187. \n  188. \n  189. \n  190. \n  191. \n  192. \n  193. \n  194. \n  195. \n  196. \n  197. \n  198. \n  199. \n  200. \n  201. \n  202. \n  203. \n  204. \n  205. \n  206. \n  207. \n  208. \n  209. \n  210. \n  211. \n  212. \n  213. \n  214. \n  215. \n  216. \n  217. \n  218. \n  219. \n  220. \n  221. \n  222. \n  223. \n  224. \n  225. \n  226. \n  227. \n  228. \n  229. \n  230. \n  231. \n  232. \n  233. \n  234. \n  235.\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nAlso by Virginia and Lee McAlester\n\nA Field Guide to America's Historic Neighborhoods and Museum Houses: The Western States\n\nGreat American Houses and Their Architectural Styles\n\nBy Virginia Savage McAlester, Willis Cecil Winters FAIA, and Prudence Mackintosh; photography by Steve Clicque\n\nThe Homes of the Park Cities, Dallas: Great American Suburbs\n\nTHIS IS A BORZOI BOOK  \nPUBLISHED BY ALFRED A. KNOPF\n\nCopyright \u00a9 1984, 2013 by Virginia Savage McAlester\n\nAll rights reserved. Published in the United States by Alfred A. Knopf, a division of Random House, LLC, New York, and in Canada by Random House of Canada Limited, Toronto, Penguin Random House Companies. Originally published in the United States in different form by Alfred A. Knopf, a division of Random House, Inc.  \nwww.aaknopf.com\n\nKnopf, Borzoi Books, and the colophon are registered trademarks of Random House LLC.\n\nLibrary of Congress Cataloging-in-Publication Data  \nMcAlester, Virginia Savage, [date]\n\nA field guide to American houses \/ by Virginia Savage McAlester ; revised and expanded from the orginal edition written by Virginia and Lee McAlester ; with drawings by Lauren Jarrett and model house drawings by Juan Rodriguez-Arnaiz ; revision drawings by Suzanne Patton Matty and photographs by Steve Clicque.\u2014Second edition.  \npages cm  \nIncludes bibliographical references and index.  \nISBN 978-1-4000-4359-0  \neBook ISBN: 978-0-385-35387-8  \n1. Architecture, Domestic\u2014United States\u2014Guidebooks. 2. United States\u2014Guidebooks. I. McAlester, A. Lee (Arcie Lee), [date] II. Jarrett, Lauren, illustrator. III. Rodriguez-Arnaiz, Juan, illustrator. IV. Title.\n\nNA7205.M35 2013  \n728.0973\u2014dc23 2013018432\n\neBook ISBN: 9780385353878\n\nCover illustrations by Juan Rodriguez-Arnaiz and Lauren Jarrett  \nCover design by Linda Huang  \nBook design by Cassandra Pappas, adapted for eBook\n\nFirst Edition published June 12, 1984  \nSecond Edition\n\nv3.1\nTO | Carty Talkington  \nMartine McAlester  \nAmy Talkington  \nKeven McAlester  \nClementine Adams  \nVirginia Adams  \n---|---\n\n# Contents\n\n_Cover_\n\n_Other Books by This Author_\n\n_Title Page_\n\n_Copyright_\n\n_Dedication_\n\nHow to Use This Book\n\nPreface\n\nPictorial Key\n\nPictorial Glossary\n\nLooking at American Houses\n\n[STYLE:  \nThe Fashions of American Houses](Mcal_9780000000000_epub_c01_r1.htm)\n\n[FORM:  \nThe Shapes of American Houses](Mcal_9780000000000_epub_c02_r1.htm)\n\n[STRUCTURE:  \nThe Anatomy of American Houses](Mcal_9780000000000_epub_c03_r1.htm)\n\n[NEIGHBORHOODS:  \nThe Groupings of American Houses](Mcal_9780000000000_epub_c04_r1.htm)\n\nFolk Houses\n\nNative American\n\nPre-Railroad\n\nNational\n\nManufactured\n\nColonial Houses (1600\u20131820)\n\nPostmedieval English\n\nDutch Colonial\n\nFrench Colonial\n\nSpanish Colonial\n\nGeorgian\n\nFederal\n\nEarly Classical Revival\n\nRomantic Houses (1820\u20131880)\n\nGreek Revival\n\nGothic Revival\n\nItalianate\n\nExotic Revivals\n\nOctagon\n\nVictorian Houses (1860\u20131900)\n\nSecond Empire\n\nStick\n\nQueen Anne\n\nShingle\n\nRichardsonian Romanesque\n\nFolk Victorian\n\nEclectic Houses (1880\u20131940)\n\nEnglish and Anglo-American Period Houses\n\nColonial Revival\n\nNeoclassical\n\nTudor\n\nFrench Period Houses\n\nChateauesque\n\nBeaux Arts\n\nFrench Eclectic\n\nMediterranean and Spanish Period Houses\n\nItalian Renaissance\n\nMission\n\nSpanish Revival\n\nMonterey\n\nPueblo Revival\n\nModern Houses (1900\u2013present)\n\nEarly Modern\n\nPrairie\n\nCraftsman\n\nModernistic\n\nBankers Modern\n\nMinimal Traditional\n\nRanch\n\nSplit-Level\n\nMainstream Modern\n\nInternational\n\nContemporary\n\nShed\n\nOther 20th-Century Modern\n\n21st-Century Modern\n\nStyled Houses Since 1935\n\nMansard\n\nStyled Ranch\n\nMillennium Mansion\n\nNew Traditional\n\nAmerican Vernacular\n\nAppendix\n\nApproaches to Construction in the 20th and 21st Centuries\n\nNotes\n\nFor Further Reference\n\nAcknowledgments\n\nAbout the Illustrations\n\nPhoto Credits\n\nIndex\n\n# How to Use This Book\n\nMost of the chapters in this book treat one of the major architectural fashions, or _styles,_ that have been popular over our country's history. These chapters are arranged roughly chronologically, with the earliest styles first. The opening page of each chapter features a large drawing showing the three or four most important _identifying features_ which differentiate that style from others. The most common shapes, or _principal subtypes,_ of each style are also pictured on the opening page, along with references to pages of photographs in the chapter that allow the reader to see quickly the common features in a range of examples from each particular style and subtype. Most chapters also include drawings that show typical smaller details\u2014for example, windows, doors, and roof-wall junctions\u2014that cannot easily be seen in full-house photographs. Text supplementing the drawings and photographs discusses the _identifying features, principal subtypes, variants and details,_ and _occurrence_ of each style. Concluding _comments_ provide a brief introduction to the origin and history of the style. A few later chapters have a less detailed treatment of a style or styles. Generally these cover either revivals of earlier styles or less common ones.\n\nConfronted with an unfamiliar house to be identified, the reader may approach the problem three different ways. The simplest is to flip through the many pages of house photographs, looking for examples similar to the unidentified house. Here one should pay particular attention to such large-scale features as roof form (gabled or hipped, low or steeply pitched?) and facade balance (symmetrical or asymmetrical?). When a similar photograph is located, the unknown example should be compared in smaller-scale features of architectural detailing: windows, doors, roof-wall junctions, porches, etc. The additional photographs and drawings provided in each chapter will aid in this process, which can be repeated until a final identification is made.\n\nA second and more systematic approach is to turn to the Pictorial Key and Pictorial Glossary. This illustrates a variety of different types of such common architectural features as windows, doors, and roofing materials, with a listing of the styles in which each type commonly occurs. Features are listed beginning with the roof\u2014and roof-level details such as dormers\u2014and move down the house to cornices, windows, doors, porches, and, at ground level, even foundation details. Using the Key, the reader will find that a house with a red tile roof, for example, will most likely be found in either the Spanish Colonial, Spanish Revival, or New Traditional styles. Photographs and drawings for these styles can then be compared with the unknown house as in the first approach.\n\nA final approach is to become familiar with the relatively few historical precedents on which American house styles are based. These are reviewed in the introductory chapter on _Style._ With this background, one can learn to quickly determine if a house is of Modern, Medieval, Renaissance Classical, or Ancient Classical inspiration. With a bit of further practice, it becomes easy to distinguish between the half-dozen or so principal American styles that have been based on each of these traditions. With this knowledge, style identification can become almost automatic. The book then becomes a useful backup reference for identifying stylistic subtypes and subtleties.\n\n# How to Use This eBook\n\nThis eBook edition has been optimized for screens, and contains images that can be enlarged to better focus on the house details described. Color photographs have been included when available, for e-reading platforms that support color. The content has been slightly reordered from the print edition for the best presentation in a reflowable eBook format.\n\nIn this presentation, illustrations and photographs immediately follow the text that describes them. The most visible difference is that the eBook format allows the photographs for each style subtype to be grouped early in each style chapter, immediately after the description of the subtype\u2014rather than located by references to a photo spread towards the end of the chapter as one experiences in the printed book.\n\nThe greatest delight is that double-clicking enlarges each photograph and on some devices pinching the screen can make parts of the images even larger. This allows details of windows, doors, porches and cornices to be examined more closely.\n\n# Preface\n\nLooking at houses today is even more interesting and challenging than it was in 1984, the year this guide was first published. Almost 80 percent of the houses in the United States today have been built since 1940, the date the first edition concluded its full coverage. Surprisingly, during the last twenty-five years there has been a reinterpretation and new building of most American house styles that existed before 1940. Knowledge of earlier architectural styles\u2014both traditional and modern\u2014is now a necessity for understanding new houses, not just historic ones.\n\nThe first edition of _A Field Guide to American Houses_ ended very naturally at 1940. The Great Depression of the 1930s had effectively ended construction of Eclectic house styles, and when construction resumed at the end of World War II, both house styles and lifestyles had shifted dramatically. An appendix to the original book gave a brief overview of styles built from 1940 to 1984. It was my intention at the time to expand the 1984 publication to fully cover these, particularly the mid-century modern homes gradually reaching their fiftieth birthday, making them eligible for National Register listing. This updated revision intends to fulfill that original goal. While many of the homes built soon after World War II are now eligible for listing on the National Register of Historic Places, they are nevertheless among the most endangered of houses\u2014primarily because of their choice locations and generally smaller size. With this book I hope to help in their preservation.\n\nTwo additional topics were added during the course of revision: an overview of the extraordinary variety of styles built during the millennial housing boom (1990\u20132008) and a section on neighborhoods that covers the typical ways American houses are grouped together. The new chapter on neighborhoods provides a brief introduction to typical American communities. Just as _A Field Guide to American Houses_ concentrated on typical homes, not architectural landmarks, the neighborhoods section of this new edition concentrates on typical neighborhoods, not landmarks of planning. It presents the simple patterns of development repeated in cities throughout the United States, rather than offering a study of mostly one-of-a-kind planning icons.\n\nWhen it came time to write a preface for this expanded edition, it was suggested I might write an entirely new one. However, after rereading the original I found it so concise I could see no way to improve upon it. Thus, it follows here, because I believe it is as relevant to this new edition as it was in 1984. In updating this work, I have tried to adhere to the same approaches to research and presentation as the 1984 edition, described below in the original preface.\n\nThis book grew from the authors' efforts to identify the houses found in typical American neighborhoods. Many excellent guides are available to detail the features of our country's monumental dwellings; other works deal with the everyday houses of specific towns and neighborhoods. What was lacking, we discovered, was a guide that related the architectural landmarks to their far more numerous cousins throughout the country\u2014the common houses that make up most of our nation's built environment. Our book attempts to fill this gap by treating the entire spectrum of American domestic building, from the most modest folk houses to the grandest mansions, but with a heavy emphasis on the familiar dwellings that lie between these extremes. It is intended not as a scholarly treatise on architectural history, but as a practical field manual for identifying and understanding the changing fashions, forms, and components of American houses. In treating this broad subject we have imposed certain limitations on the coverage. Because it emphasizes field identification, it concerns primarily the _exterior_ appearance of houses. The important subjects of interior planning, design, and detailing are given only the most superficial treatment. The book also concentrates on _styled_ houses built before 1940. Unstyled folk houses and post-1940 houses are included, but are treated in considerably less detail. The principal focus is on _single-family houses,_ which may be either detached or, like attached urban houses, built with common walls. A few duplexes or triplexes built in the form of single-family buildings have been illustrated, but larger multi-unit dwellings are not included.\n\nThe book is organized chiefly by the chronology of changing architectural styles. In one sense these may be considered a merely ephemeral and somewhat superficial series of fashions. More fundamentally, however, they reflect the tastes and sensibilities of our forebears over three centuries of dynamic history. Looking still farther, most have deep roots in European history, whence they draw on Renaissance, Medieval, and Classical models for inspiration. An understanding of these stylistic traditions as they have repeatedly reappeared during our nation's history is, we believe, the most practical framework for identifying and understanding American houses.\n\nA principal difficulty in stylistic analysis involves recognizing underlying similarities in style when buildings differ in size, shape, and degree of formality. Many Greek Revival houses, for example, bear little resemblance to Greek temples, but almost all show certain key features that can, with a bit of practice, be easily recognized. Such features are emphasized in schematic drawings placed on the opening pages of every chapter. Beneath are placed sketches of Principal Subtypes, usually based on roof form, to which the characteristic features of the style are most commonly applied. Photographs provide a variety of typical houses of each subtype as a further aid to recognizing a style in its many guises, while drawings illustrate typical architectural details.\n\nThe many photographs of typical houses are, we feel, the heart of the book. In choosing these and preparing their descriptive captions, we have attempted to follow certain guidelines for consistency. Most of the houses illustrated are still in existence; photographs of examples known to have been destroyed are included for only a few rare styles of subtypes. All houses are identified by the state and town (or county for rural houses) in which they are located; where possible, the year construction was completed, the last name of the first owner (\"Johnson House\"), and the architect are also noted. We have usually relied on secondary sources for this information and have not attempted the enormous task of documentation from original sources. Precise addresses have been omitted to protect the occupants, since most of the houses illustrated are privately owned and not open to the public. Where secondary documentation of the date of construction was lacking, as was the case for the bulk of the examples illustrated, we have estimated the dates, using the forms \"late 19th century,\" \"1920s,\" and \"ca. 1905\" to indicate increasing certainty of attribution. Familiar names of landmark houses (Biltmore, House of Seven Gables) are sometimes used instead of, or in addition to, the name of the original owner.\n\nMost of the styles of American houses have been previously recognized and described, and we have thus drawn heavily on the works of others in preparing the book. Defining stylistic subtypes and characteristic details has, however, required much original research. To this end we have reviewed and analyzed photographs of more than a hundred thousand houses. In addition, we have traveled to almost every state to study and photograph the spectrum of American houses first hand. Even so, we can only claim to have scratched the surface of stylistic analysis. Most styles, we discovered, provided problems enough to occupy the energies of architectural historians for many years. We very much hope that our preliminary efforts will lead to such refinement and correction.\n\nIt has since been very gratifying to find a great deal of research that was not available while working on the 1984 edition of this book. Much of the new information has been included here.\n\nFor readers wishing to know more about houses that interest them, internet research has made much new information available to even casual observers. Of particular value are several websites that offer information on most houses, including when they were likely first built and much more detailed information derived largely from county property tax records. Or one can directly use the county property tax records available online to determine an approximate date of construction. The ability to quickly and easily research these most basic facts for something approximating 80 percent of the houses in the United States built after 1920 has led to simpler, yet far more informed, house watching and house watchers. Do a web search on a single address and see what you find. You might be surprised.\n\n#  Pictorial Key\n\n# Pictorial Glossary\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\nFIVE-PART PLAN\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\nTHREE-PART PLAN\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\n**ROOF-WALL JUNCTION**\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\n**WINDOWS**\n\nPICTORIAL GLOSSARY\n\n**A FEW COMMON DESCRIPTIVE HOUSE TERMS**\n\n**URBAN HOUSE TYPES**\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF ENTABLATURES: CORNICES, FRIEZES & ARCHITRAVES**\n\nANCIENT CLASSICAL MODELS\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF ENTABLATURES: CORNICES, FRIEZES & ARCHITRAVES**\n\nDOORS\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF ENTABLATURES: CORNICES, FRIEZES & ARCHITRAVES**\n\nWINDOWS\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF ENTABLATURES: CORNICES, FRIEZES & ARCHITRAVES**\n\nWINDOW OR DOOR\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF ENTABLATURES: CORNICES, FRIEZES & ARCHITRAVES**\n\nROOF-WALL JUNCTIONS\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF PEDIMENTS**\n\nFULL-FACADE PORCH\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF PEDIMENTS**\n\nROOF-WALL JUNCTION, GABLE END\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF PEDIMENTS**\n\nCENTERED GABLE\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF PEDIMENTS**\n\nENTRY PORCH\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**USE OF PEDIMENTS**\n\nDOOR WINDOW\n\nPICTORIAL GLOSSARY\n\n**PARTS OF THE CLASSICAL ORDERS APPLIED TO HOUSES**\n\n**TYPES OF PEDIMENTS**  \nOgee and segmental are only common over doors and windows\n\n# Looking at American Houses\n\n* * *\n\nSTYLE\n\nThe Fashions of American Houses\n\nFORM\n\nThe Shapes of American Houses\n\nSTRUCTURE\n\nThe Anatomy of American Houses\n\nNEIGHBORHOODS\n\nThe Groupings of American Houses\n\n# LOOKING AT AMERICAN HOUSES\n\n# Style\n\n# The Fashions of American Houses\n\nDomestic buildings are of two principal sorts: folk houses and styled houses. Folk houses are those designed without a conscious attempt to mimic current fashion. Many are built by their occupants or by non-professional builders, and all are relatively simple houses meant to provide basic shelter, with little concern for presenting a stylish face to the world. Most surviving American houses are not folk houses but are styled; that is, they were built with at least some attempt at being fashionable. As such, they show the influence of shapes, materials, detailing, or other features that make up an architectural style that was currently in vogue. The bulk of this book is organized by the changing chronology of these American architectural fashions or styles, for they provide the most effective framework for identifying and understanding American houses.\n\nThe majority of styled American houses are loosely modeled on one of four principal architectural traditions: Ancient Classical, Renaissance Classical, Medieval, or Modern. The earliest, the Ancient Classical tradition, is based upon the monuments of early Greece and Rome. The closely related Renaissance Classical tradition stems from a revival of interest in classicism during the Renaissance, which began in Italy in the 15th century. The two classical traditions, Ancient and Renaissance, share many of the same architectural details.\n\nThe third tradition, the Medieval, separated the Ancient Classical and the Renaissance Classical in time. The Medieval tradition includes architecture based on the formal Gothic style used for church buildings during the Middle Ages, as well as that based upon the simpler domestic buildings of the same era. Most of the Medieval architecture that has influenced American houses originated in England and France.\n\nThe fourth tradition, the Modern movement, began in the late 19th century and continues to the present. It is based primarily on the lack of applied historically influenced ornamentation and a resulting external simplicity or \"honesty,\" as well as on spatial variation and manipulation made possible by new materials and construction techniques.\n\nOther traditions that have influenced American houses are mostly of Spanish origin. Both the simple structures built during the Spanish Colonial era in the United States and the more elaborate architecture of Spain and Latin America have inspired American domestic buildings. In addition, Oriental and Egyptian models have occasionally provided patterns for American house design.\n\nAs we shall see in the following pages, each of these traditions has produced several different styles of American houses as they have been interpreted and re-interpreted during different building eras. Stylistic mixtures are also common in American domestic architecture. These have been created both by those who knowingly sought the unusual, as well as by those who unwittingly combined historical precedents.\n\nThe following short summary of styles is illustrated with houses found in the U.S. before 1935. In the seventy-five years since, a forty-year interlude (1935\u20131975) dominated by Modernism, both Bankers and Mainstream have been followed by almost five decades (1965\u20132010) during which the broadest variety of styles ever built during a half century have been constructed. Almost every style illustrated here\u2014whether Ancient Classical, Renaissance Classical, Medieval, Modern, or Other\u2014has been revived in one or more forms. At the same time new approaches to modernism have been explored, although by a small percentage of the population.\n\n## Ancient Classical Styles\n\nThe monumental architecture of Greece from about the 9th to the 4th century B.C., and that of Rome from the 1st century B.C. to the 5th century A.D., provide the models for those styles based upon Ancient Classical traditions. An entry or full-facade porch supported by large columns, frequently two-story, or \"colossal,\" columns, is the feature most commonly associated with American houses patterned after Ancient Classical buildings. The columns are frequently in one of the classical orders, but simple squared and octagonal interpretations are also found. The houses usually have symmetrical facades and, most commonly, a low-pitched roof.\n\nThree styles of American houses are based on Ancient Classical precedents. The first is the Early Classical Revival style (1780\u20131830). These are loosely based upon Roman models and are simple side-gabled or hipped roof houses with full-height entry porches, usually with a classical pediment above.\n\nThe next Ancient Classical style is the Greek Revival (1820\u20131860). Greek Revival houses normally have a very wide band of trim beneath the eaves, mimicking the entablature of Greek temples. Many have full-facade or full-height entry porches with large columns, but some have only a small entry porch and others have no porch or columns\u2014although in this case there are usually pilasters. Both side-gabled and hipped roofs are common. In some regions, many of the houses built in this style are oriented so that the gable end becomes the front facade; these most closely resemble their prototype Greek temples.\n\nThe third Ancient Classical style is the Neoclassical (1895\u20131955). These are normally two-story houses with prominent full-height columns. In early examples, the columns usually have very elaborate capitals of either Roman or Greek inspiration; later examples often have slender squared columns reminiscent of those on Mount Vernon's facade facing the Potomac River. The forms of Neoclassical houses are varied, and secondary details characteristic of the later Renaissance Classical movement are often used.\n\nSTYLE\n\n **ANCIENT CLASSICAL**\n\n## Renaissance Classical Styles\n\nThe Renaissance Classical styles are based upon buildings built during the revival of interest in Ancient Classical models which began in Italy in the early 15th century and gradually worked its way northward to France, where it arrived in the mid-16th century, and to England, where it had little influence until the early 17th century. From England the Renaissance traveled still later to America, where it profoundly influenced 18th-century building in the English colonies. Each country developed somewhat different interpretations of Renaissance Classical ideals and each of these has inspired several later American styles. All, however, share certain features: they usually have balanced, symmetrical facades and typically have such decorative details as pedimented (crowned) doors and windows, dentils, quoins, and pilasters. Two-story columns are rare in American interpretations, although colonnaded, one-story entrance porches are frequent.\n\nIn America, the Italian version of the Renaissance tradition inspired both the Italianate (1840\u20131885) and Italian Renaissance (1890\u20131935) styles. Round arches (widely used by the early Romans) and cornice-line brackets are the two elements that most consistently mark American houses as having Italian Renaissance roots.\n\nThe French Renaissance tradition inspired the Second Empire (1855\u20131885) and the Beaux Arts (1885\u20131930) styles as well as some subtypes of the French Eclectic (1915\u20131945) style. A steeply pitched hipped roof, or dual-pitched mansard roof, is a characteristic feature of many of these French Renaissance-inspired houses.\n\nThe English version of the Renaissance tradition was exported to the American colonies as the Georgian (1700\u20131780) and Federal (1780\u20131820) styles. Note that these are original Renaissance buildings directly inspired by the earlier Italian Renaissance. Other American houses based on Renaissance traditions are later revivals of these same influences. These are included in the long-lived Colonial Revival style (1880\u20131955), which draws heavily on American Georgian and Federal precedents. All of these English-inspired Renaissance houses are united by an emphasis on elaborated front door surrounds. These frequently have side pilasters supporting an entablature or pediment (Georgian) or a fanlight above the door (Federal).\n\nSTYLE\n\n **RENAISSANCE CLASSICAL**\n\nSTYLE\n\n **RENAISSANCE CLASSICAL**\n\n**ITALIAN**\n\nSTYLE\n\n **RENAISSANCE CLASSICAL**\n\n**FRENCH**\n\nSTYLE\n\n **RENAISSANCE CLASSICAL**\n\n**ENGLISH**\n\n## Medieval Styles\n\nMedieval architecture is that built during the Middle Ages, the era from the end of the Ancient Classical period until the beginning of the Renaissance (roughly encompassing the 6th to the 15th centuries A.D.). Throughout most of this period, the primary focus of European, styled architecture was on ecclesiastical buildings\u2014churches, cathedrals, and abbeys. The dominant style for these was Romanesque from the end of the 9th century until the 12th century and Gothic from the mid-12th century until the beginning of the Renaissance. The Romanesque, modified from the Roman Ancient Classical tradition, is characterized by the extensive use of rounded arches. It has inspired only one American domestic style, the Richardsonian Romanesque (1880\u20131900). Gothic buildings are easily identified by their characteristic pointed arches, used over doors and windows and in interior vaulting. These inspired the Gothic Revival style (1840\u20131880) and, in addition, influenced the Stick style (1860\u2013ca. 1890).\n\nLarge-scale domestic architecture during most of the Middle Ages took the form of fortified castles. These have inspired relatively few American houses, although castellated parapets are sometimes used on Gothic Revival and Tudor houses. More modest European domestic buildings from Medieval times usually had roofs covered with thatch; such roofs had to be very steeply pitched in order to shed water properly. During much of the Middle Ages, chimneys were not yet in common use; attics under the steep roofs were left open so that smoke could escape through small openings in the roof. The chimney was a crucial invention, for it allowed second stories and attics to be floored to provide additional living space. Postmedieval houses with prominent chimneys typically have steeply pitched roofs, asymmetrical facades, and, frequently, wall surfaces elaborated either by exposed half-timbered construction or by protective clay tiles hung upon such construction. They also frequently have an overhanging second story that added structural stability to the original heavy timber framing. Such houses were imported directly to the New World as the Postmedieval English style (1600\u20131700) built by the earliest English colonists. They also provided the inspiration for many informal examples of the later Tudor style (1890\u20131940).\n\nWhen Renaissance influence began to spread to France and England from Italy, it first took the form of classical detailing being applied to houses of Medieval form. In France, this era produced many of the great chateaux that provided the inspiration for the American Chateauesque style (1880\u20131910). In England, such mixed Medieval-Renaissance buildings inspired the more formal examples of the American Tudor style, as well as some examples of the Queen Anne style.\n\nEnglish architects during the latter half of the 19th century, tiring of two hundred years of buildings dominated by classical influences, began to turn back to their Medieval heritage for inspiration. The result, the English Queen Anne movement, was quickly followed by the related American Queen Anne style (1880\u20131910), which used Medieval forms both with and without the addition of classical detailing. The Shingle style (1880\u20131900), which was also inspired by the English Queen Anne movement, introduced simplified exterior surfaces and open interior planning, and thus foreshadowed the Modern phase of architectural styling.\n\nSTYLE\n\n **MEDIEVAL**\n\nSTYLE\n\n **MEDIEVAL**\n\nSTYLE\n\n **MEDIEVAL**\n\n## Modern Styles\n\nThe Modern movement in domestic architecture developed in two stages during the years from 1900 to 1940. The first phase, the Arts and Crafts (or Early Modern) movement, deliberately turned its back on the use of historical precedent for decoration and design. Ornamentation was not eliminated but merely \"modernized\" to remove most traces of its historic origins. Low-pitched roofs with wide eave overhangs were favored. Although there were many variations within the movement, it led to two distinctive styles of American houses. The first was the Prairie style (1900\u20131920), which began in Chicago under the leadership of Frank Lloyd Wright, who designed many houses in the style during the period from 1900 to 1913. These elegantly simplified buildings by Wright and his followers were to have a profound influence on the beginnings of Modernism both here and in Europe. The second style inspired by the Arts and Crafts movement is the Craftsman style (1905\u20131930), begun in southern California in about 1903 by the Greene brothers and others. It emphasizes exposed structural members and wood joinery and, like the Prairie style, eschews formal historic precedents.\n\nA second phase of the Modern movement began after World War I as a full-scale reaction against all previous architectural tradition. The emphasis in this phase was on design that was clearly of the Machine Age, with standardization of parts, absence of all non-functional decoration, and structural \"honesty\" as hallmarks. Houses were to become \"machines for living.\" Flat roofs and smooth wall surfaces were favored. Both the Modernistic style (1920\u20131940) and the International style (1925\u2013present) are products of this more austere modernism that marked the beginning of Mainstream Modern.\n\nThe Mainstream Modern movement continued to evolve in the decades following 1940 as new stylistic outgrowths of the Arts and Crafts and the Machine Age movements arose. It was joined by a third phase, Bankers Modern, after 1935 in response to FHA's lending preferences. These were to dominate American domestic building during the 1950s and 1960s before beginning to be largely replaced, in the 1970s, by a return to stylistic adaptations loosely based on Classical or Medieval prototypes. Today, Modern movement houses are still being built but constitute a relatively small portion of new home construction.\n\nSTYLE\n\n **MODERN before 1935**\n\nSTYLE\n\n **MODERN before 1935**\n\n**ARTS AND CRAFTS MOVEMENT**\n\nSTYLE\n\n **MODERN before 1935**\n\nSee Modern Houses for Modern-style houses after 1935.\n\nSTYLE\n\n **MODERN before 1935**\n\n**MACHINE AGE MOVEMENT**\n\n## Other Stylistic Traditions\n\nMany of the other traditions that have influenced American domestic architecture are of Spanish or Spanish Colonial origin. In the New World, Spanish colonists blended the adobe building traditions of the Native Americans with similar Spanish housing traditions originally brought to Spain from North Africa. Both the Spanish Colonial style (1600\u20131850) and the Pueblo Revival style (1910\u2013present) use adobe construction techniques which show this mixing of Spanish and Native American precedents. Spanish Colonial ecclesiastical buildings of the American Southwest provided the inspiration for the Mission style (1890\u20131920). This was followed by the Spanish Revival style (1915\u20131940), which broadened the precedents to include the entire spectrum of Spanish and Spanish-American architecture, thus making it an unusually varied style. Some Spanish Revival houses have elaborate decorative detailing patterned after formal Spanish Renaissance buildings. Others show Moorish and Islamic influences, while still others are based upon rural Spanish folk houses with little or no decorative detailing. The most recent style in the Spanish tradition is the Monterey style (1925\u20131955), which is loosely based on certain houses of the American Southwest that show a mixing of Spanish and English Colonial influences.\n\nOther architectural traditions have influenced several minor styles of American houses. Oriental, Egyptian, and Swiss Chalet prototypes inspired the Exotic Revivals (1835\u2013ca. 1890), while the Octagon style (1850\u20131870) developed from one man's enthusiastic sponsorship of houses designed with unorthodox ground plans of octagonal shape.\n\nSTYLE\n\n **OTHER TRADITIONS**\n\n**SPANISH**  \ntiled roofs, usually red, commonly asymmetrical\n\nSTYLE\n\n **OTHER TRADITIONS**\n\n**OTHER**\n\n## Stylistic Mixtures\n\nMost American houses have been built in one of the many architectural styles outlined in the preceding sections. Some, however, do not fit neatly into one of these stylistic categories but, instead, have characteristics of two or more styles. Such houses may have been originally built as stylistic mixtures or may have resulted from later attempts to alter the style through remodeling.\n\nPrior to about 1840, American architectural styles were rather widely separated by time or by location; that is, only one fashion usually prevailed in a region over a long interval of time. Most early stylistic mixtures occurred during the transitional periods when these persistent fashions were changing. Thus, some transitional houses in the English colonies share Georgian and Federal features, while others blend Federal with Greek Revival detailing. Similarly, Dutch, French, and Spanish Colonial buildings began to show Federal or Greek Revival detailing as Anglo influence increased with the expansion of the country. These originally built combinations of styles increased after about 1840 when pattern books, particularly A. J. Downing's influential _Cottage Residences, Rural Architecture and Landscape Gardening,_ published in 1842, presented several choices of fashionable building styles. Downing, for example, advocated both Gothic and Italianate modes of design. As might be expected, some readers and builders avoided the choice by combining features of both. Another popular mixture of the romantic era added Italianate detailing to the previously dominant Greek Revival form.\n\nHouses of the Victorian era seldom show such dramatically obvious mixtures of style. Most Victorian styles are closely interrelated and draw heavily on Medieval precedents for inspiration. Thus they naturally tend to blend into one another. Steeply pitched roofs and textured wall surfaces are common to most. Stick-style structural members are found on many Queen Anne houses; Richardsonian arches occur on Shingle-style houses, wood-shingled walls may dominate on either Queen Anne or Shingle houses, and so on. Thus the separation of the Victorian styles sometimes becomes a matter of degree, whereas the dominant Greek-Gothic-Italianate modes of the preceding romantic era were unmistakably different.\n\nDuring the early years of the Eclectic era, experimental combinations of styles were common. From about 1890 to 1915, styles as different as the Colonial Revival, Neoclassical, Prairie, Tudor, Mission, and Craftsman were being built simultaneously. Many architects and builders experimented with fanciful combinations of these styles, sometimes adding a touch of Victorian detailing as well. Some early eclectic neighborhoods contain whole streets of such marvelously experimental stylistic combinations. These innovative houses were, however, far less common than more correct stylistic interpretations, even in the early eclectic years, and by 1915 they had all but disappeared. Thus began the era of the Period House, an interval in which stylistic combinations became rare. This lasted until the late 1930s, when architects and builders began to experiment with restrained mixtures of Tudor, Colonial Revival, and Mediterranean influences.\n\nA second and far more common category of stylistic mixture results not from original design, but from later alteration of an existing house. Most houses more than a few years old have had at least some exterior alterations, and it is important to be able to recognize these changes in order to identify a house's original style and appearance.\n\nMost exterior alterations are undertaken for one or more of the following purposes:\n\n\u2022 To update the appearance of the house (stylish updates)\n\n\u2022 To add additional living space to the house\n\n\u2022 To minimize exterior maintenance of the house\n\n\u2022 To take advantage of code exemptions\n\nEach of these changes tends to produce somewhat different exterior appearances.\n\nThe most common means of updating the appearance or style of a house is to add, remove, or alter a porch. The most frequent porch alteration, because it is the simplest, is to replace the original porch roof supports. For example, many a Queen Anne house had its spindlework supports replaced by heavy masonry piers during the early years of this century when the Prairie style was in vogue. The massive piers of Prairie and Craftsman houses, in turn, have often been replaced by narrow wooden or metal posts which dramatically alter the appearance of the house. Entire new porches are also commonly added to houses that originally had none, or as replacements for porches of quite different character. Many Queen Anne houses, for example, now sport elaborate Neoclassical porches added in the early 1900s. Equally startling are those houses which have had the original porches removed without replacement.\n\nFollowing porches, the most commonly altered facade elements are doors, windows, and wall materials, changed in an attempt to mimic current fashion. Most of these updating changes in facade details, including porches, are easily recognized with a little practice, for the new additions appear out of context with the overall shape, form, and materials of the remainder of the house.\n\nStill more fundamental changes occur when a smaller earlier house is incorporated into a larger later house of a different style. All housewatchers have had the experience of visiting, for example, an Italianate museum house that the guide insists was built in 1798. Upon close questioning it transpires that a log house was built on the site in 1798 and that in 1855 the original owner's granddaughter's husband came into some money, leading to a remodeling (we would say \"rebuilding\"). For purposes of identification, such houses can be considered to have been built in 1855; the fact that an earlier house is buried within is of historical significance but does not affect the exterior appearance. When the rebuilding is less complete, however, traces of the earlier house may be revealed by a careful look at roof, walls, and\u2014especially\u2014the roof-wall junctions, which quite often provide the best clues to large-scale stylistic updating.\n\nThe second principal reason for exterior alteration is to add living space to an existing house. This is usually accomplished in one of two ways: either by adding an entire extension or wing to the existing house or, at less expense, by converting a house's under-utilized areas (attic, basement, porch, or garage) into living space. In the first instance, extensive additions can be undertaken in the style of the original house and, depending on the skill of the architect involved, leave the original lines of the house undiminished or, occasionally, enhanced. Lyndhurst in Tarrytown, New York, is an outstanding example of this. Carefully done additions and alterations that use the same materials and style as the existing house can be very difficult to detect. More commonly, additions differ in some fundamental way\u2014form, materials, or details\u2014from the original house and are easily recognized.\n\nConverting existing spaces to living areas does not usually affect the exterior appearance of the house when it involves only a basement or attached garage (although the method of handling the garage door opening is important). An attic remodeling usually involves the addition of dormers or skylights but otherwise leaves the facade intact. The decision to gain space by enclosing all or part of a front or side porch, on the other hand, can drastically change the character of a house. Rarely can such alterations be done in a manner which maintains the stylistic integrity of the facade.\n\nThe third principal reason for altering house exteriors is to minimize maintenance. The need regularly to paint wooden walls, caulk drafty windows, and repair leaky roofs can cause a house owner to make drastic changes that are designed to decrease this routine maintenance. Some of these maintenance-minimizing alterations even promise to do away with these problems \"forever.\" Owners tired of painting wooden walls frequently add a layer of pre-finished siding (aluminum, asbestos, etc.). In addition to hiding the texture of the original walls, such siding usually obscures such details as window and door surrounds, cornices, and moldings and thus quite dramatically changes a house's appearance. It also may hasten the deterioration of the wooden walls it covers; fortunately it can often be removed without serious damage. For those tired of caulking, patching, and painting older wooden windows, replacing them with pre-finished metal windows can be tempting. Such changes generally alter the house's appearance significantly. Original slate, tile, tin, or wood-shingle roofs are commonly replaced with composition roofing materials. Sometimes these materials closely resemble the original roof but more often, like changes of wall materials or windows, they fundamentally alter the nature of the facade.\n\nMost maintenance-minimizing alterations do not completely obscure a house's stylistic origins but are apparent as inappropriate changes in external appearance through the loss of wood molding profiles and the pattern of window lights. If the changes are extensive, however\u2014for example, siding added and windows and roofs replaced\u2014the original character of the house may be undiscernible.\n\nThe fourth principal reason for major house alterations is to take advantage of the original house's exemptions from new and more strict building codes. In some cases, maintaining the footprint of an existing structure allows exemptions from stringent new construction standards. Among these are wetlands regulations along the East Coast, earthquake standards along the West Coast, and more stringent lot coverage or setback requirements added with changes in zoning codes. Any of these can cause a house to be radically redesigned\u2014even creating a new house for all practical purposes.\n\n**MIXTURES: ROMANTIC AND VICTORIAN**\n\nSTYLISH MIXTURES: ROMANTIC AND VICTORIAN  \nCleveland, Ohio; mid-19th century. Otis House. A mixture of Gothic and Italianate design. The shape of the house with its low-pitched hipped roof, symmetrical facade, and centered gable is Italianate, as is the bracketed cornice. The pointed arches over upper-story windows and the flattened arches of the front porch and lower-story windows are Gothic.\n\nSTYLISH MIXTURES: ROMANTIC AND VICTORIAN  \nCanton, Mississippi; mid-19th century. A mixture of Greek Revival, Gothic, and Italianate influences. The doors and porch with classical columns are borrowed from Greek Revival. The bracketed cornice is Italianate, and the flattened arch and jigsaw-cut wood detailing of the upstairs porch show Gothic Revival influences.\n\nSTYLISH MIXTURES: ROMANTIC AND VICTORIAN  \nNew Albany, Indiana; 1866. McCord House. This mixture of Gothic and Italianate design is a reversal of the Otis House. The house form with its gabled roof and steeply pitched, front-facing gables shows Gothic influence. The bracketed cornice and the windows with heavy rounded crowns are Italianate.\n\nSTYLISH MIXTURES: ROMANTIC AND VICTORIAN  \nRolla, Missouri; late 19th century. Romanesque and Shingle elements combined with square Prairie-style porch supports.\n\nSTYLISH MIXTURES: ROMANTIC AND VICTORIAN  \nRichmond, Virginia; late 19th century. A mixture of Queen Anne (patterned masonry subtype), Romanesque, and Exotic influences.\n\n**MIXTURES: ECLECTIC**\n\nSTYLISH MIXTURES: ECLECTIC  \nEmporia, Kansas; early 20th century. A mixture showing a Prairie-influenced wide roof-overhang combined with a Shingle-style front gambrel and a rusticated stone arch borrowed from the Richardsonian Romanesque.\n\nSTYLISH MIXTURES: ECLECTIC  \nDallas, Texas; 1928. Wade House. Here a Tudor front-facing gable with half-timbering is mixed with Italian Renaissance roof tiles and arched windows.\n\nSTYLISH MIXTURES: ECLECTIC  \nCleveland, Ohio; 1910. Mather House; Charles Schweinfurth, architect. A mixture of Tudor (windows, lintels, and chimneys) and Colonial Revival details (cornice-line balustrade, low-pitched hipped roof).\n\nSTYLISH MIXTURES: ECLECTIC  \nLouisville, Kentucky; 1922. This example combines the side-gabled Craftsman form with Italian Renaissance tiled roofing and arched entries.\n\nSTYLISH MIXTURES: ECLECTIC  \nLouisville, Kentucky; early 20th century. This house combines elements of several stylistic traditions (Prairie, Craftsman, and Germanic).\n\n**ALTERATIONS: STYLISH UPDATES**\n\nALTERATIONS: STYLISH UPDATES  \nCooperstown, New York; 1804 (roof, late 19th century). Lakeland; John M. Bowers. A Federal house with a Victorian roof added (the porch is likely an intermediate addition). Cover the roof to see the symmetrical house; uncover it to see how completely the roof form changes the feel of the house.\n\nALTERATIONS: STYLISH UPDATES  \nGranbury, Texas; late 19th century (porch, early 20th century). A Queen Anne house (note hipped roof with cross gable and tower) behind an added Neoclassical porch. This was a common update.\n\nALTERATIONS: STYLISH UPDATES  \nCooperstown, New York; mid-19th century (addition, late 19th century). Holt House. A Second Empire house (on the right) has had a major half-timbered Queen Anne addition to the left.\n\nALTERATIONS: STYLISH UPDATES  \nDallas, Texas; 1914. Arrington House. A relatively unaltered Prairie-style example. Both Dallas, Texas; early 20th century (remodeling ca. 1975) and Dallas, Texas; early 20th century (remodeling 1982; Teddy Taylor, designer) originally resembled this house, although without the porte cochere to the side.\n\nALTERATIONS: STYLISH UPDATES  \nDallas, Texas; early 20th century (remodeling ca. 1975). This remodeling of a house similar to the Arrington House removed the porch and added a small broken pediment to the door, which is poorly scaled and out of character with the form and roof of the house.\n\nALTERATIONS: STYLISH UPDATES  \nDallas, Texas; early 20th century (remodeling 1982; Teddy Taylor, designer). This remodeling of a house similar to the Arrington House, although obscuring the earlier Prairie character, picks up on the Mediterranean tiled roof by adding a Palladian entry motif. The changed brick pattern shows the line of the original porch roof. When painted, the brickwork will appear uniform and only very careful observation would reveal the modifications.\n\n**ALTERATIONS: MAINTENANCE**\n\nSt. Genevieve, Missouri; ca. 1920. In a development of several dozen almost identical Craftsman houses built in the 1920s, only the example shown in the first image that follows remains essentially unaltered. The photographs show a series of modifications typical of those that obscure the original stylistic details of many older American houses.\n\nALTERATIONS: MAINTENANCE  \nIn the unaltered example, note the wood shingles in the gable, narrow bands of wood siding, triangular knee braces at the cornice, original porch supports, and original window surrounds, sash, and vertical light pattern.\n\nALTERATIONS: MAINTENANCE  \nThis example is original except for the addition of asbestos shingles covering the wood shingles and siding.\n\nALTERATIONS: MAINTENANCE  \nAnother type of added siding, this time with metal awnings.\n\nALTERATIONS: MAINTENANCE  \nIn addition to new siding and awnings, the original wood porch supports and railing have been replaced by iron, the triangular knee braces sheathed, and shutters added.\n\nALTERATIONS: MAINTENANCE  \nThese alterations are similar to those in the previous image, but without the shutters and awnings.\n\nALTERATIONS: MAINTENANCE  \nHere the windows have been changed, two types of siding added, the triangular knee braces removed, the porch supports changed, and a rear side porch added.\n\nALTERATIONS: MAINTENANCE  \nIn this example, siding has been added and the front porch screened.\n\nALTERATIONS: MAINTENANCE  \nThe entire porch area has been enclosed and a new window shape and two types of siding added.\n\nALTERATIONS: MAINTENANCE  \nHere most of the front porch has been enclosed for additional interior space, the triple window on the right has been reduced to one, and siding has been added.\n\n# LOOKING AT AMERICAN HOUSES\n\n# Form\n\n# The Shapes of American Houses\n\nThe chronology of changing architectural fashions or styles provides the fundamental framework for identifying American houses. A second basic feature of houses is form or shape. House form is not endlessly varied. Instead, a few fundamental shapes, and relatively minor variations on them, tend to be used again and again through a range of changing architectural styles.\n\nHouse shape is best analyzed by dividing the three-dimensional house into two separate two-dimensional components. The first is the ground plan, the pattern made by the exterior walls when viewed from directly above. The second two-dimensional component is elevation, the pattern made by wall, roof, and details when viewed, as they normally are, from ground level. In theory, ground plans and elevations can be varied to form an infinite number of house shapes. In practice, there are only a relatively few common variants of both plan and elevation. These combine to make several fundamental families of house shapes that dominate American domestic architecture.\n\nCertain uncomplicated house shapes have been continuously used since the first colonists arrived. On the other hand, technological changes over the past three hundred years have permitted greater flexibility and freedom of design and have led to important innovations in house shape.\n\n## Ground Plans\n\nA ground plan shows the shape of a house as if one were viewing it from directly above with the roof and the upper floors (if any) removed to leave only the ground-floor walls. The shape of the ground plan can usually be broken down into a pattern of smaller, room-sized modules or structural units. The simplest folk houses have only a single such room-sized unit. As larger houses were required, folk builders developed techniques for combining two or more units into multi-unit plans. These room-sized structural units may or may not correspond exactly to the placement of interior walls to form actual rooms. In general, the earlier and more modest the house, the more likely it is that each principal room will correspond to a structural unit (although even in the simplest houses, one or more units are commonly partitioned into smaller rooms). Since in this book we are concerned with the external appearance of houses, these differences in internal room arrangement can be discounted. The important point is that ground plans are almost always made up of rather simple combinations of room-sized units. The basic house sizes and shapes defined by these combinations have persisted in all but the largest and most pretentious houses from early colonial times to the present day.\n\nIn the simplest case, square or rectangular room-sized units are combined into larger squares or rectangles known as simple plans. These are of two principal types: (1) linear plans, made up of units aligned into single rows one unit wide or deep, and (2) massed plans, which have both a width and depth of more than one unit. The rectangles and squares of simple plans can, in turn, be combined to make compound plans, the most frequent of which resemble in shape the letters L, T, or U. Many other compound arrangements are also possible; some of the most frequent are shown in the illustrations. Note that simple plans with minor wall projections may resemble compound plans but can usually be distinguished by whether or not the projections from the principal mass of the house are room-sized or smaller. Such minor wall projections were impractical with heavy timber construction, but became relatively simple and inexpensive with later balloon and platform framing. Starting about 1990 even greater use was made of compound plans and also more complex roof forms, such as hip-on-hip, which created additional shallow wall projections. See Plans: Regular to Irregular with Shallow Extensions.\n\nFORM\n\n **GROUND PLAN COMPONENTS**\n\nFORM\n\n **SIMPLE PLANS (RECTANGLES & SQUARES)**\n\nFORM\n\n **COMPOUND PLANS**\n\nFORM\n\n **COMPOUND PLANS & MINOR IRREGULARITIES DISTINGUISHED**\n\n## Elevations\n\nElevations show the shapes of houses as viewed normally from the ground at eye level. Elevations are generally straight-on views that show the appearance of a single wall with its overlying roof and architectural details. Four such elevation views\u2014one each of the front, the sides, and the rear walls\u2014are required to describe fully all details of shape. On the other hand, the general form of a house can sometimes be understood from just the front elevation (also called the principal elevation or facade) and the ground plan which, together, define the basic three-dimensional shape of the house.\n\nThe most fundamental factor in analyzing elevations is wall height. American houses are normally either one or two stories high. Less common are heights of one and one-half, two and one-half, and three stories (a half-story has less than full-height external walls; the remaining headroom is developed from attic space beneath the roof line). Houses more than three stories high are rare except in densely populated urban settings where narrow town houses sometimes have four or more stories.\n\nMost simple-plan houses have facades either one, two, or three structural units in width. Each width normally has a characteristic pattern of symmetry in the arrangement of door and window details. These patterns frequently allow identification of the width of the underlying ground plan. Two asymmetrical ranks of window and door openings normally occur on one-unit widths; three symmetrical or four asymmetrical ranks on two-unit widths; and five symmetrical ranks on three-unit widths. In symmetrical three-ranked facades the two principal front rooms behind the facade are normally of unequal size, the entrance door opening into the larger of the two. Less commonly the door opens into a narrow entrance hallway or vestibule. The middle unit of three-unit widths is normally a central hallway and may also be somewhat narrower than the two flanking units. The presence of five, rather than three, ranks of window openings normally distinguishes this plan from symmetrical two-unit plans. In urban houses, a one-and-one-half-unit width is typical, with a hallway occupying the half unit and the principal rooms lined up in the full unit.\n\nElevations reveal not only wall height, width, and symmetry, but also the varying relative proportions of roof and wall. If the roof is flat or of low pitch, the wall will dominate the facade. Conversely, steeply pitched roofs dramatically dominate their underlying walls. Roofs of normal pitch show about equal dominance of roof and wall.\n\nFORM\n\n **FACADE WALL HEIGHT**\n\nFORM\n\n **FACADE WIDTH & SYMMETRY**\n\nFORM\n\n **ROOF-WALL PROPORTIONS**\n\n## Families of Shapes\n\nGround plan and elevation combine to make several persistent and recurring patterns or families of shapes that are characteristic of most American houses. Much of the history of American domestic architecture involves the varying patterns of details\u2014roofs, doors, windows, chimneys, porches, and decoration\u2014applied to these relatively few basic shapes. Note that several simple-plan families have such distinctive shapes that they have familiar names (saltbox, shotgun, town house, etc.). Others have less common names (I-house, massed side-gable, etc.) but all are easily recognized with a little practice. The principal compound-plan families are also easily recognized, but they show more variation in details of shape. Wings of varying ground plan and elevation can be combined to form many variants of the basic L, T, and U plans.\n\nMost of the fundamental styles of American houses display several of these shape patterns. In a few styles, however, a single family tends to dominate. Thus, Georgian houses are principally box-house or saltbox shapes, Federal mostly box-house, the Prairie often four-square, and so on. In many styles, shape families provide useful criteria for defining one or more of the principal subtypes within the style. The distinctive town-house shape, in particular, makes up a characteristic urban subtype in several styles.\n\nThe principal use of shape families in house identification applies not to styled houses but to folk houses. Such houses generally lack the architectural detailing that characterizes and differentiates styled buildings. In these, shape becomes a principal criterion for distinguishing types.\n\nFORM\n\n **SIMPLE-PLAN FAMILIES**\n\nFORM\n\n **COMPOUND-PLAN FAMILIES**\n\n## Shape Innovations\n\nSeveral important technological advances have influenced the shapes of American houses over the three and a half centuries that have passed since the earliest European colonization.\n\nMASSED PLANS\u2014The first advance was the introduction of massed plans into the English colonies during the 18th century. All but a rare handful of 17th-century English colonial houses were of linear plan (one room deep) with high, steeply pitched roofs. These traditional roofs were of ancient origin and were designed to be covered with thatch, which sheds water only if the surface to which it is applied slopes very steeply. Such steep roofs became impossibly high when applied to large massed plans; as a result, most modest English dwellings were of linear plan. The rigors of the New World climate soon led to the abandonment of thatch as a roof covering in favor of more durable wooden planks or shingles. At the same time, the long severe winters of the northern colonies made additional interior space desirable. During the period from about 1700 to 1750, many houses were built, or were expanded, to a one-and-one-half-unit depth. The roofs on these, although now usually shingled, retained the steep framing of Medieval origin. The increased depth was accommodated either by a lower-pitched shed roof over the half-unit extension (most common in one-story houses) or by a rearward continuation of the main roof slope to give a saltbox shape (most common in two-story houses). This shape limited the rearward extension to the relatively shallow depth covered by the downward projection of the steeply pitched roof line. It also truncated the rearward extension at the second-floor level, which could be used only for storage rather than as living space. These disadvantages were overcome through the development of lower-pitched roof framing, which could span a full two-unit depth without excessive height. This transition was virtually completed by 1750; since that time, massed-plan houses have been a dominant feature of American architecture. Note, however, that linear plans, descended from the earlier tradition, have also persisted, particularly in rural and folk building.\n\nHEATING\u2014Two separate technological innovations during the 19th century had profound effects on house shape. The first relates to improvements in heating. Massive fireplaces for burning wood or coal were the principal heating devices until the 1830s, when the first practical cast-iron stoves were introduced. These were vented to the exterior either through metal stovepipes or through small masonry flues. Both of these were far easier to install than massive fireplaces and thus permitted the wider use of larger\u2014and less regular\u2014house plans. Compound plans, in particular, now became more common. Still further improvement came with the introduction of central furnaces that burned wood or coal. These came into common use in the colder northern sections of the country after about 1880. In this system, heat is transferred from the furnace to individual rooms by means of heated water, steam, or air; only a single masonry flue is required to serve the furnace. This development further accelerated the trend toward compound and irregular plans. Yet with each of these heating innovations, the earlier systems were not completely abandoned. Many 19th- and 20th-century houses retain one or more fireplaces, along with stoves or coal-burning furnaces, as a sort of nostalgic interior ornament without essential function. In addition, stoves (now most commonly burning natural gas rather than wood or coal) and fireplaces have continuously remained the principal means of heating many modest houses, particularly in the milder southern part of the country.\n\nIRREGULAR PLANS\u2014A second 19th-century innovation affecting house shape was balloon-frame construction. This relatively rapid and inexpensive method of wooden framing was developed in the Chicago area during the 1830s. In earlier wooden framing systems (post-and-girt; braced-frame), as well as in solid masonry construction, outside corners are particularly difficult to fashion. Masonry is especially susceptible to erosion and failure at corner junctions and usually requires carefully shaped stones\u2014or strengthened brick-bonding patterns\u2014to make secure and permanent corners. Similarly, the heavy timbers of post-and-girt or braced frames require complex, hand-hewn corner joints and braces to give them rigidity. For these reasons, unnecessary outside corners were traditionally avoided in all but the most pretentious houses built with these wall systems. In contrast, corners in balloon (and, later, platform) framing are readily constructed with only a few two-inch boards and wire nails. Balloon framing thus freed house shapes from their traditional plane-walled patterns by allowing for easily constructed irregular plans with many extensions and re-entrants. Since the mid-19th century, such irregular wall forms have been commonly superimposed upon both simple and compound plans as balloon framing became the standard construction technique. Beginning in the late 1980s a new fashion for complex roof forms in some house styles was often accompanied by small extensions added to already irregular ground plans. An added squared bay, for example, could have a small secondary roof above it (also see Cascading Hipped Roofs).\n\nINTEGRAL GARAGES\u2014One other technological innovation that affected house shape relates not to building techniques but to transportation: the rise of the automobile as the principal means of personal travel in the 20th century. When automobiles first became common in the decade between 1910 and 1920 they were universally housed, as had been carriages and horses before, in detached garages. Such garages have persisted to the present day, but since the 1920s there has been an accelerating trend to house automobiles in extensions to, or within portions of, the main house. This trend has dramatically affected the overall size and shape of some houses constructed between 1920 and 1950, and of almost all constructed since 1950. The illustration shows graphically the changing average amount of space devoted to automobile storage during the period from 1930 to 1960 (in comparison with a standard six-unit plan of 1,000-square-foot area). Whether placed within the principal mass of the house, as shown, or added as attached units, these automobile shelters have affected the style and form of many 20th-century house facades. By the turn of the millennium three- and four-car garages were not uncommon and automobile access and storage had become an important factor in new home design and subdivision planning. See Housing the Automobile. A three-car garage built in 2010 can be 750 square feet, a space three-quarters as large as many homes built in 1950.\n\nFORM\n\n **SHAPE INNOVATIONS IN AMERICAN HOUSES**\n\nFORM\n\n **LINEAR TOMASSED PLANS 18th century**\n\nFORM\n\n **HEATING INNOVATIONS 19th century**\n\nFORM\n\n **PLANS: REGULAR TO IRREGULAR WITH SHALLOW EXTENSIONS (extensions may begin above the ground plan)**\n\nFORM\n\n **HOUSING THE AUTOMOBILE 20th century**\n\n# LOOKING AT AMERICAN HOUSES\n\n# Structure\n\n# The Anatomy of American Houses\n\nIn addition to style (fashion) and form (shape), there is a third and somewhat more technical element that is useful for identifying and understanding American houses. This is structure, which can be defined as the several individual components of houses that give them their characteristic forms and styles.\n\nAll houses are composed of three basic structural units. First come walls, the vertical units that serve both to screen the interior spaces and to support the second basic unit, the roof, which shields the interior spaces from weather and completes the enclosure. In very simple houses\u2014for example, tipis or modern A-frames\u2014roof and wall may be a single unit. Far more commonly, each is made of different materials combined into separate structural systems. The most important materials and systems used for walls and roofs in American houses are described in the pages that follow. A house made up only of walls and a roof, with an entrance opening in the walls, can be a fully functional shelter. Many simple folk houses have little more. Most American houses, however, have added architectural details to the basic walls and roof, including some or all of the following components: windows, to provide light and ventilation to the interior; doors, to permit the entranceway to be closed against the weather; chimneys, to confine and eliminate smoke from interior fires; porches, partially exposed areas having roofs but lacking one or more walls; and, finally, decorative details, which function to enrich the external appearance of the house.\n\n## Walls\n\nThe walls of houses have two separate and distinct functions; first, they provide support for the roof and for any upper floors that may be present; second, they screen the house interior from weather and intrusion. In some types of wall structure\u2014for example, those made up entirely of stone or brick\u2014the same materials serve both functions. In others the functions are separated: one material provides the structural support and another the screening. The most familiar example is the wood-frame house, in which vertical wooden members provide structural support while an exterior covering, or cladding, screens the interior. This first section describes the principal structural support systems used for walls of American houses. The principal cladding materials used with these wall structures are treated below.\n\nWALL FOUNDATIONS\u2014Walls of very modest houses are sometimes built directly on the ground with little or no underlying foundation. Such walls rest on the surface soil, which makes a very poor base for most types of construction. Wooden walls tend to rot when in direct contact with damp earth, while masonry tends to be undercut by rainwater erosion of the soil beneath. For these reasons, most house walls are set upon foundations designed to protect them by raising them above the underlying soil. Simplest are wooden walls set upon wooden posts of some rot-resistant variety such as oak, cedar, or bois d'arc. (Sometimes the posts are, themselves, set directly on the ground surface, but more commonly with this and all other foundation systems, the soil is removed to a depth ranging from several inches to several feet and the base of the foundation is \"buried\" to provide firmer support.) Columns of brick or stone masonry known as masonry piers provide a similar supporting system for wooden walls, without the danger of rotting. On the other hand, failure of mortar joints can lead to equally serious problems that can be avoided by the use of monolithic piers, sometimes of metal but usually of concrete reinforced by steel rods.\n\nThe strong basal timbers of wooden walls can be supported by separated posts or piers; masonry walls, on the other hand, require continuous underlying support. In earlier masonry houses, soil has typically been excavated beneath the proposed wall and the first courses of stone or brick laid on the firm base of the trench. For additional stability, this underlying masonry wall is usually wider and of heavier materials than is the masonry of the overlying walls. When a basement is desired, some or all of the space between the exterior walls is excavated and the foundation walls constructed around the margins of the pit. Similar masonry wall foundations are also common beneath wooden walls, particularly in larger houses or in smaller houses requiring a basement. Foundation walls of masonry, like masonry piers, are subject to erosion and failure of the mortar joints and thus require periodic repair. This problem is avoided by monolithic concrete walls made of concrete beams poured in place and reinforced with internal steel rods. Such foundations first became common in the late 19th century; by the mid-20th century they had generally replaced wooden and masonry foundations beneath all types of wall construction. Note that in all the foundation systems mentioned so far, the internal floors and walls are supported by piers of wood, masonry, concrete, or metal even when the external walls have a continuous masonry or concrete foundation. One additional foundation system, developed in this century, eliminates these internal piers. In such concrete slab foundations, a relatively thin sheet of monolithic poured concrete underlies the entire house. This system completely eliminates floor framing and support at the first-floor level, and has become increasingly common since the 1950s.\n\nWOODEN STRUCTURAL SYSTEMS\u2014Most American houses (probably well over 90 percent) use pieces of wood to support the upper floors and roof. Simplest are walls of horizontal logs, either left round or hewn square, which serve to provide both structural support and, when the cracks between the logs are filled with clay or other materials, weather screening as well. The principal structural support of a log wall is provided by the notched corners, where adjacent logs are in close contact. Several systems of log corner notching have been developed to strengthen this crucial junction. Simplest to construct but least rigid is the saddle joint; progressively more rigid are square, V-notched, and half-dovetail joints; while complex full-dovetail joints provide the strongest structure of all (see also the treatment of log houses).\n\nFar more common than horizontal log walls are those in which spaced vertical members provide structural support. Earliest is the Medieval post-and-girt system, imported from England and France by the first colonists. In this system, upper loads are borne by heavy corner posts and widely spaced intervening posts; heavy cross timbers carry upper floors which are unsupported by the thin internal walls below. Typically, all structural joints in post-and-girt houses are laboriously hewn into interlocking shapes and held fast by wooden pegs. Post-and-girt houses dominated the English and French colonies and persisted until well after the American Revolution. In the early 19th century, however, the increasing abundance of commercially sawed lumber, together with the development of relatively inexpensive wire nails, led to a modification of the traditional post-and-girt system known as braced-frame construction. This system still employs heavy corner posts connected by heavy horizontal timbers, generally with hewn joints. But within this heavy skeleton, loads are carried not by widely spaced and equally massive intervening posts and cross members, but by light, closely spaced vertical studs nailed between the horizontal timbers. Internal walls constructed entirely of light studs also now become strong bearing walls which help support the floors and roof above. This system takes its name from diagonal corner braces used to give lateral stability to the wooden framework. Note, however, that such braces are by no means unique to the system, but are common in all types of wooden framing.\n\nBy the early 19th century, braced frames were replacing post-and-girt construction throughout the former English colonies of the Atlantic seaboard; in this region braced-frame houses persisted well into the 20th century. Westward migration from these states also made this a common mode of construction throughout the country during the 19th century. By the time of the Civil War, however, another still more simplified method of frame construction was coming to dominance in the rapidly developing midwestern states. This was the balloon-frame system, begun in Chicago in the 1830s. This system eliminated altogether the tedious hewn joints and massive timbers of braced-frame and post-and-girt construction, for balloon-frame houses are supported entirely by closely spaced two-inch boards of varying widths (two-by-two, two-by-four, two-by-six, two-by-twelve, etc.) joined only by nails. Corner posts and principal horizontal members are made of two or more two-inch boards nailed together. As in braced-frame houses, the principal supporting members are the closely spaced two-by-four or two-by-six vertical studs of both the exterior and key interior walls. This system allowed both cheaper and more rapid construction by eliminating the need for skilled hand-hewing of the principal wall timbers. With slight modification it remains the dominant method of American house construction today. The most common modification, known as platform framing, relates primarily to the wall studs and flooring. In balloon-frame construction, the studs are continuous from foundation to roof and the floors are hung upon the studs. In platform framing the floors are constructed as independent units, like thin, flat platforms; the shorter wall studs are then erected upon these platforms to support the overlying platform or roof. This system is both simpler and more rigid than balloon framing, which it had largely replaced by World War II.\n\nMASONRY STRUCTURAL SYSTEMS\u2014Although wooden framing has always dominated American house construction, European immigrants to the New World brought with them an intimate knowledge of masonry techniques as well. Indeed, in colonial times, just as today, masonry houses far outnumbered those made of wood throughout most of western Europe. (For this reason first-time visitors from Europe are always surprised to find the United States to be a land of wooden houses.) Although making up only a few percent of American houses, those with masonry walls show almost all variations of masonry building technique. Spanish colonists brought traditions of building in uncut stone and unfired adobe brick. The English, French, and Dutch had elaborate techniques of building with harder, fired brick and cut stone, as well as more modest folk traditions of building with sod (blocks of earth held together by grass roots) and uncut stone. These traditions tended to dominate certain regions during the colonial period; most persisted through the 19th century and a few survive even today. Beginning in the early 20th century, technology has added two more masonry materials to the traditional repertoire: hollow, fired clay tiles and hollow concrete blocks. These new materials are as strong as fired brick or stone, but are both lighter and cheaper. They have thus come to dominate 20th-century masonry construction, either alone or combined with an exterior layer of brick or stone to make composite masonry walls.\n\nOTHER STRUCTURAL SYSTEMS\u2014Only a very small fraction of one percent of American houses rely on structural systems other than wood or masonry. A few houses, mostly built in colonial times, used both wood and masonry in combination for structural walls.1 Typically, end-chimney bearing walls were of masonry, the other walls of post-and-girt frames. In another variation, favored in the French colonies, the first-floor walls were of masonry and the overlying floor was post-and-girt.\n\nWooden and masonry walls are both composite, that is, they are made up of many small units linked together to make a wall system. Walls can also be of massive or monolithic construction, where only one or, at most, a very few units make up the entire wall. The simplest such walls are made of earth, either mixed with water to make mud and then built up in layers, or pressed into layers while only slightly damp (rammed earth). Such walls are found in both European and Native American folk houses, but are rare in post-colonial America. Somewhat more common are monolithic walls of poured concrete, usually reinforced with iron or steel rods. Such walls can either be poured in place or pre-cast and then transported to the building site. They are common in 20th-century commercial buildings but are only rarely found in houses, most of which date from the late 19th and early 20th centuries.\n\nWALL CLADDING\u2014Relatively few systems of wall structure are immediately evident from looking at the exteriors of houses. In masonry walls the structural units of stone or brick _may_ be exposed, but they also may be covered with a protective and decorative layer of stucco which masks the underlying structure. As a further complication, wood-frame buildings are often covered with an external layer of brick or stone, which gives them a superficial resemblance to masonry construction. These cladding materials can, however, provide clues to the underlying structure, which is almost always evident on close examination of foundations, basements, and attics.\n\nAll wood-frame houses _must_ have external cladding. Traditionally the cladding is also of wood, either boards or shingles; since these materials are rarely applied to masonry walls, they indicate an underlying wooden frame. It is usually difficult to tell the exact system of wood framing unless some of the cladding is removed (although, again, a careful examination of wall openings, foundations, and attics may reveal the underlying framing). Such modern cladding materials as plywood or fiberboard panels, metal or plastic strips, or asbestos, asphalt, metal, or shingles are also seldom applied to masonry walls, and thus indicate an underlying wood frame. Brick and stone veneers were introduced in the early 20th century. By 1920, brick veneering in particular was in general use. These veneers may be difficult to distinguish from solid masonry, except that veneers are far more common; thus the first suspicion should be that _any_ house showing external masonry\u2014particularly if constructed within the last century\u2014has a veneered wooden frame. An additional clue is that most brick veneers use only a running (stretcher) bond, since no headers are necessary to lock together the multiple rows required in a solid brick wall. By the turn of the millennium even thinner masonry veneers were in use\u2014half-thickness bricks, thin stones, and lightweight \"manufactured\" stone. Stucco walls can be the most enigmatic of all, for stucco finishes are commonly applied to both wood-frame and masonry buildings. Simple tapping to see if the walls sound hollow will sometimes distinguish between underlying wood or masonry. Likewise, areas of thin or failing stucco may reveal the structure beneath.\n\nHALF-TIMBERING (FILLED WOOD FRAMES)\u2014American wood-frame houses normally have cladding added to the exterior of the frame as a continuous covering that conceals the underlying structure. European framed houses of post-and-girt construction have, since Medieval times, commonly used another system of wall enclosure in which the spaces between the heavy supporting timbers are _filled_ rather than covered. Such fillings normally leave the sides of the supporting timbers exposed and are known as half-timbered construction. The most frequent filling material is clay (daub), which is usually applied over a lath of short wooden sticks or woven basketwork (wattle). Brick or stone are also commonly used as filling materials; these are generally covered with stucco and thus closely resemble wattle-and-daub fillings. Early colonists in America first built half-timbered houses, but the rigorous New World climate made it difficult to keep the exposed fillings uncracked and weather-tight. As a result, half-timbering was generally abandoned for wooden claddings or full stuccoing, both of which completely covered the underlying frame. Some surviving English and French colonial houses of post-and-girt construction retain the filled frames of Medieval tradition; in most, the filled frame was either originally\u2014or very early in the house's history\u2014covered by continuous external cladding of wood or stucco. (Some of these have, in this century, been overzealously restored to an exposed, half-timbered exterior.) The half-timbered tradition, although unsuited to the American climate, has persisted as an applied surface decoration (false half-timbering) on 19th- and 20th-century houses that mimic the earlier technique.\n\nSTRUCTURE\n\n **WALLSTRUCTURAL SYSTEMS**\n\nWOODEN\n\nSTRUCTURE\n\n **WALL STRUCTURAL SYSTEMS**\n\nMASONRY\n\nSTRUCTURE\n\n **WALL STRUCTURAL SYSTEMS**\n\nOTHER\n\nSTRUCTURE\n\n **WALL FOUNDATIONS**\n\nBENEATH WOODEN WALLS\n\nSTRUCTURE\n\n **WALL FOUNDATIONS**\n\nBENEATH WOODEN OR MASONRY WALLS\n\nSTRUCTURE\n\n **LOG CORNER NOTCHING SYSTEMS**\n\nSTRUCTURE\n\n **WOOD-FRAMING SYSTEMS**\n\nSTRUCTURE\n\n **WOOD-FRAMING SYSTEMS**\n\nSTRUCTURE\n\n **MASONRYSTRUCTURAL SYSTEMS**\n\n**ADOBE BRICK**\n\nSTRUCTURE\n\n **MASONRY STRUCTURAL SYSTEMS**\n\n**FIRED BRICK**\n\nPOSITION IN ROWS (COURSES)\n\nSTRUCTURE\n\n **MASONRY STRUCTURAL SYSTEMS**\n\n**FIRED BRICK**\n\nMORTAR JOINTS\n\nSTRUCTURE\n\n **MASONRY STRUCTURAL SYSTEMS**\n\n**FIRED BRICK**\n\nBONDING SYSTEMS, SOLID WALL\n\nSTRUCTURE\n\n **MASONRY STRUCTURAL SYSTEMS**\n\n**FIRED BRICK**\n\nBONDING SYSTEMS, CAVITY WALL\n\nSTRUCTURE\n\n **MASONRY STRUCTURAL SYSTEMS**\n\n**STONE**\n\nSTRUCTURE\n\n **WALL CLADDING MATERIALS**\n\nOVER WOOD-FRAME\n\nSTRUCTURE\n\n **WALL CLADDING MATERIALS**\n\nOVER MASONRY\n\nSTRUCTURE\n\n **CLADDING DETAILS**\n\nBOARDS, HORIZONTAL\n\nSTRUCTURE\n\n **CLADDING DETAILS**\n\nBOARDS, VERTICAL\n\nSTRUCTURE\n\n **CLADDING DETAILS**\n\nSHINGLE PATTERNS\n\nSTRUCTURE\n\n **CLADDING DETAILS**\n\nTYPICAL STUCCO FINISHES\n\nSTRUCTURE\n\n **HALF-TIMBERING (FILLED WOOD FRAME)**\n\n## Roofs\n\nRoofs, the second principal structural component of houses, occur in three fundamental shapes: the first is gabled\u2014that is, with two sloping planes supported at their ends by triangular, upward extensions of two walls known as gables. In gabled roofs, the junction of roof and wall occurs as a single horizontal plane, only on two facades; on the other two facades the wall plane continues up to the roof line, generally forming a triangular shape. In the other two roof shapes, hipped and flat, the roof meets the walls in a single horizontal plane. In hipped roofs, four sloping surfaces form the roof, while only a single horizontal or slightly sloping surface occurs in flat roofs. Each of these three principal shapes has several subtypes. These patterns of roof shape are among the most dominant features in determining the external appearances of houses. Thus they provide the basis for many of the stylistic subtypes defined throughout this book. After shape, the most apparent roof feature is pitch\u2014the angle the sloping roof planes make with the horizontal. Both gabled and hipped roofs show marked changes in character as they pass from low pitches (under 30\u00b0) through \"normal\" pitches (30\u00b0\u201345\u00b0) to steep pitches (over 45\u00b0).\n\nROOF FRAMING\u2014Roofs in all but a very small number of American houses are supported by wooden frameworks, for wood combines suitable strength and relatively light weight (as does metal, which is rarely used for this purpose in houses), thus making possible the long spanning members required for roof support. Two principal roof-framing systems have been used in American houses.2 The earlier employed heavy principal rafters with hewn joints as the principal supporting members. Lighter members (either common rafters or common purlins) were placed between the principal rafters to provide a base for attaching the roofing material. This system is analogous to post-and-girt wall framing and, indeed, is found on most early post-and-girt\u2014as well as masonry\u2014houses. With the rise of lighter braced, balloon, and platform wall framing it was discovered that light, closely spaced common rafters, joined by nails, provided adequate roof support without the need for intervening heavy timbers. Such common rafter roof-framing systems are almost universal in American houses built after the mid-19th century.\n\nPrincipal rafters of heavy timber require little additional bracing to support the weight of the overlying roof, particularly if the roof is steeply pitched and spans a relatively narrow space below. Rafters of lighter weight, lower pitch, or longer span require underlying supporting systems of joists or trusses.\n\nSimple gabled roofs always require the least complex underlying framing; the additional roof planes of gambreled or hipped roofs demand additional framing members.\n\nROOF-WALL JUNCTIONS\u2014The lines of junction between roof and wall are crucial features of house design, both esthetically and structurally, for they join differing roof and wall materials in a junction that must be watertight to protect the underlying structure from damaging moisture. There are various systems of attaching roof to wall (and for enclosing this important junction).\n\nDORMERS\u2014Finally, roof slopes may be interrupted by dormers, subunits resembling miniature houses with their own walls, roofs, and windows. These are added to provide space, light, and ventilation to the attic, thus making it a functional part of the house. Dormers are most easily characterized by their roof shapes.\n\nROOFING MATERIALS\u2014Resting upon the wooden framework is the watertight covering, which adds texture and color to the sloping roof planes and thus has a dominant effect on the external appearance of a house. Four principal kinds of materials are used for roofing.\n\nThe first are organic coverings. Of these, thatch, closely packed bundles of reeds or straw, is the most common roofing for folk houses throughout the world. Although thatch is commonly used on modest European dwellings, it was quickly abandoned by American colonists because it was particularly vulnerable to the high winds, driving rains, and severe winters of the New World. Long boards of split wood were sometimes substituted for thatch in early colonial houses but these, too, were rather quickly abandoned for roofs of wooden shingles\u2014thin wedge-shaped rectangles that were either rough-split or sawed from oak, cedar, or other durable woods. Shingles could be closely aligned and generously overlapped to give an impervious and weather-resistant roof; since colonial times, wooden shingles have remained a dominant roofing material of American houses.\n\nRoofs of mineral materials also have a long history. Simplest are roofs of earth, or of earth bound by grass roots to make sod; both are common on folk dwellings everywhere. Both the earliest New World colonists and 19th-century settlers in the treeless western half of the country commonly used earth or sod roofs on temporary dwellings. They are also used for the roofs of permanent Spanish-influenced dwellings in the American Southwest. Roofs of thin, flat pieces of natural stone, tightly overlapped as with wooden shingles, were common in the larger dwellings of Medieval and Postmedieval Europe. An abundance of wood for making shingles\u2014and a relative scarcity of quality slate, the most easily split and durable type of stone\u2014made such roofs uncommon in this country until the late 19th century, when they began to be used in houses that simulated earlier European traditions. A third type of mineral roof, composed of thin, shaped units of baked clay tiles, was developed in classical times and has since remained a continuous feature in European architecture. Several systems of interlocking tile units have been developed through this long history. Most of these systems have been employed on monumental New World houses since colonial times but, like slate roofs, they have been common only since the late 19th century. In the 20th century, tiles made from concrete and other composite ceramic materials have been developed which simulate clay tile. (Note also that metal and composition roofs are often shaped and colored to resemble ceramic tile.)\n\nMetal roofs also have a long history, for sheets of lead or copper have been used as roofing since classical times. A few landmark colonial houses of the New World used such roofs, but metal became a common roofing material only in the early 19th century when sheet iron (usually coated with zinc, tin, or lead to prevent rust) first became relatively inexpensive and plentiful. Usually metal roofs are applied as large sheets joined with standing seams, which help prevent leaks. Later in the 19th century, stronger corrugated panels of iron became common for roofing of commercial and modest domestic buildings. Their rigidity gives such panels the advantage of requiring less underlying support than do most roofing materials. In the 20th century, panels of corrugated aluminum are sometimes used for the same reason. Other metal roofs are made up of smaller units shaped to resemble shingles or ceramic tiles.\n\nThe fourth type of roofing is based on bitumen, natural semi-solid petroleum residues such as tar and asphalt. Since colonial times natural tar deposits have been used\u2014along with tar-impregnated sheets of cloth, felt, or paper\u2014to make built\u2011up roofs. Unlike roofs made up of smaller units, which must be pitched upward to prevent water from entering the joints between units, monolithic roofs of tar (or earth) can remain impervious when almost flat (many flat-roofed Spanish Colonial houses of the Los Angeles area had built\u2011up tar roofs, the material coming from nearby natural tar pits). Although most common on commercial buildings, built\u2011up roofs have also been a standard technique of house roofing since the mid-19th century. Tar normally has to be heated to make it liquid enough to spread on built\u2011up roofs. It also must be protected from the sun's rays, which make it hard and brittle, by gravel or other material. By the late 19th century, techniques had been developed to convert tar or asphalt into \"cold\" roofing by impregnating sheets or shingles of felt, paper, or cloth with bitumen. Such composition roofs had the advantage of being easy to apply, relatively inexpensive, and fire-resistant. They have become the dominant roofing (or re-roofing) material for American houses in the 20th century. Other materials, in addition to petroleum-based bitumen, have also been used for making composition shingles. In particular, shingles of asbestos fibers bound together by concrete were widely used in the early decades of the 20th century.\n\nSTRUCTURE\n\n **ROOF SHAPES**\n\n**GABLED FAMILY**\n\nSTRUCTURE\n\n **ROOF SHAPES**\n\n**HIPPED FAMILY**\n\nSTRUCTURE\n\n **ROOF SHAPES**\n\n**FLAT FAMILY**\n\nSTRUCTURE\n\n **ROOF PITCH**\n\nSTRUCTURE\n\n **ROOF FRAMING**\n\n**RAFTER SYSTEMS**\n\nSTRUCTURE\n\n **ROOF FRAMING**\n\n**RAFTER SUPPORT SYSTEMS**\n\nSTRUCTURE\n\n **ROOF FRAMING**\n\n**RAFTER TRUSSING SYSTEMS**\n\nSTRUCTURE\n\n **ROOF FRAMING**\n\n**TYPICAL GAMBREL AND HIPPED FRAMING**\n\nSTRUCTURE\n\n **ROOF-WALL JUNCTIONS**\n\nRAFTER-PLATE JOINTS\n\nSTRUCTURE\n\n **ROOF-WALL JUNCTIONS**\n\nEAVE DETAILS\n\nSTRUCTURE\n\n **ROOF-WALL JUNCTIONS**\n\nGABLE-END (RAKE) DETAILS\n\nSTRUCTURE\n\n **DORMERS**\n\n**DORMER SHAPES AND PLACEMENT**\n\nSTRUCTURE\n\n **DORMERS**\n\n**ROOF DORMERS**\n\nSTRUCTURE\n\n **DORMERS**\n\n**DORMERS AND GABLES DISTINGUISHED**\n\nSTRUCTURE\n\n **ROOFING MATERIALS**\n\nSTRUCTURE\n\n **ROOFING DETAILS**\n\nBoth slates and wood shingles are sometimes cut into decorative shapes\n\nSTRUCTURE\n\n **ROOFING DETAILS**\n\nCERAMIC TILE SYSTEMS\n\nSTRUCTURE\n\n **ROOFING DETAILS**\n\n## Architectural Details\n\nIn addition to walls and roofs, many kinds of architectural details contribute to the external appearance of houses. The most important of these are windows and doors; chimneys; porches; and decorative details.\n\nWINDOWS AND DOORS\u2014Windows are wall openings that provide light and ventilation for the house interior. The word itself derives from \"wind-holes,\" early openings that served principally to supply draft, and emit smoke, from internal fires. Early windows were without glass, which was a rare and expensive luxury until the 17th century. When ventilation wasn't required, the openings were covered with fabric or skins or by solid wooden sashes or shutters. Many schemes have been devised for opening and closing such shutters, and later glazed sashes; most have been in continuous use since at least Medieval times.\n\nTo admit light through the closed window, frames covered with translucent oiled cloth or paper came to be used instead of solid shutters in prosperous Medieval houses. Many such windows were used in colonial America, but glass glazing was also becoming widespread in England, Holland, and France at about the time of the first New World colonization. These 17th-century window sashes were glazed with many small panes of glass, usually either square or diamond-shaped, held in a wooden or metal frame by narrow strips of soft lead. Throughout the 18th and early 19th centuries, window sashes came to be glazed with panes of increasing size, as glass-making techniques improved and costs decreased. By the mid-19th century, panes large enough to glaze sashes in only one or, at the most, two units became widely available. Since then, multi-paned sashes have been used only because of historical precedent rather than technological necessity.\n\nAlthough some 17th-century window frames and sashes were of iron, windows with wooden frames, sashes, and glazing bars (muntins) became almost universal in the 18th and early 19th centuries. Beginning in the mid-19th century, industrialization made available mass-produced metal windows. These remained relatively rare until the 1930s; since then they have progressively increased in use and are now the dominant type of window in American houses.\n\nWindows are an architectural luxury lacking in some modest folk houses. At least one exterior doorway is essential, however, to permit entrance and exit of the occupants. Originally the doorway served also as the principal \"wind-hole\" for regulating light and ventilation (hence the phrase: \"Never again darken my doorway\"). Doors for closing off the doorway are almost universally made of wood in American houses. Because single pieces of wood are never large enough to cover a full door opening, doors are always composite\u2014that is, made up of many small pieces of wood. In the earliest and simplest form of the door, vertical planks are held together with horizontal strips called battens, which are nailed or screwed to the surface of the larger planks.\n\nBy the 18th century, more elaborate doors were becoming common. These paneled doors consisted of an exterior framework of relatively thick planks, carefully joined and interlocked, which supported thinner internal planks (or panels). Such doors combine the virtues of strength, light weight, and decorative appearance; they remain the most common type of door in American houses, although they were increasingly replaced in the mid-20th century by the flush door. Flush doors appear to be single, flat pieces of wood but are, in fact, of veneered construction. They are made up of large, thin sheets of wood that are first peeled from a log with a razor-sharp knife, then glued together to make a strong, composite unit (this same process also produces plywood panels, which have replaced wooden planks for many construction uses since the late 20th century). In more modest flush doors, single thin sheets of veneer are applied to the exterior of a solid or hollow framework of joined planks to make a sort of sandwich structure.\n\nIt is usually desirable for external doorways, when closed, to admit light into entrance rooms or hallways. Thus many doors are partially glazed with fixed glass panes, which are found in all the principal types\u2014batten, paneled, and flush. Note also that additional glazing is often provided around the door in the form of side or overhead lights.\n\nCHIMNEYS\u2014Chimneys are hollow columns of masonry that provide a restricted exit for the smoke and fumes of internal cooking and heating fires. Houses have sheltered fires since the dawn of human civilization, yet chimneys are a relatively recent innovation, having only become widespread in modest English houses at about the time of the first American colonization. This innovation was brought to the New World, where chimneys became a standard feature of American houses. The simplest chimneys are constructed of wooden frameworks covered with a hardened coating of clay. Such chimneys require constant repair and become a serious fire hazard if the coating fails. Thus they are usually replaced by solid masonry chimneys as quickly as circumstances permit. Both brick and stone masonry are widely used in chimney construction, but brick is the preferred material since the regular shape decreases the chance of joint failures, and thus hidden chimney fires.\n\nAlthough internal fires for heating and cooking are all but universal in American houses, chimneys are not. The first cause of their decline was the development of practical iron stoves and ranges. Stoves made of iron plates were known in Medieval Europe and were introduced in the United States by Benjamin Franklin in the 18th century. These sheet-iron stoves were, however, leaky and inefficient; the widespread adoption of iron stoves and ranges did not begin until the 1830s, when relatively cheap and airtight units of cast iron were introduced. These required only a metal pipe to vent smoke and fumes to the outside and led to widespread abandonment of large masonry chimneys. In modest houses the stoves were often vented only by metal pipes extended through roof or wall. More commonly, massive fireplaces were replaced with narrow masonry flues, to which the metal stovepipes were connected. These led to small chimneys and provided a safer, fireproof escape for the hot concentrated fumes of stove and range.\n\nA second cause of the decline of external chimneys was the widespread adoption of gas, oil, and electricity for heating and cooking in the 20th century. These, too, require only metal pipes for external venting and thus the once essential chimney and fireplace have become only nostalgic luxuries in most 20th-century houses.\n\nPORCHES\u2014In British usage the word \"porch\" means sheltered entranceway, either partially open or enclosed on all sides to make a small room. Porches in the American sense\u2014that is, roofed but incompletely walled living areas\u2014are rare in Europe, where such spaces are known by other names: _verandah_ or _piazza_ (Britain), _galerie_ (France), _portale_ (Spain), or _loggia_ (Italy). The origin and inspiration of the far more common _porch_ of American houses has been much debated. It was clearly adopted because of the oppressive heat and frequent thundershower deluges of the New World summers, but its exact sources remain uncertain.\n\nPorches are normally constructed in either of two ways: one or more external walls can be omitted under the principal house roof to give an inset porch; or, an additional roof can be added onto the principal roof to give a smaller porch roof which is relatively independent of the main roof. Both types are common in American houses. Roofs are normally supported by the external house walls; when some of these are deleted to make a porch, columns or other roof-supporting devices are required. In addition, when porches occur much above ground level, a railing or low wall, usually with an open framework to admit breezes, is required for safety. These supporting members and enclosures can be of wood, masonry, or metal and provide rich opportunities for decorative embellishment of the house facade.\n\nEven in colonial times, porches were becoming common in the New World: both French and English colonists in the warmer, southern colonies commonly added verandahs or galeries to their houses. The use of large porches expanded until, by the late 19th and early 20th centuries, they had become an almost universal, and quite distinctive, feature of American domestic architecture. These showed an enormous variety of size, shape, and placement; many houses had several porches, or extended porches covering several walls. By the mid-20th century, this trend was completely reversed. Changing fashions\u2014and the development of air-conditioning for summer cooling\u2014all but eliminated this once dominant feature of the American house facade until the late 1980s, when it was re-introduced in traditional neighborhood developments (TNDs).\n\nDECORATIVE DETAILS\u2014Architectural details such as windows, doors, chimneys, and porches all serve important practical functions. One other category of detail has no such obvious use but is, instead, added principally to enhance the beauty of the house exterior. Such decorative details are of two main types: in the first, the principal coverings of the house exterior\u2014the wall cladding or roofing\u2014are decoratively elaborated. Shaped shingles or patterned masonry are examples of this kind of decorative detail. Still more common is the second type of decorative elaboration, in which neither the roof nor walls but rather some smaller functional detail is elaborated with decorative trim. Door and window openings are commonly embellished in this way; door surrounds are particularly favored since they are closely observed by all who enter the house. Indeed, certain eras of American house building are largely characterized by their distinctive elaborated door surrounds. Windows are commonly embellished by decorative surrounds or crowns, by shaped window openings, or, most commonly, by differing shapes and sizes of glass panes. Roof-wall junctions are another favored site for the addition of decorative detail: elaborate moldings or trim, commonly matched to those of doors and windows, are frequently added beneath eaves and rakes. Chimneys, too, provide decorative opportunities; decorative shapes and patterns in brick, stone, or stucco are common. Finally, as noted previously, porches provide a wealth of decorative opportunity; roof-support columns and protective railings have been elaborated in a nearly endless variety of decorative patterns.\n\nSTRUCTURE\n\n **WINDOWS**\n\nCOMPONENTS\n\nSTRUCTURE\n\n **WINDOWS**\n\nSASH OPERATION\n\nSTRUCTURE\n\n **WINDOWS**\n\nTYPICAL SASH GLAZING PATTERNS\n\nSTRUCTURE\n\n **DOORS**\n\nDOORWAY COMPONENTS\n\nSTRUCTURE\n\n **DOORS**\n\nTYPICAL EXTERIOR DOORS\n\nSTRUCTURE\n\n **CHIMNEYS**\n\n**PLACEMENT**\n\nSTRUCTURE\n\n **CHIMNEYS**\n\n**TYPICAL CONSTRUCTION**\n\nSTRUCTURE\n\n **PORCH ROOF CONSTRUCTION**\n\nSTRUCTURE\n\n **TYPICAL DECORATIVE DETAILS**\n\n* * *\n\n1Note that many houses have exterior wall _claddings_ of both wood and masonry. Houses with true structural walls of both materials are, however, rare.\n\n2A third system uses horizontal logs rather than a wooden framework and is employed in many Spanish Colonial houses.\n\n# LOOKING AT AMERICAN HOUSES\n\n# Neighborhoods\n\n# The Groupings of American Houses\n\n_N_ _eighborhood_ is a word used in many different ways. For the purposes of this guide, it refers to a geographic area of a town or city that was developed as a whole and\/or is generally filled with similar types of houses. Neighborhoods may also include additional housing types and other compatible uses or distinctive community characteristics. In the United States, neighborhoods rarely have a public administrative function but may have private responsibilities, such as enforcing deed restrictions or maintaining shared open space. The boundaries of neighborhoods sometimes follow those of one or more original subdivisions, but frequently boundaries are determined by residents.\n\nThis chapter is written for those simply looking at a neighborhood and trying to understand it. A neighborhood develops its distinct visual appearance from the cumulative effect of multiple factors, beginning with the general neighborhood type\u2014rural, urban, suburban, post-suburban, or a subcategory of one of these. Next it considers the kind of street plan utilized, whether rectilinear or curvilinear, and some of the overlay patterns that mold this plan\u2014green space, circulation, and block and lot size. There is an overview of the role that development influences\u2014developer\/builder, financing, governance, and growth rate\u2014played in producing individual houses and neighborhoods. These factors all blend together to create understandable streetscapes.\n\nNeighborhoods are the building blocks of cities. One can appreciate the history of a town or city by understanding its neighborhoods and how they interacted with or resulted from the growth and development of commercial, office, civic, and industrial uses.\n\n## Types\n\nNeighborhood groupings of houses fall into four general types: rural, urban, suburban, and post-suburban. Just as transportation played a major role in the location of American cities, it has similarly helped govern the location and configuration of American neighborhoods. Only rural and urban neighborhood types were found in the United States in 1800. At that time walking was the primary means of transportation, and according to the census only 5 percent of the population lived in \"urban\" areas\u2014the thirty-three cities and towns that then had a population over 2,500.\n\nNEIGHBORHOODS\n\n **RURAL NEIGHBORHOODS  \n(ca. 1750\u20131940)**\n\nlot widths vary  \nshallow front yards   \nirregular placement of house on lots   \nsize and style of house vary   \noutbuildings common\n\nRURAL NEIGHBORHOODS (1750\u20131940)\u2014A rural _neighborhood_ (with a grouping of homes near a congregation of stores and sometimes small-scale industry) must be clearly distinguished from a widespread and less concentrated rural _area._ Except for New England (where a village settlement pattern was typical), prior to the Civil War a majority of the United States was settled with a pattern of widely spaced individual farmsteads that created rural areas. Farmers estimated their \"neighborhoods\" to include fifty square miles. Trading centers, sometimes as small as a single general store, might serve an early rural area.\n\nThe majority of early rural _neighborhoods_ were the result of the slow expansion of trading centers for this very low-density agricultural economy. These neighborhoods are characterized by a relatively random mixture of lot sizes as well as varied placement of the houses and outbuildings on the lot. This was made possible by a general absence of regulation (zoning, subdivision, or deed restrictions), by slow-growing economies (rarely producing large subdivisions), by the gradual sale of parcels of original large lots, and by many individual builders and self-built homes. Because everyone walked within the center, generally there was a relatively compact arrangement of free-standing houses. The edges of town usually quickly transitioned to fields and open space. Rural neighborhoods, particularly those built before railroads, often have a distinct regional character.\n\nIn 1840 only 131 U.S. towns (commonly ports or on other waterways) had a population exceeding 2,500. By 1880, hundreds of new small towns had been formed and older villages had grown; 872 towns had more than 2,500 inhabitants. This was the result of forty years of intense railroad expansion, including the first transcontinental rail line in 1869. By 1910 a fast-paced golden age of electric trolley transportation had quickly and inexpensively connected still more small towns, and 1,801 towns had passed the 2,500-population mark.\n\nIt was the ready supply of rail-delivered goods, products of the expanding industrial revolution, arriving to stock many Main Street stores that created numerous small American rural neighborhoods. Not only did existing villages grow, but the companies themselves laid out hundreds of new small towns in the lands they had been granted along their tracks. It was not until after World War II that the relatively compact and almost organic rural neighborhood pattern began to be altered by the adoption of big-city standards for regulating subdivisions, street width, lot size, and house placement.\n\nNEIGHBORHOODS\n\n **URBANNEIGHBORHOODS: Early Core  \n(ca. 1750\u20131850)**\n\nnarrow houses, typically attached or semi-detached   \nlot width generally from twelve to twenty-five feet   \nno front yards   \nvaried house size and age   \nnonresidential uses may be present   \nadditional houses sometimes face alley\n\nURBAN NEIGHBORHOODS (1750\u20131920)\u2014Urban groupings are found where the economic base required high population density, historically related to water transportation routes and later water-powered industry. In 1800, the size of an urban settlement was limited by the distance one could comfortably walk and generally did not exceed a one-mile radius. Within an older urban area, a high percentage of the land is covered with structures that housed a variety of income levels and accommodated a mixture of land uses, since a lot could be developed for any purpose.\n\nUrban neighborhoods are characterized by narrow houses (attached or closely spaced) sited close to the street, with little or no front yard. This form allowed the construction of as many homes as possible within a short walking radius. Owners lived close to or above their businesses, and dense neighborhoods of small houses for workers could grow quickly in close proximity to a dock, mill, or factory. In areas where generous rear yards had been originally platted, houses facing onto alleys were sometimes added as demand for housing grew. Even within this diverse mix, areas somewhat differentiated by employment or socioeconomic level often evolved.\n\nDaily lives were tied to a small walkable area. In order to oversee any aspect of a business or factory, one had to walk there. Nothing allowed supervision from afar. No telephone allowed one to consult or even set up a meeting time with a partner or foreman. Furthermore, workers could not escape for a weekend away. They could only walk to a small diversion and then walk back home. Only the very wealthy could afford a horse or carriage in a city.\n\nThe 1829 introduction of omnibuses\u2014twelve- to twenty-seat horse-drawn carriages utilized for regularly scheduled public transportation along fixed routes\u2014allowed for small expansions of the tightly packed urban fabric. In addition, the omnibus encouraged more detached houses located just beyond a city's urbanized edge in the \"borderlands,\" where free-standing single-family houses were sited on large tracts of land\u2014a precursor to the preferred _form_ of suburbs.  A second transportation advance, the steam ferry, allowed the development of neighborhoods remote from a city center, separated by a river or other body of water. Brooklyn, across the East River from Manhattan, was a rural area until regular ferry service made the development of Brooklyn Heights possible. First advertised in 1823, this early real estate development was built at a distance from the urbanized core\u2014a precursor to the preferred _location_ of suburbs. Despite its separation from Manhattan, Brooklyn Heights was primarily developed with the urban form of attached houses on narrow lots\u2014because, after the ferry ride, homes were still accessed primarily on foot.\n\nNEIGHBORHOODS\n\n **URBAN NEIGHBORHOODS: Urban Extensions made possible byhorse-drawn streetcars\u2014also omnibuses, ferries, cablecars  \n(ca. 1830\u20131900)**\n\nnarrow houses (attached, semi-detached, or detached)   \nlot width generally from sixteen to twenty-five feet   \nfront yards commonly absent   \nsimilar size and age houses often grouped together  \nresidential uses may occupy entire block face\n\nHORSE-DRAWN STREETCARS\u2014Made possible by the 1852 invention of a method for sinking a track into the street that left a level street surface, horse-drawn streetcars (horsecars) were a huge improvement over the earlier \"trackless\" omnibuses. The wheel moving over a flat track allowed faster speeds (six to eight miles per hour), smoother rides, and larger-capacity cars (thirty to forty passengers). These provided the first true urban transit system and facilitated the exponential expansion of new urban neighborhoods outward from the center of the city. By 1860, horsecars were in use in New York, Baltimore, Philadelphia, Pittsburgh, Chicago, Cincinnati, and Boston. Coinciding with the rapid industrialization of the U.S.\u2014accompanied by massive immigration from 1850 to 1890, exploding growth in cities\u2014the urban population of the United States rose from 3.5 million in 1850 to 22 million in 1890.\n\nHorsecars facilitated three kinds of development in cities. First and foremost, they allowed extensions of the existing urban fabric. New urban neighborhoods of row houses could be conveniently built farther away from other uses. As larger buildings and more industrial uses were built in the original mixed-use urban cores, many of those who could afford it preferred to move to these quieter residential, yet still urban, settings.\n\nSecondly, horsecars created some very early streetcar suburbs, groupings of detached houses that became widespread with faster electric streetcars. Third, toward the end of a line, they allowed access to the \"borderlands.\" Not confined to cities, a horse- or mule-car line in a small town could make it convenient to live on large-lot mini farms away from the village center\u2014the _rural_ equivalent of borderlands.\n\nBy the mid-1880s there were almost six thousand miles of horsecar track and the cars were carrying about 188 million passengers a year. In a city, however, horses were finicky, expensive, and, most of all, messy. They required tons of hay, left urine and excrement on the streets, required multi-level stables in large cities, could only work a four-hour shift, were hard to pair evenly into teams, and when they died\u2014not infrequently on the job\u2014the disposal of their bodies presented a challenge. Small wonder that the introduction of electricity to power streetcars was so quickly adopted.\n\nSUBURBAN NEIGHBORHOODS\u2014Suburban neighborhoods are a varied type of house grouping. Rather than resulting from economic necessity, suburbs grew from the utopian vision of living in a pastoral setting close enough to a city to enjoy its jobs and pleasures but removed from the unpleasant aspects of urban life. New transportation methods made a daily commute feasible and were absolutely necessary in the creation of suburbs. In addition, three other innovations made suburban living practical: the rapid spread of light wood balloon-frame construction (making it fast and affordable to build free-standing houses, 1840\u20131870); the proliferation of gas and electric utility systems (producing inexpensive heat for free-standing houses, 1880\u20131920); and the expansion of telephone service (allowing remote two-way communication, 1880\u20131910).\n\nThe strong desire to move to the suburbs resulted from both a _push_ and a _pull._ A city's older urban neighborhoods could be unpleasant for reasons that included open sewage, shared privies, disease, fire, garbage left to be eaten by pigs or dogs rather than collected, horse and industrial odors, and later the sound pollution from trains and trolleys, and the growing size of post-industrial building types. These conditions helped push those who could afford it out of urban neighborhoods. At the same time a flood of popular mid-19th-century magazines and books extolled the morality and healthfulness of living in the country and the joy of connecting to nature and your own piece of land. The desire for this lifestyle helped pull those who could afford it out of the city as soon as new kinds of transportation made it feasible.\n\nSuburban neighborhoods typically consist of free-standing houses on lots large enough to provide a desirable landscaped setting. These are divided into categories based upon the transportation innovation that connected the suburb to the city, each of which produced a somewhat different scale and look. North America has some of the world's largest and most diverse suburban groupings. It is important to understand that what was first developed as a distinct early suburb has today frequently been encircled by the city it was built to serve. Thus many cities, particularly those that expanded rapidly in the 20th century by annexing additional land, have former suburbs located in what is today a central part of the city.\n\nNEIGHBORHOODS\n\n **SUBURBAN NEIGHBORHOODS:Railroad Suburbs  \n(ca. 1850\u20131930)**\n\ndistinct groupings of houses near rail stops   \nhouses typically detached   \nlarge-scale lots, commonly with houses of similar size   \nage and style of homes vary   \nuniform front yards often required by deed restrictions   \nresidential uses typically occupy entire block face   \nstreet trees common   \ngarages typically detached\n\nRAILROAD SUBURBS (1850\u20131930)\u2014The steam railroad was the first form of transportation that made it possible to escape the city center and live in a rural setting while still commuting to work in the city. After the first U.S. steam locomotive train was introduced in Baltimore in 1830, the expansion of rail lines was very rapid. The miles of laid track grew from about 2,750 in 1849 to 87,800 in 1880. Railroads fueled the growth of new cities in the Midwest and West, and expanded older East Coast cities. The earliest tracks were laid to expand the trade areas of established ports, but soon longer lines were introduced to traverse long distances, crossing the continent in 1869 and opening vast new areas for settlement. As this system was built, it became obvious that increasing daily commuters around urban areas would bring additional income on a regular daily basis.\n\nThere were two distinct kinds of railroad towns near large cities. The most well known and publicized were new railroad suburbs planned specifically for commuters\u2014bedroom communities for men who went into the city to work each day. Far more common, however, were fully functional nearby towns with rail stops. These self-contained rail-stop towns provided the city with a rich array of needed services\u2014and also housed the workers that supported them. Among these were farming towns, suburban industrial towns, recreational destinations, and towns based around institutions such as universities. These communities had populations that generally lived and worked in the same place and utilized their rail stops in different ways\u2014for the daily transport of agricultural or industrial products into the city and as transportation for city dwellers out to enjoy a recreational weekend.\n\nThe primarily residential railroad suburbs, however, garnered most of the press. Often renowned for their creative land use and other innovations, planned railroad suburbs bragged of their existence, while their working-class rail-stop brethren served in the background. The cost of a daily suburban commute was steep and tended to make planned railroad suburbs bastions of the well-to-do. But not all residents of classic railroad suburbs were wealthy. Approximately one-third of inhabitants were servants whose families often lived in nearby neighborhoods of small houses.\n\nLlewellyn Park, New Jersey (1857\u20131859), twelve miles west of New York City, was one of the first and more influential of the planned railroad suburbs. It had a large communal green space in its center, the Ramble, a remarkable innovation in the days before public parks. It featured curved roads that followed the topography of its hillside site, something uncommon in new developments. There were many other well-known railroad suburbs\u2014chief among them Chicago's 1869 Riverside.\n\nRailroad suburbs were usually built several miles from the cities they were designed to serve. Since trains were slow to gather speed as well as come to a stop, the space between suburban stations was generally a mile or more, forming railroad towns in the 1800s into individual communities rather than the continuous development that would characterize later suburb types. This led to strings of railroad suburbs, such as the Main Line along the Pennsylvania Railroad's route west of Philadelphia. Prior to the automobile, railroad suburbs had remained small in area due to the walking distance to the station.\n\nNEIGHBORHOODS\n\n **SUBURBAN NEIGHBORHOODS: Streetcar Suburbs  \n(ca. 1890\u20131930)**\n\nlinear neighborhoods developed along streetcar lines   \nhouses generally detached with narrow end toward street   \nnarrow lots, generally thirty to fifty feet wide   \nshallow side yards   \nfront-yard setbacks and front porches are typical   \npaved sidewalks   \nlocations closest to streetcar line preferred   \ngarages typically detached, if present\n\nELECTRIC STREETCAR SUBURBS (CA. 1890\u20131930)\u2014After their introduction in 1887, streetcars powered by electricity rapidly revolutionized transit in U.S. cities. Track originally built for horse-drawn cars was converted to use by electric streetcars. The length of streetcar track in the United States skyrocketed from 5,783 miles in 1890 to 34,404 in 1907, as the country's urban population increased by 50 percent. Electric streetcars were much faster than horsecars and thus allowed large tracts of open land well beyond the edge of the city to be developed.\n\n_Trolley_ and _streetcar_ are terms often used interchangeably today. However, from 1890 to 1920, they were considered streetcars when they operated in an urban area and ran down city streets. Outside a city, they were more often called trolleys. In addition to allowing suburban expansion, their speed in open countryside (twenty to thirty miles per hour and up) allowed them to efficiently connect many small towns to each other as well as to cities. In this role, electric cars were sometimes heavier duty and the line was called an interurban.\n\nThe speed of electric cars facilitated a new real estate development process. A typical pattern was to build a trolley line into vacant countryside, often terminating at a recreational destination\u2014a park, a fairground, an amusement park, or a large cemetery (which, in the 19th century, functioned as tranquil open space). This planning helped attract riders immediately. House lots were platted adjacent to the line, subdivision improvements were added (sidewalks, utility connections, etc.), and the vacant lots placed on the market. Signs advertising \"Home Sites for Sale\" greeted passengers traveling along the line. As lots were sold and homes built, the new residents increased the number of daily commuters. The streetcar line added value to the vacant land, and the development of the land brought value to the streetcar. Often the owner of a trolley line and its adjacent property was either the same or connected in some way. By 1900 trolley lines and streetcar suburbs had become the primary factor in the development of new urban neighborhoods throughout the country.\n\nHouses built in streetcar suburbs were typically free-standing. City dwellers wanted houses in neighborhoods with lots large enough to allow for front lawns and a green suburban feeling. The spacing of houses was governed by the fact that one walked home from the streetcar stop, thus side yards were shallow and houses tended to have a narrow facade on the street. Entire neighborhoods of new, detached structures were built, filled with various styles of American four-square houses, bungalows, and front-gabled houses, each with its narrow front turned toward the street. With the exception of a few large older metropolitan areas, new attached houses became relatively rare after about 1900.\n\nStreetcars created long narrow neighborhoods along their tracks. Because these neighborhoods depended on homes having easy pedestrian access to the streetcar line, they extended only a few blocks to each side. By 1920 these straight narrow corridors of development expanding out from the urban center resembled the spokes of a wheel, with vacant land in between.\n\nThe introduction of electric trolleys, with their five-cent fare, had opened up a world to middle-class Americans where they not only dreamed of buying a house but also easily escaped the small walking radius in which they had previously lived and worked. For the first time, the electric trolley allowed day trips to a beach or a park, and citizens could see their town as they traveled. It introduced an entirely new kind of freedom and leisure that would soon be multiplied by the automobile.\n\nNEIGHBORHOODS\n\n **SUBURBAN NEIGHBORHOODS: Early Automobile Suburbs  \n(ca. 1915\u20131940)**\n\nhouses detached   \nwider lots, forty to one hundred feet or more   \nconsistent front-yard setbacks   \ngenerally long blocks   \nnarrow front driveway off paved streets   \nstreet trees common  \nsidewalks may be omitted in later developments   \ngarages typically detached\n\n**EARLY AUTOMOBILE SUBURBAN STREETSCAPE, CA 1930**  \n(in cities throughout the United States)\n\nEARLY AUTOMOBILE SUBURBS (1915\u20131940)\u2014At first, automobiles were only an avocation for the wealthy, stored in central places and delivered to the door for an afternoon drive, much as a carriage horse would be delivered by the livery. They were expensive, and when kept at home they were stored in a detached garage so gasoline fumes and the threat of fire were well removed. It was not until 1910 that automobile ownership became affordable to the middle class, due to Henry Ford's revolutionary mass production of his Model T. The result was extraordinary. In 1918, only one in thirteen families owned a car. Sales skyrocketed, and by 1929 four out of five families owned one. By 1920 it was feasible to design a new type of suburban neighborhood\u2014one dependent on automobiles for access. This made the vacant land between the spokes formed by streetcar suburbs accessible for new development. These new early automobile suburbs were not only accessed by automobiles, they were planned to accommodate them. They spread rapidly, and by 1941 there were 2,100 communities in the country with populations between 2,500 and 50,000 that did not have any public transportation.\n\nThe automobile initiated a number of changes in neighborhoods. Streets were paved. Blocks were often planned longer than in streetcar suburbs. The longer blocks were tolerable because one did not have to walk them. Long blocks were doubly profitable\u2014less street paving to pay for and more land available for lots to build upon. Sidewalks could be narrower because they were now optional. Curb cuts were added for driveways leading to a garage that was generally detached. It was feasible for individual lots to be wider than those in streetcar suburbs because walking home was no longer the norm. Setbacks from the street often became deeper.\n\nEarly automobile suburbs did not view the automobile as an enemy to be kept out. The plans for many upscale neighborhoods utilized a warped street grid\u2014a grid pattern with subtle curves\u2014sometimes with a handsome central avenue or boulevard adding interest to the interior of the neighborhood, which was considered a desirable location for the largest homes. However, by 1940 traffic in the interior of a neighborhood plan was considered undesirable.\n\nEarly automobile suburban development flourished all across the United States during the 1920s. There was so much land platted that in some places numerous vacant lots remained when the housing industry virtually shut down in the early 1930s.\n\nNEIGHBORHOODS\n\n **SUBURBAN NEIGHBORHOODS: Post\u2013World War II  \n(ca. 1940\u20131980)**\n\nhouses detached   \nwider lots, sixty to one hundred feet or more   \nconsistent front-yard setbacks   \ngenerally long blocks   \nfront driveway often doubles as front walkway   \nsidewalks frequently omitted   \nloop roads and\/or cul-de-sacs   \nhouses generally do not face main roads   \ngarages typically attached\n\n**POST WORLD-WAR II SUBURBAN STREETSCAPE, CA 1955**  \n(in cities throughout the United States)\n\nPOST\u2013WORLD WAR II SUBURBS (1940\u20131980)\u2014The federal legislative changes that fueled the development of post\u2013World War II suburbs began in 1934 with the creation of the Federal Housing Administration (FHA), which initiated strict subdivision planning guidelines. However, due to the sluggish economy of the 1930s, along with the prohibition of nonessential construction during World War II, these guidelines did not widely influence the form of suburbs until after the war ended. In 1945, after sixteen years with little residential construction, there was a massive pent\u2011up demand for new homes and FHA guidelines began to exert a staggering force on the creation of new neighborhoods. In 1940 the FHA published a bulletin titled _Successful Subdivisions_ that in twenty-eight pages explained and illustrated what was expected in developments utilizing their new mortgage insurance. In addition to well-located, dry land with a strong market, the agency required a competent professional plan and suggested it include good streets of prescribed width, well-shaped building lots with driveways, protection of the neighborhood from through traffic, and the use of protective covenants. The desire to avoid direct through traffic produced profound changes in neighborhood design. The FHA exploited the leverage of their mortgage insurance to implement completely new approaches to neighborhood street patterns that eliminated easy automobile access. (See  \"FHA-guided Subdivisions,\" and discussion of arterials in Primary Roads.)\n\nMost post\u2013World War II neighborhoods were located beyond the developed edges of cities where many municipalities were planning or beginning to build an expandable network of federally subsidized highways that fed into a system of arterials\u2014new broad city streets designed to carry substantial traffic. The FHA guidelines encouraged post\u2013World War II subdivisions both to take advantage of and to protect themselves from this new system of major streets and thoroughfares. It was recommended that new subdivisions nestle beside an arterial for easy access but with few entrances from this major road into the neighborhood. Street patterns inside the neighborhood were carefully designed to prevent use of neighborhood streets as short cuts, utilizing devices such as loop roads and cul-de-sacs. A second defining feature of post\u2013World War II suburbs is the widespread use of curvilinear planning and long blocks.\n\nWithin the plan, lots sometimes became wider in order to accommodate new Ranch or Split-Level houses with their long facade facing the street. Sidewalks became less relevant and were often omitted, as were front walks leading from the sidewalk to the front door. Wide driveways that led to an attached front-facing garage or carport often served as the front walk. Later upscale subdivisions might feature side-entry or alley-entry garages to avoid garage doors on the front facade.\n\nMass-construction techniques, perfected during the war years, were utilized for both economy of scale and rapid construction. In some cases, entire neighborhoods were built from only a few home plans with similar size and layout. The FHA encouraged small home size by limiting the maximum loan amount they would insure. This gave more people the ability to purchase a home and helped fulfill a national goal of providing homeownership for the sixteen million servicemen who had served in World War II (see Home Financing).\n\nToday the popular belief is that post\u2013World War II suburbs were isolated residential areas, but the model they were encouraged to follow\u2014and seen in postwar suburbs from the late 1940s into the 1970s\u2014was a more integrated approach that included schools, churches, nearby retail, parks, and community facilities based in large part on Clarence Perry's suggestions for a \"Neighborhood Unit.\" The elements were perhaps farther apart than in the streetcar and early automobile suburbs, but requirements for a day-to-day family life were generally nearby. Specific FHA suggestions for subdivisions included school and church sites, commercial sites where needed, and parks as an asset.\n\nNEIGHBORHOODS\n\n **POST-SUBURBAN NEIGHBORHOODS  \n(ca. 1970\u2013present)**\n\n**LEFT: \"PEDVILLE,\" POP. 12,500**  \nwalking distance of rapid transit   \none-quarter- to half-mile radius of higher-density development  \nhigh residential density supports varied retail  \nrequires green space and community amenities  \nwalking and biking connections common\n\n**CENTER: \"SALUTATION,\" POP. 1,250**  \nre-creates visual qualities of rural, urban, and pre-1940 suburban streetscapes  \nwalking and biking connections common   \nemphasis on green space and sense of community  \ndaily shopping needs hopefully nearby\n\n**RIGHT: \"WAYOUT ACRES,\" POP. 125**  \nsmall developments, often gated   \nshopping at a distance  \ncommunity amenities may not exist (large house accommodates private amenities for each household)  \ncar essential  \nmay be interspersed with agriculture\n\nNEIGHBORHOODS\n\n **POST-SUBURBANNEIGHBORHOODS  \n(ca. 1970\u2013present)**\n\n**CA. 2010**\n\nPOST-SUBURBAN (1970\u2013PRESENT)\u2014Suburban neighborhoods were built to serve as residential areas connected to and serving a downtown. By contrast, many developments after about 1970 are post-suburban, built to serve \"edge nodes\" located beyond the ring of post\u2013World War II suburbs and interacting primarily with other post-suburbs and suburbs. Three types of neighborhoods being built today are easy to identify.\n\n**SLUGS:** The most prevalent post-suburban neighborhood type is the SLUG (an area of spread-out, low-density, unguided growth). These may be small subdivisions or individual home sites scattered amid forest or farmland far from the kinds of amenities typical of postwar suburbs. Planning for less through traffic has frequently escalated into the formation of gated communities. The FHA's effort to \"protect lots from adjacent non-conforming uses\" has morphed into huge tracts of residential or apartment use without the parks or community uses found in most postwar neighborhoods. Local shopping has been swept aside by distant big-box retailers that require a huge market area to reach them by automobile.\n\nGovernmental policies enabled and continue to encourage out-migration from older areas. After the first ring of post\u2013World War II suburbs were built, we entered an era of post-suburban sprawl, including the last two decades of far-flung SLUGs. The 1956 Federal-Aid Highway Act created a dedicated gas-tax funding stream for the accelerated construction of more than 41,000 miles of interstate highways. This was supplemented with both earlier and later federal subsidies for well over a million miles of local highways, farm-to-market roads, inner loops, outer loops, and city-wide networks of arterials\u2014providing the vast roadway network that has enabled sprawl and SLUGs. In addition, federal tax policy has favored new commercial construction\u2014including incentivizing construction of big-box stores at the lucrative nodes where the federally funded roads intersect. In 1970 the Federal National Mortgage Association (\"Fannie Mae\") was listed on the New York Stock Exchange and removed from the FHA's decades-long oversight of residential construction\u2014involving home size, quality, location, and subdivision amenities.\n\n**TODs:** Today's forward-thinking national efforts are focused on directing new growth into creating a second post-suburban type, the resource-efficient TODs (transit-oriented developments) of compact neighborhoods within walking distance of rapid transit. TODs often include re-creating new walkable urban and early suburban streetscapes, as well as denser mid- to high-rise condominiums and apartments. Unfortunately, some TODs do not yet include essential pedestrian and bicycle connections and amenities.\n\n**TNDs:** The third prevalent post-suburban type is the visually inviting TND (traditional neighborhood development) that salutes the form and scale of earlier urban, rural, streetcar, and early automobile suburban types with its more compact scale and pedestrian-friendly streetscapes along with an enlightened emphasis on connectivity and green space. In the best of all worlds, traditional neighborhood developments are served by rapid transit (making them also TODs) and include the amenities needed for a complete neighborhood, along with carefully placed higher-density housing choices.\n\n## Neighborhood Plan Types and Density\n\nThere is great variety in the plans used for groupings of each neighborhood type. The following illustration shows a common plan type that occurs in each of six neighborhood types. The first diagram is a rural neighborhood with unplanned roads. Next clockwise is an urban neighborhood with a grid plan of small blocks, an early automobile suburb with a warped grid, a post-suburban SLUG with one house per acre, a post\u2013World War II suburb with an internal loop road, and a streetcar suburb with a rectilinear grid. The factors contributing to these plan types are explored in the following section.\n\nThis illustration clearly shows the wide variety of population density (often expressed as the number of persons or dwelling units on an acre of land) that can make up a neighborhood of single-family houses. Approximately two hundred acres of land are covered by each plan, with two hundred dwelling units shown placed on each. Comparing plans, it is easy to see vast differences in the amount of land utilized for the same number of dwelling units.\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**RURAL**\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**URBAN**\n\nEARLY CORE\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**SUBURBAN**\n\nSTREETCAR\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**SUBURBAN**\n\nEARLY AUTOMOBILE\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**SUBURBAN**\n\nPOST\u2013WORLD WAR II\n\nNEIGHBORHOODS\n\n **NEIGHBORHOOD TYPES** sample plans (approximately two hundred acres and two hundred dwelling units per plan)\n\n**POST-SUBURBAN**\n\nSLUG\n\nEXISTING NEIGHBORHOODS\u2014More than one-third of the total U.S. economic asset is invested in our built environment. The preservation of existing neighborhoods\u2014rural, urban, and suburban, whether or not \"historic\"\u2014is the most economic, green, and conservation-minded action.\n\nThis fact was not understood in the 1950s and early 1960s, when it was believed that the old parts of towns could be effectively wiped clean and rebuilt. There were three major assaults. First, the Housing Act of 1954 provided two-thirds of the money needed to buy up and tear down large areas of inner cities for \"urban renewal\"\u2014sadly, efficiently removing large areas of existing fabric. Second, the 1956 Federal-Aid Highway Act provided 90 percent of the funds, but gave states and municipalities almost complete control over the routes of interstates, many of which plowed through the hearts of neighborhoods and cities. Third, the FHA's financial guidelines withheld home loans from older neighborhoods through a practice called redlining. These three powerful forces left older neighborhoods in grave shape. For example, about one-third of the old part of Boston was demolished for an interstate and urban renewal\u2014a massive project so destructive that billions of dollars have been spent burying the interstate portion during the last ten years and restoring the street grid. Other cities barely escaped. New Orleans became the first city in the nation to stop an interstate highway for environmental reasons, one that would have plowed through the Vieux Carr\u00e9 right by Jackson Square in the manner that the Boston freeway had impacted the Faneuil Hall Marketplace. At the same time that demolitions were taking place for freeways, large areas of cities were being cleared for urban-renewal projects. In the midst of these controversies Jane Jacobs's pivotal _The Death and Life of Great American Cites_ was published, beautifully describing and defending mixed-use urban neighborhoods.\n\nAs the high cost of this widespread devastation of existing neighborhoods became obvious, the Historic Preservation Act of 1966 was enacted to survey and protect historic resources; it no longer allowed federal funds to be used to demolish valuable older areas without a thorough professional review. It was another eleven years before lending policies caught up (through the Community Reinvestment Act of 1977) and older neighborhoods had mortgage and home-improvement funds easily available to them.\n\nNew high-density areas today can be carefully developed in a way that does not negatively impact or attempt to replace two centuries of public and private investment\u2014and use of natural resources\u2014thus avoiding the urban fabric disruptions that marked the 1950s and 1960s.\n\nNEIGHBORHOODS\n\n **DESTRUCTIVE USE OF FEDERAL FUNDS: REDLINING**  \nfederal loan funds were witheld from many older neighborhoods, a practice that ended in 1977\n\nMUNGER PLACE, DALLAS, TEXAS  \nBefore Fannie Mae's 1976 Munger Place project\n\nNEIGHBORHOODS\n\n **DESTRUCTIVE USE OF FEDERAL FUNDS: REDLINING**  \nfederal loan funds were witheld from many older neighborhoods, a practice that ended in 1977\n\nMUNGER PLACE, DALLAS, TEXAS  \nAfter Fannie Mae's 1976 Munger Place project. This demonstrated that loans could be successfully made in older, dilapidated neighborhoods. This pilot project helped lead to the Community Reinvestment Act of 1977 that opened up lending in older neighborhoods that had previously been redlined.\n\nNEIGHBORHOODS\n\n**DESTRUCTIVE USE OF FEDERAL FUNDS: REDLINING**  \n **EXAMPLE OF REDLINED MAP**  \nshowing downtown Philadelpia with inner-city neighborhoods redlined to indicate where federal programs would not insure loans\n\nNEIGHBORHOODS\n\n **DESTRUCTIVE USE OF FEDERAL FUNDS**  \nin many cities led to 106 review process for impact of federal dollars on important resources\n\nMultiple blocks of urban fabric cleared for \"urban renewal\" with use of federal funds authorized by the Housing Act of 1954\n\nNEIGHBORHOODS\n\n **DESTRUCTIVE USE OF FEDERAL FUNDS**\n\nHeavy black line shows projected route for New Orleans interstate, hugging river, and elevated\u2014cutting Vieux Carr\u00e9 and Jackson Square off from river. First interstate highway to be stopped for environmental reasons. Led to the Historic Preservation Act of 1966 and the Section 106 review process for use of federal funds that would have an adverse impact on historic buildings\n\nNEIGHBORHOODS\n\n **DESTRUCTIVE USE OF FEDERAL FUNDS**\n\nView of historic Jackson Square and the St. Louis Cathedral from the Mississippi River\u2014had the elevated New Orleans interstate not been stopped\n\n## Ground Plans\n\nGround plans for neighborhoods can be divided into two general groups\u2014rectilinear and curvilinear. Towns with rectilinear plans have been common for thousands of years and were joined by curvilinear plans in the 19th century. Both had interconnected streets. There was an abrupt change about 1940, following the rising popularity of automobiles. For the first time, neighborhoods were planned with streets designed specifically _not_ to interconnect. Their plans were generally curvilinear and included looping interior streets, cul-de-sacs, and few entries into the neighborhood interior. This is an approach not favored by planners today.\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS** (most common until ca. 1940)\n\n**EARLY TOWN PLANS**\n\ngenerally prepared by a town's colonizer or promoter  \nregular grid plans with short blocks most typical  \ncommonly centered around a central plaza or square\n\nSavannah, Georgia, town plan\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS** (most common until ca. 1940)\n\n**EARLY TOWN PLANS**\n\nplan based on Laws of the Indies (would be laid out at 45-degree angle)\n\nRECTILINEAR PLANS\u2014Until 1940 the vast majority of the United States was developed with a rectilinear street-grid pattern, as rectilinear grids are a quick, efficient way to survey land and subdivide it for sale.\n\nThe grid system has been used in planning towns for more than four thousand years. However, when the United States passed the Land Ordinance of 1785, undertaking to divide much of North America with a strict north-south\/east-west one-mile-square grid system of \"sections,\" it was by far the most massive project of a rigid rectangular grid survey in history. Public roads were planned to run along the one-mile boundaries. The Homestead Act of 1862 reinforced this pattern when it distributed square homesteads cut from this grid to qualifying individuals. This national grid system was also used to grant land to railroad companies as an incentive to build cross-country rail lines. As the railroads platted new towns and sold lots along their tracks, they created hundreds of additional small towns with a north-south grid-system plan.\n\nEARLY TOWN PLANS\u2014A great many American towns were built upon an early plan, often drawn up by the original person or group promoting the settlement. The majority of these were grid plans. The plan often aligned with the waterway the town was built adjacent to or, later, followed the north-south land ordinance grid or a rail line. Squared open spaces were typically included. In the case of towns colonized by the Spanish, the Laws of the Indies prescribed a grid system with a central plaza to be laid out from southwest to northeast.\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS**\n\n**GRIDIRON PLAT EXTENSIONS**\n\nmultiple angles of street grids   \nold survey lines can be intuited where street grids change   \nmany streets end at T intersection   \nmany streets have slight jogs\n\nRochester, New York\n\nGRIDIRON PLAT EXTENSIONS\u2014Taking a piece of land and placing a rectangular grid on it was the simplest and most cost-effective way to ready smaller lots for sale. It little mattered if the larger piece of land was a squared section of the western states or an irregular metes-and-bounds survey from an early settlement\u2014a grid was almost always the fastest way to divide it. Generally roads were extended from adjacent developments or from the edge of an early town plan. Where a new survey introduced an angle, roads changed direction. Where roads did not completely align, small jogs were left in place. These gridiron extensions were utilized in creating a majority of neighborhoods prior to 1940.\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS**\n\n**PLANNED RECTILINEAR SUBURBS**\n\nplanned street pattern throughout  \nstraight linear green spaces occur   \nlonger rectilinear blocks are common\n\nWoodruff Place, Indianapolis, Indiana\n\nPLANNED RECTILINEAR SUBURBS\u2014Beginning in 1887, the popularity of the electric trolley spurred the major expansion of trolley lines. Tracks were most efficiently built in a straight line, making it easy for developers to plat their streetcar suburbs in a grid pattern. These fast-moving trolleys, however, also made it economical to purchase large vacant areas beyond the city's built\u2011up edge, where it was feasible to create more highly designed grid plans.\n\nThese frequently incorporated City Beautiful elaborations. Long, straight, landscaped central medians (often called boulevards, avenues, or parkways) were favorites, and sometimes the trolley tracks were placed within this linear space. Broad, elongated medians were also easily inserted into the prevalent grid street systems of the western states and thus provided a relatively affordable way to add drama and green space into pre\u2013World War II plans for suburban expansion.\n\nStreets featuring broad, landscaped park-like medians were also included in Olmsted's influential plan for Riverside and remained popular in curvilinear suburbs, particularly those planned for early automobiles. Less commonly, a communal green space was located in the center of a block.\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS**\n\n**CITY BEAUTIFUL ELABORATIONS**\n\nWoodruff Place, Indianapolis, Indiana\n\nNEIGHBORHOODS\n\n **RECTILINEAR PLANS**\n\n**CITY BEAUTIFUL ELABORATIONS**\n\nMunger Place, Dallas, Texas\n\nCITY BEAUTIFUL ELABORATIONS\u2014The 1893 World's Columbian Exposition in Chicago spurred a nationwide interest in creating the City Beautiful. The dramatic axial Beaux Arts plan of this world's fair, followed by later expositions in cities throughout the U.S., stimulated a national interest in planning that transformed parts of cities with a grand plan. Planning elements of civic design\u2014such as a street terminating at a focal point (monument, church, or civic building) and streets divided by a landscaped median park\u2014became desirable for both rectilinear and curvilinear plans. And civic art might be added (elaborate piers, pylons, or gates marking a neighborhood entrance; decorative light fixtures, a fountain, or a statue, etc.). Each of these elements had historical roots, and their use in the U.S. pre-dated\u2014but generally received a huge impetus from\u2014the influential expositions of the era.\n\nCURVILINEAR PLANS\u2014Although some towns have early curvilinear streets laid out as dictated by topography, neighborhoods with designed curvilinear street plans were not developed in the United States until the mid-19th century. Two disparate factors\u2014one aesthetic and one practical\u2014helped introduce the use of curvilinear plans. On the aesthetic side, Andrew Jackson Downing's widely read books promoted the ideal of curvilinear planning for both large and small properties. This theme began in 1841 with his first book, _A Treatise on the Theory and Practice of Landscape Gardening,_ and continued throughout his publishing career. His _Treatise,_ for example, visually contrasts an angular \"Common Farm\" with an \"improved\"\u2014and curvilinear\u2014\"Country Seat.\" Beginning in the 1850s, the principles for residential landscape espoused in Downing's books (based on English influences) were incorporated first into the designs for cemeteries and later into neighborhood plans.\n\nOn the practical side, the application of these principles to suburbs required large vacant tracts of land, and it was the rapid spread of a railroad network in the mid-19th century that allowed access to undeveloped tracts well beyond the city center. Curvilinear plans were favored for affluent suburbs throughout the late 19th and early 20th centuries, but they were not commonly used for affordable neighborhoods until the advent of the FHA and its guidelines.\n\nThere are two typical places that curved streets appear without being part of a plan. First, some early settlements had neither a plan nor a grid survey. Sometimes roads in these were reported to have followed animal trails or Native American paths, typically along ridges of high ground. Although curving, the street patterns in these places do not have a planned appearance. Second, early development was generally limited to flat or rolling terrain, as steep hillsides and canyons were not easily accessible. The widespread use of private automobiles by about 1920 made it possible to build homes on steep and previously inaccessible sites reached by irregularly curving streets dictated by the contours of the steep topography. These challenging locations did not allow the smooth curves favored by Frederick Law Olmsted.\n\nNEIGHBORHOODS\n\n **CURVILINEAR PLANS** (more common after 1940)\n\n**OLMSTEDIAN PLANS (1869\u20131940)**\n\npublic green space, often along waterways  \nsmoothly curving streets   \nfew right-angle intersections  \nstreet trees  \nminimum setback of house from road  \neach block face has similar lot size broad range of lot size within development  \nmaintains natural drainage system   \nhouses at higher grade than street   \nutility systems preceded building  \ndeed restrictions\n\nGeneral Plan of Riverside, Illinois; Olmsted and Vaux; 1869\n\nOLMSTEDIAN PLANS\u2014Although there were earlier planned railroad suburbs such as Llewellyn Park, it was Olmsted's well-publicized 1869 plan for Riverside, Illinois, that provided a complete model for subdivision design and introduced curvilinear planning to a broad audience. Olmsted intended the design of his roads to \"suggest and imply leisure, contemplativeness, and happy tranquility\" rather than the \"eagerness to press forward\" he felt was inherent in rectilinear plans. He believed Riverside's curves had \"a controlled sweep and continuity of their own, unlike any precedent.\" This iconic plan\u2014with smoothly curving streets, generous street width, lack of corners, and ribbons of parkland\u2014served as a prototype for curvilinear neighborhood planning through the early 20th century. Naturalistic Olmstedian plans required relatively large tracts of land to achieve their rural atmosphere with curving streets.\n\nNEIGHBORHOODS\n\n **CURVILINEAR PLANS**\n\n**WARPED GRIDS (CA. 1915\u20131940)**\n\ncommon in early automobile suburbs   \nstreet-grid pattern modified with gentle curves   \nsmall triangular landscaped islands at angled intersections  \nbroad street sometimes used as central planning feature\n\nSt. Francis Wood, San Francisco, CA; Olmstead Brothers; 1912\n\nWARPED GRIDS\u2014In many early automobile suburbs the streets are effectively laid out in a grid pattern, but one that is warped into gentle arcs. The grid was an efficient way to carry automobile traffic, but developers and the public alike had come to appreciate the visual effect of a curving street. A grid modified to gently curve provided both. Sometimes a somewhat wider street, either an arterial or a collector, was introduced into this plan and flowed through the neighborhood to gather the traffic from local streets. These often were planned as focal streets.\n\nNEIGHBORHOODS\n\n **CURVILINEAR PLANS**\n\n**FHA-GUIDED SUBDIVISIONS (CA. 1940\u20131980)**\n\n**BEFORE:**  \nSUBDIVISION GRID PLAN SUBMITTED TO THE FHA, CA. 1940\n\nNEIGHBORHOODS\n\n **CURVILINEAR PLANS**\n\n**FHA-GUIDED SUBDIVISIONS (CA. 1940\u20131980)**\n\n**AFTER:**  \nFHA REVISION INTO A CURVILINEAR PLAN  \nloop road  \nminimal access to main streets  \ncul-de-sacs  \nlots face into subdivision\n\nFHA-GUIDED SUBDIVISIONS\u2014The FHA changed the plan of affordable American neighborhoods from typically rectilinear to generally curvilinear. Prior to 1940, curvilinear plans were more often used in upscale neighborhoods, but under the aegis of the FHA they were introduced into neighborhoods for all income levels. The agency accomplished this first by establishing a set of guidelines for planning new subdivisions, and secondly by their review and approval of plans for neighborhoods where they insured home mortgages. _Successful Subdivisions_ (1940), the FHA's first land-planning bulletin, was heavily illustrated with curvilinear plans to achieve many of the agency's planning objectives\u2014such as \"natural features preserved for improved appearance\" and \"streets should fit the contours of irregular land.\" This publication encouraged the use of long blocks as more economical than short blocks and advised against placing residential streets along tract boundaries. The bulletin featured three pairs of submitted plans along with the revisions advised. The pair illustrated here has a loop road and a cul-de-sac in the revised plan\u2014both favored by the agency as ways to discourage heavy through traffic. This guideline was very specific: \"Minor streets should be so arranged as to make fast through traffic impossible.\" Developers gradually became adept at providing few entrances into a subdivision and designing an almost completely internal street system that made it difficult for a nonresident to locate a specific house, much less find his way back out to a main street.\n\nNEIGHBORHOODS\n\n **CURVILINEAR PLANS**\n\n**GOLF COURSE DEVELOPMENTS**  \nnew way of providing shared green space   \ngolf course creates broad green view for Mid-century Modern's rooms with views   \ncul-de-sacs very common   \nfrequently private gated communities\n\nGOLF COURSE NEIGHBORHOODS\u2014Developments overlooking golf courses gained popularity in the post\u2013World War II period, in part because they provided a scenic overlook for the \"rooms with a view\" that were integral to Mid-century Modern houses. The course offered a large privately maintained green space that raised the value of the land surrounding it while providing a quiet recreation and exercise opportunity. In the past few decades, golf-course communities, often with gated entrance, have been widely used in post-suburban neighborhoods.\n\nNEIGHBORHOODS\n\n **COMBINED PLANS** rectilinear with curvilinear\n\n**GEOMETRIC COMBINATIONS**\n\noften in bulls'-eye form\n\nNEIGHBORHOODS\n\n **COMBINED PLANS** rectilinear with curvilinear\n\n**SIDE-BY-SIDE COMBINATIONS**\n\n\"signature\" curvilinear area with later, often more affordable, rectilinear areas\n\nCOMBINED PLANS\u2014Rectilinear and curvilinear plans have been combined in a number of ways. Symmetrical geometric shapes were used for subdivisions by a few developers, particularly during the 1910s and 1920s. These combine straight and curved streets within one plan and often have a distinctive bull's-eye appearance. More common were side-by-side combinations, where a developer planned a curvilinear area of more expensive homes that transitioned to a more economical rectilinear street and lot plan of more modest homes.\n\nNEIGHBORHOODS\n\n **BENEATH THE PLAN** interaction of ground plans with topography\n\nBENEATH THE PLAN\u2014A primary factor in the character of a neighborhood is the topography and original vegetation upon which it is built. Cities and towns occupy only a fraction of the vast and varied land area of the United States. Normally their founders chose relatively level sites near dependable water supplies. On sites that were not completely flat, neighborhood street plans interacted with surface relief to add three-dimensional variety to the town's neighborhoods. Much diversity results from the interaction of street patterns and topography. In general, simple-grid streets are appropriate on both flat areas and low foothills, while Olmstedian curved streets are most natural on low foothills, but can be used on flatlands. Contour curves were historically the only affordable solution to development on steep hills and in canyons.\n\nNeighborhoods that have preserved their original ecology, vegetation, and natural drainage are unusual and to be treasured. Retention of native plants not only conserves resources such as water and nutrients, it can also make a neighborhood more distinctive than one where original vegetation was scraped during development and the site then \"landscaped\" with commonly cultivated and widely available plants. Utilizing an area's natural drainage, rather than burying storm drains, costs less and facilitates native landscape.\n\n## Overlay Patterns\n\nLooking at a large map of a city or town, one rectilinear neighborhood appears almost identical to the next. When driving the streets, however, there is great diversity added by the patterns formed by green space, street profile, block size, and lot width. These same overlays add equal diversity to curvilinear plans.\n\nNEIGHBORHOODS\n\n **GREEN SPACE**\n\n**CURVILINEAR**\n\nNEIGHBORHOODS\n\n **GREEN SPACE**\n\n**RECTILINEAR**\n\nGREEN SPACE\u2014Green space provides a highly visible and attractive organizing element for neighborhoods of every type. In urban neighborhoods, small open spaces are essential to provide relief from the built environment. Hence a square, green, or plaza was incorporated into the original plan for many American towns. In the late 19th century elongated stretches of green space (either curvilinear alongside streams or long rectangular medians or spines) began to be used and were frequently part of well-planned early suburban expansions and park systems. New York's huge Central Park (1857\u20131873) spurred many cities to develop large reserves of more natural green space or acquire preserves of wilderness land on their periphery as an antidote to rapidly increasing urban population growth.\n\nGreen space can be in a curvilinear or rectangular shape, often influenced by the overall neighborhood plan, varying widely in size, location, and detail. Inspired by Olmsted's Emerald Necklace chain of parks in Boston (1878\u20131900), small compact green spaces can be thought of as \"beads\" (spaces small enough that surrounding buildings read as an enclosure surrounding the green space), elongated linear parks as \"necklaces\" (discernable strands of green running through a neighborhood or part of a city), and quite large green spaces as \"bodies\" (generally with a naturalistic interior design, where one feels completely surrounded by green rather than a small \"clearing\" in the built environment).\n\nNEIGHBORHOODS\n\n **DEVELOPMENT PATTERNS ALONG PRIMARY ROADS**\n\n**EARLY \"PIKES\" OR \"HIGHWAYS\" (PRE-1850)**  \ndesirable location for farmsteads in early rural areas, road necessary   \nto access trade centers or cities\n\nNEIGHBORHOODS\n\n **DEVELOPMENT PATTERNS ALONG PRIMARY ROADS**\n\n**HORSECAR AND STREETCAR SUBURBS (1850\u20131920)**  \nconvenient location on streetcar line preferred, larger homes often closest to line\n\nNEIGHBORHOODS\n\n **DEVELOPMENT PATTERNS ALONG PRIMARY ROADS**\n\n**EARLY AUTOMOBILE SUBURBS (1920\u20131940)**  \nenhanced road often used as focal planning element, largest home sites often face enhanced road\n\nNEIGHBORHOODS\n\n **DEVELOPMENT PATTERNS ALONG PRIMARY ROADS**\n\n**POST\u2013WORLD WAR II SUBURBS (1940\u20131980)**  \nprimary arterials exterior to neighborhood, homes rarely face primary road, smaller slip street used when necessary\n\nSTREETS AND CIRCULATION\u2014Access into and through a neighborhood is a critical function that has evolved with neighborhood types and related transportation. The overview is to determine the types of circulation systems present in a neighborhood (pedestrian, automobile, bicycle, streetcar, or mass transit), how each is designed to flow within the neighborhood, how the different systems relate and connect to one another, and how these systems within a neighborhood connect to exterior circulation systems and plans. Discrete elements of these systems\u2014such as street width, vehicular lanes, and curb and sidewalk patterns\u2014are easy to observe. They are each important contributors to neighborhood character. Understanding how these relate to larger area plans requires research.\n\nPRIMARY ROADS\u2014There have been dramatic changes in how streets providing main transportation routes have been viewed, used, planned, and built along. The role of primary roads evolved from a favored location for homes in early urban and rural neighborhoods to a shunned location for homes in postwar suburbs.\n\nIn early urban and rural neighborhoods there were few main roads. Generally unpaved, they only carried light traffic. In both urban and rural areas, a home located close to a road was desirable. This preference continued during the early streetcar era, when a location on the streetcar line meant convenience. Many cities had at least one major street lined with late 19th- or early 20th-century mansions whose occupants were easily able to walk out their door and step onto a nearby streetcar. In modest neighborhoods the finest homes were often right on the streetcar line.\n\nBy about 1915 there were enough automobiles on city streets that planners knew additional provisions had to be made for them. An early step in accommodating cars was to convert the wider streets that had contained streetcar lines into arterials for automobiles. Due to higher traffic volumes, the desirability for residential uses fell and their attraction for commercial uses increased. The net effect was that many of the finest rows of late 19th-century and very early 20th-century American homes were either converted to commercial uses or demolished. A common early automobile suburb answer was to simply embrace the automobile and introduce an arterial or collector running _through_ the neighborhood to gather the traffic from the local streets. Such a street might be curved, have a green median or other City Beautiful feature, and create something of an automobile promenade for Sunday afternoon drives. The overall neighborhood plan was generally still a grid, perhaps warped. As the number of automobiles increased in the 1920s, adding a showy street through the middle of a neighborhood was quickly abandoned\u2014and replaced by planning both regional road systems and individual subdivisions to keep traffic outside a neighborhood. This new approach to subdivision design was encouraged by the widely publicized 1929 model community of Radburn, New Jersey, called \"A Town for the Motor Age.\" A project of the Regional Planning Association of America, Radburn separated pedestrian and automobile traffic by utilizing changes in grade and introducing large \"superblocks\"\u2014a planning term for huge blocks in both directions\u2014and cul-de-sacs. Implementation of complete separation was not feasible on a widespread basis, but planners and home buyers eagerly embraced the concept of minimizing traffic within neighborhoods.\n\nAs an affordable solution, the FHA planners strongly encouraged a new way to relate neighborhoods to primary roads like arterials that carried heavy traffic. They encouraged less traffic by nestling new neighborhoods _next to_ arterials (whether major or minor), with few entrances into the neighborhood. Where the edge of the neighborhood abutted a major street, two solutions were suggested. Preferably, lots on the outer edge faced inward with an arterial-screening wall behind. If this was not possible, it was suggested that a small secondary access road (or slip street) be added for edge houses to face onto or for neighborhood streets to end in. Beginning around 1980, the more definitive solution of a gated neighborhood with single entrances began to be used.\n\nNEIGHBORHOODS\n\nCURBS AND SIDEWALKS\u2014Curbs, or lack thereof, contribute to the character of a neighborhood. Streets without curbs seem more rural and like a small town. Streets with curbs seem more urban and suburban. Although concrete curbs of varied profiles are the norm today, before about 1900 local stone was often used.\n\nSidewalks are similar in their effect. Their absence feels rural and their presence urban. Post\u2013World War II neighborhoods frequently do not have sidewalks, presuming that almost everyone would arrive via automobile. Although curbs and sidewalks are generally found together, some moderate-income streetcar suburbs had sidewalks installed but not curbs. Sidewalks were considered essential for the many people walking, while curbs were a luxury.\n\nThere are many fine early automobile suburbs, however, where the omission of both curbs and gutters was more a neighborhood-character decision than an economic one. Such areas typically have large lots and a rolling pastoral quality. Tiny bridges over grassy drainage channels create interest, and curbs would negatively impact the neighborhood's appearance.\n\nNEIGHBORHOODS\n\n **VARIETY WITHIN GRIDIRON PLANS**\n\n**BLOCK SHAPE  \nLOT SIZE AND BLOCK SHAPE**\n\nNEIGHBORHOODS\n\n **VARIETY WITHIN GRIDIRON PLANS**\n\n**EFFECT OF INCREASING BLOCK LENGTH**  \n(shown in forty-acre parcel with dimensions in feet)\n\n200-FOOT BLOCK LENGTH PRODUCES:  \nhigher connectivity in neighborhood   \nfewer lots to sell (5,000 front feet)  \nmore paving (200,000 square feet extra)   \nperceived as more walkable\n\n1,200-FOOT BLOCK LENGTH PRODUCES:  \nlower connectivity in neighborhood  \nmore lots to sell (6,200 front feet)   \nless paving perceived as less walkable\n\nBLOCKS AND LOTS\u2014A surprising amount of diversity is possible within a simple grid street pattern, despite the fact that in plans they may look identical. The size and shape of the individual blocks is the first place variety is introduced. Rectangular blocks are the most common, and the houses built on these often face out in only two directions. The length and depth of rectangular blocks can differ significantly, with the most distinct alternative being unusually long blocks. Square blocks may have deep lots with houses facing only two directions, or they may have houses facing all four directions, the least common pattern.\n\nBlocks of identical size and shape may be platted with different-size lots, creating a varied visual interest from block to block or neighborhood to neighborhood. Lots can be narrow, such as the 13- to 30-foot-wide lots found in urban environments and some working-class neighborhoods; or of medium width, such as the 40- to 60-foot-wide lots found in many late 19th- and early 20th-century streetcar suburbs; or very wide, such as the 65- to more than 150-foot-wide lots found in many upper-income and postwar neighborhoods. An advantage of elongated rectangular blocks is that there is far less area to be paved and, at the same time, more land area available to be divided into salable lots. These same kinds of block and lot differentials can be found in curvilinear plans, where there are even more ways of introducing variety into the plan through design.\n\nPeople generally prefer to walk shorter blocks. In addition, narrower streets invite a stroll and give a friendlier neighborhood feeling. The neighborhood-savvy planners designing new TNDs and TODs frequently recommend both of these elements.\n\n## Development Influences\n\nA broad range of economic and governmental factors have had a powerful effect on neighborhoods through the evolution of government regulation, financial control, and publicly funded infrastructure. These have influenced, and in some cases controlled, the type of neighborhood built, who builds them, where they are located, and if they can survive.\n\nDEVELOPER\/BUILDER\u2014A subdivision\u2014the basic landscape unit of residential development\u2014is created when a large parcel of undeveloped land such as a farm is cut into smaller lots suitable for building a home. When subdivision occurs, improvements such as streets, storm drainage, utilities, and sewers may be added. Each lot is sold either as vacant or with a house already built. A neighborhood generally contains one or more subdivisions.\n\nThe entity that subdivides and improves land today is usually the developer. Until about 1850, however, vacant land was simply subdivided into lots by \"subdividers\" with few, if any, improvements. A builder\u2014generally different from the subdivider\u2014constructed the house or building. Every lot sold could be employed for any use or structure\u2014the genesis of the old-fashioned, mixed-use city.\n\nBeginning in the mid-19th century a few early prototypes of today's developers purchased large tracts of land outside cities and subdivided them into lots intended exclusively for expensive residences and necessary support uses. Improvements such as landscaping, roads, and utility systems were frequently added. Private deed restrictions often governed what could be built. Houses were most often constructed by individual builders\u2014some for a specific family and others on speculation that a buyer would be found. The majority of them constructed only a few houses each year. Even in these controlled neighborhoods, there was no guarantee the neighborhood would develop as intended. Far more lots were platted for sale than there was a demand for. Nonetheless, the pattern of a developer subdividing land, adding restrictions, and selling vacant lots continued to grow until the beginning of the Depression. The development of these restricted neighborhoods spread throughout the United States and reached maturity in the 1920s, when they effectively governed the design and regulation of a great many _high_ -end residential neighborhoods.\n\nThe national trauma of the Great Depression initiated changes at the _moderate_ end. By 1933 housing construction had come to a virtual standstill. The National Housing Act of 1934, adopted to revive home building, also dramatically changed how American neighborhoods were developed. It created the FHA, with the broad charge to \"improve nationwide housing standards, provide employment and stimulate industry, improve conditions with respect to mortgage financing, and realize a greater degree of stability in residential construction.\" The FHA did not lend mortgage money. Instead, it insured the home loans banks made against losses. The FHA adopted strict standards for the loans it would insure and demanded both lower-priced homes and a broad set of development practices designed to ensure long-term neighborhood stability. For economic efficiency the agency encouraged the same developer to build houses on all lots of a subdivision so that an entire neighborhood would be constructed at once, lending both economy of scale and predictability. The earlier practice of separating land subdivision from home building became far less common and one development profession emerged, sometimes called \"community building.\" Now large-scale neighborhoods, rather than individual homes, were built. No longer did one purchase a vacant lot in an area with an uncertain future. Instead, one purchased a home within a completed neighborhood.\n\nHOME FINANCING\u2014The revolutionary shift affected in neighborhood development by the FHA was accompanied by a major change in how one purchased a home. Earlier, potential home buyers commonly had to save the entire cost of a residential site and purchase it before contracting with a builder to construct the house. Local banks made short-term mortgage loans to pay for construction costs. They required periodic interest payments, with the entire principal amount falling due at some future date, generally about five years later.\n\nBy the early 20th century, a few large home-building enterprises had begun to offer custom-designed homes that could be purchased with the lot along with a long-term, installment-payment mortgage that included both interest and principal reductions. Americans now take for granted this method of home financing\u2014an amortizing loan paid over a period of up to thirty years\u2014but it is barely a century old, a revolutionary innovation that spurred the development of countless 20th-century neighborhoods.\n\nAs a result of the financial crisis of the early 1930s, many Americans were unable to make payments on their homes, no matter which financing method was used. Banks repossessing the houses could not resell them at a reasonable price due to the depressed economy. Soon the banks themselves were suffering. An entire industry ground to a halt.\n\nThe FHA renewed private mortgage lending by insuring banks and other lenders against losses from the new long-term amortizing mortgage loans. In return, the FHA required the houses so insured to meet their subdivision design and construction standards.\n\nThe FHA's programs were a great success in stimulating new-house construction and homeownership. During World War II (1941\u20131945), only housing necessary to the war effort was permitted built. After the war, however, the pent\u2011up demand for housing\u2014along with the GI Bill of 1944, which financed the down payment of a new home for all veterans\u2014set the stage for an enormous housing boom when the war ended. For decades the agency's policies guided neighborhood development across the United States.\n\nIt was through its requirements for insuring loans for both builders and home purchase that the FHA exerted a strong influence on the development of American neighborhoods. The agency encouraged professional plans for subdivisions and then reviewed these plans, pushing the nation toward curvilinear plans and new street circulation patterns. In addition, their mortgage-insurance guidelines (in effect the maximum amount they would insure any given year) encouraged smaller and more standardized house designs, making it less expensive to buy a house than to rent it.\n\nUnfortunately, the FHA's focus on new construction left the country's older neighborhoods without mortgage sources, resulting in a devastating effect throughout the 1950s, 1960s, and 1970s. The FHA's policies went even further and actively redlined some areas as \"undesirable for lending.\" The FHA (through their arm Fannie Mae, the Federal National Mortgage Association) participated in its first inner-city lending project in 1976 in Munger Place, a streetcar suburb of Dallas, Texas, that had previously been redlined, meaning it had been deemed not secure for home lending. In 1977 Congress passed the Community Reinvestment Act, requiring lending institutions to reinvest in older neighborhoods and thus enabling the widespread reclamation of the nation's historic neighborhoods.\n\nGOVERNANCE\u2014There are three primary ways to control how a neighborhood looks: deed restrictions, zoning, and subdivision-platting regulations. The first two are most relevant to the house built on a lot, while the third governs the design of the lots themselves. None of these were generally used prior to about 1870, when deed restrictions began to be utilized.\n\nNEIGHBORHOODS\n\n **DEED RESTRICTIONS**\n\n**UNIFIED STREETSCAPES**  \nopen front lawns and uniform setback of houses result from original deed restrictions; note level changes from house on terrace to street level, broad sidewalk, tree-planting strip, and green median\n\nNEIGHBORHOODS\n\n **DEED RESTRICTIONS**\n\n**HOUSE SETBACKS FROM STREET**  \n(later mandated by zoning)\n\nBELOW: NO DEED RESTRICTIONS:  \nallows random front and side lot setbacks\n\nABOVE: WITH DEED RESTRICTIONS:  \nmandates uniform front and side lot setbacks\n\nDeed Restrictions\u2014In the 19th century municipalities rarely regulated construction, and only private protections were available. These private protective covenants (often called deed restrictions) could govern a broad variety of things. First used in early upscale suburbs, they typically might require houses to be built with uniform setbacks from the street, generous side yards, open front lawns, and street trees\u2014all intended to create a pleasing visual unity. An unfortunate additional provision often excluded minorities from ownership. Additional requirements might include a restriction to residential use, and a minimum building cost ensured that the most expensive homes were grouped on certain blocks. A broad range of housing prices might be incorporated into a single development on a block-by-block or street-by-street basis. For example, a development of eighteen square blocks, laid out along six parallel streets, might require houses on the most prime street to cost a minimum of $10,000 to build, with the minimum dropping to $8,000 on the next street, then $5,000, then $3,500, and to $2,500 on the last two streets.\n\nPlanned neighborhoods where developers put protective covenants in their deeds were widespread by the early 1900s. Early sets of deed restrictions expired after a certain number of years and were often difficult to renew. By the early 20th century J. C. Nichols had produced refined wording so deed restrictions would renew automatically.\n\nZoning\u2014Deed restrictions proved so successful that some began to wonder if municipalities could successfully legislate protection for all their neighborhoods, not just those with deed restrictions. The idea evolved quickly. Protective covenants had commonly relied on homeowners for enforcement, and implementing them could be difficult. Lower-income neighborhoods were often not protected by such covenants. If zoning was adopted by municipal law, the burden of enforcing compliance would fall to the city and, in theory, the same protections would be provided to all. Two initial steps were taken when Boston imposed height restrictions during 1904\u20131905, and Los Angeles followed by dividing itself into residential and industrial sections in 1909.\n\nThen, in 1914, the New York state legislature adopted a statute allowing towns to provide more comprehensive restrictions on development. In 1916 New York City became the first city in the country to adopt a zoning code. The idea of zoning legislation spread quickly. The U.S. Department of Commerce prepared a draft of the Standard State Zoning Enabling Act (SZEA) in 1922, and by the end of that year twenty states had adopted statutes allowing their cities to make and enforce zoning restrictions, and fifty cities had already taken advantage of this right. By 1926 twenty-three more state legislatures had passed zoning statutes, and cities and towns across the country were rushing to adopt zoning codes. That same year the U.S. Supreme Court upheld the legality of such efforts in a landmark decision.\n\nToday, zoning laws are almost ubiquitous in American cities, and few people realize how recent they are. While some rhapsodize about the wonderful years when everyone agreed what a city should look like, that was before the industrial revolution rapidly changed the scale of buildings as well as the kinds of transportation that made up a city. The requirements of railroads carved the downtowns of many cities into sections. New structural systems and elevators allowed large factories and high-rise buildings. The advent of automobiles brought road widening, freeways, and parking lots. These dramatic changes began to evolve rapidly in about 1850 and were hastened by the proliferation of automobiles during the 1920s.\n\nThe old agreement of how cities should look was swept aside during these seventy-five years of massive change. No wonder the ability to regulate the location of these new uses through zoning ordinances swept the country in the ten short years between 1920 and 1930. In the mid-20th century a new kind of zoning code began to be written that allowed older neighborhoods to become historic districts with zoning to protect the original historic buildings and their features. In addition, new zoning laws were written that allowed the creation of conservation districts where design elements such as scale or architectural style were protected\u2014rather than the historic homes themselves.\n\nBut the location of uses and the height of a building were only a small part of planning a quality built environment for a city. Because SZEA came first (it was finally published in 1924) and its governance of uses was so specific and desired at the time, zoning is sometimes thought of as planning. In actuality zoning is only a small part of planning. It was two years later, in 1926, that the Standard City Planning Enabling Act (SCPEA) was published by the Department of Commerce, making it easy for states to undertake more broad-based land-use planning. Although the Act covered farsighted topics like master plans and regional planning, at the neighborhood level the effects are most specifically seen in subdivision regulations.\n\nSubdivision Regulations\u2014Because there had been an oversupply of lots platted, often with no improvements, many towns began to limit the supply of lots to be built upon through new subdivision regulations. Historically simply a way of tracking land ownership and street continuation, the Standard City Planning Enabling Act made subdivision design an important part of directing the growth of cities. It ensured coordination with overall plans (such as highways and parks), required developers to build good roads and infrastructure at their expense, and controlled the amount of land available for development at any one time to avoid oversupply of vacant lots. Subdivision regulations generally control the size of lots, the width of streets, and the location of utilities. These controls govern a subtle but large part of the streetscape.\n\nThe enforcement of zoning laws and platting restrictions has proved almost as challenging for neighborhoods as enforcing deed restrictions. A more far-reaching effect is that both subdivision and zoning regulations have today evolved in ways that can legislate urban sprawl. New urbanist planners emphasize the need to rewrite these codes as an important step in preventing future sprawl. In order to avoid an unintended consequence of harming intact older residential neighborhoods, a recommended approach is to lasso new densification, keeping it compact and concentrating it on the nation's large supply of underutilized commercial and industrial land.\n\nGrowth Rate\u2014The economic origins of an area\u2014and the income level and rate of population growth these produced through time\u2014are a major factor in shaping neighborhoods and their streetscapes. The fluctuations of the resident population determined the size of homes that could be supported and how large new neighborhoods could be. Fast and steady growth typically produced large areas of similar houses. A new industry with many workers might produce a nearby neighborhood of small homes. In fast-growing cities, middle-class streetcar suburbs could be built by the mile, but they extended only a few blocks in slower-growth areas. An area that experienced a single boom has houses almost solely from that period, sometimes widely spaced, in the belief that intervening lots would soon sell. Because small towns typically grew slowly, they often have neighborhoods with a mixture of house sizes and styles reflecting different eras. However, this was not always the case, and a small mining town in Colorado, for example, might be built seemingly overnight and then continue to exist almost as a time capsule. Neighborhoods and their homes can often be tied directly to an area's economy and the varied factors that influenced its growth.\n\n## Reading Streetscapes\n\nThe streetscapes that have been created over time reveal much about a neighborhood's history. They give you an idea about the period of time during which houses were originally built, the care the original developer took, and the current condition of the homes\u2014thus recording how economic conditions changed.\n\nNEIGHBORHOODS\n\n **HOUSES ON THE BLOCK\n\nREADING STREETSCAPES**\n\nMEDLEY STREETSCAPES  \nfound in small towns and early suburb with log growth periods  \nsee here\n\nUNIFIED STREETSCAPES  \nfound in areas with deed restrictions or zoning and relatively rapid development  \nsee here\n\nCHANGING STREETSCAPES  \ncan be caused by a broad range of circumstances  \nsee here\n\nHOUSES ON THE BLOCK\u2014The interaction of growth rate, governance, and method of development with the age and styles of individual houses establishes a rhythm on the street. One can have a smooth rhythm of uniform lots with similar houses or a syncopated rhythm of varied lot sizes with dissimilar houses.\n\nThe smooth rhythm of a unified streetscape was produced by uniform lot sizes, deed restrictions mandating similarities (lot placement, two-story houses, etc.), and rapid growth. Where a neighborhood developed quickly, the houses are more likely to be of similar age and form (urban extension, streetcar, and post\u2013World War II suburbs). Where development was spread over a long period, as in many smaller towns that never had a major growth period, the streetscape is more of a medley (rural neighborhoods and some early railroad suburbs). Many planners and even the FHA recommend slight differences in spacing and placement to create variety.\n\nInteresting rhythmic changes can be seen in early automobile suburbs that began developing with two-story period houses before World War II and then did not fill up. After the war, one-story Ranch houses are likely to have been constructed on the remaining lots. Beginning in the 1990s many desirable older neighborhoods experienced rapid change in their previously unified streetscape by the introduction of houses far larger than those that make up the original neighborhood. Placing a few massive houses into a smaller-scale neighborhood can make the remaining houses seem like potential teardowns\u2014with \"smaller-scale\" applying to what would generally be considered spacious houses. A street's rhythm is greatly changed by the addition of just one or two intrusive out-of-scale houses, and these are even more overwhelming if two lots have been combined into one.\n\nAreas with zoning changes or expired deed restrictions may have newer apartment buildings that greatly change the neighborhood rhythm with their greater lot coverage and\/or height. Historic neighborhoods in less desirable areas, or those with shrinking populations, may have their street rhythm interrupted by vacant lots created by fire (or code-enforcement efforts without any restoration funds or incentives attached), producing devastating changes.\n\nSome early homes are widely spaced because they were built originally as large-lot mini farms and gradually unneeded land was sold off for new construction. Or they might have been built in a large subdivision that was designed and platted with the belief that a building boom would continue unabated.\n\nNEIGHBORHOODS\n\n **STREET TREES**\n\nNEIGHBORHOODS\n\n **STREET ENCLOSURES**\n\n**CONTINUOUS ENCLOSURE IN EARLY URBAN CORE**\n\n(narrow street, narrow sidewalks, no tree-planting strips)\n\n**CONTINUOUS ENCLOSURE IN URBAN EXTENSION**\n\n(wider street, wide sidewalks, tree-planting strips)\n\nNEIGHBORHOODS\n\n **STREET ENCLOSURES**\n\n**DISCONTINUOUS ENCLOSURE IN STREETCAR SUBURB**\n\n(narrow street, tree-planting strip, sidewalk, front fence, porch layer)\n\n**DISCONTINUOUS ENCLOSURE IN EARLY AUTOMOBILE SUBURB**\n\n(wider street, tree-planting strip, sidewalk, no fence, street lowered, deep house setback)\n\nSTREET TREES AND STREET ENCLOSURE\u2014Of the many amenities that add character to a neighborhood, street trees are perhaps the most important. Nothing makes a stronger impression when looking at a streetscape than the absence or presence of street trees. Jefferson understood this when he lined Pennsylvania Avenue with poplar trees in the early 19th century. It was not until the late 19th century that residential neighborhoods began to receive the same attention. Many early developers worked with landscape architects to lay out their subdivisions, fully understanding the importance of landscape on neighborhood character. Frequently advertised as important neighborhood amenities, street trees were planted at regular intervals along a planting strip between the sidewalk and the street. Typically a single species of tree was planted. While the downside of single-species planting is that the trees will be more vulnerable to disease (as in the case of Dutch elm disease), a single species can have dramatic effects depending on the appearance and growth habits. Broad, spreading trees planted along mid- to narrow-width streets will eventually form an attractive tree tunnel. Columnar trees create a more formal all\u00e9e. Palm trees give a tropical flair. Beverly Hills strikes a balance between monoculture and all\u00e9es by planting each individual street with its own species of tree. Intermixing multiple tree species along a single street produces a soft and varied look.\n\nA second street-enhancing device\u2014changing the grade between the house and street\u2014was used by Olmsted at Riverside and is sometimes found in suburbs developed between about 1870 and 1940. The house was built on a flat site one to four feet higher than street level, and the transition between the two grades was most commonly accomplished by a short section of lawn that sloped up at an angle from the sidewalk.\n\nStreet enclosure refers to the effect created by the structures that line a street. Historically this was the wall created by the front facades of attached buildings in an urban neighborhood. The height of the wall (i.e., the building height) and the width of the space it encloses visually (i.e., street, sidewalk, and setbacks) are important elements of a streetscape, forming a room-like space. Varying the ratios of width to height creates different effects in urban neighborhoods.\n\nIn suburban neighborhoods the street enclosure is discontinuous, but the setback of the houses can create a somewhat room-like feeling (or lack thereof, in some post\u2013World War II suburbs). The quality of enclosure can be modified by other continuous or semi-continuous elements, such as front fences, porches, or steps.\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nDowning's influence on rural and borderlands properties, ca. 1830\u20131860\n\n**RURAL SOUTHERN FARMSTEAD**  \n(CA. 1835) BEFORE DOWNING'S _TREATISE_  \nvery large lot with geometric front garden  \ntrees and shrubs in rows   \nrectilinear planting of ornamental trees  \nformal parterre of ornamental plants  \nfood grown on lot\u2014vegetables, orchards, etc.   \nfront garden seen from street\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nDowning's influence on rural and borderlands properties, ca. 1830\u20131860\n\n**RURAL OR \"BORDERLANDS VILLA\"**  \n(CA. 1850) FOLLOWING DOWNING'S _TREATISE_  \nvery large lot with curvilinear plan  \ntrees and shrubs used to screen from street  \nnaturalistic planting of ornamental trees  \nfood grown on lot\u2014vegetables, orchards, etc.\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1880\u20131920\n\n**MINI FARM  \n(CA. 1870)**\n\nLOCATED IN RURAL NEIGHBORHOOD, RAILROAD SUBURB, OR \"BORDERLANDS\"  \ngrounds occupy large lot, as much as one square block\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1880\u20131920\n\n**RAILROAD OR HORSECAR SUBURB  \n(CA. 1880)**\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1880\u20131920\n\n**STREETCAR SUBURB  \n(CA. 1910s)**\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1915\u20131970\n\n**EARLY AUTOMOBILE SUBURB  \n(CA. 1920s)**\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1915\u20131970\n\n**POST\u2013WORLD WAR II SUBURB  \n(CA. 1950s)**\n\nNEIGHBORHOODS\n\n **HOUSE ON THE LOT**  \nca. 1915\u20131970\n\n**POST\u2013WORLD WAR II SUBURB  \n(CA. 1960s)**  \n(a less common Mid-century Modern approach)\n\nHOUSE ON THE LOT\u2014Landscape patterns around homes add character to many neighborhoods. These patterns result from the public streetscape, from legislated private responsibilities and regulations, and from individual private practices. While some patterns have evolved over time, there are many variations, including those related to climate, native landscape, regional practices, legislation, ethnic origin, and economic requirements.\n\nPrior to the mid-19th century, American gardens, whether herb, flower, or vegetable, tended to be laid out in geometric patterns. There was more than enough wild countryside offering a completely naturalistic effect. It was not until 1841, with the publication of Andrew Jackson Downing's _A Treatise on the Theory and Practice of Landscape Gardening_, that more naturalistic landscape design was first advocated for larger house grounds.\n\nVictorian-era neighborhoods often had low or open fences at the sidewalk line to keep out wandering chickens, pigs, and other farm animals. Victorian houses rarely had original plantings placed along their foundations because these plantings were thought to inhibit crawl space ventilation, thus creating mold or harboring germs. Instead, beds of blooming plants were set in the middle of the yard. The ground plane was the most dominant part of the garden, and plants might be placed in an elaborate design. Individual elements of the landscape were visually separated from each other and from the house, which stood alone in splendid isolation. Outdoor living was on the front and side porches, and rear yards were typically used as service areas. In less populated areas much food was still grown at home and, in a small town, one who could afford it might purchase a quarter of a block or more for a small farm with chickens, a cow, a fruit orchard, and a vegetable garden.\n\nIn the early 20th century broad expanses of front lawn became favored; they were an expensive status symbol that demonstrated that an area was more civilized, having no need to fence out livestock, and presented a lavish park-like space for passersby. Foundation plantings around houses often consisted of a mixed-shrub border, designed to visually connect the house to its site and to hide the raised foundation. Porches that were generally in front of the house in streetcar suburbs might be located to one side in early automobile suburbs. Garages were typically detached and backyards still functioned as service areas. Some neighborhoods followed the advice of _Craftsman Magazine,_ which promoted hedges or low fences at the sidewalk, distinctive front garden gates, and bountifully planted front and side yards.\n\nPostwar suburbs continued the practice of front lawns and foundation plants, but garages were now generally attached to the house, often opening to the front, with porches moved to the rear. With the advent of electric appliances (particularly washers and dryers) and supermarkets, backyards could be converted from service yards to private outdoor living areas. Architect-designed postwar homes sometimes went further in their use of outdoor space. A house might have a relatively blank facade facing the street yet incorporate views of side and rear gardens. In some examples, small gardens were tucked behind front walls, providing views for rooms across the front of the house. In addition, the entry to the house might be deeply recessed, creating a side-entry courtyard and eliminating the need for much interior hall space.\n\n## Underlying Survey\n\nThis continent's initial occupants, the many tribes of Native Americans, had a very different concept of land and land ownership than the European colonists. The Indians did not believe that individuals owned the land, rather that humans shared the land with all other living things, both plant and animal. Although tribes fought over the right to hunt or gather in certain localities, these disputes were not over ownership.\n\nIn contrast to Native Americans, the early colonists brought with them the European tradition of individual ownership of land. The United States and its neighborhoods bear traces of the original land-survey practices, not only of our own government but also of England, Spain, France, and Holland. Many settlers received land grants from those sovereign entities.\n\nEach colonial power surveyed portions of land, establishing a system of boundaries to accurately trace the ownership of parcels as they changed hands. These initial surveys provide the basis upon which the lands of today's United States were first granted to individuals and offered accurate physical descriptions that allowed them to later be sold. These surveys are often reflected today in the street patterns visible on maps, still shaping towns and neighborhoods.\n\nNEIGHBORHOODS\n\n **UNDERLYING SURVEY**\n\n**METES-AND-BOUNDS SURVEY**\n\nMETES-AND-BOUNDS SURVEY\u2014The earliest surveys used landscape features to establish the boundaries of landownership. The use of natural landmarks\u2014such as mountain ridges, the ocean, rivers, or even rocks, large trees, or river bends\u2014was typical of some early land grants. Using major landmarks was appropriate for large grants of what were then low-value lands, but in more settled areas like New England the distances between landscape features were measured in more formalized metes-and-bounds (measurements-and-boundaries) surveys. This type of survey was used in the original thirteen colonies, and spread to Tennessee, Kentucky, and Texas.\n\nNEIGHBORHOODS\n\n **UNDERLYING SURVEY**\n\n**LONG-LOT SURVEY**\n\nLONG-LOT SURVEYS\u2014The Spanish, French, and Dutch employed a second and closely related method, the long-lot survey, for apportioning relatively small tracts along waterways such as the Mississippi, Rio Grande, and Hudson rivers. Governed by the colonists' need for reliable water in order to survive, long-lot surveys had a short and carefully measured dimension along the valuable waterway, while the longer and less valuable dimension at right angles to the waterway might be less carefully defined. This longer stretch commonly terminated at the crest of the hills that defined the valley of the waterway, creating a long narrow shape that included water from the river as well as woodlands on the hillside. Crops could be located waterside to allow for irrigation, and sheep could graze the areas in the hills. In prairie states, valuable woodlands were in the river floodplain. Long lots were also laid out along Spanish _acequias_ (irrigation canals).\n\nNEIGHBORHOODS\n\n **UNDERLYING SURVEY**\n\n**IRREGULAR RECTANGULAR SURVEY**\n\nIRREGULAR RECTANGULAR SURVEYS\u2014In many parts of the country, settlers arriving in advance of the land's becoming U.S. territory simply surveyed town sites and neighborhoods in a grid pattern, but their grids rarely conformed to the U.S. government's prescribed north-south\/east-west orientation. Texas, in particular, has many areas initially divided into what cultural geographer Terry Jordan describes as \"irregular rectangular surveys\"\u2014places where \"the individual land holdings are roughly rectangular or square, as in the rigid rectangular system, but they vary greatly in size and lack an orderly grid pattern.\"\n\nThese irregular surveys evolved in many different ways. Where adjacent to a river or port, the early grids usually aligned with the waterfront. Others align with early railroads or important wagon roads. Still others are relatively arbitrary choices by individual entrepreneurs.\n\nNEIGHBORHOODS\n\n **UNDERLYING SURVEY**\n\n**RIGID RECTANGULAR SURVEY**  \n(in one-mile-square pattern when under Land Ordinance of 1785)\n\nRIGID RECTANGULAR SURVEYS\u2014In 1785, the Continental Congress of the United States passed the farsighted Northwest Ordinance, also known as the Land Ordinance. Today the system it created is called the Public Land Survey System (PLSS). Drafted by Thomas Jefferson, it defined a systematic way of surveying new lands, inspired by the need to sell land in Ohio to raise money for the new government. The Land Ordinance prescribed a system of continuous square-mile sections, each containing 640 acres. These were surveyed on a strict north-south\/east-west grid system and organized into thirty-six-square-mile townships. Public roads were to be established at one-mile intervals along the section lines, and smaller private roads were encouraged at the half-mile and half-section markers. Beginning with Ohio, federal land managers efficiently surveyed, identified, and mapped most newly acquired territory before it was opened for settlement. As U.S. government surveys were completed, newly available lands were granted or sold in north-south square-mile sections or fractions thereof. Some of these newly surveyed lands were made available to early settlers under various homestead acts beginning in 1862. Others were assigned to entrepreneurs to encourage the building of railroads across the West. No matter who first owned the land, the mark of the federal survey system has been indelibly stamped across the western states. Today many parts of the United States have roads running due north-south and due east-west at one-mile intervals\u2014a visible legacy of these early U.S. surveys. The north-south\/east-west grid remains even where the original one-mile pattern has been lost.\n\nIn California, New Mexico, and Texas\u2014all under Spanish rule prior to being annexed by the United States\u2014another grid orientation was required by Spanish law. The Laws of the Indies prescribed a 45\u02da southwest-by-northeast angle for laying out streets. It was believed that this was the most desirable orientation for maximum light and ventilation. A plaza (small public square) was prescribed at the center of a Spanish grid town. This orientation is seen today in the downtown area of older cities in these states.\n\nNEIGHBORHOODS\n\n **RIGID RECTANGULAR SURVEYS**  \nA Visible Legacy\n\noriginal survey pattern of Land Ordinance of 1785 reflected in agricultural fields viewed from air\n\nNEIGHBORHOODS\n\n **RIGID RECTANGULAR SURVEYS**  \nA Visible Legacy\n\noriginal survey pattern of Land Ordinance of 1785 platted into \"Planned Rectilinear\" plan\n\nSalt Lake City, Utah\n\nNEIGHBORHOODS\n\n **RIGID RECTANGULAR SURVEYS**  \nA Visible Legacy\n\nNEIGHBORHOODS\n\n **RIGID RECTANGULAR SURVEYS**  \nA Visible Legacy\n\nNEIGHBORHOODS\n\n **METES-AND-BOUNDS SURVEYS**  \nA Visible Legacy\n\noriginal metes-and-bounds survey pattern reflected in agricultural fields viewed from air\n\nNEIGHBORHOODS\n\n **METES-AND-BOUNDS SURVEYS**  \nA Visible Legacy\n\noriginal metes-and-bounds surveys platted as \"gridiron extensions\"\n\nSyracuse, New York\n\nNEIGHBORHOODS\n\n **LONG-LOT SURVEYS**  \nA Visible Legacy\n\noriginal long-lot surveys along Mississippi; later platted as \"gridiron extensions\"\n\nNew Orleans, Louisiana\n\nTHE MODERN LEGACY OF EARLY SURVEYS\u2014The differing patterns of early land surveys are often obvious on modern city and town maps. This is particularly noticeable where lands assigned by different surveys collide. The original survey method can still contribute to today's neighborhood identity and character. Every city and town is unique, and the following examples are intended only to give a hint of the history revealed by analyzing the quilt pattern formed by a town's neighborhoods.\n\nDallas, Texas, has three easily distinguishable street angles. One is from a land grant based on the 45\u02da angle prescribed by the Laws of the Indies. In a grid related to the bank of the Trinity River, another angle was surveyed by the Dallas founder, who simply laid out his own grid and began selling lots. The third is a huge area of north-south\/east-west streets surveyed at this orientation after Texas became a U.S. state. In the post\u2013World War II northern suburbs of Dallas, freeway entrances are located at one-mile intervals feeding broad arterial streets built along the original section-line boundaries, as prescribed in 1785.\n\nSchenectady, New York, shows a grid pattern imposed on an early metes-and-bounds survey, with a resulting web of grid extensions as small land holdings were platted and joined to the existing grid. In contrast, Salt Lake City, Utah, has a very regular gridiron plan placed onto land surveyed by the Northwest Ordinance.\n\nNew Orleans, Louisiana, was originally divided by the French into long-lot holdings called _faubourgs_ that extended north from the Mississippi River. Different _faubourgs_ formed fan and pie shapes as they followed the meandering river. These were eventually platted with grids at differing angles, thus maintaining the legacy of the original land division.\n\n# Folk Houses\n\n* * *\n\n**Native American**\n\n**Pre-Railroad**\n\n**National**\n\n**Manufactured**\n\nMost American houses are built in one of the many architectural fashions, or styles, that have been popular during our country's long history. These changing fashions, which are the subject of the succeeding sections of this book, either incorporate earlier architectural images (for example, Classical porch columns or Medieval half-timbering) or consciously eschew the past to create modern styles with their own fashionable images. There is, however, another and less familiar type of dwelling that lacks this concern for architectural taste. These folk houses are built to provide basic shelter with little regard for changing fashion.\n\nEarly folk houses were constructed of materials found near the building site\u2014rock, clay, logs, and timbers\u2014and prepared by the builders themselves, rather than in distant mills, kilns, or factories. Usually the future occupants of the house, these builders were often aided by part-time local craftsmen. Unlike fashionable styles, folk building traditions, handed down from generation to generation, show relatively little change with time; they are, however, more strongly influenced by _geography_ than are architectural styles. The local availability of building materials, as well as the building traditions of the earliest settlers in an area, can lead to strong contrasts in the structure and form of folk houses from region to region.\n\nOur country's first folk dwellings were, of course, those built by the Native American inhabitants. Early European explorers found them using a remarkable variety of complex building techniques that had developed over many millennia. The European colonists, in turn, imported their own folk building techniques but adapted these to the same local materials used by the indigenous dwellers: wood in the heavily forested eastern half of the country and stone or clay in the more arid West. These European folk traditions persisted, with minor modifications, from the earliest colonies of the 17th century until the spread of the railroads in the mid-19th century. Rail transportation made inexpensive building materials\u2014principally lumber from large mills located in timber-rich areas\u2014readily available over much of the nation. This in turn led to a change in folk building traditions, as local materials, such as logs, heavy timbers, and crude masonry, were replaced by light and inexpensive sawn lumber in most folk dwellings. In spite of this change in building materials, many traditional Folk _shapes_ persisted into the 20th century, when they were joined, and eventually replaced, by still other forms of National Folk housing.\n\nA relatively high percentage of houses built before 1890 are vernacular folk houses. Daniel Reiff's survey of Fredonia, New York, showed that 40 percent of houses built before 1840 were vernacular houses \"without even the 'tiniest elusive gesture' to high-style movements.\"\n\nAfter World War II, few folk houses were built. Mobile homes and Manufactured houses fulfilled the need for basic housing beginning soon after World War II.\n\n# FOLK HOUSES\n\n# Native American\n\n# to ca. 1900\n\nThe most truly American folk houses are those that were built by the native inhabitants before Europeans discovered and occupied the North American continent. Native peoples have occupied North America for thousands of years, and during that long interval many diverse building traditions arose and evolved. The first Europeans thus found cultures with widely differing patterns of dwelling construction. These are known primarily from written descriptions, drawings, and early photographs, for most tribes have long abandoned their traditional dwellings in favor of European-influenced houses. Modern reconstructions with varying degrees of authenticity are, however, found in museum villages scattered throughout the country, and these provide a glimpse of the complex building traditions of our native forebears. In addition, a few traditional dwellings survive in isolated areas of the American West, where a handful of native cultures have persisted relatively intact to the present day.\n\nThe first Europeans found native buildings constructed with both wood-frame and masonry techniques. The wood-frame structures were particularly remarkable since they were fashioned not with the Europeans' iron axes, saws, and other implements but only with simple stone tools. These wood-frame houses were of two general sorts. Round-plan houses, such as the familiar tipi, were relatively small and partially or fully portable. These tended to be associated with nomadic or semi-nomadic hunting cultures. Rectangular-plan houses, on the other hand, tended to be larger, permanent dwellings built by sedentary, usually agricultural, societies. Masonry houses, usually with earth-wall construction, were common in the more arid regions of the American West. These ranged from simple dugouts to the familiar multi-storied pueblos of some southwestern tribes.\n\nNATIVE AMERICAN\n\n **DOMINANTNATIVE AMERICAN HOUSE TYPES**  \n(adapted from Driver and Massey, 1957)\n\n## Round-Plan, Wood-Frame Family\n\nWood-frame houses with rounded ground plans were generally constructed with a framework of relatively light poles or branches fastened together with leather cords. Various materials were then used to cover the framework and make it weatherproof. Among the most frequent were tanned animal skins, sheets of bark, which were sometimes sewn together to make larger mats, thatch (tied bundles of straw), and woven mats of vegetable fiber. The houses were built in two principal shapes: some were dome-shaped with rounded tops; others were conical with pointed tops. The dome shape dominated in the midwestern transition zone between woodlands and plains and tended to be larger and more permanent than dwellings of conical shape. Crude conical shelters of sticks and brush were commonly built on hunting expeditions, and these were the principal shelters of some nomadic tribes. Such shelters were generally abandoned and rebuilt elsewhere as needed. From them probably evolved the tipi, a conical shelter that could be transported from place to place by being dragged behind domestic dogs or, after the beginning of European colonization, by captured wild horses. Simple tipis were the principal dwellings of many hunting tribes of the northern woodlands and arid Great Basin regions. These consisted of a number of straight poles joined at the top to form a cone and covered with many separate hides, mats, or pieces of bark, which were generally held down by having additional poles laid upon them. From this simple tipi evolved a still more efficient portable dwelling, the Plains tipi. This was covered not by separate units of hide, bark, or fiber, but by a tailored covering made by sewing together many carefully tanned buffalo hides. As a further improvement, two flaps, or ears, extending outward from the top of the cover, could be adjusted by attached poles to make a sort of controlled chimney that eliminated smoke from the interior (most round-plan dwellings have only a simple smoke hole at the top). Before the introduction of the horse, such tipis were probably used only during short seasons when the buffalo herds could be easily followed for hunting on foot. Most plains tribes then returned to villages where they practiced agriculture and lived in permanent wood-frame or earth-wall dwellings. This pattern changed after the introduction of the horse, which escaped from 16th-century Spanish expeditions to become wild on the vast grasslands of the American West. The plains tribes domesticated some of these wild horses and evolved a new culture based on year-round nomadic hunting of the buffalo, which was easily killed from horseback. Horses could also drag heavy tipi poles and covers from place to place. As a result, both the size and refinements of the Plains tipi increased during the centuries between the introduction of the horse and the first extensive European contacts with the Plains tribes in the 19th century.\n\n**ROUND-PLAN, WOOD-FRAME FAMILY**\n\nROUND-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nWisconsin(?); photo ca. 1890. Winnebago Tribe. Bark-covered, dome-shaped dwelling.\n\nROUND-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nUintah Valley, Utah; photo ca. 1875. Ute Tribe. Crude conical shelter.\n\nROUND-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nOklahoma(?); photo 1898. Wichita Tribe. Thatch-covered, dome-shaped dwelling. Note the wooden framework exposed at the top.\n\nROUND-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nIdaho(?); photo ca. 1900. Umatilla Tribe. Tipi covered by woven mats and canvas cloth introduced by Europeans. Note the pole holding down the mats at right. Such tipis normally had many of these surface poles.\n\nROUND-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nKansas(?); photo ca. 1870. Kiowa Tribe. Plains tipi with fitted cover of tanned buffalo hides. Note the adjustable flaps at the top for controlling drafts to eliminate smoke from interior fires. The ends of the hide covering are joined by the horizontal wood pins above and below the entrance.\n\nNATIVE AMERICAN\n\n **ADAPTABILITY OF THE PLAINS TIPI**\n\n## Rectangular-Plan, Wood-Frame Family\n\nWood-frame houses of rectangular ground plan were generally large, permanent dwellings housing several related families. These were the first native houses encountered by European colonists along the Atlantic coast, for they were the principal dwelling type of the many woodlands tribes of the eastern United States. Regrettably, they are also among the most poorly known because the natives of that region were the first to be displaced and their villages and traditional cultures destroyed by European colonization. Two principal types of houses appear to have been present in the eastern woodlands. To the north occurred large arched-roof dwellings (longhouses) made of light wooden frames. These were usually covered with bark; woven mats, thatch, and hides were less common coverings. In construction, these houses resembled larger and more refined versions of the round-plan, domed dwellings found farther west. In the southern woodlands these arched-roof forms were replaced by gabled (or, less commonly, hipped) roofs covered with thatch. Unlike the northern longhouses, where the same material was generally used for both roof and walls, these southern rectangular houses had varying wall materials beneath the thatched roofs. Woven mats and hides were sometimes used, as was a sort of half-timbering in which a light wooden framework was laced with basketry and covered with clay. In the warmer regions thatched summer houses without enclosing walls were also common. All of these rectangular houses of the eastern woodlands were associated with agricultural traditions based on the cultivation of corn. Some anthropologists believe that the house form spread northward from Middle America where corn was first domesticated and where somewhat similar houses are found.\n\nAn entirely different tradition of rectangular, wood-frame building was found along the humid Pacific Coast from northernmost California to southern Alaska. There the natives built large dwellings with heavy timber frames having carefully fitted joints. The frames were covered with large softwood planks split with stone tools from the abundant local timber. These houses most commonly had gabled roofs, although shed roofs were used in the Puget Sound area. The more pretentious houses were sometimes adorned with decorative and ceremonial carvings, particularly the familiar totem pole. This building tradition is believed to be a relatively recent introduction from northeastern Asia, where similar native dwellings occur.\n\n**RECTANGULAR-PLAN, WOOD-FRAME FAMILY**\n\nRECTANGULAR-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nWashington, District of Columbia (reconstruction); photo 1899. Abnaki-Passamaquoddy Tribe; northern New England. Bark-covered, arched-roof longhouse.\n\nRECTANGULAR-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nFort Sill, Oklahoma, vicinity; photo ca. 1875. Caddo Tribe; displaced from Louisiana and eastern Texas. Thatch- and bark-roofed dwellings. Note the half-timbered walls on the enclosed buildings.\n\nRECTANGULAR-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nFort Lauderdale, Florida; photo ca. 1917. Seminole Tribe. Thatch-roofed, open-sided dwellings.\n\nRECTANGULAR-PLAN, WOOD-FRAME FAMILY: NATIVE AMERICAN  \nAlbert Bay, British Columbia, Canada; photo ca. 1889. Kwakiutl Tribe. Gabled, plank-walled houses. The siding behind the totem pole and the small building in the foreground were introduced by Europeans.\n\nNATIVE AMERICAN\n\n **NORTHWESTERN PLANKHOUSE FRAMING**\n\n## Earth-Wall Family\n\nThe third basic type of Native American dwelling used earth, rather than organic materials (hides, bark, straw), for covering the walls and roof. These earth-wall buildings ranged from crude dugouts to the magnificent multi-unit pueblos of the American Southwest. All use some system of wooden support for their earth-covered roofs\u2014most commonly, heavy timbers spanned by smaller timbers and then covered with sticks, straw, or sod. The earth-covered walls were also sometimes supported by such a wooden framework. Alternatively, the walls were excavated below ground level or, as in the massive pueblos, built of sun-dried mud applied in successive layers (this is called puddled adobe; sun-dried adobe bricks were introduced by the Spaniards, although there is some archeological evidence for their use much earlier, along with complex stone masonry, by native peoples of the American Southwest). The most typical earth-wall dwelling was the round, partially excavated earth lodge, which generally housed several related families. These dwellings were usually supported by a carefully constructed wooden framework. An opening at the top permitted light to enter and smoke to escape. Sometimes the top opening was the only entrance, with descent inside by means of a log ladder; other earth lodges were entered by ground-level openings or tunnels. In the 18th and early 19th centuries these earth lodges were the dominant dwellings over much of the east-central plains and Columbia Plateau regions. They are also common among the native peoples of northeastern Asia and are thought to be among the earliest New World dwellings. There is some suggestion they were far more widespread before the introduction of the horse made possible year-round nomadic buffalo hunting over much of the West.\n\nThe most familiar of all Native American dwellings are the monumental pueblos of the Southwest, which are believed to have evolved from simple earth lodges (many incorporate round ceremonial chambers similar to lodges in shape and arrangement). The pueblos are multi-storied, communal structures made up of many rectangular rooms; exterior rooms are used for living quarters and interior rooms for food storage. The earthen roofs are supported by massive horizontal timbers placed on top of the thick adobe walls. Because of the difficulty of cutting the roof timbers to precise lengths with stone tools, the ends were normally allowed to project somewhat beyond the wall surface. This is the principal difference between these native buildings and related Spanish Colonial buildings built of adobe, which have otherwise similar roof-support systems. Several of the New Mexico pueblos have been continuously occupied since pre-Columbian times and thus have the distinction of being the most authentic surviving Native American dwellings.\n\n**EARTH-WALL FAMILY**\n\nEARTH-WALL FAMILY: NATIVE AMERICAN  \nLoup Fork, Nebraska; photo 1871. Pawnee Tribe. Village made up of several earth lodges. Note the timber frame visible in the entrance tunnel and the bundles of tipi poles used on hunting expeditions.\n\nEARTH-WALL FAMILY: NATIVE AMERICAN  \nZuni Pueblo, New Mexico; photo 1879. Zuni Tribe. Close\u2011up view of upper stories. Note roof-support timbers and access ladders.\n\nEARTH-WALL FAMILY: NATIVE AMERICAN  \nArizona?; photo ca. 1895. Navaho Tribe. Earth-walled dugout supported by conical frame of timbers (such Navaho dwellings are called hogans).\n\nEARTH-WALL FAMILY: NATIVE AMERICAN  \nAcoma Pueblo, New Mexico; photo 1899. Acoma Tribe. General view showing earth roofs in foreground.\n\nNATIVE AMERICAN\n\n **TYPICAL EARTH-LODGE CONSTRUCTION**\n\n# FOLK HOUSES\n\n# Pre-Railroad\n\n# before ca. 1850\u20131890; locally to ca. 1920\n\nThe first period of American folk architecture built by European colonists spanned the long interval between the earliest permanent settlements of the 17th century and the growth of the railroads as an efficient national transportation network in the last half of the 19th century. Throughout these two hundred years many modest dwellings were, of necessity, constructed of local materials without stylistic embellishment. Before the railroads, the only means of efficiently transporting bulky goods of relatively low value, such as lumber, brick, and quarried stone, was by water. Coastal towns and villages thus had access to a variety of domestic or imported construction materials, as did those inland farms and villages located near canals or the few dependably navigable rivers. Even modest houses in these areas tended to follow current architectural fashion and thus were generally styled, rather than folk, houses. Elsewhere the costs and difficulties of horse-and-wagon transport\u2014the only alternative to boats and barges\u2014restricted all but the most affluent to folk dwellings built with materials found on, or very near, the construction site. The eastern half of the country was covered with a seemingly endless supply of virgin forests; there, wooden folk building became the rule. The early English and French colonists were familiar with wooden building principally in the form of massive frameworks of hewn timber (post-and-girt construction) which, in the New World, were generally covered by thinner strips of wood to make a watertight exterior. These traditions dominated early folk building both in New England, where frame, massed-plan (more than one room deep) houses became the norm, and in the early settlements of the Tidewater South, where frame houses of linear plan (one room deep) dominated, probably because of the shorter and less confining southern winters. As settlement expanded to the West, a more distinctive tradition of wooden folk building evolved from a blending of the linear plans of the Tidewater South with techniques of construction using horizontal log walls brought to the middle colonies by immigrants from the heavily timbered areas of central and northern Europe. This Midland tradition of log building is the most familiar and well-studied aspect of American folk architecture. Still farther west, the vast woodlands gave way to grassy plains where timber was scarce. Large-scale settlement did not reach this area until well into the 19th century, but in the relatively brief interval before the arrival of the railroads a new folk tradition developed in this region: as a result of the shortage of wood, folk houses were constructed of primitive masonry. In a few areas the skills and materials were available to construct rough stone dwellings. More commonly, houses were made of sod\u2014earth held together by fibrous grass roots. Still other masonry traditions developed in the southwestern United States, which was part of Spain or Mexico throughout most of the pre-railroad era. In this Hispanic Southwest, Spanish folk traditions of building with sun-dried adobe bricks or with rough stonework were dominant.\n\nPRE-RAILROAD\n\n **PRE-RAILROAD FOLK TRADITIONS to ca. 1850\u20131890 (locally to ca. 1920)**\n\n## New England Tradition\n\nThe first New England colonists of the 17th century built primarily linear-plan houses having heavy timber frames covered with boards or shingles. These were commonly of the two-story, I-house form, although single-story, hall-and-parlor houses were also built. Both were among the commonest folk forms in 17th-century England. In the early 18th century, these plans were expanded to the rear to give increased interior space; this resulted in the one-and-one-half-room-deep saltbox and Cape Cod forms, which were better adapted to the severe and confining New England winters. From these evolved, by the mid-18th century, massed-plan houses that were to dominate New England building throughout the following century. These houses at first had two-room widths and central chimneys, as had their saltbox and I-house precursors. By the time of the Revolution, center-hall plans with paired end chimneys were common. One more change came in the early 19th century when the Greek Revival style made accentuated front gables fashionable. This trend ultimately led to simple gable-front folk houses, which became increasingly common after about 1825.\n\nThis full sequence of change in building traditions took place only in those parts of New England that were settled first\u2014coastal Massachusetts, Connecticut, and Rhode Island. There, most surviving examples reflect the early affluence of the region in the predominance of Postmedieval, Georgian, or Federal stylistic detailing. As New Englanders spread northward and westward from this core area in the late 18th and 19th centuries, they tended to build less pretentious folk houses in the same forms as the more fashionable houses then being built farther east. Saltboxes and I-houses thus occur only in the coastal region; box houses occur farther afield; and gable-front houses are the most widespread of all. An exception was the 19th-century revival of one-story forms having both the linear-plan, hall-and-parlor shape as well as the deeper Cape Cod plan. These smaller houses were well suited for initial settlement in remote areas; they were replaced or adjoined by two-story forms as prosperity increased.\n\nSettlers from New England first moved westward along the Mohawk corridor of central New York. From there they dominated the Western Reserve area around Lake Erie and spread beyond into the upper Midwest, where their building traditions became diluted by others from farther south. New England folk houses are scattered today throughout this large area, but intact survivors are quite rare. Because they were relatively large and substantially built, many of these houses had original stylistic details that took them beyond the folk threshold. Even when built as pure folk forms, they have usually been modified beyond recognition by later stylistic alterations. The New England tradition is, however, reflected in large gable-front-and-wing folk houses that became common throughout the Northeast after the expansion of the railroads.\n\n**NEW ENGLAND TRADITION**\n\nNEW ENGLAND TRADITION: PRE-RAILROAD  \nNewbury, Massachusetts; 1696. Jackman House (restoration). Rare northern hall-and-parlor without stylistic detailing.\n\nNEW ENGLAND TRADITION: PRE-RAILROAD  \nMelrose, Massachusetts; 1703. Upham House (restoration). Early saltbox without stylistic detailing.\n\nNEW ENGLAND TRADITION: PRE-RAILROAD  \nTruro, Massachusetts; late 18th century. Isaac Small House. A well-preserved original Cape Cod house. This house form has continued to be built for two centuries. See One-Story and Side-Gabled Roof (Cape Cod) for similar forms with additional details.\n\nNEW ENGLAND TRADITION: PRE-RAILROAD  \nMeriden, Connecticut; late 18th century. Redfield House. Preclassical box house (note central chimney) with modest Georgian doorway.\n\nNEW ENGLAND TRADITION: PRE-RAILROAD  \nNew Harmony, Indiana; 1815 (photo, 1903). Schnee House. Gable front without stylistic detailing. This example was built by German immigrants; similar forms were spread throughout the Midwest by settlers from New England.\n\nPRE-RAILROAD\n\n **DEVELOPMENT & WESTWARD EXPANSION OF THE PRINCIPAL NEW ENGLAND FOLK HOUSES**   \nto ca. 1850 (adapted from Kniffen, 1965; Pillsbury and Kardos, 1970)\n\n## Tidewater South Tradition\n\nLike their countrymen to the north, the earliest English colonists of the coastal South built primarily linear-plan, hall-and-parlor houses or I-houses. In contrast to the northern colonies, however, a tradition of building with brick masonry was established early in the South. The exact reason for this is unclear, although an abundance of brick clay in the region and differing English backgrounds of many of the southern colonists are probably responsible. Because of the expense of masonry construction, most of these brick houses were built with Postmedieval, Georgian, or Federal stylistic detailing and thus were not folk houses. Massive timber-frame construction, like that in the northern colonies, was also used in the South, and these early wood-frame houses were more commonly modest folk dwellings; unfortunately very few 17th- or 18th-century examples survive intact. Instead, these early southern folk houses are known primarily from their modified descendants built in the first half of the 19th century, many of which survive.\n\nBecause of the milder winters of the southern colonies, there was less emphasis on enlarging the early linear plans to create more interior space. One-story houses are far more common than in the North and true massed plans (more than one room deep) are rare. Instead, one-story shed extensions were typically added to the rear of both one- and two-story, linear-plan houses as more space was needed. By the late 18th century, another innovation was becoming universal in the southern folk house. This was the full-width, shed-roofed front porch, which provided a cool shelter in summer from the scorching sun and frequent sudden thunderstorms. Tidewater hall-and-parlor and I-house forms were the prototypes for similar pre-railroad shapes executed with log walls; these became far more widely distributed as a part of the Midland folk tradition. These Tidewater forms also persisted into the railroad era and were the dominant folk architecture throughout the rural South until well into the 20th century.\n\n**TIDEWATER SOUTH TRADITION**\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nRocky Mount, North Carolina, vicinity; late 18th century(?). Wilkins House. Rare surviving example of early single-room frame house.\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nGuilford, Virginia; ca. 1820. Clayton House. A late example of the traditional British hall-and-parlor plan, without a porch. The hipped roof of a later rear wing is barely visible; the dormer windows are probably also a later addition.\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nPerquimans County, North Carolina; ca. 1825. Winslow House. Typical extended hall-and-parlor plan. The rear flue and decorative shutters are later additions.\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nNewlin, North Carolina; ca. 1830. Allen House. Hall-and-parlor plan, with added shed porch. The metal roofing and door are also later modifications; the original chimney has been removed.\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nWrendale vicinity, North Carolina; 1789. Early I-house with added shed porch and rearward extension and later metal roof.\n\nTIDEWATER SOUTH TRADITION: PRE-RAILROAD  \nIngold, North Carolina, vicinity; 1840. Johnson House. Typical extended I-house; note the additional shed extension at the rear.\n\nPRE-RAILROAD\n\n **PRINCIPAL TIDEWATER SOUTH FOLK HOUSES to ca. 1850**\n\nHeavy lines show approximate periods of dominance; (adapted from: Pillsbury and Kardos, 1970; Glassie, 1975; Swain, ed., 1978)\n\n## Midland Tradition\n\nThis third principal folk building tradition, like those of New England and the Tidewater South, originated with early colonization along the Atlantic seaboard. It began in the middle colonies (Pennsylvania, New Jersey, Delaware, and Maryland), where Germanic immigrants from heavily wooded areas of central and northern Europe introduced techniques of building with logs hewn square and then placed horizontally, one on top of the other, to make a solid wooden wall. This massive structure was held together by various systems of carefully interlocking or notching the squared timbers where they joined at the corners of the buildings. Such construction contrasted sharply with the frame buildings of the adjacent English colonies to the north and south, where open frameworks of hewn timbers were covered by lighter planks or shingles to make them weatherproof. This framing technique used far less wood than did solid log walls and was originally inspired by the relative scarcity of timber in westernmost Europe, where the virgin forests had largely been cleared by the late Middle Ages.\n\nThe early Germanic settlers in the core area of Pennsylvania and adjacent colonies built large log houses with an almost square, three-room plan and a central chimney (the Continental log house). This pattern persisted as settlement spread westward from the core area to central Pennsylvania and then southward along the forelands and valleys in front of the Appalachian Mountain barrier which loomed to the west. In this secondary core area (see Pre-Railroad Folk Traditions map) the Germanic settlers were joined by Scotch-Irish and English pioneers who quickly adopted the log-building techniques, which were much simpler than constructing complex hewn frameworks to be covered with laboriously split planks or shingles. These settlers from the British Isles, however, modified the shape of the three-room Continental house into the familiar one-room-deep linear plan with external chimney that dominated in the Tidewater South. Thus was born the Midland log house, a tradition that was carried across the Appalachians by frontiersmen to become the dominant Pre-Railroad Folk housing over much of the heavily wooded eastern half of the country. Because of their strong, massive walls, many early log houses survive relatively intact, particularly in out-of-the-way rural areas. For this reason, Midland log houses are the most familiar and thoroughly studied Pre-Railroad Folk dwellings.\n\nA principal problem of log-wall houses is the difficulty of expanding them as additional space is required. Because the strength of the structure depends on the four corner joints, log houses are generally made up of room-sized square or rectangular units called \"pens.\" The simplest log houses have only a single unit, usually with a loft area above used for sleeping. Two-unit plans, joined in various ways, are very common; three-unit and two-story forms were also developed. Framed additions and porches were commonly added to log houses as local sawmills provided nearby sources of cut lumber. Similarly, many log houses were later covered with weatherboards, both to provide an additional seal and to make them appear more up-to-date. The tradition of building with horizontal log walls persisted in many areas long after cut lumber was locally available. Usually a framework of roughly squared and notched logs was constructed to be originally covered with either shingles or weatherboard. These second-generation log houses can sometimes be distinguished from those originally built with exposed log walls by imprecise squaring of the logs, which resulted in relatively large, irregular gaps between timbers.\n\nA distinction is usually made between log _houses,_ such as those discussed so far, which have walls of square-hewn logs joined by carefully hewn corner notching, and log _cabins,_ in which the timbers are left round and are joined at the corners by overlapping saddle notches. Such walls are difficult to chink\u2014that is, to fill the spaces between the rounded logs with clay or other material to make them weatherproof. For this reason they were generally used only for temporary shelters in the woodlands of the eastern United States. In wooded areas of the western mountains, however, folk traditions of building with rounded logs became established in the 19th century and persist to the present day in isolated areas.\n\n**MIDLAND TRADITION**\n\nMIDLAND TRADITION: PRE-RAILROAD  \nSummers, Missouri, vicinity; photo 1880. Typical single-pen (one room). Note the absence of windows and the stick chimney, which is lined on the inside with clay. Ladders were usually kept handy so that chimney fires could be easily reached. A later frame extension is visible to the rear.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nSpringfield, West Virginia; photo ca. 1947. Urban single-pen (one room) with upper half-story and window openings fitted with early, small-paned sashes. The original roof was probably shingled.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nDuchesne County, Utah. Primitive log cabin (rounded logs and saddle notching) typical of those built in wooded areas of the western mountains.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nHale County, Alabama. Dogtrot (note central passage) with later roof and rough-sawn siding added over log walls.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nMercer County, Kentucky; photo ca. 1895. Typical saddlebag. Note the difference in corner notching on the two units. The right-hand pen was probably added later, around the chimney of the pen to the left.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nVersailles, Kentucky, vicinity; 1783. Crittenden House. Early double-pen with primitive corner notching. The original roof would have been shingled.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nBoundary County, Idaho. Typical log house of the western mountains.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nJessamine County, Kentucky; 1796. Peyton House. Dogtrot with upper half-story. Note the recessed doorway added to the central passage.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nWarrensville, North Carolina, vicinity. Gentry House. Double-pen with upper half-story and typical Tidewater porch.\n\nMIDLAND TRADITION: PRE-RAILROAD  \nLouisiana, Missouri; ca. 1830. Stark House. Restored example of unusual double-pen with units of differing size.\n\nPRE-RAILROAD\n\n **ORIGIN & DEVELOPMENT OF THE MIDLAND LOG HOUSE**\n\nPRE-RAILROAD\n\n **ENLARGING THE LOG HOUSE**\n\n## Plains Tradition\n\nFolk building traditions based on wood-frame or log construction dominated the pre-railroad era in the heavily timbered eastern half of the country. As settlement spread into the treeless plains of the West in the mid-19th century, new building techniques had to be developed (see map). Only the most arid western regions lacked wood altogether. Over much of the plains, rivers and streams were bordered by at least small trees that provided short timbers for roof support and other essential construction details. Walls and roofs made entirely of wood were, however, rare and expensive luxuries on the plains before the expansion of the western railroad network in the late 19th century. Like their Native American predecessors, early settlers on the plains generally solved the shortage of wood by building with crude masonry. Many of these settlers were undoubtedly familiar with brick construction but, although suitable brick clays were widely distributed in the West, the fuel required to fire the bricks was not. Brick buildings were thus confined to areas near rail or water transport. In some regions local stone could be gathered or quarried without elaborate equipment; in these areas crude stone dwellings were common. Much of the best agricultural land of the plains was covered with thick soils that prevented access to the underlying rock for use as building stone. In these regions, which included most of Kansas, Nebraska, and the Dakotas as well as eastern Colorado, Wyoming, and Montana, pioneer settlers developed techniques of building with sod. In sod construction the uppermost few inches of soil, along with the interlocking roots of the tough plains grasses, were cut into brick-like units with a special plow. These were then laid like bricks to make thick earthen walls that provided excellent insulation from both summer heat and winter cold. The exact origins of this building tradition are obscure. Some writers suggest that the earliest settlers borrowed it from the somewhat similar earth lodges of local native tribes. Others point out that similar construction is used for simple folk dwellings in some treeless parts of Britain. Whatever their sources, folk houses made of sod quickly became the standard plains dwelling and were built at all levels of refinement, from simple dugouts to elaborate two-story mansions.\n\nA principal difficulty with early sod dwellings was the roof. Lacking wood shingles as a roof covering, roofs were typically also made of sod blocks set on a framework of wooden poles, sticks, and brush. Such roofs were notoriously unpleasant in wet weather, as they normally leaked and dripped muddy water into the interior for several days after a rain. A high priority for plains dwellers thus became watertight roofing. Roofs were supported by wooden planks, when available, on which sod was placed for insulation. As affluence increased, fully framed roofs covered with shingles were sometimes added. Most sod houses of the plains have long disappeared, but a few with these improved roofs are still in use as comfortable, energy-efficient dwellings.\n\n**PLAINS TRADITION**\n\nPLAINS TRADITION: PRE-RAILROAD  \nCuster County, Montana. Typical plains half-dugout. Partial walls of rounded logs with primitive notching were commonly used when nearby stream valleys provided sufficient timber. Note the earth roof.\n\nPLAINS TRADITION: PRE-RAILROAD  \nJackson County, Oklahoma; 1888. Perryman House. Partly restored plains half-dugout with partial walls of primitive stonework laid without mortar. Such stonework was common in the eastern and southern plains where exposures of bedrock are frequent.\n\nPLAINS TRADITION: PRE-RAILROAD  \nCuster County, Nebraska; photo 1887. Barnes House. Plains half-dugout with partial walls of sod blocks. Note the sod roof.\n\nPLAINS TRADITION: PRE-RAILROAD  \nPennington County, South Dakota. Sod house with improved roof and framed rear addition. The house was still in use in 1936 when the photo was made.\n\nPLAINS TRADITION: PRE-RAILROAD  \nCuster County, Nebraska; photo 1886. Reeder House. Typical plains sod house with sod roof. Timbers support the sod blocks above the windows.\n\nPLAINS TRADITION: PRE-RAILROAD  \nCuster County, Nebraska; 1884. Haumont House. Elaborate two-story sod house which survived into the 1970s.\n\n## Hispanic Southwest Tradition\n\nThe houses of the Hispanic Southwest are treated in the chapter on the Spanish Colonial style, which also outlines Hispanic building techniques of the region.\n\n# FOLK HOUSES\n\n# National\n\n# after ca. 1850\u2013ca. 1930\n\nThe nature of American folk housing changed dramatically as railroads mushroomed across the continent in the decades from 1850 to 1890. Modest dwellings built far from water transport were no longer restricted to local materials. Instead, bulky items used for construction, particularly lumber from distant sawmills in heavily forested areas, could now be moved rapidly and cheaply over long distances. As a result, large lumberyards quickly became standard fixtures in the thousands of new towns which sprouted as trade centers along the railroad routes. Soon folk houses built with logs, sod, or heavy hewn frames were being abandoned for wooden dwellings constructed with light balloon or braced framing covered by wood sheathing. The railroads thus changed the traditional building materials and construction techniques of folk dwellings over much of the nation. By the turn of the century, pre-railroad building traditions survived only in isolated areas, far from the nearest rail service.\n\nThe railroad-inspired era of national folk housing did not completely erase the earlier traditions, however, for many of the previous folk shapes persisted even though now built by different techniques. These, along with some new shape innovations, make up six distinctive families of house shapes that dominated American folk building through the first half of the 20th century. Only recently have these generally been abandoned for still other forms of folk dwellings.\n\nAfter the expansion of the railroads, gable-front houses remained common in the northeastern region formerly dominated by the New England folk tradition, as did similar massed plans with an added extension known as gable-front-and-wing houses. In much of the remaining eastern half of the country, hall-and-parlor and I-house shapes, both descended from the Tidewater South tradition by way of the Midland log adaptations, remained the dominant folk dwellings. All of these later folk forms, however, tend to show much less geographic restriction than did their pre-railroad predecessors, for as transportation and communication improved, each shape became distributed beyond its area of traditional dominance. Light framing techniques also led to new folk forms which grew in popularity through the early decades of this century. These were generally massed-plan houses that were now relatively simple to construct because light wooden roof framing could easily be adapted to span two-room depths. Such houses, when of rectangular shape, normally had side-gabled roofs and are called massed-plan, side-gabled folk houses. More nearly square plans typically had pyramidal (equilateral hipped) roofs.\n\nNATIONAL\n\n **EXPANSION OF PRINCIPAL RAILROAD NETWORKS 1850\u20131890**\n\n## Gable-Front Family\n\nThe Greek Revival movement, which dominated American styled houses during the period from 1830 to 1850, commonly used the front-gabled shape to echo the pedimented facade of typical Greek temples. This form was particularly common in New England and the adjacent northeast region where simple gable-front folk houses also became popular during the pre-railroad era. This shape persisted with the expansion of the eastern railroad network in the 1850s and became a dominant folk form until well into the 20th century. Gable-front houses were particularly suited for narrow urban lots in the rapidly expanding cities of the Northeast. There, many late 19th- and early 20th-century neighborhoods are dominated by both styled and simple folk examples built in this form. Most are narrow, two-story houses with relatively steep roof pitches. A related one-story urban form first became common in expanding southern cities in the late 19th century. This is the shotgun house, narrow gable-front dwellings one room wide that dominated many modest southern neighborhoods built from about 1880 to 1930. Some are elaborately styled but most are simple folk houses. The origin of these southern shotgun houses has been much debated. Some scholars note that similar forms are common in the West Indies and trace them from Africa to early Haitian influences in New Orleans, whence they became popular with Black freedmen migrating to southern urban centers following the Civil War. A less complex theory is that they are simply the familiar one-room-deep, hall-and-parlor plan of the rural South turned sideways to accommodate narrow urban lots.\n\nAn additional wave of interest in the gable-front shape grew from styled houses of the early 20th-century Craftsman movement, which were typically built in this form. Many modest folk houses without stylistic detailing were inspired by such Craftsman houses in the decades from 1910 to 1930. These are usually one-story, double-width forms with low-pitched roofs; they are most common in rural areas and occur throughout the country.\n\n**GABLE-FRONT FAMILY**\n\nGABLE-FRONT: NATIONAL  \nCuba, New York; late 19th century. Typical urban two-story example. The spindlework porch detailing and patterned shingles in the gable are borrowed from the contemporary Queen Anne style.\n\nGABLE-FRONT: NATIONAL  \nBuffalo, New York; ca. 1907. An urban one-and-one-half-story example with modest Queen Anne detailing. The door and windows are later additions.\n\nGABLE-FRONT: NATIONAL  \nCleveland, Ohio; late 19th century. Urban example executed in masonry.\n\nGABLE-FRONT: NATIONAL  \nCarteret County, North Carolina; 1864. Thomas House. Early example showing Greek Revival influence in the pedimented gable and double porch, which was common in the coastal Carolinas. Metal doors and storm windows are later additions.\n\nGABLE-FRONT: NATIONAL  \nBiloxi, Mississippi; ca. 1905. Typical shotgun house of the urban South. This example has integral porch and modest Queen Anne detailing.\n\nGABLE-FRONT: NATIONAL  \nLouisville, Kentucky; ca. 1910. Shotgun with Greek Revival\u2013like entry porch.\n\nGABLE-FRONT: NATIONAL  \nWabash County, Indiana; ca. 1935. Late example inspired by the Cape Cod shape of the Colonial Revival movement.\n\nGABLE-FRONT: NATIONAL  \nThomaston, Louisiana; ca. 1938. Typical example inspired by similarly shaped Craftsman houses.\n\n## Gable-Front-and-Wing Family\n\nWhile two-story gable-front houses dominated urban folk building in the Northeast, a related shape, also descended from styled Greek Revival houses, became common in rural areas. In this form, an additional side-gabled wing was added at right angles to the gable-front plan to give a compound, gable-front-and-wing shape. A shed-roofed porch was typically placed within the L made by the two wings. Because these were relatively large and complex houses, most built in the pre-railroad era had Greek Revival detailing and were not folk houses. With the coming of the railroads, however, abundant lumber and balloon framing led to an expansion of unstyled folk houses with this form. Some grew in stages as two-story, front-gabled wings were added to simple hall-and-parlor and I-house plans. These were typically stepped in shape\u2014that is, the roof ridge of the gable-front portion was higher than the adjacent wing. More commonly, the entire structure was built as a unit with a roof ridge of uniform height.\n\nTwo-story houses of gable-front-and-wing plan became common only in the northeastern and midwestern states. In the South, however, traditional one-story, hall-and-parlor plans were frequently built with an added one-story, gable-front wing. These one-story, gable-front-and-wing houses had more flexible interior spaces than the typical southern hall-and-parlor plan, which they steadily replaced during the early decades of this century. These one-story forms also became common, along with larger two-story examples, in adjacent areas of the expanding Midwest and are the most widely distributed of the gable-front-and-wing family of shapes.\n\n**GABLE-FRONT-AND-WING FAMILY**\n\nGABLE-FRONT-AND-WING: NATIONAL  \nHartwick, New York; late 19th century, photo 1914. Gardner House. Stepped example. Note the small attic windows, common on Greek Revival houses, on the right-hand wing, which was probably built first, with the gable-front portion added later.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nNorth Collins, New York; late 19th century. Stepped example.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nSeaford, Delaware; ca. 1947. Temple House. Late example inspired by the Minimal Traditional style of the 1940s.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nLawtons, New York, vicinity; late 19th century.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nBelmont, New York; ca. 1900.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nFayetteville, Arkansas; ca. 1910.\n\nGABLE-FRONT-AND-WING: NATIONAL  \nDallas, Texas; 1916. This variation has a hipped rear wing that extends to each side, rather than a gable extending in one direction.\n\n## Hall-and-Parlor Family\n\nSimple side-gabled, hall-and-parlor houses (two rooms wide and one room deep) are a traditional British folk form which, when expanded by a front porch and rearward addition, became the dominant Pre-Railroad Folk housing over much of the southeastern United States. Hall-and-parlor houses were first executed with heavy timber framing in the Tidewater South and then with hewn log walls over the vast Midland region. After the expansion of the railroad network, this form, now executed with light framed walls, remained the dominant folk housing over much of the rural Southeast until well into the 20th century. This folk form is thus a persistent survivor which has shown relatively little change since colonial times. The principal variations in extended hall-and-parlor houses involve differing chimney placements, porch sizes, porch roof shapes, and differing patterns of rearward extensions for enlarging the interior space.\n\n**HALL-AND-PARLOR FAMILY**\n\nHALL-AND-PARLOR: NATIONAL  \nGadsden County, Florida; late 19th century. Note the open shuttered window without a glass sash and the discontinuous siding on the rearward extension, added after the main house was built. Early hall-and-parlor houses had separate front doors leading to the two principal rooms, a pattern that survives in this example and those in McAlester, Oklahoma; ca. 1890, Smithfield, North Carolina; ca. 1910, and New Roads, Louisiana, vicinity; late 19th century.\n\nHALL-AND-PARLOR: NATIONAL  \nSmithfield, North Carolina; ca. 1910.\n\nHALL-AND-PARLOR: NATIONAL  \nCarteret County, North Carolina; ca. 1898. The gabled entry porch is probably a later addition.\n\nHALL-AND-PARLOR: NATIONAL  \nMcAlester, Oklahoma; ca. 1890. Note the vertical, board-and-batten siding, which is less expensive than horizontal weatherboarding and is commonly seen on modest folk houses.\n\nHALL-AND-PARLOR: NATIONAL  \nSalisbury, North Carolina; ca. 1900. Note the central chimney and double rearward extension.\n\nHALL-AND-PARLOR: NATIONAL  \nSmithfield, North Carolina; ca. 1910. Note the central chimney and ornamental front gable. A full rear wing replaced the traditional shed-roofed rearward extension on many later examples.\n\nHALL-AND-PARLOR: NATIONAL  \nNew Roads, Louisiana, vicinity; late 19th century. Early example expanded by adding a room to the right of the original house. The metal roof, now covering both, is a later addition.\n\nHALL-AND-PARLOR: NATIONAL  \nCrocketville, South Carolina; ca. 1890. Front-porch rooms were often added to increase interior space.\n\nHALL-AND-PARLOR: NATIONAL  \nLexington, Kentucky; ca. 1870. Dolan House. A one-and-one-half-story example on its way to becoming an I-house.\n\n## I-House Family\n\nLike the one-story, hall-and-parlor plan, two-story I-houses (two rooms wide and one room deep) are traditional British folk forms that were common in pre-railroad America, particularly in the Tidewater South. Similar forms occurred in the Midland area of log construction but were uncommon, probably because of the difficulty of constructing two-story walls made of solid, hewn logs. With the arrival of the railroads, however, I-houses again became a popular folk form over much of the eastern half of the country. They were particularly favored as modest folk dwellings in the midwestern states where the relatively long and confining winters made large houses more of a necessity than farther south. Post-railroad southern examples are also common, but these were usually the more pretentious houses of affluent local gentry. For this reason, many of these later southern I-houses have added stylistic detailing to make them appear fashionable. Like their hall-and-parlor relatives, post-railroad I-houses were elaborated with varying patterns of porches, chimneys, and rearward extensions.\n\n**I-HOUSE FAMILY**\n\nI-HOUSE: NATIONAL  \nBeaumont, Texas; 1845. John Jay French House. The basic I-house form is extended with original one-story shed-roofed extension on the rear and one-story full-width porch across the front (there are simplified Greek Revival doors).\n\nI-HOUSE: NATIONAL  \nMason County, West Virginia; ca. 1860. Porchless central chimney examples, such as this, are most frequent in the midwestern states.\n\nI-HOUSE: NATIONAL  \nClintonville, Kentucky; mid-19th century. Sidener House. An early post-railroad example. The windows and porch are later additions. Note the inside end chimneys and absence of side windows.\n\nI-HOUSE: NATIONAL  \nHelton, North Carolina, vicinity; ca. 1890. Blevins House. This example was expanded from a small log house, the walls of which are barely visible beneath the porch roof.\n\nI-HOUSE: NATIONAL  \nSalisbury, North Carolina; ca. 1898. Cannon Mill House.\n\nI-HOUSE: NATIONAL  \nCabarrus County, North Carolina; ca. 1900.\n\nI-HOUSE: NATIONAL  \nPerquimans County, North Carolina; mid-19th century. Skinner House. An early example with Greek Revival detailing and large rear wing.\n\n## Massed-Plan, Side-Gabled Family\n\nMassed-plan (more than one room deep) folk houses were common in the pre-railroad era only in parts of the Northeast where the early New England building tradition developed roof-framing techniques for spanning large, two-room depths. With the expansion of the railroad this tradition evolved into the massed-plan versions of the gable-front and gable-front-and-wing families previously discussed. Lightweight lumber made widely available by the railroads permitted still simpler methods of light roof framing, and these, in turn, led to other types of modest folk dwellings with two-room depths. These massed-plan houses, normally constructed with either side-gabled or pyramidal hipped roofs (see next section), had relatively large and flexible interior plans and thus slowly replaced the traditional one-room-deep hall-and-parlor and I-house forms.\n\nSide-gabled folk houses with massed plans are usually one-story forms that vary principally in roof pitch and in the size and placement of porches. Earlier examples, particularly in the South, commonly had full-width, shed-roofed porches. From the front, these resemble their extended hall-and-parlor predecessors, but lack the latter's rearward extensions and resultant broken rear roof line. Examples from the 1930s and later commonly have only small entry porches, or no porch at all, probably in imitation of the then popular Cape Cod shape of the Colonial Revival style. These were recommended by the FHA in its _Principles for Planning Small Houses_ and widely built in the years 1935 to 1950. They are included in the Minimal Traditional section of this revision.\n\n**MASSED-PLAN, SIDE-GABLED FAMILY**\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nYanceyville, North Carolina, vicinity; ca. 1930.\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nAbbeville, Louisiana, vicinity; late 19th century. The larger house to the right illustrates an early tradition of massed-plan, side-gabled folk building brought to Louisiana by French Canadian (Acadian) immigrants with a knowledge of long-span roof-framing techniques. Such Louisiana houses are known as Creole Cottages; they normally have the front wall moved back to make an integral porch under the steep roof line. Note how it dwarfs the traditional linear-plan hall-and-parlor to the left.\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nNew Madrid County, Missouri; 1940. The roof pitch of this example is lower than is typical of closely related Minimal Traditional houses. The hint of exposed roof rafters is a holdover from building practice during the preceding Craftsman era.\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nRolla, Missouri; ca. 1920. Example inspired by the contemporary Craftsman movement.\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nKey West, Florida; 1829. \"The Oldest House in South Florida\"; Richard W. Cussans, builder. It was moved to its current location in about 1836. The dormer on the far left was enlarged in 1936. Note raised foundation.\n\nMASSED-PLAN, SIDE-GABLED: NATIONAL  \nIrwinville, Georgia, vicinity; ca. 1920. Board-and-batten example similar to traditional hall-and-parlor plan, but with full, two-room depth. Note the lack of a broken rear roof line to cover a rearward extension.\n\n## Pyramidal Family\n\nMassed-plan folk houses of rectangular shape are normally covered by side-gabled roofs. Those with more nearly square plans, in contrast, are commonly built with pyramidal (equilateral hipped) roofs, which require more complex roof framing but need fewer long-spanning rafters, and thus are less expensive to build. Such roofs appeared on modest folk houses earlier in the post-railroad era than did the side-gabled form. In the South, one-story, pyramidal houses became a popular replacement for the less spacious southern-type hall-and-parlor house during the early decades of the 20th century. One-story pyramidals are less common in the northern and midwestern states but are joined there by two-story examples, which similarly began to replace the traditional but less spacious rural I-houses of the region in the years from about 1905 to 1930. During the same period these two-story, pyramidal houses also became a popular urban house form throughout the country. Most urban examples were called  American four-square houses  and were enhanced with Colonial Revival, Neoclassical, Folk Victorian, Prairie, or Craftsman stylistic detailing; but many also remained simple folk forms which lacked such fashionable details.\n\nLike their side-gabled relatives, pyramidal folk houses differ principally in roof pitch and in the size and placement of porches.\n\n**PYRAMIDAL FAMILY**\n\nPYRAMIDAL: NATIONAL  \nGreene County, Georgia; ca. 1900. Note the very steeply pitched roof. Such roofs are common on early southern examples; they may have been influenced by earlier roofs of similar shape built by French descendants in the Gulf Coast region.\n\nPYRAMIDAL: NATIONAL  \nStillwater, Oklahoma; ca. 1935.\n\nPYRAMIDAL: NATIONAL  \nCoffee County, Alabama; ca. 1905.\n\nPYRAMIDAL: NATIONAL  \nGwinnett County, Georgia; ca. 1920. Many one-story pyramidals have full or partial integral porches included under the principal roof.\n\nPYRAMIDAL: NATIONAL  \nEmporia, Kansas; ca. 1915. Such two-story pyramidals were a dominant urban form in the early decades of the century. Most urban examples had stylistic detailing but some, like this one, were unadorned folk houses. The metal storm windows are a later addition.\n\nPYRAMIDAL: NATIONAL  \nCabarrus County, North Carolina; ca. 1900. An unusually large two-story example.\n\n# FOLK HOUSES\n\n# Manufactured\n\n# ca. 1930\u2013present\n\nThe years since 1930 have witnessed an extraordinary change in the nature of American folk housing as the production of relatively inexpensive prefabricated, factory-built dwellings has largely replaced new construction of traditionally built wood-frame folk houses. Manufactured housing is now the dominant folk house of contemporary America. These are loosely grouped into mobile homes (their original use, and name, prior to 1980); manufactured housing: single-wide (one unit wide and the most common shape); and manufactured housing: double-wide (two units wide and gaining popularity). Today around 10 percent of all new homes in the United States are manufactured\u2014in 2009, one manufactured home was produced for every nine standard housing starts. The map shows areas where more than 20 percent of existing dwellings were manufactured housing in 1990.\n\nManufactured houses are dispersed throughout rural areas and grouped in parks that are as varied as neighborhoods of conventional houses. And just as neighborhoods often cluster houses by size, some entire parks are limited to single-wides or double-wides exclusively. These parks are generally located near cities and towns nationwide; there are also concentrations in resort and retirement communities and near worker-intensive rural industries.\n\nBecause manufactured housing is factory built, with all appliances and sometimes even the furnishings included, it is now the cheapest and simplest means of acquiring basic shelter, as its wide popularity attests.\n\nManufactured homes differ from closely related \"modular homes\" in two ways. A manufactured house is exempt from local building codes, while modular houses must meet local codes. In addition, a manufactured house must maintain the steel chassis that makes it technically movable. Newer models of manufactured houses often appear identical to modular houses, may be produced in the same factories and, except for code requirements, are pretty much interchangeable.\n\n## Mobile Homes\n\nThe earliest examples, designed for use as campers and travel trailers, appeared in the 1930s. These early mobile homes were eight feet wide and could easily be pulled behind ordinary automobiles. During World War II these were utilized as semi-permanent housing for large influxes of workers relocating near wartime industries. Although these provided small rudimentary dwellings, the concept of a complete home that could be moved onto a site was appealing, and after the war designs more suitable for permanent housing were introduced.\n\nIn 1954 the first ten-foot-wide model was produced; in 1969, this grew to twelve feet wide, and today fourteen to sixteen feet wide is more typical. Widths over eight feet allow a narrow side hall for privacy (rather than walking through a middle room to reach the rear). Lengths grew as well, and the new size allowed bathrooms, absent in most early campers, to be included. As these larger designs, more suitable for longer periods of habitation, became popular, the units lost most of their mobility and are now moved by special trucks with warning vehicles ahead and behind. In the early years, the mobile home industry was largely unregulated. Many small manufacturers built homes of varying quality and safety.\n\n## Manufactured Housing\n\n**MOBILE AND SINGLE-WIDE**\n\nSINGLE-WIDES\u2014In 1974, Congress demanded that safety standards be adopted for the popular mobile homes. In 1976, the U.S. Department of Housing and Urban Development (HUD) completed a set of uniform national standards for their construction that has since been strengthened. HUD's 1980 Housing Act mandated that the term _mobile home_ henceforth be replaced by _manufactured housing._ By the 1990 census, manufactured homes made up one-fifth of the housing stock in rural areas.\n\nSingle-wides are by far the most prevalent manufactured home. They typically come in widths up to sixteen feet and their length can now reach seventy feet or more. Single-wides are transported to a site essentially complete and are placed on foundations constructed onsite. They can be customized by adding decks, porches, carports, garages, and even room extensions. These are infrequently moved once sited, and about 90 percent remain permanently in place.\n\nLike many earlier folk forms, the prevalent single-wide manufactured house is of linear plan\u2014that is, made up of a single file of rooms. This shape allows them to be placed either long side to the road, as in the traditional hall-and-parlor plan\u2014or, where land is more expensive, on narrow lots with the short end to the road, as in the traditional shotgun plan. About 75 percent of manufactured housing is placed on private property and more likely to be sited in the hall-and-parlor configuration, lining rural roads and often interspersed among conventional houses. The remaining 25 percent of manufactured housing is placed on rented sites, often sited in a shotgun configuration. Manufacturers typically offer many different designs for units that will be placed broad side to the road, generally adding shallow gables or changes in cladding often designed to mimic a small Ranch house.\n\nMANUFACTURED\n\n **LOCATIONS WITH HIGH PERCENTAGE OF MANUFACTURED HOMES**\n\nAreas of United States where manufactured housing and mobile homes constituted more than 20 percent of all housing units in 1990\n\n**MOBILE AND SINGLE-WIDE**\n\nSINGLE-WIDE: MANUFACTURED  \nLos Angeles (hillside), California; ca. 1960s. A mobile home on a steep rural hillside is sited on a sturdy deck and has a picturesque roof added.\n\nSINGLE-WIDE: MANUFACTURED  \nAthens, Texas; ca. 2012. A very basic single-wide configured to face the road.\n\nSINGLE-WIDE: MANUFACTURED  \nAthens, Texas; ca. 2010. Nested gables on both the side and front facade, and a change of color enhancing the door, give this single-wide a more stylish look.\n\nSINGLE-WIDE: MANUFACTURED  \nKey West, Florida; 1970s. This single-wide is in an urban trailer park. The covered trailer mount for the required chassis is visible in front.\n\nSINGLE-WIDE: MANUFACTURED  \nBoulder City, Nevada; ca. 1969. Located in a neighborhood filled with manufactured houses, this has a carport added on one side and porch on the other.\n\nSINGLE-WIDE: MANUFACTURED  \nAnywhere, U.S.A.; ca. 2010. The narrow end faces the road and is enhanced by a paired window with a small roof, windowbox, and shutters.\n\nSINGLE-WIDE: MANUFACTURED  \nAnywhere, U.S.A. A pre-1984 single-wide.\n\nSINGLE-WIDE: MANUFACTURED  \nNew Orleans, Lousiana; ca. 2010. The Barrone. A single-wide styled like a Folk Victorian and configured with the entrance in the narrow end like the shotgun folk houses of the south.\n\n**FOLK HOUSES\u2014MANUFACTURED HOUSING: DOUBLE-WIDE**\n\nDOUBLE-WIDES\u2014In the 1970s, double-wides (two linear units, each without one side wall and designed to be joined together, see illustration) began to be produced. Like single-wides, these are placed on built-in-place foundations, but they have to join their two open-side walls together to create a single broad structure. These are unlikely to be moved once sited. Double-wides are growing in popularity; they have increased from about 20 percent of units sold in 2000 to about 50 percent in 2010. They may have exterior style elements that can make them harder to differentiate from other new homes.\n\nToday, as manufactured housing is trending toward these larger double-wide models, there is a nascent movement to create small basic housing from widely available used shipping containers.\n\nMANUFACTURED\n\n **TYPICAL SITING OF MANUFACTURED HOMES**\n\nURBAN PARK SITING  \nnarrow and faces the road  \n(similar to a row of urban shot-gun houses; this early configuration for single-wides has a slight slant so units can be moved in and out of park with ease)\n\nMANUFACTURED\n\n **TYPICAL SITING OF MANUFACTURED HOMES**\n\nURBAN PARK SITING  \nnarrow end faces the road; newer park is limited to double-wides, not intended to be moved and sited without the slant.\n\nMANUFACTURED\n\n **TYPICAL SITING OF MANUFACTURED HOMES**\n\nRURAL SITING  \nlong side faces the road  \n(similar to a rural hall-and-parlor house)\n\n**FOLK HOUSES\u2014MANUFACTURED HOUSING: DOUBLE-WIDE**\n\nDOUBLE-WIDE: MANUFACTURED  \nLas Vegas, Nevada; 1981. A double-wide without additions.\n\nDOUBLE-WIDE: MANUFACTURED  \nCarlsbad, California; 1985. An added side porch and carport help give this double-wide a Contemporary feel.\n\nDOUBLE-WIDE: MANUFACTURED  \nAthens, Texas; ca. 2012. The accented entry on the long side is favored when a double-wide is placed sideways to the road in rural areas.\n\nDOUBLE-WIDE: MANUFACTURED  \nLas Vegas, Nevada; 1970. A double-wide with a Craftsman-style front.\n\nDOUBLE-WIDE: MANUFACTURED  \nAnywhere, U.S.A.; ca. 2010. The Picasso. Note how the roof overhang accents the nested front facing gables above the entry. Compared to similar Athens, Texas; ca. 2012 without the overhang.\n\nDOUBLE-WIDE: MANUFACTURED  \nKey West, Florida (vicinity); ca. 1980s. A manufactured house in an idyllic setting.\n\nDOUBLE-WIDE: MANUFACTURED  \nDallas, Texas; ca. 2010. A triple-wide with its three units placed running from front to back. Note the pleasant living porch.\n\nDOUBLE-WIDE: MANUFACTURED  \nDallas, Texas; ca 2010. A triple-wide with its three units placed running from side-to-side.\n\n# Colonial Houses\n\n1600\u20131820\n\n* * *\n\n**Postmedieval English**\n\n**Dutch Colonial**\n\n**French Colonial**\n\n**Spanish Colonial**\n\n**Georgian**\n\n**Federal**\n\n**Early Classical Revival**\n\nThe early colonists arriving in the New World from Europe brought with them the prevailing architectural styles and building practices of their native countries. At first these were of late Medieval inspiration, for the new classicism of the Renaissance had not yet spread beneath the grandest palaces and mansions of their homelands. Indeed, most Colonial dwellings built during the 1600s lacked all Medieval decorative detailing and might be classified as folk houses if they did not so strongly reflect\u2014in form and structure rather than stylistic detail\u2014the distinctive building traditions of their countries of origin.\n\nHigh-style Dutch, French, and Spanish dwellings remained rare in our area of the New World, for the Netherlands soon lost their colonies to England, and the centers of French and Spanish colonization were concentrated elsewhere. By contrast, the prospering English colonies of the eastern seaboard began in the early 1700s to import Renaissance-inspired Georgian fashion, which was to dominate these colonies for almost a century before being replaced by the closely related English Adam style (called Federal in the United States) just as the American Revolution brought an end to British rule.\n\nIt should be noted that Old World building practices persisted in each of these colonial empires well beyond the end of European rule. The Dutch continued to settle and build traditional dwellings in the Hudson River area for more than a hundred years after the formal loss of their colony to England. Likewise, French and Spanish influence lasted for many decades after their former territories became a part of the United States. Finally, the Federal style and a general taste for English fashion persisted in the English colonies for many decades after the Revolution.\n\nBooks on architecture and architectural detail, of the type that were to have a major influence on later periods of home building, were rarely a factor in Colonial houses. Only the two English styles, Georgian and Federal, evidence the use of books in their design. Before 1740 less than a handful of architecture books were even available in the English colonies, but their numbers increased rapidly; by 1776, approximately one hundred titles had been published or imported and were in wide use by carpenters and builders. Another one hundred or more books were published during the Federal era and played the major role in bringing the work of the Adam brothers, the latest British architectural fashion, to the New World.\n\nExtant Colonial houses are relatively rare and generally owe their continued existence to active preservation efforts. These homes have an irreplaceable character that visibly demonstrates the colonial and immigrant history of the United States. Today most of those that remain have some sort of legal protection without which their survival would be in jeopardy.\n\nWhile original examples of Colonial houses are relatively rare, the forms and detailing are abundantly familiar because, beginning with the Centennial celebrations of 1876, they have been repeatedly copied during various stylistic revivals. When faithfully followed, the copies may be difficult to distinguish from originals. Two principal clues are of help here.\n\nFirst, each original style was built in a rather limited area of the country, while revival copies are widespread. For this reason, maps of the areas of original occurrence are provided for each Colonial style; outside these areas, _any_ example is almost certain to be a later copy. Secondly, Colonial houses were built before the era of industrialization, and unaltered examples thus have, on close inspection, a characteristic handmade quality in such details as doors, windows, brickwork, or siding that is always lacking on revival examples.\n\n# COLONIAL HOUSES\n\n# Postmedieval English\n\n# 1600\u20131700; locally to ca. 1740\n\n## Identifying Features\n\nSteeply pitched, side-gabled roof with little or no rake or eave overhang and no cornice detailing; massive central or end chimneys of brick or stone, often formed into decorative shapes; small windows, originally with narrow surrounds and fixed or casement sashes having many diamond-shaped panes (these were universally replaced by larger double-hung sashes during the 18th and 19th centuries; when the earlier-type windows are present today, they are modern restorations); most were originally one room deep (linear plan) with batten (vertical board) doors.\n\n## Principal Subtypes\n\nTwo distinct traditions became established in the 17th-century English colonies:\n\n**NORTHERN TRADITION** \u2014In the northern colonies wood-frame walls covered with weatherboard or wood shingles were the dominant mode of construction. These houses most commonly had two stories and a single large central chimney.\n\n**NORTHERN TRADITION**\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nMedfield, Massachusetts; 1680. Peak House (restoration). One of the few surviving northern one-story examples.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nWatertown, Massachusetts; 1694\u20131701. Browne House (restoration).\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nTopsfield, Massachusetts; 1683. Parson Capen House (restoration). The original windows were probably smaller casements.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nLincoln, Rhode Island; ca. 1687. Arnold House (restoration). A typical Rhode Island stone-ender.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSaugus, Massachusetts; ca. 1686. Boardman House. The saltbox rearward extension was added before 1696; the double-hung sash windows were probably added in the 18th century.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSalem, Massachusetts; ca. 1668. Turner House (House of the Seven Gables, restoration). The original windows were probably smaller casements.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSaugus, Massachusetts; ca. 1680. Appleton House (Ironworks House, restoration), taken about 1900, shows the house after two hundred years of modifications.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSaugus, Massachusetts; ca. 1680. Appleton House (Ironworks House, restoration). This image shows a later restoration to its probable 17th-century cross-plan form.\n\nNORTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSalem, Massachusetts, ca. 1698. Hunt House. An early photo; the house was demolished in 1863. The double-hung sash windows are probably an 18th-century addition.\n\nSOUTHERN TRADITION\u2014Separated from the northern colonies by the Dutch in New York and New Jersey, the southern English colonies emphasized one-story forms with paired end chimneys. Most surviving examples have brick walls.\n\n**SOUTHERN TRADITION**\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nVirginia Beach, Virginia, vicinity; mid-18th century. Hudgins House. A rare survivor of the once common wood-frame hall-and-parlor folk houses in the tidewater South. This example has high-style Postmedieval chimneys, which may remain from a partially destroyed earlier brick house.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nNorfolk, Virginia, vicinity; mid-17th century. Thoroughgood House. Note the decorative shape of the exterior end chimney.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nHollywood, Maryland, vicinity; ca. 1660 (left two-thirds only). Resurrection Manor. The double-hung sash windows were probably added when the house was expanded from two or three rooms in the 18th century.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nVirginia Beach, Virginia; 1680. Keeling House. The windows and cornice-line dentils are probably 18th-century additions.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSalisbury, North Carolina, vicinity; chimney late 18th century. Long House. A late example of a composite Postmedieval chimney (the house is a much modified restoration). Most such chimneys date from the late 17th and early 18th centuries when houses two rooms deep first replaced the earlier southern linear plans. The space between the two principal chimneys was commonly enclosed and covered by a shed roof to make a small interior room or closet.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nNewport News, Virginia, vicinity; early 18th century (as modified). Jones House. An earlier one-story house (note the trace of the original steep roof) altered to a two-story cross plan in the 18th century. The windows are modern.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSurry County, Virginia; mid-17th century. Bacon's Castle. Although somewhat altered, this is the only high-style house surviving from the 17th century. This photo shows the front facade.\n\nSOUTHERN TRADITION: POSTMEDIEVAL ENGLISH  \nSurry County, Virginia; mid-17th century. Bacon's Castle. Rear view.\n\n## Variants and Details\n\nIn Massachusetts and Connecticut a characteristic second-floor wall overhang is commonly present on the front facade; this is sometimes ornamented with decorative brackets or pendants. Similar wall overhangs at attic level are common beneath the end gables. Full-height cross gables were frequently used on the steeply pitched roofs to add space and light to the tall attic (few of these have escaped later roof modifications). In Rhode Island, stone end walls and chimneys were common on timber-frame houses; few of these stone-enders survive. In one-story southern examples, small dormers were sometimes used to provide attic light (many seen today are later additions).\n\nOriginally most Postmedieval houses were one room deep and symmetrical from front to back; later, lean- to rear projections were added to increase first-floor space. By around 1700 these rear additions were usually included under a single main roof in new construction, or under reframed and lowered roofs on earlier houses, to give the familiar saltbox roof form (see this page and this page). In all colonies, both timber-frame and masonry examples sometimes showed small, projecting wings or towers centered on the front or rear facades. In front, these typically served as entry areas, with a bedroom above; in the rear, they housed the stairway. When both were present, they gave the house a characteristic cross-shaped plan; few of these projections survive except in restorations.\n\nPOSTMEDIEVAL ENGLISH\n\n **CHIMNEYS northern tradition**\n\nPOSTMEDIEVAL ENGLISH\n\n **CHIMNEYS southern tradition**\n\nPOSTMEDIEVAL ENGLISH\n\n **DECORATIVE CHIMNEY TOPS both traditions**\n\nPOSTMEDIEVAL ENGLISH\n\n **DOORS & WINDOWS both traditions**\n\nPOSTMEDIEVAL ENGLISH\n\n **SHAPE VARIANTS**\n\nNORTHERN TRADITION  \ntwo-story, wood\n\nPOSTMEDIEVAL ENGLISH\n\n **SHAPE VARIANTS**\n\nSOUTHERN TRADITION  \none-story, brick\n\nPOSTMEDIEVAL ENGLISH\n\n **SHAPE VARIANTS**\n\nPOSTMEDIEVAL ENGLISH\n\n **OVERHANGS northern tradition only**\n\n## Occurrence\n\nThis was the only style in the English colonies from their founding (1607\u20131620) to about 1700, when their population had grown to 220,000 and occupied the areas shown on the map. Only a few hundred houses remain of the many thousands built in this period. Most are in Massachusetts and Connecticut, where about a hundred are preserved as museum houses and at least that many more are in private hands. Fewer examples survive in Maryland, Virginia, and the middle colonies. After 1700, early Georgian houses with less steep roof slopes, smaller chimneys, large double-hung windows having one fixed and one movable sash, and classical door surrounds rapidly replaced this style throughout the English colonies. Postmedieval houses survived longest in the South, where scattered examples with Postmedieval details were built throughout the 18th century.\n\nPOSTMEDIEVAL ENGLISH\n\n **PRINCIPAL SETTLED AREAS 1700**\n\n## Comments\n\nThese earliest English Colonial houses are New World adaptations of modest English domestic buildings which, in the decades immediately preceding colonization, had begun to undergo a transition from Medieval to Renaissance structural details. The steeply pitched roofs were a surviving Medieval development for thatch covering, which must be steep to shed water. In America the earliest roofs were also of thatch, but the ice, snow, thunderstorms, and high winds of the more severe New World climate soon made wooden shingles the preferred roofing material. The high pitch, now without function for relatively impervious shingle roofs, persisted for nearly a century. The roof pitch has been lowered in later alterations of most examples, including many restorations.\n\nThe chimney stack, replacing the open fire of Medieval vernacular houses, was the crucial Postmedieval improvement. Attic space, formerly unenclosed so that smoke could escape through roof openings, could be floored over to provide sleeping rooms. In the New World, large chimneys were used on all but the most modest 17th-century houses. In the northern colonies, central chimney placement was preferred, probably to conserve heat during the severe winters. In the southern colonies, the end chimneys allowed a central hall to improve cross-ventilation and may have helped to dissipate the heat of cooking fires during the oppressively hot summers.\n\nAlthough only a few Postmedieval timber-frame houses survive in the southern colonies, they were probably far more common originally. With the growth of the southern plantation economy in the 18th century, many early wooden houses were converted to slave quarters or storage; most were ultimately abandoned and razed. As a result, the houses that survived were primarily early masonry examples. Most of these have also been lost due to indifference and neglect, some in only the past few decades.\n\nThe few surviving Postmedieval houses have generally been in continuous use for almost three hundred years. During this long period they have been modified, improved, remodeled, and rebuilt, with the result that few reached this century in anything approaching their original form. Beginning about 1900, concern for our earliest colonial heritage led to modern restoration of many examples. When based on precise architectural and historical research, these restored houses closely approach the appearance of the 17th-century originals. When less carefully done, such restoration has produced bastard buildings with combinations of features that never existed.\n\n# COLONIAL HOUSES\n\n# Dutch Colonial\n\n# 1625\u2013ca. 1840\n\n## Identifying Features\n\nOne story (less commonly one and one-half stories, rarely two stories) with side-gabled or side-gambreled roof having little or no rake (side) overhang; most originally with entrance doors divided into separately opening upper and lower halves (in about half the surviving examples, these have been replaced by later single-unit doors).\n\n## Principal Subtypes\n\nNew World colonists from the Netherlands constructed three distinctive types of houses:\n\nURBAN TRADITION\u2014Among the earliest were brick urban houses of Medieval inspiration having steeply pitched and parapeted gable roofs and paired end chimneys. This type dominated the 17th-century Dutch trading settlements that grew at each end of the region's principal navigation route, the Hudson River: New Amsterdam (later New York) to the south and several outposts in the Albany area to the north. These towns became increasingly Anglicized in the 18th century with the result that few Dutch urban houses were built after about 1730.\n\n**URBAN TRADITION**\n\nURBAN TRADITION: DUTCH COLONIAL  \nSchenectady, New York; early 18th century. Yates House. The only surviving example in an urban setting with the entrance in the narrow gable end. Note the dagger-shaped wrought-iron anchors for roof and wall beams, a characteristic feature of urban Dutch houses. This example has wood-frame side walls. The windows and Classical doorway are later additions.\n\nURBAN TRADITION: DUTCH COLONIAL  \nEast Greenbush, New York; 1723. Breese House. Typical urban house built in rural setting, with entrance on long, non-gabled wall. The windows and porch are later additions.\n\nURBAN TRADITION: DUTCH COLONIAL  \nKinderhook, New York; 1737. Van Alen House (restoration). Urban house in rural setting expanded by adding an adjacent unit in typical Dutch fashion.\n\nURBAN TRADITION: DUTCH COLONIAL  \nCohoes, New York; mid-18th century. Van Schaick House. A transitional example with urban-style brickwork and newly fashionable gambrel roof. The porch is a late 19th-century addition.\n\nRURAL TRADITION, UNFLARED EAVES\u2014Dutch building traditions persisted far longer in rural areas. Brick, the preferred Dutch building material, was replaced by coursed stone in most rural houses. The shaping and finish of the stonework became increasingly refined as colonial inhabitants grew more affluent during the 18th century. Early rural examples had side-gabled roofs and little or no eave overhang. After about 1750 gambrel roofs became common in this type.\n\n**RURAL TRADITION, UNFLARED EAVES**\n\nRURAL TRADITION, UNFLARED EAVES: DUTCH COLONIAL  \nRotterdam, New York; early 18th century. Mabie House. One of the few surviving rural Dutch houses that preserve the steep Medieval roof pitch. The walls are of whitewashed stone; the frame extension to the right is a later addition.\n\nRURAL TRADITION, UNFLARED EAVES: DUTCH COLONIAL  \nHurley, New York; 1725. Restoration.\n\nRURAL TRADITION, UNFLARED EAVES: DUTCH COLONIAL  \nNew Paltz, New York; ca. 1692. Bevier House. The original house (left-hand portion) was expanded by early additions. The gable-end entrance is unusual.\n\nRURAL TRADITION, UNFLARED EAVES: DUTCH COLONIAL  \nCoeymans, New York; ca. 1761 (altered ca. 1790). Coeymans House. An early example expanded and altered by the early addition of a fashionable gambrel roof.\n\nRURAL TRADITION, UNFLARED EAVES: DUTCH COLONIAL  \nUlster County, New York; mid-18th century. Ten Broeck House. Note the expansion by adding two units to the portion to the left with casement windows, which is the earliest. The middle unit was added in 1751 and the right-hand unit in 1765.\n\nRURAL TRADITION, FLARED EAVES\u2014This tradition is similar to the rural subtype described just above, but has flared, overhanging eaves, which became common on both gable- and gambrel-roofed examples after about 1750 in the southern Hudson River area (see maps).\n\n**RURAL TRADITION, FLARED EAVES**\n\nRURAL TRADITION, FLARED EAVES: DUTCH COLONIAL  \nOld Tappan, New Jersey; ca. 1751. Haring House. Note the more carefully finished stonework on the front facade.\n\nRURAL TRADITION, FLARED EAVES: DUTCH COLONIAL  \nDumont, New Jersey; ca. 1760 (smaller portion in foreground), ca. 1810 (larger portion). Zabriskie House. Note the absence of flared eaves on the small early portion. The wide-end overhang of the gambrel roof is a 19th-century development.\n\nRURAL TRADITION, FLARED EAVES: DUTCH COLONIAL  \nBrooklyn, New York; ca. 1676. Schenck House. A very early wood-frame example preserving the steep Medieval roof pitch. The dormers, porch, and flared eaves are 18th-century modifications.\n\nRURAL TRADITION, FLARED EAVES: DUTCH COLONIAL  \nMahwah vicinity, New Jersey; late 18th century. Van Horn House.\n\nRURAL TRADITION, FLARED EAVES: DUTCH COLONIAL  \nCloster, New Jersey; ca. 1800. Durie House. The dormers and porch supports are later additions, as is the smaller frame wing added in 1854.\n\n## Variants and Details\n\nAs in the adjacent English colonies, the pitch of rural Dutch roofs decreased during the early 18th century as wood shingles replaced thatch, tile, and slate as the preferred roofing material. Steeply pitched, Medieval-style roofs survive on only a few rural Dutch houses, all built before about 1720. (The dating of roofs and architectural details in rural Dutch houses is unusually difficult because early stone walls were typically incorporated into expansions and modifications throughout the 18th and 19th centuries. For this reason, most houses of supposed early date show later features, particularly roof, door, and window details.) In many rural examples the stone walls do not extend into the side gables, which are instead constructed of either brick or, more commonly, shingle- or weatherboard-covered wooden framing. After about 1750, distinctively shaped gambrel roofs with short, flattened upper slopes became common, along with gable roofs of normal pitch which continued to be built. In the southernmost areas of Dutch influence, around present-day New York City and adjacent New Jersey, distinctive flared eaves were usual on both gable and gambrel roofs after about 1750. Where the Dutch colonists were in close contact with English building traditions, particularly on western Long Island, timber-frame rural houses with weatherboard or shingle siding replaced the more usual stone construction. Most existing Dutch Colonial houses have double-hung sash windows which may be original or replacements of earlier types. Like their English counterparts, 17th-century Dutch houses apparently had leaded casement windows. In the English colonies these were supplanted by wooden, double-hung windows with one movable sash early in the 18th century. The Dutch, however, apparently used outward-swinging wooden casements, sometimes hung in side-by-side pairs, during an early 18th-century transitional period between leaded casements and wooden double-hung sashes. Few of these early casement windows, either leaded or wooden, survive. The Dutch double door was probably developed to keep out livestock (with the bottom section closed) while allowing in light and air through the open top. This style of door is found in about half of the surviving houses. From the early 18th century, the treatment of the door surround commonly reflected the Georgian and subsequent Federal styles of the English colonies. It had previously been thought that porches were likely later additions to rural Dutch Colonial houses. However, recent research has demonstrated that porches were found in the Low Countries (today's Netherlands and Belgium) from the 16th to the 18th centuries. Further, written evidence establishes that porches were part of original Dutch Colonial houses in the Hudson River Valley, making them some of the earliest examples in America.\n\nDUTCH COLONIAL\n\n **DEVELOPMENT OF THE RURAL DUTCH ROOF**\n\nDUTCH COLONIAL\n\n **DOOR & WINDOW DETAILS**\n\nWINDOWS\n\nDOORS  \nnot all Dutch doors were divided\n\n## Occurrence\n\nFormal control by the Dutch of their New World colonies was remarkably brief. Dutch fur traders founded settlements near Albany in 1614 and at New Amsterdam (New York) in 1626. Centered in these areas, and along the Hudson River which connected them, Dutch colonization proceeded for only fifty years before expanding English colonies on either side led to English control in 1664. Thus New Netherland became New York and Dutch influence began to fade in the principal towns of the colony. All Dutch urban buildings have long vanished from New Amsterdam (the tip of Manhattan Island in what is now New York City) but a very few\u2014probably less than a half-dozen\u2014urban houses still survive in the Albany region. These are among the rarest of American domestic buildings; regrettably, several of the finest remaining examples have been lost during only the last few decades. In contrast, several hundred Dutch rural houses survive in various states of preservation and modification throughout the area of former Dutch influence. The English permitted feudal Dutch landholders, some of whom controlled enormous tracts along the Hudson, to retain their property. These landlords, in turn, continued to encourage rural immigration from Holland during the 18th century. As a result, most Dutch Colonial rural houses post-date by many years the era of Dutch ownership of the colony.\n\nThe building traditions brought by these Dutch immigrants survived, in isolated examples, into the early decades of the 19th century. Today, concentrations of Dutch rural houses are found principally in Bergen County (New Jersey), adjacent Rockland County (New York), and farther up the Hudson, in Ulster and Dutchess counties (New York). A handful of formerly rural houses are preserved within the bounds of present New York City, particularly in Queens County and Staten Island. Scattered examples are found throughout the area of former Dutch influence (see maps).\n\nDUTCH COLONIAL\n\n **MAXIMUM EXTENT OF EARLY DUTCH SETTLEMENT (1624\u2013ca. 1820)**\n\nDUTCH COLONIAL\n\n **PRINCIPAL SURVIVING CONCENTRATIONS OFDUTCH COLONIAL HOUSES**\n\n## Comments\n\nDutch Colonial houses of the urban type are quite similar to their Old World counterparts built in the prosperous mercantile cities of 17th-century Holland. In sharp contrast, the origins of the American Dutch rural house are uncertain and have attracted much speculation. The earliest were simple stone-walled, gable-roofed folk houses. Similar houses are found in the rural building traditions of Flanders (which includes the coastal regions of modern Belgium and immediately adjacent France) but are rare in the Netherlands, which generally lacks both stone and abundant timber for building (hence the Dutch emphasis on brick construction). Most rural Dutch immigrants, however, were persecuted Protestants from Flanders, France, and elsewhere who sought refuge first in Holland and then in her New World colony. With them most probably came the tradition of stone folk building. The principal controversy thus centers not on the stone walls of the rural Dutch Colonial house, but on its distinctive roof features: the unusual gambrel and, especially, the flared eaves, both of which became common after about 1750. The gambrel is perhaps the easiest to explain, since gambrels of somewhat different shape were also becoming common, as a means of increasing both roof span and useful attic space, throughout the English colonies at the same time. More difficulty attaches to the origin of the flared eaves. These have been considered to be: (1) a distinctive New World innovation, or (2) an adaptation of a French-Flemish tradition of protecting plastered walls under steeply sloping, thatched roofs by adding a more gently sloping extension of tiles at the eaves. Pending actual research on the question, the latter explanation appears more likely, particularly since many French-influenced Colonial buildings show similar eaves (see French Colonial houses).\n\nDutch rural houses, with their substantial stone walls, were less easily expanded than were their wooden English counterparts. Although there are many examples of early Dutch stone walls being incorporated into later, and larger, houses, these thrifty colonists generally favored another method of house expansion. When a house became too small, a larger version was built immediately beside the smaller, which then became a kitchen or bedroom wing of the new dwelling. Dutch Colonial houses thus often show a linear sequence of two or three (rarely more) units built at different times. Although the smallest unit is normally the oldest, this is not invariably the case, for small kitchen or bedroom wings were also added to larger houses long after they were built.\n\n# COLONIAL HOUSES\n\n# French Colonial\n\n# 1700\u20131830; to ca. 1860 in New Orleans\n\n## Identifying Features\n\nOne story with many narrow door and window openings having paired shutters (these openings originally had paired French doors and paired casement windows which have commonly been altered to single doors and double-hung sash windows); steeply pitched roof, either hipped or side-gabled; walls of stucco, usually over a half-timbered frame.\n\n## Principal Subtypes\n\nFrench Colonial houses are of two basic types:\n\nURBAN TRADITION\u2014In New Orleans there remain many French urban cottages which lack porches and are built right up to the adjacent sidewalk. These normally have side-gabled (sometimes hipped) roofs and flared eaves that overhang the front facade.\n\n**URBAN TRADITION**\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; 1820. Dolliole House. Hipped-roof example of unusual shape. The roof is covered with the original flat tiles; note the projecting tiles at the eaves and the four doorways.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; mid-19th century. Late hipped-roof example with wood-frame walls and modest Italianate detailing.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; ca. 1850. Mansion House. Late wood-frame example with Greek Revival doorway. The three-ranked facade is unusual.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; 1824. Gaillard House.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; ca. 1806. Font-Juncadella Building. A well-preserved example of a typical early 19th-century French shop-residence of the Vieux Carr\u00e9. Note the delicate wrought-iron balcony railing.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; 1836. Gardette House. Large Vieux Carr\u00e9 town house with elaborate cast-iron porches added in the mid-19th century.\n\nURBAN TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; ca. 1828. Boutin House. Early parapeted example with Federal doorway and pilasters. The Victorian eave brackets and trim are later additions.\n\nRURAL TRADITION\u2014More familiar than the urban cottages of New Orleans are French rural houses with extensive porches supported by slender wooden columns under the main roof line. These usually have steeply pitched, hipped roofs and are commonly raised on high masonry foundations, the porch area above being supported by massive masonry columns. They may be raised a full story above grade, allowing the main living level of the house to escape seasonal floods.\n\n**RURAL TRADITION**\n\nRURAL TRADITION: FRENCH COLONIAL  \nSt. Genevieve, Missouri; late 18th century. Bolduc House. Photo shows the side and rear of the house before restoration.\n\nRURAL TRADITION: FRENCH COLONIAL  \nSt. Genevieve, Missouri; late 18th century. Bolduc House. Photo shows the rear of the house as restored with a dual-pitched hipped roof and exposed wall timbers.\n\nRURAL TRADITION: FRENCH COLONIAL  \nHahnville, Louisiana, vicinity; early 19th century. Lehman House. A modest example built without the usual Louisiana high basement. The dormer and metal roof are later additions.\n\nRURAL TRADITION: FRENCH COLONIAL  \nNew Orleans, Louisiana; 1820. Olivier House. Large example with dual-pitched hipped roof. Note the brick columns supporting the wood porch above and the outside stairway. Originally a plantation house beyond the city, this example survived the urban growth around it until 1950, when it was demolished.\n\nRURAL TRADITION: FRENCH COLONIAL  \nSt. Genevieve, Missouri; late 18th century. Amoureaux House. This example has been much modified over the years. The roof framing suggests that the original roof was of pavilion-shaped hipped form without porches.\n\nRURAL TRADITION: FRENCH COLONIAL  \nHahnville, Louisiana, vicinity; early 19th century. Fortier House (Homeplace Plantation). The front stairway replaces an earlier one beneath the right corner of the porch. The metal roof is a later addition.\n\nRURAL TRADITION: FRENCH COLONIAL  \nNew Roads, Louisiana, vicinity; early 19th century. Riche House. The elaborate stairway probably replaces an earlier one beneath the porch, like that in the Olivier House. Note the partial side porches. In low flood-prone areas the main living levels were raised a full story above grade level. Storage and workshops were at grade. This produced the raised house form seen in Olivier House, Amoureaux House, Fortier House (Homeplace Plantation), Riche House, and Houssaye House (Acadian House).\n\nRURAL TRADITION: FRENCH COLONIAL  \nSt. Martinville, Louisiana, vicinity; 1765. D'Autrive Chevalier de St. Louis. Houssaye House (Acadian House, restoration). An early example, probably built originally with side-gabled roof as shown.\n\n## Variants and Details\n\nAs in their 17th-century English and Dutch counterparts, early French Colonial houses had very high, steeply pitched roofs, following the Medieval tradition of constructing thatched roofs at a very steep pitch in order to shed water. Early French examples usually had a characteristic pavilion roof form, which is steeply hipped with the side roof planes sloping even more steeply than the front and back planes. Very few of these survive. The addition of wide porches around such houses, a mild-climate tradition that probably originated in the West Indies, was accomplished by extending the hipped roof out over the porch but at a gentler pitch, giving it a distinctive, dual-pitched form. As this tradition developed, such roofs were even used occasionally on urban houses without porches. Somewhat later, simple hipped roofs, lower and with uniform slopes on all sides, came to dominate. Original side-gabled roofs are uncommon in rural houses, although many early hipped forms have been modified to this shape. In New Orleans, side-gabled roofs were dominant on urban cottages built after about 1830, probably to reduce roof drainage to narrow passageways between the closely spaced cottages of the expanding city. These later urban cottages also typically have extended and flared eaves, a characteristic that they share with Flemish-inspired Dutch Colonial houses of New York and New Jersey. In all roof forms, tall and narrow gabled dormers were sometimes used to provide attic light.\n\nMost French Colonial houses originally had paired French doors, with small glass panes set above wooden panels. The doors sometimes had a line of transom lights above; in later examples these were often supplanted by a Federal fanlight. Originally the doors were framed by a simple, narrow surround. Vertical board shutters hung on strap hinges covered the doors and transom (but not the fanlight, if present). The interior surface of the shutter was sometimes paneled; the shutters usually swung outward and the doors inward. In later examples, Federal or Greek Revival door surrounds are common. Early French windows were paired wooden casements which swung inward. These were generally glazed with small panes of glass and were covered by vertical board shutters which had horizontal battens on the interior and swung outward on iron strap hinges. The window surround was narrow and simple. In later examples these French-style casements were supplanted by English double-hung sashes.\n\nFRENCH COLONIAL\n\n **POSSIBLE ORIGIN OF RURAL TYPES**\n\nFRENCH COLONIAL\n\n **ROOF FORMS**\n\nFRENCH COLONIAL\n\n **WINDOW DETAILS**\n\nFRENCH COLONIAL\n\n **DOOR DETAILS**\n\nfanlight (English influence)\n\n## Occurrence\n\nIn the 18th century, France occupied much of eastern North America by means of military outposts and settlements scattered along the principal waterways, particularly the St. Lawrence, Great Lakes, and Mississippi valleys (see map). After Jefferson's purchase of Louisiana in 1803, French building traditions began to fade, although they persisted in New Orleans for half a century more and survive today in French Canada. In the United States, only a few concentrations of French-influenced buildings remain from this vast empire. All of these are in Louisiana and adjacent Mississippi save one, the little-known French Colonial houses of St. Genevieve, Missouri (see map). Several hundred houses, all of the rural type, survive in these areas; a handful of others are scattered elsewhere. Most date from the late 18th and early 19th centuries. The early outskirts of New Orleans had many similar rural houses but only a few survived the later growth of the city. Until at least the 1860s, however, French-style urban cottages were built in the Creole suburbs of New Orleans, to the north and east of the original town, or Vieux Carr\u00e9.\n\nMore than half of the country's surviving French Colonial houses are found in these New Orleans neighborhoods, but are seldom seen by visitors to that city. Only a handful remain in the Vieux Carr\u00e9, which was all but destroyed by fires in 1788 and 1791. As the commercial center of town, the Vieux Carr\u00e9 was rebuilt during the early 19th century largely with mixed-use structures having shops on the first floor and living quarters above. Many of these survive and show strong French influences, especially in the door and window treatments, but most were altered as the city grew during the later 19th century. For example, small and delicate wrought-iron balconies were originally common under full-length upper windows (Font-Juncadella Building). The expansion of cast-iron technology in the mid-1800s led to the replacement of many by elaborate systems of iron porches extending over the sidewalks and around the building at each upper level (Gardette House). These provided outdoor living areas for residents of the upper floors, but they also dramatically altered the facades. Paradoxically, it is these American-made additions, almost unknown in France, that have come to characterize French New Orleans.\n\nWith the growth of the city as a principal United States seaport after the Louisiana Purchase, a flood of American immigrants built principally upriver (southwest) of the Vieux Carr\u00e9. At the same time, the more slowly expanding French population built new suburbs in the opposite or downriver direction. Most of the city's surviving French houses are urban cottages built in these neighborhoods from 1810 to about 1860.\n\nFRENCH COLONIAL\n\n **FRENCH NORTH AMERICA AT ITS MAXIMUM EXTENT (ca. 1760)**\n\nFRENCH COLONIAL\n\n **PRINCIPAL SURVIVING CONCENTRATIONS OF FRENCH COLONIAL HOUSES**\n\n## Comments\n\nEnglish houses are usually directed inward; they have few external entrances and emphasize internal halls and stairways for access to the rooms. French houses, on the other hand, typically look outward; each room is likely to have its own exterior doorway and the stairways are commonly on exterior porches, rather than within the main body of the house. Hallways are also normally absent, the interior rooms opening instead directly into each other. These traditions are reflected in the many exterior doors and in the external stairways of French Colonial houses. Even small urban cottages, usually built in a square four-room plan, have at least four external doors, two of which lead from the two front rooms directly onto the adjacent sidewalk.\n\nMost French Colonial houses were constructed, at least in part, with half-timbered walls. Earliest, and most primitive, was post-in-ground construction, with closely spaced vertical timbers buried in the ground and filled in between with clay mixed with such binding materials as hair or straw. Later, typical timber framing, using a sill set on a foundation, was adopted; often such walls had soft brick infilling. Both the post-in-ground and framed types were originally covered with stucco or, in later examples, weatherboarding, to protect the timbers and infilling. In raised rural houses the foundation is commonly of stuccoed brick, which supports the half-timbered walls of the main floor above.\n\nThe several hundred surviving French Colonial houses are among the rarest and least appreciated American buildings. While comparable English houses of the eastern seaboard have long been revered landmarks, only a few French examples are similarly esteemed. Recent preservation efforts in New Orleans have renewed interest in its urban cottages, but unique rural houses are still being lost through indifference and neglect.\n\n# COLONIAL HOUSES\n\n# Spanish Colonial\n\n# 1600\u20131850; locally to ca. 1900\n\n## Identifying Features\n\nOne story (less commonly two stories) with low-pitched or flat roof; thick masonry walls of adobe brick or rubble stone (usually covered with protective stucco); originally with multiple external doorways and few small window openings lacking glass (bars or grilles of wood or wrought iron covered the exterior openings, which were closed from the interior by solid wooden shutters; except in reconstructions, most such early windows have been altered to accommodate double-hung, glazed sashes and trim).\n\n## Principal Subtypes\n\nSpanish Colonial houses are of solid masonry construction but show two fundamentally different roof types which are found both in Spain and in her New World colonies:\n\nPITCHED ROOF\u2014The first basic type includes pitched-roof houses with traditional European roof framing. These, in turn, are of three kinds: The first consists of steeply pitched, usually side-gabled forms in which the wooden framing supports a covering of thatch; in the United States, this tradition survives principally in Hispanic folk houses with steeply pitched, shingled roofs. In the second and most familiar type, the roofs are low-pitched with a covering of half-cylindrical tiles. These tile roofs are usually of shed- or side-gabled form, less commonly hipped. A third variant, originally found in the Los Angeles area but now very rare, consists of an almost flat, tar-covered shed roof with overhanging eaves.\n\n**PITCHED ROOF**\n\nPITCHED ROOF: SPANISH COLONIAL  \nSalinas vicinity, California; early 19th century. Sherwood House. Courtyard view of a little-altered rural survivor of modest scale. The wide rake overhang of the gable is unusual.\n\nPITCHED ROOF: SPANISH COLONIAL  \nSan Diego, California; 1829. Estudillo House (restoration). Exterior view of a part of an extended, U-shaped example that survives relatively intact. Note the intersecting shed roofs.\n\nPITCHED ROOF: SPANISH COLONIAL  \nSonoma County, California; 1836. Petaluma Adobe; Alpheus B. Thompson, designer\/builder. Large adobe house included ranch headquarters. Note extensive dual-level porches and semi-enclosed courtyard.\n\nPITCHED ROOF: SPANISH COLONIAL  \nTempleton vicinity, California; early 19th century. Blackburn House. Courtyard view of a little-altered rural survivor. Note the intersecting roof planes.\n\nPITCHED ROOF: SPANISH COLONIAL  \nRio Grande City vicinity, Texas; 19th century. A modest rural two-story example with a Monterey-style cantilevered porch. Note the absence of windows.\n\nPITCHED ROOF: SPANISH COLONIAL  \nSan Juan Bautista, California; 1841. Castro House. A very large example of the Monterey style. Note the adobe bricks exposed beneath the falling stucco. The house has been restored since the photograph was taken.\n\nPITCHED ROOF: SPANISH COLONIAL  \nMonterey, California; 1835. Larkin House; Thomas Oliver Larkin, owner\/builder. The first of the California adobes with Anglicized details and broad two-level sheltering porches.\n\nFLAT ROOF WITH PARAPET\u2014The second basic type consists of flat-roofed houses without traditional European roof framing. Instead, massive horizontal timbers are embedded in parapeted masonry walls to support an extremely heavy roof of earth or mortar. Cylindrical rainspouts of wood, tin, or tile project through the parapet along one or more walls to provide drainage. This type, introduced into Spain from North Africa by the Moors, was also developed independently by several groups of Native Americans, and was well established in Mexico and the southwestern United States when the Spaniards arrived.\n\nFrom early Spanish Colonial times, each of the two basic roof types has tended to dominate in different parts of Mexico and adjacent Hispanic areas of the United States (see map). The reasons for this pattern are uncertain; the flat, earthen roof would appear to be more suitable for very hot and dry regions, yet each type dominates through a range of climates. Most probably the building traditions of the original colonial settlers, interacting with those of the local natives, determined the patterns.\n\n**FLAT ROOF WITH PARAPET**\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nTucson, Arizona; ca. 1875. Verdugo House. In this example the original flat roof, revealed by the rainspouts, has been covered by a later pitched roof. The doors and windows are probably also later additions.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nRanchos de Taos, New Mexico; early 19th century. Courtyard view; note the recessed porch with simple, bracketed roof supports. The Anglo door and windows are probably later additions.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nSan Ygnacio, Texas; ca. 1851 (later additions). Trevino House. Extended example built from right to left in three progressively larger units (defined by the doors and rainspouts). Note the stone walls and adjacent _\"jacal\"_ folk house with crude half-timbering and thatch roof.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nSan Pablo, Colorado; mid-19th century. An extended example with Anglo doors and windows.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nSanta Fe, New Mexico; 1851. Tully House. A Territorial example with the exterior surviving as originally built.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nSanta Fe, New Mexico; early 19th century (later additions). Borrego House (restoration). A Territorial example modified from an earlier house by the addition of front porch, brick coping along the parapet, and Anglo window crowns.\n\nFLAT ROOF WITH PARAPET: SPANISH COLONIAL  \nSan Antonio, Texas; ca. 1772. Spanish Governor's Palace.\n\n## Variants and Details\n\nThe earliest houses in areas of the United States that were formerly Spanish territories showed few decorative or stylistic details when compared with more imposing Spanish or Mexican prototypes. Built in remote and impoverished colonial outposts, these houses were simple by necessity. Only with the opening of trade with the United States in the 1830s did increased prosperity come to these regions; along with the new wealth came Anglo immigrants with their own building traditions. First came wooden decorative details, principally in the Greek Revival style, and glazed, double-hung sash windows. In areas with the pitched-roof tradition, shingled roofs were introduced. In other areas, flat-roofed houses became modified by the addition of framed, shingled roofs above the parapeted walls. These innovations quickly became fashionable with both Anglo and Hispanic residents, who superimposed them upon the traditional adobe construction. Such Anglo-Spanish\u2013Greek Revival houses, in two-story variants with cantilevered second-floor porches, have come to be called Monterey style, after the colonial capital of California where many survive (Texas, 19th century; Castro House; and Larkin House). In New Mexico, western Texas, and Arizona, related flat-roofed, single-story forms, usually with a protective topping of fired brick crowning the roof parapet, are known as Territorial style houses (Tully House and Borrego House). The spread of the western railroads in the 1880s provided ready access to quantities of milled lumber and led to the final decline of Hispanic building styles as adobe construction was abandoned for wood-frame houses in all but a few remote pockets of surviving tradition.\n\nPORCHES\u2014Spanish domestic buildings commonly have long, narrow porches (the _corredor_ or _portale_ ) that open onto internal courtyards and function as sheltered passageways between rooms, which usually lacked internal connecting doorways. In more pretentious Spanish and Mexican prototypes, the porches often took the form of colonnaded arcades with elaborate masonry arches supporting the roof. In more modest examples, which include all that survive in the United States, porch roofs were supported by hewn logs, usually capped by distinctive carved brackets. In pitched-roof houses these columns supported either extensions of the main roof or separate shed roofs abutting the main walls. In flat-roofed houses, porches were normally recessed into the main structure, with the principal adobe walls supporting the ends of the porch-roof timbers. Upon these were built a lower and somewhat thinner version of the main earthen roof. Because they faced internal courtyards, traditional porches are seldom evident on the external facades. With the arrival of Anglo influence, however, front-facade porches became fashionable. Particularly characteristic were cantilevered second-floor porches on two-story houses. These usually show delicate wooden balustrades and were probably inspired, at least in part, by the cantilevered balconies common on the upper floors of traditional Spanish town houses. Alpheus B. Thompson and Thomas O. Larkin, builders of early examples with full-width porches, were familiar with this form from travels in the southeastern United States and Caribbean (Petaluma Adobe and Larkin House). Anglo influence also led to the traditional massive roof supports being abandoned for more delicate wooden columns of vernacular Greek Revival inspiration.\n\nSPANISH COLONIAL\n\n **ROOF FORMS**\n\nPITCHED\n\nSPANISH COLONIAL\n\n **ROOF FORMS**\n\nFLAT\n\nSPANISH COLONIAL\n\n **DOOR & WINDOW DETAILS**\n\nSPANISH COLONIAL\n\n **ORIGINAL PORCHES Rarely on front facade**\n\nSPANISH COLONIAL\n\n **ANGLO-INFLUENCED PORCHES Commonly on front facade**\n\nSPANISH COLONIAL\n\n **PLAN OF THE SPANISH HOUSE**\n\n## Occurrence\n\nMost of what is now the southwestern United States was Spanish from the 17th century until 1821, when Mexico gained its independence; it remained part of Mexico until it was ceded to the United States in the late 1840s following the Mexican War. Spanish Texas gained independence from Mexico in 1836 and was a separate country until it was annexed to the United States in 1845. Florida was Spanish from 1565 until it was ceded to the United States in 1821, with a brief interruption of British rule from 1763 to 1783. (A similarly brief interval of Spanish control of French Louisiana, from 1762 until 1800, resulted in a few Spanish public buildings but little change in the local French housing traditions.) This vast territory was a sparsely settled frontier region during the Spanish and Mexican periods. Forts and missions to convert the Native Americans were established at many places, but few led to permanent settlements with substantial domestic architecture: St. Augustine in Florida; around San Antonio, Texas; scattered along the length of the Rio Grande from southern Texas to northern New Mexico; around Tucson, Arizona; and along the California coast from San Diego northward to around San Francisco (see map). Today significant concentrations remain only in St. Augustine, Tuscon, Santa Fe, Taos, Mesilla, San Diego, Santa Barbara, Monterey, and a few rural communities in Texas and New Mexico. Almost all surviving examples show Anglo-influenced modifications from the mid- and late 19th century. Many have also suffered from 20th-century renovation and overly zealous restoration, this particularly for those in modern urban centers; relatively unaltered examples from the 19th century survive principally in rural areas, from which they are fast disappearing through neglect and decay.\n\nSPANISH COLONIAL\n\n## Comments\n\nUnlike their English counterparts, larger Spanish Colonial domestic buildings were not usually conceived as multi-roomed wholes but grew, instead, as series of independent rooms. Modest households had but a single room. As affluence increased, one-room units were added to make extended dwellings whose size was limited only by the wealth of the builder. Typically, the first two or three rooms were joined end-to-end to make a linear row; units were then added single file but at right angles to make an L or U. In the largest houses, the rooms made rectangular masses, enclosing an inner courtyard (the _patio_ or _placita_ ). In smaller houses, masonry walls, rather than rooms, usually completed the enclosure of similar courtyards. Traditionally, few internal openings existed between rooms; each was entered through its own door opening onto the courtyard. Long, narrow porches commonly provided sheltered passageways between rooms. The external facades of extended houses were usually austere, revealing only small windows and a single entrance door or gateway.\n\nSpanish Colonial buildings are unusually durable when executed in stone. Although the wooden roof framing quickly decays if neglected, walls often survive many decades, even centuries, of abandonment. Spanish mission buildings in Texas, Arizona, and California, some constructed in the 17th and early 18th centuries, have mostly been reconstructed in this century upon such remaining wall segments. In sharp contrast, adobe walls are unusually susceptible to deterioration; if the roofs are not continually repaired, rainwater literally melts them into a formless mass of mud. As a general rule, adobe buildings abandoned for more than twenty-five years are beyond repair. Because most Spanish Colonial houses had adobe walls, the only survivors are those that have had continuous care. Regrettably, many of the most authentic examples were abandoned just during the past thirty years in favor of frame dwellings. This is particularly true in rural New Mexico, where irreplaceable examples have been, and are being, lost.\n\n# COLONIAL HOUSES\n\n# Georgian\n\n# 1700\u20131780; locally to ca. 1830\n\n## Identifying Features\n\nPaneled front door, usually centered and capped by an elaborate decorative crown (entablature) supported by decorative pilasters (flattened columns); usually with a row of small rectangular panes of glass beneath the crown, either within the door or in a transom just above; cornice usually emphasized by decorative moldings, most commonly with tooth-like dentils; windows with double-hung sashes having many small panes (most commonly nine or twelve panes per sash) separated by thick wooden muntins; windows aligned horizontally and vertically in symmetrical rows, never in adjacent pairs, usually five-ranked on front facade, less commonly three- or seven-ranked.\n\n## Principal Subtypes\n\nThe Georgian house is usually a simple one- or two-story box, two rooms deep, with doors and windows in strict symmetry. Five principal subtypes can be distinguished:\n\nSIDE-GABLED ROOF\u2014About 40 percent of surviving Georgian houses are of this type, which is the most common in the northern and middle colonies, but also occurs in the southern colonies.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF: GEORGIAN  \nDeerfield, Massachusetts; 1749. Barnard House. Note the wide-board cladding and double door. The window screens are a later addition.\n\nSIDE-GABLED ROOF: GEORGIAN  \nProvidence, Rhode Island; ca. 1743. Hopkins House. Part of this house was built in 1707, with an expansion in ca. 1743; the door was added still later. Four-ranked examples such as this are sometimes called three-quarters houses.\n\nSIDE-GABLED ROOF: GEORGIAN  \nSouthport, Connecticut; late 18th century. Osborn House. A simple five-ranked, saltbox form with a central chimney.\n\nSIDE-GABLED ROOF: GEORGIAN  \nDeerfield, Massachusetts; 1760. Williams House. Note the broken pediment over the door and the triangular pediments above the windows. The 6\/6 window sashes are probably later additions.\n\nSIDE-GABLED ROOF: GEORGIAN  \nMedford, Massachusetts; 1737. Royall House. Enlarged from a 17th-century brick house; an equally elaborate rear facade was added in 1747. Note the chimney stacks connected by a parapet and the unadorned side, the detailing being concentrated on the front facade.\n\nSIDE-GABLED ROOF: GEORGIAN  \nPortsmouth, New Hampshire; 1718\u20131723. MacPheadris-Warner House. This fine high-style example is built of brick, unusual in early New England. It has a segmental pediment over the entry, a belt course, and dormers with alternating gabled and segmental pediments. Note the cupola, roof-line balustrade, twelve-panel door, and 9\/6 windows.\n\nSIDE-GABLED ROOF: GEORGIAN  \nSurry County, Virginia; 1652, rebuilt early 18th century. Warren House. This simple one-story example is a Georgian remodeling of a Postmedieval house. As is commonly the case in southern masonry examples, there is not an elaborated door surround but only a segmental arch in the brickwork above the paneled door.\n\nSIDE-GABLED ROOF: GEORGIAN  \nAnnapolis, Maryland; 1773. Brice House. A fine example of the five-part plan with \"hyphens\" connecting the main house and dependencies.\n\nSIDE-GABLED ROOF: GEORGIAN  \nPhiladelphia, Pennsylvania; 1772. Deshler House. Note the arched dormer windows.\n\nSIDE-GABLED ROOF: GEORGIAN  \nPhiladelphia, Pennsylvania; 1768. Johnson House. Note the rubblestone side walls and the more regular ashlar facade. The pent roof above the first floor is a common middle-colonies feature.\n\nSIDE-GABLED ROOF: GEORGIAN  \nPhiladelphia, Pennsylvania; ca. 1715. The hood over the front door and the cornice carried beneath the side gable are features found primarily in the middle colonies.\n\nGAMBREL ROOF\u2014This roof form is found primarily in the northern colonies where it is characteristic of about 25 percent of surviving Georgian houses. Few gambrels survive in the middle or southern colonies, although restoration research in Williamsburg indicates they may have formerly been common on one-story southern examples. The shape is an adaptation of the gable form, which provides more attic space for storage or sleeping.\n\n**GAMBREL ROOF**\n\nGAMBREL ROOF: GEORGIAN  \nSalem, Massachusetts; mid-18th century. Nathaniel Hawthorne Birthplace. This example was built around a smaller 17th-century house, a common practice. The 2\/2 windows are later additions.\n\nGAMBREL ROOF: GEORGIAN  \nNewport, Rhode Island; ca. 1748. Nichols House (restoration).\n\nGAMBREL ROOF: GEORGIAN  \nDeerfield, Massachusetts; ca. 1725. Dwight House. Moved from Springfield, Massachusetts, this house has a fine door pediment and pedimented windows.\n\nGAMBREL ROOF: GEORGIAN  \nWoodbury, Connecticut; 1760. Bacon House. Note the slight overhang of the second story, a holdover from Postmedieval building practices.\n\nGAMBREL ROOF: GEORGIAN  \nAlexandria, Virginia; ca. 1780. Robert Townshend Hooe House. An unusual high-style urban example with stone belt courses and delightful lintels with winged keystones.\n\nHIPPED ROOF\u2014About 25 percent of surviving Georgian houses have hipped roofs (some are dual-pitched hipped). This is the most common type in the southern colonies, but is not unusual in the middle and northern colonies, where it occurs principally on high-style landmark examples.\n\n**HIPPED ROOF**\n\nHIPPED ROOF: GEORGIAN  \nRutland, Massachusetts; ca. 1750. Putnam House.\n\nHIPPED ROOF: GEORGIAN  \nPortsmouth, New Hampshire; 1760. Wentworth-Gardner House. Note the rusticated wall cladding (wood cut to look like stonework), wood quoins (also wood imitating stone), and the variety of pediment shapes used on door, windows, and dormers.\n\nHIPPED ROOF: GEORGIAN  \nCharles City, Virginia, vicinity; 1734. Westover. An early high-style example with steeply pitched roof, double-paired interior end chimneys, belt course, and broken pediment door surround. This photo was taken before the modern addition of adjacent wings.\n\nHIPPED ROOF: GEORGIAN  \nRichmond, Virginia, vicinity; ca. 1753. Wilton. This finely detailed example was moved in 1933.\n\nHIPPED ROOF: GEORGIAN  \nPhiladelphia, Pennsylvania; 1734. Stenton House. Note the segmental arches in the brickwork over the door and windows, used in lieu of more elaborate wood crowns.\n\nHIPPED ROOF: GEORGIAN  \nClarksville, Virginia, vicinity; ca. 1765. Prestwould. Built of coursed stone ashlar with a seven-ranked facade. Both the front and side porches are later additions.\n\nHIPPED ROOF: GEORGIAN  \nLancaster County, Virginia; ca. 1754. Belle Isle. A three-ranked central block with lower wings forming a three-part plan. Two detached dependencies are not visible in the photo.\n\nCENTERED GABLE\u2014Less than 10 percent of surviving Georgian houses have a gable (pediment) centered on the front facade. The facade beneath the gable may either remain in the same plane as the rest of the wall or be extended slightly forward for emphasis as a pavilion. This subtype became common only after 1750, and is found in high-style examples in all the former colonies.\n\n**CENTERED GABLE**\n\nCENTERED GABLE: GEORGIAN  \nCambridge, Massachusetts; 1759. Longfellow House. The side porches were added in 1793, as likely were the 6\/6 windows with slender muntins. Note that the centered gable crowns a shallow projection set nine inches forward from the front facade of the house. Two-story pilasters are added for decorative effect.\n\nCENTERED GABLE: GEORGIAN  \nCharleston vicinity, South Carolina; 1738\u20131742. Drayton Hall. An unusually sophisticated early Georgian design that survives without alteration. The two-story recessed portico was inspired by the designs of Palladio; most American Georgian houses simply simulate such porticos with centered gables, as in the other centered gable examples. See the Early Classical Revival style for similar but much later buildings inspired by Palladio.\n\nCENTERED GABLE: GEORGIAN  \nPhiladelphia, Pennsylvania; 1763\u20131767. Cliveden. The front facade is of coursed ashlar; the stone urns on the roof are original.\n\nCENTERED GABLE: GEORGIAN  \nAnnapolis, Maryland; ca. 1774. Hammond House; William Buckland, architect. A high-style example in the Palladian five-part plan. The fanlight over the entrance was a feature found only occasionally in very late high-style Georgian houses; it became almost universal in the subsequent Federal style. Buckland owned James Gibbs's _Book of Architecture_ and Robert Morris's _Select Architecture,_ and both served as sources of house form and detail for this house.\n\nTOWN HOUSE\u2014The earliest surviving urban houses with narrow front facades and linear plans date from the Georgian period. These were originally built in all the pre-Revolutionary urban centers of the Atlantic Coast (see map), but only a few examples remain today, principally in Philadelphia and Boston, and in Alexandria, Virginia.\n\n**TOWN HOUSE**\n\nTOWN HOUSE: GEORGIAN  \nAlexandria, Virginia; ca. 1830s. Edward Sheehy, builder. Simple examples such as this continued to be built locally long after Federal style had become common.\n\nTOWN HOUSE: GEORGIAN  \nAlexandria, Virginia; 1786\u20131810. Gentry Row. An exceptional grouping of late Georgian town houses.\n\nTOWN HOUSE: GEORGIAN  \nBaltimore, Maryland; 1765. Robert Long House. Note the pent roof, a typical middle-colonies feature.\n\nTOWN HOUSE: GEORGIAN  \nPhiladelphia, Pennsylvania; 1765. Powell House. A high-style late Georgian town house. Note the stone belt courses, keystone lintels, and door surround with fanlight.\n\nTOWN HOUSE: GEORGIAN  \nPhiladelphia, Pennsylvania; mid-18th to early 19th centuries. Elfreth's Alley. These twenty-nine adjoining brick houses have been continuously occupied and have had relatively little exterior alteration. They make up one of our few urban streetscapes surviving from the 18th century.\n\n## Variants and Details\n\nThe structure and detailing of Georgian houses show distinct regional variations:\n\nNORTHERN COLONIES\u2014Wood-frame construction with shingle or clapboard walls and central chimneys dominated, as in the preceding Postmedieval English houses of the region.\n\nMIDDLE COLONIES\u2014Brick or stone construction dominated here. Some examples have details not found elsewhere, notably the pent roof separating the first and second floors, and the hooded front door, in which elements of the decorative crown project forward to form a small roof over the entryway.\n\nSOUTHERN COLONIES\u2014Brick was the dominant building material in surviving southern examples; red brick was most common. End chimneys continued to be common, as in Postmedieval English houses. Shapes were more varied in the South than elsewhere; dependencies were sometimes in separate connecting wings or detached from the main house in separate buildings. Some southern examples are raised off the ground on high foundations. On southern brick examples doors were sometimes accentuated only by changes in the surrounding brick pattern, rather than by an enframement of wooden pilasters and crown.\n\nPOST-1750, ALL COLONIES\u2014After 1750, a few well-documented examples have the entire door enframement extended forward to form an entrance porch. Most such porches are, however, post-Georgian innovations. Dormers and decorative quoins became common after 1750 in all colonies. In later brick examples the separation between floors is usually marked by a change in the masonry pattern (belt course). Still more elaborate detailing appears in some high-style examples after 1750. Among these are two-story pilasters, centered gables, and roof balustrades. A cupola projecting above the roof, while common on Georgian public buildings, is found on only a handful of surviving houses. Door and window detailing is discussed in the following chapter, on the closely related Federal style.\n\nGEORGIAN\n\n **REGIONAL VARIATIONS: Overview**\n\nGEORGIAN\n\nGEORGIAN\n\n ****\n\nGEORGIAN\n\n **ROOF SHAPE**\n\nGEORGIAN\n\n **LIGHTS**\n\nGEORGIAN\n\n **DOOR SURROUND VARIANTS** Similar entablatures, pediments, and broken pediments may be found atop Georgian windows and on Colonial Revival houses, both as door surrounds and window crowns\n\n[GEORGIAN\n\n **TYPICAL WINDOW**](Mcal_9780000000000_epub_c14_r1.htm#page223)\n\n[GEORGIAN\n\n **TYPICAL CORNICE**](Mcal_9780000000000_epub_c14_r1.htm#cornice)\n\nGEORGIAN\n\n **CHIMNEY PLACEMENT**\n\nGEORGIAN\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nGeorgian was the dominant style of the English colonies from 1700 to about 1780, when the population had grown to almost three million and covered the area shown on the map. In this area many thousands of Georgian houses survive today. Most have been lost from those colonial cities, such as Boston, New York, and Philadelphia, that grew rapidly in the 19th and 20th centuries. In sharp contrast are other colonial seaports (all the larger 18th-century towns had direct water communication with England; only villages occurred inland) that declined sharply in importance with the expansion of railroads in the 19th century. Examples are Portsmouth, New Hampshire; Newport, Rhode Island; New Castle, Delaware; Annapolis, Maryland; New Bern, North Carolina; and Charleston, South Carolina. Having had relatively little population growth since colonial times, these towns today preserve much of their Georgian heritage. In addition to the Georgian houses preserved in such coastal towns, many village and rural residences survive, particularly in New England. Landmark plantation houses are the principal southern survivors. With the end of the Revolution and independence (1781\u201383), the country began to develop new building styles (Federal and Early Classical Revival) based on changing European fashions. Although scattered Georgian houses were built for many decades after independence, even these usually showed some details of the newer styles.\n\nGEORGIAN\n\n **PRINCIPAL SETTLED AREAS**\n\n## Comments\n\nGeorgian is among the most long-lived styles of American building, having dominated the English colonies for most of the 18th century. The style grew from the Italian Renaissance, which emphasized classical details and reached remote England only in the mid-16th century. There, Renaissance classicism first flourished during the period 1650\u20131750 under such master architects as Inigo Jones, Christopher Wren, and James Gibbs. The style did not, however, begin to replace Postmedieval traditions in the American colonies until about 1700, when an expanding and increasingly prosperous population began to seek more fashionable buildings. The Georgian style was first brought to the New World by those interested in architecture, particularly carpenters and builders. They were well trained, and the forms they built relatively standardized, so the minimal amounts of added architectural detail were fairly simple to vary. Since only five architecture titles were available in this country before 1740, early Georgian style resulted in large part from the collective knowledge of its builders. After 1740, the number of architecture books increased rapidly and reached approximately one hundred titles by 1776. There was a similar rise in the documented instances of details and house forms copied from books. These influential volumes ranged from treatises stressing Italian models\u2014the same books from which fashionable British architects such as Inigo Jones (1573\u20131652), Christopher Wren (1632\u20131723), Colin Campbell (1676\u20131729), and William Kent (1685\u20131748) had received much of their inspiration\u2014to carpenter manuals showing how to construct fashionable doorways, cornices, windows, and mantels.\n\n# COLONIAL HOUSES\n\n# Federal\n\n# 1780\u20131820; locally to ca. 1840\n\n## Identifying Features\n\nSemi-circular or elliptical fanlight over front door (with or without sidelights); fanlight often incorporated into more elaborate door surround, which may include a decorative crown or small entry porch; cornice usually emphasized by decorative moldings, most commonly with tooth-like dentils; windows with double-hung sashes usually having six panes per sash and separated by thin wooden supports (muntins); windows aligned horizontally and vertically in symmetrical rows, usually five-ranked on front facade, less commonly three-ranked or seven-ranked; windows never in adjacent pairs, although three-part Palladian-style windows are common.\n\n## Principal Subtypes\n\nThe Federal house, like the preceding Georgian, is most commonly a simple box, two or more rooms deep, with doors and windows arranged in strict symmetry. More frequently than in Georgian houses, however, the box may be modified by projecting wings or attached dependencies; indeed, the style is perhaps best known for elaborate, but rather atypical, high-style examples having curved or polygonal projections to the side or rear. Five principal subtypes can be distinguished:\n\nSIDE-GABLED ROOF\u2014This is the most common Federal roof form, occurring in over 40 percent of surviving examples from all regions.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF: FEDERAL  \nProvidence, Rhode Island; ca. 1830. Seamans House.\n\nSIDE-GABLED ROOF: FEDERAL  \nSavannah, Georgia; 1808. William House. Note the exterior chimney, a common feature in the South.\n\nSIDE-GABLED ROOF: FEDERAL  \nPowhatan County, Virginia; early 19th century. Keswick. The simple three-ranked facade masks an elaborate H-shaped plan behind.\n\nSIDE-GABLED ROOF: FEDERAL  \nSomerset, Massachusetts; ca. 1800. Pettis House. This five-ranked wooden example has been elaborated with corner quoins. The slight overhang of the gable end is a holdover from Postmedieval building practice.\n\nSIDE-GABLED ROOF: FEDERAL  \nLongmeadow, Massachusetts; 1796. Colton House; attributed to Asher Benjamin, architect. Note the elaborate cornice-line railing and Palladian window.\n\nSIDE-GABLED ROOF: FEDERAL  \nAlexandria, Virginia; ca. 1798. Lloyd House. An elaborate brick example with paired end chimneys, keystone lintels, and arched dormer windows.\n\nSIDE-GABLED ROOF: FEDERAL  \nNew Castle, Delaware; 1801. Read House. A high-style example with roof balustrade, cornice-line modillions and dentils, keystone lintels, and elaborate Palladian window and door surround.\n\nHIPPED ROOF, TWO-STORY\u2014Hipped roofs of moderate to very low pitch (the latter may appear to be almost flat) are particularly common in New England, where they slightly outnumber side-gabled examples.\n\n**HIPPED ROOF,TWO-STORY**\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nAuburn, Massachusetts; late 18th century. Chapin House. A simple wood-frame example. The 2\/2 windows are later additions.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nColumbia Falls, Maine; ca. 1818. Ruggles House; Aaron Sherman, architect. Note the flush, horizontal boards on the front facade and the swagged window heads.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nMappsville, Virginia, vicinity; ca. 1800. Wharton Place. The interior chimneys are separated by a flat roof deck with balustrade.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nProvidence, Rhode Island; ca. 1815. Burroughs House. Note the quoins. There are two balustrades on the roof.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nProvidence, Rhode Island; ca. 1801. Halsey House. The curved bays were added ca. 1825.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nDamariscotta Mills, Maine; 1803. Kavanaugh House; Nicholas Codd, architect. This example has flush wood sheathing, a semi-circular entry porch, and a large octagonal cupola.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nGreenfield, Massachusetts; 1796. Coleman House; Asher Benjamin, architect. Note the elaborate pilasters, Palladian window, and inset panels with swags.\n\nHIPPED ROOF, TWO-STORY: FEDERAL  \nWashington, District of Columbia; 1815. Tudor Place; William Thornton, architect. The garden front (shown) has a domed portico and full-length three-part windows in blind arches. The portico forms a circular space extending half inside and half outside the plane of the front facade. Thornton, like most architects at that time, gained design inspiration from books. This circular space was no exception; it appeared six times in George Richardson's _The New Vitruvius,_ volumes I and II, both in Thornton's library.\n\nHIPPED ROOF, THREE-STORY\u2014The three-story hipped-roof Federal house is usually large and of landmark quality; it survives primarily in New England, with an unusually important concentration in the town of Salem, Massachusetts.\n\n**HIPPED ROOF,THREE-STORY**\n\nHIPPED ROOF, THREE-STORY: FEDERAL  \nPortsmouth, New Hampshire; early 19th century. Barnes House. A wood-frame example with a minimum of exterior elaboration.\n\nHIPPED ROOF, THREE-STORY: FEDERAL  \nWashington, District of Columbia; 1819. Decatur House; Benjamin Latrobe, architect. A three-ranked brick example with flat lintels over the windows. The window height varies in the three stories.\n\nHIPPED ROOF, THREE-STORY: FEDERAL  \nBoston, Massachusetts; 1797. Otis House; attributed to Charles Bulfinch, architect. A high-style brick example with belt courses and keystone lintels. The Palladian and semi-circular windows align above the elaborate doorway.\n\nHIPPED ROOF, THREE-STORY: FEDERAL  \nSalem, Massachusetts; 1821. Phillips House. Note the quoins, Palladian window, and elaborated window crowns.\n\nHIPPED ROOF, THREE-STORY: FEDERAL  \nProvidence, Rhode Island; 1806. Ives House; Caleb Ormsbee, architect. This example has roof-line and entry porch balustrades; there are matching fanlights on the first and second stories.\n\nCENTERED GABLE\u2014Less than 10 percent of surviving Federal houses have gables (pediments) centered on the front facade. The facade beneath the gable may either remain in the same plane as the rest of the wall or be extended slightly forward for emphasis as a pavilion.\n\n**CENTERED GABLE**\n\nCENTERED GABLE: FEDERAL  \nBaltimore, Maryland; ca. 1782. D'Annemours House. A three-ranked masonry example of a simple I-house plan (one room deep) with a rear wing.\n\nCENTERED GABLE: FEDERAL  \nCanterbury, Connecticut; ca. 1805. Payne House. This wood-frame example has notable pilasters set on pedestals at the corners and beside the slightly projecting center gable.\n\nCENTERED GABLE: FEDERAL  \nFrankfort, Kentucky; 1800. Liberty Hall. The centered gable covers three ranks of windows; gables of this width are frequent in high-style Federal houses but unusual in the preceding Georgian style.\n\nCENTERED GABLE: FEDERAL  \nSavannah, Georgia; 1819. Richardson House; William Jay, architect. Note the undulating entry porch roof and curved entry steps. The front door is also recessed into a curved niche.\n\nCENTERED GABLE: FEDERAL  \nCharleston, South Carolina; ca. 1822. Bennett House. Built in the typical Charleston-house form with the narrow end turned to the street. The principal facade, with a full-length porch, faces a side garden. Entry from the street is through a doorway leading to the porch.\n\nCENTERED GABLE: FEDERAL  \nCazenovia, New York; 1807. Lorenzo House. The facade is elaborated by pilasters supporting decorative arches.\n\nCENTERED GABLE: FEDERAL  \nClarkston, New York; ca. 1825. Palmer House. Full front-gabled Federal houses like this are uncommon; they are transitional to the Greek Revival style, which popularized the front-gabled form. Most Federal examples occur in western New York and in Ohio.\n\nCENTERED GABLE: FEDERAL  \nMount Vernon, Virginia, vicinity; 1805. Woodlawn; William Thornton, architect. In this five-part compound plan, a large central house block is connected to two side dependencies by hyphens. This house has no projection or elaboration below the centered gable.\n\nTOWN HOUSE\u2014Many Federal town houses survive; these include both attached row houses and narrow detached urban houses. Important concentrations remain in Boston; Philadelphia; Georgetown, District of Columbia; and Alexandria, Virginia. A few projects were built that treated individual houses as part of a larger unit, thus rendering the entire facade as a total composition rather than a collection of individual elements. Regrettably, only a few parts and pieces of these ambitious schemes survive.\n\n**TOWN HOUSE**\n\nTOWN HOUSE: FEDERAL  \nEaston, Maryland; 1820\u20131821. Attached urban houses.\n\nTOWN HOUSE: FEDERAL  \nSavannah, Georgia; ca. 1820. Clark Houses. The paired doors on the left entrance are a later addition (note the glass panels).\n\nTOWN HOUSE: FEDERAL  \nFrederick, Maryland; 1799. Taney House. Note the side walls without windows to increase privacy in an urban setting. The 6\/6 windows with lintels above and the curved patterning in the rectangular light over the door mark this example as a Federal house.\n\nTOWN HOUSE: FEDERAL  \nLibertytown, Maryland; ca. 1800. Jones House. Note the parapeted gables with double interior end chimneys.\n\nTOWN HOUSE: FEDERAL  \nCharleston, South Carolina; ca. 1809. Russell House. An unusual high-style example of compact town house form. Note the polygonal projection visible to the left of the house, the roof-line balustrade of alternating panels and balusters, and the full-width iron balcony and blind arches at second-floor level.\n\n## Variants and Details\n\nAs with the Georgian and Postmedieval English styles, northern house builders continued to show a preference for frame construction with clapboard siding, and southern for brick construction. Stucco and stone occur infrequently in all regions. Smooth wooden siding was sometimes used for the front facade with weatherboards, or even bricks, used for the less conspicuous walls. Chimney placement is less predictable than in Georgian houses, probably as a result of interiors with more complex room arrangements. Nevertheless, central or interior chimneys still tend to dominate in the North, while end chimneys are most common in the South.\n\nThe exteriors of most Federal houses have few elaborations other than the fanlight and accentuated front door (which often includes an entry porch). Among the elaborations that sometimes occur are roof-line balustrades, particularly favored in the North; the use of a Palladian-style window in the second story over the main entrance; the use of flat or keystone lintels above the windows with prominent sills below. Elliptical, half-circular, or Palladian windows are sometimes used in side or front gables; dormers typically have arched windows; in brick examples, windows may be slightly recessed into arches built into the facade. A number of decorative details from the Georgian period continue to be found occasionally: quoins, two-story pilasters (which disappeared about 1800), belt courses (now sometimes in stone), and dentils. Typical Georgian dentils are usually supplanted by less blocky, more refined versions called modillions.\n\nHouses of the Federal style are often characterized as having a lightness and delicacy in comparison with their close Georgian relatives. This generalization needs to be interpreted with care, however, for while the scale is smaller in many Federal _details_ (moldings, columns, etc.), the scale of many _structural parts_ (windows, ceiling heights, etc.) is enlarged.\n\nIn attached Federal town houses (row houses) such typical features as roof-line balustrades, Palladian windows, and entry porches are rare. Although the doorway remains the most important identifying feature, fanlights are a less consistent guide than in detached houses. Not only do fanlights occur in both late Georgian and early Greek Revival row houses, but they are often omitted in the simpler and earlier Federal examples. Front stair rails of iron were usual; iron balconies and curved front bays were particularly common in Boston.\n\nThe interiors of many Federal houses contain graceful decorative ornament, either carved in wood or cast in plaster, applied to mantels, walls, ceilings, and elsewhere. Less commonly, the external facade shows similar decorative detailing on door surrounds or entry porches, over windows, along the cornice, or in paneled wall insets. Typical decorative motifs include swags, garlands, urns, and classic geometric patterns (most commonly elliptical, circular, or fan-like shapes formed by fluted radiating lines).\n\nFEDERAL AND GEORGIAN WINDOWS\u2014Both Federal and Georgian houses have double-hung sash windows placed singly but in symmetrical rows. These windows have fixed upper sashes and movable lower sashes, the latter held open by metal pins (the familiar system of counterbalancing with weights had not yet been invented). Such windows first began to be used in the English colonies in about 1700, and by 1720 had almost completely supplanted the earlier casement-style window. Facilitating this transition was an increasing availability of larger panes of glass. Prior to the Revolution, the standard size of these panes was approximately 6 inches by 8 inches; afterward, the size increased to approximately 8 inches by 12 inches. Georgian houses thus generally have smaller windowpanes than do Federal houses. In Georgian houses these are most frequently arranged with 12 panes in each sash (12\/12) in the northern colonies and 9 panes (9\/9) in the southern colonies; 9\/6, 6\/9, 8\/12, and other combinations are also found occasionally. Federal houses, with their larger panes, most commonly have 6\/6 windows, although the earlier types also persist.\n\nThe wooden supporting moldings (or muntins) which hold the individual panes in place also differ in Georgian and Federal houses. In early Georgian houses these tend to be 1\u00bc inches wide and quite shallow; in later Georgian houses they remain shallow but are usually 1 inch wide; by Federal time they are more likely to be quite deep and narrower than 1 inch.\n\nBoth windowpanes and muntins are easily modified and thus many early houses now have 6\/6, 1\/1, 2\/2, or other patterns of glazing. Occasionally the earlier windows will be left on rear or side walls where they reveal the original pattern. As a further complication, some houses were originally built with up-to-date windows on the front facades and with older, less fashionable (and probably less expensive) types on the rear.\n\nWindows in Georgian and Federal wooden houses sometimes have elaborate decorative crowns placed above them. In Georgian houses these might be either a formal pediment, usually in one of the same patterns seen over front doors, or a decorative molding similar to those found on Georgian cornices. Federal windows more commonly have an elaborately decorated frieze above the window. These are sometimes topped by a cornice mold, but full-scale pediments are uncommon (pedimented or otherwise elaborated windows are most common on later, Colonial Revival houses).\n\nDecorative window crowns are far less usual on masonry houses. Such Georgian houses often have changed brick patterns, or simple arches, above the windows. Federal masonry houses commonly have a flat lintel, keystone lintel, or keystone without a lintel set over the windows. These are usually of stone; stone sills are also sometimes used beneath the windows of Federal houses. Both Georgian and early Federal windows set in masonry are generally surrounded by a wooden frame (or architrave); these are usually omitted from Federal houses built after about 1800.\n\nFEDERAL\n\n **DOOR SURROUND & ENTRY PORCH VARIANTS**\n\nFEDERAL\n\n **TYPICAL CORNICES: GEORGIAN & FEDERAL**\n\nA - shallow-molded, unadorned cornice, very common on Georgian and Federal houses\n\nB,C - typical Georgian cornices with modillions and \/or dentils, also found on Federal houses\n\nD,E,F,G,H,I - typical decorative Federal cornices\n\nFEDERAL\n\n **WINDOW COMPARISONS:GEORGIAN & FEDERAL**\n\nFEDERAL\n\n **MUNTINS: GEORGIAN& FEDERAL**\n\nFEDERAL\n\n **PALLADIAN WINDOW VARIANTS commonly located above the front door, occasionally in the gable end of a side-gabled roof**\n\nCLASSIC PALLADIAN WINDOWS\n\nFEDERAL\n\n **PALLADIAN WINDOW VARIANTS commonly located above the front door, occasionally in the gable end of a side-gabled roof**\n\nMODIFIED PALLADIAN WINDOWS\n\nFEDERAL\n\n **PALLADIAN WINDOW VARIANTS commonly located above the front door, occasionally in the gable end of a side-gabled roof**\n\nTHREE-PART WINDOWS\n\nFEDERAL\n\n **TYPICAL DECORATIVE DETAILS**\n\nFEDERAL\n\n **HIGH-STYLE ELABORATIONS**\n\n## Occurrence\n\nFederal was the dominant style of the new United States from about 1780 to 1820, a period in which the population grew from 3 million to about 10 million and expanded to cover the area shown on the map. The style reached its zenith in the prosperous port cities of the eastern seaboard, particularly Boston, Salem, Newburyport, and Marblehead in Massachusetts; Newport, Providence, Warren, and Bristol in Rhode Island; Portland and Wiscasset in Maine; New Castle, Delaware; Portsmouth, New Hampshire; Philadelphia, Pennsylvania; New York, New York; Charleston, South Carolina; and Savannah, Georgia. Alexandria, Virginia, and Georgetown, District of Columbia, both near the newly developing national capital, also prospered during this period. High-style Federal houses are mostly concentrated in these areas, although scattered examples occur elsewhere. Thousands of vernacular examples survive throughout the settled areas; they are least common at the westward edges of expansion, where vernacular Georgian houses persisted throughout the period (occasional Georgian hangovers, some of landmark quality, occur in all regions). By the 1820s a more strictly classical style, the Greek Revival, was supplanting the Federal style. Still earlier, by about 1800, the related Early Classical Revival style was replacing Federal houses in the South.\n\nFEDERAL\n\n **PRINCIPAL SETTLED AREAS 1820**\n\n## Comments\n\nThe Federal style was a development and refinement of the preceding Georgian style. Established first by wealthy merchants along the New England seaboard, it drew on contemporary European trends, particularly the work of the Adam brothers who, at that time, had the largest architectural practice in Britain. The eldest, Robert, had traveled to Italy and the Mediterranean to study classical buildings for himself. These studies, as well as those of others who reported on first-hand viewing, introduced a new interest in the early Greek and Roman monuments themselves, rather than as interpreted through the buildings of the Italian Renaissance. Adam popularized a number of design elements (swags, garlands, urns, and various stylized geometric designs) that he had seen in his travels. He also incorporated into his interiors a diversity of spatial planning found in some classical ruins. Because of the breadth of his influence the Federal style is considered the Adam style by some American architectural historians. Among the many architectural books that helped spread the Federal style was the Adam brothers' _Works in Architecture of Robert and James Adam_ (1779) and Asher Benjamin's _American Builder's Companion_ (Boston, 1806), which was so popular it was reprinted five times.\n\nIt was during this era that the first true architects appeared on the American scene. Among the most notable of these, with their principal areas of work, were: Charles Bulfinch (Boston); William Jay (Savannah, Georgia); Benjamin H. Latrobe (his early work in Philadelphia and Virginia); Gabriel Manigault (Charleston, South Carolina); John McComb (New York); Samuel McIntire (Salem, Massachusetts); and Alexander Parris (Maine).\n\n# COLONIAL HOUSES\n\n# Early Classical Revival\n\n# 1770\u20131830; locally to ca. 1850\n\n## Identifying Features\n\nEntry porch (portico) dominating the front facade and normally equaling it in height; porch roof usually supported by four simple columns (Roman Doric or Tuscan types) each with a shallow square base (plinth); the columns support a prominent centered gable; a semi-circular or elliptical fanlight normally occurs above the paneled front door; windows aligned horizontally and vertically in symmetrical rows, usually five-ranked on front facade, less commonly three-ranked or seven-ranked.\n\n## Principal Subtypes\n\nEarly Classical Revival houses are of three principal types:\n\nTWO-STORY\u2014Similar to one-story type, but with more imposing, two-story facades and entry porches.\n\n**TWO-STORY**\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nBeaufort, South Carolina; 1786. Tabby Manse. Built of tabby, a concrete-like mixture of oyster shell and lime mortar, covered with stucco. Note the delicate two-tiered entry porch with more slender columns above.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nBerryville, Virginia; 1790. Annefield. Built of stone, this house has a two-tiered entry porch with a fanlight in the pediment. Note the upper balustrade formed in a Chinese Chippendale pattern.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nMilledgeville, Georgia, vicinity; 1830. Boykin Hall; Daniel Pratt, architect. The relatively simple exterior contrasts with elaborate Federal detailing in the interior. The spindlework balustrade is probably a later addition.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nConetoe, North Carolina, vicinity; ca. 1820. Wilkinson House. A simple three-ranked wood-frame example with the exterior chimneys typical of the South. Note the lunette in the pediment; the porch balustrades are probably later additions.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nNatchez, Mississippi, vicinity; 1812. Auburn; Levi Weeks, architect. Weeks built this impressive home for Lyman Harding and described it as \"the first house in the territory on which was ever attempted any of the orders of [classical] architecture.\" The twin side wings were added after 1827.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nCharleston, South Carolina; ca. 1816. Nicholson House; attributed to William Jay, architect. A unique example on a high, rusticated foundation. Note the unusual Gothic (pointed-arch) windows in the pediment, which contrast with the round-arch windows of the second story.\n\nTWO-STORY: EARLY CLASSICAL REVIVAL  \nWarsaw, Virginia; ca. 1730 (entry porch ca. 1830). Sabine Hall. A well-documented case of a full-height entry porch with colossal columns added to an earlier (1730) house in an extensive exterior remodeling, which included lowering the roof.\n\nONE-STORY\u2014These are simple, rectilinear houses with side-gabled or low-pitched hipped roofs having the characteristic full-height entry porch. They are commonly built several feet off the ground on tall foundations, which exaggerates the height of the front facade.\n\n**ONE-STORY**\n\nONE-STORY: EARLY CLASSICAL REVIVAL  \nStrasburg, Virginia; 1794. Belle Grove. Built of coursed ashlar limestone on a high basement.\n\nONE-STORY: EARLY CLASSICAL REVIVAL  \nBaltimore, Maryland; 1803. Homewood. Note the five-part compound plan with a large central block connected to distant wings by lower \"hyphens.\" Decorative detailing in the gable is atypical.\n\nONE-STORY: EARLY CLASSICAL REVIVAL  \nStaunton, Virginia, vicinity; ca. 1818. Folly. Note the matching entry porch on the side facade.\n\nONE-STORY: EARLY CLASSICAL REVIVAL  \nCharlottesville, Virginia, 1770\u20131809. Monticello; Thomas Jefferson, architect. This view is of the garden facade and is dominated by an octagonal dome. There are two different styles of railing on the roof: a Chinese Chippendale railing is around the higher roof deck, and a cornice-line balustrade surrounds the main roof.\n\nONE-STORY: EARLY CLASSICAL REVIVAL  \nBremo Bluff, Virginia, vicinity; ca. 1819. Bremo. A unique house with entry porches on each facade and dependencies connected by raised terraces.\n\nGABLE FRONT AND WINGS\u2014Most Early Classical Revival houses are of the first two types with side-gabled or hipped roofs. In this less common variant, a two-story, front-gabled central block dominates the facade; this two-story unit is flanked by one-story wings on either side, making a three-part composition. This plan was introduced by Palladio in his 16th-century Italian pattern book and is called the Palladian three-part plan. In this subtype the dominant central block may have either a full-height, two-story entry porch or a smaller one-story entry porch.\n\n**GABLE FRONT AND WINGS**\n\nGABLE FRONT AND WINGS: EARLY CLASSICAL REVIVAL  \nHalifax County, North Carolina; early 19th century. A small example, with careful detailing, built in the typical three-part plan.\n\nGABLE FRONT AND WINGS: EARLY CLASSICAL REVIVAL  \nWilliamsburg, Virginia; ca. 1775. Semple House; attributed to Thomas Jefferson, architect. One of the earliest examples of the three-part plan, this house survives essentially as originally built.\n\nGABLE FRONT AND WINGS: EARLY CLASSICAL REVIVAL  \nFayette, Missouri; 1833. Morrison House. Note the Palladian window and keystone lintels.\n\nGABLE FRONT AND WINGS: EARLY CLASSICAL REVIVAL  \nLexington, Virginia; 1818. Stono. Note the front-gabled roof extended into a temple-form porch.\n\nGABLE FRONT AND WINGS: EARLY CLASSICAL REVIVAL  \nLouisville, Kentucky; ca. 1805\u20131808. Spring Station; Norborne Beale, builder and original owner. The porch on the right and porte cochere on the left are later additions. The columns on the porch have also been changed; the originals would have been thinner and more delicate.\n\n## Variants and Details\n\nEarly Classical Revival houses closely resemble those of the succeeding Greek Revival period; the doorway, cornice line, and type of column are the three principal distinguishing features. In Early Classical Revival houses, the columns were generally of the Roman type (see this page and this page), although later transitional examples may have columns with Greek details. The Early Classical Revival house also usually lacks the wide band of trim at the cornice line seen on most Greek Revival houses; frequently a narrow line of dentils or modillions adorns the cornice. Like their Federal-style contemporaries, most Early Classical Revival examples have a prominent fanlight over the front door, a feature that became very rare during the subsequent Greek Revival. In addition, Early Classical Revival houses are likely to have Federal interior detailing.\n\nThe characteristic Early Classical entry porch shows considerable variation in detail. The porches may differ in: (1) the number of columns: four is most common, two is frequent, and five, six, eight, and ten occur less often; (2) the spacing of columns: even spacing is most common, but a frequent variation has a wider space between the two central columns framing the front entrance; (3) the treatment of columns and second-story porches on two-story examples: four variations occur in about equal overall abundance. The porches are most commonly placed only on the front facade as entry porches but may also appear both on the front and the back, or on any combination of front, back, and side. Occasionally the entry porch is recessed inward (called a portico _in antis_ ). In all variants the centered gable (pediment) may be embellished with a semi-circular window (lunette); occasionally a round or oval window replaces the lunette or is found elsewhere on the facade. The entablature, or horizontal band above the columns and below the centered gable, is most often plain. If elaborated, it is with triglyphs, three closely spaced vertical lines repeated at intervals.\n\nWall materials may be either wood, brick, stucco, or stone, in order of decreasing frequency. Wall projections, which are only occasionally present, are never curved as in some contemporaneous Federal houses. Roof balustrades are rare, as are original dormers; when present, these are usually later additions. In most other details, Early Classical Revival houses resemble those of the contemporaneous Federal style (see Federal doors and windows). Many Georgian and early Federal houses, as well as simple folk houses, were updated during the early 19th century by the addition of a full-height Early Classical Revival entry porch. These modified examples may be difficult to distinguish from homes originally built in this configuration.\n\nEARLY CLASSICAL REVIVAL\n\n **COMMON PORTICO VARIATIONS**\n\n**EARLY CLASSICAL REVIVAL\n\nROMAN COLUMNS** (see also Column Types)\n\nEARLY CLASSICAL REVIVAL\n\n **HIGH-STYLE ELABORATIONS**\n\n## Occurrence\n\nThis is a relatively uncommon style found in isolated examples throughout the areas settled by 1820 (see map). It is rare north of Pennsylvania; most examples occur in the southern states, particularly Virginia, where it had its most vocal champion in Thomas Jefferson. A handful of houses in the style were built in Virginia just before the Revolution, but most examples were constructed between 1790 and 1830. By 1830 the subsequent and more universally popular phase of classical revivalism, the Greek Revival, had replaced Early Classical models even in Virginia.\n\n## Comments\n\nFollowing the Revolution there was an immediate need for public buildings to house the newly organized government at both the state and national levels. It was natural to have taken Rome as a model, with its Republican ideas and monumental architecture, a choice that symbolized the mood and politics of the new country. Roman Revival architecture thus became fashionable. This was further emphasized by a concurrent neoclassical movement in France, the new country's principal ally in its fight for independence. The United States was not a mere follower in this movement, but led the way by erecting the first large public building in the new style, the Bank of Pennsylvania, completed in 1800 at Philadelphia. Among the prominent architects working in the style were William Jay (Savannah, Georgia); Benjamin H. Latrobe (Philadelphia, Baltimore, and Washington); Robert Mills (Charleston, South Carolina, and elsewhere); and William Thornton (Washington). Most influential of all, however, was Thomas Jefferson, who not only designed Roman Revival buildings himself, but used the influence of his political office, and his considerable powers of personal persuasion, to push the United States toward his classical ideal. Jefferson thus shaped early Washington, D.C., the Virginia capitol at Richmond, and, almost single-handedly, the University of Virginia at Charlottesville. Because of his influence, the style is sometimes referred to as Jeffersonian Classicism.\n\nApart from Jefferson's influence on public architecture, his home, Monticello, his summer home, Poplar Forest, and other houses he designed for friends set the stage for Early Classical domestic architecture. These high-style examples probably sprang from Jefferson's familiarity with the writings of the Italian Renaissance architect Palladio (1508\u20131580). Inspired by such high-style landmarks, typical Palladian entry porches were soon being built throughout the South, some at the time of original construction, but many also as additions to earlier houses.\n\nA national sympathy for the Greek War of Independence (1821\u20131830) and an increasing archeological understanding of the Greek roots of Roman architecture and culture made the years 1820\u20131830 a transition to the succeeding Greek Revival style. Although relatively uncommon and primarily southern, the Early Classical Revival movement provided the background for this more pervasive classicism which dominated the new country for the next thirty years.\n\n# Romantic Houses\n\n1820\u20131880\n\n* * *\n\n**Greek Revival**\n\n**Gothic Revival**\n\n**Italianate**\n\n**Exotic Revivals**\n\n**Octagon**\n\nDuring the preceding Colonial era, a single architectural style tended to dominate in each colony for long periods of time; Georgian houses, for example, were the fashion in the English colonies through most of the 18th century. Likewise, the first popular Romantic style, Greek Revival, dominated the newly independent United States through much of the first half of the 19th century. Architectural models evocative of Greek democracy were thought to be especially appropriate in the new republic, as it rejected traditional ties to England in the decades following the War of 1812.\n\nBy the 1840s, a new trend toward competition among _several_ acceptable architectural fashions was taking shape. The harbinger of this movement was the publication in 1842 of the first popular pattern book of house styles with full-facade drawings\u2014Andrew Jackson Downing's _Cottage Residences._ Downing illustrated several new fashions he considered suitable alternatives to the prevailing Greek classicism. Medieval precedents were recommended in models that were to lead to the Gothic Revival style. Likewise, Italian Renaissance traditions were freely adapted in Downing's Italianate cottages. Now, for the first time, builders and home buyers had a choice. Soon, neighborhoods of alternately Greek, Gothic, and Italianate houses became commonplace. Still more exotic fashions, based on Egyptian and Oriental precedents and on Swiss Chalets or octagonal shapes, also came to be advocated by Downing and others, but these never achieved wide acceptance. The simultaneous popularity of several architectural styles with differing antecedents was to persist as a dominant theme throughout the later history of American housing.\n\nArchitecture books were the primary means by which a carpenter increased his knowledge sufficiently to become a designer or architect. The architects who produced homes in the Romantic styles were almost all self-trained. Among them were Alexander Parris, Ithiel Town, Minard Lafever, Henry Austin, Samuel Sloan, Richard Upjohn, Thomas U. Walter, A. J. Davis, and Alexander Downing\u2014several of whom wrote the pattern books that broadly influenced Greek Revival, Gothic Revival, and Italianate homes. Indeed, Benjamin Henry Latrobe was the only formally trained architect in the United States in 1840.\n\nAll the Romantic styles originated and grew to popularity in the decades before 1860. The Greek Revival was dominant from about 1830 to 1850 (to 1860 in the South) and the Italianate from about 1850 until 1875. Gothic Revival houses were more complex to construct and were always less common than their Greek and Italian contemporaries. The Civil War marked the end of Greek Classicism, but both Gothic and Italianate houses remained popular into the 1880s, sometimes in more elaborate versions than had appeared before the war. These later examples have been separated as High Victorian Gothic or High Victorian Italianate styles, but this distinction is difficult to recognize in field identification, particularly in houses. Highly detailed early examples are not uncommon, nor are late survivors of the earlier, less elaborate interpretations.\n\n# ROMANTIC HOUSES\n\n# Greek Revival\n\n# 1825\u20131860\n\n## Identifying Features\n\nGabled or hipped roof of low pitch; cornice line of main roof and porch roofs emphasized with wide band of trim (this represents the classical entablature and is usually divided into two parts: the frieze above and architrave below); most have porches (either entry or full-width) supported by prominent square or rounded columns, typically of Doric style; front door surrounded by narrow sidelights and a rectangular line of transom lights above, door and lights usually incorporated into more elaborate door surround.\n\n## Principal Subtypes\n\nSix principal subtypes can be distinguished on the basis of porch and roof configurations:\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT\u2014About 20 percent of Greek Revival houses have small entry porches which do not extend the full height of the facade. In some examples the entry porch is recessed _into_ the facade. About 5 percent lack porches altogether.\n\n**ENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT**\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nStafford, New York; 1835. Harmon House. A small wood-clad example with nicely detailed tripartite and frieze-band windows.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nBeaufort, North Carolina; 1866. Croft House.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nMarshall, Michigan; 1850. Montgomery House. This transitional example combines Italianate brackets under a wide eave overhang with Greek Revival door and window detailing.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nSan Augustine, Texas; 1839. Cartwright House; August Phelps, architect. Note how the very deep band of trim forms a pediment on the gable end.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nAshville, New York; 1835. Smith-Bly House. This lavishly detailed house has four Ionic two-story pilasters across the front facade, carved window lintels, and an architrave at the entrance. The door is recessed with two small columns in front.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nNew London, Connecticut; mid-19th century. Note the small windows in the frieze band and the unusual Corinthian capitals on the pilasters and columns.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nRochester, New York; 1838\u201341. Woodside; Alfred M. Badger, architect-builder. This large five-ranked masonry example has a square cupola, topped by a round turret, which lights an interior stairway. Note the matching balusters at four different levels and the tripartite window lintels.\n\nENTRY PORCH LESS THAN FULL HEIGHT, OR ABSENT: GREEK REVIVAL  \nNew Castle, Indiana; 1847. Murphy House. A simplified but nicely proportioned masonry example. Note the square columns on the entry porch and the absence of capitals on both columns and pilasters.\n\nFULL-HEIGHT ENTRY PORCH\u2014This subtype has a dominant central porch extending the full _height,_ but less than the full _width,_ of the facade; it thus resembles the Early Classical Revival style from which the Greek Revival sprang. The Greek Revival version can usually be distinguished from its predecessor by the typical band of cornice trim and the rectangular lights, rather than a curving fanlight, over the entrance. As in the earlier style, many Greek Revival examples have a traditional classical pediment above the entry porch. In contrast to the earlier style, however, many Greek examples have flat-roofed entry porches. As in the entry porch less than full height, this type of entry porch also occurs recessed _into_ the facade. About one-fourth of Greek Revival houses are of this subtype; like Early Classical Revival houses, these are most common in the southern states.\n\n**FULL-HEIGHT ENTRY PORCH**\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nSan Felipe, Texas; ca. 1838. Lambart House. A very simple one-story example. Note the very slender columns and pilasters. Such simplified details were usual in houses built far from centers of population.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nBastrop, Texas; ca. 1860. Reding House. A simple wood-clad two-story example.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nMeriwether County, Georgia; 1852. Dr. James Stinson House, Mark Hall. This house, similar to the Lambart House, has heavier moldings and more substantial columns and pilasters.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nPittsford, New York; 1840. Kirby House. The entry porch without a pediment or hipped roof above is unusual. Note the Doric columns without bases.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nScott County, Kentucky; 1842. Glencrest. Note the exaggerated depth of the frieze and architrave over the entry porch.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nMilledgeville, Georgia; 1838. Executive Mansion; Charles B. Clusky, architect-designer.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nNatchez, Mississippi, vicinity; ca. 1855. Homewood Plantation.\n\nFULL-HEIGHT ENTRY PORCH: GREEK REVIVAL  \nBelfast, Maine; 1840. White House; Calvin A. Ryder, architect. This full-height entry porch is a complex and subtle variation of the usual type. Note the elaborate cupola.\n\nFULL-FACADE PORCH\u2014In this configuration, the colonnaded porch occupies the full width and height of the facade. No pediment occurs above the porch, which is covered either by the main roof or, less commonly, by a flat or shed-style extension from it. In a few examples, the full-facade porch also extends around one or both sides of the house. This subtype makes up about one-fourth of Greek Revival houses. Like the preceding type, it is most common in the southern states.\n\n**FULL-FACADE PORCH**\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nMadison, Georgia; ca. 1859. Massey House. A simple one-story, wood-clad example.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nMobile, Alabama; 1851. Lowary House. This one-story cottage is more elaborated than the Massey House. Note the pedimented dormers and full-length windows with transoms.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nColumbus, Mississippi; ca. 1836. Homewood.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nSparta vicinity, Georgia; ca. 1853. Smith House (Glen Mary). The main part is raised above a high masonry basement story, a pattern borrowed from earlier French Colonial houses of the rural South.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nGeismar, Louisiana; 1841. Ashland; James Gallier, Sr., architect. The columns of this plantation house are four feet square and thirty feet high. The massive entablature hides a low-hipped roof.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nNew Iberia, Louisiana; 1834. Shadows-on-the-Teche. This handsome plantation house has a stairway exterior to the house and no interior halls, reflecting earlier French Colonial influences.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nSelma, Alabama; 1853. Sturdivant Hall; Thomas Helm Lee, architect.\n\nFULL-FACADE PORCH: GREEK REVIVAL  \nNashville, Tennessee; 1853. Belle Meade. Note the parapet with cresting crowning the full-facade porch. The cresting on the top of the roof parapet is modeled on a honeysuckle leaf and called an anthemion. More typically found over windows and doors, it was illustrated in Villa No. III in _The Architectural Instructor_ by Minard Lafever.\n\nFRONT-GABLED ROOF\u2014All of the preceding subtypes have side-gabled or hipped roofs. In this subtype the gable end is turned 90 degrees to make the principal facade. In some high-style examples a full-width, colonnaded porch is present beneath the front gable, giving the house the appearance of a miniature Greek temple with its traditional classical pediment, a variation sometimes called temple-form. Smaller entry porches are common on vernacular examples. This subtype is more common in the northeastern and midwestern states.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  GREEK REVIVAL  \nBreckinridge, Colorado; mid-19th century. A very simple house with just a hint of Greek Revival influence in its front-gabled form and pedimented door and windows.\n\nFRONT-GABLED ROOF:  GREEK REVIVAL  \nMarietta, Georgia; ca. 1851. Brumbly House.\n\nFRONT-GABLED ROOF:  GREEK REVIVAL  \nProvidence, Rhode Island; mid-19th century. Front-gabled examples commonly lack colonnaded porches but usually have pilasters and elaborate door and cornice details as in this example.\n\nFRONT-GABLED ROOF:  GREEK REVIVAL  \nMiddletown, Connecticut; 1828. Russell House; Ithiel Town, architect. This house is built of masonry covered with stucco scored to look like stone. Note the elaborate Corinthian capitals.\n\nFRONT-GABLED ROOF:  GREEK REVIVAL  \nAndalusia, Pennsylvania; 1836 (porch and pediment). Andalusia. An earlier house to which a three-sided porch was added.\n\nGABLE FRONT AND WING\u2014In this subtype a front-gabled roof, as in the type just described, has a side wing (less commonly two wings) added; these are typically lower than the dominant front-gabled portion. This subtype rarely occurs outside of the northeastern states and is particularly common in western New York and Ohio.\n\n**GABLE FRONT AND WING**\n\nGABLE FRONT AND WING:  GREEK REVIVAL  \nMonroeville, Ohio; mid-19th century. This example, although quite small, has a colonnaded gable front and elaborate frieze-band windows.\n\nGABLE FRONT AND WING:  GREEK REVIVAL  \nWellington vicinity, Ohio; mid-19th century. Here the frieze band is discontinuous across the gable front.\n\nGABLE FRONT AND WING:  GREEK REVIVAL  \nBuffalo, New York; mid-19th century. This imposing example has pilasters across the gable front and a two-story wing.\n\nGABLE FRONT AND WING:  GREEK REVIVAL  \nJessamine County, Kentucky; mid-19th century. Bryant House. This brick example has a colonnaded gable front and two wings. Note that the wings have pilasters rather than the more common porch with columns as seen in the other examples.\n\nTOWN HOUSE\u2014A sixth subtype consists of narrow urban houses with Greek Revival detailing. These occur both with and without porches. They are most common in those port cities of the Atlantic and Gulf coasts that were expanding in the decades from 1830 to 1860. These include Boston; New York; Philadelphia; Washington; Richmond, Virginia; Savannah, Georgia; Mobile, Alabama; New Orleans, Louisiana; and Galveston, Texas.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  GREEK REVIVAL  \nNew Orleans, Louisiana; mid-19th century. Note the exceptionally wide frieze band with windows.\n\nTOWN HOUSE:  GREEK REVIVAL  \nSavannah, Georgia; 1845. Constantine House. A wood-clad example with simplified details.\n\nTOWN HOUSE:  GREEK REVIVAL  \nMadison, Indiana; 1851. Costigan House. A small, two-ranked example with strong porch and window detailing.\n\nTOWN HOUSE:  GREEK REVIVAL  \nSt. Louis, Missouri; 1850. Sherrick House. Town houses with full-facade porches like this example were common along the Gulf Coast, particularly in New Orleans (this one slipped upriver). They may also have two tiers of one-story columns rather than the colossal ones illustrated.\n\nTOWN HOUSE:  GREEK REVIVAL  \nRichmond, Virginia; 1847 and 1853. Lindon Row. A series of attached town houses.\n\nTOWN HOUSE:  GREEK REVIVAL  \nBrooklyn, New York; ca. 1830. Tillary House. Note the tripartite windows and door lintels, and elaborated frieze band. The curved top dormer windows are holdovers from the Federal style.\n\nTOWN HOUSE:  GREEK REVIVAL  \nCharleston, South Carolina; 1838. Roper House; attributed to E. B. White, architect. An outstanding example of the Charleston house, with narrow end turned to the street and wide porch along the side. An upper-story porch (omitted in this example) is commonly present.\n\nTOWN HOUSE:  GREEK REVIVAL  \nNew York, New York; 1833. Colonnade Row; attributed to Robert Higham, architect. A highly unusual row of town houses unified by freestanding columns; this example shows Renaissance influence in the placement of the columns above a rusticated lower story.\n\n## Variants and Details\n\nThe principal areas of elaboration in Greek Revival houses are cornice lines, doorways, porch-support columns, and windows:\n\nCORNICE LINES\u2014The wide band of trim beneath the cornice of both the main roof and the porch roofs is an almost universal feature of Greek Revival houses. Commonly the band is made up of undecorated boards, but complex incised decorations also occur. In gabled houses the trim band may be variously treated along the gabled walls. Post-1850 examples, particularly in the South, often have Italianate brackets added at the cornice line.\n\nDOORWAYS\u2014As in the preceding Georgian, Federal, and Early Classical Revival styles, elaborated door surrounds are a dominant feature of Greek Revival houses. The door itself is either single or paired and is most frequently divided into one, two, or four panels. The door is usually surrounded on sides and top by a narrow band of rectangular panes of glass held in a delicate, decorative frame. Door and glazed surround, in turn, are usually encased in a larger decorative enframement of wood or masonry. Not uncommonly door and glass are recessed behind the front wall, thus creating complex three-dimensional effects; free-standing columns are sometimes added to the inset portion.\n\nCOLUMNS\u2014Classical columns for the support of porch roofs are a prominent feature of most Greek Revival houses. In some examples they dominate the entire facade; others retain only smaller entry porch columns. Although many Greek Revival houses have \"correct\" Greek columns, many also have Roman details; still more have vernacular adaptations with _no_ clear classical precedent. The following guide to Classical Column Identification must therefore be used in combination with other typical features when identifying Greek Revival houses:\n\nClassical columns are distinguished principally by their capitals (tops) and bases. Both Greek and Roman columns share three principal types of capitals which define the three familiar orders of classical architecture: Doric (plain capitals), Ionic (capitals with scroll-like spirals called volutes), and Corinthian (capitals shaped like inverted bells decorated with leaves). All three types are found in Greek Revival houses, as well as in most other classically influenced American styles. Greek and Roman examples of these three orders are distinguished by subtle differences in either the capitals or the bases.\n\nAll columns of classical antiquity were round, as are many Greek Revival columns. Vernacular Greek Revival houses, on the other hand, commonly have _square_ (and occasionally octagonal) columns, which were simple and inexpensive to construct from boards and moldings. Such columns generally lack classical capitals. About 40 percent of columns found on Greek Revival houses are square; the remaining 60 percent include about 40 percent Doric, 15 percent Ionic, and 5 percent Corinthian. Note that the Greek Doric column has no base, while the Roman version does. This distinction frequently will distinguish Greek Revival Doric columns from the Roman Doric columns of the Early Classical Revival. Note, however, that many Greek Revival houses retained Roman columns, particularly in the southern states, so that column type alone is seldom sufficient to identify the style. Round columns were most commonly constructed of wood staves; occasionally bricks formed in the shape of a slice of pie and covered with stucco were used.\n\nPilasters are also frequent Greek Revival features. They are most commonly used on the corners of frame houses but are occasionally found across the entire facade in lieu of free-standing columns.\n\nWINDOWS\u2014As in the preceding Federal style, Greek Revival window sashes most commonly had six-pane glazing. The rounded, three-part Palladian windows of Federal houses disappeared, to be replaced only occasionally by rectangular, tripartite examples. Small frieze-band windows, set into the wide trim beneath the cornice, are frequent. These are often covered with an iron or wooden grate fashioned into a decorative Greek pattern. Window surrounds were generally far less elaborate than doorways.\n\nGREEK REVIVAL\n\n **TYPICAL CORNICE DETAILS**\n\nGREEK REVIVAL\n\n **TYPICAL DOORWAYS**\n\nGREEK REVIVAL\n\n **COLUMN TYPES**\n\nROUND  \nGreek Revival copies often omit fluting in column shaft\n\nGREEK REVIVAL\n\n **COLUMN TYPES**\n\nSQUARE  \nnever found in Greedk and Roman prototypes\n\nGREEK REVIVAL\n\n **CLASSICAL CAPITALS**\n\nGREEK REVIVAL\n\n **TYPICAL WINDOW DETAILS**\n\nGREEK REVIVAL\n\n **DECORATIVE DETAILS** These designs may be found on door and window crowns, columns, pilasters, beneath cornice, etc.\n\nGreek key design, fret\n\nGREEK REVIVAL\n\n **DECORATIVE DETAILS** These designs may be found on door and window crowns, columns, pilasters, beneath cornice, etc.\n\ncresting, found on door and window crowns\n\nmutules, found beneath cornice, are placed above triglyphs on frieze\n\nGREEK REVIVAL\n\n **DECORATIVE DETAILS** These designs may be found on door and window crowns, columns, pilasters, beneath cornice, etc.\n\negg-and-dart design\n\nhoneysuckle-leaf design, anthenium\n\n## Occurrence\n\nGreek Revival was the dominant style of American domestic architecture during the interval from about 1830 to 1850 (to 1860 in the Gulf Coast states), during which its popularity led it to be called the National Style. It occurs in all areas settled by 1860, as noted on the map, and especially flourished in those regions that were being rapidly settled in the decades of the 1830s, '40s, and '50s. The style moved with the settlers from the older states as they crossed into Kentucky, Tennessee, and the Old Northwest Territory (today's Midwest). It enhanced towns built along the route of the Erie Canal. It followed the southern planters as they moved westward from the Old South into Alabama, Mississippi, and Louisiana. It even arrived on the west coast, sometimes disassembled into packages and shipped by way of Cape Horn! Each of the principal subtypes of the style shows geographic differences in frequency of occurrence, as noted above and in the maps.\n\nNot surprisingly, the largest surviving concentrations of Greek Revival houses are found today in those states with the largest population growth during the period from 1820 to 1860. These are, in descending order of growth: New York, Pennsylvania, Ohio, Illinois, Virginia, Massachusetts, Indiana, Missouri, Tennessee, Alabama, Wisconsin, Georgia, Mississippi, Michigan, Texas, Kentucky, and Louisiana. New York gained about 2\u00bd million persons during the interval, while Louisiana gained about \u00bd million.\n\nGREEK REVIVAL\n\n **DISTRIBUTION OFGREEK REVIVAL HOUSES**\n\nGREEK REVIVAL\n\n **DISTRIBUTION OF GREEK REVIVAL HOUSES**\n\n## Comments\n\nThe final years of the 18th century brought an increasing interest in classical buildings to both the United States and western Europe. This was first based on Roman models (see the Early Classical Revival chapter), but archeological investigation in the early 19th century emphasized Greece as the Mother of Rome, which, in turn, shifted interest to Grecian models. Two additional factors enhanced Greek influence in this country. Greece's involvement in a war for independence (1821\u20131832) aroused much sympathy in the newly independent United States; at the same time, the War of 1812 diminished American affection for British influence, including the still dominant Federal style in domestic architecture.\n\nThe Greek Revival began and ended in this country with public buildings built in Philadelphia. Among the first examples was the Bank of the United States (1818, William Strickland), and one of the last monuments was the Ridgeway Branch of the Philadelphia Library (1870, Addison and Hutton). Most domestic examples date from the period from 1830 to 1860. Among the earliest was a Greek remodeling of the Custis-Lee House in Arlington, Virginia, completed in 1820. The style was spread by carpenter's guides and pattern books, the most influential of which were written by Asher Benjamin ( _The Practical House Carpenter; The Builder's Guide_ ) and Minard Lafever ( _The Modern Builder's Guide; The Beauties of Modern Architecture_ ). These illustrated building details rather than views of overall houses.\n\nIn addition to these guides for local carpenter-builders, there were a growing number of trained architects in America, some educated abroad, who designed high-style buildings in the fashionable Grecian mode. Among the most prominent were Benjamin H. Latrobe and his pupils Robert Mills and William Strickland; Strickland's own pupils Thomas U. Walter and Gideon Shryock; Ithiel Town, Alexander Jackson Davis (early work), John Haviland, Alexander Parris, and Isaiah Rogers.\n\nOne of the most familiar stereotypes in American architecture is the full-colonnaded Greek Revival mansion of the southern states. In this century these are sometimes called Southern Colonial houses, a historical inaccuracy since most were built long after American independence. This particular Greek Revival subtype does, however, have a little recognized colonial background, for it sprang, at least in part, from French colonial building practices. Early in their colonial expansions both the French and the English appended broad living porches, a rarity in Europe, to houses built in tropical regions. The origins of these large _galeries_ or verandahs are obscure, yet they appear wherever British or French colonists encountered warm climates, including the West Indies, Africa, India, and Australia. In the United States, most were built by the French in subtropical Louisiana. With the waning of French influence after the Louisiana Purchase in 1803, these forms slowly evolved in the Gulf Coast states into the full colonnaded Greek Revival form now sometimes known as Southern Colonial.\n\nThe decline of Greek Revival influence was gradual. In the more fashion-conscious urban centers of the Atlantic seaboard it began to be replaced by the Gothic Revival and Italianate movements in the 1840s. In the interior states, and in rural areas everywhere, it remained a dominant style for domestic buildings until the early 1860s.\n\nAn important and enduring legacy of the Greek Revival to American domestic architecture is the front-gabled house. Popularized during the ascendance of the Greek Revival style in the early 19th century, this became the predominant form for detached urban houses in cities of the Northeast and Midwest until well into the 20th century. There it occurs in unadorned folk versions, as well as in styled Gothic, Italianate, Queen Anne, and Shingle houses. In rural areas, the form of Greek Revival known as gable front and wing likewise remained a popular form for folk houses until the 1930s.\n\n# ROMANTIC HOUSES\n\n# Gothic Revival\n\n# 1840\u20131880\n\n## Identifying Features\n\nSteeply pitched roof, usually with steep cross gables (roof normally side-gabled, less commonly front-gabled or hipped; rarely flat with castellated parapet); gables commonly have decorated vergeboards; wall surface extending into gable without break (eave or trim normally lacking beneath gable); windows commonly extend into gables, frequently having pointed-arch (Gothic) shape; one-story porch (either entry or full-width) usually present, commonly supported by flattened Gothic arches.\n\n## Principal Subtypes\n\nSix principal subtypes can be distinguished on the basis of roof form, ground plan, or detailing:\n\nCENTERED GABLE\u2014These are symmetrical houses with side-gabled or hipped roofs having a prominent central cross gable. The plane of the cross gable may be either the same as the front wall or projected forward to make a small central wing. Smaller cross gables, or gable dormers, sometimes occur on either side of the dominant central gable. In some examples these are enlarged to give three identical cross gables. This subtype makes up over one-third of Gothic Revival houses.\n\n**CENTERED GABLE**\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nSanta Clara, California; 1875. Landrum House. A small wood-clad example; the triangular pediments over the first-story windows are out of character.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nDenison, Texas; ca. 1883. Eisenhower Birthplace. A small and simplified example. The centered gable has a matching gable on each side.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nJackson, Mississippi; 1857. Manship House. Note how the centered gable is extended forward from the main plane of the front facade to form a covered entrance.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nBrownwood, Texas; ca. 1875. Adams House. This sandstone example has windows with flattened Tudor arches and drip-molds. The porch may have been modified.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nWoodstock, Connecticut; 1846. Roseland. A landmark example with board-and-batten wood cladding, elaborate porch supports, oriel windows, and two facades elaborated with gables or gable dormers.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nSalem, Massachusetts; 1851. Brooks House. An elaborately detailed house with foil windows, diamond-shape window panes, drip-molds, and castellations above the porch.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nWernersville, Pennsylvania; mid-19th century. A combination of the Gothic Revival form with Italianate cornice brackets and arched windows.\n\nCENTERED GABLE:  GOTHIC REVIVAL  \nRushford, Minnesota; ca. 1875. Note the decorative trusses at the apex of the gable and gable dormers (see also the Landrum House); these are common on post-1865 examples.\n\nPAIRED GABLES\u2014Similar to the preceding subtype but with two, rather than one or three, cross gables. The two gables are sometimes extended forward into projecting wings. About 5 percent of Gothic Revival houses are of this type.\n\n**PAIRED GABLES**\n\nPAIRED GABLES:  GOTHIC REVIVAL  \nAshe County, North Carolina; ca. 1880. McGuire House. This very simple example has wood cladding that dramatically follows the lines of the paired gables. The porch shows later modifications.\n\nPAIRED GABLES:  GOTHIC REVIVAL  \nDemopolis, Alabama; 1858. Ashe House. Both this house and the Episcopal Rectory have very delicate lace-like porches and vergeboard details.\n\nPAIRED GABLES:  GOTHIC REVIVAL  \nColumbus, Mississippi; 1880. Episcopal Rectory.\n\nPAIRED GABLES:  GOTHIC REVIVAL  \nBrunswick, Maine; 1849. Boody House; Gervase Wheeler, architect. This house has some applied stickwork (not visible in the photo) and is transitional from the Gothic Revival to the Stick style. A very similar design by Wheeler was published in Downing's 1850 book, _The Architecture of Country Houses,_ figure 130.\n\nFRONT-GABLED ROOF\u2014About 10 percent of Gothic Revival houses are simple gabled rectangles rotated so that the narrower gable end makes up the front facade. Some have additional cross gables added to the roof slope over the _side_ walls, but many lack such cross gables.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  GOTHIC REVIVAL  \nGeorgetown, Colorado; mid-19th century. A very modest example complete with pointed arch window and drip-molds on all front windows and door.\n\nFRONT-GABLED ROOF:  GOTHIC REVIVAL  \nCleveland, Ohio; mid-19th century.\n\nFRONT-GABLED ROOF:  GOTHIC REVIVAL  \nNew Orleans, Louisiana; ca. 1869. Rountree House. An unusual example with a two-tiered porch and the full-length windows often found in Gulf Coast houses. The Tudor arches between the lower-story porch supports are carefully detailed.\n\nFRONT-GABLED ROOF:  GOTHIC REVIVAL  \nCuba, New York; mid-19th century. Note the wraparound porch and matching side gable. The elaborate window crown shape featured on the gable windows was rarely used prior to ca. 1860.\n\nASYMMETRICAL\u2014About one-third of Gothic Revival houses are of compound asymmetrical plan. L-shaped plans with cross-gabled roofs are the most common form, but there are many less regular variations. Small secondary cross gables, or gable dormers, were commonly added to one or more wings. After 1860, square towers were occasionally used.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nHartford, Connecticut; mid-19th century. Although the Gothic decorated gable clearly dominates, a hodgepodge of secondary influences is evident\u2014Italianate brackets, Second Empire tower, Queen Anne porch supports, and pedimented windows.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nSouthport, Connecticut; mid-19th century. Bulkley House. Although similar to the house in Hartford, the details here are mostly of Gothic inspiration.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nSelma, Alabama; mid-19th century. A carefully detailed board-and-batten example.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nBrown's Valley, Minnesota, vicinity; ca. 1885. Similar in form to the house in Selma, the ornate trussed gables identify it as a later example.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nIowa City, Iowa; 1877. Jackson House.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nRochester, New York; 1878. This house shows clearly the transition from the Gothic to the Stick style. Note the Gothic windows and door shapes with stickwork in the main gable and as supports under the upstairs bay windows. Also note the wide roof overhang with open eaves, visible on the right.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nNewport, Rhode Island; 1841. Kingscote; Richard Upjohn, architect (rear addition by Stanford White). A handsomely detailed house with entrance canopy, castellations over the bay window, drip-molds, and diamond-pane windows.\n\nASYMMETRICAL:  GOTHIC REVIVAL  \nNew Castle, Delaware; 1852. Lesley Home; Thomas and James Dixon, architects.\n\nCASTELLATED OR PARAPETED\u2014The four preceding subtypes all have normal roof-wall junctions in which the eaves project outward beyond the wall. A fifth subtype, more closely based on English Medieval models, has either flat roofs with scalloped (castellated) parapets, or gabled roofs ending in high parapeted walls rather than overhanging eaves. Frequently both of these roof types occur on different parts of a single house. About 5 percent of Gothic Revival houses are of this type. These features are far more common on Gothic Revival churches and public buildings; most surviving houses are high-style landmarks, typically constructed of masonry.\n\n**CASTELLATED OR PARAPETED**\n\nCASTELLATED OR PARAPETED:  GOTHIC REVIVAL  \nAberdeen, Mississippi; ca. 1884. The Castle. A relatively modest wood-clad example, unusual for this subtype.\n\nCASTELLATED OR PARAPETED:  GOTHIC REVIVAL  \nFayette County, Kentucky; 1852. Ingelside; John McMurtry, architect.\n\nCASTELLATED OR PARAPETED:  GOTHIC REVIVAL  \nBrookneal, Virginia, vicinity; 1848. Staunton Hill; John E. Johnson, designer. Note the symmetrical facade with its almost classical feeling.\n\nCASTELLATED OR PARAPETED:  GOTHIC REVIVAL  \nTarrytown, New York; 1838, major addition 1865. Lyndhurst; Alexander Jackson Davis, architect for both. This marble example is the finest Gothic Revival house surviving in this country\u2014the result of a major 1865 addition to an earlier 1838 structure. Note the multiple parapets, castellations, pinnacles, foil windows, grouped chimneys, window tracery, and castellated tower. The interiors are equally elaborate.\n\nPOLYCHROMED\u2014A final 5 percent of surviving Gothic Revival houses show distinctive linear patterns in masonry wall surfaces. These decorative polychrome patterns are produced by bands of contrasting color or texture in the brick or stonework, and occur principally around windows and as horizontal bands on wall surfaces. This feature is particularly characteristic of the last phase of the Gothic Revival, from about 1865 to 1880. It is sometimes treated as a separate style called High Victorian Gothic. Like the castellated or parapeted form, it is most common on churches and public buildings. The complex masonry construction was suitable only for high-style, landmark houses. These were once far more common in the prosperous industrial cities of the northeastern and midwestern states, but most have been destroyed.\n\n**POLYCHROMED**\n\nPOLYCHROMED:  GOTHIC REVIVAL  \nPhiladelphia, Pennsylvania; 1894. Moore House; Wilson Eyre, architect. A late example with strong early Eclectic influences.\n\nPOLYCHROMED:  GOTHIC REVIVAL  \nDetroit, Michigan; 1876. Gillis House; Brush and Mason, architects. Note the banded surround above the pointed arch window to the right. This was a favorite polychrome motif (see also the Governor Tilden House).\n\nPOLYCHROMED:  GOTHIC REVIVAL  \nCleveland, Ohio; ca. 1878. Winslow House. This example combines the towered Second Empire form with elaborate polychromed Gothic detailing.\n\nPOLYCHROMED:  GOTHIC REVIVAL  \nBrooklyn, New York; 1848. The Gothic influence is seen in the door surround and drip-mold over the windows of these town houses. This example is not polychromed and is included to show a rare early Gothic town house. A roof addition and cornice modifications are evident in the house to the right.\n\nPOLYCHROMED:  GOTHIC REVIVAL  \nNew York, New York; 1874. Governor Tilden House; Calvert Vaux, architect. This town house example has elaborate polychromed detailing.\n\n## Variants and Details\n\nFanciful decorative ornamentation, cut from wood by the newly perfected scroll saw, is a dominant feature in most Gothic Revival houses. Windows, roof-wall junctions, porches, and doors were the principal sites for such decorations.\n\nWINDOWS\u2014Most Gothic Revival houses have at least one window with Gothic detailing. When only a single window is elaborated in this manner, it usually occurs in the most prominent gable. Such windows might have a pointed-arch shape or might consist of two or three such arches clustered together, or might even be designed as small projecting bay windows (oriels). Full-scale bay windows are also common on the first-floor level. In less elaborate houses, cut-out patterns were frequently used on or above rectangular windows to give a pointed-arch effect. A characteristic window crown called a drip-mold is found above many Gothic windows, both arched and square. Originally designed to protect windows from water running down the face of the building, this molding covers the top of the window and continues downward along the side before turning outward so that water will be deflected away from the window frame.\n\nROOF-WALL JUNCTIONS\u2014Decorative vergeboards, making an inverted V beneath the eaves of the steep gables, are a distinctive feature of most wooden Gothic houses and came in almost as many designs as there were Gothic carpenter-builders. After about 1865 this feature became less popular and was generally replaced by decorative trusses at the apex of the gables. Gothic cornice detailing showed fundamental changes from the preceding classical styles (Georgian, Federal, Greek Revival, etc.). The latter usually have boxed cornices with the rafters enclosed, while most Gothic Revival houses have open cornices with the rafters either exposed or sheathed parallel to the overlying roof.\n\nPORCHES\u2014One-story porches are found on about 80 percent of Gothic Revival houses.\n\nDOORS\u2014Doors commonly show pointed arches or other Gothic motifs as well as decorative crowns similar to those found on windows. Elaborate paneled doors are common but simple batten doors, mimicking modest Medieval prototypes, also occur.\n\nWALL CLADDING AND DECORATIVE DETAILING\u2014Gothic Revival houses are of both wooden and masonry construction but wood-frame Carpenter Gothic examples predominate. These were usually covered with horizontal cladding, but vertical board-and-batten siding was also common. The latter material was widely advocated by contemporary pattern books for its verticality, which was considered suitably Gothic.\n\nGOTHIC REVIVAL\n\n **TYPICAL WINDOW ELABORATIONS**\n\nGOTHIC REVIVAL\n\n **TYPICAL ROOF-WALL JUNCTIONS** (see also this page)\n\nGOTHIC REVIVAL\n\n **TYPICAL PORCH DETAILS**\n\nSpace between porch supports most commonly has a flattened arch or side brackets that mimic such an arch\n\nGOTHIC REVIVAL\n\n **TYPICAL DOORS**\n\nGOTHIC REVIVAL\n\n **HIGH-STYLE ELABORATIONS**\n\n## Occurrence\n\nMost Gothic Revival houses were constructed between 1840 and 1870; examples from the 1870s are less frequent. The style was never as popular as were houses in the competing Greek Revival or Italianate styles, yet scattered examples can still be found in most areas of the country settled before 1880. Surviving Gothic Revival houses are most abundant in the northeastern states, where fashionable architects originally popularized the style. They are less common in the South, particularly in the New South states along the Gulf Coast. In this region Greek Revival houses dominated the expansions of the 1840s and 1850s, while the Civil War and Reconstruction all but halted building until the waning days of Gothic influence.\n\n## Comments\n\nThe Gothic Revival began in England in 1749 when Sir Horace Walpole, a wealthy dilettante, began remodeling his country house in the Medieval style, complete with battlements and multiple pointed-arch windows. Over the next century, others followed his lead and such Picturesque country houses became common in England. Although a handful of earlier houses with Gothic detailing were built, the first documented, fully developed domestic example in America (Glen Ellen in Baltimore, Maryland) was designed by Alexander Jackson Davis in 1832. Davis was the first American architect to champion Gothic domestic buildings; his 1837 book, _Rural Residences,_ was dominated by Gothic examples. This was also the first house plan book published in this country. Previous publications had shown details, parts, pieces, and occasional elevations of houses, but Davis's was the first to show three-dimensional views complete with floor plans. Davis's book had only a small circulation but his ideas were picked up by his friend Andrew Jackson Downing, who expanded them in pattern books published in 1842 ( _Cottage Residences_ ) and 1850 ( _The Architecture of Country Houses_ ). Downing's writings were far more successful, because the author promoted them with tireless public speaking and personal energy. Downing thus became the popularizer of the style.\n\nThis style was seldom applied to urban houses for two reasons. First, the writings of Davis and Downing stressed its suitability as a _rural_ style, compatible with the natural landscape; it was not promoted as appropriate for urban dwellings. Secondly, its emphasis on high, multiple gables and wide porches did not physically lend itself to narrow urban lots. A few urban examples with Gothic door, window, or cornice detailing survive (see the house in Brooklyn or the Governor Tilden House), but most urban houses of the era are in the contemporaneous Greek Revival or Italianate styles.\n\nGothic Revival was in declining favor for American domestic buildings after 1865, although a small rebirth of interest during the 1870s was stimulated by the writings of the English critic John Ruskin, who emphasized continental rather than English examples as models. This High Victorian Gothic phase was principally applied to public and religious buildings, although a few surviving landmark houses reflect its influence (see the paragraph on the polychromed subtype.)\n\n# ROMANTIC HOUSES\n\n# Italianate\n\n# 1840\u20131885\n\n## Identifying Features\n\nTwo or three stories (rarely one story); low-pitched roof with moderate to widely overhanging eaves having decorative brackets beneath; tall, narrow windows, commonly arched or curved above; windows frequently with elaborated crowns, often of inverted U shape; many examples with square cupola or tower.\n\n## Principal Subtypes\n\nSix principal subtypes can be distinguished:\n\nSIMPLE HIPPED ROOF\u2014These are square or rectangular box-shaped houses with hipped roofs that are uninterrupted except, in about half of the surviving examples, by a central cupola (these have been called cube and cupola houses). Facade openings are typically three-ranked, less commonly five-ranked, rarely two- or four-ranked. This is the most common subtype, making up about one-third of Italianate houses.\n\n**SIMPLE HIPPED ROOF**\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nSalisbury, North Carolina; mid-19th century. A simple masonry two-ranked example.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nHartford, Connecticut; mid-19th century.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nSan Jose, California; ca. 1858. Fallon House.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nAustin, Texas; 1877. Tips House.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nCape May County, New Jersey; 1863. George Allen House; Samuel Sloan, architect. A cube and cupola Italianate form with a full-width porch and deep-bracketed cornice that incorporated attic-story windows.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nProvidence, Rhode Island; 1853. Bowen House; Thomas Tefft, architect. This and the Frost House reflect the more formal Renaissance town house tradition; note the cubic form, pedimented window, quoins, and restrained entry porch.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nSavannah, Georgia; 1860. Mercer House; John S. Norris, architect. A lavishly detailed example with iron balconies, paired windows with elaborately bracketed hoods, and cornice brackets with double drops.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nCleveland, Ohio; 1862. Sanford House. Rear bay windows, seen here, and rear wings, one window wide (as in the Tips House), were common methods of bringing extra light and ventilation to narrow examples built on small city lots. Note the heavy arched window hoods.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nSt. Louis, Missouri; mid-19th century. Frost House. Note the rusticated first story (see additional comments under the Bowen House).\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nBenicia, California; ca. 1880. A very simple, one-story, wood-clad example.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nAustin, Texas; mid-19th century. A more elaborate one-story masonry example with quoins accenting the door, windows, and corners.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nLouisville, Kentucky; mid-19th century. Note the use of segmentally arched windows on the first story with fully arched windows above.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nBloomington, Wisconsin; 1877. Ballantine House. The matching pair of bay windows on the front facade is unusual outside of northern California.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nSalisbury, North Carolina; 1868. Murdock-Wiley House. Occasionally the side-gabled roof form was substituted for the more typical Italianate hipped roof as in this example. Note the unusual use of paired brackets placed on top of pilasters.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nFort Smith, Arkansas; 1868. Bonneville House; David McKibben, builder and original owner. Note the handsome paired brackets and windows in the elaborated frieze band.\n\nSIMPLE HIPPED ROOF:  ITALIANATE  \nMacon, Georgia; 1855. Johnston House; James B. Ayres, architect. A large and elaborate three-story example raised on a full basement. An octagonal cupola and small round windows light the third story. Note the unusually heavy bracketed pediments above second-story windows on the front facade.\n\nCENTERED GABLE\u2014These are houses of both simple and compound plan having a front-facing centered gable. The usually rather small gable projects from a low-pitched hipped or side-gabled roof. Frequently the front wall beneath the gable extends forward as a prominent central extension. About 15 percent are of this type.\n\n**CENTERED GABLE**\n\nCENTERED GABLE:  ITALIANATE  \nSelma, Alabama; mid-19th century. White House. A simple wood-clad example.\n\nCENTERED GABLE:  ITALIANATE  \nOxford, Mississippi; 1878. Howry House. The two-story porches are unusual, as is their recessed position under the main roof of the house.\n\nCENTERED GABLE:  ITALIANATE  \nColumbus, Indiana; 1864. Storey House; James Perkenson, architect. This three-ranked masonry example has simplified, but refined, detailing.\n\nCENTERED GABLE:  ITALIANATE  \nRichmond, Virginia; 1858. Haxall House. Contrast the ornate detailing here with the restrained detailing of the Storey House. Note the curved centered \"gable,\" the cupola, and the pattern of paired and single windows.\n\nCENTERED GABLE:  ITALIANATE  \nRaleigh, North Carolina; 1873. Andrews House; G. S. H. Appleget, architect. The small room on the left is a later addition.\n\nCENTERED GABLE:  ITALIANATE  \nOakland, California; 1868. Pardee House; John J. Newsom and William C. Hoagland, architects.\n\nCENTERED GABLE:  ITALIANATE  \nRichmond, Kentucky; 1864. Whitehall; Lewinski and McMurtry, architects. Note the unusual three-part design\u2014a front-gabled central section is flanked by two wings, each with a centered gable; note also the full-height pilasters used across the entire facade.\n\nCENTERED GABLE:  ITALIANATE  \nTarboro, North Carolina, vicinity; 1859. Coolmore Plantation. Rather than the more common centered gable, this unusual example has paired side gables. Note the cupola with centered gable here and in the Pardee House.\n\nCENTERED GABLE:  ITALIANATE  \nLouisville, Kentucky; ca. 1870, porch ca. 1885. Field House. Bracketed and pedimented windows as well as quoins are used on both the main facade and the gabled central projection.\n\nASYMMETRICAL\u2014These are compound-plan houses, usually L-shaped, without towers. Roofs are cross-hipped or cross-gabled. In a few examples the addition of a second forward-facing wing makes a U-shaped plan. About 20 percent of Italianate houses are of this type.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  ITALIANATE  \nSelma, Alabama; mid-19th century. The simple gable front and wing is a common Italianate form. The Crim House and the house in Salisbury are also examples of this type.\n\nASYMMETRICAL:  ITALIANATE  \nClintonville, Kentucky; 1881. Crim House.\n\nASYMMETRICAL:  ITALIANATE  \nSalisbury, North Carolina; late 19th century.\n\nASYMMETRICAL:  ITALIANATE  \nFort Smith, Arkansas; late 19th century. An unusual example showing Italianate detailing superimposed on a steeply pitched roof with a dominant front gable and dormer borrowed from the Gothic Revival.\n\nASYMMETRICAL:  ITALIANATE  \nSalem, Oregon; 1880. Port House. This narrow, deep house form was designed for a narrow urban lot.\n\nASYMMETRICAL:  ITALIANATE  \nRaleigh, North Carolina; ca. 1872. Merrimon House. This example has a second, more shallow wing and porch (left) added to the basic gable front and wing form.\n\nASYMMETRICAL:  ITALIANATE  \nOakland, California; 1876. Camron House. Many asymmetrical Italianate examples are formed by the addition of a single large bay window to the basic cubic shape as in this example and Hill and Hollow.\n\nASYMMETRICAL:  ITALIANATE  \nPenfield, New York; 1878. Hill and Hollow. A smaller version of the Camron House, this time executed in masonry with a cupola.\n\nTOWERED\u2014Only about 15 percent of Italianate houses have the square tower that is often considered to be characteristic of the Italian Villa. The tower is sometimes centered on the front facade or placed alongside it; more commonly, it occupies the position where the wing joins the principal section of an L-plan house. Typically, such towers have narrow paired windows with arched tops. Tower roofs are most commonly low-pitched and hipped; occasionally, steep mansard roofs are used instead.\n\n**TOWERED**\n\nTOWERED:  ITALIANATE  \nSouth Stockton, New York; mid-19th century. This house, the house in Lexington, and the Morse-Libby House illustrate progressively more elaborate versions of a tower embraced by the wings of a gable-front-and-wing plan, a favorite Italianate arrangement.\n\nTOWERED:  ITALIANATE  \nCherry Creek, New York; mid-19th century. Frost House. An unusual composition which builds from a one-story wing (in shadow at left) to the two-story central block to a three-story tower.\n\nTOWERED:  ITALIANATE  \nHartford, Connecticut; mid-19th century.\n\nTOWERED:  ITALIANATE  \nLexington, Kentucky; mid-19th century.\n\nTOWERED:  ITALIANATE  \nMarshalltown, Iowa; 1875. Although classical columns are frequently used as porch supports on Italianate houses (see the Morse-Libby House and the Norton House), this large porch is probably a later addition (paired columns and columns raised on pedestals were uncommon before the 1890s).\n\nTOWERED:  ITALIANATE  \nRaleigh, North Carolina; ca. 1880. Centered towers are common in Second Empire houses, but Italianate examples, such as this, are unusual. Note how this placement produces a balanced, classical appearance even with the asymmetrical porch.\n\nTOWERED:  ITALIANATE  \nPortland, Maine; 1859; Morse-Libby House; Henry Austin, architect. An exceptional landmark example with numerous formal details. Note the segmental and the pedimented window crowns with brackets, as well as the quoins, classical columns, and balustrade.\n\nTOWERED:  ITALIANATE  \nSan Antonio, Texas; ca. 1876, 1882, 1890. Norton House. In its final form this is a most unusual Italianate house based on formal Renaissance models.\n\nFRONT-GABLED ROOF\u2014In this subtype, Italianate detailing is added to the simple front-gabled rectangular box popularized by the Greek Revival style. This subtype, about 10 percent of surviving examples, is common on narrow lots in large cities.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  ITALIANATE  \nChicago, Illinois; mid-19th century.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nBuffalo, New York; 1870\u201378. Tifft Houses. These examples show how the front-gabled form is well adapted to narrow urban lots.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nNew Orleans, Louisiana; mid-19th century. The bracketed wooden canopy over the second-story porch is a common New Orleans innovation on both one- and two-story houses. The full-length windows hark back to the earlier French Colonial building traditions of the region.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nUnion Springs, Alabama; mid-19th century.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nNew London, Connecticut; mid-19th century.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nWashington, District of Columbia; mid-19th century. Although full-width porches are found throughout the country, they are most common in areas with hot summers.\n\nFRONT-GABLED ROOF:  ITALIANATE  \nMeriden, Connecticut; ca. 1868. Smith House. Cross gables extending outward the width of a single window or door are often added for light and ventilation toward the rear of front-gabled examples (see also the Koza House).\n\nFRONT-GABLED ROOF:  ITALIANATE  \nIowa City, Iowa; 1882. Koza House. Note the unusually robust detailing of the bracketed cornice, flattened-arch window crowns, and door hood.\n\nTOWN HOUSE\u2014Italianate styling, along with the related Second Empire style, dominated urban housing in the decades between 1860 and 1880. Italianate town houses are characterized by wide, projecting cornices with typical brackets; the cornice conceals a flat or low-pitched roof behind. Typical Italianate windows further distinguish these examples.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  ITALIANATE  \nBenicia, California; ca. 1880. The cornice and parapet form a false front on this small, wood-clad example (see also the Stadtmuller House). In California flat-fronted Italianate town houses like this generally predated those with bay windows like the Stadtmuller House.\n\nTOWN HOUSE:  ITALIANATE  \nSavannah, Georgia; 1877.\n\nTOWN HOUSE:  ITALIANATE  \nPhiladelphia, Pennsylvania; ca. 1865; Weightman House. An example inspired by formal Renaissance models.\n\nTOWN HOUSE:  ITALIANATE  \nRichmond, Virginia; mid-19th century.\n\nTOWN HOUSE:  ITALIANATE  \nSan Francisco, California; ca. 1880. Stadtmuller House; P. R. Schmidt, architect. Most early Italianate California town houses have flat fronts, as in Benicia, California; ca. 1880. Later examples more often have elaborate ornamentation and large bay windows, as in this example.\n\nTOWN HOUSE:  ITALIANATE  \nRichmond, Virginia; 1861 and 1859. Putney Houses. Note how these detached town houses are closely spaced with full windowless walls.\n\nTOWN HOUSE:  ITALIANATE  \nPittsburgh, Pennsylvania; mid-19th century. Note the incised Eastlake detailing in the door surround and window crowns.\n\nTOWN HOUSE:  ITALIANATE  \nNew York, New York; mid-19th century. Residential New York City was once dominated by blocks of attached Italianate brownstone town houses such as these; some neighborhoods still have many surviving examples.\n\n## Variants and Details\n\nThe principal areas of elaboration in Italianate houses are windows, cornices, porches (including porch-support columns), and doorways. Most American examples show a free intermixing of details derived from both informal rural models as well as formal Renaissance town houses.\n\nWINDOWS\u2014Italianate window sashes most commonly have one- or two-pane glazing. For the first time, arched and curved (segmentally arched) window tops became common, along with the traditional rectangular top. Window enframements show exuberant variation: U-shaped crowns, often with brackets, are most common; simple or pedimented crowns and complete decorated surrounds also occur and appear to be more common in the western states. Paired and triple windows are frequent, as are one- and two-story bay windows that allow far more light.\n\nCORNICES\u2014Large eave brackets dominate the cornice line of Italianate houses. These show an almost infinite variety of shapes and spacings. They are usually arranged either singly or in pairs, and are commonly placed on a deep trim band that is, itself, frequently elaborated with panels or moldings.\n\nPORCHES AND PORCH-SUPPORT COLUMNS\u2014Porches, although almost universally present, are relatively restrained in elaboration and are of single-story height. Small entry porches are most common; full-width porches are also frequent, although many of those seen today are later expansions or additions. The most common type of porch support is a square post with the corners beveled.\n\nDOORWAYS\u2014Paired as well as single doors are common. Large-pane glazing in the door itself, rather than small panes in a frame surrounding the door, first became common in Italianate houses. Doors occur in the same shapes as windows (rectangular, arched, segmentally arched); elaborate enframements above doors are similar to those over windows.\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**SHAPES**\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**SASHES**\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**CROWNS**\n\n**HOODED**  \nmore common on full-arch and segmental-arch shapes\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**CROWNS**\n\n**BRACKETED AND \/ OR PEDIMENTED**  \nmore common on rectangular shapes\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**CROWNS**\n\n**FRAMED**\n\nITALIANATE\n\n **TYPICAL WINDOW DETAILS**\n\n**GROUPINGS**\n\nstilted arches\n\nITALIANATE\n\n **TYPICALBRACKETED CORNICES**\n\nITALIANATE\n\n **TYPICAL PORCH DETAILS**\n\nSUPPORTS\n\nLOCATIONS\n\nITALIANATE\n\n **TYPICAL DOORS** Similar single doors are also found\n\nITALIANATE\n\n **TYPICAL COMBINATIONS OF DETAILS**\n\n**FORMAL RENAISSANCE REVIVAL**\n\nITALIANATE\n\n **TYPICAL COMBINATIONS OF DETAILS**\n\n**INFORMAL ITALIAN VILLA**\n\n## Occurrence\n\nThe Italianate style dominated American houses constructed between 1850 and 1880. It was particularly common in the expanding towns and cities of the Midwest as well as in many older but still growing cities of the northeastern seaboard. In these decades San Francisco grew from a village to a principal American port; most of its earliest town houses were constructed of wood in this style (see the Stadtmuller House). Many of these escaped the 1906 earthquake and fire to survive today. Italianate houses are least common in the southern states, where the Civil War, Reconstruction, and the 1870s depression led to little new building until after the style had passed from fashion.\n\n## Comments\n\nThe Italianate style, along with the Gothic Revival, began in England as part of the Picturesque movement, a reaction to the formal classical ideals in art and architecture that had been fashionable for about two hundred years. The movement emphasized rambling, informal Italian farmhouses, with their characteristic square towers, as models for Italian-style villa architecture. Note that other, more formal, Italian models from the Renaissance or ancient Rome had led to the previous era of classicism. Italy, rather paradoxically, thus remained a principal source of artistic nurture during the reaction against the earlier ideals it had inspired.\n\nItalianate houses built in the United States generally followed the informal rural models of the Picturesque movement. In America these Old World prototypes were variously modified, adapted, and embellished into a truly indigenous style with only hints of its Latin origin. Far less commonly, the formal Italian Renaissance town house, rather than the rural folk house, served as model; these were sometimes imported relatively intact. In purest form such Renaissance Revival houses are austere square or rectangular boxes with little decorative detailing save for formal window crowns (most typically a triangular pediment) and restrained cornice moldings. They are always of masonry (typically stone ashlar or stucco) and typically have horizontal belt courses and corner quoins. As in the originals, most American examples were town houses. Relatively few were built and only a handful survive. More commonly, one or more characteristics of the Renaissance town house were mixed with the general Italianate vernacular.\n\nThe first Italianate houses in the United States were built in the late 1830s; the style was popularized by the influential pattern books of Andrew Jackson Downing published in the 1840s and 1850s (see the preceding chapter on Gothic Revival). Other books published plans for Italianate houses, with the most widely used being the two volumes of Samuel Sloan's _The Model Architect_ that introduced the cube-and-cupola house. By the 1860s the style had completely overshadowed its earlier companion, the Gothic Revival. Most surviving examples date from the period 1855\u20131880; earlier examples are rare. The decline of the Italianate style, along with that of the closely related Second Empire style, began with the financial panic of 1873 and the subsequent depression. When prosperity returned late in the decade, new housing fashions\u2014particularly the Queen Anne style\u2014rose quickly to dominance.\n\nSome East Coast writers have distinguished two chronological phases of Italianate styling: an earlier phase from the 1840s and 1850s with relatively simple detailing and a later, more highly decorated phase from the 1860s and 1870s (High Victorian Italianate). For domestic buildings, at least, this seems a rather artificial division. While the few surviving examples from the 1840s do have rather simple detailing, a survey of pattern book models and surviving examples shows a wide variation in decorative exuberance nationwide, with highly elaborated examples found from at least the early 1850s and simpler examples persisting through the 1870s. Local practices can produce a clearly discernible evolution. For example, town houses in San Francisco and northern California have clear differences between those built in ca. 1860\u20131870 and those from ca. 1870\u20131880 (see this page and this page).\n\n# ROMANTIC HOUSES\n\n# Exotic Revivals\n\n# 1835\u2013ca. 1915\n\nThree principal types of exotic decorative ornament were occasionally used on romantic era houses: Egyptian, Oriental, and Swiss Chalet. These define three very rare styles, which, for convenience, will be treated here as subunits of a single Exotic Revival movement. The Egyptian and Oriental revivals were patterned after similar movements taking place in 19th-century Europe. The Swiss Chalet style, in contrast, was a romantic borrowing from contemporary Swiss domestic practice.\n\n## Egyptian\n\nThe handful (probably fewer than a dozen) surviving domestic examples superimpose Egyptian columns on otherwise Greek Revival or Italianate forms. These columns resemble massive bundles of sticks tied together at the top and bottom and flared at the top. Column tops were quite varied\u2014and commonly were inspired by plants such as lotus, palms, and papyrus in either a bud or a flowering shape.\n\nThe European Egyptian Revival sprang from Napoleon's Egyptian campaign (1798\u201399), coupled with a subsequent scholarly interest in Egypt as a source for the more familiar architecture of classical Greece and Rome. In Europe, as in this country, Egyptian motifs were most often applied to public buildings.\n\n**EGYPTIAN**\n\nEGYPTIAN:  EXOTIC REVIVALS  \nNew Haven, Connecticut; 1837. Apthorp House; Alexander J. Davis, architect. Later additions surround the original cube house. The porch with its Egyptian columns is diminished by the additions above it.\n\nEGYPTIAN:  EXOTIC REVIVALS  \nRichmond, Virginia; 1847. Cabell House. The original house lacked both the bay windows and the porch balcony, making the two-story Egyptian columns still more prominent.\n\nEXOTIC REVIVALS\n\n **TYPICAL DECORATIVE DETAIL**\n\n**EGYPTIAN**\n\n## Oriental\n\nThe several dozen surviving examples are mostly hipped-roof Italianate cubes with ogee arches (sometimes with scalloped edges added) and Oriental trim. Another favored feature was the Turkish (onion) dome; few of these survive on domestic buildings.\n\nThe Oriental Revival was inspired by increasing exploration and trade in the Far East during the late 18th and early 19th centuries. Numerous detailed accounts of travels in India and China led to a new appreciation of the complexities of Oriental architecture. The resulting use of Far Eastern motifs in Europe and America was long-lived; occasional examples were built into the early 20th century.\n\n**ORIENTAL**\n\nORIENTAL:  EXOTIC REVIVALS  \nRochester, New York; 1849. Brewster-Burke House. Oriental porch motif and window details on a cube-house form typical of the Italianate (see the Apthorp House for an Egyptian counterpart).\n\nORIENTAL:  EXOTIC REVIVALS  \nLouisville, Kentucky; 1901. Biscoe House. A very late example.\n\nORIENTAL:  EXOTIC REVIVALS  \nChurch Hill, New York; 1874. Olana; Frederic E. Church, designer, Calvert Vaux, architect. An individualistic interpretation of the Exotic Revival by landscape painter Church.\n\nEXOTIC REVIVALS\n\n **TYPICAL DECORATIVE DETAIL**\n\n**ORIENTAL**\n\n## Swiss Chalet\n\nMost of the few dozen surviving examples have low-pitched front-gabled roofs with wide eave overhangs. A second-story porch or balcony with flat, cut-out patterned balustrade and vergeboard trim is characteristic, as is patterned stickwork decoration on exterior walls; the latter is virtually identical to that found on closely related and contemporaneous Stick houses. Some examples superimpose Swiss porches and trim on Greek or Gothic Revival forms; a few late examples were superimposed on Craftsman forms.\n\nThe style was introduced into the United States by the romantic popularizer Andrew Jackson Downing, whose pattern book, _The Architecture of Country Houses_ (1850), showed several Swiss models suitable for \"bold and mountainous\" sites. It experienced a revival of interest in the early 20th century.\n\n**SWISS CHALET**\n\nSWISS CHALET:  EXOTIC REVIVALS  \nChautauqua; New York, 1875; Miller House. Reported to have been precut in Akron, Ohio, and assembled on site. The applied stickwork seen here (diagonals in rectangular frames) clearly shows the close connection that could exist between the Stick style and the Swiss Chalet.\n\nSWISS CHALET:  EXOTIC REVIVALS  \nHartford, Connecticut; late 19th century.\n\nSWISS CHALET:  EXOTIC REVIVALS  \nBarrytown, New York, vicinity; 1867. Montgomery Place; Alexander Jackson Davis, architect.\n\nSWISS CHALET:  EXOTIC REVIVALS  \nPasadena, California; ca. 1910.\n\nEXOTIC REVIVALS\n\n **TYPICAL DECORATIVE DETAIL**\n\n**SWISS CHALET**\n\n# ROMANTIC HOUSES\n\n# Octagon\n\n# 1850\u20131870\n\nThe Octagon house is easily recognized by the eight-sided shape of the exterior walls. Most are two-story with low-pitched hipped roofs and wide eave overhangs; eave brackets are common. Occasional examples show six-, ten-, twelve-, or sixteen-sided forms; a few are round. About half have an octagonal cupola and most have porches. Many show Greek Revival, Gothic Revival, or Italianate decorative details; others lack detailing.\n\nThis is a very rare style; approximately two thousand of these survive, mostly in New York, Massachusetts, and the Midwest. Most were built in the decades of the 1850s and 1860s.\n\nThe style owed its popularity to Orson S. Fowler, a lecturer and writer from Fishkill, New York, who in 1849 published an elaborate defense of its virtues entitled _The Octagon House: A Home for All._ Following Fowler, at least seven other pattern books of the 1850s also illustrated Octagon houses. Fowler stressed that an octagon encloses more floor space per linear foot of exterior wall than does the usual square or rectangle, thereby \"reducing both building costs and heat loss through the walls.\" He also maintained that Octagons were superior to square houses in \"increasing sunlight and ventilation\" and in \"eliminating dark and useless corners.\" As can be seen in the two typical plans shown in the accompanying drawings, he conveniently ignored interior room shapes, which were _not_ octagonal and therefore still had \"useless\" corners, including triangular spaces not found in conventional shapes. Furthermore, much of this \"increased sunlight and ventilation\" went into pantries and closets; most rooms, in fact, have only a single exposure rather than the two commonly found in conventional houses. Such practical problems are undoubtedly responsible for the only modest success of the Octagon movement.\n\nFowler also advocated other improvements such as indoor plumbing, central heating, \"board walls\" made of lumber scraps, and \"gravel walls\" of poured concrete. He was not generally concerned with decorative treatment beyond \"the beauty of the octagon form itself,\" although many Octagons were built with decorative detailing. Fowler claimed his domestic use of the Octagon to be original, but there were scattered earlier examples, including Thomas Jefferson's summer house, Poplar Forest, completed in 1819. Octagonal wings and projections were also common in Federal houses (1780\u20131820).\n\n**OCTAGON**\n\nOCTAGON  \nEyota, Minnesota; ca. 1865. Mattison House. A small one-story example.\n\nOCTAGON  \nWiscasset, Maine; 1855. Scott House. A two-and-one-half-story brick example with simple entry porch.\n\nOCTAGON  \nWilliamson, New York; 1850. Sperry House. This house has twenty-inch-thick cobblestone walls that are covered with plaster. The porch is a later addition.\n\nOCTAGON  \nGeneva, New York; 1853. Moore House. Entrance steps, porch supports, and first- and second-story railings are all ornamental cast iron. Note the raised basement.\n\nOCTAGON  \nHoosick Falls, New York; 1855. Estabrook House. The balustrade design is repeated in different scales on the roofs of the porch, the cupola, and the main house.\n\nOCTAGON  \nMonroe, Wisconsin; 1861. West House. This unusual house is a combination of several smaller octagonal sections rather than the usual large single octagon. Note the octagonal cupola, Italianate cornice detailing, and wraparound porch with cut-out balusters above.\n\nOCTAGON  \nNatchez, Mississippi, vicinity; 1862. Longwood; Samuel Sloan, architect. This landmark example shows Exotic Revival influence in the large onion dome and Italianate influence in the porch and the cornice detailing.\n\nOCTAGON\n\n **OCTAGON FLOOR PLANS**\n\nOCTAGON\n\n **OCTAGON FLOOR PLANS**\n\n# Victorian Houses\n\n1860\u20131900\n\n* * *\n\n**Second Empire**\n\n**Stick**\n\n**Queen Anne**\n\n**Shingle**\n\n**Richardsonian Romanesque**\n\n**Folk Victorian**\n\nThe long reign of Britain's Queen Victoria lasted from 1837 to 1901 and, in the most precise sense, this span of years makes up the Victorian era. In American architecture, however, it is those styles that were popular during the last decades of her reign\u2014from about 1860 to 1900\u2014that are generally referred to as \"Victorian.\" During this period, rapid industrialization and the growth of the railroads led to dramatic changes in American house design and construction. The balloon frame, made up of light, two-inch boards held together by wire nails, was rapidly replacing heavy-timber framing as the standard building technique. This, in turn, freed houses from their traditional box-like shapes by greatly simplifying the construction of corners, wall extensions, overhangs, and irregular ground plans. In addition, growing industrialization permitted many complex house components\u2014doors, windows, roofing, siding, and decorative detailing\u2014to be mass-produced in large factories and shipped throughout the country at relatively low cost on the expanding railway network. Victorian styles clearly reflect these changes through their extravagant use of complex shapes and elaborate detailing, features previously restricted to expensive, landmark houses.\n\nThe way houses were designed underwent a similar transformation. Not only were there now many more pattern books published showing house facades, there was also a new kind of book that began to be published, showing large, easy-to-read drawings of smaller-scale architectural detail for craftsmen to follow. The number of new titles published increased from 88 during the 1850s to 192 during the 1880s. Further, there was a major expansion in design services available, as house plan books began to offer sets of plans for sale by mail. The pattern books published by Bicknell & Comstock; Palliser, Palliser & Company; R. W. Shoppell's Co-operative Building Plan Association, and George F. Barber were particularly successful. All four sold plans to go with their designs.\n\nIt was during this era that a formal education in architecture first became possible in the New World. The first architecture program was begun at the Massachusetts Institute of Technology in 1865. Others followed in rapid succession, and by 1898 there were programs at Columbia, Cornell, Syracuse, Illinois, and Pennsylvania Universities, Harvard, and Notre Dame, among others. These formal programs were reinforced by a multitude of architecture journals. The most successful, _American Architect and Building News_ ( _AABN_ ), began in 1876. It not only featured illustrations and articles on current buildings designed by architects, it also included frequent features on architecture history, including both America's colonial heritage and its European precedents.\n\nMost Victorian styles are loosely based on Medieval prototypes. Multi-textured or multicolored walls, strongly asymmetrical facades, and steeply pitched roofs are common features. Little attempt is made, however, at historically precise detailing. Instead, stylistic details are freely adapted from both Medieval and Classical precedents. These exuberant mixtures of detailing, superimposed on generally Medieval forms, mean that most Victorian styles tend to overlap one another without the clear-cut stylistic distinctions that separate the Greek, Gothic, and Italianate modes of the preceding Romantic era. This architectural experimentation continued beyond Victorian times and reached a climax in the early decades of the 20th century when the first truly modern styles\u2014Craftsman and Prairie\u2014rose to popularity.\n\nA second trend that was to end the Victorian era turned toward more precise copies of earlier styles, especially those of colonial America. This movement began with the Centennial celebration of 1876 and was greatly aided by the architecture history articles and illustrations in the _AABN._ It picked up momentum throughout the 1880s and 1890s and became dominant in the 20th century. Its influence is evident in the borrowed Georgian and Federal details seen in many late Victorian houses built in the Shingle and Queen Anne styles.\n\n# VICTORIAN HOUSES\n\n# Second Empire\n\n# 1855\u20131885\n\n## Identifying Features\n\nMansard (dual-pitched hipped) roof with dormer windows on steep lower slope; molded cornices normally bound the lower roof slope both above and below; decorative brackets usually present beneath eaves.\n\n## Principal Subtypes\n\nFive principal subtypes can be distinguished:\n\nSIMPLE MANSARD ROOF\u2014These are symmetrical, either square or rectangular houses with the mansard roof uninterrupted except by dormers. Facade openings are typically three-ranked, less commonly five-ranked, and rarely two- or four-ranked. A few examples have central cupolas. This subtype makes up about 20 percent of Second Empire houses.\n\n**SIMPLE MANSARD ROOF**\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nUnion Springs, Alabama; mid-19th century. The brackets above the central dormer and under the molded cornice at the roof top are an unusual feature.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nPeru, Indiana; 1865\u20131870. Kilgore House. This small example has unusually elaborate detailing. Note the rusticated stone basement facade.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nCambridge, Massachusetts; 1860.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nCambridge, Massachusetts; 1864. Charles Wellington, builder. This example has paired first-story windows, plus matching entry and side porches.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nVirginia City, Nevada; ca. 1860s. Savage Mining Company. Often referred to as a \"mansion,\" in fact this large residential-looking building was an office on the first floor with related living quarters on the two upper floors.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nSt. Louis, Missouri; mid-19th century. Kayser House. An unusual stone facade.\n\nSIMPLE MANSARD ROOF:  SECOND EMPIRE  \nBaltimore, Maryland; 1845. In 1868 architect Edmund G. Lind updated a three-story brick house with a fourth floor and a fashionable mansard roof.\n\nCENTERED WING OR GABLE\u2014These are similar to the type just described but have either a centered gable, which usually echoes the mansard silhouette, or a mansard-roofed extension or wing centered on the front wall. About 20 percent of Second Empire houses in America are of this type.\n\n**CENTERED WING OR GABLE**\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nHallettsville, Texas; ca. 1880. Lay House. Note the patterned roof shingles and the metal cresting at the roof line.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nPort Townsend, Washington; 1883. Bartlett House. This example has side wings rather than the more typical centered one.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nCambridge, Massachusetts; 1863. Note how the shape of the centered gable echoes the mansard roof line.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nFredonia, New York; ca. 1857. Pringle House. A small, earlier porch was replaced in 1929 by the one shown.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nSavannah, Georgia; 1873. Hamilton House; J. D. Hall, architect. Note the quoins and the slender, paired windows.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nSouthport, Connecticut; mid-19th century. Pomeroy House. The paired windows in the dormers are an unusual feature.\n\nCENTERED WING OR GABLE:  SECOND EMPIRE  \nSt. Paul, Minnesota; 1872. Ramsey House; Monroe Sheire, architect. The hooded windows here and in the Pringle House show the strong Italianate detailing often seen in this style.\n\nASYMMETRICAL\u2014These are compound-plan houses, usually L-shaped, which lack towers. The forward-facing portion of the L may be either a full wing or merely a single strongly projecting bay window. About 20 percent of Second Empire houses are of this type.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  SECOND EMPIRE  \nMeriden, Connecticut; 1879. Renton House. In this example and in the Knight House, forward-facing wings create the asymmetrical facade.\n\nASYMMETRICAL:  SECOND EMPIRE  \nCorning, New York; mid-19th century. In this example, two-story bay windows create an asymmetrical facade.\n\nASYMMETRICAL:  SECOND EMPIRE  \nDenver, Colorado; 1885. Knight House.\n\nASYMMETRICAL:  SECOND EMPIRE  \nFort Smith, Arkansas; mid-19th century. Here and in the Failing House a forward-facing wing is combined with a two-story bay window to create asymmetry.\n\nASYMMETRICAL:  SECOND EMPIRE  \nAustin, Texas; 1886. John Bremond House; John Fiegal, architect. The architect's New Orleans background shows in the exquisite decorative ironwork on the wrap-around porches.\n\nASYMMETRICAL:  SECOND EMPIRE  \nCleveland, Ohio; mid-19th century. Roche House. Note the strongly hooded windows borrowed from the contemporary Italianate style.\n\nASYMMETRICAL:  SECOND EMPIRE  \nPortland, Oregon; ca. 1870. Failing House. An elaborately detailed example with two principal facades, one with a porte-cochere and the other with a partial porch and two-story bay window.\n\nTOWERED\u2014About 30 percent of Second Empire houses have a rectangular or square tower. Sometimes it occupies the position where the wing joins the principal section of an L-plan house, but it is more commonly centered on the front facade. Occasionally it is placed off-center on the front or side facades. Typically the tower has a mansard roof with small dormer windows in each side.\n\n**TOWERED**\n\nTOWERED:  SECOND EMPIRE  \nSalem, Virginia; 1882. Evans House. The Evans House, the Jordan House, and the Cornish House show symmetrical examples with a central tower. Note the contrasting shapes of the tower roof and main roof in this example.\n\nTOWERED:  SECOND EMPIRE  \nRaleigh, North Carolina; ca. 1875. Heck House. Above, the Wager House, the Morris House, and the Green House show examples with a tower embraced between two wings of the house. Note the concave tower roof and the convex main roof in this example.\n\nTOWERED:  SECOND EMPIRE  \nRhinebeck, New York; mid-19th century. Wager House. Note window surrounds similar to one of the typical dormers illustrated. First-story windows are triple-hung.\n\nTOWERED:  SECOND EMPIRE  \nAuburn, Maine; ca. 1880. Jordan House; Charles A. Jordan, architect-builder. Note the arcaded front porch, also seen in the Green House.\n\nTOWERED:  SECOND EMPIRE  \nIndianapolis, Indiana; 1862. Morris House; Diedrich A. Bohlen, architect. Note the differing window designs in the four-and-one-half-story tower, as well as the absence of the elaborate window surrounds commonly seen in the style.\n\nTOWERED:  SECOND EMPIRE  \nDes Moines, Iowa; 1869. Terrace Hill; William W. Boyington, architect. An extremely large and elaborately detailed example with two towers. This is now the Iowa Governor's Mansion.\n\nTOWERED:  SECOND EMPIRE  \nOmaha, Nebraska; 1886. Cornish House. Note the projecting bays on each side of the centered tower.\n\nTOWERED:  SECOND EMPIRE  \nWoodbury, New Jersey; mid-19th century. Green House. The mansard roof of this house shows the rare S-curve shape. Note the extensive metal cresting on the roof line. Similar cresting, once present in many examples, has usually deteriorated and been removed. Note also the unusual cupola atop the tower.\n\nTOWN HOUSE\u2014Second Empire styling, along with the related Italianate style, dominated urban housing in the decades between 1860 and 1880. The mansard roof was particularly adapted to town houses, for it provided an upper floor behind the steep roof line, and thus made the structure appear less massive than most other styles with comparable interior space.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  SECOND EMPIRE  \nRichmond, Virginia; mid-19th century. A detached urban example with a full-width porch.\n\nTOWN HOUSE:  SECOND EMPIRE  \nSt. Louis, Missouri; mid-19th century.\n\nTOWN HOUSE:  SECOND EMPIRE  \nSt. Louis, Missouri; mid-19th century. A row showing three different interpretations of the Second Empire detached town house. The roof on the center example may have been modified.\n\nTOWN HOUSE:  SECOND EMPIRE  \nWashington, District of Columbia; mid-19th century. A row of three attached town houses.\n\nTOWN HOUSE:  SECOND EMPIRE  \nRichmond, Virginia; mid-19th century. Note the slightly projecting entrance wing and side-facing bay window, an unusual feature in town houses.\n\nTOWN HOUSE:  SECOND EMPIRE  \nNew Haven, Connecticut; ca. 1871. A row of four attached town houses. Originally all matching, they have been somewhat modified by later alterations. The doorways, window hoods, and dormers are all heavily carved.\n\n## Variants and Details\n\nThe style is characterized principally by its distinctive roof; five principal mansard silhouettes occur. Decorative patterns formed by different colors or shapes of roofing material are common, as is iron cresting above the upper cornice. Slate is a common roofing material. If a tower is present, it may have a roof silhouette different from that of the main house; the convex and ogee (S-curve) shapes, in particular, are more common on towers than on houses. Dormers and dormer windows appear in a great variety of styles. Beneath the distinctive roof line, Second Empire houses commonly have details that are similar to those of the closely related Italianate style. Many show Italianate brackets at the cornice line; note, however, that Second Empire houses normally have less eave overhang than do Italianate examples. Window, door, and porch details are similar to those used in the Italianate style, as are the varied porch locations (see the drawings of those details in the Italianate chapter). Unelaborated windows, usually arched above, are also common on Second Empire houses, but are rare in Italianate examples.\n\nSECOND EMPIRE\n\n **ROOF SHAPES**\n\nSECOND EMPIRE\n\n **TYPICAL DORMERS & WINDOW SURROUNDS** Scroll at base of surround is common\n\nSECOND EMPIRE\n\n **TYPICAL CORNICES**\n\nSECOND EMPIRE\n\n **TYPICAL ELABORATIONS**\n\n(see Italianate illustrations for typical window details, porch details, and doors)\n\n## Occurrence\n\nSecond Empire was a dominant style for American houses constructed between 1860 and 1880, although the first examples were built in the 1850s and late examples were not uncommon in the 1880s. During the decade of the 1870s it was perhaps the most fashionable and widely built house style. The style was most popular in the northeastern and midwestern states. It is less common on the Pacific Coast and relatively rare in the southern states, although scattered examples survive in all regions settled before 1880.\n\n## Comments\n\nThe contemporaneous Italianate and Gothic Revival styles were part of a Picturesque movement which looked to the romantic past for inspiration. In contrast, the Second Empire style was considered very modern, for it imitated the latest French building fashions. The distinctive roof was named for the 17th-century French architect Fran\u00e7ois Mansart. Its use was extensively revived in France during the reign of Napoleon III (1852\u20131870), France's Second Empire, from which the style takes its name. Exhibitions in Paris in 1855 and 1867 helped to popularize the style in England, from whence it spread to the United States. The first American pattern book examples of Second Empire were found in Calvert Vaux's _Villas and Cottages_ (1857), which featured three Second Empire designs. Two important public buildings followed, both designs of James Renwick Jr.\u2014Charity Hospital in New York City (begun in 1858) and Corcoran Art Gallery in Washington, D.C. (begun in 1859). For the next twenty years almost every American house pattern book published had at least one or two Second Empire designs. The boxy roof line was considered particularly functional because it permitted a full upper story of usable living area or attic space. For this reason the style became popular for the remodeling of earlier buildings as well as for new construction. The Second Empire style was used for many public buildings in America during the Grant administration (1869\u20131877) and has been facetiously called the General Grant style. It rapidly passed from fashion following the panic of 1873 and the subsequent economic depression.\n\n# VICTORIAN HOUSES\n\n# Stick\n\n# 1860\u2013ca. 1890\n\n## Identifying Features\n\nOne or more front-facing roof gables on steeply pitched roof; gables commonly show decorative trusses at apex; overhanging eaves, usually with exposed rafter ends (normally replaced by brackets in town houses); wooden wall cladding interrupted by patterns of horizontal, vertical, or diagonal boards (stickwork) raised from wall surface for emphasis; porches commonly show diagonal or curved braces. (Few houses have all of these features in combination.)\n\n## Principal Subtypes\n\nFour principal subtypes occur:\n\nGABLED ROOF\u2014This subtype includes the broad range of gabled variations that occur in the Stick: roofs may be either side-, front-, cross-, or multi-gabled; roof pitch varies from high to low; secondary cross gables are common. This subtype is the most common (except in northern California).\n\n**GABLED ROOF**\n\nGABLED ROOF:  STICK  \nNew Bedford, Massachusetts; 1870. Smith House. The gable end has vertical, horizontal, and diagonal stickwork.\n\nGABLED ROOF:  STICK  \nHartford, Connecticut; late 19th century. A very simple example with stick detailing only in the trussed gables and porch.\n\nGABLED ROOF:  STICK  \nSalem, Oregon; 1887. Collins House. This West Coast Stick\u2013style house has a deep vertical frieze under the eaves. Note the squared box-bay window and squared porch supports.\n\nGABLED ROOF:  STICK  \nHoneoye Falls, New York; ca. 1875. Note the trusses supporting small roofs over the first-floor windows and the trussed balcony in the gable.\n\nGABLED ROOF:  STICK  \nBismarck, North Dakota; 1884. Former Governor's Mansion.\n\nGABLED ROOF:  STICK  \nCambridge, Massachusetts; 1878. The cut-away bay windows are precursors of a popular Queen Anne feature. The stickwork is confined to bands below the windows. Additional stickwork may well have originally been present on this and all other examples; such applied ornamentation is quite susceptible to deterioration and was commonly removed entirely rather than repaired or replaced.\n\nGABLED ROOF:  STICK  \nNewport, Rhode Island; 1863. Griswold House; Richard Morris Hunt, architect. A well-preserved landmark example; all the style's identifying features are present.\n\nTOWERED\u2014Houses in this subtype are generally similar to the gabled-roof subtype above, but with the addition of a square or rectangular tower.\n\n**TOWERED**\n\nTOWERED:  STICK  \nRochester, New York; late 19th century. The picket-fence pattern is used as trim under the eave line and across the gable end.\n\nTOWERED:  STICK  \nStony Creek, Connecticut; 1878. Villa Vista; Henry Austin, architect. Note the unusual two-tiered porch and the elaborate curving porch supports and decorative trussing.\n\nTOWERED:  STICK  \nFergus Falls, Minnesota; 1882. Clement House. Here the picket-fence pattern is used as frieze below the porch roof and below the tower eave.\n\nTOWERED:  STICK  \nRichfield Springs, New York; late 19th century. Hinds House. Note the very high pitch of the tower roof. Square towers are typical of Stick houses, whereas round towers are most common in the related Queen Anne style.\n\nTOWERED:  STICK  \nSan Diego, California; 1887. Sherman House; Comstock and Trotsche, architects.\n\nTOWERED:  STICK  \nHartford, Connecticut; 1873. Mark Twain House; Edward Tuckerman Potter, architect. This is a high-style brick example where the pattern and colors of the brickwork are substituted for applied Stick ornament.\n\nTOWERED:  STICK  \nOakland, California; 1884. Cohen-Bray House. The cutaway bay window is transitional to the Queen Anne style.\n\nHIPPED\u2014Houses in this subtype have hipped roofs, usually steeply pitched and occasionally even mansard. They are often symmetrical. Late examples may have a hipped roof with asymmetrical lower cross gables, a form that is most characteristic of the succeeding Queen Anne style. In the Pacific Northwest, a hipped-roof house sometimes has a box-bay with vertical stickwork (as seen on San Francisco town houses) as a primary identifying feature.\n\n**HIPPED ROOF**\n\nHIPPED ROOF:  STICK  \nPortland, Oregon; late 19th century. Welty House. An example with vertical wood cladding in the gable, horizontal cladding on the main wall surfaces, and diagonal cladding below the first-story windows.\n\nHIPPED ROOF:  STICK  \nColumbus, Texas; 1860 and 1890s. Seftenberg-Brandon House. A one-story Greek Revival house completely updated with a second story and full-width Stick-style porches that wrap around one side.\n\nHIPPED ROOF:  STICK  \nWichita, Kansas; ca. 1878. Miller House. An example that is transitional from the Stick to the Queen Anne. The hipped roof has cross gables and the diagonal porch supports are turned spindles.\n\nHIPPED ROOF:  STICK  \nYankton, South Dakota; 1886. Cramer-Kenyon Heritage Home.\n\nHIPPED ROOF:  STICK  \nBenicia, California; ca. 1890. This California house has low roof pitch and dominantly vertical stickwork.\n\nTOWN HOUSE\u2014This subtype is comprised of flat-roofed urban houses commonly found in San Francisco. Most have box-bay windows with vertical stickwork that terminates in cornice-line brackets. There is sometimes a false-gable roof over the bay, towers also occur.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  STICK  \nSan Francisco, California; late 19th century. A double house (two identical attached houses).\n\nTOWN HOUSE:  STICK  \nSan Francisco, California; late 19th century. Westerfeld House. An unusual example of a narrow town house with a massive tower.\n\nTOWN HOUSE:  STICK  \nSan Francisco, California; late 19th century.\n\n## Occurrence\n\nAlthough pattern books of the day show many examples of the Stick style, the contemporaneous Italianate and Second Empire styles are more common. It is likely that deterioration and removal of the characteristic stickwork wall patterns\u2014and loss of other detailing\u2014has obscured many examples. Houses on the East Coast survive principally in the northeastern states and date from the 1860s and 1870s. The style was promoted as being appropriate for suburban locations and for summer cottages; it is frequently found in resort towns and was rarely built in urban locations.\n\nSurviving town houses are concentrated in San Francisco, where rapid growth and an abundance of lumber favored wooden urban houses. The Stick tradition developed its own distinctive urban idiom in San Francisco, where it did not peak until the 1880s, after the style was passing from fashion in the Northeast.\n\n## Variants and Details\n\nThe style is defined primarily by decorative detailing\u2014the characteristic roof trusses and multi-textured wall surfaces whose stickwork faintly mimics the exposed structural members of Medieval half-timbered houses. Varied patterns of wood siding and shingles were applied in the square and triangular spaces created by the raised stickwork. This fairly simple palette of details was applied to a variety of mid-19th-century house shapes; most show one-story porches, either entry- or full-width. Where dual-level porches occur, they can be visually dramatic. Towers are typically squared or rectangular, not rounded or polygonal as is more typical of Queen Anne, Romanesque, and Shingle houses. Some high-style houses are constructed of brick, and while these may incorporate stickwork infilled with brick, similar picturesque detail was often created by the pattern or color of the brickwork.\n\nSTICK\n\n **DECORATIVE TRUSSES  \nMost common in gabled roof**\n\n**COMMON TYPES**\n\nSTICK\n\n **DECORATIVE TRUSSES  \nMost common in gabled roof**\n\n**WHERE FOUND**\n\nSTICK\n\n **GABLED ROOF ELABORATIONS**\n\n## West Coast Stick (ca. 1880\u2013ca. 1895)\n\nIt is hard to recognize West Coast Stick\u2013style houses simply from knowledge of their eastern counterparts. Stick houses in northern California have their own unique set of identifying features. Most prominent is an almost universal square-sided bay window (box-bay), which was probably simpler to construct than its slant-sided Italianate predecessor.\n\nDecorative cornice-line brackets align with the side framing of the box-bay windows, and these two elements are connected with vertical strips of trim. In addition, cornice-line brackets usually line up with the corners of the house and long vertical strips extend from these down the corner boards.\n\nCommonly, the rectangular areas above and below the windows are also filled with ornament (such as panels), and the box-bay window ensemble thus forms a continuous decorative element from window base to cornice line. A characteristic pattern of short verticals is sometimes found beneath the cornice.\n\nThe box-bay window often has a false-gabled roof above it, and upper-story windowpanes are likely to have flat tops rather than the slight arch often found in Italianate designs.\n\nIn contrast to the fairly simple palette of details in the rest of the country, San Francisco examples of Stick town houses offer a plethora of factory-produced decorative architectural detailing. During the 19th century many houses in northern California and the Pacific Coast were built of redwood. This remarkable wood is resistant to rot, termites, and many of the other ills that affect wooden houses\u2014and a ready supply was available in the vast redwood forests of the northern California coast. These qualities have allowed elaborate redwood details to survive relatively unscathed up to the present day. Automatic lathes and milling machines made possible the inexpensive mass production of details such as turned spindles and incised floral designs that previously would have required time-consuming handcrafting. This detailing, often called Eastlake, is also found on furniture. In addition to the dry woods being mechanically shaped, redwood was sometimes soaked, and incised designs were simply stamped into it.\n\nSTICK\n\n **WEST COAST STICK TOWN HOUSE ELABORATIONS**\n\n## Differentiating Italianate and Stick Styles on the West Coast\n\nIn northern California, and occasionally elsewhere along the Pacific Coast, both Italianate and Stick houses have decorative brackets along the cornice line. These brackets, a principal identifying feature for Italianate designs in the rest of the country, are not sufficient to distinguish Italianate from Stick style in this region.\n\nTo further complicate matters, the Stick style in northern California (often called West Coast Stick) generally looks quite different from its appearance in the rest of the United States.\n\nBay windows in two distinctive shapes become a primary distinguishing feature between Italianate and Stick designs. Elaborate box-bay windows are a principal identifying feature for Stick-style houses, while slanted-side bay windows are a principal identifying feature for Italianate houses. In addition, in West Coast Stick, the brackets commonly have long, vertical extensions down the facade. The illustrations here summarize these differences.\n\nSTICK\n\n **DIFFERENTIATING ITALIANATE ANDSTICK STYLES IN CALIFORNIA**  \n **Primarily Northern California**\n\n**ITALIANATE (1860\u2013ca. 1885)**  \nBay windows with slanted sides  \nDecorative brackets at cornice line (do not always align with corner boards and sides of windows)  \nHipped roof (if townhouse, may have false-hipped roof above bay window)  \nSegmental arch common on upper windowpane\n\n**STICK (1880\u2013ca. 1895)**  \nBay windows with straight (squared) sides (box-bay)  \nBrackets align with window sides and corner boards  \nVertical strips along sides of windows and on corner boards  \nFalse-gabled roof above bay window  \nFlat tops common on upper windowpanes  \nMay have pattern of short verticals along cornice line\n\n## Comments\n\nThe Stick is a transitional style that links the preceding Gothic Revival with the subsequent Queen Anne; all three styles are free adaptations of Medieval English building traditions. Unlike early Gothic Revival houses, the Stick style stressed the wall surface _itself_ as a decorative element rather than merely as a plane with the principal decorative detailing applied at the doors, windows, or cornices. The emphasis on patterned wood walls seen in the Stick style was developed further still in the succeeding Queen Anne style.\n\nThe Stick style grew from the Picturesque Gothic ideals of Andrew Jackson Downing (see Comments) and flourished in house pattern books of the 1860s and 1870s. Downing's earliest pattern book examples were Swiss Chalets. Richard Morris Hunt, the first American architect to design Stick-style houses, had studied extensively in Europe, where the Picturesque movement was inspired by timber-built houses throughout northern Europe\u2014particularly those in France, England, Switzerland, and Germany, where architecture periodicals in the 1850s were promoting these Picturesque models for new houses. Hunt's first fully developed Stick example was the Griswold House in Newport, Rhode Island (1861\u20131863).\n\nAlthough its proponents lauded the structural honesty of the style, the visible stickwork (unlike true half-timbering) was merely applied decoration bearing no structural relation to the underlying balloon-frame construction. During the era in which it was built, the term \"Stick\" was not used.\n\n# VICTORIAN HOUSES\n\n# Queen Anne\n\n# 1880\u20131910\n\n## Identifying Features\n\nSteeply pitched roof of irregular shape, usually with a dominant front-facing gable; patterned shingles, cutaway bay windows, and other devices used to avoid a smooth-walled appearance; asymmetrical facade with partial or full-width porch which is usually one story high and extended along one or both side walls.\n\n## Principal Subtypes\n\nQueen Anne houses are most conveniently subdivided into two sets of overlapping subtypes. The first is based on characteristic variations in _shape;_ the second on distinctive patterns of _decorative detailing._ The following illustrations will show the four decorative detailing subtypes first. These will be followed by photographs of these different patterns of detailing applied to the typical shape subtypes.\n\n## Decorative Detailing Subtypes\n\nFour principal subtypes can be distinguished on the basis of decorative detailing:\n\nSPINDLEWORK\u2014About 50 percent of Queen Anne houses have delicate turned porch supports and spindlework ornamentation made possible by machine lathes. This most commonly occurs in porch balustrades or as a frieze suspended from the porch ceiling. Spindlework detailing is also used in gables and under the wall overhangs left by cutaway bay windows. Lacy, decorative spandrels and knob-like beads are also common ornamental elements in this subtype as is incised decorative detail. Spindlework detailing is sometimes referred to as gingerbread ornamentation, or as Eastlake detailing (after Charles Eastlake, an English furniture designer who advocated somewhat similar design elements).\n\nQUEEN ANNE\n\n **TYPICAL DECORATIVE DETAILS**  \nSPINDLEWORK\n\nFREE CLASSIC\u2014About 35 percent of Queen Anne houses use classical columns, rather than delicate turned posts with spindlework detailing, as porch supports. These columns may be either the full height of the porch or raised on a pedestal to the level of the porch railing; the railings normally lack the delicate, turned balusters of the spindlework type of Queen Anne house. Porch-support columns are commonly grouped together in units of two or three. Palladian windows, cornice-line dentils, swags and garlands and other classical details are frequent. This subtype became common after 1890 and has much in common with some early (asymmetrical) Colonial Revival houses.\n\nQUEEN ANNE\n\n **TYPICAL DECORATIVE DETAILS**  \nFREE CLASSIC\n\nHALF-TIMBERED\u2014About 5 percent of Queen Anne houses have decorative half-timbering in gables or upper-story walls. Porch supports in this subtype are usually heavy turned posts with solid spandrels. Groupings of three or more windows are a common characteristic. This subtype occurs principally in the northeastern states and shares certain features with the early Tudor house (see this page and this page).\n\nQUEEN ANNE\n\n **TYPICAL DECORATIVE DETAILS**  \nHALF-TIMBERED\n\nPATTERNED MASONRY\u2014About 5 percent of Queen Anne houses have masonry walls with patterned brickwork or stonework and relatively little wooden detailing. Terra-cotta and stone decorative panels are frequently inset into the walls. Gable dormers, sometimes parapeted and shaped, are frequent. Examples of this subtype are usually high-style architect-designed houses which exhibit a wide variation in shape and detail or are attached row houses. Most were built in large cities, particularly New York, Chicago, and Washington, D.C.; few have escaped subsequent demolition.\n\nQUEEN ANNE\n\n **TYPICAL DECORATIVE DETAILS**  \nPATTERNED MASONRY\n\n## Shape Subtypes\n\nFour principal shape subtypes can be distinguished:\n\nHIPPED ROOF WITH LOWER CROSS GABLES\u2014Over half of all Queen Anne houses have a steeply hipped roof with one or more lower cross gables. Most commonly there are two cross gables, one front-facing and one side-facing, both asymmetrically placed on their respective facades. Unlike most hipped roofs, in which the ridge runs parallel to the front facade, Queen Anne hipped ridges sometimes run front-to-back, parallel to the side of the house. Others have pyramidal roofs with no ridge or merely a small flat deck crowning the hip. The hipped portion of the roof may have a gable-on-hip added; dormers and additional gables are common. A tower, when present, is most commonly placed at one corner of the front facade. The roof form of this subtype is among the most distinctive Queen Anne characteristics and occurs in examples ranging from modest cottages to high-style landmarks.\n\n**SPINDLEWORK:HIPPED ROOF WITH LOWER CROSS GABLES**\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nBiloxi, Mississippi; ca. 1900. A very simple example. Additional corner-bracket detailing was probably once present above the cutaway bay window, but is now missing (see the corner brackets still present on the house in Santa Clara and the Miller House). The low roof pitch indicates a late construction date.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSanta Clara, California; late 19th century. Note the gable-on-hip roof (also present in the house in Biloxi and the Miller House); these were most common on one- and one-and-one-half-story examples.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nCripple Creek, Colorado; 1896. Miller House. This one-and-one-half-story example has unusually fine detailing.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nClement, North Carolina, vicinity; ca. 1912. Autry House. The symmetrical placement of the two gables is unusual, as is the steep roof pitch in such a late example.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nAtlanta, Georgia; ca. 1893. Martin Luther King birthplace.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nGreensburg, Indiana; 1885\u201390. Woodfill House.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nMeriden, Connecticut; c. 1890. Cahill House. Note the second-story porch over the entrance; the central hipped roof is mostly obscured behind the front gable.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSan Antonio, Texas; 1886. This masonry example has a two-tiered porch and unusually low-pitched gables, which hide a low-pitched hipped roof behind.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nUnion Springs, Alabama; late 19th century.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nWaxahachie, Texas; 1893. Williams-Erwin House, based on George F. Barber's Design 56. A fine example of Barber's exuberant spindlework design. Note the four different gable treatments (on house, porch, and two dormers) and the rounded \"moon gate.\"\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nFleischmanns, New York; 1895. Note the integral upstairs porch beneath the principal roof line and the turreted lower porch roof.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nNapa, California; ca. 1890. Note the variety of surface textures and elaborate detailing.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nLovelady, Texas; ca. 1895. Nelms House. Note the curved roof on the tower; such roofs are far less common than straight-sided examples.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nCambridge, Massachusetts; 1889. Note the absence of the typical front-facing gable; a hipped dormer is used here instead.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSanta Cruz, California; 1891. Gray House; LaBaron Oliver, architect. Note how the tower is interrupted at the first-story level by a band of shingles and at the second-story level by a band of roofing. This clearly illustrates the typical Queen Anne aversion to smooth wall surfaces. The house in Union Springs and the Haas House also show interrupted towers.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nUnion Springs, Alabama; late 19th century. Note the S-shaped curve of the tower roof.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nLaurens, South Carolina; ca. 1896. Davis House. Note the porch-roof turret and the delicate beaded spindlework frieze extending around the entire porch.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSan Francisco, California; 1886. Haas House. The central hipped roof is hidden by gables and tower. Note the unusually elaborate details of the wall surfaces.\n\nSPINDLEWORK: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nWaxahachie, Texas; 1890. Dunlap House. Margaret Culbertson's _Texas Houses: Built by the Book_ documents that this was built from Design 438 in _Shoppell's Modern Houses_ 2 (January 1887), 18.\n\n**FREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES**\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSalisbury, North Carolina; late 19th century. Brown House.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nEutaw, Alabama; late 19th century. Note the Palladian window and recessed arch under the gable.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSalisbury, North Carolina; late 19th century. Gaskill House.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nDallas, Texas; ca. 1900. Arnold House. The shingled porch-support arches are unusual.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nCleveland, Ohio; late 19th century. Note the upper-story window sashes with a single central pane of glass surrounded by smaller panes.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nJacksonville, Oregon; 1893. Nunan House. The hipped central roof is hidden by the front gable in this photograph. Note the dominant, elaborately detailed front chimney.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nDallas, Texas; ca. 1885. Daron Tapscott, restoration architect. The most typical form of a \"hipped roof with lower cross gables\" Queen Anne house. No extras added; compare with introductory sample house. Classical porch supports could be early replacements for spindlework ones.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nConcord, North Carolina; late 19th century.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nWarsaw, Indiana; ca. 1894. Wood House. Note the transoms above the windows, a frequent Queen Anne feature that often had decorative beveled or colored glass glazing.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nUnion Springs, Alabama; late 19th century.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nDallas, Texas; ca. 1899. Wilson House.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nSanta Clara, California; late 19th century.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nNew Haven, Connecticut; late 19th century.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nKirksville, Missouri; 1899. Still House. Note the shingled gable wall curving into the gable window, a motif that is more common in the Shingle style. Although of masonry, this house lacks patterning in the brick-wall surfaces. This and the classical columns differentiate it from the patterned masonry subtype. The porte cochere is a later addition, as is the ribbon of four windows under the front-facing gable. These enclose what was once a small open porch.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nNew London, Connecticut; late 19th century. Note the unusual flared eaves and the decorative frieze beneath the gable.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nMontgomery, Alabama; late 19th century. Note the dramatically exaggerated S-curved roof of the tower.\n\nFREE CLASSIC: HIPPED ROOF WITH LOWER CROSS GABLES:  QUEEN ANNE  \nConcord, North Carolina; late 19th century. Although asymmetrical, this house has a centered entry and a suggestion of classical balance; it is transitional to some early examples of the Colonial Revival style.\n\nCROSS-GABLED ROOF\u2014About 20 percent of Queen Anne houses have simple cross-gabled roofs without a central, hipped unit. These are normally of L-shaped plan; a tower, when present, is usually embraced within the L.\n\n**SPINDLEWORK:CROSS-GABLED ROOF**\n\nSPINDLEWORK: CROSS-GABLED ROOF:  QUEEN ANNE  \nBiloxi, Mississippi; ca. 1900. Even this small example has an ornamented and textured gable and cutaway bay window to avoid smooth wall surfaces. Compare this to the Folk Victorian example of similar shape in Bartlett, Texas.\n\nSPINDLEWORK: CROSS-GABLED ROOF:  QUEEN ANNE  \nHartford, Connecticut; late 19th century.\n\nSPINDLEWORK: CROSS-GABLED ROOF:  QUEEN ANNE  \nNew Haven, Connecticut; late 19th century.\n\nSPINDLEWORK: CROSS-GABLED ROOF:  QUEEN ANNE  \nHillsboro, Texas; late 19th century. Note the wide gable overhang. The gable detailing and square tower are transitional to the closely related Stick style.\n\nSPINDLEWORK: CROSS-GABLED ROOF:  QUEEN ANNE  \nOrange, New Jersey; ca. 1880. Dodd House. Note the roof cresting, patterned chimney, and heavy turned porch supports. This early East Coast example resembles many houses of the half-timbered Queen Anne subtype, but lacks half-timbered detailing.\n\n**FREE CLASSIC:CROSS-GABLED ROOF**\n\nFREE CLASSIC: CROSS-GABLED ROOF:  QUEEN ANNE  \nCambridge, Massachusetts; 1890. J. Merrill Brown, architect. The siding and shutters are later additions.\n\nFREE CLASSIC: CROSS-GABLED ROOF:  QUEEN ANNE  \nMarshall, Michigan; 1884. Page House. Note the matching front and side porches with grouped columns set on pedestals. The short, broad tower is less common than more slender versions.\n\nFRONT-GABLED ROOF\u2014About 20 percent of Queen Anne houses have a full-width front gable which dominates the front facade. This form occurs most frequently in detached urban houses. A tower, when present, is usually placed at one corner of the front facade.\n\n**SPINDLEWORK:FRONT-GABLED ROOF**\n\nSPINDLEWORK: FRONT-GABLED ROOF:  QUEEN ANNE  \nChicago, Illinois; 1898.\n\nSPINDLEWORK: FRONT-GABLED ROOF:  QUEEN ANNE  \nNew Haven, Connecticut; late 19th century.\n\nSPINDLEWORK: FRONT-GABLED ROOF:  QUEEN ANNE  \nRochester, New York; late 19th century. Front-gabled forms may have shallow cross gables extending outward the width of a single door or window, as seen in this example.\n\nSPINDLEWORK: FRONT-GABLED ROOF:  QUEEN ANNE  \nSt. Paul, Minnesota; ca. 1896. Stevens House. This early photograph shows the elaborate wall-texture detailing that was originally present on many modest houses but rarely survives today.\n\nSPINDLEWORK: FRONT-GABLED ROOF:  QUEEN ANNE  \nCambridge, Massachusetts; 1886. Parry House. Note the brick first story with wood shingling above. The porch has been modified.\n\n**FREE CLASSIC:FRONT-GABLED ROOF**\n\nFREE CLASSIC: FRONT-GABLED ROOF:  QUEEN ANNE  \nNew Haven, Connecticut; late 19th century.\n\nFREE CLASSIC: FRONT-GABLED ROOF:  QUEEN ANNE  \nHartford, Connecticut; late 19th century. Note the use of board siding on the first story, with shingles above, a common pattern. The two front doors indicate that this is a \"two-decker\" duplex with separate dwelling units on the first and second floors.\n\nFREE CLASSIC: FRONT-GABLED ROOF:  QUEEN ANNE  \nDenver, Colorado; 1892. Balcomb and Rice, architects. Note the elaborate Palladian window with decorative swags that recur above the second-story porch.\n\nFREE CLASSIC: FRONT-GABLED ROOF:  QUEEN ANNE  \nSan Francisco, California; late 19th century. Many San Francisco Queen Anne houses combine classical columns with elaborate spindlework detailing used elsewhere on the facade. These closely spaced narrow urban houses are the West Coast version of Queen Anne town houses.\n\n**HALF-TIMBERED**\n\nHALF-TIMBERED:  QUEEN ANNE  \nBuffalo, New York; late 19th century. Note the row of multiple windows in the gable; such window rows are common only in this subtype (see also the Miller House, the Day House, the Watts-Sherman House, the Baldwin House, and the Cutler House).\n\nHALF-TIMBERED:  QUEEN ANNE  \nChicago, Illinois; 1888. Miller House; G. A. Garnsey, architect. Shows the close relationship of this subtype to the Tudor style that grew from it. The multiple wall materials (stone, stucco, wood, and wood shingles) and many changes in wall plane mark this house as Queen Anne.\n\nHALF-TIMBERED:  QUEEN ANNE  \nBrookline, Massachusetts; late 19th century; E. A. P. Newcomb, architect.\n\nHALF-TIMBERED:  QUEEN ANNE  \nHartford, Connecticut; 1884. Day House; Francis Kimball, architect. Note the light-colored limestone walls banded with brownstone.\n\nHALF-TIMBERED:  QUEEN ANNE  \nBrookline, Massachusetts; ca. 1880. Toby House. Note the paneled brick chimney (see also the house in Buffalo, the Day House, the Watts-Sherman House, the Baldwin House, and the Cutler House); these are most common on half-timbered and patterned brick examples, although simply decorated chimney _tops_ are seen in all subtypes.\n\nHALF-TIMBERED:  QUEEN ANNE  \nNewport, Rhode Island; 1876. Watts-Sherman House; H. H. Richardson, architect. This is regarded as the first American Queen Anne house.\n\nHALF-TIMBERED:  QUEEN ANNE  \nNewport, Rhode Island; 1878. Baldwin House; Potter and Robinson, architects.\n\nHALF-TIMBERED:  QUEEN ANNE  \nRochester, New York; ca. 1880. Cutler House.\n\n**PATTERNED MASONRY**\n\nPATTERNED MASONRY:  QUEEN ANNE  \nNew Haven, Connecticut; 1886. Treat House. It is hard to photograph the textured patterns in the dark red brick, typical of this subtype, in a manner that shows up well in reproduction.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nHerkimer, New York; late 19th century. Suiter House.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nNew Haven, Connecticut; ca. 1895. Note the cornice patterns formed by the brickwork.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nHartford, Connecticut; late 19th century. George Keller, architect. The extensive full-width or wrap-around porches common on the spindlework and free classic subtypes are rare on patterned masonry houses; usually only an entry porch is present, as seen here.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nHartford, Connecticut; late 19th century. The one-story wooden projection on the right is a later addition.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nCincinnati, Ohio; 1882. Bell House; S. Hannaford, architect. An example with stone, rather than brick, walls. Note the shaped, parapeted gables, a rare but characteristic feature of this subtype.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nChicago, Illinois; ca. 1884. Wells House; Wheelock and Clay, architects. Note extensive roof cresting here and in George Keller's house in Hartford.\n\nPATTERNED MASONRY:  QUEEN ANNE  \nRochester, New York; ca. 1883. Harvey Ellis, architect. An American copy of a design by the English proponent of the Queen Anne style Richard Norman Shaw.\n\nTOWN HOUSE\u2014Detached Queen Anne urban houses usually have front-gabled roofs (as in the type just described). Attached row houses are uncommon but occur in both gabled and flat-roofed forms. Each attached unit may be individually distinguishable on the facade or may be part of a larger facade design.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  QUEEN ANNE  \nBoston, Massachusetts; 1880. Note the false gable with a mansard roof behind. Sculptured terra-cotta tiles add richness to this facade.\n\nTOWN HOUSE:  QUEEN ANNE  \nSavannah, Georgia; 1892. McMillan Houses. A row of attached town houses behind one unified facade.\n\nTOWN HOUSE:  QUEEN ANNE  \nSavannah, Georgia; 1892. McMillan Houses. Close-up detail of the patterned brickwork executed in two colors of brick.\n\nTOWN HOUSE:  QUEEN ANNE  \nRochester, New York; 1870s. These are half-timbered examples.\n\nTOWN HOUSE:  QUEEN ANNE  \nHartford, Connecticut; 1888.\n\nTOWN HOUSE:  QUEEN ANNE  \nCamden, New Jersey; 1886. Taylor House; Wilson Eyre, architect. A unique town house of limestone and brick with a large, shaped, parapeted gable.\n\nTOWN HOUSE:  QUEEN ANNE  \nWashington, D.C.; late 19th century. Many miles of these simplified, patterned brick row houses were built in eastern cities. Note the false gable roof; these have deteriorated and been removed from many remaining examples.\n\nTOWN HOUSE:  QUEEN ANNE  \nCleveland, Ohio; ca. 1890.\n\n## Variants and Details\n\nThe Queen Anne, like the Stick style, uses wall surfaces as primary decorative elements. This is accomplished in two ways: (1) by avoiding plain flat walls through such devices as bays, towers, overhangs, and wall projections, and (2) by using several wall materials of differing textures wherever expanses of planar wall do occur.\n\nDEVICES FOR AVOIDING FLAT WALL SURFACES\u2014Irregularities in ground plan were facilitated by the widespread adoption of balloon framing techniques in the late 19th century (see this page). Queen Anne houses make full use of this freedom by incorporating frequent bay windows and towers, as well as through the use of wall insets or projections which provide random changes in the horizontal continuity of the wall plane. Other devices are used to avoid planar walls in _elevation\u2014_ that is, to provide a similar discontinuity of the wall plane _vertically;_ these devices usually mimic the Medieval use of overhanging gables and upper stories. Particularly characteristic features are roof gables that overhang bay windows shaped into the wall below (cutaway bay windows). These occur in over half of all Queen Anne houses. In high-style examples entire gables or second stories are sometimes cantilevered out beyond the plane of the walls below. Many modest examples have less elaborate false overhangs; these are formed from moldings or pent roofs applied directly to the flat wall surfaces.\n\nWALL TEXTURE VARIATIONS\u2014Differing wall textures are a hallmark of Queen Anne houses. These are most commonly achieved with patterned wood shingles shaped into varying designs (note that this original shinglework patterning has been replaced by other materials on many surviving examples). In masonry houses, texture is obtained by using differing patterns of brick courses or brick of different colors. Other materials also provide texture, including terra-cotta panels molded into a wide variety of low-relief designs. Contrasting materials are also commonly used on the different stories of Queen Anne houses (shingle over clapboard or over brick is most common).\n\nTOWERS\u2014Towers are a common Queen Anne feature and may be round, square, or polygonal (the square form is the least common). These are of varying height and may rise from ground level, be cantilevered out at the second floor, or show other variations in position. In later examples the tower may appear to be less a separate design element than a mere bulge growing from the main mass of the house (see also the Shingle style). Round or polygonal wooden towers are particularly characteristic of the Queen Anne (round masonry towers may be Richardsonian Romanesque; square towers are more common on Stick, Italianate, or Second Empire houses). Towers placed at a front facade corner are most often Queen Anne, whereas those embraced within an L or centered on the front facade are equally common in several other styles.\n\nPORCHES\u2014Extensive one-story porches are common and accentuate the asymmetry of the facade. These always include the front entrance area and cover part or all of the front facade; they also commonly extend along one or both sides of the house. Second-story porches may be present; recessed porches sometimes occur in gables, second stories, or towers.\n\nOTHER DETAILS\u2014Door and window surrounds tend to be simple in Queen Anne houses. Window sashes usually have only a single pane of glass; a frequent elaboration has a single large clear pane surrounded by additional small rectangular panes on one or more sides. These small panes are often of colored glass. Some later examples have curved glass in tower windows. Doors commonly have delicate incised decorative detailing and a single large pane of glass set into the upper portion. Gables are commonly decorated with patterned shingles or more elaborate motifs.\n\nQUEEN ANNE\n\n **PRINCIPAL ROOF FORMS**\n\nQUEEN ANNE\n\n **TYPICAL ROOF-PITCH CHRONOLOGY**\n\nQUEEN ANNE\n\n **PORCHES**\n\nPLACEMENT\n\nTYPICAL EMBELLISHMENTS\n\nQUEEN ANNE\n\n **DEVICES TO AVOID FLAT WALL SURFACES**\n\nQUEEN ANNE\n\n **TYPICAL WALL-TEXTURE PATTERNS**\n\nWOOD SHINGLES\n\nThese and numerous other variations may be used either singly or in combination; shingle patterns are most frequent in gables and in horizontal bands between floors, but may occur anywhere.\n\nQUEEN ANNE\n\n **TYPICAL WALL-TEXTURE PATTERNS**\n\nBRICK\n\nThese and other patterns of brick texture and color are found in chimneys and in masonry walls.\n\nQUEEN ANNE\n\n **TYPICAL GABLE DETAILING**\n\n## Occurrence\n\nThis was the dominant style of domestic building during the period from about 1880 until 1900; it persisted with decreasing popularity through the first decade of this century. In the heavily populated northeastern states the style is somewhat less common than elsewhere. There, except for resort areas, it is usually more restrained in decorative detailing and is more often executed in masonry. Moving southward and westward the style increases steadily in dominance and ebullience; California and the resurgent, cotton-rich states of the New South have some of the most fanciful examples.\n\n## Comments\n\nThe style was named and popularized by a group of 19th-century English architects led by Richard Norman Shaw. The name is rather inappropriate, for the historical precedents used by Shaw and his followers had little to do with Queen Anne or the formal Renaissance architecture that was dominant during her reign (1702\u20131714). Instead, they borrowed most heavily from the late Medieval models of the preceding Elizabethan and Jacobean eras. The half-timbered and patterned masonry American subtypes are most closely related to this work of Shaw and his colleagues in England. The spindlework and free classic subtypes are indigenous interpretations.\n\nThe half-timbered Watts Sherman House, built at Newport, Rhode Island, in 1874, is generally considered to be the first American example of the style. A few high-style examples followed in the 1870s, and by 1880 the style was being spread throughout the country by a host of pattern books (many selling plans for their illustrated designs) and by the leading architecture magazine, _American Architect and Building News_ (through illustrated examples, but with faint textual praise).\n\nDespite having been introduced by architects and illustrated in _AABN,_ the Queen Anne style was not widely favored by architects, who preferred the contemporaneous Shingle and experimentation with early Eclectic styles. Instead, the style owed its popularity to the public's enthusiastic embrace and the pattern books and mail-order house plans that allowed them to build a Queen Anne house. The expanding railroad network expedited this process by making pre-cut architectural details conveniently available throughout much of the nation. Hudson Holly's 1878 _Modern Dwellings in Town and Country_ was likely the first pattern book specifically to promote Queen Anne style. But it was soon joined by dozens of others offering Queen Anne plans, among them those of William Comstock, the Palliser brothers, Robert Shoppell, and George Barber.\n\nWhile the earliest American examples followed Shaw's early half-timbered designs, during the 1880s the inventive American spindlework interpretation became dominant. Throughout the 1880s and 1890s a relatively few high-style urban examples imitated Shaw's later English models, which were executed in masonry. In the decade of the 1890s, encouraged by the Classical theme of Chicago's Columbian Exposition of 1893, the free classic adaptation became widespread. It was but a short step from these to the early asymmetrical Colonial Revival houses and the symmetrical Neoclassical houses, which, along with other competing styles, fully supplanted the Queen Anne style after about 1910.\n\n# VICTORIAN HOUSES\n\n# Shingle\n\n# 1880\u2013ca. 1910\n\n## Identifying Features\n\nWall cladding and roofing of continuous wood shingles (shingled walls may occur on second story only; original wooden roofing now replaced by composition shingles on most examples); shingled walls without interruption at corners (no corner boards); asymmetrical facade with irregular, steeply pitched roof line; roofs usually have intersecting cross gables and multi-level eaves; commonly with extensive porches (may be small or absent in urban examples).\n\n## Principal Subtypes\n\nFive principal subtypes can be distinguished:\n\nHIPPED ROOF WITH CROSS GABLES\u2014About 15 percent of Shingle houses have hipped roofs with lower cross gables. Asymmetrical gable arrangements, similar to the typical Queen Anne shape, are most common, but Shingle houses may also show paired, symmetrical cross gables.\n\n**HIPPED ROOFWITH CROSS GABLES**\n\nHIPPED ROOF WITH CROSS GABLES:  SHINGLE  \nDallas, Texas; late 19th century. Bookhout House. A transitional house with Queen Anne form, but Shingle porches and porte cochere.\n\nHIPPED ROOF WITH CROSS GABLES:  SHINGLE  \nNew London, Connecticut; late 19th century. Note the varied dormer shapes\u2014hipped, eyebrow, and gabled; the Palladian motif created above the line of gable windows; and the extensive porch with Romanesque arches.\n\nHIPPED ROOF WITH CROSS GABLES:  SHINGLE  \nBrookline, Massachusetts; late 19th century. E. A. P. Newcomb, architect. Note the rounded bay and oriel windows.\n\nHIPPED ROOF WITH CROSS GABLES:  SHINGLE  \nHelena, Montana; 1889. Boardman House. Note how the tower roof on the upstairs porch blends into the house with a continuous roof line (rather than being emphasized as a separate conical element, as is typical in Queen Anne towers).\n\nHIPPED ROOF WITH CROSS GABLES:  SHINGLE  \nBlue Ridge Summit, Pennsylvania; late 19th century. Menz House. A large, symmetrical example.\n\nSIDE-GABLED ROOF\u2014About 20 percent of Shingle houses have side-gabled roofs; many of these have asymmetrically placed towers on the front facade.\n\n**SIDE-GABLEDROOF**\n\nSIDE-GABLED ROOF:  SHINGLE  \nEmporia, Kansas; late 19th century. The wide shingles on this house are probably later additions.\n\nSIDE-GABLED ROOF:  SHINGLE  \nLexington, Kentucky; late 19th century. Note the three different dormer shapes crowded into the front roof.\n\nSIDE-GABLED ROOF:  SHINGLE  \nBuffalo, New York; ca. 1885. Note the elaborate detailing of the side gable and the way the enormous tower seems to grow from the roof.\n\nSIDE-GABLED ROOF:  SHINGLE  \nBrookline, Massachusetts; late 19th century. This early photograph shows the elaborate original wall detailing that has been lost over the years in most examples.\n\nSIDE-GABLED ROOF:  SHINGLE  \nSan Francisco, California; ca. 1908. Alto Plaza. This urban version (the left half of a double house) lacks the porches of suburban examples. Note Palladian-type window in the dormer and wide roof overhang. Known locally as the First Bay Tradition, it is the West Coast's version of the Shingle Style.\n\nFRONT-GABLED ROOF\u2014About 20 percent of Shingle houses have a front gable which dominates the main facade; subordinate cross gables and towers may be added.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  SHINGLE  \nLouisville, Kentucky; late 19th century. A modest urban example.\n\nFRONT-GABLED ROOF:  SHINGLE  \nMontauk, New York; late 19th century. Benson House. A seaside example with extensive porches.\n\nFRONT-GABLED ROOF:  SHINGLE  \nNewport, Rhode Island; late 19th century. Richardson House.\n\nFRONT-GABLED ROOF:  SHINGLE  \nNew Haven, Connecticut; 1889. A massive front gable that clearly dominates the small, shallow side gable marks this example as the front-gabled, rather than cross-gabled, subtype of the style.\n\nFRONT-GABLED ROOF:  SHINGLE  \nCorning, New York; late 19th century. The tower and front facade are united by the extended front wall and the upper band of windows.\n\nFRONT-GABLED ROOF:  SHINGLE  \nCorning, New York; late 19th century. Note the paneled chimney and patterned shingles.\n\nFRONT-GABLED ROOF:  SHINGLE  \nDallas, Texas; 1909. Miller House.\n\nFRONT-GABLED ROOF:  SHINGLE  \nBristol, Rhode Island; 1887. Low House; McKim, Mead & White, architects. A now-demolished landmark example of the style.\n\nCROSS-GABLED ROOF\u2014About 20 percent of Shingle houses have cross-gabled roofs; most are of L or T plan and have secondary cross gables and dormers intersecting the principal roof line. Subordinate hipped sections may also be added.\n\n**CROSS-GABLED ROOF**\n\nCROSS-GABLED ROOF:  SHINGLE  \nBaltimore, Maryland; late 19th century. The Gothic (pointed arch) windows are unusual in Shingle houses.\n\nCROSS-GABLED ROOF:  SHINGLE  \nMeriden, Connecticut; ca. 1890. Hale House. Note the Romanesque arched entry porch.\n\nCROSS-GABLED ROOF:  SHINGLE  \nKansas City, Missouri; late 19th century.\n\nCROSS-GABLED ROOF:  SHINGLE  \nBrookline, Massachusetts; late 19th century. The half-timbered detailing seen here and in Tuxedo Park is unusual in Shingle houses.\n\nCROSS-GABLED ROOF:  SHINGLE  \nCleveland, Ohio; late 19th century. McNairy House. This house, with its colonnaded porch, Palladian window, and overall symmetry shows a strong classical influence.\n\nCROSS-GABLED ROOF:  SHINGLE  \nNewport, Rhode Island; 1883. Bell House; McKim, Mead & White, architects.\n\nCROSS-GABLED ROOF:  SHINGLE  \nTuxedo Park, New York; late 19th century. This large house lacks the unified facade seen in most examples of the style.\n\nGAMBREL ROOF\u2014About 25 percent of Shingle houses have gambrel roofs. Normally a full second story is incorporated into the steeper, lower slope of the gambrel, giving a one-story appearance. Gambreled cross gables are usually present.\n\n**GAMBREL ROOF**\n\nGAMBREL ROOF:  SHINGLE  \nNebraska City, Nebraska; 1902. Morton House. The off-center doorway and the asymmetrical upper story emphasize the unusual roof form: the left half is gambreled, the right gabled.\n\nGAMBREL ROOF:  SHINGLE  \nCincinnati, Ohio; late 19th century. The walls of these two houses have unusually large shingles.\n\nGAMBREL ROOF:  SHINGLE  \nKansas City, Missouri; 1890. Alderson House. An uncommon three-story example.\n\nGAMBREL ROOF:  SHINGLE  \nSalisbury, North Carolina; late 19th century. The cantilevered balcony over the entry is unusual.\n\nGAMBREL ROOF:  SHINGLE  \nWichita, Kansas; 1887.\n\nGAMBREL ROOF:  SHINGLE  \nNew Haven, Connecticut; late 19th century. Note the dramatic use of windows of varying shape in the dominant front gambrel (see also previous image).\n\nGAMBREL ROOF:  SHINGLE  \nEast Hampton, New York; 1898. Quakenbush House. Cyrus L. W. Eidlitz, architect. Note that both front-facing gambrels have a three-part window (with a blind-arch above to simulate a Palladian window). In the larger slightly asymmetrical gambrel on the left this is grouped with a small triple window above and a small elliptical window to the side. In the smaller gambrel on the right it stands on its own.\n\nGAMBREL ROOF:  SHINGLE  \nGainesville, Texas; late 19th century. Although this house has a side-gabled roof, the dominant front-facing gambrel places it in the gambrel subtype.\n\n## Variants and Details\n\nUnlike most of the 19th-century styles that preceded it, the Shingle does not emphasize decorative detailing at doors, windows, cornices, porches, or on wall surfaces. Instead it aims for the effect of a complex shape enclosed within a smooth surface (the shingled exterior) which unifies the irregular outline of the house. Most variants and details are designed to enhance either the irregularity of the shape or the uniformity of its surface. Decorative detailing, when present, is used sparingly. It is the first style that begins to emphasize the volumetric spaces within the house more than exterior surface details.\n\nTowers, found in about one-third of Shingle houses, are more likely to appear as partial bulges or as half-towers rather than as fully developed elements. Tower roofs are frequently blended into the main volume of the house by a continuous roof line. Porch supports are most commonly either slender, unadorned wooden posts or massive piers of stone or shingle cladding. Window surrounds are simple; bay windows, multiple windows, and walls curving into windows are common. Massive Romanesque or Syrian arches may be used on porches or entrances. Palladian windows and simple classical columns, both borrowed from the contemporaneous early phases of the Colonial Revival, are the most common decorative details.\n\nSHINGLE\n\n **TYPICAL PORCH SUPPORTS**\n\nSHINGLE\n\n **TYPICAL WINDOWS**\n\nSHINGLE\n\n **TYPICAL DORMERS**\n\nSHINGLE\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nMost Shingle houses were built between 1880 and 1900, with a relatively few examples dating from the late 1870s and from the first decade of the 20th century. The style began and reached its highest expression in seaside resorts of the northeastern states. Fashionable summer destinations such as Newport, Cape Cod, eastern Long Island, and coastal Maine had numerous architect-designed cottages in the style, many of which survive today. From this fashionable base, well publicized in contemporary architectural magazines, the style spread throughout the country, and scattered examples can be found today in all regions. It never gained the wide popularity of its contemporary, the Queen Anne style, and thus Shingle houses are relatively uncommon except in coastal New England.\n\n## Comments\n\nThe Shingle style, like the Stick and spindlework Queen Anne, was a uniquely American adaptation of other traditions. Its roots are threefold: (1) from the Queen Anne it borrowed wide porches, shingled surfaces, and asymmetrical forms; (2) from the Colonial Revival it adapted gambrel roofs, rambling lean- to additions, classical columns, and Palladian windows; (3) from the contemporaneous Richardsonian Romanesque it borrowed an emphasis on irregular, sculpted shapes, Romanesque arches, and, in some examples, stone lower stories (some scholars consider the Shingle to be merely the wooden phase of the masonry Richardsonian Romanesque, but the styles also have many dissimilarities).\n\nThe Shingle style was an unusually free-form and variable style; without the ubiquitous shingle cladding it would be difficult to relate many of its different expressions. One reason for this great range of variation is that it remained primarily a high-fashion, architect's style, rather than becoming widely adapted to mass vernacular housing, as did the contemporaneous Queen Anne. Among the innovative designers working in the style were Henry Hobson Richardson and William Ralph Emerson of Boston; John Calvin Stevens of Portland, Maine; McKim, Mead & White, Bruce Price, and Lamb and Rich of New York; Wilson Eyre of Philadelphia; and Willis Polk of San Francisco.\n\n# VICTORIAN HOUSES\n\n# Richardsonian Romanesque\n\n# 1880\u20131900\n\n## Identifying Features\n\nRound-topped arches occurring over windows, porch supports, or entrance; masonry walls, usually with rough-faced, squared stonework; most have towers which are normally round with conical roofs; facade usually asymmetrical.\n\n## Principal Subtypes\n\nThree principal subtypes can be distinguished:\n\nHIPPED ROOF WITH CROSS GABLES\u2014About two-thirds of Richardsonian Romanesque houses have hipped roofs with one or more lower cross gables. Most commonly there are two cross gables, one front-facing and the other side-facing, each asymmetrically placed on its respective facade. This shape is similar to the typical Queen Anne roof form.\n\n**HIPPED ROOF WITH CROSS GABLES**\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nSt. Charles, Missouri; 1885. Atkinson House. This example is unusual in lacking rough-faced stonework in the facade above the foundation level.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nRichmond, Virginia; late 19th century.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nLexington, Kentucky; late 19th century. Dark red brick and white stone detailing provide startling contrast in this exuberant example. Note the tiny Romanesque-arched basement window at the left, the inventive open arch of the porch, and the exaggerated width of the stone window arches.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nProvo, Utah; 1892. Reed House. Note the eyebrow dormer.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nRichmond, Virginia; late 19th century. The polychromed walls show contrasting light brick and dark stonework.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nSavannah, Georgia; 1891. Tiedeman House; A. Eichberg, architect. Note the contrasting trim of light colored stone. As in the previous image, there are wall dormers around the tower roof.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nPueblo, Colorado (vicinity); 1890. Orman House; William Lang, architect. There are three towers, each with a different roof form and window type. Note the tall vertically modeled chimney.\n\nHIPPED ROOF WITH CROSS GABLES:  RICHARDSONIAN ROMANESQUE  \nLouisville, Kentucky; 1893. Conrad House; Clark and Loomis, architects. Two towers with differing roof shapes are used here. Note the floral detailing in the gable and elsewhere on the facade.\n\nOTHER ROOF TYPES\u2014A variety of other roof forms also occur on Richardsonian Romanesque houses. Among the most frequent are side-gabled, cross-gabled, mansard, and simple hipped roofs.\n\n**OTHER ROOF TYPES**\n\nOTHER ROOF TYPES:  RICHARDSONIAN ROMANESQUE  \nSt. Louis, Missouri; 1886. Lionberger House; Henry Hobson Richardson, architect. One of Richardson's fortress-like designs. The towers have very low pitched roofs and barely protrude from the house block.\n\nOTHER ROOF TYPES:  RICHARDSONIAN ROMANESQUE  \nSt. Louis, Missouri; 1890. Huse House; Eames and Young, architects. The wide roof overhang is unusual in the style.\n\nOTHER ROOF TYPES:  RICHARDSONIAN ROMANESQUE  \nChicago, Illinois; 1886. Glessner House; Henry Hobson Richardson, architect. Another of Richardson's few domestic designs, this presents a fortress-like face to the street.\n\nOTHER ROOF TYPES:  RICHARDSONIAN ROMANESQUE  \nKerrville, Texas; ca. 1895. Shreiner House; Alfred Giles, architect. In this example a full Romanesque facade with a two-tiered porch and end towers has been added to two earlier houses to make a single larger dwelling. Note squared openings in railing design and in basement windows of illustration of  and .\n\nOTHER ROOF TYPES:  RICHARDSONIAN ROMANESQUE  \nWashington, D.C.; 1880. Heurich House, J. G. Myers, architect. The facade rises a full three stories in this landmark example.\n\nTOWN HOUSE\u2014Richardsonian Romanesque was frequently used for detached urban houses, which typically have front-gabled or mansard roofs; attached row houses in this style are less common.\n\n**TOWN HOUSE**\n\nTOWN HOUSE:  RICHARDSONIAN ROMANESQUE  \nLouisville, Kentucky; 1900. A simple, front-gabled example. It would be hard to miss the Romanesque arches.\n\nTOWN HOUSE:  RICHARDSONIAN ROMANESQUE  \nLouisville, Kentucky; late 19th century. Groups of columns support the Romanesque arches in this front-gabled example.\n\nTOWN HOUSE:  RICHARDSONIAN ROMANESQUE  \nRichmond, Virginia; late 19th century. Here the mansard roof provides a background for the wall dormer and tower.\n\nTOWN HOUSE:  RICHARDSONIAN ROMANESQUE  \nSt. Paul, Minnesota; 1887. Riley Row; Wilcox and Johnston, architects. Attached Romanesque town houses are unusual; groups with many uniform units, such as this, are very rare.\n\n## Variants and Details\n\nRichardsonian Romanesque houses are always of masonry and usually show at least some rough-faced, squared (ashlar) stonework. Frequently two or more colors or textures of stone or brick are combined to create decorative wall patterns. Wide, rounded (Romanesque) arches are a key identifying feature of the style. These may occur above windows or porch supports or over entryways. Most commonly the arches rest on squat columns, but some are supported on massive piers or are incorporated directly into wall surfaces. Column capitals and wall surfaces may be ornamented with floral or other decorative details. Windows are usually deeply recessed into the masonry wall and have only a single pane of glass per sash. The characteristic arched windows sometimes have small decorative columns (colonnettes) on each side. Groupings of three or more arched or rectangular windows occur in over half of the examples.\n\nTowers occur in about 75 percent of free-standing Richardsonian Romanesque houses; these are most commonly round, although polygonal and squared versions are found. A second tower occurs in about 15 percent. Tower roofs are usually conical, but may be convex. Dormers are present in about half of Richardsonian Romanesque houses. Most commonly these are parapeted and gabled wall dormers, but hipped dormers, eyebrow dormers, and other variations occur.\n\nRICHARDSONIAN ROMANESQUE\n\n **ROUNDARCHES**\n\nRICHARDSONIAN ROMANESQUE\n\n **TYPICAL DECORATIVE DETAIL**\n\nUsually floral and interlacing\n\nRICHARDSONIAN ROMANESQUE\n\n **COMMON ELABORATIONS**\n\n## Occurrence\n\nThe innovative Boston architect Henry Hobson Richardson (1838\u20131886) designed houses during the 1860s and 1870s in the then fashionable Second Empire, Queen Anne, or Stick styles; in 1879\u20131880 he executed the first of his few Romanesque houses, the rectory for his monumental Trinity Church in Boston. Richardson's Romanesque adaptations became very popular for large public buildings during the 1880s but he completed only a few more houses in the style before his premature death in 1886. Other architects, following Richardson's lead, also designed houses in the 1880s, but these were uncommon. In 1888, a sympathetic monograph on Richardson's life and work was published which greatly increased interest in the style. Most domestic examples are an outgrowth of this revival and were built in the 1890s. Because they were always of solid masonry construction (masonry veneering techniques were not yet perfected), Richardsonian Romanesque houses were much more expensive to build than were those Late Victorian styles which could be executed in wood. For this reason, they are mostly architect-designed landmarks and were never common. Scattered examples occur throughout the country but are most frequent in the larger cities of the northeastern states.\n\n## Comments\n\nIn the middle decades of the 19th century, European Romanesque models were sometimes used for American public and commercial buildings (the Romanesque Revival style), but these precedents reached American _houses_ only in a later 19th-century form shaped by the powerful personality and talent of Henry Hobson Richardson. Born in Louisiana, Richardson attended Harvard and then studied architecture at the prestigious \u00c9cole des Beaux-Arts in Paris (he was only the second American to do so). He returned to the United States after the Civil War and opened an office in New York, which he subsequently moved to Boston. During the 1870s he evolved his strongly personal style, which incorporated Romanesque forms and which, like its mid-century predecessor, was applied principally to large public buildings. Unlike the earlier and more correct Romanesque revival, Richardson borrowed from many sources. He incorporated the polychromed walls seen in the contemporary late Gothic Revival. His arches are frequently not truly Romanesque but Syrian, an early Christian form which springs from ground level rather than from a supporting pedestal. Most importantly, he stressed unusual, sculpted shapes which give his buildings great individuality. His followers were usually less inventive; most houses in this style merely add Romanesque detailing to the typical hipped-with-cross-gables shape of the then dominant Queen Anne style.\n\n# VICTORIAN HOUSES\n\n# Folk Victorian\n\n# ca. 1870\u20131910\n\n## Identifying Features\n\nPorches with spindlework detailing (turned spindles and lace-like spandrels) or flat, jigsaw cut trim appended to National Folk (post-railroad) house forms; symmetrical facade (except gable-front-and-wing subtype); cornice-line brackets are common.\n\n## Principal Subtypes\n\nFive principal subtypes occur. These are closely related to the subtypes of National Folk (post-railroad) houses.\n\nFRONT-GABLED ROOF\u2014Like their pure folk counterparts, two-story, front-gabled forms with Victorian detailing are most common in the northeastern states, while one-story, narrow shotgun forms are generally found in the urban South.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  FOLK VICTORIAN  \nNew Orleans, Louisiana; late 19th century. A shotgun form, one room wide, with Italianate windows and Queen Anne spindlework porch and gable detailing.\n\nFRONT-GABLED ROOF:  FOLK VICTORIAN  \nNew Orleans, Louisiana; late 19th century. A narrow, two-ranked shotgun, with Italianate windows and brackets. Note the hipped front roof that replaces the more usual front gable.\n\nFRONT-GABLED ROOF:  FOLK VICTORIAN  \nFernandez, Florida; late 19th century. A large two-story front-gabled form with a two-tiered porch. This example has flat jigsaw cut upper balustrade and gable trim.\n\nFRONT-GABLED ROOF:  FOLK VICTORIAN  \nNew London, Connecticut; late 19th century. An inventive craftsman's interpretation of spindlework porch detailing.\n\nGABLE FRONT AND WING\u2014Both one- and two-story Victorian versions of this popular folk form are found throughout the country but are particularly common in the southern states.\n\n**GABLE FRONT AND WING**\n\nGABLE FRONT AND WING:  FOLK VICTORIAN  \nBartlett, Texas; late 19th century. This example has modest spindlework porch detailing, with a typical Queen Anne pent roof beneath the gables. The porches in this subtype are usually confined within the L formed by the gable and wing as seen here and in the following two images.\n\nGABLE FRONT AND WING:  FOLK VICTORIAN  \nHillsboro, Texas; late 19th century. An inventive craftsman's version of a spindlework porch. Note the shingled gable and bracketed window hood to the right. Slight differences in roof pitch and height between the gable front portion and the wing, as well as differing gable and window detailing in the two portions, probably indicate that they were built at different times. Gable-front-and-wing forms were often the result of such melding.\n\nGABLE FRONT AND WING:  FOLK VICTORIAN  \nSinclairville, New York, vicinity; late 19th century.\n\nGABLE FRONT AND WING:  FOLK VICTORIAN  \nLaurens, South Carolina; late 19th century. Huff House. This example has unusually elaborate spindlework detailing and eave brackets. The high-pitched gable is less common than the lower-pitched ones seen in the other examples. Note how the porch roof has been extended upward to simulate a full-scale Victorian tower.\n\nGABLE FRONT AND WING:  FOLK VICTORIAN  \nLittle River, South Carolina; ca. 1910. Ellis House. Note the contrast between the body of the house, a simple two-story folk form with little detailing, and the elaborate spindlework porch with paired gables.\n\nSIDE-GABLED ROOF, ONE-STORY\u2014This common subtype includes Victorian versions of both the hall-and-parlor (one room deep) and massed, side-gabled (two or more rooms deep) folk forms. It is widely distributed through the country.\n\n**SIDE-GABLED ROOF, ONE-STORY**\n\nSIDE-GABLED ROOF, ONE-STORY:  FOLK VICTORIAN  \nBastrop, Texas; 1890. Elzner House.\n\nSIDE-GABLED ROOF, ONE-STORY:  FOLK VICTORIAN  \nRosin, North Carolina; ca. 1890. McPhail House. Centered front gables are common in this subtype (see also Kuhn House and Graves House).\n\nSIDE-GABLED ROOF, ONE-STORY:  FOLK VICTORIAN  \nSan Antonio, Texas; ca. 1880. Kuhn House.\n\nSIDE-GABLED ROOF, ONE-STORY:  FOLK VICTORIAN  \nGalveston, Texas; 1892. Eimar House. Note the typical Queen Anne cutaway corners to the right. First stories raised high above the ground are common along the Gulf Coast.\n\nSIDE-GABLED ROOF, ONE-STORY:  FOLK VICTORIAN  \nSt. Francois County, Missouri; late 19th century. Graves House. The projecting central wing of this carefully detailed example is unusual.\n\nSIDE-GABLED-ROOF, TWO-STORY\u2014Most examples of this subtype are I-houses (one room deep) to which Victorian detailing in varying degrees of exuberance was added. They are common in all parts of the country.\n\n**SIDE-GABLED ROOF, TWO-STORY**\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nLaurens, South Carolina; late 19th century. Easterby House. Like the Howell House, this I-house also has Italianate brackets and bay windows.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nClinton, Missouri, vicinity; ca. 1879. Noble House. An I-house with modified Victorian porch detailing.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nCanton, Mississippi; late 19th century. Two-tiered, full-facade porches such as this are common in the South (see also the house in Waveland).\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nHampton, South Carolina; ca. 1880. Here a full-width, one-story porch is combined with a two-story entry porch.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nAnnapolis, Maryland; late 19th century. A simple side-gabled town house with modest spindlework porch detailing.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nWaveland, Mississippi, late 19th century. The elaborate jigsaw cut porch detailing of this example shows Gothic influence in the paired gables.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nLansing, North Carolina, vicinity; ca. 1890. Howell House. Here a two-tiered, full-height entry porch is added to an I-house form. Full-height entry porches on Folk Victorian houses are always two-tiered; two-story columns would indicate a classically influenced, styled house.\n\nSIDE-GABLED ROOF, TWO-STORY:  FOLK VICTORIAN  \nHenderson County, North Carolina; 1877. Elliott House. An unusually elaborate I-house. If this transitional example had arched windows or other Italianate detailing in addition to the eave brackets, it would clearly belong to the Italianate, rather than the Folk Victorian, style.\n\nPYRAMIDAL\u2014Both one- and two-story versions of this folk form were often given Victorian detailing in the southern states but are less common elsewhere.\n\n**PYRAMIDAL ROOF**\n\nPYRAMIDAL:  FOLK VICTORIAN  \nSan Antonio, Texas; 1903. Pancoast House. Note the flat, jig-saw cut porch frieze elaborated with stars, a frequent motif in the Lone Star state.\n\nPYRAMIDAL:  FOLK VICTORIAN  \nMidway, North Carolina, vicinity; ca. 1880. McLamb House. A five-ranked, hipped-roof I-house with eave brackets and modest spindlework porch detailing.\n\nPYRAMIDAL:  FOLK VICTORIAN  \nBiloxi, Mississippi, ca. 1900.\n\nPYRAMIDAL:  FOLK VICTORIAN  \nMcPhersonville, South Carolina; late 19th century. Gregorie House. Two-tiered, full facade porches, such as this, are common throughout the South.\n\nPYRAMIDAL:  FOLK VICTORIAN  \nBrunson, South Carolina; ca. 1875. Brunson House. This example has small eave brackets and flat, jig-saw cut porch decoration.\n\nPYRAMIDAL:  FOLK VICTORIAN  \nWoodville, Texas; ca. 1880. Shivers House. This example adds centered gables to the low-pitched hipped roof. It is quite large for a Folk Victorian; most houses of this size and detailing more closely followed one of the stylish modes of the day.\n\n## Variants and Details\n\nThe style is defined by the presence of Victorian decorative detailing on simple folk house forms, which are generally much less elaborated than the Victorian styles that they attempt to mimic. The details are usually of either Italianate or Queen Anne inspiration; occasionally the Gothic Revival provides a source. The primary areas for the application of this detailing are the porch and cornice line. Porch supports are commonly either Queen Anne\u2013type turned spindles or square posts with the corners beveled (chamfered) as in many Italianate porches. In addition, lace-like spandrels are frequent and turned balusters may be used both in porch railings and in friezes suspended from the porch ceiling (see this page and this page). The roof-wall junction may be either boxed or open. When boxed, brackets are commonly found along the cornice. Centered gables are often added to side-gabled and pyramidal examples. Window surrounds are generally simple or may have a simple pediment above. Most Folk Victorian houses have some Queen Anne spindlework detailing but are easily differentiated from true Queen Anne examples by the presence of symmetrical facades and by their lack of the textured and varied wall surfaces characteristic of the Queen Anne.\n\n## Occurrence\n\nThe style is common throughout the country; the five subtypes show differing patterns of distribution as noted in the descriptions of each above. New Orleans has a particularly rich heritage of Folk Victorian houses. This includes many distinct local forms such as Double Shotguns and Camelbacks (a shotgun house with a two-story section at the rear that resembles the hump on a camel's back).\n\n## Comments\n\nLike that of the National Folk forms on which they are based, the spread of Folk Victorian houses was made possible by the railroads. The growth of the railroad system made heavy woodworking machinery widely accessible at local trade centers, where they produced inexpensive Victorian detailing. The railroads also provided local lumber yards with abundant supplies of pre-cut detailing from distant mills. Many builders simply grafted pieces of this newly available trim onto the traditional folk house forms familiar to local carpenters. Fashion-conscious homeowners also updated their older folk houses with new Victorian porches. These dwellings make strong stylistic statements and are therefore treated here as distinctive styled houses, rather than pure folk forms. After about 1910 these Symmetrical Victorian houses, as they are sometimes called, were replaced by the Craftsman, Colonial Revival, and other fashionable eclectic styles.\n\n# Eclectic Houses\n\n1880\u20131940\n\n* * *\n\n**ENGLISH AND ANGLO-AMERICAN PERIOD HOUSES**\n\n**Colonial Revival**\n\n**Neoclassical**\n\n**Tudor**\n\n**FRENCH PERIOD HOUSES**\n\n**Chateauesque**\n\n**Beaux Arts**\n\n**French Eclectic**\n\n**MEDITERRANEAN AND SPANISH PERIOD HOUSES**\n\n**Italian Renaissance**\n\n**Mission**\n\n**Spanish Revival**\n\n**Monterey**\n\n**Pueblo Revival**\n\nThe Eclectic movement draws on the full spectrum of Western architectural tradition\u2014Ancient Classical, Medieval, and Renaissance Classical\u2014for stylistic inspiration. Unlike the free stylistic mixtures that had dominated the preceding Victorian era, the Eclectic movement stresses relatively pure copies of domestic architecture as originally built in various European countries and their New World colonies. This movement lasted from the 1880s until 1940 and occurred in two phases. The first, and smaller, arrived at the end of the 19th century and receded around 1900. The second phase arrived around 1920 and spread rapidly across the United States, dominating house styles between the two world wars. Throughout the Eclectic era many different styles engaged in a sort of friendly competition.\n\nThe first phase began quietly in the last decades of the 19th century as fashionable, European-trained architects designed landmark houses for wealthy clients in historic styles\u2014mostly Italian Renaissance, Chateauesque, Beaux Arts, Tudor, or Colonial Revival\u2014a trend that gained momentum with the Chicago Columbian Exposition of 1893, which stressed historical styles. In the first two decades of the 20th century this first phase was interrupted and almost overwhelmed by the first wave of architectural modernism presented by Craftsman and Prairie styles with their purely American origins. Many of the Eclectic homes built from 1900 through 1920 incorporated aspects of these two early modern styles\u2014primarily broad overhangs, exposed roof rafters, front porches, and grouped windows.\n\nAt the end of World War I, fashions in domestic architecture quickly shifted back toward traditional styles. This was undoubtedly encouraged by the two million American soldiers returning from Europe, where they had observed original historic homes first hand. The homes built in this second phase, sometimes called \"period houses,\" strived to present exteriors with relatively \"correct\" architectural detail.\n\nTwo major changes facilitated widespread and more authentic copies of European styles. First, new inexpensive methods of reproducing photographs allowed architects and clients to peruse illustrated books showing historic houses and their architectural details. Second, by the early 1920s, a technique was perfected for adding a thin veneer of brick or stone to any exterior. The new affordability of \"masonry\" houses revolutionized the design of small homes. Previously, it had been difficult to copy historic European styles, as these were built of solid masonry, often with decorative stone- or brickwork patterns. Now, masonry veneers allowed modest cottages to mimic Old World dwellings. Traditional homes dominated domestic building during the 1920s and 1930s. Although the Great Depression led to simpler houses with less architectural detail, period fashions remained dominant until the end of World War II, when modernism swept aside historical styles. (Paradoxically, European reactions were exactly the opposite. After World War I, Europeans embraced modernism, while following World War II they embarked on precise restoration and rebuilding of the past.)\n\n# ECLECTIC HOUSES\n\n# Colonial Revival\n\n# 1880\u20131955\n\n## Identifying Features\n\nAccentuated front door, normally with decorative crown (pediment) supported by pilasters, or extended forward and supported by slender columns to form entry porch; doors commonly have overhead fanlights or sidelights; facade normally shows symmetrically balanced windows and center door (less commonly with door off-center); windows with double-hung sashes, usually with multi-pane glazing in one or both sashes; windows frequently in adjacent pairs.\n\n## Principal Subtypes\n\nNine principal subtypes can be distinguished. Some examples may be almost identical to their colonial (particularly Georgian and Federal) prototypes. Clues for distinguishing Revival copies from early originals are given below under Variants and Details.\n\nASYMMETRICAL\u2014About 10 percent of Colonial Revival houses have asymmetrical facades, a feature rarely seen on their colonial prototypes. These asymmetrical examples range from rambling, free-form houses resembling the free classic  Queen Anne style to simple boxes with asymmetrical window or porch arrangements. Prior to 1900 this subtype accounted for about one-third of all Colonial Revival houses. After 1910 few examples were constructed until the 1930s, when irregular facades reappeared with less elaborate detailing. These were, in part, inspired by the desire for attached garages, which were difficult to incorporate within a balanced facade.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1897. Note the exaggerated broken pediments on the dormers.\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nBrookline, Massachusetts; ca. 1900. Note the two-story wing set at an angle to the left and the almost centered gable with Palladian window below to the right.\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nSalisbury, North Carolina; 1898. The line between some late free classic Queen Anne houses and some early Colonial Revival examples is not a sharp one: compare this photo with the similar transitional example shown in Concord. The Palladian window and Federal swags link this house more closely with the Colonial Revival movement.\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nHartford, Connecticut; ca. 1900. This early example shows Craftsman influence in its open eaves with exposed rafters and porch supports.\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nDallas, Texas; 1937. Hershfelt House. The tall, broad ground floor windows are almost as tall as the front door. Large-scale, double-hung, multi-pane windows like this were popular in the 1930s.\n\nASYMMETRICAL:  COLONIAL REVIVAL  \nDallas, Texas; 1939. Bowers House. Later asymmetrical examples mostly date from the 1930s and commonly have either gabled roofs with side wings, as in this house, or forward-facing gable-fronted wings as in the Hershfelt House. One-story bay windows, as seen here, were also popular in the 1930s examples.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH\u2014About one-third of Colonial Revival houses built before about 1915 are of this subtype, which is sometimes called the Classic Box. These have a one-story, full-width porch with classical columns, which is added to a symmetrical, two-story house of square or rectangular plan. Sometimes these are American Four-Square in form, while others strongly resemble the four-square but have an added central hall. Two-story pilasters are common at the corners; dormers, hipped or gabled, are usually present. Doors may be centered or placed to the side. These houses have both Neoclassical and Colonial Revival influences, but lack the full-height porches of typical Neoclassical houses.\n\n**HIPPED ROOF WITH FULL-WIDTH PORCH**\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nGalveston, Texas; ca. 1910. Lawrence House. On narrow urban lots a front-gabled roof occasionally replaces the more common hipped roof.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nDallas, Texas; ca. 1910. This early, two-ranked house with an off-center entrance is adapted from the simple four-square folk plan with a pyramidal roof. When embellished with a central dormer and front porch as seen here, this form is called American Four-Square. Compare with the Foster House, a narrower and more simply detailed American Four-Square.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nAshe County, North Carolina; ca. 1920. Livesy House. This example, like the houses in Union Springs, Winston-Salem, and Brooklyn, has a centered entrance and a three-ranked facade, indicating the likelihood of a central-hall plan rather than the simple four-square plan seen in Dallas and the Foster House.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nBuffalo, New York; ca. 1900. Foster House. A simple, early two-ranked example; note the corner pilasters.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nUnion Springs, Alabama; ca. 1910. Note the elaborate pedimented entranceway moved to the front of the porch, rather than around the doorway as in Colonial examples. Less grand pediments are seen in Dallas, Texas and the Livesy House.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nWinston-Salem, North Carolina; ca. 1910. Note the grouped columns on pedestals. This pattern of porch supports was uncommon before about 1900.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nBrooklyn, New York; 1900. John J. Petit, architect. Paired windows and a front door with sidelights, but no fanlight, are common Revival details seen clearly in this example.\n\nHIPPED ROOF WITH FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nBuffalo, New York; ca. 1900. White House. An unusually elaborate example with roof and upper-porch balustrades, upper-story bay windows, and a heavily detailed cornice with a solid railing above.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH\u2014About 20 percent of Colonial Revival houses are simple two-story rectangular blocks with hipped roofs; porches are usually absent or, if present, are merely small entry porches covering less than the full facade width. This subtype, built throughout the Colonial Revival era, predominates before about 1915. On early examples, the colonial detailing tended to be highly exaggerated and of awkward proportions; fanciful, pedimented dormers were particularly favored. Eaves often have a broad overhang, and front-facade windows were grouped together. After about 1920 detailing became more \"correct\" by closely following Georgian or Federal precedents.\n\n**HIPPED ROOF WITHOUT FULL-WIDTH PORCH**\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nSan Francisco, California; ca. 1900. A fine early example with exaggerated pediments on the dormers, an overwide entablature above the entry-porch columns, an overwide belt course, and grouped windows with decorative transoms on the first story. Such details never appear on more correct, later examples.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nDallas, Texas; 1924. Stickle House. This house demonstrates several features common from about 1900 to 1925. Note the broadly overhanging eaves, absence of dentils and modillions, grouped windows, and use of a single pane of glass in the bottom window sashes. These were not found in original Georgian or Federal houses and show the influence of contemporaneous Prairie houses. Compare with the O'Neil House.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nCincinnati, Ohio; ca. 1920s. Such austere examples emphasize the shift to more correct copies of Colonial dwellings.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nPittsburgh, Pennsylvania; ca. 1915. Note the exaggerated center dormer, the deep rounded entry porch, and the matching two-story curved window bays.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nDallas, Texas; 1927. O'Neil House. This more \"correct\" example has a shallow roof overhang with modillions below and single windows with 6\/9 pane sashes. It is more typical of the \"period houses\" of the 1920s. Compare with the Stickle House.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nDallas, Texas; 1938. Varner House. This Regency variation of the Colonial Revival, loosely based on English rather than American precedents, was popular in the 1930s. The octagonal window, simplified door surround, and low-parapeted roof-wall junction are typical of this variant.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nBrookline, Massachusetts; ca. 1900. E. A. P. Newcombe, architect. Another early example with exaggerated detailing. Note the dormers with Palladian windows and broken pediments, the two colonnaded upstairs balconies, and the very deep entry porch.\n\nHIPPED ROOF WITHOUT FULL-WIDTH PORCH:  COLONIAL REVIVAL  \nAtlanta, Georgia; 1922. McDuffie House. Hentz, Adler, and Schutze, architects. A very large and elaborate example. Note the pilasters, roof-line balustrade, and arched windows.\n\nSIDE-GABLED ROOF\u2014About 30 percent of Colonial Revival houses are simple, two-story rectangular blocks with side-gabled roofs. As in the type just described, the details tend to be exaggerated prior to 1910 and more \"correct\" afterward. This subtype was built throughout the Colonial Revival era but predominates after about 1915 and was widely built after 1930.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nBaltimore, Maryland; ca. 1910. A very simple, two-ranked example.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nLouisville, Kentucky; ca. 1920s. Side porches are common on Colonial Revival houses (the next two examples and the Thomson House).\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nLouisville, Kentucky; ca. 1930s. Another Regency house (see also the Varner House). This type of metal entry porch, with a canopy roof is a characteristic Regency feature.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nKansas City, Missouri; ca. 1910s. Although at first glance this looks like an accurate copy, the roof overhang is too wide and the windows too broad for an original colonial house.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nCleveland, Ohio; ca. 1920s. The entry porch with a curved underside is a favored Revival detail.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nDallas, Texas; 1941. Young House. Large multi-pane double-hung windows, extending to the first-floor level, and small round windows (above the entry here and in the gable end of the 1930s Louisville house) were widely used in the 1930s, '40s, and early '50s on Colonial Revival houses. This kind of small round accent window was also found on other styles from ca. 1930 to ca. 1955.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nDallas, Texas; 1919. Thomson House. This example was inspired by the Middle Colonies Georgian house. Note the pent roof and the hood over the entry. The side porch to the left has a summer sleeping porch above with windows on three sides. These were especially favored in the South, where they appear in many early 20th-century styles.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1903. John W. Ames, architect. This house demonstrates that reasonably accurate Colonial copies were being designed in the early years of the Revival; those with exaggerated detailing were, however, far more common.\n\nSIDE-GABLED ROOF:  COLONIAL REVIVAL  \nLouisville, Kentucky; ca. 1920s. The garden facade of a very large example. Note the door surround with the pediment extending over the sidelights but lacking a fanlight. Although this combination was never used in colonial houses, the example here faithfully captures the spirit of a Georgian or Federal doorway.\n\nCENTERED GABLE\u2014Less than 5 percent of Colonial Revival houses have a centered front gable added to either a hipped or side-gabled roof. These uncommon Revival houses mimic high-style Georgian or Federal prototypes. Scattered examples were built throughout the Colonial Revival era.\n\n**CENTERED GABLE**\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nBuffalo, New York; ca. 1900. Harrover House. Centered gables that cover three ranks of window or door openings (here and in Tatton Hall) are less common than those that are only one or two ranks wide.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nBuffalo, New York; ca. 1910s. Note the fine detailing: the entrance with a rounded door, the sidelights without fanlight, the wide classical pediment, the Palladian and bay windows, and the carefully executed dormers.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nBuffalo, New York; ca. 1920s.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nDes Moines, Iowa; 1905. Witmer House; Liebbe, Nourse and Rasmussen, architects. Note the first-story windows crowned with broken pediments (see also the Ely House). Although common above the main entrance, such pediments became rare on windows and dormers after about 1910.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nCleveland, Ohio; ca. 1910s. The open overhanging eaves and the entry porch with trellised roof are borrowed from the Craftsman movement. Note how the door area is recessed into the main body of the house.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nDallas, Texas; 1938. Lincoln House. The simplicity of detailing on this house is typical of examples from the 1930s and 1940s.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nMadison, Wisconsin; 1896. Ely House. Round windows were sometimes found when the triangular roof pediment was closed at the bottom, as here and in the Harrover House and Tatton Hall.\n\nCENTERED GABLE:  COLONIAL REVIVAL  \nRaleigh, North Carolina; 1935. Tatton Hall; William Lawrence Bottomley, architect. A five-ranked central block is flanked by one-story wings (obscured by trees and shadows in the photograph) in this landmark example. Typically the gabled portion of the house projects slightly out from the front wall plane as is seen here and in most of the other centered gable examples.\n\nGAMBREL ROOF\u2014About 10 percent of Colonial Revival houses have gambrel roofs. Most are one story with steeply pitched gambrels containing almost a full second story of floor space; these have either separate dormer windows or a continuous shed dormer with several windows. A full-width porch may be included under the main roof line or added with a separate roof. This subtype is known as Dutch Colonial, but very few examples closely follow early Dutch precedent. From about 1895 to 1915 the most common form has a front-facing gambrel roof, occasionally with a cross gambrel at the rear. These are influenced by the typical gambrels of the earlier Shingle style, but their narrower, front gambrel form fit onto narrow streetcar suburb lots. Side gambrels, usually with long shed dormers, became the predominant form in the 1920s and 1930s.\n\n**GAMBREL ROOF**\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nLouisville, Kentucky; ca. 1920s. This and the house in Cincinnati are typical examples of the popular Dutch Colonial house of the 1920s and 1930s. The side-gambrel shape, most often with a full-width shed dormer (see also the houses in St. Louis and Durham), is the most common form.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nLexington, Kentucky; ca. 1910. This cross-gambrel form, with wood cladding, was a popular pattern-book design during the period from about 1905 to 1915.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nKittery, Maine, ca. 1910s. Note the grouped windows in the upper half-story with decorative diamond-pattern muntins in the upper sashes.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nCincinnati, Ohio; ca. 1920s.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nUnion, South Carolina; ca. 1910. This and the houses in Cleveland and New Haven are all early gambrel-roof designs showing varying degrees of adventuresomeness. They are clearly descendants of the free-form gambrel designs of the preceding Shingle style.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nWashington, D.C.; ca. 1900. An early example with a full-front gambrel. Note the swags, often found on Federal houses, on the porch frieze.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nSt. Louis, Missouri; ca. 1920s.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nCleveland, Ohio; ca. 1910.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nNew Haven, Connecticut; 1910. Brown and Von Beren, architects.\n\nGAMBREL ROOF:  COLONIAL REVIVAL  \nDurham, North Carolina; ca. 1920s. This is a less common cross-gambrel form of the Dutch Colonial. Note the flared eaves, here and in the houses in Louisville, Cincinnati, St. Louis, and Cleveland. These mimic the Flemish eaves of many Dutch Colonial originals.\n\nSECOND-STORY OVERHANG\u2014This subtype is loosely based on Postmedieval English prototypes commonly built with the second story extended slightly outward to overhang the wall below. The subtype was relatively rare until the 1930s, when stylized, side-gabled examples (called Garrison Colonial houses) became very popular. These persisted into the 1950s. Unlike their early prototypes, these typically have masonry-veneered first stories with wooden wall claddings above. Georgian- or Federal-inspired doorways are commonly mixed with decorative pendants or other Postmedieval details.\n\n**SECOND-STORY OVERHANG**\n\nSECOND-STORY OVERHANG:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1940. C. Crowell, architect. This Garrison Colonial subtype was especially popular in the latest phases of the Colonial Revival, from about 1935 to 1955. The overhang required a wood-sided second floor since cantilevered brick veneering was very difficult to construct. Here brick is used only on the front facade of the first story.\n\nSECOND-STORY OVERHANG:  COLONIAL REVIVAL  \nDallas, Texas; 1953. Wilson House. Unlike their Colonial precedents, most Revival examples have a brick-sided first story.\n\nSECOND-STORY OVERHANG:  COLONIAL REVIVAL  \nMission Hills, Kansas; ca. 1930s. This example shows more detailing (door surround, wall dormers, centered gable) than is typical for the subtype.\n\nSECOND-STORY OVERHANG:  COLONIAL REVIVAL  \nDallas, Texas; 1951. Voss House. Note the two tall bay windows; these were very popular during the 1950s.\n\nONE-STORY\u2014The preceding subtypes are all based on familiar two-story prototypes, but one-story Colonial Revival houses are also common. These are generally Cape Cod cottages, loosely patterned after early wooden Cape Cod folk houses of eastern Massachusetts, usually with the addition of Georgian- or Federal-inspired doorways. These were built throughout the Colonial Revival era but were particularly common in the 1940s. Cape Cods with added Colonial Revival detailing appear in this chapter. Very modest Cape Cod\u2013shaped houses with little added architectural detail (popular during the 1940s) are treated as side-gabled Minimal Traditionals.\n\n**ONE-STORY**\n\nONE-STORY:  COLONIAL REVIVAL  \nGreeleyville, South Carolina; ca. 1910. Wilder House. Note modest pediment over door and more exaggerated pediment over dormers.\n\nONE-STORY:  COLONIAL REVIVAL  \nLouisville, Kentucky; ca. 1920s. This is a typical example of the Cape Cod cottage. The Wilder House is an earlier Cape Cod, which lacks the proportions of the Colonial originals (note the lower roof pitch, oversized dormers, and extra width and height of the front facade). The Cape Cod is the most common form of one-story Colonial Revival house. As a form, it originated in the early 18th century and continued with few changes through the 1950s. Note lights in transom over door.\n\nONE-STORY:  COLONIAL REVIVAL  \nDallas, Texas; 1929. Randall House. This house has a formal, Federal-inspired entry porch and doorway.\n\nONE-STORY:  COLONIAL REVIVAL  \nDallas, Texas; 1925. This was a common pattern book design in the 1920s. In some versions the entry porch is exaggerated in scale, occupying up to one-third of the front facade.\n\nONE-STORY:  COLONIAL REVIVAL  \nMacon, Georgia; 1912. Stetson House; Hentz and Reid, architects. Note the lower one-story wings; this finely detailed example, like the Randall House, was inspired by more pretentious Colonial antecedents than the typical Cape Cod examples shown in the Wilder House and the house in Louisville. See here for related side-gabled Minimal Traditionals that do not typically have modest stylistic details added.\n\nTHREE-STORY\u2014A small percentage of Colonial Revival houses are three stories high. These include both narrow urban houses and more typical forms modeled after three-story Federal prototypes, common in parts of New England (see this page). These typically have low-pitched, hipped roofs which appear almost flat; Federal fanlights are usual over entrances. In the early decades of this century, narrow urban houses were becoming less common in all but the largest cities. In those populous cities where urban houses persisted, Colonial Revival detailing remained popular through the 1920s.\n\n**THREE-STORY**\n\nTHREE-STORY:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1900. John A. Hasty, architect. Not a single-family house, but a triple-decker (one dwelling unit on each of three floors), this has the exuberant detailing associated with early examples. Note the broken pediments on the roof and over the central second-story window.\n\nTHREE-STORY:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1916. J. W. Ames, architect. This and the President's House are both modeled after the three-story Federal subtype that was popular in New England (see this page).\n\nTHREE-STORY:  COLONIAL REVIVAL  \nRichmond, Virginia; ca. 1910s. A three-story detached urban house with full-width porch.\n\nTHREE-STORY:  COLONIAL REVIVAL  \nCambridge, Massachusetts; 1911. President's House, Harvard University; G. Lowell, architect.\n\nTHREE-STORY:  COLONIAL REVIVAL  \nWashington, D.C.; 1915. Woodrow Wilson House.\n\nTHREE-STORY:  COLONIAL REVIVAL  \nBuffalo, New York; 1894\u2013ca. 1900. A portion of The Midway block of row houses. On left is the 1894 Bartow House, Marling and Johnson, architects. Note the exaggerated broken pediments on the original dormers and the original window lintels (a refined two-story shopfront window was added in the 1920s). In the center is the ca. 1900 Curtiss House, Marling and Johnson, architects. Note the high-style Federal-era detailing. On the right is the ca. 1900 Birge House, Green and Wicks, architects. It has an elaborate entry porch, blind arches with swags, and a cornice-line parapet.\n\nTHREE-STORY:  COLONIAL REVIVAL  \nNew York, New York; 1909\u20131926. Pyne, Filley, Sloane, and Davison Houses; McKim, Mead & White, Delano and Aldrich, and Walker and Gillette, architects for various houses. A remarkable surviving row of large attached town houses with detailing drawn from Georgian and Federal precedents.\n\nBUILT-IN GARAGE\u2014In 1940, an FHA bulletin illustrated two ways to integrate a garage into the main block of a side-gabled Colonial Revival house. Variations of this appeared in house pattern books until ca. 1980, with the house size growing over the decades.\n\n**BUILT-IN GARAGE**\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nChicago, Illinois; 1949. One of the FHA's three recommendations for built-in garage placement was to put the entrance in a plane in front of that of the house. This approach is used here and in the houses in Minneapolis, Bowie, Chicago, and Lisle.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nMinneapolis, Minnesota; 1965. Using a porch to visually connect a front-plane garage with the front door was quite popular (see also the houses in Bowie and Lisle). Note how the generally strict ranks of windows and doors of earlier Colonial Revival houses is lost here.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nChicago, Illinois; 1960. Here the garage is in the same plane as the house facade and interlocks with the main house block.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nBowie, Maryland; ca. 1964. This is the only one of these examples that does not have either a second-story overhang or a change in wall material between the two stories.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nChicago, Illinois; 1963. Note the use of two wall-cladding materials on most of the houses shown here. Brick is used for the lower story\u2014as would have been structurally appropriate in earlier eras\u2014and frame, shingles, or other less expensive cladding material such as vinyl is used in the upper story and garage.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nLisle, Illinois; 1979.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nAlexandria, Virginia; 1965. Here the typical Tri-Level Split spatial pattern is carried out on the level.\n\nBUILT-IN GARAGE:  COLONIAL REVIVAL  \nSeattle, Washington; ca. 1978. This later example places the garage on a third level.\n\n## Variants and Details\n\nAs in their Georgian and Federal prototypes, the principal areas of elaboration in Colonial Revival houses are entrances, cornices, and windows.\n\nENTRANCES\u2014The illustrations of Georgian and Federal entrances on this page and on this page include most variants found on colonial prototypes; some common additional variations favored on Colonial Revival houses are illustrated here. Broken pediments, rare on colonial originals, were particularly favored by the Revivalists. Entrance details on careful Colonial Revival copies can be distinguished from originals only by their regular, machine-made finish, which contrasts with the slightly irregular hand finishes of early examples. On less precise Colonial Revival copies, door surrounds are typically flatter than the originals; that is, less wood and fewer and shallower moldings are used to gain a similar frontal effect but less depth and relief are apparent when viewed from the side.\n\nCORNICES\u2014In original Georgian and Federal houses the cornice is an important identifying feature. It is almost always part of a boxed roof-wall junction with little overhang, and is frequently decorated with dentils or modillions (see this page and this page). These are also typical of many Colonial Revival examples. Some, however, have open eaves and rake, or even exposed rafters, features never found on original colonial houses.\n\nWINDOWS\u2014As in the originals, most Colonial Revival windows are rectangular in shape with double-hung sashes. In the more accurate copies, windows stand alone as single units (i.e., they are not paired or grouped) and each sash has six, eight, nine, or twelve panes. Equally common are multi-pane upper sashes hung above lower sashes that have only a single large pane, a pattern never seen on colonial originals. Where bay, paired, grouped, or triple windows (except the Federal Palladian type) are present, they clearly signify a Colonial Revival house rather than an original.\n\nOTHER DETAILS\u2014All common wall materials were used, but masonry predominates in high-style examples. Brick is typically red\u2014the color most commonly used for Georgian and Federal originals. Vernacular examples were generally of wood before about 1920, with masonry progressively more common as veneering techniques became widespread in the 1920s. High-style elaborations of Georgian and Federal originals may also occur on landmark Colonial Revival copies.\n\nCOLONIAL REVIVAL\n\n **DISTINGUISHING THECOLONIAL REVIVAL HOUSE FROM GEORGIAN, FEDERAL & DUTCH ORIGINALS**\n\n**ORIGINAL EXAMPLES**\n\nCOLONIAL REVIVAL\n\n **DISTINGUISHING THE COLONIAL REVIVAL HOUSE FROM GEORGIAN, FEDERAL & DUTCH ORIGINALS**\n\n**REVIVAL EXAMPLES**\n\nBe certain to check range map for Dutch Colonial houses. Originals occur _only_ within the range shown on this map\n\nCOLONIAL REVIVAL\n\n **COLONIAL REVIVAL SUBTYPES**\n\nCOLONIAL REVIVAL\n\n **REVIVAL ENTRANCES: SOME COMMON WAYS THEY VARY FROM ORIGINALS**\n\n## Occurrence\n\nThis was _the_ dominant style for domestic building throughout the country during the first half of the twentieth century. It was built in relatively small numbers from 1880 until about 1910, years when the Queen Anne was more dominant. During the next two decades (1910\u20131930) about 40 percent of the houses built were in the Colonial Revival style. Unlike most other Eclectic styles, the Colonial Revival was not completely eclipsed by World War II, but continued to be built, albeit in far less elaborate forms, into the 1950s and early 1960s. The different subtypes were not, however, equally common throughout this long period, but shifted with changing fashion (see each subtype above and on the chart). Today the Colonial Revival style is used for many New Traditional houses.\n\n## Comments\n\nThe term \"Colonial Revival,\" as used here, refers to the entire rebirth of interest in the early English and Dutch houses of the Atlantic seaboard. The Georgian and Federal styles form the backbone of the Revival, with secondary influences from Postmedieval English and Dutch Colonial prototypes. Details from two or more of these precedents are freely combined in many examples so that pure copies of colonial houses are far less common than are eclectic mixtures.\n\nIn the years between 1880 and 1900 the Colonial Revival movement also influenced two other architectural styles: Queen Anne and Shingle. In the Queen Anne this produced the free classic subtype, which grades into the closely related asymmetrical Colonial Revival house. In the Shingle style, the shingled walls and rambling forms were thought to evoke early New England shingled houses with shed and lean- to additions. Moreover, colonial details such as Palladian windows were used in many examples.\n\nThe Philadelphia Centennial of 1876 is credited with first awakening an interest in our colonial architectural heritage. In 1877 the fashionable architects McKim, Mead, White, and Bigelow took a widely publicized tour through New England to study original Georgian and Federal buildings at first hand. By 1886 they had executed two landmark houses in the style\u2014the Appleton House (1883\u20131884) in Lennox, Massachusetts, and the Taylor House (1885\u20131886) in Newport, Rhode Island. These important examples typify the two subtypes that were most common before 1910: the asymmetrical form with superimposed colonial details and the more authentic symmetrical hipped roof shape; details in both tended to have exaggerated proportions.\n\nThese early examples of Colonial Revival were rarely historically correct copies but were instead free interpretations with details inspired by colonial precedents. During the first decade of this century, Colonial Revival fashion began to shift toward carefully researched copies with more correct proportions and details. This was encouraged by new methods of printing that permitted wide dissemination of photographs in books and periodicals. In 1898 _The American Architect and Building News_ began an extensive series called \"The Georgian Period: Being photographs and measured drawings of Colonial Work with text.\" This was joined in 1915 by the _White Pine Series of Architectural Monographs,_ which was dominated by photographs of colonial buildings. These and similar ventures led to a wide understanding of the prototypes on which the Revival was based. Colonial Revival houses built in the years between 1915 and 1935 often reflect these influences by more closely resembling early prototypes than did those built earlier or later\u2014with those constructed between 1920 and 1930 often having the most accurate details. The restoration of historic Williamsburg and the writings of Royal Barry Wills helped maintain an interest in building Colonial Revival houses from the 1930s into the 1950s. However, the economic depression of the 1930s, World War II, and changing postwar fashions led to a simplification of the style in the 1940s and 1950s. These later examples are most often of the side-gabled type, with simple stylized door surrounds, cornices, or other details that merely suggest their colonial precedents rather than closely mirroring them. In addition, Colonial Revival Ranch houses were built from the 1940s through the 1960s, many of them inspired by Wills's books and well-publicized designs. Since the Colonial Revival began in the 1880s there has seldom been a time when some version of it was not being built somewhere in the United States. Although new Colonial Revival houses slowed to a trickle in the late 1950s, their production never completely disappeared. The Built-in Garage subtype appeared in pattern books until about 1980, when New Traditional versions of the Colonial Revival were already beginning to appear.\n\n# ECLECTIC HOUSES\n\n# Neoclassical\n\n# 1895\u20131955\n\n## Identifying Features\n\nFacade dominated by full-height porch with roof supported by classical columns; columns typically have Ionic or Corinthian capitals; facade shows symmetrically balanced windows and center door.\n\n## Principal Subtypes\n\nFive principal subtypes can be distinguished:\n\nFULL-HEIGHT ENTRY PORCH\u2014This common subtype has a dominant central entry porch extending the full height, but less than the full width, of the facade. It closely resembles certain Early Classical Revival and Greek Revival subtypes. As in both of these earlier styles, the entry porch may have a classical pediment and gabled roof above or, as in the Greek Revival only, the porch roof may be flat. Some Neoclassical examples have curved, semi-circular entry porches with flat roofs, a variation unusual on earlier prototypes.\n\n**FULL-HEIGHT ENTRY PORCH**\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nDallas, Texas; 1936. Bell House. A typical later example with side-gabled roof, roof-line balustrade, and simple square columns.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nDallas, Texas; 1940. Musgrove House. A late example with a side-gabled roof and slender columns. The somewhat awkward-looking projection of the entry porch results from the omission of an entablature under the pediment.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nRaleigh, North Carolina; 1903. Goodwin House; William P. Rose, architect. This early example has fluted columns with Roman Ionic capitals. Note the variety of pediments\u2014segmental over the lower-story windows, broken-ogee over the entrance fanlight, and triangular over the arched upper-story window.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nMontgomery, Alabama; 1906. Governor's Mansion; Moses Sabel, architect. An elaborately detailed early example. Note the composite capitals on the pilasters and columns. These and the arched and double windows are never seen on earlier classical styles.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nDallas, Texas; 1933. Lee House. Here the entry porch (or portico) has been recessed into the body of the house; this is known as a portico in antis (see also the following image). Note large double-hung multi-pane windows.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nKansas City, Missouri; ca. 1930s. Slender columns, side-gabled roof, and simplified Chinese Chippendale railing along the roof-line mark this as a late example.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nNew London, Connecticut; ca. 1910. An unusual gambrel-roof example. The triple windows, broken pediment occurring with full-height columns, and the curved entry porch with balustrade all mark this as a Neoclassical example.\n\nFULL-HEIGHT ENTRY PORCH:  NEOCLASSICAL  \nLouisville, Kentucky; ca. 1910s. This entry porch is both recessed into the body of the house and extended outward. Note the keystone lintels over the upper-story windows and three-part lintels over the lower-story triple windows and the roof-line balustrade.\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH\u2014In this relatively uncommon subtype, a full-width, one-story porch is added to the full-height entry porch just described. This dual-level entry porch is without precedent in the earlier classical styles. Most examples were built from 1895 to 1915; few, after World War I.\n\n**FULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH**\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH:  NEOCLASSICAL  \nDallas, Texas; ca. 1900. Harris House. In many examples the full-width lower porch is built independently, passing behind the tall entry porch as here and the house in Taylor. In this example the lower porch even has a pediment at the entry.\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH:  NEOCLASSICAL  \nDallas, Texas; 1905. Baird House. In this example, the lower porch attaches to the tall entry porch; where they meet, both are supported by the same columns. Note large cottage windows on first floor.\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH:  NEOCLASSICAL  \nDallas, Texas; 1911. Squared masonry pillars were sometimes substituted for classical columns.\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH:  NEOCLASSICAL  \nEufaula, Alabama; ca. 1910. This example is somewhat unusual in that no colossal columns are used for the full-height entry; instead, a second tier of one-story columns occurs above the lower porch.\n\nFULL-HEIGHT ENTRY PORCH WITH LOWER FULL-WIDTH PORCH:  NEOCLASSICAL  \nTaylor, Texas; ca. 1910. This brick example has a tiled roof; the flat roof of the full-height porch is less common than the triangular pediments on the Harris House, the Baird House, and the house in Eufaula; the triple columns supporting it are unusual.\n\nFRONT-GABLED ROOF\u2014In this uncommon subtype, the full-facade, colonnaded porch beneath the front-facing gable gives the house the appearance of a miniature Greek temple. This form was very common in Greek Revival houses, but makes up only a small percentage of Neoclassical examples.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  NEOCLASSICAL  \nJamestown, New York; 1906. The fanlight over the front door distinguishes this example and the one shown next from a Greek Revival original.\n\nFRONT-GABLED ROOF:  NEOCLASSICAL  \nBrooklyn, New York; 1905. George Hitchings, architect. This house has a cross-gabled roof behind the dominant front gable. Note the unusual use of corner quoins next to the pilasters.\n\nFRONT-GABLED ROOF:  NEOCLASSICAL  \nLittle Rock, Arkansas; 1906. Haliburton Houses. Although not front-gabled, the hipped roof with front-to-back ridge gives these houses a similar look. As with most examples of this subtype, they were designed for narrow urban lots.\n\nFULL-FACADE PORCH\u2014In this subtype, as in the one just described, a colonnaded porch occupies the full width and height of the facade. Here, however, the porch is not covered by a traditional pedimented gable but instead either by the principal (side-gabled or hipped) roof or by a flat or shed extension from such a roof. This subtype became particularly popular in the period from about 1925 to 1950. These later examples normally have slender columns without elaborate capitals or fluted surfaces.\n\n**FULL-FACADE PORCH**\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nBuffalo, New York; ca. 1910. Geir House. The elaborate columns and exaggerated entablature mark this as an early example.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nBuffalo, New York; ca. 1930. The porch is recessed under the main roof of the house. The small entry porch with balustrade is an unusual addition. The slender square columns mark this as a late example.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nJackson, Mississippi, ca. 1910. A richly detailed house with tiled roof, paired balconies, and multiple swags.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nDallas, Texas; 1919. Warren House; Henry B. Thomson, architect. Note the full-length casement windows opening onto the porch and recessed side wings. The Chinese Chippendale cornice-line railing seen here and on the house in Union Springs was once used on Mount Vernon's river facade. Inspired by the nation's first museum house, these appeared on many homes (and other structures) from about 1920 to 1950. This eye-catching, and for a time iconic, feature was later discovered to be an addition. Now removed from Mount Vernon, it lives on in miniature tributes throughout the country, where it is an important character-defining feature. See also the Bell House, the example in Kansas City, Ohrum House, Marsh House, and the Harrison House.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nMission Hills, Kansas; ca. 1930. The side-gabled form and slender, simplified columns indicate a late date for this example.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nBuffalo, New York; 1895. Williams House; Stanford White, architect. A green copper roof accents this landmark house.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nSalisbury, North Carolina; ca. 1910. The forward extension in the central part of a full-facade porch (seen here, in Jackson, and in the Williams House) is a Neoclassical feature not used during the earlier classical revivals.\n\nFULL-FACADE PORCH:  NEOCLASSICAL  \nUnion Springs, Alabama; ca. 1920s. Note the one-story side wing to the left with a roof-line balustrade matching that of the main house.\n\nONE-STORY\u2014One-story Neoclassical cottages, a common subtype, usually have hipped roofs with prominent central dormers. The colonnaded porch may be either full- or partial-width and may be included under the main roof or have a separate flat or shed roof.\n\n**ONE-STORY**\n\nONE-STORY:  NEOCLASSICAL  \nLouisville, Kentucky; ca. 1910. An unusual early example.\n\nONE-STORY:  NEOCLASSICAL  \nLouisville, Kentucky; ca. 1910. Kettig House.\n\nONE-STORY:  NEOCLASSICAL  \nDallas, Texas; 1929. Ohrum House. A late example with slender, square porch supports.\n\nONE-STORY:  NEOCLASSICAL  \nSmithfield, North Carolina; ca. 1910. This example, the following one, and the Gordon House are all variations of the most common early form of the subtype. Note that all have hipped roofs, centered dormers, and full-facade porches. This example has a porch under a separate roof.\n\nONE-STORY:  NEOCLASSICAL  \nSelma, Alabama; ca. 1910.\n\nONE-STORY:  NEOCLASSICAL  \nDallas, Texas; 1947. Marsh House. The entry porch is recessed into the house (portico in antis). A cupola has been added to suggest a miniature Mount Vernon.\n\nONE-STORY:  NEOCLASSICAL  \nDallas, Texas; 1914. Gordon House. Note how the porch is under the main roof and the columns are set on pedestals.\n\nONE-STORY:  NEOCLASSICAL  \nDallas, Texas; 1939. Harrison House. A one-story version of the most common late Neoclassical house with side-gabled roof, full-facade porch, and Chinese Chippendale cornice-line railing, inspired by the one once on Mount Vernon's river facade. Similar railings are on the Ohrum House and the Marsh House.\n\n## Variants and Details\n\nThe principal areas of elaboration in Neoclassical houses are porch-support columns, cornices, doorways, and windows.\n\nPORCH-SUPPORT COLUMNS\u2014In Neoclassical houses built before about 1920, the columns are generally more ornate than those of Early Classical Revival or Greek Revival prototypes. Corinthian or Ionic capitals, or mixtures of the two, are found in about 75 percent of Neoclassical houses but in less than 20 percent of Greek Revival examples. This change was made possible by the introduction of mass-produced capitals prefabricated of molded plaster or composition materials. Fluted column shafts are common in early houses. After about 1925, very slender, unfluted (often square) columns began to be used, primarily on houses with full-facade porches. Those usually lack capitals, and their proportions readily distinguish them from earlier Neoclassical and Greek Revival examples.\n\nDOORWAYS\u2014Doors commonly have elaborate, decorative surrounds based on Greek Revival, Federal, or even Georgian precedents (see here, here, and here). Those with Georgian or Federal doorways are easily distinguished, because original examples very rarely had full-height, two-story columns (although some have had such porches added later). Greek Revival\u2013type doorways may be more easily confused with original examples.\n\nCORNICES\u2014Neoclassical houses usually have a boxed eave with a moderate overhang, frequently with dentils or modillions beneath; a wide frieze band is occasionally found beneath the cornice. These are loosely based on Federal or Greek Revival precedents.\n\nWINDOWS\u2014Windows are rectangular with double-hung sashes. Examples following early precedent have six or nine panes to each sash; others have a multi-pane or single-pane upper sash and a single-pane lower sash. The presence of bay windows, paired windows, triple windows (except the Palladian type), transomed windows, or arched windows differentiate Neoclassical from Greek Revival or Early Classical Revival examples. Cottage windows were used from ca. 1895 to 1915, and very large double-hung multi-pane windows were popular from ca. 1930 to 1955.\n\nOTHER DETAILS\u2014Many elaborations found on Early Classical Revival and Greek Revival houses also occur in Neoclassical examples. Roof-line balustrades, in particular, are much more common in Neoclassical houses than in the earlier styles. These are commonly found on top of full-height entry porches with flat roofs (without a pediment above), above full-facade porches, along the cornice line of the roof, or around the roof deck. Another type of railing, inspired by the detail found on Chinese Chippendale furniture, is also typical. A Chinese Chippendale railing consists of a repetitive geometric line pattern, generally contained in a series of rectangular frames. It is more commonly found in later examples built after ca. 1920 (see caption for the Warren House).\n\nNEOCLASSICAL\n\nORIGINAL EXAMPLES\n\nNEOCLASSICAL\n\n **DISTINGUISHING THENEOCLASSICAL HOUSE FROM EARLY CLASSICAL REVIVAL & GREEK REVIVAL ORIGINALS**\n\nNEOCLASSICAL\n\n **TYPICALENTRANCES**\n\nHouses with a broken pediment at the entrance or above a window and two-story columns are always Neoclassical; houses with an unbroken pediment at the entrance and two-story columns are usually Neoclassical (a few Greek Revival originals have unbroken triangular pediments)\n\nNEOCLASSICAL\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nNeoclassical was a dominant style for domestic building throughout the country during the first half of the 20th century. Never quite as abundant as its closely related Colonial Revival contemporary, it had two principal waves of popularity. The first, from about 1900 to 1920, emphasized hipped roofs and elaborate, correct columns. The later phase, from about 1925 to the 1950s, emphasized side-gabled roofs and simple, slender columns. During the 1920s, the style was overshadowed by other Eclectic fashions.\n\n## Comments\n\nThis revival of interest in classical models dates from the World's Columbian Exposition, held in Chicago in 1893. The exposition's planners mandated a classical theme, and many of the best-known architects of the day designed dramatic colonnaded buildings arranged around a central court. The exposition was widely photographed, reported, and attended; soon these Neoclassical models became the latest fashion throughout the country.\n\nThe central buildings of the exposition were of monumental scale and inspired countless public and commercial buildings in the following decades. The designs of smaller pavilions representing each state of the Union were more nearly domestic in scale and in them can be seen the precedents for most Neoclassical houses. Those of Ohio, Utah, and South Dakota, for example, all had semi-circular, full-height entry porches. Nebraska and Kentucky were represented by more traditional full-height porches with triangular pediments. The Connecticut pavilion had a full-height entry porch with a lower full-width porch. All of these drew heavily on the country's previous interest in the Early Classical Revival and Greek Revival styles. The Virginia pavilion was a copy of Mount Vernon, whose full-facade porch, among the first in the country, had been added in 1777 to an earlier Georgian house. The presence of this replica at the fair, and the original's wide familiarity as the nation's premier museum house, contributed to the incorrect impression that such porches were colonial or perhaps \"Southern colonial.\" Thus did Georgian, Federal, Early Classical Revival, and Greek Revival traditions, which originally spanned a century and a half of the nation's history, become fused into the eclectic Neoclassical style.\n\n# ECLECTIC HOUSES\n\n# Tudor\n\n# 1890\u20131940\n\n## Identifying Features\n\nSteeply pitched roof, usually side-gabled (less commonly hipped or front-gabled); facade dominated by one or more prominent front-facing gables, usually steeply pitched; tall, narrow windows, usually in multiple groups, with multi-pane glazing; massive chimneys, sometimes crowned by decorative chimney pots; front door and\/or entry porch with round or Tudor arch; decorative (i.e., not structural) half-timbering present on about one-third of examples.\n\n## Principal Subtypes\n\nEight principal subtypes can be distinguished:\n\nFRONT-GABLED ROOF\u2014About 15 percent of Tudor houses have a full-width front gable that dominates the front facade. Pre-1920 examples may include a full-width one-story front porch, and details from the contemporaneous Craftsman style, while later examples often have a massive roof form, sometimes with large side-facing dormers.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  TUDOR  \nBuffalo, New York; ca. 1910s. An early symmetrical example with a full front-gabled roof. Note the open eaves with exposed rafters, borrowed from contemporaneous Craftsman houses.\n\nFRONT-GABLED ROOF:  TUDOR  \nKansas City, Missouri; ca. 1930s. An unusually tall and steeply pitched front-gabled roof forms the principal facade. Note the pedimented entry; varying interpretations of Classical doorways were added to Tudor house forms beginning in the 1920s.\n\nFRONT-GABLED ROOF:  TUDOR  \nLouisville, Kentucky; ca. 1910s. Such examples, with the dominant front gable capped by a hip, suggest Continental, rather than British, precedents. They were sometimes referred to as Germanic Cottages by Eclectic builders.\n\nSYMMETRICAL PAIRED GABLES\u2014Only about 5 percent of Tudor houses have a pair of symmetrical gables. Early examples sometimes had a lower-pitched roof and details from the contemporaneous Craftsman style.\n\n**SYMMETRICAL PAIRED GABLES**\n\nSYMMETRICAL PAIRED GABLES:  TUDOR  \nDuluth, Minnesota; 1924. The symmetrical form seen here is common in pre-1920s examples, but unusual in later examples like this one.\n\nSYMMETRICAL PAIRED GABLES:  TUDOR  \nSeattle, Washington; 1911. The overall symmetry almost masks the different window and half-timbering pattern in the second-story gables. Note how the windows fit within the half-timbering.\n\nSYMMETRICAL PAIRED GABLES:  TUDOR  \nPhiladelphia, Pennsylvania; ca. 1920s. Stone was locally available and widely utilized in Philadelphia; it is used here for the first story and for the entry area with small parapeted gable above.\n\nSYMMETRICAL PAIRED GABLES:  TUDOR  \nLouisville, Kentucky; ca. 1910s. Open eaves with exposed rafters indicate a house with Craftsman influence. These were generally built before about 1915. The early date of this example is confirmed by the symmetrical form. Such elaborate half-timbered effects are also less common on later examples.\n\nSYMMETRICAL PAIRED GABLES:  TUDOR  \nHartford, Connecticut; ca. 1910s. A brick lower story with wood-shingled walls above.\n\nSINGLE DOMINANT MID-FACADE GABLE\u2014About 20 percent of Tudor homes are side-gabled or hipped form with a single front-facing gable added in the middle as the dominant facade element. This gable is rarely centered and generally includes the entry composition.\n\n**SINGLE DOMINANT MID-FACADE GABLE**\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nDuluth, Minnesota; 1947. What is basically a side-gabled Minimal Traditional house assumes a Tudor air when a small brick composition of entry with asymmetrical roof, prominent chimney and single casement window is placed asymmetrically in front.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nMinneapolis, Minnesota; 1940. A simple side-gabled house is easily visible behind a large full-facade front-facing gable with arched door, swooping roof, and prominent chimney. Note use of informal decorative stone work around entrance and on chimney.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nLexington, Kentucky; ca. 1930s. One step up from the house in Duluth. Note how the distinctive form (side-gabled roof with the facade dominated by a prominent, steep cross gable and a massive chimney) marks this as a Tudor house even with little additional detailing.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nCleveland, Ohio; ca. 1920s. Note the finely detailed entry gable with very tall leaded glass windows, vergeboard, and decorative paneling. The curved roof line over the bay window is a distinctive but relatively rare Tudor feature.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nChicago, Illinois; 1928. The half-timbering pattern is unusual.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nSalisbury, North Carolina; ca. 1910s. An early symmetrical example with exposed rafters.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nMendon, Utah; ca. 1935. Note the high foundation and basement windows on the side.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nChicago, Illinois; 1927. Note the one-car attached garage that served even a grand house prior to World War II. The garage wing may be a 1930s addition with a door that appears original and with leaded glass across the top.\n\nSINGLE DOMINANT MID-FACADE GABLE:  TUDOR  \nCleveland, Ohio; ca. 1920s. This example appears to retain the original roof of rough-cut slate. Note the unusually low eave line and the massive front chimneys.\n\nFRONT-FACING GABLE WITH WING\u2014About 40 percent of Tudor houses are of this subtype, with a front-facing gable placed on one side of a side-gabled, or occasionally hipped, main house block. The front-facing gable sometimes includes a smaller gable nested inside.\n\n**FRONT-FACING GABLE WITH WING**\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nPortland, Oregon; 1927. This house is similar in size and plan to the house in Lexington, but has neither entry nor sitting porch.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nHartford, Connecticut; ca. 1910s. The Craftsman-influenced front porch indicates the early date.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nLexington, Kentucky; ca. 1920s. Note how a small rounded English entry porch is used in conjunction with a broad American sitting porch on the right.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nDuluth, Minnesota; 1907. Note the Craftsman detailing on the front porch. The porch begins at the ground and slopes inward with exposed wood beams over the entry.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nHomewood, Alabama; ca. 1920s. Hollywood Historic District. Note the unusual half-timbering pattern.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nMinneapolis, Minnesota; ca. 1920s. The welcoming entry features a nested gable, decorative stonework and a front door with vertical lights.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nSeattle, Washington; ca. 1930. Note sweeping curve of the front-facing gable.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nDallas, Texas; 1929. Compare this 1920s \"period house\" example, with a side porch under the main roof, to the house in Duluth, a 1900s example with a Craftsman-influenced porch.\n\nFRONT-FACING GABLE WITH WING:  TUDOR  \nSeattle, Washington; 1931. A two-story front-facing gable was often combined with a one-story wing. Here it has a large shed dormer.\n\nMULTIPLE-FACADE GABLES\u2014This picturesque subtype is found in about 15 percent of Tudor houses. It has two or more dominant cross gables placed randomly on the front facade and often includes gabled wall or roof dormers.\n\n**MULTIPLE-FACADE GABLES**\n\nMULTIPLE-FACADE GABLES:  TUDOR  \nDallas, Texas; ca. 1930s. The trio of front-facing gables has an unusually steep pitch.\n\nMULTIPLE-FACADE GABLES:  TUDOR  \nHomewood, Alabama; ca. 1920s. Hollywood Historic District. Each gable has a different pattern of half-timbering.\n\nMULTIPLE-FACADE GABLES:  TUDOR  \nToledo, Ohio; ca. 1920s. A landmark example with multiple gables and chimneys and a Renaissance-inspired door surround.\n\nMULTIPLE-FACADE GABLES:  TUDOR  \nCleveland, Ohio; remodeled 1924. S. Weringen House; Philip L. Small, architect. This landmark example has a three-story bay window with castellations above and an irregular roof shape.\n\nFALSE THATCHED ROOF\u2014This rare but distinctive subtype mimics the picturesque thatched roofs of rural England with modern materials. Typically, composition roofing is rolled or wood shingles steamed around eaves and rakes, suggesting a thick layering of thatch. Originally these materials often had irregular textures, suggesting thatch, but usually these have been replaced with new materials of a more uniform texture. Such roofs were occasionally used on both symmetrical and asymmetrical forms of Tudor houses, from modest to grand.\n\n**FALSE THATCHED ROOF**\n\nFALSE THATCHED ROOF:  TUDOR  \nPortland, Oregon; ca. 1920s. This steeply pitched roof with curved dormer and curved gable roof gives a convincing imitation of thatch.\n\nFALSE THATCHED ROOF:  TUDOR  \nCleveland, Ohio; ca. 1920s. Note how the windows in the front-facing gable are carefully inserted into the overall half-timbering pattern and the side porch on the left, under the main roof of the house.\n\nFALSE THATCHED ROOF:  TUDOR  \nColumbus, Georgia; 1930.\n\nFALSE THATCHED ROOF:  TUDOR  \nSan Francisco, California; 1909. This house has distinctive massing with four easily discernible elements: a rounded sunroom, a wing, an unusual narrow and tall half-timbered central gable, and an unusual broad and long main front-facing gable.\n\nFALSE THATCHED ROOF:  TUDOR  \nCedarhurst, New York; ca. 1910s. This atypical example lacks the front-facing gable usually found on Tudor houses. The trellised front entry shows some Craftsman influence. The original composition roofing is shown; note the textured pattern, which closely simulates thatch. There are a number of examples of similar side-gabled houses with false thatched roofs, possibly inspired by Anne Hathaway's cottage in Warwickshire, England.\n\nPARAPETED GABLES\u2014This less common subtype is based on the more formal building traditions of early Renaissance England and generally has masonry walls. The walls of the characteristic front-facing gables rise in a parapet above the roof behind. Elaborate facade detailing of Gothic or Renaissance inspiration is typical and includes windows with transoms and surrounds, and castellated parapets on flat-roofed towers, porches, and bays. Shaped Flemish gables are also found. This subtype was used in architect-designed landmarks built from about 1890 to 1915, particularly in the Northeast. Parapeted gables do not have half-timbering, but occasionally a non-parapeted half-timbered gable is added to a house of this subtype. After World War I, less formal and more picturesque early English models dominated, although scattered parapeted landmarks were built through the 1930s.\n\n**PARAPETED GABLES**\n\nPARAPETED GABLES:  TUDOR  \nCleveland, Ohio; ca. 1910s. An unusual interpretation with flat roof and battlements all around.\n\nPARAPETED GABLES:  TUDOR  \nSt. Louis, Missouri; ca. 1910s. Note the shaped Flemish gables and tabbed window and door surrounds.\n\nPARAPETED GABLES:  TUDOR  \nRichmond, Virginia; ca. 1910s. This finely detailed example was designed for a relatively narrow urban lot.\n\nPARAPETED GABLES:  TUDOR  \nTuxedo Park, New York; ca. 1910s. Mitchell House. Note the flat-roofed tower with battlements, shaped Flemish gables, and the Renaissance-influenced door surround with columns and pediments.\n\nPARAPETED GABLES:  TUDOR  \nSeattle, Washington; 1914. This adds an air of informality by combining a half-timbered gable with a parapet gable.\n\nPARAPETED GABLES:  TUDOR  \nSan Francisco, California; ca. 1910s. The narrow end is turned toward the street.\n\nPARAPETED GABLES:  TUDOR  \nBuffalo, New York; 1910s. Albright House. The multiple chimneys with paired flues and the lines of casement windows with stone transoms are features seen in many examples of this subtype.\n\nPARAPETED GABLES:  TUDOR  \nChicago, Illinois; 1895. Hyde Park.\n\nPARAPETED GABLES:  TUDOR  \nConcord, North Carolina; ca. 1920s. This and the Albright House are symmetrical interpretations that are relatively common in parapeted examples.\n\n## Variants and Details\n\nThe Tudor style is loosely based on a variety of early English building traditions, ranging from simple folk cottages to early Renaissance palaces. Decorative detailing is varied. In addition to its early English roots, the style may incorporate details from America's contemporaneous Craftsman houses.\n\nGABLES\u2014The composition and roof form of the prominent cross gables are diverse. While Tudor houses are almost always asymmetrical after about 1920, the composition of each individual front-facing gable element is more often symmetrical. The composition within a single-gabled section may include only windows, only the entry, or, more frequently, both. Compositions including a chimney are less common. The most common gable variant is a smaller gable nested within a larger one; these share the same roof pitch, and the smaller gable generally extends forward. Other gable-roof variations include: one eave much shorter than the other; one eave curving or sweeping outward; very steep or exaggerated sloops; clipped gables (a small hipped section replaces the gable peak); and gables that overlap. Gables without parapets are by far the most common and generally have a slight overhang of the gable roof. Plain vergeboards are typical, but decorative verges may have a simple scallop cut into the edge or be embossed with a floral or geometric design.\n\nHALF-TIMBERING\u2014Decorative (i.e., false) half-timbering, mimicking Medieval infilled timber framing, is found on about one-third of Tudor houses. It is generally a wood layer of two to three inches attached to the material below. Many different designs are found; most have stucco infilling between the timbers, but brick is also used, often laid in decorative patterns.\n\nDOORWAYS AND PORCHES\u2014Simple round-arched doorways with heavy wide-plank or board-and-batten wood doors are most common; small windows are often present in door. Similarly constructed rectangular and Tudor-arched doors are also used. The door surround is a favorite place for added detail. Small tabs of cut stone may project into surrounding brickwork, giving a quoin-like effect, or more informal patterns of stone or brickwork are used. A small entry porch is often present, commonly round or Tudor arched. Some examples have a deep one-story American sitting porch. Before about 1920, these often appear on the front facade, supported by squared piers. Later, sitting porches are generally located to the side, often under the main roof form.\n\nWINDOWS\u2014Tudor homes almost always have one or more groupings of wood or metal casement windows, generally gathered into series of three or more. Small transoms are sometimes present, delineated with cast-stone mullions in high-style examples. Focal groups of casements may have small rectangular or diamond-shaped glass panes held in place by thin grooved lead strips rather than wood muntins. Original English Tudor-era houses had only casement windows (double-hung wood-sash windows had not been invented). American Tudor houses commonly use a combination of casement and double-hung windows, usually with very simple or no surrounds. Windows on half-timbered walls are typically integrated into the half-timbering pattern.\n\nWALL CLADDING\u2014The English precedents for this style were almost always stone, brick, or stucco, and masonry is therefore the preferred cladding for Tudor homes. Wood cladding of shingles or weatherboard was used primarily for less expensive homes.\n\nBefore about 1920, stone and brick walls had to be built of expensive solid masonry and were used for landmark examples. More modest early houses might have masonry first floors, but upper walls clad with weatherboard, shingles, or stucco (applied over wood lath), thus avoiding the expense of complete solid-masonry construction. By 1920, brick veneering was perfected and immediately became the preferred wall finish for even the most modest Tudor cottages. Stone walls, inspired by homes in the Cotswold area of England, were also popular and dominated a few stone-rich locales. Stone houses became more widespread as stone cladding was refined.\n\nBefore 1920 the least expensive way of achieving a \"masonry\" house was to apply stucco over a wood frame; stucco over hollow tile cost far more. In addition, stucco was favored by English architect C. F. A. Voysey for cladding his early Tudor homes in the English Arts and Crafts tradition. After 1920, solid stucco remained the choice for those desiring a \"modern English home\"\u2014eschewing half-timbering and masonry patterns for a smooth-wall appearance.\n\nOTHER DETAILS\u2014Large, elaborate chimneys are commonly placed in prominent locations on the front or side of the house. The lower part of the chimney may be decorated with complex masonry or stone patterns and the chimney may step inward as it rises from the ground. Multiple chimney shafts and ornamental chimney pots are common but often only decorative, rather than expressing the number of flues contained within, as in English originals. Upper stories and gables may overhang lower stories\u2014a feature that made the heavy-timber structural skeleton stronger in original Tudor homes. Roofs were often of slate\u2014with thick or irregularly shaped slates preferred by many architects.\n\nBoth gabled and hipped-roof dormers and wall dormers are typical, as well as shed and eyebrow dormers. These facilitated additional half-stories of living space and are often found under the high-pitched Tudor roofs. It was easy space to construct, made good use of the roof volume, and added both varied eave heights and dormers to the composition. In front-gabled roof examples, broad side-facing shed dormers often add an almost complete second story under the massive roof. In the front-facing gable-with-wing subtype, a two-story front-facing gable often has a one-and-one-half-story side wing lighted either by multiple dormers or by a single large shed dormer.\n\nTUDOR\n\n **TYPICAL GABLE DETAILS**\n\nTUDOR\n\n **GABLE ROOF VARIATIONS**\n\nTUDOR\n\n **GABLE COMPOSITION**  \n(individual gable elements are most often symmetrical)\n\nTUDOR\n\n **HALF-TIMBERING**\n\nLOCATION\n\nTUDOR\n\n **HALF-TIMBERING**\n\nTYPICAL PATTERNS  \ninfill usually stucco, but occasionally brick\n\nTUDOR\n\n **TYPICALENTRANCES**\n\nTUDOR\n\n **TYPICAL WINDOWS**\n\nTUDOR\n\n **CHIMNEYS**\n\n**PLACEMENT**  \nMay have more than one\n\nTUDOR\n\n **CHIMNEYS**\n\n**TYPICAL ELABORATIONS**\n\nTUDOR\n\n **STRAPWORK**\n\nTUDOR\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nThis dominant style of domestic building was used for a large proportion of early 20th-century suburban houses throughout the country\u2014probably comprising about 25 percent of houses built during the 1920s. Colonial Revival is the only style that surpassed Tudor in popularity between 1900 and the late 1920s. Early landmark examples were built around 1890 and by 1900 were joined by slightly more modest versions. The style was particularly fashionable during the 1920s, when it was sometimes called Stockbroker's Tudor. Tudor houses were simplified by the Depression, and during the 1930s French Eclectic houses began to supplant Tudor in popularity. Both Tudor and French were replaced with more modern styles after World War II.\n\n## Comments\n\nThe popular name for the style is historically imprecise, since relatively few examples closely mimic the architectural characteristics of Tudor (16th-century) England. Instead, the style is loosely adapted from a variety of late Medieval and early Renaissance English prototypes, ranging from thatch-roofed folk cottages to grand manors. This broad variety provided the basis for a well-publicized English domestic architecture revival that began around 1850 and lasted until 1930. British architects such as Phillip S. Webb (1831\u20131915), C. F. A. Voysey (1857\u20131941), M. H. Baillie Scott (1865\u20131945), and Sir Edwin Lutyens (1869\u20131944) produced homes that were imitated both in the U.S. and Great Britain. Beginning about 1880, the publisher B. T. Batsford of London advanced this revival by publishing numerous books containing photographs, measured drawings, and drawings of old English homes, which were distributed in the U.S. In 1911 they published _The Domestic Architecture of England During the Tudor Period_ by Thomas Garner and Arthur Stratton, perhaps the style's best sourcebook.\n\nThus, many sources are freely combined in their American revivalist expressions but united by an emphasis on steeply pitched, front-facing gables that, although absent on many original English prototypes, are almost universally present as a dominant facade element in American Tudor houses.\n\nThe earliest American houses in the style date from the late 19th century. These tended to be architect-designed landmarks that might closely copy English models. Many were patterned after late Medieval buildings with Renaissance detailing that were popular during the reigns of Elizabeth I (1558\u20131603) and James I (1603\u20131625), the Elizabethan and Jacobean eras of English history. Architecture historians have proposed the contracted term \"Jacobethan\" for these early Tudor landmarks. Most fall into the parapeted-gable subtype described above.\n\nThe Tudor landmarks of the 1890s were joined in the decades from 1900 to 1920 by less pretentious Tudor houses, which superimposed half-timbering and other typical detailing upon symmetrical facades or simple gables-with-wing forms. Still relatively uncommon before World War I, the style exploded in popularity during the 1920s as masonry veneering techniques allowed even the most modest examples to mimic closely the brick and stone exteriors seen on English prototypes. They display endless variations in overall shape and roof forms. It is not uncommon for the character of Tudor-style homes to vary geographically. Wall materials greatly influence the appearance style, and nearby availability of a particular stone or brick can impart a distinct local character. Second, respected local architects and their preferred approach to Tudor design can affect the typical appearance of the style in a neighborhood or an entire town.\n\nIn addition to the variety in materials and detailing, the picturesque and asymmetrical Tudor offered architects great versatility in floor planning. The house plan could rule the design rather than its being dictated by symmetry. This freedom allowed rooms to be oriented in any direction and windows to be placed where needed; it allowed simple inclusion of wings only one room deep; rooms that were two stories high and wings placed at an angle; and it allowed studios, service rooms, and, later, attached garages, to be effortlessly incorporated into the design.\n\n# ECLECTIC HOUSES\n\n# Chateauesque\n\n# 1880\u20131910\n\n## Identifying Features\n\nSteeply pitched hipped roof; busy roof line with many vertical elements (spires, pinnacles, turrets, gables, and shaped chimneys); multiple dormers, usually wall dormers extending through cornice line; walls of masonry (usually stone).\n\n**chateauesque**\n\nCHATEAUESQUE  \nBoston, Massachusetts; 1890s. An attached town house; note the through-the-cornice wall dormer.\n\nCHATEAUESQUE  \nLouisville, Kentucky; 1890s. The paired wall dormers begin above the cornice in this detached town house.\n\nCHATEAUESQUE  \nMontgomery, Alabama; 1906. Sabel House. The pyramidal hipped roof is also seen in the Byram House and the house in Cincinnati.\n\nCHATEAUESQUE  \nChicago, Illinois; ca. 1880. Byram House; Burnham and Root, architects. The entrance door is in a basket-handle arch surrounded by ogee-arch molding.\n\nCHATEAUESQUE  \nMilwaukee, Wisconsin; ca. 1900. Goldberg House. Note the delicate tracery above the dormers.\n\nCHATEAUESQUE  \nSt. Louis, Missouri; 1892. Bixby House. White paint covers original brick and the colors of the terra-cotta trim. Note the elaborately sculpted dormer detailing.\n\nCHATEAUESQUE  \nCincinnati, Ohio; ca. 1890s. The rough-faced stone of this house is more common in Richardsonian Romanesque houses; Chateauesque examples are more often of brick or smooth-faced stone.\n\nCHATEAUESQUE  \nNew Haven, Connecticut; 1896.  Ingersoll House. Joseph W. Northrop, architect.\n\nCHATEAUESQUE  \nAsheville, North Carolina; 1895. Biltmore; Richard Morris Hunt, architect. This, the ultimate Chateauesque landmark, is located in a setting of gardens worthy of the French originals.\n\nCHATEAUESQUE  \nRedlands, California; 1897. Kimberly Crest; Dennis and Farwell, architects. Constructed of frame walls with smooth stucco cladding and wood trim, rather than of the stone construction more typical of Chateauesque.\n\n## Variants and Details\n\nThe steeply pitched hipped roofs are sometimes truncated above by a flat roof deck; others rise to a high pyramidal apex or hipped ridge. Towers and turrets have steep candle-snuffer roofs. Dormer roofs are usually steep, parapeted gables. Ornamental metal cresting is sometimes used along roof ridges or above cornice lines; the latter generally have elaborate moldings. Gables, doorways, windows, and other facade elements are commonly ornamented with shallow relief carving or Gothic tracing. Windows are usually divided by stone mullions into narrow vertical units with smaller transoms above. Windows and doorways may be arched; the arches often have a characteristic Gothic basket-handle shape.\n\nCHATEAUESQUE\n\n **HIPPED ROOF VARIATIONS**\n\nCHATEAUESQUE\n\n **PYRAMIDAL OR CONE-SHAPED PINNACLES**  \nOn gable tops, roof or tower tops, sides of dormers\n\nCHATEAUESQUE\n\n **WALL DORMERS**\n\nCHATEAUESQUE\n\n **TYPICAL DOORS**\n\nCHATEAUESQUE\n\n **TYPICAL WINDOWS**\n\nCHATEAUESQUE\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nChateauesque is a rare style used primarily for architect-designed landmark houses. Scattered examples are found throughout the country but are most frequent in the larger cities of the northeastern states. Most of these date from the late 1880s and 1890s. Elsewhere the fashion persisted through the first decade of this century.\n\n## Comments\n\nThe Chateauesque is loosely based on monumental 16th-century chateaus of France, which combined earlier Gothic elements with that century's increasingly fashionable trend toward Renaissance detailing. As in the originals, Chateauesque houses show varying mixtures of Gothic and Renaissance detail. The style was popularized in this country by Richard Morris Hunt, the first American architect to study at France's prestigious \u00c9cole des Beaux-Arts. In France, a mid-19th-century revival of buildings in the chateau (or Fran\u00e7ois I) style undoubtedly influenced Hunt, who returned to advocate similar buildings for his wealthy clients. Among these were the Vanderbilts, for whom he designed several Chateauesque houses, culminating in Biltmore, George W. Vanderbilt's North Carolina country house completed in 1895, which rivaled its early French prototypes in size and splendor. The Chateauesque style required massive masonry construction and elaborate, expensive detailing and was therefore unsuitable for vernacular imitation. It thus remained a relatively rare, architect-designed fashion throughout its brief period of popularity.\n\n# ECLECTIC HOUSES\n\n# Beaux Arts\n\n# 1885\u20131930\n\n## Identifying Features\n\nWall surfaces with decorative garlands, floral patterns, or shields; facade with quoins, pilasters, or columns (usually paired and with Ionic or Corinthian capitals); walls of masonry (usually smooth, light-colored stone); first story typically rusticated (stonework joints exaggerated); facade symmetrical.\n\n## Principal Subtypes\n\nTwo principal subtypes can be distinguished:\n\nFLAT OR LOW-PITCHED HIPPED ROOF\u2014This more common of the two subtypes is based on Italian or northern European Renaissance models. Examples lacking full-height, two-story columns are similar to some landmark Italian Renaissance houses, which, however, lack the elaborate decorative detailing typifying the Beaux Arts. This detailing also generally serves to distinguish colonnaded Beaux Arts houses from closely related Neoclassical examples, which, in addition, seldom have the paired columns typical of the Beaux Arts. In brief, look for paired columns and\/or elaborate decorative detailing to most easily help distinguish Beaux Arts houses from similar Italian Renaissance and Neoclassical examples.\n\n**FLAT OR LOW-PITCHED HIPPED ROOF**\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nLouisville, Kentucky; 1901. Ferguson House.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nCincinnati, Ohio; ca. 1910. Note the wide frieze with windows and floral swags.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nDallas, Texas; 1906. Alexander House. This example lacks the floral ornamentation typical of the style, but the paired columns with bands and flat roof with attic windows indicate a Beaux Arts, rather than Neoclassical, inspiration.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nCleveland, Ohio; 1914. Tremaine House; Frederic W. Striebinger, architect. This example, Rosecliff, and The Breakers, reflect a common Renaissance form in which a main block is flanked by symmetrical front-projecting wings. As with The Breakers, an arcaded porch is set here between the wings.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nFrankfort, Kentucky; 1914. Governor's Mansion.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nNewport, Rhode Island; 1902. Rosecliff; McKim, Mead & White, architects.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nSt. Louis, Missouri; 1899. Hills House. An example with a low-pitched hipped roof. Note the flat-roof portico on the left, pedimented portico on the front, and semi-circular portico to right.\n\nFLAT OR LOW-PITCHED HIPPED ROOF:  BEAUX ARTS  \nNewport, Rhode Island; 1892. The Breakers; Richard Morris Hunt, architect (seaside facade). Low-pitched hipped roofs, usually covered with tiles as seen here, also occur in Beaux Arts houses. In other details these are similar to flat-roof examples. Unfortunately the photograph cannot show the numerous small floral details in this Beaux Arts landmark.\n\nMANSARD ROOF\u2014This subtype is loosely based on 17th- and 18th-century French Renaissance models that have distinctive mansard (dual-pitched hipped) roofs with dormer windows on the steep lower slope. These may resemble earlier Second Empire houses with similar roofs, which are, however, generally of smaller scale with walls of wood or brick, rather than stone, and without the distinctive facade decoration of Beaux Arts examples.\n\n**mansard roof**\n\nMANSARD ROOF:  BEAUX ARTS  \nWashington, District of Columbia; 1907. Bliss House; A. Goenner, architect. Attached Beaux Arts town houses such as this are found principally in the larger cities of the Northeast. The mansard roof is particularly favored for urban houses because it reduces the apparent height of upper-floor living space.\n\nMANSARD ROOF:  BEAUX ARTS  \nWashington, District of Columbia; 1909. Fahnestock House; Nathan C. Wyeth, architect.\n\nMANSARD ROOF:  BEAUX ARTS  \nWashington, District of Columbia; 1901. Walsh House; Henry Andersen, architect.\n\nMANSARD ROOF:  BEAUX ARTS  \nMontgomery, Alabama; ca. 1910. Sabel House. A rare example built far from the urban centers of the Northeast.\n\nMANSARD ROOF:  BEAUX ARTS  \nSan Francisco, California; ca. 1915. James F. Dunn, architect. Dunn's distinctive Beaux Arts houses have a sinuous Art Nouveau influence in their details.\n\nMANSARD ROOF:  BEAUX ARTS  \nBuffalo, New York; ca. 1920s. The curved central wing is unusual in this late example.\n\nMANSARD ROOF:  BEAUX ARTS  \nWashington, District of Columbia; 1909. Moran House; George Oakley Totten, Jr., architect. The entryways of this subtype rarely have the colonnaded entry porches seen in the flat-roof subtype. Here only a simple glass canopy is used over the entrance door.\n\nMANSARD ROOF:  BEAUX ARTS  \nNew York, New York; 1905. DeLamar House; C. P. H. Gilbert, architect. This example has the paired side wings that are seen more frequently in the flat-roof subtype.\n\n## Variants and Details\n\nThe Beaux Arts is a classical style and has many of the same details found in other styles of Renaissance classical inspiration, which, however, seldom have the exuberant surface ornamentation that characterizes the Beaux Arts. Entry porches with roofs supported by classical columns are common. Cornice lines are accented by elaborate moldings, dentils, and modillions. Roof-line balustrades and balustraded window balconies are common, as are elaborated window crowns and surrounds. Classical quoins, pilasters, and columns are almost universal. Walls are usually smooth stone or brick; decorative details and elaborations are often made from terra cotta or cast stone.\n\nBEAUX ARTS\n\n **TYPICAL DECORATIVE DETAILS (often made ofterra cotta)**\n\nBEAUX ARTS\n\n **TYPICAL DECORATIVE DETAILS (often made of terra cotta)**\n\nBEAUX ARTS\n\n **TYPICAL ELABORATIONS**\n\nBEAUX ARTS\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nHouses in the Beaux Arts style are usually architect-designed landmarks and were built principally in the prosperous urban centers where turn-of-the-20th century wealth was concentrated. New York, Chicago, Washington, D.C., St. Louis, and San Francisco had many examples, as did Newport, Rhode Island, a favorite summer playground for the affluent. Most domestic examples were built before 1915 but the style persisted until the economic depression of the 1930s. Isolated examples occur throughout the country.\n\n## Comments\n\nThe term \"Beaux Arts\" (the approximate French equivalent of \"Fine Arts\") is used by architectural historians in two different senses. Some use it to describe the entire 1885\u20131920 period of elaborate eclectic styles because these tended to be advocated by Americans who studied at France's \u00c9cole des Beaux-Arts, the era's premier school of architecture. A more limited meaning, followed here, stresses only one Eclectic tradition among the many that were then popular. This is based on Classical precedents elaborated by lavish decorative detailing, and was perhaps the most typical of the many styles inspired by study at the \u00c9cole. More than any other style (except perhaps the Chateauesque), the Beaux Arts expressed the taste and values of America's industrial barons at the turn of the 20th century. In those pre-income tax days, great fortunes were proudly displayed in increasingly ornate and expensive houses. Many were of such a size that they were impossible to maintain in later eras of economic recession and higher taxes. Most of the grandest have been destroyed, but some are preserved as schools, club houses, or museums. The Preservation Society of Newport County, in particular, has saved several of the most elaborate examples as public museums.\n\nAnother concern of the \u00c9cole des Beaux-Arts was for formal planning of the spacial relationships between buildings. This influence provided the impetus for the City Beautiful movement which was prevalent at the turn of the 20th century. The first major example was the planning, supervised by Richard Morris Hunt, for the World's Columbian Exposition in Chicago in 1893. Soon cities such as Cleveland, Philadelphia, and Washington, D.C., implemented monumental planning for their city centers. On a domestic scale this interest in formal design expressed itself in planned suburbs with extensive parks and boulevards lined with landmark houses.\n\n# ECLECTIC HOUSES\n\n# French Eclectic\n\n# 1915\u20131945\n\n## Identifying Features\n\nTall, steeply pitched hipped roof (occasionally gabled in towered subtype) without dominant front-facing cross gable; eaves commonly flared upward at roof-wall junction; commonly has segmental arch on door, windows, or dormers; brick, stone, or stucco wall cladding, sometimes with decorative half-timbering.\n\n## Principal Subtypes\n\nThree principal subtypes can be recognized; each shows a great variety of detailing and wall materials:\n\nSYMMETRICAL\u2014In this subtype, the massive hipped roof, normally with the ridge paralleling the front of the house, dominates a symmetrical facade with centered entry. Facade detailing is usually rather formal, inspired by smaller French manor houses rather than grand chateaus or modest farmhouses. Wings are frequently added to the sides of the main block.\n\n**SYMMETRICAL**\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nBuffalo, New York; ca. 1920s. This example has been turned 90\u00b0 to adapt to a narrow urban lot. What would ordinarily be the side facade faces the street and has been elaborated with shutters and a dormer.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nDallas, Texas; 1941. Evans House.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nDallas, Texas; 1924. Hall House. Henry B. Thomson, architect. Note small scale of the roof dormers here and in the Lewis-Aldredge House.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nCleveland, Ohio; ca. 1920s. Although the main block of this house appears symmetrical, a close look will reveal the right side to be narrower than the left. The open eave with exposed rafters is uncommon in French Eclectic houses.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nBuffalo, New York; ca. 1920s. This house has two identical forward-facing wings; the left one is hidden behind a tree in the photograph. The through-the-cornice wall dormers have windows placed higher in the wing than those in the taller main block.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nDallas, Texas; ca. 1917. Lewis-Aldredge House; Henry B. Thomson, architect. This house has Renaissance detailing borrowed from the Beaux Arts movement. Note the columns beside the door and in the side wings, the pediment over the entry at roof level, the balustrades on the porch, over the door, and in the roof section. The pitch of the hipped roof is also lower than in most examples of the style.\n\nSYMMETRICAL:  FRENCH ECLECTIC  \nSt. Louis, Missouri; 1914. Mallinckrodt House; James P. Jamieson, architect. A strong Chateauesque influence is evident in the door surround, dormers, and roof ornaments of this early example.\n\nASYMMETRICAL\u2014This is the most common subtype and includes both picturesque examples based on rambling French farmhouses as well as more formal houses similar to the symmetrical subtype, but with off-center doorways and asymmetrical facades.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nLouisville, Kentucky; ca. 1920s. Note the irregular quoins around the door and windows.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nRichmond, Virginia; ca. 1920s. A formal example with a shallow-projecting wing that is difficult to distinguish in the photograph. The house is designed for a narrow urban lot.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nCincinnati, Ohio; ca. 1920s. Note the varied eave-line heights, massive chimney, and two types of through-the-cornice wall dormers. Compare this informal, picturesque house with the Graff House, a formal, Renaissance-inspired example.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nCleveland, Ohio; ca. 1920s. Examples based on informal French models sometimes affect Medieval half-timbering as here.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nDallas, Texas; ca. 1930. Note the five through-the-cornice windows.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nSt. Louis, Missouri; 1891. Meysenburg House; Eames & Young, architects. A very early example with Beaux Arts influence in the decorative detail at the entrance and windows.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nWashington, District of Columbia; 1904. Graff House; Jules Henri de Sibour, architect. This house has formal Renaissance detailing; note the regular quoins, keystone lintels, cornice-line dentils, and pedimented dormers with a row of smaller dormers above.\n\nASYMMETRICAL:  FRENCH ECLECTIC  \nDallas, Texas; 1929. Owens House; Greene, LaRoche & Dahl, architect. Note the decorative brick pattern of the entryway and the open front balustrade terrace.\n\nTOWERED\u2014This common subtype is immediately identifiable by the presence of a prominent round tower with a high, conical roof. The tower generally houses the principal door or stairway. Decorative half-timbering is particularly common in this subtype, which is loosely patterned after similar farmhouses from the province of Normandy in northwestern France; Eclectic builders often called these Norman Cottages.\n\n**TOWERED**\n\nTOWERED:  FRENCH ECLECTIC  \nRaleigh, North Carolina; ca. 1930s. Note the three slightly different dormers.\n\nTOWERED:  FRENCH ECLECTIC  \nDallas, Texas; 1937. Gilliland House; Charles S. Dilbeck, architect. This example is unusual in not having the entrance in the tower (see also the Kent House). Note the two chimneys of differing shapes and materials and also that none of the five windows are identical.\n\nTOWERED:  FRENCH ECLECTIC  \nMission Hills, Kansas; ca. 1930s.\n\nTOWERED:  FRENCH ECLECTIC  \nKansas City, Missouri; ca. 1930s.\n\nTOWERED:  FRENCH ECLECTIC  \nBuffalo, New York; ca. 1930s. The regular, formal placement of the windows is not typical of this subtype.\n\nTOWERED:  FRENCH ECLECTIC  \nKansas City, Missouri; ca. 1930s. Note the multi-colored slate roof, tower overhang, massive chimney, and tiny band of half-timbering on the tower.\n\nTOWERED:  FRENCH ECLECTIC  \nCleveland, Ohio; ca. 1930s. Here a stone tower is combined with walls of half-timbered stucco or brick. Note the curving secondary hipped roof, simulating thatch, above the bay windows.\n\nTOWERED:  FRENCH ECLECTIC  \nTuxedo Park, New York; ca. 1930s. Kent House. This landmark example has several towers and an unusually tall roof\u2014note the double row of dormers.\n\n## Variants and Details\n\nBased upon precedents provided by many centuries of French domestic architecture, the style shows great variety in form and detailing but is united by the characteristic roof. (Only the Spanish Revival style, similarly based upon a long and complex architectural tradition, approaches it in variety.) Informal domestic building in northwestern France (particularly Normandy and Brittany) shares much with Medieval English tradition. The use of half-timbering with a variety of different wall materials, as well as roofs of flat tile, slate, stone, or thatch, are common to both. As a result, French Eclectic houses often resemble the contemporaneous Tudor style based on related English precedent. French examples, however, normally lack the dominant front-facing cross gables characteristic of the Tudor. In contrast to these generally informal, rural prototypes, many French Eclectic houses show formal Renaissance detailing resembling that of the English Georgian.\n\nDoors in informal examples are usually set in simple arched openings; doors in symmetrical and formal houses may be surrounded by stone quoins or more elaborate Renaissance detailing (pilasters, pediments, etc.). Windows may be either double-hung or casement sashes, the latter sometimes with small leaded panes. Full-length casement windows with shutters (French doors) are sometimes used. The dominant high-pitched roof may be enhanced by one of these distinctive features that indicate a house is likely to be French Eclectic: (1) through-the cornice dormers or windows and (2) roof dormers that are relatively small in scale or that form a high second tier of dormers on a steep roof.\n\nFRENCH ECLECTIC\n\n **TYPICAL DORMERS**  \nAll types may be either roof or through-the-cornice; roof dormers typically small in scale\n\nFRENCH ECLECTIC\n\n **TYPICALENTRANCES**\n\nmore common in towered and picturesque asymmetrical\n\nmore common in symmetrical and formal asymmetrical\n\nFRENCH ECLECTIC\n\n **TYPICAL ELABORATIONS**\n\n**MORE COMMON IN SYMMETRICAL SUBTYPE**\n\nFRENCH ECLECTIC\n\n **TYPICAL ELABORATIONS**\n\n**MORE COMMON IN TOWERED & ASYMMETRICAL SUBTYPES**\n\n## Occurrence\n\nFrench Eclectic houses were rarely built before World War I. Those that occur before 1920 are commonly of the symmetrical subtype, and are more formal examples\u2014perhaps inspired by the Chateauesque of Beaux Arts traditions. The style began to be somewhat fashionable in the early 1920s, and in 1925 about 5 percent of the new homes built were French, according to a study of houses published in architectural journals that year. By 1930, French Eclectic houses were overtaking Tudor to become the second most popular Eclectic style during the 1930s (behind the always leading Colonial Revival). As with most other Eclectic styles, the French faded from favor after World War II.\n\n## Comments\n\nMany Americans\u2014among them both architects and builders\u2014served in France during World War I, and gained a firsthand familiarity with the broad spectrum of smaller French houses upon which this style is based. During the following decade, the 1920s, Americans were entranced by France, having helped rescue it during the war. Much press was given to the reconstruction of France's historic villages that had been damaged. In addition, a number of photographic studies of modest French houses were published in the 1920s, giving a wide variety of models to draw from.\n\nArthur Meigs and George Howe were among the architects who served in France and subsequently designed French houses; they designed the well-publicized Arthur Newbold house in Pennsylvania (1923). Another veteran was Walter Davis, who built \"The French Village\" in Hollywood. Architect Frank Joseph Forster began building award-winning French Eclectic houses in 1920s, and also published articles on the style. By the late 1920s even architects who had long embraced the Tudor, such as Harrie T. Lindeberg, were designing French houses, and continued to do so into the 1930s. The attention and architectural awards given to high-style French Eclectic houses made the style also desirable to local architects and builders during the 1930s.\n\n# ECLECTIC HOUSES\n\n# Italian Renaissance\n\n# 1890\u20131935\n\n## Identifying Features\n\nLow-pitched hipped roof (flat in some examples); widely overhanging eaves supported by decorative brackets; roof typically covered by ceramic tiles; upper-story windows smaller and less elaborate than windows below; commonly with round arches above doors, first-story windows, or porches; entrance area usually accented by small classical columns or pilasters; facade most commonly symmetrical.\n\n## Principal Subtypes\n\nFour principal subtypes can be distinguished:\n\nSIMPLE HIPPED ROOF\u2014Over half of Italian Renaissance houses have a simple hipped roof with a flat, symmetrical front facade. Full-width porches, often with massive square piers as porch supports, are frequent in examples built before 1920.\n\n**SIMPLE HIPPED ROOF**\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nKansas City, Missouri; ca. 1910. A simple stucco interpretation with Spanish influence in the door surround.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nMontgomery, Alabama; ca. 1920s. An unusual one-story example with an elaborate entry porch bounded by paired pilasters.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nMontgomery, Alabama; ca. 1910s. This example has paired cornice-line brackets and a triple-arched entry porch with balusters above.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nShelbyville, Kentucky; ca. 1920. Note the cornice-line brackets and full-length arched windows on the first story.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nWashington, District of Columbia; 1910. MacVeagh House; Nathan Wyeth, architect. This landmark town house now has a carriage porch added at the entry.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nNew Orleans, Louisiana; ca. 1905. Weis House; MacKenzie and Goldstein, architects. This example shows Prairie influence in its heavy square brick piers and Beaux Arts influence in the paired columns and the elaborated shields ornamenting the brick piers.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nCincinnati, Ohio; ca. 1910. Note the attic-story windows, the quoins, and the second-story windows with triangular pediments above and balustraded balconies below.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nSt. Louis, Missouri; 1906. Pendleton Investment Company. This simple four-square example is built of then-innovative concrete blocks. It adds pronounced Italian Renaissance brackets and the suggestion of an attic story to an otherwise Prairie house.\n\nSIMPLE HIPPED ROOF:  ITALIAN RENAISSANCE  \nLouisville, Kentucky; 1908. Rostrevor; Loomis and Hartman, architects. A large five-ranked stone example with an unusual recessed entry porch. Note that the shape of the porch echoes that of a Palladian window (see also the example of the first house in Montgomery).\n\nHIPPED ROOF WITH PROJECTING WING(S)\u2014Many Italian Renaissance houses have either a small central wing projecting forward from the front facade, or two small wings at either end of the facade with a recessed central block in between.\n\n**HIPPED ROOF WITH PROJECTING WING(S)**\n\nPROJECTING CENTRAL WING\n\nPROJECTING CENTRAL WING:  ITALIAN RENAISSANCE  \nMontgomery, Alabama; ca. 1920. A simple three-ranked brick example. Note the recessed entry with Palladian motif, full-length French doors set in blind arches, and one-story side wings.\n\nPROJECTING CENTRAL WING:  ITALIAN RENAISSANCE  \nBuffalo, New York; ca. 1920. This example is unusual in lacking wide overhanging eaves.\n\nPROJECTING CENTRAL WING:  ITALIAN RENAISSANCE  \nLos Angeles, California; ca. 1920s. Hancock Park. Red brick is a less typical cladding for Italian Renaissance houses than are stucco, stone, or brick in stone colors.\n\nPROJECTING CENTRAL WING:  ITALIAN RENAISSANCE  \nNew Orleans, Louisiana; ca. 1910. This elaborate stucco example has quoins and recessed upper and lower porches in the front wing.\n\nPROJECTING SIDE WINGS\n\nPROJECTING SIDE WINGS:  ITALIAN RENAISSANCE  \nWichita, Kansas; ca. 1920. The tall window to the left of the front door probably lights the stairway.\n\nPROJECTING SIDE WINGS:  ITALIAN RENAISSANCE  \nKansas City, Missouri; 1913. Halpin House; John W. McKecknie, architect. Made of concrete clad with cut stone, this house is unusual in having elaborated, arched windows in the upper story rather than the lower.\n\nPROJECTING SIDE WINGS:  ITALIAN RENAISSANCE  \nSaratoga, California, vicinity; 1912. Villa Montalvo; William Curlett, architect. This large house has little decorative detailing. The raised terrace with balustrade, a typical feature, is also seen in the houses in Buffalo and Hancock Park.\n\nPROJECTING SIDE WINGS:  ITALIAN RENAISSANCE  \nRichmond, Virginia; ca. 1920. This unusual example has a three-story central block with two-story wings projecting forward.\n\nPROJECTING SIDE WINGS:  ITALIAN RENAISSANCE  \nDallas, Texas; 1924. Ragland House; Clarence C. Bulger, architect.\n\nASYMMETRICAL\u2014A relatively small proportion of Italian Renaissance houses have unbalanced, asymmetrical facades. Usually the asymmetry involves only door and window placement on an otherwise symmetrical building of simple square or rectangular plan. Less commonly L plans or more complex shapes are used.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nDallas, Texas; 1922. This one-story example has Craftsman influence in the exposed rafters and in the muntin pattern on the front door and sidelights. But the tall arched windows, and the balustrade along the front terrace, show its Italian Renaissance aspirations.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nDurham, North Carolina; ca. 1920s. This stucco example is a simpler version of its neighbor shown in the next example.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nDurham, North Carolina; ca. 1920s. A seven-ranked brick example with asymmetrical recessed porch.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nLouisville, Kentucky; ca. 1920s. This example has Beaux Arts detailing over the door and arched windows.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nKansas City, Missouri, ca. 1910s. An unusual feature for this style is the prominent front chimney.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nMontgomery, Alabama; ca. 1910s. The taller, projecting section recalls the towers of the Italianate style.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nSt. Louis, Missouri; 1926. Tom Barnett, architect. Note the prominent Palladian-motif entry and angled side wings.\n\nASYMMETRICAL:  ITALIAN RENAISSANCE  \nMontgomery, Alabama, ca. 1910s.\n\nFLAT ROOF\u2014Many high-style Italian Renaissance houses have flat roofs, usually with a prominent, dentiled cornice and roof-line balustrade. Typically the first story is rusticated (finished as exaggerated stonework courses), while the floors above have smooth wall finishes. Facades are symmetrical. These are almost always architect-designed landmarks built of stone; they are closely related to flat-roofed, Beaux Arts\u2013style houses, which are similar but add more elaborate facade detailing.\n\n**FLAT ROOF**\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nRichmond, Virginia; ca. 1910. A detached, three-ranked town house. There is actually a low-pitched roof behind the roof-line balustrade.\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nDallas, Texas; 1909. Hill House; C. D. Hill, architect. A Prairie\u2013Italian Renaissance hybrid. Remove the brackets and add a hipped roof with overhanging eaves and you would have a house similar to those on this page.\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nPasadena, California; 1905. Fenyes Mansion; Robert D. Farquhar, architect. It is less common to have arched windows in the second story.\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nSt. Louis, Missouri; ca. 1910. Note the solid roof parapet, rather than the more usual balustrade, and the recessed Palladian entry.\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nWashington, District of Columbia; ca. 1910.\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nSavannah, Georgia; 1917. Armstrong House; Henrik Wallis, architect. A colonnaded, curved side porch or gallery is visible to the right. Arcaded and colonnaded porches such as these are sometimes called \"loggias\" (see this page for other examples).\n\nFLAT ROOF:  ITALIAN RENAISSANCE  \nSt. Louis, Missouri; ca. 1910. Note the rusticated first story, the triangular and segmental pediments above the second-story windows, and the upper-story pilasters. An unusual note is the unaccented entry at the left side of this absolutely symmetrical house; a bay window is used where one would expect the entry.\n\n## Variants and Details\n\nDetails are borrowed more or less directly from the Italian originals. Among the most characteristic are recessed entry porches and full-length first-story windows with arches above. The roof, except when flat, commonly has broadly overhanging, boxed eaves; normally the eaves have decorative brackets beneath. These features of the roof-wall junction are helpful in distinguishing Italian Renaissance houses from related Mediterranean styles with tiled roofs. Mission houses usually have wide eave overhangs, but these are commonly open rather than boxed-in. Spanish Revival houses normally have little or no eave overhang. Eave brackets are rare on both Mission and Spanish Revival houses. Common decorative details include quoins, roof-line balustrades, pedimented windows, classical door surrounds, molded cornices, and belt courses. Walls are constructed of or clad with stone, stucco, or brick (generally stone-colored); wooden wall claddings are not used. Note that similar details appear in several earlier styles with Renaissance roots, particularly the Georgian, Federal, and Italianate. Because of these similarities, Italian Renaissance houses sometimes resemble Georgian- or Federal-inspired examples of the contemporaneous Colonial Revival.\n\nITALIAN RENAISSANCE\n\n **TYPICALARCHED OPENINGS**\n\nDOORS\n\nWINDOWS\n\nPORCHES  \ncommonly recessed\n\nITALIAN RENAISSANCE\n\n **TYPICAL ROOF-WALL JUNCTIONS**\n\nUSUALLY BOXED\n\nOCCASIONALLY OPEN  \nmore common in Mission\n\nITALIAN RENAISSANCE\n\n **COMMON HIGH-STYLE ELABORATIONS**\n\n## Occurrence\n\nThe Italian Renaissance style is found in early 20th-century houses throughout the country but is considerably less common than the contemporaneous Craftsman, Tudor, or Colonial Revival styles. Primarily a style for architect-designed landmarks in major metropolitan areas prior to World War I, vernacular interpretations spread widely with the perfection of masonry veneering techniques; most of these date from the 1920s. The style steadily declined in popularity through the 1930s, and post-1940 examples are rare.\n\n## Comments\n\nThe latest revival of interest in Italian Renaissance domestic models began with the landmark Villard Houses in New York (McKim, Mead & White, 1883). Other fashionable architects used the style in the late 1880s and 1890s as a dramatic contrast to the Gothic-inspired Shingle or Queen Anne styles. These Second Renaissance Revival houses tended to mimic more closely their Italian predecessors than did the free interpretations of the preceding Italianate style. There are several reasons for this increased authenticity. By the late 19th century a great many American architects, and their clients, had visited Italy and thus had some first-hand familiarity with the original models. Furthermore, improved printing technology provided ready access to excellent photographic documentation of these models. The earlier Italianate style, in contrast, was usually based on pattern book drawings by professionals with no first-hand knowledge of Italian buildings. In addition, many houses of the earlier Italianate style had wooden wall cladding, whereas later examples invariably mimic the stuccoed or masonry walls of their original Italian prototypes. The perfection of masonry veneering techniques after World War I made this possible in even the most modest examples of the style.\n\n# ECLECTIC HOUSES\n\n# Mission\n\n# 1890\u20131920\n\n## Identifying Features\n\nMission-shaped dormer or roof parapet (these may be on either main roof or porch roof); commonly with red tile roof covering; widely overhanging eaves, usually open; porch roofs supported by large, square piers, commonly arched above; wall surface usually smooth stucco.\n\n## Principal Subtypes\n\nTwo principal subtypes can be distinguished:\n\nSYMMETRICAL\u2014About half of Mission houses have balanced, symmetrical facades. These are most commonly of simple square or rectangular plan with hipped roofs.\n\n**SYMMETRICAL**\n\nSYMMETRICAL:  MISSION  \nDallas, Texas; 1912. Bianchi House; Lang & Witchell, architects.\n\nSYMMETRICAL:  MISSION  \nHammond, Louisiana; ca. 1910. Preston House. The wood wall cladding is unusual. Although open eaves are most common in the style, boxed eaves also occur, usually with brackets below as seen here and in the Harris-Savage House.\n\nSYMMETRICAL:  MISSION  \nDallas, Texas; 1917. Harris-Savage House. This house has had the original entry arch with the shaped parapet above restored. The materials used here\u2014brown wirecut brick, Vermont slate roof\u2014are unusual for a Mission house.\n\nSYMMETRICAL:  MISSION  \nKansas City, Missouri; ca. 1910. This house shows the four-square shape that was popular in various styles built from about 1900 to 1915.\n\nSYMMETRICAL:  MISSION  \nOklahoma City, Oklahoma; ca. 1910.\n\nSYMMETRICAL:  MISSION  \nWashington, District of Columbia; 1902. Barney House; Waddy B. Wood, architect. A rare example of a Mission town house.\n\nSYMMETRICAL:  MISSION  \nLouisville, Kentucky; ca. 1910. Caperton House.\n\nSYMMETRICAL:  MISSION  \nRedlands, California; 1901. Burrage House; Charles Bingham, architect. This landmark house is a full-scale copy of a Spanish mission, complete with bell towers and arcaded side wings.\n\nASYMMETRICAL\u2014The remaining half of Mission houses have asymmetrical facades of widely varying form. Most typically the facade asymmetry is superimposed on a simple square or rectangular plan. Elaborate, rambling compound plans are found on some landmark examples.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  MISSION  \nFort Smith, Arkansas; ca. 1910. In this example the entire side gable of the main roof is covered by a shaped parapet.\n\nASYMMETRICAL:  MISSION  \nOklahoma City, Oklahoma; ca. 1910. Here a flat roof is surrounded by a parapet with a projecting visor roof beneath; the porch roof repeats this pattern on a smaller scale.\n\nASYMMETRICAL:  MISSION  \nSalisbury, North Carolina; ca. 1910. The recessed arcaded porch on the second story creates asymmetry on this otherwise balanced facade.\n\nASYMMETRICAL:  MISSION  \nKansas City, Missouri; ca. 1910. The bell tower is unusual on relatively modest examples such as this.\n\nASYMMETRICAL:  MISSION  \nDallas, Texas; 1915. Green House; Daren Tapscott, restoration architect. This house has a flat roof surrounded by a parapet, with a projecting tiled visor roof beneath. The porch roof repeats this pattern. Note the carefully restored original details\u2014including grouped windows with the original muntin pattern, porch railing, and ornament below the shaped parapets.\n\nASYMMETRICAL:  MISSION  \nFullerton, California; ca. 1915. Note the Exotic influence in the recessed porch over the entry, and in the porch columns. This house looks symmetrical, but the right and left sides of the facade are quite different. Compare it to the following example, where the house looks asymmetrical because of its wings, but has a symmetrical central block.\n\nASYMMETRICAL:  MISSION  \nKansas City, Missouri; ca. 1910. Few Mission houses are built of stone; brick and stucco are the most common materials.\n\nASYMMETRICAL:  MISSION  \nWhite Plains, New York; ca. 1910. Scholz House.\n\nASYMMETRICAL:  MISSION  \nRedlands, California; 1903. Holt House; Frederick Thomas Harris, architect. The decorative detailing is unusually exuberant for the Mission style, which normally stresses smooth, flat wall surfaces.\n\n## Variants and Details\n\nA great variety of shaped dormers and roof parapets mimic those found on some Spanish Colonial mission buildings. Few are precise copies of the original models. Most examples have prominent one-story porches either at the entry area or covering the full width of the facade; these sometimes have arched roof supports to simulate the arcades of Hispanic buildings. Mission-like bell towers occur on a few landmark examples. Quatrefoil windows are common; decorative detailing is generally absent, although patterned tiles, carved stonework, or other wall surface ornament is occasionally used. Windows are typically double-hung and are sometimes grouped together. Some examples have unusual visor roofs. These are narrow, tiled roof segments cantilevered out from a smooth wall surface (similar to the pent roofs seen in some Georgian or Colonial Revival houses). They most commonly occur beneath the parapets of flat roofs.\n\nMISSION\n\n **TYPICAL ELABORATIONS**\n\nMISSION\n\n **TYPICAL REVIVAL DORMERS & PARAPETS**\n\nMISSION\n\n **PARAPETS OF ORIGINALMISSIONS** Most are restorations\n\n## Occurrence\n\nCalifornia was the birthplace of the Mission style and many of its landmark examples are concentrated there. The earliest were built in the 1890s; by 1900 houses in this style were spreading eastward under the influence of fashionable architects and national builders' magazines. They also appeared in house plan books such as those of Sears Roebuck and Co. that sold plans for a \"Mission type\" house called the Alhambra in the late 1910s. Although never common outside of the southwestern states, scattered examples were built in early 20th-century suburbs throughout the country. Most date from the years between 1905 and 1920.\n\n## Comments\n\nOne scholar has noted that the style \"is the Californian counterpart\" of the Georgian-inspired Colonial Revival that was then gaining popularity in the northeastern states. Rather than copy the East's revival of its own colonial past, California turned to its Hispanic heritage for inspiration. Several California architects began to advocate the style in the late 1880s and early 1890s. This resulted in the large California Building at the 1893 World's Columbian Exposition being built in a Mission-inspired style that gained attention and admiration. Mission Revival received further impetus when the Santa Fe and Southern Pacific railways adopted the style for stations and resort hotels throughout the West. Most commonly, typical Hispanic design elements (shaped parapets, arches, quatrefoil windows, etc.) were borrowed and freely adapted to adorn traditional shapes. It was not uncommon for a typical four-square shape, with massive square porch supports and double-hung windows, to have tile roofs and one or more shaped parapets added to create a simplified Mission house. In a few landmark examples, however, the forms of the early missions, including twin bell towers and elaborate arcades, were faithfully followed in domestic designs. In still other examples, innovative architects designed Mission buildings with many features borrowed from the contemporary Craftsman and Prairie movements; some even anticipate the simplicity of the subsequent International style. The style quickly faded from favor after World War I as architectural fashion shifted from free, simplified adaptations of earlier prototypes to more precise, correct copies. From this concern grew the Spanish Revival style which drew inspiration from a broader spectrum of both Old and New World Spanish buildings.\n\n# ECLECTIC HOUSES\n\n# Spanish Revival\n\n# 1915\u20131940\n\n## Identifying Features\n\nLow-pitched roof, usually with little or no eave overhang; red tile roof covering; typically with one or more prominent arches placed above door or principal window, or beneath porch roof; wall surface usually stucco; wall surface extends into gable without break (eave or trim normally lacking beneath gable); facade normally asymmetrical.\n\n## Principal Subtypes\n\nFive principal subtypes can be distinguished:\n\nSIDE-GABLED ROOF\u2014About 20 percent of Spanish Revival houses have side-gabled roofs. Many of these are multi-level with taller, side-gabled sections bounded by lower, side-gabled wings.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nMission Hills, Kansas; ca. 1920s. This house shows symmetrical Renaissance influences in its centered doorway with quoined door surround.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nWichita, Kansas; ca. 1930s. The unadorned main block, with its short broad chimney and paucity of windows, resembles a smaller version of El Hogar.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nNew Orleans, Louisiana; ca. 1920s. The doorway is surrounded by low-relief carving of Plateresque inspiration. Note the elaborate window grilles.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nDallas, Texas; 1927. Hollywood Heights\/Santa Monica Conservation District. Symmetrical examples like this are far less common than asymmetrical ones.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nSanta Barbara, California; 1916. El Hogar; George Washington Smith, architect. Inspired by the houses of southern Spain, this example presents an austere facade to the world (few windows and little ornamentation) but opens into an elaborate garden behind.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nDallas, Texas; ca. 1930s. Note the spiraled columns beside the entry and the lower-story windows.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nWichita, Kansas; ca. 1930s. The projecting door surround is atypical.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nDallas, Texas; 1926. Green House; Thomson and Swaine, architects. Although similar to El Hogar in basic form, this example adds numerous double-hung windows to the front facade, giving it a less authentic look. Note the strongly textured stucco walls.\n\nSIDE-GABLED ROOF:  SPANISH REVIVAL  \nSanta Barbara, California; 1925. Casa del Herrera; George Washington Smith, architect. This entry facade shows Smith's ease with picturesque asymmetry. This entry facade suggests the gradual accumulation of informal additions that was typical of Spanish folk dwellings.\n\nCROSS-GABLED ROOF\u2014About 50 percent of Spanish Revival houses have cross-gabled roofs with one prominent, front-facing gable. These are usually L-plan houses; one-story and two-story forms are both common, as are examples with wings of differing heights.\n\n**CROSS-GABLED ROOF**\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nDelano, California; ca. 1930s. Simple one-story examples similar to this dominate many 1930s neighborhoods in Florida and California.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nSanta Barbara, California; 1923. Burke House; George Washington Smith, architect. Note the restrained facade with large expanses of windowless wall. The small house-shaped chimney capping at the right is a favorite Spanish Revival detail.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nLouisville, Kentucky; ca. 1930s. Note the strong textured pattern of the stucco walls.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nDallas, Texas; 1936. Baty House. This small example is complete with a bell tower, a focal window with stained glass, and a front entry court enclosed by a low stone wall.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nSt. Louis, Missouri; ca. 1930s.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nOklahoma City, Oklahoma; ca. 1930s.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nLos Angeles, California; ca. 1930s. The arched entry is accented with four spiraled columns. Note the large parabolic art glass window.\n\nCROSS-GABLED ROOF:  SPANISH REVIVAL  \nLos Angeles, California; ca. 1920s.\n\nCOMBINED HIPPED-AND-GABLED ROOFS\u2014Some landmark examples have rambling, compound plans in which different units have separate roof forms of varying heights arranged in an irregular, informal pattern. Typically both hipped and gabled roofs are used in combination, a pattern which mimics the varied roof forms of Spanish villages.\n\n**COMBINED HIPPED-AND-GABLED ROOFS**\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nDallas, Texas; 1938. Turner House. Note the overhanging balcony and enclosed entry court.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nSan Antonio, Texas; ca. 1929. Monte Vista Historic District; Adams & Adams, architects.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nMontecito, California; 1916. Bliss House; Carleton Winslow, Sr., architect. Note the bell tower and multiple-arched chimney crowns.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nSanta Barbara, California; ca. 1930. Villa Eseanado. Note the ornate Renaissance-inspired entryway and the differing roof heights of the three wings, which enclose an interior courtyard.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nSanta Barbara, California; 1925. Dreyfus House; W. Maybury Somervell, architect. This landmark example, with its varying roof forms, resembles an entire block of a Spanish village.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nPalm Beach, Florida; 1927. Mar-A-Lago; Addison Mizner, architect. A major landmark of the style.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nMontecito, California; 1930. Dieterich House; Addison Mizner, architect. This photograph and the next illustrate the elaborate courtyards found in most landmark examples. This photo shows an automobile entry court.\n\nCOMBINED HIPPED-AND-GABLED ROOFS:  SPANISH REVIVAL  \nMontecito, California; 1930. Dieterich House; Addison Mizner, architect. This photo shows off an interior courtyard. Note the fountain, arcaded gallery, and decorative paving.\n\nHIPPED ROOF\u2014About 10 percent of Spanish Revival houses have low-pitched hipped roofs. These are generally two-story forms with simple rectangular plans.\n\n**HIPPED ROOF**\n\nHIPPED ROOF:  SPANISH REVIVAL  \nPalo Alto, California; ca. 1930s. Kennedy House.\n\nHIPPED ROOF:  SPANISH REVIVAL  \nMorgan Hill, California; ca. 1930s. Fountain Oaks.\n\nHIPPED ROOF:  SPANISH REVIVAL  \nCorning, New York; ca. 1930s.\n\nHIPPED ROOF:  SPANISH REVIVAL  \nDallas, Texas; 1942. Luse House; Fooshee & Cheek, architects. Note the elaborate door surround, the two focal window areas, and the corner quoins.\n\nHIPPED ROOF:  SPANISH REVIVAL  \nLos Angeles, California; ca. 1920s. Hancock Park Historic District.\n\nFLAT ROOF\u2014About 10 percent of Spanish Revival houses have flat roofs with parapeted walls. These typically show combinations of one- and two-story units. Narrow, tile-covered shed roofs are typically added above entryways or projecting windows. This subtype, loosely based on flat-roofed Spanish prototypes, resembles the Pueblo Revival house.\n\n**FLAT ROOF**\n\nFLAT ROOF:  SPANISH REVIVAL  \nSanta Barbara, California; ca. 1930. This house and the other example in Santa Barbara are typical of smaller examples built by the thousands in California suburbs during the 1920s and 1930s. The flat roof with decorative tiles along the parapet is typical, as is the arched entryway with either gabled or flat roof.\n\nFLAT ROOF:  SPANISH REVIVAL  \nSt. Louis, Missouri; ca. 1930s. This house and the examples in Independence and Los Angeles combine both one- and two-story sections. Note the small shed roofs over the windows and the shed-roof entryways.\n\nFLAT ROOF:  SPANISH REVIVAL  \nSanta Barbara, California; ca. 1920.\n\nFLAT ROOF:  SPANISH REVIVAL  \nIndependence, Missouri; ca. 1930s.\n\nFLAT ROOF:  SPANISH REVIVAL  \nLos Angeles, California (vicinity); ca. 1920s.\n\n## Variants and Details\n\nThe style uses decorative details borrowed from the entire history of Spanish architecture. These may be of Moorish, Byzantine, Gothic, or Renaissance inspiration, an unusually rich and varied series of decorative precedents. The typical roof tiles are of two basic types: Mission tiles, which are shaped like half-cylinders, and Spanish tiles, which have an S-curve shape. Both types occur in many variations depending on the size of the tiles and the patterns in which they are applied. Dramatically carved doors are typical of Spanish architecture; these are more common on high-style Spanish Revival houses but also occur on modest examples. Doors are often emphasized by adjacent spiral columns, pilasters, carved stonework, or patterned tiles. Less elaborate entrance doors of heavy wood panels, sometimes arched above, are also common. Doors leading to exterior gardens, patios, and balconies are usually paired and glazed with multiple panes of rectangular glass. Many examples have at least one large focal window. These are commonly of triple-arched or parabolic shape and may be filled with stained glass of varying design. Decorative window grilles of wood or iron are common, as are similar balustrades on cantilevered balconies, which occur in a variety of shapes and sizes. Stucco walls could be smooth or have various rough or tooled finishes. Other typical details include tile-roofed (and otherwise decorated) chimney tops; brick or tile vents; fountains; arcaded walkways (usually leading to a rear garden); walled entry courtyards; twisted spiral columns (officially Solomonic columns and informally barbershop-pole columns); and round or square towers.\n\nSPANISH REVIVAL\n\n **TYPICAL TILE ROOF PATTERNS**\n\nSPANISH REVIVAL\n\n **TYPICAL DOORS & DOOR SURROUNDS**\n\nSPANISH REVIVAL\n\n **TYPICAL WINDOWS**\n\nFOCAL WINDOWS\n\nSPANISH REVIVAL\n\n **TYPICAL WINDOWS**\n\nCASEMENT WINDOWS\n\nSPANISH REVIVAL\n\n **TYPICAL WINDOWS**\n\nWINDOW GRILLES\n\nSPANISH REVIVAL\n\n **COMMON ELABORATIONS**\n\nSPANISH REVIVAL\n\n **COMMON ELABORATIONS**\n\n## Occurrence\n\nSpanish Revival is most common in the southwestern states, particularly California, Arizona, and Texas, and in Florida, all regions where original Spanish Colonial building occurred and continued into the 19th century. Landmark houses in this style are rare outside of Florida and the Southwest but, as in the related Mission style which preceded it, scattered vernacular examples are found in suburban developments throughout the country. During the 1920s, many new communities in Florida and southern California were planned in the Spanish Revival style, and older towns (such as Santa Barbara, California) also sought to affect a Spanish Colonial image in new construction. Los Angeles, in particular, added large neighborhoods of Spanish Revival homes.\n\n## Comments\n\nBefore about 1920, houses of Hispanic precedent were based on simple early Spanish missions. It was the Panama-California Exposition held in San Diego in 1915 that introduced the elaborate Spanish prototypes found in other countries. This exposition was designed by Bertram Grosvenor Goodhue (1869\u20131924), who emphasized the richness of Spanish Colonial precedents seen in the major buildings of other countries. The exhibition was widely publicized and well received. Soon other architects were inspired to look directly to Spain for source material. World War I (1914\u20131918) caused architects wishing to study and sketch in Europe to concentrate on Spain. There they found a centuries-long and very rich sequence of architecture traditions that they could meld into the quite varied Spanish Revival.\n\nThe range of decorative detail found in Spain was extraordinarily diverse. But of equal interest was the way buildings were composed and massed, a subject studied particularly in rural Andalusian houses. These vernacular dwellings resulted when homes were gradually expanded in a very informal, additive way. Facades generally had little decorative detail and instead emphasized their varied massing. This approach was favored by some high-style Spanish Revival architects such as George Washington Smith (1876\u20131930) in southern California.\n\nIn Florida, Spanish Revival was introduced by industrialist turned developer Henry Flagler to promote tourism. It developed its own Florida flavor, epitomized by the designs of Addison Mizner (1872\u20131933) and Maurice Fatio (1897\u20131943). The style reached its apex on both coasts during the 1920s and early 1930s and passed rapidly from favor during the 1940s.\n\n# ECLECTIC HOUSES\n\n# Monterey\n\n# 1925\u20131955\n\n## Identifying Features\n\nTwo stories, with low-pitched gabled roof (occasionally hipped); broad, dominant second-story balcony, usually cantilevered and covered by principal roof.\n\n**MONTEREY**\n\nMONTEREY  \nDallas, Texas; 1937. Braly House. This example, with its tiled roof and parabolic focal window, is transitional to the Spanish Revival style.\n\nMONTEREY  \nMontecito, California; 1929. Morphy House; Roland Coate, Sr., architect. Note the asymmetrical placement of doors and windows and the shaped ends of the heavy beams supporting the balcony.\n\nMONTEREY  \nKansas City, Missouri; ca. 1930s. Many examples of the style have three-ranked facades. When a broader facade was desired, a separate parallel wing was added, as here.\n\nMONTEREY  \nSan Marino, California; ca. 1930s. The full-width balcony wraps around the side.\n\nMONTEREY  \nDallas, Texas; ca. 1930s.\n\nMONTEREY  \nDallas, Texas; 1951. Bywaters House. In the late 1940s and 1950s the simple wood balcony railings and roof supports of the style were commonly replaced by lacy cast iron, leading to a variant called the Creole French style by its builders. These were, of course, inspired by the iron balconies of New Orleans. Asymmetrical interpretations of these Creole French houses, usually with front-facing gable wings, were also popular.\n\nMONTEREY  \nPasadena, California; 1941. Jordan House; Whitney R. Smith, architect. The upper story has board-and-batten cladding.\n\n## Variants and Details\n\nRoofs are usually covered with wooden shingles but are occasionally tiled or standing seam metal in later examples. Wall cladding materials are either stucco, brick, or wood (weatherboard, shingle, or vertical board-and-batten). The first and second stories frequently have different cladding materials, with wood over brick being the most common pattern. Door and window surrounds sometimes mimic the Territorial examples of their Spanish Colonial prototypes; paired windows and false shutters are common. Doors may show Colonial Revival influences. One variant substitutes balcony columns and balustrades of cast iron for the more typical wooden detailing. These are sometimes called Creole French houses. Balconies are also typical in Spanish Revival houses, where they are often narrower and joined by exuberant Hispanic decorative detail on the rest of the house. In Monterey houses the balcony is generally the primary decorative feature on the house and extends the full width of the house\u2014or the full width of the wing in a gabled-front-and-wing form.\n\nMONTEREY\n\n **COMMON DETAILS**\n\n## Comments\n\nThe Monterey is a free interpretation of the Anglo-influenced Spanish Colonial houses of northern California. These blended Spanish adobe construction with pitched-roof, massed-plan English shapes brought to California from New England. However, the key identifying feature of the Monterey\u2014the full-width cantilevered balcony (or occasionally two-story porch)\u2014was derived from house forms built in the southeastern United States, the Caribbean, and the Bahamas. The 1834 Larkin House in Monterey, California, is generally identified as the first of this distinctive form of early pitched-roof Spanish Colonial houses. Thomas Oliver Larkin, the owner-designer of it, had traveled in South Carolina and Bermuda, where similar houses were found. See the Larkin House and other examples of this Anglicized form of Spanish Colonial house.\n\nCalifornia architect Roland E. Coate, Sr., played a crucial role in reviving the style. Between 1929 and 1932 he wrote about the early Monterey homes, designed at least two homes in the Monterey, and won a prestigious Better Homes in America Award for an elegant Santa Barbara example. His efforts helped bring the style to the attention of both professionals and the public.\n\nSome early examples from about 1925 to 1940 have Spanish detailing, while those from the 1940s and 1950s generally include only English Colonial details. A third, less common, variation was sometimes called Creole French; these had decorative iron balconies inspired by the upper-level balconies found in the Vieux Carr\u00e9 in New Orleans. Though scattered examples of Monterey houses occur throughout the country in suburbs built during the second quarter of the 20th century, they are perhaps most common in California and Texas.\n\n# ECLECTIC HOUSES\n\n# Pueblo Revival\n\n# 1910\u2013present\n\n## Identifying Features\n\nFlat roof with parapeted wall above; wall and roof parapet with irregular, rounded edges; projecting wooden roof beams (vigas) extending through walls; stucco wall surface, usually earth-colored.\n\n**PUEBLO REVIVAL**\n\nPUEBLO REVIVAL  \nSante Fe, New Mexico; ca. 1930s.\n\nPUEBLO REVIVAL  \nSanta Fe, New Mexico; ca. 1920s. John Gaw Meem, architect. The closeup shows the irregular rounded edges of the roof parapet and walls, the irregular stucco texture, and the inset window lintels. Note the difference between the two projecting roof beams (vigas) and the rainwater gutter (canale) immediately to the left. In the full view the stepped\u2011up effect (emphasized by the heavy shadows) begins with the garden wall at lower left and continues to the first- and second-story sections. Note the inward curving profile of the walls at right.\n\nPUEBLO REVIVAL  \nSante Fe, New Mexico; ca. 1930s. Note adobe chimney here and in both the previous two and the following examples.\n\nPUEBLO REVIVAL  \nSanta Fe, New Mexico; 1918. Vierra House; Carlos Vierra, architect. This seminal example has stepped roofs and strongly rounded edges to walls and roofs. Note the ladder, the only stairway of the original Pueblos.\n\nPUEBLO REVIVAL  \nPhoenix, Arizona; 1928. El Encanto Historic District.\n\n## Variants and Details\n\nPueblo Revival houses imitate the hand-finishes of their Native American prototypes. Corners are blunted or rounded and wall surfaces are given irregular, stuccoed textures. In addition, rough-hewn vigas (roof beams), window lintels, and porch supports carry out the hand-built theme. The stepped-back roof line of the original pueblos is often used, and larger examples may also feature their picturesque additive massing. This effect is more easily imitated with roof parapets and garden walls that have irregularly rounded, stepped\u2011up tops. The actual structure of the house can vary from stabilized adobe or concrete block to balloon frame. Windows are double-hung or casements of wood or metal.\n\nPUEBLO REVIVAL\n\n **TYPICAL ELABORATIONS**\n\nPUEBLO REVIVAL\n\n **PORCHES (VERANDAHS OR PORTALES)**\n\n## Comments\n\nLike the contemporary Mission movement, the Pueblo Revival draws on local historical precedents for inspiration. The buildings are a mixture of influences from both flat-roofed Spanish Colonial buildings and Native American pueblos. For this reason some architectural historians have proposed the name \"Pueblo\u2013Spanish Revival\" for the style. The earliest examples were built in California around the turn of the 20th century. The style became most popular, however, in Arizona and New Mexico, areas where the original prototypes survive. It is particularly common in Tucson, Albuquerque, and Santa Fe, where it persists today, in part because of the requirements of special design control districts. Scattered examples occur throughout the southwestern states, most of which date from the 1920s and 1930s. A variation, introduced in New Mexico about 1920, could be more aptly termed the Territorial Revival. Although executed in adobe, it picked up the form and detail of the Anglo-Spanish Colonial Territorial house (the Tully House and the Borrego House) with roof parapets topped by fired brick and wooden decorative details influenced by the Greek Revival movement.\n\n# Modern Houses\n\n1900\u2013present\n\n* * *\n\n**EARLY MODERN**\n\n**Prairie**\n\n**Craftsman**\n\n**Modernistic**\n\n**BANKERS MODERN**\n\n**Minimal Traditional**\n\n**Ranch**\n\n**Split Level**\n\n**MAINSTREAM MODERN**\n\n**International**\n\n**Contemporary**\n\n**Shed**\n\n**Other 20th-Century Modern**\n\n**21st-Century Modern**\n\nInnovative American-born architect Frank Lloyd Wright exercised his prodigious talent and powerful personality first in shaping the early Modern movement in the United States and then in disseminating it throughout Europe. In 1893, after working for pioneering modernists Adler & Sullivan, Wright opened his own architecture practice, and by 1902 he had created the first of an entirely new kind of house\u2014the Prairie\u2014with free-flowing interior spaces, new spatial effects, and an innovative vocabulary of ornament that did not mimic historic forms. Wright believed in architectural ornament\u2014as did Greene and Greene, his West Coast counterparts who perfected the Craftsman style. These architects and their followers spread early modernism and its American Arts and Crafts ornamentation throughout the United States in the form of Craftsman and Prairie houses.\n\nBut it was Wright alone who carried American modernism abroad. While self-exiled in Europe after a personal scandal, Wright prepared renderings of his early work for publication. Titled the _Wasmuth Portfolio_ (1910), it dispersed his concept of organic architecture and modern spatial design to European architects.\n\nHis message was eagerly received, but European architects dramatically changed the Modern movement by rejecting ornament and designing instead \"machine age\" buildings reduced to basic functional forms\u2014the beginning of mainstream modernism. Their new approach was exported back to the United States by pre-1925 \u00e9migr\u00e9s like Rudolf Schindler, Richard Neutra, and William Lescaze. In the 1930s they were joined by the elite of Europe's great Bauhaus School\u2014Walter Gropius, Mies van der Rohe, and Marcel Breuer\u2014all fleeing Hitler. Their more austere modernism, christened the International style, swept the United States during the following two decades (1930s\u20131950s). Its stark forms, however, proved to be more beloved for skyscrapers than for homes.\n\nInstead, another mainstream Modern style, originally called simply Contemporary, dominated avant-garde domestic architecture at mid-century. Favored by architects into the mid-1960s, Contemporary houses were greatly influenced by Wright's Usonian houses and his prolific writing for shelter magazines and professional journals.\n\nThe American banking system, however, through the practices of the Federal Housing Administration's mortgage insurance program, effectively regulated the kind of modern home that could be broadly built in typical neighborhoods at mid-century. The nation had adopted a goal that all returning World War II veterans would be able to own a home, and vast single-family suburban neighborhoods were built to accomplish this. The FHA did not believe that neighborhoods of starkly modern houses were a good investment for veterans\u2014or for anyone else\u2014and therefore lenders primarily financed a more conservative branch of modernism: the less daring \"Bankers Modern\" styles, consisting of basic Minimal Traditionals and casual Ranch houses. Described as \"middle-of-the-road modern\" and \"modern inside, traditional outside,\" the Ranch style was considered more acceptable to American home buyers than International or Contemporary homes\u2014and thus a safer investment.\n\nBy the mid-1960s home finance was no longer a problem. In addition, an entire generation of postwar American architects trained only in modernism were seeking new approaches to home styles. During the last half-century, a broad and ever-expanding array of architect-designed late-modern styles have appeared primarily in high-style homes designed by architects. With the exception of the Shed style, beginning in the 1960s, few examples of other modern styles made their way into typical neighborhoods. In recent decades, even more varied and dramatic modern house designs have had little effect upon typical American homes, which have mostly returned, once again, to historic and vernacular architectural roots.\n\n# MODERN HOUSES\n\n# Prairie\n\n# 1900\u20131920\n\n## Identifying Features\n\nLow-pitched roof, usually hipped, with widely overhanging eaves that typically are boxed; two stories, with one-story wings, porches, and porte cocheres; eaves, cornices, and facade detailing emphasizing horizontal lines; often with massive, square porch supports.\n\n## Principal Subtypes\n\nFour principal subtypes can be distinguished:\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY\u2014This subtype, which is sometimes called the Prairie Box or American Foursquare, has a simple square or rectangular plan, low-pitched hipped roof, and symmetrical facade. One-story wings, porches, or carports are clearly subordinate to the principal two-story mass. The entrance, which may be centered or off-center, is a conspicuous focal point of the facade. This was the earliest Prairie form and developed into the most common vernacular version. In vernacular examples, hipped dormers are common, as are full-width, single-story front porches and double-hung sash windows. Many show Mission or Italian Renaissance secondary details, such as tiled roofs or cornice-line brackets.\n\n**HIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY**\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nDallas, Texas; ca. 1920. Jones House. Aluminum siding has been added to the walls and porch supports.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nDallas, Texas; 1910. Harrison House. This example has a door surround of stylized Sullivanesque floral ornament (not visible in the photograph). Note the Wrightian column capitals.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nLouisville, Kentucky; ca. 1910. One-story Prairie examples like this are uncommon; one-story houses of the period were usually built in the Craftsman style.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nDallas, Texas; ca. 1910. The four-square plan, which was very popular during the period from about 1900 to 1920, is indicated by the two ranks of windows and the off-center entrance.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nTulsa, Oklahoma; 1915\u20131917. O'Rouke House. This house resembles some of the homes designed by Prairie architect George W. Maher and featured in the March 1914 _Western Architect._ The handsome entrance was likely inspired by Maher's Seymour House entry.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nGowanda, New York; ca. 1910. Houses of concrete blocks simulating stone, as seen here, were widely advocated by early 20th-century pattern books as a novel new building method. Note the through-the-cornice wall dormer.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nRiver Forest, Illinois; 1893. Winslow House; Frank Lloyd Wright, architect. This is Wright's first Prairie house and is much simpler than his later examples, most of which have asymmetrical hipped roofs. This house, and similar examples, provided the model for most later pattern book and builder interpretations of the style. Note how the horizontal effect is emphasized by the thin bricks and trim band above (subtle decorative patterning in the dark upper wall is not visible in the photograph).\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nLexington, Kentucky; ca. 1910. Note the uncoursed stone used for the lower two-thirds of the house, with stucco walls above.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nFort Dodge, Iowa; 1903. Butler House; Nourse and Rasmussen, architects.\n\nHIPPED ROOF, SYMMETRICAL, WITH FRONT ENTRY:  PRAIRIE  \nChicago, Illinois; ca 1951. Chicago bungalows like these were contemporaneous with Craftsman bungalows but had stylistic details influenced by the Prairie.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY\u2014Similar to the type just described but with inconspicuous entrances and facades dominated by horizontal rows of casement windows having sharply defined vertical detailing. This is a favorite form for smaller, architect-designed Prairie houses and also for those built on narrow urban lots.\n\n**HIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY**\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nWichita, Kansas; ca. 1910.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nBuffalo, New York; ca. 1910. Note the flat flower pots at the entrance to the right.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nSt. Louis, Missouri; ca. 1910. The entry is in the right wing; a porte cochere, in the left.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nPortland, Oregon; 1911. Bennes House; John Virginius Bennes, architect. This house deftly combines Italian Renaissance with Prairie, a fairly common pairing.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nOak Park, Illinois; 1904. Mrs. Thomas H. Gale House; Frank Lloyd Wright, architect. This Wright design\u2014with two levels of broad horizontal solid railings\u2014was influential for the next century.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nRacine, Wisconsin; 1905. Hardy House; Frank Lloyd Wright, architect. Note the wide, low chimney with dark coping at the top. The vertical emphasis here is particularly pronounced.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nLake Minnetonka, Minnesota; 1913. Decker House (garden facade); Purcell, Feick and Elmslie, architects. Note the ribbon of leaded casement windows across the second story.\n\nHIPPED ROOF, SYMMETRICAL, NO FRONT ENTRY:  PRAIRIE  \nMilwaukee, Wisconsin; 1916. Bogk House; Frank Lloyd Wright, architect. This house, done after Wright's principal Prairie years, has the simple box-like shape of some of his earlier houses.\n\nHIPPED ROOF, ASYMMETRICAL\u2014Most high-style examples are of this form. Typically a single two- or three-story, hipped-roof mass is contrasted with equally dominant, but lower, wings, porches, or carports with hipped roofs. The front entrance is usually inconspicuous, the facade being dominated by horizontal rows of casement windows having sharply defined vertical detailing. Many variations occur, but in all cases the facade is asymmetrical; most have masonry walls.\n\n**HIPPED ROOF, ASYMMETRICAL**\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nKansas City, Missouri; ca. 1910. One-story examples such as this are very uncommon.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nMission Hills, Kansas; ca. 1915. This house has a strong horizontal emphasis in the linear pattern of the lower-story masonry, the dark band above, and the differing wall material of the upper one-third.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nSt. Louis, Missouri; ca. 1910. Note the massive chimney with narrow side toward the street (see also the house in Minneapolis).\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nDallas, Texas; 1910. Parker House. This wood-clad example shows Craftsman influence in the open eaves with exposed rafters and in the stickwork between the porch supports. Note the decorative transom over the first-story windows.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nWichita, Kansas; 1917. Allen House; Frank Lloyd Wright, architect. In this late example, Wright includes an interior garden not found in earlier Prairie houses.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nLexington, Kentucky; ca. 1915. Note the similarity in form between this example and the Parker House. This one, however, is pure Prairie with boxed eaves and ribbons of casement windows.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nMinneapolis, Minnesota; 1913. Purcell, Feick and Elmslie, architects. This house has particularly dramatic ribbons of decorative casement windows.\n\nHIPPED ROOF, ASYMMETRICAL:  PRAIRIE  \nBuffalo, New York; 1904. Martin House; Frank Lloyd Wright, architect. Note the variety of hipped roofs and heavy masonry piers on this major Prairie landmark. The copings along low walls provide horizontal emphasis. Greenery overflows from planting boxes and planters placed at several levels.\n\nGABLED ROOF\u2014In this subtype, gables replace the more typical hipped roofs. High-style examples typically have both front-facing and side gables, each with exaggerated eave overhangs. In some, the gables have swept-back profiles with the peaks projecting beyond the lower edges. The pitch of the roof edges may also be flattened to give a pagoda-like effect. Vernacular examples usually have simple front- or side-gabled roofs. Tudor secondary influences are common, particularly false half-timbering in gables.\n\n**GABLED ROOF**\n\nGABLED ROOF:  PRAIRIE  \nKansas City, Missouri; ca. 1910. Above and the Jeremiah House are front-gabled examples. This one has a stone lower story, with wooden shingles above. Narrow lot size dictates few extensions.\n\nGABLED ROOF:  PRAIRIE  \nDallas, Texas; ca. 1920. Jeremiah House. This late example is dominated by a wide front gable, although there is a hipped roof unit hidden behind. The wide lot allows for a one-story porte cochere and porch wings not present in the previous example.\n\nGABLED ROOF:  PRAIRIE  \nOak Park, Illinois; 1912. Eastabrook House; Tallmadge and Watson, architects. This house, the MacMillan House, and the Amberg House are cross-gabled examples; this is the simplest. Note the double-gabled roof on the left wing.\n\nGABLED ROOF:  PRAIRIE  \nDallas, Texas; 1918. Peck House. This side-gabled example has a band of Sullivanesque trim below the porch eave.\n\nGABLED ROOF:  PRAIRIE  \nEl Paso, Texas; 1908. Henry Trost House; Trost and Trost, architects. This clearly shows a wide eave overhang that is open\u2014not boxed\u2014and with the rafters covered, not exposed. Note the band of trim beneath the cornice.\n\nGABLED ROOF:  PRAIRIE  \nKansas City, Missouri; ca. 1910.\n\nGABLED ROOF:  PRAIRIE  \nMontgomery, Alabama; ca. 1910. Note the four heavy piers ending two-thirds the way up the walls.\n\nGABLED ROOF:  PRAIRIE  \nPine Bluff, Arkansas; 1903. MacMillan House; Hugh Mackie Gorden Garden, architect. This photo shows clearly the open eave with enclosed rafters that is typical of this subtype (particularly those examples designed by architects). This contrasts with the open eave having exposed rafters that is found in contemporaneous Craftsman houses.\n\nGABLED ROOF:  PRAIRIE  \nGrand Rapids, Michigan; 1910. Amberg House; Marion Mahony and Frank Lloyd Wright, architects.\n\n## Variants and Details\n\nMassive square or rectangular piers of masonry used to support porch roofs are an almost universal feature of high-style examples. They remain common in vernacular examples, which also show squared wooden imitations. The characteristic horizontal decorative emphasis is achieved by such devices as: (1) contrasting caps on porch and balcony railings, (2) contrasting wood trim between stories, (3) horizontal board-and-batten siding, (4) contrasting colors on eaves and cornice, and (5) selective recessing of only the horizontal masonry joints. Other common details in both landmark and vernacular examples include window boxes or flattened pedestal urns for flowers; geometric patterns of small-pane window glazing (usually in leaded casement windows in high-style examples and upper sashes of wooden-muntin, double-hung windows in vernacular houses); broad, flat chimneys; contrasting wall materials or trim emphasizing the upper part of the upper story; and decorative friezes or door surrounds consisting of bands of carved geometric or stylized floral (Sullivanesque) ornamentation.\n\nPRAIRIE\n\n **TYPICAL ELABORATIONS**\n\nPRAIRIE\n\n **TYPICAL ELABORATIONS**\n\nPRAIRIE\n\n **TYPICAL DOORS**\n\nPRAIRIE\n\n **TYPICAL WINDOW GLAZING & SURROUNDS** casement windows common on Prairie high-style examples\n\nPRAIRIE\n\n **DECORATIVE DETAIL**\n\nPRAIRIE\n\n **AMERICAN FOUR-SQUARE**  \nthe most common vernacular form\n\nAmerican Four-Square, a house shape, has a squared floor plan with four rooms upstairs and four rooms down; central hall is absent; found in many styles, including Prairie, Colonial Revival, Neoclassical, and Craftsman\u2014and as folk houses\n\nPRAIRIE\n\n **COMMON DORMER VARIANTS (ON AMERICAN FOUR-SQUARES)**\n\nPRAIRIE\n\n **COMMON PORCH VARIANTS (ON AMERICAN FOUR-SQUARES)**\n\nPRAIRIE\n\n **COMMON PORCH SUPPORT VARIANTS (ON AMERICAN FOUR-SQUARES)**\n\n## Occurrence\n\nThe Prairie style originated in Chicago and landmark examples are concentrated in that city's early 20th-century suburbs, particularly Oak Park and River Forest, and in other large midwestern cities. Vernacular examples were spread widely by pattern books and popular magazines; they are common in early 20th-century suburbs throughout the country. Most were built between 1905 and 1915; the style quickly faded from fashion after World War I.\n\n## Comments\n\nThis is one of the few indigenous American styles. It was developed by an unusually creative group of Chicago architects that have come to be known as the Prairie School. Frank Lloyd Wright's early work is in this style and he is the acknowledged master of the Prairie house. Wright was unusual in that he early turned his creative genius toward the problems of domestic architecture rather than public buildings. His 1893 Winslow House was perhaps the first Prairie house; it is a symmetrical rectangle (see this page). It was not until about 1900 that he began to use the asymmetrical hipped form, which he continued to develop until about 1913. As Wright explained, \"Democracy needed something basically better than the box.\" Many of the other Prairie architects worked either with Wright himself or with his earlier employer and teacher, Louis Sullivan. Others absorbed Wright's and Sullivan's influence simply by being in Chicago. Among the most important were George W. Maher, Robert C. Spencer, Jr., Thomas E. Tallmadge, John S. Van Bergen, Vernon S. Watson, Charles E. White, Jr., Eben E. Roberts, Walter Burley Griffin, Marion Mahony Griffin, William Drummond, F. Barry Byrne, George G. Elmslie, and William G. Purcell.\n\nOutside of the Chicago area, numerous local architects produced creditable and sometimes outstanding Prairie houses throughout the midwestern states and, less commonly, in other regions. The style in its vernacular form was spread throughout the country by pattern books published in the Midwest. It is among the more short-lived styles, having grown, flourished, and declined in the years between 1900 and 1920. Wright's ideas about domestic architecture\u2014first explored in Prairie homes\u2014were refined in his Usonian houses (1936\u20131959) and had an even greater effect on typical American houses through the Contemporary style.\n\n# MODERN HOUSES\n\n# Craftsman\n\n# 1905\u20131930\n\n## Identifying Features\n\nLow-pitched, gabled roof (occasionally hipped) with wide, unenclosed eave overhang; roof rafters usually exposed; decorative (false) beams or braces commonly added under gables; porches, either full- or partial-width, with roof supported by tapered square columns; columns or piers frequently extend to ground level (without a break at level of porch floor); commonly one or one and one-half stories high, although two-story examples occur in every subtype.\n\n## Principal Subtypes\n\nFour principal subtypes can be distinguished:\n\nFRONT-GABLED ROOF\u2014About one-third of Craftsman houses are of this subtype. Porches, which may either be full- or partial-width, are almost evenly divided between those sheltered beneath the main roof and those with separate, extended roofs. Most examples of this subtype are one story, but one-and-a-half- and two-story examples are not uncommon; dormers are found in only about 10 percent of this subtype.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nHolmes County, Florida; ca. 1910s. Here a Craftsman porch is attached to a simple folk form.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nCanton, Mississippi; ca. 1910s. The porch roof is a separate gabled element in this very common version of the subtype.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nLexington, Kentucky; ca. 1910s. Note the doubled porch supports set on a closed porch railing. There is a section of hipped roof in the front with a gable above.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nKansas City, Missouri; ca. 1910s. This stucco example has three front-facing gables, all with half-timbered detailing.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nJackson, Mississippi; ca. 1910s. This photograph emphasizes the triangular knee braces commonly used in the gable ends of Craftsman houses. The slightly tapered porch-roof supports, extending from ground level, are of irregular brick masonry. Note how the main roof extends over the porch.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nKansas City, Missouri; ca. 1910s. A large two-story example of stone and stucco. The gable encompassing the entire second story is unusual.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nEmporia, Kansas; ca. 1910s. This is a more typical two-story form than the previous example. Note the matching roof-support columns and gables over the entry and porte cochere.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nSanta Monica, California; 1911. Note the striking interlocking porch support detailing and the roof rafters that extend beyond the edge of the roof.\n\nFRONT-GABLED ROOF:  CRAFTSMAN  \nSanta Monica, California; 1911. Milbank House. Protruding roof beams (single, paired, and tripled), all with multiple setbacks that read almost like saw teeth.\n\nCROSS-GABLED ROOF\u2014Cross-gabled examples make up about one-fourth of Craftsman houses. Of these, three-quarters are one-story examples; dormers occur on about 20 percent. Porches are varied, but by far the most common type is a partial-width, front-gabled porch, its roof forming the cross gable.\n\n**CROSS-GABLED ROOF**\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nAbbeville, South Carolina; ca. 1920s. Modest examples with Craftsman detailing, such as this, were common in outlying areas into the early 1930s.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nSan Jose, California; ca. 1910s. The two picture windows in this house are obvious later alterations. Note how the vergeboards here and in the house in Louisville extend a bit beyond the roof edge to give a visual effect that the rafter tails extend beyond the edge of the roof.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nArdmore, Oklahoma; ca. 1910s. Note the similarity between this and the house in Santa Barbara. Examples with the single room on the second story were called airplane bungalows, presumably because they afforded a panoramic view.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nKansas City, Missouri; ca. 1910s. Note the triple front-facing gables.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nLouisville, Kentucky; ca. 1910s. Brick Craftsman houses were less common than wood; most occur in the larger cities of the Northeast and the Midwest. Fire codes in some cities\u2014Denver and Chicago, for example\u2014prohibited wooden exteriors.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nSanta Barbara, California; ca. 1910. Note the intentional omission of center porch supports here, in the house in San Jose, and in the previous example. This was done to allow an unobstructed view from front rooms.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nWichita, Kansas; ca. 1910s. Note the tapered porch supports that rise from ground level and are made of rough-faced stone.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nBellingham, Washington; 1908. Roeder House; Alfred Lee, architect. Triangular knee braces are used under the rafters, not just the roof beams, as is typical.\n\nCROSS-GABLED ROOF:  CRAFTSMAN  \nPasadena, California; 1908. Gamble House; Greene and Greene, architects. A garden view of one of the great landmarks of the style. Note the numerous low-pitched gables, open porches, and exposed wooden structural elements. (In this case they _are_ structural, not just added decoration as in most Craftsman houses.) Here the rafter tails extend beyond the edge of the roof, a typical high-style elaboration that is also visible in the house in Santa Barbara.\n\nSIDE-GABLED ROOF\u2014About one-third of Craftsman houses are of this subtype. Most are one and one-half stories high with centered shed or gable dormers. Porches are generally contained under the main roof, sometimes with a break in slope. Two-story examples commonly have added, full-width porches. This subtype is most common in the northeastern and midwestern states.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDallas, Texas; 1915. Lorrimer House. The typical exposed rafter ends show clearly here.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nSalisbury, North Carolina; 1913. Rock House. Entry porches such as this are less common than full-width porches.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nLexington, Kentucky; ca. 1910s.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nLouisville, Kentucky; ca. 1910s. Side-gabled Craftsman houses frequently have the attic area finished for bedrooms. Light comes from windows in the gable and from large centered dormers (see also the Rock House and the houses in Lexington, Kansas City, and Durham).\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nKansas City, Missouri; ca. 1910s. The elaborate shed dormer with twin gables gives this example a Swiss Chalet feel.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDenver, Colorado; ca. 1910s. Note the peaked Oriental influence in the gables.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDurham, North Carolina; ca. 1910s. The wide expanse of porch without porch supports allows an unrestricted view from the front windows (see also the houses in Lexington, Louisville, and Kansas City).\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDallas, Texas; 1920. Clem House. Note the half-timbering in the gables and the use of paired, tapering porch supports atop the wide pedestals.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDallas, Texas; 1917. Wheaton House. Large round columns such as this are seen in Craftsman pattern books, but are uncommon in actual examples.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nSanta Monica, California; ca. 1910s.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDallas, Texas; 1914. Cranfill House.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nDallas, Texas; 1911. Defreese House. Note the full-width two-tiered porch. The typical triangular knee braces are clearly visible along the side gable. Derived from _Associated Architects Fifty House Plans_ (published in 1910).\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nWichita, Kansas; ca. 1920. Lewis House.\n\nSIDE-GABLED ROOF:  CRAFTSMAN  \nBuffalo, New York; ca. 1910s. Note the contrasting stonework of the first and second stories and the shed dormers with matching shed-roof porch.\n\nHIPPED ROOF\u2014These make up less than 10 percent of Craftsman houses; they are almost equally divided between one- and two-story examples. This subtype is similar to some simple Prairie houses, which normally lack the exposed rafters and other typical Craftsman details.\n\n**HIPPED ROOF**\n\nHIPPED ROOF:  CRAFTSMAN  \nDallas, Texas; ca. 1910s.\n\nHIPPED ROOF:  CRAFTSMAN  \nWashington, District of Columbia; ca. 1910s. Note the trellised entry porch. Similar porches were also used as side or wing porches in many examples of the style.\n\nHIPPED ROOF:  CRAFTSMAN  \nDallas, Texas; 1912. Gibbs House. Note the porte cochere with a sleeping porch above. This was a typical addition to the main-house block in early 20th-century houses.\n\nHIPPED ROOF:  CRAFTSMAN  \nDallas, Texas; 1917. Burgoyne House. This house shows the close relationship of the subtype with simple Prairie houses built in the four-square shape. The unenclosed eaves distinguish this example from similar Prairie forms; the porch supports are clearly Craftsman, but these are also used frequently on Prairie houses.\n\n## Variants and Details\n\nPORCH ROOF SUPPORTS\u2014Columns for supporting the porch roofs are a distinctive and variable detail. Typically short, square upper columns rest upon more massive piers, or upon a solid porch balustrade. These columns, piers, or balustrades frequently begin directly at ground level and extend without break to a level well above the porch floor. Commonly the piers or columns have sloping (battered) sides. Materials used for piers, columns, and solid balustrades are varied. Stone, clapboard, shingle, brick, concrete block, or stucco are all common; they frequently occur in combination. Small rounded stones, such as those found in the arroyos of southern California, were particularly favored.\n\nROOF-WALL JUNCTIONS\u2014Among the most distinctive features of the style are the junctions where the roof joins the wall, which are almost never boxed or enclosed. The roof has a wide eave overhang; along _horizontal_ edges the actual rafter ends are exposed, or false rafter ends are added. These are sometimes cut into decorative shapes and rafter tails may extend beyond edge of roof. Along the sloping, or rake, edges, three or more beams (usually false) extend through the wall to the roof edge. These are either plain or embellished by a triangular knee brace.\n\nOTHER DETAILS\u2014Craftsman doors and windows are similar to those used in vernacular Prairie houses (see this page). Two or more windows are often grouped together in one assembly; a narrow window on each side of a broad center window is common. Dormers are commonly gabled or shed, with exposed rafter ends and braces such as are found at the main roof-wall junction. The most common wall cladding is wood clapboard; wood shingles rank second. Stone, brick, concrete block, and stucco are also used, most frequently in the northern and midwestern states. Secondary influences such as Tudor false half-timbering, Swiss balustrades, or Oriental roof forms are also sometimes seen.\n\nCRAFTSMAN\n\n **SOME TYPICAL PORCH SUPPORTS AND PORCH RAILINGS** Low piers without columns above are common\n\nCRAFTSMAN\n\n **COMMON PORCH SUPPORT VARIANTS**\n\nCRAFTSMAN\n\n **TYPICALROOF-WALL JUNCTIONS**\n\nCRAFTSMAN\n\n **TYPICAL ELABORATIONS**\n\nCRAFTSMAN\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nThis was the dominant style for smaller houses built throughout the country during the period from about 1905 until the early 1920s. The Craftsman style originated in southern California and most landmark examples are concentrated there. Like vernacular examples of the contemporaneous Prairie style, it was quickly spread throughout the country by pattern books and popular magazines. The style rapidly faded from favor after the mid-1920s; relatively few were built after 1930.\n\n## Comments\n\nCraftsman houses were inspired primarily by the work of two California brothers\u2014Charles Sumner Greene and Henry Mather Greene\u2014who practiced together in Pasadena from 1893 to 1914. About 1903 they began to design simple Craftsman-type bungalows; by 1909 they had designed and executed several exceptional landmark examples that have been called the \"ultimate bungalows.\" Several influences\u2014the English Arts and Crafts movement, an interest in Oriental wooden architecture, and their early training in the manual arts\u2014appear to have led the Greenes to design and build these intricately detailed buildings. These and similar residences were given extensive publicity in such magazines as the _Western Architect, The Architect,House Beautiful, Good Housekeeping, Architectural Record, Country Life in America,_ and _Ladies' Home Journal,_ thus familiarizing the rest of the nation with the style. As a result, a flood of pattern books appeared, offering plans for Craftsman bungalows; some even offered completely pre-cut packages of lumber and detailing to be assembled by local labor. Through these vehicles, the one-story Craftsman house quickly became the most popular and fashionable smaller house in the country. High-style interpretations are rare except in California, where they have been called the Western Stick style. One-story vernacular examples are sometimes called bungalows. However, during the early 20th century, the term \"bungalow\" could refer to small, one-story examples of other styles\u2014for example, a Spanish or a Tudor bungalow. Craftsman examples were often called California bungalows. In Chicago there are blocks of small houses heavily influenced by the Prairie style and called Chicago bungalows.\n\n# MODERN HOUSES\n\n# Modernistic\n\n# 1920\u20131940\n\n## Identifying Features\n\nART MODERNE\u2014Smooth wall surface, usually of stucco; flat roof, usually with small ledge (coping) at roof line; horizontal grooves or lines in walls and horizontal balustrade elements give a horizontal emphasis; facade usually asymmetrical.\n\n**MODERNISTIC: ART MODERNE**\n\nART MODERNE:  MODERNISTIC  \nSan Antonio, Texas; ca. 1930s. Note the entry area with its round window, curved glass-brick window, and curved porch roof with coping above the entry.\n\nART MODERNE:  MODERNISTIC  \nFargo, North Dakota; ca. 1930s. Note the curved front with two glass-block windows wrapping around it and the small octagonal window.\n\nART MODERNE:  MODERNISTIC  \nToledo, Ohio; ca. 1930s. This popular Art Moderne design is almost symmetrical.\n\nART MODERNE:  MODERNISTIC  \nPalm Springs, California; 1936. \"Ship of the Desert\"; Wilson and Webster, architects. Rebuilt from its original plans after a fire.\n\nART MODERNE:  MODERNISTIC  \nDes Moines, Iowa; 1937. Butler House; Kraetsch and Kraetsch, architects. This landmark example is built of poured-in-place structural concrete with metal windows. It has several curved sections and a dramatic frieze of horizontal grooves. Note also the varied angles and corner windows in the section to the left.\n\nART MODERNE:  MODERNISTIC  \nFergus Falls, Minnesota; 1937. Lee House; Foss and Co., architects. The brick wall cladding is atypical of the style. Note the extensive horizontal railings and curved one-story section.\n\nART MODERNE:  MODERNISTIC  \nDaly City, California; ca. 1950. Westlake District; Henry Doelger, builder. This late example has an atypical overhang added.\n\nART MODERNE:  MODERNISTIC  \nDenver, Colorado; ca. 1930s. Three horizontal lines in graduated lengths wrap the corners, giving this house a Moderne feeling. They are made of elongate brick embedded into stucco.\n\n## Variants and Details\n\nART MODERNE\u2014One or more corners of the building may be curved; windows frequently are continuous around corners; glass blocks are often used in windows, or as entire sections of wall; small round windows are common.\n\nMODERNISTIC\n\n **ART MODERNE: TYPICAL ELABORATIONS**\n\nART DECO\u2014Smooth wall surface, usually of stucco; zigzags, chevrons, and other stylized and geometric motifs occur as decorative elements on facade; towers and other vertical projections above the roof line give a vertical emphasis.\n\n**MODERNISTIC:ART DECO**\n\nART DECO:  MODERNISTIC  \nSan Francisco, California; ca. 1930. A rare example of an Art Deco house; note the geometric decorative details and vertical roof projections.\n\nART DECO:  MODERNISTIC  \nLos Angeles, California; 1930. Smith House; J. C. Smale, architect. Note the fluting along the parapet and the vertical elements created by the windows and deco ornament.\n\nART DECO:  MODERNISTIC  \nBoise, Idaho; ca. 1930s.\n\nMODERNISTIC\n\n **ART DECO: TYPICAL DECORATIVE MOTIFS**\n\n## Occurrence\n\nThe Modernistic styles were built from about 1920 to 1940. The earlier form was the Art Deco, which was common in public and commercial buildings in the 1920s and early 1930s. It was, however, extremely rare in domestic architecture; we know of only a few surviving houses, although it was frequently used for apartment buildings. After about 1930, Art Moderne became the prevalent Modernistic form. Although never common, many houses were built in the style; scattered examples can be found throughout the country.\n\n## Comments\n\nThe Modernistic styles received their first major impetus in 1922 when the Chicago _Tribune_ held a world-wide competition for a headquarters building in Chicago. Although the first prize went to a Gothic design, the second prize went to an Art Deco design by a young Finnish architect, Eliel Saarinen. His design was widely publicized and much of the architectural profession felt that he deserved the first prize; the style quickly became the latest architectural fashion. The style gained its name from the Paris Exhibition of 1925\u2014the Exposition Internationale des Arts D\u00e9coratifs et Industriels Modernes. Shortly after 1930 another, more diffuse influence affected the Modernistic style\u2014the beginning of streamlined industrial design for ships, airplanes, and automobiles. The smooth surfaces, curved corners, and horizontal emphasis of the Art Moderne style all give the feeling that airstreams could move smoothly over them; thus, they were streamlined. In most building types, both the horizontal, streamlined Art Moderne and the vertical, zigzagged Art Deco influences occur in combination. In houses, however, the streamline influences predominate. Many examples resemble the contemporaneous International style, in which decorative detailing was reduced to the barest minimum.\n\n# MODERN HOUSES\n\n# Minimal Traditional\n\n# ca. 1935\u20131950\n\n## Identifying Features\n\nLow- or intermediate-pitched roof, more often gabled; small house, generally one-story in height; roof eaves usually have little or no overhang; double-hung windows, typically multi-pane or 1\/1; minimal amounts of added architectural detail; rarely has dormers.\n\n## Principal Subtypes\n\nGABLE-AND-WING ROOF\u2014This subtype has a low-pitched front-facing gable added on one side of a side-gabled roof. Typically the front-facing gable does not protrude very far in front of the side gable and consists only of a small extension added to one room of the house. Two-story examples are sometimes found.\n\n**GABLE-AND-WING ROOF**\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nHamptons, New York; ca. 1940s. Note the double-hung windows and overall cladding. Door faces to side.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nPortland, Oregon; ca. 1930. This early example is unusual in having the entry in the gable.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nChicago, Illinois; ca. 1950. This is beginning to transition to the Ranch with the broader form and the beginnings of a picture window.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nAlexandria, Virginia; 1950.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nPhoenix, Arizona; 1940.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nSouth San Francisco, California; ca. 1940s. Compare to the following example in the same area. The second-story gable could be original or an addition. The window in the first-floor gable appears modified.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nSouth San Francisco, California; ca. 1940s. As the FHA suggested, in order to make the house look larger the secondary material in the gable is painted the same color as the house. Note the design of the porch rail.\n\nGABLE-AND-WING ROOF:  MINIMAL TRADITIONAL  \nDallas, Texas; 1946. Two-story examples occur.\n\nSIDE-GABLED ROOF (COMMONLY CALLED CAPE COD)\u2014Comprised of simple one-story side-gabled houses, this subtype was widely published in mass-market shelter magazines in the 1930s and 1940s. Besides being a beloved early New England folk-house form and symbol of colonial America, the Cape Cod provides some of the most economical cubic space that can be built.\n\nAlthough many examples were symmetrical, in keeping with the original folk form, there was much experimentation with asymmetrical variations, including varied window placement and small porches or carports; some had an extra half-story finished under the roof. Two-story side-gabled examples are found, but those would not be called Cape Cod, a term properly confined to the one-story form.\n\n**SIDE-GABLED ROOF (CAPE COD)**\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nBurbank, California; 1940. Note the paired double-hung windows placed toward the corners of the house.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nLevittown, New York; ca. 1946. This was one of Levitt's alternative Cape Cod designs and is in the shape of a three-quarter house.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nBurbank, California; 1941 (renovated 1953). Windows on the main house block are not double-hung and are likely later additions.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nLevittown, New York; ca. 1947. Reportedly the most intact remaining example of Levittown's early Cape Cod houses. These were 750 square feet and sold for $7,500.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nPrairie Village, Kansas; 1947. The tall double windows on the left might first have been a garage as in the following example.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nPrairie Village, Kansas; 1947. Note the uncommon built-in garage.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nDallas, Texas; 1939. Note the large double-hung windows. This house was financed by the FHA.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nSeattle, Washington; 1940.\n\nSIDE-GABLED ROOF (CAPE COD):  MINIMAL TRADITIONAL  \nAtlanta, Georgia; ca. 1940s. A neighborhood of Minimal Traditional houses.\n\nOTHER ROOF\u2014Hipped-roof and front-gabled houses are also found, with hipped-roof versions the more widespread. These variations appear to be less common than other subtypes.\n\n**OTHER ROOF**\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nBurbank, California; 1939.\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nBurbank, California; ca. 1940. This, the previous house, and the other Burbank example were built to house employees of the Lockheed Vega aircraft plant and the Burbank airfield. The entire area was camouflaged during World War II to look like a pastoral rural setting rather than a factory and housing for workers.\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nFairway, Kansas; 1941. Compare to the slightly larger example from the same Fairway neighborhood.\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nBurbank, California; ca. 1940. Note the shingle cladding, and board shutters with cutouts (even on corner windows).\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nPortland, Oregon; ca. 1940s. The corner-window placement was among the FHA's recommended designs.\n\nOTHER ROOF:  MINIMAL TRADITIONAL  \nFairway, Kansas; 1942. Note use of second wall-cladding material and scalloped cornice board on front-hipped wing.\n\n## Variants and Details\n\n\"Simplicity in exterior design gives the small house the appearance of maximum size.\" So states the FHA's 1940 version of _Principles for Planning Small Houses,_ which uses the word \"simple\" four times in its first five sentences describing exterior design. The pamphlet recommends a simple composition, simple roof lines, and simple variations and materials (thus avoiding a \"restless appearance\"). It also suggests the overall appearance can be improved by avoiding unnecessary gables and dormers, breaks in the roof form, and over-elaborate cornices. Indeed, \"all nonessential features can profitably be omitted.\" The recommendation was to concentrate on scale, carefully place the windows and doors, and make the house look larger by cladding it with one material\u2014or, if two materials are used, they should be the same color. Porches, bay windows, and platform steps were the only additions the FHA suggested. However, in practice, Minimal Traditionals commonly had a traditional paneled front door (sometimes with a multi-pane window in the upper half), perhaps accompanied by shutters or a chimney. Occasionally, Minimal Traditionals have other bits of stylistic detailing added\u2014such as a scalloped detail across the base of a gable or elements of Tudor or Colonial Revival.\n\n## Occurrence\n\nMinimal Traditional homes can be found throughout the United States. During the early 1940s, concentrations were rapidly built where new sites for World War II production plants created an urgent local need for worker housing. After the war, developers built instant communities\u2014such as Levittown, New York, on Long Island, and Brentwood in Denver, Colorado\u2014filled with Minimal Traditional houses, sometimes using only a few designs in a subdivision. These were sometimes located beyond the city's built\u2011up edge, where large tracts of land were available and new broad highways and arterials were planned for easy automobile access. In postwar subdivisions, the style is found with early Ranch houses (sometimes called Minimal Ranches or Ranchettes).\n\n## Further Comments\n\nThe Minimal Traditional house was \"the little house that could.\" It was the small house that could be built with FHA-insured loans in the midst of the Great Depression between 1935 and 1940; the house that could be built quickly to accommodate millions of relocating World War II production-plant workers (1941\u20131945); and the house that could be built rapidly during the late 1940s in large post\u2013World War II developments (1946\u20131949). These late-1940s developments were necessary to begin to fulfill the wartime GI Bill promise that every returning serviceman would be able to purchase a home.\n\nThe Minimal Traditional was a well-studied and thoughtful response to the most challenging conditions ever to affect home construction in the United States. In the early 1930s, the Great Depression virtually shut down the home-building industry. Housing starts fell from 330,000 in 1930 to an unprecedented 93,000 in 1933. Banks were going under, mortgages were past due, and there were no funds for new construction. The urgent first step was the creation of a new method for insuring long-term, low-interest mortgages. This was accomplished in 1934 through the creation of the FHA, whose goal was to produce small homes the average working American could afford (see financing, below). The FHA not only provided insurance that covered the mortgage loan a bank made, it also prepared publications that showed how to most effectively design a small house.\n\nArchitects, desperate for work after 1930, had enthusiastically turned their attention to the design of the small house. Large portions of professional journals were devoted to this subject beginning in the mid-1930s. It was of paramount importance to design the most efficient floor plans, kitchens, and baths, since every extra square foot added to the cost. A higher home cost both limited the market and made it harder to qualify for the all-important FHA loan insurance. At that time, the FHA, along with its associated Fannie Mae, limited the maximum sales price of homes they would insure so that the average home size and cost remained within the reach of a broad market. Nonetheless, a little known bonus of FHA loans was that they allowed all major appliances to be included in the home loan amount.\n\nA veritable flood of house plans and pattern books for small houses featuring Minimal Traditionals was published between 1935 and 1950\u2014and the book contents often included careful descriptions of the FHA loan programs available. The most influential publications were the FHA's own bulletins, _Principles of Planning Small Houses._ The original 1936 version had only a few drawings, but the revised and expanded 1940 edition illustrated most of the basic shapes and variations of Minimal Traditional houses. Builders knew that following the guidelines in this bulletin was the quickest way to ensure construction funds and home-purchase mortgages for their projects.\n\nNo sooner had the orderly construction of homes gotten under way in the late 1930s than there was an immediate pressing need for small houses to accommodate the unprecedented relocation of workers to ramp up wartime production. Approximately 2.3 million war and defense homes, the vast majority of them Minimal Traditional, were built between 1940 and 1945. The builder-developers who quickly erected these large developments of necessary wartime housing taught themselves, on the job, how to speed up construction through both offsite and onsite prefabrication of needed components and then applying assembly-line techniques to construction.\n\nWhen the war ended in 1945, there arose the immediate problem of housing ten million returning servicemen and their families, promised the right to purchase a new home with no down payment. There were no new homes available, so these same builder-developers went to work, applying their wartime experience in rapid construction to large subdivisions of homes available for purchase. In an explosion of home building at the war's end, 5.1 million homes were built between 1946 and 1949. Minimal Traditionals made up a significant portion of these. The fact that the FHA had already blessed these designs meant a developer could get speedy approval of loans for building to begin. Those who strived for a more modern look often encountered obstacles and costly time delays.\n\nBy 1950 the Minimal Traditional was being replaced by Ranch homes. Postwar prosperity meant that larger homes could be built and financed, and the Ranch was a perfect fit for the tastes of a new decade.\n\n# MODERN HOUSES\n\n# Ranch\n\n# ca. 1935\u20131975\n\n## Identifying Features\n\nBroad one-story shape; usually built low to ground; low-pitched roof without dormers; commonly with moderate-to-wide roof overhang; front entry usually located off-center and sheltered under main roof of house; garage typically attached to main facade (faces front, side, or rear); a large picture window generally present; asymmetrical facade.\n\n## Principal Subtypes\n\nFour principal subtypes can be distinguished:\n\nHIPPED ROOF\u2014About 10 percent of one-story Ranch houses have a simple hipped roof with a long roof ridge running parallel to the front facade. These are more common in rural areas and in neighborhoods of smaller houses. Very occasionally, as in the side-gabled subtype, a large example with broadly angled wings occurs.\n\n**HIPPED ROOF**\n\nHIPPED ROOF:  RANCH  \nChicago, Illinois; 1957. One of a row of Ranch houses built on narrow lots with garages behind. Note wide eave overhang, picture window with a different cladding at base, and entry inset under main roof with wrought-iron porch support.\n\nHIPPED ROOF:  RANCH  \nSan Marino, California; 1948 (renovated 1955).\n\nHIPPED ROOF:  RANCH  \nMiami, Florida; 1954. This minimal Ranch (or Ranchette) is part of a small subdivision. The \"shutters\" are actually tiles laid in the stucco walls, and the windows have the horizontal 2\/2 muntin pattern favored in Ranch houses.\n\nHIPPED ROOF:  RANCH  \nDuluth, Minnesota; 1950. The roof form appears as a single hip, despite variation in the front-facade plane.\n\nHIPPED ROOF:  RANCH  \nPortland, Oregon; 1953. Note the stylish ribbon windows on contrasting wall cladding to the left of the picture window.\n\nHIPPED ROOF:  RANCH  \nAtlanta, Georgia; ca. 1950s. Collier Heights Historic District. This Ranch takes advantage of the change in grade with a side-down garage.\n\nHIPPED ROOF:  RANCH  \nLos Angeles, California; ca. 1960. The wall cladding has a horizontal pattern.\n\nHIPPED ROOF:  RANCH  \nDallas, Texas; ca. 1960s. Fritch House (demolished). This elegant Ranch house makes clear the relation of many Ranch houses to Prairie style. Note the ribbon window, broad roof overhang, and low wide chimney.\n\nHIPPED ROOF:  RANCH  \nDallas, Texas; 1950. This large Ranch has wings that angle backward. Not visible is a rear-down section that includes the garage.\n\nCROSS-HIPPED ROOF\u2014About 40 percent of one-story Ranch houses have a cross-hipped roof. Typically these are one-story houses with a long roof ridge running parallel to the front facade with a single hipped extension. Occasionally a second hipped front extension is also present.\n\nSometimes the cross-gabled and cross-hipped types have a combination roof with a front hip on a side-gabled roof or, conversely, a front-facing gable on a hipped roof. Very large examples may feature rather complex roof forms with a combination of roof heights and types.\n\n**CROSS-HIPPED ROOF**\n\nCROSS-HIPPED ROOF:  RANCH  \nSt. Petersburg, Florida; ca. 1950. Kenwood Historic District. The entry door faces sideways and is recessed under the right corner of the front hip. In addition a picture window, horizontal window muntins, and low roof pitch with broad overhang make this small house a Ranch. (Compare with the following example.)\n\nCROSS-HIPPED ROOF:  RANCH  \nSan Francisco, California; ca. 1950s. A front-gabled garage is used with a hipped-roof house block. The metal porch supports are an abstracted vine pattern, with the \"vines\" escaping to the side at the tops.\n\nCROSS-HIPPED ROOF:  RANCH  \nSeattle, Washington; 1953. This appears to have the same sections as the house in Prairie Village, but at a smaller scale. Sleeping section at left, living in the center, and cars on the right.\n\nCROSS-HIPPED ROOF:  RANCH  \nChicago, Illinois; 1958. This has two corner windows and a row of three small windows near the entry. The long, low masonry planter adds horizontal emphasis to entry area.\n\nCROSS-HIPPED ROOF:  RANCH  \nPasadena, California; ca. 1950s. Board-and-batten siding and thick wood shake roof are part of the \"California Ranch\" vocabulary. Note horizontal stonework, the small cupola above the garage wing, and the exposed rafters with rounded ends.\n\nCROSS-HIPPED ROOF:  RANCH  \nPrairie Village, Kansas; 1959. This Tri-Level Split has a good example of the small cupolas (often prefabricated of metal) that are popular in Ranch-style houses.\n\nCROSS-HIPPED ROOF:  RANCH  \nDallas, Texas; ca. 1952. A classic Ranch-house arrangement with a sleeping section at the left (marked by high ribbon windows), a living section in the middle (marked by a picture window), and a section for cars on the right (marked by garage doors). It was not uncommon for front-facing garages like this to have a utility room and kitchen area located behind.\n\nCROSS-HIPPED ROOF:  RANCH  \nCarlsbad, California; 1972. Board-and-batten siding, exposed rafters, and a roof clad with thick wood shingles are typical of the \"California Ranch\" house (found in many parts of the country). Note house in U shape around entry court and low-pitched gable on hip roofs.\n\nCROSS-HIPPED ROOF:  RANCH  \nBaltimore, Maryland; 1962. A carport, as seen here, is less common in Ranch houses than in Contemporary. There is a side-down section on the left.\n\nCROSS-HIPPED ROOF:  RANCH  \nSan Marino, California; 1951. Note the ribbon windows and corner windows.\n\nSIDE-GABLED ROOF\u2014About 10 percent of one-story Ranch houses have side-gabled roofs with a long roof ridge running parallel to the front facade. These are more common in rural areas and in neighborhoods of smaller houses. Some high-style examples have slight angles in the front (or other) facade, giving the appearance of widespread welcoming arms.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  RANCH  \nLevittown, New York; ca. 1950. This was one of Levitt's Ranch designs, first offered in 1949. It sold for just under $8,000 and featured a sixteen-foot picture window on the rear facade.\n\nSIDE-GABLED ROOF:  RANCH  \nLos Angeles, California; 1958. Double-hung windows have the diamond-shape window pattern that is common in southern California\u2014as are windows that are grouped, enframed by siding, and slightly project from the wall plane (seen at left).\n\nSIDE-GABLED ROOF:  RANCH  \nDuluth, Minnesota; 1965. This minimal Ranch has a roof overhang in front and a corner picture window.\n\nSIDE-GABLED ROOF:  RANCH  \nPasadena, California; 1961.\n\nSIDE-GABLED ROOF:  RANCH  \nPhoenix, Arizona; 1938. Edwin G. Smith House. Early for a Ranch and with quite interesting shutters.\n\nSIDE-GABLED ROOF:  RANCH  \nSt. Paul, Minnesota; 1960. Slightly angled wings were often used in larger Ranch houses.\n\nSIDE-GABLED ROOF:  RANCH  \nKansas City, Missouri; 1951. The full-width front porch has simple porch supports with diagonal braces.\n\nSIDE-GABLED ROOF:  RANCH  \nDuluth, Minnesota; 1970. Note the outward slant of the gable roof overhang and the rear-down story.\n\nSIDE-GABLED ROOF:  RANCH  \nDallas, Texas; 1950. McNatt House; George Marble, local architect. This is Cliff May's award-winning 1948 design for _House Beautiful_ 's Pace Setter House Program.\n\nSIDE-GABLED ROOF:  RANCH  \nDuluth, Minnesota; 1961. This large Tri-Level Split has a triple diamond pattern on the garage door that is echoed in other doors\u2014but at this scale only the storm doors with a similar theme are visible. Note sunburst ornament on chimney. Window patterns give clue to use of different levels. Short windows indicate bedrooms on upper level. Curved bay indicates some sort of living area on lower level, likely informal and connecting to the garage wing. And full-height windows and fireplace on mid-level indicate this likely is the formal living level.\n\nSIDE-GABLED ROOF:  RANCH  \nWestlake, Texas; 1938 (restored 1977). Dealey House; Charles Dilbeck, architect, Nancy McCoy, restoration architect. This large and elegant house sprawls across the Texas prairie.\n\nCROSS-GABLED ROOF\u2014About 40 percent of one-story Ranch houses have a broad side-gabled form, with a long roof ridge parallel to the street, and a single prominent, front-facing gable extension. Occasionally a second such gable is present.\n\n**CROSS-GABLED ROOF**\n\nCROSS-GABLED ROOF:  RANCH  \nKansas City, Missouri; 1952. A minimal Ranch with a picture window and masonry below the window on the front gable.\n\nCROSS-GABLED ROOF:  RANCH  \nPhoenix, Arizona; 1959. Built-in planter at entry is accented by a slanting architectural element.\n\nCROSS-GABLED ROOF:  RANCH  \nUniversity Park, Texas; 1938. Dilbeck Home and Studio; Charles S. Dilbeck, architect. Located on a prominent corner, this advertised the architect's proficiency in the California Ranch to all who passed by.\n\nCROSS-GABLED ROOF:  RANCH  \nPasadena, California; 1936. A minimal Ranch similar to the house in Wyndmoor but with a partial porch and metal porch supports.\n\nCROSS-GABLED ROOF:  RANCH  \nPasadena, California; 1958. Note very long window on the left wing, exposed roof beams on the front-facing gable, and board and batten siding.\n\nCROSS-GABLED ROOF:  RANCH  \nWyndmoor, Pennsylvania; 1963. A Tri-Level Split with a stone first story and handsome rounded bay window. Slightly thicker muntins on four bottom panes indicate that these open for circulation. The siding above appears to be newer, perhaps a vinyl overlay.\n\nCROSS-GABLED ROOF:  RANCH  \nLas Vegas, Nevada; 1973.\n\nCROSS-GABLED ROOF:  RANCH  \nWashington, D.C.; 1966. Note change in wall cladding around entry area.\n\nCROSS-GABLED ROOF:  RANCH  \nKansas City, Missouri; ca. 1950s. The overlapping front-facing gables are unusual in Ranch houses.\n\nCROSS-GABLED ROOF:  RANCH  \nLos Angeles, California; 1951. A breezeway connects the garage wing on the left to the house. One can guess that a powder room, or perhaps a coat closet, is located behind the single high window to the left of the entry. The door sidelights and the leaded glass in this window make an interesting contrast with the three large picture windows.\n\nSPLIT-LEVELS\u2014Split-level versions of all Ranch subtypes are common. This variation is discussed on this page.\n\n## Variants and Details\n\nWINDOWS\u2014A remarkable variety of sizes and types of pre-manufactured windows were available to builders during the Ranch era, and most Ranch houses exhibit several different sizes and\/or types of windows. After World War II, factories used for war production were quickly adapted to the manufacture of domestic products and a profusion of window types was made possible by applying production methods perfected during the war. These were manufactured in standardized sizes newly regulated by the industry's trade associations. Metal (aluminum, steel, or bronze) and wood versions of double-hung, casement, and sliding windows were manufactured, as were awning styles. Metal sliding windows and jalousie windows (very common in southern Florida) also occurred.\n\nMore than 50 percent of Ranch houses have at least one picture window on the front facade, and some examples have more. These large focal windows commonly had sections that could be opened from side or top hinges for ventilation. In later examples, groups of tall fixed vertical panes were often used instead of picture windows. Very short windows were often utilized, sometimes grouped into ribbons and placed high on the facade. This allowed for light and ventilation without loss of interior privacy and accommodated flexibility in furniture arrangement below the high windows.\n\nIn the rest of the house, traditional window lengths were typically used\u2014either in a casement or double-hung design. In the latter, horizontal light patterns (2\/2 or 3\/3) and multi-pane patterns (9\/9, 8\/8, 6\/6, 12\/12, or 1\/1) were common. Generally, several window sizes or shapes occur on a house, typically made of the same material and in the same design family (meaning that the details and pattern of lights are matched). Corner windows sometimes occurred in early examples, usually with a corner support (in contrast to the mitered glass corners on some Mid-century Modern homes).\n\nFRONT ENTRIES\u2014The front entry is almost always sheltered by the main roof of the house. At its simplest, the front entry is only recessed, with extra shelter provided by the overhanging front-facade roof. Alternatively, the door is set into the L formed by the cross-hipped or cross-gabled roof, providing two overhangs for additional shelter. In about half the examples, entry or partial-width porches occur, almost always contained under the main roof of the house, making them relatively inconspicuous. Two porch forms are common. In one, a portion of the front-facing cross gable or hip has inset walls that form a roofed entry area. In the other, a partial-width porch occurs, often in the L created by the cross-hipped or gabled-roof form; occasionally a porch is full-width. It is common for the material cladding the entry area to differ from that of the main body of the house.\n\nPorch supports are most often simple wood posts\u2014sometimes with triangular braces to each side\u2014or wrought iron in a wide variety of designs, from simple modern forms to more traditional patterns that often feature vine and leaf motifs. Occasionally, porch supports are omitted and the roof spans wall to wall.\n\nAn unusually wide variety of entry details appear. The front door itself may be single or paired and occurs in many diverse designs. The simplest is a resolutely plain flush door. Some designs are \"modern\" and may feature multiples of three (three small windows, three raised horizontal panels). Other designs feature panels\u2014either distinctive curvilinear panels that were then called French Provincial or multiple squared panels more typical of Colonial Revival; these often have glass panes above. Doors may have a single sidelight or matched sidelights or side panels. Occasionally the entry door faces to the side rather than toward the street.\n\nROOF-WALL JUNCTIONS\u2014The overhanging eave was either boxed or open. When boxed, it had either a simple, unadorned cornice board or no cornice at all. When open, the rafter tails were typically either sheathed with plywood or exposed, commonly with smooth-rounded rafter tails that did not extend beyond the roof edge.\n\nOTHER DETAILS\u2014The new emphasis on standardization seen in windows produced the Ranch style's very common eight-foot ceiling height, since sheetrock, gypsum board, and two-by-four lumber were now generally produced at this uniform length. Concrete-slab foundations, used in rapid prefab construction during World War II, became common and allowed the Ranch house to move even closer to the ground as slabs replaced the higher masonry-pier foundations of earlier homes.\n\nGarages were generally attached and could face to the front, rear, or side. The typical one-car attached garage (1930s to early 1950s) soon became a two-car garage, and later even a three-car one. Houses built early in the era, and those squeezed onto older, narrow lots, might have a detached garage or one connected by a covered breezeway. Carports were also sometimes found but were more common in Contemporary houses.\n\nWood, brick, stone, asbestos and wood shingles, concrete blocks, and stucco wall cladding were all used. Board-and-batten, used in Cliff May's early Ranch designs, was a favored wood-siding pattern. Frequently two or more materials were combined. Cladding might vary on whole sections of a wall (such as the front entry area), in the top of gable ends, or in horizontal sections (such as between the bottom third of a wall, typically below the base of the windows, and the upper two-thirds). The predominant wall cladding material used sometimes varies regionally (such as red brick in Georgia or stucco in Arizona) and can differ from subdivision to subdivision.\n\nSimple, low masonry planters were favorite elaborations and could be small and located near the entry, horizontal and stretched along the front facade of the house, or free-standing and enclosing an entry courtyard.\n\nThe FHA discouraged a pronounced modern appearance in the homes they helped finance. Thus builders frequently added modest bits of traditional detailing, usually loosely based on Spanish, French, or English Colonial precedents. Decorative window shutters are the most common of these. Window boxes were often used, and small roof cupolas (sometimes of prefab metal) and pieces of metal decoration at gable ends (such as eagles) are not unusual. It was rare to find a house that featured details from a single style\u2014most of them freely mixed and matched. Ranch houses that exhibit one distinct style are treated under Styled Ranch.\n\nEarly, small examples of the Ranch are sometimes called Ranchette, Minimal Ranch, or Transitional Ranch. These generally lack the broader overhang of later examples and many of the elaborations that became common as house size increased. Ranchettes are commonly found in neighborhoods that contain or are located close to Minimal Traditionals. The line between Minimal Traditional and Ranchette is a matter of judgment. However, the intent was likely a Ranch house if a picture window and other Ranch elaboration is present (such as a corner window or wall cladding that differs at the base of the windows). While Ranch houses commonly have a broader profile than Minimal Traditionals, neighborhoods platted with narrow lots before World War II may have Ranch-style houses adapted to these lot sizes.\n\nREAR-YARD ELABORATIONS\u2014Outdoor patios are common features at the rear of the house, often reached through sliding glass doors or a double French door, and sometimes with built-in free-standing masonry grills. These private outdoor living areas are a direct contrast to the large front or side porches of most late 19th- and early 20th-century styles. Large view windows faced, and patios and covered verandahs opened onto, the more private rear of the house. Front porches, when present, were generally shallow and rarely served as the outdoor living spaces of earlier styles.\n\nRANCH\n\n **TYPICAL WINDOWS** commonly steel, aluminum, or wood\n\nRANCH\n\n **TYPICAL SHELTERED ENTRIES**\n\nRANCH\n\n **TYPICAL SHELTERED ENTRIES**\n\nRANCH\n\n **TYPICAL SHELTERED ENTRIES**\n\nRANCH\n\n **TYPICAL SHELTERED ENTRIES**\n\nRANCH\n\n **TYPICALGARAGE DOORS** single or double most common\n\nRANCH\n\n **TYPICAL ROOF-WALL JUNCTIONS**\n\nRANCH\n\n **TYPICAL ELABORATIONS**\n\ndormers not present\n\n**DIFFERENT WALL-CLADDING MATERIALS**  \nwall cladding changes in entry area  \nwall cladding changes at base of window  \nwall cladding changes in gable end (usually wood boards, applied horizontally or vertically)  \nboard-and-batten cladding (for California-rustic look)\n\n**OVERALL**  \npatterns of three (such as three small windows in doors)  \npartial-width porch at entry (occasionally full-width)  \nmultiple window shapes and sizes  \nbroad low chimney  \ncorner window\n\n**INTEGRATED \"SLANTS\"**  \ngable-roof overhang slants outward  \narchitectural elements that slant\n\nRANCH\n\n **TYPICAL ELABORATIONS**\n\n**TRADITIONAL ARCHITECTURAL FEATURES**  \n\"Colonial\" roof cupola  \nshutters  \ndecorative wrought iron\n\nRANCH\n\n **TYPICAL ELABORATIONS**\n\n**INTEGRATED PLANTERS**  \nsquared masonry planter near entry  \nlong low masonry planters add horizontal emphasis  \nwindow-box planters\n\nRANCH\n\n **REAR-YARD ELABORATIONS**\n\n## Occurrence\n\nThe Ranch style originated in southern California in the mid-1930s, after a few earlier precursors. During the 1940s, it was only one of the small house types built under FHA financing guidelines. As the financial controls that mandated very small houses were gradually lifted following World War II, the Ranch style began to gain in popularity. During the decades of the 1950s and 1960s it became by far the most popular house style built throughout the country. Often located in large subdivisions, post\u2013World War II Ranch-house suburbs form a dominant part of many American cities\u2014particularly those that grew in the postwar Sunbelt Boom of the 1950s and 1960s, such as Dallas, Houston, Phoenix, Los Angeles, and Atlanta. As well as being nearly ubiquitous across most southern, southwestern, and western states during those decades, Ranch was also found in the northern half of the nation. The northern states, however, also favored split-level and even two-story houses. This may have been due to relatively more expensive land and the heating savings offered by more compact houses. It is important to understand that FHA financing guidelines similar to those that made construction of houses possible in the 1930s continued to exert a strong influence on new houses built through the 1950s and 1960s. The size of a Ranch was quite small in the late 1940s, but the typical size gradually increased as builders actively lobbied for higher loan limits and FHA guidelines were revised upward. The FHA also encouraged the building of entire neighborhoods at once by a single large developer. A topic of great concern to developers was how to inexpensively vary the appearance of similar house plans in a neighborhood in order to avoid a monotonous appearance.\n\n## Comments\n\nThe popularity of \"rambling\" Ranch houses was made possible by the country's increasing use of automobiles. Streetcar suburbs in the early 20th century had relatively compact houses on narrow lots to facilitate walking home from the stop. As the automobile became the principal means of transportation after World War II, houses with narrow fronts could be replaced by sprawling designs on wider lots. Never before had it been possible to be so lavish with land, and the rambling form of the Ranch house emphasizes this by maximizing facade width (often broadened farther by attached garages).\n\nThe style is very loosely based on early Spanish Colonial precedents in the American Southwest\u2014primarily the larger pitched-roof homes that featured private courtyards and covered inward-facing porches.\n\nCliff May, the innovative southern California builder, designer, and promoter, who had joined with California-based _Sunset Magazine_ to introduce Ranch houses to a larger audience, was a sixth-generation Californian and intimately familiar with Spanish Colonial houses. In 1946, May and the editorial staff of _Sunset_ wrote, and the magazine published, _Western Ranch Houses._ It featured a history of Spanish Colonial houses and a number of plans and drawings on how to build new Ranch houses based on them. Highly successful, it was quickly reprinted for a broader national audience. In 1958, _Sunset_ published a second version, mainly featuring photographs of May's work. National shelter magazines like _House Beautiful_ and _House and Garden_ began publishing Ranch-house articles in the 1950s.\n\nThese same magazines also contained numerous articles promoting a casual family-oriented lifestyle for postwar families. In the hands of builders throughout the country, the Ranch became the favored way of experiencing this informality. Described as \"middle-of-the-road modern\" and \"modern inside, traditional outside,\" the Ranch style was also considered, both by lending institutions and builders, to be more acceptable to the American home-buying public than more dramatic modern designs. While in the 1950s and early 1960s architects occasionally designed Ranch-style homes, most favored Mid-century Modern styles such as the Contemporary. Home buyers, however, embraced Ranch houses and greatly enjoyed their modern amenities and large family rooms that were available for affordable prices.\n\nIn the 1970s the pitch of the Ranch-house roof began to rise, and traditional stylistic detailing was added in greater quantity. While deliberately \"styled\" Ranch houses had been built throughout the Ranch era, these variations became ever more popular. By the 1980s, rising land prices nationwide, and changing tastes, meant an abandonment of the one-story house and a return to two-story houses.\n\n# MODERN HOUSES\n\n# Split-Level\n\n# ca. 1935\u20131975\n\nSplit-Level is the name of a new and distinctive _form_ of house\u2014rather than _style_ \u2014that rose to popularity during the 1950s for both practical and aesthetic reasons. The shape is found in many different styles\u2014primarily Ranch, Styled Ranch, and Contemporary\u2014and photographs of Split-Level houses appear in this section for these styles.\n\nA Split-Level house has three or more separate levels that are staggered and separated from each other by a partial flight of stairs rather than a full flight (six to eight steps rather than twelve to sixteen). There are two primary types: the Tri-Level Split, with three distinct living stories each a half-level apart, and the Bi-Level Split, with two distinct living stories and a split-entry level staggered in between.\n\nStudies of how floor plans\u2014particularly those of small houses\u2014could best serve modern living needs were undertaken beginning in the 1930s. The Small Homes Council encouraged such studies, and in 1960 they published a pamphlet on Split-Level homes outlining their advantages and disadvantages. Split-Levels take up less room on a lot, making them ideal where land is more expensive. They also provided some construction economies. The floor plan offers the advantages of privacy, noise control, and good interior circulation. The half-flight of stairs is less daunting than a full flight, but a disadvantage of Split-Levels is that they are not ideal for the elderly or those with disabilities.\n\nHome buyers liked the fact that Split-Levels looked big\u2014more like large two-story homes. Aesthetically, it was a plus that the garage was tucked under part of the house. Splits were particularly suited to sloping ground, where they could accommodate to fit the terrain. The form proved so popular, however, that it was also constructed on level ground, where it was less at home. Split-Level houses were built throughout the country but are less common in the southern and southwestern states.\n\nThe Tri-Level Split offered a practical way to incorporate a location for two new family possessions, the automobile and the television, and an elaborate theory of interior spatial planning grew around this form. It was felt that families needed three types of interior spaces: quiet living areas, sleeping areas, and noisy living\/service areas. In the Tri-Level Split form, each of these generally occurs on a separate level. The mid-level wing contained the \"quiet\" traditional living and cooking functions, while the upper level was reserved for private bedrooms. A third and lower level was then added to house \"noisy\" living\u2014the automobile in a garage and a multipurpose recreation room for the family television, which was rapidly becoming an almost universal possession during the 1950s. The public entry was generally located on the middle, quiet-living level. An additional basement and\/or attic level was sometimes added, another half-story up or down, normally used for storage.\n\nRudimentary Tri-Levels had appeared in the _Sears Honor Bilt Home_ catalogue by 1933, but it was not until after World War II that this shape gained broad popularity. By 1954 Split-Levels were outselling one-story Ranch houses four to one on Long Island. It was not uncommon for a large-scale developer to build entire subdivisions of Split-Level homes. (The Tri-Level Split was widely popularized by the ABC sitcom _The Brady Bunch,_ which ran from 1969 to 1974.)\n\nBi-Levels came later and were widely built from the 1960s into the 1980s. The Bi-Level had its entry at an intermediate level, placed between two full living stories, one of which was partially underground. The first appears to have been Carl Koch's \"New Kind of Two-Story House\"\u2014a 1954 prefab Package House erected in Weston, Massachusetts, by Techbuilt, in just six days. With a highly flexible interior floor plan, it was widely publicized in magazines as well as in a two-part NBC Special funded by the Ford Foundation. The Bi-Level was also called a \"raised Ranch\" since it resembled a one-story Ranch house elevated half a story above ground. Bi-Levels were also called split foyers because they popularized a new kind of spacious split-entry foyer with high ceilings\u2014typically one and one-half stories high. Some foyers led immediately into a full two-story stair hall. These may have played a role in the widespread use of tall entry halls beginning around 1990.\n\nThere are related variations, such as \"side-down\" (where one side of the house has a lower story, often with the garage) and \"rear-down\" (where the rear of the house has a lower story)\u2014both of these typically used on sloping sites. Neither, however, is technically a Split-Level, as the added partial story is separated by a full story and flight of steps, but they can have a similar appearance at first glance.\n\nSplit-Levels are most commonly found in the Ranch, Styled Ranch, and Contemporary styles. Splits have the same windows, wall cladding, and other details typical of the style they emulate. Roof forms are similar, but the change in height gives a slightly different look. Split-Level houses helped make multiple levels within a house desirable, and partial level changes became a major part of the design of both Contemporary and Shed houses.\n\n**SPLIT-LEVEL:**\n\nSPLIT-LEVEL  \nBoston, Massachusetts; ca. 1950. A Tri-Level Split with an extra basement story.\n\nSPLIT-LEVEL  \nDuluth, Minnesota; ca. 1960s. A Bi-Level Split.\n\nSPLIT-LEVEL\n\n **TRI-LEVEL SPLIT**\n\nSPLIT-LEVEL\n\n **NOT ASPLIT-LEVEL**\n\n(also can have a rear-down)\n\nThis is a Ranch with a full-story side-down\n\nSPLIT-LEVEL\n\n **BI-LEVEL SPLIT**\n\n# MODERN HOUSES\n\n# International\n\n# ca. 1925\u2013Present\n\n## Identifying Features\n\nFlat roof, usually without ledge (coping) at roof line; windows set flush with outer walls; smooth, unornamented surfaces with no decorative detailing at doors or windows; facade composition commonly includes large window groupings, often linear, and expanses of windowless wall surface; unified wall cladding, generally white stucco; commonly asymmetrical.\n\n**1920\u20131950**\n\n1920\u20131950:  INTERNATIONAL  \nLos Angeles, California; 1938. Richard Lind, architect. Note the blank expanses of wall in three different planes, the garage incorporated into the facade design, and the cantilevered section of roof.\n\n1920\u20131950:  INTERNATIONAL  \nCambridge, Massachusetts; 1937. Carl Koch and Edward T. Stone, architects. Note the ribbon windows in an otherwise blank expanse of wall and the cutaway section at right with a recessed entry and courtyard screened by a pictorial stone wall.\n\n1920\u20131950:  INTERNATIONAL  \nCohasset, Massachusetts; 1938. Josephine Hagerty House; Walter Gropius, architect. This house clearly shows the simple and boxier quality of many New England homes designed by the European \u00e9migr\u00e9s of the 1930s.\n\n1920\u20131950:  INTERNATIONAL  \nLos Angeles, California; 1936. Fitzpatrick House; Rudolf M. Schindler, architect. Note the lack of overhang on the right-side facade coupled with multiple, deep, boxy cantilevered roof sections, and large expanses of glass window walls on the front facade.\n\n1920\u20131950:  INTERNATIONAL  \nCambridge, Massachusetts; 1935. Bowers House; Howard T. Fisher, architect.\n\n1920\u20131950:  INTERNATIONAL  \nLincoln, Massachusetts; 1939. Walter Bogner House. Uncommon natural wood cladding on a pre\u2013World War II example.\n\n1920\u20131950:  INTERNATIONAL  \nLos Angeles, California; 1934. Buck House; Rudolf M. Schindler, architect. Schindler and Neutra's West Coast International houses were influenced by their work with Frank Lloyd Wright and by the sunny California climate, making use of broad roof overhangs and cantilevered elements. Their designs here and in the Lind example, the Fitzpatrick House, and the Kaufman Desert House typically are more three-dimensional than those on the East Coast.\n\n1920\u20131950:  INTERNATIONAL  \nPalm Springs, California; 1946. Kaufman Desert House; Richard Neutra, architect. One of Neutra's masterpieces recently restored, this is a view from the street.\n\n1920\u20131950:  INTERNATIONAL  \nLincoln, Massachusetts; 1938. Gropius House; Walter Gropius, architect. Gropius related his home to nearby New England houses by using wood-clad walls. His home incorporated many standard elements ordered from catalogues. As with the Josephine Hagerty House, this is a simplified house form, but here an angled glass brick entry element is added.\n\n**1950\u20131980**\n\n1950\u20131980:  INTERNATIONAL  \nPalm Springs, California; 1962. \"All-Steel-House\"; Alexander and Alexander, builders, Donald Wexler, architect. This is transitional to the Contemporary. It has a carport and deep recessed entry with small garden\u2014but with only a small selective section of roof overhang.\n\n1950\u20131980:  INTERNATIONAL  \nChicago, Illinois; 1951. Farnsworth House; Mies van der Rohe, architect. The Farnsworth House is one of the most significant of Mies van der Rohe's works. Note that the entire house is raised above the ground on a podium; this was intended to protect it from seasonal flooding of the nearby river.\n\n1950\u20131980:  INTERNATIONAL  \nHamptons, New York; 1987.\n\n1950\u20131980:  INTERNATIONAL  \nNew Canaan, Connecticut; 1949. Glass House; Philip Johnson, architect. This was Philip Johnson's tribute to the Farnsworth House (that had been designed earlier but not yet built). A solid rounded masonry section contains the bathroom and almost bolts the house to the ground\u2014accentuating that Johnson built his house directly on the ground rather than on a podium like Farnsworth House.\n\n1950\u20131980:  INTERNATIONAL  \nNew Canaan, Connecticut; 1954. Noyes House 2; Eliot Noyes, architect. This is a pure bi-nuclear house with living spaces in the foreground, sleeping spaces at the rear, and a courtyard between. The stone wall faced the road, and each wing had a glass wall facing non-public views.\n\n1950\u20131980:  INTERNATIONAL  \nLos Angeles, California; 1949. Hale House; Craig Elwood, architect. This house has a steel structural skeleton that is partially exposed, much in the Miesian manner.\n\n1950\u20131980:  INTERNATIONAL  \nSarasota, Florida; 1952 (Lido Shores). Philip Hiss Studio; Philip Hiss, architect and developer. Here, the living level is raised a story above ground level, allowing movements beneath it, a principle promoted by Corbusier, who also suggested the use of slender columns (piloti), as seen here, to raise the house.\n\n1950\u20131980:  INTERNATIONAL  \nLa Jolla, California; 1960. Case Study \"Triad\" House; Killingsworth, Brady & Smith, architects. Marcel Breuer introduced the bi-nuclear house in 1945, and many almost symmetrical examples like this were modeled after it.\n\n1950\u20131980:  INTERNATIONAL  \nDuluth, Minnesota; 1955. Starkey House; Marcel Breuer, architect. Originally more of a raised one-level house, space has now been enclosed on the ground level.\n\n1950\u20131980:  INTERNATIONAL  \nNew Canaan, Connecticut; 1952. The Campbell House; John Johansen, architect. This bi-nuclear house is raised on a low podium.\n\n**1970\u2013PRESENT**\n\n1970\u2013PRESENT:  INTERNATIONAL  \nLos Angeles, ca. 1970.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDallas, Texas; ca. 1980.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDallas, Texas; ca. 2010. Urban Reserve; Robert Meckfessel, architect. The thin lines visible on the walls allow the rainscreen covering to better circulate air.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nLos Angeles, California; 1963, remodeled 2000s; Roger Sherman, architect. Note private entry courtyard and the two-story vertical window that highlights the staircase.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDeephaven, Minnesota; 1973. Rappaport House; Milo H. Thompson, architect. Note the very long ribbon window.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDallas, Texas; ca. 2010. Lionel Morrison, architect. Urban Reserve.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nLos Angeles, California; 1992.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDarien, Connecticut; 1965\u20131967. Smith House; Richard Meier, architect. Meier's crisp white box houses helped usher in a new enthusiasm for the International design elements that had dominated prior to World War II, albeit with larger expanses of glass on rear-view facades like this one. The chimney is brick and the house is clad with smooth vertical wood boards\u2014all painted white to resemble stucco.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nDestin, Florida; 2005. Corbusier's preference of raising the house above ground level is very practical on flood-prone beachfronts like this. The house is placed on the site rather than integrated into it, producing the \"house as a piece of sculpture\" appearance that Frank Lloyd Wright detested.\n\n1970\u2013PRESENT:  INTERNATIONAL  \nVenice Beach, California; 1992. International has been used for town-house-type developments in recent decades.\n\n## Variants and Details\n\nIn 1932, New York's Museum of Modern Art mounted the influential \"Modern Architecture: International Exhibition.\" In the accompanying book, _TheInternational Style,_ authors Henry-Russell Hitchcock and Philip Johnson identified three principles common to early International buildings: architecture as volume, regularity, and avoiding the application of ornament.\n\nA fundamental difference between International style structures and earlier buildings was that previously the walls, commonly of solid masonry, had provided structural support. By contrast, International buildings were supported by a lightweight structural skeleton (often metal). Walls were freed simply to enclose a volume or architectural space\u2014and could have more flexible fenestration (windows). House facades could feature a three-dimensional composition rather than consisting primarily of ornament applied to an essentially flat surface, as with earlier styles.\n\nThe structural skeleton allowed walls and windows to be arranged asymmetrically to reflect interior needs, but their placement was regulated and given subtle order by the structural system beneath. Asymmetrical facades gained coherence by this visual \"regularity.\" These first two principles applied primarily to the grandest houses, as most homes were too small to make steel-skeleton construction economical; in modest houses, a similar look might be more inexpensively achieved with balloon framing. Other materials also were used for structural support, most commonly concrete\u2014whether poured in place, gunnite, tilt wall, or hollow block.\n\nFreeing exterior walls from structural demands allowed facade treatments not feasible earlier. Now windows could wrap around building corners. Where interior functions did not require windows, or where privacy was needed, solid windowless expanses of exterior wall were used. Cantilevered projections were possible\u2014sections of roof, balcony, or second stories extending outward and dramatizing the non-supporting nature of the walls\u2014and these were more common on the West Coast. Removing walls as the support system also meant that interior walls could become mere partitions that allowed great flexibility in room layout, and floor plans further explored the open free-flowing spaces pioneered by Frank Lloyd Wright.\n\nSimple, underplayed windows\u2014often casements\u2014were preferred. These could be metal or simply detailed wood, generally placed flush with the wall and often grouped into horizontal ribbon windows. In the 1930s this fenestration style was joined by picture windows, and after World War II the affordability of plate glass made floor-to-ceiling window walls possible. The windows were no longer placed in a rigid manner governed by exterior symmetry but related instead to the interior plan or were arranged primarily to create a pleasing facade design.\n\nAdded exterior decoration was avoided, maintaining the spirit of the building as a smoothly sheathed volume. Exterior walls were clad with a single surface\u2014commonly white stucco in the 1920s and 1930s\u2014to clearly differentiate them from load-bearing masonry and emphasize the interior volume. By the late 1930s smooth board and plywood cladding were also used\u2014as these could be applied to a more common and less expensive balloon frame\u2014and Marcel Breuer had introduced the concept of adding a picturesque brick or stone wall.\n\nAfter World War II, International houses entered a new phase. While builders and many architects gravitated to the more \"natural\" Contemporary style, International houses were not completely abandoned. In 1945, Breuer introduced the bi-nuclear plan, clearly separating quiet, private sleeping spaces from public living spaces with a courtyard or entry hall in between. This inspired a new exploration of symmetrical domestic examples. Courtyards of all types\u2014entry, side, rear, and interior (either enclosed as atria or open)\u2014gained in popularity and provided an excellent way to create private views (see illustrations this page, this page). Also in the late 1940s, Mies van der Rohe's glass-clad Farnsworth House\u2014and Philip Johnson's tribute to it, his own Glass House\u2014created a new interest in glass as cladding. This \"Miesian\" interpretation of the International style achieved its broadest appeal in commercial buildings. From circa 1950 to 1970 symmetrical, or near-symmetrical, versions of the International, often with large areas of glass window wall opening onto private areas or views, were favored by architects. At the same time, wood-clad asymmetrical examples occurred, similar to pre\u2013World War II designs but clad in the natural materials promoted by Wright and used for Contemporary houses.\n\nIn the 1970s, a group known as the New York Five (Charles Gwathmey, Michael Graves, John Hejduk, Richard Meier, and Peter Eisenman) began designing new homes based on the early white stucco International house. This sparked a strong revival of \"the Whites\" that has continued to the present day. These typically have a far greater percentage of glass on their private or view facades than did their early 20th-century predecessors.\n\nINTERNATIONAL\n\n **ROOF-WALL JUNCTIONS**\n\nINTERNATIONAL\n\n **WINDOWS**\n\nINTERNATIONAL\n\n **REGULARITY** Asymmetrical facade superimposed upon a regular structural system\n\nINTERNATIONAL\n\n **COMMON ELABORATIONS**\n\nDistinguish the International's cantilevered _sections_ of roof overhang from the more continuous overhanging _eaves_ of the flat-roofed Contemporary\n\nINTERNATIONAL\n\n **COMMON ELABORATIONS**\n\nINTERNATIONAL\n\n **BI-NUCLEAR HOUSE** (common 1950\u20131965)\n\n## Occurrence\n\nThis avant-garde and primarily architect-designed style is relatively rare. Landmark examples from the 1930s occur principally in fashionable suburbs in the northeastern states and in California. The early experimental nature of the style is highlighted by how many of the well-publicized early houses were designed for the architect, the architect's mother, or a professor.\n\nDuring the 1950s and 1960s International houses continued to be built, but Contemporary houses were a far more common modern house style. In the 1970s a revival of interest in the original white stucco-clad houses began. This revival gained momentum as the turn of the millennium spurred publication of many books looking back on a century of modernism. _DWELL_ magazine, launched in 2000 with a focus on modern homes, helped further spread this new wave, and the International continues to enjoy a strong resurgence today.\n\n## Comments\n\nIn the decades separating World War I and World War II\u2014while Americans were building neighborhoods of period houses\u2014European architects were busy creating dramatic new modern homes and buildings. Le Corbusier in France, Oud and Rietveld in Holland, and Walter Gropius and Mies van der Rohe in Germany were all working without historic precedent, trying to exploit the materials and technology of the day. In addition, as these pioneers explained in the first book published by the Bauhaus, they wished to create an International architecture \"independent of specific materials, sites or cultural tradition\"\u2014thus avoiding the regional differences they felt had led to World War I\u2014and specifically chose white stucco as a uniting material to achieve these ideals. Their designs came to be known as International Style when the 1932 Museum of Modern Art exhibition christened the movement with this name.\n\nThe International style was brought to the United States by architects who emigrated from Europe. Three early \u00e9migr\u00e9s were already at work in the 1920s: William Lescaze in New York City and Rudolf Schindler and Richard Neutra in Los Angeles. They were joined in the 1930s by elite Bauhaus architects Walter Gropius, Mies van der Rohe, and Marcel Breuer, all fleeing Hitler. These later arrivals had a profound influence because their theories and concepts were widely disseminated through architectural education. Gropius became the dean at Harvard's Graduate School of Design and Mies the dean of architecture at Illinois Institute of Technology. Their new curricula swept Beaux Arts precepts out\u2014and those of the Bauhaus in\u2014as other schools of architecture quickly followed their lead.\n\nAnother theory of the Internationalists\u2014Le Corbusier's view of the house as a \"machine for living\"\u2014was appealing in a world of rapidly advancing technology. All superfluous ornament could be stripped away, while the latest domestic machinery was installed in kitchens and bathrooms, and true efficiency brought to the home. Functionalism, emphasizing how a building served its inhabitants, was of prime importance; traditional elements of the house that were merely decorative were to be discarded. International houses built today typically dwarf those of the 1920s to 1950s. And while added decoration is avoided, the mantra of functionalism can result in greatly enlarged kitchen, bath, and closet areas designed to the hilt and enjoyed with impunity.\n\n# MODERN HOUSES\n\n# Contemporary\n\n# ca. 1945\u20131990\n\n## Identifying Features\n\nLow-pitched gabled roof (sometimes flat) with widely overhanging eaves; roof beams commonly exposed; windows generally present in gable ends (or just below roof line in non-gabled facades); built with natural materials (wood, stone, brick, or occasionally concrete block); broad expanse of uninterrupted wall surface typically on front facade; entry door may be recessed or obscured; asymmetrical.\n\n## Principal Subtypes\n\nFive principal subtypes can be distinguished:\n\nFRONT-GABLED ROOF\u2014This subtype includes a broad range of simple front-gabled forms. These are ideal for showing off triangular gable-end windows that indicate a high \"vaulted\" ceiling inside. Two variations are common: a broad one-story form with a very low-pitched roof (popularized by California developer Joseph Eichler) and an asymmetrical gable front often found on Split-Level houses (nicknamed the \"wounded dove\").\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nPalo Alto, California; ca. 1955. Joseph Eichler, developer. This is a variation\u2014a front gable with side wings\u2014that was a favorite design of Eichler but less common elsewhere. Note the windows placed between roof beams on the garage wing and recessed entry under centered gable.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nLos Angeles, California; 1965. The entry feature soars on three slender piloti and is open to the sky. The relatively symmetrical front-gabled roof makes it hard to discern that this is likely a Tri-Level Split.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nSilver Spring, Maryland; 1959. The triangular window groupings in the gable here (and in the houses in Palo Alto, San Diego, East Hampton, Phoenix, Baltimore, and Las Vegas) advertise that there is a vaulted ceiling in the room behind.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nSan Diego, California; 1960. Broad one-story houses with very low-pitch front-gable roofs were the most common builder form (see also the houses in Phoenix and Las Vegas).\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nEast Hampton, New York (vicinity); ca. 1960. Rear-facade view. A deep gable-roof overhang shelters the deck facing the ocean, while the roof overhang on right side of the house is quite shallow.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nBaltimore, Maryland; ca. 1959. Here both gable end and sides of roof have broad overhangs.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nPhoenix, Arizona; 1959. Note the entry placed inside the carport on the right and the roof opening with exposed structure.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nBaltimore, Maryland; 1956. The free-standing exposed \"post-and-beam\" structure is uncommon.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nDaly City, California; ca. 1950s. This has an original diamond-pattern garage door. Note how vertical elements extend up to the roof beams.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nLas Vegas, Nevada; ca. 1961. Here a fence of concrete screen block is used to create a private front garden.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nBeverly Hills, California; 1968. This clearly shows the deeply recessed entry with exposed roof structure above.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nAtlanta, Georgia; 1960. Collier Heights Historic District. The white section at left is likely a later alteration to the original garage. The driveway ends right in front of it.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nFalcon Heights, Minnesota; 1966. Note how the wall cladding is applied in vertical strips with the windows set into a separate material. This was often a cost-saving device.\n\nFRONT-GABLED ROOF:  CONTEMPORARY  \nSeattle, Washington; 1959. Even in the rain, the \"wounded-dove\" gabled roof is quite distinctive and perfectly adapted to this Tri-Level Split.\n\nSIDE-GABLED ROOF\u2014The least common of the gabled forms, this subtype is more often found in later examples.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  CONTEMPORARY  \nLos Angeles, California; 1963. This hillside adaptation has geometric texture on retaining wall at right, V-shaped supports for the main house level, and view of exposed structural beams.\n\nSIDE-GABLED ROOF:  CONTEMPORARY  \nSilver Springs, Maryland; 1951. Note how broad this chimney is. The front door is a later modification and its oval shape is an almost shocking contrast with the rectilinear design of the adjoining windows.\n\nSIDE-GABLED ROOF:  CONTEMPORARY  \nMinneapolis, Minnesota; 1951. Niels House; Frank Lloyd Wright, architect. The large chimney is a central anchor with two gable-roof forms extending out from it. Each ends with an outward slant like a ship's prow.\n\nSIDE-GABLED ROOF:  CONTEMPORARY  \nPortland, Oregon; 1969. A Tri-Level Split with cladding change between ranks of windows.\n\nGABLED-ROOF VARIATIONS\u2014Most commonly this subtype consists of a front-gabled roof with a side-gabled extension. Other combinations include houses with two wings or two front-facing gables. From the street it is sometimes difficult to discern the roof form of low side wings.\n\n**GABLED-ROOF VARIATIONS**\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nDuluth, Minnesota; 1961. Different shapes of windowpanes are used to create windows in the gables.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nDallas, Texas; 1960s. This and one of the examples in Kansas City are the broad horizontal Ranch form, but with Contemporary detailing.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nKey West, Florida; ca. 1960s. This house substitutes a second front gable for the wing.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nSt. Paul, Minnesota; 1967. Note the broad, low stone chimney.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nSeattle, Washington; 1961. This Bi-Level Split has original garage doors.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nKansas City, Missouri; 1967. Putting windows in a separate vertical strip of cladding was more cost-efficient than having to build brick headers and footers above and below each window. It also kept trades separate (the bricklayer did not have to be coordinated with the window installer).\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nDallas, Texas; 1962.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nSeattle, Washington; 1976. As in the example from Daly City, vertical supports align with the roof beams.\n\nGABLED-ROOF VARIATIONS:  CONTEMPORARY  \nKansas City, Missouri; 1954. A Tri-Level Split needs only vertical elements and a roof overhang with patterned surface beneath to achieve a Contemporary look. Note the slight outward slant of balcony railing.\n\nFLAT ROOF\u2014This subtype includes flat-roof variations; these have long, continuous broad roof overhangs, and exposed roof beams that differentiate them from contemporaneous International-style houses. Some have shallow horizontal windows located just below the roof line, sometimes set in the space between roof beams.\n\n**FLAT ROOF**\n\nFLAT ROOF:  CONTEMPORARY  \nBernardsville, New Jersey; 1940. Christie House; Frank Lloyd Wright, architect. This large L-shaped Usonian house has a carport on the right and a broadly overhanging flat roof.\n\nFLAT ROOF:  CONTEMPORARY  \nLos Angeles, California; 1965.\n\nFLAT ROOF:  CONTEMPORARY  \nEnglewood, Colorado; 1955. Arapahoe Acres Historic District. Edward B. Hawkins, developer. Note corner windows, obscured entry, and carport.\n\nFLAT ROOF:  CONTEMPORARY  \nSt. Paul, Minnesota; 1961. Broad shallow windows are set between roof beams.\n\nFLAT ROOF:  CONTEMPORARY  \nSan Francisco, California (vicinity); ca. 1960. This \"across the valley\" view shows how a Contemporary house could be inserted on a very steep site.\n\nFLAT ROOF:  CONTEMPORARY  \nLos Angeles, California; 1960. Silver Lake; Richard J. Neutra, architect. Somewhat similar to Wright's 1904 Gale House in Oak Park, this symmetrical house has two full-width balconies with solid railings and a deep roof overhang.\n\nFLAT ROOF:  CONTEMPORARY  \nPalm Springs, California; 1955. Wexler House; Donald Wexler, architect. Wexler described plywood as a treasured new material that made many Contemporary features affordable.\n\nFLAT ROOF:  CONTEMPORARY  \nHighland Park, Texas; 1967. E. G. Hamilton, architect. This house is surrounded by a brick wall and softened with raised garden beds. This defines several courtyards and allows use of window walls with complete privacy. Note the generous roof overhang.\n\nFLAT ROOF:  CONTEMPORARY  \nKansas City, Missouri; 1952. Note the five distinct sections of this house: high windows at left (bedrooms?), broad chimney, window wall, projecting section with high ribbon windows, and a set-back garage.\n\nFLAT ROOF:  CONTEMPORARY  \nPalo Alto, California; 1954. Joseph Eichler, developer. Some of Eichler's most popular small houses combined a single slanted roof with a flat roof.\n\nBUTTERFLY AND SLANT ROOFS\u2014This subtype of less common roof types includes a slanted roof (with more reliable drainage than a flat roof), an upside-down gable called a butterfly roof (this creates a flat valley prone to leaking), and exaggerated gable roofs that might extend even to the ground. All of these could be combined with flat or gabled roofs, but the eye-catching butterfly- or slant-roof element tends to dominate visually.\n\n**BUTTERFLY AND SLANT ROOFS**\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nPalm Springs, California; 1962. \"House of Tomorrow\" (Vista Las Palmas); Palmer and Krisal, architects. Futuristic design for what was originally a model home built on three levels.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nPalo Alto, California; 1954. Note how the concrete block wall on the right wing is laid in a regular grid pattern\u2014a 1950s favorite.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nLos Angeles, California; 1959. Silver Lake. Two railing designs are used on the full-width balcony.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nMinneapolis, Minnesota; 1958. Paul and Helen Olfelt House; Frank Lloyd Wright, architect. The carport forms an integral part of the design.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nSan Francisco, California; 1948. The carport extension with added slanted elements exaggerates the simple inverted gable-roof form.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nPalm Springs, California; ca. 1960s. \"Swiss Miss\" House; Charles Dubois, designer, Alexander Construction, builder.\n\nBUTTERFLY AND SLANT ROOFS:  CONTEMPORARY  \nAtlanta, Georgia; ca. 1960. The broad wall surfaces have an integral horizontal pattern incorporated into the brick pattern. Note how the entire roof floats above an almost continuous ribbon of windows that separates the roof from the walls of the house.\n\n## Variants and Details\n\nEarlier styles were generally defined by the types of decorative detail applied to their exteriors\u2014on doors, windows, porch supports, wall surfaces, dormers, and roof-wall junctions (see this page). The Contemporary style rejects this approach and is instead more concerned with the spaces inside the house and the way in which each space relates to the outdoors. Therefore, the design is created from the inside out, with the attention not on details visible as one approaches the house but rather on the functionality of the interior space and the integration of outdoor views. House plans were no longer simply diagrams of room layouts; they now expanded to cover a large portion of the site (for small lots, the entire site) and included the various outdoor spaces\u2014and views\u2014associated with each room. This approach confers a very spacious quality to the house, particularly important because of the small size of houses in the 1940s and 1950s. The indoor-outdoor quality is achieved in two basic ways: by adding varied exterior living spaces or view gardens, and by using courtyards to bring the outside into the house. Enclosed and semi-enclosed interior courts had been used for millennia\u2014most familiarly in the houses of ancient Greece and Rome. View gardens also had a long history, particularly in China and Japan, where for centuries small gardens had been planned specifically to create pleasing views for teahouses and interior spaces. Hundreds of thousands of U.S. military personnel were exposed to this Oriental style during the occupation of Japan (1945\u20131952). During 1954\u20131955, New York's Museum of Modern Art built and displayed a carefully constructed Japanese house and garden that clearly demonstrated how Japanese gardens were designed from an interior perspective. The seamless integration of house and landscape was unlike the pristine white International house that often appeared more like a piece of sculpture distinct from the landscape.\n\nEntry facades typically reveal little of the house itself. A large area of uninterrupted wall surface is common and might have a design or pattern within the finish. Screening fences protect private side and front gardens. Grilles, most commonly concrete screen block, in see-through designs forming an overall pattern, might be used for visual screening or as brises-soleil to create shade.\n\nThe entrance is downplayed and sometimes deeply recessed. In modest houses it might be tucked deep inside a carport and in others approached through an entry garden or court. Entering the house in the mid-section creates space-saving floor plans by omitting a central hall for circulation. Frank Lloyd Wright advocated carports as a cost-saving device\u2014and low broad masonry chimneys to service large open fireplaces in the heart of the house.\n\nThe Contemporary house could be built on steep hillsides where Ranches\u2014and even Split-Levels\u2014are difficult to place. These sites were now easily accessible by the automobile, and architects devised many creative ways the Contemporary could adapt to a challenging site. From the street, houses can look completely different on the downhill side than on the uphill side. Downhill might reveal only a carport or garage and perhaps a small entry area, while uphill might show broad roof overhangs and balcony railings, or the underside of decks facing the view\u2014perhaps with a glimpse of the window walls that embraced it.\n\nOverhangs generally have open eaves (i.e., not boxed). Rafters can be exposed (with their ends covered by a single board) or disappear (smoothly covered, with affordable new exterior-grade plywood). High-style examples might incorporate subtle ornament into the design of the overhang. Openings in the roof are relatively common and allow natural light into carports or provide light and rain to the beds in a covered garden area.\n\nContemporary house design was aided by three 1930s building innovations: thick plate glass, allowing window walls; exterior-grade plywood, allowing inexpensive wall cladding and sheathed-roof overhangs; and new glues, allowing engineered wood to be used for wood paneling and post-and-beam construction.\n\nJapanese influence was also present in wood construction with exposed timbers and a panelized appearance. Post-and-beam construction was often used and might be topped by a planked roof or enclosed with a variety of solid panels or window units forming the walls. This influence came from the West Coast, where it was sometimes referred to as the Second Bay Tradition\u2014although architects in the Pacific Northwest such as Pietro Belluschi and John Yeon also used the same vocabulary. There, a nearby supply of redwood and a strong tradition of timber construction\u2014seen in the earlier Craftsman joinery of Greene and Greene and timberwork of Bernard Maybeck\u2014inspired mid-century architects to design extraordinary wood detailing that united interior and exterior design.\n\nWindow walls appeared in an almost infinite variety and were composed primarily of large single fixed-glass panes, with a few sections that opened, and often some sort of door. The spatial effect of the inside flowing out might be enhanced by allowing the floor material, the roof beam, or the ceiling material to continue from the inside to the outside, making the glass window wall seem to disappear.\n\nBy the late 1950s builders had learned that an economical way to build Contemporary houses with brick cladding was to end the masonry at the sides of windows and run a vertical panel of wood cladding from the ground to the roof line. This avoided the expense of laying window headers and bases in the masonry and created an easily recognizable vertical look.\n\nCONTEMPORARY\n\n **TRADITIONAL DESIGN V. MODERN DESIGN AS ANALYZED BYFRANK LLOYD WRIGHT**\n\nCONTEMPORARY\n\n **OUTDOOR SPACES EXPAND THE HOUSE**\n\nCONTEMPORARY\n\n **COURTYARDS BRING OUTDOORS INSIDE**  \n(can be used in town houses and urban houses, also suburban houses)\n\nCONTEMPORARY\n\n **ELABORATIONS COMMONLY VISIBLE FROM STREET**\n\nCONTEMPORARY\n\n **OTHER TYPICAL ELABORATIONS**\n\nCONTEMPORARY\n\n **POST-AND-BEAM CONSTRUCTION (WOOD OR STEEL)**\n\nCONTEMPORARY\n\n **SOME SPATIAL CONFIGURATIONS**\n\nCONTEMPORARY\n\n **WINDOW WALLS**\n\n## Occurrence\n\nThe Contemporary style was favored most by American architects from about 1945 to 1965, when it filled architectural journals and dominated awards. While Ranch-style houses proliferated in most builder subdivisions, a few successful developers built Contemporary subdivisions\u2014among them, Joseph Eichler in California, Charles M. Goodman in suburban Washington, D.C., and Edward Hawkins in Denver. The most prolific was Joseph Eichler (1900\u20131974), who built close to ten thousand homes in the San Francisco Bay area. A wholesale grocer early in life, Eichler rented a Usonian house in 1942, where he and his wife fell in love with modern architecture. Changing professions, Eichler devoted the remainder of his life to building well-designed Contemporary houses that were affordable to a large segment of the economic spectrum. In the late 1960s the Contemporary style began to lose popularity.\n\n## Comments\n\nThe Contemporary had two practical design advantages over the contemporaneous Ranch. In addition to being adaptable to steep hillsides, the Contemporary was as appropriate for a two-story house as for a one-story house, unlike the Ranch. Thus, a larger house could be built on a smaller footprint with the Contemporary, leaving more exterior green space.\n\nA major strike against the Contemporary was that lending institutions preferred to avoid financing more avant-garde designs in favor of the widely popular Ranch style. The FHA in particular did not like houses with unusual roof forms (even slanted) and felt more comfortable with homes sporting \"traditional\" exterior details such as shutters.\n\nThe moving force behind Contemporary houses was Frank Lloyd Wright (1867\u20131957) and his Usonian houses. Wright had been formulating his ideas about houses for almost fifty years when he designed his first Usonian in 1937. These small and affordable houses were constructed from natural materials, built low to the ground, included open floor plans with a free flow of interior space (only the bedrooms and baths had walls), and had broad sheltering roof overhangs. They also featured a significant spatial and visual interplay of indoor and outdoor spaces. For Wright, glass was a magical super-material, allowing window walls with views of private gardens partially enclosed by an L-shaped house. Wright also favored \"continuity\" in a house and did not admire the post-and-beam construction used in many Contemporary homes.\n\nWright tirelessly promoted his theories\u2014to the general public in _House Beautiful_ magazine, to architects and builders in _House and Home_ magazine, and to all in his many books, most notably _The Natural House_ in 1954. By the early 1950s, Wright's philosophy had entered the mainstream of American architectural design and swept the country in the form of Contemporary houses. Contemporary houses are sometimes called simply Mid-century Modern, but, as used here, this is a broader category that includes other modern styles being built at mid-century.\n\n# MODERN HOUSES\n\n# Shed\n\n# ca. 1965\u20131990\n\n## Identifying Features\n\nShed-roof forms, generally multi-directional and occasionally coupled with a gable roof; wood wall cladding (vertical, diagonal, horizontal, or shingles), occasionally with brick veneer; smooth roof-wall junction commonly with little or no overhang; asymmetrical.\n\n**SHED**\n\nSHED  \nAmagansett, New York; ca. 1970s. This is uniformly clad with smooth vertical boards.\n\nSHED  \nAmagansett, New York; 1982. A single high-pitched shed roof juts upward, forming a dramatic angular shape.\n\nSHED  \nFayetteville, Arkansas; 1999. Coudren House; James Lambeth, architect. A 21st-Century Modern house that incorporates stepped wood sheds.\n\nSHED  \nDuluth, Minnesota; 1979. When half of a gable roof is slightly \"slipped\" upward it forms a line of clerestory windows (as on the left of this house)\u2014excellent for light and passive cooling. Note diagonal siding on front facade.\n\nSHED  \nSea Ranch, California; ca. 1970s. A low roof pitch and a broad house mass allow this house to settle into the landscape.\n\nSHED  \nMinneapolis, Minnesota; ca. 1970. This house is composed of three distinct volumes (one is behind). Note how the volume on the right almost seems to float above its dark recessed base.\n\nSHED  \nAmagansett, New York; ca. 1970s. Like the example in Sea Ranch, three main shed forms are visible. Here window placement, a simple but contrasting cornice board, and a very slight overhang, give greater individual definition to each shed form.\n\nSHED  \nDallas, Texas; 1979. Here shed elements are arranged in a formal five-part composition.\n\n## Variants and Details\n\nThe overall effect of a Shed house is that of bold diagonals, counterpointed shapes, and multiple massing. The form of the house imparts its style, not the smaller elements. The shed roof is often multi-directional and used in ways that give the effect of colliding geometric shapes. The front door and entry area is generally inconspicuous, and may even be obscured.\n\nThere is little added exterior detail; elaborations are primarily simple window variations. There are few window openings on walls that face public areas and those that occur are generally quite varied and asymmetrically placed. As in the Contemporary, large fixed panes of plate glass are typically used; these are generally set flush with the exterior wall. Ribbons of clerestory windows are found high on facades or above lower roof forms, often operable for ventilation. Lower windows are often composed of vertical sections with a tall, narrow upper pane above a short lower pane. Window tops are either flat or sloped with the angle of the roof. Elaborations include a \"boxed\" enframement that partially surrounds a window grouping, and deep box-bay windows (sometimes called saddlebags).\n\nTypically no more than a single board is used as a cornice at the roof-wall junction. The chimney is typically rectangular, unelaborated, and often clad in wood or plywood. Tall metal chimney flues may be exposed and extend above the chimney top.\n\nThe architects who originated the style generally preferred wood-shingle wall cladding, but later interpretations of the style often used wood board siding (applied either horizontally, vertically, or diagonally), T1-11 plywood (that imitated wood siding), and\/or brick veneer.\n\nSHED\n\n **HOW A GABLE ROOF IS \"SLIPPED\" TO CREATE SHEDS**\n\nSHED\n\n **SOME WAYS TO ORGANIZESHED-ROOF ELEMENTS AND ROOMS**\n\nSHED\n\n **TYPICAL ELABORATIONS**\n\n## Occurrence\n\nWhile never common, this late-century modern style is the primary one that evolved beyond one-of-a-kind architect-designed homes into the widespread house pattern books used by builders. First presented as appropriate for second homes, Shed-style homes were also built as primary residences in many suburban settings. They can be found throughout the United States.\n\n## Comments\n\nAlthough there were precursors, the Shed style was launched in earnest by the publicity surrounding development of the Sea Ranch, a carefully thought-out second-home community 125 miles north of San Francisco. Sea Ranch introduced innovative ways of designing a dwelling and placing it into native landscapes. The initial projects, the first phases of the Sea Ranch Condominiums and the Hedgerow Houses, were built of vernacular forms clad with unpainted wood materials\u2014inspired by the structures found on old farms and western mines.\n\nTwo creative architecture firms were involved: Charles Moore (Moore Lyndon Turnbull Whitaker\u2014MLTW) sited and designed the Sea Ranch Condominiums, and Joseph Esherick (Joseph Esherick and Associates) directed the Hedgerow Houses. Reportedly, the designs were generated separately in 1963, however, the projects appeared strikingly similar in their approach, each featuring shed roofs\u2014the distinctive element adopted throughout the country. Even as the initial Sea Ranch structures were being completed, the Shed style burst upon the architectural scene. It seemed that young architects could hardly wait to move beyond boxy International forms to create more interesting interior spaces. The new Shed houses allowed a dynamic flow of different spatial experiences. In the _Architectural Record_ \"Record Houses 1967,\" six houses featured the new Shed style.\n\nThe style's rapid spread was undoubtedly aided by Moore's appointment as dean of the Yale School of Architecture from 1965 to 1970. In addition to the spatial qualities, Shed reflected broad societal factors at work from the mid-1960s through the 1970s. Its genesis from useful and simply built structures\u2014such as barns, mining structures, and folk houses\u2014was in sync with a new interest in \"architecture without architects.\" The vertical shed shapes with high clerestory windows could facilitate passive solar cooling, an important tenet of early energy conservation. Finally, Sea Ranch's requirements that houses be placed lightly on the land\u2014leaving natural meadows and forests intact\u2014fit with the emerging environmental movement of the 1970s. By the 1980s, Shed, along with the activism of the 1970s, was fading away and being replaced by traditional styled houses.\n\n# MODERN HOUSES\n\n# Other 20th-Century Modern\n\nModern houses are divided into early modern houses (the creatively decorated Prairie, Craftsman, and Modernistic styles); banker's modern (generally grouped into developments of Minimal Traditional, Ranch, and Split-Level houses) and mainstream modern (the modern styles favored by architects). This last category of modern styles is called mainstream not because it constitutes the bulk of modern houses built in America (an honor belonging to the banker's modern styles) but rather because it has dominated architectural education, awards, criticism, and even histories of 20th-century houses.\n\nMany histories of 20th-century modernism separate styles built during the mid-20th century\u2014Mid-century Modern\u2014from those being designed towards the end of the twentieth century. Most Mid-century Modern houses were either Contemporary or International style (each covered in a separate chapter). However, a small group of architects rejected the orthogonal nature of these two styles and instead designed Organic houses, an approach to house design united by the use of natural shapes or non-rectilinear geometries that includes an extraordinarily broad range of designs.\n\nOther devices created three other Mid-century styles. Adding a new roof form to an otherwise simple house was discovered to be a good way to create innovative modern designs. High-style variations included airplane, paraboloid, umbrella, and multi-arched roofs\u2014all rarely encountered in real life. However a very simple roof-form, called the A-frame, became widespread in Mid-century, particularly for casual vacation houses. At the opposite end of the spectrum, Formalism could create a more dignified, high-style appearance by adding slender attenuated arches to modern boxes. Brutalism was yet another Mid-century Modern style, where concrete was used to create a more rugged and insular look.\n\nBy 1965 even these variations were not enough and, following publication of the critically acclaimed book, _Complexity and Contradiction in Architecture_ by Robert Venturi (New York: Museum of Modern Art, 1966), house style moved beyond Mid-century Modern and into other directions. Postmodernism came first, adding historical elements onto houses in new and different ways. Following this the elements of houses were exploded and rearranged in a movement called Deconstructivism.\n\nAs the millennium approached, the modern movement gained a new impetus\u2014leading to exploration of both new interpretations of modernism and to revivals of past modern styles. This is treated in a final appendix on 21st-century Modern.\n\n## Organic Houses (1950s\u2013present)\n\nOrganic architecture is based on the coalescence of the built environment with nature, allowing the design to respond to the natural environment rather than impose on it. While other modern movements more often championed straight lines and orthogonal designs, Organic modernism favored natural shapes and interesting geometries. Designs\u2014conceived as reactive both to the environment and to the building material\u2014were developed\"organically\" into one harmonious unit. Organic architects utilized new technologies and building materials but rejected them as sources of stylistic inspiration, believing they imposed a design from outside sources. An Organic architect would instead carefully study the exact site for a house and then create a design that grew from within, a careful relationship between all of the parts of a house. This might begin with a geometric shape used for the floor plan and then extend into the design of all parts of the house\u2014fixtures, furnishings, and even window shapes.\n\nAccording to Frank Lloyd Wright, America's premier Organic architect, form no longer _followed_ function; it was one and the same. This principle began its evolution in Wright's Prairie houses and had matured in the mid-1930s when he demonstrated it magnificently on both the high and low ends. In 1936, Wright designed his most famous and intrinsically Organic house, Fallingwater in Pennsylvania. Constructed on a waterfall, the cantilever design complements the surrounding rock formations. The interior incorporates an original ledge rock, a small stream, and expansive windows that reconnect the interior to the exterior. During the same time that he was creating Fallingwater, Wright was also preoccupied with creating homes for middle-class America and began to design his Usonian houses. His affordable single-story homes showcased natural construction materials and utilized solar heating and natural cooling. Furniture was often built in and the house plan featured a free-flowing floor plan, specifically arranged to incorporate views of outdoor gardens. Many of Wright's Usonians also incorporated his geometrically based patterns of window design and ornament that is frequently called Wrightian.\n\nWright's designs had a threefold effect on American houses at mid-century. First, his Usonian houses helped spawn the Contemporary house. Second, Wright's body of work inspired a large diverse group of unique Organic houses by other architects (including Bruce Goff, and Wright's and Goff's students and followers), in effect making Organic a dramatic style of architecture. Finally, and of most lasting importance, Wright\u2014through his publications, his works, and his lectures\u2014instilled American architects with Organic as a way to approach a design. He railed against the International style with its orthogonal structures deliberately designed almost as a piece of white sculpture able to be placed anywhere in the world, and not, like the Organic, designed for a specific location, with local materials, adapted to the conditions of a site.\n\nOrganic modernism of the 1960s also appropriately reflected the ideals of the environmentalist movement. Adapting a house specifically to its site allowed full use of passive solar building design. While design was still meant to reflect the natural setting, eco-friendly elements were an ideal addition. This further development of Organic architecture was also characterized by more free-form designs. Some were still inspired by the landscape, others by various organic shapes found in leaves, shells, flowers, and fauna.\n\n**ORGANIC**\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nMill Run, Pennsylvania; 1936\u20131939. Fallingwater; Frank Lloyd Wright, architect.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nNorth Carolina; ca. 1990s. Mountainside House; James Fox, architect. This house, which includes a path through it to a valley view, incorporates Wrightian rectilinear ornament executed in many materials.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nNorth Carolina; ca. 2000s. Water Fall House; James Fox, architect. View from creek. Note pattern created in the concrete on bottom of house.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nNorth Carolina; ca. 2000s. Water Fall House; James Fox, architect. Entry facade.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nCarmel, California; 1948. Walker House; Frank Lloyd Wright, architect. The only house on the ocean side of Monterey Bay's scenic road, this house follows Wright's Organic design principles. It is stone and looks like part of the cliff it is built into. The low copper roof mirrors the ocean's color. This view shows the wall of windows facing the private courtyard.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nPleasantville, New York; 1948. Friedman House; Frank Lloyd Wright, architect. The round carport, resembling a large mushroom, is paired with rounded house and entry roof (in background).\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nPalm Springs, California (vicinity); ca. 1960s. This hexagonal house rotates so view and sunlight can be controlled.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nNorth Carolina; ca. 1980s. Steel House; James Fox, architect. Here curvilinear Organic elements are incorporated.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nPasadena, California; 1948. A rounded house with organic elements found in a typical post\u2013World War II neighborhood.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nDewey, Oklahoma; 1957. C. A. Comer House; Bruce Goff, architect. This house is an angular example.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nMountain Lake, Minnesota; 1970. Glen Harder House; Bruce Goff, architect. This combines huge piers growing out of the ground with an almost paper-thin roof above house and chimneys.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nMinneapolis, Minnesota, vicinity; 1969, John Howe, architect, who became Frank Lloyd Wright's chief draftsman. Curves meet at a point, providing an unusually broad angle of views.\n\nORGANIC:  OTHER 20th-CENTURY MODERN  \nLos Angeles, California; 1957. Silvertop: Reiner-Burchill Residence; John Lautner, architect. Sweeping curves with broad overhangs angle and overlap.\n\n## A-Frame (1950s\u20131970s)\n\nThe A-frame was a trendy house design that emerged in the 1950s and lasted through the 1970s. Resembling the letter A, the design is comprised of two sloping roofs creating an overall triangular shape. Architects learned that the standard form of a modern house could be visually transformed through simple experimentation with unusual roof shapes. The sloping roof of an A-frame house extends all the way down to the ground, in effect becoming a continuous wall. This bold, yet whimsical, design was seen as an appealing choice for vacation homes because it was suitable for a variety of terrains. Some featured a front deck, which provided a pleasant sitting and viewing area for both mountain and waterfront properties.\n\nThe notion of a weekend retreat was new to middle-income Americans. Travel tended to be lengthy and expensive prior to the proliferation of the automobile. But the combination of a new highway interstate system and the opening of shorelines for recreational use by the Bureau of Reclamation in the 1960s made travel to a vacation home more attractive. As a result of increased free time and disposable income, the middle class became the target market for the A-frame house. Advertised as affordable and aesthetically refreshing, the A-frame was presented as an exciting second-home option for those desiring a weekend retreat from city life. Stylistically, it also provided an exotic architectural alternative to traditional primary dwellings. Aside from enjoying this weekend retreat, the consumer was encouraged to take part in its construction as do-it-yourself guidebooks and house kits were made readily available by the 1960s.\n\nAs people became more conscious of efficient living, they realized the A-frame had several inherent design problems, including awkward unused space, lack of adequate natural light, and inefficient heating and cooling solutions. Variations of the A-frame attempted to solve some of these structural problems. A gambrel-roof A-frame created more usable space, an A-frame with wings provided a more open floor plan, and a double standard A-frame (two A-frame plans placed perpendicular to each other) allowed more natural light.\n\nDespite its practical flaws, by the mid-1960s the A-frame had become a cultural icon as the style was incorporated into commercial buildings, restaurants, churches, and even backyard playhouses. As the once-distinctive style became rather ubiquitous, its popularity declined, and the form fell out of favor by the 1970s.\n\n**A-FRAME**\n\nA-FRAME:  OTHER 20th-CENTURY MODERN  \nKey West, Florida (vicinity); ca. 1960s. Here the A-frame appears to serve as a base camp for boating in the Gulf.\n\nA-FRAME:  OTHER 20th-CENTURY MODERN  \nUnidentified House; ca. 1960s. Raised, perhaps anticipating high water, and with a deck to enjoy what one assumes is a water view.\n\nA-FRAME:  OTHER 20th-CENTURY MODERN  \nHamptons, New York; 1964. Higher ground and a relatively flat site allow placement on the ground.\n\n## New Formalism (1950s\u20131970s)\n\nIn the early 1950s the International style\u2014in the Miesian glass-and-steel aesthetic\u2014was the style of choice for most non-domestic buildings. But when Edward Durrell Stone received the commission to design an American embassy in New Delhi in 1954, International did not seem appropriate for such a symbolic building. Stone's solution was to create a new and more ceremonial modern style based on classical elements but utilizing building materials and technologies of the International style. The embassy celebrated monumental, rather than minimalist, forms. It featured an ornamental concrete screen grille, free-standing slender columns, and an interior courtyard. Other architects then began to use this as a model, while some preferred an even more classical version that omitted the screen.\n\nThe style is most commonly characterized by a symmetrical facade with columnar arched supports. New Formalism concentrated on updating, rather than re-creating, Classical forms, providing a visible and theoretic distinction from the Neoclassical style. An overall massiveness is achieved through a concrete-block-like structure set on an elevation and typically crowned with a flat slab roof. Wall surfaces, commonly made of stone, brick, and marble, are normally smooth and unadorned; columns generally take on a slender and attenuated form.\n\nWhile New Formalism was readily applied to civic and commercial buildings, it was far less common in residential architecture. Philip Johnson's 1964 Beck House and Stone's 1956 Graf House in Dallas, Texas, are rare high-style examples. The Beck House exterior visibly references Johnson's small Lake Pavilion built at Glass House, while the interior dining area recalls his Guest House. The Graf House seems inspired by the embassy at New Delhi. Most Formalistic houses are a bit more simplified.\n\n**NEW FORMALISM**\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nDallas, Texas; 1965. This row of slender, outward-slanting porch supports seems almost like seven ballet dancers executing an _arabesque en pointe_ \u2014making it easy to see how the name \"ballet classicism\" came to be used for Philip Johnson's work in this style.\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nHouston, Texas; ca. 1960s.\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nDallas, Texas; 1971. Harwood K. Smith, architect.\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nBeverly Hills, California; ca. 1960s. Side wings with decorative grille are combined with a classicized modern entry.\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nDallas, Texas; 1957. Oak Court; Edward Durell Stone, architect; 2007, Russell Buchanan, restoration architect. A screen grille (brise-soleil) of white terrazzo lends privacy to and shades the public, south-facing facade.\n\nNEW FORMALISM:  OTHER 20th-CENTURY MODERN  \nDallas, Texas; 1964. Beck House; Philip Johnson, architect. The formalistic arches are not just an exterior feature; they continue into interior spaces. Johnson had used this same attenuated arch profile for the pond pavilion at his Glass House.\n\n## Brutalism (1950s\u20131970s)\n\nBrutalism began as an aesthetic philosophy favoring the exposure of building materials, namely rough concrete and structural supports. The name is derived from the French term for raw concrete, _b\u00e9ton brut,_ illustrating its devotion to using materials in a direct and visible way. In opposition to the glass curtain wall, Brutalism favored bulky and angular designs with fewer visible glass surfaces. In a house, a simple juxtaposition of vertical and horizontal blocks might produce an exterior of slits and slabs\u2014broad expanses of solid material with narrow vertical slits of glass surface\u2014for a low ratio of void (glass windows) to solid (wall surface). This style has attracted criticism by those who perceive it as unappealing, but today it is regaining favor as a new generation comes to appreciate its bold design and monumentality.\n\nOne of the earliest examples of Brutalist architecture in the United States is the Yale Art and Architecture building designed by the school's dean, Paul Rudolph, in 1963. The deep recessions and sharp projections exemplified the grammar of this style; however, it was these same elements that were later frequently criticized.\n\nThis style is more often found in civic buildings than in residential architecture. Houses, however, can retain the main components, which include a bulky angular exterior, unornamented facades, recessed windows, often in vertical slits, exposed ducts, and exposed concrete (although domestic examples may instead use brick, stucco, or, rarely, wood).\n\n**BRUTALISM (SLITS & SLABS)**\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nDallas, Texas; ca. 1970. Narrow slits for windows combined with brick slabs for a neighborhood-friendly interpretation of Brutalism.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nPalm Springs, California; ca. 1956. Williams House; E. Stewart Williams, architect. Made of brick laid vertically to resemble one of the Brutalist materials of choice\u2014concrete with small ridges broken off to resemble rough corduroy.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nKenwood, Minnesota; 1970. Horty, Elving and Associates, architects. Note the stepped effect of the massing in this example; similar massing occurs in the Pueblo Revival style. Although clad in wood, narrow vertical windows and large areas of cladding relate it to the Brutalist style.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nPasadena, California; 1990. Condos on Grand Avenue combine a heavy Brutalist building with a Craftsman trellis to relate it to the historic side of the street.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nPasadena, California; 1988. Buff House; Conrad Buff III of Buff & Hensman & Assoc., architect. Vertical massing\u2014composed of rectilinear living modules\u2014is punctuated by narrow window slits and clad in gunnite stucco.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nChicago, Illinois; 1982. Kirsch House (\"The Bunker\"); Earl Jay Kirsch Architects. A high-style Brutalist residence.\n\nBRUTALISM (SLITS & SLABS):  OTHER 20th-CENTURY MODERN  \nVenice, California; 1999. Brutalist concrete slabs are turned sideways for an ocean view. Note vertical window slits on right.\n\n## Postmodern (late 1960s\u2013present)\n\nPostmodern houses appropriate and imitate elements of traditional styles, while incorporating these with new forms and materials. The result is both familiar and original. It is common to reference several different historical styles within one design, creating an interesting juxtaposition of period and regional elements.\n\nAdvocates of this style believed it was more important to absorb and reformulate traditional components than to continue constructing glass boxes void of heritage. Robert Venturi, a leading proponent of Postmodern architecture, believed a reference to traditional designs spawned thought-provoking interpretations. His 1966 book _Complexity and Contradiction in Architecture_ illustrated two thousand years of traditional designs and promoted new design that incorporated elements borrowed from buildings of the past. Responding to the modern idiom\"Less is more,\" Venturi rebutted with\"Less is a bore.\" Simplicity in design does not generate complex resolutions, which he believed to be the ultimate aim.\n\nWhile paying homage to the past, many houses within this stylistic movement feature playful designs and ironic combinations. It is as if a repository of architectural elements existed and the architect could select any combination of forms. The elements selected could additionally be exaggerated, manipulated, and even distorted. This idea is illustrated in Vanna Venturi's house, in which Venturi dissected the proverbial arch, removing the keystone that provides the primary structural function of this architectural element. With a Postmodern style, the result can be a house that suits the desire for historic reference, fresh design, or contradictory interpretations.\n\nA number of Postmodern architects, among them Robert Stern and Albert, Righter, & Tittmann, have now come to design New Traditional houses, often bringing to their designs the ability to slightly modify historic design elements and make them of today.\n\n**POSTMODERN**\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nCambridge, Massachusetts; 1991. \"Checkerboard House\" (Albert House); Albert, Righter, and Tittmann, architects. Asphalt shingles form a black-and-gray diamond pattern that can be found providing a maintenance alteration in many parts of Boston. With its combination of subtle classical details and a Main Street false front, one might need a second look driving down the street to realize this is a new house, not a much modified one.\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nPhiladelphia, Pennsylvania; 1962. Vanna Venturi House; Robert Venturi, architect. One of the early Postmodern houses designed by the author of _Complexity and Contradiction in Architecture_ shows how Postmodernism could play with traditional elements. Here the blind arch above the entrance has a void where the keystone would have been historically placed for support. Note the square windows, one of the earliest uses of this distinctive window shape.\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nCarlsbad, California; 1982 (remodeled ca. 2010). A Postmodern remodeling of an earlier stucco town-house row. Each is getting its own front that dominates the appearance.\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nEast Quogue, New York; 1979\u20131981. Lawson House; Robert A. M. Stern, architect. This iconic house combines a compact form with carefully placed historicist details. This entry facade has classical columns and an oversized eyebrow dormer window.\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nRosemary Beach, Florida; 2003. George Israel, architect. An interesting Postmodern house with shaped parapets that also incorporate Deconstructivist features. The stucco wall on the right is not supported from below, and faux bullet holes add an almost destructive flare.\n\nPOSTMODERN:  OTHER 20th-CENTURY MODERN  \nPasadena, California; ca. 1990s. Stair-stepped gables combined with Postmodern porch supports and entry surrounds.\n\n## Deconstructivism (1980s\u2013present)\n\nDeconstructivism is an architectural style that challenges the idea of oneness or unity in a structure and instead embraces the individuality of parts, producing designs that are fragmented. Deconstructivist philosophy regarded the parts of a whole more revealing than the whole itself, generating a multiplicity of interpretations. Applied to residential architecture, the form of a house is commonly divided into separate volumes that appear to have little coherent connection to one another. Each part is visually distinct from the others.\n\nDeconstructivist houses have a wide range of influences but no visible single source or predictability in design, other than unusual geometries. Exteriors can look unfinished, complicating usual boundaries between indoor and outdoor space. The standard 90\u02da-angle wall is no longer the default option. Windows can extend from walls to roof lines, creating an unexpected continuation of space.\n\nThe Frank Gehry house in Santa Monica illustrates this fragmentation of form and space, resulting in a visually unsettled design. In 1978 he first wrapped the original Dutch Colonial Revival house with unusual materials, obscuring any semblance of the original design. An uneven and disjointed corrugated-metal wall envelops the facade, while chain-link panels are used near roof lines and glass windows. The house appears unfinished, following Gehry's belief that a\"finished building has security, and it's predictable. I want to try something different. I like playing at the edge of disaster.\"\n\nThese unusual designs presented engineering and construction challenges until the spread of CAD (computer-aided design) and CAM (computer-aided manufacturing). Used together, these programs can effectively visualize unusual three-dimensional spaces and electronically tool and manufacture corresponding work pieces needed for the design. Computer programs like these have allowed architects to more efficiently create and control their chaotic designs.\n\n**DECONSTRUCTIVISM**\n\nDECONSTRUCTIVISM:  OTHER 20th-CENTURY MODERN  \nOrona, Minnesota (moved from Lake Minnetonka); 1983\u20131986. Winton Guest House; Frank Gehry, architect. A still life of forms created a small guesthouse.\n\nDECONSTRUCTIVISM:  OTHER 20th-CENTURY MODERN  \nVenice, California; 1981. Hopper House; Brian Murphy, architect. An exterior staircase covered in roof shingles is set into the inset on the right.\n\nDECONSTRUCTIVISM:  OTHER 20th-CENTURY MODERN  \nVenice, California; 1984. (Venice Beach) Norton House; Frank Gehry & Associates, architects. The owner had been a lifeguard in earlier years, and this provided a theme\u2014complete with the lifeguard tower and turquoise-blue pool tile popular in the 1950s.\n\nDECONSTRUCTIVISM:  OTHER 20th-CENTURY MODERN  \nLos Angeles, California; 1982. Petal House; Eric Owen Moss, architect. A Minimal Traditional house has a Deconstructivist remodeling and additions to house and detached garage. The stepped gable at right is a free-standing wall; this stepped motif is echoed by the wall cladding above the upstairs entry to the garage. The two-story gabled addition exposed the wood stud and plywood connection (visible behind the front gable of the house), and stepped windows reveal there is a roof deck to observe the adjacent freeway.\n\nDECONSTRUCTIVISM:  OTHER 20th-CENTURY MODERN  \nSanta Monica, California; 1903 (first renovated 1978, with later changes). Gehry House; Frank Gehry, architect. The shape of the gambrel roof of the original house is barely visible behind the extensive remodeling. To add more space, Gehry formed a secondary exterior skin around the old house using common materials\u2014plywood, corrugated steel, and chain-link fencing\u2014to create an experimental design that he has likened to a jazz riff.\n\n# MODERN HOUSES\n\n# 21st-Century Modern\n\nAs the turn-of-the-millennium drew near, a multitude of books catalogued and examined the modern homes of the previous century, offering a compelling synthesis of the variety and depth of the modern movement. During the 1980s and 1990s the sophistication of CAD (computer-aided design) and CAM (computer-aided manufacturing) programs developed rapidly\u2014giving architects an unprecedented ability to model complex designs, to display how they would look in three dimensions, and to easily make small changes. Thus a generation of young architects was now working on computers and inculcated with the rich possibilities offered by CAD. By 1990, designing new iterations of smooth-surfaced white stucco homes no longer presented a sufficient challenge, and many different approaches to modern house design began to be explored. As a result of this growing interest a new shelter magazine, _DWELL,_ launched in 2000, focused on modern home design and further spread this new wave of modernism.\n\nThis short appendix on 21st-Century Modern houses does not in any way attempt to be exhaustive. It cannot begin to cover the landmark modern houses that are being built today and are so well treated in monographs, series of books, and architectural periodicals. Instead the examples presented here are based almost solely on modern houses encountered while driving through typical American neighborhoods\u2014built as infill, as replacement houses, or, more recently, in neighborhood groupings limited to modern design. The homes here are loosely gathered into a few visual groupings that have been encountered frequently enough that they may have emerged as an approach to design. The majority slightly relax the International in some manner. Some add segmental vaults; others utilize several materials in combination for a decoupage effect rather than a smooth united surface. Some take advantage of the many new materials now available to clad a house, wrapping the house in new ways. Others are only a little askew\u2014slight deviations from the orthogonal, just enough to tease the eye and show off how easy this is to do today. Finally, there are the singular statements\u2014one-of-a-kind architect-designed landmarks, rarely encountered in the field (unless one has a special invitation) but comprising a great many of today's design awards and included in books on new houses.\n\nThe construction method most often observed in neighborhood examples is balloon- or platform-framed wood. In contrast, many landmark examples and experimental small houses utilize a broad range of structural possibilities.\n\n21ST-CENTURY MODERN: SEGMENTAL VAULTS\u2014A segmental vaulted roof is added to a house that otherwise would be thought of as either a late International or one of the other 21st-Century Modern house groupings. The vaulted element is often canted and contrasts significantly with the rectilinear lines of the rest of the house\u2014and thus tends to become the dominant visual element. This feature is also found today on other building types.\n\n**SEGMENTAL VAULTS**\n\nSEGMENTAL VAULTS:  21st-CENTURY MODERN  \nSeattle, Washington; 2007. A segmental vaulted roof, set at an angle, came to symbolize modernism on many building types during the last two decades.\n\nSEGMENTAL VAULTS:  21st-CENTURY MODERN  \nLos Angeles, California; 1969. The tilted arched roof is likely part of a relatively recent remodeling.\n\nSEGMENTAL VAULTS:  21st-CENTURY MODERN  \nChicago, Illinois; ca. 2005. This is part of a large grouping of similar town houses where the arched roofs are level rather than canted.\n\nSEGMENTAL VAULTS:  21st-CENTURY MODERN  \nEast Hampton, New York; 1995. Highway House; Rafael Vi\u00f1oly, architect. This house is a curved roof resting on a wall set at a right angle. Three poured-in-place concrete chimneys add a rhythmic element to the composition.\n\n21ST-CENTURY MODERN: D\u00c9COUPAGE\u2014It appears that the most common 21st-Century Modern house is an orthogonal box, or boxes, designed with two, three, or more wall-cladding textures and materials. These are chosen from the great variety available today\u2014metal siding, wood applied in many differing ways, concrete panels, HardieBoard, concrete block, and occasionally brick\u2014all in combination with glass. The pattern created by these materials becomes a primary element of the home's design. Usually each material is in a slightly different plane, adding a three-dimensional element to even modest examples. High-style examples can have elaborate modeling of the house, approaching the look of an orthogonal sculpture.\n\n**D\u00c9COUPAGE**\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nSeattle, Washington; 2006. Note the use of smooth horizontal wood; concrete textured in a grid pattern; and vertical metal siding\u2014each in a different wall plane. Both entrance facade and the side (view) facade  are shown.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nLos Angeles, California; 2006. Horizontal boards, vertical metal and stucco (it appears)\u2014in different planes, but less three-dimensional than in the previous example.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nSeattle, Washington; 2010.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nSeattle, Washington; 2006. Note the broad roof overhang and walled garden on the side (view) facade of the first example of this style.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nBoston, Massachusetts; ca. 2000. Big Dig House. Multiple wall materials (salvaged when Boston's main interstate was buried in a tunnel) are used to create a D\u00e9coupage effect.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nDallas, Texas; 2012. A row of townhouses set perpendicular to the street.\n\nD\u00c9COUPAGE:  21st-CENTURY MODERN  \nDallas, Texas; 2011. Cheatham House, Urban Reserve. There is a plan to eventually hang a shade screen from the boxed-in side of the roof. In the meantime, one can enjoy this sculptural form. The second floor forms a \"T\" at the rear and has a pool and large entertainment area overlooking a pond and the setting sun.\n\n21ST-CENTURY MODERN: UNIFYING MATERIAL\u2014White stucco was in and of itself considered an important element of the International house. It was thought appropriate for any location, thus promoting International architectural unity rather than regional loyalties. While some mid-century examples used other materials, there was a 1970s and 1980s return to the white stucco roots that give center stage to the house form alone. By about 1990, architects had tired of exclusively using white stucco and instead began to design with the wide variety of other wall claddings that had been introduced. These included various types of enhanced glass, concrete panels, wood application techniques, polycarbonates, and many textures and colors of metals\u2014including easily visible informal corrugated metal. Many of these could be applied as rain screens, a new method of applying sidings that increased air circulation and insulation (discussed on this page).\n\nAny of these could also provide a theme for an entire house, and architects have adopted and used these as an integral part of many designs. Sometimes the goal of using a material is to blend in, but equally likely the cladding is used almost as gift wrapping, enhancing and changing a boxy house with the decorative surface.\n\n**UNIFYING MATERIAL**\n\nUNIFYING MATERIAL:  21st-CENTURY MODERN  \nFayetteville, Arizona; ca. 2005. \"L-stack House\"; Marlon Blackwell, architect. Responding to an irregular site, Blackwell utilized two rectangular boxlike shapes placed at right angles connecting them with a glassed-in stairway. The lower one bridges a small stream and forms a roof garden (maintaining the site's natural drainage). The upper box shelters an outdoor living space in the rear. The exterior cladding is a Brazilian redwood rainscreen with EPDM behind. The house is insulated with soy-based foam.\n\nUNIFYING MATERIAL:  21st-CENTURY MODERN  \nChicago, Illinois (Hyde Park); 1996. Davis House; John Vinci, architect. Note entry deep in side of house and use of overall brick cladding. The traditional brick helps relate to its historic surroundings.\n\nUNIFYING MATERIAL:  21st-CENTURY MODERN  \nLong Island, New York; ca. 2008. House at Sagaponac. Flat board siding, no roofline cornice, and almost no windows on the front facade.\n\nUNIFYING MATERIAL:  21st-CENTURY MODERN  \nLos Angeles, California; 2005. Finkelstein Residence; Antony Unruh of Unruh Boyer, architect. The ways of building on a slope illustrated earlier are still being utilized today. The auto entry is at street level and is connected to the house inserted lower on the hill. The view facade faces away from the street. The house is unified by a handsome cladding of galvanized aluminum with a zinc coating called Corrugated Galvalume.\n\nUNIFYING MATERIAL:  21st-CENTURY MODERN  \nVenice, California (Venice Beach); 1988. Snipper House; Miguel Angelo Flores & Associates, architects. The stair-stepped podium visible at left is lavender. The house has a glass brick and stucco cylinder for its core.\n\n21ST-CENTURY MODERN: SLIGHTLY ASKEW\u2014While Deconstructivist houses are still being built today, one of the side effects of this movement has been that more simple rectilinear forms have been slightly loosened up. CAD programs make this far easier to do. This can be quite subtle\u2014one tilted or canted wall, or even a prominent overhang with a slight angle. But the eye is challenged and the point is made\u2014rich design opportunities are possible today because the house does not have to be completely rectilinear.\n\n**SLIGHTLY ASKEW**\n\nSLIGHTLY ASKEW:  21st-CENTURY MODERN  \nPasadena, California; 1996. Dumbacher House; Hagy Belzberg, architect. Slight angles are introduced in the shape and upward tilt of the broad overhang above the second-floor outdoor living area.\n\nSLIGHTLY ASKEW:  21st-CENTURY MODERN  \nChicago, Illinois; 2005. Here there are three distinct sections of the house in three planes\u2014the section on the left uses a diagonal parapet to create the appearance of a slanted roof; the central section is orthogonal and filled with windows; and rear one appears to have an outward slanting wall.\n\nSLIGHTLY ASKEW:  21st-CENTURY MODERN  \nBeverly Hills, California; 1932 (rebuilt ca. 2012). Three materials on varied planes are accented by the slanted roof marking the entry\u2014a combination of D\u00e9coupage and Slightly Askew.\n\nSLIGHTLY ASKEW:  21st-CENTURY MODERN  \nLos Angeles, California (Silver Lake); 1955 (later remodeling). Note the subtle wall cants and meticulous changes in material.\n\nSLIGHTLY ASKEW:  21st-CENTURY MODERN  \nLos Angeles, California; 1948 (remodeled). The colored house walls (blue and brown), the textured wall finish, and the slightly Pueblo-esque elements mark this as a \"desert\" modern house. The cactus garden is added to further the theme.\n\n**SINGULAR STATEMENTS**\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nLos Angeles, California; 1988. Robert Bridges, architect. The street view gives little hint of what lies behind.\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nLos Angeles, California; 1988. Robert Bridges, architect. Looking uphill one can see smooth wood sheathing meet huge concrete pylons. This two-sided house is located on an almost unbuildable site (see Los Angeles, California; 1988).\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nLa Jolla, California; 1990. The street facade gives not a hint of the facade that faces toward downtown La Jolla and the ocean view shown in the next images.\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nLa Jolla, California; 1990. Ocean view facade of the house in the previous image.\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nMalibu, California; 2011. The 747 Wing House; David Hertz Architects. A new, and unique, example of how an unusual roof form can completely dominate a house\u2014such houses were sometimes referred to as \"roof architecture\" in the 1950s and 1960s.\n\nSINGULAR STATEMENTS:  21st-CENTURY MODERN  \nDallas, Texas; 1994. Rose House; Antoine Predock, architect. Morning sunlight shines through translucent doors in this two-sided house. The street side appears to grow out of the limestone bed of Turtle Creek, with stepped ledges planted with native plants. The private side opens up and steps up to the heavily shaded creek complete with a bridge for bird watching.\n\n21ST-CENTURY MODERN: A NEIGHBORHOOD GROUPING\u2014Throughout the 20th century, modern houses have generally been placed in neighborhoods of traditional houses, where the tendency is to think of them simply as a \"modern\" house in contrast to adjacent homes of historical styles. Modern homes have rarely been built in compelling modern streetscapes\u2014in large enough concentration to easily display many different examples together. Thus it was difficult to appreciate either the many different kinds of modern houses or the subtle differences in related houses.\n\nUrban Reserve, a fourteen-acre project in Dallas, Texas, presents a timely example not only of how modern houses can be grouped together, but also of a green approach to site plan, landscape, and long-term sustainability\u2014a rarity in the world where burying storm water underground in culverts is considered normal and large development sites are routinely cut and filled, scraped clean, wiping out native vegetation and obliterating wildlife habitat.\n\nDeveloper Diane Cheatham assembled the site (located along a hike and bike trail and close to a DART stop, yet for half a century a landfill and illegal dumping ground) and hired Robert Meckfessel of DSGN as the master planner and Kevin Sloan as the landscape architect. The goal was to create a neighborhood of fifty modern houses in a green and sustainable manner. All houses would be required to have some level of LEED certification.\n\nFirst the team devised a pioneering approach to storm water. Runoff from about thirty-five offsite acres already drained here pours onto the site through a large culvert. The plan collected this into a pond, now used permanently to water the landscape of the development. A second pond collects any overflow as well as the runoff from the development itself, much of this flowing through rain gardens on its way. In all, fifty acres of storm water is naturally filtered on its way to White Rock Creek.\n\nSecond, there is an innovative street pattern. The street is but twenty-two feet wide, the minimum to handle a fire truck, rather than the thirty-foot city standard. A gently rounded curb line on the dry side eliminates curb cuts. The wet side has head-in parking spaces for guests, interspersed with long rectangular rain gardens filled with horsetail that not only filter water but also hide the cars. Native desert willows are planted on the dry side of the street and bald and pond cypress on the wet side.\n\nThird, there is an inventive approach to lot size, house siting, and design criteria. The dry side of the street consists of forty-foot lots\u2014twenty feet for construction and twenty for a side garden in the Charleston pattern (see illustration). The wet side of the street has larger lots set behind the rain gardens and pond. But how to create design criteria to regulate \"modern\" houses without tying architect hands? Eventually, it was decided simply to ban all historical styles, including Mid-century Modern (Contemporary in this guide) and create an architecture-review committee with a \"we'll know it when we see it\" definition of what constitutes a modern house. The result is that the houses built from 2007 to the present represent a broad range of the 21st-century modern houses groupings in this appendix.\n\n**GROUPINGS**\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; ca. 2008. Wise House; Jim Wiley, architect.\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; 2010. Cube House; Russell Buchanan, architect. Built of integrally colored stucco, the struts in the awning highlight the slightly askew element.\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; 2011. \"Thyme to Be\"; Robert Meckfessel, design architect. This LEEDS Platinum house has an eco shell of western red cedar shakes on a plastic backing\u2014a cladding that saves labor by properly aligning the shakes, and creates a rain screen. Client and architect visited Sea Ranch and have captured some of its spirit in natural material and gently sloped shed roof. Solar panels on the roof and geothermal heat and cooling helped gain Platinum status.\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; ca. 2010. This is a D\u00e9coupage house. Note the wide boxed-in roof overhang used on the southwest corner.\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; ca. 2010. This has a rust-tone metal cladding and a two-story plant-covered screening wall.\n\nGROUPINGS:  21st-CENTURY MODERN  \nDallas, Texas; ca. 2008. Chris Craft House; Vincent Snyder, architect. A singular statement clad in slate located on the wet side of the street.\n\n# Styled Houses Since 1935\n\n* * *\n\n**Mansard**\n\n**Styled Ranch**\n\n**Millennium Mansion**\n\n**New Traditional**\n\n**American Vernacular**\n\nStyled houses with traditional de-signs that mimic the past have alternated with modern designs throughout the 20th century. World War II swiftly ended the Eclectic movement of the 1920s and 1930s as postwar tastes enthusiastically embraced modern styles throughout the 1950s and 1960s. Despite this overwhelming preference for modern styles during these decades, the very early beginnings of some late-20th-century-styled homes (such as Styled Ranch and American Vernacular) came as early as 1935. However, most of the late 20th-century styles did not emerge until the 1970s and 1980s, when a renewed taste for period styles based on earlier architectural traditions emerged, nurtured by the historic nostalgia created by the U.S. Bicentennial and a burgeoning nationwide preservation movement. At first there was little attempt to closely copy European or Colonial prototypes. Instead, distinctive historical details (for example, Tudor half-timbering, Georgian doorways, and Queen Anne spindlework porches) were applied to Ranch-house forms, producing the Styled Ranch. Those desiring a more formal look turned to the Mansard style.\n\nBut as the new millennium approached, home building boomed, and simply grafting historic details onto mid-century forms no longer sufficed. Three new kinds of styled houses emerged\u2014Millennium Mansions, New Traditional houses, and American Vernacular\u2014the broadest range of home styles ever before constructed at the same time. This was made possible by the Internet and its exploding content, giving immediate access to view historic precedents and to order reproduction details.\n\nMillennium Mansions were an entirely new form with an even more complex version of a roof form seen only once before, on exuberant Queen Anne and Romanesque houses of the late 19th century. The roofs of Millennium Mansions practically explode above the house. It is almost as if the creative multi-part roof forms seen in both the late 19th and late 20th centuries presaged sensational fireworks displays to celebrate a new century mark. The exteriors of Millennium Mansions almost always include details borrowed from one or more historic styles and have a new signature feature\u2014prominent (and largely non-historic) one-and-one-half- or two-story entries.\n\nWhile some builders were mass-producing Millennium Mansions, New Traditional houses began gaining favor elsewhere. Architect and educator Robert Stern led the way in making the design of New Traditional houses acceptable within the architecture profession.\n\nWith simple roof forms and unified sets of stylistic detailing, these houses now display an extraordinary range that includes almost all home styles found during the preceding three centuries of American home building. Designing and building New Traditional houses presented an enormous challenge when the public first became interested in the late 1970s. The architects, craftsmen, and builders who had been responsible for the design and construction of America's rich heritage of Eclectic-era styles were no longer active, if indeed they were still alive. The monographs of their work\u2014and the great books with photographic views of European architecture they had used for ideas and guidance\u2014were no longer in print. Architecture schools no longer taught historic architecture (even today, less than a handful of universities teach classical design). Thus, almost every New Traditional home one sees is designed by someone who decided to train himself or herself in historic architectural styles. The best of these architects, such as Stern, understand historic precedent well enough to reinterpret or generate new details, keeping the purpose of the original form but giving it a new integrity for today.\n\nSome present-day residential developments include custom-written pattern books that designate architectural styles and specify architectural details. Other developments utilize the approach of regulating house form, generally through form-based zoning. This method relies on creating a streetscape unified by similar size and shape, rather than governing a specific style. It is also possible to combine these two approaches for a high degree of regulation. Architect and planner Andr\u00e9s Duany of Duany Plater-Zyberk & Co. (DPZ) has pioneered both concepts, designing style pattern books for some traditional neighborhood developments and introducing form-based zoning to create simple harmonious new development in cities and towns.\n\nA third \"style\"\u2014American Vernacular (sometimes called Regionalism)\u2014is favored by both builders and architects. Unadorned folk houses make up a large part of the nation's built environment, and American Vernacular houses emulate their simple forms. Because these new homes are carefully designed, they leave the realm of folk houses. Generally built of natural materials, American Vernacular houses emphasize the basic shape of any of the folk-house families, or traditions, and avoid style-associated decorative elements. The result is an understated yet familiar appearance created through careful choice of material and window placement, and the avoidance of high-style embellishment.\n\nTogether these three groups of houses, all with some historic precedent, make up the great bulk of homes built during the millennial housing boom that began about 1990 and was almost ended by the financial crisis of 2008.\n\n# STYLED HOUSES SINCE 1935\n\n# Mansard\n\n# ca. 1940\u20131985\n\n## Identifying Features\n\nMansard roof (dual-pitched hipped roof) generally with dormer windows on steep lower slope; segmental arch over entrance, windows, or dormers (round arch substituted in many later examples); one story with a second story often contained under mansard roof; wall surface normally brick veneer.\n\n## Principal Subtypes\n\nTwo principal subtypes can be distinguished:\n\nSYMMETRICAL\u2014In this subtype the mansard roof dominates a symmetrical facade with centered entry. There are normally three or five ranks of windows; occasionally a side wing is added. In elaborate examples a small central wing may project forward from the front facade or two small wings project forward at either side.\n\n**SYMMETRICAL**\n\nSYMMETRICAL:  MANSARD  \nAtlanta, Georgia; ca. 1960s. The entry and the windows on the central house block have segmental arches.\n\nSYMMETRICAL:  MANSARD  \nHighland Park, Illinois; 1937. The central entry pavilion gives this a symmetrical feeling that is not completely correct. Each side wing has a different window and dormer pattern. Note the outward flare of the roof and the faux corner quoins.\n\nSYMMETRICAL:  MANSARD  \nLos Angeles, California; 1958. Reynolds House; John Woolf, designer. This tall, narrow entry design was widely imitated, leading to a new approach to Mansard as seen on the examples in Houston and Dallas, and perhaps even to the two-story arched entries found on Millennium Mansions.\n\nSYMMETRICAL:  MANSARD  \nBeverly Hills, California; 1949. This early example neither has dormers nor contains a second story under the roof.\n\nSYMMETRICAL:  MANSARD  \nHouston, Texas; ca. 1980s. For a brief period two-story front porte cocheres were popular, imparting a bit of a country club look.\n\nASYMMETRICAL\u2014This is the most common subtype; its asymmetry is usually based on an L-shaped front-facade plan. The entrance is typically located on the recessed portion of the facade, with a main living space thrust toward the street. Elaborate examples may have a much more complex asymmetry.\n\n**ASYMMETRICAL**\n\nASYMMETRICAL:  MANSARD  \nPrairie Village, Kansas; 1968. Note through-the-cornice windows and the recess of the paired front doors.\n\nASYMMETRICAL:  MANSARD  \nLake Forest, Illinois; 1928 (and later). This is a high-style example with roof dormers on the left wing and wall dormers on the right wing. It has almost certainly been modified from its recorded construction date of 1928 and converted into a ca. 1960 Mansard house.\n\nASYMMETRICAL:  MANSARD  \nOak Park, Illinois; 1940. Three similar, but not matching, Mansards, on the site of a former Congregational church.\n\nASYMMETRICAL:  MANSARD  \nPrairie Village, Kansas; 1958. A Tri-Level Split with the Mansard roof only on the two-level section.\n\nASYMMETRICAL:  MANSARD  \nDallas, Texas; 1970. Dilbeck House; Charles S. Dilbeck, architect. The use of a boxy curb at the bottom of the Mansard roof was unusual.\n\nASYMMETRICAL:  MANSARD  \nSan Marino, California; 1939. This appears to have had a major stylish update adding dramatic surrounds to the front door and windows.\n\nASYMMETRICAL:  MANSARD  \nDallas, Texas; 1977. In this late example, John Woolf\u2013influenced door and windows all are round-arched and extend through the cornice.\n\n## Variants and Details\n\nEntries, which are generally arched, grew in prominence and height from the style's inception circa 1940 until its demise circa 1985. Most early examples have an entry with a simple segmental arch above, often recessed into the main body of the house for shelter. A tall, slender, exaggerated entrance (sometimes called a Pullman door) was used by Los Angeles architectural designer John Elgin Woolf beginning in 1946, becoming widespread by 1970. These extended up through the cornice to form a one-and-one-half-story feature, with a one-story door below and tall arched element above. An unelaborated but deep entry surround provided shelter for the door.\n\nPaired front doors are the most common, usually wood panels in curvilinear design that occasionally incorporated a circular or elliptical panel. Stone quoins, or faux quoins shaped from brick, are often used at the corners of the house, or side wings.\n\nWindow features may extend to ground level or have an ornamental wood panel beneath the glass window. Roof dormers are almost always found in early versions and provide light to two-story sections of the house. After about 1975 roof dormers were sometimes replaced by very tall windows that extend upward through the cornice. Roof cladding was typically composition, wood shingles, or standing-seam metal.\n\nMansard houses of the 20th century typically are one story in height (not including whatever floor space might be behind the tall roof) and lack the molded cornices and decorative eave-line brackets frequently found on 19th-century Second Empire houses.\n\nMANSARD\n\n **TYPICAL ELABORATIONS**\n\n## Comments\n\nThe Mansard style was the primary formal and \"historic\" house style built during an era dominated by more informal Ranch-style homes and Contemporary designs. Many of the subdivisions developed from the 1940s through the 1970s had deed restrictions and\/or zoning ordinances that dictated one-story structures or low roof heights. The Mansard style could follow many of these one-story, low-height regulations and still provide a two-story home, as up to a full story of living space was possible under the massive roof. Mansard roofs had historically provided an opportunity to squeeze in an extra story of living space; it is reported that original Second Empire houses in France allowed an extra story that was not taxed because of wording in the tax code.\n\nArchitect John Elgin Woolf (1908\u20131980) helped popularize the style with his Los Angeles\u2013area homes, among them his 1958 Reynolds House in Hancock Park. His designs were embraced by those who yearned for traditional formality\u2014and even drama\u2014and rejected the more prevalent informality of the era. Shunned by the architectural press, Woolf's homes were built for a Who's Who of movie stars and were popularized by articles in shelter magazines.\n\nThe Mansard could be a relatively inexpensive style to build, as only one story of masonry veneer was required, while the remainder of the wall surface was clad in roofing material. As such, it proved very popular for apartment complexes and other small-scale commercial construction.\n\nThe style fell from favor in the 1980s, but its exceptionally tall entry door topped by an arch, popularized by Woolf, was likely the genesis of the almost ubiquitous one-and-one-half- to two-story, round-arched entries found on Millennium Mansions until the mid-2000s.\n\n# STYLED HOUSES SINCE 1935\n\n# Styled Ranch\n\n# 1935\u20131985\n\nStandard Ranch houses frequently incorporate one or more common historic elements\u2014shutters, wrought iron, paneled doors\u2014because lending institutions felt comfortable with the traditional feeling these details impart. What sets a Styled Ranch apart is the presence of a more complete and unified set of stylistic details that spell out a distinct style, such as Spanish. It is a matter of judgment where the Ranch ends and the Styled Ranch begins. If one still wonders, after careful scrutiny, whether one is looking at a Styled Ranch, the answer is probably not.\n\nStyled Ranch houses were built intermittently during the Ranch-house era (1935 to ca. 1975), but they became increasingly common during the 1970s and dominated new one-story homes in the 1980s. Five main styles are common: Spanish, French, Tudor, Colonial Revival, and Neoclassical. All of these styles have features similar to those found on the Eclectic versions of each style but were adapted to a wide, low, one-story form. There were also more exotic Character styles. Each of these could be found throughout the country, but some prevailed in a particular geographic location or time period.\n\nAlthough Styled Ranches can be found from 1935 onward, before 1970 they were only common where a specific local building tradition or developer promoted their use\u2014for example, in a Spanish Ranch subdivision in Arizona or Colonial Revival Ranches in Massachusetts. By 1970, however, buyers had begun to tire of simple Ranch homes and sought a different look. The 1976 U.S. Bicentennial celebration and a growing nationwide preservation movement were both generating nostalgia for older styles. The easiest way to achieve a different look was to continue building the familiar one-story Ranch (the low height often required by deed restrictions) while adding strong stylistic details to the exterior.\n\nStyled Ranch designs were prevalent in house pattern books of the 1970s and 1980s. The _Book of Successful Home Plans_ by Richard Pollman (Structures Publishing Company, 1976), for example, had multiple plans for five subtypes (Tudor, Spanish, Colonial Revival, Neoclassical, and French). Other home plan books were devoted to a single house style (such as _Colonial Home Plans,_ a Bantam\/Hudson Plan Book first published in 1977, and _Tudor Houses,_ by Home Planner, Inc., in 1989). These pattern books also included the beginnings of the New Traditional House, along with designs for regular Ranch houses.\n\nIn general, Styled Ranch houses lack the broad overhanging eaves found on many Ranch houses and are more likely than the Ranch to have a dominant entry and traditional multi-paned windows, and omit short windows, corner windows, and often picture windows. Each Styled Ranch subtype has its own distinctive features, as seen on the following pages. One-and-one-half-story forms had roofs with a higher pitch than the earlier Ranch houses, and the dormers that provided light for the upper half-story were perfectly adapted to Ranch houses styled in the Tudor, Colonial Revival, Neoclassical, and French styles. The extra half-story helped create larger homes (to house master bedroom suites, large bathrooms, and closets) all within the same lot size. As with the Ranch style, split-level variations occur in all subtypes.\n\nRising land prices, desire for larger homes, and changing tastes gradually led to a new era of two-story houses. The year 1988 was the first time since the beginning of the Federal Housing Administration that a higher percentage of two-story houses were completed. As the one-story Ranch fell from favor during the 1990s, it was replaced by New Traditional houses and Millennium Mansions.\n\nSPANISH RANCH\u2014The earliest Styled Ranch subtype was Spanish. Cliff May, the prolific house designer and builder credited with popularizing the Ranch style, was a sixth-generation Californian. His 1946 book _Sunset Western Ranch Houses_ identified the pitched-roof Spanish Colonial homes of southern California as the genesis of the Ranch house and included designs for them. Built throughout the entire Ranch era and into the 1980s, Spanish Ranches are most common in California and the Southwest.\n\nSpanish Ranches are generally clad in stucco (or buff-colored brick) and topped with a tiled roof, most often red and of the types described on this page. One or more round or parabolic arches are often present, usually at the front entry or porch, principal windows, or courtyard entry. Other decorative elements may include exposed roof rafters and beams, wood or metal window grilles and balconettes, and inward-slanting chimneys or wing walls.\n\n**SPANISH RANCH**\n\nSPANISH RANCH:  STYLED RANCH  \nSan Diego, California; 1945. Cliff May, architect. This house has strong details, including the boxed wood window grille, deep diagonal-patterned molding on the front door, inward-slanting chimney, and tile roof with a handmade look.\n\nSPANISH RANCH:  STYLED RANCH  \nMiami, Florida; 1970.\n\nSPANISH RANCH:  STYLED RANCH  \nMontgomery, Alabama; ca. 1950s. Arched windows and openings to entry courtyard are coupled with \"portales.\"\n\nSPANISH RANCH:  STYLED RANCH  \nClaremont, California; ca. 1935. An early example that includes a picturesque water well in its imagery (in front of far-right window).\n\nSPANISH RANCH:  STYLED RANCH  \nDallas, Texas; 1947. Cox House. This includes a small two-story wing at far left.\n\nSPANISH RANCH:  STYLED RANCH  \nCarlsbad, California; 1960. A Tri-Level Split with the entrance in a stair tower with two stepped windows.\n\nCOLONIAL REVIVAL RANCH\u2014This subtype evolved from the broadly popular Minimal Traditional Cape Cod homes of the 1940s. Architect Royal Barry Wills helped keep alive interest in many forms of Colonial Revival with his _Houses for Good Living_ (1946) and _More Houses for Good Living_ (1968), which include examples of the one-story Ranch. Colonial Revival Ranch houses are often symmetrical or include a symmetrical central house block; the Ranch form may be achieved by adding clear house elements, often with slightly varied roof heights (as seen in examples in Baltimore, Boston, and Prairie Village). The main house block is most often side-gabled or hipped. Colonial Revival Ranches are commonly clad in one material, usually red brick or wood siding, while attached wings may be of a secondary material. The front door is typically prominent and may be centered and\/or enhanced with a Colonial Revival surround or entry porch. Other Colonial Revival detailing (such as dentils) may be present, as may dormers. The Colonial Revival Ranch was found throughout the Styled Ranch era and is more common along the Atlantic seaboard from Massachusetts south to Georgia.\n\n**COLONIAL REVIVAL RANCH**\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nSan Marino, California; 1948 and 1953. The recessed door has a single line of lights above and is crowned by a small segmental pediment.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nMission Hills, Kansas; 1955. Each of the pair of front-facing gables has a small bull's-eye window enhanced with large floral swags; oversized dentils line the cornice.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nBaltimore, Maryland; 1957. This house has four distinct sections: a three-bay main block with a six-panel door, two-bay side wing, one-car garage, and wood-clad section connecting the house with the garage.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nPhoenix, Arizona; 1936.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nBoston, Massachusetts; 1962. Four side-gabled shapes of graduated sizes combine to create a very broad Ranch-house form.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nColumbus, Georgia; 1953. A symmetrical five-bay center block has an entry with sidelights, a rounded entry porch, and oversized multi-pane double-hung windows.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nKansas City, Missouri; 1961. This Tri-Level Split has a second-story overhang clad in wood shingles that were laid close together and in straight lines.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nBoston, Massachusetts; 1955. The side-facing garages in this wood-clad house are in a side-down section.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nPrairie Village, Kansas; 1956.\n\nCOLONIAL REVIVAL RANCH:  STYLED RANCH  \nBoston, Massachusetts; 1965. Bi-Level Splits like this one are well adapted to centered entries and symmetrical (or almost symmetrical) facades.\n\nNEOCLASSICAL RANCH\u2014A less common subtype found more often in southern states, the Neoclassical Ranch style is characterized by a one-story porch supported by Classical columns. The porch may be present only at the entry (entry porch) or extend the full width of the house. Generally there is a symmetrical main block. Often clad in brick, it is sometimes also accompanied by sections of wood frame. Traditional multi-pane windows are typically used, and roof dormers may be present.\n\n**NEOCLASSICAL RANCH**\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nDallas, Texas; ca 1960s. This achieves the look of a five-part plan although it is actually a broad, single house block. The symmetrical gables have broken pediments over the windows.\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nKansas City, Missouri; 1956.\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nColumbus, Georgia; 1948. A triangular-pedimented portico and double-hung windows dominate this front facade.\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nSan Marino, California; 1947. Note the full-width porch and prominent chimney. The picture window is not common on Neoclassical Ranch houses.\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nKansas City, Missouri; 1959. The red brick main house block has a full-width porch and is flanked by wood-clad side wings.\n\nNEOCLASSICAL RANCH:  STYLED RANCH  \nAlexandria, Virginia; 1963. A Bi-Level Split with a centered pair of front doors and a full-height, full-width porch. The columns are quite slender.\n\nFRENCH RANCH\u2014Called \"French Provincial\" and \"formal\" in pattern books, French Ranches were most popular during the 1970s and 1980s. Generally, at least one portion of the house (commonly the central block) is topped by a high-pitched hipped roof. One or more segmental arches are usually present (on doors, windows, or dormers). Windows are usually tall and narrow, sometimes full height. The front entry is typically prominent and features a paneled front door (both paired doors and curvilinear panels are common). Tall narrow shutters generally enhance windows and\/or entry. Walls are frequently clad with brick veneer.\n\nFrench Ranches may be either symmetrical or asymmetrical. Symmetrical examples may have small side wings, either extending forward or set back from the main facade. (Some early survey books on Ranch houses assign the category French Provincial to most cross-hipped Ranch houses. In this guide, we look for a more complete set of style characteristics to categorize hipped and cross-hipped Ranch houses as French Ranch.)\n\n**FRENCH RANCH**\n\nFRENCH RANCH:  STYLED RANCH  \nSan Marino, California; 1947. The front door has matching sidelights.\n\nFRENCH RANCH:  STYLED RANCH  \nKansas City, Missouri; 1957.\n\nFRENCH RANCH:  STYLED RANCH  \nSt. Paul, Minnesota; 1972. This five-part composition has a symmetrical central block and hipped-roof wings. Note the tall vertical windows; these are often found on French Ranches.\n\nFRENCH RANCH:  STYLED RANCH  \nKansas City, Missouri; ca. 1960s. It is rare for a Tri-Level Split to have a single large hipped roof sheltering the entire house.\n\nTUDOR RANCH\u2014The Tudor Ranch style relied on half-timbering as a stylistic element and it is almost always present, generally attached to the exterior. _Tudor Houses_ explained that half-timbering \"evolved into a decorative device applied after the framing had been covered by a continuous wraparound surface of brick, wood, stucco or stone.\" In other words, the wood \"timbers\" were simply attached to the finished surface, rather than slightly recessed into the exterior, as in the Eclectic era. While roughly one-third of Eclectic-era Tudors had half-timbering, it appears on almost all Tudor Ranches.\n\nA second stylistic element often present are casement windows, sometimes with diamond-shaped panes. Occasionally, decorative garage doors are used. The roof form is typically gabled or cross-gabled.\n\nThe pattern books of the day referred to these as \"Tudor adaptations\" and \"English styling.\" Like the French Ranches, these were most popular during the 1970s and 1980s.\n\n**TUDOR RANCH**\n\nTUDOR RANCH:  STYLED RANCH  \nPasadena, California; 1940.\n\nTUDOR RANCH:  STYLED RANCH  \nDallas, Texas; ca. 1968. This broad Styled Ranch has a group of three leaded glass windows placed to the left of the nested gables with entry.\n\nTUDOR RANCH:  STYLED RANCH  \nDallas, Texas; 1978. Landsberger House. The entry is under the nested gables. At far right a small through-the-cornice dormer has leaded glass windows below.\n\nTUDOR RANCH:  STYLED RANCH  \nSeattle, Washington; 1979. A less common kind of Split\u2014the garage is a half-story down, but a third level above it\u2014that would make this house a Tri-Level Split\u2014is not obvious.\n\nOTHER RANCH\u2014There were many decorative effects the builders of Ranch homes occasionally used to make a home recognizably different. One of the earliest was a \"Hansel and Gretel\" storybook look, resembling the Exotic Revival's Swiss Chalet, with an added vergeboard of deep scallops that swooped down across the front facade. Often accompanied by diamond-shaped windowpanes and copious window boxes, the Storybook Ranch sold well in the early 1950s but soon dropped from favor.\n\nAnother early set of less common detailing was Oriental. Some were Japanese, with a pagoda form to the roof. Other Oriental-themed details could be added to enhance the effect\u2014moon-gates, cast-iron hardware, Japanese-style gardens, etc. A Polynesian theme and details were also used.\n\nOne of the last of these to appear was Victorian. Growing out of the infatuation with old houses that emerged from the growing neighborhood preservation movement in the 1970s, it is easily identified by the broad low Ranch form with a spindlework porch and sometimes even a turret.\n\n**OTHERSTYLED RANCH**\n\nOTHER STYLED RANCH:  STYLED RANCH  \nPhoenix, Arizona; 1960. There was a brief fad for storybook details, such as the entry roof and scalloped windows on this minimal example.\n\nOTHER STYLED RANCH:  STYLED RANCH  \nPasadena, California; 1960. This large, cohesive Storybook Ranch shows how successful the fairy tale motif could be. Note the diamond-pane windows, and the pigeonnier in the peak of the asymmetrical gabled roof.\n\nOTHER STYLED RANCH:  STYLED RANCH  \nChicago, Illinois; 1959. This house has dramatic upturned \"pagoda\" corners on the roof and calligraphic details on the garage.\n\nOTHER STYLED RANCH:  STYLED RANCH  \nPrairie Village, Kansas; 1957. More subtle Oriental imagery than in the previous example, this Tri-Level Split has a moon-gate picture window, an Oriental post-and-beam entry, and balcony railings with bamboo imagery.\n\n# STYLED HOUSES SINCE 1935\n\n# Millennium Mansion\n\n# 1985\u2013Present\n\n## Identifying Features\n\nComplex high-pitched roof, with lower cross gables or hips; tall entry features, one and one-half to two stories high and often arched; dormers; multiple wall-cladding materials, may be applied to single surfaces like wallpaper; differing window sizes and shapes, sometimes arched; commonly asymmetrical and with tall vertical appearance.\n\n## Principal Subtypes\n\nThe roofs of Millennium Mansions are quite varied. Here they are loosely grouped into hipped roof with lower cross gables (appears to be the most common), hipped roof with lower cross-hipped roofs, and gabled roof with lower cross gables.\n\n**HIPPED ROOF WITH LOWER CROSS GABLES**\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nDuluth, Minnesota; 1994. Note the three levels of the ridge of the main hipped roof (compare to the single long ridge on the Houston example). As in the Chicago example nested gables highlight both the garage entry and the front-door entry. However, in this example they are appropriate to the Tudor theme.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nKansas City, Missouri; 1997. This house appears to be clad in stucco board but with seams left exposed. This could indicate the boards are being used as a rain screen and the cracks are actually providing ventilation.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nNaperville, Illinois; 2006. Millennium Mansions often incorporate Palladian-inspired windows as seen here above the entry and right wing; window on right is patched from three identical windows with an over-large round arch window above. The Palladian window motif, red brick and keystone lintels\u2014even above the double garage door\u2014evoke the Colonial Revival (see the example in Alexandria). Note very closely spaced balusters (likely required by current building code).\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nChicago, Illinois; 1996. Stone quoins, originally used to reinforce corners, assume a Postmodern look when used for two-story tall porch supports and to support gables on a garage wing. Note how nested gables relate to both garage bays and a two-story entry.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nNaperville, Illinois; ca. 2006. Side automobile courts have recently become utilized as a way to handle multiple cars when alleys are not present to provide a rear automobile entrance.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nSeattle, Washington; 2003. One-story Millennium Mansions are less common. Note how garages are placed in two different planes. At least eight roof levels and elements are present.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nDuluth, Minnesota; ca. 2000. Note the five jerkin-head roofs.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nChicago, Illinois; ca. 1990s. Note the use of three completely different kinds of arched windows.\n\nHIPPED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nNaperville, Illinois; 2007. Each front-facing gable is different.\n\n**HIPPED ROOF WITH LOWER CROSS HIPS**\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nDuluth, Minnesota; 1995. Cascading hipped roof, but otherwise calm details for a Millennium Mansion.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nDallas, Texas; ca. 1980. When this photo was published in the 1984 edition, the authors had no idea the cascading hip and two-story entry were harbingers of a style that would sweep the country.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nLeawood, Kansas; 2005. The one-and-a-half-story entry seems short next to the adjacent two-story tower.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nLeawood, Kansas; 2007. Part of the same developments as the two photos on this page. This shows the shift to Mediterranean-style details with squared entries that occurred in some Millennium Mansion developments\u2014even beyond Florida and the Southwest\u2014in the 2000s.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nNaperville, Illinois; 2005. An unusual stucco rectangle\u2014with rounded arch in the center\u2014provides the design theme for this house. Three different sizes are cut out and the fourth extends upward through a cornice. There is likely no historic structural precedent for this use of a rounded arch.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nNaperville, Illinois; 2005. Nine separate hipped-roof forms are visible. Note the matching round arches above the entry and wall dormer, and the two large contrasting bay windows.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nLake Oswego, Oregon; 1989. Garages comprise more than one-half of the one-story facade. The roof is five cascading hips and is twice the height of the facade.\n\nHIPPED ROOF WITH LOWER CROSS HIPS:  MILLENNIUM MANSION  \nDallas, Texas; 2005. This house has three easily visible hip-on-hip roof forms\u2014one on each of the side wings and one above the front entry. Relatively consistent detailing, a single wall cladding, and an underplayed one-story entrance make this transitional to New Traditional French.\n\n**GABLED ROOF WITH LOWER CROSS GABLES**\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nDuluth, Minnesota; 1996. Brick is used like wallpaper on the left side wing, the two-story entry, and the gabled double garage.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nBaltimore, Maryland; ca. 1990s.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nAlexandria, Virginia (vicinity); 2005. Small keystones are used throughout\u2014over single windows, triple windows, garage doors, and two-story entry arches. These, along with red brick cladding on the front facade, impart a Colonial Revival flavor to this Millennium Mansion, despite the three front-facing gables.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nHouston, Texas; 2004. Three heights are visible across the main roof ridge, and there are nine gables (including a small gabled roof dormer atypically nested in the left roof gable) and eight different sizes and types of windows. However, the single wall cladding, one-story entry, and similar segmental arched openings make this transitional to a New Traditional Tudor.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nCarlsbad, California; 1970 with later additions. It appears that a 1970s house has had a Millennium Mansion remodeling. The original cross-gabled Ranch is in the middle. It seems to have been expanded by a double-gabled addition and pop\u2011up on right, three dormers on the Ranch form, and new wing on left\u2014attached to what may have been the original garage wing.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nPhoenix, Arizona; ca. 2000. Multiple roof forms and a one-and-one-half-story-high entry mark this as a Millennium Mansion; the tile roof and stucco add a Mediterranean influence.\n\nGABLED ROOF WITH LOWER CROSS GABLES:  MILLENNIUM MANSION  \nChicago, Illinois; 2008. A familiar, yet unusual house. The complex cross-gabled roof and front porch might put it in New Traditional Queen Anne. The single wall material and lack of elaborate detail in the gables or along the porch could be a \"more complex roof form\" American Vernacular. But by the time one adds up nine different roof elements, it seems to approach being a Millennium Mansion\u2014perhaps one marketed as a \"new farmhouse.\"\n\n## Variants and Details\n\nAfter half a century dominated by low, broad, one-story houses with simple uncluttered roof lines and underplayed entries, a dramatic new form of house virtually exploded from the drawing boards of building designers and spread quickly across the country during the 1990s. Sporting a roof form seen only once before\u2014on exuberant Queen Anne and Romanesque houses just before the turn of the 20th century\u2014Millennium Mansion roofs added even more complexity, as if to celebrate the new millennium with still more drama. Many Millennium Mansions have the high-pitched hipped roof with lower cross gables of the late 19th century (but with more gables per house), while others sport a roof rarely seen before\u2014a hip-on-hip roof that sometimes expanded into cascading hips-on-hips-on-hips. Roof and wall dormers are both common and roof ridges are often discontinuous, adding more complexity to the roof line. These complicated roofs can be thought of as crowns, or, more satirically, as the Future Roofers of America Relief Act. The complex roof forms sometimes relate to bay windows, or other small extensions in the ground plan, that are expressed in the elevation of the house and continue into the roof form.\n\nThe style's vertical appearance is not only a result of the typical two-story height. It also reflects taller interior ceiling heights (ten feet is now standard, and higher heights can be used) and taller floor joists engineered to allow more room for mechanical systems between the floors. Millennium Mansions are frequently built out to the limits of the zoning envelope (meaning they contain as much square footage as is legally allowed on a lot).\n\nThe entry into many Millennium Mansions assumed a new dominance\u2014generally one-and-a-half or two stories tall and arched\u2014a feature introduced by Hollywood architect John Wolff in his dramatic Mansard and Regency designs. Often an arched window is placed above the front door. This one-and-a-half- or two-story entry was not simply an exterior feature; it generally signals the presence of a two-story interior entry foyer that often leads into a two-story great room\u2014a pair of interior features that became popular beginning about 1985. Decades of the Ranch style's standardized eight-foot ceilings had led to a craving for very high ceilings. Ground-floor master bedroom suites injected a bit of interior practicality. By 2005, a tower\u2014sometimes part of or adjacent to the entry\u2014was being touted in real estate ads for new homes, and the earlier two-story entries were beginning to lose their broad appeal.\n\nWalls are often clad in several materials\u2014sometimes applied as a thin sheet on a single surface, treating the cladding almost like wallpaper. Windows are typically of varied size and shape, and may not even be from the same design family. Traditional detailing\u2014French, Mediterranean, and English in derivation\u2014is generally present on the exterior, and details may be a mixture of historic styles. As this guide is going to press, Millennium Mansions appear to be growing less exuberant. In addition to entries shrinking to one story, windows are more likely to be from the same family, producing a more related design. Detailing is more frequently becoming associated with a single historic style\u2014such as Tudor or French\u2014with Mediterranean now gaining favor in Texas, California, and the Southwest. The more tame and \"proper\" New Traditional house might gradually subsume the exuberant Millennium Mansion.\n\nMILLENIUM MANSION\n\n **CASCADING HIPPED ROOFS: indicative ofMillennium Mansions (uncommon before ca. 1985)**\n\nSeven cascading hipped roofs are visible from side facade\n\nMILLENIUM MANSION\n\n **CASCADING HIPPED ROOFS: indicative of Millennium Mansions (uncommon before ca. 1985)**\n\nFive cascading hipped roofs (also called hip-on-hip roofs) and three lower cross-gables are visible from street\n\nMILLENIUM MANSION\n\n **GROUND PLANS RELATED TO ROOF FORMS**\n\nMILLENIUM MANSION\n\n **TALL ENTRY PORCHES (most popular ca. 1990\u20132005)**\n\n## Comments\n\nThe Millennium Mansion is the predominant style in many large subdivisions built from the late 1980s up to the present. They can be built on higher-priced land because their vertical massing allows far more square feet per lot than the horizontal Ranch house. It has been used as infill for older neighborhoods and, for a time, was also favored for huge custom-designed homes of five thousand to ten thousand square feet and up. These are generally built as a single massive structure, rather than broken down into visually distinct smaller additive elements, as is often the case with other contemporaneous styles (see illustration). This penchant for size, particularly when interspersed amid small, earlier neighboring houses\u2014or scattered on super-large far-flung rural lots\u2014led to the nickname _McMansion_ , initially used for Millennium Mansions but now referring to any new house deemed to be either oversized in comparison with adjacent homes or disjointed in style.\n\n# STYLED HOUSES SINCE 1935\n\n# New Traditional\n\nAfter almost half a century dominated by modernism, the 1970s saw a renewed interest in historical styles that produced what today are called New Traditional houses. The first houses little resembled the earlier styled homes they sought to emulate. The 1984 edition of this guide had a brief chapter, \"Neoeclectic,\" that illustrated builder examples of both Styled Ranch houses and two-story homes with relatively awkward proportions and details. Early architect-designed examples often featured abstracted Postmodern historic details. By the 1990s, however, New Traditional houses with more historically accurate proportions, forms, and details were being sought by clients and designed in nearly all the earlier styles.\n\nHomes are commonly based on styles popular in the early 20th century\u2014Colonial Revival, Tudor, Neoclassical, French, Italian Renaissance, Spanish, Craftsman, and Prairie. Romantic- and Victorian-era styles are also found, with Shingle style being by far the most common of these. The many styles of New Traditional are geographically spread throughout the United States; some styles, however, are favored locally, often with a bow to earlier traditions (for example, New Traditional Colonial Revival in New England, New Traditional Shingle on Long Island, and New Traditional Mediterranean in Arizona and southern California).\n\nIn traditional house design, both the overall composition and the individual details of each style are important. The earlier chapters of this guide have illustrations and photographs of the kinds of homes that have inspired New Traditional houses. Prior to 1930, traditional houses were most often constructed by builders who were familiar with these details. Today this is generally not the case\u2014and careful study of precedents is important.\n\nThe following pages of illustrations are not desirable details to emulate. Instead they are details that usually signal a recent construction date, making it easy to distinguish the majority of turn-of-the-millennium New Traditional houses from their earlier precedents. _It is to be emphasized that these are not desirable details to be used in constructing a new home or in a design review process;_ rather, these are generally details to be avoided, as each immediately signals new construction. In the deftest of hands, it is difficult to distinguish a New Traditional from an earlier construction simply by looking at the exterior. The location and size of the house and the garage both provide clues, as do slightly inventive details\u2014and the rear facade, if visible.\n\n## Identifying Features for Simplified or Poorly Detailed New Traditional Houses\n\nFRONT-FACING GARAGE INCORPORATED INTO MAIN BODY OF HOUSE\u2014Side wings with garages might have been added later, but if you see a main house block that includes a front-facing garage, or a garage wing extending to the front, it generally indicates a post-1935 house.\n\nFEW OR NO WINDOWS IN SIDE FACADES\u2014Pre-1935 houses generally lacked air-conditioning and required many windows that opened for summer temperature control and air circulation. Today's homes have no such restrictions, and although windows can be expensive to buy and install, when builders are urged to include more side windows they rarely mention cost and instead cite the advantages of arranging interior furnishings without interference from windows\u2014providing a \"sofa wall\" and \"bed walls\"\u2014and dismiss assurances from those who live in older homes that furniture actually can be arranged in front of windows.\n\nPORCHES TOO SHALLOW TO BE REALLY USABLE\u2014These look like a porch from the street (and on elevations presented for design review) but are not deep enough to accommodate porch swings, chairs, or chaises with space to walk past them\u2014like the true living porches of homes built before air-conditioning.\n\nBUILT ON SLAB FOUNDATION WITH FIRST FLOOR AT GROUND LEVEL; FEW OR NO STEPS UP TO FIRST FLOOR OF THE HOUSE\u2014Slab foundations for houses gained popularity during the 1950s and 1960s and worked well for Ranch and Mid-century Modern houses where indoor spaces flowed outside. Even here, the slab was often poured high enough to raise the house one or two shallow steps above grade. Many New Traditional houses do not even have this small amount of elevation.\n\n## New Traditional House Styles (links to original style)\n\nNEW TRADITIONAL SHINGLE\u2014Generally based on precedents found in the Shingle chapter.\n\n**SHINGLE**\n\nSHINGLE:  NEW TRADITIONAL  \nFishers Island, New York; 1985. Berkowitz House; Albert, Righter & Tittman, architects. Also inspired by the Low House, this design angles the main facade, bending it backward on the right.\n\nSHINGLE:  NEW TRADITIONAL  \nMedina, Washington (vicinity); 2008. A rectilinear interpretation of a Palladian window in the gables. Note how the side of the garage wing (on right) is detailed as if it were simply part of the house.\n\nSHINGLE:  NEW TRADITIONAL  \nFishers Island; 1981. \"Gable in a Square\"; Albert, Righter & Tittmann, architects. The broad front-facing gable of the Low House has inspired this creative geometry.\n\nSHINGLE:  NEW TRADITIONAL  \nBoston, Massachusetts (vicinity); 2001. Note the shingles curving into the recessed window in the gable.\n\nSHINGLE:  NEW TRADITIONAL  \nEast Hampton, New York; 2005. Mullen House; John Mullen, architect. A home with handsome, understated details. The entry is actually onto a split-level a few feet below the first floor.\n\nSHINGLE:  NEW TRADITIONAL  \nEast Hampton, New York; 1999. Shope Rheno Wharton Architects. This house has a broad symmetrical facade. Note the bank of paired doors leading from the house to the garden.\n\nSHINGLE:  NEW TRADITIONAL  \nSeattle, Washington; 2005. A front-gabled Shingle with a 21st-century interpretation of cottage windows on the first floor.\n\nSHINGLE:  NEW TRADITIONAL  \nSt. Paul, Minnesota; ca. 2000. This has a full-width porch and multiple shed dormers on a gable roof.\n\nSHINGLE:  NEW TRADITIONAL  \nEast Hampton, New York; 2004. A builder's interpretation of the entry porch Robert Stern used in the Residence at Calf Creek. The dead space in between the round arch and the gambrel roof could hold the family archives.\n\nSHINGLE:  NEW TRADITIONAL  \nAnnapolis, Maryland; ca. 2000.\n\nSHINGLE:  NEW TRADITIONAL  \nWater Mill, New York; 1984\u20131987. Residence at Calf Creek; Robert A. M. Stern, architect. This gambrel roof example has deft detailing in the arched fanlight over the entrance.\n\nSHINGLE:  NEW TRADITIONAL  \nBedminster Township, New Jersey; 1985\u20131988. Residence at Pottersville; Robert A. M. Stern, architect. The faux window (opening without glass) in the pediment is a postmodern salute to the round and elliptical (oculus) windows sometimes found on pediments from the Federal style forward.\n\nNEW TRADITIONAL COLONIAL\u2014Generally based on precedents found in the Postmedieval English, Dutch Colonial, Georgian, Federal, and Colonial Revival.\n\n**COLONIAL REVIVAL**\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nKansas City, Missouri; ca. 2006. The over-scaled broad triangular pediment gives what is actually a three-bay house with side wing the initial appearance of being a symmetrical five-bay house.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nChicago, Illinois; 1996. In this example the two-story entrance comes from the contemporaneous Millennium Mansion. The symmetry and simple roof are why it has been placed in the New Traditional category.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nDallas, Texas, ca. 1980. Note the wide spacing of the balusters above the front door and on the side wing\u2014typical of 1980s examples. This saved money by using fewer balusters than the correct spacing seen in Eclectic-era houses. Today, stricter building codes often require extremely close spacing that looks equally out of place. See the example in Naperville\u2014no space can be wider than four inches.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nBoston, Massachusetts; ca. 2000. This, like the house in Chicago, has a strong Millennium Mansion influence. Here the two front-facing gables with red brick \"wallpaper\" are an instant tip-off.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nKentlands, Maryland; ca. 1992. This 1990s side-gabled example had the benefit of Kentlands' building guidelines to keep it simple, like most original Colonial houses.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nStillwater, Minnesota; ca. 2003. A side-gambrel example with full-width porch and broad shed dormer.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nDallas, Texas; 2000. Weathers House; Ann Abernathy, architect. A small and understated main house block is part of a much larger house designed in an additive manner\u2014as is partially visible at right.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nBoston, Massachusetts; 2004. Here triple wall dormers above the garage echo triple roof dormers on the house. The house and garage connect in an additive manner.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nKansas City, Missouri; 1998. This house was closely modeled after Virginia Georgian mansions with the use of red brick, high foundation, steeply pitched hipped roof, and very understated entry.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nKentlands, Maryland; ca. 2000s. At Kentlands, a subtle mix of house size within a block face combines with slightly different setbacks of houses on the lots to create pleasingly irregular streetscapes.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nNew England; 2009. Federal House; PPA, Peter Pennoyer and Elizabeth Graciolo, architects. Designed for a historic district, this elegant house has historic details that are carefully and subtly reinterpreted in some places.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nPennsylvania; ca. 2005, Stone House (public facade); Albert, Righter & Tittmann, architects. This house is an excellent example of an additive composition; it has more stone and smaller windows on this public side.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nPennsylvania, ca. 2005, Stone House (meadow facade); Albert, Righter & Tittman, architects. Far more windows, including two bays, are on this private facade.\n\nCOLONIAL REVIVAL:  NEW TRADITIONAL  \nStillwater, Minnesota; ca. 2003. The additive stepped-roof form is interesting.\n\nNEW TRADITIONAL CLASSICAL\u2014Generally based on precedents found in the Neoclassical, Early Classical Revival, and Greek Revival.\n\n**CLASSICAL**\n\nCLASSICAL:  NEW TRADITIONAL  \nAlexandria, Virginia; 2006. The two-story entry feature signals a house that is New Traditional rather than from the Eclectic era. Note that the entry porch and pediment are slightly out of scale.\n\nCLASSICAL:  NEW TRADITIONAL  \nSeaside, Florida (vicinity); ca. 2000. High foundation is required for possible floods. The small window size is less common on New Traditional houses.\n\nCLASSICAL:  NEW TRADITIONAL  \nDallas, Texas; 2000\u20132004. Muse House; Quinlan Terry, architect. Larry Boerder, local architect. Solid masonry construction.\n\nCLASSICAL:  NEW TRADITIONAL  \nDallas, Texas; 1975. This early example lacks an adequate entablature above the full-width porch and uses large windows more typical of Ranch houses.\n\nCLASSICAL:  NEW TRADITIONAL  \nCarillon Beach, Florida; 2000.\n\nCLASSICAL:  NEW TRADITIONAL  \nSeaside, Florida (vicinity); ca. 1990s. This front-gabled house is very similar to the previous example. Compare the deeper entablature and more heavily molded cornice here to the much simpler one in Carillon Beach.\n\nCLASSICAL:  NEW TRADITIONAL  \nHighland Park, Texas; 2007. Gilliland House; Larry Boerder, architect. This house utilizes so many details of early 20th-century Neoclassical houses\u2014including the South's beloved full-height entry porch with lower full-width porch subtype\u2014that many neighbors think the house has been there for a century. Note the grouped windows, front door surrounded by lights, and the well-proportioned classical porches.\n\nCLASSICAL:  NEW TRADITIONAL  \nCarlsbad, California; 2006. Squared porch supports substitute for classical columns.\n\nCLASSICAL:  NEW TRADITIONAL  \nCape Porch House; Albert, Righter & Tittmann, architects. This could be the country cousin of AR&T's Temple House shown in the next example. The Greek Revival elements here are simplified squared columns, corner pilasters, and a wide band of trim in the gable.\n\nCLASSICAL:  NEW TRADITIONAL  \nConcord, Massachusetts; 1991. Temple House; Albert, Righter & Tittmann, architects. This striking new interpretation of the Greek Revival has dramatic pilasters on the gable front and an exceptionally wide frieze band with windows.\n\nCLASSICAL:  NEW TRADITIONAL  \nSanta Rosa Beach, Florida; 1997. This house, inspired by the full-facade porch of the Greek Revival style, has a small temple-form room on the roof rather than a more typical cupola.\n\nNEW TRADITIONAL ITALIAN RENAISSANCE\u2014Generally based on precedents found in the Italian Renaissance chapter.\n\n**ITALIAN RENAISSANCE**\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nDestin, Florida; ca. 1990s. Note the triple arched opening across the front entry porch.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nCarillon Beach, Florida; 1998. Lloyd Vogt, architect. A very simple beach-house design with a triangular pediment. Note the checkerboard patterned paving and squared fence supports topped with round globe.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nBeverly Hills, California; 1985. Styled houses with two-story entry features and large windows (as seen here) are almost certain to be post-1980 New Trads.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nAtlanta, Georgia; 1995. Sloane House; William T. Baker, architectural designer. It would take close examination to distinguish this refined design from an Eclectic-era example.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nDallas, Texas; 2001. Bramblett House; Allen Oliver, architect. This handsome but restrained house with projecting side wings likely inspired the larger more elaborate Dallas example, which is built nearby.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nDallas, Texas; 1988. Frank Welch, architect. Welch, renowned for his American Vernacular designs (called Texas Regional locally), distilled this simplified design to fit into a streetscape with many elaborate original 1910s and '20s Italian Renaissance houses.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nPasadena, California; 2005. There is a dramatic difference in the size of the small upper-story windows and tall arched first-floor windows. The seven-bay design requires a broad lot frontage.\n\nITALIAN RENAISSANCE:  NEW TRADITIONAL  \nDallas, Texas: ca. 2010. Larry Boerder, architect. This larger relative of the Bramblett House has a broader central section with room for a pair of oval windows, corner quoins, and a broad side balustraded terrace connecting to a two-story side wing on the right.\n\nNEW TRADITIONAL TUDOR\u2014Generally based on precedents found in the Tudor chapter.\n\n**TUDOR**\n\nTUDOR:  NEW TRADITIONAL  \nLee's Summit, Missouri; 2005. A simple two-story side-gabled house has nested Tudor gables placed in front\u2014similar to a previous example in Minneapolis.\n\nTUDOR:  NEW TRADITIONAL  \nPortland, Oregon; 2004. The garage wing forms a front automobile court that is as prominent from the street as the house proper.\n\nTUDOR:  NEW TRADITIONAL  \nDalton, Georgia; ca. 2000. McEntire House; William T. Baker, architectural designer.\n\nTUDOR:  NEW TRADITIONAL  \nPortland, Oregon; 2009. A broad driveway and garage in the main body of the house dominate this Tudor design; the development likely lacks alleys.\n\nTUDOR:  NEW TRADITIONAL  \nNashville, Tennessee; 1998\u20132000. Allen House; William T. Baker, architectural designer. This house utilizes clinker bricks and a roof clad with terra-cotta tile to blend with the older houses in the neighborhood.\n\nTUDOR:  NEW TRADITIONAL  \nAtlanta, Georgia; 2000\u20132003. Burfitt House; William T. Baker, architectural designer. An almost symmetrical stone-clad house with parapeted gables.\n\nTUDOR:  NEW TRADITIONAL  \nHillsboro, Oregon; ca. 1996. Multiple-facade gables are far more common in today's New Traditional houses than they were in Eclectic-era Tudor houses.\n\nTUDOR:  NEW TRADITIONAL  \nDallas, Texas; ca. 2000. Richard Drummond Davis, architect. The house as village\u2014an approach seen in both Tudor and Colonial New Trad houses. Here one can imagine a half-timbered house at right, a parapeted gable house in the middle, and a Shingle house at left.\n\nTUDOR:  NEW TRADITIONAL  \nDallas, Texas; 2011. Note the very high roof pitch and half-timbered box bay window. The owners voluntarily redesigned this infill house to conform with the historic setback line of the rest of the block, after they learned their new home would project 15 feet in front of it. This was very public-spirited, since they had a legal building permit and the forms for their foundation were already built.\n\nNEW TRADITIONAL FRENCH\u2014Generally based on precedents found in the French Eclectic. More recently houses have been based on French Colonial, Chateauesque, and Beaux Arts.\n\n**FRENCH**\n\nFRENCH:  NEW TRADITIONAL  \nPasadena, California; 2000. The high-pitched hipped roof and through-the-cornice segmental arched windows identify this house's French origins.\n\nFRENCH:  NEW TRADITIONAL  \nAtlanta, Georgia; 1997. Smidt House; William T. Baker, architectural designer. This one-story example is the result of a total remodeling of a Neoclassical Ranch.\n\nFRENCH:  NEW TRADITIONAL  \nDuluth, Minnesota; 1993. The sloping site allows the garage of this one-story French to step down from the main house block.\n\nFRENCH:  NEW TRADITIONAL  \nDallas, Texas; 2005. Larry Boerder, architect. This symmetrical house has a central block with side pavilions. Note the segmental pediment and diminutive roof dormers on the main block and segmental arched through-the-cornice dormers on the pavilions.\n\nFRENCH:  NEW TRADITIONAL  \nNaperville, Illinois; ca. 2005. Beginning about 2000, turrets and towers became popular on New Trads.\n\nFRENCH:  NEW TRADITIONAL  \nSan Diego, California; ca. 2000s. A New Traditional version of a towered French Eclectic house.\n\nFRENCH:  NEW TRADITIONAL  \nDallas, Texas; 2009. This New Traditional Chateauesque house has wall dormers with pinnacles and a turret.\n\nFRENCH:  NEW TRADITIONAL  \nMedina, Washington; 2007. Formal French houses favored stucco or stone walls with gray-slate roofs as seen here and in houses in Pasadena and Dallas.\n\nFRENCH:  NEW TRADITIONAL  \nDallas, Texas; 2009. Larry Boerder, architect. This formal house has Beaux Arts roots. Note the roof-line balustrades, paired pilasters, and oval cartouche with swags in the triangular pediment.\n\nFRENCH:  NEW TRADITIONAL  \nDallas, Texas; 2011. Larry Boerder, architect. The symmetrical facade of this French house is made less formal by the roof and wall materials.\n\nNEW TRADITIONAL VICTORIAN\u2014Most commonly based on precedents in the Queen Anne and Folk Victorian, but also occasionally in Second Empire, Stick, and Richardsonian Romanesque.\n\n**QUEEN ANNE AND OTHER VICTORIAN**\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nSeaside, Florida; 1985. This is based on Folk Victorian precedents, specifically, the pyramidal subtype that was prevalent in the South.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nDallas, Texas; 1976. Bicentennial House. Originally an American Four-Square, this was expanded and extensively remodeled in celebration of the nation's Bicentennial.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nMinneapolis, Minnesota; 2003. A duplex with little added detail but two Romanesque arches and a broad, squat tower. This and the example in Dallas both have lower automobile courts with arched entries.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nHillsboro, Oregon; ca. 2005. This and the house in Naperville are both inspired by Free Classic Queen Annes with towers. Here the tower is broad and round.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nDallas, Texas; 2005. This elaborate house has many intricate masonry Richardsonian Romanesque details.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nStillwater, Minnesota; ca. 2003. Note the typical Queen Anne hipped roof with lower cross-gabled roof. This late 19th-century roof rarely is built today in this pure form and is quite simple when compared to the roofs of many new Millennium Mansions.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nHighland Park, Chicago; 2003. This house has Queen Anne as its inspiration\u2014the Free Classic subtype. Note the paired columns and the Palladian window above the entry.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nNaperville, Illinois; 2004. Here the tower and the porch roof turrets both are hexagonal.\n\nQUEEN ANNE AND OTHER VICTORIAN:  NEW TRADITIONAL  \nHillsboro, Oregon; 1995. The huge porch roof turret dominates this facade and makes a _Mutt and Jeff_ pair with the tall, narrow tower. The wing on the right, with an almost solid wall, appears to be the garage.\n\nNEW TRADITIONAL CRAFTSMAN\u2014Generally based on precedents found in the Craftsman chapter.\n\n**CRAFTSMAN**\n\nCRAFTSMAN:  NEW TRADITIONAL  \nSeattle, Washington; 2000. This house has very shallow roof overhangs and diminutive exposed rafters with triangular braces. Unusual windows on the main floor mimic cottage windows but are double-hung and were photographed in the fully open position.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nPortland, Oregon; 2006. A much higher percentage of New Traditional Craftsman houses are two-story than were their Early Modern predecessors.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nSanta Rosa Beach, Florida; 2001. A one-story front-gabled New Trad Craftsman bungalow with a screened front porch.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nLee's Summit, Missouri; ca. 2005. Modeled after cross-gabled Craftsman houses, the porch support bases would likely have begun at ground level in an original.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nPortland, Oregon; 2003. A side-gabled example with three dormers lighting the upper half-story.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nChicago, Illinois; 2005.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nCarlsbad, California; 2001. Small one-car detached garages such as that on the left were very common in the 1910s and 1920s. This small TND has used these as a new way to get a third garage and at the same time create a small enclosed front garden behind it.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nLos Angeles, California; ca. 2000. Look past the front garage and this New Trad Craftsman has a nice feel to it.\n\nCRAFTSMAN:  NEW TRADITIONAL  \nPortland, Oregon; 2001. Another newer house form\u2014two wings extend almost to the street and are detailed with windows to look like part of the house even though both are garages.\n\nNEW TRADITIONAL PRAIRIE\u2014Generally based on precedents found in the Prairie chapter.\n\n**PRAIRIE**\n\nPRAIRIE:  NEW TRADITIONAL  \nSeattle, Washington; ca. 2000. A duplex with both entry porches recessed into the main body of the house.\n\nPRAIRIE:  NEW TRADITIONAL  \nLisle, Illinois; 1989.\n\nPRAIRIE:  NEW TRADITIONAL  \nDallas, Texas; ca. 2000. Note the use of corner windows.\n\nPRAIRIE:  NEW TRADITIONAL  \nDallas, Texas; 2011. The broad roof overhang at the front distracts the eye from the many areas where the overlay is actually quite shallow.\n\nPRAIRIE:  NEW TRADITIONAL  \nRaytown, Missouri; ca. 2005. This house has the American Four-Square look of many more modest Prairie-era houses. The simple hipped roof with a single centered hipped dormer is quite typical, as is the full-width porch supported by heavy squared piers.\n\nPRAIRIE:  NEW TRADITIONAL  \nDallas, Texas; ca. 2010.\n\nPRAIRIE:  NEW TRADITIONAL  \nChicago, Illinois; 2001. Ribbon windows on both levels, combined with a change in wall cladding below the upstairs window line, give this a Prairie look.\n\nPRAIRIE:  NEW TRADITIONAL  \nKansas City, Missouri; 2000. The broad overhang, the change in wall cladding, and the horizontal stone impart a Prairie feel to this house. The many single-spaced small square windows are atypical.\n\nNEW TRADITIONAL MEDITERRANEAN\u2014Based on precedents found in the Spanish Revival, joined by rural Italian elements, such as the squared towers in Italianate, and in this form sometimes called Tuscan.\n\n**MEDITERRANEAN**\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nCarlsbad, California; 2005. Examples with twisted columns and rounded towers have Spanish roots.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nDallas, Texas; 2001. New Mediterranean houses with an added squared tower, as seen here, are often called \"Tuscan.\"\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nDallas, Texas; 1996. Richardson Robertson III, architect. Note the raised terrace and the heavy squared tower.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nSan Antonio, Texas; 2000. Daniell House; Michael G. Imber, architect. This house is sited on a front motor court with a large fountain in the center.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nHighland Park, Texas; 2011\u20132012. An unusually shaped corner lot facilitated this rambling shape.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nDallas, Texas; 2010. Larry Boerder, architect.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nCarlsbad, California; 1989. The developer also finished this same house with a version of modest French details.\n\nMEDITERRANEAN:  NEW TRADITIONAL  \nDallas, Texas; 2008. This almost symmetrical example has Moorish details.\n\n## Variants and Details for Simplified or Poorly Detailed New Traditional Houses\n\nThe previously mentioned items are the most obvious elements to look for. Finding under-scaled or missing elements is more challenging and may require referring back to the chapter on the style of the house being examined. Careful reproductions are almost impossible to distinguish from earlier homes, particularly from the street. Some of the most common differences follow.\n\nWINDOWS\u2014The majority of new windows are made from vinyl, fiberglass, aluminum, or metal-clad wood. In general, these are much flatter (have less depth) than earlier wood windows. This flatness can be observed in every part of the window: it is recessed from the front wall surface (this space is called the \"reveal,\" because in old construction it revealed how thick the wall was), the thickness of the sliding sashes (if present), and the depth of the muntin bar that holds (or appears to hold) smaller glass panes in place. Window sashes originally had to be manufactured from many small panes of glass, and first leading, then later wood muntins, were used to assemble these panes into a single window sash. Since the late 19th century, when large panes of glass became readily available, windows with multiple lights have been a stylistic choice, not a necessity. Today, insulated double-glass panes are proportionately far less expensive in larger sizes, while there is an added cost for reproducing small-pane window patterns. Muntins can be emulated cheaply by tape or a shallow grid inserted between the double panes. Neither of these techniques produces a significant shadow or depth. More convincingly, molding is applied to the exterior and interior of a large double-glass pane to simulate individual panes (with an added charge for each pane thus).\n\nSHUTTERS\u2014Today a shutter of wood or a composite that resembles wood can cost ten times as much as a vinyl copy; thus the temptation to use vinyl is high. The least expensive vinyl shutters are only exterior \"shells,\" applied with screws to the wall surface. Because these are easily available in prefab sizes, they often are slightly too short, too long, too narrow, or too wide to actually close over the window (which their method of attachment makes impossible anyway). Exposed attachment screw heads, visible vinyl bubbles covering attachment screws, and\/or flexible shutter surface curving inward (from screws having been too tightly applied) are all giveaways.\n\nDORMERS\u2014Dormers are widely used, particularly where building codes define house height as the \"mid-point\" of the roof. Constructing a higher roof pitch and inserting multiple dormers for light and space can effectively add an extra story to a house. Simplified or poorly detailed new dormers include windows far smaller than the dormer they are placed in, wall cladding used on the front surface of a dormer, and dormers placed very close to the edge of the roof to provide more interior space. A \"pork chop\" eave return is very common on gabled dormers and is also used with other gabled elements of the house. This easily constructed detail, named for its pork-chop shape and widely used beginning in the 1950s, is today routinely substituted for a traditional gable return.\n\nCLASSICAL DETAILS\u2014A New Traditional house with Classical roots has many historic details. Each represents an opportunity to err; entire books illustrate these (see note 15). Most commonly, columns are too skinny, too few, or poorly spaced. The entablature (the horizontal-beam element placed atop columns supporting the triangular pediment or cornice above) may be omitted or incorrectly proportioned. Pilasters (flattened columns) may be omitted from common locations. These include placement against the back wall of a colonnaded porch or portico to visually connect it to the house, on each side of a door with a pediment or entablature above, and on elaborate Palladian windows. Most disconcertingly, houses that look symmetrical initially may be slightly asymmetrical in many small ways.\n\nMILLENNIUM MANSION\u2013INFLUENCED FEATURES\u2014These relate to common identifying features of Millennium Mansions: tall entries, one and one-half to two stories in height and often with an arched element; varied wall cladding materials, often applied to single surfaces like wallpaper; windows in many different shapes and sizes; and a complex roof form with nonessential elements. If there are two of these present, consider Millennium Mansion as the style.\n\nHALF-TIMBERING CLUES\u2014Today half-timbering is sometimes placed to cover the joints in sheets of four-foot-wide siding and does not relate to window placement or gable design. True half-timbering originally provided a home's structure and therefore naturally related to story changes, window placement, and gable design. Late 19th- and early 20th-century Tudor houses generally applied half-timbering to mimic historic structural patterns.\n\nWALL CLADDING\u2014Wall cladding materials are often intermixed, particularly in builder's houses. The development of thinner and less expensive stone cladding, shallow half-depth bricks for cladding, manufactured stone veneers, stucco board, and a wide variety of fabricated materials have led to greater use of masonry and stucco, often combined with imitation wood such as HardiePlank, HardieShingles, or vinyl siding. Today wall materials are often applied in vertical sections (a thin brick gable front might be found on an otherwise HardiePlank structure\u2014a bit like applying brick wallpaper on the gable end). Historically, materials were built in horizontal layers (a solid structural masonry first floor with a wood second story).\n\nGARAGES\u2014The number and size of garages in new homes has grown dramatically, not uncommonly to three or more\u2014with one often oversized. Garage access and placement is one of the first things that must be addressed when designing a New Traditional house. Garages facing the street deaden the public space; rear garages accessed from the alley or a front drive are ideal. When the illustrated street-accessed garage patterns appear, it likely means a New Traditional house rather than one from the Eclectic era.\n\nREAR FACADES\u2014Unlike front facades, rear facades vary significantly from historic precedents. Freed from many of the needs for utilitarian areas (such as a back porch with icebox and space for washtubs) or access to exterior work areas (such as clotheslines), they can instead offer full interaction with rear gardens and views through large windows, complete window walls, porches, balconies, decks, and terraces.\n\nADDITIONAL DETAILS\u2014The most obvious clue is a chimney that is not masonry but rather a rectangular box shape clad in wood or siding sometimes with metal flue caps exposed above. Arches also provide clues, among them three (or more) shapes of arch on the same house or arches not properly completed where they meet a pier or wall. Larger houses may be massive rather than composed of smaller additive sections (see illustration).\n\nNEW TRADITIONAL\n\n **IDENTIFYING FEATURES FOR SIMPLIFIED OR POORLY DETAILEDNEW TRADITIONAL HOUSES**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **COMPARISON OF DETAILING INNEW TRADITIONAL HOUSES**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\n**SIMPLIFIED OR POORLY DETAILED NEW TRADITIONAL HOUSE**\n\n**WELL-DETAILED NEW TRADITIONAL HOUSE**\n\nNEW TRADITIONAL\n\n **SHUTTERS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nLEAST EXPENSIVE SHUTTERS ARE HOLLOWED-OUT VINYL\n\nWOOD AND WOODLIKE COMPOSITES APPEAR HISTORIC\n\n**WELL-DETAILED REPRODUCTION SHUTTERS ARE HUNG BY HINGES, CORRECTLY SIZED, AND APPEAR TO BE WOOD**\n\nNEW TRADITIONAL\n\n **WINDOWS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\n\"FLAT\" APPEARANCE OF MANY NEW TRADITIONAL WINDOWS  \n(window not wood)\n\nNEW TRADITIONAL\n\n **WINDOWS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nMORE \"DEPTH\" IN HISTORIC AND WELL-DETAILED REPRODUCTION WINDOWS  \n(windows generally wood or metal-clad)\n\nNEW TRADITIONAL\n\n **DORMERS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **PORTICOS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **PILASTERS**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **MORE \"CLASSICAL\" DETAILS FOR SIMPLIFIED OR POORLY DETAILEDNEW TRADITIONAL HOUSES**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **MILLENNIUM MANSION FEATURES  \n(if two or more are present, consider Millennium Mansion as possible style)**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **HALF-TIMBERING: not related to structure, like originals**\n\n**The illustrations in this section are not intended to be used for any type of design or preservation review.**\n\nNEW TRADITIONAL\n\n **ATTACHEDGARAGE VARIATIONS**\n\n## Comments\n\nArchitect Robert A. M. Stern (b. 1939), a faculty member of Columbia University and later dean of the Yale School of Architecture, has been a pivotal leader in this movement, beginning with his first Shingle-style houses in the late 1970s. His preeminent position and prolific publication has given his work a broad influence. Stern authored numerous books on his designs that by 1990 already included houses inspired by the Shingle, Italian Renaissance, and Georgian Revival styles; his stylistic inspirations have continued to broaden. It is important to note that Stern's work, and that of other talented New Traditional architects, is not completely derivative. Rather, they understand classical principles and architectural style well enough to subtly alter or rearrange elements to create New Traditional home designs, not copies\u2014houses instantly familiar yet subtly different from the homes that inspired them. Architectural historian Vincent Scully describes this as a \"conversation across the generations.\"\n\nAt the neighborhood level, architect and planner Andr\u00e9s Duany (b. 1949; Duany Plater-Zyberk & Company) and others have included stylistic guidelines in plans for traditional neighborhood developments since the 1980s\u2014generally based upon local building traditions and styles. These can be very specific in requirements (or incentives) for use of certain forms or styles, for elements such as front porches, and\/or for certain colors or materials.\n\nToday, New Traditional houses are constructed throughout the United States\u2014as country houses on large estates, as infill for older neighborhoods, and in new developments, many of which require historic house styles. But there is an extraordinary disconnect between the styles of these widely built homes and the homes featured in most professional architecture periodicals. While prior to 1970 there was a general lineage from architecture-award-winning homes to those being built in typical American neighborhoods, today the AIA and time-honored publications such as _Architectural Record_ rarely give even passing notice to New Traditional homes\u2014despite the fact that a nationwide cadre of architects and building designers are creating them to fulfill strong consumer demand.\n\nThis disconnect led to the founding of the Institute of Classical Architecture and Art in 2002. In addition, two periodicals originally focused on the preservation of older houses have specifically advocated New Traditional houses: _Clem Labine's Period Homes_ and _Old House Journal's New Old House.Architectural Digest,_ with its audience of affluent tastemakers and primary focus on interiors, has featured many New Traditional houses\u2014as have a host of other widely distributed shelter magazines, such as _House Beautiful._\n\nDesigning and building New Traditional houses presented an enormous challenge when interest in historic styles reemerged in the 1970s. When Robert Venturi's _Complexity and Contradiction in Architecture_ was published in 1966, the body of knowledge needed to build houses with historic precedent had largely disappeared. It had been more than thirty years since such houses were being built, and the architects and builders who had produced them were no longer active. The monographs of Eclectic architects' work and the books illustrating European precedents and details were no longer in print. Architecture schools were only teaching modernism\u2014even today, with the exception of less than a handful of universities, this is still true. Some architects of New Traditional houses used the 1984 edition of this book to communicate with clients about the kind of house they would like.\n\nEvery early New Traditional house was designed by someone self-trained in appropriate forms and details. However, in the last fifteen years architecture books have been published, period reproduction products developed, and new websites launched to make building New Traditional houses easier. As a result, almost every earlier house style was built somewhere in the United States during the millennium housing boom.\n\nShingle was the first New Traditional style broadly adopted by architects. Two late-modern styles, Shed and Postmodern, were often clad in shingles and sometimes published as new Shingle houses. By 1990, Shingle houses with simpler forms and detailing were being designed by architects. In 1994, _Life_ magazine began a new House of the Year program\u2014because, they explained, so many people were having to just \"settle\" for a house, rather than really loving it. Their initial design was a fine New Traditional Shingle house, and the architect was, appropriately, Robert Stern. The Shingle continues to be a favorite.\n\nWhile Colonial Revival homes continued to be built occasionally during the 1950s and 1960s (primarily the Built-in Garage subtype), by the 1970s houses based on the complete range of English Colonial and Colonial Revival precedents sharply increased in popularity with builders. At first these New Traditional Colonial homes utilized a wide range of free adaptations and were often oddly proportioned (called Neocolonial in the 1984 edition of this book). By the 1980s better proportioned and architect-designed Colonial Revival houses were being built. Kentlands in Gaithersburg, Maryland, is an excellent example of a 1988 traditional neighborhood development that required Colonial Revival through its building codes.\n\nNew Traditional Classical homes generally have the full-height columns or pilasters of earlier Classical styles; these were at first freely applied to a variety of house forms by builders with little concern for historically accurate detailing. By the 1990s, architects favored houses with Classical precedents and were designing a wide variety of beautifully detailed homes based on ancient Classical styles\u2014as well as Renaissance Classical, seen primarily in New Traditional Italian Renaissance homes.\n\nNew Traditional Tudors have been a favorite since the late 1970s. Like their more correctly detailed pre-1935 predecessors, these have dominant, steeply pitched front gables, and far more commonly than their predecessors they have half-timbered detailing.\n\nNew Traditional French houses also appeared in the late 1970s and by the mid-1980s were among the most fashionable throughout the country, reaching a level of popularity never achieved by their quite varied pre-1935 French Eclectic forebears. These are typically characterized by high-hipped roofs and through-the-cornice-dormers (often with segmental arches). More recently these have been joined by houses inspired by French Colonial and more elaborate Chateauesque and Beaux Arts precedents. These all have been facilitated by stucco exterior boards and new types of stone and faux-stone cladding.\n\nAround 1980 New Traditional versions of the various Victorian styles began to be built. Most New Traditional Victorian houses are Folk Victorian or Queen Anne with spindlework porch detailing. Other Victorian styles can also be found. In the 1990s New Traditional Craftsman began to gain in popularity, as did New Traditional Prairie houses.\n\nPerhaps the most recent to be widely built are New Traditional Mediterranean homes. These include homes based on the Spanish Revival, as well as a newer source of inspiration, Tuscan (based loosely on the rural houses of Italy, often with squared towers). Most New Traditional Mediterranean houses have rounded arches, red tile roofs, and stucco walls that may include large sections clad with stone, a combination rarely if ever used in the Eclectic era.\n\n# STYLED HOUSES SINCE 1935\n\n# American Vernacular\n\n# ca. 1930\u2013Present\n\n## Identifying Features\n\nSimple geometric forms (the kinds of houses you could build with a set of blocks); covered porches and balconies, with unadorned porch supports and railings; uncomplicated roofs; walls clad with one dominant material\u2014generally wood, stone, or brick; stylistic details not present.\n\n## Principal Subtypes\n\nFRONT-GABLED ROOF\u2014Simple rectangular block with front-gabled roof; may have uncomplicated one- or two-story porch.\n\n**FRONT-GABLED ROOF**\n\nFRONT-GABLED ROOF:  AMERICAN VERNACULAR  \nOcean Springs, Mississippi; ca. 2005. Katrina Cottage; Marianne Cusato, architect. With plans and materials easily available, many homes similar to this one have recently been built across the Gulf South. This plan features a 544-square-foot home. Groupings of similar small folk houses were found across the South until well after World War II.\n\nFRONT-GABLED ROOF:  AMERICAN VERNACULAR  \nWyoming; ca. 2000. Duncker House; Paul and Peggy Duncker, architects. The windows are evidence that this is architect-designed and not a folk house. The decision to build a detached garage lets the house maintain a simple, straightforward shape.\n\nFRONT-GABLED ROOF:  AMERICAN VERNACULAR  \nStillwater, Minnesota; ca. 2003. The front porch and the sunroom on the side balance each other.\n\nFRONT-GABLED ROOF:  AMERICAN VERNACULAR  \nKey West, Florida; ca. 1996. This is part of a large group of American Vernacular houses built on a former military base in the Truman Annex, near Harry Truman's Little White House. See also this other example in Key West.\n\nSIDE-GABLED ROOF\u2014Simple rectangular block with side-gabled roofs; may have uncomplicated one- or two-story porch.\n\n**SIDE-GABLED ROOF**\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nKey West, Florida; ca. 1996. Here the porch is under the main roof of the house. Contrast this with the added porch in the South Carolina example.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nEast Hampton, New York; ca. 2005. Single windows are carefully spaced. A shed porch in front is joined by an apparent shed dormer in back.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nDuluth, Minnesota; 2001. David Salmala, architect. A side-gabled house is joined to its front-gabled garage by an open breezeway. Note the square windows.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nSea Ranch, California; 1984. Binker Barn; MLTW Moore Turnbull, architects. Sea Ranch gave a huge impetus to American Vernacular houses inspired by the simple gable-roof shapes of farm buildings. The Binker Barn was developed as a stock plan that could be successfully sited in the different habitats at Sea Ranch. The large gable shape dominates but can be expanded and modified with the subordinate sheds. Construction was a heavy timber frame and block walls clad with redwood. Seventeen variations were built between 1968 and 1971. See also the Binker Barn.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nPanama City Beach, Florida; 2000.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nCharleston, South Carolina (vicinity); ca. 2000. Raised cottages like this are being built again to avoid floods in the Carolina Low Country. Note the small size of the Palladian windows in the center gable.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nSea Ranch, California; ca. 1970. Binker Barn; MLTW Moore Turnbull, architects. It sits in the right foreground of an American Vernacular grouping.\n\nSIDE-GABLED ROOF:  AMERICAN VERNACULAR  \nDallas, Texas; 1939. Bromberg House, O'Neil Ford, architect. Ford was the father of Texas Regionalism. This sophisticated house with paired chimneys and a three-bay main house block feels symmetrical but with casually added additions. Porches and breezeways make up half the perimeter of the house. Note the use of large screened front porches downstairs on the left and upstairs on the right.\n\nHIPPED ROOF\u2014Simple hipped-roof form; may have uncomplicated one- or two-story porch.\n\n**HIPPED ROOF**\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nDallas, Texas; 2001. Dan Shipley, architect. This variation has a hipped-roof main block with asymmetrical side additions. The brick color and right-side stone wall are colors and materials used locally.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nDestin, Florida; 2003. Cottages with full-width porches and pyramidal hipped roofs were widely built in the South and are now being built again along Florida's Gulf Coast.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nRaytown, Missouri (vicinity); ca. 2005. Note the irregular size and placement of the upstairs windows.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nKey West, Florida; ca. 1994. The form of the house is exaggerated by the full-width porch extending out on each side. Compare to the previous example.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nMississippi; ca. 2000. Ken Tate, architect. This house was carefully planned to look like several smaller structures that were built earlier and then joined together.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nDallas, Texas; 1965. Field House; Frank Welch, architect. This house has an interior atrium. The roof was originally wood shingles, long since replaced with the standing-seam metal roofing now widely used on Texas regional houses.\n\nHIPPED ROOF:  AMERICAN VERNACULAR  \nCarillon Beach, Florida; ca. 2000. A row of American Vernacular houses with two-level porches.\n\nCROSS-GABLED ROOF\u2014Simple gable-and-wing form; may have straightforward one- or two-story porch.\n\n**CROSS-GABLED ROOF**\n\nCROSS-GABLED ROOF:  AMERICAN VERNACULAR  \nStillwater, Minnesota; ca. 2003.\n\nCROSS-GABLED ROOF:  AMERICAN VERNACULAR  \nRhode Island; 1998. Wolf House; Estes\/Twombly, architects. Scaled to fit into a neighborhood of small houses, a two-story gabled form sits behind a broad shed roof and chimney. The L-shape created by the two main elements holds a private garden.\n\nCROSS-GABLED ROOF:  AMERICAN VERNACULAR  \nDallas, Texas; ca. 1980. John Mullen, architect. This house is designed with additive massing. The roof has understated parapets on the gable ends. Note how the chimney is subtly incorporated into the right wing.\n\nCOMPLEX ROOF FORM\u2014This subtype includes a broad range of forms designed to evoke the qualities of slightly more complex American farmhouses. Some are designed to appear as if the house has grown over time, with basic shapes gathered or lined up together. Others add screened porches or incorporate familiar rural shapes like barns or silos. Some simply add roof or wall dormers, or a gable, to more basic forms.\n\n**COMPLEX ROOF FORMS**\n\nCOMPLEX ROOF FORMS:  AMERICAN VERNACULAR  \nAnnapolis, Maryland (vicinity); ca. 1970s. Paired windows on two front-gabled forms sheathed with what appear to be smooth wood boards.\n\nCOMPLEX ROOF FORMS:  AMERICAN VERNACULAR  \nPortland, Oregon; 1996. Note the three matching through-the-cornice gables.\n\nCOMPLEX ROOF FORMS:  AMERICAN VERNACULAR  \nStillwater, Minnesota; ca. 2003.\n\nCOMPLEX ROOF FORMS:  AMERICAN VERNACULAR  \nAnnapolis, Maryland; 1956. If the 1956 listed date is correct this might be a house that actually did grow in stages, rather than built to look like one.\n\nCOMPLEX ROOF FORMS:  AMERICAN VERNACULAR  \nNorth Carolina; ca. 2000. Dail Dixon, architect. Inspired by the tobacco barns of the region, this house resembles three barns connected with glass-walled links. The varying windows on each front-gabled element let you know this is an architect-designed house and also hint at the differing functions inside. An open breezeway pavilion connects the house to the side-gable garage.\n\n## Variants and Details\n\nPorches are frequently the primary exterior embellishment on an American Vernacular house. In addition to the type of porch (one-story, two-story, partial-width, full-width, etc.), an interesting detail to note is the way a porch attaches to the house, as this can have a subtle but important visual effect. American Vernacular houses typically have porches deep enough to accommodate outdoor activity, not truncated ones that only appear to be real from the street. A screened porch is common. See illustrations and photographs of Pre-Railroad Folk and National Folk on this page through this page for a few of the wide variety of sources for American Vernacular. In addition, the porch variations of Folk Victorian on this page provide sources for variations in form. American Vernacular examples, however, would generally have simple porch details and omit spindlework porch detailing or under-eave brackets.\n\nWindows are typically double-hung, have simple frames, and may be multi-paned. The manner in which windows are arranged on the facades can sometimes distinguish an architect-designed example from one created from a pattern book or for a traditional neighborhood development. Architects are more likely to experiment with window size, spacing, and placement, while houses created by pattern books and design review are more likely to have window placement similar to that found on original folk houses.\n\nADDITIVE VS. MASSIVE\u2014American Vernacular houses commonly achieve a large size in an \"additive\" manner\u2014that is, with many smaller elements joined together\u2014like a house that grew over time or one that was built as a central house block with hyphens and\/or wings. By contrast, Millennium Mansions most often achieve their large size as a single massive element. Both approaches can be found in New Traditional houses, but using the additive avoids an overly large main-house block.\n\nAMERICAN VERNACULAR\n\n **WAYS TO ACHIEVE A LARGER HOUSE**\n\nAMERICAN VERNACULAR\n\n **WINDOW PATTERNS**\n\nAMERICAN VERNACULAR\n\n **TYPES OF PORCHES**\n\n**WAYS PORCH CAN CONNECT TO HOUSE**\n\nAMERICAN VERNACULAR  \nSEA RANCH, CALIFORNIA In addition to folk houses (discussed in the Pre-Railroad and National chapters), the simple shapes of barns and outbuildings, such as these at Sea Ranch, provided inspiration for American Vernacular houses.\n\n## Occurrence\n\nFor decades the American Vernacular movement was limited to cerebral architects consciously designing a quiet house with good bones for clients preferring to be understated. Most were one-of-a-kind designs that fit easily into almost any setting. Beginning in the 1980s some TNDs had design guidelines written with the express purpose of creating new homes that recalled an area's architectural heritage\u2014in effect legislating particular kinds of houses. Some of these guidelines concentrated on the form of the region's vernacular house (rather than a style) and created neighborhoods of American Vernacular houses.\n\n## Comments\n\nIn the late 1920s most American architects were designing Eclectic English-, Spanish-, and French-influenced houses. A few architects, however, began to study regional folk-house traditions and emulated them in the homes they designed\u2014simplifying houses rather than complicating them. Chief among these were William Wurster in California and O'Neil Ford and Dave Williams in Texas. As William Wurster explained the process, \"Design up from the log cabin, instead of trying to compress the mansion.\" In 1928, Wurster designed the Gregory Farmhouse in Scotts Valley, California\u2014a well-publicized early American Vernacular house\u2014and his architecture continued to use a process of simplifying rather than creating new shapes and forms.\n\nIn Texas, O'Neil Ford and his mentor and employer, Dave Williams, found inspiration in the folk houses that had been built in the Texas Hill Country by early German settlers. These sturdy, straightforward homes were simple shapes, built of native materials, and designed to take advantage both of their sites and the prevailing winds. Williams wrote two articles for the _Southwest Review_ advocating a new regional architecture based on these Texas precedents. In doing this, Williams and Ford joined Wurster as early leaders of a nascent regionalist movement where architects looked to the folk architecture of their own regions and designed homes based on them. In 1965 the Sea Ranch reinforced this with its use of simple vernacular forms, such as MLTW's Binker Barns with their predominant gabled shape. In recent decades a growing number of architects have been designing these simple commonsense houses.\n\nSince the early 1980s, TNDs have adopted style- and\/or form-based guidelines for new construction, many intended to re-create a region's heritage of vernacular architecture. Recent pattern books have sometimes included regional patterns. An excellent example is _Louisiana Speaks: Pattern Book_ by Urban Design Associates and Raymond L. Gindroz, FAIA, written after Hurricane Katrina destroyed so much of the rich vernacular architectural heritage of the Gulf Coast states. In addition, architects anxious to preserve the area's strong regional identity, such as Marianne Cusato and Duany Plater-Zyberk & Company, have designed plans for quite basic American Vernacular houses, such as Cusato's \"Katrina Cottage.\"\n\nFinally, perhaps as a reaction to the spread of Millennium Mansions, \"Farmhouse\" became a favored style during the 1990s and 2000s and was featured in pattern and other books.\n\n# APPENDIX\n\n# Approaches to Construction in the 20th and 21st Centuries\n\nIt seems astonishing that during the 20th century\u2014when man went to the moon and lived on space stations circling the earth, and when computers became commonplace and phones were made mobile\u2014the basic construction technique used for the vast majority of American houses continued to be wood, either 19th-century balloon-frame or a platform-frame variation of it.\n\nOnsite stick construction has continued to dominate the construction of new American houses, despite an interest in prefab houses that was already under way in 1900, continued through the early 20th century, and that reached a near fever pitch after World War II, when multiple attempts were made to transform the factories left idle at war's end into housing production. And although a cadre of major architects\u2014among them Frank Lloyd Wright, Carl Koch, Buckminster Fuller, and Le Corbusier\u2014worked hard to design attractive and affordable prefab houses during the early and mid-20th century, the structural frames of houses in the U.S. are usually built onsite, and primarily with wood.\n\nA concern for green, energy-efficient construction blossomed in the late 1960s. Yet forty years later wood construction, most of it not eligible for even the most basic of energy-efficiency ratings, continues to dominate. The need for green is great and this quest continues in full force today, with promise for the future.\n\nWhat did evolve\u2014and became a part of most American houses built today\u2014are many of the materials that encase the stick structural frames, along with a great many technological innovations that enhance quality of life. Some of these changes are so fundamental and widespread that it is hard to imagine that at the beginning of the 20th century none were in place.\n\n## Prefabricated Construction\n\nA prefabricated house is one that is made of pre-cut and pre-assembled parts manufactured offsite. The parts are delivered and expeditiously assembled on location. Some houses are constructed from individual pre-cut boards, while others are panelized or modular, meaning that entire wall panels or modular boxes are factory-built and then joined together on location. Any style of house can be prefabricated, either in part or in whole; in the United States houses were being shipped from New England to the West Coast as early as the mid-19th century.\n\nBy the early 20th century, mail order retailers like Montgomery Ward and Sears Roebuck were selling \"kit houses\" of pre-cut lumber, available in a wide range of styles and virtually indistinguishable from other houses of the same style that were not pre-cut.\n\nIt was the early-20th-century modernists, however, who began to seriously consider prefabrication as a way to meet the demand for affordable housing\u2014a need particularly evident in Europe after World War I. Le Corbusier, Walter Gropius, Frank Lloyd Wright, and Buckminster Fuller had all designed prefabricated houses before 1929, when the Depression affected all types of construction. Although the housing market began to recover slowly during the 1930s, Sears stopped production in 1940.\n\nDuring World War II, prefabrication methods were eagerly applied to defense and worker housing, with about 200,000 units of factory-built homes erected during the war. In addition, the war led directly to one of the most widely used prefabricated units ever\u2014the Quonset hut\u2014built for housing and other purposes. The efficacy of prefab in construction was even more clearly demonstrated after the war. The need to produce sixteen million homes for returning servicemen brought about a golden age in prefabrication, and it was utilized in four primary ways.\n\nFirst, there was a broad postwar movement of individuals and companies trying to design and produce complete prefab houses\u2014many of which mimicked traditional homes. These ran the gamut from Lustron, which manufactured innovative metal houses in a retooled World War II aircraft plant (supported by government grants) to Macy's, the revered New York City department store, which sold standardized homes completely furnished right down to color-coordinated bath towels and Melmac dinnerware.\n\nSecond, there were those experimenting with very distinctive house forms that could be erected quickly and inexpensively, such as Buckminster Fuller's early Dymaxion House and his geodesic domes, the latter of which can still be ordered.\n\nA third approach used standardized _parts,_ as well as parts available from commercial catalogues, to build one-of-a-kind houses, such as the Eames House. This approach offered almost infinitely changeable designs, instead of a few standardized ones. These were rarely intended to mimic a traditional house but, rather, to look quite modern. Carl Koch's Techbuilt post-and-beam panelized houses were among the more successful of this approach.\n\nFourth, and undoubtedly most commonly, builders introduced prefabricated elements into traditional construction. For example, they set up shop areas to prefabricate large components, such as wood roof joists and wall panels, that were then used to construct large developments of balloon-frame houses.\n\nPrefabricated houses experienced a decline in the 1980s and 1990s due to the increase in suburban developments and the public's association of prefab with inexpensive manufactured design. During this period, modular houses\u2014almost identical to manufactured housing except for not being required to keep the moving chassis in place and being required to meet local building codes\u2014filled a need for inexpensive prefab houses.\n\nRecently, however, there has been a resurgence in high-end prefab housing due to affordability, an increase in companies offering this type of construction, the availability of more house designs (including both modern and traditional styles), and the inclusion of eco-friendly elements. New technologies affording energy-efficient designs include: structural insulated panels (panels containing a foam core), insulated concrete forms (concrete molds that create a continuous surface), and photovoltaic systems (that generate electricity). Prefabricated houses not only facilitate a shorter construction time but also produce half the material waste of onsite construction. In addition, prefabrication avoids exposure of building materials to natural elements, since production is in climate-controlled factories.\n\n**PRE-FAB**\n\nPRE-FAB: APPENDIX  \nMontauk, New York; ca. 1964. Leisurama Home. Exhibited as the \"Typical American House\" at the American National Exhibition in Moscow in 1959, Macy's sold about two hundred of these for a second-home development in Montauk. The purchase included all of the furnishings, right down to Melmac dishes and color-coordinated sheets and towels.\n\nPRE-FAB: APPENDIX  \nUnidentified house; ca. 1960s. Buckminster Fuller, architect. Fuller patented this design consisting of a rigid geometric frame of metal or plastic covered by a structural skin on rigid panels. Fuller's own dome is one of the National Trust's \"America's Treasures.\"\n\nPRE-FAB: APPENDIX  \nBoston, Massachusetts; 1999. Neighbors report that this Prefab house was manufactured in New Hampshire and erected in just a few days. It is New Traditional Colonial in style.\n\nPRE-FAB: APPENDIX  \n1945 (designed 1920). Dymaxion House; Buckminster Fuller, architect. The only remaining example is at the Henry Ford Museum.\n\nPRE-FAB: APPENDIX  \nMadison, Wisconsin; 1957. Frank Lloyd Wright, architect. Wright believed that a well-designed Prefab house was possible and designed four plans for the Marshall Erdman Company. Only seven examples were built.\n\n## Green Construction\n\nGreen or eco-friendly houses focus on environmental impact and sustainability. While newly designed houses can incorporate green elements, the preservation and retrofitting (or \"greening\") of existing houses is the most eco-friendly action a community can take. The most efficient way to demonstrate sustainability is to preserve houses that are already built, as debris from building construction accounts for one-third of all waste generated in the United States. The amount of energy exhausted when a house is demolished and reconstructed can never be recovered within the generally shorter life span of new construction, regardless of green elements used and the amount of materials recycled.\n\nNew discoveries and experiments with green architecture were largely a result of the environmental movement of the 1960s and 1970s, as well as the OPEC oil embargo of 1973. The first efforts toward new green construction generally relied on either the sun's energy or the stable temperature and insulating effects of the earth (or both) to reduce energy requirements. Some are passive solar houses (using no mechanical systems, only natural air flow; the vertical design of the Lake Placid example was intended to facilitate this); others are active solar houses with mechanical distribution of heat and cooling, while still others stress thick coverings of earth for insulation. The solar collectors, air-flow systems, heavy insulation, lack of windows, and earth coverings used in these techniques sometimes created unique facade designs that bear little resemblance to traditional houses. These might be partially buried (as in the Sea Ranch and Phoenix examples). Other green homes, often those built in the Contemporary and Shed styles, were designed with careful consideration to orientation, placement of window openings, shade structures, and plantings that facilitated passive solar energy.\n\nBy the 1990s, it was obvious that the real solution was to find ways to incorporate \"green\" into every style and type of house. Therefore, formal programs were set in place to promote ecological designs and materials, including the Committee on the Environment created by the American Institute of Architects and the Energy Star program by the EPA and the U.S. Department of Energy. Currently there are more than five hundred regional and national systems that rate energy efficiency and environmental impact. In 2000, the U.S. Green Building Council developed LEED (Leadership in Energy and Environmental Design), an excellent and inclusive rating system that evaluates energy use, water efficiency, air quality, overall design, and site selection for future construction. This program awards a silver, gold, or platinum rating to exemplary projects.\n\nWhile an eco-friendly house of the mid- to late-20th century might be identifiable by a \"space age\" design, today green houses can appear traditional in style and still maintain equal environmental standards. Elements of green or eco-friendly houses may include Energy Star appliances, low-flow toilets, a tank-less water heater, solar-generated power, and native landscaping, all of which can be applied to both new and old houses. When retrofitting an older home, a green solution to aged windows is repair instead of replacement. New wood, vinyl, or PVC windows have a shorter life expectancy and cannot be recycled or restored effectively. Original windows that have been properly restored could last for another one hundred years.\n\n**GREEN**\n\nGREEN: APPENDIX  \nLake Placid, Minnesota; ca. 1970s. Milo Thompson, architect. The vertical design was intended to facilitate natural air flow.\n\nGREEN: APPENDIX  \nSea Ranch, California; ca. 1970. Houses with earth roofs were designed to both fit into the landscape and provide ample natural insulation.\n\nGREEN: APPENDIX  \nPhoenix, Arizona; 1979. Houses built underground in the 1970s were considered an ideal way to maintain year-round temperature without fuel expenditures.\n\n## Alternative Construction Methods\n\nThere are a number of new construction methods that combine elements of prefabricated construction and green construction. They substitute different systems for the wood stick framing, assembled on site, that is most commonly used to build new houses today. Three are illustrated here.\n\nThe most familiar is Manufactured houses (or Modular)\u2014completed houses built in factories, shipped to a site, and assembled. These can be either an entire pre-constructed house (a single-wide) or combinable sections of pre-constructed houses (double- and triple-wides). They are either placed on a foundation or have a skirt wrapped around their chassis, and the parts are sealed together.\n\nA second method is to construct walls and roofs with large, relatively solid panels that have insulation as their core. Pre-cut panels are shipped and assembled onsite; doors and windows are added, and interior and exterior finishes are added.\n\nA third method is to construct the walls from concrete forms that will remain permanently in place and have insulation built in. The forms are assembled and concrete is poured into them. Windows and doors are then added.\n\nAll three of these can have front or side porches, carports, or even enclosed garages added to the basic house.\n\n## Construction Changes in the 20th Century\n\nWhile the basic way of constructing a house\u2014balloon or platform framing created with two-by-four wood sticks and nailed joints\u2014remained the same through the 20th century, much of what encased and served that wood structural skeleton evolved.\n\nThe most far-reaching transformation has been to the \"guts\" of houses: the interior systems that circulate heat, air conditioning, water, natural gas, and electricity; that drain sewage and waste water; and that\u2014in the last few decades\u2014allow various forms of television and internet access. Many of these internal systems were in turn connected to much larger networks built and maintained by municipalities or utility companies. The room types and fixtures these various systems enabled have changed spatial allocations and costs of new construction in major ways. For example, bathrooms and kitchens as we know them today are quite new and their evolution in size and complexity during the 20th century is remarkable.\n\nElectrifying houses was perhaps the most momentous change, as it enabled so many other improvements. In 1907 only 8 percent of houses in the United States even had access to electric service, and not all of those homes were wired to take advantage of it. The spread of electric systems throughout the United States was amazingly rapid. By 1920, 34 percent of households had access to an electric supply, 63 percent by 1927 and 80 percent by 1941. And while at first the new power source was used mainly for light, within only a few decades a legion of electric appliances, large and small, were introduced\u2014and air conditioning became affordable.\n\nTwo major appliances\u2014the refrigerator and the clothes dryer\u2014greatly changed the character of rear facades and gardens in the mid-20th century. In 1923, there were only twenty thousand electric refrigerators in the United States (thus the almost ubiquitous exterior rear service porches that accommodated iceboxes and ice delivery). Made practical by Freon in the early 1930s, electrified refrigerators were quickly adopted. Only 5,000 refrigerators were manufactured in 1921, with that figure rising to 1 million in 1931 and then to 6 million in 1937. This was an astonishing number given that this was in the midst of the Depression. The fact that the FHA would include the cost of major appliances in the home mortgages they insured undoubtedly helped; and a careful cost accounting would likely show that the refrigerator was no more costly than adding a back porch for the icebox. Similarly, the much later spread of clothes dryers allowed large areas of rear yards, previously dedicated to clothes lines, to be converted to other uses.\n\nAs municipal water systems and sanitary sewer systems were built throughout the country, it gradually became feasible to bring water into houses. Interior bathrooms, for example, are a remarkably modern innovation. In marveling over how many and how large bathrooms are in new houses built today compared to houses built in 1950, it is easy to lose sight of the real headline\u2014in 1940, 45 percent of homes in the United States still lacked complete plumbing. Almost one-half of the houses in the United States did not even have all the _elements_ common to a bathroom located anywhere inside. Complete plumbing was defined as hot and cold running water, a bathtub or shower, and a flush toilet\u2014and it was _not_ a requirement that these all be available in a single room, only somewhere within the house or dwelling unit. By 1980 all but 2.7 percent of homes in the United States had complete plumbing, a remarkable transformation. Cable television and hard-wired internet access are recent examples of the rapid transformative spread of a service.\n\nLess dramatic than the lifestyle changes enabled by new technical systems has been the gradual evolution of ways to encase the wood stick structural frame with new cladding and insulating materials for both interior and exterior walls. Originally, frame houses had lumber wall sheathing, sub floor, and roof deck. Brick and other veneers were applied directly to the wood sheathing. In the mid-20th century this lumber sheathing was replaced by plywood or various types of fiberboard on the outside and sheetrock on the inside. In the 1980s a product made from wood scraps, oriented strand board (OSB), began to be commonly used to sheath the exterior of the wood frame. It was typically wrapped with a breathable and waterproof membrane before the exterior cladding was attached. Today a fundamental change is being experimented with\u2014particularly in 21st-Century Modern houses\u2014the use of a rain screen that is not directly attached. First a waterproof membrane is applied to a structurally sound box, often OSB. Then a water-shedding (note, water _-shedding,_ not necessarily waterproof) panel of cladding is added that is separated from the structural wall's waterproof membrane with spacers that allow an inch or more of space, providing air circulation. This allows the building to breathe, as well as providing an insulating space. It can also free the design of the exterior shell of a house from the actual structural systems, opening up possibilities that have begun to be explored in some civic buildings.\n\nAdvances in glass technology have opened up yet another avenue of change: allowing large panes of plate glass and thermal-insulated glass to create larger and larger windows. And as the price of producing these products fell, soon entire walls of windows became feasible. The first house to demonstrate window walls was the House of Tomorrow exhibited at the 1933 Century of Progress (George Frederick Keck, architect)\u2014where Keck discovered he had to use all kinds of screening devices on sunny days\u2014from Venetian blinds to curtains. However, it was not until interior air-conditioning became affordable after World War II that large window walls with sunny exposures were feasible. The ability to maintain the temperature of a house year-round not only allowed window walls to be placed without regard to orientation but also allowed solid walls without windows. It allowed house designs to abandon operable windows evenly placed on most wall surfaces to insure cross ventilation and instead made it feasible to design houses that located windows with greater attention to views, to privacy, and to furniture placement.\n\nAPPENDIX\n\n **ALTERNATIVE CONSTRUCTION METHODS:**  \nThese combine elements of both prefabricated and green construction\n\n**MANUFACTURED**  \n(also Modular)\n\nAPPENDIX\n\n **ALTERNATIVE CONSTRUCTION METHODS:**  \nThese combine elements of both prefabricated and green construction\n\n**STRUCTURAL INSULATED PANELS**  \n(panels contain an insulating foam core)\n\nAPPENDIX\n\n **ALTERNATIVE CONSTRUCTION METHODS:**  \nThese combine elements of both prefabricated and green construction\n\n**INSULATED CONCRETE FORMS**  \n(concrete molds create a permanent insulating surface)\n\n# Notes\n\n## Looking at American Houses\n\n1. Strict building codes enacted in wetlands or in earthquake-prone areas often have exceptions for a remodeled house (rather than a new house). This can lead to \"remodeling\" that essentially builds a new house on an old foundation. The same sort of rebuilding can be triggered by new and more restrictive zoning codes that apply only to a new house, not one deemed \"remodeling\" because it is built on the old foundation.\n\n2. There are also business and retail neighborhoods, and these are generally beyond the scope of this book.\n\n3. The census of 1800 defined an \"urban\" area as one with a population over 2,500. There were twenty-seven towns with populations between 2,500 and 10,000\u2014and some, such as Newburyport, Massachusetts, and Alexandria, Virginia, still retain portions of their early rural and urban streetscapes. Of these, five already had individual populations exceeding 10,000 (New York, Philadelphia, Baltimore, Boston, and Charleston). By 1810, there were thirty-five towns with populations between 2,500 and 10,000.\n\n4. John R. Stilgoe, _Common Landscape of America, 1580 to 1845_ (New Haven, CT: Yale University Press, 1982), 82.\n\n5. In much of the English Colonial South and Mid-Atlantic a pattern of larger and more self-contained plantations was common. Philip Pregill and Nancy Volkman, _Landscapes in History: Design and Planning in the Western Tradition_ (New York, NY: Van Nostrand Reinhold, 1993), 329\u201330. See pages 328\u201332 for other colonial patterns.\n\n6. Some small towns\u2014particularly company towns or towns with origins in mining, forestry, and fishing\u2014grew with great rapidity.\n\n7. Regional patterns might include central green spaces (such as a town green in New England, a courthouse square in Texas, or a central plaza in Spanish towns), extensive use of regional materials, or distinctive patterns of streets, house sites, or block size (such as is found in Mormon-planned towns). For a few western towns with distinct regional variations see Virginia McAlester and A. Lee McAlester, _A Field Guide to America's Historic Neighborhoods and Museum Houses: The Western States_ (New York, NY: Alfred A. Knopf, 1998): Butte, Montana, 372; Manti, Utah, 642; Nevada City, California, 127; Monterey, California, 115\u2013123; Santa Fe, New Mexico, 441\u2013452; Fredericksburg, Texas, 593\u2013596; and others.\n\n8. Only a few hundred inhabitants occupied many of the early rural neighborhoods. Today, areas with populations up to 10,000 are generally considered small towns. Even larger towns may still have a remaining core with \"rural neighborhood\" streetscapes. There is nothing magic about the figure 2,500 except that it was used by the census and is easily traceable as to how many towns had reached this size in census years. In 1880, there were 872 towns between 2,500 and 10,000. By 1890, there were 1,150. By 1900, there were 1,445, showing a steady increase of 270\u2013300 towns established per decade.\n\n9. Pregill and Volkman, 491\u201394. It is reported that between 1850 and 1870 approximately 7 percent of the total land area of the United Stated (129 million acres) was awarded to various railroad companies to incentivize the construction of transcontinental rail lines. \"Land and Freedom,\" accessed August 15, 2010, <http:\/\/www.landandfreedom.org\/ushistory\/us13.htm>. By 1870, there were about 53,000 miles of railway in the United States. See table featured in _One Hundred Years of American Commerce 1795\u20131895,_ ed. Chauncey Depew (New York, NY: D. O. Haynes), 143. It was essential to the railroads to establish small towns along these tracks\u2014to add potential passengers and also to have maintenance facilities throughout their systems.\n\n10. Dolores Hayden, _Building Suburbia_ (New York, NY: Pantheon Books, 2003), 181\u201397; Julie Campoli, Elizabeth Humstone, and Alex MacLean, _Above and Beyond: Visualizing Change in Small Towns and Rural Areas_ (Chicago, IL: Planners Press, American Planning Association, 2002), 37\u201345, 74\u201376, 153\u201361. In some cases, this has led to an old compact rural neighborhood adjacent to a newer neighborhood that looks like a post\u2013World War II suburb. This was both enabled and exacerbated by the interstate highway system which connected more small towns and helped change the pattern not only of residential but also of small-town Main Streets by adding commercial nodes at highway interchanges. In addition, an arterial with strip development, often anchored by a WalMart, might develop along a road between Main Street and the interchange.\n\n11. The distance considered \"walkable\" is often used in defining neighborhoods and their size. For a short illustrated history of neighborhood scale and walking radius, see \"Sustainable Neighborhood Planning for the Region\" (Treasure Coast Regional Planning Council, 2004), accessed September 22, 2011, <http:\/\/www.tcrpc.org\/orientation\/02_neighborhood_scale\/2_neighborhood_scale_print.pdf>. The area considered walkable is sometimes called a pedestrian shed and is typically considered today to be a five-minute walk or about a quarter-mile radius. In other countries and eras half-mile and one-mile radii have been used. For additional walkability planning elements, see Dhiru A. Thadani, _The Language of Towns and Cities: A Visual Dictionary_ (New York, NY: Rizzoli, 2010), 731\u201335. To determine if an address is in a walkable neighborhood today visit www.walkscore.com (although changing retail patterns mean that this score may not accurately reflect the era in which a neighborhood was first planned).\n\n12. Kenneth T. Jackson, _Crabgrass Frontier: The Suburbanization of America_ (New York, NY: Oxford University Press, 1985), 33\u201335. Introduced in France, the first omnibus in the U.S. ran along Broadway in New York City beginning in 1829. Despite not running on tracks, and providing a rather rough ride over oft-rutted streets, by 1853 there were 653 omnibuses in New York City alone, and they were found in many other urban areas. Also see Clay McShane, \"The Centrality of the Horse to the Nineteenth-Century American City,\" in _The Making of Urban America,_ ed. Raymond Mohl (New York: SR Publishers, 1997), 110.\n\n13. Hayden, 21\u201325; Henry C. Binford, _The First Suburbs: Residential Communities on the Boston Periphery, 1815\u20131860_ (Chicago, IL: University of Chicago Press, 1985), 83\u2013149.\n\n14. Jackson, 25\u201333. In 1814, the first steam ferry service opened up locations for development across water bodies. Brooklyn Heights (ca. 1823), sometimes described as a suburb because of its remote location\u2014a ferry ride across the East River from Manhattan\u2014was actually developed as an urban-neighborhood type with narrow lots and attached houses accessed by foot. \"Situated directly opposite the southeast part of the city, and being the nearest country retreat, and easiest of access from the center of business that now remains unoccupied; the distance not exceeding an average fifteen to twenty-five minute walk, including the passage of the river; the ground elevated and perfectly healthy at all seasons; as a place of residence all the advantages of the country with most of the conveniences of the city. Gentlemen whose business or profession require daily attendance into the city, cannot better, or with less expense secure the health and comfort of their families than by uniting in such an association\" (32).\n\n15. Jackson, 39\u201342; David L. Ames and Linda Flint McClelland, _Historic Residential Suburbs: Guidelines for Evaluation and Documentation for the National Register of Historic Places,_ National Register Bulletin (Washington, DC: U.S. Department of the Interior, National Park Service, National Register of Historic Places, 2002), 17. Also see McShane, 111\u201312. In Boston, commuters rose from 6 percent in 1846 to 18 percent in 1860. Under ideal conditions, the area one could commute (about three square miles if walking for thirty minutes, and about twelve square miles with a four mile-per-hour omnibus) increased exponentially to about twenty-eight square miles with a horsecar going eight miles per hour.\n\n16. For rural and urban population, see \"Selected Historical Decennial Census Population and Housing Counts,\" U.S. Census Bureau, Table 4, <http:\/\/www.census.gov\/population\/www\/censusdata\/hiscendata.html>. Pittsburgh, for example, went from 47,000 in 1850 to 156,000 in 1880, Milwaukee from 20,000 to 116,000, Chicago from 30,000 to 503,000, and Boston from 137,000 to 363,000. Towns that had barely existed in 1850, like Denver and Kansas City, had populations of over 100,000 by 1890\u2014thanks to the possibilities opened up by long-distance rail travel and shipping. The population of cities skyrocketed. Campbell Gibson, \"Population of the 100 Largest Cities and Other Urban Places in the United States: 1790 to 1990,\" U.S. Census Bureau, _Population Division Working Paper No. 27,_ Tables 8 and 12, June 1998, accessed July 14, 2010, <http:\/\/www.census.gov\/population\/www\/documentation\/twps0027\/twps0027.html>\n\n17. Horsecars facilitated urban extensions in cities throughout the United States, creating areas like Dupont Circle and Capitol Hill in Washington, D.C., Lafayette Square in St. Louis, and Union Square and Franklin Square in Baltimore, Maryland.\n\n18. Sam Bass Warner, _Streetcar Suburbs: The Processof Growth in Boston, 1870\u20131900_ (Cambridge, MA: Harvard University Press, 1962), 76\u201378.\n\n19. McAlester and McAlester, xxxii\u2013xxxiii, 578\u201380.\n\n20. Jackson, 41. New York City, where these were introduced in 1853, is a documented example of how rapidly the horsecar network grew. The city had 23 miles of track in 1856, expanding to 142 miles of track by 1860 when ridership reached about 100,000 per day. For an extensive survey, see Clay McShane, \"The Centrality of the Horse to the Nineteenth-Century American City,\" in _The Making of Urban America,_ ed. Raymond Mohl (New York, NY: SR Publishers, 1997), 105\u201330.\n\n21. McShane, 116\u201325; Jackson, 105\u201307.\n\n22. These three essentials were highlighted in an essay the author read but has not been successful in locating for this footnote.\n\n23. Ames and McClelland, 34. Two primary books were Andrew Jackson Downing's _Cottage Residences, or a Series of Designs for Rural Cottages_ (1842 and 1850) and Beecher's _Treatise on Domestic Economy_ (1841). Both were widely read, frequently reprinted, and together had a huge effect on both the popular press and public consciousness. Hayden, 26\u201344; Jackson, 45\u201352.\n\n24. \"During the decade 1830\u20131840, the total length of completed railroad lines increased from 23 to 2,808 miles, and during the next ten years, more than 6,200 miles of railroad were opened, bringing the total network up to 9,021 miles in 1850. The most intensive growth during this period was in the Atlantic and Seaboard states.\" \"American Railroads: Their Growth and Development,\" _Central Pacific Railroad Photographic History Museum,_ last updated October 24, 2004, http:\/\/cprr.org\/Museum\/RR_Development.html. As a specific example, by 1850 Boston had eighty-three commuter stops within a fifteen-mile radius. Fifty-nine separate trains served these stops in 1849. As railroads expanded westward, track miles grew up to 28,920 by 1860, 49,168 by 1870, 87,801 by 1880, and 163,562 by 1890. Ames and McClelland, 16\u201317; Jackson, 35\u201338; Ann Durkin Keating, _Chicagoland: City and Suburbs in the Railroad Age_ (Chicago, IL: University of Chicago Press, 2005), 7; United States Census Bureau, _Report on Transportation Business in the United States at the Eleventh Census 1890,_ 4, and United States Census Bureau, _Report on the Agencies of Transportation in the United States at Tenth Census 1880,_ 308\u201309.\n\n25. McAlester and McAlester, xxv, xxvi\u2013xxix. Each transcontinental railroad created new towns along their routes. The railroads also expanded existing cities; the first railroad in the country, the Baltimore & Ohio, was built to enlarge the trade area for Baltimore's harbor\u2014in an attempt to counter the dramatic advantage that goods shipped along the Erie Canal had given the port of New York.\n\n26. Keating, 7.\n\n27. Jackson, 99\u2013100.\n\n28. Hayden, 54\u201361; Jackson, 76\u201386.\n\n29. Jackson, 101\u201302.\n\n30. Ibid., 87\u2013100.\n\n31. Ames and McClelland, 18; Jackson, 111; a shift from eight million in 1890 to twelve million in 1910, see U.S. Census Bureau, Table 4, <http:\/\/www.census.gov\/population\/www\/censusdata\/hiscendata.html>.\n\n32. A related form of transit, cable cars had been invented in 1873 and proved particularly useful in areas with steep hills such as San Francisco. However, they were very expensive to install (requiring a central power plant) and never gained the broad popularity of horse-drawn and electric streetcars. See Jackson, 103\u201305, and McAlester and McAlester, 217\u201318, 514, 517.\n\n33. John Stilgoe, _Metropolitan Corridor: Railroads and the American Scene_ (New Haven, CT: Yale University Press, 1983), 289\u201393.\n\n34. These could extend up to one hundred miles from a large city. The longest interurban line in the South connected Dallas and Fort Worth to a string of small towns located along the fertile blackland prairie\u2014and once ran a few blocks from the author's home. Buildings associated with its relatively widely spaced stops still remain. See Hayden, 94, and Robert A. Rieder, \"Electric Interurban Railways,\" _Handbook of Texas Online,_ accessed January 18, 2011, <http:\/\/www.tshaonline.org\/handbook\/online\/articles\/eqe12>.\n\n35. Hayden, 93\u201394; Jackson, 112\u201313; McAlester and McAlester, 313, 578, 671\u201372. The St. Charles Streetcar Line in New Orleans is the oldest continuously operated electric trolley in the country. After operating from ca. 1873 to 1893 as a horse-drawn line, it was electrified in 1893. In 1964 it became the only operating line in New Orleans. <http:\/\/www.railwaypreservation.com\/vintagetrolley\/neworleans.htm> (accessed 4\/4\/2013).\n\n36. Trolleys also facilitated development around rural settlements by connecting them with cities. See Jackson, 118\u201320; Ames and McClelland, 18\u201320; McAlester and McAlester, xxxii\u2013xxxiii, 45\u201346, 68, 106, 156, 191, 313, 403, 480, 519, 578\u201379, 584\u201385, 671\u201372.\n\n37. See Prairie and Craftsman styles (particularly the American Four-Square), this page, for examples of these house styles.\n\n38. Ames and McClelland, 20. Streetcar lines facilitated shopping downtown for major purchases (producing a golden age for downtown department stores). Shopping for daily necessities was accommodated by small commercial buildings located at intermittent streetcar stops.\n\n39. Stilgoe, 299\u2013305; Jackson, 112\u201313.\n\n40. Jackson, 189.\n\n41. Joel Tarr and Josef Konvitz, \"Patterns in the Development of Urban Infrastructure,\" in _American Urbanism: A Historiographical Review,_ ed. Howard Gillette, Jr., and Zane L. Miller (Westport, CT: Greenwood Press, 1987), 211; Ames and McClelland, 22; Virginia McAlester, Willis Winters, and Prudence Mackintosh, _The Homes of the Park Cities, Great American Suburbs_ (New York, NY: Abbeville Press, 2008), 48, 87\u201389. Paving, as we know it today, was relatively rare prior to 1900. \"Paving\" at that time generally meant a gravel road, possibly with the dust subdued by oiling the gravel. New asphalt paving for streets of existing and developing neighborhoods was undertaken between 1910 and 1930. Post\u2013World War II neighborhoods were almost always paved.\n\n42. Examples include Highland Park West, see McAlester, Winters, and Mackintosh, 102; McAlester and McAlester, 76, 699 (Beverly Hills), 234\u201335 (St. Francis Wood), 575 (Highland Park West).\n\n43. McAlester and McAlester, 416 (Fairacres in Omaha) and 337\u201338 (North End in Boise).\n\n44. Joseph B. Mason, _History of Housing in the U.S. 1930\u20131980_ (Houston, TX: Gulf Publishing Company, 1981), 13.\n\n45. Ibid., 44\u201347.\n\n46. FHA, _Successful Subdivisions: Principles of Planning for Economy and Protection Against Neighborhood Blight,_ Land Use Planning Bulletin No. 1 (Washington, DC: GPO, n.d., ca. 1940).\n\n47. Most common in large areas of previously vacant land that was not subdivided until after World War II. Post\u2013World War II houses were built in many other settings: on still-vacant lots platted in earlier grid neighborhoods, and also in various subdivision patterns on small farms and large homesteads that were still present in built\u2011up areas.\n\n48. Cynthia L. Girling and Kenneth I. Helphand, _Yard, Street, Park_ (New York, NY: John Wiley & Sons, 1994), 97\u201398.\n\n49. Clarence Perry's principles for his \"Neighborhood Unit,\" first published in 1929, included schools, parks, and nearby retail as important parts of a neighborhood. This had a significant effect first on the 1931 President's Conference on Housing and through this on the FHA's land-planning standards, in particular their recommendation for inclusion of these amenities in large-subdivision plans. Ames and McClelland, 47\u201348.\n\n50. FHA, 13\u201314.\n\n51. Hayden, 154\u201397.\n\n52. Ames and McClelland, 23\u201324. During 1921\u20131936 alone more than 420,000 miles of road were built in the U.S., including intercity highways and farm-to-market roads, much of it with 50 percent federal subsidy from the 1916 Federal-Aid Highway Act. Jackson, 248\u201350. The interstate system was planned in the 1940s but did not receive their 90 percent federal funding until 1956. Earlier highways had been generally only 50 percent funded by the federal government.\n\n53. Hayden, 168\u201371; www.alliemae.org\/historyoffanniemae.html.\n\n54. Thadani, 666. Also called DOT (development-oriented transit). For an excellent overview of \"walkable urban\" neighborhoods see Christopher B. Lionberger, _The Option of Urbanism: Investing in a New American Dream_ (Washington, DC: Island Press, 2008), 86\u2013137. While not all accessible by transit, a goal for sustainable development is to create walkable urban areas at select rapid-transit stops. This book also describes alternative \"drivable suburban areas\" and separates these two viable alternatives from the \"Neverlands\" that make up many edge nodes and are not enjoyable to either walk or drive.\n\n55. Thadani, 664; Hayden, 201\u201329.\n\n56. Thadani, 209\u201310; see illustrations on pages 211\u201312. See page 604 for density needed to support commercial uses; for example, it takes one thousand dwelling units for a corner store, two thousand dwelling units for a convenience store, six to eight thousand dwelling units for a supermarket plus ten to fifteen smaller stores.\n\n57. Lionberger, 7. This is of a total package of U.S. assets that includes the value of natural resources, corporate equity, government bonds, and individual-citizen cash and even art collections.\n\n58. Mason, 70; The interstate portion was engineered prior to strict interstate guidelines.\n\n59. Richard O. Baumbach, Jr., and William E. Borah, _The Second Battle of New Orleans: A History of the Vieux Carr\u00e9 Riverfront-Expressway Controversy_ (Tuscaloosa, AL: University of Alabama Press, 1981).\n\n60. Jane Jacobs, _The Death and Life of Great American Cities_ (New York, NY: Random House, 1961). It is a loss not to yet have books similarly describing early suburbs\u2014streetcar, early automobile, and even post\u2013World War II first-ring suburbs. These are not the dense, mixed-use urban neighborhoods of the Jacobs book but are quite distinct from the far-flung later post-suburban SLUG development (called \"Neverlands\" by Lionberger) that has proved so costly to the economy and the environment. Preserving the unique qualities of the excellent early suburban environments has high value, as they are often well-located and livable, drivable suburban neighborhoods.\n\n61. National Historic Preservation Act of 1966, Community Reinvestment Act of 1977, Urban Renewal and Housing Act of 1949, National Housing Act of 1934. The National Historic Preservation Act of 1966 spurred surveys to list neighborhoods and landmarks on the National Register of Historic Places\u2014the officical listing that gave them protection against the use of federal funds for projects that would demolish them (or at least required a study of the alternatives to demolition). Because so little of the country had been fully surveyed, determining that an area was _eligible_ for listing also gave some protection.\n\n62. citiwire.net. Neal Peirce, Washington Post Writers Group, March 10, 2012. \"What's easily forgotten is how the deep gashes in America's city fabric occurred. The highway planners of 1950s and 1960s seemed unfazed by pushing massive highways straight through cities, devastating black and other low-income neighborhoods as well as cutting off waterfronts. It's a dark chapter in the American story. As my friend Peter Harnik writes in his Island Press book _Urban Green_ : 'Waterfronts were blockaded in Portland, Cincinnati, Hartford, Cleveland, Philadelphia and San Francisco. Nooses of concrete were wound tightly around the downtowns of Dallas and Charlotte. Trenches of noise and smog cut through Boston, Detroit, Seattle and Atlanta. Stupendous elevated structures threw shadows over Miami and New Orleans. And wide strips of land were taken from large, iconic parks in Los Angeles, St. Louis, Baltimore and San Diego ( _and San Antonio_ ).' Yet, as some of the intrusive roadways have collapsed, they've actually triggered amazingly rapid recovery and new prosperity. The collapse of a chunk of New York City's elevated West Side Highway in 1973 didn't, for example, cause the traffic Armageddon anticipated. As Streetsblog founder Aaron Naparstek noted at the Philadelphia Forum, substitution of a ground-level, more modest urban boulevard 'has made some of Lower Manhattan into some of the world's most valuable real estate.' \" And the same has happened in Boston, Massachusetts, where the Big Dig put an elevated freeway underground. The Congress for the New Urbanism has a list of the Top Ten Possible Freeway Teardowns.\n\n63. Michael Southworth and Eran Ben-Joseph, _Streets and the Shaping of Towns and Cities_ (New York, NY: McGraw Hill, 1997), 84.\n\n64. Hannah B. Higgins, _The Grid Book_ (Cambridge, MA: MIT Press, 2009), 79\u201398; \"Grid Plans,\" last modified on January 9, 2011, http:\/\/en.wikipedia.org\/wiki\/Grid_plan.\n\n65. The original 1862 version gave out 160 acres, or a one-quarter section. Later versions of the act addressed less productive land moving west, and they gave away half-sections and full-sections. Lee Ann Potter and Wynell Schamel, \"The Homestead Act of 1862,\" _Social Education_ 61, no. 6 (October 1997): 359\u201364. \"Teaching with Documents: The Homestead Act of 1862,\" National Archives, accessed July 3, 2010, <http:\/\/www.archives.gov\/education\/lessons\/homestead-act\/>; Milestone Documents (Washington, DC: National Archives and Records Administration, 1995), 55\u201356.\n\n66. Government policy was to distribute sections of land to the railroads in a checkerboard pattern as incentive for private companies to build rail lines, particularly in the West.\n\n67. For a remarkable overview of early town plans, see John William Reps, _The Making of Urban America: A History of City Planning in the United States_ (Prince-ton, NJ: Princeton University Press, 1965).\n\n68. Pregill and Volkman, 443\u201347. Denver, Kansas City, Boston, Brooklyn, Baltimore, Dallas, and Memphis are among the many cities that have streets with green medians linking neighborhoods together. In addition, many individual neighborhoods had streets with green medians included in their neighborhood plans\u2014a few examples are Riverhills in Houston, Munger Place in Dallas (today the portion with the median is part of the Swiss Avenue Historic District), Woodruff Place in Indianapolis, St. Francis Wood in San Francisco, Highland Park West in Dallas, and Guilford in Baltimore. After about 1940 planning generally shifted away from grand boulevards and turned to more utilitarian arterial streets coupled with large \"efficient\" highway systems and interior streets that did not invite outsiders. For a discussion of grand boulevards and side access lanes, see Allan B. Jacobs, Elizabeth Macdonald, and Yodan Rof\u00e9, _The Boulevard Book: History, Evolution, Design of Multiway Boulevards_ (Cambridge, MA: MIT Press, 2002), 44\u201353.\n\n69. Pregill and Volkman, 528\u201330.\n\n70. Andrew Jackson Downing, _A Treatise on the Theory and Practice of Landscape Gardening, Adapted to North America, with a View to the Improvement of Country Residences_ (New York, NY: Wiley and Putnam, 9th ed., 1875), 92\u201393. Downing contrasts the angular \"Plan of a Common Farm, before any improvements\" with a curvilinear \"Plan of the foregoing grounds as a Country Seat, after ten years improvement.\" These same theories were further promoted by his equally successful Victorian cottage residences (Dover Architectural Series [New York, NY: Dover Publications, 1842]) and in _The Horticulturist,_ a journal he edited during 1846\u20131852.\n\n71. Downing's book was the popular American voice of a romantic landscape movement that began in England and was first implemented in the United States in Mount Auburn Cemetery outside Boston, as well as other cemetery projects that followed. Also see Pregill and Volkman, 394\u201398, and Reps, 325\u201331.\n\n72. McAlester and McAlester, 698\u201399.\n\n73. Norman T. Newton, _Design on the Land: The Development of Landscape Architecture_ (Cambridge, MA: Belknap Press of Harvard University Press, 1971), 465\u201368; Hayden, 61\u201365; Jackson, 79\u201381. Olmsted's \"Report\" on Riverside also introduced the use of deed restrictions, tree planting, and other staples of neighborhood planning still used today. Olmsted (1822\u20131903) was widely considered the father of landscape architecture, and his son, Frederick Law Olmsted, Jr. (1870\u20131957), organized the country's first landscape design program at Harvard University in 1900. \"Olmsted planted no fewer than 7,000 evergreens, 32,000 deciduous trees, and 47,000 shrubs in Riverside.\" The Frederick Law Olmsted Society of Riverside Illinois, <http:\/\/www.olmstedsociety.org\/>.\n\n74. Ames and McClelland, 37; McAlester and McAlester, 77.\n\n75. FHA, 15, 18, 26\u201327.\n\n76. Mason, 113. _The Community Builders Handbook: Prepared by the Community Builder's Council of the Urban Land Institute, 1947_ ( _revised 1968_ ). 1968. 218\u201322.\n\n77. McAlester and McAlester, 698. In special circumstances, such as mining towns or the deepwater port of San Francisco, it was necessary to build towns on less regular topography.\n\n78. This might be a central town square, market square, courthouse square, neighborhood square, or, rarely, multiple squares. Circles also occurred. More irregular town greens were typical in parts of New England. A central plaza was considered so important by the Spanish that they actually legislated this space in their rules for building colonial settlements, the Laws of the Indies. See Reps for in-depth discussions of these and other plan types.\n\n79. Pregill and Volkman, 423\u201347. Parks featuring active athletic areas such as baseball fields and tennis courts became increasingly common during the early 20th century but are not included in this simple matrix of green areas.\n\n80. What happened to America's streetcar systems is an interesting topic with two hotly contested sides. It seems likely, based on both GM's desire to sell buses and the need to have broad arterial streets in existing cities, that GM played a strong, varied, and behind-the-scenes role in the conversion from electric streetcar to motor bus. By removing streetcar lines (a distinct advantage to motor bus sales), they gained more than ten thousand miles of broad paved city streets perfectly located to become automobile arterials with no expense beyond the removal of the tracks. There was no more cost-efficient or resource-conserving way to add automobile traffic into developed areas. This motivation is not discussed in the articles cited here that present two differing sides. \"General Motors Streetcar Conspiracy,\" accessed August 18, 2010, <http:\/\/www.fact-index.com\/g\/ge\/general_motors_streetcar_conspiracy.html>; Bradford C. Snell, \"American Ground Transport: A Proposal for Restructuring the Automobile, Truck, Bus and Rail Industries,\" report presented to the Committee of the Judiciary, Subcommittee on Antitrust and Monopoly, United States Senate, February 26, 1974 (Washington, DC: U.S. Government Printing Office, 1974), 16\u201324; Cliff Slater, \"General Motors and the Demise of Streetcars,\" _Transportation Quarterly_ 51 (1997). General Motors seems to have used a revolving-fund concept, buying streetcar companies, converting the cars to buses, reselling them with guarantees they would never be converted back to streetcars, and then taking the funds and buying the next company.\n\n81. McAlester and McAlester, 704\u201305.\n\n82. Ibid., 699\u2013700; \"Variety of American Grids,\" _Discovering Urbanism,_ last modified May 25, 2010, http:\/\/discoveringurbanism.blogspot.com\/2010\/05\/variety-of-american-grids.html.\n\n83. Ames and McClelland, 26; Marc A. Weiss, _The Rise of the Community Builders: The American Real Estate Industry and Urban Land Planning_ (New York, NY: Columbia University Press, 1987), 1\u20136. Land intended for uses other than residential is also subdivided.\n\n84. Warner, 37\u201338, 126\u201332; Hayden, 132.\n\n85. Weiss, 2.\n\n86. Ibid.\n\n87. McAlester and McAlester, 8\u201310.\n\n88. Weiss, 1\u20136.\n\n89. McAlester and McAlester, 8\u201310; Jackson, 203\u201306.\n\n90. United States Department of Veteran Affairs, \"G.I. Bill History,\" last modified November 6, 2009, <http:\/\/www.gibill.va.gov\/benefits\/history_timeline\/index.html>.\n\n91. Mason, 12\u201314; McAlester, Winters, and Mackintosh, 163\u201364. The inclusion of all major appliances with the mortgage made purchase even more attractive.\n\n92. Jackson, 197\u2013218; Kristen B. Crossney and David W. Bartelt, \"Residential Security, Risk, and Race: The Home Owners' Loan Corporation and Mortgage Access in Two Cities,\" _Urban Geography_ 26, no. 8 (2005): 707; William J. Wilson, _When Work Disappears: The World of the New Urban Poor_ (New York, NY: Knopf, 1996).\n\n93. This was called \"redlining\" because maps were annotated with red ink, showing the undesirable areas. Inner-city neighborhoods throughout the United States were redlined, effectively starving them of funds for home purchases or restoration. In 1975 Munger Place, an early-20th-century streetcar suburb in Dallas, Texas, was effectively redlined. Preservation Dallas formed a nonprofit revolving fund (this was a volunteer project chaired by the author) and the fund purchased two dozen houses, most slated for demolition by the City of Dallas because of their poor condition. The question was how to resell the houses. We were able to work with Fannie Mae's national office and with the Lakewood Bank and Trust Company, then the area's neighborhood bank, to create a demonstration lending project. The goal was to show that such areas could be restored if mortgages were made available. The Lakewood Bank made construction loans to bring the houses up to code, and Fannie Mae agreed to provide \"take-out\" letters guaranteeing to purchase the banks' construction loans and provide typical 20- and 30-year mortgages to the home-owners. The program pioneered a completely new way of lending for Fannie Mae and became available nationally. The new program was actively promoted to lending institutions throughout the United States by Don Wright, who had been Lakewood Bank's president.\n\n94. Board of Governors of the Federal Reserve System, \"Community Reinvestment Act of 1977,\" last modified May 4, 2011, <http:\/\/www.federalreserve.gov\/communitydev\/cra_about.htm>.\n\n95. This specific example is drawn from the deed restriction for the first plan of Munger Place in Dallas, Texas. Today many new high-end residential neighborhoods are also the most heavily governed, their zoning enhanced by deed restrictions sometimes to obtain a certain look.\n\n96. Ames and McClelland, 36. In 1924 the Standard State Zoning Enabling Act was written by Secretary of Commerce Herbert Hoover's Advisory Committee on Zoning, both to make it simpler for a state to allow zoning and to try to provide some legal continuity between the statutes of different states. Stuart Meck, AICP, _Model Planning and Zoning Enabling Legislation: A Short History,_ vol. 1 of _Modernizing State Planning Statutes: The Growing Smart Working Papers_ (American Planning Association, 1996) is a concise and understandable essay on this topic.\n\n97. Meck, 2\u20134. William I. Goodman and Eric C. Freund, _Principles and Practices of Urban Planning_ (Washington, DC: International City Managers' Association, 1968), 443\u201384. Comprehensive subdivision regulation spread almost as fast as zoning. By 1934, some 425 American cities had made subdivision regulation part of their planning departments.\n\n98. Christopher Leinberger, \"Sustainable Development and the Value of Urban Design,\" Dallas City Council briefing, January 2009. For a group of new walkable neighborhoods mostly built on former industrial sites, see Nate Berg, \"Cutting Car Use at the Neighborhood Level,\" _Atlantic Cities Place Matters_ , October 3, 2011, accessed September 27, 2011, <http:\/\/www.theatlanticcities.com\/commute\/2011\/10\/cutting-car-use-new-developments\/226\/>.\n\n99. McAlester and McAlester, vi. This approach is introduced on page vi and used throughout.\n\n100. Pregill and Volkman, 516\u201317.\n\n101. McAlester and McAlester, 703. Evergreen trees with branches that grow to the ground, such as southern magnolias, are not desirable as street trees as they obliterate any view of historic houses.\n\n102. Thadani, 239\u201340; Randall Arendt and Elizabeth A. Brabec, _Rural by Design: Maintaining Small Town Character_ (Chicago, IL: Planners Press, American Planning Association, 1994), 10.\n\n103. Girling and Helphand, 21\u201334; Residential Yard Typology chart, 29.\n\n104. Ames and McClelland, 52\u201353.\n\n105. Pregill and Volkman, 503\u201310.\n\n106. Ibid., 594\u201396; Leonard H. Johnson, _Foundation Planting_ (New York, NY: A. T. DeLa Mare Company, Inc., 1927), x\u2013xi.\n\n107. Ames and McClelland, 57\u201359; Rick Darke, _In Harmony with Nature: Lessons from the Arts and Crafts Garden_ (New York, NY: Friedman\/Fairfax Publishers, 2000); McAlester and McAlester, 705.\n\n108. Ames and McClelland, 60 and 69.\n\n109. According to geographer Terry G. Jordan's _Texas: A Geography_ (Boulder, CO: Westview Press, 1984), there were four original types of western land surveys; see, in addition, accessed July 3, 2010, http:\/\/en.wikipedia.org\/wiki\/Public_Land_Survey_System, for a good overview; McAlester and McAlester, 695\u201398; Pregill and Volkman, 328\u201344.\n\n110. Jordan, 200\u201304. He goes on to say that southerners knew no other survey method prior to about 1800.\n\n111. Under Spanish rule, Texas had land on waterways divided into long lots as early as 1731, according to Jordan, and in the 1830s independent Texas made \"this pattern of survey mandatory for lands lying adjacent to major streams.\" The long-lot survey was not confined to the Spanish colonies; it also was used by the French in parceling out land in the Mississippi River Valley. In addition, the Dutch distributed land along the Hudson River in large \"patroonships,\" with a sixteen-mile frontage along the Hudson and stretching as far back from the river as was \"reasonable.\" Also see Pregill and Volkman, 331\u201332.\n\n112. \"The Public Land Survey System (PLSS),\" last modified January 26, 2011, http:\/\/nationalatlas.gov\/articles\/boundaries\/a_plss.html.\n\n113. Where roads have been maintained along the one-mile lines, they are often obvious from the air. In some western cities the roads are broad arterials, lit at night with multiple rows of streetlights and clearly displaying the section lines first surveyed in accordance with the Land Ordinance of 1785.\n\n114. California and New Mexico still had large areas of public lands that were surveyed under the Land Ordinance system after they became part of the United States in 1848. Texas surveyed its own land and (for lands not lying along rivers) at first followed Spanish\/Mexican practice with an angled 45\u02da southwest\/northeast grid, later changing to a north-south system similar to the PLSS.\n\n115. Jordan, 202.\n\n116. Dell Upton, _Another City: Urban Life and Urban Spaces in the New American Republic_ (New Haven, CT: Yale University Press, 2008), 33.\n\n117. We understand this because of an unpublished 1990 paper by our friend Constance Adams titled \"The Visible City: Tracing Memory and Character in Dallas' Historical Landscape.\" While at Yale Architecture School, she was able to trace the varying angles of in-town Dallas streets back to the original land grants. It should be noted that when Texas entered the U.S. it was an independent republic and was allowed to retain its right to assign out its own public lands; the state eventually chose to voluntarily follow the general precepts of the Land Ordinance of 1785.\n\n## Folk Houses\n\n1. Daniel D. Reiff, _Houses from Books: Treatises, Pattern Books, and Catalogs in American Architecture, 1738\u20131950: A History and Guide_ (University Park, PA: Pennsylvania State University Press, 2000), 304. The statement of continuity into later years is observational by the author.\n\n2. Jay Dearborn Edwards, \"Shotgun: The Most Contested House in America,\" _Buildings and Landscapes: Journal of the Vernacular Architecture Forum_ 16 (2009): 62\u201396. Based on the aftermath of Hurricane Katrina, this paper traces the evolution of the shotgun houses of New Orleans.\n\n3. John Fraser Hart, Michelle J. Rhodes, and John T. Morgan, _The Unknown World of the Mobile Home_ (Baltimore, MD, and London: John Hopkins University Press, 2002), 1\u201335. This short history explains why definitive figures regarding manufactured housing vary from source to source. It includes excellent documented information that is used throughout this chapter, along with current statistics from the Manufactured Housing Institutes. See, accessed September 18, 2011, <http:\/\/www.allmanufacturedhomes.com\/html\/vintage_mobile_homes.htm>, for dated photos of mobile homes from the 1930s to 1975.\n\n## Colonial Houses\n\n1. Reiff, 27\u201333.\n\n2. A handful of urban-style houses were built in rural settings in the Albany region. These had the entrance on the long, non-gabled wall; in contrast, houses built in rows had the entrance in the narrow, gabled end, which always faced the street to conserve space on narrow, urban lots. Ironically, only these \"rural\" urban houses have survived, those of the larger towns having been destroyed by three centuries of urban growth.\n\n3. Joseph Manca, \"On the Origins of the American Porch, Architectural Persistence in Hudson Valley Dutch Settlements,\" _Winterthur Portfolio_ 40 (2005): 95\u201396, 99\u2013111.\n\n4. Inspired by Native American traditions, folk houses with crude, half-timbered walls and thatched roofs are also common in Spanish America. Known as _jacal_ or _palisado_ construction, the walls consist of vertical posts set in the ground to provide support for a framework of twigs covered with clay. An example appears to the right of the masonry house in Figure 3.\n\n5. David Gebhard, \"Some Additional Observations on California's Monterey Tradition,\" _Journal of the Society of Architectural Historians_ 46, no. 2 (1987): 170. This article traces the origin of these broad porches\u2014both cantilevered and two-story\u2014to the similar porches found in the \"southeastern United States, the Gulf of Mexico, the Caribbean, and Bermuda.\" Gebhard suggests Monterey's porches be considered Southern\/Caribbean elements, not \"New England\" ones, as has frequently been the case.\n\n6. David Gebhard, \"Some Additional Observations on California's Monterey Tradition,\" _Journal of the Society of Architectural Historians_ 46, no. 2 (1987): 169\u201370. See also Monterey (the revival of this type of Spanish Colonial house) on this page.\n\n7. Daniel D. Reiff, _Houses from Books: Treatises, Pattern Books, and Catalogs in American Architecture, 1738\u20131950: A History and Guide_ (University Park, PA: Pennsylvania State University Press, 2000), 24. In some cases carved architectural details were imported from England.\n\n8. Reiff, 23\u201333, 42\u201344.\n\n9. Reiff, 25. This was the only house built before 1750 in the U.S. that closely followed a design published in a book (Palladio's Book II, plate 56).\n\n10. Reiff, 31\u201332.\n\n11. Fanlights are found in only a few high-style Georgian houses but become almost universal in the Federal house; they also occur in the closely related Early Classical Revival houses.\n\n12. Reiff, 138\u201343.\n\n13. The full flowering of this new concern for archeological classicism came in the contemporaneous Early Classical Revival and subsequent Greek Revival houses.\n\n14. Reiff, 40\u201341.\n\n## Romantic Houses\n\n1. Reiff, 79\u201384. Latrobe also trained two pupils as architects, William Strickland and Robert Mills.\n\n2. Reiff, 66.\n\n3. A number of early-20th-century periodicals and books featured Swiss Chalets\u2014and considered their exposed wood structure to be related to Craftsman houses. See <http:\/\/www.dahp.wa.gov\/styles\/swiss-chalet>.\n\n4. John H. Martin, \"Orson Squire Fowler: Phrenology and Octagon Houses 1809\u20131887,\" _Crooked Lane Review_ 137 (fall 2005).\n\n## Victorian Houses\n\n1. Reiff, 79.\n\n2. Reiff, 133\u201335.\n\n3. George L. Hersey, \"Godey's Choice,\" _Journal of the Society of Architectural Historians_ 18 (1959): 104\u201311. This survey of the architectural styles of the house design offered by _Godey's Lady's Book_ and _Lady's Magazine_ from 1851 to 1891 shows Mansard as most common during the 1870s.\n\n4. Reiff, 86, 92\u201393.\n\n5. The refinement and spread of factory techniques to shape raw wood into endless intricate variations made it progressively easier to provide an elaborated front facade as the late 19th century progressed.\n\n6. Reiff, 86. Henry Hudson Holly, _Holly's Country Seats Containing Lithographic Designs for Cottages, Villas, Mansions, etc., with Their Accompanying Outbuildings, also country Churches, City Buildings, Railway Stations, etc., etc._ (New York, NY: D. Appleton, 1863). Designs Number 1 and 7 show applied horizontal and vertical stickwork. Robert P. Guter and Janet W. Foster, _Building by the Book: Pattern Book Architecture in New Jersey_ (New Brunswick, NJ: 1992), 141\u20137.\n\n7. For a detailed discussion of this movement and how it influenced Hunt and other American architects, see Sarah Bradford Landau, \"Richard Morris Hunt, the Continental Picturesque, and the Stick Style,\" _Journal of the Society of Architectural Historians_ 42 (1983): 272\u201389.\n\n8. This term was introduced in Vincent Scully, _The Shingle Style and the Stick Style: Architectural Theory and Design from Richardson to the Origins of Wright_ (New Haven, CT: Yale University Press, 1971).\n\n9. Janet W. Foster, _The Queen Anne House: America's Victorian Vernacular_ (New York, NY: Abrams, 2006), 16\u201317.\n\n10. Henry Hudson Holly, _Modern Dwellings in Town and Country Adapted to American Wants and Climate with a Treatise on Furniture and Decoration_ (New York, NY: Harper and Brothers, 1878). Holly began with a chapter titled \"The Queen Anne Style\" and proceeded not only to illustrate the style with plans but also to cover a range of topics related to the style, including a definitive illustration of the new type of entry\u2014a living stair hall\u2014a large roomy space, often with a fireplace and stairway designed so railings could be easily admired (pages 42 and 117).\n\n11. Foster, 18. Guter and Foster, 147\u201356.\n\n12. This preference was greatly encouraged by the 1893 World's Columbian Exposition in Chicago. Nothing was easier, after falling in love with the Classical columns that dominated the heart of the Exposition, than adding Classical detail to the shape and form of a house that was already being built with spindlework details.\n\n13. Jay Dearborn Edwards, \"Shotgun: The Most Contested House in America,\" _Buildings and Landscapes: Journal of the Vernacular Architecture Forum_ 16 (2009): 62\u201396. This presents a scholarly overview of the evolution of all types of shotguns in New Orleans. Lloyd Vogt, _New Orleans Houses: A House-Watcher's Guide_ (Gretna, LA: Pelican Pub. Co., 1985). This has excellent illustrations of the most common New Orleans Folk Victorian house forms and typical bracketed detailing.\n\n## Eclectic Houses\n\n1. FHA. _Principals of Planning Small Houses_ (1940), 28\u201329.\n\n2. Gavin Edward Townsend, \"The Tudor House in America: 1890\u20131930\" (Ph.D. dissertation, University of California, Santa Barbara), 1, 264\u201365. Two studies are reported here. First, the results of a study in _Architectural Record_ 's November 1925 issue that counts the styles of 571 houses published in architectural yearbooks and periodicals from 1923 to 1925. Second, Townsend's own count of the styles of 779 houses illustrated in _Architectural Record_ 's annual country house issue between 1910 and 1932.\n\n3. W. Barksdale Maynard, \"Best, Lowliest Style! The Early Nineteenth-Century Rediscovery of American Colonial Architecture,\" _Journal of the Society of Architectural Historians_ 59 (2000): 338\u201357. Maynard carefully traces the intellectual and nostalgic rediscovery of America's Colonial past during the early and mid-19th-century. He links it to both England's Picturesque movement and the 1876 Centennial.\n\n4. See David Gebhard, \"The American Colonial Revival in the 1930s,\" _Winterthur Portfolio_ 22, no. 2\/3 (summer\/autumn 1987), 109\u201348; and David Gebhard, \"Royal Barry Wills and the American Colonial Revival,\" _Winterthur Portfolio_ 27, no. 1 (spring 1992), 45\u201374.\n\n5. In England the word porch was used for an _entry_ porch\u2014a covered approach to the front door. In America the word porch has a far more inclusive meaning and often refers to a wider and deeper porch meant for use as an outdoor living space.\n\n6. T. Matlack Price, \"The Inherent Qualities of Building Materials: An Exposition of Considerations Governing Choice\" _Arts and Decoration,_ volume 5, number 11 (September 1915), 416. A builder estimated the cost of building a house from different materials. He began with a balloon-frame house clad with wood shingles that could be built at the base price of $10,000. Were the same house clad in clapboard it would cost $10,040. If it were covered in wire-lathe and stucco the cost would be $10,226. If it were constructed of hollow tile covered with stucco the price would be $10,681. If the house was instead constructed with solid brick walls the cost would be $11,272. Rough-finished stone walls would be $12,046\u2014and true half-timbered construction would cost $12,546.\n\n7. Gavin Edward Townsend, \"The Tudor House in America: 1890\u20131930\" (Ph.D. dissertation, University of California, Santa Barbara), 192\u20133.\n\n8. Gavin Edward Townsend, \"The Tudor House in America\" (Ph.D. thesis, University of California, Santa Barbara, 1986), 1, 265. Townsend includes two separate studies that count the number of times houses of the most popular styles appeared in architectural publications during the 1920s. These are primarily large architect-designed houses. It seems likely that when one-story builder houses are also included that the Tudor style would likely be closer to the higher number given here.\n\n9. Townsend, 103\u201310. This has a comprehensive discussion of the many different types and titles of books that were available.\n\n10. Townsend, 3\u201313 discusses the origins of these terms and others that are commonly used to describe Tudor houses.\n\n11. For example, stone homes dominate suburbs near Philadelphia (where Wissahickon schist was quarried locally) and homes of ochre brick with ironstone trim dominate Dallas (where local clays produced an ochre brick generally trimmed with reddish ironstone from nearby quarries).\n\n12. McAlester and McAlester, 518, 674. In Seattle, Washington, the homes designed by Bebb and Mendel with stone first floors and extensive half-timbering in the upper stories likely led to the popularity of heavy half-timbering there. In contrast, in Portland, Oregon, architect Wade Pipes (1877\u20131961) designed many simple stucco-clad homes, unadorned by half-timbering, an approach widespread in that city. Pipes had studied in England and was familiar with, and favored designing, the \"modern English home\" designs of Voysey and Lutyens. Individual details, such as false thatched roofs or clipped gables, could also be locally favored. In the hands of master architects, such as Frank Lloyd Wright's Nathan Moore House in Oak Park, Illinois, and Greene and Greene's Robinson House in Pasadena, California, the style assumed intriguing Arts and Crafts qualities. See Lee Goff, _Tudor Style_ (New York, NY: Universe Publishing, 2002), 177, 113. Most American architects practicing from 1890 to 1930 designed one or more houses in the Tudor style. This included both architects with a national reputation who were designing in many locations as well as architects whose work was generally confined to a single city or region.\n\n13. Although a Beaux Arts house may look like it is all constructed of stone (both walls and decorative trim) or like it has brick walls with contrasting stone decorative trim, chances are very high that the trim and decorative elements are made of glazed terra cotta or possibly cast stone. Glazed terra cotta is a hard-fired clay product that could be glazed to look like most of the stones that were typically used on houses. Cast stone is a concrete product with less versatility in its finishes. Both offered relatively inexpensive ways to provide door and window surrounds and a broad range of decorative details that resembled carved stone. They were not limited to the Beaux Arts, but were also used for the decorative and trim elements of masonry houses in many other Eclectic and late Victorian styles. In fact, glazed architectural terra cotta is so versatile in its finishes, and can so closely resemble stone, that few people are aware of how widespread its use was in the early 20th century. It was not limited to trim, but was also used to clad entire buildings. It was particularly useful for disguising the steel skeletons of tall buildings built before World War II. According to the National Park Service's Preservation Brief 7, below, many owners and architects today are \"surprised to discover that what they presumed to be a granite or limestone building is glazed architectural terra cotta instead.\" The National Park Service's Technical Preservation Services Publications are available online and give excellent short histories of the materials and topics they cover\u2014in addition to discussing preservation techniques. For more on terra cotta and cast stone see: Preservation Brief 7: De Teel Patterson Tiller, \"The Preservation of Historic Glazed Architectural Terra Cotta,\" Technical Preservation Services of the National Park Service (<http:\/\/www.nps.gov\/tps\/how-to-preserve\/briefs\/42-cast-stone.htm>); and Preservation Brief 42: Richard Pieper, \"The Maintenance, Repair and Replacement of Historic Cast Stone,\" Technical Preservation Services of the National Park Service (<http:\/\/www.nps.gov\/tps\/how-to-preserve\/briefs\/7-terra-cotta.htm>).\n\n14. Townsend, 1, 264\u20136.\n\n15. Townsend, 253\u201379. McAlester, Winters, and Mackintosh, 259\u201365, 325\u201329. Townsends's thesis has an excellent chapter on French Eclectic houses. _Homes of the Park Cities, Dallas_ has excellent examples of how the style was interpreted by Dallas architects in the 1930s\u2014as it likely was in other areas where construction continued during the 1930s. George N. Marble designed many French Eclectic houses for Dallas builders Dines & Kraft in the 1930s. In addition, Charles S. Dilbeck, who was, according to Willis Winters, \"the most prodigious and popularly admired residential architect ever to practice in Dallas,\" designed many creative houses. He first designed French houses in Tulsa, Oklahoma, from 1929 to 1932. He then moved to Dallas in 1932 and partnered with several different builders as he designed a wonderful repertoire of French houses. Dilbeck also designed many creative Ranch houses.\n\n16. Goodhue had encouraged the publication of and prepared the plans for _Spanish-Colonial Architecture in Mexico_ (Boston, MA: J. B. Millet, 1901) and had already designed buildings with Mexico's elaborate Spanish Colonial precedents (Sant\u00edsima Trinidad Cathedral in Havana, Cuba, and Hotel Col\u00f3n in Panama).\n\n17. Then called the Great War, it involved France, England, Italy, Austria, Russia, Germany, and others. Spain was not involved, protected by the Pyrenees Mountains. It made an excellent destination for a sketching trip after architecture school. This had been an important rite of passage for aspiring architects in the 19th century, when reproductions of photographs were prohibitively expensive. Traditionally, they filled sketchbooks with details they could use later in buildings they designed. Although in the 1910s books of photographs were becoming more widely available, such a trip was still of importance.\n\n18. Elizabeth McMillian, _California Colonial: The Spanish and Rancho Revival Styles_ (Atglen, PA: Schiffer Publishing, Ltd., 2002), 37\u201357, has an overview of the Spanish architectural eras found in the quite varied Spanish Eclectic. For many years, this style was referred to as Spanish Colonial Revival, a correct term for buildings based on those of Mexico, Cuba, and South America. However, because most of the 1920s publications were of Spain itself and formed the basis of many houses, the 1984 edition of this book used the term Spanish Eclectic. Today, architecture historians are beginning to use the more inclusive Spanish Revival.\n\n19. It is not unusual for Spanish features to be combined with Italian and even southern French details, creating houses that do not fit neatly into Spanish Revival, Mission, or Italian Renaissance. These are often called Mediterranean Revival, an even more inclusive name that covers a still wider range of variations and mixtures.\n\n20. David Gebhard, \"Some Additional Observations on California's Monterey tradition,\" _Journal of the Society of Architectural Historians_ 46, no. 2 (June 1987): 157\u201370.\n\n21. McMillian, 57\u201358; Mason, 14\u201315; Roland Coate, Sr., \"The Early California House: Blending the Colonial and California Forms,\" _California Arts and Architecture_ 35, no. 3 (1929): 22\u201323.\n\n## Modern Houses\n\n1. Larry Paul Fuller, ed., _The American Institute of Architects Guide to Dallas Architecture_ (New York: McGraw-Hill Construction Information Group, 1999): 89\n\n2. \"The Cape Cod Cottage,\" _Architectural Forum_ (March 1949), 101\u201306, author unknown. This example uses traditional balloon-frame wood construction.\n\n3. Minimal Traditional houses are also called \"small houses,\" \"Depression-era cottages,\" \"War Years cottages,\" \"Victory cottages,\" \"economical small houses,\" \"FHA Houses,\" and \"FHA Smalls.\" See \"The American Small House,\" a presentation of The Historic Preservation Division of the Georgia Department of Natural Resources (pdf available at georgiashpo.org\/historic\/housing).\n\n4. Joseph B. Mason, _History of Housing in the U.S. 1930\u20131980_ (Houston, TX: Gulf Publishing Company, 1982), 6\u20137. The 1933 starts were down a stunning 90 percent from 1925.\n\n5. McAlester, Winters, and Mackintosh, 164.\n\n6. Among these were: Editors of _Architectural Forum, The Book of Small Houses_ (New York, NY: Simon and Schuster, 1936); L. F. Garlinghouse Company, _Sunshine Homes_ (Topeka, KS: L. F. Garlinghouse, 1938); National Plan Service, Inc., _Homes of Moderate Cost_ (1941); _Home Builders Book of Low Cost Homes_ (St. Paul, MN: Brown-Blodgett Company, 1941); _House-of-the-Month Book of Small Houses,_ ed. Harold C. Group (Garden City, NJ: Garden City Publishing Co., Inc., 1946); John P. Dean and Simon Breines, _The Book of Houses_ (New York, NY: Crown Publishers, 1946); John S. Burrows, _Your New Home_ (New York, NY: Archway Press, Inc., 1948).\n\n7. _Principles of Planning Small Houses, Federal Housing Administration Technical Bulletin No. 4,_ revised July 1, 1940, 12\u201325. There were many versions of this booklet. Its 1940 edition shows multiple ways to arrange interiors and slightly vary the exteriors of similarly shaped small houses. It was a less illustrated May 1, 1936, version that became available as construction of Minimal Traditional houses heated up in 1936. Starts grew to 336,000 in 1937, 406,000 in 1938, 515,000 in 1939, and 603,000 in 1940. This bulletin also showed how to efficiently add heating, plumbing, kitchens, and bathrooms to a house. These elements had not previously been a requirement for very small homes\u2014and indeed had often not been the norm for them in earlier decades. Examples from the 1930s are shown in McAlester, Winters, and Mackintosh, 159\u201364, and discussed in McAlester and McAlester under Fairview, 10.\n\n8. Mason, 31\u201341. Builders closed up their current projects to take up residence near planned industry and build large developments of urgently needed defense housing for the workers relocating to wartime industrial complexes. Wartime production was extraordinary: \"71,000 naval ships; 300,000 aircraft; 100,000 tanks; 2.5 million trucks; 370,000 artillery pieces; 5.9 million bombs; 20 million small arms; 42 million rounds of ammunition; 5,400 cargo ships\" were built in the U.S. for the Allies to use during World War II (Mason, 34). All of this varied production utilized the fastest production methods possible.\n\n9. Mason, 41\u201344. Experience in rapid construction not only was taking place at home, it was also being gained on the front lines. The Seabees (the nickname of the Navy's Construction Battalion) had one thousand architects among its ranks. They were building bridges and assembling entire bases overseas in days to accommodate landing battalions, create instant airfields, and such. They used many ingenious approaches.\n\n10. Mason, 46. Ten million servicemen returned right at the end of the war. The number of servicemen eligible for these no-down-payment GI Bill loans eventually totaled sixteen million.\n\n11. Mason, 47, 49, 51, 53.\n\n12. Levittown, New York, on Long Island, is the best known of the rapidly built postwar developments. Its developer, William Levitt, had built housing for defense workers during the war and learned to improvise rapid building techniques. When he applied these to his first postwar development, he was building side-gabled Minimal Traditional houses and small Ranchettes (early Ranch houses). While Levitt garnered much of the publicity, many other developers were doing the same thing in other parts of the United States. Mason, 48\u201349.\n\n13. A very slightly revised version of _Principles of Planning Small Homes_ was published in 1946, signifying continued support for these houses. The story of a Dallas couple who tried for a slightly more \"modern\" house is told in \"The Mortgage Pattern Poses Problems: How a Texas Couple Stuck to Their Guns,\" _Living for Young Homemakers_ (January 1951): 108\u201314.\n\n14. The percentages stated here were derived from Ranch-style pattern book examples and from photographing suburbs of large cities. The following book counted house types in certain small towns and shows that the simple side-gabled and hipped-roof subtypes may be far more common particularly in rural areas. John A. Jakle, Robert W. Bastian, and Douglas K. Meyer, _Common Houses in America's Small Towns: The Atlantic Seaboard to the Mississippi Valley_ (Athens, GA, and London: University of Georgia Press, 1989), 186\u201390.\n\n15. Sometimes a grouping of several traditional double-hung windows is used in place of a picture window.\n\n16. Patrick Sullivan, Mary Beth Reed, and Tracey Fedor, _The Ranch House in Georgia: Guidelines for Evaluation_ (Stone Mountain, GA: New South Associates, 2010), 61. Richard Cloues has linked the widespread use of vine wrought-iron patterns to the real vines often photographed growing on the porch supports of California examples.\n\n17. Exposed rafter tails in Craftsman homes often extended beyond the edge of the roof and might have decorative shapes cut into them. Exposed rafters in contemporary houses often had a single board attached that hid the rafter ends, producing a more sleek and unified appearance.\n\n18. David Bricker, \"Ranch Houses Are not All the Same,\" California Department of Transportation, accessed August 18, 2011, www.nps.gov\/nr\/publications\/bulletins\/suburbs\/Bricker.pdf.\n\n19. The cap on the loan amount that the FHA would insure determined the square footage of speculative homes.\n\n20. _Principles of Planning Small Houses_ , 39\u201343; Alan Hess, _The Ranch House_ (New York, NY: Harry N. Abrams, Inc., 2004), 45. In addition, the _American Builder Magazine_ had regular features in the 1940s and 1950s showing how a few designs could be easily modified to produce a varied appearance\u2014by rotating the house orientation, flipping plans, varying the roof form, and using a variety of materials and\/or stylistic detail. The FHA also recommended not siting houses in a row but rather varying the setbacks.\n\n21. The Editorial Staff of _Sunset_ magazine in collaboration with Cliff May, _Sunset Western Ranch Houses_ (Menlo Park, CA: California Lane Publishing Company, 1946), 42. One of the magazine's illustrated house plans was presented with the headline, \"Where there's room, let the house ramble.\"\n\n22. Architects intensely studied the design of new, efficient floor plans for small houses during the 1930s and 1940s (there was little else to do during the Depression). Ranch-house interiors were designed for a more informal home life\u2014with kitchens opening onto casual eating areas, generally with an adjoining family room or den, and a sliding glass patio door or French doors opening onto a rear patio. Formal living and dining rooms were generally minimized in size or simply omitted.\n\n23. Among those who designed Ranch-style houses were California architects Cliff May, Paul R. Williams and William Wurster, and Dallas architect Charles S. Dilbeck. McAlester, Winters, and Mackintosh, 259\u201372.\n\n24. Hess, 14\u201317. The introduction to _The Ranch House_ eloquently expresses the need to fully understand and appreciate Ranch houses and the large neighborhoods they comprised. Builders played a far greater role than architects in designing Ranch houses, yet these houses were popular, and in many cases beloved, as family homes. The infrastructure and neighborhoods they created comprise a large part of our country's usable built environment.\n\n25. Building Research Council, \"The History of the Building Research Council,\" accessed October 2, 2011, http:\/\/brc.arch.illinois.edu\/history.htm. The council was founded in 1944 at the University of Illinois to consider how the university could help meet the housing needs of returning World War II servicemen. They undertook decades of research and created the concept of the \"kitchen work triangle,\" invented air-conditioning, and were the first to advocate roof truss construction. They have published more than two hundred highly respected pamphlets on many aspects of construction.\n\n26. G. Lewis Craig and Rudard A. Jones, AIA, for Small Homes Council-Building Research Council, \"Split-Level Houses,\" _University of Illinois Bulletin_ 58 no. 24 (1960). Circular Series Index Number C2.5.\n\n27. In 1946, there were only 17,000 televisions in the U.S. By 1949, the number of TVs purchased each month was 250,000. By 1953, 66 percent of Americans owned a TV; by 1960, 87 percent. See http:\/\/profcatcurrenthistory.wordpress.com\/2011\/05\/03\/television-ownership-drops-in-u-s-nielsen-reports-nytimes-com\/.\n\n28. See James Massey and Shirley Maxwell, \"Split Decisions,\" _Old House Journal_ (March\u2013April 2002): 78; and also \"On Long Island, Splits Outsell Ranches Four to One,\" _House and Home_ (April 1954): 111\u201321.\n\n29. \"Here Is a New Kind of Two-Story House,\" _House and Home_ (February 1954): 106\u201313, in a periodical for home builders, and in _Living for Young Homemakers_ for the popular market. This design was a part of architect Carl Koch's (1912\u20131998) efforts to create a prefab building system for houses. It was promoted as an entirely new kind of two-story house that had the long, low silhouette of a Ranch, included all the economies of a basement (but was livable as the ground floor), and had all the savings of an attic but was lighted and usable. This promotion was likely the reason that bi-level houses are often called \"raised ranch\" houses.\n\n30. Henry-Russell Hitchcock and Philip Johnson, _The International Style: Architecture Since 1922_ (New York, NY: Museum of Modern Art, 1932).\n\n31. Steel structural systems were also used in Contemporary houses where the steel structural _skeleton_ was more often emphasized, rather than the architectural _volume_ of the early International. The Post-and-Beam illustration (this page) shows how Contemporary houses were sometimes panelized\u2014featuring panels set into the structural system rather than having a smooth volumetric stucco covering. International houses from the 1950s and 1960s were far more likely to have steel structure exposed in the Miesian manner.\n\n32. Le Corbusier had incorporated much of the above in his Five Points of the New Architecture: 1) pilotis (columns) to raise the house above the ground, leaving free movement beneath; 2) a roof garden to replace the land lost beneath the house; 3) using the pilotis or columns to allow a free plan (plan libre) for the interior; 4) a free facade, responding to the needs of the interior; and 5) long, horizontal ribbon windows allowing even, generous exterior light and views outside.\n\n33. Charles Moore, Gerald Allen, and Donlyn Lynch, _The Place of Houses_ (New York, NY: Holt, Rinehart and Winston, 1974), 152\u201375.\n\n34. The Case Study house program, sponsored by John Entenza and the _Arts and Architecture_ magazine he edited, advanced similar principles\u2014including the use of industrial materials, steel, glass, courtyards, and view windows\u2014from 1945 until 1965. However, Case Study houses also were built of wood. Some Case Study houses were International, but others were Contemporary in style; they would all be called Mid-century Modern.\n\n35. Richard Weston, _Twentieth-Century Residential Architecture_ (New York, NY: Abbeville Press, 2002), 52. The term International architecture had been in use prior to the MOMA exhibit. Early International style architects, rebuilding Europe after World War I, believed the use of white stucco for a wall cladding material was very important. They were anxious to avoid the regional differences that they believed had led to that war. White stucco could be produced anywhere and was equally available internationally. Its use avoided the visible regional differences that were lent by nearby stone quarries, by brick colors produced from local clays, and by wood logged in timber-rich locales.\n\n36. Henry-Russell Hitchcock and Philip Johnson were responsible for the instant dissemination of the name and when they used it as the title of their exhibition catalogue, _The International Style._\n\n37. Perhaps the first house in the United States to capture the essence of the America's International style was Rudolf Schindler's King's Road House, built in 1922. Although the house has a bit of added modern ornament, its complete indoor-outdoor integration through full window walls with view gardens outside (originally sliding glass was canvas doors so the canvas could be removed in the summer), its ground-level foundation permitting the indoors to flow out, and the tilt-slab concrete structure allowing an unconventional open floor plan all express the essential spirit of Mid-century Modernism, both International and Contemporary.\n\n38. All of these industrial age \"guts\" of the building were relatively new in 1920, when only a fraction of the industrialized world even had access to electricity.\n\n39. At the same time, some working in this style are minimizing everything in an attempt to lower consumption of energy and products.\n\n40. The solid glass and steel columns of the Glass House and the Farnsworth House presaged the direction of International skyscrapers but rarely houses. The Glass House had a 50-acre site to provide the privacy needed to have such expanses of glass in a residence. The similar lack of privacy at the Farnsworth House was a constant trauma for Farnsworth, because her weekend home was far more open to public view.\n\n41. Neil Jackson, _California Modern: The Architecture of Craig Ellwood_ (New York: Princeton Architectural Press, 2002); 44\u201348. This was Craig Elwood's first house with a strong expression of structure; and architectural historian Reyner Banham considered it one of California's three seminal post\u2013World War II houses.\n\n42. Additional variations occur in architect-designed examples.\n\n43. Massey and Maxwell, \"Split Decisions,\" 78\u201383.\n\n44. Sandy Isenstadt, _The Modern American House: Spaciousness and Middle-Class Identity_ (Cambridge: Cambridge University Press, 2006), 198\u2013214.\n\n45. According to the exhibit, the Japanese garden itself was designed to be viewed from the house rather than to be used as an outdoor living area. Arthur Drexler, _The Architecture of Japan_ (New York, NY: Museum of Modern Art, 1955), 262. Drexler explained: \"A Japanese building was chosen by the museum for its third house in the garden because traditional Japanese design has a unique relevance to modern Western architecture. The characteristics which give it this relevance are post and lintel frame construction; flexibility of plan; close relation of indoor and outdoor areas; and the decorative use of structural elements.\" On the West Coast, Maybeck's early 20th-century description of architecture as \"landscape gardening around a few rooms\" anticipated the philosophy of the Contemporary style. Weston, 30.\n\n46. Open grille materials such as metal and unglazed terra cotta could be found, but concrete screen block was strong and the easiest to manufacture. Anthony Rubano, \"The Grille Is Gone: The Rise and Fall of Screen Block,\" _Preserving the Recent Past 2_ (Washington, DC: Historic Preservation Education Foundation, National Park Service, 2000), 3, 89\u201399.\n\n47. It is believed that Wright created the name \"carport,\" and that it was used first in connection with his first Usonian house, the Jacobs House.\n\n48. While split-levels worked well with gently sloping sites, where a site had more than an eight- or ten-foot drop, or access only at the very base or very top of a steep slope, these required a great deal of site preparation. The many varied approaches to Contemporary houses on steep slopes were less expensive to build and allowed much of the rock and natural vegetation on a site to remain.\n\n49. The width of roof overhang varies depending upon orientation and house design. Ideally, above window walls it is scaled to provide passive solar heating in winter and shading of glass surfaces in summer. Solid facades might not require this width.\n\n50. All three had banner dates in 1934. This was the year Libbey-Owens-Ford purchased the patent for double-glass insulating Thermopane windows; this made it practical to have an entire window wall, rather than just a large window _placed in_ a wall. In addition, thick plate glass was now manufactured by a more affordable method, placed in a variety of large view windows, and heavily promoted, particularly by Libbey-Owens-Ford, in print advertisements in professional journals. Their additional placement in shelter magazines was an early example of a \"pull\" ad campaign designed to get consumers to demand this product from design and building professionals. The campaign often emphasized the \"view\" as an important part of a room, and for the first time the view out of a house from primary windows began to have a monetary value. Sandy Isenstadt, _The Modern American House: Spaciousness and Middle-Class Identity_ (Cambridge: Cambridge University Press, 2006), 198\u2013210.\n\nSecond, the first plywood suitable for exterior use was introduced in 1934. See <http:\/\/www.fs.fed.us\/t-d\/pubs\/htmlpubs\/htm07732308\/index.htm>. By the 1950s plywood was affordably available in a smooth finish (often used for overhangs) and in a K1\u201311 finish that mimicked wood siding and became a popular inexpensive wall cladding. And finally, 1934 marked the first glued laminated timber (GLULAM) structure in the United States, a research lab for USDA Forest Products. New glues soon enabled many engineered wood products, including: structural plywood; oriented strand board; GLULAM, which allows versatility in the shaping of beams; structural composite lumber; and prefabricated wood I-joists, which are commonly used in floor and roof framing. For more detailed definitions see <http:\/\/www.apawood.org\/level_b.cfm?content=srv_med_new_bkgd_gloss>. Also see Thomas C. Jester, _Twentieth Century Building Materials: History and Conservation_ (New York, NY: McGraw-Hill, 1995), 132, 137.\n\n51. Pietro Belluschi had built a Contemporary house for a bachelor outside Portland that had a handsome front-facing gabled window wall and post-and-beam porch by 1937 and followed with others of the same ilk. This tradition followed him east when he became dean of MIT's architecture school in 1950. Pietro Belluschi and Jo Stubblebine, _The Northwest Architecture of Pietro Belluschi_ (New York, NY: F. W. Dodge Corp., 1953), 66.\n\n52. The feeling of interior spatial volume was accomplished through varying the ceiling's height, adding high clerestory windows, and using room dividers that did not reach the ceiling.\n\n53. Paul Adamson, Marty Arbunich, and Ernest Braun, _Eichler: Modernism Rebuilds the American Dream_ (Layton, UT: Gibbs Smith, 2002).\n\n54. Many sources have noted that the FHA chose not to finance Contemporary houses for various reasons. One well-documented case is discussed in Patsy Swank's \"The Mortgage Pattern Poses Problems: How a Texas Couple Stuck to Their Guns,\" _Living for Young Homemakers_ (January 1952): 108\u201314.\n\n55. James Duff Law coined the word \"Usonian\" in an attempt to create a term that, unlike \"American,\" was more truly linked to the United States. Thus the suggested term \"Usonian,\" derived from U.S.-onian, for the United States. Frank Lloyd Wright adopted the term, and it became alternatively identified with a particular set of his small houses.\n\n56. He preferred carports, and a somewhat L form to the house that turned its back to the street, sheltering a private courtyard with the narrow end and allowing a long expanse of glass onto the courtyard on the long end. There was no dining room\u2014instead incorporating this function into the large living area that was the heart of the house. The kitchen was in the heart of the house and allowed various degrees of interaction between the cook (now considered to be the mother), the main room, and views of the garden. He was to build more than sixty Usonians over the next twenty years in a wide variety of materials and design but always with the same principles.\n\n57. In the 1910s and 1920s, the Prairie house was modifying floor plans of large houses to include more T and L plans. These plan types created more spaces that were one room deep and end rooms with light from many directions. In the 1930s the later one-story Usonian incorporated these principles and began to influence Ranch houses and then the Contemporary.\n\n58. Frank Lloyd Wright, _The Natural House_ (New York, NY: Horizon Press, 1954), 54\u201358. The interiors of Wright's small houses generally had a dominant solid wall on one side (with windows floating above it) with open window walls on the other.\n\n59. In some areas, overhanging eaves were used, often with exposed rafters. The use of an overhang was more practical in areas with snow.\n\n60. Donlyn Lyndon and Jim Alinder, _The Sea Ranch_ (New York, NY: Princeton Architectural Press, 2004), 13\u201331. Located on a former sheep ranch on the coast along California Highway 1, its four-thousand-acre site had been used for grazing and as timberland. The sheep ranch portion was divided into numerous separate pastures that swept from the sea cliff up to the forested area. Planted cypress hedgerows had been planted as windbreaks along the fence lines. Innovative landscape planning was designed to leave large portions of the meadows in their natural states and group structures toward the hedgerows. The forested parts of the sites had houses carefully placed within them. Natural materials (unpainted wood primarily) are used, and nonnative plants are only allowed inside a home's small fenced enclosure.\n\n61. Ann Abernathy, AIA, interview, 2012 on recollections of MIT's design studio. \"In the late '60s, early '70s, shed roof followed naturally from working with wood systems, unit masonry, even unit concrete systems, and so all our design projects had them. Only reinforced concrete did not imply the shed roof. MIT was not in the camp of Corbusier (Harvard) or Kahn (Penn). If anything, there was more of a Wrightian connection, but we studied vernacular architecture from all over, and primitivism in cultures from Mesoamerica and Africa. Pietro Belluschi (early work), Saarinen, Aalto, Mies (early work), Paul Rudolph, and the beloved Kevin Lynch had a strong influence. From California, we were influenced by Greene and Greene and Bernard Maybeck. We looked for examples of shed roofs in New England vernacular residential, farm, maritime, and warehouse buildings. In the Midwest and West, we looked at stockyards and mining structures, and grain elevators. Everyone was looking at 'architecture without architects.' \"\n\n62. Gerald Allen, _Charles Moore_ (New York, NY: Whitney Library of Design, 1980), 30\u201337. Moore dramatized interior space within the tall slanted-roof vertical forms through use of varied levels and light.\n\n63. \"Building Types Study: Record Houses of 1967,\" _Architectural Record_ ( _New York City: The Record and Guide: 1967_ ): 42\u201347, 58\u201361, 72\u201375, 94\u201397, 100\u201303, 112\u201315.\n\n64. John Mullen, FAIA, and Ann Abernathy, AIA, have independently reported the almost omnipresence of the Shed style and its formal principles during their respective graduate careers at Yale in the late 1960s and MIT in the 1970s.\n\n65. \"Architecture Without Architects\" was an influential 1964 exhibition at New York's Museum of Modern Art. It was organized by architect Bernard Rudofsky, who also published a book on the same subject: _Architecture Without Architects: An Introduction to Non-Pedigreed Architecture._\n\n66. The organization Friends of Kebyar was founded in 1983. Its tours and its seventy-seven journal issues are one of the best ways to learn about the full breadth of the Organic architecture movement.\n\n67. Chad Randl, _A-frame_ (New York, NY: Princeton Architectural Press, 2004), 9\u201329. The A-frame house has many antecedents found in other cultures. It was Rudolf Schindler, however, who brought this form into the modern era with his Bennati House near Lake Arrowhead, California, in 1934\u201337. In this house Schindler introduced many features that would become typical of postwar A-frames, chief among them fully glazed gable ends.\n\n68. While the A-frame was by far the most common of the alternative roof forms, other shapes occurred. Adding a new roof form to otherwise simple houses was discovered to be a good way to create innovative modern designs\u2014and win architectural awards! High-style roof variations included airplane, hyperbolic paraboloid, paper-fold, umbrella, inverted umbrella, and multi-arched roofs. While frequently included in architecture books, examples of these are rarely encountered in real life.\n\n69. This was a challenge for all ceremonial buildings during the 1950s. Oscar Niemeyer used Formalism in many different forms for the buildings of Brasilia in 1956. His Itamaraty Palace there has attenuated arches of the type that came to adorn many different building types in the United States. Philip Johnson used this style for the Amon Carter Museum of Art (Fort Worth, 1961), where the segmented arches and \"chicly tapered columns\" helped coin the term \"ballet classicism\" for his Formalistic buildings. See Marcus Whiffen, _American Architecture Since 1780: A Guide to the Styles_ (Cambridge, MA: MIT Press, 1969), 256\u201361.\n\n70. See Whiffen, 275\u20139 for the midcentury view of Brutalism. Today there is a reaction against this style, making it endangered. Prince Charles has been very vocal about his disdain for Brutalist architecture. Speaking at the 150th anniversary of the Royal Institute of British Architects, he stated, \"For far too long, it seems to me, some planners and architects have consistently ignored the feelings and wishes of the mass of ordinary people in this country.... To be concerned about the way people live, about the environment they inhabit, and the kind of community that is created by that environment, should surely be one of the prime requirements of a really good architect.\" To see the entire speech, see, \"A Speech by HRH The Prince of Wales at the 150th anniversary of the Royal Institute of British Architects (RIBA), Royal Gala Evening at Hampton Court Palace, 29th May 1984,\" <http:\/\/www.architecture.com\/TheRIBA\/175thAnniversary\/AnnualLecture\/speech.aspx#.UdW1JBZYndc>.\n\n71. A gambrel roofline visible from a distance does allow one to orient to the original house form.\n\n72. K. Michael Hays, _Architecture Theory Since 1968_ (Cambridge, MA: MIT Press, 2000), 379.\n\n73. Frank Gehry, \"A Jazz Riff, in Corrugated Steel,\" _The Wall Street Journal,_ November 22, 2012.\n\n74. Victoria Ballard Bell's _Materials for Design_ (New York: Princeton Architectural Press, 2006) has sections on how glass, concrete, wood, metals, and plastics are used on building exteriors today.\n\n75. In some places groupings developed by chance\u2014such as second-home areas where experimentation seemed natural, near universities with architecture schools, or near an influential architect. In the 1950s and 1960s, some subdivisions were built by developers with an individual interest in modern design who chose to build and sell modern spec houses (mainly Contemporary). Today there are a number of new developments under way that are limited to modern houses. Incomplete citations from the Urban Reserve's website follow. David Sokol, \"The Sagaponac Effect: Modernist Subdivisions Multiply,\" _Architectural Record_ (March 25, 2008); Peter Hellman, \"Modern Developments,\" _Metropolitan Home_ (December 2005\u2013January 2006); Gregory Ibanez, \"Suburban Revolution,\" _Texas Architect;_ David Dillon, \"Urban Oasis,\" _Dallas Morning News,_ FDluxe.com (October 2008), 17.\n\n76. McAlester and McAlester, 24; Christine Hunter, _Ranches, Rowhouses & Railroad Flats_ (New York and London: W. W. Norton, 1999), 117\u201319.\n\n## Styled Houses Since 1935\n\n1. Enforcement is generally through a public or private design review body, or occasionally comprised of city staff.\n\n2. John Chase, _Glitter, Stucco & Dumpster Diving: Reflections on Building Production in the Vernacular City_ (New York, NY, and London: Verso, 2000), 87\u201392.\n\n3. Matt Tyrnauer, \"Glamour Begins at Home,\" _Vanity Fair_ (March 2009), accessed April 18, 2012, <http:\/\/www.vanityfair.com\/style\/features\/2009\/03\/john-woolf200903?currentPage=1>.\n\n4. The transition to Styled Ranch and New Traditional Houses can be seen in pattern books in the early 1970s. For example, Andy Lang's _101 Select Dream Houses_ (published in 1972 by Hammond, Inc., for the Associated Press) presents approximately 14 percent Styled Ranch designs along with 22 percent Ranch, 22 percent Contemporary, 22 percent two-story New Traditional, 8 percent Split-Level, 4 percent A-frame, 4 percent Mansard, 2 percent Cape Cod, 1 percent Shed, 1 percent Brutalist, and one unidentifiable house.\n\n5. Cooper\/Roberts Architects, AIA, and Phoenix (AZ), _Historic Homes of Phoenix: An Architectural & Preservation Guide_ (Phoenix, AZ: City of Phoenix, 1992), 111.\n\n6. Michael Walsh and Richard Toglia, _Tudor Houses_ (Farmington Hills, MI: Home Planners, 1989), 9. This is particularly obvious when the timbers are used to cover over the seams left by four-foot-wide wall sheathing. It is important to remember that half-timbering in original early English precedents was created from the actual structural timbers that were infilled and left exposed. The patterns thus created were logical and structural.\n\n7. Mason, 71.\n\n8. Home Planners, LLC, _European Dream Homes_ (2nd printing, 2001), and Don Sater, _European Classics_ (Sater Design Collection, 2007). _European Classics_ and Sater's other house plan books illustrate the specialized floor plans\u2014with numerous angular, rounded, and squared bays\u2014that are produced, and enabled by, the complex roof forms. Complex floor plans that produce complex roof forms are not limited to pattern-book examples. Even a few architects designing custom houses draw \"ideal\" floor plans first and then worry about the roof forms later. The author was sitting in a design review for a new home when suddenly the client turned to the architect and said, \"Oh dear, I forgot to get you to include space for my exercise bike in the library.\" The architect said, \"No problem, we will just make this little addition to the floor plan for it\"\u2014and proceeded to draw a small extension to the floor plan, just enough for an exercise bike. The architect then turned to the builder and said, \"Can you just frame up some sort of roof over it?\" The builder said, \"Sure, I can frame up a little roof over it.\" And thus another cascading hip was added to an already complex roof plan\u2014with the roof form being shaped by the complex floor plan.\n\n9. David Dillon, \"Big Mess on the Prairie,\" _Dallas Morning News_ (October 2, 1994).\n\n10. In many homes without a tall, arched one-and-one-half- or two-story added entry feature, it is possible to see the same general pattern expressed in the front door and a \"feature\" window above it. A _Wall Street Journal_ real estate article reported on the new popularity of towers, and actually counted the number of references to towers in current real estate ads. Dan F. Sater, II, _European Luxury Homes_ (Bonita Springs, FL: Sater Design Collection, 2003), and _Mediterranean Home Plans_ (Bonita Springs, FL: Sater Design Collection, 2005), feature Millennium Mansion designs with towers. Square or rectangular towers are more common on Mediterranean designs; round or octagonal towers on others.\n\n11. Historically, varied-size windows on houses are related by their details. Today, new window manufacturers typically offer a broad range of sizes for each window design they offer. Using windows from this same design family gives a unity even when sizes vary. Some early Millennium Mansions used windows from different design families, giving them a less related overall appearance.\n\n12. This free mix of historic details\u2014that includes the application of flat details (such as a single-layer \"wallpaper\" wall of brick) and use of oversized details (such as overlarge Palladian windows)\u2014may reflect builder attempts to produce postmodern architecture, with its witty use of symbolic architectural details.\n\n13. It might soon be possible to have a Styled Millennium Mansions section in addition to simply having Millennium Mansions\u2014much in the same way that Styled Ranch can be distinguished from Ranch. Tudor is easily applied to the subtypes of hipped roof with lower cross gables and gabled roof with lower cross gables, and French to the subtypes of hipped roof with lower cross-hipped roofs. Spanish Revival, with its highly varied roof forms, is also easily applied. The most problematic are Classical and Colonial styles\u2014yet elements of these are clearly visible on some Millennium Mansions, particularly Palladian windows, red-brick keystone lintels, and occasionally a two-story entry with Classical columns.\n\n14. Some municipalities have taken steps to ensure that this happens. The City of Beverly Hills, California, for example, in 2004 adopted the _Residential Design Style Catalogue_ which describes a number of appropriate styles and incentivizes New Traditional houses rather than Millennium Mansions. The catalogue specifically stresses the importance of proper roofs, materials, and window design. See City of Beverly Hills, _Residential Design Style Catalogue_ (Planning Division, March 2004), 87\u201388. Ironically the incentive for a \"calmer\" style is a slightly larger house.\n\n15. Marianne Cusato, Ben Pentrath, Richard Franklin Sammons, and Leon Crier's _Get Your House Right_ (New York, NY: Sterling, 2007) and Stephen A. Mouzon and Susan M. Henderson's _Traditional Construction Patterns_ (New York, NY: McGraw-Hill, 2004) are two very useful references for those trying to understand or reproduce historically correct details for a new house or addition. Some of the early 20th-century books used by Eclectic era architects are available today as well.\n\n16. An important exception are homes inspired by Colonial-era house styles, in particular early wood-clad New England homes, which would properly be placed quite low to the ground.\n\n17. Dan Cooper, _New Classic American Houses: The Architecture of Albert, Righter & Tittmann_ (New York, NY: Vendome Press, 2009), 10. The entire book is filled with examples of this process, often with excellent explanation (11, 17\u201324).\n\n18. ICAA was originally two separate organizations that merged in 2002. Today it has strong education, publication, and advocacy programs that promote the Classical tradition in architecture and urbanism. It also works with non-architect home builders and Habitat for Humanity International to bring traditional design skills to a broad cross-section of the country's housing and neighborhoods.\n\n19. It should be noted that Venturi was trying to add new dimensions of thought and ornamentation to modernism, likely not to encourage New Traditional architecture.\n\n20. It is almost impossible to know the actual distribution of New Traditional styles. The following paragraphs are based on the author's observations; on notes from Larry Boerder, AIA; and on a survey of house styles built in Highland Park, Texas. McAlester, Winters, and Mackintosh, 216.\n\n21. Vincent Scully, _The Shingle Style Today or the Historian's Revenge_ (New York, NY: G. Braziller, 1974), illustrations 1, 63, 107, 112, 115. Scully writes about these early beginnings.\n\n22. Many of the monographs of New Traditional architects include one or more Shingle houses, and handsome full-color books continue to appear on the Shingle style. For early examples see Peter Arnell and Ted Bickford, editors, _Robert A. M. Stern: Buildings and Projects: 1965\u20131980_ (New York, NY: Rizzoli, 1981), 18, 72, 178; Leland M. Roth and Bret Morgan, _Shingle Styles: Innovation and Tradition in American Architecture 1874 to 1982_ (New York, NY: H. N. Abrams, 1999), 212.\n\n23. The Colonial Revival style works well with codes as there is a relatively simple set of rules, followed from the 17th until the early 20th centuries, that if followed ensure that such a house will fit in well with its predecessors. Kentlands was planned by Andrus Duany and Elizabeth Plater-Zyberk, and the neighborhood's first model homes were ready in 1990.\n\n24. Willis Cecil Winters, FAIA, author of many books on Texas architects, developed a set of similar criteria that he shared with the author and on which these identifying features were based. His list includes the standing-seam metal roofs favored on American Vernacular houses in Texas.\n\n25. Urban Design Associates, Louisiana Recovery Authority, and LRA Support Foundation, _Louisiana Speaks: Pattern Book_ (Baton Rouge, LA: LRA Support Foundation, 2006). See also Urban Design Associates, Ray Gindroz, and Rob Robinson, _The Architectural Pattern Book: A Tool for Building Great Neighborhoods_ (New York, NY: W. W. Norton and Company, 2004), 41\u201346, 77\u201378, 92\u201395, 112\u201313, 124\u201327.\n\n26. Marc Treib, _An Everyday Modernism: The Houses of William Wurster_ (Berkeley, CA, Los Angeles, CA, London: University of California Press, 1995), 98. Some have called this an early Ranch house, but it is more understated than the later Ranch style.\n\n27. The articles were \"An Indigenous Architecture\" (1928), which featured Ford's sketches of German folk houses and Williams's designs inspired by them; and \"Towards a Southwestern Architecture\" (1931).\n\n28. In addition to Texas and California, Louisiana, the Carolinas, and the Pacific Northwest all had early regional architecture movements.\n\n29. In addition to introducing what became the Shed style, with its prominent use of shed elements inspired by vernacular structures, simpler shapes were also built at Sea Ranch.\n\n30. Among them, Frank Welch, John Mullen, William McDonald, and LakeFlato (particularly with their Porch Houses) in Texas; David Salmela and Sala Architects in Minnesota; Lloyd Vogt, Marianne Cusato, and Duany Plater-Zyberk & Company in the Gulf Coast states, and Estes\/Twombly and Jeremiah Eck in New England. Hugh Newell Jacobsen, FAIA, of Washington, D.C., a master of American Vernacular forms, utilizes a highly additive approach.\n\n31. Thadani, 267\u201368.\n\n32. \"Cusato Cottages,\" accessed November 16, 2011, http:\/\/cusatocottages.com.\n\n33. Books include Jean Rehkamp Larson, _The Farmhouse: New Inspiration for the Classic American Home_ (Newtown, CT: Taunton Press, 2004); William Morgan, _Yankee Modern: The Houses of Estes\/Twombly_ (New York, NY: Princeton Architectural Press, 2010); and Jeremiah Eck, _The Face of Home: A New Way to Look at the Outside of Your House_ (Newtown, CT: Taunton Press, 2006). A pattern book example is Looney Ricks Kiss Architects, _Traditional Neighborhood Home Plans_ (Tucson: Home Planners, LLC: 2000).\n\n34. David Dillon, _The Architecture of O'Neil Ford: Celebrating Place_ (Austin: University of Texas Press, 1999), 41.\n\n## Appendix: Approaches to Construction in the 20th and 21st Centuries\n\n1. Mason: 56.\n\n2. A Navy team had designed the versatile prefab Quonset huts in a one-month effort as the war was beginning. More than 150,000 Quonset huts were subsequently produced and used for a broad range of purposes during the war. Gwendolyn Wright, _USA: Modern Architecture in History_ (London: Reakton Books, Ltd., 2008), 142\u201344.\n\n3. A reported three hundred home builders of prefabricated homes were operating in 1946, many with federal subsidies as efforts were made to convert wartime plants to peacetime uses. See Andrew Blauvelt's essay for the Yale University School of Architecture's exhibition \"Some Assembly Required: Contemporary Prefabricated Houses,\" October 27, 2006\u2013February 2, 2007; Paul Sahre, _Leisurama Now: The Beach House for Everyone (1964\u2013 )_ (New York, NY: Princeton Architectural Press, 2008); Alastair Gordon, _Weekend Utopia: Modern Living in the Hamptons_ (New York, NY: Princeton Architectural Press, 2001).\n\n4. Aaron Britt, Diana Budds, Jaime Gross, Jordan Kushins, and Miyoko Ohtake, \"Special Report: The Next Generation of American Prefab,\" _DWELL_ (December\/January 2012): 72\u2013114.\n\n5. Donovan Rypkema, \"Sustainability and Historic Preservation,\" from the Economic Benefits of Preservation Session at the Historic Districts Council Annual Conference in New York City, March 10, 2007, accessed October 5, 2011, <http:\/\/www.preservation.org\/rypkema.htm>.\n\nSee <http:\/\/www.census.gov\/hhes\/www\/housing\/census\/historic\/plumbing.html>.\n\n6. David P. Handlin, _The American Home: Architecture and Society\u20141815\u20131915_ (Boston: Little, Brown: 1979): 452 ff. Most of these systems, in turn, connected a house to a larger system of distribution such as sewers, electric power lines, water service, cable TV, and internet. \"By connecting a house to the surrounding community and eventually to the world at large, they did their part in destroying... isolation.... They disrupted age-old relationships and brought people into contact with one another in ways that they were not accustomed to or did not understand. The result was often not mutual enlightenment but consternation and even conflict.\"\n\n7. Merritt Ierley, _The Comforts of Home: The American House and the Evolution of Modern Convenience_ (New York: Clarkson Potter, 1999): 172. \"Most Houses built before 1900 contained little if anything of twentieth-century technology. Well into the twentieth century many of these houses relied on kitchen and parlor stoves for heat, kerosene lamps for light, portable tubs for bathing and laundering, and outdoor privies for use as bathrooms.\"\n\n8. Gerrylynn K. Roberts and Phillip Steadman, _American Cities & Technology: Wilderness to Wired City_ (New York: Routledge, 1999): 99.\n\n9. Barbara Krasner-Khait, \"The Impact of Refrigeration,\" <http:\/\/www.history-magazine.com\/refrig.html>. Accessed January 21, 2013.\n\n10. <http:\/\/www.census.gov\/hhes\/www\/housing\/census\/historic\/plumbing.html>.\n\n11. Richard Weston, _100 Ideas That Changed Architecture_ (London: Laurence King, 2011): 186.\n\n12. Lisa D. Schrenk, _Building a Century of Progress_ (Minneapolis: University of Minnesota Press, 2007): 163\u20135.\n\n# For Further Reference\n\nThe selected works included here are those that I believe will be among the most helpful to readers seeking additional information about houses of a particular style or region, or about the neighborhoods in which they are located. The first section lists important references on the general topic of American houses. This is followed by nine lists of works treating more specific subjects (such as \"House Form and Structure,\" \"Colonial Houses,\" and \"Styled Houses Since 1935\"); these are arranged to parallel the generally chronological organization of the book. Four succeeding sections list selected \"Regional and Local Guides,\" which cover a range of architectural styles as found in a particular part of the country. (Note that local works treating only a single style or period may be listed in the preceding chronological sections rather than with these guides of broader scope.) Only a few important examples of pattern books\u2014works showing sample house plans and elevations\u2014have been included from among the many thousands of such publications. These are labeled \"[pattern book]\" in the lists. A comprehensive survey of all such works published before 1895 can be found in Henry-Russell Hitchcock's _American Architectural Books_ rev. ed. (Mansfield Centre, CT; Martino Press 2003). Daniel D. Reiff's _Houses from Books: Treatises, Pattern Books, and Catalogs in American Architecture, 1738\u20131950\u2014A History and Guide_ (University Park, PA: Pennsylvania State University Press, 2000) provides an excellent overview of the role such books have played in American domestic architecture; its appendices include selections of the numerous books covering the first half of the twentieth century. A final section, \"House Preservation and Restoration,\" guides those wishing to conserve the original architectural character of older houses as they adapt them to contemporary living.\n\n## General Works\n\nAlexander, Christopher. _The Timeless Way of Building._ New York: Oxford University Press, 1980.\n\nAlexander, Christopher, Sara Ishikawa, and Murray Silverstein. _A Pattern Language: Towns, Buildings, Construction._ 1979, Reprint, New York: Oxford University Press, 2010.\n\nAndrews, Wayne. _Architecture, Ambition, and Americans._ Rev. ed. New York: Free Press, 1979.\n\nBlumenson, John J.-G. _Identifying American Architecture._ 2nd ed. Nashville, TN: American Association for State and Local History, 1995.\n\nBrand, Stewart. _How Buildings Learn: What Happens After They're Built._ Rev. ed. New York, Viking, 1997.\n\nCarley, Rachel. _The Visual Dictionary of American Domestic Architecture._ New York: Owl Book, 1997.\n\nChudacoff, Howard P., Judith E. Smith, and Peter C. Baldwin. _The Evolution of American Urban Society._ 7th ed. Boston: Prentice Hall, 2010.\n\nClark, Clifford Edward. _The American Family Home, 1800\u20131960._ Chapel Hill: University of North Carolina Press, 1987.\n\nCorkin, Caitlin. _Surveying the Recent Past: The Challenge of Creating and Defining Context._ Paper presented at Looking Forward: Preservation in New England in the Twenty-first Century, Rhode Island, 2011.\n\nDavidson, Marshall B. _Notable American Houses._ New York: American Heritage Publishing Co., 1971.\n\nEggener, Keith. _American Architectural History: A Contemporary Reader._ London and New York: Routledge, 2006.\n\nFitch, James Marston. _American Building 1: The Historical Forces That Shaped It._ 2nd ed. New York: Oxford University Press, 1999.\n\nFoley, Mary Mix. _The American House._ New York: Harper & Row, 1981.\n\nFoster, Gerald L. _American Houses: A Field Guide to the Architecture of the Home._ Boston: Houghton Mifflin, 2004.\n\nGelernter, Mark. _A History of American Architecture: Buildings in Their Cultural and Technological Context._ Hanover, NH: University Press of New England, 2001.\n\nGottfried, Herbert, and Jan Jennings. _American Vernacular Buildings and Interiors, 1870\u20131960._ New York: W. W. Norton, 2009.\n\nGowans, Alan. _Images of American Living._ New York: Harper & Row, 1983.\n\n\u2014\u2014\u2014. _The Comfortable House: North American Suburban Architecture, 1890\u20131930._ Cambridge, MA: MIT Press, 1989.\n\nHandlin, David P. _The American Home Architecture and Society, 1815\u20131915._ Boston: Little, Brown & Co, 1979.\n\n\u2014\u2014\u2014. _American Architecture._ London: Thames & Hudson, 2004.\n\nHarris, Cyril M. _American Architecture: An Illustrated Encyclopedia._ New York: W. W. Norton, 2009.\n\nHarrison, Henry S. _Houses: The Illustrated Guide to Construction, Design, and Systems._ 3rd ed. New York: Kaplan, 1998.\n\nHilowitz, Beverley, and Susan Eikov Green. _Historic Houses of America._ Rev. ed. New York: Simon and Schuster, 1980.\n\nHopkins, Owen. _Reading Architecture: A Visual Lexicon._ London: Laurence King, 2012.\n\nHowe, Barbara J. _Houses and Homes: Exploring Their History._ Walnut Creek, CA: AltaMira Press, 1997.\n\nHull, Brent, and Christine G. H. Franck. _Traditional American Rooms: Celebrating Style, Craftsmanship, and Historic Woodwork._ East Petersburg, PA: Fox Chapel, 2009.\n\nIerley, Merritt. _The Comforts of Home: The American House and Evolution of Modern Convenience._ New York: Three Rivers Press, 1999.\n\n\u2014\u2014\u2014. _Open House: A Guided Tour of the American Home, 1637\u2013Present._ New York: Henry Holt and Co., 1999.\n\nJakle, John A., Robert W. Bastian, and Douglas K. Meyer. _Common Houses in America's Small Towns: The Atlantic Seaboard to the Mississippi Valley._ Athens: University of Georgia Press, 1989.\n\nJandl, H. Ward, John A. Burns, and Michael Auer. _Yesterday's Houses of Tomorrow: Innovative American Homes, 1850 to 1950._ Washington, DC: Preservation Press, 1991.\n\nKahn, Lloyd. _Home Work: Handbuilt Shelter._ Bolinas, Calif: Shelter Publications, 2004.\n\n\u2014\u2014\u2014. _Tiny Homes: Simple Shelter: Scaling Back in the Century._ Bolinas, CA: Shelter Publications, 2012.\n\nKoeper, Frederick, and Marcus Whiffen. _American Architecture._ 2 vols. Cambridge, MA: MIT Press, 1981, 1984.\n\nLounsbury, Carl, and Vanessa Elizabeth Patrick. _An Illustrated Glossary of Early Southern Architecture and Landscape._ Rev. ed. New York: Oxford University Press, 1999.\n\nMaddex, Diane. _Master Builders: A Guide to Famous American Architects._ Washington, DC: Preservation Press, 1996.\n\nMartinson, Tom. _The Atlas of American Architecture: 2,000 Years of Architecture, City Planning, Landscape Architecture and Civil Engineering._ New York: Rizzoli, 2009.\n\nMason, Joseph B. _History of Housing in the U.S., 1930\u20131980._ Houston: Gulf Publishing, 1982.\n\nMassey, James C., and Shirley Maxwell. _House Styles in America: The Old-House Journal Guide to the Architecture of American Homes._ New York: Penguin Studio, 1999.\n\nMcAlester, Virginia, and Lee McAlester. _Great American Houses and Their Architectural Styles._ New York: Abbeville Press, 1994.\n\nMorgan, William, and Radek Kurzaj. _The Abrams Guide to American House Styles._ New York: Harry N. Abrams, 2008.\n\nPackard, Robert T. _Encyclopedia of American Architecture._ 2nd ed. New York: McGraw-Hill, 1995.\n\nParadis, Thomas W. _The Greenwood Encyclopedia of Homes Through American History._ Westport, CT: Greenwood Press, 2008.\n\nPickering, Ernest. _The Homes of America._ New York: Thomas Y. Crowell, 1951.\n\nPollman, Richard B. _Book of Successful Home Plans_ [pattern book]. Farmington, MI: Structures Publishing, 1976.\n\nPoppeliers, John, S. Allen Chambers, and Nancy B. Schwartz. _What Style Is It?_ Rev. ed. Washington, DC: Preservation Press, 2003.\n\nReiff, Daniel D. _Houses from Books: The Influence of Treatises, Pattern Books, and Catalogs in America, 1738\u20131950\u2014A History and Guide._ University Park, PA: Pennsylvania State University Press, 2001.\n\nRoth, Leland M. _A Concise History of American Architecture._ 2nd ed. Boulder, CO: Westview Press, 2003.\n\n\u2014\u2014\u2014. _Understanding Architecture: Its Elements, History, and Meaning._ 2nd ed. Boulder, CO: Westview Press, 2007\n\nSaylor, Henry H. _Dictionary of Architecture._ 1967. Reprint: New York: John Wiley & Sons, 1994.\n\nSchafer, Gil, Marc Kristal, and Bunny Williams. _The Great American House: Tradition for the Way We Live Now._ New York: Rizzoli, 2012.\n\nSchweitzer, Robert, and Michael W. R. Davis. _America's Favorite Homes: Mail-Order Catalogues As a Guide to Popular Early 20th\u2013Century Houses._ Detroit: Wayne State University Press, 1990.\n\nSmith, Peter D. _City: A Guidebook for the Urban Age._ London: Bloomsbury, 2012.\n\nSusanka, Sarah, and Kira Obolensky. _The Not So Big House: A Blueprint for the Way We Really Live._ 10th ed. Newtown, CT: Taunton Press, 2009.\n\nU.S. Department of Commerce, Bureau of the Census. _Historical Statistics of the United States: Colonial Times to 1970._ Washington, DC, 1975.\n\nWalker, Lester. _American Shelter._ Rev. ed. Woodstock, NY: Overlook, 1997.\n\nWeston, Richard. _100 Ideas That Changed Architecture._ London: Laurence King, 2011.\n\nWhiffen, Marcus. _American Architecture Since 1780: A Guide to the Styles._ 1969. Rev. ed. Cambridge, MA: MIT Press, 1993.\n\nWilliams, Henry Lionel, and Ottalie K. Williams. _A Guide to Old American Houses, 1700\u20131900._ San Diego, CA: Oak Tree, 1977.\n\nWiseman, Carter. _Twentieth-Century American Architecture: The Buildings and Their Makers._ New York: W. W. Norton, 2000.\n\nWright, Gwendolyn. _USA: Modern Architectures in History._ London: Reaktion Books, 2008.\n\nIn addition, a thorough bibliography of general works, \"American Architecture, Architectural History & Historic Preservation: A Selected Bibliography,\" has been compiled by the Portland Bureau of Planning and Sustainability. It is available as a PDF file at www.portlandoregon.gov\/bps\/article\/147430\n\n## House Form and Structure\n\nAllen, Edith Louise. _American Housing._ Peoria, IL: Manual Arts Press, 1930.\n\nAnderson, L. O. _Wood-Frame House Construction._ Rev. ed. Carlsbad, CA: Craftsman Book Company, 1992.\n\nBlackburn, Graham. _The Parts of a House._ New York: Richard Marek, 1980.\n\n\u2014\u2014\u2014. _The Illustrated Book of Housebuilding and Carpentry._ Omnibus reprint edition of _Illustrated Housebuilding_ and _Illustrated Interior Carpentry._ Woodstock, NY: Overlook Press, 2003.\n\nDiDonno, Lupe, and Phyllis Sperling. _How to Design and Build Your Own House._ 2nd ed. New York: Knopf, 1987.\n\nDietz, Albert G. H. _Dwelling House Construction._ 5th ed. Cambridge, MA: MIT Press, 1992.\n\nKauffman, Henry J. _The American Fireplace._ New York: Galahad Books, 1972.\n\nMcKee, Harley J. _Introduction to Early American Masonry: Stone, Brick, Mortar and Plaster._ Washington, DC: Preservation Press, 1980.\n\nNewcomb, Rexford, and William A. Foster. _Home Architecture._ New York: John Wiley & Sons, 1932.\n\nPeterson, Charles E., ed. _Building Early America._ Radnor, PA: Chilton, 1992.\n\nTownsend, Gilbert, and J. Ralph Dalzell. _How to Plan a House._ 3rd ed. Chicago: American Technical Society, 1965.\n\n## Neighborhood Groupings\n\nAdams, Thomas. _The Design of Residential Areas: Basic Considerations, Principles, and Methods._ New York: Arno Press, 1934.\n\n_The American Collection: Craftsman Style: 165 New Home Plans in the Arts & Crafts Tradition of Fine Craftsmanship, Natural Materials, and Simple Elegance_ [pattern book]. Washington, DC: Hanley Wood, 2006.\n\nAmes, David L., and Linda Flint McClelland. _Historic Residential Suburbs: Guidelines for Evaluation and Documentation for the National Register of Historic Places._ National Register bulletin. Washington, DC: U.S. Department of the Interior, National Park Service, National Register of Historic Places, 2002. Available at www.nps.gov\/nr\/publications\/bulletins\/pdfs\/Suburbs.pdf.\n\nArcher, John. _Architecture and Suburbia: From English Villa to American Dream House, 1690\u20132000._ Minneapolis: University of Minnesota Press, 2005.\n\n_The Architectural Pattern Book: A Tool for Building Great Neighborhoods._ New York: W. W. Norton, 2004.\n\nArendt, Randall, and Elizabeth A. Brabec. _Rural by Design: Maintaining Small Town Character._ Chicago: Planners Press, American Planning Association, 1994.\n\nBaumbach, Richard O., and William E. Borah. _The Second Battle of New Orleans: A History of the Vieux Carr\u00e9 Riverfront Expressway Controversy._ Tuscaloosa: University of Alabama Press, 1981. Published for the Preservation Press, National Trust for Historic Preservation in the United States.\n\nBeasley, Ellen. _The Alleys and Back Buildings of Galveston: An Architectural and Social History._ Houston: Rice University Press, 2007.\n\nBeveridge, Charles E., and Paul Rocheleau. _Frederick Law Olmsted: Designing the American Landscape._ Rev. ed. New York: Rizzoli, 2005.\n\nBinford, Henry C. _The First Suburbs: Residential Communities on the Boston Periphery, 1815\u20131860._ Chicago: University of Chicago Press, 1988.\n\nCampoli, Julie, and Alex S. MacLean. _Visualizing Density._ Cambridge, MA: Lincoln Institute of Land Policy, 2007.\n\nCampoli, Julie, Elizabeth Humstone, and Alex S. MacLean. _Above and Beyond: Visualizing Change in Small Towns and Rural Areas._ Chicago: Planners Press, American Planning Association, 2002.\n\nChurch, Thomas D., Michael Laurie, and Grace Hall. _Gardens Are for People._ 1955. 3rd ed. Berkeley: University of California Press, 1995.\n\nCigliano, Jan, and Sarah Bradford Landau. _The Grand American Avenue, 1850\u20131920._ San Francisco: Pomegranate Artbooks, 1994.\n\nClay, Grady, ed. _Landscapes for Living._ New York: McGraw-Hill, 1980.\n\nCondon, Patrick M. _Seven Rules for Sustainable Communities: Design Strategies for the Post-Carbon World._ Washington, DC: Island Press, 2010.\n\nCooney, Loraine M., Hattie C. Rainwater, Florence N. Marye, and Phillip T. Marye. _Garden History of Georgia, 1733\u20131933._ 1933. Reprint, Atlanta, GA: Peachtree Garden Club, 1976.\n\nCory, Gregory L. _Golf Course Development in Residential Communities._ Washington, DC: Urban Land Institute, 2001.\n\nCrawford, Margaret. _Building the Workingman's Paradise: The Design of American Company Towns._ London: Verso, 1995.\n\nDarke, Rick. _In Harmony with Nature: Lessons from the Arts & Crafts Garden._ New York: Friedman\/Fairfax, 2001.\n\nDowning, A. J. _Landscape Gardening and Rural Architecture._ 1849. Reprint, New York: Dover Publications, 1991.\n\n\u2014\u2014\u2014. _A Treatise on the Theory and Practice of Landscape Gardening, Adapted to North America; with a View to the Improvement of Country Residences._ 1844. Reprint, New York: Wiley and Putnam, 2010.\n\nDuany, Andres, Elizabeth Plater-Zyberk, and Robert Alminana. _The New Civic Art: Elements of Town Planning._ New York: Rizzoli, 2011.\n\nDuany, Andres, Jeff Speck, and Mike Lydon. _The Smart Growth Manual._ New York: McGraw-Hill, 2010.\n\nEasterling, Keller. _American Town Plans: A Comparative Time Line._ New York: Princeton Architectural Press, 1993.\n\nEckbo, Garrett. _Landscape for Living._ 1950. Reprint, New York: Architectural Record with Duell, Sloan, & Pearce, 2009.\n\nFishman, Robert. _Bourgeois Utopias: The Rise and Fall of Suburbia._ New York: Basic Books, 1999.\n\nFord, Larry R. _Cities and Buildings: Skyscrapers, Skid Rows, and Suburbs._ Baltimore: Johns Hopkins University Press, 1995.\n\nFox, Pamela W., and Sarah B. Gilman. _Farm Town to Suburb: The History and Architecture of Weston, Massachusetts, 1830\u20131980._ Portsmouth, NH: Peter E. Randall, 2002.\n\nGarvin, Alexander. _The American City: What Works, What Doesn't Work._ 2nd ed. New York: McGraw-Hill, 2002.\n\nGirling, Cynthia L., and Kenneth I. Helphand. _Yard, Street, Park: The Design of Suburban Open Space._ New York: J. Wiley and Sons, 1994.\n\nHayden, Dolores. _Building Suburbia._ New York: Pantheon Books, 2004.\n\nHayden, Dolores, and Jim Wark. _A Field Guide to Sprawl._ New York: W. W. Norton, 2006.\n\nHayward, Mary Ellen. _Baltimore's Alley Houses: Homes for Working People Since the 1780s._ Baltimore: Johns Hopkins University Press, 2008.\n\nHegemann, Werner, and Elbert Peets. _The American Vitruvius: An Architect's Handbook of Civic Art._ 1922. Reprint, s.l.: De Facto Publishing, 2008.\n\nHerman, Bernard L. _Town House: Architecture and Material Life in the Early American City, 1780\u20131830._ Chapel Hill: University of North Carolina Press, 2005. Published for the Omohundro Institute of Early American History and Culture, Williamsburg, VA.\n\nHiggins, Hannah B. _The Grid Book._ Cambridge, MA: MIT Press, 2009.\n\nHise, Greg. _Magnetic Los Angeles: Planning the Twentieth-Century Metropolis._ Baltimore: Johns Hopkins University Press, 1999.\n\nHunter, Christine. _Ranches, Rowhouses, and Railroad Flats: American Homes: How They Shape Our Landscapes and Neighborhoods._ Rev. ed. New York: W. W. Norton, 2005.\n\nJackson, Kenneth T. _Crabgrass Frontier: The Suburbanization of America._ New York: Oxford University Press, 2008.\n\nJacobs, Allan B., Elizabeth Macdonald, and Yodan Rof\u00e9. _The Boulevard Book: History, Evolution, Design of Multiway Boulevards._ Cambridge, MA: MIT Press, 2003.\n\nJacobs, Jane. _The Death and Life of Great American Cities._ 1961. Rev. ed. New York: Random House, 2011.\n\nJohnson, Leonard H. _Foundation Planting._ New York: A. T. De La Mare Co., 1937.\n\nJordan, Terry G. _Texas: A Geography._ Boulder, CO: Westview Press, 1984.\n\nKeating, Ann Durkin. _Chicagoland: City and Suburbs in the Railroad Age._ Chicago: University of Chicago Press, 2005.\n\nKlaus, Susan L., and Frederick Law Olmsted. _A Modern Arcadia: Frederick Law Olmsted Jr. & the Plan for Forest Hills Gardens._ Amherst: University of Massachusetts Press, 2002.\n\nKunstler, James Howard. _The Geography of Nowhere: The Rise and Decline of America's Man-Made Landscape._ Rev. ed. New York: Simon & Schuster, 2004.\n\nLampl, Elizabeth Jo, and Kimberly Prothro Williams. _Chevy Chase: A Home Suburb for the Nation's Capital._ Crownsville, MD: Montgomery County Historic Preservation Commission, 2009.\n\nLangdon, Philip. _A Better Place to Live: Reshaping the American Suburb._ Rev. ed. Boston: University of Massachusetts Press, 1997.\n\nLeinberger, Christopher B. _The Option of Urbanism: Investing in a New American Dream._ Washington, DC: Island Press, 2009.\n\nMcAlester, Virginia, and Lee McAlester. _A Field Guide to America's Historic Neighborhoods and Museum Houses: The Western States._ New York: Knopf, 1998.\n\nMcAlester, Virginia, Willis Winters, and Prudence Mackintosh. _Great American Suburbs: The Homes of the Park Cities\u2014Dallas._ New York: Abbeville Press, 2008.\n\nMcKeever, J. R. _The Community Builders Handbook._ 1968. 6th ed. Washington, DC: Urban Land Institute, 1988.\n\nMcKelvey, Blake. _The Urbanization of America, 1860\u20131915._ New Brunswick, NJ: Rutgers University Press, 1963.\n\nMcShane, Clay. \"The Centrality of the Horse to the Nineteenth-Century American City.\" In _The Making of Urban America,_ edited by Raymond Mohl. New York: SR Publishers, 1997.\n\nNewton, Norman T. _Design on the Land: The Development of Landscape Architecture._ Cambridge, MA: Belknap Press of Harvard University Press, 1971.\n\nO'Malley, Therese. _Regional Garden Design in the United States_ Dumbarton Oaks Colloquium on the History of Landscape Architecture XV. Washington, DC: Dumbarton Oaks Research Library and Collection, 1995.\n\nPeterson, Jon A. _The Birth of City Planning in the United States, 1840\u20131917._ Baltimore: Johns Hopkins University Press, 2003.\n\nPettis, Emily. _A Model for Identifying and Evaluating the Historic Significance of Post\u2013World War II Housing._ Washington, DC: Transportation Research Board, 2012.\n\nPregill, Philip, and Nancy Volkman. _Landscapes in History: Design and Planning in the Western Tradition._ 2nd ed. New York: Van Nostrand Reinhold, 1999.\n\nPunch, Walter T., and William Howard Adams. _Keeping Eden: A History of Gardening in America._ Boston: Bulfinch Press, 1992.\n\nRae, John Bell. _The Road and the Car in American Life._ Cambridge, MA: MIT Press, 1971.\n\nReps, John William. _The Making of Urban America: A History of City Planning in the United States._ 1966. Reprint, Princeton, NJ: Princeton University Press, 1997.\n\nRowe, Peter G. _Making a Middle Landscape._ Cambridge, MA: MIT Press, 1992.\n\nSchroeder, Fred E. H. _Front Yard America: The Evolution and Meanings of a Vernacular Domestic Landscape._ Bowling Green, OH: Bowling Green State University Popular Press, 1993.\n\nSchuyler, David. _Apostle of Taste: Andrew Jackson Downing, 1815\u20131852._ Baltimore: Johns Hopkins University Press, 1999.\n\nScott, Frank J. _Victorian Gardens: The Art of Beautifying Suburban Home Grounds\u2014A Victorian Guidebook of 1870._ Rev. ed. Watkins Glen, NY: American Life Foundation, 1982.\n\nSouthworth, Michael, and Eran Ben-Joseph. _Streets and the Shaping of Towns and Cities._ New York: McGraw-Hill, 2003.\n\nStern, Robert A. M., and John M. Massengale. _The Anglo American Suburb._ London: Architectural Design, 1981.\n\nSteuteville, Robert. _New Urbanism: Best Practices Guide._ 4th ed. Ithaca, NY: New Urban News Publications, 2009.\n\nStilgoe, John R. _Common Landscape of America, 1580 to 1845._ New Haven: Yale University Press, 1985.\n\n\u2014\u2014\u2014. _Metropolitan Corridor: Railroads and the American Scene._ New Haven: Yale University Press, 1985.\n\n\u2014\u2014\u2014. _Borderland: Origins of the American Suburb, 1820\u20131939._ New Haven: Yale University Press, 1990.\n\nTeyssot, Georges. _The American Lawn._ New York: Princeton Architectural Press, 1999.\n\nThadani, Dhiru A. _The Language of Towns & Cities: A Visual Dictionary._ New York: Rizzoli, 2010.\n\nTishler, William H. _American Landscape Architecture: Designers and Places._ New York: John Wiley & Sons, 1996.\n\n\u2014\u2014\u2014. _Midwestern Landscape Architecture._ Urbana, IL: University of Illinois Press, 2004.\n\nTreib, Marc. _The Architecture of Landscape, 1940\u20131960._ Philadelphia: University of Pennsylvania Press, 2002.\n\nUnited States. _Successful Subdivisions: Principles of Planning for Economy and Protection Against Neighborhood Blight._ Washington, DC: Federal Housing Administration, 1940.\n\nUpton, Dell. _Another City._ New Haven: Yale University Press, 2008.\n\nWarner, Sam Bass. _Streetcar Suburbs: The Process of Growth in Boston, 1870\u20131900._ 3rd ed. Cambridge, MA: Harvard University Press, 1980.\n\nWeiss, Marc A. _The Rise of the Community Builders: The American Real Estate Industry and Urban Land Planning._ Washington, DC: Beard Books, 2002.\n\nWilson, William H. _The City Beautiful Movement._ Baltimore: Johns Hopkins University Press, 1994.\n\nWorley, William S. _J. C. Nichols and the Shaping of Kansas City: Innovation in Planned Residential Communities._ Columbia, MO: University of Missouri Press, 1993.\n\n## Folk Houses\n\nBrunskill, R. W. _Houses._ London: Collins, 1982.\n\n\u2014\u2014\u2014. _Vernacular Architecture: An Illustrated Handbook._ 4th ed. Retitled. London: Faber and Faber, 2000.\n\nDriver, Harold E., and William C. Massey. \"Comparative Studies of North American Indians.\" _American Philosophical Society Transactions_ 47 (new series), pt. 2 (1957): 165\u2013456.\n\nEdwards, Jay Dearborn. \"Shotgun: The most contested house in America.\" _Buildings and Landscapes: Journal of the Vernacular Architecture Forum_ 16 (2009): 62\u201396.\n\nFinley, Robert, and E. M. Scott. \"A Great Lakes-to-Gulf Profile of Dispersed Dwelling Types.\" _Geographical Review_ 30, no. 3 (July 1940): 412\u201319.\n\nGlassie, Henry. _Pattern in the Material Folk Culture of the Eastern United States._ 1968. Reprint, Philadelphia: University of Pennsylvania Press, 1993.\n\nHart, John Fraser, Michelle J. Rhodes, and John Morgan. _The Unknown World of the Mobile Home._ Baltimore: Johns Hopkins University Press, 2003.\n\nHutslar, Donald A. _The Log Architecture of Ohio._ Columbus: Ohio Historical Society, 1977.\n\nJett, Stephen C., and Virginia E. Spencer. _Navajo Architecture: Forms, History, Distributions._ Tucson: University of Arizona Press, 1981.\n\nJohnson, Wes. _The Manufactured Home Buyer's Handbook._ Jefferson, NC: McFarland & Co, 2005.\n\nJordan, Terry G. _Texas Log Buildings._ 1978. Reprint, Austin: University of Texas Press, 1994.\n\nKniffen, Fred. \"Folk Housing: Key to Diffusion.\" _Annals of the Association of American Geographers_ 55, no. 4 (Dec. 1965): 549\u201377.\n\nKniffen, Fred, and Henry Glassie. \"Building in Wood in the Eastern United States: A Time-Place Perspective.\" _Geographical Review_ 56, no. 1 (Jan. 1966): 40\u201366.\n\nLewis, Peirce F. \"Common Houses, Cultural Spoor.\" _Landscape_ 19, no. 2.\n\nMarshall, Howard Wight. _Folk Architecture in Little Dixie._ Columbia: University of Missouri Press, 1981.\n\nMontell, William L., and Michael L. Morse. _Kentucky Folk Architecture._ Lexington: University Press of Kentucky, 1995.\n\nPillsbury, Richard, and Andrew Kardos. _A Field Guide to the Folk Architecture of the Northeastern United States._ Hanover, NH: Geography Publications at Dartmouth, 1970.\n\nShortridge, James R. \"Some Relationships Between External Housing Characteristics and House Types.\" _Pioneer America_ 13, no. 2 (Sept. 1981): 1\u201328.\n\nSwain, Doug, ed. _Carolina Dwelling._ North Carolina State University School of Design Student Publication, vol. 26. Raleigh: North Carolina State University, 1978.\n\nTaylor, Steven V. _Manufactured Homes: The Buyer's Guide\u2014How to Realize Your Dream in a Manufactured Home._ San Francisco: Cycle\/Van der Plas, 2004.\n\nWaterman, T. T. \"North American Indian Dwellings.\" _Geographical Review_ 14 (1924): 1\u201325.\n\nWelsch, Roger L. _Sod Walls: The Story of the Nebraska Sod House._ 1968. Reprint. Lincoln, NE: J&L Lee Company, 1991.\n\nWeslager, C. A. _The Log Cabin in America._ New Brunswick, NJ: Rutgers University Press, 1969.\n\nWilson, Eugene M. _Alabama Folk Houses._ Montgomery: Alabama Historical Commission, 1975.\n\n## Colonial Houses\n\nArchitects' Emergency Committee. _Great Georgian Houses of America._ 2 vols. 1933\/1937. Reprint, New York: Dover Publications, 1970.\n\nBaer, Morley, and Augusta Fink. _Adobes in the Sun._ San Francisco: Chronicle Books, 1972.\n\nBailey, Rosalie Fellows. _Pre-Revolutionary Dutch Houses and Families in Northern New Jersey and Southern New York._ 1936. Reprint, New York: Dover Publications, 1968.\n\nBunting, Bainbridge. _Early Architecture in New Mexico._ Albuquerque: University of New Mexico Press, 1976.\n\nBunting, Bainbridge, Jean Lee Booth, and William R. Sims Jr. _Taos Adobes: Spanish Colonial and Territorial Architecture of the Taos Valley._ Fort Burgwin Research Center Publication no. 2, 1964. Reprint, Albuquerque University of New Mexico, 1992.\n\nCummings, Abbott Lowell. _The Framed Houses of Massachusetts Bay, 1625\u20131725._ 2nd ed. Cambridge, MA: Harvard University Press, 1979.\n\nForman, Henry Chandlee. _The Architecture of the Old South: The Medieval Style, 1585\u20131850._ Cambridge, MA: Harvard University Press, 1948.\n\n\u2014\u2014\u2014. _Early Manor and Plantation Houses of Maryland._ 2nd ed. Baltimore: Bodine & Associates, 1982.\n\nGarvan, Anthony N. B. _Architecture and Town Planning in Colonial Connecticut._ 1951. Reprint, New Haven: Yale University Press, 1982.\n\nGebhard, David. \"Some Additional Observations on California's Monterey Tradition.\" _Journal of the Society of Architectural Historians_ 46, no. 2 (June 1987): 157\u201370.\n\nGiffen, Helen S. _Casas and Courtyards: Historic Adobe Houses of California._ Oakland, CA: Biobooks, 1955.\n\nGross, Geoffrey, Susan Piatt, Roderic H. Blackburn, and Harrison Frederick Meeske. _Dutch Colonial Homes in America._ New York: Rizzoli, 2002.\n\nGuinness, Desmond, and Julius Trousdale Sadler Jr. _Mr. Jefferson, Architect._ New York: Viking Press, 1973.\n\nHamlin, Talbot. _Benjamin Henry Latrobe._ 1955. Reprint, New York: Oxford University Press, 1969.\n\nHannaford, Donald R., and Revel Edwards. _Spanish Colonial or Adobe Architecture of California, 1800\u20131850._ 1931. Reprint, Lanham, MD: Taylor Trade Publishing, 2012.\n\nHarris, Eileen. _The Country Houses of Robert Adam: From the Archives of Country Life._ London: Aurum Press, 2007.\n\nHowells, John Mead. _The Architectural Heritage of the Piscataqua: Houses and Gardens of the Portsmouth District of Maine and New Hampshire._ 1937. Reprint, Washington, DC: Whalesback Books, 1988.\n\n\u2014\u2014\u2014. _The Architectural Heritage of the Merrimack._ New York: Architectural Book Publishing, 1941.\n\nJohnston, Frances Benjamin, and Thomas Tileston Waterman. _The Early Architecture of North Carolina._ Chapel Hill: University of North Carolina Press, 1947.\n\nKatz, Ron, and Arielle de La Tour d'Auvergne. _French America: French Architecture from Colonialization to the Birth of a Nation._ New York: French Heritage Society, 2005.\n\nKelly, J. Frederick. _The Early Domestic Architecture of Connecticut._ 1924. Reprint, New York: Dover Publications, 2012.\n\nKimball, Fiske. _Domestic Architecture of the American Colonies and of the Early Republic._ 1922. Reprint, New York: Dover Publications, 2001.\n\nKirker, Harold. _The Architecture of Charles Bulfinch._ 1969. Reprint, Cambridge, MA: Harvard University Press, 1998.\n\nLane, Mills. _Architecture of the Old South: Colonial & Federal._ Savannah, GA: Beehive Foundation, 1996.\n\nManca, Joseph. \"On the Origins of the American Porch.\" _Winterthur Portfolio_ 40.23 (2006): 91\u2013132.\n\nMcCall, Elizabeth B. _Old Philadelphia Houses on Society Hill, 1750\u20131840._ New York: Architectural Book Publishing, 1966.\n\nMillar, John Fitzhugh. _The Architects of the American Colonies._ Barre, MA: Barre Publishers, 1968.\n\nMorgan, William. _The Cape Cod Cottage._ New York: Princeton Architectural Press, 2006.\n\nMorrison, Hugh. _Early American Architecture: From the First Colonial Settlements to the National Period._ 1952. Reprint, New York: Dover Publications, 2011.\n\nOverdyke, W. Darrell. _Louisiana Plantation Homes._ New York: American Legacy Press, 1981.\n\nParissien, Steven. _The Georgian House in America and Britain._ 2nd ed. New York: Rizzoli, 2008.\n\nPierson, William H., Jr. _American Buildings and Their Architects: The Colonial and Neoclassical Styles._ New York: Oxford University Press, 1986.\n\nPorterfield, Neil H. \"Ste. Genevieve, Missouri.\" In _Frenchmen and French Ways in the Mississippi Valley,_ edited by John Francis McDermott. Urbana, IL: University of Illinois Press, 1969.\n\nReiff, Daniel Drake. _Small Georgian Houses in England and Virginia: Origins and Development Through the 1750s._ Newark: University of Delaware Press, 1986.\n\nReynolds, Helen Wilkinson. _Dutch Houses in the Hudson Valley Before 1776._ 1929. Reprint, New York: Dover, 1965.\n\nRichmond, Arthur P. _The Evolution of the Cape Cod House: An Architectural History._ Atglen, PA: Schiffer, 2011.\n\nSchuler, Stanley. _The Cape Cod House._ Atglen, PA: Schiffer, 1982.\n\nStoney, Samuel Gaillard. _Plantations of the Carolina Low Country._ 1938. Reprint, New York: Dover Publications, 1990.\n\nTatum, George B. \"Architecture.\" In _The Arts in America: The Colonial Period_ by Louis B. Wright et al. New York: Charles Scribner's Sons, 1966.\n\nToledano, Roulhac, and Mary Louise Christovich. _New Orleans Architecture, Volume VI: Faubourg Trem\u00e9 and the Bayou Road._ Gretna, LA: Pelican, 2003.\n\nToledano, Roulhac, Sally Kittredge Evans, and Mary Louise Christovich. _New Orleans Architecture, Volume IV: The Creole Faubourgs._ Gretna, LA: Pelican, 1996.\n\nWaterman, Thomas Tileston. _The Mansions of Virginia, 1706\u20131776._ Chapel Hill: University of North Carolina Press, 1947.\n\n\u2014\u2014\u2014. _The Dwellings of Colonial America._ Rev. ed. New York: W. W. Norton, 1999.\n\nWhiffen, Marcus. _The Eighteenth-Century Houses of Williamsburg,_ 2nd ed. Charlottesville, VA: University Press of Virginia, 1988.\n\n## Romantic and Victorian Houses\n\nAndrews, Wayne. _American Gothic._ New York: Random House, 1975.\n\nBarber, George F. _Victorian Cottage Architecture: An American Catalog of Designs, 1891_ [pattern book]. Reprint of _The Cottage Souvenir No. 2._ Mineola, NY: Dover Publications, 2004.\n\n\u2014\u2014\u2014. _Barber's Turn-of-the-Century Houses: Elevations and Floor Plans_ [pattern book]. Reprint of _Modern Dwellings: A Book of Practical Designs and Plans, 3rd ed.,_ 1901. Mineola, NY: Dover Publications, 2008.\n\nBenjamin, Asher. _The American Builder's Companion_ [pattern book]. 1827. Reprint, New York: Dover Publications, 1969.\n\n\u2014\u2014\u2014. _The Architect, or Practical House Carpenter_ [pattern book]. 1830. Reprint, New York: Dover Publications, 1988.\n\nBicknell, A. J. _Bicknell's Village Builder and Supplement_ [pattern book]. 1872. Reprint, Watkins Glen, NY: The American Life Foundation, 1976.\n\nBrettell, Richard R. _Historic Denver, 1858\u20131893._ Denver: Historic Denver, 1973.\n\nCameron, Christina, and Janet Wright. _Second Empire Style in Canadian Architecture._ Canadian Historic Sites: Occasional Papers in Archaeology and History no. 24. Ottawa: National Historic Parks and Sites Branch, 1980.\n\nCampen, Richard N. _Architecture of the Western Reserve, 1800\u20131900._ Cleveland: Case Western Reserve University Press, 1971.\n\nCochran, Gifford A. _Grandeur in Tennessee: Classical Revival Architecture in a Pioneer State._ New York: J. J. Augustin, 1946.\n\nComstock, William T. _Country Houses and Seaside Cottages of the Victorian Era_ [pattern book]. Reprint of _American Cottages._ 1883. New York: Dover Publications, 1985.\n\nConover, Jewel Helen. _Nineteenth-Century Houses in Western New York._ Albany: State University of New York Press, 1971.\n\nCooper, J. Wesley. _Ante-Bellum Houses of Natchez._ Natchez, MS: Southern Historical Publications, 1983.\n\nCummings, M. F., and C. C. Miller. _Victorian Architectural Details._ Reprint of _Architecture_ [pattern book], 1865, and _Architectural Details,_ 1873. Watkins Glen, N.Y.: American Life Foundation, 1980.\n\nDana, William Sumner Barton. _The Swiss Chalet Book; A Minute Analysis and Reproduction of the Chalets of Switzerland, Obtained by a Special Visit to That Country, Its Architects, and Its Chalet Homes._ New York: William T. Comstock Co., 1913.\n\nDelehanty, Randolph, and Richard Sexton. _In the Victorian Style._ San Francisco: Chronicle Books, 2006.\n\nDenison, Allen, and Wallace Huntington. _Victorian Architecture of Port Townsend, Washington._ Seattle: Hancock House, 1978.\n\nDowning, A. J. _The Architecture of Country Houses_ [pattern book]. 1850. Reprint, New York: Dover Publications, 1969.\n\n\u2014\u2014\u2014. _Cottage Residences_ [pattern book]. 1873. Reprint, New York: Dover Publications, 1981.\n\nEarly, James. _Romanticism and American Architecture._ New York: A. S. Barnes and Co., 1965.\n\nFoster, Janet W. _The Queen Anne House: America's Victorian Vernacular._ New York: Abrams, 2006.\n\nFowler, Orson S. _A Home for All_ [pattern book]. 1853. Reprint, New York: Dover Publications, 1973.\n\n\u2014\u2014\u2014. _The Octagon House: A Home for All_ [pattern book]. 1853. Reprint, New York: Dover Publications, 1973.\n\nGarvin, James L. \"Mail-Order House Plans and American Victorian Architecture.\" _Winterthur Portfolio_ 16 (1981): 309\u2013334.\n\nGillon, Edmund V., Jr., and Clay Lancaster. _Victorian Houses._ New York: Dover Publications, 1973.\n\nGirouard, Mark. _Sweetness and Light: The \"Queen Anne\" Movement, 1860\u20131900._ New Haven: Yale University Press, 1990.\n\nHackett, Cheryl, and Kindra Clineff. _Newport Shingle Style._ London: Frances Lincoln, 2010.\n\nHall, John, and Thomas Gordon Smith. _John Hall and the Grecian Style in America: A Reprint of Three Pattern Books Published in Baltimore in 1840._ New York: Acanthus Press, 1996.\n\nHamlin, Talbot. _Greek Revival Architecture in America._ 1944. Reprint, New York: Dover Publications, 1985.\n\nHammond, Ralph. _Ante-Bellum Mansions of Alabama._ 1955. Reprint, New York: Architectural Book Publishing Co., 1988.\n\nHersey, George L. \"Godey's Choice.\" _The Journal of the Society of Architectural Historians_ 18, no. 3 (1959): 104\u2013111.\n\nHitchcock, Henry-Russell. _The Architecture of H. H. Richardson and His Times,_ Rev. ed. Cambridge, MA: M.I.T. Press, 1970.\n\nHolly, Henry Hudson, and Michael Tomlan. _Country Seats & Modern Dwellings: Two Victorian Domestic Architectural Stylebooks_ [pattern book]. 1863 and 1878. Reprint, Watkins, Glen, NY: American Life Foundation, 1980.\n\nHussey, E. C. _Victorian Home Building_ [pattern book]. 1875. Reprint, Watkins Glen, NY: American Life Foundation, 1976.\n\nKennedy, Roger G., and John M. Hall. _Greek Revival America._ New York: Stewart, Tabori & Chang, 2010.\n\nKeyes, Margaret N. _Nineteenth Century Home Architecture of Iowa City._ Rev. ed. Iowa City: University of Iowa Press, 1993.\n\nLafever, Minard. _The Beauties of Modern Architecture_ [pattern book]. 1835. Reprint, New York: Da Capo Press, 1968.\n\n\u2014\u2014\u2014. _The Modern Builder's Guide_ [pattern book]. 1846. Reprint, New York: Dover Publications, 1969.\n\nLancaster, Clay. _Architectural Follies in America._ Rutland, VT: Charles E. Tuttle, 1960.\n\nLandau, Sarah Bradford. \"Richard Morris Hunt, the Continental Picturesque, and the 'Stick Style.' \" _The Journal of the Society of Architectural Historians_ 42, no. 3 (1983): 272\u2013289.\n\nLane, Mills, and Van J. Martin. _Architecture of the Old South: Greek Revival & Romantic._ Savannah: Beehive Foundation, 1996.\n\nLewis, Arnold. _American Country Houses of the Gilded Age (Sheldon's \"Artistic Country-Seats\")._ 1886\u201387. Reprint. New York: Dover Publications, 1982.\n\nLewis, Arnold, and Keith Morgan. _American Victorian Architecture._ 1886. Reprint, New York: Dover Publications, 1975.\n\nLoth, Calder, and Julius Trousdale Sadler Jr. _The Only Proper Style: Gothic Architecture in America._ Boston: New York Graphic Society, 1975.\n\nMaass, John. _The Victorian Home in America._ Reprint, New York: Dover Publications, 2000.\n\nMajor, Howard. _The Domestic Architecture of the Early American Republic: The Greek Revival._ Philadelphia: J. B. Lippincott, 1926.\n\nMartin, John H. \"Orson Squire Fowler: Phrenology and Octagon Houses, 1809\u20131887.\" _The Crooked Lane Review_ 137 (2005).\n\nMcArdle, Alma de C., and Deirdre Bartlett McArdle. _Carpenter Gothic: Nineteenth-Century Ornamented Houses of New England._ New York: Whitney Library of Design, 1978.\n\nMontgomery, Gladys. _Storybook Cottages: America's Carpenter Gothic Style._ New York: Rizzoli, 2011.\n\nOchsner, Jeffrey Karl. _H. H. Richardson: Complete Architectural Works._ Cambridge, MA: MIT Press, 1985.\n\nOlwell, Carol, and Judith Lynch Waldhorn. _A Gift to the Street._ San Francisco: Antelope Island Press, 1978.\n\nPalliser, George, and Charles Palliser. _The Palliser's Late Victorian Architecture_ [3 pattern books]. 1878. Reprint, Watkins Glen, NY: American Life Foundation, 1978.\n\n\u2014\u2014\u2014. _Late Victorian Architecture_ [pattern book]. Reprint of _Palliser's New Cottage Homes and Details,_ 1887, and _American Architecture, or Every Man a Complete Builder._ 1888. Watkins Glen, NY: American Life Foundation, 1978.\n\nPeat, Wilbur D. _Indiana Houses of the Nineteenth Century._ Indianapolis: Indiana Historical Society, 1969.\n\nPierson, William H., Jr. _American Buildings and Their Architects Volume 2: Technology and the Picturesque\u2014The Corporate and the Early Gothic Styles._ New York: Oxford University Press, 1986.\n\nPlymat, William, Jr. _The Victorian Architecture of Iowa._ 2nd ed. Mason City, IA: Palladian, 1997.\n\nRobinson, Annie. _Peabody & Stearns: Country Houses and Seaside Cottages._ New York: W. W. Norton, 2010.\n\nRoth, Leland M., and Bret Morgan. _Shingle Styles: Innovation and Tradition in American Architecture, 1874 to 1982._ New York: Abrams, 1999.\n\nSaylor, Henry H. _Architectural Styles for CountryHouses: The Swiss Chalet Type._ New York: Robert M. McBride & Co., 1919.\n\nSchmidt, Carl F. _The Octagon Fad._ Scottsville, NY: publ. by author, 1958.\n\nSchmitt, Peter J., and Balthazar Korab. _Kalamazoo: Nineteenth-Century Homes in a Midwestern Village._ Kalamazoo, MI: Kalamazoo City Historical Commission, 1976.\n\nScully, Vincent J., Jr. _The Shingle Style Today, or the Historian's Revenge._ New York: Braziller, 1978.\n\n\u2014\u2014\u2014. _The Shingle Style and the Stick Style._ Rev. ed. New Haven: Yale University Press, 1979.\n\nShoppell, R. W. _Complete Collection of Shoppell's Modern Houses: Fifteen Hundred Illustrations_ [pattern book]. New York: Co-operative Building Plan Association, 1886.\n\n\u2014\u2014\u2014. _Turn-of-the-Century Houses, Cottages and Villas: Floor Plans and Line Illustrations of 118 Homes from Shoppell's Catalogs_ [pattern book]. Reprint, New York: Dover Publications, 1984.\n\nSkjelver, Mabel Cooper. _Nineteenth Century Homes of Marshall, Michigan._ 3rd ed. Marshall, MI: Marshall Historical Society, 1982.\n\nSloan, Samuel. _The Model Architect_ [pattern book]. 1852. Reprint, New York: Dover Publications, 1980.\n\nThomas, George E., and Carl Doebley. _Cape May, Queen of the Seaside Resorts._ Philadelphia: Art Alliance Press, 1976.\n\nVaux, Calvert. _Villas and Cottages_ [pattern book]. 1864. Reprint, New York: Dover Publications, 1970.\n\n\u2014\u2014\u2014. _Villa and Cottage Architecture: The Style-Book of the Hudson River School._ 1867. Reprint, New York: Dover Publications, 1991.\n\nWilson, Samuel, Jr., and Bernard Lemann. _New Orleans Architecture, Volume 1: The Lower Garden District._ Gretna, LA: Pelican, 1991.\n\nWoodward, George Everston. _Woodward's National Architect_ [pattern book]. 1869. Reprint, American Life Foundation, 1977.\n\n\u2014\u2014\u2014. _Victorian City and Country Houses: Plans and Designs_ [pattern book]. 1877. Reprint, New York: Dover Publications, 1996.\n\n## Eclectic Houses\n\nAmerican Builder Publishing Corporation. _Modern Homes: Their Design and Construction_ [pattern book]. Chicago: American Builder Publishing Corporation, 1930.\n\nAppelbaum, Stanley. _The Chicago World's Fair of 1893._ New York: Dover Publications, 1980.\n\nArchitectural Forum. _The Book of Small Houses_ [pattern book]. New York: Simon and Schuster, 1936.\n\nAxelrod, Alan. _The Colonial Revival in America._ New York: Norton, 1985.\n\nBarnstone, Howard. _The Architecture of John F. Staub: Houston and the South._ Austin: University of Texas Press, 1979.\n\nBlackburn, Bob L., and Jim Argo. _Heritage Hills: Preservation of a Historic Neighborhood._ Oklahoma City: Western Heritage Books, 1990.\n\nBricker, Lauren Weiss. _The Mediterranean House in America._ New York: Abrams, 2008.\n\nCardwell, Kenneth H. _Bernard Maybeck: Artisan, Architect, Artist._ Santa Barbara, CA: Peregrine Smith, 1977.\n\nClarke, Ann Brewster. _Wade Hampton Pipes: Arts and Crafts Architect in Portland, Oregon._ Portland: Binford & Mort, 1986.\n\nCoate, Roland E. \"The Early California House.\" _California Arts and Architecture_ 35 (March 1929): 21\u201330.\n\nCoffin, Lewis A., and James Ford. _American Country Houses of the Thirties, with Photographs and Floor Plans._ Mineola, NY: Dover Publications, 2007.\n\nEdgell, G. H. _The American Architecture of To-day._ 1928. Reprint, New York: AMS Press, 1970.\n\nEmbury, Aymar II. _The Dutch Colonial House._ New York: McBride, Nast & Company, 1913.\n\nFox, Stephen, and Richard Cheek. _The Country Houses of John F. Staub._ College Station: Texas A&M University Press, 2007.\n\nFrazer, Susan Hume. _The Architecture of William Lawrence Bottomley._ New York: Acanthus Press, 2007.\n\nGarrison, James B. _Mastering Tradition: The Residential Architecture of John Russell Pope._ New York: Acanthus Press, 2006.\n\nGebhard, David. \"The American Colonial Revival in the 1930s.\" _Winterthur Portfolio_ 22, nos. 2\u20133 (summer\u2013autumn 1987): 109\u2013148.\n\n\u2014\u2014\u2014. \"Royal Barry Wills and the American Colonial Revival.\" _Winterthur Portfolio_ 27, no. 1 (spring 1992): 45\u201374.\n\nGebhard, Patricia. _George Washington Smith: Architect of the Spanish Colonial Revival._ Salt Lake City, UT: Gibbs Smith, 2005.\n\nGellner, Arrol, and Douglas Keister. _Storybook Style: America's Whimsical Homes of the Twenties._ New York: Viking Studio, 2001.\n\n\u2014\u2014\u2014. _Red Tile Style: America's Spanish Revival Architecture._ New York: Viking Studio, 2002.\n\nGoff, Lee, and Paul Rocheleau. _Tudor Style: Tudor Revival Houses in America from 1890 to the Present._ New York: Universe, 2002.\n\nGordon\u2013Van Tine Co. _117 House Designs of the Twenties_ [pattern book]. 1923. Reprint, Philadelphia: Athenaeum, 1992.\n\nGrady, James. _Architecture of Neel Reid in Georgia._ Athens: University of Georgia Press, 1973.\n\nHewitt, Mark A. _Carr\u00e8re & Hastings Architects._ New York: Acanthus Press, 2006.\n\nHoffstot, Barbara. _Landmark Architecture of Palm Beach._ 3rd ed. Pittsburgh: History & Landmarks Foundation, 1991.\n\nHunter, Paul Robinson, and Walter L. Reichardt, eds. _Residential Architecture in Southern California._ 1939. Reprint, Santa Monica: Hennessey & Ingalls, 1998.\n\nJackson, Allen W. _The Half-Timber House._ 1912. Reprint. New York: Robert M. McBride, 2012.\n\nJ. D. Loizeaux Lumber Company. _Classic Houses of the Twenties_ [pattern book]. 1927. Reprint, Philadelphia: Athenaeum, 1992.\n\nJones, Robert T. _Authentic Small Houses of the Twenties: Illustrations and Floor Plans of 254 Characteristic Homes_ [pattern book]. 1929. Reprint, New York: Dover Publications, 1987.\n\nJordy, William H. _American Buildings and Their Architects, Volume 4: Progressive and Academic Ideals at the Turn of the Twentieth Century._ New York: Oxford University Press, 1986.\n\nKathrens, Michael C. _Newport Villas: The Revival Styles, 1885\u20131935._ New York: W. W. Norton, 2009.\n\nKathrens, Michael C., Richard C. Marchand, and Eleanor Weller. _American Splendor: The Residential Architecture of Horace Trumbauer._ Rev. ed. New York: Acanthus Press, 2011.\n\nKidney, Walter C. _The Architecture of Choice: Eclecticism in America, 1880\u20131930._ New York: George Braziller, 1974.\n\nKohler, Sue A., and Jeffrey R. Carson. _Sixteenth Street Architecture,_ vol. 1. Washington, DC: Commission of Fine Arts, 1978\u20131988.\n\nMcKim, Mead & White. _A Monograph of the Works of McKim, Mead & White, 1879\u20131915._ 1915. Reprint, New York: Dover Publications, 1990.\n\nMcMillian, Elizabeth Jean, and Matt Gainer. _California Colonial: The Spanish and Rancho Revival Styles._ Atglen, PA: Schiffer, 2002.\n\nMizner, Addison. _Florida Architecture of Addison Mizner._ Ann Arbor: University of Michigan Press, 1993.\n\nMockler, Kim I. _Maurice Fatio: Palm Beach Architect._ New York: Acanthus Press, 2010.\n\nMorrison, William. _The Work of Dwight James Baum._ New York: Acanthus Press, 2008.\n\nMurphy, Kevin. _Colonial Revival Maine._ New York: Princeton Architectural Press, 2004.\n\nNewcomb, Rexford. _The Spanish House for America._ Philadelphia: J. B. Lippincott, 1927.\n\n\u2014\u2014\u2014. _Mediterranean Domestic Architecture in the United States._ 1928. Reprint, New York: Acanthus Press, 1999.\n\nPennoyer, Peter, and Anne Walker. _The Architecture of Delano & Aldrich._ New York: Norton, 2003.\n\n\u2014\u2014\u2014. _The Architecture of Warren & Wetmore._ New York: W. W. Norton, 2006.\n\nPennoyer, Peter, Jonathan Wallen, and Robert A. M. Stern. _The Architecture of Grosvenor Atterbury._ New York: W. W. Norton, 2009.\n\nPlatt, Frederick. _America's Gilded Age: Its Architecture and Decoration._ Cranbury, NJ: A. S. Barnes, 1976.\n\nSaylor, Henry H., ed. _Architectural Styles for Country Houses._ 1912. Reprint, Ithaca, NY: Cornell University Library, 2009.\n\nSclare, Liisa, and Donald Sclare. _Beaux-Arts Estates: A Guide to the Architecture of Long Island._ New York: Viking Press, 1980.\n\nSkinner, Tina. _Radford's Artistic Homes: 250 Designs_ [pattern book]. 1908. Reprint, Atglen, PA: Schiffer Publishing, 2002.\n\nSmith, Henry Atterbury. _The Books of a Thousand Homes, Volume 1: 500 Small House Plans_ [pattern book]. New York: Home Owners Service Institute, 1923.\n\n\u2014\u2014\u2014. _500 Small Houses of the Twenties_ [pattern book]. 1923. Reprint, New York: Dover Publications, 1990.\n\nStamp, Gavin, and Andr\u00e9 Goulancourt. _The English House, 1860\u20131914: The Flowering of English Domestic Architecture._ Chicago: University of Chicago Press, 1986.\n\nStevens, John Calvin, and Earle G. Shettleworth. _John Calvin Stevens, Domestic Architecture, 1890\u20131930._ Portland, ME: Greater Portland Landmarks, 1995.\n\nStevenson, Katherine H., and H. Ward Jandl. _Houses by Mail: A Guide to Houses from Sears, Roebuck, and Company._ Washington, DC: Preservation Press, 1996.\n\nSully, Susan. _Casa Florida: Spanish-Style Houses from Winter Park to Coral Gables._ New York: Rizzoli, 2005.\n\nTownsend, Gavin Edward. \"The Tudor House in America, 1890\u20131930.\" Ph.D. diss. University of California, 1986, 1988.\n\nUnderwood, Francis H. _The Colonial House Then and Now._ Rutland, VT: Charles E. Tuttle, 1977.\n\nWalsh, Michael, and Richard Toglia. _Tudor Houses_ [pattern book]. Farmington Hills, MI: Home Planners, 1989.\n\nWare, William R. _American Vignola: Guide to the Making of Classical Architecture._ 1903. Reprint, Dover Publications, 1995.\n\nWeitze, Karen J. _California's Mission Revival._ Santa Monica: Hennessey & Ingalls, 1984.\n\nWhite, Samuel G., and Elizabeth White. _Stanford White, Architect._ New York: Rizzoli, 2008.\n\nWhite, Samuel G., and Jonathan Wallen. _The Houses of McKim, Mead & White._ New York: Universe, 2004.\n\nWhite, Samuel G., Elizabeth White, and Jonathan Wallen. _McKim, Mead & White: The Masterworks._ New York: Rizzoli, 2003.\n\nWilson, Henry L. _The Bungalow Book_ [pattern book]. 1910. Reprint, New York: Dover Publications, 2006.\n\nWilson, Richard Guy. _McKim, Mead & White, Architects._ New York: Rizzoli, 1983.\n\n\u2014\u2014\u2014. _The Architecture of McKim, Mead & White in Photographs, Plans, and Elevations._ New York: Dover Publications, 1990.\n\nWilson, Richard Guy, and Noah Sheldon. _The Colonial Revival House._ New York: Abrams, 2004.\n\nWyllie, Romy. _Bertram Goodhue: His Life and Resi-dential Architecture._ New York: W. W. Norton, 2007.\n\n## Modern Houses\n\nAdamson, Paul, Marty Arbunich, and Ernest Braun. _Eichler: Modernism Rebuilds the American Dream._ Layton, UT: Gibbs Smith, 2002.\n\nAladdin Company. _Aladdin Homes Built in a Day,_ catalog no. 31 [pattern book]. 1919. Reprint, Watkins Glen, NY: American Life Foundation, 1985.\n\nAllen, Gerald. _Charles Moore._ New York: Whitney Library of Design, 1981.\n\nBaker, Geoffrey Harold, and Bruno Funaro. _Windows in Modern Architecture._ New York: Architectural Book Publishing Co., 1948.\n\nBlake, Peter. _Marcel Breuer, Architect and Designer._ New York: Architectural Record [u.a.], 1954.\n\nBottomley, Myrl Elijah. _The Design of Small Properties; A Book for the Homeowner in City and Country._ New York: Macmillan Co., 1929.\n\nBrooks, H. Allen. _The Prairie School: Frank Lloyd Wright and His Midwest Contemporaries._ 1972. Reprint, New York: W. W. Norton, 2006.\n\nBrooks, Turner, et al. _Turner Brooks: Work._ New York: Princeton Architectural Press, 1995.\n\nBrown-Blodgett Company (Saint Paul, MN). _100 Small Houses of the Thirties._ 1936. Reprint, Mineola, NY: Dover Publications, 2005.\n\nBurrows, John S. _Your New Home._ New York: Archway Press, 1950.\n\nCallender, John Hancock. _Before You Buy a House._ New York: Crown, 1953.\n\nCerwinske, Laura. _Tropical Deco: The Architecture and Design of Old Miami Beach._ New York: Rizzoli, 1991.\n\nClausen, Meredith L. _Pietro Belluschi: Modern American Architect._ Cambridge, MA: MIT Press, 1999.\n\nLe Corbusier, and Frederick Etchells. _Towards a New Architecture._ 1946. Reprint, London: Architectural Press, 2011.\n\nCraig, G. L., Rudard A. Jones, and William H. Kapple. _Split-level Houses._ Champaign, IL: Small Homes Council-Building Research Council, University of Illinois at Urbana-Champaign, 1960.\n\nCurrent, William R., and Karen Current. _Greene & Greene: Architects in the Residential Style._ Dobbs Ferry, NY: Morgan & Morgan, 1977.\n\nCygelman, Ad\u00e8le, and David Glomb. _Palm Springs Modern: Houses in the California Desert._ New York: Rizzoli, 2006.\n\nDavies, Colin. _Key Houses of the Twentieth Century: Plans, Sections and Elevations._ London: Laurence King, 2006.\n\nDietsch, Deborah. _Classic Modern: Midcentury Modern at Home._ London: Simon & Schuster, 2001.\n\nDillon, David. _The Architecture of O'Neil Ford: Celebrating Place._ University of Texas Press, 1999.\n\nDitto, Jerry, Lanning Stern, and Marvin Wax. _Eichler Homes: Design for Living._ San Francisco: Chronicle, 1995.\n\nDoan, Mason C. _American Housing Production, 1880\u20132000: A Concise History._ Lanham, MD: University Press of America, 2000.\n\nDoubilet, Susan, and Daralice D. Boles. _American House Now: Contemporary Architectural Directions._ Rev. ed. New York: Universe, 2002.\n\nDuchscherer, Paul, and Douglas Keister. _The Bungalow: America's Arts and Crafts Home._ New York: Penguin Studio, 1995.\n\nEarls, William D. _The Harvard Five in New Canaan: Midcentury Modern Houses by Marcel Breuer, Landis Gores, John Johansen, Philip Johnson, Eliot Noyes & Others._ New York: W. W. Norton, 2006.\n\nEhrlich, Doreen. _Usonian Houses._ London: PRC, 2002.\n\nEtter, Don D. _Denver Going Modern: A Photographic Essay on the Imprint of the International Style on Denver Residential Architecture._ Denver: Graphic Impressions, 1977.\n\nFaibyshev, Dolly. _Palm Springs Mid-Century Modern._ Atglen, PA: Schiffer, 2010.\n\nFlagg, Ernest. _Flagg's Small Houses: Their Economic Design and Construction, 1922._ Reprint of _Small Houses: Their Economic Design and Construction,_ 1921. Mineola, NY: Dover Publications, 2006.\n\nFord, James, and Katherine Morrow Ford. _The Modern House in America._ New York: Architectural Book Publishing Co., 1940.\n\nFrampton, Kenneth, and David Larkin. _The Twentieth Century American House: Masterworks of Residential Architecture._ London: Thames and Hudson, 1995.\n\nFriedman, Avi. \"The Evolution of Design Characteristics During the Post-Second World War Housing Boom: The U.S. Experience.\" _Journal of Design History_ 8, no. 2 (1995): 131\u2013146.\n\nGebhard, David, and Patricia Gebhard. _Purcell & Elmslie: Prairie Progressive Architects._ Salt Lake City, UT: Gibbs Smith, 2006.\n\nGeorge, Mary Carolyn Hollers. _O'Neil Ford, Architect._ College Station: Texas A&M University Press, 1992.\n\nGordon, Alastair. _Weekend Utopia: Modern Living in the Hamptons._ New York: Princeton Architectural Press, 2001.\n\nGorlin, Alexander, and Geoffrey Gross. _Tomorrow's Houses: New England Modernism._ New York: Rizzoli, 2011.\n\nGraf, Jean, and Don Graf. _Practical Houses for Contemporary Living._ New York: F. W. Dodge, 1953.\n\nGregory, Daniel Platt, and Cliff May. _Cliff May and the Modern Ranch House._ New York: Rizzoli, 2008.\n\nGroup, Harold E. _Small Houses of the Forties, with Illustrations and Floor Plans_ [pattern book]. 1946. Reprint, Mineola, NY: Dover Publications, 2007.\n\nHess, Alan. _Palm Springs Weekend: The Architecture and Design of a Mid-Century Oasis._ San Francisco: Chronicle Books, 2001.\n\n\u2014\u2014\u2014. _Organic Architecture: The Other Modernism._ Layton, UT: Gibbs Smith, 2006.\n\n\u2014\u2014\u2014. _Forgotten Modern: California Houses 1940\u20131970._ Layton, Utah: Gibbs Smith, 2007.\n\n\u2014\u2014\u2014. _Frank Lloyd Wright: Mid-century Modern._ New York: Rizzoli, 2007.\n\n\u2014\u2014\u2014. _Frank Lloyd Wright: The Houses._ New York: Rizzoli, 2007.\n\n\u2014\u2014\u2014. _The Ranch House._ New York: H. N. Abrams, 2008.\n\nHistoric Preservation Division, Georgia Department of Natural Resources. \"American Small Houses.\" N.d.] Retrieved from [http:\/\/georgiashpo.org\/sites\/uploads\/hpd\/pdf\/American_Small_House.pdf\n\n\u2014\u2014\u2014. \"The Ranch House in Georgia: Guidelines for Evaluation.\" N.d.] Retrieved from [http:\/\/georgiashpo.org\/sites\/uploads\/hpd\/pdf\/Ranch_House_Evaluation_revSept2010.pdf\n\nHitchcock, Henry-Russell. _In the Nature of Materials, 1887\u20131941: The Buildings of Frank Lloyd Wright._ 1942. Reprint, New York: Da Capo Press, 1975.\n\nHitchcock, Henry-Russell, and Philip Johnson. _The International Style: Architecture Since 1922._ 1932. Rev. ed. New York: W. W. Norton & Co., 1997.\n\nHodgson, Fred T. _Practical Bungalows and Cottages for Town and Country_ [pattern book]. Chicago: Frederick J. Drake, 1916.\n\nHome Planners, Inc. _The Essential Guide to Contemporary Home Plans_ [pattern book]. Farmington Hills, MI: Home Planners, 1987.\n\nIsenstadt, Sandy. _The Modern American House: Spaciousness and Middle-class Identity._ Cambridge: Cambridge University Press, 2006.\n\nJackson, N. _California Modern: The Architecture of Craig Ellwood._ New York: Princeton Architectural Press, 2002.\n\nJencks, Charles. _The Language of Post-Modern Architecture._ 6th ed. New York: Rizzoli, 1991.\n\n\u2014\u2014\u2014. _The Story of Post-Modernism: Five Decades of the Ironic, Iconic and Critical in Architecture._ 2nd ed. John Wiley & Sons, 2012.\n\nJensen, Robert, and Patricia Conway. _Ornamentalism._ New York: Clarkson Potter, 1982.\n\nJohnson, Eugene J. _Charles Moore: Buildings and Projects, 1949\u20131986._ New York: Rizzoli, 1991.\n\nJones, Robert T. _Small Houses of Architectural Distinction_ [pattern book]. 1929. Reprint, New York: Dover Publications, 1987.\n\nJunior League of Tulsa. _Tulsa Art Deco: An Architectural Era, 1925\u20131942._ Tulsa, OK: Junior League, 1980.\n\nKeil, Rob. _Little Boxes: The Architecture of a Classic Midcentury Suburb._ Daly City, CA: Advection Media, 2006.\n\nKhan, Hasan-Uddin. _International Style: Modernist Architecture from 1925 to 1965._ K\u00f6ln (Germany): Taschen, 1998.\n\nLambin, Jeanne, and Janine Duncan. \"The Recent Past Is Groovy: Researching American Architectural Styles after World War II.\" _Alliance Review,_ July\u2013August 2008. <http:\/\/www.landmarksil.org\/pdfs\/alliance_review.pdf>\n\nLancaster, Clay. _The American Bungalow, 1880\u20131930._ 1983. Reprint, New York: Dover Publications, 1995.\n\nLevine, Neil, and Frank Lloyd Wright. _The Architecture of Frank Lloyd Wright._ Princeton, NJ: Princeton University Press, 1998.\n\nLuce, Henry R. _Form Givers at Mid-Century._ [New York]: Time, Inc. for American Federation of Arts, 1959.\n\nLyndon, Donlyn, James Alinder, Donald Canty, and Lawrence Halprin. _The Sea Ranch._ New York: Princeton Architectural Press, 2004.\n\nLyon, Hortense. _American Contemporary Houses._ Paris: Telleri, 1998.\n\nMaddex, Diane. _Frank Lloyd Wright's House Beautiful._ New York: Hearst Books, 2000.\n\n\u2014\u2014\u2014. _Wright-Sized Houses: Frank Lloyd Wright's Solutions for Making Small Houses Feel Big._ New York: Abrams, 2003.\n\nMaddex, Diane, and Alexander Vertikoff. _Bungalow Nation._ New York: Harry N. Abrams, 2003.\n\nMaine Historic Preservation Commission. \"Post World War II Architecture in Maine: A Guide for Surveyors.\" <http:\/\/www.maine.gov\/portal\/index.html>\n\nMakinson, Randell L. _Greene & Greene._ Rev. ed. Salt Lake City: Gibbs Smith, 2001.\n\nMartin, Harry, Dick Busher, and Warren Winther. _Contemporary Homes of the Pacific Northwest._ Seattle: Madrona Publishers, 1980.\n\nMaster Plan Service. _Encyclopedia of Home Designs._ Series. Mineola, NY: Master Plan Service, 1967\u201384.\n\nMay, Cliff, and Paul C. Johnson. _Western Ranch Houses by Cliff May._ 1946. Reprint, Santa Monica, CA: Hennessey & Ingalls, 1997.\n\nMay, Cliff, and Sunset Magazine. _Sunset Western Ranch Houses._ 1958. Reprint. Santa Monica, CA: Hennessey & Ingalls, 1999.\n\nMcCarthy, Muriel Q. _David R. Williams, Pioneer Architect._ Dallas: Southern Methodist University Press, 1984.\n\nMcCoy, Esther. _Five California Architects._ 2nd ed. Santa Monica, CA: Hennessey & Ingalls, 1975.\n\n\u2014\u2014\u2014. _The Second Generation._ Salt Lake City: Peregrine Smith Books, 1984.\n\nMoore, Charles Willard, Gerald Allen, and Donlyn Lyndon. _The Place of Houses._ 1979. Reprint, New York: Holt, Rinehart and Winston, 2011.\n\nMorand, Fran\u00e7ois C. _Small Homes in the New Tradition_ [pattern book]. New York: Sterling, 1959.\n\nNational Plan Service, Inc. _Homes of Moderate Cost_ [pattern book]. Chicago: National Plan Service, 1941.\n\nNelson, George, and Henry Wright. _Tomorrow's House: A Complete Guide for the Home-Builder._ New York: Simon and Schuster, 1946.\n\nNoever, Peter, and Regina Haslinger. _Architecture in Transition: Between Deconstruction and New Modernism._ Munich: Prestel, 1997.\n\nOlsberg, R. Nicholas, and Jocelyn Gibbs. _Carefree California: Cliff May and the Romance of the Ranch House._ New York: Rizzoli International, 2012.\n\nPascal, Patrick, Julius Shulman, and David Sadofski. _Kesling Modern Structures: Popularizing Modern Design in Southern California, 1934\u20131962._ Los Angeles: Balcony Press, 2002.\n\nPearson, Clifford A., and Thomas Hine. _Modern American Houses: Four Decades of Award-winning Design in Architectural Record._ New York: Harry N. Abrams, 1996.\n\nPearson, David. _New Organic Architecture: The Breaking Wave._ Berkeley: University of California Press, 2001.\n\nPeterson, Gary G. \"Home off the Range: The Origins and Evolution of Ranch Style Architecture in the United States.\" _Design Methods and Theories_ 23 (1989): 1040\u201359.\n\nRandl, Chad. _A-frame._ New York: Princeton Architectural Press, 2004.\n\nRawlins, Christopher Bascom. _Fire Island Modernist: Horace Gifford and the Architecture of Seduction._ New York: Metropolis Books, 2013.\n\nReisley, Roland, and John Philip Timpane. _Usonia, New York: Building a Community with Frank Lloyd Wright._ New York: Princeton Architectural Press, 2001.\n\nRiera Ojeda, Oscar, and Lucas H. Guerra. _The New American House: Innovations in Residential Design and Construction: 30 Case Studies._ 4 vols. New York: Whitney Library of Design, 1995\u20132003.\n\nRifkind, Carole. _A Field Guide to Contemporary American Architecture._ New York: Penguin, 2001.\n\nRosenbaum, Alvin. _Usonia: Frank Lloyd Wright's Design for America._ Washington, DC: Preservation Press, National Trust for Historic Preservation, 1993.\n\nSamon, Katherine Ann. _Ranch House Style._ New York: Clarkson Potter, 2003.\n\nSchulze, Franz. _Philip Johnson: Life and Work._ New York: Knopf, 2009.\n\nScully, Vincent Joseph, and Neil Levine. _Modern Architecture and Other Essays._ Princeton, NJ: Princeton University Press, 2005.\n\nSears, Roebuck and Co. _Sears, Roebuck Catalog of Houses_ [pattern book]. 1926. Reprint, New York: Dover Publications, 1991.\n\nSergeant, John. _Frank Lloyd Wright's Usonian Houses: The Case for Organic Architecture._ 1976. Reprint, New York: Whitney Library of Design, 2009.\n\nSerraino, Pierluigi. _NorCalMod: Icons of Northern California Modernism._ San Francisco: Chronicle Books, 2006.\n\nSerraino, Pierluigi, Julius Shulman, and Peter G\u00f6ssel. _Julius Shulman: Modernism Rediscovered._ Los Angeles: Taschen, 2009.\n\nShulman, Julius, and Gary Gand. _Julius Shulman: Chicago Mid-Century Modernism._ New York: Rizzoli, 2010.\n\nShulman, Julius, Michael Stern, and Alan Hess. _Julius Shulman: Palm Springs._ New York: Rizzoli, 2008.\n\nShulman, Julius, Sam Lubell, Douglas Woods, and Judy McKee. _Julius Shulman, Los Angeles: The Birth of a Modern Metropolis._ New York: Rizzoli, 2011.\n\nSlaton, Deborah, and Rebecca A Shiffer, eds. _Preservingthe Recent Past._ Washington, DC: Historic Preservation Education Foundation, 1995.\n\nSlaton, Deborah, and William G. Fouls, eds. _Preserving the Recent Past 2,_ Washington, DC: Historic Preservation Education Foundation, National Park Service, and Association for Preservation Technology International, 2000.\n\nSmith, Bruce, and Alexander Vertikoff. _Greene & Greene: Masterworks._ San Francisco: Chronicle Books, 1998.\n\nSmith, Elizabeth A. T., and Peter G\u00f6ssel. _Case Study Houses, 1945\u20131966: The California Impetus._ K\u00f6ln (Germany): Taschen, 2009.\n\nSmith, Herbert L. _Twenty-five Years of Record Houses._ New York: McGraw-Hill, 1981.\n\nStickley, Gustav. _Craftsman Homes_ [pattern book]. 1909. Reprint, New York: Dover Publications, 1979.\n\n\u2014\u2014\u2014. _Gustav Stickley's Craftsman Homes and Bungalows_ [pattern book]. 1912. Reprint, New York: Skyhorse, 2009.\n\nStorrer, William Allin. _The Architecture of Frank Lloyd Wright: A Complete Catalog._ 3rd ed. Chicago: University of Chicago Press, 2007.\n\nStubblebine, Jo, ed. _The Northwest Architecture of Pietro Belluschi._ New York: F. W. Dodge Corp., 1953.\n\nStubblebine, Ray, and Gustav Stickley. _Stickley's Craftsman Homes: Plans, Drawings, Photographs._ Layton, UT: Gibbs Smith, 2006.\n\nSutro, Dirk. _West Coast Wave: New California Houses._ New York: Van Nostrand Reinhold, 1994.\n\nTrapp, Kenneth R., and Leslie Greene Bowman. _The Arts and Crafts Movement in California: Living the Good Life._ Oakland, CA: Oakland Museum, 1993.\n\n_A Treasury of Contemporary Houses._ New York: F. W. Dodge, 1954.\n\nTrulove, James Grayson, and Il Kim. _The New American Cottage: Innovations in Small-Scale Residential Architecture._ New York: Whitney Library of Design, 1999.\n\nU.S. Federal Housing Administration. _Principles of Planning Small Houses._ Washington, DC: U.S. Federal Housing Administration, 1936, 1940.\n\nVenturi, Robert. _Complexity and Contradiction in Architecture._ 1966. 2nd ed. New York: Museum of Modern Art, 2011.\n\nVon Holst, H. V. _Country and Suburban Homes of the Prairie School Period, with 424 Photographs and Floor Plans_ [pattern book]. N.d. Reprint, New York: Dover Publications, 1982.\n\n\u2014\u2014\u2014. _Modern American Homes, Etc._ [pattern book]. 1913. Reprint, American School of Correspondence: Chicago, 2008.\n\nWagner, Walter F. _A Treasury of Contemporary Houses._ New York: McGraw-Hill, 1978.\n\nWebb, Michael, and Roger Straus. _Modernism Reborn: Mid-Century American Houses._ New York: Universe, 2001.\n\nWeingarten, David, Lucia Howard, and Joe Fletcher. _Ranch Houses: Living the California Dream._ New York: Rizzoli, 2009.\n\nWeston, Richard. _Twentieth-Century Residential Architecture._ New York: Abbeville Press, 2002.\n\nWickes, Molly, and Kate Irvin. _A Guide to Oak Park's Frank Lloyd Wright and Prairie School Historic District._ Oak Park, IL: Oak Park Historic Preservation Commission, Village of Oak Park, 1999.\n\nWilson, Henry L. _California Bungalows of the Twenties_ [pattern book]. N.d. Reprint, New York: Dover Publications, 1993.\n\nWinter, Robert. _The California Bungalow._ Los Angeles: Hennessey and Ingalls, 1980.\n\n\u2014\u2014\u2014. _Toward a Simpler Way of Life: The Arts & Crafts Architects of California._ Berkeley: University of California Press, 1997.\n\nWinter, Robert, and Alexander Vertikoff. _American Bungalow Style._ New York: Simon & Schuster, 1996.\n\nWolfe, Tom. _From Bauhaus to Our House._ 1981. Reprint, New York: Farrar Straus Giroux, 2012.\n\nWright, Frank Lloyd. _The Natural House._ 1954. Reprint, New York: Horizon Press, 1982.\n\nZaleski, Caroline Rob. _Long Island Modernism, 1930\u20131980._ New York: Society for the Preservation of Long Island Antiquities, in association with W. W. Norton & Company, 2012.\n\nIn addition, Richard Longstreth's comprehensive \"A Historical Bibliography of Architecture, Landscape Architecture, and Urbanism in the United States Since World War II\" is updated regularly and is available through the following link: <http:\/\/www.recentpast.org\/bibliography\/RPPNbib2012.pdf>\n\n## Styled Houses Since 1935\n\nBaker, William T., James R. Lockhart, and Beverly Means Dubose. _New Classicists._ Mulgrave (Australia): Images Publishing Group, 2004.\n\nBassenian, Aram, and Laura Hurst Brown. _Pure California: 35 Inspiring Houses in the New California Tradition._ Newport Beach, CA: Bassenian\/Lagoni Architects, 2004.\n\nBrostrom, Caitlin Lempres, William Wilson Wurster, and Richard C. Peters. _The Houses of William Wurster: Frames for Living._ New York: Princeton Architectural Press, 2011.\n\nCarter, Thomas, and Elizabeth C. Cromley. _Invitation to Vernacular Architecture: A Guide to the Study of Ordinary Buildings and Landscapes._ Knoxville: University of Tennessee Press, 2005.\n\nChase, John. _Glitter Stucco and Dumpster Diving: Reflections on Building Production in the Vernacular City._ London: Verso, 2004.\n\nCooper, Daniel Robert. _New Classic American Houses:The Architecture of Albert, Righter & Tittmann._ New York: Vendome Press, 2009.\n\nCrosbie, Michael J., and John R. DaSilva. _Architecture of the Cape Cod Summer: The Work of Polhemus Savery DaSilva._ Mulgrave (Australia): Images Publishing Group, 2008.\n\nCusato, Marianne, Ben Pentreath, Richard F. Sammons, and L\u00e9on Krier. _Get Your House Right: Architectural Elements to Use & Avoid._ New York: Sterling, 2011.\n\nDaSilva, John R. _Shingled Houses in the Summer Sun: The Work of Polhemus Savery DaSilva._ Mulgrave (Australia): Images Publishing Group, 2011.\n\nDillon, David. \"Big Mess on the Prairie.\" _Dallas Morning News,_ October 2, 1994.\n\nDowling, Elizabeth Meredith. _New Classicism: The Rebirth of Traditional Architecture._ New York: Rizzoli, 2004.\n\nEck, Jeremiah. _The Face of Home: A New Way to Look at the Outside of Your House._ Newtown, CT: Taunton Press, 2006.\n\nFerguson, Mark, Oscar Shamamian, and Joseph Giovannini. _New Traditional Architecture: Ferguson & Shamamian Architects\u2014City and Country Residences._ New York: Rizzoli, 2011.\n\nGabriel, J. Fran\u00e7ois. _Classical Architecture for the Twenty-first Century: An Introduction to Design._ New York: W. W. Norton\/Institute of Classical Architecture and Classical America, 2004.\n\nHome Planners, Inc. _Encyclopedia of Home Designs: 450 House Plans_ [pattern book]. Farmington Hills, MI: Home Planners, 1987.\n\n\u2014\u2014\u2014. _Luxury Dream Homes_ [pattern book]. Farmington Hills, MI: Home Planners, 1989.\n\n\u2014\u2014\u2014. _Arts & Crafts Home Plans: Showcasing 85 Home Plans in the Craftsman, Prairie, and Bungalow Styles_ [pattern book]. Tucson, AZ: Home Planners, 2004.\n\nHopkins, George D. _Creating Your Architectural Style: Designing and Building a Fine Home._ Gretna, LA: Pelican, 2009.\n\n_House Plans: Custom-Designed Homes for the South._ Birmingham, AL: Oxmoor House, 1988.\n\nIke Kligerman Barkley (firm), Robert A. M. Stern, and Marc Kristal. _Ike Kligerman Barkley: Houses._ New York: Monacelli Press, 2010.\n\nJacobsen, Hugh Newell, Paul Goldberger, Massimo Vignelli, and Robert C. Lautman. _Hugh Newell Jacobsen, Architect: Works from 1993 to 2006._ New York: Rizzoli International, 2007.\n\nKemp, Jim. _American Vernacular: Regional Influences in Architecture and Interior Design._ Washington, DC: American Institute of Architects Press, 1990.\n\nLang, Andy. _101 Select Dream Houses_ [pattern book]. Maplewood, NJ: Hammond, 1972.\n\nLarson, Jean Rehkamp, and Ken Gutmaker. _The Farmhouse: New Inspiration for the Classic American Home._ Newtown, CT: Taunton Press, 2006.\n\nLooney Ricks Kiss Architects. _Traditional Neighborhood Home Plans: 170 Designs for Living in Villages & Towns_ [pattern book]. Tucson, AZ: Home Planners, 2000.\n\nMcNamara, Sarah. \"The Rise and Fall of the Mansard Roof.\" _The Old House Journal_ 12, no. 7 (August\u2013September 1984): 152\u201354.\n\nMiers, Mary. _American Houses: The Architecture of Fairfax & Sammons._ New York: Rizzoli, 2006.\n\nMorgan, William. _Yankee Modern: The Houses of Estes\/Twombly._ New York: Princeton Architectural Press, 2010.\n\nMouzon, Stephen A., and Susan M. Henderson. _Traditional Construction Patterns._ New York: McGraw-Hill, 2004.\n\nMulvin, Paulette. _Western Home Plans: Over 200 Home Plans Specially Designed for California, Pacific Northwest, Rocky Mountains, Texas & Western Plains, Desert Southwest, Western Lovers Everywhere_ [pattern book]. Tucson, AZ: Home Planners, 1992.\n\nPrideaux, Jan. _European Dream Homes: 200 French, English and Mediterranean Designs_ [pattern book]. Tucson, AZ: Home Planners, 2002.\n\nRoyal Barry Wills Associates. _More Houses for Good Living._ 1968. Reprint, New York: Architectural Book Publishing, 1976.\n\n\u2014\u2014\u2014. _Houses for Good Living._ 1946. Reprint, New York: Architectural Book Publishing, 1993.\n\nSagharchi, Alireza, and Lucien Steil. _New Palladians: Modernity and Sustainability for 21st-Century Architecture._ London: Artmedia, 2010.\n\nSater, Dan F. _European Luxury Homes_ [pattern book]. Bonita Springs, FL: Sater Design Collection, 2003.\n\n\u2014\u2014\u2014. _Mediterranean Home Plans_ [pattern book]. Bonita Springs, FL: Sater Design Collection, 2005.\n\n\u2014\u2014\u2014. _European Classics_ [pattern book]. Bonita Springs, FL: Sater Design Collection, 2007.\n\nSkurman, Andrew. _Contemporary Classicism: The Architecture of Andrew Skurman._ New York: Princeton Architectural Press, 2012.\n\nStern, Robert A. M. _New Directions in American Architecture._ Rev. ed. New York: G. Braziller, 1982.\n\nStern, Robert A. M., and Clive Aslet. _The American Houses of Robert A. M. Stern._ New York: Rizzoli, 1991.\n\nStern, Robert A. M., and Elizabeth Kraft. _Buildings and Projects, 1987\u20131992._ New York: Rizzoli, 1992.\n\nStern, Robert A. M., and Raymond Gastil. _Modern Classicism._ New York: Rizzoli, 1988.\n\nStern, Robert A. M., Peter Arnell, and Ted Bickford. _Robert A. M. Stern 1965\u20131980: Toward a Modern Architecture After Modernism._ New York: Rizzoli, 1987.\n\nToplin, Jim. _The New Cottage Home._ Newtown, CT: Taunton Press, 2000.\n\nTreib, Marc, and David Gebhard. _An Everyday Modernism: The Houses of William Wurster._ San Francisco: San Francisco Museum of Modern Art, 1999.\n\nUrban Design Associates, Louisiana Recovery Authority, and LRA Support Foundation. _Louisiana Speaks: Pattern Book._ [Baton Rouge, LA]: LRA Support Foundation, 2006. <http:\/\/www.urbandesignassociates.com\/services_patternbooks_pg1.html>\n\nVersaci, Russell, and Erik Kvalsvik. _Creating a New Old House: Yesterday's Character for Today's Home._ Newtown, CT: Taunton Press, 2007.\n\nVetter, Cyril E., and Philip Gould. _The Louisiana Houses of A. Hays Town._ Baton Rouge: Louisiana State University Press, 1999.\n\nWalker, Anne. _Peter Pennoyer Architects: Apartments, Townhouses, Country Houses._ New York: Vendome Press, 2010.\n\nWatkin, David. _Radical Classicism: The Architecture of Quinlan Terry._ New York: Rizzoli, 2006.\n\n## Approaches to Building in the Twentiety Century\n\nBergdoll, Barry, Peter Christensen, and Ron Broadhurst. _Home Delivery: Fabricating the Modern Dwelling._ New York: Museum of Modern Art, 2008.\n\nBletter, Rosemarie Haag. _Remembering the Future: The New York World's Fair from 1939 to 1964._ New York: Rizzoli, 1989.\n\nCarroon, Jean. _Sustainable Preservation: Greening Existing Buildings._ Hoboken, NJ: Wiley, 2010.\n\nCowan, Ruth S. _A Social History of American Technology._ New York: Oxford University Press, 1997.\n\nGianino, Andrew. _The Modular Home._ North Adams, MA: Storey, 2005.\n\nGraff, Raymond K., Rudolf A. Matern, and Henry Lionel Williams. _The Prefabricated House: A Practical Guide for the Prospective Buyer._ Garden City, NY: Doubleday, 1947.\n\nHerbert, Gilbert. _The Dream of the Factory-Made House: Walter Gropius and Konrad Wachsmann._ Cambridge, MA: MIT Press, 1986.\n\nJester, Thomas C. _Twentieth-Century Building Materials: History and Conservation._ [New York]: McGraw-Hill, 1995.\n\nJohnston, David, and Scott Gibson. _Green from the Ground Up: Sustainable, Healthy, and Energy-Efficient Home Construction._ Newtown, CT: Taunton Press, 2008.\n\nKaufmann, Michelle, and Catherine Remick. _Prefab Green._ Layton, UT: Gibbs Smith, 2009.\n\nKoones, Sheri. _Prefabulous and Sustainable: Building and Customizing an Affordable, Energy-Efficient Home._ New York: Abrams, 2010.\n\nMouzon, Stephen A., and Robert Francis Kennedy. _The Original Green: Unlocking the Mystery of True Sustainability._ Miami: Guild Foundation Press, 2010.\n\nRydell, Robert W., Laura B. Schiavo, Robert Bennett, and Matthew Bokovoy. _Designing Tomorrow: America's World's Fairs of the 1930s._ Yale University Press, 2010.\n\nSchrenk, Lisa Diane. _Building a Century of Progress: The Architecture of Chicago's 1933\u201334 World's Fair._ Minneapolis: University of Minnesota Press, 2007.\n\n## Regional and Local Guides: Northeastern States\n\nAmerican Institute of Architects, Long Island Chapter, and Society for the Preservation of Long Island Antiquities. _AIA Architectural Guide to Nassau and Suffolk Counties, Long Island._ New York: Dover Publications, 1992.\n\nAndrews, Wayne. _Architecture in New England: A Photographic History._ New York: Harper & Row, 1980.\n\nBassett, William B. _Historic American Buildings Survey of New Jersey._ Newark: New Jersey Historical Society, 1977.\n\nBronxville Centennial Celebration. _Building a Suburban Village: Bronxville, New York, 1898\u20131998._ Bronxville, NY: Bronxville Centennial Celebration, Inc., 1998.\n\nBrown, Elizabeth Mills. _New Haven: A Guide to Architecture and Urban Design._ New Haven, CT: Yale University Press, 1976.\n\nBuffalo Architectural Guidebook Corporation. _Buffalo Architecture: A Guide._ Cambridge, MA: MIT Press, 1981.\n\nBunting, Bainbridge. _Houses of Boston's Back Bay: An Architectural History, 1840\u20131917._ Cambridge, MA: Harvard University Press, 1967.\n\nBurke, Bobbye, and Trina Vaux. _Historic Rittenhouse: A Philadelphia Neighborhood._ Philadelphia: University of Pennsylvania Press, 1985.\n\nCambridge Historical Commission. _Survey of Architectural History in Cambridge, Reports 1\u20135._ 5 vols. Cambridge, MA, 1965\u20131977.\n\nCandee, Richard M. _Atlantic Heights: A World War I Shipbuilders' Community._ Portsmouth, NH: published for the Society by P. E. Randall, 1985.\n\nConry, Jaci. _A History Through Houses Cape Cod's Varied Residential Architecture._ Charleston, SC: History Press, 2010.\n\nDennis, Stephen Neal, and William J. Penberthy. _Historic Houses of the Sewickley Valley._ Sewickley, PA: White Oak Pub, 1996.\n\nDiamonstein, Barbaralee. _The Landmarks of New York._ New York: Monacelli Press, 2005.\n\nDorsey, John, and James D. Dilts. _A Guide to Baltimore Architecture,_ 3rd ed. Centreville, MD: Tidewater Publishers, 1997.\n\nDowning, Antoinette F., and Vincent J. Scully Jr. _TheArchitectural Heritage of Newport, Rhode Island, 1640\u20131915._ Rev. ed., New York: American Legacy Press, 1982.\n\nEberlein, Harold Donaldson, and Cortlandt Van Dyke Hubbard. _Historic Houses of the Hudson Valley._ 1942. Reprint, New York: Dover Publications, 1990.\n\n\u2014\u2014\u2014. _Historic Houses and Buildings of Delaware._ Dover, DE: Public Archives Department, 1962.\n\nFishman, David, Thomas Mellins, and Robert A. M. Stern. _New York: Architecture and Urbanism._ 5 vols. New York: Monacelli Press, 1983\u20132009.\n\nFoerster, Bernd. _Architecture Worth Saving in Rensselaer County, New York._ Troy, NY: Rensselaer Polytechnic Institute, 1965.\n\nForman, H. Chandlee. _Maryland Architecture: A Short History from 1634 through the Civil War._ Cambridge, MD: Tidewater Publishers, 1968.\n\nFox, Pamela W. _North Shore Boston: Houses of Essex County, 1865\u20131930._ New York: Acanthus Press, 2005.\n\nGallery, John Andrew. _Philadelphia Architecture: A Guide to the City._ 3rd ed. Philadelphia: Paul Dry Books, 2009.\n\nGarrison, James B., John Andrew Gallery, and William Morrison. _Houses of Philadelphia: Chestnut Hill and the Wissahickon Valley, 1880\u20131930._ New York: Acanthus Press, 2008.\n\nGarvin, James L. _A Building History of Northern New England._ Hanover, NH: University Press of New England, 2001.\n\nGoldberger, Paul. _The City Observed: New York\u2014A Guide to the Architecture of Manhattan._ New York: Vintage Books, 1989.\n\nGoldstone, Harmon H., and Martha Dalrymple. _History Preserved: A Guide to New York City Landmarks and Historic Districts._ New York: Schocken Books, 1974\/1976.\n\nGowans, Alan. _Architecture in New Jersey._ The New Jersey Historical Series, vol. 6. Princeton, NJ: D. Van Nostrand, 1964.\n\nGreiff, Constance M., Mary W. Gibbons, and Elizabeth G. C. Menzies. _Princeton Architecture._ Princeton, NJ: Princeton University Press, 1975.\n\nGuter, Robert P., Janet W. Foster, and Jim DelGiudice. _Building by the Book: Pattern Book Architecture in New Jersey._ New Brunswick, NJ: Rutgers University Press, 1992.\n\nHardin, Evamaria, and Jon Crispin. _Syracuse Landmarks: An AIA Guide to Downtown and Historic Neighborhoods._ New York: Onondaga Historical Association, 1993.\n\nHartford Architecture Conservancy Survey. _Hartford Architecture,_ vols. 1\u20133. Hartford, CT, 1978\u20131980.\n\nHayward, Mary Ellen, and Charles Belfoure. _The Baltimore Rowhouse._ New York: Princeton Architectural Press, 2001.\n\nHistoric American Buildings Survey. _Historic Buildings of Massachusetts._ New York: Charles Scribner's Sons, 1976.\n\nHitchcock, Henry-Russell. _Rhode Island Architecture._ 1939. Reprint, Cambridge, MA: MIT Press, 1968.\n\nHowland, Richard Hubbard, and Eleanor Patterson Spencer. _The Architecture of Baltimore._ 1953. Reprint, Baltimore: Johns Hopkins Press, 1970.\n\nHubka, Thomas C. _Big House, Little House, Back House, Barn: The Connected Farm Buildings of New England._ Rev. ed. Hanover, NH: University Press of New England, 2004.\n\nHuxtable, Ada Louise. _The Architecture of New York: A History and Guide._ 3 vols. Garden City, NY: Doubleday, 1964.\n\nJackson, Richard S., and Cornelia Brooke Gilder. _Houses of the Berkshires, 1870\u20131930._ Rev. ed. New York: Acanthus Press, 2011.\n\nJacobs, Stephen W. _Wayne County_ [New York]: _The Aesthetic Heritage of a Rural Area._ New York: Publishing Center for Cultural Resources, 1979.\n\nJunior League of Kingston. _Early Architecture in Ulster County_ [New York]. Kingston, NY, 1974.\n\nKathrens, Michael C. _Great Houses of New York, 1880\u20131930._ New York: Acanthus Press, 2005.\n\nLancaster, Clay. _The Architecture of Historic Nantucket._ New York: McGraw-Hill, 1972.\n\n\u2014\u2014\u2014. _Old Brooklyn Heights: New York's First Suburb._ 2nd ed. New York: Dover, 1979.\n\nLancaster, Clay, Robert A. M. Stern, and Robert J. Hefner. _East Hampton's Heritage: An Illustrated Architectural Record._ New York: W. W. Norton, 1982.\n\nLandscape Research (firm) and Somerville, MA. _Beyond the Neck: The Architecture and Development of Somerville, Massachusetts._ Rev. ed. Somerville: City of Somerville, 1990.\n\nLanier, Gabrielle M., and Bernard L. Herman. _Everyday Architecture of the Mid-Atlantic: Looking at Buildings and Landscapes._ Baltimore: Johns Hopkins University Press, 1997.\n\nLarew, Marilynn M. _Bel Air_ [Maryland]: _The Town Through Its Buildings._ Annapolis: Maryland Historical Trust, 1995.\n\nLawrance, Gary, and Anne Surchin. _Houses of the Hamptons, 1880\u20131930._ New York: Acanthus Press, 2009.\n\nLockwood, Charles. _Bricks & Brownstone: The New York Row House, 1783\u20131929._ 2nd ed. New York: Rizzoli, 2003.\n\nLongstreth, Richard W. _Housing Washington: Two Centuries of Residential Development and Planning in the National Capitol_ [sic] _Area._ Chicago: Center for American Places at Columbia College Chicago, 2010.\n\nMalo, Paul. _Landmarks of Rochester and Monroe County._ Syracuse, NY: Syracuse University Press, 1974.\n\nMaryland Historical Trust. _Inventory of Historic Sites in Calvert County, Charles County, and St. Mary's County._ Rev. ed. Annapolis, MD, 1980.\n\nMateyunas, Paul J. _North Shore Long Island: Country Houses, 1890\u20131950._ New York: Acanthus Press, 2007.\n\nMcGowan, Robert Harold. _Architecture from the Adirondack Foothills: Folk and Designed Architecture from Franklin County, New York._ Malone, NY: Franklin County Historical and Museum Society, 1977.\n\nMorrison, William, and Michael C. Kathrens. _The Main Line: Country Houses of Philadelphia's Storied Suburb, 1870\u20131930._ New York: Acanthus Press, 2006.\n\nMyers, Denys Peter. \"The Historic Architecture of Maine.\" _Maine Catalog, Historic American Buildings Survey,_ pp. 1\u2013198. Augusta, ME: Maine State Museum, 1974.\n\nNew York State Office of Planning Coordination. _Long Island Landmarks._ Albany, NY, 1969.\n\nOnorato, Ronald J., Warren Jagger, and Richard Guy Wilson. _AIA Guide to Newport._ Providence, RI: AIA RI Architectual Forum, 2007.\n\nPancoast, Patricia McGraw, and Josephine H. Detmer. _Portland._ Portland, ME: Greater Portland Landmarks, 1972.\n\nPratt, Richard. _A Guide to the Architecture of Hightstown Houses._ Hightstown, NJ: Stockton Street Solutions, 2012.\n\nPreservation Worcester, Elliott B. Knowlton, and Sandra Gibson-Quigley. _Worcester's Best: A Guide to the City's Architectural Heritage._ Rev. ed. Worcester, MA: Preservation Worcester, 1996.\n\nProkopoff, Stephen S., and Joan C. Siegfried. _The Nineteenth-Century Architecture of Saratoga Springs._ New York: State Council on the Arts, 1970.\n\nRaymond, Eleanor. _Early Domestic Architecture of Pennsylvania._ 1931. Reprint, Exton, Pennsylvania: Schiffer, 2007.\n\nReiff, Daniel D. _Architecture in Fredonia, 1811\u20131972: From Log Cabin to I. M. Pei._ Fredonia, NY: White Pine Press, 1997.\n\nRettig, Robert Bell. _Guide to Cambridge Architecture: Ten Walking Tours._ Cambridge, MA: MIT Press, 1986.\n\nRifkind, Carole, and Carol Levine. _Mansions, Mills, and Main Streets: Buildings and Places to Explore Within Fifty Miles of New York City._ New York: Schocken Books, 1975.\n\nSanchis, Frank F. _American Architecture: Westchester County, New York._ Croton-on-Hudson, NY: North River Press, 1977.\n\nSchiffer, Margaret Berwind. _Survey of Chester County, Pennsylvania, Architecture: 17th, 18th, and 19th Centuries._ Exton, PA: Schiffer, 1997.\n\nSchull, Diantha Dow. _Landmarks of Otsego County_ [New York]. Syracuse, NY: Syracuse University Press, 1980.\n\nSchwartz, Helen, and Margaret Morgan Fisher. _The New Jersey House._ New Brunswick, NJ: Rutgers University Press, 1990.\n\nShand-Tucci, Douglass. _Built in Boston: City and Suburb, 1800-2000._ Rev ed. Amherst: University of Massachusetts, 1999.\n\nSouthworth, Susan, and Michael Southworth. _AIA Guide to Boston._ 3rd ed. Guilford, CT: Globe Pequot, 2008.\n\nStotz, Charles Morse. _The Early Architecture of Western Pennsylvania: A Record of Building Before 1860._ 1936. Reprint, Pittsburgh: University of Pittsburgh Press, 1995.\n\nTatum, George Bishop. _Penn's Great Towns: 250 Years of Philadelphia Architecture Illustrated in Prints and Drawings._ Philadelphia: University of Pennsylvania Press, 1961.\n\nTeitelman, Edward, and Richard W. Longstreth. _Architecture in Philadelphia: A Guide._ Cambridge, MA: MIT Press, 1981.\n\nThompson, Deborah, ed. _Maine Forms of American Architecture._ Camden, ME: Downeast Magazine, 1976.\n\nTolles, Bryant F. Jr., and Carolyn K. Tolles. _New Hampshire Architecture: An Illustrated Guide._ Hanover, NH: University Press of New England, 1979.\n\n\u2014\u2014\u2014. _Architecture in Salem: An Illustrated Guide._ Salem, MA: Essex Institute\/Historic Salem, 2004.\n\nUniversity of Vermont Historic Preservation Program. _The Burlington Book: Architecture, History, Future._ Burlington, VT, 1980.\n\nVan Trump, James D., and Arthur P. Ziegler Jr. _Landmark Architecture of Allegheny County, Pennsylvania._ Pittsburgh: Pittsburgh History and Landmarks Foundation, 1967.\n\nWarner, Sam Bass. _The Private City Philadelphia in Three Periods of Its Growth._ 2nd ed. Philadelphia: University of Pennsylvania Press, 1996.\n\nWebster, Richard J. _Philadelphia Preserved: Catalog of the Historic American Buildings Survey._ 2nd ed. Philadelphia: Temple University Press, 1981.\n\nWeeks, Christopher. _The Building of Westminster in Maryland._ 2nd ed. Annapolis, MD: Fishergate Publishing Company, 1979.\n\nWeeks, Christopher, and Alan Karchmer. _AIA Guide to the Architecture of Washington, D.C._ 3rd ed. Baltimore: Johns Hopkins University Press, 1994.\n\nWhite, Norval, Elliot Willensky, and Fran Leadon. _AIA Guide to New York City._ 5th ed. Oxford: Oxford University Press, 2010.\n\nWoodward, William McKenzie. _PPS\/AIA RI Guide to Providence Architecture._ Providence, RI: Providence Preservation Society, 2003.\n\nYarnall, James L. _Newport Through Its Architecture: AHistory of Styles from Postmedieval to Postmodern._ Newport, RI: Salve Regina University Press in association with University Press of New England, 2005.\n\n## Regional and Local Guides: Southern States\n\nAmerican Institute of Architects, New Orleans Chapter. _A Guide to New Orleans Architecture._ New Orleans, 1974.\n\nAmerican Institute of Architects, Winston-Salem Section. _Architectural Guide, Winston-Salem, Forsyth County._ Winston-Salem, NC, 1978.\n\nAndrews, Wayne. _Pride of the South: A Social History of Southern Architecture._ New York: Atheneum, 1979.\n\nBehar, Roberto M., and Maurice Culot. _Coral Gables: An American Garden City._ Paris (France): Norma Editions, 1997.\n\nBiloxi, City of. _The Buildings of Biloxi: An Architectural Survey._ Biloxi, MS, 2000.\n\nButchko, Tom. _An Inventory of Historic Architecture, Sampson County, North Carolina._ Clinton, NC: City of Clinton, 1981.\n\nCaemmerer, Alex. _Houses of Key West._ Sarasota, FL: Pineapple Press, 2009.\n\nCenter for Planning Excellence. \"Louisiana Speaks Deliverables.\" n.d.] Retrieved from [http:\/\/cpex.org\/downloads\/louisiana-speaks-deliverables\n\nChambers, S. Allen Jr. _Lynchburg_ [Virginia]: _An Architectural History._ Charlottesville, VA: University Press of Virginia, 1981.\n\nCox, Ethelyn. _Historic Alexandria, Virginia, Street by Street: A Survey of Existing Early Buildings._ Alexandria: Historic Alexandria Foundation, 1976.\n\nCox, Warren Jr., and others. _A Guide to the Architecture of Washington, D.C._ 2nd ed. New York: McGraw-Hill, 1974.\n\nCrocker, Mary Wallace. _Historic Architecture in Mississippi._ Jackson: University Press of Mississippi, 1988.\n\nDulaney, Paul S. _The Architecture of Historic Richmond_ [Virginia]. 2nd ed. Charlottesville, VA: University Press of Virginia, 1976.\n\nEufaula Heritage Association. _Historic Eufaula: A Treasury of Southern Architecture, 1827\u20131910._ Eufaula, AL, 1972.\n\nGleason, David King. _Virginia Plantation Homes._ Baton Rouge, LA: Louisiana State University Press, 1989.\n\nGould, Elizabeth Barrett. _From Fort to Port: An Architectural History of Mobile, Alabama, 1711\u20131918._ Tuscaloosa, AL: University of Alabama Press, 1988.\n\nGournay, Isabelle, Paul G. Beswick, Dana F. White, and Gerald W. Sams. _AIA Guide to the Architecture of Atlanta._ Athens, GA: University of Georgia, 1993.\n\nGross, Steve, Sue Daley, John H. Lawrence, and James Conaway. _Creole Houses: Traditional Homes of Old Louisiana._ New York: Abrams, 2007.\n\nHarris, Linda L. _An Architectural and Historical Inventory of Raleigh, North Carolina._ Raleigh: City of Raleigh, 1978.\n\nIseley, N. Jane. _Beaufort._ Beaufort, SC: Historic Foundation, 2003.\n\nIseley, N. Jane, William P. Baldwin, and Agnes LeLand Baldwin. _Plantations of the Low Country: South Carolina, 1697\u20131865._ Rev. ed. Greensboro, NC: Legacy Publications, 1987.\n\nJeane, D. Gregory, and Douglas Clare Purcell. _The Architectural Legacy of the Lower Chattahoochee Valley in Alabama and Georgia._ University, AL: University of Alabama Press, 1978.\n\nLancaster, Clay. _Eutaw: The Builders and Architecture of an Ante-Bellum Southern Town._ Eutaw, AL: Greene County Historical Society, 1979.\n\nLane, Mills, and Van Jones Martin. _Architecture of the Old South: Mississippi & Alabama._ New York: Abbeville Press, 1997.\n\nLane, Mills, Van Jones Martin, and Gene Carpenter. _Architecture of the Old South: Georgia._ Rev. ed. Savannah, GA: Beehive Press, 1996.\n\nLane, Mills, Van Jones Martin, Calder Loth, and Gene Carpenter. _Architecture of the Old South: Virginia._ Savannah, GA: Beehive Press, 1996.\n\nLane, Mills, Van Jones Martin, Gene Waddell, and Gene Carpenter. _Architecture of the Old South: South Carolina._ Savannah, GA: Beehive Press, 1996.\n\nLinley, John. _Architecture of Middle Georgia: The Oconee Area._ Athens, GA: University of Georgia Press, 1972.\n\n\u2014\u2014\u2014. _The Georgia Catalog, Historic American Buildings Survey: A Guide to the Architecture of the State._ Athens, GA: University of Georgia Press, 1982.\n\nLittle-Stokes, Ruth, and Tony P. Wrenn. _An Inventory of Historic Architecture, Caswell County, North Carolina._ Yanceyville, NC: Caswell County Historic Association, 1979.\n\nLoth, Calder. _Virginia Landmarks Register._ 4th ed. Richmond: Univerity of Virginia Press, 1999.\n\nLyle, Royster Jr., and Pamela Hemenway Simpson. _The Architecture of Historic Lexington_ [Virginia]. Charlottesville, VA: University Press of Virginia, 1977.\n\nMaddex, Diane. _Historic Buildings of Washington, D.C._ Pittsburgh: Ober Park Associates, 1973.\n\nMillas, Aristides J., and Ellen J. Uguccioni. _Coral Gables, Miami Riviera: An Architectural Guide._ Miami: Dade Heritage Trust, 2004.\n\nMitchell, William R., and Van Jones Martin. _Classic Atlanta Landmarks of the Atlanta Spirit._ New Orleans: Martin\u2013St Martin Publishing, 1991.\n\nMobile, City of. _Nineteenth-Century Mobile Architecture._ Mobile, AL, 1974.\n\nMorrison, Mary L., ed. _Historic Savannah._ 2nd ed. Savannah, GA: Historic Savannah Foundation, 1979.\n\nNichols, Frederick Doveton, and Frances Benjamin Johnston. _The Early Architecture of Georgia._ Chapel Hill, NC: University of North Carolina Press, 1957.\n\n\u2014\u2014\u2014. _The Architecture of Georgia._ Savannah, GA: Beehive Press, 1976.\n\nOssman, Laurie, and Steven Brooke. _Great Houses of the South._ New York: Rizzoli, 2010.\n\nOverdyke, W. Darrell. _Louisiana Plantation Homes._ New York: Architectural Book Publishing, 1965.\n\nPatrick, James. _Architecture in Tennessee, 1768\u20131897._ Knoxville, TN: University of Tennessee Press, 1990.\n\nPoston, Jonathan H. _The Buildings of Charleston: A Guide to the City's Architecture._ Columbia, SC: University of South Carolina Press, 1997.\n\nReeves, F. Blair. _A Guide to Florida's Historic Architecture._ Gainesville, FL: University of Florida Press, 1990.\n\nSchezen, Roberto, and Shirley Johnston. _Palm Beach Houses._ New York: Rizzoli International, 1991.\n\nSchuler, Stanley. _Mississippi Valley Architecture: Houses of the Lower Mississippi Valley._ Exton, PA: Schiffer Pub, 1984.\n\nSchwartz, Nancy B. _District of Columbia Catalog, 1974, Historic American Buildings Survey._ Charlottesville, VA: University Press of Virginia, 1976.\n\nSeverens, Kenneth. _Southern Architecture: 350 Years of Distinctive American Buildings._ New York: E. P. Dutton, 1981.\n\nShulman, Allan T., Randall C. Robinson, and James F. Donnelly. _Miami Architecture: An AIA Guide Featuring Downtown, the Beaches, and Coconut Grove._ Gainesville, FL: University Press of Florida, 2010.\n\nSimons, Albert, and Samuel Lapham Jr. _The Early Architecture of Charleston_ [SC]. 1927. Reprint, Columbia, SC: University of South Carolina Press, 1990.\n\nStarr, S. Frederick. _Southern Comfort: The Garden District of New Orleans._ Rev. ed. New York: Princeton Architectural Press, 2005.\n\nStoney, Samuel Gaillard. _This Is Charleston: A Survey of the Architectural Heritage of a Unique American City,_ Rev. ed. Charleston, SC: Carolina Art Association, 1990.\n\nToledano, Roulhac. _A Pattern Book of New Orleans Architecture._ Gretna, LA: Pelican, 2010.\n\nUrban Design Associates. _A Pattern Book for Gulf Coast Neighborhoods: Mississippi Renewal Forum._ Pittsburgh, PA: Urban Design Associates, 2005. <http:\/\/www.mississippirenewal.com\/documents\/Rep_PatternBook.pdf>\n\nVirginia Historic Landmarks Commission. _Virginia Catalog: Historic American Buildings Survey._ Charlottesville, VA: University Press of Virginia, 1976.\n\nVogt, Lloyd. _New Orleans Houses: A House-Watcher's Guide._ Gretna, LA: Pelican, 1989.\n\nWhitwell, W. L., and Lee W. Winborne. _The Architectural Heritage of the Roanoke Valley._ Charlottesville, VA: University Press of Virginia, 1982.\n\n## Regional and Local Guides: Midwestern States\n\nAmerican Institute of Architects, Kansas City Chapter. _AIA Guide to Kansas City._ Kansas City, MO: AIA\/Highwater Editions, 2000.\n\nAndrews, Wayne. _Architecture in Chicago and Mid-America._ New York: Harper and Row, 1973.\n\nBach, Ira J. _A Guide to Chicago's Historic Suburbs on Wheels and on Foot._ Chicago: Swallow Press, 1981.\n\nBenjamin, Susan S., and Stuart Earl Cohen. _Great Houses of Chicago, 1871\u20131921._ New York: Acanthus, 2012.\n\nBigott, Joseph C. _From Cottage to Bungalow: Houses and the Working Class in Metropolitan Chicago, 1869\u20131929._ Chicago: University of Chicago Press, 2001.\n\nBlock, Jean F. _Hyde Park Houses, An Informal History, 1856\u20131910._ Chicago: University of Chicago Press, 1978.\n\nBryan, John Albury. _Missouri's Contribution to American Architecture._ St. Louis: St. Louis Architectural Club, 1928.\n\nCampen, Richard N. _Ohio: An Architectural Portrait._ Chagrin Falls, OH: West Summit Press, 1973.\n\nCohen, Stuart Earl, and Susan S. Benjamin. _North Shore Chicago: Houses of the Lakefront Suburbs, 1890\u20131940._ New York: Acanthus Press, 2006.\n\nDarbee, Jeffrey T., and Nancy A. Recchie. _The AIA Guide to Columbus._ Athens, OH: Ohio University Press, 2008.\n\nEhrlich, George. _Kansas City, Missouri: An Architectural History, 1826\u20131990._ Rev. ed. Columbia: University of Missouri Press, 1992.\n\nFerry, W. Hawkins. _The Buildings of Detroit,_ Rev. ed. Detroit: Wayne State University Press, 2012.\n\nGebhard, David, and Gerald Mansheim. _Buildings of Iowa._ New York: Oxford University Press, 1993.\n\nGebhard, David, and Tom Martinson. _A Guide to the Architecture of Minnesota._ Minneapolis: University of Minnesota Press, 1977.\n\nHill, Eric J., and John Gallagher. _AIA Detroit: The American Institute of Architects Guide to Detroit Architecture._ Detroit: Wayne State University Press, 2003.\n\nHunter, Julius K., Robert C. Pettus, and Leonard Lujan. _Westmoreland and Portland Places: The History and Architecture of America's Premier Private Streets, 1888\u20131988._ Columbia, MO: University of Missouri Press, 1988.\n\nIndiana Architectural Foundation. _Indianapolis Architecture._ Indianapolis, 1975.\n\nJohannesen, Eric. _Cleveland Architecture, 1876\u20131976._ Rev. ed. Cleveland: Western Reserve Historical Society, 1981.\n\nJohnson, Carl H. Jr. _The Building of Galena: An Architectural Legacy._ 3rd ed. Galena, IL: publ. by author, 1997.\n\nJones, Elizabeth F., and Mary Jean Kinsman. _Jefferson County: Survey of Historic Sites in Kentucky._ Louisville: Jefferson County Office of Historic Preservation and Archives, 1981.\n\nJunior League of Evansville. _Reflections Upon a Century of Architecture: Evansville, Indiana._ Evansville, IN, 1977.\n\nKansas City, Missouri Landmarks Commission. _Kansas City: A Place in Time._ Kansas City, MO, 1977.\n\nKennedy, Roger. _Minnesota Houses._ Minneapolis: Dillon Press, 1967.\n\nKentucky Heritage Commission. _Ballard County: Survey of Historic Sites in Kentucky._ Frankfort, KY, 1978.\n\n\u2014\u2014\u2014. _Jessamine County: Survey of Historic Sites in Kentucky._ Frankfort, KY, 1979.\n\nKidney, Walter C. _Historic Buildings of Ohio: A Selection From the Records of the Historic American Buildings Survey._ Pittsburgh: Ober Park Associates, 1972.\n\nKoeper, Frederick. _Illinois Architecture from Territorial Times to the Present: A Selective Guide._ Chicago: University of Chicago Press, 1975.\n\nLancaster, Clay. _Vestiges of the Venerable City: A Chronicle of Lexington, Kentucky._ Lexington: Lexington\u2013Fayette County Historic Commission, 1978.\n\nMagness, Perre. _Good Abode: Nineteenth-Century Architecture in Memphis and Shelby County, Tennessee._ Memphis, TN: Junior League of Memphis, 1983.\n\nMason City, City of. _Mason City, Iowa: An Architectural Heritage._ Mason City, IA, 1977.\n\nMcArthur, Shirley du Fresne. _North Point Historic Districts\u2014Milwaukee._ Milwaukee: North Point Historical Society, 1981.\n\nMcCue, George. _The Building Art in St. Louis: Two Centuries._ 3rd ed. St. Louis, MO: Knight Publishing Co., 1981.\n\nMeyer, Katharine Mattingly, ed. _Detroit Architecture: AIA Guide._ Rev. ed. Detroit: Wayne State University Press, 1980.\n\nMillett, Larry. _AIA Guide to the Twin Cities: The Essential Source on the Architecture of Minneapolis and St. Paul._ St. Paul: Minnesota Historical Society Press, 2007.\n\n\u2014\u2014\u2014. _AIA Guide to the Minneapolis Lake District._ St. Paul, MN: Minnesota Historical Society Press, 2009.\n\n\u2014\u2014\u2014. _AIA Guide to St. Paul's Summit Avenue and Hill District._ St. Paul, MN: Minnesota Historical Society Press, 2009.\n\nMillstein, Cydney, and Carol Grove. _Houses of Missouri, 1870\u20131940._ New York: Acanthus Press, 2008.\n\nNewcomb, Rexford. _Architecture of the Old Northwest Territory._ Chicago: University of Chicago Press, 1950.\n\n\u2014\u2014\u2014. _Architecture in Old Kentucky._ Urbana, IL: University of Illinois Press, 1953.\n\nPerrin, Richard W. E. _Historic Wisconsin Architecture._ Rev. ed. Milwaukee: Wisconsin Society of Architects, 1976.\n\n\u2014\u2014\u2014. _Historic Wisconsin Buildings: A Survey in Pioneer Architecture, 1835\u20131870,_ 2nd ed. Milwaukee: Milwaukee Public Museum, 1981.\n\nSandeen, Ernest R. _St. Paul's Historic Summit Avenue._ 1978. Reprint, Minneapolis: University of Minnesota Press, 2004.\n\nSchofield, Mary-Peale. _Landmark Architecture of Cleveland._ Pittsburgh: Ober Park Associates, 1976.\n\nScott, James Allen. _Duluth's Legacy,_ vol. 1: _Architecture._ Duluth, MN: City of Duluth, 1974.\n\nShank, Wesley I. _Iowa Catalog: Historic American Buildings Survey._ Iowa City: University of Iowa Press, 1979.\n\nSinkevitch, Alice. _AIA Guide to Chicago._ 2nd ed. Orlando, FL: Harcourt, 2004.\n\nWindle, John T., and Robert M. Taylor. _The Early Architecture of Madison, Indiana._ Madison, IN: Historic Madison, 1986.\n\n## Regional and Local Guides: Western States\n\nAlexander, Drury Blakeley, and Todd Webb. _Texas Homes of the Nineteenth Century._ Austin: University of Texas Press, 1966.\n\nAmerican Institute of Architects, Dallas Chapter. _Dallasights: An Anthology of Architecture and Open Spaces._ Dallas, 1978.\n\nAmerican Institute of Architects, and Stephen Fox. _Houston Architectural Guide._ 3rd ed. Houston: American Institute of Architects, Houston Chapter, 2012.\n\nAndree, Herb, and Noel Young. _Santa Barbara Architecture: From Spanish Colonial to Modern,_ 3rd ed. Reprinted with corrections. Santa Monica: Hennessey & Ingalls, 2005.\n\nBarnstone, Howard. _The Galveston_ [TX] _That Was._ 1966. Reprint, College Station: Texas A&M Press, 1999.\n\nBernhardi, Robert. _The Buildings of Berkeley_ [CA]. Berkeley: Lederer Street & Zeus, 1991.\n\nBracken, Dorothy Kendall, and Maurine Whorton Redway, _Early Texas Homes._ 1956. Reprint, Dallas: Southern Methodist University Press, 1982.\n\nBrettell, Richard R., and Willis Winters. _CraftingTraditions: The Architecture of Mark Lemmon._ Dallas: Meadows Museum, 2005.\n\nBruce, Curt, and Thomas Aidala. _The Great Houses of San Francisco._ New York: Knopf, 1974.\n\nBurkholder, Mary V. _The King William Area_ [San Antonio]: _A History and Guide to the Houses._ 2nd ed. San Antonio, TX: King William Association, 1977.\n\nButler, Phyllis Filiberti, and Junior League of San Jose. _The Valley of Santa Clara_ [CA] _: Historic Buildings, 1792\u20131920._ 2nd ed. Novato, CA: Presidio Press, 1981.\n\nCarter, Thomas, and Peter L. Goss. _Utah's Historic Architecture, 1847\u20131940: A Guide._ Salt Lake City: University of Utah Press, 1991.\n\nChase, John. _The Sidewalk Companion to Santa Cruz Architecture._ 3rd ed. Santa Cruz, CA: Museum of Art & History, 2005.\n\nClark, Anne. _Historic Houses of San Augustine_ [TX]. Austin, TX: San Augustine Historical Society, 1972.\n\nCooper\/Roberts Architects, AIA, and City of Phoenix (AZ). _Historic Homes of Phoenix: An Architectural & Preservation Guide._ City of Phoenix, 1992.\n\nCulbertson, Margaret. _Texas Houses Built by the Book: The Use of Published Designs, 1850\u20131925._ College Station: Texas A&M University Press, 1999.\n\nDallas, Sandra. _Colorado Homes._ Norman, OK: University of Oklahoma Press, 1986.\n\nDuchscherer, Paul, and Douglas Keister. _Victorian Glory in San Francisco and the Bay Area._ New York: Viking Studio, 2001.\n\nFairfax, Geoffrey W. _The Architecture of Honolulu._ Sydney (Australia): Island Heritage, 1972.\n\nFuller, Larry Paul, ed. _The American Institute of Architects Guide to Dallas Architecture, with Regional Highlights._ New York: McGraw Hill Construction Information Group, 1999.\n\nGebhard, David, and Robert Winter. _An Architectural Guidebook to Los Angeles._ Rev. ed. Salt Lake City: Gibbs Smith, 2003.\n\n\u2014\u2014\u2014. _A Guide to Architecture in Los Angeles and Southern California._ Santa Barbara, CA: Peregrine Smith, 1977.\n\nGebhard, David, et al. _A Guide to Architecture in San Francisco and Northern California._ Rev. ed. Salt Lake City: Peregrine Smith, 1985.\n\nGleye, Paul. _The Architecture of Los Angeles._ Los Angeles: Rosebud Books, 1981.\n\nGoeldner, Paul. _Utah Catalog: Historic American Buildings Survey._ Salt Lake City: Utah Heritage Foundation, 1969.\n\n\u2014\u2014\u2014. _Texas Catalog: Historic American Buildings Survey._ San Antonio, TX: Trinity University Press, 1975.\n\nGoins, Charles R., and John W. Morris. _Oklahoma Homes, Past and Present._ Norman, OK: University of Oklahoma Press, 1980.\n\nHart, Arthur A. _Historic Boise: An Introduction to the Architecture of Boise, Idaho, 1863\u20131938._ Auckland, NZ: Caxton Press, 1993.\n\nHawkins, William John, and William F. Willingham. _Classic Houses of Portland, Oregon, 1850\u20131950._ Portland, OR: Timber Press, 2005.\n\nHenry, Jay C. _Architecture in Texas, 1895\u20131945._ Austin: University of Texas Press, 2010.\n\nHistoric Denver. \"HD Guides and Publications.\" http:\/\/store.historicdenver.org\/store\/historic-denver-guides-series\/\n\nKirker, Harold. _California's Architectural Frontier: Style and Tradition in the Nineteenth Century._ 3rd ed. Santa Barbara, CA: Peregrine Smith, 1986.\n\nLenggenhager, Werner, and Lucile McDonald. _Where the Washingtonians Lived._ Seattle, WA: Superior Publishing, 1969.\n\nMasson, Kathryn, and Paul Rocheleau. _The California House: Adobe, Craftsman, Victorian, Spanish Colonial Revival._ New York: Rizzoli, 2011.\n\nMcDonald, William L. _Dallas Rediscovered: A Photographic Chronicle of Urban Expansion, 1870\u20131925._ Dallas: Dallas Historical Society, 1978.\n\nMcGrew, Patrick, and Robert Julian. _Landmarks of Los Angeles._ New York: Abrams, 1994.\n\nMoore, Charles Willard, Peter Becker, and Regula Campbell. _The City Observed\u2014Los Angeles: A Guide to Its Architecture and Landscapes._ Santa Monica, CA: Hennessey and Ingalls, 1998.\n\nNeil, J. Meredith. _Saints and Oddfellows: A Bicentennial Sampler of Idaho Architecture._ Boise, ID: Boise Gallery of Art Association, 1976.\n\nNoel, Thomas Jacob. _The Montclair Neighborhood._ Denver: Historic Denver, 1999.\n\nOlmsted, Roger, and T. H. Watkins. _Here Today: San Francisco's Architectural Heritage._ San Francisco: Chronicle Books, 1995.\n\nPaglia, Michael, Diane Wray Tomasso, and Kathleen Roach. _The Mid-Century Modern House in Denver._ Denver: Historic Denver, Inc, 2007.\n\nPolyzoides, Stefanos, Roger Sherwood, and James Tice. _Courtyard Housing in Los Angeles: A Typological Analysis._ 2nd ed. New York: Princeton Architectural Press, 1992.\n\nRegnery, Dorothy F. _An Enduring Heritage: Historic Buildings of the San Francisco Peninsula._ Stanford, CA: Stanford University Press, 1976.\n\nSchwarzer, Mitchell. _San Francisco\u2014Architecture of the San Francisco Bay Area: A History & Guide._ San Francisco: William Stout Publishers, 2007.\n\nShay, James, and Christopher Irion. _New Architecture San Francisco._ San Francisco: Chronicle Books, 1989.\n\nStoehr, C. Eric. _Bonanza Victorian: Architecture and Society in Colorado Mining Towns._ Albuquerque: University of New Mexico Press, 1975.\n\nSwope, Caroline T. _Classic Houses of Seattle, High Style to Vernacular, 1870\u20131950._ Portland, OR: Timber Press, 2005.\n\nUniversity of Kansas Museum of Art. _Nineteenth Century Houses in Lawrence, Kansas._ Lawrence, KS: University of Kansas Museum of Art, 1968.\n\nVaughan, Thomas, and Virginia Guest Ferriday, eds. _Space, Style and Structure: Building in Northwest America._ 2 vols. Portland: Oregon Historical Society, 1974.\n\nWatters, Sam. _Houses of Los Angeles._ New York: Acanthus Press, 2007.\n\nWebb, Michael. _Brave New Houses: Adventures in Southern California Living._ New York: Rizzoli, 2003.\n\nWelch, Frank D., Paul Hester, and Philip Johnson. _Philip Johnson & Texas._ Austin: University of Texas Press, 2000.\n\nWiberg, Ruth Eloise. _Rediscovering Northwest Denver._ 1976. Reprint, Niwot: University Press of Colorado, 1995.\n\nWilk, Diane. _A Guide to Denver's Architectural Styles and Terms._ Denver: Historic Denver\/Denver Museum of Natural History, 1995.\n\nWilliamson, Roxanne Kuter. _Austin, Texas: An American Architectural History._ San Antonio, TX: Trinity University Press, 1973.\n\nWoodbridge, Sally B., ed. _Bay Area Houses._ Salt Lake City: Peregrine Smith, 1988.\n\nWoodbridge, Sally B., and Chuck Byrne. _San Francisco Architecture: An Illustrated Guide to the Outstanding Buildings, Public Artworks, and Parks in the Bay Area of California._ Rev. ed. Berkeley, CA: Ten Speed Press, 2005.\n\nWoodbridge, Sally B., and Roger Montgomery. _A Guide to Architecture in Washington State._ Seattle: University of Washington Press, 1980.\n\nWoods, Douglas, Melba Levick, and D. J. Waldie. _Classic Homes of Los Angeles._ New York: Rizzoli, 2010.\n\n## House Preservation and Restoration\n\nAbramovitch, Ingrid. _Restoring a House in the City._ New York: Artisan, 2009.\n\nAnderson Notter Associates. _The Salem Handbook: A Renovation Guide for Homeowners._ Salem, MA: Historic Salem, 1977.\n\nCarroon, Jean. _Sustainable Preservation: Greening Existing Buildings._ Hoboken, NJ: John Wiley and Sons, 2010.\n\nFerro, Maximilian L. _How to Love and Care for Your Old Building in New Bedford._ New Bedford, MA: City of New Bedford, 1977.\n\nHarris, Kip. _Confronting the Older House: A Homeowner's Guide._ Salt Lake City: Utah Heritage Foundation, 1979.\n\nHewitt, Mark Alan, and Gordon Bock. _The Vintage House: A Guide to Successful Renovations and Additions,_ 2011.\n\nHistoric New England. \"Surveying the Recent Past: The Challenge of Creating and Defining Context.\" <http:\/\/www.historicnewengland.org\/preservation\/regional-resources\/looking-forward-symposium\/papers\/surveying-the-recent-past-final.pdf>\n\nHoward, Cynthia. _Your House in the Streetcar Suburb._ Medford, MA: City of Medford, 1979.\n\nHoward, J. Myrick. _Buying Time for Heritage: How to Save an Endangered Historic Property._ Raleigh, NC: Preservation North Carolina, 2007.\n\nHull, Brent. _Historic Millwork: A Guide to Restoring and Re-creating Doors, Windows and Moldings of the Late Nineteenth Through Mid-Twentieth Centuries._ New York: Wiley, 2003.\n\nHutchins, Nigel. _Restoring Old Houses._ Rev. ed. Toronto: Firefly Books, 1997.\n\nJackson, Albert, and David Day. _The Complete Home Restoration Manual: An Authoritative, Do-It-Yourself Guide to Restoring and Maintaining the Older House._ New York: Simon & Schuster, 1992.\n\nKitchen, Judith L. _Caring for Your Old House: A Guide for Owners and Residents._ Washington, DC: Preservation Press, 1995.\n\nLabine, Clem, and Carolyn Flaherty, eds. _The Old-House Journal Compendium._ 1980. Reprint, Woodstock, NY: Overlook Press, 2007.\n\nLeeke, John C. _Save America's Windows._ CreateSpace Independent Publishing Platform, 2009.\n\nLegner, Linda. _City House: A Guide to Renovating Older Chicago-Area Houses._ Chicago: Commission on Chicago Historical and Architectural Landmarks, 1979.\n\nMaddex, Diane, ed. _The Brown Book: A Directory of Preservation Information._ Washington, DC: Preservation Press, 1983.\n\nMoss, Roger W. _Century of Color: Exterior Decoration for American Buildings, 1820\u20131920._ Watkins Glen, NY: American Life Foundation, 1981.\n\nMurtagh, William J. _Keeping Time: The History and Theory of Preservation in America._ 3rd ed. Hoboken, NJ: John Wiley & Sons, 2006.\n\nNash, George. _Renovating Old Houses: Bringing New Life to Vintage Homes._ Newtown, CT: Taunton Press, 2003.\n\nNational Park Service\u2013U.S. Department of the Interior. \"Historic Preservation: Action Transforming our Thinking.\" <http:\/\/www.nps.gov\/history\/history\/resedu\/historic_preservation.pdf>\n\n\u2014\u2014\u2014. \"Technical Preservation Services: Preservation Briefs.\" <http:\/\/www.nps.gov\/tps\/how-to-preserve\/briefs.htm>\n\nNew York Landmarks Conservancy. _Repairing Old and Historic Windows._ Washington, DC: Preservation Press, National Trust for Historic Preservation, 1992.\n\nPhillips, Morgan W. _The Eight Most Common Mistakes in Restoring Houses (and How to Avoid Them)._ AASLH Technical Leaflet 118. _History News_ 34, no. 8 (August 1979).\n\nPoore, Patricia. _The Old-House Journal Guide to Restoration._ New York: Dutton, 1992.\n\nPrentice, Helaine Kaplan. _Rehab Right: How to Realize the Full Value of Your Old House._ Berkeley: Ten Speed Press, 1986.\n\nRypkema, Donovan D. _The Economics of Historic Preservation: A Community Leader's Guide._ 2nd ed. Washington, DC: National Trust for Historic Preservation, 2008.\n\nSeale, William. _Recreating the Historic House Interior._ Nashville, TN: American Association for State and Local History, 1979.\n\nShopsin, William C., and Grania Bolton Marcus, eds. _Saving Large Estates._ Setauket, NY: Society for the Preservation of Long Island Antiquities, 1977.\n\nShull, Carol D., and Beth L. Savage. \"From the Glass House to Stonewall: National Register Recognition of the Recent Past.\" National Register of Historic Places Workshop. NCSHPO Annual Meeting, Washington, DC, March 25, 2001.\n\nStanforth, Deirdre, and Martha Stamm. _Buying and Renovating a House in the City._ Rev. ed. New York: Hippocrene Books, 1985.\n\nStephen, George. _Remodeling Old Houses Without Destroying Their Character._ New York: Knopf, 1972.\n\nStipe, Robert E. _A Richer Heritage: Historic Preservation in the Twenty-first Century._ Chapel Hill, NC: University of North Carolina Press, 2007.\n\nTyler, Norman, Ted J. Ligibel, and Ilene R. Tyler. _Historic Preservation: An Introduction to Its History, Principles, and Practice._ 2nd ed. New York: W.W. Norton, 2009.\n\nU.S. Department of the Interior. _The Preservation of Historic Architecture: The U.S. Government's Official Guidelines for Preserving Historic Homes._ Guilford, CT: Lyons Press, 2004.\n\nU.S. Department of the Interior, National Park Service, Preservation Assistance Division. _The Secretary of the Interior's Standards for Rehabilitation._ Rev. ed. Washington, DC, 1990.\n\nU.S. National Park Service, Technical Preservation Services. _Respectful Rehabilitation: Answers to Your Questions About Old Buildings._ Washington, DC: Preservation Press, 1998.\n\nWagner, Richard. _Buyer's Guide to Older and Historic Houses._ Washington, DC: National Trust for Historic Preservation, 2000.\n\nYoung, Robert A. _Historic Preservation Technology._ Hoboken, NJ: J. Wiley & Sons, 2008.\n\n## Web Sites\n\nArchitectural Record. archrecord.construction.com\n\nAtlas Mobile Home Museum. Mobile Home \/Trailer Coach Museum Archives. www.allmanufacturedhomes.com\/html\/vintage_mobile_homes.htm\n\nCarnegie Mellon University. \"Remaking Cities Institute.\" <http:\/\/www.cmu.edu\/rci\/>\n\nCongress for the New Urbanism. <http:\/\/www.cnu.org\/>\n\nDOCOMOMO. www.docomomo-us.org\n\nInstitute of Classical Architecture & Classical America. <http:\/\/www.classicist.org\/>\n\nMarilyn and Ray Gindroz Foundation. <http:\/\/www.marilynandraygindrozfoundation.org\/>\n\nNational Alliance of Preservation Commissions. napc.uga.edu\/resources-links\/preservation-links\n\nNational Park Service. \"National Register of Historic Places Program: State Historic Preservation Officers (SHPO).\" www.nps.gov\/nr\/shpolist.htm\n\nNational Trust for Historic Preservation. www.preservationnation.org\n\nOld House Journal. www.oldhousejournal.com\n\nPeriod Homes. www.period-homes.com\n\nReconnecting America. <http:\/\/www.reconnectingamerica.org\/>\n\nSociety of Architectural Historians. www.sah.org\n\nVernacular Architecture Forum. \"Vernacular Architecture Newsletter Bibliography.\" <http:\/\/resources.umwhisp.org\/vafbib.htm>\n\n# Acknowledgments\n\nMany people have helped me in the preparation of this revision. Two were involved in almost every aspect\u2014Steve Clicque, who took the vast majority of the new photographs, and Amanda Olson McCoy, who was my research assistant. Amanda did a wide range of things, chief among them preparing the preliminary layouts for the new materials and putting all of the corrections into a PDF of the original book. She also reviewed the materials I'd gathered for \"Other 20th-Century Modern\" and pulled the information together into a rough draft; some of her lively wording is found in the final text. Steve drove up and down streets of neighborhoods across the United States to gather photographs of visible examples; he cataloged and kept track of all the line art and photographs, doggedly pursuing some of the toughest final photo permissions. In addition, he restored many old photographs, kept the computers and programs running, providing virtually all IT-related tasks. I could not have completed the book without him.\n\nAleida Rodr\u00edguez, an experienced copy editor with a specialty in museum catalogs, helped edit all the text of the new changes, guiding me toward consistent style.\n\nAfter the book was in production, both Jane Griffith Quinn and Julie Travis reviewed specific parts of the proofs, including the Notes, For Further Reference, and the numbering of photographs and captions. Sharon Dorsey, my office manager, kept everything else organized so that I could actually focus on writing the book.\n\nArtist and architect Suzanne Patton Matty cheerfully and expertly prepared the line art for the new book chapters. She was recommended to me by William Seale and stuck with the project for over five years, even after she had returned to school full-time to earn a graduate degree. Dallas preservation architect Gary Skotnicki provided the additional drawings added to the Pictorial Key.\n\nI particularly appreciate my longtime editor at Knopf, Jane Garrett, her belief that this revision was needed and her patience with the length of time it took to actually complete. When she retired from Knopf, Andy Hughes, who had overseen the production of the original edition in 1984, became my new editor. His expertise, friendship, and enthusiasm made the production of the revision a joy. Kevin Bourke was production editor of the new edition, and Kathleen Fridella and Maralee Youngs helped to round up some of the final pieces. Cassandra Pappas conceived and executed the beautiful, fresh design of the book's interior. North Market Street Graphics, the typesetter\u2014Dennis Bicksler, Tracy Pitz, Rhonda Stough, and Cindy Szili\u2014contributed much to the quality of the book you see.\n\nBecause such a high percentage of mainstream modern homes are designed not to reveal themselves to the street, it was necessary to visit as many modern homes as possible. One of the best ways to do this is on organized tours. The Vernacular Architecture Forum hosts an annual conference that features numerous tours. Richard Longstreth organized a particularly helpful one of neighborhoods in the Washington, D.C., area that stressed the evolution of neighborhood planning during and after World War II and produced a well-researched companion book, _Housing Washington: Two Centuries of Residential Development and Planning in the National Captial Area._ The Frank Lloyd Wright Conservancy offers tours of Wright and Wright-influenced houses. I enjoyed attending their tours in Minneapolis\u2013St. Paul, in Los Angeles, and in Palm Springs. The Palm Springs tour was led by Alan Hess, co-author of _Palm Springs Weekend: The Architecture and Design of a Midcentury Oasis_ (and many other books on Mid-century Modern houses). The Society of Architectural Historians sponsored a tour of Paul Rudolph's work (as well as the work of other architects) in and around Sarasota, Florida, led by Joseph King, co-author of _Paul Rudolph: The Florida Houses._ Ken Turino organized an excellent conference with tours of mid-century modern houses for Historic New England. Dr. Mardges Bacon, Matthews Distinguished University Professor in the Department of Architecture at Northeastern University, gave the introductory speech. I was heavily influenced by her clear view of how the modern movement has evolved in America. Field trips to homes designed by Walter Gropius and Marcel Breuer near Boston and to iconic houses in New Canaan, Connecticut, made this tour invaluable\u2014as did the talk by William D. Earls, author of _The Harvard Five in New Canaan._ And last but definitely not least, two tours of organic architecture offered by the Friends of Kebyar and led by its president, architect Nelson Brackin\u2014one concentrating on Bruce Goff in Oklahoma and the other focusing on the work of Fay Jones in Arkansas\u2014had a stunning cross-section of modern houses.\n\nGood friends offered shelter and helped plan photo routes in their neighborhoods. Alice Ingraham and Tony Davies helped with Weston, Massachusetts, along with their friend Pamela W. Fox, author of _Farm Town to Suburb: The History and Architecture of Weston, Massachusetts, 1830\u20131980._ Nelson Brackin planned a day in Atlanta, Georgia, along with a two-day \"Fox Trot,\" visiting the quite varied organic homes of James Fox in North and South Carolina. Dick and Kathy Kleinsasser Morris helped with Portland, Oregon, and James Sodeman shared his expertise on San Francisco, California. Anne and John Mullen invited me to spend several days in the New Traditional Shingle home that John had recently designed for the summers they spend in East Hampton, New York. The visit included the Museum of Modern Art's \"Home Delivery: Fabricating the Modern Dwelling\"; a tour of the original Levittown on Long Island; seeking out a new modern neighborhood being built in Sagaponack, New York; looking at Macy's 1950s pre-fab houses in Montauk; and visiting neighborhoods throughout the Hamptons.\n\nAnd finally, great thanks to those who took the time to read large portions of the text and give me criticism and comments\u2014New Orleans land-use attorney and preservationist William Borah, John Mullen FAIA, preservation architect Daron Tapscott, Willis Winters FAIA, and Ann Abernathy, architect in charge of the restoration of the Frank Lloyd Wright home and studio who, happily for me, ended up in Dallas. Marcel Quimby FAIA, Nancy McCoy FAIA, of Quimby McCoy Preservation Architecture, Larry Boerder, Kevin Sloan, Robert Meckfessel FAIA, Katherine Seale, and William Seale.\n\nMany people have asked over the years about the roles that Lee and I respectively played in writing the 1984 edition of the book. It is unlikely that there would have been a 1984 _Field Guide_ were it not for Lee McAlester. He had recently written two geology textbooks, _The Earth_ and _The History of Life_ , and firmly maintained that if he could research, write, and synthesize such massive topics certainly I could do the same for something as simple as houses\u2014at least with his oversight and assistance. I needed this confidence to plunge forward with a book I very much wanted to write. I did the primary research, counted characteristics, wrote the initial drafts, gathered photographs, and prepared illustrations. He edited everything and pointed out the flaws in my research and logic (as I know he had done for the many Ph.D. students who studied under him at Yale). He also researched and wrote a few of the early chapters that particularly interested him. I do believe he soon discovered houses are not quite so simple.\n\nThe following were instrumental in obtaining photo permissions, digital prints, and pertinent information from their organizations and institutions:\n\n**Biltmore Hotel and Gardens, Asheville, North Carolina**\n\nErica Walker\n\n**Bonneville House Association**\n\nJeanette Falkner\n\n**Buffalo and Erie County Historical Society**\n\nCynthia Van Ness\n\n**Catskill Center for Conservation**\n\nLisa Rainwater\n\n**Cuyahoga County Archives**\n\nJudith G. Cetina\n\n**City of Dallas**\n\nCarol Roark\n\n**Dallas Historical Society**\n\nJack Bunning\n\nSusan Richards\n\n**Friends of the Cabildo**\n\nKaitlin Ryan\n\n**Galveston Historical Foundation**\n\nBrian Davis\n\nDwayne Jones\n\n**The Gamble House**\n\nBobbi Mapstone\n\n**Genesee Valley Council on the Arts**\n\nKathryn Hollinger\n\n**Historic New England**\n\nJeanne Gamble\n\n**Indiana Historical Society**\n\nSusan Sutton\n\n**Library of Congress, Historic American Building Survey**\n\n**Louisiana State Museum**\n\nTony Lewis, Curator of Visual Arts\n\n**Maryland Historical Society**\n\nChristopher Becker\n\n**Missouri History Museum**\n\nEllen Thomasson\n\n**Museum of the City of New York**\n\nRobbi Siegel\n\n**National Anthropological Archives, Smithsonian Institution**\n\nLeanda Gahegan\n\nDaisy Njoku\n\n**National Association of Historic Buildings**\n\nPaul Lopez\n\n**National Register of Historic Places and National Historic Landmarks**\n\nJim Gabbert\n\nLinda McClelland\n\nBarbara Wyatt\n\nRicah Marquez\n\n**National Trust for Historic Preservation**\n\nPeter Brink\n\n**New York State Historical Association**\n\nWayne Wright\n\n**Oak Court**\n\nBuchanan Architecture\n\n**Ohio Historical Society**\n\nLisa Wood\n\n**Palm Harbor Homes**\n\nHoward Broughton, V. P. of Sales & Marketing\n\n**Preservation Dallas**\n\nKatherine Seale and Carol Roark, former Directors\n\n**Still National Osteopathic Museum & A. T. Still University**\n\nJason Haxton\n\nRobert Clement\n\nBarb Magers\n\n**Syracuse University Press**\n\nKelly Balenske\n\n**Texas Historical Commission**\n\nMark Wolfe\n\n**University of Louisville, Ekstrom Library Special Collections: Photographic Archives**\n\nBill Carner\n\n**University of Minnesota Libraries**\n\nJennifer C. Torkelson\n\nBarbara A. Bezat\n\n**University of Oklahoma**\n\nAlexandra Shadid\n\n**Westchester County Historical Society**\n\nDiana D. Deichert\n\n**Western Pennsylvania Conservancy**\n\nClinton Piper\n\nFallingwater, PO Box R, Mill Run, PA 15464 fallingwater@paconserve.org\n\n**Western Reserve Historical Society**\n\nAnn Sindelar\n\nVicki Catozza\n\nThe following assistants, photographers, architects, and independent professionals provided additional photographs, assistance, and information:\n\nAlbert, Righter & Tittmann Architects, Inc.\n\nWilliam T. Baker & Associates\n\nBrian Clicque\n\nChelsea Clicque\n\nHeather Clicque\n\nBonnie Cochrane\n\nRick Brettell\n\nIan Cole\n\nAl Cox\n\nMarianne Cusato\n\nRichard Drummond Davis\n\nJacqueline Decter\n\nEve Epstein\n\nStephen Fox\n\nPatricia Gebhard\n\nMark Gunderson\n\nThomas Hahn\n\nTom Martinson\n\nKeven McAlester\n\nAlex McLean\n\nCraig Melde, Architexas\n\nEvelyn Montgomery\n\nRobert A.M. Stern, Architects\n\nAmy Talkington\n\nCarty Talkington\n\nBonnie Wheeler\n\nWillis Winters\n\nGloria Wise\n\nDavid Woodcock\n\nAlthough I very much appreciate the help and input of all of these individuals, they shouldn't be held responsible for my opinions stated within the text.\n\n## City of Hope\n\nMy last acknowledgment is one I never dreamed I would need to make. However, without City of Hope in Duarte, California, I could not have finished this book. A few months after turning in the manuscript, I was diagnosed with myelofibrosis, an unusual chronic leukemia. At first I didn't worry because the prognostic factors indicated I had about fourteen-years-plus life expectancy. But after correcting my first set of layouts, I suddenly felt much worse. The severe anemia produced by my rapidly dropping red-blood-cell count made it hard to focus (not enough oxygen makes it to the brain). During a consultation at City of Hope, I learned that in their opinion I needed an immediate stem-cell transplant. By some miracle, my sister, Dorothy \"Dotsy\" Savage, was a perfect match and willing to be my donor. I had hoped to be able to finish the second, third, and fourth layouts (not to mention the related e-book and documentary projects), but my primary physician, Dr. Eileen Smith, strongly encouraged that I schedule the transplant immediately.\n\nIt was an arduous process\u2014including a two-week stint in the ICU\u2014but knowing that I needed to finish this book helped get me through, along with the attentive care of Transplant Team 5\u2014Dr. Stephen Forman, Dr. Smith, Dr. Ravi Bhatia, Dr. David Snyder, and Dr. Ahmed Aribi. Countless other generous nurses, caregivers, administrators, housekeepers, and indeed everyone I came in contact with at City of Hope, treated me with the highest level of professionalism and caring I possibly could imagine.\n\nOne special feature at City of Hope is The Village\u2014two wonderful groupings of Mid-century Modern cottages that serve as housing for recuperating patients. So, as luck would have it, I found myself literally inhabiting my work. I finished this book while living in a Shed style cottage, the exact kind of space and grouping written about in the newly added Modern section of this book.\n\nI had expected to spend the last four months (June to September 2013) gathering acknowledgments and double-checking facts, but instead I've had to spend great amounts of time simply recuperating. Despite my best efforts, I know there must be omissions to these acknowledgments and credits. I apologize to those of you who have helped me write this book and whom I may have omitted. I do intend to make corrections in the next printing.\n\nThus, my highest thanks go to City of Hope and, of course, to the production team at Knopf\u2014always a joy to work with and incredibly understanding. Miraculously, the book is _almost_ on schedule.\n\nHope Village, City of Hope, Duarte, California; 1958. This Contemporary (Mid-century Modern) was built for patients needing easy access to the hospital.\n\nParsons Village, City of Hope, Duarte, California; 1987. A Shed style similar to my temporary accommodations since July 2013.\n\nParsons Village, City of Hope, Duarte, California; 1987. Sidewalks wind through beautifully landscaped and friendly grounds.\n\n# About the Illustrations\n\nAll the final illustrations for the new chapters in this revision were drawn by architect and artist Suzanne Patton Matty. The author made a rough sketch for the artist to work from and in many cases provided additional visual materials. Often there were several rounds of corrections, with the final responsibility for the illustration content falling to the author.\n\nThe majority of the illustrations were based on photographs of houses or architectural details taken by the author or photographer Steve Clicque. In some cases, however, specific plans were only slightly modified, while in other cases a published plan or a specific location provided general inspiration for the illustration.\n\nThe only exception to the above is that preservation architect Gary Skotnicki drew all the small additions to the Pictorial Key.\n\nListed below are illustrations and locations that were particularly helpful to the author and artist.\n\n## Looking at American Houses\n\nill.1 inset map inspired by Woodstock, Vermont\n\nill.2 \"New England village\" inspired by Nantucket, Massachusetts\n\nill.3 \"midwestern small town\" inspired by Athens, Ohio\n\nill.4 inset map inspired by 19th-century plans of Philadelphia, Pennsylvania\n\nill.5 \"Mid-Atlantic ship building district\" inspired by Fells Point, Baltimore, Maryland\n\nill.6 \"Upper South river port\" inspired by South Lee Street, Alexandria, Virginia\n\nill.7 \"Mid-Atlantic seaport\" inspired by Union Park neighborhood, Baltimore, Maryland\n\nill.8 \"midwestern river port\" inspired by Lafayette Square, St. Louis, Missouri\n\nill.9 \"along Chicago rail line\" inspired by River Forest, Illinois\n\nill.10 \"TOD\" inspired by _Sprawl Repair Manual_ by Galina Tachieva (Washington, DC: Island Press, 2010), 167\n\nill.11 \"TND\" inspired by Seaside, Florida\n\nill.12 \"Rural Plan\" inspired by _Rural by Design_ by Randall Arendt (Chicago: American Planning Association, 1994), 39\n\nill.13 \"Urban Plan\" inspired by Philadelphia, Pennsylvania\n\nill.14 \"Suburban: streetcar\" inspired by _Seven Rules for Sustainable Communities: Design Strategies for the Post-Carbon World_ by Patrick M. Condon (Washington, DC: Island Press, 2010), 26\n\nill.15 \"Suburban: early automobile\" inspired by Highland Park West, a Dallas suburb planned by George Kessler in 1918. Kessler was succeeded by planners Hare & Hare in the early 1920s.\n\nill.16 \"Suburban: post\u2013World War II\" inspired by Levittown, Pennsylvania\n\nill.17 \"Post-suburban: SLUG\" inspired by _Rural by Design_ by Randall Arendt (Chicago: American Planning Association, 1994), 39\n\nill.18 \"Olmstedian plans\" is a reproduction of the original plan for Riverside, Illinois (Frederick Law Olmsted, 1869).\n\nill.19 \"Warped grids\" modified from plan of St. Francis Wood, San Francisco, California (Olmsted Brothers, ca. 1912)\n\nill.20 \"FHA-guided subdivisions\" inspired by _Successful Subdivisions_ , FHA, 1940, 12\u201321\n\nill.21 \"Golf course developments\" based on _A Field Guide to Historic Neighborhoods and Museum Houses_ by Virginia and Lee McAlester (New York: Alfred A. Knopf, 1998), 9. Square-mile section developed with golf course green space\n\nill.22 \"Geometric combinations\" from McAlester and McAlester, _Field Guide to Historic Neighborhoods,_ xxxix, based on plan of El Encanto subdivision, ca. 1928, Tucson, Arizona\n\nill.23 \"Side-by-side combinations\" based on Kessler Park and Kessler Square, Dallas, Texas\n\nill.24 \"Effect of increasing block length\" based on http:\/\/en.wikipedia.org\/wiki\/Grid_plan\u2014Block Size and Street Lengths\n\nill.25 \"Deed restrictions\" based on Swiss Avenue and Munger Place Historic Districts, Dallas, Texas, 1905\n\nill.26 \"Street enclosures\" inspired by _Rural by Design_ by Randall Arendt (Chicago: American Planning Association, 1994), 10\n\nill.27 \"Rural southern farmstead\" based on _Garden History of Georgia, 1733\u20131933_ by Loraine Meeks Cooney, Hattie C. Rainwater, Florence Nisbet Marye, and P. Thornton Marye (Atlanta: Garden Club of Georgia, 1976), 79. Athens house built by John Thomas Grant\n\nill.28 \"Rural villa or 'borderlands,' \" inspired by Andrew Jackson Downing, _Cottage Residences,_ 3rd ed. (New York and London: Wiley and Putnam, 1847), 50, fig. 16\n\nill.29 \"Mini farm\" from McAlester and McAlester, _Field Guide to Historic Neighborhoods,_ xxxii, fig. 17\n\nill.30 \"Railroad or horsecar suburb\" inspired by _American Shelter: An Illustrated Encyclopedia of the American Home_ by Lester Walker (New York: Overlook Press, 1981), 149\n\nill.31 \"Post\u2013World War II suburb (ca. 1950s)\" inspired by _Yard, Street, Park: The Design of Suburban Open Space_ by Cynthia L. Girling and Kenneth I. Helphand (New York: Wiley, 1997), 82\n\nill.32 \"Post\u2013World War II suburb (ca. 1960s)\" inspired by _Case Study Houses_ by Elizabeth Smith (Los Angeles: Taschen, 2009), 48\n\nill.33 \"Underlying survey,\" from McAlester and McAlester, _Field Guide to Historic Neighborhoods,_ 695. Adapted from _Texas: A Geography_ by Terry G. Jordan Bychkov et al., (Boulder, CO: Westview Press, 1984)\n\nill.34 \"Rigid rectangular surveys: Salt Lake City, Utah,\" http:\/\/upload.wikimedia.org\/wikipedia\/en\/b\/b5\/Platslc.jpg\u2014plan of Salt Lake City, Utah, Brigham Young University, Harold B. Lee Library, Special Collections\n\nill.35 \"Rigid rectangular surveys: Dallas, Texas\" map from McAlester and McAlester, _Field Guide to Historic Neighborhoods,_ 698. Inspired by \"The Visible City: Tracing Memory and Character in Dallas' Historical Landscape\" by Constance Adams (unpub.)\n\nill.36 \"Long-lot surveys\" inspired by historic map of New Orleans reproduced in _The Second Battle of New Orleans: A History of the Vieux Carr\u00e9 Riverfront Expressway Controversy_ by Richard O. Baumbach Jr. and William E. Borah (Tuscaloosa: University of Alabama Press, 1981)\n\n## Folk Houses\n\nill.37 \"Mobile home, before World War II\" based on http:\/\/Allmanufacturedhomes.com\u20141937 Silver Dome. See also _The Unknown World of the Mobile Home_ by John Frase-Hart, Michelle J. Rhodes, and John T. Morgan (Baltimore: The Johns Hopkins University Press, 2002), 34\n\nill.38 \"Typical Siting of Manufactured Homes\": \"Rural siting,\" Key West, Florida vicinity, ca. 1980s \"Urban park siting,\" Phoenix, Arizona, ca. 1960s \"Urban park siting,\" Carlsbad, California vicinity, ca. 1980s\n\n## Victorian Houses\n\nill.39 Queen Anne sample house inspired by Martin Luther King, Jr., birthplace at 501 Auburn Avenue, Atlanta, Georgia\n\n## Eclectic Houses\n\nill.40 Mission sample house inspired by the Alhambra, 1923 Sears Roebuck Modern Homes (The Mission Type)\n\nill.41 Monterey sample house inspired by Williams House in Dallas, Texas, 1932, Dave Williams, architect\n\n## Modern Houses\n\nill.42 Minimal Traditional sample house based on \"The Phelps,\" in _Homes of Moderate Cost_ (National Plan Service, 1941), 12\n\nill.43 \"Bi-Level Split\" based on Design 47\u20131159 in _Contemporary Home Plans_ (National Plan Service, 1990) p. 612 \"Tri-Level Split\" based on Plan 9783, \"split level planning,\" Lester Cohen, architect, in _Foremost Home Plans_ 4 (1970)\n\nill.44 \"Bi-nuclear house,\" based on Bi-Nuclear House III, 1945, Marcel Breuer, architect, in _Marcel Breuer, Architect and Designer_ by Peter Blake (New York: Architectural Record\/Museum of Modern Art, 1949), 86\n\nill.45 Contemporary sample house based on houses in Surrey Downs development, Bellevue, Washington\n\nill.46 \"Elaborations commonly visible from street\" modeled on real-estate ad for Kimmelman House, 8973 Wonderland Park Avenue, Los Angeles, 1957, Philip Kimmelman, architect (chief project architect for many Weldon Becket structures)\n\nill.47 \"Decorative grilles, commonly concrete screen block\" inspired by \"Behold the Lowly Concrete Block... It Isn't Lowly Anymore,\" _House and Home_ , March 1956, 142\u201349\n\nill.48 \"Other typical elaborations\" inspired by \"Nine Hillside Houses,\" _House & Home_, April 1952, 82\u201383, 97\n\nill.49 \"Some spatial configurations\" modeled on Deck House, Inc., Series 5000 Marketing Kit\u2014Interiors\n\nill.50 \"Post-and-beam construction\" inspired by \"Post-Beam-Plank Construction,\" _House & Home_, June 1954, 98\u2013115\n\nill.51 Shed sample house based on East Texas Residence, 1969, John Mullen, architect. Awarded Dallas AIA Best House of the Decade\n\nill.52 \"How a gable roof is 'slipped' to create sheds\" based on a sketch by Ann Abernathy, AIA\n\nill.53 \"Some ways to organize shed-roof elements and rooms\" lower left house based on Lake Dallas residence, Lewisville, Texas, John Mullen, architect\n\nill.54 \"Some ways to organize shed-roof elements and rooms\" third to last house based on house in Athens, Texas, by John Mullen, architect\n\nill.55 \"Some ways to organize shed-roof elements and rooms\" inspired by \"some ways to organize rooms,\" _Charles Moore: Buildings and Projects 1949\u20131986,_ edited by Eugene J. Johnson (New York: Rizzoli, 1986), 18\n\nill.56 \"Typical elaborations\" based on _Contemporary Home Plans_ (New York: Bantam\/Hudson Plan Books, 1977), 47\n\nill.57 Brutalism sample house modeled on _American Architecture Since 1780_ by Marcus Whiffen (Cambridge, Mass: MIT Press, 1993)\n\nill.58 Deconstructivism sample house modeled on front facade of Blades House, Santa Barbara, California, 1995, Morphosis Architects (William Morgan, _The Abrams Guide to American House Styles_ , 2004), 390\n\nill.59 Postmodern sample house modeled on Vanna Venturi House, Philadelphia, Pennsylvania, 1962, Robert Venturi, architect; Organic sample house modeled on Glen Harder House, Mountain Lake, Minnesota, 1970, Bruce Goff, architect\n\nill.60 D\u00e9coupage sample house modeled on house in Urban Reserve, Dallas, Texas, ca. 2010\n\nill.61 Unifying Material sample house modeled on L-stack House, Fayetteville, Arizona, ca. 2005, Marlon Blackwell, architects\n\nill.62 Slightly Askew sample house modeled on Dumbacker House, Pasadena, California, 1996, Hagy Belzberg, architect\n\nill.63 \"Urban Reserve site plan\" modified from original site plan, provided courtesy Kevin Sloan (July 2009)\n\n## Styled Houses Since 1935\n\nill.64 \"Typical elaborations\" house on right based on Dilbeck House, Dallas, Texas, 1970, Charles S. Dilbeck, architect\n\nill.65 \"Cascading hipped roofs\" house photographed in new development near Kansas City, Missouri, ca. 2005\n\nill.66 \"Tall entry porches\" inspired by HPT 290139 by Fillmore Design Group, _Luxury Dream Homes_ (Tucson, AZ: Home Planners, LLC, 2001), 151\n\nill.67 \"Tall entry porches\" inspired by Italian 8010 \"Capucin,\" Don Sater, _Don Sater's European Classics_ (Bonita Springs, FL: Sater Design Collection, 2007), 129\n\nill.68 \"Comparison of detailing in new traditional houses\" top, house in Dallas, Texas, ca. 2010\n\nill.69 \"Comparison of detailing in new traditional houses\" bottom, house in Munger Place Historic District, Dallas, Texas, 1985\n\nill.70 American Vernacular sample house modeled on Three Sides Farm in Bowie County, Texas, 1979, John Mullen, architect\n\nill.71 \"Ways to achieve a larger house\" additive house on left, from _New Classic American Houses: The Architecture of Albert_ , _Righter & Tittmann_ (New York: Vendome, 2009), Stone House, 99\n\nill.72 \"Window patterns,\" first drawing, from _The Farmhouse: New Inspiration for the Classic American House_ by Jean Rehkemp Larson (Newton, CT: Taunton Press, 2004), 3. Based on the Marek House. In person, the full-width front porch includes a slightly deeper and handsome screened porch that would have been confusing at small scale.\n\nill.73 \"Window patterns,\" second drawing, from _Yankee Modern_ by William Morgan (New York: Princeton Architectural Press, 2010), 105. Based on the Osprey House in Jamestown, Rhode Island, 2005, Estes\/Twombly, architects.\n\nill.74 \"Window patterns,\" third drawing, from author's photo of the Arvold House in Duluth, Minnesota, David Salmela, architect\n\n## Appendix\n\nill.75 \"Green construction,\" Green Building Pyramid, copyright _Green Builder Magazine_\n\nill.76 \"Alternative construction methods\" modeled on \"Alternative Construction Methods,\" in _Louisiana Speaks: Pattern Book_ , 26, prepared by Urban Design Associates for the Louisiana Recovery Authority and the LRA Support Foundation\n\n# Photo Credits\n\nAnn Abernathy, 45.29; 47.5; 47.1; 52.15; 52.88; app.5\n\nAllison Abraham, 37.1, 37.2, 37.4, 37.5, 37.3\n\nState of Alabama Bureau of Publicity and Information, 28.4\n\nAlabama Department of Archives and History, 31.12\n\nArchitektursammlung, Technische Universitat Munchen from _L'Architecture Americaine,_ 17.33; 23.24; 23.57, 23.60; 23.66, 23.67; 23.69; 30.4\n\nArkansas Historic Preservation Program, 28.16; 38.33. Bob Dunn, 18.15\n\nAtlanta Historical Society, Peachtree-Cherokee Trust Collection, 27.22. Kenneth Kay, 27.58\n\nWilliam T. Baker, 52.41; 52.48, 52.50, 52.51\n\nCity of Biloxi, 23.1; 23.20; 26.25\n\nBentz-Thompson Architects, app.6. Eric Sutherland, 44.24\n\nJean F. Block, _Hyde Park Houses_ (Chicago: University of Chicago Press, 1978). Samuel W. Block Jr., 23.54\n\nLarry Boerder, 52.33; 52.58, 52.63; 52.96\n\nBowdoin College Library, Special Collections, 17.12\n\nBrian Vanden Brink, courtesy of Peter Pennoyer Architects, 52.19\n\nR. Bruhn (\u00a9 1978), 21.28\n\nBuffalo and Erie County Historical Society, 27.10, 27.14; 27.32; 29.38; 38.25. Roy Nagle Collection, 28.17\n\nCambridge Historical Commission, 27.59, 27.62; 44.2. Richard Cheek, 21.3; 21.12; 23.29; 23.47; 27.50; 44.5. Roger Gilman, 22.6. B. Orr, 21.4; 23.13; 27.1; 27.30; 27.60\n\nCatskill Center for Conservation and Development, Arkville, New York. Mark Zeek, 23.11\n\nJohn Chase, _The Sidewalk Companion to Santa Cruz Architecture_ (Paper Vision Press), 23.14, 23.15\n\nSteve Clicque, 7.15; 7.36; 7.43a; 8.1; 8.2, 8.3, 8.4, 8.5; 8.9, 8.10, 8.11, 8.12, 8.14, 8.15; 13.6, 13.8; 13.27; 13.30; 18.42; 23.25; 23.40; 27.16, 27.19; 27.66, 27.67, 27.68, 27.69, 27.70, 27.71, 27.72, 27.73; 28.11; 29.4, 29.5; 29.9, 29.10; 29.12; 29.18, 29.21, 29.23, 29.24, 29.26; 29.27; 29.33, 29.34; 29.36, 29.37, 29.39; 32.12; 33.19; 34.13; 35.17; 35.29; 36.4, 36.7; 37.6; 38.7; 38.14; 39.29; 40.7, 40.10; 41.2, 41.4, 41.6, 41.7; 41.9, 41.11, 41.13, 41.14, 41.15, 41.16; 41.18, 41.19, 41.20, 41.21, 41.22, 41.23; 42.3, 42.4, 42.5, 42.7, 42.9; 42.11, 42.12, 42.14, 42.15, 42.16, 42.17, 42.18; 42.21, 42.22, 42.23, 42.25, 42.27, 42.28, 42.29, 42.30; 42.31, 42.32, 42.33, 42.34, 42.35, 42.36, 42.37, 42.38, 42.40; 43.1, 43.2; 44.6, 44.8; 44.10, 44.13, 44.14, 44.15, 44.16, 44.17; 44.22, 44.23, 44.25, 44.26, 44.29; 45.1, 45.5, 45.6, 45.8, 45.9, 45.10; 45.14; 45.15, 45.16, 45.18, 45.19; 45.20, 45.22, 45.23, 45.24, 45.25, 45.27, 45.28; 45.30, 45.32, 45.33, 45.34, 45.35, 45.36, 45.38; 45.39, 45.40, 45.41, 45.42, 45.43, 45.44, 45.45; 46.3, 46.4, 46.5, 46.6; 47.2, 47.5, 47.7; 47.14; 47.17, 47.18, 47.19, 47.22; 47.23, 47.24, 47.27, 47.28, 47.29; 47.32, 47.34, 47.35; 47.37, 47.38, 47.40; 48.1, 48.2, 48.3; 48.6, 48.7, 48.10, 48.11; 48.12, 48.14, 48.16; 48.17, 48.18, 48.19, 48.20, 48.21; 48.23, 48.24, 48.25, 48.27; 48.28, 48.29, 48.30, 48.31, 48.32, 48.33; 49.1, 49.2, 49.3, 49.4, 49.5; 49.6, 49.7, 49.8, 49.9, 49.10, 49.11; 50.1, 50.2, 50.4, 50.6; 50.7, 50.8, 50.9, 50.10, 50.11, 50.12, 50.13, 50.14, 50.15, 50.16; 50.17, 50.18, 50.19, 50.20, 50.21, 50.22; 50.23, 50.24, 50.25, 50.26; 50.27, 50.28; 50.31, 50.32, 50.33, 50.34; 51.1, 51.3, 51.4, 51.5, 51.6, 51.7, 51.9; 51.10, 51.12, 51.13, 51.14, 51.15, 51.16, 51.17; 51.18, 51.19, 51.20, 51.21, 51.22, 51.23, 51.24; 51.24a; 51.24b; 52.2, 52.4, 52.7, 52.8, 52.10; 52.13, 52.14, 52.16; 52.17, 52.18, 52.22; 52.27, 52.28, 52.29, 52.31, 52.34; 52.38, 52.39, 52.42, 52.43, 52.44, 52.45; 52.46, 52.47, 52.49, 52.52, 52.53, 52.54; 52.55, 52.57, 52.59, 52.60, 52.61, 52.62, 52.64; 52.67, 52.68, 52.69, 52.70, 52.71, 52.72, 52.73; 52.74, 52.75, 52.76, 52.77, 52.78, 52.79, 52.80, 52.81, 52.82; 52.83, 52.84, 52.85, 52.86, 52.87, 52.89, 52.90; 52.91, 52.92, 52.93, 52.95, 52.97, 52.98; 53.1a; 53.3, 53.4; 53.5, 53.7, 53.8, 53.9, 53.11, 53.12; 53.13, 53.14, 53.15, 53.16, 53.18, 53.19; 53.20, 53.21; 53.23, 53.25, 53.26; app.3; app.7, app.8\n\nCommission of Fine Arts. Jack E. Boucher, 31.11, 31.15; 32.14\n\nConnecticut Historical Commission. Susan Babbitt, 22.9. Brian Pfeiffer, 6.4; 18.48; 21.15. Ellen Rosebrock, 23.7; 24.20\n\nCourtesy of Biltmore House and Gardens, Ashville, North Carolina, 30.9\n\nCourtesy of Palm Harbor Homes, 8.6, 8.8; 8.13\n\nAl Cox, 13.16; 13.28, 13.29\n\nPaul Crews, 26.6\n\nMary Wallace Crocker, _Historic Architecture in Mississippi_ (1973). 17.11; 18.18\n\nMarianne Cusato, 53.1\n\nCuyahoga County Archives, 23.34. David M. Thurn, 18.6\n\nDallas Historical Society, 23.33\n\nDallas Public Library, Texas\/Dallas History and Archives Division, 24.1\n\nDetroit Public Library, Burton Historical Collection, 17.30\n\nJames Fox, 47.1, 47.2, 47.3; 47.8\n\nState University College at Fredonia, New York, Jewel Conover Archives in the Reed Library, 28.14\n\nGalveston Historical Foundation, 26.13; 27.7\n\nDavid Gebhard, 34.8; 34.14; 44.5\n\nRobert C. Giebner, 12.8\n\nGeoffrey Gross, 44.27\n\nJulia Guice, editor, _The Buildings of Biloxi: An Architectural Survey,_ 7.5\n\nKen Gutmaker, 53.2; 53.17; 53.27\n\nThomas Hahn, 16.6; 17.17; 18.2; 18.36; 18.46; 19.7; 22.2; 23.21, 23.22; 23.26; 23.44; 23.49, 23.50; 23.64, 23.65; 23.73 24.2; 26.4; 27.4; 28.7; 29.8; 29.19\n\nCarol M. Highsmith, 44.11\n\nHistoric American Buildings Survey Office. Jack E. Boucher, 13.32\n\nHistoric Landmarks Foundation of Indiana, 21.26\n\nhttp:\/\/creativecommons.org\/licenses\/by\/2.0\/deed.en, 25.9\n\nMichael G. Imber, architects, 52.94\n\nIllustration Inspirations, 7.43a\n\nIndiana Historical Society Library, 1.3; 16.8; 21.2; 23.6; 23.38\n\nWarren Jagger, 53.21\n\nKansas City Landmarks Commission, 24.28\n\nKentucky Historical Society, 6.17\n\nMargaret M. Keyes, _Nineteenth Century Home Architecture of Iowa City_ (University of Iowa Press), 17.21; 18.49\n\nThe King William Association, San Antonio, Texas. Mary V. Burkholder, 26.23\n\nKirksville College of Osteopathy and Surgery, 23.43\n\nCarleton Knight III, 13.25; 18.19; 18.47; 20.7; 21.17, 21.18; 22.12; 23.51; 25.12\n\nCarson Leh, 48.22\n\nLexington-Fayette County Historic Commission, 7.24; 7.27; 18.27\n\nLibrary of Congress, 16.19; 19.8; 26.16. American Press Assn., 33.5. George S. Cook, 13.19. Francis B. Johnston Collection, 9.10; 11.11; 13.22; 14.30; 15.7; 16.21; 16.40. National Photo Co., 39.34. W. H. Sutton, 21.29. Witteman Collection, 22.11; 29.35; 34.2. Marion Post Wolcott, app.4\n\nLibrary of Congress, Farm Security Administration Archives. Walker Evans, 6.15; 26.3; 26.20. Lange, 12.5. Russell Lee, 6.18; 29.11; 35.10. Carl Mydans, 7.37. Arthur Rothstein, 6.22, 6.25; 7.7; 12.11. John Vachon, 7.34. Marion Post Wolcott, 7.8; 7.26; 7.32; 7.38, 7.40; 29.11\n\nLibrary of Congress, Historic American Buildings Survey, 7.39; 9.11; 12.6; 13.9, 13.10, 13.11; 15.9; 23.5; 23.75; 27.35; 31.8; 38.28. L. D. Andrew, 15.3. W. Harry Bagby, 9.13. Nelson E. Baldwin, 10.2; 10.5. John M. Beckstrom, 14.3. John O. Bostrup, 14.6. Jack E. Boucher, 6.7; 9.16; 10.1; 13.13; 13.21; 13.31; 14.33; 16.36; 17.18, 17.23; 18.5; 22.7; 24.13; 25.13; 25.17; 31.9. Branzetti, 6.1, 6.2; 13.12; 13.24; 14.19. James Butters, 15.5; 16.15; 17.25. Richard Cheek, 9.6. Clinedinst, 34.6. W. Collins, 41.5. C. O. Greene, 9.15, 9.17; 13.20, 13.23. Arthur C. Haskell, 14.1, 14.5; 14.8. Cortlandt Hubbard, 13.26. John A. Huffman, 26.15. Lester Jones, 6.5; 11.10, 11.13. Richard Koch, 11.1, 11.6; 11.14. R. Merritt Lacey, 10.10, 10.11, 10.13, 10.14. Leslie, 11.4. Jane Lidz, 17.1; 23.2; 23.41; 33.16; 35.26, 35.27; 39.11. E. P. MacFarland, 10.12; 16.39. Stanley P. Mixon, 10.6; 14.27. Eric Muller, 17.6. Frederick D. Nichols, 12.9, 12.13. Paul Piaget, 16.37. E. H. Pickering, 14.21, 14.23, 14.24; 14.26. James Rainey, 13.3. Cervin Robinson, 6.3; 13.4; 14.13, 14.15; 16.16; 20.2; 24.18. Sirlin Studies, 18.10; 18.53; 22.19. Roger Sturtevant, 12.1. Robert Thall, 21.27; 38.9; 40.8. Laurence E. Tilley, 13.2; 14.1; 14.11, 14.12; 18.8. Josiah Tully, 14.9. Thomas Waterman, 9.12; 10.9; 13.7. Carl F. White, 16.30. Henry F. Withey, 12.2\n\nJames R. Lockhart, courtesy of William T. Baker, 52.56\n\nLouisiana State Museum, 17.15\n\nUniversity of Louisville Photographic Archives, 29.3. Brown-Doherty Collection, 25.15; 30.2. Caufield and Shook Collection, 1.10 (neg. 36593); 1.9 (neg. 41271); 7.6 (neg. 3909?); 15.17 (neg. 38025); 18.12 (neg. 40954); 24.11 (neg. 37097); 27.24 (neg. 72429); 27.25 (neg. 72425); 27.31 (neg. 41518); 27.40 (neg. 68196); 28.5 (neg. 79590); 28.25 (neg. 72648); 28.26 (neg. 5774A); 29.7 (neg. 41252); 32.8 (neg. 72687); 34.7 (neg. 2614); 35.12 (neg. 89283); 39.14 (neg. 41267); 39.22 (neg. 541259). Potter Collection, 27.55 (neg. 2523.5). A. B. Rue, 6.16. Standard Oil of New Jersey Collection, 6.13 (neg. 52483); 7.22 (neg. 48981); 7.33 (neg. 49832); 17.5 (neg. 51912); 17.7 (neg. 53953)\n\nLouisville Landmarks Commission, 18.25; 19.4; 25.8; 25.14; 31.1; 33.9; 33.22\n\nJ. Lynch\u2014City of Bellingham Planning Division, 39.17\n\nMadison County Historical Society. Howard L. Colyer, 18.23\n\nMaine Historic Preservation Commission, 21.25\n\nManufactured Housing Institute, 8.7\n\nMarshall Historical Society, 16.3; 23.48\n\nVan Jones Martin, 16.11; 16.14; 16.17, 16.20; 16.26; 18.7, 18.16\n\nTom Martinson, 17.8; 17.20; 40.9; 44.18; 47.6; 47.25; 47.31; 47.36\n\nMaryland Historical Society, 21.7\n\nVirginia and Lee McAlester, prints by Doug Tomlinson, 1.2, 1.4, 1.5; 1.6, 1.7; 1.12, 1.14, 1.15, 1.16; 1.17, 1.18, 1.19, 1.20, 1.21, 1.22, 1.23, 1.24, 1.25; 7.1, 7.2, 7.3; 7.10, 7.12, 7.13, 7.14; 7.19, 7.20; 7.29, 7.30; 7.35; 7.42, 7.43; 11.2, 11.3, 11.7; 11.9; 13.1; 13.14, 13.15; 14.2, 14.7; 14.22; 14.29; 15.1, 15.6; 16.23; 16.32; 16.34, 16.35, 16.38; 17.2, 17.3; 17.10; 17.14, 17.16; 17.19, 17.24; 18.1, 18.11; 18.17, 18.20; 18.26, 18.28, 18.29, 18.32; 18.34, 18.35, 18.37, 18.39; 18.43, 18.44, 18.45; 18.51, 18.53, 18.55; 19.1, 19.2, 19.6; 21.1; 21.12, 21.13; 21.16; 21.23; 21.30, 21.31, 21.33, 21.34, 21.35; 23.9, 23.18; 23.23; 23.31, 23.32, 23.39, 23.42, 23.45, 23.46; 23.53, 23.56; 23.61; 23.70, 23.71; 24.6, 24.7, 24.8; 24.14, 24.15, 24.16, 24.17; 24.19, 24.21, 24.24; 24.29, 24.30, 24.31, 24.33; 25.2, 25.3, 25.5, 25.6; 25.10; 25.16; 26.1, 26.2; 26.5, 26.7; 26.17, 26.19; 27.3, 27.5, 27.6; 27.8, 27.11, 27.12; 27.15, 27.20; 27.23, 27.26, 27.27, 27.28; 27.34, 27.35, 27.36, 27.37, 27.39; 27.41, 27.45, 27.47, 27.48, 27.49; 27.51, 27.52, 27.53; 27.56; 27.61, 27.64; 28.1, 28.2, 28.3, 28.5, 28.6; 28.10, 28.12, 28.13; 28.18, 28.19, 28.20, 28.21, 28.22, 28.23, 28.24; 28.27, 28.29, 28.30, 28.31, 28.32; 29.1, 29.2; 29.10, 29.13; 29.12; 29.20, 29.22; 29.31, 29.32; 29.36, 29.37, 29.38, 29.40; 30.3, 30.6, 30.7, 30.8; 31.2, 31.3, 31.4, 31.6; 31.14; 32.1, 32.2, 32.3, 32.4, 32.5, 32.6, 32.7; 32.9, 32.10, 32.11, 32.13, 32.15; 32.16, 32.17, 32.18, 32.19, 32.20, 32.21, 32.22; 33.1, 33.2, 33.3, 33.4, 33.6, 33.7; 33.10, 33.11, 33.12, 33.14, 33.15, 33.17, 33.18; 33.20, 33.21, 33.23, 33.24, 33.26; 33.27, 33.28, 33.30, 33.31, 33.32, 33.33; 34.1, 34.4, 34.5; 34.9, 34.10, 34.11, 34.12, 34.15; 35.1, 35.2, 35.3, 35.6, 35.7, 35.8; 35.13, 35.14, 35.15; 35.18, 35.19; 35.28; 35.32, 35.34; 36.1, 36.3, 36.5, 36.6; 38.1, 38.2, 38.3, 38.4, 38.6, 38.8; 38.10, 38.11, 38.12; 38.18, 38.19, 38.20, 38.21, 38.23; 38.26, 38.27, 38.29, 38.31, 38.32; 39.2, 39.3, 39.4, 39.5, 39.6, 39.7; 39.12, 39.13, 39.16; 39.19, 39.21, 39.23, 39.25, 39.26, 39.27, 39.30, 39.32; 39.33, 39.35, 39.36; 40.1; 41.8; 42.8; 44.21; 45.21, 45.26; 46.8; 49.12; 50.5; 50.29, 50.30; 52.30; 52.66\n\nVirginia Savage McAlester, 4.1, 4.2; 7.25; 8.16; 12.3, 12.7; 16.4; 18.3; 18.22; 18.32; 19.9; 21.5; 21.9; 21.19; 22.3, 22.5; 22.14; 22.16, 22.18; 23.10, 23.12, 23.19; 23.40; 24.4; 24.10; 24.32; 25.7; 27.18; 27.42; 29.6; 29.25; 30.10; 31.13; 33.13; 33.29; 34.3; 34.17; 35.4; 35.16; 35.19; 35.30; 35.35; 38.5; 38.22; 38.30; 39.24; 40.2, 40.3; 40.5, 40.11; 41.4; 41.10, 41.12; 42.1, 42.2; 42.13, 42.19; 42.20, 42.26; 42.39; 44.12; 44.20, 44.28; 45.1, 45.2, 45.3; 45.4, 45.7; 46.1, 46.2, 46.7; 47.4; 47.16; 47.20; 47.26; 47.39; 48.4; 48.5, 48.8, 48.9; 48.13, 48.15; 48.26; 51.1a; 51.2, 51.8; 52.5; 52.6; 52.9; 52.32; 52.37; 52.40; 52.65; 53.6; app.1\n\nAlex McLean, 12.14; 14.25; 22.13; 25.11; 35.9\n\nMinnesota Historical Society, 20.1; 21.14; 22.10; 23.28\n\nUniversity of Minnesota Libraries, 38.24; 41.7\n\nThe State Historical Society of Missouri, Columbia, Missouri, 6.12, 6.21; 11.8, 11.12; 15.15; 25.1; 26.14\n\nMissouri Historical Society, 18.9; 21.6; 21.32; 27.46; 31.7; 33.25; 33.29\n\nCity of Mobile Planning Commission. Thigpen Photography, 16.18\n\nMonterey Domes, app.2\n\nEvelyn Montgomery, 33.8\n\nJohn Mullen, 41.1; 53.22\n\nNational Archives, 7.11; 12.4; 41.3\n\nNational Association of Home Builders, 47.15; 50.3; 51.11; 52.15\n\nNational Trust for Historic Preservation, 11.5; 11.15; 12.12; 14.14; 14.16, 14.17; 14.31; 15.9; 15.13; 17.29; 18.4; 18.56; 19.5; 23.59; 31.5; 35.23. Jack E. Boucher, 15.8; 16.5; 16.29; 35.23. Louis H. Frohman, 17.28. Gleason, 16.22. Carleton Knight III, 17.29. Marler, 27.63. Cervin Robinson, 19.5; 23.59. Russell Zimmerman, 30.5\n\nNebraska State Historical Society, Solomon D. Butcher Collection, 6.24, 6.26, 6.27; 24.26\n\nCity of New Bedford, Office of Historic Preservation, 22.1\n\nThe New Haven Preservation Trust, _New Haven Heritage_ (1974). Brooks Mather Kelley, 23.63\n\nNew Jersey Office of Cultural and Environmental Services Department of Environmental Protection, Trenton, New Jersey. Jack E. Boucher, 23.74\n\nMuseum of the City of New York, 17.32. The Byron Collection, 24.25; 29.39. E. Pallme, 16.41. Wurtz Archives, 18.57\n\nNew York City Landmarks Preservation Commission, 27.65; 31.16. Andrew S. Dolkart, 27.13; 28.15\n\nNew York State Historical Association, Cooperstown, 1.11, 1.13; 7.9\n\nNew York State Office of Parks, Recreation and Historic Preservation. Lynn A. Beebe, 24.12. L. E. Gobrecht, 10.3. Herkimer County Historic Society, 23.62. David Horton, 32.23. Chester Liebs, 10.4. NYSPIX-Commerce, 10.7. Rheinbeck Historical Society, 21.24. R. W. Smith, 20.5. Harry A. Thayer, 10.8\n\nLandmark Society of Western New York. Stephen W. Baldwin, 23.27. Steven Haag, 19.3. Hans Padelt, 18.33; 22.8; 23.68; 23.72. The Walrus, 17.22. John A. Wenrich, 14.32; 22.4. Gary Whelpley, 20.4\n\nNorth Carolina Division of Archives and History, Raleigh, North Carolina, 6.6, 6.8, 6.9, 6.10, 6.11; 6.20; 7.4; 7.17, 7.18, 7.21; 7.28, 7.31; 15.4; 15.15; 17.9; 18.21, 18.24; 23.4; 26.11; 26.21, 26.22; 26.24; 27.9; 28.28\n\nThe Ohio Historical Society, Inc., Archives-Library Division, 7.16; 7.41; 12.10; 16.31; 21.20; 24.27; 27.17; 27.27; 27.43; 29.29; 39.1; 40.6\n\nUniversity of Oklahoma, Western History Collections. Bob Goins, 6.23\n\nAmanda Olson-McCoy, 27.57\n\nOregon Historical Society, 18.30; 21.21; 22.15; 23.35\n\nPennsylvania Historical & Museum Commission, 24.5\n\nFree Library of Philadelphia, 18.52\n\nHistoric Pittsford, Inc. Richard L. Turner, 16.12\n\nWilliam Plymat, Jr., _The Victorian Architecture of Iowa,_ 18.38\n\nGreater Portland Landmarks, Inc., Portland, Maine. Nicholas Dean, 18.40\n\nDaniel D. Reiff, _Architecture in Fredonia_ (Michael Rockefeller Arts Center Gallery), 21.11\n\nAlbert Righter & Tittmann Architects Inc, 47.30; 52.1, 52.3; 52.20, 52.21; 52.35, 52.36\n\nRobert P. Ruschak, courtesy of Western Pennsylvania Conservancy, 47.4\n\nRussell House Museum, 16.28\n\nHistoric Salisbury Foundation, Inc. Wayne Hinshaw, 9.14; 18.14; 23.30; 39.20\n\n_Santa Barbara Architecture_ (copyright \u00a9 1975, 1980 by Capra Press, Santa Barbara, CA 93120). Wayne McCall, 35.5; 35.11; 35.20, 35.21, 35.22, 35.24, 35.25; 35.31, 35.33; 36.2; 39.15\n\nDorothy Harris Savage, 28.9\n\nBen Schnall, 44.19\n\nMary-Peale Schofield, _Landmark Architecture of Cleveland_ (Ober Park Assoc., Inc., Pittsburg, Pennsylvania), 23.76\n\nJulius Shulman, 44.4, 44.7; 47.8\n\nRoger B. Smith, _The Early Architecture of the Genesee Valley_ (courtesy Milne Library, State University College of Arts and Sciences, Genesee, New York), 16.1; 20.3\n\nSmithsonian Institution, National Anthropological Archives, 5.1, 5.2, 5.3, 5.4, 5.5; 5.6, 5.7, 5.8, 5.9; 5.10, 5.11, 5.12, 5.13\n\nThe Society for the Preservation of New England Antiquities, 9.1, 9.2, 9.3, 9.4, 9.5, 9.7, 9.8, 9.9; 13.5; 13.17; 14.18; 23.55, 23.58; 24.3; 24.9; 24.22; 27.2; 27.21; 30.1\n\nJames T. Sodeman, 18.54; 22.20, 22.21, 22.22\n\nRoman Sokal, 39.8, 39.9; 39.28\n\nSouth Carolina Department of Archives and History, 7.23; 23.16, 23.17; 26.9; 26.13; 26.18; 26.26, 26.27; 27.44; 27.54; 39.10\n\nRobert A. M. Stern, Architects. Peter Aaron, 47.33. Steven Brooke, 52.12.\n\nC. Eric Stoehr, _Bonanza Victorian_ (University of New Mexico Press), 16.25; 17.13; 23.3\n\nSyracuse University. Hans Padelt, 16.7\n\nThe Association for the Preservation of Tennessee Antiquities, Nashville Chapter, 16.24\n\nTexas Historical Commission, 16.9, 16.10; 17.4; 18.41; 21.8; 23.8; 26.10, 26.12; 26.28; 40.4\n\nTransylvania University, Lexington, Kentucky, The J. Winston Coleman Kentuckiana Collection, Francis Carrick Thomas Library, 6.19; 14.28; 16.13; 16.33; 17.26\n\nUnited States Department of Housing and Urban Development, 14.20; 16.27; 38.34. David Valdez, 6.14\n\nUnited States Department of the Interior\u2014National Park Service, National Register of Historic Places, 13.18; 29.22; 29.28; 42.10; 44.3. W. Collins, 41.1. G. G. George, 42.24. James R. Lockhart, 41.17; 42.6; 45.2. Paul Seder, 45.17. Mariann Seriff, 45.3. Ruth Williams, 44.3. Diane Wray, 45.31.\n\nUrban renewable with Federal funds acquired by Knopf, 4.3\n\nUtah State Historical Society, Historic Preservation Office, 25.4\n\nAlexander Vertikoff, courtesy of The Gamble House, 39.18\n\nVirginia Historic Landmarks Commission, 14.10; 15.2; 15.10, 15.12; 15.14, 15.16; 17.27; 21.22\n\nWestchester County Historic Society by permission of The North River Press, Inc. Frank Sanchis. 34.16\n\nWestern Reserve Historical Society, 1.1; 1.8; 17.31; 24.23\n\nWichita\/Sedgwick County Historical Museum, Wichita, Kansas, 22.17; 39.31\n\nJames F. Wilson, courtesy of Buchanan Architects, 47.21\n\nState Historical Society of Wisconsin, 38.15. E. C. Hamilton, 20.6. James A. Sewell, AlA, 18.13. Mary Ellen Young, 38.17\n\nLes Wollam photograph, courtesy of Daron Tapscott, 23.36\n\n# Index\n\nNumbers in _italics_ refer to illustrations.\n\nAbernathy, Ann\n\nAdam, Robert, 14.1, 14.2\n\nAdams & Adams, _35.1_\n\nAdams House, _17.1_\n\nAddison and Hutton\n\nadditive vs. massive size increase\n\nAdler & Sullivan\n\nadobe, 1.1, _3.1_ , 3.2, _3.3, 3.4_, 5.1, 5.2, 6.1, 12.1, 12.2, 12.3, 36.1, 37.1, 37.2\n\nA-frame houses, 47.1, 47.2, _47.3_\n\nair-conditioning, app1.1, app1.2\n\nAlabama\n\nColonial houses, _13.1_\n\nEclectic houses, _27.1, 28.1, 28.2, 28.3, 28.4, 29.1, 29.2, 30.1, 31.1, 33.1, 33.2, 33.3_\n\nfolk houses, _6.1, 7.1_\n\nModern houses, _38.1_\n\nRomantic houses, 16.1, 16.2, _16.3, 17.1, 17.2, 18.1, 18.2_\n\nstyled houses since 1935, _50.1_\n\nVictorian houses, _21.1, 23.1, 23.2, 23.3, 23.4_\n\nAlbert, Righter & Tittmann, 47.1, _47.2, 52.1, 52.2, 52.3, 52.4_\n\nAlbright House, _29.1_\n\nAlderson House, _24.1_\n\nAlexander and Alexander, builders, _44.1_\n\nAlexander Construction, _45.1_\n\nAlexander House, _31.1_\n\nAlexandria, Va.\n\nColonial houses, 13.1, _13.2, 13.3, 14.1, 14.2_\n\nEclectic houses, _27.1_\n\nModern houses, _41.1_\n\nstyled houses since 1935, _50.1, 51.1, 52.1_\n\nAllen House, Nashville, _52.1_\n\nAllen House, Newlin, _6.1_\n\nAllen House, Wichita, _38.1_\n\n\"All-Steel House\", _44.1_\n\nalterations, 1.1, 1.2, _1.3, 1.4_, 1.5, 6.1, 9.1, _39.1, 47.1_\n\nalternative construction methods, app1.1, _app1.2_\n\nAlto Plaza house, _24.1_\n\nAmberg House, _38.1_\n\n_American Architect and Building News_ ( _AABN_ ), 20.1, 20.2, 23.1, 27.1\n\n_American Builder's Companion_ (Benjamin)\n\nAmerican embassy building, New Delhi, 47.1, 47.2\n\nAmerican Four-square houses, _see_ four-square houses\n\nAmerican Institute of Architects, Committee on the Environment\n\nAmerican Revolution, 8.1, 15.1, 15.2\n\nAmerican Vernacular style, _52.1_ , 53.1, _53.2, 53.3, 53.4_\n\ncomplex roof form, 53.1, _53.2_\n\ncross-gabled, 53.1, _53.2_\n\nfront-gabled, 53.1, _53.2_\n\nhipped-roof, 53.1, _53.2_\n\nside-gabled, 53.1, _53.2_\n\nAmes, John W., _27.1, 27.2_\n\nAmoureaux House, _11.1_\n\nAncient Classical styles, influence of, 1.1, 1.2, _1.3_\n\nAndersen, Henry, _31.1_\n\nAndrews House, _18.1_\n\nAnnefield house, _15.1_\n\nAppleget, G. S. H., _18.1_\n\nAppleton House, Lennox\n\nAppleton House, Saugus, _9.1_\n\nappliances\n\nApthorp House, _19.1_\n\nArapahoe Acres Historic District house, _45.1_\n\narched doorways\n\nChateauesque\n\nFederal, _14.1_\n\nFrench Eclectic, 32.1, 32.2, _32.3_\n\nItalianate\n\nItalian Renaissance, 33.1, _33.2_\n\nMansard, 49.1, _49.2, 49.3, 49.4_, 49.5\n\nMillennium Mansion, 49.1, _50.1_ , 51.1, _51.2_ , 52.1\n\nNew Traditional, _52.1_\n\npictorial key to\n\nSpanish Revival, 35.1, 35.2, _35.3, 35.4_\n\nTudor, 29.1, _29.2_\n\narched porches\n\nGothic Revival\n\nItalian Renaissance, 33.1, _33.2, 33.3_\n\nMission, _33.1_ , 34.1\n\nNew Traditional, _52.1, 52.2_\n\npictorial key to\n\nRichardsonian Romanesque, 25.1, _25.2_\n\nRomanesque, _24.1_\n\nShingle\n\nSpanish Colonial\n\nSpanish Revival, 35.1, _35.2_\n\nStyled Ranch\n\nTudor\n\narched windows\n\nBeaux Arts, _29.1_\n\nChateauesque\n\nColonial Revival, _27.1_\n\nEarly Classical Revival, _15.1_\n\nFederal, 14.1, 14.2\n\nFrench Eclectic, 32.1, _32.2_\n\nGeorgian, _13.1, 14.1_\n\nGothic Revival, 1.1, _1.2, 16.1_, 17.1, 17.2, _17.3, 17.4_\n\nItalianate, _17.1, 17.2_, 18.1, _18.2, 18.3, 18.4_\n\nItalian Renaissance, _1.1, 32.1_, 33.1, 33.2, _33.3, 33.4, 33.5, 33.6, 33.7_\n\nMansard, 49.1, _49.2, 49.3_\n\nMillennium Mansion, 51.1, 51.2, _51.3, 51.4, 51.5, 52.1_\n\nNeoclassical, 28.1, _28.2, 28.3_\n\nNew Traditional, _52.1, 52.2, 52.3, 52.4_\n\npictorial key to\n\nRichardsonian Romanesque, _24.1_ , 25.1, _25.2, 25.3, 25.4_\n\nSecond Empire\n\nShingle, _24.1, 24.2_\n\nSpanish Revival, _34.1_ , 35.1\n\nStyled Ranch, 50.1, _50.2_ , 50.3\n\narches, _25.1, 33.1_\n\n_Architect_\n\n_Architectural Digest_\n\n_Architectural Instructor, The_ (Lafever), _16.1_\n\n_Architectural Record_ , 39.1, 46.1, 52.1\n\n_Architecture of Country Houses, The_ (Downing), 17.1, 17.2, 19.1\n\nArizona\n\nColonial houses, 12.1, _12.2_ , 11.1, _12.3_ , 12.4\n\nEclectic houses, 35.1, 37.1, _37.2_\n\nfolk houses, _5.1_\n\nModern houses, _41.1_ , 42.1, _42.2, 42.3, 45.1, 48.1_\n\nstyled houses since 1935, 50.1, _50.2, 50.3, 51.1_, 52.1, _app1.1_\n\nArkansas\n\nEclectic houses, _28.1, 34.1_\n\nfolk houses, _7.1_\n\nModern houses, _38.1, 46.1_\n\nRomantic houses, _18.1, 18.2_\n\nVictorian houses, _21.1_\n\nArmstrong House, _33.1_\n\nArnold House, Dallas, _23.1_\n\nArnold House, Lincoln, _9.1_\n\nArrington House, _1.1_\n\nArt Deco, _39.1_ , 40.1, _40.2_\n\nArt Moderne, _39.1_ , 40.1, 40.2, _40.3, 40.4_\n\nArts and Crafts (Early Modern) style, 1.1, _1.2_ , 1.3, 29.1, 37.1, 39.1, 39.2\n\nAshe House, _17.1_\n\nAshland house, _16.1_\n\nasphalt or tar roofing\n\n_Associated Architects Fifty House Plans,39.1_\n\nasymmetrical houses\n\nColonial Revival, 27.1, _27.2_\n\nFrench Eclectic, 32.1, _32.2_\n\nGothic Revival, 17.1, _17.2_\n\nhipped roof, 38.1, _38.2_\n\nItalianate, 18.1, _18.2_\n\nItalian Renaissance, 33.1, _33.2_\n\nMansard, 49.1, _49.2_\n\nMission, 34.1, 34.2\n\nSecond Empire, 21.1, _21.2_\n\nAtkinson House, _25.1_\n\nattic space\n\nAuburn house, _15.1_\n\nAustin, Henry, 15.1, _18.1, 22.1_\n\nAutry House, _23.1_\n\nAyres, James B., _18.1_\n\nBacon House, _13.1_\n\nBacon's Castle, _9.1_\n\nBadger, Alfred M., _16.1_\n\nBaird House, _28.1_\n\nBaker, William T., _52.1, 52.2, 52.3_\n\nBalcomb and Rice, _23.1_\n\nbalconies\n\nAmerican Vernacular\n\nBeaux Arts, 31.1, _31.2_\n\nChateauesque, _30.1_\n\nColonial Revival, _27.1_\n\nContemporary, _45.1_ , 45.2, _45.3, 45.4, 45.5, 45.6_\n\nCraftsman, _39.1, 39.2_\n\nexotic revival, _18.1, 19.1_, 19.2\n\nFederal, 14.1, _14.2, 14.3_\n\nFrench Colonial, 11.1, _11.2_\n\nFrench Eclectic, _32.1_\n\nInternational, 44.1, _44.2_\n\nItalianate, _18.1_\n\nItalian Renaissance, _33.1_\n\nMonterey Revival, _35.1_ , 36.1, _36.2, 36.3_, 36.4\n\n_Neoclassical,28.1_\n\nPrairie, 38.1, _38.2_\n\nRichardsonian Romanesque, _25.1_\n\nShingle, _24.1, 24.2_\n\nSpanish Colonial\n\nSpanish Revival, 35.1, _35.2, 35.3, 35.4_\n\nStick style, _22.1_\n\nStyled Ranch, _50.1_\n\nBaldwin House, _23.1_\n\nBallantine House, _18.1_\n\nballoon-frame system, 2.1, 2.2, _3.1_ , 3.2, _3.3_ , 3.4, 3.5, 4.1, 7.1, 7.2, 20.1, 44.1, 44.2, 48.1, app1.1\n\nin Queen Anne houses\n\nBaltimore, Md., 4.1, 4.2\n\nColonial houses, _13.1, 13.2, 14.1, 14.2_, 15.1, _15.2_\n\nEclectic houses, _27.1_\n\nModern houses, _42.1, 45.1_\n\nRomantic houses\n\nstyled houses since 1935, _50.1, 51.1_\n\nVictorian houses, _21.1, 24.1_\n\nbalustrades\n\nexotic revival\n\nQueen Anne\n\nbalustrades, on porches, balconies, and terraces\n\nBeaux Arts, 31.1, _31.2_\n\nChinese Chippendale, fm2.1, _15.1_\n\nColonial Revival, _27.1_\n\nCraftsman\n\nEarly Classical, _15.1_\n\nexotic revival, _18.1_\n\nFederal, _14.1_\n\nFolk Victorian, _26.1_\n\nFrench Eclectic, _32.1, 32.2_\n\nItalianate, _18.1_\n\nItalian Renaissance, _33.1, 33.2, 33.3_\n\nMonterey Revival\n\nNeoclassical, _28.1, 28.2, 28.3_\n\nNew Traditional, _52.1, 52.2_\n\npictorial key to\n\nQueen Anne\n\nSpanish Colonial, 12.1, _12.2_\n\nSpanish Revival\n\nSwiss Chalet, _18.1, 19.1_, 19.2\n\nbalustrades, roof- or cornice-line\n\nBeaux Arts, 31.1, _31.2_\n\nChinese Chippendale, fm2.1, _15.1_ , 28.1, _28.2, 28.3, 28.4_\n\nColonial Revival, _1.1, 27.1_\n\nEarly Classical, 15.1, _15.2, 15.3_\n\nFederal, 14.1, _14.2, 14.3, 14.4, 14.5, 14.6_\n\nFrench Eclectic, _32.1_\n\nGeorgian, 13.1, _13.2, 13.3_\n\nItalian Renaissance, 33.1, 33.2, _33.3, 33.4_\n\nNeoclassical, 28.1, _28.2, 28.3, 28.4_\n\nNew Traditional, _52.1_\n\npictorial key to, fm2.1, fm2.2\n\nBank of Pennsylvania building\n\nBank of the United States building\n\nBarber, George F., 20.1, _23.1_ , 23.2\n\nbargeboards, _see_ vergeboards\n\nBarnard House, _13.1_\n\nBarnes House, Custer County, _6.1_\n\nBarnes House, Portsmouth, _14.1_\n\nBarnett, Tom, _33.1_\n\nBarney House, _34.1_\n\nBarrow House, _27.1_\n\nBartlett House, _21.1_\n\nbathrooms\n\nBaty House, _35.1_\n\nBauhaus\n\nbay windows\n\nColonial Revival, _27.1, 27.2, 27.3, 27.4, 27.5_\n\nFolk Victorian, _26.1_\n\nFrench Eclectic, _32.1_\n\nGothic Revival, 17.1, _17.2, 17.3_\n\nItalianate, 18.1, _18.2, 18.3, 18.4, 18.5, 18.6, 22.1_\n\nItalian Renaissance, _33.1_\n\nMillennial Mansion, 51.1, _51.2, 52.1_\n\nMinimal Traditional\n\nNeoclassical, 28.1, _28.2_\n\nQueen Anne, 23.1, 23.2, _23.3_ , 23.4, _23.5, 23.6_\n\nRanch, _42.1_\n\nSecond Empire, 21.1, _21.2, 21.3, 21.4_\n\nShed, 46.1, _46.2_\n\nShingle, 24.1, _24.2_\n\nStick style, 22.1, 22.2, _22.3_ , 22.4, _22.5, 22.6, 22.7_\n\nTudor, _29.1_ , 29.2\n\n_see also_ oriels\n\nBeale, Norborne, _15.1_\n\n_Beauties of Modern Architecture, The_ (Lafever), 16.1\n\nBeaux Arts style, _30.1_ , 31.1, _31.2, 31.3, 31.4, 31.5_\n\ndistinguishing from Italian Renaissance and Neoclassical\n\nflat or low-pitched hipped roof, 31.1, _31.2_\n\nmansard roof, 31.1, _31.2_\n\norigins and development of, 1.1, _1.2_ , 31.1\n\nBeck House, _47.1_ , 47.2\n\nBell, Victoria Ballard\n\nBelle Grove house, _15.1_\n\nBelle Isle house, _13.1_\n\nBelle Meade house, _16.1_\n\nBell House, Cincinnati, _23.1_\n\nBell House, Dallas, _28.1_\n\nBell House, Newport, _24.1_\n\nbell towers, 34.1, _34.2, 34.3, 34.4_, 34.5, _35.1, 35.2_\n\nBelluschi, Pietro\n\nBelzberg, Hagy, _48.1_\n\nBenjamin, Asher, _14.1_ , 14.2, 16.1\n\nBennes, John Virginius, _38.1_\n\nBennes House, _38.1_\n\nBennett House, _14.1_\n\nBenson House, _24.1_\n\nBerkowitz House, _52.1_\n\nBetter Homes in America Award\n\nBeverly Hills, Calif.\n\nBevier House, _10.1_\n\nBianchi House, _34.1_\n\nBicentennial House, _52.1_\n\nBicknell & Comstock pattern books\n\nBig Dig House, Boston, _48.1_\n\nBiltmore, 30.1, _30.2_\n\nBingham, Charles, _34.1_\n\nBinker Barns, 53.1, _53.2_\n\nBiscoe House, _19.1_\n\nBixby House, _30.1_\n\nBlackburn House, _12.1_\n\nBlackwell, Marlon, _48.1_\n\nBlevins House, _7.1_\n\nBliss House, Montecito, _35.1_\n\nBliss House, Washington, D.C., _31.1_\n\nblocks and lots, 4.1, _4.2_\n\nBoardman House, Helena, _24.1_\n\nBoardman House, Saugus, _9.1_\n\nBoerder, Larry, _52.1, 52.2, 52.3, 52.4, 52.5_\n\nBogk House, _38.1_\n\nBohlen, Diedrich A., _21.1_\n\nBolduc House, _11.1_\n\nBonneville House, _18.1_\n\nBoody House, _17.1_\n\nBookhout House, _24.1_\n\n_Book of Architecture_ (Gibbs), _13.1_\n\n_Book of Successful Home Plans_ (Pollman)\n\nBorah, William, _nts.1_\n\nBorrego House, _12.1_\n\nBoston, Mass., 4.1, 4.2, 4.3, 4.4, _50.1, 52.1, 52.2, 52.3, app1.1_\n\nColonial houses, _9.1_ , 13.1, 13.2, _13.3_ , 14.1, _14.2_ , 14.3, _14.4_ , 14.5\n\nEclectic houses, _30.1_\n\nEmerald Necklace parks (1878\u20131900)\n\nModern houses, _43.1, 47.1, 48.1_\n\nstyled houses since 1935, _50.1, 52.1, 52.2, 52.3, app1.1_\n\nVictorian houses, _23.1_ , 24.1, 25.1\n\nBottomley, William Lawrence, _27.1_\n\nBoutin House, _11.1_\n\nBowen House, Cambridge, _44.1_\n\nBowen House, Providence, _18.1_\n\nBowers, John M., _1.1_\n\nBowers House, _27.1_\n\nBoyington, William W., _21.1_\n\nBoykin Hall, _15.1_\n\nBrady & Smith, _44.1_\n\n_Brady Bunch_ (TV show)\n\nBraly House, _36.1_\n\nBramblett House, _52.1_\n\nBreakers house, _31.1_\n\nBreese House, _10.1_\n\nBremo house\n\nBrentwood community, Denver\n\nBreuer, Marcel, 37.1, 44.1, _44.2, 44.3_\n\nBrewster-Burke House, _19.1_\n\nBrice House, _13.1_\n\nbrick cladding, 3.1, _3.2_ , 12.1, _12.2_ , 23.1, _23.2, 23.3_, 27.1, 33.1, 36.1, 38.1, 39.1, _40.1_ , 42.1, 46.1, 47.1, _51.1_\n\nAmerican Vernacular, _52.1_ , 53.1, 53.2\n\nBrutalism, 47.1, _47.2_\n\nColonial Revival, _27.1, 27.2_, 27.3, _27.4, 27.5_\n\nContemporary, 45.1, _45.2, 45.3_\n\nFrench Eclectic, _32.1_ , 32.1, _32.2, 32.3, 32.4_\n\nItalian Renaissance, 33.1, _33.2_\n\nMansard, _48.1_ , 49.1, 49.2\n\nNew Traditional, 52.1, 52.2, _52.3, 52.4_\n\nStyled Ranch, 50.1, _50.2_ , 50.3\n\nTudor, _29.1_ , 29.2, _29.3_ , 29.4, _29.5_\n\n21st-Century, 48.1, _48.2_\n\nbrick construction, 3.1, 3.2, _3.3_ , 3.4, _3.5, 3.6_, 3.7, 6.1, 6.2, 15.1, _16.1_ , 17.1, _20.1, 21.1, 28.1_, 29.1, _30.1_ , 31.1, _39.1_ , 45.1\n\nColonial Revival, _27.1, 27.2_\n\nDutch Colonial, _9.1_ , 10.1, 10.2, 10.3, _10.4_\n\nFederal, 14.1, _14.2, 14.3_\n\nGeorgian, 13.1, _13.2, 13.3, 13.4, 13.5_\n\nItalian Renaissance, 33.1, _33.2, 33.3_\n\nPostmedieval English, _8.1_ , 9.1, _9.2_\n\nQueen Anne, 23.1, 23.2, _23.3, 23.4, 23.5, 23.6_\n\nRichardsonian Romanesque, 25.1, _25.2_\n\nStick style, 22.1, _22.2_\n\nBridges, Robert, _48.1_\n\nBrookline, Mass., _23.1, 24.1, 24.2, 24.3_\n\nBrooks House, _17.1_\n\nBrown, J. Merrill, _23.1_\n\nBrown and Von Beren, _27.1_\n\nBrowne House, _9.1_\n\nBrown House, _23.1_\n\nBrumbly House, _16.1_\n\nBrunson House, _26.1_\n\nBrush and Mason, _17.1_\n\nBrutalist style, 47.1, 47.2, _47.3_\n\nBryant House, _16.1_\n\nB. T. Batsford publishers\n\nBuchanan, Russell, _47.1, 48.1_\n\nBuck House, _44.1_\n\nBuckland, William, _13.1_\n\nBuff, Conrad, III, _47.1_\n\nBuffalo, N.Y.\n\nEclectic houses, _27.1, 27.2, 27.3, 28.1, 29.1, 29.2, 31.1, 32.1, 32.2, 33.1_\n\nfolk houses, _7.1_\n\nModern houses, _38.1, 38.2, 39.1_\n\nRomantic houses, _16.1, 18.1_\n\nVictorian houses, _23.1, 24.1_\n\nBuff & Hensman & Associates, _47.1_\n\nBuff House, _47.1_\n\n_Builder's Guide, The_ (Benjamin), 16.1\n\nbuilt-in garage houses, 2.1, _2.2_ , 27.1, _27.2_\n\nBulfinch, Charles, _14.1_ , 14.2\n\nBulger, Clarence C., _33.1_\n\nBulkley House, _17.1_\n\nbungalows, 4.1, 39.1, _39.2_ , 39.3, _52.1_\n\nBureau of Reclamation\n\nBurfitt House, _52.1_\n\nBurgoyne House, _39.1_\n\nBurke House, _35.1_\n\nBurnham and Root, _30.1_\n\nBurrage House, _34.1_\n\nBurroughs House, _14.1_\n\nButler House, Des Moines, _40.1_\n\nButler House, Fort Dodge, _38.1_\n\nbutterfly and slant roof houses, 45.1, _45.2_\n\nByram House, _30.1_\n\nByrne, F. Barry\n\nBywaters House, _36.1_\n\nCabell House, _19.1_\n\nC. A. Comer House, _47.1_\n\nCAD (computer-aided design), 47.1, 48.1, 48.2\n\nCahill House, _23.1_\n\nCalifornia\n\nColonial houses, _12.1, 12.2_\n\nEclectic houses, _30.1, 33.1, 33.2_, 34.1, _34.2, 34.3_, 35.1, _35.2, 35.3, 35.4, 35.5, 35.6_, 35.7, 36.1, _36.2_ , 36.3, 37.1\n\nfolk houses, _8.1_\n\nModern houses, 39.1, _39.2, 39.3, 39.4_, 39.5, _40.1, 41.1, 41.2, 41.3_, 42.1, 42.2, _42.3, 42.4, 42.5, 42.6_, 44.1, _44.2, 44.3, 44.4_, 45.1, _45.2, 45.3, 45.4, 45.5, 46.1, 47.1, 47.2, 47.3, 47.4, 47.5, 47.6, 48.1, 48.2, 48.3_\n\nRomantic houses, _17.1, 18.1, 18.2, 18.3, 18.4_, 18.5, _19.1_\n\nstyled houses since 1935, _49.1, 49.2_, 50.1, _50.2, 50.3, 50.4, 50.5, 50.6, 50.7, 51.1_, 52.1, _52.2, 52.3, 52.4, 52.5, 52.6_, 53.1, _53.2, 53.3_\n\nVictorian houses, 22.1, 22.2, 22.3, _22.4, 22.5, 22.6_, 23.1, _23.2, 23.3, 23.4, 23.5_\n\n_see also_ Los Angeles; San Francisco\n\nCAM (computer-aided manufacturing), 47.1, 48.1\n\nCambridge, Mass.\n\nColonial houses, _13.1_\n\nEclectic houses, _27.1, 27.2, 27.3, 27.4_\n\nModern houses, _44.1, 47.1_\n\nVictorian houses, _21.1, 21.2, 22.1, 23.1, 23.2, 23.3_\n\nCampbell, Colin\n\nCampbell House, _44.1_\n\nCamron House, _18.1_\n\nCannon Mill House, _7.1_\n\nCape Cod houses, 6.1, 7.1, 27.1, 27.2\n\nside-gabled, 41.1, _41.2_\n\n_see also_ Minimal Traditional style\n\nCaperton House, _34.1_\n\ncapitals\n\nGreek Revival\n\nEarly Classical Revival\n\nNeoclassical\n\ncarports, 8.1, 38.1, 41.1, 42.1, 45.1, 45.2\n\nCartwright House, _16.1_\n\nCasa del Herrera, _35.1_\n\ncascading hipped roofs, _51.1_\n\nCase Study \"Triad\" House, _44.1_\n\ncastellations\n\nGothic Revival, _16.1_ , 17.1, 17.2, _17.3, 17.4, 17.5_\n\nTudor, 29.1, _29.2, 29.3_\n\nCastle house, _17.1_\n\nCastro House, _12.1_\n\ncentered-gable houses\n\nColonial Revival, 27.1, _27.2_\n\nFederal, 14.1, _14.2_\n\nGeorgian, 13.1, _13.2_\n\nGothic Revival, 17.1, _17.2_\n\nItalianate, 18.1, _18.2_\n\nSecond Empire, 21.1, _21.2_\n\ncentered wing or gable houses, 21.1, _21.2_\n\nCentral Park (1857\u201373)\n\nCentury of Progress (1933)\n\nChapin House, _14.1_\n\nCharity Hospital, New York City\n\nCharleston, S.C.\n\nColonial houses, 13.1, _13.2, 13.3, 14.1_, 14.2, _14.3, 14.4_, 14.5, 15.1, _15.2_\n\nModern houses, _45.1_ , 48.1\n\nRomantic houses, _16.1_\n\nstyled houses since 1935, _53.1_\n\nChateauesque style, _30.1_ , 30.1, _30.2, 30.3_, 30.4\n\norigins and development of, _1.1_ , 1.2, 30.1\n\nCheatham, Diane\n\nCheatham House, Urban Reserve, _48.1_\n\n\"Checkerboard House\" (Albert House), _47.1_\n\nChicago, Ill., 2.1, 3.1, 4.1, 4.2, _4.3_ , 4.4\n\nEclectic houses in, 26.1, _27.1_ , 28.1, _29.1, 29.2, 30.1_, 31.1\n\nModern houses in, 1.1, 38.1, 38.2, _39.1_ , 40.1, _41.1, 42.1, 42.2, 44.1, 47.1, 48.1, 48.2, 48.3_\n\nRomantic houses, _18.1_\n\nstyled houses since 1935, _50.1, 51.1, 51.2, 52.1, 52.2, 52.3, 52.4_\n\nVictorian houses, 23.1, _23.2, 23.3, 23.4_, 23.5, _25.1_\n\nChicago Columbian Exposition (1893), 4.1, 23.1, 26.1, 28.1, 31.1, 34.1\n\nChicago _Tribune_\n\nchimneys, fm2.1, fm2.2, 1.1, _2.1_ , 3.1, _3.2_ , 3.3, _15.1, 24.1, 25.1, 33.1, 37.1, 39.1, 44.1, 48.1_, 49.1, 52.1\n\nAmerican Vernacular, 53.1, 53.2\n\nChateauesque, _30.1_ , 30.1, _30.2_\n\nContemporary, 45.1, _45.2, 45.3, 45.4, 45.5_\n\ndecorative details of, 3.1, _23.1, 23.2, 23.3_, 23.4, _28.1_ , 29.1, 29.2, _29.3, 29.4_, 35.1, _35.2, 42.1_\n\nDutch Colonial, _9.1_ , 10.1\n\nFederal, 14.1, _14.2, 14.3, 14.4_\n\nof folk houses, _7.1_ , 7.2, _7.3_\n\nFrench Eclectic, _32.1, 32.2, 32.3_\n\nGeorgian, 13.1, _13.2, 13.3, 13.4_\n\nGothic Revival, _17.1_ , 17.2\n\npictorial key to\n\nPostmedieval English, _8.1_ , 9.1, 9.2, _9.3, 9.4_\n\nPrairie, 38.1, _38.2, 38.3, 38.4_\n\npre-railroad, 6.1, _6.2, 6.3_, 6.4, _6.5_\n\nQueen Anne, _23.1, 23.2, 23.3, 23.4, 23.5_\n\nRanch, _42.1, 42.2, 42.3_\n\nShed, 46.1, _46.2_\n\nSpanish Revival, 35.1, _35.2, 35.3, 35.4, 35.5_\n\nStyled Ranch, 50.1, _50.2, 50.3_\n\nTudor, _28.1_ , 29.1, 29.2, _29.3_ , 29.4, _29.5, 29.6, 29.7, 29.8, 29.9, 29.10_\n\nChinese Chippendale railings, fm2.1, _15.1, 15.2_, 28.1, _28.2, 28.3, 28.4_\n\nChris Craft House, _48.1_\n\nChristie House, _45.1_\n\nChurch, Frederic E., _19.1_\n\n\"City Beautiful\" movement, 4.1, _4.2_ , 4.3, 31.1\n\nCivil War, U.S., 7.1, 18.1\n\ncladding, _see_ wall cladding\n\nClark and Loomis, _25.1_\n\nClark Houses, _14.1_\n\nClassical New Traditional houses, _52.1, 52.2_, 52.3\n\nClayton House, _6.1_\n\nClement House, _22.1_\n\nClem House, _39.1_\n\n_Clem Labine's Period Homes_\n\nCleveland, Ohio\n\nEclectic houses, _1.1, 27.1, 27.2, 27.3, 29.1, 29.2, 29.3, 29.4, 29.5_, 31.1, _31.2, 32.1, 32.2, 32.3_\n\nfolk houses, _7.1_\n\nRomantic houses, _1.1, 17.1, 17.2, 18.1_\n\nVictorian houses, _21.1, 23.1, 23.2, 24.1_\n\nCliveden house, _13.1_\n\nClusky, Charles B., _16.1_\n\nCoate, Roland, Sr., _36.1_ , 36.2\n\nCodd, Nicholas, _14.1_\n\nCoeymans House, _10.1_\n\nCohen-Bray House, _22.1_\n\nColeman House, _14.1_\n\nCollier Heights Historic District house, _42.1, 45.1_\n\nCollins House, _22.1_\n\n_Colonial Home Plans_ (Bantam-Hudson)\n\nColonial Houses\n\nDutch Colonial, _9.1_ , 10.1, _10.2, 10.3, 10.4, 10.5_\n\nEarly Classical Revival, _14.1_ , 15.1, _15.2, 15.3, 15.4, 15.5_\n\nFederal, _13.1_ , 14.1, _14.2, 14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 14.9, 14.10_\n\nFrench Colonial, _10.1_ , 11.1, _11.2, 11.3, 11.4_\n\nGeorgian, 12.1, 13.1, _13.2, 13.3, 13.4, 13.5, 13.6, 13.7, 13.8, 13.9_\n\nPostmedieval English, _8.1_ , 9.1, _9.2, 9.3, 9.4_\n\nSpanish Colonial, _11.1_ , 12.1, _12.2, 12.3, 12.4, 12.5_\n\nColonial Revival Ranch houses, 50.1, _50.2_\n\nColonial Revival style, _26.1_ , 27.1, _27.2, 27.3, 27.4, 27.5, 27.6, 27.7, 27.8, 27.9, 27.10, 27.11, 27.12, 27.13_, 27.14\n\nasymmetrical, 27.1, _27.2_\n\nbuilt in garage, 27.1, _27.2_\n\ncentered gable, 27.1, _27.2_\n\ndistinguishing from: Dutch Colonial, _27.1_ ; Georgian and Federal, _27.2_ , 27.3, _27.4_ , 27.5\n\ngambrel roof, 27.1, _27.2_\n\nhipped roof with full-width porch, 27.1, _27.2_\n\nhipped roof without full-width porch, 27.1, _27.2_\n\ninfluence on Queen Anne and Shingle, 27.1, 27.2\n\n\u2013New Traditional, 52.1, _52.2, 52.3, 52.4_\n\none-story, 27.1, _27.2, 27.3_\n\norigins and development of, 1.1, _1.2_ , 27.1, 27.2\n\nsecond-story overhang, 27.1, _27.2_\n\nside-gabled roof, 27.1, _27.2_\n\nthree-story, 27.1, _27.2_\n\nColonnade Row houses, _16.1_\n\nColorado\n\nColonial houses, _12.1_\n\nModern houses, _39.1, 40.1_, 41.1, _45.1_\n\nRomantic houses, _16.1, 17.1_\n\nVictorian houses, _21.1, 23.1, 23.2, 25.1_\n\nColton House, _14.1_\n\nColumbia University\n\ncolumns, fm2.1, _32.1, 38.1_\n\nBeaux Arts, _30.1_ , 31.1, _31.2, 31.3_\n\nClassical orders, 16.1, 16.2, _16.3_\n\nEarly Classical Revival, _14.1_ , 15.1, 15.2, _15.3, 15.4_\n\nEgyptian, _13.1_ , 19.1, _19.2, 19.3_\n\nFrench Colonial, 11.1, _11.2_\n\nGreek Revival, _15.1, 16.1_, 16.2, _16.3_ , 16.4, _16.5, 16.6, 16.7, 16.8, 16.9, 16.10, 16.11, 17.1_\n\nItalian Renaissance, _32.1_ , 33.1, 33.2\n\nNeoclassical, 1.1, _27.1_ , 28.1, _28.2_ , 28.3, _28.4, 28.5, 28.6, 28.7_\n\nNew Traditional, _52.1_ , 52.2, 52.3, _52.4, 52.5, 52.6_\n\nRoman, _15.1_ , 16.1\n\nSpanish Revival, 35.1, _35.2, 35.3, 35.4_\n\n_see also_ capitals; entry porches; pilasters; porches\n\ncombined hipped-and-gabled roofs, 35.1, _35.2_\n\nCommerce Department, U.S., 4.1, 4.2\n\nCommunity Reinvestment Act (1977), 4.1, 4.2\n\n_Complexity and Contradiction in Architecture_ (Venturi), 47.1, 47.2, 52.1\n\ncomplex roof form houses, 53.1, _53.2_\n\nComstock, William, _22.1_ , 23.1\n\nComstock and Trotsche, _22.1_\n\nconcrete\n\nblocks, _3.1_ , 3.2, _33.1_ , 37.1, 38.1, 39.1, 42.1, 44.1, 45.1, _45.2_ , 47.1\n\nas cladding, fm2.1, 47.1, _47.2_ , 48.1, _48.2_\n\ncurbs and sidewalks, _4.1_ , 4.2, _4.3, 4.4_\n\nfoundations, 3.1, _3.2_ , 42.1, 52.1, _52.2_ , 52.3\n\npoured, fm2.1, _3.1_ , 3.2, 20.1, _40.1_ , 44.1, 48.1, app1.1, _app1.2_\n\nscreen blocks, fm2.1, 45.1, _45.2, 45.3, 47.1_\n\ntiles\n\nConnecticut\n\nColonial houses, 9.1, 9.2, _13.1, 13.2, 14.1_\n\nEclectic houses, _27.1, 27.2, 28.1_, 28.2, _29.1, 29.2, 30.1_\n\nfolk houses, _6.1_ , 6.2\n\nModern houses, _44.1, 44.2_\n\nRomantic houses, _16.1, 16.2, 17.1, 17.2, 18.1, 18.2, 18.3, 19.1_\n\nVictorian houses, _21.1, 21.2, 21.3, 22.1, 22.2, 23.1, 23.2, 23.3, 23.4, 23.5, 23.6, 23.7, 23.8, 24.1, 24.2, 24.3, 24.4, 26.1_\n\nConrad House, _25.1_\n\nConstantine House, _16.1_\n\nContemporary style, _44.1_ , 45.1, _45.2, 45.3, 45.4, 45.5, 45.6, 45.7, 45.8, 45.9, 45.10_, 45.11, app1.1\n\nbutterfly and slant roofs, 45.1, _45.2_\n\ncross-gabled roof\n\nflat roof, 45.1, _45.2_\n\nfront-gabled roof, 45.1, _45.2, 45.3_\n\ngabled-roof variations, _45.1_\n\nside-gabled roof, 45.1, _45.2_\n\nContinental Congress, U.S.\n\nCoolmore Plantation, _18.1_\n\nCooperative Building Plan Association\n\nCorcoran Art Gallery\n\ncornices, 3.1, 15.1, 17.1, 24.1, 28.1, _50.1_\n\nBeaux Arts, 31.1, _31.2_\n\nbracketed, 1.1, _1.2_ , 16.1, _17.1_ , 18.1, _18.2, 18.3, 18.4_, 21.1, _21.2, 21.3_, 22.1, _22.2_ , 22.3, _22.4_ , 26.1, 26.2, _33.1, 33.2_, 38.1\n\nColonial Revival, 27.1, _27.2_ , 27.3\n\nFederal, _13.1_ , 14.1, 14.2, _14.3_\n\nFolk Victorian, 26.1, 26.2\n\nGeorgian, _12.1_ , 13.1, _13.2, 13.3, 13.4, 14.1_\n\nGreek Revival, _15.1_ , 16.1, 16.2, _16.3, 16.4_\n\nItalianate, _1.1, 17.1_, 18.1, _18.2, 18.3, 18.4, 18.5_, 18.6, _20.1, 22.1_\n\nItalian Renaissance, 1.1, 33.1, 33.2, _33.3, 33.4_\n\nNeoclassical\n\nNew Traditional, _52.1_ , 52.2, _52.3_\n\npictorial key to, fm2.1, fm3.1\n\nPrairie, 38.1, 38.2, _38.3, 38.4, 38.5_\n\nQueen Anne, 23.1, _23.2_\n\nRanch, 42.1, _42.2_\n\nSecond Empire, _20.1_ , 21.1, 21.2, _21.3, 21.4_\n\nShed, 46.1, _46.2_\n\nStick style, 22.1, _22.2_ , 22.3, _22.4_\n\nCornish House, _21.1_\n\nCostigan House, _16.1_\n\n_Cottage Residences_ (Downing), 15.1, 17.1\n\nCoudren House, _46.1_\n\n_Country Life in America_\n\ncourtyards, 4.1, _37.1, 47.1_, 47.2\n\nContemporary, 45.1, _45.2, 45.3_\n\nInternational, 44.1, _44.2, 44.3, 44.4, 44.5_\n\nRanch, 42.1, 42.2\n\nSpanish Colonial, 12.1, 11.1, _12.2, 12.3, 12.4_\n\nSpanish Revival, 35.1, _35.2, 35.3_\n\nStyled Ranch, 50.1, _50.2_\n\nCox House, _50.1_\n\n_Craftsman Magazine_\n\nCraftsman style, _38.1, 38.2_, 39.1, _39.2, 39.3, 39.4, 39.5, 39.6, 39.7_, 39.8\n\ncomments about, 39.1, 39.2\n\ncross-gabled roof, 39.1, _39.2_\n\nfront-gabled roof, 39.1, _39.2_\n\nhipped roof, 39.1, _39.2_\n\nidentifying features\n\ninfluence on: Mission, 34.1; Tudor, 29.1\n\n\u2013New Traditional, _52.1_ , 52.2\n\noccurrence\n\noriental influences on, 39.1, _39.2_ , 39.3, 39.4\n\norigins and development of, 1.1, _1.2_ , 39.1, 39.2\n\nside-gabled roof, 39.1, _39.2, 39.3_\n\nvariants and details, 39.1, _39.2_\n\nCramer-Kenyon Heritage Home, _22.1_\n\nCranfill House, _39.1_\n\nCreole cottages, _7.1_ , 11.1\n\nCreole French houses, 36.1, _36.2_ , 36.3\n\nCrim House, _18.1_\n\nCrittenden House, _6.1_\n\nCroft House, _16.1_\n\ncross-gabled houses\n\nAmerican Vernacular, 53.1, _53.2_\n\nContemporary\n\nCraftsman, 39.1, _39.2_\n\nMillennium Mansions, _51.1, 51.2_\n\nQueen Anne, 23.1, 23.2, _23.3, 23.4, 23.5, 23.6, 23.7, 23.8_\n\nRanch, 42.1, _42.2_\n\nRichardson Romanesque, 25.1, _25.2_\n\nShingle, 24.1, _24.2, 24.3_\n\nSpanish Revival, 35.1, _35.2_\n\ncross-hipped roof houses, 42.1, _42.2, 42.3_\n\nCrowell, C., _27.1_\n\nCube House, _48.1_\n\nCulbertson, Margaret, _23.1_\n\ncupolas, fm2.1, _28.1, 52.1_\n\nFederal, _14.1, 14.2_\n\nGeorgian, 13.1, _13.2_\n\nGreek Revival, _16.1, 16.2_\n\nItalianate, _17.1_ , 18.1, _18.2, 18.3, 18.4, 18.5, 18.6_, 18.7\n\nOctagon, 20.1, _20.2_\n\nRanch, 42.1, _42.2, 42.3_\n\nSecond Empire, 21.1, _21.2, 21.3_\n\ncurbs and sidewalks, _4.1_ , 4.2\n\nCurlett, William, _33.1_\n\ncurvilinear plans, 4.1, _4.2_ , 4.3, _4.4_\n\nOlmsted and, 4.1, _4.2_ , 4.3\n\nCusato, Marianne, 53.1, _53.2_\n\nCussans, Richard W., _7.1_\n\nCustis-Lee House\n\nCutler House, _23.1_\n\nDallas, Tex.\n\nColonial houses, _8.1_\n\nEclectic houses, _1.1, 27.1, 27.2, 27.3, 27.4, 27.5, 27.6, 27.7, 28.1, 28.2, 28.3, 28.4, 29.1, 29.2, 31.1, 32.1, 32.2, 32.3, 33.1, 33.2, 33.3, 34.1, 34.2, 35.1, 35.2, 35.3, 35.4, 36.1_\n\nfolk houses, _7.1_\n\nModern houses, _1.1, 38.1, 38.2, 38.3, 39.1, 39.2, 39.3, 41.1, 41.2_, 42.1, _42.2, 42.3, 42.4, 44.1, 45.1, 46.1, 47.1_, 47.2, _47.3, 48.1, 48.2_, 48.3, _48.4_\n\nMunger Place project (1976), _4.1, 4.2_, 4.3\n\nstyled houses since 1935, _49.1, 50.1, 50.2, 50.3, 51.1, 52.1, 52.2, 52.3, 52.4, 52.5, 52.6, 52.7, 52.8, 52.9, 52.10, 52.11, 53.1, 53.2, 53.3_\n\nVictorian houses, _23.1, 23.2, 24.1, 24.2_\n\nDaniell House, _52.1_\n\nD'Annemours House, _14.1_\n\nDavid Hertz Architects, _48.1_\n\nDavis, Alexander Jackson, 15.1, 16.1, 17.1, _17.2_ , 17.3, _19.1_\n\nDavis, Richard Drummond, _52.1_\n\nDavis, Walter\n\nDavis House, Chicago, _48.1_\n\nDavis House, Laurens, _23.1_\n\nDay House, _23.1_\n\nDealey House, _42.1_\n\n_Death and Life of Great American Cities, The_ (Jacobs), 4.1\n\nDecatur House, _14.1_\n\nDecker House, _38.1_\n\nDeconstructivist style, 47.1, 47.2, _47.3_\n\nd\u00e9coupage houses, 48.1, _48.2_\n\ndeed restrictions, 4.1, _4.2_\n\nDefreese House, _39.1_\n\nDeLamar House, _31.1_\n\nDelano and Aldrich, _27.1_\n\nDelaware\n\nColonial houses, 13.1, 14.1, _14.2_\n\nfolk houses, 6.1, _7.1_\n\nRomantic houses, _17.1_\n\nDennis and Farwell, _30.1_\n\ndentils, fm2.1, _1.1_ , 1.2, 15.1, 28.1, 31.1, _32.1, 52.1_\n\nColonial Revival, 27.1, _27.2_\n\nFederal, _13.1_ , 14.1, 14.2, _14.3, 14.4_\n\nGeorgian, _12.1_ , 13.1, _13.2, 14.1_\n\nQueen Anne, 23.1, _23.2_\n\nStyled Ranch, 50.1, _50.2_\n\nDeshler House, _13.1_\n\nDieterich House, _35.1_\n\nDilbeck, Charles S., _32.1, 42.1, 42.2, 49.1_\n\nDilbeck Home and Studio, _42.1_\n\nDilbeck House, Dallas, _49.1_\n\nDixon, Dale, _53.1_\n\nDixon, James, _17.1_\n\nDixon, Thomas, _17.1_\n\nDodd House, _23.1_\n\nDoelger, Henry, _40.1_\n\ndogtrot houses, _6.1, 6.2_\n\nDolliole House, _11.1_\n\n_Domestic Architecture of England During the Tudor Period, The_ (Garner), 29.1\n\ndoors and doorways, _3.1_ , 3.2, 3.3\n\narched, _see_ arched doorways\n\nChateauesque, _30.1_\n\nDutch Colonial, 10.1, _10.2_\n\nFederal, _14.1_\n\nFrench Colonial, 11.1, 11.2\n\nGeorgian Colonial, _13.1_\n\nGothic Revival, 17.1, _17.2_\n\nGreek Revival, 16.1, _16.2_\n\nItalianate, 18.1, _18.2_\n\nMillennium Mansion, 50.1, 51.1\n\nNeoclassic\n\npictorial key to\n\nPrairie, _38.1_\n\nSpanish Colonial, _12.1_\n\nSpanish Revival, _35.1_\n\nTudor, 29.1, _29.2_\n\ndormers, _3.1_ , 3.2, _3.3, 10.1_, 11.1, 15.1, _16.1, 18.1, 22.1_, 31.1, 45.1, _47.1_\n\nAmerican Vernacular, 53.1, _53.2_\n\nChateauesque, _30.1_ , 30.1, _30.2, 30.3, 30.4_\n\nColonial Revival, 27.1, 27.2, _27.3, 27.4, 27.5, 27.6, 27.7, 27.8, 27.9, 27.10, 27.11_\n\nCraftsman, 39.1, 39.2, _39.3, 39.4, 39.5_\n\ndistinguishing from gables, _3.1_\n\neyebrow, _3.1, 24.1, 24.2_, 25.1, _25.2, 25.3_, 29.1, 47.1\n\nFederal, 14.1, _14.2, 16.1_\n\nFrench Eclectic, _32.1_ , 32.1, _32.2, 32.3, 32.4, 32.5, 32.6_\n\nGeorgian, 13.1, _13.2, 13.3, 13.4, 13.5_\n\nGothic Revival, 17.1, 17.2, _17.3_\n\nMansard, _48.1_ , 49.1, 49.2, _49.3, 49.4, 49.5_\n\nMillennium Mansion, _50.1_ , 51.1, _51.2, 51.3_\n\nMinimal Traditional, _41.1_ , 41.1, 41.2\n\nMission, _34.1_ , 34.1, _34.2_\n\nNeoclassical, 28.1, _28.2_\n\nNew Traditional, 52.1, _52.2, 52.3_, 52.4, 52.5, _52.6, 52.7, 52.8, 52.9, 52.10_\n\npictorial key to\n\nPostmedieval English, 9.1, _9.2_\n\nPrairie, 38.1, _38.2, 38.3_\n\nQueen Anne, 23.1, 23.2, _23.3, 23.4, 23.5_\n\nRanch, _42.1_ , 42.1, _42.2_\n\nRichardsonian Romanesque, 25.1, _25.2, 25.3, 25.4_\n\nroof, _3.1_ , 11.1, _13.1, 13.2, 14.1_, 15.1, _16.1, 16.2, 18.1, 20.1_, 21.1, _21.2, 21.3, 21.4, 21.5, 24.1, 27.1, 27.2, 27.3, 28.1_, 29.1, 29.2, 29.3, _30.1_ , 31.1, 32.1, _32.2, 32.3, 34.1_, 34.1, _34.2, 38.1_, 39.1, 39.2, _39.3, 39.4, 39.5, 48.1_, 49.1, 49.2, _49.3, 50.1_, 51.1, 52.1, _52.2, 52.3_, 52.4, _52.5, 52.6, 52.7_\n\nSecond Empire, _20.1_ , 21.1, 21.2, _21.3, 21.4, 21.5, 21.6, 21.7_\n\nShingle, 24.1, _24.2, 24.3, 24.4_\n\nStyled Ranch, 50.1, _50.2_ , 50.3\n\nTudor, 29.1, 29.2, _29.3_ , 29.4, _29.5, 29.6_\n\nwall, fm2.1, fm2.2, _3.1_ , 25.1, _25.2_ , 25.3, _27.1, 28.1_, 29.1, 29.2, _29.3_ , 30.1, _30.2, 30.3_, 32.1, _32.2, 32.3, 32.4, 32.5, 38.1, 38.2, 49.1, 50.1_, 51.1, _51.2_ , 52.1, _52.2, 52.3_, 53.1\n\nDowning, Andrew Jackson, 4.1, 4.2, 15.1, 17.1, 17.2, 18.1, 19.1, 22.1\n\nDrayton Hall, _13.1_\n\nDreyfus House, _35.1_\n\ndrip-molds, fm2.1, 17.1, _17.2, 17.3, 17.4, 17.5, 17.6_\n\nDrummond, William\n\nDSGN\n\nDuany, Andr\u00e9s, 48.1, 52.1\n\nDuany Plater-Zyberk & Company (DPZ), 48.1, 52.1, 53.1\n\nDubois, Charles, _45.1_\n\nDumbacher House, _48.1_\n\nDuncker, Paul, _53.1_\n\nDuncker, Peggy, _53.1_\n\nDuncker House, _53.1_\n\nDunlap House, _23.1_\n\nDunn, James F., _31.1_\n\nDurie House, _10.1_\n\nDutch Colonial style, _9.1_ , 10.1, _10.2, 10.3, 10.4, 10.5_, 10.6\n\ndistinguishing from Colonial Revival, _27.1_\n\ndistribution of, 10.1, 10.2\n\nmaps, _10.1_\n\n_DWELL_ , 44.1, 48.1\n\nDwight House, _13.1_\n\nDymaxion House, app1.1, _app1.2_\nEames and Young, _25.1, 32.1_\n\nEames House\n\nEarl Jay Kirsch Architects, _47.1_\n\nEarly Classical Revival style, _15.1_ , 15.1, _15.2, 15.3, 15.4, 15.5_, 15.6\n\ndistinguishing from: Greek Revival, 15.1, 28.1, _28.2_ ; Neoclassical, 28.3, _28.4_\n\ngable front and wings, 15.1, _15.2_\n\none-story, 15.1, _15.2_\n\norigins and development of, 1.1, _1.2_ , 15.1\n\ntwo-story, 15.1, _15.2_\n\nEastabrook House, _38.1_\n\nEasterby House, _26.1_\n\nEastlake detailing, _18.1, 22.1_, 22.2, 23.1, _23.2_\n\n_see also_ spindlework\n\neaves, _1.1_ , 19.1, 20.1, _23.1_ , 28.1, _36.1, 44.1_, 47.1, 50.1, 52.1, 53.1\n\nbracketed, fm2.1, _11.1, 16.1, 18.1_, 18.1, 18.2, 20.1, _21.1_ , 21.2, 22.1, _26.1, 26.2, 26.3, 26.4, 33.1_, 33.2, 33.3, _33.4, 34.1_, 49.1, 53.1\n\nColonial Revival, 27.1, 27.2, _27.3, 27.4, 27.5, 27.6_\n\nContemporary, _45.1_ , 45.2, 45.3, _45.4_\n\nCraftsman, _39.1_ , 39.2, 39.3, 39.4\n\nDutch Colonial, _10.1_ , 10.2, 10.3, _10.4_ , 10.5, _10.6, 10.6_\n\nflared, _10.1, 10.2_\n\nFolk Victorian, _26.1, 26.2, 26.3, 26.4_\n\nFrench Colonial, _11.1_ , 11.2, 11.3, _11.4_\n\nFrench eclectic, _32.1_ , 32.2, _32.3, 32.4_\n\nGothic Revival, 17.1, _17.2, 17.3_\n\nGreek Revival, 16.1, _16.2_\n\nItalianate, _18.1_ , 18.2, 18.3\n\nItalian Renaissance, _33.1_ , 33.2, 33.3, _33.4, 33.5_\n\nMinimal Traditional, _41.1_ , 41.2\n\nMission, _34.1_ , 34.2, _34.3_\n\noverhanging, fm2.1, 1.1, _1.2, 10.1, 11.1_, 11.2, 12.1, _12.2, 16.1, 17.1_, 18.1, 19.1, 20.1, 21.1, _22.1_ , 22.2, _22.3_ , 27.1, _27.2, 27.3, 33.1_, 33.2, 33.3, _33.4, 34.1_, 34.2, _37.1_ , 38.1, 38.2, _38.3, 39.1_, 39.2, 39.2, _42.1, 42.2, 45.1_, 45.2, 45.3, _45.4_\n\npictorial key to, fm2.1, fm2.2\n\nPrairie, _37.1_ , 38.1, 38.2, _38.3, 38.4_\n\nRanch, 42.1, _42.2, 42.3_\n\nSecond Empire, _21.1_ , 21.2, 21.2\n\nShingle, _24.1_ , 24.2\n\nSpanish Colonial, 12.1, _12.2_\n\nSpanish Revival, _35.1_ , 35.2\n\nStick style, _22.1_ , 22.2, _22.3, 22.4, 22.5_\n\nTudor, 29.1, _29.2, 29.3, 29.4, 29.5, 29.6, 29.7_\n\nEclectic styles\n\nBeaux Arts, _31.1_ , 31.2, _31.3, 31.4, 31.5, 31.6_\n\nChateauesque, _30.1_ , 30.2, _30.3, 30.4, 30.5_, 30.6\n\nColonial Revival, _26.1_ , 27.1, _27.2, 27.3, 27.4, 27.5, 27.6, 27.7, 27.8, 27.9, 27.10, 27.11, 27.12, 27.13_, 27.14\n\nFrench Eclectic, _32.1_ , 32.2, _32.3, 32.4, 32.5, 32.6, 32.7_, 32.8\n\nItalian Renaissance, _33.1_ , 33.2, _33.3, 33.4, 33.5, 33.6, 33.7, 33.8_, 33.9\n\nMission, _34.1_ , 34.2, _34.3, 34.4, 34.5, 34.6_, 34.7\n\nmixtures of, 1.1, _1.2_\n\nMonterey Revival, _36.1_ , 36.2, _36.3, 36.4_, 36.5\n\nNeoclassical, _28.1_ , 28.2, _28.3, 28.4, 28.5, 28.6, 28.7_, 28.8\n\nPueblo Revival, _37.1_ , 37.2, _37.3, 37.4_, 37.5\n\nSpanish Revival, _35.1_ , 35.2, _35.3, 35.4, 35.5, 35.6, 35.7, 35.8, 35.9, 35.10_, 35.11\n\nTudor, _29.1_ , 29.2, _29.3, 29.4, 29.5, 29.6, 29.7, 29.8, 29.9, 29.10, 29.11, 29.12, 29.13, 29.14_, 29.15\n\n\u00c9cole des Beaux-Arts, Paris, 25.1, 30.1, 31.1\n\nEdwin C. Smith House, _42.1_\n\nEgyptian Revival style, 19.1, _19.2, 19.3_\n\nEichberg, A., _25.1_\n\nEichler, Joseph, 45.1, _45.2, 45.3_\n\nEidlitz, Cyrus L. W., _24.1_\n\nEimar House, _26.1_\n\nEisenhower Birthplace, _17.1_\n\nEisenman, Peter\n\nelectric streetcar suburbs, 4.1, _4.2_\n\nelectrification of houses\n\nEl Encanto Historic District house, _37.1_\n\nelevations, 2.1, 2.2, _2.3_ , 2.4\n\nElfreth's Alley houses, _13.1_\n\nEl Hogar house, _35.1_\n\nElizabeth I, queen of England\n\nElliott House, _26.1_\n\nEllis, Harvey, _23.1_\n\nEllis House, _26.1_\n\nElmslie, George G.\n\nElwood, Craig, _44.1_\n\nEly House, _27.1_\n\nElzner House, _26.1_\n\nEmerald Necklace parks (1878\u20131900)\n\nEmerson, William Ralph\n\nEnergy Department, U.S.\n\nenergy efficiency, _see_ green construction\n\nEnergy Star program\n\nentablatures, fm3.1, 1.1, 1.2, 13.1, _13.2, 14.1_, 15.1, 16.1, _16.2, 16.3, 27.1, 52.1, 52.2_, 52.3, 52.4, _52.5_\n\nentrances, 27.1, _27.2, 28.1, 29.1, 32.1_\n\nentry porches (porticos), 1.1, 1.2, 7.1, _13.1, 18.1, 19.1, 20.1, 23.1, 24.1, 34.1, 39.1_, 42.1, _50.1_\n\nBeaux Arts, 31.1, _31.2, 31.3_\n\nColonial Revival, 27.1, _27.2, 27.3, 27.4, 27.5, 27.6, 27.7_\n\nEarly Classical Revival, _15.1_ , 15.1, 15.2, _15.3, 15.4, 15.5, 15.6_, 15.7\n\nFederal, _14.1_ , 14.1, 14.2, _14.3_ , 14.4, _14.5, 14.6, 14.7_\n\nfull-height, fm2.1, 1.1, 15.1, _16.1_ , 16.2, _16.3, 16.4, 26.1, 28.1_, 28.2, 28.3, _28.4, 28.5_, 28.6, _52.1_\n\nGothic Revival, _17.1_ , 17.2, _17.3_\n\nGreek Revival, _16.1_ , 16.2, _16.3, 16.4_\n\nItalian Renaissance, 33.1, _33.2, 33.3_\n\nNeoclassical, _28.1_ , 28.2, 28.3, _28.4, 28.5, 28.6, 28.7_, 28.8\n\nNew Traditional, _52.1_ , 52.2\n\nTudor, _29.1_ , 29.2, _29.3_ , 29.4, _29.5, 29.6_\n\nEnvironmental Protection Agency (EPA)\n\nEpiscopal Rectory, _17.1_\n\nEsherick, Joseph\n\nEstabrook House, _20.1_\n\nEstes\/Twombly, _53.1_\n\nEstudillo House, _12.1_\n\nEvans House, _18.1_\n\nExecutive Mansion, Milledgeville, Ga., _16.1_\n\nexotic revival styles, _19.1_ , 19.2, _19.3, 19.4_, 19.5\n\norigins and development of, _1.1_ , 1.2, 19.1\n\nEyre, Wilson, _17.1, 23.1_, 24.1\n\nfacades, _2.1, 2.2_\n\nFahnestock House, _31.1_\n\nFailing House, _21.1_\n\nFallingwater house, 47.1, 47.2\n\nFallon House, _18.1_\n\nfalse thatched-roof houses, 29.1, _29.2_\n\nFaneuil Hall Marketplace\n\nfanlights, fm2.1, _52.1_\n\nColonial Revival, _26.1_ , 27.1, 27.2, _27.3, 27.4, 27.5, 27.6_\n\nEarly Classical Revival, _15.1_ , 15.1, 15.2, _15.3_\n\nFederal, 1.1, _14.1_ , 14.2, 14.3, _14.4, 14.5, 14.6_\n\nFrench Colonial, 11.1, _11.2_\n\nGeorgian, _13.1, 13.2, 13.3_\n\nNeoclassical, _28.1, 28.2_\n\nFarnsworth House, 44.1, _44.2_\n\nFarquhar, Robert D., _33.1_\n\nFatio, Maurice\n\nFederal-Aid Highway Act (1956), 4.1, 4.2\n\nFederal House, _52.1_\n\nFederal Housing Administration (FHA), 4.1, 4.2, 4.3, 4.4, 4.5, 4.6, 7.1, 37.1, 41.1, 41.2, 42.1, 42.2, 45.1, app1.1\n\n-guided subdivisions\n\nFederal National Mortgage Association (\"Fannie Mae\"), 4.1, 4.2, 41.1\n\nFederal style, 14.1, 14.2, _14.3, 14.4, 14.5, 14.6, 14.7, 14.8, 14.9, 14.10, 14.11_, 14.12\n\ncentered gable, 14.1, _14.2_\n\ndistinguishing from: Colonial Revival, _27.1_ , 27.2, _27.3_ , 27.4; Georgian, 14.1, 14.2, _14.3_ ; Neoclassical, 28.1\n\nhipped roof, 14.1, _14.2, 14.3_\n\nmap, _14.1_\n\norigins and development of, 1.1, _1.2_\n\nside-gabled roof, 14.1, _14.2_\n\ntown house, 14.1, _14.2_\n\nFeigal, John, _21.1_\n\nFenyes Mansion, _33.1_\n\nFerguson House, _31.1_\n\nField House, Dallas, _53.1_\n\nField House, Louisville, _18.1_\n\nFinkelstein Residence, _48.1_\n\nFisher, Howard T., _44.1_\n\nFitzpatrick House, _44.1_\n\nFlagler, Henry\n\nflat or low-pitched hipped roof, 31.1, _31.2_\n\nflat-roof houses, 33.1, _33.2_ , 45.1, _45.2_\n\nwith parapet, 12.1, _12.2, 12.3_\n\nFlorida\n\nColonial houses, 11.1, _13.1_\n\nEclectic houses, 35.1, _35.2, 35.3_, 35.4\n\nfolk houses, _5.1, 7.1, 7.2, 8.1, 8.2_\n\nModern houses, _39.1_ , 42.1, _42.2, 42.3, 42.4, 44.1, 44.2, 45.1, 47.1, 47.2_\n\nstyled houses since 1935, _50.1, 51.1, 52.1, 52.2, 52.3, 52.4, 53.1, 53.2, 53.3_\n\nVictorian houses, _26.1_\n\nfolk houses\n\nmanufactured, _8.1_ , 8.2, _8.3, 8.4_, app1.1, _app1.2_\n\nnational, _7.1_ , 7.2, _7.3, 7.4, 7.5, 7.6, 7.7, 7.8_\n\nNative American, _5.1_ , 5.2, _5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10_\n\npre-railroad, _6.1_ , 6.2, _6.3, 6.4, 6.5, 6.6, 6.7, 6.8, 6.9, 6.10_\n\nFolk Victorian style, _26.1_ , 26.2, _26.3, 26.4, 26.5, 26.6, 26.6_\n\ndistinguishing from Queen Anne\n\nfront-gabled roof, 26.1, _26.2_\n\ngable front and wing, 26.1, _26.2_\n\norigins and development of\n\npyramidal, 26.1, _26.2_\n\nside-gabled roof, 26.1, _26.2, 26.3_\n\nspindlework on, _26.1_ , 26.2, 26.3, _26.4, 26.5_\n\nFolly (house), _15.1_\n\nFont-Juncadella Building, _11.1_\n\nFooshee & Cheek, _35.1_\n\nFord, O'Neil, 53.1, _53.2_\n\nFord Foundation\n\nform, architectural, _2.1_ , 2.2, _2.3, 2.4, 2.5, 2.6, 2.7, 2.8, 2.9, 3.1_\n\nelevations, 2.1, 2.2, _2.3_\n\nfamilies of shapes, 2.1, _2.2_\n\nground plans, 2.1, _2.2_\n\nshape innovations, 2.1, _2.2, 2.3_\n\nForster, Frank Joseph\n\nFortier House, _11.1_\n\nFoss and Company, _40.1_\n\nFoster House, _27.1_\n\nfoundations, 3.1, _3.2_ , 42.1, 52.1, _52.2_ , 52.3\n\npictorial key to\n\nFountain Oaks house, _35.1_\n\nfour-square houses, fm2.1, 2.1, _2.2_ , 4.1, 7.1, 27.1, _27.2, 33.1_, 34.1, 38.1, _38.2, 38.3, 39.1, 52.1, 52.2_\n\nFowler, Orson S.\n\nFox, James, _47.1, 47.2_\n\nFrank Gehry & Associates, _47.1_\n\nFredonia, N.Y., survey\n\nFrench Colonial style, _11.1_ , 11.2, _11.3, 11.4, 11.5_, 11.6\n\nmaps, _11.1_\n\norigins and development of\n\nFrench Eclectic style, 32.1, _32.2, 32.3, 32.4, 32.5, 32.6_, 32.7\n\nasymmetrical, 32.1, _32.2_\n\ncomments about\n\nidentifying features\n\noccurrence\n\norigins and development of, 1.1, _1.2_ , 32.1, 32.2\n\nsymmetrical, 32.1, _32.2_\n\ntowered, 32.1, _32.2_\n\nvariants and details, 32.1, _32.2_\n\nFrench New Traditional style, _52.1_ , 52.2\n\nFrench North America, 11.1, _11.2_\n\nFrench Ranch style, _50.1_ , 50.2\n\n\"French Village,\" Hollywood\n\nFriedman House, _47.1_\n\nFritch House, _42.1_\n\nfront entries, 42.1, _42.2_\n\nfront-facing gable with wing houses Folk Houses, National, 7.1, Tudor, 29.1, _29.2_\n\nfront-gabled roof houses\n\nAmerican Vernacular, 53.1, _53.2_\n\nContemporary, 45.1, _45.2, 45.3_\n\nCraftsman, 39.1, _39.2_\n\nFolk Houses, National\n\nFolk Victorian, 26.1, _26.2_\n\nGothic Revival, 17.1, _17.2_\n\nGreek Revival, 16.1, _16.2_\n\nItalianate, 18.1, _18.2_\n\nNeoclassical, 28.1, _28.2_\n\nQueen Anne, 23.1, 23.2\n\nShingle, 24.1, _24.2_\n\nTudor, 29.1, _29.2_\n\nFrost House, Cherry Creek, _18.1_\n\nFrost House, St. Louis, _18.1_\n\nFuller, Buckminster, app1.1, app1.2, _app1.3_\n\nfull-facade porch houses, 16.1, _16.2_ , 28.1, _28.2_\n\nfull-height entry porch house, 16.1, _16.2_ , 28.1, _28.2_\n\nfull-height entry porch with lower full-width porch houses, 28.1, _28.2_\n\ngable-and-wing houses, Minimal Traditional, _41.1_ , 41.2, _41.3_\n\ngabled roof houses, 22.1, _22.2, 22.3_, 38.1, _38.2, 46.1_\n\ngable-front-and-wing houses, prf.1, _2.1_ , 6.1, 7.1, 7.2, _7.3_ , 7.4\n\nEarly Classical Revival, _15.1_ , 15.1, _15.2_\n\nFolk Victorian, _26.1_ , 26.2, _26.3_\n\nGreek Revival, 16.1, _16.2_ , 16.3\n\nItalianate, _18.1, 18.2_\n\n\"Gable in a Square\" house, _52.1_\n\ngables, _23.1_ , 29.1, _29.2_\n\nGaillard House, _11.1_\n\nGallier, James, _16.1_\n\nGamble House, _39.1_\n\ngambrel roof houses, 13.1, _13.2_ , 24.1, _24.2_ , 27.1, _27.2_\n\ngarages, 2.1, _2.2_ , 27.1, _27.2_ , 42.1, _42.2_ , 52.1, 52.2\n\nGarden, Hugh Mackie Gorden, _38.1_\n\ngardens and landscaping, 4.1, 4.2, 4.3, 4.4, _4.5, 4.6_, 4.7, 47.1\n\nAmerican Vernacular, _53.1_\n\nChateauesque\n\nCity Beautiful, 4.1, _4.2_ , 4.3\n\nContemporary, 45.1, _45.2_ , 45.3, _45.4, 45.5, 45.6_, 45.7\n\nFederal, _14.1_\n\nGolf course developments, 4.1, _4.2_\n\nInternational, _44.1_\n\nJapanese-style, 45.1, 50.1\n\nNew Traditional, 52.1, _52.2, 52.3, 52.4_\n\nOlmstedian, 4.1, _4.2_ , 4.3\n\nparks and green space\n\nPrairie, _38.1_\n\nShed, 46.1, _46.2_\n\nSpanish Revival, 35.1, _35.2, 35.3_\n\n21st-Century Modern, _48.1, 48.2, 48.3_, 48.4\n\nGardette House, _11.1_\n\nGarner, Thomas\n\nGarnsey, G. A., _23.1_\n\nGaskill House, _23.1_\n\nGehry, Frank, 47.1, _47.2_\n\nGehry House, 47.1, _47.2_\n\nGentry House, _6.1_\n\nGentry Row houses, _13.1_\n\ngeodesic domes, app1.1, _app1.2_\n\nGeorge Allen House, _18.1_\n\nGeorgia\n\nColonial houses, _15.1_\n\nEclectic houses, _27.1, 27.2, 29.1_\n\nfolk houses, _7.1, 7.2_\n\nModern houses, _41.1, 42.1, 45.1, 45.2_\n\nRomantic houses, _16.1, 16.2, 16.3, 18.1_\n\nstyled houses since 1935, _49.1, 50.1, 50.2, 52.1, 52.2, 52.3_\n\nVictorian houses, _23.1_\n\nGeorgian style, _13.1_ , 13.2, _13.3, 13.4, 13.5_, 13.6, _13.7, 13.8, 13.9, 13.10, 13.11_\n\ncentered gable, 13.1, _13.2_\n\ndistinguishing from: Colonial Revival, _27.1_ , 27.2, _27.3_ , 27.4; Federal, 14.1, 14.2, _14.3_ ; Neoclassical, 28.1\n\ngambrel roof, 13.1, _13.2_\n\nhipped roof, 13.1, _13.2_\n\nmap, _13.1_\n\norigins and development of, 1.1, _1.2_ , 13.1, 14.1, 14.2\n\nregional variations, 13.1, 13.2, _13.3_\n\nside-gabled roof, 13.1, _13.2, 13.3_\n\ntown house, 13.1, _13.2_\n\nGibbs, James, _13.1_ , 13.2\n\nGibbs House, _39.1_\n\nGI Bill (1944), 4.1, 41.1\n\nGilbert, C. P. H., _31.1_\n\nGiles, Alfred, _25.1_\n\nGilliland House, Dallas, _32.1_\n\nGilliland House, Highland Park, _52.1_\n\nGillis House, _17.1_\n\nGindroz, Raymond L.\n\ngingerbread, _see_ spindlework\n\nGlass House, 44.1, _44.2_ , 47.1\n\nGlencrest house, _16.1_\n\nGlen Ellen house\n\nGlen Harder House, _47.1_\n\nGlessner House, _25.1_\n\nGoff, Bruce, _47.1_ , 47.2\n\nGoldberg House, _30.1_\n\ngolf courses, 4.1, _4.2_\n\n_Good Housekeeping_\n\nGoodhue, Bertram Grosvenor\n\nGoodman, Charles M.\n\nGoodwin House, Dallas, _32.1_\n\nGoodwin House, Raleigh, _28.1_\n\nGordon House, _28.1_\n\nGothic Revival style, _17.1_ , 17.2, _17.3, 17.4, 17.5, 17.6, 17.7, 17.8, 17.9, 17.10_, 17.11\n\nasymmetrical, 17.1, _17.2_\n\ncastellated or parapeted, 17.1, _17.2_\n\ncentered gable, 17.1, _17.2_\n\ncomments about, 17.1, 17.2\n\nfront-gabled roof, 17.1, _17.2_\n\nidentifying features\n\noccurrence\n\norigins and development of, 1.1, _1.2_ , 1.3, 17.1, 17.2\n\npaired gables, 17.1, _17.2_\n\npolychromed, 17.1, _17.2_\n\nvariants and details, 17.1, _17.2, 17.3_\n\nGovernor's Mansion, Former, Bismarck, _22.1_\n\nGovernor's Mansion, Frankfort, _31.1_\n\nGovernor's Mansion, Montgomery, _28.1_\n\nGovernor Tilden House, _17.1_\n\nGraciolo, Elizabeth, _52.1_\n\nGraff House, _32.1_\n\nGraf House, _47.1_ , 47.2\n\nGraves, Michael\n\nGraves House, _26.1_\n\nGray House, _23.1_\n\nGreat Depression, 4.1, 26.1, 41.1, app1.1\n\nGreek Revival style, _16.1_ , 16.2, _16.3, 16.4, 16.5, 16.6, 16.7, 16.8, 16.9, 16.10, 16.11_, 16.12\n\ncomments about, 16.1, 16.2\n\ndistinguishing from Early Classical, 15.1, 28.1, _28.2_\n\nentry porch less than full height, or absent, 16.1, _16.2_\n\nfront-gabled roof, 16.1, _16.2_\n\nfull-facade porch, 16.1, _16.2_\n\nfull-height entry porch, 16.1, _16.2_\n\ngable front and wing, 16.1, _16.2_\n\nidentifying features\n\nmaps, _16.1_\n\noccurrence, 16.1, _16.2_\n\norigins and development of, 1.1, _1.2_ , 16.1, 16.2, 16.3\n\ntown house, 16.1, _16.2_\n\nvariants and details, 16.1, _16.2, 16.3_\n\nGreek War of Independence\n\ngreen construction, app1.1, app1.2, _app1.3_ , app1.4\n\nGreene, Charles Sumner, 39.1, 39.2\n\nGreene, Henry Mather, 39.1, 39.2\n\nGreene, LaRoche & Dahl, _32.1_\n\nGreene and Greene, 37.1, 39.1, _39.2_ , 45.1\n\nGreen House, Dallas (1915), _34.1_\n\nGreen House, Dallas (1926), _35.1_\n\nGreen House, Woodbury, _21.1_\n\ngreen spaces, _4.1_ , 4.2\n\nGregorie House, _26.1_\n\nGregory Farmhouse\n\nGriffin, Marion Mahony\n\nGriffin, Walter Burley\n\nGriswold House, _22.1_\n\nGropius, Walter, 37.1, 44.1, 44.2, _44.3_ , app1.1\n\nGropius House, _44.1_\n\nground plans\n\nand house shape, 2.1, _2.2, 2.3_\n\nfor neighborhoods, 4.1, _4.2, 4.3, 4.4, 4.5_\n\ngrowth rate, neighborhood\n\nGwathmey, Charles,\n\nHaas House, _23.1_\n\nHale House, Los Angeles, _44.1_\n\nHale House, Meriden, _24.1_\n\nhalf-timbering, _3.1_ , 3.2, 11.1, 11.2, 23.1, 29.1, _29.2_ , 52.1\n\nHaliburton Houses, _28.1_\n\nHall, J. D., _21.1_\n\nhall-and-parlor houses, _2.1_ , 6.1, _6.2_ , 6.3, _6.4, 6.5_, 6.6, 7.1, 7.2, 7.3, _7.4, 7.5, 7.6, 7.7_, 8.1, _8.2, 9.1_, 26.1\n\nHalpin House, _33.1_\n\nHalsey House, _14.1_\n\nHamilton, E. G., _45.1_\n\nHamilton House, _21.1_\n\nHammond House, _13.1_\n\nHancock Park Historic District house (Spanish Revival), _35.1_\n\nHancock Park house (Italian Renaissance), _33.1_\n\nHannaford, S., _23.1_\n\nHarding, Lyman, _15.1_\n\nHarding House, _38.1_\n\nHaring House, _10.1_\n\nHarmon House, _16.1_\n\nHarris, Frederick Thomas, _34.1_\n\nHarris House, _28.1_\n\nHarrison House, Dallas (1910), _38.1_\n\nHarrison House, Dallas (1939), _28.1_\n\nHarris-Savage House, _34.1_\n\nHarrover House, _27.1_\n\nHartford, Conn.\n\nColonial houses, _13.1_\n\nEclectic houses, _27.1, 29.1, 29.2_\n\nRomantic houses, _17.1, 18.1, 18.2, 19.1_\n\nVictorian houses, _22.1, 22.2, 23.1, 23.2, 23.3, 23.4, 23.5_\n\nHarvard's Graduate School of Design\n\nHasty, John A., _27.1_\n\nHaumont House, _6.1_\n\nHaviland, John\n\nHawkins, Edward B., 45.1, _45.2_\n\nHaxall House, _18.1_\n\nheating innovations, 2.1, _2.2_\n\nHeck House, _21.1_\n\nHedgerow Houses\n\nHejduk, John\n\nHenry Trost House, _38.1_\n\nHentz, Adler, and Schutze, _27.1_\n\nHentz and Reid, _27.1_\n\nHershfelt House, _27.1_\n\nHeurich House, _25.1_\n\nHigham, Robert, _16.1_\n\nHighway House, _48.1_\n\nHill, C. D., _33.1_\n\nHill and Hollow house, _18.1_\n\nHill House, _33.1_\n\nHills House, _31.1_\n\nHinds House, _22.1_\n\nhipped roof houses, 41.1, _41.2_\n\nAmerican Vernacular, 53.1, _53.2_\n\nasymmetrical, 38.1, _38.2_\n\nChateauesque, 30.1, _30.2_\n\nCraftsman, 39.1, _39.2_\n\nwith cross gables, 24.1, _24.2_ , 25.1, _25.2, 51.1_\n\nwith full-width porch, 27.1, _27.2_\n\nwithout full-width porch, 27.1, _27.2_\n\nGeorgian, 13.1, _13.2_\n\nwith lower cross hips, _51.1_\n\nMinimal Traditional, 41.1, _41.2_\n\nwith projecting wings, 33.1, _33.2_\n\nRanch, 42.1, _42.2_\n\nsimple, 18.1, _18.2, 18.3_, 33.1, _33.2_\n\nSpanish Revival, 35.1, _35.2_\n\nStick style, 22.1, _22.2_\n\nsymmetrical, with front entry, 38.1, _38.2_\n\nsymmetrical, with no front entry, 38.1, _38.2_\n\nthree-story, 14.1, _14.2_\n\ntwo-story, 14.1, _14.2_\n\n_see also_ four-square houses\n\nHiss, Philip, _44.1_\n\nHistoric Preservation Act (1966)\n\nHitchcock, Henry-Russell\n\nHitchings, George, _28.1_\n\nHoagland, William C., _18.1_\n\nHolly, Hudson\n\nHolt House, Cooperstown, _1.1_\n\nHolt House, Redlands, _34.1_\n\nhome financing\n\nHomestead Act (1862)\n\nHomewood house, Baltimore, _15.1_\n\nHomewood house, Columbus, _16.1_\n\nHomewood house, Natchez, _16.1_\n\nHopkins House, _13.1_\n\nHopper House, _47.1_\n\nhorse-drawn streetcars\n\nHorty, Elving and Associates, _47.1_\n\n_House and Garden_\n\n_House and Home_\n\n_House Beautiful_ , 39.1, 42.1, 45.1, 52.1\n\n_Houses for Good Living_ (Wills)\n\nHouse of Tomorrow (1933)\n\n\"House of Tomorrow\" (Vista Las Palmas), _45.1_\n\nhouses on the lot, 4.1, _4.2, 4.3, 4.4_\n\nHousing Act (1954)\n\nHousing Act (1980)\n\nHousing and Urban Development Department, U.S. (HUD)\n\nHoussaye House, _11.1_\n\nHowe, George\n\nHowell House, _26.1_\n\nHowry House, _18.1_\n\nHudgins House, _9.1_\n\nHuff House, _26.1_\n\nHunt, Richard Morris, 22.1, _22.2_ , 30.1, _30.2_ , 31.1\n\nHunt House, _9.1_\n\nHuse House, _25.1_\n\nIdaho, _5.1, 6.1, 40.1_\n\nI-house, 6.1, 7.1\n\nIllinois\n\nEclectic houses, _27.1_\n\nModern houses, _38.1, 38.2, 38.3_\n\nstyled houses since 1935, _49.1, 49.2, 51.1, 51.2, 52.1, 52.2_\n\n_see also_ Chicago\n\nIllinois Institute of Technology\n\nImber, Michael G., _52.1_\n\nIndiana\n\nfolk houses, _6.1, 7.1_\n\nRomantic houses, _1.1_ , 16.1, _16.2, 16.3, 18.1_\n\nVictorian houses, _21.1, 21.2, 23.1, 23.2_\n\nIngleside house, _17.1_\n\nInstitute of Classical Architecture and Art\n\ninsulated panels, app1.1, _app1.2_\n\nInternational style, _44.1_ , 44.2, _44.3, 44.4_\n\norigins and development of, _1.1_ , 1.2, 44.1, 44.2\n\n_International Style, The_ (Hitchcock and Johnson), 44.1\n\nIowa, _17.1, 18.1, 18.2, 21.1, 27.1, 38.1, 40.1_\n\nIsaac Small House, _6.1_\n\nIsrael, George, _47.1_\n\nItalianate style, _18.1_ , 18.2, _18.3, 18.4, 18.5, 18.6, 18.7, 18.8, 18.9, 18.10_, 18.10\n\nasymmetrical, 18.1, _18.2_\n\ncentered gable, 18.1, _18.2_\n\ncomments about, 18.1, 18.2\n\ndistinguishing from: Italian Renaissance, 33.1; Second Empire, 21.1; Stick, 22.1, _22.2_\n\nfront-gabled roof, 18.1, _18.2_\n\nidentifying features\n\noccurrence\n\norigins and development of, _1.1_ , 1.2, 18.1, 18.2\n\nsimple hipped roof, 18.1, _18.2_\n\ntowered, 18.1, _18.2_\n\ntown house, 18.1, _18.2_\n\nvariants and details, 18.1, _18.2, 18.3, 18.4_\n\nItalian Renaissance style, _33.1_ , 33.2, _33.3, 33.4, 33.5, 33.6, 33.7_, 33.8\n\nasymmetrical, 33.1, _33.2_\n\ndistinguishing from: Beaux Arts, 31.1; Italianate, 33.1; Mission and Spanish Revival, 33.2\n\nflat roof, 33.1, _33.2_\n\nhipped roof with projecting wings, 33.1, _33.2_\n\n\u2013New Traditional, _52.1_ , 52.2\n\norigins and development of, _1.1_ , 1.2, 33.1, 33.2\n\nsimple hipped roof, 33.1, _33.2_\n\nIves House, _14.1_\n\nJackman House, _6.1_\n\nJackson House, _17.1_\n\nJacobs, Jane\n\nJames I, king of England\n\nJamieson, James P., _32.1_\n\nJay, William, _14.1_ , 14.2, 15.1, _15.2_\n\nJefferson, Thomas, 4.1, 4.2, 15.1, 15.2, _15.3, 15.4_, 15.5, 20.1\n\nJeremiah House, _38.1_\n\nJohansen, John, _44.1_\n\nJohn Bremond House, _21.1_\n\nJohn Jay French House, _7.1_\n\nJohnson, John E., _17.1_\n\nJohnson, Philip, 44.1, 44.2, _44.3, 47.1_, 47.2\n\nJohnson House, Ingold, _6.1_\n\nJohnson House, Philadelphia, _13.1_\n\nJohnston House, _18.1_\n\nJones, Inigo\n\nJones House, Dallas, _38.1_\n\nJones House, Libertytown, _14.1_\n\nJones House, Newport News, _9.1_\n\nJordan, Charles A., _21.1_\n\nJordan House, Auburn, _21.1_\n\nJordan House, Pasadena, _36.1_\n\nJoseph Esherick and Associates\n\nJosephine Hagerty House, _44.1_\n\nKansas\n\nEclectic houses, _1.1, 27.1, 28.1, 32.1_, 33.1, 35.1\n\nfolk houses, _5.1_ , 6.1, _7.1_\n\nModern houses, _38.1, 38.2, 39.1, 39.2, 39.3, 41.1, 41.2, 42.1_\n\nstyled houses since 1935, _49.1, 50.1_, 50.2, _51.1_\n\nVictorian houses, _22.1, 24.1, 24.2_\n\nKansas City, Mo.\n\nEclectic houses, _27.1, 28.1, 29.1, 32.1, 33.1, 33.2, 33.3, 34.1, 34.2, 36.1_\n\nModern houses, _38.1, 38.2, 39.1, 39.2, 39.3, 42.1, 42.2, 45.1, 45.2_\n\nstyled houses since 1935, _50.1, 50.2, 50.3, 51.1, 52.1, 52.2, 52.3_\n\nVictorian houses, _24.1, 24.2_\n\nKatrina Cottage, 53.1, _53.2_\n\nKaufman Desert House, _44.1_\n\nKavanaugh House, _14.1_\n\nKayser House, _21.1_\n\nKeck, George Frederick\n\nKeeling House, _9.1_\n\nKeller, George, _23.1_\n\nKennedy House, _35.1_\n\nKent, William\n\nKent House, _32.1_\n\nKentucky\n\nColonial houses, _14.1_\n\nEclectic houses, _26.1, 31.1_\n\nfolk houses, _6.1, 7.1_\n\nRomantic houses, _16.1, 16.2, 17.1, 18.1, 18.2_\n\n_see also_ Lexington; Louisville\n\nKeswick house, _14.1_\n\nKilgore House, _21.1_\n\nKimball, Francis, _23.1_\n\nKimberly Crest, _30.1_\n\nKingscote house, _17.1_\n\nKirby House, _16.1_\n\nKirsch House (\"The Bunker\"), _47.1_\n\nkit houses\n\nKnight House, _21.1_\n\nKoch, Carl, 43.1, _44.1_ , app1.1, app1.2\n\nKoza House, _18.1_\n\nKraetsch and Kraetsch, _40.1_\n\nKuhn House, _26.1_\n\n_Ladies' Home Journal_\n\nLafever, Minard, 15.1, 16.1, _16.2_\n\nLake Pavilion at Glass House\n\nLamb and Rich, designers\n\nLambart House, _16.1_\n\nLambeth, James, _46.1_\n\nLand Ordinance (1785)\n\nLandrum House, _17.1_\n\nLandsberger House, _50.1_\n\nLang, William, _25.1_\n\nLang & Witchell, _34.1_\n\nLarkin, Thomas O., 11.1, _12.1_ , 36.1, 36.2\n\nLarkin House, 36.1, 36.2\n\nLatrobe, Benjamin H., _14.1_ , 14.2, 15.1, 15.2, 16.1\n\nLautner, John, _47.1_\n\nLawrence House, _27.1_\n\nLaws of the Indies\n\nLawson House, _47.1_\n\nLay House, _21.1_\n\nLe Corbusier, 44.1, 44.2, _44.3_ , app1.1, app1.2\n\nLee, Alfred, _39.1_\n\nLee, Thomas Helm, _16.1_\n\nLEED (Leadership in Energy and Environmental Design) rating system\n\nLEEDS Platinum house, _48.1_\n\nLee House, Dallas, _28.1_\n\nLee House, Fergus Falls, _40.1_\n\nLehman House, _11.1_\n\nLescaze, William, 37.1, 44.1\n\nLesley Home, _17.1_\n\nLevittown, N.Y., 41.1, _41.2, 42.1_\n\nLewinski and McMurtry, _18.1_\n\nLewis-Aldredge House, _32.1_\n\nLewis House, _39.1_\n\nLexington, Ky.\n\nEclectic houses, _27.1, 29.1, 29.2_\n\nfolk houses, _7.1_\n\nModern houses, _38.1, 38.2, 39.1, 39.2_\n\nRomantic houses, _18.1_\n\nVictorian houses, _24.1, 25.1_\n\nLiberty Hall, _14.1_\n\nLiebbe, Nourse and Rasmussen, _27.1_\n\n_Life_\n\nlights, in doors and door surrounds, 3.1, 11.1, _13.1, 14.1_, 16.1, _16.2, 27.1, 28.1_, 29.1, _50.1, 52.1_\n\n_see also_ fanlights\n\nLincoln House, _27.1_\n\nLind, Edmund G., _21.1_\n\nLind, Richard, _44.1_\n\nLindeberg, Harrie T.\n\nLindon Row houses, _16.1_\n\nLionberger House, _25.1_\n\nLivesy House, _27.1_\n\nLlewellyn Park, N.J., 4.1, 4.2\n\nlog construction, _3.1_\n\nLongfellow House, _13.1_\n\nLong House, _9.1_\n\nLongwood house, _20.1_\n\nLoomis and Hartman, _33.1_\n\nLorenzo House, _14.1_\n\nLos Angeles, Calif.\n\nEclectic houses, _33.1, 35.1, 35.2, 35.3_\n\nfolk houses, _8.1_\n\nModern houses, _40.1, 42.1, 42.2, 42.3, 44.1, 44.2, 44.3, 45.1, 45.2, 45.3, 45.4, 47.1, 47.2, 48.1, 48.2, 48.3, 48.4, 48.5_\n\nstyled houses since 1935, _49.1, 52.1_\n\nLouisiana\n\nColonial houses, 11.1, _11.2_\n\nEclectic houses, _34.1_\n\nfolk houses, _7.1, 7.2, 7.3_\n\nRomantic houses, _16.1_ , 16.2\n\n_Louisiana Speaks: Pattern Book_ (Urban Design Associates and Raymond L. Gindroz), 53.1\n\nLouisville, Ky.\n\nColonial houses, _15.1_\n\nEclectic houses, _1.1, 27.1, 27.2, 27.3, 28.1, 28.2, 29.1, 29.2, 30.1, 31.1, 32.1, 33.1, 33.2, 34.1, 35.1_\n\nfolk houses, _7.1_\n\nModern houses, _38.1, 39.1, 39.2_\n\nRomantic houses, _18.1, 18.2, 19.1_\n\nVictorian houses, _24.1, 25.1, 25.2_\n\nLowary House, _16.1_\n\nLowell, G., _27.1_\n\nLow House, _24.1_\n\n\"L-Stack House\", _48.1_\n\nlunettes, 15.1, _15.2, 15.3_\n\nLuse House, _35.1_\n\nLustron\n\nLuytens, Edwin\n\nLyndhurst, _17.1_\nMabie House, _10.1_\n\nMacKenzie and Goldstein, _33.1_\n\nMacMillan House, _38.1_\n\nMachine Age movement, 1.1, _1.2_ , 1.3\n\nMacPheadris-Warner House, _13.1_\n\nMacVeagh House, _33.1_\n\nMacy's, app1.1, _app1.2_\n\nMaher, George W., _38.1_ , 38.2\n\nMahony, Marion, _38.1_\n\nMaine\n\nColonial houses, _13.1_ , 14.1, _14.2_ , 14.3\n\nEclectic houses, _27.1_\n\nfolk houses, _8.1_\n\nRomantic houses, _13.1_ , 14.1, _14.2_ , 14.3, _16.1, 17.1_\n\nVictorian houses, _21.1_ , 24.1, 24.2\n\nMallinckrodt House, _32.1_\n\nManigault, Gabriel\n\nmansard roofs, 1.1, 18.1, _22.1, 23.1_\n\nBeaux Arts, _31.1_ , 31.2, _31.3, 31.4_\n\nMansard style, _48.1_ , 49.1, 49.2, _49.3_\n\nRichardsonian Romanesque, 25.1, _25.2_\n\nSecond Empire, _21.1_ , 21.2, 21.3, _21.4, 21.5, 21.6_\n\nMansard style, 21.1, _21.2_ , 31.1, _31.2, 48.1_, 49.1, _49.2, 49.3_, 49.4\n\nMansart, Fran\u00e7ois\n\nManship House, _17.1_\n\nMansion House, _11.1_\n\nmanufactured houses, 4.1, _8.1_ , 8.2, _8.3, 8.4_, app1.1, app1.2\n\ndouble-wides, 8.1, _8.2_\n\nmap, _8.1_\n\nmobile homes, 4.1, _8.1_ , 8.2, _8.3, 8.4_\n\nsingle-wides, 8.1, _8.2_\n\nsiting of, _8.1_\n\nMar-A-Lago house, _35.1_\n\nMarble, George, _42.1_\n\nMark Twain House, _22.1_\n\nMarling and Johnson, _27.1_\n\nMarsh House, _28.1_\n\nMartin House, _38.1_\n\nMartin Luther King birthplace, _23.1_\n\nMaryland\n\nColonial houses, _9.1, 13.1, 13.2, 14.1, 27.1_\n\nEclectic houses, _27.1_\n\nfolk houses\n\nModern houses, _45.1, 45.2_\n\nstyled houses since 1935, _52.1, 52.2, 53.1_\n\nVictorian houses, _26.1_\n\nmasonry\n\nQueen Anne patterned, 23.1, _23.2_\n\nstructural systems, 3.1, _3.2_\n\nMassachusetts\n\nColonial houses, 9.1, 9.2, _9.3, 13.1, 13.2, 13.3, 14.1, 14.2, 14.3_\n\nEclectic houses, _27.1, 27.2_\n\nfolk houses, _6.1_ , 6.2\n\nModern houses, _44.1_\n\nRomantic houses, _17.1_\n\nstyled houses since 1935, _52.1_\n\nVictorian houses, _22.1, 23.1, 24.1, 24.2, 24.3_\n\n_see also_ Boston; Cambridge\n\nmassed plans, 2.1, _2.2_\n\nMassey House, _16.1_\n\n_Materials for Design_ (Bell)\n\nMattison House, _20.1_\n\nMay, Cliff, 42.1, 42.2, _42.3_ , 50.1, _50.2_\n\nMaybeck, Bernard\n\nMcComb, John\n\nMcCoy, Nancy, _42.1_\n\nMcDuffie House, _27.1_\n\nMcEntire House, _52.1_\n\nMcGuire House, _17.1_\n\nMcIntire, Samuel\n\nMcKecknie, John W., _33.1_\n\nMcKibben, David, _18.1_\n\nMcKim, Mead & White, _24.1, 24.2_, 24.3, _27.1_ , 27.2, _31.1_ , 33.1\n\nMcLamb House, _26.1_\n\nMcMillan Houses, _23.1_\n\nMcMurtry, John, _17.1, 18.1_\n\nMcNairy House, _24.1_\n\nMcNalt House, _42.1_\n\nMcPhail House, _26.1_\n\nMeckfessel, Robert, _44.1_ , 48.1, _48.2_\n\nMedieval styles, influence of, 1.1, 1.2, _1.3_ , 1.4, 20.1, 29.1, 29.2, 32.1\n\nMediterranean New Traditional style, _52.1_ , 52.2\n\nMeem, John Gaw, _37.1_\n\nMeier, Richard, 44.1, _44.2_\n\nMeigs, Arthur\n\nMenz House, _24.1_\n\nMercer House, _18.1_\n\nMerrimon House, _18.1_\n\nMeysenburg House, _32.1_\n\nMichigan, 16.1, _16.2, 17.1, 23.1, 38.1_\n\nMid-century Modern style, 4.1, _4.2, 4.3_, 42.1, 42.2, 47.1, 48.1, 52.1\n\n_see also_ Contemporary style\n\nMies van der Rohe, 37.1, 44.1, _44.2_\n\nMiguel Angelo Flores & Associates, _48.1_\n\nMilbank House, _39.1_\n\nMillennium Mansions, _51.1_ , 51.2, _51.3, 51.4, 51.5, 51.6_\n\ncascading hipped roofs, _51.1_\n\ngabled roof with lower cross gables, _51.1_\n\nhipped roof with lower cross gables, _51.1_\n\nhipped roof with lower cross hips, _51.1_\n\ntall entry porches, _51.1_\n\nMiller House, Chautauqua, _19.1_\n\nMiller House, Chicago, _23.1_\n\nMiller House, Cripple Creek, _23.1_\n\nMiller House, Dallas, _24.1_\n\nMiller House, Wichita, _22.1_\n\nMills, Robert, 15.1, 16.1\n\nMinimal Traditional style, _41.1_ , 41.2, _41.3, 41.4, 41.5_\n\ngable and wing, 41.1, _41.2_\n\nhipped roof, 41.1, _41.2_\n\norigins and development of\n\nside-gabled (Cape Cod) houses, 41.1, _41.2_\n\nMinnesota\n\nEclectic houses, _27.1, 29.1, 29.2, 29.3_\n\nModern houses, _38.1, 38.2, 40.1, 42.1, 42.2, 43.1, 44.1, 44.2, 45.1, 45.2, 45.3, 45.4, 45.5, 46.1, 47.1, 47.2, 47.3_\n\nRomantic houses, _17.1, 17.2, 20.1_\n\nstyled houses since 1935, _50.1, 51.1, 51.2, 51.3, 52.1, 52.2, 52.3, 52.4, 52.5, 53.1, 53.2, 53.3, 53.4_\n\nVictorian houses, _21.1, 22.1, 23.1, 25.1_\n\nMission style, _34.1_ , 34.2, _34.3, 34.4, 34.5, 34.6_, 34.7\n\nasymmetrical, 34.1, _34.2_\n\ncomments about, 34.1, 34.2\n\ndistinguishing from: Italian Renaissance, 33.1; Spanish Colonial, 34.1\n\nidentifying features\n\ninfluenced by Craftsman and Prairie\n\noccurrence\n\norigins and development of, _1.1_ , 1.2, 34.1\n\nsymmetrical, 34.1, _34.2_\n\nvariants and details, 34.1, _34.2_\n\nMississippi\n\nColonial houses, _15.1_\n\nEclectic houses, _28.1_\n\nfolk houses, _7.1_\n\nModern houses, _39.1_\n\nRomantic houses, _1.1_ , 16.1, _16.2, 16.3, 17.1, 17.2, 17.3, 18.1, 20.1_\n\nstyled houses since 1935, _53.1, 53.2_\n\nVictorian houses, _23.1, 23.2, 26.1, 26.2_\n\nMissouri\n\nColonial houses, 11.1, _11.2, 15.1_\n\nEclectic houses, _35.1_\n\nfolk houses, _6.1, 7.1_\n\nModern houses, _1.1_\n\nstyled houses since 1935, _52.1, 52.2, 52.3, 53.1_\n\nVictorian houses, _1.1, 23.1, 25.1, 26.1, 26.2_\n\n_see also_ Kansas City; St. Louis\n\nMitchell House, _29.1_\n\nmixtures, stylistic, 1.1, 1.2, 1.3, _1.4_ , 1.5, _1.6_ , 18.1, 20.1, 27.1, 30.1, 37.1, 42.1, 51.1\n\nMizner, Addison, _35.1_ , 35.2\n\nMLTW, _see_ Moore Lyndon Turnbull Whitaker\n\nmobile homes, 4.1, _8.1_ , 8.2, _8.3, 8.4_\n\n_see also_ manufactured folk houses\n\n_Model Architect, The_ (Sloan), 18.1\n\n\"Modern Architecture: International Exhibition\" (MoMA)\n\n_Modern Builder's Guide, The_ (Lafever), 16.1\n\n_Modern Dwellings in Town and Country_ (Holly), 23.1, 23.2\n\nModernistic style, _40.1_ , 40.2, _40.3, 40.4_\n\nArt Deco, 40.1, _40.2, 40.3_\n\nArt Moderne, 40.1, _40.2_\n\ncomments about\n\nidentifying features\n\noccurrence\n\norigins and development of, _1.1_ , 1.2, 40.1\n\nvariants and details, 40.1, _40.2_\n\nModern movement, 1.1, 1.2, 1.3, _1.4_ , 1.5, 37.1, 48.1, app1.1\n\nModern styles\n\nArts and Crafts (Early Modern), 1.1, _1.2_ , 1.3, 29.1, 37.1\n\nContemporary, _45.1_ , 45.2, _45.3, 45.4, 45.5, 45.6, 45.7, 45.8, 45.9, 45.10, 45.11, 45.12_, 45.13\n\nCraftsman, 1.1, _1.2, 39.1_, 39.2, _39.3, 39.4, 39.5, 39.6, 39.7, 39.8_, 39.9\n\nInternational, _44.1_ , 44.2, _44.3, 44.4_\n\nMinimal Traditional, _41.1_ , 41.2, _41.3, 41.4, 41.5_\n\nModernistic, _40.1_ , 40.2, _40.3, 40.4_\n\norigins and development of, 1.1, _1.2_ , 1.3\n\nother 20th-century modern, _47.1_ , 47.2, _47.3, 47.4_, 47.5, _47.6, 47.7, 47.8, 47.9, 47.10, 47.111_\n\nPrairie, 1.1, _1.2, 37.1_, 38.1, _38.2, 38.3, 38.4, 38.5, 38.6, 38.7, 38.8_, 38.9\n\nRanch, _42.1_ , 42.2, _42.3, 42.4, 42.5, 42.6, 42.7, 42.8, 42.9, 42.10_\n\nShed, _46.1_ , 46.2, _46.3, 46.4, 46.5_\n\nSplit-Level, 42.1, _42.2, 42.3, 42.4, 43.1_, 43.2, _43.3_\n\n21st-century modern, _48.1_ , 48.2, _48.3, 48.4, 48.5, 48.6, 48.7_, 48.8\n\nmodillions, fm2.1, 14.1, _14.2, 14.3_, 15.1, 27.1, _27.2_ , 28.1, 31.1\n\nmodular houses, 8.1, app1.1, app1.2, app1.3, _app1.4_\n\nMontana, 6.1, _6.2, 24.1_\n\nMonterey style, _36.1_ , 36.2, _36.3, 36.4_, 36.5\n\norigins and development of, _1.1_ , 1.2, 36.1, 36.2\n\nMonte Vista Historic District house, _35.1_\n\nMontgomery House, _16.1_\n\nMontgomery Place house, _19.1_\n\nMontgomery Ward\n\nMonticello, _15.1_ , 15.2\n\nMoore, Charles\n\nMoore House, Geneva, _20.1_\n\nMoore House, Philadelphia, _17.1_\n\nMoore Lyndon Turnbull Whitaker (MLTW), 46.1, 53.1, _53.2_\n\nMoran House, _31.1_\n\n_More Houses for Good Living_ (Wills)\n\nMorphy House, _36.1_\n\nMorris, Robert, _13.1_\n\nMorris House, _21.1_\n\nMorrison, Lionel, _44.1_\n\nMorrison House, _15.1_\n\nMorse House, _18.1_\n\nMorton House, _24.1_\n\nMoss, Eric Owen, _47.1_\n\nMountainside House, _47.1_\n\nMount Vernon\n\nMullen, John, _52.1, 53.1_\n\nMullen House, _52.1_\n\nmultiple-facade gabled houses, 29.1, _29.2_\n\nMunger Place project, Dallas (1976), _4.1, 4.2_, 4.3\n\nmuntins, 14.1, 14.2\n\nMurdock-Wiley House, _18.1_\n\nMurphy, Brian, _47.1_\n\nMurphy House, _16.1_\n\nMuse House, _52.1_\n\nMuseum of Modern Art (MoMA), 44.1, 44.2, 45.1, 47.1\n\nMusgrove House, _28.1_\n\nMyers, J. G., _25.1_\n\nNapoleon I, emperor of France\n\nNathaniel Hawthorne birthplace, _13.1_\n\nnational folk houses, _7.1_ , 7.2, _7.3, 7.4, 7.5, 7.6, 7.7, 7.8_\n\ngable-front-and-wing family, 7.1, _7.2_\n\ngable-front family, 7.1, _7.2_\n\nhall-and-parlor family, 7.1, 7.2, _7.3, 7.4, 7.5_\n\nI-house family, 7.1, _7.2_\n\nmassed-plan, side-gabled family, 7.1, _7.2_\n\npyramidal family, 7.1, _7.2_\n\nrailroad expansion and, 7.1, _7.2_\n\nNational Housing Act (1934)\n\nNational Trust's \"America's Treasures\"\n\nNative American folk houses, 1.1, _5.1_ , 5.2, _5.3, 5.4, 5.5, 5.6, 5.7, 5.8, 5.9, 5.10_\n\nearth-wall family, 5.1, _5.2_\n\nmap, _5.1_\n\nrectangular-plan, wood-frame family, 5.1, _5.2_\n\nround-plan, wood-frame family, 5.1, _5.2, 5.3, 5.4_\n\nNative Americans\n\n_Natural House, The_ (Wright), 45.1\n\nNebraska, _5.1_ , 6.1, _6.2, 21.1, 24.1_, 28.1\n\nneighborhoods, _4.1_ , 4.2, _4.3, 4.4, 4.5, 4.6_, 4.7, _4.8, 4.9, 4.10, 4.11, 4.12, 4.13, 4.14, 4.15, 4.16, 4.17, 4.18, 4.19, 4.20, 4.21, 4.22, 4.23, 4.24, 4.25, 4.26, 4.27, 4.28_, _4.29_\n\ndevelopment influences on, 4.1, _4.2_\n\nearly automobile suburbs (1915\u20131940), 4.1, 4.2\n\nelectric streetcar suburbs, 4.1, _4.2_\n\ngolf courses, 4.1, _4.2_\n\ngovernance\n\nground plans, 4.1, _4.2, 4.3, 4.4, 4.5_\n\ngroupings, 48.1, _48.2_\n\ngrowth rate in\n\nhorse-drawn streetcars\n\nhouses on the block\n\noverlay patterns and, 4.1, _4.2, 4.3, 4.4_\n\nplan types and density, 4.1, _4.2_\n\npost-suburban (1970\u2013present), 4.1, _4.2_\n\npost\u2013World War II (1940\u20131980), 4.1, _4.2_ , 4.3\n\nrailroad suburbs (1850\u20131930)\n\nreading streetscapes in, 4.1, _4.2, 4.3, 4.4, 4.5, 4.6_\n\nredlining in older, 4.1, _4.2_ , 4.3\n\nrural, 4.1, _4.2_\n\ntopography and vegetation\n\nunderlying surveys, 4.1, _4.2, 4.3, 4.4, 4.5_\n\nurban (1750\u20131920), 4.1, _4.2, 4.3_\n\nNeighborhood Unit, _4.1_\n\nNelms House, _23.1_\n\nNeoclassical Ranch style, 50.1, _50.2_ , 50.3\n\nNeoclassical style, _28.1_ , 28.2, _28.3, 28.4, 28.5, 28.6, 28.7_, 28.8\n\nChinese Chippendale railings on, 28.1, _28.2_\n\ndistinguishing from: Beaux Arts, 31.1; Early Classical Revival, 28.1, _28.2_ ; Georgian and Federal, 28.3; Greek Revival, 28.4, _28.5_\n\nfront-gabled roof, 28.1, _28.2_\n\nfull-facade porch, 28.1, _28.2_\n\nfull-height entry porch, 28.1, _28.2_\n\nfull-height porch with lower full-width porch, 28.1, _28.2_\n\none-story, 28.1, _28.2_\n\norigins and development of, 1.1, _1.2_ , 28.1, 28.2\n\nNeutra, Richard, 37.1, 44.1, _44.2, 45.1_\n\nNevada\n\nfolk houses, _8.1, 8.2_\n\nModern houses, _42.1, 45.1_\n\nVictorian houses, _21.1_\n\nNewbold house\n\nNewcomb, E. A. P., _24.1, 27.1_\n\nNew Formalism style, 47.1, 47.2, _47.3_\n\nNew Haven, Conn.\n\nColonial houses, _13.1, 14.1_\n\nEclectic houses, _27.1, 30.1_\n\nRomantic houses, _19.1_\n\nVictorian, _21.1, 23.1, 23.2, 23.3, 23.4, 23.5, 24.1, 24.2_\n\nNew Jersey, 4.1, 4.2, 6.1, 9.1, 10.1, 10.2, _10.3_ , 11.1, _18.1, 21.1, 23.1, 23.2, 45.1, 52.1_\n\nNew Mexico, 4.1, 37.1\n\nColonial houses, 12.1, _12.2_ , 11.1, _12.3_ , 12.4\n\nEclectic houses, 37.1, _37.2_ , 37.3\n\nfolk houses, _5.1_ , 5.2\n\nNew Orleans, La., 4.1, _4.2, 4.3_, 4.4, 7.1, _33.1, 33.2, 35.1, 36.1_, 36.2\n\nColonial houses, 11.1, _11.2, 11.3, 11.4_, 11.5, _14.1_\n\nEclectic houses, _33.1, 33.2, 35.1, 36.1_, 36.2\n\nfolk houses, _8.1_\n\nRomantic houses, 16.1, _16.2, 17.1, 18.1_\n\nVictorian houses, _21.1_ , 26.1, _26.2_\n\nNewport, R.I.\n\nColonial houses, 13.1, _13.2, 13.3_, 14.1\n\nEclectic houses, 27.1, 31.1, _31.2_\n\nRomantic houses, _17.1_\n\nVictorian houses, 22.1, _22.2_ , 23.1, _23.2, 24.1, 24.2_\n\nNewsom, John J., _18.1_\n\nNew Traditional styles, _14.1, 52.1_, 52.2, _52.3, 52.4, 52.5, 52.6, 52.7, 52.8, 52.9_, 52.10\n\nClassical, _52.1, 52.2_, 52.3\n\nColonial, 52.1, _52.2, 52.3, 52.4_\n\nCraftsman, _52.1_ , 52.2\n\nFrench, _52.1_ , 52.2\n\nItalian Renaissance, _52.1_ , 52.2\n\nMediterranean, _52.1_ , 52.2\n\nPrairie, _52.1_ , 52.2\n\nQueen Anne and other Victorian, _52.1_ , 52.2\n\nShingle, 52.1, _52.2, 52.3_\n\nTudor, _52.1_ , 52.2\n\nNew Traditional styles, clues to identification of, 52.1, _52.2_ , 52.3, 52.4, _52.5_\n\nClassical details, _52.1_ , 52.2\n\ndormers, 52.1, _52.2_\n\nhalf-timbering, 52.1, _52.2_\n\nMillennium Mansion elements, _52.1_ , 52.2\n\nwindows, 52.1, _52.2_ , 52.3\n\n_New Vitruvius, The_ (Richardson), _14.1_\n\nNew York City, 4.1, 4.2\n\nCentral Park (1857\u201373)\n\nColonial houses, 10.1, 10.2, 10.3, 13.1, _13.2_ , 14.1, 14.2\n\nEclectic houses, _27.1, 27.2, 28.1, 31.1, 31.2_\n\nRomantic houses, 16.1, _16.2, 17.1, 18.1_\n\nVictorian houses, 21.1, 23.1\n\nNew York Five\n\nNew York State, _4.1_ , 4.2, _4.3_ , 4.4\n\nColonial houses, _1.1_ , 10.1, 10.2, _10.3, 10.4, 14.1_\n\nEclectic houses, _28.1, 29.1, 29.2, 32.1, 34.1, 35.1_\n\nfolk houses, 4.1, 6.1, _7.1, 7.2_\n\nModern houses, _38.1, 41.1, 41.2, 42.1, 44.1, 45.1, 46.1, 47.1, 47.2, 47.3, 48.1, 48.2_\n\nRomantic houses, _16.1, 16.2, 17.1, 17.2, 17.3, 18.1, 18.2, 19.1, 20.1_\n\nstyled houses since 1935, _52.1, 52.2, 53.1, app1.1_\n\nVictorian houses, _1.1, 21.1, 21.2, 21.3, 22.1, 22.2, 23.1, 23.2, 23.3, 23.4, 23.5, 24.1, 24.2, 24.3, 26.1_\n\n_see also_ Buffalo; New York City\n\nNew York Stock Exchange\n\nNichols, J. C.\n\nNichols House, _13.1_\n\nNicholson House, _15.1_\n\nNiels House, _45.1_\n\nNoble House, _26.1_\n\nNorman Cottages\n\nNorris, John S., _18.1_\n\nNorth Carolina\n\nColonial houses, _9.1_ , 13.1, _13.2, 15.1, 15.2_\n\nEclectic houses, _27.1, 27.2, 27.3, 28.1, 28.2, 29.1, 29.2, 30.1_, 30.2, _33.1, 34.1_\n\nfolk houses, _6.1, 6.2, 7.1, 7.2, 7.3, 7.4, 7.5_\n\nModern houses, _39.1, 47.1, 47.2, 53.1_\n\nRomantic houses, _16.1, 17.1, 18.1, 18.2, 18.3, 18.4_\n\nstyled houses since 1935, _53.1_\n\nVictorian houses, _23.1, 23.2, 23.3, 24.1, 26.1, 26.2, 26.3_\n\n_see also_ Raleigh\n\nNorth Dakota\n\nModern houses, _40.1_\n\nVictorian houses, _22.1_\n\nNorthrop, Joseph W., _30.1_\n\nNorthwest Ordinance (1785), 4.1, 4.2\n\nNorton House, San Antonio, _18.1_\n\nNorton House, Venice, _47.1_\n\nNourse and Rasmussen, _38.1_\n\nNoyes, Eliot, _44.1_\n\nNoyes House, _44.1_\n\nNunan House, _23.1_\n\n_Octagon House, The: A Home for All_ (Fowler), 20.1\n\nOctagon houses, _1.1_ , 1.2, _20.1_ , 20.2, _20.3_\n\nOhio\n\nEclectic houses, _27.1, 27.2, 30.1, 31.1, 32.1, 33.1_\n\nModern houses, _40.1_\n\nRomantic houses, _16.1_\n\nVictorian houses, _24.1_\n\nOhrum House, _28.1_\n\nOklahoma, _5.1, 5.2, 6.1, 7.1, 7.2, 34.1, 34.2, 35.1, 38.1, 39.1, 47.1_, nts.1 _n_\n\nOlana house, _19.1_\n\n_Old House Journal's New Old House_\n\nOliver, Allen, _52.1_\n\nOliver, LaBaron, _23.1_\n\nOlivier House, _11.1_\n\nOlmsted, Frederick Law, 4.1, 4.2, 4.3, 4.4\n\nO'Neil House, _27.1_\n\none-story houses, 15.1, _15.2_ , 27.1, _27.2, 27.3_, 28.1, _28.2_\n\n_see also_ Cape Cod houses, Minimal Traditional style, Ranch style, Styled Ranch houses\n\nOPEC oil embargo\n\nOregon\n\nEclectic houses, _29.1, 29.2_\n\nModern houses, _38.1, 41.1, 41.2, 42.1, 45.1_\n\nRomantic houses, _18.1_\n\nstyled houses since 1935, _51.1, 52.1, 52.2, 52.3_\n\nVictorian houses, _21.1, 22.1, 22.2, 23.1_\n\nOrganic houses, 47.1, 47.2, _47.3_\n\noriels, fm2.1, 17.1, _17.2, 17.3, 24.1, 29.1_\n\nOriental Revival style, 19.1, _19.2, 19.3_\n\ninfluence on: Craftsman, 39.1, _39.2_ , 39.3, 39.4; Styled Ranch, _50.1_ , 50.2\n\nOrman House, _25.1_\n\nOrmsbee, Caleb, _14.1_\n\nO'Rourke House, _38.1_\n\nOsborn House, _13.1_\n\nOtis House, _14.1_\n\nOud, Jacopus\n\noutdoor spaces, _45.1_\n\noverhanging upper stories and gables, 1.1, 23.1, _29.1, 32.1_\n\nOwens House, _32.1_\n\nPage House, _23.1_\n\npaired gables houses, 17.1, _17.2_\n\nPalladian windows\n\nAmerican Vernacular, _53.1_\n\nColonial Revival, _27.1, 27.2_, 27.3\n\nEarly Classical Revival, _15.1_\n\nFederal, 14.1, _14.2, 14.3, 14.4, 14.5_, 16.1\n\nGreek Revival\n\nItalian Renaissance, _29.1, 33.1_\n\nMillennium Mansion, _51.1_\n\nNew Traditional, 52.1, _52.2, 52.3_\n\nQueen Anne, 23.1, _23.2, 23.3, 23.4_\n\nShingle, 24.1, _24.2, 24.3, 24.4_\n\nPalladio, Andrea\n\nPalliser, Palliser & Company pattern books\n\nPalliser brothers\n\nPalmer and Krisal, _45.1_\n\nPalmer House, _14.1_\n\nPancoast House, _26.1_\n\nparapeted gables houses, 29.1, _29.2, 29.3_\n\nparapets, fm2.1, _3.1, 3.2, 9.1, 11.1, 13.1, 14.1, 16.1, 18.1_, 30.1, _40.1, 47.1, 48.1, 52.1, 53.1_\n\nColonial Revival, _27.1, 27.2_\n\nDutch Colonial, _10.1_ , 10.2\n\nGothic Revival, _17.1_ , 17.2, 17.3, _17.4, 17.5_\n\nItalian Renaissance, _33.1, 33.2_\n\nMission, _34.1_ , 34.2, _34.3, 34.4, 34.5_, 34.6\n\nPueblo Revival, _37.1_ , 37.2, _37.3, 37.4_, 37.5\n\nQueen Anne, 23.1, _23.2, 23.3, 23.4_\n\nRichardsonian Romanesque, 25.1, _25.2_\n\nSpanish Colonial, _12.1_ , 12.3, 12.3, _12.4, 12.5_\n\nSpanish Revival, 35.1, _35.2_\n\nTudor, 1.1, _1.2_ , 1.3, _29.1_ , 29.2, _29.3_ , 29.4, _29.5, 29.6, 29.7_\n\nPardee House, _18.1_\n\nParis Exhibition, 1925 (Exposition Internationale des Arts D\u00e9coratifs et Industriels Modernes)\n\nParker House, _38.1_\n\nParris, Alexander, 14.1, 15.1, 16.1\n\nParry House, _23.1_\n\nParson Capen House, _9.1_\n\nPaul and Helen Olfelt House, _45.1_\n\nPayne House, _14.1_\n\nPeak House, _9.1_\n\nPeck House, _38.1_\n\npediments, fm3.1, 1.1, _1.2_ , 1.3, _1.4_ , 7.1, _7.2_ , 14.1, _23.1_ , 26.1\n\nBeaux Arts, _31.1, 31.2_\n\nColonial Revival, 14.1, 27.1, 27.2, _27.3_ , 27.4, _27.5, 27.6, 27.7, 27.8, 27.9, 27.10, 27.11_\n\nEarly Classical Revival, 15.1, _15.2, 15.3_\n\nFrench Eclectic, 32.1, _32.2, 32.3_\n\nGeorgian, 1.1, 13.1, _13.2, 13.3, 13.4, 13.5, 13.6, 13.7_, 14.1, _14.2_\n\nGothic Revival, _17.1, 17.2_\n\nGreek Revival, 16.1, 16.2, _16.3, 16.4, 16.5, 16.6_\n\nItalianate, 18.1, _18.2, 18.3, 18.4, 18.5, 18.6, 18.7_, 18.8\n\nItalian Renaissance, 33.1, _33.2, 33.3, 33.4_\n\nNeoclassical, 28.1, _28.2, 28.3, 28.4_, 28.5\n\nNew Traditional, _52.1_ , 52.2, _52.3, 52.4, 52.5, 52.6, 52.7_\n\npictorial key to, fm2.1, fm2.2, fm2.3, fm2.4, fm3.1\n\nStyled Ranch, _50.1, 50.2_\n\nTudor, _29.1, 29.2_\n\nPendleton Investment Company house, _33.1_\n\nPennoyer, Peter, _52.1_\n\nPennsylvania\n\nEclectic houses, _27.1_\n\nfolk houses, 6.1, 6.2\n\nModern houses, _42.1, 47.1_\n\nRomantic houses, _16.1, 17.1, 18.1_\n\nstyled houses since 1935, _52.1_\n\nVictorian houses, _24.1_\n\nPerkenson, James, _18.1_\n\nPerryman House, _6.1_\n\nPetal House, _47.1_\n\nPetit, John J.\n\nPettis House, _14.1_\n\nPeyton House, _6.1_\n\nPhelps, August, _16.1_\n\nPhiladelphia, Pa., 4.1, 4.2\n\nColonial houses, _9.1_ , 13.1, 13.2, _13.3, 13.4, 13.5, 13.6, 13.7_, 14.1, _14.2_ , 14.3, 14.4, 15.1\n\nEclectic houses, 27.1, _29.1_ , 31.1, _47.1_\n\nModern houses, _47.1_\n\nRomantic houses, 16.1, 16.2, _17.1, 18.1_\n\nVictorian houses\n\nPhiladelphia Centennial (1876)\n\nPhiladelphia Library, Ridgeway Branch building\n\nPhilip Hiss Studio, _44.1_\n\nPhillips House, _14.1_\n\npictorial glossary\n\npictorial key\n\narched doors, windows, porch openings\n\nchimneys\n\ncornices, fm2.1, fm3.1\n\ndate-related clues\n\ndecorative elements\n\ndoors and doorways\n\ndormers\n\nfoundations\n\nhouse shapes\n\nporches, fm2.1, fm2.2, fm2.3, fm2.4, fm3.1\n\nrailings (around porch, roofline, rooftop)\n\nroof elaborations\n\nroof forms\n\nroof-wall junctions\n\nwalls\n\nwindows\n\npicture windows, fm2.1, _39.1, 41.1_, 44.1\n\nRanch, _42.1_ , 42.2, 42.3, _42.4_ , 42.5, _42.6, 42.7, 42.8, 42.9_\n\nStyled Ranch, 50.1, _50.2, 50.3_\n\npilasters, fm2.1, fm2.2, fm3.1, 1.1, _1.2_ , 1.3, _11.1_ , 35.1\n\nBeaux Arts, _31.1_ , 31.2, 31.3, _31.4_\n\nColonial Revival, _26.1_ , 27.1, _27.2, 27.3, 27.4_\n\nFederal, 14.1, _14.2, 14.3, 14.4_\n\nGeorgian, _13.1_ , 13.2, 13.3, _13.4, 13.5_\n\nGreek Revival, _16.1_ , 16.2, _16.3, 16.4, 16.5, 16.6_\n\nItalianate, _18.1, 18.2_\n\nItalian Renaissance, _33.1_ , 33.2, _33.3, 33.4_\n\nNeoclassical, _28.1, 28.2_\n\nNew Traditional, _52.1_ , 52.2, 52.3, _52.4, 52.5, 52.6_\n\n_see also_ columns\n\npitched-roof houses, 12.1, _12.2, 12.3_\n\nPlains tipis, 5.1, _5.2_\n\nplumbing, app1.1, app1.2\n\nPolk, Willis\n\nPollman, Richard\n\npolychromed houses, 17.1, _17.2_\n\nPomeroy House, _21.1_\n\nPoplar Forest (Jefferson summer home), 15.1, 20.1\n\nporches, 3.1, 3.2, _28.1, 31.1, 32.1, 45.1, 47.1_\n\nalteration of, 1.1, 1.2, _1.3, 1.4_\n\nAmerican Vernacular, _53.1_\n\nColonial Revival, _26.1_ , 27.1, 27.2, _27.3, 27.4, 27.5, 27.6, 27.7, 27.8, 27.9_\n\nCraftsman, _39.1_ , 39.2, 39.3, _39.4, 39.5, 39.6, 39.7, 39.8, 39.9_\n\nDutch Colonial\n\nentry, _see_ entry porches\n\nexotic revival, _19.1, 19.2_, 19.3\n\nof folk houses, _7.1_ , 7.2, 7.3, _7.4, 7.5_, 7.6, _7.7_ , 7.8, _7.9, 8.1_\n\nFolk Victorian, _26.1_ , 26.2, 26.3, _26.4, 26.5, 26.6, 26.7_\n\nFrench Colonial, _11.1_ , 11.2, 11.3, _11.4_ , 11.5\n\nFrench Eclectic, _32.1, 32.2_\n\nfull-facade, 1.1, 6.1, _7.1_ , 11.1, 16.1, _16.2, 16.3, 16.4_, 16.5, _17.1, 18.1, 18.2, 21.1_, 23.1, 23.2, 23.3, 27.1, 27.2, _27.3, 27.4, 27.5, 27.6, 28.1_, 28.2, _28.3_ , 28.4, _28.5, 28.6, 28.7_, 29.1, 33.1, 39.1, _39.2, 39.3, 42.1_\n\nGeorgian, 13.1, _13.2_\n\nGothic Revival, _17.1_ , 17.2, 17.3, _17.4, 17.5, 17.6_, 17.7\n\nInternational, _44.1, 44.2_\n\nItalianate, 18.1, 18.2, _18.3_ , 18.4, 18.5, _18.6, 18.7_\n\nItalian Renaissance, _33.1_ , 33.2, _33.3_ , 33.4, _33.5, 33.6_\n\nMinimal Traditional, 41.1, 41.2, _41.3_\n\nMission, _34.1_ , 34.2, _34.3_ , 34.4\n\nNeoclassical, 1.1, _28.1_ , 28.2, 28.3, _28.4_ , 28.5, _28.6, 28.7, 28.8, 28.9_, 28.10\n\nNew Formalism, 47.1, _47.2_ , 47.3\n\nNew Traditional\n\nOctagon, 20.1, _20.2, 20.3_\n\npictorial key to, fm2.1, fm2.2, fm2.3, fm2.4, fm3.1\n\nPrairie, _37.1_ , 38.1, 38.2, _38.3, 38.4, 38.5, 38.6, 38.7_\n\npre-railroad, _6.1_ , 6.2, _6.3, 6.4_, 6.5\n\nPueblo Revival, 37.1, _37.2_\n\nQueen Anne, _22.1_ , 23.1, 23.2, _23.3, 23.4_, 23.5, _23.6, 23.7, 23.8, 23.9, 23.10, 23.11, 23.12, 23.13_\n\nRanch, 42.1, _42.2_ , 42.3, 42.4, 42.5, _42.6_ , 42.7, _42.8, 42.9, 42.10, 42.11_\n\nRichardsonian Romanesque, _25.1_ , 25.2, _25.3, 25.4, 25.5, 25.6_\n\nroof construction, _3.1_\n\nSecond Empire, 21.1, _21.2, 21.3, 21.4, 21.5, 21.6_\n\nShingle, _24.1_ , 24.2, 24.3, _24.4, 24.5, 24.6, 24.7_\n\nSpanish Colonial, 12.1, 11.1, _12.2, 12.3, 12.4_\n\nSpanish Revival, _35.1_ , 35.2, _35.3_\n\nStick style, _22.1_ , 22.2, 22.3, _22.4, 22.5, 22.6, 22.7_\n\nstructure of, _3.1_ , 3.2\n\nStyled Ranch, 50.1, _50.2_\n\nTudor, 29.1, 29.2, 29.3, _29.4, 29.5, 29.6_\n\nwraparound, _17.1, 20.1, 21.1, 22.1_, 23.1, 23.2\n\nPorter and Robinson, _23.1_\n\nPort House, _18.1_\n\nporticos, _see_ entry porches\n\nPortsmouth, N.H., Colonial houses, _13.1, 13.2, 14.1_\n\npost-and-beam construction, _45.1_\n\nPostmedieval English style, _9.1_ , 9.2, _9.3, 9.4, 9.5_, 9.6\n\nmap, _9.1_\n\norigins and development of, _1.1_ , 1.2, 9.1\n\nPostmodern style, 47.1, 47.2, _47.3_\n\nPotter, Edward Tuckerman, _22.1_\n\nPowell House, _13.1_\n\n_Practical House Carpenter, The_ (Benjamin), 16.1\n\nPrairie style, _37.1_ , 38.1, _38.2, 38.3, 38.4, 38.5, 38.6, 38.7_, 38.8\n\ngabled roof, 38.1, _38.2_\n\nhipped roof, symmetrical, with front entry, 38.1, _38.2_\n\nhipped roof, symmetrical, with no front entry, 38.1, _38.2_\n\ninfluence on Mission\n\n\u2013New Traditional, _52.1_ , 52.2\n\norigins and development of, 1.1, _1.2_ , 38.1, 38.2\n\n_see also_ four-square houses\n\nPratt, Daniel, _15.1_\n\nPredock, Antoine, _48.1_\n\nprefabricated construction, app1.1, _app1.2_ , app1.3\n\npre-railroad folk houses, _6.1_ , 6.2, _6.3, 6.4, 6.5_, 6.6, _6.7, 6.8, 6.9, 6.10, 6.11_, 7.1\n\nHispanic Southwest tradition, _see_ Spanish Colonial houses\n\nmap, _6.1_\n\nMidland log house tradition, 6.1, _6.2, 6.3_, 7.1, 7.2, 7.3\n\nNew England tradition, 6.1, _6.2, 6.3_, 6.4, 7.1, 7.2\n\nPlains tradition, 6.1, _6.2_\n\nTidewater South tradition, 6.1, _6.2, 6.3_, 6.4, 7.1, 7.2, 7.3\n\nPresident's House, Harvard University, _27.1_\n\nPreston House, _34.1_\n\nPrestwould house, _13.1_\n\nPrice, Bruce\n\nprimary roads, 4.1, _4.2_\n\n_Principles for Planning Small Houses_ (FHA), 7.1, 41.1\n\nPringle House, _21.1_\n\nProvidence, R.I.\n\nColonial houses, _13.1, 13.2, 14.1_, 14.2, _14.3, 14.4, 14.5_\n\nRomantic houses, _16.1, 18.1_\n\nPublic Land Survey System (PLSS)\n\nPueblo Revival style, _37.1_ , 37.2, _37.3, 37.4_\n\norigins and development of, _1.1_ , 1.2, 37.1\n\npueblos, 5.1, _5.2, 5.3_, 5.4\n\nPurcell, Feick and Elmslie, _38.1, 38.2_\n\nPurcell, William G.\n\nPutnam House, _13.1_\n\nPutney Houses, _18.1_\n\nPyne, Filley, Sloane, and Davison Houses, _27.1_\n\npyramidal houses, 7.1, _7.2_ , 26.1, _26.2_\nQuakenbush House, _24.1_\n\nQueen Anne style, _22.1_ , 23.1, _23.2, 23.3, 23.4, 23.5, 23.6, 23.7, 23.8, 23.9, 23.10, 23.11, 23.12, 23.13, 23.14, 23.15_, 23.16\n\ncross-gabled roof, 23.1, _23.2, 23.3_\n\ndistinguishing from Folk Victorian\n\nfree classic columns on, 23.1, _23.2, 23.3, 23.4, 23.5_\n\nfront-gabled roof, 23.1, _23.2, 23.3_\n\nhalf-timbered, 23.1, _23.2_\n\nhipped roof with lower cross gables, 23.1, _23.2, 23.3, 23.4, 23.5_\n\ninfluenced by Colonial Revival, 27.1, 27.2\n\n\u2013New Traditional, _52.1_ , 52.2\n\norigins and development of, _1.1_ , 1.2, 23.1, 23.2\n\npatterned masonry, 23.1, _23.2, 23.3_\n\nspindlework on, 23.1, _23.2, 23.3, 23.4, 23.5, 23.6_\n\ntowers on, 23.1, 23.2\n\ntown house, 23.1, _23.2_\n\nquoins, fm2.1, 1.1, _28.1_ , 29.1, _51.1, 52.1_\n\nBeaux Arts, _31.1_ , 31.2, 31.3, _31.4_\n\nFederal, 14.1, _14.2, 14.3, 14.4_\n\nFrench Eclectic, 32.1, _32.2, 32.3_\n\nGeorgian, 13.1, _13.2, 13.3_\n\nItalianate, _18.1, 18.2, 18.3, 18.4, 18.5_, 18.6\n\nItalian Renaissance, 33.1, _33.2, 33.3, 33.4_\n\nMansard, 49.1, _49.2_ , 49.3\n\nSecond Empire, _21.1, 21.2_\n\nSpanish Revival, _35.1, 35.2_\n\nQuonset huts,\n\nRadburn, N.J.\n\nRagland House, _33.1_\n\nrailings\n\nChinese Chippendale, fm2.1, _15.1_ , 28.1, _28.2, 28.3, 28.4_\n\ncornice- and roof-line, _14.1, 2.1o, 27.1_, 28.1, _28.2, 28.3, 28.4_\n\npictorial key\n\nporch and balcony, _1.1_ , 3.1, 3.2, _11.1, 20.1_, 23.1, _25.1_ , 26.1, 32.1, 34.1, _35.1, 35.2, 36.1_, 38.1, _38.2, 39.1, 39.2, 40.1_, 45.1, _45.2, 45.3, 45.4, 45.5, 45.6, 50.1, 53.1_, 53.2\n\nrailroad suburbs, 4.1, _4.2_\n\nrain screens, fm2.1, 48.1, 48.2, 51.1, app1.1\n\nRaleigh, N.C.\n\nEclectic houses, _27.1, 28.1, 32.1_\n\nRomantic houses, _18.1, 18.2, 18.3_\n\nVictorian houses, _21.1_\n\nRamsey House, _21.1_\n\nRanchettes, 41.1, 42.1, 42.2, _42.3_\n\nRanch houses, _42.1_ , 42.2, _42.3, 42.4, 42.5, 42.6, 42.7, 42.8, 42.9, 42.10_\n\ncross-gabled roof, 42.1, _42.2_\n\ncross-hipped roof, 42.1, _42.2, 42.3_\n\ndistinguishing from Styled Ranch\n\nhipped roof, 42.1, _42.2_\n\nside-gabled roof, 42.1, _42.2_\n\nsplit levels, 42.1, _42.2, 42.3, 42.4_\n\n_see also_ Styled Ranch houses\n\nRandall House, _27.1_\n\nRappaport House, _44.1_\n\nRead House, _14.1_\n\nrear facades\n\nrear-yard elaborations\n\nrectilinear grid plans, 4.1, 4.2, _4.3_ , 4.4, _4.5_\n\nRedfield House, _6.1_\n\nReding House, _16.1_\n\nredlining, 4.1, _4.2_ , 4.3\n\nReeder House, _6.1_\n\nReed House, _25.1_\n\nRegional Planning Association of America\n\nReiff, Daniel\n\nRenaissance Classical styles, influence of, 1.1, _1.2_ , 1.3, _1.4_ , 13.1, 18.1, 18.2, 31.1, 32.1\n\nRenton House, _21.1_\n\nRenwick, James, Jr.\n\nResidence at Calf Creek, _52.1_\n\nResidence at Pottersville, _52.1_\n\nResurrection Manor, _9.1_\n\nReynolds House, 49.1, _49.2_\n\nRhode Island\n\nColonial houses, 9.1, _9.2, 9.3_\n\nfolk houses\n\nstyled houses since 1935, _53.1_\n\nVictorian houses, _24.1_\n\nribbon windows, _38.1, 42.1, 42.2, 42.3_, 44.1, _44.2, 44.3, 44.4, 45.1, 52.1_\n\nRichardson, George, _14.1_\n\nRichardson, Henry Hobson, _23.1_ , 24.1, 25.1, _25.2_ , 25.3\n\nRichardson House, Newport, _24.1_\n\nRichardson House, Savannah, _14.1_\n\nRichardson Romanesque style, _25.1_ , 25.2, _25.3, 25.4, 25.5, 25.6_, 25.7\n\nhipped roof with cross gables, 25.1, _25.2_\n\norigins and development of, 1.1, _1.2_ , 25.1, 25.2\n\ntown house, 25.1, _25.2_\n\nRiche House, _11.1_\n\nRichmond, Va.\n\nColonial houses, _10.1, 13.1, 14.1_, 15.1\n\nEclectic houses, _27.1, 29.1, 32.1, 33.1, 33.2_\n\nRomantic houses, _1.1_ , 16.1, _16.2, 18.1, 18.2, 19.1_\n\nVictorian houses, _1.1, 21.1, 25.1, 25.2, 27.1, 29.1, 32.1, 33.1, 33.2_\n\nRietveld, Gerrit Thomas\n\nRiley Row houses, _25.1_\n\nRiverside, Ill. (1869 plan), 4.1, 4.2, _4.3_ , 4.4\n\nRobert Long House, _13.1_\n\nRoberts, Eben E.\n\nRobertson, Richardson, III, _52.1_\n\nRobert Townshend Hooe House, _13.1_\n\nRoche House, _21.1_\n\nRock House, _39.1_\n\nRoeder House, _39.1_\n\nRogers, Isaiah\n\nRomantic style\n\nexotic revivals, _19.1_ , 19.2, _19.3, 19.4_\n\nGothic Revival, _17.1_ , 17.2, _17.3, 17.4, 17.5, 17.6, 17.7, 17.8, 17.9, 17.10_, 17.11\n\nGreek Revival, _16.1_ , 16.2, _16.3, 16.4, 16.5, 16.6, 16.7, 16.8, 16.9, 16.10, 16.11_, 16.12\n\nItalianate, _18.1_ , 18.2, _18.3, 18.4, 18.5, 18.6, 18.7, 18.8, 18.9, 18.10, 18.11_, 18.12\n\nmixtures of, _1.1_ , 1.2\n\nOctagon, _20.1_ , 20.2, _20.3, 20.4_\n\nroofs, 3.1, _3.2, 3.3_\n\nDutch Colonial, 10.1, _10.2_ , 10.3\n\nFrench Colonial, 11.1, _11.2_\n\nmaterials for, 3.1, _3.2_\n\npictorial key to, prf.1, fm2.1\n\nPostmedieval English\n\npyramidal (equilateral hipped)\n\nQueen Anne, 23.1, _23.2, 23.3_\n\nSecond Empire, _21.1_\n\nSpanish Colonial, 12.1, _12.2_\n\nstructure, 3.1, _3.2, 3.3, 3.4, 3.5_, 3.6\n\n_see also_ hipped roof houses; tile roofs\n\nroof-wall junctions\n\nContemporary\n\nCraftsman, 39.1, _39.2_\n\nGothic Revival, 17.1, _17.2_\n\nGreek Revival\n\nInternational, _44.1_\n\nItalian Renaissance, _33.1_\n\npictorial key to\n\nRanch, 42.1, _42.2_\n\nRoper House, _16.1_\n\nRose, William P., _28.1_\n\nRosecliff house, _31.1_\n\nRose House, _48.1_\n\nRoseland house, _17.1_\n\nRostrevor house, _33.1_\n\nRountree House, _17.1_\n\nRoyall House, _13.1_\n\nRudolph, Paul\n\nRuggles House, _14.1_\n\nrural neighborhoods, 4.1, _4.2_\n\n_Rural Residences_ (Davis)\n\nRuskin, John\n\nRussell House, Charleston, _14.1_\n\nRussell House, Middleton, _16.1_\n\nRyder, Calvin A., _16.1_\n\nSaarinen, Eliel\n\nSabel, Moses, _28.1_\n\nSabel House (mansard roof), _31.1_\n\nSabel House (pyramidal hipped roof), _30.1_\n\nSabine Hall, _15.1_\n\nSt. Louis, Mo.\n\nEclectic houses, _27.1, 29.1, 30.1_, 31.1, _31.2, 32.1, 32.2, 33.1, 33.2, 33.3, 35.1, 35.2_\n\nModern houses, _38.1, 38.2_\n\nRomantic houses, _16.1, 18.1_\n\nVictorian houses, _21.1, 21.2, 25.1, 27.1, 29.1, 30.1_, 31.1, _31.2, 32.1, 32.2, 33.1, 33.2, 33.3, 35.1, 35.2, 38.1, 38.2_\n\nSalmala, David, _53.1_\n\nSalt Lake City, Utah\n\nSan Antonio, Tex.\n\nColonial houses, _12.1, 11.1, 12.2_\n\nEclectic houses, _34.1, 35.1_\n\nModern houses, _40.1_\n\nRomantic houses, _18.1_\n\nstyled houses since 1935, _52.1_\n\nVictorian houses, _23.1, 26.1, 26.2_\n\nSan Diego Panama-California Exposition (1915)\n\nSanford House, _18.1_\n\nSan Francisco, Calif.\n\nColonial houses, _12.1_ , 11.1\n\nEclectic houses, _27.1, 29.1, 29.2_, 31.1, _31.2_\n\nModern houses, _40.1, 41.1, 42.1_, 45.1, _45.2, 45.3_, 46.1\n\nRomantic houses, 18.1, _18.2_ , 18.3\n\nVictorian houses, 22.1, 22.2, _22.3, 23.1, 23.2, 24.1_, 24.2\n\nSavage Mining Company \"mansion\", _21.1_\n\nSavannah, Ga., _4.1_\n\nColonial houses, _13.1_ , 14.1, _14.2, 14.3, 14.4_, 14.5, 15.1\n\nEclectic houses, _33.1_\n\nRomantic houses, 16.1, _18.1, 18.2, 18.3, 21.1, 23.1, 25.1_\n\nVictorian houses, _21.1, 23.1, 25.1_\n\nSchenck House, _10.1_\n\nSchenectady, N.Y.\n\nSchindler, Rudolf, 37.1, 44.1, _44.2_\n\nSchmidt, R. R., _18.1_\n\nSchmidt House, _52.1_\n\nSchnee House, _6.1_\n\nScholz House, _34.1_\n\nScott House, _20.1_\n\nScott, M. H. Baillie\n\nSeamans House, _14.1_\n\nSea Ranch, Calif., _53.1, 53.2, app1.1_\n\nSea Ranch Condominiums\n\n_Sears Honor Bilt Home_ catalogue\n\nSears Roebuck\n\nSears Roebuck house plan books\n\nSeattle, Wash.\n\nEclectic houses, _27.1, 29.1, 29.2, 29.3_\n\nModern houses, _41.1, 42.1, 45.1, 45.2, 48.1, 48.2_\n\nstyled houses since 1935, _50.1, 51.1, 52.1, 52.2, 52.3_\n\nSecond Bay Tradition\n\nSecond Empire style, _21.1_ , 21.2, _21.3, 21.4, 21.5, 21.6, 21.7, 21.8_, 21.9\n\nasymmetrical, 21.1, _21.2_\n\ncentered wing or gable, 21.1, _21.2_\n\ndistinguishing from Italianate\n\norigins and development of, 1.1, _1.2_ , 21.1\n\nsimple mansard roof, 21.1, _21.2_\n\ntowered, 21.1, _21.2_\n\ntown house, 21.1, _21.2_\n\nsecond-story overhand houses, 27.1, _27.2_\n\nSeftenberg-Brandon House, _22.1_\n\nsegmental vaults houses, 48.1, _48.2_\n\n_Select Architecture_ (Morris), _13.1_\n\nSemple House, _15.1_\n\n747 Wing House, _48.1_\n\nShadows-on-the-Teche house, _16.1_\n\nshape innovations, 2.1, _2.2, 2.3_\n\nShaw, Richard Norman, 23.1, 23.2\n\nShed style, _46.1_ , 46.2, _46.3, 46.4, 46.5_, app1.1\n\nSheehy, Edward, _13.1_\n\nSheire, Monroe, _21.1_\n\nSherman, Aaron, _14.1_\n\nSherman, Roger, _44.1_\n\nSherman House, _22.1_\n\nSherrick House, _16.1_\n\nShingle style, _24.1_ , _24.2_ , _24.3_ , 24.4, 24.5, 24.6, 24.7, 24.8, 24.9, 24.10, _24.11_ , _24.12_\n\ncross-gabled roof, 24.1, _24.2_\n\nfront-gabled roof, 24.1, _24.2_\n\ngambrel roof, 24.1, _24.2_\n\nhipped roof with cross gables, 24.1, _24.2_\n\ninfluenced by Colonial Revival, 27.1, 27.2\n\n\u2013New Traditional, 52.1, _52.2, 52.3_\n\norigins and development of, _1.1_ , 1.2, 24.1\n\nside-gabled roof, 24.1, _24.2_\n\nshingles, wood, as roofing, 1.1, 2.1, 3.1, _3.2_ , 3.3, _10.1_ , 12.1, 12.2, _12.3, 21.1_, 29.1, 36.1, _36.2, 42.1_, 49.1\n\nshingles, wood, as wall and porch cladding, fm2.1, 1.1, _1.2_ , 3.1, _3.2, 7.1_, 10.1, 13.1, _26.1, 27.1_, 36.1, _38.1_ , 39.1, _41.1_ , 42.1, _50.1, 53.1_\n\nNew Traditional, 52.1, _52.2_\n\nPostmedieval English, 9.1, 9.2\n\npre-railroad, 6.1, 6.2, 6.3, _6.4_ , 6.5, 6.6\n\nQueen Anne, _22.1_ , 23.1, 23.2, _23.3_ , 23.4, _23.5, 23.6, 23.7, 23.8, 23.9_\n\nShed, _46.1_ , 46.2\n\nShingle style, _24.1_ , 24.2, 24.3, _24.4, 24.5, 24.6, 24.7_\n\nStick style, _21.1_ , 22.1\n\nShipley, Dan, _53.1_\n\n\"Ship of the Desert\", _40.1_\n\nShivers House, _26.1_\n\nShope Rheno Wharton Architects, _52.1_\n\nShoppell, R. W., 20.1, 23.1\n\n_Shoppell's Modern Houses,23.1_\n\nShreiner House, _25.1_\n\nShryock, Gideon\n\nshutters\n\nSibour, Jules Henri de, _32.1_\n\nside-gabled (Cape Cod) houses, 41.1, _41.2_\n\nside-gabled roof houses\n\nAmerican Vernacular, 53.1, _53.2_\n\nColonial Revival, 27.1, _27.2_\n\nContemporary, 45.1, _45.2_\n\nCraftsman, 39.1, _39.2, 39.3_\n\nFederal, 14.1, _14.2_\n\nFolk Victorian, 26.1, _26.2, 26.3_\n\nGeorgian, 13.1, _13.2, 13.3_\n\nRanch, 42.1, _42.2_\n\nShingle, 24.1, _24.2_\n\nSpanish Revival, 35.1, _35.2_\n\nSidener House, _7.1_\n\nSilver Lake house, _45.1, 45.2_\n\nSilvertop: Reiner-Burchill Residence, _47.1_\n\nsimple hipped roof houses, 18.1, _18.2, 18.3_, 33.1, _33.2_\n\nsimple mansard roof houses, 21.1, _21.2_\n\nsingle dominant mid-facade gable houses, 29.1, _29.2, 29.3_\n\nSkinner House, _7.1_\n\nslab foundations, 3.1, 52.1, _52.2_ , 52.3\n\nslightly askew houses, 48.1, _48.2_\n\nSloan, Kevin\n\nSloan, Samuel, 15.1, _18.1_ , 18.2, _20.1_\n\nSloane House, _52.1_\n\nSLUG (spread-out, low-density, unguided growth), 4.1, _4.2_\n\nSmale, J. C., _40.1_\n\nSmall, Philip L., _29.1_\n\nSmith, George Washington, _35.1, 35.2_, 35.3\n\nSmith, Harwood K., _47.1_\n\nSmith, Whitney R., _36.1_\n\nSmith-Bly House, _16.1_\n\nSmith House, Darien, _44.1_\n\nSmith House, Glen Mary, _16.1_\n\nSmith House, Los Angeles, _40.1_\n\nSmith House, New Bedford, _22.1_\n\nSnipper House, _48.1_\n\nSnyder, Vincent, _48.1_\n\nsod construction, 6.1, _6.2_\n\nsolar houses\n\nSomervell, W. Maybury, _35.1_\n\nSouth Carolina, _7.1_ , 13.1, _13.2, 13.3_, 14.1, _14.2, 14.3_, 14.4, 15.1, _15.2, 16.1, 23.1, 26.1, 26.2, 26.3, 27.1, 27.2_, 36.1, _39.1, 45.1, 53.1_\n\nSouth Dakota\n\nfolk houses, _6.1_\n\nVictorian houses, _22.1_\n\n_Southwest Review_\n\nSpanish Colonial style, 1.1, _12.1_ , 12.2, _12.3, 12.4, 12.5, 12.6_, 12.7\n\nAnglo influence on, 12.1, _12.2_ , 11.1, _12.3_\n\ndistinguishing from: Mission, 34.1; Native American pueblos, 5.1, 5.2\n\nflat roof with parapet, 12.1, _12.2, 12.3_\n\nfloor plan, _12.1_\n\nmap, _12.1_\n\norigins and development of, _1.1_ , 1.2, 12.1\n\npitched roof, 12.1, _12.2, 12.3_\n\nSpanish Governor's Palace, _12.1_\n\nSpanish Ranch style, 50.1, _50.2_\n\nSpanish Revival style, _35.1_ , 35.2, _35.3, 35.4, 35.5, 35.6, 35.7, 35.8, 35.9_, 35.10, 35.11,\n\ncombined hipped-and-gabled roofs, 35.1, _35.2_\n\ncomments about, 35.1, 35.2\n\ncross-gabled roof, 35.1, _35.2_\n\ndistinguishing from Italian Renaissance\n\nflat roof, 35.1, _35.2_\n\nhipped roof, 35.1, _35.2_\n\nidentifying features\n\noccurrence\n\norigins and development of, _1.1_ , 1.2, 35.1, 35.2\n\nside-gabled roof, 35.1, _35.2_\n\nvariants and details, 35.1, _35.2, 35.3, 35.4_\n\nSpanish southwestern United States, _12.1_ , 11.1\n\nSpencer, Robert C., Jr.\n\nSperry House, _20.1_\n\nspindlework\n\nFolk Victorian, _26.1_ , 26.2, 26.3, _26.4, 26.5_\n\nQueen Anne, 23.1, _23.2, 23.3, 23.4, 23.5, 23.6_\n\n_see also_ Eastlake detailing\n\nSplit-Level houses, 42.1, _42.2, 42.3, 42.4, 43.1_, 43.2, _43.3_\n\nSpring Station house, _15.1_\n\nStadtmuller House, _18.1_\n\nStandard City Planning Enabling Act (SCPEA; 1926)\n\nStandard State Zoning Enabling Act (SZEA; 1922)\n\nStarkey House, _44.1_\n\nStark House, _6.1_\n\nStaunton Hill house, _17.1_\n\nSteel House, _47.1_\n\nStenton House, _13.1_\n\nStern, Robert A. M., 47.1, _47.2_ , 48.1, 52.1, 52.2, _52.3_\n\nStetson House, _27.1_\n\nStevens, John Calvin\n\nStevens House, _23.1_\n\nStickle House, _27.1_\n\nStick style, _22.1_ , 22.2, _22.3, 22.4, 22.5, 22.6, 22.7, 22.8_\n\ndifferentiating from Italianate, 22.1, _22.2_\n\ngabled roof, 22.1, _22.2, 22.3_\n\nhipped, 22.1, _22.2_\n\norigins and development of, 1.1, _1.2_ , 22.1\n\ntowered, 22.1, _22.2_\n\ntown house, 22.1, _22.2_\n\nWest Coast, 22.1, _22.2_\n\nStill House, _23.1_\n\nStinson House, _16.1_\n\nstone\n\nchimneys, _3.1_ , 3.2, 9.1, _9.2, 9.3, 39.1_\n\ncurbs and sidewalks, 4.1, _4.2_\n\nroofing, _3.1_ , 3.2, 32.1\n\nstructural, 3.1, _3.2_ , 3.3, _3.4, 3.5_, 3.6, 6.1, _6.2_ , 6.3, 6.4, _6.5_\n\nStone, Edward Durrell, 47.1, _47.2_ , 47.3\n\nStone, Edward T., _44.1_\n\nstone cladding, fm2.1, fm2.2, _14.1, 17.1, 20.1, 21.1_, 26.1, 45.1, 47.1, _47.2_ , 50.1\n\nAmerican Vernacular, _53.1_ , 53.2, _53.3_\n\nBeaux Arts, _31.1_ , 31.2, 31.3\n\nChateauesque, _30.1_ , 30.2, _30.3_\n\nCraftsman, 39.1, _39.2, 39.3_\n\nDutch Colonial, _10.1_ , 10.2, 10.3, _10.4_ , 10.5, _10.6, 10.7_\n\nEarly Classical Revival, _15.1, 15.2_\n\nFrench Eclectic, _32.1_ , 32.2, _32.3, 32.4_\n\nGeorgian, 13.1, _13.2, 13.3, 13.4, 13.5_\n\nInternational, 44.1, _44.2, 44.3_\n\nItalianate, _18.1_ , 18.2\n\nItalian Renaissance, 33.1, 33.2, _33.3, 33.4_\n\nNew Traditional, 52.1, 52.2, _52.3, 52.4, 52.5, 52.6_\n\nPostmedieval English, 9.1, _9.2, 9.3_\n\nPrairie, _38.1, 38.2_\n\nQueen Anne, 23.1, _23.2, 23.3, 23.4, 23.5_\n\nRanch, 42.1, _42.2, 42.3_\n\nRichardsonian Romanesque, _25.1_ , 25.2, _25.3, 25.4_\n\nSpanish Colonial, _12.1_ , 12.2, _12.3_ , 12.4\n\nTudor, 29.1, _29.2_ , 29.3, _29.4, 29.5, 29.6_\n\nStone House, _52.1_\n\nStono house, _15.1_\n\nStorey House, _18.1_\n\nstreets and circulation, 4.1, _4.2_\n\nstreet trees and street enclosures\n\nStrickland, William\n\nStriebinger, Frederic W., _31.1_\n\nstructure, _3.1_ , 3.2, _3.3, 3.4, 3.5, 3.6, 3.7, 3.8, 3.9, 3.10, 3.11, 3.12, 3.13, 3.14, 3.15, 3.16_\n\narchitectural details, 3.1, _3.2, 3.3_\n\nroofs, 3.1, _3.2, 3.3, 3.4, 3.5_, 3.6\n\nwalls, 3.1, _3.2, 3.3, 3.4, 3.5, 3.6, 3.7_\n\nstucco, fm2.1, 3.1, _3.2_ , 3.3, 3.4, 14.1, 18.1, _30.1_ , 36.1, _38.1, 47.1_\n\nBrutalism, 47.1, _47.2_\n\nCraftsman, 39.1, _39.2_\n\nEarly Classical Revival, 15.1, _15.2_\n\nFrench Colonial, _11.1_ , 11.2, 11.3\n\nFrench Eclectic, _32.1_ , 32.2, _32.3_\n\nGreek Revival, 16.1, _16.2_\n\nwith half-timbering, _3.1_ , 3.2, _11.1_ , 11.2, _29.1_\n\nInternational, 44.1, 44.2, 44.3, _44.4_\n\nItalian Renaissance, 33.1, _33.2, 33.3, 33.4_, 33.5\n\nMillennium Mansions, _51.1, 51.2, 51.3_\n\nMission, _34.1, 34.2_, 35.1\n\nModernistic, _40.1_ , 40.2, _40.3_\n\nNew Traditional, 52.1, 52.2, _52.3_\n\nPueblo Revival, _37.1_ , 37.2, _37.3_\n\nRanch, 42.1, _42.2_\n\nSpanish Colonial, _12.1_ , 12.2, _12.3_\n\nSpanish Revival, _35.1_ , 35.2, 35.3, _35.4, 35.5, 35.6_\n\nStyled Ranch, 50.1, 50.2\n\nTudor, _29.1_ , 29.2\n\n21st-century Modern, 48.1, 48.2, _48.3, 48.4, 48.5_\n\nSturdivant Hall, _16.1_\n\nStyled houses since 1935\n\nAmerican Vernacular, _53.1_ , 53.2, _53.3, 53.4, 53.5_\n\nMansard, _48.1_ , 49.1, _49.2, 49.3_, 49.4\n\nMillennium Mansions, _51.1_ , 51.2, _51.3, 51.4, 51.5, 51.6_\n\nStyled Ranch, _50.1_ , 50.2, _50.3, 50.4, 50.5, 50.6, 50.7, 50.8_, 50.9\n\n_see also_ New Traditional style\n\nStyled Ranch houses, _50.1_ , 50.2, _50.3, 50.4, 50.5, 50.6, 50.7, 50.8_, 50.9\n\nColonial Revival Ranch, 50.1, _50.2_\n\ndistinguishing from Ranch\n\nFrench Ranch, _50.1_ , 50.2\n\nNeoclassical Ranch, 50.1, _50.2_ , 50.3\n\noriental influences on, _50.1_ , 50.2\n\nSpanish Ranch, 50.1, _50.2_\n\nTudor Ranch, _50.1_ , 50.2\n\nstyles, _1.1_ , 1.2, _1.3, 1.4, 1.5, 1.6, 1.7, 1.8, 1.9_\n\nalterations and, 1.1, 1.2, _1.3, 1.4_, 1.5\n\nmixtures of, _1.1_ , 1.2, _1.3, 1.4_\n\n_see also specific styles_\n\nsubdivisions, 4.1, 4.2\n\nsuburban neighborhoods, 4.1, _4.2_\n\nearly automobile (1915\u20131940), 4.1, 4.2\n\nelectric streetcar, 4.1, _4.2_\n\nplanned rectilinear, 4.1, _4.2_\n\npost-suburban (1970\u2013present), 4.1, _4.2_\n\npost\u2013World War II (1940\u20131980), 4.1, _4.2_ , 4.3\n\nrailroad suburbs\n\n_Successful Subdivisions_ (FHA; 1940)\n\nSuiter House, _23.1_\n\nSullivan, Louis, 38.1, 38.2\n\n_Sunset Magazine_\n\n_Sunset Western Ranch Houses_ (May), 42.1, 50.1\n\nSupreme Court, U.S.\n\nsurveys, types of, 4.1, _4.2, 4.3, 4.4, 4.5_\n\nS. Weringen House, _29.1_\n\nSwiss Avenue Historic District\n\n\"Swiss Miss\" House, _45.1_\n\nsymmetrical houses, 32.1, _32.2_ , 34.1, _34.2_ , 49.1, _49.2_\n\nsymmetrical paired gables houses, 29.1, _29.2_\n\nTabby Manse, _15.1_\n\nTallmadge, Thomas E.\n\nTallmadge and Watson, _38.1_\n\nTaney House, _14.1_\n\nTapscott, Daron, _23.1, 34.1_\n\nTate, Ken, _53.1_\n\nTatton Hall, _27.1_\n\nTaylor, Teddy, _1.1_\n\nTaylor House, Camden, _23.1_\n\nTaylor House, Newport\n\nTefft, Thomas, _18.1_\n\nTemple House, Concord, _52.1_\n\nTemple House, Seaford, _7.1_\n\nTen Broeck House, _10.1_\n\nTennessee\n\nRomantic houses, _16.1_\n\nstyled houses since 1935, _52.1_\n\nTerrace Hill house, _21.1_\n\nterra cotta, _23.1_ , 23.2, _23.3, 30.1_, 31.1, _31.2, 38.1, 52.1_\n\nTerry, Quinlan, _52.1_\n\nTexas, 4.1, _4.2_ , 4.3\n\nColonial houses, _12.1, 12.2_\n\nEclectic houses, _1.1, 27.1_, 35.1\n\nfolk houses, _5.1, 7.1, 8.1, 8.2_\n\nModern houses, _42.1, 42.2, 45.1, 47.1_\n\nRomantic houses, _16.1, 16.2, 17.1, 18.1, 18.2_\n\nstyled houses since 1935, _49.1, 51.1, 52.1, 52.2_, 53.1\n\nVictorian houses, _1.1, 21.1, 21.2, 22.1, 23.1, 23.2, 23.3, 24.1, 25.1, 26.1, 26.2, 26.3_\n\n_see also_ Dallas; San Antonio\n\n_Texas Houses: Built by the Book_ (Culbertson), _23.1_\n\nThomas House, _7.1_\n\nThompson, Alpheus B., 11.1, _12.1_\n\nThompson, Milo, _app1.1_\n\nThompson, Milo H., _44.1_\n\nThomson, Henry B., _28.1, 32.1_\n\nThomson House, _27.1_\n\nThornton, William, _14.1, 14.2_, 15.1\n\nThoroughgood House, _9.1_\n\nthree-story houses, 27.1, _27.2_\n\n\"Thyme to Be\" house, _48.1_\n\nTiedeman House, _25.1_\n\nTifft Houses, _18.1_\n\ntile, decorative, _23.1, 35.1, 35.2, 35.3, 42.1_\n\ntile roofs, fm2.1, _3.1_ , 3.2, _11.1_ , 31.1, 32.1, 38.1, _51.1_\n\nDutch Colonial, 10.1, 10.2\n\nItalian Renaissance, _33.1_ , 33.2, 33.3, _33.4_\n\nMission, _34.1_ , 34.2, _34.3, 34.4_, 34.5\n\nMonterey Revival, 36.1, _36.2, 36.3_\n\nNeoclassical, _28.1, 28.2_\n\nNew Traditional, _52.1, 52.2_\n\nSpanish Colonial, 12.1, _12.2_\n\nSpanish Ranch, 50.1, _50.2_\n\nSpanish Revival, _35.1_ , 35.2, _35.3, 35.4_\n\nTillary House, _16.1_\n\nTips House, _18.1_\n\nTNDs, _see_ traditional neighborhood development\n\nToby House, _23.1_\n\nTODs (transit-oriented developments), _4.1_ , 4.2, 4.3\n\nTotten, George Oakley, Jr., _31.1_\n\ntowers\n\nChateauesque, 30.1, _30.2, 30.3_\n\nFrench Eclectic, _32.1_ , 32.2, _32.3, 32.4_\n\nGothic Revival, 17.1, _17.2, 17.3, 17.4_\n\nItalianate, _18.1_ , 18.2, 18.3, _18.4, 18.5_\n\nMillennium Mansions, 51.1, _51.2_\n\nMission, 34.1, _34.2, 34.3, 34.4_, 34.5\n\nModernistic, _40.1_ , 40.2\n\nNew Traditional, 52.1, _52.2, 52.3, 52.4_, 52.5\n\nQueen Anne, 23.1, 23.2, _23.3_ , 23.4, 23.5, _23.6, 23.7, 23.8, 23.9_\n\nRichardsonian Romanesque, _25.1_ , 25.2, 25.3, _25.4, 25.5, 25.6_\n\nSecond Empire, _21.1_ , 21.2, 21.3, _21.4_\n\nShingle, 24.1, 24.2, _24.3, 24.4, 24.5, 24.6_\n\nSpanish Revival, 35.1, _35.2, 35.3, 35.4_\n\nStick style, 22.1, _22.2, 22.3_\n\nTudor, 29.1, _29.2_\n\nTown, Ithiel, 15.1, 16.1\n\ntown houses\n\nFederal, 14.1, _14.2_\n\nGeorgian, 13.1, _13.2_\n\nGreek Revival, 16.1, _16.2_\n\nItalianate, 18.1, _18.2_\n\nQueen Anne, 23.1, _23.2_\n\nRichardson Romanesque, 25.1, _25.2_\n\nSecond Empire, 21.1, _21.2_\n\nStick style, 22.1, _22.2_\n\ntraditional neighborhood development (TND), _4.1_ , 4.2, 4.3, 53.1, 53.2\n\ntransit-oriented developments (TODs), _4.1_ , 4.2, 4.3\n\nTreat House, _23.1_\n\n_Treatise on the Theory and Practice of Landscape Gardening, A_ (Downing), 4.1, 4.2\n\nTremaine House, _31.1_\n\nTrevino House, _12.1_\n\nTrinity Church, Boston, Mass.\n\nTrost and Trost, _38.1_\n\nTruman Annex, Key West, _53.1_\n\n_Tudor Houses_ (Home Planner, Inc.), 50.1\n\nTudor Place house, _14.1_\n\nTudor Ranch style, _50.1_ , 50.2\n\nTudor style, _29.1_ , 29.2, _29.3, 29.4, 29.5, 29.6, 29.7, 29.8, 29.9, 29.10, 29.111, 29.12, 29.13_, 29.14\n\nfalse thatched roof, 29.1, _29.2_\n\nfront-facing gable with wing, 29.1, _29.2_\n\nfront-gabled roof, 29.1, _29.2_\n\ninfluenced by Craftsman\n\nmultiple-facade gables, 29.1, _29.2_\n\n\u2013New Traditional, _52.1_ , 52.2\n\norigins and development of, _1.1_ , 1.2, 29.1, 29.2, 29.3\n\nparapeted gables, 29.1, _29.2, 29.3_\n\nsingle dominant mid-facade gable, 29.1, _29.2, 29.3_\n\nsymmetrical paired gables, 29.1, _29.2_\n\nTully House, _12.1_\n\nTurner House, Dallas, _35.1_\n\nTurner House, Salem, _9.1_\n\n20th-century modern houses, _see specific types_\n\n21st-century modern houses, _48.1_ , 48.2, _48.3, 48.4, 48.5, 48.6, 48.7_, 48.8, app1.1\n\nd\u00e9coupage, 48.1, _48.2_\n\nneighborhood grouping, 48.1, 48.2, _48.3_\n\nsegmental vaults, 48.1, _48.2_\n\nsingular statements, _48.1_\n\nslightly askew, 48.1, _48.2_\n\nunifying materials, 48.1, _48.2_\n\ntwo-story style, 15.1, _15.2_\n\nUnifying materials style, 48.1, _48.2_\n\nUnruh, Antony, _48.1_\n\nUnruh Boyer, _48.1_\n\nUpham House, _6.1_\n\nUpjohn, Richard, 15.1, _17.1_\n\nUrban Design Associates\n\nurban neighborhoods (1750\u20131920), 4.1, _4.2_\n\nUrban Reserve house, _44.1, 48.1, 48.2_\n\nU.S. Green Building Council\n\nUsonian houses\n\nUtah, _4.1_ , 4.2, _5.1, 6.1, 25.1_, 28.1, _29.1_\n\nVan Alen House, _10.1_\n\nVan Bergen, John S.\n\nVanderbilt, George W., 30.1, 30.2\n\nVan Horn House, _10.1_\n\nVanna Venturi House, 47.1, _47.2_\n\nVan Schaick House, _10.1_\n\nVarner House, _27.1_\n\nVaux, Calvert, _17.1, 19.1_, 21.1\n\nVenturi, Robert, 47.1, 47.2, _47.3_ , 52.1\n\nVerdugo House, _12.1_\n\nvergeboards\n\nCraftsman, _39.1_\n\nexotic revival, _19.1_ , 19.2\n\nGothic Revival, _17.1_ , 17.2, _17.3, 17.4_, 17.5\n\nStyled Ranch\n\nTudor, 29.1, _29.2, 29.3_\n\nVictorian houses\n\nFolk Victorian, _26.1_ , 26.2, _26.3, 26.4, 26.5, 26.6, 26.7_\n\nmixtures of, _1.1_ , 1.2\n\norigins and development of, 1.1, _1.2_ , 1.3\n\nQueen Anne, _22.1_ , 23.1, _23.2, 23.3, 23.4, 23.5, 23.6, 23.7, 23.8, 23.9, 23.10, 23.11, 23.12, 23.13, 23.14, 23.15_, 23.16\n\nRichardson Romanesque, _25.1_ , 25.2, _25.3, 25.4, 25.5, 25.6_, 25.7\n\nSecond Empire, _21.1_ , 21.2, _21.3, 21.4, 21.5, 21.6, 21.7, 21.8_, 21.9\n\nShingle, _24.1_ , 24.2, _24.3, 24.4, 24.5, 24.6, 24.7, 24.8_, 24.9\n\nStick style, _22.1_ , 22.2, _22.3, 22.4, 22.5, 22.6, 22.7, 22.8_\n\nVierra, Carlos, _37.1_\n\nVierra House, _37.1_\n\nVigas, 37.1, 37.2\n\nVilla Escanado, _35.1_\n\nVilla Montalvo, _33.1_\n\nVillard Houses\n\n_Villas and Cottages_ (Vaux)\n\nVilla Vista, _22.1_\n\nVinci, John, _48.1_\n\nVi\u00f1oly, Rafael, _48.1_\n\nVirginia\n\nColonial houses, _9.1, 13.1, 13.2, 14.1, 14.2, 14.3_, 15.1, _15.2, 15.3, 15.4, 27.1_\n\nEclectic houses\n\nfolk houses, _6.1_\n\nRomantic houses, _17.1_\n\nVictorian houses, _1.1, 21.1_\n\n_see also_ Alexandria; Richmond\n\nVirginia, University of\n\nVogt, Lloyd, _52.1_\n\nVoss House, _27.1_\n\nVoysey, C. F. A.,\n\nWager House, _21.1_\n\nWalker and Gillette, _27.1_\n\nWalker House, _47.1_\n\nwall cladding, 3.1, _3.2_ , 17.1, 29.1, 52.1\n\nChateauesque, _30.1_\n\nColonial Revival, 27.1, _27.2, 27.3_\n\nContemporary, 45.1, _45.2, 45.3, 45.4_\n\nCraftsman\n\nDeconstructivist, _47.1_\n\nFrench Eclectic, _32.1_ , 32.2\n\nGeorgian, _13.1_ , 13.2\n\nGothic Revival, 17.1, _17.2, 17.3_\n\nInternational, 44.1, 44.2, _44.3, 44.4_\n\nItalian Renaissance, 33.1, _33.2_ , 33.3\n\nMillennium Mansion, _51.1_ , 51.2, 51.3, _51.4, 51.5_\n\nMinimal Traditional, 41.1, _41.2, 41.3_\n\nMission, _34.1_\n\nModernistic, _40.1_\n\nMonterey Revival, 36.1, _36.2_\n\nnew materials for\n\nNew Traditional, _52.1_ , 52.2, _52.3_ , 52.4, 52.5, _52.6_\n\nRanch, 42.1, 42.2, _42.3_ , 42.4, _42.5_ , 42.6\n\nShed, _46.1_ , 46.2, _46.3_\n\nShingle, _24.1_ , 24.2, 24.3\n\nSplit-Level\n\nStick style, _22.1_ , 22.2, 22.3\n\nTudor\n\n21st-Century Modern, 48.1, _48.2, 48.3_\n\nwindows as\n\n_see also_ brick cladding; stone cladding; stucco\n\nWallis, Henrik, _33.1_\n\nwalls\n\npictorial key to\n\nstructure, 3.1, _3.2, 3.3, 3.4, 3.5, 3.6, 3.7_\n\nwall texture variations, 23.1, _23.2_\n\nWalpole, Horace\n\nWalsh House, _31.1_\n\nWalter, Thomas U., 15.1, 16.1\n\nWalter Bogner House, _44.1_\n\nWarren House, Dallas, _28.1_\n\nWarren House, Surry County, _13.1_\n\nWashington, D.C.\n\nColonial houses, _14.1, 14.2, 14.3_, 15.1, 15.2\n\nEclectic houses, _27.1, 27.2_, 31.1, _31.2, 32.1, 33.1, 33.2, 34.1_\n\nfolk houses, _5.1_\n\nModern houses, _39.1, 42.1_, 45.1\n\nRomantic houses, 16.1, _18.1_\n\nVictorian houses, 21.1, _21.2_ , 23.1, _23.2, 25.1_\n\nWashington State\n\nModern houses, _39.1_\n\nVictorian houses, _21.1_\n\n_see also_ Seattle\n\n_Wasmuth Portfolio_ (Wright)\n\nwater and sewer systems, app1.1, app1.2\n\nWater Fall House, _47.1_\n\nWatson, Vernon S.\n\nWatts Sherman House, 23.1, _23.2_\n\nWeathers House, _52.1_\n\nWebb, Phillip S.\n\nWeeks, Levi, _15.1_\n\nWeightman House, _18.1_\n\nWeis House, _33.1_\n\nWelch, Frank, _52.1, 53.1_\n\nWellington, Charles, _21.1_\n\nWells House, _23.1_\n\nWelty House, _22.1_\n\nWentworth-Gardner House, _13.1_\n\nWesterfield House, _22.1_\n\n_Western Architect,38.1, 39.1_\n\nWest House, _20.1_\n\nWestlake District house, _40.1_\n\nWestover house, _13.1_\n\nWest Virginia, folk houses, _6.1, 7.1_\n\nWexler, Donald, _44.1, 45.1_\n\nWexler House, _45.1_\n\nWharton Place house, _14.1_\n\nWheaton House, _39.1_\n\nWheeler, Gervase, _17.1_\n\nWheelock and Clay, _23.1_\n\nWhite, Charles E., Jr.\n\nWhite, E. B., _16.1_\n\nWhite, Stanford, _17.1, 28.1_\n\nWhitehall house, _18.1_\n\nWhite House, Belfast, _16.1_\n\nWhite House, Buffalo, _27.1_\n\nWhite House, Selma, _18.1_\n\n_White Pine Series of Architectural Monographs_\n\nWilcox and Johnston, _25.1_\n\nWilder House, _27.1_\n\nWiley, Jim, _48.1_\n\nWilkins House, _6.1_\n\nWilkinson House, _15.1_\n\nWilliam House, _14.1_\n\nWilliams, Dave\n\nWilliams, E. Stewart, _47.1_\n\nWilliams-Erwin House, _23.1_\n\nWilliams House, Buffalo, _28.1_\n\nWilliams House, Deerfield, _13.1_\n\nWilliams House, Palm Springs, _47.1_\n\nWills, Royal Barry, 27.1, 50.1\n\nWilson House, Dallas (1953), _27.1_\n\nWilson House, Dallas (ca. 1899), _23.1_\n\nWilson and Webster, _40.1_\n\nWilton house, _13.1_\n\nwindow crowns, _12.1, 16.1_\n\nBeaux Arts, 31.1, _31.2_\n\nFederal, 14.1, _14.2_\n\nGeorgian, _13.1_ , 14.1\n\nGothic Revival, 17.1, _17.2, 17.3_\n\nItalianate, _18.1, 18.2, 18.3_, 18.4\n\nwindows, _3.1_ , 3.2\n\nAmerican Vernacular, _53.1_\n\nBeaux Arts, _29.1_\n\nChateauesque, 30.1, _30.2_\n\nColonial Revival, _27.1_ , 27.2, 27.3, _27.4, 27.5, 27.6, 27.7, 27.8_, 27.9\n\nDutch Colonial, 10.1, _10.2_\n\nEarly Classical Revival, _15.1, 15.2_\n\nFederal, 14.1, 14.2, _14.3, 14.4, 14.5, 14.6, 14.7_, 16.1\n\nFederal and Georgian, 14.1, _14.2_\n\nFolk Victorian, _26.1_\n\nFrench Colonial, 11.1, _11.2_\n\nFrench Eclectic, 32.1, _32.2, 32.3_\n\nGeorgian, 13.1, _13.2_ , 14.1, _14.2, 14.3_\n\nGothic Revival, 1.1, _1.2, 17.1_, 17.2, _17.3_ , 17.4, _17.5_ , 17.6, _17.7_\n\nGreek Revival, 16.1, _16.2_\n\nInternational, _44.1_\n\nItalianate, _17.1, 18.1_, 18.2, _18.3, 18.4, 18.5, 18.6, 18.7, 18.8, 22.1_\n\nItalian Renaissance, _29.1, 33.1, 33.2_\n\nMillennium Mansion, 51.1, _51.2, 51.3, 52.1_\n\nMinimal Traditional\n\nNeoclassical, 28.1, _28.2, 28.3_\n\nNew Traditional, 52.1, _52.2_ , 52.3, 52.4, _52.5, 52.6, 52.7, 52.8, 52.9_\n\npictorial key to, fm2.1, fm2.2\n\nPrairie, _38.1_\n\nQueen Anne, 23.1, 23.2, _23.3_ , 23.4, _23.5, 23.6, 23.7, 23.8_\n\nRanch, _42.1_ , 42.2, 42.3, _42.4_ , 42.5, _42.6, 42.7, 42.8, 42.9_\n\nRichardson Romanesque, _25.1_ , 25.2, _25.3, 25.4, 25.5_\n\nSecond Empire, 21.1, 21.2, _21.3, 21.4, 21.5_\n\nShed, 46.1, _46.2_\n\nShingle, 24.1, _24.2, 24.3, 24.4, 24.5_\n\nSpanish Colonial, _12.1_\n\nSpanish Revival, _35.1_ , 35.2, _35.3_\n\nStyled Ranch, 50.1, _50.2, 50.3_\n\nTudor, 29.1, _29.2, 29.3_, 29.4\n\nas wall cladding\n\n_see also_ arched windows; bay windows; Palladian windows; picture windows; ribbon windows\n\nwindow walls, _45.1_\n\nWinslow, Carleton, Sr., _35.1_\n\nWinslow House, Cleveland, _17.1_\n\nWinslow House, Perquimans County, _6.1_\n\nWinslow House, River Forest, 38.1, _38.2_\n\nWinton Guest House, _47.1_\n\nWisconsin, _5.1_ , 16.1, _18.1, 20.1, 27.1, 30.1, 38.1, app1.1_\n\nWise House, _48.1_\n\nWitmer House, _27.1_\n\nWolf House, _53.1_\n\nWood, Waddy B., _34.1_\n\nwooden structural systems, 3.1, 3.2, _3.3, 3.4_\n\nWoodfill House, _23.1_\n\nwood-framing systems, _3.1, 3.1_, app1.1\n\n_see also_ balloon-frame system\n\nWood House, _16.1_\n\nWoodlawn house, _14.1_\n\nWoodrow Wilson House, _27.1_\n\nWoodside house, _16.1_\n\nWoolf, John Elgin, 49.1, 49.2, _49.3, 49.4_, 49.5, 51.1\n\n_Works in Architecture of Robert and James Adam_ (Adam brothers)\n\nWorld's Columbian Exposition (1893), _see_ Chicago Columbian Exposition\n\nWorld War, I, II, 8.1, 26.1, 26.2, 27.1, 27.2, 29.1, 32.1, 33.1, 34.1, 35.1, 37.1, 38.1, 41.1, 41.2, 42.1, 43.1, 44.1, 44.2, 48.1\n\nWren, Christopher\n\nWright, Frank Lloyd, 1.1, 37.1, 38.1, _38.2, 38.3, 38.4, 38.5_, 38.6, 44.1, _44.2_ , 45.1, _45.2, 45.3, 45.4, 45.5_, 45.6, 47.1, _47.2_ , 47.3, app1.1, app1.2, _app1.3_\n\nWurster, William, 53.1, 53.2\n\nWyeth, Nathan C., _31.1, 33.1_\n\nYale Art and Architecture building\n\nYale School of Architecture, 46.1, 52.1\n\nYates House, _10.1_\n\nYeon, John\n\nYoung House, _27.1_\n\nZabriskie House, _10.1_\n\nzoning\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nCopyright \u00a9 2014 by Peter Reinhart  \nPhotographs copyright \u00a9 2014 by Paige Green\n\nAll rights reserved.  \nPublished in the United States by Ten Speed Press, an imprint of the Crown Publishing Group, a division of Random House LLC, a Penguin Random House Company, New York.  \nwww.crownpublishing.com  \nwww.tenspeed.com\n\nTen Speed Press and the Ten Speed Press colophon are registered trademarks of Random House LLC\n\nLibrary of Congress Cataloging-in-Publication Data\n\nReinhart, Peter.  \nBread revolution : world-class baking with sprouted and whole grains, heirloom flours, and fresh techniques \/ Peter Reinhart; photography by Paige Green.  \npages cm  \n1. Bread. 2. Cooking (Cereals) 3. Cookbooks. lcgft I. Title. II. Title: World-class baking with sprouted and whole grains, heirloom flours, and fresh techniques.  \nTX769.R4148 2014  \n641.81\u20325\u2014dc23\n\n2014015090\n\nHardcover ISBN: 978-1-60774-651-5  \neBook ISBN: 978-1-60774-652-2\n\nFood stylist: Karen Shinto  \nFood styling assistant: Jeffrey Larsen  \nPhotography assistant: Debbie Wilson  \nPhotography assistant: Morgan Bellinger  \nProp stylist: Tessa Watson  \nProp stylist assistant: Alexis Scarborough  \nSurfaces donated by Heritage Salvage, Petaluma\n\nv3.1\n\nIntroduction\n\nCHAPTER 1\n\nTUTORIAL\n\nCHAPTER 2\n\nA SOURDOUGH PRIMER\n\nCHAPTER 3\n\nSPROUTED FLOUR BREADS\n\nSprouted Wheat Pancakes\n\nSprouted Wheat or Spelt Quick Bread\n\nSprouted Wheat Quick Bread or Muffins\n\nSprouted Whole Wheat Bread\n\nSprouted Pain au Levain\n\nSprouted Rye Bread\n\nSprouted Wheat Pizza Dough\n\nGluten-Free Sprouted Flour Pizza Dough\n\nSprouted Wheat Focaccia\n\nSprouted Wheat Breakfast Focaccia\n\nSprouted Wheat Bagels\n\nSprouted Struan Bread\n\nSprouted Wheat Soft Rolls or Sandwich Bread\n\nSprouted Sandwich Rye Bread\n\nGluten-Free \"Do No Harm\" Sprouted Grain Bread\n\nSprouted Vollkornbrot\n\nSprouted Wheat Challah\n\nSprouted Multigrain Crackers\n\nGluten-Free Sprouted Grain Crackers\n\nSprouted Corn Bread\n\nGluten-Free Sprouted Corn Bread with Teff\n\nFlaky Sprouted Wheat Biscuits\n\nSprouted Wheat Sweet Potato Brioche\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\nSprouted Wheat Croissants\n\nCHAPTER 4\n\nSPROUTED PULP BREADS\n\nMultigrain Sprouted Wheat Pulp Bread\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Wheat Pulp Bread with Sprouted Multigrain Flour\n\nSprouted Kamut Pulp Bagels\n\nCHAPTER 5\n\nWHOLE GRAINS AND WHOLE MILLING\n\nWhole-Milled Lean Dough French Bread\n\nWhole-Milled Whole Wheat Ciabatta\n\nWhole Wheat Currant Pretzels\n\nWhole Wheat and Raisin English Muffins\n\nHigh-Extraction Pain au Levain\n\nNaturally Leavened Carolina Wheat Hearth Bread\n\nGluten-Free Many-Seed Toasting Bread\n\nGluten-Free Holiday Cookies\n\nGluten-Free Holiday Biscotti\n\nCHAPTER 6\n\nTHE NEXT NEW BREAD FRONTIER\n\nSyrah Grape Skin Flour Olive Bread\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\nGluten-Free Focaccia\n\nGrape Skin Flour Crackers\n\nCascara Seca Lean Bread\n\nProBiotein Lean Bread\n\nEPILOGUE\n\nIS THE ROAD LESS TRAVELED THE ROAD AHEAD?\n\nFragrant Peach, Apple, and Pear Bread with \"Peach Trap\" Starter\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nMozzarella, Milk, and Pear Bread with \"Coffee-Bean Trap\" Starter\n\nResources\n\nAcknowledgments\n\nAbout the Author\n\nIndex\n\n# INTRODUCTION\n\nBread\u2014well, actually wheat\u2014is once again in the crosshairs. And not just wheat, but other grains too, depending on who you believe and what your struggles are. There are a lot of theories about diet, wheat, grains, and carbohydrates floating around and, not surprisingly, they all seem plausible. And like it or not, bread is getting the blame for a lot of ills. I have baker friends who say, \"It will pass, just like the last scare.\" But I'm not so sure.\n\nFifteen years ago the Atkins diet was very popular, followed by the South Beach diet and other low-carb plans. Collectively, they took a big bite out of the bread market, and the immediate result was that Wonder Bread lost a lot of sales but somehow recovered\u2014until recently, that is, when the parent company declared bankruptcy. In 2003, when numerous reporters at a bread conference asked me, \"Is bread dead?\" my reply was, \"No. Bread has been with us for six thousand years; I don't think it's going away.\" But my less public response to my baker friends was, \"There's an opportunity here. It's time to focus on whole grain breads and make them as good as the artisan loaves. This is the future.\" And so they did\u2014not because of anything I said, but because it was the logical, intuitive, necessary thing to do. Yet here we are, ten years later, and thanks to the growing (and important) gluten-free movement and some recent popular books, even whole grain breads have a big bull's-eye on them. What on earth is going on? Is it possible, after six thousand years, that bread really is dead? I still say no, but once again we bakers are at a crossroads and need to ask, \"What is the opportunity within all of this concern?\"\n\n## ROOTS OF A REVOLUTION\n\nI've been thrilled and privileged to be in the midst of the American artisan bread revolution that began in the mid-1980s. Actually, its roots go back even further, as I've chronicled in earlier books, but it wasn't until the 1980s that things really took off. I remember the excitement that many of us felt as we metaphorically and literally sat at the feet of our mostly European bread heroes and learned their tricks of pre-ferments and soakers and how the relationship of time and temperature work on ingredients. The excitement of discovery was palpable as bakers and millers took field trips together to meet farmers and learn about the differences among wheat varieties and the influences of terroir.\n\nSoon, schools of thought emerged, with disciples of various bread masters working their way through dogmatic beliefs, arguing about the virtues of poolish versus biga, yeast versus wild yeast, mixing versus folding, and high-protein versus low-protein flours. They faced off at competitions and in the marketplace and railed against the mainstream. Then they softened\u2014integrating, expanding, and sharing their repertoires with each other and creating new schools of thought. American teams excelled at the Coupe du Monde de la Boulangerie (the World Cup of Bread or, as we call it, the Bread Olympics). American bakers, who had been cross-pollinating each other's approaches with growing stores of knowledge and expertise, became internationally influential. The home baking movement grew exponentially through the advent of baking websites and award-winning bread books, each adding a new technique or breakthrough method and posing questions that were previously taboo, such as, \"Do we really need to knead?\" and, \"Can bread that's partially baked and frozen and then rebaked be as satisfying as a freshly baked loaf?\"\n\nOver the years I've become friends with a lot of bakers, millers, and farmers. It's a wonderful community of earthy, spiritual, generous, and above all hardworking people. It's also a community of creative, resourceful, and resilient people. Six thousand years is a long lineage; bread and makers of bread are not going away. But I will say this: during bread's six-thousand-year saga, bread bakers have always been reinventing themselves and their craft.\n\nThis brings us to the current moment\u2014the opportunity at hand. If you've followed me through my literary journey with bread, you know that I'm fascinated by new frontiers and revolutionary turning points, whether cold fermentation, new ways to make whole grain breads, or even unconventional methods for making gluten-free bread. Early on, I learned that answers come by asking the right questions: what-if questions and questions that others are too timid or narrowly focused to ask. Some people have the tenacity to do one thing over and over again until they do it better than anyone else. They establish benchmarks and signposts for those who follow in their path. Others, more restless in spirit, step onto paths less traveled and forge new frontiers. Sometimes they go too far and disappear for a time\u2014or forever. But sometimes they stumble upon fertile ground and become the pioneers for the next wave.\n\nWhile I admire beyond words those who can relentlessly drill down deeper and deeper in their Zen-like quest for the perfect loaf, I tend to be even more fascinated by and drawn toward those adventurous souls who yearn for something not yet seen. I've lived in each world at different times, and I believe both are essential aspects of the journey. But at this crucial time and crossroads in the history of bread, I especially delight in exploring the as-yet-unknown and in meeting others who, each in his or her own way, expand the boundaries of what is possible. In this book, I've applied some of what I've learned from them to create new recipes and formulas, and I also share some of their recipes, insights, and stories.\n\nSome of the things these bakers are exploring address current questions related to health and nutrition, some focus on flavor, and some are responses to global, environmental, and holistic concerns. Each is a piece of the puzzle of how bread, glorious in its tradition, symbolism, and significance, is relevant at this time. As you'll see in the following pages, I think it is. In fact, I think bread is having, as it has so often throughout history, yet another revolutionary moment.\n\n## THE FRONTIER AT HAND: SPROUTED GRAIN FLOUR\n\nIn fall 2009 I got a call from Joe Lindley, the owner of Lindley Mills, located in Graham, North Carolina. I knew of Lindley Mills mainly as an independent, private-label organic mill whose most well-known client was King Arthur Flour. I was already using Lindley Mills flours at a pizza restaurant in Charlotte where I was a partner, and we were very happy with them. Lindley's multigrain blend was unique in that it was milled into a very fine powder, which gave it the ability to form fairly strong gluten bonds despite containing a number of gluten-free grains. For pizza dough, having a strong gluten network is critical for allowing the dough to stretch without ripping, so this flour was a revelation. However, I did have one concern: it resulted in a crust that was slightly drier, lacking the creamy texture of classic white dough. That said, it was still the best whole grain pizza dough I'd had to that point.\n\nToward the end of that restaurant's time, Joe Lindley called and asked if I'd be willing to try a new flour made with sprouted wheat that he was developing, called Super Sprout. He'd also developed a sprouted gluten-free flour blend that he called Sprouted Ancient Grain, made with sprouted amaranth, quinoa, millet, sorghum, and buckwheat. Like many people, I'm a fan of Ezekiel and Alvarado Street breads, which are both made with sprouted wheat kernels, so I asked Joe if his new flour was like what they used.\n\nHe said, \"No. At those places they sprout the wheat, then grind the sprouts into a wet pulp and then add other ingredients and mix it into a dough. The grain never actually becomes flour. With mine, I sprout the grain, then stabilize and dry it, and then mill those sprouted kernels into flour that can be bagged, stored, and shipped just like regular flour. It's a totally different product.\"\n\n\"But doesn't sprouting the wheat compromise the gluten and damage the starch?\" I asked. After all, millers had often warned me about this kind of starch damage. Although all flour has some starch damage that arises during harvest and storage, and also from the pounding the grain takes during the milling process, it falls within an acceptable range. Sometimes, during overly wet growing seasons or if stored wheat kernels are exposed to too much moisture prior to milling, starch damage can exceed acceptable levels. The resulting flour is either ruined or is considered inadequate for bread, as determined by its falling number (see Glossary). Using sprouted wheat, or any sprouted grain, to make flour seems to go against the conventional wisdom. In fact, the way Ezekiel and other bakeries that use sprouted wheat pulp get around this is by adding a relatively large amount of pure gluten, called vital wheat gluten, to the dough to provide structure. This allows the dough to bake up into what looks and tastes like bread made from regular flour.\n\nJoe said, \"You'd think sprouted wheat flour wouldn't work for bread, but for some reason it does, and I'm not totally sure why. I need you to try this and tell me if I'm crazy, but the breads I've made from it are really good, and I haven't needed to add any vital wheat gluten to it to make it work.\"\n\nA few days later, I received two boxes from Joe: one containing twenty-five pounds of Super Sprout wheat flour, and the other containing the Sprouted Ancient Grain blend. Joe advised me that the Super Sprout flour required greater hydration than regular whole wheat flour. \"It really sucks up the water,\" he said, then added, \"I think the key to what makes it work for bread is that I'm using the best-quality high-protein wheat I can find. And that isn't always easy, especially in the organic realm.\"\n\nAs most bakers know, all wheat is not created equal. Plus, during growth and processing it's subject to a number of factors that can create differences even in the same strain of wheat, such as amount of rainfall or irrigation, temperature, humidity, and soil quality. Hard wheat, aka high-protein wheat, can also vary in performance depending on whether the protein balance in the kernels is tilted more toward gliadin or glutenin, the two proteins that ultimately create gluten.\n\nIt was time for me to play with this flour and see for myself what Joe was getting so excited about. I mixed up a small batch of basic dough with about 85% water to Super Sprout flour. The water was quickly absorbed, and within a few minutes the dough seemed fairly firm\u2014a little too firm actually. So I worked in some more water and ended up with a very soft, sticky dough that felt similar to ciabatta dough. When I did the math, I had used 14.6 ounces (416 g) of water, which, by weight, is 91.25% of the 16 ounces (454 g) of flour. That's a lot! A typical white flour ciabatta has only about 75% to 80% water. Then, at five-minute intervals, I did a version of kneading involving four stretches and folds. Little by little, the dough firmed up into a supple, very tacky, pillow-like beauty. It had what I like to call bounce.\n\nAbout three hours later\u2014after a ninety-minute first rise, shaping, and a sixty-minute final rise, followed by thirty-five minutes of baking in my home oven on a baking stone, I tasted quite possibly the best 100% whole wheat bread I'd ever had. No sugar or honey, no oil, no pre-ferment, and no long, extended fermentation\u2014just flour, water, salt, and yeast. Suddenly, the artisan playbook no longer applied, and this was just my first attempt. I had been prepared to add oil and honey, and maybe milk, to the second go-round, as I would for a standard 100% whole wheat dough, but even without these the bread was soft, moist, and creamy or, as some bakers say, custard-like. This mouthfeel, which I prize in bread, is usually the result of long fermentation and a very hot oven. It can also be accomplished by including fats, sugar, and eggs in the dough, but the holy grail of artisan baking is to get these qualities without resorting to enrichments, as is sometimes achieved in the best baguettes, levains, and ciabattas. It's difficult to accomplish, though not impossible, with 100% whole wheat flour, and doing so usually entails using ample pre-ferments.\n\nLater, I made several doughs using a combination of Lindley's gluten-free Ancient Grain blend and the Super Sprout flour to create a multigrain version, finally settling on 20% Sprouted Ancient Grain to 80% Super Sprout. Eventually, I even came up with ways to use the ancient grain blend without any wheat at all, resulting in 100% gluten-free dough. In all cases, the natural sweetness and tenderness of the sprouted grain obviated the need for sweeteners or oils, though for loaf pan breads, soft dinner rolls, and sweet doughs, I did add some enrichments.\n\nI was having a lot of fun with this flour, and I began to realize that I was standing on the threshold of the next frontier in bread. In the pages that follow, I'll take you on my journey of discovery into this bread frontier. Along the way, I followed sprouted grain flour as if it were a breadcrumb trail, leading me to pulp made from sprouted grains, to millers and bakers who had controversial perspectives on baking with whole grains and wild yeast, and even back into the gluten-free world, which I explored in The Joy of Gluten-Free, Sugar-Free Baking. Along the way, I visited some arcane corners populated by unusual flours made from grape skins and seeds or from coffee cherries (the fruit that encloses coffee beans). It all adds up to an exciting time for bakers, ushered in by the emergence of sprouted grain flour and proving, once again, that bread is far from dead. Welcome to the new bread revolution!\n\nCHAPTER 1\n\n# TUTORIAL\n\nThis chapter provides basic information that you'll need to make the recipes throughout this book, including definitions and a step-by-step photo tutorial of the techniques and methods used. The recipes will sometimes refer back to these pages, but once you've made a few of them, you'll internalize most of the information here.\n\nAlthough the instructions in this book are easy enough for even beginning bakers to understand, the recipes and formulas do presuppose that you have some prior bread baking experience and understanding of baking terminology. For example, in my earlier book Peter Reinhart's Artisan Breads Every Day, I introduced a technique called the stretch and fold. This method is also used throughout the present book (and described in detail). I didn't invent the stretch and fold; it has been used by bakers for many years, probably for centuries. It has recently become popular, with artisan bakers using it in place of long mixing cycles, and now it shows up in many bread books. If you haven't encountered this or some of the other mixing, shaping, and baking techniques called for in the recipes, please refer to Working with Dough for details on handling, shaping, and proofing dough and setting up your oven for hearth baking.\n\nYou'll also notice that the recipes make frequent use of a technique that I call \"the oil slick method.\" As simple as it is, it nevertheless took twenty years of professional baking before I awoke to it and started lightly oiling the work surface rather than dusting it with flour. It isn't required, and you can still use flour if you prefer, but it will make your life a lot easier. Again, refer to the section Stretch and Fold.\n\nChapter 2 is devoted to naturally leavened (aka wild yeast) sourdough starters, and you should read through it before making any of the breads that call for a starter. As you become familiar with this approach, you may become a sourdough fanatic and possibly a mad scientist. In each of my recent books, as my own knowledge has grown, I've added to the ongoing conversation about how to make and use sourdough\u2014a conversation that may continue through the next six thousand years of bread making.\n\nThroughout the book are responses to recipe testers' most frequently asked questions, some of which may be questions of yours as well:\n\nWhat's the best way to scoop flour so that a cup is likely to match the listed weight?\n\nIf I don't have a scale, how do I accurately measure the salt?\n\nHow important is it to have the exact size of loaf pan called for?\n\nDo I need to adjust baking temperatures if using a convection oven?\n\nHow can I achieve a crisper crust?\n\nDoes it matter whether I keep a wet sponge mother starter or a firm dough mother starter?\n\nWhere can I get sprouted grain flour?\n\nWill pre-ferments, such as a poolish, biga, or sponge, improve breads made with sprouted flour? How about the overnight cold fermentation technique?\n\nCan I use the Dutch oven baking method for some of these breads even if you don't call for it?\n\nWhat are ancient grains?\n\nAre ancient grains gluten-free? And if they are, how does this impact bread making?\n\nCan I substitute one kind of ancient grain flour for another?\n\nCan I use regular whole grain flour in place of the sprouted flour in these recipes?\n\nCan sprouted and nonsprouted flours be combined in some recipes?\n\nCan I make a sourdough starter using sprouted flour?\n\nWhy do gluten-free breads often call for xanthan gum?\n\nCan these recipes be made using standard, nonsprouted gluten-free flours, rather than the sprouted versions?\n\nWhy bother with an egg wash? And why is egg wash sometimes made with egg whites and other times with whole eggs?\n\nIf a recipe calls for vital wheat gluten, can I make the bread without it?\n\nCan sprouted, gluten-free flours made from grains be substituted for the nut and seed flour in these recipes?\n\nCould a juicer be used to make the fruit waters?\n\nWill the recipe work if I use only apple or only peach or only pear water?\n\nWhy use bottled spring water?\n\nCan I use my own mother starter instead of Mike's \"trap\" starters?\n\n## TOOLS AND INGREDIENTS\n\nYou don't need a lot of tools to make great bread at home, but there are a few that are essential and others that are useful and can ease the process. You definitely don't need to run out and purchase every nice-to-have tool in order to get started. You really only need some mixing bowls, baking pans, a large mixing spoon, and rubber spatulas or something similar.\n\nMy two absolutely favorite tools are a plastic bowl scraper and a metal pastry blade. Of course, you'll also need an oven, but the type of oven doesn't matter\u2014all ovens can bake bread as long as they are fairly accurate. Almost everything else you'll need is probably already in your kitchen. You can add tools over time, but build your inventory slowly and consciously, as there are few things lonelier than fancy equipment sitting idle and unused.\n\nIngredients can also be divided into need to have and nice to have. Flour is obviously necessary, as is salt and, for many of the breads, yeast. Oil, milk, sweeteners, vegetable spray oil, and the like are called for fairly often, but these are usually present in home pantries. Some of the recipes do call for specialty ingredients, so read the ingredient list closely before starting. In addition, this book often calls for ingredients that may not be readily available to you, such as sprouted grain flour and whole-milled wheat flour. While I believe that many of these will become increasingly available in the near future, for now you may need to order some of these ingredients. The Resources section will direct you to suppliers for these.\n\nI've covered equipment and standard ingredients extensively in my previous books, so here I'll just share a few pointers and briefly address ingredients new to this book.\n\n### TOOLS\n\nBAKING STONE. There are many brands and thicknesses of baking stones. Typically, the thicker the stone, the more heat it will absorb and radiate back into the oven\u2014and any baked goods therein. You can now even find versions made from solid steel or cast aluminum, and both work very well.\n\nBANNETONS. These proofing baskets, made from bent wood, are expensive and can be hard to find, but they do make for nicely shaped loaves.\n\nBOWL SCRAPER. A flexible plastic bowl scraper is very inexpensive, yet it's also one of the most useful tools. It's especially handy for easing dough out of bowls and off of work surfaces without tearing the dough.\n\nCLOCHE. A cloche is a covered baking dish that simulates hearth baking, trapping moisture to create big oven spring. You can improvise one using a cast-iron Dutch oven or something similar.\n\nCOUCHE. Also known as proofing cloths, couches, which are usually made of linen, are used to gently support shaped loaves as they rise and help them retain their shape. You can improvise using lint-free sheets, pillowcases, or a white tablecloth.\n\nDIGITAL SCALE. I highly recommend using a digital scale to take the guesswork out of measuring. Opt for one that displays both ounces and grams.\n\nINSTANT-READ THERMOMETER. There are many brands of instant-read thermometers available, some with a digital display and others with a temperature dial. The probe should reach into the center of the loaf and quickly determine the internal dough temperature.\n\nLOAF PANS. These come in various sizes, usually 4 to 5 inches wide and 8 to 10 inches long. The loaf pan breads in this book will work in any of these sizes, and either metal or glass is fine.\n\nMEASURING CUPS AND SPOONS. I recommend measuring most ingredients by weight. But for small amounts, measuring spoons may be more useful than a scale.\n\nMIXING BOWLS. It's best to have a number of mixing bowls in various sizes to accommodate different volumes, and for ease in mixing dry and wet ingredients separately. I prefer using stainless steel bowls, but glass bowls are also fine, albeit more fragile.\n\nMIXING SPOONS. Either wooden or metal mixing spoons are fine for bread.\n\nOVEN. Of course, an oven is essential for baking, and just about any type will work: gas or electric, conventional (radiant heat) or convection (with moving air), and even wood-fired. If you have a convection oven you'll generally need to lower the baking temperature in these recipes by 25\u00b0F to 50\u00b0F (14\u00b0C to 28\u00b0C).\n\nPASTRY BLADE. Also called a bench blade or bencher, this flat, rectangular stainless steel blade is ideal for dividing doughs and scraping work counters.\n\nPEEL. A long, flat-handled wooden or metal paddle, a peel is used for sliding dough onto a baking stone and removing it once baked.\n\npH PAPER. When you are making sourdough starters and breads, pH paper can help determine the acidity level of your dough. It is available at many pharmacies and from baking supply companies. And while nice to have, it's not essential.\n\nPLASTIC WRAP. Yes, it's a disposable product, but plastic wrap will prevent a skin from forming on dough.\n\nPROOF BOX. Although these doughs can rise nicely at room temperature in a covered container, a temperature- and humidity-controlled proof box offers more control and consistency. There are now small portable proofers available, but you can also improvise your own using ice chests or microwave boxes with a cup of boiling water to provide warmth and humidity. Even a dishwasher can be used to create a warm, humid box, as can an oven with a pilot light or light bulb.\n\nSCORING KNIFE. Many implements can be used to score loaves before baking, so you don't really need a dedicated scoring knife (lame in French). You can use a double-edge razor blade mounted on coffee stirring sticks, or a sharp steak knife or bread knife.\n\nSEED OR SPICE GRINDER. For grinding small seeds, a dedicated electric seed or spice grinder works better than a blender. In a pinch, you can use an electric coffee grinder\u2014if you clean it well both before and after use.\n\nSHEET PANS. The most useful size for home ovens is about 12 by 17 inches.\n\nSILICONE BAKING MATS. Reusable nonstick pan liners are a nondisposable alternative to parchment paper for lining sheet pans. However, I recommend against using them on baking stones because the very high heat will wear them out quickly.\n\nSTAND MIXER. Many good brands are available. For bread making, it's best to choose one with a heavy-duty motor to decrease the chances of overworking it.\n\nWIRE COOLING RACKS. These are helpful to have on hand as bread cools more quickly on racks, which prevents moisture from being trapped underneath the loaf.\n\n### INGREDIENTS\n\nBREAD FLOUR. Choose unbleached bread flour if possible, preferably between 12% and 13% protein.\n\nBUTTER. The recipes in this book (and almost all baking recipes) call for unsalted butter, as this allows closer control of the amount of salt in a recipe.\n\nEGGS. The standard for baking is to use large eggs, which weigh approximately 1.75 ounces (47 g). I usually figure that one egg is approximately 1.25 ounces (35 g) egg white and 0.5 ounce (14 g) yolk, but this can vary from egg to egg.\n\nMILK. Milk serves a number of functions in baking, from hydration to contributing to tenderness, flavor, crust color, and nutrition. As a general rule, for most of the formulas in this book you may substitute nondairy milks such as rice, almond, flaxseed, or hemp milk, and can also use any richness level from skim to whole, unless otherwise specified. In many cases, you can even use water in place of milk for a leaner version.\n\nOILS. Any high-quality vegetable oil will work in most of the recipes. I prefer olive as well as grape seed oil, which is generally GMO-free and easy to digest, and tastes good. However, if you have a favorite oil whose flavor complements the bread you are making, feel free to use it, since it will provide the same functionality in terms of fat.\n\nSALT. The size of salt grains determines the weight per teaspoon, with finer grinds weighing more per teaspoon. Even kosher salt comes in various sizes depending on the brand, so weighing the salt is essential. Most of the recipes in this book were developed using standard table salt, but you can use any type as long as you weigh it or know the equivalency in teaspoons.\n\nSPROUTED GRAIN FLOURS AND SPROUTED GRAINS. These products are becoming increasingly available at natural food stores and even well-stocked supermarkets. If they aren't yet available to you locally, see the Resources section for online vendors.\n\nSWEETENERS. Syrups, such as honey and agave nectar, are heavier than dry sugar, so keep that in mind when substituting. Liquid stevia is highly concentrated, with 1 teaspoon (3 g) equaling 1 cup (227 g) of sugar or a sucralose sweetener such as Splenda. When substituting for sugar, I prefer liquid stevia over sucralose. You can also substitute honey or agave nectar unless otherwise specified, and many of the recipes provide quantities for these substitutions.\n\nVEGETABLE SPRAY OIL. For the recipes in this book, you can use pan spray made with either vegetable or olive oil.\n\nWHOLE WHEAT FLOUR. An exciting new development in the world of whole wheat flour is the recognition of the difference between whole-milled versus reconstituted whole wheat flour. (For more details, see chapter 5.)\n\nYEAST. The three types of yeast most commonly available are compressed fresh yeast (which is moist and crumbly), active dry yeast (which is granular and must be dissolved in warm water first), and instant yeast (which has the smallest granules and can usually be added directly to the flour in a recipe). The recipes in this book specify instant yeast. If you want to substitute active dry yeast, increase the amount by 25%. If substituting with fresh yeast, use three times as much by weight.\n\n## GLOSSARY\n\nThis isn't an exhaustive glossary of baking terms; rather, it's a list of some of the important terms used in this book. In some instances, where terms may have a number of definitions or have been used in other books in a different way, the following definitions clarify how the terms are used in this book.\n\nANCIENT GRAINS. This term can apply to both wheat and nonwheat grains and refers to strains that existed prior to the development of modern hybrids. For wheat, this means strains such as spelt, emmer, einkorn, and Khorasan wheat (aka Kamut, a trademarked name). Nonwheat ancient grains include amaranth, barley, buckwheat, millet, quinoa, sorghum, teff, and even corn, oats, and rice if they are heirloom, non-hybrid strains. More information is in this Q&A.\n\nBACTERIAL FERMENTATION. In contrast to yeast fermentation, this refers to the action of various strains of bacteria as they feed on sugars, breaking them down to release flavorful acids and esters. It is the primary source of sourdough flavor.\n\nBAKER'S MATH. See the sidebar on Recipes, Formulas, Master Formulas, and Baker's Math.\n\nENZYMES. Chemical proteins naturally found in all plants and animals, enzymes function to catalyze reactions, often breaking down complex molecules into more basic forms. Some enzymes break down proteins (protease enzymes), and some break down starches or carbohydrates (amylase enzymes). These two types of enzymes are critical to fermentation and digestive processes.\n\nFALLING NUMBER. This refers to how long it takes for a stick to fall through a mixture of flour and water, a measure used to determine flour quality in terms of level of enzyme activity and starch breakdown. The lower the number, the faster the stick falls. For bread, most bakers and millers prefer a falling number over 250 and even as high as 400. Below 250 usually means there's too much starch damage, making the flour unsuitable for creating good bread structure.\n\nFERMENTATION. Fermentation is a metabolic process in which sugars are converted into gases, alcohols, and acids. (See Bacterial fermentation and Yeast fermentation.) Bakers often use ferment to refer to the first, bulk rising of dough, whereas proof is used to refer to the rising of shaped loaves.\n\nFLOAT TEST. A method for determining whether enough gas has been created during proofing is to see if a piece of dough, such as a shaped bagel, will float when placed in a bowl of water. It can also be used to test a piece of sourdough starter to see whether it has enough activity to leaven a dough.\n\nFORMULA. See the sidebar on Recipes, Formulas, Master Formulas, and Baker's Math.\n\nGLUTEN. A strong protein made by the bonding of two smaller proteins\u2014glutenin and gliadin\u2014gluten gives dough its elasticity and extensibility and also traps gases to create air bubbles in the rising dough and baked bread. Gluten occurs mainly in wheat, rye, and, to a smaller extent, barley.\n\nLEAVEN. As a noun, leaven refers to any ingredient that causes a dough to rise. It can be biological (yeast), chemical (baking soda or baking powder), or physical (gas or air). As a verb, leaven means to cause to rise or, as its root implies, to enliven.\n\nLEVAIN. Levain is a French term for a natural leaven or starter made with wild yeast and bacteria, rather than commercial yeast (which the French call levure). It can be either firm or wet (if wet, it's a sponge). Levain may also refer to a specific type of bread, pain au levain, which is raised primarily with a levain starter.\n\nMALTING. When seeds of any type are soaked in water, they begin to sprout. In the process, natural enzyme activity releases maltose and other sugars stored as starches in the seed. Therefore, malting is closely akin to sprouting. (Because the enzymes are still active at this point, this type of malted grain is known as diastatic, because the diastase, or amylase, enzymes are still viable.) But in baking and brewing, malting may also refer to roasting the partially sprouted grain to various degrees of darkness to create other flavors. Roasting denatures the enzymes, resulting in nondiastatic malt, which has no enzyme activity.\n\nMASTER FORMULA. See the sidebar on Recipes, Formulas, Master Formulas, and Baker's Math.\n\nMOTHER STARTER. This is a natural leaven that's kept and fed, or refreshed, perpetually and from which other, smaller amounts of starter can be made for use in specific doughs.\n\nPREBIOTICS. Prebiotics are foods that serve as nourishment for healthful bacteria (probiotics) in the digestive system, such as fiber and enzyme-rich products.\n\nPRE-FERMENT. A generic term, pre-ferment refers to any dough that's made in advance to be added to a subsequent dough to improve flavor and texture. Types of pre-ferments include biga (a firm dough made with a small amount of yeast but without salt, of Italian origin); poolish (a wet sponge made with a small amount of yeast but no salt, of Polish origin and named and adopted by the French); sponge (a faster method using a higher amount of yeast in a wet batter); p\u00e2te ferment\u00e9e (a piece of old dough from a previous batch added to a subsequent dough, named by the French); and wild yeast or sourdough starter, aka levain (a naturally leavened pre-ferment).\n\nPROBIOTICS. Probiotics are beneficial bacteria and microorganisms in the digestive system. They are also present in yogurt and other cultured foods.\n\nPROOFING. Proofing is a term that refers to proving yeast is alive when activating the yeast in warm water or when raising (fermenting) dough. Bakers usually use the term fermentation for the first, or bulk, rise, and proofing for the final rise. I follow that convention in this book, but the two terms are interchangeable.\n\nRECIPE. See the sidebar on Recipes, Formulas, Master Formulas, and Baker's Math.\n\nRECONSTITUTED MILLING. This is a method of making whole wheat flour in which the bran and germ are separated from the endosperm early in the process and then added back in later. There is controversy as to whether all of the recombined portions are from the original batch of wheat that was separated or from other batches, and also as to whether the wheat performs the same as it would if left whole (see Whole milling).\n\nROOM TEMPERATURE. Room temperature is generally considered to be 70\u00b0F (21\u00b0C). If your home's ambient temperature is higher or lower than that, the fermentation times of your doughs may be either faster or slower than indicated in the instructions.\n\nSCOOP AND SWEEP MEASURING. This method of measuring dry ingredients by volume rather than weight, such as with cups, involves scooping the ingredient up in a measuring cup, overfilling it, and then using a flat tool, such as a bowl scraper, to sweep off the excess and obtain an even measure. (See this Q&A for more information.)\n\n## Q & A\n\n### What's the best way to scoop flour so that a cup is likely to match the listed weight?\n\nVolume measures are never as accurate as weights because of variations in how people scoop flour (and other ingredients), and also because the coarseness of the flour, the brand, and even humidity can all affect density and therefore volume accuracy. But if you're going to use volume measures, here's the method that should be most accurate: First, lightly aerate or fluff the flour with a whisk or even just a large spoon to break it up a bit so it isn't densely packed (don't sift, as that will overdo it). Then scoop the flour into your measuring cup, filling it over the rim. Sweep off the excess with a straight edge, such as a bowl scraper, a metal pastry blade, or even the back of a table knife. Don't pack the flour in this step; just let the sweeping do the job. The amount should be pretty close to the weight measure, but you may have to adjust the dough with more flour or more liquid as you mix. Therefore, when using volume measures it's especially important to follow the cues in the recipe instructions. The rule of thumb is always to let the dough dictate what it needs. In fact, even if you weigh ingredients accurately, the variability between types and brands of flour may necessitate adjustments.\n\n### If I don't have a scale, how do I accurately measure the salt?\n\nThere are many different forms of salt, and they vary in density. Therefore, weight measurements are always more accurate than teaspoons because a weighed ounce or gram will yield the same amount of sodium chloride regardless of how large or dense the salt crystals are. This will ensure that the ratio of salt to flour is correct. Further, a teaspoon is only as accurate as the spoon and the person doing the scooping, so it will always be approximate at best. However, if you must use volume measures, you need to know that the weights for salt listed in this book are for table salt, which typically weighs about 0.25 ounces (7 g) per 1 teaspoon. Scoop and sweep salt for the best accuracy, just as when measuring flour. Unless you are weighing it, I don't recommend using flaky sea salt, because these vary so widely depending on brand, crystal size, and density. If you'll be using a different form of salt, here are some details that can help you decide how much to use:\n\n\u2022 Table salt or fine sea salt (most brands):  \n1 teaspoon = 0.25 ounce or 7 grams\n\n\u2022 Morton kosher salt:  \n1 teaspoon = 0.21 ounce or 6 grams\n\n\u2022 Diamond kosher salt:  \n1 teaspoon = 0.11 ounce or 3 grams\n\nIf you use volume measures for salt, always taste the dough and make notes to help you settle on standard adjustments to use in the future based on the type of salt used. Too much or too little salt can adversely affect the dough's flavor, fermentation time, and structure of the dough. That said, for people on salt-restricted diets, it's okay to decrease the amount of salt by as much as 50%, but if you do so, also decrease the yeast slightly (by about 10%) and note the resulting fermentation times, since yeast acts more quickly in the absence of salt.\n\nSPROUTED GRAIN. This refers to grain of any type that has been sprouted only just until the shoot begins to show. Then it is either dried or pulped. Once dried, it can be milled into flour. (See also Malting and Sprouted pulp, below.)\n\nSPROUTED PULP. Once grain has been germinated and sprouted, it can be milled into a pulp that can be used to make bread dough. This is the method used for Ezekiel and Alvarado Street breads, as well as Giusto's Vita-Grain bread. (For more details, see chapter 4.)\n\nSTARTER. Starter is another word for a pre-ferment made with either commercial or natural, wild yeast.\n\nWHOLE MILLING. This is a milling method in which all components of the whole grain are milled to create flour, with minimum separation of bran, germ, and endosperm.\n\nYEAST FERMENTATION. Yeast is a microscopic fungus that causes fermentation by acting upon sugars, mainly glucose, to release carbon dioxide and ethyl alcohol. The most common bread yeast is from the species Saccharomyces cerevisiae, though many other species show up in sourdough starters.\n\n## WORKING WITH DOUGH\n\nMixing and shaping dough can be done in many ways, whether by hand or machine. The current rage is to \"properly undermix\" the dough and then gently strengthen it by performing a series of stretches and folds followed by intervals of resting the dough. The intervals can range from just a few minutes up to an hour, depending on the dough and the philosophy of the baker. Commercial artisan bakeries often use what is called the improved mixing method, first articulated by the great baking teacher Raymond Calvel. It begins with mixing slowly and gently for 4 to 6 minutes to hydrate the ingredients and begin developing the gluten. This is followed by mixing more quickly and intensively for 2 to 4 minutes to fully organize the gluten. Other popular techniques include the autolyse, which consists of a first mixing cycle without salt\u2014and sometimes even without yeast\u2014followed by a brief rest, perhaps 20 minutes, before adding the remaining ingredients and mixing at medium speed for 4 to 8 minutes.\n\nMost of the breads in this book are made using a short first mix followed by a series of stretch and fold sequences with short intervals between them. These methods aren't necessarily set in stone. The formulas are very flexible in terms of how they can be prepared. The photos that follow demonstrate the mixing and shaping methods I suggest, but if you have a good foundation in baking science and a preferred method of your own, feel free to use alternative techniques.\n\n## Recipes, Formulas, Master Formulas, and Baker's Math\n\nAlthough the terms recipe and formula are sometimes used interchangeably in cookbooks, for bakers they have distinct meanings, and even among bakers opinions differ as to what those meanings are. Here's what I mean when I use those words in this book: Recipe refers to the measurements of ingredients for a specific batch size and includes instructions. Formula refers to the ratio of each ingredient, by weight, to the total weight of flour, regardless of the batch size. Master formula refers to a formula template that can be used to create a number of bread variations. Examples in this book include Sprouted Whole Wheat Bread, and Sprouted Wheat Quick Bread or Muffins, which can be modified by adding a variety of ingredients or shaping the dough differently. In fact, almost all of the recipes in this book are so versatile that they could qualify as master formulas, providing a starting point for exploration and variations.\n\nThe ingredients for each recipe are presented in a table that includes a column for percentage, indicating the ratio between each ingredient and the total weight of flour. Armed with this information, you can use baker's math to calculate different batch sizes. Bakers are vitally interested in these ratios because they provide a concise synopsis of the type of bread being made. These ratios can't be determined based on volume measures because different ingredients have different densities.\n\nIn this book, you will see both recipes and formulas. If you decide to expand or even tweak a recipe, you can use the formula to help you make adjustments. I've explained this in great depth in some of my previous books, so rather than repeating all of the details here, I'll just recap the most important points:\n\n\u2022 In formulas, the most important reference point is the total weight of the flour, which is always 100%: the number on which the ratios of all other ingredients are based. To determine this ratio, divide the weight of the ingredient by the total weight of the flour. If the recipe includes more than one type of flour, the total weight of all the flours equals 100%. For example, in a recipe that uses 1,000 grams of flour (35.27 oz) and 20 grams of salt (0.7 oz), dividing 20 by 1,000 reveals that the ratio of salt is 0.02, or 2%.\n\n\u2022 Based on the formula, you can easily scale up a recipe. Say you'd like to triple the batch of the hypothetical bread in the previous bullet point. So you'd use 3,000 grams of flour. Then, you can multiply the total flour weight by 2% to determine how much salt you'll need: 3,000 \u00d7 0.02 = 60. Therefore, you'd need 60 grams of salt.\n\n\u2022 In formulas that use more than one kind of flour, such as a combination of whole wheat, rye, and bread flour, each flour is treated as an ingredient, with their total equaling 100%. So if the total weight of flour should be 1,000 grams and you want to use 700 grams of whole wheat flour, the other two flours would have to add up to 30% of the total flour. If you choose to use 50 grams of rye flour (5% of the total flour weight, or 50\/1,000), then you would use 250 grams of bread flour (25% of the total flour weight, or 250\/1,000).\n\n\u2022 Some bread books present the ratios based on all of the ingredients, including those in any pre-ferments and starters. This is a good way to do it, especially for professional bakers, but for home bakers I think it's easier to consider the pre-ferments as separate ingredients, rather than breaking out their flour and water and aggregating it with the total amounts of those ingredients. The formulas in this book are set up that way, with only the flour in the final dough constituting 100%, and pre-ferments and starters being listed like salt, yeast, and other ingredients, with a ratio to the flour in the final dough.\n\n### STRETCH AND FOLD\n\nLightly oil the work surface and place the dough on it. With wet or oiled hands, reach under the front end of the dough and stretch it out, then fold it back onto the top of the dough. Do this from the back end and then from each side. Then flip the entire mass of dough over and tuck it into a ball. The dough should be significantly firmer, though still very soft and fragile. Place the dough back in the bowl, cover the bowl, and let it sit at room temperature for 5 to 10 minutes. Repeat the stretch and fold process, then return the dough to the bowl again, cover the bowl, and let it sit at room temperature for 5 to 10 minutes more. Do this twice more. The entire process should be completed in less than 40 minutes. After the final stretch and fold, you can begin timing the first fermentation cycle or place the covered dough in the refrigerator, depending on the specific instructions in the recipe.\n\n### Stretch and fold\n\n### SHAPING\n\nOnce the dough rises, you'll need to shape it before letting it proof at room temperature. I suggest these common loaves as a starting point, but there are many options and I encourage you to experiment once you are comfortable with these shapes.\n\n#### Boules\n\nTo shape a boule, gently pat the risen dough into a rectangle, then bring all four corners together in the center. Squeeze the corners to seal them and tighten the skin of the dough to create surface tension. Use your hands to rotate the dough on the counter and make a tight, round ball. Using either the edge of your hand or your thumbs, press firmly on the bottom crease to exert pressure and tighten the surface. Flour a proofing basket or line a sheet pan with parchment paper, then dust it with flour, semolina, or cornmeal. Transfer the dough to the prepared proofing basket, seam side up, or place it on the prepared pan, seam side down, to proof.\n\n### Shaping a boule\n\n#### B\u00e2tards\n\nTo shape a classic b\u00e2tard, gently pat the risen dough into a thick rectangle. Fold the bottom half to the center and press with your fingertips to hold the dough in place and seal the seam. Fold the top half to the center and once again press with your fingertips to seal the seam. Roll the top half of the dough over the seam to create a new seam on the bottom of the loaf. Pinch the new seam closed with your fingertips or the edge of your hand to create surface tension on the outer skin, making a tight loaf. Gently rock the loaf back and forth to extend it to the desired length, typically 6 to 12 inches. To create a torpedo shape, taper the loaf slightly at each end with increased hand pressure while rocking the loaf. Transfer the shaped loaf to a floured proofing cloth or an oiled sheet pan, seam side down, cover, and proof.\n\n### Shaping a b\u00e2tard\n\n#### Baguettes\n\nTo shape a baguette, start by making a b\u00e2tard, as described above, then let it rest for 5 to 10 minutes. Repeat the same folding process: bottom to center, top to center, then pinching to create a seam. Seal the new seam with your fingers, thumbs, or the heel of your hand. It should create a tight surface tension. Then, with the seam side underneath, gently rock the loaf back and forth with your hands, moving out toward both ends and increasing the pressure at the ends to slightly taper the loaf. Repeat this rocking as needed until the baguette is the length of the sheet pan or baking stone. Transfer the shaped baguette to a floured proofing cloth or sheet pan that has been lightly misted with vegetable spray oil, cover, and proof.\n\n#### Sandwich Loaves\n\nTo shape a sandwich loaf, flatten the dough into a 5 by 8-inch rectangle. Working from a 5-inch side of the dough, roll up the length of the dough. Pinch the final seam closed using your fingertips or the back edge of your hand. Gently rock the loaf to even it out. Don't taper the ends; keep the top surface of the loaf even. Put the loaf in an oiled pan, seam side down so it touches both ends of the pan; cover and proof.\n\n### Shaping a sandwich loaf\n\n#### Rolls and Buns\n\nFor round dinner rolls: Place a 2-ounce (56.5 g) piece of dough on the work surface, cup your hand around it, then rapidly rotate the dough in a circular motion, as if trying to push it through the work surface. If need be, wipe the work surface with a damp towel to create traction to help you round the dough into a tight, smooth ball. Transfer the rolls to a parchment-lined sheet pan, cover, and proof. For pull-apart rolls, assemble any number of rounded balls of dough on a parchment-lined pan, just touching, so they'll rise into one another. After they bake, the rolls will easily pull apart.\n\n## Q & A\n\n### How important is it to have the exact size of loaf pan called for?\n\nSome of the recipes call for 4\u00bd by 8-inch loaf pans. It isn't advisable to use smaller pans, unless making mini loaves, but if you only have larger loaf pans, they should work just fine. But the loaves may take longer to bake than specified and will probably not be as tall (or you can slightly increase the weight of the dough).\n\n### Do I need to adjust baking temperatures if using a convection oven?\n\nConvection ovens are much more efficient than conventional ovens. However, many new convection ovens have two speeds of convection, so there is no single rule that assures a proper adjustment. My suggestion is that for convection ovens that offer a low setting, use that setting and decrease the temperature by 25\u00b0F (14\u00b0C). If the convection is very powerful, decrease the temperature by 50\u00b0F (28\u00b0C). The baking time is usually shorter in a convention oven than in a conventional oven, even with the reduced temperature, so you'll have to monitor the process and use the cues in the recipe to determine doneness.\n\n### How can I achieve a crisper crust?\n\nAfter baking, turn off the oven and leave the bread in the oven for 5 to 10 minutes longer to drive off more moisture. If the crust is getting too dark, cover the bread loosely with aluminum foil. Alternatively, you can put baked, cooled bread in a hot oven (450\u00b0F \/ 232\u00b0C) for 5 minutes to recrisp the crust prior to serving.\n\nFor hoagies: Divide the dough into 4-ounce (113 g) pieces for 7-inch rolls or up to 8-ounce (227 g) pieces for foot-long rolls. Flatten each piece of dough with your hand, then form it into a 4-inch torpedo shape, or a 7-inch torpedo shape for foot-long rolls, much as you would a b\u00e2tard. Let each piece of dough rest as you move on to the other pieces. When you return to the first torpedo, gently roll it back and forth to extend it out to about 7 inches, or 13 inches for a foot-long roll. The roll should have only a very slight taper at the ends. Place the rolls on a sheet pan lined with parchment paper or a silicone mat, spacing them about 2 inches apart. (The rolls may shrink back about 1 inch as you pan them.) Mist the tops of the rolls with vegetable spray oil, cover, and proof.\n\n#### Bagels\n\nTo shape bagels, prepare a sheet pan by lining it with parchment paper or a silicone mat, then mist it with vegetable spray oil or lightly coat it with oil. Divide the dough into 6 to 8 equal pieces. (A typical bagel is about 4 ounces or 113 grams before baking, but you can make them smaller or larger. If you make more than 6 bagels, you may need to prepare two sheet pans.) Form each piece into a loose ball by rolling it on a clean, dry work surface with a cupped hand. (Don't use any flour on the work surface. If the dough slides around and won't ball up, wipe the surface with a damp paper towel and try again.) There are two methods to shape the balls into bagels.\n\nThe first method is to poke a hole through the center of the ball to create a donut shape. Holding the dough with both thumbs in the hole, rotate the dough with your hands, gradually stretching it to create a hole about 2 inches in diameter.\n\nThe second method, preferred by professional bagel makers, is to use both hands (and a fair amount of pressure) to roll the ball into a rope about 8 inches long on a clean, dry work surface. (Again, wipe the surface with a damp towel if necessary to create sufficient friction on the work surface.) Taper the rope slightly at each end and moisten the last inch or so of the ends, if needed. Place one end of the dough in the palm of your hand and wrap the rope around your hand to complete the circle, going between your thumb and forefinger and then all the way around. The ends should overlap by about 2 inches. Squeeze the overlapping ends together by closing your hand, then press the seam into the work surface, rolling it back and forth a few times to seal. Remove the dough from your hand, squeezing it to even out the thickness, if need be, and creating a hole about 2 inches in diameter.\n\nPlace each shaped bagel on the prepared sheet pan, then mist with vegetable spray oil or brush with a light coating of oil. Cover the entire pan with plastic wrap and refrigerate overnight or for up to 2 days. (You can also proof the unshaped dough in an oiled bowl overnight and then shape the bagels on baking day.) Proof until they pass the float test, approximately 60 to 90 minutes, before boiling and baking.\n\n## Float Test for Bagels\n\nRemove the bagels from the refrigerator 60 to 90 minutes before you plan to bake them to test for readiness. Immediately check whether the bagels are ready for baking using the float test: Place one of the bagels in a small bowl of cold water. If it sinks and doesn't float back to the surface, shake it off, return it to the pan, and wait for another 15 to 20 minutes, then test it again. When one bagel passes the float test, they're all ready to be boiled. If they pass the float test before you are ready to boil and bake them, return them to the refrigerator so they don't overproof.\n\n### PROOFING\n\nAfter the dough is shaped, it usually needs anywhere from 1 to 2 hours at room temperature to rise, or proof (longer for some sourdough breads). You can accelerate the rising process by placing the dough in a warm place (see Proof Box). But if the dough warms up too quickly, the yeast ferments at a wildly uncontrollable pace. This can easily overferment the dough and ruin both the flavor and the color of the bread, so be careful! Whichever proofing method you use, when the dough has fully risen, gently transfer the dough onto a peel or the back of a sheet pan dusted with semolina, flour, or cornmeal, then proceed with scoring and baking.\n\n#### Bannetons and Couches\n\nBannetons are baskets used to provide structure for dough as it proofs, and couches (which literally means \"bed\" or \"diaper\") are the linen proofing cloths many bakeries use for freestanding loaves. If you don't have professional bannetons or couches, you can improvise with standard kitchen equipment such as stainless steel or glass mixing bowls for bannetons and a white tablecloth or old pillowcase for a couche. The size of the bowl depends on the size of the loaf, but as a rule of thumb, the bowl needs to be twice as large as the piece of dough going into it to accommodate the rise.\n\nTo proof in a bowl, line a stainless steel or glass mixing bowl with a smooth, lint-free cloth napkin, scrap of fabric, or towel. Mist the fabric with vegetable spray oil, then lightly dust it with flour. Place the loaf in the bowl, seam side up, and mist the top with vegetable spray oil. Cover with flaps of the fabric or a separate cloth.\n\nTo proof in a cloth, lightly mist the surface with vegetable spray oil and dust with flour before transferring your loaves onto the cloth, spacing them about 3 inches apart. With the loaves placed, bunch up the fabric between the loaves to make walls to support the dough, then cover with more cloth or plastic wrap.\n\n#### Sheet Pans\n\nTo proof on a sheet pan, line the pan with parchment paper, mist with vegetable spray oil, and dust with cornmeal, semolina, or flour. Mist the shaped dough with vegetable spray oil and loosely cover with plastic wrap or a towel.\n\n### SCORING\n\nScoring bread releases some of the trapped gas, which helps create proper crumb, and it makes for attractive loaves. To score loaves, wait until just prior to baking, then place the shaped dough on the back of a floured sheet pan or baking peel. Using a razor blade, lame, or serrated knife, score the loaf about \u00bd inch deep. If using a razor blade, keep the back end up so it doesn't drag through and tear the dough. I often tell my students to say the word slit when they make the cut to emphasize an action like slitting open an envelope over any other notion of cutting. The cut shouldn't go straight down but rather be on an angle, so that it's almost parallel to the surface of the bread. If you prefer to use a sharp serrated knife or another type of blade, remember to let the knife do the work and resist the urge to press down on the dough.\n\n### HEARTH BAKING AND CREATING STEAM\n\nA number of the breads in this book are designed to be baked at high temperatures, preferably on a hearth of some type. A baking stone is the most popular option for home hearth baking, but not everyone has one. If you don't have a baking stone, it's perfectly okay to bake on a sheet pan. You also need steam to enhance oven spring and put a shiny, crackly crust on the bread. There are many ways to create a blast of steam, but my preferred method is to use a steam pan.\n\nTo prepare your oven for hearth baking, put a baking stone or sheet pan on the middle rack of the oven and a steam pan on the bottom. For the steam pan, you can improvise by using a sheet pan with a 1-inch rim, a casserole pan with taller sides, or a cast-iron frying pan. Preheat the oven for at least 45 minutes to fully heat the baking stone.\n\nWhen you're ready to bake, slide the shaped dough onto the preheated baking stone, then lay a kitchen towel over the oven's glass window to protect it from any potential backsplash. Wearing an oven mitt to prevent burns, pour about 1 cup of hot water into the preheated steam pan. (I like using a watering can because of the control and distance the spout provides.) Using a spray bottle such as a plant mister, you can also spritz the oven walls a couple of times to create additional steam.\n\nCHAPTER 2\n\n# A SOURDOUGH PRIMER\n\nEvery time I write a bread book, I find it necessary to update my previous instructions on how to make and work with sourdough starters. One reason is that I keep learning more about the world of natural leavening, and another reason is the natural variability in the world of microorganisms. What worked in the past for a starter or what works in some regions may not work in your neighborhood.\n\nAfter perfecting what I thought was the simplest, most foolproof method for my book The Bread Baker's Apprentice, I discovered that it didn't always work. For many readers, the first stage of preparing the starter, creating a seed culture, wasn't proceeding as I'd said it would. I never had that problem, but based on emails I received (and continue to receive even now), the problem is apparently pretty widespread. In response, a team of amateur sleuths and passionate home bakers, led by chemist Debra Wink of Columbia, Missouri, tackled the problem on the King Arthur Flour website a few years ago in a discussion group called the Baking Circle. (This discussion is preserved in the King Arthur archives online, if you'd like to read it in full.)\n\nThe short story is that Debra put the starter under a microscope and came up with the theory that the cause of the problem was leuconostoc, a strain of lactic acid bacteria that generates a lot of carbon dioxide (as yeast does) but also temporarily interferes with the growth of wild yeast. Based on this discovery and feedback that I received from hundreds of home sourdough bakers, I've continued to tweak the method to make it as reliable as possible while also being flexible enough to work with different flours and in both firm starters and wet sponges.\n\nThis chapter offers my latest update, built upon what I've written before. However, if you use a different method or already have an established mother starter on hand, feel free to stick with what works for you.\n\n## UNDERSTANDING SOURDOUGH STARTERS\n\nBy sourdough, I mean dough leavened naturally, with wild yeast and bacteria, as opposed to dough leavened by commercial yeast. Breads leavened with sourdough starters have a number of appealing qualities, some related to flavor and some that are more romantic and evocative. There's something compelling about capturing wild yeast and bacteria and harnessing their ability to raise dough; it's very craft-like and artisanal.\n\nWild yeast starters go by any number of names, such as levain, chef, mother, madre, barm, wild yeast sponge, or simply sourdough starter. Because a sourdough starter must be fermented in advance, it functions very much like other types of pre-ferments as a flavor enhancer, while also often carrying most or all of the responsibility for leavening. However, the flavor of bread made with a sourdough starter is distinctive because a sourdough starter has a more complex flavor profile than a pre-ferment made with commercial yeast, such as a sponge, biga, or poolish. Therefore, sourdough starters are sometimes used in conjunction with commercial yeast to serve as a flavor booster more than as leaven, such as in some rye breads.\n\nThe acidity generated in a sourdough starter is important for both flavor development and control of enzymatic activity in the dough. Many people also believe sourdough starters help with digestibility, as the naturally occurring yeast and bacteria essentially predigest the grains, making it easier for the body to break them down further.\n\nThere are many classic versions of sourdough bread, though they aren't always referred to as sourdough. Pain au levain, for instance, a classic French bread, is typically made using both natural leaven (levain) and a small spike of commercial yeast, which helps speed the process just enough to reduce the sourness. (Unlike Americans, the French prefer a less sour flavor profile and one Frenchman told me he thought of American sourdough as \"ruined bread.\")\n\n## The Science of Sourdough Starters\n\nBefore discussing considerations about sourdough starters when baking with sprouted flours, and before getting into the details of making a starter, I'd like to hone in on the science within a sourdough starter so you can understand the forces at play. The most common misperception about sourdough starters is that wild yeast is what causes the sour flavor, but there's actually a more complex microbial drama taking place. Various strains of wild yeast are living side by side with various strains of bacteria, and the bacteria cause the sour flavor as they metabolize sugars and convert them into lactic acid or acetic acid. I use the plural strains because wild yeast is not just one specific species, such as Saccharomyces cerevisiae, an organism in both commercial yeasts and the family of wild yeasts. \"Wild yeast\" refers to an indeterminate and variable number of strains, mostly within the species category Saccharomyces exiguus.\n\nFrom a functional standpoint, the role of the wild yeasts is to leaven and slightly acidify the bread via production of carbon dioxide and ethyl alcohol, while the function of the bacteria is to acidify and flavor the dough and, to a lesser degree, create some carbon dioxide. The acidifying work of the bacteria lowers the pH level of the dough to create an environment ideal for the growth of the desired strains of wild yeast. As the dough becomes more acidic, commercial yeast, cultivated in the controlled conditions of a laboratory, doesn't survive, whereas wild yeast does. (In addition, acidity also controls the population of certain strains of bacteria, including leuconostoc.) Thus, the yeasts and bacteria exist in a state of symbiotic reciprocality that allows them to harmoniously share the same environment and food source while supplementing each other's work. In the process, different strains of bacteria interact with various strains of yeast to create different flavor compounds. This explains why breads made in various parts of the world may have different flavors even when made with the exact same formula and ingredients.\n\nIn essence, a sourdough starter is simply a medium in which microorganisms colonize and multiply, all the while creating important by-products, including alcohol, carbon dioxide, lactic acid, and acetic acid. In other words, dough fermentation. To harness this power, bakers use the amount of starter necessary to raise the final dough, which can range from 15% up to, in some instances, 100% of the weight of flour in the final dough. In most bakeries, it's usually in the range of 20% to 33%.\n\nAs you can see, this microbial world is very complicated. It's also very fortunate, as this complexity manifests itself in the final flavor of well-made bread, with a similar dynamic being at play in the creation of great cheeses, beers, and wines.\n\n## USING SOURDOUGH YEAST STARTERS WITH SPROUTED GRAIN FLOUR\n\nAs we explore the next frontier of bread, we must find the best ways to utilize sourdough starters with sprouted grain flours, which are inherently high in enzyme activity. This opens up a host of new questions: in particular, whether it's possible to make a sourdough starter with sprouted grain flour; and if so, whether this results in a superior final dough. I'll address those issues here, and what I have to say might surprise you.\n\nAs noted earlier, doughs made with sprouted grain flour don't benefit from the use of a pre-ferment as much as conventional doughs do. The flour is already preconditioned (though not technically pre-fermented) by the sprouting process, releasing its flavor. So why use a sourdough starter in sprouted flour breads? The main reason is that bacterial fermentation produces complex acidic flavors that are often beloved in bread. (See The Science of Sourdough Starters.) Therefore, a sourdough starter provides a level of complexity that sprouted flour can't attain on its own.\n\nAs mentioned, fermentation by bacteria and wild yeasts acts upon conventional, nonsprouted flour in a way that makes it more digestible. Sourdough bread is generally considered more healthful and more easily assimilated than commercially yeasted bread, and it seems that the longer the fermentation, the easier it is to digest. There's a growing community of serious sourdough bread enthusiasts who fervently believe that sourdough bread made via a long, super slow rising process is the only healthy bread, and they're probably correct, to a point. But when bread is made from sprouted flour, we don't need that benefit; the enzymatic activity initiated by the sprouting process accomplishes the same goal. So, in the end, it's all about flavor, as is often the case when it comes to food.\n\nYou may wonder whether you can make a wild yeast starter using sprouted grain flour. I've experimented with this, and I've found that it is possible, but it's tricky because there's so much enzyme activity in sprouted flour that it doesn't stand up well over the long duration required to building up the population of microorganisms. Although it certainly feeds and supports the microorganisms, sprouted flour more quickly loses its structural integrity. Therefore, I suggest beginning with nonsprouted flour and then later, when the time comes to make the final dough, possibly switching to sprouted grain flour to build an intermediate or final starter.\n\nAll of this science may seem unnecessary, but I assure you that understanding the underlying details will help you feel more confident in making and using your own starter. So let's begin. First I'll provide instructions for an all-purpose mother starter, one that can be used in any of the breads that call for natural starter or levain. Then, at the end of this chapter, I'll take you on a brief journey into another new frontier: a raisin wild yeast starter. Later, in the Epilogue, I'll introduce you to yet another method for making starters\u2014one that's so far on the outer edge of the bread revolution that it needs its own chapter.\n\n## MAKING A SEED CULTURE\n\nThere are many ways to make a seed culture. The simplest is with just flour and water, and while it generally works, it doesn't always develop on a predictable schedule. I've seen methods online that call for everything from onion skins, wine grapes, plums, or potatoes to milk, buttermilk, or yogurt. These can all serve as fuel for the microorganisms, and all will work for making a seed culture. But in the end, a starter (and bread in general) is really about fermented flour. So my goal is to use flour to create the conditions in which the appropriate organisms can flourish.\n\nThe following method produces a versatile starter that can be used to make both 100% sourdough breads and mixed-method breads, leavened with a combination of wild yeast starter and commercial yeast. The starter can be made from whole wheat flour, white flour, or whole rye flour. (Rye bread fanatics tend to keep a rye-only starter, but I've found that a wheat starter works just as well in rye breads.)\n\nAt this first stage, you aren't making the starter that goes into the final dough, but rather a \"seed\" starter used to create another starter: the mother starter. In most bakeries, the mother is used to build yet another starter, the levain, also called the chef in some bakeries, or simply the final starter. That's the one that goes into the final dough. For the purposes of small-batch home baking, you can generally use your mother starter as the final starter as long as it's been refreshed within the past 2 to 3 days.\n\n### THE PINEAPPLE JUICE SOLUTION\n\nTo avoid the timing problems of some of my earlier instructions, and to reflect the work of Debra Wink and her many cohorts in the King Arthur Baking Circle, I now use what I call \"the pineapple juice solution\" when creating a seed culture. Pineapple juice isn't the only acidic ingredient that can control leuconostoc bacteria, but it is reliable and can also reduce the time required to get a seed culture going by a couple of days. Other acidic fruit juices can do the same. Some of my recipe testers have reported great success with orange juice or diluted lemon juice.\n\nOf course, you can also use water, but the pineapple juice introduces enough acidity to encourage the growth of wild yeast earlier in the process. If you prefer to use water, it's best to go with filtered or spring water to minimize chlorine, but I've made it with regular tap water too. Aerating the starter at least once every 8 hours by stirring it vigorously is helpful for stimulating the growth of wild yeast, as yeast multiplies more rapidly in the presence of oxygen. Being conscientious about stirring a starter can do wonders for a seed culture that's slow to develop.\n\nOpinions are divided on whether it's necessary to use pineapple juice after Day 1. It probably isn't, but it won't hurt to use it on Day 2, and in some instances, it may serve as added insurance againist overgrowth of leuconostoc. That said, it's fine to forgo the pineapple juice and simply use water after Day 1.\n\nThe four phases of seed culture (clockwise from top left)\n\n### A FEW WORDS ON TIMING\n\nAs you embark on the process below, be aware that, even with the pineapple juice solution, the timing of the various phases is approximate and can be greatly affected by the time of year, ambient temperature, and aeration. At cool temperatures, it may take twice as long for the seed culture to become active, and at warm temperatures it can happen very quickly. If the process seems to be taking longer than predicted, don't abandon it. Carry on, faithfully stirring the mixture at least once every 8 hours.\n\nAnother benefit of aeration is that it protects the seed culture from invasion by microorganisms that cause molds, which tend to settle on the surface and grow on the sides of the container. But if you stir regularly, the desired microorganisms will be distributed throughout the seed culture, enhancing their ability to destroy or control the invaders. This is especially important during the dormant phase, when it appears that nothing is happening.\n\nSpeaking of the dormant phase, don't be fooled by bubbling during the first day or two. This doesn't mean the starter is active; rather, it's probably evidence of leuconostoc or other bacteria. The starter will soon go through a seemingly dormant phase in which you might think something has gone wrong. This is just a transitional period during which the starter is slowly acidifying and the wild yeast and lactic acid bacteria are slowly multiplying. Again, continue to aerate faithfully, and wait for the signs of well-established fermentation, probably 5 to 9 days into the process.\n\nIf progress remains slow, you can repeat the Phase 4 step, or you can simply extend the waiting period between phases by a day or two. Often the key is to give the seed culture more time. In the end, the microbial good guys prevail.\n\n### PHASE 1 SPONGE (DAY 1)\n\nNOTES:\n\n\u2022 You can substitute white flour, but if you do, increase the amount by 1 tablespoon (0.3 oz \/ 8.5 g). This also applies to Phase 2.\n\n\u2022 The temperature of the liquid doesn't matter (as long as it's not hot), and this remains true throughout the process.\n\n\u2022 To make lemon water, combine 3 parts water with 1 part lemon juice.\n\nwhole wheat or whole rye flour (see at left): 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nunsweetened pineapple juice (or filtered or spring water, orange juice, or lemon water; see at left): \u00bc cup (2 oz \/ 56.5 g)\n\nTOTAL: 3 oz \/ 85 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole wheat or whole rye flour| 100  \nunsweetened pineapple juice| 200  \nTOTAL| 300\n\nBecause the aim is to create a population of desirable microorganisms, all tools and bowls should be sanitized in advance, either in boiling water or in a dishwasher. In a small bowl, stir together the flour and juice with a spoon or whisk until the flour is fully hydrated, resulting in a sponge with the consistency of thin pancake batter. Cover the bowl with plastic wrap or a lid (not too tightly, so that the mixture can breathe). Leave the mixture out at room temperature for 48 hours, stirring it with a spoon or whisk several times each day, at 8-hour intervals, to aerate it. There will be few or no bubbles or other signs of fermentation during the first 24 hours. Bubbles should begin to appear within 48 hours, but even if they don't, move on to Phase 2.\n\n### PHASE 2 SPONGE (DAY 3)\n\nwhole wheat or whole rye flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nunsweetened pineapple juice or filtered or spring water: 2 tablespoons (1 oz \/ 28.5 g)\n\nPhase 1 Sponge: all (3 oz \/ 85 g)\n\nTOTAL: 5 oz \/ 142 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole wheat or whole rye flour| 100  \nunsweetened pineapple juice or filtered or spring water| 100  \nPhase 1 Sponge| 300  \nTOTAL| 500\n\nAdd the new, Phase 2 ingredients to the Phase 1 sponge and stir with a spoon or whisk to distribute and fully hydrate the flour. Again, cover with plastic wrap or a lid and leave out at room temperature for 24 to 48 hours, stirring with a spoon or whisk several times each day. There should be signs of fermentation (bubbling and expansion) during this period. When the mixture becomes very bubbly or foamy, continue to Phase 3. If there isn't any bubbling after 48 hours, continue to stir at 8-hour intervals until the mixture begins to bubble. This may take a couple of extra days, but in most circumstances the bubbling will occur within 48 hours.\n\n### PHASE 3 DOUGH (DAY 4 OR 5)\n\nwhole wheat or unbleached bread flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nfiltered or spring water: 2 tablespoons (1 oz \/ 28.5 g)\n\nphase 2 Sponge: all (5 oz \/ 142 g)\n\nTOTAL: 7.5 oz \/ 213 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole wheat or unbleached bread flour| 100  \nfiltered or spring water| 67  \nphase 2 Sponge| 334  \nTOTAL| 501\n\nAdd the new, Phase 3 ingredients to the Phase 2 sponge and stir with a large spoon. Because of the reduced percentage of water, the sponge will become thicker, but it will still be very wet and sticky. Again cover with plastic wrap or a lid and leave out at room temperature for 24 to 48 hours, stirring with a wet spoon at least twice each day. Within 48 hours the mixture should be very bubbly and expanded. If not, let it develop for another day or two, stirring at 8-hour intervals, until it becomes active. If the Phase 2 sponge was very bubbly prior to this feeding, it could become active and bubbly in less than 24 hours. If so, proceed to Phase 4.\n\n### PHASE 4 DOUGH (DAY 5 OR LATER)\n\nwhole wheat or unbleached bread flour: 7 tablespoons (2 oz \/ 56.5 g)\n\nfiltered or spring water: 3 tablespoons (1.5 oz \/ 42.5 g)\n\nPhase 3 Dough: half (3.75 oz \/ 106 g)\n\nTOTAL: 7.25 oz \/ 205 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole wheat or unbleached bread flour| 100  \nfiltered or spring water| 67  \nPhase 3 Dough| 188  \nTOTAL| 355\n\nDiscard or give away half of the Phase 3 dough. Add the new, Phase 4 ingredients to the remaining Phase 3 dough and stir with a large spoon. The seed culture will become firmer, but still be slightly sticky or very tacky. Again, cover with plastic wrap or a lid and leave out at room temperature until bubbly and foamy, with an aroma similar to apple cider vinegar. It should register between 3.5 and 4.0 if tested with pH paper, and should swell and nearly double in size, though it may fall when jostled. Completion of this phase can take anywhere from 4 to 24 hours. If there are few signs of fermentation after 24 hours, stir the seed culture as before or knead it, and continue to leave it out at room temperature, covered, until it rises. Once fermentation has reached this point, you can proceed to the next step, building the mother starter, or you can refrigerate the seed culture for up to 2 days before proceeding.\n\n## MAKING A MOTHER STARTER\n\nOnce you've established a seed culture, you need to convert it into a mother starter. This is the starter you'll keep perpetually in your refrigerator and use to make doughs. (In previous books I sometimes referred to this starter as a barm, but I've since learned from my wild yeast guru, Monica Spiller, that the term barm is more properly used for starters made with mashed [scalded] grain as the growth medium.)\n\nTo build your seed culture into a mother starter, you'll use a portion of it to inoculate a larger batch of flour and water and make a firm piece of dough. Although the seed culture is full of wild yeast and bacteria, its structure has been weakened by acidity and ongoing enzyme activity. To make the mother starter strong enough to function in the final dough, you'll use three times as much flour, by weight, as seed culture. Hydration will be 67% the weight of the new flour, to create a firm mother starter. (You could also use up to 100% water for a sponge starter, but the formulas in this book are based on using a firm mother starter at 67% hydration.)\n\nA little seed culture goes a long way, so the following instructions call for discarding or giving away about half of the seed culture. (You can save somebody else a few days of work by giving them half of yours.) If you'd prefer to keep a larger mother starter on hand, perhaps if you bake often or tend to bake larger batches, you can convert the entire seed culture into a mother starter by doubling the weight of the new flour and water. This will give you more than 3 pounds (1.36 kg) of starter. However, most home bakers prefer keeping a smaller batch.\n\n## Q & A\n\n### Does it matter whether I keep a wet sponge mother starter or a firm dough mother starter?\n\nSome people prefer a wet sponge because it's easy to rebuild: simply stir in new flour and water with a spoon. I used to follow this approach, but now I prefer to keep a firm mother starter. Despite requiring some kneading or mixing with a dough hook, it's also fairly easy to rebuild. The reason for my preference is that it holds its structural integrity longer than a sponge. Also, I like the flavor of a firmer starter; it seems slightly tangier and more acidic. However, if you prefer keeping a sponge starter at, say, 100% hydration (equal parts water and flour by weight), then you'll need to decrease the amount of water in the final dough accordingly. Since most of the formulas in this book use high-hydration ratios, the amount of adjustment will probably be only a matter of a tablespoon or two of water.\n\n### MOTHER STARTER\n\nunbleached bread flour or whole wheat flour: 2\u2154 cups (12 oz \/ 340 g)\n\nfiltered or spring water: 1 cup or 1 cup plus 2 tablespoons if using whole wheat flour (8 or 9 if using whole wheat flour oz \/ 227 or 255 if using whole wheat flour g\n\nPhase 4 Seed Culture: about half (3.25 oz \/ 92 g)\n\nTOTAL: 23.25 or 24.25 oz \/ 659 or 687 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nunbleached bread flour or whole wheat flour| 100  \nfiltered or spring water| 67 or 75  \nPhase 4 Seed Culture| 27  \nTOTAL| 194 or 202\n\n1 Combine all the ingredients in the bowl of a stand mixer fitted with the dough hook or in a large bowl. If using a stand mixer, mix on low speed for 1\u00bd to 2 minutes; if mixing by hand, stir with a large spoon and then knead by hand until the flour is hydrated and a coarse, shaggy ball of dough forms. It will be slightly sticky.\n\n2 Let the starter rest, uncovered, for 5 minutes, then mix or knead for 1 to 2 minutes, until the starter is fairly smooth and all the ingredients are evenly distributed. It should feel like soft bread dough; if using unbleached bread flour, the starter will be softer and more supple than if using whole wheat flour, which is more absorbent.\n\n3 Mist a large bowl or container with vegetable spray oil and put the starter in the bowl. (The bowl or container should be large enough to contain the starter after it doubles in size.) Cover loosely with plastic wrap or with a lid and leave out at room temperature for 4 to 12 hours, until the starter is nearly double in size. It should have a pleasant aroma similar to apple cider vinegar and register 4.0 or less if tested with pH paper. If the starter hasn't risen much within 12 hours, reshape it into a ball, return it to the covered bowl, and leave it out for 4 to 12 hours longer. Eventually, it will spring to life.\n\n4 When the starter is about double in size, degas it by kneading it for a few seconds. Then shape it back into a ball, return it to the container, cover, and refrigerate. It is now ready to use for building final dough or intermediate starters, such as levains. After a few hours, once the mother starter has cooled, vent any carbon dioxide buildup by opening the lid briefly, then cover the container tightly. The mother starter can be used directly as a final starter for up to 3 days or for building an intermediate starter for up to 7 days. To use it after these times, refresh all or part of it to make a new mother starter as described below.\n\n## REFRESHING THE MOTHER STARTER\n\nIf you haven't used the mother starter in the time frames outlined above, if it starts to get mealy, or if you're using it and running low, you'll need to refresh it. (This is also called feeding or rebuilding.) To do so, discard all but 3.25 ounces (92 g), and use the saved portion as a seed culture, repeating the process above. You can use more of the old mother starter if you want; just be sure to increase the amounts of new flour and water by the same ratio. For that matter, you can even begin with as little as 1 ounce (28.5 g) of starter, rebuild in increments over a couple of feedings, at a ratio of 4 to 1 or even 5 to 1 flour to starter. This is a good thing to do when you haven't used your mother starter for a while and plan to start baking bread again.\n\nFor example, you can use 1 ounce (28.5 g) of mother starter, 1 cup (4.5 oz \/ 128 g) of flour, and 6 tablespoons (3 oz \/ 85 g) of water. This will produce about 8 ounces (227 g) of dough that you can ferment as described above to create a new mother starter. You can then build all or part of that into an even larger piece by using the same ratios: 100% flour, 67% to 70% water (or up to 75% if using whole wheat flour), and 20% to 25% starter. For example, to build 7 ounces (198 g) of newly refreshed starter into a larger starter, you can add up to 35 ounces (992 g) of flour and about 24.5 ounces (695 g) of water, resulting in over 4 pounds (1.8 kg) of starter, probably more than you need unless you're making a number of loaves.\n\nAfter a mother starter spends a few weeks in the refrigerator unrefreshed, its proteins and starches start to break down. The result is a starter with the structural strength of potato soup. This is okay, because the microorganisms are still viable though somewhat dormant (and maybe even a little drunk). Discard all but 1 ounce (28.5 g) of the old starter and build it back in stages using the 4-to-1 or 5-to-1 process described above. In a day or two, you'll have a strong and refreshed mother starter.\n\nOne final note on refreshing: The acidity of a mother starter should be between 3.5 and 4 on the pH scale. If you have pH paper, you can use it to test the pH of a small piece of starter. However, acidity is caused mainly by bacteria, and it doesn't serve one of the key functions of a starter: to provide leavening. For recipes leavened using solely sourdough starter, you must also see signs of growth in the starter, indicating that the yeasts are active and capable or reproducing. In a healthy starter, the yeast and bacteria are in a state of coexistence, with the exact composition being largely determined during the seed culture stage, and then maintained in the mother starter. However, temperature variations and long dormancy periods can affect the yeast more adversely than the bacteria, so it may be necessary to refresh an old starter with a few feedings to sufficiently bolster the yeast population.\n\n## AN ALTERNATIVE: RAISIN WATER\n\nI've recently discovered an interesting method for making a wild yeast starter using raisins and water but no flour. This unique liquid starter is sometimes referred to as yeast water or raisin water. Jeffrey Hamelman, author of Bread: A Baker's Book of Techniques and Recipes, wrote about it in the Bread Bakers Guild of America newsletter (a great resource; see here for details). After reading his article, I started looking into it on the Internet and found a number of variations for how to make it. The idea is to create a starter that has plenty of wild yeast but not a lot of bacterial fermentation, allowing for naturally leavened breads without a lot of lactic acid. This starter creates a distinct flavor profile that's very pleasant.\n\nInterestingly, raisin skins do harbor a lot of wild yeasts, but few of these strains are used in bread baking. Debra Wink, who's establishing herself in the American baking community as a go- to person on matters of yeast and bacteria biochemistry, commented in the same newsletter that the yeasts found on raisins and grapes are primarily wine-producing yeasts, such as Hanseniaspora (aka Kloeckera) and other obscure strains, and very little Saccharomyces cerevisiae (the species used for commercial bread yeast). According to Debra, the wine yeasts require more vitamins than the raisin water can provide, and they also don't like acidity, which wild strains of S. cerevisiae can tolerate. As a result, there's a little fermentation from the wine yeasts at the beginning, and then the activity settles down a bit until S. cerevisiae and other wild bread yeasts catch up and the wine yeasts die off. By the fifth or sixth day, the raisin water starts to get potent with the right kinds of yeast, and the lactic acid bacteria, which have been held at bay because there's no flour for them to feed on, don't have much of a chance to get in on the action.\n\nAfter reading about how others create these starters, I experimented and came up with my own method. In the end, it's all about raisins, water, and time. The method below indicates a certain timeline for development of sufficient microbial activity. However, in cooler temperatures, it could take as long as 8 or even 10 days for the yeast to become established. Just keep stirring or shaking as directed until the raisin water is bubbly.\n\nThe resulting raisin water can be used to create a firm starter that you can use in any recipe calling for a natural sourdough starter. You'll get all of the rise without the sour tang. You can also keep some or all of the firm starter as a mother starter. However, as you use this mother starter and refresh it with flour, it will gradually become more acidic, transitioning into a regular (tangy) sourdough starter.\n\n### RAISIN WATER\n\norganic raisins: (\u00bc cup oz \/ 1.5 g \/ 42.5%)\n\nwater (preferably filtered or spring): (1 cup plus 3 tablespoons oz \/ 9.5 g \/ 269%)\n\nTOTAL: 11 oz \/ 311.5 g\n\n1 Put the raisins and 3 tablespoons (1.5 oz \/ 42.5 g) of the water in a sterilized jar or container and stir or shake to combine. Cover loosely with a lid, plastic wrap, or a towel. Put the jar in a warm place or a sunny part of the house and stir or shake it every 8 hours for 2 days.\n\n2 On the third day, add \u00bd cup (4 oz \/ 113 g) of the water and continue shaking or stirring at 8-hour intervals.\n\n3 On the fourth day, add the final \u00bd cup (4 oz \/ 113 g) of water and continue shaking or stirring at 8-hour intervals.\n\n4 On the fifth or sixth day, when the water is bubbly, the raisins are floating to the top, and a whitish mold has formed on them, the seed culture is ready to use. Strain the liquid into a bowl, discarding the raisins and any sediment that has settled to the bottom. Use immediately, or refrigerate for up to 3 days before building the final starter.\n\n### FINAL STARTER\n\nRaisin Water: \u00be cup (6 oz \/ 170 g)\n\nwhole wheat or unbleached bread flour or sprouted whole wheat flour: 2 cups or 1\u2154 cups (9 or 7 oz \/ 255 or 198 g)\n\nTOTAL: 15 or 13 oz \/ 425 or 368 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nRaisin Water| 66  \nwhole wheat or unbleached bread flour or sprouted whole wheat flour| 100  \nTOTAL| 166\n\nMix and ferment as for Mother Starter.\n\nCHAPTER 3\n\n# SPROUTED FLOUR BREADS\n\nIn the spring of 2010, a few months after my initial experiments with Joe Lindley's sprouted grain flour, I met Peggy and Jeff Sutton at a baking summit in Denver, Colorado. The Suttons also have a sprouted mill: To Your Health Sprouted Flour Co. Unlike Joe, who I'd characterize as a miller who happens to sprout grain, the Suttons impressed me as sprouters who happen to mill. They operate out of Fitzpatrick, Alabama, a small farming community on the outskirts of Montgomery, and their expanding business is rapidly adding new jobs to the local economy. Unlike Lindley Mills, which currently makes only two sprouted grain flours (though one of them is a blend of five ingredients), the Suttons produce a line of about twenty different sprouted grain and bean flours, including spelt, yellow corn, blue corn, rye, amaranth, barley, black bean, and quinoa.\n\nWhen I visited the Suttons in Fitzgerald in 2013, Peggy told me, \"At first, I was baking for friends and farmers' markets out of my home. But the supply of sprouted grain flour was limited, so I created my own sprouting system using canning jars and tubs with screened lids. I dedicated one room of our new barn to sprouting, another to drying the sprouts, and another to grinding the sprouts into flour with a small stone mill. Before long, demand for my sprouted flour outgrew this system, so I took over Jeff's barn completely. Eventually he built another new barn behind the house, and we moved the operation over there. Three years ago we ran out of room again, so we took the plunge and built this facility on some of our farmland. A few weeks ago we held a dedication ceremony for yet another new building, which will house an entirely gluten-free milling and packaging operation. The demand just keeps growing, and so does our operation.\"\n\nAs Peggy toured me around the two buildings, I saw millet soaking in dozens of large buckets, hundreds of pounds of corn kernels revolving in mechanically operated sprouting tumblers, and perhaps thousands of pounds of sprouted wheat and spelt air-drying in rooms dedicated to that purpose. Compared to some mills I've visited, the Suttons' operation is relatively modest, processing about 5,600 pounds of sprouted flour per day, but it's miles beyond where they were when I first met them in Denver. These days, they're providing flour for large bakeries, such as the Whole Foods Market Bakehouses, and they also supply the flour for a few prestigious private-label packagers. In addition, Internet sales via their website continue to enjoy rapid growth, spurred on by word of mouth and Peggy's blog.\n\nWhen the new facility is complete, they'll be able to increase production to 10,000 pounds per day, nearly double their current volume. I asked if that will be enough to keep up with the demand, and Jeff said, \"Probably for a short while, but we're already looking at building out more of the property. We'll just keep putting up new buildings as the demand rises.\"\n\nI told the Suttons about my concern that large flour companies will simply wait for them to create the market and then swoop in with deeper pockets and try to steal it away. While they share that concern, they pointed out that it takes a lot of work to do it right. They're hoping that the big companies will just buy the flour from them, rather than try to make it themselves. Time will tell. It's difficult to know what will happen, especially since sprouted flour is still a niche market. But, as we've seen with craft beer, farmstead cheese, and other artisan producers, there is a pattern of big companies sweeping aside innovators.\n\n## FROM SPROUT TO MASS MOVEMENT\n\nTo Your Health Sprouted Flour Co. and Lindley Mills aren't the only producers of sprouted flour, though they are among the larger companies at the moment. Other players already in the game include Arrowhead Mills, Essential Eating Sprouted Foods, and Shiloh Farms, as well as Canadian companies such as One Degree Organic Foods and Anita's Organic Mill. (See the Resources section for other suppliers and contact information.) Clearly, sprouted foods are no longer a cult phenomenon or my own secret little discovery, and it's only going to get bigger.\n\nJoe Lindley has been milling high-quality organic flour for nearly forty years. His first sale was made on July 4, 1976, a year after he bought back and restored the family's abandoned mill, which sits on the site of the 1781 Battle of Lindley's Mill, which took place during the Revolutionary War. He told me that he's been seeking better ways of milling flour to achieve maximum flavor and nutrition ever since he reestablished the mill, but it wasn't until about 2007 that he discovered the possibilities of sprouted flour. He now believes that this will be his most important contribution to the world of flour, and he's prepared to stake his future on it, though he will continue to mill traditional organic flour for his regular customers.\n\nI think the Suttons feel the same way. Peggy Sutton and Joe Lindley have only met each other once, at the annual Asheville Bread Festival, where I introduced them in 2011. However, I view them as kindred spirits, even though their personalities are very different. Like every other miller I know, both are genuinely nice, and both are driven by their sense of mission yet are not evangelical or pushy. I think their patience must be governed by an inner knowing that their products speak for themselves and will eventually win the day. Besides, both are growing their businesses as fast as they can while, wisely, not getting too far ahead of themselves.\n\nThis brings me to the obvious question: Why are they so willing to stake their future on sprouted flour, and why have I become such an advocate of this new approach? As usual, it starts with flavor. Flavor always rules when it comes to economic viability in the world of food. However, there's much more to commend sprouted grain flours.\n\n## WHY SPROUTED GRAIN FLOUR IS THE NEXT FRONTIER\n\nThe statement with which I am most associated in the bread world is probably this: \"The baker's mission is to evoke the full potential of flavor trapped in the grain.\" In previous books, I've described how bakers use various fermentation and mixing methods to accomplish this. In brief, the flavor trapped in the grain mostly resides in the sugars within the starch, and other potential flavors can be evoked from the grain through fermentation, both fungal (via yeast) and bacterial.\n\nTwenty years ago, few American bakers knew about tricks such as using pre-ferments and harnessing enzyme activity to release simple sugars from starches. Now it seems as if these tricks are common knowledge, and the current generation of superb bread books continues to advance the ongoing discovery process. But sprouted flour is new territory; it's different from anything previously seen in the artisan bread movement. And what I especially love about it is that it can accomplish everything we strive to achieve through the use of pre-ferments and extending fermentation time, and it does so before the grain is even turned into flour!\n\nAnother benefit of using sprouted flour is that it means using the whole kernel\u2014the entire grain or seed. And by germinating grains and seeds we actually enhance their nutritional benefits, allowing us receive the maximum nutritional benefit inherent in the grain. When a seed is germinated by soaking in water, this activates enzymes that initiate the transformation of the germ into a viable growing organism. To support that process, the nutritional value of the seed improves. In addition, sprouting softens the bran, reduces its phytic acid, renders it less bitter, and makes the grain's minerals more bioavailable. And when those grains or seeds are ground into flour, the resulting bread tastes better, is higher in nutrients, and is also easier to digest, allowing the body to access more of its nutrients.\n\nAnecdotal reports also indicate that some people who are usually sensitive to wheat can tolerate sprouted wheat without typical reactions, such as indigestion or rashes. However, this is not yet hard science and must be subjected to rigorous testing. And it definitely doesn't apply to people with celiac disease or others who are highly sensitive to wheat.\n\n## Sprouting and Drying Grains for Flour\n\nBefore the recipes, I'll briefly review the process of sprouting seeds and preparing them for milling. If you've ever grown bean or alfalfa sprouts you'll instantly recognize the early steps in this method:\n\n1. Rinse the seeds in fresh, cool water. Some seeds, including quinoa, are coated with a bitter resin, and these require two or three rinses.\n\n2. Cover the seeds in fresh water and let them sit at room temperature for about 3 to 5 hours.\n\n3. Drain the water and rinse and drain the seeds again. The germination process has begun.\n\n4. Rinse the seeds again before retiring for the night, and let them drain in either a strainer, a jar fitted with a screened lid, or any container that allows the water to drain away.\n\n5. The following morning (or later if the room is cool), you should see the beginning of a sprout: a nub breaking through the skin, or bran, of the seed. Technically, the seed is a sprout as soon as the nub appears, but the ideal time to move to the drying step is when the nub splits into two thin shoots. At this point the seeds have become vegetables and are much more digestible.\n\n6. Dry the sprouts on a screen or perforated pan, setting up a fan to blow air across the surface. Alternatively, you can use a food dehydrator, but don't put them in a gas oven, as even just the pilot light can generate too much heat and denature the enzymes. Moving air is really the key, whether at room temperature or slightly warmer. The shoots will shrivel back into the seeds as they dry out. About 12 hours later, the seeds will be as dry as they were before you soaked them. They are now ready to be milled into flour. It's as simple as that.\n\n## TECHNICAL CONCERNS\n\nAlthough the nutritional profile of sprouted flour is clearly superior, artisan bakers have rightly been concerned about whether the gluten quality of the wheat is compromised by the enzyme activity and release of natural sugars caused by sprouting. Would the resulting dough hold together when rising and still provide enough starch to fill the body of the loaf?\n\nAs I'll discuss later in the book, there are already methods for using sprouted grain pulp (also known as sprouted mash) to make popular, delicious, and commercially successful breads. However, those breads usually include a substantial amount of vital wheat gluten. Independently, both Peggy Sutton and Joe Lindley discovered that if they source their grain carefully and select high-quality, high-protein wheat, their sprouted whole wheat flour performs well with no need for supplemental gluten. (That said, some commercial bakeries do spike their sprouted flour doughs with a bit of vital wheat gluten to ensure consistent performance and loaf size.)\n\nIn addition, the natural enzyme activity triggered by soaking and germinating the seeds can enhance the performance of dough. Some of the enzymes remain active in the dough and continue to release sugars from the starches while the dough is fermenting, in the process feeding yeast and bacteria and promoting rapid fermentation. So whereas a dough made with standard, nonsprouted flour might require many hours of slow fermentation or supplementation with a pre-ferment to release so much flavor, similar results can be achieved much more quickly with sprouted flour. In most cases, all that's required is about 1\u00bd hours for the first rise and 1 to 1\u00bd hours for a final rise after shaping. I think the resulting flavor is as good as that of breads made with long-rising doughs\u2014and actually even better, as you'll discover when you make some of the recipes in this chapter.\n\n## Q & A\n\n### Where can I get sprouted grain flour?\n\nSprouted grain flours are showing up on baking shelves in supermarkets nationwide, usually in small bags, from brands including Arrowhead Mills and One Degree. For more information on buying these flours directly from millers by mail order, see the Resources section.\n\n### Will pre-ferments, such as a poolish, biga, or sponge, improve breads made with sprouted flour? How about the overnight cold fermentation technique?\n\nThe purpose of pre-ferments is to condition flour through long, slow fermentation, coaxing more flavor from the grain and developing acidity. This preconditioned dough is added to the final dough to, in essence, age it and imbue it with more flavor. Overnight cold fermentation is another way to lengthen the development time to evoke more flavor. Because sprouted flour is already preconditioned, with flavor being released during the sprouting process, breads made with sprouted flour will taste much the same whether made with or without pre-ferments or cold fermentation. There's no harm in using pre-ferments and overnight cold fermentation, but they aren't necessary. The one exception is wild yeast sourdough starters (which in essence are another type of pre-fermented dough). One of the functions of a sourdough starter, aside from raising the dough, is to introduce specific acidic, tangy flavors created by bacterial fermentation. So when making sprouted flour sourdough breads, you do need to use a sourdough starter to achieve the flavor complexity that it provides. (For more on starters, see Understanding Sourdough Starters.)\n\n### Can I use the Dutch oven baking method for some of these breads even if you don't call for it?\n\nYes. This method of baking, made popular by Jim Lahey (My Bread) and Chad Robertson (Tartine Bread), is always an option, especially for breads made with wet, high-hydration doughs. I'm fond of baking in a cloche (clay baking pot), rather than a Dutch oven. The recipes in this book most suited for that baking method are Sprouted Whole Wheat Bread, Sprouted Pain au Levain, Whole-Milled Lean Dough French Bread, and High-Extraction Pain au Levain.\n\n### What are ancient grains?\n\nThe term ancient grain is a little confusing, since all grains are ancient or come from ancient strains. However, the term has come to signify heirloom varieties or grains that have undergone little or no intentional crossbreeding or hybridization. For wheat, this includes einkorn, emmer, spelt, and Kamut (aka Khorasan wheat). However, the term ancient grain also refers to a whole gamut of other grains, such as buckwheat, quinoa, amaranth, millet, sorghum, and even rice, corn, barley, and oats\u2014some of which aren't even technically grains!\n\n### Are ancient grains gluten-free? And if they are, how does this impact bread making?\n\nBecause many so-called ancient grains are gluten-free (only wheat, rye, and barley naturally contain gluten-forming proteins), the term ancient grain often gets lumped in with gluten-free, but this is imprecise at best. Amaranth, buckwheat, corn, millet, oats, quinoa, rice, sorghum, teff, and wild rice are all gluten-free. Because they don't contain gluten, they tend to weaken the gluten structure of doughs made with wheat. For this reason, unless you're purposely making a gluten-free product, I suggest using no more than 20% gluten-free flour in place of wheat flour in a recipe. For more on gluten-free baking, see Gluten-Free Breads.\n\n### Can I substitute one kind of ancient grain flour for another?\n\nFor the most part, yes. Joe Lindley has created a five-grain ancient grain blend, but other companies, including To Your Health Sprouted Flour Co., sell a variety of individual sprouted flours, and you can use these to create your own blends. Feel free to mix and match or make substitutions as long as you stay within the same proportional ratios.\n\n### Can I use regular whole grain flour in place of the sprouted flour in these recipes?\n\nIn theory, nonsprouted flour can be substituted for sprouted, but the two don't perform in exactly the same way. If you study chapter 5, on whole grain and whole-milled bread, you'll see that the recipes using nonsprouted flours almost always utilize pre-ferments or cold fermentation to release the full flavor potential from the grain. Therefore, I don't recommend substituting nonsprouted for sprouted flour in any of the recipes in chapter 3 without also modifying the fermentation process.\n\n### Can sprouted and nonsprouted flours be combined in some recipes?\n\nYes, and some bakeries are already doing exactly this. But for the purposes of this book, I've mostly separated the two types of flour, using only one or the other in any given recipe. However, there's no reason why you can't add sprouted flour to a nonsprouted mix. One of the goals of this book is to give you the tools to create your own variations. After you experiment with the recipes here, you can apply similar methods to create an infinite number of variations.\n\n### Can I make a sourdough starter using sprouted flour?\n\nWhile you can make a mother starter using sprouted flour, I recommend against it. Sprouted flour has so much enzyme activity that it tends to overripen quickly, breaking down into a mush within a few days. The yeast and other microorganisms are still viable, but the dough has no gluten structure, so it has to be rebuilt over a couple of feedings before it can be used. Therefore, I suggest keeping a traditional mother starter, made with bread flour or whole wheat flour, and converting a portion of that into a sprouted starter shortly before you'll use it. For more on sourdough starters, see chapter 2.\n\nSprouted Wheat Pancakes\n\n# SPROUTED WHEAT PANCAKES\n\nMAKES 5 LARGE PANCAKES OR 16 SILVER DOLLAR-SIZE PANCAKES\n\nI thought it best to start off with a recipe that's very simple and quickly demonstrates the attributes of sprouted flour. If I'm not mistaken, once you try it, you'll immediately seek out a bulk supplier for your sprouted flour pantry. These are the best pancakes I've ever eaten, period. They're so naturally sweet and creamy that you really don't need butter or maple syrup on top. But since that might be tampering with sacred ritual, I'll leave it up to you. This same batter can be used in a waffle iron to make fabulous waffles; see the variations at the end of the recipe for details.\n\n### BATTER\n\nsprouted whole wheat flour: 1 cup plus 1 tablespoon (4.5 oz \/ 128 g)\n\nsalt: \u00bc teaspoon (0.06 oz \/ 1.5 g)\n\nbaking soda: \u00bd teaspoon (0.11 oz \/ 3 g)\n\nsugar, honey, or agave nectar: 1 teaspoon (0.25 oz \/ 7 g)\n\nbuttermilk: 1\u00bd cups (12 oz \/ 340 g)\n\negg, slightly beaten: 1 (1.75 oz \/ 50 g)\n\nunsalted butter, melted: 2 tablespoons (1 oz \/ 28.5 g)\n\nTOTAL: 19.67 oz \/ 558 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.1  \nbaking soda| 2.3  \nsugar, honey, or agave nectar| 5.5  \nbuttermilk| 266  \negg| 39  \nunsalted butter| 22  \nTOTAL| 435.9\n\n1 In a medium bowl, stir together the flour, salt, baking soda, and sugar (if using honey or agave nectar, add it to the buttermilk in the next step). In a separate bowl, whisk together the buttermilk, egg, and melted butter, then pour into the flour mixture. Stir with a large spoon just until the flour is hydrated; don't overmix. The result will be a fairly thin, pourable batter; it will thicken slightly as it sits, so don't add more flour. Transfer the batter to a measuring cup with a pouring spout (or leave it in the bowl and portion it with a ladle).\n\n2 Preheat a nonstick skillet or griddle over medium heat.\n\n3 Put about 1 teaspoon of butter or oil in the hot pan, just enough to thinly coat the surface. Lower the heat to just below medium. Pour in batter to make pancakes of the desired size. You may need to tilt the pan to spread the batter into an even circle. Cook until the bottom is rich golden brown and bubbles form on the top, 2\u00bd to 3 minutes for larger pancakes, and less for smaller pancakes. Flip and cook until the other side is golden brown, about 2\u00bd minutes.\n\n4 Serve hot, or keep the pancakes in a warm oven at about 200\u00b0F (93\u00b0C) while cooking the remaining pancakes.\n\n### Variations\n\nSPROUTED WHEAT FLOUR WAFFLES: You can cook this batter in a waffle iron. If making waffles, I recommend doubling the recipe, since the waffle iron gobbles up a lot of batter. Also, separate the eggs. Add the yolks to the batter, and whip the whites until stiff, then fold them into the batter for additional aeration. To cook, follow the instructions for your waffle iron.\n\nSPROUTED MULTIGRAIN PANCAKES: Replace about \u00bc cup (1.1 oz \/ 30 g) of the sprouted whole wheat flour with an equal amount of any combination of other sprouted flours, such as corn, buckwheat, millet, sorghum, or quinoa.\n\nBLUEBERRY PANCAKES: Add 1 cup of fresh or frozen blueberries (or other fresh berries) to the batter.\n\nRosemary Walnut Quick Bread (variation of Sprouted Wheat or Spelt Quick Bread)\n\n# SPROUTED WHEAT OR SPELT QUICK BREAD\n\nMAKES 1 LOAF\n\nPeggy Sutton, founder of To Your Health Sprouted Flour Co., has been making variations of this bread for years, both for her family and to sell at local farmers' markets. You can make it with any kind of sprouted flour, even gluten-free sprouted flours, as long as the total weight of flour remains the same. This version showcases sprouted wheat or sprouted spelt flour, the latter one of Peggy Sutton's best-selling alternative flours. Because this bread is leavened only with baking soda, it falls in the category of chemically leavened breads, more commonly known as quick breads. The acidity of the buttermilk is necessary to activate the baking soda and generate carbon dioxide to raise the dough. If you prefer a more highly leavened bread, you can also add 1 tablespoon of baking powder, but try it first with only the baking soda. This master recipe lends itself to a wide range of variations, and you'll find a number of suggestions at the end of the recipe. While you can bake this in muffin pans, it will not be as sweet as you may be hoping. Try the next recipe for muffins.\n\n### BATTER\n\nsprouted wheat flour (or sprouted spelt flour): 3 cups plus 3 tablespoons (or 3 cups plus 6 tablespoons) (13.5 oz \/ 383 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\nbaking soda: 2\u00bd teaspoons (0.5 oz \/ 14 g)\n\nbuttermilk: 2 cups (16 oz \/ 454 g)\n\neggs, slightly beaten: 3 (5.25 oz \/ 149 g)\n\nhoney: 6 tablespoons (4 oz \/ 113 g)\n\nunsalted butter, melted: \u00bc cup (2 oz \/ 56.5 g)\n\nTOTAL: 41.5 oz \/ 1,176.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted wheat flour| 100  \nsalt| 1.8  \nbaking soda| 3.7  \nbuttermilk| 119  \neggs| 39  \nhoney| 30  \nunsalted butter| 15  \nTOTAL| 308.5\n\n1 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Mist a 4\u00bd by 8-inch loaf pan with vegetable spray oil; alternatively, brush the pan with melted butter, then dust with sprouted flour. You can also bake this bread in an 8-inch or larger round cake pan. In that case, cut a round of parchment paper to fit into the bottom of the pan and then prepare the pan as above.\n\n2 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and baking soda (on low speed if using a stand mixer). In a separate bowl, whisk together the buttermilk, eggs, honey, and melted butter, then pour into the flour mixture. Mix or stir until the flour is hydrated and all the ingredients are well combined, about 1 minute. The result should be a thick, soupy batter. Transfer the batter to the prepared pan.\n\n3 Bake for 45 minutes, then rotate and bake for 30 to 45 minutes longer, until the top is golden brown and springy to the touch and a toothpick inserted in the center comes out clean. If the top is golden brown but the center seems doughy or too soft, tent the pan with aluminum foil, lower the temperature to 325\u00b0F (163\u00b0C), and continue baking until a toothpick comes out clean.\n\n4 Let cool in the pan for at least 30 minutes. Run an icing spatula or something similar around the edges to separate the bread from pan, then transfer the bread to a wire rack. Let cool for at least 30 minutes longer before slicing and serving.\n\n### Variations\n\nAPRICOT ALMOND QUICK BREAD: When mixing the final batter, add 1 tablespoon (0.42 oz \/ 12 g) almond extract, 1 teaspoon (0.14 oz \/ 4 g) vanilla extract, and about \u00bd cup (3 oz \/ 85 g) chopped dried apricots. Sprinkle about \u00bd cup (1.65 oz \/ 47.5 g) sliced or slivered almonds over the loaf before baking.\n\nCINNAMON RAISIN QUICK BREAD: When mixing the final batter, add 2 tablespoons (0.55 oz \/ 15.5 g) ground cinnamon, 1 tablespoon (0.42 oz \/ 12 g) vanilla extract, 1 teaspoon (0.07 oz \/ 4 g) cinnamon oil (optional), and 1 cup (6 oz \/ 170 g) raisins.\n\nLEMON POPPY SEED QUICK BREAD: When mixing the final batter, add 1\u00bd tablespoons (0.55 oz \/ 15.5 g) lemon extract, 1 teaspoon (0.07 oz \/ 4 g) vanilla extract, 1 tablespoon (0.75 oz \/ 21.5 g) ground dried lemon peel or the grated zest of 1 lemon, and 1 tablespoon (0.32 oz \/ 9 g) poppy seeds, or more to taste.\n\nROSEMARY WALNUT QUICK BREAD (PICTURED HERE): When mixing the final batter, add 1\u00bd teaspoons (0.16 oz \/ 4.5 g) ground dried rosemary 1\u00bd teaspoons (0.03 oz \/ 1 g) rosemary leaves (fresh or dried), and \u00bd teaspoon (0.05 oz \/ 1.5 g) ground sage. Sprinkle \u00bd to \u00be cup (2 to 3 oz \/ 56.5 to 85 g) coarsely chopped walnuts over the loaf before baking.\n\nHERBED QUICK BREAD: When mixing the final batter, add 1\u00bd teaspoons (0.03 oz \/ 1 g) dried dill weed, 1 teaspoon (0.02 oz \/ 0.5 g) dried tarragon, and \u00bd teaspoon (0.007 oz \/ 0.2 g) each oregano, basil, and thyme.\n\nCranberry Quick Muffins (a variation of Sprouted Quick Wheat Bread or Muffins)\n\n# SPROUTED WHEAT QUICK BREAD OR MUFFINS\n\nMAKES 1 LARGE LOAF OR 12 TO 18 MUFFINS | MASTER FORMULA\n\nQuick breads and muffins are essentially the same product in two different formats. There are many ways to make them, but in the end, they're all chemically leavened, using baking powder, baking soda, or both, rather than yeast. The recipe below is a template you can use to make any number of variations. I've included several suggestions for variations, but feel free to experiment and come up with your own creations once you become comfortable with the recipe. You'll notice that I've included sprouted spelt flour and sprouted wheat pastry flour as options. Either will work well, and both are becoming increasingly available.\n\n### BATTER\n\nsprouted wheat flour (or sprouted wheat pastry flour or sprouted spelt flour): 3\u00be cups (or 4 cups) (16 oz \/ 454 g)\n\nbaking powder: 1 tablespoon plus 1 teaspoon (0.75 oz \/ 21.5 g)\n\nbaking soda: 1\u00bc teaspoon (0.25 oz \/ 7 g)\n\nsalt: \u00be teaspoon (0.19 oz \/ 5.5 g)\n\nbrown sugar (or honey or agave nectar): 1 cup plus 2 tablespoons (or \u00be cup plus 1\u00bd tablespoons) (9 oz \/ 255 g)\n\nbuttermilk: 2 cups (16 oz \/ 454 g)\n\neggs, slightly beaten: 3 (5.25 oz \/ 149 g)\n\nvegetable oil or melted unsalted butter: \u00be cup (6 oz \/ 170 g)\n\nvanilla extract: 2 teaspoons (0.28 oz \/ 8 g)\n\nStreusel Topping (optional)\n\nTOTAL: 53.72 oz \/ 1,524 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted wheat flour| 100  \nbaking powder| 4.7  \nbaking soda| 1.5  \nsalt| 1.2  \nbrown sugar| 56  \nbuttermilk| 100  \neggs| 33  \nvegetable oil or melted unsalted butter| 37  \nvanilla extract| 1.8  \nTOTAL| 335.2\n\n1 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Prepare a 4\u00bd by 8-inch loaf pan or a 12-cup muffin pan. For a loaf pan, generously mist with vegetable spray oil; alternatively, brush the pan with melted butter, then dust with sprouted flour. For a muffin pan, line the muffin cups with paper liners, then mist the liners with vegetable spray oil.\n\n2 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, baking powder, baking soda, and salt (on low speed if using a stand mixer). Stir in the sugar (if using honey or agave nectar, add it to the buttermilk in the next step). In a separate bowl, whisk together the buttermilk, eggs, oil, and vanilla extract, then pour into the flour mixture. Mix or stir until the flour is hydrated and all the ingredients are well combined, about 1 minute. The result should be a thick batter, which will thicken further as it sits. Transfer the batter to the prepared pan, filling a loaf pan to within \u00be inch of the rim and muffin cups to just below the rim. Sprinkle the streusel topping over the top if desired.\n\n3 Bake for 20 minutes for muffins or 30 minutes for a loaf, then rotate and bake for 15 to 20 minutes longer for muffins, or 25 to 35 minutes longer for a loaf, until the top is golden brown and springy to the touch and a toothpick inserted in the center comes out clean.\n\n4 Let cool in the pan for at least 15 minutes for muffins or 30 minutes for a loaf. Turn out onto a wire rack and let cool for at least 10 minutes longer for muffins or 30 minutes longer for a loaf before serving.\n\n## Streusel Topping\n\n\u00bd cup (about 2.1 oz \/ 60 g) sprouted flour (any type)\n\n\u00bd cup (4 oz \/ 113 g) white or brown sugar\n\n\u215b teaspoon (0.03 oz \/ 1 g) salt\n\n\u00bc teaspoon (0.02 oz \/ 0.5 g) ground cinnamon (optional)\n\n\u00bc cup (2 oz \/ 57.5 g) unsalted butter\n\nIn a small bowl, stir together the flour, sugar, salt, and cinnamon. Melt the butter, then pour it in and stir until evenly distributed. Use your fingers to break the mixture into fine crumbs. If it's too warm to crumble, wait for it to cool, then crumble it again. Alternatively, use cold butter and pulse all the ingredients in a food processor until the texture resembles fine cornmeal.\n\n### Variations\n\nWhen making some of these variations, you may end up with extra batter. You can bake it in an additional muffin pan, a mini loaf pan, or even ovenproof ramekins. Stored in a covered container, the batter will keep for 3 days in the refrigerator or 3 months in the freezer.\n\nBLUEBERRY QUICK BREAD OR MUFFINS: When mixing the final batter, add up to 3\u00bd cups (about 18 oz \/ 510 g) fresh or frozen blueberries to the final batter. If using frozen blueberries, toss them in \u00bc cup (about 1.1 oz \/ 30 g) sprouted flour before adding them, to prevent clumping and sinking.\n\nCRANBERRY QUICK BREAD OR MUFFINS (PICTURED HERE): Replace \u00bd cup (4 oz \/ 113 g) of the buttermilk with \u00bd cup (5 oz \/ 142 g) thawed orange juice concentrate, and, optionally, add \u00bd teaspoon (0.07 oz \/ 2 g) lemon or orange extract to the buttermilk mixture. When mixing the final batter, add 2\u00bd cups (about 9 oz \/ 255 g) fresh or frozen cranberries.\n\nBANANA QUICK BREAD OR MUFFINS: Mash 3 or 4 ripe bananas and add them to the buttermilk mixture, along with 1 teaspoon (0.14 oz \/ 4 g) banana extract, if you have it on hand. Optionally, when mixing the final batter, add 1\u00bd cups (6 oz \/ 170 g) coarsely chopped toasted walnuts or pecans (see Toasting Seeds and Nuts for toasting instructions).\n\nPOPPY SEED QUICK BREAD OR MUFFINS: Add up to 1 cup (5 oz \/ 142 g) poppy seeds to the flour mixture. Optionally, add 1 teaspoon (0.14 oz \/ 4 g) lemon extract to the buttermilk mixture.\n\nZUCCHINI QUICK BREAD OR MUFFINS: Add 4 cups (16 oz \/ 454 g) grated zucchini to the buttermilk mixture. Optionally, when mixing the final batter add 1\u00bd cups (9 oz \/ 255 g) raisins and 1\u00bd cups (6 oz \/ 170 g) coarsely chopped walnuts or pecans (toasted if you like; see Toasting Seeds and Nuts).\n\nCARROT QUICK BREAD OR MUFFINS: Add 2 teaspoons (0.18 oz \/ 5 g) ground cinnamon to the flour mixture. Add 4 cups (16 oz \/ 454 g) grated carrots and \u00bc cup (2.5 oz \/ 71 g) molasses to the buttermilk mixture. When mixing the final batter, add 1\u00bd cups (9 oz \/ 255 g) raisins and, optionally, 1\u00bd cups (6 oz \/ 170 g) coarsely chopped walnuts or pecans (toasted if you like; see Toasting Seeds and Nuts).\n\nCHOCOLATE CHIP QUICK BREAD OR MUFFINS: When mixing the final batter, add 2\u00bd cups (15 oz \/ 425 g) chocolate chips.\n\nCHOCOLATE CHERRY QUICK BREAD OR MUFFINS: When mixing the final batter, add 2\u00bd cups (15 oz \/ 425 g) chocolate chips and 3 cups (26.5 oz \/ 751 g) fresh or frozen sour pie cherries, or 2 cups (12 oz \/ 340 g) dried cherries.\n\nAPPLE BRAN QUICK BREAD OR MUFFINS: Add \u00bd cup (2 oz \/ 56.5 g) wheat bran or oat bran to the dry ingredients. Increase the buttermilk to 2\u00bd cups (20 oz \/ 567 g) and add \u00bc cup (2.5 oz \/ 71 g) molasses to the buttermilk mixture. When mixing the final batter, add 2 large apples, peeled and chopped into \u00bd-inch pieces, and 1 cup (6 oz \/ 170 g) raisins.\n\nSprouted Whole Wheat Bread\n\n# SPROUTED WHOLE WHEAT BREAD\n\nMAKES 1 LARGE LOAF, 2 SMALLER LOAVES, OR UP TO 15 ROLLS | MASTER FORMULA\n\nThis master dough can be used to make bread in any shape or size. It showcases the natural sweetness and tenderness of sprouted whole wheat flour without any added oil, fat, or other enrichments, such as milk, eggs, or sweeteners. Sprouting the wheat changes it so much that many of the \"rules\" for artisan breads, such as using pre-ferments and long, slow rising times, are unnecessary. The aims of those techniques can be achieved in less time with sprouted flour because the sprouting phase has already accomplished what pre-ferments and long fermentation typically do.\n\nI suggest that you make this bread before attempting any of the more elaborate recipes that follow. This will familiarize you with the flavors and performance of sprouted whole wheat flour. In fact, it may be the only recipe you need for everyday breads, as it works equally well as a loaf pan bread and a crusty hearth bread.\n\n### DOUGH\n\nsprouted whole wheat flour: 3\u00be cups (16 oz \/ 454 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\ninstant yeast: 1\u00bd teaspoons (0.16 oz \/ 4.5 g \/ 1%)\n\nwater, at room temperature: 3\u00bc cups (14.5 oz \/ 411 g)\n\nTOTAL: 30.91 oz \/ 876.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.6  \ninstant yeast| 1  \nwater| 90  \nTOTAL| 192.6\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Add the water and mix or stir until the flour is hydrated and a coarse, wet dough forms, about 1 minute. Don't add more flour, as the dough will thicken while it rests.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should be smooth but still very soft and sticky (similar to ciabatta dough). Add flour or water only if necessary to achieve that texture; the dough will firm up as you continue to work it.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Oil the work surface again and use an oiled bowl scraper or rubber spatula to transfer the dough to the oiled area. For hearth loaves, prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule or b\u00e2tard, then put the shaped loaves in the prepared proofing vessels. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the dough into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan.\n\n6 Mist the top of the dough with vegetable spray oil, then cover it loosely with plastic wrap. Proof for 1 to 1\u00bd hours at room temperature, until the dough increases in size by 1\u00bd times. When poked with a finger, it should spring back within a few seconds; if it holds the dimple, it's risen for too long. (Because the dough is so hydrated, it's fragile and will fall if you proof it until double in size. It's better to bake it while it's still on the rise.)\n\n7 To bake a hearth loaf, about 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan, then preheat the oven to 450\u00b0F (232\u00b0C). Transfer the shaped loaf to a floured peel (or keep it on the sheet pan for baking). Score. Transfer the loaf onto the baking stone (or put the sheet pan on the baking stone). Pour about 1 cup of hot water into the steam pan. Bake for 15 minutes, then rotate and bake for 15 to 20 minutes longer, until the loaf is golden brown on all sides and sounds hollow when thumped on the bottom. The internal temperature should be about 200\u00b0F (93\u00b0C). Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n8 To bake pan loaves, preheat the oven to 375\u00b0F (191\u00b0C); steam is optional. Bake for 25 minutes, then rotate and bake for 25 to 40 minutes longer, until the bread is golden brown all around, the side walls are firm and not squishy, and the loaf sounds hollow when thumped on the bottom. The internal temperature should be at least 190\u00b0F (88\u00b0C). Let cool in the pans for at least 10 minutes, then transfer to a wire rack and let cool for at least 20 to 30 minutes longer before slicing and serving.\n\n9 To bake rolls, preheat the oven to 400\u00b0F (204\u00b0C); steam is optional. Bake for 12 minutes, then rotate and bake for about 10 to 15 minutes longer, until the rolls are golden brown and sound hollow when thumped on the bottom (they will soften as they cool). The internal temperature should be about 190\u00b0F (88\u00b0C). Transfer to a wire rack and let cool for at least 10 minutes before serving.\n\nNOTE: If it is more convenient for you to use an overnight method, put the covered bowl of dough in the refrigerator immediately after the final stretch and fold. The next day, remove it from the refrigerator 2\u00bd hours before you plan to bake. Shape the cold dough and proof it at room temperature until it increases in size by 1\u00bd times, then bake as directed.\n\n# SPROUTED PAIN AU LEVAIN\n\nMAKES 1 LARGE LOAF OR 2 SMALL LOAVES\n\nClassic French pain au levain is a naturally leavened crusty hearth bread typically made with a combination of white bread flour and a small amount of whole grain flour. Therefore, you'll currently find nothing like the following recipe under the name pain au levain in France, unless the sprouted flour movement hits Paris while this book is being printed. However, this version captures the full spirit of its namesake while also being the closest you're likely to get to producing a San Francisco sourdough bread using sprouted whole wheat flour.\n\nThe main difference between pain au levain and San Francisco sourdough is the absence of whole grain flour in the latter. In addition, San Francisco sourdough bread is, well, more sour. There are two reasons for this. First, in San Francisco (and a few other places around the world), there is a strong presence of a particular lactic acid bacteria\u2014Lactobacillus sanfranciscensis\u2014that lends a very tangy flavor to the dough. It's too tangy, it seems, for the French, whom I've heard refer to it as \"spoiled\" or \"ruined\" bread. The second reason is a trick that French bakers use to minimize sour flavor notes: spiking the dough with a small amount of commercial yeast\u2014just enough to boost the levain, or natural leaven starter. In the following recipe, I offer two options: a pure, starter-only version (a true sourdough bread) and a version spiked with commercial yeast. Also, you can either build a fresh starter, as described below, or you can use a piece of your mother starter as long as it's been refreshed within 3 days of making the final dough.\n\n### LEVAIN\n\nNOTE: The amount of water required will vary depending on the type of flour used. For whole wheat flour you may need only 6 tablespoons (3 oz \/ 85 g), and for bread flour you may need only 5\u00bd tablespoons (2.75 oz \/ 78 g).\n\nMother Starter, recently refreshed (see sidebar): 2 tablespoons (1 oz \/ 28.5 g)\n\nsprouted whole wheat flour (or whole wheat flour or unbleached bread flour): \u00be cup plus 3 tablespoons (or \u00be cup plus 2 tablespoons) (4 oz \/ 113 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C; see note): 7 tablespoons (3.5 oz \/ 99 g)\n\nTOTAL: 8.5 oz \/ 240.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 25  \nsprouted whole wheat flour| 100  \nwater| 88  \nTOTAL| 213\n\nPut the starter in a small bowl and add the water to soften it. Add the flour and stir until a coarse ball of dough forms. Turn it out onto a lightly floured work surface and knead by hand for about 1 minute, until a smooth dough forms and the starter is evenly distributed. Mist a medium bowl with vegetable spray oil and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Ferment the dough at room temperature for 4 to 8 hours, until double in size. Use immediately or refrigerate for up to 3 days.\n\n### FINAL DOUGH\n\nNOTE: You can increase the amount of yeast to as much as 1 teaspoon (0.11 oz \/ 3 g) if you want a faster-rising, less sour bread. Generally, the slower the rise, the more complex the flavor. For a full-on sourdough bread, omit the yeast altogether.\n\nLevain or recently refreshed Mother Starter: \u00be cup (3.25 oz \/ 92 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00bc cups (14 oz \/ 397 g)\n\nsprouted whole wheat flour: 3\u00be cups (16 oz \/ 454 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g \/ 2%)\n\ninstant yeast (optional; see note): \u00bc teaspoon to 1 teaspoon (0.03 to 0.11 oz \/ 1 to 3 g \/ 0.2 to 0.7%)\n\nTOTAL: 33.6 to 33.68 oz \/ 953 to 955 g \/ 209.2 to 209.7%\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nLevain or recently refreshed Mother Starter| 20  \nwater| 87  \nsprouted whole wheat flour| 100  \nsalt| 2  \ninstant yeast| 0.2 to 0.7  \nTOTAL| 209.2 to 209.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the starter and water to distribute the starter. Add the flour, salt, and yeast. Mix or stir until a coarse, wet dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should be smooth but still very soft and sticky. Add a bit more flour or water if necessary to achieve this texture; the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at 20-minute intervals, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature until double in size, 3 to 5 hours, or 1\u00bd to 2 hours if using the optional yeast. (Either time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.) If the dough hasn't doubled in size in the suggested time, you may need to extend the fermentation to as much as 8 hours, but it will eventually rise.\n\n5 Oil the work surface again and use an oiled bowl scraper to transfer the dough to the oiled area. For hearth loaves, prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule or b\u00e2tard, then put the shaped loaves in the prepared proofing vessels. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans.\n\n6 Mist the top of the dough with vegetable spray oil and cover it loosely with plastic wrap. Proof at room temperature until the dough increases in size by 1\u00bd times, 2 to 3 hours, or 1 to 2 hours if using the optional yeast. When poked with a finger, the dough should spring back within a few seconds; if it holds the dimple, it's risen for too long. (Because the dough is so fully hydrated, it's fragile and will fall if you proof it until double in size. It's better to bake it while it's still on the rise.)\n\n7 To bake a hearth loaf, about 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan, then preheat the oven to 450\u00b0F (232\u00b0C). Transfer the shaped loaf to a lightly floured peel (or keep it on the sheet pan for baking). Score. Transfer the loaf to the baking stone (or put the sheet pan on the baking stone). Pour about 1 cup (240ml) of hot water into the steam pan. Bake for 20 minutes, then rotate and bake for 15 to 30 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be 200\u00b0F (93\u00b0C). Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n8 To bake pan loaves, preheat the oven to 375\u00b0F (191\u00b0C); steam is optional. Bake for 30 minutes, then rotate and bake for 25 to 40 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be 190\u00b0F (88\u00b0C). Let cool in the pans for at least 10 minutes, then transfer to a wire rack and let cool for at least 30 minutes longer before slicing and serving.\n\nNOTE: Because sourdough breads can take a long time to rise, it may be more convenient to spread the process over two days using an overnight method. To do so, put the covered bowl of dough in the refrigerator immediately after the final stretch and fold (for pure sourdough with no added yeast, give the dough about 2 hours to begin fermenting before chilling it). The next day, remove it from the refrigerator 3\u00bd hours before you plan to bake. Shape the cold dough and proof it at room temperature until it increases in size by 1\u00bd times, then bake as directed.\n\n### Variations\n\nMULTIGRAIN: You can replace up to 20% of the sprouted whole wheat flour, by weight, with other sprouted flours. If you do so, decrease the amount of water by about 2 to 4 tablespoons (1 to 2 oz \/ 28.5 to 56.5 g), as needed.\n\nNONSPROUTED: You can replace some of the sprouted whole wheat flour with an equal amount, by weight, of conventional, nonsprouted flour, but you'll need to decrease the water amount accordingly, with the amount depending on what kind of flour you substitute. If using whole-milled wheat flour, you shouldn't need to adjust the amount of water very much, if at all.\n\n## Refreshing and Using Your Own Mother Starter\n\nFor those of you who already have a starter, whether whole wheat, rye, or white, you can use it in this recipe and its variations. Simply use your mother starter (the one you keep perpetually in your refrigerator) to make a fresh \"final\" starter, or, if your mother starter has been refreshed within the past three days, you can use it as your starter in the final dough. The key is to build the final starter using a 70% ratio of water to fresh bread flour (75% water if you are using whole wheat flour, or 88% if you are using sprouted wheat flour). In the sourdough primer I recommended not using sprouted flour for your mother starter because I found that it broke down more quickly than a regular starter, but for your final starter you can introduce sprouted flour if you desire. Since any starter has already achieved a full breakout of sugars and flavor esters through long, slow fermentation, it is not necessary to use a sprouted flour starter, but it is an option if you prefer. For single loaf recipes, the amount of starter is so small that it is usually easier to use a piece from your refreshed mother starter.\n\nSprouted Rye Bread\n\n# SPROUTED RYE BREAD\n\nMAKES 1 LARGE LOAF, 2 SMALL LOAVES, OR UP TO 18 ROLLS | MASTER FORMULA\n\nGreat rye bread is always more challenging to make than wheat bread because rye flour has less gluten and much more pentosan gum, which makes it difficult to get an open crumb structure. Historically, rye is an important grain because it's heartier than wheat and can survive and thrive in many places where wheat can't. It certainly holds a venerable place in Eastern European and Scandinavian bread making. Although rye breads have perhaps waned in popularity in the United States in recent decades, the bread world is seeing a revival on the rye front, with bakers such as Chad Robertson and Jeffrey Hamelman demonstrating just how versatile and wonderful rye can be. Now, with sprouted rye flour becoming easier to obtain, it's time for a new rye revolution too. Here's a sprouted version of a seigle rye, a bread in which rye constitutes more than 50% of the final flour. Feel free to adjust the ratios up or down as you explore the potential variations of this master formula.\n\nAlso, note that adding instant yeast is an option. As with many sourdough recipes, you can spike the dough with commercial yeast to shorten fermentation times and reduce the sour flavor. In the case of rye bread, the acidity of the sourdough starter is important for controlling the higher enzyme activity of the rye and preventing collapse of the dough during baking (referred to by bakers as starch attack). Therefore, the total amount of starter in the following formula is higher than in the sourdough levain. As with that recipe, you can either build a fresh starter, as described below, to use in the final dough or you can use a piece of your mother starter as long as it's been refreshed within 3 days of making the final dough.\n\n### SPROUTED RYE FINAL STARTER\n\nMother Starter, recently refreshed: 2 tablespoons (1 oz \/ 28.5 g)\n\nsprouted rye flour (or sprouted whole wheat flour or whole wheat flour or rye flour): 1 cup plus 2 tablespoons (or \u00be cup plus 3 tablespoons or \u00be cup plus 2 tablespoons) (4 oz \/ 113 g)\n\nwater: 7 tablespoons (3.5 oz \/ 99 g)\n\nTOTAL: 8.5 oz \/ 240.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 25  \nsprouted rye flour| 100  \nwater| 88  \nTOTAL| 213\n\nIn a small bowl, stir all the ingredients with a large spoon just until a coarse ball of dough forms. Turn it out onto a lightly floured work surface and knead by hand for about 1 minute, until a smooth dough forms and the mother starter is evenly distributed. If using sprouted rye flour, the dough won't be as firm as with sprouted whole wheat flour, but it should come together to make a thick, sticky dough. This is fine. Mist a medium bowl with vegetable spray oil and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Ferment the dough at room temperature for 4 to 8 hours, until it doubles in size or swells noticeably (the expansion can vary depending on which flour you use, as rye flour won't rise as much as wheat). Use immediately or refrigerate for up to 3 days.\n\n### FINAL DOUGH\n\nSprouted Rye Final Starter or recently refreshed Mother Starter: all (8.5 oz \/ 240.5 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00bd cups plus 2 tablespoons (13 oz \/ 369 g)\n\nsprouted rye flour: 2\u00bd cups plus 1 tablespoon (9 oz \/ 255 g)\n\nsprouted whole wheat flour: 1\u2154 cups (7 oz \/ 198 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g \/ 2%)\n\ninstant yeast (optional): 1 teaspoon (0.11 oz \/ 3 g)\n\nTOTAL: 37.93 oz \/ 1,074.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nSprouted Rye Final Starter or recently refreshed Mother Starter| 53  \nwater| 81  \nsprouted rye flour| 56  \nsprouted whole wheat flour| 44  \nsalt| 2  \ninstant yeast| 0.7  \nTOTAL| 236.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the starter and warm water to distribute the starter (on low speed if using a stand mixer). Add the flour and salt, and for a faster-rising, less tangy version, the yeast. Mix or stir until a coarse, wet dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should be smooth but still be very soft and sticky. Add a bit more flour or water if necessary to achieve that texture; the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but will still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at 20-minute intervals, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 3 to 5 hours, or 1\u00bd to 2 hours if using the optional yeast, until double in size. (Either time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.) If the dough hasn't doubled in size in the suggested time, you may need to extend the fermentation to as much as 8 hours, but it will eventually rise.\n\n5 Oil the work surface again and use an oiled bowl scraper or rubber spatula to transfer the dough to the oiled area. For hearth loaves, prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule or b\u00e2tard, then put the shaped loaves in the prepared proofing vessels. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the dough into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan.\n\n6 Mist the top of the dough with vegetable spray oil and cover it loosely with plastic wrap. Proof at room temperature until the dough increases in size by 1\u00bd times, 2 to 3 hours, or 1 to 2 hours if using the optional yeast. When poked with a finger, the dough should spring back within a few seconds; if it holds the dimple, it's risen for too long. (Because the dough is so fully hydrated, it's fragile and will fall if you proof it until double in size. It's better to bake it while it's still on the rise.)\n\n7 To bake a hearth loaf, about 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan, then preheat the oven to 450\u00b0F (232\u00b0C). Transfer the shaped loaf to a lightly floured peel (or keep it on the sheet pan for baking). Score. Transfer the loaf to the baking stone (or put the sheet pan on the baking stone). Pour about 1 cup of hot water into the steam pan. Bake for 20 minutes, then rotate and bake for 15 to 20 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be 200\u00b0F (93\u00b0C). Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n8 To bake pan loaves, preheat the oven to 375\u00b0F (191\u00b0C); steam is optional. Bake for 15 minutes, then rotate and bake for 30 to 40 minutes longer, until the crust is golden brown all over and the bread sounds hollow when thumped on the bottom. The internal temperature should be 190\u00b0F (88\u00b0C). Let cool in the pans for at least 10 minutes, then transfer to a wire rack and let cool for at least 30 minutes longer before slicing and serving.\n\n9 To bake rolls, preheat the oven to 400\u00b0F (204\u00b0C); steam is optional. Bake for 12 minutes, then rotate and bake for about 10 to 15 minutes longer, until the rolls are golden brown and sound hollow when thumped on the bottom (they will soften as they cool). The internal temperature should be about 190\u00b0F (88\u00b0C). Transfer to a wire rack and let cool for at least 10 minutes before serving.\n\nNOTES:\n\n\u2022 If you play with the ratio of wheat and rye flour, note that the more rye flour, the more difficult the challenge, especially if you want to try to achieve an open crumb structure. If the bread is to open up in the oven, the dough must be fairly wet, and you'll need to handle it carefully. Also, the oven heat must be quite high for hearth breads; you might start the oven at its highest setting and then lower it to the designated temperature right after the steaming process. This can help create initial oven spring, especially with wet doughs.\n\n\u2022 For an overnight method, put the covered bowl of dough in the refrigerator immediately after the final stretch and fold (or after 2 hours if not using commercial yeast). The next day, remove it from the refrigerator 3\u00bd hours before you plan to bake. Shape the cold dough and proof it at room temperature until it increases in size by 1\u00bd times, then bake as directed.\n\n### Variations\n\nFeel free to add these ingredients with the dry ingredients to create variations.\n\nSEEDED RYE: Add 1 to 2 tablespoons (0.32 to 0.65 oz \/ 9 to 18 g) caraway or nigella seeds.\n\nONION RYE: Add 1 to 2 tablespoons (0.18 to 0.35 oz \/ 5 to 10 g) dried onion flakes, or use \u00bc to \u00bd cup (1.5 to 3 oz \/ 42.5 to 85 g) finely diced fresh onion.\n\nSWEDISH-STYLE LIMPA RYE: Add the grated zest of 1 orange and 1 to 2 tablespoons (0.32 to 0.65 oz \/ 9 to 18 g) anise seeds.\n\nSprouted Wheat Pizza Dough\n\n# SPROUTED WHEAT PIZZA DOUGH\n\nMAKES 5 INDIVIDUAL PIZZA CRUSTS\n\nThis dough elevates 100% whole grain pizza to the next level. Unlike most pizza doughs, it can be mixed and baked on the same day and still achieve its maximum flavor potential, thanks to the sprouted flour. However, you can also hold the dough in the refrigerator for up to three days or in the freezer for three months. Feel free to top this dough with any of your favorite cheeses and sauces, but for the best results, be sure to bake it at a high temperature.\n\n### DOUGH\n\nsprouted whole wheat flour: 5\u2154 cups (24 oz \/ 680 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: 1\u215d teaspoons (0.18 oz \/ 5 g)\n\nwater, at room temperature: 2\u00bd cups plus 3 tablespoons (21.5 oz \/ 610 g)\n\nolive oil: 2 tablespoons (1 oz \/ 28.5 g \/ 4%)\n\nTOTAL: 47.18 oz \/ 1,337.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 2.1  \ninstant yeast| 0.7  \nwater| 90  \nolive oil| 4  \nTOTAL| 196.8\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Add the water and olive oil and mix or stir until the flour is hydrated and a coarse, wet dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should smooth out and thicken slightly.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will become firmer and less sticky with each stretch and fold. After the final fold it should be very tacky and supple and have a springy or bouncy quality when patted. (Note: If holding the dough overnight, put the dough in the refrigerator immediately after the final stretch and fold cycle instead of letting it rise.)\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Line a sheet pan with parchment paper or a silicone mat. Mist the surface with vegetable spray oil or lightly coat it with olive oil. Oil the work surface again, and transfer the dough to the work surface with an oiled bowl scraper or rubber spatula. Divide the dough into 5 equal pieces, each weighing about 9 ounces (255 g), with an oiled metal pastry blade or plastic bowl scraper.\n\n6 With lightly oiled hands, form each piece into a boule. Put the dough balls on the prepared pan, spacing them evenly. Mist with vegetable spray oil, then loosely cover the pan with plastic wrap or put it in a large plastic bag. Proof for 1 to 2 hours; the dough won't double in size, but it should show signs of swelling and expansion. If you won't be making pizzas immediately, refrigerate the dough, then remove it from the refrigerator about 1\u00bd hours before you plan to make the pizzas.\n\n7 To shape a pizza, see Shaping Pizza Dough. Preheat the oven to the highest it will go.\n\n8 When the crust is ready to be topped, place it on a floured peel. Be sure to use flour rather than cornmeal or semolina, as it doesn't burn as quickly in the oven. Top the pizza as desired, then slide it onto the baking stone. If you aren't using a baking stone, just put the panned pizza in the oven. While the pizza is baking, shape your next pizza.\n\n9 Bake for about 4 minutes, then use the peel or a spatula to rotate the pizza. It will take anywhere from 5 to 7 minutes for the pizza to fully bake, depending on the oven. The edge should puff up and be a deep golden brown, perhaps even slightly charred.\n\n10 Remove the pizza, garnish as desired, then let it cool for 1 minute before slicing and serving. Continue baking as many pizzas as you'd like (the dough will hold up to an hour out of the refrigerator.)\n\n### Variation\n\nMULTIGRAIN SPROUTED PIZZA DOUGH: You can replace up to 20% of the sprouted whole wheat flour with an equal amount, by weight, of other sprouted or nonsprouted flours, such as rye, corn, buckwheat, or millet. Depending on the grain, you'll probably need to decrease the water by about 3 tablespoons (1.5 oz \/ 42.5 g).\n\n## Shaping Pizza Dough\n\nTo shape pizza dough, press the ball of dough into a flat disk using your fingertips. Slide the backs of your hands under the dough, then lift it and begin to rotate it, using your thumbs to coax the edges of the dough into a larger circle. Don't stretch the dough with the backs of your hands or your knuckles; let your thumbs and gravity do all of the work. Your hands and knuckles merely provide a platform to support the dough. (If the dough starts to shrink back, set it on the floured work surface and let it rest for a minute or two. You can move on to another dough ball, repeating the same gentle stretching.) Work from the edges only, not from the center of the dough, and continue stretching until you have a 9- to 12-inch disk. Place the shaped dough on a floured or parchment-lined peel or back of a sheet pan. Patch any holes in the dough so the sauce and other toppings don't go through the dough, then add toppings.\n\n# GLUTEN-FREE SPROUTED FLOUR PIZZA DOUGH\n\nMAKES 8 TO 10 INDIVIDUAL PIZZA CRUSTS\n\nBecause gluten-free sprouted flours generally work better in a batter than a stiff dough, they are ideal for this innovation: twice-baked gluten-free pizza crusts! After all, a pizza crust is thin, it doesn't have to hold a loaf shape, and it's best when crispy on the bottom yet moist and creamy inside. In other words, it's like a big pancake\u2014and I have, in fact, used these crusts as pancakes, drizzling them with maple syrup rather than using savory toppings. There's no sweetener in the dough because sprouted flours are naturally sweet, and the number and variety of flours you can use is limited only by what you have in your pantry, with just one caveat: If using sprouted bean flour, include no more than 20% in your blend, as its stronger flavor tends to take over. As you become familiar with the flavors of different sprouted gluten-free flours, you can create a blend that suits your preferences. Whatever flours you use, I guarantee that this is better tasting (and better for you) than any commercial gluten-free pizza crust you can buy.\n\nTo bake the crusts, you can use any size of round glass or metal pan, but my suggestion is to use disposable aluminum pie pans. Although they make small pizzas, they do a great job of releasing the crust when properly oiled, and because they're lightweight, you can even stack and store the crusts in the pans. Plus, if you have enough of them, you can bake the entire batch of dough in one round of baking, especially if you have three or four shelves in your oven; or store them in the freezer to pull out as needed. They thaw very quickly for an instant pizza.\n\n### DOUGH\n\nsprouted gluten-free flour blend (any combination of sprouted gluten-free grain and bean flours): about 8\u00bd cups (28 oz \/ 794 g)\n\nground sunflower seeds (optional): \u00bc cup (1 oz \/ 28.5 g \/ 3%)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nxanthan gum powdered or psyllium husks (optional): 2\u00bc teaspoons (0.25 oz \/ 7 g)\n\ninstant yeast: 1\u00bc teaspoons (0.14 oz \/ 4 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 5\u00bd cups (44 oz \/ 1,247 g)\n\nTOTAL: 73.89 oz \/ 2,094.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted gluten-free flour blend| 97  \nground sunflower seeds| 3  \nsalt| 1.7  \nxanthan gum powdered or psyllium husks| 0.9  \ninstant yeast| 0.5  \nwater| 152  \nTOTAL| 255.1\n\n1 In a large bowl, stir together the flour blend, sunflower seeds, salt, xanthan gum, and yeast. Add the water and stir or whisk for 30 to 60 seconds to hydrate the flour and make a thin, batter-like dough; the consistency should be similar to pancake batter (it will thicken as it sits). Scrape down the bowl with a rubber spatula and cover the bowl with plastic wrap. Ferment the dough at room temperature for 1 to 1\u00bd hours, until it begins to bubble. If making the crusts right away, you can proceed as soon as the dough bubbles. If not, refrigerate the dough for up to 36 hours.\n\n2 For each crust, brush a disposable aluminum pie pan or other round pan with at least \u00bd teaspoon of olive or vegetable oil. Ladle or pour in about \u00be cup (177ml) of the batter (more for larger pans), using enough to cover the bottom of the pan to a thickness of \u00bc inch. Tilt the pan to evenly distribute the batter. Ferment at room temperature for about 1 hour, until signs of bubbling appear.\n\n3 Preheat the oven to 350\u00b0F (177\u00b0C).\n\n4 Put as many pans as will comfortably fit into the oven (refrigerate any others and bake them, directly from the refrigerator, after the first round of baking). Bake for 20 minutes, then rotate and bake for 10 to 20 minutes longer, until the crusts are golden brown around the edges and springy to the touch.\n\n5 If not using disposable pie pans, transfer the baked crusts to a wire rack. You can use them immediately; or, let them cool completely, then wrap well in plastic wrap and store in the refrigerator for up to 1 week or the freezer for up to 3 months.\n\n6 When you're ready to make pizzas, preheat the oven to 450\u00b0F (232\u00b0C). Top the crusts as you would any pizza. Bake them either in the pie pans, on sheet pans, directly on the oven racks, or on a fully heated baking stone. Bake for 6 to 8 minutes, until the toppings and cheese look done. The crust will crisp up and the edges will turn dark brown.\n\n## Q & A\n\n### Why do gluten-free breads often call for xanthan gum?\n\nXanthan gum, which is produced from fermented seeds, can play the role of gluten, to a limited extent, by providing stretchability. Some bakers prefer to use guar gum, but I find it can cause stomach upset for a small percentage of people. Another option is to use half guar gum and half xanthan gum; some gluten-free bakeries do it this way and believe the combination produces a stronger structure. A more recent innovation is to use powdered psyllium husks in place of xanthan gum. If making this substitution, use twice as much psyllium, by volume, as you would xanthan gum.\n\nSprouted Wheat Focaccia\n\n# SPROUTED WHEAT FOCACCIA\n\nMAKES ONE 18 BY 13-INCH PAN OF FOCACCIA; 12 SERVINGS\n\nI may be even more well known for my focaccia recipes than I am for pizza, at least at Johnson & Wales University, where I'm asked to do focaccia demonstrations more than any other baked good. I think a key reason for this is that most Americans have never had truly great focaccia, so they experience it as a revelation when they do. Focaccia is akin to what is sometimes referred to as Sicilian-style pizza, and the following dough can be used for either. Sicilian-style pizza can be every bit as satisfying as the thin-crusted pizzas that hail from southern Italy\u2014if you follow a few important but simple rules. Basically, success lies in a wet, sticky dough; long, slow fermentation (usually overnight); and a very hot oven. Those rules still apply to dough made with sprouted flour, with the exception of long, slow fermentation, which isn't necessary for reasons explained in this book.\n\nConditioning the dough overnight is certainly an option with this recipe, but sprouting the grain prior to milling flour has already accomplished the task of flavor development, so it's an unnecessary step, though it may offer convenience on bake day. Welcome to the next frontier of focaccia!\n\n### DOUGH\n\nsprouted whole wheat flour: 4\u00be cups (20 oz \/ 567 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g)\n\ninstant yeast: 1\u00bc teaspoons (0.14 oz \/ 4 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2\u00bc cups plus 1 tablespoon (18.5 oz \/ 524 g)\n\nolive oil: 1 tablespoon (0.5 oz \/ 14 g)\n\nadditional olive oil for the pan and for dimpling: about 3 tablespoons\n\nherb oil for topping (optional): about 3 tablespoons\n\nTOTAL: 39.46 oz \/ 1,118 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.5  \ninstant yeast| 0.7  \nwater| 92  \nolive oil| 2.5  \nTOTAL| 196.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Add the water and olive oil and mix or stir until the flour is hydrated and a coarse, wet, almost soupy dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should smooth out and thicken slightly, but it will still be very wet and sticky.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will become firmer and less sticky with each stretch and fold. After the final fold it should be just slightly sticky. It will barely hold together when lifted, but it should have structure.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Line an 18 by 13-inch rimmed sheet pan with parchment paper or a silicone mat. Generously oil the surface, including the sides of the pan, with 1\u00bd tablespoons (0.75 oz \/ 21 g) of the additional olive oil. Using an oiled bowl scraper or rubber spatula, transfer the dough to the center of the pan. Drizzle the remaining 1\u00bd tablespoons (0.75 oz \/ 21 g) of additional olive oil evenly over the dough. Dip your fingertips in a bit of olive oil, then dimple the dough until it fills the pan. You may need to do this in two or three passes, allowing the dough to rest for 5 minutes between passes so it can relax enough to spread into the corners without shrinking back. If you like, drizzle the herb oil evenly over the dough. Cover the pan loosely with plastic wrap and let the dough rise at room temperature for 1 to 1\u00bd hours, until it rises to the top of the pan. If using other toppings, add them after the dough has risen; see Assembling a Focaccia for ideas.\n\n6 Position a rack in the middle of the oven and preheat the oven to 450\u00b0F (232\u00b0C).\n\n7 Bake for 15 minutes, then rotate the pan and bake for 10 to 15 minutes longer, until the top of the focaccia is golden brown.\n\n8 Run a metal pastry blade or metal spatula around the edges to separate the focaccia from the pan. Transfer the entire focaccia to a cutting board. If the parchment paper or silicone mat adheres to the bottom of the focaccia, carefully remove it by slightly lifting one edge of the focaccia and sliding it out after the focaccia cools for a few minutes. If any oil remains in the pan, drizzle it over the focaccia. Let cool for 5 minutes before cutting and serving.\n\nNOTE: For an overnight method, you can refrigerate the dough at one of two stages\u2014after the final stretch and fold or after panning\u2014for later baking. If you refrigerate it after the final stretch and fold, remove the bowl from the refrigerator 4 hours before you plan to bake the focaccia, and proceed with panning and baking as described here. If you refrigerate the pan overnight, remove it from the refrigerator 3 hours before you plan to bake the focaccia, do any final dimpling necessary to fill the pan with the dough, and proof and bake as directed.\n\n### Variations\n\nSICILIAN-STYLE PIZZA: Bake the pan of dough without toppings after it fills the pan but before it rises, for 10 to 15 minutes, until it just starts to set and brown. You can then use this as a crust at any time, finishing it with sauce, cheese, and other toppings, then baking as directed above.\n\nMULTIGRAIN FOCACCIA: You can replace up to 20% of the sprouted whole wheat flour with an equal amount, by weight, of other sprouted or nonsprouted flours, such as rye, corn, buckwheat, or millet. Depending on the grains, you'll probably need to decrease the amount of water by about 3 tablespoons (1.5 oz \/ 42.5 g).\n\n## Herb Oil\n\nThis herb oil can also be used to flavor focaccia toppings like thinly sliced tomatoes or potatoes. In addition, it's great as a dipping oil for bread or as a base for salad dressings.\n\n2 cups (16 oz \/ 454 g) olive oil\n\n2 tablespoons (0.14 oz \/ 4 g) dried basil\n\n2 tablespoons (0.14 oz \/ 4 g) dried parsley\n\n1 tablespoon (0.07 oz \/ 2 g) dried oregano\n\n1 tablespoon (0.07 oz \/ 2 g) fresh rosemary leaves\n\n1 teaspoon (0.02 oz \/ 0.5 g) dried thyme\n\n2 tablespoons (0.7 oz \/ 22 g) granulated garlic, or 10 cloves fresh garlic, pressed and lightly saut\u00e9ed in \u00bd cup (4 oz \/ 113 g) of the olive oil\n\n1 tablespoon (about 0.5 oz \/ 14 g) kosher salt or coarse sea salt\n\n\u00bc teaspoon (0.02 oz \/ 0.5 g) freshly ground black pepper\n\n1 teaspoon (0.11 oz \/ 3 g) dried chile flakes (optional)\n\n1 teaspoon (0.09 oz \/ 2.6 g) sweet or hot paprika (optional)\n\nIn a bowl, whisk together all the ingredients. Use immediately, or keep any extra in the refrigerator, where it will keep for a month.\n\n## Assembling a Focaccia\n\nIt may be tempting to think you can top focaccia just as you would a pizza, with everything going on top before you bake it. But this is only partially true. While a pizza may bake in 5 to 7 minutes, a focaccia, with its thicker crust, typically takes about 20 to 25 minutes to bake. Many toppings, such as cooked vegetables and tomatoes or tomato sauce, moist cheeses like feta and blue cheese, or meats like bacon, salumi, chicken, and sausage, can go on the top just prior to baking. Drier ingredients, like sun dried tomatoes or dried fruits like raisins, are best worked into the dough itself, where they will be protected from burning in the hot oven. But semi-moist cheeses, such as mozzarella, provolone, Cheddar, and Swiss, take only a few minutes to melt and caramelize, so they should be reserved and then added during the final 5 to 7 minutes of baking. Dry aged cheeses, like Parmesan, Romano, Asiago, and the like will begin to burn after only 2 to 3 minutes in the oven, so they should be held back until the focaccia looks fully baked, then added and returned to the oven for 2 additional minutes of baking, or as needed.\n\nThe general rule of thumb for how much topping to use is to keep everything in balance; too much will weigh down the dough; too many toppings will create a muddle of flavors. It's best to use a small amount of high quality, flavorful ingredients rather than to overload the dough. One or two cups of saut\u00e9ed or grilled vegetables, a cup or so of cheese, \u00bd cup (120ml) of pizza sauce, and just enough specialty ingredients like nuts or meats to assure that every bite will contain a small amount of every topping is the key. Restraint will pay off. When using herbs, spices, and salt on the top, remember that a little goes a long way.\n\nThere are many inventive toppings that have been developed for focaccia, many of them easily found online. But the most popular versions in my classes are the following:\n\nPOTATO, BACON, PARMESAN: While the pan of focaccia dough is rising, cook up about 6 to 8 slices of bacon till crisp, then chop or crumble them into bits after they cool. Grate about 1 cup of Parmesan or other dry aged cheese and set it aside. Thinly slice any type of potato as if making potato chips, paper-thin (I use a small Kyocera ceramic slicer that I bought at Sur la Table, but there are other models available, as well as more sophisticated mandoline style slicers). It only takes one or two potatoes to make enough slices to cover the surface of a pan of focaccia. You can also use a knife but it is difficult to get the slices paper-thin. If the slices are more than \u215b inch thick, parboil them in simmering, salted water for about 60 seconds to soften. Toss the slices in 2 tablespoons (30ml) of olive oil or, even better, herb oil (see recipe). When the focaccia dough is fully risen and ready for baking, cover the surface with the bacon pieces and then cover the bacon with the oiled potato slices. Bake as directed, until the potatoes turn golden brown, then add the grated cheese and return to the oven and bake for 2 additional minutes, or until the cheese melts and just begins to brown. Allow the focaccia to cool for at least 10 minutes before serving.\n\nTOMATO PESTO: Slice 6 to 8 fresh plum tomatoes about \u00bc inch thick and toss with 3 tablespoons (45ml) of olive oil or, even better, herb oil. Prepare pesto using your favorite recipe, or use commercial pesto. Cover the surface of the risen focaccia with the oiled tomatoes and bake as directed. Because of the moistness of the tomatoes, the focaccia will take about 5 to 10 minutes longer to bake than a standard focaccia. When the dough under the tomatoes is golden brown and springy to the touch and the tomatoes are beginning to slightly char, the focaccia is baked. Immediately after removing it from the oven, drizzle 1 cup of pesto over the top. Let the focaccia cool for at least 10 minutes before serving.\n\n# SPROUTED WHEAT BREAKFAST FOCACCIA\n\nMAKES ONE 18 BY 13-INCH PAN OF FOCACCIA; 12 SERVINGS\n\nThis focaccia with fruit is one of my favorite recipes in this book. Once you start eating it, it's very difficult to stop. When you first make the dough it will seem too wet, but don't overcompensate by adding more flour. It will thicken up overnight as the dried fruit plumps and absorbs some of the water. The glaze is just an option, but it will elicit encore calls. In addition to being a great breakfast bread, it can also be served as a dessert or sweet treat. Note that this is a two-day process, so plan accordingly.\n\n### DOUGH\n\nsprouted whole wheat flour: 4\u00be cups (20 oz \/ 567 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g)\n\ninstant yeast: 2\u00bc teaspoons (0.25 oz \/ 7 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2\u00be cups plus 2 tablespoons (23 oz \/ 652 g)\n\ndried cranberries: about \u00be cup (5 oz \/ 142 g)\n\ngolden raisins: about \u00be cup (5 oz \/ 142 g)\n\nolive oil for the proofing bowl, the pan, and topping: about 4 tablespoons\n\norange or lemon glaze (optional)\n\nTOTAL: 53.57 oz \/ 1,519 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.6  \ninstant yeast| 1.2  \nwater| 115  \ndried cranberries| 25  \ngolden raisins| 25  \nTOTAL| 267.8\n\n1 In the bowl of a stand mixer fitted with the paddle attachment or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Add the water and mix or stir until the flour is hydrated and a very wet, soupy batter forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Add the dried cranberries and raisins. Switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer, until the fruit is evenly distributed. The dough should thicken slightly, but it will still be very wet, with the consistency of a thick soup. Let it rest for 5 minutes, then stir for 30 seconds.\n\n3 Use 1 tablespoon (0.5 oz \/ 14 g) of the olive oil to coat a clean bowl large enough to hold the dough if it doubles in size. (It probably won't double, but it will expand and then fall.) Transfer the dough to the bowl, mist the top with vegetable spray oil, and cover the bowl with plastic wrap. Let the dough rest at room temperature for 15 minutes. Put the bowl in the refrigerator for a cold, overnight fermentation. (At this point, the dough will keep for up to 3 days.)\n\n4 Remove the dough from the refrigerator 3\u00bd hours before you plan to bake. It will be much thicker now, though still somewhat sticky. Line an 18 by 13-inch rimmed sheet pan with parchment paper or a silicone mat. Generously oil the liner and the sides of the pan with 1\u00bd tablespoons (0.75 oz \/ 21 g) of the olive oil. Using an oiled bowl scraper or rubber spatula, transfer the cold dough to the center of the pan. Drizzle the remaining 1\u00bd tablespoons (0.75 oz \/ 21 g) of olive oil evenly over the dough. Dip your fingertips in a bit of olive oil, then dimple the dough until it fills the pan. You may need to do this in two or three passes, allowing the dough to rest for 5 minutes between passes so it can relax enough to spread into the corners without shrinking back. Cover the pan loosely with plastic wrap and let the dough rise at room temperature for about 3 hours, until it rises to the top of the pan.\n\n5 Position a rack in the middle of the oven and preheat the oven to 450\u00b0F (232\u00b0C).\n\n6 Bake for 15 minutes, then rotate the pan and bake for 15 to 20 minutes longer, until the top of the focaccia is golden brown and firm to the touch. The exposed pieces of fruit will get very dark.\n\n7 Run a metal pastry blade or metal spatula around the edges to separate the focaccia from the pan. Transfer the entire focaccia to a cutting board. If the parchment paper or silicone mat adheres to the bottom of the focaccia, carefully remove it by slightly lifting one edge of the focaccia and sliding it out.\n\n8 Drizzle the glaze evenly over the focaccia, then use a rubber spatula or icing spatula to spread it evenly over the top. (If you don't use the glaze, brush enough olive oil over the surface to create a shiny top.) Let the focaccia cool for at least 45 minutes, until the glaze sets up, before cutting and serving. (If you don't use the glaze, you only need to wait 10 minutes.)\n\n## Orange or Lemon Glaze\n\n2 cups (6 oz \/ 170 g) confectioners' sugar, sifted\n\n\u00bd teaspoon (0.07 oz \/ 2 g) orange or lemon extract, 2 teaspoons (0.42 oz \/ 12 g) thawed orange juice concentrate, or 1 tablespoon (0.5 oz \/ 14 g) lemon juice\n\n\u00bc cup to 6 tablespoons (2 to 3 oz \/ 56.5 to 85 g) water or milk\n\nIn a medium bowl, stir together the confectioners' sugar and orange or lemon extract. Whisk in \u00bc cup (2 oz \/ 56.5 g) of the water or milk, adding more only if necessary to make a thick but spreadable glaze. It should be thick enough to hold a spoon upright for a few seconds. If it seems thin, add more sifted confectioners' sugar.\n\n# SPROUTED WHEAT BAGELS\n\nMAKES 6 OR 7 BAGELS OR 12 MINI BAGELS\n\nIt seems as though everybody loves bagels, even folks who don't eat them anymore out of concerns about carbs or gluten. Perhaps creating healthier versions will allow more people to enjoy them once again. In this book you'll find two different recipes for bagels, each following different formulas and using different ingredients; both are excellent. Aside from amazing flavor, the biggest advantage of this version made with sprouted flour is that, unlike many bagel recipes, this one doesn't require an overnight method to develop optimum flavor. You can find barley malt syrup at most natural food stores and many supermarkets. If you'd like to use crystal malt, it's available at beer making supply stores or online (see Resources).\n\n### DOUGH\n\nsprouted whole wheat flour: 4\u00bc cups (18 oz \/ 510 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g)\n\ninstant yeast: 1\u00bc teaspoons (0.14 oz \/ 4 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00be cups (14 oz \/ 397 g)\n\nbarley malt syrup (see notes) or crystal malt: 3\u00bd teaspoons or 1 teaspoon (0.75 or 0.18 oz \/ 21.5 or 5 g \/ 4.2 or 1%)\n\nbaking soda or honey for boiling the bagels: 2 tablespoons\n\ncornmeal or semolina flour: about 2 tablespoons\n\nTOTAL: 33.21 or 32.64 oz \/ 941.5 or 925 g \/ 184.8 or 181.6%\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.8  \ninstant yeast| 0.8  \nwater| 78  \nbarley malt syrup or crystal malt| 4.2 or 1  \nTOTAL| 184.8 or 181.6\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast, on low speed if using a stand mixer. (If using crystal malt, add it at this point.) In a separate bowl, whisk together the water and barley malt syrup, then pour into the flour mixture. Mix or stir for 30 to 60 seconds or until the flour is hydrated and a coarse, shaggy dough forms. It will be much stiffer than any other dough in the book; this is necessary for the bagels to stand up to the boiling process without collapsing.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook and mix on low speed, or turn the dough out onto a work surface and knead, for 2 minutes. The dough should be smooth but still fairly firm and only slightly tacky. Add a bit more flour or water if necessary to achieve this texture, then mix or knead for 1 minute longer. The dough should be firm but supple, and dry and satiny rather than tacky to the touch.\n\n3 Form the dough into a ball. Mist a large bowl with vegetable spray oil and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Ferment the dough at room temperature for 1\u00bd to 2 hours, until the dough begins to swell and increases in size by about 1\u00bd times.\n\n4 Line a sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil. Spread a bit of vegetable oil on the work surface and transfer the dough to the oiled area. Divide the dough into pieces of the desired size; I recommend 4.5 ounces (128 g), but you can make larger or smaller bagels if you prefer. Shape the bagels and put them on the prepared pan, spacing them at least 1 inch apart. Mist the tops with vegetable spray oil and cover the pan loosely with plastic wrap.\n\n5 Proof at room temperature for 30 to 60 minutes, until the bagels just begin to swell and rise.\n\n6 About 20 minutes before you plan to bake the bagels, position a rack in the middle of the oven and preheat the oven to 425\u00b0F (218\u00b0C). Do a float test: Place one bagel in a small bowl of room-temperature water. If it doesn't float within 15 seconds, return it to the pan and check again every 20 minutes until a bagel passes the float test. Line a sheet pan with parchment paper or a silicone mat, mist with vegetable spray oil, and dust with cornmeal.\n\n7 Bring 4 to 6 inches of water to a boil in a wide pot. Add the baking soda or honey to the water (both will promote deeper browning as well as add flavor) and lower the heat to maintain a simmer. Boil 2 to 4 bagels at a time for 30 seconds, then flip and boil for 30 seconds longer. Using a slotted spoon, transfer the bagels to the newly prepared baking pan, spacing them evenly, rounded side up. Garnish with selected toppings (see below) as soon as they emerge from the water.\n\n8 Bake for 12 minutes, then rotate the pan and bake for 8 to 12 minutes longer, until golden brown all over. If the bottoms are getting too dark during baking, slide a second pan under the first for insulation.\n\nNOTES:\n\n\u2022 You can substitute an equal amount of honey, agave nectar, sorghum syrup, or molasses for the malt syrup. The flavor will be slightly different with each.\n\n\u2022 For an overnight method, instead of fermenting the dough after mixing it, immediately divide it into the desired number of pieces. Line a sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil. Shape the pieces into bagels and put them on the prepared pan. Mist the tops with vegetable spray oil and cover the pan with plastic wrap or a large plastic bag. Refrigerate overnight. About 1 hour before you plan to boil and bake the bagels, remove them from the refrigerator. After 1 hour, do a float test, then boil and bake as directed.\n\n### Variations\n\nTOPPINGS: Sesame seeds and poppy seeds are the most common bagel toppings, but other options include black onion seeds (also called nigella seeds), caraway seeds, granulated garlic, dried garlic flakes (soaked in just enough water to hydrate them), coarse kosher or pretzel salt, dried onion flakes (soaked in just enough water to hydrate them), or diced fresh onion.\n\nEVERYTHING BAGELS: Go for broke and use 1 tablespoon (about 0.18 oz \/ 5 g) granulated garlic or rehydrated dried garlic flakes, 1 tablespoon (0.32 oz \/ 9 g) poppy seeds, 1 tablespoon (0.32 oz \/ 9 g) sesame seeds, 1 teaspoon (about 0.16 oz \/ 4.5 g) coarse salt, \u00bd teaspoon (0.05 oz \/ 1.5 g) paprika or smoked paprika (sweet or hot, depending on your taste), and, optionally, 1 teaspoon (0.11 oz \/ 3 g) caraway or anise seeds.\n\nCINNAMON-SUGAR BAGELS: Whisk together \u00bd cup (3 oz \/ 85 g) sugar and 2 tablespoons (0.55 oz \/ 15.5 g) ground cinnamon. When the bagels are just out of the oven, immediately brush the tops with melted butter, then dip into the cinnamon sugar to coat the top. As the bagels cool, the cinnamon sugar will set on the crust.\n\nMULTIGRAIN BAGELS: You can substitute up to 20% other sprouted grain or sprouted bean flours for an equal amount of sprouted whole wheat flour to make a multigrain version. You can also substitute nonsprouted grain flours but, for best flavor, I suggest sticking with the sprouted flours.\n\nRAISIN OR CINNAMON RAISIN BAGELS: For raisin bagels, add 1\u00bd cups (9 oz \/ 255 g) raisins to the dough during the final minute of mixing or until they are evenly distributed. You may also need about 1 additional tablespoon (0.5 oz \/ 14 g) of water, but let the dough dictate whether or not it needs the added water. You can also add 1 teaspoon (0.09 oz \/ 2.6 g) ground cinnamon.\n\nSprouted Struan Bread\n\n# SPROUTED STRUAN BREAD\n\nMAKES 2 LOAVES OR UP TO 24 ROLLS\n\nIn each of my bread books I include a new recipe for struan. It's my all-time favorite bread, and I keep coming up with new ways to make it because, although it's a traditional bread, there has never been a specific recipe for it other than the ones I've published. Its origins are in the Michaelmas harvest celebration of Western Scotland, where it was made only one time a year, on September 28, the eve of the feast, using whatever ingredients could be harvested. The name struan comes from the Gaelic sruthan, which means \"a convergence of streams.\"\n\nAt Brother Juniper's Bakery this was our signature loaf, and our top-selling bread by far. I've come to think of it as a metaphor as much as a bread; the metaphor of me (and all of us, really)\u2014yes, a convergence of streams. One of the reasons why it remains so popular is that it makes exquisite toast. For a number of years, I pretty much lived on a breakfast of two slices of buttered struan toast slathered with local blueberry or raspberry jam. It's a deeply embedded taste memory; it is my madeleine.\n\n### DOUGH\n\nsprouted whole wheat flour: 3\u00be cups (16 oz \/ 454 g)\n\nsprouted corn flour or cornmeal: 6\u00bd tablespoons (1.6 oz \/ 45.5 g)\n\nsprouted rolled oats: \u2153 cup (1.6 oz \/ 45.5 g)\n\nsprouted brown rice flour (or sprouted rye flour): 2 tablespoons plus 2 teaspoons (or 3 tablespoons plus 2 teaspoons) (0.8 oz \/ 22.5 g)\n\nsalt: 1\u00bd teaspoons (0.35 oz \/ 10 g)\n\ninstant yeast: 5 teaspoons (0.55 oz \/ 15.5 g)\n\nbrown sugar (or honey or agave nectar): 6 tablespoons (or 4\u00bd tablespoons) (3 oz \/ 85 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00bc cups (10 oz \/ 283 g)\n\nbuttermilk, lukewarm (95\u00b0F \/ 35\u00b0C): \u00be cup (6 oz \/ 170 g)\n\ncooked brown rice: \u2153 cup (2 oz \/ 56.5 g)\n\negg white wash: 1 egg white whisked with 1 tablespoon water\n\npoppy or sesame seeds (optional): 2 to 3 teaspoons\n\nTOTAL: 41.9 oz \/ 1,187.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 87  \nsprouted corn flour or cornmeal| 8.7  \nsprouted rolled oats| 8.7  \nsprouted brown rice flour| 4.3  \nsalt| 1.9  \ninstant yeast| 2.9  \nbrown sugar| 16  \nwater| 54  \nbuttermilk| 33  \ncooked brown rice| 11  \nTOTAL| 227.8\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the sprouted whole wheat flour, corn flour, rolled oats, rice flour, salt, and yeast, on low speed if using a stand mixer. (If using sugar rather than honey or agave nectar, add it at this point.) In a separate bowl, stir together the water and buttermilk, then stir in the honey and cooked rice. Add the buttermilk mixture to the flour mixture and mix or stir until the flour is hydrated and a coarse, sticky dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 2 minutes, on medium-low speed if using a stand mixer. The dough should become more cohesive, but it will still be very soft and sticky, though not as loose as a batter. (The various grains take a while to fully hydrate.) Add a bit more flour or milk if necessary to achieve this texture, but bear in mind that the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Oil the work surface again. Using an oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the loaves in the prepared pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the dough into the desired number of pieces (24 for small dinner rolls) and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan.\n\n6 Brush the tops of the dough with the egg white wash and sprinkle with poppy seeds if you wish. Mist with vegetable spray oil, then cover loosely with plastic wrap or place the pans in a large plastic bag. Proof for 1 to 1\u00bd hours at room temperature, until the dough increases in size by 1\u00bd times.\n\n7 To bake pan loaves, position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Bake for 20 minutes, then rotate and bake for 25 to 40 minutes longer, until the crust is golden brown on all sides and the loaves sound hollow when thumped. The internal temperature should be about 190\u00b0F (88\u00b0C).\n\n8 To bake rolls, position two racks in the oven an equal distance apart and preheat the oven to 400\u00b0F (204\u00b0C). Bake for 10 minutes, then rotate the pans front to back and between racks. Bake for 10 to 12 minutes longer, until the crust is golden brown and the rolls sound hollow when thumped. The internal temperature should be about 190\u00b0F (88\u00b0C).\n\n9 Transfer from the pans to a wire rack. Let loaves cool for at least 1 hour before serving. Rolls can be served after 10 minutes.\n\nNOTE: You can also form this dough into 1 large or 2 smaller freestanding boules. Proof each on a sheet pan lined with parchment paper or a silicone mat, then bake as described for rolls. Large boules will take 35 to 45 minutes to bake; small boules 30 to 40 minutes.\n\n### Variation\n\nYou can replace any or all of the nonwheat flours with an equal amount, by weight, of a sprouted flour multigrain blend, such as Lindley Mills Ancient Grain (see Resources), or individual flours, such as sprouted millet, quinoa, or spelt. You can also use nonsprouted flours or grains, such as conventional cornmeal or rolled oats, if you wish.\n\n# SPROUTED WHEAT SOFT ROLLS OR SANDWICH BREAD\n\nMAKES 2 LOAVES OR UP TO 36 ROLLS\n\nMany people find the Sprouted Whole Wheat Bread sufficiently sweet and soft to use for sandwiches and rolls. However, sandwich loaves and dinner rolls typically include enrichments such as milk, oil or butter, and sweeteners, so I also offer this recipe, which is formulated to more closely approximate the soft, sweet texture and flavor of enriched white bread while still using 100% sprouted whole wheat flour. This kind of dough is generically referred to as milk dough because it uses milk rather than water as the liquid. If you like a little tang, go for the buttermilk option, which provides some acidity for enhanced flavor.\n\n### DOUGH\n\nsprouted whole wheat flour: 6\u2154 cups (28 oz \/ 794 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: 2\u00bc teaspoons (0.25 oz \/ 7 g)\n\nsugar (or honey or agave nectar): 5 tablespoons (or 3 tablespoons) (2 oz \/ 56.5 g \/ 7%)\n\nmilk or buttermilk, lukewarm (95\u00b0F \/ 35\u00b0C): 3 cups (24 oz \/ 680 g)\n\nvegetable oil: 3 tablespoons (1.5 oz \/ 42.5 g)\n\negg wash: 1 egg whisked with 1 tablespoon water\n\nsesame or poppy seeds for topping (optional)\n\nTOTAL: 56.25 oz \/ 1,594 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.8  \ninstant yeast| 0.9  \nsugar| 7  \nmilk or buttermilk| 86  \nvegetable oil| 5.4  \nTOTAL| 201.1\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Stir in the sugar (if using honey or agave nectar, add it to the milk in the next step). Add the milk and oil and mix or stir until the flour is hydrated and a coarse, sticky dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should be smooth but still very soft and sticky. Add a bit more flour or milk if necessary to achieve this texture, but bear in mind that the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or a rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold sequence. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Oil the work surface again and use an oiled bowl scraper or rubber spatula to transfer the dough to the oiled area. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the dough into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan.\n\n6 Mist the exposed surface of the dough with vegetable spray oil. Cover loosely with plastic wrap or place the pans in a large plastic bag. Proof for 1 to 1\u00bd hours at room temperature, until the dough increases in size by 1\u00bd times. If making pan loaves, the dough should dome up above the pan by 1 inch or more.\n\n7 To bake pan loaves, position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Brush the tops of the loaves with the egg wash and sprinkle with sesame or poppy seeds if you wish. Bake for 20 minutes, then rotate and bake for 25 to 40 minutes longer, until the crust is golden brown on all sides and the loaves sound hollow when thumped. The internal temperature should be about 190\u00b0F (88\u00b0C).\n\n8 To bake rolls, position two racks in the oven an equal distance apart and preheat the oven to 400\u00b0F (204\u00b0C). Brush the tops of the rolls with the egg wash and sprinkle with sesame or poppy seeds if you wish. Bake for 12 minutes, then rotate the pans front to back and between racks. Bake for 8 to 12 minutes longer, until the crust is golden brown and the rolls sound hollow when thumped. The internal temperature should be about 190\u00b0F (88\u00b0C).\n\n9 Transfer from the pans to a wire rack. Let loaves cool for at least 1 hour before slicing and serving. Rolls can be served after 10 minutes.\n\nSprouted Sandwich Rye Bread\n\n# SPROUTED SANDWICH RYE BREAD\n\nMAKES 1 LARGE OR 2 SMALL LOAVES\n\nSandwich rye breads generally contain far less rye flour than wheat flour, mainly because the gluten content of rye is only about half that of wheat. In many commercial rye breads, the dough is reinforced with vital wheat gluten to ensure a tall, airy loaf. Another characteristic of sandwich rye bread is that sourdough starter is often used in addition to commercial yeast. It isn't the primary leavening, but it does provide some acidity and flavor and also helps control enzyme activity. Therefore, this recipe doesn't require building a final starter; rather, you can simply use a freshly rebuilt mother starter.\n\nSprouted rye flour isn't as easy to find as sprouted whole wheat flour, so you may need to purchase it online (see Resources). You can also use regular, nonsprouted rye flour, as long as the primary flour is sprouted wheat. Either way, this is a great bread for grilled cheese sandwiches and, of course, corned beef or pastrami sandwiches.\n\n### DOUGH\n\nsprouted whole wheat flour: 4\u00be cups (20 oz \/ 567 g)\n\nsprouted rye flour: 1\u00bc cups plus 3 tablespoons (5 oz \/ 142 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g \/ 2%)\n\ninstant yeast: 1\u215d teaspoons (0.18 oz \/ 5 g)\n\ntoasted caraway seeds or black nigella seeds (optional; see Toasting Seeds and Nuts for toasting instructions): about 1 tablespoon (0.32 oz \/ 9 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2\u00bd cups (20 oz \/ 567 g)\n\nmolasses: 1\u00bd tablespoons (1 oz \/ 28.5 g \/ 4%)\n\nMother Starter, recently refreshed: \u2154 cup (5 oz \/ 142 g)\n\nTOTAL: 52 oz \/ 1,474.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 80  \nsprouted rye flour| 20  \nsalt| 2  \ninstant yeast| 0.7  \ntoasted caraway seeds or black nigella seeds| 1.3  \nwater| 80  \nmolasses| 4  \nMother Starter| 20  \nTOTAL| 208\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, stir together the flours, salt, yeast, and caraway seeds (on low speed if using a stand mixer). In a separate bowl, combine the water, molasses, and mother starter and stir for about 20 seconds to soften the starter. Add the starter mixture to the flour mixture and mix or stir until the flour is hydrated and a coarse, sticky dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 to 2 minutes, on medium-low speed if using a stand mixer. The dough should be smooth but still slightly soft and sticky. Add a bit more flour or water if necessary to achieve this texture, but bear in mind that the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be firm and bouncy when slapped, and only slightly tacky.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Oil the work surface again. Using an oiled bowl scraper, transfer the dough to the oiled area. For a hearth loaf or 2 freestanding loaves, prepare two bannetons or a couche or line an 18 by 13-inch sheet pan with parchment paper then mist with vegetable spray oil. Shape the dough into one large boule or b\u00e2tard, or divide the dough in half and shape each piece into a small boule or b\u00e2tard. Put the shaped loaves in the prepared proofing vessel(s) or on the sheet pan. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans.\n\n6 Mist the exposed surface of the dough with vegetable spray oil. Cover loosely with plastic wrap or put the shaped loaves in a large plastic bag. Proof for 1 to 1\u00bd hours at room temperature, until the dough increases in size by 1\u00bd times. If making pan loaves, the dough should dome up above the pan by 1 inch or more.\n\n7 To bake 1 or 2 freestanding loaves, about 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan, then preheat the oven to 450\u00b0F (232\u00b0C). Transfer the dough to a floured peel (or keep it on the sheet pan for baking). Score the top with 3 or 4 parallel cuts, about \u00bd inch deep, across the top. (Rye bread is typically sliced straight across as opposed to diagonally, but this is really the baker's choice.) Transfer the dough to the middle shelf of the oven and, if baking hearth bread, pour about 1 cup of hot water into the steam pan. Bake for 5 minutes, then lower the temperature to 425\u00b0F (218\u00b0C). Bake for 20 minutes, then rotate and bake for 15 to 20 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be about 195\u00b0F (91\u00b0C). Transfer to a wire rack and let cool for at least 1 hour before slicing and serving.\n\n8 To bake pan loaves, preheat the oven to 375\u00b0F (191\u00b0C). Bake for 25 minutes, then rotate and bake for 25 to 35 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be about 190\u00b0F (88\u00b0C). Transfer the loaves from the pans to a wire rack and let cool for at least 1 hour for before slicing and serving.\n\n### Variations\n\nDARK SANDWICH RYE: Add 3 tablespoons (1 oz \/ 28.5 g) cocoa powder to the flour mixture and increase the amount of water by 1 tablespoon (0.5 oz \/ 14 g).\n\nONION RYE: Add 2 tablespoons (0.35 oz \/ 10 g) dried onion flakes or 1 onion, finely chopped, to the flour mixture.\n\n## Toasting Seeds and Nuts\n\nToast in a dry skillet over medium heat, stirring constantly, until they begin to darken and smell nutty. Immediately remove from the heat and transfer to a bowl to cool. An alternative method is to lay them out on a sheet pan and toast in a preheated oven 350\u00b0F \/ 177\u00b0C. Stir them every 3 minutes until they begin to darken and smell nutty. Whole nuts will take longer than chopped nuts or seeds.\n\n# GLUTEN-FREE \"DO NO HARM\" SPROUTED GRAIN BREAD\n\nMAKES 1 LOAF\n\nThere are a lot of gluten-free sandwich bread recipes and products out there, but to the best of my knowledge, no one has yet marketed a bread made solely with sprouted gluten-free flour, as this one is. This recipe employs a hybrid leavening of both yeast and baking powder. It also uses xanthan gum, which provides enough structure to hold the loaf together and to create a slightly open crumb. The dough is more like a batter, which is about the only way I've found to make a moist loaf with gluten-free flour. If the dough were firm enough to be moldable, it would taste like cardboard by the time it came out of the oven. Therefore, it can't be made as a hearth loaf, but that's about the only drawback to this recipe. It's free from the most common allergens, such as dairy, eggs, nuts, and wheat. It's why I call it Do No Harm Bread. The internal texture of this bread is soft and creamy, the crust is crisp and nutty, and it tastes great, especially when toasted. To maximize the flavors and texture, toast this bread until crisp and golden brown, which may take a few plunges of the toaster. If you have a corn sensitivity, you can replace the corn with more sprouted brown rice flour or a comparable increase in the other sprouted grain or bean flours in the recipe (all of which are available online, see Resources). One final note: If you have a set of mini loaf pans, this is a good time to use them, although you can also make a full size loaf. The mini loaves take only half as long to bake and are easy to store in the freezer, which is why I like them. Don't worry if the top of the loaf flattens out during baking due to the absence of gluten; it's a very small tradeoff for the moist, creamy interior.\n\n### DOUGH\n\nsprouted corn flour: 1\u00be cups (7 oz \/ 198 g)\n\nsprouted brown rice flour: 1\u00bd cups (7 oz \/ 198 g)\n\nsprouted amaranth, quinoa, black bean, garbanzo, or lentil flour: about \u00bd cup (2 oz \/ 56.5 g)\n\nflaxseeds or sesame seeds (optional): about 4 teaspoons (0.5 oz \/ 14 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\ninstant yeast: 1\u00bd teaspoons (0.16 oz \/ 4.5 g \/ 1%)\n\nbaking powder: 1\u00bd teaspoons (0.28 oz \/ 8 g)\n\nxanthan gum: 1 tablespoon (0.32 oz \/ 9 g \/ 2%)\n\nsugar or liquid stevia: 2 tablespoons or \u215b teaspoon (about 10 drops) (1 or 0.04 oz \/ 28.5 or 1 g \/ 6.3 or 0.2%)\n\nwater or unsweetened nondairy milk, lukewarm (95\u00b0F \/ 35\u00b0C): 3 cups plus 2 tablespoons (25 oz \/ 709 g)\n\nTOTAL: 43.51 or 42.55 oz \/ 1,232.5 or 1,205 g \/ 272.7 or 266.6%\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted corn flour| 44  \nsprouted brown rice flour| 44  \nsprouted amaranth, quinoa, black bean, garbanzo, or lentil flour| 12  \nflaxseeds or sesame seeds| 3.1  \nsalt| 1.5  \ninstant yeast| 1  \nbaking powder| 1.8  \nxanthan gum| 2  \nsugar or liquid stevia| 6.3 or 0.2  \nwater or unsweetened nondairy milk| 157  \nTOTAL| 272.7 or 266.6\n\n1 In a large bowl, stir together the sprouted flours, flaxseeds, salt, yeast, xanthan gum, and sugar. (If using liquid stevia, add it with the water in the next step.) Add the water and stir for about 1 minute to make a thick, sticky batter-like dough. Cover the bowl with plastic wrap and let the dough rest at room temperature for about 1 hour, until the dough swells slightly and shows signs of fermentation. It will thicken as it sits.\n\n2 Line the bottom of a 5 by 9-inch (or 4\u00bd by 8-inch) loaf pan with parchment paper and generously mist the pan with vegetable spray oil (if using mini pans you do not need to line them with parchment). Transfer the dough to the prepared pan; it should be about three-quarters full. Mist the top with vegetable spray oil and pat the dough into the pan to spread it evenly and smooth the top. Cover loosely with plastic wrap and proof at room temperature for about 45 minutes, until the dough rises almost to the top of the pan (the dough has very little oven spring once baked, so it shouldn't spill over the sides of the pan).\n\n3 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C).\n\n4 Bake for 40 minutes, then rotate and bake for 40 to 45 minutes longer. Remove the loaf from the pan and put it on a sheet pan or directly on the oven rack, then continue baking for about 10 minutes, until the top is hard when tapped, the loaf sounds hollow when thumped on the bottom, and the crust is dark golden brown (though there may be a whitish hue over some of the top crust from the long bake and dry oven). The internal temperature of the loaf should be at least 190\u00b0F (88\u00b0C).\n\n5 Transfer to a wire rack and let cool for at least 1 hour before slicing and serving.\n\n## Q & A\n\n### Can these recipes be made using standard, nonsprouted gluten-free flours, rather than the sprouted versions?\n\nYes, but the nutritional and digestive benefits are greater when using sprouted flour.\n\nSprouted Vollkornbrot\n\n# SPROUTED VOLLKORNBROT\n\nMAKES 2 LARGE LOAVES OR 6 TO 8 SMALL LOAVES\n\nDense, chewy German-style breads are coming into vogue, with increasing numbers of bakeries featuring their own versions under various names. This recipe is for vollkornbrot, which simply means \"whole grain bread.\" It's a type of bread designed to be dense and packed with intense flavor and fiber, so a thin slice goes a long way. Toasted or untoasted, it's a great cocktail or appetizer bread for serving with spreads, cheeses, and p\u00e2t\u00e9s.\n\nAlthough vollkornbrot is usually associated with whole rye flour, there are many ways to make it. In Peter Reinhart's Whole Grain Breads, I offered a version that called for making a scalded mash of rye flour to heighten the enzyme activity, along with a pre-ferment. In this new version, neither of those approaches is needed, thanks to the high enzyme activity of sprouted flour. However, including some recently refreshed mother starter is an option if you'd like to add more acidic flavor (see the variation). Because there are so many ways to make this bread, you'll see that a number of the ingredients are optional. Although the cooked rye or wheat berries are among the optional ingredients, they do add a wonderful, chewy texture. If you use them, be sure to cook them thoroughly so they don't dry out during baking, especially those near the surface.\n\nAlthough these breads are often made in long loaf pans and baked for a long time at low temperatures, I now prefer using mini loaf pans and making smaller loaves. The advantage of this is that the breads take far less time to bake and you can freeze the extra mini loaves for future use. This kind of dense bread is often at its most flavorful the day after baking. If storing, let the loaves cool completely before wrapping them in aluminum foil or plastic wrap.\n\n### DOUGH\n\nNOTES:\n\n\u2022 You can use all sprouted rye flour or all sprouted whole wheat flour, or any combination of those flours, as long as the total weight is 28 ounces (793 g).\n\n\u2022 The amount of water needed may vary depending on which optional ingredients you use. View the amount listed as a guideline and adjust as needed, following the cues in the recipe.\n\n\u2022 To cook the rye or wheat berries, combine \u00bd cup (about 6.5 oz \/ 185 g) grain with 1\u00bd cups (12 oz \/ 340 g) water in a saucepan and bring to a boil. Lower the heat, cover, and simmer until the grains are soft, about 1 hour. Drain, reserving the liquid if you'd like to use it as part of the water in the recipe. Let the grains cool completely before use. Because of the time needed for cooling, you might prefer to cook the grains a day or two ahead and keep them in the refrigerator.\n\nsprouted rye flour: 5 cups plus 2 tablespoons (18 oz \/ 510 g)\n\nsprouted whole wheat flour: 2\u2153 cups (10 oz \/ 283 g)\n\nsunflower seeds, lightly toasted (see Toasting Seeds and Nuts for toasting instructions): about \u00bc cup (1.5 oz \/ 42.5 g)\n\nwhole flaxseeds: about 3 tablespoons (1 oz \/ 28.5 g)\n\ncocoa powder (optional): 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nwater, at room temperature (see notes): 3\u00bd cups (28 oz \/ 794 g)\n\nmolasses (optional): 3 tablespoons (2 oz \/ 56.5 g)\n\ninstant yeast: 5 teaspoons (0.55 oz \/ 15.5 g \/ 2%)\n\ncooked rye or wheat berries (optional; see notes): 1 cup (6 oz \/ 170 g)\n\nTOTAL: 68.05 oz \/ 1,928 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted rye flour| 64  \nsprouted whole wheat flour| 36  \nsunflower seeds| 5.4  \nwhole flaxseeds| 3.6  \ncocoa powder| 1.8  \nsalt| 1.8  \nwater| 100  \nmolasses| 7.1  \ninstant yeast| 2  \ncooked rye or wheat berries| 21  \nTOTAL| 242.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the sprouted flour, toasted sunflower seeds, flaxseeds, cocoa powder, and salt (on low speed if using a stand mixer). In a separate bowl, whisk together the water and molasses, then stir in the yeast to dissolve it (it need not bloom or bubble). Add the molasses mixture and cooked rye berries to the flour mixture and mix or stir until the flour is hydrated and a coarse, wet dough forms. The texture will be similar to mud. Mix for 2 more minutes; if using a stand mixer, start on low speed and increase to medium for the last 15 seconds. The dough should be quite wet and just firm enough to hold its shape if squeezed. Add a bit more water or flour if necessary to achieve this texture, but bear in mind that the dough will thicken as it sits, so a wet and somewhat loose dough is better than dry and stiff. All of that said, there's a fair amount of leeway regarding the texture as long as the dough can be molded by hand.\n\n2 Spread a bit of vegetable oil on a work surface. Using an oiled bowl scraper, transfer the dough to the oiled area. Lightly oil your hands and press the dough into a rounded mound. It won't feel like any other kind of dough, and it will have little or no gluten structure. Mist a bowl with vegetable spray oil and cover the dough with the bowl, or transfer the dough to the bowl, mist the top with vegetable spray oil, and cover the bowl with plastic wrap. Ferment at room temperature for 1 hour. The dough will hardly rise, but the yeast will begin working.\n\n3 Generously mist two 4\u00bd by 8-inch loaf pans or 6 to 8 mini loaf pans with vegetable spray oil or rub them with vegetable oil; if you'd like to make freestanding loaves, line a sheet pan with parchment paper or a silicone mat and mist the surface with vegetable spray oil.\n\n4 Generously dust the work surface with flour. Transfer the dough onto the floured work surface and divide into the appropriate number of pieces. Roll each piece in the flour on the work surface until completely coated, adding more flour to the work surface if necessary. Shape each piece into a b\u00e2tard; don't worry about cracks in the surface, as these will be part of the finished look. Put the pieces in the prepared pans or on the sheet pan and cover loosely with plastic wrap. Proof at room temperature for 1 hour. Again, the dough will hardly rise.\n\n5 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 500\u00b0F (260\u00b0C).\n\n6 Put the loaves in the oven and pour about 1 cup of hot water into the steam pan. Immediately lower the temperature to 375\u00b0F (191\u00b0C) for freestanding loaves or 350\u00b0F (177\u00b0C) for pan loaves. Bake for 30 minutes, then rotate and bake until the crust is hard and the loaves sound hollow when thumped on the bottom, about 30 minutes longer for mini loaves and freestanding loaves, or 45 to 50 minutes longer for large pan loaves. The internal temperature should be at least 190\u00b0F (88\u00b0C). The loaves won't rise very much in the oven and will feel very heavy and dense, but they will rise enough for the dusting of flour on the surface to create an interesting pattern.\n\n7 Transfer from the pans to a wire rack and let cool completely before slicing and serving\u2014at least 1 hour for small loaves, and at least 2 hours for large loaves.\n\n### Variation\n\nTo make this with natural leaven, add 7 ounces (198 g) recently refreshed Mother Starter and an extra \u00bc teaspoon (0.06 oz \/ 1.5 g) salt. You can decrease the amount of instant yeast by as much as half if you'd like a slightly tangier loaf. In that case, increase the rising time by 1 hour during both the first fermentation and the final proofing.\n\n# SPROUTED WHEAT CHALLAH\n\nMAKES 1 LARGE OR 2 SMALL LOAVES\n\nOver time, I continue to develop new variations of challah, and one of the things I've learned over the years is that challah dough, which is defined primarily by enrichment with eggs, always seems better with more yolks and fewer whites. Although egg whites are very nutritious, they have a drying effect on foods. Egg yolks, on the other hand, are rich in natural oils and lecithin and therefore help keep bread moist and give it a creamy mouthfeel. So these days I use only yolks, saving the whites for other purposes, such as meringues, egg white omelets, or as an egg white wash on breads. This dough can be made in any number of shapes (including as an alternative dough for the cinnamon buns), but braided challah is the most well-known version. There are many traditions associated with the braids, but my research hasn't yet revealed any rule governing the number of strands. The tradition that makes the most sense to me is that the braided loaf should have twelve sections, symbolizing the twelve tribes of Israel, but this is just one among many interpretations. New Year's challah (for Rosh Hashanah) is typically coiled rather than braided, symbolizing the eternal cycles of life. Note that this recipe calls for vegetable oil, rather than butter, because butter wouldn't be compatible with a meat-based meal according to kosher laws. If you aren't keeping a kosher diet, you can certainly substitute melted unsalted butter for the oil. However, even though I personally love the flavor of butter, I find that vegetable oil is actually more functional in this bread, contributing to a softer texture.\n\n### DOUGH\n\nsprouted whole wheat flour: 4\u00bc cups (18 oz \/ 510 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g)\n\ninstant yeast: 2\u00bc teaspoons (0.25 oz \/ 7 g)\n\nsugar (or honey or agave nectar): 3 tablespoons (or 2 tablespoons plus \u00be teaspoon) (1.5 oz \/ 42.5 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00bc cups plus 2 tablespoons (11 oz \/ 312 g)\n\negg yolks, slightly beaten: 5 (3.75 oz \/ 106 g)\n\nvegetable oil: 3 tablespoons (1.5 oz \/ 42.5 g)\n\negg wash: 1 egg whisked with 1 tablespoon water\n\nsesame or poppy seeds for topping (optional)\n\nTOTAL: 36.32 oz \/ 1,029 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nsalt| 1.8  \ninstant yeast| 1.4  \nsugar| 8.3  \nwater| 61  \negg yolks| 21  \nvegetable oil| 8.3  \nTOTAL| 201.8\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Stir in the sugar (if using honey or agave nectar, add it to the water in the next step.) In a separate bowl, whisk together the water, egg yolks, and oil, then pour into the flour mixture and mix or stir until the flour is hydrated and a coarse, sticky dough forms, about 1 minute.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 to 2 minutes, on medium-low speed if using a stand mixer. The dough should be smooth but still slightly soft and sticky (however, it should not be as soft as the Sprouted Whole Wheat Bread). Add a bit more flour or water if necessary to achieve this texture, but bear in mind that the dough will firm up during the stretch and fold process.\n\n3 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be soft and slightly sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be only slightly tacky. It will be firmer than many doughs, because it must hold up to the braiding and retain the braided appearance when baked. That said, it must also be supple and extensible enough to roll out for braiding.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n5 Oil the work surface again and transfer the dough to the oiled area. Divide it evenly into the desired number of pieces. The number of strands in the braid is up to you, with the main rule being that the strands should each be the same weight and be rolled out to the same length. Flatten each piece with the palms of your hands, then roll each into a cigar shape about 3 inches in length. Let rest for 2 minutes, then roll each piece into a tapered strand 10 to 12 inches in length. (If making 2 smaller loaves, cut the pieces in half but roll them out to about 8 inches in length.) Braid the strands until you get to the ends, then pinch the tips together to seal. (See sidebar on Braided Loaves for more information.)\n\n6 Line a sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil. Transfer the braided dough to the baking pan. If making 2 smaller loaves, lay them side by side crosswise on the pan, allowing at least 2 inches between them so they won't touch when they rise. Brush the entire top and side surfaces with about half of the egg wash. Mist the top with vegetable spray oil and cover loosely with plastic wrap. Let proof for about 1 hour at room temperature. Brush with the remaining egg wash and generously sprinkle with seeds if you like. Proof for 10 to 15 minutes longer, until the dough increases in size by nearly 1\u00bd times. (It will rise further in the oven.)\n\n7 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C).\n\n8 Bake for 20 minutes, then rotate and bake for 10 minutes longer for small loaves, or 15 to 20 minutes longer for a large loaf, until the crust is rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be about 190\u00b0F (88\u00b0C).\n\n9 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n## Braided Loaves\n\nYou can make braided breads with 2, 3, 4, or 5 strands\u2014or more. The most important principle is to be sure each strand is the same weight and length. If you don't have a scale, estimate the size as closely as possible. Also keep in mind that the position numbers refer to the actual position of the strands on the counter, starting from your left, rather than to the particular strands; in other words, the number of a given strand changes as it's moved during the braiding process. To form the strands, use the same gentle rocking motion as for shaping baguettes. For all braids, place the prettiest side up when you transfer to the baking sheet, then cover and proof.\n\nTO SHAPE A 2-BRAID LOAF, lay 2 strands of equal weight and length on the work surface, perpendicular to one another and crossed in the center. Take both ends of the strand that's underneath and cross them over to the opposite sides. Cross the ends of the other strand in the same way. Continue crossing and alternating until you get to the ends of the strands, then pinch the tips together at each end to seal off the ends. Lay the braid on its side.\n\nTO SHAPE A 3-BRAID LOAF, lay 3 equal strands side by side, parallel to one another. Beginning in the middle of the loaf, overlap one of the outside strands over the middle strand, then take the opposite outside strand and cross it over the new middle strand. Continue this pattern until you get to the ends of the strands, then pinch the tips together to seal. Rotate the loaf so the unbraided side is facing you, then repeat the pattern on that end.\n\nTO SHAPE A 4-BRAID LOAF, connect 4 strands of equal weight and length at one end, spreading the other ends out with the tips facing you. From the left, number the strands 1, 2, 3, 4. Follow this pattern: 4 over 2, 1 over 3, and 2 over 3. Repeat until you get to the ends of the strands, then pinch the tips together to seal.\n\nTO SHAPE A 5-BRAID LOAF, connect 5 strands of equal weight and length at one end, spreading the other ends out with the tips facing you. From the left, number the strands 1, 2, 3, 4, 5. Follow this pattern: 1 over 3, 2 over 3, and 5 over 2. Repeat until you get to the ends of the strands, then pinch the tips together to seal.\n\nSprouted Multigrain Crackers\n\n# SPROUTED MULTIGRAIN CRACKERS\n\nMAKES UP TO 48 CRACKERS, DEPENDING ON SIZE | MASTER FORMULA\n\nCrackers are like chips: it's very hard to eat just one. If I were to go back into the baking business, it would be to make crackers, specializing in making them with sprouted flour and seed and nut flours. This recipe is versatile and allows for many variations. It calls for ground seeds (or ground nuts in one of the variations). Seed and nut flours, or meals, are increasingly available for purchase, but you can easily make your own if you have a small burr mill, spice grinder, or food processor, and it can even be ground by hand using a mortar and pestle. While a regular blender will work, it's tricky; you have to be careful not to blend for too long, as the friction can easily turn the seeds from meal into butter as their natural oils heat up. Short pulses can help prevent this problem. Food processors work well for some seeds and nuts, but not hard or tiny ones, such as flax, chia, or sesame seeds; for those, use a seed or spice grinder.\n\nBaking powder is optional in this recipe. If you omit it, the crackers won't rise and will be more like chips. With baking powder they will rise slightly and tend to be more flaky. Both methods make excellent crackers, so you may want to try it both ways to determine your own preference. Either way, I bet you can't eat just one.\n\n### DOUGH\n\nsprouted whole wheat flour: 1 cup plus 1 tablespoon (4.5 oz \/ 128 g)\n\nsprouted rye flour (or sprouted corn flour or other sprouted grain flours): 1\u00bc cups plus 1 tablespoon (or 1 cup plus 2 tablespoons or about 1 cup plus 1 tablespoon) (4.5 oz \/ 128 g)\n\nground sunflower seeds, pumpkin seeds, or flaxseeds: about 1\u00bc cups (5 oz \/ 142 g)\n\nsalt: \u00bc teaspoon (0.06 oz \/ 1.5 g)\n\nbaking powder (optional): 1\u00bd teaspoons (0.28 oz \/ 8 g \/ 2%)\n\nwater: 1 cup (8 oz \/ 227 g)\n\nvegetable oil or melted unsalted butter: 2 tablespoons (1 oz \/ 28.5 g)\n\nhoney or agave nectar: 3\u00bd teaspoons (0.75 oz \/ 21.5 g)\n\negg wash: 1 egg whisked with 1 tablespoon water\n\nTOTAL: 24.09 oz \/ 684.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 32  \nsprouted rye flour| 32  \nground sunflower seeds, pumpkin seeds, or flaxseeds| 36  \nsalt| 0.3  \nbaking powder| 2  \nwater| 57  \nvegetable oil or melted unsalted butter| 7.2  \nhoney or agave nectar| 5.4  \nTOTAL| 171.9\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flours, ground seeds, salt, and baking powder (on low speed if using a stand mixer). In a separate bowl, whisk together the water, oil, and honey, then pour into the flour mixture. Mix or stir until all the ingredients gather into a coarse ball, about 1 minute. If all of the flour isn't picked up by the dough ball, add a few drops of water.\n\n2 Lightly dust a work surface with sprouted flour or lightly oil the work surface. Transfer the dough to the work surface and knead by hand for about 30 seconds to ensure that all the ingredients are evenly distributed and the dough holds together. It should be pliable and only slightly tacky, not sticky, with a texture similar to Play-Doh.\n\n3 Preheat the oven to 300\u00b0F (149\u00b0C).\n\n4 Divide the dough into 4 equal pieces. (If you don't want to bake all of the crackers at once, wrap any pieces you won't be baking in plastic wrap; they can be stored in the refrigerator for up to 1 week or in the freezer for up to 3 months.) For each piece of dough you plan to bake, line a sheet pan with parchment paper or a silicone mat.\n\n5 Generously mist two sheets of parchment paper or waxed paper with vegetable spray oil. Transfer one piece of the dough to the center of one of the prepared pieces of parchment paper. Mist the top with vegetable spray oil, then use your hands to gently pat the dough down into a thick rectangle or oblong with a fairly even surface. Flip the other piece of parchment paper over and position the oiled side atop the dough. Using a rolling pin, very gently roll the dough out until about \u215b inch thick. When rolling the dough, use short, gentle strokes, rather than pressing hard or making long strokes. Try to ease or coax the dough into a wider and wider circle or rectangle, always rolling from the center to the corners, and then to each of the sides. Carefully peel off the top piece of parchment paper.\n\n6 Brush with one-fourth of the egg wash. Use a pizza cutter to cut the dough into rectangles, diamonds, or any shape you like. You can also use a small biscuit cutter, dipped in flour, to make round crackers. If so, you'll have leftover scraps that you'll need to roll out again. The crackers not need all be the same size or shape. Transfer the cut pieces to the lined pan. They can be nearly touching, as they won't spread.\n\n7 Multiple pans of crackers can be baked at the same time. Put the pans on different racks and bake for 10 minutes, then rotate the pans front to back and between racks. Bake for 10 minutes longer. Rotate the pans once more and bake for 5 to 20 minutes longer, depending on thickness, until rich golden brown and fairly dry. For a bit more browning, increase the temperature to 325\u00b0F (163\u00b0C) for the final few minutes of baking.\n\n8 Remove from the oven and leave the crackers on the pans; they'll become more crisp as they cool. If they don't snap cleanly after cooling, return the pan to a hot oven (375\u00b0F \/191\u00b0C) for a few minutes, until they dry sufficiently to snap when broken. Cool for at least 15 minutes before serving. The cooled crackers can be stored in an airtight container at room temperature for up to 8 days, or in a zip-top bag in the freezer indefinitely.\n\n### Variations\n\nReplace any or all of the sprouted flours with an equal amount, by weight, of conventional wheat or rye flour, or any other flour, including gluten-free and ancient grain flours. Depending on which flours you use, you may have to decrease the water slightly.\n\nWhile ground sunflower and pumpkin seeds are often used in the dough, you may also substitute any ground seed or nut flour, such as hemp, chia, flax, pecan, walnut, or almond.\n\nPoppy and sesame seeds are generally the best toppings because their light flavor doesn't compete with the crackers as stronger spices, like cumin or anise, would. Flavored salt blends, such as garlic or lemon pepper, and spice rubs also work nicely, but use them with a light touch. Chopped pumpkin seeds or nuts are another possibility. For all of these options, sprinkle them over the surface after applying the egg wash.\n\nReplace the egg wash with a flavored oil, such as Herb Oil or perhaps a blend made with salt, smoked paprika, and herbes de Provence or other herbs to taste. Alternatively, go ahead and brush the crackers with the egg wash before baking, then brush with flavored oil as soon as they come out of the oven, then return the crackers to the oven for a few minutes to set the glaze.\n\nCHEESE CRACKERS: Grated cheese can either be worked into the dough during kneading or sprinkled on top. In either case, use about 1 cup (3 to 4 oz \/ 85 to 113 g) grated cheese. Given that the baking temperature is so low, most cheeses can be sprinkled on top prior to baking. However, dry aged cheeses like Parmesan will brown more quickly than semi-moist cheeses like Cheddar, Swiss, and provolone and therefore should be sprinkled on top when you rotate the pans the first time.\n\n# GLUTEN-FREE SPROUTED GRAIN CRACKERS\n\nMAKES ABOUT 60 CRACKERS, DEPENDING ON SIZE\n\nThere are so many ways to make crackers, and that extends to gluten-free crackers. Now, sprouted grain and sprouted bean flours open yet another new realm of possibilities. The key is to have the right balance of flour, fat, and sweetness, plus good rolling technique. The rolling method here is just one of many ways to roll out crackers, but it works especially well for gluten-free versions, which are typically difficult to transfer onto a pan without falling apart. The combination of flours in this recipe provides a starting point, but feel free to experiment with other flours as desired, as long as the total flour weight remains the same. You can also use nonsprouted flour in this recipe, but the sprouted flour offers exceptional flavor. These crackers are tender and flaky; if you prefer crisper crackers, decrease the amount of oil by half. The egg whites serve as a binder; you could use a whole egg instead for both binding and more richness from the yolk. One final note: Any of the variations of the Sprouted Multigrain Crackers recipe will also work well with this recipe.\n\n### DOUGH\n\nsprouted brown rice flour: 1\u00bc cups (6 oz \/ 170 g)\n\nsprouted corn flour: 1\u00bd cups (6 oz \/ 170 g)\n\nother sprouted gluten-free flour, such as quinoa, teff, lentil, black bean, garbanzo, millet, sorghum, or amaranth: about 5 tablespoons (1 oz \/ 28.5 g \/ 8%)\n\nsalt: \u00bd teaspoon (0.12 oz \/ 3.5 g)\n\nbaking powder (optional): 1\u00bd teaspoons (0.28 oz \/ 8 g)\n\nflaxseeds or sesame seeds: about 3 tablespoons (1 oz \/ 28.5 g \/ 8%)\n\nhoney (or agave nectar or sugar): 1\u00bd tablespoons (or 2 tablespoons) (1 oz \/ 28.5 g \/ 8%)\n\negg whites, slightly beaten: \u00bc cup (2 oz \/ 56.5 g)\n\ngrape seed oil, olive oil, or melted unsalted butter: 6 tablespoons (3 oz \/ 85 g)\n\nwater: 1 cup (8 oz \/ 227 g)\n\negg white wash: 1 egg white whisked with 1 tablespoon water\n\nTOTAL: 28.4 oz \/ 805.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted brown rice flour| 46  \nsprouted corn flour| 46  \nother sprouted gluten-free flour| 8  \nsalt| 0.9  \nbaking powder| 2.2  \nflaxseeds or sesame seeds| 8  \nhoney| 8  \negg whites| 15  \ngrape seed oil, olive oil, or melted unsalted butter| 23  \nwater| 62  \nTOTAL| 219.1\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the sprouted flours, salt, baking powder, and flaxseeds (on low speed if using a stand mixer). (If using sugar rather than honey or agave nectar, add it at this point.) In a separate bowl, whisk together the honey, egg whites, oil, and water, then pour into the flour mixture. Mix or stir for 1 to 2 minutes to form a thick, sticky, batter-like dough, adding a bit of flour or water if necessary to achieve this texture. The dough will thicken somewhat as it sits.\n\n2 Generously mist two sheets of parchment paper or silicone mats with vegetable spray oil. Transfer half of the batter to the center of one of the prepared pieces of parchment paper. Mist the top with vegetable spray oil, then use your hands to gently pat the dough down into a thick rectangle or oblong with a fairly even surface. Flip the other piece of parchment paper over and position the oiled side atop the dough. Using a rolling pin, very gently roll the dough out until \u215b to \u00bc inch thick, always working from the center to the corners and then from the center to the sides and ends. Carefully peel off the top piece of parchment paper. Keeping the dough on the lower piece of parchment paper, carefully lift and transfer it to a sheet pan.\n\n3 Position two racks in the oven an equal distance apart and preheat the oven to 300\u00b0F (149\u00b0C).\n\n4 Roll out the remaining dough in the same way, between two sheets of parchment paper or silicone mats that have been generously misted with vegetable spray oil. Transfer the rolled dough to a second sheet pan. Brush the egg wash evenly over the dough. Use a pizza cutter to the dough into rectangles or diamonds; they need not all be the same size. Don't try to move the cut pieces to separate them; they'll snap apart after they bake.\n\n5 Bake for 15 minutes, then rotate the pans front to back and between racks. Bake 15 minutes longer, then remove any browned crackers to a wire rack (the crackers on the edges of the pan tend to bake faster). Rotate the pans once again and bake for 10 to 15 minutes longer, depending on thickness, until rich golden brown and crisp. For a bit more browning, increase the temperature to 325\u00b0F (163\u00b0C) for the final few minutes of baking.\n\n6 Remove from the oven and leave the crackers on the pans; they'll become more crisp as they cool. If they don't snap after cooling, return the pan to a hot oven (400\u00b0F \/ 204\u00b0C) for a few minutes, until they dry sufficiently to snap when broken.\n\n7 Cool for at least 15 minutes before serving. The cooled crackers can be stored in an airtight container at room temperature for up to 8 days, or in a zip-top bag in the freezer indefinitely.\n\nSprouted Corn Bread\n\n# SPROUTED CORN BREAD\n\nMAKES 8 SERVINGS\n\nWhen my wife, Susan, and I had our first restaurant, Brother Juniper's Bakery and Cafe, in Forestville, California, we were well known for a few signature dishes in addition to our bread. We learned that side dishes can go a long way in making a restaurant memorable, and in our case, some of the menu items that brought people back were the coleslaw, barbecue sauce, salad dressings, and corn bread. When we later moved to the Carolinas, I learned that our corn bread was what, around here, would be called Yankee-style because it was sweet and more cakelike than Southern-style corn bread, which is simpler, drier, and designed to be served with a sweetener like molasses or honey drizzled over the top. I still prefer my Yankee-style corn bread, but now I generally make it with sprouted corn flour and sprouted whole wheat flour. Sprouted corn flour (also sold as sprouted cornmeal) retains a lot more of the corn flavor than conventional cornmeal. In fact, in the past I've suggested adding fresh or frozen corn kernels to boost the corn flavor, but with sprouted corn flour that isn't necessary. Still, it never hurts to add corn kernels for the moist pop of flavor they provide, or to sprinkle crumbled bacon over the top, as suggested in the variations below.\n\n### DOUGH\n\nNOTE: You can vary the ratio of sprouted corn flour to sprouted wheat flour if you prefer more corn flavor. For instance, you can use 75% corn flour and only 25% wheat flour, or even 100% corn flour and no wheat flour.\n\nsprouted corn flour: 1\u00bd cups plus 3 tablespoons (6.75 oz \/ 191 g)\n\nsprouted whole wheat flour: 1\u00bd cups plus 1\u00bd tablespoons (6.75 oz \/ 191 g)\n\nbaking powder: 4 teaspoons (0.72 oz \/ 20.5 g)\n\nbaking soda: \u00bd teaspoon (0.11 oz \/ 3 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\nsugar (or honey or agave nectar): 6 tablespoons (or 4\u00bd tablespoons) (3 oz \/ 85 g)\n\nbuttermilk: 2\u00bd cups (20 oz \/ 567 g)\n\neggs, lightly beaten: 2 (3.5 oz \/ 99 g)\n\nunsalted butter, melted: 2 tablespoons (1 oz \/ 28.5 g)\n\nbacon fat or unsalted butter, melted: 2 tablespoons (1 oz \/ 28.5 g)\n\nTOTAL: 43.08 oz \/ 1,220.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted corn flour| 50  \nsprouted whole wheat flour| 50  \nbaking powder| 5.4  \nbaking soda| 0.8  \nsalt| 1.8  \nsugar| 22  \nbuttermilk| 148  \neggs| 26  \nunsalted butter| 7.5  \nbacon fat or unsalted butter| 7.5  \nTOTAL| 319\n\n1 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C).\n\n2 In a large bowl, stir together the flours, baking powder, baking soda, salt, and sugar (if using honey or agave nectar, add it to the buttermilk in the next step). In a separate bowl, whisk together the buttermilk, eggs, and 2 tablespoons (1 oz \/ 28.5 g) of melted butter, then pour into the flour mixture. Stir with a large spoon to make a smooth, pourable batter.\n\n3 Grease a 9-inch round cake pan or 8-inch square baking pan (or a larger pan for a thinner corn bread) with either the bacon fat or the butter. Put the pan in the oven for about 2 minutes, until the bacon fat almost starts to smoke (or, if using butter, until it starts to brown). Remove the pan from the oven and pour in the batter, spreading it in an even layer.\n\n4 Bake for 20 minutes, then rotate and bake for 15 to 20 minutes longer, until the surface is golden brown and firm and springy when poked in the center and a toothpick inserted in the center comes out clean.\n\n5 Let cool in the pan for 20 minutes before cutting and serving.\n\n### Variations\n\nAdd up to 1\u00bc cups (about 6 oz \/ 170 g) fresh or frozen corn kernels when mixing the final batter. Alternatively, or in addition, add \u00bd to 1 cup (about 2.5 to 5 oz \/ 71 to 142 g) diced vegetables (uncooked or frozen) such as red bell peppers, onion, green onions, or even red jalape\u00f1o or Fresno chiles. Another alternative, alone or in combination with the others, is adding up to 2 cups (about 8 oz \/ 227 g) shredded or grated soft cheese, such as Cheddar, Monterey Jack, mozzarella, or provolone. With any of these additions, you'll probably need to bake the corn bread for about 5 minutes longer.\n\nBACON CORN BREAD: Cook about 5 slices of bacon until crisp and let cool. (Use the bacon fat to grease the pan.) Crumble the bacon and sprinkle it over the batter before baking.\n\nSOUTHERN-STYLE CORN BREAD: Decrease the amount of sugar or sweetener to taste. Also reduce the amount of sprouted wheat flour by half, replacing it with an equal amount of sprouted corn flour.\n\n# GLUTEN-FREE SPROUTED CORN BREAD WITH TEFF\n\nMAKES 8 SERVINGS\n\nTeff is probably one of the most ancient grains. This tiny grain, about the size of a poppy seed, is a nutritional powerhouse, high in iron, calcium, and protein but containing no gluten. Teff is most closely associated with Ethiopia and Eritrea, where it's the main flour used to make injera, the flat, sourdough-type bread used for scooping up wonderfully spicy dishes. I find the flavor of teff a little too strong to feature as the primary flour in a bread, but I enjoy blending it with other flours. While teff can be sprouted and used in sprout form, it's difficult to dry sprouted teff and mill it into flour because it's so small. Therefore, I use standard, nonsprouted teff flour in this delicious corn bread. All of that said, you can certainly substitute other ancient grains, as well as bean flours, for the teff in this recipe. One of my favorite variations is to replace the teff with sprouted lentil flour, which makes this bread the perfect accompaniment to hearty soups. One final note: Any of the variations of the Sprouted Corn Bread recipe will also work well with this recipe.\n\n### DOUGH\n\nsprouted corn flour (or cornmeal): 2\u00be cups plus 1 tablespoon (11.75 oz \/ 333 g)\n\nteff flour (or another ancient grain or bean flour): \u00bd cup (2.25 oz \/ 64 g)\n\nbaking powder: 4 teaspoons (0.72 oz \/ 20.5 g)\n\nbaking soda: \u00bd teaspoon (0.11 oz \/ 3 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\nsugar (or honey or agave nectar): 6 tablespoons (or 4\u00bd tablespoons) (3 oz \/ 85 g)\n\nbuttermilk: 2\u00bd cups (20 oz \/ 567 g)\n\neggs, slightly beaten: 2 (3.5 oz \/ 99 g)\n\nunsalted butter, melted: 2 tablespoons (1 oz \/ 28.5 g)\n\nbacon fat or melted unsalted butter: 2 tablespoons (1 oz \/ 28.5 g)\n\nTOTAL: 43.58 oz \/ 1,235.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted corn flour| 84  \nteff flour| 16  \nbaking powder| 5.2  \nbaking soda| 0.8  \nsalt| 1.8  \nsugar| 21  \nbuttermilk| 143  \neggs| 25  \nunsalted butter| 7.2  \nbacon fat or melted unsalted butter| 7.2  \nTOTAL| 311.2\n\n1 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C).\n\n2 In a large bowl, stir together the flours, baking powder, baking soda, salt, and sugar (if using honey or agave nectar, add it to the buttermilk in the next step). In a separate bowl, whisk together the buttermilk, eggs, and 2 tablespoons (1 oz \/ 28.5 g) of melted butter, then pour into the flour mixture. Stir or whisk for about 1 minute to make a smooth, pourable batter.\n\n3 Grease a 9-inch round cake pan or 8-inch square baking pan (or a larger pan for a thinner corn bread) with either the bacon fat or the melted butter. Put the pan in the oven for about 2 minutes, until the bacon fat almost starts to smoke (or, if using butter, until it starts to brown). Remove the pan from the oven and pour in the batter, spreading it in an even layer.\n\n4 Bake for 25 minutes, then rotate and bake for 25 minutes longer, until the surface is firm and springy when poked in the center and a toothpick inserted in the center comes out clean.\n\n5 Let cool in the pan for 20 minutes before cutting and serving.\n\n# FLAKY SPROUTED WHEAT BISCUITS\n\nMAKES 12 TO 24 BISCUITS\n\nThe distinguishing characteristic for a great biscuit is that it be either tender or flaky. This method for making biscuits was a big hit in Peter Reinhart's Artisan Breads Every Day because it produces biscuits that are both tender and flaky. But I didn't know about sprouted whole wheat flour back then, and this updated version is even better. Sprouted pastry flour is just beginning to become available, and if you can get your hands on some, the biscuits will be even more flaky. If not, regular sprouted whole wheat flour will be just fine.\n\n### DOUGH\n\nNOTE: You can substitute buttermilk for the cream. If you do so, increase the quantity to 1\u00bc cups (10 oz \/ 283 g) and omit the vinegar.\n\nwhipping cream or half-and-half, cold (see note): 1 cup plus 2 tablespoons (9 oz \/ 255 g)\n\nwhite vinegar, apple cider vinegar, or lemon juice: 2 tablespoons (1 oz \/ 28.5 g)\n\nsprouted whole wheat flour or sprouted whole wheat pastry flour: 2 cups plus 2 tablespoons (9 oz \/ 255 g)\n\nsalt: \u00bd teaspoon (0.12 oz \/ 3.5 g)\n\nbaking powder: 3\u00bd teaspoons (0.6 oz \/ 17 g)\n\nbaking soda: \u00bc teaspoon (0.05 oz \/ 1.5 g)\n\nsugar (or honey or agave nectar): 1 tablespoon (or 2\u00bc teaspoons) (0.5 oz \/ 14 g)\n\nunsalted butter, cold: \u00bd cup plus 2 tablespoons (5 oz \/ 142 g)\n\nTOTAL: 25.27 oz \/ 716.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhipping cream or half-and-half| 100  \nwhite vinegar, apple cider vinegar, or lemon juice| 11  \nsprouted whole wheat flour or sprouted whole wheat pastry flour| 100  \nsalt| 1.4  \nbaking powder| 6.7  \nbaking soda| 0.6  \nsugar| 5.5  \nunsalted butter| 56  \nTOTAL| 281.2\n\n1 In a small bowl or measuring cup, stir together the cream and vinegar. Let sit for at least 10 minutes.\n\n2 In a large bowl or a food processor fitted with the metal blade, combine the flour, salt, baking powder, baking soda, and sugar. (If using honey or agave nectar, whisk it into the cream.) Process or stir briefly to evenly distribute the ingredients. Cut the butter into small bits and use your fingertips or use quick pulses with the food processor to break the butter into small, pea-size bits and distribute it evenly throughout the flour mixture. Work quickly to prevent the butter from softening excessively.\n\n3 Add all the cream-vinegar mixture and stir or briefly pulse just until the flour is hydrated and a soft, wet, and shaggy dough forms. If it seems stiff or the liquid doesn't adequately hydrate the flour, drizzle in a bit more cream.\n\n4 Generously dust a work surface with sprouted whole wheat flour and use a bowl scraper or rubber spatula to transfer the dough to the work surface. Sprinkle more flour on the top of the dough. With floured hands, gently press the dough into a rectangle or oblong about \u00be inch thick, with a long edge facing you. Use a rolling pin to gently roll out the dough, using short strokes from the center of the dough outward, sprinkling more flour under the dough if it seems to be sticking, which it is likely to do. Continue rolling to form a rectangle about \u00bd inch thick. Using a metal pastry blade, separate the dough from the work surface, then fold it in thirds, as if folding a letter.\n\n5 Dust the work surface with flour again and rotate the dough 90 degrees. Once again gently roll it out into a rectangle about \u00bd inch thick, then fold it in thirds, like a letter. Repeat the process one more time, rotating the dough 90 degrees, rolling it out into a rectangle about \u00bd inch thick, and folding it in thirds. The dough should be fairly firm and smooth by this point and easy to handle. Continue to use dusting flour as needed to prevent sticking.\n\n6 Let the dough rest for about 15 minutes. If the butter seems to be softening or smearing, put the dough on a sheet pan lined with parchment paper and misted with vegetable spray oil and refrigerate it until the butter firms up.\n\n7 Line a sheet pan with parchment paper or a silicone mat. Spread about 1 teaspoon of vegetable oil on a work surface and transfer the dough to the oiled area. Gently roll the dough out into a square or oval just under \u00bd inch thick. Cut the dough with a floured biscuit cutter, metal pastry blade, or pizza cutter. A 2-inch biscuit cutter will produce 20 to 24 biscuits, but you can make them larger or smaller, and they need not all be the same size. Press any scraps back together and cut more biscuits to use up all the dough.\n\n8 Transfer the biscuits to the prepared pan, placing them \u00bd inch apart. Let rest for 15 to 30 minutes to relax the gluten and allow for a more even rise. Alternatively, you can refrigerate the pan of biscuits for up to 2 days and bake them later.\n\n9 Position a rack in the middle of the oven and preheat the oven to 500\u00b0F (260\u00b0C).\n\n10 Put the biscuits in the oven and lower the temperature to 450\u00b0F (232\u00b0C). Bake for 8 minutes, then rotate the pan and bake for 6 to 10 minutes longer, until both the tops and bottoms of the biscuits are rich golden brown. The biscuits should rise about 1\u00bd times in height.\n\n11 Transfer the biscuits to a wire rack and let the biscuits cool for at least 5 minutes. Serve while warm, or reheat them later if desired.\n\nSprouted Wheat Sweet Potato Brioche\n\n# SPROUTED WHEAT SWEET POTATO BRIOCHE\n\nMAKES 2 LARGE LOAVES OR 24 SMALL BRIOCHE \u00c0 T\u00caTE\n\nIn rich doughs, of which brioche is the poster child, the combined weight of fats, eggs, and sugar is above 25% of the weight of the flour, and that percentage can climb to as high as 90%. Rich dough is fit for special occasion breads and can also elevate pedestrian fare like hamburgers into a gourmet feast. Sometimes it serves as the foundation for a dish, such as the crust of a tart or clafoutis, or for wrapping meat, fish, and other savory foods in dough to be served en cro\u00fbte, and this dough can be used for both of those purposes.\n\nBrioche has a very high butter content and therefore is more highly appreciated in Europe, where there is less fear of butter than in the United States. With all the eggs and butter, it ought to be the best bread you ever tasted, and when properly made, it can be. However, in my experience, brioche is a bread that promises much but often underdelivers. This can happen if the dough isn't properly mixed or isn't adequately chilled before baking. Brioche dough should always be refrigerated overnight before baking so the butter can firm up the dough. Therefore, note that making this recipe is a two-day process, so plan accordingly. Another common problem is overbaking, which results in an overly dry final product.\n\nA fellow instructor at Johnson & Wales University, Harry Peemoeller, developed this version to celebrate both sprouted whole wheat flour and North Carolina's most iconic vegetable: sweet potatoes. It has a creamy flavor that magnifies the buttery richness and makes beautiful classic brioches \u00e0 t\u00eate, (\u00e0 t\u00eate means \"with heads,\" in reference to the little top knot that distinguishes this bread). Alternatively, it can be baked in pan loaves and used for toast, sandwiches, and fantastic French toast. Although this dough can be mixed by hand, using a stand mixer is advisable because it requires a lot of mixing to work in the butter.\n\n### DOUGH\n\nNOTES:\n\n\u2022 Don't be alarmed by the high percentage of yeast. This is often necessary for proper rising in doughs with high amounts of fat and other enrichments. There is a type of yeast, known as osmotolerant, specially designed for both rich and acidic doughs. The most well-known brand to me is SAF-Instant Gold. If you use one of these yeasts, the fermentation times will be more reliable and the amount of yeast will remain the same. However, this recipe works fine with any type of regular instant yeast.\n\n\u2022 To make sweet potato puree, skin a sweet potato (or yam), cut it into 4 to 6 pieces, and steam for about 15 minutes, until very soft. Mash with a fork or potato masher to make a coarse puree. Let cool to room temperature or lukewarm before using.\n\nsprouted whole wheat flour: 5 cups (21.25 oz \/ 602 g)\n\nbrown sugar: 5 tablespoons (2.5 oz \/ 71 g)\n\nsalt: 1\u215d teaspoons (0.4 oz \/ 11.5 g)\n\ninstant yeast (see notes): 5 teaspoons (0.55 oz \/ 15.5 g)\n\negg: 1 (1.75 oz \/ 50 g)\n\negg yolks: 3 (2.25 oz \/ 64 g)\n\nsweet potato puree (from about 1 medium to large sweet potato; see notes): about 1 cup plus 3 tablespoons (10.5 oz \/ 298 g)\n\nwhole milk, at room temperature: \u00be cup (6 oz \/ 170 g)\n\nunsalted butter, at room temperature: 1\u00bc cups plus 1 tablespoon (10.5 oz \/ 298 g)\n\negg wash: 1 egg whisked with 1 tablespoon water or milk\n\nTOTAL: 55.7 oz \/ 1,580 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted whole wheat flour| 100  \nbrown sugar| 12  \nsalt| 1.9  \ninstant yeast| 2.6  \negg| 8.3  \negg yolks| 11  \nsweet potato puree| 50  \nwhole milk| 28  \nunsalted butter| 50  \nTOTAL| 263.8\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, stir together the flour, brown sugar, salt, and yeast on low speed. Add the egg, egg yolks, sweet potato puree, and milk and mix on low speed for 1 minute, until a soft, coarse dough forms. Switch to the dough hook, increase the speed to medium-low, and mix for 3 to 4 minutes, until the dough begins to pull away from the bowl and show signs of elasticity, an indication of gluten development.\n\n2 Cut the butter into 4 pieces. While mixing on medium-low speed, add the butter one piece at a time, waiting until each piece is fully incorporated before adding the next. The dough should be soft, tacky, and supple and should feel bouncy when patted. If it's very sticky, add a bit more flour.\n\n3 Spread a small amount of vegetable oil on a work surface (or dust the work surface with flour) and use a bowl scraper to transfer the dough to the oiled area. With either oiled or floured hands, stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides.\n\n4 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Let the dough rest at room temperature for 30 minutes. Put the bowl in the refrigerator for at least 12 hours or overnight. (It's best to use it within 2 days because of the high yeast content.)\n\n5 About 3 hours before you plan to bake, remove the dough from the refrigerator. It will be much more solid because the butter is cold, and therefore easy to handle and shape. Transfer the dough to an oiled or floured work surface. For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Divide the dough in half and shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. You can also divide the dough into 4 pieces and round each piece into a ball. Place two balls in each prepared pan, where they will grow into a two-part loaf with rounded domes. For brioches \u00e0 t\u00eate, mist small brioche molds or muffin cups with vegetable spray oil. Divide the dough into the desired size. For small brioches \u00e0 t\u00eate, each piece should weigh about 2 ounces (56.5 g), for a yield of about 24. To shape, roll one end of a small ball of dough into a cylindrical cone. Poke a hole in the thick end, then slip the tip of the cone through it so that a nub of dough pokes through to make a \"head.\" Transfer the shaped dough to the oiled brioche molds.\n\n6 Mist the tops with vegetable spray oil, cover loosely with plastic wrap, and let proof at room temperature until the dough swells and rises to fill the brioche pans or crests above the loaf pans by at least 1 inch. This will take 1\u00bd to 2 hours for brioches \u00e0 t\u00eate and up to 4 hours for loaves. Brush the tops with the egg wash, using a light touch, as the dough will be very soft and fragile after warming to room temperature.\n\n7 Position a rack in the middle of the oven and preheat the oven to 400\u00b0F (204\u00b0C).\n\n8 Bake for 13 to 15 minutes for brioches \u00e0 t\u00eate, or 50 to 60 minutes for loaves (rotating after 30 minutes for even baking), until the crust is golden brown and firm to the touch on all sides and the bread sounds hollow when thumped on the bottom.\n\n9 Turn out onto a wire rack. Let brioches \u00e0 t\u00eate cool for at least 15 minutes before serving, and let loaves cool for at least 40 minutes before slicing and serving.\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\n# SPROUTED WHEAT CINNAMON BUNS AND SWEET ROLLS\n\nMAKES 18 CINNAMON BUNS OR 30 SWEET ROLLS\n\nThe more I try to create healthful bread options, the more I get asked for yet another recipe for sweet cinnamon buns. It seems like, in the end, comfort food always comes out ahead. So in my cookbooks, I always offer my latest version of cinnamon buns and their even more decadent cousins, sticky buns, and everyone is happy. This time around, the recipe has the paradoxical combination of extremely good-for-you 100% sprouted whole wheat dough, filled with a generous amount of cinnamon sugar, which is not so healthful, yet so deeply satisfying. You can, of course, use sucralose substitutes like Splenda if you want to replace the sugar. But unless you're diabetic, I think this is one case where it might be okay to give yourself a pass and just use classic cinnamon sugar.\n\nThis dough is like challah in that it's enriched. However, in this case the enrichments include milk but not eggs, whereas challah is enriched with eggs but never milk. If you wish, you can also make cinnamon buns with challah dough\u2014or for that matter, brioche dough. The final result will be similarly satisfying no matter which dough you use. That said, for cinnamon buns and sweet rolls, this is my go- to dough.\n\n### DOUGH\n\nmilk (any kind), lukewarm (95\u00b0F \/ 35\u00b0C): 2\u00bc cups (18 oz \/ 510 g)\n\ninstant yeast: 4 teaspoons (0.44 oz \/ 12.5 g)\n\nvegetable oil: 6 tablespoons (3 oz \/ 85 g)\n\nhoney (or agave nectar or granulated sugar): \u00bc cup (or 5 tablespoons) (2.5 oz \/ 71 g)\n\nsprouted whole wheat flour: 5\u2154 cups (24 oz \/ 680 g)\n\nsalt: 1\u215d teaspoons (0.4 oz \/ 11.5 g)\n\nsugar for filling: 1 cup(8 oz \/ 227 g)\n\ncinnamon for filling: \u00bc cup ground cinnamon (1.12 oz \/ 32 g)\n\nvegetable oil or melted unsalted butter for filling: 2 teaspoons (0.35 oz \/ 10 g)\n\nQuick Fondant Glaze\n\nTOTAL: 57.81 oz \/ 1,639 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nmilk| 75  \ninstant yeast| 1.8  \nvegetable oil| 12.5  \nhoney| 10  \nsprouted whole wheat flour| 100  \nsalt| 1.7  \nTOTAL| 201\n\n1 In a medium bowl, whisk together the milk and yeast. Add the oil and honey and whisk until evenly combined. Set aside for 2 to 5 minutes to activate the yeast. (The mixture need not become foamy; it just needs to show signs of bubbles. This helps the yeast to get a head start in the presence of a sweeter dough.)\n\n2 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour and salt (on low speed if using a stand mixer). Pour in the milk mixture and mix or stir until the flour is hydrated and a coarse, sticky dough forms, about 1 minute.\n\n3 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook or use a wet spoon or wet hands and mix for 1 minute, on medium-low speed if using a stand mixer. The dough should be smooth but still very soft and sticky. Add a bit more flour or milk if necessary to achieve this texture, but bear in mind that the dough will firm up during the stretch and fold process.\n\n4 Spread about 1 teaspoon of vegetable oil or olive oil on a work surface. Using a wet or oiled bowl scraper or rubber spatula, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will firm up slightly but still be very soft and somewhat sticky. Cover the dough with the mixing bowl and then, at intervals of 5 minutes or up to 20 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit more with each stretch and fold. After the final fold it should be soft, supple, and tacky and have a springy or bouncy quality when patted.\n\n5 Oil a large bowl and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with a lid or plastic wrap; if using plastic wrap, stretch it tightly over the bowl rather than laying it directly on the dough. Ferment the dough at room temperature for 1\u00bd to 2 hours, until double in size. (This time can be shortened by using a warm proof box set at about 90\u00b0F \/ 32\u00b0C.)\n\n6 Oil the work surface again and use an oiled bowl scraper or rubber spatula to transfer the dough to the oiled area. Divide the dough in half and form each piece into a ball. Cover loosely with plastic wrap or a towel and let rest for 10 minutes to allow the gluten to relax.\n\n7 In a small bowl, whisk together the sugar and cinnamon to make the filling. Line two sheet pans with parchment paper or silicone mats.\n\n8 Working on the oiled work surface, use your hands or a rolling pin to gently press the dough into a rectangle measuring about 12 by 15 inches, with the long side facing you. It should be a bit over \u00bc inch thick. If using a rolling pin, roll out from the center to each of the four corners, and then to the sides. If the dough sticks to the rolling pin, rub a bit of vegetable oil on the surface of the dough.\n\n9 Rub or brush 1 teaspoon of vegetable oil or melted unsalted butter over the dough. Sprinkle up to half of the cinnamon sugar over the dough, leaving a \u00bc-inch bare border on all four sides. Evenly roll up the dough like a rug, from the bottom to the top, to form a tight log. Use your hands to gently squeeze and form the log so it's even across and not tapered toward the ends. Cut the log into equal-size slices at least 1 inch thick and up to 1\u00be inches for very large cinnamon buns; don't cut them thinner than 1 inch, or the baked buns will be tough. Put them on one of the lined pans, spacing them at least 1 inch apart. If any cinnamon sugar falls out of the buns, scoop it up and sprinkle it over the buns. Repeat the process with the other ball of dough, or return it to the bowl, cover, and refrigerate for use the next day.\n\n10 Mist the top of the dough with vegetable spray oil and cover loosely with plastic wrap. Proof at room temperature for 1\u00bd to 2 hours, until nearly double in size.\n\n11 To bake one pan of buns, position a rack in the middle of the oven; to bake two, position two racks an equal distance apart. Preheat the oven to 350\u00b0F (177\u00b0C).\n\n12 Bake for 10 minutes, then rotate the pans (switching shelves if baking two pans) and bake for 5 to 15 minutes longer, until the tops are golden brown.\n\n13 Let cool on the baking pan or on a wire rack for at least 5 minutes before glazing and serving. As the buns cool, the glaze will harden.\n\n## Quick Fondant Glaze\n\nThe corn syrup is optional in this recipe, but be aware that it will help keep the glaze smooth and prevent crystallization.\n\n4 cups (12 oz \/ 340 g) confectioners' sugar, sifted\n\n2 tablespoons (1.33 oz \/ 37.5 g) light corn syrup (optional)\n\n1 teaspoon (0.14 oz \/ 4 g) vanilla, lemon, or orange extract\n\n\u00bd to \u00be cup (4 to 6 oz \/ 113 to 170 g) milk (any type) or water\n\nIn a large bowl, stir together the sugar, corn syrup, and vanilla extract. While whisking, gradually pour in the milk, adding just enough to make a thick but creamy glaze that ribbons off the end of the whisk. Adjust with more confectioners' sugar or milk as needed. You can make it thinner if you prefer to brush the glaze over the buns, or thicker for drizzling if you want it to form a design of streaks or squiggles on the top.\n\n### Variations\n\nMULTIGRAIN: You can replace up to 20% of the sprouted whole wheat flour with an equal amount, by weight, of other sprouted flours. If you do so, decrease the amount of milk by 2 to 4 tablespoons (1 to 2 oz \/ 28.5 to 56.5 g).\n\nRAISIN NUT: Before rolling the dough into a log, sprinkle about \u00bd cup (3 oz \/ 85 g) raisins and about \u00bd cup (2 oz \/ 56.5 g) coarsely chopped walnuts or pecans over the surface.\n\nFILLED SWEET ROLLS: Instead of dividing the dough in half, divide it into about 30 pieces, each weighing about 1\u00bd ounces (43 g). Roll up each piece into a tight ball, and distribute the balls evenly on two sheet pans that have been lined with parchment paper or silicone baking mats. Cover loosely and proof at room temperature for about 1\u00bd hours, until they increase in size by 1\u00bd times. Make an indentation in the top of each roll with your thumb, pressing all the way to the bottom of the dough, but not through the dough. Use your fingertips to widen the indentation until there's just a \u00bc-inch rim of dough surrounding it. Fill the indentation with store-bought fruit pie or Danish filling (do not use fruit preserves or jelly, which are too thin), using about 1 heaping teaspoon (0.25 oz \/ 7 g) per bun. The rolls will swell and close around the filling as they bake, so you need to use enough filling that it doesn't disappear. Bake and glaze as for cinnamon buns.\n\nSTICKY BUNS: There are a number of sticky bun glazes available on the Internet and in my previous books that could be used. The main thing to know is that you should use a taller pan for baking to prevent the glaze from bubbling over. I use 9-inch round cake pans and bake only 5 buns per pan so they aren't crowded as they rise and bake. Coat the bottom of the pan with a \u00bc-inch thickness of glaze. Sprinkle nuts, raisins, or other toppings over the glaze, then put the buns on top of the glaze. Remember that the top surface will be the bottom of the buns when they're turned out of the pan, and because it's exposed, it will brown before the side in the glaze. It's important to let the glaze fully caramelize so the side in the glaze will brown slightly. If need be, cover the pans with foil toward the end of the baking time to prevent the top from getting too brown. Wait about 5 minutes before flipping the pan over onto a platter; this will give the glaze a chance to set up somewhat so that it doesn't slide off the top of the buns. Be very careful when doing this tricky maneuver, because the glaze will be molten hot. Oven mitts are essential, and a rubber spatula will be very helpful.\n\nSprouted Wheat Croissants\n\nSprouted Wheat Croissants\n\n# SPROUTED WHEAT CROISSANTS\n\nMAKES 10 CROISSANTS\n\nAlong with brioche, perhaps the most popular rich breads are croissants and their kissing cousins, Danish pastries. Yes, sweet indulgence! Whenever people tell me they want to open a bakery caf\u00e9, I tell them that if they offer nothing more than great baguettes, ciabattas, and whole grain breads, along with the best croissants in town, they'll be successful. (Of course, excellent coffee and tea are also a must.) Some bakers have tried to formulate whole wheat croissants in an effort to create a more guilt-free treat, but until recently I haven't had one that got me excited.\n\nEnter Harry Peemoeller, who is quite well known in the artisan bread world, having represented the United States at the 2012 Coupe du Monde de la Boulangerie (aka the International Bread Championship or Bread Olympics), where Team USA won the silver medal. In addition to his wonderful brioche, he's developed an amazing croissant recipe using sprouted whole wheat flour. In his bakeshop classroom at Johnson & Wales University, Peemoeller has the distinct advantage of having a reversible sheeter, which is kind of like a giant pasta rolling machine through which dough can travel in both directions through a set of adjustable rollers. Of course, this is something very few home bakers are likely to own\u2014except in their dreams\u2014so here I provide my own lamination technique. Note that this is a two-day technique, so plan accordingly.\n\n### DOUGH\n\nmilk (whole or low-fat), at room temperature: 1\u00bd cups plus 3 tablespoons (13.5 oz \/ 383 g)\n\ninstant yeast: 2\u00bc teaspoons (0.28 oz \/ 8 g)\n\nsugar: 2 tablespoons plus 1 teaspoon (1.25 oz \/ 35.5 g)\n\nsprouted whole wheat flour: 3\u00bd cups plus 1\u00bd tablespoons (15.2 oz \/ 431 g)\n\nsalt: 1\u00bd teaspoons (0.35 oz \/ 10 g)\n\nunsalted butter, at room temperature: 2 teaspoons (0.35 oz \/ 10 g)\n\nunsalted butter, slightly chilled but not ice-cold: 1 cup plus 1\u00bd tablespoons (8.8 oz \/ 250 g)\n\negg wash (optional): 1 egg whisked with 1 tablespoon water\n\nTOTAL: 39.73 oz \/ 1,127.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nmilk| 89  \ninstant yeast| 1.9  \nsugar| 8.2  \nsprouted whole wheat flour| 100  \nsalt| 2.3  \nunsalted butter| 2.3  \nunsalted butter| 58  \nTOTAL| 261.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the milk and yeast (on low speed if using a stand mixer). Add the sugar, flour, and salt and the 2 teaspoons of room-temperature butter. If using a stand mixer, mix on low speed about 1 minute, then switch to the dough hook and increase the speed to medium and mix for 2 to 3 minutes, until the dough is very smooth, supple, and tacky. If mixing by hand, stir with a large spoon for about 5 minutes, then briefly knead by hand until the dough is very smooth, supple, and tacky. Form the dough into a ball and return it to the bowl. Cover the bowl and let the dough rest at room temperature for 1 hour. It will rise slightly.\n\n2 Line a sheet pan with parchment paper or a silicone mat. Mist generously with vegetable spray oil or coat lightly with vegetable oil. Spread about 1 teaspoon of vegetable oil on a work surface. Transfer the dough to the oiled area and flip it over so both sides of the dough are slightly oiled. Use your hands or a rolling pin to press or roll the dough into a rectangle about \u00bd inch thick, then transfer it to the prepared pan. Cover with plastic wrap and refrigerate for at least 12 hours or overnight.\n\n3 About 1 hour before you plan to continue, remove the chilled butter from the refrigerator so it will be cool but not cold. It should be slightly pliable; 60\u00b0F (16\u00b0C) is the ideal butter temperature for lamination.\n\n4 When ready to begin laminating, spread about 1 teaspoon of vegetable oil on the work surface, covering an area about the same size as the sheet pan, and transfer the cold dough to the oiled area. Using your hands or a rolling pin, gently form or roll it into a rectangle measuring 8\u00bd by 14 inches.\n\n5 Put a sheet of parchment paper or a silicone mat on the work surface and mist it with vegetable spray oil, then sprinkle about 1 tablespoon (0.26 oz \/ 7.5 g) of flour over the surface. Transfer the butter to the floured parchment paper. Mist the top with vegetable spray oil, then cover with plastic wrap. Working through the plastic wrap, use your hands to flatten and press the butter into a rectangle or square large enough to cover half of the dough rectangle, leaving just a \u00bc-inch margin; it should measure about 8 by 6\u00be inches. Alternatively, use a rolling pin to tap the surface of the butter (through the plastic wrap) to achieve that same size. Lift the butter, parchment and all, onto the dough to determine whether the size is right. Once you're certain that the butter will cover half of the dough, remove the plastic wrap and, holding on to the parchment paper, invert the butter and fit it onto the dough, covering the entire left half and leaving a small, \u00bc-inch border around the edges. Fold the right half of the dough over the butter. Pinch the top and bottom edges together to seal the butter inside. Then use a rolling pin to gently tap or roll the dough into a larger rectangle measuring about 8 by 24 inches. Always roll in short strokes from the center of the dough outward, first to the corners and then to the sides, to ease the dough out, rather than forcing it. If the dough is sticky, dust it and the surface beneath it with flour or oil the work surface again. Once the dough is rolled out, fold it in thirds, like a letter, and pat it back into a rectangle or square with your hands. Put the dough on the prepared pan lengthwise, cover with plastic wrap, and refrigerate for 30 to 45 minutes so the butter can firm up again.\n\n6 Once again spread about 1 teaspoon of vegetable oil on the work surface. Turn the dough 90 degrees (a one-quarter turn) on the oiled work surface, then repeat the steps above, rolling the dough out into a rectangle measuring about 24 by 8 inches, folding it in thirds, like a letter, then patting into a rectangle. Return the dough to the refrigerator for 30 to 45 minutes, then repeat the process one final time, rotating the dough, rolling it out, folding it, and patting it into a rectangle. Return the dough to the refrigerator for 1 hour.\n\n7 Again, spread about 1 teaspoon of vegetable oil on the work surface. Place the dough on the oiled area and roll it out into an 8 by 14-inch rectangle; it should be about \u00bd inch thick. Return the dough to the refrigerator for 30 minutes.\n\n8 Oil the work surface once again, or dust it lightly with flour, then transfer the dough to the work surface. Roll it out into a 10 by 24-inch rectangle; it should be about \u00bc inch thick. Use a metal pastry blade or pizza cutter to cut the dough from top to bottom at 4\u00bd-inch intervals, forming 5 rectangles each measuring 10 by 4\u00bd inches. Then, cut each rectangle diagonally into two triangles, each with a 4\u00bd-inch base.\n\n9 Cut a \u00be-inch notch in the center of each 4\u00bd-inch base. Slightly spread out the two halves of the base at the notch and roll the dough into a crescent shape. Repeat with all of the pieces. Line two sheet pans with parchment paper or silicone mats and put 5 croissants on each pan, spacing them about 2 inches apart. Brush them evenly with the egg wash if you like, then cover loosely with plastic wrap. Proof at room temperature for 1\u00bd to 2 hours, until the croissants swell noticeably; they won't rise, but they will expand around the roll-up.\n\n10 Preheat the oven to 400\u00b0F (204\u00b0C).\n\n11 Bake one pan at a time. For each pan, bake at the initial temperature for 5 minutes, then lower the temperature to 375\u00b0F (191\u00b0C). and bake for another 13 minutes, until golden brown all over, with no white showing in the folds. If the bottoms are getting too dark during baking, slide a second pan under the first for insulation.\n\n12 Transfer the croissants to a wire rack and let cool for at least 30 minutes before serving.\n\nNOTE: You can freeze unbaked shaped croissants, either individually in zip-top freezer bags or on a covered or wrapped tray. Once they're frozen solid, they can be transferred to a larger bag. To bake them, remove from the freezer 5 to 6 hours before baking and proof at room temperature, or transfer from the freezer to the refrigerator the night before and proof at room temperature for 3 to 4 hours before baking.\n\n### Variation\n\nCHOCOLATE CROISSANTS: Instead of cutting the rectangles into triangles, cut them in half crosswise to make squares. Cover the bottom edge of each square with about 1 ounce (28.5 g) of semisweet or bittersweet chocolate chips or chocolate strips (batons) and roll the dough over the chocolate to form a log. Place the croissants seam side down on a lined pan, then proof and bake as directed.\n\n## Q & A\n\n### Why bother with an egg wash? And why is egg wash sometimes made with egg whites and other times with whole eggs?\n\nBoth types of egg washes add to the eye appeal of baked goods, and either can help toppings adhere to the dough. Egg washes made with whole eggs tend to create a dark and glossy surface, while those made with egg whites yield a shinier, more translucent gloss. Some people like to add a pinch of salt or sugar, or sometimes milk is used instead of water, but I find that a simple version with just egg and water makes for the best all-purpose egg wash.\n\nCHAPTER 4\n\n# SPROUTED PULP BREADS\n\nI've known Keith Giusto for more than twenty-five years. We first met when he dropped in at Brother Juniper's Bakery, in Santa Rosa, and introduced me to the line of flours milled by his family sixty-five miles down Highway 101 in South San Francisco, Giusto's Vita-Grain. It worked perfectly for us, and I used that flour until the day I sold the bakery. I often called Keith when I had questions about flour or dough formulations because, in addition to being trained as a cereal chemist, he was also one of the best bakers I knew. He was far ahead of many of his competitors in recognizing the coming artisan bread movement and thinking about creating the kinds of flours the new breed of bakers would want. When his family sold Giusto Mills, Keith gathered some new partners and started his own flour company, Central Milling, in Petaluma, California.\n\nIn 2010 we reconnected in Denver while attending a baking symposium, and he introduced me to his nephew Nicky, whom he'd brought in with the intent of training him to be his eventual successor. This freed Keith to focus on what he loves to do: working with growers and millers, and baking.\n\nThree years later, I visited Central Milling and was stunned by how big the business had become. As I pulled into the parking lot, I saw four huge shipping trucks emblazoned with the Central Milling logo. Nicky, looking a little older than when I first met him in Denver, greeted me in the reception area. He gave me the grand tour, through the offices, past the break room and company pool table, and into a large, open warehouse well stocked with a variety of Central Milling flours and other baking products waiting to be loaded onto trucks and shipped.\n\nWe worked our way over to the small bakeshop lab, with scaling and molding workbenches, a Kemper spiral mixer, a proofer\/retarder, custom-built wooden forms (called Parisians) for proofing loaves, and an impressive WP Matador deck oven. There were also a number of bins filled with various Central Milling flours and ingredients.\n\nNicky showed me a beautiful, crusty hearth loaf with a large, open crumb, laced with golden raisins, and said, \"I'm working on a bread formula for the annual California Raisin Marketing Board's raisin bread competition. I'm calling it Raisin de Soleil. Just a few more tweaks and I think it will be ready to submit.\"\n\n\"It seems like you've gone beyond the business side, Nicky,\" I said. \"I didn't realize you were seriously into baking too.\"\n\n\"I've learned a lot from Keith and, hey, what can I say? Baking is in my blood.\"\n\nNicky filled me in on other developments, including that, since we met up in Denver, Keith had gone into semiretirement and essentially put the operation in Nicky's hands. Nicky had stepped up and assimilated it all, including the craft of bread baking. He is, after all, a Giusto.\n\n\"And here's something you may not know,\" he told me. \"Remember those sprouted Vita-Grain breads that Giusto's used to make way back when? Well, we're bringing them back.\"\n\n## THE ROOTS OF SPROUTED PULP BREAD\n\nI did, in fact, have an inkling that this was happening, and this was the very reason I'd come to Petaluma\u2014to hear more about the revival of one of the Giusto family's most iconic products: sprouted grain breads dating back to the 1940s. These weren't breads made with sprouted whole wheat flour, which didn't exist back then, but breads made in much the same way as Ezekiel Bread and Alvarado Street Bakery do it, using sprouted wheat berries and other sprouted grains, which are ground into a pulp (or mash) and then mixed with other bread ingredients, such as yeast, salt, honey, and, importantly, vital wheat gluten. The result is a dough that bakes up just like regular bread even though, amazingly, the grain is never made into flour; it goes straight from sprouts to pulp, and then to dough. In a very real way, the Giusto Vita-Grain breads were the prototype for baking with sprouted grain pulp, predating by a couple of decades Ezekiel and Alvarado Street, the two most well-known sprouted grain bakeries today, But Vita-Grain Bread was sold primarily in the San Francisco Bay Area, and in the late 1990s, it faded out of production.\n\nBut by then, others had entered the game and risen to prominence. In about 1964, Max Flores, a baker in Southern California, started Food For Life, a company that produced sprouted grain products that eventually became best known under the name Ezekiel Bread. Ezekiel followed a slow but steady growth curve and then shot up in the late 1990s, when various low-carb diet plans pointed to Ezekiel as their favorite healthy bread option. And Alvarado Street Bakery, established in 1981 in Rohnert Park, California, about halfway between Petaluma and Santa Rosa, grew from a small hippie co-op bakery to a large, successful producer of a full line of organic sprouted grain products. So, the concept of breads made from the pulp of sprouted grains can't really be considered the next frontier; it was the next frontier decades ago. (And, as Ezekiel's biblical roots in Genesis 1:29 indicate, it's really a throwback to ancient times.) But what is new and exciting is that Keith and Nicky Giusto are now making it possible for other bakeries, as well as home bakers, to get into the game by providing sprouted grain pulp.\n\nAs Nicky explained, Central Milling has begun producing sprouted pulp and freezing it in vacuum-sealed cylinders called chubs. \"We can ship it overnight to anyone who wants to use it, and we can even help bakers with their dough formulation.\" When I asked him if anyone was using the pulp yet, he told me, \"We're already shipping large quantities to bakeries, and I'm currently studying how the meat industry handles shipping. So by the time your book comes out, we should be able to ship small orders to home bakers.\" Of course, you can also make your own sprouted pulp (see below).\n\n## Homemade Sprouted Grain Pulp\n\nSee Sprouting and Drying Grains for Flour for instructions on how to sprout and dry grain. For sprouted pulp, instead of drying the sprouted grain, run the sprouts through a meat grinder (I use the meat grinder attachment on my KitchenAid mixer). You can also process them with a bit of water in a high-powered blender, such as a Vitamix, but it's not as effective as the grinder.\n\nThe challenge when baking with sprouted grain is that when sprouts are ground into pulp rather than being dried and milled into flour, the grain's capacity to form gluten bonds is compromised. To compensate for this, sprouted pulp doughs often contain vital wheat gluten, made by washing the starch away from flour and drying the result into a powder. For most professional bakers, vital wheat gluten is a familiar ingredient, often used to boost the stretchability of breads made with weaker flours that have a lower percentage of gluten, such as rye, low-protein wheat, or multigrain blends. However, the amount of vital wheat gluten usually used for, say, rye bread is typically between 1% to 2% of the total flour weight, whereas the amount used in commercially produced sprouted pulp breads is often closer to 10%. This produces a bread that looks and feels a lot like bread made from regular flour, and it's full of the benefits of sprouted grain as long as you can tolerate the added gluten. When you think about it, it's a very clever way to reengineer the whole notion of bread. Imagine, bread made without flour. Ingenious!\n\nWhen my wife, Susan, and I owned Brother Juniper's Bakery, just twenty minutes up Highway 101 from Alvarado Street Bakery, I became friends with some of the founders and co-op members. They even helped us get our bakery off the ground by giving us their used loaf pans so we could save money for other essentials (like flour and yeast and paying our staff!). I was (and still am) a big fan of their products and have long been impressed by how they grew their business while adhering to the tricky co-op business model in which every employee is also an owner. They've made it work, and in the process, they grew from making just a few loaves a day to well over twenty thousand.\n\nBakers tend to be generous by nature, probably because they know how unbelievably hard it is to operate a bakery and produce consistent goods day after day. I'm so grateful to Keith and Nicky Giusto for providing some of the following formulas, and for making sprouted grain pulp available to home bakers. As Keith told me, \"I'm just glad to find people who care as much as I do about baking and who are interested in hearing about some of the things I've learned along the way.\" And by the way, Nicky won the grand prize in the Professional Artisan category for his magnificent Raisin de Soleil (a link to his formula is in the Resources section).\n\n## Q & A\n\n### If a recipe calls for vital wheat gluten, can I make the bread without it?\n\nYou can, but it will be denser and take longer to bake. You should also lower the oven temperature by about 10% to allow more time for the loaves to bake without burning the crust; for example, if the instructions specify 380\u00b0F (193\u00b0C), bake at 340\u00b0F (171\u00b0C) instead. You can also replace the vital wheat gluten with xanthan gum or ground psyllium husks, which are often used in gluten-free baked goods. They aren't quite as effective but will help provide some structure. If using xanthan gum, the amount should be about 0.5% of the total sprouted grain weight. So for each 10 ounces (283 g) of sprouted pulp, that would be about 0.05 ounce (1.5 g), or \u00bd teaspoon. For psyllium, use 1 teaspoon.\n\n# MULTIGRAIN SPROUTED WHEAT PULP BREAD\n\nMAKES 2 LOAVES OR UP TO 24 ROLLS | MASTER FORMULA\n\nHere's a master formula for sprouted pulp bread, perfected by Keith and Nicky Giusto. As you add the ingredients to the mixing bowl, you may wonder if they'll truly transform into bread. They will, and the bread will taste great! You'll also notice the method is quite different from the breads in the previous chapter, which stands to reason, since this is an entirely different product. It contains quite a bit of yeast and doesn't utilize much bulk fermentation, yet turns into delicious bread in a relatively short amount of time, especially when compared to long, slow fermentation methods of making artisan bread.\n\nTo make your own sprouted pulp, see the sidebar on Homemade Sprouted Grain Pulp. The soaker calls for a nine-grain cereal blend, and you should be able to find something along these lines in local markets. However, feel free to create your own combination using sprouted grains, rye or oat flakes, coarse cornmeal, grits, or seeds such as flax, chia, or hemp\u2014whatever you have on hand, really. While it's only a small fraction of the final loaves, it adds great texture and character. You can also use sprouted grains or flours in the soaker. As you'll see, the soaker uses only a bit of water\u2014just enough to soften the grains and get some enzyme activity going for flavor development.\n\n### SOAKER\n\ncracked nine-grain cereal mix or any combination of grain meals or sprouted grains: 1 cup (4.5 oz \/ 128 g)\n\nwater, at room temperature: \u2153 cup (2.7 oz \/ 77 g)\n\nTOTAL: 7.2 oz \/ 205 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \ncracked nine-grain cereal mix| 100  \nwater| 59  \nTOTAL| 159\n\n1 Put the nine-grain cereal in a small bowl and stir in the water. Cover with plastic wrap and leave out at room temperature for 12 hours or overnight; if your kitchen is very warm, refrigerate the soaker after a few hours.\n\n### FINAL DOUGH\n\nsprouted wheat pulp: 6 cups (32 oz \/ 907 g)\n\nwater, at room temperature: \u00bd cup (4 oz \/ 113 g)\n\nvital wheat gluten: \u00bd cup plus 1 tablespoon (3 oz \/ 85 g)\n\ninstant yeast: 2 tablespoons (0.66 oz \/ 19 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nhoney: 1\u00bd tablespoons (1 oz \/ 28.5 g)\n\nmolasses: 1\u00bd tablespoons (1 oz \/ 28.5 g)\n\nsugar: 2 tablespoons (1 oz \/ 28.5 g)\n\nvegetable oil: 1 tablespoon plus 1 teaspoon (0.65 oz \/ 18 g \/ 2%)\n\nSoaker: all (7.2 oz \/ 205 g)\n\nTOTAL: 51.01 oz \/ 1,446.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted wheat pulp| 100  \nwater| 12  \nvital wheat gluten| 9.4  \ninstant yeast| 2.1  \nsalt| 1.5  \nhoney| 3.1  \nmolasses| 3.1  \nsugar| 3.1  \nvegetable oil| 2  \nSoaker| 23  \nTOTAL| 159.3\n\n1 Put all the dough ingredients in a mixing bowl or the bowl of a stand mixer fitted with the paddle attachment, in the order listed. Stir or mix, on low speed if using a stand mixer, until a coarse, thick, slightly sticky dough forms, about 3 minutes. (Once the dough forms a coarse ball, you can knead it by hand if you prefer.)\n\n2 Let the dough rest, uncovered, for 5 to 10 minutes. Then switch to the dough hook and mix on medium speed or knead by hand for 3 to 4 minutes, until the dough is tacky but supple and feels springy when poked. The internal temperature of the dough should be about 75\u00b0F (24\u00b0C). Knead it by hand for a few seconds, then form it into a ball.\n\n3 Mist a bowl with vegetable spray oil and put the dough in the bowl. Mist the top with vegetable spray oil, and cover the bowl with plastic wrap or a lid. Let rest at room temperature for 15 minutes. It won't rise much in that time.\n\n4 Divide the dough in half and form each piece into a ball. Mist with vegetable spray oil and cover loosely with plastic wrap. Let rest at room temperature for 20 minutes.\n\n5 For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. For hearth loaves, line two sheet pans with parchment paper or silicone mats. Shape the pieces into boules or b\u00e2tards, then put the shaped loaves on the lined pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the two dough balls into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan, spacing them at least 2 inches apart.\n\n6 Mist the tops with vegetable spray oil and cover loosely with plastic wrap. Proof at room temperature until nearly double in size, about 1\u00bd hours.\n\n7 To bake pan loaves, position a rack in the middle of the oven and preheat the oven to 380\u00b0F (193\u00b0C). Bake for 25 minutes, then rotate and bake for 20 to 25 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 190\u00b0F (88\u00b0C). Turn out onto a wire rack and let cool for at least 45 minutes before slicing and serving.\n\n8 To bake hearth loaves or rolls, position two racks in the oven an equal distance apart and prepare the oven for steaming. (A baking stone isn't used.) Preheat the oven to 380\u00b0F (193\u00b0C). Put the bread in the oven and pour about 1 cup of hot water into the steam pan. Bake rolls for about 10 minutes and hearth loaves for about 20 minutes, then rotate the pans front to back and between racks. Bake for 10 to 12 minutes longer for rolls or 20 to 25 minutes longer for hearth loaves, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 190\u00b0F (88\u00b0C). Transfer from the pans to a wire rack. Let loaves cool for at least 1 hour before slicing and serving. Rolls can be served after 10 minutes.\n\nSprouted Emmer Pulp Power Bread\n\n# SPROUTED EMMER PULP POWER BREAD\n\nMAKES 2 LOAVES OR UP TO 24 ROLLS\n\nI remember when Keith Giusto showed me the first version of this bread over twenty-five years ago. I was so impressed that I started making my own versions, using seed flours and plumped raisins instead of sprouted pulp. Back then, Keith told me that the original intent was to create a loaf especially designed for long-distance runners and people doing other intensive physical training. This latest version, Keith's best yet, incorporates sprouted emmer wheat, an ancient strain sometimes referred to as farro and still used in Italian cooking. (The definition of farro is debated, and in Italy, the term sometimes refers to spelt or even einkorn.) It has a sweet and nutty flavor that's wonderful in this bread, but you can certainly substitute other types of sprouted wheat pulp, including spelt, einkorn, or Kamut (aka Khorasan wheat).\n\nAs you'll see, this formula calls for making a levain. If your mother starter has been refreshed within the last three days, you can use 1 cup (6.75 oz \/ 191 g) of that instead. However, if the hydration of your starter is different from that in the recipe, you may need to adjust the amount of flour or water in the final dough accordingly. Also, the dough includes both sprouted pulp and standard bread flour. This combination approach can be used in many recipes, including any in this book, once you become familiar with the functionality of sprouted pulp. Note that this is a two-day process, so plan accordingly.\n\n### LEVAIN\n\nMother Starter: \u00bc cup (1.6 oz \/ 45.5 g)\n\nwater, at room temperature: \u00bc cup (2 oz \/ 56.5 g)\n\nunbleached bread flour: \u00bc cup (3.5 oz \/ 99 g)\n\nTOTAL: 7.1 oz \/ 201 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 46  \nwater| 57  \nunbleached bread flour| 100  \nTOTAL| 213\n\nPut the mother starter in a small bowl and add the water to soften it. Break the starter up into smaller pieces, then add the flour and stir until a coarse ball of dough forms. Transfer to a work surface and knead by hand for about 30 seconds, until the starter is evenly distributed and a tacky dough forms. Mist a small bowl with vegetable spray oil and put the dough into the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Let the levain rest at room temperature for 6 to 12 hours, until double in size. If it doubles in size before you're ready to make the final dough, put it in the refrigerator.\n\n### SOAKED RAISINS\n\nraisins: (about \u2154 cup oz \/ 4 g)\n\nwater, at room temperature: (\u00be cups oz \/ 6 g)\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nraisins| 113  \nwater| 170\n\n1 Put the raisins in a small bowl and pour in the water. Cover the bowl with plastic wrap and leave out at room temperature for at least 6 hours or overnight.\n\n2 To make raisin water for the final dough, drain the raisins in a strainer set over a bowl, pressing gently to extract more liquid. This will yield more than the \u00bc cup (2 oz \/ 56.5 g) needed for the dough. You can discard the excess or use it in something else, such as a smoothie, or you can simmer it in a saucepan until reduced to the amount needed (cool it before using), to add more raisin flavor to the bread.\n\n### FINAL DOUGH\n\nLevain: all (7.1 oz \/ 201 g)\n\nwater, at room temperature: 1\u00bc cups plus 2 tablespoons (11 oz \/ 312 g)\n\nsprouted emmer pulp or other sprouted wheat pulp (see Homemade Sprouted Grain Pulp): 2 cups (11 oz \/ 312 g)\n\nunbleached bread flour: 3\u00bd cups plus 1 tablespoon (16.25 oz \/ 461 g)\n\nvital wheat gluten: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\ninstant yeast: \u00bd teaspoon (0.05 oz \/ 1.5 g)\n\nsalt: 2\u215c teaspoons (0.6 oz \/ 17 g)\n\nraisin water: \u00bc cup (2 oz \/ 56.5 g)\n\nSoaked Raisins: about \u2154 cup (4 oz \/ 113 g)\n\nwalnuts, coarsely chopped: \u00bd cup plus 3 tablespoons (2.75 oz \/ 78 g)\n\nalmonds, coarsely chopped: \u00bd cup plus 3 tablespoons (2.75 oz \/ 78 g)\n\nTOTAL: 58 oz \/ 1,644 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nLevain| 26  \n---|---  \nwater| 40  \nsprouted emmer pulp or other sprouted wheat pulp| 40  \nunbleached bread flour| 60  \nvital wheat gluten| 1.8  \ninstant yeast| 0.2  \nsalt| 2.2  \nraisin water| 7.3  \nSoaked Raisins| 15  \nwalnuts| 10  \nalmonds| 10  \nTOTAL| 212.5\n\n1 Cut the levain into about 5 pieces and put it in the bowl of a stand mixer fitted with the dough hook or in a large bowl. Add the water and stir (on low speed if using a stand mixer) for a few seconds to soften the levain. Add the sprouted emmer pulp, bread flour, vital wheat gluten, yeast, salt, and raisin water. Mix on low speed or knead by hand for about 3 minutes, until a coarse dough forms.\n\n2 Let the dough rest, uncovered, for 5 minutes. Add the raisins, walnuts, and almonds and mix on medium-low speed or knead by hand for up to 6 minutes, until the dough is tacky but supple. The raisins will break down and disappear into the dough, making it wet at first, but it will eventually firm up. Add more flour if the dough remains sticky. Form the dough into a ball. Mist a large bowl with vegetable spray oil and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Ferment the dough at room temperature for 1\u00bd hours, until nearly double in size.\n\n3 Divide the dough in half and round each piece into a ball. Mist with vegetable spray oil and cover loosely with plastic wrap. Let rest at room temperature for 30 minutes.\n\n4 For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Shape each piece of dough into a sandwich loaf, then put the shaped loaves in the prepared pans. For b\u00e2tards, line a sheet pan with parchment paper or a silicone mat. Shape each piece of dough into a b\u00e2tard. Put the shaped loaves on the lined pan, spacing them at least 3 inches apart. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the two dough balls into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan, spacing them at least 2 inches apart.\n\n5 Mist the tops with vegetable spray oil and cover loosely with plastic wrap. Proof at room temperature for about 1\u00bd to 2 hours, until nearly double in size.\n\n6 To bake pan loaves, position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Bake for 20 minutes, then rotate and bake for 35 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 195\u00b0F (91\u00b0C). Turn out onto a wire rack and let cool for at least 1 hour before slicing and serving.\n\n7 To bake b\u00e2tards or rolls, position a rack in the middle of the oven and prepare the oven for steaming. (A baking stone isn't used.) Preheat the oven to 460\u00b0F (238\u00b0C). If making b\u00e2tards, score them as desired. If making rolls, bake one pan at a time, holding the second pan in the refrigerator to slow the rising if necessary. Put the bread in the oven and pour about 1 cup of hot water into the steam pan. Bake rolls for 10 minutes and b\u00e2tards for 20 minutes, then rotate and bake for 5 to 10 minutes longer for rolls or 20 to 25 minutes longer for b\u00e2tards, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 195\u00b0F (91\u00b0C). Transfer from the pans to a wire rack. Let b\u00e2tards cool for at least 45 minutes before slicing and serving. Rolls can be served after 10 minutes.\n\n# SPROUTED WHEAT PULP BREAD WITH SPROUTED MULTIGRAIN FLOUR\n\nMAKES 2 LOAVES OR UP TO 24 ROLLS\n\nSo, what happens when you mix sprouted wheat pulp with sprouted grain flour? Wonderfulness! In collaboration with others, I developed this formula at the end of a Bread Baker's Guild of America workshop in Atlanta, Georgia, held at Alon's Bakery. After making everything on our workshop list, we found ourselves with leftover sprouted wheat pulp and a variety of sprouted flours from To Your Health Sprouted Flour Co. and Lindley Mills, so we created this formula on the spot. We nailed it on the first try: a balanced, flavorful, and hearty yet soft bread that worked in loaf pans or as a freestanding boule or b\u00e2tard. Yes, it does call for vital wheat gluten to achieve the ideal texture, but it can also be made without it if you prefer a less airy loaf. Because all of the flour is sprouted, there's no need to make a soaker or pre-ferment. Feel free to swap in your own favorite sprouted flours, even sprouted bean flours, such as sprouted lentil or garbanzo flour, or a mixture, such as the Lindley Ancient Grain blend.\n\n### DOUGH\n\nsprouted wheat pulp or any variety of sprouted pulp, such as emmer, Kamut, or spelt: 6 cups (32 oz \/ 907 g)\n\nsprouted corn flour: \u00bd cup (2 oz \/ 56.5 g \/ 5%)\n\nsprouted brown rice or sprouted spelt flour: 6\u00bd tablespoons or \u00bd cup (2 oz \/ 56.5 g \/ 5%)\n\nsprouted rolled oats or sprouted oat flour: \u00bd cup (2 oz \/ 56.5 g \/ 5%)\n\nsprouted sorghum, millet, quinoa, or amaranth flour: about \u00bd cup (2 oz \/ 56.5 g \/ 5%)\n\nwater, at room temperature: 1\u00bc cups plus 2 tablespoons (11 oz \/ 312 g)\n\nvital wheat gluten: \u00bd cup plus 1\u00bd teaspoons (2.8 oz \/ 79.5 g \/ 7%)\n\ninstant yeast: 2 tablespoons (0.66 oz \/ 19 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nhoney: 1\u00bd tablespoons (1 oz \/ 28.5 g)\n\nmolasses: 1\u00bd tablespoons (1 oz \/ 28.5 g)\n\nsugar: 2 tablespoons (1 oz \/ 28.5 g)\n\nvegetable oil: 1 tablespoon plus 1 teaspoon (0.65 oz \/ 18 g)\n\nTOTAL: 58.61 oz \/ 1,661 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsprouted pulp| 80  \nsprouted corn flour| 5  \nsprouted brown rice or sprouted spelt flour| 5  \nsprouted rolled oats or sprouted oat flour| 5  \nsprouted sorghum, millet, quinoa, or amaranth flour| 5  \nwater| 28  \nvital wheat gluten| 7  \ninstant yeast| 1.7  \nsalt| 1.2  \nhoney| 2.5  \nmolasses| 2.5  \nsugar| 2.5  \nvegetable oil| 1.6  \nTOTAL| 147\n\n1 Put all the ingredients in the order listed in a mixing bowl or the bowl of a stand mixer fitted with the paddle attachment. Stir or mix, on low speed if using a stand mixer, until a coarse, thick, slightly sticky dough forms, about 3 minutes.\n\n2 Let rest, uncovered, for 5 to 10 minutes. Then switch to the dough hook and mix on medium speed or knead by hand for 3 to 4 minutes, until the dough is tacky but supple and feels springy when poked, and its temperature is about 75\u00b0F (24\u00b0C). Knead it by hand for a few seconds, then form it into a ball.\n\n3 Mist a bowl with vegetable spray oil and put the dough in the bowl. Mist the top with vegetable spray oil and cover the bowl with plastic wrap or a lid and let rest at room temperature for 15 minutes. It won't rise much in that time.\n\n4 Divide the dough in half and form each piece into a ball. Mist with vegetable spray oil and cover loosely with plastic wrap. Let rest at room temperature for 20 minutes.\n\n5 For pan loaves, mist two 4\u00bd by 8-inch loaf pans with vegetable spray oil. Shape the pieces into sandwich loaves, then put the shaped loaves in the prepared pans. For hearth loaves, line two sheet pans with parchment paper or silicon mats. Shape the pieces into boules or b\u00e2tards, then put the shaped loaves on the lined pans. For rolls, line two sheet pans with parchment paper or silicone mats. Divide the two dough balls into the desired number of pieces and shape as desired (see Rolls and Buns). Put half of the rolls on each lined pan, spacing them at least 2 inches apart.\n\n6 Mist the tops with vegetable spray oil and cover loosely with plastic wrap. Proof at room temperature until nearly double in size, about 1\u00bd hours.\n\n7 To bake pan loaves, position a rack in the middle of the oven and preheat the oven to 380\u00b0F (193\u00b0C). Bake for 25 minutes, then rotate and bake for 20 to 25 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 190\u00b0F (88\u00b0C). Turn out onto a wire rack and let cool for at least 45 minutes before slicing and serving.\n\n8 To bake hearth loaves or rolls, position two racks in the oven an equal distance apart and prepare the oven for steaming. (A baking stone isn't used.) Preheat the oven to 380\u00b0F (193\u00b0C). Put the bread in the oven and pour about 1 cup of hot water into the steam pan. Bake rolls for about 10 minutes and freestanding loaves for about 20 minutes, then rotate the pans front to back and between racks. Bake for 10 to 12 minutes longer for rolls or 20 to 25 minutes longer for freestanding loaves, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 190\u00b0F (88\u00b0C). Transfer from the pans to a wire rack. Let loaves cool for at least 1 hour before slicing and serving. Rolls can be served after 10 minutes.\n\nSprouted Kamut Pulp Bagels\n\n# SPROUTED KAMUT PULP BAGELS\n\nMAKES 8 BAGELS\n\nThe name Kamut is trademarked by Mack and Bob Quinn, who first started cultivating this ancient variety of wheat in the United States in the 1980s. Kamut is a strain of Triticum turanicum, known more generically as Khorasan wheat, as this type of wheat is believed to have originated in the Persian province of Khorasan. Whole Kamut is fairly widely available, and other brands of Khorasan wheat are now appearing in the marketplace, with Central Milling selling it in sprouted pulp form (see Resources). As with spelt, einkorn, and emmer, it is reported that some people who are sensitive to conventional wheat can better tolerate this strain.\n\nAs you'll see, this formula calls for making a small piece of whole wheat levain. If your mother starter has been recently refreshed, you can simply use an equal amount of that instead. However, if the hydration of your starter is different from that in the recipe, you may need to adjust the amount of flour or water in the final dough accordingly. You can substitute any type of sprouted wheat pulp but this Khorasan version produces a beautiful, golden bagel. Note that this is a two- to three-day process, so plan accordingly.\n\n### WHOLE WHEAT LEVAIN\n\nMother Starter: 1 teaspoon (0.25 oz \/ 7 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2 tablespoons plus 1\u00bc teaspoons (1.2 oz \/ 34 g)\n\nwhole wheat flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nsalt: pinch (0.02 oz \/ 0.5 g)\n\nTOTAL: 2.97 oz \/ 84 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 16  \nwater| 80  \nwhole wheat flour| 100  \nsalt| 0.1  \nTOTAL| 196.1\n\nPut the starter in a small bowl and add the water to soften it. Add the flour and salt and stir until a coarse ball of dough forms. Transfer to a work surface and knead by hand for about 30 seconds, until the starter is evenly distributed and a soft, tacky dough forms. Mist a small bowl with vegetable spray oil and put the dough in the bowl. Mist the top of the dough with vegetable spray oil and cover the bowl with plastic wrap. Let the levain rest at room temperature for 6 to 12 hours, until double in size. If it doubles in size before you're ready to make the final dough, put it in the refrigerator.\n\n### FINAL DOUGH\n\nWhole Wheat Levain: all (2.97 oz \/ 84 g)\n\nwater: \u00bd cup (4 oz \/ 113 g)\n\nsprouted Kamut (Khorasan wheat) pulp: 2\u00be cups (15 oz \/ 425 g)\n\nwhole-milled whole wheat flour or sprouted whole wheat flour: 2 \u00bc cups or 2\u2153 cups (10 oz \/ 283 g)\n\ninstant yeast: \u00be teaspoon (0.08 oz \/ 2.5 g)\n\nvital wheat gluten: 1\u00bd tablespoons (0.5 oz \/ 14 g \/ 2%)\n\nsalt: 1\u215d teaspoons (0.4 oz \/ 11.5 g)\n\nhoney or barley malt syrup: 3\u00bd teaspoons (0.75 oz \/ 21.5 g \/ 3%)\n\nbaking soda for boiling the bagels: about 2 tablespoons\n\nhoney for boiling the bagels (optional): about \u00bc cup\n\ncornmeal or semolina flour: about 2 tablespoons\n\nTOTAL: 33.7 oz \/ 954.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nWhole Wheat Levain| 12  \nwater| 16  \nsprouted Kamut pulp| 60  \nwhole-milled whole wheat flour or sprouted whole wheat flour| 40  \ninstant yeast| 0.4  \nvital wheat gluten| 2  \nsalt| 1.6  \nhoney or barley malt syrup| 3  \nTOTAL| 135\n\n1 Cut the levain into about 5 pieces and put it in the bowl of a stand mixer fitted with the dough hook or in a large bowl. Add the water and stir (on low speed if using a stand mixer) for a few seconds to soften the levain. Add the sprouted Kamut pulp, flour, yeast, vital wheat gluten, salt, and honey in that order. Mix on low speed or knead by hand for 4 to 5 minutes to make a firm, coarse dough.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then mix on medium-low speed or knead by hand for 3 to 4 minutes. The dough should be smooth, firm, and satiny, Form the dough into a ball and return it to the oiled bowl. Cover the bowl with plastic wrap and let the dough rest for 20 minutes.\n\n3 Divide the dough into eight 4-ounce (113 g) pieces (you can also make larger bagels if you prefer). Form each piece into a ball. Cover with plastic wrap and let rest for 10 minutes. Line a sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil.\n\n4 Shape the pieces into bagels and put them on the prepared pan, spacing them at least 1 inch apart. Mist the tops with vegetable spray oil and cover the pan loosely with plastic wrap. Proof the dough at room temperature for 30 minutes, then refrigerate for at least 12 hours or overnight.\n\n5 Follow the instructions for boiling and baking the bagels.\n\n6 Transfer the baked bagels to a wire rack and let cool for at least 20 minutes before slicing and serving.\n\nCHAPTER 5\n\n# WHOLE GRAINS AND WHOLE MILLING\n\nWhole grain breads have long been seen as the healthier option, a source of important fiber, and therefore the most guilt-free choice. But as we dive deeper into the next frontier of bread, we enter a realm of subtle, nuanced distinctions. Putting aside controversies surrounding wheat and grains in general, a new question is whether all whole wheat flour is equal.\n\nCraig Ponsford, one of my longtime baking heroes, believes that some serious misinformation is floating around, and that much of what we believe to be 100% whole wheat flour actually isn't. He told me that when he decided to focus on baking healthy breads with whole grain flours after an award-winning career in making classic French-style white breads, he was stunned to learn that some whole wheat flours might be as unhealthful as white flour.\n\nAs he told me, \"It comes down to how the wheat is milled. I learned that most so-called whole wheat flour is actually fractionated flour, reconstituted during milling. As a result, the body responds to it differently than true whole wheat flour. Reconstituted whole wheat flour is not the same as whole-milled wheat flour.\"\n\nMy first thought was What on earth is he talking about? Whole wheat flour, as I understood it, meant that when the wheat kernels go into the mill, all of their components get ground into flour: 100% in, 100% out. I asked Craig about this and he explained: \"You could say that if 100 pounds of wheat goes into the mill and 100 pounds of flour comes out, then yes, that's whole wheat flour. That's called 100% extraction.\"\n\nThat's how I understood it too, just as I understood that if only 85 pounds of flour is collected, 15 pounds of bran and germ must have been sifted out, resulting in an 85% extraction rate, which bakers refer to as high-extraction flour. And as the extraction rate lowers to between 70% and 75%, the result is white flour, containing just the endosperm of the wheat\u2014the white, starchy part that also contains the proteins gliadin and glutenin, which eventually bond to make gluten. As for the remaining 25% to 30%, it may be sold or used as wheat bran or wheat germ or used in animal feed. And sometimes it's added back into white flour to create flours with various degrees of extraction.\n\n\"But here's what most people don't know,\" Craig continued. \"If you run the wheat through a stone mill, or even a stainless steel roller mill, and just collected it at the end, you'll end up with a lot of large bran and germ particles, resulting in a very coarse whole wheat flour, which most bakers don't really want. So mills typically sift out the germ and bran early in the process and then run them through again, separately from the endosperm, to break them down into smaller particles. Then, usually after at least two such siftings and millings, it's added back into the flour, reconstituting it to, in theory, 100% of its original weight.\"\n\n\"So why is that a bad thing?\" I asked.\n\n\"Once it's separated and added back, it's somehow different. Something changes, and we don't get all of the nutritional benefits. It doesn't perform the same way in baking, either, or taste as good, and it also doesn't keep as well. Although I'm not a scientist and I don't know all the intricacies, I can tell you that ever since I switched to whole-milled flour, my products have improved dramatically and no one complains about digestibility. I think these are some of the best whole grain baked products you will ever taste.\"\n\nHe was right about that last part, as I found out a few nights later when I stayed up with Craig during his nighttime baking stint at Ponsford's Place, the little bakery he operates as a test laboratory in San Rafael, California. Craig only bakes for the public two nights a week, Thursday and Friday (or, more accurately, Friday and Saturday mornings, from about midnight until a few hours after opening at 8 a.m.). He usually sells out of everything by midafternoon. Every few weeks, he and his girlfriend, Diana Benner, also offer a Wednesday pizza night featuring his whole wheat dough and sell hundreds of slices and many whole pies to the loyalists who know where to find Ponsford's Place.\n\n## GETTING INTO THE NITTY-GRITTY\n\nI spent a number of days with Craig, visiting various people he arranged for me to meet. One was Joe Vanderliet, the owner of Certified Foods in Woodland, about an hour east of San Rafael in California's Central Valley. Craig told me that Joe was the person who convinced him to focus totally on whole-milled flour.\n\nJoe is a tall, strapping Dutchman who, during his fifty-plus years in the business, has worked for some of the largest flour companies in the country, such as Archer Daniels Midland and Bay State Milling. He's continued to educate himself along the way by studying at baking and brewing schools and, over the years, his frustration at seeing so many nutrients wasted in the milling process spurred him to dedicate himself to whole-milled flour, ground primarily with stone mills rather than steel rollers. Eventually, his company established Joseph's Best as their flagship brand, and at the time of my visit, they were milling and packaging more than thirty products, including four types of whole wheat flour, two types of rye flour, nine types of rice flour, and sixteen specialty flours, such as amaranth, buckwheat, emmer, spelt, teff, quinoa, and various legume flours.\n\nAs Joe took us on a tour of the mill, where the whirr of large stone mills grinding hard red wheat into flour dominated the soundscape of the high-ceilinged facility, he was clearly adamant about not separating the endosperm from the bran and germ in the early stages of milling. He told me, \"Once the kernel is broken, it can't be put back together,\" and went on to say that the larger companies who dominate the flour industry reconstitute their sifted flour by adding bran and maybe the germ back into it\u2014and not necessarily the bran or germ from the same wheat kernels. \"It just isn't the same, and it's definitely not as good for you,\" he insisted.\n\nPrior to this trip, I'd never thought about whole wheat flour not really being whole wheat flour. And though I knew the big particles might get sifted out and then run back through the mill to break them down into finer particles before adding them back into the flour, I couldn't see much harm in that, so I asked Joe about it.\n\nHe replied, \"It really depends on when it gets sifted out. The longer you can keep it all together, the better it will be.\" And the fact is, in order to make his fine- and medium-grind whole wheat flour, even Joe sifts out the bran and germ and regrinds them before adding them back to the same batch of flour. But as he told me, most companies have two breaks, or points where the sifting occurs, and the first one comes early on. He said, \"That's where the real problem is. I always wait until the second break, and believe me, those extra seconds where all three parts of the grain stay together make a huge difference.\"\n\nI whispered to Craig, \"Boy, that's kind of like inside baseball.\"\n\nBut Craig said, \"Wait until we get back to my bakery. You'll see there really is a difference.\"\n\n## THE GLUTEN CONNECTION\n\nIt's important to clarify a few things here. First, as I was writing this book, Certified Foods was sold to a larger company, Bay State Milling, who retained Joe and his team, and they have assured me they're committed to honoring and using Joe's methods. Second, many small regional mills already use a whole milling process, especially for their coarse flour, which is usually a straightforward, whole-milled 100% extraction with little or no sifting and reconstituting, though they may not do it with as much precision as Vanderliet. If they're using a stone mill, it's likely that they're sifting out the larger particles at some point for regrinding when making fine- or medium-grind flour. And finally, I have to add that Joe's declarations are theories. When I contacted larger mills, they either refuted or sidestepped Joe's claims and wouldn't actually tell me how they reconstitute beyond vague statements like, \"Everyone has to sift and reconstitute if they want a finer flour.\"\n\nBut Joe and Craig are both convinced that one of the causes of increasing rates of gluten sensitivity is conventional modern milling practices. And customers often tell both of them that Joe's is the only wheat flour that doesn't give them digestive problems. To me, this makes sense intuitively, as do some of the other hypotheses floating around regarding the causes of gluten sensitivity. One is the idea that ancient strains of wheat, such as spelt, einkorn, emmer, and Kamut (aka Khorasan wheat) are tolerable for some people who are otherwise sensitive to gluten, perhaps because of the genetic makeup of these strains, or maybe because the gluten isn't as complex or strong as in more recent, crossbred strains. Another is that locally grown grains are often easier to digest because they're grown in the same soil and climate where consumers live, so their bodies are more predisposed to respond positively. In any case, I generally support practices that connect us with the things we eat.\n\n## THE LOCAL CONNECTION\n\nOther bakers, millers, and grain visionaries have been on quests similar to Craig's and Joe's. There are far too many to profile here, so I'll cover just a few.\n\nA prime example is Bob Klein, the visionary owner of Oliveto, a restaurant in Oakland, California. He's a proponent of a new concept that's been gaining traction recently: identity-preserved grain. A few years ago he started a venture called Community Grains, using California-grown grain to make gourmet products such as pastas and polenta. He's passionate about commissioning farmers to grow specific strains of wheat and other grains that are well suited to the local soil and climate. In addition, Community Grains whole wheat flour trumpets the value of whole milling right on the packaging. Bob, who has long been a major barometer for coming culinary trends, told me, \"Whole milling, rather than reconstituted, is a real difference maker. It would be a shame to take this beautiful, distinctive, carefully grown grain and then screw it up at the mill. Plus, it's the right thing to do. And identity-preserved farming supports the local economy, allows me to work closely with local farmers, and showcases both the terroir and the craft.\"\n\nIn Peter Reinhart's Whole Grain Breads, I wrote about Jennifer Lapidus, a baker in Asheville, North Carolina, who was making outstanding Flemish-style desem breads at her Natural Bridges Bakery. Since then, she's closed the bakery and established the milling company Carolina Ground. She recently told me, \"I got into milling in response to the soaring cost of wheat, artificially driven up by futures traders and other people who never even touched the wheat. So I decided to focus on locally milled flour, bought from local wheat farmers and milled for local bakers.\" As it turns out, other farmers and millers around the country felt the same, and now numerous local mills are making flour from locally grown grain, including Farmer Ground Mill (Trumansburg, New York), Somerset Grist Mill (Skowhegan, Maine), Camas Country Mill (Eugene, Oregon), and Fairhaven Organic Flour Mill (Burlington, Washington).\n\nDr. Stephen Jones, of the Washington State University Agricultural Research Center, put it this way when we spoke: \"Because of the commodity system, there used to be more bakers than wheat farmers in places like Vermont, but now we're seeing a lot more small-scale farmers growing wheat that's been developed specifically for their region and soil.\" Among the various projects he leads is a plant breeding program in which heirloom wheat strains are crossbred with disease-resistant wheat to create strains that are well suited to particular climates and soils.\n\nAccording to Jones, this can even result in superior flavor: \"If you plant wheat that's designed for wet regions in an area like the coastal Pacific Northwest, it will produce chocolate, spicy, and pleasant grassy flavor tones. But if you plant that same wheat in dry regions, where so much commodity wheat is grown, it's almost flavorless.\" He went on to tell me about Heather Darby, at the University of Vermont, who's been working with Jack Lazor, of Butterworks Farm, to develop a Vermont farmers breeding club with the same goal as his program: developing strains of wheat appropriate for their region, climate, and soil. \"The same thing is happening in many other places, and the best outcome will be that the availability of locally farmed wheat creates an infrastructure that includes millers, malters, bakers, and even livestock farmers. Essentially, it restores a local grain economy and it keeps the value where it's created.\"\n\nAs for Carolina Ground, Jennifer told me that she's currently whole milling 100% whole wheat bread flour, and also high-extraction (85%) flour, whole wheat and high-extraction pastry flour, and a variety of rye flours, along with lighter, lowerextraction flours for bakers who want them. She also likes to focus on milling single-variety wheats, as opposed to the blends associated with commodity flour.\n\nWhen I asked Jennifer where she finds her wheat, she said, \"I try to source my grain from North Carolina or nearby regions, but sometimes I have to spread the net wider to find the wheat that will yield the performance bakers need. But my mission is also to support local wheat farmers and help them find buyers for their grain, even if it isn't appropriate for bread flour. One example is Asheville's Riverbend Malt House, where they sprout and roast wheat, then send it to me for milling before it's shipped out to customers for beer making. And this past season, when growing conditions produced a less-than-ideal crop, I was thrilled that many bakers were willing to step up and accept a flour that was more challenging to work with. For example, Harry Peemoeller used it in his baking classes at Johnson & Wales University, and Lionel Vatinet used it in his breads at La Farm Bakery.\"\n\nThis kind of community support and networking reflects a wider national movement, and I've compiled a small list of likeminded farmers, millers, and bakers in the Resources section.\n\nIf the mission of the baker is to evoke the full potential of flavor trapped in the grain, it follows that the mission of farmers and millers is to provide the best-quality wheat, ground into the highest-quality flour, so that it can be transformed into the best possible bread. The recipes in this chapter, which feature whole-milled flours and flours made from heritage varieties of wheat, showcase these congruent missions of craft and quality.\n\nWhole-Milled Lean Dough French Bread\n\n# WHOLE-MILLED LEAN DOUGH FRENCH BREAD\n\nMAKES 1 LARGE LOAF, UP TO 4 SMALLER LOAVES, OR 6 MINI BAGUETTES\n\nThis foundational dough, developed by Craig Ponsford, can be used to make any number of shapes. It's a simple, lean dough made with a poolish and featuring 100% whole-milled wheat flour. Craig is adamant that it won't be as good if made with standard, reconstituted whole wheat flour, so see the Resources section for online purveyors if you can't find whole-milled wheat flour locally. Be aware that the hydration ratios could vary depending on the brand of flour you use. Let the dough guide you regarding adjustments to the amount of water or flour (or, as I tell my students, let the dough dictate what it needs). That said, whole wheat flour hydrates more slowly than white flour, so try to resist the temptation to add more flour. The dough will appear too sticky at first, but it will strengthen and firm up during the stretch and fold process. Craig calls for a small amount of salt in his poolish, which is unconventional, but as he explains, it's helpful because whole wheat flour has more enzyme activity than white flour, so it can easily overferment even with only a small pinch of yeast.\n\n### POOLISH (DAY 1)\n\nwhole-milled wheat flour: 1\u2153 cups (6 oz \/ 170 g)\n\ninstant yeast: pinch (0.007 oz \/ 0.2 g)\n\nsalt: pinch (0.02 oz \/ 0.5 g)\n\nwater, at room temperature: \u00be cup (6 oz \/ 170 g)\n\nTOTAL: 12.027 oz \/ 340.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole-milled wheat flour| 100  \ninstant yeast| 0.1  \nsalt| 0.3  \nwater| 100  \nTOTAL| 200.4\n\nIn a small bowl, stir together the flour, yeast, and salt. Add the water and stir until the flour is hydrated and a thick, batter-like dough forms, about 1 minute. Use a wet bowl scraper to scrape down the sides of the bowl. Cover the bowl with plastic wrap and let the poolish rest at room temperature for 12 hours or overnight, until it gets very bubbly and swells in size. If it gets very bubbly before you're ready to make the final dough, put it in the refrigerator. If you do so, use lukewarm water, at about 95\u00b0F (35\u00b0C), when mixing the final dough.\n\n### FINAL DOUGH (DAY 2)\n\nwater, at room temperature (lukewarm if poolish was refrigerated): 1\u00be cups plus 1 tablespoon (14.5 oz \/ 411 g)\n\npoolish: all (12.027 oz \/ 340.7 g)\n\nwhole-milled hard wheat flour (red or white wheat, or a blend): 4 cups (18 oz \/ 510 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: \u00be teaspoon (0.08 oz \/ 2.5 g)\n\nTOTAL: 45.107 oz \/ 1,278.2 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwater| 81  \npoolish| 67  \nwhole-milled hard wheat flour| 100  \nsalt| 2.7  \ninstant yeast| 0.5  \nTOTAL| 251.2\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the water and poolish (on low speed if using a stand mixer). Add the flour, salt, and yeast and mix or stir for 20 to 30 seconds or until the flour is hydrated and a coarse, sticky dough forms. If using a stand mixer, switch to the dough hook, using wet hands to get all the dough off the paddle. Mix, or stir with a wet spoon for 2 to 3 minutes. If the dough is difficult to stir with a spoon, turn it out onto a lightly floured work surface and knead it with floured or oiled hands. The dough should be somewhat shaggy and sticky. Fold the dough into a ball.\n\n2 Mist a large bowl with vegetable spray oil or lightly coat it with vegetable oil. Transfer the dough to the bowl, mist the top with vegetable spray oil, and cover the bowl with plastic wrap (or leave the dough on an oiled work surface and cover with the bowl). Ferment the dough at room temperature for 30 minutes.\n\n3 Working either in the bowl or on a lightly oiled work surface, stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Return the dough to the bowl, smooth side up, and cover the bowl (or again leave the dough on the work surface and cover with the bowl). Let the dough rest for 30 minutes. Repeat the process, stretching and folding, then returning the dough to the bowl (or covering it with the bowl). The dough will become smoother and more supple with each stretch and fold, and after the second stretch and fold, it will be very supple and tacky but not sticky. Ferment at room temperature for 30 to 60 minutes, until it increases in size by about 1\u00bd times.\n\n4 Oil the work surface or dust it with flour. Using a wet or oiled bowl scraper, transfer the dough to the oiled or dusted area. Using a metal pastry blade, divide the dough into the desired number of pieces\u20146 for mini baguettes or 2 to 4 for smaller loaves\u2014or use the entire amount of dough for one large loaf. Gently form each piece into a boule, b\u00e2tard, or baguette. Cover loosely with a clean towel or plastic wrap.\n\n5 Mist the tops with vegetable spray oil and cover loosely with plastic wrap or a clean towel. Proof at room temperature for about 1 hour, or until the dough increases in size by 1\u00bd times.\n\n6 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 500\u00b0F (260\u00b0C). About 10 minutes before baking, uncover the dough to let the surface dry a little bit.\n\n7 Transfer the dough to a peel and score as desired. Transfer onto the baking stone and pour about 1 cup of hot water into the steam pan. Immediately lower the temperature to 450\u00b0F (232\u00b0C). Bake for 14 minutes, then rotate and bake for 10 minutes longer for mini baguettes, 15 minutes longer for small loaves, and 20 minutes longer for a large loaf, until the crust is rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n8 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\nWhole-Milled Whole Wheat Ciabatta\n\n# WHOLE-MILLED WHOLE WHEAT CIABATTA\n\nMAKES 1 LARGE CIABATTA OR 2 SMALL CIABATTAS\n\nThis wet, rustic dough is a small-scale version of the one Craig Ponsford uses at Ponsford's Place for both his ciabatta and his Wednesday night pizza events. (To use it for pizza, see the sidebar on Whole-Milled Whole Wheat Pizzas.) I also use it for focaccia. It requires a poolish pre-ferment for structure, acidity, and maximum flavor development. The dough also contains a bit of coarse whole rye flour, also known as pumpernickel flour; it adds texture and contributes to the flavor complexity. Between the poolish and the water in the final dough, the total hydration of this dough is very close to 100% water to flour, and that's one of the secrets to making whole wheat products at the quality level to which Craig aspires. It takes practice to learn to handle such wet dough, whether using oiled hands as I do or floured hands as many bakers, including Craig, prefer. You may be tempted to add more flour when mixing it to make the dough feel more \"normal.\" Try to resist that urge, as whole-milled flour will eventually absorb a lot of water. And with the help of the three stretch and fold sequences, the dough will firm up enough to handle. Still, it may seem wetter than you're used to. This is where handling technique can make or break the final result. Especially when this dough for ciabatta is used, it must be gently cradled.\n\n### POOLISH (DAY 1)\n\nwhole-milled wheat flour: 1 cup plus 2 tablespoons (5 oz \/ 142 g)\n\nwater, at about 75\u00b0F (24\u00b0C): \u00bd cup plus 2 tablespoons (5 oz \/ 142 g)\n\nsalt: pinch (0.02 oz \/ 0.5 g)\n\ninstant yeast: pinch (0.007 oz \/ 0.2 g)\n\nTOTAL: 10.027 oz \/ 284.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole-milled wheat flour| 100  \nwater| 100  \nsalt| 0.3  \ninstant yeast| 0.1  \nTOTAL| 200.4\n\nIn a small bowl, stir together the flour, yeast, and salt. Add the water and stir until the flour is hydrated and a thick, batter-like dough forms, about 1 minute. Use a wet bowl scraper to scrape down the sides of the bowl. Cover the bowl with plastic wrap and let the poolish rest at room temperature for 12 hours or overnight, until it gets very bubbly and swells in size. If it gets very bubbly before you're ready to make the final dough, put it in the refrigerator. If you do so, use lukewarm water, at about 95\u00b0F (35\u00b0C), when mixing the final dough.\n\n### FINAL DOUGH (DAY 2)\n\nwater, at about 75\u00b0F (24\u00b0C), or 95\u00b0F (35\u00b0C) if poolish was refrigerated: 1 cup plus 3 tablespoons (9.5 oz \/ 269 g)\n\nPoolish: all (10.027 oz \/ 284.7 g)\n\nwhole-milled hard red winter wheat flour: 2\u00bc cups (10 oz \/ 283 g)\n\npumpernickel or coarse whole rye flour: 1 tablespoon (0.3 oz \/ 8.5 g)\n\nsalt: 1\u00bc teaspoons (0.32 oz \/ 9 g)\n\ninstant yeast: \u00bd teaspoon (0.05 oz \/ 1.5 g)\n\nTOTAL: 30.677 oz \/ 869.2 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwater| 92  \nPoolish| 97  \nwhole-milled hard red winter wheat flour| 97  \npumpernickel or coarse whole rye flour| 2.9  \nsalt| 3.1  \ninstant yeast| 0.5  \nTOTAL| 292.5\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the water and poolish (on low speed if using a stand mixer). Add the flours, salt, and yeast and mix or stir for 1 minute. Scrape down the bowl with a spatula or bowl scraper, then mix or stir for 3 minutes. If using a stand mixer, increase the speed to medium-low, and if mixing by hand, use a large spoon or your hands, dipped in water from time to time to minimize sticking, and mix for 3 minutes longer. The dough should be soft, wet, and very sticky.\n\n2 Mist a bowl with vegetable spray oil or lightly coat it with vegetable oil. Using a wet or oiled bowl scraper, transfer the dough into the bowl. Working either in the bowl or on a lightly oiled work surface, stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Return the dough to the bowl, smooth side up, and cover the bowl (or again leave the dough on the work surface and cover with the bowl). Then, at intervals of 1 hour, perform two additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. After the final stretch and fold, cover the dough and let it rest at room temperature for 15 to 20 minutes (or up to 1 hour for more fermentation if the dough seems sluggish). The dough will become firmer with each stretch and fold. After the third stretch and fold, it will be a bit bouncy when patted but will still be soft and sticky.\n\n3 Use the whole piece of dough for a large ciabatta, or divide it in half for 2 smaller ciabattas. Dust a sheet pan with about 3 tablespoons (0.85 oz \/ 24 g) of whole wheat flour or a blend of whole wheat and rye flour. With floured hands, lift, cradle, and transfer the dough pieces to the pan and gently fold each piece in thirds, like a letter, using the flour on the pan to coat the outer surface as you fold. (Alternatively, leave the pieces unfolded, coat with flour, and place them on the pan; this yields what the French would call pain rustique, eliminating the need to flip them over during proofing, as for ciabatta.) Roll the shaped dough in the flour to coat it again, then lay the dough seam side down on the floured pan, gently forming each piece into a slipper-like oblong or rectangle. Proof, uncovered, at room temperature for 30 minutes. Gently flip so the seam side is up, then proof for 15 to 25 minutes longer.\n\n4 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 550\u00b0F (288\u00b0C) or as high as it will go.\n\n5 Just before baking, dust a peel with flour, or cover it with parchment paper and mist with vegetable spray oil. Gently transfer the shaped dough to the peel, cradling it in your hands for support and laying it seam side up. Coax the dough back into a slipper shape with floured hands. Slide the dough (and the parchment if using) onto the baking stone and pour about 1 cup of hot water into the steam pan. Lower the temperature to 400\u00b0F (204\u00b0C). Bake for 20 minutes, then rotate (if using parchment, slip it out from under the dough and discard). Bake for about 10 to 15 minutes longer for 2 small loaves or 20 to 30 minutes longer for a large loaf, until the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n6 Transfer to a wire rack and let cool for at least 30 minutes before slicing or serving.\n\n## Whole-Milled Whole Wheat Pizzas\n\nTo make individual pizzas from this dough, divide it into 3 or 4 pieces after the final stretch and fold. Gently shape each piece into a ball, as if shaping a boule. Put the balls on a floured sheet pan, mist with vegetable spray oil, and cover loosely with plastic wrap or a clean towel. Refrigerate for at least 1 hour before making the pizzas. Because the dough is so wet, it will be easier to shape when cold than when warm. But if the dough is very elastic and hard to stretch to the desired diameter of 10 to 12 inches, let it rest at room temperature for about 45 minutes so it can relax enough to stretch out. See Shaping Pizza Dough for guidance on pizza shaping, topping, and baking.\n\nWhole Wheat Currant Pretzels\n\n# WHOLE WHEAT CURRANT PRETZELS\n\nMAKES 10 LARGE PRETZELS\n\nThis is my small-batch variation of Craig Ponsford's unique 100% whole wheat pretzel recipe, made with whole-milled flour. A tricky part of this recipe is using a lye dipping solution to create the distinctive dark brown sheen associated with soft pretzels, so the method also includes instructions for using a baking soda solution. Baking soda won't create the same intensity of sheen and may even leave white streaks on the dough, so if you use this method, you should also brush the tops of the pretzels with an egg white wash. Also, pretzels can lose their shape when handling and dipping. To deal with this, the method below includes a freezer trick I learned from Harry Peemoeller. This dough can also be used to make pretzel rolls, bread sticks, and any number of other shapes, and you can also make it without the soaked currants or with raisins, if you prefer.\n\n### BIGA (DAY 1)\n\nwhole-milled wheat flour: 1\u00bd cups plus 2 teaspoons (7 oz \/ 198 g)\n\nsalt: pinch (0.02 oz \/ 0.5 g)\n\ninstant yeast: pinch (0.007 oz \/ 0.2 g)\n\nwater, at room temperature: \u00bd cup plus 2 tablespoons (5 oz \/ 142 g)\n\nTOTAL: 12.027 oz \/ 340.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole-milled wheat flour| 100  \nsalt| 0.3  \ninstant yeast| 0.1  \nwater| 71  \nTOTAL| 171.4\n\nIn a medium bowl, stir together the flour, salt, and yeast. Add the water and stir until the flour is hydrated and a coarse ball of dough forms, about 1 minute. Transfer to a lightly floured work surface and knead by hand until slightly smoother and firm, about 1 minute. Mist a clean bowl with vegetable spray oil, put the biga in the bowl, and cover the bowl with plastic wrap. Let the biga rest at room temperature for at least 12 hours or overnight, until it increases in size by 1\u00bd to 2 times. If it approaches this size before you're ready to make the final dough, put it in the refrigerator.\n\n### CURRANTS (DAY 1)\n\ncurrants (or raisins): (about 1 cup oz \/ 6 g)\n\nwater, at room temperature: (\u00be cup oz \/ 6 g)\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \ncurrants| 170  \nwater| 170\n\nPut the currants in a small bowl and pour in the water. Cover the bowl with plastic wrap and leave out at room temperature for at least 12 hours or overnight.\n\n### FINAL DOUGH (DAY 2)\n\nNOTE: You can also use low-fat milk or nondairy milk instead of milk powder. To do so, replace \u00be cup (6 oz \/ 170 g) of the water with an equal amount of milk.\n\nwhole-milled wheat flour: 3\u00be cups (17 oz \/ 482 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: 1 teaspoon (0.11 oz \/ 3 g)\n\nnonfat dry milk powder (see note): 4 tablespoons (0.75 oz \/ 21.5 g)\n\nbrown sugar (or honey or agave nectar): 2 tablespoons (or 1\u00bd tablespoons) (1 oz \/ 28.5 g)\n\nwater, at room temperature, or 95\u00b0F (35\u00b0C) if biga was refrigerated: 1\u00bd cups plus 2 tablespoons (13 oz \/ 369 g)\n\nsoaked currants (or raisins): all (12 oz \/ 340 g)\n\nBiga: all (12.027 oz \/ 340.7 g)\n\nunsalted butter, at room temperature: 1\u00bd tablespoons (0.75 oz \/ 21.5 g)\n\negg white wash (optional): 1 egg white whisked with 1 tablespoon water\n\npretzel salt for topping: about 1 tablespoon or to taste\n\nTOTAL: 57.137 oz \/ 1,620.2 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nwhole-milled wheat flour| 100  \n---|---  \nsalt| 2.9  \ninstant yeast| 0.6  \nnonfat dry milk powder| 4.5  \nbrown sugar| 5.9  \nwater| 77  \nsoaked currants| 71  \nBiga| 71  \nunsalted butter| 4.5  \nTOTAL| 337.4\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, yeast, and milk powder (on low speed if using a stand mixer). Stir in the sugar (if using honey or agave nectar, add it with the water in the next step). Add the water and currants, including their soaking water. Mix or stir for 30 to 60 seconds, until the flour is hydrated. Switch to the dough hook. Break up the biga into about 8 pieces and add it to the bowl. Mix on medium-low speed or turn out onto a floured work surface and knead for 3 minutes. Add the softened butter and mix on medium speed or knead until the dough is somewhat firm yet supple and tacky to the touch, about 1 minute. The currants will be broken into bits and dispersed throughout the dough.\n\n2 Mist a large bowl with vegetable spray oil, put the dough in the bowl, and cover the bowl with plastic wrap. Ferment the dough at room temperature for about 1 hour.\n\n3 Spread about 1 teaspoon of vegetable oil on the work surface and transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. The dough will strengthen and firm up somewhat. Return the dough to the bowl and cover the bowl with plastic wrap, or leave it on the work surface and cover it with the bowl. Ferment at room temperature for 1 to 1\u00bd hours, until the dough increases in size by about 1\u00bd times.\n\n4 Spread about \u00bd teaspoon of vegetable oil on the work surface and transfer the dough to the oiled area. Divide the dough into 10 equal pieces, each about 5.5 ounces (156 g). (Or make a greater quantity of smaller pretzels if you prefer.) Flatten each piece and then roll it into a tight cigar-shaped cylinder 4 to 6 inches long. Cover the dough with plastic wrap or a clean towel and let it rest for 20 minutes.\n\n5 Line two sheet pans with parchment paper or silicone mats and mist with vegetable spray oil. Use your hands to roll each piece of dough into a tapered strand 24 to 28 inches long (shorter if making smaller pretzels). If a piece resists or shrinks back, move on to the next one and return to it later. Shape each piece into a pretzel shape, then lay it on one of the prepared pans, 5 per pan. When you fill each pan, mist the top of the dough with vegetable spray oil, cover the pan with plastic wrap, and put it into the refrigerator or freezer for 1 hour to firm the dough and make the pretzels easier to handle. (The shaped pretzels can be stored in the refrigerator for up to 36 hours or in the freezer for up to 3 weeks; for long-term freezer storage, once the pretzels are frozen, transfer them to zip-top freezer bags.)\n\n6 About 1 hour before you plan to bake, remove the pretzels from the refrigerator or freezer. Bring about 2 cups (16 oz \/ 454 g) of water to a boil, then pour it into a stainless steel or glass bowl or pan. Add 1\u00bd tablespoons (0.65 oz \/ 18 g) of lye or \u00bc cup (2.5 oz \/ 71 g) of baking soda and whisk gently to dissolve. If using lye, be careful not to splash any on the counter or your skin, and please refer to the sidebar for tips on working with lye. Let cool.\n\n7 Position two racks in the oven an equal distance apart and preheat the oven to 400\u00b0F (204\u00b0C). While the pretzels are still cold, remove them from the sheet pans and generously mist the pan liners with vegetable spray oil; or just line two fresh sheet pans with parchment paper or silicone mats and mist generously with vegetable spray oil. (Once dipped, in the next step, the dough will stick to parchment paper that's only lightly misted.)\n\n8 There are two options for coating the pretzels in the lye or baking soda. The standard approach is to wear latex or other food handling gloves, and dip one pretzel at a time into the lye or baking soda solution, submerging it for 4 to 5 seconds; if the solution isn't deep enough to cover the pretzel, flip the pretzel over. Lift the pretzel out and let the solution drip back into the bowl, and then place the pretzel on one of the prepared pans; the pretzels shouldn't touch, but they won't expand much, so they can be fairly close. A new approach I learned from Craig is to use a spray bottle to mist the prepared lye solution over the tops of the pretzels as they rest on the sheet pan.\n\n9 Sprinkle the pretzel salt over the pretzels.\n\n10 Bake for 7 minutes, then rotate the pans front to back and between racks, and bake for 5 to 15 minutes longer, until the crust has a rich, nutlike brown color and is hard to the touch.\n\n11 Transfer them to a wire rack immediately. The pretzels will soften slightly and be ready to eat in 5 to 10 minutes.\n\n### Variations\n\nMULTIGRAIN PRETZELS: Replace up to 30% of the whole-milled flour, by weight, with other whole grain flours, such as barley, buckwheat, or rye (nonsprouted or sprouted). Depending on the flours you use, you may need to adjust the amount of water to achieve the textures indicated in the method.\n\nSWEET PRETZELS: Forgo the pretzel salt and instead sprinkle coarse sugar or cinnamon sugar over the pretzels before baking.\n\nCHEESE PRETZELS: When 5 to 7 minutes of baking time remains, sprinkle 2 tablespoons (0.5 oz \/ 14 g) of grated semi-moist cheese, such as Cheddar, Monterey Jack, Swiss, or mozzarella, to taste, over each pretzel. If using a hard, aged cheese, such as Parmesan, sprinkle it over the pretzels when 2 minutes of baking time remains.\n\n## Working with Lye\n\nLye (sodium hydroxide) is the secret ingredient that gives pretzels their unique color and taste. Bakers use a food quality lye crystal (or pellets), which is not the same as the technical grade version used in drain cleaners. It is caustic, so caution should be taken when using it. I always advise my students to wear disposable latex or food handling gloves when dipping their pretzels in lye solution to protect their hands, and then immediately discard the gloves.\n\nThe typical ratio of lye to water is between 3% to 4%. I prefer 4%, which is about 2\u00bd tablespoons (1.25 ounces or 35 g) dissolved in 1 quart of warm or hot water. Always add the lye to the water rather than pouring the water over the lye. Use a stainless steel bowl or pot, not aluminum, which can cause a reaction. Lye is highly alkaline (about a 14 on the pH scale), much more so than baking soda, which is only 8 to 9 on the scale. If you are nervous about using lye, you can dissolve baking soda instead (use about 22% soda to water, or 14 tablespoons per quart), but it won't create the deep nut brown color of lye and can sometimes leave whitish streaks on the dough. When using a baking soda solution, I also immediately brush the pretzels with an egg white wash to get rid of the streaks. The noted food science writer Harold McGee suggests \"baking\" the baking soda on a dry sheet pan for 1 hour at approximately 275\u00b0F (135\u00b0C) to chemically change it through moisture evaporation from sodium bicarbonate into sodium carbonate. This increases the pH and causes it to perform more like lye (you can access his New York Times article for more on the subject). As a precaution, you might also want to consider wearing goggles to protect your eyes from any splashes whenever working with lye or baking soda solutions.\n\nThere is divided opinion as to whether the dipping solution should be hot or cold. I find that the lye dissolves faster in hot or warm water, but once it's dissolved, warm or cold solutions work equally well. Work on a surface that is protected from the bleaching effects of the lye, such as ceramic tile, formica, plastic cutting boards (not wood), or stainless steel. I always line my baking pans with baking parchment and then generously mist the parchment with vegetable spray oil, and then dust the surface with either corn meal or semolina flour to prevent the pretzels sticking to the paper. Craig Ponsford, on the other hand, bakes his pretzels directly on the metal pans without any problems. While I prefer the total immersion method, dipping my pretzels into the bowl of lye solution, Craig uses a spray bottle and mists them after he pans them. He told me that the lye doesn't affect the plastic spray bottle or his sheet pans (they actually look cleaner after they come out of the oven than when they went in), but unless you have a stainless steel pan I'd go with baking parchment. The lye is completely cooked off during baking, leaving behind a shiny, rich brown crust and distinct flavor.\n\nOne final note: You can reuse lye solution over and over if you have a safe means of storing it, but I prefer to discard it and start fresh each time. You can safely pour it down your sink drain (lye is, after all, used as a drain cleaner) or down the toilet, but be careful to avoid splashing when doing so. You can easily find sources for food grade lye on the Internet, or check the Resources section for links.\n\n# WHOLE WHEAT AND RAISIN ENGLISH MUFFINS\n\nMAKES 6 TO 8 ENGLISH MUFFINS\n\nThis is a modified version of Craig Ponsford's hearty English muffins, which he typically makes with raisins and walnuts. The recipe calls for a small amount of applesauce in addition to soaked raisins. As the notes indicate, a bit of sweetener may be substituted for these two ingredients, but both the applesauce and the raisins help keep the English muffins soft and moist, and also add flavor. Using these enhancements is an excellent trick to incorporate into your own whole grain baking repertoire; it works well in many other bread recipes. The use of pre-ferments, in this case a biga, is one of Craig's signature techniques for capitalizing on the flavor potential in his whole grain doughs. As a bonus, it makes it possible to bake these English muffins on the same day as you mix the final dough.\n\nEnglish muffins are first cooked on a griddle or skillet before finishing in the oven. If you have a large griddle, you can cook and bake them all in one batch. If you only have a smaller griddle or skillet, make them in small batches, as it's important to get them into the oven as soon as they come out of the pan. After that, all you'll need is a toaster, butter, and jam!\n\n### BIGA (DAY 1)\n\nwhole-milled wheat flour: \u00bd cup plus 3\u00bd tablespoons (3.25 oz \/ 92 g)\n\ninstant yeast: pinch (0.007 oz \/ 0.2 g)\n\nsalt: pinch (0.02 oz \/ 0.5 g)\n\nwater, at room temperature: 5 tablespoons (2.5 oz \/ 71 g)\n\nTOTAL: 5.777 oz \/ 163.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole-milled wheat flour| 100  \ninstant yeast| 0.2  \nsalt| 0.5  \nwater| 76  \nTOTAL| 176.7\n\nIn a small bowl, stir together the flour, yeast, and salt. Add the water and stir until the flour is hydrated and a coarse ball of dough forms. Transfer to a lightly oiled work surface and knead by hand until slightly smoother and firm, about 1 minute. Mist a medium bowl with vegetable spray oil and put the biga in the bowl. Mist the top with vegetable spray oil and cover the bowl with plastic wrap. Let the biga rest at room temperature for at least 12 hours or overnight, until double in size.\n\n### SOAKED RAISINS (DAY 1)\n\nraisins: (about 7 tablespoons oz \/ 2.5 g)\n\nwater, at room temperature: (\u2153 cup oz \/ 2.7 g)\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nraisins| 71  \nwater| 77\n\nPut the raisins in a small bowl and pour in the water. Cover the bowl with plastic wrap and leave out at room temperature for at least 6 hours (or overnight).\n\n### FINAL DOUGH (DAY 2)\n\nNOTES:\n\n\u2022 You can also use low- fat milk or nondairy milk instead of milk powder. To do so, replace \u00bd cup plus 2 tablespoons (5 oz \/ 142 g) of the water with an equal amount of milk.\n\n\u2022 You can replace the applesauce and raisins with 1 tablespoon (0.5 oz \/ 14 g) sugar or 2\u00bc teaspoons (0.67 oz \/ 19 g) honey. You may need to add a bit more water to the final dough.\n\nBiga: all (5.777 oz \/ 163.7 g)\n\nwhole-milled wheat flour: 2\u00bc cups (10 oz \/ 283 g)\n\nnonfat dry milk powder (see notes): 3\u00bd tablespoons (0.65 oz \/ 18 g)\n\nsalt: 1 teaspoon (0.25 oz \/ 7 g)\n\ninstant yeast: 1 teaspoon (0.11 oz \/ 3 g)\n\nSoaked Raisins (see notes): all (5.2 oz \/ 148 g)\n\nwater, at room temperature: 1 cup plus 3 tablespoons (9.5 oz \/ 269 g)\n\napplesauce (see notes): 1 tablespoon plus 1 teaspoon (0.65 oz \/ 18 g)\n\nunsalted butter, at room temperature: 1 tablespoon (0.5 oz \/ 14 g \/ 5%)\n\ncornmeal for dipping the muffins before cooking them: about \u00bd cup\n\nTOTAL: 32.637 oz \/ 923.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nBiga| 58  \n---|---  \nwhole-milled wheat flour| 100  \nnonfat dry milk powder| 6.4  \nsalt| 2.5  \ninstant yeast| 1.1  \nSoaked Raisins| 52  \nwater| 95  \napplesauce| 6.4  \nunsalted butter| 5  \nTOTAL| 326.4\n\n1 Cut the biga into about 8 pieces and put it in the bowl of a stand mixer fitted with the dough hook, or in a large bowl. Add the flour, milk powder, salt, and yeast and mix or stir briefly to combine (on low speed if using a stand mixer). Strain the raisin soaking water and add it to the bowl, along with the water and applesauce. Mix or stir for about 2 to 3 minutes. Add the butter mix or stir for 2 to 3 minutes, on medium-low speed if using a stand mixer. Add the raisins and mix or stir until evenly distributed. The dough will be wet and sticky.\n\n2 Using an oiled bowl scraper or rubber spatula, transfer the dough to an oiled bowl and cover the bowl with plastic wrap. Ferment at room temperature for 40 minutes, until the dough just begins to expand. Lightly oil your hands, then stretch and fold the dough in the bowl, folding it over itself four times: once each from the top, bottom, and sides. Cover and let rest for 40 minutes. Stretch and fold again, then cover the bowl and let the dough rest for 20 minutes. The dough will become tighter and firmer with each stretch and fold, but it will remain somewhat sticky.\n\n3 Dust a work surface generously with whole wheat flour and use an oiled bowl scraper or rubber spatula to transfer the dough to the floured area. Scatter a bit of flour over the dough for easier handling. Use a metal pastry blade or plastic bowl scraper to divide the dough into 6 to 8 pieces (3.5 to 4.5 oz \/ 99 to 128 g each). Toss the pieces in the flour to coat them. Gently form each piece into a ball, then put it on a floured spot on the work surface, spacing the pieces at least 2 inches apart. Mist the tops with vegetable spray oil and cover loosely with plastic wrap or a clean towel. Proof at room temperature for 1 hour. The dough will swell but won't double in size.\n\n4 Preheat the oven to 350\u00b0F (177\u00b0C). Sprinkle the cornmeal on the work surface or put it in a small bowl. Line a sheet pan with parchment or a silicone mat and mist with vegetable spray oil. Heat a griddle or large skillet over medium-low heat.\n\n5 Dust your hands with flour. Dip and press one piece of dough in the cornmeal, turning to coat, and flatten both sides. When you've dipped enough pieces to fill your griddle, lay and press them in the hot pan about 1 inch apart and cook until a rich golden brown on the bottom, about 5 minutes. Flip and cook the other side until golden brown, about 5 minutes. Transfer to the prepared pan and bake for 8 to 10 minutes, until firm and springy to the touch. If you couldn't fit all of the muffins on the griddle in the first round, griddle the remaining muffins while the first set is baking.\n\n### Variations\n\nFLAVORED ENGLISH MUFFINS: Omit the raisins and replace 2 tablespoons (1 oz \/ 28.5 g) of the water with an equal amount of amaretto, orange liqueur or other liqueur, rum, brandy, or a sweet Italian dessert wine such as vin santo or Moscato.\n\nRAISIN NUT ENGLISH MUFFINS: Add up to 1 cup (about 4 oz \/ 113 g) coarsely chopped toasted walnuts or pecans when you add the raisins (see Toasting Seeds and Nuts for toasting instructions).\n\nHigh-Extraction Pain Au Levain\n\n# HIGH-EXTRACTION PAIN AU LEVAIN\n\nMAKES 1 LARGE LOAF OR 2 SMALL LOAVES\n\nThis version of pain au levain, a French type of sourdough, was developed by my colleague Harry Peemoeller. Whereas classic pain au levain is usually made with just a small amount of whole grain flour, here Harry pushes the bar toward a higher-extraction flour: type 85 (meaning that 85% of the total wheat is included in the final flour, putting it somewhere between whole wheat and white flour in terms of bran and germ content). Harry, who competes in international bread competitions, works closely with Jennifer Lapidus of the milling company Carolina Ground to, as he puts it, \"Test drive the flour to see what it can do.\" You may not be able to get the exact same flour where you live, but you might be able to find a miller in your region who's producing something similar.\n\nThe flour used to develop this recipe was made from a heritage strain of wheat known as Turkey Red, and various millers sometimes offer such flour via the Internet (see Resources). If you can't get it, blend one-third whole wheat flour with two-thirds unbleached bread flour, using the best-quality flour you can get your hands on. You can push the ratio of whole wheat flour to 50% if you prefer.\n\nInstant yeast is optional in this recipe; including it will decrease the rising times and result in a bread that's less tangy in flavor. Also, Harry's levain is closer to a sponge-style poolish than the type of firm starter that most sourdough breads in this book use. The method here replicates the texture of his starter by building a whole wheat levain at a ratio of 1 part mother starter to 4.5 parts new flour and water. In theory, this bread could also be made with a firmer levain. Once you've mastered this recipe, feel free to use whatever type of starter you like, adjusting the final dough hydration as needed.\n\n### WHOLE WHEAT LEVAIN (DAY 1)\n\nwhole wheat flour (preferably from Turkey Red wheat; see above): 1 cup (4.5 oz \/ 128 g)\n\nMother Starter: 2 tablespoons (1 oz \/ 28.5 g)\n\nwater: \u00bd cup plus 1 tablespoon (4.5 oz \/ 128 g)\n\nTOTAL: 10 oz \/ 284.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwhole wheat flour| 100  \nMother Starter| 22  \nwater| 100  \nTOTAL| 222\n\nIn a small bowl, stir together all the ingredients for about 1 minute to evenly distribute the starter and make a smooth, sticky, batter-like dough. (The levain will thicken as it sits.) Scrape the bowl down with a wet spatula and cover with plastic wrap or a lid. Let the levain rest at room temperature for about 12 hours or overnight, until it begins to swell and bubble.\n\n### FINAL DOUGH (DAY 2)\n\ntype 85 flour (preferably from Turkey Red wheat): 2\u2154 cups (12 oz \/ 340 g)\n\nwhole wheat flour (preferably from Turkey Red wheat): 3\u00bd tablespoons (1 oz \/ 28.5 g \/ 8%)\n\nwater, at room temperature: 1\u00bd cups (12 oz \/ 340 g)\n\nWhole Wheat Levain: all (10 oz \/ 284.5 g)\n\nsalt: 1\u00bd teaspoons (0.35 oz \/ 10 g)\n\ninstant yeast (optional): \u00bd teaspoon (0.05 oz \/ 1.5 g)\n\nTOTAL: 35.4 oz \/ 1,004.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \ntype 85 flour| 92  \nwhole wheat flour| 8  \nwater| 92  \nWhole Wheat Levain| 77  \nsalt| 2.7  \ninstant yeast| 0.4  \nTOTAL| 272.1\n\n1 In the bowl of a stand mixer with the paddle attachment, or in a large bowl, stir together the flour, water, and levain until the flour is hydrated, the levain is evenly distributed, and a coarse, very wet dough forms, about 1 minute. Cover the bowl with plastic wrap and let the dough rest at room temperature for 30 minutes.\n\n2 Add the salt and yeast (if using). If using a stand mixer, switch to the dough hook. Mix (on medium-low speed if using a stand mixer), stir, or knead by hand for 2 to 3 minutes, until a smooth, sticky dough forms.\n\n3 Mist a large bowl with vegetable spray oil. Spread 1 teaspoon of oil on a work surface. Using a wet or oiled bowl scraper, transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Form the dough into a ball. Put it in the oiled bowl and cover the bowl with plastic wrap, or leave the dough on the work surface and cover it with the bowl. At intervals of 40 minutes, perform two additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up a bit after each stretch and fold, but after the final fold it will still be very soft and somewhat sticky.\n\n4 Put the dough back in the bowl and cover the bowl with plastic wrap. Ferment the dough at room temperature until it increases in size by at least 1\u00bd times, 3 or 4 hours, or just 60 to 90 minutes if using the optional yeast.\n\n5 Oil the work surface once again or dust it with flour. Transfer the dough to the work surface. For two smaller loaves, divide the dough in half and form each piece into a ball; otherwise leave it whole and form it into a ball. Mist the top with vegetable spray oil, cover loosely with plastic wrap or a clean towel, and let rest at room temperature for 10 to 20 minutes to relax the gluten.\n\n6 Prepare one or two bannetons or a couche, or line an 18 by 13-inch sheet pan with parchment paper and then mist it with vegetable spray oil. Shape the dough into one large boule or b\u00e2tard, or divide the dough in half and shape each piece into a small boule or b\u00e2tard. Put the shaped loaves in the prepared proofing vessel(s) or on the sheet pan.\n\n7 Loosely cover the dough with plastic wrap or a clean towel and proof at room temperature until it increases in size by 1\u00bd times, 2 to 4 hours or just 1 to 1\u00bd hours if using the optional yeast.\n\n8 About 45 minutes before you bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 500\u00b0F (260\u00b0C).\n\n9 Just before baking, dust a peel with flour. Transfer the dough to the peel and score as desired. Slide the dough onto the baking stone and pour about 1 cup of hot water into the steam pan. Lower the temperature to 470\u00b0F (243\u00b0C). Bake for 15 minutes, then rotate and bake for 15 to 25 minutes longer, until the crust is rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n10 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\nNaturally Leavened Carolina Wheat Hearth Bread\n\n# NATURALLY LEAVENED CAROLINA WHEAT HEARTH BREAD\n\nMAKES 1 LARGE LOAF OR 2 SMALL LOAVES\n\nThis bread was created by Harry Peemoeller, winner of numerous gold medals in international baking competitions. He's been pioneering new ways to use flours made from heirloom and local varieties of wheat, and developed this recipe to showcase heritage Red Fife wheat flour from Anson Mills, of Columbia, South Carolina, developers of regional heritage grains and flours. Of course, it can also be made with other whole wheat flours, so you can make it in the spirit intended and support your own local or regional mills (or, as increasing numbers of people are doing, mill your own flour). Red Fife is a high-protein wheat, so look for something similar grown closest to home, or order Red Fife bread flour from Anson Mills (see Resources). The amount of hydration needed may vary slightly depending on the variety of wheat. This version is leavened primarily by a medium-firm levain, with only a little boost from a spike of commercial instant yeast. The resulting loaf is a seriously delicious bread with a tangy complexity created by the natural leaven and a touch of honey.\n\n### LEVAIN (DAY 1)\n\nMother Starter: 1 tablespoon (0.4 oz \/ 11.5 g)\n\nwater: \u00bc cup plus \u00bd teaspoon (2.1 oz \/ 60 g)\n\nwhole wheat flour (preferably from Red Fife wheat): \u00bd cup plus 1 tablespoon (2.5 oz \/ 70 g)\n\nTOTAL: 5 oz \/ 141.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 16  \nwater| 86  \nwhole wheat flour| 100  \nTOTAL| 202\n\nPut the starter in a small bowl and add the water to soften it. Break the starter up into smaller pieces, then add the flour and stir until a smooth, sticky dough forms, about 1 minute. Use a rubber spatula to scrape down the sides of the bowl. Cover the bowl with plastic wrap and let the levain rest at room temperature for at least 12 hours or overnight, until it becomes bubbly and swells in size.\n\n### FINAL DOUGH (DAY 2)\n\nLevain: all (5 oz \/ 141.5 g)\n\nwater: 1\u00bd cups plus 1 tablespoon (12.5 oz \/ 354 g)\n\nhoney or agave nectar: 1\u00bd tablespoons (1 oz \/ 28.5 g)\n\nwhole wheat flour (preferably from Red Fife wheat): 3\u00bd cups plus 1 tablespoon (16.25 oz \/ 461 g)\n\nsalt: 1\u215d teaspoons (0.4 oz \/ 11.5 g)\n\ninstant yeast: \u00bc teaspoon (0.03 oz \/ 1 g)\n\nTOTAL: 35.18 oz \/ 997.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nLevain| 31  \nwater| 77  \nhoney or agave nectar| 6.1  \nwhole wheat flour| 100  \nsalt| 2.5  \ninstant yeast| 0.2  \nTOTAL| 216.8\n\n1 Break the levain into 4 or 5 pieces and put it in the bowl of a stand mixer fitted with the dough hook or in a large bowl. Add the water and honey and stir (on low speed if using a mixer) for a few seconds to soften and break up the levain. Add the flour, salt, and yeast and mix or stir for 4 minutes. Then, if using a mixer, increase the speed to medium-low, or transfer the dough to a lightly oiled work surface and knead by hand, for 2 to 3 minutes, until a smooth, slightly sticky dough forms.\n\n2 Oil a work surface and transfer the dough to the oiled area. Also oil a large bowl. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Form the dough into a ball. Put it in the oiled bowl and cover the bowl with plastic wrap, or leave the dough on the work surface and cover it with the bowl. At intervals of 40 minutes, perform two additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will feel stronger and less sticky after each stretch and fold, and after the final fold, it should be tacky rather than sticky.\n\n3 Put the dough back in the bowl, cover the bowl with plastic wrap, and ferment the dough at room temperature for 1 to 1\u00bd hours, until it nearly doubles in size.\n\n4 Oil the work surface once again and transfer the dough to the oiled area. For two smaller loaves, divide the dough in half and form each piece into a ball; otherwise leave it whole and form it into a ball. Mist the top with vegetable spray oil, cover loosely with plastic wrap, and let rest at room temperature for 10 to 20 minutes to relax the gluten.\n\n5 Prepare one or two bannetons or a couche, or line an 18 by 13-inch sheet pan with parchment paper and then mist it with vegetable spray oil. Shape the dough into one or two boules or b\u00e2tards. Put the shaped loaves in the prepared proofing vessel(s) or on the sheet pan.\n\n6 Loosely cover with plastic wrap or a clean towel and proof at room temperature for 1\u00bd to 2 hours, until the dough increases in size by 1\u00bd times. (Alternatively, put the dough in the refrigerator after the first 30 minutes of proofing and bake it the next day.)\n\n7 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 475\u00b0F (246\u00b0C).\n\n8 Just before baking, dust a peel with flour. Transfer the dough to the peel and score as desired. Slide the dough onto the baking stone and pour about 1 cup of hot water into the steam pan. Bake for 15 minutes, then rotate and bake for 15 to 25 minutes longer, until the crust is rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n9 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n### Variations\n\nSEEDED CAROLINA WHEAT HEARTH BREAD: Add up to 2 cups (10 oz \/ 283 g) toasted sunflower seeds or pumpkin seeds to the final dough. (See Toasting Seeds and Nuts for toasting instructions.)\n\nCAROLINA WHEAT HEARTH BREAD WITH NUTS: Add up to 2 cups (8 oz \/ 227 g) coarsely chopped toasted nuts. (See Toasting Seeds and Nuts for toasting instructions.)\n\n## Gluten-Free Breads\n\nNot that long ago, gluten-free baking was considered to be a fairly arcane corner of the bread world. Then, in just a few short years, it became one of the most discussed and debated food issues. Dozens of new books on gluten-free baking have been published, usually with an emphasis on replacing wheat, rye, and barley\u2014the primary sources of gluten\u2014with flours made from rice, potatoes, cassava, quinoa, sorghum, buckwheat, and various beans. This former side street has become a vast new frontier populated with countless products, recipes, and spokespersons.\n\nIn 2012 even I chimed in, teaming with Denene Wallace to coauthor The Joy of Gluten-Free, Sugar-Free Baking, with a totally new take on gluten-free baking in which we offered products that were safe not only for people with gluten sensitivities (perhaps as much as 5% of the population), but also for people with diabetes and prediabetes (about 30% to 35% of the population). Denene, who has type 2 diabetes, completely weaned herself off of insulin by revamping her diet, and her approach to baked goods is reflected in that book's recipes. Basically, all starch flours are replaced with nut and seed flours, and all sugars are replaced with stevia and other alternatives.\n\nAs I've often said, when it comes to baking, flavor rules over everything else. One of the benefits of the recent interest in these lesser known grains and ingredients is that it allows us to expand our flavor horizons, escaping the confines of the monoculture farming that dominates our era. We are freeing ourselves from the monotony of reliance on just a few types of plants that are easily mass produced. From my perspective, any interest in the preservation and cultivation of plants that are less common gives us hope for diversity, along with more well-rounded nutrition. But, and here's the reality check, interest in gluten-free baking will thrive only if gluten-free products taste good.\n\nFortunately, we are now learning how delicious gluten-free grains can be, and we've figured out ways to make them function similarly to flours that contain gluten.\n\nThere is great debate regarding the causes of increasing sensitivity to gluten. One argument is that alterations in the genetics and protein structure of modern hybrid wheat, especially the prolific short straw dwarf variety promoted by Norman Borlaug and his many Green Revolution disciples over the past fifty years, have made mass-produced wheat less digestible. (Though it must be said that those innovations are also credited with saving countless people from death by starvation.) Other arguments focus on wheat processing, as mentioned in chapter 5, on whole-milled grains.\n\nThe jury is still out on these and other hypotheses, and it will be interesting to see what new research into this topic reveals. In the meanwhile, the good news is that the ongoing debate stimulates more diversity of thinking, more attention to our health, more variety in our diets, and more awareness of the chain of events that lead from the earth to the table, from farm to fork, or from seed to slice. Anything that makes us more aware of the bigger picture is a good thing.\n\nEven if you aren't sensitive to gluten, I encourage you to try the following recipes. They will broaden your palate and can contribute to more well-rounded nutrition. They also break the mold in regard to baking, so they're inherently revolutionary. Plus, they all fulfill the fundamental flavor rule: that flavor rules!\n\n# GLUTEN-FREE MANY-SEED TOASTING BREAD\n\nMAKES 1 LARGE LOAF OR 4 TO 6 MINI LOAVES\n\nThis quick bread recipe, leavened with baking powder rather than yeast, goes to show that there are many ways to make a great loaf of bread. Based on the approach Denene Wallace pioneered in our collaboration, The Joy of Gluten-Free, Sugar-Free Baking, it's high in fiber yet low in carbs. I like making this recipe into mini loaves, which bake up much faster than a full-size loaf. The mini loaves can be stored in the freezer, and they thaw fairly quickly at room temperature, in just 30 minutes or so. I love the variety of flavors and textures the nut and seed meals provide, but feel free to substitute other nut or seed flours for those listed, as long as the overall weight remains the same.\n\n### DOUGH\n\nflaxseed meal: 1 cup (4 oz \/ 113 g)\n\npecan flour: 1 cup (4 oz \/ 113 g)\n\nalmond flour: 1 cup (4 oz \/ 113 g)\n\nground sesame seeds: \u00bd cup (2 oz \/ 56.5 g)\n\nground chia or hemp seeds: \u00bd cup (2 oz \/ 56.5 g)\n\nsesame seeds: 3 tablespoons (1 oz \/ 28.5 g)\n\nflaxseeds (optional): 3 tablespoons (1 oz \/ 28.5 g)\n\nsalt: \u00bd teaspoon (0.12 oz \/ 3.5 g)\n\nbaking powder: 2 tablespoons (1 oz \/ 28.5 g)\n\nxanthan gum: 1\u00bd teaspoons (0.38 oz \/ 11 g)\n\nsugar or liquid stevia (optional; see Replacing Sugar with Stevia): \u00bc cup or \u00bc teaspoon (2 or 0.05 oz \/ 56.5 or 1.5 g \/ 12.5 or 0.3%)\n\nmilk (any type): 1\u00bd cups (12 oz \/ 340 g)\n\negg whites or liquid egg whites: 9 or 1\u00bd cups (11.25 oz \/ 319 g)\n\nTOTAL: 44.75 or 42.8 oz \/ 1,267.5 or 1,212.5 g \/ 280.6 or 268.4%\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nflaxseed meal| 25  \npecan flour| 25  \nalmond flour| 25  \nground sesame seeds| 12.5  \nground chia or hemp seeds| 12.5  \nsesame seeds| 6.3  \nflaxseeds| 6.3  \nsalt| 0.8  \nbaking powder| 6.3  \nxanthan gum| 2.4  \nsugar or liquid stevia| 12.5 or 0.3  \nmilk| 75  \negg whites or liquid egg whites| 71  \nTOTAL| 280.6 or 268.4\n\n1 Position a rack in the middle of the oven and preheat the oven to 375\u00b0F (191\u00b0C). Line the bottom of a 4\u00bd by 8-inch loaf pan with parchment paper and generously mist the pan with vegetable spray oil; if making mini loaves, forgo the parchment and simply mist the pans generously with vegetable spray oil.\n\n2 In a large bowl, stir together the flaxseed meal, pecan flour, almond flour, ground sesame seeds, ground chia seeds, sesame seeds, flaxseeds, salt, baking powder, xanthan gum, and sugar. (If using stevia, add it to the milk in the next step.) In a separate bowl, whisk together the milk and egg whites, then pour into the flour mixture. Stir vigorously for about 2 minutes to make a thick, slightly aerated batter. Pour the batter into the prepared pan; for mini pans, fill them to about \u00bd inch from the top.\n\n3 To bake a large loaf, put the bread in the oven and lower the temperature to 350\u00b0F (177\u00b0C). Bake for 45 minutes, then rotate and bake for 35 to 45 minutes longer, until the top is golden brown and springy when pressed in the center. To bake mini loaves, don't lower the oven temperature. Bake for 30 minutes, then rotate and bake for 25 to 30 minutes longer, until the top is golden brown and springy when pressed in the center.\n\n4 Put the pan on a wire rack and let the bread cool for at least 20 minutes. Run an icing spatula or something similar around the edges to separate the bread from the pan, then transfer the bread to the rack and let cool for at least 45 minutes, until cool to the touch, before slicing and serving. (See the note for storage tips.)\n\nNOTE: Baked goods made with nut and seed flour should always be stored in the refrigerator or freezer. If freezing, you might want to preslice the loaf so you can take slices out of the freezer as you need them. To maintain the freshness, store this bread in a plastic container or zip-top bag lined with a paper towel to absorb any oil that seeps out of the loaf.\n\n## Q & A\n\n### Can sprouted, gluten-free flours made from grains be substituted for the nut and seed flour in these recipes?\n\nYes and no. The premise of many of these recipes is to replace high-carb starch flour with nut and seed flours because they're much lower in carbohydrates and have a more favorable glycemic index. The resulting recipes are safe for both diabetics and those who are sensitive to gluten. Although it appears that sprouted flour may spike blood sugar less than its nonsprouted counterpart, it's still higher in carbs than nut and seed flours. If you have diabetes or prediabetes, I suggest a cautious approach. You can try swapping in sprouted gluten-free flours, but eat only a small amount to begin with, and test your blood sugar levels to see how your body responds.\n\nGluten-Free Holiday Cookies\n\n# GLUTEN-FREE HOLIDAY COOKIES\n\nMAKES 24 COOKIES\n\nI developed these cookies, a variation of the pecan sandies from The Joy of Gluten-Free, Sugar-Free Baking, for a Christmas presentation. They are a symbolic tribute to the birth of Jesus and the gifts of the magi: gold (almond flour and walnuts or pecans); frankincense (cinnamon and allspice); and myrrh (orange oil or extract). Their flavor profile is similar to classic Middle Eastern ma'amoul cookies. And as a bonus, they're gluten-free, making them a healthy anytime gift. Be sure to bake them until hard and golden brown, and then let them cool completely before eating; then they'll shatter and crumble in your mouth like the best flaky pie crust.\n\n### DOUGH\n\nalmond flour: 2\u00bc cups (9 oz \/ 255 g)\n\ncoarsely chopped walnuts or pecans: about \u00be cup (3.25 oz \/ 92 g)\n\nsugar: 1\u00bc cups (10 oz \/ 283 g)\n\nground cinnamon: 1\u00be teaspoons (0.16 oz \/ 4.5 g)\n\nground allspice: \u00bc teaspoon (0.03 oz \/ 1 g)\n\nsalt: \u00bc teaspoon (0.06 oz \/ 1.5 g)\n\nbaking soda: \u00bd teaspoon (0.11 oz \/ 3 g)\n\nunsalted butter, melted: \u00be cup (6 oz \/ 170 g)\n\negg, slightly beaten: 1 (1.75 oz \/ 50 g)\n\nvanilla extract: 1 tablespoon (0.42 oz \/ 12 g)\n\nalmond extract: 2 teaspoons (0.28 oz \/ 8 g)\n\norange extract: 1 teaspoon (0.14 oz \/ 4 g)\n\nTOTAL: 31.2 oz \/ 884 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nalmond flour| 100  \ncoarsely chopped walnuts or pecans| 36  \nsugar| 111  \nground cinnamon| 1.8  \nground allspice| 0.4  \nsalt| 0.6  \nbaking soda| 1.2  \nunsalted butter| 67  \negg| 20  \nvanilla extract| 4.7  \nalmond extract| 3.1  \norange extract| 1.6  \nTOTAL| 347.4\n\n1 Position a rack in the middle of the oven and preheat the oven to 350\u00b0F (177\u00b0C). Line an 18 by 13-inch sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil.\n\n2 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the almond flour, walnuts, sugar, cinnamon, allspice, salt, and baking soda. In a separate bowl, whisk together the butter, egg, vanilla, almond extract, and orange extract, then pour into the flour mixture. Mix or stir, on medium-low speed if using a stand mixer for 1 minute to make a thick batter (it will thicken as it sits).\n\n3 Drop small spoonfuls of dough onto the prepared cookie sheet to make 24 cookies, spacing them evenly (they don't spread so they need be only \u00bd inch apart). With wet fingers, pat down the top of each cookie to make a round patty.\n\n4 Bake for 10 minutes, then rotate and bake for 9 to 10 minutes longer, until the cookies are firm when tapped in the center and a rich golden brown. If the bottoms are getting too dark during baking, slide a second pan under the first for insulation.\n\n5 Transfer the cookies to a wire rack and let cool for at least 10 minutes before serving.\n\n## Replacing Sugar with Stevia\n\nLiquid stevia is an outstanding alternative sweetener because it's all natural and has no long-term side effects. If you'd like to replace the sugar in any of these recipes with liquid stevia, use about 1 teaspoon (0.25 oz \/ 7 g) of liquid stevia per 1 cup (8 oz \/ 227 g) of sugar. But because sugar (and many alternative sweeteners, like Splenda and Stevia in the Raw) also provide bulk and mass to a recipe, you must also add \u00bc cup (about 1 oz \/ 28.5 g) of any nut flour or meal per cup of sugar (or Splenda or Stevia in the Raw) replaced with liquid stevia. If you prefer to use powdered stevia, for most brands you can replace 1 cup (8 oz \/ 227 g) of sugar with \u00bd teaspoon (0.09 oz \/ 2.6 g) of powdered stevia, again, adding \u00bc cup (about 1 oz \/ 28.5 g) of nut flour or meal per cup of sugar replaced.\n\n# GLUTEN-FREE HOLIDAY BISCOTTI\n\nMAKES 20 TO 24 SMALL BISCOTTI\n\nBiscotti, or twice baked biscuits, are beloved because they are crisp, flaky, and have a long shelf life. They're the perfect travel companion and were probably originally developed for use on long journeys. This version is a variation of one originally published in The Joy of Gluten-Free, Sugar-Free Baking, embellished with holiday spices like cinnamon, powdered ginger, and allspice, which I associate with celebrations. This recipe calls for sugar but it can also be made with liquid stevia for a truly low-carb version; see the sidebar on Replacing Sugar with Stevia. (I now use liquid stevia almost always in place of sugar when cooking at home.)\n\n### DOUGH\n\nwalnut flour or pecan flour: 1 cup (4 oz \/ 113 g)\n\nalmond flour: 1 cup (4 oz \/ 113 g)\n\ncoconut flour: \u00bc cup (1 oz \/ 28.5 g)\n\nsugar: 1 cup (8 oz \/ 227 g)\n\nground cinnamon: 2\u00bc teaspoons (0.21 oz \/ 6 g)\n\ndried ginger powder or ground allspice: \u00bd teaspoon (0.05 oz \/ 1.5 g)\n\nsalt: \u00bc teaspoon (0.07 oz \/ 2 g)\n\nbaking powder: 2 teaspoons (0.5 oz \/ 14 g)\n\negg whites: 4 (5 oz \/ 142 g)\n\nvanilla extract: 1 tablespoon (0.42 oz \/ 12 g)\n\nunsalted butter, melted: \u2153 cup (2.7 oz \/ 77 g)\n\ncoarsely chopped or slivered almonds: 1\u00bd cups (7.5 oz \/ 213 g)\n\nChocolate Glaze (optional)\n\nTOTAL: 33.45 oz \/ 949 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwalnut flour or pecan flour| 44.5  \nalmond flour| 44.5  \ncoconut flour| 11  \nsugar| 89  \nground cinnamon| 0.4  \ndried ginger powder or ground allspice| 0.6  \nsalt| 0.8  \nbaking powder| 5.5  \negg whites| 56  \nvanilla extract| 4.7  \nunsalted butter| 30  \ncoarsely chopped or slivered almonds| 84  \nTOTAL| 371\n\n1 Position a rack in the middle of the oven and preheat the oven to 375\u00b0F (191\u00b0C). Line a sheet pan with parchment paper or a silicone mat and mist with vegetable spray oil. Have a cooling rack at hand.\n\n2 In a large bowl, stir together the flours, sugar, cinnamon, ginger, salt, and baking powder. In a separate bowl, whisk together the egg whites and vanilla. Add the butter and whisk just until blended. Pour the egg white mixture into the flour mixture and stir for about 1 minute to make a thick, moist batter similar to cookie dough.\n\n3 Use a rubber spatula to transfer the dough to the center of the prepared pan. Mist the top with vegetable spray oil. Using wet hands or the spatula, press the dough into either one large oval about 6 inches wide and 10 to 12 inches long or two smaller ovals, each about 4 inches wide and 8 inches long; regardless of size, the dough should be about \u00be inch thick, and no thicker.\n\n4 Bake for 12 minutes, then rotate and bake for 10 to 20 minutes longer, until the top is golden brown and the center is firm and springy to the touch.\n\n5 Remove the pan from the oven, leaving the oven on, and let the dough rest for 1 minute to firm up slightly. While the dough is still hot, use a chef's knife or a metal pastry blade to cut the oval in half lengthwise. Then cut crosswise at \u00be-inch intervals to create strips. If you let the dough cool too much, it will harden and the pieces will shatter and fall apart when sliced, so work quickly. Use a metal spatula and transfer the pieces to the wire rack, flat side down, spacing them about \u00bd inch apart.\n\n6 Lower the temperature to 275\u00b0F (135\u00b0C). Put the rack of biscotti strips on a sheet pan to catch any loose crumbs and bake for 40 to 45 minutes, until all sides of the biscotti are hard to the touch.\n\n7 Remove from the oven and let the biscotti cool on the rack until just slightly warm, then decorate with the chocolate glaze if desired. As for glazing style, you have a few options; see notes. To harden the glaze more quickly, put the biscotti in the refrigerator or freezer. Once the glaze has set, transfer the biscotti to an airtight container and store in the refrigerator for up to 2 weeks, or in the freezer for up to 6 months.\n\nNOTES:\n\n\u2022 Use a rubber spatula to transfer the glaze into a plastic sandwich bag. Work the glaze into one corner of the bag and cut off a tiny bit of the corner. Pipe thin streaks of the glaze over the biscotti, crisscrossing it over each. (You can also drizzle the glaze off the end of a spoon or spatula in squiggles.)\n\n\u2022 Dip one or both ends of the biscotti into the glaze to make chocolate tips, then return them to the wire rack until the glaze hardens.\n\n\u2022 Use a pastry brush, rubber spatula, or icing spatula to spread glaze across one of the flat surfaces of each biscotti.\n\n\u2022 If you have leftover glaze, you can apply more or cover it and refrigerate for another use. (Or just lick the bowl!)\n\n## Chocolate Glaze\n\n1 cup (6 oz \/ (170 g) semisweet chocolate chips\n\n\u00bc cup (2 oz \/ 56.5 g) unsalted butter\n\nCombine the chocolate and butter in the top of a double boiler and heat over simmering water, stirring occasionally, until melted and smooth. Alternatively, combine them in a microwave-safe bowl and microwave at medium heat. The chocolate will melt quickly in the microwave and be damaged if it overheats, so use short intervals (about 30 seconds) and stir in between. Once the chocolate is almost melted, use even shorter intervals.\n\nCHAPTER 6\n\n# THE NEXT NEW BREAD FRONTIER\n\nAs the preceding chapters of this book amply demonstrate, the next frontier of bread is already here. Sprouted grain flours, sprouted grain pulps, whole-milled grains, regional milling, gluten-free flours, nut flours, and seed meal\u2014all are rapidly advancing in the marketplace. In this chapter, I want to look even further ahead, where there always seems to be yet another new wave forming, even if it's too early to see it clearly or know what its impact will be. Here are some things I'm tracking on the wave beyond the current wave.\n\n## PREBIOTICS: THE NEW PROBIOTICS\n\nA few years ago I was teaching a bread class at Reno's Nothing To It! Culinary Center when one of the attendees introduced himself. Bob Thornberg had flown in from Walhalla, North Dakota, because of something he'd read in one of my books: a reference to beer as liquid bread, and bread as solid beer. Bob was intrigued because he was in the business of fermenting grain to create highly nutritious animal feed. The notion of beer as liquid bread had triggered an idea that connected a series of dots for him, and he wanted to talk about it.\n\nOne of the breads we made during that class utilizes spent grain, a by-product of beer making, as an ingredient. It's a fun technique because so many types of beer are being made these days, and in the process, malted and roasted grains are used in a seemingly infinite number of variations. After brewing, the grain is considered spent because most of its sugars and flavors have been brewed into the wort, which is eventually transformed into beer. However, the process does leave some residual flavor in the fiber-rich husks, which are usually given or sold to livestock farmers or just discarded.\n\nTo me, this seemed like a waste, so I started experimenting with using spent grain in bread dough to bump up the flavor and fiber content. As a side benefit, this increased my bread repertoire dramatically, because I could use one master formula to create many different flavors of bread depending on the type of spent grain used; if I collected the spent grain from four different beers, I could make four different breads using the same master dough. Bob realized that what I was doing with the spent grain was essentially the same as what he was doing: creating a high-powered, nutritionally loaded food source, in his case by fermenting a blend of wheat, oats, barley, and flaxseed.\n\nAfter grain is fermented, what's left behind is about 33% protein and 30% fiber, along with starches and water. Unlike humans, cows and other ruminants can make good nutritional use of that fiber because they have multichambered digestive systems with a rumen (sort of a \"pre-stomach\") leading to the true stomach. In the rumen, fiber is broken down by beneficial bacteria, or probiotics. Cows, then, are built for a high-fiber diet. Therefore, a fermented grain blend, served up as a wet mash snack, is a very healthful supplement to livestock feed, being full of prebiotics, material that serves as nourishment for probiotics, the good bacteria already in their guts. For twenty-five years, Bob has marketed this product under the name SweetPro, and he's received numerous testimonials about its benefits.\n\nIn Reno, Bob told me, \"I've been wondering: if this is so beneficial for cows and horses, why not humans? So I started using a bit of it in my morning smoothies, and I felt great. Then, when I read what you wrote about beer being liquid bread, I started to wonder what effect the product might have on actual bread.\" He gave me a jar of dried SweetPro, which he now markets in its dried form as ProBiotein. \"I've been having good success with it in my breads, but I'm no baker. I hope you'll try it and tell me what you think.\"\n\nI promised to play around with it when I got home. My initial thought was that it probably wouldn't do much more than boost the fiber content of the bread. Then I made a couple of loaves of regular French bread adding in a small amount of the ProBiotein, just 2% of the flour weight. The crusty bread baked up as nicely as ever. The real surprise was the flavor, which was quite similar to a very tangy San Francisco sourdough bread. I made another batch using the same recipe, this time with only 1% ProBiotein, and the results were perfect! I've since tried it with other recipes, and even in breads made with sprouted flour, I achieved the same complex sourdough flavor.\n\nWhen I wrote to Bob to tell him the results, he seemed pleased and told me, \"I've decided to explore selling an organic version of this for human consumption. I'm setting up a separate facility since it will need to be certified as food-grade. And beyond making great bread, I think it has even bigger implications in the world of nutrition and health.\"\n\nBob and I stayed in touch over the next few years. He sent me new samples as he developed the product, and I continued to add it to bread dough whenever I wanted to add some zing to the flavor. There seemed to be just enough viable, dormant organisms in the dried grain to both enhance the fermentation of the dough and also generate sufficient lactic acid so that it acted almost like an instant sourdough starter. Many large commercial operations use sourdough powders, but I always considered it kind of like cheating. However, this stuff worked, and there was nothing in it beyond pre-fermented dried grain!\n\nAs for the nutritional and health benefits, I'll leave that explanation to Bob: \"ProBiotein is full of prebiotics\u2014fiber and other nutrients that probiotics, or beneficial gut bacteria, need to flourish. Even people who are taking probiotic supplements may not have the gut conditions to allow the good bacteria to thrive. The good bacteria in your gut love this mixture of dried, fermented grains and it helps them do what they do to keep us healthy.\"\n\nIntrigued by this, I had a conversation with a nutrition researcher Craig Ponsford had introduced me to. When I asked him for his take on the seemingly rampant increase of food sensitivities, he said, \"Of course, people can stop eating the things that cause reactions, but that doesn't get at the cause. I'm convinced that many of these sensitivities are indications of poor gut health, which is almost like a national epidemic. Good health starts in the gut, and I have a feeling that if people heal their gut, a lot of their sensitivities will fade away.\"\n\nI think of this every time I put ProBiotein in my smoothies, sprinkle it on my food, or put it in my bread dough, and I've developed a theory of my own. Maybe there's a connection between the effect ProBiotein has on dough, turning it into an instant sourdough, and anecdotal evidence that sourdough breads in general seem to cause fewer digestive problems and reactions than commercially yeasted bread. Could it be that sourdough bread might serve as a prebiotic?\n\nTo explore this further, I spoke with Dr. Harold Dowse, who many in the bread world know as Dusty Dowse. An avid bread enthusiast, Dusty is a professor of biology, mathematics, and statistics at the University of Maine and also serves as the director of the Maine Artisan Bread Fair and on the board of the Maine Grain Alliance. One of his hypotheses is that people with celiac disease and also general gluten sensitivity may lack the enzymes or microorganisms necessary to digest gluten molecules, which are very complex and contain hundreds of amino acids. He told me, \"Sourdough breads that are fermented over a long period of time and contain the right kind of bacteria seems to cut through the folds of the gluten molecule and break it down into smaller pieces.\"\n\nThat certainly aligns with what some sourdough advocates have long maintained. It's too soon to know whether ProBiotein does indeed provide a similar benefit when added to dough, but wouldn't it be nice if it did? I'll be tracking it, maybe for my next book...\n\n## PILLAGING THE POMACE PILE\n\nMeanwhile, in Northern California, a related and highly interesting endeavor has been unfolding: a collaboration among vintners interested in making use of grape skins and grape seeds, by-products of wine making that are often discarded or composted. Craig Ponsford, who serves as a consultant on the project, set up a meeting for me with some of the principal players in this experiment, who operate under the name WholeVine Products.\n\nWholeVine is the creation of Peggy Furth and Barbara Banke. Peggy is a cofounder of the winery Chalk Hill, and Barbara is head of Jackson Family Wines, which she helped grow into one of the world's largest and most successful wine brands in partnership with her late husband, Jess Jackson. These two wine luminaries, Peggy and Barbara, believe that the contents of the pomace pile, where \"spent grapes\" are discarded after wine making, can be used to make valuable and important new products. In addition to Peggy, I met Torey Arvik, director of applied and research science, and general manager Paul Novak. Over the course of the meeting, I learned that the juice in wine grapes represents about 80% of the total weight of the harvest. Once the juice is extracted, the remaining 20%, primarily skins and seeds, is typically either composted or fed to dairy cattle, or, sometimes, sent to landfills. But what if these constituents still have functional and nutritional value, like spent grain?\n\nAs it turns out, they do, and Torey Arvik showed me the results of several studies that prove this.\n\nPaul Novak summed up WholeVine's mission well: \"We want to feed more of the world, and it begins at the pomace pile.\"\n\nAs I was to discover, nutritional value lies not just in the seeds, which can be used to make grape seed oil, but also in the skins, which can be dried to make a kind of flour. As for the seeds after the oil has been expelled, even they need not go back to the pomace pile. They too can be dried and milled into flour. (In fact, Craig Ponsford is the person who connected WholeVine with Joe Vanderliet, who now mills their grape seed flour.) Flour made from grape skins and grape seeds? That's definitely next, new-frontier stuff.\n\nWhile I loved the idea of this flour, I had to wonder how it would perform. Craig had experimented with it, and he told me, \"You can't use it exclusively, in place of regular flour, but it works well as a supplemental flour. A small amount of grape seed and grape skin flour adds value.\" Of course, my next question was what kind of value. Enter Eric Frischkorn, chef de cuisine for Kendall-Jackson wines, which hosted our lunch meeting. In addition to being a skilled chef, Eric is a serious bread baker, and during the meeting he served us a couple of excellent breads, as well as semolina crackers and some of WholeVine's locally famous chocolate chip cookies, all containing some combination of grape seed oil, grape seed flour, and grape skin flour. I was especially impressed at how the hole structure of the breads opened up so nicely. They were equal to the products of an artisan bakery, with rich color, a moist and creamy mouthfeel, and, of course, that large, open crumb. Because the grape skin flour is used in such small proportions, its contribution to the flavor is very subtle, but it does add a lovely color and a slight undercurrent of fruitiness. The grape seed flour, on the other hand, adds a more distinctive flavor that's pleasant, and reminiscent of hemp or sunflower seeds.\n\nOver the course of a long and lavish lunch, I learned a great deal about WholeVine, its products, and their health benefits. Much of this information is available on their website (see Resources), so here I'll just recap some of the highlights:\n\n\u2022 Grape skin flour is naturally antimicrobial due to its malic, tartaric, ferulic, and caffeic acids, as well as phytoalexins (including resveratrol, well known for its purported antiaging effects). Grapes naturally produce these compounds to fight off microorganisms on the vine. In addition to enhancing flavor and nutrition, these compounds confer antimolding protection and give bread a longer shelf life.\n\n\u2022 Grape skin and grape seed flours are rich in fiber, pectin, minerals, flavonols such as epicatechin (which has remarkable health benefits), and phytosterols (known to block the formation of cholesterol).\n\n\u2022 Grape seed flour contains 30% to 50% fiber and 14% to 17% protein and is also high in calcium. Syrah seed flour has the most fiber, but the seeds of all varietals contain more fiber than a comparable amount of raspberries, broccoli, or cooked lentils. And amazingly, just 0.11 ounce (3 g) of Chardonnay seed flour has as much antioxidant capacity as 3.5 ounces (99 g) of fresh blueberries.\n\n\u2022 Grape skin flour contains 25% to 36% dietary fiber and 7% to 13% protein, as well as calcium, iron, potassium, and nearly a full complement of amino acids.\n\n\u2022 Both types of flour are gluten-free.\n\nA number of studies are currently looking into the health benefits of grape seed and grape skin flours, and early results are promising. Of course, these products have a lot to commend them beyond fantastic nutrition, such as contributing interesting flavors and allowing us to make excellent use of material that was formerly discarded. So although I'm calling them the new next frontier, they are already available. I encourage you to get your hands on them and start riding the next wave!\n\n## BREAD WITH COFFEE TEA, REALLY?\n\nBefore we get to the recipes, I must introduce one final \"next frontier\" product: cascara seca\u2014the dried pulp of coffee \"cherries,\" the fruits that contain coffee beans. Cascara seca has been available for a while, but I hadn't discovered it until David Haddock, the head roaster at Boquete Mountain Coffee, in Charlotte, North Carolina, told me about it.\n\nAs David explained it, until recently, the only part of the coffee cherry thought to be valuable was the greenish bean inside, which eventually goes through a process of fermentation, drying, and roasting prior to being brewed into coffee. The dried (seca) fruit, which looks more like a husk (cascara), is usually composted or discarded (sound familiar?). But not anymore. People have figured out that cascara seca can be steeped like tea to produce a naturally sweet, translucent, dark amber brew that contains only 10% of the caffeine of coffee, but a host of beneficial antioxidants and polyphenols. As with many \"discoveries,\" this isn't entirely new knowledge. A similar dried husk brew, called qishr, has long been enjoyed in Yemen, where it's sometimes steeped with cinnamon and spices, kind of like a cascara chai.\n\nDried coffee cherries are now also being sold as a nutritional supplement and being used as an ingredient in cosmetics because they're believed to have antiwrinkling properties\u2014not unlike the resveratrol in grape skins and seeds. Recent studies indicate that coffee cherries have more than twice the antioxidant capacity of dark chocolate and ten times more antioxidant power than green tea, pomegranate, and a\u00e7ai, all currently in vogue as superfoods.\n\nSince my conversation with David, I've been brewing cascara seca tea as an alternative to black tea or coffee\u2014and a nice complement to my morning ProBiotein smoothie! Of course, I quickly started thinking about how it might perform in baking. After all, if other formerly discarded products like spent grain and pomace can add nutritional value to bread, why not cascara seca? So I ground up some cascara seca into a grainy powder in my little burr mill. The smell is nothing like coffee because it isn't roasted (nor is it actually coffee); it smells more like mild tobacco with subtle aromatic undertones of blueberries, rose hips, and even a hint of orange marmalade. I added a few tablespoons of the powder to a small batch of simple French bread dough and ended up with a very pretty, slightly speckled bread that looked like it had bits of vanilla bean seeds floating throughout the crumb. The flavor was subtle yet noticeable, kissed with a hint of fruited green tea flavors.\n\nFor the next batch, I brewed some cascara tea and substituted it for the water in that same recipe. This had a more dramatic influence, with the tea's brownish amber color giving the dough a light caramel color. Again, its contribution to the bread's flavor was subtle and delicate, almost floral, with a long, pleasant aftertaste. I felt like I was onto something. In a third batch, I used both the cascara tea and the steeped cascara used to make it. Ahhh, success!\n\nAs always in my books, it ultimately comes down to flavor. If an ingredient doesn't add flavor value, it will be a hard sell to convince people to use it. They might take a chance on it based on health claims, but they're unlikely to stick with it if the flavor doesn't deliver. Plus, I think the amount of cascara we could actually consume in bread would probably be negligible from a nutritional or health standpoint. Fortunately, I can assure you that it does add flavor value; the breads I've made with cascara have tasted really good. Like classic sourdough breads, they have a long finish, far longer than bread leavened only with commercial yeast, but without the sour tang. And although the health benefits of cascara are only just now being studied, I think it holds great potential. In fact, I predict that we'll be hearing a lot more about it in the coming years. Hopefully you'll already be well ahead of the wave, drinking cascara tea and using it in your breads.\n\n# SYRAH GRAPE SKIN FLOUR OLIVE BREAD\n\nMAKES 2 LOAVES\n\nGrape skin flour intensifies both the flavor and the color of olive bread and also helps prevent the bread from becoming stale. Here, it's a simple addition to a basic mixed-method bread, leavened with both sourdough starter and commercial yeast in the style of a French pain au levain, but it yields a totally different loaf. You can use any type of grape skin flour, but Eric Frischkorn, the Kendall-Jackson chef who created this recipe, recommends the Syrah grape skin flour called for here (available from WholeVine; see Resources) because it best complements the flavor of the olives. Note that although this version uses mostly regular bread flour and whole wheat flour, you can also make it with sprouted wheat flour instead.\n\n### DOUGH\n\nMother Starter, recently refreshed: \u00bd cup (4.5 oz \/ 128 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2 cups plus 1\u00bd tablespoons (16.75 oz \/ 475 g)\n\nunbleached bread flour: 4\u00bd cups plus 1 tablespoon (20.5 oz \/ 581 g)\n\nwhole wheat flour (preferably whole-milled): \u00bd cup plus 1 tablespoon (2.5 oz \/ 71 g)\n\ngrape skin flour (preferably Syrah): \u00bc cup (1 oz \/ 28.5 g \/ 4%)\n\ninstant yeast: \u215b teaspoon (0.02 oz \/ 0.5 g)\n\nkalamata olives, pitted and halved: 1\u00bd cups (7.25 oz \/ 206 g)\n\nunbleached bread flour for the olives: 1 tablespoon (0.3 oz \/ 8.5 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nTOTAL: 53.32 oz \/ 1,512.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nMother Starter| 19  \nwater| 70  \nunbleached bread flour| 86  \nwhole wheat flour| 10  \ngrape skin flour| 4  \ninstant yeast| 0.1  \nkalamata olives| 30  \nunbleached bread flour for the olives| 1.2  \nsalt| 2.1  \nTOTAL| 222.4\n\n1 Break the starter into 5 or 6 pieces and put it in the bowl of a stand mixer fitted with the dough hook or in a large bowl. Add the water and stir (on low speed if using a stand mixer) for a few seconds to soften and break up the starter. Add the flours and mix or stir for about 2 minutes, until the flour is hydrated and the starter is evenly distributed. Continue mixing for about 2 minutes, until a coarse, sticky dough forms. Let the dough rest in the bowl at room temperature for 20 minutes.\n\n2 Add the yeast and mix on medium-low speed if using a stand mixer or knead by hand for 2 to 3 minutes, until the dough is firm and slightly tacky. In a separate bowl, toss the olives with the 1 tablespoon (0.3 oz \/ 8.5 g) of bread flour. Add the olives to the dough and mix on low speed or knead by hand for 1 minute. Add the salt and mix or knead for 1 to 2 minutes, until the salt and olives are evenly distributed and the dough is very soft and supple and slightly sticky.\n\n3 Oil the work surface and transfer the dough to the oiled area. Also oil a large bowl. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Form the dough into a ball. Put it in the oiled bowl and cover the bowl with plastic wrap, or leave the dough on the work surface and cover it with the bowl. At intervals of 45 minutes, perform two additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will become firmer and smoother after each stretch and fold, and after the final fold, it should be tacky and very supple and bouncy when patted.\n\n4 Put the dough back in the bowl and cover the bowl with plastic wrap. Ferment the dough at room temperature for 30 minutes.\n\n5 Dust two bannetons with flour and lightly dust the work surface with flour. Transfer the dough to the work surface. Divide it in half and shape into two balls. Mist the tops with vegetable spray oil and cover loosely with plastic wrap or a clean towel. Let the dough rest for 20 minutes longer.\n\n6 Form each piece into a boule. Place the shaped loaves seam side up in the bannetons and cover loosely with plastic wrap or a clean towel. Proof at room temperature for 1\u00bd to 3 hours, until the dough has nearly doubled in size.\n\n7 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 450\u00b0F (232\u00b0C).\n\n8 Just before baking, dust a peel with flour. Transfer the dough to the peel and score as desired. Slide the dough onto the baking stone and pour about 1 cup of hot water into the steam pan. Bake for 4 minutes, then lower the temperature to 425\u00b0F (218\u00b0C) and bake for 15 minutes longer, then rotate the loaves and continue baking for an additional 15 to 20 minutes, until the crust is rich golden brown and the loaves sound hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n9 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\n# GRAPE SKIN FLOUR RUSTIC SOURDOUGH WITH SUN-DRIED TOMATOES, ROASTED GARLIC, AND CARMODY CHEESE\n\nMAKES 2 LARGE LOAVES\n\nThis bread showcases both grape seed oil and grape skin flour in a very inventive, fully loaded hearth bread. Eric Frischkorn, the Kendall-Jackson chef who created this recipe, recommends Chardonnay seed oil and Chardonnay grape skin flour (both available from WholeVine; see Resources) for this bread, but you can use any varietal for either. The add-ins include some interesting ingredients, including herbes de Provence and Carmody cheese. Eric sifts out the lavender from his herbes de Provence (which is kind of odd since it is the lavender that defines this blend, but it does tend to assert itself); therefore, you could use a blend of Italian herbs instead. Carmody cheese is named for a road that runs next to Bellwether Farms, one of America's finest farmstead creameries, located in Sonoma County. It's a creamy, buttery cheese similar to a rich Monterey Jack, so you could use that instead, or perhaps Fontina or even Muenster. It may seem odd to prepare the add-ins a day in advance, but overnight storage allows the cheese to become infused with the flavors of the garlic and herbs.\n\n### ADD-INS (DAY 1)\n\ngrape seed oil (preferably Chardonnay): (\u00bc cup oz \/ 2 g \/ 56.5%)\n\ngarlic cloves, peeled and quartered: (about 6 oz \/ 1 g \/ 28.5%)\n\nherbes de Provence: (1 tablespoon oz \/ 0.07 g \/ 2%)\n\nCarmody cheese, cut into \u00bd-inch pieces: (1\u00bd cups oz \/ 5.75 g \/ 163%)\n\nsun-dried tomatoes (not oil-packed), quartered: (\u00bc cup oz \/ 1.5 g \/ 42.5%)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): (\u00bc cup oz \/ 2 g \/ 56.5%)\n\nTOTAL 12.27 oz \/ 347.7 g\n\n1 Warm the oil in a small saucepan over low heat, add the garlic, and saut\u00e9 until golden brown, 10 to 15 minutes. Drain the garlic, reserving the oil. Put the oil in a container, cover, and refrigerate. Put the garlic in a medium bowl.\n\n2 Sift the herbes de Provence to remove the lavender, then add it to the garlic. Stir in the cheese. Cover the bowl and refrigerate.\n\n3 Pour the water over the sun-dried tomatoes and let soak for at least 1 hour, then cover and refrigerate overnight.\n\n### FINAL DOUGH (DAY 2)\n\nAdd-Ins: all (12.27 oz \/ 347.7 g)\n\nunbleached bread flour: 4 cups plus 1 tablespoon (18.25 oz \/ 517 g)\n\nMother Starter, recently refreshed: 1 cup (4.5 oz \/ 128 g)\n\nwater, at room temperature: 2 cups (16 oz \/ 454 g)\n\nwhole wheat flour: 1 cup plus 1 tablespoon (4.75 oz \/ 135 g)\n\ngrape skin flour (preferably Chardonnay): \u00bc cup (1 oz \/ 28.5 g \/ 4%)\n\ninstant yeast: \u215b teaspoon (0.02 oz \/ 0.5 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\nTOTAL: 57.29 oz \/ 1,624.7 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nAdd-Ins| 51  \nunbleached bread flour| 76  \nMother Starter| 19  \nwater| 67  \nwhole wheat flour| 20  \ngrape skin flour| 4  \ninstant yeast| 0.1  \nsalt| 2.1  \nTOTAL| 239.2\n\n1 Remove the add-ins from the refrigerator: the oil; herbs, garlic, and cheese; and the tomatoes. Add the tomatoes (and any water that wasn't absorbed) and 2 tablespoons (0.6 oz \/ 1.5 g) of the bread flour to the garlic and toss until evenly distributed.\n\n2 Break the starter into 5 or 6 pieces and put it in the bowl of a stand mixer fitted with the dough hook or in a large bowl. Add the water and stir (on low speed if using a stand mixer) for a few seconds to soften and break up the starter. Add the remaining bread flour and the whole wheat and grape skin flours and mix or stir for 2 to 3 minutes. Toward the end of the mixing time, drizzle in the oil and continue mixing until the flour is hydrated and a coarse ball of slightly sticky dough forms. Let the dough rest in the bowl, covered, at room temperature for 20 minutes.\n\n3 Add the yeast and mix on medium-low speed or knead by hand for 2 to 3 minutes, until the dough is soft, supple, and tacky. Add the cheese mixture and mix on low speed or knead by hand for 1 minute. Add the salt and continue to mix or knead for 1 minute, until the cheese is evenly distributed. The dough should be soft, tacky or slightly sticky, and very supple.\n\n4 Oil the work surface and transfer the dough to the oiled area. Also oil a large bowl. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Form the dough into a ball. Put it in the oiled bowl and cover the bowl with plastic wrap, or leave the dough on the work surface and cover it with the bowl. At intervals of 45 minutes, perform two additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will firm up slightly after each stretch and fold, and after the final fold, it should be very supple and bouncy when patted.\n\n5 Put the dough back in the bowl and cover the bowl with plastic wrap. Ferment the dough at room temperature for 30 to 60 minutes, until it shows signs of rising.\n\n6 Lightly flour a work surface. Transfer the dough to the work surface and divide it in half. Gently pat each piece into a 1-inch-thick rectangle. Cover the dough loosely with plastic wrap or a clean towel and let it rest for 20 minutes.\n\n7 Line a sheet pan with parchment paper or a silicone mat, mist with vegetable spray oil, and, optionally, dust lightly with semolina flour or another coarse flour. Working with one piece of dough at a time, gently press to form a larger rectangle, about \u00bd inch thick. Fold the top edge over, bringing it to the center of the dough, then do the same with the bottom edge, bringing it to the center so it touches the edge of the top section, creating a seam. Fold the dough over the seam to make a narrower, thicker rectangle. Pinch the open sides together along the seam and the ends to seal them and tighten the surface of the dough. Gently pull both ends to make a longer rectangle about 9 to 12 inches long. Put both shaped loaves on the prepared pan. Mist with vegetable spray oil and cover loosely with plastic wrap or a clean towel. Proof at room temperature for 1\u00bd to 3 hours, until they increase in size by at least 1\u00bd times.\n\n8 About 45 minutes before you plan to bake, position a rack in the middle of the oven and prepare the oven for steaming. (A baking stone isn't used.) Preheat the oven to 450\u00b0F (232\u00b0C).\n\n9 Put the pan in the oven and pour about 1 cup of hot water into the steam pan. Bake for 4 minutes, then lower the temperature to 425\u00b0F (218\u00b0C) and bake for 15 minutes. Rotate and bake for 10 to 20 minutes longer, until the crust is golden brown all over and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 195\u00b0F (91\u00b0C).\n\n10 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving.\n\n# GLUTEN-FREE FOCACCIA\n\nMAKES ONE 18 BY 13-INCH PAN; 8 TO 12 SERVINGS\n\nIn this focaccia, grape skin flour makes up only a small percentage of the total flour, but it makes a major contribution to both color and flavor. If you can't find quinoa flour, simple grind whole quinoa in a seed or spice mill to make your own, or use more brown rice flour or another gluten-free flour. It's also fine to substitute sprouted brown rice flour and sprouted quinoa flour for the nonsprouted versions called for here. It's also fine to use generic grape seed oil or even olive oil, though Chardonnay grape seed oil heightens flavor. This recipe was created by Eric Frischkorn, and he uses a number of interesting tricks. One is using carbonated water, which gives additional aeration to the crumb. Another is using whipped egg whites, again to provide aeration. The acidity of the vinegar helps both with flavor and with preserving some of the aeration. Plus, the high amount of yeast contributes to both flavor and rapid leavening. All in all, very innovative.\n\n### BATTER\n\nbrown rice flour: 4\u00bd cups (20 oz \/ 567 g)\n\nquinoa flour: 1\u00be cups (8 oz \/ 227 g)\n\ngrape skin flour (preferably Chardonnay): \u00bd cup (2 oz \/ 56.5 g)\n\nxanthan gum: 3 tablespoons (1 oz \/ 28 g)\n\nsugar: 4 tablespoons (2 oz \/ 57 g \/ 7%)\n\ninstant yeast: \u00bc cup (1.32 oz \/ 37.5 g)\n\nsalt: 4 teaspoons (1 oz \/ 28.5 g)\n\ndried basil: 1 tablespoon (0.07 oz \/ 2 g)\n\ndried oregano: 1 tablespoon (0.12 oz \/ 3.5 g)\n\ngrape seed oil (preferably Chardonnay): \u00bc cup (2 oz \/ 56.5 g)\n\nclub soda or plain seltzer water: 3 cups (24 oz \/ 680 g)\n\nwhite wine vinegar: 2 teaspoons (0.35 oz \/ 10 g)\n\negg whites: 6 (7.5 oz \/ 213 g)\n\nadditional grape seed oil (preferably Chardonnay) for the pan and for topping: about \u00bc cup\n\nParmesan cheese, grated: about \u00bd cup\n\nTOTAL: 69.36 oz \/ 1,966.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nbrown rice flour| 67  \nquinoa flour| 26.4  \ngrape skin flour| 6.6  \nxanthan gum| 3.3  \nsugar| 7  \ninstant yeast| 4.4  \nsalt| 3.4  \ndried basil| 0.4  \ndried oregano| 0.4  \ngrape seed oil| 6.6  \nclub soda or plain seltzer water| 80  \nwhite wine vinegar| 1.2  \negg whites| 25  \nTOTAL| 231.7\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flours, xanthan gum, sugar, yeast, salt, basil, and oregano (on low speed if using a stand mixer). In a separate bowl, whisk the \u00bc cup (2 oz \/ 56.5 g) of grape seed oil with the club soda and vinegar. In a third bowl, vigorously whisk the egg whites for about 1 minute to make a light to medium foam.\n\n2 With the mixer on low speed or while mixing with a large spoon, drizzle in the egg whites and mix until fully incorporated. While continuing to mix or stir, slowly pour in the club soda mixture. Mix on medium speed or by hand for 2 minutes. Scrape down the bowl, then mix for 2 more minutes to make a thick, slightly foamy batter.\n\n3 Line an 18 by 13-inch rimmed sheet pan with parchment paper or a silicone mat. Generously oil the surface, including the sides of the pan, with 2 tablespoons (1 oz \/ 28.5 g) of the grape seed oil. Pour the batter into the center of the pan. Drizzle the remaining 2 tablespoons (1 oz \/ 28.5 g) of oil over the batter and use a rubber spatula or icing spatula to spread the batter evenly into the pan, filling all four corners. Sprinkle the cheese over the top. Cover the pan loosely with plastic wrap and let the dough rise in a warm place (80\u00b0F \/ 27\u00b0C) for at least 1 hour, until nearly double in size; you can proof it in a cooler location, but it will take longer to rise.\n\n4 Position a rack in the middle of the oven and preheat the oven to 375\u00b0F (191\u00b0C).\n\n5 Bake for 12 minutes, then rotate the pan and bake for 12 to 18 minutes longer, until golden brown and springy to the touch when pressed in the center.\n\n6 Let cool for at least 10 minutes before cutting and serving.\n\n# GRAPE SKIN FLOUR CRACKERS\n\nMAKES 15 LARGE CRACKERS\n\nThis cracker recipe provides a great format for tasting the differences in flavor between grape skin flour and grape seed oil made from different grape varietals. If you prefer, feel free to use grape skin flour, grape seed oil, or both\u2014or even grape seed flour\u2014in any other cracker recipe. For the grape skin flour, keep the amount to 4% of the total flour weight; for grape seed flour, keep it to under 10% of the total flour weight. For grape seed oil, substitute it for some or all of the oil or other fat in the recipe, using an amount between 12% and 15% of the total flour weight. This cracker recipe is unleavened. For flakier crackers that rise a bit, add 1\u00bd teaspoons (0.28 oz \/ 8 g) of baking powder when mixing the dry ingredients. The egg white wash is optional, but it will help salt and any other toppings adhere to the crackers. One final note: Feel free to substitute sprouted whole wheat flour for either the semolina flour or bread flour or both. However, you'll probably need to increase the amount of water depending on how much sprouted flour you substitute.\n\n### DOUGH\n\nsemolina flour: 3\u2153 cups (15 oz \/ 425 g)\n\nunbleached bread flour: 2\u2154 cups (12 oz \/ 340 g)\n\ngrape skin flour: \u00bc cup (1.12 oz \/ 32 g \/ 4%)\n\nsalt: 1\u00bd teaspoons (0.35 oz \/ 10 g)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 1\u00bd cups (12 oz \/ 340 g)\n\ngrape seed oil: \u00bd cup (4 oz \/ 113 g)\n\negg white wash (optional): 1 egg white whisked with 1 tablespoon water\n\ncoarse salt for topping: to taste\n\nTOTAL: 44 oz \/ 1,260 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nsemolina flour| 53  \nunbleached bread flour| 43  \ngrape skin flour| 4  \nsalt| 1.4  \nwater| 43  \ngrape seed oil| 14  \nTOTAL| 158.4\n\n1 In the bowl of a stand mixer fitted with the dough hook, or in a large bowl, stir together the flours and salt (on low speed if using a stand mixer). In a separate bowl, whisk together the water and oil, then slowly pour it into the flour mixture while mixing on low speed or stirring. Mix for about 3 minutes, until the flour is hydrated and a firm, coarse ball of dough forms. Add a bit more water or flour if needed to achieve this texture. Mix or stir for about 2 or 3 more minutes, until the dough feels supple and satiny.\n\n2 Lightly flour a work surface with either semolina flour or bread flour. Transfer the dough to the work surface and knead by hand for about 30 seconds, until the dough is a smoother and more supple ball, adjusting the flour or water as needed.\n\n3 Mist a bowl with vegetable spray oil, put the dough in the bowl, and cover the bowl with plastic wrap. Refrigerate for at least 1 hour or up to 2 days.\n\n4 Position two racks in the oven an equal distance apart and preheat the oven to 325\u00b0F (163\u00b0C). Line three sheet pans with parchment paper or silicone mats and mist with vegetable spray oil.\n\n5 Divide the dough into 3 equal pieces. (If you'd like to save some of the dough to bake later, wrap the pieces in plastic wrap and refrigerate for up to a week.) Divide each piece into 5 smaller pieces of equal size; they should be just under 3 ounces (85 g) each. Shape each piece into a ball and let rest for about 5 minutes to allow the gluten to relax.\n\n6 Dust the work surface with flour again or spread about \u00bc teaspoon of vegetable oil on the work surface. Place one dough ball on the floured or oiled area and use a rolling pin to roll out the dough into a rectangle measuring about 5 by 3 inches and just under \u00bc inch thick. Transfer to one of the prepared pans. Repeat, placing 5 rolled pieces of dough on each pan and spacing them evenly; they can be close because they won't spread.\n\n7 Brush the tops with water or the egg white wash, then lightly sprinkle coarse salt over the top to taste. Use the tines of a fork to prick the surface of each cracker several times to prevent bubbling of the dough during baking.\n\n8 Bake for 8 minutes, then rotate the pans front to back and between shelves and bake for 5 to 8 minutes longer, until the tops just begin to brown around the edges.\n\n9 Leave the crackers on the pans and let cool for at least 10 minutes before serving. They will be soft, not hard and crisp.\n\n### Variations\n\nCRISP CRACKERS: After rotating the pans during baking, lower the temperature to 300\u00b0F (149\u00b0C) and bake for 10 to 15 minutes longer, until the entire upper surface is golden brown.\n\nHERB CRACKERS: After sprinkling with salt, dust the surface with any combination of herbs and spices, bearing in mind that a little goes a long way. Recommendations include basil, oregano, thyme, marjoram, rosemary, herbes de Provence, sweet paprika, cayenne, and black pepper.\n\nCascara Seca Lean Bread\n\n# CASCARA SECA LEAN BREAD\n\nMAKES 1 VERY LARGE BOULE, 2 TO 6 SMALLER LOAVES, OR UP TO 24 ROLLS\n\nTea has been used for many years by many bakers and pastry chefs as a way to infuse unique, subtle flavors into their food. Here's a basic cascara-infused lean dough (aka French bread) based on the overnight method in my book Artisan Breads Every Day. This recipe is designed to give you a fairly neutral flavor foundation upon which to taste the contribution that cascara seca tea can add to a bread. You can use the same method of brewing to make a tea for use as the liquid in any number of other bread formulas once you get the hang of it. You can also use other teas and even brewed coffee to hydrate doughs in the same manner. But cascara seca is the new kid on the block, and one well worth getting to know. I find the flavor and long finish that it adds to bread delightful and intriguing.\n\n### DOUGH\n\nwater: 2\u00bc cups (18 oz \/ 510 g)\n\nground cascara seca: 2 tablespoons (0.5 oz \/ 14 g)\n\nunbleached bread flour: 5\u2153 cups (24 oz \/ 680 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: 2\u00bc teaspoons (0.25 oz \/ 7 g \/ 1%)\n\nTOTAL: 43.25 oz \/ 1,225 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nwater| 75  \nground cascara seca| 2.1  \nunbleached bread flour| 100  \nsalt| 2.1  \ninstant yeast| 1  \nTOTAL| 180.2\n\n1 Bring the water to a boil. Pour it over the cascara seca and let steep for at least 15 minutes, until it cools to lukewarm (95\u00b0F \/ 35\u00b0C). It can then be used immediately or stored in the refrigerator for later use. If stored in the refrigerator, warm it to at least room temperature before mixing the dough.\n\n2 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, and yeast (on low speed if using a stand mixer). Add the cascara infusion, grounds and all. Mix or stir for 2 minutes, until well blended; if mixing by hand and the spoon gets too doughy, dip it in a bowl of warm water or turn the dough out onto a lightly oiled work surface and knead by hand. The dough should be sticky and shaggy.\n\n3 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook and mix for 1 to 2 minutes, on medium-low speed if using a stand mixer. The dough will smooth out slightly but still be very soft and sticky.\n\n4 Mist a bowl with vegetable spray oil or lightly coat it with vegetable oil. Use a wet bowl scraper or spatula to transfer the dough to the bowl, then cover the bowl with plastic wrap (or leave the dough on the work surface and cover with the bowl). Let the dough rest for 5 minutes.\n\n5 Oil the work surface and transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Cover the dough with the bowl and then, at intervals of 5 to 10 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will become smoother and significantly firmer with each stretch and fold. After the final fold the dough should be soft, supple, and still very tacky, but it will have a bouncy feel when patted.\n\n6 Return the dough to the bowl, cover the bowl tightly with plastic wrap, and refrigerate for at least 12 hours or up to 4 days. The dough will double or possibly triple in size within 12 hours.\n\n7 Remove the dough from the refrigerator about 2 hours before you plan to bake. Oil the work surface and use a wet or oiled bowl scraper to transfer the dough to the oiled area. Using a metal pastry blade, divide the dough into the desired number of pieces\u20142 to 6 for smaller loaves, or more for rolls\u2014or use the entire amount of dough for one large loaf. For one or more hearth loaves, prepare the appropriate number of bannetons or a couche, or line an 18 by 13-inch sheet pan with parchment paper and then mist it with vegetable spray oil. Shape each piece of dough into a boule or b\u00e2tard. Put the shaped dough in the prepared proofing vessel(s) or on the sheet pan. For rolls, line two sheet pans with parchment paper or silicone mats. Shape the rolls as desired (see Rolls and Buns) and put half of them on each lined pan.\n\n8 Mist the top of the dough with vegetable spray oil, then cover loosely with plastic wrap. Proof at room temperature for 1 hour. Uncover and proof for 40 to 60 minutes longer, until the dough increases in size by 1\u00bd times. The surface will dry out slightly and the dough will spring back slowly when poked with your finger.\n\n9 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan as. Preheat the oven to 550\u00b0F (288\u00b0C) or as high as it will go.\n\n10 To bake loaves, just before baking, dust a peel with flour. Transfer the dough to the peel and score as desired. Slide the dough onto the baking stone and pour about 1 cup of hot water into the steam pan. Lower the temperature to 450\u00b0F (232\u00b0C). Bake for 10 to 12 minutes, then rotate and bake for 10 to 15 minutes longer, until the crust is hard and rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be 200\u00b0F to 205\u00b0F (93\u00b0C to 96\u00b0C). Transfer to a wire rack and let cool for at least 1 hour before slicing and serving.\n\n11 To bake rolls, bake one pan at a time. As soon as you put the pan in the oven, pour about 1 cup of hot water into the steam pan and lower the temperature to 450\u00b0F (232\u00b0C). Bake for 8 minutes, then rotate and bake for 8 to 12 minutes longer, until the crust is golden brown and the rolls sound hollow when thumped on the bottom. The internal temperature should be 200\u00b0F to 205\u00b0F (93\u00b0C to 96\u00b0C). Transfer to a wire rack and let cool for at least 10 minutes before serving.\n\n## Including Cascara Seca and ProBiotein in Dough\n\nCascara seca or ProBiotein can be used in any of the recipes in this book, but I suggest you make these lean versions Cascara Seca Lean Bread and Probiotein Lean Bread first to familiarize yourself with the flavor. Steeped cascara seca (or any tea for that matter) can be substituted for water in any bread formula, and ProBiotein can be used to add a sourdough tang to any bread where you think it would be appropriate.\n\nFor cascara seca, calculate the total flour weight in the recipe and multiply by 0.02 (2%). Weigh out that amount of cascara seca, then grind it into a powder in a spice or seed grinder, coffee mill, or high-speed blender. Measure out the amount of water called for in the dough and bring it to a boil. Pour it over the cascara seca and let steep for at least 15 minutes or as long as 2 hours. When you mix the dough, use this cascara tea, including the grounds, in place of the water. When mixing, you may need to adjusting the amount of liquid or flour a bit to achieve the target texture.\n\nFor ProBiotein, calculate the total flour weight and multiply by 0.01 (1%), or 0.015 (1.5%) for a stronger tang. Add that amount of ProBiotein to the dry ingredients. Because of the process used to make this supplement, it will function in the dough like a powdered pre-ferment.\n\n# PROBIOTEIN LEAN BREAD\n\nMAKES 1 VERY LARGE BOULE, 2 TO 6 SMALLER LOAVES, OR UP TO 24 ROLLS\n\nThe same lean dough formula in the recipe for Cascara Seca Lean Bread can be varied to feature ProBiotein instead of the dried coffee husks. In addition to the nutritional benefits provided by this dried spent grain powder, it also contributes a tangy flavor and long, pleasant finish on the palate. This version calls for just 1% ProBiotein against the total flour weight, but if you prefer a sourer flavor you can double the amount. Once you determine the percentage you prefer, feel free to add ProBiotein in that same ratio to any bread dough.\n\n### DOUGH\n\nunbleached bread flour: 5\u2153 cups (24 oz \/ 680 g)\n\nsalt: 2 teaspoons (0.5 oz \/ 14 g)\n\ninstant yeast: 2\u00bc teaspoons (0.25 oz \/ 7 g \/ 1%)\n\nProBiotein: 2 teaspoons (0.25 oz \/ 7 g \/ 1%)\n\nwater, lukewarm (95\u00b0F \/ 35\u00b0C): 2\u00bc cups (18 oz \/ 510 g)\n\nTOTAL: 43 oz \/ 1,218 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nunbleached bread flour| 100  \nsalt| 2.1  \ninstant yeast| 1  \nProBiotein| 1  \nwater| 75  \nTOTAL| 179.1\n\n1 In the bowl of a stand mixer fitted with the paddle attachment, or in a large bowl, stir together the flour, salt, yeast, and ProBiotein (on low speed if using a stand mixer). Add the water and mix or stir 2 minutes, until well blended. If mixing by hand and the spoon gets too doughy, dip it in a bowl of warm water or turn the dough out onto a lightly oiled work surface and knead it by hand. The dough should be sticky and shaggy.\n\n2 Let the dough rest, uncovered, for 5 minutes. Then switch to the dough hook and mix or stir for 1 to 2 minutes, on medium-low speed if using a stand mixer. The dough will smooth out slightly but still be very soft and sticky.\n\n3 Mist a bowl with vegetable spray oil or lightly coat it with vegetable oil. Use a wet bowl scraper or spatula to transfer the dough to the bowl, then cover the bowl with plastic wrap (or leave the dough on the work surface and cover with the bowl). Let the dough rest for 5 minutes.\n\n4 Oil the work surface and transfer the dough to the oiled area. Lightly oil your hands, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Cover the dough with the bowl and then, at intervals of 5 to 10 minutes, perform three additional sequences of stretching and folding. For each stretch and fold sequence, lightly oil your hands to prevent sticking. The dough will become smoother and significantly firmer with each stretch and fold. After the final fold the dough should be soft, supple, and very tacky and bouncy when patted.\n\n5 Return the dough to the bowl, cover the bowl tightly with plastic wrap, and refrigerate for at least 12 hours or up to 4 days. The dough will double or possibly triple in size within 12 hours.\n\n6 Remove the dough from the refrigerator about 2 hours before you plan to bake. Oil the work surface and use a wet or oiled bowl scraper to transfer the dough to the oiled area. Using a metal pastry blade, divide the dough into the desired number of pieces\u20142 to 6 for smaller loaves, or more for rolls\u2014or use the entire amount of dough for one large loaf. For one or more hearth loaves, prepare the appropriate number of bannetons or a couche, or line an 18 by 13-inch sheet pan with parchment paper and then mist it with vegetable spray oil. Shape each piece of dough into a boule or b\u00e2tard. Put the shaped dough in the prepared proofing vessel(s) or on the sheet pan. For rolls, line two sheet pans with parchment paper or silicone mats. Shape the rolls as desired (see Rolls and Buns) and put half of them on each lined pan.\n\n7 Mist the top of the dough with vegetable spray oil, then cover loosely with plastic wrap. Proof at room temperature for 1 hour. Uncover and proof for 40 to 60 minutes longer, until the dough increases in size by 1\u00bd times. The surface will dry out slightly and the dough will spring back slowly when poked with your finger.\n\n8 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 550\u00b0F (288\u00b0C) or as high as it will go.\n\n9 To bake loaves, just before baking, dust a peel with flour. Transfer the dough to the peel and score as desired. Slide the dough onto the baking stone and pour about 1 cup of hot water into the steam pan. Lower the temperature to 450\u00b0F (232\u00b0C). Bake for 10 to 12 minutes, then rotate and bake for 10 to 15 minutes longer, until the crust is hard and rich golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be 200\u00b0F to 205\u00b0F (93\u00b0C to 96\u00b0C). Transfer to a wire rack and let cool for at least 1 hour before slicing and serving.\n\n10 To bake rolls, bake one pan at a time. As soon as you put the pan in the oven, pour about 1 cup of hot water into the steam pan and lower the temperature to 450\u00b0F (232\u00b0C). Bake for 8 minutes, then rotate and bake for 8 to 12 minutes longer, until the crust is golden brown and the rolls sound hollow when thumped on the bottom. The internal temperature should be 200\u00b0F to 205\u00b0F (93\u00b0C to 96\u00b0C). Transfer to a wire rack and let cool for at least 10 minutes before serving.\n\nEPILOGUE\n\n# IS THE ROAD LESS TRAVELED THE ROAD AHEAD?\n\nBaker Mike Pappas lives near me and has some radical ideas about natural wild yeast starters. One day Mike came to visit me and brought three loaves, along with three containers, each holding a small piece of sourdough starter. \"They look alike, but each one is different,\" he asserted. Then he added another wrinkle, producing a small piece of Parmesan cheese, a ripe peach, and a few tablespoons of coffee beans. \"I can change the flavor profile of my starters simply by using a piece of fruit, coffee beans, or cheese, almost like a magnet, to trap different strains of yeast,\" he said.\n\n\"Really?\" I asked, skeptical but intrigued.\n\nMike had me uncover each of the starters in turn and take a whiff. Sure enough, each had a slightly different aroma. I figured he must have submerged those foods in the starters, but his actual method more than defied that expectation. Here's how he explained it: \"All I do is leave one of these on the top of the bowl when I first make my starter, and, well, something happens. Each seems to draw different strains of wild yeast or bacteria to the flour mixture, resulting in different flavors even though these ingredients never actually touch the dough.\" He told me he'd noticed that coffee beans, for example, draw microorganisms that produce a robust, deeper flavor, whereas those drawn to pears or stone fruits create flavors and aromas that tend to be more floral or smooth and less acidic.\n\nThen we tasted the breads, and I was surprised at how different each one was, and also impressed by how great each one tasted.\n\nDuring subsequent meetings, Mike explained more of his bread philosophy and theories, some of which seem highly speculative, but they do follow an intuitive logic. He told me, \"My goal from the start has been to create breads with excellent crumb and crust, great flavor, and also long shelf life. I'm looking for an A-plus crust, not only when the bread first comes out of the oven, but also after three or four days, if the bread is properly stored. I think the highest-quality gluten bonds are formed when the dough molecules first touch. That's why resting the dough is so important to my method. It minimizes mixing, which typically overorganizes the gluten. It preserves the initial, high-quality bonds and, in the end, releases more flavor.\"\n\n## Q & A\n\n### Could a juicer be used to make the fruit waters?\n\nYes, but Mike Pappas doesn't think the liquid would be as \"pure,\" which is why he still does it by hand (squeezing the fruit through a porous cloth).\n\n### Will the recipe work if I use only apple or only peach or only pear water?\n\nYes, but that's not the point. This is a bread method that symbolizes the depths of the bread quest. It doesn't just push the envelope, it shatters the envelope!\n\n### Why use bottled spring water?\n\nIt's all about controlling the flavor and purity of the final result. However, you can use flitered water if necesary.\n\n### Can I use my own mother starter instead of Mike's \"trap\" starters?\n\nIf you have any seed culture, sponge starter, or mother starter left over from the other recipes in this book, they could certainly be used in this chapter too. Would a different \"trap\" produce a noticeably different flavor? Mike Pappas definitely thinks so. Does anyone need to go through all these steps to make great bread? No, but it's all part of the discovery process.\n\nDuring the past few years, there's been a lot of interest in short, gentle mixing methods, including so-called no-knead mixing. But this was the first time I'd talked with someone who had hypotheses about why that method makes a difference. But mixing is only one part of Mike's overall methodology. He also believes strongly in what I'll call \"limited use\" natural starters, which is why he developed his wild yeast \"traps.\" Mike believes that to achieve the purest flavor, it's best to make a sourdough culture that finishes its life after a single round of baking.\n\nAs he explained it, \"The longer we perpetuate a starter, the more susceptible it becomes to elements we can't control\u2014other yeasts and bacteria that will affect the results, including the structure-building ability of the flour bonds. An older starter that has gone through many feedings and rebuildings may make fine bread, but there's an aging process that can introduce properties that aren't beneficial. I don't want this element of risk if I'm going for a very specific flavor and structure.\"\n\nWhen I asked him if it's possible to make his breads with a long-term, perpetuated starter, he said, \"Of course, but there will always be a measure of biological process within the starter's culture that, while natural to the starter, is unnecessary for baking the bread.\"\n\nI took this to mean that Mike's goal is to engage in a process that's as uncorrupted as possible while still working with natural wild yeast leaven. Every baker is, of necessity, a control freak; this is just a more extreme version. You may wonder whether it's necessary to exert this much control when, as so many artisan bakers have shown, it's possible to make consistently great bread with a perpetuated starter that's well maintained. For example, Boudin Bakery in San Francisco has been using the same starter, refreshed day after day, for over 160 years, and no one is complaining.\n\nStill, what Mike is doing, as he pushes the envelope of all the artisan techniques that we've discussed in this and other books to the outer edges of obsessiveness, reminds me of a similar approach in the world of beer (aka liquid bread), in the niche known as extreme brewing. As that niche amply demonstrates, willingness to ask what-if questions and boldly follow intuitive hunches can have extraordinary (and sometimes also negative) results. I think it's exciting that Mike and others like him are testing the limits in the bread world in a similar fashion. I encourage you to consider taking at least a brief ride on this next, next new wave in the bread revolution. Who knows where it will take you, but you're bound to expand your horizons considerably in the process.\n\n# FRAGRANT PEACH, APPLE, AND PEAR BREAD WITH \"PEACH TRAP\" STARTER\n\nMAKES 2 LOAVES\n\nThis formula showcases Mike Pappas's \"peach trap\" method, along with the use of apple, peach, and pear water to draw in particular microorganisms and fine-tune the flavor of the final bread. As you'll see, it's a complex and lengthy process (at least four days) and asks a lot from the baker, including hand squeezing not just one but three \"fruit waters.\" Given all that, you may wonder whether it's worth the effort. There's only one way to find out!\n\nYou may also quite reasonably ask whether, after all of the work of making the starter for this bread, it would be okay to perpetuate the starter rather than using it up? You can certainly perpetuate the starter to shortcut some of the preliminary steps next time you make the bread. In many ways, it's like the seed culture method described. Although Mike likes to start each batch of bread with a new culture in his quest for purity of flavor, most of us would be thrilled with loaves made from the original's refreshed progeny, batch after batch. You'll have to try it for yourself to see if you agree with his assessment.\n\nBecause this bread requires such a lengthy and unusual process, I'll reiterate my usual advice: The predicted time schedule for the various starters and stages is approximate and depends on the ambient temperature and other conditions in your kitchen. Adjust the times as needed, using the visual prompts and fermentation indicators as a guideline.\n\n### SEED CULTURE (DAY 1)\n\nlarge peach: 1\n\npeach water: 6 tablespoons (3 oz \/ 85 g)\n\nwhole wheat flour: 7 tablespoons (2 oz \/ 56.5 g)\n\noverly ripe peach: 1\n\nTOTAL: 5 oz \/ 141.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \npeach water| 150  \nwhole wheat flour| 100  \nTOTAL| 250\n\nLine a fine-mesh sieve with cheesecloth and place it over a bowl. Grate the peach into the lined sieve, and lift the cheesecloth to make a pouch. Squeeze the pouch directly over the sieve to extract as much peach juice as possible into the bowl. Put 6 tablespoons (3 oz \/ 85 g) of the liquid in a small bowl. (Discard, or eat, the pulp; if any liquid remains, store it in a covered container in the refrigerator and use it in the Firm Starter on Day 3.) Add the flour and stir to make a smooth, wet dough. Cover about three-quarters of the bowl tightly with plastic wrap and set the overly ripe peach on top of the plastic. Let rest at room temperature for 48 hours.\n\n### SPONGE STARTER (DAY 3)\n\nlarge apple: 1 oz\n\napple water: \u00bc cup (2 oz \/ 56.5 g)\n\nwhole wheat flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nwhole rye flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nSeed Culture: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nTOTAL: 4.5 oz \/ 127.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \napple water| 99  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nSeed Culture| 15  \nTOTAL| 214\n\nLine a fine-mesh sieve with cheesecloth and place it over a bowl. Grate the apple over the lined sieve, and lift the cheesecloth to make a pouch. Squeeze the pouch directly over the sieve to extract as much apple juice as possible into the bowl. Put \u00bc cup (2 oz \/ 56.5 g) of the juice in a small bowl (discard the pulp and drink, save, or discard any remaining juice). Add the flours and the seed culture and stir to make a wet, sponge-like batter. Cover the bowl with plastic wrap and ferment at room temperature for 24 hours. The starter should show signs of bubbling the next day.\n\n### FIRM STARTER (DAY 3)\n\nleftover peach water from Day 1: \u00bc cup (2 oz \/ 56.5 g)\n\nSeed Culture: 2 tablespoons (0.75 oz \/ 21.5 g)\n\nwhole wheat flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nwhole rye flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nTOTAL: 5.75 oz \/ 163 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nleftover peach water from Day 1| 67  \nSeed Culture| 17  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nTOTAL| 184\n\nIf you don't have enough leftover peach water from Day 1, make more using the same method. In a medium bowl, stir together all the ingredients to make a firm ball of dough. Transfer to a work surface and knead for about 30 seconds to evenly distribute all the ingredients. Return the dough to the bowl, cover the bowl with plastic wrap, and ferment at room temperature for 24 hours. The starter should swell slightly by the next day.\n\n### FINAL DOUGH (DAY 4)\n\napples: 2\n\npears: 2\n\napple water: \u00be cup (6 oz \/ 170 g)\n\npear water: \u00be cup (6 oz \/ 170 g)\n\nbottled spring water: 1\u00bc cups plus 2\u00bd tablespoons (11.25 oz \/ 319 g)\n\nSponge Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nFirm Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nunbleached bread flour: 3\u00bd cups (16 oz \/ 454 g)\n\nwhole wheat flour: 2\u00be cups plus 2 tablespoons (13 oz \/ 369 g)\n\nwhole rye flour: \u2154 cup (3 oz \/ 85 g)\n\nsalt: 1 tablespoon (0.75 oz \/ 21.5 g)\n\nTOTAL: 62.4 oz \/ 1,770.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \napple water| 19  \npear water| 19  \nbottled spring water| 35  \nSponge Starter| 10  \nFirm Starter| 10  \nunbleached bread flour| 50  \nwhole wheat flour| 40.6  \nwhole rye flour| 9.4  \nsalt| 2.4  \nTOTAL| 195.4\n\n1 Line a fine-mesh sieve with cheesecloth and place it over a bowl. Grate the apples over the sieve, and lift the cheesecloth to make a pouch. Squeeze the pouch over the sieve and bowl to extract as much liquid as possible. Put \u00be cup (6 oz \/ 170 g) of the juice in a large bowl (discard the pulp and drink, save, or discard any remaining juice). Follow the same procedure with the pears, then add \u00be cup (6 oz \/ 170 g) of the pear juice to the bowl. Add the water and both starters. Stir for about 15 seconds, just long enough to distribute the starters. Add the flours and stir for about 1 minute, until the flour is hydrated and a coarse, shaggy dough forms. Cover the bowl with plastic wrap and let the dough rest for 1 hour and 45 minutes.\n\n2 Oil a work surface and transfer the dough to the oiled area. Sprinkle half of the salt over the dough, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Sprinkle the remaining salt over the dough and perform two more sequences of stretching and folding without allowing the dough to rest between sequences. Return the dough to the bowl and cover the bowl with plastic wrap, or leave the dough on the work surface and cover it with the bowl. Then, at 1-hour intervals, perform two additional sequences of stretching and folding. The dough should be supple and tacky, with a bouncy quality when patted. After the final fold, cover the dough and let it rest at room temperature for 1 hour.\n\n3 Prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule or b\u00e2tard. Put the shaped loaves in the prepared proofing vessels. Proof at room temperature for about 5 hours, until the loaves have increased in size by more than 1\u00bd times but haven't doubled in size.\n\n4 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 435\u00b0F (224\u00b0C).\n\n5 Just before baking, dust a peel with flour. Transfer the loaves to the peel and score as desired. Slide the loaves onto the baking stone and pour about 1 cup of hot water into the steam pan.\n\n6 Bake for 25 minutes, then rotate and bake for about 25 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n7 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving, but note that the subtle flavor nuances can be more easily discerned after the bread cools completely, about 2 hours.\n\n# SEEDED MULTIGRAIN HEARTH BREAD WITH \"PARMESAN TRAP\" STARTER\n\nMAKES 2 LOAVES\n\nLike all of Mike Pappas's breads, this one involves a lot of steps and requires pretty much an all-day commitment on the final day. But if you're reading this, you must be intrigued. And, like Mike, you probably feel that the discovery process is as satisfying as the bread itself\u2014which is delicious, by the way, with a complex nutty flavor and just a hint of umami from the cheese. Despite how much is going on in this loaf, in the end it's about simple, pure flavor evoked from fermented grain. Mike's mantra, which he told me was inspired by Steve Jobs, is \"Traveling into the depths of complexity will allow us to offer our findings with grace and simplicity.\" That's certainly reflected in this bread.\n\nIf you don't have all the ingredients required for the seed and coating mixture, feel free to substitute other ingredients, such as sesame or poppy seeds, as long as the flavors are compatible. Also, be aware that potato flour is not the same as potato starch.\n\nBecause this bread requires such a lengthy and unusual process, I'll reiterate my usual advice: The predicted time schedule for the various starters and stages is approximate and depends on the ambient temperature and other conditions in your kitchen. Adjust the times as needed, using the visual prompts and fermentation indicators as a guideline.\n\n### SEED CULTURE (DAY 1)\n\napple: 1\n\napple water: 6 tablespoons (3 oz \/ 85 g)\n\nwhole wheat flour: \u2154 cup (3 oz \/ 85 g)\n\nrind of Parmesan or other hard, aged cheese (whole, not grated): about matchbook size\n\nTOTAL: 6 oz \/ 170 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \napple water| 100  \nwhole wheat flour| 100  \nTOTAL| 200\n\nLine a fine-mesh sieve with cheesecloth and place it over a bowl. Grate the apple over the lined sieve, and lift the cheesecloth to make a pouch. Squeeze the pouch directly over the sieve to extract as much juice as possible through the sieve into the bowl. Put 6 tablespoons (3 oz \/ 85 g) of the juice in a small bowl. (Discard, or eat, the pulp; if any juice remains, store it in a covered container in the refrigerator and use it in the Firm Starter on Day 3.) Add the flour and stir to make a smooth, wet dough. Cover about three-quarters of the bowl tightly with plastic wrap and set the cheese rind on top of the plastic. Let rest at room temperature for 48 hours.\n\n### SPONGE STARTER (DAY 3)\n\napple: 1\n\napple water: \u00bc cup (2 oz \/ 56.5 g)\n\nwhole wheat flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nwhole rye flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nSeed Culture: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nTOTAL: 4.5 oz \/ 127.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \napple water| 99  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nSeed Culture| 25  \nTOTAL| 224\n\nGrate the apple and make apple water again using the method in the Day 1 Seed Culture. Put \u00bc cup (2 oz \/ 56.5 g) of the liquid in a small bowl. Discard the pulp but reserve any remaining juice for use in the Firm Starter, below. Add the flours and seed culture and stir to make a wet, sponge-like batter. Cover the bowl with plastic wrap and ferment at room temperature for 24 hours. The starter should show signs of bubbling the next day.\n\n### FIRM STARTER (DAY 3)\n\nleftover apple water: \u00bc cup (2 oz \/ 56.5 g)\n\nSeed Culture: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nwhole wheat flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nwhole rye flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nTOTAL: 5.5 oz \/ 155.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nleftover apple water| 66  \nSeed Culture| 16  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nTOTAL| 182\n\nIf you don't have enough leftover apple water from Day 1 and the sponge above, make more using the same method. (If any juice remains, store it in a covered container in the refrigerator and use it in the final dough.) In a medium bowl, stir together all the ingredients to make a firm ball of dough. Transfer to a work surface and knead for about 30 seconds to evenly distribute all the ingredients. Return the starter to the bowl, cover the bowl with plastic wrap, and ferment at room temperature for 24 hours. The starter should swell slightly by the next day.\n\n### SEED AND COATING MIXTURE (DAY 4)\n\nsunflower seeds, toasted (see Toasting Seeds and Nuts for toasting instructions): (about 6 tablespoons oz \/ 2 g \/ 56.5%)\n\nflaxseeds: (about 6 tablespoons oz \/ 2 g \/ 56.5%)\n\nmillet: (about 6 tablespoons oz \/ 2 g \/ 56.5%)\n\nchia seeds: (about 6 tablespoons oz \/ 2 g \/ 56.5%)\n\nwheat bran: (1 cup oz \/ 4 g \/ 113%)\n\nrolled oats: (\u00bd cup oz \/ 3 g \/ 85%)\n\nTOTAL 15 oz \/ 424 g\n\nPut the sunflower seeds, flaxseeds, millet, chia seeds, and bran in a medium bowl and stir until evenly combined. Transfer one-quarter of the mixture to a separate bowl and stir in the rolled oats.\n\n### FINAL DOUGH (DAY 4)\n\napples: 2\n\napple water: 1\u00bc cups plus 2 \u00bd tablespoons (11.25 oz \/ 319 g)\n\nbottled spring water: 1\u00bc cups plus 2\u00bd tablespoons (11.25 oz \/ 319 g)\n\nSponge Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nFirm Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nParmesan or other hard, aged cheese, grated: about 2 tablespoons (0.5 oz \/ 14 g)\n\nunbleached bread flour: 3\u00bd cups (16 oz \/ 454 g)\n\nwhole wheat flour: 2\u00be cups plus 2 tablespoons (13 oz \/ 369 g)\n\nwhole rye flour: \u00bd cup (2.25 oz \/ 64 g \/ 7%)\n\npotato flour: 3 tablespoons (1 oz \/ 28.5 g \/ 3%)\n\nsalt: 1 tablespoon (0.75 oz \/ 21.5 g)\n\nSeed and Coating Mixture: all (15 oz \/ 424 g)\n\nTOTAL: 77.4 oz \/ 2,195 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \napple water| 35  \nbottled spring water| 35  \nSponge Starter| 9.9  \nFirm Starter| 9.9  \nParmesan or other hard, aged cheese| 1.5  \nunbleached bread flour| 50  \nwhole wheat flour| 40  \nwhole rye flour| 7  \npotato flour| 3  \nsalt| 2.3  \nSeed and Coating Mixture| 46.3  \nTOTAL| 239.9\n\n1 Grate the apples and make apple water, again using the same method. Put 1\u00bc cups plus 2\u00bd tablespoons (11.25 oz \/ 319 g) of the juice in a large bowl (discard the pulp and drink, save, or discard any remaining juice). Add the water, both starters, and the cheese. Stir for about 15 seconds, just long enough to distribute the starters. Add the flours and stir for about 1 minute, until the flour is hydrated and a coarse, shaggy dough forms. Cover the bowl with plastic wrap and let the dough rest for 1 hour and 45 minutes.\n\n2 Oil a work surface and transfer the dough to the oiled area. Sprinkle half of the salt and half of the seed mixture without the rolled oats over the dough. Stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Sprinkle the remaining salt and seeds over the dough and perform two more sequences of stretching and folding without allowing the dough to rest between sequences. Return the dough to the bowl and cover with plastic wrap, or leave the dough on the work surface and cover it with the bowl. Then, at 1-hour intervals, perform two additional sequences of stretching and folding. The dough will be supple and tacky, with a bouncy quality when patted. After the final fold, cover the dough and let it rest at room temperature for 1 hour.\n\n3 Prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule. Brush the top and sides of each ball with water, then roll the brushed surface in the rolled oat mixture to coat. Put the dough in the prepared proofing vessels, top (seeded) side down. Proof at room temperature for about 5 hours, until the dough increases in size by 1\u00bd times.\n\n4 About 45 minutes before you bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 420\u00b0F (216\u00b0C).\n\n5 Just before baking, dust a peel with flour. Transfer the loaves to the peel and score as desired. Slide the loaves onto the baking stone and pour about 1 cup of hot water into the steam pan.\n\n6 Bake for 25 minutes, then rotate and bake for about 25 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n7 Transfer to a wire rack and let cool for at least 30 minutes before slicing and serving, but note that the subtle flavor nuances can be more easily discerned after the bread cools completely, about 2 hours.\n\n# MOZZARELLA, MILK, AND PEAR BREAD WITH \"COFFEE-BEAN TRAP\" STARTER\n\nMAKES 2 LOAVES\n\nThis formula uses coffee beans as the \"yeast trap\" with pear water to evoke a slightly more robust flavor than the \"peach trap\" and \"Parmesan trap\" create. The flavor differences are subtle, but this process is all about possibilities and what-ifs. This bread has the wonderful addition of mozzarella cheese and milk, softening the texture to create a different kind of hearth bread from the ones to which we are accustomed, with a softer, rounder flavor followed by the long finish provided by the natural leaven with its own pear and coffee influence. Whether or not you ever make this bread, I think you'll get excited just reading about it. (But then, I do hope you'll make it.)\n\n### SEED CULTURE (DAY 1)\n\npear: 1\n\npear water: 6 tablespoons (3 oz \/ 85 g)\n\nwhole wheat flour: \u2154 cup (3 oz \/ 85 g)\n\nwhole coffee beans (any roast): about 3 tablespoons\n\nTOTAL: 6 oz \/ 170 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \npear water| 100  \nwhole wheat flour| 100  \nTOTAL| 200\n\nLine a fine-mesh sieve with cheesecloth and place it over a bowl. Grate the pear over the lined sieve, and lift the cheesecloth to make a pouch. Squeeze the pouch directly over the sieve to extract as much pear juice as possible into the bowl. Put 6 tablespoons (3 oz \/ 85 g) of the juice in a small bowl. (Discard the pulp; if any juice remains, store it in a covered container in the refrigerator and use it in the Firm Starter on Day 3.) Add the flour and stir to make a smooth, wet dough. Cover about three-quarters of the bowl tightly with plastic wrap. Put the coffee beans in a small plastic container and set it on top of the plastic. Let rest at room temperature for 48 hours.\n\n### SPONGE STARTER (DAY 3)\n\npear: 1\n\npear water: \u00bc cup (2 oz \/ 56.5 g)\n\nwhole wheat flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nwhole rye flour: 3\u00bd tablespoons (1 oz \/ 28.5 g)\n\nSeed Culture: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nTOTAL: 4.5 oz \/ 127.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \npear water| 99  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nSeed Culture| 25  \nTOTAL| 224\n\nGrate the pear and make pear water again using the method in the Day 1 Seed Culture. Put \u00bc cup (2 oz \/ 56.5 g) of the juice in a small bowl. (Discard the pulp but reserve any remaining juice for use in the Firm Starter, below.) Add the flours and seed culture and stir to make a wet, sponge-like batter. Cover the bowl with plastic wrap and ferment at room temperature for 24 hours. The starter should show signs of bubbling the next day.\n\n### FIRM STARTER (DAY 3)\n\nleftover pear water: \u00bc cup (2 oz \/ 56.5 g)\n\nSeed Culture: 1\u00bd tablespoons (0.5 oz \/ 14 g)\n\nwhole wheat flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nwhole rye flour: \u2153 cup (1.5 oz \/ 42.5 g)\n\nTOTAL: 5.5 oz \/ 155.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \nleftover pear water| 66  \nSeed Culture| 16  \nwhole wheat flour| 50  \nwhole rye flour| 50  \nTOTAL| 182\n\nIf you don't have enough leftover pear water from Day 1 and the sponge above, make more using the same method. (However, you will probably have enough. If any liquid remains, store it in a covered container in the refrigerator and use it in the final dough.) In a medium bowl, stir together all the ingredients to make a firm ball of dough. Transfer to a work surface and knead for about 30 seconds to evenly distribute all the ingredients. Return the starter to the bowl, cover the bowl with plastic wrap, and ferment at room temperature for 24 hours. The dough should swell slightly by the next day.\n\n### FINAL DOUGH (DAY 4)\n\npears: 2\n\npear water: \u00bd cup plus 3 tablespoons (5.65 oz \/ 160 g)\n\nbottled spring water: 1 cup plus 6\u00bd tablespoons (11.25 oz \/ 319 g)\n\nmilk (whole or low-fat): \u00bd cup plus 3 tablespoons (5.65 oz \/ 160 g)\n\nSponge Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nFirm Starter: \u00bd cup (3.2 oz \/ 91 g)\n\nunbleached bread flour: 7 cups (32 oz \/ 907 g)\n\nmozzarella cheese, grated: about 3 cups (13 oz \/ 369 g)\n\nsalt: 1 tablespoon (0.75 oz \/ 21.5 g)\n\nTOTAL: 74.7 oz \/ 2,118.5 g\n\n### BAKER'S PERCENTAGE FORMULA\n\nIngredient| %  \n---|---  \n|   \npear water| 18  \nbottled spring water| 35  \nmilk| 18  \nSponge Starter| 10  \nFirm Starter| 10  \nunbleached bread flour| 100  \nmozzarella cheese| 41  \nsalt| 2.4  \nTOTAL| 234.4\n\n1 Grate the pears and make pear water again using the same method. Put \u00bd cup plus 3 tablespoons (5.65 oz \/ 160 g) of the pear water in a large bowl (save, drink, or discard any remaining juice). Add the water, milk, and both starters and stir for about 15 seconds, just long enough to distribute the starters. Add the flour and cheese and stir for about 1 minute, until the flour is hydrated, the cheese is evenly distributed, and a coarse, shaggy dough forms. Cover the bowl with plastic wrap and let the dough rest for 1 hour and 45 minutes.\n\n2 Oil a work surface and transfer the dough to the oiled area. Sprinkle half of the salt over the dough, then stretch and fold the dough, folding it over itself four times: once each from the top, bottom, and sides. Sprinkle the remaining salt over the dough and perform two more sequences of stretching and folding without allowing the dough to rest between sequences. Return the dough to the bowl and cover with plastic wrap, or leave the dough on the work surface and cover it with the bowl. Then, at 1-hour intervals, perform two additional sequences of stretching and folding. The dough will be supple and tacky, with a bouncy quality when patted. After the final fold, cover the dough and let it rest at room temperature for 1 hour.\n\n3 Prepare two bannetons or a couche. Divide the dough in half and shape each piece into a boule or b\u00e2tard. Put the shaped loaves in the prepared proofing vessels. Proof at room temperature for about 5 hours, until the loaves have increased in size by 1\u00bd times.\n\n4 About 45 minutes before you plan to bake, prepare the oven for hearth baking with a baking stone and steam pan. Preheat the oven to 420\u00b0F (216\u00b0C).\n\n5 Just before baking, dust a peel with flour. Transfer the loaves to the peel and score as desired. Slide the loaves onto the baking stone and pour about 1 cup of hot water into the steam pan.\n\n6 Bake for 25 minutes, then rotate and bake for about 25 minutes longer, until the crust is golden brown and the bread sounds hollow when thumped on the bottom. The internal temperature should be at least 200\u00b0F (93\u00b0C).\n\n7 Transfer to a wire rack and let cool for at least 30 minutes before serving, but note that the subtle flavor nuances can be more easily discerned after the bread cools completely, about 2 hours.\n\n# RESOURCES\n\nBecause this is such a new and open frontier, the amount of tools, ingredients, and general information is on an accelerating upward trajectory. What is printed here may well be out of date before this book ever hits the shelves, but at least it will provide you with some resources to get started.\n\nI've organized these resources into three categories: articles, books, and links; ingredients; and tools. Given the ever-expanding state of the Internet and the dynamic nature of commerce, I make no promises that these links are still neither viable nor exhaustive, but they have proven useful to me as I explored the depths of the current bread revolution. I will post these and ongoing updates on my Facebook page.\n\n## ARTICLES, BOOKS, AND LINKS\n\n#### THE BREAD BAKERS GUILD OF AMERICA\n\nEvery serious bread baker should be a member. The quarterly newsletters and technical reports are worth the price of admission, as is access to the many workshops and classes.\n\nbbga.org\n\n#### FLOUR POWER: A GUIDE TO MODERN HOME GRAIN MILLING BY MARLEETA F. BASEY (JERMAR PRESS)\n\nAn excellent reference for anyone interested in home milling.\n\n#### IN SEARCH OF THE PERFECT LOAF BY SAMUEL FROMARTZ (VIKING)\n\nA well-written memoir and survey of some of the newest findings in bread science and technique.\n\n#### THE FRESH LOAF\n\nA wonderful website with helpful information on home milling:\n\nthefreshloaf.com\/node\/24640\/home-milling\n\n#### \"FOR OLD-FASHIONED FLAVOR, BAKE THE BAKING SODA\" BY HAROLD MCGEE (NEW YORK TIMES)\n\nHarold McGee's article on how to make baking soda perform like lye for pretzels:\n\nnytimes.com\/2010\/09\/15\/dining\/15curious.html\n\n#### NICKY GIUSTO'S AWARD WINNING RAISIN BREAD RECIPE\n\n(Raisin de Soleil):\n\ncalraisins.org\/recipe\/raisin-de-soleil-grape-of-the-sun\n\n#### \"NUTRITIONAL IMPROVEMENT OF CEREALS BY SPROUTING\" BY J.K. CHAUAN AND S.S. KADAM (CRITICAL REVIEWS IN FOOD SCIENCE AND NUTRITION)\n\nThis article from 1989 shows evidence that \"sprouting of grains for a limited period causes increased activities of hydrolytic enzymes, improvement in the contents of certain essential amino acids, total sugars, and B-group vitamins, and a decrease in dry matter, starch, and antinutrients. The digestibility of storage proteins and starch are improved due to their partial hydrolysis during sprouting.\"\n\nncbi.nlm.nih.gov\/pubmed\/2692609\n\nThe following links all lead to articles on the health benefits of sprouted grains:\n\nonegreenplanet.org\/vegan-health\/sprouted-grains-are-they-a-healthy-choice\n\nonegreenplanet.org\/vegan-health\/flour-power-10-reasons-you-should-bake-with-sprouted-whole-grain-flour\n\nonegreenplanet.org\/natural-health\/is-going-grain-free-better-for-your-health\n\n#### \"WHEAT BELLY\u2014AN ANALYSIS OF SELECTED STATEMENTS AND BASIC THESES FROM THE BOOK\" BY JULIE JONES (CEREAL FOODS WORLD)\n\nThe most widely cited refutation of the book Wheat Belly by Dr. William Davis:\n\naaccnet.org\/publications\/plexus\/cfw\/pastissues\/2012\/OpenDocuments\/CFW-57-4-0177.pdf\n\nOther articles that rebut the Wheat Belly claims:\n\nthebestgrains.com\/wheat-is-not-unhealthy-a-rebuttal-to-recent-claims\n\nforksoverknives.com\/the-smoke-and-mirrors-behind-wheat-belly-and-grain-brain\n\nAlso, go to the Whole Grains Council website for all current news on the subject: wholegrainscouncil.org\n\n## INGREDIENTS\n\n#### Sprouted Flour\n\nIn this book we focused primarily on two millers, Lindley Mills and To Your Health Sprouted Flour Co., but there are other producers and, most likely, new names will continue to join this list. The websites below also provide incredibly useful information:\n\n#### ANITA'S ORGANIC MILL\n\nanitasorganic.com\n\n(604) 823-5547\n\n#### ARROWHEAD MILLS\n\narrowheadmills.com\n\n(800) 434-4246\n\n#### ESSENTIAL EATING SPROUTED FOODS\n\nessentialeating.com\n\n#### LINDLEY MILLS\n\nLindleyMillsInc.com\n\n(336) 376-6190\n\n#### ONE DEGREE ORGANIC FOODS\n\nonedegreeorganics.com\n\n#### SHILOH FARMS\n\nshilohfarms.com\n\n#### TO YOUR HEALTH SPROUTED FLOUR\n\nhealthyflour.com\n\n(877) 401-6837\n\n#### Sprouted Pulp (mash)\n\nSprouted pulp is the key ingredient in Ezekiel and Alvarado Street breads. You can either make it yourself or it is now available in many varieties of grain to both professional and home bakers from Central Milling. Their grain mills are located in Logan, Utah, but the office headquarters is in Petaluma, California. Write to them at flour@centralmilling.com or call (707) 778-1073. The following website lists all their products: centralmilling.com\n\n#### Food Grade Lye (Sodium Hydroxide)\n\nThere are many sources on the Internet, including Amazon.com and essentialdepot.com.\n\n#### Cascara Seca (dried coffee cherry husks, aka coffee flour)\n\nThis product is now increasingly available from many sources, but the two I personally know are:\n\n#### BOQUETE MOUNTAIN COFFEE\n\nboquetemountaincoffee.com\n\n(704) 243-8900\n\n#### COFFEE FLOUR\u2122\n\ncoffeeflour.com\n\n#### Community Grains\n\nIdentity preserved whole grain products.\n\nTheir website is communitygrains.com.\n\n#### Grape Seed and Grape Skin Flour\n\nThis promises to become a growth industry as more and more wineries learn how to recycle their seeds and skins into healthful, functional products. For now, however, there is one company leading the charge:\n\n#### WHOLE VINE PRODUCTS\n\nwholevine.com\n\n#### ProBiotein (prebiotic fiber and protein supplement)\n\nprobiotein.com\n\n#### Whole Milled and Regional Flour Mills\n\nThere are many more than what's listed here, with new ones opening all the time, so use this list as a starting point and seek out mills closer to where to you live:\n\n#### ANSON MILLS (RED FIFE, ETC.)\n\nColumbia, South Carolina\n\nansonmills.com\n\n#### BOB'S RED MILL (POTATO FLOUR, ETC.)\n\nPortland, Oregon\n\nbobsredmill.com\n\n#### CAMAS COUNTRY MILL\n\nAlvadore, Oregon\n\ncamascountrymill.com\n\n#### CAROLINA GROUND\n\nAsheville, North Carolina\n\ncarolinaground.com\n\n#### CENTRAL MILLING\n\nPetaluma, California\n\ncentralmilling.com\n\n#### CERTIFIED FOODS, INC (BAY STATE MILLING)\n\nWoodland, California\n\ncertifiedfoods.com\n\n#### FAIRHAVEN ORGANIC FLOUR MILL\n\nBellingham, Washington\n\nfairhavenflour.com\n\n#### FARMER GROUND FLOUR\n\nTrumansburg, New York\n\nfarmergroundflour.squarespace.com\n\n#### HEARTLAND MILL\n\nMarienthal, Kansas\n\nheartlandmill.com\n\n#### MAINE GRAINS AT THE SOMERSET GRIST MILL\n\nSkowhegan, Maine\n\nmainegrains.com\n\n#### RIVERBEND MALT HOUSE\n\nMany micro malt houses are opening across the country and partnering with local, small flour mills, such as Riverbend and Carolina Ground.\n\nAsheville, North Carolina\n\nriverbendmalt.com\n\n## TOOLS\n\n#### Sprouting Equipment\n\nThere are many companies, too many to list here, selling small and large sprouting equipment for both home and commercial application. Two that can get you started for setting up sprouting systems are:\n\n#### THE MAIL ORDER CATALOG FOR HEALTHY EATING\n\nhealthy-eating.com\/category\/75\n\n#### SPROUT PEOPLE\n\nsproutpeople.org\/sprouting-supplies\n\n#### Home Milling and Pulping Equipment\n\nTo turn sprouted grain into mash or pulp, you can use any brand of meat grinder, hand cranked or electric, set to a medium grind. I use my Kitchen Aid mixer with its detachable meat grinder, but you will find many variations at stores such as Lowe's and Home Depot for hand-cranked units, as well as attachments for brands such as Cuisinart and Waring.\n\n#### General Baking Tools\n\nSur la Table (surlatable.com) and The King Arthur Flour Baking Catalog (kingarthurflour.com) are still the gold standards for general baking supplies, but CHEFS (chefscatalog.com) also has almost anything you will need. Stores (and catalogs) like Bed, Bath, and Beyond; Williams-Sonoma; and even Walmart and Target have most of the basics too.\n\n# ACKNOWLEDGMENTS\n\nAs always, my biggest thanks goes to my wife, Susan, who supported me throughout this two-year process of research trips, all-night writing, and baking sessions with patience, advice, and great soups. She also made sure I got up from the computer every half-hour to walk around and, for the first time in memory, my back did not go out during the final push. For that alone, I will be eternally grateful!\n\nThe Ten Speed Press team was fabulous, as they always are. This is the third book I've done with my editor Melissa Moore, who is a wonderful collaborator. I wish her well as she moves into her new advocacy work in the non-profit sector, but hope we get to work together again in one way or another. Thank you also to editorial assistant Ali Slagle, who brought the final manuscript all the way home, as well as associate art director Katy Brown, who really grasped the essence of this book and came up with a beautiful presentation. My thanks also go to publisher Aaron Wehner, Michele Crim in marketing, and Kara van de Water in publicity.\n\nThank you to the photo team, headed by photographer Paige Green, who miraculously turned a bakery warehouse into a photo studio; food stylist Karen Shinto (this is our third book together!); food styling assistant Jeffrey Larsen; photography assistants Debbie Wilson and Morgan Bellinger; prop stylist Tessa Watson; and prop assistant Alexis Scarborough. Thank you to Heritage Salvage, who donated the photography surfaces. And a special thanks to copyeditor Jasmine Star on our fourth collaboration.\n\nThis was a tricky book when it came to testing recipes since many of the ingredients were not, at the time, locally available and needed to be shipped. I was fortunate to find the perfect tester, home baker Paul Scivetti, who went far beyond the call of duty, not only testing recipes but also catching measurement inconsistencies, making valuable suggestions, and using these formulas as a launching point for his own bread frontier explorations.\n\nPeggy and Jeff Sutton, as well as Joe Lindley, were extremely generous in providing us with ample supplies of sprouted flour, as were Nicky and Keith Giusto, who not only provided sprouted grain mash but also allowed us to use their baking lab in Petaluma for the week-long photo shoot. Craig Ponsford was an invaluable connector and, of course, his breads totally rock!\n\nThanks also to the many experts, bakers, millers, and \"next frontier\" revolutionaries who let me interview them and pick their brains for information and recipes, including Michael Pollan, Mike Pappas, Harry Peemoeller, Bob Thornberg, David Haddock, Dr. Steve Jones, Dr. Harold \"Dusty\" Dowse, Janice Cooper, Mark Shigenaga, Joe Vanderliet, Bob and Maggie Klein, Peggy Furth, Paul Novak, Torey Arvik, Justin Wangler, Eric Frischkorn, Tucker Taylor, Jennifer Lapidus, and Alan Balshan.\n\nFinally, thanks to Johnson & Wales University in Charlotte for providing a nurturing environment that allows me to forge these uncharted territories, especially President Art Gallagher, Vice President Tarun Malik, Culinary Dean Mark Allison, Baking and Pastry Department Chair Amy Felder, Laura Benoit (proofreader extraordinaire), and my many inspired colleagues on all four JWU campuses.\n\n# About the Author\n\nPETER REINHART is a baking instructor and faculty member at Johnson and Wales University in Charlotte, North Carolina. He was the cofounder of Brother Juniper's Bakery in Santa Rosa, California, and is the author of ten books on bread baking, including Crust and Crumb, The Bread Baker's Apprentice (winner of the 2002 James Beard Cookbook of the Year and IACP Cookbook of the Year), and the 2008 James Beard Award\u2014winning Peter Reinhart's Whole Grain Breads.\n\nPeter is the founder and host of the popular website PizzaQuest.com, where he continues to chronicle his never-ending search for the perfect pizza through videos, essays, and recipes. He also has created two instructional video courses, on artisan bread and on pizza, for Craftsy.com.\n\n# INDEX\n\nA\n\nAlmonds\n\nApricot Almond Quick Bread\n\nGluten-Free Holiday Biscotti\n\nGluten-Free Holiday Cookies\n\nGluten-Free Many-Seed Toasting Bread\n\nSprouted Emmer Pulp Power Bread\n\ntoasting\n\nAncient grains\n\ndefinition of, 1.1, 3.1\n\ngluten and\n\nsubstituting\n\nApples\n\nApple Bran Quick Bread or Muffins\n\nFragrant Peach, Apple, and Pear Bread with \"Peach Trap\" Starter\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nApricot Almond Quick Bread\n\nArvik, Torey\n\nAutolyse method\n\nB\n\nBacon\n\nBacon Corn Bread\n\nPotato, Bacon, and Parmesan Focaccia\n\nBagels\n\nCinnamon Raisin Bagels\n\nCinnamon-Sugar Bagels\n\n\"Everything\" Bagels\n\nfloat test for\n\nMultigrain Bagels\n\nRaisin Bagels\n\nshaping\n\nSprouted Kamut Pulp Bagels\n\nSprouted Wheat Bagels\n\nBaguettes, shaping\n\nBaker's math\n\nBanana Quick Bread or Muffins\n\nBanke, Barbara\n\nBannetons, 1.1, 1.2\n\nB\u00e2tards, shaping, 1.1, 1.2, 1.3\n\nBeer, as liquid bread\n\nBenner, Diana\n\nBiscotti, Gluten-Free Holiday\n\nBiscuits, Flaky Sprouted Wheat\n\nBlueberries\n\nBlueberry Pancakes\n\nBlueberry Quick Bread or Muffins\n\nBorlaug, Norman\n\nBoules, shaping, 1.1, 1.2\n\nBread flour\n\nBreads. See Quick breads; Sourdough breads; Yeasted breads\n\nBrioche, Sprouted Wheat Sweet Potato\n\nBuns\n\nshaping\n\nSprouted Wheat Cinnamon Buns\n\nSticky Buns\n\nButter\n\nC\n\nCalvel, Raymond\n\nCarrot Quick Bread or Muffins\n\nCascara seca, 6.1, 6.2\n\nCascara Seca Lean Bread\n\nChallah, Sprouted Wheat\n\nCheese\n\nCheese Crackers\n\nGluten-Free Focaccia\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\nMozzarella, Milk, and Pear Bread with \"Coffee Bean Trap\" Starter\n\nPotato, Bacon, and Parmesan Focaccia\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nCherry Quick Bread or Muffins, Chocolate\n\nChia seeds\n\nGluten-Free Many-Seed Toasting Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nChocolate\n\nChocolate Cherry Quick Bread or Muffins\n\nChocolate Chip Quick Bread or Muffins\n\nChocolate Croissants\n\nChocolate Glaze\n\nCiabatta, Whole-Milled Whole Wheat\n\nCinnamon\n\nCinnamon Raisin Bagels\n\nCinnamon Raisin Quick Bread\n\nCinnamon-Sugar Bagels\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\n\"Coffee Bean Trap\" Starter, Mozzarella, Milk, and Pear Bread with\n\nConvection ovens\n\nCookies, Gluten-Free Holiday\n\nCorn flour, sprouted\n\nBacon Corn Bread\n\nGluten-Free \"Do No Harm\" Sprouted Grain Bread\n\nGluten-Free Sprouted Corn Bread with Teff\n\nGluten-Free Sprouted Grain Crackers\n\nSouthern-Style Corn Bread\n\nSprouted Corn Bread\n\nCouches, 1.1, 1.2\n\nCrackers\n\nCheese Crackers\n\nGluten-Free Sprouted Grain Crackers\n\nGrape Skin Flour Crackers\n\nHerb Crackers\n\nSprouted Multigrain Crackers\n\nCranberries\n\nCranberry Quick Bread or Muffins\n\nSprouted Wheat Breakfast Focaccia\n\nCroissants\n\nChocolate Croissants\n\nSprouted Wheat Croissants\n\nCrust, technique for crisper\n\nCurrant Pretzels, Whole Wheat\n\nD\n\nDarby, Heather\n\n\"Do No Harm\" Sprouted Grain Bread, Gluten-Free\n\nDowse, Harold\n\nDutch oven baking method\n\nE\n\nEggs\n\nsize of\n\nwash\n\nEmmer Pulp Power Bread, Sprouted\n\nEnglish muffins\n\nFlavored English Muffins\n\nRaisin Nut English Muffins\n\nWhole Wheat and Raisin English Muffins\n\nEnzymes, definition of\n\n\"Everything\" Bagels\n\nF\n\nFalling number\n\nFermentation\n\nbacterial\n\ndefinition of\n\nyeast\n\nFlaxseeds\n\nGluten-Free Many-Seed Toasting Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nSprouted Multigrain Crackers\n\nFloat test, 1.1, 1.2\n\nFlores, Max\n\nFlour\n\nin formulas\n\nmeasuring\n\nreconstituted milling method for\n\nwhole milling method for\n\nSee also individual flours\n\nFocaccia\n\nassembling\n\nGluten-Free Focaccia\n\nMultigrain Focaccia\n\nPotato, Bacon, and Parmesan Focaccia\n\nSprouted Wheat Breakfast Focaccia\n\nSprouted Wheat Focaccia\n\nTomato Pesto Focaccia\n\nFondant Glaze, Quick\n\nFormulas\n\ndefinition of\n\nmaster\n\nFrench Bread, Whole-Milled Lean Dough\n\nFrischkorn, Eric, 6.1, 6.2, 6.3\n\nFurth, Peggy\n\nG\n\nGiusto, Keith and Nicky, 4.1, 4.2, 4.3, 4.4, 4.5\n\nGlazes\n\nChocolate Glaze\n\nLemon Glaze\n\nOrange Glaze\n\nQuick Fondant Glaze\n\nGluten\n\ndefinition of\n\nsensitivity, 5.1, 5.2, 6.1\n\nGluten-free baking\n\nancient grains and\n\nGluten-Free \"Do No Harm\" Sprouted Grain Bread\n\nGluten-Free Focaccia\n\nGluten-Free Holiday Biscotti\n\nGluten-Free Holiday Cookies\n\nGluten-Free Many-Seed Toasting Bread\n\nGluten-Free Sprouted Corn Bread with Teff\n\nGluten-Free Sprouted Flour Pizza Dough\n\nGluten-Free Sprouted Grain Crackers\n\nGrape skin and grape seed flour\n\nGluten-Free Focaccia\n\nGrape Skin Flour Crackers\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\nSyrah Grape Skin Flour Olive Bread\n\nH\n\nHaddock, David\n\nHamelman, Jeffrey, 2.1, 3.1\n\nHearth baking\n\nHearth breads\n\nCarolina Wheat Hearth Bread with Nuts\n\nNaturally Leavened Carolina Wheat Hearth Bread\n\nSeeded Carolina Wheat Hearth Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nHerbs\n\nHerb Crackers\n\nHerbed Quick Bread\n\nHerb Oil\n\nI\n\nIdentity-preserved grain\n\nJ\n\nJones, Stephen\n\nK\n\nKamut Pulp Bagels, Sprouted\n\nKlein, Bob\n\nL\n\nLahey, Jim\n\nLapidus, Jennifer, 4.1, 5.1, 5.2\n\nLazor, Jack\n\nLeaven, definition of\n\nLemons\n\nLemon Glaze\n\nLemon Poppy Seed Quick Bread\n\nLevain, definition of\n\nLimpa Rye, Swedish-Style\n\nLindley, Joe, itr.1, 3.1, 3.2, 3.3, 3.4\n\nLye, working with\n\nM\n\nMalting\n\nMaster formulas, definition of\n\nMcGee, Harold\n\nMeasuring, scoop and sweep method for, 1.1, 1.2\n\nMilk\n\nMixing methods\n\nMozzarella, Milk, and Pear Bread with \"Coffee Bean Trap\" Starter\n\nMuffins\n\nApple Bran Muffins\n\nBanana Muffins\n\nBlueberry Muffins\n\nCarrot Muffins\n\nChocolate Cherry Muffins\n\nChocolate Chip Muffins\n\nCranberry Muffins\n\nPoppy Seed Muffins\n\nSprouted Wheat Muffins\n\nZucchini Muffins\n\nSee also English muffins\n\nMultigrain recipes\n\nMultigrain Bagels\n\nMultigrain Focaccia\n\nMultigrain Sprouted Pizza Dough\n\nMultigrain Sprouted Wheat Pulp Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nSprouted Multigrain Crackers\n\nSprouted Multigrain Pancakes\n\nSprouted Pain au Levain\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\nSprouted Wheat Pulp Bread with Sprouted Multigrain Flour\n\nN\n\nNovak, Paul\n\nNuts\n\nCarolina Wheat Hearth Bread with Nuts\n\ntoasting\n\nSee also individual nuts\n\nO\n\nOat bran\n\nApple Bran Quick Bread or Muffins\n\nOils\n\nHerb Oil\n\ntypes of\n\nOil slick method\n\nOlive Bread, Syrah Grape Skin Flour\n\nOnion Rye, 3.1, 3.2\n\nOrange Glaze\n\nP\n\nPain au levain\n\nHigh-Extraction Pain au Levain\n\nSprouted Pain au Levain\n\nPancakes\n\nBlueberry Pancakes\n\nSprouted Multigrain Pancakes\n\nSprouted Wheat Pancakes\n\nPan sizes\n\nPappas, Mike, epl.1, epl.2\n\nPeach, Apple, and Pear Bread, Fragrant, with \"Peach Trap\" Starter\n\nPears\n\nFragrant Peach, Apple, and Pear Bread with \"Peach Trap\" Starter\n\nMozzarella, Milk, and Pear Bread with \"Coffee Bean Trap\" Starter\n\nPecans\n\nBanana Quick Bread or Muffins\n\nCarrot Quick Bread or Muffins\n\nGluten-Free Holiday Biscotti\n\nGluten-Free Holiday Cookies\n\nGluten-Free Many-Seed Toasting Bread\n\nRaisin Nut English Muffins\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\ntoasting\n\nZucchini Quick Bread or Muffins\n\nPeemoeller, Harry, 3.1, 3.2, 5.1, 5.2, 5.3\n\nPineapple juice, starter with, 2.1, 2.2\n\nPizza dough\n\nGluten-Free Sprouted Flour Pizza Dough\n\nMultigrain Sprouted Pizza Dough\n\nshaping\n\nSicilian-Style Pizza\n\nSprouted Wheat Pizza Dough\n\nWhole-Milled Whole Wheat Pizzas\n\nPonsford, Craig, 5.1, 5.2, 5.3, 5.4, 5.5, 5.6, 6.1, 6.2\n\nPoppy seeds\n\nLemon Poppy Seed Quick Bread\n\nPoppy Seed Quick Bread or Muffins\n\nPotato, Bacon, and Parmesan Focaccia\n\nPrebiotics\n\nPre-ferments\n\ndefinition of\n\npurpose of\n\nwith sprouted flour breads\n\nPretzels, Whole Wheat Currant\n\nProBiotein, 6.1, 6.2\n\nProBiotein Lean Bread\n\nProbiotics\n\nProofing, 1.1, 1.2\n\nPulp breads, sprouted\n\nhomemade pulp for\n\nMultigrain Sprouted Wheat Pulp Bread\n\nroots of\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Kamut Pulp Bagels\n\nSprouted Wheat Pulp Bread with Sprouted Multigrain Flour\n\nPumpkin seeds\n\nSeeded Carolina Wheat Hearth Bread\n\nSprouted Multigrain Crackers\n\ntoasting\n\nQ\n\nQuick breads\n\nApple Bran Quick Bread\n\nApricot Almond Quick Bread\n\nBacon Corn Bread\n\nBanana Quick Bread\n\nBlueberry Quick Bread\n\nCarrot Quick Bread\n\nChocolate Cherry Quick Bread\n\nChocolate Chip Quick Bread\n\nCinnamon Raisin Quick Bread\n\nCranberry Quick Bread\n\nGluten-Free Many-Seed Toasting Bread\n\nGluten-Free Sprouted Corn Bread with Teff\n\nHerbed Quick Bread\n\nLemon Poppy Seed Quick Bread\n\nPoppy Seed Quick Bread\n\nRosemary Walnut Quick Bread\n\nSouthern-Style Corn Bread\n\nSprouted Corn Bread\n\nSprouted Spelt Quick Bread\n\nSprouted Wheat Quick Bread, 3.1, 3.2\n\nZucchini Quick Bread\n\nR\n\nRaisins\n\nCarrot Quick Bread or\n\nMuffins\n\nCinnamon Raisin Bagels\n\nCinnamon Raisin Quick\n\nBread\n\nRaisin Bagels\n\nRaisin Nut English Muffins\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Wheat Breakfast Focaccia\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\nstarter with\n\nWhole Wheat and Raisin English Muffins\n\nWhole Wheat Currant Pretzels\n\nZucchini Quick Bread or Muffins\n\nRecipes, definition of\n\nReconstituted milling\n\nRice flour\n\nGluten-Free \"Do No Harm\" Sprouted Grain Bread\n\nGluten-Free Focaccia\n\nGluten-Free Sprouted Grain Crackers\n\nRobertson, Chad, 3.1, 3.2\n\nRolls\n\nCascara Seca Lean Bread\n\nMultigrain Sprouted Wheat Pulp Bread\n\nProBiotein Lean Bread\n\nshaping\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Rye Bread\n\nSprouted Struan Bread\n\nSprouted Wheat Cinnamon Sweet Rolls\n\nSprouted Wheat Pulp Bread with Sprouted Multigrain Flour\n\nSprouted Wheat Soft Rolls\n\nSprouted Whole Wheat Bread\n\nRoom temperature, definition of\n\nRosemary Walnut Quick Bread\n\nRye berries\n\ncooking\n\nSprouted Vollkornbrot\n\nRye flour, sprouted\n\nDark Sandwich Rye\n\nOnion Rye, 3.1, 3.2\n\nSeeded Rye\n\nSprouted Rye Bread\n\nSprouted Sandwich Rye Bread\n\nSprouted Vollkornbrot\n\nSwedish-Style Limpa Rye\n\nS\n\nSalt\n\nmeasuring\n\ntypes of\n\nSandwich loaves, shaping, 1.1, 1.2\n\nScoop and sweep measuring, 1.1, 1.2\n\nScoring\n\nSeeded Rye\n\nSeeds\n\nGluten-Free Many-Seed Toasting Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\ntoasting\n\nSee also individual seeds\n\nShaping techniques\n\nSicilian-Style Pizza\n\nSourdough breads\n\nCarolina Wheat Hearth Bread with Nuts\n\ndefinition of\n\nFragrant Peach, Apple, and Pear Bread with \"Peach Trap\" Starter\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\nHigh-Extraction Pain au Levain\n\nMozzarella, Milk, and Pear Bread with \"Coffee Bean Trap\" Starter\n\nNaturally Leavened Carolina Wheat Hearth Bread\n\nOnion Rye\n\nSeeded Carolina Wheat Hearth Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nSeeded Rye\n\nSprouted Pain au Levain\n\nSprouted Rye Bread\n\nSwedish-Style Limpa Rye\n\nSyrah Grape Skin Flour Olive Bread\n\nSee also Starters\n\nSouthern-Style Corn Bread\n\nSpelt Quick Bread, Sprouted\n\nSprouted grain flours\n\nbenefits of, itr.1, 3.1\n\ngluten and, itr.1, 3.1\n\nmaking\n\npre-ferments with\n\nproducers of, itr.1, 3.1, 3.2\n\nstarters with, 2.1, 3.1\n\nsubstituting nonsprouted for\n\nSee also individual flours\n\nStarters\n\ndefinition of\n\nfunctions of\n\nmother, 1.1, 2.1\n\nnames for\n\nwith pineapple juice, 2.1, 2.2\n\nwith raisin water\n\nrefreshing\n\nscience of\n\nseed cultures for, 2.1, 2.2\n\nwith sprouted flour, 2.1, 3.1\n\n\"trap\"\n\nSteam, creating\n\nStevia, 1.1, 5.1\n\nSticky Buns\n\nStretch and fold, 1.1, 1.2\n\nStreusel Topping\n\nStruan Bread, Sprouted\n\nSugar, alternatives to, 1.1, 5.1\n\nSunflower seeds\n\nSeeded Carolina Wheat Hearth Bread\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nSprouted Multigrain Crackers\n\nSprouted Vollkornbrot\n\ntoasting\n\nSutton, Jeff\n\nSutton, Peggy, 3.1, 3.2, 3.3, 3.4\n\nSwedish-Style Limpa Rye\n\nSweet Potato Brioche, Sprouted Wheat\n\nT\n\nTeff, Gluten-Free Sprouted Corn Bread with\n\nThornberg, Bob\n\nTomatoes\n\nGrape Skin Flour Rustic Sourdough with Sun-Dried Tomatoes, Roasted Garlic, and Carmody Cheese\n\nTomato Pesto Focaccia\n\nTools\n\nV\n\nVanderliet, Joe, 5.1, 6.1\n\nVatinet, Lionel\n\nVital wheat gluten, 4.1, 4.2\n\nVollkornbrot, Sprouted\n\nW\n\nWaffles, Sprouted Wheat Flour\n\nWallace, Denene, 5.1, 5.2\n\nWalnuts\n\nBanana Quick Bread or Muffins\n\nCarrot Quick Bread or Muffins\n\nGluten-Free Holiday Biscotti\n\nGluten-Free Holiday Cookies\n\nRaisin Nut English Muffins\n\nRosemary Walnut Quick Bread\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\ntoasting\n\nZucchini Quick Bread or Muffins\n\nWheat berries\n\ncooking\n\nSprouted Vollkornbrot\n\nWheat bran\n\nApple Bran Quick Bread or Muffins\n\nSeeded Multigrain Hearth Bread with \"Parmesan Trap\" Starter\n\nWheat flour, high-extraction, 5.1, 5.2\n\nHigh-Extraction Pain au Levain\n\nWheat flour, sprouted\n\nApple Bran Quick Bread or Muffins\n\nApricot Almond Quick Bread\n\nBanana Quick Bread or Muffins\n\nBlueberry Pancakes\n\nBlueberry Quick Bread or Muffins\n\nCarrot Quick Bread or Muffins\n\nChocolate Chip Quick Bread or Muffins\n\nCinnamon Raisin Quick Bread\n\nCranberry Quick Bread or Muffins\n\nFlaky Sprouted Wheat Biscuits\n\nHerbed Quick Bread\n\nLemon Poppy Seed Quick Bread\n\nPoppy Seed Quick Bread or Muffins\n\nRosemary Walnut Quick Bread\n\nSprouted Corn Bread\n\nSprouted Pain au Levain\n\nSprouted Struan Bread\n\nSprouted Vollkornbrot\n\nSprouted Wheat Bagels\n\nSprouted Wheat Breakfast Focaccia\n\nSprouted Wheat Challah\n\nSprouted Wheat Cinnamon Buns and Sweet Rolls\n\nSprouted Wheat Croissants\n\nSprouted Wheat Flour Waffles\n\nSprouted Wheat Focaccia\n\nSprouted Wheat Muffins\n\nSprouted Wheat Pancakes\n\nSprouted Wheat Pizza Dough\n\nSprouted Wheat Quick Bread, 3.1, 3.2\n\nSprouted Wheat Soft Rolls or Sandwich Bread\n\nSprouted Wheat Sweet Potato Brioche\n\nSprouted Whole Wheat Bread\n\nZucchini Quick Bread or Muffins\n\nWheat flour, whole-milled\n\nFlavored English Muffins\n\nRaisin Nut English Muffins\n\nWhole-Milled Lean Dough French Bread\n\nWhole-Milled Whole Wheat Ciabatta\n\nWhole-Milled Whole Wheat Pizzas\n\nWhole Wheat and Raisin English Muffins\n\nWhole Wheat Currant Pretzels\n\nWink, Debra, 2.1, 2.2, 2.3\n\nX\n\nXanthan gum\n\nY\n\nYeast\n\nfermentation\n\ntypes of\n\nYeasted breads\n\nCascara Seca Lean Bread\n\nDark Sandwich Rye\n\nGluten-Free \"Do No Harm\" Sprouted Grain Bread\n\nMultigrain Sprouted Wheat Pulp Bread\n\nOnion Rye\n\nProBiotein Lean Bread\n\nSprouted Emmer Pulp Power Bread\n\nSprouted Sandwich Rye Bread\n\nSprouted Struan Bread\n\nSprouted Vollkornbrot\n\nSprouted Wheat Challah\n\nSprouted Wheat Pulp Bread with Sprouted Multigrain Flour\n\nSprouted Wheat Sandwich Bread\n\nSprouted Wheat Sweet Potato Brioche\n\nSprouted Whole Wheat Bread\n\nWhole-Milled Lean Dough French Bread\n\nWhole-Milled Whole Wheat Ciabatta\n\nZ\n\nZucchini Quick Bread or Muffins\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Neville Allen, Malcolm Farmer and the Online\nDistributed Proofreading Team at http:\/\/www.pgdp.net\n\n\n\n\n\n\n\n\n\n  PUNCH,\n\n  OR THE LONDON CHARIVARI.\n\n  VOLUME 147.\n\n  AUGUST 19th 1914.\n\n       *       *       *       *       *\n\nIllustration: A QUICK CHANGE OF FRONT.\n\n       *       *       *       *       *\n\nTHE NATURE OF A MORATORIUM.\n\n\"It's a big ship\" (I could overhear Ethel's voice through the open\nnursery window). \"I know perfectly well it is. It's one of the\nCunarders.\"\n\n\"Well, you're quite wrong then,\" (this from Jack). \"It was passed\nthrough Parliament. You can't pass a ship through Parliament.\"\n\n\"It's the sister ship to the _Lusitania_--so there!\"\n\nJoan's thoughtful voice intervened.\n\n\"I can tell you what it is,\" she said. \"It's a place for burying\npeople--a sort of big tomb where they put dead kings. There's one at\nWindsor.\"\n\nCuriously enough I was myself at the moment rather puzzled as to what it\nwas and how it worked.\n\n\"Do you know, William,\" I said to my host, \"that you are owing me ten\npounds and I've got to get home to-day, and I've no money?\"\n\n\"Oh, but I shan't pay it now,\" he replied shamelessly.\n\n\"Why not?\"\n\n\"I'm going to put a Moratorium on you. I don't know, of course, if\nthat's quite the correct phrase. The thing is new to me. But at least I\ncan see how it works. You had better try James. He owes you five, and he\nnever reads the papers, so he may not have heard of it.\"\n\nI went at once into the library, where I found James making up a parcel\nof three half-sovereigns to send to his bank. No one is going to accuse\nJames of hoarding gold.\n\n\"About that fiver,\" I began.\n\n\"Ah, yes. I was just coming out to talk to you about that before you\nwent,\" said he. \"Now that I'm sending all this stuff to the bank I'm\njust afraid I may be a bit short. I'll tell you what I think we ought to\ndo, you and I, I think we ought to enter into a temporary Moratorium.\nAll the best people are doing it. Of course I don't know if that's the\nright phrase. But I begin to see how it works.\"\n\n\"It doesn't apply to sums under five pounds,\" said I severely.\n\n\"That's true. I admit it's a pretty narrow squeak. I just managed to get\non board, so to speak. Still, as the debt is five pounds----\"\n\n\"I'll take L4 19_s_. 11_d_.,\" said I, and held out my hand.\n\n\"That's not playing the game,\" said James. \"Can't you see you're going\nto encourage all sorts of panic if you go about reducing debts in that\nsort of way? What is to become of British credit if a man in your\nposition shows himself willing to accept sweeping reductions for the\nsake of getting hold of cash? I'm just a little ashamed of you.\"\n\n\"Well, I've got to get home to-day. The ticket costs over five pounds,\nand I've only got sixteen shillings.\"\n\n\"Nothing simpler, my dear fellow,\" said James cheerfully. \"You ask the\nbooking-clerk for a ticket--pick it up--cover him with a Moratorium (if\nthat's the proper phrase) and hop into the train. The sixteen bob will\ncome in for tips.\"\n\nI went back to William and sat down. \"The upshot of it is, William,\" I\nsaid, \"that I can't go. You had better consider pretty carefully what\nyou're doing. I don't think the Moratorium was intended to work in this\nsort of way. I've got to report myself at the War Office, and I can't\ngo. You may think you're acting as a good citizen should. You may not be\nhoarding gold or hoarding food, but you are hoarding _me_.\"\n\n\"It doesn't apply to National Insurance payments,\" said William\nbrightly, \"if that's any help to you.\"\n\n\"It only goes on till the 4th of September,\" I reminded him, \"and the\nbank rate was recently as high as ten per cent. and may easily go up\nagain. You've got to pay interest on it, you know.\"\n\nThat was where I had him. \"How will you take it?\" he asked, thrusting a\nhand into his pocket.\n\n\"In new pound notes,\" said I.\n\n       *       *       *       *       *\n\nDIES IRAE.\n\n_To the GERMAN KAISER._\n\n  Amazing Monarch! who at various times,\n    Posing as Europe's self-appointed saviour,\n  Afforded copy for our ribald rhymes\n          By your behaviour;\n\n  We nursed no malice; nay, we thanked you much\n    Because your head-piece, swollen like a tumour,\n  Lent to a dullish world the needed touch\n          Of saving humour.\n\n  What with your wardrobes stuffed with warrior gear,\n    Your gander-step parades, your prancing Prussians,\n  Your menaces that shocked the deafened sphere\n          With rude concussions;\n\n  Your fist that turned the pinkest rivals pale\n    Alike with sceptre, chisel, pen or palette,\n  And could at any moment, gloved in mail,\n          Smite like a mallet;\n\n  Master of all the Arts, and, what was more,\n    Lord of the limelight blaze that let us know it--\n  You seemed a gift designed on purpose for\n          The flippant poet.\n\n  Time passed and put to these old jests an end;\n    Into our open hearts you found admission,\n  Ate of our bread and pledged us like a friend\n          Above suspicion.\n\n  You shared our griefs with seeming-gentle eyes;\n    You moved among us cousinly entreated,\n  Still hiding, under that fair outward guise,\n          A heart that cheated.\n\n  And now the mask is down, and forth you stand\n    Known for a King whose word is no great matter,\n  A traitor proved, for every honest hand\n          To strike and shatter.\n\n  This was the \"Day\" foretold by yours and you\n    In whispers here, and there with beery clamours--\n  You and your rat-hole spies and blustering crew\n          Of loud Potsdamers.\n\n  And lo, there dawns another, swift and stern,\n    When on the wheels of wrath, by Justice' token,\n  Breaker of God's own Peace, you shall in turn\n          Yourself be broken.\n\n    O. S.\n\n       *       *       *       *       *\n\nA DETERMINED ISLAND.\n\nII.\n\nI continue this record of our daily lives at Totland Bay on August 12th.\nBefore it appears in _Mr. Punch's_ columns great and decisive events may\nhave happened, but at present, except for such slight distractions as I\nshall relate, we are still calm and peaceful. When we think or speak of\nBelgium our faces glow, and we are all resolved, should the need arise,\nto do as Belgium has done, and to do it in the same resolute and\nunconquerable spirit. In the meantime we rush for the newspapers with a\nconstantly increasing eagerness. At about 11 A.M. the whole of Totland\nBay is filled with people reading their papers in the open air.\nEverybody bumps into everybody else, but nobody minds. A gentleman the\nother day set out in a canoe and read the morning's news to a party of\nswimmers, who appeared to be much invigorated by what they heard.\n\nOn Sunday night, just as we had finished dinner, we suddenly heard the\nreport of a great gun from the fort at the Needles. The explosion was\nfollowed by three plaintive answering notes from a fog-horn. \"They're\nfiring at a ship,\" said someone, and out we all rushed to the nearest\nvantage-point, and even as we ran another gun went off and again the\nfog-horn answered with its bleat. The searchlights were striking great\nshafts of light along the Solent, and far away their beams outlined the\nshape of a big ship. She was still advancing on her course, when--Bang!\nanother violent explosion shattered the night. This time it came from\nthe fort just over the pier of Totland Bay. The echoes reverberated and\nrumbled, and the shot tore past close to the ship. Now she took the\nwarning. There were no more appeals from the fog-horn. Slowly she turned\nand disappeared into the darkness. Possibly she had been at sea for a\nlong time and knew nothing of the war. How she must have marvelled at\nthis strange and dreadful welcome from the Isle of Wight. We went to our\nbeds that night with a feeling of perfect security.\n\nOn land, too, we have had our excitements. Yesterday afternoon, when the\nheather-clad <DW72>s of Headon Hill were crowded with picnickers, there\nwas a sudden alarm of spies. Some men, reported to have been conversing\nin German, were said to have been peering into cracks in the ground and\notherwise behaving in a most suspicious manner. The alarm was given, and\nalmost instantly, springing as it were from the very bowels of the\nearth, came some half-dozen soldiers running with rifles and fixed\nbayonets. Amid the shouts of the children they spread about the heather\nin their hunt, but nothing came of it, for the \"spies,\" though they were\ncaught, turned out to be some Italians resident in Totland Bay and\nfervently British in their sympathies.\n\nI mentioned last week that we had a children's maid, a German, in our\nhousehold. Since then, in obedience to the Act, she has been registered\nas an \"alien enemy.\" I took her by train to Newport for that purpose. On\narriving at the station I hailed a fly. \"Where to, Sir?\" said the\ndriver. \"To the police-station,\" I answered, and the man broke out into\na grin. \"It isn't a serious offence,\" I added, but I doubt if he\nbelieved me. At the police-station, however, they were quite prepared\nfor us, and in a very few minutes Maria Hasewitz--that is her eminently\nGerman name--had had all the particulars of her birth-place, her age,\nher height, and her personal appearance entered on a blue form by a\njocose and affable sergeant. \"Brown eyes, I _think_,\" said the sergeant;\n\"height, five feet four inches; no beard _or_ moustache, ha-ha. Now sign\nhere and make a mark with your left thumb in this space. That'll pin you\ndown; no escape after that, ha-ha.\" He produced a board covered with\nsome black sticky substance, dabbed her thumb in it, dabbed it hard on\nthe paper, and, lo, Maria Hasewitz had been registered and had\nundertaken not to move five miles from Totland Bay without a special\npermit.\n\nAt present this particular alien enemy is engaged, together with all the\nother available female members of the household, in making pyjamas for\nour soldiers. Wonderful deeds are being done all round me with scissors\nand needle and thread. A sewing-machine has been requisitioned.\nButton-holes are being manufactured with immense expedition. A good deal\nof \"basting\" is being got through. In my illimitable ignorance I had\nhitherto imagined that basting was something that you did to a joint of\nmeat with a big ladle and some gravy. If you did it sufficiently the\njoint came out succulent, if not it became dry and you abused the\nbutcher. However, we live and learn. Part, at any rate, of three suits\nof pyjamas that are to go to the Red Cross to-day has been severely and\ncompletely basted without either gravy or a ladle.\n\n    R. C. L.\n\n       *       *       *       *       *\n\nIllustration: WELL MET!\n\nGREAT BRITAIN JOINS HER ALLIES IN THE FIELD.\n\n       *       *       *       *       *\n\nCHARIVARIA.\n\nEven war has its humours. \"In the midst of perfect peace the enemy\nsurprises us,\" is a sentence from a proclamation not by the King of the\nBELGIANS but by the GERMAN KAISER.\n\n       ***\n\nWILHELM II. is said to be extremely annoyed in his capacity as a British\nAdmiral that he is not being kept fully informed as to the movements of\nour Fleet.\n\n       ***\n\nThe danger, of course, of a fondness for a place in the sun is that one\nmay get burnt.\n\n       ***\n\nThe coming generation would certainly seem to be all right. Even\nchildren are taking part in the fray. The Boy Scouts are helping\nmanfully here, and at Liege the Germans, we are told, used nippers for\ncutting wire entanglements.\n\n       ***\n\nA vivid idea of the horrors of the return journey from the Continent to\nEngland after the declaration of war may be gained from the fact that a\nlady, in recounting her experiences in a contemporary, states that she\nwas thankful to get back to Battersea.\n\n       ***\n\nGeneral VILLA, it is stated, has now virtually proclaimed his\nindependence of General CARRANZA, and hostilities are said to be\nimminent. We caution these gentlemen, however, that we are not prepared\nat this juncture to take a great deal of interest in their little war,\nand, if they take our advice, they will postpone it.\n\n       ***\n\nAt the present moment, fortunately, one does not hear much of the sex\nwar, but sex-pride compels us to draw attention to an account in _The\nLiverpool Echo_ of a recent agricultural show, from which we learn that\n\"in a class for cows, in which there was a score of entries, Mr. S.\nSanday won with pedigree dairy bulls.\"\n\n       ***\n\nThe news that a large number of yachts had been placed at the disposal\nof the Admiralty was, no doubt, responsible for a statement in _The\nBirkenhead News_ of the 8th inst., to the effect that the Hoylake Town\nBand, consisting of Bavarians, in a moment of patriotic fervour during\nthe crisis struck up \"_Der Yacht am Rhein_.\"\n\n       *       *       *       *       *\n\nIllustration: _GERMAN KAISER._ \"Donnerwetter! No wonder I've missed my\nappointment. The silly idiots have given me an 1870 time-table.\"\n\n       *       *       *       *       *\n\nOverheard in the heather of a grouse moor:--\"What ho! The Moratorium.\"\n\n       ***\n\nIn feline circles it is being pointed out with some pride that not only\nare there Dogs of Wars but that Active Service Kits are being advertised\nvery freely.\n\n       *       *       *       *       *\n\nIllustration: AT THE OFFICIAL PRESS BUREAU.\n\n_MR. F. E. SMITH_ (_against his gallant instincts_). \"Permit me, Madam.\"\n\n       *       *       *       *       *\n\n\"We, as a party,\" says Mr. KEIR HARDIE in _The Labour Leader_, \"surely\nhave a right to make a special protest against this altogether useless\nand unnecessary conflict.\" The KAISER'S address, KEIR, is Potsdam,\nBerlin (Germany).\n\n       ***\n\nWe rejoice to hear that the thousand fresh herrings which a certain\ncosmopolitan financier purchased at the outbreak of the war to store up\nhave one and all gone bad.\n\n       ***\n\nParis now has a \"Rue de Liege.\" And, in order to obviate any feeling of\njealousy, a certain virulent microbe which has just been discovered by a\nBelgian scientist is, we hear, to be called the \"Wilhelm Germ.\"\n\n       ***\n\nWe trust that the Dutch are taking every precaution to protect the\nPalace of Peace at the Hague.\n\n       ***\n\nBrick-box, the Irish Guards' pet terrier, has been sent for the present\nto a dogs' home. In the event of their going abroad the Irish Guards\nhope to bring back with them a certain other dog who seems to have gone\nmad.\n\n       ***\n\nThe British Isles have been defeated at Lawn Tennis, but we really\nshan't mind so long as we win the war.\n\n       ***\n\n\"On shop after shop in Paris,\" says _The Evening News_, \"is the notice,\n'Maison fermee a cante du de depart du patron et les employes sous les\ndrapeaux Francais.'\" Sorry, _Evening News_, but we cannot believe your\nstatement in its entirety. We are afraid you did not get it confirmed by\nthe Official Press Bureau.\n\n       ***\n\nAccording to the St. Petersburg _Gazette_ the Germans have arrested the\nGrand Duke CONSTANTINE CONSTANTINOVITCH at Badwildungen. The Russian\nGovernment admits that the GRAND DUKE has published several volumes of\nverse.\n\n       ***\n\nAccording to a statement in _The Globe_ \"the German liner, _Belgia_,\nhaving run short of coal, put in at Newport (Mon.) to-day, and was\nseized as a prize. She has over L250,000 worth of food on board,\nincluding 400 tons of cheese, 73 German reservists, and also a large\namount of specie.\" The last two items must, of course, be regarded as\nemergency rations.\n\n       ***\n\nAn unfortunate misprint:--\n\n\"WAR NEWS IN A FEW LIES.\"\n\n       *       *       *       *       *\n\nIllustration: THE MONOPOLIST.\n\n_Late Arrival_ (_wishing to put his machine in bicycle rack_). \"WELL,\nUPON MY WORD, THIS IS PREPOSTEROUS! CADDIE, WHO PUT HIS BICYCLE LIKE\nTHAT?\"\n\n_Caddie._ \"CAN'T SAY FOR SURE, SIR. THE KAISER, I SHOULD THINK.\"\n\n       *       *       *       *       *\n\nHOW WAR IS \"MADE IN GERMANY.\"\n\n(_Extract from the KAISER'S Diary._)\n\nLetter captured bearing mark of Venezuela Consulate at Berlin. Stamp not\nput on straight. Insult to me--therefore to the flag. Proceed to issue\nultimatum to Venezuela. Venezuela omits to concede one of the 421 points\nraised. Declare war on Venezuela and publish address to my\npeople:--\"Owing to this wicked and determined challenge to Our nation,\nWe have been forced, greatly against Our wish, into a quarrel with a\npowerful and designing enemy,\" etc., etc.\n\nConsignment of Chicago sausages, arriving Hamburg, is found to bear\nlabel \"The Best.\" Deliberate blow at German supremacy. Germany is the\nSausage Queen. Ultimatum to United States. Reply unsatisfactory, so\ndeclare war. Speech to my people:--\"Owing to this wicked,\" etc.\n\nDespatch from Pomeranian farming district to effect that a Cochin-China\nhen has peaked at representation of German Eagle in picture-book. At\nonce issued ultimatum to Cochin-China demanding humble and complete\napology, otherwise war would be declared. Received immediate reply,\nstating that as Cochin-China belongs at present to France I may save\nmyself the trouble of a fresh declaration of war. Do so.\n\nRead statement that \"heat in neighbourhood of equator surpasses that of\nany other part of the world.\" See in this a direct challenge to our\nsovereignty. _We_ are the hottest stuff in the world. Declare war on all\ncountries abutting on equator. Speech to my people:--\"Owing to this\nwicked,\" etc.\n\nHear South Pole Republic showing signs of activity. Involves serious\nmenace to our pacific plans. Issue ultimatum. Hear later that President\nis a penguin. As, however, withdrawal of ultimatum is out of the\nquestion, have despatched warships. Speech to my people:--\"Owing to this\nwicked,\" etc.\n\nHaving five minutes before lunch, declare war on Spain, Portugal, Tibet,\nLapland and the Principality of Monaco. Reasons and ultimata to follow.\n\nDeclare war on Bosnia and Herzegovina, but subsequently remember that\nthese territories were recently absorbed by my ally. Undignified to\ncancel ultimatum, so declare war on said ally.\n\nMake painful discovery that, in spite of overtime at Imperial printing\nworks, I am out of ultimatum forms. Urgent instructions have been sent\nto hasten delivery of forms, which are of course so printed that only\nthe name of the offending country has to be filled in.\n\n       *       *       *       *       *\n\nApparently no more countries remain to be challenged. Must find some at\nall costs.\n\nSudden inspiration. Have issued ultimatum to my own country that, if she\ndoes not find fresh countries for me to fight before midnight, war will\nensue.\n\n_Midnight._ No new countries found. I declare war on Germany.\n\n       *       *       *       *       *\n\nThe Journalistic Manner.\n\n\"Every inch of Belgium will be fought for foot by foot.\"--_Daily\nTelegraph._\n\n       *       *       *       *       *\n\nTHE OLD ORDER CHANGES.\n\nA thousand years ago I won a cup for jumping. It was not a very good\ncup, but then it was not a very good jump. Such as the cup is, however,\nit stands on a shelf in my library, and I have ways of directing the\nattention of visitors to it. For instance, if a collector of old prints\nis coming to dinner, I hang my oldest print just above the cup, ready\nfor him; we take our--or better, his--cigars into the library, and I\nsay, \"Oh, look here, I picked this print up last week; the man said it\nwas a genuine Eyre and Spottiswoode; you might give me your opinion.\" He\ngives me his opinion ... and then his eye wanders down. I see him\nreading the inscription on the cup.\n\nThe inscription says: \"Long Jump, 1739,\" or some such date. \"First\nPrize, won by ----\" and then my name very big and splendid. Underneath\ncomes the school crest, followed by the motto, \"_Dat Deus Incrementum_,\"\nthough I have never jumped any further since. Its shape is the ordinary\nsherry-glass shape. It is my only cup, and I am proud of it.\n\nI look up as I write, and I see the--by the way, I don't know if you\nhave ever tried \"looking up as you write.\" It is a common thing for\nreflective writers to say they do, but you should never believe them. It\nis impossible to write properly when looking somewhere else. What we do\nis to stop and slew our necks round, and then take a fresh dip in the\nink. Well, slewing my neck round as I stop writing, I see my precious\ncup standing on its shelf, and ... horror! It is standing upside down!\n\nThis comes as a surprise to you, but it is no surprise to me. The thing\nhas been going on for months. It is months ago that I first spoke to\nCelia about it.\n\n\"It's Jane,\" she said. \"She always puts it like that when she's been\ndusting.\"\n\n\"Yes, but what for? Just to catch the eye?\"\n\n\"I suppose because you always stand glasses upside down when you've\ncleaned them--to keep the dust out.\"\n\n\"But if she'd only think a moment she'd see that I don't drink out of\nthis, and that glasses don't have 'First Prize, won by ----'\"\n\n\"Jane isn't here to think, she's here to work.\"\n\nThis seemed to be a distinction drawn between Jane and me.\n\n\"You see what I mean,\" I said, \"don't you? It's very difficult to read\nthe cup upside down. A stranger mightn't know who--er--who had won it.\"\n\n\"But don't you always turn it back again? I do, if ever I see it.\"\n\n\"Yes, but--but---- Oh, well, it doesn't matter.\"\n\nI went back to the library. It was difficult to explain why I minded;\nbecause, after all, to fill a pipe, light it and sit down to work every\nmorning is very little less trouble than to turn a cup round, fill a\npipe, light it and sit down to work every morning. Anything regular soon\ngets taken for granted. And yet I was annoyed. I think it was the\n_silliness_ of standing a First Prize upside down which annoyed me. That\nand the apparent difficulty of getting into communication with Jane\nabout it.\n\nFor it was difficult. One day I went very humbly to Celia and said--\n\n\"I know I'm a baby about it. Forgive me. But it's getting on my mind. Do\ntell Jane about the cup.\"\n\n\"It's awfully hard,\" she said, after a little thought. \"You see, it's\nsuch a very, very small thing that it never seems quite the right moment\nfor it. And if, after I'd told her, she said 'What?' I couldn't possibly\nsay it again.\"\n\n\"You must be very articulate the first time. Lead the conversation\nslowly round to long-jumping or the difficulty of reading on your head,\nand then casually but articulately----\"\n\n\"Well, we'll see,\" said Celia. \"Of course, if I ever caught her doing\nit, I'd tell her. Perhaps I shall.\"\n\nWell, we saw. We saw that the thing still went on. The direct approach\nto Jane was evidently impossible. So I tried sarcasm.\n\nSarcasm, directed into the blue in the hope of hitting the person you\nwant, may not be effective, but it does relieve the feelings. I had a\nthoroughly sarcastic morning all to myself. My deadly irony took the\nform of turning _everything_ in the library upside-down. The cup was in\nposition already; I turned up two pewter mugs (third prizes in\nConsolation Races), the flower bowls, the cigarette box, the lamp, a\nstool, half-a-dozen pictures, two photographs and the mahogany clock.\nThey all stood on their heads and sneered at Jane. \"Why don't you do the\nthing properly while you're about it?\" they said to her. I felt\nextremely well after I had finished.\n\nCelia stood in the door and gurgled to herself.\n\n\"You baby,\" she smiled.\n\n\"On the contrary,\" I said, \"I have made a dignified yet subtle protest.\nYou wouldn't move in the matter so I had to do something. I flatter\nmyself that a sense of her past silliness will rush over Jane like a\nflood when she comes in here to-morrow morning.\"\n\n\"If Jane's flooded at all,\" said Celia, \"it will be with the idea that\nthe master's mad. But I don't think she'll notice it particularly.\"\n\nNext morning everything was right side up again--except the cup.\n\n\"It's no good,\" I told Celia; \"she is obviously determined. Perhaps it\nmeans more than we think to her to have that cup upside-down. Its\nbeauty, the memories it brings back, the symbolism of it, these things\ntouch some hidden spring ... Still I _am_ master in my own house.\" And I\nturned the cup round again....\n\nAnother month passed and I could bear it no longer. Yesterday I made up\nmy mind. I would speak to Jane myself. I turned my First Prize the right\nway up, and then looked for Celia.\n\n\"Celia,\" I said firmly, \"where is Jane?\"\n\n\"She's gone out,\" said Celia softly. \"Her--her man goes off to-day.\"\n\n       ***\n\nAn hour later, with bands playing and people cheering, they wheeled out\nof barracks, brown and businesslike. Jane was in the front somewhere,\nwaving her handkerchief--not such a silly Jane, after all. And at the\nback, very proud for her, Celia and I stood silent, with a something in\nthe throat that had come there suddenly....\n\nAnd this morning the cup was upside-down again. Well, well, if she likes\nit that way, that way let it be.\n\nBut take warning, O Jane! When your man--here's luck to him!--comes\nback, then I shall assert myself once more. My cup, \"Long Jump, 1739.\nFirst Prize,\" shall stand the right way up; either that or you leave my\nservice. I am determined about this....\n\nMeanwhile we can share the daily paper.\n\nA. A. M.\n\n       *       *       *       *       *\n\n\"Dear _Mr. Punch_,--You may remember that QUEEN VICTORIA recorded in her\n_Journal in the Highlands_ that 'Vicky sat down on a wasps' nest.'\n'VICKY,' of course, was destined later to be the mother of WILHELM II.\nCan we not see in the present situation rather a remarkable example of\nheredity?--Yours, etc., MEDICO.\"\n\n       *       *       *       *       *\n\nFrom a _Daily Chronicle_ special correspondent--\n\n    \"A little meat and plenty of vegetables take one a long\n    way--lettuce, soup, eggs, en surprise, peas, dessert, voila--even\n    the very poor can afford such a dinner in Brussels.\"\n\nA seven-course dinner is certainly more than we can afford in England.\n\n       *       *       *       *       *\n\nIllustration: \"IT'S AN ILL WIND ...\"\n\n_Old Cock Grouse._ \"I SEE THEY'VE ALL GONE SHOOTING EAGLES.\"\n\n       *       *       *       *       *\n\nTHE PRIVATE VIEW.\n\nI take train home every evening from one of our best stations. Crowned\nheads fairly tumble over one another there in their anxiety to get a\nfirst glimpse of London. Personages are matters of daily arrival.\n\nThe other night I reached my station just as a Personage was due. A\ndrive led from his platform to the outside world. On one side of it were\nlined up the public six deep. On the other side of it was the left\nluggage office. Four policemen saw to it that no person crossed to the\nother side except on business.\n\nI began crossing.\n\n\"Not that side,\" said Robert, \"unless you want the left luggage.\"\n\n\"The left luggage,\" I explained, \"is my one desire.\"\n\nI crossed.\n\nThe clerk was unusually prompt.\n\n\"What's yours?\" he said.\n\n\"Since you ask,\" I replied, \"I could do with a small stout; or,\nalternatively, a sherry and bitters.\"\n\nHe kept silence, but with a touch of urgency in it. It is hard to\ntemporize when confronted with a businesslike silence. Yet my view of\nthe drive was worth fighting for.\n\n\"I might leave my watch,\" I continued after a brief hesitation, \"but the\nfact is I left it last week with my only godson. Have you a godson? You\nknow what they are--always wanting something.\"\n\n\"Come along, now,\" said the official brusquely. Robert, too, was\nbecoming restive.\n\n\"Very well; I will deposit my hat. You will be careful with it, won't\nyou?\"\n\nHe accepted my hat untenderly.\n\n\"What name?\"\n\n\"George,\" I said; \"but they call me 'Winkles' at home.\"\n\nHe was a man not easily moved. He wrote down \"George\" without hesitation\non a bit of pink paper and asked for twopence as he gave it to me.\n\nJust then, to my great relief, the Boat Express arrived. I searched in\nall my pockets and at last found half-a-sovereign.\n\nI told you he was a man not easily moved. He gave me nine-and-tenpence\nwithout a word, but with more halfpennies than was quite nice.\n\nThere was a stir in the crowd. I must hang on yet a little, or give it\nup, or stand six deep. I cannot stand standing six deep. But it is the\nduty of every citizen to welcome Personages.\n\nThen I bethought me of my pink paper.\n\nI summoned the man who was not easily moved and presented it. \"The\ndeposit,\" I explained, \"was a hat--a felt hat--I cannot be sure of the\nsize, but at a guess I should put it somewhere between 7 and 8.\"\n\nBut he had already retrieved it.\n\nI took it and replaced it on my head as I turned in the nick of time to\ntake it off to the Personage. He gave me a very sweet smile, the memory\nof which I cherish so fondly that I am loth to attribute it to the\nfashionable dent I subsequently discovered in my bowler.\n\n       *       *       *       *       *\n\nIn the present restriction of Sport we sympathize with that section of\nthe Press which makes it a speciality. However, there are outlets; and\none of our Sporting contemporaries has burst forth into history, as\nfollows:--\n\n    \"Once again England is faced with a crisis. There has been nothing\n    like it since Alexander the Great burned his boats and crossed the\n    Rubicon.\"\n\nAn Infant Prodigy.\n\n    \"Although only in his 41st year Mr. F. E. Smith is a Master of Arts\n    ...\"\n\n    _Pall Mall Gazette._\n\n       *       *       *       *       *\n\nIllustration: _Medical Officer._ \"SORRY I MUST REJECT YOU ON ACCOUNT OF\nYOUR TEETH.\"\n\n_Would-be Recruit._ \"MAN, YE'RE MAKING A GRAN' MISTAKE. I'M NO WANTING\nTO BITE THE GERMANS, I'M WANTING TO SHOOT 'EM.\"\n\n       *       *       *       *       *\n\nA FIRST CHARGE.\n\n_Mr. Punch's_ appeal is once more for the children. Most earnestly, and\nwith great confidence, he begs his readers to care for those little ones\nwhose fathers and brothers are serving under the Flag for our country's\nhonour and the defence of our homes, or may suffer through loss of work.\nAll gifts to the National Relief Fund should be addressed to H.R.H. The\nPrince of Wales, at Buckingham Palace.\n\n       *       *       *       *       *\n\nA PLEA FOR PEGASUS.\n\n  Ye mobilisers of that other arm\n    Whose might is famed superior to the sabre's,\n  Who furnish forth the wherewithal to charm\n    The Special Correspondent to his labours,\n  And by whose enterprise we're daily fed on\n  Reports of Armageddon.\n\n  List to my plaint. It is not that I tire\n    Of those despatches--picturesque effusions--\n  Which by the witness of a later wire\n    Are proved to rank among the Great Illusions;\n  Though much to be deplored, such news, I'm willing\n  Freely to own, is thrilling.\n\n  But when your pages, shrunken through the scare\n    Of that worst blow of all, a paper famine,\n  Dispense exclusively Bellona's fare,\n    And, failing battle tales, you simply cram in\n  Facts about spies, commodities and prices,\n  I writhe beneath this crisis.\n\n  I can support the other pains of war:\n    Transport disorganised and credit shaken,\n  The fear of hunger knocking at the door,\n    And threepence extra on a pound of bacon;\n  In fact, I'd be the most resigned of creatures\n  If you'd compose your \"features.\"\n\n  Could you not lift a corner of the mask\n    That makes these solemn days so much more solemn?\n  A very little ray is all I ask\n    To light the utter darkness--say a column\n  Of \"stories\" which your slang describes as \"snappy;\"\n  With these I could be happy;\n\n  With these my topic Muse I might entice;\n    But war has left her mute, and me despairing.\n  They call for horses; must I sacrifice\n    The steed with whom I've taken many an airing?\n  Poor Pegasus--and none too well-conditioned!\n  Must _he_ be requisitioned?\n\n       *       *       *       *       *\n\nFrom parallel columns in The Evening News:---\n\n    \"Haelen is forty-five miles northwest  | \"The centre of the battle was\n    of Liege; it is fifty miles            | at Haelen (thirty miles\n    east of Brussels.\"                     | northwest of Liege\n                                           | and thirty miles\n                                           | from Brussels).\"\n\n\nThis is simply to deceive the Germans.\n\n       *       *       *       *       *\n\nIllustration: THE WORLD'S ENEMY.\n\nTHE KAISER. \"WHO GOES THERE?\"\n\nSPIRIT OF CARNAGE. \"A FRIEND--YOUR ONLY ONE.\"\n\n       *       *       *       *       *\n\nIllustration: _Fond Mother_ (_full of war news_). \"DON'T GO TOO FAR\nOUT, GIRLS. YOU CAN'T BE TOO CAREFUL WITH ALL THIS FIGHTING GOING ON.\"\n\n       *       *       *       *       *\n\nMR. PUNCH'S HOLIDAY STORIES.\n\nII.--THE ISLAND CUP RACE.\n\nCowes week was drawing near to its brilliant climax. Through the blue\nwaters of the Solent a swarm of palatial steam yachts, saucy outriggers,\ngraceful cutters and wasp-like motor boats jostled one another in their\nefforts to gain safe anchorage after the strenuous excitement of the\nday's racing. Everywhere could be heard the clank of mooring chains,\nmingled with the full-flavoured oaths of sailor men.\n\nGradually silence fell upon the scene, broken only by the melodious\nmurmur of numberless gramophones and the soft strains of the band of the\nRoyal Yacht Squadron.\n\nAs the sun descended lower beneath the horizon the dusk deepened, and\npresently thousands of Chinese lanterns twinkled through the gloom from\nmast and yard-arm. Lady Margaret Tamerton, leaning idly against the\nbarnacle of her brother's yacht, the _Seamaid_, drank in the beauty of\nthe night with deep inhalations.\n\nThe voice of young Lord Tamerton at her side at last broke the spell of\nsilence.\n\n\"Madge,\" he said softly, \"Wonderson has not yet arrived. If he doesn't\ncome, our chances of winning the Island Cup to-morrow are practically\nhopeless.\"\n\n\"Don't worry, Fred,\" replied Lady Margaret. \"Ralph never fails....\nListen, he is coming now.\"\n\nAnd, indeed, the muffled beat of oars was heard approaching from the\ndarkness. Soon a slim white boat came gliding up to the prow of the\n_Seamaid_. Ralph Wonderson, a tall athletic figure in his immaculate\nflannels and straw boater, poised himself on the gunwale, gathered\nhimself for a spring, and leaped with the agility of a cat to the\nbowsprit of the yacht. Sliding rapidly down this, he nodded easily to\nLord Tamerton and clasped the beautiful figure of Lady Margaret in his\narms.\n\n\"S-sh!\" he whispered warningly, laying his fingers on her lips, as she\nwould have spoken. \"Nobody must know I am here till to-morrow. That is\nwhy I came aboard like that. Listen. Your cousin, Sir Ernest Scrivener,\n_alias_ Marmaduke Moorsdyke, is here, and is plotting to kidnap you.\nThere is a traitor somewhere on this yacht who supplies him with all\ninformation. The attempt is to be made to-night.\"\n\n\"To-night!\" murmured Lady Margaret in horror. \"What am I to do? His\ningenuity is dev--er--fiendish.\"\n\n\"It shall be baffled,\" replied Ralph reassuringly. \"I have thought it\nall out. It would be dangerous for you to leave the yacht because, in\nview of to-morrow's race, neither your brother nor I could accompany\nyou. There is only one place on board where you can pass the night in\nassured safety--the crow's-nest.\"\n\n\"The crow's-nest,\" repeated Lady Margaret, clapping her hands. \"What\nfun! I shall be rocked to sleep beautifully, and of _course_ they will\nnever think of looking for me there.\"\n\n\"Come,\" said Ralph, taking her hand. \"There is no time to lose, and none\nof the crew must be allowed to see you. We don't know whom we can\ntrust.\"\n\nSnatching her in his arms, he carried her easily up the frail rigging,\nhis mountain training showing in every step he took. Five minutes later\nhe returned alone and dropped noiselessly to the deck. He looked round\ncautiously; there was nobody in view except Lord Tamerton.\n\n\"It's all right, Fred,\" he whispered. \"Let us turn in.\"\n\nThey descended the broad staircase arm-in-arm. No sooner had they\ndisappeared than a dark figure crept with a low chuckle from underneath\na coil of rope and dropped silently over the yacht's counter.\n\nA phosphorescent gleam disturbed the darkness of the water.\n\n       ***\n\nEarly next morning Ralph Wonderson ran nimbly up the rigging of the\n_Seamaid_, carrying a tray loaded with toast, eggs, tea and marmalade.\nHe tapped at the door of the crow's-nest. There was no response. After a\npause he tapped again and cautiously pushed open the door. The\ncrow's-nest was empty!\n\n\"Betrayed,\" cried Ralph, clapping his hand to his forehead. A moment\nlater two soft-boiled eggs devastated the snowy whiteness of the\n_Seamaid's_ deck.\n\nDespite their precautions, Lady Margaret had been spirited away during\nthe night. As soon as he had recovered from the shock of the discovery,\nRalph ran to Lord Tamerton and acquainted him with the terrible news.\nThere was a period of agonised and fruitless discussion.\n\n\"Wait! I have an idea,\" exclaimed Ralph presently. He pressed an\nelectric bell, and a steward appeared almost simultaneously.\n\n\"Jenkins, fetch me a race card,\" said Ralph.\n\n\"Yes, Sir,\" replied the steward. \"I anticipated your request and have it\nhere.\"\n\nRalph and Lord Tamerton bent their heads over the card.\n\n\"See,\" said the former. \"It is as I hoped. Among the entries for the\nIsland Cup we have the _Watersnake_, owner Sir Ernest Scrivener. He will\nsail her himself, that is certain. It is equally certain that he has\nMadge on board. If I know anything of him he will not let her out of his\nsight. Fred, by yonder centreboard I swear that before the race is over\nwe will win her back.\"\n\n_Bang!_ It was the signal for the competitors to line up for the great\nrace for the world-famous Island Cup.\n\n       ***\n\nOf all the thousands who pressed themselves against the straining booms\nnone realised that the race was for a prize far more precious than a\nmere cup of gold valued at two thousand guineas.\n\nThe _Watersnake_ was in front, a clear hundred yards separating her from\nthe pursuing _Seamaid_. All the other yachts lagged hopelessly in the\nrear.\n\nScattering the foam at their bows, the two boats rushed along the blue\nlane of clear water which lay between the booms. Ralph, at the wheel of\nthe _Seamaid_, gazed anxiously forward. Could they do it?\n\n\"Let loose the spinnaker,\" he commanded gruffly. \"Haul on the signal\nhalyard. Lower the keelson.\"\n\nThe orders were swiftly executed, and the _Seamaid_ leaped forward with\na bound. The distance between the two vessels rapidly lessened.\n\n\"Fred,\" said Ralph, \"you must take the wheel for a time. I'm going\nforward to board the _Watersnake_.\"\n\nLord Tamerton obediently grasped the wheel, while Ralph ran forward and\ncrept along the bowsprit. The intervening space was now very small.\nBracing himself for the effort, he shot through the air and landed upon\nthe deck of the _Watersnake_. The first object which met his gaze was\nLady Margaret, her wrists bound, lying beside the barnacle.\n\nSir Ernest Scrivener uttered a horrible oath as he recognised the\nfeatures of his successful rival. For an instant he loosened his grasp\non the wheel. The vessel yawed in her course and he was compelled to\nseize the spokes again.\n\n       ***\n\nBefore Scrivener could command his wits sufficiently to shout an order\nto his crew, Ralph had caught up Lady Margaret in his arms and dashed to\nthe side of the vessel. Deprived of his skilled command, the _Seamaid_\nhad dropped behind; it was impossible to leap back to her decks.\n\nWithout hesitation, Ralph dived into the water, and still supporting the\nnow unconscious form of Lady Margaret, swam rapidly towards the yacht.\nTwo minutes later he was gripping the wheel and concentrating all his\nimmense will power upon the task of winning the race.\n\nInch by inch the _Seamaid_ crept up to her rival. Despite all\nScrivener's efforts, the gap grew less and less.\n\nAnd now the winning post was close at hand. Could it be done? Could it\nbe done? The frantic spectators behind the boom shouted themselves\nhoarse. Lord Tamerton bit his thumbs till the blood ran.\n\nNearer drew the _Seamaid_. Nearer and nearer. Nearer still. At the\ncritical moment, Ralph, with a mighty effort, pushed down the wheel.\n\nA bare three inches parted the _Watersnake_ from the winning post when\nthe slight shudder ran through her which told that the prow of the\n_Seamaid_ had touched her stern. The bump had been made; the race was\nwon.\n\n       ***\n\nRalph Wonderson stood with the magnificent Island Cup in his hand,\nfilled to the brim with bubbling champagne.\n\n\"To the restoration of the fortunes of the house of Tamerton,\" he said\nas he raised it to his lips.\n\n       *       *       *       *       *\n\nIllustration: _The Turkey Buzzard_ (_to the Sea Eagle_). \"You may call\nyourself a Turkey Buzzard if you like, but they'll still know you by\nyour white feather.\"\n\n       *       *       *       *       *\n\nTHE VIKING SPIRIT.\n\n    [\"The week-end was dull and much rain fell, but this did not spoil\n    the visitors' pleasure. The sight of the sea in a turbulent mood was\n    a great attraction.\"--_Seaside note in daily paper._]\n\n  It has rained for a week down at Shrimpton;\n    'Tis zero or less in the shade;\n  You can paddle your feet in the principal street\n    And bathe on the stony parade;\n  But still on our holiday pleasures\n    No thoughts of discomfort intrude,\n  As we whisper, \"This sight is a bit of all right,\"\n    For the sea's in a turbulent mood.\n\n  There's nobody harks to the pierrots;\n    For music we don't care a straw;\n  And the \"comic\" in vain chants the usual strain\n    Concerning his mother-in-law.\n  Unbought are the beach's bananas;\n    Our souls are all far above food;\n  Not a man of us dreams of consuming ice-creams\n    When the sea's in a turbulent mood.\n\n  You may prate of the fervour of Phoebus\n    Of days that are calm and serene,\n  When a tint as of teak is imposed on the cheek\n    That is commonly pallid (when clean);\n  But _we_ have a taste that's aesthetic;\n    Mere sunshine seems vulgar and crude,\n  As we gather to gaze with artistic amaze\n    On the sea in a turbulent mood.\n\n       *       *       *       *       *\n\n_The Beekeepers' Record_, referring to the photograph of a group of\nprominent beekeepers, says:--\"Mr. Dadant's well-known features are\neasily spotted.\" We are sorry, but a little cold cream will sometimes do\nwonders.\n\n       *       *       *       *       *\n\nANSWERS TO CORRESPONDENTS.\n\n\"FOR NUTS.\"--The origin of this curious phrase to indicate incompetence\nin any pursuit or pastime--_e.g._, \"He can't play for nuts,\" etc.--is\nobscure; but its antiquity is incontestable. Thus one of the fragments\nof ENNIUS runs: \"_Nucibus non ludere possum_.\" Perhaps the most\nplausible theory is that which views the phrase as a heritage from our\nsimian ancestors, among whom nuts were the common medium of exchange. On\nthis assumption a monkey--whether gorilla, chimpanzee, baboon or\norangutan--who was described as unable to do anything \"for nuts,\"\n_i.e._, for pecuniary remuneration, was obviously inefficient. Another\nexplanation, which we believe is supported by Mr. EUSTACE MILES, scouts\nthe notion of an ancient origin of the phrase and fixes the _terminus a\nquo_ by the recent introduction of vegetarian diet. Nuts being a prime\nstaple of the votaries of this cult, a person who cannot do anything\n\"for nuts\" means, by implication, a carnivorous savage who is incapable\nof progress. Lastly, there remains the ingenious solution that the\nphrase as commonly employed involves a misspelling. It ought to be \"four\nnuts,\" and playing four nuts was an ancient but simple game, which may\nbe connected with the cognate phrase about knowing or not knowing \"how\nmany beans make five.\"\n\n       *       *       *       *       *\n\nPOLLY PERKINS: WAS SHE A REAL PERSON?--A careful search in the registers\nof Paddington in the early and mid-Victorian period reveals so many Mary\nPerkinses as to render the task of identification peculiarly difficult.\nIt will be remembered, however, that the heroine of the famous ballad is\ndescribed as not only \"little,\" but \"pretty;\" indeed, she is spoken of\nas being \"as beautiful as a butterfly and as proud as a queen.\" So far,\nhowever, these clues to her appearance have yielded no solid results.\nThe representatives of the famous family of brewers have been unable to\nthrow any light on the subject, and an application to the managing\ndirector of the London and General Omnibus Company has also proved\nunproductive. (Polly Perkins \"married the conductor of a twopenny\n'bus.\") Her brilliant appearance suggests a possible relationship with\nDr. PERKINS, the famous pioneer of the aniline dye industry; but this,\nas well as the theory that she was a descendant of PERKIN WARBECK, is\nmere surmise.\n\n       *       *       *       *       *\n\nTHE FIRST MAN WHO ATE AN OYSTER.--The most widely circulated account of\nthis feat is that which ascribes it to the notorious Roman epicure\nPublius Esurius Gulo, who was nicknamed Bellipotens from the rotundity\nof his figure. According to the account given in the _Gastronomica_ of\nVoracius Bulbo (ii. 18) Gulo was always making daring experiments, and,\nwhen bathing at Baiae on a very hot day, and seeing a bivalve which had\nrashly opened its jaws in the sun, he dexterously inserted a stone and\nconveyed the contents to his mouth on the point of the pin of his\n_fibula_. He was subsequently created a proconsul by NERO. The only\ndrawback connected with this account is the fact that oysters were\nrecognised as delicacies in Rome at least a hundred years before NERO.\nIt is right to add that the genuineness of Bulbo's _Gastronomica_ has\nbeen seriously impugned, the best authorities (including FRANCATELLI)\nbeing convinced that the treatise was the work of a sixteenth-century\n_farceur_ who belonged to the royal house of Paphlagonia.\n\n       *       *       *       *       *\n\nPARLOUR PATHOS, SPECIMENS OF.--The best specimens of this interesting\nemotional product are to be found in the words of Royalty Ballads. A\ngood instance is to be found in the following choice quatrain:--\n\n  Nature cares not whence or how,\n    Nature asks not why;\n  'Tis enough that thou art thou,\n    And that I am I.\n\n       *       *       *       *       *\n\nCOMPARATIVE COUPLETS.--The correct form of this literary disease is as\nfollows:--\n\n  A chair without a leg\n  Is like a hen without an egg.\n\nBut it is emphatically not to be encouraged, as excessive indulgence in\nthe habit has been known to lead to the break-up of happy homes.\n\n       *       *       *       *       *\n\nNAMES OF GOLF CLUBS.--The latest addition to the list is, so far as we\nare aware, the \"Sammy,\" but efforts are being made to induce the St.\nAndrews authorities to sanction the \"Biffy,\" a combination of the\njigger and the baffy, and the \"Duncher,\" a powerful weapon for\nextricating the ball out of rushes, tar and other viscous lies.\n\n       *       *       *       *       *\n\nTHE JUGGINS FAMILY.--This family claims descent from Joskin ap Gwyggan,\nthe last native prince who ruled in Dwffryn. The earlier lines in the\ndescent are doubtful. The various families claiming to spring from\nJoskin adopted different patronymics in the fifteenth and succeeding\ncenturies, amongst which may be noted Joskins, Gherkin, Guggenheimer,\nand Gaga.\n\n       *       *       *       *       *\n\nIllustration: THE OLD REFRAIN.\n\n_First Old Lady._ \"MY DEAR, WHAT _DO_ YOU THINK OF THIS WAR? ISN'T IT\nTERRIBLE?\"\n\n_Second Old Lady._ \"AWFUL! BUT IT CAN'T LAST LONG; _THE POWERS WILL\nSURELY INTERVENE_.\"\n\n       *       *       *       *       *\n\nIllustration: _The Patriot._ \"HOARD MY GOLD! I'D STARVE FIRST!\"\n\n       *       *       *       *       *\n\nMIDDLECOMBE _v_. PADDLEWICK.\n\nI.\n\n    _Philip Renwick to Charles Holcombe._\n    Room 99, X.Y.Z. Offices,\n    Whitehall,\n    _8th August, 1914._\n\nDEAR CHARLIE,--Can you possibly turn out for us on Thursday next _v_.\nPaddlewick? We lost to them rather heavily in May last and are anxious\nto give them a sound beating. Their fast bowler is playing for them\nagain, I hear, and we absolutely rely on your help. Can you get off for\nthe day?\n\n    Yours ever, P. R.\n\nII.\n\n    _Charles Holcombe to Philip Renwick._\n    Room 83, P.Q.R. Offices,\n    Lombard Street,\n    _9th August, 1914._\n\nMY DEAR PHIL,--Thanks for yours. Will try to manage it next Thursday,\nbut am doubtful. My chief, though a capable official, is no sport, and I\nanticipate difficulties. I had a day off only two weeks ago for cricket.\nWill do my best.\n\n    Thine, C. H.\n\nIII.\n\n    _Charles Holcombe to Philip Renwick._\n    P.Q.R.\n    _10th August, 1914._\n\nMY DEAR PHIL,--Awfully sorry; no luck _re_ Thursday. Boss hopeless. I\nbroached the matter this morning (without actually asking for\npermission), but I fear the worst. You had better get another man for\nthe Paddlewick match. So sorry.\n\n    Yours ever,\n    CHARLIE HOLCOMBE.\n\nIV.\n\n    _Philip Renwick to Charles Holcombe._\n    X.Y.Z.\n    _10th August, 1914._\n\nMY DEAR CHARLIE,--We shall be absolutely in the cart without you.\nThey've got an awfully hot fast bowler. Bartram now tells me he can't\npossibly turn out, and you are the only really decent bat I know. We\nsimply _can't_ lose to Paddlewick again--we shall never hear the last of\nit. (No one need know that you don't play regularly for Middlecombe.) Do\ntry your best, old man. Mightn't your Aunt Martha be seriously ill?\n\n    Yours ever, PHIL.\n\nV.\n\n    _Charles Holcombe to Philip Renwick_\n    (_wire._)\n\nAunt Martha dying. All well. Boss absent Thursday, so can explain to him\nafterwards. HOLCOMBE.\n\nVI.\n\n    _Philip Renwick to Charles Holcombe_\n    (_wire._)\n\nGood boy. Funeral 11.30. Train Paddington 10.5. Lunch 1.30. Draw 6.30.\nPHILIP.\n\nVII.\n\n    _Charles Holcombe to Philip Renwick._\n    Room 83, P.Q.R. Offices,\n    _14th August._\n\nMY DEAR PHIL,--I regret that I was forced to leave somewhat hurriedly\nafter the game last night. I have nothing to add to what I told you at\nlunch as to the identity of the Paddlewick Spofforth with my chief, of\nwhose sporting talent I was in ignorance. But if you should hear of a\ngood berth going anywhere I should be extraordinarily grateful.\n\n    Yours ever,\n    CHARLIE HOLCOMBE.\n\nP.S.--It was doubly unfortunate (in a way) that I should have scored a\nsix and three fours in one over from his bowling.\n\n       *       *       *       *       *\n\nOLD STYLE AND NEW.\n\nI.--OLD STYLE.\n\n_He._ Has anyone seen the paper?\n\n_She._ I haven't.\n\n_He._ Didn't it come this morning?\n\n_She._ Very likely not. The boy often forgets it. You're the only person\nwho ever looks at it.\n\n_He._ Well, I suppose I must wait till I get to the Club; but I dare say\nthere isn't anything that matters in it.\n\n_Or_\n\n_She._ Have you done with that paper, my dear?\n\n_He._ Absolutely; there's nothing in it. There never is. I can't think\nwhy we waste money in taking it.\n\n_She._ Then perhaps I may have it for a pattern?\n\n_He._ Why, certainly. I've no use for it.\n\nII.--NEW STYLE.\n\n_The whole family_ (all together).\n\n{Has the paper come yet?\n{What's the news?\n{Where's the paper?\n{What about Liege?\n{I say, where's the paper?\n{Isn't the paper here yet?\n{What's the matter with the people?\n\n_Or_\n\n_The whole family_ (all together again).\n\n{I say, father, you might read quicker.\n{Can't you tear it in half?\n{Do tell us the news.\n{Do read it out loud.\n{What about Liege? Quick!\n{Oh dear, why don't we have ten copies of it?\n\n       *       *       *       *       *\n\n    \"The 'Daily Telegraph' Algeciras correspondent, wiring yesterday,\n    says news from Gibraltar reports a naval fight off the Canaries. One\n    of the latter was sunk and the other captured and brought to\n    Gibraltar.\"\n\n    _Liverpool Evening Express._\n\nOur own canary protests indignantly at this treatment of its allies.\n\n       *       *       *       *       *\n\nIn order to be in the very admirable fashion the L.C.C. has decided, we\nunderstand, to change the name of Jermyn Street to Jellicoe Lane.\n\n       *       *       *       *       *\n\nIllustration: THE LOCAL TOUCH. _East Anglian._ \"TELL YOW WHAT THAT IS,\nSIR: THAT THERE KAISER 'E 'ONT NEVER BE SATISFIED UNTIL 'E'S RUINED\nMUDBOROUGH.\"\n\n       *       *       *       *       *\n\nIllustration: A BRAVE MAN. \"LARGE LAGER, WAITER.\"\n\n       *       *       *       *       *\n\nOUR BOOKING-OFFICE.\n\n(_By Mr. Punch's Staff of Learned Clerks._)\n\nMR. DORNFORD YATES, whose name I seem to recall as a contributor to the\nmagazines, has written a book of the most agreeable nonsense which he\nhas called _The Brother of Daphne_ (WARD, LOCK). For no specially\napparent reason, since _Daphne_ herself plays but a small part in the\nargument, which is chiefly concerned with the brother and his love\naffairs. This brother, addressed as _Boy_, was a bit of a dog, and an\nuncommonly lucky dog at that. The adventures he had! He apparently could\nnot go out for the simplest walk without meeting some amiable young\nwoman, divinely fair and supernaturally witty, with whom he presently\nexchanged airy badinage and, towards the end of the interview, kisses.\nWhat distressed me a little at first, till I tumbled to the spirit of\nthe thing, was the discovery that the charmer was always a fresh one,\nand in consequence that these osculations had, so to speak, no\nmatrimonial significance. Perhaps, however, _Boy_ recognised an\nessential similarity in each of his partners. He may, for example, have\nbeen deceived by the fact that they all talked exactly the same Dolly\ndialogue--light, frothy and just a little more neatly turned than is the\ncommon intercourse of mortals. You know the kind of speech I mean. It is\nvastly pleasant and easy to read; but I must decline to believe that any\nyoung man could have the amazing fortune to meet fifteen pretty girls\nwho all had the trick of it. Still, that by no means lessened my\nenjoyment of an entertaining volume, notice of which would be incomplete\nwithout a word of praise for the illustrations of Mr. C. W. WILMSHURST,\na favourite black-and-white artist of mine, whose name is unaccountably\nomitted from the title-page.\n\n       *       *       *       *       *\n\nIf DOROTHEA CONYERS knew as much about English syntax as she does about\nIrish, and were as certain in the handling of a story as she is in the\nconduct of a horse, _Old Andy_ (METHUEN) might be taken at a single\nrefreshing gallop. As it is, I advise the reader to tackle it piecemeal,\na brisk run here and there, followed by a considerable breather. For the\nnovel is put together in a scrambling fashion, being full of repetitions\nof almost identical scenes and making very little definite way in a\nforward direction. There are the usual Irish peasantry and farmers who\nworship the horse for pecuniary and sentimental reasons, as the\nIsraelites worshipped the golden calf; the usual hunting people, who\neither ride straight and are grimly sarcastic or talk very big and go\nfor the gates; and the usual English visitors, who astound by their\nguilelessness and simplicity when confronted by aboriginal horse-copers\nand native bogs and stone-walls. If cubbing be included, I should be\nafraid to say how many meets are described in this book, or how many\nhunt-breakfasts and heavy teas in Irish interiors--interiors of\ncottages, of course, I mean--resulting in how many tricky deals and\nharmless tosses in the heather and the mud. But if you follow my lead\nthere is plenty of pure joy in _Old Andy_, and the most and the best of\nit perhaps is to be found in the remarks of grooms, servant-girls and\ncasual country folk, who as often as not have no kind of connection with\nthe thread of the tale. \"'If meself an' the Masther wasn't rowlin' rocks\nall the day yestherday, he would be within long ago,' replied the\ncovert keeper.\" \"If there is one rabbit with a skinned nose there's a\nhundther, an' they runnin' by mistake to the door they're used to be\nat.\" Such scattered flowers of speech abound in a book whose very want\nof construction is perhaps symbolical and a reflection of the charming\nincoherence of the Irish mind.\n\n       *       *       *       *       *\n\nIt is my painful experience that, when a novelist sets out to write a\ntale of English country life, the better he is at the job the more\nsombre is the finished product. Mr. GEORGE STEVENSON is very good indeed\nat his job; he has sincerity and power, and a certain austere aloofness\nthat will take him far; and the result is that _Jenny Cartwright_ (LANE)\nis about as gloomy a story as ever I read. Above everything else, what I\nnoticed about this book was its freedom from all straining after effect.\nWhatever takes place, I fancy Mr. STEVENSON saying, do not let us be\nsentimental about it. Half the characters in the book seem to come by\nviolent ends; of the two chief women, one commits suicide and the other\nis hanged. Mr. STEVENSON, one can only suppose, speaks of life as he\nfinds it. There are really two stories, that of _Beatrice Barrington_,\nthe faithless wife of _Sir Philip_, and the dreary mockery of life up at\nThe Court, with its hatreds and subterfuges, its crippled master,\nfrightened children and spying servants. This is the county as the\nauthor sees it. Linked with this is the life of the farm, where _Jenny_\nis brought up by an uncle who hates her; where she tends his bedridden\nwife; where her cousin _Beatrice_ goes wrong; where _Beatrice's_\nbetrayer is killed in an accident, and her baby falls into the fire; and\nwhere finally the dour uncle himself, after shooting the young squire\nwho has offered dishonourable addresses to _Jenny_, allows her to pay\nthe penalty of his crime. There is undeniable strength about the book\nand it holds the attention; but I dispute the right of anyone to call it\ncheerful.\n\n       *       *       *       *       *\n\nCYNTHIA STOCKLEY has the writing quality in her; she can both see and\nfeel; she can do man-talk with a plausibility beyond the reach of most\nof her sex; and she works with a refreshing dash and freedom. With a\ncertain carelessness also sometimes; as thus: \"The other, turning to\nrun, got a shot in his leg that put him out of business, but in spite of\nwhich he managed to crawl away.\" And there are little kakophonies, such\nas: \"He was loved, openly and gladly, back.\" The work is good enough to\nmake worth while the cleansing of these defects. The author certainly\nputs into a short story more thought and characterisation than is common\nin these days of half-hours with even the best authors through the\nmedium of magazine pot-boilers. _Wild Honey_ (CONSTABLE) is the title of\nthe first (not quite the best) of an excellent bunch. It sums up the\nbitter-sweet of South Africa, which is the setting of all these stories\nof love, adventure, horror and the wild. They give a strong impression\nof fidelity of draftsmanship, though here we know so little that is\nintimate of the dark continent that we cannot judge how far actual\noccurrences are based on fact or probability. But CYNTHIA STOCKLEY has\nsome of the mysterious qualities of a possible South African laureate.\nPerhaps she will contrive to put away a little weakness for tall and\nscornful aristocratic women; but, in any case, I can commend her book\nconfidently to all intelligent beach-haunters.\n\n       *       *       *       *       *\n\n    \"The price of bread has just been fixed by the authorities at 32\n    centimes the kilometre.\"--_Globe._\n\nSo you can get a couple of yards of French roll for about\nhalf-a-farthing. Not bad for war-time.\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Punch, or the London Charivari, Vol.\n147, August 19th, 1914, by Various\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n**Praise for _The Doom of Kings..._**\n\n_\"Bassingthwaite skillfully balances the high adventure common to the_ DUNGEONS AND DRAGONS _novels with some tender and believable character moments. The grief over a lost sword-brother is given equal weight to intense battles, as is Ashi's frustration at the regimentation of her life amongst the Dragonmarked House of Deneith. My favourite touch however, was that rarity of rarities, a non-human culture that felt true without borrowing slavishly from an existing or ancient people of our own world. I'm very excited that there will be two more novels with which the author can showcase his goblin peoples..._ The Doom of Kings _is also an excellent starting point for a fantasy reader unfamiliar with_ EBERRON _and the other worlds of_ DUNGEONS AND DRAGONS.\"\n\n\u2014 **Chadwick Ginther, writing forMcNallyRobinson.com**\n\nGeth reached inside himself and shifted. The familiar feeling of invincibility burned through his veins. The pain in his rope-burned hand and his aching shoulder seemed to grow distant, than to vanish altogether. His skin felt like hide, his hair like thick, coarse bristles.\n\nAnd he pushed himself further, pouring everything he had into the shifting. Hide, hair, flesh, bone\u2014he was as hard and dense as the heaviest oak. Wild power flooded him and thought vanished. This was how a charging bear or a rampaging boar felt. Geth drew in his legs, pressed himself against the wall, and kicked out with all his strength, roaring as he unleashed the coiled power.\n\nHe let go of the rope just before it snapped taut. The plaza rushed up at him. so did the guards.\n\nLEGACY OF DHAKAAN  \nBY DON BASSINGTHWAITE\n\n_The Doom of Kings_\n\n_Word of Traitors_\n\n_The Tyranny of Ghosts  \nJune 2010_\n\n# Contents\n\n_Cover_\n\n_Other Books by This Author_\n\n_Title Page_\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\n_Glossary_\n\n_About the Author_\n\n_Copyright_\n\n_Raat shi anaa_.\n\n\"The story continues.\"\n\n\u2014Traditional opening to hobgoblin legends\n\n#\n\n# THE EVENTS  \nOF THE DOOM  \nOF KINGS\n\nOnce a savage hunter, now the bearer of a powerful dragonmark, Ashi d'Deneith found herself isolated in the city of Karrlakton and at odds with her mentor, Vounn d'Deneith. Vounn had been tasked with transforming Ashi into an elegant, educated asset of House Deneith, a process which generated considerable resentment between the two women. Storming out of Sentinel Tower after a particularly bad argument, Ashi discovered and fought with a thief attempting to steal an artifact from a Deneith memorial. The thief, however, turned out to be a friend, Ekhaas, a hobgoblin _duur'kala_ , or \"dirgesinger.\"\n\nEkhaas was in Karrlakton as part of a delegation to Deneith from Darguun, the goblin nation that was a major source of mercenaries for the house. The leader of the delegation, Tariic\u2014nephew of Lhesh Haruuc Shaarat'kor, high warlord and founder of Darguun\u2014carried an invitation for Vounn, Deneith's premiere diplomat, to come to Haruuc's court as a special envoy. Haruuc, a cunning but aging ruler, had seen the need for a closer bond between Darguun and Deneith.\n\nVounn agreed to allow Ashi to accompany them to Darguun. Their journey took them through the city of Flamekeep, where they discovered that Ekhaas and Tariic had not been entirely truthful. The mission to Karrlakton, while genuine, had served to disguise their real purpose: making contact with the shifter Geth, another friend of Ashi and Ekhaas. Accompanied by Chetiin, an elder of the goblin assassin clan, the _shaarat'khesh_ , Geth joined them. Tariic explained that Haruuc wished the shifter brought into Darguun covertly. The canny lhesh knew that although closer ties between Deneith and Darguun might aid his nation, his eventual successor would need something more. Geth, as the bearer of Wrath, the legendary hobgoblin Sword of Heroes forged thousands of years before when the goblin Empire of Dhakaan ruled the land, was the key to lasting stability in Darguun.\n\nReaching the goblin nation, Geth, Ashi, Ekhaas and the others\u2014joined now by Midian Mit Davandi, a gnome scholar of the famed Library of Korranberg and an expert on the history of Dhakaan\u2014found themselves caught in simmering unrest as warlords chafed at Haruuc's commands of peace between Darguun and surrounding nations. One clan, the Gan'duur, moved to open rebellion. In the company of a select group of Haruuc's allies that included the aged warlord Munta the Gray and Ekhaas's superior, Senen Dhakaan, ambassador of the traditionalist Kech Volaar clan, Geth, Ashi, and Vounn finally learned the truth behind Haruuc's need for Geth.\n\nThe lhesh understood that whomever he eventually named as his heir would need a strong symbol of power to bring the fractious Darguul clans into line. Haruuc had been in negotiations with the Kech Volaar, for many generations keepers of the lore of fallen Dhakaan, to bring the isolated clan into alliance with Darguun. From their stories, he had learned of an artifact passed down from emperor to emperor during the Dhakaani Age. This artifact, the Rod of Kings, had been forged from the same vein of the purple byeshk as the Sword of Heroes and a third artifact, the Shield of Nobles. The shield had been shattered and the rod and sword lost at different times before the fall of Dhakaan, but Geth's recovery of the sword offered hope that the rod could be found as well. The rod would provide a potent symbol that could unite Darguun, and Haruuc had asked Geth to undertake the quest to recover it.\n\nGeth accepted. Undergoing a ritual of _duur'kala_ magic to wake the power dormant in the sword, he experienced memories of the great hobgoblin heroes who had wielded it during the time of the empire and found that through the sword he could sense the direction in which the rod lay. He set out from Haruuc's capital of Rhukaan Draal to find it, accompanied by Ekhaas, Ashi, Chetiin, Midian, and Dagii of Mur Talaan, a young warlord in the service of Haruuc. The sword guided them to a hidden valley in the Seawall Mountains. Bypassing the camp of a savage tribe of Marguul bugbears, they descended into the valley where they discovered an untouched forest and a mysterious stone staircase of pre-Dhakaani design.\n\nBefore they could investigate, however, they were attacked by trolls and forced to flee\u2014a flight that resulted in the capture of Ashi, Dagii, and Ekhaas by the bugbear tribe. Invoking the names of Haruuc and House Deneith brought no respect from Makka, the leader of the bugbears, and the trio was held under threat of being either sacrificed or sold as slaves. Geth, Chetiin, and Midian had managed to escape, however, and with the help of Chetiin's cunning wolf-like mount, Marrow, they rescued the others, burning the bugbears' camp in the process and leaving the tribe in disarray. Now armed with fire to ward off the trolls, they returned to the valley and descended the stairs to find an ancient shrine.\n\nWithin the shrine, a crevice led down to a strange cave where the withered corpse of a richly-dressed hobgoblin sat on a throne, clutching the rune-carved byeshk rod that they sought. As Geth tried to take the Rod of Kings, however, the apparent corpse moved and spoke, revealing that he was Dabrak Riis, the very emperor who had vanished with the rod thousands of years before. The cave was a forgotten place of power, the Uura Odaarii, or \"Womb of Eternity,\" where time did not pass and to which Dabrak Riis had retreated from the world with no idea of Dhakaan's fall.\n\nAttempts to persuade the lost emperor to give them the rod only made Dabrak angry, and revealed a power of the Rod of Kings unmentioned in any Kech Volaar legend: commands issued by Dabrak could not be resisted. The Sword of Heroes protected Geth from the power of the rod, but Dabrak had also mastered the strange power of the Uura Odaarii and used it to defeat the shifter. Geth's heroic resistance gave Ashi the inspiration she needed to invoke her dragonmark, rendering her impervious to any form of mental domination or attack. Dabrak couldn't affect her with the powers of either the rod or the cavern, but in the timeless place, she couldn't harm him physically.\n\nShe found the answer to her dilemma in the skills that Vounn had driven into her, and struck a bargain with Dabrak for the rod and the release of the others. When Dabrak went back on his word, however, it took a clever trick to make him hurl the rod at Geth in a fit of anger. The heroes fled the cavern with the rod in Geth's grasp. Dabrak pursued them but burned to ashes when he attempted to pass beyond the shrine.\n\nGeth and the others found that none of them felt comfortable handing the rod over to Haruuc once they knew the terrible power it contained. Not giving it to the lhesh, however, could spell the end of Darguun. They decided on a compromise proposed by Chetiin. Dabrak Riis had said that it had taken generations for the Dhakaani emperors to unravel the full potential of the rod. As the rod now seemed to have fallen into an inactive state, they would give it to Haruuc to act as the symbol he needed, but keep its power a secret, acting only if Haruuc or his successor should discover the rod's true nature.\n\nIn Rhukaan Draal, meanwhile, Vounn also faced danger, kidnapped by a changeling disguised as one of her guards. Although she was rescued, the kidnapping could have embarassed and weakened Haruuc at a time when increased raids by the rebellious Gan'duur had cut off food supplies and left the city starving. Vounn and a loyal guard, Aruget, also discovered that Tariic, expecting to be named as heir by Haruuc, was gathering his own power by making secret alliances with a number of warlords, including the devious Daavn of Marhaan.\n\nThe heroes' return, and the rod they carried, was greeted with celebration. As Tariic took the rod from Geth and presented it to Haruuc in the throne room of Haruuc's fortress, Khaar Mbar'ost, the heroes waited uneasily in case its power should manifest. But the rod did nothing except lend Haruuc a more majestic presence. Relieved, the heroes accepted Haruuc's gratitude. The lhesh offered a special reward to Geth, an invitation to become his _shava_ or sacred \"sword brother.\" Early in his life, Haruuc had taken three warriors as _shava_ , though two of them\u2014Dagii's father and Tariic's father\u2014had since died. Geth agreed and joined Vanii, Haruuc's surviving _shava_ , at his side.\n\nDagii, meanwhile, was given the glory of leading an attack against the Gan'duur and their wily warlord, Keraal. Backed by an army assembled from loyal clans and accompanied by Vanii for guidance, Dagii defeated the Gan'duur and captured Keraal. Vanii, however, was slain in the attack\u2014a tragedy that brought out an unexpected brutality in Haruuc, who ordered the captive Gan'duur warriors left to die slowly along the road to Rhukaan Draal. When Dagii\u2014who had disagreed with Haruuc's dishonorable execution of the Gan'duur and secretly given each of them a swift death\u2014presented Keraal to him, the former warlord of the Gan'duur found the strength to taunt Haruuc, accusing him of cowering on the doorstep of humans.\n\nAngered, Haruuc responded that he cowered before no one, that Darguun cowered before no nation, and the people of Darguun would reclaim the heritage of Dhakaan. On their return to Khaar Mbar'ost, Geth, Munta, Tariic, and Chetiin all confronted the lhesh about what amounted to a threat of war. Haruuc forced Tariic and Munta to yield to his authority, but Chetiin would not yield and spoke harshly of the danger of acting rashly\u2014for which advice Haruuc exiled him. Chetiin departed but warned that Haruuc would destroy everything he had built unless he was stopped.\n\nWhen he and Geth were alone, Haruuc offered his _shava_ the chance to leave his service if he would perform the final act of standing with Haruuc to honor Vanii. Geth agreed, but he was dismayed to discover that Haruuc had found and erected an ancient Dhakaani grieving tree, a magical stone construction upon the thorny branches of which its victims suffered an agonizing death. This was the punishment Haruuc had for Keraal. In shock, Geth confronted Haruuc. Haruuc admitted that he did not like what he was doing, but nonetheless did what a king must.\n\nGeth realized that just as the Sword of Heroes sometimes showed him visions of how a hero should act, the Rod of Kings showed Haruuc visions of how a king\u2014an emperor\u2014should act. What was for Geth a gift, however, was a curse for Haruuc. The visions, memories carried by the rod of an ancient time when Dhakaan's power was unlimited, were driving the lhesh to become a tyrant. He begged Geth for help to save his nation once more.\n\nGeth guessed that Ashi's dragonmark might be able to shield Haruuc. Leaving Haruuc to face the honoring of Vanii and the punishment of Keraal alone, Geth ran in search of Ashi. He found her in her quarters, preparing to leave Darguun under Vounn's orders. She accompanied Geth at his urging. As they made their way back to the throne room, Geth's explanation of the situation was overheard by Aruget. Assuring Geth that he could keep a secret, the guard guided them to a side entrance to the throne room.\n\nThey arrived to hear the assembly of warlords calling on Haruuc to lead them in war\u2014and to see Haruuc almost give in to them and to the curse of the rod, before cannily suggesting the elves of Valenar as a target. Geth understood that by starting a small war with the elves, ancient enemies of Dhakaan and, like Darguun, mistrusted by other nations of Khorvaire, Haruuc was preventing a larger war that would surely have destroyed his nation. Once Ashi used the power of her dragonmark to shield him from the rod's curse, even that small war might still be averted.\n\nAs Haruuc raised his arms in triumph, however, a crossbow bolt from a balcony struck him. Geth and Dagii ran to his aid, but the bolt was poisoned and the assassin was already descending from the balcony. It was Chetiin. Avoiding Geth and Dagii's attempts to defend the wounded lhesh, the traitor killed Haruuc with a dagger thrust through his eye, then made his escape.\n\nAs the throne room erupted into chaos, Geth saw two things. First, that Chetiin had believed Haruuc had discovered the rod's power of command when in fact all the power Haruuc had ever needed had been his own charisma. And second, that the rod remained a danger and that whoever sought to succeed Haruuc would surely fall to its curse as well. Acting to buy what time he could and knowing he was protected by the magic of his sword, Geth seized the rod and proclaimed that, until an heir could be determined, it was his sacred duty as Haruuc's _shava_ to hold the throne of Darguun.\n\n#\n\n# CHAPTER  \nONE\n\n# 19 Sypheros, 999 YK (mid-autumn)\n\nNoise battered Geth loud enough that he could feel it in his belly. It swelled out from pipes fashioned from brass stems and inflated bags of leopard skin; from the rhythm of big drums beaten with short, thick rods; from the voices of hundreds\u2014thousands\u2014of goblins, hobgoblins, and bugbears as they crammed the streets of Rhukaan Draal and shouted a final farewell to Lhesh Haruuc Shaarat'kor.\n\nThe corpse of the assassinated lhesh sat in a throne carried on the shoulders of six strong bugbears. Haruuc had been dressed in a suit of heavy armor decorated with the claws and fangs of great cats, his hands curled around the hilt of the famous red sword that had carved out Darguun's destiny. Protective magic had held off decay for the ten day period of mourning. When Geth had knelt before the throne in the last ritual submission demanded by goblin tradition, it had seemed to him that the lhesh might have been resting except for the ruin of the eye socket through which Chetiin's dagger had plunged. Goblin tradition put the fatal wound on display for all to see, though Geth knew that the greatest wound was invisible. The dagger, straight and ugly with a blue-black crystal winking from its blade like the eye of a great cat, was called Witness. When it killed, it consumed its victim's soul. Powerful magics could return life to the dead, but Chetiin had made certain that Haruuc was beyond even their power.\n\nFor ten days, no fires had burned in Rhukaan Draal. For ten days, the streets had been empty between dawn and dusk, and even between dusk and dawn, they had been quiet. The infamous Bloody Market was virtually deserted, most of its stalls shuttered. For ten days, no one had entered or left the city without permission from the fortress of Khaar Mbar'ost\u2014no easy command to enforce, but the guards who patrolled Rhukaan Draal's ragged fringes and stood watch at the barricades that sealed its entrances were not above using fists and clubs to keep order. Wagons carrying food for a city still recovering from the raids of the rebellious Gan'duur clan were permitted to enter, but they did not leave.\n\nTen days without fire, ten days of silence, ten days of isolation. By goblin tradition, a warlord was mourned within his clanhold for five days, but Haruuc Shaarat'kor had been more than just a warlord.\n\nThe morning of the eleventh day had come. Soon the people would be released to the spectacle of games commemorating the dead lhesh. But first, Haruuc's tomb waited for him. The mingled sound of pipes, drums, and goblin voices was discordant and terrifying, halfway between a lament and a call to battle, a primal roar to accompany a king to his grave.\n\nOr, Geth thought as he marched behind the moving throne, to sound the doom of a shifter who was in over his head.\n\nHis hands, already clenched around the Rod of Kings, tightened even more. The rune-carved byeshk shaft seemed colder and heavier that it had any right to be. He glanced at it and thought for the hundredth time in the last ten days, This is your fault.\n\nIf the rod made any response, he couldn't hear it. At his side, Wrath, the Sword of Heroes fashioned by the same ancient hands from the same vein of byeshk as the rod, murmured its own subtle song of inspiration. Not so long ago, he'd only been vaguely aware of the sword's influence as it urged him toward the deeds of a hero. Now, knowing where the rod had led Haruuc, the sword's very weight was an uncomfortable reminder of its influence. Would it someday guide him to his doom as the rod had guided Haruuc to his?\n\nA crooked smile pulled on Geth's lips, baring sharp teeth. Maybe it already had. For ten days, a shifter mercenary had been the ruler\u2014in name if not in practice\u2014of a goblin kingdom. Why? Because it was the heroic thing to do?\n\nOn Geth's right, Tariic, who had been Haruuc's nephew, leaned close and spoke over the noise of the crowd. \"You look uneasy.\"\n\nGeth forced the smile away, but he couldn't completely hide how he felt. \"I feel uneasy,\" he growled back.\n\n\"I could help you,\" Tariic said. \"With the ceremony. You don't have to do it all yourself.\" His eyes darted to the rod and his ears flicked. \"It would be within my right\u2014\"\n\nThere was sudden movement at the edge of Geth's vision, and he turned his head to see another hobgoblin, his broad shoulders made even broader by two thick leopard pelts worn as a mantle, his cheeks marked with ritual scars like clawmarks, pushing out of the packed mass of warlords who followed close behind the throne. Other warlords looked at him, but his eyes were on Tariic, and he had the look of a zealous magistrate watching for the slightest violation of the law. Geth shifted his grip on the rod and dropped one hand to Wrath's hilt. He might mistrust the sword's guidance, but while it was in his grasp it allowed him to understand the harsh sounds of the Goblin language as if he'd been born to it. The warlord's words became clear in his ears.\n\n\"Have care if you seek to advance your status, Tariic. Haruuc is not yet in his tomb!\"\n\nHe spoke louder than he needed to. Even against the noise of the crowd, his voice carried to the warlords nearby. Geth knew him: Aguus, warlord of the Traakuum clan, and like Tariic, one of those in contention to take Haruuc's place as lhesh. The other claimants were close, too. Garaad of Vaniish Kai, lean as a spear and just as deadly by reputation, walked with his supporters on the left. Iizan of Ghaal Sehn, wealthy and willful, looking more like a merchant than a warrior, watched from the right.\n\nHaruuc had been deeply concerned with selecting the perfect heir, with finding a successor who would build on the foundations that he had built. Unfortunately, death had found Haruuc before he had named that heir. It could have been worse, Geth knew. If there had been a clear heir, he wouldn't have had the chance to take control of the rod. It had killed Haruuc as surely as Chetiin. But sooner or later there _would_ be an heir, and he'd have no choice but to hand over the rod\u2014and the dangerous secrets within it.\n\nTariic looked over his shoulder as he walked. He met Aguus's challenge with a practiced calm. \"I speak to a friend, Aguus.\"\n\n\"So long as he holds the throne, he is no friend of yours,\" Aguus snapped. \"The assembly of warlords swore to respect the terms of mourning. We do not seek to advance ourselves until Haruuc is laid to rest. You already force the terms by marching in a place of pride.\"\n\nTariic's ears pressed back against his skull. \"Haruuc was warlord of the Rhukaan Taash clan before he was lhesh. The traditions of the Rhukaan Taash are clear. By those traditions, I am Haruuc's heir as warlord. A place of pride is my right.\"\n\n\"Haruuc's status as lhesh takes precedence over his status as warlord. You try to influence the assembly by putting yourself on display before the allotted time.\"\n\n\"And what do you do by challenging him, Aguus? Why do you challenge Tariic's place when no one else does?\"\n\nThe words came from the young warlord who walked on Geth's left. Dagii of Mur Talaan's gray eyes flashed, and his ears shivered as he glared back at Aguus. Geth had asked him to walk at his side as an advisor\u2014a good choice. In the battle-scarred ancestral armor, the three long horns of a massive tribex mounted on its shoulders and back, of the warlords of Mur Talaan, Dagii made a commanding figure. Warlords who might have mistrusted the shifter who had seized the Rod of Kings could respect the hobgoblin who had defeated the Gan'duur for Haruuc.\n\nAguus's glare shifted from Tariic to Dagii, but other warlords were rumbling their agreement with the chief of Mur Talaan. Aguus dropped his gaze. \"Haruuc must be honored,\" he said. \"He leaves us a great legacy.\"\n\n\"He is honored,\" said Dagii. \"No more challenges. Let the people see that the lords of Darguun are united in their respect for Haruuc.\" With a fluid motion, he drew his sword and thrust it into the air. \"Haruuc!\" he shouted. \"Haruuc!\"\n\nThe other marching warlords followed his example, drawing their weapons and holding them aloft. Tariic, Geth noticed, was the fastest to put his sword in the air. For a moment, it seemed as if the watching crowd paused in their own shouting\u2014then the air shook with a roar that was louder than ever. Light flashed from polished blades as the gesture was imitated in a ripple along the procession of the powerful and important\u2014warlords and chiefs, ambassadors and envoys from hobgoblin clans that stood apart from the lhesh's authority, from the nations beyond Darguun, and from the great dragonmarked houses\u2014that followed behind the throne.\n\nHaruuc's dead ears heard nothing, but the bugbears carrying his throne stood taller and the day seemed a little brighter.\n\nGeth's hand squeezed the rod until his fingers ached. He glanced at Dagii and found the young warlord looking back at him. He nodded grimly and Dagii nodded back. Geth drew breath between his teeth and turned his eyes forward again. Somewhere back in the procession, Ekhaas and Ashi would be feeling the same thing he did.\n\nIf they failed in what lay before them, Haruuc's only legacy would be chaos and the collapse of the nation he had founded. Darguun would die. And so might they.\n\n## Ten days earlier\u20149 Sypheros\n\nSmoke rose in a column over Rhukaan Draal, illuminated by flames below and moonlight above. From the window at which Ekhaas stood, high in Khaar Mbar'ost, she could see only the one fire, but others were probably burning. Word of Haruuc's assassination had spread into the city. The night would be violent.\n\nWord had almost certainly spread beyond Rhukaan Draal. Beyond Darguun, as well. Across the city, ambassadors and envoys would be employing any means at their disposal to rush news of the lhesh's death to their lords. House Sivis's network of speaking stones would be alive with whispers. House Orien's legion of couriers would be flashing across vast distances. In every nation of Khorvaire, in palaces and seats of power, sovereigns and dragonmarked patriarchs would be called from tables, desks, and beds to hear of events in Darguun.\n\nShe had just helped to do the same, weaving her _duur'kala_ magic together with that of Senen Dhakaan, ambassador of the Kech Volaar to Haruuc's court, to send a message to Volaar Draal. Now the wind brought the ghost of a song back to her ears\u2014a reply, but not one that was meant for her.\n\nEkhaas turned to look at Senen Dhakaan. The chamber in which they stood was hers, simply decorated in a way that imitated a stark style popular during a middle period of the Dhakaani Empire. Senen's ears stood high and quivering as she listened to a song that had been sung far away in the mountain caverns of their clan. The song faded and she nodded.\n\n\"The visit has been cancelled,\" she said. \"Tuura Dhakaan and Kurac Thaar will remain in Volaar Draal.\"\n\n\"And the alliance with Darguun?\" Ekhaas asked. Like the other Dhakaani Clans that revered the old ways of the ancient Empire of Dhakaan, the Kech Volaar had stood apart when Haruuc founded his new nation. The Dhakaani Clans lived within Darguun but were not a part of it. Haruuc had recently persuaded the leaders of the Kech Volaar, Tuura Dhakaan and her warlord consort Kurac Thaar, that Darguun and the Kech Volaar had more to gain in working together. By joining Darguun, the Kech Volaar would have a voice in the assembly of warlords and the means to spread the stories of Dhakaan that they had collected for thousands of years\u2014and Haruuc would have access to the clan's hoarded secrets.\n\nTuura and Kurac had been planning a journey from Volaar Draal to Rhukaan Draal to formalize the alliance. But like so much that Haruuc had accomplished during his reign, the nascent alliance had been built on the force of his personality. With his death...\n\nSenen shook her head. \"They will wait to see who comes to the throne.\" She turned a slow stare on Ekhaas. \"They aren't certain what to make of Geth's actions.\"\n\nEkhaas's clenched teeth ground a little tighter before she answered. \"He is Haruuc's _shava_. It was his duty to take charge of Haruuc's affairs until an heir is determined. He follows tradition.\"\n\nSenen pursed her lips and her ears flicked. \"Which Tuura Dhakaan respects. She also recognizes that by taking up Guulen, the Rod of Kings, he staves off a more serious battle between the prospective heirs. But he is a shifter, not one of the _dar_. Why is he doing these things?\"\n\n_Dar_ \u2014the ancient term for the goblin races. Ekhaas had heard the word\u2014and the Goblin words for the three races\u2014more frequently in recent days than she ever had before, as if Haruuc had woken a new pride in the triple races before his death and the people were throwing off names pressed on them by human domination. Bugbears were once again _guul'dar_ , the strong people; goblins were _golin'dar_ , the quick people; hobgoblins, _ghaal'dar_. The mighty people. But in uncertain times, maybe it was good to have such things to cling to.\n\n\"Geth is _shava_ to Haruuc,\" Ekhaas said. \"He bears the Sword of Heroes.\" She shrugged. \"He has a respect for tradition.\"\n\n\"How do you know?\"\n\n\"He is my friend.\"\n\n\"You also counted Chetiin as one of your friends.\" Senen's eyes narrowed. \"What do you know, Ekhaas?\"\n\nA knock on the door of the chamber and a call from the corridor beyond saved her from a lie. \"Senen Dhakaan! Are you there? I'm looking for Ekhaas.\"\n\nIt was Dagii. Senen's ears twitched and her hard eyes took on a knowing, triumphant gleam. She turned away from Ekhaas to open the door. _\"Saa'atcha_ , lord of the Mur Talaan.\"\n\nDagii didn't enter the room. He stood in the doorway, his gray eyes moving from her to Senen and back again as if he could sense the brewing tension between them. Those pale eyes, combined with shadow-gray hair and a naturally somber face, made him look older than he really was. In fact, he was no older than Ekhaas herself, and young for a warlord. \"Geth sent me,\" he said. \"He wants to talk to you.\"\n\nWorse words couldn't have been spoken. Senen's ears rose sharply. Ekhaas walked past her without saying anything. Once the door was closed and Dagii was ushering her along the corridor, she said, \"Senen suspects something.\"\n\nDagii gave her a grim smile. \"This is a time for suspicion.\"\n\nThe halls of Khaar Mbar'ost were quiet. In the moments after Haruuc's death, they had been chaotic, but Ekhaas guessed that as the shock of the assassination had ebbed, people had stumbled back to their rooms or out into the city, seeking reflection or mindless violence, as they preferred. She glanced at Dagii. The aged metal of his ancestral armor carried shiny, fresh scrapes and dents. In attempting to drive off Chetiin, he had come within reach of the grieving tree Haruuc had erected in his throne room. The lhesh's final words had stilled the tree and forced it to release Dagii. His face bore the beginnings of bruises and the limp in his walk\u2014the result of a broken ankle healed in a rush by her magic during the quest for the Rod of Kings\u2014was even more pronounced than usual.\n\n\"Did the tree hurt you?\" she asked.\n\n\"No. It wasn't able to get past my armor. Falling out of it hurt more.\" His face tightened. \"Keraal is worse off.\"\n\nEkhaas's ears rose. In the madness of Haruuc's death, she had forgotten the rebel lord. The command that had released Dagii had also released Keraal. \"He's alive?\"\n\n\"For now.\"\n\nThere was noise ahead, noise that grew louder as they approached the antechamber outside of the throne room. It seemed that not everyone had retreated to grieve. A final turn in the passage revealed the antechamber\u2014and Geth, along with two old hobgoblins, under siege by a small mob of shouting warlords. The two old hobgoblins looked to be in their element.\n\nOne, a thin woman, met the shouts of the warlords with calm, firm answers. \"It is tradition! Would you do it differently in your clan hold? The period of mourning is dedicated to the dead\u2014there will be no discussion of succession until Haruuc is in his tomb. Ten days of mourning Haruuc's death, five days of games to celebrate his life, and then an heir will be selected. Until then the _shava_ holds Haruuc's power.\"\n\nThe other, a vastly old, rotund, yet vigorous man who wore the ceremonial armor of a warlord himself, replied to shouts as if he were on the battlefield. \"The assembly will meet, Garaad! Tomorrow. _Maabet_ , decide for yourself, Iizan! Honor him as you choose.\"\n\nThe thin woman was Razu, who had been Haruuc's mistress of rituals. The fat old warlord was Munta the Gray, Haruuc's closest ally for more than thirty years. Between them, shielded by them, stood Geth. The shifter gripped the Rod of Kings with one hand and the hilt of Wrath, the Sword of Heroes, with the other. Wrath was still sheathed but Geth looked ready to draw it. Ekhaas knew that animal instincts ran strong in Geth's veins. His large eyes were wide and fever-bright, and his thick, coarse hair was almost standing on end. He looked like a wild thing backed into a corner.\n\nHis expression brightened slightly when he saw Ekhaas and Dagii. He nodded toward a door across the antechamber, then turned to face the warlords. \"Enough!\" he said, his voice a snarl that cut through the clamor. He spoke the human language but many of the warlords in the antechamber had fought in the mercenary armies of House Deneith during the Last War and understood the tongue. They fell silent. Geth glared at them all. \"Leave. You heard Razu. I'm not going to talk about the succession.\"\n\n\"Maybe not about the succession.\" Ekhaas recognized the warlord who spoke\u2014Aguus of Traakuum. \"But what about the war?\"\n\n\"The war?\" asked Geth.\n\nAguus grunted. \"It is as Haruuc said. We should turn our blades against the Valenar. If we must wait ten days or more before we make a decision, the Valenar will have an advantage over us.\"\n\n\"There is no war. How can there be a war when the man who spoke of it was cut down moments later?\" Geth swept his gaze across the silent warlords. \"Think about that during the period of mourning.\"\n\nAguus wasn't finished. \"Haruuc would have wanted it!\"\n\nGeth froze for a moment, then growled, \"You're wrong.\" He turned away from them, gesturing with the rod. \"Leave now. A _shava_ needs to mourn, too.\"\n\nHe pushed away from them without waiting for a response. Munta followed for a few moments, exchanging words with him, then turned back to block any of the more aggressive warlords from following. Geth came to join Ekhaas and Dagii at the door.\n\n\"How are you?\" Ekhaas asked him.\n\n\"How do you think I am?\" He opened the door and led them through into a short corridor.\n\nAlong the corridor was another door and a hobgoblin warrior wearing the red corded armband that signified service to Khaar Mbar'ost standing outside it. Aruget, one of the loyal warriors assigned by Haruuc to act as a personal guard to Vounn and Ashi d'Deneith. No one would interrupt them as long as he stood guard.\n\nThe chamber beyond was the same luxurious waiting room for dignitaries where they\u2014she, Dagii, Geth, Ashi, the gnome scholar Midian Mit Davandi, and Chetiin\u2014had been deposited on their return to Khaar Mbar'ost bearing the Rod of Kings. Ashi was waiting for them, pacing the room like a cat. Her dark gold hair had been drawn back, revealing the intricate lines of the powerful dragonmark, a rare Siberys Mark, that patterned her neck\u2014and her shoulders, back, arms, her entire body save the palms of her hands and a narrow strip between her cheeks and her brow. Two small gold rings that pierced her lower lip were the only sign that less than a year ago, Ashi had been a savage hunter of the Shadow Marches before discovering her heritage in House Deneith, and before the manifestation of the mark that allowed her to shield minds and block powerful divinations.\n\nAt the opening of the door, she looked up, her mouth framing a greeting. Geth didn't give any of them time to speak. With a snarl of frustration, the shifter twisted his torso and hurled the Rod of Kings into a large, ornately framed mirror that hung above a sideboard.\n\nThe glass exploded out in a shower of sharp fragments. The rod bounced off the thin wooden backing, hit the sideboard hard enough to gouge it, and fell to the floor with the heavy clang of solid metal. Ashi's words turned into a yell of surprise and the door slammed open as Aruget charged in.\n\nGeth turned on him. \"Out!\"\n\nThe guard stepped back and closed the door without a word. Ekhaas went over and looked down at the rod. The rune-carved shaft was unmarked. She would have picked it up, but Geth stopped her.\n\n\"Don't touch it,\" he said. \"Don't anybody touch it.\" Feet crunching on the broken mirror, he retrieved the rod, handling it as if it were a snake. Geth grimaced and dropped back into one of the room's chairs. His voice was a growl. \"Chetiin said something to me after he killed Haruuc, just before he escaped. He said, 'We swore we would do what we had to.'\"\n\n\"He thought Haruuc had discovered the power of the rod,\" said Ekhaas. Memory of the rod's hold over her mind brought her ears low. There had been as much hope of resisting it as of holding back a flood with a leaking bucket.\n\n\"He should have talked to the rest of us.\" Dagii's face was grim.\n\n\"I wish he had,\" said Geth. \"Because if that's what he was thinking, he was wrong.\" He turned the rod in his hands, and for a moment Ekhaas thought he might throw it again, but then he stood and drew Wrath. Forged from the same twilight-purple byeshk as the rod, the sword was massive and heavy, sharp on one edge and deeply notched on the other, a _dar_ design that hadn't changed much in thousands of years. \"Wrath... talks to me. In a way. Sometimes it gives me a nudge, pushing me to act the way a hero would. Occasionally it puts heroic words in my mouth. It's not just the Sword of Heroes. It's a sword that makes heroes.\"\n\nIn the other hand, he hefted the Rod of Kings. \"The rod is the same. Just before Keraal's punishment, Haruuc and I had an argument. All the time that we were carrying the rod back to Rhukaan Draal, I was the only one to touch it. Wrath doesn't just protect me from the rod's power of command\u2014it shields me from all of the rod's power. When Haruuc took up the rod, he didn't have that protection. A hero inspires. A king rules.\" He bared his teeth. \"From the moment he first held the rod, it fed him the memories of the Dhakaani emperors.\"\n\nDagii's ears rose. _\"Maabet_. The rod has been pushing Haruuc to act like a king?\"\n\n\"Not a king. An emperor.\" Geth lowered both sword and rod. \"Haruuc said the rod responds to the touch of anyone with the will to rule. I think he was strong enough to hold its influence off for a time, but when Vanii was killed in battle against the Gan'duur, it was too much.\"\n\n\"He gave in,\" said Ashi. \"What he did to Keraal and the Gan'duur, his talk of war...\"\n\n\"The rod,\" Geth agreed.\n\n\"But not his talk of war with Valenar,\" Ekhaas said. She remembered the look of desperate cunning on Haruuc's face when he'd spoken of the elves as ancient enemies of the hobgoblin empire. \"Two upstart nations carved out of territory seized during the Last War. The rod might have been driving him to war, but he was trying to choose a target that wouldn't bring all of Khorvaire down on Darguun.\"\n\n\"And if Chetiin hadn't taken matters into his own hands it might have worked,\" Geth said. \"I could have gotten Ashi to him. We might have been able to block the rod's influence.\" He sheathed Wrath. \"Haruuc and Chetiin had an argument. Chetiin tried to point out to Haruuc that he was heading for conflict. Haruuc ordered him out of Khaar Mbar'ost. The last thing Chetiin said to Haruuc was that he would destroy everything he had built unless he was stopped. Maybe he thought he was acting to preserve Darguun. He's only made it worse, though.\" He held out the rod. \"Whoever succeeds Haruuc is going to claim this\u2014and it will claim them. Haruuc had the strength of will to resist it. I don't think anyone who comes after him is going to have that.\"\n\n\"Destroy it,\" Ashi said. \"Steal it. Hide it. Just get rid of it.\"\n\nEkhaas's ears rose. \"We can't,\" she said. \"Haruuc's plan to use the rod as a symbol of power was good. Too good. If it disappears now, there's never going to be a clear line of succession. It may be possible for the assembly of warlords to agree on a new lhesh, but the rod links him to Haruuc and Darguun.\"\n\n\"We could warn the new lhesh,\" suggested Dagii.\n\n\"Warn him how? 'Be careful. The rod will try to make you an emperor.'\" Geth said. \"Is that going to scare a new lhesh or make him curious?\"\n\n\"There's still the danger of the rod's real power, too,\" Ekhaas said. \"Haruuc did everything that he did with the force of his own personality. The rod may have enhanced his presence, but he didn't need its help. If he hadn't resisted it, would the rod have revealed itself to him eventually?\" She looked around the room at the others. \"We're holding a sword by the blade.\"\n\n\"So what do we do then?\" asked Ashi.\n\n\"I don't know,\" said Geth. \"But we've got ten days of mourning and five of games to think of something.\" He pointed the rod at Dagii, then grimaced and lowered it, gesturing instead with his free hand. \"Dagii, I'm going to give you charge of keeping order in Rhukaan Draal. Use it to get a message out to Midian and tell him to return to the city. He's clever. I want his help.\"\n\n_\"Mazo,\"_ Dagii said.\n\nEkhaas nodded, too\u2014Rhukaan Draal was technically the territory of the Mur Talaan clan, but Dagii's father Feniic had been one of Haruuc's original _shava_ and had ceded it to the lhesh as neutral territory for Darguun's capital city. Putting Dagii in charge of it during the period of mourning would meet with the approval of Darguun's warlords. Bringing Midian back to the city was a good idea as well. The gnome scholar had left Rhukaan Draal days before to explore Dhakaani ruins in the south, a reward from Haruuc for his role in recovering the Rod of Kings. She wondered if news of the assassination had even reached Midian yet.\n\n\"What about Ashi and me?\" she asked.\n\n\"Listen,\" said Geth. \"You're in a position to hear what's being said among the warlords\u2014especially the possible heirs like Tariic and Aguus. Ashi, you listen to what's happening among the ambassadors and envoys. They'll be coming to me, but I want to know what they're really thinking.\"\n\nThere was still one worry lurking in Ekhaas's mind, though. \"What about Chetiin? He's out there somewhere.\"\n\nGeth hesitated, then bared his teeth. \"If he's smart, he won't show his face again.\"\n\n#\n\n# CHAPTER  \nTWO\n\n# 19 Sypheros\n\nStorm at dawn, I think I'm going to go deaf!\" said Sindra d'Lyrandar above the roar of the crowd as a fresh wave of noise honoring Haruuc rose to meet the morning sun.\n\n\"You've said that already!\" Pater d'Orien shouted back at her.\n\n\"That's because I mean it. Imagine how bad it must be at the _front_ of the procession. Where's the dignity? How much longer is this going to go on?\"\n\nAshi clenched her teeth and tried to ignore the running litany of complaints from the viceroys of Houses Orien and Lyrandar. The crowd that packed the windows and rooftops of Rhukaan Draal was bad enough\u2014she could almost feel the weight of hundreds of gazes on her\u2014but the viceroys' comments just made it worse. All around her, the envoys who represented the affairs of the dragonmarked houses in Darguun marched in their appointed place in Haruuc's funeral procession, yet only Pater and Sindra felt the need to make their opinions known. It was almost as if they were in competition, something Ashi could believe far too easily. There was no love lost between the two viceroys. Their houses had a long-established rivalry, with Orien controlling overland transport across Khorvaire while Lyrandar's sailing vessels dominated the sea and their wondrous airships the skies, but Pater and Sindra took it personally. Haruuc's funeral wasn't the place for it, though. If they wanted to talk about dignity...\n\nAt Ashi's side, Vounn said calmly, \"Ashi, we're here to show our respect for Haruuc, not start a fight.\" The gray-haired lady seneschal of House Deneith\u2014special envoy of the House to the court of Haruuc Shaarat'kor, and Ashi's mentor in the ways of civilization\u2014nodded pointedly downward.\n\nAshi realized her hand had gone to the hilt of her sword.\n\nHer first thought, as it always was when she looked at the weapon was, no, not _my sword_ , only _the sword I wear_. Her sword, the honor blade that had belonged to her grandfather, Kagan, and that had been the first clue she was more than just a hunter of the savage Bonetree clan, had been lost in the wilderness of the Seawall Mountains in the race to recover the Rod of Kings. It had been a terrible bargain. She wished that the rod had stayed lost. She would have Kagan's sword and Haruuc would be alive.\n\nBut she understood what Vounn was trying to tell her. She couldn't let herself get caught up in Pater and Sindra's argument. She might not have been walking near the front of the procession as Ekhaas, or Dagii, or Geth were, but she still represented House Deneith. She let her hand fall back to her side.\n\nVounn leaned closer so that only Ashi could hear her words. \"The mourning period has been difficult\u2014for all of us.\"\n\nShe caught Ashi's eye as she said it.\n\n\"I appreciate that, Vounn,\" said Ashi.\n\nVounn drew back, eyes still on her. Ashi waited a moment before she glanced away, then didn't look at her mentor again. She knew Vounn had guessed she was holding something back, just as Senen Dhakaan knew Ekhaas wasn't telling everything. The difference, Ashi thought, was that she wished she could tell Vounn what she knew. When Vounn had first become her mentor in House Deneith, they had been in conflict: a barbarian bearing a rare and powerful dragonmark and the diplomat tasked with turning her into a lady. Talent, knowledge, and poise counted as much or more than an individual's mark for respect within the house\u2014the mark Vounn bore was small, capable of creating a shield against physical blows for a time, yet she had the ear of Deneith's patriarch himself\u2014but Ashi's deeds in recovering the Rod of Kings had opened a new path of respect between the two women. Ashi had taken some of Vounn's lessons to heart and Vounn had begun to trust her more, treating her as the capable woman she was, not merely as an asset of House Deneith. If Ashi could have returned that trust by telling her the truth about the rod she would have. She didn't dare. The rod's secret had to be kept.\n\nThey were going to need help soon, though. There had been no word yet from Midian or even from the messenger Dagii had sent to find him. She, Geth, and the others still hadn't figured out a solution for the problem posed by the rod\u2014as the days left ahead of them shrank, theft or destruction looked like the only options. If they wanted to save Darguun, they might have to risk destroying it.\n\nAt least she'd had nothing to report to Geth of unrest or rumors among the envoys of the dragonmarked houses or the ambassadors of the nations of Khorvaire. She'd listened to them as Geth had asked, but most were either angered at being trapped in Rhukaan Draal for the mourning period or uncertain what to make of the shifter who had seized Haruuc's power. Only the nations of Breland and Zilargo shared a direct border with Darguun. The Brelish ambassador hinted that they were watching events closely. The Zil ambassador, a gnome like most Zils, put on a show of being timid and flighty, but Ashi could tell she was as sharp as a knife. Against Geth's prediction, almost none of the ambassadors or the envoys were interested in making deals with someone they considered just a figurehead. They preferred to wait until the proper heir was determined.\n\nPerhaps the only other good thing that could be said of the mourning period was that there had been no sign of Chetiin. If the goblin was still in the city, he wasn't making himself known. Nor were any members of the _shaarat'khesh_ , the Silent Blades, or their cousins, the _taarka'khesh_ , the Silent Wolves. The Silent Clans might just as well have vanished from Rhukaan Draal. It was probably a good thing, too. Public anger had turned against the secretive goblins. City guards charged with enforcing the terms of mourning under Dagii's authority had kept things quiet so far, but those terms ended with Haruuc's entombment. Envoys and ambassadors had been quietly visiting House Deneith to arrange the hire of additional mercenaries to supplement their security in the coming days.\n\nIt occurred to Ashi suddenly that Vounn had made sure she was in attendance during those discreet visits. She'd thought it was just further training in the business of the House, but what if it hadn't been? Much of the information\u2014nothing damning or private, only rumors and descriptions of mood\u2014that she'd passed along to Geth had come from those meetings. Ashi looked back to the lady seneschal and found her still watching her. Vounn's eyebrow rose again and she smiled briefly before turning away.\n\nIf she hadn't been in the midst of the procession, Ashi would have stopped cold with surprise. As it was, she stumbled. Maybe Vounn didn't know everything that was happening, but her mentor was helping her anyway!\n\n\"Ashi?\" Pater d'Orien caught her arm. \"Is something wrong?\"\n\n\"No. Thank you.\" She smiled at the heavy-set viceroy of House Orien. In spite of his rivalry with Sindra d'Lyrandar, Ashi liked Pater. He had the manners and appearance of a caravan master in a fancy coat but behind his blunt exterior was a keen and cunning mind. \"Excuse me, please\u2014I need to catch up to Vounn.\"\n\n\"She won't be going far,\" Pater said, nodding ahead. \"This is the end of the\u2014\"\n\nA swelling of pipes and drums drowned out his words. Ashi looked around and realized that they had reached the edge of Rhukaan Draal. The city dwindled away into a scattering of huts and shacks, though the road continued. The crowd was as thick as ever, but suddenly it was silent. When the crash of the pipes and drums ended, Ashi realized that she could hear another kind of thunder: the noise of the first cataract of the mighty Ghaal River, the steep stretch of whitewater west of Rhukaan Draal that prevented ships from progressing upstream beyond the city.\n\nAhead, the crowd ended, held back by a low wall of stone topped by a black iron fence, while the procession marched on through an imposing heavy arch as big as a good-sized house and built of the same red stone that had been quarried for Khaar Mbar'ost. Ashi heard Sindra snort. She was a half-elf and her fine-featured nose turned up in disdain. \"A Karrnathi victory arch. Not very original.\"\n\n\"Get closer,\" Pater told her. \"I think you'll find it's more original than you think.\"\n\nAshi knew victory arches\u2014House Deneith was based in the ancient Karrnathi city of Karrlakton, where monuments were nearly as common as hovels\u2014and as they approached the arch in the wall, she could see what Pater meant. Karrnathi arches were typically decorated simply with fluted columns and a band of relief sculpture around the top, crowned perhaps with a memorial statue. This arch was different. Reliefs crawled across the red stone walls: hobgoblins and bugbears and goblins in battle and on the hunt. It was difficult to distinguish what they were fighting or hunting, but the scenes of struggle were clear. Nor could the arch properly be called a \"victory arch,\" because at least as many of the goblins depicted in the fantastic carvings were dead or dying as were triumphant. The higher up the walls the reliefs went, the more dead _dar_ there seemed to be, until just beneath the crown of the arch, where rows of curved spikes jutted out like sinister horns, the carved bodies were piled in heaps. Ashi stole a glance at Sindra. The Lyrandar viceroy looked vaguely unsettled.\n\nThe carvings continued in the shadows of the arch as well, though here the dead stood in a parade of figures pierced with swords and crushed with hammers, bristling with arrows and ravaged by monsters, burned, tortured, decapitated, and dismembered. Ashi stared at a bugbear who appeared to be marching onward as if in ignorance of the massive ballista bolt that pierced his belly, and felt recognition. \"Baargaar Seven Axes,\" she said. \"These are the heroes of _dar_ history.\"\n\n\"Aye,\" said Pater. \"And now they have one more.\" He pointed up.\n\nAbout halfway through the vault of the arch, the parade of figures gave way to smooth stone and on the edge of the empty space was a figure freshly carved\u2014a hobgoblin wearing a spiked crown, one eye socket empty, a sword in his right hand and a rune-carved rod in his left.\n\n\"Haruuc,\" said Ashi. She couldn't help noticing that the stonecarver had taken some liberties. Under Haruuc's feet lay the broken body of a goblin dressed in the clothes of the _shaarat'khesh_ and holding two daggers, one wickedly curved, the other straight and plain. Chetiin. Ashi wondered if the others were having the same difficulty as she was in reconciling the quiet, wise goblin who had traveled with them to recover the rod with the treacherous assassin who had cut down Haruuc. That Chetiin was a killer\u2014yes, even an assassin\u2014there was no doubt, but which of them wasn't? Ashi had been a hunter and briefly the huntmaster of the Bonetree, the most feared and savage clan in the Shadow Marches. Chetiin was an elder in an ancient clan of assassins and skilled in ways Ashi could only hope to imitate. When he moved, he was a whisper. When he fought, he was the blade of a dagger. When he spoke, his strained voice carried the lessons of a lifetime. He carried their loyalty and the loyalty of Haruuc and the loyalty of the Silent Clans. Yet he had turned against and slain in cold blood the greatest leader the _dar_ had known in generations, someone who had trusted him and called him a friend. If Chetiin were standing in front of her right now, Ashi didn't know if she would talk to him or try to put her sword through him.\n\nBeyond the arch, the only sounds were the movement of bodies and the crash of the cataract. The road formed the only level surface across rocky and irregular ground and even it ended within a dozen paces of the arch. The funeral procession walked through tall grass, dry with the end of autumn, heading toward a ridge of weathered rock\u2014the same ridge that formed the cataract in the river. Haruuc's tomb waited within the shelter of a fold of the ridge, a low structure with a peaked roof that sank back into the rock and a larger underground chamber. It had been built of the local gray stone and seemed stark in its simplicity. The door gaped open, ready to receive its occupant. Ashi felt her flesh crawl. The arch, the vanishing road, the ridge, and the eternal crash of the cataract created a vista to haunt the soul of anyone who approached. Haruuc's was the only tomb here, yet Ashi felt as though she walked through a graveyard that already held the royal dead of centuries.\n\nThe strict hierarchy of the funeral procession broke down in the field of grass. Goblins crowded forward as Haruuc's body, still borne aloft on its throne, was carried up a set of steep stairs carved into the rock. Humans, half-elves, and other races stayed back, clustering together with unspoken wariness. Ashi caught up to Vounn. Pater and Sindra joined them, rivalry set aside for the moment.\n\nThree hobgoblins waited beside the open door of the tomb. Priests, Ashi thought. Goblins by tradition offered their prayers and sacrifices to the gods of the Dark Six, hoping to appease the harsh deities. Haruuc, however, had embraced the worship of the Sovereign Host, the gentler faith followed by the majority of the nations of Khorvaire. Over the years of his reign, acceptance of the Host had filtered down from warlords and courtiers eager to emulate their lhesh to average warriors, merchants, and farmers. The Darguuls, however, had put their own mark on the worship of the Host. Ashi guessed that the three priests spoke for Dol Dorn, Dol Arrah, and Balinor, the gods of strength, honor, and the hunt that Haruuc had venerated above the other Sovereigns. All three wore snowy robes, but their faces had been smeared with dirt of different colors, and beneath their robes they wore armor\u2014plate for Dol Arrah, chain for Dol Dorn, a hunter's leathers for Balinor.\n\nAs soon as the gathered warlords had grown still, the priest of Dol Dorn stepped forward. His face was darkened with gray soil that could have been scooped from among the rocks of the ridge. He called out in Goblin, but Ashi had been studying with Ekhaas and the harsh sounds of the language were familiar to her now. \"Who comes to the gates of death?\"\n\nGeth stepped forward from the crowd to mount the steps up to the tomb. \"Geth, who bears the sword Aram, who carries the honor of Kuun, who killed the dragon Dah'mir, comes.\" His accent was thick, but he spoke the Goblin words carefully.\n\nThe priest of Dol Arrah moved forward. His face was yellow with the dust of Rhukaan Draal. \"You will not pass, Geth who bears Aram. Death has not claimed you.\"\n\n\"I follow one whom death has claimed. I seek passage for him.\"\n\nBalinor's priest came forward as well. His face was stained with ochre, red as the blood spilled during the hunt. \"For whom do you seek passage? Who will enter the gates of death?\"\n\n\"I seek passage for Haruuc of Rhukaan Taash, who was son of Tiraan, who was Haruuc Shaarat'kor, Lhesh of Darguun. He will enter the gates of death.\"\n\n\"By what right do you seek passage for him?\" The priest's voice was deliberately disdainful.\n\nGeth met his eyes in ritual defiance. \"I was _shava_ to him.\"\n\nBalinor's priest paused before responding solemnly, \"You have the right.\"\n\nDol Arrah's priest put out his hand. \"Bring forward the treasures that will pass with Haruuc through the gates of death!\"\n\nThere was movement behind Ashi, and she stepped aside as those who had walked through Rhukaan Draal at the end of the funeral procession came forward. Bearers representing all three races of the _dar_ passed among ambassadors, envoys, and warlords, the sun's light gleaming on what they carried in their arms. Caskets full of gold and jewelry. Chests packed with bright weapons and armor. Fine goods from across Khorvaire, from exotic Xen'drik, and from distant Sarlona as well. If Ashi hadn't known better, she would have thought that Khaar Mbar'ost had been stripped to fill Haruuc's tomb, but this was only a fraction of the treasures that Haruuc had amassed and some of what would follow Haruuc were offerings from other clans\u2014even from dragonmarked houses. She saw the crest of Deneith on a polished shield and the sign of Orien on a small but exquisite sculpture of a horse cast in silver. There was also a bearer who walked apart from the others with nothing more than a small open chest in his hands. A single dagger rested inside the chest, the deadly weapon, left behind by Chetiin, that had struck the fatal blow. By _dar_ custom, it would rest with its victim.\n\nOne by one, the bearers climbed the ridge, bowed once before the priests and twice before Haruuc's corpse, then vanished briefly into the tomb before reappearing without their burdens. It took a long time, but when all of the bearers had returned to the field below, the priest of Dol Dorn spoke again.\n\n\"Traditions tell that the People were born in caverns and lived there before we emerged to fight beneath the sun and the sky. When we pass through the gates of death, we return to caverns, the womb and the grave. Life continues. Tradition continues.\"\n\nHe gestured for the bugbears holding Haruuc's throne to kneel. When they had, he reached up and pulled free Haruuc's sword, then turned to Geth. \"The People have mourned Haruuc's passing. Free them to continue their lives.\"\n\nGeth nodded. Tucking the Rod of Kings between his arm and his body, he knelt before the priest and the sword. The other two priests came forward. Balinor's priest now carried a broad tray. Dol Arrah's stood close with an unlit torch.\n\nFrom the tray, Geth took a copper bowl lined with thick shreds of fiber and a flint, laying the Rod of Kings in their place. He set the bowl on the ground. The priest of Dol Dorn grounded Haruuc's sword in it. Carefully angled so that all could see what he did, Geth leaned forward and struck the flint on the blade of the sword.\n\nSparks jumped from the sword into the tinder. Smoke curled up in a fine wisp, growing thick as Geth bent close to blow gently onto the bowl. When he sat back, the torch was ready for him. He took it and touched it to the burning tinder. Flames leaped into life. Geth lifted the torch and stood to face the warlords.\n\n\"From this will fire return to Rhukaan Draal,\" he said. \"From this will life continue.\"\n\nHe took back the rod. The priest of Dol Dorn returned the sword to Haruuc's grasp, then looked out over the crowd.\n\n\"This was Haruuc!\" the priest shouted, his ears high and quivering. \"Haruuc who founded Darguun and Rhukaan Draal, whose boldness made a new homeland for the People. He will never be forgotten! He will live in stories and legends as long as mighty, quick, and strong people draw breath. He will inspire us as long as the Ghaal and Torlaac Rivers flow from mountain to sea. Let all who would be great follow in the path that his red sword has carved!\"\n\nThe roar that rose from the gathered warlords was as loud as anything Ashi had heard in the city. Weapons were drawn and flashed against the sky. Ashi felt the fervor of the moment sweep her up\u2014she drew the sword at her side and raised it high as well. \"Haruuc!\" she shouted along with the Darguuls. \"Haruuc!\"\n\nThe priest of Dol Arrah nodded and the bugbears lifted the throne again. Turning underneath it, they reversed their positions so that they could walk forward. With his warlords shouting his name, Haruuc was borne into his tomb facing outward and the shadows of the grave closed over his face like dark water.\n\nThe three priests followed him in. Geth moved to stand before the doorway and the shouts of the warlords died away. In the silence, the noise of the cataract seemed louder than ever.\n\nAfter a moment, Pater spoke softly. \"The first time I met Haruuc, I expected him to talk about taxes or tolls. Instead he asked me if the trade roads were in good enough condition to allow easy trade between Darguun and Breland.\"\n\nThe humble memory startled Ashi, but then she realized that throughout the crowd, warlords and ambassadors were whispering, sharing quiet reminiscences.\n\n\"He asked me about distant ports,\" said Sindra. \"He knew what was happening across Khorvaire.\"\n\nThe Zil ambassador, Esmyssa Entar ir'Korran, stood nearby. \"He always spoke to me as if I was as tall as he was.\"\n\n\"We discussed philosophy,\" said Vounn.\n\nAshi tried to find her own most significant memory of Haruuc. It took her a moment\u2014his presence, even before he held the rod, had been inspiring and it was hard to pick out just one memory. She finally found the one she sought. Something that probably would have seemed small to anyone else but made a great difference to her. \"He saw me as more than my dragonmark.\"\n\nVounn raised a slim eyebrow at the statement, but said nothing.\n\nThe throne-bearers were the first to return to the surface, followed by the three priests. The priest of Balinor stepped around to the side of the tomb and, straining hard, pushed. The massive door swung shut. Stone as thick as a dagger's length moved ponderously but, once set in motion, smoothly. The dark doorway became a narrow gap, then a sliver. Then the great door closed with a solid boom\u2014and a hollow crack as the pivots that had allowed it to swing broke as they had been designed to do. The tomb of Haruuc Shaarat'kor was sealed. From the front of the door, an effigy of Haruuc glared down, a fierce warrior and a mighty king at the height of his power, the guardian of his own grave.\n\nStanding under that gaze, Geth raised the torch and the rod high above his head. \"The time for mourning is done. We remember Haruuc's death\u2014but now we celebrate his life. Now the games begin!\"\n\nThe cheer that rose from the warlords was not as powerful as that which had honored Haruuc, but it was joyful and enthusiastic. Ashi was certain that she even heard an echo of it from the common people who waited beyond the arch.\n\nGeth began the descent of the ridge, new fire for the city held out before him, and warlords shifted in an attempt to be among the first to accompany him back into Rhukaan Draal. Ashi tried to catch his eye, to give him a simple gesture of encouragement and show him he had done well, but it was no good. There were too many people vying for his attention. She started to turn back to Vounn\u2014\n\n\u2014and felt the back of her neck prickle with the sensation of being watched. An old hunter's instinct. She glanced around. This wasn't like it had been while the funeral procession moved through Rhukaan Draal. No crowd watched her here.\n\nNo crowd except the one beyond the arch. Ashi spun sharply.\n\nAnd the feeling was gone. The crowd was already breaking up and streaming back into the city, eager for the start. If someone had been watching her, they'd needed only to turn away to lose themselves in the crowd. Ashi found her hand was back on the hilt of her sword. She trusted her instincts. Someone _had_ been watching her. Who?\n\nHer hand tightened on the sword. Chetiin?\n\n\"Ashi?\"\n\nThe sound of her own name made her whirl. Her sword was halfway out of its sheath before she realized it was Ekhaas. The _duur'kala's_ ears rose at the sight of her blade. \"Is something wrong?\"\n\nAshi felt blood rush to her face and she shoved the blade back. \"No.\"\n\nEkhaas's ears didn't drop but she nodded. \"Would you like to watch the start of the games together?\"\n\nThe thought of a fight after the long period of mourning was good\u2014it certainly seemed like a fitting tribute to Haruuc. Ashi looked back to the tomb, then nodded. \"I'll tell Vounn.\"\n\nShe glanced at the arch and the moving crowd beyond one last time. Someone had been watching her. That didn't mean it was Chetiin. With such a large crowd, it could have been anyone.\n\n#\n\n# CHAPTER  \nTHREE\n\nWhen the bugbears of the White Stone tribe, savaged in the dark woods by a huge wolf that could only have been part demon, returned to their camp to find every hut ablaze, their children alive only by the dark gods' whim, and the prisoners who had started it all escaped, their only choice was to flee. The wolf-demon had settled for the moment to feast on the bodies of fallen warriors. Their camp was on fire. The trolls that lived in the cursed valley below, that had for years kept to themselves in return for carcasses thrown down from the bugbear camp, were raging. Overnight, their haven in the Seawall Mountains had become too dangerous to hold.\n\nSo they fled, slipping away down the western approach to their high vale with the flames of their burning camp leaping high into the sky behind them. Then they started looking for someone to blame.\n\nMakka, who had been the _chib_ , the leader, found himself on the other side of their anger.\n\n\"Makka kept the lowlanders that the trolls were chasing!\" said Guun, who had been his closest friend. \"He should have sent them back, but he kept them! He was greedy for the treasure they sought in the valley!\"\n\n\"When the lowlanders' friends challenged us from the woods, Makka led us out,\" said Utaa, whose skill in a fight was a close second to Makka's. \"The human woman had a dragonmark\u2014her magic must have summoned the wolf. Now our camp is gone, seven warriors are dead, and we have nothing!\"\n\nIn the tradition of the Marguul tribes of the Seawall Mountains, Makka could have defended his honor in combat against Guun and Utaa, but the wolf had torn into his right arm. It would heal, but until it did all he had were threats. \"Guun and Utaa are cowards! They are foxes circling an injured lion. I am Makka! I am your _chib!_ I am\u2014\"\n\nThe first stone struck him between the shoulders. Roaring, Makka whirled to lash out with his trident at the one who dared attack him from behind, but the blow was weak. More stones, mud, and branches pelted him, splattering the bear hide vest that he wore. The White Stone tribe howled its rage, Tuneer and Wiraar, the mothers of his children, among them. Guun hefted his big mace, Utaa his sword. The two warriors came forward through the rain of debris.\n\nMarguul tradition valued honor, but not so much as it valued staying alive. Followed by the jeers of his tribe, Makka fled for the second time that night.\n\nFor three days he hid, thoughts of vengeance festering in his mind just as his wounded arm festered. On the fourth day, he went to a brook upstream from the White Stone's new camp and washed the pus from his arm, praying to the Dark Six that the sluggish water would carry the infection to his former tribe. Then he packed the burning wound with moss and spiderwebs and made his way back to the burned-out camp. Down the mountainside, he found tracks. The footprints of humans and hobgoblins, the hoofprints of horses, the pawprints of wolves, all leading out of the mountains and back to the lowlands.\n\nThe ones responsible for his shame were the three prisoners, two hobgoblins and a human that he had rescued from the trolls\u2014and whatever allies had appeared to help them to escape and lay waste to the White Stone camp. He had learned something about the three prisoners while they were his captives. The hobgoblin warrior had named himself Dagii of Mur Talaan and claimed the hobgoblin woman as a follower, though Makka doubted the claim. The human woman carried the dragonmark of Deneith across her face and arms, and the bright sword that now hung at Makka's side like an oversize dagger had been hers until he had taken it.\n\nHis shame had been begun by the three. On a mountain side during a thunderstorm, Makka swore by the passion of the Fury, dark goddess of rage, that his vengeance would start with them.\n\nWhere would he find them? They had tried to bargain for their safety by invoking the name of the lowland king, Haruuc. Whether they had his favor or not was questionable, but if they'd known to try invoking his name, there was only one place to begin his search.\n\nMakka turned his footsteps toward Rhukaan Draal.\n\n## 19 Sypheros\n\nHe arrived just before dawn on a fine cool day to find the entrances to Haruuc's city clogged with lowlanders, all dressed as if for a festival. Makka waded into the throng, shouldering past hobgoblins, goblins, and bugbears alike. There were only _dar_ present, he noticed. That was strange. Usually humans forced their way into everything.\n\nAs he got closer to the front of the crowd, he saw what was holding it back: guards wearing armbands fashioned of red cords stood at makeshift barricades. He grabbed the nearest goblin and spun the little creature around, forcing him to look up into his face. \"What's happening here, _taat?\"_\n\nThe goblin stared at him with wide eyes. \"The end of mourning,\" he said, his voice cracking. \"Haruuc goes to his tomb today.\"\n\nMakka grunted and looked back to the guards. The goblin, he noticed, took the opportunity to shrink back into the crowd. Makka didn't try to stop him.\n\nHe'd heard about Haruuc's death during his journey. A goblin in the southland had squeaked out the news, not that it had meant much to Makka. The end of a warrior\u2014may the Keeper treasure his soul\u2014but also a fitting end for a lowland sop who had turned dog for the humans. Makka had his vengeance to think about. Other goblins and hobgoblins along the way had confirmed the passage of the three he sought. Yes, two hobgoblins and a human with a dragonmark across her face had traveled that way. Yes, they had been going toward Rhukaan Draal. Yes, they had traveled with others: a shifter, a gnome, and a goblin who rode on a great black wolf. Makka had added three more deaths to his oath of vengeance.\n\nNames had come later\u2014those who had passed were heroes, acclaimed by Haruuc. Dagii of Mur Talaan, Ekhaas of Kech Volaar, Chetiin of the _shaarat'khesh_ , Ashi of Deneith, Midian the gnome, Geth the shifter and _shava_ to Haruuc. Later still, the news that Chetiin was Haruuc's assassin and Geth now held the throne of Darguun. That had made Makka smile. His targets were _chib_. Killing them would bring him much honor.\n\nAnd the Fury, it seemed, approved of his vengeance, because he found four of his targets with an ease that could only be an omen.\n\nThey were out in the open, marching in Haruuc's funeral procession, unsuspecting of the death that had come for them. Dagii of Mur Talaan and a shifter that could only be Geth walked directly behind Haruuc\u2014the lowland king made a wrinkled, pathetic corpse. Ekhaas of Kech Volaar was further back, and Ashi of Deneith walked with humans and other races near the rear of the procession. The riches borne at the very end of the parade of mourners gave Makka pause. So much wealth! He ground his teeth, closing out greed inspired by the Keeper, and turned back to his targets in the procession. They were the reason he had come.\n\nThey were well-protected. He would have to wait, but he had waited for prey many times. Makka chose his target. The human woman would be first. He would slay her with her own sword and leave the weapon in her steaming guts, letting the others know who was hunting them.\n\nHe shadowed the procession through the stinking crowds of Rhukaan Draal. When the city ended, he pushed forward until he was right against the wall beyond which the procession had passed. Heaving himself up against the iron fence atop the wall, Makka could see the marchers, but not Ashi in particular. That didn't bother him. He'd pick up her trail again.\n\nHe had a good view of the tomb along the ridge and of the three hobgoblins standing alongside it. Only when the ceremony had started did he realize they were priests. The tips of his ears curled and a growl rumbled out of his belly. He'd heard that Haruuc had abandoned the Dark Six and the old ways of the _dar_ to follow the Sovereign Host, but to bury a warlord or a chief without the sanction of the Keeper, god of death? As Haruuc's throne-bearers descended into the tomb, Makka thought maybe the priests would at least make a sacrifice of them, but all of the bugbears returned to the surface and the tomb was sealed without blood being shed. Makka's hand went to the _muu'kron_ , the talisman of six knotted cords on his belt, and his fingers closed around the fang that was the token of the Keeper.\n\nAlong the wall, he could see a few others doing the same thing, though furtively. Makka felt a surge of disgust\u2014Haruuc hadn't turned all lowlanders to the worship of gentle gods, but he'd forced those who'd kept to the old traditions to hide their beliefs. He caught the token of the Fury, a bit of wood carved like a snake and polished smooth by his touch, in his fingers and prayed for a swift resolution to his vengeance so that he could return to the freedom of the mountains.\n\nHis prayer was answered. As the blasphemous ceremony ended and the crowd below the tomb broke up, he caught sight of Ashi again. He smiled.\n\nThe human woman stiffened, as if catching his scent, and turned.\n\nMakka dropped away from the wall, slipping into the sea of people turning away from the tomb to rush on to the games. He didn't want Ashi to know he was here. Not yet.\n\nFrom behind a moving screen of _dar_ , he watched Ekhaas of Kech Volaar distract Ashi. The two women spoke together, then to an old human woman, then they moved back to the road that led away from the tomb and joined the crowd streaming into the city. Alone.\n\nMakka hadn't considered taking two of his victims at once, but when the Six gave a sign, only a fool ignored it.\n\nAshi waited until they were away from the tomb, away from the remains of the procession and safely anonymous among the crowds of Rhukaan Draal before she spoke. \"I had an idea last night. A way to deal with the rod.\"\n\nEkhaas looked at her, ears flicking gently. \"How?\" she asked.\n\n\"As long as I'm in Rhukaan Draal, I can use my power to protect whoever holds the rod from its influence. We tell the new lhesh that it's dangerous, so he knows he needs my help, but we don't tell him about the power of command. The protection of my dragonmark lasts for about a day\u2014I could renew it every morning.\"\n\n\"You'd be trapping yourself in Rhukaan Draal,\" said Ekhaas.\n\n\"I'm willing to do that,\" Ashi said.\n\nEkhaas's ears stood up straight. \"Why? Ashi, this isn't your fight. You could leave Darguun any time and forget all about us.\"\n\n\"Darguun provides mercenaries to House Deneith, but it doesn't want anything from us except gold. Vounn came to Darguun to find a way to change that balance. If the lhesh needs me to protect him from the rod, that changes the balance.\"\n\nThe _duur'kala_ stared at her as if she were crazy. Ashi's confidence in her argument crumbled under that gaze. \"It sounded good in my head,\" she said.\n\n\"Leaving aside the insanity of binding yourself into service,\" said Ekhaas, \"what if something happened to you? What if you missed renewing the protection? What about when you get old and die\u2014the rod isn't going to go away.\" She grimaced. \"It would also mean telling the lords of Deneith about the rod. How do you think they'd react to that news?\"\n\n\"Not well.\" Ashi clenched her jaw tight. \"We're going into battle armed with a spoon and shoe leather, aren't we?\"\n\n\"We've got five more days. We'll think of something.\"\n\n\"My head hurts from thinking about it.\" She hesitated, then added, \"We should ask Vounn for help. She already suspects something.\"\n\n\"No.\" Ekhaas's answer was immediate. \"Ashi, too many people already know about this. The fewer people who know the rod's secret, the better\u2014and the less Vounn knows, the better for her.\" She glanced sideways at Ashi. \"If something happens and this all goes wrong, Vounn will be able to say honestly that she didn't know anything. Tell her and you could even risk the honor of Deneith.\"\n\n\"The same for Senen Dhakaan and the Kech Volaar?\" Ashi asked her.\n\nEkhaas's ears folded back and she made a sour expression, an answer that spoke as loud as words. If she had actually intended to reply, however, her words were lost in the commotion that broke out around them.\n\nThe street they followed narrowed at a point where a building pushed out into the street like an overly bold merchant. If the crowd on the street had been moving in two directions, the spot would have been almost impassable, but with everyone heading in the direction of the games, movement was merely slow\u2014at least until someone on the rooftop began throwing stones and debris down onto the milling crowd below. Yelping and cursing, those on the street tried to push through, pull back, or just take cover. The crawl of movement turned to a wall of chaos and more people were still pushing into them from behind.\n\n_\"Khaavolaar,\"_ muttered Ekhaas. \"This way.\"\n\nAn alley opened off to one side, narrower than the street, but the space at its opposite end seemed wide and open, presumably with another exit. Some people were already turning down it and they showed no sign of returning. Ashi and Ekhaas followed them.\n\nBehind them, the shower of stones shifted as whoever was on the rooftop moved to cut off the escape route. Only a few quickwitted goblins made it into the alley on their heels before the crowd pulled back.\n\n\"Children looking for sport,\" Ekhaas said. \"We should be lucky it was only stones\u2014\"\n\nThere was an ominous grinding from overhead. Ashi glanced up just in time to see the rough bricks of a chimney tilt and topple into the alley. The bricks fell several paces behind them, but the goblins who had followed them into the alley weren't so lucky. A couple were down, clutching heads covered in dust, bits of mortar, and blood. Those who had escaped the falling bricks were shaking their fists and shrieking in fury at their unseen tormentors.\n\nA creeping sensation ran along Ashi's spine. Stones had stopped falling at the mouth of the alley as if those on the rooftop had tired of their game, but with the collapse of the chimney, no one else was venturing down the alley.\n\n\"I don't like this,\" she said. She looked at Ekhaas and the hobgoblin nodded. The fallen chimney and injured goblins blocked the way back along the alley. Ekhaas jerked her head and they continued on, moving more quickly, one eye on the bright gap of sky above.\n\nThe alley opened into a yard of packed earth. Aside from some chickens clucking with dim curiosity and three hobgoblin children, too young to have possibly climbed up to the roofs, staring from a patch of mud in one corner, the yard was empty. Those who had entered the alley before Ashi and Ekhaas were gone, down a second alley that gaped at the other end of the yard. Ashi's neck prickled again. They were in the open. She whirled, running backward so she could look up at the rooftops.\n\nAnd just in time. A mighty roar split the air as the bulky silhouette of a bugbear came swinging out of the sunlight, the trident clenched in one massive fist stabbing down at her.\n\nAshi flung herself aside. Their attacker drew his legs under himself and landed in a crouch, soaking up the hard impact and pushing off again to lunge at Ashi. She rolled, and the trident stabbed packed earth. He kept after her, thrusting again and again as she scrambled away. Chickens flapped and leaped around the clear space. The hobgoblin children were pressed up into a corner, screaming.\n\nThe bugbear spun the trident in his grip and swiped at her with the butt of the shaft. It caught her across the face and blood sprayed from her mouth as she went sprawling sideways. She twisted, catching herself, and stared up at him, anger at the unprovoked attack flooding through her\u2014then giving way to shock.\n\nShe knew the bugbear. She recognized the bear hide vest, the trident, his face, his sneer.\n\nMakka.\n\nShe didn't have a chance to say anything though. \"Ashi!\" shouted Ekhaas. \"Move!\"\n\nThe warning might have been intended for her, but Makka understood it too. Even as Ashi sprang away, he spun to face Ekhaas.\n\nThe _duur'kala_ stood at a distance, sword drawn but held low. Ashi saw Makka blink with confusion. Then Ekhaas drew a breath\u2014and sang.\n\nDust jumped from the ground in a thin yellow cloud as waves of battering sound burst around Makka. Ashi caught the very edge of the magical attack and it was as if all of the noise that the crowd had made in cheering Haruuc's funeral procession were focused on her.\n\nMakka reeled under the assault of the song, then roared defiance and charged Ekhaas on unsteady legs.\n\nAshi ripped her sword from its scabbard and leaped to meet him, knocking aside the trident before he could reach Ekhaas. Makka jumped away from the follow-through and dropped into a crouch, weapon at the ready.\n\nAshi crouched across from him, every hard breath sending a little spray of blood from a broken lip. \"You!\" she panted.\n\nEkhaas recognized him, too. \"Makka,\" she said. She moved in cautiously, trying to flank him. Makka moved the trident between them, pointing the weapon first at one, then the other, as he shifted backward to stay clear. Chickens and children had both settled into frightened silence along the walls. Ashi watched his feet. Ekhaas watched his face. \"What are you doing here?\" the _duur'kala_ asked in Goblin.\n\nMakka didn't answer. Instead, he surged forward with a howl. Ekhaas brought her sword up, but he batted it aside with a swing of the trident's shaft, the same swing that caught Ekhaas across her shoulder on the down stroke. She staggered and went to her knees.\n\nAshi hissed and darted around her, launching a flurry of blows to drive Makka back before he could strike a second time. He tried to hold her at the length of his trident, but she twisted like a weasel, slipped inside his reach, and slashed at him. His heavy bear hide vest turned the blow. He kicked and managed to catch her. Ashi stumbled back, sucking breath through her teeth\u2014and saw him draw a sword from a sheath at his belt.\n\nThe sword he had taken from her. Kagan d'Deneith's bright honor blade.\n\nHer eyes went wide and Makka grinned, exposing sharp teeth. \"This sword will kill you,\" he snarled in Goblin. \"This is my vengeance!\"\n\nMaybe he mistook what he saw in her face for fear, but if he thought the threat or the sight of the stolen sword could frighten her, he was wrong.\n\nA tremble ran through her. The world seemed to sharpen as blood roared in her ears. Her lips twisted to expose teeth stained red with her own blood. Goblin words tore themselves out of her, starting as a snarl and ending as a scream. \"Give. That. _Back_!\"\n\nAshi had moved like a weasel before. She moved like a tiger now, attacking with a ferocity that put Makka on the defensive. The bugbear stood a full head taller than her. He weighed probably twice as much as she did. She still forced him back. Her attacks fell with such speed and force that it was all he could do to block them, first with the bright Deneith sword, then with the shaft of the trident. Desperation started to show on his face\u2014vengeance seemed forgotten and all of his attention was focused on Ashi as she pushed him step by step across the space between the buildings.\n\nShe chopped down with her sword. Makka raised his trident to block it\u2014and the stout wooden shaft, already deeply gouged, snapped. Deflected, the sword nicked his arm. They staggered apart, but rage surged in Ashi and she lunged. This time her blade cut across his side. She felt it grate along Makka's ribs, then catch flesh and plunge deep. He howled and dropped the broken trident to lash out blindly with a fist.\n\nThe blow was lucky. It caught her on the shoulder and threw her to one side, breaking her attack.\n\nBut Ekhaas was there. In the instant that Ashi fell, she heard the _duur'kala's_ song swell. The music was different this time though, not hard and battering, but strangely thick. Ashi saw Makka struggle, his movements slow and dragging as if he were underwater. For a moment, it seemed as if his muscles would lock, betraying him\u2014\n\n\"No!\" he roared\u2014and smashed his arm into his wounded side.\n\nAshi saw pain cross his face and his limbs moved freely again, magic's grip broken by the sudden agony. Ekhaas scowled and let her song fade, but raised her sword. Ashi rolled to her feet and faced Makka, ready to carry on the fight. The bugbear's small eyes narrowed and his ears curled. Ashi slid close, blade ready.\n\nMakka spun around and took two long strides to the corner where the hobgoblin children huddled and cried. He grabbed the first one his fingers found and dragged it up in front of him, the Deneith sword held across its squirming body. At the touch of the steel, the child went still and silent.\n\nAshi stopped. From the corner of her eye, she saw Ekhaas freeze, too.\n\nMakka didn't say anything. Moving slowly and leaving a bloody trail behind him, he slid sideway across the wall. The alley leading out of the yard was close. Ashi started to shadow him, but he pulled the bright blade tighter and the child whimpered.\n\nShe stopped. He would kill the child. She knew it in her gut.\n\nMakka stepped into the alley, walking backward until he was out of sight. A moment later, the hobgoblin child came running out of the alley, face taut with terror.\n\nAshi was waiting for that. She leaped to the alley mouth, but the shadowed passage was empty. A swirling crowd filled the street at its far end.\n\nThe only signs of Makka were the drops of blood that ended where the crowd began, treading feet wiping away the trail as surely as flowing water. Ashi bit off a curse.\n\n#\n\n# CHAPTER  \nFOUR\n\n# 19 Sypheros\n\nTradition dictated the opening battle of Haruuc's games\u2014two junior warriors of the warlord's clan fighting each other to honor the departed chief. Now one of the junior warriors of the Rhukaan Taash lay with his life's blood soaking into the sand of the arena floor, while the other faced the raised box where Geth sat and thumped a trembling fist against his chest in the hobgoblin salute. Geth stood, raising the Rod of Kings, and slowly the cheers that filled the three tiers of the arena subsided.\n\n\"Faalo of Rhukaan Taash, you honor Haruuc Shaarat'kor,\" Geth called into the quiet. \"Name your reward.\" The Goblin words weren't as elaborate as tradition demanded, but they were what he could manage. It had taken all of his concentration just to get the responses of Haruuc's funeral right.\n\nThe young hobgoblin lowered his head for a moment, then looked up, his ears standing tall. Wrath translated his answer: \"I want to lead a squad in battle against the Valenar!\"\n\nThe crowd erupted again, but Geth felt his belly flip. \"Boar and Bear!\" he growled quietly in the human tongue.\n\nSeated beside him as the new warlord of the Rhukaan Taash, Tariic leaned a little closer. \"Geth\u2014\"\n\nGeth glanced down at him. \"There's not going to be a war!\"\n\nTariic's jaw tightened. \"I was going to say give him a rank and be done,\" he said. \"Make no promises about elves if you don't want to. If there's no war, he'll lead border patrols.\"\n\nIt was good advice. Geth felt heat flood his face. \"What rank?\"\n\nTariic's ears flicked as he thought, then said, _\"Lhikor.\"_\n\nGeth raised his voice against the noise of the crowd, speaking Goblin once more. \"In the name of Haruuc, Faalo of Rhukaan Taash, you will be _lhikorl\"_\n\nThe crowd answered with a wall of sound. A smile spread against Faalo's face and the young warrior thumped his chest once more, then turned to face the crowd, raising his arms in victory. A pair of honor guards appeared to escort him through one of the arena's two gates. From the other, a troop of goblins appeared, throwing down fresh sand and dragging off the body of the warrior who had not been so fortunate. From a raised platform, an announcer using a speaking trumpet called out descriptions of the spectacles that would follow through the first day of the games. Geth dropped back into his seat and didn't listen.\n\n\"I would have thought you'd enjoy this,\" said Tariic, sitting down as well.\n\n\"If I was sitting out there\"\u2014Geth gestured around the arena\u2014\"instead of up here, I probably would.\"\n\nTariic laughed. \"Geth, you realize that if the king of Breland had chosen to attend Haruuc's funeral, he would be here beside you right now? You've put yourself on a level with monarchs and you'd rather be sitting with the people.\" He stopped laughing when he realized Geth wasn't smiling. \"This isn't where you thought you'd find yourself, is it?\"\n\n\"No.\" There was no point in lying about that.\n\n\"You did an honorable thing. It confused me at first, but then I realized that if you hadn't claimed the duty of a _shava_ , Darguun might have fallen into civil war. And after the chaos of the Gan'duur raids, with just one clan in rebellion, I don't know if we could have survived. You've given us a chance to calm down.\" Tariic considered him with a serious expression. \"You think quickly.\"\n\n\"Not quickly enough.\" The words slipped out and Geth tried not to wince. \"This isn't my place,\" he said. \"I'm a fighter, not a talker.\"\n\n\"That's to be expected. You wield the Sword of Heroes, not the Rod of Kings.\" On Tariic's other side, another hobgoblin leaned forward so he could look at Geth. \"But seizing control was the act of a hero. You'll be remembered after you leave us.\"\n\nThe first time Geth had encountered Daavn of Marhaan, the warlord had been trying to persuade Haruuc to allow his clan to raid into Breland. Haruuc had embarrassed him before his allies with a clever ruse, but since then, Daavn had found new favor with Tariic. Even before Haruuc's death, the two had become close, as Tariic, confident his uncle would eventually name him as heir, looked for supporters. Vounn d'Deneith had suspicions that Daavn was even more ambitious than Tariic\u2014and more ruthless. She believed Daavn might have been behind an attempt to kidnap her from Khaar Mbar'ost\u2014an attempt that had been widely blamed on Keraal of the Gan'duur and that would have shamed Haruuc had it succeeded. They had no proof to confront Daavn with, however, and so Vounn's suspicions remained just that.\n\nWhenever Daavn spoke, though, his words left Geth with a sense that he was up to something. The shifter had fallen into a habit of turning them over in his mind, trying to find the hidden danger. \"I... hadn't thought about leaving yet,\" he said cautiously.\n\n\"You hadn't?\" Daavn asked. \"Then you're a true friend to Darguun. A lesser man would have left at the first opportunity. But when your duty as a _shava_ ends, what reason will there be for you to stay?\"\n\n\"Don't pressure him, Daavn,\" said Tariic. His ears twitched as he smiled again. \"As you say, Geth is a hero. He'll always have a place of honor in Khaar Mbar'ost.\" He rose. \"But you have other duties to see to, don't you, Geth?\"\n\nHe did, but he raised his eyebrows and looked at Tariic. \"How did you know\u2014?\"\n\n\"I asked Razu. These games honor my uncle. I want to know what's happening. Lead on. I'll come with you.\" He gestured for Geth to go ahead of him.\n\n\"You're not going to stay for the games?\" asked Geth. \"I thought you'd want to be seen.\"\n\nTariic bent his head. \"My presence isn't strictly necessary. Daavn will be here.\"\n\nThe gates of the arena opened and two bugbears advanced across the sands in the second bout of the games. \"Pesh of Ghaal Cave and Riil of Thunder Gap fight open-handed,\" called the announcer. \"To the victor of this match, Tariic of Rhukaan Taash promises a chalice of gold from his own table! Hail to Tariic, nephew of Haruuc Shaarat'kor!\"\n\nThe crowd bellowed its approval as Daavn produced a shining goblet. Tariic turned and waved. The bugbears looked up at him, then at each other\u2014then roared and came together like twin juggernauts.\n\nKhaar Mbar'ost was less than thirty years old. Built by the humans of House Cannith under commission from Haruuc, it was a blend of human and _dar_ styles. It was also the tallest building in Rhukaan Draal. A mighty fist of a structure, it rose against the sky in a demonstration, to both Darguun and other nations, of the strength of the lhesh.\n\nIt also still felt almost new when compared to any other fortress Geth had been in. Most were many decades\u2014or even many centuries\u2014old, their stones worn and stained. The stones of Khaar Mbar'ost, however, still had the sharp corners put on them by masons' chisels. Their surfaces were dry and clean. In places where the odor of living hadn't permeated the air, Geth sometimes thought he could still smell the dusty, fresh-cut stone.\n\nEven the dungeons sunk into the rock beneath the fortress still had a crisp new feel to them, though they smelled nearly as bad as Geth had expected. It felt strange to step into an almost pristine corridor lit by everbright lanterns while grubby faces peered through the barred windows cut into the cell doors on either side, the interiors of the cells lost in stinking darkness.\n\nThe noise that the prisoners made was startling as well, echoing in the closed space until it seemed as loud as the crowd in the arena. Prisoners yelped and cursed as they fought to get a look out at those who had descended to their world: Geth, Munta the Gray, Tariic, and a large number of guards. Geth had left the Rod of Kings in his chamber, locked safely away and with guards posted outside the door. It felt good to be rid of it for a time. He looked back at the prisoners and tried to guess how many were packed into each cell. \"It's crowded,\" he said in halting Goblin.\n\nThe dungeon keeper, a big hobgoblin with numerous scars and only one ear, looked at him blankly. Geth had to repeat himself twice more, speaking carefully, until he was understood and the scarred hobgoblin grumbled a response that Wrath's magic translated perfectly and instantly.\n\n\"We've been keeping them for a while instead of enacting punishment. Bringing them in from across the city.\" He strode up to one of the cells. The prisoners inside backed away as the keeper ran through a catalogue of crimes. \"The usual thieves and murderers stupid enough to get caught. Cheats. Profiteers who tried to get rich when the Gan'duur raids starved the city. Rioters. _Taat_ caught violating the terms of mourning\u2014\"\n\n\"Mercy!\" came a shout from one of the cells. There was a commotion and a human face pressed up against the bars of the window. The man looked like he had been beaten. He spoke better Goblin than Geth. \"I needed light! It was only a lamp!\"\n\nA guard's club hit the bars and the human jumped back. \"The terms of mourning applied to everyone!\" roared the keeper. \"You disrespected Haruuc. Now you have another chance to honor him!\" He looked to Geth, who clenched his teeth and gave a nod to Munta.\n\nThe old warlord stepped forward as if what he were about to do came as naturally as rallying troops. \"Condemned! Be glad! In the tradition of the People and in memory of Haruuc Shaarat'kor, you are given the chance to win your freedom. The games wait for you.\" He paused, letting his words settle into the prisoners. \"Win in the arena and your crimes are forgiven!\"\n\nThe dungeon filled with a new cacophony that almost made Geth cringe back. Many of the voices raised in the cells proclaimed an eagerness to see the arena. Only a few, mostly non-goblins it seemed to Geth, begged for an alternative. Munta came back to him. \"Just as I told you,\" he said with confidence. \"Offer a _dar_ the chance to die before a crowd and he'll take it.\"\n\nGeth nodded numbly. At last count, over five hundred combatants had signed up of their own free will to fight in the games, but the arena was hungry. Letting prisoners fight for their freedom was an ancient tradition. He'd tried telling himself that most of them would have died for their crimes anyway\u2014Darguun's few laws carried harsh punishments\u2014but there was still something that seemed deeply wrong about forcing them to fight for the amusement of the crowds. And yet, as Munta said, many of the prisoners were eager to take the chance.\n\n\"Who first?\" asked the keeper.\n\nThere were six big cells, as if the builders of Khaar Mbar'ost had foreseen the need of the games. Geth pointed at the cell whose occupants seemed most enthusiastic for the fight.\n\nThe keeper and the guards descended on the cell like vultures on carrion. A few prisoners tried to rush the door when it was opened, but the guards' clubs beat them down and they were first to be locked into a long chain of shackles. Guards began pulling other prisoners out one by one, adding them to the line, while those in the other cells whooped and yelled.\n\nAs the gang was assembled, Tariic stood ready, giving each prisoner a swift inspection. He had a goblin with him as an assistant, and the little creature crept forward to swipe green paint onto the leg of those Tariic indicated. The first prisoner, a mangy bugbear still dazed from the guards' clubs, tried to kick him away. The goblin squealed and jumped back. Tariic planted himself in front of the bugbear, hand on sword, and glared up at him.\n\n\"Take the mark and you get a weapon in the arena. No mark, no weapon. Understand?\"\n\nThe bugbear's lips curled back from his teeth.\n\nTariic shrugged. \"No weapon.\" He pointed at the next prisoner in line, a hobgoblin. \"You get his weapon. All you have to do is dedicate the fight to me\u2014Tariic of Rhukaan Taash\u2014if you win.\"\n\nThe hobgoblin stuck out his leg eagerly. Other prisoners did the same, though Tariic had the goblin mark only the strongest-looking of them. The bugbear at the front of the chain began to look like he regretted his decision.\n\n\"Clever,\" murmured Munta. \"The audience in the arena will remember this, and any of these scum who survive will tell the story. The other potential heirs are going to copy this.\"\n\n\"They won't be able to,\" Geth said. \"Tariic paid a heavy bribe to the keeper for the privilege.\"\n\nThere were fifteen prisoners shackled together now. Most seemed ready for, or at least resigned to, the arena. A few struggled and pleaded with the guards as they were pulled from the cell. The human who had called out for mercy. An elf woman dressed in rags that had once been quite fine. A hobgoblin who cradled one arm to his chest and looked feverish and ill. A dwarf who glared at the guards surrounding him, thick fists opening and closing in barely restrained anger. Tariic made sure he got a stripe of paint.\n\n\"They don't want to go,\" said Geth.\n\nMunta's ears flicked. \"They don't have a choice. Don't interfere, Geth.\"\n\nThe shifter ground his teeth together and held his silence\u2014until a guard emerged from the cell leading the last of the prisoners. Geth's breath burst out of him. \"Grandmother Wolf, no. That's enough.\"\n\nThe final prisoner was an elderly goblin woman, so hunched and wizened that she stood no taller than Geth's thigh. He wasn't sure that she would have been able to walk easily on her own without the guard's support. She looked up at his outburst and he saw that her eyes were milky. Geth fought back a growl and turned to Munta. \"She's not going. She doesn't stand a chance. What's she even doing in here?\"\n\nMunta looked to the keeper, repeating Geth's words in Goblin. The keeper grunted. \"She led a famine march during the Gan'duur raids. Her followers damaged the Bloody Market.\"\n\n\"A famine march is a rite of the Dark Six,\" Munta said. \"Marchers make sacrifices to the Devourer in an attempt to ward off further suffering.\"\n\n\"I saw the march, Geth,\" said Tariic. \"I remember Haruuc's anger at it. He ordered her thrown into the dungeons.\"\n\n\"He didn't order her to fight in the arena, did he?\" He faced Tariic. \"How will the audience react to that? An old woman in the arena?\"\n\nTariic's ears went back, but after a moment he nodded. Geth glanced at Munta. The old warlord frowned but nodded as well. Geth went over and knelt before the goblin woman. \"Old mother,\" he said in Goblin, \"what is your name?\"\n\n\"Pradoor.\" Her voice was sharp and shrill. She reached out a gnarled hand and put it on his face, feeling his features. Her big ears twitched and her mouth curled in recognition. \"So, shifter\u2014am I going to the arena to honor one who turned his back on the gods of the Six?\"\n\nThe confidence in her sharp voice almost took him aback, but he shook his head. \"No. You're going free.\" He looked up at the guard who held her. \"Give her food and take her out of Khaar Mbar'ost.\"\n\nHis words opened a floodgate. Abruptly all of the prisoners who had been struggling to stay out of the arena\u2014as well as those who now saw the possibility of release\u2014were calling out to him.\n\n\"Me, shifter! Release me!\"\n\n\"I don't stand a chance!\"\n\n\"Look at me!\"\n\n\"Have mercy!\"\n\nSome of the harsher goblin prisoners just laughed. A few of the cries for help ended in sharps gasps of pain and more urgent, realistic cries as those in the cell were dragged down and shown just how little of a chance they stood. The keeper and some of the guards started banging on the cell doors. Geth clenched his jaw and made sure Pradoor and the guard escorting her got out of the dungeon and started the climb up the stairs leading to the fortress above. He didn't look at Munta, Tariic, or the keeper, and he did his best to ignore the slowly dying pleas for aid.\n\nThen one shout cut through the din. \"Hey\u2014brother! Here!\"\n\nThe voice had a distinctive gravelly roughness, the accent of Geth's own home in the Eldeen Reaches. He turned around in time to see a shifter clinging to the bars of one door, struggling to get his attention while holding off the jeering prisoners who shared his cell. Wide animal eyes met Geth's own. \"Let me go, brother,\" the other shifter pleaded. \"Or at least make sure I get a sharp weapon for the arena!\"\n\n\"Quiet, you!\" said the keeper, slamming a fist onto the fingers that gripped the bars.\n\nThe shifter held on, though. Geth charged across the dungeon hall and grabbed the keeper's arm as he raised it again. \"You didn't say there was a shifter here!\" he said.\n\n\"There isn't.\" The keeper tried to shake himself free but Geth hung on. \"This _taat_ is a changeling.\"\n\n\"No! He's wrong!\" protested the prisoner. \"I'm a shifter like you, brother!\"\n\nA hand inside the cell stopped trying to pull the shifter away and instead slammed his head forward against the bars. The shifter jerked and sagged. A big hobgoblin pushed him aside and peered out. \"I've been in this cell for seven days. Until just now, there was no shifter here.\" The hobgoblin dragged the shifter upright. \"He's a _gaa'ma.\"_\n\n_Gaa'ma_ \u2014a wax baby, the Goblin term for changeling. Geth let his hand drop from the keeper's arm and the scarred hobgoblin snorted. \"I told you. His name's Ko. He's the changeling who tried to kidnap the envoy of House Deneith by murdering one of her guards and taking his place.\"\n\n\"Shifter...\" the prisoner said feebly, but the accent of the Eldeen had slipped.\n\nGeth stared at him, surprise quickening the beat of his heart. The changeling who had tried to kidnap Vounn. If anyone could provide the evidence that would link Daavn of Marhaan to the plot and show Tariic just what kind of serpent he was dealing with, it might be this Ko.\n\n\"Take him out,\" said Geth. \"Put him in the empty cell. He's not going to the arena.\"\n\nThe storm that command produced made his sparing of Pradoor seem like the smallest act of charity. Howls of outrage sprang from the other prisoners\u2014and from the guards, who this time did nothing to silence their charges. Munta and Tariic both came forward to press Geth. The big hobgoblin prisoner who still held Ko bellowed in anger. \"You're sparing this cowardly piece of filth?\"\n\nHe punctuated his words by driving Ko against the door of the cell with bone-shaking force, forcing gasps out of his captive. The thick hair on Geth's arms and on the back of his neck rose. The hand that gripped Wrath's hilt tightened and he wrenched the sword free of its sheath. \"Silence!\" he roared in Goblin, thrusting the weapon high.\n\nWhen she had first seen the sword, before she had guessed at its true nature as the Sword of Heroes, Ekhaas had proclaimed it a _lhesh shaarat_ , a blade so fine that any descendant of Dhakaan instinctively recognized it as a weapon of kings and warlords. Just the act of drawing a _lhesh shaarat_ proclaimed the wielder's might.\n\nWrath didn't have the power of the Rod of Kings to force obedience, but it could command respect.\n\nSilence fell over the dungeon. The big hobgoblin released his hold on Ko, who slid down out of sight. Geth gestured to the keeper with Wrath. \"Get him out of there.\"\n\nThe keeper moved to obey him.\n\n\"What are you doing, Geth?\" Munta asked softly. \"You can't keep all the prisoners from the arena.\"\n\n\"He might know something about Vounn's kidnapping.\"\n\n\"He's been questioned,\" said Tariic. \"A hobgoblin in a mask and using a false name hired him. We know that was Keraal. The changeling has outlived his usefulness.\"\n\nGeth wanted to ask how Tariic could be certain the masked hobgoblin had really been Keraal, but he didn't. Instead, he said, \"He tried to hurt a friend.\"\n\n\"Then send him to the arena,\" Munta said. \"You're causing unrest!\"\n\nThe keeper had the cell door open. Waving his club to keep those inside back, he kicked and dragged Ko clear. Geth stood over the battered prisoner and looked down at him. Ko's eyelids flickered, then his face seemed to blur and run. A shifter's dark hair turned pale, animal eyes became blank and white. His features grew soft and strangely ill-defined, his skin turned a dusky gray, and his body became a little taller and a little leaner. The scrapes and bruises on his face didn't disappear, though. \"Thank you,\" he said, his voice thick.\n\n\"Don't thank me yet,\" said Geth. He turned to the keeper. \"Watch him. I'll be back to talk with him when I can.\"\n\nThe scarred hobgoblin grumbled something under his breath but grabbed Ko by the front of his shirt and shoved him into the recently emptied cell. The other prisoners jeered and grumbled again. The keeper slammed the door on Ko, then pointed at the chain of prisoners destined for the arena that day. \"Take them away!\" he ordered the guards. His eyes swept the other cells. \"You shut your mouths. You'll all have your turns.\"\n\nThe prisoners just laughed and shook the cell doors. The keeper's remaining ear went back flat. \"No food tonight, then! Maybe you'll fight better on empty bellies!\"\n\n\"Nicely done,\" Tariic whispered in Geth's ear. \"You're right. You aren't much of a talker.\"\n\nGeth didn't answer that, sliding Wrath back into its sheath instead. \"Are we done?\"\n\n\"I wish we were,\" said Munta. \"There's one more prisoner we need to see.\" He gave Geth a hard look. \"Don't pardon him.\"\n\nKeraal had a small cell to himself. The defeated rebel sat on a heap of straw, arms restrained by a length of chain running through rings set into the wall, and stared at them with dead eyes as they entered. He still wore only the loincloth in which he had been led into Haruuc's throneroom. His red-brown skin was covered in tiny, freshly healed scars\u2014the marks of the grieving tree. Geth waited for him to say something, but Keraal didn't speak. After a long moment, the former warlord of the Gan'duur clan lowered his head, and his face disappeared behind thick, dark hair.\n\n\"The problem is what to do with him,\" said Munta in the human tongue. \"He should already be dead, killed on the grieving tree, but Haruuc's final words spared him.\"\n\n\"Haruuc's final words saved Dagii,\" Geth said. \"They just happened to set Keraal free at the same time.\" He looked down at Keraal. The hobgoblin was entirely broken. Keraal had fought against a lhesh he thought had gone too far in seeking the acceptance of human nations\u2014and been crushed when Haruuc actually became the ruthless warlord Keraal had sought. Geth hardened his heart. \"We can't return him to the grieving tree,\" he said. \"Haruuc was the only one who knew the words that controlled it. Without them, it's only so much stone.\"\n\nHe thought he saw Keraal's ears, protruding through his lank hair, tremble in relief at the news, a movement so slight it could have been his imagination. Secretly Geth was glad the words had been lost as well. The tree might be a Dhakaani artifact, but it was a device for torture and slow death.\n\nMunta also shook his head at mention of the tree. \"Even if we knew the words we couldn't use them for the same reason this _gaa'taat_ is still alive,\" the old hobgoblin said heavily. \"By tradition, when a warlord spares an enemy, no one under his command may seek his death. Haruuc may not have intended to spare Keraal, but he did. Yet tradition also holds that _all_ prisoners in a warlord's stronghold face their judgment in the arena.\"\n\nThe glance he gave Geth was harsh, and the shifter felt heat spread across his face. \"Why not send him to the arena, then?\" he asked.\n\n\"A prisoner who wins in the arena walks free,\" said Tariic. \"As much as the people would love to see Keraal forced to fight, do you want him to have his freedom?\"\n\n\"Put me in the arena and let me die there.\"\n\nTariic actually jumped at Keraal's sudden words. The chained hobgoblin raised his head and looked at them all. His voice was as flat and dead as his eyes. He spoke in the human language. \"What reason is there for me to live? The warriors of my clan rot in trees along the road to Rhukaan Draal. The women and children have been sold into slavery. I am alive by chance. The Gan'duur are destroyed. My clan paid the price for my arrogance. I failed them. _Chiit gath'muut. Chiit gath'atcha. Chiit gath'piir.\"_\n\nI am without duty. I am without honor. I have nothing.\n\nGeth looked at Munta and Tariic. Munta nodded. \"It is decided.\"\n\nTariic's ears bent. \"The people won't like it if he doesn't fight back.\"\n\nA growl escaped Keraal. \"Do I care what the people think? Give me a sword and I will fall on it in front of them. Will they find amusement in that?\"\n\n\"If you give yourself a coward's death, then you will truly be without _muut_ or _atcha.\"_ A shadow fell across the light that entered the cell from the hall outside. Geth looked over his shoulder to find Dagii standing in the doorway, his face hard and his gray eyes narrow. \"The Gan'duur fought hard\u2014I know this because I fought them. They followed you willingly. Let yourself die cheaply and what pride remains in the name of the Gan'duur will die with you.\" His ears rose tall. \"My victory over you will have no meaning.\"\n\nKeraal stared up at him for a moment, then stood with a rattling of chains. \"The honor of the Gan'duur will last beyond my death. The Gan'duur starved Rhukaan Draal and eluded all the troops that Haruuc Shaarat'kor sent against us.\"\n\n\"All except the last,\" Dagii said grimly, \"and that's how I want my victory remembered: a triumph over a strong enemy. But I promise you that if Keraal of Gan'duur dies without fighting in the arena, then all that Darguun remembers of the Gan'duur will be a warlord who passed from life as a coward. It would be better for your legacy if you had died in agony on the grieving tree.\"\n\nThe dead look had left Keraal's eyes. They were bright and angry, and his chest heaved with each breath. Tariic let out a furious hiss. \"Dagii!\" he said. \"What do you think you're doing?\"\n\n\"He reminds me that I owe _muut_ to my clan even if I no longer have a clan, even if I no longer have _atcha.\"_ Keraal straightened and turned to face Munta. \"I will fight in the arena.\"\n\nMunta nodded again, slowly and with a look of approval on his face.\n\nTariic's features twisted with frustration, however. \"And if he wins, he walks free? What kind of punishment is that for my uncle's enemy?\"\n\nGeth felt a strange pressure creep into his mind, a vague memory that wasn't his own, and shivered. He recognized the sensation: it was Wrath. This was what Haruuc had experienced and what had almost driven him mad. The Sword of Heroes had been created to protect and inspire, though, not to command. Geth pushed it away\u2014and it retreated, but not without leaving an idea behind.\n\n\"He won't fight just any fight,\" Geth said. \"He'll fight a battle each day. If he wins all of them, he wins his freedom.\"\n\nTariic spun around to give him an ugly look, but Keraal stood tall and nodded. \"I accept these conditions,\" he said.\n\n\"No!\" said Tariic. He looked to Munta. \"What about honoring Haruuc?\"\n\n\"Keraal didn't kill Haruuc,\" Munta said. \"Chetiin did. Keraal can fight. He _must_ be allowed to fight.\"\n\n\"No weapons, then!\" snapped Tariic. \"He fights with nothing more than he has now.\"\n\nDagii looked Tariic over and nodded. \"I agree.\" Tariic seemed relieved\u2014relief that turned into renewed fury as Dagii added, \"Let him fight with the chains he wears.\"\n\n\"I won't allow this!\" Tariic said. \"It can't be allowed to happen.\"\n\n\"You're not lhesh, Tariic.\" said Munta. \"The decision is Geth's. He is Haruuc's _shava.\"_\n\nThe old warlord looked to him. So did Tariic. And Dagii. And Keraal. Geth drew a breath and let it out.\n\n\"Keraal fights. Five battles wielding the chains he wears.\"\n\nKeraal bent his head in acceptance. Tariic's eyes flashed. He turned and strode out of the cell, pushing past Dagii. Munta frowned after him, then looked to the others. \"I'll summon the keeper and make the arrangements. We'll need to have a fight added to the games today.\"\n\nMunta left, leaving Keraal with Geth and Dagii. The chained hobgoblin glanced between them, then bent his head again to Geth. \"Ta _muut,\"_ he said. You do your duty\u2014the simplest way of saying \"Thank you\" in Goblin.\n\nThen he turned to Dagii. _\"Paatcha_!\" An offering of honor.\n\nDagii made no response\u2014none was needed. He stepped out of the cell as the dungeon keeper, grumbling about warlords changing their minds, and a pair of guards arrived. Geth glanced once more at Keraal as he stood still for the keeper to unlock his chains, then went after Dagii. \"Honor between enemies?\" he asked the young warlord.\n\n\"A good enemy is better than a bad friend,\" Dagii said.\n\n\"You didn't come down here to shame Keraal into fighting, though.\"\n\nA smile flickered across Dagii's face, then was gone. \"No. That was just luck. I came looking for you.\" He leaned close for a moment. \"Midian has returned to Rhukaan Draal.\"\n\nGeth's gut twisted. \"Get messages to Ashi and Ekhaas and let them know. We'll meet tonight.\"\n\n#\n\n# CHAPTER  \nFIVE\n\n# 19 Sypheros\n\nThey met in the small room high in Khaar Mbar'ost where they had once met with Haruuc and where he had revealed his plan to seek out the Rod of Kings. Once again, Aruget stood guard outside the door. After Haruuc's assassination, Ekhaas had wondered if the warrior could be trusted, but he already knew some of the truth behind the rod and had kept his silence. He could be trusted.\n\nEkhaas saw Aruget's ears rise slightly as she and Ashi approached. _Duur'kala_ magic had erased the aches and cuts of their battle against Makka, but there were still bruises. In addition, Ashi walked like an angry cat, full of rage and ready to lash out. Aruget said nothing, though; just opened the door for them.\n\nGeth and Dagii, on the other hand, weren't so restrained. \"Rat!\" Geth said, leaping up out of a chair by the room's only window. \"Were you in a fight?\"\n\n\"A fight that ended too early,\" said Ashi. She snapped her teeth on the words.\n\n\"Makka is in Rhukaan Draal,\" Ekhaas explained.\n\nDagii's ears lay back. \"The bugbear Marguul chief from the mountains? What's he doing here?\"\n\n\"I'd say looking for revenge. He looked like he'd been traveling and he seemed to be alone.\" She described their fight with the bugbear\u2014and his escape.\n\nGeth growled. \"That's all we need,\" he said. \"An enraged bugbear trying to kill you.\"\n\n\"Trying to kill us,\" Dagii corrected him. \"He had Ashi, Ekhaas, and me captive. I doubt if he'd end an attempt at vengeance with just Ashi and Ekhaas. He may even be looking for you\u2014it wouldn't take much to link us together.\"\n\n\"How would he have found us?\" Geth asked.\n\n\"He knows my name. He might have started with that.\"\n\nGeth's eyebrows jumped. \"You _told_ him? You told him your real name?\"\n\nDagii shrugged. \"There was no reason to hide it.\"\n\n\"He may not be coming after anyone anymore,\" said Ekhaas. \"Ashi left him with a bad wound. If he doesn't get to a healer, it could kill him.\"\n\n\"Let's hope,\" Geth said.\n\n\"Let's not.\" Ashi's hands opened and closed as she moved around the room. \"He's still got my sword\u2014Kagan's honor blade.\"\n\n\"We'll find it,\" Ekhaas assured her. \"One way or another, we'll\u2014\"\n\nA knock on the door interrupted her. The door opened and Midian Mit Davandi slipped through. Geth gave a genuine smile, probably the first one Ekhaas had seen from him in ten days. \"Midian.\"\n\nThe gnome's sun-browned face was flushed and his pale hair damp with sweat as if he'd been running. \"Sorry,\" he said. \"I had to call on the Zil ambassador. She's trying to keep track of all the Zils currently in Darguun in case there's trouble.\"\n\nEkhaas had never been to Zilargo, but she'd heard it was a strange place, ruled by a blend of gossip, co-operation, and subtle coercion. Then again, an entire race that was no bigger than goblins, without larger hobgoblin and bugbear cousins to rely on for physical might, probably would develop different ways of dealing with the world. Certainly it showed in their history\u2014the gnomes of Zilargo had never fought a war, preferring to hide behind policies of conciliation and neutrality. It seemed to work. The nation still existed in a pocket between humans and _dar_ when by all rights it should have been overrun long ago.\n\nMidian caught her looking at him and his blue eyes flashed. \"I have something for you, Ekhaas. I found it in the ruins I was investigating at Bloodrun.\" He produced a small object and tossed it to her. It was a Dhakaani coin, black with age and badly corroded. A hole had been punched through one edge. Once it would have been threaded on a cord, the face of the emperor on the coin looking outward, to make a kind of simple amulet. They were common artifacts in all eras of the empire. She looked for the dynasty name on the coin, frowned, and glanced at Midian.\n\n\"Koolt Dynasty. Early empire. The ruins at Bloodrun are late empire.\"\n\n\"Wrong, _duur'kala,\"_ Midian crowed with delight. \"Dig down and you find that the late empire ruins are built on top of early empire ruins. Did the Kech Volaar know that? I don't think so!\"\n\nEkhaas glowered at him. Tariic had hired Midian to join the quest for the Rod of Kings without Haruuc's knowledge. The gnome had proved to be clever and resourceful, but it had taken time for Ekhaas to admit respect for him. He was a researcher for the Library of Korranberg, a scholar, and a historian\u2014and as such a bitter rival to the _duur'kala_ of the Kech Volaar. Her clan kept the glorious history of the Empire of Dhakaan alive through tales and the careful collection of artifacts. Scholars like Midian turned vibrant history into dusty reports and stole Dhakaani artifacts from their rightful keepers. In fact, when Senen Dhakaan had first learned Midian was in Darguun, she'd demanded his death as a grave robber and thief. Haruuc had overruled her, and when he was handing out rewards for the recovery of the rod, had granted Midian official permission to investigate some of Darguun's many ruins. Ekhaas had eventually come to like the gnome. Most of the time.\n\nShe flicked the coin back to him and his fingers snapped it out of the air. He tucked it away and turned to Geth and Dagii, his expression sober. \"Your message was slow reaching me\u2014the messenger showed up at the ruins suffering from dust fever. I tried to treat him, but we didn't have what he needed. I stayed with him until he died, then came to Rhukaan Draal.\" His face darkened. \"It wasn't easy to hear about Chetiin. If Haruuc had discovered the rod's power, though, I suppose he did the right thing.\"\n\n\"No,\" Geth said, \"he didn't. He only made the situation worse.\" Midian looked at him with startled curiosity and Geth let his breath out in a hiss. \"There were things we couldn't trust to the messenger. The power of command isn't the rod's only secret\u2014\"\n\nAs Geth laid the whole truth about the Rod of Kings before Midian, the gnome's face grew first pale, then hard. Ekhaas pushed a chair at him. It had been designed for a larger creature and for Midian it was like jumping up to sit on a table, but he did it anyway, never taking his eyes from Geth. There was a strange intensity about him, Ekhaas thought. The light-hearted researcher who had gloated over an ancient coin was gone, replaced by someone who grasped immediately just what kind of trouble they\u2014and Darguun\u2014faced. When Geth had finished, Midian sat in silence for a long moment.\n\n\"Chetiin needs to answer for this,\" he said finally.\n\n\"He will,\" said Dagii. \"If we find him, he will.\"\n\n\"And you wouldn't trust any of the potential heirs with the rod?\"\n\n\"No,\" Ekhaas answered. She counted the names off on her fingers. \"Aguus, Garaad, Iizan\u2014definitely not. Tariic...\" She hesitated.\n\n\"Not even Tariic,\" said Geth. \"I'd rather see him on the throne than any of the other three and maybe he would have been Haruuc's choice, too. But if Haruuc couldn't stand up under what the rod was urging him to do, how can Tariic?\"\n\nThe gnome wrinkled his nose. \"I agree. And the only plans you've come up with are stealing the rod or destroying it?\"\n\nAshi looked up as if about to repeat her suggestion of using her dragonmark to block the rod's power, but Ekhaas shook her head sharply and said, \"Or both. Something stolen can be recovered. I don't think we want anyone to get their hands on the rod. But stealing the rod presents its own problems. Darguun needs it as a symbol of stability.\"\n\nMidian's lips twitched. \"By Aureon's blue quill, it's a good thing none of you were born a gnome. You would have had to be locked up for your own safety.\" He sat back in the chair and spread his hands. \"Replace the rod with a fake.\"\n\nEkhaas stared at him. They all stared at him. Midian looked back at them then rolled his eyes. \"You can't all be that high-minded, can you? Replace the true rod with a false rod. Darguun has its symbol, the lhesh is safe, and we can take the true rod somewhere and destroy it without anyone ever suspecting.\"\n\n\"But it's the Rod of Kings,\" said Geth. \"How do you create a fake? Someone will notice.\"\n\n\"Nobles across Khorvaire walk around with paste gems all the time, and no one can tell. Half of the nobles probably don't even know they're wearing fakes.\" Midian sat forward again. \"How many people besides the five of us and Haruuc have ever examined the rod closely?\"\n\n\"Chetiin,\" Ashi said.\n\nMidian waved the name away. \"He's not likely to get close to the rod again, is he? Anyone else?\"\n\n\"Senen Dhakaan wanted to look at it, but Haruuc wouldn't let her,\" said Ekhaas. \"Maybe he already realized there was a danger in handling the rod.\"\n\nGeth pressed his lips together in thought. \"Most of the warlords have seen it, but never up close. Razu has been close to me and to Haruuc, though.\"\n\n\"Do you think she would suspect anything?\"\n\n\"Probably not.\"\n\n\"Wait.\" Dagii looked uncertain. He rose from the chair he had been sitting in and paced around the small room. \"The rod is made out of byeshk. That's not exactly a common metal.\"\n\nMidian gestured toward the window. \"We're in Rhukaan Draal. You can buy anything at the Bloody Market.\"\n\nDagii frowned. \"Maybe so. But the rod is more than just a piece of metal. Even without its power of command, you could feel something when Haruuc held it. He had a greater presence. He seemed more majestic.\"\n\n\"Any artificer worth his fee could create the same effect\u2014and work the byeshk, too.\" The gnome shifted. \"The only problem might be finding an artificer we can take into our confidence. If you're willing to try this.\"\n\nOnce again, they looked at each other. What Midian had suggested was, Ekhaas thought, dangerously simple. It wasn't without risks, but it was the only plan they'd come up with that met all of their needs.\n\n\"I'm willing,\" she said.\n\n\"So am I,\" said Ashi.\n\nDagii nodded his agreement.\n\nGeth opened his hands. \"We'll do it. So we need to find an artificer we can trust and who can create a replica of the rod in five days before the end of the games.\"\n\n\"Four days,\" Ekhaas said. \"We've lost a day now. I'll take care of that\u2014of all of us, I can move around Rhukaan Draal without attracting attention.\"\n\n\"Move fast.\" Geth leaned his head back against the wall behind his chair. \"Grandfather Rat's naked tail. This could actually work.\" He looked at Midian. \"You're brilliant.\"\n\nThe gnome's smile flashed. \"Say that again. I don't get tired of hearing it.\"\n\nThe wound in Makka's side was an agony. He'd tried to staunch the bleeding, but every movement tore the wound open again. Blood matted the thick hair of his body and left a spattering of big drops on the ground wherever he stopped.\n\nWhen the wolf had savaged his arm on the mountainside, he'd been in familiar territory and\u2014for a short time at least\u2014among friends. There had been someone with sure hands to bandage the wound. There had been herbs to treat it. Rhukaan Draal was strange and alien. There were no allies. Makka had tried to find a healer, but everyone he'd demanded aid from had fled.\n\nWhen he staggered and fell against the wall of a building, he knew the wound was too deep. This was the end of him\u2014the end of his search for vengeance. The jackals of this accursed city would circle him, and when he was dead they would strip the flesh from his bones. He felt along the wall until the building became an ally. He slipped into the cool shadows, found shelter behind an abandoned cart, and lay down to wait.\n\nMemories and dreams came to him. Hunting deer at dusk in the mountains. Feasting on liver cut fresh from the steaming carcass. Gorging on hot, dripping meat roasted over a fire. Creeping up behind Ashi of Deneith and plunging her bright sword through her belly, laughing as she turned in astonishment to face her killer, as he wrenched the sword sideways to tear through her flesh. Catching Ekhaas the _duur'kala_ and cutting the tongue from her mouth, then using her mewling cries to lure Dagii of Mur Talaan. Stringing him up like a deer and butchering his still living flesh, blood falling with a _drip-drip-drip\u2014_\n\n_\u2014tap-tap-tap_. Slow shuffle of feet. _Tap-tap_ , shuffle again. _Tap-tap-tap-tap-tap_.\n\nMakka opened his eyes. Full darkness had fallen, though not yet the darkness of death. The constant noise of Rhukaan Draal was a din in the distance.\n\nAn elderly goblin woman made her way along the alley, tapping before her with a stick. After feeling in front of her with the stick, she would slide forward a few steps and repeat the procedure. Her old eyes were milky white.\n\nThe stick found the cart and she came around it.\n\n\"Go away,\" growled Makka. She moved to the other wall of the alley but kept coming. \"I said, go away!\"\n\nHer answer was a thin chuckle. Makka snarled and lashed out at her. He still had strength in his arm, if not in his body. The attempted blow pulled him off balance and he sprawled to the side. His weight fell against the cart, sending it rolling forward a short distance with the protesting squeal of a rusted axle. Makka fell against the ground and lay there choking on a new burst of pain. His arm stretched out across the alley\u2014the old _golin'dar_ was just out of reach.\n\nHer tapping stick encountered his hand, then rapped down hard across his hairy knuckles. \"You have fallen,\" the old woman said in a shrill voice. \"One of the hunters lies wounded. The order of the world is reversed.\"\n\n\"If you're going to try and rob me, get on with it,\" said Makka. \"There's still enough strength in these hands to drag you into the Keeper's domain with me!\"\n\nHer chuckle turned into a cackling laugh. Makka roared and thrust himself forward, sliding on a slick of his own blood, ignoring the pain in his side. \"By the Fury, you'll meet the Keeper before I will\u2014and by the Mockery, you'll suffer more too!\"\n\nHis push wasn't quite enough. Somehow she was still just beyond his reach, though her cackle was dropping now. Her big ears cupped as if she was listening to something more than his threats and curses, and her wrinkled face creased in a thin smile.\n\n\"Oh, you would be a terror at your full strength!\" Her stick rapped his knuckles again. He grabbed for her and missed again. Her smile grew tight. \"You honor the old ways,\" she said. \"You wear the _muu'kron.\"_\n\nIt was a statement, not a question. An eerie feeling penetrated Makka's anger. How had the goblin known? She couldn't have seen the six knotted cords on his belt. Had she guessed? There was too much confidence in her voice. His hands dropped and he pulled back a little bit.\n\n\"I wear the _muu'kron,\"_ he said.\n\n\"You call on the Dark Six. The Keeper. The Fury. The Mockery.\" She paused and cocked her head. \"The Devourer?\"\n\n\"When thunder rolls and my belly is empty,\" said Makka.\n\n\"The Shadow?\"\n\nHe shivered. \"I have little dealing with dark magic.\"\n\n\"But you honor him?\"\n\n\"I fear his power.\"\n\nThe goblin tapped her stick against the ground. \"That is as much as honoring him. And the Traveler?\"\n\nMakka's hair rose as stories of the trickster god came creeping out of his childhood. Stories that told of how the Traveler had remained on Eberron after the age of creation, wandering the world to spread chaos and test the faithful, never appearing in the same guise twice. Stories rejected by the adult mind of a hunter that had no room for chance, but broke down on the edge of death. Makka looked at the old goblin woman with new eyes and a growing wariness.\n\nHis silence must have given away his thoughts. The _golin'dar_ laughed. \"You think I could be the Traveler? Your wounds must be bad! And yet surely the Traveler led me to you.\"\n\nMakka found his voice again. \"Or the Mockery, to increase my torment.\"\n\nShe laughed again, a sound like some night-hunting bird, and moved closer to settle down on her haunches. She was close enough to seize now, but Makka didn't move. It wasn't just because of the growing darkness that drained the strength from him. There was something odd in this fearless woman.\n\n\"I am Pradoor,\" she said.\n\n\"Makka.\" He didn't add the name of his tribe, partly because he no longer had a tribe, partly because a strange sensation had fallen over him. He stood on a cliff. With one step, such things as tribes would no longer matter.\n\n\"You are called, Makka,\" said Pradoor. \"The change beloved of the Traveler is coming. Haruuc fled to the gods of the Sovereign Host in the belief that it would make Darguun strong, but only the strength of the old ways can make Darguun great. There will be a new lhesh, and he will respect the Six. The people still believe. The order of the world will be set right.\"\n\nHer blind eyes faced him. \"You want revenge. The Dark Six\u2014the Fury\u2014will place it in your path. Serve me, and your strength will be greater than anything you had before. Refuse and die.\"\n\nThe words were blunt. It wasn't a threat, only the truth. \"When the Six give a sign, only a fool ignores it,\" Makka said. \"I choose vengeance.\"\n\n\"When the Six call, you have no choice.\" Pradoor stretched out a gnarled hand\u2014she was somehow even closer than he had thought\u2014and stroked his head the way his mother had when he'd been a pup. \"You are their instrument. The Keeper will not take you this night.\"\n\nWaves of fatigue and weakness washed over Makka. For a moment, he thought that Pradoor was wrong, that the Keeper had claimed him, then fatigue and weakness were both gone. He trembled, as if just recovering from a long and terrible fever, but the fire of the wound in his side had cooled. He stood up and pulled apart the gap that Ashi's sword had made in his bear hide vest. The skin beneath was crusted in dried blood, but the wound had closed, leaving a tender pink scar behind.\n\n\"Stare later,\" said Pradoor. She poked her stick into his leg. \"Up.\"\n\nMakka lifted her up to his left shoulder where she settled herself comfortably, one hand curled around the back of his neck. The old goblin weighed almost nothing. She tapped his head and pointed to the mouth of the alley. \"The people wait for us. A king waits for us.\" Her fingers stroked his head again. \"Your revenge waits for you.\"\n\nMakka bared his teeth. \"Praise the Six,\" he said, and walked out to face the jackals of Rhukaan Draal.\n\n#\n\n# CHAPTER  \nSIX\n\n# 20 Sypheros\n\nThe most rumored match of the second day of the games had already begun as Ekhaas stepped into the sun of the warlords' box. Below, Keraal circled the ring, back to the wall, taking the measure of his opponent\u2014an ettin captured somewhere in the northern hills of Darguun and forced into the arena. Ekhaas shaded her eyes against the afternoon's brightness and studied the creature just as Keraal did. The ettin stood nearly twice as tall as the hobgoblin. Its limbs were thick and its features angry, with fleshy lips and ragged ears on each of the two heads that sprouted from its massive shoulders.\n\nBoth pairs of the ettin's tiny black eyes were fixed on Keraal, watching intently as the warlord of the Marhaan spun the chain that was his appointed weapon in a slow circle before him. The ettin had been provided with arms of a sort as well: a club fashioned from a length of heavy building timber and a shield made out of a door.\n\nThe spectators in the stands were cheering, most of them for the ettin, a vocal few for Keraal. Ekhaas saw one of the ettin's heads murmur something to the other, then the creature let loose a bloodcurdling dual-pitched cry and charged. The club swept down and Keraal slid aside, but it was a feint. The ettin pushed its makeshift shield into the path of Keraal's spinning chain. A few chunks of wood flew free, then the chain crashed into a tangle. Keraal tried to leap away, but he was slow. The club dealt him a glancing blow. He staggered and turned the stagger into a desperate lunge under the ettin's arm. He kicked hard into its calf, then sprinted away as it hopped in pain, yelping with two voices. On the other side of the arena, Keraal shook out his chain and began to spin it again.\n\nA few more voices were cheering for Keraal.\n\nGeth sat alone at the front of the box. Ekhaas slipped in beside him. \"None of the heirs are putting themselves on display?\" she asked.\n\n\"They are. Just not here.\" Geth pointed to either end of the box, then out into the stands. \"They claimed their own territory.\"\n\nEkhaas followed his gesture. At one end of the raised box, Tariic stood cheering with Daavn at his side. A good number of other warlords clustered around him. Aguus of Traakuum and the warlords who supported him had claimed the other end of the box. Garaad of Vaniish Kai had taken a populist approach, sitting in the stands surrounded by warriors of lower ranks\u2014many of them looking vaguely wistful as they stared across the arena at the section of benches taken over by Iizan of Ghaal Sehn and his supporters where wine was flowing freely and boxes of sweet _shaat'aar_ were being handed around.\n\n\"He'll empty the vaults of the Ghaal Sehn,\" Ekhaas said.\n\n\"They're all spending money,\" Geth told her. \"Iizan's just being more showy about it.\"\n\nEkhaas spotted Midian among the Ghaal Sehn, trying to get Iizan's attention and apparently not having much luck. \"What's he doing?\"\n\n\"Trying to get all the heirs to let him continue digging at Bloodrun, I think. So far he's only got Tariic to agree. I don't think any of the others care much for gnomes\u2014or history.\"\n\nEkhaas's ears lay back.\n\nIn the arena, the ettin charged again. This time, Keraal flicked his arm and the chain broke out of its spin to fly low at the ettin. At the last moment, the hobgoblin twisted into the chain, wrapping it around his waist. The flying end of the chain, weighted with shackles, changed direction sharply and jumped up to slam into the shoulder of the ettin's sword arm. The creature's shield dipped and it pulled its charge up short\u2014though not short enough. Keraal jerked on the chain and the shackles cracked across one of the ettin's faces, tearing its lips and cheek. A roar went up from the crowd at the sight of first blood.\n\n\"Who chooses his opponents?\" asked Ekhaas.\n\nThe ettin swatted Keraal with its shield as he tried to get past and a cheer went up from Tariic and his supporters. \"Who do you think?\" said Geth. \"He's let it out that there's a fat reward to whoever defeats\u2014and kills\u2014the rebel who defied Haruuc.\"\n\nAnother successful blow by Keraal brought another, louder roar for the rebel. Ekhaas's ears flicked upright again. \"That could bite him,\" she said. \"The longer Keraal stays alive, the more the crowd will treat him like a hero.\"\n\n\"Haruuc told me once that Darguuls want blood and it doesn't matter who sheds it. I think the crowd will be just as happy if he dies.\" Geth grimaced. \"I haven't figured out what I'm going to do if he lives.\"\n\n\"There's nothing to figure out. He goes free. We're bound by tradition.\" She looked sideways at him. \"I've found our artificer,\" she said quietly.\n\nThe shifter sat up. Ekhaas put a hand on his arm and pulled him back down in his seat. \"Don't make a fuss!\" She glanced around. No one seemed to have noticed Geth's excitement. \"He'll see us tonight. You, me, and Dagii.\"\n\n\"Just the three of us?\"\n\nEkhaas nodded. \"I think he suspects something, so he wants to be discreet, which is what we want, too. I'm his contact, you're the only one who can handle the rod, and Dagii will provide us with extra protection. Bringing Ashi and Midian would attract too much attention in the streets at night.\"\n\n\"How did you get him to agree to it?\"\n\nShe drew breath through her teeth. \"He hasn't entirely. Not yet. He knows that he's going to be copying a Dhakaani artifact. That's all. I even asked in the Bloody Market about the availability of byeshk. We'll get it to him when he needs it.\" Ekhaas sat back. \"He's fascinated by the lost knowledge of the _daashor_ , the Dhakaani artificer tradition. That's what hooked him. Bring Wrath with you tonight. If he hesitates when he finds out what we really want, I think the opportunity to examine two artifacts created by the legendary Taruuzh will help him make up his mind.\"\n\n\"I'd bring Wrath anyway.\" Geth touched the sword's hilt. \"I'm not going anywhere without it right now.\"\n\n\"Dagii and I will come for you when the first moon rises,\" Ekhaas said. \"Wrap the rod in something to disguise it\u2014and try to disguise yourself. Too many people recognize you now.\" She rose.\n\nAn loud whistling sound from the arena drew her attention back to the sands. Keraal had doubled up his chain and was swinging it hard and fast overhead. The ettin, uncertain what the hobgoblin was up to, backed away slowly. Maybe too slowly\u2014Keraal dropped suddenly to one knee and loosened his grip. The chain slid through his fingers with a long rattle and swept around in an expanding arc. The end of it wrapped around the ettin's ankle, the shackles catching in the chain and tangling.\n\nInstantly, Keraal was back on his feet and racing around the ettin even as the two-headed creature tried to shake the chain loose. Keraal caught its other leg inside the loop of chain, and hauled back with all of his strength. Muscles strained under his scarred skin.\n\nThe ettin's legs were pulled together, then wrenched out from under it. Arms flailing, it crashed faces first to the ground. Keraal dropped the chain and charged, jumping up onto its broad shoulders and leaping high.\n\nHe came down with both feet together on the back of the creature's right head. Even with the shouts of the crowd ringing in the arena, Ekhaas thought she heard a distinct crack as the head's face was smashed into the sand. The left head bellowed in shared agony. The ettin heaved, trying to turn itself over.\n\nKeraal dragged the club out of its slack right fist, wrapped both of his hands around the heavy timber, and lifted it. The ettin saw the raised club and tried to get its shield up, but its left arm was supporting most of its weight. It dropped as if it could roll away, but Keraal swung first. The club came down square on the left head. Bone crunched. The ettin managed a feeble roar. Keraal, silent, swung the club again.\n\nThe skull shattered and collapsed. Blood and brain spattered the hobgoblin. The ettin's right head jerked and screamed. Keraal turned to it and swung the heavy club a third time.\n\nThe crowed roared in delight. Keraal flung down the club and turned to look up at Geth. Shifter and hobgoblin faced each other in silence, neither moving for a moment, then Keraal turned to Tariic and thumped his chest with his fist in a mocking salute. Tariic's ears pressed back flat against his skull.\n\nThe arena resounded to the cheers and applause of the crowd.\n\nGeth knew Dagii and Ekhaas had arrived when he heard song in the hall outside his door. It was a beautiful, soft song, strange and broken as all _dar_ songs were to someone who hadn't grown up with them, but still soothing. It reminded Geth of warm nights looking up at the stars and Eberron's twelve moons and the hazy glow of the Ring of Siberys that dominated the southern sky.\n\nHe jerked back to alertness as the door of his chamber opened and Ekhaas entered. The _duur'kala_ was wearing plain, drab clothes and a loose scarf that hid half her face but left room for her large and mobile ears. \"Geth? Are you ready? The guards won't sleep for\u2014\" Her eyes landed on him and she stopped. \"What's that?\"\n\nGeth pulled the enveloping cloak that he wore more tightly about himself. The hood hid his face better than Ekhaas's scarf hid hers\u2014in fact, he could barely see to either side. \"You said to disguise myself because too many people recognize me.\"\n\n\"So we wouldn't attract attention. Now you look like someone _trying_ to disguise himself. Get that off and we'll try something else.\"\n\nGrowling, he shook the cloak off\u2014then growled again as Ekhaas squeezed her eyes shut and bent her ears down. \"What?\"\n\n\"We're not going to be able to cover that.\" She rapped the great gauntlet that covered his right arm.\n\nA sleeve of magewrought black steel plates, studded with flat spikes on the forearm and short hooks on the back of the hand, the gauntlet was all the armor he needed. It was light. It was strong. Paired with a sword in his left hand, it made a second weapon that had surprised many opponents over the years. What it wasn't was inconspicuous.\n\n\"I was wearing the cloak to cover it and Wrath!\" He slapped the sword at his side.\n\n\"Hurry!\" Dagii's voice came low from the hall.\n\nEkhaas's ears bent even further. \"There isn't time to take it off. Just hold still.\" She concentrated on his face for a moment, her amber eyes intense, then raised a hand and sang a brief rippling passage of song. Geth had experienced Ekhaas's _duur'kala_ magic many times before, and each time it felt as though he had been dipped in some wild spring that bubbled with the primal music of the world's creation. This time the music still had that primal energy, but it was strangely muted, almost as if it were only an echo. Ekhaas lowered her hand and nodded. \"It will do.\"\n\n\"What will do?\"\n\n\"I'll tell you when we're out. Where's the rod?\"\n\n\"Here.\" Geth had the Rod of Kings wrapped up in the soft oiled leather that normally protected his gauntlet. He reached for the anonymous bundle and was startled to see an unfamiliar hand pick it up\u2014a hobgoblin's orange-red hand on the end of a slender arm wearing a black wool sleeve. He glanced at Ekhaas.\n\n\"Illusion,\" she said. \"Just take the rod. We need to leave.\"\n\nHe didn't ask anything more, but grabbed the wrapped rod and followed Ekhaas out the door, closing it behind him. The guards that stood on either side of the door\u2014an honor for the _shava_ of Haruuc more than anything else\u2014were leaning back against the walls, both of them lightly dozing under the subtle effect of Ekhaas's magic. They wouldn't know he had slipped out of his chamber.\n\nDagii was waiting just along the hall. For the first time Geth could recall, the young warlord wore no armor, though he did carry a sword. His shadow-gray hair had been shaken loose and brushed forward over his face. Dagii barely looked like himself. His ears rose at the sight of them, but he said nothing and fell into step alongside them.\n\nAs soon as they were away from the hall outside his chamber, Geth looked down at himself. He wore\u2014or seemed to wear\u2014a robe of black wool and a wide girdle of red leather tooled in the angular patterns favored by hobgoblin design.\n\nHe also appeared to have breasts.\n\n\"A woman?\" he said under his breath. \"You made me a woman?\"\n\n\"My older sister, actually,\" Ekhaas whispered. \"I was in a hurry. I had to choose someone I knew well but that no one in Khaar Mbar'ost was likely to recognize. The illusion won't last long\u2014that's why I prefer non-magical disguise. Try not to talk. The magic doesn't change your voice.\"\n\n\"You don't know any hobgoblin _men_?\"\n\nDagii's voice was thick with restrained laughter. \"If it's any consolation, Ekhaas is clearly the beauty in the family.\" He nodded to the _duur'kala_. Ekhaas's ears flicked and she returned the nod gracefully, a smile playing around her lips.\n\nGeth ground his teeth together.\n\nIf it was embarrassing, the illusion was effective. They tried to move through the least busy parts of Khaar Mbar'ost, but even at night the fortress was an active place. Still, no one within the walls gave them the slightest glance except for one hobgoblin warrior whose gaze lingered on Geth. They couldn't sneak around the guards at the gates, but there Dagii simply looked one of the guards in the eye. The guard, startled by his unexpected appearance, snapped up straight. If anyone cared to question him, the hobgoblin might report that the warlord of Mur Talaan had left Khaar Mbar'ost, but Geth doubted if he would remember the two women who passed by with him.\n\nBeyond Khaar Mbar'ost, the daytime revelry of the games spilled over into the night. Bonfires burned in the middle of some of the wider streets, and the people of Rhukaan Draal gathered around them to sing and dance and drink. The guards that had patrolled the city during the period of mourning were all but gone. Those few Geth spotted as they passed through the shadows were celebrating as heartily as the rest of the crowd. Here and there, vast casks of ale stood open and whole roast hogs were laid out courtesy of those with aspirations to the throne. Geth recognized the chiefs of a few lesser clans calling out the virtues of Aguus or Iizan or Garaad. The warlords had called on their allies to help support the costs of their campaigns of popularity.\n\n\"I don't see anyone touting for Tariic,\" he said.\n\n\"He's being clever,\" said Dagii. \"Here all the food and drink blur together. He's concentrating his attention on the arena and on winning over the unallied warlords.\"\n\n\"He may win over some with a previous allegiance, too, if the others push their client-clans too far,\" Ekhaas added.\n\nThe street where the artificer lived was far from the busy parts of the city. Here no light but moonlight illuminated the crooked streets. In spite of the silence and the darkness\u2014or maybe because of them\u2014Geth felt a distinct unease. When Ekhaas's illusion finally dissipated in a flickering of ghostlight, he was more than glad to be rid of it. The sight of Wrath and his great gauntlet might hint at his identity to anyone watching from the shadows, but they would also be a greater deterrent to attack than a gown and a girdle.\n\nThe buildings in the area were, for the most part, dark and in poor repair, leaning on each other like a bunch of drunkards. Ekhaas guided them to a low stone building that was doing most of the work of propping up its neighbors. It had the look of human construction rather than _dar_ , and Geth guessed that it predated Haruuc's creation of Darguun and the establishment of Rhukaan Draal. With a wide double door and only a few small, high windows, he suspected it had originally been a barn or a dairy or some other outlying farm building. Light shone around the door and the shutters. Ekhaas gestured for him and Dagii to hang back along the stone wall, then she went forward and knocked in a short rhythm on the double door.\n\nA moment later, light stabbed her face as a peephole opened up. Geth heard a murmuring voice. Ekhaas gave an answer. The peephole closed again, then bolts rattled and one half of the door opened wide. Ekhaas waved for them to join her and stepped inside. Holding the bundle containing the Rod of Kings close, one hand on Wrath, Geth went after her.\n\nBarn, he decided as he stepped through the door, the stone building had definitely once been a barn. The central room of it was lined with the remains of stalls and the cobble floor showed channels where filth had once been sluiced away. Any animal odor was gone, however, replaced by a strange smell like hot copper. The stalls had been lined with shelves which were in turn filled with books and papers and strange implements. Everbright lanterns hung from the rafters, illuminating the space with an unwavering magical light.\n\nEkhaas was in conversation with their host\u2014a horn-browed tiefling. As Dagii closed the door behind himself, both turned to face them. \"This,\" said Ekhaas, \"is Tenquis.\"\n\nThe tiefling was tall and lean, his skin of his face smooth and dark brown. Geth would have said that he was probably a little bit older than either Ekhaas or Dagii, possibly even around his own age. It was difficult to tell because his eyes were solid golden orbs without white, iris, or pupil. Descendants of ancient mages who had bargained with devils, tieflings showed the taint of their ancestors' bid for power. Tenquis's strange eyes were echoed in gold flecks on the polished heavy black horns that curled back from his forehead above dark, wavy hair. Horny spikes edged his chin in imitation of a sharp goatee. He wore a kind of long vest embroidered in an intricate labyrinthine pattern over a much-laundered shirt and brown leather trousers; in the back, vest and trousers were cut to make room for a thick fleshy tail, brown like his skin.\n\nAnd as Tenquis stared at Geth staring at him, the tail rose and lashed the air. \"I know you,\" he said. His teeth, sharp as a shifter's, flashed white with each word. His voice had a husky rasp. He glared at Ekhaas. \"A Dhakaani artifact? I know what you want! By the sorcerer-kings, get out before you ruin me!\"\n\n\"Tenquis, wait,\" said Ekhaas, raising her hands. \"Everything I told you this afternoon is true. Listen to us before you answer. You don't know what we want\u2014\"\n\n\"I can guess.\" He pointed a finger at Geth. His nails were the same color as his horns, black flecked with gold. \"You want a copy made of a Dhakaani artifact. What artifact does Haruuc's _shava_ hold? The rod of the Dhakaani emperors. I'm no idiot. I am not bloody copying the symbol of the lhesh's rule! Get out!\"\n\nHe strode toward the door as if he could throw them out. Dagii stepped in front of it, crossing his arms over his chest. The tiefling pulled up short and looked at Dagii, then back at Ekhaas. \"Don't try to intimidate me,\" he said, his voice sinking into his chest as his tail whipped from side to side. \"I don't appreciate intimidation.\"\n\n\"We're not trying to intimidate you,\" Ekhaas said. Her tone was soothing yet beguiling at the same time. \"If you know who Geth is, then you can make a good guess at who we are. We're the ones who found the rod for Haruuc and we're trying to fix a mistake before it destroys Darguun. We need your help to do that. The rod is cursed.\"\n\n\"Cursed?\" The word dripped with disbelief.\n\n\"Believe what you want,\" said Dagii grimly. \"It's the truth.\"\n\nTenquis looked at the young warlord, then at Ekhaas, then at Geth. \"And how do you know this?\"\n\nEkhaas opened her mouth, but Geth spoke first. \"Haruuc told me,\" the shifter answered, meeting the golden-eyed gaze. \"The rod drove Haruuc to hang the warriors of the Marhaan on along the road to Rhukaan Draal. It drove him to sell their women and children into slavery. It drove him to torture Keraal on a Dhakaani grieving tree. It almost drove him to lead Darguun to war.\"\n\n\"I know a lot of Darguuls who were happy with all of that.\"\n\n\"Haruuc wasn't. He knew that what the rod wanted would destroy Darguun.\" Geth hesitated, then plunged on with the truth\u2014or at least part of it. \"It was created to guide the emperors of Dhakaan, but this isn't the world the Dhakaani knew. Whoever holds the rod sees the memories of the emperors.\"\n\n\"The memories of emperors?\" Tenquis's eyes opened a little wider.\n\nBeyond Tenquis, Geth saw Ekhaas's ears stand tall and remembered what she had said in the arena, that the artificer was fascinated by the lost knowledge of the ancient Dhakaani _daashor_. He fumbled with the ties that held the bundle closed. The leather fell open to reveal the Rod of Kings. Geth lifted it free.\n\nTenquis stared at it, his lips open just a little bit, his tongue running across the tips of his teeth. He reached out with one hand. Geth pulled the rod back. \"Don't touch it! That's how it passes on the memories.\"\n\nTenquis drew back, but just a little bit. \"How can you hold it then?\" he asked.\n\nGeth dropped the leather and drew Wrath with his other hand. Behind Tenquis, Ekhaas smiled. \"The histories preserved by the Kech Volaar,\" she said in the tones of a trained storyteller, \"tell of three artifacts created by Taruuzh _daashor_ from the vein of byeshk he named _Khaar Vanon_ , the Blood of Night. The first was Aram, or Wrath, the Sword of Heroes that was lost by Rakari Kuun in Jhegesh Dol when he killed the daelkyr lord of that place. The second was Muut, or Duty, the Shield of Nobles that was shattered as Dhakaan slid toward the Desperate Times. The third was Guulen, or Strength, the Rod of Kings. Three great artifacts, each the equal of the others in power.\"\n\n\"Wrath protects me,\" Geth said simply. \"I'm the only one who can safely touch the rod.\"\n\nTenquis's gaze moved from the rod to the sword and back again. He swallowed. \"Taruuzh made these.\"\n\n\"You'll be the first artificer to have the chance to study them,\" said Ekhaas. \"You won't get this opportunity again.\"\n\nTenquis looked longingly at the rod once more, then his lips pressed together into a thin line and he turned to Ekhaas. \"What exactly do you need from me?\"\n\nGeth took a long breath of relief. Dagii relaxed a little, too. Ekhaas's face remained impassive, however. \"An exact copy of the rod, enchanted to enhance the presence of the one who holds it\u2014\"\n\nTenquis snorted. \"Easy enough.\"\n\n\"\u2014and ready in three days.\"\n\nThe snort turned into a twitch. \"Three days? That's not possible. This isn't like forging a horseshoe. Six, maybe. Byeshk is hard to work with and I'd need to find some first\u2014\"\n\n\"We can have the byeshk here in the morning,\" said Ekhaas.\n\n\"I'll need more than byeshk. Other materials. They won't be cheap.\"\n\nDagii stepped past Geth and tossed a fat, clinking pouch to Tenquis. \"That should cover the price of anything else you need.\"\n\nGeth wondered where he had come up with the money. The Mur Talaan clan was highly respected, but it wasn't wealthy. Tenquis rolled the pouch between his fingers, looking both startled and pleased. \"I'll need to study the rod,\" he said. \"Make sketches, take measurements.\"\n\n\"You have tonight,\" Ekhaas said. \"No touching it. Geth can hold it for you.\"\n\nThe tiefling flicked his fingers dismissively. \"Gloves,\" he said. \"Has anyone tried holding the rod while wearing gloves?\"\n\nGeth looked at Ekhaas and Dagii, and felt a flush of embarrassment warm his face. They'd been too worried about the danger of the rod to consider it, but the priest at Haruuc's funeral had held the rod on a tray and felt nothing.\n\nTenquis raised an eyebrow at their silence. \"I didn't think so. I'll want to study the sword, too.\"\n\n\"Another time,\" said Ekhaas. \"Can you do better than six days?\"\n\nTenquis pursed his lips and glanced at Geth and the rod. \"I may be able to make it in five.\" He held up a finger. \"I have conditions. Nobody outside of this room can know about my involvement. Not any allies, not your mothers. I'm not losing my head for you.\"\n\n\"That's fair,\" said Ekhaas.\n\nTenquis held up a second finger. \"There will be a... fee.\"\n\nDagii's ears pressed back. \"The pouch has more than you'll need for materials. The rest is yours.\"\n\n\"More than money.\" Tenquis looked at Ekhaas again. \"I want to know what the Kech Volaar know about the _daashor_. Histories, stories, legends\u2014anything.\"\n\n\"I'll tell you everything I know,\" Ekhaas promised. \"After the rod is copied. Five days, no more. If the Rod of Kings passes into the hands of an heir, we've run out of time. Do we have an agreement?\"\n\nTenquis smiled. \"We have an agreement.\" He held out his hand. Ekhaas drew a knife. The tiefling's smile faltered for a moment, then returned. \"By your people's custom, then.\" He pulled an ornate dagger from his belt and touched the blade to Ekhaas's, sealing the deal in _dar_ fashion. Then he dropped the dagger onto a nearby table and pointed at Geth. \"Bring the rod, here. If I only have tonight to examine it, I need to get started.\"\n\nTenquis worked quickly, clearing a large table and directing Geth to set the rod on it. Multiple everbright lanterns with arrangements of mirrors and lenses directed bright light onto the rod, lighting it up as if the sun were shining into the converted barn. The artificer produced paper, pens, and ink, a measuring stick, calipers, and more lenses. He began with a careful examination of the byeshk shaft\u2014with Geth turning it as he instructed\u2014then took calipers and began transferring the dimensions of the rod onto paper. As quickly as he worked, though, the copying took time. Ekhaas found a chair among the shelves and books, stretched out, and dozed off. Dagii simply lay down on a carpet that covered a section of the stone floor.\n\nGeth didn't have that luxury. He could move about, stretch, occasionally sit down, but it was never long before Tenquis called on him to turn the rod\u2014the tiefling had tried handling the rod with thick gloves, but while they protected him, they were also clumsy. Geth's hands were more dexterous. At Tenquis's request, he laid Wrath alongside the rod, so the artificer could compare the runes scribed on the two artifacts. Half the night seemed to pass and Geth watched him fill page after page with careful sketches of the rod in the most minute detail. Tenquis was a talented artist\u2014the drawings he made were vivid and fine. His dark hands, calloused and nicked in ways that set them apart from a fighter's hands, moved with swift certainty from pen to calipers and back again, adjusting a lens on a lantern, grabbing for a fresh piece of paper, or flipping back to consult a previous sketch.\n\nAfter a time, he realized that Tenquis was glancing up at him as well. He twitched his gaze away. \"Sorry.\"\n\n\"Don't be.\" Tenquis set his pen down and stretched his hands. The joints popped. He leaned back on the stool he occupied and looked at him. \"The rod and the sword aren't the only artifacts you carry.\"\n\nGeth touched the collar of black stones, each one roughly polished and marked with a symbol, that he wore around his neck. \"You mean this.\"\n\n\"Yes. Those are orc runes.\" He rose and leaned across the table, reaching out to lift one of the stones and rub his thumb across it. His fingers were warm and dry. \"Druidic tradition. Very old. Powerful too.\"\n\n\"The collar is a Gatekeeper artifact,\" said Geth. \"It protects my mind from the forces of Xoriat, the plane of madness. It belonged to a... close friend.\"\n\n\"Who died?\"\n\nIt was difficult to read the tiefling's featureless eyes, but up close Geth could see the tiny creases in the skin around them, the slight movement of the brow below the heavy horns. He half-expected to see himself reflected in the golden orbs, but they gave back no reflection at all. \"He was killed,\" he said.\n\n\"And the person who killed him?\" Tenquis let the stone fall back against Geth's neck and stood straight.\n\n\"Dead.\"\n\nTenquis smiled, showing sharp teeth again. \"My grandmother has a saying: the way through the maze is clear for the wise and the lucky, but the rest of us have to fight.\" He sat down and stared at the rod for a moment before looking up at Geth again. \"The symbols on the rod and the sword\u2014you don't understand them, do you?\"\n\n\"They're not Goblin.\"\n\n\"No. They're not any kind of language you could read, really.\" He picked up the pen and tapped it against the rod. \"They bind magic, shaping it. The methods used by the Dhakaani _daashor_ are different from what modern artificers use, but there are similarities if you know what to look for.\" The pen touched one carved rune, then another. \"These are strong runes. Taruuzh bound incredible power into the rod.\" Tenquis lifted his eyes. \"There's more to the rod's power than just the memories of emperors and enhancing the presence of the holder, isn't there?\"\n\nThe hair on Geth's arms and neck rose. He didn't say anything. For a moment, neither did Tenquis. Then the tiefling nodded slowly and set the pen down. \"I might be able to figure it out myself, but I'm not sure I want to,\" he said. He paused before asking, \"When I've made the copy, what are you planning to do with the true rod?\"\n\n\"Who says we're planning to do anything with it?\" said Geth.\n\n\"The voice of experience. When people want something copied, they usually have plans for the original. They want to sell it if it belongs to them or they want to steal it if it doesn't.\"\n\nGeth's throat felt dry. \"We're going to destroy it,\" he said.\n\nTenquis's lips curled. \"Really?\" he said.\n\nQuick as a thought, he reached down and caught up a heavy smith's hammer. Before Geth could stop him, he had raised the hammer and brought it down on Wrath's blade with a crash that jolted both Ekhaas and Dagii from their sleep.\n\n\"Grandfather Rat!\" shouted Geth. \"What do you think you're\u2014\"\n\nHe stopped, staring at Wrath. By rights the blade, heavy as it was, should have been smashed or at least bent, but there wasn't a mark on the purple metal. Tenquis dropped the hammer. \"It's not so easy to destroy powerful artifacts like these,\" he said. \"You might want to come up with another plan.\"\n\n_\"Maabet!\"_ cursed Dagii. \"What was that about?\" He had his sword out, ready for a fight.\n\nGeth looked at Tenquis for a moment longer, then turned to Dagii. \"Tenquis was showing me how tough Wrath and the rod are.\"\n\n\"You couldn't have been quieter about it?\" Ekhaas looked at the sea of papers and sketches that covered the table. \"Are you finished?\"\n\n\"I think so.\" Tenquis began to gather up his drawings. \"Be sure the byeshk gets here. I'll send word if there are problems. Otherwise you'll hear from me when the false rod is completed.\"\n\nAs soon as Geth returned Wrath to its sheath and wrapped the rod back into the bundle of leather, they took their leave. Tenquis gave Geth one last golden-eyed glance, then his door closed behind them. The night was almost over, the first hints of dawn glowing in the east, though the streets were still dark enough to have left a human blind. Rhukaan Draal was, briefly, at its most silent. Ekhaas didn't bother trying to renew the illusion that had cloaked Geth\u2014instead she brushed his shaggy hair forward to hide his face. Dagii shed the jacket he wore and draped it over Geth's shoulder to conceal both his great gauntlet and the bundled rod. A rough disguise, but it would do. The weariness of a night without sleep weighed on Geth. There was something he knew he had to say, though.\n\n\"Tenquis figured out that there's something more to the rod.\"\n\nEkhaas let out a soft curse. \"Can we trust him?\" asked Dagii.\n\n\"I think so,\" said Geth. \"He also guessed we were planning to do something with the true rod. I told him we were going to destroy it. That banging was him showing me that's going to be harder than we thought.\"\n\n\"I was afraid it might be,\" said Ekhaas grimly. \"In legends, great artifacts are either ridiculously fragile or impossibly durable. The cursed ones are never fragile. We'll think of something\u2014\"\n\nThe skittering of a stone in the shadows interrupted her. Instantly, all three of them were on the alert, hands on their swords. \"Rat?\" breathed Dagii.\n\n\"No,\" Geth said. There was another sound, this time from the shadows on their other side. They moved back to back. Almost immediately, a third stone rattled on the street, louder than the others. They all looked toward it.\n\n\"Whoever is out there,\" said Ekhaas, \"is either incredibly clumsy or wants us to know we're not alone.\"\n\n\"The latter,\" said a voice that was thick and strained like a scar. A small shadow detached itself from high on a wall and dropped softly to the street, then moved forward so they could clearly see an old goblin with thin cobweb hair, the parchment-like skin of his face stained dark to match the black clothing of the _shaarat'khesh_.\n\n\"We need to talk,\" said Chetiin.\n\n#\n\n# CHAPTER  \nSEVEN\n\n# 21 Sypheros\n\nThe fatigue Geth had felt dropped away, burning into rage. \"You!\" he snarled. He shrugged off Dagii's jacket and thrust the bundle containing the rod at Ekhaas. Wrath rang as he drew the twilight blade, then he charged at the goblin.\n\nChetiin had picked a good spot for a fight, a small open square, little more than a broadening of the street with the shutters on the surrounding buildings closed tight. Four running strides closed the distance between Geth and the goblin. Chetiin sank back into a crouch, hands raised in loose fists. Geth swung Wrath on the last step. The heavy sword arced down\u2014and Chetiin slid out of the way.\n\nGeth spun, letting the momentum of the blow carry him around, and turned Wrath in another blow that forced Chetiin to dodge with a little less grace. Dagii joined him, moving in on Chetiin's other side with his sword at the ready. Chetiin slipped back into his crouch and rocked gently from one side to the other, his hands weaving along with his body and making it difficult to tell where he was going to go next.\n\n\"Geth,\" he said, his strained voice low, \"listen to me\u2014\"\n\nThe shifter answered with a fast backhanded swipe of his gauntlet. Magewrought steel swept the air. Chetiin stepped aside\u2014and vanished. Geth and Dagii whirled, searching for him.\n\nHe reappeared behind Dagii. Geth caught the flash of a fist as Chetiin punched into the warlord's leg at mid-thigh before darting back. It was a hard blow, delivered with precision. Dagii grunted and nearly went down, his leg numbed, but Ekhaas appeared to support him. Eyes flashing with anger, she drew breath to sing.\n\nChetiin's hand dipped into a pouch at his waist and emerged to fling ash into her face. Fine gray particles floated in an expanding cloud. Ekhaas's eyes went wide and her song turned into a fit of choking. Chetiin leaped away from her and Dagii and turned to face Geth again, dropping into a crouch once more. His wide eyes were hard, his big ears cupped, his teeth bared\u2014and through them he spat, \"Geth, I didn't kill Haruuc!\"\n\nThe protest was so feeble, so ridiculous, that the only answer Geth could manage was a growl that rolled out of his belly. Dropping into a matching crouch, he reached deep inside himself, drew on his fury... and shifted.\n\nThe lycanthrope ancestors of the shifter race had the power to become animals, but as their bloodlines had tangled together, that ability had been diluted. Shifters couldn't take animal form, but they could assume the animal traits of their ancestors. Some shifters could manifest sharp claws or put on a burst of feral speed. Geth's gift was sheer toughness.\n\nThe shifting spread through him like a heat in his blood. His skin thickened into leathery hide. His hair turned even more coarse and heavy. A sense of invulnerability washed over him. The growl turned into a shout and he surged forward, spinning Wrath in deadly arcs, holding his gauntlet ready to block any blow Chetiin attempted or to bash him should he try to skip aside again.\n\nBut the _shaarat'khesh_ elder fell back before the attack. Light and quick, he swayed back and forth, looking for a way around, but Geth kept him contained. He pressed hard, forcing Chetiin backward until the goblin was caught in a corner of the square. His face tightened. Geth lunged.\n\nAnd Chetiin turned to push off one wall onto the next and back to the first, each leap taking him higher. Wrath struck stone, grating out a shower of orange sparks. Geth twisted around to follow Chetiin as the goblin made a final leap from the wall and soared over his head.\n\nHe landed like a cat. Geth charged again, whirling as he swung. When Chetiin tried to dodge, he was ready for him. His gauntlet caught him with a punch that sent him tumbling. Chetiin rolled to his feet with blood trickling from his mouth. He leaped away as Geth came in again, drawing a dagger from one of the sheaths on his forearms and circling well away from the shifter.\n\n\"Why would I kill Haruuc?\" Chetiin asked. \"He was my friend.\"\n\n\"You said he needed to be stopped. You thought he'd discovered the power of the rod.\" Geth flung the words Chetiin had spoken over Haruuc's body back at him. \"You did what you had to do.\"\n\n\"I didn't.\"\n\nGeth saw his eyes flick toward Dagii, standing on his own again, and Ekhaas, able to breathe once more. In a moment, they would have the advantage. He slid forward. Chetiin's eyes snapped to him and he slid back.\n\n\"You're an assassin,\" Geth growled. \"An assassin from a clan of assassins!\"\n\nChetiin's ears pulled back. \"Then why would I kill Haruuc in full view of every warlord in Darguun? Why would I leave Witness, a treasure of the _shaarat'khesh_ , behind? Why would I put my clan in danger?\"\n\nHe threw the questions like knives and they found their target. Geth hesitated. Why would Chetiin have let himself be seen? He could have struck from the shadows in some quiet corridor. He could have slipped into Haruuc's own chambers. He could have arranged a quiet death for the lhesh so that it didn't look like an assassination at all\u2014\n\nAnd in the instant that he hesitated, Chetiin struck. He threw himself forward in a swift tumble. His leg swept around and, small as the goblin was, kicked the shifter's legs out from under him. Geth slammed down onto his back. Chetiin jumped on his chest, one heel kicking the wind out of him as he landed. The point of his dagger pricked the skin beneath Geth's chin.\n\n\"I wasn't the one who killed Haruuc,\" Chetiin said.\n\nHe held the dagger under Geth's chin a moment longer, then vaulted off to land well away from him, Ekhaas, and Dagii. Geth thrust himself up, wheezing as he sucked in air.\n\n\"Geth,\" said Ekhaas, \"I think we should listen to him.\"\n\n\"Why?\" Geth croaked.\n\n\"Because we're alive. He could have killed any of us from the shadows before we even knew he was there, but he didn't. He could have killed you just now, but he didn't.\"\n\nGeth looked at her and at Dagii, then at Chetiin. \"I saw you. You killed Haruuc right in front of me.\"\n\n\"It wasn't me. A disguise. An illusion, maybe.\" Chetiin remained in his crouch, dagger ready. \"Blood of six kings, I swear it.\" He gestured. \"Put away Wrath.\"\n\n\"No.\"\n\n\"Geth\u2014\" Dagii said. The hobgoblin sheathed his sword. So did Ekhaas. Geth grimaced and put Wrath away as well. Chetiin made his dagger disappear into the sheath on his forearm with a smooth gesture. Geth glared at him with narrowed eyes. A part of him wanted to trust the goblin elder. He'd been a wise and capable friend. The rest of him still stung from Haruuc's death and Chetiin's betrayal of their friendship.\n\n\"Why should we believe you?\" he demanded.\n\n\"For the reasons I said. If I wanted to kill Haruuc, I wouldn't have done it the way that it happened. It was far too public. If you believe nothing else I say, believe that.\" He spread his hands. \"You know I believed in Haruuc's vision of a stable Darguun as much as anyone else. Such an obvious killing, especially when Haruuc had not named an heir, would have served no purpose.\"\n\n\"Except to separate him from the rod if he'd discovered its true power,\" said Geth.\n\nChetiin scowled and jerked his head at the windows around the square. \"This isn't the place to discuss secrets,\" he said. \"Come with me. There's a place we can talk.\" His face softened. \"Trust me. On Haruuc's memory and my word, I'm telling you the truth.\"\n\nGeth glanced at Ekhaas and Dagii. Ekhaas's ears flicked and she nodded curtly. He looked back to Chetiin. \"No tricks,\" he said.\n\nThe goblin nodded. \"After the last two weeks I don't think I have any tricks left.\"\n\nThe place he led them to was a fragment of brick wall standing in the middle of large swath of burned-out ruins. The nearest surviving buildings were some distance away.\n\n\"One of the fires that burned on the night of Haruuc's death destroyed this area,\" said Chetiin. \"It will be reclaimed, but between the mourning period and the games, no one has had time. We won't be overheard.\" He squatted down in the shelter of the wall and looked up at them. \"I've been watching you for the last few days, waiting for a chance to speak with you alone,\" he said. \"I saw you leave Khaar Mbar'ost tonight and I followed you to the tiefling's house. His shutters fit badly. I know your plans. Who thought of creating a false rod?\"\n\nEkhaas answered. \"Midian. When he returned.\"\n\nChetiin nodded. \"I heard what you told Tenquis about the rod's influence on Haruuc. _Maabet_ , I should have seen it. Nothing he did in the last days of his life was like him.\"\n\n\"You said as much when you argued with him just before his death,\" said Geth. \"You said, 'You're not the Haruuc I've known for so many years.'\"\n\n\"If we knew truth when we spoke it, we'd all be seers.\" Chetiin's mouth tightened. \"I didn't guess that the rod had a hold on him. If anything I would have guessed that he had begun to grasp the rod's power.\"\n\n\"That's what we thought you believed,\" Geth said. \"You also told Haruuc that he would destroy what he'd built unless he was stopped.\"\n\n\"And you thought that I had followed through and stopped him.\" Chetiin fell silent for a moment before adding, \"I would have considered it. We swore to keep the rod's powers secret by any means necessary.\" He glanced at each of them in turn. \"But we all swore to that. I wouldn't have acted on my own. I would have come to you.\"\n\n\"Then what happened?\" asked Dagii.\n\nChetiin made a face and his ears folded down. \"I was careless. When I left Haruuc after our argument\u2014after he ordered me out of Khaar Mbar'ost\u2014I was angry. I forgot the first law of the _shaarat'khesh:_ watch and listen. Someone put a crossbow bolt in my back. Poisoned.\"\n\nBreath hissed between Geth's teeth. \"Just like Haruuc. Did you see who did it?\"\n\n\"No.\" Chetiin shook his head. \"The bolt was bad. The poison was worse. It worked fast, burning in me. It made me weak and blind. Strandpine sap, I think. A big dose, maybe as much as they used on Haruuc.\"\n\n\"And you're smaller than he was. How come you're not dead?\"\n\n\"An assassin who works with poisons for many years develops a resistance to them,\" the goblin said. \"As it was, I was as good as dead. That probably saved me. I think if I had been any more alive, my attacker would have been certain to finish the job.\"\n\n\"Lucky,\" Ekhaas said.\n\n\"It wasn't my only piece of luck. I drifted in and out of consciousness after that while the poison ran its course. When I finally woke up, I realized that I'd been stuffed up onto a ledge in one of the chimneys of Khaar Mbar'ost. If not for the ban on fires during the period of mourning, smoke in the chimney would have suffocated me.\" He spread his hands. \"The bolt had been pulled out of me, maybe to make it easier to hide my body. That spared me further exposure to the poison. My daggers had been taken, of course. When I managed to climb down, I discovered the ledge I'd been hidden on was above the fireplace in my own chamber.\"\n\n\"Your chamber was searched,\" said Dagii. \"I was there.\"\n\n\"Whoever attacked me did a good job. The ledge was invisible from below. Smoke and heat would have preserved my body. What little smell there might be would have been whisked up the chimney. I might never have been found.\"\n\nHe said it with a chilling bluntness, as if talking about his own murder was the most natural thing in the world. Geth held back a shudder. \"And we would have kept on thinking you'd gone into hiding. Why didn't you come to us?\"\n\n\"I overheard that Haruuc was dead and that I had done the deed. I didn't know who had attacked me, but it was clear that anyone who found me wasn't going to let me live long enough to explain. I could barely move, much less defend myself. It was two days after Haruuc's death. I bandaged my wound, disguised myself, got out of Khaar Mbar'ost, and made my way to a _shaarat'khesh_ house. It was deserted\u2014those of my clan had fled the city or moved to more secure shelter\u2014but it was enough for me. I spent eight days recovering there until the end of the mourning period.\" His expression darkened. \"With the beginning of the games, I was able to move around Rhukaan Draal in disguise. I learned everything that had happened\u2014and I started to watch for my chance to talk to you. You gave it to me tonight.\" He spread his hands. _\"Sit ya toomiish_ \u2014those are events as they happened to me. I didn't kill Haruuc, but someone wants it to look like I did.\"\n\n\"If you didn't,\" asked Geth, \"who did? And why?\"\n\n\"You believe me then?\"\n\n\"I\u2014\" Geth hesitated. He wanted to believe the goblin. What Chetiin had told them made at least as much sense to him as the idea that Chetiin could have turned on Haruuc. And yet the time he had spent since Haruuc's death cursing Chetiin's name wouldn't leave him so easily. Both versions of events were incredible.\n\nDagii spoke into the silence. \"I believe you,\" he said firmly. \"You have too much _muut_ , too much _atcha_ to have acted against Haruuc on your own. I know that you would have come to us first.\"\n\nGeth glanced at Ekhaas. The _duur'kala's_ ears were cocked. Slowly, she nodded. \"I think you're telling the truth,\" she said to Chetiin. She smiled. _\"Khaavolaar_ , I hated thinking of you as a traitor!\"\n\nGeth's gut felt a little hollow, as if he was a traitor now. He forced a smile to his face, though. \"I believe you, too. It's good to have you back.\"\n\nChetiin didn't smile. His dark-stained face remained pensive. \"I'm not back,\" he said. \"Too many people think I was the one who put Witness into Haruuc's eye.\" He shook his head. \"I've thought about it ever since I escaped, but I don't know the answer to your question, Geth. I don't know who actually killed Haruuc.\"\n\n\"A changeling in your shape?\" suggested Dagii.\n\n\"A changeling can't imitate a goblin,\" said Chetiin. \"We're too small. A changeling child might be able to, but no child could have done what I am told I did.\" The muscles of his jaw tightened. \"I can only come up with one plausible explanation. It was another of the _shaarat'khesh,\"_\n\n\"One of your own clan?\" Geth asked. \"They would do that?\"\n\n\"Not all of them. _Shaarat'khesh_ can refuse a request, but there are those who would have considered it a challenge.\"\n\n\"And they would have tried to kill you to do it?\"\n\n\"I have my rivals,\" Chetiin said grimly, \"though so far as I knew, none of them were in Rhukaan Draal at the time. But yes, they would have.\"\n\nDagii looked doubtful. \"Why use your identity to kill Haruuc? The honor of his death would fall on you.\"\n\nChetiin gave him a thin smile. \"You fight open battles, Dagii. Among the Silent Clans, the assassin would be twice-honored: once for killing Haruuc, once for concealing his true identity.\" The smile faded from his face. \"But I've been in touch with _shaarat'khesh_ I trust who remain in hiding in Rhukaan Draal. None of our clan has claimed Haruuc's death. Maybe no one will\u2014because of the trouble that it brought down on us, the clan is angry. I had to talk fast to get even old friends to believe my story. In any case, if Haruuc truly was killed by one of the _shaarat'khesh_ , the blame doesn't fall on the assassin, but on the one who hired him.\"\n\n\"Then we need to think about why Haruuc was killed,\" said Ekhaas. \"What did anyone have to gain from Haruuc's death?\"\n\n\"They could stop him from becoming a tyrant,\" Geth said. \"They could prevent a war\u2014they would have had their plans in place before he tried to turn the warlords away from Breland toward Valenar.\"\n\nDagii shook his head. \"Those are the reasons we've been chasing since Haruuc's death. Except that, as Tenquis pointed out, most Darguuls liked the way Haruuc was acting under the rod's influence. They would still welcome a war with Valenar. Or with anyone else, never mind the consequences.\" His ears flicked. \"Could a non-Darguul have hired one of the _shaarat'khesh?\"_\n\n\"It would be difficult,\" said Chetiin, \"but not impossible. If one of my clan wanted the honor badly enough\u2014\"\n\n\"We've forgotten something.\"\n\nThey all turned to look at Ekhaas. Her ears were flat against her skull and her eyes were narrow. She pointed at Geth. \"You said that after he killed Haruuc, the assassin looked at you and said 'We swore we would do what we had to.' That's what made us all think Chetiin believed Haruuc had discovered the power of the rod.\"\n\n\"And we were wrong,\" Geth said, but Chetiin's eyes opened wide and he drew a long hard breath. Dagii's lips peeled back from his teeth. A moment later, Geth understood what Ekhaas had seen as well.\n\nOnly the six of them who had recovered the rod and understood its terrible secret knew about the oath they had made. The idea left him cold. He looked out across the burned ruins to the eastern sky. Khaar Mbar'ost stood as a silhouette between them and the coming dawn. \"One of us hired the assassin? Who would have done that? Who _could_ have done that?\"\n\nDagii spoke through clenched teeth. \"Midian.\"\n\nGeth whirled to question this blunt accusation, but Ekhaas was already talking, building up evidence. \"He was the only one of us who wasn't there. You, Geth, and you, Dagii were with Haruuc on the dais. Ashi was just off of the dais in the side room. I was with Senen Dhakaan on the floor of the throne room. Chetiin was lying wounded. But Midian was conveniently out of Rhukaan Draal. He could have hired the assassin, told him what to say to make his disguise as Chetiin even more convincing, and left for the ruins at Bloodrun.\" She began to pace back and forth among the ashes as she thought. \"You knew that Haruuc hadn't really discovered the power of the rod, Geth. You told me and you told Ashi. If one of us had hired the assassin, we had time to stop the assassination.\"\n\n\"I didn't hire an assassin!\" Geth growled.\n\n\"I'm not saying you did. Or that Ashi did.\" She looked to Dagii. \"Or you.\"\n\n\"I didn't know about the danger of the rod until after Haruuc was dead,\" Dagii said stiffly.\n\n\"No, but I know you.\" Ekhaas's ears rose and flicked. \"When you returned with Keraal as your prisoner, you were as dirty as a farmhand and your hands were blistered because you had insisted on binding the Gan'duur warriors into the grieving trees along the road yourself. You took responsibility for their deaths. Someone with such _muut_ wouldn't hire an assassin to kill his lhesh.\"\n\nGeth thought he saw something pass between them, a meeting of amber and gray eyes, then Dagii lowered his head in acknowledgement and Ekhaas turned back to him and Chetiin.\n\n\"None of that proves Midian is the one behind it, only that he wasn't there,\" she said. \"Haruuc had enemies\u2014any one of them could have hired an assassin. But the false Chetiin knew the words of our pledge. Midian has to be our suspect.\"\n\n_Only if Chetiin is telling the truth_ , whispered a voice inside Geth. He swallowed it, sending it down into the cold feeling that swirled in his gut. He remembered how pale Midian had been when they had told him about the danger of the rod. If Geth had just realized he'd made a horrible mistake, surely he would have reacted in the same way. \"It still leaves us with the question of why he would do it,\" he said, \"and why he wouldn't have come to the rest of us before hiring an assassin.\"\n\n\"For any and all of the reasons we guessed Chetiin might have done it,\" said Ekhaas. \"We've got Midian. Do we confront him?\"\n\n\"No,\" Chetiin said quietly. The goblin elder rose from his crouch. \"Solve the problem of the rod first. When Haruuc's heir is lhesh with the false rod in his hand and the true rod has been dealt with, then we confront Midian.\"\n\n\"You think he'd betray us.\"\n\n\"I'm not certain what to think, but I know that accusing him of orchestrating the murder of Haruuc will not go well right now. We can't involve other people without revealing the secret of the rod. We need to deal with one problem before we move onto the next. You're close to getting the true rod away. Will rushing to confront Midian change anything?\"\n\nEkhaas flicked her ears. \"No. I don't think we should tell him any more than he needs to know though. We've already promised Tenquis we wouldn't reveal his identity. We should keep our meeting with you a secret, too.\"\n\n\"It's better if no one knows you've seen me,\" said Chetiin. \"Not even Ashi\u2014the more people who know, the more people who could give me away. There's someone out there who believes I'm dead. We should let them continue to believe it. If it's Midian, Ashi's trust will hide our suspicion. I'll listen for rumors of the _shaarat'khesh_ who performed the assassination. We may be able to learn something more.\" He nodded to the east. \"Dawn is almost here. You should return to Khaar Mbar'ost.\"\n\n\"What if we need to talk to you again?\" asked Dagii. \"How do we contact you?\"\n\n\"Hang something from your window in Khaar Mbar'ost. I'll come to you.\" He smiled. \"I'm pleased that we were able to speak. It's good to know I can trust you.\"\n\nSuspicion seethed in Geth's belly. He struggled to keep it from his face as Chetiin vaulted up onto a charred beam that slanted down into the ruins from a broken roofline. They lost sight of him among the shadows for a moment\u2014a moment that stretched out longer and longer until they all realized he had gone.\n\nTheir own return through Rhukaan Draal was quiet. There were a few more people moving on the streets now, though they were still able to pass without attracting notice. The chill sense of suspicion was still with Geth as they paused a short distance from Rhukaan Draal. \"I'll catch the attention of the guards at the gates,\" said Dagii. \"You two go in. There will be other people arriving at Khaar Mbar'ost now. Once we're inside, there's no way to avoid being seen at this time of day, but there also won't be anything unusual in us moving around.\"\n\n\"How do I get past the guards outside my chamber?\"\n\n\"Just walk in,\" Ekhaas said with a smile. \"It's your chamber. They'll think you're already inside and they'll be too surprised and embarrassed to say later that you weren't.\"\n\nGeth made a face. \"That sounds too easy.\"\n\nIt was. They turned a corner and came into sight of the gates of Khaar Mbar'ost. Instead of only a few people passing into a fortress still stirring in the early morning, the gates were a rush of messengers, warriors, and warlords.\n\n\"Rat,\" said Geth.\n\n_\"Khaavolaar,\"_ said Ekhaas.\n\n\"Something has happened,\" said Dagii grimly.\n\n\"What do we do?\" Geth looked at the two hobgoblins, who looked at each other. Ekhaas's ears bent forward.\n\n\"Keep going,\" she said. \"We may be able to slip through in the chaos. We'll find out what's going on once we're inside.\"\n\nThe guards at the gate, however, were alert. Even as they approached, trying to move causally in the wake of a warlord's entourage, one of the guards straightened and shouted, \"He's here! Send the message\u2014he's here!\"\n\nInstantly, Geth was the center of attention as guards came pouring out into the courtyard of the fortress. For a moment, he feared that they were there to arrest him, but then he realized that they were forming up as an honor guard. He shoved the bundle containing the Rod of Kings at Ekhaas. They couldn't be found with it. \"Get that away! Get it back to my chamber.\"\n\nShe nodded and melted away into the moving crowd. Dagii stayed by Geth's side. A moment later, Munta and Tariic appeared. _\"Maabet!\"_ cursed Munta. \"Where have you been? We've been looking for you.\"\n\n\"I wanted to see the celebrations that took place at night after the games ended,\" Geth said. \"I wanted to go without anyone knowing who I was.\"\n\nMunta blinked. \"Rhukaan Draal? At night? With only Dagii for protection?\"\n\n\"He's not entirely helpless, Munta,\" Dagii said. \"What's going on?\"\n\n\"You picked a bad night to go out.\" Tariic's voice was dark. \"A messenger falcon arrived in Khaar Mbar'ost during the third watch. It was carrying word from the village of Zarrthec. Villages and clanholds to the east have been attacked by raiders.\"\n\nDagii's ears rose sharply. \"Someone sympathetic to the Gan'duur?\"\n\nTariic shook his head. \"Valenar.\" He fixed his gaze on Geth. \"The war that you said wouldn't happen is here.\"\n\n#\n\n# CHAPTER  \nEIGHT\n\n# 21 Sypheros\n\nThe throne room had been pressed into use for the first time since Haruuc's death. Leaning forward from the viewing gallery above, Ashi could still see the stain of the lhesh's blood on the dais. A _dar_ tradition, like leaving the death wound visible. As long as the stain remained, people would remember that a great leader had died on that spot.\n\nThe white bulk of the Dhakaani grieving tree still stood to the side of the dais as well, blocking one of the tall windows behind the throne. The tree made a sinister presence, but the carved stone limbs were harmless now. The words that commanded them had died with Haruuc. No one had tried to dismantle and remove the tree, though. Maybe Haruuc had been the only one who knew that secret, too.\n\nNoise filled the rest of the throne room and the gallery. The assembly of warlords gathered in the throne room beneath walls lined with tall statues of fierce hobgoblin warriors and banners carrying clan crests. Exiled to the gallery were all those who had no place among the warlords and clan chiefs: ambassadors and envoys from the other nations of Khorvaire, the dragonmarked houses, and those clans like the Kech Volaar that dwelled in Darguun but had not sworn allegiance to Haruuc. Today the gallery was more crowded than the throne room. Everyone was in attendance. Word of the messenger falcon's arrival during the night, and of the news it carried, had spread quickly.\n\nGeth sat at the head of everything on Haruuc's blocky throne, the Rod of Kings in his hand. Ashi might have expected Dagii or Munta to stand with him, but they were on the floor among the other warlords. Instead, the four contending heirs stood around Geth\u2014Tariic and Garaad to his right, Aguus and Iizan to his left. The heirs took turns glaring at each other and nodding to supporters among the assembly. Geth just looked uncomfortable and exhausted. Ashi felt for him. All of his insistences that there would be no war had collapsed under him. She wondered if he had really believed what he was saying.\n\n\"I've heard that when the falcon arrived last night, they couldn't find Geth and when they finally did, he was returning from the city with Dagii at dawn.\" Esmyssa Entar ir'Korran, the ambassador of Zilargo, raised herself up a little to speak in Ashi's ear. Normally the gnome sat on a cushion that lifted her up on chairs made for larger beings. There was no room for a servant to get through the crowd with a cushion today. In fact, only Esmyssa's small size had enabled her to squeeze through and claim the seat. Pale blue eyes flashed. \"I wonder where he was.\"\n\nThe comment was so probing she might as well have asked the question outright. \"I don't know,\" said Ashi. It was no lie for a change, though she could guess at where Geth had been. As she had Vounn had made their way to their seats, Ashi had caught a glimpse of Ekhaas standing with Senen Dhakaan. The _duur'kala_ looked just as tired as Geth. Ashi wondered if that meant they had found an artificer to create a false rod.\n\nThat at least would be welcome news. A war was bad, but the danger of the Rod of Kings was potentially even greater.\n\nThe blunt rebuff brought a flicker of disappointment to Esmyssa's eyes, but it didn't silence her. She looked back to the dais. \"This business of a _shava_ and four heirs is messy. The sooner the Darguuls have a lhesh again, the better. It's so much easier to deal with one stable leader. With four possible lheshes, it's hard to know where you stand\u2014\"\n\n\"Isn't Zilargo ruled by three leaders?\" Ashi asked, trying to head off the inevitable question of who she thought might stand closest to the throne.\n\n\"The Triumvirate speaks with a single voice backed by the wisdom of three minds working in concert,\" Esmyssa said proudly. \"Haruuc's heirs struggle against each other. And the assembly of warlords...\" She made a dismissive, if quiet, noise. \"Without a strong, thoughtful lhesh to lead them, Darguun's clans are a danger as much to themselves as the rest of Khorvaire. If this was Zilargo, we wouldn't be facing the possibility of a war.\"\n\nThe Zil ambassador was right about Darguun's clans, of course, but her self-righteous arrogance was like a dull blade dragged across Ashi's skin. \"And what would Zilargo do if Valenar elves were raiding its borders?\"\n\n\"Negotiate,\" said Esmyssa. \"Emissaries of the Triumvirate would be sent to talk to them and find a solution before conflict arises. Words are the armies of Zilargo.\"\n\nAshi's teeth ground together. _\"Razh toch tao gi_ ,\" she said in the language she had spoken growing up among the barbarian clans of the Shadow Marches. \"A sword has no ears.\"\n\nEsmyssa's smile tightened as well. \"Pithy,\" she said, and turned away to speak to the Brelish ambassador. Ashi turned as well, righteous anger warm in her belly. Vounn was seated on her other side with Pater d'Orien beyond her and the two dragonmarked envoys were having a much more practical discussion.\n\n\"\u2014content of the message directly from Tariic,\" Vounn was saying. \"It only mentioned two clanholds by name: Tii'ator and Ketkeet. Both small and not too far from Zarrthec. The messages the falcons can carry are short by necessity.\"\n\nPater grunted and scratched under the collar that stretched tight around his thick neck. \"I know something of that part of the country,\" he said. \"Orien wagons make a market circuit there. Settlements east of Zarrthec are sparse. The land can be good and a fair amount of it was cleared by Cyre before the founding of Darguun, but go too far east and you get uncomfortably close to the Mournland.\"\n\nVounn pressed her lips together. Ashi thought she could guess what her mentor was thinking. What had remained of the human nation of Cyre after Darguun and Valenar had seized their pieces of it had been consumed in the massive catastrophe that had come to be known as the Mourning. The terrible event had happened only five years before, but already mention of it evoked a legendary dread in most people. All that was left of Cyre was a cursed wasteland inhabited by dangerous monsters and surrounded by borders of dead-gray mist. The Mournland was a blight on central Khorvaire, and by unfortunate coincidence Darguun shared the longest border of any nation with it, from the long inlet of Kraken Bay that became the mouth of the Ghaal River all the way to the spur of mountains marking the northern boundary with Breland. It also formed a deadly barrier hundred of miles wide between Valenar and Darguun. If Valenar elves were raiding in Darguun, they had either managed to cross the Mournland\u2014not entirely impossible for a people with a reputation for almost supernatural horsemanship\u2014or they had slipped unnoticed up the inlet of Kraken Bay to the mouth of the Ghaal.\n\nThe same thing must have occurred to Pater. \"There's a town\u2014Rheklor\u2014that stands on a peninsula with Kraken Bay on both sides. Haruuc placed a garrison there. They watch all ships traveling inland from the ocean. They would have seen anything unusual.\"\n\n\"Perhaps.\" Vounn glanced at Ashi, then dropped her voice so that only the three of them could hear. \"Have either of you seen Sindra this morning?\"\n\nAshi raised her head and looked around the gallery. From where she was sitting, there was no sign of the viceroy of House Lyrandar, but Sindra could easily have been lost in the crowd. \"No,\" she said, \"but\u2014\"\n\nPater's face had turned red. \"There are no Lyrandar ships at the Rhukaan Draal docks right now!\" he said. \"I noticed that yesterday.\"\n\nVounn raised an eyebrow. Pater turned a deeper shade of red.\n\n\"Lyrandar wouldn't aid Valenar against Darguun, would they?\" Ashi asked. \"They'll have to come back to Rhukaan Draal to do business.\"\n\n\"I think Sindra would try to claim her absence was just a coincidence,\" said Vounn. \"But it strikes me as a very fortunate coincidence when she might otherwise have to answer some awkward questions. Lyrandar ships travel up the Ghaal all the time without attracting attention. Ships for the Lhazaar Principalities are common, too\u2014but the Lhazaars haven't built ties with Valenar the way that House Lyrandar has.\"\n\n_\"Khyberit gentis,\"_ muttered Ashi. \"Do we tell\u2014?\"\n\n\"No,\" Vounn said with a quick shake of her head. \"We can't be seen to inform on another house. The Darguuls will figure it out on their own\u2014if they haven't already. They can investigate if they want to.\" She pursed her lips. \"But if Lyrandar is already involved in the conflict, then we should be too.\"\n\nThe color in Pater's face broke and Ashi realized he hadn't been holding in anger but a huge, greedy grin. \"By Kol Korran's golden bath,\" he said, his cheeks jiggling with the effort of keeping a straight face. \"I've missed war!\"\n\nAshi stared at them both, but before she could say anything, there was movement down below. A lean old hobgoblin spoke to Geth, then moved to a tall pole on the floor below the dais. Two young hobgoblin warriors stood beside the pole and at a nod from the older hobgoblin, they attached a black banner to ropes hanging from the pole's top and raised it. The warlords and clan chiefs fell silent and turned their attention to the dais. A second banner was raised, this one bearing the sword and crown symbol that had been Haruuc's crest. Geth stood.\n\n\"A message has been received by messenger falcon,\" he said in his heavily accented Goblin. \"This is that message.\" He produced a piece of paper too large and stiff to have been carried by one of the hobgoblin-trained falcons. Someone must have translated the original Goblin runes for him and coached him in reading it. \"To Khaar Mbar'ost,\" he read. \"A runner from Ketkeet clanhold has arrived in Zarrthec. Valenar raiders have struck at Tii'ator, are advancing on Ketkeet, and are believed to have struck at more locations. Survivors of Tii'ator seeking refuge at Ketkeet report seeing smoke in the direction of other clan- and farmholds. Other runners and falcons sent by Tii'ator and Ketkeet have not arrived. I believe they have been brought down by Valenar. Zarrthec stands to defend itself.\"\n\nGeth looked up. \"It is signed by the _chib_ of Zarrthec and dated the evening of 20 Sypheros. Yesterday.\"\n\nWords of anger and frustration swelled from the warlords, but the old hobgoblin by the pole rapped a staff against the floor. \"Respect the order of assembly! The _shava_ of Haruuc continues.\"\n\nWhen silence had returned, Geth looked back at his paper. \"Two other falcons have since arrived. One comes from Baar Kai clanhold, along the border of the Mournland south of the ruins of Lyrenton. The message it carried reads only, 'Baar Kai falls. Elves burn our fields and kill all who stand against them.'\" He hesitated, then said, \"The third message was written in Elven.\"\n\nThe room erupted in outrage. Ashi saw those ambassadors and envoys with elf blood\u2014the half-elf viceroy of House Medani, the entertainer who served as the spokesperson for House Thurani in Rhukaan Draal, an aide to the Aundairian ambassador\u2014sitting in the gallery flinch at the anger below. The old hobgoblin slammed his staff down and called for order repeatedly. Geth shouted for calm in Haruuc's name. There was no response until Tariic's voice rang over the chaos. \"Darguuls! We give victory to our enemy if there is not order!\"\n\nAnd the warlords listened to him. Many took their seats again, dragging more boisterous neighbors with them. \"He sounds like his uncle,\" murmured Pater.\n\n\"The warlords who responded are all in his camp,\" said Ashi. \"Daavn was one of the first to sit.\"\n\nVounn glanced at her and nodded at the observation. They weren't the only ones to notice. Garaad, Aguus, and Iizan glared at Tariic, but the new lord of the Rhukaan Taash had already nodded to Geth. The shifter drew a breath and read from his paper.\n\n\"The Valaes Tairn\"\u2014Ashi recognized the Valenar elves' name for themselves\u2014\"manswer the challenge of Lhesh Haruuc Shaarat'kor. If the blades of Darguun would fall on Valenar, Darguuls must face the warband of Kaelan Vaerian! Kaelan Vaerian will defeat all who come against her!\"\n\nSomeone below laughed loudly. \"A warband? A single warband? They must be junior warriors out to make honor for themselves! They can't be much of a threat.\"\n\n\"Be quiet, you fool!\" Tariic's voice cracked like the thunder of a nearby lightning strike. He stood forward again, glaring down among the assembled warlords. \"This isn't just one warband. The territory of the Baar Kai is too far from Ketkeet or Tii'ator. There are at least four or five warbands working together, probably more. We know they're a threat because they've destroyed at least three clanholds\u2014and there has been no further word from Zarrthec.\" He turned, snatched the paper out of Geth's hand, and shook it in the air. \"This is no missive from a true war leader. This is a boast sent out to the enemy by an inexperienced warrior drunk on a fleeting victory. If a junior warrior under my command did such a thing, I'd have him whipped.\"\n\nThe old hobgoblin's staff tapped the floor. \"Tariic of Rhukaan Taash, respect the order of assembly.\"\n\nTariic nodded. \"Arbiter, there is more I have to say, but I will wait for recognition.\" He turned to look at Geth. Before the shifter could open his mouth, though, Aguus had pushed forward.\n\n\"Tariic has already spoken. I will speak now!\"\n\nGaraad and Iizan weren't far behind with their own protests and demands. Geth turned a bewildered gaze to the arbiter, then to Munta and Dagii. Munta pointed at one of the heirs. Ashi couldn't be sure which one. Neither could Geth, it seemed, because Munta grimaced when he called out \"Tariic of Rhukaan Taash will speak.\"\n\nThe banner of the lhesh came down from the pole and a banner bearing the crest of the razor crown rose in its place. Tariic nodded to Geth. The other heirs backed down\u2014reluctantly, Ashi thought. Geth returned to the throne as Tariic moved forward. He stood straight and proud as he addressed the assembly.\n\n\"Chiefs, warlords, elder warlords\u2014warriors, heed me. Lhesh Haruuc was struck down at the height of his glory. We all know the last words he spoke. 'Ancient blood demands an ancient enemy. As it was in the age of Dhakaan, the People shall go into battle against elves. Let our blades fall on Valenar.' Haruuc's murder robbed us of a great leader, one who deserves all of the honor that we give to him.\" His voice dropped and he lowered his eyes. \"But perhaps in honoring him with mourning and games, we have denied him the greatest honor of all. We have failed to follow his final words.\"\n\nHe paused for a moment and the hall was silent between his words. \"'Ancient blood demands an ancient enemy.' We have let our recent history cover our glorious past. Haruuc signed the Treaty of Thronehold that brought an end to the Last War and recognition as a sovereign nation to Darguun\u2014and yes, to Valenar. Yet fear of this document, fear of war, has held us back. Some have even claimed there could be no war.\"\n\nAshi saw Geth sit up, anger crossing his face. Tariic ignored him and continued.\n\n\"Our hesitation has cost us. Valenar has struck the first blow. They have seen what Haruuc surely saw: that the Treaty of Throne-hold is a document drawn up by _chaat'oor_ , defilers, strangers to the shores of Khorvaire.\"\n\nHis voice rose again, filling the throne room. \"'As it was in the age of Dhakaan'\u2014our People ruled this continent before humans even dreamed of its existence. And in those days, only one enemy met us as an equal on the field of battle: the Tairnadal, honored ancestors of the Valaes Tairn. Those who wrote the Treaty of Thronehold are lovers of peace. They see war as unnatural. A temporary condition. There is a reason we do not number elves among the _chaat'oor_. Like us, they know that war is eternal, that struggle, not peace, is the true way of the world. Since the age of the Empire of Dhakaan, they have been an honored enemy.\"\n\n\"Their attack reminds us of who we are, of who we were meant to be. Now is the time to throw off illusions of peace. Now is the time to meet their attack. Now is the time to remember and honor Haruuc. The arch-traitor Chetiin silenced his voice, but he could not silence his spirit for it is the spirit of the People.\" Tariic drew his sword and thrust it into the air. \"Let our blades fall on Valenar!\"\n\nThere was not even a moment's silence before a roar of approval rose from the warlords. Light reflecting from drawn blades flashed around the throne room and the thumping of fists on chests in the _dar_ salute was like the sound of drums. The arbiter didn't try to restore order, but simply stood and offered applause in the form of an open hand slapped against his chest. Aguus, Garaad, and Iizan looked sour, but they roared and cheered along with the other warlords\u2014Ashi knew there was nothing else they could do now. Anything they said would sound like hollow imitation.\n\nEven those _dar_ in the gallery were applauding. Ashi saw Senen beating a hand against her chest, eyes wide and ears high in admiration. At her side, Ekhaas applauded as well, though when her eyes met Ashi's there was worry in them.\n\nAmong the ambassadors and envoys\u2014the _chaat'oor_ \u2014in the gallery, there was only mechanical applause, if that. Even Pater who had moments before greeted the prospect of war with greed looked stunned. \"Lords of the Host, I've never seen Tariic like this before. He always talked about how Haruuc brought Darguun into the world but that he would bring the world into Darguun.\"\n\n\"He wants the throne,\" said Vounn. \"Haruuc told me once that Tariic valued _atcha_ over _muut_. He's willing to play any game of politics to reach it.\"\n\n\"Including starting a war?\" asked Ashi.\n\nVounn sat back in her chair. \"He didn't start it, Ashi. He's only using it\u2014just as all of us will.\"\n\nAshi looked back down into the throne room. Geth had stood up, an expression that mixed fury and confusion on his face. His hand was tight on the Rod of Kings and Ashi suspected that if it had been possible for him to use its power, he might have done so at that moment. Tariic spoke a few quiet words to him, but the shifter just jerked away. The warlords were growing quiet again, and the arbiter rapped his staff on the floor. Tariic turned away from Geth to give the old hobgoblin a nod. The banner of the Rhukaan Taash descended on the pole of order.\n\nBut another shout rose from the crowd. Ashi recognized Daavn of Marhaan's voice. \"Who will lead us?\"\n\nThe descending banner paused. Tariic turned back to the warlords.\n\nThis time Garaad was the first heir to protest. \"No!\" The lean warlord still had his sword drawn. The blade rose to point at Tariic. \"You will not take this honor! There are two days of games left. Haruuc's successor will only be chosen then. You are not lhesh yet!\"\n\nThe words sounded hollow and desperate even to Ashi, but Tariic only bent his head to Garaad. \"It is as you say,\" he said. \"I wouldn't dare to act as lhesh while we still honor Haruuc. But we can't sit on our swords while Valenar raiders sweep Darguun. We must act now, don't you agree?\"\n\nHe looked at each of the other contending heirs in return and once again Ashi knew that there was nothing they could say. Garaad's ears went back flat against his skull. \"Yes,\" he said, lowering his sword. \"But it will not be you leading the battle.\"\n\n\"Not any of us. We've been too busy fighting each other. But there is one who has recently fought on behalf of Darguun.\" Tariic looked down among the assembled warlords. \"Dagii of Mur Talaan, named _lhevk'rhu_ by Haruuc, will you recall the army that defeated the Gan'duur?\"\n\nThe warlords around Dagii seemed to draw back a little bit, leaving him to stare back at Tariic like a sentry caught off guard. Then his face hardened and he stood straight. _\"Mazo,\"_ he said.\n\nTariic looked back at the other heirs and at Geth. \"Dagii's troops will be a vanguard, moving swiftly to meet the threat presented by the Valenar. His loyalty to Darguun cannot be questioned. Does this satisfy you all?\"\n\nAshi could see the heirs' ears flick as they thought through the merits of the suggestion. One by one, they nodded. Tariic glanced at Geth. \"And you, _shava_ of Haruuc? You hold the throne in trust. Do you grant your approval?\"\n\nGeth's expression was hard as he looked down at Dagii. The young warlord gave him back a slight nod. Geth looked up at Tariic and bared his teeth. \"How can I refuse?\"\n\nTariic met his bared teeth with a fierce smile, but didn't bother to answer the question. There was, Ashi knew, no need to. Instead, he gestured for Iizan, Aguus, and Garaad to stand forward with him. \"It is decided!\" he said. \"The reign of a new lhesh will begin with glory and war on Valenar as Haruuc wanted. The assembly of warlords is ended!\"\n\nThe gathered warlords cheered again, some closing on Dagii to congratulate him, but most saluting the four heirs gathered on the dais. Ashi wondered if she was the only one to notice the old arbiter look to Geth for confirmation of Tariic's words and to see the shifter nod like a weary fighter defeated in battle.\n\nAshi turned away and founded herself facing Esmyssa Entar ir'Korran once more. The Zil ambassador's face pinched up into a smile. \"Better to have Darguun and Valenar at each other's throats than ours, at least,\" she said. \"If they want to break the Treaty of Thronehold, I don't imagine the other nations of Khorvaire will rush to aid either of them.\"\n\nAshi couldn't hold back a snort. \"And that's what Haruuc really wanted.\"\n\nEsmyssa looked puzzled.\n\nAshi shook her head. \"Never mind,\" she said and stood up. \"Please excuse me.\"\n\nShe managed to spot Ekhaas before the hobgoblin left the gallery. Ekhaas and Senen Dhakaan were standing in a corner, arguing with quiet words. By the time Ashi reached the spot, the two _duur'kala_ had separated and Ekhaas was on her own, glaring at Senen's retreating back.\n\nAshi caught her arm. \"What do we do now?\"\n\nEkhaas's amber eyes flickered to her, then went back to Senen. \"Carry on as planned, I think,\" she said, her voice hard.\n\n\"You think?\" Ashi tried to keep her own voice both level and quiet. \"Ekhaas, we're losing Dagii. He's going to be...\" She stopped, following Ekhaas's gaze after Senen. \"What's wrong?\"\n\n\"Senen wants me to go with Dagii,\" said Ekhaas. Her ears were back against her head. \"She wants a representative of the Kech Volaar along. She wants a _duur'kala_ to witness the great clash of hobgoblins and elves.\"\n\nAshi's hand fell away from Ekhaas's arm. \"No. She can't.\"\n\n\"Unfortunately, she can,\" said Midian.\n\nAshi twisted around to find the gnome leaning against a chair.\n\nMidian thrust out his jaw, tipped back his head, and said in an arrogant tone, _\"You serve the Kech Volaar and the memory of Dhakaan, daughter of the dirge. You will do as I say!\"_\n\nIt was a flawless imitation of Senen Dhakaan. \"How long have you been listening?\" Ekhaas asked.\n\n\"Long enough. Ashi's right. This is bad timing.\" He raised an eyebrow \"But you look as tired as Geth and Dagii this morning and I don't think they were just out seeing the city last night. Did you find\u2014?\"\n\nEkhaas flicked a finger to silence him, glanced around, then jerked her head toward the stairs leading down from the gallery. The antechamber outside the throne room was hardly less crowded when they reached it, but the envoys, ambassadors, and their various assistants who frequented the galleries were always listening for bits of information. The warlords who had flooded into the antechamber were thinking of nothing but war. Ekhaas found a momentarily quiet corner, beckoned them into it, and whispered, \"We found an artificer who can make what we need\u2014in five days.\"\n\nMidian wrinkled his nose. \"Three days after the end of the games.\"\n\n\"Two if we count today.\"\n\n\"Even if you do, Tariic won't want to wait that long for his coronation.\" The gnome folded his arms around himself. \"He'll want to take the throne immediately.\"\n\n\"You think it will be him?\" Ashi asked.\n\n\"Do humans hit their heads in gnome houses? Even if it's not, it doesn't matter. We need to find a way to delay the coronation.\" Midian tapped his fingers against the wall. \"I'll talk to Razu. Maybe I can put a knot in her plans or persuade her that the ceremony needs to be more elaborate.\"\n\n\"Anything to buy time.\" Ekhaas's ears stood tall suddenly and her already soft voice dropped even more. \"Ashi, Vounn is here.\"\n\nAshi looked over her shoulder. Vounn stood a discreet distance away with Aruget and another hobgoblin guard, Krakuul, just behind her. As Ashi met her gaze, she raised one eyebrow. \"I'm returning to our chambers,\" she said. \"Would you like to come with me?\"\n\nHer tone said clearly that it was neither a question nor an invitation. Only a month ago, Ashi would have rebelled against the command, but the new understanding that had grown between her and Vounn was stronger than that. She looked briefly at Ekhaas. \"I'll find you tonight,\" she said, then went to fall in beside Vounn.\n\nHer mentor said nothing as they swept out of the antechamber and nothing as they climbed the stairs up through Khaar Mbar'ost. Not until they had reached the chambers Haruuc himself had assigned to them and were inside with Aruget and Krakuul standing outside the heavy door did she speak\u2014and when she did, it was without turning to look at Ashi.\n\n\"I think,\" she said, \"it's time you told me what's going on.\"\n\nAshi glanced sharply at the gray-haired lady seneschal. Vounn ignored her, instead walking to look out of a window that presented a wide view of the city below. \"I know there's more going on than you're telling me,\" she said. \"You know that I know. I am your mentor and your superior in the house. If there's something happening that could put either of us or the operations of Deneith in danger, I should know about it.\"\n\nAshi tried to think of what to say. Perhaps mistaking her silence for reluctance, Vounn continued. \"Just before Haruuc's assassination, Geth summoned you away with no words other than, 'Haruuc needs your dragonmark.' Whatever he needed it for, I presume you were too late. I have not asked for an explanation.\"\n\nShe finally looked at Ashi. \"The night Haruuc died, I was prepared to send you out of Darguun with the next Orien caravan. Circumstances prevented it. War has arrived\u2014between Darguun and Valenar perhaps, but still war. When I report what happened here today to the head of the house, I expect that he will demand that you, as the bearer of the Siberys Mark of Sentinel be moved out of Darguun for your own safety. Give me a reason to keep you here.\"\n\n\"I... I can't.\" Ashi clenched her teeth and lowered her voice. There was no lie she could think of that would cover everything, not with the looming possibility of being sent away from Rhukaan Draal just when the others needed her most. \"Vounn, something is going on, but I can't tell you what. I've made an oath. All I can say is that it doesn't present a danger to you or Deneith. You have to believe that. It may even save\u2014\"\n\nVounn held up a finger, silencing her. \"If it's so important you've taken an oath to keep it secret, don't say anything more.\" The older woman studied her. \"Whatever is going on doesn't present a danger to me or Deneith. What about you? Does it present a danger to you?\"\n\n\"Yes,\" Ashi said bluntly. \"But it's a danger I'm willing to face.\" She raised her chin. \"It's a danger I have to face.\"\n\nVounn's eyebrow twitched. \"Will it advance House Deneith when it becomes known?\"\n\nAshi felt an ache like a punch in the gut. \"If it's successful, it can never become known. No one will ever know about it.\" She drew a breath and stepped closer. \"Vounn, you know what I'm capable of. I can look after myself. I've proved that.\"\n\n\"I know,\" Vounn said. \"That's why I'm not sending you back. Baron Breven d'Deneith can rant all he likes. You're in my charge.\" She reached out and took Ashi's hands. \"But promise me that if you ever need my help, then you will come to me. I can keep secrets, you know.\"\n\nRelief put a smile on Ashi's face. \"I know.\"\n\n\"Good.\" Vounn held her hands a moment longer. \"I want you to do one thing though. Whenever you can, take Aruget with you. Two swords are better than one, and I know that he can keep a secret, too.\"\n\n\"I will, Vounn.\" Ashi fought to keep her smile from getting wider. She didn't think her mentor would appreciate knowing that her guard already knew more than she did.\n\n#\n\n# CHAPTER  \nNINE\n\n# 21 Sypheros\n\nMoving through Rhukaan Draal with Pradoor on his shoulders gave Makka a feeling like nothing he had felt before. On the one hand, he felt very small and very humbled. The wizened goblin had saved his life. Now he was her beast of burden, guided by her taps, and sometimes her punches, on the side of his head. He strode along streets not knowing when Pradoor would catch some clue\u2014a familiar odor or sound, he still wasn't sure how the blind woman found her way\u2014that would prompt her to pull on his ear and command him, \"Turn!\" At times it was almost as if he had a god or some divine spirit on his shoulders directing the order of his existence, a sensation intensified by Pradoor's spontaneous recitation of stories and lore of the Dark Six.\n\nOn the other hand, he felt as large as if he had blundered into the middle of a _duur'kala'_ s tale. At times, he didn't carry a divine spirit\u2014he _was_ a divine spirit. Everywhere that Pradoor directed him, people stepped aside. That, Makka was used to, but what he wasn't used to was the respect with which they looked at the pair of them. People nodded to them and lowered their eyes. _Dar_ who wore the _muu'kron_ touched it as they passed. They knew Pradoor by name and while they might not have known Makka, they included him in their greetings not with a casual _saa_ but a formal _saa'atcha_.\n\nBut it wasn't just goblins and hobgoblins and bugbears who moved out of their way. In the rush of his first day in Rhukaan Draal, Makka hadn't understood how many different races inhabited the city. Nor had he appreciated the range of those who made prayers to the Six. A stout dwarf merchant dressed in fine silk bent his head to Pradoor. A shifter with feral eyes and matted hair simpered and whined like a dog. A rangy gnoll, as tall as Makka and with a head like a hyena, bowed low and forced the coffle of slaves she led to their knees as well. Even a fey eladrin, wrapped in a silvery cloak that the ilth of the streets did not seem to touch, lowered pearly eyes as they passed.\n\nAnd there were offerings! From the moment that he and Pradoor had left the alley where he had come close to death, people had thrust gifts upon them. Sometimes they requested blessings\u2014\n\n\"A new sword, Pradoor!\"\n\n\"My children, Pradoor!\"\n\n\"I fight in the games tomorrow, Pradoor!\"\n\n\u2014but often they were given up with only a word of praise for one or all of the Six. Meat, bread, wine, beer, a fine knife, coins of copper and silver, so many offerings that Pradoor directed Makka to take a sack from a shop. The merchant bowed and scraped as if the theft was an honor.\n\nMuch of what they were given was handed out again to beggars, but Makka's belly was full and the sack was never empty. The knife found a place at his belt. When Pradoor was tired, it seemed like all they had to do was turn a corner and they were met with an offer of a place to rest.\n\n\"Pradoor,\" Makka said as his third day in Rhukaan Draal drew toward dusk, \"are there other priests of the Six in Rhukaan Draal?\"\n\nPradoor laughed her shrill cackle. \"Ah, it speaks! I wondered if the Keeper had kept hold of your voice when I snatched you back from him.\" She tugged on one of Makka's ears. \"Yes, there are other priests. Haruuc spread the faith of the Sovereign Host, but the old faith of the people never went away. Priests of the Six are like mice in Rhukaan Draal\u2014and most hide like mice too! They scurry about in shrines and temples, afraid of Haruuc's pet cats. They'll grow bold sometimes, but only Pradoor has the faith to walk the streets!\"\n\n\"You don't walk, I carry you,\" said Makka.\n\nPradoor pulled his ear hard enough to make him wince. \"You serve as I serve!\" she said. \"I keep the old ways alive. The faithful may seek out shrines from time to time, but they see me and they remember the hold of the Six on their lives. When the city starved, I led the famine march. When the Night of Long Shadows falls, I tell the stories that make the faithful roar and nonbelievers tremble. I feel the mood of the streets and the people.\" Her voice sank into a harsh croak. \"The age turns. Rhukaan Draal is the axle and I am the pin.\"\n\n\"Do you mean the war with the elves?\" Makka asked.\n\nRumors had spread through the streets all day, growing wilder and wilder with each telling. Raiders had destroyed clanholds. Fires had consumed eastern Darguun, the smoke blotting out the sun at dawn. Valenar cavalry had crossed the Mournland and were riding on Rhukaan Draal. Darguun would follow Dhakaan into the dust of ages. All of the elves of Eberron had risen to war, determined to exterminate the _dar_ \u2014unless _dar_ marched to destroy them first, which they undoubtedly would because Haruuc himself had returned from the dead to reclaim the Rod of Kings and lead Darguun to victory!\n\nMakka believed less than a quarter of what he heard. Something was happening, there was no doubting it, but it could have been anything from a pitched battle to a mere skirmish. Still, he had seen hobgoblins and bugbears with the look of seasoned warriors checking armor, sharpening weapons, and glaring murder at any elves they saw. War it was, then.\n\nPradoor's ears twitched. \"The war is a part of it as I am a part of it and you are a part of it,\" she said. \"The Six give straw to some, clay or steel to others. What we are given makes no difference\u2014we are judged by what we make of it.\"\n\nSudden certainty uncoiled in Makka's mind. \"I am given steel,\" he said.\n\n\"Yes,\" said Pradoor. \"You are a warrior called to serve.\"\n\nMakka twisted his head so that he could see Pradoor out of the corner of his eye. \"What were you?\" he asked. \"What are you given?\"\n\nPradoor laughed again, her cackle rising above the noise of the street. \"No one who has served me has ever dared to ask that question!\"\n\n\"Are you going to answer it?\"\n\nBlind eyes turned to a red sky and the setting sun. \"I was a midwife,\" said Pradoor. \"I am given souls.\" Then she pulled back her hand and smacked the back of his head. \"Now turn here!\" she commanded. \"And hurry. We are expected.\"\n\nMakka turned and strode along another narrow crooked street. A hobgoblin working the edge of a well-used sword with a whetstone glanced up and gave him a nod. Makka returned it.\n\nFull dark had fallen and the streets had come to life when the tight-packed buildings fell away. A crowd stood in the space beyond. Makka instinctively held back to assess what lay ahead. Pradoor smacked his skull. \"Keep going.\"\n\nHe stepped out from the shadow of the buildings and into an unpaved square over which arched the first trees he'd seen since before he entered Rhukaan Draal. They were twisted, spindly things, much like the _guul'dar_ who lived in the city, with smooth trunks he could have circled with his hands and thin canopies that barely iltered the moonlight. Torches\u2014real burning torches and not harsh magical imitations\u2014had been hammered into the ground around them and wedged into their lower branches. Figures roamed the square in small groups, talking quietly and casting shadows against the smoky flames. There was something at the heart of the square among the trees, something that looked like a dark jumble, though he could make out no more against the torchlight and the shifting shadows.\n\n\"What is this place?\" he asked Pradoor.\n\n\"Somewhere older than Rhukaan Draal, a place that was here before the city and that the city surrounded but could not fully consume. People come here when they are uncertain or when they're afraid. I always find them here.\" She tapped his head again, gently this time, and he continued on toward the dark jumble among the trees.\n\nHe had taken only a few paces before some of those in the square noticed him\u2014or rather, noticed Pradoor. A pair of hobgoblins talking with their heads together looked up. Their ears rose, then they bent their heads and murmured, \"Pradoor.\"\n\nTheir voices drew the attention of others, who bent their heads and spoke Pradoor's name in turn. The respect that the wizened goblin woman had received on the streets of Rhukaan Draal had left Makka amazed. The respect she received as they passed through the square came close to adoration. The bending heads were like grass in a windy field; the chorus of her name was like the whispering of a breeze. \"Pradoor.\" \"Pradoor.\" \"Pradoor.\"\n\nThey moved under the branches of the trees, and for a moment they were alone. The dark jumble resolved itself into a pile of weathered, lichen-covered rocks. A gaping hole among them plunged into the ground, and Makka thought he could hear the rush of water. The rocks were an ancient well, he realized, and the water below some hidden branch of the Ghaal River\u2014and yet there was something more here, as if a vast and unseen presence had focused its attention on this spot.\n\nHe knew the feeling. The cursed valley that had lain below the camp of the White Stone tribe, the valley that Dagii of Mur Talaan, Ekhaas of Kech Volaar, and the rest of their party had disturbed, had felt like this. No bugbear of the tribe had ever gone further than the edge of the ancient trees that covered the valley floor, but all of them had gone at least that far, if only so they understood why the valley should be left alone and the trolls that lived there kept sated.\n\nBut Pradoor seemed to have no fear of the strange presence. Her fingers on his head urged him forward until he stood beside the rocks and above the hole. \"Turn,\" she said in his ear, and he did.\n\nThose who wandered the square had moved in among the trees, crowding in on all sides. Torchlight illuminated faces even more diverse in race and rank\u2014though Makka could see no elves among them\u2014than those that had greeted Pradoor on the streets, as if the cover of night had drawn to the Six those who by day professed only a faith in the Sovereign Host.\n\nThey were silent, then Pradoor spoke. \"What troubles you?\"\n\nThose gathered stayed quiet, glancing nervously at each other, until a bugbear found the nerve to speak. \"War comes,\" he said.\n\nA goblin called, \"The Valenar are riding against us!\"\n\nAnd suddenly all those who stood among the trees seemed to find their voices at once. Makka heard all of the wild rumors he had heard during the day and more beside. He heard fear for safety in Rhukaan Draal and fear for the safety of sons and husbands called to ight. He turned his head to glance at Pradoor and found her listening to everything with cupped ears. She let the babble run, then raised her hands. The gathering fell silent again.\n\n\"Your fear,\" she said, \"shames you. Your fear shames me. You're afraid of war? Why?\"\n\nNo answer came. Pradoor gave a mirthless laugh. \"You dread even giving your fear a name when you of all the peoples of Eberron have the least to fear. The Dark Six smile on Darguun. You who will ight\u2014do you fear that the Fury will not give you strength to smite your enemies?\"\n\nThis time a chorus came back. \"No.\"\n\n\"Do you fear that the Mockery will not give you the skill to make your enemies suffer?\"\n\n\"No!\" Louder and stronger than before.\n\n\"Do you fear that the Shadow will not give power to the spell-casters who march at your side?\"\n\n_\"No!_ \"\n\nPradoor raised her shrill voice to match the volume of the crowd. \"You who will remain\u2014do you fear that the Devourer will not protect the supplies stored against attack?\"\n\n\"No!\"\n\n\"Do you fear that the Traveler will lead your enemy past those who defend you?\"\n\n_\"No!\"_\n\nHer voice rose so loud that it seemed impossible it should come out of her small, trembling body. \"Do any of you fear that the Keeper will break his pact with those who have faith, that if you fall your souls will wither like forgotten fruit?\"\n\n_\"No!\"_\n\n_\"Then why do you fear war?\"_\n\nMakka felt his heart stir to Pradoor's words as the hearts of her audience did. For all of his life he had feared and venerated the gods of the Six. They were the primal forces of the worlds\u2014hunger and passion and pain and death and power and change. But as Pradoor spoke, he found fear and veneration coming together with the sense of service that had hung over him since the goblin woman had healed his wounds. What did a true servant of the Six have to fear from war\u2014or from anything in life?\n\nPradoor let her words hang among the trees for a moment before she continued. \"You who are of the People understand _muut_ and _atcha_ , duty and honor\u2014you who are not should learn. There is _muut_ in serving the Six, just as there is _muut_ in serving a warlord. There is _muut_ in faith. But duty is like the two halves of a mill stone. There is _muut_ in serving a warlord, but for a warlord there is _muut_ in protecting those who serve him. So it is with the Six, who protect those who serve them and keep their faith!\"\n\nShe spoke the words with a power and confidence far larger than her frail old body. The words lifted Makka up\u2014lifted him beyond those others who stood before the old goblin woman. He could see the wonder in their expressions, but he knew in his gut that none of them could feel the way he did. Some were crying. Some had dropped to their knees. His spirit struggled within him as if it was ready to burst free.\n\n\"The Six protect those who serve them and keep their faith,\" repeated Pradoor, her tone softening slightly. \"Don't fear war. If you must fear something, fear defeat, because it will mean you have failed the Six\u2014and Darguun. This is a time to celebrate. You have the chance to prove yourselves.\" Her voice rose again. \"Are you strong?\"\n\n\"Yes!\" answered the crowd in one unified voice.\n\n\"Are you fierce?\"\n\n\"Yes!\"\n\n\"Are you faithful?\" \"Yes!\"\n\nPradoor raised her hand in blessing. \"Then return to your homes and prepare for what the Six bring you, but leave your fear behind. It will only drag you down. Now go from this place in awe of the Six who rule our lives!\"\n\nThe crowd broke apart, its unified voice splintering into babble. Most of those who had gathered melted away into the shadows and torchlight, renewed confidence showing in the way they held themselves and in the tone of their voices. A few came forward to kneel before Pradoor and Makka. Pradoor tapped him on the top of his head, and he lowered her to the ground. It was strange to see hobgoblins and other bugbears humbling themselves before the blind old woman, but it almost seemed that, in that moment, she was bigger than they were. Drawing a worn _muu'kron_ from her belt, she pressed it against the foreheads of those who knelt and murmured a blessing to each of them.\n\nWith each blessing, Makka felt his own belly grow tighter. His head seemed to throb. When the last kneeling igure had risen and left the shelter of the trees, he spoke, his voice cracking in his throat. \"Pradoor\u2014\"\n\nShe stopped him with a raised hand. \"There are others who want to speak.\" She turned her blind eyes out to the shadows among the trees. \"You. Come forward now.\"\n\nTwo hobgoblins detached themselves from the shadows. One walked barefaced. The other had a scarf wound around his neck, and he tucked his face down into it. In a crowd he might not have drawn attention, but alone he seemed familiar. As he drew closer, Makka recognized him. He had walked beside the shifter Geth in Haruuc's funeral procession, and Makka suspected that made him important. He tried to push back his own urgent desire to speak with Pradoor and studied the intruders more closely. The one who had walked beside Geth carried himself with confidence in spite of his attempt at disguise. His eyes were bright and a shade of dark brown so intense they were almost red. He looked back at Makka, raising his face from the concealment of his scarf to study him and Pradoor just as he was studied. A sense of wariness seeped into Makka.\n\nThe unknown hobgoblin, however, had the look of a schemer. His eyes went to Pradoor and stayed there. He tried to walk tall, but his shoulders hunched as he drew close. \"Pradoor,\" he said, \"I am Liirt\u2014\"\n\nPradoor interrupted him with absolute confidence. \"No, you're not. Speak the truth to me or do not speak to me at all.\"\n\nThe schemer looked at his companion, who nodded. The schemer looked back to Pradoor and his hunched shoulder became a little bow\u2014useless, Makka thought, before someone who was blind. \"You are as perceptive as you are eloquent, Pradoor. I am Daavn of Marhaan. My companion is Tariic of Rhukaan Taash.\"\n\nPradoor smiled at the second name. _\"Saa'atcha_ , lhesh of Darguun.\"\n\nThe second hobgoblin smiled slightly in return, an honest reflex and not some vain attempt to ingratiate himself. _\"Saa'atcha_ , Pradoor, but I am not lhesh. Yet.\"\n\n\"You know you will be though\u2014or at least you believe you could be.\" Pradoor raised her face in the direction of Tariic's voice. \"You remember me now, don't you?\"\n\nTariic answered without hesitation. \"The dungeons of Khaar Mbar'ost. You were spared from the games.\"\n\nShe cackled. \"Your agreement with the decision was grudging at the time. I know it. You don't begrudge me my freedom now, though.\"\n\n\"No,\" said Tariic bluntly. Pradoor turned to Daavn.\n\n\"Why do you come to me tonight?\" asked Pradoor. \"Surely those who would rule Darguun have no fear of war.\"\n\n\"We don't,\" Daavn told her, \"but I knew there would be those who did\u2014that's why I brought Tariic to see you. Your words are inspiring.\"\n\nPradoor snorted and Daavn's ears flicked. Makka saw his eyes dart to Pradoor as if he thought she might be mocking him. The hobgoblin continued with more care and less flattery. \"The assembly of warlords will give their support to Haruuc's heir in two days' time. The voice of the people can sway their decision. For three days, our rivals have been attempting to buy the people with food and drink\u2014\"\n\n\"As Tariic buys them with contests in the arena,\" Pradoor interrupted.\n\nDaavn squirmed but carried on. \"The announcement of war with Valenar gives us a new opportunity. The assembly supports the war but we need a way to reach the people. If they embrace the war, they will embrace the man who called for it. I knew of your popularity in the streets. I knew that if people had fears, they would come to you.\"\n\n\"And you came to watch me perform.\" Pradoor's voice was dry. \"You want me to lead the people to you so that they forget the comforts of food and drink and see only the glory of war.\"\n\n\"You already do it,\" said Daavn. \"Your talk of _muut_ between the people and the lhesh, between the faithful and the Six\u2014\"\n\nPradoor's ears cupped and her eyes narrowed. \"I do not _talk_ of _muut_ , Daavn of Marhaan,\" she said.\n\nDaavn almost seemed to wither at her words and Makka came close to smiling. Tariic, however, scowled at his companion and waved him back. He stepped forward in his place\u2014and knelt before Pradoor.\n\n\"Guide the people,\" he said, \"and when I am lhesh, the old ways will be restored to Darguun. You will speak from the dais of Khaar Mbar'ost and warlords will be your audience. You call me lhesh and say that I believe it will be so. I believe that it will not happen without the favor of the Six.\"\n\n\"Ah.\" A smile spread across Pradoor's face. \"Flattery is sweet, but the truth is sweeter, and reward sweeter still. When you are crowned, where will I stand?\"\n\n\"At my left hand, as those who speak for the Six have always stood at the left hand of warlords.\"\n\n\"And the priests that Haruuc raised?\"\n\nTariic bent his head. \"When the lhesh praises the Dark Six, I think they'll find that the worship of the Sovereign Host has not found such deep roots in Darguun as they think.\"\n\nPradoor's hand rose, feeling for Tariic's face, and she pressed her _muu'kron_ to his forehead. \"The might of the Six be yours, Tariic of Rhukaan Taash. I will give you the people.\"\n\n_\"Ta muut_ , Pradoor.\" Tariic caught her hand, holding the _muu'kron_ close for a moment longer. When he released Pradoor, he rose, nodded to Makka, then gestured for Daavn. The scheming warlord of Marhaan followed him away among the trees without a word or a nod to either Pradoor or Makka.\n\n#\n\n# CHAPTER  \nTEN\n\n# 22 Sypheros\n\nAruget stood outside the map room of Khaar Mbar'ost. \"Ashi's here?\" Ekhaas asked.\n\nThe guard nodded. \"Geth and Dagii, too.\" He hesitated, then added, \"Is it true you ride east with Dagii's army?\"\n\nEkhaas flicked her ears. \"You know?\"\n\n\"By command of Haruuc and Lady Vounn, I remain close to Lady Ashi. But I keep my secrets.\" His ears rose high. \"Swift travel and great glory, Ekhaas _duur'kala.\"_\n\nEkhaas bent her head to him, knocked once on the door, then went in.\n\nMidday light flooded through a window and illuminated a room hung with maps, with cabinets containing more maps lining the walls. In the middle of the room stood a large table. Dagii, Geth, and Ashi looked up from it. Under the sunlight, Dagii's face was creased and weary, though his eyes brightened at the sight of her.\n\n_\"Saa_ , Ekhaas,\" he said. \"Come look.\"\n\nShe joined them at the table. A large map had been laid out on it, a fine map depicting the whole of Darguun in good detail. The Seawall Mountains on the west had been drawn in slate blue, Kraken Bay off the southeastern coast and the Thunder Sea off the southern in brighter shades. The great rivers Ghaal and Torlaac divided the land into thirds with shining threads of silver. Roads were red, lightly drawn for lesser roads, heavily for the fine trade roads maintained by House Orien. The names and locations of town and villages showed in crisp black, except for Rhukaan Draal, which had been marked with gold. The border of the Mournland, running more than half of Darguun's length and pressing against its entire northeastern side was colored in stormy, featureless gray.\n\nAll of the text was written in Goblin. This was no human map, drawn over and recolored to suit a conquered territory. It was a fresh new map, made by _dar_ to show a nation of _dar_. Her heart stirred at the sight of it.\n\nMarkers had been placed on the map, short sticks and round counters of polished wood, tracing a route across the Ghaal River then east and slightly south to the black dot that was Zarrthec. Ekhaas looked up at Dagii. The warlord of Mur Talaan twitched his ears.\n\n\"The advance regiments of the army have already started marching east,\" said Dagii. \"We're using a staggered deployment to reach the area hit by the Valenar raids quickly.\" He reached out with a thin wooden wand and traced a line from Zarrthec to the wide scattered dots, close to the gray blotch of the Mournland, that were the eastern clanholds.\n\n\"Senen will be angry I'm not riding with the first troops,\" Ekhaas said. Dagii's ears twitched again and stiffened.\n\n\"If Senen wants to record the heroic tale of soldiers marching to camp, she's welcome to ride in their dust herself.\" He drew back the wand and tapped Zarrthec. \"You and I will leave tomorrow after the games finish\u2014I need to stay for the naming of Haruuc's heir. We will be able to catch up to the advance regiments before they pass Zarrthec. Your story can begin there.\"\n\nGeth bared his teeth and gave a little growl. \"Grandfather Rat. It's bad enough that Tariic pushed you into leading this army. You don't have to sound like you're enjoying it.\"\n\n\"Tariic turned the situation to his advantage, but I would have taken command in any case. Darguun must be defended. The Valenar raids must be answered.\" Dagii stood back from the table. He smiled. \"We'll be in more danger than you will be here.\"\n\n\"That's what I'm afraid of.\" Geth looked at Ekhaas. \"What about the false rod? How are we going to get it if you and Dagii are gone? T\u2014\" He cut himself off before he said the artificer's name and his eyes darted to Ashi.\n\nThe human woman wrinkled her nose. \"I know. You promised not to reveal the artificer's involvement. Whoever he is, you're not going to be able to go on the day of the coronation and get it from him, and he's not likely going to deliver it to Khaar Mbar'ost.\"\n\nEkhaas had already weighed the problem. \"He'll just have to deal with one more person knowing his secret,\" she said. \"Ashi, you're going to have to get it from him. His name is Tenquis. I'll take you to him today to show you how to get to his workshop and to tell him about the change in plans.\"\n\n\"What about Midian?\" Ashi asked. \"Should we tell him?\"\n\nBeyond Ashi, Dagii tensed and Geth twitched and Ekhaas knew they were thinking of what Chetiin had hinted at. The same thoughts had occurred to her, but she had an answer ready. \"No,\" she told Ashi smoothly. \"We did make a promise to Tenquis. We'll bend it out of necessity, but we won't break it. One more person needs to know but not two.\"\n\nAshi grimaced, but nodded. \"I suppose. It feels strange to be keeping secrets from Midian, though\u2014especially when we're being broken up. We've already lost Chetiin.\"\n\n\"We're always going to have secrets,\" Dagii added. \"Whatever we do with the Rod of Kings, we'll have to keep a secret.\"\n\n\"That's not the same as keeping secrets from each other.\"\n\nThe irony of Ashi's words put a slightly sick feeling into Ekhaas's gut\u2014they were already keeping the secret of their suspicion of Midian from her\u2014but then Ashi pressed her lips together for a moment and added, \"There's something I've been holding back. Vounn and Pater didn't want me to say anything, but Sindra d'Lyrandar wasn't in the gallery yesterday, and there are no Lyrandar ships at the docks\u2014the Valenar may have used House Lyrandar to get their raiders into Darguun.\"\n\nEkhaas raised her ears at the news but Dagii only nodded. \"Some of the warlords already guessed that.\"\n\nAshi's face turned red.\n\nDagii shook his head. \"No, thank you for telling me, Ashi. I appreciate it.\" His ears flattened. \"Lyrandar knows we'll need to welcome them into Darguun again eventually. This war is only business for them\u2014as it is for all the dragonmarked houses. Vounn and Pater have come to me. Orien wagons will follow our army and form our supply lines. House Deneith has contracted a regiment of our own mercenaries back to us.\"\n\n\"Bastards!\" said Geth. \"I served House Deneith as a mercenary during the Last War. They'll do anything for a profit. Sorry, Ashi.\"\n\nAshi shook her head. \"No, I know Vounn and Deneith. You're right. Darguun has always provided Deneith with mercenaries, but they've never needed anything but gold in return. Vounn came to Darguun to try and reverse that.\"\n\n\"She's found her opportunity,\" said Dagii. \"The mercenaries were waiting at the Deneith enclave at the Standing Stone for a chance to work outside Darguun. Deneith was able to offer them to us more quickly than we could raise another regiment of our own. Vounn has even offered us mercenaries from other nations if we want them.\"\n\nThe idea pinched Ekhaas like a dissonant note. \"Will you take them?\" she asked.\n\nDagii's smile was thin. \"Right now I couldn't if I wanted to. Tariic has made sure this will be a war of _dar_ against elves. Even people in the streets are saying the war is _muut_ , our duty.\"\n\nGeth looked at the young warlord of Mur Talaan. \"What do you think?\"\n\n\"I think it's a war.\" Dagii's smile vanished entirely. _\"Dar_ or mercenaries, it has to be fought, and it has to be won. That is my _muut_ to Darguun.\"\n\nThe arrival of Munta and other warlords marked the end of their privacy and they separated, leaving Dagii to plan strategy while Ekhaas and Ashi went out to pay a visit to Tenquis.\n\nGeth headed back to his own chamber vaguely envious of Dagii's role in what was to come\u2014not of the warlord's command of Darguun's armies but of his excursion onto the battlefield. Sneaking, conspiring, and playing at politics weren't for him. The assembly of warlords the day before had been an embarrassment, but there had been no way to escape. He had his _muut_ , Dagii might have said. One heroic act had bound him more closely than being thrown into a dungeon.\n\nAt his side, Wrath seemed to stir with reassurances that he had done the right thing. \"Easy for you to say,\" he muttered back at it.\n\nBut it would all be over soon. As soon as the false rod was in the new lhesh's hand\u2014and, for all of his ambition and posturing, Geth hoped it would be Tariic\u2014he would be free to leave Darguun and spirit the true Rod of Kings away with him. What they would do with it after that was another matter\u2014one he didn't want to consider just yet.\n\nFirst he had to put in another appearance at the games. He hadn't lied to Tariic when he'd said he would have enjoyed the contests in the arena more if he'd been sitting in the stands rather than the warlords' box, but he wasn't sure even that would be enough anymore. Watching other people fight only made him want to draw Wrath and leap into the fray himself. Fighting was fast and real. No lies, no waiting. Your enemies were right there in front of you and all it took to deal with them was a sharp sword.\n\nThe closest thing he'd come to a good fight in weeks was against Chetiin.\n\nThe thought of the goblin pulled a groan out of him that produced a look of concern from a passing hobgoblin guard. Geth waved her away. In the chaos of the news of the Valenar raiders and the assembly of warlords, it had been easy to put Chetiin out of his mind, but he couldn't ignore the problem forever. He still wasn't sure how he felt about the goblin. Part of him was hurt and suspicious and clung to the idea that Chetiin had been the one to murder Haruuc, like a child clinging to an old fear.\n\nBut another part wanted to believe his story of being attacked and left for dead, his identity usurped for the assassination. The proof Chetiin had offered might have been nebulous, but it made sense\u2014there was no reason for him to have made such a public killing.\n\nAt the same time, though, if he believed Chetiin, it meant there was a different traitor among them, that Midian had conspired in Haruuc's death. And that was just as hard to take. He needed some proof, something to tell him which of his friends he could trust.\n\nIt came to him that he knew exactly where to look for that proof. He turned at the next corridor and moved off into a different part of Khaar Mbar'ost.\n\nHe found what he was looking for without too much effort. The door was the same as many on private chambers throughout Khaar Mbar'ost, with two handles\u2014one high for hobgoblins and creatures their size and one low for goblins\u2014and a lock set midway between them. This door, however, had been marked with Haruuc's sword and crown crest above a short phrase written in Goblin. Geth could have grasped Wrath and ordered the sword to translate the angular runes for him, but he didn't need magic to have a good idea of what the phrase said. There was something about Keep Out that looked the same in any language.\n\nFor a change, though, it didn't apply to him. He probably could have found someone willing to open the door for Haruuc's _shava_ but it was easier to keep his visit a secret. Geth checked up and down the corridor, then drew his knife, slipped it between the door and the frame, and slid the blade up until he encountered the bolt. For a long period of his life, he'd lived on the run and he'd learned a number of tricks to survive. One of them was how open a locked door. With a swift motion, he kicked the knife up and gave it a twist to one side.\n\nThe blade broke. Geth was left staring at the hilt end of the knife while the tip clattered to the ground on the other side of the door. \"Rat,\" he muttered. Checking the corridor again, he stepped back and slammed his foot against the door just above the lock. With a sharp crack of wood, the door flew open. Geth waited for a moment to see if the noise brought anyone to investigate, then stepped inside, closed the door, retrieved the broken knife blade, and studied the room that had been Chetiin's.\n\nHe had trouble imagining that those who had searched it after Haruuc's death could have needed very long. Chetiin must have lived simply. A slashed pack, a few articles of discarded black clothing, a broken vial\u2014these were all of the personal items left in the room. Maybe the searchers had taken anything else but Geth thought it was equally likely there had been little more to take.\n\nThe furniture in the room did show the unmistakable signs of a search, however. The bed had been pulled apart, a chest overturned, the stuffed seats of a pair of narrow chairs slit like throats. Geth walked around the wreckage and over to the fireplace.\n\nIf Chetiin's story was true, there would be a ledge in the chimney\u2014and presumably some sign of the goblin's escape from death.\n\nAshi was a far better tracker than he was, but Geth knew he wasn't completely useless. He scanned the floor as he went, searching the thin carpet that covered it for signs of ash that might have been scattered when Chetiin emerged from the fireplace. He found nothing. His gut twisted more than he would have expected. No sign\u2014no escape. Chetiin had been lying.\n\n\"No,\" he whispered to himself. It didn't mean that he had been lying. Chetiin was as wily as anyone Geth had ever met. He would have taken care to leave no trail behind.\n\nGeth squatted in front of the fireplace and stared at the cold ashes of the last fire to burn there. The charred remains of logs had been tumbled about, the poker that had been used to stir them still protruding from the ashy heap. Someone searching the room might have stirred the dead fire, but it seemed to Geth that the fire had been stirred too well. Ash lay in a soft gray blanket, as evenly turned as the soil in a kitchen garden.\n\nHe rose and squeezed into the fireplace, trying not to step right in the ash. Fortunately, flat stones a double handspan wide ran along the sides and back of the firebox, making a space to set pots or kettles or big feet. Geth had to crouch a bit to avoid the sloping upper surface that fed into the chimney, but by straddling the firebox and twisting his neck, he could peer up into the dark shaft.\n\nThe shadows were so thick and blended so closely with the soot-covered walls that even shifter eyes had trouble seeing through them. He waited, letting his vision adjust. After a long moment, he saw a stray wisp of gray, like a cloud scudding across a moonless sky. Smoke from a neighboring fireplace. With that wisp as reference, other vague details made sense\u2014the flat planes of shadow that were the walls of the chimney, the smoky darkness that was the rising central flue. There was something not quite right about the junction of shadow and smoke, though. Geth twisted his head around the other way. There was another plane in the dark.\n\nHe grimaced and shifted one foot right into the middle of the ireplace, digging down into the ash until there was solid stone beneath his sole, then stretched an arm up into the chimney.\n\nHis questing ingers caught the lip of a ledge. The stone was warm and dry, heated by air rising from unseen ires. Geth moved his hand back and forth. He couldn't reach far enough too feel how deep the ledge was, but it was wide. Wide enough to accommodate the body of a goblin.\n\nThe twist in his belly unraveled. \"Wolf and Tiger,\" he murmured. A ledge in the chimney, just as Chetiin had said.\n\nThen the twist came back. Chetiin had been telling the truth, but that meant Midian Mit Davandi had brought about Haruuc's death.\n\nGeth brought his arm down and stepped out of the ireplace. A bit of discarded clothing made a rag to brush the ash from his foot and wipe the soot from his hands. He stirred the remains of the ire again, then stuffed the sooty rag among the remains of the bed. With the door broken in, there was no hiding that someone had been in the room, but he could at least disguise what he had done here. He backed out of the room and closed the door behind him, brushing away splinters of wood and securing the ruined latch as best he could. Gut aching with a mixture of relief and anger, he headed back through the corridors of Khaar Mbar'ost to his chamber.\n\nHe was nearly there when he turned a corner and found himself facing Midian.\n\n\"Geth!\" The gnome's face curved into a smile. \"I was looking for you. When you weren't in your chamber, I thought I might have to go ind you at the arena.\"\n\nGeth forced himself to smile back. Not too much of a smile. Not too little. He couldn't give away what he knew. \"I haven't been yet today. I was looking in on Dagii. I'm just on my way back to my chamber now.\"\n\n\"I'll walk with you.\" Midian turned and fell into step at his side. \"I spoke to Razu.\"\n\n\"What about?\"\n\nMidian lowered his voice as a hobgoblin came along the corridor toward them. \"I told her I'd try to ind a way to get Razu to delay the coronation of the lhesh so that your artificer would have the time he needs to finish... what's he working on. Razu wasn't aware of the importance that the Dhakaani emperors placed in having an auspicious conjunction of moon phases on the day of their ascension to the throne.\"\n\n\"Did they really care that much about the phases of the moons?\"\n\nMidian looked hurt. \"I'm not known as a noted researcher of Dhakaani history for nothing, Geth. Fortunately for Razu and the new lhesh there will be just such a conjunction just two days after the end of the games.\"\n\n\"Really?\"\n\nThe gnome's lips twitched. \"Let's just say it's not a position I'd try to put forward in a research paper for the Library of Korranberg. But there's only ever been one coronation of a lhesh, and Razu is desperate for ideas to build the ceremony around. You've noticed how _dar_ love tradition?\" His blue eyes twinkled. \"You can tell me I'm brilliant again.\"\n\nThey had almost reached the door of his chamber. Geth knew he should end the conversation and get away from the gnome, but Midian's smugness was like vinegar in his mouth. \"What if Razu asks Senen Dhakaan about this?\" he said sharply. \"Senen will know you've made it up.\"\n\n\"There's an odd thing.\" Midian's voice turned serious. \"Razu did ask Senen\u2014and Senen said I was right. I may be brilliant, but I'm not that brilliant. Senen hates me. She'd contradict hard evidence just to spite me. I wonder if she knows we're up to something.\"\n\nThe answer gave Geth a moment of real surprise, and he glanced down at Midian. \"Maybe she does.\"\n\n\"It would be better if she didn't. Aren't we trying to make sure the secret of the rod _stays_ a secret?\"\n\nMaybe it was his imagination, the shock of having Chetiin's story confirmed, but Geth thought he heard a chilling ruthlessness in Midian's words. He tried to hide the shiver that raised the hair in his neck. \"You're being suspicious,\" he said.\n\n\"That's what keeps gnomes alive.\" Midian stopped beside the guards who stood outside Geth's door. \"Your chamber. I'll see you at the arena?\" His face brightened considerably. \"Keraal has developed a popular following. He defeated three Kech Shaarat bladedancers yesterday. There's a rumor that he's ighting four Marguul berserkers today.\"\n\n\"I'll be there.\" Geth stepped up to his door\u2014the guards put fists to chests in a salute\u2014then glanced back at Midian for a moment as the gnome bounced away down the corridor. Maybe it wasn't so difficult to see him orchestrating Haruuc's death. He wondered how long they'd be able to keep Chetiin's survival a secret.\n\nGeth pushed open his door, stepped into his chamber, and closed the door behind him, then looked around the room. Hang something out of your window if you need to talk to me, Chetiin had said. Geth's eye landed on a bright green blanket across his bed. Dragging it off, he took it over to the open window and wedged one end firmly around the hinge of a shutter. The other he tossed out of the window. The wind caught it and blew it out like a woolen banner.\n\nIt was all he could do for now. Hopefully Chetiin would see the signal and come to him. Geth turned away to prepare for his appearance at the games\u2014and stopped as he caught sight of himself in a mirror that hung on his wall.\n\nOne cheek was streaked with black. The whole time he'd been talking to Midian he'd had soot on his face. He cursed and looked more closely. The patch of soot was small and narrow, left behind by the careless touch of a inger maybe. Geth turned his face back and forth, then tilted his head back, trying to guess how much a short person like Midian could really have seen. The soot was close to the thick hair of the sideburns that traced his jaw and easy to mistake for a shadow. Maybe the gnome hadn't even noticed it. And what if he had? It was only a smear of soot. It could have come from anywhere.\n\nYou're worrying over nothing, Geth told himself. He drew a deep breath, blew it out again, and scrubbed the soot away with the heel of his palm.\n\n#\n\n# CHAPTER  \nELEVEN\n\n# 22 Sypheros\n\nTwilight came, and with it an end to the flow of supplicants to the grove around the ancient well. Pradoor had chosen to remain beneath the twisted branches for the day, letting the faithful come to her rather than wandering the streets to meet them. Normally Makka would have chafed at a day forced to sit and do nothing, but he found the time slipping past like a fast stream. Like the unseen water that rushed in the depths of the well.\n\nThe dark presence that lingered in the grove didn't vanish with daylight, but seemed to grow stronger the longer Makka sat above the rock-rimmed hole.\n\nWhen the last of the faithful had left the grove and the setting sun outlined the wizened branches with red light, Pradoor let out her breath in a long hiss of triumph. \"Rhukaan Draal is the axle and I am the pin. The order of the world will be set right. The old ways will be given their proper place once more.\" She turned to face Makka. \"You have been silent.\"\n\n\"I have been thinking,\" he said.\n\n\"Have you?\" Pradoor asked. Her thin lips twitched.\n\nThe old goblin's speech the night before had lit a fire in Makka's belly. He dropped to his knees in front of her. \"The Six call. Show me how I may serve them.\"\n\n\"You already serve.\"\n\n\"Show me how to serve them better. Show me how to serve as you serve.\"\n\nPradoor smiled, showing her teeth. \"The Six marked you as theirs before they guided me to you, Makka. You cannot serve as I serve\u2014I am given souls, you are given steel. But I can show you how to serve in your own way.\" She pointed. \"Turn.\"\n\nMakka shifted around and found himself staring into the ancient well among the jumbled rocks. \"What do I do?\"\n\n\"Look,\" Pradoor said. She reached up and pushed his head forward. \"Look and learn to see. The age turns, and you have your own part to play in the order of the world.\"\n\nMakka leaned out over the hole and peered down into the echoing darkness. At first it seemed there was nothing to see, but then shapes moved, and the sound of rushing water became the thunder of blood in his veins. Makka's eyes widened and he _saw_.\n\nHe saw Ashi of Deneith dying on her own sword.\n\nHe saw the shifter, Geth, crushed and in agony.\n\nHe saw Dagii of Mur Talaan impaled on an elf spear.\n\nHe saw Ekhaas of Kech Volaar with her throat torn out.\n\nHe saw the White Stone tribe wasting away from a plague that afflicted their camp and no other, but that pursued them no matter where they fled.\n\nHe saw every person he had ever sworn vengeance against brought low. He saw every person who had wronged him in even the slightest way met with a swift and terrible justice.\n\nHe saw himself, filled with anger and power and strength, mercy wiped away by rage, bringing divine wrath down upon the world.\n\n\"Who do you serve?\" asked Pradoor's voice.\n\n\"I serve the Six,\" Makka said.\n\n\"How do you serve?\"\n\n\"With steel.\" More. Understanding rose up inside him. \"With steel and faith, and a will to bring the old ways back to the _dar!\"_\n\n\"Who is your patron?\"\n\nHe knew the answer. It throbbed with the beating of his heart and raced through him with all of life's ecstasy. He saw it before his eyes. The calm that had guided him through the day vanished in a wave of frenzy. He drew the knife that had been an offering to the Six from his belt and pushed the tip against his chest, carving what he saw into his flesh. \"The Fury,\" he roared. \"I belong to the Fury and her power belongs to me!\"\n\n#\n\n# CHAPTER  \nTWELVE\n\n# 23 Sypheros\n\nGeth.\"\n\nThe call intruded on Geth's sleeping mind, and it seemed to him that he had been hearing it for quite some time. It came again. \"Geth.\"\n\nHe curled more tightly under his blankets.\n\n\"Geth!\" The speaker, his voice strained and thick, sounded irritated. Something flicked Geth's nose.\n\nAwareness, if not alertness, burst over him like a war wizard's spell over a battlefield. He moved by instinct. A figure stood close to him and he punched at it in the same movement that brought him out of bed. The figure simply tumbled away, landing in a crouch on the sill of the open window. Geth dropped into a defensive stance, hands and arms raised, shocked mind calculating how he could reach Wrath before his attacker came at him again\u2014\n\nRecognition intruded on the rush of battle. The figure on the window sill was Chetiin. The goblin watched him with careful intensity in his black eyes. The sky behind him was gray with dawn. Geth shook his head and lowered his hands. \"Sorry,\" he mumbled.\n\n\"That's why I tried to wake you from a distance first,\" said Chetiin. He reached down to the back of a chair below the window and tossed Geth's pants to him. \"Cover yourself.\"\n\nGeth struggled into the pants, swaying as the dizziness of sudden waking crashed down on him. \"I thought you might have come sooner.\"\n\n\"Sneaking into Khaar Mbar'ost isn't easy.\" The _shaarat'khesh_ elder looked at him. \"You summoned me. You need to talk?\"\n\nOver the afternoon and evening of the day before, Geth had worked out what he would say to Chetiin. How he would describe the misgivings and shameful suspicions he'd had as smoothly as Ekhaas might have. Sleep, however, had stolen the words from him. \"I...\" he bumbled, then clenched his teeth and said simply, \"I doubted you. I'm sorry. I've seen the ledge\u2014\"\n\nChetiin shook his head. \"Don't speak of it,\" he said. \"Just tell me that wasn't the only reason I climbed up here.\"\n\n\"You climbed?\" Geth went over to the window and peered past him. His window was easily seven or eight floors up. Khaar Mbar'ost was far shorter than even a minor tower in the city of Sharn, but it was still tall enough. There was nothing under his window except a drop to the stones of the plaza around the fortress. He looked at Chetiin with new respect.\n\nThe goblin just shrugged casually. \"It was the easiest way to reach you. Now talk. Is it something about the war?\"\n\n\"Have you learned anything more about who attacked you?\"\n\n\"No.\" Chetiin's ears twitched. \"And that troubles me. If no one has stepped forward to claim the assassination by now, they may never come forward. They may not be able to. Whoever hired them may have betrayed them.\"\n\n\"Midian.\"\n\n\"If he could betray Haruuc, he could betray a hired assassin.\"\n\n\"We are talking about someone who tried to kill you, Chetiin.\"\n\n\"Who breaks _muut_ with one member of the Silent Clans breaks _muut_ with all of us, Geth.\" The muscles of his jaw tightened. \"And if the true assassin is dead, I can't clear my name. I convinced you, but I doubt I could convince others. _Volaar kapaa'taat kesha do haan_ \u2014the word of traitors is written on air.\"\n\n\"We could vouch for you,\" Geth said.\n\nChetiin shook his head. \"And reveal the secret of the rod? I don't think so. My honor may become a sacriice.\"\n\nIt was as good an opening as he was going to get. Geth took a breath and pushed out the idea that had prompted him to hang the blanket from his window. \"If you're not having any luck tracking down the assassin, maybe there's something else you can do,\" he said. \"I'd like you to go to war with Dagii and Ekhaas.\"\n\nHe didn't think he'd ever seen Chetiin look surprised. The expression survived only moments on the goblin's wrinkled face, though, then it was gone. \"Why?\" he asked.\n\n\"I want to be certain they make it back. I don't think I'll be able to deal with the Rod of Kings on my own.\"\n\nChetiin looked past him. Geth turned and followed his gaze to the small chest that rested on\u2014or rather was bolted to the top of\u2014a heavy table. The chest was bound in iron and had three magewrought locks, the keys to which hung around Geth's neck. There were other defenses, too, invisible to his eyes, but Ekhaas assured him they were there. In truth, though, Geth didn't see the chest as protecting the rod from others so much as protecting others from the rod.\n\n\"You've done well so far,\" said Chetiin.\n\n\"And the switch with the false rod should be easy,\" Geth continued for him. \"No, it's after that I'm worried about. I won't be Haruuc's _shava_ any more. I want Ekhaas and Dagii\u2014and you\u2014here to help me and Ashi.\" He echoed what Ashi had said. \"Our group is being broken up. We need to stay together.\"\n\nChetiin looked at him for a long moment. \"Ekhaas and Dagii are capable. They can take care of themselves,\" he said. \"If I go, only you and Ashi will be here. I don't think Midian can be trusted.\"\n\n\"Neither do I. He still doesn't know we've figured him out, but if we have to deal with him after the false rod is in the hands of the new lhesh, he may guess. That's why I need you to make sure Dagii and Ekhaas make it back. War isn't predictable. I want someone watching over them.\"\n\nChetiin's ears twitched. \"You trust me.\"\n\n\"I do now.\"\n\nChetiin actually smiled. \"That pleases me. I'll go.\" His smile jerked a little higher on one side. \"It isn't often that one of the _shaarat'khesh_ is asked to protect lives rather than take them.\"\n\n\"I'm not used to sending other people out to fight in my place,\" said Geth with a grunt. \"I'm surprised Wrath hasn't pushed me to go myself.\"\n\n\"Fighting isn't always the hero's part,\" Chetiin said. He twisted around and swung his legs out the window, then nodded toward the glowing horizon. \"The last day of Haruuc's games. The beginning of the end. Good luck, Geth.\"\n\n\"Rat and Tiger dance for you, Chetiin.\" Geth leaned out and watched the goblin start his climb down the wall of Khaar Mbar'ost like some big shadowy spider.\n\nThe end began as Haruuc's funeral had begun\u2014with a procession.\n\nAt first there was little to see from where Ekhaas sat in the stands of the arena, but she could hear the waves of cheering that accompanied the progress of the procession through Rhukaan Draal. It was like listening to the approach of a violent storm. The excited murmur of those lucky enough to have found a place in the arena itself\u2014and there wasn't a spare place to be had, even among the sections reserved for dignitaries\u2014was wind in storm-tossed trees. The blurred susurrus grew louder and louder until the storm was upon them. One pair of the arena's great gates opened and lightning might have struck. The roar of the crowd was thunder.\n\nWhere Haruuc's corpse had led the funeral procession, Geth led the way onto the blood-damp sand of the arena floor. Somehow, Ekhaas thought, the shifter managed to look even more grim than the dead lhesh had. Ashi, sitting at her side, took her hand and squeezed it. The very human gesture was embarrassing, but Ekhaas didn't pull away. It felt good to share her anxiety.\n\nThe four claimants to Haruuc's throne followed Geth, smiling like victorious soldiers and waving to their supporters. Among the deafening echoes that illed the arena, it was impossible to tell who received the loudest cheers. Iizan looked just as conident as Tariic, and Garaad looked just as conident as Aguus. All four were dressed in splendid armor that flashed in the sunlight. All four walked as if they strode the polished stones of a throne room rather than an arena that had seen five days of combat and bloodshed. Not that any of them had any choice now, even if they doubted their true chances. To abandon their claim would be a stain on their honor.\n\nThe warlords and clan chiefs of Darguun came last. They entered as a group, more solemn than the contending heirs, though not so grim as Geth, and took their places for the final ritual of Haruuc's mourning.\n\nGeth moved to stand against one wall of the arena, the rival claimants against the others. The warlords\u2014Dagii among them, the three tribex horns mounted to shoulders of the ancestral armor of Mur Talaan rising over his head\u2014spread themselves out on either side, leaving a broad pathway between them.\n\nDrums took up a deep rolling heartbeat that sounded even above the noise of the crowd. Slowly, sound in the arena died away until only the drums remained. Then they, too, fell silent and Razu stepped onto the raised platform that had recently been occupied by the announcer of the games. Her staff rapped the platform three times.\n\nOn the other side of Ekhaas from Ashi, Senen leaned close. \"Did you know that Razu moved the date of the coronation by two days on the advice of your friend Midian Mit Davandi?\"\n\n\"I didn't,\" Ekhaas lied.\n\n\"Tariic isn't happy.\"\n\n\"I don't imagine he is.\"\n\nRazu spoke, her voice ringing. \"By tradition, when a warlord of the Ghaal'dar Clans dies without declaring an heir, any senior warrior of his clan who believes he can hold the position may seek it. Rivals must pass the judgment of the other senior warriors and meet the approval of the members of the clan. If they cannot, they are not strong enough, and only the strong may take a place as warlord.\" The beat of a single drum began again, a counterpoint to the old hobgoblin's words. \"Lhesh Haruuc Shaarat'kor died without declaring an heir. We look to tradition to guide us!\"\n\nShe thrust out her staff. \"Iizan of Ghaal Sehn. Aguus of Traakuum. Garaad of Vaniish Kai. Tariic of Rhukaan Taash. You believe you have the strength to take the throne of Darguun as lhesh. With the warlords of Darguun to judge you and before the people of Darguun\"\u2014her staff swung around to encompass the crowd in the arena\u2014\"step forward and claim it!\"\n\nThe sound of all the unseen drums surged together into a powerful, throbbing beat that Ekhaas felt in her belly. The crowd remained very nearly silent, however. Ekhaas could see the four claimants studying Geth and each other.\n\nIizan moved, taking a slow step forward. His eyes swung around the assembled warlords.\n\nOne... two... three of them met his gaze. The rest looked down or up or simply away, anywhere but into his face. Among the stands, there were only scattered shouts. Iizan trembled and took another, tentative step, still searching the faces of the warlords for support, but even those who had met his gaze before looked away. The shouts from the stands gave way to mocking hoots. Iizan stopped where he stood, his face pale, his ears sagging.\n\nGaraad, Tariic, and Aguus didn't wait to take turns. As Iizan stood in shame, they stepped out almost at the same time. The noise in the stands exploded and once again it was impossible to tell who had the greatest support from the people. It was simpler, though, to tell who had support among the warlords. Aguus and Garaad advanced across the sand step by step, finding and then losing the gazes of the lords of Darguun. Garaad made it several paces beyond Iizan before the last of his supporters looked away. Aguus stopped a short way beyond that.\n\nTariic's stride was casual, almost arrogant. He didn't pause to search for support, he paused to accept it. As Garaad and then Aguus stopped, it became abundantly clear whom the people of Darguun\u2014or at least those of Rhukaan Draal\u2014favored as well. One loud voice broke free to rise above the others. _\"Tariic and victory over the Valenar!\"_\n\nTariic lifted his head and pointed out into the crowd in the general direction of the shout. The noise in the arena seemed to double. Tariic strode across the sand, stopped in front of Geth, bowed his head, then stepped to the shifter's side and raised his hand.\n\nWarlords and people alike roared their approval of the next lhesh of Darguun.\n\nHe's done it, thought Ekhaas. Ashi squeezed her hand. Ekhaas squeezed back.\n\nRazu didn't appear again to make any kind of formal announcement of Tariic's succession to the throne. There really wasn't any need. Aguus, Garaad, and Iizan came forward and knelt before him in acknowledgement of his triumph\u2014a true pledge of allegiance would come after his coronation. The rhythm of the drums shifted into something almost festive, and the warlords, Tariic, and Geth retreated from the arena to the cheers of the crowd. A few moments later, Tariic and Geth appeared together at the rail of the warlords' box. Tariic waved once more to the crowd, allowed them to cheer a little longer, then nodded to the announcer, who had reclaimed his platform. The drums fell silent, and so did much of the crowd. The event, aside from the selection of the lhesh, that had driven their anticipation through much of the day had arrived.\n\nThe announcer raised his speaking trumpet to his mouth. \"By tradition, the final match of the games honors both the old warlord and the new with the finest fighters available. Today we honor Lhesh Haruuc\"\u2014a cheer from the crowd\u2014\"and Lhesh Tariic\"\u2014a second cheer\u2014\"with the final battle of the only man to fight matches on all five days of the games!\"\n\nA rumble grew in the audience. The announcer gestured with a flourish to one of the gates of the arena. \"I give you Keraal!\"\n\nAs the gates swung open to reveal a chain-draped figure and an enthusiastic roar leaped from the audience, Ekhaas couldn't help wondering if the people in the stands remembered that this was the man who had led a rebellion against Haruuc. Who had tried to starve Rhukaan Draal by burning crops and storehouses. She leaned forward to get a better look at him.\n\nScars covered much of the one-time warlord's skin, but Keraal had far fewer visible injuries than she would have expected. Someone must have been giving him magical healing in preparation for his next match. He looked worn and weary, though, anticipation for the battle only a faint flicker in his eyes. His ears were down. He didn't respond at all to the crowd's cheering or to the closing of the gates behind him.\n\n\"He looks like a knife that's been sharpened too many times,\" Ashi said in her ear. \"I don't think Tariic likes that he's still alive.\"\n\nEkhaas looked up at the warlords' box. Tariic barely seemed to be watching the arena, but she caught him in a glance, and in that glance was anger fit to kill. She remembered what Geth said about Tariic choosing Keraal's opponents. \"He's got one more chance to kill him,\" she said.\n\n\"He's fought a tiger, an ettin, bladedancers and berserkers,\" said Ashi. \"What else can Tariic throw at him?\"\n\nThe answer came from the announcer's speaking trumpet. \"Keraal fights the Five Homas, hunters of the Talenta Plains!\"\n\n\"The Talenta Plains?\" Ashi asked. \"Halflings?\"\n\nThe gates opposite Keraal shuddered and jerked as something on the other side bumped them in its eagerness for combat. Ekhaas felt her throat clench, partly in amazement, partly in involuntary anticipation of inevitable bloodshed. \"No,\" she said, her voice feeling thick. \"Not just halflings, I think.\"\n\nThe gates shuddered again, then were forced wide as Keraal's opponents emerged. Those gathered in the arena fell into a stunned silence\u2014then shouted even louder than they had for Keraal. _\"Rond betch!\"_ gasped Ashi.\n\n_\"Khaavolaar!\"_ said Ekhaas.\n\nHalflings, yes. The same size as goblins, though with finer limbs and human-like features. And five of them, dressed in leathers painted in bright pigments, embellished with colored stones, and stitched in elaborate patterns. Their hair, coated with some kind of pale clay, rose in wild ridges and matted clumps above the bone masks that concealed their faces. Their weapons were glaives\u2014wide, sharp blades pointed like a spear, edged like an axe, and set on long, curved poles.\n\nThey were mounted on the great lizards of the Talenta Plains, all of the creatures decorated like their riders so that it was difficult to tell where the shimmering colors of scales ended and vibrant paint began. Four of the halfling hunters rode lizards as tall as a hobgoblin that strode upright on their hind legs, powerful heads balanced by a thick tail. The beasts' forelegs were small and grasping, their jaws terrifying, but their hind legs were the danger. They were massively muscular and the great toe of each foot carried a heavy claw as sharp as a sickle. The lizards prowled out into the arena, heads darting and nostrils flaring.\n\nThe fifth lizard came out more slowly. It was big, nearly twice the height of a hobgoblin in the length of its body alone, and easily twice that again from the small hook-beaked head on its lowslung neck to the tip of its long, powerful tail. A double row of bony plates rose from its back, but the beast had weapons as well as armor: the massive tail ended in a cluster of four long bony spikes. Its rider, nestled between the plates over its shoulders, used a small hook on the butt of his glaive's shaft to prod the giant lizard just behind its skull. The creature raised its head as far as its neck would allow and let out a rumbling honk. The other four lizards answered with whistling shrieks imitated by their riders.\n\n\"What are they?\" Ashi asked in awe.\n\n\"Clawfoots and a daggertail,\" said Ekhaas. \"Halflings may not put a lot of imagination into naming things, but they go straight for the important details.\"\n\n\"They look hungry!\"\n\n\"They likely are\u2014the clawfoots at least. The daggertail eats plants.\"\n\n_\"Khyberit ghentis.\"_ Ashi shook her head. \"Keraal can't fight all of them, can he?\"\n\nDown on the arena floor, the lone warrior seemed to be thinking the same thing. Ears flat back against his head, he watched the halflings and their mounts just as they watched him. The announcer gave the audience a moment longer to drink in the sight of the magnificent lizards, then shouted through his speaking trumpet, \"For the honor of Lhesh Haruuc Shaarat'kor and the glory of Lhesh Tariic _\u2014begin!\"_\n\nKeraal let his chain slip down his arms, then whirled it up into a shield of spinning, flashing metal. The clawfoot riders urged their mounts forward with kicks and piercing whistles. They spread out to the sides of the arena as the daggertail lumbered to take a position closer to the center. Its rider prodded at it with his long goad, turning it around so the powerful spiked tail could be brought into play. It looked nervous to Ekhaas. All of the lizards did, in fact. One clawfoot rider even seemed to be struggling with his beast. Keraal shifted a few paces to the left, then back again as if gauging the reactions of his opponents.\n\nThen he moved, sprinting at the nearest clawfoot on his right. The rider whooped and the clawfoot surged into a long leap, thick leg muscles bunching and releasing to send it high. The crowd in the arena gasped in unison and even Keraal looked startled. The lizard's terrible claws slashed down, but Keraal dived to the ground, throwing himself away in a spray of bloody sand. The clawfoot missed him, instead landing in a heavy crouch. Another clawfoot leaped, forcing Keraal to scramble across the sand on all fours with his chain dragging behind him. The halfling on the daggertail laughed wildly and his mount's tail slapped the ground\u2014nowhere near Keraal but enough to intimidate even Ekhaas. They were toying with him, she realized, keeping him off-balance and weak. The other pair of clawfoots stalked closer. The first halfling whooped again and his clawfoot leaped\u2014\n\nKeraal flung himself aside once more, but this time as he rolled, he whipped out one end of his chain so that it wrapped around the clawfoot's leg. In the same moment that the lizard landed, Keraal came up to his knees and jerked back hard on the chain. Pulled off balance, the lizard bleated like a sheep and smashed forward. Its rider swayed in the saddle, momentarily stunned\u2014long enough for Keraal to free his chain with a flick of his wrist, turn, and send the chain arcing at another rider. Halfling and lizard both ducked instinctively, but neither was Keraal's target. His chain tangled around the shaft of the halfling's glaive and Keraal yanked it out of his hand. Another fast tumble across the sand and Keraal rose with the weapon in his grasp.\n\nThe crowd howled in approval. With a flip and a twist, Keraal wrapped the chain around his torso and gripped the glaive\u2014sized for a halfling but still useable by a hobgoblin\u2014with both hands. Three of the four clawfoot riders were circling him now. It was impossible for Ekhaas to see their expressions behind the bone masks, but they no longer seemed interested in toying with Keraal. The hobgoblin had landed the first blow\u2014the downed clawfoot seemed reluctant to get up in spite of the cajoling and curses of its rider.\n\nThen the three circling clawfoots broke and moved back. Those in the stands saw what was coming and their reaction may well have saved Keraal, who turned and lurched to one side as the daggertail's spikes came swinging at him. He didn't quite manage to get out of the way. The point of one spike gashed his chest.\n\nBut it also caught in the chain wrapped around him and instead of being thrown back to where the clawfoots waited, he was dragged along with the tail. As the tail curved back, he flew free, rolling across the sand and ending up near the daggertail's head. The lizard's rider stared at him for an instant, then started to goad his mount into a turn, heavy forelegs smashing into the ground, hooked beak snapping, small eyes wild.\n\n\"Plant-eaters?\" said Ashi.\n\n\"Just the daggertail,\" said Ekhaas. She pointed. \"Look there!\"\n\nThe rider of the fallen clawfoot had his mount up again. The creature's pointed tongue licked at a muzzle covered in sand and the clawfoot might even have pressed itself back to the arena floor if the halfling hadn't vaulted into the saddle and hauled back on the reins that guided it. The lizard shrieked in protest.\n\n\"What's it doing?\" Ashi asked, then her eyes widened as she realized what Ekhaas already had. \"The blood out of the sand!\"\n\n\"Five days' worth of blood from the games,\" said Ekhaas. To a hungry predator, the smell in the arena must have been intoxicating. No wonder the clawfoots seemed hard to control\u2014and no wonder the daggertail seemed skittish and wild. To a plant-eater, the arena would stink only of death and danger.\n\nKeraal must have realized it too. As the daggertail's head and forelegs came closer, he pushed himself to his feet and ran\u2014not away from the lizard but _at_ it. The halfling hunter saw him and shifted his grip on his glaive, abandoning efforts to goad his mount in order to defend himself. Sticking between the daggertail's double row of plates, he darted back along its spine, trying to keep pace with Keraal.\n\nThe hobgoblin was quicker, though. With a shout, he leaped high and thrust the head of his stolen glaive into the daggertail's side with all of his weight and strength behind it.\n\nThe great lizard let out a terrible honking screech and reared up on its hind legs. The halfling on its back, trying to stab down between the plates at Keraal, had to drop his weapon and hang onto one of the plates with both hands. Keraal just clung to the shaft of his glaive as his weight dragged the sharp blade inexorably through the creature's flesh. Blood spurted out, spraying him. The daggertail crashed back to all four legs. Its head twisted around to try and bite at the source of its agony and its spiked tail thrashed wildly, but Keraal had chosen well: neither neck nor tail were flexible enough to reach him where he clung. He kept digging with the glaive, forcing the wound deeper and wider.\n\nThe slapping tail kept the clawfoot riders back, too. The half-ling on the daggertail whistled and waved, gesturing for one of them to throw him a glaive, but the other riders were busy trying to control their hungry clawfoots as they caught the scent of fresh blood\u2014and a moment later, the daggertail decided to act on its own. Turning suddenly, it made a lumbering run for the nearest wall of the arena. Spectators sitting in the lower rows fled for higher elevation, even though the spiked tail couldn't have done more than splinter the wall below them. The clawfoot riders scattered. Keraal seemed to reach into the wound before he flung himself away from the beast, leaving the glaive behind.\n\nThe daggertail's rider wasn't fast enough. Yelling at the beast as if he could calm it by voice alone, he hung onto one of the plates along its back right up until the moaning daggertail slammed its side against the wall and began to rub like a cow against a tree. The impact jarred the halfling loose. He screamed as he lost his grip and went sliding down between the great lizard and the wall.\n\nThe audience winced and groaned with one voice.\n\nKeraal had his chain free again. He let it swing loose in one hand as he stalked closer to the nearest clawfoot, a blue-streaked monster somewhat larger than the others. The halflings were wary now. The others circled as the hobgoblin warrior's target backed his mount away\u2014but Keraal's eyes were on the lizard, not the halfling. He stretched out his free hand, in it a big chunk of bloody flesh torn from the body of the daggertail.\n\nThe blue-streaked clawfoot cocked its head like some enormous bird.\n\nIts rider saw the chunk of flesh and stiffened, then bent over his mount's neck, maybe trying to whisper to it, to control it. No luck. Keraal took two quick paces forward then flung the meat to one side of the beast.\n\nThe clawfoot whirled and lunged, snapping for the food. So did the next nearest clawfoot as both riders fought to control their mounts. In the moment of chaos, Keraal darted close. His chain snapped up and curled around the throat of the unfortunate halfling whose clawfoot had betrayed him. Keraal stepped back, heaving the struggling rider out of his saddle. The halfling hit the sand of the arena and an instant later, Keraal was on him, one arm around his neck, the opposite hand gripping his skull. The hobgoblin's shoulders tensed and the halfling's head twisted around on his shoulders, the snap of his breaking neck completely lost in the roar of the crowd.\n\nThe now-riderless clawfoot turned to stare at Keraal, who froze with the body of the halfling hunter still in his hands. The clawfoot lowered its head, taking a slow stride forward, and even from where she sat, Ekhaas could see a kind of feral intelligence and loyalty on its reptilian face. It knew its rider, knew he was dead, and knew that Keraal had killed him. Behind it, the remaining three riders spread out. Keraal let the corpse slip from his grasp and backed up slowly, swinging his chain.\n\nIt might have been his final act if the daggertail hadn't at that moment staggered away from the arena wall, honking in pain and distress. It left a long smear of blood on the wall behind it, along with the broken body of its rider and the shattered shaft of the glaive\u2014the head of which, Ekhaas guessed, must have broken off inside the wound, now even larger and uglier than before.\n\nAll four clawfoots turned to look at it. All of the surviving riders tried to rein in their mounts and control them just as the dead rider had tried to control his, but with no greater success. The distraction that Keraal had set up by wounding the daggertail was too strong. The halflings had left their clawfoots hungry before the battle. The sight and smell of the injured daggertail\u2014natural prey for such predators\u2014was too powerful.\n\nThe clawfoot that Keraal had initially brought down was the first to break. Twitching its head against the pull of the reins, it stalked out to confront the daggertail. The wounded lizard's eyes fixed on it. The fearsome tail swung back and forth, but the clawfoot stayed well back. The other clawfoots moved in, forcing the daggertail to try and watch all of them. Ekhaas saw one of the lizards turn its head and fix its rider with an ugly stare. The halfling stiffened and whistled to the other halflings before he leaped to the ground to let his mount hunt. The other two hunters followed his example and the clawfoots, all riderless now, circled the daggertail.\n\nThe blue-streaked clawfoot looked once more at Keraal and threw back its head to let out a bone-chilling shriek. The daggertail swung toward the sound\u2014and the other clawfoots pounced on it. The big spiked tail caught one in mid-air, bashing it to the ground with deep wounds in its flank, but the others were on it, trying to find a grip in its flesh with their claws and their teeth. The blue-streaked clawfoot shrieked again and leaped join in.\n\nKeraal picked up a lost glaive, snapped the shaft over his knees to create a weapon that he could wield in one hand, and went after the three surviving halflings.\n\nFor a warrior who had defeated a tiger, a two-headed ettin, three Kech Shaarat bladedancers, and four Marguul berserkers armed only with the chains that had once bound him, they were no challenge. The clawfoots fought their own battle and the deaths of the hunters of the Talenta Plains were accompanied first by the screams of the daggertail and then by the sounds of the feasting clawfoots.\n\nAnd by the roar of the cheering crowd, a roar that died away only when Keraal stood below the warlords' box and let the head of the last halfling fall to the sand.\n\nTariic rose slowly and glared down. His face was dark and tight with anger, but somehow it didn't reach his voice. \"Keraal, who was warlord of the Gan'duur,\" he called out\u2014and if there was no anger in his voice, there was at least malice. \"Who defied Lhesh Haruuc Shaarat'kor. Who led his clan to defeat and watched it die. Who is now Keraal of _nothing.\"_ He paused, his ears trembling, before continuing. \"You have fought in the arena and you have triumphed. By the tradition of the People, you have won your freedom.\"\n\nHe gestured and one of the gates of the arena, the one farthest from the still feasting clawfoots, opened just a crack. Ekhaas was fairly certain that whoever stood on the other side was keeping a very close watch on the great beasts. Keraal, however, ignored both lizards and gate. He just looked up at Haruuc's successor as Tariic pointed and said, \"Go.\"\n\n\"No,\" said Keraal.\n\nExcept for the tearing and gulping of flesh, the silence in the arena was complete. \"No?\" asked Tariic, a hint of fury finally creeping into his voice.\n\n\"No,\" Keraal repeated. He stood tall. \"Where is Dagii of Mur Talaan?\"\n\nPartway along the box, just behind a wide-eyed Geth, Dagii rose and came forward. \"I'm here.\"\n\nKeraal looked up at him. \"You ride to war against the Valenar?\"\n\n\"Yes.\" Dagii glanced at Tariic, then back at the fallen warlord. \"I leave Rhukaan Draal before twilight. I wait only for the blessing of the lhesh.\"\n\nDown on the blood-soaked sand, Keraal bowed his head, \"You taught me the meaning of _muut_ , Dagii of Mur Talaan. You defeated me in battle, but gave me respect. When my warriors were to be hung in trees, you bound them there yourself. When I would have died in shame, you forced me to fight and live.\" He raised his chains in one hand and the broken glaive in the other. \"If you will have me, I pledge myself to your service, to follow you and obey your commands. On blood and graves, I swear it.\"\n\nFor a moment, Dagii just stared at him, then he drew his sword and held it high. \"Keraal, I accept your service!\"\n\nThe applause began with the thumping of a single fist against a single chest\u2014and Ekhaas was surprised to see that it came from Senen Dhakaan. She was alone for only a moment, though. Up in the warlords' box, Munta the Gray began to applaud. Then Garaad of Vaniish Kai. And Aguus of Traakuum. And Geth. And abruptly the applause was taken up all around the arena, no shouts, no cheers, only the beating of fists on chests so loud that even the clawfoots took notice and looked up from their grisly meal. Dagii lowered his sword and gave Keraal a curt nod. The warrior let his weapons fall and walked out of the arena with his ears held high.\n\nDagii turned to face Tariic, lifting his sword again, and the applause slowly died. \"Lhesh,\" said Dagii, \"I go to meet the enemies of Darguun at your command.\"\n\nSome of the rage at Keraal's defiance faded from Tariic's face. He drew his sword as well and touched the blade to Dagii's. \"Swift travel and great glory, Dagii of Mur Talaan. Show the Valaes Tairn that Darguun fears no invader!\"\n\nThey let their swords fall, the blades sliding against each other with the slow ring of metal on metal, then Dagii bent his head and turned away without looking back. Tariic turned to face the arena and thrust his sword into the air. \"The games are over! Lhesh Haruuc is remembered with honor and Darguun is strong!\"\n\nThe cheer that rose in the arena was the loudest one yet, so loud it seemed to shake the stands. Tariic simply stood and soaked it all in. The clawfoots hunched and stared around in fear.\n\nEkhaas turned at looked at Ashi. \"That's my sign too,\" she said over the noise. \"Dagii will be waiting for me. Great glory, Ashi.\"\n\nThe human woman's mouth tightened for a moment, then she spread her arms and threw them around her. Ekhaas stiffened, shame at the public embrace spreading through her, then she relaxed and returned it\u2014very briefly. \"Warriors in victory are permitted such displays,\" she said in Ashi's ear.\n\n\"I feel like we've won a victory,\" Ashi answered. \"Two days to the coronation. Come back as soon as you can.\"\n\nShe released her and Ekhaas turned to Senen. The ambassador nodded to her. \"Swift travel and great glory, Ekhaas _duur'kala_. Craft your tale carefully\u2014I have a feeling it will be one for the ages.\"\n\nEkhaas returned her nod and, like Dagii, walked to the exit from the stands without looking back.\n\n#\n\n# CHAPTER  \nTHIRTEEN\n\n# 25 Sypheros\n\nThe next visitor Geth had to his chamber didn't arrive with the same stealth as Chetiin. There was a knock at the door and Ashi entered. Midian slipped in after her and Geth caught a glimpse of Aruget speaking with the guards before the gnome closed the door again. Both wore tense expressions. Geth was certain Ashi's was genuine; he wasn't so sure about Midian's.\n\n\"I ran into Midian in the entrance courtyard,\" said Ashi.\n\n\"Ran into, nothing. I was looking for you,\" Midian said with the kind of desperate cheerfulness people used to cover up stress. \"This is the day, isn't it? I would have waited in the hall outside Geth's door if his guards didn't look at me funny every time I walk by.\" He turned to Geth. \"Sage's quill, you're dressed up like kings and queens are coming to call. No, wait\u2014they are. Or at least one is.\"\n\n\"Shut it, Midian,\" Geth told the gnome. He knew how he looked. The mirror in his chamber told him that. During his time fighting in the Last War, he'd gotten into the habit of preparing early on the days that he would see battle. Sometimes very early. His comrades had mocked him for it until they'd realized that the earlier Geth rose, the worse his temper before the battle was likely to be. Tariic's coronation was a kind of battle and Geth had risen _very_ early. His thick hair was washed and brushed and tied back. His clothes\u2014fine trousers and a crimson shirt, a close-fitting vest of black leather stitched with polished bronze plates in the hobgoblin style\u2014were all new, chosen by Razu and tailored to fit him. The great gauntlet on his right arm was as polished and bright as the black steel could ever be. Wrath hung at his side. He'd been ready since before dawn, and the coronation wouldn't take place until the sun had passed noon.\n\nUnlike human courts, _ghaal'dar_ tradition not only permitted but required that arms and armor be worn in the presence of rulers as a sign of service and respect. Wrath and the gauntlet were a comforting weight, even if they weren't the weapons he would need today. He looked to Ashi.\n\nHe didn't need to say anything. She held out an innocent-seeming bundle wrapped in coarse sackcloth and tied with rough cords. Geth took it and laid it on the table beside the chest that held the Rod of Kings. The cords were intricately knotted. Geth simply cut them. More sackcloth had been wadded up around an inner wrapping of fine linen that reminded him disturbingly of a shroud. He folded it back.\n\nPurple byeshk forged into a shaft as long as his forearm, as thick as his wrist, and traced with strange symbols winked up at him. The rod that lay among linen and sackcloth might have been the true rod instead of the false.\n\nA slip of paper had been wrapped around it. He pulled it free and read the crisp, flowing script upon it. _Balance owing: Kech Volaar tales of the_ daashor, _Geth to bring the sword for my examination. You hold an exceptional piece of work. I should charge you more. Don't tell anyone else my name!_\n\nGeth smiled at an image of Tenquis writing the brief note. Midian tried to peer at the message. \"What's that?\"\n\n\"Nothing.\" Geth folded the paper and tucked it inside his vest, then glanced at Ashi. \"Did you have any trouble getting it from him?\" he asked, taking care not to mention Tenquis by name.\n\nAshi was just as cautious. She shook her head. \"He wasn't expecting Aruget, though. He made him wait outside.\" She looked down at the false rod. \"It looks perfect, doesn't it?\"\n\n\"Put them together,\" urged Midian.\n\nGeth nodded and drew the keys to the chest up from inside his shirt. The three locks made heavy clicks as they opened. The true Rod of Kings lay like a slug among folds of black silk. Geth picked it out and held it next to the false rod. Tenquis's work really was exceptional. The two rods were identical.\n\nMidian whistled, his blue eyes wide. \"You wouldn't want to get those mixed up.\"\n\n\"Our man thought of that,\" Ashi said. \"There's an extra mark carved on the end of the false rod so it's possible to tell them apart.\"\n\nGeth pushed the linen and sackcloth down so he could inspect the end of the rod. A faint spiral marked the byeshk, unmatched on the true rod. \"What about the magic?\" he asked.\n\nAshi grinned, reached down, and picked up the false rod.\n\nSomething about her changed almost instantly. Geth couldn't have said exactly what it was. She seemed taller somehow. The blue-green colors of her dragonmark seemed brighter, the dark gold of her hair richer. Something stirred in him\u2014he felt like he was in the presence of greatness. The effect was subtle but strong. Her words, when she spoke, were as stirring as one of Ekhaas's stories.\n\n\"Concentrate,\" she said, \"and you can fight it. It's not as powerful as you think.\"\n\nGeth blinked and pushed back. The illusion of glory and greatness slipped away and Ashi was herself again. He whistled. \"Grandfather Rat! Even Haruuc would have been satisfied with that.\"\n\n\"You could never feel the effect of the rod when Haruuc held it, but it was almost exactly like that.\" Ashi handed the false rod to Geth. It felt no different in his hand than the true rod, a heavy bar of cold metal, but Ashi and Midian's eyes turned to him like a needle to lodestone. Midian's smile faded, however. \"That's bad,\" he said. \"The true rod doesn't have that effect when you hold it. People might be suspicious.\"\n\n\"Rat.\" He was surprised Tenquis hadn't thought of that.\n\nOr maybe he had. Geth replaced the true rod in the chest and moved the false rod to his gauntleted hand. Ashi's eyes refocused. Midian shook his head. Geth nodded in satisfaction at Tenquis's work and a lightness he hadn't felt since before Haruuc's death settled over him. Their plan was going to work! \"Just like the true rod,\" he said. \"You need to touch it with bare skin.\"\n\n\"Brilliant,\" said Midian. \"Now, what about the true rod?\"\n\nGeth reached out and closed the lid of the chest. The triple locks snapped closed. \"It will be safe here for now,\" he said. \"We'll find another place for it after the coronation. And after that\u2014\"\n\n\"Ekhaas and Dagii's return?\" Midian asked. He made a pinched face. \"We're putting an awful lot of faith in their survival.\"\n\n\"I'd rather assume their survival than count on their deaths,\" Ashi said hotly. \"If they don't come back, we'll deal with the rod on our own\u2014until then, I'm happy knowing that the danger is past.\" Her lips twitched and curled. _\"Rond betch_ , we did it. Tariic will take the throne with the symbol of rulership that Haruuc wanted his successor to have. Is there a better tribute than that?\"\n\n\"Maybe not going to war with Valenar?\" asked Midian. But he sighed and his face unwound into a smile as innocent as if he hadn't plotted Haruuc's death. \"Lords of the Host, I guess it could be worse, couldn't it? Haruuc wanted Darguuls to be united and they are. Maybe Dagii will spank the elves hard enough that they'll ride home with pillows on their saddles.\"\n\nGeth forced a smile onto his face. Maybe they still had to deal with the gnome's treachery and maybe Ekhaas and Dagii were still at risk\u2014even if they did have Chetiin to back them up\u2014but Midian and Ashi were right about one thing. Darguun was safe from the danger that had brought down Haruuc. He closed his armored fist around the false rod and felt a little pulse from Wrath.\n\nEven if no one else would ever know the truth of what they had accomplished, the Sword of Heroes approved.\n\nThe plain little room that opened onto one side of the dais in the throne room had memories attached to it\u2014not good ones. Here Geth had witnessed the argument that had broken the friendship between Haruuc and Chetiin. From here and out onto the dais, Geth had followed Haruuc in the wake of that argument and discovered the terrible influence that rod held over its wielder. Into this room, he had led Ashi in a desperate effort to reach Haruuc and use her dragonmark to break the rod's hold on him, only to watch as he was struck down.\n\nIt was still too easy to think of the assassin as Chetiin. Another of the _shaarat'khesh_ , Geth reminded himself, services paid for by Midian.\n\nHe also tried to remind himself that the small room would soon also have a more triumphant memory attached to it. From here, Tariic's reign as lhesh of Darguun would begin\u2014although it was hard to be optimistic when the air in the room was stifling from the bodies crowded into it. Tariic, wearing bright armor of brass-chased steel, the chestplates worked into the pattern of a skull, the helmet riveted with rows of sharp blades. Razu, staff in hand, fussing as she awaited the arrival of the priests of Dol Arrah, Dol Dorn, and Balinor. A hobgoblin servant, likewise awaiting the appearance of the priests, held the spiked crown of Darguun on a velvet cushion. Daavn of Marhaan, grasping Tariic's sword. Aguus of Traakuum, carrying a heavy cape of tiger skin edged in the soft white fur of a tiger's belly. Munta the Gray, balancing a tray holding a pitcher of water and a silver basin.\n\nAnd Geth, holding the false Rod of Kings. The shifter who had claimed\u2014for nearly three weeks\u2014the throne of a goblin nation and in doing so had saved it. His mouth curved into a grin.\n\n\"You look pleased with yourself,\" said Munta. \"Ready to give up the rod?\"\n\n\"More than you know.\"\n\nMunta laughed. \"I'll tell you something Haruuc told me,\" the old hobgoblin said. \"Sometimes he wanted to leave the throne behind and go back to being the warlord of Rhukaan Taash or even just a warrior of the clan. He couldn't, though. The throne held him tight.\"\n\n\"He told me something like that once, too.\"\n\nMunta's ears flicked and he smiled. \"You're luckier than most warriors who leave the battlefield to take a throne, Geth. You've tasted power but you have the chance to walk away\u2014and without anybody trying to kill you!\" He laughed again.\n\nGeth laughed with him. Heads around the room turned to look at the pair of them. The stares didn't bother Geth. He felt a flush of confidence. Beyond one of the room's two doors, the throne room was full of all the warlords of Darguun and all the ambassadors and envoys in Rhukaan Draal. He could hear them. Soon the responsibility for Darguun would be in Tariic's hands. All he had to do was keep the true rod hidden for a little longer. For a moment, he even dared to dream about what he'd do after they'd found a way to deal with the true rod. He had friends in Fairhaven in Aundair and in Zarash'ak in the Shadow Marches that he could trust to keep a secret. The stories he'd be able to tell them...\n\nAcross the room, Daavn said something to Tariic. The new lhesh laughed at it, but Daavn's eyes darted toward Munta. The old warlord didn't seem to notice, but there was something in Daavn's gaze that Geth didn't like. Something cunning. Something scheming.\n\nThe confidence he felt coalesced into a need to act. He'd held off telling Tariic about Vounn d'Deneith's suspicions of Daavn for lack of any hard evidence. He'd never gotten the chance to bring Daavn and Ko the changeling face-to-face to see if there was any recognition between the two of them. Maybe there was one last thing he could do before he passed power on to Tariic.\n\nHe left Munta and crossed the room to the two warlords. \"Tariic,\" he said, ignoring Daavn, \"I need to talk to you for a moment. Alone.\"\n\nHe tipped his head to the door that opened into a corridor beyond the little room.\n\nUnder his helmet, Tariic smiled. \"Of course.\" He nodded to Daavn\u2014who shot Geth an angry glare\u2014and led the way out the door. Once they were in the corridor, he sighed extravagantly. _\"Maabet_ , if you think it's hot in there, you should try wearing this.\" He rapped his helmet. \"What did you need to talk about?\"\n\nHe seemed more relaxed than Geth had seen him since Haruuc's death, but then Geth felt more relaxed, too. It almost seemed wrong to spoil that. He did it anyway. \"It's Daavn,\" he said. \"I think he's been getting close to you so that he comes into power when you take the throne. Some of us think it may actually have been him, not Keraal, behind the attempt to kidnap Vounn. We don't have anything more than guesses right now, but the changeling in the dungeon who made the attempt might be able to\u2014\"\n\n\"Wait.\" Tariic held up his hand and Geth stopped with the explanation still on his tongue. Tariic smiled. \"I know.\"\n\nGeth almost choked. \"You... knew?\"\n\n\"I'm not stupid, Geth. I grew up in Haruuc's court. I've known politics all my life.\" He lowered his hand. \"I didn't know about the kidnapping, but I'll ask him about it after the coronation.\"\n\n\"But why let him get close?\" Geth asked. \"He's using you.\"\n\n\"No. I'm using him.\" Tariic's ears, poking out through holes in the helmet, twitched. \"A king\u2014a lhesh\u2014needs someone he can trust. My uncle had Munta, then his three _shava_ , and then you. I'd never take Daavn as _shava_ , but as the saying goes in Sharn, you can always trust a greedy man to watch out for himself. It's handy to have someone like Daavn around.\"\n\n\"Oh.\" Geth's confidence fell as limp as an empty wineskin.\n\nTariic knocked his knuckles against the steel of his great gauntlet. \"Don't worry, Geth. I keep an eye on him. I know what he's doing and I won't let him get beyond my control. I appreciate that you tried to warn me.\" He nodded at the rod. \"I appreciate that you took care of that for me, too.\"\n\nGeth forced a smile. \"It might not have been you that the warlords chose as lhesh.\"\n\nTariic's ears stiffened and his eyes turned hard. \"No,\" he said. \"It was always going to be me. I was always going to be _lhesh.\"_\n\nThe hair on Geth's arms and on the back of his neck rose. He didn't have a chance to say anything, though. Tariic's eyes shifted to look past him and the new lhesh said, \"Finally. You're here.\"\n\n\"Your guards wouldn't let us in,\" answered a thin, shrill voice that struck Geth as strangely familiar.\n\n\"That won't be a problem again.\" Tariic opened the door into the little room and called, \"Razu, join us.\"\n\nGeth turned around\u2014and stared in shock at the bugbear who filled the corridor and the old, blind goblin woman who sat on his shoulder. The hair on his arms and neck rose even higher. Pradoor still wore the same ragged dress she had when he'd set her free from the dungeons of Khaar Mbar'ost, but now she was wrapped in a fine, dark green mantle as well. Makka wore the bear hide vest Geth remembered from the Marguul camp in the mountains. Apparently he'd survived the mortal wound Ashi had dealt him after all. The thick hair of his chest had recently been gashed in a savage design: a serpent with the outstretched wings of a bat.\n\nMakka looked at him and his black eyes narrowed. His hand moved to the sword\u2014Ashi's bright Deneith honor blade!\u2014that hung from his belt but Pradoor slapped the back of his head and his hand dropped.\n\nGeth heard the tap of Razu's staff on the floor, then he heard the mistress of rituals gasp.\n\n\"Razu,\" said Tariic, \"there's been a change of plan. The priests of the Sovereign Host won't be participating in the coronation. This is Pradoor. She'll be taking their place. If everything else is ready, we can proceed.\"\n\nAshi shifted her weight from foot to foot in an almost imperceptible movement. Vounn had tried to teach her the technique as an indispensable skill of courtly manners, a way to make standing through long speeches and parties bearable. At the time, Ashi had been amused\u2014it was the same trick she had learned as a hunter, a way to keep legs and feet from aching as she waited for prey. Now, after months as a part of House Deneith, she knew better. Hunting and attending court weren't so very different after all.\n\nThe throne room of Khaar Mbar'ost was filled and everyone was standing. The carved wood benches that provided seating for the assembly of warlords had been moved out. Dust had been shaken from the clan banners that covered the walls. Braziers had been heaped with incense that gave off the resinous smell of cedar. The tall windows behind the blocky throne showed a blue sky and a city at peace, though Ashi knew that the streets around and the plaza before Khaar Mbar'ost were actually packed with a lively crowd. The common people of Rhukaan Draal didn't attend the coronation except in the form of a delegation of nine individuals plucked from the street and deposited in a corner of the throne room to gawk at the power gathered around them.\n\nEven the grieving tree that still stood on one side of the dais looked strangely beautiful: white and gleaming, a piece of strange sculpture rather than an ancient device of torture.\n\nBuzzing excitement drifted through the crowd, but Ashi doubted if anyone could be quite as excited as she was\u2014after all, no one else knew what had been at stake leading up to this moment. Not even the dire whispers that passed between those she stood with could darken her spirits.\n\n\"I've had a letter from friends in House Lyrandar,\" said Pater d'Orien. \"They confirm there are factions within Lyrandar that see a greater profit in committing their services to Valenar than in selling to both sides.\"\n\n\"Sindra among them?\" Vounn asked. Her lips barely moved.\n\nPater snorted. \"What do you think?\"\n\nEsmyssa Entar ir'Korran raised an eyebrow. \"Orien and Deneith were quick to sell their services to Darguun,\" she pointed out. Ashi wondered why the ambassador of Zilargo had bothered to stand with them. When the ceremony started, the little gnome wouldn't be able to see anything\u2014Midian had paused to greet them earlier, then passed on to get closer to the dais. The conversation must have been worth more to Esmyssa than the view.\n\nPater just snorted again. \"Selling cartage to Valenar elves is like selling stone to dwarves. Their warbands carry everything they need. Our routes in Valenar are limited to runs between a few established fortresses.\"\n\n\"Deneith's relationship with Valenar is nearly as important as our relationship with Darguun,\" said Vounn. \"An offer was made, of course. Neutrality saw Deneith through the Last War. More, I don't know. Details of forces contracted to opposing sides in a conflict are kept secret.\"\n\n\"And if you were to speculate, Lady Vounn?\" asked Esmyssa.\n\nVounn pressed her lips together for a moment before she said. \"If I were to speculate, I would say that the Valaes Tairn declined our offer. This war is as much a point of honor for them as it is for the Darguuls. We were only able to contract to Darguun because the mercenaries were their own people. The war is a test of ancient blood against ancient blood.\" She bent her head to the fifth member of their group.\n\nSenen Dhakaan dipped her head in return, but added, \"My blood, but not yet my people. The Kech Volaar will watch the war, though. An alliance with Darguun may still be a possibility.\"\n\nEsmyssa's eyes flashed with delight. \"I've heard,\" she said, \"that the Kech Shaarat clan have embraced the war and have already approached Tariic about sending warriors to ight.\"\n\nSenen's ears lay back. \"The Kech Shaarat would fight pigs in a wallow and call it a rout. I wouldn't put much value to their boasting\u2014\"\n\nThe wail of Darguul war-pipes burst over the throne room, followed a moment later by the throbbing of drums. Conversations ended instantly and all heads turned to the dais. As the martial music rose to a pitch, a door opened and a procession emerged, one by one, to take up positions behind the throne. Razu came first\u2014and Ashi's curiosity stirred. The old mistress of rituals looked shaken.\n\nMunta, a pitcher and basin on a tray in his hands, followed her. His face was dark and troubled. Ashi glanced at Vounn. Her mentor was frowning.\n\n\"What is it?\" asked Esmyssa. \"What's going on?\"\n\n\"I don't know,\" Ashi told her.\n\nAguus came next, then Daavn. The warlord of Marhaan seemed smug. Then\u2014\n\nThe breath caught in Ashi's throat. Her hand went to the sword at her side, gripping the hilt and ready to draw. _Makka_ stood on the dais with the spiked crown of Darguun in his hands. Her reaction, however, was lost in the chaos that gripped the throne room. Many people\u2014warlords and ambassadors alike\u2014gasped. A piper's instrument struck a screechingly bad note.\n\nA very few warlords, after a moment of shock, shouted out, \"Praise the Six!\"\n\n\"Quill and staff, what's happening?\" Esmyssa finally gave up and squirmed forward through the audience as only a gnome or goblin could.\n\nAshi ignored her, spinning to face Vounn. \"That's Makka!\" she said.\n\n\"That's the goblin who led the famine march!\" Vounn stared at the old goblin woman, her eyes milky white, Makka carried on his shoulder. \"What is this?\"\n\nPater's round face was tense. \"Host shield us,\" he said. He pointed at the goblin woman, then at Makka. \"Dark green is the color of the Devourer. The winged wyrm is a symbol of the Fury. Tariic has gone back to the Dark Six.\"\n\nMakka reached the throne and shifted the crown into one hand so he could lower the goblin woman to the ground with the other, then handed the crown to her. It was larger than her whole head. The effect should have been comical, but Ashi didn't feel like laughing.\n\nGeth appeared, the false rod raised in front of him. His gaze swept the audience and found her. His eyes were hard. All Ashi could do was nod to him, then he had taken a position beside Makka. He glanced at the bugbear. Makka looked back at him and bared his teeth.\n\nIf it had been her up there, Ashi didn't think she would have been able to stop herself from running Makka through. She was surprised Geth didn't. Instead, he just stiffened and glared back.\n\nHe was going through with the coronation, she realized. Makka's presence, Tariic's unexpected embrace of the Dark Six\u2014neither mattered. They had to get the false rod into Tariic's hands and Geth would make sure it happened.\n\n\"Ekhaas told me about Makka,\" Senen said. \"Geth bears an insult.\"\n\n\"He has a duty, Senen Dhakaan,\" Ashi told her tersely.\n\nThe music swelled again. Tariic entered, the armor he wore flashing in the light of the hall. The cheers and applause that greeted him were half-hearted at best, the crowd uncertain what to make of the appearance of Makka and the goblin woman. Tariic didn't break his stride, but his ears went back. Across the dais, Daavn jerked his head at someone in the crowd and instantly a renewed cheer rose up. Tariic stopped in front of the throne, faced the crowd, and raised his hands.\n\nThe pipes and drums stopped. The cheers died out. For a moment, there was silence, then Razu cried out in Goblin, \"Behold Tariic of Rhukaan Taash, brave warrior and mighty warlord!\"\n\nShe rapped her staff twice against the floor. Tariic pulled off his gauntlets, then reached up and removed his helmet. Beneath it, his red-brown skin was shiny with sweat, His hair was lank and damp. Munta came forward, holding out the tray he carried. Tariic raised the pitcher on it and poured a long stream of water into the basin. Returning the pitcher to the tray, he plunged his hands into the basin and splashed water onto his face and through his hair. Munta lifted a square of thick white cloth from the tray and offered it to him. Tariic dried himself and returned the cloth. Munta stepped back to his place.\n\n\"He is puriied in the mighty waters!\" said Razu. Her staff rapped the floor again, and this time Aguus stepped up to lay a magnificent long cloak of tiger skin across Tariic's shoulders, fastening it with thick gold chains to rings on his armor. \"He is clothed in the strength of beasts!\" Her staff rapped the floor a third time and Daavn came before Tariic with a self-confident smile on his face. He went down one knee and held Tariic's sword up to him. The new lhesh took it and favored Daavn with a smile and a nod as he sheathed the weapon. Daavn returned to his place, like a dog who had been thrown a scrap from his master's table.\n\n\"He is armed,\" said Razu, \"with his own skill and cunning! He is become more than Tariic of Rhukaan Taash.\" She half-turned to Tariic, encompassing him with a sweep of her staff while still facing the crowd below. \"High warlord, how will you be known?\"\n\nTariic raised his head high. _\"Kurar'taarn,\"_ he said and a murmur of approval swept through the throne room. It took Ashi a moment to understand the phrase in human terms.\n\nThe death of elves.\n\n\"He embraces the event that deines his reign,\" said Senen.\n\n\"He's slapping the Valenar in the face,\" said Vounn.\n\nRazu's staff hit the floor again. The murmur of the crowd slipped away\u2014and became an eerie quiet as Makka guided the blind goblin woman forward before retreating. The goblin stood alone on the dais, facing Tariic, with the crown of Darguun held out before her.\n\nAt Ashi's side, Senen let out a soft hiss. Ashi looked at her. \"What is it?\"\n\n\"The ritual humbling,\" said Senen. \"By tradition, warlords of the Ghaal'dar Clans are confirmed in their position by priests of the Dark Six, but first they must kneel before the priest to show their respect for the Six. She won't raise the crown to put it on his head. He'll have to lower himself.\"\n\n\"But she's a goblin. Tariic will have to practically lie on the floor!\"\n\n\"It is the tradition,\" Senen said with a certain satisfaction.\n\nOn the dais, Tariic stepped before the goblin woman and said in a ringing tone, \"Pradoor, I honor the Six and crave their blessing. You will stand at my side and I will listen to your guidance.\" He paused and a wry smile crept across his face. \"But the emperors of Dhakaan did not crawl before priests, and neither will I.\"\n\nHe reached down and plucked the crown from her hands. Turning to face the assembled warlords and ambassadors, he placed it on his head. \"I name myself Lhesh Tariic Kurar'taarn!\"\n\nOnce again, confusion swept through the throne room.\n\n\"Tradition, you say?\" Pater asked Senen.\n\nThe ambassador of the Kech Volaar actually looked both surprised and strangely pleased. \"He embraces a tradition older than the Ghaal'dar Clans,\" she said with amazement in her voice. \"Until the empire began to decline into the Desperate Times, the Dhakaani emperors acknowledged no power greater than their own. I didn't think it was something widely known or respected outside of the Dhakaani clans.\"\n\nAshi watched Makka's face twist with rage, and the face of the goblin woman, Pradoor, go from confusion to anger... to amusement. Her voice rose, thin and shrill but more powerful than Ashi would have expected. \"May your reign last as long as your strength and cunning, lhesh, and the Six show you their favor all your days!\"\n\nThere was something in the blessing that brought a chill to Ashi's skin, but the Darguuls seemed to pay it no mind. Tentatively at first, then in a great rush, applause and cheering put an end to the silence. Pradoor turned and groped her way back to Makka and her place behind the throne while Tariic turned and stretched his hands out over the crowd in a blessing of his own.\n\nRazu rapped her staff against the floor, but the sound was almost inaudible and she was forced to gesture for Geth to come forward. Ashi's heart seemed to slow. This was the moment they had waited days for. Giving Makka and Pradoor a wide berth, Geth approached Tariic with the false rod, grasped in his gauntleted hand, held out before him. Tariic turned to face him, triumph and eagerness written on his face. Shifter and hobgoblin nodded to each other, and Geth knelt down and extended the rod. Tariic drew a slow breath, preparing himself for the final ritual of his coronation, then he reached down and closed his fingers around the byeshk shaft.\n\nHe froze. His face tightened. He leaned close to Geth and whispered something to him. The shifter stiffened.\n\nAshi's heart might have stopped altogether. She felt Vounn's hand on her arm and heard the lady seneschal ask, \"Ashi?\"\n\nWords felt thick on her tongue. \"Something's wrong,\" she said.\n\nGeth could see the frustration in Razu's eyes. The old hobgoblin lived for ritual and the coronation, her shining moment, had been spoiled, first by Makka and Pradoor's unexpected appearance, then by Tariic's startling crowning of himself. When the crowd drowned out the sound of her staff, he half-expected her to delay the ceremony until the cheers faded.\n\nDon't, he willed the mistress of rituals. Just keep going. Finish it!\n\nWhen she turned and gestured for him to go ahead, he almost gasped with relief. If he hadn't been holding the false rod in his armored hand, it probably would have slid right out of his sweating palm.\n\nMakka's glares had been redirected to Tariic, but Geth still stepped wide around him and Pradoor, then fixed his eyes on the new lhesh and crossed the dais. His mouth was as dry as his palms were wet. Tariic, eyes bright and ears high, bent his head to him. Geth nodded in return and lowered himself to his knees.\n\nThe dais under him was marked with a dark stain. He knelt, he realized, on the spot where Haruuc had died. The circle of succession was complete. Power passed from Haruuc to his _shava_ to a new ruler. He looked up into Tariic's face again and held out the rod. Tariic's chest swelled as he breathed in. He reached down and grasped the rod\u2014\n\n\u2014and his eyes widened, then narrowed. He bent closer and the whisper that came out between his sharp teeth was hot in Geth's ear.\n\n\"This,\" snarled Tariic, \"is not the Rod of Kings!\"\n\n#\n\n# CHAPTER  \nFOURTEEN\n\n# 25 Sypheros\n\nGeth twisted his head to stare at Tariic, but the hobgoblin was already straightening and sliding the false rod out of his slack grasp. The moment it was in Tariic's grip alone, Geth felt the magic Tenquis had crafted into it take hold, enhancing the new lhesh's presence. The cheers of the crowd died into exclamations of amazement. Tariic seized his free hand and drew him up to stand at his side, raising their joined hands high as if they were two warriors united in victory.\n\nRazu's staff slammed twice against the floor and her voice rose in ringing, triumphant tones. _\"Behold, Darguun! Behold Tariic Kurar'taarn, second lhesh of Darguun!\"_\n\nThis time there wasn't a moment's hesitation in the wild cheers that erupted. Arms beat against chests in a flurry of applause. Tariic lifted the rod and waved it. His other hand tightened on Geth's in a crushing grip.\n\nShock numbed Geth. Tariic knew the rod was false. He'd recognized the truth as soon as he had touched it. _How?_ The false rod was a perfect duplicate except for the spiral Tenquis had added and that end of the rod had been concealed in Geth's hand. Tariic couldn't have seen, much less felt, it. It was almost as if he'd known there was more to the true rod, known that he should have felt the glory of the emperors of Dhakaan in his mind as Haruuc had.\n\nBut that wasn't possible. Haruuc couldn't have told him about it. None of those who knew the rod's secret would have told him. Tariic couldn't have known unless he touched the rod before and he had never\u2014\n\nA memory rose in Geth's mind of the day that they had brought the Rod of Kings back to Rhukaan Draal and stood in triumph before the dais in the throne room, basking in Haruuc's gratitude.\n\nThe day that Tariic had taken the rod from him and climbed the dais to kneel and present it to Haruuc.\n\nMere moments of contact. Small enough to forget in the wash of events but long enough. Haruuc had told Geth that the rod had been in his head since the moment he held it, and that once the rod's power had gripped him, it fed him its memories of Dhakaan's glories whether it was in his grasp or not.\n\nThe Rod of Kings answered to those with the will to rule, the old lhesh had said. And Tariic had the will to rule. He'd always had the will. _It was always going to be me_ , Tariic had said. _I was always going to be lhesh_.\n\nThey'd tried to save Haruuc's successor from the curse of the rod, but it had already been too late.\n\nBut it might not be too late to save Darguun. The true rod was still safe in his chambers. For now.\n\nGeth's gut tightened, determination slipping past shock and pushing aside the glamour of the false rod. The thin armor under Tariic's upraised arm made a tempting target. A hard punch there would certainly force the lhesh to ease his grip. If he could escape and retrieve the rod, he could run. Tariic would rule, but he wouldn't have the true rod.\n\nHe curled his free hand, his gauntleted hand, into a fist.\n\nTariic caught the movement and squeezed tighter. \"Attack me,\" he said into Geth's ear, \"and I'll denounce you as a traitor. I may not have the Rod of Kings but I have the warlords on my side now. You'll die before you can leave this hall. Continue with the ceremony.\"\n\nA few stairs led down from the dais to the floor of the throne room and a wide aisle clear to the hall's great doors. Tariic, pulling him along at his side, descended them. The music of pipes and drums began again. Geth knew what would happen next\u2014or at least what was expected to happen next: Tariic would pass through the crowd of warlords in triumph, then proceed out of Khaar Mbar'ost to greet the people who had gathered before the fortress. The final act of the coronation spectacle. Once it was complete, there would be nothing, no interruption, that could stop Tariic from taking the rod by force.\n\nHe had to get away from the new lhesh before then.\n\nHe raised his gauntlered arm and waved to the crowd in imitation of Tariic. The hobgoblin glanced at him and growled, \"What are you doing?\"\n\n\"The same thing you are,\" Geth said through a false smile. He tried to find Ashi, but it was harder to see through the mass of waving arms from the floor than it had been from the dais.\n\n\"Where is the rod?\"\n\n\"Somewhere safe.\"\n\n\"You knew, didn't you?\" Tariic kept waving. \"So did Chetiin. He did us all a favor by killing my uncle.\"\n\nGeth couldn't find a reply to that but Tariic didn't give him a chance to answer.\n\n\"Haruuc couldn't master the rod\u2014I felt it trying to reach him and I felt him holding it back. That was his mistake. Embrace the glories of Dhakaan and you become the master of the rod. I'm not going to make the mistake Haruuc did. Give me the true rod, Geth, and I'll tame it. I'll unlock its secrets.\"\n\nA chill ran through Geth. \"That's what I'm afraid of, Tariic.\"\n\nThe doors of the throne room were getting closer. An honor guard waited on the other side, ready to escort Tariic\u2014or obey his commands to whisk a treacherous shifter out of sight. Geth glanced back over his shoulder. The others who had participated in the coronation ritual had followed them down from the dais. Munta and Razu, the two he might have counted on for some kind of aid, were last and too far away. Aguus was paying more attention to the crowd than to the others in the procession. Pradoor and Makka\u2014out of the question.\n\nBut Daavn darted ahead of the goblin and the bugbear. He walked just behind Tariic and Geth, strutting and waving as if he had taken the crown himself. Geth looked ahead once more, then, as they reached the doors, called back to the warlord of the Marhaan. \"Daavn, go before us and announce the lhesh's approach!\"\n\nThose lining the aisle heard. Makka and Pradoor heard. Daavn's face tightened in suspicion. Tariic turned to look at his ambitious friend\u2014and Geth twisted around, reaching between himself and the new lhesh to grab a fistful of fur and drag the trailing edge of the long tiger skin cloak forward.\n\nIt was a ridiculous, desperate trick, but it worked. The heavy cloak tangled between Tariic's legs. Caught off guard and off balance, he stumbled. His raised arm came down, his grip eased momentarily, and Geth wrenched his hand free. Leaping ahead of the hobgoblin, he turned back and said loudly, \"No? I'll do it myself!\"\n\nHe whirled again, ducking between the startled guards. The antechamber to the throne room flashed past him, then a corridor, then he was bounding up stairs two at a time, racing for his chamber like a fox before hounds.\n\nThe moment that Tariic slid the false rod out of Geth's grasp, Ashi narrowed her eyes, focused her concentration, and drew on the power of her dragonmark. Warmth flashed through the colorful lines that patterned her body and a sharp clarity wrapped around her. The mark had protected her from the influence of the tainted dragon Dah'mir, the alien madness of one of the terrible daelkyr, and the commanding power of the true Rod of Kings\u2014the magic Tenquis had woven into the false rod didn't have a chance. She could even have fought it off on her own, but she needed all of her wits about her.\n\nSuspicion made a hard lump in her belly. Something was _very_ wrong. Geth's eyes had a startled, hunted look to them as Tariic pulled him up to the edge of the dais. And when Razu proclaimed Tariic as the new lhesh, the shifter should have looked triumphant\u2014but he didn't. Protected by the power of her mark, Ashi knew she was probably the only one to notice that for all of Tariic's apparent goodwill and pleasure, he gripped Geth's hand with the strength of someone holding a prisoner.\n\nShe turned and grasped Vounn's shoulder, drawing on her dragonmark once more, but this time channeling its protection into her mentor. Heat like a fever flashed on her skin, then Vounn blinked and looked at her, the influence of the false rod banished. \"Ashi, what\u2014?\"\n\n\"Don't trust Tariic,\" Ashi said. \"Whatever happens, don't trust him.\" She released Vounn and turned away.\n\nThe lhesh had come down from the dais and was passing through the cheering crowd with Geth at his side. The shifter's gaze was sharper now. His free arm rose and he started to wave along with Tariic. What was he doing? Ashi cursed under her breath and tried to push forward. The crowd resisted. She cursed again and wished that she were the size of a gnome\u2014Midian could have slid through easily. Hopefully he had seen things as she had.\n\nMakka passed along the aisle with Pradoor on his shoulder and she crouched down to avoid being seen. Through a gap among arms and elbows, Ashi watched her grandfather's sword swing at the bugbear's side. Her hands clenched and she forced her eyes away. So close to the stolen weapon and yet she couldn't risk stealing it back. Not here. Not now.\n\nWhen she looked up again, Tariic and Geth had reached the throne room doors. She was losing them! She shoved the nearest hobgoblin hard and his gaze finally shifted from Tariic. \"Watch yourself, _taat!\"_ he snapped at her.\n\n\"Yes, sorry,\" she said, pushing past him to the next obstacle in her way. \"My fault. Excuse me\u2014\"\n\nA commotion interrupted her and she looked up just in time to see Tariic batting aside his cloak. She heard Geth's voice over the noise of the pipes, drums, and crowd. \"No? I'll do it myself.\"\n\nPast Tariic, she caught a glimpse of the shifter, free of the lhesh's grasp, breaking past the guards that stood outside the throne room. Tariic did not look pleased. He summoned Daavn to his side with a sharp gesture and spoke to him in tones that did not carry. Daavn nodded and slipped ahead of Tariic, pointing to four guards who fell in behind him. Tariic turned back to the crowd in the throne room and waved the rod, raising another cheer, then resumed his progress as if nothing was wrong.\n\nAshi paused, uncertain of what to do. Try and find Geth or catch up to Tariic and watch him? Either option would be slow\u2014\n\nThe hobgoblin she had shoved gave her a push. \"Don't just stand there!\"\n\nShe glared at him, lips peeling back from her teeth in a snarl\u2014then looked past him and realized that the whole crowd was moving, following Tariic out of the throne room. She could ride the current out.\n\nOr she could swim against the stream. Snarl turning to a fierce smile, Ashi thrust herself against the moving crowd, forcing hobgoblins aside. The progress of the crowd toward the throne room doors was a slow shuffle, but in just a few moments she had reached the thin edge of the mass and vaulted onto the dais. With no one to block her way, she dashed across the dais to the door and the small room on its other side, and then into the corridor beyond.\n\nWhere had Geth gone? He wasn't one to run from a fight, which, Ashi guessed, meant that he was running _to_ something\u2014his chamber and the rod. There would be no following him through the crowded antechamber outside the throne room, but there was always more than one way through Khaar Mbar'ost.\n\nShe started running.\n\nFlight after flight of stairs passed under Geth's feet. He had darted past three floors of the fortress before the sounds of pursuit echoed after him. He recognized Daavn's voice calling more guards to the chase. He smiled without humor. A shifter was still faster than a hobgoblin. Thankfully, he met no one coming down the stairs\u2014anyone who was of importance in Khaar Mbar'ost had been at the coronation and anyone who wasn't would either be at work preparing for the feast to follow or in the streets celebrating. Another floor passed and another. His breath rasped in his lungs but he drove on. Another floor, then only one more to go. His chamber and the rod were close. Did he have time to grab a pack? Maybe. He dragged the keys to the locked chest up from beneath his shirt\u2014\n\n\u2014and came to a halt so abrupt he almost fell over. A hobgoblin guard wearing the red corded armband of Khaar Mbar'ost sprawled across the stairs. Blood ran in the thin streams down the stone steps.\n\nGeth sucked air between his teeth and moved closer, quick but cautious. The guard was dead, no doubt about it. The blood poured from a slit throat and, strangely, slashed legs. He'd been hamstrung, his legs rendered useless. He would have been kneeling or fallen when his throat was slashed.\n\nThere were no smears or handprints or any other signs to indicate that anyone else had already seen the corpse. He was the first. Stepping around the dripping blood, Geth touched his fingers to the body. It was still almost as warm as life. Death had come recently. The hair on his arms and the back of his neck rose. Who would have done this? And why?\n\nThere was a yelp and a crash as one of his pursuers missed a step. Geth's head snapped around and a growl forced its way out of him. Every moment he spent standing still was a moment in which pursuit grew closer. He glanced once more at the nameless guard, and bounded past.\n\nHe moved more slowly, and where the central corridor of his floor opened off of the stairs, he stopped and peered around the corner before moving on. Whoever had killed the guard might have gone anywhere in Khaar Mbar'ost, up the stairs or down. But so close to his floor? That couldn't be a coincidence.\n\nGeth glanced at each door along the corridor as he came to it. Closed tight\u2014except for his. He put his back to the wall. The hard breathing of his race up the stairs became cold and shallow.\n\nThe door to his chamber stood ajar. The guards who had stood outside his door since Haruuc's death were gone\u2014with the Rod of Kings passing to Tariic's possession, at least in theory, there had been no reason for them to remain. If they had still been on duty, Geth suspected he would have found them on the floor with their throats cut.\n\nFrom the stairwell, sounds of pursuit became sounds of surprise and anger. His pursuers had found the dead body. With their cries as his cover, Geth moved, leaping in front of the door and kicking it wide.\n\nChetiin froze in the middle of the room. There was a thick tube of stiff leather slung across his back. The chest that contained\u2014 _had_ contained\u2014the Rod of Kings lay open, all locks and magical protections defeated, a pair of blacksmith's tongs and a bloody dagger with a wicked curved blade abandoned beside it.\n\nAnd Geth froze as well, caught in the doorway as the black-clad goblin's treachery crashed down on him.\n\nNo, he tried to remind himself, it had been another of the _shaarat'khesh_ who'd killed Haruuc. If another goblin had taken Chetiin's place before, why not again?\n\nHis mind told him that. His heart fell back on the betrayal he'd felt after Haruuc's assassination. Then Chetiin spoke and Geth knew his heart was right.\n\n\"Give my regards to Tenquis.\"\n\nOnly four other people knew that name. Three of them Geth would have trusted with his life. The fourth was Chetiin.\n\n\"No,\" Geth choked.\n\nThe proof of innocence that Chetiin had offered in his story of a second assassin and an attempt on his own life, of blame laid on Midian, shattered like glass on stone. A lie. It had all been an enormous lie, right down to his promise to watch over Ekhaas and Dagii. The _shaarat'khesh_ elder had waited for the one moment that all eyes had been elsewhere to make his final move on the true rod.\n\nChetiin gave him a smile that seemed almost pitying\u2014then he moved. In one jump, he went from the floor to a chair, and in another from the chair to the windowsill. A thin rope had been secured to a shutter in the same place Geth had secured the blanket he had used to signal the goblin. Chetiin grabbed it and whipped it around his body in a smooth motion. Then, with a last look to Geth, he pushed himself back into space.\n\n\"No, _you bastard!\"_ screamed Geth. He ran to the window. Chetiin was already halfway down the wall, gliding in long arcs of descent slowed by brief brushes with the wall. A few people in the plaza below looked up and pointed, their attention drawn by Geth's scream, but most were moving to the fringes of the vast crowd at the front of the fortress. Tariic must have emerged, displaying the false rod even as Chetiin made off with the true.\n\n_The word of traitors is written on air_.\n\nRage burst inside Geth. He grabbed for Wrath. One blow would sever the rope and send Chetiin plummeting\u2014\n\n\"Stop!\"\n\nWrath half-drawn, Geth whirled around. Daavn and three hobgoblin guards stood across the room with more guards crowding the corridor outside. All of them had their swords out.\n\nAshi emerged from a flight of narrow stairs onto the floor where Geth had his chamber to the sound of angry cries and running footsteps. She drew her sword and hurried along a side corridor, moving with the practiced silence of a hunter of the Shadow Marches. She had almost reached the main corridor when a scream pushed her back against the wall.\n\n_\"No, you bastard!\"_\n\nGeth's voice. Her breath caught in her throat. She slid along the wall, then peered around the corner into the main hall just in time to see Daavn and three guards push into Geth's chamber while five more guards crowded around the door.\n\n\"Stop!\" ordered Daavn from inside the room.\n\nAshi pulled back and tightened her grip on her sword. Nine to one if Geth was forced to fight Daavn and the guards alone. The odds would be far better if he had some help.\n\nShe drew breath and tensed, ready to spring around the corner and charge.\n\nArms grabbed her, one around her neck with a hand covering her mouth, the other catching the crook of her sword arm and forcing it back. A soft voice rasped in her ear. \"Don't do something stupid.\"\n\nShe moved on instinct, punching back with her elbow, flinging back her head, and biting down hard on the hand over her mouth. Her assailant avoided her elbow and her head bash with surprising grace and endured her bite with remarkable discipline, even though she tasted blood. \"It's Aruget!\" the voice said, pinched with pain. \"Aruget!\"\n\nThe guard forced her around so she could glimpse his face. Ashi blinked, opened her mouth\u2014the hobgoblin guard snatched away a bleeding hand\u2014and snapped, \"They're attacking Geth! Let me go!\"\n\n\"No, you\u2014\"\n\nA loud gasp and a curse of _\"Maabet!\"_ from Geth's chamber interrupted him. Ashi's heard rapid footsteps and another curse, then Daavn's voice speaking in Goblin. \"No, alive! Tariic wants him alive!\"\n\nAshi growled and strained toward the corner in an attempt to see around it again. Aruget's hold on her loosened. She pulled forward\u2014and something hit her hard across the back of the head. Dark blotches swam before her eyes, then the world turned upside down as Aruget heaved her over his shoulder and trotted with her back to the narrow stairs.\n\nDaavn and the guards spread out around the chamber, more of them squeezing in. Geth sank down into a crouch, Wrath still only half-drawn. Daavn had the advantage of numbers and he had\u2014\n\nNothing. Not even the Rod of Kings now.\n\nDaavn's eyes narrowed as he circled closer. \"Come quietly, Geth. Tariic just wants to talk to you.\"\n\n\"Boar's snout, he does,\" Geth said.\n\nEight guards and the warlord of the Marhaan. Daavn held his sword with a veteran's ease. Geth had a nasty feeling that even Wrath and his great gauntlet weren't going to be enough to get him past them. In the close quarters of the chamber, they'd pile on him and the fight would be over. He also had a feeling that any \"talk\" with Tariic wasn't something he'd likely survive. He could try and fight his way out\u2014or he could attempt the same route Chetiin had.\n\nGeth slammed Wrath down in his scabbard, spun, and swung himself up to straddle the window sill. Down below, the treacherous goblin had vanished. Geth grabbed the rope, wrapped it once around his gauntleted forearm, then gripped it hard with both hands. One of the guards gasped and Daavn jumped forward, but the warlord was too late. Geth swung both legs over the sill, braced himself for a moment then pushed out and let himself drop.\n\nThe thin cord sang with tension as it raced around his gauntlet and the shutter creaked, but both held. His unprotected left hand burned, skin rubbed away by the rope, but when he swung back toward the wall of Khaar Mbar'ost, he'd dropped almost a full floor.\n\n\"No, alive!\" came Daavn's voice from above. \"Tariic wants him alive!\"\n\nGeth looked up to see the flash of light on polished blade as a sword was drawn away from the rope. A chill passed through him\u2014he'd planned the same fate for Chetiin. He started lowering himself as fast as he could, hand under hand under hand. The movement bumped him back and forth against the hard wall of Khaar Mbar'ost. The thin rope swung and lashed below him with every motion. He hooked it with his foot, twisting it around his leg in an effort to keep it steady. The ground approached at a snail's crawl, those few people who had not deserted the plaza below after Chetiin's descent staring up at him.\n\nThen the rope jerked with such force that it almost pulled his arm out of his socket. He shouted as pain lanced through his shoulder and chest. His burned hand jumped free of the rope. The grip of his good hand failed. He dropped and the rope screeched as it slid around the metal of his gauntlet\u2014and stopped short with another jerk as the rope twisted around his leg and closed tight on his flesh. For a long, long moment, Geth dangled above the plaza. He shivered uncontrollably, watching the stones of the plaza twenty paces below spin in his vision.\n\nThe rope jerked again. With a yelp, he closed his metal-clad fingers on it in a hold tighter than the Keeper's. His other hand joined it, pain forgotten in fear. He held very still.\n\nAnd yet the wall in front of his nose was still moving past him\u2014in the wrong direction. Geth craned his head back. Hands were on the rope where it emerged from the window of his chamber. Daavn and the guards were hauling him back up! An armslength of wall slid past in fits and stops, then he felt the pull grow steadier and stronger. Another guard had joined in.\n\nWith eight guards heaving at the rope, it would be like drawing up a fish hooked on a line.\n\nHeart trembling, he kicked at the twist that had saved him, loosening it so that he could descend again. He forced his right hand to open, shake the loop from around his gauntlet, and move below his left, then his left to move below the right. The surface of the rope was stained red where his left hand gripped it. He was moving again\u2014but not for long. Climbing as fast as his shaking body would allow, he was still only barely holding position against the hobgoblins pulling him up. And he was running out of rope. A knot-weighted tail that had dangled no more than the height of a tall man above the plaza was now nearly ten paces up and rising fast.\n\nGeth scanned the walls for windows, but for all that Khaar Mbar'ost might have been a palace at the heart of a busy city, it was still a fortress. Only the upper floors had windows worthy of the name. The lower floors were largely featureless except for narrow slits for light and defense. They offered no hope at all of escape. If he waited for the guards to pull him up higher, maybe he could swing to an upper window and make his way back into\u2014\n\nShouts from below caught his attention and he looked down again. Two guards had appeared, attracted by the commotion. No. Three guards. One was running off, probably to find support. One of the others was dragging at the crossbow slung across his back. Geth's heart jumped.\n\n\"No!\" Daavn shouted from above. \"The lhesh wants him alive. Hold your bolt! Hold your bolt!\"\n\nEven Geth heard only half the warlord's words clearly. To the hobgoblins on the ground, he would have been all but unintelligible. The guard with the crossbow planted the nose of the weapon on the ground, held it with a foot, and hauled up on the cord, trying to draw it into position.\n\n\"Bear and Boar,\" breathed Geth. He was out of time. He couldn't go up, he couldn't go to the side, and certainly couldn't stay where he was. He looked down at the knotted tail of the rope and loosened his grip.\n\nHe dropped just as the guard below raised his crossbow to his shoulder and let the bolt fly. The missile spanged off the stone where his back had been just an instant before. Geth squeezed his hands tight again and jerked to a stop. There were loud grunts and curses from above as the sudden force yanked the guards pulling on the rope off balance. Geth sucked in a rasping breath. The tail of the rope pressed against his belly.\n\nOnly about fifteen paces below, the guard with the crossbow was rearming his weapon. The other guard stood beside him, hand on the hilt of his sword, waiting for his own chance.\n\nGeth reached inside himself and shifted. The familiar feeling of invincibility burned through his veins. The pain in his ropeburned hand and his aching shoulder seemed to grow distant, then to vanish altogether. His skin felt like hide, his hair like thick, coarse bristles.\n\nAnd he pushed himself further, pouring everything he had into the shifting. Hide, hair, flesh, bone\u2014he was as hard and dense as the heaviest oak. Wild power flooded him and thought vanished. This was how a charging bear or a rampaging boar felt. Geth drew in his legs, pressed himself against the wall, and kicked out with all his strength, roaring as he unleashed the coiled power.\n\nHe let go of the rope just before it snapped taut. The plaza rushed up at him. So did the guards. He had the briefest glimpse of two terrified faces and of a crossbow snapped up toward him.\n\nBlack pain stole his sight, but there was no blocking out the sounds that came with the moment of impact. The clash of metal on metal. The hollow thump of flesh on stone. Moist crunches and wet tearings. A cry that ended in bubbling gurgles.\n\nLight returned like thunder, and with it came the urge to vomit. Geth held down his gorge. He felt numb, almost separated from his body. He lay on his side, stone under his cheek. He sat up slowly\u2014or tried to. His left arm buckled when he tried to put weight on it. He looked at it and saw an unnatural bend between wrist and elbow. He rolled over instead, felt a burst of pain in his side, ignored it, and pushed himself up with a right arm that wouldn't bend properly but at least wasn't broken.\n\nKhaar Mbar'ost towered over him, an angry giant. By Geth's legs lay the guards. Both looked as if the fortress-giant had raised a hand and swatted them like flies. Both were sprawled with the joints of their limbs at odd angles. One lay still and silent, his skull broken against the stones of the plaza, while the other twitched and gurgled, his rib cage crushed.\n\nA few _dar_ stood around, not too close, staring at them and at him. Geth looked up to the rope, still swaying against the wall, and the distant window of his chamber. The red-brown faces of hobgoblins gaped at him for a moment, then pulled back and vanished.\n\nDaavn and his guards were coming. Geth stood, slowly and carefully, the worst of the pain kept at bay by the shifting, though he no longer felt invincible. Left arm broken. Right arm bent\u2014his gauntlet was dented and locked. Pain in his side\u2014broken ribs. Something ground against his left hip\u2014the final bolt from the crossbow, deeply embedded. Wrath still hung at his side, through one of the leather loops fastening scabbard to belt had been torn free. He suspected that later he'd find an imprint of the sword's length stamped into the flesh of his leg. His right knee pulsed with every step. One side of his face felt strangely soft, and his head was buzzing. He could feel a loose tooth wobbling in his mouth.\n\nThe staring _dar_ jumped away as he turned, putting his back to Khaar Mbar'ost. The fortress Haruuc had built was no haven for him anymore. The maze of Rhukaan Draal lay before him. Limping and weaving, he fled for it. Ramshackle buildings swallowed him up, a mob concealing him from the gaze of the giant behind him.\n\nWhen the grinding of the crossbow bolt in his hip threatened to stop his flight, he found a niche and thrust himself into it. Clenching his teeth tight, he wrenched the crossbow bolt out of his leg and pressed his left hand against the wound as best he could. Bright sparks danced in his vision. Greater pain was coming, though. Geth braced himself and pushed back the shifting.\n\nComforting numbness vanished, laying bare his injuries. A scream tore out of him and left him gasping. His entire body ached and nausea hit him in waves. The end of the shifting brought a gift, though. Sharp pains pinched him here and there as the very worst of his injuries healed themselves\u2014which wasn't saying much. Arm, ribs, and face still hurt, but his knee throbbed a little less and when he lifted his hand, the deep hole in his hip had become a paper-thin, paper-smooth scar.\n\nThe wound Chetiin's betrayal had dealt to him didn't seem likely to ever close.\n\nHe wanted to sit. He wanted to sleep. He didn't dare. Daavn would be hunting him. Geth stumbled out of his hiding place and on through half-familiar streets, trying to stay in the shadows. He turned the scraps of his attention to going unnoticed and let his feet guide him away from Khaar Mbar'ost. Once, he thought he heard the commotion of searching soldiers and dived into a stinking alley, then hastened along its crooked length to emerge onto another street. He pressed on, the sense that hunters were close behind raising the hair on his neck and arms, until the city became a string of streets and alleys.\n\nIcy cold seemed to creep into him. Some part of his mind recognized the shock that descended on a body after great injury. Even if he found some hiding place where Daavn wouldn't find him, his own body might kill him. He needed help. He needed refuge\u2014but his only allies still in Rhukaan Draal were behind him in Khaar Mbar'ost. Ashi. Midian, redeemed now. Vounn. Senen. Munta.\n\nHe didn't think he'd ever felt more alone, a fugitive in a city where he had been a reluctant king. He tried to laugh but it hurt. He stumbled and spat blood as he rose. All of his friends, practically everyone he knew in Rhukaan Draal were in Tariic's power.\n\nNo. Maybe all his friends, but not everyone he knew. Geth looked around himself at buildings that leaned on each other like drunks and realized where his staggering footsteps had taken him. He turned around and found the low stone building that had once been a barn, though for some reason, it and the rest of the world insisted on spinning around him. He reeled up to it and banged his gauntlet against the double doors.\n\nThere was no response.\n\nIt took the last of his strength but he hammered his fist on the wood a second time. His legs gave out from under him and he slid down to lean against the stone wall. It was enough, though. The door opened and Geth managed a smile as he looked up into Tenquis's startled gold-eyed, black-skinned face.\n\n\"You wanted to have another look at the sword?\" Geth said.\n\nThen the tiefling, his door, the street and all of Rhukaan Draal blurred together and he squeezed his eyes closed to shut them out.\n\n#\n\n# CHAPTER  \nFIFTEEN\n\n# 25 Sypheros\n\nAway from the city, the number of stars visible in the sky multiplied ten-fold. By turning her back on the few campfires that had been lit, Ekhaas could see far up into the depths of the night. Only a few of the moons had risen and most of those were thin crescents, bringing even greater life to the stars and, drawn like a veil across the southern sky, the bright band of the Ring of Siberys. She drew a deep breath of cool night air, so much fresher than that of Rhukaan Draal\u2014except for the lingering stink of cold, wet ashes.\n\nThe burned shell of Tii'ator clanhold clawed the sky above the camp. Around the hill on which clanhold and camp stood, burned fields scarred the gently rolling landscape. In the eastern distance, the dead gray mists of the Mournland rolled under the starlight.\n\nEkhaas tried to fix the scene in her memory, a beginning to the story she would take back to Senen Dhakaan and to the Kech Volaar. Burned clanhold and shifting mists, pale starlight and bright campfires. Forty warriors drawn from Dagii's army to accompany their commander forward to a scouting position.\n\nNo sign of the Valenar raiders save for ruined clanhold, fields turned to ash, and bodies _\u2014dar_ and animal\u2014left to rot. When they'd reached the hill around noon, Dagii had ordered two pits dug below Tii'ator, one for people and one for beasts. The unfortunate soldiers conscripted to the duty were only just now heaping soil back on the mass graves.\n\nEkhaas put her back to the stars and the distant mists and looked over the well-ordered lines of the camp. After the games, a swift boat had borne her and Dagii\u2014and Keraal\u2014down the Ghaal River from Rhukaan Draal to a point closer to the village of Zarrthec. With some hard riding, they'd caught up to the bulk of the army. The Valenar hadn't tested Zarrthec yet, but raiders had been seen from the walls, watching and studying before riding away. The town was filled with refugees from clan- and farmholds like Tii'ator; the tales they carried told of swift strikes, slaughter, and miraculous survival.\n\n\"Not miraculous,\" Dagii had muttered to her. \"The Valenar are letting enough live to spread fear and become a burden on Zarrthec.\"\n\n\"I know,\" Ekhaas had told him in return. \"The strategy appears in stories of war preserved by the Kech Volaar.\"\n\nHe'd given her a slightly mocking smile. _\"Ban_ \u2014and now you'll get to be a part of those stories yourself.\"\n\nAt Dagii's command, the body of the army had established its camp about half a day's ride east of Zarrthec\u2014the better part of two days' march back from Tii'ator. Small squads of scouts, mostly goblins and bugbear ambushers, had left the forward camp at dusk, fading into the long shadows. Some had gone north toward Baar Kai and Lyrenton. Others had gone south. Most would be back by dawn. Elves saw as well by night as _dar_ , but elf bows would be less deadly in the shifting shadows. Darkness was be a fickle ally in this conflict.\n\nBack in Rhukaan Draal, the coronation should have taken place. Tariic would be sitting in the throne, Ekhaas thought, or more likely presiding over a great feast, all with the false rod in hand. She wondered what Geth and Ashi were doing: joining in the well-deserved revelry or standing nervous guard over the true rod?\n\nA figure detached itself from the firelight and came to her. She recognized Dagii easily from his stance and the limp in his walk. He'd set aside his ancestral armor for plain scalemail\u2014there was little point in letting the Valenar know that the commander of the enemy forces had left the safety of the army. Dagii's _lhevk'mor_ , the warlord who served him as a second-in-command, had suggested he remain at the main camp, but Dagii wouldn't be persuaded. He insisted on seeing for himself what damage the elves had wrought on Darguun.\n\nEkhaas suspected his motives weren't so noble as he claimed. He wore detachment as he wore his armor, but she knew his blood stirred at the call of battle just as much as any hobgoblin's.\n\nFor a moment, another figure followed him from the fires, but Dagii waved him back. Ekhaas waited until the lord of the Mur Talaan was closer, then said, \"Keraal has attached himself to you.\"\n\n\"More tightly than he needs to,\" Dagii muttered, ears flicking. \"I took his pledge as a warrior, not a servant.\"\n\nEkhaas looked back to the camp. Keraal had moved into the firelight and the flames glinted on the armor he now wore\u2014and on the chains that still swung from his hip. He had adopted the makeshift weapon as his own. \"Legends of the Koolt Dynasty,\" she said, \"tell of the Marhu Dresin Koolt, who was sold into slavery as a child but fought his way to freedom when he grew and seized the throne of Dhakaan from a wicked cousin. Yet even when he was an emperor, he still displayed his slave brand openly. The expression 'I would rather be the sum of what I endure than of what I deny' is attributed to him. Keraal has been your enemy and your prisoner. Now he's trying to be your ally. Give him time to find his own pride again.\"\n\n\"I am,\" said Dagii, \"but I don't need him at my side constantly.\" He glanced at her. \"Walk with me a little ways. There is... something you should know.\"\n\nShe kept her face and ears still, but her heart and belly trembled a little bit at his words. An instant later, she felt like twice a foolish girl as Dagii's strong face turned dark with embarrassment.\n\n\"I'm sorry,\" he said with a stammer. \"That came out poorly. I meant that there's something I need to tell you, but not in front of Keraal\u2014\" He winced and his ears went back. \"Just come with me.\" He walked past her, around the hill and down its slope.\n\nEkhaas followed, embarrassment turning to secret amusement at the thought that Dagii was as flustered as she was. They passed a sentry and Dagii's was so curt, the warrior ended up saluting her instead. There was something odd about the cuff of leather he wore on the hand that remained still at his side. It seemed thick and strangely knobby. It took her a moment to realize that the thick leather was sewn with soot-blackened bells. She caught up to Dagii and fell into step behind him as they strode through tall grass. \"Your sentry is wearing bells,\" she said.\n\n\"All of them are,\" he said. \"It's my idea. A knife or an arrow might kill silently, but even if a sentry falls, he'll make noise.\"\n\n\"Clever.\"\n\nHis ears flicked. \"I still have the reliquary of Duural Rhuvet. I sleep with it near my bed. I like to think it inspires me.\"\n\nHer tremble and her embarrassment returned. The reliquary\u2014a sealed casket a handspan long and wide made of age-darkened iron bound with gold\u2014had languished for generations as an anonymous trophy in a House Deneith memorial until Ekhaas had rescued it. In fact, it contained the brain and tongue of the last great strategist of the Empire of Dhakaan. Ekhaas had recovered it with the intention of placing it in the vaults of Volaar Draal, but on the day Dagii had ridden out of Rhukaan Draal to attack the rebellious Gan'duur, she'd pressed it on him. She hadn't told Senen. Ashi was the only other person who knew of the gift.\n\n\"I am pleased,\" she said. \"But no legend of Dhakaan tells of sentries equipped with bells.\"\n\nDagii smiled slightly. \"The Dhakaani didn't think of everything, Ekhaas.\"\n\nA thick stand of trees rose ahead, remnants of an ancient wood long cleared but surrounded by a fringe of new growth sprung up since Cyre had abandoned this land to Darguun. Dagii led her into it, following a narrow game trail until the wood gave way to an open space cleared by the death of an old tree. He paused, gesturing for silence. For a long moment, they waited.\n\nFinally, a shadow stirred alongside the great fallen tree\u2014and a huge black wolf rose to its feet. Her feet, Ekhaas corrected herself, and not a wolf, but a worg. She knew the beast. \"Marrow?\" she said. The worg's tail swished once. Ekhaas looked at Dagii. \"Then\u2014?\"\n\n\"Yes,\" came Chetiin's scarred voice.\n\nEkhaas turned to find the _shaarat'khesh_ elder crouched beside the game trail as if he'd been there the whole time. And maybe he had.\n\nHe stood and smiled at her, teeth flashing in a face stained dark. _\"Saa_ , Ekhaas.\"\n\n\"He revealed himself to me in Zarrthec,\" said Dagii. They all sat or crouched in the shadow of the fallen tree, their voices low. \"He told me not to tell you until we were here.\"\n\n\"You should have told me anyway,\" Ekhaas said indignantly\u2014and saw something like a smile pull on Marrow's muzzle and creep into her coldly cunning eyes. She glanced away. The worg understood what they said, even if she couldn't speak their language.\n\nChetiin reached up and scratched behind his mount's ears. \"There was no reason to tell you,\" he said. \"You would have tried to seek me out.\"\n\n\"I wouldn't have!\" she protested. He just continued to look at her as he scratched Marrow. After a moment, Ekhaas flicked her ears. \"Maybe I would have.\"\n\nChetiin's dark eyes seemed to flash. \"When you march with an army, you have little need of additional protection.\"\n\n\"Protection?\" Ekhaas asked. She saw Dagii wince slightly and Chetiin's ears twitch. She looked from one of them to the other. \"What are you talking about?\"\n\n\"Geth asked me to watch over you during the war,\" Chetiin said. \"He wants to be sure you make it back to Rhukaan Draal to help dispose of the Rod of Kings.\"\n\n\"I hope you told him we don't need watching over!\"\n\nMarrow raised her big head at her angry tone and woofed softly as if in warning.\n\n\"Ekhaas,\" said Dagii. There was command in his voice, the sort of tone she'd heard him use with his soldiers. She closed her mouth. He nodded to Chetiin. \"In war, there is a place for any aid that is offered.\"\n\nChetiin returned the nod. Marrow snorted and lowered her head back onto crossed paws. Ekhaas glowered. \"When we get back to Rhukaan Draal, I'm going to take Geth up Khaar Mbar'ost and throw him off it.\"\n\n\"He was worried for you,\" said Chetiin. \"We spoke the night before the final day of the games. I think he was feeling powerless\u2014a strange thing to say about someone who sat on the throne of a king, but still true. He needs allies. We should make sure we all return to him.\" He left off scratching Marrow and reached into a pouch at his waist. \"Marrow and I didn't stray far from your marching route, but we had an interesting encounter today. We could have had more if we'd gone looking for them. You know you've been watched?\"\n\nHe held out two severed elf ears, fleshy and lobed like human ears but tapering to long tips. Marrow whined and thumped the ground with her tail. He shook his head at her, holding out the ears to Ekhaas and Dagii.\n\n\"Know it?\" asked Dagii. \"I expected it.\"\n\nEkhaas glanced from the ears\u2014both left ears, so cut from two elves\u2014to him. \"You expected it?\"\n\n\"It would be difficult to move a company of soldiers through territory controlled by the enemy without being noticed.\" He sat back. \"I've never fought the Valenar before. I've only heard stories. If we're going to beat them, I want to see them fight firsthand. I want to see their tactics.\"\n\n\"I could tell you a dozen stories about battle with the Valaes Tairn,\" Ekhaas said.\n\n\"Stories,\" said Dagii. \"I need to fight them myself.\"\n\n\"You approve of this?\" she asked Chetiin. The old goblin frowned.\n\n\"I didn't know about it until now. It's not the _shaarat'khesh_ way.\" His frown became thoughtful. \"It makes some sense\u2014if the Valenar don't decide to overwhelm you in numbers. Their idea of honor is not _atcha_ or _muut_. To them, a victory is a victory, no matter how it's achieved. The only things they hold sacred are their ancestors and their horses. You can't count on a fair fight.\"\n\nDagii's face was calm, his voice assured. \"They respond to the same challenge Haruuc put to the warlords of Darguun. _Dar_ and elves are ancient enemies. I am told the Valenar venerate their ancestors by testing themselves. Their ancestors fought our ancestors. This will be a matter of honor\u2014such as their honor is\u2014for them. They won't try to crush us, I think, but only send a force equivalent in strength to ours. Just enough to best us.\" A spark entered his voice and his ears stood up straight. \"I want to see how many they consider 'enough.'\"\n\nChetiin's expression tightened. \"Geth was right to send me to protect you. You're mad.\" He flicked the elf ears to Marrow, who snapped them out of the air and began to chew at the tough cartilage.\n\nA chill passed through Ekhaas. \"If their force is evenly matched to ours, both sides could suffer heavy losses.\"\n\n\"Then it will be a test of strategy and will. If we can't beat them here, how will we beat them if they come against the main army?\"\n\n\"The main army is bigger,\" Ekhaas said.\n\nDagii smiled\u2014and put his hand on her knee. The touch was hesitant but it wasn't fleeting or accidental. It lingered and Ekhaas could feel it even through the stout steel-studded leather she wore. It took her breath away. Dagii's face had turned a little dark, but he kept his voice steady. \"We'll beat them, Ekhaas. I promise you.\" When he moved his hand, he slipped it away as if reluctant to break the moment of contact.\n\nEkhaas tried to find words. Chetiin had looked away, politely ignoring the awkward, intimate moment.\n\n\"What about the scouts?\" Ekhaas managed to say.\n\nDagii's smile faded. \"They have orders to go back to the main army if they return and we're not here,\" he said. \"Not all of them are truly scouting either\u2014a few have stayed and concealed themselves to watch Tii'ator. They'll carry word back to the army if we're defeated.\"\n\nEkhaas's ears flicked. \"I begin to doubt your promise of victory. If we're defeated, will the scouts be able to evade the Valenar?\"\n\n\"I sent out the best available. Only _taarka'khesh_ of the Silent Clans would have been better,\" he said.\n\nChetiin looked back at that and even Marrow perked up in the midst of her chewing. Ekhaas recalled what Chetiin had once told her, that the worg's pack had an ancient alliance with the legendary scouts of the _taarka'khesh_ and that she traveled with him as a favor.\n\nDagii nodded at them both. \"I tried to make contact with the _taarka'khesh,\"_ he said. \"I had no response in either Rhukaan Draal or Zarrthec.\"\n\n\"The Silent Wolves stay away from Rhukaan Draal at the best of times,\" said Chetiin. \"They've abandoned the city altogether in the wake of Haruuc's assassination. It isn't safe for any of the Silent Clans. In Zarrthec, they avoided you because your ties to Haruuc and to Tariic stain you. You'll have to convince the _taarka'khesh_ of your good will before they'll work for you.\"\n\n\"I thought that might be the case,\" Dagii said. \"I have faith in those I've chosen, though. How we proceed will depend on what information they bring back.\" He scraped a patch of dirt clear and leaned forward to draw in it with a stick. \"The border of the Mournland, the Ghaal River, Zarrthec, Tii'ator, Ketkeet, Lyrenton, the Orien trade road\"\u2014the stick made two crooked lines, one straight line, and four dots in the black soil\u2014\"these mark the limits of Valenar raids so far as we know. A worst-case scenario is that the raiding warbands are operating independently, forcing us to chase them down separately.\"\n\n\"The Gan'duur did the same thing in their rebellion against Haruuc,\" said Ekhaas. \"Could Keraal give you insight into that tactic?\"\n\n\"I asked him. The Gan'duur sought to starve Rhukaan Draal, It's not clear what the Valenar are trying to accomplish by burning fields and clanholds in an isolated area.\" Dagii tapped his crude map. \"Ideally, the scouts will find some kind of central staging area and uncover some clue of the elves' plans.\"\n\n\"An attack on Zarrthec.\" Chetiin touched one of the points on the map.\n\nDagii nodded. \"Maybe, but it seems wrong to me. It's too obvious.\" He made a new mark on the ground, extending the Orien trade road and poking a hole into the middle of it. \"The Gathering Stone,\" he said. \"House Deneith's primary enclave in Darguun. All of our trade in mercenaries with Deneith flows through there. It would be a difficult target, but a strike against it would be incredibly disruptive.\"\n\n\"You think they would attack a Deneith holding?\" asked Ekhaas. \"They contract mercenaries to Deneith, too.\"\n\n\"Attacking the Gathering Stone would hurt us far more than it would hurt them. I'm expecting anything of the Valaes Tairn. Chetiin is right. To them, a victory is a vic\u2014\"\n\nMarrow's head rose suddenly, her nostrils flaring. She gave a low growl that could almost have been a word and climbed to her feet. Chetiin stiffened and turned around. \"Elves!\" he said.\n\nDagii twisted around. Ekhaas's head snapped up.\n\nShe was just in time to see a flash of color among the trees and to catch the blur of motion as arrows flew from singing bowstrings. Chetiin twisted and seemed to vanish into the shadows. With a solid thunk, an arrow sprouted from the trunk of the fallen tree. Dagii grunted and turned his twist into a dive in front of Marrow. A second arrow rang on the plates of his armor, bunched up across his shoulders. A third sank into Marrow's hindquarters, bringing out a yelp of pain, but at least the one meant for her chest had been deflected by Dagii's actions.\n\nAt almost the same moment, bells jangled loudly up the slope of the hill and closer to the camp. There were other bells as well and voices raised in alarm. _\"Toh! Toh! Itaa!_ \"\n\nSomeone cried out orders but all of Ekhaas's attention was on the small clearing and the wood around it. Three arrows. At least three elves, trying to take down their unarmored enemies first. She had to make sure they didn't get another opportunity. Surging to her feet, Ekhaas focused her will, drew song up from inside of her, and flung it at the trees where color had flashed.\n\nSound burst outward in a wave of dissonance. Leaves stiffened and fluttered as if struck by a strong wind, some of them tearing free to dance on the air. For an instant, she caught a glimpse of two elves still clutching short bows as hands covered ears, dark red leggings and short, close-fitting robes worn over light armor fluttering as the leaves fluttered.\n\nOnly two. Where was the third?\n\nA figure leaped out of the trees to her left. Somewhat shorter than a hobgoblin but far more slender, the elf moved with a grace that made Ekhaas feel heavy and slow. Violet eyes blazed beneath a cloth-swaddled, cap-like helm and above a concealing veil. It was difficult to tell if the elf was male or female beneath the veil and the wrapped robes, but there was a delicacy to the brow above the fierce eyes that made Ekhaas guess female.\n\nShe held a curved scimitar, broad, but elegant as a hawk's wing, already raised.\n\nEkhaas dragged at her own sword, trying to free the blade and knowing she was already too late.\n\nThen Chetiin leaped out of the shadows, launching himself at the elf's shoulders and head. He tackled her from the left, where she could not easily bring her scimitar around, and swung across to her back. The dagger he carried flashed\u2014and screeched across a metal gorget hidden beneath the veil. The elf ducked and whirled like a dancer. Chetiin's legs swung free. He jerked his dagger up. Flesh and fabric tore, then Chetiin leaped clear to land in a cat's crouch.\n\nThe elf straightened. Her veil hung askew and blood ran from a wound that sliced across jaw and cheek to a pointed ear. She swung her curved sword at Chetiin and missed but continued around in a scything strike at Ekhaas.\n\nBut the _shaarat'khesh's_ attack had given Ekhaas the moment she needed to draw her own sword. The elven blade met the deep-toothed edge of the heavy hobgoblin blade with a crash that jolted Ekhaas's arm. She wouldn't last long in a fight against this warrior! Twisting her sword, she locked the two weapons for an instant and kicked out under them desperately. Her boot sank into the elf's stomach. The elf staggered back. Chetiin leaped again, this time catching the elf around her neck from behind and using his weight to drag her off balance and backward.\n\nEkhaas stepped back, turned her sword, and swung the weapon in a flat arc. She felt the sharp edge of the blade shear through mail, into the flesh underneath, and back out through mail again. The elf wailed\u2014the first sound she had made\u2014then fell back, letting her scimitar drop and groping feebly for the terrible wound in her torso before sliding to the ground, eyes wide in death. Ekhaas spun to face the remaining elves.\n\nThe powerful burst of song had shaken them, but they were pressing in now, bows abandoned for scimitars in the close fight. Lacking the shield he usually carried in combat, Dagii had drawn his sword and waited for their attack. Marrow snarled and circled to their side, limping on three legs, but still moving like a deadly shadow. One of the elves turned to face her.\n\nDagii roared and charged, sweeping his sword out and driving the two elves apart. Marrow darted in at one, teeth snapping, to force him back. Chetiin jumped atop the fallen tree and ran along its mossy trunk, joining Marrow. Ekhaas moved to fight at Dagii's side. The warlord's sword was swinging and hacking in deadly blows, but the elf managed to parry each one, his curved sword a blur of bright metal. He'd learned from the dead elf's mistake and was careful not to put his blade in a position where Dagii could bind it. Dagii, however, gave him no room to return his blows. They turned around and around each other, locked in a deadly dance.\n\nAs Ekhaas joined in, however, the elf's eyes darted at her, then his free hand dipped into his close-fitting robes and emerged holding a rough ceramic flask no bigger than her fist. Ekhaas's ears rose sharply and Chetiin's words came back to her. _To them, a victory is a victory, no matter how it is achieved_.\n\nShe whipped up with her sword, aiming for the elf's wrist.\n\nDagii did the same, striking down.\n\nThe elf, perhaps thinking to seize this opening, thrust his blade forward.\n\nDagii's blow struck first, slashing through the elf's forearm\u2014and driving down the hand that held the flask. A fraction of a heartbeat later, Ekhaas's blow caught the severed limb and spun it up. Dagii twisted in close and the elf's thrust skimmed past his back, shock only just registering on the elf's face. Dagii shoved him hard with his elbow and sent him reeling back half a dozen paces.\n\nThe ceramic flask fell free of the spinning hand. Hardly thinking, Ekhaas snatched at it in midair and flung it after the dazed elf.\n\nIt hit his armored chest and shattered. Green smoke burst out, writhing up around his shoulders and head. The elf wheezed, shuddered once, and collapsed. The green smoke dissipated, leaving only thin threads of gray drifting from smoldering hair, robes, and veil.\n\nA snarl and a broken wail brought Ekhaas's attention back to the last of the elves. Marrow had her jaws around the elf's sword arm and was shaking her shaggy head. Sword already lost, the elf jerked back and forth\u2014then, with a wet tearing, the arm pulled free of its socket. Armor held the limb in place, but the elf's face went white and his body limp. Marrow shook her head once more and flung him away\n\nChetiin was on him instantly, drawing his knife expertly across an exposed throat.\n\nThe sounds of combat from the camp were growing. Dagii turned for the game trail they had followed into the wood, pushing his way through tree branches and undergrowth. For the first time, Ekhaas saw the arrow\u2014meant for Marrow\u2014that protruded from high on his shoulder, lodged in armor and flesh. \"Dagii!\" she called after him. \"You're wounded. Let me heal you.\"\n\nHe glanced back at her, then reached over his shoulder and snapped the shaft of the arrow between his fingers, breaking it off short. He threw the fletched wood to the ground. \"Heal Marrow,\" he said. \"Follow when you're able. Chetiin, stay with Ekhaas. Watch for more ambushers.\" Then he turned again and plunged on through the trees.\n\nEkhaas looked at Chetiin, but the goblin elder only jerked his head at Marrow. Her ears laid back flat, Ekhaas turned to the panting worg, pressed her hand against the beast's wounded flank and sang as she tugged on the arrow embedded there.\n\n#\n\n# CHAPTER  \nSIXTEEN\n\n# 25 Sypheros\n\nThere's no sign of the rod,\" said Daavn. \"And no sign of Geth. _Maabet_ , Tariic, he shouldn't have been able to walk away from the fall he took, but he did. The guards I have searching haven't found him. No one has seen him. The streets were practically empty this afternoon\u2014anyone who was out had gathered to see you after your coronation.\" He pursed his lips and added, \"If we could be more specific in our description, it might help. 'A wounded shifter wearing a black steel gauntlet' might jog more memories than just 'a wounded shifter.'\"\n\n\"No,\" Tariic said.\n\nThe lhesh stared out of the window into the night. Unlike Tariic, Makka found more to look at inside the chamber than out. The final transition of power in Khaar Mbar'ost seemed to find a reflection here. What had been Haruuc's royal quarters were now Tariic's. Old trophies of war had been shuffled out and luxuries brought in. Makka couldn't have guessed where the rich goods came from other than somewhere beyond Darguun's borders. Thick carpets in strange patterns. Furniture carved with delicate vines and flowers. Small chests of hammered metal inlaid with bright stones. Sweet-scented candles of uncommonly smooth wax in stands of fine ironwork. All had been haphazardly placed or tumbled about the room, abandoned when Tariic had ordered the servants out.\n\nA grin of pleasure spread across Makka's face. He belonged to the Fury. He knew the currents of vengeance. When Tariic had told Pradoor about Geth's treacherous theft of the Rod of Kings, asking if she knew any prayers or divinations to locate lost objects, he'd recognized the hands of the Six. Pradoor knew no such prayers.\n\nAs if sensing the smile, Tariic turned and met his eyes. His ears went back. \"Pradoor, I permit your servant's presence. I won't suffer his insolence.\"\n\n\"He isn't my servant, Tariic,\" said the old goblin. Pradoor perched on top of a spindly little table, her fingers idly tracing the deep carvings of the dark wood. \"He serves the Six. Surely his insolence is no greater than yours.\"\n\nTariic bared his teeth, speaking between them. \"Have care, Pradoor!\"\n\n\"Or what?\" Pradoor turned white eyes in the direction of Tariic's voice. \"Perhaps you don't believe you need to humble yourself before the Six, but you need me. My words brought you the people. My words can take them away.\" She smiled and her blind gaze softened. \"But there is nothing in that for me, lhesh,\" she added. \"Continue to show favor to the Six as you promised and I will be your most loyal councilor.\"\n\nTariic's eyes narrowed, but his ears and face relaxed a little. \"You use me, Pradoor.\"\n\n\"As you use me, lhesh,\" said Pradoor, inclining her head. \"Consider this my best advice: why do you seek the Rod of Kings with such vigor when you possess what you need? The rod you hold has power even I can feel. All accept it as if it were the true rod. Rule with it and find Geth in your own time.\"\n\n\"The rod was a triumphant gift from my uncle to the nation. It is my duty to recover the true rod. It would be a shame upon him if I didn't.\" A harsher tone crept into his voice. \"And as long as I don't possess the true rod, there is the risk that the false rod will be revealed. I must have the Rod of Kings in my hands as quickly as possible.\"\n\nIf Makka hadn't been looking directly at Tariic\u2014and if Tariic hadn't been looking at Pradoor as he spoke, his reactions attuned more to her blindness than to anything else\u2014he would have missed the momentary tightening of the lhesh's face and the darting of his eyes to the false rod where it rested alongside the spiked crown of Darguun on a velvet covered sideboard.\n\nThe grin on Makka's face slipped away. Tariic's glance had the look of greed, of a hunter who had made a good kill, but still wanted more. Makka felt a twinge of unease.\n\nTariic seemed to regard the fading of his smile as nothing more than proper concern. The hobgoblin's ears rose and he nodded to Makka. \"Yes,\" he said, \"there's nothing amusing in that, is there?\" He gathered the tiger skin cloak that was still fastened around his shoulders and sat down in a nearby chair. \"Until the rod has been retrieved, this matter is a secret. No one outside of this room is to know that Geth is being hunted. Daavn, find another explanation for the death of the guard he murdered on the stairs. The guards who were with you when he jumped\u2014where are they?\"\n\n\"Out in the streets. Searching for him.\"\n\n\"Deal with them.\"\n\nThere was a hard finality in his words. _\"Mazo,\"_ said Daavn. \"But people will start to wonder what's become of Geth.\"\n\nTariic sat back. \"I have a solution ready,\" he said, ears twitching. \"One that Geth himself made possible and inspired.\" He raised his voice. \"You can come in now.\"\n\nA door opened and Geth stepped into the room.\n\nMakka held back his rage, just as he had when he had faced the shifter before the coronation. To be so close to one of those he had sworn to kill and yet be forced to _cooperate_ with him...\n\nYet something was different. Geth looked nervous, but not startled or ready to attack as he had before. He looked at them all in turn before his eyes finally settled on Tariic and he gave a little bow. Pradoor slapped Makka's thigh.\n\n\"What's this?\" she demanded. \"Who's there?\"\n\n\"Geth,\" Makka growled. \"But not Geth.\"\n\nTariic frowned. \"Perceptive.\" He looked at Geth. \"Well?\"\n\n\"I only met him once,\" Geth muttered. \"I don't have much to go on. It would be best if I stayed away from people who know him well. You think this is easy?\"\n\n\"It's easier than dying in a corner of my dungeon,\" Tariic said. \"Show them.\"\n\nGeth wrinkled his nose\u2014then his face flowed and changed, becoming dusky-skinned and softly formed with wide eyes milkier than Pradoor's. Makka's lips pulled back from his teeth. \"Wax baby,\" he spat and Pradoor cackled.\n\nThe changeling looked more uncomfortable now than he had as Geth. He didn't look any more uncomfortable, however, than Daavn. The hobgoblin's ears flicked furiously, almost pulling back flat. He stepped in close to Tariic and tried to whisper in his ear. Makka caught some of his words. \"You can't trust a changeling, Tariic. They're treacherous\u2014\"\n\nTariic pushed him back. \"Daavn,\" he said coldly, \"this is Ko. Have you ever met before?\"\n\nDaavn drew a breath, then spread his hands. \"If we have, I didn't know it. You know what they say about changelings: they all look the same or else completely different\u2014\"\n\nThe lhesh cut him off. \"Ko, have you ever met Daavn of Marhaan before?\"\n\n\"Not as such,\" Ko said without hesitation. \"But I met a masked hobgoblin named Wuud once who sounded a lot like him. He hired me to do a job. That job landed me in your dungeon.\"\n\nDaavn's ears flattened. \"I don't know what he's talking about.\"\n\n\"I do,\" said Tariic. \"You tried to undermine my uncle by having Vounn d'Deneith kidnapped, Daavn. Somehow Vounn guessed it. She told Geth. Geth tried to warn me about you.\"\n\nThe warlord of the Marhaan was still and silent for a long moment. Finally, he bowed his head. \"I schemed against Haruuc, lhesh. But remember that I also guided you to power.\"\n\n\"You guided me as a boatman without oars or rudder guides his boat down the Ghaal\u2014I brought you with me.\" Even without crown or rod, it seemed to Makka that Tariic radiated command. \"The relationship between us is changed, Daavn. Remember that.\" He turned his head. \"Have you learned from this, Pradoor?\"\n\nPradoor sat for the space of five heartbeats, as if listening to some distant voice only she heard, then ducked her head as well. \"I have, lhesh.\"\n\n\"I am pleased.\" He gestured to Ko. Makka watched, his skin creeping, as the changeling's features once again shifted into those of Geth. Tariic spread his arms on the arms of the chair, sitting as if it were the blocky throne of Darguul. \"Now,\" he said, \"the real Geth could possibly be hiding anywhere in Rhukaan Draal. Our chances of finding him are slim. However, I'm certain that there must be someone who knows where he is.\"\n\nHe rose and strode to the false rod, plucking it from its velvet resting place and turning it in his hands. \"There could be several reasons Geth might want the rod. Perhaps to sell to another nation. Perhaps as some remembrance of Haruuc. In any case, the scheme isn't something he could have created on his own.\" His smile exposed his teeth. \"He's brave and stubborn, a good fighter, but not a schemer. He must have had help.\"\n\n\"Ashi d'Deneith,\" Daavn said.\n\n\"Ekhaas of Kech Volaar. Dagii of Mur Talaan. Munta the Gray. Any of those close to him.\" Tariic seated himself again, holding the rod at an angle against one outstretched knee. \"However, Ekhaas and Dagii are beyond our reach\u2014for now. Munta, if he is involved, is nothing. An old man with fading power. Ashi... Ashi is of interest.\" He flicked his ears. \"And protected by House Deneith.\"\n\n\"You've declared that the gods are not above you, Tariic,\" said Pradoor. \"Why should the dragonmarked houses be?\"\n\nTariic smiled and contemplated the false rod again. \"They'll fall in time,\" he said, \"but not yet. We need another way to reach Ashi.\"\n\n\"Or we forget her,\" Daavn suggested. \"What about Midian Mit Davandi? He's only here because you hired him. You're his only protector.\"\n\nBlood seemed to thunder in Makka's skull, driven by the recitation of the hated names\u2014and an abrupt understanding of what Pradoor meant when she talked about the turning of ages. Fate seemed to focus on him at that moment, as if it was the will of the Fury that he should be here, in this room, at this moment. He stepped forward, feeling like he walked through water. \"I can reach Ashi,\" he said. \"I can reach any of them. I hunt them already\u2014the Fury knows my oath.\"\n\nKo and Daavn flinched back. Pradoor smiled, her ears twitching. Tariic's eyes narrowed. \"Why do you hunt them?\" he asked. \"Who _are_ you?\"\n\n\"They destroyed my tribe's camp and turned my tribe against me. I have taken an oath of vengeance. The Fury guides me. She blesses this hunt.\" He crossed his arms over the bat-winged serpent on his chest. \"I am Makka.\"\n\nThe lhesh's ears pricked up. \"Makka?\" His eyes went to the sword at Makka's side and the bugbear knew he recognized it now. A thoughtful expression passed over Tariic's face, then he smiled. \"I have heard of you, Makka, though the stories I've heard are from another point of view.\"\n\nDaavn moved closer to him. \"Tariic, you can't let him\u2014\"\n\nTariic waved him to silence, gaze still on Makka. \"Priests of the Fury aren't known for their subtlety,\" he said.\n\n\"I'm no priest,\" said Makka, showing his teeth. \"I am the Fury's warrior. When I fight, I fight. When I stalk, I stalk. I can reach Ashi of Deneith for you\u2014and her house will know nothing of it.\"\n\n#\n\n# CHAPTER  \nSEVENTEEN\n\n# 25 Sypheros\n\nThe healing song took only moments to work through Marrow's flesh, but they were moments in which Ekhaas could hear commands, screams, and curses from beyond the trees. The instant the deep puncture closed, leaving a hairless patch the size of her thumb among Marrow's dense fur, Ekhaas rose and crept cautiously up the game trail that led out of the grove. Chetiin and Marrow came after her\u2014or at least she thought they did. Goblin and worg vanished in the shadows. Ekhaas had a sense that they were still close, but she couldn't see or hear them.\n\nAt the edge of the trees, she paused, sword ready, and looked out.\n\nThe tall grass of the hillside had been trampled by a fight. The sentry who had been so confused by Dagii's curt nod lay dead a little way up the slope. Blood from a slashed throat soaked the ground, but his bell-covered wrist remained outstretched.\n\nLithe forms in red garb and veils flowed over the hill like cats racing across a field. At the edge of the camp, hobgoblins and bugbears formed up into a perimeter two ranks deep. Shields and spear points flashed in the firelight. Ekhaas recognized the voices of Keraal and the two _lhurusk_ who commanded the soldiers shouting orders\u2014contradictory orders, she thought. The surprise of their attack spoiled by Dagii's strategy of bells, the Valaes Tairn prowled just beyond spear's reach, looking for an opening.\n\n\"The ranks are too thin,\" said Chetiin. \"They'll crumble.\" His scarred voice seemed to come from right beside her, but she had to look twice to find him and Marrow.\n\n\"At least the elves aren't riding.\"\n\nHis ears twitched. \"Not all of the Valaes Tairn fight from horseback. It doesn't make them any less deadly.\"\n\nEkhaas scanned the hillside for Dagii and found him hugging the wall of the ruined clanhold. A hobgoblin on his own\u2014the elves would turn on him as soon as they spotted him. He needed a distraction to give him the chance he needed to join his soldiers.\n\nHe looked down the hill and saw her. As if he guessed her thoughts, his ears went back and he shook his head. She raised her ears in response and bared her teeth. Then she focused her attention on a point of the hillside below the camp and behind the prowling elves. A breath drew up her song.\n\nThe magic passed her lips as a whisper, but burst out behind the elves with the roar of a tiger.\n\nThe attackers whirled. Whatever misgivings Dagii might have had about her causing a diversion, he took advantage of the moment's confusion. He charged, howling a battle cry. His sword slashed one elf across the back of the leg, taking him down, then cut deep into the belly of another who spun to meet him. Realizing they been tricked, more elves turned to him. The ranks of Darguul soldiers strained as warriors took an unthinking step, ready to defend their commander.\n\n\"Hold position!\" Dagii roared. \"Archers, give cover!\"\n\nFrom high up in the ruins of Tii'ator clanhold, arrows spat down on the ground between Dagii and the elves. One took an elf warrior, pinning veil to throat. Another struck a shoulder, but most only forced the elves to check their advance. A few arrows hissed up from the base of the hill\u2014the Valaes Tairn had archers in concealment as well\u2014but they fell short. The ranks of hobgoblins opened like a parting curtain and Dagii plunged through. He disappeared, but his voice rose. \"Archers, loose!\"\n\nArrows fell again, this time carefully aimed. The elf who had taken an arrow in his shoulder took a second in his chest. Other elves danced back, some struck, others simply avoiding the deadly rain. They were on all sides of the camp though, and the archers in the ruins were forced to divide their efforts. Ekhaas saw one of the hobgoblins silhouetted for a moment against the starry sky; the heavy bow he carried seemed too thick to bend, yet bend it he did, and another elf died.\n\n\"Form up\u2014double fortress!\"\n\nThe thin lines of the Darguul perimeter dissolved and reformed into two solid rectangles of soldiers parallel to each other with camp and campfires between them. Ekhaas caught another glimpse of Dagii. He'd claimed a helmet and a shield. Three other hobgoblins clustered around him for a moment: Keraal with his chain and the two _lhurusk_. Keraal and one of the _lhurusk_ were nodding, but the other one seemed inclined to argue. His hand thrust toward the second formation of soldiers.\n\nDagii's fist, still wrapped around the hilt of his sword, punched out and cracked him in the jaw. The struck hobgoblin staggered, then ducked his head and joined the distant column. Dagii sent the other _lhurusk_ with him, then he and Keraal melted into the first formation.\n\n\"Archers, hold!\"\n\nThe arrows stopped. For a moment, the night was still, the Valenar waiting for the Darguuls to move, the Darguuls waiting for the Valenar.\n\nThen a scimitar flashed up, whirling around the head of its wielder as she let out a high, musical war cry\u2014and abruptly the night was filled with war cries. The elves ran at the defenders, not as disciplined _dar_ might, but singly, each elf fighting alone. They darted and cut and dodged, their red garb like dancing flames in the night.\n\n\"Hold position!\" Dagii commanded. \"Hold!\"\n\nThe wave of elves broke and receded for a moment, and Ekhaas saw that for all its apparent ferocity, that attack had been a show. She couldn't see the faces of the veiled elves, but their posture was stiff and their weapons trembling. They were disappointed, she guessed. The mock attack had been intended to break the enemy formation and draw them out. The Darguuls had resisted.\n\nAn elf voice screamed. The wave crashed forward again.\n\nThis time, the scimitars flashed out in earnest. \"Forward ranks, attack! Archers, loose!\" roared Dagii.\n\nSpears thrust at the elves. A few found flesh, but the elves were clearly used to this tactic and many slid or ducked to come up inside the reach of the first rank\u2014only to encounter spears jabbed forward by the second rank underneath the arms of their comrades. At the same time, the first rank of Darguuls dropped their now useless spears and drew swords. Arrows rained down on those elves who hung back or tried to pull away, making retreat almost as dangerous as staying close.\n\nAnd yet a band of elves had leaped into the gap between the two rectangles of _dar_ , leaping bedrolls and campfires. They didn't, however, attack the defenders, and Ekhaas knew immediately what they were up to. They were going for the gaping charred doors of Tii'ator. They would try to take the ruined clanhold and seize the high ground from the hobgoblin archers.\n\nShe started to rise, to shout a warning, but Chetiin grabbed her hand and pulled her back down. \"Dagii knows!\" he rasped\u2014just as the warlord of the Mur Talaan shouted out, \"Rear ranks, close!\"\n\nThe rear ranks of each rectangle spun around and slammed together like the jaws of a vise. The running elves found themselves trapped. Scimitars turned against heavy _dar_ swords as they tried to fight their way clear.\n\nChetiin drew breath through his teeth. \"The elves will call for their archers soon. Marrow, with me.\" He turned black eyes on Ekhaas. \"Stay hidden! The elves will be looking for a spellcaster now. Hopefully they'll think you're fighting among the ranks in the camp.\"\n\nHe turned and disappeared without another sound into the shadows. Marrow padded along with him, cold vengeance in her eyes. Ekhaas looked after them for an instant, then up at the struggling elves and _dar_ , and made her decision. She wasn't going to stay out of the fight like a coward.\n\nShe rose to her knees, watching the battle and at the same time listening to the darkness. After a long moment, she heard a muffled thump, like a falling body, from the direction Chetiin had gone. At least one of the unseen elf archers was no longer a danger. She crept forward in the long grass, then stood, singing as she moved. Magic danced along her skin. She felt it as a tingling, a kind of scraping as if her flesh were being gently drawn apart; then, like a soap bubble, the feeling burst. The song flicked away from her and took on another form.\n\nThree more Ekhaases rose out of the grass alongside her. Thin echoes of song tied them to her\u2014what she did, they imitated. She strode up the slope of the hill, closer to the battle. Elves who had managed to escape the close fighting and retreat for the moment cried out as they saw her. Or rather, as they saw them.\n\nThe warriors acted just as she'd hoped they would, in the heat of battle seeing four figures, lightly-armed scouts perhaps, instead of just one surrounded by magical illusions. Three of the elves broke off and came gliding toward her, eager for an easy kill to dishearten their opponents. Ekhaas smiled and eased sideways a little so the Valaes Tairn were across the hill from her rather than uphill with Darguuls at their back\u2014\n\nOne of the elves brought up a bright throwing knife, hurling it with a snap of his arm.\n\nShe threw herself aside, a move that was both too slow and completely unnecessary. The knife flashed in the air and plunged through one of her illusory duplicates. Ekhaas felt as much as she saw her double wink out of existence.\n\nEven if the knife hadn't struck it, she'd given herself away. Not even the most coordinated troops would dive for cover with the same movement. She spoke a little Elven, more than enough to understand one of the elf warriors as he pieced together what had just happened. \"Spellcaster!\" he shouted. \"She's a spellcaster!\"\n\nNo time now to draw them closer to her. Ekhaas cursed, rolled back to her feet\u2014the elves were charging, bounding across the hill\u2014and sang again. Not an illusion this time. Not a diversion. Not a stunning burst of sound. The song that rippled from her lips was dark and deep, a haunting song that played across hearts like footsteps in an empty room or the distant cry of carrion birds. The strides of the elves faltered. Above their veils, their eyes grew wide. One began to tremble, his scimitar falling from his hand.\n\nThey turned and fled, gripped in the terror inspired by her song. They weren't the only ones\u2014two more, caught by the edge of the magic, fled with them. Another two, perhaps sensing a shift in the tide of battle, went too.\n\nSeven of the Valaes Tairn taken out of the fight. Perhaps not dead, but as good as until they stopped running. Her side of the hill was nearly empty of living elves. Only a handful were still on their feet. The fighting had shifted to the other side of the hill, beyond the remains of the Darguul camp, and even that sounded like it was growing less frantic.\n\nThe same commanding elf warrior whose whirling scimitar had signaled the Valenar attack seemed to realize the same thing. She thrust away from the hobgoblin she had been fighting, raised her scimitar again, and this time Ekhaas heard distinctly the orders she cried. \"Arrows! Feather me these dogs!\"\n\nBut no arrows fell out of the darkness except those loosed from the ruined clanhold. The elf's veil had been dragged aside in the fighting and Ekhaas saw her fine-featured face twist in rage. Her scimitar fell as the hobgoblin she'd thrust away came at her again and he went down with his shoulder half cut from his body. Ekhaas ran forward, but the elf was turning away, hand reaching into a pouch to produce a ceramic flask very much like the one the elf in the trees had carried.\n\nWith a swift motion, she hurled it into a campfire burning at the back of a knot of hobgoblins.\n\nThe fire erupted into a column of gold-white flame that blasted all those nearby, defenders and attackers alike, off their feet. The Darguul defenders took the worst of it though: their backs smoldered and two hobgoblins lay where they fell, unmoving.\n\nThe elf commander turned again, a second flask in her hand, and took aim at another fire. \"Stop her!\" Ekhaas shouted.\n\nA chain hissed out of the battle and wrapped around the elf's raised forearm. The whipping metal spun up her wrist and hand, hitting the flask\u2014and shattering it. Shards and golden dust rained down over half of the elf's face. Her eyes opened wide and she shrieked in agony. Red welts streaked her skin wherever the dust touched. She dropped her scimitar and groped at her face, but Keraal still held the end of his chain. With a tremendous heave, he yanked the elf off her feet to squirm on the ground. His free hand grabbed the fallen scimitar and drove it into her back.\n\nShrieks and squirms ended. Once again silence fell on the night as Valaes Tairn and Darguuls stared.\n\nThen the _dar_ were shouting in victory and the last of the elves were fighting to escape. Hobgoblins moved to pursue the red-garbed forms that darted into the night but Dagii's stern voice called them back. \"Let them go! The battle is ours!\"\n\nA cheer rose. A trio of bugbears grabbed Keraal where he stood, panting for breath, over the body of the elf commander, and hoisted him onto their shoulders. Keraal stared around in surprise and his eye fell on Ekhaas. He grabbed one of the bugbears and tried to point at her, but Ekhaas just shook her head and stepped back.\n\nDismissing her remaining illusory duplicates with a whisper of song, she went to find Dagii.\n\nShe found him walking among the victims and the survivors of the attack. He saw her and nodded, but stopped first beside a young warrior crouched over the body of a fallen elf, flipping through the folds and pockets of her clothing. The warrior glanced up, saw who it was, and sprang to his feet, thumping his chest in a salute. Dagii looked him up and down. \"Who are you?\"\n\n\"Faalo of Rhukaan Taash, _thevk'rhu.\"_\n\n\"You killed this elf?\"\n\nFaalo straightened. \"Yes. My first kill in real combat.\"\n\n\"A good clean blow.\" Dagii examined the wounds on the body. \"Well done.\" He clapped Faalo on the shoulder, a moment of contact between two comrades in victory. Faalo seemed to stand even straighter, his ears high and proud. Dagii released him and came to Ekhaas.\n\n\"I saw what you did,\" he said.\n\n\"Driving off seven elves or giving you a chance to join your soldiers?\" she asked him.\n\n\"I was thinking of the elves.\" His gray eyes narrowed. \"The diversion was not so well done. I could have made it back on my own. You put yourself at risk.\"\n\n\"At more of a risk than facing seven elves?\" Amber eyes met gray.\n\n\"Chetiin shouldn't have let you do that either.\"\n\n\"Chetiin went to deal with the elf archers.\" She dropped her eyes and looked him over. His armor had new dents and scratches. The links of mail protecting one side of his torso were broken and his stance favored that side, though no blood seeped through the padding beneath the armor. A thin bloody scratch traced the line of his jaw just beneath his helmet. She stepped around him, examined the stump of the arrow that still stuck out from the back of his shoulder, and snorted. \"I'll give you healing now.\"\n\n\"There are warriors who need it more than me.\"\n\n\"You are their leader. They look to you for command. You need to be healthy.\" She pushed him over to one of the remaining campfires. \"Take off your armor so I can get the arrowhead out.\"\n\nHis face flushed. \"Not in front of the troops!\"\n\n\"Why? I'm a _duur'kala_. I'm offering you healing.\"\n\nThe muscles of Dagii's jaw tightened and his mouth pressed into a thin line. He reached up\u2014a little awkwardly because of his side and his shoulder\u2014and pulled off his helmet. The shadow-gray hair that had come early to him fell lank and sweaty. Ekhaas helped him remove his mail coat and the padding beneath. Ekhaas started to peel away the light linen shirt he wore beside his orange-red skin but Dagii caught her hand. \"Leave it on,\" he said with a little embarrassment in his voice.\n\n\"It will be ruined.\"\n\n\"I have others.\"\n\nShe nodded. Dagii sat down on an abandoned pack and she went around behind him. Slowed by his armor, the arrow hadn't penetrated deep, but it had dragged bits of padding and linen with it into the wound. Ekhaas tore the hole in the shirt a little wider, then took a firm hold of the broken shaft and pulled. Crusted blood broke and fresh blood seeped out. Dagii grunted softly, but she could feel the tension in the broad muscles beneath her fingers. A leather flask had also been abandoned by the fire. She opened it, sniffed and tasted the contents, then sluiced water over the wound until it was clean. Then she pressed one hand over the hole and sang a healing song.\n\nDagii drew a short breath as the magic worked on him. Ekhaas could feel a little of the song as well, vibrant and energizing. She shifted the song, sending it deep into his flesh, and reached around him with her other hand to touch the place where an elf scimitar had broken his armor. He flinched slightly at the second touch, then relaxed into it.\n\nMaybe she allowed herself to sing slightly longer than was absolutely necessary.\n\nThe clearing of a throat made both her and Dagii jump a little bit. Keraal and the two _lhurusk_ stood a discreet distance away, carefully looking anywhere but directly at them. Ekhaas ended her song and stepped back. Dagii rose and the three waiting hobgoblins came forward as if they had only just seen the two of them standing there. All three were smiling, even the _lhurusk_ that Dagii had struck. \"A triumph, Dagii!\" he said.\n\n\"You fought well, Uukam\u2014and you, Biiri.\" He nodded to both _lhurusk_ , then to Keraal. \"And you, Keraal. _Ta muut.\"_\n\nKeraal didn't bend his head. \"Ekhaas _duur'kala_ turned the tide,\" he said. \"Her song started them running. But a triumph?\" His ears lowered and he shook his head. \"I wouldn't call it that.\"\n\nBiiri and Uukam looked ready to protest but Dagii raised a hand to them. \"I agree with Keraal. How many warriors did we lose?\"\n\n\"About half,\" said Biiri. \"Twenty or so. It could have been worse.\"\n\n\"Yes,\" said Dagii, \"but it could have been better. I count ten dead elves.\"\n\n\"Five more fled at the end,\" said Keraal. \"Ekhaas forced seven away.\"\n\n\"Four archers lie dead in the dark. Plus three who tried to ambush us.\" Chetiin came strolling past Ekhaas.\n\nThe reaction from Keraal, Uukam, and Biiri was immediate. They grabbed for their weapons and dropped into defensive crouches, their ears back and their teeth bared. _\"Shaarat'khesh!\"_ snarled Uukam.\n\n\"Easy!\" Dagii said. \"He's a friend. He's\u2014\"\n\n\"I'm Maanin,\" said Chetiin smoothly. \"I'm with Dagii to redeem the honor of the Silent Clans.\" He crossed his arms and waited. Slowly the three warriors lowered their weapons, though Keraal was the last to do so. His ears twitched and he looked to Dagii and Ekhaas, then nodded.\n\n\"Maanin,\" he said. He looked back to Dagii. \"One of the Silent Blades instead of one of the Silent Wolves?\"\n\n\"Do you want to argue with four more elves dead?\" Ekhaas asked him.\n\nKeraal's eyes narrowed but he bent his neck in the slightest of nods.\n\n\"Maanin's place here is not the issue,\" Dagii said. \"Ten elves dead here, four and three dead below the hill, twelve fled in fear or defeat.\" He put his hands on his hips and looked around at all of them. \"Twenty-nine Valaes Tairn sent against forty Darguuls. If not for Ekhaas's song, I think that more than half our number would be dead right now. Don't claim a victory here\u2014claim a lesson learned.\"\n\nThe others had no response.\n\nDagii nodded. \"Uukam, Biiri, give the warriors a short time to celebrate, then order them back to discipline. The camp needs to be restored and sentries set again. It's possible the elves may try their luck again. Keraal, pick out those who fought worst in the battle\u2014they're to collect the dead and bury them in the morning.\"\n\nKeraal's ears flicked. \"Those who fought worst are already among the dead,\" he said with the ghost of a smile.\n\nDagii returned the smile, then jerked his head dismissing all three. When they had gone, he looked down at Chetiin. \"Maanin?\"\n\nThe goblin seated himself on the pack by the fire. \"You don't want to be seen with Haruuc's assassin, do you? Trying to defend my innocence to all your warriors would only raise more questions. Better that I be someone else for a while.\"\n\n\"You could have stayed in hiding,\" said Ekhaas. \"Keraal knows something isn't right.\"\n\n\"Hiding isn't always an advantage. Tell Keraal the truth later. When there are fewer things to concern him\u2014and you.\" Chetiin glanced up at them. \"I followed the fleeing elves a short way. I doubt that they'll return tonight, but the odd thing is that they had no horses.\"\n\nEkhaas narrowed her eyes. \"You told me that not all Valaes Tairn fight from horseback.\"\n\n\"They don't,\" said Dagii, \"but all of them use horses for transportation. If they didn't ride, their camp must be close.\" His smile became grim. \"We can scout them out.\"\n\n\"Marrow can track them by scent,\" Chetiin said.\n\nDagii nodded. \"Let me find some light armor. Something that won't give us away.\" He looked at Ekhaas. \"You'll come?\"\n\n\"Try to stop me.\"\n\n\"You should find some light armor and a less rattling weapon for Keraal and bring him too,\" said Chetiin.\n\nDagii's ears rose at the suggestion. So did Ekhaas's.\n\n\"He's already suspicious of you,\" she said.\n\n\"Suspicions are like gardens\u2014left untended, they grow wild.\" The goblin's thin lips pressed together for a moment. \"But in this case, I like the idea of an extra sword at my side. The Valaes Tairn are cunning.\"\n\n\"I'll find Keraal,\" said Dagii.\n\nKeraal, outfitted in leather with a sword at his side, reacted to Marrow with surprise at first, then gave her a deep, respectful nod. The worg growled something to Chetiin, who smiled.\n\n\"What did she say?\" asked Keraal.\n\n\"She appreciates your gesture of submission but says that only pups present the back of the neck.\"\n\nKeraal's ears flicked and he addressed himself to Marrow, \"I doubt I would survive your tenderness, mother.\"\n\nMarrow's tail waved rapidly, her ears flipped forward, and her mouth opened so that her tongue hung out. She looked, Ekhaas decided, amused.\n\n\"Humor, Keraal?\" asked Dagii.\n\nThe other warrior's mouth set in a firm line. \"It happens sometimes,\" he said.\n\nMarrow led them into the night. The campfires faded behind them, obscured by trees and the rolling landscape until only the sharp finger of the ruined clanhold was visible against the sky. Biiri and Uukam had orders to break camp and return to the main army if Dagii didn't return by mid-morning. They had tried to persuade him not to go, but Dagii had insisted with the same argument he had given Ekhaas: he needed to see the Valaes Tairn forces for himself.\n\nFor a while, the trail of the fleeing elves was so easy to see that Ekhaas could have followed it herself. She supposed that the elf warriors she had frightened with her song had made it, driven by their fear without a thought for stealth. Here and there, blood made a smear on the ground or on a leaf, evidence that at least one of the elves had been wounded in the battle. As the obvious trail of broken branches and crushed grass faded, Marrow moved to the fore. She cast about, sniffing, then stopped, whuffed sharply, and growled at two trees.\n\nChetiin found a long branch on the ground and approached the trees cautiously, tapping ahead with the branch. It caught something. Chetiin peered at the trees though Ekhaas could see nothing. Taking a few steps back, the goblin flung the branch.\n\nThere was a snap and a short hiss. The branch jerked and fell apart in three pieces, the leafiest piece somehow remaining suspended and bobbing gently in the air. \"Come look,\" Chetiin said. \"It's safe now.\"\n\nEkhaas ventured forward. Three thin dark wires curled up close to one of the tree trunks. The leafy branch was caught in the embrace of one. A broken tripwire showed how the trap had been triggered. \"They were stretched between the trees,\" said Chetiin. \"A goblin walking into that trap would have been seriously injured.\"\n\n\"Will there be more traps?\" asked Keraal.\n\n\"There might be,\" Chetiin admitted. \"But I think it's more likely this was set as a warning, to deter pursuers or at least make them wary and slow them down. We should be fine.\"\n\n\"Should be?\" Keraal said.\n\nChetiin shrugged.\n\n\"Keep alert,\" ordered Dagii. \"Marrow, show us the way.\"\n\nThe elves must not have anticipated the presence of a scent-tracker\u2014the worg was able to follow their trail with ease, even when there was absolutely no visible sign of their passage. Once or twice, false trails appeared, seemingly accidental traces indicating that the elves had turned this way or that, but Marrow led them right past. Just as Chetiin had suggested, there were no more traps. Accounting for variations forced by the landscape, it seemed to Ekhaas that they were heading consistently to the east.\n\nThe realization brought a chill to her flesh. She leaned close to Dagii. \"We're heading for the Mournland.\"\n\n\"I know.\" His voice was taut. \"They must make their camp close to the border. No one would be likely to wander this close.\"\n\nThe guess was proved wrong as they came around the shoulder of a hill. Across a broad, very shallow valley the dead-gray mists of the Mournland's border rose into the sky. Ekhaas had been close to the mists before, close enough to hear the screams and roars of the unseen monsters that made the cursed land beyond their home. Tonight, in this place, the mists were quiet, hanging like a drifting, billowing curtain. The valley, marked by the small, dry riverbed, lay empty but for a few withered trees under the moonlight. There was no elf camp.\n\nThey all stopped and stared. Ekhaas looked away to the north and the south. \"Maybe they turned aside here,\" she said.\n\nMarrow's hackles rose and she growled. \"They didn't,\" Chetiin translated.\n\n\"Who would want to make camp in the Mournland?\" asked Keraal with a grimace.\n\n\"Someone who wanted to hide from prying eyes or magics,\" said Dagii. \"Someone desperate or frightened enough might flee there to throw off pursuit.\"\n\n\"Do you think the Valaes Tairn were _that_ frightened of us?\"\n\n\"No,\" Dagii said. \"All the more reason to believe they've camped there.\" He slipped off down the gentle slope into the valley, moving from stunted tree to stunted tree.\n\n\"He is mad, isn't he?\" muttered Chetiin, but he moved down after the young warlord.\n\nOne by one, they followed Dagii in to the valley. Only Marrow didn't stick to the dubious cover of the trees, instead flowing like a sleek black shadow along the faint rise and fall of the valley floor. Nose to the ground, she trotted all the way to the very edge of the mists before returning to join them in the shadow of the crumbling riverbed. She snarled and whimpered, and Chetiin said, \"That's the way they went, but the mists smell\"\u2014he paused, searching for the right word to translate the worg's language\u2014\"wrong. Unnatural.\"\n\nEkhaas searched her memory for anything she'd heard of the Mournland. \"They say that laws of life and death are suspended there\u2014that wounds don't heal and dead flesh doesn't decay. Water, plants, and animal life are tainted.\"\n\n\"It's true,\" said Chetiin, his scarred voice unexpectedly soft. \"I've been there. Don't count on your healing songs, Ekhaas. Don't count on anything\u2014nothing is as it seems. We'll need to be careful. If the Valenar raiders have made camp inside the border, they'll be extra vigilant because of the Mournland's dangers.\" His face tightened. \"The mists may be a problem. They're disorienting.\"\n\n\"Won't Marrow be able to track through them?\" asked Dagii.\n\nChetiin gave him a curt nod, \"Yes, but they confuse more than just your sense of direction. If you feel anything... odd, if you feel like you just want to lie down and sleep, _fight it.\"_\n\n\"We're going in and out,\" Dagii said. \"We won't stay long and we won't fight unless we have to. We see what we need to of the Valenar camp and then we leave.\" He looked around at each of them, then nodded to Marrow. The worg loped up the bank, Dagii close behind.\n\nThere was no need for a warning to stay together. Ekhaas knew that they all understood it implicitly. The wall of mist drew closer and closer as they climbed the valley's far slope\u2014then all at once, they were inside it, as if the Mournland had reached out to claim them.\n\nMoons and stars were completely cut off. By rights, she shouldn't have been able to see any better than a human in the dark, but somehow she could. A dim radiance seemed to permeate the mists, as if they caught the moonlight, rendered it thick and opaque, and smeared it through the air. She could see no more than two paces in front of her. Chetiin was a shadow and Dagii, walking beyond him, a ghost. Ekhaas felt no shame in reaching ahead to put one hand on Chetiin's shoulder and reaching back so that Keraal could grasp the other.\n\nThe mists were slightly cool, but not cold. If she stopped moving and the heat of her body warmed the air around her, she probably wouldn't feel anything at all. Sounds were at once magnified and muffled as if she held a great glass vessel around her head. Her footfalls on the ground\u2014which was dry in spite of the mists\u2014were as quiet as if she walked on green grass, yet her breathing was loud in her ears. She swallowed and heard it like a big stone dropped from a height into a still pond.\n\nIt was impossible to tell if they were moving. The mists were constant, the rise of the land\u2014or maybe its fall\u2014so gradual that it could have been level. She understood what Chetiin had meant when he said the mists could be disorienting. It would be easy to wander in circles. Easy too to simply stop and stand still...\n\n\"Ekhaas.\" Keraal's voice. A push from behind her. Startled, she stumbled. Her hand left Chetiin's shoulder. Instantly, the goblin's small hand seized hers in a hard, rough grip.\n\n\"Keep walking,\" he said.\n\n\"I thought I was walking.\"\n\n\"It's the mists.\" He sounded tired.\n\nThere was a muffled sob from ahead. \"Dagii?\" Ekhaas called.\n\n\"It's nothing.\" His voice was thick.\n\n\"Nothing?\" Keraal now. Ekhaas looked over her shoulder. His face was drawn and wracked with guilt. \"My clan is dead. I led them to their destruction. You know my grief, Dagii. Tell me yours.\"\n\n\"No, I can't. I... can't.\" Dagii struggled. \"I\u2014\"\n\n\"Fight it,\" Chetiin murmured like a distant echo. \"You must fight it.\"\n\nEkhaas ground her teeth together and dragged up a song from inside her. There was magic in it, but not the focused magic of a spell. Rather it was a simple magic, just as it was a simple song, the kind of tune heard in every _dar_ drinking hall\u2014or the drinking halls of any other race for that matter. Into it she poured all of the bawdy joy that she could, singing it as loud as she dared.\n\n_\"Ahhh, when I was a baby, my mother gave me suck_.\n\n_She changed my clothes and wiped my nose and tied my hair for luck_.\n\n_But now that I'm a warrior, I hold other things more dear_.\n\n_I love my sword, I love my song, but most I love my beer!\"_\n\nShe heard Keraal snort in amusement. She squeezed his hand and Chetiin's. \"Sing!\" she said, and launched into the chorus.\n\n_\"Beer! I love my beer! Beer I love! I love my beer! Be-eer-eer-beer!\"_\n\nSlowly and dirge-like at first, the men joined in, but their song gained strength until even Dagii sang _\"Be-eer-eer-beer\"_ with an offkey lustiness. By the time she launched into the second verse, their joined hands were swinging back and forth in time to the song.\n\n_\"When I was a child, my father gave me sticks_.\n\n_He told me they were spears and blades and taught me many tricks_.\n\n_But now that I'm a warrior, I keep my weapons near_.\n\n_I have my sword, I have my shield, I also have my beer!_\n\n_Beer! I have my beer\u2014_ \"\n\nIn no story that Ekhaas had ever told or even heard had the heroes crept up on their enemy while simultaneously singing a drinking song. In fact, she was fairly confident that no _duur'kala_ had ever heard of such a thing. There was no dignity to it. There was precious little stealth. If there had been elves lurking in the mist\u2014though she couldn't imagine that they would linger here\u2014they probably would have dismissed the whole spectacle as an illusion too odd to be believed.\n\nAnd yet it was ridiculously fun. By the time Marrow came to a halt and huffed at them in warning, they were all laughing softly, the terrors of the mist banished. Up ahead, the mists were thinning and honest moonlight filtering through. Marrow sat down on her haunches and growled at them. Keraal, wiping tears out of his eyes, choked, \"Yes, mother! As you say, mother!\"\n\nChetiin chuckled. Marrow actually looked bewildered.\n\nDagii drew a deep breath, steadying himself. \"Move to the side,\" he said, gesturing. \"If the elves come this way frequently, they may have sentries posted.\"\n\nThey followed him, each of them struggling to suppress the lingering humor of the song that had seen them through the mists. Ekhaas gulped lungfuls of air, pride warming her belly. Dagii caught her eye and gave her a thin smile that was as rewarding as gold.\n\nAs they emerged from the mists, she could feel the wrongness of the Mournland that Marrow, through Chetiin, had tried to describe. The air felt too thin, the moonlight too harsh. The stars didn't twinkle but instead seemed hard as ice. There was a smell in the air that reminded her of a lightning strike or certain powders burned in an alchemist's furnace. Even the land had changed\u2014somehow they stood just below the rocky brow of a steep slope, though she was certain that they hadn't climbed anything more than a gentle grade. Looking back along the brow, she could see a gap, probably the start of a way down the other side and likely the way that the elves had gone.\n\nThe boulders lining the brow of the slope made climbing easy, but Chetiin still reached the top before any of them. Staying low to avoid presenting a betraying silhouette, he stuck his head up over the edge, froze for an instant, then ducked back down, his eyes very wide. With one hand, he waved them all forward. With the other, he gestured for absolute silence.\n\nDagii reached the edge next. Ekhaas watched his ears stand before falling back flat against his head. Then she was at the edge, too, and peering between two boulders down into another wide valley\u2014\n\n\u2014at a camp that stretched from one side of the valley to the other. Tents made a small town. Horses picketed together at the center of the camp made a herd that could have raised a noise like thunder if they'd been running. Next to the picketed horses stood a pavilion flying a long swallow-tailed banner with a pattern of stars. There was activity at the pavilion. The survivors of the attack on Tii'ator were likely reporting their defeat. Ekhaas tried to guess at how many elves moved beneath the harsh moonlight and how many more might be asleep in those tents. Far more than the four or five warbands Tariic had anticipated in his rousing speech in Khaar Mbar'ost.\n\nDagii touched Ekhaas's hand and motioned for her to go back. All of them slipped carefully to the ground and joined Marrow back at the edge of the mists.\n\n_\"Maabet!\"_ cursed Keraal. \"That's a full Valaes Tairn warclan. They're hiding an entire warclan in the Mournland! How did they get them all through the mist?\"\n\n\"I don't know,\" Dagii said tightly, \"but they must have some trick. How many do you think there are?\"\n\n\"Three hundred,\" said Keraal at the same moment as Ekhaas and Chetiin said, \"Four hundred.\"\n\nDagii nodded. \"We'll assume the worst. Four hundred Valaes Tairn warriors. Based on our experience tonight, enough to crush our troops without lathering their horses.\"\n\n\"What do we do?\" asked Ekhaas.\n\n\"We laugh our way back through the mists,\" Dagii told her in tones that brooked no laughter whatsoever. \"We return to Tii'ator, dispatch all of our messenger falcons in the hope that at least one makes it to Khaar Mbar'ost, then we run back to the main army, make a stand outside Zarrthec, and hope we can slow them down.\"\n\nKeraal grunted agreement. Chetiin nodded. Ekhaas looked at all three of them. \"Slow them down? If they get through us, they won't have far to go to reach Zarrthec.\"\n\nDagii bared his teeth. \"They're not interested in Zarrthec, Ekhaas. They didn't bring that many warriors to attack a village, and if they wanted to harry the countryside they would be doing it already instead of hiding here. A force of that strength is meant for a big target. They're planning an attack on Rhukaan Draal.\"\n\n#\n\n# CHAPTER  \nEIGHTEEN\n\n# 27 Sypheros\n\nSafe,\" said Aruget and led Ashi and Vounn across a small square. Krakuul watched their rear. Ashi could have told Aruget the square was clear of assassins, assailants, thugs, thieves, or any other danger. Darkness was falling, but there was still enough light for her to see the streets of Rhukaan Draal. The houses that lined the streets here were built of stone with high, windowless outer walls that could have seen off a casual attack. It was as wealthy and peaceful an area as the city could boast\u2014a pleasant place for a dinner party, a terrible place for an ambush. She scowled at Aruget's back.\n\nVounn poked her in the ribs. Ashi forced her face to return to neutrality and wondered what Krakuul thought of Aruget's new caution. She hoped he was as frustrated as she was. The lump at the back of her skull, two days old now but still tender, throbbed as if to torment her.\n\nDown the street, guards stood before a door marked with the crest of House Cannith. A warforged, metal body swathed in a gown stiff with intricate embroidery, appeared from inside the doorway. \"Lady Seneschal Vounn, Lady Ashi, welcome to Cannith enclave.\"\n\nIn marked contrast to a mask-like face and green glass eyes, the warforged's voice was lively, warm, and surprisingly delicate. Even more than the gown, the voice and a certain way of moving made Ashi think instinctively of the warforged as a woman in spite of her muscular frame and bald head. The effect was disconcerting, and she had to force herself to nod when the warforged offered a graceful curtsy.\n\nVounn didn't even blink. \"Thank you, Stitch,\" she said. \"We're late. Has dinner begun?\"\n\nThe warforged ushered them into a narrow, high-ceilinged entry hall. \"Lady Dannel has waited for you, lady. I'll show you to the library. If your guards care to go to the kitchen, they'll find refreshment.\" She indicated an unobtrusive door just inside the main entrance.\n\nAruget looked at Vounn, scowling, but when she gestured he and Krakuul vanished through the door.\n\nThe lines of the house were clean and fine, the walls and floors faced with polished stone, yet there was a strange echo about the place as if it was more than half empty. Ashi tried to sneak a look around as Stitch led them through the entry hall.\n\nThe warforged caught her curious glances. \"The enclave in Rhukaan Draal was built at the same time that House Cannith was constructing Khaar Mbar'ost and other projects for Lhesh Haruuc Shaarat'kor,\" Stitch said. \"At the time, there were many more members of the house in Rhukaan Draal than there are now. But we cling to our pride, don't we?\"\n\n\"Uh... yes,\" said Ashi, but Stitch had already turned away to open a fine wooden door ornamented with nothing more than its natural grain.\n\n\"Lady Seneschal Vounn d'Deneith and Lady Ashi d'Deneith,\" she announced.\n\nIn a library with walls lined with velvet drapes and dark bookshelves, nearly a dozen people looked back at them. Ashi saw Pater d'Orien and Esmyssa Entar ir'Korran. She recognized most of the others: the ambassadors of Breland, Karrnath, and Aundair, the envoys of Houses Vadalis and Medani. Dannel d'Cannith, envoy of her house, strode up to Vounn, welcoming her with a smile.\n\nJust the wealthy and powerful of Darguun's visiting dignitaries gathered for dinner\u2014as if there was absolutely nothing wrong.\n\nTwo days before, with the sounds of the city celebrating Tariic's coronation drifting in the window, Ashi had sat with a cold cloth pressed to her head in the chambers she shared with Vounn and listened to the hobgoblin and the lady seneschal argue.\n\n\"You hit her?\"\n\n\"She would have tried to attack Daavn and his guards, Lady Vounn.\"\n\n\"So you _hit_ her?\"\n\n\"Lhesh Tariic sent Daavn after Geth. If Ashi had attacked him, she would have been interfering with the lhesh's orders.\" Aruget gave Vounn a level look. \"Would Deneith have been able to protect her?\"\n\nThat won the argument for him. Vounn gave him cold thanks for his discretion, sent him away to find out what had happened to Geth\u2014and turned her attention to Ashi.\n\n\"You protected me with your dragonmark. You told me not to trust Tariic. Geth has clearly done something to anger him. You're clearly involved.\" The lady seneschal's expression, normally as calm and controlled as still water, was like a storm. \"No more evasions, Ashi. What's going on?\"\n\nThere was no way around it. Ashi had given too much away already and even when she tried to hold back, it all came rushing out. She confessed everything, from the pact that she and the others had made to keep the rod's power a secret to Haruuc's discovery of the curse, to Geth's decision to seize the rod after the assassination and Midian's idea to present the new lhesh with a false rod, to the fear that had pierced her at Tariic's reaction during the coronation. The only thing she managed to keep secret was Tenquis's name.\n\nRed spots of color appeared high on Vounn's cheeks. She sat down stiffly and didn't move or speak for a long, long time. When she did finally speak, it was to say, \"You're leaving Darguun.\"\n\nAshi's head snapped up. \"I won't! Geth needs me now more than ever!\"\n\n\"You would rather be arrested for conspiracy?\"\n\n\"It's not a conspira\u2014\"\n\n\"It is,\" Vounn said harshly. \"You may have had the best of intentions, but what you have done is conspire against the throne\u2014and in every nation of Khorvaire, that's a crime. If you were a Darguul, it would be treason. Is there evidence? You're Geth's friend, so suspicion will fall on you, but is there hard evidence?\"\n\nHer gut felt numb. \"Geth's word, but Geth would never betray me or any of us.\"\n\n\"If Tariic is serious about rooting you out, he may not give Geth a choice.\" Vounn closed her eyes for a moment, then opened them and put her hands on Ashi's shoulders. Her voice was tense but not so angry. \"If you had attacked Daavn, Tariic would have had a reason to arrest you\u2014you should thank the Host that Aruget stopped you. He had the right of it. Unless he has evidence that you've done something wrong, Tariic would be putting himself against Deneith if he tried to take you captive.\"\n\n\"And if he has a reason to arrest me?\" Ashi asked.\n\n\"Then I'll have to give you to him.\" Vounn looked into her eyes as she said it. \"He won't want to anger us, but we don't want to anger him. Deneith values our contracts too highly. You've dug yourself a grave, Ashi. We need to get you out of Darguun before you're forced into it.\"\n\n\"Abandon my friends or you'll abandon me?\" Ashi gave her a bitter smile. \"What if I don't give Tariic any reason to arrest me? What if Geth doesn't betray me?\" The smile twisted a bit. \"What if he escaped Daavn? I'll have left my friends in danger for nothing.\"\n\n\"You want to take a chance on that?\"\n\n\"It's what you're doing.\" She lifted her chin stubbornly. \"My friends took a chance on me once, Vounn. They gave me the strength to leave the Bonetree Clan.\"\n\n\"Some people would say you repaid that debt by giving yourself up to House Deneith when you had to.\"\n\n\"Are you one of them?\"\n\nVounn's lips pressed into a thin white line and she looked away\u2014but anything she might have said was interrupted by a knock on the door and Aruget's entrance. The hobgoblin must have sensed the tension in the room. His ears flicked. \"Should I come back?\"\n\n\"No,\" said Vounn. \"Report. What have you found out?\"\n\n\"I've seen Geth,\" Aruget said. Ashi's heart gave a lurch. \"He walks with Tariic in the hall of honor in advance of the coronation feast.\"\n\n\"How did he look? Was he a captive?\" Ashi asked.\n\nAruget shook his head. \"He didn't walk like a captive. He looked uninjured, though he had changed his clothes since the coronation. There was a crowd around Tariic\u2014I didn't want to get too close.\"\n\n\"Was Makka\u2014the bugbear from the coronation\u2014there?\"\n\n\"No.\" Aruget hesitated, then added, \"I went past Geth's chamber but there were guards outside his door and I didn't try to go in.\"\n\n\"It would have been suspicious if you had,\" Vounn said, nodding. \"What about Daavn? Did you see him at all?\"\n\n\"No.\"\n\nAshi stood up. Between her argument with Vounn and Aruget's news, her blood seemed to boil. Her head felt light. Geth hadn't been captured\u2014or had he? If it had been anyone else, she might have been afraid that Tariic had used the Rod of Kings to command that they stay by his side, but thanks to his bond with the Sword of Heroes, Geth was immune to the rod's influence. There had to be some reason he'd stayed with Tariic, though. \"I need to talk to Geth. We need to know what happened.\"\n\n\"Sit down, Ashi.\" Vounn's eyebrows drew together as she thought. \"Is it possible that there's another explanation for what happened at the ceremony? Geth doesn't like spectacles\u2014and he did seem friendly with Tariic as they left the throne room.\"\n\n\"At which point he bolted for his room like a rabbit.\" Ashi shook her head, teeth clenched. \"It's not my imagination. Something is wrong, Vounn. If I can talk to Geth\u2014\"\n\n\"You can't,\" Vounn said with a note of finality. \"You are going to stay in these chambers while we figure out the status of things.\" Ashi started to protest, but her mentor silenced her with a raised finger. \"I will grant you that the situation doesn't seem as dire as I thought. Maybe Geth is on good terms with Tariic and there's nothing to worry about.\"\n\nAshi saw Aruget's ears twitch and rise at the hint of trouble.\n\nVounn ignored him. Lips drawn down at the corners, she said, \"You can stay in Darguun for now\u2014at least until we know whether you're in danger or not. But there are two conditions.\" She held up a finger again, then added a second one. \"First, don't give Tariic any reason to take an interest in you. Stay in our chambers and keep your head down\u2014I'll pass word that you're not feeling well\u2014while Aruget and I make a few discreet inquiries. Second, we make arrangements to get you out of Darguun, and when I tell you to leave, you _leave.\"_\n\nShe lowered her hand. \"Your friends took a chance on you. You're taking a chance on your friends, and now so am I. Does that satisfy you?\"\n\nIt would have been impossible to be completely happy at that moment. Ashi still had too much anger and fear inside her and Vounn's face was set in spite of her offer of compromise. It was also, Ashi knew, the best aid she could hope for.\n\nShe nodded.\n\nThe long table in the Cannith dining room had been set with delicate porcelain plates ringed with silver and remarkable goblets of cut glass. The big candelabras that marched down the center of the table were likewise cut glass. The flickering candlelight\u2014no cold, steady magical light here\u2014danced through them. Esmyssa Entar ir'Korran, seated in a special tall chair, leaned between two of the dazzling pillars. \"We haven't seen you in Tariic's court since the coronation,\" the Zil ambassador said.\n\n\"I've been ill,\" said Ashi.\n\nEsmyssa gave an understanding nod. \"I'm so glad that you were able to come tonight, though,\" she said. \"You weren't in Rhukaan Draal long enough to join one of our dinners before Haruuc sent you off. Then of course, the business with the Gan'duur and mourning afterward. I'm sure you'll find the evening interesting.\"\n\n\"I hope so,\" Ashi said with a politeness that she didn't really feel. \"Vounn insisted that I make the effort to attend.\" She shifted aside slightly as a servant's arm reached past to place a shallow bowl of soup in front of her. It was creamy, fragrant, and golden. Seated on Ashi's left, between her and Vounn, Pater d'Orien breathed deep.\n\n\"Ah,\" he sighed. \"Aundairian. Beautiful. Proper food from the Five Nations\u2014not that I have anything against _dar_ cuisine, but sometimes you want to sit down with friends over a meal that reminds you of home. I'm sure you understand what I mean, Senen Dhakaan.\"\n\nOn the other side of the table, Senen's ears bent slightly. \"More than you know, Lord Pater,\" she said. She picked up a spoon and stirred the soup dubiously. \"It seems rich.\"\n\n\"Yes, well, give it a try. I've never met anyone who doesn't like it.\"\n\nSenen started to scoop up a little of the soup, then glanced at Dannel d'Cannith, chatting casually with Vounn while the rest of the table was served. The ambassador of the Kech Volaar set her spoon down again and sat back stiffly. Ashi understood her discomfort in the situation\u2014it reminded her strongly of her own first experience with the etiquette of formal dining in the style of the Five Nations. To tell the truth, she preferred _dar_ feasts and the fashion of eating as food was presented. But seeing Senen here at all was a surprise. It had taken quite some time for the strongly traditionalist hobgoblin to find common ground to even talk with Vounn. To find her eating with non-Darguuls was a sign that she was opening up a little bit more.\n\nIf having Senen there was a surprise, though, it was nothing compared to the other gnome who had stood at Esmyssa's back when they'd entered the library. Midian now sat at the far end of the table, chattering about Dhakaani history to the half-elf viceroy of House Medani. As they'd passed from the library into the dining room, Ashi had managed to exchange a few words with Midian\u2014a very few words.\n\n\"What are you doing here?\"\n\n\"I suspected you'd be coming, so I talked Esmyssa into bringing me as a guest.\"\n\n\"Have you seen Geth since\u2014?\"\n\nHe'd interrupted her, dropping his voice. \"We'll talk later.\"\n\nThe last guest was served. Dannel d'Cannith picked up her spoon, and everyone at the table began eating. The soup was indeed rich. Pater slurped it up with a look of pleasure on his round, rough face. Ashi watched Senen work through her bowl with the dogged determination of someone set to a chore.\n\nConversation flowed around Ashi, light and casual. Esmyssa attempted to engage Senen with questions about the ancient history preserved by the Kech Volaar and the other Dhakaani clans. On Ashi's right, the envoy of House Vadalis, a lean man named Kavrin, struck up a question about the wildlife of the Shadow Marches. Ashi described the beasts and monsters she had seen as a hunter of the wild swamps, a pleasant distraction as the soup bowls were removed and replaced with fish poached in Brelish stock. Fish gave way to pork roasted and sauced in the Karrnathi style. Ashi kept her eyes and ears alert. Around the meat course, Vounn had said\u2014that was when it would happen.\n\nAnd, just as she picked up her fork, it did. Seated beside Dannel and across from Vounn, Laren Roole, the ambassador of Breland, leaned a little forward and asked Pater, \"How goes the process of supplying Darguun's army?\"\n\nPater sipped a little wine\u2014Ashi saw abruptly that the servants who had stood behind the table, ready to refill empty glasses, had departed and that only the warforged Stitch remained\u2014and said, \"It goes well. Orien has lent an aide to Dagii's quartermaster and he sends me reports. There was some expectation that the Valenar might attack the supply wagons, but there haven't been any attacks yet.\" He raised his glass. \"Pray Kol Korran keeps it so!\"\n\nLike the first punch thrown in a brawl, the question changed the tone of conversation around the table. Abruptly Ashi found herself in a bubble. All around her, everyone was talking about the war and Tariic, but perhaps believing that a junior envoy would have nothing to add, no one talked to her. She didn't dissuade them. Not being talked to gave her a better chance to listen. Esmyssa turned to the Aundairian ambassador on her other side to ask his opinion of the impact of the war on Khorvaire at large; the Aundairian was tight-lipped and grim. Kavrin d'Vadalis discussed the Valenar cavalry capabilities with the Karrnathi ambassador on his right. Senen and Midian both became sought-after partners in the discussion as they related stories and histories of past conflicts between _dar_ and elves, sometimes glaring at each other and arguing when what they told clashed. Dannel, Laren, Vounn, and Pater quietly spoke of Tariic's new power and what he might do with it. In the two days since his coronation, he had divided his time between public appearances and the assembly of warlords, stirring pride in the Darguuls and whipping up sentiments against the Valenar\u2014and all elves by extension.\n\nA game was in play around the table, with each ambassador and envoy offering up a little bit of what they knew in return for new knowledge form the others. Here and there, hints revealed plans. Vadalis hoped to convince Tariic to purchase their strong and tough magebred mounts for his army. Breland would increase patrols along their side of the Seawall Mountains in case Darguul warlords were stirred up too much, while Karrnath, having more direct experience with the Valenar, wondered if the distracted elves might pull away from their northern territories. Zilargo had hopes that the war would be brief and Tariic would settle into the role of peaceful, _predictable_ ruler. Everyone talked about House Lyrandar and Sindra d'Lyrandar's conspicuous absence from Rhukaan Draal.\n\n\"We gather with our own,\" Vounn had told Ashi as they'd prepared for the dinner. \"We know what we're doing, we know what we trade. We each bring a thread and from them weave a tapestry none of us could have woven alone. Everyone leaves on even footing.\"\n\nAnd Vounn, Ashi saw quickly, was one of the most able players of the game. She spoke only a little, but listened with intensity, and Ashi felt sure that if a tapestry was being woven tonight, Vounn sat at the loom and threw the shuttle.\n\nSenen, surprisingly, was another able player. She might not have been familiar with the table manners of the Five Nations, but she was surely a veteran of intrigues among the Dhakaani clans. Her ears flicked rapidly back and forth, as if she was listening to several conversations at once. Ashi realized that with a lhesh now on the throne, the Kech Volaar and Darguun might draw closer to an alliance again. Ekhaas had confided in her that Senen suspected that she was involved in something. Ashi wondered if the ambassador could have suspicions of Tariic as well.\n\nPlates were cleared and glasses emptied. A lull fell into the conversation, as if the diplomats had sated their need for information along with their hunger. Ashi saw Dannel give a glance and a nod to Stitch. The warforged stepped out of the room. Dannel smiled at those seated around the table. \"Let us adjourn to the library. We have sweet wine and cheese to finish.\"\n\nShe rose and her guests rose with her, Vounn, however, caught Pater's hand and held him back as the others left the room. Ashi, waiting for the signal, stayed as well. \"Pater,\" said Vounn, \"I need a favor.\"\n\nThe eyebrows on Pater's round face rose slightly. \"There are worse things than having House Deneith in your debt.\" His voice was pleasant but his gaze was suspicious. \"What do you need?\"\n\n\"I think Baron Breven will recall Ashi to Sentinel Tower in Karrlakton soon. I've put him off before, but I don't think I can delay again. Ashi will have to leave Darguun.\"\n\nThe lie was simple and completely believable because it was mostly true. When Pater glanced at her, Ashi didn't need to feign her frustration.\n\nVounn continued her appeal smoothly. \"Unfortunately, with conditions as they are in the country, the route could be dangerous. We've had no word from Dagii's army or from Zarrthec. It's possible raiders could be scouting the trade road.\"\n\n\"I haven't heard anything from our caravans,\" Pater grunted.\n\n\"Yet you don't rule out an attack on your supply wagon,\" Vounn reminded him. \"If I move a force of mercenaries guarding one person up the trade road, I expect the Valenar might find that too tempting a target to resist.\" She gestured to Ashi. \"Can you take her? I mean, are you capable of taking her should the need arise?\"\n\nPater screwed up his face and cast an eye over Ashi. She felt as if she was being sized up as cargo\u2014which, strictly speaking, she was. House Orien bore the Mark of Passage. Pater's dragonmark, though not the most powerful of Orien's marks, allowed him to step instantly across vast distances. Vounn had told her it was an ability that the viceroy seldom used and then only to carry urgent letters and parcels, but that it was theoretically possible for him to transport a passenger on his long-distance jaunts.\n\nShe found herself holding her stomach in, as if that would make her look like a lighter load.\n\nPater just grunted again. \"Aye. I won't do it for free, though. Standard Orien fee.\"\n\n\"Of course,\" said Vounn. \"I regard it as a favor that you're willing to do it at all.\" She clasped his hands and smiled. \"Thank you.\"\n\nPater's face didn't relax. \"One step follows another, Vounn. You wouldn't ask me like this if it was a simple transport.\"\n\n\"And you wouldn't agree if it was just a simple transport. I know you, Pater.\" Her smile took on an edge. \"Not all dangers wait along the road. No one will suspect your involvement. I intend to organize a mercenary escort and send Ashi out of Rhukaan Draal with them\u2014you meet her outside the city, on the other side of the bridge over the Ghaal River, and take her from there. Attention will remain on the escort, which will disband a reasonable distance from the city with no sign of Ashi\u2014or you. Will you do it?\"\n\nPater glanced from her to Ashi. \"Aye,\" he said. \"I will. Give me a day's notice when you need me.\"\n\n\"Thank you, Viceroy Pater,\" Ashi said. She stepped forward and bowed slightly. \"I appreciate this.\"\n\nThey were the only words she'd been allowed to say\u2014Vounn had told her specifically to keep her mouth shut while she spoke. Pater's assistance had been far from guaranteed and even Ashi had seen it. The words of thanks were the polish on the sword, though. Pater puffed up like a rooster strutting before hens. \"You're welcome, Lady Ashi,\" he said.\n\nVounn shifted her grasp from his hand to his arm and gave him another open smile. \"Wonderful. Now, let's catch up to Dannel and the others before the wine and cheese are gone.\"\n\nThey strolled out of the dining room. Ashi followed in their wake along the short passage that connected library and dining room.\n\nShe didn't even see Midian until he grabbed her wrist and tugged her back into the shadow of a large decorative urn.\n\n\"You're leaving?\" he demanded.\n\n\"You heard that?\"\n\n\"I hear a lot more than people think I do.\" His eyes glittered. \"I know about Tenquis.\"\n\nHe'd learned the name. \"How\u2014?\"\n\nHe scowled. \"Finding an artificer was my idea. You don't think I could ask the same questions as Ekhaas? Now it's your turn. You're leaving?\"\n\nAshi looked around. The urn concealed Midian completely, but what hid a gnome didn't hide her. \"Not here,\" she said. \"Somewhere private.\"\n\nServants had entered the dining room to clear away the dinner plates. Midian, still holding tight to Ashi's hand, led her the other way along the passage, away from the library and up a flight of stairs. A door opened onto an airy gallery with ornate filigree screen panels forming a long wall open to the night. They were up high, well above the street. The gallery was unlit and dim to her eyes, though Midian moved with confidence.\n\n\"Don't touch the screens,\" he warned her. \"They're Cannith gearwork, trapped to keep out thieves.\"\n\n\"How did you know this was here?\"\n\n\"I had a look around before dinner.\" He let go of her hand and turned to face her. \"Let's try this again. You're leaving?\"\n\n\"Not if I can help it,\" Ashi told him stubbornly. She described her attempt to reach Geth after Tariic's coronation\u2014and her subsequent conversation with Vounn. When she had finished, Midian let out a hiss of frustration.\n\n\"I wondered why you hadn't left your chambers for the last two days. Your guard Aruget told me you were ill whenever I came around.\"\n\n\"What did you make of what happened at the coronation?\"\n\n\"I couldn't see anything. A fat lump of a bugbear pushed in front of me. Not that I was all that eager to be seen once Makka strolled onto the dais.\" Ashi could make out Midian's face\u2014he was chewing nervously on a thumbnail. \"Sage's quill. Tariic may know about the false rod.\" He glanced at her, his blue eyes flashing in the moonlight. \"So if you leave, what happens to me, Ekhaas, and Dagii?\"\n\nAshi shook her head. \"I'm not leaving yet. Vounn just wanted to make arrangements. She spoke with Tariic yesterday on business for Deneith, and she says that he doesn't act like he suspects anything. Or at least he doesn't suspect us. She hasn't spoken with Geth yet, though. Aruget hasn't gotten close to him either.\"\n\n\"Neither have I. I wanted to talk to him, but I couldn't find him. I have seen him with Tariic a lot though.\" He hesitated for a moment, then added. \"What if Tariic has the true rod? What if he's found some way to dominate Geth?\"\n\n\"He can't. Wrath protects Geth.\"\n\n\"Here's the thing, though\u2014whenever I've seen Geth, he's not wearing Wrath.\"\n\n\"I don't think that matters,\" Ashi said. \"When we recovered the rod, he was disarmed, but the rod still couldn't affect him.\"\n\n\"Then why won't he talk to us, and why isn't he wearing Wrath? What's going on?\"\n\n\"I don't know.\" Uncertainty and fear stirred in Ashi's gut\u2014along with grim determination. \"But we're going to find out. We need to talk to Geth. Come see me tomorrow. We'll decide what to do.\"\n\nMidian nodded, then said, \"We should get into to the library. Vounn has probably missed you by now, and she'll know something is up when we come in together.\"\n\n\"That doesn't bother me,\" Ashi said. \"She's had me out tonight. She can't confine me to our chambers now. Maybe we are in danger, but we need to get answers while we can.\" She clenched her jaw. \"And Geth is the only one who has them.\"\n\n\"And Geth is the only one who has them.\"\n\nStretched out on top of the thick outer wall, Makka hugged a clenched fist to his chest and bared his teeth. _When I fight, I fight. When I stalk, I stalk_. The first time he had confronted Ashi and Ekhaas, he'd made the mistake of fighting without properly stalking his prey. He'd been too hasty. He'd forgotten the lessons of the hunt. The Fury seemed to appreciate revenge well-savored, though. Patience and stalking\u2014even with the pathetic caution of Deneith's hobgoblin guards\u2014had paid off.\n\nHis decision to scale the walls of the building Ashi visited had paid off, too. As had his accidental touch of the great screen an armslength above his head. Whirring metal springing to life had gouged the skin of his fingers and palms, but the instinct of freezing in the shadows rather than running had both saved him from discovery by guards and put him in exactly the place he needed to be.\n\nNot just Ashi of Deneith but the gnome Midian too.\n\nThe door in the screened chamber closed. Makka offered a silent prayer of thanks to the Fury, rose to his feet, and moved with silent steps back to the deep shadows where he had climbed up.\n\nCareful stalking was one of the lessons of the hunt. Choosing proper bait for the trap was another.\n\n#\n\n# CHAPTER  \nNINETEEN\n\n# 28 Sypheros\n\nHe'd had this dream before.\n\nAdolan sat across the fire from him. The druid's face was calm under his red-brown beard. His eyes were the same color as the fresh oak leaves tied to the heavy shaft of his spear, with pupils as black and shining as the collar of stones around his neck. \"Are you just passing through?\" he asked.\n\nGeth could smell the stink from his own body. It had been a long time since he'd bathed. The rank odor blended with smoke from the fire, the sizzling juices of the chicken that charred on the rough spit above it, and the cool damp scent of the deep forests of the Eldeen Reaches. There was another smell, too, like hot copper. It seemed out of place, but Geth ignored it.\n\n\"Maybe,\" he answered the druid. \"Maybe not.\"\n\nAdolan's eyes bored through him. \"You should move on.\"\n\nGeth looked at him. That wasn't right. He repeated the words he'd said to Adolan the night that the druid had confronted a wandering, chicken-stealing shifter. \"Yes. Just passing through.\"\n\n\"Good. Be on your way.\" Adolan rose, supporting himself on his spear.\n\n\"What?\" Geth dropped the chicken he'd been holding and jumped to his feet. \"No!\"\n\n\"Why not?\" The face across the fire looked genuinely surprised. \"You want to stay here?\"\n\n\"I'm supposed to,\" Geth said. \"That's what happens, Ado. You convince me to stay in Bull Hollow.\"\n\n\"Not this time. This time you have to keep going.\" Adolan turned away.\n\nThe pain of the rejection was a giant fist wrapped around Geth's chest. He felt a piercing ache in his side, like broken ribs. The hot copper smell grew stronger. He grabbed for his sword, drawing Wrath\u2014and a part of him knew that was wrong, too. He'd still carried a plain Deneith service blade when he'd encountered Adolan. That wrongness didn't stop him from pointing the twilight blade and shouting \"Stop!\"\n\n\"Or what?\" The figure that turned was small and dressed in black. It spoke with a strained, scarred voice. Chetiin. In his hand, he held the Rod of Kings. \"I'm your friend, Geth. What are you going to do?\"\n\nChetiin turned around again and leaped through the window that grown in the middle of the forest. \"No!\" Geth screamed. He sprinted after the goblin.\n\nVoices shouted for him. Daavn commanding him to stop. Tariic demanding the true rod. Haruuc begging his aid. Figures flashed in the periphery of his vision. Ashi reaching out for him. Ekhaas, doing the same thing. Dagii, stern and reserved. Chetiin, his large eyes somber and wise.\n\nAdolan, watching. And maybe a little sad.\n\nGeth tried to stop, to turn back, but it was too late. He plunged through the window, and the stones of the plaza below Khaar Mbar'ost rushed up to meet him\u2014\n\nHe jerked and snapped upright, a roar tearing itself from his throat. From somewhere close, there was a yelp, the crash of shattering glass, and a stream of curses. Body trembling, Geth stared around. He sat in a high bed, threadbare sheets twisted around him. A low, raftered ceiling was close overhead. To either side of the bed rose stone walls that stopped well short of the ceiling and the opposite wall. The hot copper smell of his dream filled the air.\n\nA moment later, Tenquis peered around the corner of one of the short walls. The tiefling's expression, at first cautious, hardened and he stepped around to stand at the foot of the bed, glaring at Geth with his golden eyes. \"Horns of Ohr Kaluun, can you do anything quietly?\"\n\nGeth squeezed his eyes shut for a moment, then opened them again and the strange bedchamber made sense. The last moments of his flight from Khaar Mbar'ost came back to him. He was in Tenquis's home, the one-time barn. The short walls closing in the bed were the sides of stalls. Cows had once slept where the bed stood. Geth lay back, wringing a sickening ache out of his side that echoed the agony of his dream. He raised his head enough to look down at himself. His chest was wrapped in bandages. More bandages swaddled his left arm, and there was something thick and crusty smeared across his face over his cheekbones. He reached up with his right hand and whatever was on his face crumbled into a lumpy powder that left his fingertips dark and glittery. The flesh underneath was tender.\n\n\"A healing compound,\" Tenquis said. The anger faded from his face, replaced by a certain self-satisfaction. \"Faster than a body on its own, slower than true magic. A good artificer needs to know something about anatomy, as well as alchemy and artifacts. You had several broken ribs, a broken arm, a broken cheekbone, and were badly bruised all over. Your left hip had a deep wound that was just barely healed\u2014magic or some shifter gift, I assume. The bruising is gone. Your hip is completely healed. The broken bones are likely mended, although you'll want to be careful of them. They'll be like green wood for a few days yet.\"\n\nGeth bent his bandaged arm experimentally. More dark, glittering powder ran out between the fabric strips. \"I... Twice tak, Tenquis.\"\n\nThe tiefling wrinkled his nose. \"You did pass out on my doorstep. It would have attracted attention if I'd left you to die in the street\u2014and once you were inside, I had to do something or I would have had to get rid of the body.\"\n\nIt was hard to tell if he was joking. Geth waited for him to laugh or smile, but he didn't. Finally Geth broke the silence. \"There's been trouble.\"\n\n\"I suspected it.\" Tenquis's voice was flat. His fleshy tail snaked slowly back and forth through the air. \"I really was tempted to leave you in the street, but since I'd guess this has something to do with the false rod, whatever happened to you puts me in danger, too.\" He cocked his head and his gold-flecked black horns flashed in the light of the lanterns that lit the barn. \"Lhesh Tariic discovered the deception.\"\n\n\"Good guess,\" said Geth.\n\n\"I wish it wasn't. Given that you came on the night of his coronation, it seemed obvious, though.\"\n\n\"Wait.\" The tiefling's words settled into his head like leaves drifting to the ground. \"On the night of Tariic's coronation? How long have I been here?\"\n\n\"Three nights.\"\n\n\"Three?\" He sat up again, threw his legs over the side of the bed, and stood\u2014almost. Tenquis darted in and grabbed him before his legs folded completely.\n\n\"Give yourself a moment,\" he advised.\n\nGeth nodded numbly. He wore only his smallclothes. Tenquis dragged a sheet off the bed and draped it around him, then helped him out into his workshop in the main room of the barn.\n\nThe bright light of noon shone around the edges of the shuttered windows. \"Tiger, Wolf, and Rat,\" Geth muttered. He looked at Tenquis. \"I need to go.\"\n\n\"Easy.\" Tenquis guided him to a table and pulled out a straight-backed chair so he could sit. Geth grabbed the back of the chair and hung on. Tenquis shrugged and let go. He left him holding onto the chair, fetched a broom, and began to clean up a mess of broken glass and thin, smoking liquid.\n\nAcross the workshop, Geth's clothes lay on another table together with Wrath and his great gauntlet. Geth shook out his legs, took a deep breath, and walked\u2014wobbling only slightly\u2014to the other table. Tenquis paused in his sweeping to watch him. \"In a hurry?\"\n\n\"I've been in a bed for three days. I need to get out. I need to find out what's been happening.\" Geth held onto the edge of the other table and let his breath catch up with him.\n\n\"As far as I can tell,\" Tenquis said, \"not much.\"\n\nGeth stared at him. \"What do you mean 'as far as you can tell?' And what do you mean 'not much?'\"\n\nTenquis ignored him, tipping a bucket on its side, sweeping the glass and smoking liquid into it, then shaking sand from a second bucket over the remaining liquid. When he'd finally finished, he looked up at Geth and flashed sharp white teeth in a sly grin.\n\n\"Did you think I was sitting by your bedside? I may make good guesses, but there's only so much I can glean from someone who's unconscious. And I'm not the kind of person who waits for trouble to come creeping up on him. While you were sleeping, I went out to see what kind of danger I was in.\"\n\nHe set the broom aside and moved around the workshop, gathering things\u2014a basin, a brush, hot water from a small iron stove\u2014as he spoke.\n\n\"That first night, there were guards from Khaar Mbar'ost looking for a shifter in parts of the city, though not around here and not for long. By dawn, there was no more search. Tariic's been putting himself on display to the people for the last two days. He's getting them ready for war with the Valenar\u2014not that _dar_ need much encouragement. I want to see one of his speeches so I could get a look at him, and I noticed two interesting things. The first was that whatever he might know about it, he was still using the false rod. I recognize my own magic.\" He stopped in front of Geth, basin held in his arms. \"The second thing is you shouldn't be here because for all appearances, you were there, standing with Tariic.\"\n\nGeth's lips peeled back from his teeth. \"How\u2014?\"\n\n\"A changeling? An illusion disguising someone else?\" Tenquis shrugged again. \"How it's been done doesn't matter so much as that it's been done at all. Someone, whether it's Tariic or someone else, is trying to cover up the fact that you're missing. And given that neither you nor Tariic has the true Rod of Kings right now, I'm going to guess that someone else has it. Am I right?\"\n\nGeth blinked, then nodded slowly.\n\n\"Boiled down, yes.\" He looked at Tenquis for a long moment and the tiefling looked back. Questions raced through Geth's head. What had happened to Ashi and Midian? Where was Chetiin now? Had Dagii and Ekhaas engaged the Valenar? What was Tariic up to?\n\nThe one that made it to his tongue, though, was, \"Why are you doing this? You didn't want anyone to know your name. You didn't want to be involved.\"\n\n\"I told you\u2014whatever happened to you puts me in danger too. If Tariic knows about the false rod, he could find a way to track it back to me. I think we've just been lucky that he hasn't.\"\n\n\"No,\" said Geth, shaking his head. He felt ashamed at the suspicions that gnawed at him. Tenquis had hidden and healed him. Why shouldn't he trust him? But he'd trusted Chetiin, too. \"I mean why are you still here and not halfway out of Darguun? Why go out looking for answers?\"\n\n\"Like I said, I'm not one for letting trouble creep up on me.\"\n\n\"Trouble's not going to creep up on you if you're in Breland or Aundair.\"\n\n\"If I were Tariic and I was looking for the true Rod of Kings, a little thing like distance wouldn't stop me. But your point is taken.\" Tenquis grimaced and set the basin, brush, and water down on the table. \"I thought about running\u2014for about the time it takes water to boil. The thing is, I don't run easy. Most tieflings have the fury of ancient devils in their blood. I've got the other side of our heritage too: the curiosity of the sorcerer-kings who made deals with those devils.\"\n\nHis tail lashed from side to side. \"You know I'm interested in the lost lore of the Dhakaani _daashor_. Darguun is the place to be to look for that lore. Ekhaas still owes me the stories preserved by the Kech Volaar. You still owe me time with your sword and I have a feeling that if I stay with you, I'll find out even more. Why would I give that up by running?\" Tenquis crossed his arms. \"When I was young, my grandmother said to me, 'Quiso, curiosity has consequences\u2014if you're going to ask questions, you need to be ready for the answers, or they'll take you down hard.' I'm all for self-preservation, but outright fear is something else, and I haven't seen anything yet that makes me think I should be afraid of helping you.\"\n\nGeth looked at him again, suspicion turning into a strange camaraderie for the feisty artificer. \"Are you sure about this?\" he asked. \"You really don't know what you're getting into.\"\n\n\"Do I need to remind you that I'm already in it?\"\n\nGeth felt a crazy grin spread across his face. \"You've got guts, Tenquis. You're like rolling over a log and finding a wolverine underneath.\"\n\nTenquis snorted. \"You don't know many tieflings, do you? I'm practically a coward.\" He picked up the brush from the table and thrust it at him. \"Clean yourself before you get dressed. Brush off the healing compound first or it will turn to mud when it gets wet. Are you hungry?\"\n\nHe was. \"Ravenous. Like I haven't eaten for three days.\"\n\n\"I'll cook something and you can tell me all about what I'm really getting into.\" Tenquis stepped behind Geth and the shifter felt him unfastening the bandages around his chest. \"Let's start with how you ended up looking like you found the bad side of an angry ogre.\"\n\n\"I jumped out of Khaar Mbar'ost.\"\n\nThe hands on his bandages paused. \"You're joking.\"\n\nGeth thought of the stones of the plaza rushing at him. His belly clenched and rose at the memory. \"I wish I was,\" he said.\n\nThe tiefling was not a good cook. The stove on which he made their food seemed to be the same one on which he heated various experiments. The spices and herbs he used came out of jars plucked from among others that Geth felt certain contained alchemical ingredients. The bowl he eventually put in front of Geth contained balls of starchy _noon_ bobbing in an over-spiced broth alongside uncertain meats, mushy vegetables, and bits of black loosened from the bottom of the pot. More _noon_ balls, slightly stale, were served as bread, with hard cheese, slightly moldy, on the side.\n\nBut with one bite, Geth's hunger seemed to explode and he ate everything. As he spun out the story of the Rod of Kings, however, Tenquis's appetite appeared to shrivel. Finally, he pushed his bowl away. \"So the true rod wants to make its wielder an emperor bent on conquest and can give him the power to make it happen. Tariic knows about the true rod and is aligned with Makka, who wants to kill all of you. Chetiin has betrayed you twice and currently has possession of the rod. Two of your allies are off to fight the Valenar and the other two are likely under Tariic's watchful eye.\" The artificer slumped on his stool. \"Is there anything else I should know?\"\n\n\"I killed a dragon once.\"\n\nTenquis's gold eyes opened very wide.\n\nGeth shrugged. \"I thought you might find that reassuring.\"\n\n\"It disturbs me that you were in a position where you had to fight a dragon.\"\n\n\"No dragons this time. At least not so far.\" Geth lifted his bowl and swallowed the last of the broth. \"I want to find out what's happened to Ashi and Midian. I need to know if Tariic has harmed them.\"\n\n\"I have a better idea,\" said Tenquis. He leaned forward. \"Track down the true rod first.\"\n\nGeth growled as he set the bowl down. \"How are we supposed to do that? Chetiin could have taken it anywhere, and he's hard enough to keep track of when you're looking right at him.\"\n\n\"The same way you found it before, of course. Use Wrath.\"\n\nGeth blinked and looked at the tiefling. Tenquis spread his hands. \"It stands to reason, doesn't it? You say the _duur'kala_ opened a connection between them so you could locate the rod in the wilderness of Darguun. Is there any reason the connection shouldn't still exist?\"\n\n\"I... hadn't thought of it.\" Geth shook his head. \"I haven't tried it since we brought the Rod of Kings back to Haruuc. I just thought the magic would end when we found it.\"\n\nTenquis snorted. \"The sword is still the sword. The rod is still the rod. Try it.\"\n\nGeth rose, went to the table where Wrath lay, and drew the sword from its sheath. It felt good to hold the twilight-purple blade in his hand again. The grip, the weight and feel of the sword\u2014even the sense of it at the edge of his awareness\u2014had become familiar to him. Geth moved to the open center of Tenquis's workshop, held out the sword, and opened himself to it.\n\nHe felt the pull immediately, a draw toward the sibling artifact that had been crafted from the same vein of byeshk and by the same as hands as Wrath. The same pull he had felt when they'd first sought the rod. Turning slowly, he found the direction in which the pull was strongest. \"That way,\" he said. \"I can't tell how far it is, but the rod is that way!\" He grinned. \"Grandmother Wolf, that's perfect!\"\n\n\"Chetiin probably assumed the same thing you did,\" said Tenquis. He got up from the table. \"I assume you're ready for a little scouting?\"\n\n\"Almost.\" Geth sheathed Wrath again, then picked up his great gauntlet and slid his arm into it. Tenquis had repaired the dented metal and the joints of the armored sleeve moved freely once more. Geth tightened the straps that held it in place, flexed his arm, and bared his teeth. \"Now I'm ready.\"\n\n#\n\n# CHAPTER  \nTWENTY\n\n# 28 Sypheros\n\nThey walked out of Tenquis's workshop into a day that was cool and bright, though gray clouds were piling up in the east. Geth could smell rain on the wind, faint behind the odor of the kitchen and workshop scraps that Tenquis had used to disguise him. Egg and carefully drawn soot gave him wrinkles while ash streaked his hair. Walking with a hunch and hanging onto Tenquis's arm completed the appearance of an aged shifter. Not so impenetrable a disguise as the illusion Ekhaas had created, but it would do the job. If Tariic had people watching for Geth, they wouldn't give him a second glance.\n\n\"You should hope they don't,\" Tenquis said when Geth admired his handiwork. \"This kind of trick works better at night. Keep your face down so no one gets a good look at you.\"\n\nGeth had checked the direction Wrath pointed before they left and they went that way, west and somewhat north. The twisting, crowded streets of Rhukaan Draal forced them make frequent detours and left them guessing that they were going in the correct direction. Fortunately, the winding path didn't require them to pass close to Khaar Mbar'ost.\n\nThe influence of the mighty fortress\u2014or at least of its new master\u2014seemed to have grown, though. Geth could feel a new edginess to the crowds in the streets, a new aggression and confidence among the _dar_. People of other races stayed out of their way. Weapons, not uncommon before, were even more apparent. Small groups of militia drilled wherever there was open space.\n\n\"You can thank Pradoor as well as Tariic,\" Tenquis said. \"She's well-known. People put a lot of store in her words. The last few days she's been preaching in support of war with Valenar and in support of Tariic.\" He rubbed the horny spikes that edged his chin. \"If she's joined with Tariic, it would explain a few things.\"\n\n\"Like?\" Geth growled.\n\nTenquis pointed as they turned a corner and Geth followed his finger\u2014not that what he pointed at would have been hard to miss. They stood on one side of a crossroads where five streets came together. At the center of the intersection stood a flat-sided pillar of white stone surmounted by a sculpture of four metal arms, three vertical and one horizontal, a very large version of the eight-pointed Octogram, symbol of the Sovereign Host. Carved into the pillar below it were the symbols of Dol Arrah, Dol Dorn, and Balinor, the gods Haruuc had chosen to venerate over those of the Dark Six. The monument seemed familiar. It took Geth a moment to remember why. Haruuc's funeral procession had come this way, pausing briefly before the column. There were words written on the column, Geth remembered now. Lhesh Haruuc Shaarat'kor gives thanks for Darguun's victory at the Battle of Marguul Pass.\n\nThe dedication was obscured by the naked body of a skinny elf that hung from a rope lashed around one of the arms of the Octogram. Dried blood made new symbols on the white stone, those of the Fury, the Mockery, and the Keeper.\n\n\"Bear and Boar,\" muttered Geth. \"Was he a Valenar?\"\n\n\"Maybe. Maybe not. It's hard to tell now.\"\n\nThe people of Rhukaan Draal passed around the desecrated monument without looking up, although Geth noticed they did give it a wide berth. \"Why hasn't he been cut down?\"\n\n\"I suspect it suits Tariic to leave him up,\" Tenquis said tightly. \"Maybe you're right when you say Haruuc picked a fight with Valenar to avoid fighting all of Khorvaire, but what he's started is going to be very bad.\"\n\nGeth stared at the body as it rocked against the stone, pushed back and forth by the breeze\u2014then started forward.\n\nTenquis grabbed him and held him back. \"No. You can't do anything for him and you'll only draw attention to yourself.\"\n\nBreath hissed in and out of Geth's mouth, but he turned away. \"Tonight,\" he said. \"It feels like there's going to be a storm tonight. I'm going to come out and do it then.\"\n\n\"Fair enough.\" Tenquis steered him out of the intersection and into the temporary shelter of a stopped wagon. \"Check our direction.\"\n\nThe sword was disguised just as he was, wrapped up in leathers and carried under his arm as an anonymous bundle. Geth slipped a hand through the leathers, grasped the hilt, and held it out just enough to tell which way it pointed. They'd drifted only a little from their course. They left the shelter of the wagon just as the carter, a fat hobgoblin, appeared, ready to curse them for thieves. Tenquis flicked his tail at him as they strode off.\n\nThe farther they went, the more Geth was certain that he'd been this way before, though he didn't recognize the buildings or the shops that lined the streets. The twists and turns that he and Tenquis followed, though... those seemed somehow familiar, except that the crowds were out of place.\n\nThen he saw why their route was familiar. Empty the streets and put the crowds in front of the shops and on top of the buildings and the scene made sense to him just as the monument had.\n\n\"This is the same way Haruuc's funeral procession came,\" he said.\n\n\"It's just a coincidence,\" said Tenquis. \"We're staying on the larger streets and the funeral procession couldn't very well have gone through alleys.\" His nose wrinkled and his tail lashed. \"You don't think Chetiin could have taken the Rod of Kings out of the city, do you?\"\n\n\"Maybe,\" Geth said. His mind was only half on the answer. A coincidence? He had to work to make himself believe it. Belief came even harder as the crowd thinned out and the buildings of Rhukaan Draal became shacks and huts along the side of the road, thick at first, then scattered, and finally nonexistent. The roar of the first cataract of the Ghaal filled empty space.\n\nAhead of them, the great red stone arch cast a long shadow on the road as the sun, chased by clouds, set behind it. Tenquis's steps slowed. \"Check Wrath,\" he said.\n\nThe sword pointed through the arch.\n\nGeth didn't put Wrath away. They walked up to the arch in silence and stole into its shadow. Geth paid no more attention to the carvings within than he had the first time he'd passed through. A tall iron gate closed the far end of the arch, but it wasn't locked. It opened at a touch with something like a soft sigh and they slipped through. Geth walked with Wrath extended before him. The road ended and they paced through long, dry grass. Twilight was settling over the ridge of weathered rock that Haruuc had intended to be the graveyard of kings.\n\nThe great lhesh, a young warlord again, glared down at them from the heavy door of his tomb.\n\nWrath didn't waver. It pointed at the ridge below the outer structure of the tomb. Geth climbed the steep stairs up to the carved door and felt the sword dip in his hands. He walked around the tomb, just to be certain. Wrath moved like an iron needle drawn to a lodestone and what he felt was so far beyond amazement that it left him stunned.\n\n\"It's inside,\" he said. \"The Rod of Kings is inside Haruuc's tomb.\"\n\nTenquis joined him and ran dark fingers around the seam of the door. \"It hasn't been opened.\"\n\n\"It can't be opened,\" said Geth. \"The pivots were meant to crumble once the door was closed.\" He stared into Haruuc's stone face. \"Grandfather Rat, how did Chetiin manage to get it inside?\"\n\n\"Magic,\" Tenquis suggested. \"Or just another entrance. Hobgoblins prefer to bury their dead in caves. The underground portion of the tomb was originally a cave, wasn't it?\"\n\nGeth nodded and the tiefling stepped back, dusting off his hands and nodding around them at the folds and cracks of the ridge.\n\n\"It won't be the only one. The builders would likely have walled off any connections, but a goblin wouldn't need much space to wiggle through.\"\n\nGeth looked around at the ridge as well. \"It would take days to find the entrance and a connection.\"\n\n\"Then that leaves magic.\" Tenquis patted the door. \"I might be able to open this. Not now\u2014I'd need to prepare\u2014but I have an idea how it could be done.\" He met Geth's eyes. \"If you think it's necessary.\"\n\n\"What do you mean 'if I think it's necessary?'\" Geth asked. \"Of course it's necessary!\"\n\nTenquis held up his hands. \"Think about it,\" he said. \"We only found it because we have Wrath. No one else is going to think to look here, are they? The tomb is sealed. Unless we open it, no one is going to have any reason even bothering to try and look inside. The rod is safe with Haruuc. Maybe that's what Chetiin intended.\"\n\n\"Then why steal it from me?\" Geth demanded. \"We could have worked together. Maybe Chetiin just wanted to hide the rod somewhere safe for a while. I don't know. I can't even guess what he's up to anymore.\" He pulled his lips back in a snarl. \"And where one goblin can go, so can others. No tomb is unlootable\u2014and there's a lot of loot in Haruuc's tomb. Anyone who breaks in looking for treasure isn't going to present much of a challenge to the rod if they pick it up by mistake.\"\n\nHe closed his eyes, took a deep breath, and opened them again. \"How long will it take you to get what you need ready?\"\n\n#\n\n# CHAPTER  \nTWENTY-ONE\n\n# 28 Sypheros\n\nIn the afternoons at Khaar Mbar'ost, many of the warlords, councilors, and courtiers could be found walking and talking in the hall of honor. The hall occupied the full length of one of the fortress's upper floors. Statues of _dar_ heroes stood against the walls, and stained glass windows depicting scenes of famous battles dominated the distant ends. The air in the hall was generally soft with murmured conversation, though a few times Ashi had heard it ring with the sound of steel on steel as conversation erupted into argument and a brief duel.\n\nToday it was quiet. The curse Ashi muttered under her breath as she entered seemed like the loudest exclamation in the big room.\n\nThere was no sign of Geth here, either. She looked back at Aruget, waiting beyond the door\u2014in spite of Vounn's insistence that the hobgoblin warrior accompany her everywhere, there were some places guards weren't permitted\u2014and said, \"I'll be back.\"\n\nAruget's ears pulled back just a little bit. Ashi had the distinct impression that he didn't appreciate being dragged through Khaar Mbar'ost in her search for Geth, but Geth wasn't with Tariic in the throne room today. He was finally alone. If she could find him, they'd finally be able to talk.\n\nAt the east end of the hall of honor, Munta the Gray leaned against the wall beside one of the tallwindows. The stained glass had been tilted open to allow a cool breeze into the room. More than half the sky was covered in heavy clouds, beautiful day slipping into threatening evening. The old warlord held a goblet and sipped from it frequently. Furs had been bundled around his shoulders, but he faced into the wind with a drawn look on his face. He turned as she came near and his pensiveness faded a little.\n\n_\"Korluaat_ , Lady Ashi?\" he asked her. He gestured and a goblin servant offered Ashi a goblet before she had a chance to answer. The beverage inside was strong enough to make her nose twitch at the alcoholic fumes. She'd been served it once or twice at feasts in Khaar Mbar'ost\u2014the name of the stuff translated as \"hero's blood\"\u2014but it wasn't a drink she enjoyed. She smiled and mimed sipping a little of it before asking, \"Munta, have you seen\u2014\"\n\nBut Munta spoke before she could finish, gesturing expansively at the horizon beyond the window. \"Dagii's out there somewhere,\" he said and Ashi could smell the _korluaat_ rolling off of him. \"Chasing down elves, maybe being chased down himself.\" He took a swallow from his goblet. \"Tariic assembles a new army, ready to fight. I've never seen Ghaal'dar warlords so eager to work together. I've even heard that one of the Dhakaani clans have asked to march with him. The Kech Shaarat\u2014the Blade Bearers. Have you met one of them yet?\"\n\n\"Their ambassador is Kroon Dhakaan,\" Ashi answered. \"A hobgoblin with the shoulders of a bugbear.\"\n\n_\"Cho_. That's him.\" Munta looked out the window again. \"Real war, Ashi. And I won't fight it. Tariic's 'honor' to an aged warlord. I was old when Haruuc first proposed his dream of a land for the _dar!_ What I've done for Darguun, what I've experienced, and Haruuc never denied me the chance to take up my sword again. Just recently in Droaam\u2014\" His ears flattened and he took another drink. \"But Tariic won't even let me take a place in the command tent. He gives me gold and grants favored rank to the warriors of my clan, but he won't allow me to see the battlefield. And while he said it, I just bowed like a gnome. _Maabet_ , he has a presence. He could be greater than Haruuc.\"\n\nAshi pressed her lips together for a moment, resisting the urge to warn Munta of the danger that faced them, before she spoke. \"Munta, have you seen Geth today?\"\n\n\"Ah!\" Munta gulped the rest of his _korluaat_ , the loose skin of his throat folding in on itself as he swallowed. \"I ramble like a gnome, too. I saw Geth this morning. He seemed to be in a hurry, but he said that if I saw you, I should tell you to talk to Razu if you hadn't already.\"\n\nAshi's heart leaped. \"Did he say why?\"\n\n\"No. Some ritual. I suppose.\" Munta looked down the length of the hall of honor and nodded his head. \"She's there.\"\n\nAshi saw the head of Razu's staff before she saw the mistress of rituals. \"Ta _muut_ , Munta,\" she said. \"I'm sorry I have to leave you.\"\n\nThe old warlord just grunted. \"Come back when you want,\" he said. \"I'm not going anywhere.\" He waved his empty goblet at the nearest servant.\n\nRazu must have caught a glimpse of Ashi's approach, because she ended her conversation with the hobgoblin she'd been speaking to and fell into step alongside Ashi, guiding her to an empty part of the hall. \"I have a message for you, Lady Ashi,\" she said. \"Geth told me to give it to you or to Midian, whoever I saw first. At the change of the second watch tonight, you're both to meet him where the _duur'kala_ sang and the sword woke.\" She looked a little confused by her own words. \"He said you'd understand.\"\n\nAshi kept her face carefully neutral, even though her heart was now jumping like a child at play. \"When did he give you the message?\" she asked.\n\n\"This morning. I told him I'm no messenger, but he said I was the only one he trusted.\" Her thin face flushed with pleasure.\n\n\"You did well, Razu.\" Ashi hesitated for a moment, then asked, \"Has it seemed to you that Geth has been acting strangely lately? Nervous and tense, maybe?\"\n\nRazu's ears flicked and her mouth pursed primly. \"Lady, he has held the throne as Lhesh Haruuc's _shava_ and now he guides Lhesh Tariic. If he seems tense, surely he has reason to be.\"\n\n\"I see.\" Maybe that made sense. Ashi bent her head to Razu. _\"Ta muut.\"_\n\nThe mistress of rituals bent her head in return and moved off into the hall. Ashi turned to the door where Aruget waited, holding herself to a sedate pace so that it wouldn't look too much like she was rushing away. The goblet of _korluaat_ , the level of the beverage undiminished since she'd picked it up, she left with a servant near the door. Aruget's ears rose when he saw her.\n\n\"You found something,\" he said softly as they walked away.\n\nAshi led him down the stairs before she whispered the message Razu had passed her. \"I know where the _duur'kala_ sang and the sword woke,\" she said. \"It's the roof of Khaar Mbar'ost where Ekhaas, Senen, and another _duur'kala_ worked a spell to wake Wrath so Geth could locate the Rod of Kings.\"\n\nAruget's ears dropped again. \"The roof?\" he asked.\n\nShe could guess what he was thinking. \"A good place for a secret meeting, but also a good place for a trap. Maybe too good. If someone was planning an ambush, there are places that would create a lot less suspicion.\"\n\n\"Maybe,\" Aruget said doubtfully. \"I'm coming with you.\"\n\nShe glanced at him. His face was hard. \"It is my _muut.\"'_\n\nAshi hesitated, then nodded. \"We'll go early. Just in case.\"\n\n\"It would be safest not to go at all,\"\n\n\"Safe hasn't gotten us any answers so far. Let's find Midian.\"\n\nWhen the time came for sleep, Ashi stretched out fully clothed on her bed, stared out the open window of her bedchamber, and waited. In her own chamber on the other side of the sitting room, Vounn would be pulling on nightclothes, slipping between sheets, and drifting off to sleep.\n\nThe night was dark\u2014the gathered lights of Rhukaan Draal cast a thin glow on the underside of low clouds. The wind had dropped as if the clouds had choked it off. There would be rain before dawn. Ashi kept her mind alert with the same games she had played while stalking the Shadow Marches. Naming the constellations hidden by the clouds. Counting the bones in her hands against the ancient rhyme of the broken blade. Reciting her lineage, a task that, when she was part of the Bonetree Clan, she was only able to do one side of. Since coming to Deneith, she'd learned to fill in the other as well, eldest child of eldest child. \"Ashi, daughter of Ner,\" she murmured to the rafters of her ceiling, \"son of Kagan, son of Tyman, son of Joherra, daughter of Wroenna, daughter of Maal...\"\n\nEach name was like a charm. In the tradition of the clans of the Shadow Marches, a bit of the power of each ancestor filtered down to her. The bloodline of House Deneith was the same, deeds accumulating in a heritage of honor, the magic of the Mark of Sentinel ebbing and flowing over time. As a girl, sitting and pointlessly sharpening the eternally keen blade of the sword that her father had inherited from his, had she ever dreamed that she would come so far and see so much?\n\nShe guessed that it was approaching the changing of the watch. Time to leave. She rose, slipped sword into scabbard by touch, and closed the shutters on the gathering storm. Hand on the door, she paused, gathered her will, and invoked her dragonmark. Warmth flashed through the pattern that covered her skin. She could almost imagine she saw a faint glow in the darkness, then the clarity that the mark granted settled over her like a splash of cold water. Silent as a shadow, she opened the door and crossed the sitting room.\n\nOf the two guards on duty outside the chamber, only one\u2014a young hobgoblin, new to the service of Khaar Mbar'ost\u2014started when she emerged. The other was Aruget. He glanced at Ashi, then at the younger guard. \"Remain,\" he said in Goblin. \"We will return.\"\n\n_\"Mazo.\"_ The young guard thumped his chest with a fist.\n\nThey avoided the main stairs, instead taking the narrower flight of stairs that Ashi had raced up after Tariic's coronation. Midian was waiting for them there and joined them in their silent climb. When she'd told the gnome about Geth's message, he'd had the same suspicions as they had. Like them, he'd come ready for a fight. Just in case. In one hand he gripped a polished metal baton\u2014a snap of his wrist, Ashi knew, would pop a slim curved head out of the shaft, transforming the baton into a deadly little pick. On his belt he wore several large pouches, a more convenient version of the backpack he'd worn during their quest for the rod and from which he'd produced a number of cunning magical devices.\n\n\"What are you carrying tonight, Midian?\" she asked.\n\n\"What aren't I carrying?\" was his grim response. \"I don't think I entirely like this, Ashi.\"\n\n\"I know I don't,\" said Aruget.\n\n\"Tell that to Geth when we see him,\" Ashi told them both.\n\nThe stairs ended as they approached the top of the fortress's central tower. They were forced onto the main stairs, but they encountered no one. When even the central stairs ended, only a dark, tightly wound spiral staircase remained. Midian produced a tiny everbright lantern from a pouch. A rotating cover allowed him to release a narrow slit of light, just enough for Ashi to see.\n\nShe put her foot on the first stair, only to have Aruget push her aside and take the lead, sword ready and ears high.\n\nAshi clenched her teeth.\n\nMidian just nudged her. \"If he wants to go first, let him!\"\n\nA trap door\u2014closed\u2014covered a final set of steep open steps above a small landing at the head of the stairs. Aruget waited until she and Midian were with him, then gestured for Midian to close the lantern. Darkness cloaked the cramped space. Ashi's first hint that Aruget had opened the trap door was a sudden cool draft that carried the smell of imminent rain. The hinges had been well-oiled. After a moment, she could make out the slightly less dark gap of the raised trap, partly blocked by Aruget's head.\n\nThe hobgoblin's vague silhouette turned. \"He's already here,\" he whispered. \"He's early, too.\"\n\n\"I don't blame him,\" said Ashi. She drew her sword. \"Go.\"\n\nAruget threw the trap door open violently and surged through so quickly he was out and looking around before it had even crashed down on the stones of the roof. Ashi came up hard after him, alert as well. About fifteen paces away, on the far side of the roof, a figure whirled around. A covered lantern snapped open, its light glaring into their eyes for a moment.\n\nGeth's voice came out of the dark. \"Ashi!\"\n\nShe felt a burst of relief. The roof was clear. Beyond the light of the lantern, she could make out the distinctive crouch of the shifter's body, his thick hair blowing in the wind. He didn't move.\n\nShe took a few steps toward him. \"Geth! _Rond betch_ , what's been going on? We've been trying to talk to you since the coronation.\" She shaded her eyes with her free hand.\n\n\"I know. I'm sorry. Things have gotten... difficult.\" Geth shifted the lantern and opened its other sides so that light spilled across the roof, catching him in its glow. He was dressed in clothing Ashi recognized, but just as Midian had told her, he wasn't wearing Wrath. She opened her mouth ready to ask him where the sword was, when two more things struck her.\n\nHe carried a lantern\u2014but he didn't need one. Geth could see in the dark.\n\nAnd not only wasn't he wearing Wrath, he wasn't wearing the collar of black stones that had belonged to his fallen friend Adolan. He might have left the sword somewhere. He would never have taken off the collar.\n\nHer sword snapped up. \"This isn't Geth.\"\n\nToo late.\n\nOut of the corner of her eye, she saw Aruget jerk, his hand grabbing for his throat as something looped around his neck. A strangler's noose of fine braided leather. The hobgoblin's fingers were too slow and the noose tightened before they could close on it. Ashi spun, following the dark line of the leather cord.\n\nAs if he'd materialized out of the night itself, Makka stood behind the trap door. Where had he\u2014? She found the answer before she'd even thought the question. A waist-high wall surrounded the roof. He must have been hanging over it.\n\nThe bugbear's muscles bulged and he heaved hard on the cord. Aruget flew back, dragged by the neck, and slammed down onto the roof. His sword went skittering away. Ashi had a brief glimpse of his contorted face, hands still grabbing for the noose, and of Makka, dropping the cord. Her sword hung at his side, but he ignored it and instead snatched a thick club from a thong hanging down his back.\n\nThen whoever played at being Geth snapped shut the lantern.\n\nAgain darkness, this time overlaid by the bright afterimages of the lantern. The trap door banged shut, an obstacle to easy escape, and she could imagine Makka coming for her. \"Midian, I need light!\" she shouted, as she threw herself blindly aside. Just in time. Something heavy whistled past her. She lashed out with her sword, but found nothing except empty air.\n\n\"Close your eyes!\"\n\n\"That won't help!\" A bulky shape shifted in the shadows. She struck again and came closer. This time her blow was blocked. The shape moved, angling for an advantage.\n\n\"Do it now!\"\n\nShe squeezed her eyes shut\u2014just as something shattered at her feet and an intense flash lit up the inside of her eyelids in shades of red. Makka roared. Ashi forced her watering eyes open again. Midian had his tiny lantern open and its light swung across the rooftop. Aruget, still down, dragged at the noose around his throat. Makka shook his head, blinked furiously, and backed away, club held warily in a guard position. Even though the glare of the flash still danced in her vision, she lunged for him.\n\nHe sensed her. The club slapped at her sword in an awkward parry. Ashi twisted and the blade skipped around to rip a thin line up Makka's forearm. He growled and pulled back further. Dropping his lantern, Midian moved in on the bugbear's other side with his pick held low.\n\nMakka's squinting eyes swung between them and he shouted, _\"Ko!\"_\n\nAt the far end of the roof, the figure that was not Geth bent and came up with a crossbow aimed at Midian. The bow steadied and snapped in the same movement.\n\nMidian tumbled backward. The steelhead of the crossbow bolt made sparks as it hit the stone of the roof.\n\n\"Take him!\" Ashi ordered the gnome, but she hardly needed to have bothered. Midian was already darting across the rooftop to prevent another attack. Ko... Ashi realized she knew the name. The changeling Geth had found imprisoned in Khaar Mbar'ost's dungeons.\n\nShe spun back to Makka, teeth clenched. \"Where's Geth?\" she demanded, thrusting at the bugbear. _\"Baano a Geth?\"_\n\nHe snarled in response and stopped her blow with a solid block. He was still blinking but his eyes were clear. She pressed her attack, feeling her anger grow inside her. _\"Baano a Geth?\"_ she screamed again.\n\n\"Waiting for you!\" Makka roared in Goblin. His club swung back. \"Fury give me strength!\"\n\nFor the barest instant, the thin shadows of the rooftop seemed to pull tight around Makka, making the livid scar of the bat-winged serpent on his broad chest leap out in contrast. Then his club came around hard and fast, too fast for Ashi to avoid. It hit with all of the bugbear's strength and the impact of it drove agony into her left shoulder. The hot rage behind the blow was like a physical force, throwing her across the rooftop. Her back hit the low wall, bringing another burst of pain and sending her toppling back.\n\nEmpty space and the sparse lights of Rhukaan Draal swung around her. Her sword dropped from her hand and vanished in the darkness below. She tried to throw her weight forward, back to safety.\n\nA massive hairy hand seized her forearm, yanking her back and whipping her across the roof. Ashi caught a glimpse of Makka, then she was tumbling across the stones. She ended up flat on her back and gasping for breath. A single fat drop of water hit the center of her forehead. The rain had come.\n\n\"Alive,\" she heard Makka growl. \"For now.\"\n\nShe forced herself to twist over onto her belly, then up onto her knees. On the far side of the roof, Midian fought the changeling who wore Geth's face, pick flashing and clashing against a long, heavy knife. The changeling looked panicked. He gave ground with each exchange. The gnome had him overmatched. Much closer, however, Makka stalked across the rooftop. In the up-cast light of Midian's lantern, his face was a demonic mask.\n\nAruget's dropped sword lay within reach to her right. Ashi grabbed for it but her throbbing left shoulder gave out and she tumbled back to the rain-dappled stones. Makka raised his club.\n\nAnother figure launched itself out of the shadows, slamming into him shoulder first. Aruget. The tackle sent Makka sprawling across the roof. Ashi sucked breath between her teeth and lurched to her feet, bringing up the hobgoblin's sword. Aruget, however, seized her arm and wrenched her around, dragging her toward the trap door.\n\n\"Aruget! No\u2014\"\n\n\"Down!\" he rasped. His throat showed deep red marks from Makka's noose. \"Somewhere we can't fall.\"\n\nAshi twisted in his grasp. Makka was rising. Midian still battered at Ko's desperate defense\u2014but as Aruget snatched up the lantern and sent shadows dancing across the rooftop, he glanced away. Ashi saw surprise flicker across his face.\n\nKo seized the moment of distraction. His blade licked past Midian's pick and slashed down his side. The gnome jumped back, blood mixing with rain on the stones.\n\n\"Midian!\" Ashi tried to pull away, but Aruget held her tight as he stooped to yank open the trap door.\n\nMakka roared and charged them. Ashi screamed fury in return and tore herself free.\n\nAruget grabbed her again, spun her around, and dropped her through the door. Her scream turned into a yelp. She just missed catching her feet on the steep steps, and her knees buckled as she hit the small landing below with a jarring impact. The lantern clattered down beside her, the metal shield breaking off but the magical core shining steady. She looked up just in time to see Aruget pull the trap door closed after himself. A heartbeat later, the door jerked ferociously as Makka heaved at it from the other side, but Aruget had his feet hooked into the steps. He held the handle of the door with one hand and dug into a pouch at his belt with the other. It emerged holding some kind of nail or spike. The hobgoblin slammed the nail up into the wood of the door. There was a shimmer of blue light. Aruget released his hold and dropped down beside her.\n\nThe door continued to shake under Makka's strength, but it didn't open. Aruget took Ashi's arm and pulled her to her feet. \"Go!\"\n\n\"But Midian\u2014\"\n\nAruget's face was strangely harsh. \"Midian can look after himself.\" He forced her onto the spiral stairs.\n\nAshi pushed against him. \"We have to go back for him!\"\n\n\"We can't go back. That door isn't opening again until it's broken in,\" Aruget said. \"We have to get you to safety. Makka couldn't have set this trap without Tariic's knowledge and aid. Khaar Mbar'ost isn't safe for you anymore, Ashi.\"\n\nShe started to argue, to accuse him of betraying Midian and leaving him for Makka, but the words didn't reach her mouth.\n\nNormally Aruget spoke the human language with a thick accent. All trace of that accent had suddenly vanished. She stared at him. His ears flicked and lay back.\n\nThen before either of them could say anything, there were other sounds on the spiral stairs. The tread of climbing footsteps. The rattle of armor. Many footsteps and much armor. A hobgoblin's voice floated up. \"Alive! She must be taken alive!\"\n\n_\"Khyberit gentis,\"_ Ashi cursed.\n\nAruget let out a sharp hiss, then he grabbed Ashi once again, gripping her tightly. \"Ashi,\" he whispered in her ear, \"you have to trust me. Forget Midian\u2014this is about you and me now. I think I can get us out of this, but you have to trust me. I'm on your side. Understand?\"\n\n\"I\u2014\"\n\n\"Good.\" He pulled the sword from her hand. \"Act defeated and frightened.\"\n\nHe pushed her on down the stairs. The light of the lantern, left behind on the landing, faded quickly and Aruget pushed her a little too fast so that she stumbled and groped in the dark. Between her stumbling feet and Aruget's sword and armor they made more noise than the soldiers coming up. A voice echoed along the stairs, asking in Goblin, \"Who's there?\"\n\n\"I have her!\"\n\nAshi flinched and almost fell. The voice, also speaking Goblin, that came from over her shoulder sounded nothing like Aruget. It was deeper and much rougher. The hand on her arm tightened, holding her up, and the voice continued. \"I've got the Deneith woman, but Makka needs help. She managed to seal the roof door somehow. Makka's stuck up there.\"\n\nA dim glow came up the stairs now and Ashi could see the silhouettes of hobgoblin guards. \"Who are you?\" the voice from below demanded. \"Daavn said Makka wouldn't let any guards come with him.\"\n\n\"Do you want to question what Makka said, or do you want to make Daavn happy?\"\n\nThey'd reached the first of the ascending guards. The stairs were narrow. There was barely room to pass. Aruget pushed by the first guards. Ashi could smell their sweat and the sour vinegar smell of _dar_ food on them. They glared at her with unfriendly eyes as she squeezed along. They muttered among themselves. One of them called down, \"It's her, _chib!\"_\n\nThe voice from below cursed, then shouted. \"Pass her along! We'll take her to Daavn.\" Ashi could see the speaker, a larger hobgoblin at the foot of the spiral stairs, the light from the main stairs illuminating him.\n\n_\"Maabet,\"_ said Aruget. \"I'm taking her to Daavn. I captured her. This is my _muut.\"_ He paused just above the large hobgoblin. \"He wanted her brought right to him, didn't he? Do you want to keep Makka waiting out in the rain while you take her\"\u2014he shook Ashi\u2014\"to Daavn? He's already angry.\"\n\nThe other hobgoblin hesitated, his ears up. Makka's attacks on the door were loud. Finally the hobgoblin's ears dropped. \"Take her,\" he said, jerking his head at the main stairs. Then he slapped the hobgoblin ahead of him. \"Get moving!\" he shouted.\n\nHe spared one last hostile glance at Ashi as she pushed past him, then she was off the spiral stairs and back in the open. Aruget stayed at her back. \"Keep going,\" he said in that strange deep voice. \"Remember you're my prisoner.\"\n\nThey made it down only a few steps before a call came back to them from above. \"You!\" It was the large hobgoblin again. Ashi felt Aruget stiffen but he kept pushing her onward. \"You can't take her by yourself. She could get away from you. You better let me help you.\"\n\n\"I can handle her,\" Aruget growled. \"Go back to your men.\"\n\n\"They can open a door.\" The other hobgoblin's footsteps closed on them. Aruget paused. \"Yes,\" said the hobgoblin. There was ambition in his voice. \"Enough _muut_ in this to share\u2014\"\n\nAruget's hand left her arm as he whirled. The other hobgoblin's words ended in the sound of ripping flesh, a slight wheeze, and the clatter of a falling body. Ashi spun around\u2014and froze, staring.\n\nNot at the hobgoblin sprawled with his throat slashed open, but at the other hobgoblin on the stairs. The one who carried Aruget's bloody sword and wore Aruget's armor, but who didn't wear his face.\n\nHe looked back at her for an instant, then his features blurred and reshaped themselves until Aruget faced her again.\n\nAshi's voice almost caught in her throat, but she forced it out. \"You're a changeling?\"\n\n#\n\n# CHAPTER  \nTWENTY-TWO\n\n# 28 Sypheros\n\nNot the place for explanations.\" Aruget took her hand and tugged her down the stairs. As soon as she was moving, he let her go and opened his stride, jumping down two and three steps at a time. When they reached a floor with access to the back stairs they had climbed on the way up, he led the way to them\u2014then off again only a few floors down. The same floor where Tariic had his quarters.\n\n\"This isn't a good place to stop,\" Ashi said.\n\n\"It's the last place anyone would look for us.\" He walked with light steps down the corridor, selected a dusty looking door, and tried the handle. The door was unlocked, the hinges stiff. Aruget eased it open a little way and slipped inside, beckoning Ashi to follow.\n\nShe hesitated.\n\nHe frowned. \"Trust me,\" he said. He slipped through the door. Ashi grimaced and followed.\n\nThe room beyond had a musty smell, and by the light that leaked in from the corridor, she could make out fabric-draped bundles. Aruget pushed the door shut a little ways, leaving only enough of a gap to allow a thread of light into the room. He stayed close to it so that the glow fell across his face. Ashi had a gut feeling he did that deliberately, as if it to ease some of her fears.\n\nThen he did something completely unexpected. He bent his head and his features melted and reformed. His entire body shifted in stature and bulk. When he looked up again, he had red-blond hair and the fine features of a young half-elf. A young half-elf woman.\n\nA young half-elf woman that Ashi knew. _\"Benti?\"_ she asked as softly as she could manage. _\"Benti Morren?\"_\n\nBenti smiled. \"Hello, Ashi. It's been a while\u2014for you, at least.\" Hard, cunning eyes narrowed. \"You understand now, don't you? I'm on your side. You can trust me. You _must_ trust me.\"\n\nThe urge to sit down washed over Ashi, but she didn't trust any of the dusty bundles in the dark room. She and other friends had encountered the half-elf\u2014or at least the person they had all assumed to be a half-elf\u2014in the city of Sharn almost a year ago. At the time, Benti had been posing as a renegade member of House Lyrandar selling her services as an airship pilot. After she had aided them\u2014and they'd aided her in return\u2014they'd discovered that she was more than she seemed. In fact, she was an agent of the King's Citadel of Breland, one of the so-called Dark Lanterns. In short, a spy.\n\nAnd it seemed that hadn't been her only secret.\n\n\"How?\" Ashi asked. \"Why? _When?_ \"\n\nBenti held up a hand. \"Fast answers,\" she said. \"We don't have time. Let's start with when: I've been Aruget since the night you were attacked by Gan'duur raiders on your journey from Sterngate in Breland to Rhukaan Draal.\"\n\n\"I remember that. We found you\u2014Aruget\u2014with a bashed skull after the attack. We thought the raiders had knocked you out.\" Ashi pressed her lips together. \"Where's the real Aruget?\"\n\n\"Buried under a collapsed sandbank near the spot where you camped. The scalp wound was self-inflicted. The raiders\"\u2014she shrugged\u2014\"a coincidence.\"\n\nThe cool detachment in her voice made Ashi shiver. \"Why?\" she asked again.\n\nBenti made a strange expression, as if her face was straining to move in a way it wasn't meant to. Her lips twitched and she put fingers to one slightly pointed ear. \"Spend too long as one race and you forget how other bodies work,\" she said. Her hand fell. \"Why? Because the King's Citadel was suspicious when we discovered Tariic was returning from a diplomatic mission with two ladies of House Deneith and a wandering shifter carrying an artifact sword of Dhakaan. I knew both you and Geth, so I was assigned to investigate.\" Her eyebrows twitched. \"It's turned into quite the assignment.\"\n\nConfusion churned in Ashi's belly. \"You couldn't have told us earlier?\"\n\nBenti's voice went cool again. \"I shouldn't be telling you now, but it seems to be my only choice. You have information I need. I've put together almost all the pieces of the puzzle.\" Her green eyes met Ashi's, and Ashi felt like they were looking right through her.\n\n\"The Rod of Kings,\" Benti said, \"tries to make its wielder into an emperor of Dhakaan. I heard Geth tell you that the night Haruuc died. Chetiin killed Haruuc to prevent a war, but when Tariic brings that war to life, no one tries to stop him. Instead, you, Geth, Ekhaas, and Dagii go to a tiefling artificer and have\u2014what? a copy of the rod?\u2014made with the power to enhance Tariic's presence. Geth tries to pass the copy of the rod to Tariic at the coronation. Tariic discovers the substitution and sends Daavn to arrest Geth. Daavn fails, or so I assume, and Geth is replaced with a changeling to keep his disappearance quiet. Now Makka, who previously wanted to kill you, is trying to capture you\u2014\"\n\nAshi's lips curled back. \"Wait. How long have you known that wasn't Geth?\"\n\n\"I had my doubts since the evening after the coronation, but I wasn't sure. Kill us and we return to our true form, but changelings can't recognize each other on sight any more than you can tell what color smallclothes another human is wearing.\" She flicked her hand. \"Let me finish. Makka is now trying to capture you, probably because of something you know or something that Tariic thinks you know.\"\n\n\"Like where Geth might be,\" Ashi said, the idea coming on her like a blossoming flower. \"And whether he has the true Rod of Kings.\"\n\n\"That's what I thought,\" said Benti. \"Which leads to the missing pieces of my puzzle. One\"\u2014she held up a finger\u2014\"where would Geth be? And two\"\u2014she held up another\u2014\"why go to so much trouble for the rod when ambition and history books can show any ruler how to be a tyrant?\"\n\nAshi shook her head. \"I don't know where Geth is. He might not even be in Rhukaan Draal anymore. We thought about running with the rod at one point. He might have done that. And the rod\u2014\"\n\nShe hesitated before saying anything more. Could she trust Benti with the ultimate secret of the rod? What if she couldn't escape Tariic's grasp? Benti knew everything else\u2014and as long as they were conspiring against a throne, as Vounn had pointed out, a spy seemed like a good ally to have.\n\n\"The rod doesn't just show its wielder how to behave like an emperor, it gives him the power to become one,\" she said. \"The wielder of the rod can force people to obey his commands. The magic Tenquis put in the false rod is just an imitation of the true rod's power. The true rod is irresistible. My dragonmark can block its power and Geth is immune because of his connection with Wrath, but those are the only defenses we know. When we found it, the rod was used against us. Its power crushes your will.\"\n\nEven talking about the power of the rod, just contemplating what might happen if Tariic got his hands on it, left her feeling cold. Benti, eyes narrowed in thought, didn't, but just narrowed her eyes in thought. Finally she said, \"Midian knows all this?\"\n\nAshi nodded. \"He was there. He felt the power of the rod. He took the same oath to keep it a secret that all of us did. It was his idea to substitute a false rod for the real one.\"\n\n\"Was it?\" Her eyebrows came together. \"Do you think he knows where Geth might be?\"\n\n\"If I don't, he doesn't.\" Ashi's lips curled back from her teeth. \"And you left him to be captured by Makka. Tariic could find out the power of the rod.\"\n\n\"Don't worry about Midian. Given how much interest Tariic is showing in the Rod of Kings, Ashi, I'd think he may already know.\"\n\nAshi stared at her. \"That's not possible. We kept it a secret.\"\n\n\"Possible or not, we should assume it's a fact.\" Benti drew a deep breath. \"Thank you for your help. Now I think it's time we got you to safety\u2014and preferably out of Darguun.\"\n\nShe concentrated and her face blurred once more, taking on the familiar coarse features, ruddy tones, and long mobile ears of Aruget. Ashi found herself continuing to stare at the changeling, no longer in shock at the fluid transformation but at the harshness of her tone.\n\n\"You're going to leave Midian to Tariic? And what about finding Geth? You need me!\"\n\n\"I don't need you.\" Aruget's voice warbled like the voice of a boy becoming a man, then settled into its normal pitch and accent. \"I can find Geth. And I told you, Midian can take care of himself.\" Eyes that had been green and were now deep brown flecked with orange studied her. \"You're vulnerable, Ashi. Both Makka and Tariic are after you now. If you really want to help Geth\u2014and Ekhaas and Dagii\u2014you run, and you keep yourself alive. If you want to stay, though, you're on your own. I can't hold your hand anymore.\" He adjusted his armor on a once-more bulky frame. \"Does Vounn have a plan for getting you to safety?\"\n\nChurning confusion and boiling anger settled into a sick feeling in the pit of Ashi's stomach. Benti... Aruget... whoever the changeling was, there was hard truth in his words. Geth had vanished, Midian was gone, Ekhaas and Dagii were far away. Vounn wouldn't be able to help her either. She'd already told her what she would have to do if Tariic's soldiers came for her. Aruget would look for Geth and the rod. Her usefulness was over\u2014it was time to retreat from the fight.\n\n\"Pater d'Orien,\" she said. \"Vounn told him I may be called back to Karrlakton. He's agreed to use his dragonmark to take me there.\"\n\n\"Then I hope he's willing to see late night visitors.\" Aruget eased the door of the room open again and peered out cautiously.\n\nAshi caught his arm. \"Wait. What's your real name?\" she asked.\n\nHe smiled and his ears flicked. \"Whatever one belongs to the face I'm wearing,\" he said.\n\nThey moved swiftly between floors, dashing down the stairs and ducking into doors whenever a guard appeared\u2014and there were more guards roaming the halls than Ashi had ever seen before.\n\nThere was no point going back to her chambers. Daavn would have anticipated that. Vounn had probably been woken already. Ashi was doubly glad she'd kept the meeting a secret. The lady seneschal would at least be able to tell the truth in saying she had no idea where Ashi had gone.\n\n\"How do we get out? The exits are going to be guarded.\" Ashi murmured in Aruget's ear as they crouched in another dusty room, waiting for a guard to pass.\n\n\"If you can't be silent,\" Aruget answered with a sly smile, \"make a lot of noise.\"\n\nHe led her away from the grand areas of the castle into a region of narrow corridors thick with the smell of cooking. They were near the kitchens. \"When Ko kidnapped Vounn, he brought her this way,\" said Ashi. \"Tariic won't forget to guard the back gate.\"\n\nAruget's ears twitched. \"Kitchens contain many interesting things.\"\n\n\"Knives.\"\n\n\"Cauldrons. Kettles. _Noon_ paste. _Korluaat.\"_\n\nShe looked at him questioningly. He shook his head. \"This is something I'll do more quickly alone.\" He hurried her past a wide, high vaulted passage that led to the even wider caverns of the kitchens and down another. A pair of big doors, plain and scarred from frequent use, emerged from the gloom. Wet footprints showed on the stone of the floor\u2014the doors opened to the outside of Khaar Mbar'ost and they'd been used recently. There would be guards on the other side.\n\nAruget went to a smaller door in the wall of the corridor and pushed it open, scanning the darkness inside. \"Storeroom. Wait for me here and be ready to run. If I'm wearing a different face, I'll wink. If I don't come, get out on your own.\"\n\nAshi stepped into the storeroom and was enveloped in the smell of unseen vegetables. She glanced back at Aruget. \"Ko couldn't see in the dark as Geth. How can you see in the dark as Aruget?\"\n\n\"Let me keep some secrets.\" He closed the door on her, leaving it open only a finger's width. His footsteps, so quiet that if he hadn't been wearing armor she probably wouldn't have heard them at all, went back along the passage.\n\nAshi squatted down in the shadows and tried to recapture the same patience and alertness she'd felt while waiting for the meeting on the rooftop. \"Ashi, daughter of Ner,\" she murmured under her breath, \"son of Kagan, son of Tyman, son of Joherra, daughter of Wroenna, daughter of Maal...\"\n\nTime stretched out. Patience didn't come and there was no need to stave off weariness\u2014Ashi didn't feel like she'd ever sleep again. She should have asked Aruget to bring a knife from the kitchen for her. A crude blade was better than no weapon at all.\n\nThere would be questions when she returned to Karrlakton. The lords of House Deneith would want to know why she had come back so suddenly. What would she tell them? What _could_ she tell them? What would she do\u2014?\n\nRunning footsteps echoed in the passage. Ashi pushed herself to her feet and backed into the darkness of the storeroom. But when the door swung open, the dim lights beyond shone on Aruget. He dived inside, pulling the door almost closed behind him.\n\n\"Aruget,\" she whispered, \"what\u2014?\"\n\nAnything she might have said was lost in an incredible _boom_ that sounded as if someone had thrown a massive bell against the wall of the fortress. It was followed by a tremendous crash of collapsing metal. A quavering shriek started up, stretching on and on without pause.\n\nOne of Aruget's hands found hers. The other pressed itself over her mouth. Out in the passage, the big outer doors were flung open and booted feet raced by. The opening of the outer doors nudged the door of the storeroom a little wider and Ashi caught a glimpse of hobgoblin guards running for the kitchen with drawn swords. The open doors also drew warm air from inside the fortress. Suddenly she could smell smoke, weirdly scented and stinging.\n\n\"Go!\" said Aruget. He released her and leaped for the storeroom door. She followed close behind.\n\nTwo unlucky guards lingered in the open doors. One saw them and opened his mouth to shout. Aruget's sword took him across the belly and then back across the throat in two swift cuts. Ashi dealt the other one a hard punch to the jaw that spun him around and dropped him to his knees. Aruget turned and swung his sword a third time. The guard's head leaped from his neck and rolled back into the passage. His body toppled to the side. Ashi started.\n\n\"Did you have to kill him?\"\n\n\"Yes.\" Aruget jumped for the doors, dragging them shut. \"There's an outer gate\u2014open it!\"\n\nBeyond a jutting roof that gave shelter to the doors, rain came down in cascades, turning the small courtyard beyond into a vast black puddle. Ashi splashed her way through it. The outer wall of the courtyard was simple brick, meant to keep out trespassers more than to repel attackers. There were no guards\u2014they must have all been huddled near the inner doors to stay dry. Heavy wooden doors in the brick wall were held shut only by a thick beam. Ashi grasped the rain-slicked wood, clenched her teeth, and hauled the beam free of one door before letting it drop. They didn't need both doors open to escape. The freed door swung wide with only a tug. Aruget joined her and, side by side, they dashed through the gate and ran out into Rhukaan Draal.\n\nThe night was very dark. Wind-driven rain soaked Ashi's clothes entirely. Aruget led her through a maze of sidestreets and alleys, always away from Khaar Mbar'ost but never in a straight line. Ashi flipped wet hair out of her face, wiped water from her eyes, and stayed with him. Once she thought she heard the sounds of pursuit, but they were gone as quickly as they appeared.\n\n\"What did you do in the kitchen?\" she said.\n\n_\"Korluaat_ poured inside a cauldron,\" Aruget gasped. His breathing was more labored than hers, but he carried the weight of his armor and of a larger body. \"Another cauldron jammed over top with _noon_ paste to seal the gap. Set over the fire with a rag for a fuse.\" He drew a ragged breath. \"More pots piled around them. Kettle stopped up with cork and a bit of metal. Spice jars on the coals. No more questions.\" He sucked in more air. \"Bloody hobgoblins\u2014not built for running!\"\n\nThey splashed through puddles that stank like sewers and others that were already as deep as her shins. She stumbled over unseen obstacles and Aruget dragged her to her feet. They didn't slow. Cold from her wet clothes numbed her skin, beaten back temporarily by the heat of exertion. Faces flicked past in the shadows beneath eaves and stairs\u2014people without homes or simply those caught in the storm, seeking out any shelter they could.\n\nIt seemed like they'd been running half the night before the alley they followed opened onto a broad street running with water. Rain and shadow rendered it as anonymous as all of the others at first, but then Aruget slowed to a stumbling, wheezing walk and pointed ahead. Ashi looked\u2014and a complex of buildings enclosed by a stout wall seemed to resolve out of the night. Beside tall iron gates, a crest depicting the head of a unicorn was illuminated by a muted but steady magical light.\n\nThe Orien compound. Ashi grabbed Aruget's arm and all but dragged him the rest of the way. \"Enough,\" he gasped. \"Let me go!\"\n\nShe released him and seized the gates by their heavy bars instead. They were locked, of course. The compound beyond was broad and lit by a few everbright lanterns that shone steady through the rain. Empty wagons were drawn up against outbuildings, and she could smell the animal odor of horses and tribex. No people\u2014human, hobgoblin, or otherwise\u2014were visible, however. They were probably taking shelter.\n\nAshi rattled the gates. \"Is anyone there?\" she shouted. \"I need to see Viceroy Pater d'Orien! It's urgent!\"\n\nAn iron rod was chained at one side of the gate. She seized it and hammered on the gates until they rang like chimes. \"I am here on an urgent matter concerning Deneith and Orien!\"\n\nThere was no response. None at all. She lifted the rod again and struck the gates harder. \"Pater! _Pater!_ \"\n\nA door opened, light flooding across the compound. A stiff-looking man in the uniform of a servant darted out into the rain and ran to the gates. Ashi recognized him from visits to the Orien compound\u2014he was Tars, Pater's manservant. His eyes were frightened and his mouth set tight. He slid to a stop at the gate and thrust a paper through the bars at her. \"No one will answer you,\" he said.\n\nHe glanced over his shoulder and stiffened. Another figure stood in the doorway. A hobgoblin warrior, armed and armored. For a moment it looked like he might come out into the compound but the fat form of Pater d'Orien appeared and drew him back. The viceroy threw a glance over his shoulder. Ashi couldn't tell if it was meant for her or his manservant. Tars shuddered. The paper slipped from his fingers and he fled back inside.\n\nAshi clung to the gate, cold metal pressing wet clothes against her skin. An armed hobgoblin inside the Orien compound. One of Tariic's soldiers? Almost certainly. Now that she'd seen him, she saw other things. A horse, still wearing saddle and bridle, tucked into the shelter of an overhanging eave when every other beast was in a stable. A comfortably dry gatehouse that stood empty. Faces that peered from darkened windows but made no move to answer her call.\n\nAruget bent and scooped the paper Tars had brought out of a puddle. His eyes skimmed over it. His ears lay flat. \"Ashi.\" He pushed the paper into her hands.\n\nA gust of wind tore at it and she had to stretch it tight, leaning into the torchlight that came through the gate to read it. Water was already making the ink run but she could make out what it said easily enough.\n\n_By decree of Lhesh Tariic Kurar'taarn, Ashi d'Deneith is accused of the murder of a soldier of Darguun. To offer her aid or interfere in her arrest is an offense to the throne and the people of Darguun_.\n\nThe warning was repeated in the dark letters of Goblin, but it had been written in the script of humans first. A deliberate warning to Pater.\n\n\"He knew,\" Ashi breathed. \"Tariic knew! Who told him?\"\n\nHoofbeats sounded over the patter of rain and the rush of wind. Ashi's head snapped up. Fear punched into her gut. One soldier sent to the compound as a messenger to prevent her escape\u2014and a whole squad sent after to trap her. She whirled. \"Aruget, we have to\u2014\"\n\nShe stood alone. Aruget was gone.\n\nSix hobgoblins on horseback burst out of the darkness, swords drawn, the hooves of their mounts sending up sprays of water. They came to stop in a semi-circle around her, trapping her against the gates. One of them walked his horse forward a little and pointed his sword at her.\n\nHe didn't have to say anything. Wet, shivering, and unarmed, Ashi crumpled the paper in her hand as she raised her head to meet his gaze.\n\n\"Geth.\" Hands shook him hard. \"Geth, wake up! There are horsemen outside.\"\n\nSleep burned away like shadows in the sun. Geth opened his eyes and sat up. Tenquis's workshop spun around him for a moment as his mind made the leap from drowsing to alertness. He'd fallen asleep in a big stuffed chair. The workshop was still brightly lit. Tenquis was still dressed. The table where they had eaten earlier in the day was now carefully laid with the tools for tomb breaking.\n\nAnd the sound of rain was overwhelmed by the clatter of hooves.\n\nGeth jumped up. He still wore his great gauntlet, though Wrath had been laid aside. He seized it. The blade seemed alert and happy, ready for its chosen hero's moment of glory. He cursed the ancient sword. \"How late is it?\" he asked Tenquis.\n\n\"Most of the way through the second watch, I think.\" The tiefling dashed around his workshop with quick movements, stuffing papers and trinkets into the pockets of his long embroidered vest. His tail lashed furiously. \"This is Tariic, isn't it? He figured you out\u2014or someone gave you away.\"\n\nGeth didn't answer that. Outside, hoofbeats had given way to footsteps. He pointed at the tool-covered table. \"Get rid of those!\"\n\nTenquis leaped to the table. His eyes flicked over it and he added a few more things to his pockets\u2014then took up a heavy steel pry bar and jabbed it into an inner pocket of his vest as well. The massive shaft slid out of sight without even shifting the fabric. Tenquis gripped the collar of his vest, whispered a word, and the labyrinthine pattern of embroidery that decorated the garment seemed to writhe. Any hint of bulging pockets vanished. \"Safe,\" Tenquis hissed between his teeth, then he seized the edge of the table and heaved, overturning it and sending the remaining tools skittering across the floor in an anonymous jumble.\n\nThe crash brought an exclamation from those outside\u2014and a command to attack. \"Get out one of the back windows!\" Geth shouted at Tenquis.\n\n\"They don't open!\"\n\nThe twin doors of the old barn burst in a shower of splinters under the shoulders of two big bugbears. Geth roared and charged to meet them, sweeping Wrath ahead of him. The twilight blade tore into the flesh of one of the bugbears, but the other managed to duck aside. A hairy fist wrapped in rings of scarred brass punched at him. Geth snapped up his gauntlet and brass screeched across black steel. Geth kicked the bugbear's shins and followed up with another swing of Wrath that forced the Darguul to jump back.\n\nBut more soldiers were pushing through the door, and hobgoblin hands were tearing at the shutters over the front windows of the barn. Geth saw Tenquis bare his teeth and snatch a slim wand from a workbench. Shifting to one side of the fight, he flicked the wand with one hand and, with the other, dashed the contents of a tiny vial into the air. Pale liquid leaped like something alive, flying farther than it should have and splashing in a ragged line under the windows and before the door. Thick greenish vapors rose up from it, a smoky curtain that brought shrieks of pain from the hobgoblins who thrust arms and faces through the broken windows.\n\n_\"Paaldaask!_ \" someone shouted. Spellcaster!\n\nFour hobgoblins had made it through the door before Tenquis's curtain had risen. Two charged for the tiefling while the other two moved warily to aid the bugbears menacing Geth. The shifter growled and made a low feint at the bugbear he'd wounded before. The soldier stumbled back, getting in the way of one of the hobgoblins, and Geth turned the feint into a whirling attack that brought him up inside the reach of the other bugbear. His armored fist drove hard into the Darguul's gut. The bugbear wore a heavy leather jerkin but the blow still doubled him over and sent him reeling.\n\nGeth stayed with him, pressing the attack. His foot came down on something hard and round\u2014one of the spilled tools from the overturned table. Already pulled off balance by the swinging weight of Wrath, he staggered.\n\nThe doubled-over bugbear lunged at him, big arms spread wide. Geth tried to twist out of the way, but the bugbear crashed into him and slammed him to the floor. Wrath flew from his hand. Instantly, the other soldiers were on him as well. They all carried clubs or weighted saps and didn't hesitate to mix their blows with hard kicks. Geth tried to ward them off with a sweep of his gauntlet, but a bugbear caught his arm and held it back.\n\nGeth caught a glimpse of Tenquis, wand stripped from his grasp and struggling with his own assailants, before a well-placed blow from a leather-wrapped club set his ears ringing and dark spots dancing before his eyes. Waves of nausea rolled through him, and he barely felt the pain as both arms were jerked behind his back and bound.\n\nThe door of the lhesh's chambers opened, and Daavn, dripping water onto the rich carpets, strode in. \"Geth and Ashi are captured, along with the tiefling artificer,\" he said. \"We found no sign of Aruget. He may have fled in shame.\"\n\n\"He doesn't matter.\" Tariic sat in a vast chair, fully dressed in spite of the hour. \"Geth and Ashi are the ones I wanted. Especially Geth. Did he have the rod?\"\n\n\"He wasn't carrying it. I searched the tiefling's workshop, but I couldn't find it. I have guards standing watch over the place, ready for you.\" Daavn paused and added. \"You don't have to search yourself, Tariic. You're the lhesh. There are soldiers I trust, clever goblins\u2014\"\n\n\"No!\" Tariic sat forward and his voice cracked like a whip. \"I will search for the Rod of Kings. It's mine. No one else is to so much as touch it.\"\n\nDaavn flinched, then ducked his head. _\"Mazo_ , lhesh.\"\n\nTariic sat back in his chair. He glanced at Makka\u2014and Makka glanced down at the prisoner he held by one shoulder.\n\nMidian was pale, with crusted blood on his mouth and one eye swollen, but his voice was bold. \"We had a deal.\"\n\nRage burned in Makka's gut. He squeezed and Midian squirmed. He squeezed harder, and the gnome gasped.\n\nPradoor, seated again on the spindly carved table she favored as a perch, gave him a poke with her stick. He glared at her, then at Tariic. Pradoor might not have been able to see him, but Tariic could\u2014and the lhesh didn't even blink. Makka eased his grip. Midian slumped a little but Makka held him upright. \"I swore an oath of vengeance,\" he growled. \"Will you _ever_ let me keep it?\"\n\n\"My needs come before yours. Let the royal historian go.\"\n\nDaavn started at the title and his mouth dropped open. Makka gave Midian another hard squeeze, then lifted his hand. For a moment, the gnome stood like a startled deer, ready to bolt at a moment's notice. Then his little pink slip of a tongue darted out and licked his bloodied lips. \"Thank you, lhesh. I promise, you won't regret this.\"\n\n\"You betrayed me and then your friends, Midian,\" Tariic said coolly. \"You're an opportunistic little rodent, but don't think you can dig your burrow a third time.\"\n\nMidian gave a wretched, scraping bow. \"Never. Lhesh, when you came to me and said you were looking for a scholar to join your uncle's search for the Rod of Kings\u2014\"\n\n\"Strange,\" said Tariic, \"I seem to recall that you came to me looking for a way to get into Darguun so you could pursue your research.\" He rose so that he towered over the gnome. \"I'll be watching you. Remember that you've already had your chance to run but that you chose to bargain for a chance to stay.\"\n\n\"I'm yours, lhesh!\" He scuttled out of the room.\n\nDaavn found his voice. \"Tariic, I don't like this.\"\n\n\"It's my decision, Daavn. Midian didn't let Ashi slip out of Khaar Mbar'ost.\" Tariic turned his eyes on Makka. \"Or allow her to escape a trap he promised was inescapable. Pradoor, will Ko recover?\"\n\n\"I have prayed over him.\" Her wrinkled face hardened. \"The Dark Six speak to me, Tariic. I agree with Daavn and Makka. Don't trust the gnome.\"\n\n\"I don't.\" The lhesh seated himself again. \"But, like Ko, he's useful. Both have their price. Ko loves money. Midian loves history\u2014and himself.\" His ears flicked and he looked at Makka again. \"You have the watching of him. If he turns against us, fulfill your oath.\"\n\n\"What about Ashi and Geth?\"\n\n\"If I don't find the Rod of Kings at the tiefling's workshop, I'll ask them. And when they've answered me, you can have what's left.\"\n\nMakka's lips drew back from his teeth. \"That's a bad trade.\"\n\n\"It's the only one you're getting.\"\n\nThere was a tentative rap at the door of the chamber and one of the guards of Khaar Mbar'ost entered. He bent his head diffidently, not raising his eyes to Tariic. \"Lhesh, you asked not to be disturbed, but this has arrived. The falcon carrying it was delayed by the storm.\"\n\nHe held out a metal tube smaller than a goblin's finger. A band of copper sealed it. Tariic took the thing and gestured for the guard to leave. When the door was closed, he examined the copper band and the design stamped into it, so small Makka saw it only as a darker dent.\n\n\"Dagii,\" he said. He broke the seal, pulled the tube apart, and extracted a tightly wound bit of paper. He spread it out. His eyes narrowed and his ears went flat. \"Daavn, all warriors in Rhukaan Draal are to be drafted in the city's defense. Warriors of all clans within a day's ride are to be summoned.\" Tariic flung the paper down. \"The Valaes Tairn have brought an entire warclan into Darguun. Dagii will meet them at Zarrthec.\"\n\nDaavn's ears rose. \"An entire warclan? Dagii's army can't face that. We'll reinforce him?\"\n\n\"We'll defend Rhukaan Draal.\" A wolf's smile spread across Tariic's face. \"And Dagii of Mur Talaan will find _muut_ and his death at Zarrthec.\"\n\n#\n\n# CHAPTER  \nTWENTY-THREE\n\n# 1 Aryth\n\nDawn turned the horizon gray. Cold air trapped smoke and steam from quenched fires close to the ground. Ordered chaos stretched around Ekhaas as Dagii's army broke camp. The commands of officers were sharp but unnecessary. Every soldier knew what to do. She watched, making note of the expressions on the faces of goblins, hobgoblins, and bugbears as they pulled down tents and packed away gear. Grim. Determined. Excited. Eager.\n\nHer own belly trembled with anticipation, the bloodlust of the _dar_ rising in her. Fear was a distant emotion. Battle was coming.\n\nFootsteps, muffled by ground still wet from the previous evening's storm, emphasized by the rattle of armor, approached from behind her. \"No word from Khaar Mbar'ost,\" said Dagii.\n\n\"Did you expect any?\" She turned around\u2014and stared for a moment.\n\nDagii wore the ancestral armor of the warlords of Mur Talaan once more, but it had been cleaned and polished until it flashed even in dawn's half light. The dents and scratches in the heavy plates were scars of honor. The three tall tribex horns mounted behind his head and shoulders could have been banners.\n\nDagii's ears flicked under her gaze, and he bent his head before he answered. \"Time is short. Tariic will do better to prepare Rhukaan Draal. Where will you watch the battle?\"\n\n\"From the command hill. If the _lhevk'rhu_ permits it.\"\n\n\"He will if the _duur'kala_ of Kech Volaar promises to retreat to safer ground when the battle turns against us.\"\n\nEkhaas scowled at him. _\"Duur'kala_ of the Kech Volaar can take care of themselves. You don't have to worry about me, Dagii.\"\n\n\"I'm not worried about you.\" Dagii's ears flicked again and his face tightened as he heard his own words. He lowered his voice. \"I am worried about you, Ekhaas, but I'm worried about what you'll carry as well. Senen asked you to record the story of our fight against the Valaes Tairn when we thought there were only warbands in Darguun. Now we're facing a warclan. You have to survive to carry the tale of what happens today.\"\n\nShe looked into his eyes, amber meeting gray. \"When the time comes,\" she said, \"I'll retreat with a sword in my hand and elf blood on my teeth.\"\n\nDagii's lips twitched, though he managed to keep a stern face. _\"Ban,\"_ he said, but Ekhaas could hear a fierce pride in his voice. Her belly trembled again.\n\nThe sun climbed two handspans into the sky.\n\nBelow the low hill Dagii had chosen for his command, below the earthen ramparts thrown up to give a measure of cover, warriors waited in close lines. Ekhaas could pick out the individual companies by their crests and their colors, simple strips of cloth tied to armor or polearms. Seven companies of infantry. Two more companies of cavalry, some mounted on horses, others\u2014in the ancient _dar_ tradition\u2014on battle-trained great cats. Leopards for goblins, tigers for hobgoblins. The cats were the only things that moved, pacing back and forth under the guidance of their riders, always kept carefully distant from the horses.\n\n\"They're good and hungry,\" said one of the handful of warlords Dagii had picked to stand with him on the hill. \"Always go into battle on a hungry cat.\"\n\n\"Hungry enough to keep them keen,\" said another, \"but not so hungry they stop to tear into prey. That's one good thing about horses.\"\n\nBeyond the lines of the army stretched the rolling grassy plain that would be the battlefield. Short grass waved\u2014except where it had been trampled down in a broad swathe before the army\u2014like a long green carpet running for leagues into the east between hills on the north and a low ridge to the south. A well-worn dirt track along its center, passing right under the feet of the waiting army. The plain was a natural passage through this part of the land; both it and the track ended at Zarrthec, and together they were much of the reason for the village's existence.\n\nTii'ator lay at the far end of the track. Ekhaas's shoulders itched at the memory of the retreat along the plain after the skirmish with the Valaes Tairn and their discovery in the Mournland. The entire way, she'd expected to feel an elven arrow in her back, though none had come. \"They'll know we've found them out,\" Dagii had said. \"They won't bother trying to catch us. They'll launch their attack\u2014and this is the path they'll travel to reach Rhukaan Draal. They won't try to hide themselves. They'll just move fast.\"\n\n\"We won't have much time to prepare,\" Chetiin had commented.\n\n\"No. We won't.\"\n\nSome time later, Ekhaas had realized that Chetiin and Marrow were no longer with them. She hadn't seen the _shaarat'khesh_ elder or the worg since.\n\nScouts left behind had confirmed Dagii's prediction. If it hadn't been for the storm that had rolled through, the warclan that rode under the swallow-tailed banner of stars would have been on top of them in the night.\n\nOn the track far out in front of the army, on the very edge of the trampled area, a solitary figure waited on a fine bay horse. Keraal.\n\nThe clear sound of a horn broke the air\u2014and ended in a discordant honk as if the scout who'd blown it had succumbed to a sudden, fatal wound. The first death of the day, thought Ekhaas.\n\n\"They're close,\" said Dagii. \"Drummers and pipers, as I ordered.\"\n\nA drum just behind Ekhaas began a low, slow beat. Warpipes droned. More drums and pipes scattered through the companies below joined in.\n\nThe first elves appeared over a rise in the distance, slim red-robed forms on white horses.\n\nSome paused to stare at the Darguuls gathered before them. Others turned and raced back, maybe to alert their leaders. Still others continued to ride on until Ekhaas could see eyes above veils and arrows nocked on bows, but even they stopped just out of bow range of the waiting army.\n\nThe Valaes Tairn rode in clusters or alone. They had no structure, no formations, no discipline. As increasing numbers appeared and rode up to the edge of range, their lines remained ragged and shifting. If there were officers among them, Ekhaas couldn't pick them out\u2014or even detect their influence. When the lord of the warclan finally appeared, it was almost a surprise. Only the large, dark crystal that sparkled in his helmet set him apart from other warriors. Even the bearer of the swallow-tailed starry banner seemed more to linger near him than to ride at his side. Ekhaas saw the lord of the warclan speak to one, then another, of the elves close to him, then for a long time, he simply sat and watched the _dar_.\n\nDrummers and pipers played on.\n\n\"I have heard that one of the most difficult things about fighting the Valaes Tairn,\" Dagii had said the previous night, \"is drawing them into a battle. They fall back before a charge. They ride around a stand. They come to a fight on their own terms. Victory is victory.\"\n\n\"How do you intend to engage them, then?\" a warlord had asked.\n\n\"We make them curious,\" Dagii had said, ears flicking. \"Then we give them a reason to fight.\"\n\nOn the battlefield, Keraal shook out and raised a banner. Tall and narrow, it had until last night been the red silk lining of a warlord's fine cloak. Now it carried a crest of three black rings, one above the other, each with three stretched slashes along the outside, like a sword blade bent into a circle with the notched edge out. Ekhaas's heart soared, not just because she'd been the one to supply the design, but for sheer awe at the sight of a banner that had not been raised over a _dar_ army since the beginning of the Desperate Times. Atop the command hill and on the battlefield, she saw warlords and warriors alike stand straighter as they gazed on the crest, ears rising proudly, as instinct stirred in them.\n\nWords rose in her, and her voice rang over the sound of pipes and drums. \"As the armies of the Dhakaani emperor fought, so shall we! Behold the _Riis Shaarii'mal!_ Behold the Three Tearing Wheels of Dhakaan!\"\n\n\"Give honor!\" shouted Dagii from her side.\n\nNine companies of disciplined _dar_ warriors responded in unison, fists striking chests in a single salute sharp as a crack of thunder.\n\nThe Valaes Tairn shifted warily. Ekhaas bared her teeth. The elves knew their ancient enemy\u2014and they knew the symbols of Dhakaan. The _Riis Shaarii'mal_ had flown above countless battles between _dar_ and elves in the time of the empire. To bring it forth again was a challenge, a declaration of rivalry. The leader of the warclan leaned over and spoke with one of the elves beside him. The rider nodded and urged his horse down to meet Keraal.\n\n\"Only one?\" growled a warlord on the hill.\n\n\"Patience,\" said Dagii.\n\nDrums and pipes fell silent. The elf reined in his horse a few paces from Keraal. The hobgoblin raised the red banner. \"I speak for Dagii of Mur Talaan, _lhevk-rhu_ of these warriors!\" he roared in Goblin.\n\nBehind his veil, the elf's eyes narrowed in disdain. He answered in the musical tones of Elven, lilting nonsense to many of the _dar_ on the battlefield but clear to Ekhaas. \"I speak for Seach Torainar, high warleader of the Sulliel warclan\u2014\"\n\n\"Now,\" said Dagii.\n\nThe drummer behind Ekhaas brought a stick down hard on the skin of his instrument.\n\nOn either side of Keraal, the trampled sod rose and flew back. Goblins hidden in camouflaged pockets popped up onto their knees, raised compact crossbows, and sighted. The waiting elves jerked in surprise. The elf who had ridden forward cried out and started to wheel his horse, but he was too slow. Ten bowstrings sang and ten bolts flew\u2014\n\n\u2014and ten scarlet flowers blossomed on the white hide of the elf's horse. The animal wore light barding, but it wasn't armored everywhere and the goblin bolts sought any exposed flesh. The horse screamed and bucked at the pain, fighting its rider's attempts at control. The goblins were up and running from their hiding places, racing up the dirt track to join the army. Keraal turned his horse and retreated as well, but slowly, mockingly. Bows rose among the waiting elves and arrows buzzed but they were too far and their own man, struggling with his horse, blocked true aim.\n\nThe wounded horse shrieked again. It stumbled once and its hindquarters collapsed as the strandpine sap that had coated the goblins' crossbow bolts burned into its blood. The elf leaped from the saddle and stared in horror at his crippled mount trying to drag itself with its forelegs. Its hind legs kicked uselessly. Then the elf wailed nearly as loudly as his horse, drew his scimitar and leaped forward to slash the beast's throat. Blood sprayed him, dark against scarlet robes. The horse's screams fell silent, but the elf's did not. Still wailing like a madman, he turned and raced after Keraal.\n\nHis cries were joined by others as the ragged lines of the Valenar broke and elves streamed into battle.\n\n_The only things they hold sacred_ , Chetiin had once said, _are their ancestors and their horses_.\n\nKeraal put spurs to his horse and galloped up the track, the _Riis Shaarii'mal_ snapping in the wind of his passage. Arrows loosed by the charging Valenar traced a dark line behind him. Ekhaas stole a glance at Dagii. The young warlord's face was hard, his eyes intent. He held his left hand in the air, waiting... waiting...\n\nHe brought it down. The piercing voice of the warpipes rose.\n\nAnd out at the edge of the battlefield, just behind the holes left by the emergence of the goblin snipers, long ropes snapped up from among the grass, pulled tight by teams of bugbears lurking in the thin woods on either side of the plain. Shards of broken glass worked in among the fibers glittered in the sunlight.\n\nCharging horses hit the ropes hard. The trees to which they had been lashed thrashed violently, and the heavy stakes that anchored them on the battlefield leaped. The long ropes sagged and snapped, but their damage was done. Horses whinnied and fell, tumbling like toys. They screamed at broken and slashed legs, and their struggling bodies brought down more horses that weren't quick enough to turn sharply or jump high. Some struggled back to their feet. Others didn't rise at all. The first charge of the Valaes Tairn had been broken. But there would be another.\n\nAn outraged voice roared in Elvish for an attack, and those elves who had held back surged forward.\n\n\"Form up!\" ordered Dagii and the great drum beat the signal. Lesser drums took it up and the Darguul lines folded and split. Seven squares formed up on the battlefield, shields locked together. Dagii snapped more commands, and five of the squares moved to meet the oncoming elves, warpipes wailing in their midst. Archers among the two companies that remained in the rear sent clouds of arrows arcing down ahead of the marching companies. The elves answered with arrows of their own\u2014arrows that rattled as harmlessly as hail on the locked shields.\n\nAbove the din of battle, Ekhaas didn't hear Keraal approach, but suddenly he was there. In his hand was clenched the _Riis Shaarii'mal_. He dropped to one knee and held it out to Dagii. The warlord clapped a fist to his chest.\n\n\" _Ta muut,\"_ he said.\n\nKeraal's ears flicked. He rose and stabbed the banner's shaft into the soft ground of the earthworks at the brow of the hill.\n\nThe first wave of Valaes Tairn closed on the Darguuls. The beat of the drum changed, and the squares stopped, two ahead, three behind. Shields parted slightly and spears thrust through. The armored turtles of the squares abruptly became bristling porcupines. Charging elves screamed and howled, demons in their flying red robes. They slashed at the air with their scimitars. Even on the hill, Ekhaas thought she could feel the earth trembling beneath the driving hooves of their horses. She had heard stories that during the Last War, human armies that had faced a Valenar charge had often crumbled before a blow was struck, their lines broken by sheer terror.\n\nThe Darguul lines stood strong. Valenar wheeled away, forced aside by the unyielding spear points. Crowded by the rush behind them, though, they had little room to turn. Many were forced around the two leading squares, splitting like a stream around stones.\n\nThe charge of those who swung too wide to the outside foundered as their horses' hooves found the third trap hidden under the trampled grass: all of the baskets in Zarrthec and all of the cages of willow switches that an army could weave, hidden under all the loose branches that the forest could provide. Sticks and switches and baskets closed around hooves and legs, doing no damage but fouling the charge as surely as pits of mud. And as the charge slowed, more goblin crossbowmen concealed among the autumn-brown trees at the edge of the plain loosed their bolts\u2014not poisoned this time, but enough to bloody the elves and drop them in their saddles.\n\nThe Valenar who flowed between the squares fared no better. They found themselves in broad aisles between the forward and back formations with nowhere to go except out to the sides where branches grabbed hooves or onward through narrower aisles between the three rear squares.\n\nSo they raced on\u2014directly into the arrows of the two companies that had stayed behind. More elves fell, but others escaped the trap of the squares.\n\n\"Cavalry!\" Dagii ordered and the command drum changed its beat. The rain of arrows stopped and the Darguul cavalry took the field, sweeping in front of the standing companies to meet the elves that had made it through the gauntlet. The Darguul horse thundered across the plain, a moving wedge composed of lances, but now the elves had room to move. They melted away before the charge and answered in kind. They streamed along the sides of the horse formation, scimitars slashing, and where they passed, hobgoblins died.\n\nBut behind the horse cavalry, they met the cats.\n\nThere were no formations for them to evade. The tiger and leopard riders fought as the Valenar did, alone and in clusters. The already-trampled grass became a mass of green and brown as hooves and paws chewed into the ground underneath. Ekhaas watched an elf rider dart close to a hobgoblin, lash out with a scimitar, and wheel away out of sword's reach. The hobgoblin nudged his tiger mount. The great cat coiled and sprang. Massive claws seized the haunches of fleeing horse and bore it down. The impact flung the elf out of her saddle. A goblin-ridden leopard pounced on her before she could even regain her feet.\n\nThe beat of the drums changed again. The infantry companies pressed forward once more, squares altering shape to become blunt-nosed wedges with the brute strength of bugbears at their head. The horse company reformed and drove in behind them.\n\n\"We have command of the field!\" cried one of the warlords on the hill.\n\nDagii's ears pressed flat back against his head. \"A blow has been struck,\" he growled, \"but the battle continues.\"\n\nBeyond the foremost of the charging companies, Ekhaas saw one of the Valaes Tairn raise his hand. An orb of violet glass flashed in his grip.\n\nPurple-tinged flames exploded at the head of the nearest Darguul company. Bugbears became writhing torches an instant before dropping. Hobgoblins scattered. The marching wedge cracked like a bone thrown into a fire. With a wild cheer, elves wheeled their horses into the broken company as drums sounded and the Darguuls fought to recover their formation.\n\nOther wizards rode with the elves. Yellow vapors engulfed trees where goblin snipers perched. When they faded, dry leaves and goblins alike lay on the ground beneath. The tiger that Ekhaas had watched bring down a horse fell at the touch of an icy blue ray. Lightning crackled and arced among the spear points of another company. Across the battlefield, flame and frost and lightning ravaged the Darguuls, tearing openings for the Valenar to exploit. Here and there, hobgoblin warcasters responded with blasts of rippling force, but their spells were weak and few _\u2014dar_ were born to war, but elves were born to magic. Lightning-scorched shields parted and a warcaster thrust his staff at an elf wizard. The air shimmered between them, tearing at the elf's red robes, but he held his ground and responded with a golden bolt that spun the warcaster around and sent him sprawling.\n\nArrows followed like a swarm of bees, tearing a hole in the Darguul company. Shields closed again, but the damage had been done and the formation was left with a gouge in its side. The company drew itself together, leaving the corpses of the warcaster and half a dozen warriors behind.\n\nThe early triumph of _dar_ discipline over elven disarray was gone. In the moments when the battle shifted, Ekhaas could count more Darguul bodies on the churned ground than she could Valenar. Elven skill reasserted itself. Many of the Valaes Tairn had abandoned their horses to engage the hobgoblins on foot. The bristling spears of the surviving squares, so effective in warding off cavalry attacks, were little use against lithe foot soldiers who weaved between the shafts and wedged shields apart with scimitars. Armored squares dissolved into thick knots of hobgoblins fighting back-to-back, sword and shield against whirling scimitars. Though the great cats still stalked the battlefield, the remains of the Darguul horse cavalry had been reduced to a handful of mounted units fighting in shrinking clusters.\n\nAt the far edge of the battle, the starry banner of the Sulliel warclan moved into the thick of the fighting.\n\nDagii bared his teeth and retrieved his helmet, polished as bright as the rest of his armor. \"There is no more command,\" he said. \"When war calls, all fight.\" He scanned the three warlords who had stood with him on the command hill, then thumped his chest with his fists.\n\nThe warlords straightened like junior warriors on parade and saluted in return, then turned and hurried down the back of the hill. Keraal moved to retrieve the _Riis Shaarii'mal_ , but Dagii shook his head. \"No.\" He looked at Ekhaas. \"Take it when you leave.\"\n\nEkhaas nodded and forced her ears to stand high as she met his gray-eyed gaze. He nodded to her, then slid his helmet over his head. A visor of brass hid the upper half of his face, as if in reverse of veils worn by the Valaes Tairn, and gave him a cold, merciless look.\n\nHe turned to the final four warriors on the hill: the command drummer, the command piper, and Biiri and Uukam, the _lhurusk_ who had accompanied them to Tii'ator. \"You will protect Ekhaas _duur'kala_ with your lives. Our _muut_ survives with her.\"\n\nAll four answered together. _\"Mazo, lhevk'rhu!\"_\n\nThen Dagii and Keraal were gone, following the warlords down the hill. Ekhaas looked up at the sun. Somehow it had already climbed another handspan and stood now at its zenith, an unblinking witness to the battle.\n\nA voice shouted below. Ekhaas leaned past the earthworks and gazed down. The two companies that had been held in reserve stood straight. The one on the left, marching beneath the standard of an iron fox, moved forward, ready to take to the field. The lines of the other shifted and spread out, a frail bulwark to protect the command hill. From around the side of the hill came the three warlords, Keraal, and Dagii. The warlords were mounted on good horses and Keraal was on his bay once more, but Dagii rode on one of the finest tigers Ekhaas had ever seen. It was enormous, large even for a tiger. Its striped fur shone like fire and brass flashed from the high-cantled saddle strapped across its back. Its eyes didn't hold the sinister intelligence of Marrow's, but there was cunning there and a lust for blood. Under Dagii's guidance, it loped out to take a place at the head of the company.\n\nDagii raised his sword\u2014then whirled it around his head and let loose a battle cry. \"Bring honor to Darguun! Attack! _Attack!\"_\n\nThe roar of his tiger nearly drowned out the cry. In an instant, the beast was racing across the battlefield for the nearest cluster of elves. Keraal followed close behind, chain whistling around his head. The warlords and the warriors of the company followed, too, warriors trotting at a pace that ate up the ground, yet kept them in formation. The elves that Dagii had chosen as a target were still just turning when the _lhevk'rhu_ reached them. His sword cut the head from one. The tiger's claws tore another from his saddle. The weighted end of Keraal's chain cracked the skull of a third.\n\nThen they were through and plunging on. The dazed survivors they left behind lived only a few moments longer as the company of the Iron Fox rolled over them. Elves turned to meet the new charge like ants drawn to a dollop of honey. In the midst of them came the swallow-tailed banner of stars, and Ekhaas picked out the crystal-set helmet of the high warleader, Seach Torainar. Dagii must have seen the banner, too. He leveled his sword and shouted again, urging his tiger and his followers on.\n\nThe cheer that answered him came not just from the Iron Fox Company, but from all the Darguuls surviving on the field. New vigor seemed to flow into them, and they struck back at the elves. Formations drew back together. One piper, then another, and another, then even the command piper beside Ekhaas, began to play a fierce, determined tune.\n\nEkhaas looked down on the battlefield and clenched her teeth tight. The resurgence was glorious\u2014and almost certainly doomed. Slim scimitars outnumbered heavy swords now. For every two elves that Dagii struck down, three more were ready to meet him.\n\n\"He turns _muut_ into _atcha,\"_ said Biiri in awe. \"If I survive, I will praise him before Lhesh Tariic!\"\n\n\"Storytellers will chant his name,\" said Uukam. He looked at Ekhaas, his ears high. \"Stay as long as you can, Ekhaas _duur'kala_. Witness the end of a hero!\"\n\nShe wasn't certain she trusted herself to speak. She nodded silently instead\u2014a nod that froze as a slow wind stirred, and the ghost of a song touched her ears. At first it was thin, so thin it could have been an echo, but then it swelled with Senen Dhakaan's voice, as full and distinct as if the ambassador of the Kech Volaar stood beside her.\n\n_The ghaal'dar Aruget brings news. Ashi d'Deneith has been arrested at Tariic's orders. A changeling takes Geth's place. Makka hunts. Beware, daughter of the dirge!_\n\nBreath caught in Ekhaas's throat. Ashi arrested? A changeling replacing Geth? What was happening in Rhukaan Draal? Her gut twisted. Dagii rode to his doom. The friends she had left behind were in trouble. Chetiin had vanished once more. For the first time since they had found the Rod of Kings and discovered the dangerous secrets it held, she felt very, very alone.\n\n\"Ekhaas?\" Uukam asked. \"Is something wrong?\"\n\nShe raised her head\u2014and shook it. She'd been alone before. She'd traveled and fought alone for years before she'd met Geth, Ashi, and the others. Dagii faced his death. She could do nothing less. Whatever had happened to Ashi and Geth, she might be their only hope. Ekhaas pushed out the knots in her belly and forced herself to breathe again. She could still hear Senen's song. As long as it lingered, the magic would carry back a reply. She looked down at the struggle on the battlefield, gathered her thoughts, and opened her mouth to sing\u2014\n\nBehind her, the music of the warpipes ended with a sudden moan from the pipes and a gasp from the piper. She whirled around, Biiri and Uukam moving with her.\n\nThe piper, throat slashed, hung in the arms of an elf. The drummer stood with the scimitar of a second elf at his chest. Six more of the Valaes Tairn crouched on the back edge of the hill, weapons ready.\n\nSenen's song faded.\n\nThe elves screamed and charged.\n\n#\n\n# CHAPTER  \nTWENTY-FOUR\n\n# 1 Aryth\n\nUukam and Biiri drew their swords and moved in front of her with the reflexes of trained soldiers. Strategy flickered in Ekhaas's mind. The elves must have made their way through the woods to the back of the hill. They were already too close to blast them with sound or sing fear into their souls. There would be no disabling all of them. This fight would be toe-to-toe and they'd need every sword at hand.\n\nThe elf whose scimitar hovered over the drummer's chest drew back his blade, ready to strike, as the other elves charged. Ekhaas focused her will on him, drew on her magic, and let loose a wail like a ghost rising from the grave. The elf's head snapped around, eyes wide. His scimitar paused, trembled.\n\nAnd the hobgoblin drummer struck back. The short thick rods of brass that had beaten the great drum knocked the scimitar out of the elf's hand, and cracked across his face. The elf reeled back but the hobgoblin stayed with him, raining blows with lethal rhythm.\n\nThen the other elves were on her, Biiri, and Uukam. Curved blades crashed on the warriors' shields. Two couldn't hold back six\u2014seven, as the elf who had killed the piper dropped his victim and joined the attack. Ekhaas snatched her sword from its sheath and thrust her free hand into the face of the first Valenar to come at her. Song burst from her throat, and light as bright as a sliver of the noon sun flared from her palm. Dazzled, the elf fell back a pace. Ekhaas stepped after him, chopping down with her sword. It screeched off of armor beneath the red robes, but the elf cried out and retreated.\n\nAnother took his place, swinging backhand as he whirled past. Ekhaas twisted her sword and the scimitar caught among the heavy teeth on the back of the blade. Biiri struck past her in a blow that sheared through metal and flesh. One elf down.\n\nTwo\u2014Uukam cut through the torso of another, leaving her grasping at a terrible wound. Three\u2014the drummer's opponent didn't rise as the hobgoblin straightened, drumsticks red with blood, chest heaving with exhilaration.\n\n\"Call for aid!\" Ekhaas shouted at him. He dashed for his drum, but jerked and spun around between one step and the next. Two arrows sprouted from his back.\n\nWhere eight Valenar could go, more could surely follow.\n\nClenching her teeth, Ekhaas threw herself across the hilltop. Uukam and Biiri shifted instantly to protect their own backs. Four elves still faced them. One went after Ekhaas and hooked her foot with his. Ekhaas sprawled face first into grass and dirt. The impact knocked the wind out of her, but she forced herself to roll.\n\nA scimitar gouged the soil a handspan behind her. She kicked blindly and felt her boot connect. The elf staggered back. Ekhaas got up on her hands and knees and scuttled to look over the back edge of the hill.\n\nSix more of the Valaes Tairn were climbing the slope, two covering the ascent with arrows nocked. One swung his bow toward her. Ekhaas sucked air into her lungs, swept her free hand through the air, and forced out a thin, liquid song. The ground beneath and around the elves turned dark and shiny as greased leather. Their feet slipped out from under them, and they went tumbling back down the hill in a flurry of robes, scimitars, and wildly-loosed arrows. Ekhaas grinned\u2014\n\n\u2014and pain like fire seared down her back with a force that knocked her to the ground. Gasping, she twisted. The Valenar she had kicked stood over her, raising his scimitar for another strike. Ekhaas kicked again, but he avoided her easily this time and shifted to get a better blow.\n\nHis foot came down on the pipes dropped by the slain piper. The sagging leopard skin bag collapsed under his heel and the pipes released one final startling bleat of sound. The elf leaped in surprise. Ekhaas pushed herself away from the edge of the hill and back to her feet. She could feel blood running down her back underneath the leather armor. Her back burned with every movement, but she could still raise her sword. The elf moved around her, picking the moment of his next attack.\n\n\"Ekhaas! With us!\" shouted Biiri. From the corner of her eye, she saw him cut down one of the Valenar who menaced him and Uukam, just as Uukam struck out with his shield, forcing back the other two elves. The two soldiers turned as one, offering her a chance to reach their side.\n\nEkhaas took it. She feinted at the elf, then dodged past him to join Biiri and Uukam. Three hobgoblins fighting together now\u2014but four elves circling them, death in the eyes that shone above red veils.\n\n\"We'll force an opening,\" Uukam rasped, his voice thick with exertion. \"You run. Get to the woods.\"\n\nFierce determination rose from Ekhaas's gut up into her throat. Her ears stiffened. There would be no escape through the woods, but she also felt no desire to flee. Eight Valenar had taken them by surprise and they had still brought half of them down. They could take the other four. They _would_ take the other four. \"No!\" she spat. \"No running. We fight! We fight as the Dhakaani fought! We fight and win!\"\n\nHer voice rose in a song, harsh and martial, a song she'd heard Dagii's soldiers chanting as they marched to the devastated clanhold of Tii'ator. It was no spell, but she wove magic into it, invoking the ferocity and discipline of an entire army prepared for battle. Biiri and Uukam stood straight as the song caught them up. New strength seemed to enter their arms, and Ekhaas's as well. Her grip on her sword grew steady; the throbbing pain of her wounded back grew distant.\n\nThe Valaes Tairn appeared to feel the song, too. They glanced at each other, scimitars wavering slightly. Then one of them cried out in Elven, \"For the glory of Kaelan!\" and leaped forward.\n\nBiiri's shield snapped up to catch the falling scimitar\u2014and Uukam's sword cut beneath to tear open the elf's belly. Without even a pause, he swung himself over the falling body, slicing with the lower lip of his shield at the next Valenar. The elf ducked under the sudden blow, only to meet Biiri's rising sword. Ekhaas stayed with them, her back to theirs. The two remaining elves shouted and plunged at her as if to end the song. She dropped down and struck back in a low, wide arc. They moved like dancers, evading her sweep with ease.\n\nUukam and Biiri whirled, closed in on either side of Ekhaas, and brought their heavy swords around with a speed and force that turned elven grace into bloody tatters. Biiri's blade cut the head from one elf. Uukam's shattered the scimitar of the other and drove on into his chest. Red robes fluttered to the ground and torn flesh thumped down on top of them. Ekhaas's song rose then faded away. For a moment, it seemed that the only sound was her and her companions' breathing, the sound of triumph more certain and primal than any cheer.\n\nThen the sound of the battle that still raged below the hill burst over them. The clash of metal. The shrieks of horses, the roars of great cats, the screams of the dying. Biiri's sword, still raised, dropped. \"Now we run, Ekhaas _duur'kala,\"_ he said. \"Dagii commands it. You carry our _muut.\"_\n\nHe moved to the earthworks at the brow of the hill to seize the crimson banner of the _Riis Shaarii'mal_ , but when he reached it, he froze, staring down. Ekhaas rose from her crouch. Both she and Uukam went to stand beside him. Her ears flicked then went flat.\n\nWhile they had fought atop the hill, the battle had shifted and surrounded them.\n\nThe reserve company that protected the bottom of the hill was a thin wall fighting raging opponents. The squares of the five companies that had first marched into battle were like sputtering flames, clinging to life. Even as she watched, two were extinguished completely as the elves surged and overwhelmed them. Bursts of lightning and fire flared here and there, wiping out more _dar_. What remained of the Darguul cavalry fought either with the reserve company or with the loose formation of the final company to enter battle, the Iron Fox. Maybe half of the Valaes Tairn warclan lay dead\u2014but so did more than three-quarters of the Darguuls.\n\nAlong the seething border between _dar_ and elves, Dagii fought Seach Torainar. The tall tribex horns mounted to the shoulders of Dagii's armor marked the _lhevk'rhu_ as surely as the flashing crystal in the high warleader's helm did him. Bounding tiger leaped around wheeling horse. Hobgoblin sword and shield met deadly Valenar double-scimitar\u2014two curved blades joined end to end through a single long hilt. Keraal fought close by, his whirling chain warding off any of the Valaes Tairn who tried to take Dagii from behind.\n\n_\"Paatcha!_ \" said Uukam in awe. \"This is a battle worth dying in!\"\n\n\"It's a battle one of us must survive,\" Biiri said. \"Ekhaas _duur'kala_ , if we don't leave we'll be caught.\" He grabbed her arm and tried to pull her away.\n\nShe shook him off. \"We're already caught. There are elves at the back of the hill.\"\n\nUukam cursed and raced across the hilltop to the back edge, then cursed again. \"A squad of our cavalry fight them, but more come. Our riders won't hold for long.\"\n\n\"We could fight our way through,\" said Biiri, baring his teeth.\n\n\"And be struck down by Valenar arrows while we ran.\" Trembling eagerness ran through Ekhaas. \"Dagii makes a stand. The battle isn't over.\" She spun and thrust her finger at Biiri. \"Bring the drum! Play!\"\n\n\"I can't!\"\n\nUukam snatched the bloody brass rods from the hands of the dead drummer. \"I can.\"\n\n\"Then do it. Beat a pace like a loping tiger. Biiri, watch our backs.\"\n\nEkhaas stepped up onto the earthworks so that she stood beside the _Riis Shaarii'mal_. She took three deep, slow breaths, then one that was very deep. She reached down inside herself, drew up the power of the song that twined around her soul, and sang as she never had before.\n\nThere had been words to the song she had sung for Biiri and Uukam. The tune had been familiar to them. What she sang now had no words, and the tune was ancient. Another _duur'kala_ or one of the dedicated lorekeepers of the vaults of the Kech Volaar might know it. Ekhaas was certain that no one on the battlefield below had ever heard it, yet she was equally certain that it would fire the spirit of every _dar_ , every true child of ancient Dhakaan, who did. The greatest glories of the past could never truly be forgotten.\n\nSong rolled out of her belly and her chest as it had rolled out of _duur'kala_ of old, but freshly infused with her magic\u2014and from the first note, it seemed that what she sang was even older than Dhakaan, that it grew out of a primal need to fight for life and to triumph over death. The sensation was dizzying, but Ekhaas poured everything she felt into the song. Long hours of training in Volaar Draal had taught her how to project her voice, but even a _duur'kala's_ voice couldn't carry unaided over the battlefield. A slim thread of magic stolen from the magnificent whole amplified and tied it to the rhythm Uukam beat out on the great drum. Both song and drumbeat rose into the sky and echoed across the plain, pure as sunlight and as powerful as a storm.\n\nThe battle paused, all eyes looking up at her in amazement, elven bright and _dar_ dark. In the elves she could see a sudden tension, an instinctive mistrust of this martial music. In the _dar_ , though, she saw exactly what she had expected. Wonder. Longing. Awe. Rage.\n\nShe sought out Dagii. The brass half-mask of his helm was raised to her\u2014then he thrust his sword in the air. \"Attack!\" he roared. \"Attack!\"\n\nThe battle crashed back into motion. Dagii urged his tiger at Seach Torainar, but a ripple in the currents of fighting forced the two leaders apart. Quick-thinking elves launched a volley of arrows up at Ekhaas but the angle of the hill and a fold of the earthworks offered her protection. Or perhaps the song itself deflected the arrows. Ekhaas's voice soared.\n\nIn her mind's eye, she saw the glories of Dhakaan portrayed in the stories and artifacts that had come down through the ages to the Kech Volaar.\n\nCarved cities and mighty fortresses.\n\nVast armies sweeping enemies before them\u2014gnolls, elves, the dread daelkyr and their foul minions. Works of staggering majesty.\n\nThe magical songs of _duur'kala_ and the wondrous creations of _daashor_.\n\nThe deeds of emperors and generals and warriors\u2014heroes of the _dar_.\n\nShe channeled what she saw into what she sang, and her song was the song of the Empire of Dhakaan.\n\nThe beat of Uukam's drum became the beat played by all the drums that had survived on the battlefield. The rhythm anchored Ekhaas to the fighting soldiers of Darguun. It carried her vision into living hearts, bringing a renewed energy to the goblin who rode his leopard against a Valenar horse, to the bugbear who laid about him with a steel mace that reaped lives as a scythe reaped grain, to the hobgoblin warcaster who beat his broken staff against the bleeding head of a staggering elf wizard. Ekhaas watched the reserve regiment at the base of the hill gather itself and throw back the Valaes Tairn. She watched the fragments of Darguul companies flow together, form themselves into wedges and force their way through their enemies to reach each other. She watched the Iron Fox Company, under Dagii's command, take a position at the heart of the swirling battle.\n\nAnd Valaes Tairn died\u2014but so did Darguuls. Slowly, in spite of the valor of the _dar_ , the battle turned back to the favor of the elves. A newly reformed company collapsed. A rush of elven cavalry drove the reserve company back to the base of the hill and shattered their lines. Beneath the swallow-tailed banner of stars, Seach Torainar whirled his double scimitar around his head and gathered elves for a rush on the Iron Fox. Dagii shouted and roared, urging his warriors into stronger lines to meet the attack.\n\n_\"Maabet!\"_ Biiri cried from behind her. \"They're through. They're coming\u2014the elves are climbing!\"\n\nThe beat of Uukam's drum faltered, then faded, and Ekhaas knew he had gone to aid in whatever defense Biiri could muster.\n\nShe sang alone. Her throat was raw. Her jaw and chest ached. She knew that the veins and muscles of her neck must be standing out, straining like the rigging on a ship in a storm. She looked down on the defeat of Dagii's army and almost\u2014almost\u2014the sorrow of a dirge, of the fall of Dhakaan and the beginning of the hard Desperate Times, crept into her song.\n\nNo. She wouldn't let Darguuls die with the sounds of mourning in their ears. She wouldn't sing defeat before the Valaes Tairn.\n\nEkhaas reached deep into herself, flung her arms wide, and sang defiance. Her song soared again as she built on the old music, weaving a new vision into it. A vision of a homeland for an ancient people, restored to pride after long millennia. A vision of a red tower above a sprawling, bustling city; of clan chiefs and warlords gathered in unity to take back the land that belonged to the _dar;_ of a new age for hobgoblins, goblins, and bugbears. Haruuc's vision.\n\nShe threw back her head and howled Darguun's pride to the sky.\n\nAnother howl answered her.\n\nAnd another.\n\nAnd another.\n\nAnd the elves of the Sulliel warclan were shouting in alarm and fear. Ekhaas's gaze snapped back to the battlefield.\n\nFrom the hills and woods on the northern side of the plain poured a river of black and gray and white. Small, lithe forms hunched over powerful shoulders that bunched and surged beneath thick fur. Goblins and wolves, the larger shapes of worgs among them as well. _Taarka'khesh_. The Silent Wolves, silent no longer.\n\nHope gripped her and her song rose, wild and triumphant. She howled and wolves howled back. Ekhaas heard startled shouts from Biiri and Uukam and elven screams from the backside of the hill, and she knew that the Valenar who had sought to capture them had new problems. Across the battlefield, the _taarka'khesh_ crashed into the elves, throwing them into disarray. Teeth snapped at the legs of horses. Short blades and crossbows bit into elf flesh.\n\nAnother rider joined Keraal at Dagii's side. Dressed in black, he was all but invisible against the black fur of the great worg he rode, but Ekhaas knew him. Chetiin\u2014and Marrow\u2014had returned.\n\nThe double scimitar of Seach Torainar dipped and spun, gutting a goblin with one blade and piercing his wolf mount with the other. Then the weapon rose high and whirled. The high warleader raised a slim horn to his lips and blew a long, wailing note\u2014and he wheeled his mount and raced from the battlefield. Everywhere, elves broke off their fighting to follow his retreat. Ekhaas might have been standing at the ocean's edge, watching the tide turn. Victory had turned into a rout.\n\nThose Valenar who were still mounted offered a hand to comrades on foot, or else turned to cover their escape. The _taarka'khesh_ didn't pause. Elves who stood to fight and horses too slow in their flight were overwhelmed by snapping, tearing jaws and stabbing blades. The retreat was a flurry of red robes and white flanks galloping away along the plain; the pursuit was a rushing shadow, night chasing day into the east instead of the west.\n\nThe desolation of the battlefield was revealed. Corpses of elf and _dar_ , horse and great cat and wolf. Of the proud Darguul army, only the Iron Fox Company remained in any numbers.\n\nEkhaas's song swirled to a final ringing note that filled the sky in triumph. The power of it faded from her body and left her trembling. Her right hand found a shoulder of the earthwork for support. Her left found the _Riis Shaarii'mal_. She plucked the banner from the ground and thrust it high.\n\nA cheer rose from the survivors of the battle, muted and small amid the carnage, yet deafening in its own way. A rider turned his tiger, racing it across torn corpses and churned grass, and urging it up the steep slope of the hill's front in mighty bounds. When the beast reached the earthworks, Dagii slid from the saddle, pulled his helmet from sweat-drenched hair, and dropped to his knees in front of Ekhaas. His ears stood tall. His shadow-gray eyes were wide with pride and adoration.\n\n_\"Taarka'nu,\"_ he rasped. Wolf woman.\n\nEkhaas's strained throat could barely work but she forced her voice through it. _\"Ruuska'te,\"_ she whispered. Tiger man.\n\nHe rose, put his hand over hers on the shaft of the _Riis Shaarii'mal_ , and they turned to face the battlefield together. The survivors roared again, even louder than before.\n\nThe long shadows of late afternoon reached along the plain. The survivors of Dagii's army gathered _dar_ corpses. A pit would be dug and a cairn built over it. Later a proper monument might be erected to the heroes who had expected only to slow an attacking army and had instead defeated it.\n\nThe bodies of elves and their horses were left where they lay. Carrion eaters were already circling in the sky and gathering in the woods.\n\nMost of the _taarkakhesh_ and a number of the surviving great cat cavalry still pursued the fleeing Valenar. The howls of wolves and worgs echoed out of the east, relayed over great distances. \"The elves make no stand,\" Chetiin said, listening to the howls along with Marrow. \"They may run all the way to the Mournland. Horses can outpace cats and wolves over long distance but the _taarka'khesh_ will patrol the border for a time to make sure they don't try to come back.\"\n\n\"They'll find their way back to Valenar,\" said Dagii. He sat on a log, one bandaged leg thrust out before him. A scimitar had pierced a weak point in his armor. One of the _taarka'khesh_ had offered him magic to heal it, but Dagii had directed him to chant his spells over Ekhaas's wounded back instead. \"Through the Mournland or down to Kraken Bay for passage on a sympathetic Lyrandar ship.\"\n\nKeraal, standing with his arms crossed, grunted agreement and added, \"Lhesh Tariic owes a debt to you. He should greet you in Rhukaan Draal as a hero.\"\n\nCaught in the middle of a sip of numbingly hot herbal tea intended to soothe her throat, Ekhaas grimaced. She looked around. The four of them were, for the moment, alone. She wasn't sure she wanted to speak in front of Keraal, but she didn't think she had much choice.\n\n\"That may not happen,\" she croaked. Her voice sounded as strained as Chetiin's. \"A song message from Senen Dhakaan came to me during the battle. Tariic has arrested Ashi, and a changeling has taken Geth's place. She said Makka is hunting, too. I don't know what that means, but\u2014\"\n\n\"Something is wrong,\" Dagii said.\n\n\"I'm going back,\" Ekhaas told him. \"Ahead of the returning troops. I may be able to slip into the city.\"\n\nDagii nodded. \"I'm going with you.\"\n\n\"No,\" said Chetiin. The _shaarat'khesh_ elder's ears cupped. \"Your place is with your soldiers. I'll go with her.\"\n\nKeraal's eyes had narrowed in suspicion. Dagii bared his teeth and flattened his ears. \"There's more happening than you know, Keraal. This may be a test of your oath to serve me.\"\n\n_\"Ban,\"_ the other warrior snarled. \"I see as well as a hawk by day.\" He looked at Chetiin. \"Your name's not Maanin. You're Chetiin.\"\n\nMarrow growled. The goblin's ears flicked, but he nodded.\n\nKeraal looked back to Dagii. \"Did you hire him to assassinate Lhesh Haruuc? I hated him, but I wouldn't have done something like that. If you did, you don't have the _muut_ or the _atcha_ I believed you did.\"\n\nBreath hissed between Dagii's teeth and his ears pressed back even further. \"I didn't\u2014and Chetiin wasn't the assassin. Suggest something like that again and I'll pit my sword against your chains.\"\n\nTension pulled at the air between the two warriors. Ekhaas's hands tightened around her mug, but before she could say anything, Keraal bent his head. \"I am without honor in this, _lhevk'rhu,\"_ he said in apology. \"I doubted you.\"\n\nDagii said nothing for a moment, then his ears rose slightly. \"I have a story to tell you, Keraal, but it will wait for the journey back to Rhukaan Draal.\" He looked at Ekhaas and Chetiin. \"If Geth and Ashi are in trouble, we should hurry. It would take three days if we travel at the pace of the slowest survivor. A small company could make it in two. Tariic would think nothing of that\u2014a victorious warlord isn't slow to share his news.\"\n\n\"Your arrival can serve as a distraction, then,\" said Ekhaas. \"Chetiin and I will go ahead. I know magic that can speed our travel.\"\n\n\"Save your voice.\" Chetiin scratched Marrow's head. \"You're not the only one who knows something of swift travel. The Silent Clans will aid us. I guarantee that no one will know we've returned to Rhukaan Draal.\" He glanced at her. \"If you're not too tired to leave tonight.\"\n\nEkhaas's ears stood tall. \"I'm ready.\"\n\n#\n\n# CHAPTER  \nTWENTY-FIVE\n\n# 3 Aryth\n\nGeth could smell burned flesh. It was his.\n\nHe could smell scorched hair and stale sweat, old blood and hot metal, charcoal and, curiously, the hint of sweet spices.\n\n_\"Te laloo kaanii.\"_ Daavn's voice. The warlord of the Marhaan spoke Goblin. Without Wrath, Geth only caught the roughest meaning of his words. Something about healing quickly.\n\n_\"Chiit so shiftaa,\"_ said Tariic. Geth felt something poke at the skin of his side. With his arms stretched and bound over his head, his body was exposed and vulnerable. _\"Toma piisho,\"_ Tariic added. _\"So kaas te vusrii.\"_\n\n_Vusrii_. To burn.\n\nThe touch of red-hot metal seared his other side and Geth jerked and screamed. His eyes snapped open to the same small, brazier-lit chamber lit chamber he had seen for... who knew how long. He managed to evade the burning metal for a brief instant, but the it was back, pressed firmly against his skin. He howled and thrashed but the iron stayed on him. The gnarled hands of the waxy-fleshed goblin who held it were steady. Dark eyes flashed with greedy pleasure.\n\nFinally the metal pulled away. Geth collapsed back against the inclined table on which he was stretched. Tariic moved close and clamped a hand over his forehead, holding him still. Eyes so brown they were almost red stared into his.\n\n\"Where is the Rod of Kings?\" he asked.\n\nGeth fought the haze of pain and forced out the same answer he had given again and again. \"I don't have it!\"\n\n_\"Te kuur doovol.\"_ He tells the truth.\n\nThis time it was Pradoor's shrill voice. Geth twisted his head under Tariic's palm. He could just see the old goblin woman crouched beside a heavy rack of knives, white eyes shining like the sharpened blades. Symbols had been scrawled in a rough arc on the filth-crusted floor in front of her. At the center of the arc, smoke shifted from a metal bowl filled with coals.\n\nTariic cursed. \"You're certain?\"\n\nPradoor's fingers twined through a bunch of cords knotted together and strung with small, flashing tokens. \"The Six lend me the wisdom to hear lies,\" she said in the human tongue, accented but clear. \"He tells the truth.\" Her wrinkled face split in a smile. \"But he doesn't answer the question, does he? Ask another.\"\n\nTariic's ears went back and he looked at Geth again. \"Who has the Rod of Kings, then?\"\n\n\"I told you!\" Geth groaned. The evasions came easily. \"Chetiin stole it!\"\n\n\"He tells the truth.\"\n\nWith a growl, Tariic gestured and pointed at Geth's belly. The goblin torturer nodded and turned to the brazier. Metal grated on metal as he exchanged the cooling iron for a fresh one. He didn't speak. Tariic had shown Geth that he couldn't\u2014his tongue had been cut out\u2014and that he couldn't hear pleas, questions, or answers either. Deaf and mute, he was the perfect tool for extracting secret information.\n\nThe chamber was well-used. Had the torturer plied his trade for Haruuc?\n\nHot metal swiped across Geth's stomach like a knife. He screamed again and strained against his bonds. Ropes creaked. Tariic slammed him back.\n\n\"What did Chetiin do with it?\"\n\n\"He ran! He climbed down the wall of Khaar Mbar'ost and disappeared into Rhukaan Draal. I haven't seen him since!\" His voice cracked in an involuntary sob. Deep inside him, an inner voice was stronger. _Hold out! He must not find it_.\n\n\"He tells the truth,\" said Pradoor again.\n\nGeth looked up into Tariic's eyes. \"Just kill me,\" he said. \"Get it over with.\"\n\nTariic roared and seized the collar of black stones that still hung around Geth's neck\u2014the torturer hadn't been able to break or remove it, a strange property that even Geth hadn't been aware of\u2014in one hand and wrenched him up by it. \"I have your sword,\" he snarled into Geth's face. \"I have Aram. It hangs on the wall of my quarters as a trophy. I know you're hiding something. If you don't tell me where to find the Rod of Kings, you will die by the Sword of Heroes!\"\n\nGeth bared his teeth. \"You can't wield Wrath. The Sword of Heroes won't bear the touch of a coward.\" Up close he could see the red burns on Tariic's palms. \"You've found that out already.\"\n\nTariic's ears went back flat and he shifted his grip on the collar, twisting it around his fist until the stones bit into Geth's throat. Shadows swam in Geth's vision. He saw the torturer's face. The goblin looked disappointed. Geth sank down into warm oblivion\u2014until the pressure on his neck eased. Air came rushing back. He thought he saw Tariic step back, fury on his face, and he thought he heard the lhesh say, \"He won't break easily. I don't have time for this. Take him away.\"\n\nHe gestured at the torturer, communicating instructions with signs. The goblin produced a leather bag and pulled it over Geth's head. Something coated the leather, making his vision whirl again. His last sight as the bag came down was Tariic turning to Daavn. The last thing he heard were the words, \"Bring me the tiefling.\"\n\nHobgoblin guards marched Ashi, hands bound behind her back, through a stout door and into a room with high, narrow windows. Sunlight pierced the windows, bright enough to blind her after the darkness of her cell. Fire warmed the room. Carpets cushioned the hard floor and soft chairs waited for her.\n\nVounn stood across the room, before the fire. Ashi couldn't hold back a gasp at the sight of her. She pulled away from the guards. \"Vounn!\"\n\nHer mentor turned, crossed the room in three swift strides\u2014and slapped her hard.\n\n\"You fool!\" she said, her tone seething. She stormed past her to confront the guards. \"Get out!\" she said in Goblin. The hobgoblins looked baffled. Vounn flicked a hand at them imperiously. \"Get out, I said! Wait outside the door. Where is she going to go?\"\n\nThe guards glanced at each other, then bent their heads and retreated. They didn't untie her hands. As the door closed behind them, Vounn whirled on Ashi again and thrust her furious face close.\n\n\"We're being watched,\" she whispered. Her voice rose again. \"You killed a guard of Khaar Mbar'ost! Explain yourself!\"\n\nAshi blinked. The first words out of her mouth were no act. \"I didn't kill any guard! It was\u2014\"\n\nVounn slapped her again. \"You bring shame on Deneith!\" she spat, then cried out and pulled Ashi into an embrace like a mother crying over a willful child\u2014and whispered in her ear, \"Aruget came to me. He told me what happened. Keep his name out of it.\"\n\nNow Ashi understood what she was doing. _Dar_ culture was uncomfortable with touching\u2014especially embracing\u2014in public. Any Darguul watching them would more than likely look away at least briefly from this human affectation. They had a few moments of privacy.\n\nThere was one question she needed answered more than any other, one thing that had haunted her while she waited in her cell. Only three people knew about their arrangement with Pater d'Orien to transport her away from Rhukaan Draal and could have told Tariic to send a warning to the Orien compound. Vounn had no reason to give her away to Tariic. Pater wouldn't have sent a warning to himself. That left only one person who could have betrayed her.\n\n\"Have you seen Midian?\" she asked Vounn.\n\nMaybe Vounn had guessed the same thing. \"Yes,\" she said without hesitation. \"He's been avoiding me, but I've seen him. Tariic has appointed him royal historian.\"\n\nAshi stiffened, rage flashing through her. Vounn held her still. \"You look better than I thought you might after three days in a dungeon. Tariic wouldn't let me see you. Has he questioned you about the Rod of Kings?\"\n\n\"No,\" Ashi said tightly. That was something she didn't understand. She hadn't seen Tariic\u2014or Daavn or Makka\u2014since her arrest. Had it really been three days? She'd spent a long time afraid that Tariic might torture her to find out what she knew or that Makka would come seeking his frustrated vengeance. There'd been nothing, only a little food and water shoved through a hatch in the door of her cell. \"Where's Aruget?\"\n\n\"I don't know. I only spoke with him once, but he leaves me messages. He knew I was coming here.\" She hesitated, then added, \"He told me to tell you he'd get you out.\" She looked into Ashi's eyes. _\"Don't go with him.\"_\n\nBefore Ashi could respond, the older woman pushed her away and her voice grew angry again. \"You don't deserve it, but I'm working to have you released and banished rather than executed,\" she said, a little loud for the benefit of anyone watching. \"House Deneith supports you even if you've dishonored it.\" Her eyes caught Ashi's again and she said pointedly, \"The negotiations are delicate. We're risking much, but the evidence against you is weak.\"\n\nAshi blinked again. Vounn still played the same balancing game against Tariic that had kept her free after the coronation. Maybe that was why Tariic hadn't tried to question her about the rod. Darguun still needed House Deneith and a charge of murder, especially one that couldn't be proved, wasn't the same as one of conspiracy. Escaping with Aruget would leave her a fugitive; if Vounn could negotiate her release, she wouldn't be pursued.\n\nIf her negotiations were successful.\n\nMindful of watching eyes, Ashi bent low. \"I don't deserve this.\"\n\n\"No,\" Vounn said, \"you don't, but our lives belong to Deneith.\" She turned and sat down in one of the chairs. \"There's something you might like to know,\" she continued casually. \"Word has come from Zarrthec. Dagii's army met a Valenar warclan in battle and defeated them.\"\n\nAshi couldn't hold back a gasp. \"An entire warclan? But they\u2014\" She swallowed. \"Dagii? Ekhaas?\"\n\n\"Dagii's company returns to Rhukaan Draal today. Word comes that Ekhaas travels with him.\" Vounn put disdain into the answer, although her eyes were actually bright and warm. She gestured at the windows and Ashi became aware of a distant buzz of activity somewhere beyond it. \"A heroes' welcome is being prepared. I'll tell them of your shame.\"\n\nFear knotted Ashi's belly\u2014not for herself, but for Ekhaas and Dagii. They were walking into a storm of danger. At least Vounn might be able to warn them. She bent her head, trying to make it look like she was suitably dismayed. Vounn raised her voice and called in Goblin. \"Guards! I am finished. Take her away.\"\n\nThe door opened, and the guards returned, laying rough hands on her. Ashi managed a last glance back at Vounn. Her mentor's face was hard and cold but her lips were pressed together into a thin, pale line of concern.\n\nThere were few prisoners left in the larger cells of the dungeon. First the games, then the war, had emptied them. A deeper level of the dungeon held other prisoners. Ashi had heard screams, muffled by distance and stone, more than once while she sat contemplating her own fate. Her gut felt hollow as the guards marched her back through the dungeon, and not just because she was hungry. Could Vounn find a way to have her released or would Tariic finally come and take her down into the deep dungeon? Or would Aruget try to free her and should she let him?\n\nThe guards pushed her around a corner into the dead end corridor where her cell and a handful of others\u2014empty\u2014waited, doors half open like pits of shadow.\n\nSomething moved suddenly in one of them and a glittering vial of dark glass flew through the air.\n\nIt hit the stone floor just in front of Ashi and shattered. At least, she thought it shattered. In the same instant that it struck the stones, light flared from it, blue-white and bright. There was no sound of breaking glass, as if even sound had been overwhelmed by the blinding light. Ashi was certain that she gasped, but she didn't hear her own voice. She felt the guards' hands leave her, but heard nothing of them. She felt, rather than saw, a rush of movement. Out of instinct, she jumped back, only to collide with a heavy object. One of the guards. His falling body knocked her off balance. With her hands still tied, she stumbled hard into a wall.\n\nThis time she heard herself grunt. Vision and hearing were already returning. Against the fading glare of the bright light, she made out the figure of the second guard flailing helplessly as if at an unseen attacker before abruptly collapsing.\n\nHer first thought was of Aruget\u2014but then a small figure resolved out of the glare.\n\n\"Ashi?\" asked Midian.\n\n#\n\n# CHAPTER  \nTWENTY-SIX\n\n# 3 Aryth\n\nRage swelled in her and she kicked out at him. \"Midian, you double-crossing\u2014!\"\n\n\"Easy!\" The gnome skipped aside, avoiding her kick. He raised his hands, but stayed well out of range. \"And be quiet! There are more guards around. I can explain!\"\n\nAs her vision cleared, Ashi got a better look at the bodies of the guards. The hobgoblins weren't dead, only unconscious. Strange glass spikes, like small knives with a point but no edge, stuck out of them, expertly thrust through gaps in their light armor into flesh beneath. The points seemed to be hollow, their insides smeared with the remnants of something thick and green-black. Midian followed her gaze, and said, \"Bloodspikes. I got them from an inquisitive of House Medani.\" He pointed at a patch of dark, shattered glass surrounded by a slick of shimmering fluid. \"That was mine. Light to blind, improved with a burst of silence to cover up any sounds. You can tell me I'm clever.\"\n\n\"You betrayed me!\"\n\n\"And now I'm rescuing you!\" His blue eyes hardened. \"What was I supposed to do? Aureon's quill, you left me out on the roof with Makka! I'm lucky Tariic wanted us taken alive!\"\n\nEven through her anger, she winced at the rebuke. Aruget had said Midian could take of himself\u2014and he had\u2014but he was right. They'd left him behind.\n\nMidian was watching her warily, his hand hovering close to a stiff pouch on his waist. Ashi wondered if he had another of the bloodspikes in there. She pushed her fury away and sat up, her bound hands making the movement awkward.\n\n\"I'm sorry,\" she said. \"We shouldn't have done that.\"\n\n\"Well, I'm sorry I gave you away\u2014and that I had to wait so long to come for you. Tariic, Daavn, and Makka have been watching me.\"\n\nHe moved around behind her. Ashi heard him draw a knife and cut at her bonds. \"I've just seen Vounn. She said Tariic has appointed you royal historian,\" she said.\n\n\"I thought it would make him trust me so I'd have a chance to get to you,\" said Midian. \"It didn't work quite as well as I hoped. I'm here now, though.\"\n\nThe ropes parted. Ashi stretched her shoulders and rubbed her wrists. So much for worrying whether to wait for Vounn to negotiate a release or go with Aruget when he came to rescue her. She was already rescued and once more a fugitive. \"How long will the guards stay unconscious?\"\n\n\"Long enough, I hope.\" Midian stuffed rags into the hobgoblin's mouths and bound them hand and foot with rope from a coil produced from the cell where he had hidden. \"Put them in your cell.\"\n\nShe did. Even heaving them over her shoulders and dropping them onto the cold floor didn't disturb their slumber. She pulled the sword belt off one and buckled it around her waist. \"Ekhaas and Dagii are returning to Rhukaan Draal today,\" she said as Midian closed and barred the cell door.\n\n\"I know,\" said the gnome. \"That's the reason I was able to get away from Tariic. It's going to be a shame to miss the party, but it will provide a distraction.\"\n\n\"We need to warn them.\"\n\nHe flinched at the idea. \"There isn't time.\"\n\n\"We make time!\" Ashi said, snapping her teeth on the words. \"I sat for three days afraid Tariic was going to come and question me about the Rod of Kings. I don't know why he didn't, but I don't want him to have that chance with Ekhaas or Dagii.\"\n\nMidian flinched again, and the color drained from his face. \"I... uh, I know why Tariic didn't come for you,\" he said.\n\nAshi looked at him sharply.\n\nHe turned his eyes away. \"I gave him Geth and Tenquis too.\"\n\n_\"What?\"_ Ashi stepped back in shock. \"How could you\u2014? Wait. You knew where Geth was?\"\n\n\"I didn't!\" Midian said, his voice rising like that of a protesting child. \"After Makka caught me, I tried giving Tenquis's name to Tariic first. I thought he'd be interested in the artificer who made the false rod, but he wanted more, and that's when I had to turn you over to him. When Tariic's men went to arrest Tenquis, they found Geth too. He'd been hiding with him.\"\n\nFear struck Ashi like a cold blade. _\"Rond betch_. So Tariic has the true rod?\"\n\n\"No.\" Midian shook his head. \"They couldn't find it at Tenquis's. Geth must have hidden it.\"\n\n\"And where's Geth now?\"\n\nMidian held out his hand\u2014and pointed down. Ashi remembered the muffled screams she'd heard from the deeper dungeon. \"No,\" she breathed. \"Do you know how to get down there?\"\n\nThe gnome nodded. \"Back to the main cells and down from there.\"\n\nAshi spun around and raced back up the hall. The stone floors of Khaar Mbar'ost weren't the soft ground of the Shadow Marches but she could still run silently enough. She burst into the wide room lined with the doors of the large cells.\n\nAt the foot of the stairs that led into the dungeon, a hobgoblin guard wearing the red corded armband of Khaar Mbar'ost froze in surprise. Ashi snarled and threw herself at him.\n\nThe impact slammed the hobgoblin back against the stairs. The guard got his hands on her wrists, but the wind had been knocked out him and Ashi was stronger. Her fingers closed around his throat even as he sucked in new breath and wheezed\u2014\n\n\"Ashi! It's me!\"\n\nEkhaas's voice. Ashi jerked back. The guard beneath her pushed her hands away and narrowed his eyes in concentration. Ghostlight flickered and the illusion vanished, revealing Ekhaas's familiar face. Ashi stared at her in astonishment. \"What are you doing here?\"\n\n\"Senen sent me a message. I'm rescuing you!\"\n\n\"There's a lot of that happening.\" Ashi jumped up and helped Ekhaas stand. The few remaining prisoners in the large cells were staring out at them. Midian was staring too. The gnome stood in an archway of another, darker corridor, the one that led, Ashi guessed, to the lower level of the dungeon. \"Midian beat you to it. Come with us\u2014Geth and Tenquis are being held below.\" She grabbed Ekhaas's hand to drag her along.\n\nThe _duur'kala_ resisted. \"The real Geth?\n\n\"You know about the changeling?\"\n\n\"Aruget told Senen.\"\n\nAruget had gotten around. Ashi wondered where he was now. \"It's the real Geth.\" She dropped her voice as she drew Ekhaas across to the dark corridor, and murmured the story the gnome had told her. \"He didn't know Geth was with Tenquis. I've heard screams. I think Tariic's been torturing Geth to find out where he hid the rod. Will you be able to heal him if he needs it?\"\n\nEkhaas's ears flicked back. She gave Midian a long look. \"I'll try,\" she said.\n\n\"You don't need to,\" said Midian. \"I brought healing potions.\"\n\n\"You knew Geth had been tortured?\" asked Ekhaas.\n\n\"I guessed,\" Midian said. \"Him or Ashi.\"\n\n\"Wait!\" cried a voice in Goblin. Ashi turned to see a hobgoblin's face thrust against the bars of a cell door. \"Free us!\" he hissed.\n\nA soft babble of pleas for release joined his. The other prisoners had finally figured out what was going on. Maybe they didn't speak the human tongue. Ashi glanced at Ekhaas and Midian. Both of them shook their heads, Midian instantly, Ekhaas after a moment's consideration.\n\nThe first prisoner's face turned hard. \"Let us out or we call the guards. There's a dozen of them below!\"\n\n\"If we let them go,\" Ekhaas said in the human language, \"they'll cause chaos in Khaar Mbar'ost. People will notice the escape. We'll be found either way.\"\n\nThe hobgoblin prisoner looked angry. He might not have understood the language of humans but he must have read their meaning in Ekhaas's expression. His voice rose into a shout. \"Guards! Escape! _Escape!\"_\n\nOther prisoners joined in, the din echoing in the dungeon. A moment later, the sound of rapid footsteps rose up the stairs from the lower level.\n\n_\"Khyberit ghentis!\"_ Ashi raised her stolen sword and turned to the stairs just as a hobgoblin guard emerged from them. His eyes landed on her and Ekhaas and widened.\n\nBut Midian was waiting. He dived between the guard's legs and the guard went crashing to the ground. Ekhaas leaped to his aid but Midian was already on his feet and jamming another glass spike into the guard's neck. Ashi whirled on the prisoner who had started the commotion. He tried to pull back but she pushed her hand through the bars, grabbed the front of his shirt, and jerked him forward. Hard.\n\n\"Do that again,\" she snarled, \"and you'll die before you face the Valenar!\"\n\nShe let him drop. The other prisoners fell back from the cell doors.\n\nEkhaas was looking down the stairs, her ears pricked forward. \"There's no one else coming,\" she said. \"Only one guard?\"\n\nAshi glanced into the cell. \"Where are all the guards?\" she asked in Goblin.\n\n\"Tuuk was bluffing,\" answered one of the prisoners tremulously. \"There were never that many and those there were left a while ago. Tariic led them out.\"\n\nAshi didn't like the sound of that. She turned away, pushed past Midian, and trotted down the stairs to another corridor. There was light at the bottom, but no sound. On the left and right were doors like those of the cells above, small barred windows set in heavy wood. At the end of the corridor was a larger, even heavier door.\n\nThe floor of the corridor was streaked with smeared trails of dried blood connecting the cell doors with the heavy door at the end. Ashi's gut knotted. Her hand tightened on the sword.\n\nEkhaas touched her arm and pointed. Only one of the cell doors was closed and barred. Ashi crept closer. Through the barred window, she could hear a strangely pathetic sound. A soft, whimpering growl. But the cell was too deeply shadowed for her to see anything. Midian nudged her and held out a tiny everbright lantern, even smaller than the one that had been lost on the roof. She opened it only slightly and held it up to the window.\n\nThe cold, magical light fell on Geth's curled and trembling form. He lay on his side, facing the door. He'd drawn his shirt over himself like a thin blanket. Where it gaped open, Ashi could see big patches of bare skin amid the thick hair of the shifter's torso and arms. The hair had been burned away. The flesh beneath was ravaged by healed and healing scars. She directed the light to his face, frightened at what she might find.\n\nDirty. Haggard. But intact. Her heart skipped and her breath came sharp.\n\nAs if he'd heard it, Geth's eyes snapped open, shining like an animal's. The growl rose sharply and he pulled back, rolling up into a crouch. \"Geth!\" Ashi said. \"Geth! It's us!\"\n\nHe jerked. \"Ashi!\" He jumped to his feet and charged at the door, sharp teeth bared. \"Let me out! They've taken Tenquis!\"\n\nAshi dropped the lantern to the floor and dragged at the heavy bolts that held the door. Geth stood on the other side, shaking the door and making the job harder. The thick nails of his fingers gouged at the wood. His eyes were very wide, the pupils as large and dark as a madman's. Ashi could smell sweat, blood, and burned flesh on him. His savage growl rumbled and broke.\n\nShe pulled the last bolt just as he slammed especially hard against the door. It burst open, throwing her back into Ekhaas and Midian. Geth didn't even glance at them, but just hurled himself down the corridor to the heavy door. He hit the door with a muscular shoulder and whatever latch held it closed on the other side tore right out of the frame. Running after him, Ashi caught a glimpse into the room beyond\u2014\n\nShadows in deep corners. Dim, ruddy light glinting on sharp metal. A pale goblin jerking away from an angled table to which a dark-skinned tiefling was bound, arms and legs outstretched. Blood dripping from the table and pooling on the floor under it.\n\n\u2014then Geth slammed the door behind himself. A bestial roar rattled the broken wood. Metal crashed on stone. There was a hiss like hot coals spilled into water and a sudden, awful stench, then a thin, wordless cry.\n\nGeth didn't look at Tenquis\u2014at what had been done to the tiefling\u2014a second time. Flayed skin, glistening muscle, and exposed bone slid through his mind but the images were buried in the hot fury that fell over him. Fury at what Tenquis had suffered. Fury at what _he_ had suffered.\n\nHis charge into the room had sent a brazier crashing to the ground. Blood on the floor quenched the hot coals, raising a haze of stinking smoke. The torturer cried out and tried to flee into the depths of the chamber but Geth's hand closed on his skinny neck. The shifter whirled him off his feet and slammed him hard into the stained boards of an upright rack.\n\nHis free hand groped for and found one of the torturer's knives. With another roar, he drove it through the pale goblin's shoulder and into the wood beneath. The impact jolted his hand. The goblin cried out against the grip that strangled him, his pain as wordless as Geth's rage. Another knife went into his other shoulder, sharp blade grating on bone, and Geth released his hold on the goblin's throat, seized his hand, and wrenched his arm out against the board. The goblin struggled and flailed, but Geth ignored his kicks and flapping arm. He grabbed a third knife. The goblin's fingers clenched convulsively. Geth punched the blade through them and into the board, and jabbed a fist into the torturer's belly before forcing his other hand out and impaling it too.\n\nPinned on the rack, the goblin squirmed and flopped. His heels and the back of his head beat against the wood. His screaming mouth stretched so wide Geth could see the ragged root of his tongue.\n\nRage threatened to give way to disgust, but memories of agony twisted in him. Vengeance for himself\u2014for Tenquis\u2014rose to choke him. The torturer became Tariic.\n\nHot fury turned cold as death. One of the irons the goblin had used to burn him lay on the floor, smoking in Tenquis's blood. The metal was still hot enough to sting Geth's hand when he picked it up.\n\nHe didn't think the struggling goblin even saw him as he brought the heavy end of the iron down on his head. The screaming stopped after the first blow, but Geth beat the iron against the torturer's skull until bone cracked and sagged like a half-empty wineskin. Then he turned away, hurled the bloody iron across the chamber, and raised his voice.\n\n\"Ekhaas!\"\n\nAshi caught the handle of the door at the first heavy impact of body against wood and would have thrown it open if Midian hadn't caught her arm.\n\n\"Don't,\" he said. His face was pale.\n\n\"Get off me!\" Ashi spat at him, but Ekhaas seized her shoulders and dragged her back.\n\n\"No,\" she said. Her ears pressed against her head. \"He's right. Let Geth do what he needs to do.\"\n\nAshi still strained toward the door. \"Geth! _Geth!\"_\n\nThe thin wail continued, punctuated by short thumps and Geth's terrible snarls, only to end abruptly with the wet, pulpy crack of breaking bone. There was a clatter of metal. Geth's snarl faded into a guttural groan\u2014then rose in a call. \"Ekhaas!\"\n\nThe hobgoblin raced to the door and pushed it open. She froze in the doorway for a moment before she stepped through and swiftly closed it after herself. This time the only thing Ashi glimpsed was Geth standing at Tenquis's side.\n\nShe heard him, though. His words were hoarse. \"What can you do for him?\"\n\nEkhaas didn't answer immediately, but then she said, \"I'll need a knife.\"\n\nAnd she started to sing. The song was soft and soothing, with echoes of energy to it. Ashi could hear Geth's voice through it, murmuring something that might be comfort and encouragement\u2014until another voice, Tenquis's, leaped high in a wail before trailing back into a series of sobs. Ekhaas kept singing.\n\nMidian gave Ashi a nudge and put a flask into her hand. \"Drink?\" he asked quietly. Ashi nodded and lifted the flask, though she didn't drink from it. She could only stare at the closed door. Fabric tore in the room beyond and she could imagine a shirt or a cloak being shredded for bandages. Midian bumped her elbow, reminding her of the flask in her hand. She raised it again.\n\nA hand came past her and plucked it from her grasp.\n\nShe whirled around, drawing her sword as she turned, to find Aruget putting the flask to his lips. The disguised changeling paused, unflinching in spite of the sword at his belly. \"Don't let your guard down,\" he said, lowering the flask untasted. \"You're not out of danger yet.\"\n\nHeart racing, Ashi returned her sword to its sheath. \"I could have killed you.\"\n\n\"I could have killed _you.\"_ His ears flicked. \"You've already been rescued.\"\n\n\"Twice. Vounn told me you'd be coming but Midian, then Ekhaas, got here first.\"\n\n\"I had to stop for something.\" Aruget swung a sack from his shoulder and opened it so she could peer inside. Peeking out from among the muffling folds of a cloak were Geth's great gauntlet and the hilt of Wrath. \"I knew Geth wouldn't leave without them. Tariic had them displayed as trophies in his quarters. He'll miss them. We need to be gone before he does.\"\n\n\"You knew Geth was here too?\"\n\n\"I investigated.\" He looked at Midian and nodded in cool greeting. _\"Saa.\"_\n\nThe gnome's eyes narrowed. His lips pressed tight together. Ashi could guess what he was thinking. \"None of us would be here if we hadn't left Midian on the roof with Makka,\" she told Aruget. \"He made a deal to save his life. You owe him an apology.\"\n\n\"He didn't have to make a deal.\" His ears pressed back. \"He could have given up his life to save three.\"\n\n\"And you'd do that?\" Midian asked.\n\n_\"Mazo.\"_ Aruget's eyes stayed on Midian but he turned his face toward Ashi. \"I told you he'd be able to care of himself, didn't I?\"\n\nAshi ground her teeth. \"Apologize to\u2014\"\n\nThe door of the torture chamber opened. Ekhaas emerged first. Her red-brown face was drawn and her amber eyes haunted.\n\nGeth and Tenquis followed, the tiefling leaning so heavily on the shifter that Geth might as well have been carrying him. Blood spattered Geth, matting the patchy, half-burned hair of his bare chest. Tenquis shook with every step as if his legs might give out under him. His dark face was ash-pale and carried a sheen of sweat. His golden eyes were dull and seemed to stare off into some private nightmare. Breath came in shudders. His clothes\u2014shirt, leather pants, labyrinth-patterned vest\u2014hung awkwardly on his body, as if someone else had dressed him. They were mostly clean, though. No blood soaked through to betray an injury. His face was bruised and scraped as if a coarse gag had been bound into his mouth, but that was all. Geth seemed to have suffered more. For a moment, Ashi wondered what had been done to him or if Ekhaas's magic had somehow healed him of any wound.\n\nThen she realized that where the tiefling's long, sinuous tail should have been was only a thick, bandaged stump.\n\nGeth's mouth was set in a hard line. His gaze fell on Aruget. \"He knows everything,\" Ashi said quickly, but Geth didn't seem to hear her. His eyes settled on Midian. He shifted Tenquis's weight onto Ekhaas and went to the gnome, dropping to his knees in front of him. Midian had two vials of pale blue liquid ready in his hands\u2014the healing potions he had mentioned, Ashi guessed\u2014but Geth ignored them.\n\n\"I have to tell you something,\" he said hoarsely. \"And I'm sorry about it. On the night that Ekhaas, Dagii, and I went to see Tenquis, Chetiin found us.\"\n\nAshi stiffened. \"You've seen Chetiin?\" She looked to Ekhaas, but the _duur'kala_ was watching Geth and Midian, her ears flicking rapidly. Aruget's hand grasped her arm, urging her to silence. Geth's attention was entirely on Midian, whose eyes, in turn, were darting rapidly between each of them.\n\nGeth continued without even a pause, as if he was determined to speak his piece before they left the dungeon. \"He convinced us that you were actually the one behind Haruuc's assassination\"\u2014Ashi couldn't hold back a gasp, but Geth still didn't stop\u2014\"and that it was a _shaarat'khesh_ assassin you hired who pretended to be Chetiin. I thought I found evidence that confirmed it, but I was wrong.\" He half-turned his head to speak over his shoulder. \"Ekhaas, Chetiin lied to us. He was supposed to go with you and Dagii to fight the Valenar, but he stayed in Rhukaan Draal.\" The shifter stood. He turned to look at all of them, fury twisting his face. \"On the day of Tariic's coronation, when I rushed up to my chambers, I caught him in the middle of stealing the true Rod of Kings. He betrayed us again!\"\n\nMidian's eyes opened wide. Aruget stood frozen. Ashi's stomach felt like it had turned over inside her. \"So Chetiin has the Rod of Kings now?\"\n\nGeth bared his teeth and nodded. He might have added something, but Ekhaas spoke first. Her ears went back and she said, \"Geth, Chetiin _did_ fight with us.\"\n\n\"He stole the rod! I saw him!\"\n\n\"He was with us!\" Ekhaas insisted. \"He's with me now\u2014he's standing guard outside the dungeon!\"\n\n\"He's here,\" said the _shaarat'khesh_ elder's strained voice. All of them turned to follow it. Chetiin crouched at the foot of the stairs, a dagger in his hand, and his black eyes glittering. Ashi's stomach flipped again.\n\n_\"You_!\" Geth roared\u2014and charged at the goblin.\n\nChetiin slid aside, flattening himself against the wall. Geth sprawled past him, twisted around, and came back up. The terrible, savage growl was back. Chetiin kept his back to a wall, dagger ready. \"I did what you asked,\" he said tightly. \"I went with Ekhaas and Dagii. I kept them safe.\"\n\n\"Liar,\" Geth snarled. _\"Traitor!\"_\n\n\"He was with us,\" Ekhaas said again. \"He couldn't have stolen the rod!\"\n\nGeth paced forward, stalking Chetiin. The old goblin held his ground. \"Then maybe he did what he accused Midian of doing,\" said Geth. \"Maybe he hired another of the _shaarat'khesh_ to do the job while he went with you as a cover!\"\n\n\"Or maybe,\" said Aruget, \"Chetiin was right the first time he approached you.\"\n\nHis heavy accent had vanished again, but Ashi wasn't certain anyone else noticed it. All eyes went to Aruget, then followed his to Midian.\n\nThe gnome looked shocked, then his expression drew together in anger. \"That's impossible! I wouldn't do something like that\u2014and even if I did, where's this _shaarat'khesh_ I'm supposed to have hired?\"\n\n\"Dead,\" said Chetiin. \"Silenced so he couldn't give you away and the assassination would be laid solely on me.\"\n\n\"And I just hired another to steal the rod.\" Midian pointed at Chetiin. \"He's trying to turn you against me again! Would the _shaarat'khesh_ agree to work with me a second time if I'd already turned on one of them?\"\n\n\"There was no _shaarat'khesh,\"_ Aruget said calmly\n\n#\n\n# CHAPTER  \nTWENTY-SEVEN\n\n# 3 Aryth\n\nPain and fury seethed in Geth. Pain from the burns inflicted on him. Fury at what had been done to Tenquis. Fury and pain both for Chetiin's treachery, as fresh and hot as if the rod had just been stolen or if Haruuc were newly struck down. The goblin torturer was dead, though, and with every breath, Geth promised himself that soon Chetiin would be, too.\n\nBut Aruget's words broke through the rush of blood that roared in his ears. \"There was no _shaarat'khesh.\"_\n\nIf there was no _shaarat'khesh_ involved in Midian's scheme, that meant...\n\nHe looked at Midian again.\n\nOn the night they had first gone to see Tenquis, Ekhaas had used magic to disguise Geth as a hobgoblin woman. Midian's pack and pouches were filled with strange and useful magical devices\u2014one of them might easily have been capable of disguising a gnome as a goblin. The two races were about the same size and Midian was an excellent mimic. He fought well too. Surprisingly well for a scholar, even an adventurous one.\n\nGeth thought back to the day he had investigated Chetiin's room and confirmed that there was a ledge above the fireplace, just as the goblin had said. He'd encountered Midian in the hall afterward. He'd thought then that Midian hadn't noticed the soot smudge on his face, but what if he had? The gnome could have investigated Chetiin's room and discovered that the supposedly dead _shaarat'khesh_ elder was no longer where he'd been left. Midian would have realized that at least part of his plot had been uncovered. He would have had to act. But no...\n\n\"It's not possible,\" he said. \"Midian wasn't in Rhukaan Draal when Haruuc was assassinated.\"\n\n\"Whose word do you have for that?\" asked Aruget.\n\n\"The coin he brought back to Ekhaas from Bloodrun\u2014\"\n\n\"\u2014could have come from anywhere,\" Ekhaas said, her voice low. \"The messenger we sent to Bloodrun to fetch him supposedly died of dust fever, didn't he? What if he didn't?\"\n\nThe gnome made no reaction to the accusations. His face was expressionless.\n\nAshi spoke up. \"I saw you at the beginning of the coronation ceremony,\" she said to Midian, \"but not later. Afterward when I talked to you about what happened there, you said you didn't see anything because your view was blocked. But you didn't stay, did you? You made sure I saw you, then you left to disguise yourself again and steal the rod.\"\n\nMidian's silence was hard to ignore. He damned himself with it. He kept very still, back against a wall. There was a tension in him Geth hadn't seen before, like a blade ground so keen the touch of a whetstone would break it.\n\n\"Why?\" Geth asked him.\n\nA cold smile split Midian's face\u2014and he sprang into sudden motion. One hand hurled the potion vials at Chetiin and Geth. The other whipped a knife, the blade stained black, from his belt as he leaped at Aruget.\n\nA bad feeling about those vials gripped Geth. He stuck out his hands and dived for them, snatching them out of the air as gently as he could. Aruget swung a bulky sack up at Midian, but the gnome ducked around it with astonishing speed. His knife slashed, forcing Aruget back, then he was past him and running for the stairs. Ashi ran after him. Ekhaas shifted Tenquis so he leaned against the wall, and drew breath, ready to sing a spell at the fleeing gnome.\n\nIt was Chetiin who caught him, though. The goblin's movement was as fast as a thrown knife. He bounded past Geth and actually seemed to run along the wall itself for several steps before launching himself at Midian. His arms wrapped around the gnome's legs and brought him down. Midian's knife flashed, but Chetiin twisted and the blade struck stone. Then Aruget was at his side, tearing the knife from Midian's grasp and pinning both wrists with one large hand. Chetiin rolled free, grabbed the fallen knife, and held the stained blade to the gnome's throat.\n\nMidian went still, as if he knew further struggles were useless. His eyes were like chips of glass. He glared up at Aruget. \"Dark Lantern?\" he asked.\n\nAruget's ears flicked as he hauled the gnome upright. He nodded to Geth. \"Good catch. I doubt you would have wanted to drink\u2014or breathe\u2014those potions.\" The hobgoblin looked to Ashi. \"Or Midian's wine. I don't think you were intended to survive your rescue.\"\n\n\"He called you a Dark Lantern,\" said Chetiin. The captured knife didn't waver, but Geth saw the goblin steal a glance at Aruget. \"You're an agent of Breland?\"\n\n\"You can trust him, Chetiin,\" said Ashi. \"He's a friend.\" She caught Geth's eye, then Ekhaas's, and added, \"An old friend.\"\n\nAn old friend? Geth thought of the only other Brelish agent he'd ever known\u2014the half-elf, Benti Morren. He glanced at Ashi and silently mouthed Benti's name. Ashi nodded. Geth turned back to Aruget in confusion but the hobgoblin only held up a hand.\n\n\"Not now,\" he said.\n\nEkhaas wasn't so easily put off. \"What's a Brelish Dark Lantern doing in Darguun?\"\n\n\"Getting more involved than I should be for someone whose orders were only to watch,\" Aruget said. He gave Midian a gentle shake. \"A better question would be what's an agent of Zilargo doing in Darguun? We found out that Midian had attached himself to Tariic. We already had suspicions that Haruuc was up to something. I was sent to keep an eye on both of them, but I don't think anyone ever thought it would go this far.\"\n\nGeth looked to Midian. \"You work for Zilargo?\"\n\nSomehow Midian managed to look proud. \"I work for the Library of Korranberg,\" he said. His eyes went to Ekhaas. \"That coin was from Bloodrun. You go there and look. You'll find Koolt Dynasty ruins\u2014\"\n\nHis words ended at the touch of the poisoned knife to his throat. \"I don't care who you work for,\" said Chetiin. \"I want to know about Haruuc. I don't particularly care that you tried to kill me in the process. I want to know why you killed him.\"\n\n\"I think I know,\" Ashi said. She moved forward, her eyes wide and thoughtful. \"When Esmyssa ir'Korran told me that Zils negotiate instead of fighting wars, she said that Zilargo prefers to deal with stable rulers. Haruuc wanted the Rod of Kings to keep Darguun stable. If Midian helped him find it, Zilargo would have the stable neighbor it wanted _and_ an agent with Haruuc's trust.\" She raised a finger. \"But when Haruuc became warlike under the influence of the rod, it didn't suit Zilargo's plans, so Haruuc had to be removed. We thought he was killed to keep him from discovering the power of the rod or to prevent a war, but that was only part of it. He was also killed so that a lhesh more sympathetic and less warlike would come to the throne.\" A second finger joined the first. \"Tariic also trusted Midian. And he'd said that he wanted to bring Darguun into greater contact with the rest of Khorvaire.\"\n\n\"Tariic wasn't Haruuc's heir,\" said Chetiin. \"Any warlord could have taken the throne.\"\n\n\"But Tariic was in a better position than anyone else.\"\n\nGeth felt a chill in his belly. \"Zilargo killed Haruuc? _Gnomes_ were trying to influence the rulers of Darguun?\" He looked Midian\u2014who smiled\u2014then at Aruget. \"I thought Zilargo was supposed to be Breland's ally!\"\n\n\"In this game, there are no allies and no traitors,\" said Aruget, \"only opportunities. Haruuc started a small war to avoid a large one; Zilargo killed a king to keep their peace. I doubt if Breland would object to a more pliable lhesh either. If you ask in Zilargo, though, I don't think you'll find anyone who would admit to anything more than regret at Haruuc's assassination.\" He gave Midian another shake. \"What do you have to say to this?\"\n\nThe gnome looked at Ashi. \"I'd say you have more imagination than I thought you did.\"\n\nAshi's lips pressed tight, and for a moment it seemed that she might lose her temper. She kept her emotions in check, however, and Geth could see Vounn's influence in that. She raised a third finger. \"After the assassination, Midian slipped out of Rhukaan Draal. Whether he actually went to Bloodrun doesn't matter. He must have anticipated we'd send a messenger for him, though, and he dealt with him. When the city reopened after the mourning period, he returned\u2014and discovered that by killing Haruuc, he'd only made things worse. For a little while, we were all working toward the same goal of finding a way to keep Tariic from succumbing to the curse of the rod.\" She cocked her head to one side. \"It's even possible he came up with the idea of a false rod before he found out about the curse. We just gave him the perfect chance to propose it. He may have intended to steal the true rod all along.\"\n\nA growl escaped Geth. \"And it might have worked if Tariic hadn't recognized the false rod!\" Another idea occurred to him. \"Wait\u2014why dress as Chetiin again to steal the rod? He couldn't have known that I'd come rushing up from the coronation to get it.\"\n\n\"He would have made sure someone else saw him,\" said Chetiin. \"Saw me. He had the perfect scapegoat.\"\n\n\"I think he already knew you were still alive,\" Geth said. \"I may have given it away to him.\"\n\n\"Even better. He would have turned you against me again.\"\n\nGeth felt blood rush into his face and the hate he'd felt since Haruuc's death turned into a sick feeling of shame in the pit of his stomach. \"He did,\" he said. \"Chetiin, you're the one I owe an apology to. You're no traitor.\"\n\n\"No, I'm not,\" the goblin said, \"but I hold no anger for what you thought. You were manipulated. We all were.\" He nodded to Midian. _\"Paatcha_. Your plan was cunning.\"\n\nMidian's eyes danced although there was no warmth in them. \"What plan?\"\n\nThe gnome's smug satisfaction burned in Geth. He pushed Chetiin aside, bending down to snarl in Midian's face. \"You think you're clever, don't you? You think we can't beat you. Well, you're wrong.\" He thrust a hand at Tenquis, still leaning against the wall and following everything with weary golden eyes, though color at least had returned to his face. \"We know where you hid the Rod of Kings. Tenquis realized that the connection between Wrath and the rod was still active. We followed it.\" He stepped back and looked at the others. \"He hid the rod in Haruuc's tomb.\"\n\nSurprise passed across all their faces, but none of it was as sweet as the look of shock and anger that finally broke through Midian's mask.\n\nGeth bared his teeth at him. \"Yes,\" he said, \"it was clever, but now you're not the only one who knows. Now we all do.\"\n\n\"So does Tariic.\"\n\nTenquis's words\u2014the first he had spoken\u2014were unsteady, but angry and determined. Geth turned sharply to look at him. They all turned. The tiefling shoved himself away from the wall and stood straight, head held high. \"I'm sorry, Geth,\" he said. \"I couldn't hold back. Horns of Ohr Kaluun, I tried, but I couldn't in the end.\" He drew a shuddering breath. \"I gave Tariic what he wanted.\"\n\nThe sick feeling returned to Geth's stomach. \"He knows?\"\n\nTenquis nodded.\n\n\"Clever,\" said Midian.\n\nThey walked out through the front gates of Khaar Mbar'ost.\n\nIt was simple. It was direct. The great courtyard within the gates and the plaza beyond it were both crowded with Darguuls prepared for celebration of Dagii's victorious return. No one paid any attention to them as they passed through, moving in small groups to avoid notice. Ekhaas and Ashi went first, Ashi's face shadowed by a wide-brimmed hat that Aruget produced from his sack. Chetiin and Tenquis followed, the tiefling hunched beneath a cloak, both of them tottering like frail elders. Geth knew it wasn't much of an act on Tenquis's part at the moment. Every few steps, he staggered a little before catching himself. Geth had tried to help him at first, but Tenquis had shaken him off in a flash of temper.\n\n\"I just need to find my balance,\" he'd said through his teeth. \"I'm used to having a tail behind me.\"\n\nThere was deep pain behind the angry words.\n\nWhen the other four were safely out of Khaar Mbar'ost, Aruget and Geth struck out. Geth kept his head down and stayed close to Aruget, trusting to the hobgoblin's bulk to shield him. A cloak draped over his shoulder concealed both his great gauntlet and, held tight in his hand, Wrath. The sword seemed to echo his own simmering rage\u2014and his fear. He could feel the Rod of Kings through the blade, a distant but distinct presence. The rod wasn't moving. Tariic didn't have it. Yet.\n\nHis eyes darted to the sack Aruget carried over his shoulder. In spite of having cloaks and hat, gauntlet and sword pulled out of it, the sack was once again full and heavy.\n\nIt shifted a little bit.\n\nAruget bounced it on his shoulder to disguise the movement and hissed, \"Stop squirming.\"\n\n\"I couldn't feel my leg,\" muttered a voice from inside the sack.\n\n\"I could make that permanent, Midian,\" Geth said under his breath. \"Move again, and you'll be dead before you hit the ground.\"\n\nHe'd been in favor of killing the gnome right there in the corridor outside the torture chamber, but others\u2014primarily Midian himself, backed up by Ekhaas, Chetiin, and eventually Aruget\u2014had persuaded him otherwise. Whatever Midian might have done, they all shared an interest in keeping the Rod of Kings out of Tariic's hands. Only the gnome knew exactly where the rod was hidden in Haruuc's tomb. If they were too slow in reaching the tomb, every moment might be precious. They needed Midian.\n\nBut not one of them trusted him. He went into Aruget's sack. Geth couldn't say that he cared if the gnome ever came out again.\n\nAnd there was Aruget. He glanced at the hobgoblin\u2014or rather, at the changeling wearing a hobgoblin's face. Ashi had pulled him, Ekhaas, and Chetiin aside and whispered the truth to them.\n\nAgents of Zilargo and Breland, shadows of two nations lurking in the twilight of Darguun.\n\n\"It's more common than you know,\" Aruget said abruptly. It was vaguely disconcerting to hear him speak without his familiar accent. \"Everyone has their fingers in the jam jar.\"\n\nGeth blinked, then narrowed his eyes. \"You knew what I was thinking?\"\n\nAruget's ears twitched. \"I'm good at reading faces, and you had the look of someone who just got his first peek behind the curtain at a brothel.\" He looked down at Geth. \"We won't be the only agents in Darguun. Every nation, every dragonmarked house has its eyes and its hands here. I'm certain Haruuc knew it. I imagine it pleased him. Being a part of the Shadow War is like a rite of passage\u2014children don't play these games.\"\n\n\"Children don't assassinate kings,\" Geth said. \"When did you know it was Midian and not Chetiin?\"\n\nAruget shrugged, jostling the sack and bringing another soft grunt from Midian. \"I suspected Midian was up to something when he made arrangements to leave Rhukaan Draal. I didn't expect such a direct action, though. Assassination is a last resort. Killing the ruler of a nation is unheard of.\" His voice was cool, unflinching.\n\n\"You couldn't have warned us about him?\" Geth asked.\n\n\"My orders were to watch. I work for Breland, not Darguun.\"\n\nGeth clenched his teeth. \"Why are you helping us now, then?\"\n\n\"Once I had the final piece, the puzzle turned out to be bigger than I thought.\"\n\nThe sack chuckled quietly. \"We're cold-hearted bastards, Geth,\" said Midian.\n\nAruget turned slightly so that the sack swung against the side of a wagon. Midian let out a muffled curse that Aruget covered with a well-timed cough\u2014a cough that turned into a soft curse of his own. He twisted, giving Geth a little more cover, at the same time moving a hand across his face, as if scratching his nose. When he lowered his hand, his features had shifted just enough that he could have been any hobgoblin. \"Keep moving but get your head down,\" he said. \"Tariic's here!\"\n\nGeth heard the hoofbeats of trotting horses over the noise of the milling crowd in the plaza. He dropped his eyes, feigning great interest in his feet. His hand curled tight around Wrath. When Aruget paused, along with everyone around them, to turn and shout praise to the passing lhesh, he paused, too. He couldn't bring himself to call Tariic's name, though. The hoofbeats didn't slow. When they'd passed, he risked a glance up.\n\nTariic rode on into Rhukaan Draal without even acknowledging the cheers of his subjects. Two hobgoblin guards followed him.\n\n\"Now we're in trouble,\" said Aruget. \"If he goes to his quarters, he'll notice that Wrath and your gauntlet are missing.\" He quickened his pace.\n\nGeth risked another glance back at Tariic. Haruuc's nephew didn't look happy. \"He doesn't have the rod yet.\"\n\n\"It won't be for want of trying. If he's here, he'll have left someone at the tomb. Daavn, probably. Maybe Makka too.\"\n\nGeth allowed himself a grim smile. \"Good.\"\n\nTwo hobgoblin soldiers guarded the great red stone arch, probably intended more to raise an alarm if anyone approached than to provide actual protection. From his hiding place among the shacks that were the fading edge of Rhukaan Draal, Geth didn't even see Chetiin slip past the guards, but somehow the _shaarat'khesh_ elder was abruptly behind them. One guard went down with a knife in his back. The other turned, only to meet a second knife as Chetiin dropped down from above.\n\n\"He's got them both,\" Geth growled. \"Move.\"\n\nThey raced for the cover of the arch. Ekhaas and Ashi took the lead, Geth kept pace with Tenquis, and Aruget stayed close to Midian\u2014now freed from the sack. They all clustered in the shadows of the arch and peered through the open gates to the ridge where Haruuc's tomb lay.\n\nHalf a dozen bugbears wearing the red corded armbands of Khaar Mbar'ost attacked the dagger-thick stone of the tomb door with picks, hammers, and bars. The banging of their efforts was louder than the rush of the cataract, but so far all they'd managed to do was scar the carving of Haruuc. Daavn and three more hobgoblins watched their slow progress along with Makka and, perched on his shoulder, Pradoor. Horses picketed to a low line cropped the grass close to them.\n\nAshi scowled. \"Why so many?\" she asked quietly. \"Tariic could have hired a wizard to get into the tomb with magic, couldn't he?\"\n\n\"One more person who would learn about the rod,\" said Aruget. \"Tariic doesn't have to explain himself to servants. And I doubt if anyone except Daavn, Makka, and Pradoor is going to survive long after that door is opened.\"\n\nAshi's scowl deepened.\n\n\"There are too many for me to hold with a song,\" Ekhaas said.\n\n\"And Pradoor might resist it,\" Geth added. His skin crawled a little at the memory of the blind goblin woman chanting her spell in the torture chamber.\n\n\"We don't have to go through them,\" said Midian. \"I already have a way in, remember? I always planned on getting the rod out again the same way. I could slip in and get it without anybody knowing.\"\n\nGeth and all of the others only glared at him. The gnome shrugged. \"You don't think I'd bring it back to you? Send Chetiin with me.\"\n\nChetiin's jaw twitched. \"I don't value my life so little.\" A wounded expression crossed Midian's face and he hunched back in a sulk.\n\nA little too far back for Geth's liking. \"Stay where I can see you.\"\n\nWounded turned to frustrated. \"Do you want to hold my hand?\" He stuck out his arm.\n\n\"That's not a bad idea. Does anyone have a piece of rope?\"\n\nEkhaas did. Geth tied one end around Midian's wrist and the other around his own. Midian gave him a sour look, but submitted before slouching back.\n\n\"I have a plan.\" said Chetiin. He sketched lines in the dust to mark the course of the ridge and touched points along them. \"A few of us show ourselves among the rocks of the ridge here and here. Daavn will try to protect the rod. He'll send his men to hunt us down. Makka will certainly go as well. Once they're away, the rest of us get up to the tomb.\"\n\n\"Who takes the ridge?\" asked Geth.\n\nChetiin considered. \"Ashi, Ekhaas, and Aruget. If they stay close to the wall, they should be able to get into the shelter of the ridge without attracting attention.\"\n\nAshi peered out at the rough landscape. \"I can do it.\" Ekhaas and Aruget gave nods of agreement.\n\nGeth turned\u2014the movement bringing a tug and a grumble of complaint from Midian\u2014to Tenquis. \"You'll be able to get the door open quickly and quietly?\"\n\nThe bandaged stub of the tiefling's tail waggled and his golden eyes narrowed with hate. \"I'd rather have a long knifepoint conversation with Daavn,\" he said, \"but like Grandmother says, there's more than one way to sour milk.\" He gripped the collar of his vest and whispered a word. The embroidery of the vest writhed and the slight bulges of pockets reappeared. \"I have everything I need.\"\n\n\"Then we're ready.\" He looked to the others. \"Rat and Tiger dance for us all.\" He stretched out his hands to Ashi and Ekhaas\u2014and felt his left brought up short by the rope tied to Midian's wrist.\n\nThere was no grumble of complaint and Midian's arm didn't yield. Geth turned.\n\nMidian squatted behind him, unmoving and apparently intent on something in his hand. The rope that should have been tied to his wrist seemed to pass completely through it. Ekhaas hissed and swept a foot across the ground where the gnome crouched.\n\nAnd a glittering crystal disk no bigger than a coin bounced off the stone of the arch. Midian vanished. The rope was tied to a bar of the iron gate.\n\n\"Tiger, Wolf, and Rat, I thought we searched him!\" Geth snarled. \"Where is he?\"\n\nAshi pointed. \"There!\"\n\nThe sun shone off a shock of pale hair, just visible above the long grass on the near side of the grazing horses. Midian moved with such stealth that the grass around him barely swayed. It was easy to guess where the Zil was going: his own secret way into the tomb. Maybe he was going to fetch the rod for them just as he'd said, but Geth doubted it.\n\n\"I could stop him,\" said Chetiin.\n\n\"No.\" Geth drew Wrath. \"We're in a race now. Ekhaas, Ashi, Aruget\u2014go!\"\n\n\"Wait. Let's put honey in the trap.\" Aruget grabbed Geth's chin and turned his face toward him. He studied the shifter, then let him go. Hobgoblin features melted and reformed. Ears shrank and hair grew out. Small eyes become large and wide. In only moments, Geth stared at himself wearing Aruget's armor. Aruget-Geth smiled and said in eerie mimicry of his growl, \"Daavn will be more likely to chase a face he knows. Ko isn't the only one who can imitate you.\"\n\nHe jerked his head and he, Ashi, and Ekhaas stole out of the gate, moving along the low wall in the opposite direction to the way Midian had gone. They ran low and fast, heading for the nearest fold in the ridge. Geth realized he was holding his breath.\n\n\"Midian has seen them,\" Chetiin said. Geth swung around to look for the gnome\u2014just in time to see his arm swing as he hurled something back at the horses, then dropped into the grass.\n\n\"Down!\" Chetiin said.\n\nThey pressed themselves against the ground. Geth kept his head up just enough to see the hurled object\u2014an ordinary stone\u2014hit one of the horses on the flank. The startled beast reared just a little and danced forward a few steps. The other horses reacted to it, raising their heads and looking around.\n\nThe movement was enough to draw the eye of one of the hobgoblin guards. He swung around.\n\nEkhaas, Ashi, and Aruget-Geth were caught in the open. _\"Toh!\"_ cried the guard. Daavn, Makka, the hobgoblins and bugbears all swung to look as well.\n\nTheir friends froze for an instant, then dashed for the cover of the ridge. Geth saw Daavn's eyes go wide\u2014and Makka's narrow with bloodlust. He grabbed Wrath so he could understand the words that were shouted.\n\n\"Get them!\" Daavn ordered. \"Kill them!\"\n\nThe guards were in motion as soon as the command was ordered.\n\n_\"They're mine!\"_ roared Makka. He charged, swinging his fists and bashing hobgoblin guards aside, with Pradoor clinging to his shoulder and cackling like a mad thing. The other bugbears, picks and hammers still in their hands, stood still, obviously uncertain whom to obey. Daavn drew his sword and screamed at them to follow as he took off after Makka. The bugbears roared as Makka had and leaped to the pursuit. They didn't bother descending the steep stairs cut up to the tomb but jumped from rock to rock across the face of the ridge.\n\nThe distraction had worked, although not just for them. Midian was up and running, stealth abandoned for speed. He looked to be heading for a particularly rough section of the ridge about two long arrowflights away from the gate. Geth waited a heartbeat longer, until Daavn and his men were well away from the tomb, then pushed himself to his feet. \"Go!\" he said. \"Tenquis, can you run?\"\n\n\"You should have asked that before,\" the tiefling snapped, but he was up already up and running nearly as quickly as Geth himself, if a little more unsteadily. Geth stayed close to him. Chetiin, faster than either of them, paused at the bottom of the stairs, then darted up ahead.\n\nTenquis was grimacing in pain by the time he reached the top, but he lurched over to the tomb door and ran his hands over the scarred surface. Geth felt a fresh burst of anger for Daavn and Tariic. The fine carving of Haruuc was nearly destroyed. Only his fierce, watchful face remained. Tenquis caught his look. \"It can be repaired,\" he said. \"A good artificer or even a magewright with a little time can fix anything.\"\n\nThe bugbears had taken their tools with them, but Tenquis reached into one of the pockets of his vest and pulled out the heavy steel pry bar Geth had watched him slide into it. He threw it to Geth. \"When I tell you, work that in about there\"\u2014he pointed to the seam between the door and the frame where a bugbear's pick had already broken a hole\u2014\"and get ready to heave.\"\n\n\"That's not going to work. I told you, the pivots are broken.\"\n\n\"And I told you an artificer can fix anything.\" He pulled more objects out of his pockets: a couple of tiny flasks, a stick of bright red chalk, and several roughly polished stones. The flasks and the stones he set to the side. Taking the chalk in one hand, he spread the fingers of the other wide and touched them lightly to the door on the side where the pivots had been. His face took on a distant expression and, after a moment, he started to trace out stange lines on the door with the chalk.\n\nGeth looked to Chetiin. The goblin crouched on the edge of the space before the tomb, watching the ridge where Ekhaas and the others\u2014and their pursuers\u2014had disappeared among the age-carved rocks. Shouts and cries, the scrape and clatter of metal against rock came back over the ridge. Their friends were doing their job, keeping Daavn and his men busy. Geth still felt fear for them in his gut.\n\n\"Have you seen them?\" he asked.\n\n\"Glimpses,\" said Chetiin just as Ekhaas's voice rose in a song that ended in a crash and a short, swiftly silenced scream. Geth's hands tightened on the shaft of the pry bar.\n\n\"Geth, I'm ready,\" Tenquis said.\n\nBoth Geth and Chetiin turned around. Tenquis was dusting shimmering powders from the tiny flasks over the chalk-marked door and onto his hands. He nodded at the broken spot he had pointed out before and Geth quickly set the end of the pry bar into it. When he was ready, he nodded back to Tenquis. The tiefling took a deep breath. \"As soon as you feel the door shift, work the bar with everything you've got,\" he said. \"We only have one chance at this.\"\n\nHe picked up two of the stones, a near match in color and grain for the stone of the door, and held them against his palms with two fingers of each hand. He spread his other fingers and, stretching his arms, pressed them against the door at the top and bottom of his chalked lines. His eyes closed and his face tightened in concentration. His lips moved in a rapid, nearly inaudible whisper.\n\nThere was a clash of blades from the ridge and another sharp scream. Chetiin turned to look. Geth kept his eyes on Tenquis and his grip steady on the pry bar. He could feel sweat forming on his palms.\n\nTenquis's teeth clenched. His whispers slid between them.\n\nStone creaked.\n\nThrough the steel bar, Geth felt a distinct vibration as the door shivered and rose by the tiniest fraction. He threw his weight against the bar, hauling at it. For an instant, steel grated against stone, then the tip of the bar caught again and held. Geth strained, his muscles cracking and popping.\n\nThe door moved.\n\nGeth groaned at the weight of it. He ground his teeth together until they hurt and heaved harder on the bar. The thickness of stone that stood out from the frame grew slowly. A finger's width. A finger's length. Two fingers' lengths. A dagger's length.\n\nDarkness appeared. Chetiin seized a loose rock and shoved it into the gap. Geth drew back the pry bar and thrust it into the darkness before the stone could crack. The heavy steel squealed as it took the weight of the door. Geth drew a breath and shifted, letting the ancient heritage of his blood give new energy to his muscles before he stepped around, worked his fingers into the thin gap and pulled. Tenquis moved with him, hands resting steady on the stone, whispers rising.\n\nThe gap grew. Geth could have slipped through sideways. \"Enough!\" he gasped at Tenquis.\n\n\"All the way or it will swing closed on us,\" said the artificer, and even those few words interrupting his whispers brought new creaking from the unseen pivots. Geth groaned again and kept pulling. Step by step, back until the mouth of the tomb gaped wide. He waited for a shout from Daavn or one of his men as they caught a glimpse of what was happening and realized they'd been tricked. None came.\n\n\"Almost there!\" said Tenquis\u2014and pulled his hands away from the door, getting out from behind it. There was a crunch and a grinding sound as the magic that had held the shattered pivots together faltered. For an instant, Geth felt the unbearable weight of the door against his arms. He pulled with all his strength, trying to hold in the straining cry that threatened to escape him.\n\nThe door shifted one last time, then ground to a stop, striking the side of the tomb with a gentle tap. Geth's arms and shoulders felt heavy and numb. His legs trembled, but he limped around to the front of the tomb and the doorway.\n\nChetiin was already standing in the shadows, poised at the top of a dark staircase. Tenquis pulled a stone that glimmered pale as moonlight from his pocket. Geth drew Wrath. He could feel the presence of the Rod of Kings pulsing in the sword.\n\nThey stepped down into Haruuc's tomb together, moving away from sun and into shadow. The cold stench of cave damp and slow decay rose to meet them. Tenquis's moonstone\u2014shedding just enough light for shifter or _dar_ or tiefling eyes to see\u2014revealed walls that changed from worked stone to rough, natural rock as they descended. The stairs became rougher, too, hacked out of the floor of a steep passage wide enough for two broad-shouldered men to walk side by side. The words that the hobgoblin priests had spoken at Haruuc's funeral came back to Geth.\n\n_Traditions tell that the People were born in caverns and lived there before we emerged to fight beneath the sun and the sky. When we pass through the gates of death, we return to caverns, the womb and the grave_.\n\nThe steep passage grew taller. Glints of light shone ahead, reflections of the moonstone, and they emerged into a cave perhaps twice as big as Geth's quarters in Khaar Mbar'ost and far taller, heaped with gold and treasures.\n\nLhesh Haruuc Shaarat'kor sat on his throne in the midst of this tribute, both eye sockets empty now as they stared at the stone sky.\n\nGeth stopped at the bottom of the passage and looked on the remains of the father of Darguun, haste brushed aside by a curious sense of awe. He'd watched the corpse being carried down into the tomb, had walked with it through Rhukaan Draal. Haruuc was as dead now as he'd been then, yet somehow there was a particular solemn majesty about him. It wasn't so much the wealth that surrounded him as it was the unnatural stillness of something dead, alone in the unchanging solitude of one of Eberron's small secret places. Geth felt like an intruder. He lowered Wrath and bent his head in a nod of respect.\n\nTenquis must have felt it too. He bowed low, a flourishing gesture that was distinctly tiefling. Chetiin, however, didn't move at all for a long, long moment and it took all that time for Geth to realize that this would be the first chance he'd had to see Haruuc up close since Midian had attacked him.\n\nWhen the goblin finally moved, he walked directly up to Haruuc's seated corpse, knelt down, and opened a small chest that rested by Haruuc's feet. From inside it, he took the ugly, crystal-set dagger named Witness\u2014the dagger that had been stolen from him, the dagger that had killed Haruuc. He pressed the flat of the blade to his heart as he looked up at the lhesh and Geth heard him murmur, \"You will be avenged, my friend.\" He slid the dagger into an empty sheath on his right forearm, then turned back to Geth and Tenquis. \"Find the rod,\" he said.\n\nGeth raised Wrath again and swung it around the cave. Awareness of the Rod of Kings prickled across his senses. To the left and across the cavern\u2014the sword pointed directly at a pile of rolled carpets. Geth had to admit that it was a good hiding place. If by chance the tomb was pilfered while the rod was within, the fine but bulky carpets would almost certainly be ignored in favor of gold and gems. \"There!\" said Geth. He stepped toward the carpets.\n\nSomething flickered in the very corner of his vision, high up among the shadows of one wall of the cavern. Something pale, quickly obscured by the movement of something dark that gave a soft _snap_.\n\nHe threw himself back with a curse at the same moment a crossbow bolt hissed through the space where he'd been. It sank deep into the wood of a treasure chest. Shadows leaped across the cavern wall as Tenquis raised the moonstone. Its pale light revealed Midian, perched in the mouth of narrow crack, already sighting along the stock of a small hand crossbow once more. The gnome gave a crooked grin and winked at Geth before he squeezed the trigger.\n\n#\n\n# CHAPTER  \nTWENTY-EIGHT\n\n# 3 Aryth\n\nThe rocks of the ridge made a challenging hunting ground. Makka's initial charge carried him to the arm of the ridge behind which Ekhaas, Ashi, and Geth had sought shelter, but his quarry had already vanished into the rough gray folds. At the point where a shallow gully broke the slope he paused and tested the air with flaring nostrils. They hadn't lingered\u2014they were moving higher. He crouched. \"Get off, Pradoor,\" he said. \"Wait here.\"\n\n\"And do what?\" The old goblin woman clung tight to his shoulder. \"Admire the sounds of battle? My place is with you. The age turns.\"\n\nMakka bared his teeth for all the good that it did. Her blind eyes didn't see him. \"I can't fight with you on my back.\"\n\n\"The age turns around both of us.\" Pradoor's fingers dug into him. \"We serve\u2014\"\n\n\"\u2014the Six,\" Makka said through his teeth. \"But the Fury puts my revenge before me.\"\n\n\"Then why do you hesitate to follow her path?\" asked Pradoor. \"The Six reward those who serve.\"\n\nMakka snarled and rose to his feet just as rocks clattered along the arm of the ridge behind him. He whirled to find two of Daavn's bugbear workers clambering over the crest of the slope. The other bugbears, iron hammer and pry bar in hand, paused at the sight of him. Makka tensed his ears, thrust out his chest, and growled.\n\nThe other bugbears hunched back. Their ears flattened and they ducked their heads, offering submission in the ancient manner. When Makka turned again and continued on up the ridge, they fell into line behind him. On his shoulder, Pradoor chuckled.\n\nHe felt like the leader of a tribe again. But better. Stronger. The favor of the Fury was on him.\n\nThe age turned around him.\n\nSomewhere behind him, Daavn was shouting his name. He ignored it.\n\nThe peak of the ridge broke into a jumble of pits and crevices and more gullies, some shallow, some twice his height in depth, all of them offering hiding places. A scattering of stunted trees and bushes on the backside gave even more cover. The only figures visible were Daavn's other workers. They saw him and the pair of bugbears that followed him and, one by one, they offered their silent submission. Makka gestured to them all with crisp movements, ordering them to spread out among the broken places of the ridge. His gut told him that his enemies wouldn't have gone far\u2014they would stay near the tomb.\n\nDaavn and his hobgoblins came trotting up behind him with a clatter of armor. The warlord's ears were back and his sword was out. _\"Maabet_ , I knew we should have killed them in the dungeon.\"\n\nMakka swung around to glare at him. \"Their deaths are mine,\" he said. \"The Fury gives them to me.\"\n\nThe guards with Daavn had the sense to flinch back, but Daavn just leaned into Makka's anger. \"If I find them first, I will kill them, gift of the Fury or not. They're not going to stop Tariic from taking what's\u2014\"\n\nA weird, fluting battlecry and the sudden clash of weapons interrupted him. Makka knew the cry\u2014it belonged to Ashi of Deneith. Shouts in Goblin answered it as bugbears converged on the crevice where the dust of battle rose. Makka put his back to Daavn and raced with them, leaping across the tops of mounds and spires. One bugbear reached the crevice before the others and dropped down into it with a roar, but his cry turned to a gurgle. The swirl of dust faded even as Makka reached the crevice. Two bugbears lay dead within, one stabbed through the heart, one with his belly ripped open.\n\nTheir assailants were gone, vanished into the maze of broken rock. \"They're close!\" Makka shouted. \"Look for them from above. They can't hide from you.\"\n\nEven as the words left his mouth, one of the other bugbears yelped and fell hard on his face, his feet pulled out from under him by a loop of rope. Makka saw shock in the bugbear's eyes as he was dragged back. Big hands clutched at the rock, but as half his body disappeared down into a pit, he thrashed abruptly and went still.\n\nThe lithe form of a shifter, spattered with blood, vaulted up out of the pit. Geth flashed a grin at Makka, thumped his chest in salute, then dived into another crevice.\n\nMakka howled in rage. Daavn appeared, his guards in close formation behind him. Makka thrust a finger at the crevice. \"Geth's in there! You go after him and we'll take the top!\" He grabbed a bugbear with his other hand and dragged him forward.\n\nDaavn glanced into the crevice\u2014and jerked back as a stone whistled past his head. The warlord's face twisted in anger. He whirled his sword around his head in command, then he and his soldiers plunged after Geth. Makka raced to the edge of the crevice, shouting for the other bugbears to converge on him.\n\nOne of them didn't make it. A _duur'kala's_ keening song rose on the air. Stone cracked and the worker who had been the first to submit to Makka vanished as the rock wall on which he stood collapsed. His cry rose above the rumble of sliding rock, ending abruptly.\n\nAt the same time, a hobgoblin gasped in pain. A blur of hair and blood, Geth popped up out of the crevice and dodged into another. Makka howled again. \"It's like fighting spirits, Pradoor! They strike and run!\"\n\n\"Make them stand!\" the goblin said, slapping his head. \"The Fury favors you\"\u2014her voice rose\u2014\"as the Six favor all those who fight for Darguun!\"\n\nThe words were met with a roar from _dar_ throats, but they were more than just an inspiration. Makka felt the blessing wrap around him like the embrace of victory. Strength and confidence flowed into him. He ripped the bright sword of Deneith from its scabbard and turned, looking for a target. Any target.\n\nAcross the ridge, Ashi, a stolen hobgoblin sword poised to strike, rose up silently behind one of the two remaining bugbears. Makka pointed his sword\u2014her sword\u2014at her. \"Fight me, Ashi of Deneith!\" he bellowed. \"Fight me!\"\n\nShe jerked at the challenge, startled to be caught. Her intended target turned. Sword met steel bar with the ring of metal on metal. Weapons drew back for another exchange of blows.\n\nMakka charged along the ridge. _\"By the Fury, fight me!\"_\n\nHe felt the power of the Fury move through him, binding him to Ashi. She felt it, too. Her eyes widened, but she didn't let down her guard. The bugbear she fought swung his pry bar. Ashi turned it easily, then whipped her sword at his chest.\n\nThe bond between Makka and her tightened like a noose. Sudden pain wracked Ashi's face\u2014Makka felt an echo of it like the sweetest of stings. Her swing faltered and the killing blow became a flesh wound. Her opponent stumbled back.\n\nAshi tried to strike him again, but once more Makka felt the Fury's power pull tight. Ashi's face grew pale, the swirling lines of her dragonmark leaping out in sharp contrast.\n\nThen he was on her. Human sword in _dar_ hands clashed against _dar_ sword in human hands.\n\nThrough their bond, Makka felt Ashi's pain ease now that she submitted to the power of his challenge, but it had hurt her. He could see it in her face and feel it in her blows. She was slower than the last time they had fought\u2014even if he was now fighting with Pradoor perched on his shoulder like a cackling bird. The old goblin laughed with glee. \"For the Six!\" she cried. \"For the Six!\"\n\n\"Close your mouth!\" Ashi thrust at Pradoor.\n\nMakka beat her sword down and slashed up on the return blow. Ashi pulled back, but the tip of the bright blade sliced through her shirt and drew a thin red line across her belly. She gasped and circled away. Makka turned with her.\n\nJust in time to see Geth racing to her aid. The shifter moved low and fast like an animal, sword at the ready. Makka grinned. More sacrifices for the altar of his vengeance! He braced himself.\n\nInstead of attacking, Geth slid to a stop and took up a position a spear's length away from Ashi. Now they both threatened him\u2014attack one and he was vulnerable to the other. It was a cold, calculating strategy. The shifter's eyes were cool and hard.\n\nAnd wrong. When he'd faced Geth before, on the dais at Tariic's coronation, his eyes had been hot and alive with barely contained anger. The hobgoblins who had captured Geth had described his fiery, unflinching attacks. He hadn't held back.\n\nA strangely familiar sense intruded on Makka's lust for vengeance. A familiar sense, and a memory: Tariic's introduction of the changeling Ko, wearing Geth's face but without Geth's fear.\n\nAshi was Ashi, but whoever stood ready to attack him was no more Geth than Ko had been.\n\nPradoor still cackled on his shoulder, but the rhythm of battle was suddenly quiet in Makka's heart. He stepped back, turning slightly as he went so that his opponents were forced to move with him\u2014and so that the length of the ridge and the slope that faced Rhukaan Draal came into view.\n\nThe door of Haruuc's tomb stood open. They'd been tricked. Ashi, Ekhaas, and the false Geth were only a distraction.\n\nConflict roiled inside Makka. Ashi and Ekhaas were here on the ridge and vengeance too long denied called out to him. The Fury blessed his hunt. She'd taken him as her own. Vengeance was his sacred duty. But the Rod of Kings lay within the tomb, the key to Tariic's ambition, the key to new power for Darguun. The key to new strength for the faithful of the Fury and of all the Dark Six.\n\n_The age turns_.\n\nMakka roared and drove directly between Ashi and the false Geth with a suddenness that brought a screech from Pradoor. He slashed left and right, driving his opponents back with powerful blows, then he was past them and running with long leaping strides across the face of the ridge.\n\n\"What are you doing?\" Pradoor demanded, her words jolted by every step. \"The battle\u2014\"\n\n\"A trick to get us away from the tomb. The door is open\u2014they've come for the rod.\"\n\n\"You have to stop them!\"\n\n\"I will.\" Makka's teeth clashed together on a hard landing, but he stayed on his feet and slid down the final slope to the bottom of the stairs before the tomb. He grabbed Pradoor, dragged her from his shoulder, and planted her on the ground, turning her so that she faced the fight on the ridge. \"Find some way to guard us.\"\n\nA smile creased Pradoor's wrinkled face. She pushed his hand away, turned milky eyes to the sky, and started to chant.\n\nMakka bounded up the steps to the tomb three at a time. He was at the door of the tomb when he heard a dull drone sweep over the ridge, but he didn't stop to see what it was. Silent as grease, he slipped into the tomb.\n\nThere hadn't been time for a discussion of strategy as they fled up the ridge, only two quick rules from Aruget. Bring down as many of their enemies as possible\u2014and don't get caught in an open fight.\n\nEkhaas heard the power that rang in Makka's challenge to Ashi, and she knew immediately that Ashi had broken the second rule. Unfortunately, she couldn't _see_ what was happening. The song that had collapsed the rock wall under one of Daavn's bugbear workers, tumbling him into the gully below, had also cut off her easy way out. She turned and raced down the other way to a shadowed gap that marked the way into another of the ridge's broken passages. Had Tenquis's attempt to open the tomb worked? She could only hope that it had and that the others were already inside, searching for the rod.\n\nSwords clashed somewhere above. Ekhaas twisted around a corner\u2014and stumbled over the body of one of Daavn's guards. Her sword grated along rock as she caught herself.\n\nOn the slope leading up out of this gully, Daavn and his remaining guards stopped and turned back to stare at her. The warlord's ears went back. \"Go!\" he ordered the guards. \"Help Makka!\"\n\nThey raced past him and vanished from sight. Daavn stood ready for her, commanding the high ground, his sword held loose and easy. \"Sing, _duur'kala,\"_ he said. \"Something sad. Something for mourning.\"\n\nA knife glittered in his hand, held by the blade and ready for throwing.\n\nEkhaas didn't hesitate. A song would have brought an attack and Daavn had the advantage of higher ground. She threw herself at him, charging up the slope and sweeping her sword ahead of her, not high but low. At his feet.\n\nThe move startled Daavn. He drew back the throwing knife and released it with a snap of his wrist, but the cast was weak and wide. The slim blade rang on rock. Ekhaas's attack forced him to hop and dance backward. By the time he had the opportunity to strike back, Ekhaas had secure footing even if she was still below him. Daavn's sword swung down at her; she managed to raise her blade and block it. Metal skittered on metal, and for a moment the blades locked. Daavn's lips drew back from his teeth.\n\n\"You're not going to stop Tariic,\" he said. \"He'll have the true rod.\"\n\nHe kicked at her chest. Ekhaas felt the shift in his balance through their braced swords and twisted desperately to the side. The kick missed her and in the heartbeat that it took Daavn to recover his balance, she ran past him and up to the top of the slope.\n\nShe could see Ashi now, circling Makka and Pradoor with Aruget at her side. Daavn's guards and the last of the bugbear workers were closing in. A song might scatter them, but first she would have to deal with Daavn\u2014if she could stay away from the warlord's sword. \"How much has Tariic told you about the rod?\" she asked him. \"Has he told you that it's cursed?\"\n\n\"He's told us that Haruuc didn't know how to control its power.\" Daavn lunged, whirling his blade. Ekhaas twisted aside and tried to reach under his extended arm, but Daavn was faster and closed his stance quickly. He snapped an elbow up, catching her under the chin and sending her staggering back.\n\nThe roar that erupted from Makka gave both of them pause. Ekhaas swung around to catch a glimpse of the bugbear charging past Ashi and Aruget, and leaping down the ridge in the direction of the tomb. Fear clenched a fist in her guts. Their distraction had failed.\n\n_\"Maabet,\"_ cursed Daavn, then shouted orders. \"Move in! All of you, move in!\"\n\nEkhaas slashed her sword at his throat.\n\nHe blocked her with the ease of a practiced swordsman and forced her back across the ridge with a series of powerful, hammering blows. He smiled the next time they crossed swords.\n\n\"Did you know your friend Midian betrayed Ashi and Geth to save his own life?\"\n\nShe felt him relax a little in anticipation of her surprise. She turned it against him.\n\n\"Yes,\" she said and jerked her knee up into his groin.\n\nHe wore an armored codpiece that left her knee aching, but the force of the blow was still enough to make his ears droop and his eyes open wide. Ekhaas shoved hard against his chest with her free hand and he reeled back.\n\nRight over the edge of a crevice in the ridge. He didn't even teeter, but just went over. Ekhaas heard his armor strike rock twice as he fell. She whirled and ran to the aid of Ashi and Aruget.\n\nShe'd taken perhaps four paces when a dull drone filled the air. She looked around, trying to pinpoint the source of the sound, but it seemed to come from everywhere. Aruget, Ashi, and the guards and workers who had pinned them down broke off their attacks, looking around in confusion.\n\nThe brown body of an insect\u2014a locust\u2014landed on her arm. Its eyes were the same milky white as Pradoor's. The drone turned into a thrum that reverberated in Ekhaas's belly. The horses picketed before the tomb whinnied and shifted in fright.\n\nAnd a vast swarm of the white-eyed locusts rose up from behind the ridge.\n\nThey came whirring down like a blizzard, only a few at first, then more and more. Daavn's bugbears and hobgoblins shouted and fled, dropping tools and weapons. Ekhaas sprinted to Ashi and Aruget. \"Makka has gone into the tomb!\" she shouted over the thrumming of the locusts' wings. Daylight was growing dim as the swarm clustered around them.\n\n\"I saw him,\" said Ashi, \"and I saw Pradoor chanting. I think this swarm is\u2014\" She broke off with a hiss of pain and plucked a locust away from the thin bloody scratch across her belly. \"It bit me!\" She hissed again.\n\nSo did Aruget. Blood matted Geth's thick hair. The locusts seemed to be settling on it like leeches. The changeling could have been doing a bizarre dance as he swatted at them. Another locust settled on Ekhaas but this time it brought with it a stinging pain. She slapped it away and it left a smear of red blood behind. Her blood.\n\nTwo more locusts landed on the same spot. The thrumming of the swarm rose in pitch. Ekhaas's fear rose with it. She grabbed for Aruget and Ashi and pulled them close, ignoring the bites of the locusts as they settled on her. \"Cover your ears!\" she ordered, then drew a breath through barely parted lips and sang a burst of sound directly on top of them.\n\nThe song blew back the swarm like a stone thrown into a pond. Ekhaas felt the sound pulse through her body but it did more damage to the locusts than to her. The brown insects dropped in drifts all around them. For a moment, daylight returned and the thrumming vanished.\n\nOnly for a moment. Pradoor's shrill voice rose. \"Devourer, aid your servant! Let your hunger be manifest!\" The thrumming returned, and the sun dimmed as a new swarm settled over them.\n\n\"Ekhaas, we have to get into the tomb!\" said Ashi. \"Geth and Chetiin need us!\"\n\n\"We'd never make it,\" Aruget said, and spat as a locust crawled into his mouth.\n\nEkhaas held tight to both human and changeling and drew them down into a tight huddle. Forcing her voice deep into her throat, she started to hum with the same pitch and resonance as the swarm. It was hardly a song, but when she drew up magic from deep inside herself, it took on strength and substance.\n\nThe thrum of the swarm became a roar, but no locusts landed on them. Ekhaas felt Ashi raise her head and look around. _\"Rond betch,\"_ she said in amazement.\n\nAruget\u2014shifter's features melting back into a hobgoblin's\u2014raised his head, too, but he looked at Ekhaas. She could guess what he was thinking because she was thinking it, too.\n\nWhich would last longer: her hum or Pradoor's chant?\n\nGeth tumbled across the floor of the tomb, bashing aside treasures and scattering coins. He still felt Midian's second crossbow bolt pass uncomfortably close to him. He landed beside a broad polished shield bearing the crest of House Deneith and snatched it up, ducking behind it like a turtle retreating into its shell. The high opening of the stairs leading up out of the tomb was a half dozen paces away. Tenquis stood there, still directing the dim light of his moonstone up at Midian.\n\n\"Drop the stone, Tenquis!\" Geth shouted. \"You're a target!\"\n\nThe tiefling flinched and flicked the stone away. Shadows lurched crazily. Geth seized the moment and ran for the stairs, still covering himself with the shield. In spite of the shifting shadows and the shield, another bolt from Midian's hand crossbow glanced off the stone right beside his moving feet. Geth yelped and skipped, diving the last of the way into the cover of the passage.\n\n\"You're only making this a challenge, Geth.\" Midian's voice drifted down from his perch above. \"I'm a very good marksman.\"\n\nGeth, panting for breath, looked up at Tenquis. \"Where did he get a crossbow?\"\n\n\"If I was him,\" said Chetiin from nearby, \"I would have left a cache of weapons and supplies behind.\"\n\nBoth Geth and Tenquis flinched this time. Geth twisted around and spotted Chetiin squatting against one wall of the stairs. \"I thought you were still somewhere out there in the cave!\" he said, almost choking.\n\n\"I thought we should talk without having to shout our business.\" The _shaarat'khesh_ elder stood up and added quietly. \"I'll deal with Midian.\"\n\n\"You're not going to get close to him,\" Tenquis said. \"He's a good eight paces up and he'll have clear aim on you.\"\n\n\"You're whispering,\" Midian called out to them. \"Speak up.\"\n\nChetiin glanced up at Tenquis. \"I'll deal with him,\" he said again, then looked to Geth. \"When you have your opening, take it. Get the rod and run. Don't wait for me. Just take it and run as far from Rhukaan Draal as you can. I'll find you.\" He put a fist to his chest. _\"Paatcha_ , Geth. Swift travel.\"\n\nGeth repeated the gesture. _\"Paatcha_ , Chetiin.\"\n\nThe old goblin moved closer to the mouth of the passage and raised his strained, scarred voice. \"Midian!\"\n\nThis time the gnome made no response. Geth peered around the edge of the passage and up into the gloom near the cavern ceiling. Although the moonstone now lay on the floor somewhere beyond Haruuc's withered corpse, its light still revealed Midian's hiding place. The pale shape of his face was still there, and still partly obscured by the darkness of his small crossbow. \"He's waiting for you,\" Geth murmured to Chetiin.\n\n\"I know.\" Chetiin stepped a little further into the cavern, a shadow standing among shadows. \"Midian, we have unfinished business!\"\n\nMidian's first contact with Tariic had come in a letter. _Master Davandi_ , it had read, _mutual friends recommend your knowledge as an expert on the history of the Empire of Dhakaan_.\n\nMidian had smiled. He'd been expecting the letter. Their \"mutual friends\" had been other agents of the silent masters of the Trust, the body that served the Triumvirate\u2014or was served by them, according to some suspicions, but that in any case served the interests of Zilargo. He'd been proud to take the assignment.\n\nBoth he and the Trust had underestimated Tariic's ambitions, but regrets made poor excuses. The game wasn't over yet.\n\nMidian kept the crossbow aimed at Chetiin's chest, kept his hands still and his breathing slow. This time he would finish the job\u2014he'd cursed himself many times over the last several days for not making sure the old goblin was dead the first time. He'd had the stolen dagger, the stealer of souls, the Keeper's fang. He should have used it on Chetiin as well as Haruuc.\n\nOf the three figures in the tomb below, Chetiin was the most dangerous. Geth was fast but he couldn't evade the bolts forever. Tenquis... the tiefling barely even merited aiming.\n\nChetiin stepped out of the passage leading to the stairs and into the cavern. The shadows cast by Tenquis's moonstone gave him cover. The crossbow waved between two shadows. Midian clenched his teeth. _Wait_.\n\n\"Midian, we have unfinished business!\" Chetiin called.\n\n_Now_.\n\nHe squeezed the trigger of the crossbow. The light weapon kicked in his hand as the bent arms sprang straight and the taut string sang. His aim was good. He heard Geth try to call a warning. Too late. The bolt pierced Chetiin\u2014and porcelain shattered, spilling coins across the tomb floor.\n\nA tall vase. Not Chetiin. Midian's jaw tightened. The scholar in him remembered the vase from his explorations when he'd first wormed his way into the tomb\u2014Dhakaani, late empire, Riis dynasty. Beautiful work.\n\nThe assassin in him was already cocking the crossbow with a swift pull of the ratcheting lever, and dropping another bolt into the channel. His eyes didn't stop scanning the cavern.\n\n\"You tried to kill me,\" said Chetiin, and once more Midian thought he saw him, this time close to Haruuc's throne. \"You tried to make it seem like I'd killed Haruuc. I admire that. Among the _shaarat'khesh_ you would be honored. But\u2014\"\n\nMidian shifted his crossbow to cover the goblin, though he didn't squeeze the trigger. He wouldn't be tricked a second time.\n\nChetiin moved, the light of the moonstone shining full on his face. Again the crossbow snapped.\n\nThe bolt caught only a fold of Haruuc's cloak. Midian cursed. Fingers flickered on crossbow once more. Chetiin was good.\n\nHe was better.\n\nChetiin's voice, sourceless now, continued as if nothing had happened. \"\u2014you used my face to kill a friend. However much I respect your technique, I can't let that pass.\"\n\nThen there was silence. Geth and Tenquis peered around the edge of the passage. Geth's eyes flickered toward the rod but he didn't move. Waiting for an opening, Midian knew. He was tempted to put a bolt in the shifter's forehead.\n\n_Hold to your target_.\n\nMidian turned his head side to side, making a show of searching for Chetiin, before calling out, \"Nothing about the Rod of Kings? Nothing about breaking my oath to keep it a secret? Nothing about Zilargo?\"\n\n\"Nothing.\" Nothing nothing nothing...\n\nChetiin's answer echoed from a dozen places around the cavern at once, but Midian knew that trick. He twisted and loosed his bolt at the point where the goblin would be standing.\n\nAnd from _above_ him dropped Chetiin, breaking away from the cavern wall like some great spider. His feet struck the crossbow and forced it from Midian's grip. His hands caught the lip of the crack in which the gnome perched and his body curved back up so that his ankles hooked together behind Midian's neck.\n\nMidian threw himself back into the crack, dragging Chetiin with him. He'd taken a dagger from the cache he'd hidden in the tunnel. If he could draw it... but Chetiin didn't give him a chance. The goblin's speed and strength belied his age. He wormed around Midian and grappled with him, a primal struggle in the dark, cramped tunnel.\n\nThere was no room to maneuver. Midian glanced back to the mouth of the crack, glowing with the light of the moonstone, and kicked toward it. Maybe Chetiin had the same idea because he kicked, too.\n\nThey burst out into the open space of the cavern as if spat out of the mouth of some huge beast. Even as they fell, though, they pushed themselves apart. Midian twisted his body in mid-air and hit the cave floor in a springy crouch that absorbed the impact of the fall.\n\nSo did Chetiin.\n\nThey drew daggers at the same moment and circled each other briefly. Then Midian leaped.\n\nChetiin caught him with one hand, guiding him in a sweeping arc, but Midian grabbed the goblin's arm in return and held tight. The momentum of his body dragged Chetiin off his feet and they both crashed into piled plates of silver and gold. In an instant they were rolling and wrestling, Midian with bared teeth, Chetiin with flattened ears.\n\nMidian didn't hold back or offer mercy. He knew Chetiin wouldn't. Their business would be finished here.\n\nGeth saw his opening. He sprinted across the tomb, past Haruuc's staring eyeless face, and dived into the pile of rolled carpets, digging among them until his fingers touched stiff leather. He closed them and wrenched out the leather tube that he had seen the false Chetiin take from his quarters.\n\nThere was a weight to the tube that he knew well, but he'd been tricked one too many times. Fingers fumbling with the clasp on the tube, Geth opened it just a bit.\n\nThe dim light of the moonstone\u2014broken by the struggling shadows of Midian and Chetiin\u2014flashed on a shaft of purple byeshk, thick as his wrist, carved with ancient symbols. A touch of his hand to Wrath confirmed it. This was the Rod of Kings. The true rod.\n\n\"I have it!\" he shouted. He pushed himself to his feet and dashed back across the cavern, closing the clasp again as he went. Midian let out a howl of frustration\u2014that turned into a howl of pain. Geth resisted the temptation to turn and see who was winning the fight. He ran for the stairs, racing up them with Tenquis following close behind.\n\nHe was about halfway up, the light of day glowing in the tomb door and a strange thrumming roar growing in his ears, when something big, heavy, and hairy dropped on him.\n\nThere was no chance to think what it was or where it came from. Geth had a brief sense of something falling on him, then he was crushed flat against the stairs. His breath exploded out of his mouth. The ribs he'd broken in his dive out of Khaar Mbar'ost cracked again in a sharp burst of pain rivaled only by the blossom of fire that bloomed when his head bounced off the stone steps. He heard Tenquis yell, but the sound seemed very far away.\n\nThe leather tube jerked out of his hand. A massive, wide-shouldered figure blotted out the daylight as it bounded up the stairs.\n\n\"Geth!\" Hands grabbed him and turned him over. Tenquis's face spun above him. Geth couldn't quite focus on him. He couldn't quite breathe either. His body heaved with the effort of it. Tenquis cursed and dug in a pocket, coming out with a little leather flask that he opened, stuck under Geth's nose, and squeezed.\n\nAcrid orange dust puffed up into Geth's vision. It tickled his nose, burned his eyes, and somehow opened up his throat. He gasped and air rushed into his lungs. His head stopped spinning, though it still throbbed mercilessly. \"What happened?\" he asked.\n\nTenquis was already trying to drag him to his feet. \"Makka! He must have braced himself between the walls of the passage. Geth, he has the rod!\"\n\nThat, more than the orange dust, cleared Geth's head.\n\n\"No.\" Fighting back the pain in his side and his head, he thrust himself up the stairs. He twisted his head around as he ran to shout back into the tomb. \"Chetiin! Makka's stolen the rod!\"\n\nHe staggered out of the tomb door just in time to see the bugbear snatch Pradoor up from where she stood chanting and vault into the saddle of a terrified horse. He wheeled the horse once and his eyes met Geth's, then he spurred the beast along the road back into Rhukaan Draal.\n\n#\n\n# CHAPTER  \nTWENTY-NINE\n\n# 3 Aryth\n\nHorror mingled with rage in Geth's gut and he latched onto the edge of the stone doorway to hold himself up. Tenquis, coming up behind him, grabbed his shoulder.\n\n\"Look there!\" Tenquis pointed at a vast cloud of locusts that swarmed along the ridge.\n\nEven as Geth turned his head, the cloud dissipated, the thrum of wings fading as fast as Pradoor's chant. Some of the insects flew away, others dropped to the ground like a brown hail around the trio of bloodied figures that huddled at its heart. Geth's fear surged again\u2014then the figures stirred and Ashi, Ekhaas, and Aruget looked down at him.\n\n\"The rod?\" called Ekhaas, her voice raw.\n\nGeth pointed at Makka's retreating horse.\n\n\"They have it?\" Ashi asked. \"After all that, they have the rod?\"\n\nGrim determination settled over Geth. \"Makka has it, not Tariic,\" he said. \"We still have a chance to stop him.\"\n\nThe other horses, calmer now that the locusts had gone, were still picketed. Geth let go of the doorway and moved for the stairs down to the ground\u2014and would have pitched over if Tenquis hadn't been there to catch him. The tiefling lowered him to the ground and pulled more flasks out of his pockets.\n\nGeth pushed at him. Every moment they lingered Makka got closer to Khaar Mbar'ost. \"There isn't time for this!\"\n\n\"You can't stop him if you can't walk,\" Tenquis said.\n\n\"Listen to him, Geth. _Tapaat te nuusha ka koor te hara_ \u2014bind your wounds or bleed out your victory.\"\n\nGeth turned his head to see Chetiin climbing the stairs from the tomb. The goblin was bleeding himself. Long scratches tore the parchment-like skin of his face and a blood-soaked black sleeve clung to one arm. Far more astonishing, though, was the sight of Midian draped over the _shaarat'khesh's_ shoulder\u2014limp and unconscious.\n\nChetiin answered Geth's question before he spoke it. \"This tomb was built for Haruuc, not his killer.\" He dropped Midian. The gnome fell heavily, groaning as he hit the ground. Chetiin drew the dagger that had killed Haruuc and the blue-black crystal set in the ugly blade glittered. \"Traitors die on the doorsteps of heroes.\"\n\nA desperate idea came to Geth. \"Wait!\" he said. \"Don't kill him.\"\n\nChetiin and Tenquis both froze, the tiefling with surprise on his face, the goblin with cupped ears. Ashi, descending the ridge alongside Ekhaas and Aruget, cursed. Up close, Geth could see that the bloody appearance of the three came from dozens of tiny bites.\n\n\"Are you serious?\" Ashi demanded as she slid down the last of the slope to land before the tomb.\n\n\"Yes.\" Geth looked to Tenquis who held a flask motionless in one hand and a small heap of shimmering powders in the other. \"Finish that,\" he said, pointing at Midian, \"then get out your orange dust and wake him up. Chetiin, do you know where his crossbow is?\"\n\nThe goblin's eyes narrowed and he gave a curt nod.\n\n\"Get it,\" said Geth.\n\nChetiin didn't move.\n\nGeth saw the others exchange glances, then Ekhaas raised her voice. \"What are you doing? Midian has turned on us three times.\"\n\n\"Four,\" he corrected her.\n\n\"And you still want to let him live?\"\n\nGeth struggled to his feet. \"I'm trying to stop Tariic from getting his hands on the rod,\" he said. \"I don't think Midian and his masters in Zilargo want it to happen any more than the rest of us do. We know him now. We won't give him the chance to turn on us.\" He moved unsteadily to Midian and nudged him with his toe. The gnome groaned again and Geth said, \"If we can catch Makka, maybe we won't need him. If we can't, if Makka gets the rod to Tariic, I think we will.\"\n\nMidian's features twitched\u2014Chetiin dropped into a crouch, dagger ready to strike\u2014and one bright blue eye opened, rolling around to look at them. \"You'll need me for what?\" he croaked.\n\n\"Awake after all,\" Geth said. He crouched down beside the gnome and bared his teeth. \"If Makka gets the rod to Tariic, we're going to need all the help we can get, including you and your crossbow. We'll have to finish what you started.\" He looked up at the others. \"We have to kill a king.\"\n\nThey galloped back into Rhukaan Draal on the horses left behind. Chetiin rode with Geth, Midian with Aruget. The gnome looked as grim as any of them. Geth knew he'd guessed correctly. Even if Midian was a Zil agent, even if he wanted to capture the Rod of Kings for his own people, he didn't want the rod under Tariic's control. Or Tariic under the rod's.\n\nBeyond that, Geth didn't trust him any more than a dog with a sausage. Midian rode with his hands tied behind his back. His crossbow rode with Ashi.\n\n\"I don't like this,\" said Chetiin in Geth's ear, his voice pitched just over the thunder of the horses' hooves.\n\n\"Killing Tariic or working with Midian?\" Geth asked.\n\n\"Both. And working with an agent of Breland.\"\n\nGeth glanced at Aruget. \"We can trust him. Or at least we could trust Benti.\"\n\n\"Exactly.\" The goblin's strained voice dropped even lower. \"He did nothing until he was forced to.\"\n\n\"When Ashi was in trouble.\"\n\n\"When his source of information was in trouble. He works for Breland just as Midian works for Zilargo. Be wary, Geth.\"\n\nHe said no more.\n\nEkhaas and Tenquis, knowing the city best, rode point. Makka was long out of sight. Any hope of catching him seemed gone, but Tenquis still led them toward Khaar Mbar'ost by the route he swore Makka would most likely have followed, although he'd warned them that it might be crowded.\n\n\"The city will come out to see Dagii return,\" he'd said. \"It will slow us but it will slow Makka, too.\"\n\nExcept that the streets _weren't_ crowded. Most were less busy than when they had made their way out to Haruuc's tomb. \"Where is everyone?\" asked Ashi.\n\nA distant cheer answered her. \"Dagii has crossed the Ghaal River,\" said Ekhaas. \"Everyone has gone to watch his procession.\"\n\n\"By the sound of it, he's near the Bloody Market,\" Chetiin said.\n\nEkhaas's face went hard and for a moment she looked like she might add something, but then she closed her mouth and put her ears back. Geth could guess what she was thinking. \"Dagii needs to know what he's heading into,\" he said. \"We need to warn him.\"\n\n\"He already knows there's danger,\" said Ekhaas. Her ears flicked. \"It's best for him if he can deny any part in this.\"\n\nNo one answered that. They all knew the same thing: the time for tricks and clever plans was over. If they wanted to keep the power of the rod a secret, to prevent Haruuc's dream from destroying itself in the memories of a fallen empire, Tariic had to be the last to hold the Rod of Kings.\n\nIf they succeeded, they'd be the most famous assassins and thieves in Khorvaire\u2014and the most hunted.\n\nRiding alongside the noise of the crowd that cheered for the hero of the Battle of Zarrthec was like riding parallel to an unseen but rushing river. Along some of Rhukaan Draal's straighter streets, Geth caught glimpses of the crowd, and once the flash of sunlight on spearpoints and armor like water seen through trees. After a long while, though, the shouting fell behind. Khaar Mbar'ost rose ahead\u2014and the river fed into an ocean.\n\nPeople packed the plaza before the red fortress. Off to their left, a wide path, kept open by bugbear guards, led directly to the gates of Khaar Mbar'ost. Warlords and dignitaries stood on a raised platform of about shoulder height, waiting for Dagii's approach. As Geth and the others reined in their horses at the edge of the crowd, he saw familiar faces on the platform. Aguus of Traakuum. Garaad of Vaniish Kai. Iizan of Ghaal Sehn. Pater d'Orien. Senen Dhakaan. Munta. Vounn.\n\nIn fact, he knew all of the faces. Not so long ago, he would have been standing on the platform, too.\n\n\"I don't see Makka or Pradoor,\" said Ashi. \"Or Tariic.\"\n\n\"There,\" Chetiin said.\n\nThe people on the platform parted to reveal the lhesh climbing up stairs at the back of the platform. Makka, looking around with the wariness of a hunter on edge, and Pradoor followed close behind him. The excited murmurs of the crowd turned into a roar of approval. Tariic stepped up to the front of the platform and raised the Rod of Kings. The roar of the crowd rose even higher.\n\n\"He has it,\" said Chetiin.\n\nGeth looked to Midian. The gnome's lips pressed tight and he raised his eyes to scan the rooftops\u2014also crowded with people\u2014around them. He shook his head. \"The plaza is too wide. Even with a more powerful crossbow I'd have trouble hitting him. With my little hand bow it's impossible.\"\n\n\"Don't worry,\" said Geth, \"we're going closer.\"\n\n\"We'll be recognized!\"\n\n\"I'm counting on it.\" A plan, desperate and dangerous but possibly their only hope, had formed in his mind. He slid from his horse and gestured for the others to do the same.\n\nThe hats and cloaks Aruget had used to smuggle them out of Khaar Mbar'ost had been abandoned by Haruuc's tomb, but silver quickly procured more. Ashi kept her face down and her cowl\u2014stinking of the hobgoblin beggar who been wearing it only moments before\u2014well up as she fought her way through the crowd beside Ekhaas. A few paces away, Geth wore a similarly ragged and foul cloak. Ekhaas, along with Aruget, his features shifted to anonymity, blended into the crowd of other _dar_ faces. Chetiin and Midian\u2014arms unbound but now tethered to and closely watched by Aruget\u2014simply moved unseen among the legs of larger figures. Ashi wasn't sure if she envied Tenquis his role or not. Standing back with the horses, he didn't need to wear a disguise, but it would be his job to get them out again once they'd done what they had to do.\n\nHer palms were wet. She wiped them on the legs of her trousers.\n\n\"If we'd taken the rod and run after Haruuc died,\" she murmured to Ekhaas in human language as bodies jostled them on all sides, \"none of this would be happening.\"\n\nThe _duur'kala_ glanced at her. \"If we'd taken the rod and run,\" she answered, lips barely moving, \"the succession would have been even more chaotic and Darguun likely would still have been at war with Valenar.\"\n\n\"We're going to force a new succession _and_ we're going to steal the rod.\"\n\nEkhaas's ears flicked and drooped. \"But now we have war. _Dar_ may not understand peace, but we understand war very well. There will be a new lhesh in days, and he won't need the rod as a symbol to unify the clans. He'll continue the war with Valenar and that will be enough. Darguun will follow him. And without the rod's dreams of empire, Darguun will remain only Darguun.\"\n\n\"If the war drags on, Darguun will weaken. Other nations could decide to take it on. It could fall.\"\n\nEkhaas gave a thin smile. \"Even Dhakaan fell eventually, Ashi. We can only make certain Darguun does not fall today.\" She stopped. \"We part here.\"\n\nAshi raised her head a little, just enough to look around. They stood about fifteen paces from the platform and close to the line of guards that kept the path through the plaza clear. Geth and Chetiin had already pushed and slid their way to a position five paces nearer the platform. Aruget and Midian were a little to the right.\n\nUp on the platform, Vounn chatted casually with warlords, ambassadors, and other envoys. Ashi knew her well enough now to recognize her apparent ease for the act it was. Her mentor's back was stiff and her hands moved in small circles when she spoke. She knew something was amiss. If she didn't know Tariic had regained the true rod, she at least guessed it.\n\nAshi felt a twinge of guilt for the trouble she was about to cause both Vounn and all of House Deneith. She glanced at Makka and at the sword\u2014her sword, the Deneith honor blade\u2014he still carried. Perhaps it was just as well she hadn't recovered it. Her grandfather probably wouldn't have approved of what she was about to do. Or maybe he would have. She forced the sword out of her mind, saying a silent good-bye to it and Vounn alike, then looked at Ekhaas and gave her a silent nod. The _duur'kala_ returned it and moved to join Aruget and Midian. Ashi pushed through the crowd to Geth and Chetiin. The stench of her cowl actually helped\u2014people shifted just to get away from her.\n\nBeneath his cloak, Geth had Wrath already drawn. He turned his head and looked at her. \"Ready?\" he murmured.\n\nAshi concentrated, drawing on the power of her dragonmark. The bright lines that patterned her skin warmed and she felt the familiar clarity of the mark's protection settle over her mind. She concentrated again, the warmth becoming an almost uncomfortable heat, and reached down to touch Chetiin. The goblin shivered as the shield of the Siberys Mark of Sentinel wrapped him as well. Even if Tariic had mastered the rod's power, they would be safe from it, while Wrath protected Geth. Then she nodded and reached for her sword. \"Read\u2014\"\n\nA sudden roar erupted from behind them at the edge of the plaza. It spread quickly through the crowd. Heads turned, even among those on the platform. Ashi twisted around.\n\nAt the end of the path through the plaza, two hobgoblins rode at the head of a column of soldiers. One wore chains wrapped around his torso like a badge of honor. The other, mounted on a tiger, wore the horned armor of the warlord of Mur Talaan and raised his arms in triumph.\n\nDagii\u2014and Keraal\u2014had reached the plaza.\n\n\"Now!\" snapped Geth. His gauntleted arm rose and fell, dashing a vial of dark glass, surrendered by Midian, against the paving stones at their feet. Ashi squeezed her eyes shut.\n\nThe intense light released by the shattered vial flared even through her eyelids. All around them, cheers turned into shouts of alarm.\n\nAshi forced her eyes open again, throwing back her cowl with one hand and drawing her sword with the other. She whirled the blade around her head and let the fluting battlecry of the Bonetree Clan ripple from her lips.\n\nGeth thrust Wrath toward the platform and howled, \"Tariic Kurar'taarn! We come for you!\"\n\nShocked and dazzled by the burst of light, startled by cries and swinging swords, the crowd surged away from them. Chetiin drove them back further, darting and tumbling among them like a furious black cat, his dagger slashing at legs. The line of guards trying to hold the crowd back from the path through the plaza buckled and fell as people moved. Out of the corner of her eye, Ashi had a glimpse of Keraal's horse rearing and Dagii fighting to control his tiger, even as he caught sight of them.\n\nChaos erupted on the platform as well. Warlords flinched in surprise, then pushed forward like the trained warriors they were, struggling for a moment with the envoys and ambassadors who were trying to get back. Makka forced Pradoor behind him and drew his sword. Tariic, rod raised to greet Dagii, froze for an instant, then moved. The others on the platform had closed in behind him, blocking access to stairs. He turned, crossing the front of the platform away from her and Geth.\n\nAnd Aruget, waiting just at the edge of the open space cleared by the panicked crowd, waiting for just such an attempt at escape, lifted Midian up onto his shoulders. In one smooth movement, the agent of Zilargo braced himself, brought down his crossbow, and aimed over the heads of the crowd. Ekhaas stepped in front of the pair, drawing her blade to turn back blundering spectators. The crossbow tracked Tariic for a moment, then Midian squeezed the trigger.\n\nAshi couldn't hear the crack of the bow's release over the noise of the crowd's confusion, but she saw Tariic jerk and sprawl backward. The fletching of a crossbow bolt smeared with Midian's entire remaining supply of strandpine sap protruded from his throat.\n\nTariic's hand spasmed and the Rod of Kings fell from it.\n\nStill howling, still waving Wrath, Geth sprang for the platform. Ashi and Chetiin stayed close behind him, and the remnants of the crowd parted before them. Before the rod had even stopped rolling, before greedy warlords could do more than stare at the prize before their feet, the three of them had vaulted onto the platform. Chetiin took one side of Geth and Ashi the other, twitching her sword back and forth to keep Aguus of Traakuum and Garaad of Vaniish Kai at bay, as the shifter scooped up the rod.\n\nStanding close beside Aguus and Garaad, calmer than any other envoy, Vounn stood and stared at her. Ashi drew a breath between her teeth. \"I'm sorry,\" she murmured.\n\nVounn's eyes opened wide. Her finger came up\u2014and pointed at something behind Ashi. Aguus and Garaad stiffened as well. Ashi heard a soft curse from Chetiin. She threw a fast look over her shoulder. Like Vounn and the warlords, Geth had stopped and was staring. She followed his eyes down to Tariic's corpse.\n\nRed-brown flesh seemed to flow and turn dusky gray. Flat, harsh features became round and soft. Brown eyes so bright they were almost red turned white. Short, dark hair grew long and became pale.\n\nA changeling, returned to his true form in death.\n\n\"Ko!\" choked Geth. He flipped the Rod of Kings around to reveal the faint spiral Tenquis had made to mark the false rod.\n\nAshi gasped. \"Where's Tariic?\"\n\n\"Here!\"\n\nAshi whirled as those on the platform parted and the real Tariic stepped forward with the rod\u2014the true rod\u2014raised high. Protected by her dragonmark, she couldn't feel the power of the true rod, but she could see it in the expressions of those around Tariic. It made the effects of the false rod seem as cheap and gaudy as gilded lead. The Darguul warlords who moved aside for Tariic stood straight, ears high, proud in his presence. The ambassadors and dragonmarked envoys looked even more frightened than they already had. A startled silence spread among the crowd as they saw that Lhesh Tariic still lived\u2014and moreover that he stood before them like an emperor returned.\n\nHe swept the Rod of Kings across the platform and his voice almost trembled with eagerness. \"Seize them, Darguuls! Seize the assassins!\"\n\nWithin the arc that the rod described, every _dar_ head\u2014hobgoblin, bugbear, and goblin\u2014turned to Ashi, Geth, and Chetiin. Those few envoys and ambassadors who hadn't already retreated looked around in confusion. Vounn, still standing in front of Ashi, opened her mouth as if to speak, but anything she might have said was lost as more than a dozen of the most powerful and important warlords in Darguun surged forward.\n\n\"Run!\" shouted Geth.\n\nAshi hesitated for an instant, as if she could seize Vounn and drag her free, then she spun and followed him and Chetiin in a desperate leap from the platform.\n\nToo slow. Arms wrapped around her in a tackle that sent her sword flying from her hand and brought her crashing down.\n\n_\"Maabet!\"_ cursed Aruget.\n\nMidian froze in the act of climbing down from his shoulder. Ekhaas felt a sudden nausea sweep through her.\n\nThe changeling who'd impersonated Geth. The false rod. Tariic had anticipated an attempt to recapture the true rod. He'd prepared for it.\n\nAnd they'd failed.\n\n\"Seize them!\" Tariic shouted. \"Seize the assassins!\"\n\nThe command spread from the platform, sweeping over the crowd. Out to the limits of Tariic's voice, it gripped minds and souls. The crowd that had been scattering in panic turned and rushed back like the turning tide.\n\nThe line of Aruget's jaw tightened, and he shook Midian off his back. \"Ideas?\" he said.\n\n\"One,\" said Midian\u2014and Ekhaas heard his crossbow clatter to the stones under their feet. She turned, but the gnome was already sprinting past her and racing _into_ the crowd, darting among a forest of legs. Chaos marked his plunge, but he was fast, using his small size to evade the hands that grabbed for him.\n\nAruget looked at Ekhaas and his ears flicked. \"We tried,\" he said.\n\nThen he was diving into the crowd, too. Even as he moved, though, Ekhaas saw his face and body shift and start to change. Adult hobgoblin became youthful bugbear. A few hands grabbed for him, there was a flurry of activity, but then nothing. His disappearance was even more complete than Midian's\u2014and it left an even greater hole in Ekhaas's gut. She dragged her sword from its sheath and swung it in a wide circle, forcing the advancing crowd back for a moment, but where the blade passed, the crowd pressed in\u2014\n\n\u2014until the roar of a tiger brought them and Ekhaas around. Above the heads of the crowd, Dagii appeared, his tiger mount leaping through the mob as if through grass. In his wake, led by Keraal, came the soldiers of the Iron Fox Company. Joy and anger warred in Ekhaas. Anger that Dagii had involved himself, opened himself up to Tariic's retribution. Joy that he'd come to her rescue. The last ranks of the crowd scattered as the tiger came to a snarling stop before her. Ekhaas looked up at Dagii, her heart racing.\n\nHe stared at her with gray eyes as hard as the half-visor of his helmet and as cold as the Rod of Kings. \"Ekhaas of Kech Volaar, assassin and traitor,\" he said, \"by command of Lhesh Tariic, you are my prisoner.\"\n\nThe hole in Ekhaas's gut swallowed her.\n\nThe day in the dungeon, the day he had spared Ko from the arena, came back to Geth. Tariic's disapproval of his mercy. His own promise to the dungeon keeper\u2014\"I'll be back to talk to him when I can.\" But he'd never made it back and his mercy had returned to damn him.\n\nThere was no room in his fury\u2014at Tariic, but especially at himself\u2014even for cursing. He'd thought he was a hero. He was a fool.\n\nHis feet hit the stones of the plaza and he sank into a crouch, Wrath ready, his gauntlet up. Chetiin landed beside him. They were in the clear for the moment, but the crowd, summoned back by Tariic's command, was swarming in fast.\n\n_\"Geth!\"_\n\nAshi. He twisted to face the platform. Ashi lay near the edge of it, struggling desperately but held by half a dozen pairs of hands that tried to drag her back. Two of those pairs belonged to Aguus and Garaad.\n\nThe rest of the warlords caught in the rod's power were jumping and climbing down from the platform.\n\n\"Geth...\" Chetiin said in low warning.\n\n\"Watch the crowd,\" Geth growled. \"I'm going for Ashi.\"\n\nBefore the goblin elder could say anything else, he moved, throwing himself against Tariic's puppets. Confronted by Wrath, the Darguuls drew weapons, though they didn't strike to kill\u2014Tariic's command had been to seize. Geth lashed out with the twilight blade, trying to drive them back while using the false rod, still gripped in gauntleted hand, as a club against those who got close. For a moment, it worked\u2014until Munta the Gray thrust himself between the others. The old warlord's sword caught Geth's and held it. Dark eyes in a wrinkled face blazed. \"Traitor!\"\n\nThe hatred and ferocity in his voice made Geth bare his teeth. \"Munta, it's the rod! Tariic has\u2014\"\n\nNothing in Munta's face or posture hinted that he even heard him. \"You're mine,\" he snarled. \"When I drag you before Tariic, he will know I'm still fit for battle!\"\n\nHe threw back Wrath and swung his own sword with a strength and speed that Geth wouldn't have expected in someone of his age. The shifter blocked the blow with his gauntlet, then jabbed at Munta with the false rod.\n\nThe old warlord's sword whirled around and struck the rod at a sharp angle. The edge of the blade bit deep into the byeshk. Geth thought he felt a sting in his hand as the magic Tenquis had woven into the false rod unraveled. Munta must have felt something, too. He hissed and stumbled, dropping hard onto one knee.\n\n\"Sorry, Munta,\" Geth growled. He swung the damaged rod down onto his gray head. Munta collapsed like an empty sack.\n\nGeth let the false rod tumble onto him as he jumped over the old warlord's sagging body. More hands grabbed for him. He struck them away with his gauntlet. Ashi saw him, and her struggles intensified. She freed a leg and gave Aguus a hard kick in the chest. She freed a fist, but Garaad grabbed it again.\n\n\"Ashi, I'm coming!\" Geth roared, but the crowd was all around him now. Every step was a battle. Chetiin was fighting at his back, stabbing at knees and legs and chests whenever someone fell.\n\nSomewhere a tiger roared. \"Dagii!\" said Chetiin.\n\nGeth twisted his head around. Across the seething mob that filled the plaza, Dagii had drawn his mount up before Ekhaas\u2014but one glance at the warlord's face told him that his appearance was no rescue. \"Tariic has him. I don't see Aruget or Midian!\"\n\nUp on the platform, Tariic stood back with a look of supreme confidence on his face. To one side of him, Pradoor's head turned back and forth, ears twitching as the blind goblin listened to the sounds of the fight. To the other, Makka strained like a dog on a lead, eager to join the battle but held back by his master. Geth grimaced. Tariic didn't need Makka's strength to defeat them\u2014the sheer numbers of the crowd would drag them down. Soon.\n\nThen somewhere behind Geth, hooves beat on stone and a voice rose in a rasping shout\u2014\" _For the blood and line of Castalla!\"_ Geth turned again, the other way this time.\n\nTenquis, mounted and riding at a full gallop, plunged through the crowd, splitting it apart. In one hand, he held a rope gathering together the reins of the other horses; they followed him in a bucking, whinnying wedge of muscle and hooves. The tiefling rode with his head low over his horse's neck, gaze fixed on the platform.\n\nHands dragged at Geth as he turned his head between lhesh and artificer. Tariic raised the Rod of Kings and drew breath. One shouted command would halt the wild charge.\n\n\"Tenquis, watch out!\" Geth roared.\n\nBut Tenquis had already released the other horses and pulled back on the reins of his own. The beast reared up on its hind legs. Tenquis's free hand flicked out. Pale liquid spread out from a vial clutched in his ingers and seemed to evaporate on the air.\n\nAt the same instant, thick greenish vapor burst up in a smoky curtain around Tariic, Pradoor, and Makka. Whatever command Tariic might have issued disappeared in a strangled cough.\n\nAnd the anger and energy of the crowd seemed to drain away, as if only Tariic's concentration had sustained it. The hands that held Geth fell away. The forehooves of Tenquis's horse clattered back to the ground and he urged the animal around in a tight circle, driving the confused crowd further apart. \"Take a horse and mount up!\" he shouted at Geth and Chetiin.\n\nGeth glanced back to the platform. The warlords who had seized Ashi were as confused as the crowd below. Their grips went slack\u2014and Ashi pulled away from them, punching at one, cracking an elbow across Garaad's face, then twisting to her feet. Her eyes met Geth's and she gave him a ierce grin. Relief spread through Geth, so sharp it made him feel almost sick. He turned and leaped onto the back of the nearest horse.\n\nDagii's gaze snapped around at Tenquis's charge, but as the tiefling artificer cast pale liquid on the air and Tariic choked on green smoke, the warlord blinked and his gray eyes cleared. He stared down and Ekhaas saw horror in his face. _\"Taarka'nu_ , I didn't\u2014\"\n\nThe hole that had opened in her closed a little, but not all the way. Her legs trembled. Her head spun. Tenquis was already shouting at Geth. On the platform, Ashi was fighting free. Ekhaas looked up at Dagii and cut him off with curt words. \"Tariic has the true rod. Ride or he'll have you again!\"\n\nHis ears pressed back. \"If I ride, he'll have you. Take a horse. I'll cover your retreat.\" He whipped up his sword, whirling it around his head. \"Iron Fox, forward! Take defensive lines! Protect the lhesh!\"\n\nSoldiers already in motion changed their step with disciplined obedience, rushing toward the platform. Ekhaas stared at Dagii. His eyes narrowed and his mouth curved in a smile that was both fond and hard. \"Go, _taarka'nu!_ Fight another day.\"\n\nHer ears rose. \"Great glory, _ruuska'te,\"_ she said, then raced for Tenquis and the horses. Geth was already in the saddle with Chetiin clinging behind him. Ekhaas's foot found a stirrup and she mounted.\n\nThe green vapor that burst around Makka seared his nostrils and stung his eyes, but at least he didn't choke. Tariic, caught mid-breath, sucked the stuff in deep and doubled over in wracking coughs. Pradoor coughed as well, wheezing between gasps \"What is this? What's happening?\"\n\nMakka held his breath, tore the sword from his belt, and leaped through the smoke. Wisps of it caught in his hair and scorched his skin, but he could see again.\n\nHe could see revenge slipping away from him.\n\nGeth and Chetiin were already on horseback. Ekhaas of Kech Volaar was mounting. Dagii was rushing forward as if eager to meet his death. Ashi of Deneith\u2014\n\n\u2014was still on the platform and the way between them was open.\n\nThe rage of the Fury fell over him. Tariic had denied him his vengeance for too long! Howling the anger that had seethed in him since these _taat_ had destroyed his tribe's camp and his power with it, Makka charged. The sword of Deneith flashed in his grip. Ashi turned. She was unarmed. Defenseless.\n\nGood.\n\nHe lunged.\n\n#\n\n# CHAPTER  \nTHIRTY\n\n# 3 Aryth\n\nEkhaas saw it happen.\n\nBetween one heartbeat and the next, Makka burst through the veil of green vapor and charged with a roar like wind in the mountains. Ashi turned but the bugbear's sword\u2014the bright blade that he had stolen from her\u2014was already thrusting at her belly.\n\nThen, darting from among warlords clustered on the platform, Vounn threw herself between sword and victim.\n\nThere was a rippling in the air around her, and Ekhaas recognized the shield of force conjured by Vounn's dragonmark. Ashi's mentor twisted as she moved, using her phantom armor to deflect the blade.\n\nIt didn't work. Makka's mighty blow plunged his sword through the rippling shield into Vounn's body\u2014and through her into Ashi. The force of it slammed both women back so hard that Ashi's arms seemed to wrap around Vounn.\n\nMakka held them there for a long moment, then released the hilt of the sword. Vounn and Ashi fell together, joined by the honor blade of Deneith, hitting the platform with a hard impact that drove a cry of pain from Ashi. Ekhaas thought she saw Vounn turn her head to look at Ashi, thought she saw the lady-seneschal's lips move before blood oozed from them and her body went limp. Ashi shuddered once, then her head fell back against the rough wood of the platform.\n\nAt the side of the platform, Pater d'Orien's eyes went wide. He glanced up and met Ekhaas's gaze, then his lips pressed together and his eyes lost focus as if he was gazing into the far distance. He took a small step\u2014and vanished.\n\nSomewhere far away, someone would learn very shortly of the murder of Vounn and Ashi d'Deneith.\n\nAnd Ekhaas knew she wasn't the only one to witness Pater's disappearance. As the curtain of green vapor faded into drifting wisps, she saw Tariic staring with shock and dismay at the place where House Orien's envoy had been.\n\nMakka threw his head back and howled a second time, then spun to face Ekhaas and Geth. His body tensed, ready to leap\u2014\n\nTariic's voice cracked like thunder. \"Makka, be still!\"\n\nThe bugbear froze. Instantly. Completely.\n\nThought broke through Ekhaas's shock. Their chance at escape was passing. She turned her horse sharply and reached out to grab the bridle of Geth's horse, pulling it after her. The shifter was still staring at Ashi's body, trembling with a rage so great it seemed he might shatter.\n\n\"Ride, Geth!\" she ordered, but it was Chetiin who slid around Geth's shaking body and snatched the reins from his hands. Tenquis slapped the horse on its hindquarters, then they were all galloping across the plaza, heading back the way they had come.\n\nEkhaas clutched her reins tight, expecting at any moment to hear the command from Tariic that would summon them all\u2014or at least her and Tenquis\u2014back, but it didn't come. She heard Dagii shouting, ordering his men to protect Tariic. She heard more shouts as other warlords, knowing only that an attempt had been made on the lhesh's life, tried to break free of the defensive lines and pursue them. She heard calls from the confused crowd.\n\nAnd finally, just as they'd almost reached the edge of the plaza, she heard Tariic's voice rise in a command that made her stomach lurch as if he'd called to her directly. \"Dagii of Mur Talaan, stop them!\"\n\nShe twisted her head and looked back over her shoulder to see Dagii turning his tiger after them, his face once more hard and cold. The rapidly thinning crowd that remained in the plaza scattered at the great cat took two huge bounds, closing at least a third of the distance between them.\n\nThen they were around a corner onto an empty street and the plaza was out of sight. Ekhaas's ears pressed back flat against her scalp. A horse could outpace the tiger in a distance race, but their horses had already run halfway across Rhukaan Draal. Dagii's tiger was fresh. She looked at Tenquis, hunched over his horse's neck, then at Chetiin and Geth. The shifter seemed to have come back to himself\u2014he'd taken the reins from Chetiin, though his face was still twisted with rage and grief.\n\nThe same emotions gnawed at her. One day Makka\u2014and Tariic\u2014would pay for Ashi's murder, but this wasn't that day.\n\n\"Keep riding!\" she snapped.\n\nGeth looked at her, but she just looked away and reined in her horse, turning it around in the street. Behind her, hoofbeats slowed and Tenquis called her name. She ignored him.\n\nDagii's tiger came leaping around the corner into the street, a streak of fire striped black and orange. Its master's gray eyes fixed on her, and he raised his sword. Ekhaas's hands tightened into fists. She drew breath, reached down into herself, and drew up magic.\n\nThe song that rolled along the street was not, however, hers. The voice that sang was high and clear. It danced and rippled, like sunlight falling on water or through the leaves of trees. It seemed to move around, coming first from a doorway, then a high window, then a deep shadow.\n\nDagii blinked and shook his head, allowing his tiger to slow to a walk as he looked around in confusion for the source of the music. Even the tiger seemed dazed.\n\nEkhaas held her horse still in the middle of the street as Dagii and his tiger came within a few paces of her, then turned around and went back the other way. She might as well have been invisible.\n\nShe knew the singer, even if she didn't know the song. Looking around, she found her. Senen Dhakaan had pressed herself into a deep doorway. She met Ekhaas's eyes and nodded. When Dagii had passed her again, she left off singing\u2014though the song seemed to linger\u2014and hurried to Ekhaas's side.\n\n\"I'm sorry for Ashi's death,\" she said.\n\nEkhaas bit back her anger. \"What are you doing here?\"\n\n\"I saw you enter the plaza. When you left your horses at the street, I guessed this would be the way you'd leave. I slipped away before you attacked.\" Senen's ears flicked. \"So the Rod of Kings commands obedience.\"\n\n\"Aruget told you,\" Ekhaas said. The changeling's name tasted like dirt in her mouth.\n\n\"I guessed it just now. When the real Tariic appeared, your tiefling friend fled the plaza and covered his ears. I did the same.\" She glanced at Dagii, still caught by her magic, then up at Ekhaas. \"You'll be an outlaw now. You should have told me everything. I could have helped you.\"\n\nEkhaas bared her teeth. \"This isn't the time, Senen.\"\n\n\"I know.\" Senen stepped back. \"A _duur'kala_ should listen as much as she sings. I've been listening in Khaar Mbar'ost. Find refuge in Volaar Draal, Ekhaas. Take a message to Tuura Dhakaan: there should be no alliance. Lhesh Tariic Kurar'taarn will not be a friend to the Kech Volaar.\"\n\n\"I don't think Tariic will be a friend to anybody,\" said Ekhaas. \"What the rod shows him will destroy Darguun.\"\n\n\"The vaults of Volaar Draal are deep.\" Senen nodded toward Dagii and his tiger. \"The song will fade soon. Ride now!\"\n\nShe turned away without waiting for a response and vanished into an alley. Ekhaas realized that her hands, even clenched into fists, were trembling. She looked at Dagii one last time, then turned her horse and urged it back to a fast trot.\n\nTenquis, Geth, and Chetiin hadn't followed her instructions. They'd stopped just around the next bend in the street and as she rode up, they fell in beside her. \"Well?\" growled Geth.\n\n\"We have a hiding place.\"\n\n\"What good's hiding?\"\n\n\"It keeps us alive,\" said Tenquis.\n\n\"Ashi's dead,\" Geth snapped at him, \"and Tariic has the rod.\"\n\nEkhaas looked up at the red bulk of Khaar Mbar'ost. The sun was settling into the west and it lit the fortress of the lhesh up like a pyre. A pyre for Darguun and Haruuc's lost dream. She looked back to Geth, her ears pressed flat. \"We'll find a way to take it away from him.\"\n\n#\n\n# CHAPTER  \nTHIRTY-ONE\n\n# 3 Aryth\n\nMakka's howl came so suddenly that what followed seemed like a blur to Ashi. The spinning of the world as she turned to meet his attack. The bright blade of her sword plunging at her. Vounn stepping into its path, trying to deflect it.\n\nThe shock as Makka's blow drove Vounn's body into Ashi's and the sword through both of them. The sudden blossoming of pain\u2014sharp metal into flesh, falling body against wood, body against body against metal against wood. Darkness swam across her vision, and Ashi cried out.\n\nWhen her vision cleared, she was looking into Vounn's eyes. She managed three words.\n\n\"That was stupid.\"\n\nVounn's lips twitched. \"Our lives belong to Deneith,\" she said, her voice wheezing in her throat, \"but some things are bigger than the house. You were... right about Tariic.\" One of her hands found Ashi's where they held her. \"I'm proud of you, Ash\u2014 _ah.\"_\n\nHer words ended in a sigh and bubbling blood.\n\nDarkness swarmed over Ashi and dragged her down.\n\nShe smelled incense and heard prayers, too soft to hear the words but loud enough to recognize the language. Goblin. She could feel something warm and dry on her forehead\u2014and something cold along her belly, right where there should have been searing pain.\n\nThe prayers faded into silence and someone coughed. The warm, dry thing on her forehead\u2014a small hand\u2014moved away. \"She'll live,\" said Pradoor's shrill voice. \"So will the other one.\"\n\n\"Good.\" Tariic's voice, hard and angry. \"Get one of the gnomes from the House Sivis outpost here. I need to send a message to Breven d'Deneith in Karrlakton.\"\n\nAshi opened her eyes to the bulk of Khaar Mbar'ost against a twilight sky. A silhouette moved over her. Tariic.\n\nShe bared her teeth and tried to grab at him, but her arms moved less than a handspan. Ropes cut into her wrists. Tariic held out the Rod of Kings. \"Be still!\"\n\nThe command skittered over the shield of her dragonmark. \"Blood in your mouth,\" she snarled at him. She raised her head. She was still on the platform before the gates of Khaar Mbar'ost, though the plaza was empty now.\n\nDagii stood at her feet, standing stiffly in the presence of the lhesh, but his ears flicked a little when he met her eyes and his mouth twitched.\n\nMakka crouched next to him, kneeling, yet at the same time almost straining to stand upright. Ashi recognized the power of the rod and allowed herself a little smile. She looked back up at Tariic. \"Where's Vounn?\"\n\nTariic flicked his ears and gestured to her right with the rod. Ashi turned her head\u2014and caught her breath. Vounn lay rolled onto her side just a few paces away from her. Her eyes were wide and staring, the wound inflicted by the Deneith honor blade gaping and ugly in her belly. The sword itself lay beside her, bright blade still stained with blood.\n\nPradoor had said \"the other one\" would live too. Who? Ashi twisted her head to the other side, wondering who else had survived, dreading the possible sight of more friends dead.\n\nMidian's bright blue eyes stared back at her from a face marked by the fading bruises of a savage beating. Hatred rose like bile in Ashi's throat.\n\nTariic crouched down between the two of them. \"Now,\" he said, \"what are we going to say really happened here?\"\n_Raat shan gath'kal dor_.\n\n\"The story stops but never ends.\"\n\n\u2014Traditional closing of hobgoblin legends.\n\n#\n\n# GLOSSARY\n\n**Aguus:** The hobgoblin warlord of the Traakuum clan. A contender for the throne of Darguun upon Haruuc's death.\n\n**Aram:** see Wrath\n\n**Aruget:** A hobgoblin warrior in the service of Haruuc, assigned to guard Vounn and Ashi d'Deneith.\n\n**_atcha:_** The goblin concept of personal honor, something which is earned and carefully protected. Compare with muut.\n\n**_ban:_** Goblin expression of noncommittal agreement, roughly equivalent to \"yeah, all right,\" or \"your funeral.\"\n\n**Bloody Market, the:** The largest market in Rhukaan Draal, the Bloody Market (Khaari Batuuvk in Goblin) is a chaotic mass of tents and stalls in a milling sea of bodies. Because the laws of Rhukaan Draal and Darguun are lax, nearly anything can be found for sale in the market, including slaves.\n\n**Breland:** One of the original Five Nations of Galifar. Breland is a tolerant country with a keen interest in trade and commerce.\n\n**byeshk:** A rare metal, hard and dense with a purple sheen.\n\n**_chaat'oor:_** Goblin term for any species not native to Khorvaire, especially humans, but with the exception of elves. It is often loosely translated as \"defiler.\"\n\n**Chetiin:** A goblin and an elder of the shaarat'khesh. Chetiin carries two daggers sheathed on his forearms, one of which (the dagger named Witness carried on his right arm) is a dangerous \"Keeper's Fang\" weapon capable of capturing the soul of a slain enemy.\n\n**_chib:_** Goblin for \"boss\" or \"big man.\" Used colloquially by goblins outside of Darguun to refer to any taller humanoid, including hobgoblins, humans, and dwarves.\n\n**_Chiitgath'muut. Chiitgath'atcha. Chiitgath'piir.:_** A goblin formula of complete submission. \"I am without duty. I am without honor. I have nothing.\"\n\n**_cho:_** Goblin expression of informal agreement with or acknowledgement of a statement made, roughly equivalent to \"yes\" or \"yeah.\"\n\n**_daashor:_** A goblin artificer, especially one from the time of the Empire of Dhakaan. The secret knowledge of the daashor has largely vanished, but at one time, they were capable of creating wonders. Most daashor were male.\n\n**Daavn:** The hobgoblin warlord of the Marhaan clan and an advisor to Tariic.\n\n**Dagii:** The young hobgoblin warlord of the small but highly honored Mur Talaan clan.\n\n**_dar:_** Goblin for \"the people.\" It is the ancient collective term for the hobgoblin, goblin, and bugbear races.\n\n**Darguul:** A goblin inhabitant of Darguun. Non-goblins living in Darguun are not generally recognized as citizens.\n\n**Darguun:** The nation of goblins, founded in 969 YK by the hobgoblin warlord Haruuc of the Rhukaan Taash clan in a swift campaign that seized territory held at the time by the human nations of Cyre and Breland. Darguun was officially recognized as a sovereign nation in the Treaty of Thronehold in 996 YK. Translated, it means \"Land of the People.\"\n\n**Dark Six, the:** Deities representing the violent and threatening aspects of the world, typically shunned in more civilized nations, but widely worshipped in Darguun. Once numbered among the Lords of the Sovereign Host, myth holds that they were cast out for their evil ways. The Six are: the Devourer (god of the ocean and destruction), the Fury (god of violence and madness), the Keeper (god of death and decay), the Mockery (god of treachery and murder), the Shadow (god of dark magic), and the Traveler (god of deception and change).\n\n**Davandi, Midian Mit:** A gnome, a field researcher of the Library of Korranberg with a specialty in the history of the Empire of Dhakaan.\n\n**d'Deneith, Ashi:** A former hunter of the Bonetree Clan of the Shadow Marches, Ashi turned her back on the clan after discovering her descent from House Deneith. She bears the Siberys Mark of Sentinel, a powerful dragonmark that patterns her entire body.\n\n**d'Deneith, Vounn:** A dragonmarked heir of House Deneith, Vounn is a skilled diplomat with a distinguished career serving as the liaison between Deneith and its most important clients. She holds the title of Lady Seneschal, indicating her special responsibilities within the House.\n\n**Deneith, House:** A dragonmarked house bearing the Mark of Sentinel. House Deneith operates services offering various forms of protection, including the mercenary companies of the Blademarks and the law enforcement services of the Sentinel Marshals.\n\n**Desperate Times, The:** The dark ages of chaos after the fall of the Empire of Dhakaan. Particularly conservative members of the Dhakaani Clans might argue that the Desperate Times extend into the present, but most goblin historians agree that the Desperate Times ended with the domination of Khorvaire by humans, somewhere between 3,000 years (when the human Karrn the Conqueror established Karrnath) and 2,000 years (when Galifar I united the Five Nations in a single kingdom) before the present.\n\n**Dhakaan:** An epithet claimed or given by popular acclaim among the Dhakaani clans to indicate allegiance with the ancient Empire of Dhakaan.\n\n**Dhakaani Clans:** Clans of goblins, primarily hobgoblins but incorporating other goblin races, devoted to keeping alive the glories of the Empire of Dhakaan. Although they claim territory in the mountains of Darguun, they do not pledge allegiance to Lhesh Haruuc, nor do they hold any particular alliance among themselves. The Dhakaani Clans include the Kech Draguus, the Kech Nasaar, the Kech Shaarat, the Kech Uul, and the Kech Volaar.\n\n**Dhakaani Empire:** see Empire of Dhakaan.\n\n**d'Orien, Pater:** A dragonmarked heir of House Orien, Pater is Viceroy of his House in Darguun, in charge of his House's operations in the country.\n\n**_duur'kala:_** Among the Dhakaani Clans, particularly the Kech Volaar, duur'kala preserve the history and knowledge of past ages. Their music is the most common form of magic among the clans. Duur'kala means \"dirge singers.\" Because the magic manifests mostly in females, duur'kala are often called \"daughters of the dirge\" and elders are referred to as \"mothers of the dirge.\"\n\n**Ekhaas:** A hobgoblin woman and a duur'kala of the Kech Volaar.\n\n**Empire of Dhakaan:** An ancient empire ruled by hobgoblins, the Empire of Dhakaan stretched across southern Khorvaire millennia before the arrival of humans. Dhakaan was weakened by the Daelkyr War and collapsed about six thousand years before the present.\n\n**Esmyssa Entar ir'Korran:** Ambassador from the gnomes of Zilargo to the court of Khaar Mbar'ost.\n\n**_gaa'ma:_** Goblin pejorative term for changelings. Literally translated, it means \"wax babies.\"\n\n**_gaanu duur:_** \"Daughter of the dirge,\" an alternative term for duur'kala.\n\n**_gaa'taat:_** A highly insulting Goblin term suggesting that someone is less than a child.\n\n**Garaad:** The hobgoblin warlord of the Vaniish Kai clan. A contender for the throne of Darguun upon Haruuc's death.\n\n**Gan'duur:** \"Eaters of Sorrow,\" a rebel clan of the Ghaal'dar, now crushed and disbanded. Their banner was yellow with the crest of a snarling dog.\n\n**Gathering Stone, the:** The primary stronghold of House Deneith in Darguun, located at a major crossroad two days' ride from Rhukaan Draal.\n\n**Geth:** A shifter veteran of the Last War, rediscovering his worth after fleeing from deeds done in the past. He wields a great gauntlet, a magewrought gauntlet that is both shield and weapon, and the ancient Dhakaani blade named Wrath.\n\n**_ghaal:_** Goblin for \"mighty,\" with specific connotations of prowess in battle.\n\n**Ghaal River:** A mighty river in central Darguun. It is navigable from its mouth at Kraken Bay to the city of Rhukaan Draal, almost two-thirds of its length. Above Rhukaan Draal, the first of several cataracts breaks the river into dangerous stretches of white water.\n\n**ghaal'dar:** The ancient name for the hobgoblin race, it means \"mighty people.\" In the present time, Ghaal'dar is also the name of the loose confederacy of goblin clans living in the lowlands of Darguun, especially in the broad area around the Ghaal River. Notable Ghaal'dar clans include the Gan'duur (\"Eaters of Sorrow\"), the Gantii Vus (\"Hungry Flame\"), the Ja'aram (\"Bright Anger\"), the Mur Talaan (\"Horned Shoulders\"), the Rhukaan Taash (\"Razor Crown\"), the Marhaan, the Ghaal Sehn, and the Pin Galaac.\n\n**goblin:** A term that causes much confusion as it applies both to the small-statured goblin race and to the three related races of goblins, hobgoblins, and bugbears (as well as other less well-known races). The longstanding tradition of referring to the related races by the term \"goblinoid\" has been abandoned by forward-thinking scholars of Wynarn and Morgrave Universities, an attitude that is slowly spreading among the general population.\n\n**_golin:_** Goblin for \"quick.\" Among hobgoblins and bugbears, it refers only to speed, but goblins use it to refer to intelligence as well.\n\n**_golin'dar:_** The ancient name for the goblin race, it means \"quick people.\"\n\n**_guul:_** Goblin for \"strong.\"\n\n**_guul'dar:_** The ancient name for the bugbear race, it means \"strong people.\"\n\n**Guulen:** See Rod of Kings.\n\n**Haruuc:** Properly Lhesh Haruuc Shaarat'kor (\"High Warlord Haruuc of the Red Blade\"), founder of Darguun. As a charismatic young warlord serving as a mercenary of House Deneith, Haruuc saw an opportunity to bring the goblin races out from under the shadow of humans and give them a homeland. Finding allies in other warlords of the Ghaal'dar, Haruuc hatched a plan to turn on the nations that employed Deneith's goblin mercenaries and seize power for himself. His plan culminated in a lightning-swift conquest of southern Cyre (including areas claimed at the time by Breland) in the summer of 969 YK and the subsequent founding of Darguun. Assassinated on 8 Sypheros, 999 YK.\n\n**Iizan:** The hobgoblin warlord of the wealthy Ghaal Sehn clan. A contender for the throne of Darguun upon Haruuc's death.\n\n**_Itaa!:_** A Goblin war-command equivalent to \"Attack!\"\n\n**Karrlakton:** A city in Karrnath, ancient center of power of House Deneith. Warlords manifesting the Mark of Sentinel ruled in the area before the founding of Karrnath.\n\n**Karrnath:** One of the original Five Nations of Galifar. Karrnath is a cold, grim land whose people are renowned for their martial prowess.\n\n**Kech Volaar:** One of the smallest, but most influential of the Dhakaani Clans, the Kech Volaar devotes itself to gathering and preserving the history, knowledge, and artifacts of the Empire of Dhakaan. Duur'kala form a powerful class among them and their stronghold of Volaar Draal is known to contain deep vaults filled with the wonders of ages past.\n\n**Keraal:** Former warlord of the Gan'duur. Once a rebel against Haruuc's rule, he was defeated by Dagii of Mur Talaan.\n\n**Khaar Mbar'ost:** Lhesh Haruuc's fortress in the heart of Rhukaan Draal, constructed for him by House Cannith. Its name means literally \"blood-colored fortified dwelling,\" an allusion to the reddish stone facing used in its construction.\n\n**_Khaavolaar!:_** A Goblin curse of frustration or amazement. It is a contraction of \"Khaar volaar\" or \"blood of the word.\"\n\n**Ko:** A changeling who attempted to kidnap Vounn d'Deneith by taking the place of one of her guards.\n\n**Koolt:** A dynasty of the early Dhakaani Empire.\n\n**korluaat:** A highly alcoholic beverage favored by the hobgoblin warriors of Darguun (literally \"hero's blood\").\n\n**Krakuul:** A hobgoblin warrior in the service of Haruuc, assigned to guard Vounn and Ashi d'Deneith.\n\n**Kuun:** Surname of a line of heroes of the Dhakaani Empire, first carried by the warlord Duulan, a friend of Taruuzh. Tales of the name of Kuun are closely tied to Aram, the legendary Sword of Heroes.\n\n**_lhesh:_** Goblin for \"high warlord.\" In the time of the Dhakaani Empire, a lhesh was a general given command of the empire's armies for a set period of time. In modern times, Haruuc has adopted the term as the title of the ruler of Darguun.\n\n**_lhevk'mor:_** \"Adjunct warlord,\" a temporary rank assigned to warriors acting under the authority of a warlord in Darguun's army.\n\n**_lhevk'rhu:_** \"Skilled warlord,\" the third highest formal rank in Darguun's army. A lhevk-rhu is outranked only by a lhevket (\"elder warlord\") or the lhesh.\n\n**_lhikor:_** \"Warblade,\" a warrior commanding a squad in Darguun's army.\n\n**_lhurusk:_** \"War-leader,\" a senior warrior commanding a platoon in Darguun's army.\n\n**Library of Korranberg, the:** The greatest repository of knowledge in Khorvaire and a central institution of life in Zilargo\u2014some historians maintain that Zilargo's unique form of government and internal security were modelled on the organization of the Library.\n\n**_Maabet!:_** An extremely ancient Goblin curse word still in use today. There is no known translation.\n\n**magebred:** Any animal magically bred for enhanced characteristics (greater speed and endurance, for example), usually by House Vadalis.\n\n**Makka:** Formerly the chief a tribe of bugbears living in the southern Seawall Mountains, now a rogue hunter looking for revenge.\n\n**Marguul:** The bugbear tribes that dwell in the highlands of the Seawall Mountains in the west of Darguun. Paying only lip service at best to Haruuc's authority, they live in independent, often hostile tribes.\n\n**Marrow:** A worg from a pack allied with the taarka'khesh, but assisting Chetiin. Worgs resemble wolves the size of ponies, but are far more intelligent. They speak their own language and can understand others.\n\n**_mazo:_** Goblin affirmative, stronger than \"yes\" and used specifically when discussing plans or acknowledging orders.\n\n**Mournland, the:** A common name for the wasteland that was once Cyre, devastated by the unnatural disaster known as the Mourning. The borders of the Mournland are marked by dense banks of gray mist, behind which lurk dangerous monsters and phenomena. The Mournland forms more than half of Darguun's eastern border.\n\n**Munta the Gray:** The aged hobgoblin warlord of the powerful Gantii Vus clan.\n\n**_muu'kron:_** An ancient goblin symbol of the Dark Six consisting of six strings knotted together and bearing small talismans representing the dark gods. Philosophers note that when spread out flat, the muu'kron form the Hexagram that is the more common symbol of the Dark Six.\n\n**_muut:_** The goblin concept of ordinary honor or duty, something is gained by doing one's job properly. Compare with atcha.\n\n**_noon:_** A starchy grain, most often pressed into compact balls, that is a staple of the traditional goblin diet and that still forms an important part of goblin cuisine.\n\n**Orien, House:** A dragonmarked house bearing the Mark of Passage. House Orien operates services related to travel, shipping and communication, including a postal service and the Lightning Rail. Orien also maintains the network of trade roads across Khorvaire.\n\n**_Paatcha!:_** An offer of honor through admiration, spoken as a compliment or delivered as an imperative to troops. Literally \"to offer honor.\"\n\n**Pradoor:** An elderly goblin priestess of the Dark Six. Pradoor is blind but claims to be guided by the Six. Her patron god is the Devourer.\n\n**_Raat shan gath'kal dor:_** \"The story stops but never ends.\" The traditional closing of hobgoblin legends.\n\n**_Raat shi anaa:_** \"The story continues.\" The traditional opening to hobgoblin legends.\n\n**Razu:** An old hobgobin woman, the mistress of rituals in Khaar Mbar'ost.\n\n**Rhukaan Draal:** The capital city of Darguun (literally \"Crown city\"), founded by Lhesh Haruuc on the site of a former Cyran frontier town to serve as a neutral territory where all Ghaal'dar clans would be equal. Over the course of thirty years, it has grown into a rough and sprawling city with a population of approximately 80,000.\n\n**Rhukaan Taash:** \"Razor Crown,\" the largest and most powerful clan of the Ghaal'dar. Warriors of Rhukaan Taash receive a series of ritual scars across their brow as a sign of adulthood. Tariic inherited the position of warlord of Rhukaan Taash on Haruuc's death.\n\n**Riis:** a dynasty of the late Dhakaani Empire.\n\n**Riis Shaarii'mal:** The Three Tearing Wheels, an ancient battle standard of Dhakaan depicting three notched black rings on a red background.\n\n**Rod of Kings, the:** Properly known by its Goblin name, Guulen, the Rod of Kings was forged by the legendary Dhakaani dashoor Taruuzh during the Daelkyr War and presented to the emperors of Dhakaan. Lost for millennia, it was recovered to serve as a symbol of rulership in Darguun, but bears both a terrible curse and tremendous power.\n\n**_ruuska'te:_** A goblin term of affection, literally \"tiger man.\"\n\n**_saa:_** A casual Goblin greeting. A more formal greeting is saa'atcha, roughly equivalent to \"It is an honor to meet you.\"\n\n**Seawall Mountains:** Mountains forming the western and northwestern borders of Darguun, a natural barrier between Darguun and Breland.\n\n**Senen Dhakaan:** The ambassador of the Kech Volaar to Khaar Mbar'ost, a duur'kala, and Ekhaas's superior.\n\n**Sentinel Tower:** The primary stronghold of House Deneith, a massive keep in the city of Karrlakton.\n\n**_shaarat'khesh:_** The Silent Blades. See: Silent Clans, the.\n\n**_shaat'aar:_** A small sweet bun filled with honey cream. A common Goblin sweet.\n\n**_shava:_** One of the most honored goblin traditions, shava is best translated as \"sword brother,\" someone a warrior trusts with his life. The relationship between shava is exceptionally strong and carries with it significant responsibilities and expectations. Most goblin warriors never even consider taking a shava.\n\n**Shadow Marches, the:** A distant wilderness of marshes, home to tribes of orcs and savage humans.\n\n**Silent Clans, the:** Although technically numbered among the Dhakaani clans, the two Silent Clans stand apart. They are formed entirely of goblins and are renowned for their stealth: the taarka'khesh (\"silent wolves\") are scouts, while the shaarat'khesh (\"silent blades\") are spies and assassins. By ancient tradition, the Silent Clans do not take sides in any conflict, instead acting as mercenaries of complete impartiality and reliability.\n\n**Six Kings, the:** An allusion to the six hobgoblin warlords brought together by Jhazaal Dhakaan approximately 17,000 years before the present to found the Empire of Dhakaan.\n\n**Sovereign Host, the:** A religion found across much of Khorvaire and actively promoted in Darguun by Haruuc as a civilizing influence. The Lords of the Host are Arawai (god of agriculture), Aureon (god of law and knowledge), Balinor (god of beasts and the hunt), Boldrei (god of community and hearth), Dol Arrah (god of honor and sacrifice), Dol Dorn (god of strength at arms), Kol Korran (god of trade and wealth), Olladra (god of good fortune), and Onatar (god of artifice and the forge).\n\n**_ta muut:_** The most basic way of saying \"thank you\" in Goblin, ta muut literally means \"you have honor\" or more accurately \"you do your duty.\" Spoken as acknowledgement of a deed performed correctly, it carries no connotation of debt on the part of the speaker. For comparison, see Ya panozhii kita atcha.\n\n**_taarka'khesh:_** The Silent Wolves. See: Silent Clans, the.\n\n**_taarka'nu:_** A goblin term of affection, literally \"wolf woman.\"\n\n**_taat:_** Goblin term for someone of significantly lesser status than the speaker. Derogatory and insulting.\n\n**Tariic:** A hobgoblin warrior of the Rhukaan Taash clan and nephew of Haruuc. He has a more worldly approach to culture, politics, and economy than more conservative Darguuls. A claimant to the throne after his uncle's assassination.\n\n**Taruuzh:** A legendary Dhakaani daashor, creator of the original binding stones, the grieving trees, and the sword Wrath. His stronghold, Taruuzh Kraat, was located where Tzaryan Keep now stands in Droaam.\n\n**tying of hair:** A custom among goblin mothers, particularly hobgoblins, of tying distinctive knots in their children's hair as a luck charm\u2014not, as suggested by scholars of the Library of Korranberg, as a means to distinguish them among crowds of goblin children.\n\n**To hold a sword by the blade:** A goblin expression for being in a risky situation.\n\n**_Toh!:_** A Goblin warning cry (\"Beware!\").\n\n**Torlaac River:** A powerful river of southern Darguun. South of the river, the land rises to become the desolate and dangerous Torlaac Moor.\n\n**Treaty of Thronehold:** Signed in 996 YK, the Treaty of Thronehold marked the end of the Last War. Among its many articles, it formally recognized a number of new nations (including Darguun, Zilargo, and Valenar) and bound all signing nations in a pact of peace.\n\n**Valaes Tairn** : The proper term for the warrior elves of Valenar.\n\n**Valenar:** The lands claimed by the Valenar elves as their rightful territory by virtue of ancient ties to the land dating back to conflicts with the Empire of Dhakaan. Recognized as an independent nation by the Treaty of Thronehold.\n\n**Vanii:** A hobgoblin warrior of the Ja'aram clan and the last of Haruuc's three shava, slain in battle against the rebellious Gan'duur clan.\n\n**Wrath:** Properly known by its Goblin name, Aram, Wrath is the legendary Sword of Heroes. Forged from byeshk by the legendary Dhakaani dashoor Taruuzh during the Daelkyr War, it was wielded by the hobgoblin heroes of the line of Kuun before being lost in the slaying of the daelkyr lord of Jhegesh Dol. Legend says that Wrath will not bear the touch of a coward. Wrath was recovered from the ghostly fortress of Jhegesh Dol by Geth, who now carries the blade.\n\n**_Ya panozhii kita atcha:_** The most formal form of thanks in Goblin. Translated, it means \"I owe a debt to your honor.\" It shows a deep indebtedness on the part of the speaker and is never used casually. Compare with ta muut.\n\n**Zilargo:** The homeland of the gnomes, Zilargo has long pursued a diplomatic policy of neutrality and conciliation. Recognized as an independent nation by the Treaty of Thronehold.\n\n# ABOUT THE AUTHOR\n\nDon Bassingthwaite is the author of numerous fantasy and dark fantasy novels. His most recent book is the first book in the Legacy of Dhakaan, _The Doom of Kings_. It continues the tale of Geth, Ashi, and Ekhaas, first introduced in The Dragon Below trilogy: _The Binding Stone, The Grieving Tree_ , and _The Killing Song_.\n\nDon lives in Toronto, surrounded by gadgets, spice jars, and too many books. You can find him online at **www.dbassingthwaite.com**.\n\nLegacy of Dhakaan, Book 2  \n **WORD OF TRAITORS**  \n\u00a92009 Wizards of the Coast LLC\n\nAll characters in this book are fictitious. Any resemblance to actual persons, living or dead, is purely coincidental.\n\nThis book is protected under the copyright laws of the United States of America. Any reproduction or unauthorized use of the material or artwork contained herein is prohibited without the express written permission of Wizards of the Coast LLC.\n\nPublished by Wizards of the Coast LLC EBERRON, WIZARDS OF THE COAST, and their respective logos are trademarks of Wizards of the Coast LLC in the U.S.A. and other countries.\n\nMap by Rob Lazzaretti\n\neISBN: 978-0-7869-5652-4\n\nU.S., CANADA, | EUROPEAN HEADQUARTERS  \n---|---  \nASIA, PACIFIC, & LATIN AMERICA  | Hasbro UK Ltd  \nWizards of the Coast LLC | Caswell Way  \nP.O. Box 707 | Newport, Gwent NP9 0YH  \nRenton, WA 98057-0707 | GREAT BRITAIN  \n+1-800-324-6496 | Save this address for your records.\n\nVisit our web site at www.wizards.com\n\nv3.0_r1\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Paul J. Hollander.  HTML version by Al Haines.\n\n\n\n\n\n\n\n\n\nTHE PARENTICIDE CLUB\n\n\nby\n\nAmbrose Bierce\n\n\n\n\nCONTENTS\n\n\nMy Favorite Murder\nOil of Dog\nAn Imperfect Conflagration\nThe Hypnotist\n\n\n\n\nMY FAVORITE MURDER\n\n\nHaving murdered my mother under circumstances of singular atrocity, I\nwas arrested and put upon my trial, which lasted seven years.  In\ncharging the jury, the judge of the Court of Acquittal remarked that\nit was one of the most ghastly crimes that he had ever been called\nupon to explain away.\n\nAt this, my attorney rose and said:\n\n\"May it please your Honor, crimes are ghastly or agreeable only by\ncomparison.  If you were familiar with the details of my client's\nprevious murder of his uncle you would discern in his later offense\n(if offense it may be called) something in the nature of tender\nforbearance and filial consideration for the feelings of the victim.\nThe appalling ferocity of the former assassination was indeed\ninconsistent with any hypothesis but that of guilt; and had it not\nbeen for the fact that the honorable judge before whom he was tried\nwas the president of a life insurance company that took risks on\nhanging, and in which my client held a policy, it is hard to see how\nhe could decently have been acquitted.  If your Honor would like to\nhear about it for instruction and guidance of your Honor's mind, this\nunfortunate man, my client, will consent to give himself the pain of\nrelating it under oath.\"\n\nThe district attorney said: \"Your Honor, I object.  Such a statement\nwould be in the nature of evidence, and the testimony in this case is\nclosed.  The prisoner's statement should have been introduced three\nyears ago, in the spring of 1881.\"\n\n\"In a statutory sense,\" said the judge, \"you are right, and in the\nCourt of Objections and Technicalities you would get a ruling in your\nfavor.  But not in a Court of Acquittal.  The objection is overruled.\"\n\n\"I except,\" said the district attorney.\n\n\"You cannot do that,\" the judge said.  \"I must remind you that in\norder to take an exception you must first get this case transferred\nfor a time to the Court of Exceptions on a formal motion duly\nsupported by affidavits.  A motion to that effect by your predecessor\nin office was denied by me during the first year of this trial.  Mr.\nClerk, swear the prisoner.\"\n\nThe customary oath having been administered, I made the following\nstatement, which impressed the judge with so strong a sense of the\ncomparative triviality of the offense for which I was on trial that he\nmade no further search for mitigating circumstances, but simply\ninstructed the jury to acquit, and I left the court, without a stain\nupon my reputation:\n\n\"I was born in 1856 in Kalamakee, Mich., of honest and reputable\nparents, one of whom Heaven has mercifully spared to comfort me in my\nlater years.  In 1867 the family came to California and settled near\n<DW65> Head, where my father opened a road agency and prospered beyond\nthe dreams of avarice.  He was a reticent, saturnine man then, though\nhis increasing years have now somewhat relaxed the austerity of his\ndisposition, and I believe that nothing but his memory of the sad\nevent for which I am now on trial prevents him from manifesting a\ngenuine hilarity.\n\n\"Four years after we had set up the road agency an itinerant preacher\ncame along, and having no other way to pay for the night's lodging\nthat we gave him, favored us with an exhortation of such power that,\npraise God, we were all converted to religion.  My father at once sent\nfor his brother, the Hon. William Ridley of Stockton, and on his\narrival turned over the agency to him, charging him nothing for the\nfranchise nor plant--the latter consisting of a Winchester rifle, a\nsawed-off shotgun, and an assortment of masks made out of flour sacks.\nThe family then moved to Ghost Rock and opened a dance house.  It was\ncalled 'The Saints' Rest Hurdy-Gurdy,' and the proceedings each night\nbegan with prayer.  It was there that my now sainted mother, by her\ngrace in the dance, acquired the _sobriquet_ of 'The Bucking Walrus.'\n\n\"In the fall of '75 I had occasion to visit Coyote, on the road to\nMahala, and took the stage at Ghost Rock.  There were four other\npassengers.  About three miles beyond <DW65> Head, persons whom I\nidentified as my Uncle William and his two sons held up the stage.\nFinding nothing in the express box, they went through the passengers.\nI acted a most honorable part in the affair, placing myself in line\nwith the others, holding up my hands and permitting myself to be\ndeprived of forty dollars and a gold watch.  From my behavior no one\ncould have suspected that I knew the gentlemen who gave the\nentertainment.  A few days later, when I went to <DW65> Head and asked\nfor the return of my money and watch my uncle and cousins swore they\nknew nothing of the matter, and they affected a belief that my father\nand I had done the job ourselves in dishonest violation of commercial\ngood faith.  Uncle William even threatened to retaliate by starting an\nopposition dance house at Ghost Rock.  As 'The Saints' Rest' had\nbecome rather unpopular, I saw that this would assuredly ruin it and\nprove a paying enterprise, so I told my uncle that I was willing to\noverlook the past if he would take me into the scheme and keep the\npartnership a secret from my father.  This fair offer he rejected, and\nI then perceived that it would be better and more satisfactory if he\nwere dead.\n\n\"My plans to that end were soon perfected, and communicating them to\nmy dear parents I had the gratification of receiving their approval.\nMy father said he was proud of me, and my mother promised that\nalthough her religion forbade her to assist in taking human life I\nshould have the advantage of her prayers for my success.  As a\npreliminary measure looking to my security in case of detection I made\nan application for membership in that powerful order, the Knights of\nMurder, and in due course was received as a member of the Ghost Rock\ncommandery.  On the day that my probation ended I was for the first\ntime permitted to inspect the records of the order and learn who\nbelonged to it--all the rites of initiation having been conducted in\nmasks.  Fancy my delight when, in looking over the roll of membership,\nI found the third name to be that of my uncle, who indeed was junior\nvice-chancellor of the order!  Here was an opportunity exceeding my\nwildest dreams--to murder I could add insubordination and treachery.\nIt was what my good mother would have called 'a special Providence.'\n\n\"At about this time something occurred which caused my cup of joy,\nalready full, to overflow on all sides, a circular cataract of bliss.\nThree men, strangers in that locality, were arrested for the stage\nrobbery in which I had lost my money and watch.  They were brought to\ntrial and, despite my efforts to clear them and fasten the guilt upon\nthree of the most respectable and worthy citizens of Ghost Rock,\nconvicted on the clearest proof.  The murder would now be as wanton\nand reasonless as I could wish.\n\n\"One morning I shouldered my Winchester rifle, and going over to my\nuncle's house, near <DW65> Head, asked my Aunt Mary, his wife, if he\nwere at home, adding that I had come to kill him.  My aunt replied\nwith her peculiar smile that so many gentlemen called on that errand\nand were afterward carried away without having performed it that I\nmust excuse her for doubting my good faith in the matter.  She said I\ndid not look as if I would kill anybody, so, as a proof of good faith\nI leveled my rifle and wounded a Chinaman who happened to be passing\nthe house.  She said she knew whole families that could do a thing of\nthat kind, but Bill Ridley was a horse of another color.  She said,\nhowever, that I would find him over on the other side of the creek in\nthe sheep lot; and she added that she hoped the best man would win.\n\n\"My Aunt Mary was one of the most fair-minded women that I have ever\nmet.\n\n\"I found my uncle down on his knees engaged in skinning a sheep.\nSeeing that he had neither gun nor pistol handy I had not the heart to\nshoot him, so I approached him, greeted him pleasantly and struck him\na powerful blow on the head with the butt of my rifle.  I have a very\ngood delivery and Uncle William lay down on his side, then rolled over\non his back, spread out his fingers and shivered.  Before he could\nrecover the use of his limbs I seized the knife that he had been using\nand cut his hamstrings.  You know, doubtless, that when you sever the\n_tendo Achillis_ the patient has no further use of his leg; it is just\nthe same as if he had no leg.  Well, I parted them both, and when he\nrevived he was at my service.  As soon as he comprehended the\nsituation, he said:\n\n\"'Samuel, you have got the drop on me and can afford to be generous.\nI have only one thing to ask of you, and that is that you carry me to\nthe house and finish me in the bosom of my family.'\n\n\"I told him I thought that a pretty reasonable request and I would do\nso if he would let me put him into a wheat sack; he would be easier to\ncarry that way and if we were seen by the neighbors _en route_ it\nwould cause less remark.  He agreed to that, and going to the barn I\ngot a sack.  This, however, did not fit him; it was too short and much\nwider than he; so I bent his legs, forced his knees up against his\nbreast and got him into it that way, tying the sack above his head.\nHe was a heavy man and I had all that I could do to get him on my\nback, but I staggered along for some distance until I came to a swing\nthat some of the children had suspended to the branch of an oak.  Here\nI laid him down and sat upon him to rest, and the sight of the rope\ngave me a happy inspiration.  In twenty minutes my uncle, still in the\nsack, swung free to the sport of the wind.\n\n\"I had taken down the rope, tied one end tightly about the mouth of\nthe bag, thrown the other across the limb and hauled him up about five\nfeet from the ground.  Fastening the other end of the rope also about\nthe mouth of the sack, I had the satisfaction to see my uncle\nconverted into a large, fine pendulum.  I must add that he was not\nhimself entirely aware of the nature of the change that he had\nundergone in his relation to the exterior world, though in justice to\na good man's memory I ought to say that I do not think he would in any\ncase have wasted much of my time in vain remonstrance.\n\n\"Uncle William had a ram that was famous in all that region as a\nfighter.  It was in a state of chronic constitutional indignation.\nSome deep disappointment in early life had soured its disposition and\nit had declared war upon the whole world.  To say that it would butt\nanything accessible is but faintly to express the nature and scope of\nits military activity: the universe was its antagonist; its methods\nthat of a projectile.  It fought like the angels and devils, in\nmid-air, cleaving the atmosphere like a bird, describing a parabolic\ncurve and descending upon its victim at just the exact angle of\nincidence to make the most of its velocity and weight.  Its momentum,\ncalculated in foot-tons, was something incredible.  It had been seen\nto destroy a four year old bull by a single impact upon that animal's\ngnarly forehead.  No stone wall had ever been known to resist its\ndownward swoop; there were no trees tough enough to stay it; it would\nsplinter them into matchwood and defile their leafy honors in the\ndust.  This irascible and implacable brute--this incarnate\nthunderbolt--this monster of the upper deep, I had seen reposing in\nthe shade of an adjacent tree, dreaming dreams of conquest and glory.\nIt was with a view to summoning it forth to the field of honor that I\nsuspended its master in the manner described.\n\n\"Having completed my preparations, I imparted to the avuncular\npendulum a gentle oscillation, and retiring to cover behind a\ncontiguous rock, lifted up my voice in a long rasping cry whose\ndiminishing final note was drowned in a noise like that of a swearing\ncat, which emanated from the sack.  Instantly that formidable sheep\nwas upon its feet and had taken in the military situation at a glance.\nIn a few moments it had approached, stamping, to within fifty yards\nof the swinging foeman, who, now retreating and anon advancing, seemed\nto invite the fray.  Suddenly I saw the beast's head drop earthward as\nif depressed by the weight of its enormous horns; then a dim, white,\nwavy streak of sheep prolonged itself from that spot in a generally\nhorizontal direction to within about four yards of a point immediately\nbeneath the enemy.  There it struck sharply upward, and before it had\nfaded from my gaze at the place whence it had set out I heard a horrid\nthump and a piercing scream, and my poor uncle shot forward, with a\nslack rope higher than the limb to which he was attached.  Here the\nrope tautened with a jerk, arresting his flight, and back he swung in\na breathless curve to the other end of his arc.  The ram had fallen, a\nheap of indistinguishable legs, wool and horns, but pulling itself\ntogether and dodging as its antagonist swept downward it retired at\nrandom, alternately shaking its head and stamping its fore-feet.  When\nit had backed about the same distance as that from which it had\ndelivered the assault it paused again, bowed its head as if in prayer\nfor victory and again shot forward, dimly visible as before--a\nprolonging white streak with monstrous undulations, ending with a\nsharp ascension.  Its course this time was at a right angle to its\nformer one, and its impatience so great that it struck the enemy\nbefore he had nearly reached the lowest point of his arc.  In\nconsequence he went flying round and round in a horizontal circle\nwhose radius was about equal to half the length of the rope, which I\nforgot to say was nearly twenty feet long.  His shrieks, _crescendo_\nin approach and _diminuendo_ in recession, made the rapidity of his\nrevolution more obvious to the ear than to the eye.  He had evidently\nnot yet been struck in a vital spot.  His posture in the sack and the\ndistance from the ground at which he hung compelled the ram to operate\nupon his lower extremities and the end of his back.  Like a plant that\nhas struck its root into some poisonous mineral, my poor uncle was\ndying slowly upward.\n\n\"After delivering its second blow the ram had not again retired.  The\nfever of battle burned hot in its heart; its brain was intoxicated\nwith the wine of strife.  Like a pugilist who in his rage forgets his\nskill and fights ineffectively at half-arm's length, the angry beast\nendeavored to reach its fleeting foe by awkward vertical leaps as he\npassed overhead, sometimes, indeed, succeeding in striking him feebly,\nbut more frequently overthrown by its own misguided eagerness.  But as\nthe impetus was exhausted and the man's circles narrowed in scope and\ndiminished in speed, bringing him nearer to the ground, these tactics\nproduced better results, eliciting a superior quality of screams,\nwhich I greatly enjoyed.\n\n\"Suddenly, as if the bugles had sung truce, the ram suspended\nhostilities and walked away, thoughtfully wrinkling and smoothing its\ngreat aquiline nose, and occasionally cropping a bunch of grass and\nslowly munching it.  It seemed to have tired of war's alarms and\nresolved to beat the sword into a plowshare and cultivate the arts of\npeace.  Steadily it held its course away from the field of fame until\nit had gained a distance of nearly a quarter of a mile.  There it\nstopped and stood with its rear to the foe, chewing its cud and\napparently half asleep.  I observed, however, an occasional slight\nturn of its head, as if its apathy were more affected than real.\n\n\"Meantime Uncle William's shrieks had abated with his motion, and\nnothing was heard from him but long, low moans, and at long intervals\nmy name, uttered in pleading tones exceedingly grateful to my ear.\nEvidently the man had not the faintest notion of what was being done\nto him, and was inexpressibly terrified.  When Death comes cloaked in\nmystery he is terrible indeed.  Little by little my uncle's\noscillations diminished, and finally he hung motionless.  I went to\nhim and was about to give him the _coup de grace_, when I heard and\nfelt a succession of smart shocks which shook the ground like a series\nof light earthquakes, and turning in the direction of the ram, saw a\nlong cloud of dust approaching me with inconceivable rapidity and\nalarming effect!  At a distance of some thirty yards away it stopped\nshort, and from the near end of it rose into the air what I at first\nthought a great white bird.  Its ascent was so smooth and easy and\nregular that I could not realize its extraordinary celerity, and was\nlost in admiration of its grace.  To this day the impression remains\nthat it was a slow, deliberate movement, the ram--for it was that\nanimal--being upborne by some power other than its own impetus, and\nsupported through the successive stages of its flight with infinite\ntenderness and care.  My eyes followed its progress through the air\nwith unspeakable pleasure, all the greater by contrast with my former\nterror of its approach by land.  Onward and upward the noble animal\nsailed, its head bent down almost between its knees, its fore-feet\nthrown back, its hinder legs trailing to rear like the legs of a\nsoaring heron.\n\n\"At a height of forty or fifty feet, as fond recollection presents it\nto view, it attained its zenith and appeared to remain an instant\nstationary; then, tilting suddenly forward without altering the\nrelative position of its parts, it shot downward on a steeper and\nsteeper course with augmenting velocity, passed immediately above me\nwith a noise like the rush of a cannon shot and struck my poor uncle\nalmost squarely on the top of the head!  So frightful was the impact\nthat not only the man's neck was broken, but the rope too; and the\nbody of the deceased, forced against the earth, was crushed to pulp\nbeneath the awful front of that meteoric sheep!  The concussion\nstopped all the clocks between Lone Hand and Dutch Dan's, and\nProfessor Davidson, a distinguished authority in matters seismic, who\nhappened to be in the vicinity, promptly explained that the vibrations\nwere from north to southwest.\n\n\"Altogether, I cannot help thinking that in point of artistic atrocity\nmy murder of Uncle William has seldom been excelled.\"\n\n\n\n\nOIL OF DOG\n\n\nMy name is Boffer Bings.  I was born of honest parents in one of the\nhumbler walks of life, my father being a manufacturer of dog-oil and\nmy mother having a small studio in the shadow of the village church,\nwhere she disposed of unwelcome babes.  In my boyhood I was trained to\nhabits of industry; I not only assisted my father in procuring dogs\nfor his vats, but was frequently employed by my mother to carry away\nthe debris of her work in the studio.  In performance of this duty I\nsometimes had need of all my natural intelligence for all the law\nofficers of the vicinity were opposed to my mother's business.  They\nwere not elected on an opposition ticket, and the matter had never\nbeen made a political issue; it just happened so.  My father's\nbusiness of making dog-oil was, naturally, less unpopular, though the\nowners of missing dogs sometimes regarded him with suspicion, which\nwas reflected, to some extent, upon me.  My father had, as silent\npartners, all the physicians of the town, who seldom wrote a\nprescription which did not contain what they were pleased to designate\nas _Ol. can._  It is really the most valuable medicine ever\ndiscovered.  But most persons are unwilling to make personal\nsacrifices for the afflicted, and it was evident that many of the\nfattest dogs in town had been forbidden to play with me--a fact which\npained my young sensibilities, and at one time came near driving me to\nbecome a pirate.\n\nLooking back upon those days, I cannot but regret, at times, that by\nindirectly bringing my beloved parents to their death I was the author\nof misfortunes profoundly affecting my future.\n\nOne evening while passing my father's oil factory with the body of a\nfoundling from my mother's studio I saw a constable who seemed to be\nclosely watching my movements.  Young as I was, I had learned that a\nconstable's acts, of whatever apparent character, are prompted by the\nmost reprehensible motives, and I avoided him by dodging into the\noilery by a side door which happened to stand ajar.  I locked it at\nonce and was alone with my dead.  My father had retired for the night.\nThe only light in the place came from the furnace, which glowed a\ndeep, rich crimson under one of the vats, casting ruddy reflections on\nthe walls.  Within the cauldron the oil still rolled in indolent\nebullition, occasionally pushing to the surface a piece of dog.\nSeating myself to wait for the constable to go away, I held the naked\nbody of the foundling in my lap and tenderly stroked its short, silken\nhair.  Ah, how beautiful it was!  Even at that early age I was\npassionately fond of children, and as I looked upon this cherub I\ncould almost find it in my heart to wish that the small, red wound\nupon its breast--the work of my dear mother--had not been mortal.\n\nIt had been my custom to throw the babes into the river which nature\nhad thoughtfully provided for the purpose, but that night I did not\ndare to leave the oilery for fear of the constable.  \"After all,\" I\nsaid to myself, \"it cannot greatly matter if I put it into this\ncauldron.  My father will never know the bones from those of a puppy,\nand the few deaths which may result from administering another kind of\noil for the incomparable _ol. can._ are not important in a population\nwhich increases so rapidly.\"  In short, I took the first step in crime\nand brought myself untold sorrow by casting the babe into the\ncauldron.\n\nThe next day, somewhat to my surprise, my father, rubbing his hands\nwith satisfaction, informed me and my mother that he had obtained the\nfinest quality of oil that was ever seen; that the physicians to whom\nhe had shown samples had so pronounced it.  He added that he had no\nknowledge as to how the result was obtained; the dogs had been treated\nin all respects as usual, and were of an ordinary breed.  I deemed it\nmy duty to explain--which I did, though palsied would have been my\ntongue if I could have foreseen the consequences.  Bewailing their\nprevious ignorance of the advantages of combining their industries, my\nparents at once took measures to repair the error.  My mother removed\nher studio to a wing of the factory building and my duties in\nconnection with the business ceased; I was no longer required to\ndispose of the bodies of the small superfluous, and there was no need\nof alluring dogs to their doom, for my father discarded them\naltogether, though they still had an honorable place in the name of\nthe oil.  So suddenly thrown into idleness, I might naturally have\nbeen expected to become vicious and dissolute, but I did not.  The\nholy influence of my dear mother was ever about me to protect me from\nthe temptations which beset youth, and my father was a deacon in a\nchurch.  Alas, that through my fault these estimable persons should\nhave come to so bad an end!\n\nFinding a double profit in her business, my mother now devoted herself\nto it with a new assiduity.  She removed not only superfluous and\nunwelcome babes to order, but went out into the highways and byways,\ngathering in children of a larger growth, and even such adults as she\ncould entice to the oilery.  My father, too, enamored of the superior\nquality of oil produced, purveyed for his vats with diligence and\nzeal.  The conversion of their neighbors into dog-oil became, in\nshort, the one passion of their lives--an absorbing and overwhelming\ngreed took possession of their souls and served them in place of a\nhope in Heaven--by which, also, they were inspired.\n\nSo enterprising had they now become that a public meeting was held and\nresolutions passed severely censuring them.  It was intimated by the\nchairman that any further raids upon the population would be met in a\nspirit of hostility.  My poor parents left the meeting broken-hearted,\ndesperate and, I believe, not altogether sane.  Anyhow, I deemed it\nprudent not to enter the oilery with them that night, but slept\noutside in a stable.\n\nAt about midnight some mysterious impulse caused me to rise and peer\nthrough a window into the furnace-room, where I knew my father now\nslept.  The fires were burning as brightly as if the following day's\nharvest had been expected to be abundant.  One of the large cauldrons\nwas slowly \"walloping\" with a mysterious appearance of self-restraint,\nas if it bided its time to put forth its full energy.  My father was\nnot in bed; he had risen in his night clothes and was preparing a\nnoose in a strong cord.  From the looks which he cast at the door of\nmy mother's bedroom I knew too well the purpose that he had in mind.\nSpeechless and motionless with terror, I could do nothing in\nprevention or warning.  Suddenly the door of my mother's apartment was\nopened, noiselessly, and the two confronted each other, both\napparently surprised.  The lady, also, was in her night clothes, and\nshe held in her right hand the tool of her trade, a long,\nnarrow-bladed dagger.\n\nShe, too, had been unable to deny herself the last profit which the\nunfriendly action of the citizens and my absence had left her.  For\none instant they looked into each other's blazing eyes and then sprang\ntogether with indescribable fury.  Round and round, the room they\nstruggled, the man cursing, the woman shrieking, both fighting like\ndemons--she to strike him with the dagger, he to strangle her with his\ngreat bare hands.  I know not how long I had the unhappiness to\nobserve this disagreeable instance of domestic infelicity, but at\nlast, after a more than usually vigorous struggle, the combatants\nsuddenly moved apart.\n\nMy father's breast and my mother's weapon showed evidences of contact.\nFor another instant they glared at each other in the most unamiable\nway; then my poor, wounded father, feeling the hand of death upon him,\nleaped forward, unmindful of resistance, grasped my dear mother in his\narms, dragged her to the side of the boiling cauldron, collected all\nhis failing energies, and sprang in with her!  In a moment, both had\ndisappeared and were adding their oil to that of the committee of\ncitizens who had called the day before with an invitation to the\npublic meeting.\n\nConvinced that these unhappy events closed to me every avenue to an\nhonorable career in that town, I removed to the famous city of\nOtumwee, where these memoirs are written with a heart full of remorse\nfor a heedless act entailing so dismal a commercial disaster.\n\n\n\n\nAN IMPERFECT CONFLAGRATION\n\n\nEarly one June morning in 1872 I murdered my father--an act which made\na deep impression on me at the time.  This was before my marriage,\nwhile I was living with my parents in Wisconsin.  My father and I were\nin the library of our home, dividing the proceeds of a burglary which\nwe had committed that night.  These consisted of household goods\nmostly, and the task of equitable division was difficult.  We got on\nvery well with the napkins, towels and such things, and the silverware\nwas parted pretty nearly equally, but you can see for yourself that\nwhen you try to divide a single music-box by two without a remainder\nyou will have trouble.  It was that music-box which brought disaster\nand disgrace upon our family.  If we had left it my poor father might\nnow be alive.\n\nIt was a most exquisite and beautiful piece of workmanship--inlaid\nwith costly woods and carven very curiously.  It would not only play a\ngreat variety of tunes, but would whistle like a quail, bark like a\ndog, crow every morning at daylight whether it was wound up or not,\nand break the Ten Commandments.  It was this last mentioned\naccomplishment that won my father's heart and caused him to commit the\nonly dishonorable act of his life, though possibly he would have\ncommitted more if he had been spared: he tried to conceal that\nmusic-box from me, and declared upon his honor that he had not taken\nit, though I know very well that, so far as he was concerned, the\nburglary had been undertaken chiefly for the purpose of obtaining it.\n\nMy father had the music-box hidden under his cloak; we had worn cloaks\nby way of disguise.  He had solemnly assured me that he did not take\nit.  I knew that he did, and knew something of which he was evidently\nignorant; namely, that the box would crow at daylight and betray him\nif I could prolong the division of profits till that time.  All\noccurred as I wished: as the gaslight began to pale in the library and\nthe shape of the windows was seen dimly behind the curtains, a long\ncock-a-doodle-doo came from beneath the old gentleman's cloak,\nfollowed by a few bars of an aria from _Tannhauser_, ending with a\nloud click.  A small hand-axe, which we had used to break into the\nunlucky house, lay between us on the table; I picked it up.  The old\nman seeing that further concealment was useless took the box from\nunder his cloak and set it on the table.  \"Cut it in two if you prefer\nthat plan,\" said he; \"I tried to save it from destruction.\"\n\nHe was a passionate lover of music and could himself play the\nconcertina with expression and feeling.\n\nI said: \"I do not question the purity of your motive: it would be\npresumptuous of me to sit in judgment on my father.  But business is\nbusiness, and with this axe I am going to effect a dissolution of our\npartnership unless you will consent in all future burglaries to wear a\nbell-punch.\"\n\n\"No,\" he said, after some reflection, \"no, I could not do that; it\nwould look like a confession of dishonesty.  People would say that you\ndistrusted me.\"\n\nI could not help admiring his spirit and sensitiveness; for a moment I\nwas proud of him and disposed to overlook his fault, but a glance at\nthe richly jeweled music-box decided me, and, as I said, I removed the\nold man from this vale of tears.  Having done so, I was a trifle\nuneasy.  Not only was he my father--the author of my being--but the\nbody would be certainly discovered.  It was now broad daylight and my\nmother was likely to enter the library at any moment.  Under the\ncircumstances, I thought it expedient to remove her also, which I did.\nThen I paid off all the servants and discharged them.\n\nThat afternoon I went to the chief of police, told him what I had done\nand asked his advice.  It would be very painful to me if the facts\nbecame publicly known.  My conduct would be generally condemned; the\nnewspapers would bring it up against me if ever I should run for\noffice.  The chief saw the force of these considerations; he was\nhimself an assassin of wide experience.  After consulting with the\npresiding judge of the Court of Variable Jurisdiction he advised me to\nconceal the bodies in one of the bookcases, get a heavy insurance on\nthe house and burn it down.  This I proceeded to do.\n\nIn the library was a book-case which my father had recently purchased\nof some cranky inventor and had not filled.  It was in shape and size\nsomething like the old-fashioned \"ward-robes\" which one sees in\nbed-rooms without closets, but opened all the way down, like a woman's\nnight-dress.  It had glass doors.  I had recently laid out my parents\nand they were now rigid enough to stand erect; so I stood them in this\nbook-case, from which I had removed the shelves.  I locked them in and\ntacked some curtains over the glass doors.  The inspector from the\ninsurance office passed a half-dozen times before the case without\nsuspicion.\n\nThat night, after getting my policy, I set fire to the house and\nstarted through the woods to town, two miles away, where I managed to\nbe found about the time the excitement was at its height.  With cries\nof apprehension for the fate of my parents, I joined the rush and\narrived at the fire some two hours after I had kindled it.  The whole\ntown was there as I dashed up.  The house was entirely consumed, but\nin one end of the level bed of glowing embers, bolt upright and\nuninjured, was that book-case!  The curtains had burned away, exposing\nthe glass-doors, through which the fierce, red light illuminated the\ninterior.  There stood my dear father \"in his habit as he lived,\" and\nat his side the partner of his joys and sorrows.  Not a hair of them\nwas singed, their clothing was intact.  On their heads and throats the\ninjuries which in the accomplishment of my designs I had been\ncompelled to inflict were conspicuous.  As in the presence of a\nmiracle, the people were silent; awe and terror had stilled every\ntongue.  I was myself greatly affected.\n\nSome three years later, when the events herein related had nearly\nfaded from my memory, I went to New York to assist in passing some\ncounterfeit United States bonds.  Carelessly looking into a furniture\nstore one day, I saw the exact counterpart of that book-case.  \"I\nbought it for a trifle from a reformed inventor,\" the dealer\nexplained.  \"He said it was fireproof, the pores of the wood being\nfilled with alum under hydraulic pressure and the glass made of\nasbestos.  I don't suppose it is really fireproof--you can have it at\nthe price of an ordinary book-case.\"\n\n\"No,\" I said, \"if you cannot warrant it fireproof I won't take\nit\"--and I bade him good morning.\n\nI would not have had it at any price: it revived memories that were\nexceedingly disagreeable.\n\n\n\n\nTHE HYPNOTIST\n\n\nBy those of my friends who happen to know that I sometimes amuse\nmyself with hypnotism, mind reading and kindred phenomena, I am\nfrequently asked if I have a clear conception of the nature of\nwhatever principle underlies them.  To this question I always reply\nthat I neither have nor desire to have.  I am no investigator with an\near at the key-hole of Nature's workshop, trying with vulgar curiosity\nto steal the secrets of her trade.  The interests of science are as\nlittle to me as mine seem to have been to science.\n\nDoubtless the phenomena in question are simple enough, and in no way\ntranscend our powers of comprehension if only we could find the clew;\nbut for my part I prefer not to find it, for I am of a singularly\nromantic disposition, deriving more gratification from mystery than\nfrom knowledge.  It was commonly remarked of me when I was a child\nthat my big blue eyes appeared to have been made rather to look into\nthan look out of--such was their dreamful beauty, and in my frequent\nperiods of abstraction, their indifference to what was going on.  In\nthose peculiarities they resembled, I venture to think, the soul which\nlies behind them, always more intent upon some lovely conception which\nit has created in its own image than concerned about the laws of\nnature and the material frame of things.  All this, irrelevant and\negotistic as it may seem, is related by way of accounting for the\nmeagreness of the light that I am able to throw upon a subject that\nhas engaged so much of my attention, and concerning which there is so\nkeen and general a curiosity.  With my powers and opportunities,\nanother person might doubtless have an explanation for much of what I\npresent simply as narrative.\n\nMy first knowledge that I possessed unusual powers came to me in my\nfourteenth year, when at school.  Happening one day to have forgotten\nto bring my noon-day luncheon, I gazed longingly at that of a small\ngirl who was preparing to eat hers.  Looking up, her eyes met mine and\nshe seemed unable to withdraw them.  After a moment of hesitancy she\ncame forward in an absent kind of way and without a word surrendered\nher little basket with its tempting contents and walked away.\nInexpressibly pleased, I relieved my hunger and destroyed the basket.\nAfter that I had not the trouble to bring a luncheon for myself: that\nlittle girl was my daily purveyor; and not infrequently in satisfying\nmy simple need from her frugal store I combined pleasure and profit by\nconstraining her attendance at the feast and making misleading proffer\nof the viands, which eventually I consumed to the last fragment.  The\ngirl was always persuaded that she had eaten all herself; and later in\nthe day her tearful complaints of hunger surprised the teacher,\nentertained the pupils, earned for her the sobriquet of Greedy-Gut and\nfilled me with a peace past understanding.\n\nA disagreeable feature of this otherwise satisfactory condition of\nthings was the necessary secrecy: the transfer of the luncheon, for\nexample, had to be made at some distance from the madding crowd, in a\nwood; and I blush to think of the many other unworthy subterfuges\nentailed by the situation.  As I was (and am) naturally of a frank and\nopen disposition, these became more and more irksome, and but for the\nreluctance of my parents to renounce the obvious advantages of the new\nregime I would gladly have reverted to the old.  The plan that I\nfinally adopted to free myself from the consequences of my own powers\nexcited a wide and keen interest at the time, and that part of it\nwhich consisted in the death of the girl was severely condemned, but\nit is hardly pertinent to the scope of this narrative.\n\nFor some years afterward I had little opportunity to practice\nhypnotism; such small essays as I made at it were commonly barren of\nother recognition than solitary confinement on a bread-and-water diet;\nsometimes, indeed, they elicited nothing better than the\ncat-o'-nine-tails.  It was when I was about to leave the scene of\nthese small disappointments that my one really important feat was\nperformed.\n\nI had been called into the warden's office and given a suit of\ncivilian's clothing, a trifling sum of money and a great deal of\nadvice, which I am bound to confess was of a much better quality than\nthe clothing.  As I was passing out of the gate into the light of\nfreedom I suddenly turned and looking the warden gravely in the eye,\nsoon had him in control.\n\n\"You are an ostrich,\" I said.\n\nAt the post-mortem examination the stomach was found to contain a\ngreat quantity of indigestible articles mostly of wood or metal.\nStuck fast in the esophagus and constituting, according to the\nCoroner's jury, the immediate cause of death, one door-knob.\n\nI was by nature a good and affectionate son, but as I took my way into\nthe great world from which I had been so long secluded I could not\nhelp remembering that all my misfortunes had flowed like a stream from\nthe niggard economy of my parents in the matter of school luncheons;\nand I knew of no reason to think they had reformed.\n\nOn the road between Succotash Hill and South Asphyxia is a little open\nfield which once contained a shanty known as Pete Gilstrap's Place,\nwhere that gentleman used to murder travelers for a living.  The death\nof Mr. Gilstrap and the diversion of nearly all the travel to another\nroad occurred so nearly at the same time that no one has ever been\nable to say which was cause and which effect.  Anyhow, the field was\nnow a desolation and the Place had long been burned.  It was while\ngoing afoot to South Asphyxia, the home of my childhood, that I found\nboth my parents on their way to the Hill.  They had hitched their team\nand were eating luncheon under an oak tree in the center of the field.\nThe sight of the luncheon called up painful memories of my school\ndays and roused the sleeping lion in my breast.  Approaching the\nguilty couple, who at once recognized me, I ventured to suggest that I\nshare their hospitality.\n\n\"Of this cheer, my son,\" said the author of my being, with\ncharacteristic pomposity, which age had not withered, \"there is\nsufficient for but two.  I am not, I hope, insensible to the\nhunger-light in your eyes, but--\"\n\nMy father has never completed that sentence; what he mistook for\nhunger-light was simply the earnest gaze of the hypnotist.  In a few\nseconds he was at my service.  A few more sufficed for the lady, and\nthe dictates of a just resentment could be carried into effect.  \"My\nformer father,\" I said, \"I presume that it is known to you that you\nand this lady are no longer what you were?\"\n\n\"I have observed a certain subtle change,\" was the rather dubious\nreply of the old gentleman; \"it is perhaps attributable to age.\"\n\n\"It is more than that,\" I explained; \"it goes to character--to\nspecies.  You and the lady here are, in truth, two broncos--wild\nstallions both, and unfriendly.\"\n\n\"Why, John,\" exclaimed my dear mother, \"you don't mean to say that I\nam--\"\n\n\"Madam,\" I replied, solemnly, fixing my eyes again upon hers, \"you\nare.\"\n\nScarcely had the words fallen from my lips when she dropped upon her\nhands and knees, and backing up to the old man squealed like a demon\nand delivered a vicious kick upon his shin!  An instant later he was\nhimself down on all-fours, headed away from her and flinging his feet\nat her simultaneously and successively.  With equal earnestness but\ninferior agility, because of her hampering body-gear, she plied her\nown.  Their flying legs crossed and mingled in the most bewildering\nway; their feet sometimes meeting squarely in midair, their bodies\nthrust forward, falling flat upon the ground and for a moment\nhelpless.  On recovering themselves they would resume the combat,\nuttering their frenzy in the nameless sounds of the furious brutes\nwhich they believed themselves to be--the whole region rang with their\nclamor!  Round and round they wheeled, the blows of their feet falling\n\"like lightnings from the mountain cloud.\"  They plunged and reared\nbackward upon their knees, struck savagely at each other with awkward\ndescending blows of both fists at once, and dropped again upon their\nhands as if unable to maintain the upright position of the body.\nGrass and pebbles were torn from the soil by hands and feet; clothing,\nhair, faces inexpressibly defiled with dust and blood.  Wild,\ninarticulate screams of rage attested the delivery of the blows;\ngroans, grunts and gasps their receipt.  Nothing more truly military\nwas ever seen at Gettysburg or Waterloo: the valor of my dear parents\nin the hour of danger can never cease to be to me a source of pride\nand gratification.  At the end of it all two battered, tattered,\nbloody and fragmentary vestiges of mortality attested the solemn fact\nthat the author of the strife was an orphan.\n\nArrested for provoking a breach of the peace, I was, and have ever\nsince been, tried in the Court of Technicalities and Continuances\nwhence, after fifteen years of proceedings, my attorney is moving\nheaven and earth to get the case taken to the Court of Remandment for\nNew Trials.\n\nSuch are a few of my principal experiments in the mysterious force or\nagency known as hypnotic suggestion.  Whether or not it could be\nemployed by a bad man for an unworthy purpose I am unable to say.\n\n\n\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of The Parenticide Club, by Ambrose Bierce\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n# Contents\n\n* * *\n\nTitle Page\n\nContents\n\nCopyright\n\nDedication\n\nAfter the Hurricane\n\n\u23af\n\nIn the Supply Closet at Illing Middle\n\nFree Armchair, Worcester\n\nThe First Time\n\nNatick\n\nAgain\n\nBrothers\n\nWorkbench\n\nTap Out\n\nDeciding\n\nGraduation\n\n\u23af\n\nBlue\n\n\u23af\n\nGraffiti\n\nMy Father at 49, Working the Night Shift at B&R Diesel\n\nWhen Charlie Pulls the Colorado Over\n\nV.F.W. Post #2046\n\nFranklin Free Clinic\n\nSafety\n\nClose\n\nAfter the Attempt\n\nVows\n\nIn the Sideyard\n\nDundalk\n\nMichael\n\n\u23af\n\nNext of Kin\n\n\u23af\n\nTraining\n\nGoing\n\nDry Season\n\nKabekona\n\nFarmsitting\n\nMud Season\n\nFor Good\n\nSalvage\n\nPiecework\n\nMy Father at 23, on the Highway Side of an Overpass Fence\n\nWindow Washers\n\nInterim\n\n\u23af\n\nBehind the Eyes, & Shining\n\nAcknowledgments\n\nAbout the Author\n\nConnect with HMH\nCopyright \u00a9 2019 by Edgar Kunz\n\nAll rights reserved\n\nFor information about permission to reproduce selections from this book, write to trade.permissions@hmhco.com or to Permissions, Houghton Mifflin Harcourt Publishing Company, 3 Park Avenue, 19th Floor, New York, New York 10016.\n\nhmhbooks.com\n\n_Library of Congress Cataloging-in-Publication Data_\n\nNames: Kunz, Edgar, author.\n\nTitle: Tap out : poems \/ Edgar Kunz.\n\nDescription: Boston : Mariner Books\/Houghton Mifflin Harcourt, 2019. | \"A Mariner Original.\"\n\nIdentifiers: LCCN 2018033158 (print) | LCCN 2018033745 (ebook) | ISBN 9781328518132 (ebook) | ISBN 9781328518125 (hardcover)\n\nClassification: LCC PS3611.U59 (ebook) |\n\nLCC PS3611.U59 A6 2019 (print) | DDC 811\/.6\u2014dc23\n\nLC record available at <https:\/\/lccn.loc.gov\/2018033158>\n\nCover photograph \u00a9 Paul Anthony \/ Getty Images\n\nCover design by Mark R. Robinson\n\nAuthor photograph \u00a9 Hieu Minh Nguyen\n\nv1.0219\n> _For Noah and Luke,  \n>  my brothers_\n\n# After the Hurricane\n\n> Three hundred\n\n> miles north, my father beds down in a van by the Connecticut River.\n\n> Snow tires rim-deep in the silt. He has a wool horse blanket\n\n> tacked inside the windshield. A pair of extra pants bunched\n\n> into a pillow. He has a paper bag of partially smoked butts.\n\n> A Paw Sox cap. A Zippo. He has state-sponsored cell phone minutes\n\n> and a camo jacket hung on the sideview to dry. He can see the Costco\n\n> parking lot through the trees. Swelling and emptying out. He wants\n\n> to fix things with his wife. He wants a couch to crash on.\n\n> He wants a drink. He wants sex. He has a few cans of kidney beans\n\n> and a tin of ShopRite tuna. Wrinkled plastic piss bottles line the dash.\n\n> Sometimes he walks out to the river and lets the wind sift his lank\n\n> and matted hair. Sometimes he peels his socks and stands\n\n> in the murky current and thinks about his wife. The birthmark\n\n> on her neck. Her one toe longer than the others. Her freckled hands.\n\n> He tries to hold her hands in his mind. He tries to remember\n\n> the birth years of his sons. He tries to make sense of the papers\n\n> he signed. The icy water wetting the hem of his pants. The river stones\n\n> sharp underfoot. The wind. I hold him like this in my mind\n\n> all afternoon.\n* * *\n\n# In the Supply Closet at Illing Middle\n\n> Mike pins me to the sink, forearm\n\n> levered against my throat, flexing\n\n> the needle-nose pliers in one hand.\n\n> He and Ant examine the hole in my head\n\n> where the pencil lead snapped off, blood\n\n> leaking down my temple\n\n> and pooling in my ear. I squirm\n\n> and Mike presses harder. _Hold still._\n\n> _I know how to do this._\n\n> I know what he means: our fathers\n\n> used to salvage wrecks in Mike's sideyard.\n\n> Hammer out the paneling,\n\n> clean the fouled spark plugs\n\n> with spit. Flip them for cash or drive them\n\n> until the transmission seized.\n\n> If they didn't know where\n\n> one came from, they pulled it\n\n> into the garage, sold it off quick.\n\n> Now, Ant stands lookout\n\n> in the doorway. Half-watching\n\n> for teachers and half-watching Mike,\n\n> who rinses my hair\n\n> with floor cleaner thick\n\n> as motor oil. Eases my head\n\n> toward the weak light\n\n> of the pull-chain bulb. Presses\n\n> the pliers to my skull, and starts to dig.\n\n# Free Armchair, Worcester\n\n> He pinches the j between his first two fingers squints an eye against the ribbon of smoke sliding up and over his cheekbone. It's me my buddy Ant and Ant's stepdad Randy a half-ass house painter who's always trying to hit us up for weed or pills even though we're thirteen and don't do pills or have any idea how to get them. We're driving Randy's work van into Worcester to pick up a recliner he found in the free section of the Globe. Ant hates his guts and I don't like him much either but Ant's always doing stuff for me like asking his mom if I can stay the night or sneaking me empanadas when my dad doesn't come home so I go along Ant up front me in the back bracing myself against the wheelwells trying not to get knocked around too bad. Randy pulls up in front of the house and we try stuffing the armchair in the back but the arms are too wide. We flip it on one end heave it onto the roof. Lash it down with a tangle of rope from the glovebox and step back. It's not a bad-looking chair. Fabric ratty at the edges but sturdy. Mostly clean. Randy twists another j to celebrate and buys us sandwiches. We post up in an Arby's parking lot the three of us cracking jokes Randy belting folk songs in Spanish. Recliner strapped to the van like a prize buck. He flicks the roach into the weeds says but you skinny-asses you little faggots you could barely lift it and we stop laughing. I look over at Ant and he's sort of picking at his jeans face tight like he got caught doing something dumb like he's ashamed or something and for a second it's like what's gonna happen has already happened. Like the rope's already snapped the armchair gone headlong into the road behind us. Like we're pulled off on the shoulder Randy punching the wheel calling us dumbfucks fuckheads sons-of-bitches sending us out to wait for a lull in traffic and drag the wreckage to the median. Like we've already started to say what we'll say over and over: We knew the whole time. Chair was too heavy. Rope too frayed. Too thin. Nah we knew. No shit we knew. You think we're stupid?\n\n# The First Time\n\n> Me and Ant shirtless at the corner of Sanford\n\n> and St. Paul, straddling our bikes, watching Daryl\n\n> pace bowlegged in the gutter\u2014 _Yeah man, I mean,_\n\n> _you wouldn't_ believe _this chick, man_ \u2014scuffing\n\n> at the No Dumping plate epoxied to the curb\n\n> with the toe of his high-top\u2014 _It was like nothing_\n\n> _I ever felt_ \u2014Ant and me following the jut of the older\n\n> boy's chin to what looks like a popped balloon\n\n> lying slack at the bottom of the storm drain.\n\n> Nineteen ninety-nine and the most brutal summer\n\n> on record, the water ban parching every ball field\n\n> statewide. The old men who play rummy\n\n> in the shade of a stunted maple have folded up\n\n> their lawn chairs and gone inside. The street\n\n> is mostly empty\u2014just stillness and heat\n\n> and Daryl going on about this girl who just moved\n\n> to town and has _tits like this_ \u2014a girl who doesn't\n\n> know about Daryl yet, his conquest complex\n\n> and his big mouth\u2014a girl who doesn't know yet\n\n> about this town, the legless vets hanging around\n\n> the Army-Navy catcalling public school girls,\n\n> the True Gospel Pentecostal women handing out\n\n> pamphlets in denim skirts and turtlenecks, the fake\n\n> fifties diner on Middle Turnpike where kids get blitzed\n\n> in the parking lot and fistfight until the cops show up.\n\n> I mean it when I say I'm thirteen and already sick\n\n> to death of this place, sick of Daryl, his acned swagger,\n\n> the scuff of his hand-me-down Nikes on the curb.\n\n> But when Ant taps my shoulder and turns to go,\n\n> I don't move. I stand here at the corner, a quick ride\n\n> from home, the still-slick condom catching light\n\n> in the storm drain, the blacktop radiating heat.\n\n> I lay my bike on its side. I step closer to get a good look.\n\n# Natick\n\n> Windshield smeared with dust. Sun bedded down\n\n> in the hills. Drum of my father's hand on the dash startling\n\n> the box-nails in the ashtray. Stub he held delicately in his teeth.\n\n> Silence we passed back and forth between us, like a joke.\n\n> Knowing one day we would stop speaking for good. Knowing it\n\n> when the freeway cut ahead of us and Natick fell away\n\n> on either side. When he held up his hand to mine, palm\n\n> to palm. Nail beds packed with grease. Knuckles more scar\n\n> than skin. When he said I had piano hands,\n\n> and I was ashamed, and hid them in the pockets of my coat.\n\n# Again\n\n> Tell me how she left\n\n> that morning early left\n\n> & you two towns\n\n> over hoisting the brush-\n\n> rod all day leaning\n\n> ladder to brick & me\n\n> rapt in the crib quiet\n\n> as anything quiet\n\n> as something said\n\n> almost out loud how\n\n> you cupped my skull\n\n> in refrigerator light\n\n> groped for the whole\n\n> milk jar of tomato\n\n> how for years\n\n> after you'd startle\n\n> awake and hover\n\n> your hand to feel\n\n> for the small fact\n\n> of my breathing\n\n> though I won't pick up\n\n> for you anymore though\n\n> what's left is mostly\n\n> shame & damaged\n\n> light tell me lean\n\n> your head into\n\n> your shoulder whisper\n\n> into your hands\n\n# Brothers\n\n> _Camp Yawgoog_\n> \n> _Rockville, Rhode Island_\n\n> Lift the lid of Rico's steamer trunk\n\n> at the foot of the bunkbeds we shared\n\n> and it was all laid out, unhidden.\n\n> My glove signed by Juan Pe\u00f1a.\n\n> My hip-hop tapes. The headlamp\n\n> I made out of a bandana and a bike light.\n\n> The _Hustler_ I stole from my uncle.\n\n> Mornings I'd take back what was mine\n\n> and each night more would go missing.\n\n> Wool socks and a monkey fist. A roll\n\n> of camo duct tape. We worked the dining hall,\n\n> sweeping up food and bleaching tables.\n\n> He told me he was from Worcester\n\n> and pulled up his shirt to show the crease\n\n> in his belly where he said he was stabbed\n\n> by his brother on Farrar Avenue.\n\n> Said it didn't even hurt until later.\n\n> Told the cops it was a stranger that did it.\n\n> It went like that for the rest of the summer,\n\n> him stealing and me stealing back\n\n> when he wasn't around. When I found\n\n> the cashbox from the front office\n\n> stuffed in with his underwear\u2014told him\n\n> I knew about it, told him it wasn't right\u2014\n\n> he called me _family._ Called me _brother._\n\n> Said he knew he didn't have to worry about me.\n\n> Gripped my hand and pulled me close.\n\n# Workbench\n\n> We built it out of scrap wood pulled\n\n> from the dumpster next door,\n\n> four pressure-treated legs nailed\n\n> to a plywood sheet and propped\n\n> against a larch in the sideyard.\n\n> That winter, the Wilsinski house\n\n> had burned down. The fire swelled\n\n> above the power lines, singeing the dark\n\n> fir trees and making the block reek\n\n> of burnt plastic and insulation.\n\n> By day, men in baggy jeans\n\n> and face masks gutted the single\n\n> story ranch, trucked in drywall\n\n> and pallets of gypsum brick.\n\n> At night, we took what we needed:\n\n> three boys slipping under\n\n> the chainlink fence, passing\n\n> a flickering Maglite over the yard,\n\n> looking for the gleam of metal,\n\n> roofing tiles, fresh lengths of pine\n\n> and vinyl siding we carried back\n\n> in our arms. I can't remember, now,\n\n> what we were making. If we ever\n\n> made a single useful thing.\n\n> Mostly, I remember pounding nails\n\n> into the larch with a two-by-four.\n\n> If one bent, I hammered it sideways\n\n> until it went flat, until it was flush\n\n> with the bark. Then kept hammering.\n\n> My hand going numb, starting to feel\n\n> like someone else's hand.\n\n# Tap Out\n\n> We were vicious. Swollen cheekbones, bruised jaws.\n\n> Forearms chafed raw and weeping. The Boston\n\n> Crab. The Texas Cloverleaf. The Cross-\n\n> Face Chicken Wing. One time, Ant wrenched\n\n> my shoulder so hard I couldn't lift my arm\n\n> for a week. Another time, Mike's brother Daryl tried\n\n> a front-flip slam off the back steps, landed\n\n> face-first in the dirt. Wrist bone shot clear\n\n> through the skin and gleaming. Mike's dad worked\n\n> second shift at Pratt, so if we were loud he'd holler\n\n> out the bedroom window, but there was nothing\n\n> he could do to punish us we weren't already doing\n\n> to each other. And we knew it. Like that time\n\n> Daryl showed us his pistol, a .22 he lifted\n\n> from a friend's house. We passed it around,\n\n> weighing it in our palms. It was heavier\n\n> than it looked, but it felt good. He put the barrel\n\n> in his mouth and when we jumped up\n\n> he laughed and laughed. _Priceless!_ he said red-faced\n\n> and gasping. _You pussies almost_ _wet your pants!_\n\n> We learned new moves, new ways to shock the body\n\n> into miracles of pain. The Figure-Four Lock.\n\n> The Vise Grip. Every muscle trembling.\n\n> The Tarantula. The Camel Clutch. Mouth\n\n> pressed against their ear, hissing _Tap out dickhead_\n\n> _you're not getting out of this you're mine kid_\n\n> _tap out and it'll stop._ The Sharpshooter.\n\n> The Hammerlock. That sour-hot breath in your ear\n\n> and knowing you won't give in, you won't give him\n\n> the satisfaction, even when it hurts more\n\n> than anything, more than your dad's belt\n\n> blistering your backside, more than the night\n\n> when Daryl put that gun in his mouth and the sound\n\n> of it woke the whole block, so much you grit\n\n> your teeth against the pain, sharp kneecap\n\n> bearing down on your chest, elbow torqued\n\n> past its limit, and you swear you could bust out\n\n> of yourself and look down at your body, helpless\n\n> and small and trembling, press your mouth\n\n> to your own ear and whisper _Not you. Not you._\n\n# Deciding\n\n> Not the sirens. Not the men\n\n> dragging canvas, the canceled-\n\n> out moon, the ash windblown\n\n> and snarling in our hair. Not\n\n> the sick crack of the ridge beam\n\n> or the gun-clapped silence\n\n> after. Not the four of us, brothers\n\n> and our sometimes father,\n\n> our breath knit and drifting,\n\n> our useless hands. I mean\n\n> when I lift Noah, half-asleep,\n\n> to my chest and turn\n\n> for home. When I look up\n\n> at the windows of our own house\n\n> and see the flames\n\n> cold and writhing\n\n> in the glass. That first time\n\n> I say it out loud: _I'm_\n\n> _gonna go._ And the slow walk\n\n> up the drive. And my brother\n\n> growing heavy\n\n> in my arms.\n\n# Graduation\n\n> When you showed up drunk as hell, humming\n\n> tunelessly to yourself, and slumped against\n\n> the auditorium's faux-wood paneling\u2014when\n\n> you fumbled in the pockets of your coat,\n\n> fished out a cigarette, brought it to your lips,\n\n> then, realizing for the first time where you were,\n\n> tossed it away and said _Fuck it_ loud enough\n\n> that everyone turned in their seats and a friend\n\n> elbowed me and asked if I knew you\u2014I shook\n\n> my head and spent the next hour wondering why\n\n> I was so glad you came. You, who slept\n\n> each night in your battered van, who skipped\n\n> meetings and lied to your sponsor, who still\n\n> called your ex-wife every day, restraining order\n\n> be damned. You shouldn't have been there\n\n> either: a hundred yards was the agreement\n\n> after you gathered all the meds in the house\n\n> into a shoebox and threatened to take them.\n\n> You had come regardless. You were there.\n\n> And I was there. And when I walked the stage\n\n> you hollered my name with a kind\n\n> of wild conviction, then said it a second time,\n\n> less convinced, and I thought of that night\n\n> when the cops came and you, unashamed\n\n> of the fuss you caused, of your desperate,\n\n> public struggle for happiness, kissed me\n\n> on the head\u2014once, twice\u2014and went quietly.\n* * *\n\n# Blue\n\n> Because Craig Mathis fell two stories\n\n> through the skylight over the dining room\n\n> and lay face-up on the wood floor me\n\n> and the other waitstaff waxed on Sundays,\n\n> and because the sprinkler pipes tore out\n\n> of the ceiling when he fell, tripping\n\n> the fire alarm and spraying salt water\n\n> over the place settings and chairs, the siren\n\n> pealing over every speaker in that\n\n> tinderbox hotel with its sheetrock walls\n\n> packed with newspaper, and because all\n\n> the guests were rushed to the granite\n\n> breakwater that divides New Hampshire\n\n> from Maine and connects our island\n\n> to the island named for the man\n\n> who wrecked his ship discovering it,\n\n> it took near two hours for someone\n\n> to find him there, sprawled on his back\n\n> in a Metallica shirt and jeans, hauling in\n\n> ragged breaths and murmuring\n\n> to himself, and another hour before\n\n> the helicopter touched down in the yard\n\n> and the EMTs loaded him onto a gurney\n\n> and flew him to Boston General.\n\n> When Craig came back the next summer,\n\n> he limped into the front office\n\n> with a different face, a quad-cane\n\n> he carried everywhere, a jaw that clicked\n\n> when he talked. He said he didn't know\n\n> why he was on the roof in the first place.\n\n> Said he was glad he couldn't remember.\n\n> And so the rest of us needed\n\n> to imagine it: that bright instant\n\n> before the fall, and the long time\n\n> after, having gone through the skylight\n\n> and sprawled on his back on the waxed\n\n> and polished floor\u2014to wonder\n\n> if he looked up at the ruptured piping\n\n> and splintered glass, if he understood\n\n> it was the route his body took when it left\n\n> the charted world, if he saw with his one\n\n> undamaged eye the rails of sunlight\n\n> and the salt water pouring down,\n\n> the framed sky, not a single cloud in it.\n* * *\n\n# Graffiti\n\n> _Baltimore Rescue Mission_\n> \n> _Fairmount and Central_\n\n> I saw it on the drive up to the farm yesterday\n\n> and I see it again this morning: SCUM\n\n> in fat bubble letters. White paint livid\n\n> on the blacked-over brick. Six thirty and a line\n\n> to the middle of the block\u2014looks like\n\n> ordinary folks, mostly. Tired, sure. Hungry.\n\n> A little embarrassed. How my brothers and I\n\n> must have looked waiting outside Social Services\n\n> while our father went in to sign up for stamps.\n\n> Squatting on the curb, hoping we didn't see\n\n> anyone we knew, certain everyone was looking.\n\n> No one was looking. Nobody looks\n\n> at me now, idling at the light, or at the tag,\n\n> or at each other, even\u2014heads down,\n\n> shuffling toward the double doors that open\n\n> on rows of lawn chairs and folding tables,\n\n> plastic placemats the color of bleach.\n\n> When we ran the cattle yesterday morning,\n\n> herd of Herefords raised for beef, it was\n\n> my first time, but I recognized the sharp flanks,\n\n> the hunger and fear that moves them\n\n> from one chewed-up pasture to another.\n\n> When Dad finally came out, he had a look\n\n> we couldn't figure. He told the three of us\n\n> to stay put, then went and sat by himself\n\n> in the truck. The day before, someone had taken\n\n> a claw hammer to the steering column\n\n> and sped off with a bed full of tools\u2014\n\n> table saw, air compressor, everything. The cops\n\n> found it abandoned, empty, on the interstate.\n\n> He seemed leaner then, and dangerous.\n\n> We knew better than to speak. We practiced\n\n> hocking loogies. We took a chunk\n\n> of concrete to the side of the building,\n\n> carved our names into the flaking paint.\n\n> Then a few dirty words we knew. I'm at the light\n\n> long enough now to see the line swell to the end\n\n> of the block and disappear around the corner.\n\n> Listening to the tick of the engine.\n\n> Wondering if our names are still back there.\n\n> When we stood, a few of us, at both ends\n\n> of Rayville Road, waving off cars and driving\n\n> the cattle up toward the far pasture,\n\n> I watched a calf shuffle by. Same as the others,\n\n> but with a white spray across his flank.\n\n> Crude lettering\u2014some local kids,\n\n> I guessed. Names, or cusses. Something violent\n\n> and proud. Something about hunger.\n\n# My Father at 49, Working the   \nNight Shift at B&R Diesel\n\n> There's no one left to see his hands\n\n> lifting from the engine bay, dark and gnarled\n\n> as roots dripping river mud,\n\n> no one to see how his palms\u2014slabs of callus\n\n> from scouring the long throats of chimneys,\n\n> hauling mortar and brick\u2014move\n\n> in the fabricated light. Thumb-knuckle\n\n> thick and white as a grub where the box-\n\n> cutter bit. Split nail grown back\n\n> scalloped and crooked. The stitch-\n\n> puckered skin. And when they fold into and out\n\n> of themselves by the steaming faucet,\n\n> when they strip clean, the tap water\n\n> running black, then copper, then clear\n\n> into the grease-clotted drain,\n\n> there's no one to witness the slap\n\n> of a wet rag tossed in the break-\n\n> room sink or the champ of gravel\n\n> in the empty lot. How the stars dim\n\n> as morning comes on. How a semi downshifts\n\n> on the overpass and the shop windows rattle\n\n> as it goes.\n\n# When Charlie Pulls the Colorado Over\n\n> When Charlie pulls the Colorado over\n\n> and tells me to fuck off, says\n\n> I can ride the bed with the bales\n\n> or hitch back to Parkton with whoever\n\n> will stop but god help him he don't\n\n> give a damn if no one does, I pull\n\n> the latch and stumble down\n\n> onto the sun-scrubbed shoulder.\n\n> The passenger door hangs open\n\n> like a jaw, hinge locked up and squealing\n\n> in the Chesapeake wind that loosens\n\n> hay strands fistfuls at a time\n\n> from the twine-tied stacks,\n\n> scatters them into traffic.\n\n> I run my thumb along the mis-\n\n> matched quarter panel, thinking\n\n> about a woman I loved who called\n\n> that morning to say she's marrying\n\n> a fighter pilot, bought a place\n\n> outside Jerusalem, that she's learned\n\n> to say _his ways are ways_\n\n> _of pleasantness_ and she's chosen\n\n> to cover her hair and the freckled skin\n\n> of her arms\u2014\n\n> halfway between\n\n> Hunt Valley and Hackensack,\n\n> the Econo Lodge sign pulsing\n\n> behind the curtain, the sounds\n\n> we make mixing with the branches\n\n> lashing the window, the rain, the big-rigs\n\n> on the interstate\u2014 _the kind of love_\n\n> _that makes you forget,_ she says,\n\n> slipping from the sheets\u2014\n\n> and whatever\n\n> seized her then, whatever swept her\n\n> toward those distances, the Abrahamic plains,\n\n> a language she'd never spoken\n\n> but is learning, now, to speak,\n\n> is what lifts me, one foot on the bumper,\n\n> good arm levering my body into the bed.\n\n> What sets me down among the wind-\n\n> torn bales, pushes me upstate,\n\n> toward Monkton, Hereford, New Freedom,\n\n> the dropseed prairies, the runoff ponds\n\n> and feedcorn fields. What asks me\n\n> to try and track one straw, to hold it\n\n> with my eyes for more than a second,\n\n> and fail, then choose another\n\n> and fail again, Charlie leaning on the horn\n\n> as they vault into the wind.\n\n# V.F.W. Post #2046\n\n> _for N.M._\n\n> She fires the boiler first thing,\n\n> cursing the pilot and Christ Jesus\n\n> and the damp matches she fumbles\n\n> in the furnace room's musty light.\n\n> Sunday, bingo night, shuttered bar,\n\n> the shining slab where my father\n\n> spent the better part of a decade\n\n> conning drinks out of regulars,\n\n> sharking the table, cracking jokes,\n\n> lined with upturned stools and rags.\n\n> For years it was a couple days\n\n> a week, him coming in dusted\n\n> with sheetrock, permanent grin,\n\n> slapping a twenty on the grain.\n\n> Then it was every other, then\n\n> every, then two-three times a day\n\n> coming in dug out and feral, fat-\n\n> faced, broad hands gone thin.\n\n> She stopped serving him then.\n\n> I've always loved her for that,\n\n> though by those days nothing\n\n> could have slowed him. I was a boy\n\n> in the booth behind, watching\n\n> as the light held still over the black-\n\n> and-white portraits, the bills signed\n\n> and tacked to the rafters.\n\n> She doesn't think of him now.\n\n> Not often. She twists the key\n\n> to the storage room. Disappears\n\n> and reappears hauling a rolling cart\n\n> of folding tables. Heaves them\n\n> onto their sides, kicks out and locks\n\n> the legs. Flips them each upright\n\n> and starts setting out the chairs.\n\n# Franklin Free Clinic\n\n> She drops the tooth\n\n> in the pan, packs my cheek\n\n> to sop the blood. I'm\n\n> telling her about the mole\n\n> on my hand I'm sure\n\n> is cancer. Runs\n\n> in my family. My aunt\n\n> with the scar smeared\n\n> between her breasts.\n\n> My grandfather's femur\n\n> riddled with it.\n\n> She tells me to relax.\n\n> I'm fine. I'm not fine\n\n> and she pretends\n\n> not to hear. I try\n\n> telling her about my ex,\n\n> the pale seam\n\n> at her throat where\n\n> after months\n\n> of mysterious sickness,\n\n> after thrush, fever, bone-\n\n> deep pain, they lifted\n\n> a mass slick\n\n> as an avocado pit.\n\n> I shape my hands\n\n> to show the largeness of it.\n\n> I tell her how I'd lie\n\n> awake at night and look.\n\n> How my own throat caught.\n\n> She pulls the cotton\n\n> from my mouth, coughs\n\n> into her elbow. Hands me\n\n> two tiny tubes of toothpaste.\n\n> One soft-bristled brush.\n\n# Safety\n\n> My brother shows me\n\n> the iron sights. The dark O\n\n> of the muzzle. The grip.\n\n> Describes the caliber,\n\n> the diameter of the holes\n\n> they hollow, how the copper\n\n> jackets bloom. Presses\n\n> its weight to my palm, says\n\n> they make the real thing\n\n> in runs of a hundred\n\n> thousand. _Ideal,_ he says.\n\n> _Light, and cheap._ He lays\n\n> his hands on mine, steers\n\n> the open mouth toward\n\n> his window, the neighbor's\n\n> place beyond it, then\n\n> toward the bedroom where\n\n> our mother is sleeping.\n\n> _I want to keep us_\n\n> _safe,_ he says. I ask what\n\n> he means. Crickets\n\n> string out their thin music\n\n> between the duplexes\n\n> and brick-front ranches\n\n> of our block. A late rain\n\n> slicks the patched-\n\n> over street. Our motion\n\n> light flickers on,\n\n> and the blacktop shines.\n\n# Close\n\n> Off early from B&R Diesel, sharp\n\n> with liquor and filtered Kings, he drifts\n\n> across the double-yellow, swings\n\n> into an iced-over lot. He runs me through\n\n> the basics: K-turn, parallel, back-in.\n\n> Jerks the Sierra into reverse and eases\n\n> the bumper up against the side\n\n> of the old bank building. We meet\n\n> at the end of the loaded bed, exhaust\n\n> and brakelight pooling around our knees.\n\n> He balls the front of my coat in his fist,\n\n> pulls me close to show the distance\n\n> between bumper and brick, pulls hard\n\n> until I'm up against the slender arc\n\n> of his collarbone, the fine dark stubble\n\n> shading his jaw, his hollowed-out cheeks.\n\n> He's still beautiful, my father. Fluid.\n\n> Powerful. His bare forearms corded\n\n> with muscle, bristling in the cold. Yes,\n\n> he's drunk. Yes, I have already begun the life-\n\n> long work of hating him, a job\n\n> that will carve me down to almost\n\n> nothing. I have already begun to catalog\n\n> every way he has failed me. Yes.\n\n> And here he is. Home early from a day shift\n\n> in Fall River. Teaching me what I need\n\n> to know. Pulling me roughly toward him,\n\n> the last half-hour of sunlight blazing\n\n> in his face, saying _This is how close_\n\n> _you can get._ Asking if I can see it.\n\n> If I know what he means. Saying _This. This_\n\n> _close. Like this._\n\n# After the Attempt\n\n> They took your shoelaces,\n\n> your carabiner of tooth-\n\n> edged keys, but left you\n\n> your belt, which you cinched\n\n> over your loopless scrubs.\n\n> They shaved your scalp\n\n> for the stitches but missed\n\n> a tuft above your ear\n\n> that catches the light\n\n> from the hingeless windows.\n\n> The receptionist holds up\n\n> a small paper bag\n\n> stapled shut. Whatever\n\n> you had worth saving.\n\n> You look, then look away.\n\n> Once, hungover\n\n> on a gut-and-remodel job\n\n> in Grafton, you cracked the root\n\n> of your nose with your claw\n\n> hammer's backswing.\n\n> You stood very still after,\n\n> watching your blood scatter\n\n> on the plywood floor, alien\n\n> and bright as coins\n\n> from a distant country.\n\n# Vows\n\n> You said _I want to be married._\n\n> You said _I want to be married_\n\n> _to you._ You said _We were children_\n\n> _together. Who better?_\n\n> You said _I moved for you once_\n\n> _already._ You said _I need this._\n\n> You said _It will be quick._\n\n> _Backyard. July. My mother_\n\n> _will cook, my brother will DJ._\n\n> _Here's the date. Here's the phone._\n\n> You said _There is so much_\n\n> _to do:_ _spray the bushes_\n\n> _with repellent, bind_\n\n> _these sunflowers with twine._\n\n> _Hack this stump down_\n\n> _to a hollow, fill it with stone._\n\n> _Here._ Standing in July,\n\n> in the backyard, reciting the words\n\n> you wrote in ballpoint\n\n> on a scrap of ruled paper. _Here_\n\n> _I am._ And slowly, as if\n\n> emerging from a long sleep,\n\n> and looking around,\n\n> and confusing myself\n\n> for the cufflinks, the hushed\n\n> crowd, the white tent\n\n> billowing like a sail\u2014I take\n\n> your hand. I start to speak.\n\n# In the Sideyard\n\n> A splinter of moon lodges deep\n\n> in the limbs of the spruce.\n\n> My brother\u2014half my age, shivering\u2014\n\n> lifts the splitting maul, one\n\n> smooth hand gripping the heel,\n\n> the other at the haft. The steel\n\n> he set and honed himself glints\n\n> at the crest of his practiced arc,\n\n> dividing cold from dark,\n\n> each half-log of knotted oak\n\n> from its twin. His shadow, large\n\n> as a man's, pitches headlong\n\n> into the dirt.\n\n# Dundalk\n\n> Wayne came back late from lunch, put his head down on his desk and slept. I was finishing up a lesson on supply-side economics. Noon slumped through the half-shut blinds and the classroom was hot. I knew he was high. I clicked the next slide. Turns out he was spotted ducking out at lunch, slipping behind the Sunoco on Church. School cop asked me a few leading questions then kicked him out for good. Said it was his last strike. He punched his own ticket. Next day, it's like nothing happened: the kids of south Baltimore turn to the clock, the scuffed tile floor. Snap gum in their teeth. I go back to _deadweight loss, human capital, returns on investment_. I go back to my rented rowhome on the northside where I live with a guy who ships in airtight bricks from Washington State, breaks them into eighths and makes a killing off Hopkins kids. He says stuff like _money makes money_ and _cops here have bigger fish_. Some days, driving home, I go south, get halfway out on the Key Bridge and pull off. From here, you can see the waste plant's gold honeycomb towers, the faded terminals, dock cranes leaning out over the harbor like drunks. You could stay, if you wanted, for an hour, two, as long as you could stand it, watching the container ships drift in and churn out, the longshoremen offloading, shuffling the stacks. Long enough and you could watch what's left of the workday drain off into the Chesapeake. Watch night spread out like an oil slick and go still.\n\n# Michael\n\n> If we met up in the iced-over lot at the neighborhood's edge\n\n> we were kids in\u2014grid of low-slung ranches sunk\n\n> under the lengthening shadows of larch and pine,\n\n> each street slanted toward the state building where our folks\n\n> collected their checks on the first of each month\u2014\n\n> and if your eyes were glossed with oxys and a week\n\n> without sleep, body a loose frame of copper piping propped\n\n> under your oversized coat, and we stood, face-to-face\u2014\n\n> Michael, what would be left between us?\n\n> What would remain of tunneling under chainlink\n\n> after the Wilsinski house burned down, slipping\n\n> between the brick pallets and front-end loaders, looking\n\n> for something to claim? Or that July we worked stripping kudzu\n\n> and poison oak from your sideyard on the promise of a few bucks\n\n> from your dad, our longsleeves matted with pine pitch and sweat?\n\n> We found a yellowjacket nest, a paper lantern buried deep\n\n> in the brake. You dared me to hit it with a Wiffle ball bat\n\n> and I did and the yellowjackets stitched my chest and arms\n\n> with fire. I came back last Christmas and sat on the hard edge\n\n> of my little brother's twin bed as he showed me how to thumb\n\n> an imaginary bullet into a handgun with REPLICA etched\n\n> on the barrel. Taught me words like breechblock\n\n> and chamber-throat. Blowback and primer. Showed me how\n\n> to switch off the safety, to keep my finger away from the trigger\n\n> until I'm ready to pull. The way your brother Daryl\n\n> took himself out of this world. I thought of you, thirteen,\n\n> weighing out nickels in your bedroom at your dad's place.\n\n> Twisting a dutchie, licking it shut. You didn't give a shit,\n\n> but I stuffed a paper towel tube with dryer sheets and we blew\n\n> our smoke through to hide the smell. All I have of you now\n\n> is rumor: a few run-ins with the cops for small stuff\u2014\n\n> petty theft, possession\u2014that you knocked up a girl\n\n> from Willimantic. That you were faded on cough syrup\n\n> and drifted into oncoming traffic on 84, limped away\n\n> with a sprained ankle but otherwise fine. There was a time\n\n> when I thought I knew what swerves us from disaster,\n\n> what separates us. All I can do now, Mike, is praise the state-\n\n> cut checks and the baggies of pills. Praise the quick transaction,\n\n> the no-look pass, twenty twisted into a palm. The Robitussin-\n\n> kiss, the slow drift of the wheel. The soft shoulder.\n* * *\n\n# Next of Kin\n\n> A dim-eyed woman skips her meds and scales\n\n> a barb-wire fence outside Lake Charles. Five hundred\n\n> miles west, her daughter paces the faded blue outline\n\n> of a handicapped spot. The salt wind whips her hair\n\n> into her mouth. She says _what_ over and over\n\n> into the phone. A voice says _torsional_ and _trauma_\n\n> and _need you to decide_. The last time she saw\n\n> her mother, she found her rocking on the stoop\n\n> of her rented duplex, threading electrical wiring\n\n> through the spokes of a bicycle rim. Hair matted\n\n> and half-braided. Long fingers deftly working.\n\n> She took her inside and put her to bed. Next morning,\n\n> her mother was gone. What can you do?\n\n> The palm trees hang their heads. The wind delivers\n\n> car exhaust and brine. She has given up the parking spot.\n\n> She starts to circle the shopping cart corral.\n* * *\n\n# Training\n\n> My little brother says he's worried about me.\n\n> Asks why I left my wife. We get cut off\n\n> and he calls back, says the bottoms\n\n> of his feet are finally healing up, he's back\n\n> on base, packing on muscle. Four nights\n\n> in the brackish swamps of south Georgia,\n\n> hunger-sick, stumbling, and each time one\n\n> of his squad lay prone to consult the wilted map,\n\n> he'd have to kick them awake. _Roots,_ he says.\n\n> _Get one jammed in your ribs and the ache_\n\n> _will keep you up._ I tell him I found a studio\n\n> in Oakland, full-size stove, eucalyptuses\n\n> leaning prehistoric in the hills. That I climbed\n\n> one morning up under the overpass on Forest,\n\n> a little drunk, and pressed my palm to the cool\n\n> underside to feel the traffic rushing over.\n\n> _I was pissing brown,_ he says. _I knew if I quit_\n\n> _they'd give me a gallon of water, let me crash_\n\n> _in the jeep. But I also knew the worst of it_\n\n> _was done._ The third night, orbs of red light\n\n> circled the cottonwood trunks. The fourth,\n\n> he spotted his first girlfriend fifty, a hundred\n\n> yards ahead. She turned slowly to face him,\n\n> then bent down to hover a pale hand\n\n> above the water. It happened over and over.\n\n> She'd bend and he'd drop his rucksack and sprint\n\n> into a clearing soaked in half-light and steam,\n\n> and see her again, turning, farther off.\n\n# Going\n\n> Alone now in San Francisco. Thin cloud rusting\n\n> over Bernal Hill, garlic simmering\n\n> in the pan, lavender potted and long dead\n\n> in the breezeway. I start the water,\n\n> carry the milk crates in from the garage.\n\n> You with your mother in Los Angeles.\n\n> The lanterns we scavenged and hung\n\n> at the ceremony now a soft racket\n\n> in the magnolia. Me turning an old\n\n> summer over, the one where we slept\n\n> most nights in a park in Hartford,\n\n> bedded down in the soaked grass.\n\n> The local kids coming always after dark\n\n> to tag the pumphouse, sling rocks\n\n> at the heron cages. Their bright\n\n> startled cries and us burrowing deeper\n\n> in our bags. I start unshelving\n\n> my books, fitting them side by side\n\n> in a crate. How one time a guard\n\n> came hollering, whipping his light\n\n> over the lawn and they took off, ditching\n\n> their backpacks, the cans, their names\n\n> silvering the brick. We watched\n\n> as they tore down the moonlit hill,\n\n> headed for the coupe they stashed\n\n> at the turnoff, bare legs flashing, the guard\n\n> close behind as they vaulted the fence\n\n> and hit the blacktop sprinting,\n\n> picking up speed\u2014the two of us clutching\n\n> at each other, wincing, whispering.\n\n> You saying you hope they get busted.\n\n> Me hoping they get away clean.\n\n# Dry Season\n\n> It had ended and ended\n\n> badly so I'd stopped drinking\n\n> and started again or was\n\n> about to when an old friend\n\n> bought me a nonstop to stay\n\n> with him in Colorado this was early\n\n> October and the first snows\n\n> had driven the elk down\n\n> out of the upper ranges\n\n> backing up traffic into Lyons\n\n> and drawing crowds of tourists\n\n> who posed alone or with\n\n> their blonde polo'd families a safe\n\n> distance from their wildness I\n\n> watched disgusted I thought\n\n> there is nothing worse\n\n> than this shit knowing of course\n\n> that there was much worse and that I\n\n> had done it I lay down\n\n> on my friend's bottom bunk\n\n> and woke in the morning\n\n> and wandered into the living\n\n> room where vaulted windows\n\n> looked out on a parched field\n\n> and there was an elk there\n\n> then four five clustered\n\n> between boulders picking at\n\n> the stunted shrubs even larger\n\n> up close than I expected the night\n\n> before I left for good I slept\n\n> on the living room floor\n\n> and she came out shivering\n\n> and sobbing asking me\n\n> to hold her just for a minute and I\n\n> said no I said no because so\n\n> many times before I had said\n\n> yes and not meant it and just\n\n> like that I knew I was\n\n> small and cruel and moved\n\n> out across the placid bay\n\n> and shut myself up in my oneroom\n\n> apartment and drank\n\n> and watched spaghetti\n\n> westerns A Fistful of Dollars\n\n> Duck You Sucker the elk\n\n> meandering closer\n\n> to the window where I stood\n\n> scowling into the light I pressed\n\n> my palm to the thick sheet\n\n> of glass between us and smacked it\n\n> once hard and not one of them\n\n> turned so I hit it again both\n\n> hands this time making a sound\n\n> like an empty plastic tub a\n\n> hollowed-out thing and the closest\n\n> lifted its head ears high\n\n> tuft of white tail twitching\n\n> and looked calmly upon me\n\n> without recognition\n\n> and went on eating the wild grasses\n\n# Kabekona\n\n> Drifting in a borrowed Old Town\n\n> on a chain-lake outside Bemidji.\n\n> Bluegills nosing the surface.\n\n> Birches and their bright silences\n\n> on the shore. Two weeks gone\n\n> and nothing is easier. Remembering\n\n> her thumb tracing my hipbone,\n\n> early sun running its hands\n\n> through our hair. The shyness\n\n> gone out of us then, all sweat\n\n> and a reckless need, pressing\n\n> hard, trying to break through\n\n> into feeling. I lean out\n\n> over the gunnel and trail my fingers,\n\n> watch the walleye flicker\n\n> in their private dark. Wanting that.\n\n> To be open-mouthed and simple.\n\n> To let the cold water touch me all over.\n\n# Farmsitting\n\n> Most days the same\n\n> with minor variations. Flat blue\n\n> of the 5 a.m. kitchen. Two scoops of feed\n\n> in a plastic bucket. A small bowl\n\n> of yogurt and an hour stacking\n\n> what the ice brought down overnight.\n\n> I was happy. I slept in their bed,\n\n> I read the mysteries on their shelves.\n\n> Always something precious gone,\n\n> someone hot on the trail.\n\n> I walked in borrowed boots\n\n> across the frozen pasture and back\n\n> each morning, each morning\n\n> the feed, the spigot, the horse dragging\n\n> its bulk against the stall.\n\n> I'd walk out nights and stand\n\n> on the same trampled spot in the yard\n\n> and listen to the cold stirring\n\n> in the cheatgrass. Dull glow of a town\n\n> on the horizon. Hiss of snow.\n\n> I'd lie in their bed under three heavy\n\n> cotton blankets and worry\n\n> about the horse and the dwindling\n\n> supplies. It was a life and it was not\n\n> mine. To sleep, I imagined the great\n\n> slabs of granite buried slantwise\n\n> in the hills. To sleep, I counted\n\n> the measures ticked out\n\n> in the porcelain tub, slow drip\n\n> to keep the pipes from freezing.\n\n# Mud Season\n\n> The road leading away from the hospital\n\n> was long, tree-lined and pocked\n\n> with shallow holes from the constant freeze\n\n> and thaw so that each time I swerved\n\n> my father's jaw flexed in the passenger\n\n> seat, though he said nothing. He was wearing\n\n> a pair of loose black jeans and an off-\n\n> white T-shirt, clean, the sleeves covering\n\n> the Celtic knotwork our neighbor\n\n> had inked clumsily around both biceps.\n\n> He pointed. I turned onto a two-lane highway,\n\n> cup change chiming, ladders shuddering\n\n> against the bolted-on racks. The sun was low\n\n> in the trees and it cut across us\n\n> in flashes, bright, then dark, then bright\n\n> again, quickly. He rested his head\n\n> on the window and I realized I had no idea\n\n> what I was supposed to do with him.\n\n> He pointed. I turned. Dense neighborhoods\n\n> gave way to fields of wild grasses. A duplex\n\n> tight to the curve of the road sprang up\n\n> alongside us: pale blue siding, a porch\n\n> that slumped heavily toward the street.\n\n> There was nowhere to pull off,\n\n> so we idled in the road. _You remember_\n\n> _this place?_ he said. _We lived here, you and me._\n\n> It was getting dark. Headlights winked on\n\n> across the far field. The wheel trembled\n\n> under my hands and the dash lights\n\n> made strange angles of his face.\n\n> _Remember?_ he said. _When you were small?_\n\n# For Good\n\n> When she left\n\n> for good, turning hand\n\n> over hand\n\n> onto the interstate,\n\n> I imagine my birth\n\n> mother knew my father\n\n> wasn't bent or broke\n\n> beyond our fixing.\n\n> Just that it is enough\n\n> to want to go.\n\n# Salvage\n\n> Still somewhere in me the summer\n\n> spent driving steel into the wet earth: heft\n\n> and swing of the mattock, my blistered hands,\n\n> blackflies rising like steam. The tables\n\n> I served. The law firms I hustled\n\n> from one zipline to another, classroom\n\n> where I taught economics to the medicated\n\n> kids of bus drivers and stevedores, swept-\n\n> clean boulevards of the city that paid me\n\n> to snap a picture of every downtown\n\n> business, jot the names and hours in a spiral-\n\n> bound book. Somewhere in me the failed\n\n> industrial towns of New England\n\n> with their posh English names\u2014Weymouth,\n\n> Bridgeport, Lowell, Worcester\u2014their dead\n\n> cars, their factories and silk mills converted\n\n> and upsold to commuters, somewhere\n\n> the third-floor walkup we lived in\n\n> longest: cracked plaster and single-pane, plastic\n\n> paneling painted to look like real wood,\n\n> and my stepmother, my real mom, bending\n\n> over the glossy stack of Star Market mailers,\n\n> hands thin, approximate, bright scars\n\n> on the backs of her wrists where the surgeries\n\n> didn't take, and me, problem kid\n\n> with a mushroom cut and his shirt tucked\n\n> into his sweats, clipping the dollar-offs,\n\n> the half-offs, the buy-one-get-ones, the buy-one-\n\n> get-twos, the store-issued doublers, shoulder\n\n> to shoulder on the kitchen floor and the afternoon\n\n> stretching on into no kind of heaven\n\n> I could have understood then. Of peeling\n\n> linoleum and the drone of interstate traffic.\n\n> Of WIC checks, name-brand knockoffs, the gray\n\n> stamps card made to pass as a regular Visa.\n\n> Where we are allowed to know exactly what we\n\n> can have, and keep. And what it will cost.\n\n# Piecework\n\n> My grandpa was always afraid\n\n> of the machines in his shop\u2014plunge-\n\n> router, lathe, temperamental planer\n\n> he traded for back when he was first\n\n> getting sober. Said his buddy\n\n> was doing piecework one time\n\n> at the table saw, looked up,\n\n> looked down and his right thumb\n\n> was on the floor. Hum of the cross-\n\n> cut blade, morning swelling\n\n> in the high windows. And just now,\n\n> twenty hours in, I nod off doing 80\n\n> outside Harrisburg and this\n\n> borrowed Civic goes perfectly on\n\n> without me. Quiet. Efficient.\n\n> Hands numb on the straightaway,\n\n> shorn stalks and industrial silos\n\n> sliding. I hit the rumble strip\n\n> and pull off at the Shurfine for air.\n\n> Dry flakes swirling in the fluorescent\n\n> overheads, the lot choked with cars.\n\n> No pain, he said, until later.\n\n> I blow into my hands, crack a window\n\n> and keep going. My brother ships out\n\n> in a week. It's a wide country.\n\n> I need to tell him before he goes.\n\n# My Father at 23, on the Highway   \nSide of an Overpass Fence\n\n> The details always the same.\n\n> Salt wind tearing at his jacket.\n\n> Bootheels dug deep\n\n> in the chainlink. The two doses\n\n> slipped under his tongue\n\n> at a friend of a friend's party\n\n> and the coming-to\n\n> blinking at a half-lit stretch\n\n> of the Long Island Expressway.\n\n> He would call me down\n\n> into the bachelor pad he made\n\n> of our basement\u2014ratty couch,\n\n> knob-dial TV, mini-fridge\n\n> he bartered a tiling job for\n\n> stocked with Narragansett\u2014\n\n> and tell me again about the fence,\n\n> the wind, the semis pitching\n\n> into the dark.\n\n> Even then, I didn't believe him.\n\n> This was the man who one time\n\n> told me three punk kids\n\n> mugged him for the camo jacket\n\n> I found later in the trash.\n\n> Who said he was a SEAL,\n\n> that he was shot in Da Nang\n\n> and showed me an acne scar\n\n> on his chest. Pale crater\n\n> above his left nipple. And now\n\n> that I've stacked a decade\n\n> and the width of the country\n\n> between us, now that I've chewed\n\n> a fistful of bluish stems\n\n> and sprawled out\n\n> in the soybean, slightest\n\n> stalk-sway rippling the brilliant\n\n> surfaces of my skin\u2014\n\n> now that I know the one\n\n> where my stepmom tells him\n\n> finally to leave and he walks\n\n> calmly into the bathroom\n\n> and fills a shoebox with all\n\n> the leftover meds\n\n> and the cops pull up\n\n> and cuff him and scrawl\n\n> their badge numbers on the forms\u2014\n\n> if he was ever up\n\n> on that overpass, it wasn't\n\n> a bad trip that put him there.\n\n> I get it. In California, in the thin\n\n> middle of my twenties, I'm up\n\n> most nights on Bernal Hill,\n\n> walking the radio tower fence-line,\n\n> measuring the distances.\n\n> The three thousand miles\n\n> between San Francisco\n\n> and the town where the shadows\n\n> of my brothers grow tall.\n\n> The cash I don't wire,\n\n> the numbers I don't dial.\n\n> The marriage that didn't survive\n\n> the summer. The version\n\n> where I'm blameless and the one\n\n> where I again abandoned\n\n> what was difficult.\n\n> Holding them each up.\n\n> Testing for nerve. For weight.\n\n> Some nights are clear\n\n> and the moon lacquers everything.\n\n> Some nights a thunderhead\n\n> scuds low across the bay,\n\n> scraping over the dock cranes\n\n> and port-locked ships,\n\n> the variegated stacks.\n\n> The headlands massing behind.\n\n> Bright traffic streaking below.\n\n# Window Washers\n\n> All day they ride the long metal box up the gleaming side\n\n> of the hospital. Rags in a plastic washtub. Squeegees\n\n> and a trough of soapy water. Below, the sidewalk fenced off,\n\n> the upturned faces gawking as they rise, ratcheting up\n\n> to the next row of windows. A man sits on a bus stop bench\n\n> and watches them work until he is ready to go in and see\n\n> what's become of his father. A man he some days comes close\n\n> to loving. He has traveled a long way to get here. He's the only\n\n> one who is coming. He will sit by his father's bed in the ward\n\n> and wait for a face to appear in the window. Wet and squinting\n\n> under a white cap. He will wait for the suds to blot out the light.\n\n> The furious churning on the other side. The soap rinsed away\n\n> as someone works the ratchet. Chest. Knees. Sneakers. Sky.\n\n# Interim\n\n> On the glassed-in back porch\n\n> of a friend's house on Folsom,\n\n> I slept three weeks on a heap\n\n> of patterned wool blankets,\n\n> a large Ziploc of granola\n\n> and a jar of pistachios on the sill.\n\n> I woke to bus traffic\n\n> in the floorboards and sun\n\n> on my face, drank thin coffee\n\n> and scoured the listings\n\n> for a studio someplace more\n\n> possible. Each day nothing\n\n> and each day I paced\n\n> the bright narrow side streets\n\n> with my friend, who was taking\n\n> time off and who was an expert\n\n> in digital currencies.\n\n> I'd tell him about the collapse\n\n> of my marriage and he'd tell me\n\n> about the distant servers\n\n> that mine electronic coins\n\n> by solving complex equations.\n\n> The specialized equipment\n\n> required for this kind of work.\n\n> I would ask him basic questions\n\n> and he would answer patiently:\n\n> _The coins are encrypted code._\n\n> _The code is the currency. Value_\n\n> _is determined by speculation._\n\n> Those days, every detail\n\n> had the glimmer of potential\n\n> cruelty: hot-pink curtain\n\n> caught in a shut window,\n\n> drainpipe signed KING BABY\n\n> in white-out pen, paper bag\n\n> of potatoes rotting in the trunk\n\n> of the car I borrowed to retrieve\n\n> a crate of books from storage.\n\n> I called a man about a place\n\n> above a Thai restaurant and lied\n\n> about how much I make in a year.\n\n> He was from Pittsburgh. We talked\n\n> about rain. He said he'd call later\n\n> to tell me if I got it. On another walk\n\n> I asked my friend more questions.\n\n> Will it replace cash? _Yes._\n\n> Is it untraceable? _Yes._ What happens\n\n> when they run out of equations?\n\n> A bus hummed past, skimming\n\n> the lowest branches of the ficus tree\n\n> giving us shade. _It's not like that,_\n\n> he said. _It could go on forever._\n* * *\n\n# Behind the Eyes, & Shining\n\n> If I could say it once, clearly. If I could get it right.\n\n> If I could hold it all together in my mind: the pollen shook loose\n\n> like dander and the sapsucker punching holes\n\n> in the siding. The chainlink grown through birch and wind\n\n> where the ranch used to be. If I could pass my body\n\n> through the seam between shingle and ridge beam, linoleum\n\n> and plank. Return as termite, ditch weed. If I could go back\n\n> to that July in Northampton, blowing fiberglass\n\n> into rich folks' attics. To when me and Ant\n\n> let the blower run and smoked blunts all day in the trailer.\n\n> To when it was a scam and we knew it. If I could admit\n\n> it was a scam: my father's voice soft on the machine. Sober.\n\n> Asking me to call back. If I had to admit why I won't. If I had to reckon\n\n> with what the past asks of the present. If I am here\n\n> in his van. Stale cigarillo smoke and the heavy redolence\n\n> of the body. Windows fogged over. Blankets damp with rain.\n\n> If I squat against the wheelwell, and look at his quiet hands,\n\n> and do not turn away. If they tremble. If they're still.\n\n# Acknowledgments\n\nGratitude to the editors of the following journals and magazines in whose pages these poems first appeared, sometimes in a different form:\n\n_Adroit:_ \"Deciding.\" _AGNI:_ \"Natick,\" \"Window Washers.\" _Arcadia:_ \"Graduation.\" _Bat City Review:_ \"Free Armchair, Worcester,\" \"Dundalk.\" _Blackbird:_ \"Brothers.\" _The Cortland Review:_ \"When Charlie Pulls the Colorado Over.\" _Devil's Lake:_ \"Blue.\" _Forklift, Ohio:_ \"Behind the Eyes, & Shining,\" \"Next of Kin.\" _Gulf Coast:_ \"After the Attempt.\" _Indiana Review:_ \"After the Hurricane,\" \"My Father at 49, Working the Night Shift at B&R Diesel.\" _The Journal:_ \"The First Time.\" _The Missouri Review:_ \"Graffiti.\" _Narrative:_ \"Close,\" \"My Father at 23, on the Highway Side of an Overpass Fence.\" _New England Review:_ \"In the Supply Closet at Illing Middle.\" _Passages North:_ \"Workbench.\" _Ploughshares:_ \"Interim.\" _Redivider:_ \"Tap Out.\" _The Sewanee Review:_ \"Piecework,\" \"Franklin Free Clinic,\" \"Going,\" \"Michael,\" \"Safety.\"\n\n\"After the Hurricane\" was featured at the NEA Writer's Corner.\n\n\"My Father at 49 . . .\" was chosen by Tracy K. Smith for inclusion in _Best New Poets 2015._\n\n\"Free Armchair, Worcester\" was chosen by Natalie Diaz for inclusion in _Best New Poets 2017._\n\n* * *\n\nThis book would not have been possible without the support of the National Endowment for the Arts, the Creative Writing Program at Stanford University, the MFA program at Vanderbilt University, the Kratz Center for Creative Writing at Goucher College, the Bread Loaf Writers' Conference, the Sewanee Writers' Conference, and the MacDowell Colony. Thank you to the hardworking folks who keep these institutions alive and well.\n\nUnending thanks to my teachers & mentors Elizabeth Spires, Mark Jarman, Eavan Boland, Louise Gl\u00fcck, Ken Fields, Rick Hilles, Beth Bachmann, Pete Fairchild, Ann Christie, Edward Hirsch, Tom Sleigh, Kate Daniels, Stephen Rascher, David Lang, Adam Ross, Eduardo C. Corral, Carl Phillips, and Rick Barot\u2014and to my very first poetry teacher, Steve Straight, whose sharpness and generosity encouraged me to pursue a life in language.\n\nTo fellow writers & dear friends Tiana Clark, Will Brewer, Ryann Stevenson, Chris Drangle, Emma Catherine Perry, Noah Warren, Grady Chambers, Margaret Ross, Kai Carlson-Wee, Javier Zamora, Mikko Harvey, Matthew Kelsey, Laura Romeyn, Keith Leonard, Kien Lam, Ari Banias, Solmaz Sharif, Brian Tierney, Kara Krewer, Will Schutt, Michael Lee, Janet Thielke, Claire Jimenez, Ricardo Zamarano Baez, Anessa Ibrahim, Justin Boening, Cate Lycurgus, Chad Abushanab, Josh Kalscheur, Phillip B. Williams, Sam Ross, and Ben Naka-Hasebe Kingsley.\n\nTo my workshop-mates at Vanderbilt and Stanford, especially Chris Adamson, Cara Dees, Dan Haney, Alicia Brandewie, Anne Charlton, Sara Strong, Max McDonough, Simone Wolff, Mary Somerville, Charif Shanahan, J. Bruce Fuller, Casey Thayer, Nick Friedman, and Essy Stone.\n\nTo my family, especially Rowena and Jim, Linda and Tim, Nancy, Sharon, the King and Queen, Rob, Jason, Larry and Dora, Jessie and Tom, Evil Auntie Sue, Camma, Cheryl and Joe, Zack, Carole, Greg, Cathy and Steve, Bill and Teresa, Otto and Karen, Jim and Nivin, Meme and Pepe, Grandpa K\u2014and to all my cousins, blood or otherwise.\n\nTo my mom, Suzanne, and my brothers, Luke and Noah, whose love has saved me many times over.\n\nTo my father, Christopher, who cared for me when I was a baby.\n\nTo my brilliant editor, Jenny Xu, whose ideas shaped this book profoundly, and to the HMH crew who worked tirelessly on its behalf.\n\nTo Anders Carlson-Wee, whose steadfast friendship has meant the world. Here's to many more years working side by side in the shop.\n\nAnd to my sweetheart, the radiant Katie Moulton, without whom none of this would mean a thing.\n\n# About the Author\n\nEdgar Kunz was born and raised in New England. A former Wallace Stegner fellow at Stanford, he has been supported by the National Endowment for the Arts, the Academy of American Poets, the MacDowell Colony, and Vanderbilt University. Kunz lives in Baltimore, where he teaches at Goucher College and in the MFA program at Salve Regina University.\n\n# Connect with HMH on Social Media\n\nFollow us for book news, reviews, author updates, exclusive content, giveaways, and more.\n\n# Contents\n\n  1. Title Page\n  2. Contents\n  3. Copyright\n  4. Dedication\n  5. After the Hurricane\n  6. \u23af\n    1. In the Supply Closet at Illing Middle\n    2. Free Armchair, Worcester\n    3. The First Time\n    4. Natick\n    5. Again\n    6. Brothers\n    7. Workbench\n    8. Tap Out\n    9. Deciding\n    10. Graduation\n  7. \u23af\n    1. Blue\n  8. \u23af\n    1. Graffiti\n    2. My Father at 49, Working the Night Shift at B&R Diesel\n    3. When Charlie Pulls the Colorado Over\n    4. V.F.W. Post #2046\n    5. Franklin Free Clinic\n    6. Safety\n    7. Close\n    8. After the Attempt\n    9. Vows\n    10. In the Sideyard\n    11. Dundalk\n    12. Michael\n  9. \u23af\n    1. Next of Kin\n  10. \u23af\n    1. Training\n    2. Going\n    3. Dry Season\n    4. Kabekona\n    5. Farmsitting\n    6. Mud Season\n    7. For Good\n    8. Salvage\n    9. Piecework\n    10. My Father at 23, on the Highway Side of an Overpass Fence\n    11. Window Washers\n    12. Interim\n  11. \u23af\n    1. Behind the Eyes, & Shining\n  12. Acknowledgments\n  13. About the Author\n  14. Connect with HMH\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# **STALIN**\n\n# **STALIN**\n\n# **NEW BIOGRAPHY OF A DICTATOR**\n\n**Oleg V. Khlevniuk**\n\n_Translated by_ NORA SELIGMAN FAVOROV\n\nYale University Press gratefully acknowledges the financial support given for this publication by the Smith Richardson Foundation.\n\nPublished with assistance from the foundation established in memory of Calvin Chapin of the Class of 1788, Yale College.\n\nCopyright \u00a9 2015 by Oleg Khlevniuk.\n\nEnglish translation copyright \u00a9 2015 by Yale University.\n\nAll rights reserved.\n\nThis book may not be reproduced, in whole or in part, including illustrations, in any form (beyond that copying permitted by Sections 107 and 108 of the U.S. Copyright Law and except by reviewers for the public press), without written permission from the publishers.\n\nYale University Press books may be purchased in quantity for educational, business, or promotional use. For information, please e-mail sales.press@yale.edu (U.S. office) or sales@yaleup.co.uk (U.K. office).\n\nSet in Utopia, Bodega, and Aller type by Integrated Publishing Solutions, Grand Rapids, Michigan.\n\nPrinted in the United States of America.\n\nISBN: 978-0-300-16388-9 (cloth)\n\nCatalogue records for this book are available from the Library of Congress and the British Library.\n\nThis paper meets the requirements of ANSI\/NISO Z39.48\u20131992 (Permanence of Paper).\n\n10 9 8 7 6 5 4 3 2 1\n_In memory of my wife Katya (1961\u20132013)_\n\n## **CONTENTS**\n\nPreface\n\n_The Seats of Stalin's Power_\n\n**1. BEFORE THE REVOLUTION**\n\n_The Bulwarks of Stalin's Power_\n\n**2. IN LENIN'S SHADOW**\n\n_A World of Reading and Contemplation_\n\n**3. HIS REVOLUTION**\n\n_Trepidation in the Inner Circle_\n\n**4. TERROR AND IMPENDING WAR**\n\n_Patient Number 1_\n\n**5. STALIN AT WAR**\n\n_Family_\n\n**6. THE GENERALISSIMO**\n\n_The Dictatorship Collapses_\n\n**THE FUNERAL: THE _VOZHD,_ THE SYSTEM, AND THE PEOPLE**\n\nIllustrations\n\nNotes\n\nAcknowledgments\n\nIndex\n\n## **PREFACE**\n\nFor more than two decades, I have been studying this man and the causes and logic underlying his actions, which upended or utterly destroyed millions upon millions of lives. This work has been stressful and emotionally draining, but it is my vocation. Lately, the paradoxical turns of recent Russian history, the large-scale poisoning of minds with myths of an \"alternative\" Stalin\u2014one whose effective stewardship is held up as a model worthy of emulation\u2014have given my research more than scholarly relevance.\n\nThe literature on Stalin and his era is impossibly vast. Even scholars of Stalinism freely admit to not having seen the half of it. Within this vastness, serious, meticulously documented research coexists with slapdash pen-pushing carelessly cobbled together out of anecdotes, rumors, and fabrications. The two camps\u2014historical scholarship and lowbrow (usually pro-Stalin) ramblings\u2014rarely cross paths and have long since given up the idea of reconciling.\n\nScholarly biographies of Stalin have gone through the same stages as the historiography of the Soviet period overall. I have a high regard for some classics written at a time when Soviet archives were completely inaccessible. Two authors who stand out are Adam Ulam and Robert Tucker. Back in the 1970s, historians of the Stalin period resembled specialists in antiquity: they tended to know the few available documents and memoirs inside out and had little ability to expand their number. This dearth of documentation encouraged the painstaking study of these sources and elegant and thoughtful extrapolation. The situation was bound to change after the archival floodgates were opened in the early 1990s, and it took us some time to get our heads above water. The eventual appearance of new works informed by archival materials\u2014including scholarly biographies of Stalin, as well as other investigations of the man and the political system\u2014signal that historians have begun to cope with the inundation.\n\nThe opening of the archives gave rise to a new genre of Stalin biography that one might call \"the archival expos\u00e9.\" It's trailblazers include Dmitri Volkogonov, a former party loyalist who became a driving force for perestroika, and the Russian playwright Edvard Radzinsky. This genre favors personal accounts over \"dry\" statistics or administrative paper trails and page-turning narratives over painstaking research and historical contextualization. For many readers, the archival expos\u00e9 has played an important role in shaping Stalin's image.\n\nOne of the most successful Western authors working to feed appetites for newly available details about the Stalin era is Simon Sebag Montefiore. A notable feature of his method is the citation of a broad spectrum of sources, not only from memoirs and interviews, but also from the archives. Montefiore struck a sort of middle ground, striving to instill some scholarly discipline into the \"archival expos\u00e9s\" genre while producing readable history capable of attracting a wider audience than more scholarly texts.\n\nIn today's Russia, on the other hand, Stalin's image is primarily being shaped by pseudo-scholarly apologias. An extremely diverse array of authors, all with their own motivations, contributes to Stalinist mythology. Most of these authors blend a lack of the most elementary knowledge with a willingness to make bold assertions. Their apologias typically cite fabricated sources or shamelessly misrepresent real ones. The impact of this powerful ideological assault on readers' minds is intensified by the circumstances of Russian life, which include rampant corruption and outrageous social iniquities. When they reject the present, people are more likely to idealize the past.\n\nApologists for Stalin no longer try, as they once did, to deny the crimes of his regime. Instead they resort to more subtle rewritings of history. In their version of events, lower-level officials, such as secret police chiefs and the secretaries of regional party committees, supposedly hiding their actions from Stalin, instigated mass repression. The most cynical Stalinists take a different tack, claiming that the Terror was just and that the millions destroyed on Stalin's orders really were \"enemies of the people.\"\n\nMany Russian Stalinists find it convenient to draw on theories developed by various Western historians: that the Terror developed spontaneously, that Stalin was not deeply involved in it, and that he was a far more \"ordinary\" political leader than usually thought. It is certainly not my intention to accuse my Western colleagues of fomenting re-Stalinization. They bear no more responsibility for Russia's contemporary political battles than Marx did for the Bolshevik revolution. Still, we should be aware that our words can have bizarre reverberations.\n\nOne variety of apologia widely cultivated in Russia's intellectual and political soil is the relatively moderate idea of \"modernizing Stalinism.\" While this ideology formally acknowledges the Terror's countless victims and the high price paid for the \"great leap\" strategy, it sees Stalinism as an organic and unavoidable means of addressing the need to modernize and prepare for war. Within these postulates we can detect prejudices deeply rooted in the Russian social consciousness: that the interests of the state take absolute priority, that the individual is insignificant, that the flow of history is governed by higher-order laws. According to this paradigm, Stalin was the expression of an objective historical need. His methods were regrettable but necessary and effective. Furthermore, it is inevitable that the flywheel of history will become spattered with blood.\n\nIt would be wrong to deny that the \"long waves\" of Russian history helped shape the path toward Bolshevism and Stalinism. A strong state with authoritarian traditions, feeble private property and civil society institutions, and the colossal reach of a colonizing power that enabled, among other things, the creation of the Gulag Archipelago, all paved the way toward the Stalinist system. But elevating these factors to some sort of \"Russian destiny\" leads to the dead-end theory of \"inevitable Stalinism.\" Adherents of this theory have little interest in specific facts and prefer to recycle Stalinist interpretations of Soviet history, sometimes with a fresh twist, more often without. They adamantly dismiss questions about the price paid for transformations and military victories, alternative development paths, and the role of the dictator. They close their eyes to the fact that Stalin himself, when he brought matters to a state of crisis and ruin, was occasionally forced to soften his policies, thereby demonstrating that even within the framework of Stalinism there were multiple paths toward industrialization. They do not even try to explain how the executions of seven hundred thousand people in 1937\u20131938 alone, ordered by Stalin, served the goals of modernization. Overall, the theory of modernizing Stalinism makes no serious attempt to ascertain how effective the Stalinist system was or to evaluate Stalin's own role in the development of the USSR from the 1920s to the early 1950s.\n\nReducing history to historical imperative is the least creative way of presenting the past. Historians are compelled to deal not with simple schemes and political conjecture but with concrete facts. Working with documents, they cannot avoid noticing the intricate dance between objective factors and personalities or between pattern and random occurrence. In a dictatorship, the role of the dictator's personal predilections, prejudices, and obsessions is greatly magnified. What better medium than biography to unravel this complex tangle of problems?\n\nBiography is a unique genre of research that can, at one extreme, be reduced to the minutia of historical context or, at the other, be bloated with novelistic details of human behavior. Context without soul and soul without context\u2014these are the main pitfalls confronting the biographer. Navigating them was a challenge for me. In the end, I understood that it was simply not possible to squeeze into this book even a passing reference to every significant episode or aspect of the Stalin period. I was compelled to choose which phenomena and tendencies most deserved inclusion, selecting the facts and events that seemed to characterize Stalin, his time, and the system that bears his name with the greatest clarity and vividness. This selectivity was all the more necessary given the appearance, over the past twenty years, of so many new sources shedding light on Stalin and his period. These sources should be briefly identified.\n\nFirst, because of the opening of the state archives after the collapse of the Soviet Union, historians now may consult original firsthand documents, whereas in the past they were forced to whittle layers of distortion from official publications. A good example is the works and speeches of Stalin himself. Most were published during the leader's lifetime, but we now have the ability to work with the originals and compare what was actually said with edited versions. Furthermore, the body of Stalin's published speeches can now be supplemented with those that did not appear in print. Among the most important documents are papers generated by governmental bodies that Stalin himself chaired, such as the protocols and stenographic records of Politburo meetings and wartime State Defense Committee decrees. These dry bureaucratic documents are tremendously important in understanding Stalin's personality and life. They took up a huge portion of the dictator's time and were the tools by which he exercised power. Many resolutions bear traces of his heavy editorial hand.\n\nBy themselves, of course, the orders issued under Stalin paint only a partial picture. Why were they adopted? What were the logic and motives behind his directives? Much more revealing is Stalin's intermittent correspondence with his Politburo colleagues, conducted primarily when he was away on vacation and requiring letters to steer the actions of his fellow leaders back in Moscow. This correspondence was most prolific in the 1920s and the first half of the 1930s, before Russia had any reliable telephone service. It is a marvelous example of how sluggish technological progress can be a historian's friend. After the war, telephone communication became more reliable, and Stalin, now securely at the pinnacle of power, felt less need for detailed correspondence with subordinates. Curt directives sufficed. Despite their fragmentary nature, Stalin's letters constitute an important documentary whole and make for fascinating reading. They represent the most candid testaments he has left to posterity.\n\nHistorians have been able to glean a great deal of important information from the logs of visitors to Stalin's Kremlin office. These logs recorded visitors' names and the times they entered and left the office and thus shed light on how Stalin conducted business. Comparing them with other sources (such as memoirs or the protocols of Politburo meetings) offers important clues to the circumstances surrounding the adoption of various resolutions. Still, like his correspondence, these logs reflect only a portion of Stalin's activity. In addition to his Kremlin office, he occasionally worked in his office at Central Committee headquarters on Staraia Square and received visitors in his Kremlin apartment, as well as at his numerous dachas outside Moscow and in the south. Although we know that the service responsible for protecting Soviet leaders kept records of visits to Stalin's Kremlin apartment, researchers have yet to be given access to this archive. There appears to be no sign of analogous records for the Central Committee office or the dachas.\n\nThe visitor logs were kept by Stalin's secretariat and security team. It seems likely that these services also kept, for their own purposes, records of Stalin's movements, as well as accounts by security personnel of what happened during their shifts. It goes without saying that these materials would be of tremendous value to Stalin's biographers. At this point, there is no solid evidence that such records exist.\n\nStalin's correspondence and the log of visitors to his Kremlin office are both part of his personal archive, which was compiled under his direct supervision and apparently with an eye toward history. Many documents in this collection feature the notations \"my archive\" or \"personal archive.\" An important addition to the personal archive is an assortment of materials about Stalin gathered from various repositories. This assortment, which includes books from Stalin's library with notations by him, was concentrated in the Central Party Archive. Today both sets of materials have been brought together in the Stalin Collection of the Russian State Archive of Social and Political History (RGASPI, successor to the Central Party Archive, which comprises the bulk of its holdings), a key source of knowledge about Stalin now used extensively by historians.\n\nYet despite its importance, the Stalin Collection has serious deficiencies. It offers only limited insights into Stalin's modi vivendi and operandi. Its primary shortcoming is the absence of much of the vast array of papers that made their way to Stalin's desk on a daily basis. These include thousands upon thousands of letters, statistical compilations, diplomatic dispatches, and reports and memoranda from the various branches of state security. The lack of access to these documents hinders historians in their effort to develop a thorough understanding of how well informed Stalin was, what he knew about a given question, and thus the logic of his actions. The documents that would enable such insights have not been lost. They reside in the Presidential Archive of the Russian Federation (APRF, the former Politburo Archive), organized into \"thematic\" folders. While working on this book, I was able to examine a few of them. For the time being, the Presidential Archive does not accommodate systematic scholarly study. However, the very fact that these folders exist encourages hope. The history of Russia suggests that sooner or later the archive will open.\n\nThe most tempting sources for biographers are always diaries and memoirs. These contain the sorts of three-dimensional treatments of people and events that are hard to extract from official paperwork. Such firsthand accounts permit biographers to fill their works with attention-grabbing details, but historians are well aware of these sources' liabilities. Memoirists, even candid ones, are rarely disinterested, and they often muddle events and dates or simply lie. These perils are compounded in memoirs from the Soviet era. As far as we know, no member of Stalin's inner circle kept a diary, depriving us of the kind of detailed source that Goebbels's famous diaries provided to Hitler's biographers. The situation with memoirs is not much better. Only two people close to Stalin left detailed reminiscences: Nikita Khrushchev and Anastas Mikoyan. While these memoirs represent major contributions, both men were silent on important topics (such as their participation in the mass repression), and there was much that they simply did not know. Within Stalin's inner circle there was a strict rule: each man was privy only to information that he needed for the effective fulfillment of his duties. In the case of Mikoyan, some elements of his memoirs were distorted by his son, who prepared the manuscript for publication. He arbitrarily and without the customary disclosures simply inserted his own additions and revisions into the dictated text, supposedly based on subsequent accounts shared by his father.\n\nWe also have memoirs by Soviet and foreign officials and other prominent figures who had some\u2014usually extremely limited\u2014interaction with Stalin. These works make a minor contribution to what we know about his life. In additional, many memoirs (for example by Red Army marshals) were published during the Soviet era and were therefore subjected to censorship (including self-censorship). After the fall of the USSR, many other people whose paths had crossed with Stalin's spoke up. Freedom sparked a flood of memoirs from the children and relatives of Stalin-era leaders. This \"children's literature,\" as the Russian historian Elena Zubkova so aptly labeled the genre, was mainly motivated by commerce and a passion for self-justification, and the results are indeed juvenile. Many relatives of Stalin and his comrades concocted fairy tales and cock-and-bull stories, blending personal impressions with fantasy. Naive pronouncements on politics serve to show that these offspring had only the faintest idea of what their fathers were up to. Third-hand information, rumors, and gossip abound. The primary factor detracting from the potential value of this literature is that Stalin's underlings were obsessed with maintaining strict secrecy. They lived with unrelenting secret police surveillance and the constant fear of being provoked into a politically fatal slip of the tongue. It is difficult to imagine what could have compelled them to be candid within their own families. The price was too high.\n\nIn this book I have been restrained in my use of memoirs, even though many contain fascinating descriptions and anecdotes readers would certainly find of interest. Guided by the most basic rules of source verification, I have made every effort to compare memoir accounts with other materials, archival materials first and foremost. On one hand, memoirs that generally held up to scrutiny were given greater credence. On the other hand, numerous errors and flagrant fabrications were treated as clear signs of unreliability, even if some claims could not be proved false through other sources. Certain memoirs were put on my personal blacklist. While I do not condemn others for citing these works, I will never do so.\n\nWhen all is said and done, however, a historian endeavoring to write a biography of Stalin is in a relatively good position. The abundance of archival documents and evidence offers opportunities for prolonged, intensive, and (one can hope) fruitful work. Significant lacunae and the inaccessibility of many materials are frustrating impediments; nevertheless, it is now possible to write a genuinely _new_ biography of Stalin insofar as newly accessible archival material has forced changes in our understanding of both the man and his era.\n\nI would like to add a few final words about the size and structure of this biography. Restraints in the former have inspired innovations in the latter. Exhaustive details had to be forsaken. References and notes had to be kept to a minimum, so priority has been given to the attribution of quotes, numbers, and facts. By no means all of the worthy works of my colleagues have been mentioned, for which I offer them my apologies. Such economies leave me ambivalent. I regret the omission of many telling facts and quotes, but I am glad for the reader. I know how it feels to gaze wistfully at stacks of fat tomes that will never be conquered.\n\nAnother aspect of the book that I hope will facilitate reading, in addition to its modest size, is its structure. A conventional chapter-section chronology did not lend itself to presenting the two interdependent strata of Stalin's biography: the sequence of his life events and the most salient features of his personality and dictatorship. This difficulty gave rise to the idea of two alternating narratives, a sort of textual _matryoshka_ or Russian nesting doll. One conceptual chain examines Stalin's personality and system of rule against the backdrop of his final days. The other, more conventionally chronological, follows the main stages of his biography in sequence. As a result, the book can be read in two ways. Readers can trust my arrangement and follow the page order, or they can take one stratum at a time. I have tried to make both methods equally convenient.\n\n## **THE SEATS OF STALIN'S POWER**\n\n**The early morning hours of 1 March 1953 at the near dacha. The \"Five's\" last supper.**\n\nOn Saturday, 28 February 1953, Josef Stalin invited four of his senior associates to the Kremlin: Georgy Malenkov, Lavrenty Beria, Nikita Khrushchev, and Nikolai Bulganin. During the final six months of his life, Stalin and these four men constituted what was known as the \"ruling group\" or simply the \"Five.\" They met regularly in Stalin's home. The leader's other old friends\u2014Vyacheslav Molotov, Anastas Mikoyan, and Kliment Voroshilov\u2014were in disgrace, and he did not wish to see them. Assembling a small group of supporters to act as his right hand in ruling the country was a key element of Stalin's modus operandi. He liked to name these groups according to the number of members: the Five (Piaterka), the Six (Shesterka), the Seven (Semerka), the Eight (Vos'merka), the Nine (Deviatka). These informal groups enjoyed supreme authority while formal party and state structures functioned as regular bureaucracies handling the day-today running of the country. Dividing government into formal and informal institutions allowed the dictator to exploit the capabilities of a vast, all-encompassing bureaucratic machine while keeping a firm hold on the true levers of power. Stalin often changed the composition of the ruling group. He maintained daily, hands-on control over this central node of power, keeping its members at his constant beck and call for meetings and \"friendly\" gatherings. The dictator's approach to exercising power through a combination of bureaucratic institutions and patrimonial power inspired Yoram Gorlizki to coin the phrase \"neopatrimonial state.\"\n\nFear was the primary force behind the dictator's patrimonial power over his top associates and other highly placed officials. With the Soviet state security system under his firm control, Stalin could arrest anyone at any moment and have the person summarily shot. He did so countless times. The entire patrimonial political enterprise rested on terror.\n\nThe most important decisions were always made through direct\u2014ideally face-to-face\u2014communication with the dictator. This was the fastest and most effective way for an official to achieve personal and administrative objectives. But communication required access to the seats of power, places that for countless Soviet officials and members of the top leadership took on an almost sacred aura. Some were more sacred than others. There was an unspoken hierarchy in the various settings from which Stalin wielded power, and admission to some endowed greater status than others. Stalin spent a significant portion of his life in these seats of power. Each reflected some aspect of his personality and dictatorship.\n\nThe primary and most official seat of power was Stalin's Kremlin office. This commodious, oak-paneled study was divided into two zones: Stalin's desk and a long conference table. Other furnishings included a grandfather clock (which Stalin used to monitor the promptness with which those summoned arrived) and a plaster death mask of Lenin encased in glass and displayed on a special stand. On the walls hung portraits of Lenin and Marx. During the war, they were joined by the tsarist-era military heroes Aleksandr Suvorov and Mikhail Kutuzov. Otherwise the decor hardly changed over the many years he spent there. During the war, the bomb shelter built beneath the Kremlin contained a slightly smaller but otherwise almost exact replica of this office: the same furniture, the same pictures, the same curtains (despite the lack of windows).\n\nOver thirty years, approximately three thousand different people visited the Kremlin office. Stalin's closest associates, of course, were there frequently, but the visitors also included heads of government ministries and enterprises, academics, cultural figures, senior state security and military personnel, and foreign guests. The Kremlin office was the most accessible of the seats of Stalin's power.\n\nOn the evening of 28 February 1953, Bulganin, Beria, Malenkov, and Khrushchev, who had been called to the Kremlin by Stalin, did not linger in this office. Stalin immediately took them to the Kremlin movie theater, a much more exclusive place. The theater was a 7.5-by-17-meter space with twenty seats, installed in 1934 where Russia's tsars had once enjoyed a winter garden. Before it was built, Soviet leaders watched movies either outside the Kremlin, in the building of the cinematography directorate, or in a small Kremlin room that had been used for silent films. Stalin enjoyed watching movies with his comrades, and these viewing sessions gradually became obligatory. Thanks to detailed records kept by Boris Shumiatsky, who oversaw the Soviet film industry, we know quite a bit about how these evenings in the movie theater were spent during 1934\u20131936. Shumiatsky would bring the movies and listen to the comments of Stalin and his colleagues, as well as the decisions that were sometimes taken during a viewing. His notes provide a valuable window onto the rules of behavior within Stalin's inner circle and the atmosphere of these gatherings.\n\nAs a rule, the viewing sessions began late in the evening and extended into the early hours of the morning. Stalin sat in the front row, surrounded by members of the top leadership. There was always a great deal of discussion about the movies and newsreels, both while a film was rolling and afterward. Stalin always had the first word. He would issue instructions concerning the content of specific films, the Soviet film industry, and ideology in general. In the movie theater, he made on-the-spot decisions on everything from budgetary issues to the publication of policy-setting articles in the Soviet press to personnel matters. Filmmakers would occasionally be invited to viewings of their films. Such an invitation was a great honor. Stalin would congratulate them on their work and offer \"guidance\" on improving it. Shumiatsky's records make it clear that these get-togethers in the Kremlin movie theater were not merely relaxation for the Soviet leadership. They were informal meetings of the top level of government at which questions of ideology and cultural policy were decided. Most likely, Stalin and his colleagues also discussed other affairs of state before and after the viewings.\n\nShumiatsky's records end abruptly in early 1937. This was undoubtedly tied to the intensification of repression in the country. Shumiatsky himself was arrested in early 1938 and shot soon after. Stalin's movie viewing continued, but we know almost nothing about later sessions. It appears that toward the end of his life, only his closest associates were admitted to the Kremlin movie theater. The 28 February meeting of the Five was Stalin's last movie-theater get-together.\n\nWhen the movie was over, Stalin, as he often did, invited the others to dine with him at his dacha near the Moscow suburb of Volynskoe. The dacha was just a few minutes away, earning it the nickname of \"the near [dacha]\" _(blizhniaia)._ Occasionally the seat of power would shift to one or another of the houses or dachas around Moscow or in the south, where Stalin spent lengthy annual vacations. But the \"near\" dacha held a special place in his heart. It was an important epicenter of his life and rule.\n\nThe first house on the site of the near dacha was built in 1933. The move there was associated with upheavals in Stalin's personal and political life. The terrible famine that swept the land in the early 1930s as a result of Stalin's policies coincided with family tragedy. In November 1932 his wife, Nadezhda Allilueva, died by her own hand. Stalin started a new life in a new place.\n\nStalin personally oversaw the near dacha's many expansions and renovations. The huge house that resulted was an odd blend of the institutional and the pretentious. All the rooms resembled one another and were, in the words of Stalin's daughter Svetlana, \"impersonal.\" The second floor, for which an elevator had been installed, was rarely used. Stalin's favorite room toward the end of his life was the so-called \"small dining room\" on the first floor. This roomy space contained a rectangular table three meters long, a couch, a cupboard, an easy chair, a small telephone table, and a fireplace. A pair of binoculars, hanging from a hook, and a hunting rifle were kept next to the fireplace. A large carpet covered the floor. The room led to a glassed-in veranda and a terrace. According to Svetlana, Stalin both slept and worked in this room. The large table was always piled with papers and books. Unless he had company, he ate at one of the table's corners. He kept his medicines in the cupboard. Stalin enjoyed sitting by the fire, where he would sometimes order shashlik to be roasted. He liked to receive his visitors here. It is also where he suffered the stroke that ended his life.\n\nThe dacha was surrounded by a fifty-acre park. Stalin personally oversaw the landscaping and farming that took place on the grounds. He designed a greenhouse for citrus plants, supervised the installation of a vineyard, grew his own watermelons, and kept a pond stocked with fish. He sometimes had a portion of his watermelon crop sent to Moscow stores. There were also horses, cows, chickens, ducks, and a small apiary. His bodyguards testified that Stalin devoted a great deal of time to the running of this agricultural enterprise and kept track of even the smallest details. Hundreds of orders from Stalin to the man in charge of running the estate, Lieutenant Colonel P. V. Lozgachev, have been preserved:\n\n7 April 1950: a) Start planting watermelons and melons in raised beds on 10 May; b) In mid-July, trim the watermelon and melon vines....\n\n20 April:... Line the path from the kitchen to the pond with fir trees.... Plant corn every half meter next to the main house and between apple trees by the pond, toward the gazebo. Plant beans there too.... Plant eggplant, corn, and tomatoes along the edge of the garden.\n\nLozgachev reported that he received such instructions almost every day. In essence, Stalin was the master of a small estate that he preferred to run himself, not leaving important details in the hands of subordinates. The patriarchal way in which he ran his dacha estate is consistent with his approach to running his much larger \"estate,\" the Soviet Union. He kept track of state resources and reserves and took charge of their allocation, jotting down important pieces of information in a special notebook. He immersed himself in the details of film scripts, architectural plans, and the design of military hardware. His interest in landscaping extended beyond his personal domain to the streets of Moscow: \"People say that the square on the Arbat... has not yet been covered with paving blocks (or asphalt). This is shameful!... Put pressure on them and make them finish up the square.\"\n\nOne result of Stalin's desire to shape the spaces around him was the creation of a room that served as the dacha's social nexus: a 155-square-meter hall. The room's centerpiece was a 7-meter-long table that stood on a 6-by-12-meter rug. (The area of this rug, incidentally, equaled the average living space of sixteen Soviet city dwellers in 1953: 4.5 square meters per person.) Easy chairs and couches lined the walls. Occasionally Stalin worked at the table in this large room or on the couch or easy chairs. For the most part, however, the room was reserved for meetings and festive gatherings.\n\nA number of participants in these gatherings, which were held regularly, have left descriptions. The food was simply placed on the table, and guests helped themselves to whatever they wanted and took their plates to any free seat. Dinner stretched for many hours, ending long after midnight or even at daybreak. These meals were an opportunity to discuss and decide various matters of state. But that was not all. For Stalin, they were a way of keeping an eye on his associates and gleaning information. As one of the few forms of entertainment available to him, they also filled an important social need: they eased his sense of isolation. As Khrushchev wrote, \"He felt so alone he didn't know what to do with himself.\"\n\nPlenty of drinking went on around this table. As he aged, Stalin moderated his own consumption of spirits, but he liked to spur others to overdo it and then watch their behavior. He had several ways of forcing his guests to drink more heavily than they might have wished. Toasts were proposed in rapid succession, and failing to empty one's glass was unacceptable. \"If someone didn't participate when a toast was made, he was 'fined' by having to drink another glassful and perhaps several glasses.\" The Yugoslav politician and writer Milovan Djilas later recalled a drinking game he witnessed at Stalin's dacha during a visit in January 1948: \"Everyone guessed how many degrees below zero it was outside and then, as a penalty, downed... a glass for each degree he was off.... I remember that Beria missed by three and claimed that he had done so on purpose to get more vodka.\"\n\nThe alcohol loosened inhibitions. \"The atmosphere at these dinners was unconstrained, and jokes, many of them obscene, evoked raucous laughter.\" In addition, there were other, more \"cultured\" amusements. Sometimes they sang revolutionary and folk songs in which, the wife of Andrei Zhdanov recalled, Stalin would join with a quiet tenor. Zhdanov entertained his comrades with lewd ditties. \"Such songs could be sung only at Stalin's. You couldn't possibly repeat them anywhere,\" Khrushchev recalled. For a while a piano stood in the large room. Some remembered Zhdanov playing it, although there is no clear record of what he played or how well. After Zhdanov's death in 1948, Stalin ordered the piano moved to an adjacent room. More often, music was provided by a radiogram (a combination radio and phonograph), on which Stalin played records, both Russian folk songs and classical music. Sometimes he enjoyed listening to his impressive collection of some 2,700 albums, on his own or with guests. Occasionally there was dancing. According to Khrushchev, Mikoyan was considered the best dancer. Everyone did the best he could. Even Stalin \"would move his feet around and stretch out his arms.\"\n\nThere probably was no dancing during those early hours of 1 March. This was a quiet get-together, limited to Stalin's most trusted associates. \"We would go to Stalin's place quite often, almost every evening,\" Khrushchev recalled of that period. These dinner gatherings with the aging and unbalanced Stalin were not easy on his guests. In Khrushchev's words, \"We were supposed to work at our jobs and the posts to which we had been elected and, besides that, attend Stalin's dinners like some sort of characters in a play and entertain him. That was a difficult and painful time for us.\" But Stalin's comrades were not about to complain, and they assiduously fulfilled their dinner duties as a condition of their inclusion in the ruling circle. As usual, the gathering adjourned toward morning (Khrushchev places its conclusion around five or six a.m.). They parted on a good note. As Khrushchev described it, \"Stalin was a bit tipsy and seemed very well disposed toward everyone.\" He led his guests into the vestibule, \"joked a lot, waved his hands around, and as I recall he poked me in the stomach with his finger and called me Mikita. When he was in a good mood, he always used the Ukrainian form of my name\u2014Mikita.... We too were in a good mood when we left because nothing unpleasant had happened at the dinner, and not all these dinners ended that well.\" We have no reason to doubt Khrushchev's account. Dmitri Volkogonov claimed that Stalin was irritable and threatened his guests, but he does not cite any specific sources.\n\nStalin was equally capable of rewarding his underlings with his amiability and menacing them with threats. For almost two decades he used both the carrot and the stick (in Russian, the knout and the ginger cookie, with a good deal more of the former) to keep not only his close associates in hand, but also the many millions who lived in the USSR and, later, the entire \"socialist camp.\"\n\nOver his seventy-four-year life, the Soviet dictator fought through a stormy historical landscape to become an important factor in events not only in Russia, but also the world. Among scholars, there is more agreement than controversy on the historical and ideational antecedents that shaped him, including traditional Russian authoritarianism and imperialism, European revolutionary traditions, and Leninist Bolshevism. These influences, of course, do not diminish his major personal contribution to the formation of a uniquely Soviet totalitarian system and ideology. Ideological doctrines and prejudices were often decisive in Stalin's life and actions, but instead of receiving them passively, he adapted them to the interests of his own dictatorship and emerging superpower. His personality also played no small role in the political course he forged. He was cruel by temperament and devoid of compassion. Of all the available methods for resolving political, social, and economic conflict, he favored terror and saw no reason to moderate its use. Like other dictators, he was stubborn and inflexible. Concession and compromise were seen as a threat to the inviolability of his power. He made limited and half-hearted reforms only when socioeconomic crises were reaching the breaking point and the stability of the system was imperiled. His theoretical dogmatism lay at the root of the violence that defined his regime.\n\nUnderpinning Stalin's worldview was an extreme anti-capitalism. His hostility toward this system was unequivocal, and he rejected even the limited concessions that Lenin made in instituting the New Economic Policy (NEP). Stalin grudgingly allowed a few capitalist economic vehicles within the Soviet system, such as money, limited market relations, and personal property. After millions had died during the famine of 1932\u20131933, he agreed to allow peasants limited freedom to produce and sell outside the collective and state farm system. But to the end he believed that the concessions that had been forced on him by hard circumstances would soon be reversed and the socialist economy would be transformed into a money-free powerhouse where people would work as ordered by the state and receive in exchange the natural goods that the state decided they needed.\n\nIn Stalin's worldview, the state the Bolsheviks created was an absolute. All existence was completely and unconditionally subordinate to the state, and its highest personification was the party and its leader. Personal interests were recognized only to the extent that they served the state, which had the unquestioned right to demand from people any sacrifice, including their lives. The state was unrestricted in its actions and could never be wrong, as it represented the ultimate truth of historical progress. Any action by the regime could be justified by the greatness of its mission. Mistakes and crimes by the state did not exist; there was only historical necessity and inevitability or, in some cases, the growing pains of building a new society.\n\nThe primary tool used to compel submission to the state and suppress the individual and the social was the so-called \"class war\" against foreign and domestic \"enemies.\" In this war, Stalin was the foremost theoretician and a ruthless tactician. With the successful advance of socialism, he asserted, the class war would only intensify. This idea was a cornerstone of his dictatorship. As a means of interpreting reality, the class war theory was also a powerful propaganda tool. Inadequate political and economic outcomes, the hardships endured by the populace, and military failures could all be explained by the underhanded scheming of \"enemies.\" As a method of state repression, class war gave the Terror the breadth and brutality of an actual war. The Soviet dictator has earned the distinction of being the organizer and director of one of the most powerful and merciless terror machines known to history.\n\nStalin had no trouble reconciling Marxist and Bolshevik-Leninist dogma with great-power imperialism. In November 1937, he told his associates the following: \"The Russian tsars did many bad things. They plundered and enslaved the people. They waged wars and grabbed territory in the interests of the landowners. But they did do one good thing\u2014they created a huge state that stretches all the way to Kamchatka. We have inherited that state. And for the first time we, the Bolsheviks, have brought together and consolidated this state as a single, indivisible state... for the benefit of the workers.\" These candid words are all the more telling as they were spoken at a dinner celebrating the twentieth anniversary of the October Revolution, the country's main revolutionary holiday. In the international arena, Stalin's expansion of the empire makes him a worthy heir to the Russian tsars. Only the ideological fa\u00e7ade was different. At the Berlin train station on the eve of the Potsdam Conference in 1945, U.S. Ambassador to the Soviet Union Averell Harriman asked Stalin how it felt to arrive in the capital of a defeated enemy as a victor. Stalin replied, \"Tsar Alexander made it all the way to Paris.\" Yet Stalin arguably outdid the tsars. The Soviet empire expanded its sphere of influence to encompass huge swaths of Europe and Asia and transformed itself into one of the world's two superpowers.\n\nDid Stalin look back on his triumphs after parting with his guests for the last time in his life on 28 February? Did his thoughts take him to earlier times\u2014his childhood, youth, the revolution? Like the lives of his fellow revolutionaries, Stalin's life was cleanly divided into two parts: before and after the revolution. Conceptually and chronologically, these two periods were approximate halves of his life. The first thirty-eight of his seventy-four years were lived before the revolution, and twenty of them were spent actively working toward it.\n\n## **1 BEFORE THE REVOLUTION**\n\nAccording to his official Soviet biography, Stalin was born in 1879. In fact Ioseb Jughashvili (his birth name) was born one year earlier. Stalin knew, of course, when and where he was born: in the small Georgian town of Gori, in a far corner of the vast Russian Empire. A Gori church register (part of Stalin's personal archive) provides the exact date: 6 December 1878. This date can also be found in other documents, such as his graduation certificate from the Gori Theological School. In a form filled out in 1920, his year of birth is again given as 1878. But the year 1879 began to appear in paperwork completed by his various helpers, and that date was used in all encyclopedias and reference materials. After he had consolidated power, a grand celebration was held in honor of his fiftieth birthday on 21 December 1929. There was confusion over not only the year of his birth, but also the day, given as 9 December (Old Style) instead of 6 December. This inaccuracy came to the attention of historians only in 1990. The reason for it has yet to be determined. One thing is clear: in the 1920s, Stalin decided to become one year younger. And he did.\n\nLegends surround Stalin's parentage. Sensation seekers proclaimed Ioseb (who later became Iosif once his interactions began to be primarily in Russian) to have been the illegitimate son of a prosperous merchant, a factory owner, a prince, and even Emperor Alexander III, who supposedly was attended to by Ioseb's mother while the emperor was visiting Tiflis. The historical record suggests more prosaic origins. Ioseb was born into a humble Georgian family. His mother, Ekaterine or Keke (Yekaterina in Russian) Geladze, the daughter of serfs, was born in 1856. In 1864, after the abolition of serfdom, her family moved to Gori, where, at the age of eighteen, she was given in marriage to the cobbler Besarion or Beso (Vissarion in Russian) Jughashvili, six years her senior. Their first two children died in infancy; Ioseb (Soso) was the third.\n\nFew pieces of documentary evidence survive from Stalin's youth. The primary source of our knowledge is memoirs written after he had already attained the pinnacle of power. Even an uncritical reader will notice that these memoirists are writing about the childhood and youth of a future dictator, not the early years of Ioseb Jughashvili. This aberration magnifies the tendency, common to biographies generally, toward selective exaggeration and exclusion. Depending on the situation and the writer's politics, emphasis is placed on either Ioseb's virtues and leadership qualities or his innate cruelty and psychological abnormalities. But as Ronald Grigor Suny has shown, attempts to find the future dictator in the child Ioseb Jughashvili are highly suspect.\n\nIt is commonly believed that Ioseb had a difficult childhood. Abuse and beatings by his drunkard father, as well as material deprivation, supposedly embittered the boy and made him ruthless and vindictive. But there is plenty of evidence to support a very different picture. By many measures, Stalin's childhood was ordinary or even comfortable. A number of accounts attest that his father was not only a skilled cobbler, but also that he was able to read Georgian and converse in several languages, including Russian. His mother had received some home schooling and could also read and write in Georgian. Given the low literacy rate in Georgia at the time, this would have given the family an advantage. During Ioseb's early years, Besarion Jughashvili apparently was quite successful and his family was well provided for.\n\nLater, after Besarion began to drink heavily and then abandoned his wife and child, responsibility for Ioseb's upbringing fell on his mother's shoulders. Ekaterine was a woman of strong character and a hard worker, and, starting with odd jobs, she managed to learn the craft of dressmaking. As an only child (a circumstance that would prove significant), Soso, unlike many of his peers, did not have to work and could therefore attend school. In a letter written in 1950, requesting a meeting for old times' sake, one of Stalin's childhood friends commented, \"In 1894, when you graduated from the theological school, I graduated from the Gori Municipal School. You were accepted that same year into the Tbilisi Theological Seminary, but I wasn't able to continue my studies since my father had 8 children, so we were poor and we helped him.\" Ioseb's mother, dreaming that her son would climb the social ladder to become a priest, doggedly worked to make this dream a reality and did everything she could to facilitate his education. Such strivings are hard to reconcile with the idea of a bleak, impoverished childhood.\n\nCertainly there was discord in the family, and the drunken Besarion let loose with his fists. Soso was apparently beaten by both parents. But as Suny rightly observes, the evidence we have is insufficient either to judge whether violence within the Jughashvili family was unusual for that place and time or to assess its impact on Soso's perception of the world. Stalin's childhood and adolescence seem to have been utterly typical of the environment from which he came\u2014the world of poor, but not destitute, craftsmen and shopkeepers in a small town at the outskirts of the empire. This was a world where coarse mores coexisted with traditions of neighbor helping neighbor and periods of relative well-being alternated with hard times. Children were exposed to severity and cruelty as well as to affection and indulgence. Soso Jughashvili experienced the good and the bad\u2014his father's harshness and his mother's limitless affection\u2014in relatively balanced proportion. The family's financial difficulties, which came when Soso was in school, were eased by the help of friends and relatives. While at the local theological school and later at the seminary in Tiflis, Ioseb received assistance from the state and benefited from the intercession of sympathetic protectors. Despite their modest means, mother and son were fully accepted into their small community.\n\nDuring an interview many years later, Stalin said, \"My parents were uneducated, but they did not treat me badly by any means.\" It is possible he was not being candid or was suppressing unpleasant childhood memories. There is little evidence regarding Stalin's feelings toward his father, who died young. To all appearances, however, he felt genuine affection for his mother. His letters to her in her later years contain lines such as the following: \"Hello Mama dear! How are you getting on, how are you feeling? I haven't had any letters from you in a long time\u2014you must be upset with me, but what can I do? I'm really very busy,\" and \"Greetings dear mother! I'm sending you a shawl, a jacket, and medicines. Show the medicines to your doctor before taking them because a doctor has to set the dose.\" Despite her son's meteoric rise, Keke remained in Georgia, living in a position of respect and comfort. Stalin did not attend her funeral in 1937. Throughout that year, the height of the Great Terror, he did not set foot outside of Moscow. The dedication he wrote for a memorial wreath in both Georgian and Russian still survives: \"To my dear and beloved mother from her son Ioseb Jughashvili (from Stalin).\"\n\nStalin owed her a true debt of gratitude. She worked hard to protect her son from want and to enable him to get an education, and she nursed him through numerous illnesses, including smallpox, which pockmarked his face for the rest of his life. Soso also suffered a childhood mishap, exacerbated by poor medical treatment, that rendered his left arm severely disabled. The joints remained atrophied for the rest of his life, and the arm never functioned properly. Another physical defect was congenital: two toes on his left foot were joined. It seems unlikely that these defects remained unremarked in the often heartless company of boys. Yet Soso was not an outcast. He remained on an equal footing with his peers and took part in all of their games. He had an excellent memory, always a respected quality. It does not appear that a difficult childhood sowed in Ioseb Jughashvili the cruelty that emerged in Joseph Stalin. There is also no obvious sign of what in his childhood might have turned him into a rebel.\n\n###   **THE FAILED SEMINARIAN**\n\nIoseb's mother, whose efforts were inspired by the hope that her son would successfully overcome the social circumstances of his birth, was not the only one who noticed his intellectual abilities. When the time came to send the boy to school, Keke was able to solicit the support of well-wishers who felt strongly that the boy could profit from an education. Her aspiration that Ioseb would become a priest seemed entirely fitting. The well-wishers were the family of a priest named Khristofor Charkviani, in whose home the Jughashvilis rented a room. They helped Soso gain admission to the Gori Theological School. The Charkviani children also taught him Russian, the language of instruction. These language lessons enabled Soso to immediately enter the school's highest preparatory class\u2014undoubtedly a significant moment in the future leader's life. Ten-year-old Soso was making an important step into the Russophone world.\n\nHe spent almost six years, from 1888 to 1894, at the Gori Theological School, a period that saw dramatic changes in the Jughashvili family. After much domestic strife, Besarion left Gori, depriving his wife and son of their means of support and imperiling Soso's continued attendance at the school. Keke was able to find help, a task undoubtedly made easier by Soso's academic success. He was a model student and was even granted a stipend. The mother took care that her son would in no way feel inferior to his classmates and always ensured that he was dressed well and appropriately for the weather. According to numerous reminiscences, Soso distinguished himself at school by his diligence and hard work. He was reputed to be a fine reader of prayers and singer in the church choir, and he got along well with the teachers. The Russian teacher, whom the children called \"the gendarme\" behind his back, made Soso his assistant in charge of distributing books. Many decades later, in 1949, another former teacher at the school, S. V. Malinovsky, took the bold step of contacting his former pupil. \"In my old age,\" he wrote, \"I am proud that my humble efforts contributed to your education.\" Malinovsky requested that he be awarded a personal pension, \"so that in the twilight of my days my basic needs can be met and I can die in the happy awareness that my Great Pupil did not leave me in poverty.\" While there is evidence that this letter was placed before Stalin, the record is unclear on whether assistance was granted.\n\nIoseb graduated in May 1894. The certificate issued to him lists the courses he took and the grades he received. He earned a grade of \"excellent\" for behavior, as well as for Sacred History, Orthodox Catechism, Liturgical Exegesis and Ecclesiastical Typikon, Russian and Church Slavonic, Georgian, geography, penmanship, and liturgical chant. In Greek and arithmetic, his weakest subjects, he managed a grade of \"very good.\" His academic success yielded a recommendation for entry into a theological seminary. Despite the narrow curriculum, Soso acquired a great deal of skill and knowledge at the school in Gori and developed a passion for reading. More significant, he developed a mastery of Russian. Recollections of his time at the school paint a picture of an active child with pretentions toward leadership, pretentions undoubtedly affirmed by his standing as a top student. He seems to have had pleasant recollections of these years. Many decades later he remembered his school friends and even tried to help them. In notes dated May 1944, when he was sixty-five, Stalin wrote: \"1) To my friend Petya\u201440,000, 2) 30,000 rubles to Grisha, 3) 30,000 rubles to Dzeradze,\" and \"Grisha! Accept this small gift from me.... Yours, Soso.\" Written in Georgian, these documents hint at bursts of nostalgia felt by an old man reflecting fondly on his adolescence.\n\nThere are vague and inconsistent accounts by memoirists claiming that Ioseb Jughashvili's rebellious behavior and break with religion dated to his days in Gori. Leon (Lev) Trotsky, one of Stalin's first biographers (and hardly an impartial one), convincingly argues that Stalin's former classmates are confusing the Gori period with events that took place later, in Tiflis. The best proof of the schoolboy Soso's exemplary behavior and law-abiding attitude is the glowing assessment on his graduation certificate and the recommendation that he enroll in a seminary.\n\nIn September 1894, having successfully passed the entry examination, young Jughashvili enrolled in the Tiflis Theological Seminary. Ekaterine and her son enjoyed good fortune here as well. The seminary was more eager to have students born into the clerical estate, and others were required to pay tuition. But Ioseb's abilities, along with the intercession of friends and relatives, earned him a free room and meals in the seminary cafeteria. He was required to pay only for his courses and clothing. Did the ambitious boy perceive this as a demeaning handout to a \"poor relative\"? Perhaps. But it is equally possible that this grant-in-aid was viewed as a recognition of past achievements.\n\nStalin spent more than four and a half years in the Tiflis seminary, from the autumn of 1894 to May 1899. The move to a large city undoubtedly brought a degree of stress. However, Ioseb had not come alone but with a group of friends and acquaintances from the Gori Theological School. Furthermore, he seems to have found the course work relatively easy. He ranked eighth in his class in his first year and fifth the next year. His behavior was assessed as \"excellent.\"\n\nYet behind this promising fa\u00e7ade lurked a growing dissatisfaction and insubordination. While there is no moment that stands out as marking his departure from the path of the law-abiding and well-adjusted student, we do have two well-known pieces of evidence attesting to the unbearable living conditions at the seminary. The first such testimony belongs to Stalin himself. In 1931, in an interview with German writer Emil Ludwig, he described the seminary's role in pushing him toward insurrection: \"In protest against the outrageous regime and the Jesuitical methods prevalent at the seminary, I was ready to become, and actually did become, a revolutionary, a believer in Marxism as a really revolutionary teaching.... For instance, the spying in the hostel. At nine o'clock the bell rings for morning tea, we go to the dining-room, and when we return to our rooms we find that meantime a search has been made and all our chests have been ransacked.\" This account is supplemented by a widely cited description by one of Stalin's classmates:\n\nWe were brought to a four-story building and put in huge dormitory rooms with 20\u201330 people each.... Life in the theological seminary was repetitious and monotonous. We arose at seven in the morning. First, we were forced to pray, then we had tea, and after the bell we went to class.... Classes continued, with breaks, until two o'clock. At three we had supper. At five there was roll call, after which we were not allowed to leave the building. We felt as if we were in prison. We were again taken to vespers, and at eight we had tea, and then each class went to its own room to do assignments, and at ten it was lights out, sleep.\n\nHaving only Sundays free of this regimentation probably did not much brighten the seminarians' lives, especially as the day was partially taken up by mandatory church services. It was a regime of constant surveillance, searches, denunciations, and punishments. Although the range of disciplines was somewhat broader than in Gori\u2014in addition to scripture, church singing, Russian philology, and the Greek and Georgian languages, the curriculum included biblical and secular history and mathematics\u2014intellectual life was constrained by dogmatism. The reading of secular literature was harshly punished and Russification was crudely enforced, insulting the national pride of Georgian seminarians. The strong undercurrent of resentment and rebellion among the students was hardly surprising. A strike had erupted the year before Ioseb enrolled. The seminarians stopped attending their classes and demanded an end to arbitrariness by the teachers and the firing of some of them. In response, the authorities closed down the institution and expelled a large number of students.\n\nThe firm suppression of unrest doubtless helps account for the lack of open protest during Ioseb's years at the seminary. Any individual or group dissent was kept underground. At first the future dictator found an outlet in romantic literary heroes exemplifying the struggle for justice, especially those from Georgian literature. One of his first models came from _The Patricide,_ a novel by Alexandre Kazbegi. This was a tale of the fearless and noble avenger Koba, scourge of Russian oppressors and the Georgian aristocracy. Koba became the future leader's first pseudonym, one he treasured and allowed his closest comrades to use for him throughout his life.\n\nHis fascination with romantic rebellion flavored with Georgian nationalism predictably led young Stalin to try his hand at verse. After completing his first year at the seminary, he brought a sample of his poetry to the editorial office of a Georgian newspaper, which published five poems between June and October 1895. Another poem appeared in a different newspaper the following summer. The poems, written in Georgian, extolled service to the motherland and the people. During Stalin's leadership of the Soviet Union, his poetry was translated into Russian, but these translations were not included among his collected works. He undoubtedly understood that his undistinguished and naive verse belied the image of the single-minded revolutionary:\n\nA lark in the high clouds\n\nSang ever so sonorously.\n\nAnd a joyous nightingale said this:\n\n\"Blossom, lovely land,\n\nExult, country of Georgians.\n\nAnd you, Georgian,\n\nGladden your motherland with learning.\"\n\nAlthough such lines do nothing to soften the image of Stalin the dictator, they do attest to the pure intentions of Jughashvili the seminarian, who found inspiration in the ideas of service to the motherland and the people. During his third year at the seminary, these vague, half-formed strivings did lead to one concrete step. Ioseb joined an illegal discussion group of seminarians and apparently assumed a leadership role within it. The books read by the group were perfectly legal but forbidden by the seminary. Entries in the journal used to keep track of the seminarians' conduct record violations by Jughashvili involving the reading of forbidden books, including novels by Victor Hugo, in late 1896 and early 1897. Beginning in his third year, Ioseb's grades began to decline, and he was caught violating rules with increasing frequency.\n\nIoseb Jughashvili was growing increasingly radicalized. He stopped writing verse and developed an ardent interest in politics. Participating in the discussion group was no longer enough. He longed to get involved in something \"real,\" a desire that led him to the Social Democrats, an interest in Marxism, and attendance at illegal meetings of railway workers. According to his official biography, in August 1898, while still enrolled in the seminary, Ioseb joined a Social Democratic organization and began working as a propagandist for small groups of workers. At this point, his knowledge of Marxism must have been fairly superficial, but his fascination with it was consuming. For the young seminarian, the all-encompassing nature of Marxism, almost religious in its universality, was tremendously appealing. It filled the gap in his worldview created by his disillusionment with religion. The belief that human history was governed by a set of laws and that humanity was inexorably advancing toward the higher stages of socialism endowed the revolutionary struggle with special meaning. But this fascination with Marxism hardly set young Jughashvili apart. Belief in Marxism was a veritable epidemic.\n\nOne influence on Ioseb was the older fellow revolutionaries and rebels who came to Tiflis from other regions of Georgia. The figure most often mentioned in this context is Lado Ketskhoveli. Though still a young man, he had already advanced along the path on which young Stalin was just embarking. After being expelled from the Tiflis seminary, Ketskhoveli enrolled in the Kiev Theological Seminary, where he was arrested by the authorities for possessing illegal literature. Only a general amnesty occasioned by the coronation of Tsar Nicholas II saved him from punishment. After returning to Tiflis and then moving to Baku, this committed revolutionary immersed himself in subversive work and organized an underground printing press. In 1903 he was shot by a prison guard. Legend has it that he was killed for shouting revolutionary slogans. This was the sort of man of action Ioseb looked up to.\n\nIoseb's behavior during his final academic year at the seminary (1898\u20131899), when he was increasingly involved in the Social Democratic movement, clearly shows an intention to break with the past. All the indignation that had festered during his first years in Tiflis came to the surface. The seminary's conduct journal serves as a chronicle of his rebellion. In September he was caught reading excerpts from banned books to his comrades. In October he was confined to a punishment cell three times for failing to attend prayers, bad behavior during liturgy, and returning late after a school recess. Over the following months, periods of confinement alternated with reprimands for a variety of offenses.\n\nIn January 1899, a serious conflict with the seminary's administration resulted in Ioseb's being prohibited from leaving the seminary for a month. Historian Aleksandr Ostrovskii attributes this punishment to an incident described in the memoirs of one of Ioseb's classmates, published in 1939. According to this account, a seminary inspector searched Jughashvili's room and found forbidden books. At this point, a seminarian by the name of Kelbakiani pounced on the inspector and knocked the books out of his grasp. Helped by Jughashvili, Kelbakiani then gathered up the books and fled. Among the sources that cast doubt on this account is the seminary's conduct journal for 1899, which describes Kelbakiani's infraction quite differently. A search of Kelbakiani's own possessions turned up a notebook into which excerpts from prohibited literature had been copied. When the inspector refused Kelbakiani's request that the notebook be returned, the seminarian grabbed it and threw it into the toilet. The seminary rector was immediately informed of this incident and Kelbakiani was placed in a punishment cell for several hours.\n\nAccording to the conduct journal, \"Kelbakiani displayed strong remorse.\" He admitted his guilt and asked for indulgence. There is no mention of Jughashvili's involvement in this incident. All that is known for certain is that in January 1899 Jughashvili was deprived of the right to leave the seminary premises for one month, and Kelbakiani was expelled. The difference in punishments may indicate that Ioseb was penalized for some other infraction or that he played only a minor role in the destruction of the notebook.\n\nIn June 1951, Kelbakiani wrote the following to his former classmate:\n\nComrade Soso! If you knew how impoverished I was at the present time, I am certain you would not leave me without attention. I have grown old and have no income and I am in a state of need.... Comrade Soso, in some way you are in my debt: you probably remember how I grabbed from the seminary inspector... illegal literature that was taken during a search of your drawer, for which I was expelled from the seminary.... I am not proud of this and am not boasting, of course.... Poverty has forced me to remember this. Help me, Comrade Soso.\n\nThis letter was placed before Stalin. There is no record to show whether Kelbakiani was given any assistance, but his letter does shed light on the 1899 incident. Kelbakiani was undoubtedly familiar with the account published in 1939 describing the future Stalin's \"heroic deed,\" and he generally adheres to its details. The confiscated notebook is identified as \"illegal literature\" and is found among Jughashvili's possessions rather than Kelbakiani's. It is, however, noteworthy that Kelbakiani unequivocally states that he himself, without help from \"Comrade Soso,\" was the one to grab the confiscated notebook from the inspector. He is just as unequivocal on the subject of Soso's involvement in the incident and in suggesting that he, Kelbakiani, performed a favor for the future leader. Overall, it would appear that Ioseb really was involved. We can surmise, for example, that the notebook Kelbakiani destroyed belonged to Jughashvili. This may not have been reported in the conduct journal because it was not known at the time. It seems almost certain that Ioseb did not help Kelbakiani save the materials. This was among the more harmless of the legends that took shape to foster the cult of the leader.\n\nThe notebook incident aside, Jughashvili committed more than enough sins in the eyes of the seminary leadership to render him persona non grata. In May 1899 he was expelled, the formal cause being \"for failing to appear at examinations for unknown reasons.\" One odd detail is that the certificate he was given upon expulsion, stating he had completed four years at the seminary, gives him excellent grades for behavior. Stalin's biographers have long commented on the confusion surrounding the circumstances of his departure. He himself preferred to say that he was \"kicked out\" \"for Marxist propaganda.\" In one interview, Ekaterine claims that she took her son out of the seminary because of his poor health. There may be some truth to all these accounts\u2014both the official formulation and the statements by Jughashvili and his mother. The seminary leadership may have been eager to rid itself of a rebel while avoiding scandal. Ioseb may have withdrawn \"by mutual consent\" with a commendatory certificate on the completion of four years. If so, Ekaterine and her complaints of her son's worsening health probably played a major role. In the end, Ioseb really was \"kicked out,\" but quietly, leaving the door open for him to mend his ways.\n\n###   **UNDERGROUND, PRISON, AND EXILE**\n\nThe certificate issued to Ioseb Jughashvili by the seminary would have enabled him to work in the area of religion or teach elementary school. But a return to ordinary life did not interest him. In late 1899 Ioseb was hired, with the help of friends, to work at the Tiflis Meteorological Station. His job involved constant recording of instrument readings and therefore required him to live on the premises, taking care of his need for both money and housing.\n\nContinuing to work with revolutionary groups, he soon aligned himself with the radical wing of the Tiflis Social Democratic organization, which rejected agitation through legal propaganda and instead favored fomenting strikes and demonstrations. Given the twenty-two-year-old rebel's record at the seminary and his friendship with such revolutionaries as Lado Ketskhoveli, his turn toward radicalism is hardly surprising.\n\nThe years 1900 and 1901 saw a wave of strikes in Tiflis, followed by crackdowns. Under threat of arrest, Jughashvili left the weather station and went underground. There was no turning back; he had become a professional revolutionary.\n\nWhatever their backgrounds, Russian revolutionaries tended to have one thing in common. Their break with ordinary life and move underground took place in a moment of hatred and decisiveness: hatred for the existing order and a decision to combat it. In the Russian Empire, there was no shortage of either emotion. An authoritarian regime and social injustices created a breeding ground for rebels. The persecution to which radicals were subjected radicalized them still more. The hatred felt by Ioseb Jughashvili, aroused by the arbitrariness and obscurantism that prevailed at the seminary, was further inspired by the propaganda and actions of his more experienced comrades, those who had chosen the path of revolution before him. His decisiveness was both a feature of his character and a product of the milieu into which he was born. Anyone with social origins like his had little to lose.\n\nIn exploring the sources of Stalin's rebelliousness and ruthlessness, many historians have pointed to the atmosphere that reigned in the outlying regions of the Russian Empire. Alfred Rieber has called him a \"man of the borderlands.\" The Caucasus, a roiling cauldron of social and ethnic conflict where industrial enclaves emerged amid tribal traditions, would inevitably have played a role in shaping Stalin's character. J\u00f6rg Baberowski has written that Stalin and his comrades-in-arms \"brought into the party, both at the center and edges of the empire, the culture of violence of the Caucasian periphery, the blood feud and archaic conceptions of honor.\" Such opinions are supported by Boris Nicolaevsky, a Social Democrat who later became a well-known historian. Before the revolution, Nicolaevsky had spent time in Transcaucasia and had even met with Jughashvili. He described the future dictator as \"exceptionally vicious and vindictive\" and capable of applying \"the most extreme measures\" in his struggle to dominate the party. Yet many of Jughashvili's opponents within the Social Democratic movement were no different. Nicolaevsky said he was told that these traits resulted from \"the injection of Caucasian mores into the intraparty struggle.\"\n\nIt is not unreasonable to take into account the mentality forged by the hardships and tragic history of the Russian borderlands. Yet the entire Russian Empire was one vast borderland: between Asia and Europe, between the promises of modernization and the deteriorating traditional ways of life, between the city and the country, between authoritarianism and democratic strivings, between the obscurantism of the regime and the bloodthirstiness of many revolutionaries. Whatever features may be particular to the Caucasus must be seen within the context of the Russian culture of extremism and violence, which merely provided an outlet for the impulse. Such a context does not, of course, relieve young Jughashvili of personal responsibility for his choices.\n\nRevolutionaries are not all cut from the same cloth. Many throw themselves into the fight under the influence of youth, ardor, and thrill seeking. These factors were probably not what led Stalin onto this path, though they should not be discounted entirely. The future dictator could be described as a calculating revolutionary, the sort who doggedly and methodically\u2014even cautiously\u2014moved the revolution forward and later, when success came, had the best chance of solidifying power. He had just the right balance of decisiveness and caution, obsession and cynicism, to emerge unscathed through the revolution's countless dangers.\n\nAn overview of the activities of the Tiflis Social Democratic organization found in the files of the local gendarme administration describes Ioseb Jughashvili as \"conducting himself with complete caution and constantly looking over his shoulder as he walks.\" He managed to avoid arrest for some time, giving him a significant advantage, since many members of the Social Democratic Party were in prison, and facilitating his rise within the Tiflis party leadership. Apparently to evade arrest, he moved from Tiflis to Batum, a major center of the empire's petroleum industry. A propaganda campaign by him and his associates evidently had an effect, as Batum workers staged a spate of strikes and demonstrations. The government response was severe. On 9 March 1902, when workers stormed a prison where many of their comrades were being held, troops opened fire. At least thirteen people were killed and dozens were wounded. News of violence in Batum spread, and Jughashvili, one of the organizers of the demonstration, was arrested.\n\nIn an effort to avoid punishment, Jughashvili denied his guilt, asserting that he had been nowhere near Batum during the period leading up to the attack. In notes sent from prison, he asked his mother, friends, and relatives to give him an alibi by falsely testifying that he had arrived in Gori before mid-March. One such note fell into the hands of the police. The police in Batum still could not prove that Jughashvili was directly involved in organizing the storming of the prison, but in probing his background, they brought to light his activities in Tiflis. The investigation inched along. Languishing in prison, Ioseb did what he could to improve the outcome of his case. In October and November 1902, seven and eight months after his arrest, he sent two petitions to the offices of the administrator-in-chief for the Caucasus. Citing a \"worsening asphyxiating cough and the helpless situation of my aged mother, who has been abandoned by her husband for 12 years now and sees me as the only person she can count on in life,\" he asked to be released under police supervision. \"I beseech the office of the Administrator-in-Chief not to neglect me and to respond to my request.\" In January 1903 Ekaterine also submitted a request to the authorities that her son be freed. Her petition, written in Russian but signed in Georgian, stated that her son, \"as the breadwinner for himself and his mother, has neither the time nor the occasion to participate in conspiracies or disturbances.\"\n\nThese entreaties proved ineffective. Ioseb remained in prison for several more months, suffering deprivation and harassment. Not until the fall of 1903, one and a half years after his arrest, was he finally sent into exile in eastern Siberia. Soon, in early 1904, he escaped from his place of banishment. Such an escape was not at all unusual. Lax security enabled many revolutionaries to flee their places of exile, although such escapes demanded careful preparation, courage, and physical endurance. Jughashvili learned from his first stint in exile and later had several opportunities to put that experience to use.\n\nThere is evidence to suggest that during the first months after his return to Transcaucasia, Jughashvili was suspected of being a double agent. Social Democrats were being arrested throughout the region. Although these arrests cast a pall of suspicion over him, the lack of personnel began to facilitate his ascent within the underground movement. He rose through the ranks to the governing committee of the Transcaucasian Social Democratic organization. Other factors in his success were his active efforts in the underground and his ability to generate fiery prose. Rumors that he was collaborating with the police remained just that.\n\nDuring the two years that Jughashvili spent in prison and exile, Russia's Social Democratic Party had undergone major changes. While formally a single party, in actuality it was divided between the adherents of Lenin\u2014Bolsheviks\u2014and the more moderate Mensheviks. Lenin advocated the creation of a militant and cohesive underground party that would serve as an instrument of revolution. It was Lenin's belief that the workers, who were to be the main force in the revolution, were not capable of developing proper revolutionary thinking on their own. They had to be taught by professional revolutionaries. Lenin's teachings were aimed at hastening revolution and speeding up \"historical time.\" The Mensheviks felt that the party should be less rigid and accept among its ranks sympathizers as well as activists. The Mensheviks had greater respect for the workers and placed less emphasis on their own role as teachers. This approach was a natural byproduct of their core belief that the revolutionary process would move gradually and organically forward as the objective preconditions for socialism reached fruition. Jughashvili was temperamentally inclined to accept Lenin's viewpoint and to embrace his radicalism and calls to action. Furthermore, as a member of the party intelligentsia, Jughashvili welcomed the idea that professional revolutionaries must lead the workers' movement. To be leaders, to show the masses the way forward\u2014surely this was the intelligentsia's proper place within the revolution? Many of his articles were devoted to promoting Lenin's ideas.\n\nThe first Russian Revolution, in 1905, initially intensified discord between the Bolsheviks and Mensheviks but ultimately brought the two sides closer together. Both groups faced a common enemy\u2014the government and its supporters\u2014and both sides increasingly resorted to violence and brutality. In Transcaucasia, roiled by social and ethnic animosities, the situation was particularly dire. As usual, the government did not hesitate to use arms. In response, the revolutionaries murdered figures associated with the autocratic regime and committed arson against industrial enterprises. Ethnic pogroms fed the rush of carnage. Violence and bloodshed became commonplace. Mensheviks and Bolsheviks organized their own armed detachments and made generous use of terrorist methods. Jughashvili took an active part in these events, traveling across Georgia, helping to organize strikes and demonstrations, writing leaflets and articles, and helping set up an underground printing press and militant groups. He gradually reached the forefront of the Bolshevik leadership in Transcaucasia.\n\nIn October 1905, unrest compelled the tsar to make concessions. Russia was given its first parliament, the State Duma. Political freedoms were proclaimed: freedom of conscience, free speech and assembly, and the inviolability of the person. The revolution nonetheless continued to build, and it forced maneuvering by the Social Democrats as well as the tsar. Under pressure from the ranks, the Bolsheviks and Mensheviks agreed to a reconciliation, restoring a superficial party unity. This newfound unity, however, did not advance the interests of the Bolsheviks in Transcaucasia, Jughashvili in particular, because it put the Mensheviks in charge of the region's revolutionary organizations. The election of delegates to the party's April 1906 \"Unity Congress\" in Stockholm put the Bolsheviks' demeaning position on full display: the future dictator was the only Transcaucasian Bolshevik delegate elected. The next congress, in London the following May, was even more humiliating. At first, only Mensheviks were elected. The Bolsheviks had to arrange by-elections so they could send at least one representative. Again, they sent Jughashvili.\n\nJughashvili's trips to these congresses undoubtedly expanded his sense of the world and the party, as well as his circle of contacts. There is evidence that in 1907, while traveling to London, he met with Lenin in Berlin. Returning from London, he spent several days in Paris, where he stayed with fellow Georgian Grigory Chochia, a student there. He returned to Russia using the passport of a friend of Chochia's who had died. This arrangement enabled him to evade police surveillance and improved his personal safety. Forty years later, in May 1947, Chochia, then living in Leningrad, reminded Stalin of this: \"In mid-1907, after you stayed with me for several days, I escorted you to the St. Lazare train station in Paris. You were so kind as to say to me, 'I will never forget your help' (you were referring to my giving you the international passport). Right now, I am greatly in need of your attention. I ask to be granted a 5\u201310 minute meeting with you.\" The letter was filed away. Stalin rarely recalled his foreign travel. We do not know what he saw in Europe and how he perceived it. Did he bring any gifts to his young wife, Yekaterina Svanidze, whom he married in July 1906, or his son, Yakov, born in March 1907 (right before Ioseb left for Western Europe)? Undoubtedly, Jughashvili's mind was on the revolution.\n\nImmediately after he returned from the West, on 13 June 1907, a group of Transcaucasian Bolsheviks staged an armed robbery of money being transported to a bank in Tiflis; the robbery has become a part of the history of the Russian revolutionary movement. At the cost of several lives, it yielded a huge sum for Bolshevik coffers: 250,000 rubles. The ringleader of this \"expropriation\" was Jughashvili's good friend Simon Ter-Petrosian, nicknamed Kamo. The obvious link between the two men has led some to suggest that Stalin was involved in organizing the heist and perhaps even took part in it, but there is no hard evidence. Boris Nicolaevsky, who completed a thorough study of the case in the course of chronicling the Social Democratic movement, concluded that Jughashvili was informed of the activities of Kamo's group and \"helped conceal them from the local party organization.\" But \"he was in no regard a ringleader.\" Nicolaevsky found a document showing that Kamo was working directly with the Bolshevik center abroad, specifically an agreement between Kamo and Lenin's Bolshevik center on the details of the robbery. It was Kamo, not Jughashvili, who signed this agreement.\n\nExcept for the amount stolen, the Tiflis holdup was nothing out of the ordinary. The robbery of government institutions and private individuals was widely practiced at the time, by the Bolsheviks as well as other groups. Although such actions generated income, they undermined the morals of the revolutionaries and damaged their reputation with the public. From time to time, ordinary criminals would join forces with the revolutionaries for personal gain. In fact, ideologically motivated thieves stealing to further the revolution, even if they did not take a kopeck for themselves, were sometimes hard to distinguish from the ordinary criminals. This state of affairs must have been deeply disturbing for the leaders of the Social Democratic Party. At the 1907 congress in London the Mensheviks passed a resolution prohibiting Social Democrats from conducting such robberies. This resolution did not stop Lenin and his followers. The Tiflis operation was already being planned, and they did not cancel it. That this robbery was carried out so soon after the party congress made it look particularly cynical. Controversy spread through the ranks of the Social Democrats. Not for the first time and knowing his association with Kamo, the Tiflis Mensheviks showed Jughashvili how displeased they were with him. He was forced to leave Tiflis for Baku.\n\nIn Baku, where the Mensheviks also dominated the party, Jughashvili could still rely on a stalwart group of Leninists. This major industrial center was ripe with opportunity for both agitation among the working class and combat against political opponents. Jughashvili managed to drive a wedge through the Baku organization, and the Bolsheviks took over the party leadership. But the joy of victory was overshadowed by personal tragedy. In Baku, Ioseb's wife Yekaterina died. The couple's infant son was taken in by the mother's relatives. His father had no time for him.\n\nThe unrest surrounding the 1905 revolution frightened the ruling classes and awakened the tsarist government to the need for concessions. Russia became a freer country. Serious agrarian reform was introduced that had fundamental significance for a country in which the peasantry represented an overwhelming\u2014and explosive\u2014majority. Historians still argue over where these reforms might ultimately have led. One thing is clear: Russia was not allowed to follow the course of reform long enough to yield results. Furthermore, alongside the reforms and concessions, the authorities began to \"restore order\" and more decisively and brutally combat the revolutionary underground. One victim of this post-revolutionary crackdown was Jughashvili. In March 1908 he was arrested. As before, he denied any wrongdoing, claiming that he did not belong to any revolutionary party and had spent a long time abroad. These ploys did not work. After seven months in prison, he was sent into exile in Vologda Province, where he spent four months before fleeing. In the summer of 1909, he returned to Baku.\n\nBy this time, the Social Democratic organization in Baku had been infiltrated by undercover police. Failed operations and arrests aroused mutual suspicion and rising tempers among the revolutionaries. Jughashvili again came under scrutiny: new rumors emerged that he was working for the police. This idea has continued to be promoted, although most historians have never given credence to theories that he was a double agent. The opening of the archives has confirmed their skepticism. A key document used to bolster these accusations against Stalin has been definitively exposed as a forgery, produced within \u00e9migr\u00e9 circles after the revolution.\n\nJughashvili spent more time in prison and exile than one would expect for a double agent. In the spring of 1910 he was again arrested and this time threatened with serious punishment. The police demanded that he be sent for five years to \"the most remote reaches of Siberia.\" He resorted to a tried-and-true method: pleas for leniency, citing his poor health and the absence of serious evidence. In an attempt to demonstrate good intentions, he requested that he be allowed to marry a woman he had met while in exile and with whom he was living. It is hard to assess what effect these \"humble pleas\" had, but in October 1910, instead of the five-year sentence in Siberia initially sought, Jughashvili was returned to Vologda Province to complete his previous sentence. This was a mild punishment. His term concluded in July 1911.\n\nThe year-and-a-half between his release from this term of exile and his final arrest, in February 1913, was the peak of his career in the underground. He advanced into the ranks of the Bolshevik leadership, becoming a member of the Leninist party's Central Committee in 1912. This elevation had at least two consequences. First, he now zigzagged across Russia and often spent extended periods in the two capitals, St. Petersburg and Moscow, rather than working full time in Transcaucasia. Second, he was the target of much more intense police surveillance. He engaged in underground work in Russia, assisted in the publication of Bolshevik newspapers, wrote articles, and strategized with Bolshevik representatives in the State Duma. He also became one of Lenin's closest associates. The Bolshevik leader was still in hiding outside the country and needed loyal helpers in Russia. Several times, Jughashvili traveled to meet with Lenin abroad. Detained by circumstances in Vienna for several weeks in 1913, he began work on an article addressing the party's approach to ethnic minorities. This work was of particular interest to Lenin. In lockstep with Lenin's views, Jughashvili advocated a unified Russian Social Democratic Party and argued against the fragmentation of revolutionary forces based on ethnicity.\n\nJughashvili exemplified this sort of inter-ethnic cooperation. He considered himself an actor on the Russian imperial\u2014not just the Georgian\u2014stage. Putting his youthful nationalism and Transcaucasian Social Democratic past behind him, he consciously transformed himself into Stalin. He began to use this Russian-sounding pseudonym, which symbolized his affinity with the revolutionary movement, around the time he moved into the Bolshevik party leadership.\n\nStalin undoubtedly deserved his standing and reputation as a prominent Bolshevik. His organizational and writing abilities, daring, decisiveness, cool head, simple tastes, adaptability, and devotion to Lenin all contributed to his elevation to the top ranks. He stuck with the party even during the crisis in the Social Democratic movement that followed the crushing of the first revolution, a crisis characterized by mass arrests of underground operators, infiltration of the organization by police agents, and a severe shortage of funds. In March 1913, an agent who had penetrated the Baku Social Democratic organization reported that \"The committee is currently not undertaking any activities.\" Meanwhile, in February, in far-off Petersburg, Stalin was arrested. He had been betrayed by fellow Bolshevik leader and Lenin favorite Roman Malinovsky, who had been working for the police for several years.\n\n###   **FOUR YEARS IN SIBERIA**\n\nIn June 1913, Ioseb Jughashvili was sentenced to a four-year term of exile in Siberia's Turukhansky Krai. From the start, this last period of exile was marked by particular hardship. Turukhansky Krai was an extremely inhospitable region. Stalin's letters during the first months were filled with pleas for help and complaints that he lacked funds and was in poor health:\n\nIt seems that I have never been in such a terrible situation. My money is gone, the intensifying cold (37 below) has brought on a suspicious cough, and I'm in a general state of ill health, have no supply of bread, sugar, meat, or kerosene (all my money has gone toward day-to-day expenses, clothing, and footwear).... I understand that none of you, you in particular, have time for this, but, damn it, I don't have anyone else to turn to. And I don't want... to croak here. This has to be taken care of today and money sent by telegraph because waiting any longer means starving, and I'm already malnourished and sick.\n\nMy hardship grows by the hour, I'm in a desperate situation, and on top of it all I've fallen ill and some suspicious cough has set in. I need milk, but... money, I have no money. My dear, if you get some money, send it to me immediately via telegram. I can't stand it any longer.\n\nAt first there was a lingering hope of freedom. The party leadership adopted a resolution to arrange an escape for Stalin and his comrade in exile, Yakov Sverdlov. An escape would require money, but there were delays in sending it. Furthermore, the traitor Malinovsky informed the police of the escape plans. In March 1914, on orders from St. Petersburg, Stalin and Sverdlov were sent to the even more remote village of Kureika, not far from the Arctic Circle, and placed under the charge of personal wardens. Escape was almost impossible.\n\nStalin took this transfer as a severe blow. In late March 1914 he sent an angry letter to St. Petersburg rebuking his party comrades for their long silence and demanding to know: would there be money for an escape or not? Several weeks later he changed his plans. In April he wrote to Malinovsky: \"The new governor has relocated me to the far north and confiscated the money sent to me (60 r. total). We're still living, brother.... Someone, it turns out, has been spreading rumors that I'm not going to remain in exile for the rest of my term. Nonsense! I'm telling you and swear on the life of my dog that I will serve out my term (until 1917). At one point I thought of leaving, but now I've abandoned that idea, abandoned it for good.\"\n\nThis letter raises questions. Was Stalin's firm assertion that he did not plan to escape intended for the eyes of the police? Or was he expressing his dissatisfaction with party comrades who had failed to help him? Perhaps he recognized the fruitlessness of any hope of escape and had made a genuine decision to remain in exile. Given that the subject of escape did not arise again, it appears that he really did reconcile himself to his fate.\n\nStalin's life in Kureika was shaped by events that occurred during his first months there. First, he had a falling out with Sverdlov. Upon arriving in Kureika, the two set up house together, but this arrangement did not last long. In his letters, Sverdlov only hinted at conflict with his roommate: \"I'm living with the Georgian Jughashvili.... He's a fine fellow but too much of an individualist in practical matters. I am an adherent of some minimal order. This is a source of agitation for me at times.\" The picture is filled in by other sources. According to the reminiscences of Anna Allilueva, the sister of Stalin's second wife, Stalin later admitted that he found various pretexts for shirking his household duties\u2014cleaning, keeping the stove going, etc. Sverdlov wound up stuck with all the chores. Khrushchev offered further information:\n\nStalin told the following story: \"We would make dinner for ourselves.... The main thing we did in the way of earning a livelihood was to fish for white salmon. That didn't take any great skill. We also went hunting. I had a dog and called him Yashka. Of course for Sverdlov that wasn't pleasant; he was Yashka and the dog was Yashka, and so then Sverdlov used to wash the dishes and spoons after dinner, but I never did. I would eat and put the dishes on the dirt floor and the dog would lick everything clean. But that fellow had a passion for cleanliness.\"\n\nThese differences over hygiene were bound to provoke discord, but there may have been other sources of conflict. The animosity that developed between Sverdlov and Stalin was so strong that they not only moved into separate houses, but also broke off contact altogether. Sverdlov wrote to his wife some time later: \"After all, you know, dear one, what abominable conditions I endured in Kureika. On a personal level, the comrade I lived with there turned out to be the sort that we did not talk to one another or get together.\"\n\nSoon after his falling out with Sverdlov, Stalin moved into the home of the Pereprygin family\u2014five brothers and two sisters, all orphans. Stalin, who was thirty-five, entered into an intimate relationship with the fourteen-year-old Lidiia Pereprygina. This apparently provoked an argument between Stalin and the man in charge of guarding him, which escalated into a fistfight. The local police took Stalin's side. One circumstance that may have worked in Stalin's favor was that the police chief in Turukhansky Krai was I. I. Kibirov, an ethnic Ossetian who, like Stalin, was from Georgia. It is possible that Stalin and Kibirov came to an agreement that he would be given a degree of liberty in exchange for a promise that he would not attempt to flee. Stalin not only was not charged for his transgression with a minor, but he was also given a new guard, M. A. Merzliakov, who treated him exceptionally well. In 1930, when he was persecuted under the Soviet regime for having served in the tsarist police, Merzliakov turned to Stalin for help. \"I am asking Com. Stalin,\" he wrote, \"to inform our village soviet that I truly did have a friendly relationship with you while serving in Turukhansky and did not act against you.\" Stalin responded with a glowing recommendation: \"Mikh. Merzliakov had a formal attitude toward his police duties, without the usual police zeal; he did not spy on me, did not badger me, did not pick on me, and turned a blind eye to my frequent absences.\"\n\nTaking advantage of this obliging attitude, Stalin managed to arrange a relatively pleasant life for himself, to the extent such a thing is possible in the Arctic. He continued to live with Lidiia Pereprygina. There were rumors\u2014though muddled and contradictory\u2014that the two had a child together. Stalin devoted his copious free time to fishing, hunting, visiting fellow exiles in neighboring settlements, receiving guests, and taking part in local merrymaking. His financial situation stabilized enough to support his modest lifestyle. Most important is that his health improved. \"I'm living as before. I feel fine. I'm completely healthy\u2014I must have gotten used to the nature around here. And nature here is harsh: three weeks ago the temperature went to 45 below,\" he cheerfully reported in a letter written in late 1915.\n\nThis unusual period in Stalin's life reveals some interesting aspects of his character. He was completely unfazed by the absence of creature comforts in this harsh environment. In Kureika, with a total of eight houses and sixty-seven residents, he seems to have suffered an utter absence of suitable conversation partners. Yet he endured this lack of intellectual stimulation with equanimity. Apparently he was perfectly capable of living without the revolution and felt no need to exercise his intellect. His opponents have long accused him of wasting the time spent in Turukhansky Krai. Trotsky, for example, wrote that \"Any attempt to find traces of his spiritual life during this period of solitude and leisure would be in vain.\" Indeed, Stalin's collected works feature not a single article written between early 1913 and early 1917.\n\nStalin's correspondence from this period, however, paints a more complicated picture. During the first year of exile, either because he still hoped to escape or simply out of habit, he did try to work. He wrote a new article on nationalities problems and sent them to a journal. He asked his comrades to send him books, journals, and newspapers. In subsequent years as well, his correspondence from exile contained references to work on articles and his need for new books. But his enthusiasm was waning. In 1914, Malinovsky was exposed as a double agent. This was a crushing blow to the entire Bolshevik party, but for Stalin, who was friendly with Malinovsky and had turned to him for help, the revelation was especially painful. And there were other discouraging developments. An article that Stalin submitted to a journal was not published, his comrades failed to send him new journal issues, and he lacked the money for subscriptions. In November 1915, after two years in Turukhansky Krai, he explained his situation in a rare letter to Lenin: \"My life is not great. I'm hardly doing anything. And what is there to do when you have no or almost no serious books?... I have lots of questions and topics in my head, but as for material\u2014nothing. I'm itching to do something, but there's nothing to do.\" Stalin's communication with the party leadership in emigration gradually dropped off, and he occasionally complained in letters that they had forgotten him. Indeed, Lenin's requests in 1915 to be reminded of Stalin's last name became well known: \"Do you remember Koba's last name?\"; \"I have a big favor: find out... 'Koba's' last name (Iosef J...?? We've forgotten).\"\n\nStalin's situation reflected the general state of affairs in the Bolshevik party. Its leadership was languishing either in forced internal exile or self-imposed exile abroad. Periods of hope, dreams, and failed attempts to activate the movement alternated with quarrels, both internally and with opponents from other parties. On both the personal and political front, the future looked gloomy for the revolutionaries. How thirty-eight-year-old Stalin imagined his future at this point is hard to know. Perhaps he tried not to think about it.\n\n## **THE BULWARKS OF STALIN'S POWER**\n\n**The day and evening of 1 March 1953 at the near dacha. Consternation among the bodyguards.**\n\nAfter his guests departed in the early morning hours of 1 March, Stalin most likely went to bed. He may not have felt well. He was aged and sickly. He remained in his rooms and did not, as he usually did, summon any guards or servants toward suppertime. As of early 1952, Stalin's apartment and dacha were protected by a staff of 335 security personnel. Another 73 attended to his non-security needs. All told, 408 people, working in shifts at various sites, were devoted to taking care of Stalin. Stalin spent a significant portion of his time in these people's company. They walked behind him, stood guard under his windows, cooked, cleaned, and, if needed, entertained him. At the near dacha, a long corridor separated the staff quarters from the part of the house where Stalin lived. His rooms were equipped with buttons to summon staff members.\n\nThe deviation from Stalin's routine on 1 March alarmed his security team. The guards reported to their superiors that there was no \"movement\" within the leader's residence. Evening approached with no signs of life. The sense of alarm escalated, but if they were not summoned, nobody wanted to check on the boss. Finally, sometime after six o'clock, the guards were relieved to see a light turn on in Stalin's rooms. Everyone prepared for a call. None came. Anxiety again began to mount. The guards argued over who should go check on Stalin. Nobody volunteered.\n\nTheir hesitation was understandable. Of course, they had grown accustomed to Stalin, just as the lonely leader, for whom the hired help often served as a surrogate family, had grown accustomed to them. From time to time, Stalin and the dacha staff worked together in the garden or roasted shashlik in the fireplace. Sometimes he would come into the kitchen and lie down on the Russian brick oven to ease the pain in his back. But the distance that separated Stalin and his guards was much greater than the length of the corridor that separated their quarters from his. He was strict with his staff, and they knew better than to relax the fear they felt toward him.\n\nThe guards who protected Stalin and other members of the top leadership belonged to a special department within the Soviet security system, the Main Guard Directorate. In the early days of the regime, when the egalitarian romance of the revolution still lingered, Soviet leaders often mixed with the public. In the 1920s, Stalin's wife could still ride streetcars, and he himself walked the streets of Moscow or rode in cars with no particular precautions, though always accompanied by bodyguards. In July 1930, while vacationing in Sochi, Stalin and his wife were involved in a car crash. He was slightly injured when his head hit the windshield.\n\nTwo months after the car crash, amid growing hysteria in the struggle against \"enemies,\" the Politburo adopted a resolution \"to oblige Com. Stalin to immediately desist from walking through the city on foot.\" Stalin did not submit to this restriction. On 16 November 1931, while walking down the street, accompanied by bodyguards, from the Central Committee building to the Kremlin, he happened to run into an armed agent of an anti-Bolshevik organization who had come from abroad. The agent was so surprised that he did not have time to pull out his gun before he was arrested. A report on the incident by the Joint State Political Directorate, the OGPU (the Soviet secret police of the time), was sent to Stalin and the other members of the Politburo. Molotov made a notation on the report: \"To PB members. Com. Stalin's walking around Moscow on foot must be stopped.\" It is not known whether Stalin submitted to this demand. It is also unclear whether the encounter could have been orchestrated.\n\nOn his 1933 vacation in the south, several incidents appeared to place Stalin in danger. In August, his car was hit by a truck in Sochi. The truck's driver was drunk, and Stalin was unharmed. Another incident took place on the Black Sea coast in September when a motorboat on which Stalin was riding came under rifle fire from the shore. The bullets landed in the water, and no one on the boat was injured. An investigation determined that rifles had been fired by border guards who had not been warned that a boat would be entering the protected zone.\n\nThe murder of Sergei Kirov on 1 December 1934 was a watershed moment in attitudes toward the safety of Soviet leaders. Using it as an excuse, Stalin undertook a series of reprisals against former members of the party opposition, who were accused of orchestrating Kirov's murder and plotting other terrorist acts against the Soviet leadership. In 1936\u20131938, when terror ravaged the country, engulfing hundreds of thousands of lives, Stalin eliminated everyone suspected of disloyalty. The security apparatus was one important target of the purges, and those in charge of guarding the leaders also fell victim. In April 1937, Stalin's chief of security was arrested and swiftly executed. Of his two successors in 1937\u20131938, one shot himself and the other was executed. Finally, in late 1938, the uneducated but efficient Nikolai Vlasik was appointed to the post. Stalin took a liking to him and kept him in the job for more than thirteen years.\n\nVlasik's career even survived an incident that took place in Moscow on 6 November 1942. An official car carrying Anastas Mikoyan, one of Stalin's closest associates, came under rifle fire that day as it exited the Kremlin. No one was injured, and after a brief struggle the shooter was taken into custody. It turned out that he was a soldier from a Moscow air defense unit who was likely suffering from mental health problems. This incident was a terrible blow to the protection service under Vlasik's command: an unbalanced and armed soldier had been standing in plain sight at the Kremlin gates for some time, waiting for an official car to come out, without being questioned or apprehended. Vlasik was demoted, but the leader gave him a second chance. He continued to oversee Stalin's security.\n\nVlasik seemed to enjoy Stalin's full confidence. He followed the leader everywhere, often sat down at the same table with him to eat, and was granted the right to photograph him. Under Vlasik, the Main Guard Directorate became a powerful and influential government agency. In early 1952 it comprised 14,300 people and had an enormous budget of 672 million rubles. Vlasik's directorate was responsible not only for protection, but also for the maintenance of the apartments and dachas of top-level Soviet leaders, keeping Central Committee members supplied with consumer goods, handling the transportation and lodging of foreign guests, and overseeing the construction of new government buildings. In 1951 approximately 80 million rubles of the directorate's budget went toward maintaining the dachas and apartments of the fourteen highest-ranking Soviet leaders (including expenses for protection and servants). Stalin was, of course, the most expensive of the fourteen. A total of 26.3 million rubles were spent on his apartment and dacha in 1951. This sum probably did not include such expenses as automobile transport.\n\nServing in the Guard Directorate was both prestigious and lucrative. In 1951 the average compensation for members of Stalin's security team (including uniforms, housing, etc.) was 5,300 rubles per month, at a time when the average monthly wage throughout the Soviet Union was 660 rubles and the average per capita income for collective farm workers was approximately 90 rubles per month. In addition to material benefits, Vlasik's relationship with the leader gave him significant political influence, leading to his increasing involvement\u2014with Stalin's encouragement\u2014in the political intrigues that roiled around the _vozhd_ (leader). Having a powerful patron and sense of impunity was intoxicating. Vlasik drank and enjoyed a promiscuous love life, and so did his subordinates.\n\nStalin generally tolerated such \"weaknesses\" as a pledge of obedience and devotion. Yet he was known to put his subordinates in their place, especially if they took too many liberties. During the summer of 1947 one of the waitresses at the near dacha informed Stalin that while he had been away, the dacha commandant and his deputy threw a party with drinking and prostitutes, for which they stole refreshments from the official supply. Furthermore, the deputy commandant and his female companions looked through papers on Stalin's desk. On Stalin's orders, the deputy commandant was arrested, interrogated at length, beaten, and shot. This incident should have served as a warning to Vlasik, but it did not. Stalin continued to show a fairly relaxed attitude toward his chief bodyguard's morals. In 1950, on Vlasik's own admission, Stalin reprimanded him for \"graft\" and \"relationships with women,\" yet he remained in favor.\n\nVlasik's star waned only when the aging Stalin decided it was time for another general purge of state security. On 19 May 1952, the Politburo approved a resolution criticizing Vlasik and the entire leadership of the Ministry of State Security's Main Guard Directorate for \"criminal dissipation and the uncontrolled expenditure of resources.\" Significant cutbacks to the directorate's personnel, functions, and budget followed. Some of its members were charged with crimes. Vlasik was expelled from the party and demoted to deputy head of a labor camp in the Urals, and in December 1952 he was arrested. Running the Guard Directorate fell to the USSR minister for state security, Semen Ignatiev.\n\nThe arrests, personnel cutbacks, and reorganization of the Guard Directorate undoubtedly set its members on edge. None of them, fearing for their jobs and their lives, wanted to face the consequences that could come with taking initiative. For these reasons Stalin's bodyguards were very reluctant to check on him on 1 March 1953, even though something out of the ordinary was clearly taking place.\n\nThe branches of state security, including the branch in charge of Stalin's personal safety, were one very important set of controls regulating the huge machine that historians call the Stalinist party-state. The framework that held this machine together was the Bolshevik party, bequeathed by Lenin, but repeatedly modified to fit the needs of Stalin's dictatorship. Under Stalin, the party was a rigidly centralized organization whose power rested on its unquestioned right to hire, fire, and reassign personnel. Over many years, lists of positions were compiled (\"the nomenklatura\"). Each position came under the purview of a particular party committee, from the _raikom_ (district committee) to the TsK (the party's Central Committee). The career and fate of every official in the country depended on one of these party committees, and nobody, including the party functionaries themselves, could evade the system. Key government leaders were approved within the TsK apparat in Moscow.\n\nThe nomenklatura of TsK positions was constantly growing, a reflection of the center's pursuit of ever-greater control. In September 1952, half a year before Stalin's death, it comprised approximately 53,000 positions. Those who filled these positions were the \"cream\" of Soviet society, including high-level party and state officials, top military leaders, and the heads of the \"creative unions\" such as the Writers' Union. One step lower were officials in charge of important regional bodies: those holding nomenklatura positions within _obkoms_ (oblast or provincial committees), _kraikoms_ ( _krai_ or territorial committees), and the central committees of the Communist parties of the various republics that made up the Soviet Union. This list was also constantly growing. As of 1 July 1952 it totaled 350,000 positions.\n\nThese hundreds of thousands of functionaries were the backbone of the apparat and the pillar of the dictatorship. Of course Stalin never had direct contact with the vast majority of them. Furthermore, the party-state apparat had a life of its own and was relatively free of interference from the top leadership. In the struggle to survive, prosper, and rise through the ranks, officials sought ways to get around the strict rules aimed at centralization. They could generally act as convenience dictated so long as the paper trail they left reflected adherence to the rules. Abuses of power were common. A number of historians, exaggerating the significance of these processes, have argued that the Stalinist dictatorship was unstable, and many have attempted to explain the worst features of Stalinism\u2014mass repression especially\u2014as arising spontaneously from below.\n\nThe documentary evidence offers no support for the idea of a \"weak dictator.\" We do not know of a single decision of major consequence taken by anyone other than Stalin. We do not know of even a brief period when he did not exercise dictatorial control. The dictatorship developed extremely effective methods of manipulating and pressuring society and the apparat, and thus Stalin had a firm grip on power and the implementation of key decisions. Ongoing repression and purges of personnel kept society and the apparat in a state of mobilized tension. The archives have allowed historians to assess, in fairly precise numbers, the scale of the violence necessary to achieve such control. Official records show that approximately eight hundred thousand people were shot between 1930 and 1952. The number who perished as a result of the regime's actions, however, was much higher, insofar as Stalin's security apparat made frequent use of fatal torture techniques and the conditions prevailing in labor camps at times made them indistinguishable from death camps. Between 1930 and 1952, some 20 million people were sentenced to incarceration in labor camps, penal colonies, or prisons. During that same period no fewer than 6 million, primarily \"kulaks\" and members of \"repressed peoples,\" were subjected to \"administrative exile\": forced resettlement to a remote area of the USSR. On average, over the more than twenty-year span of Stalin's rule, 1 million people were shot, incarcerated, or deported to barely habitable areas of the Soviet Union every year.\n\nThose who were shot or sent to the camps included a fair number of ordinary criminals. But the exceptional severity of laws and the criminalization of all spheres of socioeconomic and political life meant that ordinary citizens who committed minor infractions or were swept up in various political campaigns were often classified as criminals. Furthermore, in addition to the 26 million who were shot, imprisoned, or subjected to internal exile, tens of millions were forced to labor on difficult and dangerous projects, arrested, subjected to lengthy imprisonment without charges, or fired from their jobs and evicted from their homes for being relatives of \"enemies of the people.\" Overall, the Stalinist dictatorship subjected at least 60 million people to some sort of \"hard\" or \"soft\" repression and discrimination.\n\nTo this figure we must add the victims of periodic famines or starvation, which during 1932\u20131933 alone took the lives of between 5 and 7 million people. The Stalinist famine was largely the result of political decisions. In its campaign to break peasant opposition to collectivization, the Stalinist government used famine as a means of \"punishing\" the countryside. All opportunities to relieve the situation\u2014such as purchasing grain abroad\u2014were rejected. Starving villages had their last stores of food expropriated.\n\nWe can conclude from this horrific summation that a significant proportion of Soviet citizens suffered some form of repression or discrimination during the Stalin period. It would not be an exaggeration to say that an absolute majority were brutally suppressed by a privileged minority\u2014except that many in that minority were also swept up in the terror.\n\nTo achieve its goals, including the implementation of mass repression and the extraction of grain from the starving countryside, the regime did not need its apparat to run with clocklike precision. The inability to achieve perfect centralization in such a vast country was compensated for by the widespread use of campaigns, which mostly followed a similar template. Campaigns were the cornerstone of Stalinist political practice. They all began with a set of goals and the assignment of specific tasks that originated with the center, usually Stalin himself. These steps were followed by the mobilization of the apparat to carry out the assigned tasks, using extraordinary methods and the total suspension of any sort of legality. As a result, a campaign took on the aura of a crisis, culminating at a point where retreat became necessary. This retreat took the form of a counter-campaign that eliminated some of those who had carried out the original campaign while solidifying its results and stabilizing the situation. This swinging pendulum led to the destruction of vast material resources and countless human lives. But within the context of the Stalinist system, the campaigns were an effective method of mobilizing a vast country toward a central goal.\n\nStalin himself did not need to exercise tight control over all party and government bodies in order to retain dictatorial power. It was sufficient to hold the main levers of power, the most important being control of the secret police. He understood, sooner than other Soviet leaders, that state security could be a valuable weapon in intraparty warfare. This was a key reason for his success. Once he attained control of the Soviet Union's \"punitive structures,\" he never let it slip from his hands. He continued to use state security as an instrument of power until the day he died.\n\nAs we will see, Stalin devoted much time to the hands-on management of state security, and during certain periods\u2014most notably during the Terror of 1937\u20131938\u2014the majority of his time. He personally initiated all the main repressive campaigns, devised plans for carrying them out, and painstakingly monitored their implementation. He guided the fabrication of evidence for numerous political trials and in several instances wrote detailed scripts for how trials should play out. He had a passion for reading the cascade of arrestee interrogation protocols that came before him, and the notations he made on these documents show that he read them thoughtfully and attentively. He often wrote commentaries and issued orders for additional arrests or for the use of torture to \"get to the truth.\" He personally sanctioned the shooting of many people. Some he knew personally; others he had never met.\n\nIn addition to the many \"ordinary\" functions that the chekists performed for Stalin, they also dealt with special, \"delicate\" matters.\n\nOn 5 May 1940, on Stalin's orders, a special state security group abducted Kira Kulik-Simonich, the wife of the deputy people's commissar for defense, Marshal Grigory Kulik, as she was leaving her house. She was secretly transported to prison, interrogated at length, and then quietly shot. Kulik-Simonich was the descendant of a highly placed tsarist official. Many of her relatives had been shot, and some had managed to escape abroad. She had been married before and had spent time in exile with a previous husband charged with illegal activities involving hard currency. The chekists who reported all this to Stalin embellished the story with many more transgressions, including Kulik-Simonich's affairs with foreigners. Stalin advised Kulik to divorce his wife, but when the marshal balked, Stalin ordered that Simonich be quietly done away with. When Kulik discovered his wife's disappearance, he telephoned state security chief Lavrenty Beria, who denied that his agency was involved. Kulik did not believe him and began to dig for the truth. He was summoned to the Central Committee, where he underwent a three-hour interrogation and was ordered not to \"slander\" state security. Furthermore, he was told, his wife was probably a spy who had fled under threat of exposure. Kulik relented.\n\nCases like this one, where Stalin, for political reasons, felt it was not expedient to arrest and charge people openly, were no rarity. A year before Marshal Kulik's wife was murdered, in July 1939, the Soviet ambassador to China was killed along with his wife. Specially selected chekists beat their heads with hammers and then staged a car crash. In early 1948, the Jewish civic leader and stage director Solomon Mikhoels, a popular and well-known figure in the USSR and the West, was similarly done away with. Chekists crashed into Mikhoels with a truck and presented the incident as an accident. The evidence leaves no doubt that this murder was also carried out on Stalin's direct orders. It is one of numerous acts of individual terror committed by Stalin. Such targeted killings were also perpetrated overseas. The most famous is the 1940 murder of Trotsky in Mexico.\n\nThe archives contain a huge number of documents confirming that Stalin routinely used the secret police to carry out arbitrary and brutal actions based solely on his own assumptions of guilt. They leave a clear impression that Stalin personally organized acts of terror that went far beyond any reasonable sense of \"official necessity.\" This homicidal aspect of his dictatorship obviously held special appeal for him. Immersion in a world of violence, provocation, and murder fed and intensified his pathological suspicion. Driven by fears and a certainty that he was surrounded by enemies, he felt no compunction about using violence on the grandest scale. These personal qualities were an important factor in the brutalities committed by the Soviet government from the 1920s through the 1950s.\n\nAlthough Stalin relied heavily on state security, he never became beholden to it. In assigning the secret police the dirtiest work, he did not harbor illusions about the loyalty of his \"sword of revolution\" but instead kept his chekists in rein through periodic shake-ups and purges of their ranks. In a moment of candor, he confided to State Security Minister Ignatiev that \"A chekist has only two paths\u2014advancement or prison.\" He remained true to this principle. From the 1930s through the 1950s, chekist organizations were subjected to waves of brutal repression. The new executioners destroyed the old, only to later wind up in the torture chamber themselves.\n\nFor many decades historians have been arguing over the antecedents and causes of Stalin's exceptional brutality. Many trace the source back to the 1917 Bolshevik Revolution, an event that, for Stalin, opened the door to power.\n\n## **2 IN LENIN'S SHADOW**\n\nHistorians debate the extent to which the unrest in Petrograd in late February 1917 was spontaneous. Some claim the demonstrations were organized by professional revolutionaries, but nobody can say with certainty that this was so. The revolution erupted without warning, as a result of the social destabilization caused by almost four years of war, and the tsar and his advisers did not immediately grasp the gravity of the situation. Lenin, in Switzerland, learned of the revolution by reading about it in Western newspapers. The news was also slow in reaching Stalin in Siberian exile, as the local authorities, apparently hoping the upheavals would blow over, banned their local papers from carrying reports from Petrograd.\n\nThe tsar's abdication sparked widespread jubilation. His brother, Grand Duke Mikhail, had been named Nicholas's successor, but he also relinquished the throne, thus formally ending the monarchy. Shortly thereafter, in early March 1917, a town meeting was held in Achinsk, where Stalin was exiled at the time. For some reason he was not present, but his close comrade Lev Kamenev played a major role in it. A telegram praising the grand duke's decision was sent on behalf of those gathered. In 1925, when Stalin and Kamenev wound up on different sides in the struggle for power, Stalin reminded his old friend of this warm gesture toward a member of the royal family, a gesture that now looked like a serious political blunder. It is unlikely, however, that Stalin felt this way in 1917. The telegram reflected the prevailing intoxication with hope and freedom. In this mood, Stalin, Kamenev, and other freed revolutionaries streamed toward Petrograd.\n\nIt took some time before Stalin and his fellow Bolsheviks found their bearings when they first were able to emerge from the underground and play a legitimate role in the new system. In the capital, they discovered divided political power. Russia's parliament, the State Duma, had formed a provisional government, composed primarily of members of liberal parties that favored the creation of a Western-style parliamentary republic. Yet at the same time, the Petrograd Soviet of Workers' and Soldiers' Deputies, a revolutionary body whose authority came from the support of rebelling workers and, most important, soldiers of the Petrograd garrison, exercised a significant share of actual power. The soviet was run by members of socialist parties: Menshevik Social Democrats and Socialist Revolutionaries (SRs). These two parties were the most influential forces within the revolutionary camp, and they had so far outmaneuvered the other parties, including the Bolsheviks. The SRs and Mensheviks were the ones setting the revolution's short- and long-term objectives. They considered the events of February a bourgeois revolution that would introduce a prolonged period of bourgeois-democratic development. They therefore believed that at the initial stage, a liberal bourgeois party should hold power and that it was for the Constituent Assembly to determine the shape of the new Russia. The attainment of socialism was a distant goal. Other, more developed capitalist countries\u2014not Russia\u2014would lead the way toward world socialism.\n\nAt the same time, the Russian socialists had no intention of renouncing the power that had fallen into their laps. They were not obtuse dogmatists, incapable of deviating from doctrine, but realists and pragmatists, albeit lacking in political sophistication and decisiveness. They were well aware of the dangers confronting the country. Foremost among them was civil war and the spread of a bloody rebellion that could wreak havoc and take Russia to the brink of catastrophe and collapse, not for the first time in its history. The most eloquent symbols of this danger were the millions of war-weary and embittered armed men returning from the front. In 1917, the only responsible position a politician could take was that civil war must be avoided at all costs. Maintaining civil peace was the only way to prevent massive casualties and pave the way toward a better future. The socialists leading the soviet saw it as their duty to suppress revolutionary excesses and work with the liberals and the Provisional Government. Cooperating from a position of strength, they made reasonable use of their power and placed the highest priority on maintaining peace. The official formulation of this policy of compromise was: support for the Provisional Government so long as it advanced the cause of revolution.\n\nMany Bolsheviks, usually described as \"moderate\" or \"rightist,\" endorsed essentially the same approach. Kamenev was one of this faction's leaders. He and Stalin shared a bond of long-standing friendship and party collaboration. In December 1912 Stalin wrote him, \"Greetings friend! I rub your nose in an Eskimo kiss. Dammit. I miss the hell out of you. I miss you\u2014I swear on my dog! There's nobody, nobody to have a heart-to-heart talk with, devil take you.\"\n\nThere is nothing surprising in the fact that early on, Stalin and Kamenev held similar political positions. While Lenin and many other prominent Bolsheviks remained in Switzerland, Kamenev and Stalin played an important role in leading the party in Russia. After arriving in Petrograd, they essentially took control of the Bolshevik newspaper _Pravda_ and used it to promote a moderate agenda, based on the belief that the ascent of the liberal bourgeoisie to power was in accordance with the dictates of history and that socialism was a long-term prospect. The newspaper proclaimed conditional support for the Provisional Government. As members of the Petrograd soviet leadership, Kamenev and Stalin interacted closely with other socialists. The Bolsheviks were beginning negotiations to explore joining forces with the Menshevik left wing.\n\nFrom the start, Kamenev and Stalin were forced to defend their stances. Lenin, dissatisfied with the political line being promoted by _Pravda,_ demanded different slogans. Writing from emigration, he argued for a radical course, declaring war on the Provisional Government and advocating socialist revolution. Kamenev and Stalin worked together to parry these attacks. They heavily edited an article sent by Lenin before publishing it in _Pravda._ Most likely, they truly did not understand Lenin's intentions and assumed his radicalism was simply a function of being out of touch with what was actually happening in the country.\n\nLenin's position, however, was based on meticulous political calculations. Kamenev's and Stalin's moderate positions opened the door to cooperation among the main socialist parties, but the cooperation never materialized. From the standpoint of the country's well-being, cooperation in a joint effort to keep radicalism at bay was the only correct course. From the standpoint of the ultimate goal of a Bolshevik takeover of sole power, it was ruinous. Taking part in a coalition, even as oppositionists, would tie the Bolsheviks' hands and deprive them of support from radical segments of the population. This was not what Lenin had in mind, and his disapproval ultimately sealed the fate of \"rightist\" Bolshevism.\n\nWhen news of revolution in Russia reached Lenin, he was ready with a plan of action, carefully worked out in light of past political struggles. Lenin was gambling on being able to grab power before the revolutionary situation stabilized. His historical moment would be the period of revolutionary radicalization, a period he knew well based on the experience of other revolutions. Even at the early, relatively moderate stage of the revolution, Lenin advanced an extreme program for which the revolution was not quite ready. To put it another way, knowing that a tendency toward radicalization would come, he was playing a waiting game. This strategy had obvious advantages for a party whose ultimate goal was to seize power. The advancement of radical goals that many saw as reckless put the party in a class all its own. That nobody wanted to enter a coalition with it allowed it a certain freedom. A radical program also served as a means of crushing moderate forces within the party and mobilizing its more decisive elements. Finally, such a program, despite being initially rejected by the masses, would eventually gain wider acceptance as mounting despair and impatience fostered a greater acceptance of extremism.\n\nOnce he heard about the revolution, Lenin hastily prepared to leave Switzerland for Russia. Eager to enter the fray, he negotiated an agreement with the German authorities allowing him to travel to Russia across enemy territory. In so doing, he was taking a serious risk and opening himself up to accusations of collusion with the enemy or even espionage. But the ends justified the means: he needed to get to Petrograd. As soon as he stepped off the train, he publicly announced his plan of action.\n\nLenin proclaimed that the Bolsheviks must refuse to support the Provisional Government and fight for socialist revolution and the transfer of power \"into the hands of the proletariat and the poorest segments of the peasantry\"\u2014in other words, into the hands of the Bolshevik party. The fledgling democracy that had come about after the February Revolution was never given a chance to establish itself, but for Lenin, it had already outlived its usefulness. The parliamentary republic had to be replaced with a soviet republic that, under Bolshevik leadership, would introduce socialist changes. For now, Lenin mentioned just a few of the most important changes: the nationalization of land, the transformation of large estates into model farms under the control of the soviets, and the nationalization of banks or even their merger into a single national bank. In accordance with these new objectives and to clearly distinguish the Bolshevik party from other socialist parties, Lenin proposed changing its name from the Social Democratic Party to the Communist Party.\n\nThis platform met with serious opposition, both from outside the party and within. Lenin was, in essence, proposing a vaguely articulated program for the seizure of power. How would that power be used if his plan succeeded? What would socialism mean under Russian conditions? What guarantee was there that revolution in Russia would be followed by revolution in more developed countries (without which Russia would find itself isolated)? Instead of answers, these questions were met with brazen demagoguery. For now it was clear that the Leninist course was kindling civil war.\n\nAccording to contemporary memoirs, during one of Lenin's speeches after his arrival in Petrograd, a party comrade who had once been close to him cried out from his seat, \"That's nonsense, the ravings of a madman!\" Lenin's Bolshevik associates could not abide such an outcry, even if they more or less agreed with it. Yet in early April, at meetings of the leading Bolshevik organizations, Lenin's ideas were voted down by the majority. Not only did Kamenev continue to publicly oppose Lenin's ideas, but so too did Stalin.\n\nThe sharp reaction of political opponents outside the party apparently suited Lenin's purpose. He was intentionally setting up a confrontation that would distance the Bolsheviks from the country's other political forces. Within the party, however, he would have to calm the discord. It was not possible to do so by the methods Stalin would employ later. The Bolsheviks were not yet that party. The situation in the country\u2014buffeted by the turmoil of revolution and fledgling democracy\u2014was also different. And Lenin was a different sort of leader. He used a combination of hard-line intransigence and conciliation. A particularly important maneuver was the recruiting of \"rightist\" Bolsheviks, especially Stalin and Kamenev, to his side. Lenin moved cautiously, always allowing his opponents to save face. Instead of driving them into a corner, he promoted them to top party positions. In Stalin's case, this approach worked. Whatever may have been going on in Stalin's head, he quickly threw his support behind Lenin.\n\nThe endorsement that Lenin gave Stalin during Central Committee elections at the April 1917 party conference clearly reflects their close working relationship: \"We have known Com. Koba for very many years.... He handles any responsible job well.\" This recommendation earned Stalin a spot on the Central Committee, yielding him more votes than anyone except Zinoviev and Lenin himself. Stalin saw, very directly, Lenin's huge influence within the party. After some wavering, he made a firm decision to align himself with strength.\n\nWas Stalin merely advancing his own career, or did he actually understand and accept what Lenin stood for? Identifying the source of Stalin's initial inclination toward \"moderate\" Bolshevism is of fundamental importance for anyone seeking to understand the workings of his mind. Clearly, the flexibility he exhibited in March\u2013April 1917 does not fit the image of an uncompromising, power-hungry radical. Was his apparent moderation due to Kamenev's influence? Or was he swayed by the other socialists in the Petrograd soviet, where many of the Mensheviks were fellow Georgians? Perhaps he had not yet developed the confidence to act as an independent political figure and felt he needed someone to follow. In that case, why did he not immediately fall in line behind Lenin after receiving his letter from Switzerland? Perhaps Stalin was genuinely \"moderate\" in early 1917 but, like many others, changed under the force of circumstances. Historical sources offer no clear-cut answers to these questions. What we do know is that Stalin was not always a radical Bolshevik. His \"moderation\" and \"rightism\" would emerge again after Lenin's death, when the party leaders were choosing the path toward socialism, down which they would lead their vast and isolated country.\n\n###   **STALIN IN LENIN'S REVOLUTION**\n\nThe escalation of Russia's February Revolution followed a typical pattern. The moderate revolutionaries who found themselves in power after the tsar's overthrow sought mainly to avoid civil war. But while these moderates vacillated, stumbled, and missed opportunities to consolidate their position, the increasingly impatient masses began looking to those who promised radical and immediate change. In this environment, Bolshevik propaganda found fertile ground. Calls for immediate withdrawal from the war, immediate expropriation of large estates and the turning over of land to the peasants, and immediate worker control of industry had broad appeal. As often happens in times of revolution, few demanded that the Bolsheviks spell out just how their program would be put into practice. The masses were inspired by a new faith. Among the Bolshevik rank and file, fewer and fewer were asking their leader the difficult question: What would come next? Lenin led the party with amazing energy, promising that socialism would somehow solve all problems. The banners of the Leninist party\u2014\"Most important\u2014engage the enemy\"; \"We'll see what happens\"; and \"Things couldn't be any worse\"\u2014sum up the folk wisdom that guided millions to put their faith in Bolshevik promises.\n\nStalin was among the Bolshevik leaders who supported Lenin without demanding detailed explanations. Having cast off doubts about the suitability of socialism for a predominantly agrarian country, Stalin now proclaimed that \"It is entirely possible that Russia will prove to be the country that paves the way toward socialism.... We must reject the obsolete notion that only Europe can show us the way. There is dogmatic Marxism and creative Marxism. I stand on the ground of the latter.\" The ground of \"creative Marxism\" proved so accommodating to Stalin's political needs that he settled there permanently. In 1917, having cast aside the apprehensions of \"rightist\" Bolshevism, Stalin set out on Lenin's radical course toward the seizure of power and the introduction of socialism. He never wavered in this decision. The occasional inconsistencies that scholars have noted between Lenin's and Stalin's pronouncements are quite superficial and probably show only that Stalin had trouble keeping up with Lenin's frequent tactical twists and turns. Lenin himself had trouble keeping up with them.\n\nHaving set his sights on seizing power, Lenin faced a changeable and complicated situation that made it hard to choose the right moment to strike. The Bolsheviks' strategy was to maintain revolutionary momentum while awaiting the right moment to cross the line of legality. Overt action against the Provisional Government and the soviets would undoubtedly trigger a confrontation. The time for action had to be chosen carefully, but holding back also had its risks. The only way to gauge the opposing side's strength was to probe its weaknesses. Furthermore, the Bolsheviks needed to demonstrate to the radical workers and soldiers on whom they were counting that they were capable of action, not just words. Bolshevik forces had to maintain a constant state of combat readiness through \"war games,\" one of which would turn into a real battle.\n\nIn early July 1917, armed soldiers, sailors, and workers took to the streets, marching under Bolshevik banners calling for the overthrow of the Provisional Government. Blood was spilled. The Bolsheviks did not overtly take charge of the rebels, but few were fooled. It was crystal clear to virtually everyone that they were working behind the scenes to overthrow the government. The only question\u2014and historians continue to debate it\u2014was the extent of their involvement in planning the demonstrations. The Provisional Government was able to crush these disturbances, but its efforts at counterstrikes proved haphazard and ineffective. The authorities launched an investigation into allegations that Lenin was a spy being financed by Germany to foment revolution. Charges that the Bolsheviks had organized the riots provided grounds for certain actions against them. The Bolshevik newspaper offices and headquarters were laid waste and shut down, and a few activists were arrested. The \"moderate\" Kamenev was among those arrested, while Lenin and Zinoviev remained free and went underground.\n\nStalin, less well known to the government, was not on the list of targeted revolutionaries. He felt so secure that he even proposed that Lenin hide out where Stalin was living at the time, in the apartment of his old friends, the Alliluevs. Stalin's friendship with the Alliluevs was long-standing and strong. In 1919 he married their daughter Nadezhda, still a teenager at the time.\n\nStalin accompanied Lenin and Zinoviev as they traveled from Petrograd to the suburban town of Razliv, where the two fugitives were concealed by the family of a worker, Nikolai Yemelianov, a Bolshevik sympathizer. They lived in a loft above Yemelianov's shed. Later, disguised as farm workers, they made their way to a more sparsely populated area where they took shelter in a hut. In August, Lenin moved to Finland, and from July to October Stalin did not meet with him. Nevertheless, during Stalin's dictatorship several assertions appeared claiming that he had met with Lenin not once but twice during this period. The main witness of these supposed meetings was Yemelianov.\n\nLike many other revolutionaries, Yemelianov met a tragic fate. He and three of his sons were arrested in the 1930s. Two sons were shot, and one was released after Stalin's death. The elder Yemelianov was sent into exile in Siberia. In June 1945, apparently grasping that offering an appropriately hagiographic episode for Stalin's biography represented his best chance for leniency, he appealed to Stalin for permission to return to his village: \"In 1917 you saved the life of Vladimir Ilyich Lenin by arranging for me to hide him in a hut.\" The appeal was shown to Stalin, and soon afterward Yemelianov was permitted to return to Razliv and even to work in the Lenin Museum established there. There is no doubt that his release was decided by Stalin personally. Yemelianov's \"recollection\" that Stalin twice visited Lenin became part of Stalin's official biography.\n\nWhile Lenin was in hiding in Finland, Stalin and other Bolshevik leaders continued to strengthen the party ranks. In late July 1917 they convened the Sixth Party Congress, at which Stalin delivered speeches and generally played a prominent role. The political winds were starting to favor the Bolsheviks. Having fully recovered from the Provisional Government's ineffective efforts at suppression, they began to strengthen their position, helped by Prime Minister Aleksandr Kerensky's frequent missteps. In August, Kerensky provoked a confrontation with the commander in chief of the Russian Army, General Lavr Kornilov. With Kerensky's consent, Kornilov had sent some of his most reliable units to Petrograd to help secure the city after the unrest in July. Soon, however, Kerensky began to doubt Kornilov's loyalty to the Provisional Government. In a pivotal moment of anti-Bolshevik dysfunction, he proclaimed Kornilov to be a mutineer. This conflict distracted attention from the Bolshevik threat. When the Bolsheviks sided with Kerensky against Kornilov, they obtained the release of several of their activists from prison. Lenin remained in hiding.\n\nIn September and October, the Provisional Government's hold on power was clearly weakening, as was the influence of the Menshevik and Socialist Revolutionary soviets that supported it. The Bolsheviks, meanwhile, grew increasingly active. Lenin believed that the time to revolt and seize power had come. Again he encountered opposition within the party to his call for armed insurrection, most prominently from Kamenev and Zinoviev. Most of the other party leaders, including Stalin, supported him. Understanding that his presence would help assuage doubts about the use of force, Lenin snuck into Petrograd. The final vote on the uprising was held at a Central Committee meeting on 10 October 1917. Kamenev and Zinoviev found themselves in the minority but did not back down. The following day they wrote a letter to a wider circle of members.\n\nThey had a strong case to make. They enumerated the weaknesses of Lenin's arguments, disputing the assumption that the majority of Russians supported the Bolsheviks. They reminded their comrades of the huge difference between chanting popular slogans and putting them into effect. Furthermore, Germany was apparently prepared to reject the Bolsheviks' peace terms, and Russian soldiers were clearly in no mood for a \"revolutionary war.\" \"The soldierly masses will leave us in droves.\" Kamenev and Zinoviev rejected Lenin's references to imminent revolutions in the West as hypothetical. They hoped to avoid a civil war, but such avoidance required that the Bolsheviks coexist with other political forces. Now that they had majority support in many soviets, the Bolsheviks needed to gain seats within the Constituent Assembly since \"only in the Soviets will the Constituent Assembly be able to find support for its revolutionary work. The Constituent Assembly plus the Soviets\u2014this is the combined type of state institution toward which we are moving.\" The way events were developing, the Bolsheviks were guaranteed significant or even overwhelming influence in these legal governmental bodies. On the other hand, if they launched an insurrection and it failed, the consequences would be much worse than the aftermath of the July riots.\n\nA strategy of achieving dominance through legal and peaceful means was neither utopian nor farfetched, but it did not appeal to Lenin. It is hard to know whether he truly believed that the Bolsheviks would be crushed in a counterrevolution if they failed to act first, but it is certain that Lenin did not want his party to join a coalition or take part, even as a dominant force, in the legal political process. The armed seizure of power was the best or perhaps the only means of avoiding a coalition with Mensheviks and SRs and getting rid of the Constituent Assembly, which was due to hold elections in a few weeks. Zinoviev and Kamenev's proposal that the Bolsheviks launch a serious campaign for seats in the Constituent Assembly reflected the general recognition within the country of the importance of Russia's new parliament. Officially, the Bolsheviks also recognized it. Stalin was among the party leaders running for a seat. It is telling that on 18 October 1917, amid heated preparations to seize power, he did not forget to send the Caucasus District Electoral Commission a telegram confirming his candidacy.\n\nClearly concealing his true thinking and offering eloquent editorializing and slogans in place of practical planning, Lenin stubbornly repeated his call to action: it was necessary and possible to seize power by force, and the time had come. What would happen after the revolution? This question seemed to worry everyone but Lenin, whose implacable obstinacy was the only real argument in favor of insurrection. For a party that was not monolithic but was strongly oriented toward its leader, a party that was tired of uncertainty and contention, Lenin's stubbornness was decisive. Most historians agree that without Lenin the October Revolution would probably never have happened.\n\nConvinced that they were right (and not without justification, as it turned out), Zinoviev and Kamenev made a desperate move. Having been blocked from publishing in the Bolshevik press, Kamenev submitted an article to a small non-party newspaper spelling out the opposition's views. Lenin was furious and demanded that Kamenev and Zinoviev be expelled from the party. Stalin was among those opposing this measure. He responded to Lenin by using his position as editor of _Pravda_ to publish a letter from Zinoviev, along with a conciliatory editorial characterizing the incident as having \"run its course\" and stating that \"overall, we remain like-minded.\" This is one of the few times he openly opposed Lenin on a matter of substance. What explains this mini-revolt? Was Stalin not yet free of \"rightist illusions\"? It is possible that while appearing to follow Lenin, in his heart he shared Kamenev's and Zinoviev's concerns. Other factors were probably at play as well, including Trotsky.\n\nLev (Leon) Trotsky had always played a prominent role within the Russian Social Democratic movement, but his ambitions were not limited to prominence within the party. Before the revolution, he was often at loggerheads with Lenin, and their mutual attacks often turned ugly. But as much as Lenin and Trotsky may have argued, they were also drawn to one another. Both were preoccupied with the idea of socialist revolution and fervently believed that it would soon be possible. Both were decisive and fearless of risk. Like Lenin, Trotsky learned of the revolution when he was out of the country, in the United States. He did not manage to return to Russia until May 1917, but once there he immediately entered the fray. His talents as an orator and organizer, along with his revolutionary credentials (he had been one of the leaders of the soviets during the 1905 revolution), earned him instant recognition. Upon arriving in Petrograd in 1917, Trotsky immediately understood that he and Lenin were natural allies. Their allegiance fell into place naturally, without any negotiations. Trotsky joined the Bolsheviks and Lenin immediately recognized him as a strong partner, ready to use word and deed in an unwavering battle for power. Trotsky quickly found himself at the center of events. By September he was head of the Petrograd soviet, playing a key role in plotting the insurrection.\n\nEven as they recognized Trotsky's value to the party, Lenin's long-standing comrades could not have been happy about his meteoric ascent. To them he was an ambitious interloper. Stalin would surely have felt a certain sting of envy, if only because this rising Bolshevik star was everything he was not. During the fevered lead-up to revolution, when oratorical gifts were in demand, Trotsky could keep a crowd of thousands spellbound, while Stalin was a lackluster speaker. Trotsky was a brilliant and compelling writer, while Stalin lacked the talent for inspiring slogans or mobilizing catchphrases.\n\nTrotsky's ascent prompted Lenin's long-term comrades-in-arms to close ranks, a realignment complicated by Kamenev's and Zinoviev's diminished standing. It was during these tumultuous months that the seeds of the anti-Trotsky alliance were sown; they would sprout shortly after Lenin's demise. Lenin must have understood the clashes taking place around him in 1917, but what he cared about most was party unity and, undoubtedly, a distribution of counterpoising power within the party leadership. He put up with the internal divisions. Kamenev and Zinoviev kept their posts, and events soon overtook intraparty strife. In the early hours of 26 October 1917, the Bolsheviks arrested members of the Provisional Government and formed their own Council (or Soviet) of People's Commissars, with Lenin as its chairman. Stalin was named people's commissar for nationalities.\n\nAfter Stalin achieved power, official Soviet propaganda proclaimed him and Lenin the leaders of the revolution. His political opponents, Trotsky especially, argued that his role had actually been insignificant. The truth lies somewhere between these highly politicized interpretations. Stalin did not lead the revolution, but as a senior Bolshevik, member of the party's Central Committee, and editor of its main newspaper, he filled an important role. His choice to follow Lenin determined his place within the revolution.\n\nWhat lessons did Stalin draw from his first experience in fighting to attain power? He seems to have been greatly impressed by Lenin's decisiveness, his stubborn and relentless insistence on his own program of action. Years later, when Stalin carried out his \"revolution from above,\" one of many crises in the history of long-suffering Russia, he fully demonstrated his own talent for decisive action. Borrowing from Lenin a dogged and unscrupulous political modus operandi, he strove to seize and maintain power without worrying about what effect his actions would have on others. This principle allowed him to act with maximal ruthlessness and little constraint. Pushing his own revolution in the 1920s, Stalin, like Lenin, bet on a strategy of unrestrained radicalism.\n\n###   **THE MILITARIZATION OF THE PARTY**\n\nOne aspect of Lenin's ruthlessness that put him in a particularly strong position was his utter lack of reluctance to provoke a civil war, which he saw as a natural element of the transition to socialism. There was no reason to expect that all of Russia, to say nothing of its allies, would compliantly accept the supremacy of radical Bolshevism. The unexpectedness of their uprising and the fatigue of the masses initially bought the Bolsheviks some time, but the situation soon changed. The illegitimacy of the new government, its crude and cynical actions, and social experiments that turned the existing order on its head inevitably met with mass resistance. The Provisional Government was toppled and replaced by a Bolshevik Council of People's Commissars. In January 1918, the Constituent Assembly disbanded. In March 1918, a humiliating and predatory separate peace with Germany was concluded. All these events paved the way toward a civil war that soon engulfed the country. Aligned against the Bolsheviks were members of the upper and middle classes (\"the White movement\"), persecuted socialists, and peasants angry over the confiscation of their crops. The peace with Germany also brought Russia's former allies into the Civil War. War furthermore presented opportunities for ultra-radical elements and ordinary criminals. Peasants rose up against both the Bolsheviks and the Whites, and soon innumerable groups were fighting one another. The new wave of bloodletting unleashed by the Bolsheviks grew with amazing speed and continued more or less unabated for three years, from 1918 through 1920.\n\nIn scale and loss of life, the Civil War greatly exceeded Russian casualties during World War I and the February Revolution. Of the 16 million people within the Russian Empire and Soviet Russia who demographers estimate died of wounds, hunger, or disease during 1914\u20131922, at least half (8 million) perished during the three years of the Civil War. Another 2 million fled the country. The horrific famine of 1921\u20131922, largely a by-product of the Civil War, took some 5 million lives. By comparison, \"only\" slightly more than 2 million Russians were killed in World War I (1914\u20131917). These gruesome statistics set Russia apart from the other countries ravaged by World War I. War, famine, epidemics, and civil strife persisted there twice as long and took a much greater toll.\n\nEven these awful numbers do not fully reflect the Civil War's horrors. Statistics cannot capture the pervasive misery, the numbing of human feelings, and the destruction of any sense of right and wrong. Savage murders and mass terror became commonplace. The epidemic of savagery inevitably engulfed the Bolsheviks themselves. The Civil War shaped the new state and largely determined its trajectory.\n\nStalin was a typical product of his time. As he did before the revolution, he continued to follow Lenin. Part of an exclusive group of influential Soviet functionaries, Stalin was a member of the government, a member of the party's Central Committee, and a member of the top leadership. He spoke with Lenin almost daily. In 1919 he was elected to the Politburo, the body that remained at the center of power in Soviet Russia and the USSR for the next seventy years, until the collapse of the Communist system. Stalin had his own area of expertise: smoothing relations between the Bolshevik center and the outlying ethnic entities that comprised the Russian Empire and then the Soviet Union. But as with all Bolshevik leaders, his \"portfolio\" would remain subordinate to the primary imperative of retaining power. He spent his time from 1918 to 1920 on various fronts. He was away from Moscow so often that of the fifty-one Politburo meetings held in 1919, he took part in only fourteen; in 1920 he attended thirty-three out of seventy-five.\n\nHis first mission on behalf of the Soviet government came in June 1918. As hunger swept central Russia, Stalin was sent to Tsaritsyn (later Stalingrad, now Volgograd) to acquire grain from southern Russia for the country's starving center. This economic mission quickly turned into a military one. Tsaritsyn was under attack by forces hostile to the Bolsheviks. Railway lines connecting the cities of central Russia with agricultural areas were constantly being cut. Bolshevik armed forces in Tsaritsyn were organized on a model that became widespread during the early stages of the Civil War, a model that relied primarily on poorly disciplined and unprofessional partisan detachments. Aware that no successful war could be waged without a regular army, the Bolshevik leaders in Moscow\u2014primarily Trotsky, who was in charge of the Red Army\u2014decided to use officers from the former tsarist army and place them under the control of party commissars. This policy met with serious resistance. Newly appointed revolutionary commanders had little desire to subordinate themselves to former officers, whom they did not trust. The feeling was mutual. Indignities and mistreatment drove many officers to defect to the other side. Gradually, military necessity and pressure from Moscow forced the army to become more professional and tolerant of former officers.\n\nLargely thanks to Stalin, Tsaritsyn became a model of revolutionary guerrilla warfare. He wielded his authority as a member of the government and Central Committee and enjoyed unimpeded control not only over the civilian government, but also over the forces of the North Caucasus Military District, headquartered in Tsaritsyn. He found a loyal and obedient helper in Kliment Voroshilov, commander of Red Army detachments retreating to Tsaritsyn from Ukraine, which had been captured by the Germans. The two men shared a mutual hostility and mistrust toward trained military professionals or \"specialists.\" This theme often came up in Stalin's telegrams to Moscow:\n\nSpecialists are lifeless pen-pushers, completely ill-suited to civil war.\n\nIf our military \"specialists\" (cobblers!) weren't sleeping and loafing, the [railway] line would not have been cut, and if the line is restored, it won't be because of the military men, but despite them.\n\nThey, as \"headquarters\" workers, capable only of \"drafting plans\" and submitting plans for reorganization, are absolutely indifferent to operational actions, to the matter of supplies, to the control of different army commanders and generally feel like outsiders, like guests.\n\nOur new army is being built thanks to the fact that side-by-side with new soldiers, new revolutionary commanders are being born. Imposing known traitors on them [Stalin goes on to list a number of military professionals] disrupts the entire front.\n\nThese comments (there are many more examples) accurately reflect Stalin's philosophy of how the Soviet military should be developed. His words were matched by actions. Stalin dismissed the experienced officers and took operational command into his own hands. His dispatches to the capital were filled with glowing reports of the results brought about by this decision. It is difficult to imagine, however, that Stalin, who had no military experience, had never served in the army, and was relying on dilettantes like himself for guidance, was able to quickly acquire the complicated skills needed to run an effective military force. Common sense and revolutionary fervor could have taken him only so far. Indeed, the Stalin-Voroshilov partisan army was not able to withstand attacks by the enemy's regular units.\n\nIn August 1918, after two months under his command, Tsaritsyn was on the verge of falling. Stalin responded to the threat of defeat with a maneuver that would later become his political signature: a hunt for \"counterrevolutionary plots.\" A wave of arrests in Tsaritsyn swept up former tsarist officers (including those currently serving in the Red Army), former tsarist officials, businessmen, and ordinary citizens unfortunate enough to find themselves in the path of the purge. A \"plot\" headed by an employee of the People's Commissariat for Railroads, N. P. Alekseev, was alleged to be at the center of the counterrevolutionary movement. Alekseev was a \"bourgeois specialist,\" a former nobleman and officer working for the Soviet government who had been sent to Tsaritsyn from Moscow on commissariat business. In short, he perfectly fit the preconceived profile of someone who would mastermind a counterrevolutionary conspiracy. The accusations leveled against the \"conspirators\" were boilerplate and not terribly persuasive. A case was thrown together in a matter of days, culminating in executions and an announcement in the local newspaper.\n\nThis incident might have been just another chapter in the annals of the \"Red Terror\" had Alekseev not been accompanied on his trip to Tsaritsyn by Konstantin Makhrovsky, a senior official from the Supreme Economic Council and a long-standing member of the Bolshevik party. In the heat of the moment, Makhrovsky was also arrested and imprisoned for several months. He was not shot, however, and eventually was released under pressure from Moscow. This left an unwanted witness eager to relate what he had observed. The indignant Makhrovsky wrote a long report chronicling how things were being done in Tsaritsyn. He made it clear that the Alekseev case had been fabricated by members of the secret police \"obsessed,\" he wrote, \"with hunting down counterrevolution.\" Makhrovsky's portrait of life in Tsaritsyn probably shocked some senior officials in Moscow who had been following the war from their offices:\n\nHere is the picture I saw:... N. P. Alekseev, whose face was totally covered by a mask of blood.... One eye was completely closed, and you could not tell if it had been beaten out of him or was just covered by swelling.... They were beating Alekseev with the butt of a revolver and their fists, and, after he collapsed, they trampled him with their feet....\n\nReturning to the gallery of types, in regard to those arrested and detained by the Cheka whom I happened to see, I must make the following comment: most of them were arrested by chance, shot, and some time later notices appeared in the local paper listing those who had been shot as all sorts of criminals....\n\nTwo arrestees were brought into my cell who had been held on a barge. One of them told me about the barge on the Volga holding 400 people. Using a barge as a prison started during the evacuation of Tsaritsyn. When the [anti-Bolshevik] Cossacks attacked, they put arrestees from prisons on one, and the assortment of arrestees was extremely diverse. There were 30 from a labor camp, 70 former officers, 40 members of the bourgeoisie, and the rest were arrested for a wide variety of reasons, mostly workers and peasants. The barge packed with all these people had only one latrine, and people had to stand in line for four hours and fainted. The prisoners were not given anything to eat.\n\nMakhrovsky accused not only the Cheka of abuses, but also Tsaritsyn's political leaders, including Stalin. He provided examples of people being arrested for merely arguing with Stalin. Several months later, Voroshilov confirmed Stalin's leading role in organizing the terror. \"These 'gentlemen,'\" Voroshilov said of the former officers, \"were arrested [by me] and Comrade Stalin.\" Having developed a taste for the Tsaritsyn approach, Stalin requested that it be applied in surrounding areas. On 31 August 1918 he asked Lenin to authorize a \"group of reliable people\" from Tsaritsyn to \"purge\" the city of Voronezh of \"counterrevolutionary elements.\" The request was granted.\n\nStalin apparently sent his request to Lenin before he heard that the previous day, 30 August, the Bolshevik leader had been wounded by an act of terrorism attributed to the SRs. The assassination attempt opened up new prospects for Stalin and the Bolshevik party overall: the Red Terror became official policy. In early September Stalin sent a report to Moscow on behalf of the leadership of the North Caucasus District outlining plans to organize \"open, mass, systematic terror against the bourgeoisie and its agents.\" In September and October, the Tsaritsyn Cheka, according to some sources, executed 102 people, of whom 52 were former tsarist army officers or former members of the tsarist security police.\n\nWhether the scale of the terror was due to the panic triggered by military defeat or whether it was premeditated, the threat of terror made it easier to keep the unruly Red Army in line. Furthermore, the discovery of \"plots\" offered convenient excuses for military failures and opportunities to demonstrate decisiveness and efficiency to the top leadership. Stalin used the threat of growing counterrevolution to demand special powers and justify his refusal to subordinate himself to the military authorities in his district.\n\nIt is not known through what channels and in what form information about the Tsaritsyn atrocities reached Moscow or how widely the Makhrovsky report and other firsthand accounts were circulated. There is evidence that the top leadership knew about Stalin's initiatives. Several months later, in March 1919, Lenin said at the Eighth Party Congress, \"When Stalin was shooting people in Tsaritsyn, I thought this was a mistake; I thought that they were shooting incorrectly.\" (He did not, apparently, object to the executions in principle, only that they were being carried out in a disorderly manner.) Lenin even claimed he sent a telegram to Stalin asking him to be careful, although no such telegram has been discovered. Another speaker mentioned the \"famous\" barge in Tsaritsyn \"that did so much to prevent military specialists from being assimilated.\" Apparently, Stalin's executions were no secret, but he suffered no serious consequences as a result. The Bolshevik leaders took a relaxed attitude toward excesses committed in defense of the revolution. During the same speech to the Eighth Congress, Lenin even said that in the end the Tsaritsyners were right. Why condemn comrades over a few \"holdovers of the bourgeoisie\"?\n\nWhile mass shootings did not much trouble Lenin, military setbacks did. As head of the Red Army, Trotsky took an implacable position toward the Tsaritsyn events. His feelings were influenced both by a strong personal dislike for Stalin and by pragmatic concerns. In his eyes, the measures taken in Tsaritsyn were a dangerous example of unconstrained action that would hinder the professionalization of the army through the institution of strict discipline and the recruitment of military professionals. He made his position clear to Lenin in a telegram dated 4 October 1918:\n\nI categorically insist that Stalin be recalled. Things are not going well on the Tsaritsyn front, despite an abundance of forces. Voroshilov can command a regiment, but not an army of fifty thousand soldiers.... Tsaritsyn must either submit [to its ranking commanders] or get out of the way. We are seeing success in all armies except the Southern one, especially in Tsaritsyn, where we have a colossal superiority of forces but total anarchy at the top. We could get this under control in 24 hours with your firm and decisive support; in any event, this is the only way forward I see for myself.\n\nStalin began to campaign against Trotsky. In telegrams to Lenin, he and Voroshilov accused Trotsky of making a mess of the front and behaving disrespectfully toward \"prominent members of the party to please traitors from among military specialists.\" He traveled to Moscow, hoping to talk to Lenin personally and tip the scales in his favor, but his trip was in vain. The leadership supported Trotsky's efforts to consolidate the army. In October 1918 Stalin was forced to leave Tsaritsyn. Soon thereafter, Voroshilov and other Stalin allies were also removed. From that point forward, Stalin took every opportunity to scheme against Trotsky and advance the careers of his Tsaritsyn comrades.\n\nThe experience acquired in Tsaritsyn seems to have guided Stalin throughout the remaining years of the Civil War. Although he was compelled to recognize the party policy of recruiting military professionals, Stalin apparently remained hostile toward it. He had little respect for professional military men, whom he considered politically suspect, and preferred the enthusiasm and \"common sense\" of true revolutionaries. In a 16 June 1919 telegram to Lenin from the Petrograd front, he wrote with slightly comical bravado and arrogance: \"Naval experts assert that the capture of Krasnaya Gorka [a Petrograd fort] from the sea runs counter to naval science. I can only deplore such so-called science. The swift capture of Gorka was due to the grossest interference in the operations by me and civilians generally, even to the point of countermanding orders on land and sea and imposing our own. I consider it my duty to declare that I shall continue to act in this way in future, despite all my reverence for science.\" Lenin, who knew that the fort had not, despite Stalin's claim, fallen from a naval attack, seems to have been amused by Stalin's swagger. He left a notation on the telegram: \"??? Krasnaya Gorka was taken by _land._ \"\n\nStalin's bravado stayed with him through the war's concluding stages. In the spring and summer of 1920 he was on the Southwestern Front, where the Soviet-Polish War was raging and Soviet forces were facing General Petr Wrangel, the commander of what was left of the White Army who had moved beyond his main stronghold in Crimea. At first the Polish forces dealt the Red Army crushing defeats, but the situation soon changed. The Red Army went on the offensive, made its way to Warsaw, and prepared to take it. Bolshevik leaders were euphoric. They anticipated that revolution would not only prevail in Poland, but (finally!) would also spread to other European countries. \"Through Warsaw to Berlin!\" was the watchword. On 13 July 1920, in response to Lenin's question about the advisability of concluding a truce with Poland, Stalin wrote: \"The Polish armies are completely falling apart; the Poles have lost communication lines and management; Polish orders, instead of reaching their recipient, are increasingly falling into our hands. In a word, the Poles are experiencing a breakdown from which they won't soon recover.... I don't think that imperialism has ever been as weak as it is now, at the moment of Poland's defeat, and we have never been as strong as we are now, so the more resolutely we behave ourselves, the better it will be for Russia and for international revolution.\"\n\nStalin's writings from this period are permeated with the hope that Red Army bayonets would coax along world revolution. On 24 July, in a telegram to Lenin that treated victory over Poland as a foregone conclusion, he proposed \"raising the question of organizing an insurrection in Italy and in such still precarious states as Hungary and Czechoslovakia (Romania will have to be crushed).\" Stalin backed up his words with actions. On the Southwestern Front that had been entrusted to him, he was especially anxious to capture the important city of Lvov. He pressed the leaders of the First Cavalry, urging them to make a decisive charge, but in vain: Lvov evaded his grasp. The Soviet military effort was not going well in another sector of the Southwestern Front, Crimea. Units of Wrangel's army were entrenched there, and with the Red Army busy on the Polish front, Wrangel undertook successful attacks beyond the peninsula. Stalin, as one of the main officials responsible for the failures outside Lvov and in Crimea, sent reports to Moscow citing objective difficulties and blaming the inaction of the Red Army's central command. He clearly felt uncomfortable as a military commander incapable of achieving decisive success. This failure was particularly mortifying given the rapid advance on Warsaw by the Red Army that was taking place on the neighboring Western Front.\n\nBut the situation soon took another sharp turn. The invasion of Poland bogged down, the Red Army suffered heavy casualties, and the Poles ended up imposing humiliating peace terms on the Bolsheviks. Defeat on the Polish front had a number of causes, one of which can be traced directly to Stalin. It has been suggested the Red Army spread itself too thin by carrying out offensive actions in too many areas at once. For example, the First Cavalry Army, an important force, was trying to take Lvov instead of supporting the troops marching on Warsaw. Not long before the Red Army's defeat, a decision was made to move the First Cavalry Army west from Lvov, but it was never implemented. Stalin played a part in this failure. On 13 August 1920 he sent the Red Army Main Command a telegram asserting that the redeployment of the cavalry would be harmful, in that it had already begun a new offensive against Lvov. The redeployment should have been ordered earlier, he maintained, when the army was still in reserve. \"I refuse to sign the order,\" he wrote.\n\nStalin's refusal was probably not a major factor in the Polish debacle. In 1920, when the reasons for the Red Army's defeat were dissected, most of the blame was laid on the commanders of the Western Front in charge of the invasion of Warsaw. But Stalin's willful behavior may be why he was recalled from the front just a few days after the incident with the First Cavalry Army. He left for Moscow and never returned to military action. The laurels for victory over Wrangel that soon followed were placed on other heads.\n\nThe return to the capital was hardly triumphant. On top of his failure to achieve a decisive victory either in Lvov or against Wrangel, Stalin's refusal to carry out an order could be seen as a major factor in the Warsaw defeat. It may have been fear that he would be cast as a scapegoat, together with hurt feelings, that led him to launch a characteristic preemptive attack. On 25 August 1920, when events in Poland were clearly turning catastrophic for the Red Army, he submitted a memorandum to the Politburo calling for the creation of military reserves. On the surface, this memorandum\u2014calling for a troop increase, expanded military production, and the formation of new units\u2014was fully in keeping with the priorities that had dominated Bolshevik policy throughout the Civil War. But its real importance lies in one sentence: \"The latest successes of the Poles have disclosed a fundamental defect of our armies, namely, the lack of effective fighting reserves.\" This was Stalin's attempt to place responsibility for the defeat on the shoulders of the army's main leadership. He attributed great significance to this memorandum and insisted on a response. On 29 August 1920 he again wrote to his Politburo colleagues: \"I am drawing the attention of the Central Committee to the urgency of the matter of the republic's military reserves that I raised... and which as of now (29 August) has yet to be dealt with.\"\n\nTrotsky ultimately provided a condescending explanation of the situation and proposed creating a procurement council, on which he invited Stalin to serve. It was a clever move to invite Stalin to take on the thankless job of keeping the army of their impoverished country well supplied, and Stalin seems to have been enraged by Trotsky's response. On 30 August he sent three memoranda to the Politburo, all aimed at Trotsky. In one, he characterized Trotsky's response to his previous memorandum as a \"runaround\" and demanded that the Central Committee keep a closer watch over the military\u2014in other words, over Trotsky. In a second brief but categorical note, he responded to Trotsky's proposal that he join the procurement council: \"I hereby state that I cannot and, consequently, will not work on Trotsky's planned procurement council.\" To top off these hostile pronouncements, he made a risky move. He proposed creating a commission \"to investigate the circumstances of our July offensive and August retreat on the Western Front.\" Given the context of his accusations of negligence in regard to reserves, this was a clear declaration of war against Trotsky. Was Stalin aware that he was indirectly attacking Lenin as well since Lenin had been at the forefront of those urging the Polish adventure? If, in the heat of emotion, he did not immediately realize this, he certainly was informed of it soon enough.\n\nThe next day, on 1 September, a decisive showdown took place at a Politburo meeting. The main parties to the conflict\u2014Stalin, Trotsky, and their arbitrator, Lenin\u2014were all present. The mood was somber. Much of the meeting was spent discussing the humiliating peace with Poland. Stalin's military reserves proposal was taken up toward the end and essentially rejected. The resolution adopted recognized \"Trotsky's statement that the military is taking measures in the spirit of Stalin's proposals.\" In other words, steps were being taken and Stalin's advice on the matter was no longer required. A special council on supplying the army was chaired by Trotsky and did not include Stalin, whose refusal to serve was taken with infuriating literalness. Equally insulting was the rejection of Stalin's call for an investigation into the reasons for defeat in Poland. Lenin adamantly opposed this idea.\n\nTo the great regret of historians, no detailed stenographic record was kept of this Politburo meeting (or of many other important meetings). The only documentation is a laconic record of resolutions, a poor indicator of the passions that no doubt flared, either openly or within the hearts of the participants. Stalin resigned his military duties. His resignation was accepted, depriving him of his membership in the Military Revolutionary Council. Trotsky's authority and rights were confirmed, and he was assigned to inspect the Western Front. Lenin clearly took Trotsky's side. On 20 September a Central Committee plenum adopted a decision to send Stalin \"on long-term assignment to the Caucasus.\" He was given the job of \"settling relations with highlanders\" and \"bringing order... to policy in the Caucasus and East [Soviet Asia].\" Perhaps this was an honorable exile, or perhaps it was a new and important assignment. In any case, several days later, at the ninth conference of the Russian Communist Party, a public confrontation took place between Stalin on one side and Lenin and Trotsky on the other. The recriminations over the Polish war that had been roiling in the Politburo erupted into public view.\n\nAt the conference, Lenin and Trotsky both spoke out against the charges Stalin had leveled against the commanders of the Western Front and, essentially, the entire Red Army command. Lenin took personal responsibility for a large share of the strategic miscalculations and rejected Stalin's call for an investigation. Trotsky made snide references to Stalin's optimistic anticipation of victory in Poland and his assurances that he would take Lvov. On 23 September, a deeply offended Stalin submitted a statement to the conference's presidium. He categorically denied Trotsky's and Lenin's accusations. He reiterated his charge that the commanders of the Western Front were responsible for the defeat in Poland (a jab at Trotsky) and claimed that he, Stalin, had always advocated prudence and caution. \"Comrade Lenin evidently is being merciful toward the command, but I think what is needed is mercy for the cause, not the command,\" he concluded caustically. With the benefit of currently available documents, we can state with certainty that Stalin was lying about his past advocacy of caution. Lenin nevertheless did not challenge him, probably because Stalin's calls for decisiveness and world revolution suited his interests. Ultimately, all their fates hung on the success of their common endeavor, so they preferred to put this unpleasant chapter of defeat behind them as quickly as possible. In his call for an investigation into mistakes, Stalin looked like a dissident. Furthermore, everyone knew that he was as guilty as anyone. But as in the past, he escaped this episode generally unscathed. He left for the Caucasus, but several weeks later, in late November 1920, he returned to Moscow. Stalin's conflicts with his colleagues during these years were turning into a habit. It was not a new habit, but it was becoming more pronounced and deeply rooted. His behavior reflected the objective fact that the party was plagued by conflict spawned by principled differences and personal ambitions. This circumstance inevitably led to the formation of cliques. Stalin's was comprised of veterans of Tsaritsyn, members of the First Cavalry Army, and Transcaucasians who enjoyed Stalin's patronage and support. Other Soviet leaders were also assembling followers. The seeds of future clashes and power struggles were being sown.\n\nThe Bolsheviks' first experience running the country came in a time of war. This factor shaped both their practical approach to governing and their philosophy. Experiences acquired in Tsaritsyn and Petrograd reinforced Stalin's intuitive mistrust of \"bourgeois specialists\" and his fear of conspiracies. Grain requisitions in the south and the organization of a labor army in Ukraine gave him experience using strong-arm tactics to steer the economy. The Civil War accustomed the Bolsheviks to blood and ruthlessness. Atrocities lost their horror.\n\n###   **GENERAL SECRETARY**\n\nThe Bolsheviks emerged from the Civil War as winners. But explaining to the exhausted country, or even to themselves, what they had been fighting for was no simple matter. The dream of world revolution appeared to be just that, and Lenin's idea that socialism would be immediately introduced in Russia proved catastrophically utopian, just as his opponents had warned. Attempts to abolish the market system and replace it with direct exchange under total governmental control only furthered economic collapse. Famine and devastation sparked massive anti-government protests. Huge areas were engulfed by peasant revolts. The unrest spread to cities, including such Bolshevik strongholds as Moscow and Petrograd. The rebellion by sailors of the Kronstadt garrison outside Petrograd became a symbol of the failures of the Bolshevik policy of militarized socialism. When this bastion of the 1917 revolution took up arms, \"Kronstadt\" became a highly fraught political watchword.\n\nUnder these circumstances, Lenin, who had a well-developed instinct for political self-preservation, allowed his steadfast principles a generous bend. In 1921\u20131922, Leninist socialism was replaced by the Leninist NEP (New Economic Policy). Many aspects of the Soviet economy reverted to their state before the Bolshevik revolution. The lion's share of the economy remained under state control, but certain market activities were allowed. The use of money was restored. Peasants were allowed to sell their produce after paying taxes to the state. Private small industry and trade were returned to private ownership (the entrepreneurs who ran small businesses were called \"Nepmen\"). Despised capitalism came to the Bolsheviks' rescue, saving their country and their hold on power. Thanks to the NEP, the USSR came back from the brink of disaster in just a few years. But before the recovery could be felt, the horrific famine of 1921\u20131922, a direct outcome of the Civil War, took millions of lives.\n\nSuch was the backdrop to Stalin's life during the lead-up to the death of his teacher, Lenin. The historical record does not offer evidence of any active involvement by Stalin in discussing or deciding key problems in the transition to the NEP. He followed the political course set by Lenin and was a loyal and true comrade. Lenin undoubtedly valued this loyalty. But after the Civil War, Stalin's political prominence was hardly guaranteed. Simply being a member of the Politburo assured him a certain degree of power. But in the Soviet party-political system, the degree of power a leader actually exercised was directly tied to the influence of the government agency he headed. From this standpoint, Stalin was in danger of becoming a second-tier functionary.\n\nThe conclusion of the war found Stalin running two agencies: the nationalities commissariat and the Workers' and Peasants' Inspectorate. Neither had meaningful levers of power or more than limited lobbying potential. At a closed meeting, Stalin himself characterized the nationalities commissariat as serving a purely \"agitation\" purpose without any \"administrative rights.\" He spent very little time at this agency. In November 1921 he submitted his resignation from it to the Politburo, but it was not accepted. He did everything he could to abolish the commissariat, and in 1923 he finally succeeded. Even earlier, in 1922, he had managed to shed his duties with the Workers' and Peasants' Inspectorate. He exchanged these undesirable posts for one that was much more appealing: running the Central Committee apparat. This position moved him into the upper echelons of the leadership.\n\nWhat brought about this turning point in Stalin's political career was not only his talents and energy, but also a heated battle within the Soviet leadership. The central conflict was between Lenin and Trotsky, but smaller clashes reverberated all around them. Trotsky was the only top Bolshevik who could rightfully claim to be a leader in his own right, not just a follower of Lenin. His role was more that of a partner and ally in revolution, and he behaved accordingly, earning himself a following within the party. At the end of 1920, Lenin realized that a significant portion of party functionaries, including some within the Central Committee apparat, supported Trotsky. Lenin had to respond to this challenge to his primacy. At the Tenth Party Congress in early 1921, after intense maneuvering and considerable use of his authority, Lenin made sure that his followers received a majority of votes. This outcome determined who would be chosen to run the top party organizations, and many Trotsky followers were removed from their posts. Stalin was one of Lenin's key allies in this struggle. Given Lenin's declining health, such cooperation took on new importance. Beginning in mid-1921, he was increasingly plagued by symptoms of severe cerebral arteriosclerosis. Headaches, fatigue, episodes of paralysis, and impaired speech and cognition forced him to take extended vacations.\n\nLenin's illness and clash with Trotsky, along with the reshuffling of party personnel, all helped Stalin play an increasingly important role in party affairs. In early 1922, this role was formalized when Stalin was appointed to the newly created post of general secretary of the Central Committee of the Russian Communist Party (of Bolsheviks)\u2014TsK RKP(b). The job of the general secretary included overseeing the Central Committee apparat and its \"leading structures\"\u2014the bureaucratic machine that carried out the will of the party. Two duties deserve particular mention: setting the agenda for Politburo meetings and deciding personnel matters. Countless mid-level functionaries now depended on Stalin for their careers.\n\nFor Stalin, the running of the party apparat was not a burden. His previous party experience and his personality made him well suited for this position. Later, even as dictator, Stalin seemed to enjoy routine bureaucratic work. Upon taking up the post of general secretary, he began to reorganize the work of the Politburo. On 31 August 1922 he announced at a Politburo meeting that certain institutions were tardy in submitting materials for consideration. A resolution was adopted to \"not place any matter before the Politburo unless materials are submitted by four o'clock the previous day.\" A few weeks later, the rule became even stricter: the deadline was pushed back to noon. Through these petty decisions Stalin was gradually, and with increasing confidence, shaping how the party apparat was run.\n\nSome interesting accounts survive of how this tendency was perceived within the apparat. Stalin's assistant, Amaiak Nazaretian, regularly corresponded with Sergo Ordzhonikidze, Stalin's old friend who was working in Transcaucasia in the early 1920s. This correspondence has been preserved in Ordzhonikidze's archive. In the letters written during the summer of 1922, Nazaretian described his work under Stalin:\n\nAm I happy with my job? Yes and no. On one hand, I'm getting quite an education here, I know what's going on in international and Russian life, and I'm being schooled in discipline, developing precision in my job.... On the other, this work is purely paper pushing, painstaking, not very satisfying from a subjective standpoint; it's menial work that takes such tremendous amounts of time that you can't sneeze or squirm, especially under Koba's firm hand. Do we get along? We do.... You can learn a lot from him. Now that I've gotten to know him, I have extraordinary respect for him.... Under his stern demeanor is an attentiveness to those he works with. We're creating order in the TsK.\n\nKoba has really got me trained.... He's really cunning. Hard as a nut, it takes a while to understand what he's up to.... Despite his well-reasoned savagery of temperament, if I can put it that way, he is soft, he has a heart, and he knows how to appreciate people's dignity.... Now, the work of the TsK has really changed. What we found here was indescribably awful. Now we've shaken things up.\n\nNazaretian felt Stalin was tremendously significant: \"Ilyich has fully recovered.... Yesterday, Koba went to see him. He has to keep a watchful eye over Ilyich and all of Mother Russia\"; \"Ilyich undoubtedly has a trusty Cerberus in him, fearlessly standing guard at the gates of the TsK RKP.\" Nazaretian's letters provide important details on how Stalin was perceived within the Bolshevik bureaucratic community. In Moscow, according to Nazaretian, an expression came into fashion: \"to be going under Stalin.\" This referred to officials who had been summoned to Moscow from their previous posts but had not yet been assigned new jobs and were \"hanging, so to speak, in the air.\"\n\nSuch was Stalin as he appeared to his assistant early in his tenure as general secretary. Obviously, these descriptions carry an element of exaggeration, the admiration of a loyal secretary toward his boss. But the intelligent and observant Nazaretian was conveying a certain mood within the apparat. Many members of the bureaucracy began to perceive Stalin as an experienced and confident bureaucrat who held secure positions within the hierarchy. He was coolheaded, but he could be stern and unbending in standing up for his interests and opinions. At a time when the world of the Bolshevik bureaucracy was increasingly fracturing into patron-client cliques, these qualities drew him quite a few supporters.\n\nIn Nazaretian's letters, Stalin is perceived within the party as Lenin's loyal comrade, his pillar in times of political strife. And this view was largely accurate. Long years of collaboration, marred by only a few instances of discord, had created a strong bond between Lenin and Stalin. One Bolshevik left behind an eloquent memoir of a meeting between Lenin and Stalin in September 1921 in the latter's apartment. A difficult squabble among top officials in Petrograd was being settled. Lenin tried to reconcile the feuding parties while Stalin paced the room smoking his pipe. At one point, Lenin looked at Stalin and said, \"That's an Asian for you\u2014all he does is suck on his pipe!\" Stalin knocked the pipe right out of his own mouth. This playful manner went beyond the boundaries of the boss-subordinate relationship. For Lenin, Stalin was a comrade-in-arms with whom relations were warm enough to allow for teasing. It is difficult to imagine that he would take such liberties with Trotsky, with whom he maintained a stiff, official manner, using the polite pronoun _vy_ for \"you\" rather than the familiar _ty._\n\nOn 30 May 1922, an incident occurred that further attests to the close relationship between Lenin and Stalin. Lenin, who was ill and facing the prospect of paralysis, summoned Stalin to Gorki, his residence outside Moscow. He asked Stalin to procure poison so that he could have the option of taking his own life when the time came. Stalin immediately told Lenin's sister, Maria Ilinichna Ulianova, and Nikolai Bukharin, who then happened to be staying at Gorki, about this request. According to Maria Ulianova's memoirs, they decided together to try to boost Lenin's spirits. Stalin went back to him and told him that the time to carry out his intention had not yet come, and the doctors were promising he would get better. Lenin, in Ulianova's account, \"became noticeably more cheerful and consented, although he asked Stalin, 'Are you being deceitful?' 'When have you ever seen me be deceitful?' Stalin replied.\"\n\nLenin showed his concern for Stalin in several ways. While seriously ill in Gorki in June 1922, Lenin sent a recommendation to Moscow: \"Require Com. Stalin, through the Politburo, to spend one day per week, beside Sunday, entirely at his dacha outside town.\" The Politburo adopted the resolution. In August, after Lenin's health improved, Stalin visited him regularly in Gorki. According to Maria Ulianova's memoirs, \"Ilyich greeted him in a friendly manner, with jokes and laughter, and urged me to be hospitable to Stalin and bring him wine, etc.\" Later, when he himself was in power, Stalin adopted Lenin's manner of showing concern for his subordinates.\n\nHarmony between Lenin and Stalin lasted until the fall of 1922.\n\n###   **QUARRELS WITH THE TEACHER**\n\nLenin's illness had tremendous political ramifications. The party, which was structured around a single leader, was vulnerable. The Politburo was forced to begin thinking about Lenin's successor. The \"troika\" of Zinoviev, Kamenev, and Stalin was growing in influence in its contest with its main opponent, Trotsky. This face-off was actually an outcome of Lenin's tactic of isolating Trotsky, but with Lenin's illness, Trotsky's isolation served to strengthen the troika, a dangerous prospect in Lenin's eyes. Hoping for a recovery from illness, Lenin attempted to shift the balance of power, and Stalin was the easiest target.\n\nA conflict over the program for uniting the Soviet republics can be seen as the starting point of Lenin's efforts. The Civil War had created a unified state, but in the second half of 1922 it was decided to make this union official by publicly announcing the principles on which the new state would be built. For the most part, the Bolshevik leadership saw eye to eye on this issue. Nobody entertained thoughts of breaking up what had been the Russian Empire or granting real autonomy to any areas under Moscow's control. There were arguments over the form the new union would take and the degree of independence various Bolshevik entities would enjoy, but all parties to the decision were expected to submit to the discipline of a unified party.\n\nStalin was open about his position. He proposed that the real state of affairs and Moscow's true intentions be codified in the constitution without undue ceremony or diplomacy. He favored bringing all the major republics (Ukraine, Belarus, Georgia, Azerbaijan, and Armenia) and the smaller ethnic entities into the Russian Federation with certain rights of autonomy. Overall, this proposal was in full accord with the party line and was supported by most party officials, in both Moscow and the ethnic republics. Stalin was probably surprised when Lenin opposed his proposal and advanced his own plan to proclaim a union of \"independent\" Soviet republics\u2014even though the Bolshevik leader had no intention of granting genuine independence. The motives for Lenin's position are difficult to pinpoint. Perhaps he was responding to dissatisfaction with Stalin's program among Georgian and some Ukrainian party leaders. Perhaps, with his illness receding, he simply saw this as a good opportunity to reenter the political fray.\n\nIn September 1922 Lenin began promoting his program. He criticized Stalin for being too hasty, an assessment that must have stung. Stalin resisted and made a fighting retreat, accusing Lenin of \"national liberalism.\" His feelings are easy to understand: he had been put in a humiliating position and was forced to change a stance that he had put a lot of energy into advocating. But he chose not to do serious battle with Lenin. On 28 September, an interesting exchange of notes took place between Kamenev and Stalin during a Politburo meeting:\n\nKAMENEV: Ilyich is ready to go to war to defend independence....\n\nSTALIN: I think we need to stand up to Ilyich....\n\nKAMENEV: I think so long as Vladimir Ilyich is insistent, we'd be worse off resisting.\n\nSTALIN: I don't know. Let him do as he sees fit.\n\nStalin relented. He knew Lenin well and appreciated how powerful he still was.\n\nIn October\u2013December 1922 a conflict surrounding the question of monopolizing foreign trade followed a similar script. At a plenum on 6 October, a majority within the Central Committee voted to somewhat loosen the monopoly. Lenin, who was away from Moscow, took a stand against the liberalization. Stalin, who supported the 6 October decision, was slow to relent and expressed reservations. Lenin undoubtedly was not pleased.\n\nThis dispute ended with a move by Lenin that Stalin must have found extremely upsetting. On the issue of monopolizing foreign trade, Lenin demonstratively brought Trotsky out of disfavor and recruited him as an ally. Lenin had often resorted to this sort of maneuver\u2014exploiting the conflicts ever-present at the upper echelons of the party. Now, however, the circumstances were different. Lenin was seriously ill, and the jockeying for power and influence was greatly intensified. To the alarm of Stalin, Kamenev, and Zinoviev, whose influence had been growing, Lenin proposed that Trotsky continue to work with him. On 21 December 1922, immediately after a Central Committee plenum voted to uphold his opposition to liberalization, Lenin dictated a note to Trotsky, employing his wife, Nadezhda Krupskaia, as stenographer: \"It seems that we've captured the position without firing a single shot, using a simple maneuver. I propose that we not stop here and continue the offensive.\" Lenin advised Trotsky to raise the question of foreign trade at the upcoming party congress and also to speak at the Congress of Soviets. Such a move would discredit Lenin's opponents, including Stalin, before a large assembly of party functionaries.\n\nTrotsky immediately got to work and telephoned Kamenev, who told Stalin about the call. Stalin refused to carry out Lenin's instructions to put Trotsky's speech on the schedule of the Congress of Soviets. He also called Krupskaia and reprimanded her for taking down and sending the letter to Trotsky. Apparently the reprimand was rather indelicate, or at least it seemed so to the overburdened and high-strung Krupskaia. In theory, Stalin had a legitimate grievance against Krupskaia. Just a few days previously, on 18 December, the Central Committee plenum had voted to limit contact with Lenin, who had suffered another health setback. \"Personal responsibility shall be placed on Com. Stalin to isolate Vladimir Ilyich both in regard to face-to-face dealings with officials and correspondence.\" Krupskaia had violated this directive. But Stalin had also crossed the line with his emotional outburst. The troika saw Lenin's appeal to Trotsky as dangerous and provocative.\n\nRealizing his mistake, Stalin apologized to Krupskaia. Judging by Maria Ulianova's memoirs, he also made an attempt to reconcile with Lenin. He met with Ulianova and told her how upset he was about being estranged from him:\n\nI couldn't sleep at all last night.... What does Ilyich think of me, how does he feel about me! As if I were some sort of traitor. I love him with all my heart. Find a way to tell him that.\n\nBut Lenin was implacable. Ulianova offers the following description:\n\nIlyich called me in to see him for something, and I told him, among other things, that his comrades send their respects.... \"And Stalin asked me to send you his heartfelt regards and asked me to say that he truly loves you.\" Ilyich grinned and remained silent. \"So should I send him your regards?\" I asked. \"You can send them,\" Ilyich replied rather coldly. \"But Volodia,\" I continued. \"He is, after all, very smart, Stalin.\" \"He's not smart at all,\" Ilyich replied firmly, wincing.\n\nUlianova does not say exactly when this conversation with her brother took place, but it was almost certainly in late 1922 or early 1923, when relations between Lenin and Stalin were deteriorating and threatened to rupture completely. On 24 December Lenin dictated a document to his secretary\u2014the well-known \"Letter to the Congress\"\u2014in which he expressed apprehension about divisions within the party's top leadership. Regarding Stalin, this document states, \"Com. Stalin, now that he is general secretary, has concentrated immense power in his hands, and I am not sure whether he will always be capable of exercising this power with sufficient caution.\" In another letter, dictated on 4 January, Lenin was even more categorical. He proposed removing Stalin from the post of general secretary because he was \"too rude.\"\n\nLenin's growing irritation was the backdrop against which the \"Georgian Affair\" unfolded. This episode involved a dispute between a group of Georgian Bolsheviks and the leadership of the Transcaucasian Federation, which comprised Georgia, Armenia, and Azerbaijan. The conflict was not with the entire federation leadership but with its head, Ordzhonikidze. The friendship between Stalin and Ordzhonikidze would certainly have influenced the general secretary's stance on the matter. The Georgian Bolsheviks, with variable success, were inundating Moscow with complaints about Ordzhonikidze's heavy hand. In late 1922 Ordzhonikidze gave his opponents more ammunition against him: in a fit of anger, he struck one of his adversaries. A commission headed by Feliks Dzerzhinsky was sent from Moscow to investigate. Lenin took a great interest, and when the commission turned in a report favorable toward Ordzhonikidze, he was not pleased. He believed that Dzerzhinsky and Stalin were covering for Ordzhonikidze and being unfair to his beleaguered accusers.\n\nIf it had not been for the clash between an ailing Lenin and his increasingly powerful followers, the Georgian Affair would have remained a bureaucratic squabble of the sort that was commonplace within the Bolshevik party, especially early on, when their government had yet to achieve a stable footing. In Transcaucasia, infighting among competing groups continued for many years. It was Lenin who elevated this incident\u2014artificially, one could argue\u2014to the level of fundamental political principles since it gave him a pretext for attacking his ambitious associates. Though ill, Lenin was still prepared to fight for control of the party and was obviously looking for a way to quell the dissent that threatened to undermine his power. He saw Stalin as the symbol of that dissent.\n\nAll the evidence suggests that Lenin spent the winter of 1923 preparing to launch an attack against Stalin at the Twelfth Party Congress, scheduled for March. On 5 March 1923, having assembled the necessary materials, he again approached Trotsky with a proposal that they collaborate: \"Dear Com. Trotsky! I would like to ask you to take on the defense of the Georgian Affair within the party's TsK. This matter is currently being 'pursued' by Stalin and Dzerzhinsky, and I cannot rely on their impartiality. Quite the contrary. If you agreed to defend it, I could rest assured.\" That same day, 5 March, Lenin dictated a note addressed to Stalin in regard to an old matter\u2014the reprimand Stalin had made against Krupskaia in December 1922. The note was curt. Lenin threatened to sever their relationship: \"Dear Com. Stalin! You were so ill-mannered as to call my wife to the telephone and scold her.... I have no intention of so easily forgetting what has been done against me, and it goes without saying that what has been done against my wife is also done against me. I therefore ask you to weigh whether you are amenable to taking back what was said and apologize or you prefer to break off relations with me.\"\n\nThe appearance of this letter, written two and a half months after Stalin's reprimand, has generated many hypotheses among historians. Perhaps Lenin had only just learned of Stalin's telephone call to Krupskaia. It appears more likely, however, that Lenin saw the incident as an excuse for removing Stalin from the post of general secretary, a possibility proposed by Robert Tucker. All of Lenin's objections to Stalin emphasized the same point: he was too rude. Such a charge was much more persuasive and clear-cut than any of the other possible complaints he might have lodged. Rudeness toward party comrades was completely inappropriate for someone holding the post of general secretary.\n\nThe following day, 6 March, Lenin again wrote about Stalin's abrasive manner. He dictated several lines to the beleaguered Georgian Bolsheviks, instructing that copies of the note be sent to Trotsky and Kamenev. Kamenev was scheduled to travel to Georgia and was asked to deliver the note personally. \"Dear Comrades!\" Lenin wrote. \"With all my heart I am following your case. I am outraged by Ordzhonikidze's rudeness and the connivances of Stalin and Dzerzhinsky. I am drafting a memorandum and speech for you.\"\n\nTo the Politburo, the meaning of Lenin's actions was clear: he had declared war on Stalin. Shortly before leaving for Georgia, Kamenev wrote to Zinoviev that Lenin wanted not only reconciliation in Transcaucasia, \"but also certain organizational expulsions at the top\"\u2014Soviet administrative jargon for firings. Stalin could sense the approaching storm. On 7 March he received Lenin's ultimatum threatening to sever relations. He immediately responded with a half-hearted apology: \"Although if you feel that to maintain 'relations' I have to 'take back' the words that I said... I can take them back, but I really can't understand what the point is, where my 'guilt' lies, and just what it is they want from me.\" That same day Stalin sent a strictly confidential letter to Ordzhonikidze. He warned him that Lenin had sent a letter of support to Ordzhonikidze's opponents. Stalin urged caution: \"Reach a compromise... that is natural, voluntary.\" This letter to Ordzhonikidze clearly shows that Stalin appreciated the seriousness of the situation and was maneuvering to deprive Lenin of ammunition.\n\nUntil this decade, the authenticity of Lenin's dictated correspondence and accounts of the actions he took against Stalin have never been called into question. Recently, however, there have been attempts to demonstrate that evidence of a rupture between the two men was fabricated. With no real evidence beyond an assumption of Stalin's infallibility, some revisionists have proposed that evidence of Lenin's doubts about Stalin were manufactured and placed in Lenin's archives by followers of Trotsky!\n\nThe strongest evidence of the authenticity of Lenin's dictated correspondence from this period is that nobody among Lenin's comrades-in-arms, including Stalin himself, had any doubts about it. Stalin certainly had both the cunning and wherewithal, given his control over the apparat and influence within Lenin's inner circle, to avoid falling victim to a forgery. He understood the danger of Lenin's \"testament\" and went to great pains to neutralize any evidence that he did not enjoy Lenin's full confidence.\n\nThere is no question that Lenin took steps against Stalin during the final weeks of his active life. The reasons are another matter. We must consider not only the intentions and motives of a masterful politician, but also the role played by his sense of imminent death. \"Lenin's last struggle,\" as Moshe Lewin has called it, is a clear manifestation of his single-minded will toward political domination and power\u2014his primary personality trait. Illness did not break this will but, if anything, intensified it. One can only marvel at the persistence of Lenin, racked by agonizing physical and emotional suffering, as his dogged ascent to power was interrupted by forced intervals in the background. The struggle for power sustained him, energized him, and gave purpose to his battle against affliction. This was not the first time he had taken up a challenge from a comrade-in-arms, but the gravity of his illness in 1922\u20131923 lent any such challenge a new and urgent significance.\n\nFrom the standpoint of \"the technology of power,\" Lenin's maneuvers in late 1922 and early 1923 relied on the same sources of strength that had carried him through earlier clashes: his unquestionable authority among party functionaries and rivalries among party leaders (primarily between Trotsky and the troika). That Stalin bore the brunt of Lenin's manipulations appears to be largely a matter of chance. The positions he took in regard to the organization of the USSR and the Georgian Affair represented political miscalculations and turned out to be poorly timed. Finally, he insulted the wife of the ailing leader, exhibiting behavior unbecoming a Bolshevik. Stalin had stepped under the sword himself and so provided Lenin a perfect opportunity to reassert his political authority and subdue other Bolshevik leaders. Lenin probably had no intention of removing Stalin from the party's upper echelons. Such a move would have thrown a wrench in the mechanism he used to maintain power. Within that mechanism, Stalin was the perfect counterbalance to the ambitions of other Bolshevik leaders, as well as an irreplaceable administrator. Lenin's actions were part of a rebalancing that required a dialing back of Stalin's power.\n\nThis context is important in understanding Stalin's reactions to the disfavor being shown him by his teacher. Stalin had every reason to feel genuinely hurt. When all was said and done, his sins were no worse than those he and other Soviet leaders had committed in the past. All Bolshevik leaders contradicted and argued with Lenin, and like Stalin, they all eventually relented. Sometimes Lenin punished these transgressions by removing their perpetrators from the center of power, but he later brought them back into the fold. Lenin usually punished his subordinates out of public view to avoid wounded pride. What was different now? What was behind such a provocative and demonstrative move against a man who had served Lenin so loyally? Stalin apparently found the most convenient explanation for this lashing out, both psychologically and politically, in Lenin's illness.\n\nAs it turned out, the letter to the Georgian Bolsheviks was the last document Lenin dictated. Several days later, his health took a sharp turn for the worse. He did not speak at the party congress; the Politburo swept the Georgian Affair under the carpet and later abandoned the idea of removing Stalin as general secretary. These decisions were not charity on the part of Stalin's \"friends.\" They were the outcome of a fierce power struggle that began during Lenin's final months and continued into 1924.\n\n###   **TRYING OUT COLLECTIVE LEADERSHIP**\n\nAlthough he managed to avoid the more serious dangers posed for him by the political game Lenin was playing during his final months of leadership, Stalin found himself somewhat weakened and thus more dependent on his Politburo colleagues. It is a commonly held view that the Bolshevik oligarchs who inherited power after Lenin's demise underestimated Stalin and believed him to be harmless and mediocre. This is not true. The members of the Politburo fully appreciated Lenin's concerns about Stalin and the power he held as general secretary, and they tried to limit this power. But political happenstance and, to no small degree, Stalin's skillful maneuvering undermined the plans of his rivals and enemies.\n\nThe first serious conflict that we know of within the Politburo's tightly knit opposition to Trotsky occurred during the summer of 1923. After the party congress, the successful neutralization of Lenin's attack, and the country's return to relative stability after the horrific famine, Politburo members regained enough peace of mind to take a vacation. In July 1923, while resting in the North Caucasus resort town of Kislovodsk, Grigory Zinoviev came up with a scheme to shift the balance of power within the Politburo to limit Stalin's influence. In a 30 July letter to Kamenev, who was in Moscow, he launched into a tirade against Stalin: \"If the party is destined to go through a stretch (probably a very short one) of Stalin's sole power\u2014so be it. But I, for one, have no intention of covering up this swinishness.... In reality, there is no troika, there is only Stalin's dictatorship. Ilyich was a thousand times right. Either a serious way out has to be found, or a long stretch of struggle is inevitable.\"\n\nAlthough the letter contained no detailed plan, it charged that Stalin was manipulating the Politburo and essentially making unilateral decisions. It is important to note the line \"Ilyich was a thousand times right\": Zinoviev was using Lenin's letters as ammunition against Stalin. In Kislovodsk, he discussed joint action with Bukharin, who was also upset by some of Stalin's moves, and with other prominent party figures who were vacationing in the south. No specific proposals were entrusted to paper, but Stalin was sent a \"spoken letter\" (Ordzhonikidze, who was leaving for Moscow, was supposed to convey a message). Because this communication was oral, we do not know in detail what was proposed. From statements made in subsequent years, it appears that the plan involved reorganizing the Central Committee secretariat. Stalin would remain a member, but Zinoviev and Trotsky would also be included. This reorganization would have created a new balance of power in Stalin's fiefdom: the Central Committee apparat.\n\nStalin, not surprisingly, was indignant, perhaps even furious. He responded to the grievances of his \"friends\" with a show of hurt feelings and accusations of their undermining unity. On 3 August 1923, immediately after meeting with Ordzhonikidze, he wrote to Zinoviev and Bukharin: \"Evidently you're not hesitant to make ready for a break, as if it were inescapable.... Do as you wish\u2014there must be some people in Russia who will see that for what it is and condemn the guilty.... But what fortunate people you are: you're able to dream up all sorts of fairy tales at your leisure... while I'm stuck here like a chained dog and turn out to be 'guilty' to boot. You can tell anyone you want. All that soft living has gone to your heads, my friends.\"\n\nThis half-angry, half-friendly letter attests to Stalin's relatively limited options in opposing his colleagues. For their part, Zinoviev's and Bukharin's proposals signaled that they still felt they could limit Stalin's influence. They were not impressed by Stalin's expression of injury. Calmly but firmly they let him know that the matter was not settled. Soon they would be able to meet face to face in the south, where Stalin was planning to vacation in mid-August.\n\nStalin could not have relished this prospect. His opponents held all the cards. Their proposal to reorganize the secretariat so as to promote unity and cohesion seemed perfectly reasonable. Stalin's objections would appear to confirm Lenin's warnings that he did not want to work as part of a team. Zinoviev's accusation that Stalin was violating the principle of collective leadership also put him in an awkward position. And another idea advanced by Zinoviev and Bukharin could prove particularly dangerous\u2014that Stalin's position on events in Germany was \"incorrect.\"\n\nThe political crises that had shaken Germany since early 1923 had reawakened Moscow's dream of salvation through European revolution. For the Bolsheviks, who still had trouble imagining a future for the USSR if it remained the only socialist bastion, socialism in Germany would be a great relief. But they took warning from the European revolutionary movements' recent defeats. Stalin was among the Bolshevik leaders who urged restraint, while Zinoviev and Bukharin were eager to do battle, as was Trotsky, for whom world revolution remained a precondition for the victory of socialism in Russia. Realizing that his cautious approach was becoming politically dangerous and gave his rivals ammunition against him, Stalin made an effective political move. On 9 August 1923, amid frantic letter writing with Zinoviev and Bukharin, he placed a resolution before the Politburo summoning Zinoviev, Trotsky, and Bukharin back to Moscow to discuss the prospects for revolution in Germany. Naturally, all three agreed. The meeting was set for 21 August.\n\nThis change of plans gave Stalin important advantages. He deflected charges that he was not sufficiently attentive to revolutionary developments in Germany. Also, the question of reorganizing the secretariat and the collective leadership was pushed off the agenda by the more urgent German problem. Stalin had managed to disrupt Zinoviev's and Bukharin's offensive and had forced them to follow a new script. After gathering in Moscow on 21 August, the Politburo heatedly and enthusiastically discussed the impending German revolution, the assistance the USSR would provide, and the possible responses by European powers. Everyone agreed that war was imminent. Supporting his colleagues' optimism, Stalin stated: \"If we really want to help the Germans, and we do want that and must help, we have to prepare for war seriously and thoroughly, since in the end it will be a matter of the existence of the Soviet Federation and of the fate of world revolution in the near future.... Either the revolution in Germany will collapse and they will beat us, or revolution will succeed there and everything will go well, and our situation will be assured. There is no other option.\"\n\nHere we see that Stalin and other Bolshevik leaders still shared the opinion that the USSR's fate was tied to the fate of world revolution, although the extent of this interdependence was not discussed in detail. What exactly did Stalin mean by \"they will beat us\" or \"our situation will be assured\"? What would this \"beating\" entail, and just what kind of assurance did he expect? These appear to be empty phrases, a nod to Marxist orthodoxy. When it came to tactical questions, he still sounded cautious and skeptical. He refused to support Trotsky and Zinoviev's proposal to set an exact date for the German revolution, believing it was better to make preparations and await the right moment. He also warned against hasty \"leftism\": \"Concerning the [German Communist] leftists. They are the most dangerous people for us. A premature takeover of factories, etc., would hold great dangers for us.\" On the question of setting an exact timetable for revolution, he wound up in the same camp as Bukharin and Aleksei Rykov. The latter was the most consistent adherent of caution: \"It is completely clear that everything is being bet on this one card. We are absolutely not ready.... We have to back off.\"\n\nWith war supposedly looming, the reorganization of the secretariat must have seemed inconsequential. We do not know how and when this issue, which just two weeks earlier had seemed vitally important, was finally resolved\u2014probably some agreement was reached in the corridors during breaks between meetings devoted to Germany. As a result, in September 1923 a rather pointless decision was made: Zinoviev and Trotsky were made members of the Central Committee's Organizational Bureau rather than the secretariat. This move would do nothing to solve the original problem\u2014Stalin's excessive control over decision making, to which Zinoviev and Bukharin had so hotly objected in July and August.\n\nAn event of great political significance took place at a plenary session of the Central Committee in September. The plenum adopted a decision to place Stalin and Voroshilov on the governing bodies of the military\u2014Trotsky's domain. Trotsky was being surrounded by his political opponents on his own turf. He stormed out of the plenum in indignation.\n\nHistorians still lack information on how this highly provocative attack against Trotsky was staged. It must have emerged from behind-the-scenes collusion between (at least) Stalin, Zinoviev, and Kamenev. They may have rationalized their actions with the following logic: events in Europe were coming to a head. The role of the Red Army and the military would be crucial, as it had been during the Civil War, and the influence of the Red Army's recognized leader would grow. The military therefore had to be brought under the control of Politburo members other than Trotsky before he became too powerful. It is unclear who initiated the ejection of Trotsky from the army's leadership. What is clear is that Stalin benefited significantly from this sharp escalation in the power struggle among the party's top leadership.\n\nAggrieved and isolated, in October 1923 Trotsky launched a counterattack. He submitted a letter to the membership of the party's Central Committee and Central Control Commission charging that the majority of Politburo members were conducting a misguided and unsound policy. He became a magnet for dissatisfied members. A fierce struggle broke out in which Zinoviev and other Politburo members, even those who felt that Stalin was already too powerful, were forced to join forces with him. In the coming two years this polarization\u2014the Trotsky camp versus the Stalin camp\u2014would serve Stalin well.\n\nDiscussion of Lenin's last dictated texts, about the need to remove Stalin from the post of general secretary, was shaped by this battle. Lenin died in January 1924. In May came the next party congress. During the congress, party leaders decided to disclose Lenin's \"testament.\" By general consensus this was done in such a way as to minimize the sting to Stalin. Lenin's final dictated words were not read at a general session of the congress but at the meetings of separate delegations. This procedure made it inevitable: Stalin was reelected as general secretary. Trotsky did not speak out, but it was not his silence that helped Stalin. Trotsky's very presence was enough.\n\nDespite his masterful handling of this situation, Stalin found himself in a vulnerable position. His virtues and shortcomings were a matter of public discussion. The very fact that such conversations could take place and that verdicts were being reached, however favorable, threatened to diminish his political authority. Rather than feeling gratitude toward those colleagues who had defended him before the congress's delegates, he seemed to respond with festering resentment. Their sympathy was demeaning; it looked too much like condescension, and their support felt like a favor that would have to be returned in kind. Stalin had no intention of paying off any political debts or allowing himself to be turned into a junior partner. Several weeks after the end of the congress he started biting the hands that fed him. In June 1924, _Pravda_ published a speech by Stalin in which he found fault with some rather innocuous statements by Kamenev and Zinoviev.\n\nThis outrageous breach of the anti-Trotsky leadership's united front caused consternation among top party ranks. Historians have uncovered no documents to shed light on what prompted Stalin's public scolding of Kamenev and Zinoviev, but it appears that this incident was discussed among a close circle of party leaders during the Central Committee plenum of August 1924, and Stalin found himself outnumbered. It is hard to find another explanation for Stalin's 19 August 1924 letter of resignation, a copy of which is preserved in his archive. In this remarkable document, Stalin stated that his collaboration with Kamenev and Zinoviev in the Politburo after Lenin's retirement had yielded deplorable results, demonstrating the \"impossibility of an honest and sincere political collaboration with these comrades within the framework of a single, close collegium.\" In light of this, he submitted his resignation from the Politburo and, accordingly, from the post of general secretary. He requested a two-month medical leave, after which he asked to be \"assigned to some minor post either in Turukhansky Krai, Yakutsk Oblast, or abroad.\"\n\nThis manipulative passive-aggressive outburst could hardly have been taken seriously. Nobody would have believed that Stalin actually intended to endure another Siberian exile, this time as a low-level paper-pusher! The full membership of the Central Committee, to whom the letter of resignation was addressed, never saw it. The matter was dealt with by a close-knit group of \"friends\" and allies, probably on 19 August, the day the letter appeared, or the following day. One can only assume that the establishment of an informal majority within the Central Committee took place in conjunction with the discussion of Stalin's letter. Later testimony by Zinoviev suggests that this all occurred between sessions of the Central Committee plenum, which concluded on 20 August. The majority faction, made up of the most influential anti-Trotsky members of the Central Committee, elected a _semerka,_ a group of seven, to serve as its governing body. The Seven included the chairman of the Central Control Commission and all the members of the Politburo except for Trotsky and functioned as a sort of shadow Politburo. Historians most often describe the establishment of this Central Committee majority faction and the Seven as an anti-Trotsky effort. This is partially true, but as Stalin's letter of resignation shows, the new unofficial body's primary task was to work behind the scenes to consolidate a majority within the Politburo and overcome internal disagreements. The Seven replaced the troika, which had not succeeded in this role.\n\nThis pivotal episode in the party's internal struggles reflects the balance of power in the Politburo during the summer of 1924. Stalin was apparently intentionally inciting conflict with Kamenev and Zinoviev, even though he could not yet be certain that other Politburo members, who were concerned with unity, would take his side. The letter of resignation was not only an obvious test of his own strength, but also a sign that he was still relatively weak. This incident was an important step toward Stalin's break with Kamenev and Zinoviev and his gradual alliance with Bukharin and Rykov. Having freed himself from the confines of the troika and now having the Seven to work with, he gained maneuverability.\n\nWhatever personal intentions and calculations were at play in forming the anti-Trotsky coalition in 1924\u20131925, it gave rise to a curious system of collective leadership that has been little studied as a force shaping the system of government that developed after Lenin's death. This collective leadership involved the interaction of politically equal Soviet leaders and the relatively autonomous government agencies they headed. It featured a rather well-developed division of functions between party and governmental apparats. Government policy, shaped by compromises among the competing interests represented by these leaders and agencies, became flexible and well balanced.\n\nThe period of collective leadership was a time of productive decision making and the flourishing of the NEP. The Seven overcame the crises the NEP was designed to address and adjusted the country's economic course while avoiding measures that would have caused systemic damage. Oligarchic government lent itself to relatively moderate political and economic policies. But collective leadership began to disintegrate when the government turned to a more hard-line, radical course. As historians have long believed and as recent archival research has confirmed, the seeds of conflict that put an end to collective leadership were intentionally sown by Stalin.\n\n###   **THE CRUSHING OF TROTSKY AND ZINOVIEV**\n\nUltimately, the viability of the collective leadership depended on its top leaders' willingness to adhere to the rules of their unique system of government. This system, which faced no threats beyond the personal ambitions of individual Politburo members, had marked advantages over an individual dictatorship. Whether it would survive after Lenin's demise had everything to do with the personal qualities of the three Bolshevik oligarchs: Trotsky, Zinoviev, and Stalin (in theory, these names could be given in any order\u2014their standing was supposed to be equal). These personal qualities, however, undermined collective leadership, and intrigues among these three figures inevitably drew other highly placed Bolsheviks into the fray, destabilizing the entire collective decision-making process.\n\nLacking a system for resolving personal conflicts, the collective leadership resorted to rather boorish methods to isolate Trotsky and exclude him from power. In so doing, it launched a process that destroyed the last shreds of relative democracy within the Bolshevik party. In January 1925, Trotsky was removed from his post as people's commissar for military and naval affairs, ending his hold on any real power. Zinoviev proposed that he also be removed from the Politburo. This proposal made perfect sense since Trotsky had already been excluded from the Politburo's work (as well as the unofficial deliberations of the Seven). But most members of the Politburo and Central Committee did not relish such changes, which always carried unpredictable consequences, and stood firmly under the banner of \"unity.\" Zinoviev's proposal seemed a bit bloodthirsty. The jokester Bukharin even made up an aphorism inspired by Zinoviev's anti-Trotsky zeal: \"If you see that the name Othello has been replaced with 'Grigory' [Zinoviev's first name], believe your eyes.\"\n\nStalin was well aware of these moods, and along with the rest of the Seven, he opposed Zinoviev's proposal, cunningly presenting himself as a supporter of unity and collective leadership. \"We plan to take all measures that preserve the unity of the Seven come what may,\" he wrote to Ordzhonikidze in February 1925. In actuality, the situation was coming to a head. New jabs were being exchanged between the Seven majority, on the one hand, and Zinoviev and Kamenev on the other, and Stalin's skilled hand could be seen in these intrigues. By late 1925 Zinoviev and Kamenev had formed a faction that threw down a gauntlet before Stalin, Bukharin, Rykov, and their followers.\n\nAt first the struggle for control centered on procedural matters\u2014how and by whom the Politburo's agenda should be set, as well as how the matter of Trotsky should be handled. These seemingly innocuous questions actually expressed a heated struggle for dominance within the collective leadership, but in order for this struggle to be taken beyond the bounds of the Seven, it needed a program. One could not gain the support of party functionaries, as Zinoviev and Kamenev counted on doing, with talk of winning control of the Politburo. Zinoviev, Kamenev, and their supporters chose a more respectable theme: the struggle against the \"rightist\" threat of allowing the NEP\u2014which supposedly would strengthen \"capitalist elements\" and prosperous peasants (kulaks)\u2014to become entrenched. Coming from the \"moderate\" Kamenev and Zinoviev, who were opposing the \"leftist\" Trotsky, or from Lenin's widow Krupskaia (who, out of long-standing friendship, supported Zinoviev and Kamenev over Stalin), this program looked out of place, even absurd. But they had no other choice. The Politburo majority was following a \"rightist\" course, so in order to oppose it, they were forced to move leftward. Probably Zinoviev and Kamenev also counted on recruiting to their cause the rather sizable subset of party functionaries who were inclined against the NEP.\n\nThey miscalculated. Even those party leaders who may have felt opposed to the NEP knew on which side their bread was buttered: all power flowed downstream from the Politburo. Everything was decided by this supreme body and transmitted to the local level through the top leaders' client networks. During the Fourteenth Party Congress in December 1925, when Zinoviev and Kamenev launched a determined attack against the Politburo majority in general and Stalin in particular, they were able to count on only the Leningrad delegation, which had been handpicked by Zinoviev, the region's party boss. This backing was not enough: they suffered a crushing defeat. Furthermore, the move cost Zinoviev his Leningrad fiefdom. Immediately after the congress a large group of Central Committee members was sent to Leningrad to make sure that Stalin's prot\u00e9g\u00e9, Sergei Mironovich Kirov, became Leningrad's new boss. Kirov's letters indicate that this takeover did not go particularly smoothly:\n\nThe situation is heated. There's a lot of work to be done, and even more yelling.\n\nHere, you get nothing without a battle. And what battles! Yesterday we were at Triangle [a reference to the party organization of the Triangle rubber factory], a collective of 2,200 people. The fighting was incredible. I haven't seen a meeting like that since the days of October, and I never even imagined that there could be such a meeting of party members. At times, it even came to fistfights in some corners of the meeting\n\nZinoviev's loyal party followers in Leningrad and the local party apparat were dealt with ruthlessly\u2014although by the standards of the time, \"ruthless\" did not extend beyond large-scale firings and transfers to remote regions of the country. This heavy-handed purge escalated the conflict between the opposition and the majority, which continued through 1926 and 1927. After a period of relative calm, in the spring of 1926 the majority found itself confronted with a newly unified opposition headed by Trotsky, Zinoviev, and Kamenev. This \"marriage of convenience\" (though no more so than the other alliances within the top leadership) was doomed to failure, but it made life difficult for the majority. The united opposition provided a rallying point for the dissatisfied, of whom there was no shortage. Keeping the opposition at bay demanded time, effort, and resourcefulness. Someone had to make this struggle his primary focus. By position and temperament, the best man for the job was Stalin.\n\nThe full range of intrigues perpetrated by both camps deserves a thorough study, which remains to be undertaken. Particularly worthy of attention is one basic and potent ingredient in this toxic brew: the use of state security to suppress the opposition. Gradually, with increasing frequency, the party opposition was branded the \"enemy,\" a label the Bolsheviks had previously reserved for outsiders such as the bourgeoisie, Mensheviks, or SRs. The historical record allows us to trace the origins of this practice to Stalin, who employed it not just in the mid-1930s, when the fight against the opposition reached its bloody apogee, but also much earlier.\n\nOn 6 June 1926, approximately seventy Moscow Bolsheviks with oppositionist sympathies gathered in a dacha community outside the capital. They chose this setting because they had been banned from holding meetings and needed to gather out of sight of the authorities. The gathering was addressed by a supporter of Zinoviev, Mikhail Lashevich, a longtime Bolshevik who had managed to keep his post as deputy head of the military commissariat. As might have been expected, an undercover agent was present at the meeting, possibly a specially infiltrated agent of the OGPU. The matter was placed in the hands of the party's investigative commission, which, try as it might, was not able to prove that the opposition's leaders had helped organize the meeting. This did not stop Stalin. In a 25 June 1926 letter to the Politburo, written while on vacation, he proposed using the \"Lashevich Affair\" as a pretext for destroying the Zinoviev group and expelling Zinoviev himself from the Politburo. The ideological justification for this cynical move rested on the idea that the opposition was breaking the party apart. An exceptionally stormy Central Committee plenum in July 1926, during which the opposition attempted to make a decisive stand, ended in accordance with Stalin's script. The plenum passed a resolution asserting that \"the opposition had decided to cross the line from legally advocating its views to creating an all-union illegal organization.\" The next step\u2014casting this \"all-union illegal organization\" as an \"all-union counterrevolutionary and terrorist organization\"\u2014would take Stalin another ten years, by which time his hold on power would be firm and his opponents executed.\n\nStalin's plan to expel only Zinoviev from the Politburo was a diversion, an attempt to divide the opposition and demonstrate objectivity. Just months later, in October 1926, Trotsky and Kamenev were also removed. Yet the oppositionists did not lay down their arms: they used every opportunity to do battle, denouncing the Politburo majority and its policies. The mutual animosity finally reached its pinnacle when, with no other options left to them, the oppositionists resorted to an underground propaganda campaign, to which the Politburo responded with a sting operation. In September 1927 the OGPU sent an agent posing as a former officer from Wrangel's army to a printing press that, despite the official prohibition, was still publishing opposition materials. Fabricated materials were used to charge the oppositionists with belonging to a \"counterrevolutionary organization\" that was supposedly plotting a military coup. The OGPU carried out the arrests. This police operation was organized by Stalin. While other Politburo members were vacationing in the south, he remained in Moscow and kept the others informed.\n\nIn October 1927, Zinoviev and Trotsky were removed from the Central Committee in a particularly ugly plenary session. When Trotsky attempted to address the plenum with a question, he had a book and a glass thrown at him and was forcibly pushed from the podium as shouting erupted in the hall. On 7 November, the tenth anniversary of the October Revolution, the oppositionists attempted to hold their own demonstrations in parallel with the official ones but were forcibly dispersed. These demonstrations served as an excuse for new reprisals: many opposition members were arrested and sent into exile. In December, the crushing of the opposition was officially sanctioned at the Fifteenth Party Congress. Some publicly capitulated, but Trotsky and his closest associates did not back down. Trotsky was sent to Kazakhstan and later expelled from the USSR. The majority of oppositionists, both those who had relented and those who had not, were killed during the second half of the 1930s. In 1940, on Stalin's orders, Trotsky was killed by a Soviet agent in Mexico.\n\nThe repression of the late 1920s, though relatively mild, still made a gloomy impression on the party's old guard and marked an important turning point in the party's development. As had happened during the French Revolution\u2014whose history the Bolsheviks knew well\u2014the Russian Revolution had begun to eat its own children. The similarities provoked a sense of dejection and unease. On 1 January 1928, soon after the opposition had been definitively crushed, Valerian Osinsky, one of the Old Bolsheviks, wrote an anxious letter to Stalin reflecting the sense that an injustice had been committed.\n\nDear Comrade Stalin,\n\nYesterday I learned that V. M. Smirnov is being sent somewhere in the Urals (evidently to Cherdyn District), and today, when I met Sapronov on the street, I heard that he is heading for Arkhangelsk Province for the same term. Furthermore, they have to leave by Tuesday, and Smirnov only just had half his teeth removed so they can be replaced with false teeth, and now he'll have to leave for the Ural north toothless.\n\nIn his day, Lenin kicked Martov out of the country in comfort, first making sure that he had a warm coat and galoshes. This is because Martov was once a revolutionary. Our former party comrades who are being sent away are deeply mistaken politically, but they are still revolutionaries\u2014there's no denying this.... The question therefore arises: is it really necessary to drive them all up north and essentially pursue a policy of their spiritual and physical destruction. I don't think so. And I don't understand why we can't (1) send them abroad the way Lenin did with Martov or (2) settle them within the country in places with a warm climate....\n\nThese sorts of banishments only create unnecessary bitterness.... They intensify whisperings about similarities between our current regime and the old police state.\n\nOn 3 January Stalin sent a curt response: \"Com. Osinsky! If you think about it you'll probably understand that you have no grounds, either moral or any other kind, for putting down the party or taking up the role of some sort of arbiter between the party and opposition. I'm returning your letter as insulting to the party. As for concern for Smirnov and other oppositionists, you have no grounds for doubting that the party is doing everything possible and necessary in this regard.\"\n\nWas Stalin's promise to do \"everything necessary\" for the oppositionists a kind of black humor, a hint at the coming moral and physical destruction of his opponents? There is no evidence that in 1928 Stalin was planning the purges or terror of the late 1930s. How are we to interpret the apparently genuine anger with which he responded to Osinsky? Was it merely that he was sick of talking about the opposition, worn out from years of tense struggle during which he had to watch every step, exercise unrelenting caution, make no false moves, hide his intentions, and conceal his actions? At the time he corresponded with Osinsky, Stalin was evidently making a critical decision that no opposition would be tolerated and no collective leadership was needed. Perhaps he was curt with Osinsky because he was anxious. Or perhaps he was confident and felt no hesitation in making it clear to Osinsky that they were no longer on the same level and \"heart to heart\" talks between them were no longer appropriate.\n\n###   **THE CHOICE**\n\nStalin's alliance with Rykov, Bukharin, and other Politburo members, first against Trotsky and later against Zinoviev, was a tactical move in a struggle for power and influence. It is probably safe to say that the primary forces driving this struggle were the personal ambitions of Lenin's heirs, their confrontational characters and outsized political ambitions, their nasty revolutionary habit of fighting for the sake of fighting, and a propensity to see enemies at every turn. That said, in their constant skirmishes the Bolshevik leaders were also guided by certain political ideas.\n\nThe Politburo majority, including Stalin, adhered to the so-called \"rightist course.\" This was a logical continuation of the NEP of 1921\u20131922. Once they saw that it would be impossible to immediately introduce a socialism free of money and markets, the Bolshevik leaders, with Lenin at their forefront, took a step backward. Keeping political power and heavy industry in the hands of the government, they allowed small industry and business owners (peasants first and foremost) relative freedom. Markets and money were rehabilitated. Nobody knew how or in what directions they should be moving. Only the general principles were clear: there would be a mixed economy combining market mechanisms, a strong state, and a monopoly on political power. There was also general agreement on the timetable: all shared Lenin's vision of the NEP as a long-term policy lasting through the 1920s.\n\nThe issue of the NEP was bound to become entangled in intraparty squabbles. Trotsky, later joined by Zinoviev and Kamenev, criticized the NEP strategy that had been devised by the Politburo majority. While not urging a total abandonment of the NEP, the oppositionists felt too many concessions had been made to the peasants and the urban bourgeoisie, and they called for greater emphasis on the development of major industries. This criticism was typical of the opposition movement in its struggle to undermine the power of those in charge and gain more for themselves: it exploited popular desires for greater equality and nostalgia for a \"heroic epoch.\" Most important, it was short on details. Had they achieved power, the \"leftist\" leaders, who were fundamentally pragmatic, would most likely have shifted imperceptibly onto the \"rightist\" path, abandoning their radicalism under the force of the objective need to develop the economy. This assumption is supported by the past behavior of \"leftist\" leaders. During the Civil War, did not the ultra-revolutionary Trotsky use the tsarist officer corps as a foundation for the Red Army? Did not all the Bolshevik leaders originally support the NEP? While a member of the government, Kamenev, one of the leaders of the left opposition, always gravitated toward moderation and followed a perfectly \"rightist\" course. Grigory Sokolnikov, another member of the opposition, was a brilliant finance commissar under whose leadership the country stabilized its currency. Often it was not principled programmatic differences that spawned conflict but ties of friendship, sore feelings, or ambition.\n\nThe consequences of this battle of political wills were devastating. The Bolshevik party endured irreparable losses of personnel. The disinclination to show mercy or compromise and the desire to decimate opponents not only took time and energy away from real problems, but it also undermined the collective leadership's will to conduct needed reforms and adjust social and economic policies. Every decision was examined under a magnifying glass, not only with an eye toward viability, but also to detect the slightest ideological vulnerabilities. Such an approach deprived the country's leadership of the flexibility and initiative it needed.\n\nMany of the decisions made in 1926\u20131927, a time of fierce struggle against the opposition, were politically motivated and destructive for the economy. Measures against \"capitalist elements\" were primarily targeted at relatively prosperous peasants and small-scale traders. Reckless and misguided economic decisions undermined stability. Yet these measures were not catastrophic or irreversible. The NEP, like any economic strategy, demanded constant adjustments, the elimination of mistakes, and an agile response to disparities as they arose. Lacking were the political preconditions for effective decision making. And the party infighting was only making the atmosphere worse.\n\nOne sign of the unhealthy political situation was the noisy campaign waged under the banner of fighting foreign threats. In 1927, a series of international crises was used to pump up war hysteria: a note from Britain's foreign secretary, Austen Chamberlain, objecting to Soviet anti-British propaganda in February; a raid on the Soviet embassy in Beijing in April; the breaking off of diplomatic relations with Great Britain in May; the June murder of the Soviet ambassador to Poland, Petr Voikov, who had helped organize the 1918 execution of Russia's royal family; and repression against Communists in China. Calls for vigilance and military readiness spawned rumors and panic buying of manufactured goods and food supplies \"in case of war.\" The government's fanning of martial passions was largely an attempt to counter criticism from the left, which was using foreign policy difficulties as fodder for attacks against the majority.\n\nAll of the Bolshevik leaders, both those still in power and those who had been expelled from office, took part in fanning militaristic passions. Stalin was no exception. News of Voikov's murder found Stalin vacationing in the south. In an 8 June coded telegram to Moscow he offered his take on the situation: \"Received about murder of Voikov by monarchist. Sense England's hand here. They want to provoke conflict with Poland. They want to repeat Sarajevo.\" By comparing Voikov's murder with the event generally seen as the trigger for World War I, Stalin showed that he felt war was imminent. In the coded message he urged \"maximal caution\" in regard to Poland but recommended conducting ruthless reprisals and purges within the USSR:\n\nWithout delay, all prominent monarchists in our prisons or labor camps should be proclaimed hostages. We should immediately shoot five or ten monarchists and announce that with every assassination attempt, new groups of monarchists will be shot. We should give the OGPU a directive about house-to-house searches and arrests of monarchists and any sort of White Guardists throughout the entire USSR in order to completely liquidate them using all measures. Voikov's murder gives us grounds to take revolutionary measures to completely crush monarchist and White Guard cells in all parts of the USSR. The task of fortifying our own rear demands this.\n\nThese statements foreshadow some of the hallmarks of Stalin's policies in the coming years. Relative prudence in foreign policy (\"maximal caution\") always went hand-in-hand with exceptional ruthlessness at home. The idea of \"fortifying our own rear\" through repression would be a cornerstone of Stalin's policy in the 1930s.\n\nThe Politburo members who had remained in Moscow adopted Stalin's recommendation. A wave of repressions swept the country. On 10 June 1927, _Pravda_ reported that twenty former members of the nobility\u2014\"hostages\"\u2014had been shot. The barbaric executions of innocent people severely damaged the Soviet government's reputation. The bloodthirsty behavior of the collective leadership suggested that all the top Bolsheviks were cut from the same cloth, but this is true only up to a point. On many key issues, Politburo members were capable of independent judgment. That the members of this body did not think in lockstep offered a kernel of hope that the Bolshevik authorities could govern with a degree of rationality.\n\nOne of the last glimmers of true collective leadership could be seen in the summer of 1927. This was a time of escalating crisis, and the Politburo reached its decisions on important political matters through genuine debate. A series of short letters from Molotov to Stalin, who spent that June and July vacationing in the south, offer a window onto these debates. The main points of conflict were the nation's policies toward China and Great Britain and the question of expelling Trotsky and Zinoviev from the Central Committee. Politburo members were still conducting themselves rather independently and forming surprising (in light of subsequent events) tactical coalitions. For example, Ordzhonikidze, Voroshilov, Rykov, and Rudzutak were critical of the policy toward China, where Moscow insisted, without success, on cooperation between the Kuomintang and the Communists. (Voroshilov \"has reached the point of groundless name-calling toward 'your leadership over the past few years,'\" Molotov complained in a letter to Stalin dated 4 July 1927.) Molotov and Bukharin, who enjoyed Stalin's support, defended the correctness of the policy. Opinions were evenly split on the fates of Trotsky and Zinoviev. Kalinin, Rykov, Ordzhonikidze, and Voroshilov believed that their expulsion from the Central Committee should be delayed until the party congress that fall. In telegrams from the south, Stalin unsuccessfully objected. Only after he demanded that his vote be counted in absentia and Kalinin joined those in favor of immediate expulsion did the Politburo resolve in late June to advance the timetable. Nevertheless, the implementation of this decision was delayed. The opposition leaders were not expelled during the Central Committee plenum in late July\u2013August but in October. Molotov, fresh from a contentious Politburo meeting on 4 July 1927, sent Stalin an anxious letter:\n\nThe most unpleasant thing is the situation within the Seven. In terms of questions concerning the opposition, China, and the ARK [Anglo-Russian Unity Committee], you can already see more or less distinct divisions, and over and over we're split down the middle with one deciding vote.... I'm increasingly wondering whether you'll need to come to Moscow earlier than scheduled. As undesirable as that might be in health terms, judge for yourself what the situation is.... The symptoms are bad; you can't count on stability. I haven't talked to anyone about this, but I feel the situation isn't good.\n\nHow justified were Molotov's expressions of alarm? Judging from the correspondence, Stalin took these reports in stride: \"I am not afraid of the situation in the group. Why\u2014I'll explain when I come.\" He had every reason for optimism. The clashes in the Politburo did not pose a serious threat to any of the Bolshevik oligarchs, including him. A stable balance of power was taking hold within the collective leadership. The summertime disputes Molotov described showed that the conflict within the Politburo was not among combating groups bent on crushing one another. As Stalin's follower, Molotov acted in conjunction with Bukharin. Rykov, who was close to Bukharin, was acting in coordination with Stalin's old friend Voroshilov. Kalinin, who had no strong alliances, moved from camp to camp. This sort of debate and formation of blocs was usual and helpful to the Politburo's functioning. The future of the collective leadership depended on the extent to which Bolshevik leaders were prepared to follow the rules of the oligarchy. Stalin was the weakest link in this chain.\n\nOnce the very ambitious Trotsky and Zinoviev were removed, only one power-hungry member remained in the Politburo: Stalin. The others, for a variety of reasons, were not capable of pretending to supreme power. In the pivotal post of general secretary, Stalin used the battle against the left opposition to strengthen his position. The schism within the party permitted him to play the role of preserver of Lenin's legacy and strengthened his control over the party apparat and state security. These advantages did not assure him victory, but they shifted the odds in his favor.\n\nIn December 1927, during the first plenary session of the Central Committee elected at the Fifteenth Party Congress, Stalin made a carefully calculated move: he submitted his resignation and refused to run for reelection to the post of general secretary. Now that the opposition had been crushed, he announced, it was a good time to fulfill Lenin's \"testament.\" Earlier, he modestly explained, a \"tough\" man had been needed as general secretary to wage a \"tough\" battle against the opposition. \"Now, it is no longer necessary to have tough people in such a prominent post.\"\n\nAs Stalin had surely expected, the plenum refused to accept his resignation. This move earned him important political dividends. First, once again, it diminished the relevance of Lenin's proposal that Stalin be removed as general secretary. Second, he presented himself to top party functionaries as the driving force behind the victory over the opposition: a \"tough\" leader capable of \"tough\" measures. This toughness undoubtedly enhanced his credentials in the eyes of those who favored a \"firm hand.\" Third, his show of loyalty, his stated readiness to retire, must have mollified those concerned about the breakdown of collective leadership and the emergence of a \"gravedigger of the revolution\" (as Trotsky had labeled him). Stalin had sought and found an important formal affirmation of his status. It is hard to believe that he took this risk for the sake of intraparty democracy. What came next\u2014his famous voyage to Siberia and attacks against rightists\u2014attests that he was acting with careful deliberation at the December plenum. This may well have been when he reached the fateful conclusion that he was destined to rule as dictator.\n\n## **A WORLD OF READING AND CONTEMPLATION**\n\n**Late evening of 1 March 1953. The near dacha. The mail arrives.**\n\nOnly as night approached did Stalin's bodyguards, after many hours of anxious waiting, decide to enter his quarters. They were thankful to have a pretext: the mail had arrived. A bodyguard took the packet and set out for Stalin's private rooms.\n\nWe do not know the contents of this last mail delivery, but normally Stalin received a huge number of papers. Lists of items sent to him from Moscow while he was vacationing in the south give us an idea of the types of documents the _vozhd_ dealt with on a regular basis. During a vacation extending from September through December 1946, he received an average of just under fifty letters, reports, and other materials per day. During his final southern vacation, August through December 1951, the average dropped to thirty-five documents\u2014not a small number. For obvious reasons, Stalin was regularly sent orders and draft orders by the highest governmental bodies\u2014not all, but the most important ones. Reports from the foreign and military ministries and state security and intelligence bodies regularly crossed his desk. He saw summaries of the foreign press prepared by TASS, the Soviet news agency. Some of these summaries, with his notations, have been preserved in the archives. He was also brought summaries of reports by foreign correspondents in Moscow. In keeping with a habit he had developed before the war, he regularly received daily reports on the production of planes and aircraft engines. Top aviation industry officials often wrote him on specific issues. The _vozhd_ had always taken a special interest in aviation, but he also received reports on the production of other military hardware. After the Korean War started in 1950, he received daily summaries of military actions and reactions to the war by the foreign press. He was also regularly informed about national stockpiles. On top of all this, the volume of correspondence between Stalin and China's leaders was growing. Finally, his mail included many letters from his top associates on various topics, requests from government agencies, and personnel proposals. Just reading all these letters and reports must have taken an enormous amount of time, and many of them required him to make decisions and compose some sort of response.\n\nIn addition to these official papers, Stalin found time to keep up with Soviet magazines, books, and newspapers, particularly _Pravda,_ which he studied attentively. The inventory of materials sent to him during his southern vacation in 1926 lists a large number of Soviet and \u00e9migr\u00e9 newspapers and journals, including Menshevik and White Guard publications. In later years, periodicals disappeared from the list\u2014probably not because Stalin ceased reading them but because they were delivered to him so routinely that listing them was a waste of time.\n\nAccording to some memoiristic sources, Stalin claimed to read an average of four to five hundred pages a day. It is difficult to imagine how he could keep up such a fantastic pace. Some days he may really have read that much or, more likely, scanned texts, focusing on the most interesting passages. In addition to the time he had to spend at his desk dealing with official papers, his workday was filled with hours-long conferences and meetings in his office. The dinners he hosted could extend for hours, as did his regular movie screenings. And he spent quite a bit of time writing. From what we know of his schedule, it appears that Stalin had little time to sit at his desk contemplating the steady stream of papers with which he was daily confronted.\n\nHe liked books. Reading played a major role in shaping his ideas. In the revolutionary milieu to which Stalin had been drawn as a youth, the value placed on intellectual pursuits and theorizing was tremendous, but these explorations were ideologically one-sided. This one-sidedness left a permanent mark on Stalin. He read \"socially significant\" books and studied Marx and Lenin. A literary scholar who made a thorough analysis of Stalin's writings and speeches noted the narrow scope of his erudition in literary fiction. He was well versed in literature from the Soviet period but had a poor knowledge of Russian or foreign classics. Observations regarding the political and ideological blinders that limited Stalin's reading are supported by the lists of books and journals in his library, or rather those in which he made notations. In total his archive holds 397 items. Of course his reading was not limited to these books, but his marginal comments and underscorings suggest that they are the ones that most captured his attention.\n\nThe lion's share of this collection is comprised of books and journals containing works by Lenin\u2014seventy-two items in all. Stalin was an attentive student of Lenin, and some of his own works represent a recasting or popularization of Lenin's thinking. Not surprisingly, he constantly cited Lenin in his public speeches. But he also relied on Lenin's work as a sort of bible or instruction manual when dealing with affairs of state within his close circle of associates. \"Whenever I was at Stalin's, either at a large or small meeting or talk,\" one of Stalin's commissars related, \"I'd notice the following habit. If somebody made a proposal that may have been practical but a bit out of the ordinary, he'd walk up to the shelf with Lenin's books, think a moment, and pull out a little volume. Sometimes he'd say, 'Let's have a look at what Vladimir Ilyich has to say on the matter.' Sometimes he'd read something aloud; sometimes he'd just paraphrase.\" Marx and Engels are much less evident in Stalin's articles. The archival collection of his library includes only thirteen of their works. Although Marxism was official doctrine and portraits of the bearded wise men were ubiquitous features of the Soviet landscape, Stalin occasionally allowed himself certain liberties in regard to these classics. In 1934, in memoranda to Politburo members and the ideological overseers of various party organizations, he criticized a number of Engels's works: \"Only idiots can harbor any doubts that Engels was and remains our teacher. But this by no means implies we must paper over Engels's mistakes, that we must conceal them and\u2014especially\u2014pass them off as incontestable truths.\"\n\nOne noteworthy portion of the collection consists of works\u2014thirty in all\u2014by Russian and foreign theoreticians of the Social Democratic movement, as well as prominent Bolsheviks: Aleksandr Bogdanov, Georgy Plekhanov, Bukharin, Karl Kautsky, and Trotsky, among others. Stalin also appears to have closely studied the nineteen issues of the prerevolutionary underground Bolshevik theoretical journal _Prosveshchenie_ (Enlightenment) kept in his library. The rest of the items in which he made notations largely consisted of propagandistic and educational literature written while he was in power, twenty-five of which he wrote himself. Overall, the classics of Marxism-Leninism (including his own works) and works by their propagandists comprise the vast majority of the nearly four hundred books in which Stalin made notations.\n\nAmong the remaining books, one category that deserves mention is historical works, including several courses on Russian history published before the revolution. Stalin loved history and constantly used historical examples and analogies in his articles, speeches, and conversation. He arranged for new history textbooks to be written and encouraged the production of numerous historical books and films. As is well known, he felt a particular affinity for two Russian tsars: Peter the Great and Ivan the Terrible. They consolidated and enlarged Russia, built up its military might, and fought mercilessly against internal enemies. For Stalin, history was a means of legitimizing his own policies. He was not particularly interested in scholarly discussions and actual historical evidence, choosing instead to adapt the facts to his preferred narrative. Ivan the Terrible was proclaimed a stalwart defender against the forces pulling Russia apart, saving it from a second Tatar yoke. His brutal repression, as Stalin saw it, was necessary, and if anything, it did not go far enough: \"It should have been done even more decisively.\" During the Cold War, Stalin praised Tsar Ivan for adopting \"a national perspective and not allowing foreigners into his country, shielding the country from the intrusion of foreign influence.\" He condemned his otherwise beloved Peter the Great for taking a liberal attitude toward foreigners. Even more, he molded Soviet history to justify his own policies. The falsification and rewriting of the party's history culminated in the creation of an ideological bible of the regime produced with Stalin's active participation, the _History of the All-Union Communist Party (Bolsheviks): Short Course._ Appearing in 1938, at the height of the Great Terror, this work proclaimed Stalin to be equal to Lenin as a leader of Bolshevism and the revolution. Utter fictions were inserted into many episodes of Bolshevik history; other episodes were distorted beyond recognition. The opposition leaders, who had by then been killed, were portrayed as inveterate enemies.\n\nMilitary problems particularly attracted Stalin's interest. In addition to books of military regulations, he made notations in several books on the history and theory of war, such as works by the Prussian military theorist Carl von Clausewitz and the Russian theorist Aleksandr Svechin.\n\nThe few books of non-Marxist philosophy contained in the collection include Plato and a philosophical treatise by Anatole France, _The Last Pages: Dialogues under the Rose._ The small number of books on economics is dominated by Soviet works on political economics. As for literary fiction, the collection contains only a few literary journals and works by Lev Tolstoy (the novel _Resurrection_ ), Mikhail Saltykov-Shchedrin, Maxim Gorky, and a few Soviet writers.\n\nOf course this particular collection does not tell the whole story. We know from other sources that Stalin often read literature by contemporary Soviet authors. He offered advice on plays and screenplays and made decisions about the awarding of prizes. He had his likes and dislikes, and the latter, however talented, were often targeted for repression. Even Soviet literary lions faced ideological tongue-lashings. All were made aware of their vulnerability and utter dependence on the government's favor. Yet despite his politically slanted tastes, Stalin did have a certain ability to distinguish good writing from bad. Perhaps this is why he tolerated and even protected certain talented writers who were not helpful or were even harmful to the regime, such as Mikhail Bulgakov. Still, the censors kept such writers on meager rations, just barely surviving and under constant threat of arrest. Literature and dramaturgy interested the dictator primarily as ideological tools, a means of social manipulation and brainwashing. Officially permitted writers were part of the state's vast propaganda apparatus. Amalgamated into state corporations, writers, artists, and composers were completely dependent on the state. Like state-run factories, these corporations were not very effective. They encouraged bureaucratization and mediocrity and suffocated talent. \"The time is long overdue for us to focus attention on... the irresponsible activities of the three thousand people brought together by the Writers' Union, out of which two thousand\u2014at least\u2014hardly belong in literature,\" Maxim Gorky, Stalin's choice to lead Soviet writers, lamented in a 1936 letter.\n\nStalin knew of Gorky's feelings (he kept this letter in his personal files), but he was hardly troubled by literary mediocrity. He lived and breathed political power, so works of art and literature were to be judged according to their ideological and propagandistic usefulness. \"Simplicity\" and \"accessibility\" were key literary virtues. He welcomed readability and straightforward political edification free of highbrow devices. The \"creative intelligentsia\" was called on to depict a reality that was idealized (\"correct,\" \"socialist\") rather than objective. It was to bring to the masses not that which was but that which should be, while distracting them from hardships and extolling the virtue of placing the party and the state above self-interest.\n\nThe record of conversations that took place during screenings in the Kremlin movie theater offers an interesting window onto Stalin's taste. He critiqued the films shown exclusively from the standpoint of political utility, which, he believed, called for the production of edifying and entertaining films \"that are exciting, cheerful, and fun.\" \"Just don't drive everyone into depression, into a labyrinth of psychology. There's no need for people to engage in pointless philosophizing,\" he said during one screening. He fully approved of the rollicking musical _Jolly Fellows,_ the Soviet answer to Hollywood comedies. The film was not profound and politically pointed, but, as Stalin put it, it gave people \"interesting and engaging relaxation.\" His running commentaries treated what was happening on screen as if it were real life. A few favorites were viewed over and over. _Chapaev,_ about the Civil War hero of that name, for example, was viewed thirty-eight times between late 1934 and early 1936.\n\nStalin's taste in theater and music were equally conservative. He condemned the stage director Vsevolod Meyerhold, known for provocative experimentation, for \"clownishness\" and \"gimcrackery.\" The _vozhd_ himself initiated a campaign against new musical forms, such as those being created by the great composer Dmitry Shostakovich. Such innovations were given the derogatory term \"formalism.\" A regular theatergoer, Stalin preferred classical drama, opera, and ballet. Countless official receptions at the Kremlin were accompanied by concerts featuring a strictly traditional repertoire.\n\nThere may have been a relationship between Stalin's literary tastes and his manner of writing. It has often been noted that he was not a gifted orator, a judgment that can easily be confirmed by listening to recordings of his speeches. But his written texts are much more coherent than his impromptu speeches. As a writer, he strove for a clarity and conciseness that bordered on oversimplification. He liked to drive a point home through numerous repetitions, as if he were hammering an idea into his audience's heads. Lacking the gift (possessed by many other Bolsheviks and writers) for brilliant public speaking, Stalin simply ignored this art. His texts are dull but easily understood. He was a master of slogans and clich\u00e9s. In a society where education was achieving breadth but not depth, especially in the humanities, such a public speaking style was rather effective.\n\nAs a child, Stalin used only Georgian, the language in which he composed verse and revolutionary articles in his youth. He occasionally used Georgian later in life as well. At the age of eight or nine, the future dictator began to study Russian and was able to achieve a high level of proficiency, almost to the point of making it a second native language. But until the end of his life, he spoke with a strong accent. This \"accent\" can also be felt in his written texts. Stalin's writing in Russian is grammatically correct and expressive, but he occasionally let slip jarring stylistic infelicities and mangled idioms. Students of Stalin's language have been able to assemble quite a few examples from his published works. Such examples are also found in his day-to-day writings not intended for publication. As general secretary of the Central Committee, Stalin reviewed Politburo resolutions before they were finalized and often made changes to them. In a number of cases, the fact that he was not a native Russian speaker led to errors and ambiguities.\n\nThere is scant information concerning Stalin's knowledge of other languages. He traveled abroad several times before the revolution (to Berlin, Stockholm, London, Vienna, and Krakow), but it is unlikely that he had either the time or the need for serious study of the languages spoken in those cities. These trips were made on party business, and his time was spent mainly with party comrades. His 1913 work on the nationalities question, which made use of sources in German, was written in Vienna with the help of someone who knew that language. While in exile in Turukhansky Krai in 1913\u20131917, he demonstrated a desire to improve his knowledge of languages. He asked to be sent books by German authors (although it is not clear whether he was asking for the originals or translations). In February 1914 he wrote to a society in Paris that assisted Russian exiles, requesting a French-Russian dictionary and some English newspapers. A May 1914 letter that he wrote to Zinoviev urged him to send \"some sort of (civic) English journal (old, new, it doesn't matter\u2014for reading, since here there's nothing in English and I'm afraid that without practice I'll lose what English I've learned).\" In November 1915 he again wrote to his comrades: \"I don't suppose you could send something interesting in French or English?\" In 1930, while vacationing in the south, he asked his wife to send him a textbook for learning English. How serious was Stalin's intention to study languages? How far did he advance? We cannot answer these questions. As far as we know, he never tried to demonstrate a knowledge of languages during any of his countless meetings with foreigners.\n\nIn the end, Stalin's self-education, political experience, and character formed a mind that was in many ways repellant but ideally suited to holding onto power. His oversimplification of reality, in which phenomena were explained in terms of a historic standoff\u2014between classes, between capitalism and socialism\u2014outlived his system. Whatever the sources of this simplistic worldview\u2014his religious education, his adherence to Lenin's version of Marxism\u2014its unidimensionality simplified the dictator's life. A model of the world based on the principle of class struggle permitted him to ignore complexity and despise his victims. It allowed the regime's most heinous crimes to be seen as a natural expression of historical laws and innocent mistakes to be seen as crimes. It allowed criminal intentions and actions to be attributed to people who intended and committed no crimes. In a relatively uneducated country, simplification was an excellent tool of social manipulation.\n\nStalin's theoretical model of the world was in fact tottering and unreliable. Excessively simple and ineffective, it gave rise to abundant contradictions and failures. Yet he saw any adjustments to the ideological system that might have benefited the country as threatening to the stability of his regime. So he responded to life's demands with rigid ideological and political dogmatism and agreed to limited changes only as a last resort, when crises reached a breaking point. Shielding himself from reality, he retreated\u2014and tried to bring others with him\u2014to the thickets of ideological scholasticism. The contents of his personal archive, which reflect what he thought was worthy of being kept close at hand, are almost completely devoid of documents that represent any sort of outside, expert perspective. Meanwhile, a huge country was engaged in the earnest study of Stalin's \"expert\" opinions on fields as diverse as linguistics and political economy. It followed his dictates in crushing \"formalists\" and \"cosmopolitans.\" Fearing change and the pernicious influence of the West, Stalin rejected a number of scientific advances, such as genetics. He believed only in what \"you could touch with your hands,\" what he understood and felt to be politically safe.\n\nThis dogmatism and rejection of the complex posed serious impediments to the country's development. Yet even as his life came to an end, Stalin had no intention of changing the political system that had brought him power, a system that he methodically forged throughout the 1930s.\n\n## **3 HIS REVOLUTION**\n\nBy the end of 1928, the crushing of the \"left opposition\" had been transformed into Stalin's personal victory. Cohesion among the Politburo majority, which had been easy to maintain during the fight against Trotsky and Zinoviev, began to deteriorate. The growing socioeconomic crisis was paralleled by a crisis at the upper echelons of power, a volatile mix that put the system of government in peril. This political kindling was finally ignited by the state's failure to collect sufficient grain supplies in 1927, one of many signals that the NEP was not working.\n\nThe NEP model of development was doomed by a range of factors. Allowing market forces to govern the relationship between the peasants and the state violated fundamental Bolshevik doctrine. Despite the tragic experiences of War Communism, the ruling party continued to preach radical socialism and punish private economic initiative. Furthermore, Soviet agriculture was simply incapable of immediately producing the resources the government needed to support industrialization. Every camp within the ruling party\u2014rightists, leftists, and everyone in between\u2014was aware of the need to adjust the NEP and spur industrialization. The problem was finding how to best modify the system. The fierce battle for power severely limited the available options. The economy was once again falling victim to political conflict and the need to adhere to dogma, and no one was more guilty of putting political expediency before the needs of the economy than Stalin.\n\nThe reasons for the crisis of late 1927 were perfectly familiar to the country's leadership. Pricing policy errors and a disproportionate investment in industry, among other factors, had undermined peasant incentives to sell grain to the state and disrupted the overall economic balance. In previous years, the leaders had found successful recipes for overcoming similar crises. Such a recipe was needed again. At first the Politburo searched for solutions as a unified collective. Although they considered economic stimuli, on this occasion members decided to try intensifying pressure on the peasants through \"administrative\" means. This meant a campaign to expropriate grain by force, and a key component was visits by the country's leaders to grain-producing regions to inspire greater effort on the part of local officials. Molotov, who was sent to Ukraine, reported to Stalin on the first day of 1928:\n\nDear Koba! I'm in my 4th day here in Ukraine\u2014and people say I'm doing some good. I've pumped up the lazy _khokhols_ [derogatory term for Ukrainians].... I managed to get Ukraine's \"chiefs\" and \"centers\" to travel around to local sites and to promise to work hard. Now I'm hanging around Melitopol (a gold mine!) and also arranged a pogrom here with all the usual swearing that goes with grain collection.... Lots of new impressions; I'm really glad to be able to touch earth. I'll tell you all about it when I get back. Regards to all.\n\nThe tone of Molotov's letter\u2014more lighthearted than hard-line\u2014partially reflected the relatively peaceable mood that still prevailed in the Politburo. Molotov was not yet \"unmasking opportunists\" or branding \"kulaks\" and \"wreckers.\" He asked Stalin to give Ukraine a bonus out of its grain collections to enable the purchase of farm machinery abroad: \"This is urgently needed for encouragement (plus to push production) and is expedient in all regards.\"\n\nStalin was not so jovial: he was spending his time thinking up ways to institute extreme policies. What prompted Stalin to take a sharp turn that placed him far to the left of Trotsky and Zinoviev? What drove his sudden opposition to the NEP: a belief that an ultra-leftist course was truly inevitable or self-serving political calculations? The evidence suggests a complex of motives. Some of the NEP's contradictions were indeed gradually drawing the entire top leadership leftward and leading to a restructuring of the NEP that favored more rapid industrialization. Stalin was among those who were most eager for this new direction. His political and managerial temperament inclined him toward violent measures. Furthermore, he had no expertise whatsoever when it came to dealing with the economy and probably sincerely believed it could be forced into whatever mold politics dictated. The extreme economic measures he mandated served obvious political purposes. In staking his wager on a radical course, Stalin was intentionally destroying the system of collective leadership. The battle within the Politburo that ensued permitted him to create a new majority faction that was unambiguously his to control.\n\nIn essence, Stalin was adopting Lenin's revolutionary strategy, which called for maximally spurring leftist excesses, undercutting \"moderates,\" and mobilizing radicals with extremist policies. To launch his revolutionary push, Lenin had had to come to Petrograd from emigration in April 1917. Stalin set out for Siberia in early 1928 with a similar purpose: to turn this distant and enormous region into a proving ground for new upheavals. The trip seems to reflect some scheming on his part. The plan had been for the Politburo's top troika\u2014Stalin, Rykov, and Bukharin\u2014to remain in Moscow to watch over the government, but Stalin took advantage of Ordzhonikidze's ill health to take his place on the trip to Siberia. He probably assigned Siberia to Ordzhonikidze in the first place realizing that he would not be able to go, given his poor health in late 1927. The very fact that Stalin\u2014who did not like to travel\u2014made such a long trip shows the seriousness of his intentions. After 1928 his official trips were few. He made some stops on the way to his southern vacations; in July 1933 he visited the White Sea\u2013Baltic Canal; and he made one trip to the front during World War II and three to meet with Roosevelt and Churchill in Tehran, Yalta, and Potsdam. Clearly, he had his reasons for going to Siberia in 1928.\n\nIt took three days to reach Novosibirsk by train. The general secretary spent a total of three weeks in Siberia during the latter half of January and the first days of February. Most of this time was spent in meetings with the _aktiv_ (local bosses and party stalwarts). Stalin extracted from them a pledge to fulfill an ambitious plan to supply the country with Siberian grain. He told the Siberian officials just how they would achieve this challenging objective, rolling out his plan to bring down the full force of the police state on the kulaks and charge them with the crime of \"speculation.\" In essence, this plan represented a return to War Communism. Many Siberian leaders objected. The change of course was so sudden that some even permitted themselves to argue with him. On 19 January the head of the Siberian branch of the agricultural bank, Sergei Zagumenny, wrote to Stalin to voice his concerns, saying he doubted the effectiveness of treating peasants like criminals for refusing to sell grain to the state. Peasants would see this as a return to the policy of mandatory sales of surplus grain to the state practiced during the early years of Soviet rule. It could make matters worse. \"It seems to me that we are making too sharp a turn,\" he wrote. Stalin's many notations on Zagumenny's letter (underscorings and derisive comments) attest to his irritation.\n\nStalin continued to pressure the Siberian officials and insisted that repression would be effective. At the same time, he maintained a certain restraint in his interactions. In talking about the failures of grain procurement, he stopped short of making threats and combined confident and decisive authority with displays of comradery. At a meeting in Novosibirsk, in response to a statement that he had caught _krai_ officials making mistakes, Stalin answered with a conciliatory \"No, I wasn't trying to catch anyone.\" Even the criticism leveled against Zagumenny was fairly gentle. This combination of ruthlessness toward \"enemies\"\u2014in this case grain-hiding kulaks\u2014and amiability toward his party comrades is one aspect of the strategy that helped him climb to the top of the political hierarchy. It undoubtedly made a favorable impression on local party officials and was an effective way for Stalin to reassure anyone who might have doubted the changing nature of the party under his leadership.\n\nThrough pressure and persuasion, Stalin got what he wanted. Dressed in a new sheepskin coat made for him in a local workshop, he spent several weeks crisscrossing the vast expanses of Siberia. Everywhere he demanded the same thing: give us grain. As he put it in a telegram to Moscow, he \"got everyone good and worked up.\" In a subsequent telegram sent on 2 February, the eve of his return to Moscow, he triumphantly reported that \"A turnaround in grain deliveries is beginning. During 26\u201330 January, 2.9 million poods [approximately 52,400 tons] of grain was procured, instead of the norm of 1.2 million. This is a major turning point.\" Stalin also expressed hope that the pace of grain collection would continue to grow. In a single month, Siberia had supposedly fulfilled more than a third of its annual grain quota.\n\nBehind these figures was escalating brutality in Siberian villages. Bands of agents empowered to use an iron fist in demanding the turnover of grain swept through the countryside. Disdaining even to pay lip service to legality, these agents followed a principle openly expressed by one of them: \"What kind of bureaucratism is that? Comrade Stalin gave us our motto\u2014press, beat, squeeze.\" The countryside was gripped by searches and arrests. Such large quantities of grain were confiscated that peasant families were ruined. Under Stalin's influence, Siberia received more unsparing treatment than the country's other grain-producing regions, although probably not by much. Pressure from Moscow and the active involvement of highly placed emissaries subjected villages everywhere to violence and lawlessness. But the precedent for extremism set in Siberia had special significance. Coming straight from the general secretary, the order to wage war against the kulaks was seen as a universal license.\n\nAs political theater, Stalin's Siberian trip had a complex subtext. The first thing it did was change the ideological framework of the crisis. Ignoring the official line that the government had made mistakes (a point reiterated in numerous Politburo directives), Stalin shifted the emphasis onto exposing the hostile actions of kulaks and anti-Soviet forces, thus opening the door to the broad use of repressive measures. At his suggestion (his creative contribution to the 1928 grain requisitions), confiscation was not, as previously, conducted on an extraordinary basis but as part of an ongoing effort to enforce the criminal code. \"Speculators\" were handed over to the courts for refusing to sell grain that they themselves had planted, tended, and harvested. Such actions made a mockery of justice, but they gave extraordinary measures a legal footing and made them routine and permanent. In essence, Stalin was proposing to jettison the principles that, under the NEP, had governed interactions between the state and the countryside. Finally, Stalin's trip across Siberia confronted the government's economic apparat\u2014and Rykov, as premier, personally\u2014with a serious challenge. The party, embodied by Stalin, was taking charge of the country's most important political and economic problem and thus asserting its primacy.\n\nStalin knew that some of his colleagues would raise objections to the strong-armed measures he instigated in Siberia. He was provoking conflict with careful calculation. The Siberian trip allowed him to confront his fellow leaders from a position of strength, as an energetic leader who had succeeded by applying revolutionary methods to pressing problems. The results cast moderation in an unflattering light and made radicalism look more effective. Fissures in the Politburo started to show immediately after he returned to Moscow in February 1928. But he was apparently not quite ready for all-out war. To an outside observer it might seem that by failing to force a showdown, he was letting an exceptional opportunity slip by, but Stalin probably did not see it that way. At the time, there was no clear evidence that he would emerge victorious. This was a pivotal moment in his campaign for sole power, and he turned it into a guerrilla operation, using deceit, patience, and subversion.\n\n###   **A SHIFT TO THE FAR LEFT**\n\nCircumstances prevented Stalin from quickly and openly asserting primacy over his Politburo colleagues\u2014and preventing them, in turn, from calling him to account for his recklessness. From the standpoint of his political interests, the leadership could be divided into two groups. The first consisted of potential adversaries, leaders who enjoyed a degree of independent power and influence and would oppose his rise to power. This group included Aleksei Rykov, chairman of the Council of People's Commissars (the country's premier); Nikolai Bukharin, the party's chief ideologue and editor of _Pravda;_ Mikhail Tomsky, the leader of Soviet trade unions; Nikolai Uglanov, Moscow party secretary; and Mikhail Kalinin, chairman of the Supreme Soviet, the country's parliament. These leaders, proponents of collective leadership and a gradual transformation of the NEP, were not happy about Stalin's ambitions or his extreme policies. The second group\u2014only a minority of the Politburo\u2014had close personal ties to Stalin: Vyacheslav Molotov, Central Committee secretary; Kliment Voroshilov, chief of the military commissariat; Grigory Ordzhonikidze, head of the party's Central Control Commission; and Anastas Mikoyan, head of the trade commissariat. They had looked up to Stalin and followed his lead since the revolution and Civil War. Even his friends, however, were not likely to unquestioningly support his efforts to break down the party's collective leadership and proclaim himself sole leader. In early 1928 the \"Stalin faction\" could be rallied and counted on only in time of war.\n\nWaging such a war would be complicated and risky. The fevered four-year standoff with the opposition had created a deep desire for unity. The oppositionists had been castigated as schismatics who had put their personal political ambitions before the interests of the party. Any leader who openly threatened the party's newfound unity would find himself in an unpopular position. How could Stalin fight for dominance without undermining unity? There was only one solution: to surreptitiously provoke a split and then cast himself as an injured adherent of unity and his enemies as schismatics. That is the script Stalin followed.\n\nAnother concern was that the radical measures Stalin was proposing, measures close to the hearts of party leftists, had huge destructive potential. Two dangers were immediately evident. First, the peasants, knowing that their harvest would be confiscated, might simply plant less. Second, there were worrisome signals coming from the Red Army. Letters from relatives back home complaining of mistreatment were stoking anti-government sentiment in the barracks. Young peasant recruits underwent military training at bases not far from home, and emissaries streamed from the villages to the bases pleading for help.\n\nLacking sufficient political strength to simply sweep such realities under the rug, Stalin was forced to bide his time. Evidence from the period after his return from Siberia shows him ready for compromise. Resolutions adopted around that time, while expressing approval for the extreme measures already taken, condemned \"distortions and excesses.\" Stalin's handling of objections to tactics used in Siberia foreshadowed the brand of political warfare he would favor in subsequent years, before he achieved complete victory. In essence, his approach was to \"agree and ignore.\" Wishing to avoid a showdown, he put his faith in stealthy manipulation of the bureaucratic machine and a strategic reshuffling of personnel.\n\nEverything depended on the alignment of forces within the Politburo. In 1928, with help from political intrigues, Stalin managed to weaken the Rykov-Bukharin group and strengthen unity among his friends. He also benefited from the foolish mistakes of his opponents\u2014especially Bukharin\u2014and likely from the use of blackmail. He may have made use of recently discovered compromising evidence against Mikhail Kalinin and Yan Rudzutak, unearthed in prerevolutionary police records in 1928 but never brought to light. A transcript of a February 1900 police interrogation has Kalinin stating: \"Having been called in for interrogation as a result of a request I submitted, I wish to give frank testimony on my criminal activities.\" The transcript shows that Kalinin gave the police detailed information about the operations of his underground organization. Police records also showed that Rudzutak, who was sentenced to ten years' hard labor in 1909, apparently gave interrogators the names and addresses of members of his organization. The police then conducted searches and seized weapons and propagandistic literature. Similar compromising materials Stalin could have used against other members of the top leadership may remain to be found.\n\nAlthough there is no hard evidence to show that Stalin used these discoveries in his quest for loyal supporters, his relationship with the secret police was such that he would almost certainly have been informed about them, and his using the crude but powerful tool of blackmail would have been entirely in character. Even his friends on the Politburo understood the reasons for the split within its ranks. Stalin's pontification on the \"rightist threat\" did not mask his intention of achieving dominance within the Politburo. The war he was waging was starkly personal. In an attempt to reconcile the sides, Stalin's old friend and loyal follower Ordzhonikidze wrote a frank letter to Rykov amid clashes in the fall of 1928:\n\nAny more fighting within the party is bound to lead to unbelievably bitter upheavals. That has to be our starting point. I am absolutely convinced that we'll get over this. In terms of grain and other such issues, we can argue and decide, but it shouldn't lead to fighting.... There are no fundamental disagreements, and that's the most important thing.... It seems that the relationship between Stalin and Bukharin has really deteriorated, but we need to do everything possible to reconcile them. It can be done.\n\nIt is unlikely that Ordzhonikidze was attempting to deceive Rykov in order to help Stalin. He was merely describing the moods and views then held by the majority, including many of Stalin's supporters. The Politburo's collective leadership was still a viable and functional institution. Even as authoritarian a Bolshevik as Ordzhonikidze understood that it was better to \"argue and decide\" than to engage in political name calling. All Soviet leaders recognized the need to revise economic policy in favor of accelerating industrialization. Only the details were in dispute. There was no reason friction within the Politburo had to lead to a complete rupture\u2014so long as no member of the collective leadership harbored ambitions of achieving sole power.\n\nAttuned to the prevailing mood, Stalin paid lip service to unity while using others to undermine his opponents. In 1928 he organized rebellions within Tomsky's trade union apparat and Uglanov's Moscow party organization. By orchestrating upheavals within these organizations, Stalin managed to deprive both leaders of their \"patrimonies.\" Furthermore, his opponents were weakened by a fatal political misstep by Bukharin, who in July 1928 secretly met with the disgraced Kamenev and gave him a candid account of conflicts roiling the Politburo. Kamenev's written account of this conversation was stolen and sent to followers of Trotsky, who, despising both Stalin and Bukharin, were only too glad to print it up on leaflets and distribute them publicly. The true story is still not entirely clear, but even if Stalin and the secret police, which was already under his control, had nothing to do with the theft of the notes, there is no doubt that he did everything he could to ensure that the leaflets were broadly disseminated. Bukharin and his supporters were hopelessly compromised.\n\nWhile branding Bukharin a schismatic who fraternized with the crushed opposition behind the backs of his Politburo colleagues, Stalin prepared his heavy artillery. In mid-1928, engineers from a Donetsk coal mine were subjected to a show trial based on fabricated charges\u2014the so-called Shakhty Affair. They were charged with sabotage, and their trial was accompanied by a powerful propaganda campaign. Meanwhile, as the 1928 grain collections were again turned into a war against the kulaks, Stalin proclaimed a new theory (which he made sure was borne out): the farther socialist construction progressed, the more heated the class war would become as the enemies of socialism intensified their resistance. They would also, he warned ominously, exert influence over the party. Persistently and methodically, he introduced into party documents and propaganda the idea of \"danger from the right\" and from agents of hostile influence within the party. Keeping constant pressure on \"the enemy,\" destroying him and his \"rightist\" allies within the party\u2014that was how the victory of socialism and the long-awaited overcoming of difficulties and conflicts would finally be achieved. These sinister theories may have appealed to poorly educated party functionaries, but they are not consistent with what was happening in the country.\n\nOnce he had isolated the Bukharin-Rykov group, Stalin cast his final blow by blaming the two men for the \"right deviation\" within party ranks. In an atmosphere of political hysteria and growing radicalism, the more moderate forces within the party were compelled to remain silent. When forced to take sides, most Politburo members\u2014each for his own reasons\u2014chose to support Stalin. The entire Politburo became a sort of Stalin faction. One after another in 1929 and 1930, Bukharin, Tomsky, Uglanov, and Rykov were expelled from the Politburo and relegated to the status of second-tier functionaries. None survived the Terror.\n\nStalin's victory in the Politburo was due to political intrigues and errors by his opponents. The general secretary made good use of the vast experience building and wielding power and influence he had acquired during the years of struggle against Trotsky, Zinoviev, and Kamenev. Of no small importance was Stalin's power, as general secretary, to influence appointments. He knew how to manipulate people, how to wait for the right moment and strike with just the right amount of force to avoid scaring off potential supporters or waverers. Masking his true intentions, he presented himself as a reasonable politician and loyal member of the party community, implacable only toward enemies. In a few short years, everything would be completely different. Many who supported Stalin bitterly repented their choice once their turn for destruction came. This was Stalin's genius: to ensure that his victims developed regrets only after it was too late.\n\nOne result of the Stalin faction's victory was the approval and implementation of the Great Leap policy. Largely due to Stalin's influence, \"class warfare\" and \"revolutionary spirit\" were introduced into the economic sphere. Socioeconomic constraints were discarded as so much rubbish. No objective limits were placed on industrial plans or on capital investments in manufacturing\u2014whatever industry needed, it would get. A tremendous wager was placed on large-scale purchases of Western equipment and even entire factories in the hope that these resources would be quickly up and running, producing an abundance of goods. The historical circumstances were propitious. With their economies languishing from the Great Depression, Western countries were more inclined to cooperate with the USSR than they might have been in times of plenty.\n\nThe ambitious five-year economic growth targets adopted in April 1929 were almost immediately rejected as too modest. Targets were increased by 50 percent, then doubled and tripled. The Five-Year Plan was changed to a Four- and even Three-Year Plan. Trying to outdo one another in this frenzy, party and economic functionaries pulled ever higher numbers out of the air. \"In ten years at most,\" Stalin exhorted, \"we must make good the distance that separates us from the advanced capitalist countries.... Some claim that it is hard to master technology. That is not true! There are no fortresses that Bolsheviks cannot capture.\"\n\nTreating the economy as a fortress to be captured plunged the country back into the War Communism of the Civil War period. Political campaigns, an enthusiastic minority, and the compulsion of the majority almost completely took the place of economic incentives and proven practices of manufacturing and labor management. A disordered financial and commercial system and skyrocketing inflation were explained away as predictable obstacles on the path toward socialism, toward the withering away of commodity-money relations and the introduction of product exchange between cities and the countryside. As foreseen by the more moderate party leaders, this mad race to industrialize left no place for the tracking of basic economic indicators. In December 1930 the new chief of Soviet industry, Grigory Ordzhonikidze, reported that even such key industrial sites as the Magnitogorsk and Kuznetsk Metallurgical Works, the Nizhny Novgorod Automotive Plant, and the Bobrikov Chemical Works were being built without finalized blueprints. In many cases, he wrote in a memorandum, \"money is being spent without any budget.... Accounting is exceptionally weak and muddled. No one has yet been able to say how much construction of the Stalingrad Tractor Factory has cost.\" Stalin read this memorandum; his perfunctory notations demonstrate no desire to change the way things were being done. Such an extravagant pumping up of industry needed material resources and workers. Both were taken from the countryside.\n\n###   **THE WAR ON THE PEASANTS**\n\nStalin's costly leap forward was paid for by a sharp reduction in the entire population's standard of living, but the pain inflicted on rural populations was particularly severe. The countryside was treated like a conquered colony to be exploited rather than the country's mainstay. At first no one doubted that in a primarily agrarian country, the peasantry would have to foot the bill for industrialization. The only disagreements had to do with the size of the bill and how payment would be exacted. The Bolsheviks did not like the peasantry\u2014they considered it a dying class\u2014but during the NEP, cognizant of the economic importance of agriculture, the government tried to maintain reasonable relations with the countryside, even if that meant turning a blind eye to such politically unsavory phenomena as the expanded use of private plots. In the late 1920s, however, the government abandoned such liberalism. The increase in capital investment in industry\u2014a policy the entire collective leadership supported\u2014required changing the relationship between the state and the peasantry. In late 1927 and early 1928, the still unified Politburo continued its leftward drift, mixing repression and strong-armed tactics with the economic incentives that had already been put in place to encourage agriculture. How well this mixed approach might have worked will never be known since Stalin took the initiative and turned the leftward drift into a sudden leap. The radical expropriation of grain began to look very much like the confiscations carried out under War Communism.\n\nAs Stalin's opponents had warned, these measures yielded immediate but unsustainable results. The confiscations took away the peasants' economic incentive and led to a drop in production. Each harvest was worse than the one before, leading the grain collectors to resort to increasingly ruthless methods. This vicious cycle of extraordinary measures was fraught with political crises, including mass unrest among peasants that spilled over into the army. Those dealing with these problems on the ground looked to Stalin, who had by then taken a leading position within the Politburo, for a way out of this cycle.\n\nStalin's options were limited, however, by the various ultra-leftist policies he had advocated during his political battles against the rightists. He chose what for him personally was the simplest and safest path, however ruinous it might be for the country. The fight against kulaks and the expropriation of peasant property were taken to their logical conclusion: lands were confiscated and the peasants were transformed into workers in agrarian enterprises managed by the state. The method by which these changes were achieved, labeled \"collectivization,\" involved the large-scale forcible movement of peasants to collective farms\u2014kolkhozes. Nullifying the party's previous decision to make such a transition gradually, in November 1929 Stalin proclaimed that collectivization would be universal and immediate. In December came his call to destroy the kulaks as a class.\n\nIn essence, the victorious _vozhd_ was intentionally provoking a new and deadly wave of revolution in the countryside. By brandishing slogans about the urgent need to crush the kulaks, he gave local stalwarts a free hand. A fevered and violent collectivization effort gripped the countryside even before the new kolkhoz project could receive serious discussion or be embodied in specific directives. In a signature Stalinist move, the party was confronted with a fait accompli. Collectivization supposedly began \"from below,\" leaving no alternative but to support and expand the kolkhoz movement, whatever monstrous forms it might be taking. Many party careerists and radicals, sensing Stalin's strength and decisiveness, responded enthusiastically to his call. Reports of collectivization's successes poured into Moscow.\n\nA finalized plan for collectivization was adopted in early 1930, during a special meeting of Central Committee commissions established to work out the details. Commission members\u2014functionaries fully obedient to Stalin\u2014at first expressed a certain hesitance. While they were in principle ready to support Stalin's push for wholesale collectivization, they urged that it take place over several years. Despite the atmosphere of class-war hysteria in the country, the commissioners tried to ease the fate of millions of kulaks, believing that while they were, of course, enemies of the entire kolkhoz system, they should not be driven into a corner. Repression should be reserved for those who actively resisted. The rest should be accepted into kolkhozes, albeit with certain restrictions. Taking this relatively moderate approach, the commission members made important organizational suggestions\u2014for example, that instead of the total confiscation of property, peasants should be allowed to keep small plots for their own use.\n\nThe proposals made by the Central Committee commissions were of great practical importance and probably the best that could be achieved given the political realities of 1930. They somewhat appeased party extremists while conceding something meaningful to the peasants. As the subsequent history of the Soviet Union has shown, allowing kolkhoz workers to keep their own personal plots saved the system, the peasants, and the entire country. In essence, the arrangement returned peasants to the status of serfs in pre-emancipation Russia, paying feudal homage to the state through their work on collective farms but able to retain some land for personal use. It allowed them to feed themselves\u2014and much of the country\u2014despite the poor performance of the kolkhozes.\n\nStalin preferred a different model: his idea was to turn the peasants into slaves of the state, fully dependent on their state jobs. He favored the total expropriation of peasant property and the incorporation of villages into a state economy where market forces would be allowed no influence. He subjected the commissions' conclusions to harsh criticism and undertook to correct their many errors. By the time he was done, the collectivization plan resembled a military campaign against the traditional peasant way of life. First, Stalin drastically cut the timeline for carrying out collectivization. In several of the most important agricultural regions, the task was to be completed by the fall of 1930, and the tone of his directives made it clear to local functionaries that there was not a moment to lose. Second, he put a quick stop to all talk of integrating kulaks into kolkhozes. Such a step was categorically forbidden. Kulaks and their families were to be exiled to remote areas of the USSR, arrested, placed in camps, or shot. Finally, he put an end to all proposals that kolkhozes coexist with private peasant plots. Provisions for peasants to keep any land whatsoever were adamantly deleted from the draft directives. Ultimately, \"communes\"\u2014agricultural and social utopias, the brainchild of socialist fanatics\u2014were proclaimed to be the ideal form and goal of collectivization. In the Soviet embodiment of this ideal, peasant property became the property of the community, right down to family chickens and personal items.\n\nThese insane and inevitably bloody plans fully reflected Stalin's ideas and intentions. By pushing the pace of collectivization and annihilating the most prosperous and influential segment of the peasantry, Stalin was pursuing several goals at once. Kulak property would provide land and equipment for the collective farms, and the kolkhozes themselves would serve as conduits through which resources could be rapidly and efficiently pumped out of the countryside and into industry. One factor in Stalin's calculations was his belief (shared by many party functionaries) that a moneyless form of socialism based on the exchange of goods was right around the corner. Under forced industrialization, money would cease to be an economic regulator\u2014good riddance, thought the party leftists.\n\nStalin was emboldened to wage this perilous war against the peasantry partly because he believed this population segment, despite being the country's largest, lacked the strength to pose any serious threat to the state. This assumption was only partly borne out. The peasantry really was no match for the totalitarian state, but it did offer serious resistance to collectivization and caused Stalin a good deal of trouble.\n\nIn order to fulfill Stalin's vision of a massive system of kolkhozes, the party leadership mobilized and empowered tens of thousands of people dispatched from cities, as well as local stalwarts. Spurring competition among the regions, party newspapers ( _Pravda_ first and foremost) voiced one demand: as quickly as possible and by whatever means necessary, drive the peasants into kolkhozes. Despite official optimism, the leadership was under no illusions that collectivization could be achieved voluntarily. One of the main instruments propelling it forward was the arrest and exile of kulaks. Fearing the fate of their repressed fellow villagers, peasants gritted their teeth and joined the despised kolkhozes.\n\nBrandishing the threat of \"dekulakization\" and arrest, the authorities quickly achieved stunning collectivization results\u2014at least on paper. While 7.5 percent of the country's peasant households belonged to kolkhozes as of 1 October 1929, by 20 February 1930 that percentage had reached 52.7. Underlying this statistic was a horrific and tragic reality. People sent from the city or mobilized from the local population to carry out collectivization behaved like conquering hordes toward a defeated enemy. Anyone who refused to enter the kolkhoz was arrested and beaten. The plundering of \"dekulakized\" property and the raping of women were standard. Churches were closed and clergy members arrested. \"Fervent\" members of the Komsomol\u2014the Communist Youth League\u2014desecrated churches and pranced about in church vestments.\n\nThis abuse and humiliation drove the usually docile countryside to rebellion. A wave of peasant militancy swept across the country. In all of 1926\u20131927, the authorities identified just 63 incidents of large-scale anti-government unrest in rural areas. In 1929 there were more than 1,300 such incidents, involving 244,000 participants. In January\u2013February 1930 alone, there were approximately 1,500 incidents with 324,000 participants. Stalin, though undoubtedly informed of the growing unrest, did not immediately respond. He was probably confident that the wave of rebellion was simply the inevitable resistance of an \"obsolete class.\" By late February, however, he began to think again. First came a report on 26 February from Kharkov, then the capital of Ukraine, containing news of unrest in the Shepetovka District, near the border with Poland. Crowds of peasants were demanding the reopening of churches and the abolition of the kolkhozes. Party activists were beaten. Other reports reaching Moscow around the same time described similar incidents in Kazakhstan, Voronezh, and even near the capital. Unrest broke out on 21 February in the Pitelinsky area of Riazan District outside Moscow. Peasants removed their livestock and family stores from kolkhozes and returned property to kulaks. Church bells were rung and delegations sent to neighboring villages to rally others to the cause. Peasants armed with stakes tried to prevent the arrests of kulaks. A policeman was killed and eight activists were wounded. OGPU agents responded with firearms, as a result of which three peasants were wounded and six killed, according to official reports.\n\nThe escalating disturbances and the threat that the spring sowing could be disrupted forced the authorities to pull back. On 28 February 1930 the Politburo adopted a resolution calling on Stalin to address collectivization in the press. The famous article \"Dizzy with Success\" was published on 2 March. It contained an optimistic assessment of the \"huge strides\" made in collectivization and proclaimed \"the countryside's radical turn toward socialism.\" At the same time, Stalin condemned individual \"anti-Leninist inclinations\"\u2014the spread of communes; the expropriation of all peasant property for communal use; violations of \"the principle of voluntarism and accounting for local circumstances\"; and the removal of church bells\u2014placing the blame for these excesses at the feet of local officials. On 10 March, secret Central Committee directives were sent out demanding the return of some expropriated property to peasants (poultry, livestock, the lands immediately adjacent to their homes), the correction of \"mistakes\" made during dekulakization, and a halt to the creation of communes and the closing of churches. This was a temporary retreat intended to calm the peasants and allow them to plant their crops.\n\nStalin's article and the Central Committee directives did little to calm tempers. Both failed to provide what was most sought: an explanation of what would be done with the kolkhozes that already existed. The peasants took this problem into their own hands. They forcibly destroyed the collective farms, took away confiscated property and seeds, and restored abolished property lines. The contradictory signals from Moscow only fanned the flames of anti-kolkhoz sentiment and provoked further disturbances by peasants, leaving local activists unsure of how to proceed. March 1930 marked the apex of the war in the countryside: there were more than 6,500 instances of mass unrest, almost half the total for the entire year. In all, approximately 3.4 million peasants took part in acts of rebellion in 1930. Based on that number, it can be presumed that 1.5\u20132 million revolted in March. The higher figure is more likely since the political police had an incentive to underestimate participation in anti-government unrest. Some incidents were well organized; the peasants formed detachments and took over significant territory.\n\nUprisings were especially widespread in Ukraine, the site of almost half of the March disturbances. The authorities were particularly alarmed by rebellions in border regions. As of 16 March, fifteen out of Tulchin District's seventeen administrative areas were in a state of revolt. Representatives of the Soviet government were driven out of fifty villages and replaced with _starostas,_ traditional village elders. Kolkhozes were abolished in most of the district's villages. Rebels beat members of the Communist Party and Komsomol and banished them from villages. In some places, armed rebels engaged in gun battles with OGPU punitive detachments.\n\nFor Moscow, the unrest along Ukraine's western border raised the specter of Polish intervention. On 19 March, Stalin gave Ukrainian State Political Directorate (GPU) chief Vsevolod Balitsky a dressing down, demanding that he stop \"making speeches and act more decisively.\" The wounded Balitsky replied that he was personally traveling to \"the sectors under threat\" and was not just overseeing the fight \"from a train car.\" But he did carry out Stalin's orders. Ordzhonikidze, who traveled to Ukraine for an inspection, wrote that the disorders in border areas were being put down with \"armed forces using machine guns and in some places cannons. There are 100 killed and shot and a few hundred wounded.\"\n\nHaving very little weaponry, the peasants could not withstand well-armed OGPU detachments and mobilized Communists. Their isolated attempts to join forces\u2014by sending messengers and delegations to neighboring villages or sounding the alarm using church bells\u2014were ineffective. The uprisings remained fractured and uncoordinated. Such weaknesses made the task of mobile punitive detachments easier and permitted them to control large areas at once. Mass arrests of the uprisings' ringleaders, kulaks, and the rural intelligentsia, along with the demonstrative brutality of government forces, undermined the resistance. Furthermore, the peasants' behavior was much more civilized than the government's. They generally did not kill their tormentors but merely drove them out of their villages. As a result, the government forces suffered few casualties, partly due to false promises. Another important factor in the diminishing disturbances was the spring sowing. The peasants had little time for rebellion when there were crops to be planted. The fall harvest\u2014on which life itself depended\u2014would not come unless they dropped what they were doing and headed to the fields. By the time the 1930 harvest came, ruthless collectivization had resumed, and the majority of peasants had been forced into kolkhozes.\n\nCollectivization was the cornerstone of Stalin's dictatorship, and all the other features of the Stalinist system can be seen as deriving from it. Wholesale violence against the country's largest class required a large apparatus of oppression, complete with a system of camps and places of exile. Beyond making it clear that terror was the primary instrument of government, collectivization completely and almost instantly severed countless traditional social connections, accelerated the atomization of society, and made ideological manipulation much easier. The rampant and merciless pumping of material and human resources out of the countryside enabled the pursuit of insanely ambitious economic goals.\n\nForced collectivization and ineffective industrialization dealt the country a blow from which it never fully recovered. In 1930\u20131932, hundreds of thousands of \"wreckers\" and \"kulaks\" were shot or imprisoned in camps, and more than 2 million kulaks and their family members were sent into exile. Many of those exiled were just as doomed as those who were shot. Kulak families were sent to live in barracks not suitable for habitation and sometimes simply dropped off in open fields. Terrible living conditions, backbreaking labor, and hunger brought on mass fatalities, especially among children.\n\nThe situation for peasants who were not arrested or exiled was hardly better. The Soviet village, ravaged by collectivization, was seriously degraded. Agricultural production plummeted, and the livestock sector was hit hard. Between 1928 and 1933 the number of horses dropped from 32 million to 17 million, heads of cattle fell from 60 million to 33 million and pigs from 22 million to 10 million. Despite such declining productivity, the state pumped an ever-growing share of its yield out of the countryside. And yet throughout the Soviet period, the kolkhozes were unable to adequately feed the country. Most Soviet citizens survived on meager rations. Many periods were marked by famine. One of the worst was the famine of 1931\u20131933, the predictable result of Stalin's Great Leap.\n\n###   **FAMINE**\n\nWhen the time arrived to announce the results of the First Five-Year Plan, Stalin had to be creative. Exercising the privilege of power, he did not cite a single actual figure but simply proclaimed that the emperor was indeed wearing clothes. The Five-Year Plan, he said, had been fulfilled ahead of schedule  Of course the investment of vast resources and tons of equipment purchased from the West did yield results. Many modern factories were built, and industrial production did increase significantly. But there was no miracle. The unachievable five-year targets were, predictably, not achieved. The actual production figures were not even close: 6.2 million metric tons of cast iron in 1932 instead of the desired 17 million; 21.4 million tons of petroleum instead of 45 million; 48,900 tractors instead of 170,000; 23,900 automobiles instead of 200,000. The state of consumer goods manufacturing was particularly lamentable.\n\nBut the main problem with the First Five-Year Plan was that it established a ruinously inefficient approach to industrialization. Vast sums and resources were poured into undertaking construction that was never completed; into equipment for which no use was ever found, purchased from abroad out of Soviet gold reserves; into wasteful redesigns, the inevitable result of excessive haste; and into goods so poorly produced as to be unusable. The task of arriving at an approximation of these losses rests with historians. Much better known are the statistics from another tragic result of the Great Leap\u2014the toll taken by the Great Famine.\n\nThis famine, which reached its peak over the winter of 1932\u20131933, took the lives of between 5 million and 7 million people. Millions more were permanently disabled. In a time of peace and relatively normal weather, agriculturally rich regions were ruined and desolated. Although the famine was a complex phenomenon, posterity has every right to call it the Stalin Famine. The Stalinist policy of the Great Leap was its primary cause; moreover, it was Stalin's decisions in 1932 and 1933 that, instead of easing the tragedy, made it worse.\n\nThe famine was the inevitable result of industrialization and collectivization. From a productivity standpoint, the kolkhozes were a poor substitute for the destroyed farms of those who had been branded \"kulaks.\" The only advantage of the kolkhozes was that they gave the state a convenient means of channeling resources out of the countryside. The exceptional exploitation of peasants had two effects: agricultural workers were physically weakened by hunger, and they were deprived of any incentive to work, leading to despondency and apathy. They knew in advance that everything they grew would be taken by the state, dooming them, at best, to semi-starvation. Several years of this policy led to a gradual decline in output. In 1932 the crops did not grow well and were also poorly harvested.\n\nThe state's interests and those of the peasants were diametrically opposed. The state was extremely aggressive in taking from the countryside as many resources as possible. The peasants, like famine victims all over the world, used \"the weapons of the weak.\" They sabotaged the fulfillment of their obligations to the state and tried to stash away stores to feed themselves. Stalin was well aware of the hostility of the forcibly collectivized countryside, but he placed the blame fully on the peasants' shoulders. They had declared war, he proclaimed, against the Soviet government.\n\nThe looming crisis was obvious to everyone, including Stalin, long before the famine entered its most critical phase. There were obvious steps that, if they did not prevent the famine altogether, could at least have diminished its impact. The first would have been to establish set norms for grain deliveries to the state\u2014in other words, a move from a system of confiscation to a system of taxes. This step would have given the peasants an incentive to boost production. Stalin, however, rejected this approach. He preferred to take as much as possible from the countryside without any constraints. Another step to alleviate the famine might have been to reduce grain exports or even buy grain abroad. Such purchases were made on a limited basis during the spring of 1932, so they were in principle possible. But Stalin refused to make further purchases. Any concessions that hinted at the misguidedness of the Great Leap were contrary to his nature and politically dangerous to his dictatorship. To alleviate the pressure on the peasants there would have to be a reduction in the pace of industrial growth. Reluctantly, Stalin did agree to such a reduction in 1933, but his slowness to take action cost millions of lives.\n\nBy the autumn of 1932, critical delays, stubbornness, and cruelty had led Stalin himself into a dead end. No good options remained. The harvest produced by the devastated countryside in 1932 was even worse than the poor harvest of 1931. Meanwhile, industrialization continued apace, and the Soviet Union's foreign debt for purchases of equipment and raw materials reached new heights. Given these circumstances, there was only a little room to maneuver. The government could mobilize all available resources, or dip into reserves, or appeal for international aid, as the Bolsheviks had done during the famine of 1921\u20131922. These measures came with economic and political costs, but they were possible. Stalin probably did not even consider them. Instead, the state intensified pressure on the countryside.\n\nDocuments discovered in recent years paint a horrific picture. All food supplies were taken away from the starving peasants\u2014not only grain, but also vegetables, meat, and dairy products. Teams of marauders, made up of local officials and activists from the cities, hunted down hidden supplies\u2014so-called _yamas_ (holes in the ground), where peasants, in accordance with age-old tradition, kept grain as a sort of insurance against famine. Hungry peasants were tortured to reveal these _yamas_ and other food stores, their families' only safeguard against death. They were beaten, forced out into sub-freezing temperatures without clothing, arrested, or exiled to Siberia. Attempts by peasants dying of hunger to flee to better-off regions were ruthlessly suppressed. Refugees were forced to return to their villages, doomed to slowly perish, or be arrested. By mid-1933 some 2.5 million people were in labor camps, prisons, or exile. Many of them fared better than those who starved to death \"in freedom.\"\n\nAt its peak in late 1932 and early 1933, the famine afflicted an area populated by more than 70 million people: Ukraine, the North Caucasus, Kazakhstan, and some Russian provinces. This does not mean that the remaining Soviet population of 160 million was eating normally. Many in regions not officially in a state of famine lived on the edge of starvation. The entire country was hit by epidemics, primarily typhus. Millions suffered serious illnesses, were left disabled, or died several years after the famine from the damage it had inflicted on their bodies. And no statistics can measure the moral degradation it caused. Secret OGPU and party summaries _(svodkas),_ especially during the early months of 1933, are filled with accounts of widespread cannibalism. Mothers murdered their children, and deranged activists robbed and tormented the population.\n\nWhile the entire country suffered from famine and mass repression, Ukraine and the North Caucasus were the most affected. It was in these two important regions of the USSR where the policy of punishing grain requisitions and terror were most brutally applied. Two interrelated reasons explain Stalin's focus on these areas. The first could be described as economic. Ukraine and the North Caucasus supplied as much as half of all grain collected by the state. But in 1932\u20131933 they turned over 40 percent less than the previous year. While this decline was partially compensated by Russian grain-producing areas, which despite going hungry had significantly overfulfilled their plans, they could not completely make up the shortfall. In 1932 the state collected almost 20 percent less grain than in 1931. These figures partially explain the demands Stalin placed on Ukraine and the North Caucasus. He wanted \"his\" grain and was infuriated that they were not providing it.\n\nSecond, Stalin saw the crisis of 1932 as the continuation of the war against the peasantry and as a means of consolidating the results of collectivization, and he had a point. In a letter to the Soviet writer Mikhail Sholokhov on 6 May 1933, he wrote: \"The esteemed grain growers were in essence waging a 'quiet' war against Soviet power. A war by starvation.\" He undoubtedly considered the peasantry of Ukraine and the North Caucasus to be at the forefront of this peasant army battling the Soviet government. These regions had always been hotbeds of anti-Soviet sentiment, and Ukraine had been at the forefront of the anti-kolkhoz movement in 1930. Repeated incidents of unrest flared up in both Ukraine and the North Caucasus in 1931\u20131932. A further cause for concern was Ukraine's border with Poland. Stalin feared that Poland, in its hostility toward the USSR, could exploit the Ukrainian crisis. Overall, as Hiroaki Kuromiya points out, Stalin was suspicious of all peasants, but \"Ukrainian peasants were doubly suspect both for being peasants and for being Ukrainian.\"\n\nBy proclaiming grain collection to be a war, Stalin was untying his own hands and the hands of those carrying out his orders. The ideological basis for this war was the Stalinist myth that \"food difficulties\" resulted from acts of sabotage by \"enemies\" and \"kulaks.\" Any suggestion of a link between the crisis and government policy was categorically rejected. By blaming all food shortages on \"enemies\" and on the peasants themselves while also promoting the idea that the scale of the famine was being maliciously exaggerated, Stalin relieved himself and the central government of any obligation to help the hungry. A statement by the general secretary in February 1933 at a congress of kolkhoz shock workers shows the depth of his cynicism: \"One of our achievements is that the vast masses of the poor peasants, who formerly lived in semi-starvation, have now, in the collective farms, become middle peasants, have attained material security.... It is an achievement such as has never been known in the world before, such as no other state in the world has yet made.\" This statement came at a time when thousands were dying every day.\n\nStalin could not deceive everyone. In May 1933, as the famine raged, he met with Colonel Raymond Robins, an American progressive who sympathized with Soviet Russia. Robins was famous for his meetings with Lenin as a member of the Red Cross mission to Russia in 1917\u201318. Counting on Robins's help in strengthening relations with the United States, Stalin was friendly toward the American and adopted a tone of sincerity and candor. He knew that Robins was well informed about Soviet realities and did not dare deny that his country was afflicted by famine. In response to a direct question about the poor harvest of 1932, Stalin, after some lengthy equivocation, did admit that \"some peasants are currently starving.\" The reasons he gave for the famine exhibited impressive inventiveness and imagination. Parasitically inclined peasants, he argued, who had joined the kolkhozes late and were not earning anything through them, were the ones starving. Independent peasant farmers who did not work on their own plots but lived by stealing grain from kolkhozes were also \"going terribly hungry.\" They supposedly were left with nothing to eat after the introduction of harsh penalties for theft. To top off these lies, Stalin assured Robins that the state was helping the victims of famine, even though the kolkhoz members themselves were against such aid: \"The _kolkhozniks_ are really mad at us\u2014you shouldn't help idlers, let them die. That's how they are.\" Robins was probably not convinced, but as a true diplomat, he did not press Stalin.\n\nWhile it is difficult to know how much Stalin believed of his own explanations, his conversations with Robins tell us something about his thinking. First, he apparently knew about the famine and recognized it as an actual fact, not a fiction made up by \"enemies.\" Second, he does not appear to put much store in his own accounts of underhanded plotting by enemies and wreckers. He does not mention this \"problem\" once in his talks with Robins, which may suggest an awareness of the true causes of the famine and its ties to collectivization. It is doubtful, however, that he ever admitted any mistakes, even to his closest associates. Only mythic explanations of reality served his purpose. Claims about enemies, sabotage by peasants, or mistakes by local bosses permitted him to deflect guilt and doom millions without wavering.\n\nStalin's comments do not reveal exactly what he knew about the famine. What did he have in mind when he admitted to Robins that some peasants were \"going terribly hungry\"? Did he see in his mind's eye images of walking skeletons; desperate people foraging through buried animal remains; mothers, mad from hunger, murdering their own children? Probably not. He only encountered ordinary people at orchestrated events, and Moscow, which he regularly saw from his car window, was the relatively well-fed fa\u00e7ade of Soviet power. OGPU reports that have recently come to light offer a detailed description of the famine, of cannibalism, and spreading anti-Soviet sentiments among the populace. But we do not know whether Stalin read these reports. One compelling document we do know he read is Mikhail Sholokhov's letter of 4 April 1933. In horrific detail, the appalled writer described what was taking place near his home in Veshenskaya, in the Northern Caucasus:\n\nI saw things that I will remember until I die.... During the night\u2014with a fierce wind, with freezing temperatures, when even the dogs hide from the cold\u2014families thrown out of their homes [for failure to fulfill their grain quotas] set up bonfires in the lanes and sat near the flames. They wrapped the children in rags and placed them on ground that had been thawed by the fire. The unceasing crying of children filled the lanes.... At the Bazkovsky kolkhoz they expelled a woman with a baby. She spent the night wandering through the village and asking that she and the baby be allowed inside to get warm. No one let her in [there were severe penalties for aiding \"saboteurs\"]. By morning the child had frozen to death in the mother's arms.\n\nSholokhov's letter describes how suspected hoarders were coerced into handing over their grain: mass beatings, the staging of mock executions, branding with hot irons, and hanging by the neck to induce partial asphyxiation during interrogations, among other methods. The writer did not attempt to whitewash the fact that the criminal abuses being perpetrated in the Veshensky District were part of a purposeful campaign by the regional authorities\u2014not \"deviations\" by local zealots. But for obvious reasons, he did not press this point.\n\nStalin took the news in stride. He ordered that the Veshensky District be given additional grain assistance and that an investigation be conducted into the abuses Sholokhov described. Overall, however, he supported the local authorities. In a response to Sholokhov he accused the writer of taking a one-sided view and of covering his eyes to sabotage by peasants. The local leadership, some of whom were at first condemned to harsh punishment for abuses, were ultimately acquitted. On Stalin's orders they were simply removed from their posts and given reprimands. They were not even expelled from the party. Stalin had no intention of retreating from his war against the peasants, however many innocent lives were taken in the process.\n\n###   **THE \"MODERATE\"**\n\nThe victory over the peasants had all the hallmarks of defeat. Despite the campaign's extreme ruthlessness, the grain procurement plan was not fulfilled. And the 20 percent decline in grain collections between the meager harvest of 1931 and the disastrous one of 1932, bad as this was, paled in comparison to the decimation of the livestock sector. If ruthless measures could not squeeze food out of the countryside, what should be done next? Continuing a policy of confiscation\u2014 _prodrazverstka_ \u2014would only kill off the population. Furthermore, the policy of forced industrialization was proving untenable. The mad surge of capital investment in heavy industry had reached its limit. Trotsky's call to make 1933 \"a year of capital repair\" resonated with Stalin's opponents, who called on him to reduce the pace of growth.\n\nEven the relentless terror machine was beginning to falter. By 1933 the large network of camps and prisons could not handle the growing flood of arrestees. The government took urgent steps to create remote settlements capable of accommodating 2 million internal deportees, but this program failed because of a lack of resources. In the end, only about 270,000 people were sent into internal exile. The seemingly limitless capacity for destroying and isolating \"enemies\" apparently had its limits. And while the execution, arrest, and deportation of vast numbers helped the government maintain control, even Stalin could see that these tactics were doing as much to undermine the smooth running of the system as to bolster it.\n\nAll this dysfunction weakened the USSR at a time of escalating international tension. One of the first signs of looming war was Japan's occupation of Manchuria in late 1931. \"The Japanese are certainly (certainly!) preparing for war against the USSR, and we have to be ready (we must!) for anything,\" Stalin wrote to Ordzhonikidze in June 1932. An urgent buildup of military forces was begun in the Soviet Far East. But trouble was also brewing in Europe. In January 1933, while the Soviet Union was in the throes of famine, the Nazis came to power in Germany. The Bolsheviks' European strategy, which was centered on building relations with Weimar Germany, had to be immediately revamped. Faced with growing threats from east and west, Stalin was forced to seek alliances with Western democracies. On 19 December 1933 the Politburo adopted a top secret resolution concerning the USSR's possible entry into the League of Nations and conclusion of a regional mutual defense pact against Germany with a number of Western countries, including France and Poland. Stalin understood that this new foreign policy would not be possible unless he sent clear signals that the Stalinist USSR was a \"normal\" country and not simply a convenient enemy of fascism. The Soviet regime would need to improve its reputation. Soviet leaders did not have to exchange their military service jackets for tailcoats, but they at least needed to button up.\n\nStalin had led the Bolsheviks into a dead end. The resources that had made the First Five-Year Plan possible had been used up. Too late for countless victims of his policies, he agreed to measures that could and should have been taken years before.\n\nFirst among them were some minor but critical concessions to the peasantry. Although the Stalinist state continued to rely primarily on compulsion in the countryside, there were important changes. Essentially recognizing the tremendous harm done by limitless confiscations, in January 1933 the government introduced set quotas for grain deliveries (a food tax or _prodnalog,_ in official Soviet parlance). The peasants were promised that predictable quotas would be set for the amount of produce to be taken and that they would have the right to sell the surplus. The resolution mandating this change was never put into practice, but it was a milestone in the transition from the Stalin-era War Communism of the First Five-Year Plan to the Stalin-era NEP of the Second. It was within the framework of this transition that other, more practical and effective, decisions were adopted.\n\nStalin grudgingly allowed peasants to have small private plots that they were allowed to cultivate for their own benefit, a concession of great importance to the survival of the countryside and the country overall. At the first congress of \"kolkhoznik-udarniks\" (collective farm shock workers) in February 1933, he promised that the state would help each kolkhoz household acquire a cow over the coming two years. Laws guaranteeing ownership of farm plots were gradually put into place. This expansion of private agriculture was critically important, paving the way toward a new compromise between the state and the peasants. The peasants, who earned almost nothing working on collective farms, would now be able to make ends meet by farming their private plots. Despite being subject to exorbitant taxes, these plots were exceptionally productive. Although private agriculture took up a miniscule amount of land compared with the kolkhozes, official statistics from 1937 show that it provided 38 percent of the country's vegetables and potatoes and 68 percent of its meat and dairy products. When yet another famine hit after the poor harvest of 1936, it was private agriculture that helped the country survive, once again underscoring how flawed the original collectivization plan had been. If the mad rush toward total collectivization had been adjusted to allow private plots, peasants (and Soviet agriculture) would not have been utterly ruined overnight.\n\nAlso long overdue and unavoidable were changes to industrial policy. The first limited signs that the state was being compelled to pull back from the destructive policy of forced industrialization and repression against those running the Soviet economy came in 1931\u20131932. During the Central Committee plenum of January 1933, Stalin provided a new set of slogans to go with the new policies. While proclaiming new class battles ahead, he nevertheless promised that the pace of industrial construction during the Second Five-Year Plan would be significantly reduced. Unlike many other slogans, this one did not prove empty. Alongside reduced growth for capital investment in industry, in 1934\u20131936 various experiments and reforms were introduced aimed at enhancing enterprises' economic independence and reviving financial incentives for labor. By this time, the idea of an economy based on the exchange of goods had been definitively rejected as \"leftist,\" \"money\" and \"commerce\" were no longer dirty words, and the need to strengthen the ruble was a hot topic. That Stalin was reorienting the economic signposts became apparent in his remarks during a discussion on abolishing the ration system at the November 1934 plenum:\n\nWhy are we abolishing the ration system? First and foremost it is because we want to strengthen the cash economy.... The cash economy is one of the few bourgeois economic apparatuses that we, socialists, must make full use of.... It is very flexible; we need it.... To expand commercial exchange, to expand Soviet commerce, to strengthen the cash economy\u2014these are the main reasons we are undertaking this reform.... Money will start to circulate, money will come into fashion, which hasn't been the case for some time; the cash economy will be strengthened.\n\nUnderlying this liberalization was a recognition of the importance of personal interests and material incentives. The sermons on asceticism, calls for sacrifice, and hostility toward high salaries that had characterized the First Five-Year Plan were replaced by a focus on \"culture and a prosperous life.\" Instead of the mythic images of a future of abundant socialism that had been promoted with the First Five-Year Plan, the Soviet people, especially the urban population, were now offered the prospect of tangible creature comforts: a private room, furniture, clothing, a tolerable diet, and expanded leisure. The possibility of an improved standard of living was being deliberately used to motivate the workforce.\n\nThe improved quality of life after the successful harvest of 1933 was, of course, remarkable only in contrast with the previous years' mass famine. The full store shelves seen in major cities came as some rural areas continued to starve. But compared to 1932\u20131933, these pockets of hunger were \"nothing,\" just as the ongoing arrests and deportations could be seen as \"nothing\" compared with previous years. For a while, state terror continued at a low and predictable pace. The pullback began with a special directive Stalin signed in May 1933 calling for the release of some of those arrested for \"minor crimes\" from overcrowded prisons and prohibiting the secret police from conducting mass arrests and deportations.\n\nStalin continued to demonstrate adherence to \"socialist legality.\" It was on his instigation that in February 1934 the Politburo voted to abolish the odious OGPU and place the political police under the newly formed People's Commissariat for Internal Affairs (NKVD), blending it with the more innocuous branches of law enforcement and public safety. On paper, people's rights in the regular judicial system were expanded, and the power of extrajudicial bodies\u2014the instruments of mass terror\u2014was reduced. The handling of certain legal matters in which Stalin clearly had a hand was especially significant. Within the Soviet political system, it was these signals from the _vozhd_ that showed the way forward for government officials.\n\nOne of the first such signals had to do with the conviction of Aleksei Seliavkin. During the witch hunt of the early 1930s Seliavkin, a senior heavy-industry official and decorated Civil War veteran, had been sentenced to ten years for selling classified military documents. In a petition sent from labor camp, Seliavkin stated that his interrogators had dictated a false confession and forced him to sign it under threat of being shot. This petition came at an opportune time. Stalin (without whose consent Seliavkin would never have been arrested in the first place) now signaled leniency. Not surprisingly, an investigation showed that the secret police had fabricated the evidence. On 5 June 1934 the Politburo annulled Seliavkin's sentence and demanded \"attention to serious deficiencies in the handling of the case by OGPU investigators.\"\n\nThe annulment of Seliavkin's sentence was just the start. In September 1934 Stalin ordered the Politburo to establish a commission to investigate several other cases that had been brought against \"wreckers\" and \"spies.\" He called on the commission to free the innocent, purge the OGPU of perpetrators of certain \"investigative techniques,\" and punish them \"without favoritism.\" \"In my opinion,\" he wrote, \"this is a serious matter and it has to be pursued to the end.\" Surviving documents show that this commission actually took its work seriously, assembling evidence of secret police abuses. There was no shortage of cases.\n\nThen came the murder of Leningrad party boss Sergei Kirov. The commission never completed its task.\n\nHad it not been for Kirov's murder, would there have been a serious effort to put an end to secret police abuses? The evidence suggests otherwise. Although there were fewer arrests in 1934, the victims of repression still numbered in the hundreds of thousands. Stalin himself sent contradictory signals. In September 1934, at the height of the campaign for \"socialist legality,\" the Politburo sanctioned the execution of a group of employees of the Stalin Metallurgical Factory in Siberia who were accused of spying for Japan. It was Stalin who instigated the roundup, writing: \"Everyone caught spying for Japan should be shot.\" There were other examples. The foundation of Stalin's system of oppression was never dismantled. The \"moderation\" of 1934 was nothing more than a temporary adjustment in the level of terror.\n\nAlthough this moderation was inconsistent and limited, it did imply recognition that the Great Leap policy had been misguided. In theory, this forced change-of-course might have cast an unfavorable light on Stalin and prompted dissatisfaction with him. Such apparently logical inferences have inspired historians to posit the existence of plots and intrigues against Stalin among the party ranks. One focus of these theories is Sergei Kirov, a close Stalin associate and the Leningrad party boss. The confusion surrounding the circumstances of Kirov's murder and the crackdown that followed it have led some to conclude that Kirov was actually behind the new political moderation, making him someone an anti-Stalin movement might rally around. This speculation, of which there has been a great deal, is based solely on the memoirs of people with only a second- or third-hand knowledge of the central facts in the matter.\n\nSetting aside the many discrepancies in these \"eyewitness\" accounts, we are left with the following picture. During the Seventeenth Party Congress a number of senior party officials (various names are mentioned) discussed the possibility of removing Stalin as general secretary and replacing him with Kirov. Kirov rejected this proposal, but Stalin got wind of the plans. According to some accounts, Kirov himself told Stalin what others were plotting. During Central Committee elections at the congress, many delegates supposedly voted against Stalin. On learning about this, Stalin allegedly ordered the removal of any ballots where his name was crossed out. Ten months later, he organized Kirov's murder in order to remove a dangerous rival. These contradictory accounts have never inspired much confidence, and now that the archives have been opened, they appear even less convincing. A number of painstaking searches have failed to turn up even circumstantial evidence of a plot against Stalin.\n\nThe details of Kirov's party career offer scant evidence that he enjoyed an independent political position and much to suggest that he did not. Like other Politburo members in the 1930s, Kirov was a Stalin man. His initiatives were confined to the needs of Leningrad\u2014requests for such items as new capital investment and resources or for the opening of new stores. He rarely came to Moscow to attend Politburo meetings or participated in voting on Politburo resolutions or the polling of its members. Not only was Kirov not a reformer, but the available documents do not even show that he took any serious part in developing or implementing high-level political decisions. He was Stalin's faithful comrade-in-arms and remained so to the end. Within the party he was never regarded as a political leader on a par with Stalin, and he did not promote any political programs that differed from Stalin's. His death had an incomparably greater effect on the country's development than his life. As often happens, it was his death that turned Kirov into a legend.\n\n###   **THE MURDER**\n\nKirov was killed on 1 December 1934 in Leningrad's Bolshevik headquarters in the Smolny Institute, a neoclassical building that formerly housed Russia's first educational institution for girls. In the seven decades between the 1918 attempt on Lenin's life and the end of the Soviet regime, this was the only successful assassination attempt against a senior Soviet official. But that is not what has drawn the attention of historians. The shots fired in the Smolny Institute were followed by a new intensification of repression that is often treated as a step toward the Great Terror of 1937\u20131938 and the ultimate consolidation of Stalin's dictatorship. The obvious political benefit that Stalin derived from Kirov's murder has led historians to suspect he had a hand in bringing it about. Such suspicions even became part of official propaganda during Khrushchev's de-Stalinization effort and Gorbachev's perestroika. Although it is rarely helpful when politicians involve themselves in the interpretation of past events, this case may be an exception. The numerous commissions established by Khrushchev and Gorbachev compiled and studied a great body of evidence, which gives us a rather full picture of what occurred in Leningrad on 1 December 1934 and during the murder's aftermath.\n\nOn the evening of 1 December, a meeting of party stalwarts was scheduled to take place in Leningrad's Tauride Palace. Kirov was to give a speech on the outcome of the Central Committee plenum that had taken place in Moscow the previous day. The topic at hand was the upcoming abolition of the ration system, a change that would affect virtually the entire population of the country. An announcement of the meeting had already been published in newspapers, and Kirov spent the entire day preparing his speech. At approximately four o'clock he summoned a car and headed to his Smolny office. Using the building's main entrance, he climbed to the third floor, where his office and the offices of the oblast committee were located. He walked down the third floor's main corridor to a smaller corridor to the left that led to his office. It was the job of his bodyguard, Mikhail Borisov, to keep watch over the party boss inside the building. Borisov followed Kirov at a slight distance. When Kirov turned into the small corridor leading to his office, Borisov continued down the main corridor. Kirov remained out of his sight for some moments.\n\nLeonid Nikolaev, a party member and former employee of the Leningrad Oblast Committee, was preparing to shoot Kirov that evening at the Tauride Palace. To gain entry he needed an invitation card, and he had come to Smolny to get one, counting on help from acquaintances who worked there. Because he had a party membership card, he had no trouble entering the building. While wandering its corridors, Nikolaev unexpectedly saw Kirov walking toward him. Nikolaev turned away and let Kirov pass. Since there was nobody between him and his target, Nikolaev decided to carry out his plan immediately. He followed Kirov into the corridor leading to his office, ran up to him, and shot him in the back of the head. Nikolaev then attempted to shoot himself in the temple but was prevented from doing so. Borisov and several Smolny staff members had come running at the sound of gunfire and saw Kirov lying bloody on the floor. It was all over in an instant.\n\nDoctors and the heads of the Leningrad NKVD were summoned to Smolny. Stalin was telephoned at his Kremlin office. As soon as he was told of Kirov's death, the general secretary convened a series of meetings. Early the following morning, on 2 December, he arrived in Leningrad on a special train. That same day he joined other members of the team from Moscow in interrogating Nikolaev. Stalin could hardly have failed to notice that Nikolaev was not a typical ideologically motivated terrorist.\n\nIn December 1934 Leonid Vasilyevich Nikolaev was 30 years old. He had been born into a working-class family in St. Petersburg and lost his father at an early age. His family struggled with poverty, and rickets prevented Leonid from walking until the age of eleven. The record of his recruitment for military training provides a detailed description of his physical features at age twenty: long arms that extended to the knees, an elongated torso, and a height of approximately five feet. Nikolaev was often ill and had a quarrelsome disposition, but his early professional life was nevertheless fairly successful. Since his social origins were of the \"correct\" sort, he was able to get a job working for the Komsomol and join the party, steps that opened the door to other advantageous positions, including working for the Leningrad Oblast Committee in the same building where he later killed Kirov. But being prone to conflict, he could not hold any job for long. He was unemployed during the months leading up to the murder and spent his time filing grievances with various institutions and plotting revenge. The numerous diaries, letters, and other writings that were confiscated after his arrest show him to have been mentally unstable. His letters of grievance recounted various perceived injustices, demanded a job and a resort voucher, adopted a threatening tone, and assumed the pose of a hero whose name would go down in history alongside the great revolutionaries of the past.\n\nAnother factor contributing to Nikolaev's state of mind was his relationship with his wife, Milda Draule, whom he met when they both worked for the Komsomol. Draule, age thirty-three in 1934, appears to have been an attractive woman whose career, unlike Nikolaev's, was advancing successfully. In 1930, long-standing connections led to a secretarial job at the Leningrad Oblast Committee offices. There were rumors before Kirov's death that Draule was having an affair with him, and speculation about an affair has persisted ever since. There is reason to believe that Kirov's childless marriage was an unhappy one. His wife, four years his senior, was often ill and spent months at a time away from home in sanatoriums or rest homes. Although there is no hard evidence to prove that Kirov and Draule were intimate, the possibility has to be recognized. Even if Nikolaev did not believe the rumors, one can only assume that they fostered animosity toward Kirov.\n\nSuch was the man brought before Stalin at Smolny on 2 December. The _vozhd_ was undoubtedly briefed on Nikolaev's less than sterling work and party history and may even have been discreetly informed of the rumors about Kirov and Draule. Nikolaev's appearance tended to support the idea that the shooting was the act of an embittered loner of questionable mental competence. He was brought before the Moscow commission shortly after a severe hysterical fit brought on by the murder and his own failed suicide attempt. Molotov, who was with Stalin, remembered Nikolaev as follows: \"Mousey.... Short and skinny.... I think something must have made him angry... and he looked like something had offended him.\"\n\nWhat Molotov remembers is probably what Stalin saw too, but treating Nikolaev as an unstable loner did not suit his purposes. Even before he left for Leningrad, an official account of Kirov's murder had been crafted. The following day, Soviet newspapers reported that Kirov had died \"at the treacherous hand of an enemy of the working class.\" This interpretation was entirely predictable. At who else's hand could a Politburo member perish? Something as mundane as murder by a jealous husband was unthinkable. Only a devious enemy of the people would fit the part. Any other interpretation cast not only Kirov but also the entire regime in an unfavorable light, making it look incapable of protecting its leaders from deranged loners. The agreed-upon narrative fit Stalin's extreme suspiciousness and hunger for power.\n\nBefore returning to Moscow on the evening of 3 December, Stalin ordered that a case be fabricated to show that Nikolaev belonged to an organization comprised of former oppositionists, followers of Zinoviev, who had wielded power in Leningrad in the 1920s as head of city government. This task was assigned to Moscow-based NKVD investigators and Stalin's political commissars\u2014Nikolai Yezhov and Aleksandr Kosarev, who remained behind in Leningrad. Two years later, at the February\u2013March 1937 plenum, Yezhov said the following about the task assigned him: \"Com. Stalin... called me and Kosarev and said, 'Look for murderers among the Zinovievites.'\" This assignment would, of course, require creativity and law breaking. Not only had Nikolaev never belonged to any oppositionist group, but the NKVD had also never turned up the slightest evidence of oppositionist sympathies. The only way to link Nikolaev and the Zinovievites was to manufacture evidence, so under Stalin's watchful eye, this is what the chekists did. During the investigation, Stalin was sent approximately 260 arrestee interrogation protocols and many reports. He met with senior members of the NKVD, the procuracy, and the Supreme Court's military collegium to discuss the investigation and trial. The historical record shows that he personally orchestrated the court sessions and assembled the groups of defendants in the Kirov case.\n\nIn accordance with Stalin's orders, a series of trials was held in late 1934 and early 1935. Dozens of former oppositionists, whom investigators claimed had links to Nikolaev, were sentenced to be shot or imprisoned. Political and moral responsibility for Kirov's murder was placed on the shoulders of the former opposition leaders Zinoviev and Kamenev, who were also put on trial. The evidence on which they were convicted was blatantly fabricated. Stalin was settling scores with his old political rivals and charging them with crimes they had not committed.\n\nStalin's exploitation of Kirov's murder has prompted a great deal of suspicion over the years. Many have accused Stalin of organizing the shooting itself. The first serious attempts to look into such accusations were undertaken during the Khrushchev thaw and continued with small interruptions into the early 1990s. These investigations have turned up some circumstantial evidence of Stalin's involvement but no proof. At this point, it is unlikely any will be found.\n\nUntil the early 1990s, most theories about a plot by Stalin against Kirov adhered to the same basic storyline. Displeased by Kirov's growing popularity, Stalin decided to deal with the situation and then use the murder as a pretext for mass repression. With this goal, the general secretary either directly or implicitly assigned Genrikh Yagoda, then NKVD chief, to handle the matter. Yagoda sent a trusted prot\u00e9g\u00e9, Ivan Zaporozhets, to serve as deputy in the Leningrad branch of the NKVD, where he could lay the groundwork for this supposed \"act of terrorism.\" Nikolaev was chosen to carry out the deed and was armed and taken under Zaporozhets's wing. When he was arrested by NKVD agents after trying to carry out the assassination before 1 December, Zaporozhets arranged to have him released. After Kirov's murder, those involved in the conspiracy killed the bodyguard, Mikhail Borisov, because he knew too much. On 2 December he was killed in a staged accident while being taken to Stalin by truck for questioning. Such is the basic narrative proposed by those suspecting Stalin of complicity in Kirov's death.\n\nThis narrative does not stand up to careful examination. First of all, it is unclear why Stalin would enter into a conspiracy so fraught with risk, given that Kirov was a faithful client rather than a political rival. The evidence is also not convincing. To start with, the argument that Nikolaev would not have been able to get a firearm without help is flawed. The restrictions on gun ownership that were introduced later in the decade (partly in response to the Kirov murder) did not yet exist. Nikolaev acquired his revolver in 1918, when the country was awash in firearms, and had legal possession of it for sixteen years. Such ownership was nothing out of the ordinary, especially for a party member.\n\nAs for Nikolaev's multiple detentions by the NKVD before 1 December and his \"miraculous\" release, records show only one such incident, not the several that some authors claim. On 15 October 1934, Nikolaev was detained by NKVD agents near Kirov's home but released shortly thereafter after his documents were checked. According to Nikolaev's own testimony, on that day he ran into Kirov and several companions and followed them to Kirov's house but did not work up the nerve to speak to Kirov. \"Back then I was not thinking about committing murder,\" Nikolaev stated during his 2 December interrogation. After the murder, this incident, which was recorded in the NKVD incident log, was specially investigated. The NKVD agents who freed Nikolaev had a simple and convincing explanation: he had produced his party membership card and also an old identification card showing that he had worked at Smolny. His desire to approach Kirov to ask about the possibility of a job was \"natural and did not arouse suspicion.\"\n\nA cornerstone of the theories that Kirov's murder was part of a plot is the death of the bodyguard, Borisov. During the second half of 1933, Kirov's security team had grown to fifteen people, each with his own job. Borisov was charged with meeting Kirov at the entrance to Smolny, accompanying him to his office, waiting in the reception area while Kirov worked, and accompanying him out of the building when he left. One other member of the team\u2014an NKVD agent like Borisov\u2014was N. M. Dureiko, who watched over Kirov as he moved around the third floor of Smolny. When the shot was fired, Dureiko was walking toward Kirov in the small corridor leading to his office. It could be argued that Dureiko was just as culpable in not preventing the murder as Borisov. Nevertheless, those promoting the idea of a plot have never taken an interest in Dureiko. If the plotters felt they had to do away with Borisov, why did they leave Dureiko alive?\n\nMuch importance has been assigned to the fact that Borisov did not follow Kirov when he turned toward his office, thus allowing Nikolaev to carry out his assassination, but Borisov's behavior is not as sinister as the conspiracy theorists have made it out to be. If we put ourselves in the shoes of this fifty-three-year-old bodyguard who had been protecting Kirov since he had arrived in Leningrad in 1926, his behavior seems entirely normal. All those years, day in and day out, he had to stick close to a man who, by many accounts, was not easy to guard. Kirov was reportedly annoyed when his bodyguards remained too close, and at times he even escaped from them. With his long experience working for Kirov, Borisov was surely sensitive to his boss's moods and tried not to irritate him. On 1 December in Smolny he kept his usual distance. Furthermore, as he walked down the corridor, Kirov stopped several times to have short conversations. Discretion demanded that Borisov step aside at such times. There was nothing unusual about this behavior.\n\nOn 2 December, the Moscow commission decided to question Borisov. He was escorted to Smolny by two other NKVD agents. Because no cars were available (not surprising given how many officials had suddenly descended on Leningrad from Moscow), Borisov was brought in a truck that turned out to be in disrepair. The driver lost control of the vehicle and crashed into a building. Borisov's head hit a wall of the building, and he died in the hospital without ever regaining consciousness. This is the sequence of events established by investigations and expert assessments conducted at various times, and there is no evidence to the contrary. Proponents of a plot reject the idea that the vehicle crashed by accident and claim that Borisov was murdered.\n\nThe idea that Stalin was behind Kirov's murder has all the hallmarks of a conspiracy theory. Such theories tend to rest on the idea that if an event benefits some sinister person, he must have brought it about. They tend to deny the possibility of random occurrences and ignore the fact that chance events happen all the time. The idea that Stalin conspired to kill Kirov has received far too much attention. Even if he did have a hand in Kirov's death, this possibility hardly changes our understanding of him or his era. In the annals of the dictator's crimes, Kirov's murder would have been one of the least heinous.\n\n###   **REHEARSAL FOR THE GREAT TERROR**\n\nAccording to Stalin's relative Maria Svanidze, he was extremely upset by Kirov's murder. \"He became pale and haggard, and there was a hidden suffering in his eyes.\" \"I feel so alone,\" he reportedly confided to his brother-in-law, Pavel Alliluev. There is no reason to doubt these accounts. Tyrants often combine exceptional cruelty and complete indifference to the deaths of millions with extreme sentimentality toward those near to them. In Stalin, Kirov's murder brought out both extremes. The way he used his friend's death as a pretext for a new campaign of terror is beyond cynical. Oppositionists falsely accused of plotting Nikolaev's crime were not the only ones swept up in the Kirov tributary of what would become the raging river of the Great Terror. Many thousands of Leningraders (so-called \"formers\"\u2014former members of the nobility and clergy and former tsarist officials and military officers, among others) were sent into exile and to camps. The party was purged and articles of the penal code providing for the arrest of anyone suspected of \"counterrevolutionary activities\" were put to energetic use.\n\nFor a long time it was believed that this campaign marked the beginning of the wave of repression that came crashing down on the country during the second half of the 1930s. But a closer look at the sequence of events suggests a slightly different picture. In 1935 and 1936, terror coexisted with remnants of \"moderate\" policies. On 31 January 1935, at the very height of the \"Kirov repression,\" the Politburo, on Stalin's instigation, adopted a decision to pass a new Soviet constitution. A central feature of this document was the granting of voting rights to numerous groups previously unenfranchised as \"alien elements.\" Now elections were to be direct and ballots secret rather than open, as they had been. These changes suggested the adoption of a more democratic constitutional model to replace the \"revolutionary\" one that excluded people with suspect class credentials. In a memorandum accompanying the draft Politburo resolution on the new constitution, Stalin wrote:\n\nIn my opinion, this matter of a constitution for the Union of SSRs is a lot more complicated that it might seem at first glance. First of all, the electoral system has to be changed not only in the sense of making voting more direct. It also has to be changed in the sense of replacing open voting with closed (secret) voting. We can and must see this matter through to the end and not stop halfway. The situation and alignment of forces in our country is such that we can only benefit politically from this. I am not even talking about the fact that the need for such a reform is dictated by the interests of the international revolutionary movement since such a reform will definitely serve as a mighty weapon in the fight against international fascism.\n\nThis memorandum suggests that even after Kirov's murder, Stalin counted on exploiting the advantages of the \"moderate\" course in both domestic and international affairs. International considerations were probably the main force driving his interest in liberalization. The growing threat from Germany and Japan was bringing the USSR closer to the Western democracies. In May 1935 the Soviet Union signed a treaty of mutual assistance with France and Czechoslovakia. The Seventh World Congress of the Comintern, held that summer, allowed for cooperation with socialist governments and endorsed the idea of an inclusive popular front against fascism. Hoping for leftward movement by the West European countries and a growth in pro-Soviet sentiments, Stalin saw a need to enhance the image of the \"motherland of socialism\" as a prosperous and democratic country.\n\nThe promise to restore the voting rights of those labeled socially alien was the centerpiece of a policy of reconciliation. In Stalin's mind, in addition to the vast numbers he considered true enemies in the country, there were also many more or less innocent victims of the bitter class struggle. Young people in particular had to be brought over to the regime's side. Continuing to discriminate based on family background threatened to expand the ranks of the government's potential opponents. An important signal in the reconciliation campaign was a piece of political theater Stalin performed at a meeting of combine operators in early December 1935. When a Bashkir kolkhoznik by the name of A. Tilba proclaimed from the podium, \"I may be the son of a kulak, but I will fight honorably for the cause of workers and peasants and for the building of socialism,\" Stalin interjected a phrase that became famous: \"The son does not answer for the father.\" In fact, sons and daughters did answer for their fathers, and fathers for their children, but \"alien elements\" now had a better prospect of making their way in Soviet society. The promise of equal voting rights was accompanied by other liberalizing campaigns. For example, hundreds of thousands of people convicted of nonpolitical crimes were released from prison or rehabilitated.\n\nA degree of social stability was needed to secure and promote the positive economic trends that began to appear in late 1933 and continued into 1934. The miserable experience of previous crises had taught Stalin the economic price to be paid for each new campaign of repression. In 1935 he made the most significant concession to the peasantry since the beginning of collectivization: the right to farm private plots was enshrined in law and somewhat expanded. This step enabled an improvement in the country's food situation. Similar improvements could be seen in industrial sectors in 1935\u20131936. In November 1935 Stalin invented a new slogan: \"Life has become better, life has become more cheerful!\" That year, the ration system began to be phased out, and certain limitations on salary increases were abolished. Financial incentives boosted productivity. These were good years for the Soviet economy.\n\nOne might think that the fruits of moderation would have inspired Stalin to try more of it. They did not, and a new wave of terror became increasingly evident. Historians are still trying to understand his motives for expanding repression at a time of social stability and an improving economy. Did Stalin truly believe that the country was threatened by terrorist conspiracies? Did he actually fear for his life? There is a fair amount of evidence to the contrary. Stalin commanded the NKVD to find proof that former oppositionists had gone underground and formed terrorist organizations, but try as it might, the NKVD was unable to do so. The cases that were brought did not have the ring of truth, and Stalin must have understood that they were fabricated. In any event, he did not make any changes in his daily life that would indicate a concern for his own safety. He adhered to his daily work schedule, traveled south for vacations, and occasionally went out among the people to demonstrate his solidarity.\n\nOn the evening of 22 April 1935, some of Stalin's relatives and fellow Politburo members gathered at his Kremlin apartment. Stalin was with his children. His daughter Svetlana asked permission to take a ride on the metro, which had recently opened. Stalin, in a good mood, decided to organize an excursion. Since no preparations had been made for this outing, he and his companions were surrounded by crowds of passengers at each station. Maria Svanidze wrote in her diary: \"There was an unimaginable commotion and people rushed to greet the _vozhds,_ cried 'Hurray,' and ran after us. We were all separated, and I came close to being crushed against a column.... It was a good thing that by then the police and bodyguards had arrived.\" Stalin's fourteen-year-old son Vasily \"was the most agitated of all.\" But Stalin \"was cheerful and asked the construction supervisor, who appeared out of nowhere, endless questions.\" At the next station Stalin again went onto the platform, but his relatives, including his daughter Svetlana, stayed in the metro car, \"frightened by the unrestrained delight of the crowd, which in its excitement toppled a cast-iron lamppost not far from the _vozhds_ at one station.\" After visiting the metro, Stalin went to his dacha. Vasily, traumatized by the crowds, \"threw himself onto his bed and cried hysterically\" as soon as he returned home. The adult relatives took sedatives.\n\nWould a man living in serious fear of attack venture\u2014let alone relish\u2014such an excursion? The intensification of repression that came in late 1934 was prompted by more complex calculations. Kirov's murder provided an ideal pretext for action of the sort any dictatorship relies on to promote its central task: solidifying the power of the dictator. Admittedly, by late 1934, Stalin was already a dictator, but dictatorships, like any unstable system of government, depend on the constant crushing of threats. During this period, Stalin faced two such threats, which at first glance appear unrelated. The first was the remnant of the system of \"collective leadership\" within the Politburo, and the second was the survival of a significant number of former oppositionists. These threats belonged to what might be called Bolshevik tradition. They hung over Stalin like a sword of Damocles, reminders that there were alternatives to sole dictatorship. His fellow Politburo members enjoyed significant administrative, if not political, independence. They ran the various branches of government and had a host of clients from within the party and state apparats. The bonds of institutional and clan loyalties, along with the vestiges of collective leadership and intraparty democracy, were the last impediments to sole and unquestioned power.\n\nIn a speech given in early 1937, Stalin divided senior officials into several categories. He labeled one \"the generals of the party\" (the three or four thousand most senior officials) and another \"the party's officers\" (thirty to forty thousand mid-level officials). Until the mid-1930s, the party's old guard had held a place of honor within these two groups, but Stalin had reason to distrust these respected figures. Whatever they might say from the podium, however earnestly they swore allegiance to him, he knew: these party elders well remembered that Lenin's testament at one point almost brought Stalin's political career to an end, and he had held onto power only through the support of Zinoviev and Kamenev; that in the late 1920s Stalin had managed to defeat the Rykov-Bukharin group only with the support of the Central Committee; and that party policy in the 1930s had brought about catastrophic failures. By 1937, party functionaries had every reason to regard Stalin as \"first among equals,\" but not so long ago he had been one among many jockeying for position. Stalin knew that the old guard had the clearest memory of that time.\n\nOver long years of collaboration, the Old Bolsheviks had established close relationships with each other. Stalin periodically shuffled the deck, but it was hard to disrupt the networks of personal loyalty that had formed around officials at various levels. Leaders took \"their people\" with them from job to job. The people in these networks had divided loyalties: they served the dictator, but they also had their own patrons within the Politburo or other high-level bodies. Of course all of these groups lacked formal cohesion and political power. No one has yet found evidence of a serious effort by them to oppose Stalin. At most, they expressed their dissatisfaction privately. But like any dictator, Stalin assumed the worst. He anticipated being stabbed in the back the moment the domestic or international situation worsened. Replacing the old guard with absolutely devoted younger stalwarts was a critical aspect of his program to solidify his position. The growing threat of war provoked the _vozhd_ 's anxiety and desire to secure his power in case the unexpected happened. \"The conqueror's peace of mind requires the death of the conquered.\" This phrase, attributed to Genghis Khan, was underlined in one of the books in Stalin's library.\n\nThe conquered\u2014the repentant and humiliated former oppositionists\u2014were indeed a worrisome subgroup within the community of Old Bolsheviks. Although the secret police kept a close watch over them, the former oppositionists were still party members in good standing. Many held posts within the government and even the party apparat, or they had senior positions in major economic enterprises. Most Old Bolsheviks remembered the role the oppositionists had played during the glory days of the revolution. Kirov's murder and the fabricated case alleging that followers of Zinoviev and Kamenev were involved in a terrorist plot changed everything. The former opposition was transformed overnight from comrades who had once committed political indiscretions into \"enemies\" and \"terrorists.\"\n\nThe former oppositionists were not the only ones affected by this sudden transformation. Among the old guard it was hard to find anyone who was not in some way tied to them. A significant proportion of Soviet generals had served under Trotsky, who had founded the Red Army and led it for many years. Many up-and-coming functionaries had \"erred\" in their youth. In the 1920s, either because they were not yet sure which way the winds were blowing or were simply following their hearts, many had at some point supported the opposition. Others developed friendships with future members of the opposition during their years underground and during the revolution or when they fought side by side during the Civil War. Some had recently collaborated with repentant oppositionists. In short, in striking a blow against the former oppositionists, Stalin launched a huge shake-up in the party ranks. It allowed him both to take care of political opponents who might have been lurking in the shadows and to purge the apparat overall, including getting rid of some of his Politburo comrades.\n\nBetween 1935 and early 1937, the persecution of former oppositionists was accompanied by shake-ups at the highest echelons of power. The Kirov murder strengthened the position of three enterprising young men: Nikolai Yezhov, Andrei Zhdanov, and Nikita Khrushchev. Yezhov's promotion was especially significant. It was on his shoulders that Stalin placed direct responsibility for conducting the purge. After acquitting himself well in fabricating cases during the Kirov Affair, Yezhov was entrusted with a new assignment\u2014the Kremlin Affair. In early 1935 a group of support staff working in government offices located in the Kremlin\u2014maids, librarians, and members of the Kremlin commandant's staff\u2014were arrested and accused of plotting against Stalin. Among those arrested were several relatives of Lev Kamenev, who was charged with hatching the plot. The arrestees came under the authority of Stalin's old friend Avel Yenukidze, who oversaw the running of all Kremlin facilities, and he was accused of abetting the plot. Stalin took a great interest in the Kremlin Affair. The archives show that he regularly received and read arrestee interrogation protocols, made notations on them, and gave specific instructions to the NKVD.\n\nAlthough Yenukidze was not a member of the Politburo, he was an intimate part of the system of collective leadership insofar as he was close friends with many top officials, including Stalin himself. Stalin in essence used Yenukidze to test the durability of the collective leadership system. This was the dictator's first significant strike against his inner circle. The test was successful. The Politburo offered only weak resistance, and Yenukidze was fired, arrested, and shot. For a while Stalin trod carefully, taking the operation one step at a time, but gradually the cleansing of the top nomenklatura picked up steam. A turning point was the first Moscow show trial of former opposition leaders in August 1936. After being extensively tortured, the defendants, who included Kamenev, Zinoviev, and other prominent party figures, were proclaimed terrorists and spies and then shot.\n\nThe August trial took the hunt for enemies to a new level of hysteria. Stalin appointed Yezhov to take over the NKVD, and under the _vozhd_ 's guidance, he began preparing new trials and intensified the purge of the party and state apparats. In January 1937 a second show trial was held, this time of former oppositionists who held senior positions overseeing the economy and industrial enterprises. They were charged with wrecking and espionage. Stalin's close associates, compromised by ties with supposed enemies, gave in. Only Ordzhonikidze would not allow his underlings in the heavy-industry sector to be arrested, sparking a conflict with Stalin that ended with Ordzhonikidze's suicide. This desperate act shows how helpless the Politburo members felt before Stalin, whose control of the secret police made him an indomitable force. The _vozhd_ 's long-standing comrades-in-arms, to say nothing of middle-level functionaries, were a fractured force. They fell all over one another in an effort to ingratiate themselves with Stalin, each hoping to save his own skin.\n\nSuch was the state of affairs when the already thinned ranks of the nomenklatura convened for the February\u2013March Central Committee plenum of 1937. During the plenum, Stalin ordered that repression be continued, and Yezhov made a speech calling for a case to be brought against the leaders of the \"right deviation,\" Nikolai Bukharin and Aleksei Rykov (their fellow \"rightist,\" Mikhail Tomsky, had already killed himself in August 1936). The plenum of course approved Yezhov's proposal. Bukharin and Rykov were arrested, and in March 1938 they were convicted to be shot at the third Moscow show trial. Like the other trials, this one was followed by a wave of spurious convictions across the country.\n\nThe repression that roiled the party and state apparats came down with particular force on the \"power structures,\" the NKVD and the army\u2014organizations that Stalin thought posed the greatest threat to his dictatorship. Once Yezhov took over the NKVD, he destroyed his predecessor, Yagoda, and many of his associates. In June 1937, after being tortured, a large number of senior military officers, including the deputy people's commissar for defense, Mikhail Tukhachevsky, were given death sentences based on trumped-up charges of belonging to an \"anti-Soviet Trotskyite military organization.\" Soon afterward, a wave of arrests swept through the entire army. Scholarly investigation of recently opened archives can now set decades-long debate to rest: the Tukhachevsky Affair and the entire anti-military campaign was based on evidence fabricated by the NKVD under Stalin's direct supervision. The charges brought against the military leaders had absolutely no basis in fact.\n\nAt first, repression was primarily targeted at key members of the government, party, state security services, and military and had little effect on ordinary citizens. If the terror had been limited to the party-state nomenklatura, one might agree with those who have argued that Stalin's main goal was to destroy the party's old guard and install a new generation of functionaries blindly devoted to him. He did undeniably pursue this goal. But in the second half of 1937, the terror was brought to bear on a much larger swath of the Soviet population, and this expansion is what made it \"the Great Terror.\" In terms of their scale and the number of victims, these later operations greatly overshadowed those primarily targeted at officials. After shooting a significant fraction of the nomenklatura, Stalin brought his terror to its logical conclusion. Having solidified power at the top, he undertook to purge the country of a suspected fifth column. The threat of a major war exacerbated Stalin's paranoia. Hundreds of thousands of innocent people paid the price.\n\n## **TREPIDATION IN THE INNER CIRCLE**\n\n**The initial arrival of the four at the near dacha, early morning hours of 2 March 1953.**\n\nThe bodyguard entered Stalin's apartments with the packet of mail and started looking for him. After walking through several rooms, he finally found the _vozhd_ in the small dining room. The sight must have been extremely disturbing. Stalin was lying helpless on the floor, which was wet beneath him. This last point is important not for reasons of schadenfreude or as an evocative detail but because it affected subsequent events. It appeared to the bodyguard that Stalin was unable to speak, but he did make a small hand gesture, beckoning him to approach. The bodyguard summoned his colleagues, who helped him lift Stalin onto the couch. They then rushed to telephone their immediate superior, State Security Minister Semen Ignatiev. According to the bodyguards' later accounts, Ignatiev refused to make any decisions and told them to call members of the top leadership: Beria and Malenkov.\n\nIgnatiev's reaction was perfectly understandable. He was behaving just as the bodyguards had done several hours earlier, when they were afraid to enter Stalin's rooms uninvited. Ignatiev did not want to take responsibility for a decision to summon doctors to the _vozhd_. This was a ticklish matter for a man who, just two years earlier, had been plucked from the relatively cozy position of Central Committee department head and assigned to hunt for enemies of the people as minister of state security. He must have rued the day Stalin picked him for this job, which carried a high price for failure. From then on he lived in fear. Upon hearing that Stalin had suffered some sort of stroke, his only desire was to hand decision-making responsibility to somebody else.\n\nHaving failed to get any guidance from their boss, the bodyguards managed to find Malenkov, who then informed the other members of the ruling Five: Beria, Khrushchev, and Bulganin. This made sense. Without a clear understanding of Stalin's condition, Malenkov did not want to go to the dacha by himself or be the only one to sanction the summoning of doctors. Any decisions should be made collectively. The four men agreed to meet at the dacha to assess the situation and give each other cover for whatever actions were taken.\n\nBoth Khrushchev's memoirs and the bodyguards' accounts describe the top leadership's extreme caution after arriving at the dacha in the middle of the night. They were afraid of doing anything that might provoke Stalin's wrath if he recovered. According to Khrushchev, at first they did not even enter Stalin's apartments, choosing instead to interrogate the bodyguards. What they heard made them even more nervous. That Stalin was incapacitated and had apparently urinated on himself put the leaders in a difficult position. They knew he would not want anyone to see him in such a state. What if this was just a passing episode? Stalin would not look fondly on anyone who had witnessed his humiliating helplessness. As Khrushchev describes it, once they learned from the bodyguards that Stalin \"now seemed to be sleeping, we thought that since he was in such poor shape, it would be awkward for us to appear at his side and make our presence officially known. So we went back to our homes.\"\n\nKhrushchev's memoirs apparently do not tell the whole story. According to the bodyguards, before leaving, the four designated Malenkov and Beria to enter Stalin's rooms and personally assess his condition. Such an assessment required two men for obvious reasons. If all four went, they would make unnecessary noise and risk rousing the _vozhd_. And none of them wanted to go in by himself. Khrushchev and Bulganin thus waited in the bodyguards' quarters while Beria and Malenkov snuck stealthily in to look at Stalin, terrified of waking him. The bodyguards recalled one slapstick detail: Malenkov's new shoes made a squeaking noise, so he took them off and carried them under his arm. As the two men approached, they could hear Stalin lightly snoring. After emerging, Beria berated the bodyguards for raising a fuss over nothing. Stalin was just sleeping. The bodyguards defended their actions, explaining that matters had been much worse a few hours earlier. Dismissing the bodyguards' concerns, the four men returned to Moscow.\n\nSome historians and commentators have detected conspiratorial overtones in this episode and blame Stalin's death on the decision not to call for medical help. This interpretation is doubtful. First, according to the doctors who performed the autopsy, Stalin's stroke was the result of atherosclerosis that had been developing for years. Quick intervention would not have saved him. On the other hand, his fellow leaders could not have known this. They did not understand the implications of providing or withholding medical care, and their failure to summon doctors could have contained some malicious intent. Many Soviet leaders, in their hearts, surely did not wish their abusive leader long life. Nevertheless, less sinister explanations must also be considered. Stalin's associates were simply afraid of intervening. They were not used to taking the initiative, and they knew Stalin's suspicious and capricious nature all too well. During those days in early March, everyone involved\u2014the bodyguards, Ignatiev, and the other members of the Five\u2014behaved exactly as Stalin had trained them to behave. They tiptoed nervously forward, always looking over their shoulders and trying to shift as much responsibility as possible onto each other.\n\nFor many years, even Stalin's closest associates and friends, people with whom he had shared long years of struggle, had lived under the constant threat of destruction. A dictator can only be sure of his power if those around him are at his mercy. After destroying the former opposition leaders, in 1937\u20131938 Stalin proceeded to have a significant portion of the Politburo shot. The close relatives of some of his surviving associates were also arrested or killed. The brother of Politburo member Lazar Kaganovich committed suicide, and Kalinin's wife wound up in a camp. This suppression of potential oligarchs continued after the war. The Leningrad Affair did away with Nikolai Voznesensky and Aleksei Kuznetsov, two members of the younger generation who had risen to prominence under Stalin. Molotov's wife was arrested around the same time. In the final months of his life, Stalin lashed out at Molotov and Mikoyan, essentially removing them from power. His death would provide the only guarantee against new purges.\n\nAt some point in their careers, virtually everyone in the top Soviet leadership had to endure a ritual of humiliation and repentance followed by renewed oaths of allegiance to the _vozhd_. Stalin would cast his comrades into disfavor only to later bring them back into the fold. He was generous with rebukes and liked to orchestrate verbal floggings in the press and at various meetings. And when he lost his temper, it was a horrifying sight to behold. Minister of Foreign Trade Mikhail Menshikov told of one instance when he incurred Stalin's wrath during a meeting by failing to properly hear his question. \"He gave me a furious look,\" Menshikov recalled, \"and launched a fat pencil at me as hard as he could, hurling it along the length of the table in my direction. For a moment everyone froze and waited to see what would happen next.\" After Stalin's death, Ignatiev complained about having been subjected to constant dressings-down: \"Comrade Stalin reprimanded me using fouler language than I'd ever heard in my life and called me an idiot.\" When the writer Konstantin Simonov attended the Central Committee plenum in October 1952, he was struck by the furious, \"almost ferocious\" and \"unrestrained,\" tone of Stalin's speech denouncing Molotov and Mikoyan. Stalin's temper and unpredictability, especially during his final years, were made worse by his declining health.\n\nTop Soviet officials lived a golden-cage existence. While they exercised life-and-death power over their subordinates, they were at the constant mercy of their ultimate boss. Their security, transportation, incoming and outgoing correspondence, special telephone lines, dachas, and apartments\u2014all were handled by state security, which was entirely under the dictator's control. Such control meant that Stalin knew everything about how and with whom these officials spent their time. As if that were not enough, he apparently asked the secret police to install listening devices to spy on certain Politburo members.\n\nDespite the oppression of the collective leadership, periodic manifestations of oligarchy inevitably threatened Stalin's sole power. Though very much under his thumb, his fellow leaders did enjoy a certain administrative autonomy as the heads of major government institutions, and they independently made many decisions of consequence for the running of the country. Furthermore, their authority expanded as Stalin's physical frailty diminished his involvement in day-to-day decision making. Stalin was aware of this threat. Konstantin Simonov recorded a typical comment by the _vozhd_ about his comrades, as reported by an eyewitness:\n\nEven when differences remain, they will come to some agreement on paper and present the issue to me in that form.... The managers understand that I cannot know everything; all they want from me is a stamp with my signature. Yes, I cannot know everything, so I pay attention to differences, to objections, and I try to make sense of why they come up, where the real problem lies. The managers do their best to conceal these from me; they go along with the votes but they conceal the differences, all so that they can get a stamp with my signature. What they want out of me is my stamp.\n\nStalin's method for penetrating the defenses of this mutual protection society could best be described as scattershot. The dictator's underlings never knew what question might suddenly interest him. They never knew whether Stalin would react to a particular decision and, if so, how or when. The constant threat of a random attack allowed him to keep the apparat and his close associates in a state of tension that helped to compensate for his lack of total control over them. The _vozhd_ 's effort to maximize his power over his subordinates was helped by the number of channels through which he received information. The government and party bureaucracies, the courts, and state security all kept an eye on one another and constantly tried to prove their vigilance and effectiveness by denouncing one another to Stalin, zealously exposing others' warts while concealing their own.\n\nRepression, the constant threat of punishment, and Stalin's temper and whims made the life of top Soviet officials almost as difficult as that of the powerless man or woman on the street. His \"comrades\" lived and worked under constant stress. One long-term Soviet diplomat left the following remembrance of the country's minister of foreign affairs, Andrei Vyshinsky, one of Stalin's most devoted and successful associates: \"Vyshinsky was terrified of Stalin. Every Thursday he would go and report to him, and well beforehand, in anticipation of this encounter, his mood would sour. The closer it came to Thursday, the gloomier and more irritable he got.... But by Friday, when it was all behind him, he allowed himself to relax for a day or two. Experienced people knew that this was when it was best to report to him on the most complicated matters or approach him with requests of a personal nature.\"\n\nStalin was a merciless boss. He expected total dedication from his subordinates and favored a military management style: orders had to be carried out unquestioningly and at any cost\u2014no excuses. In addition to the constant danger of arrest and the excessive workload, the lives of Stalin's close associates were made difficult by his nocturnality. To accommodate the _vozhd,_ the apparat worked both at night, when Stalin was awake, and during the day, when the rest of the country was up. The stresses of working for Stalin apparently made some stronger. A number of his closest associates lived many years. Molotov and Kaganovich, for example, nearly reached the century mark. But not everyone had the iron constitution and adaptability needed to survive the demands Stalin placed on his subordinates. A Central Committee document written in 1947 admitted that \"An analysis of the health of the party and government's leading cadres has shown that many individuals, even among the relatively young, suffer from diseases of the heart and the circulatory and nervous systems sufficiently serious to impact their ability to work. One cause of these diseases is stressful work not only during the day, but also during the night, and often even on holidays.\" As long as Stalin was alive, nothing could be done about this problem, but soon after his death a resolution was adopted requiring regular government offices to remain closed at night, and the bureaucracy began to run in a more normal way.\n\nStalin kept himself at the center of the huge machine used to manipulate officials. He initiated and guided repression, orchestrated all major reassignments, and was constantly reshuffling people so that nobody grew too comfortable in a particular job. Like any dictator, he strove to instill a sense of fear, adoration, and instinctive devotion in his underlings. Vyacheslav Molotov, a diehard follower of the dictator, described Lazar Kaganovich as a \"two hundred percent Stalinist.\" These were the sorts of people Stalin tried to cultivate.\n\nA key element of the process by which the Soviet government\u2014including its very top leadership\u2014was \"Stalinized\" was the mass purges of the 1930s. In a matter of months, the purges destroyed the party's old guard and replaced it with fresh faces, unburdened by excessive knowledge of the past or ideas about how the country might be run differently. \"New stock\" replaced officials who had earned their places in the Soviet government during the revolution. By 1940, after the Terror had receded, 57 percent of party secretaries in the regions of Russia and on the central committees of the Soviet Union's ethnic republics were under the age of thirty-five. Many ministers, generals, directors of major enterprises, and leaders of cultural unions were between thirty and forty.\n\nStalin gave these upstarts tremendous power, allowing them to preside over their own little dictatorships. The fates, even the lives, of millions were in their hands. The distribution of significant resources and the functioning of gigantic enterprises depended on them. They formed their own caste, which lived by its own laws and enjoyed its own privileged world. The members of this caste did not know hunger or material want. They were not affected by the catastrophic shortage of housing or the backwardness of the health care system. They lived in spacious apartments and dachas, protected by guards. Their cars sped past overcrowded public buses and trolleys. Whoever did their shopping did not have to line up for hours outside empty stores. Their salaries and tax-free supplemental pay (known as \"envelopes\") exceeded by orders of magnitude the meager pay of ordinary citizens. The fees paid to Soviet writers privileged to belong to the nomenklatura reached the hundreds of thousands of rubles, in some cases generating annual incomes of up to a million rubles, many thousands of times what a Soviet peasant survived on. Dazzled by the sense of belonging to an all-powerful government corporation and by their own importance, they were utterly free of compassion, self-reflection, or understanding of the \"other.\"\n\nStalin was the gatekeeper for the world of the nomenklatura. Entry could be gained only with his favor and support. For those fortunate enough to survive, the horrible fates of their predecessors and the continuing repression only intensified their gratitude toward the dictator. Stalin was twice the age of many members of this new generation of officials. Many of them knew little of the party's revolutionary period or of former leaders who were now labeled enemies. For them, Stalin was the ultimate authority, the leader of the revolution, the victorious generalissimo, and a theoretician on a par with the founders of Marxism.\n\nStalin strove variously to feed this image. He cultivated an inferiority complex in his close associates: \"You are blind like little kittens. Without me the imperialists would strangle you.\" Gradually he acquired the exclusive right to advance any initiative of significance, leaving the operational details to his comrades. His speeches, conversations, and letters were like lectures that he laced with contrived profundities. He liked to assign meaning to events and show off his vast knowledge and deep understanding of problems. The self-confident tone of his pontificating often belied the flimsiness and artificiality of his reasoning. But who would dare challenge him? For most functionaries, who tended to lack sophistication, Stalin's utterances had an almost sacred quality. However, it was not just his monopoly on theoretical pronouncements that made the _vozhd_ the voice of authority. He was well read and had a good memory, as well as a knack for pithy aphorisms. He would spend time preparing for his meetings, and it enabled him to show an impressive knowledge of detail. Such knowledge left a deep impression on many who witnessed these performances.\n\nThe primary reason that every utterance by Stalin carried such weight was that these were the words of an enormously powerful dictator who inspired both horror and adoration. To promote this image, he adopted the manner of a judge and master of destinies. During conferences he did not fraternize with other attendees but strolled around, pipe in hand. Before the spellbound gazes of onlookers, he reasoned out loud as if mulling weighty decisions. Stalin never publicly spoke of himself as a great man. It was enough that official propaganda shouted his greatness to the point of absurdity. Aware that brilliance stands out nicely against a fa\u00e7ade of modesty, Stalin presented himself as a mere disciple of Lenin and servant of the party and the people. Every opportunity was taken to highlight this \"humility.\" He feigned impatience or even embarrassment when greeted with the inevitable standing ovation. He peppered his speeches with self-deprecation and folksy humor. He helped certain visitors to his dacha with their coats. After arriving at a reception arranged by Mao Zedong during the Chinese leader's January 1950 visit to Moscow, Stalin greeted the cloakroom attendant but turned down his services. \"Thank you, but this is something even I seem to be able to manage.\" After removing his coat, he hung it on a hanger himself. This affected modesty did not prevent Stalin from asserting his own worth when warranted. In 1947 he personally edited his official biography, inserting the following: \"Masterfully performing the job of _vozhd_ of the party and people and enjoying the full support of the Soviet people, Stalin nevertheless did not allow even a shadow of self-importance, conceit, or self-admiration into anything he did.\" Thirteen million copies of this biography were printed.\n\nStalin must have believed that if he was going to hold on to power, he had to be considered infallible. On occasion he recognized that mistakes were made, but they could never be his. Misguided decisions and actions were attributed to \"the government,\" officials, or\u2014most often\u2014the plotting of enemies. The idea that he might bear personal responsibility for the country's afflictions was rejected out of hand. He was, however, willing to take credit for its achievements. Boundless power inevitably gave him, as it does any dictator, a belief that he was endowed with remarkable prescience. But unlike the mystically inclined Hitler, who believed he was following a higher calling, Stalin's belief in his infallibility probably had more to do with his untrusting nature and anxieties. He was sure that the only person he could count on was himself. Around him swarmed enemies and traitors. At times, this political paranoia was the cause of unfathomable tragedy. Such was the case in 1937\u20131938.\n\n## **4 TERROR AND IMPENDING WAR**\n\nThroughout 1937, the wave of repressions against members of the nomenklatura and former oppositionists continued to grow. In August, this wave turned into a tsunami when the ranks of the repressed were expanded from a few tens of thousands of officials to hundreds of thousands of ordinary Soviet citizens. It was at this point that the repression of 1937\u20131938 earned the name given it by Robert Conquest: \"the Great Terror.\"\n\nAfter the archives were opened, we learned that the Great Terror was actually a series of operations approved by the Politburo and aimed at different groups. The most far-ranging of these operations\u2014the one against \"anti-Soviet elements\"\u2014was carried out in fulfillment of NKVD Order No. 00447, approved by the Politburo on 30 July 1937 and planned for August through December. Each region and republic was assigned specific numerical targets for executions and imprisonments in camps. The quotas for the destruction of human lives were very much like those for the production of grain or metal. During the first stage, approximately two hundred thousand people were to be sent to the camps and more than seventy thousand were to be shot. Yet Order No. 00447 allowed for flexibility: local officials had the right to ask Moscow to increase the permitted number of arrests and executions. It was clear to everyone involved that this right was actually a duty. After expeditiously reaching initial targets, local authorities sent Moscow new \"increased obligations,\" which were almost always approved. With Moscow's encouragement, the initial plan for destroying \"enemies\" was fulfilled several times over.\n\nThe first \"anti-Soviet elements\" affected by the operation were the kulaks, who, according to Order No. 00447, had continued their \"anti-Soviet subversive activities\" after returning from camps and exile. Order No. 00447 placed so much emphasis on kulaks that it has often been called \"the kulak order.\" This is a misnomer, however, since it provided for the arrest and execution of many other population groups: former members of parties that opposed the Bolsheviks, former members of the White Guard, surviving tsarist officials, \"enemies\" who had completed their sentences and been released, and political prisoners still in the camps. Toward the end of the list came common criminals.\n\nThis list of targets suggests that the operation's purpose was the extermination or imprisonment of anyone the Stalinist leadership considered a current or potential threat. This goal was even more clear-cut in the \"nationalities\" operations that were conducted alongside the \"anti-Soviet elements\" operation. The \"nationalities\" operations were also planned in Moscow and governed by special NKVD orders approved by the Politburo. They had a catastrophic impact on the Soviet Union's ethnic Poles, Germans, Romanians, Latvians, Estonians, Finns, Greeks, Afghans, Iranians, Chinese, Bulgarians, and Macedonians. The Soviet leadership viewed all these groups as ripe for recruitment by hostile foreign powers. A special operation was also conducted against Soviet employees of the Chinese Eastern Railway, who had returned to the USSR from Harbin after the railway was sold to Japan in 1935.\n\nThe two campaigns, the \"anti-Soviet elements\" and the \"nationalities\" operations, comprised the Great Terror. It was a highly centralized effort begun in the summer of 1937 and concluded in November 1938. Based on the most recent knowledge, approximately 1.6 million people were arrested, and 700,000 of them were shot. An unknown number perished in NKVD torture chambers. Over the roughly year-and-a-half duration of the Great Terror, approximately 1,500 \"enemies\" were killed every day. None of Stalin's other crimes against the Soviet population matched the Great Terror in either scale or savagery, and human history offers few episodes that compare.\n\nThese figures explain why the Great Terror has come to symbolize Stalin's dictatorship and personal cruelty. That Stalin himself was the inspiration behind the Terror has never been disputed by serious scholars, and further evidence of his involvement was found after the opening of the archives, which revealed how closely Moscow directed the operations. Having put to rest any lingering doubts that Stalin was the instigator and organizer of the Great Terror, historians have now turned to the task of reconstructing his plans and calculations during these bloody months. Scholars have debated Stalin's motives for years. The horrific nature of his deeds has led some to think he might have been insane. Clinical proof of such a possibility is undoubtedly beyond reach at this point, but we do have extensive evidence of Stalin's mental state during this period. For the first time in many years he did not take his usual summer vacation in the south, remaining in Moscow to oversee the roundup. More telling are the many notations and instructions he left on interrogation protocols and the vast body of correspondence between him and the NKVD during this period.\n\nCom. Yezhov: Very important. You have to go through the Udmurt, Mari, Chuvash, and Mordov republics; go through them with a broom.\n\nBeat Unshlikht for not naming the Polish agents for each region.\n\nComrade Yezhov: Very good! Keep on digging and cleaning out this Polish spy filth.\n\nYou don't need to \"check,\" you need to arrest.\n\nValter (a German). Beat Valter.\n\nOne important source for understanding the fury Stalin unleashed in 1937\u20131938 is the complete transcripts of his speeches and remarks from this period; these have recently become available. Unusually convoluted and incoherent, they are filled with references to conspiracies and omnipresent enemies. In remarks to a meeting of the defense commissar's council on 2 June 1937, Stalin asserted, \"Every party member, honest non-member, and citizen of the USSR has not only the right but also the duty to report any failings that he notices. Even if only 5 percent are true, it will still be worthwhile.\" In another example, the top-performing workers in the metallurgical and coal industries, while being honored with a special reception at the Kremlin on 29 October 1937, were told by Stalin that he was not certain he could trust even them: \"I'm not even sure that everyone present, I truly apologize to you, is for the people. I'm not sure whether even among you, I again apologize, there might be people who are working for the Soviet government but at the same time have set themselves up with some intelligence agency in the West\u2014Japanese, German, or Polish\u2014for insurance.\" These words, which must surely have surprised those present, were expunged from the official record of the reception.\n\nThese examples, of which there are many, are consistent with a statement made by the commissar for foreign trade, Arkady Rozengolts, and contained in his NKVD case file. Rozengolts, who knew Stalin well, described him as \"suspicious to the point of insanity\" and felt that by 1937 he had changed. In the past, Rozengolts noted, whenever he had reported to Stalin, the _vozhd_ had calmly signed whatever papers needed his signature. Now he would fall into \"a fit, a mad fit of rage.\" This rage was undoubtedly an important factor in the huge scope and brutality of the Great Terror. By the same token, Stalin's agitated state does not fully explain the decisions he made throughout this period. Pivotal questions remain unanswered. With whom was Stalin so furious, and why did this fury emerge specifically then?\n\nTo understand the nature of Stalin and his regime it is important to keep in mind that the Soviet Union was born out of war. The country came into being as a result of World War I, established itself through victory in the Civil War\u2014a victory that involved overcoming foreign intervention\u2014and was perpetually preparing for the next war. Having come to power solely through war, Bolshevik leaders always believed their power could be taken away by the coordinated efforts of a foreign enemy and domestic counterrevolutionary forces. War readiness, for them, had two aspects: a strong military economy and a secure homeland. The latter required destroying internal enemies.\n\nThe gradual move toward terror during the second half of the 1930s coincided with growing international tensions and a growing threat of war. In addition to Japanese aggression along the Soviet Union's Far Eastern borders, events in Europe were increasingly alarming: Hitler had come to power, and Poland, which lay between the USSR and Germany, seemed in Stalin's eyes to favor relations with Germany over the USSR. Western powers were pursuing a policy of appeasement toward the Nazis, and the Rhineland had been remilitarized in 1936. Another factor influencing Stalin's foreign policy was the civil war in Spain, which convinced him that England and France were incapable of standing up to Germany. He had little faith in the Western democracies in any case. A policy of non-intervention no longer made sense for the Soviet leadership, and it decided to enter the war in support of Spain's Republicans, who were fighting Hitler's ally, General Francisco Franco. Stalin, observing the situation in Spain, became further convinced of the need to purge the homeland in the interests of military readiness. The Spanish Civil War was bringing to the fore a familiar assortment of ills, including anarchy, guerrilla warfare, sabotage, a drifting and ambiguous line dividing the front from the rear, and all manner of treachery. This was the war that gave us the concept of the fifth column. In October 1936, at a critical moment when four columns of Francoist forces were approaching Madrid, the Nationalist general Emilio Mola claimed to have a \"fifth column\" within the Republican-held city that would rise up and help his forces take it. This term quickly became embedded in the Soviet leaders' political lexicon.\n\nWar in Spain and repression in the USSR escalated in parallel. When the conflict broke out in Spain, on 18 July 1936, the Stalinist leaders initially reacted with caution. But catastrophic defeats suffered by the Republican army led them to intervene. On 29 September 1936, the Politburo adopted a plan of action. (It may be significant that this decision coincided with Yezhov's appointment as head of the NKVD.) The Spanish defeats were taking place alongside setbacks in Europe and the Far East. On 25 October 1936, Italy signed a treaty with Germany, followed on 25 November by the Anti-Comintern Pact between Germany and Japan. All of these developments seemed to heighten the danger of war.\n\nNewly available archives confirm that Stalin was heavily involved in Spanish affairs. The evidence clearly shows that he believed Republican defeats were caused by saboteurs in the ranks. He demanded that the internal enemy be dealt with decisively. On 9 February 1937 Soviet representatives in Valencia and Madrid were sent a telegram asserting that a series of failures at the front had been directly caused by treachery at headquarters: \"Make use of these facts, discuss them, observing caution, with the best of the Republican commanders... so that they may demand... an immediate investigation of the surrender of Malaga, a purge of Franco agents and saboteurs from army headquarters.... If these demands by front-line commanders do not produce immediately the necessary results, put it... that our advisers may find it impossible to continue working under such conditions.\" A few days later, he repeated these demands: \"We tell you what our firmly established opinion is: that the General Staff and other headquarters must be purged thoroughly of their complement of old specialists who are unable to understand the conditions of civil war and, in addition, are politically unreliable.... Headquarters must be reinforced with fresh people, staunch and full of fighting spirit.... Without this radical measure the Republicans will unquestionably lose the war. This is our belief.\"\n\nAt the same time that Stalin was dispatching telegrams to Spain, the notorious February\u2013March 1937 Central Committee plenum, which signaled an intensification of repression, was taking place in Moscow. Stalin, reading a draft of the speech Molotov planned to make to the plenum, made some comments in the margins. He underlined the parts where Molotov talked about Trotsky ordering his followers in the USSR \"to save their strength for the most important moment\u2014for the start of war\u2014and at that moment to strike with total decisiveness at the most sensitive points in our economy.\" Near the words \"those incapable of fighting the bourgeoisie, who prefer to cast their lot with the bourgeoisie rather than the working class, have abandoned [the party],\" Stalin made a notation: \"This is good. It would be worse if they abandoned us in time of war.\" The theme of the special danger posed by wreckers and spies in wartime ran through the speeches delivered at the plenum, including Stalin's: \"Winning a battle in time of war takes several corps of Red Army soldiers. But reversing that victory at the front requires just a few spies somewhere in army headquarters or even division headquarters, able to steal the battle plans and give them to the enemy. To build a major railway bridge would take thousands of people. But to blow it up, just a few people would be enough. There are dozens, hundreds of such examples.\"\n\nStalin took an active hand in preparing an article for the 4 May 1937 issue of _Pravda,_ titled \"Certain Insidious Recruitment Techniques Used by Foreign Intelligence.\" This lengthy piece, taking up the bottom halves of three pages, was an important element of the Great Terror's ideological underpinning. It was reprinted in various publications, actively used in propaganda, and discussed at party study groups. We can see from the initial draft, which Stalin filed in his personal archive, that he modified its headline, which originally read \"Certain Methods and Techniques Used by Foreign Intelligence,\" to give it a more sinister tone.\n\nThis article, unlike others that Stalin helped produce, was not at all theoretical. It described specific (most likely fictitious) instances in which Soviet citizens, especially those sent overseas on state business, had been recruited by foreign intelligence agencies. These examples made the article credible and persuasive. Stalin contributed almost an entire page of text describing an instance in which a Soviet official working in Japan met regularly with an \"aristocratic lady\" in a restaurant. During one such meeting, a Japanese man in a military uniform appeared, claimed to be the woman's husband, and made a scene. Another Japanese man appeared and offered to help resolve the matter, but only after the Soviet citizen agreed in writing to keep him informed of what was happening in the USSR. This \"helpful intermediary\" turned out to be an agent of Japanese intelligence, and the Soviet citizen became a spy.\n\nIn the months that followed, Stalin's suspicions were translated into massive police operations. During the spring and summer of 1937, the urgent call to expose spies and forestall potential treason became the basis for a case against a counterrevolutionary organization within the Red Army. On 2 June 1937, Stalin explained the goal of the plot to members of the defense commissar's Military Council: \"They wanted to turn the USSR into another Spain.\" Reports of treachery and anarchy in Spain were an important component of the propaganda campaign to \"intensify vigilance\" and fight against \"enemies\" within the USSR. In June and July 1937, when the government was preparing to launch large-scale operations against domestic anti-Soviet elements, Soviet newspapers were filled with articles about arrests of German spies in Madrid and of Trotskyites in Barcelona and the fall of the Basque capital Bilbao brought about by a treacherous commander in the Basque army. Also during that summer, the Spanish Republican government created a special state security agency to counteract espionage and combat the \"fifth column\"\u2014the Servicio de Investigacion Militar (SIM), which sent tentacles into all parts of Republican Spain and brutally suppressed any opposition. The methods used by this new structure prompted sharp criticism even by sympathetic leftists in Western countries. Intensified repression in the Soviet Union was being mirrored in Spain (including by Soviet agents operating there). The Spanish Republican police and the Soviet secret police each worked to crush their own \"fifth columns.\"\n\nIn July the situation in the Far East became even more tense after Japan invaded China. Two important events occurred on 21 August 1937. First, the USSR and China, both with eyes on Japan, signed a non-aggression pact. Second, a resolution was adopted by the Council of People's Commissars and the Central Committee to \"Expel the Korean Population from Border Regions of the Far Eastern Territory.\" In the fall of 1937 a massive operation was undertaken to arrest and deport Koreans from this vast region. More than 170,000 people were expelled. The expressed goal was to \"prevent the penetration of Japanese espionage into the Far Eastern Territory.\"\n\nThe idea that the country had to be purged of a potential fifth column, a recurring theme throughout the 1930s in the USSR, was an article of faith among Stalin's close associates. Even many decades later, they referred to it:\n\nNineteen thirty-seven was necessary. If you consider that after the revolution we were slashing left and right, and we were victorious, but enemies of different sorts remained, and in the face of the impending danger of fascist aggression they might unite. We owe the fact that we did not have a fifth column during the war to '37.\n\nThis was a struggle against a fifth column of Hitlerite fascism that had come to power in Germany and was preparing war against the country of the Soviets.\n\nThere is little doubt that Stalin encouraged these ideas among his fellow Politburo members. From their narrow perspective, he had a logical and convincing argument. The Soviet government had many internal enemies who might be keeping a low profile at the moment but were ready to leap into action as soon as the USSR was challenged by a foreign power. The relatively independent old party nomenklatura, which still had ties to the military and the NKVD, might seek to take charge. Former oppositionists were surely eager to take revenge after long years of humiliation and persecution. The kulaks and the perpetually starving peasants might band together with former members of the nobility, White Guard, and the clergy to follow the example of the Bolsheviks in 1917 and turn war with a foreign enemy into a civil war against a despised regime. Then there were the Soviet Union's many ethnic minorities with ties to neighboring countries\u2014Germans and Poles especially\u2014who Stalin suspected would collaborate with an enemy based on ties of blood. The way to eliminate these dangers was to destroy as many potential enemies and collaborationists as possible. Such was the logic of Stalin's fearful and ruthless mind as the threat of war grew. In the fevered imaginations of his inner circle, such a fifth column loomed orders of magnitude larger than it could possibly have been in reality. Phantom threats overshadowed the very real dangers confronting the Soviet Union.\n\n###   **WAS IT ALL YEZHOV'S FAULT?**\n\nStalin claimed to have had no part in his own atrocities. He told the renowned Soviet aeronautical engineer Aleksandr Yakovlev that it was all Yezhov's fault: \"Yezhov was a beast! A degenerate. You'd call him at the commissariat, and they'd tell you, 'He went to the Central Committee.' You'd call the Central Committee, and they'd tell you, 'He went to his office.' You'd send someone to his house, and it turns out that he's lying on his bed dead drunk. Many innocent lives were lost. That's why we shot him.\"\n\nThe winding down of the Great Terror in late 1938 and early 1939 was accompanied by a campaign to deflect suspicion away from its true perpetrators. This effort was helped by Yezhov's removal and the very public unmasking of \"slanderers\" who had submitted denunciations against honest people\u2014supposedly a major cause of the repression. Even today some are willing to argue Stalin's innocence, proposing pseudo-scholarly theories that the Great Terror erupted spontaneously on the initiative of local officials. Of course, once Moscow issued its orders, the momentum generated was bound to look elemental. In the bureaucratic language of the Stalin era, the behavior of zealous officials was labeled _peregiby_ (excesses). But it was not excesses that determined the scale and ferocity of the Terror. The documentary evidence shows that large-scale operations rarely deviated from Stalin's orders.\n\nAfter Moscow's arrest and execution quotas were received by the NKVD headquarters of each oblast (province) and _krai_ (a territory similar to a province but containing semi-autonomous administrative units), the regional NKVD chief would gather the heads of local (municipal and district) NKVD offices for a meeting, at which the regional quota would be parceled out among the administrative entities (districts, towns, villages, settlements). The first source used in compiling a list of enemies was the card files that the political police kept on various suspected \"anti-Soviet elements,\" as well as any other compromising materials that came to hand. After a victim was arrested, an investigation was conducted to expose his or her \"counterrevolutionary ties\" or uncover the existence of \"counterrevolutionary organizations.\" The necessary \"evidence\" was obtained using a variety of methods, most often torture, which was officially sanctioned by the country's top leadership. The forms of torture were brutal and sometimes caused an arrestee's death. One major goal of interrogation was to obtain testimony implicating others, thus generating a second wave of arrestees, who in turn provided more names. These police operations could, in theory, continue indefinitely, or until the potential pool of victims had been thoroughly drained. Such operations did not continue only because Stalin had full control of the state security system and party apparat and could close the spigot whenever he wanted. Every decision to sentence a presumed enemy to a labor camp or to be shot was approved in Moscow.\n\nAt first it was assumed that these large-scale operations would conclude at the end of 1937. Gradually, the date was moved back to November 1938. On 17 January 1938, Stalin sent NKVD chief Yezhov new orders:\n\nThe SR [Socialist Revolutionary Party] line (both left and right) has not been fully uncovered.... It is important to keep in mind that there are still many SRs in our army and outside the army. Can the NKVD account for the (\"former\") SRs in the army? I would like to see a report promptly. Can the NKVD account for \"former\" SRs outside the army (in civil institutions)? I also would like a report in two\u2013three weeks.... What has been done to expose and arrest all Iranians in Baku and Azerbaijan? For your information, at one time the SRs were very strong in Saratov, Tambov, and the Ukraine, in the army (officers), in Tashkent and Central Asia in general, and at the Baku electrical power stations, where they became entrenched and sabotaged the oil industry. We must act more swiftly and intelligently.\n\nThis document is one of many pieces of evidence that Stalin played the decisive role in organizing the Great Terror and that Yezhov was following his orders. Archival records clearly show Stalin to be the initiator of all key decisions having to do with purges of party and government institutions and the mass operations that swept up ordinary citizens. He not only ordered the arrest and execution of hundreds of thousands of people, but he also took a strong interest in the details. He sent telegrams about the need to make particular arrests, threatened dire consequences for insufficient vigilance, and signed lists of members of the nomenklatura to be executed and imprisoned. In many cases he personally decided whether someone would be shot or sent to a labor camp. Overseeing the large-scale operations to wipe out enemies took up a significant portion of the dictator's time in 1937\u20131938. Over a twenty-month period from January 1937 to August 1938, he received fifteen thousand _spetssoobshchenii_ (special communications) reporting on arrests and the conduct of various secret police operations or requesting approval for a particular act of repression, usually accompanied by interrogation protocols (transcripts). On a typical day, he received twenty-five documents from Yezhov, some running to many pages. Furthermore, the record of visitors to Stalin's office shows that during 1937 and 1938, Yezhov visited him almost 290 times and spent a total of 850 hours with him. The only person who visited more often was Molotov.\n\nYezhov was a capable and motivated pupil. He organized the trials of former oppositionists and conducted day-to-day oversight of the giant machine of repression. He personally participated in interrogations and issued orders to apply torture. To please Stalin, who always demanded greater efforts in the fight against enemies and constantly pointed to new threats, Yezhov encouraged his subordinates to exceed the Politburo's targets for mass arrests and executions and to fabricate new conspiracies. To encourage them, the NKVD and Yezhov personally were lavished with praise throughout 1937 and most of 1938. Yezhov was given every conceivable award and title and simultaneously held several key party and government posts. Cities, factories, and kolkhozes were named after him.\n\nDespite these signs that Stalin was pleased with his people's commissar for internal affairs, there is evidence that the _vozhd_ was maintaining a certain distance, even as Yezhov and his organization were lavished with praise for their excellent work in exposing enemies. Inevitably, Stalin eventually brought the mass extermination to a halt and blamed the \"excesses\" and \"violations of law\" on Yezhov and his subordinates. Stalin laid the groundwork for Yezhov's removal gradually and systematically. In August 1938, he appointed Lavrenty Beria, first party secretary for Georgia, to serve as Yezhov's deputy. On the surface, nothing had changed. Yezhov still seemed to enjoy power and favor. But now, by his side was a man he would never have chosen. Several months later Yezhov even alluded to Beria's appointment in a letter to Stalin, describing it as showing \"an element of mistrust toward me\" and admitting that he saw \"[Beria's] appointment as preparation for my being relieved.\" He was right. Unable to cope with the stress of the situation, he descended into alcoholism and lost control of both the NKVD and himself.\n\nTwo months after Beria's appointment, Stalin took further steps toward Yezhov's removal. On 8 October 1938 the Politburo established a commission to draft a resolution concerning the NKVD. Yezhov's subordinates began to be arrested. Beria's henchmen set to work beating testimony against Yezhov out of them, just as Yezhov's henchmen had done when he was building a case against his precedessor, Genrikh Yagoda. On 17 November the Politburo adopted a transparently hypocritical and mendacious resolution remarking on NKVD successes in destroying \"enemies of the people and foreign intelligence agencies' espionage-sabotage networks\" but also condemning \"shortcomings and perversions\" in the NKVD's work. While repeatedly demanding an intensified struggle against enemies, Stalin had never questioned the mission of mass terror that he himself had conceived and promoted. Yezhov and the NKVD now stood accused of doing what Stalin had ordered them to do. If Yezhov had been allowed to make a serious case for himself, he would have had no trouble doing so. But as he knew better than anyone, that was not how the Stalinist system worked. All he could do was hope and repent.\n\nHaving done his job, the faithful Yezhov was no longer needed. He was arrested and shot as the head of a (nonexistent) counterrevolutionary organization within the NKVD. Stalin apparently did not feel the need to goad excessive public outrage, and Yezhov's downfall was arranged without fanfare. The cautious tidiness with which he was removed shows that Stalin was reluctant to draw public attention to the activities of the NKVD and the mechanics of the Great Terror. Yezhov was Stalin's senior scapegoat. He paid the ultimate price so that his _vozhd_ could remain above suspicion. For the Soviet people, the Terror became the \"Yezhovshchina\"\u2014a term using a Russian suffix suggesting some rampant evil.\n\nThe final stage of the Great Terror\u2014its unwinding, which Stalin carefully controlled\u2014mainly targeted Yezhov's top lieutenants at the NKVD. A miniscule number of ordinary citizens swept up by the large-scale operations\u2014primarily those who had fallen into NKVD clutches during the second half of 1938\u2014were released. The machinery of terror remained in place with only minor adjustments, and ruthless repression continued until Stalin's death. The _vozhd_ never stopped believing that enemies were all around or demanding that they be unmasked, arrested, and tortured. But he never again resorted to repression on the scale seen during 1937\u20131938.\n\nStalin must have been aware of the Terror's devastating consequences, yet he never, either in public or even within his inner circle, questioned its necessity. But the consequences could not have escaped his attention. A huge number of those responsible for running the Soviet economy had been arrested. Workplace discipline suffered, and engineers were afraid to propose any changes or innovations that might later subject them to unscrupulous accusations of \"wrecking.\" The Terror led to a sizable decline in the rate of growth in industrial production. The military too suffered from a shrinking pool of experienced and competent commanders and a decline in discipline and responsibility. The Red Army was so heavily affected by repression that the Soviet leadership was forced to return many previously arrested or discharged commanders to service, at least those the NKVD had not yet had time to execute.\n\nThe Great Terror of 1937\u20131938 put huge stresses on Soviet society and caused widespread misery. Millions of people were directly affected. Many who escaped being shot, confined to labor camps, or subjected to internal deportation lost their jobs or were evicted from their apartments or even towns for the sole crime of having ties to \"enemies of the people.\" Such abuses and upheavals could not be forgiven and passively accepted. Although fear was a fairly effective means of keeping the population from expressing its displeasure, grievances were lodged. In 1937\u20131938, these grievances mainly took the form of the millions of complaints that came pouring into government and party offices. In January 1937 alone, 13,000 complaints were filed with the procuracy, and in February\u2013March 1938 the number reached 120,000. It has not yet been established how many letters and petitions were sent to Stalin himself during the Great Terror or how many actually reached his desk. The records are either inaccessible or were not preserved. We can only assume that Stalin's office was inundated with such petitions. The _vozhd_ could not have been entirely shielded from his subjects' desperation, grief, and disillusionment.\n\nWhat was Stalin's reaction to the suffering of his fellow citizens? The historical record gives no clear answer to this question, but there is no evidence that he felt the slightest remorse or pity. Nevertheless, he could not entirely ignore political realities. Although he still despised imaginary enemies and feared imaginary conspiracies, he never repeated his experiment in large-scale terror. After 1938, repression continued on a smaller scale and in a more routine manner.\n\n###   **THE SEARCH FOR ALLIES**\n\nThe Great Terror damaged the Soviet Union's international reputation. Stalin undoubtedly understood that people in the West, especially on the left, were shocked to learn that prominent revolutionaries were being put to death. In an effort to minimize the impact on public opinion, the campaign of repression was paralleled by an energetic propaganda campaign. Accounts of the Moscow trials\u2014at which Lenin's comrades-in-arms and other Old Bolsheviks admitted plotting terrorist acts against Stalin and having ties with foreign intelligence agencies\u2014were translated into European languages and widely circulated. Prominent Western intellectuals and cultural figures were invited to Moscow. The German writer Lion Feuchtwanger met personally with Stalin and then wrote a book casting the Soviet Union in a favorable light. Caught between the hammer of Nazism and the anvil of Stalinism, many were ready to delude themselves as to the regime's true nature. The West's political decision makers, however, had every reason not only to distrust Stalin, but also to see the hysteria over supposed enemies as evidence of weakness. The purge of Red Army commanders and the execution of well-known Soviet marshals in particular made the regime appear unstable. The West clearly saw the Terror in very different terms than Stalin. Obsessed with the idea of a fifth column, Stalin simply failed to understand that his moves to arrest and shoot so many of his own citizens looked more like weakness and instability than strength.\n\nTo some extent the Western observers were right. Signs of the Terror's devastating impact on Soviet military might soon became apparent. In June 1938, the NKVD general in charge of the Far East, Genrikh Liushkov, crossed the Soviet border into Manchuria and offered his services to the Japanese. This was of course a traitorous act, but Liushkov was pushed in that direction by Stalin. After faithfully serving the regime and spilling rivers of other people's blood, he realized it would soon be his own turn to bleed. When a summons came to report to Moscow, Liushkov decided that his best option was to defect. Given his years as a top NKVD official in Moscow, his experience working face-to-face with Stalin, and his role as secret police chief of the militarily critical Far Eastern region, he had a great deal to offer. He was well informed about military readiness in the Far East and the makeup and placement of Soviet troops\u2014and he shared all this information with the enemy. Stalin further undermined military preparedness in the Far East by ordering another wave of arrests within the army. Meanwhile, in July and August 1938, the Red Army clashed with Japanese forces near Lake Khasan, an area near the borders with Korea and China. Stalin closely monitored this conflict and demanded decisive action. In a conversation with the commander of the Far Eastern front, Marshal Vasily Bliukher (who had expressed his reluctance to use aviation), Stalin issued the following order: \"I don't understand your fear that bombing might hurt the Korean population or your fear that aviation won't be able to fulfill its mission because of fog. Who forbade you to hurt the Korean population in time of war with Japan? Why would you care about Koreans when the Japanese are striking at lots of our people? What do a few clouds matter to Bolshevik aviation when it wants to truly defend the honor of its Motherland?\"\n\nWhile the Battle of Lake Khasan ended favorably for the Soviet side, the clash exposed significant shortcomings in the combat readiness of Red Army troops and command structures. As usual, Stalin assumed that the army's poor performance was the result of treachery. Marshal Bliukher was arrested and died in prison after being brutally tortured.\n\nRepression and the perception of Soviet weakness were not the primary causes of Stalin's deteriorating relations with the West. The mass arrests just added to Western leaders' list of reasons for mistrusting him. A warming of relations with France in the mid-1930s did not last, despite the threat posed to both countries by the rapid rise of Nazism. In the Spanish Civil War, the Soviet Union and Western democracies found themselves in frequent disagreement. Underlying this tendency toward poor relations, despite their common collective security concerns, was the fundamental incompatibility of Stalinism with \"bourgeois\" democracy. During the second half of the 1930s Western leaders preferred to appease Hitler rather than form an alliance with Stalin, a trend that reached its climax with the Munich Agreement. On 30 September 1938, the leaders of Great Britain and France, Neville Chamberlain and \u00c9douard Daladier, signed an agreement with Hitler and Mussolini handing over Czechoslovakia's Sudetenland, an area primarily populated by German speakers, to Germany. Czechoslovakia was forced to accept this devastating pact. The Soviet Union was simply ignored, even though it and France had signed mutual assistance agreements with Czechoslovakia. Stalin was shut out of European great power politics.\n\nStalin undoubtedly took such marginalization as a personal insult. Munich only intensified his fear that the democracies and fascists were conspiring against the USSR and planning to channel Nazi aggression eastward. He could not respond from a position of strength. In addition to expressing his outrage, in late September Stalin ordered a Red Army troop buildup along the USSR's western border, a purely demonstrative move that is unlikely to have worried the Germans. In any event, just days later, in mid-October, the Politburo decided to disband the reserve units that had been mobilized in response to the events in Czechoslovakia. A total of 330,000 troops, 27,500 horses, and 5,000 vehicles and tractors were released from active duty.\n\nIn practical terms, Stalin could do little about the Munich Agreement beyond trying to drive a wedge between the Western democracies and Hitler. To this end, he made a series of statements condemning Great Britain and France, while opening the door to improved bilateral relations with Germany. The most significant overture to Germany came during a speech at the Eighteenth Party Congress in March 1939, in which Stalin warned the English and French that he had no intention of \"pulling the chestnuts out of the fire\" for them (a line that earned this address the nickname \"the chestnut speech\" in the West) and accused them of attempting to provoke conflict between the USSR and Germany. He told Germany that the Western powers had not succeeded in \"enraging the Soviet Union against Germany, poisoning the atmosphere, and provoking conflict with Germany on no apparent grounds.\" These pronouncements took on special significance several days later when Europe's fragile peace was broken. Hitler, confident that no one would stop him, seized the entire territory of Czechoslovakia. Even the most optimistic observers now realized that Munich had made world war all but inevitable. As a third party to the growing conflict, Stalin and the Soviet Union were in a position to choose sides.\n\nThe spring and summer of 1939 were a time of urgent diplomatic maneuvering and negotiation. Understanding the nature of these efforts and the actual intentions of the parties involved was difficult enough for their direct participants, to say nothing of historians today. Nobody trusted anybody, and all were trying to outsmart their adversaries and partners alike. Such confusion was surely true of the talks between the Soviet Union and the Western powers of England and France. Progress was painfully slow, despite the efforts of Soviet foreign affairs commissar Maksim Litvinov, who staked his reputation on building cohesion among anti-Hitler forces. In early May 1939, Stalin relieved Litvinov of his duties and put Molotov in charge of foreign affairs. This change was undoubtedly intended as a gesture of friendship toward Germany, but it also radically reshaped foreign policy decision making. The new arrangement allowed Stalin to take full control of foreign affairs, not only in terms of their guiding principles (as he had always done), but also their day-to-day operations. Molotov, with whom Stalin was in almost constant conversation, was a more convenient foreign-policy right hand than Litvinov, who rarely visited Stalin's office. Such practical details were important to the _vozhd_. At the top tier of Soviet power, government was adapted to Stalin's habits and rhythms, and the choice of Molotov to oversee foreign affairs at this critical time is a prime example of this adaptation.\n\nWhat was uppermost in Stalin's mind during this period\u2014putting pressure on his Western partners or exploring the possibility of an alliance with the Nazis? It is tempting to assume that he had decided to align himself with Hitler long before the fateful events of 1939. Arguments in favor of this view include the general idea of an affinity between totalitarian regimes and Stalin's mistrust of the changeable Western democracies, which seemed inclined to retreat in the face of brute force. But the foundation for a Nazi-Soviet alliance was actually flimsy. The available evidence offering insights into Stalin's thinking is open to interpretation. On one hand, Mikoyan reported that Stalin spoke approvingly of Hitler's 1934 purges. We also know that the Soviet leader initiated overtures aimed at establishing direct contact with Hitler. Most damning of all was the result: an impressive demonstration of Soviet-German \"friendship\" in the fall of 1939. But on the other hand, there is convincing evidence that Stalin had little faith in Hitler as a potential ally. If he trusted the German leader, there likely would not have been a powerful anti-Nazi propaganda campaign waged in the USSR or mass repression against Soviet Germans\u2014both of which were carried out over the strong objections of the Nazi government. Stalin's attitude toward the Germans seemed to alternate between approval and annoyance. Responding to a September 1938 NKVD memorandum about the destruction of a cemetery dating to World War I for German soldiers and officers in Leningrad Oblast, rather than replying with his usual laconic \"in favor,\" Stalin wrote, \"Correct (tear it down and fill it in).\" The German interpreter present at negotiations with Foreign Minister Joachim von Ribbentrop in Moscow also offers some insight into the Soviet leader's mindset. Stalin apparently rejected a draft of an upbeat press communiqu\u00e9 with the words: \"Don't you think that we should give more consideration to public opinion in both our countries? We've been slinging mud at one another for years now.\"\n\nWhatever Stalin's true inclinations were, it was Hitler who took the initiative in bringing about a Soviet-German non-aggression pact. As soon as the German chancellor decided that his invasion of Poland, scheduled for 1 September, would require Soviet cooperation, he took steps to promote a rapprochement between the two countries. On 21 August Stalin received a personal correspondence from Hitler hinting rather transparently at his plans for Poland and expressing the urgent desire to conclude a non-aggression pact within a few days. Hitler asked that Stalin receive von Ribbentrop in Moscow the very next day or at least on 23 August. On 21 August Molotov handed Stalin's response to the German ambassador in Moscow. Von Ribbentrop could come to Moscow on the later date.\n\nStalin and Molotov were both there to receive the German foreign minister. The meeting was cordial, even amicable. Each side got what it wanted. In addition to the non-aggression pact, Stalin insisted that a secret protocol be drawn up stipulating that Germany and the Soviet Union would divide up Eastern Europe. The eastern portion of Poland, which then included the western parts of both Ukraine and Belarus; Latvia; Estonia; and Finland were recognized as belonging within the Soviet sphere. Germany also supported Soviet pretensions to Bessarabia. Western Poland and Lithuania would go to Germany. Subsequent negotiations gave Lithuania to the Soviets. The protocol wound up being a sort of Brest-Litovsk in reverse. Hitler needed a worry-free border with the USSR, and he would pay for it with territorial concessions.\n\nStalin kept the threads of the Soviet-German negotiations in his own hands. The only other person involved was Molotov. What history calls the Molotov-Ribbentrop Pact was actually an agreement between Stalin and Hitler. Stalin took total responsibility for the \"friendship\" with Germany and doubtless had very specific motives for entering into the risky alliance. The nature of these motives is one of the most important questions facing his biographers.\n\nFirst, there were the political and moral aspects of the problem. Stalin, no doubt, was fully aware of the agreements' political and moral undesirability. We can infer this from the persistence with which the Soviet Union denied that a secret protocol existed. When copies came to light, Soviet leaders proclaimed them to be forgeries. Stalin understood that the sudden switch from hatred toward the Nazis to friendship would be ideologically disorienting, both within the USSR and in the world Communist movement. This problem was secondary, however, and could be dealt with using the boilerplate explanation: the pact was in the ultimate interests of socialism. Within the USSR, skeptics could be dealt with in the usual manner. The moral issue actually took on greater weight later, after Germany's defeat, when the international community condemned Nazism as an absolute evil.\n\nIn 1939, even the most democratic of Western politicians took a flexible approach to dealing with the Nazis\u2014anything to avoid war. Great Britain and France could hardly be proud of these policies, and it would be naive to expect Stalin to sympathize with their approach. Nobody was refusing to deal with Hitler out of principle; it was a matter of what agreements were achievable and acceptable. In terms of political pragmatism, Stalin was no worse than the Western parties to the Munich Agreement. In signing the Munich pact, Great Britain and France not only shielded themselves from Hitler's aggression\u2014or so they thought\u2014but also placed a number of small countries, not just the Sudetenland, in peril. Stalin took his self-interest a step further and joined in the division of Eastern Europe. He was sure that Munich had pushed Hitler's aggression eastward, so it only made sense for him to set the F\u00fchrer's mind at rest about the East and attempt to turn him back toward the West. From the Soviet perspective, Stalin was only trying to get back what was rightfully Russia's. Redressing a historical injustice by restoring parts of the Russian Empire that had been taken by force when the country was weakened by war and revolution must have been a part of the Soviet dictator's thinking. This motive drew sympathy not only within the USSR, but among some foreigners as well.\n\nIt is difficult to say how prominently emotional and moral considerations figured in Stalin's thinking. Surely they were far outweighed by the immediate risk of war. There is a broad spectrum of opinion on the geo-strategic reasons for the agreement with Germany. At one end are those who point to the speech Stalin allegedly gave to the Politburo on 19 August 1939, just before the pact was signed. One version of this speech, published in France in late 1939, caused a sensation as a supposed expos\u00e9 of Stalin's expectations of what war would mean for the USSR. The French publication quotes him giving the following justification for the pact with Hitler: \"We are absolutely convinced that if we conclude an agreement to ally with France and Great Britain, Germany would be forced to give up on Poland and seek a modus vivendi with the Western powers. War would be averted and the subsequent course of events would prove dangerous for us.\"\n\nThis alleged speech made it seem as if Stalin believed war was needed to weaken the West, expand the USSR's boundaries, and help spread communism in Europe. These supposed remarks compromised Stalin in Hitler's eyes and made the French Communist Party look like an agent of hostile forces. Publication of this \"top secret\" document clearly served somebody's interest.\n\nMost historians have never assigned much significance to this forgery. Neither the Politburo archive nor Stalin's own files contain even circumstantial evidence of such a speech\u2014or even that the Politburo met on 19 August. This is not surprising. Based on what is known about Stalin's dictatorship in the late 1930s, it is hard to believe he would speak so openly to his Politburo comrades, for whose opinions\u2014and even existence\u2014he felt no need whatsoever. The \"transcript of Stalin's speech,\" like many other well-known forgeries, promotes a particular viewpoint in regard to Stalin and his actions. According to this extreme view, Stalin concluded a pact with Hitler because he wanted war in Europe as a means of carrying out his plans.\n\nThe views reflected in the forgery differ sharply from statements by Stalin for which we do have a reliable source. Georgy Dimitrov, the head of the Comintern at the time, recorded in his diary the following remarks by Stalin, made at a meeting on 7 September: \"We would rather have reached agreement with the so-called democratic countries, so we conducted negotiations. But the English and French wanted to use us as field hands and without paying us anything! We, of course, would not go work as field hands, especially if we weren't getting paid.\" Nobody should feel compelled to take Stalin's words at face value. But the possibility that he was driven toward his pact with Hitler by his country's isolation and a sense that he was undervalued by his Western allies deserves serious consideration.\n\nThe diversity of opinions concerning Stalin's motives in August 1939 reflects the complexity of events and abundance of international intrigues during the lead-up to World War II. In recent times, however, pieces of historical evidence have become available that clarify the situation. The negotiations among the Soviet Union, England, and France were fraught with problems, and both the Soviet and the Western sides were to blame for their lack of progress. Stalin saw in the Western nations' obstinacy further confirmation of their intent to appease Hitler at the expense of the USSR. Most likely, he thought war between Germany and Poland was inevitable however the other powers were aligned, and he was probably right. It was difficult to predict how such a war would affect his country. The Nazis would be right on Soviet borders. Hitler was prepared to pay a fair price for a pact that would grant Soviet blessing to this arrangement. For Stalin, the pact offered nearly risk-free expansion of Soviet territory and a chance to create a buffer between his country and the war about to be unleashed on Europe.\n\nThen there were the Japanese. In the spring of 1939, clashes were already erupting between Soviet and Japanese troops in Mongolia. The first engagements did not end well for the Red Army, but by the time of the von Ribbentrop negotiations, the Soviet side was achieving significant victories. These strengthened Stalin's position in his dialogue with Germany. The signing of the pact was a diplomatic blow to Japan. At least for the near term, it could not count on its German ally in its confrontation with the USSR. There is no serious argument against assuming that Stalin was guided by all these considerations.\n\nIn August 1939, Stalin had every reason to consider himself ascendant. He had concluded an agreement with the world's strongest military power and averted a war with it, at least for the time being and possibly for a long time to come. He had won back much of the territory lost by Russia two decades earlier. He could anticipate reaping third-party benefits as the warring European countries created a new balance of power on the continent. The pact with Germany and secret protocol were morally distasteful and they diminished the Soviet Union's reputation with progressives around the world, but these were relatively minor concerns. Was Stalin looking into the distant future and plotting the creation of a Communist empire extending over a large part of Europe? Such a prospect must have been hard to envision in 1939. Did he conclude the pact in order to provoke war in Europe? Given Nazi aggression, such a provocation seems hardly necessary. It is another matter that we will never know how the war would have played out had Stalin not signed the agreement with Hitler and continued to try to make common cause with England and France.\n\nWe will also never know how the Molotov-Ribbentrop pact and secret protocol would look today had Stalin used these documents simply to restrain Germany and expand the Soviet sphere of influence. In that case, posterity would have seen the Soviet-German understanding as an unsavory but understandable and pragmatic maneuver by a savvy politician. But Stalin was the iron-fisted ruler of a totalitarian system. He used the agreement not simply to keep the Nazis out of the small countries along the USSR's border, but also to assimilate new territories. And assimilation, in Stalin's world, meant aggression and the brutal purging of society.\n\n###   **AS WAR RAGED**\n\nGermany invaded Poland on 1 September 1939. Poland's allies, Great Britain and France, responded with a declaration of war, and World War II was under way. The Nazis swept through Poland almost unopposed. The British and French forces that came to Poland's defense assembled too slowly and seemed in no great hurry to fight. The Red Army's entry into Poland, and the line dividing this country between Germany and the USSR, had been determined during the von Ribbentrop negotiations in Moscow the previous month, but Stalin was also in no hurry to begin military actions. The Soviet invasion began only on 17 September, after the outcome of Germany's Polish campaign was fully evident. Clearly, Stalin preferred to wait until the risk of an invasion was minimal and Soviet aggression would not look like it had been coordinated with Germany's. The Red Army primarily occupied the parts of western Ukraine and western Belarus that Poland had seized in 1921. The official propaganda claimed that Soviet actions were being taken on behalf of the Ukrainian and Belarusian peoples and described the invasion as an act of \"liberation.\" This interpretation suited Western politicians, who still hoped to win Stalin to their side.\n\nThe reality bore little resemblance to the image promoted by Soviet propaganda. The Soviet absorption of western Ukraine and western Belarus was not a joyous reunion of divided nations. For the first year and a half of their sovietization, the new territories underwent the same violent social engineering that the USSR had been experiencing for decades. The goal was to force them into the Soviet mold: do away with the capitalist economic system, inculcate a new ideology, and destroy any real or imagined hotbeds of dissent against the regime. The traditional methods were used. \"Suspicious\" people were shot, sent to labor camps, or exiled to the Soviet interior; private property was expropriated; and farming was brought into the kolkhoz system. The Stalinist regime was trying to eliminate, in just months, any potential for anti-Soviet collaboration. An important component of this bloody effort was the notorious Katyn massacre. On 5 March 1940 the Politburo adopted a decision to put to death many thousands of Poles held in prisoner-of-war camps or regular prisons in the western provinces of Ukraine and Belarus. The victims were largely members of the Polish elite: military and police officers, former government officials, landowners, industrialists, and members of the Polish intelligentsia. A total of 21,857 people were shot in April and May 1940. In exterminating these people, Stalin was clearly attempting to head off any movement to restore the prewar Polish leadership.\n\nStalin proceeded more cautiously and gradually in the Baltic states, which the Molotov-Ribbentrop pact had recognized as falling into the Soviet sphere of influence. Immediately after the partition of Poland and the settlement of various issues with Germany, in late September and October 1939 the Soviet leadership forced Estonia, Latvia, and Lithuania to permit Soviet military bases on their territory, including in the Baltic Sea ports. Molotov and Stalin personally took on the task of intimidating their Baltic neighbors during negotiations at the Kremlin. These meetings were tense. When the representatives of the Baltic governments insisted on preserving their sovereignty and neutrality, Molotov threatened them with war and refused to make the slightest concession. Stalin applied a softer touch and offered a few insignificant compromises, reducing, for example, the number of troops to be stationed in the Baltic countries. The intransigence of the Baltic representatives evidently irritated him, but he kept his temper. According to the Latvian foreign minister, Stalin wrote, doodled, strolled around the room, and picked up books and newspapers while others were speaking. At critical points he interrupted and went off on tangents, expounding at length on abstruse ethnographic or historical topics.\n\nThe Soviet side obviously had the advantage. Red Army units were already positioned along the Baltic nations' borders. Germany\u2014the only possible counterweight to the Soviet Union\u2014was acting in concert with the USSR. Stalin, nevertheless, did not hurry to overwhelm his victims, instead taking what he wanted a little at a time. Until Soviet troops entered Latvia, Lithuania, and Estonia, Stalin applied a tactic he shared with Comintern head Dimitrov: \"It's not good to rush ahead!... Slogans should be advanced that suit the particular stage of the war.... We think we've found in mutual assistance pacts (Estonia, Latvia, Lithuania) a form that permits us to bring a number of countries into the Soviet Union's orbit of influence. But for this, we need to hold back\u2014to strictly respect their internal regimes and independence. We won't try to sovietize them. The time will come when they'll do it themselves!\"\n\nThe prediction Stalin makes in the last sentence of this explanation betrays his ultimate goal: to sovietize and absorb the countries and territories added to his country's sphere of influence under the Molotov-Ribbentrop pact. From a historical standpoint, he could justify this goal as the reconstitution of the Russian Empire. As military strategy, it surely made sense to establish strong control over areas through which an attack might come. But the future\u2014the who, what, when, and where of the impending war\u2014was shrouded in uncertainty, and Stalin was forced to wait. For now, he preferred to play a balancing game and went out of his way to avoid unnecessarily irritating either Great Britain and France or, especially, the F\u00fchrer. There were many small signs of Stalin's caution during this period. We see it, for example, in his reaction to a report from Belarus on a speech given to the republic's parliament by army group commander Vasily Chuikov. Intoxicated by his easy victory in Poland, Chuikov told his audience in this speech, which went out over the radio, \"If the party says the word, we'll march to that tune\u2014first Warsaw, then Berlin!\" Furious, Stalin wrote Chuikov's boss, Voroshilov: \"Com. Voroshilov. Chuikov is evidently at least a fool, if not an enemy element. I say he should be given a spanking. At the very least.\" While Chuikov apparently survived, many other Soviet citizens who expressed anti-Nazi sentiments were not so lucky. Between August 1939 and the beginning of war between Germany and the USSR, expressions of anti-Hitlerism were treated as a crime in the Soviet Union.\n\nStalin's stealthy approach to expansion was bound to hit a stumbling block eventually, and that stumbling block was Finland. In October 1939, having won the concessions he wanted from Latvia, Lithuania, and Estonia, the Soviet dictator turned his attention to his Nordic neighbor, which the Nazis had recognized as part of the Soviet sphere of influence. Finland was presented with much harsher demands than the Baltic countries. In addition to the placement of Soviet military bases in Finland, the USSR demanded a large portion of Finnish territory near Leningrad in exchange for land in less populated border regions. On the surface, these demands appeared perfectly reasonable. The USSR wanted to be able to defend Leningrad\u2014the country's second capital and a major center of defense production\u2014and its approaches from the Baltic Sea. But Finland, a former province of the Russian Empire that had received its independence in 1917, suspected the USSR of imperial ambitions. The Finns remembered the horrors of the 1918 civil war, which had largely been provoked by their Communist neighbor. They also noted the recent example of Czechoslovakia, which had given up the Sudetenland only to be entirely taken over by Hitler. Finland categorically refused the Soviet demands. Stalin decided to use force.\n\nThe Red Army invaded Finland in late November, having every reason to believe that its campaign would be short and successful. Finland was a tiny country with no more than 4 million inhabitants\u2014forty times smaller than the Soviet population. The territory, economic resources, and military might of the two countries were not comparable. The 26 tanks with which Finland began the war would have to fend off 1,500 Soviet ones. Furthermore, the USSR would be able to throw significant additional troops and resources into the battle, and it did so as the conflict\u2014known as the Winter War\u2014unexpectedly continued. Staking success on overwhelming force, Stalin decided to make Finland the site of his first experiment applying a different takeover model from the one used in the Baltic states. The Red Army brought with it the \"people's government of Finland,\" consisting of Communists hand-picked in Moscow. This was the government that would be installed to rule a defeated Finland.\n\nBut the people's government of Finland never took office. The Finns showed the Red Army fierce and capable resistance. As the war dragged on, a strongly anti-Soviet mood spread throughout the rest of the world. The USSR was expelled from the League of Nations, and France and England prepared to intervene on the Finnish side. Stalin decided not to tempt fate. Despite a series of victories made possible by a major buildup of forces, in March 1940 he signed a peace treaty with Finland. Plans to sovietize the USSR's northern neighbor were set aside. The Finns wound up losing a significant portion of their territory and economy, but they maintained their independence. The Red Army lost approximately 130,000 troops, either killed in combat, dying from wounds or disease, or missing in action. More than 200,000 were wounded or frostbitten. The Finnish losses were significantly lower: 23,000 killed or missing in action and 44,000 wounded. The war, a major symbolic defeat for the USSR and Stalin personally, exposed weaknesses in every component of the Soviet military machine. Historians have proposed that it was this conflict that prompted Hitler to push forward his timetable for invading the Soviet Union.\n\nSoviet failure in Finland contrasted ominously with Hitler's triumphant advance. Soon after the Winter War, in April\u2013June 1940, Germany occupied a number of West European countries, forcing France to capitulate in just weeks. British troops were evacuated from the continent, and Italy entered the war on Germany's side. France's quick and inglorious fall radically changed the situation in the world. Khrushchev later described how upset and worried Stalin was about the French defeat, lamenting the country's inability to put up a fight. Even if Khrushchev's account is tainted by hindsight, there is no reason to doubt Stalin's general sense of alarm. The Soviet leader had lost his former maneuvering room between the warring sides. A strategy that had looked rock solid had suddenly turned to dust. Now there would be no easy way out through a mutually convenient treaty. A huge threat hung over the Soviet Union. The nation that had been its sole if unreliable ally began to look like a mortally dangerous enemy.\n\nStalin reacted feverishly. As Germany solidified its control over Western Europe in the summer of 1940, Latvia, Lithuania, and Estonia were incorporated into the USSR, as were Bessarabia and part of Bukovina, both of which had been taken from Romania. A top priority for the Stalinist leadership was the rapid sovietization of these new possessions. A large-scale expropriation of private property was accompanied by a massive purge of the population. Repression now fell on the newly integrated western regions. As usual, in addition to the arrest and execution of \"unreliable\" citizens, many were exiled to remote areas of the Soviet interior. In four relocation campaigns in 1940 and the first half of 1941, some 370,000 people were moved from western Ukraine, western Belarus, the Baltic states, and Bessarabia into the Soviet interior. This was a huge number given the small populations of these regions.\n\nBusy as he was dealing with hundreds of thousands of \"suspect\" people in the newly sovietized areas, Stalin did not forget about faraway enemies. In August 1940 Lev Trotsky was killed in Mexico on his orders. An NKVD agent who had penetrated Trotsky's inner circle killed the former opposition leader with an ice pick. Stalin had long stalked his most implacable, energetic, and eloquent foe. Was he driven by a personal thirst for revenge or concern that Trotskyites within the USSR might rally in time of war? Most likely both factors played a role.\n\nHaving subdued the territories stipulated for Soviet control under his agreements with Hitler, Stalin faced the question: What now? On one hand, the success of the German war machine made friendship with Hitler more important than ever. On the other, the growing threat that Nazi aggression posed to the USSR made such friendship increasingly dangerous. Soviet and German interests were clashing in Finland, where Germany, having occupied Norway, was making inroads as a result of the outcome of the Winter War. The two powers were also clashing in the Balkans due to Hitler's desperate need for Romanian oil. Stalin also hoped to gain a share of Romania and Bulgaria and achieve a long-standing Russian imperial goal: control over the Turkish Straits.\n\nFor Stalin, the signing of the Tripartite Pact among Germany, Italy, and Japan on 27 September 1940 was bad news. The three aggressor countries were agreeing to help each other divide up the rest of the world. Germany and Italy were recognized as dominant in Europe, and Japan in Asia. In theory, this agreement was aimed at Great Britain and the United States. But Stalin had every reason to worry.\n\nBelieving it necessary at this stage to avoid exacerbating tensions with the Soviet Union, in November 1940 Hitler made a conciliatory gesture by inviting Molotov to Berlin. During negotiations with Hitler and von Ribbentrop, the Soviet foreign minister insisted that his country's interests be recognized in Finland, the Balkans, and the Turkish Straits. Hitler was equally firm, especially when it came to Soviet claims in Finland and Romania. While avoiding making specific promises, Hitler suggested that the USSR become a fourth partner in the Tripartite Pact, take part in dividing up the British Empire, and determine exact Soviet spheres of influence through further negotiations. Both sides apparently were probing to see what such an arrangement might offer. Was this four-way alliance ever a real possibility? On one hand, we know that while these negotiations were going on, Hitler was already hatching plans to invade the USSR. We also know that Stalin was entirely aware of the threat posed by Germany. On the other hand, in August 1939, when the Molotov-Ribbentrop Pact was being concluded, the Soviet Union and Germany were just as fundamentally hostile toward one another. Everything had changed in an instant once Stalin and Hitler found a point of common interest.\n\nOn 25 November 1940, shortly after his return from Berlin, Molotov gave the German ambassador in Moscow the Soviet conditions for a four-way pact. Here, Stalin was again resorting to the tactic that had yielded success in August 1939. In exchange for the support of his partners (and with an understanding that significant amounts of Soviet raw materials would be supplied to Germany), he issued four specific demands. First, German troops must pull out of Finland. In exchange he would guarantee that Finland would remain friendly toward Germany and supply it with timber and nickel, a point on which Hitler had particularly insisted during his talks with Molotov. Second, Stalin laid claim to Soviet influence in Bulgaria, including the conclusion of a mutual assistance treaty and the establishment of Soviet military bases near the Turkish Straits. Third, the three partners must recognize the Soviet Union's right to expand southward through Iran and Turkey to the Persian Gulf. Fourth, Japan must give up claims to coal and oil concessions in North Sakhalin in exchange for \"fair compensation.\" This program, which closely mirrored the aspirations of the Russian Empire, probably included everything Stalin wanted, and he was undoubtedly prepared to bargain. The submission of these conditions to Berlin indicated, presumably, his readiness to cast his lot with the aggressor countries.\n\nIt has been asserted, however, that Stalin never seriously considered Hitler's proposal to form a four-way pact and that the demands sent to Berlin on 25 November were a delaying tactic, intentionally designed to be unacceptable to Germany. The most significant evidence cited by proponents of this view is an account of a Politburo meeting on 14 November 1940, during which Molotov supposedly reported on his negotiations in Berlin. The account has Stalin stating that Hitler could not be trusted and that the time had come to prepare for war against Germany. But there is no record of any such Politburo meeting or of Stalin making this remark. The only source of this information is Yakov Chadaev, chief of administration for the Sovnarkom (Sovet Narodnykh Kommissarov; the Council of People's Commissars\u2014the Soviet cabinet), who claimed to have been present and to have taken notes at the meeting.\n\nThere are several reasons to doubt Chadaev's account. First, Molotov could not have been in Moscow on 14 November since that is the day he boarded the train home from Berlin. Furthermore, it is hard to understand why Stalin would have wanted to hold such a meeting, especially one including people who were not Politburo members. Most other major foreign policy decisions during the prewar years (including the Nazi-Soviet Pact of 1939) were not voted on by the Politburo. Stalin kept his foreign policy cards close to the vest, at most consulting with Molotov. The talks exploring joining the Tripartite Pact were a closely held state secret.\n\nAnother piece of evidence casting doubt on the meeting is the log of visitors to Stalin's office, which shows no activity between 6 and 14 November. It is nearly certain, therefore, that Stalin spent these days at his dacha. Finally, there is no evidence of any Politburo meetings in November, and even if there had been, Chadaev is unlikely to have been allowed to attend, to say nothing of his taking notes. As chief of administration for the Sovnarkom, he gained easy access to Stalin only after the _vozhd_ became chairman of that body in May 1941. The fact remains that on 25 November 1940, Stalin responded quickly and substantively to Hitler's proposal for an enhanced alliance. Berlin did not react to Stalin's conditions, despite being prodded by Moscow. Soon after Molotov left Berlin, Hungary, Romania, and Slovakia\u2014three countries entirely dependent on Hitler's will\u2014joined the pact, followed in March 1941 by Bulgaria, which Stalin had so insistently claimed for his sphere of influence. In April Germany took over Greece and Yugoslavia.\n\nIn December 1940, Hitler approved plans to invade the USSR in May 1941. The only allies Stalin had left were his own people. The _vozhd_ spent the final months before Hitler marched into the Soviet Union consolidating his power and making extraordinary efforts to bolster the country's military strength.\n\n###   **THE CONSOLIDATION OF SUPREME POWER**\n\nOne important result of the Great Terror was the dramatic shift in the balance of power within the Politburo. Remnants of collective leadership survived into the mid-1930s, but by late 1937 the Politburo was entirely subject to Stalin's will. The Terror brought his power to new heights. He was now a full-fledged dictator in whose hands rested the lives not only of ordinary citizens, but also those of his most esteemed fellow leaders. Five Politburo members (Stanislav Kosior, Vlas Chubar, Robert Eikhe, Pavel Postyshev, and Yan Rudzutak) were shot, and one (Grigory Petrovsky) was expelled from the upper echelons and survived only because Stalin chose to show him clemency. Another name on the list of Stalin's high-ranking victims was Grigory Ordzhonikidze, driven to suicide by Stalin's ruthlessness. But even the top leaders who held onto their posts found themselves in an impotent and demeaning position, forced to carefully walk the line between power and death and unable to protect their most valued subordinates or even close friends and relatives. The names of top leaders inevitably came up in the countless confessions the NKVD extracted under torture. It was up to Stalin to decide what denunciations and incriminations should be taken seriously. Anyone could suddenly be labeled an enemy.\n\nAs Stalin's longtime comrades disappeared from the top leadership, younger faces took their place. As noted, these replacements were an important element of his consolidation of power. Lacking the revolutionary credentials of the older generation, these young leaders owed their standing directly to Stalin and were entirely dependent on him. In March 1939 Andrei Zhdanov and Nikita Khrushchev, members of this second generation, were granted full membership in the Politburo. At the same time, a member of the third generation, Lavrenty Beria, was made a candidate member. In February 1941 three other members of the third generation were added: Nikolai Voznesensky, Georgy Malenkov, and Aleksandr Shcherbakov. These appointments did not simply represent the normal advance of competent leaders up the career ladder. Stalin made a point of placing young officials in important posts, often as counterweights to his older, more deserving colleagues.\n\nChanges to the composition of the Politburo were just one manifestation of processes taking place under the surface that ultimately destroyed the formal aspects of the collective leadership and substituted new unofficial or quasi-official institutions adapted to the administrative and political needs of Stalin's dictatorship and lifestyle. The deterioration of the Politburo's meaningful role was brought to its logical conclusion when it essentially ceased to function as a formal institution. During the years of the Great Terror, it was replaced by a narrower group within the leadership, always chaired by Stalin. In early 1938 the \"Secret Five\" took shape, consisting of Stalin, Molotov, Voroshilov, Mikoyan, and Kaganovich. This group, though not an official body, largely took the place of the Politburo. The only vote that mattered was Stalin's. In addition to his deliberations during meetings of the Five, Stalin settled many questions with individual members of the leadership. These ad hoc decision-making mechanisms bore little resemblance to constitutional structures or procedures and depended purely on the will of the _vozhd_. The meetings, following Stalin's habits and nocturnal lifestyle, took the most varied forms. Matters of state could be decided day or night, in Stalin's Kremlin office or at his dacha, in the movie theater or during long hours at the dinner table.\n\nThe next level of the pyramid of power consisted of governmental bodies to which Stalin delegated particular authority while retaining overall control. This system first took shape within the party's Central Committee apparat, which had the mission of promulgating ideology and selecting and assigning senior party and state officials. These key areas were overseen personally by Stalin's prot\u00e9g\u00e9s, Zhdanov and Malenkov, who could make relatively trivial decisions on their own but had to bring more consequential ones to Stalin for approval. In January 1941, Stalin explained the Central Committee's new modus operandi: \"It's been four or five months since we in the Central Committee have convened the Politburo. All questions are prepared by Zhdanov, Malenkov, and others in separate meetings with comrades who have the necessary expertise, and the job of governing is only going more smoothly as a result.\"\n\nOn the government side, accommodating the commissariats, departments, and committees of the Sovnarkom to the dictator's needs was more difficult. The Sovnarkom oversaw the entire Soviet economy, which was then laboring under the strain of urgent preparations for war. Stalin sought to make the bureaucracy into something he could steer at will, but the sluggishness and unmanageability of its agencies sent him into fits of irritation and temper. His frustration led to numerous attempts to reorganize how the system was managed by the country's top leadership. Finally, in March 1941, a new governmental body was created: the Bureau of the USSR Sovnarkom, consisting of Sovnarkom chairman Molotov and his deputies. This bureau was created as a governing group within the Sovnarkom, much like the leading group within the Politburo.\n\nAs part of the political intrigue around the reorganization, the relatively young Nikolai Voznesensky became first deputy to the government's chairman, Molotov. His appointment to such an important post, over the heads of more senior members of the Politburo such as Mikoyan and Kaganovich, heightened tensions within Stalin's inner circle. Even in memoirs written decades later, Mikoyan could not hide his hurt feelings: \"But what struck us most of all about the composition of the Bureau leadership was that Voznesensky became first deputy chairman of the Sovnarkom.... Stalin's motives in this whole leapfrog were still not clear. And Voznesensky, being naive, was very pleased with his appointment.\" In giving this important job to Voznesensky, Stalin may have been intentionally pitting him against Molotov, hinting that the Sovnarkom chairman was not able to handle all his duties and needed a younger and more energetic deputy. In any event, the entire government reorganization came with a chorus of reprimands and accusations directed against Molotov's Sovnarkom leadership. This was a clear sign that Stalin had something up his sleeve.\n\nHis plans became evident a month after the Sovnarkom Bureau was established. On 28 April 1941 Stalin sent a memorandum to Bureau members explaining that it had been created for the purpose of straightening out government operations and bringing an end to \"chaos\" within the economic leadership, which continued to decide \"important questions related to the building of the economy through so-called 'polling.'\" As an example of the inappropriate use of polling (having members of a committee vote on a circulated document individually rather than meeting to discuss it in person), Stalin pointed to a draft resolution concerning the construction of an oil pipeline in the Sakhalin area. Molotov had signed the document, he wrote indignantly, even though it had not been discussed by the Sovnarkom Bureau. After labeling this practice \"paper-pushing and scribbling,\" he issued an ultimatum: \"I think 'management' of this sort can't go on. I propose discussing this question in the Central Committee's Politburo. And for now, I feel compelled to say that I refuse to participate in voting through polling on any draft resolution whatsoever concerning economic questions of any consequence whatsoever if I don't see the signatures of the Sovnarkom Bureau indicating that the draft has been discussed and approved by the Bureau of the USSR Sovnarkom.\"\n\nThis outburst must have taken Molotov by surprise. Polling was standard practice in Soviet decision making. As recently as January 1941, Stalin himself had criticized the Sovnarkom for \"parliamentarianism,\" by which he meant that its members were having too many meetings. As everyone involved surely noticed, Stalin offered only one example of \"incorrect\" polling\u2014and not a particularly compelling one, as the question of the Sakhalin pipeline probably did not require detailed discussion at a bureau meeting. The charges leveled in the April memorandum sounded frivolous, and Molotov and the other Politburo members must have realized that they were a pretext. The discussion of Stalin's memorandum led to a Politburo decision, dated 4 May 1941. It read in part as follows:\n\nI. In the interests of full coordination between Soviet and party organizations and the unconditional assurance of unity in their work as leaders, as well as to further enhance the authority of Soviet bodies given the current tense international situation, which demands every possible effort by Soviet agencies in the defense of the country, the Politburo unanimously resolves:\n\n1. To appoint Com. I. V. Stalin Chairman of the Council of People's Commissars [Sovnarkom] of the USSR.\n\n2. To appoint Com. V. M. Molotov Deputy Chairman of the USSR Sovnarkom and to place him in charge of the foreign policy of the USSR, leaving him in the post of People's Commissar for Foreign Affairs.\n\n3. Inasmuch as Com. Stalin, who, on the insistence of the Central Committee's Politburo, retains the position of first secretary of the TsK VKP(b) [Central Committee of the All-Union Communist Party (Bolsheviks)], will not be able to allot sufficient time to work in the TsK Secretariat, to appoint Com. A. A. Zhdanov Com. Stalin's deputy in the TsK Secretariat, relieving him of his duties overseeing the TsK VKP(b) Directorate for Propaganda and Agitation.\n\nNo documents or memoirs have been located that shed light on the discussions leading up to this resolution, but some clues are offered by its wording, which equates the reorganizations with a return to the Leninist revolutionary model of leadership. The leader of the party and the country, it states, should head the government, especially at a time of looming war. If Stalin had fully bought into the logic that it was important to adhere to the original Soviet model, he would have had to renounce the post of Central Committee secretary since Lenin was the founder and leader of the party but did not hold that post. But he chose to take both the top party and government posts for himself.\n\nAt last the dictatorial system of government was complete. At the top of the hierarchy stood the dictator himself. With the title of general secretary of the party added to that of chairman of the government, the supreme power he had been exercising for some time was made official. The Politburo's leading group\u2014a subset of its membership hand-picked by Stalin\u2014would serve as his consultative body. One step down the hierarchy were two governing bodies: the secretariat of the party's Central Committee, headed by Zhdanov, and the Sovnarkom Bureau, headed by Voznesensky. These two bodies served as the dictator's arms. They took care of the routine running of the country and brought consequential matters to Stalin for approval.\n\nThis reorganization was undoubtedly motivated by more than a desire for efficiency. Stalin's decision to give himself, the leader of the party, the added title of chairman of the government told the country and the world that at a time of international instability, the Soviet Union had consolidated its leadership. Again, Stalin's personality\u2014his hunger to possess not only real power, but also all of its accouterments and his tendency to regard even his closest comrades with suspicion\u2014also has to be taken into consideration. The latter quality was surely a factor in his decision to accelerate the advance of the younger generation and put Zhdanov and Malenkov in charge of the Central Committee apparat. Voznesensky\u2014not Molotov, the logical choice\u2014was appointed to serve as Stalin's first deputy in his role as government chairman. Beria, another member of the new generation, oversaw the network of security agencies. Stalin's old comrades, even those who remained at the upper echelons of power, suffered significantly diminished standing as they made way for their younger colleagues.\n\nMolotov was a particular target of Stalin's displeasure. After long years of devoted service and exceptional closeness with the _vozhd,_ Molotov was deprived of the Sovnarkom chairmanship and was not even appointed Stalin's first deputy. Stalin took every opportunity to demonstrate his disdain for Molotov. One of the last recorded manifestations of his irritation toward his longtime comrade occurred not long before the outbreak of war. In May 1941, at a meeting of the newly constituted Sovnarkom Bureau, Stalin took Molotov to task. Yakov Chadaev, the Sovnarkom's chief of administration, who was taking minutes at the meeting, recalls:\n\nStalin did not conceal his disapproval of Molotov. He very impatiently listened to Molotov's rather prolix responses to comments from members of the Bureau.... It seemed as if Stalin was attacking Molotov as an adversary and that he was doing so from a position of strength.... Molotov's breathing began to quicken, and at times he would let out a deep sigh. He fidgeted on his stool and murmured something to himself. By the end he could take it no longer:\n\n\"Easier said than done,\" Molotov pronounced in a low but cutting voice. Stalin picked up [Molotov's] words.\n\n\"It has long been well-known,\" said Stalin, \"that the person who is afraid of criticism is a coward.\"\n\nMolotov winced, but kept quiet\u2014the other members of the Politburo sat silently, burying their noses in the papers.... At this meeting I was again convinced of the power and greatness of Stalin. Stalin's companions feared him like the devil. They would agree with him on practically anything.\n\nWhat was behind this abusive treatment of a faithful colleague? Perhaps Stalin was taking out his frustrations over the state of Soviet foreign relations. Or perhaps, in the lead-up to war, he was making an example of his old comrade to keep the rest of the leadership in line. In any event, the result was a further centralization of power and a top leadership afraid to voice dissent. Critical questions of war and peace, concerning the fates of millions, rested solely in the dictator's hands.\n\n###   **A PREEMPTIVE STRIKE?**\n\nOn 5 May 1941, the day after his appointment as chairman of the government, Stalin went to meet with members of the Soviet military at a traditional Kremlin reception for graduates of military academies. At a similar event six years earlier, on 4 May 1935, Stalin had come out with the slogan, \"Cadres solve everything!\" This time the watchword the _vozhd_ shared with his military guests was classified and did not appear in the press. In May 1941, just six weeks before the outbreak of war with Germany, he called for a switch from a defensive to an offensive posture enabled by a powerful Red Army.\n\nWhile these remarks have attracted the particular interest of scholars, it is important to note that he had made similar comments in the past. In October 1938, for example, he told a gathering the following:\n\nBolsheviks are not just pacifists who long for peace and reach for arms only if they're attacked. That's not true. There will be times when Bolsheviks are the invaders; if the war is just, if the situation is right, and if the conditions are favorable, they will go on the offensive themselves. They are by no means against invading, against any war. The fact that we're now shouting about defense\u2014that's a veil, a veil. All countries mask their true selves: \"If you live with wolves, you have to howl like a wolf.\" [Laughter.] It would be stupid to spill your guts and lay them on the table.\n\nIn April 1940, when speaking to the military council in the aftermath of the Winter War, Stalin continued to address this topic. He spent a long time explaining to the officers that \"an army that has been cultivated not for attacking but for passive defense\" cannot be called modern.\n\nObviously when Stalin made these statements, in 1938 and early 1940, he had no intention of invading Germany. But as certain historians and commentators have pointed out, by 1941 the situation was very different. The German Army massed along the Soviet border and ready to pounce on the USSR might very well have convinced Stalin of the advisability of a preventive strike. A variety of arguments and pieces of evidence (albeit circumstantial) have been used to defend this viewpoint. For a biographer of Stalin, this question is far from secondary. Are we seeing, in 1941, a \"different Stalin\"\u2014not the cautious incrementalist who could be drawn into a fight only when he felt himself in a position of strength but a daring leader who believed the Red Army was prepared to challenge the Wehrmacht? Such an assumption is in fundamental conflict with the traditional view of the prewar Stalin, which is based on the reminiscences of Soviet marshals and evidence of his vacillating inconsistency in the months leading up to the war. Convincing evidence that Stalin was firmly resolved to go on the offensive has yet to surface. There is no serious basis for revising the traditional view that Stalin was fatally indecisive and even befuddled in the face of the growing Nazi threat.\n\nIt is, however, true that during 1940 and 1941 Stalin worked hard to strengthen the Red Army and prepare the country for the upheaval of war. In 1940, for the fourth year in a row, he did not take a vacation in the south. His primary concern was the army and the munitions industry. The accelerated buildup of heavy industry and its defense branches had been a priority since the late 1920s. The Stalinist approach to industrialization made this buildup especially costly, but in the end, the sacrifice of millions of ruined peasants and Gulag slaves and the expenditure of the vast country's significant resources did have a military and economic effect. By the time war with Germany broke out, the Soviet Union had more than twenty-five thousand tanks and eighteen thousand fighter planes, three to four times more than Germany. Such figures have inspired proponents of the theory of a \"preventive war\" to claim that the USSR was ready to take on Germany. But statistics often lie. In the Soviet case, the true story was often one of poor quality weaponry and padded figures, made worse by a shortage of well-trained military personnel. In any event, Stalin and the military leadership did not believe all this military hardware was sufficient. Having a military threat right at their doorstep demanded special measures. Ominous rumors of the might of the German Army and the quality of its weaponry were reaching the USSR from vanquished Europe. During the prewar period, the Soviet Union made a desperate attempt to increase output and modernize at the same time. By 1940, military production was two and a half times what it had been in 1937. This was an extraordinary increase. Special emphasis was placed on the production of new types of weapons, modern tanks and planes especially. Key to this modernization effort were purchases of military hardware from Germany, enabled under the Nazi-Soviet Pact.\n\nDespite the energy put into this buildup, progress was slow. There are well-known examples from the tank and aviation industry. Of the 25,000 tanks in the Soviet arsenal as of June 1941, only 1,500 were of modern design, and only a quarter of Soviet military aircraft was new. This is not to say that the remaining tanks and planes were useless. It does, however, show that the job of modernizing the Soviet military was far from complete. The leadership knew this.\n\nStalin had a much better understanding of the problems plaguing the Soviet military economy than do today's proponents of the preventive war theory, who focus exclusively on munitions-industry production statistics. The army and munitions industry were part of a huge socioeconomic machine with myriad interdependent parts. There was a limit to how much could be spent on the military buildup, especially as the prewar years coincided with yet another slowdown in the Soviet economy, associated with an imbalance between investment and resources. Such crucial resources as metal and electricity were in short supply, and the diversion of so much investment toward military production meant cutting the already scant resources put toward meeting the basic needs of Soviet citizens. Prices and taxes were rising, most of the population was getting by on a meager ration, and in some rural areas there were signs of famine. In late 1939 a ban was placed on the sale of flour and bread in the countryside. Hungry peasants rushed to cities and towns to buy these items, which were in short supply there too. The leadership in Moscow was inundated with desperate pleas for help. In February 1940, a woman wrote from the Urals, \"Joseph Vissarionovich, something really terrifying has begun.... I've so wasted away I don't know what will become of me.\" Someone in Stalingrad wrote to the Central Committee that \"We don't have time to sleep anymore. At two in the morning people begin lining up for bread, and by five or six there are already 600\u2013700\u20131,000 people standing outside the stores.... You might be interested to know what they're feeding workers in the cafeterias. What they used to give to swine they now give to us.\"\n\nThe country's top leadership was fully aware of the situation. The Politburo made repeated attempts to address the shortages, giving priority to major cities and industrial enterprises. The food crisis exacerbated the problems of employee turnover and absenteeism that had always plagued the Soviet economy. As the country mobilized for war, harsh measures were introduced to combat these problems. On 26 June 1940, as France was succumbing to the Nazis, the USSR enacted a new law lengthening the workday and work week and making it a crime to be late or to leave one's place of employment without permission. Soviet peasants had lost their freedom of movement long ago. Now factory and office workers lost theirs. In the year between the enactment of this law and the start of war, it was used to convict more than three million people. Of them 480,000 served prison terms up to four months. The rest, though not imprisoned, were forced to perform compulsory labor for up to six months. The convicted were often allowed to remain at their jobs, but a significant share of their meager pay was deducted, condemning them and their families to hunger.\n\nSuch extreme laws and the declining standard of living took a toll on Soviet society, whose suffering only increased Stalin's deeply ingrained fear of a fifth column. Whereas the purges of the prewar years had been targeted primarily at the western areas recently annexed by the USSR, Stalin now began to worry, and with reason, that people throughout Soviet society could prove disloyal to him in time of war. Too many had suffered at the hand of the government; too many had starved or eked out a meager existence. The propagandistic claims of monolithic unity at both the front and the rear were intended for the people, for foreign enemies, and for gullible posterity. Stalin was not among the gullible.\n\nSoviet propaganda described the Red Army as the people's own flesh and blood, and it was. Within the Red Army, the unique features and contradictions of the Stalinist system were manifested in concentrated form. Between January 1939 and June 1941 the Soviet armed forces more than doubled in size. This rapid increase came with the same fundamental problem that plagued Stalinist \"leaps forward\" in general, especially the rapid industrialization of the early 1930s. Ambitious attempts to calculate exactly what equipment\u2014even what entire factories\u2014had to be purchased from the West failed miserably. Young, untrained Soviet workers produced defective products, damaging factory equipment in the process. Stalin's understanding of the complex interdependence between technical and social progress was expressed in the updating of the slogan \"Cadres solve everything!\" to \"Technology solves everything!\" The rapidly growing Red Army needed not only to be armed but also trained. It is difficult to say which was the harder task.\n\nBetween 1937 and 1940, the Soviet officer corps grew more than two and a half times. As a result, a sizable proportion of commanders lacked the requisite knowledge and experience. During the war Stalin reproached one of his generals for the quality of army officers: \"You in the military in your time ruined the army by sending all sorts of junk into academies and administration.\" As usual, he was blaming others for problems that were primarily his fault. It was on his initiative that in the 1930s, tens of thousands of commanders, men who would have been capable of serving their country with distinction, were fired, sent to the camps, or shot for political reasons. But the damage to the Red Army was not measured only in numbers. Until the outbreak of war (and to a lesser extent even during it), repression had distorted the decision-making process, including promotions, making it possible for time-serving incompetents, skilled primarily in expressions of loyalty, to make successful careers. It also discouraged a commander's most important quality\u2014a willingness to take the initiative\u2014and instead encouraged excessive caution. As was well known from anti-wrecking campaigns, repression subverted the authority of those in charge and undermined discipline. The problems of rule breaking and drunkenness that had always plagued the Red Army were magnified.\n\nThe Soviet leadership could see that there was trouble within the army. The clearest signal was the Winter War with Finland. The unexpected foiling of the Red Army by an incomparably weaker enemy dealt the Soviet military's reputation a stunning blow that could not have come at a worse time. After the peace treaty was signed, Stalin conducted a review to determine what had gone wrong. Countless deficiencies in the arming and training of soldiers were discovered, along with problems in the command system. Stalin removed his old friend Kliment Voroshilov from the post of people's commissar for defense and replaced much of the military's leadership. These changes brought little improvement. In April 1941, approximately one year after the shake-up, the Politburo looked into accidents in military aviation. It turned out that even in peacetime, an average of two to three planes was lost in accidents every day. Furious, Stalin placed all the blame on the air force leadership. On the very eve of war, a new wave of arrests roiled the military command.\n\nStalin did not allow his focus on the Red Army to distract him from keeping an eye on his adversary's forces. The ruthless efficiency of the Wehrmacht was extremely alarming. Delegations of Soviet weapons experts who visited German munitions plants under a Soviet-German cooperation agreement returned home with glowing reports. Delegation members were unable to hide how impressed they were and wrote of the huge successes of the German weapons industry. In keeping with the Russian saying \"Fear has big eyes,\" Soviet intelligence and the military and economic leadership constantly exaggerated the enemy's strength. In 1940 the new people's commissar for airplane production, Aleksei Shakhurin, reported to Stalin that Germany's aviation industry had twice the capacity of its Soviet counterpart. The reports Stalin received from his intelligence agencies significantly exaggerated both the potential of German industry and the size of its armed forces. As a result of these overestimates, the enemy looked much more imposing that it actually was.\n\nThe sources of Stalin's prewar anxiety are a huge subject that cannot be fully addressed within the scope of this book. Clearly, he had good reason to fear war with Germany. One way he may have reacted to this fear was with a desire (which many believe he felt) to delay the start of war in order to give the Soviet Union time to strengthen its military capabilities and hope that international events would take a favorable turn. He certainly had reason to hope that war would be delayed. One of the most convincing reasons was the idea that Hitler would not be so foolhardy as to mire his forces on two fronts by engaging the Soviet Union while he had Great Britain and the increasingly active United States threatening his rear. Stalin was not alone in this line of reasoning. Hitler, fully aware of how much sense this theory made, took care to exploit it. Secure in the knowledge that he was preserving the element of surprise, he did indeed take the risky plunge of engaging enemies on two fronts\u2014largely because his enemies saw such a move as an impossibility. Nazi propaganda spread disinformation to perpetuate this mistaken idea. Stalin wound up the victim of his belief in Hitler's instinct for self-preservation.\n\nA few peripheral factors strengthened Stalin's faith that Hitler would not hurry to attack the USSR. For one, Soviet-German economic cooperation was thriving. Soviet exports were feeding Germany's appetite for raw materials. Goods imported into Germany from three different countries traveled across Soviet territory, so war with the USSR would undermine some of Germany's important economic ties. The intelligence reports reaching Stalin's desk were contradictory. His predisposition to believe Hitler would not attack soon influenced his intelligence agencies, who preferred to tell Stalin what he wanted to hear. Such a cause-and-effect sequence is hardly unique in world history.\n\nStalin's reaction to a 17 June 1941 intelligence report claiming that an attack was imminent is well known. Just days before the actual invasion, he wrote to the state security commissar, \"You can send your 'source' from German aviation headquarters back to his f**king mother. This is disinformation, not a 'source.'\" Even if Stalin may have been correct in this case, clearly reactions like this frightened intelligence officials and discouraged them from speaking up, rendering them much less effective. It was safer to say what Stalin wanted to hear or be silent, and those in charge of the country's security and military readiness increasingly opted for safety. Stalin got what he wanted. He alone had the right to an opinion. Everyone waited to see what the dictator had to say, hoping he knew what he was doing. Unfortunately, he did not.\n\n## **PATIENT NUMBER 1**\n\n**The summoning of the doctors to the near dacha on the morning of 2 March 1953.**\n\nBeria, Khrushchev, Bulganin, and Malenkov returned to their homes, leaving Stalin on the couch without medical attention. Perhaps out of fear, or perhaps out of unspoken ambivalence toward his recovery, Stalin's comrades rejected the idea that they were facing a medical emergency. After Malenkov and Beria checked on the _vozhd_ and found him sleeping, they proceeded to dismiss what the bodyguards had told them about his symptoms. Had he really had some sort of fit? The bodyguards were not doctors. Their imaginations could have been playing tricks on them. His colleagues probably also remembered that Stalin had recently accused his own doctors of being murderers. Who would take responsibility for calling a doctor (or summoning a murderer, as the _vozhd_ might see it) unless he were absolutely sure one was needed? A simple need for emergency medical care was transformed into a multidimensional political problem.\n\nStalin's bodyguards spent the remainder of the night in a state of anxiety. No doubt worried that they could be held accountable if Stalin died, they again asked for guidance from above and reported that things did not seem right with the boss. This time the four comrades decided to send a team of doctors to the dacha. Before doing so, however, they convened the Bureau of the Central Committee Presidium so that the summoning of medical luminaries would look like a collective decision by the party leadership. Should Stalin recover, his anger would fall on everyone at once. On the morning of 2 March the doctors arrived at Stalin's bedside.\n\nThe renowned Soviet cardiologist Aleksandr Miasnikov, one of the medical experts summoned to attend Stalin, gives a detailed description of the visit in his memoirs. \"The diagnosis,\" he wrote, \"was clear to us, thank God: hemorrhage in the left cerebral hemisphere of the brain caused by hypertonia and atherosclerosis.\" The doctors gave Stalin generous doses of various stimulants but without any real hope of preventing death. From a medical perspective, his condition was no mystery. An autopsy confirmed the initial diagnosis, revealing a large cerebral hemorrhage and severe damage to the cerebral arteries due to atherosclerosis. Stalin had been a sickly old man. He would have turned seventy-five later that year.\n\nIn totalitarian regimes, too much depends on the personality of the dictator. From the time he came to power, Stalin's health was a topic of worldwide interest. During his lifetime there was periodic speculation in the Western press that he was ill or even near death. People in the Soviet Union whispered similar rumors. Scholars and commentators looked to Stalin's physical and mental health as possible keys to understanding his personality and the brutality of his dictatorship. For a long time speculation surrounding Stalin's health was based on unfounded assumptions. Only recently have we gained access to Stalin's surviving medical records and testimony by the doctors who monitored his health and examined him after his death.\n\nThe only one of the Jughashvilis' three children to live to adulthood, the future dictator suffered a variety of ills growing up. At an early age, Ioseb came down with smallpox, which left his face permanently pockmarked. He also had a bout of malaria. Then, through some sort of accident, the details of which have never been clear (some say he was hit by a horse-drawn carriage), he severely injured his left arm. The injury caused his arm to atrophy, giving him problems for the rest of his life. In 1898 Ioseb wrote to the rector of the Tiflis Theological Seminary asking to be excused from a reexamination \"due to a disease of the chest that has long plagued me and that grew more severe during examinations.\" He sought to be released from police custody in October and November 1902 because of his \"predisposition toward pulmonary consumption\" and worsening cough. Apparently his juvenile tuberculosis eventually abated, and he did not show signs of the disease later in life.\n\nAs a professional revolutionary, Stalin had to endure many hardships: prison, exile, and an unsettled existence even in times of freedom. During one term of exile he became ill with typhus. His most difficult trial was his final exile in Turukhansky Krai, which lasted three years. He had difficulty adapting to the harsh climate, austere living conditions, isolation from \"the world at large,\" and forced idleness, and in letters to friends he complained of a \"suspicious cough\" brought on by \"intensifying cold (37 below)\" and a \"general state of ill health.\" Overall, however, the tsarist government was immeasurably kinder to convicts than the Stalinist dictatorship. Had young Stalin had to endure so many imprisonments and exiles in the sort of Gulag system he went on to create, he most likely would not have survived.\n\nThe revolution and Civil War not only put millions in their graves, but also deeply affected the Bolshevik party and undermined the health of its leaders. In March 1921 Stalin underwent an appendectomy. On 23 April 1921, out of concern for their health, the Politburo voted to grant Stalin, Kamenev, Rykov, and Trotsky extended vacations. In late May, Stalin left for the North Caucasus and did not return to Moscow until 8 August, almost two and a half months later. In 1922 he skipped his vacation, but in July the Politburo compelled him to spend three days a week out of town. Once the Civil War ended, spending time in the fresh air of Moscow's leafy suburban dacha communities became an established lifestyle for the top Bolshevik leadership. Stalin and his family commandeered the country home of a former petroleum industrialist. Later, after the death of his wife, the _vozhd_ built himself a new dacha, more convenient to Moscow. This famous country home (the \"near\" dacha in Volynskoe) was Stalin's main residence for nearly two decades and will forever be associated with him. It was here that he died.\n\nAt the dacha, Stalin would spend time with his immediate family and other relatives or get together with his comrades. In addition to the festive dinners with lots of alcohol (described above), Stalin's dacha lifestyle also included games, such as billiards or _gorodki_ (a Russian game similar to skittles), although the dictator himself was not a big lover of physical activity. \"He preferred stretching out on a deckchair with a book and his documents or the newspapers. And he could sit at the table with his guests by the hour,\" his daughter Svetlana recalled. This penchant for immobility only increased with age.\n\nAnother significant part of Stalin's life were his vacations in resort areas of southern Russia. He spent time in the south every year from 1923 to 1936 and from 1945 to 1951. These trips were working vacations. A constant stream of documents was forwarded to him, and he kept up an active correspondence with his comrades back in Moscow, a practice that generated invaluable records for historians. But there was also time for rest and relaxation. While in the south Stalin treated his numerous diseases: rheumatoid arthritis, bouts of tonsillitis, long-lasting intestinal disturbances, and neurasthenia. His ailments were also eased by therapeutic baths. \"I am getting better. The Matsesta waters (near Sochi) are good for curing sclerosis, reviving the nerves, dilating the heart, and curing sciatica, gout, and rheumatism,\" he reported to Molotov on 1 August 1925.\n\nBut Stalin was not a conscientious patient. His chronic ailments were exacerbated by his lifestyle and bad habits: smoking, drinking, rich foods, and overwork. Like most people, Stalin alternated between taking care of his body and inflicting damage. In May 1926 he left for a vacation in the Caucasus. After a brief stop in Sochi he set out with Mikoyan to travel through Georgia, where he visited his native Gori before going to stay with Ordzhonikidze in Tiflis. Letters from the head of Stalin's Sochi-based security team, M. Gorbachev, suggest that this was a boisterous trip. While \"under the influence,\" as Gorbachev put it, on a whim, Stalin suddenly summoned him from Sochi to Tiflis but then forgot he had done so. When Gorbachev showed up, Stalin was surprised to see him. When it became clear what had happened, everyone \"had a good laugh.\" Gorbachev was forced to hurry back to Sochi, covering the vast distance at breakneck speed. Continuing his spree, Stalin spent a long time driving around the Caucasus and wound up returning to Sochi in bad shape. \"I returned to Sochi today, 15 June,\" he reported to Molotov and Bukharin. \"In Tiflis I came down with a stomachache (I got food poisoning from some fish) and am now having a hard time recovering.\" Gorbachev wrote to Stalin's assistant, Ivan Tovstukha, \"Overall, the boss wound up paying quite a price for this trip across the Caucasus in terms of his health. Mikoyan and Sergo [Ordzhonikidze] turned him topsy-turvy.\" Stalin called for a doctor, went on a diet, and began to take the waters on a regular basis. The doctor who treated him in Sochi, I. A. Valedinsky, recalled that his patient complained of pain in his arm and leg muscles. When his doctors forbade him to drink, Stalin asked, \"But what about cognac?\" Valedinsky replied that \"on Saturday you can let loose, on Sunday you should rest, and on Monday you can go to work with a clear head.\" \"Stalin liked this response, and the next time he arranged a ' _subbotnik'_ [a word usually used for mandatory 'volunteer' work on Saturdays], it was very memorable for me,\" Valedinsky wrote, although he did not explain what made this particular gathering so unforgettable.\n\nReferences to his poor health are scattered throughout Stalin's later correspondence as well. While on vacation in July 1927, he wrote to Molotov: \"I'm sick and lying in bed so I'll be brief.\" According to Valedinsky, that year he also complained of pain in his arm and leg muscles. Therapeutic baths were followed by the usual _subbotnik._ Stalin invited his doctors to dine with him \"and was so generous with the cognac,\" Valedinsky wrote afterward, \"that I did not make it home until the following day, on Sunday.\" In 1928, before taking a curative bath in Sochi, Stalin again complained of pain in his arms and legs. The rheumatoid arthritis in his left arm was progressing. During a vacation in August 1929 Stalin wrote to Molotov that \"I am beginning to recuperate in Sochi after my illness in Nalchik.\" In 1930, while undergoing treatment in Sochi, he fell ill with tonsillitis. His teeth also hurt. In September 1930 he wrote to his wife that the dentist had \"sharpened\" eight of his teeth in one go, so he \"was not feeling very well.\" In 1931 he again took therapeutic baths. \"I spent about 10 days in Tsqaltubo. I took 20 baths. The water there was marvelous, truly valuable,\" he wrote to Yenukidze. That September he wrote to his wife that he was vacationing in Sochi with Kirov. \"I went one time (just once!) to the seaside. I went bathing. It was very good! I think I'll go again.\" Apparently he used the Russian word for \"bathing\" because he could not swim.\n\nThe vacation Stalin took in 1932 was one of his longest. The log of visitors to his Kremlin office shows that he did not receive anyone there between 29 May and 27 August\u2014almost three months. The apparent reason for such a long break was poor health. The following spring the foreign press was still speculating that Stalin was seriously ill. On 3 April, _Pravda_ took the unprecedented step of publishing a response by Stalin to a query by the Associated Press: \"This is not the first time that false rumors that I am ill have circulated in the bourgeois press. Obviously, there are people in whose interest it is that I should fall seriously ill and for a long time, if not worse. Perhaps it is not very tactful of me, but unfortunately I have no information to gratify these gentlemen. Sad though it may be, the fact is that I am in perfect health.\" Behind this characteristically mocking response was genuine irritation. Stalin's symptoms were serious, and rest and relaxation in the beneficial climate of southern Russia apparently did not alleviate them. \"It seems I won't be getting better anytime soon,\" Stalin wrote to Kaganovich from the south in June 1932. \"A general weakness and real sense of fatigue are only now becoming evident. Just when I think I'm beginning to get better, it turns out that I've got a long way to go. I'm not having rheumatic symptoms (they disappeared somewhere), but the overall weakness isn't going away.\" Soon, however, he felt well enough to make a 230-mile trip across the Black Sea by motor boat.\n\nRegular trips to the south inspired Stalin to build new vacation homes there. These construction projects began in 1930 and continued for the rest of his life. \"We've built a marvelous little house here,\" he wrote of his new dacha outside Sochi in August 1933. A month later he wrote of another residence: \"Today I visited the new dacha near Gagry. It's turned out (they just finished building it) to be a splendid dacha.\"\n\nIn 1933 Stalin was away from his Kremlin office from 17 August to 4 November. On 18 August he left Moscow to travel south with Voroshilov. The trip\u2014by train, boat, and automobile\u2014took seven days, during which they visited several regions of the country. Stalin spent the remainder of his vacation traveling (including by sea), entertaining guests, and, inevitably, working. This vacation was apparently among the more enjoyable. The situation in the country had somewhat stabilized after the devastating famine, putting the Soviet leadership in a good mood. Moreover, Stalin enjoyed relatively good health. \"Koba felt great the entire time,\" Voroshilov wrote to Yenukidze. His only health problem was some tooth pain.\n\nStalin's vacation the following year was less successful. In 1934 he caught influenza and returned to Moscow having lost weight. Kirov, who accompanied Stalin that summer, also did not enjoy himself. \"As fate would have it, I wound up in Sochi,\" Kirov wrote, \"which I'm not happy about\u2014the heat here isn't tropical; it's hellish.... I really regret that I came to Sochi.\" Things did not go well in 1935 either: Stalin again caught influenza and injured his finger when the head of his security team accidentally slammed a car door on it. Stopping in Tiflis toward the end of this vacation to visit his mother, he came down with a stomach ailment. In 1936, Stalin's letters to his comrades-in-arms back in Moscow during August through October are brief, harsh, and often ill humored. They contain no personal information, just orders. They are largely devoted to the topic of \"enemies of the people,\" especially arrangements for the first Moscow show trial against Zinoviev and Kamenev.\n\nNineteen thirty-seven had a gloomy start both for the country, which was succumbing to another round of repression, and for Stalin, who began the year with a bout of tonsillitis. (By 5 January he had sufficiently recovered to enjoy dinner with his comrades and doctors, followed by dancing to phonograph records.) Despite his continued poor health, for the first time in many years he did not leave Moscow on vacation. The decision to stay was undoubtedly due to his intimate involvement in the purging of Soviet society. He also stayed in Moscow the following few summers. After the winding down of the Great Terror, the impending war prevented him from relaxing down south. In 1939, for example, he spent August embroiled in difficult negotiations with Western powers and then the Nazis, resulting in the pact with Hitler. He had recently turned sixty, and his health had not improved. In records dated February 1940, Valedinsky mentions another episode of tonsillitis and a bad cold.\n\nThe outbreak of war in the summer of 1941 pushed the already hardworking leader to his limits. Unlike many Soviet citizens, of course, he was not going hungry or enduring long days of backbreaking labor, but the additional workload and responsibilities put a greater strain on his health. In September 1944 discussions with the United States ambassador to Moscow, Averell Harriman (who was attempting to arrange for the Soviet leader to meet with Roosevelt and Churchill), Stalin explained that he would not be able to leave the country because of \"increasingly frequent illnesses.\" According to one account of these talks, \"In the past Com. Stalin would have the flu for one or two days, but now it was lasting for one and a half or two weeks. He was showing his age.\" In his categorical refusal to travel by plane, Stalin may have been overdramatizing his health problems, but not by much. A number of memoirs describe Stalin's frail health during the war years. Whenever the situation at the front permitted it, the dictator retreated to his dacha and worked from there.\n\nIn October 1945, shortly after the surrender of Japan, Stalin took his first southern vacation in several years. Toward the end of his life these trips were shifted to later in the year, usually commencing in August or September and ending in December. Apparently he preferred to enjoy the peak of summer at his dacha outside Moscow and to head south when the weather up north turned cold. His vacations also grew longer. In 1946\u20131949 they extended to three or three and a half months, and in 1950 and 1951 he spent four and a half months out of town. While at his southern residences, Stalin engaged in more or less the same activities as in Moscow. He spent time working on the day's mail and writing to his comrades. He also received visitors, although fewer than in Moscow. As in Moscow, however, he enjoyed presiding over festive gatherings at the dinner table and playing billiards. But some activities were specific to his vacation lifestyle. During his visits to Russia's resort towns, he took therapeutic baths, went for walks, and traveled. In 1947 he expressed a desire to travel by car from Moscow to Crimea, although the poor quality of the roads allowed him to get only as far as Kursk, where he boarded a train. Long car trips were evidently bad for his rheumatism. A number of memoirs report that he nevertheless preferred the less comfortable jump seats to the cushioned back seat. He seldom stayed in one place very long when visiting the south, moving among his continuously growing collection of dachas. Sometimes he would invite his daughter and son to join him, occasioning a sort of family reunion that, for a number of reasons, was not possible in Moscow.\n\nAfter the war, these visits to the south alternated with long periods when Stalin barely left his Moscow dacha. Visits to his Kremlin office became increasingly rare, primarily due to his deteriorating health. He continued to suffer from stomach pain and intestinal disturbances, accompanied by fever, throat problems, colds, and influenza. His atherosclerosis was progressing. Despite scattered attempts to do so, he was by now simply incapable of changing his sedentary lifestyle. The copious fare served at his frequent late-night dinner gatherings was surely not good for him. According to Milovan Djilas, who visited Stalin's dacha several times in the 1940s, \"The selection of food and drink was huge, with an emphasis on meat dishes and hard liquor.\" The leader of the Hungarian Communist Party, Matyas Rakosi, recalled the following:\n\nThe atmosphere at these dinners was free and easy; people told jokes\u2014often even dirty ones\u2014to the raucous laughter of everyone present. Once they tried to get me drunk, but wine doesn't affect me, which earned me recognition and a bit of surprise from those in attendance. Our last dinner together was in the fall of 1952. When Stalin left the room at three in the morning, I commented to the Politburo members, \"Stalin is already 73; aren't such dinners, stretching so late into the night, bad for him?\" His comrades assured me that Stalin knew his limits.\n\nStalin brought up his age and the importance of cultivating a new generation of leaders with increasing frequency. Deep down, however, he must have hoped for the best. In November 1949, when the Albanian leader Enver Hoxha expressed the wish that Stalin would live to one hundred, the Soviet leader joked: \"That's not enough. Back home in Georgia we have old people still alive at 145.\" As Stalin's daughter Svetlana attested, \"In later years he wanted to continue in good health and live longer.\"\n\nIn 1952, Stalin did not travel south. Even though he remained in Moscow, he visited his Kremlin office only fifty times, an average of less than once a week. On 21 December 1952, for his seventy-third birthday, his daughter Svetlana made her final visit to her father's dacha. \"I was worried at how badly he looked,\" she recalled. \"He must have felt his illness coming on. Maybe he was aware of some hypertension, for he'd suddenly given up smoking and was very pleased with himself.... He'd been smoking for fifty or sixty years.\" By this time his atherosclerosis was well advanced. The autopsy performed two and a half months later showed that damage to the arteries had greatly impeded blood flow to the brain.\n\nTo what extent was Stalin's death hastened by a lack of professional care? It is widely believed that he did not see any doctors during the final months of his life due to arrests at government hospitals in connection with the Doctors' Plot (see chapter 6 below). Svetlana Allilueva writes:\n\nHe was probably aware of an increase in his blood pressure, but he hadn't any doctor to take care of him. Vinogradov [a renowned doctor who had treated Stalin], the only one he trusted, had been arrested and he wouldn't let any other doctor near him.\n\nSomewhere or other he got hold of some quack remedies, and he'd take some pills or pour a few drops of iodine into a glass of water. Moreover, he himself did a thing no doctor would ever have allowed: Two months after I last saw him and just twenty-four hours before his stroke he went to the bathhouse near the dacha and took a steam bath, as he'd been accustomed to doing ever since Siberia.\n\nAllilueva's testimony has to be taken with a grain of salt. She rarely saw her father and knew little about his life. Her reminiscences offer a subjective view of events. No archival documents have been found to clarify whether Stalin was under the care of doctors during the final months of his life. Nothing has been written about the quality of his health care at that time. Perhaps no treatment in the world would have helped.\n\nWe are equally in the dark about another complex question: the effect Stalin's ailments had on his decisions and actions. Without solid evidence, speculation on this subject remains just that. What we do know is that Miasnikov, one of the doctors summoned to his deathbed, believed that the extensive damage to Stalin's cerebral arteries uncovered during his post mortem must have affected his character and behavior:\n\nI believe that Stalin's cruelty and suspiciousness, his fear of enemies and loss of the ability to assess people and events, his extreme obstinacy\u2014all this was the result, to a certain extent, of atherosclerosis of the arteries in his brain (or rather, atherosclerosis exacerbated these traits). Basically, the state was being governed by a sick man.... Sclerosis of the blood vessels in the brain developed slowly, over the course of many years. Areas of cerebral softening that had originated much earlier were discovered in Stalin.\n\nThese observations by a distinguished doctor are entirely consistent with the testimony of Stalin's associates. Even the most devoted among them, Vyacheslav Molotov, admitted, \"In my opinion, Stalin was not quite in possession of his faculties during his final years.\" A historian, as well, would have no trouble coming up with \"oddities\" and inappropriate responses in Stalin's political behavior. But historians are not doctors. While keeping their subjects' possible ailments in mind, they try not to dwell on them.\n\n## **5 STALIN AT WAR**\n\nThe 22 June 1941 surprise attack came with plenty of warning. The previous evening Moscow's military leadership received a report: a sergeant in the German army had crossed the border with the news that an invasion would begin the following morning. Stalin was immediately informed, and the military leaders and Politburo gathered in his office to decide how to respond. People's Commissar for Defense Semen Timoshenko and Army Chief of Staff Georgy Zhukov, according to the latter's memoirs, asked for a directive allowing them to bring troops to a state of combat readiness. Stalin was doubtful: \"Could it be that the German generals sent us this defector to provoke a clash?\" After hearing out his military chiefs, he concluded, \"It would be premature to issue such an order. The matter might still be resolved peacefully. We should issue a brief order indicating that an invasion could start with provocative actions by German units. To avoid complicating matters, forces in border districts should not give in to any provocations.\" The order reached troops shortly after midnight.\n\nStalin and the Politburo continued to discuss the alarming news until they finally parted ways, exhausted, around three o'clock in the morning. It was not long before Zhukov telephoned Stalin to report that German troops had launched an invasion. After briefly trying to refuse the general's demand that the _vozhd_ be summoned to the phone, his chief bodyguard finally went to wake him:\n\nAfter about three minutes, I. V. Stalin came to the phone.\n\nI informed him of the situation and asked for permission to commence an armed response. I. V. Stalin was silent. All I heard was his heavy breathing.\n\n\"Did you understand what I said?\"\n\nAgain silence.\n\n\"Will there be orders?\" I persisted.\n\nZhukov's memoirs seem to suggest that Stalin withheld permission to respond to the attack and simply ordered Zhukov and Timoshenko to the Kremlin. But in 1956 Zhukov offered an important detail about this conversation that was never included in his memoirs. During the telephone call, he said, Stalin issued an order to the troops: \"This is a provocation by the German military. Do not open fire to avoid unleashing wider action.\" There is no reason to disbelieve this account.\n\nAccording to Zhukov, he and Timoshenko arrived at Stalin's office at 4:30 a.m. to find the Politburo already there. This timing contradicts the log of visitors to Stalin's office, which states that Timoshenko and Zhukov's first visit on 22 June occurred at 5:45. A simple explanation could be that the 4:30 meeting took place not in Stalin's office but in his Kremlin apartment. In any event, after being updated by his military chiefs, Stalin again expressed doubts: \"Couldn't this just be a provocation by German generals?... Hitler surely doesn't know about this.\" He sent Molotov to meet with Germany's ambassador, Friedrich von der Schulenburg. As Zhukov describes it, he and Timoshenko asked Stalin to order a counterstrike, but Stalin told them to wait until Molotov returned.\n\nThe idea that the attacks were a conspiracy by German generals and were unknown to Hitler fit perfectly with Stalin's thinking. Further evidence that the Soviet leaders harbored serious illusions about Hitler can be found in Molotov's behavior during his meeting with Schulenburg, which began at 5:30 that morning. Obeying instructions sent by his government, Schulenburg, clearly upset, read Molotov the following brief notification: \"In view of the intolerable threat to Germany's eastern border posed by the massive concentration and readying of all the armed forces of the Red Army, the German government feels compelled to take military countermeasures.\" Molotov's reaction suggests that he did not understand what was actually happening. He began to dispute that Soviet forces were concentrated along the border and concluded with the almost desperate question: \"Why did Germany sign a non-aggression pact only to break it so easily?\" He tried to convince Schulenburg that the USSR was innocent in this matter and that it was Germany that was being treacherous, although he must have understood that even if the German ambassador believed him, nothing could be done. Schulenberg was just the messenger.\n\nThis meeting took place right in the Kremlin, so by 5:45 Molotov was already back in Stalin's office, along with Beria, Lev Mekhlis, Timoshenko, and Zhukov. As Zhukov describes it, upon hearing from Molotov that Germany had declared war, Stalin \"silently dropped into his chair and became immersed in thought. A long and painful pause ensued.\" Stalin agreed to issue a directive ordering the destruction of the invading enemy and added, \"So long as our troops, with the exception of aviation, do not violate the German border anywhere for now.\" This order was issued to the troops at 7:15 a.m., almost four hours after the invasion began. It showed that the top leadership still did not understand what was happening. Stalin did not sign the order. It went out over the signatures of Timoshenko, Malenkov, and Zhukov.\n\nIn the hours that followed, Stalin conferred with his fellow leaders on several questions. Among the most pressing was how Soviet citizens would be informed that their country was at war. It was not just a matter of an official statement but of how the war was to be presented, what political slogans would be put into play, and what objectives were to be pursued. Stalin's comrades felt strongly that he should be the one to speak to the country, but he refused. The job fell to Molotov. Of course Stalin understood the political drawbacks of this decision, but he simply did not know what to say. The situation was fraught with uncertainty. Molotov's speech announced that the country was at war, emphasized that Germany was the aggressor, and expressed confidence that the Soviet Union would prevail. He ended with the words, \"Our cause is just. The enemy will be crushed. Victory will be ours.\" Throughout this horrific war, these watchwords were emblazoned on posters and banners and repeated over the airwaves.\n\nThe archives contain a version of the speech written and edited in Molotov's hand. The speech he actually delivered was somewhat different from this initial draft and added references to Stalin. It started with the introductory statement, \"The Soviet government and its head, Comrade Stalin, have asked me to make the following announcement.\" A paragraph was added toward the end calling on the people to \"rally their ranks\" around the party, the government, and \"our great leader Comrade Stalin.\" These references to Stalin were undoubtedly designed to preclude any doubts and rumors that might have arisen from his silence.\n\nMolotov's speech exposes a central political concern worrying Stalin during the war's early hours. The brief remarks repeatedly emphasized the idea that the German aggression was completely unprovoked and that the USSR had meticulously adhered to the non-aggression pact. As the speech put it, \"The German government was not once given grounds for complaining to the USSR that it was not fulfilling the agreement.\" Molotov emphasized that Germany \"is the invading side\" and even called the German fascists \"traitors.\" Implicit in this word choice is the idea that there was an understanding between the two countries that could be betrayed.\n\nThe English historian John Erickson has suggested that Molotov's speech exposed a sense of unease and even humiliation on the part of the Soviet leadership. It was as if Molotov were taking the German explanation for the invasion at face value and defending the Soviet Union against charges of aggressive intent. Was this insistence on Soviet adherence to the pact intended for Hitler in the faint hope that the invasion had indeed been launched by rogue generals? Or was the idea of Soviet blamelessness meant to influence public opinion in the West, in whose eyes it was suddenly important to seem a victim, rather than a partner, of Nazism? Or was the speech meant purely for the domestic audience in an effort to fan indignation toward a treacherous enemy?\n\nFive minutes after noon, Molotov left Stalin for twenty minutes, during which his voice was broadcast over the radio while Soviet officials streamed in and out of Stalin's office. A general army mobilization was announced. The situation remained ambiguous. Stalin decided to send high-ranking emissaries to the front: Zhukov, Shaposhnikov, and Kulik. The use of plenipotentiaries to represent him remained Stalin's preferred method of overseeing the war throughout its duration.\n\nAt 9:15 p.m., another directive went out to Soviet forces, again over the signatures of Timoshenko, Malenkov, and Zhukov. The results of the first day of fighting were sugarcoated. While recognizing that the German forces had achieved \"minor successes\" in a number of areas, the directive claimed that in most border sectors \"attacks were repelled with heavy enemy casualties.\" Having painted this optimistic picture, the directive went on to spell out the goal: deal a counterblow and destroy the enemy. In his memoirs, Zhukov noted his disapproval of the directive's wording and his feeling that it did not reflect the true state of affairs.\n\nIn truth, Stalin did not have accurate information about the first day of combat. Communication with frontline forces had broken down, and commanders at all levels were afraid to deliver bad news. Stalin himself had a hand in creating a distorted picture. On 23 June the first Red Army Main Command's combat overview was published in newspapers. The _vozhd_ labored over the wording of this summary himself. \"After fierce battles,\" the overview read, \"the enemy was beaten back and suffered great losses.\" Supposedly there were only two points at which the Germans were able to penetrate the border by 10\u201315 kilometers. In reality, the first day of fighting was catastrophic. According to official Soviet sources, on 22 June the Red Army lost 1,200 airplanes, many of which were destroyed while still sitting on airfields. German figures record more than 1,800 Soviet airplanes lost, of which approximately 1,500 were destroyed on the ground. In one day the Germans advanced 60\u201380 kilometers into the Baltic states, 40\u201360 kilometers into Belarus, and 10\u201320 kilometers into Ukraine.\n\nDespite lacking accurate information and his understandable desire to hope for the best, Stalin must have realized the seriousness of the situation. According to eyewitnesses, he was stunned by the outbreak of war. As Zhukov describes it, \"During the first day he was not able to really take himself in hand and get a firm grip on events. The shock to I. V. Stalin caused by the enemy invasion was so strong that his voice even became softer and his instructions on organizing the military effort were not always appropriate to the situation.\" Chadaev later recalled, \"Early on the morning of 22 June I caught sight of Stalin in the corridor. He had arrived at work after a brief sleep. He looked tired, worn out, and sad. His pockmarked face was sunken. You could see he was depressed.\"\n\nStalin's indecisiveness during the war's first hours and his refusal to make a radio address on 22 June clearly show that he was not himself. His indecisiveness continued the following day, when it came time to set up a command headquarters. He refused to formally take charge of General Command Headquarters, and Defense Commissar Timoshenko took over that responsibility. Officially, Stalin's membership in the Command Headquarters was on a par with those of Molotov, Voroshilov, Semen Budenny, Zhukov, and Admiral Nikolai Kuznetsov. A number of other Politburo members and military leaders were given the status of advisers to Command Headquarters. This system was extremely inefficient. Though officially in charge of the war effort, Timoshenko in fact had little authority among his colleagues. According to Kuznetsov, the members of Command Headquarters and the top leaders \"had no intention of subordinating themselves to the people's commissar for defense. They demanded reports and information from him, and even made him account for his actions.\" Timoshenko was certainly not able to go over Stalin's head. The chain of command became long and tangled, and the system whereby decisions were made and implemented was highly disorganized.\n\nStalin's prewar strategy had failed. He had not managed to avoid war, and furthermore, it had gotten off to a worse start than anyone might have imagined. In addition to the military catastrophe, he had suffered a devastating blow to his self-esteem. Nobody could openly criticize him for his miscalculations, but he must have known that not only his colleagues in the leadership but also tens of millions of Soviet citizens were reproaching him in their thoughts.\n\n###   **THE STATE DEFENSE COMMITTEE**\n\nStalin's actions during the war's first days were frenetic, confused, and reactive. Even though he did not grasp the situation and was not qualified to manage armies, he tried to do something simply because it was impossible to do nothing. He tried desperately (and incompetently) to strike back at the Germans. Many, if not most, of these efforts only made matters worse.\n\nStalin clearly understood the dangers facing his country. There is convincing evidence that during the war's very first days he tried to barter away Western portions of the USSR in exchange for a truce. Beria was assigned to arrange a meeting between his representative and the Bulgarian ambassador, whose country was allied with the Nazis. The Bulgarian ambassador was asked to determine what conditions might be acceptable for a peace with Berlin. What lands was Germany claiming? Just how this initiative ended is unknown. Probably the Bulgarian ambassador was reluctant to act as an intermediary. But the attempt itself speaks volumes. Whether Stalin was truly prepared to give up Soviet lands or was just hoping to break the momentum of Germany's offensive, he clearly felt less than confident about the Red Army's defensive capabilities.\n\nThis negotiation attempt was not the only sign of Stalin's pessimism. In parallel with a general mobilization and the preparation of new defensive lines in the interior, he ordered a massive evacuation campaign during the war's earliest days. Not only were people and material resources moved away from the front line, but a secret evacuation of the capital got under way, even though the Germans were nowhere near. On 27 June the Politburo approved an order to urgently (within three days) remove from Moscow the government's precious metal and gem reserves, the Soviet Diamond Fund, and valuables held in the Kremlin armory. On 28 June it was decided that currency held in Moscow's Gosbank and Gosznak depositories should be immediately relocated, and on 29 June, that the commissariat apparats and other top government offices should be moved to the rear. On 2 July the Politburo resolved to move Lenin's sarcophagus from his tomb in Red Square to Siberia, and on 5 July, to move government and Central Committee archives.\n\nAn official summoned to Stalin's office on 26 June later recalled the following: \"Stalin did not look his usual self. He didn't just look tired. He had the appearance of someone who had endured a profoundly upsetting experience. Until I met with him, I had a feeling based on various pieces of circumstantial evidence that we were taking a heavy beating along the borders. Maybe defeat was looming. After I saw Stalin, I understood that the worst had already happened.\" The next few days brought no relief. Stalin was increasingly aware of the futility of his orders and how difficult it was to manage the army.\n\nJust a week after the war began, alarming news reached Moscow about the grave situation along the Western Front and that Minsk, the capital of Belarus, had already fallen into enemy hands. Communication with the troops had largely broken down. A tense pause settled in at the Kremlin. On 29 June, for the first time since the war began, no meetings were scheduled in Stalin's Kremlin office. According to Mikoyan, that evening Molotov, Malenkov, Beria, and he gathered at Stalin's, probably at his Kremlin apartment or his dacha. Stalin telephoned Timoshenko, but the defense commissar did not seem to know anything. The military leaders were not in control of the situation. Alarmed, Stalin, in violation of long-standing practice, proposed to the Politburo members that they all go to the defense commissariat. Here, finding further confirmation that the catastrophe had become gigantic, he showered the generals with rebukes and accusations. Unable to withstand the tension, Zhukov, head of Command Headquarters, broke into tears and ran to a neighboring room. Molotov went to comfort him. This scene evidently had a sobering effect on Stalin. He understood that putting pressure on his military leadership would not help. According to Mikoyan, as he and Molotov were leaving the commissariat, Stalin said, \"Lenin left us a great legacy. We, his heirs, have pissed it all away.\" Crude language was not unusual for Stalin, but in this case it revealed an extreme state of inner turmoil. After leaving the commissariat, Stalin apparently went to his dacha.\n\nThe following day, 30 June, Stalin did not show up at his Kremlin office or anywhere else in Moscow. Given the growing crisis, this withdrawal from his duties was truly reckless. The huge machine of government had been specially designed so that it could not run without him; inevitably, it started to break down. Something had to be done, and Molotov took the initiative. He was the most senior member of the informal hierarchy within the Politburo. According to various eyewitnesses, after losing track of Stalin, Molotov began calling him at the dacha. When he was unable to get a response\u2014or, more likely, after bearing the brunt of Stalin's dark mood\u2014he concluded that Stalin was truly struggling. According to Mikoyan, Molotov said that \"Stalin is so exhausted that he doesn't care about anything; he's lost all initiative and is in a bad way.\" This account was indirectly confirmed many years later by Molotov himself in interviews conducted by the writer Feliks Chuev: \"He didn't show himself for two or three days; he was at the dacha. He was certainly suffering and a little depressed.\" Molotov's memory seems to have failed him on certain details: Stalin did not seclude himself at the dacha for even two full days, let alone three, but given the catastrophic circumstances, even a brief absence by the country's leader must have seemed an eternity.\n\nAlarmed, Molotov called a meeting with Beria, Malenkov, and Voroshilov. There was no talk of officially removing Stalin from power or even taking over his duties. Instead the group tried to figure out how to lure Stalin out of his dacha and make him do his job. This was a delicate task. One simply did not show up at Stalin's dacha without an invitation, and under the circumstances they could only imagine how he might react to an un-sanctioned visit. Furthermore, it would not be easy explaining their reason for coming to see him. Nobody wanted to be the one to tell Stalin that his breakdown was placing the entire country in jeopardy. But these men were not neophytes when it came to political maneuvering, and they devised a brilliant plan. They decided to go together (certainly nobody wanted to go alone!) and present Stalin with a proposal for creating a supreme authority to oversee the war effort: the State Defense Committee, to be headed by Stalin himself. In addition, the committee would include the four men who had come up with the plan. Molotov would serve as first deputy to the committee chairman.\n\nThe creation of the State Defense Committee solved multiple problems at once. Now Stalin's fellow leaders could visit him at his dacha without implicitly reproaching him for not showing up at the Kremlin. That the committee would be headed by Stalin demonstrated his continued leadership and the Politburo's firm support, while the fact that it was a small committee of his most faithful comrades allowed them to privately help him make decisions as he recovered his mental balance. Finally, the four men together interacting with Stalin at this delicate time helped protect each of them from the full force of Stalin's outbursts.\n\nOnce Molotov, Malenkov, Voroshilov, and Beria had agreed on the idea of the committee, Mikoyan and Voznesensky were called to Molotov's office. They were two members of the leadership group that the four men had decided not to include in the committee, but it was important that they also come to the dacha as a demonstration of unity.\n\nMikoyan left behind an account of what happened when the delegation arrived at Stalin's dacha late in the day on 30 June. The _vozhd_ was sitting in an easy chair in the small dining room. He looked at his unexpected visitors inquisitively and asked why they had come. As Mikoyan describes it, \"He looked calm, but somehow strange.\" After hearing Beria, the chosen spokesman for the delegation, present the proposal to create a State Defense Committee, Stalin raised only one objection: he wanted Mikoyan and Voznesensky included as well. Beria was ready with the argument against expanding the membership: someone had to lead the Council of People's Commissars. Stalin relented.\n\nMikoyan's memoirs were edited by his son Sergo, who took a number of liberties with his father's original text, which is preserved in the archives. In editing his father's account of this incident, Sergo clearly tried to create the impression that Stalin was frightened by his comrades' visit, inserting embellishments such as \"Upon seeing us, he [Stalin] seemed to cower in his chair\" and \"I [Mikoyan] had no doubt: he had decided that we were there to arrest him.\"\n\nWas Stalin really frightened? How should we interpret this meeting? Unquestionably, it was an exceptional moment in the history of his dictatorship. However deferential their demeanor, Stalin's associates had violated his supreme authority in at least five ways. (1) They had come unbidden to the dacha, (2) having worked out an enormously important initiative behind his back, and (3) they urged that their proposal be adopted in the form they had agreed on among themselves. (4) They had formalized Molotov's role as second-in-command in the government despite the fact that he was out of favor with the _vozhd_ , and (5) they had decided to exclude Voznesensky from the committee, even though just that May, when Stalin had taken over the chairmanship of the Council of People's Commissars, he had chosen Voznesensky over Molotov to serve as his first deputy. In essence, Stalin's closest colleagues were letting him know that in the face of an existential threat, the post-Terror leadership had to be consolidated and that he had better abandon any thought of further shake-ups at the top. This was a unique episode; in his time in power, Stalin saw nothing like it before or after. It signaled a temporary change in the nature of the dictatorship and the emergence of a wartime political compromise, a rebalancing of power within the Politburo somewhere between the flexibility Stalin had demonstrated in the early 1930s, when he was first consolidating his dictatorship, and the tyranny he was exercising when the war broke out. This arrangement endured almost until the war's end.\n\nThe day after the meeting at the dacha, the establishment of the State Defense Committee was announced in newspapers. The fact that the committee's membership was limited to Stalin, Molotov, Beria, Voroshilov, and Malenkov did not mean that the rest of the Politburo's top leadership had lost its influence. Mikoyan and Voznesensky had important jobs keeping the economy running. Zhdanov was focused on the defense of Leningrad. Given the critical nature of wartime supply and evacuation, Kaganovich's responsibilities as railway commissar were pivotal. In February 1942, Mikoyan, Voznesensky, and Kaganovich also joined the committee.\n\nThe establishment of the State Defense Committee was the first in a series of organizational changes that eventually placed supreme leadership in the Soviet war effort in Stalin's hands. On 10 July General Command Headquarters, which had been headed by Defense Commissar Timoshenko, was replaced with a Supreme Command Headquarters, headed by Stalin. On 19 July the Politburo passed a resolution making Stalin people's commissar for defense and, on 8 August, supreme commander. The customary order was restored. Stalin was once again the sole leader of both the people and the army, decisive and confident of victory. An important milestone in \"Stalin's return\" was his famous radio address on 3 July.\n\nWhereas Molotov had gone to the Central Telegraph Building, next door to the Kremlin, to make his nationally broadcast speech of 22 June, Stalin demanded that radio facilities be set up in the Kremlin itself. The telegraph service's already overwhelmed technical staff had no choice but to comply. Cables were extended to the Council of People's Commissars building. Stalin read his address sitting at a little table with microphones and a bottle of Borzhomi mineral water. From the very start it was clear that the address would not conform to his usual style. \"Comrades! Citizens! Brothers and sisters! Fighters of our army and navy! It is to you, my friends, that I speak!\" The speech, different from any other in his career, was long talked about and remembered. Glued to their radios or studying his words in the newspaper, people sought an answer to the most pressing questions: What did the future hold? When would the war be over? Stalin offered little cause for comfort. While greatly exaggerating German losses (\"The enemy's best divisions and the best units of its aviation have been smashed\"), he was forced to acknowledge that \"This is a matter of... the life and death of the Soviet state, the life and death of the peoples of the USSR.\" Ominously, he called on the people to recognize \"the full depth of danger that threatens our country,\" to organize a partisan struggle in German-occupied territories, to create militia detachments, and to remove or destroy all material resources from territories under threat from the enemy. He used two difficult-to-translate words in characterizing the war: _vsenarodny_ (of all the peoples) and _otechestvenny_ (domestic or \"of the fatherland,\" but often translated as \"patriotic\" in the context of World War II). Anyone listening could draw only one conclusion: the war would be long and hard.\n\nThe people and especially the army deserved some explanation for what had gone wrong. They deserved scapegoats, and the search did not take long. The breakdown of Soviet defenses was attributed to missteps under the leadership of General Dmitry Pavlov, commander of the Western Front. He and many of his subordinates were tried and shot. The orders, signed by Stalin, were widely circulated within the army.\n\n###   **THE BLUNDERER IN CHIEF**\n\nAccording to Soviet General Staff statistics, between the start of the war and 1 January 1942, 4.5 million members of the Red Army and Navy were killed, wounded, or captured. Of this total, 2.3 million were listed as missing in action or taken prisoner. These estimates were probably low. Nevertheless, they show that much of the army that was thrust into battle on 22 June 1941, including a large number of newly formed units, was completely wiped out. The causes of this catastrophe need further study. Clearly they included insufficient war readiness, the massive casualties resulting from the enemy's use of surprise, and the military and organizational advantages of the Wehrmacht. Despite countless examples of heroism and steadfastness, the Red Army was demoralized. Another important factor was incompetence on the part of the military and political leadership.\n\nLacking a firm grasp of the situation, Moscow was often too slow in its decision making, and many of its decisions were bad. The links in the chain of command, the General Staff especially, were not fully functional, and it took a long time to establish reliable communication with the forces in the field. \"Even the Chinese and Persian armies,\" Stalin scolded his subordinates, \"understand the importance of communication when it comes to managing an army. Are we really worse than the Persians and the Chinese? How can you manage units without communications?... We can't stand for this absurdity, this disgrace, any longer.\" During the early stages of the war, Stalin spent a great deal of time in a special room set up next to his Kremlin office conducting conferences via telegraph. This was a cumbersome means of communication, the main beneficiaries of which are the historians who today have access to tapes of the conversations. The army and the rear were largely managed using plenipotentiary \"helpers.\" These plenipotentiaries gathered information for Stalin and, with varying degrees of success, helped him deal with the never-ending bottlenecks plaguing transport, industry, and the overall war effort. This system, apparently unavoidable during this time of defeat and disorganization, was extremely inefficient.\n\nStalin, who had no experience commanding a modern army, did the best he could, relying largely on common sense rather than military science. On 27 August 1941 he sent the Leningrad leadership the following advice on organizing the city's defenses: \"Position a KV tank an average of every kilometer, in some places every 2 kilometers and in some every 500 meters, depending on the terrain. Behind these tanks or between them position other less powerful tanks and armored vehicles. Behind this line of tanks, in back of it, place heavier artillery. Infantry divisions will be immediately behind the tanks, using the tanks not only as a strike force, but also as a shield.\" To achieve this plan, Stalin was prepared to allocate 100\u2013120 KV tanks, the newest and best heavy tanks in the Soviet arsenal, a mighty force in the right hands.\n\nStalin's involvement in tactical actions, sometimes even at the platoon level, shows just how disorganized the military command was. The first months of the war offered many painful lessons in the futility of uncoordinated counterattacks. Poorly planned, they often led to huge losses and achieved little. The Red Army's leaders had scant knowledge of how to thwart an enemy advance or minimize casualties through the use of tactical retreats to positions prepared in advance. Stalin insisted on holding every inch of ground, no matter the cost. Retreat was not allowed until it was too late. The result was the encirclement of Soviet armies and their gradual destruction, one unit at a time.\n\nSeeing battlefield failures left and right often heightened Stalin's tendency to suspect treachery. Playing up to his suspicions, on 19 August 1941 Georgy Zhukov, then commanding the Reserve Front, sent Stalin the following report: \"I believe that the enemy knows our entire defensive system very well, all the operational-strategic alignments of our forces, and knows what capabilities we have at hand. It seems that the enemy has its own people among our very senior officials with immediate knowledge of the overall situation.\" Ten days later, Stalin himself wrote to Molotov, who was then in Leningrad: \"Does it seem to you that someone is intentionally paving the way for the Germans?\" This paranoia most likely had no serious consequences. Stalin, well aware of how dangerous it would be to start a witch hunt among Soviet generals in the midst of war, limited himself to accusations of cowardice. Few generals were arrested. More often they were deprived of their command or demoted and reassigned.\n\nIntangibles such as patriotic readiness for self-sacrifice and determination to defend the motherland could partly compensate for a shortage of weaponry, battlefield experience, and tactical skill. Heroism and self-sacrifice by Soviet soldiers existed side by side with the demoralization brought on by the overwhelming force of the German assault, and Stalin received abundant evidence of both. He believed in the importance of intangibles and attributed the failures of the Red Army to panic, the wholesale surrender of Soviet units, mass desertions, and the absence of a firm command. With shrinking faith in the army's ability to consolidate its own ranks, when it came time to ensure that his commanders absorbed his own ideas about leadership and discipline, he resorted to tried-and-true methods. In July 1941 he resurrected the institution of the military commissar, loyal and eagle-eyed party representatives who would be assigned to work side by side with every commander at every level. The commissars were given vast powers, to be exercised largely through \"special\" (secret police) departments within the army. According to official statistics, between the outbreak of war and 10 October 1941, 10,201 members of the Red Army were shot, 3,321 of them in front of their units. Even these numbers hardly tell the full story of repression at and around the front lines.\n\nTo ensure that the troops fought as hard as they could, Stalin made it not only shameful but also illegal to be taken prisoner. The provisions making capture by the enemy a crime were contained in the notorious Order No. 270, issued by Supreme Command Headquarters on 16 August 1941. Judging by its style, the order was mostly (if not solely) written by Stalin. It required that those taken prisoner be killed \"by any means, either from the ground or from the air.\" The families of commanders who joined the ranks of \"malicious deserters\" were to be arrested. Families of soldiers who allowed themselves to be taken prisoner were deprived of their government pensions. The order was read out loud in every unit of the army. Treating capture as treasonous doomed former Soviet prisoners of war to discrimination long after the war concluded.\n\nUsing a combination of threats and promises of reinforcements, Stalin tried to instill in his military the will to be unyielding. On 11 July 1941, when the Germans had reached the outskirts of Kiev, Stalin sent Ukrainian party secretary Khrushchev a telegram that read: \"I warn you that if you take even one step toward pulling your troops back to the left bank of the Dnieper and fail to defend the fortified districts on the right bank of the Dnieper, you will all face brutal retribution as cowards and deserters.\" On 16 July he signed a State Defense Committee order to defend Smolensk to the last. Any thought of surrendering the city was \"criminal, bordering on outright treason against the Motherland.\" Throughout the Battle of Smolensk, which lasted until September, the surrounded Red Army put up a dogged fight, delaying the German advance across the Central Front to Moscow. Hitler's decision to move a sizable portion of his forces from the Central Front to Ukraine and Leningrad also helped slow the Nazi advance toward the capital. Throughout July and August Stalin continued to hope that Soviet forces would hold the line. Beyond it stood their three major capitals: Leningrad to the north, Moscow in the center, and Kiev to the south. Time was working against the Germans. Fall was coming, with its slushy roads, and the first frosts would not be far behind.\n\nDemonstrating that the Red Army could put up a good fight was important for Stalin's negotiations with his Western allies, Great Britain and the United States. Right after the German invasion, the leaders of these countries expressed full support for the Soviet people in their fight against the Nazis. Then began the complicated process of working out relations and holding talks about what form support would take. President Roosevelt sent his adviser, Harry Hopkins, to Moscow to obtain firsthand information. Stalin gave Hopkins an exceptionally warm welcome and tried to demonstrate decisiveness and confidence of victory. When their talks were interrupted by an air raid, the Soviet leader brought Hopkins in his own car to the bomb shelter at metro station Kirovskaia, where they were met by bodyguards and Internal Affairs Commissar Beria. One Soviet official left a description of the scene:\n\n[Beria] took Stalin by the arm and tried to bring him down below, making some remark about danger. Stalin responded curtly and rudely, which is how he always spoke when he was irritated: \"Get away from me, coward!\"... Stalin stood in the middle of the dark courtyard and looked into the black sky at the German plane in the searchlight's cross beams. Hopkins stood next to him, also watching. Then something happened that did not happen very often during night raids. The German Junker started to fall uncontrollably from the sky\u2014it must have been hit. And just then the anti-aircraft artillery hit a second plane. Stalin said, and the interpreter told Hopkins:\n\n\"That's what will happen to everyone who comes to us with a sword. And anyone who comes in the name of the good will be welcomed as a dear guest.\"\n\nHe took the American by the arm and led him below.\n\nIn such demonstrations of steadfastness, together with the fierce fight put up by the Red Army, the Western allies saw something for which they were ardently hoping: Hitler's blitzkrieg was being impeded. They could and should help the Russians. On 29 September through 1 October 1941, a conference of the three powers\u2014the USSR, Great Britain, and the United States\u2014was held in Moscow. Britain's minister of supply, Lord Beaver-brook, led the British delegation, and Averell Harriman, the U.S. ambassador to the USSR, acted as President Roosevelt's personal representative. On the Soviet side, negotiations were conducted by Stalin and Molotov. The Moscow Conference concluded with important specific agreements on assisting the Soviet war effort. The scope of assistance gradually grew. Western tanks and planes supplied through Lend-Lease made a significant contribution along the Soviet-German front. By war's end, the Red Army was mostly driving American-made trucks. Lend-Lease also played a crucial role in supplying communications equipment, locomotives, railcars, and food to the Soviet Union. \"If not for Lend-Lease, victory would have been greatly hindered,\" Stalin told Roosevelt during their meeting in Crimea in February 1945.\n\nThe USSR's new allies were clearly worried about the grim situation along the Soviet-German front. Not long before the Moscow Conference, disaster had struck the Southwestern Front, where a ferocious battle was being waged over Kiev. According to Zhukov, in late July he had informed Stalin of the difficult situation and proposed abandoning Kiev and focusing on fortifying the eastern bank of the Dnieper to prevent the Germans from breaking through the Southwestern Front's right flank. Stalin responded with a gruff refusal, removed Zhukov as chief of the General Staff, and sent him to the Western Front. The situation in Ukraine continued to deteriorate. In early August the Sixth and Twelfth Armies\u2014approximately 130,000 men\u2014found themselves completely encircled by the Germans outside Uman. On 8 August, after an advance by German troops, Stalin summoned the commander of the Southwestern Front, General Mikhail Kirponos, to confer with him via telegraph. He began the meeting in his usual manipulative manner, attributing to Kirponos intentions he had not openly expressed but that might be expected. \"We have received information that the front has decided to surrender Kiev to the enemy with a light heart supposedly due to a shortage of units capable of holding Kiev. Is that true?\" Kirponos assured Stalin: \"You have been misinformed. The Front's Military Council and I are taking every measure to prevent Kiev from surrendering under any circumstances.\" Stalin ordered him to stand firm and promised help in a few weeks.\n\nIt was obvious that the Soviet armies in the vicinity of Kiev were in danger of being encircled. In early September, the Southwestern Command, with the support of the General Staff in Moscow, proposed that forces be urgently pulled back. Stalin categorically refused. \"Just the mention of the harsh necessity of relinquishing Kiev was enough to throw Stalin into a rage and cause him to momentarily lose his composure,\" Aleksandr Vasilevsky wrote in his memoirs. On 14\u201315 September, the Germans closed the ring, encircling some 452,700 Soviet troops east of Kiev, the worst defeat of the war thus far. On 20 September, Kirponos and the rest of the Southwestern Command were killed in combat. The opportunity to surrender Kiev but preserve the army had been lost. The destruction of this huge force further strengthened the Germans' strategic advantage.\n\nHistorians of every stripe, even those favorably disposed toward Stalin, place most of the blame for this catastrophe on his shoulders. Zhukov claims that Stalin implicitly acknowledged his own guilt. When putting Zhukov in charge of the Leningrad Front in September 1941, Stalin brought up the general's warning about the threat to the Southwestern Front and said, \"Your report to me back then was accurate, but I did not understand it quite correctly.\"\n\nDefeat in Ukraine heightened the danger to Leningrad. By 8 September the city was completely surrounded. The following day the Germans launched a new offensive that took the front line to its doorstep. On 11 September Zhukov replaced Voroshilov as commander of the Leningrad Front. As Zhukov later told the writer Konstantin Simonov, Stalin considered the fall of Leningrad inevitable. On 13 September the _vozhd_ received the commissar of the navy, Nikolai Kuznetsov, in his Kremlin office, where they discussed scuttling the ships docked in Leningrad if the city was taken. That very day Stalin approved a plan to destroy the fleet. Over the next two weeks, fighting in the Leningrad suburbs became particularly brutal. As the Germans fiercely battled toward the city, Soviet soldiers, in a show of mass heroism, fought tooth and nail to repel their attacks. By the end of September the advance came to a halt. The Leningrad Blockade, one of the most horrific chapters in World War II\u2014and one of the most astounding testaments to the fortitude of the Soviet people\u2014began. Over the course of the blockade, hundreds of thousands of civilians died of hunger or German shelling.\n\n###   **INSIDE BESIEGED MOSCOW**\n\nHitler's hopes of taking Moscow before winter were revived by the destruction of a huge Soviet force in Ukraine, and he reassigned a sizable part of the German Army to the Moscow offensive. On 7 October most of the Red Army's Western and Reserve Fronts were encircled in the vicinity of Vyazma, and on 9 October the Bryansk Front was also surrounded. The road to Moscow had been cleared. The fighter pilot Aleksandr Golovanov describes how he was summoned to Stalin's office around this time. He found the _vozhd_ alone, sitting silently in his chair with some untouched food before him.\n\nI had never seen Stalin like this. The silence was oppressive.\n\n\"A great misfortune, a great sorrow has befallen us,\" I finally heard Stalin's quiet but distinct voice say. \"The German has broken through our defenses outside Vyazma....\"\n\nAfter a pause, either asking me or talking to himself, Stalin said just as quietly:\n\n\"What are we going to do? What are we going to do?!...\"\n\nHe then raised his head and looked at me. Never before or after have I seen a human face express such horrible emotional anguish. We had met and spoken just two days before, but in those two days he had grown extremely haggard.\n\nAccording to Zhukov, Stalin was suffering from influenza at the time, but staying in bed was not an option. He continued to work, overseeing defensive preparations and the redeployment of all possible reserves to the outskirts of Moscow. As part of this effort, Zhukov was called from the Leningrad Front and put in command of the defense of Moscow. On 8 October Stalin signed a State Defense Committee order to prepare to destroy 1,119 plants and factories in the city and oblast of Moscow. On 14 October the Germans captured Rzhev and Kalinin. They were just kilometers from Moscow.\n\nAs Mikoyan described it, at nine in the morning on 15 October, members of the top Soviet leadership gathered (Mikoyan mentions Molotov, Malenkov, Voznesensky, Shcherbakov, and Kaganovich). Stalin informed the group that the Germans might soon breach Moscow's defenses and proposed evacuating foreign diplomatic missions and government offices. According to Mikoyan, Stalin did not want Moscow to be surrendered, even if that meant fighting within the city until reserves capable of expelling the Germans arrived. He himself would remain in the capital as long as possible. At the conclusion of discussions, Stalin signed a State Defense Committee order dated 15 October, stating that \"Com. Stalin will be evacuated tomorrow or later, depending on the circumstances.\" Provisions were made. According to Aleksandr Vasilevsky, who was among a small group of General Staff members who remained with Stalin, planes were readied for a last-minute evacuation.\n\nThe decision to evacuate Moscow prompted a brief and frantic effort to destroy or pack up files, followed by a mass exodus, primarily by party and government officials, of which there was no shortage in the capital. Even after the evacuation, \"utter chaos reigned\" in the Central Committee building: \"The locks on many desks and the desks themselves were forced open, and forms and every sort of correspondence were scattered all over the place, including classified papers.... Top secret documents that had been brought to the boiler room to be burned were left in piles, unburned.\" In the confusion, many officials abandoned the offices and enterprises with which they had been entrusted in order to save themselves, their families, and their property. A line of official vehicles snaked out of the city. There were many cases of theft of government property and valuables. According to official statistics, on 16 and 17 October more than a thousand of Moscow's Communist Party members destroyed their membership documents. The flight of government and party officials in combination with rampant rumors provoked a general panic that grew into unrest. According to documentary evidence and eyewitness accounts, this unrest lasted for several days and fell into three main categories. First was the looting of stores and warehouses, especially those stocked with liquor, often accompanied by orgies of drunkenness. Second were attacks, often involving theft, on cars leaving Moscow filled with evacuees and their property. Third were spontaneous protests at factories and plants, including defense production facilities, by workers who had not been paid their promised wages and were upset by rumors that their places of employment were about to be destroyed. Feeling betrayed and abandoned, in many cases workers prevented the removal of equipment and demanded that the factories be cleared of the explosives that had been put in place to destroy them.\n\nMost of the top leadership did not leave Moscow on 15 October, as initially planned, and on the following day Stalin summoned a number of his associates to his apartment. Aviation industry commissar Aleksei Shakhurin, who was the first to arrive, describes this meeting in his memoirs. The Kremlin, he writes, looked deserted. The anteroom into Stalin's apartment was open, and he found the _vozhd_ smoking and silently pacing the dining room. There were signs of evacuation preparations, such as empty bookshelves. Stalin was wearing his usual jacket and pants, which were tucked into boots whose creases were riddled with holes. Noticing Shakhurin's surprise on seeing such boots, Stalin explained that his other footwear had already been removed. Soon Molotov, Malenkov, Shcherbakov, and the others arrived. Stalin did not invite anyone to sit down. Pacing back and forth he asked everyone who arrived the same question: \"How are things in Moscow?\" Shakhurin reported that at one factory not all the workers had received their pay, that the streetcars and metro were not running, that bakeries and other stores were closed, and that instances of looting had been observed. Stalin responded with the following orders: fly in money using airplanes and fix the situation with public transportation and stores. He tried to calm himself and his comrades: \"Well, it's not too bad. I thought it would be worse.\" Over the next few days the situation in Moscow really did stabilize, largely because the mass detention and arrest of \"suspicious elements\" began after a state of siege was declared on 20 October.\n\nStalin's comment that he had expected worse disorder in Moscow is consistent with his way of thinking. He was undoubtedly worried about the possibility of disturbances. The danger that conflict with a foreign enemy could be used to start a civil war\u2014a formula used by the Bolsheviks in 1917\u2014greatly affected Stalin's political decision making in the late 1930s. The catastrophic start of the war could only have revived such fears. Yet anti-government and defeatist tendencies did not reach a critical level in the Soviet rear, in large part because of the secret police system put in place before the war. After 22 June 1941 this system was not relaxed; it became, if anything, more ruthless. Nevertheless it would be wrong to attribute political stability solely to repression. A blend of patriotism, growing hatred of the Nazis, a sense of duty, and a tradition of subservience led people to unite in the name of victory. The few large-scale disturbances about which historians have learned more from recently opened archives were mainly caused by the government's panicked actions and a sense of defenselessness on the part of the population.\n\nWhile Moscow offers some of the most dramatic examples of unrest, there were others. One well-documented case is the disorder that broke out in Ivanovo Oblast, northeast of Moscow. As the Germans approached, plans were being made to evacuate local textile mills. Rumors spread that the mills would be blown up, that food supplies were being trucked out, and that party and government officials were fleeing the area. Textile workers, fearing that they would be left to starvation and slaughter, erupted in spontaneous uprisings on 18\u201320 October. They tried to prevent the removal of equipment and beat some plant managers and party activists. Cries could be heard from the crowds: \"They'll take our equipment and leave us without work\"; \"All the big shots have fled the city and we've been left on our own\"; \"Makes no difference to us if we work for Hitler or Stalin.\" A combination of persuasion and arrests eventually restored calm. Furthermore, the situation at the front was improving, and it was no longer necessary to evacuate Ivanovo's textile plants.\n\nBy late October, Soviet troops had halted the enemy advance in the Central Direction. In addition to determined fighting by the Red Army, which suffered huge losses, the exhaustion of German troops and the mud and slush of autumn helped bog down the invasion. Urgent measures were now needed to prevent renewed Wehrmacht attacks on Moscow. Stalin was very involved in improving the capital's defenses, forming new fighting units, and overseeing the production of military hardware, especially tanks and aircraft. In many cases he turned his Kremlin office into a sort of master control center for dealing with logistical questions and overseeing cooperation among enterprises.\n\nHe also remained personally involved in the minute planning of combat operations. As in previous months, he closely followed the situation at the front, demanded thorough accounts of operations, and issued detailed orders in a broad array of areas. He was clearly eager to go on the offensive, whether or not the time or resources were available, in the hope that unexpected attacks would put pressure on an enemy that had spread itself thin across a huge front. His commanders did not always agree. In November Zhukov, now commanding the Western Front, objected to one such plan. Stalin demanded that counterstrikes immediately be launched in the areas of Volokolamsk and Serpukhov to disrupt German preparations for offensive action. Zhukov tried to explain that he simply lacked the forces to prepare both a defense and an attack. Stalin brought the argument to a close: \"Consider the question of a counterstrike to be settled. Submit your plan this evening.\" He then immediately called a member of the Western Front's military council, Bulganin, and threatened: \"You and Zhukov have gotten pretty full of yourselves. But even you can be called to account!\" The hastily organized offensives achieved little. Zhukov, who was trying to maintain a reserve force capable of dealing with a new German offensive, was probably right.\n\nStalin was much more effective in the area of propaganda. Taking advantage of the relative tranquility at the front in early November, he ordered that the usual celebration be held to honor the anniversary of the October Revolution. He understood that carrying on with this annual event in the besieged capital would have a tremendous propaganda impact. On the eve of the anniversary, 6 November, a huge celebratory gathering was held at the Maiakovskaia metro station. A train parked at the station was set up with a cloakroom and tables of food for party and military leaders. Speeches in honor of the revolution's anniversary were followed by a concert, but the centerpiece of the event was Stalin's address to the country, only his second public appearance since the war had begun. Clearly he was expected to provide some sort of explanation for the German forces' ability to take so much Soviet territory and to offer some idea of what lay ahead. When would the war end? This was the question on the mind of every Soviet citizen. The _vozhd_ admitted that the danger hanging over the country \"has not only not receded but has intensified.\" Overall, however, he was optimistic. Citing huge (and fictitious) German casualty statistics, he pronounced that Germany's human reserves \"are already drying up,\" while the Soviet Union's reserves were \"only now being fully deployed.\"\n\nThe following day, the anniversary itself was marked with a military parade through Red Square. This was a risky undertaking since a few days earlier, on 29 October, German planes had dropped a large bomb right on the Kremlin. A total of 146 people were injured and 41 were killed. The Luftwaffe could certainly strike again. In anticipation of this possibility, a parallel parade was held in Kuibyshev (today's Samara), the city chosen as the reserve capital should Moscow fall. In case of an attack during the Moscow parade, radio coverage of the celebration would switch to Kuibyshev. No such attack took place.\n\nStalin addressed the parading troops with a short speech delivered from atop Lenin's Mausoleum. He recalled the glorious victories of prerevolutionary commanders and of the Bolsheviks during the Civil War. Speaking of the coming German defeat, he was so bold as to speculate on the timing: \"In just a few months, just a half year, perhaps a year, Hitler's Germany will collapse under the weight of its own crimes.\" This assurance seems to reflect his understanding of the military situation, and it soon led him to demand an offensive on all fronts.\n\nThe celebrations in Moscow\u2014especially Stalin's speeches\u2014were part of a major propaganda campaign through every possible medium. The military parade on Red Square was captured on film, but for some reason Stalin's speech was not. It was decided to stage the speech in an improvised studio. A mockup of Lenin's tomb was built in one of the halls of the Great Kremlin Palace, and Stalin repeated his speech for the cameras on 15 November. In December, movie theaters began showing _The Parade of Our Troops on Moscow's Red Square on 7 November 1941,_ including the reenactment of Stalin's speech. Over seven days, beginning December 4, two hundred thousand viewers watched the film in Moscow alone. Hundreds of copies were sent to towns across the country.\n\nOn the same day Stalin reenacted his speech for the cameras, after lengthy preparations the still overwhelming forces of the Wehrmacht launched a new attempt to take Moscow. The advance covered significant ground and in some areas managed to reach the boundaries of the Soviet capital. Nevertheless, the Red Army, bolstered by a constant stream of reinforcements, was able to prevail. Just when the Germans had used up their last reserves and had come to a halt, the Red Army, almost without pause, launched a surprise counteroffensive. By January 1942 the enemy had been driven back 100\u2013250 kilometers from Moscow. Finally there was true cause for celebration.\n\n###   **THE DEFEATS OF 1942**\n\nThe offensive by Soviet troops outside Moscow, together with successes on other fronts, inspired hope throughout the entire anti-Nazi world but also exposed the Red Army's weakness and the enduring advantage of the Wehrmacht. Soviet troops demonstrated a strong will to fight but could not achieve some important objectives the Soviet leadership placed before them. Meanwhile, the Germans dug in and prepared their own counteroffensive.\n\nOn 10 January 1942, Red Army units received a letter critiquing past operations and looking ahead to upcoming ones. The tone and style of the letter suggest that much of it was written by Stalin. It was generally critical of the way in which German defenses had been breached during the December counteroffensive. The widely dispersed actions by the Red Army, which was stretched thin along the entire front, were characterized as incorrect. \"The offensive can achieve the necessary effect,\" it read, \"only if we create a force capable of overwhelming the enemy in one sector of the front.\" A second major failing was the poor use of artillery. \"We often throw the infantry into an offensive against the enemy's defensive line without artillery, without any artillery support, and then complain that the infantry is not advancing against a well-defended and dug-in enemy.... This is an offense, not an offensive\u2014an offense against the Motherland, against the troops forced to endure senseless casualties.\" The Supreme Command demanded regular artillery support for attacking units, not just during the preparatory stages of an offensive. Here too the main emphasis was on concentrating artillery where the thrust of the attack would be focused.\n\nThese were sensible and important observations on the perils of frontal attacks, which entail large casualties, and the need to concentrate forces and maneuver skillfully. But in planning the winter campaign of 1942, Stalin ignored his own warnings and insisted on attacking on all fronts at once. He wanted the swift, victorious conclusion to the war that he had promised during his 7 November 1941 address. This idea was also expressed in secret documents. Stalin's basic assumption, apparently based on the intelligence reports he was receiving, was that Germany had used up its reserves. In his 6 November 1941 speech he claimed that the Germans had lost 4.5 million men during four months of war, and the subsequent reports he received tended to support these fantastic numbers. For example, German casualties as of 1 March 1942 were estimated at 6.5 million. These figures, five or six times higher than the actual ones, were probably the result of the usual Soviet system of distortion, in which the _vozhd_ was told what he wanted to hear.\n\nThe plan for the summer campaign, approved in March 1942, provided for a shift to strategic defense and a buildup of reserves for the next offensive. Stalin wound up issuing orders that conflicted with this decision and led to the staging of offensive operations in multiple sectors. \"After reviewing the plan of action adopted for the summer of 1942, I must say that its weakest aspect is the decision to conduct defensive and offensive actions at the same time,\" Marshal Vasilevsky wrote several decades later. This opinion also prevails in scholarly literature on the subject.\n\nDuring the summer of 1942, offensives were planned for Crimea, the Central Direction, and around Kharkov and Leningrad. Stalin was heavily involved in the planning of these operations. In matters of staffing, where he was, as usual, worried about selecting leaders capable of acting decisively, his personnel choices again reveal his shortcomings as supreme commander. He sent Lev Mekhlis, the head of the Red Army's Main Political Directorate, to represent Moscow in Crimea. Mekhlis, who had served as Stalin's secretary, was fanatically loyal to the _vozhd,_ energetic, decisive, and ruthless, but he was completely ignorant of military science.\n\nVoroshilov was assigned to the Volkhovsky Front, outside Leningrad, despite having been earlier dismissed from the Leningrad Front for incompetence. His special relationship with the _vozhd_ allowed him to turn down this assignment, infuriating Stalin. On 1 April 1942 the Politburo adopted a decision, dictated by Stalin, subjecting Voroshilov to savage criticism. The disclosure of his reason for turning down this command was obviously meant to embarrass him. The former defense commissar was quoted as saying that \"The Volkhovsky Front is a difficult front\" and that he did not want to fail at the job. The Politburo resolved to \"(1) Recognize that Com. Voroshilov did not prove himself in the work assigned him at the front. (2) Send Com. Voroshilov to perform military work away from the front lines.\" This was an empty gesture: Voroshilov was not banished from Stalin's inner circle. Nevertheless, the resolution, which became known to a wide circle of top officials, may have been a warning to others.\n\nThe Southwestern Command was not a particular source of Stalin's complaints. Aware of his inclinations, the front commander, Timoshenko, and military council member, Khrushchev, proposed an offensive to retake Kharkov. After confronting objections from the General Staff, Stalin decided to maneuver. He approved the Ukrainian operation but pronounced it an internal matter for the front's commanders. This decision did not change anything, but it relieved Stalin of some responsibility for how it turned out.\n\nThe poorly conceived plans for the offensive led to more heavy losses and damaged the overall strategic situation. The first disturbing sign was defeat in Crimea. The German counteroffensive, launched on 8 May 1942, crushed Soviet troops in twelve days and sealed the fate of the Crimean city of Sevastopol, which had been under siege for eight months. Large-scale heroism was not enough to prevent catastrophe. The city fell in July after the Germans brought in significant forces from other fronts. According to the Sovnarkom's chief of administration, Chadaev, Mekhlis tried to make his excuses to Stalin in person, waiting outside the _vozhd_ 's office. Chadaev described what happened when Stalin appeared in the doorway: \"Mekhlis jumped up from his seat: 'Hello, Comrade Stalin! Permit me to report.' Stalin paused for a moment, looked Mekhlis up and down, and with a voice filled with emotion pronounced: 'Damn you!' He then headed straight into his office and slammed the door. Mekhlis slowly lowered his arms to his sides and turned toward the window in distress.\"\n\nThe following day, 4 June 1942, Stalin signed a Supreme Command directive to the military councils of all fronts and armies on the reasons for defeat in Crimea. The style of the directive, which pointed out that the Crimean forces had been crushed despite having a significant numerical advantage, suggested he had a hand in composing it. The commanders in Crimea, including Mekhlis, were accused of incompetence and inability, removed from their positions, and stripped of their rank. Nevertheless, Mekhlis did not fall out of favor with Stalin and continued to be given important posts. Zhukov later speculated that Stalin was relatively lenient in punishing those who had directed the Crimean catastrophe \"because he was aware of his own personal responsibility for it.\"\n\nThe effort to retake the eastern Ukrainian city of Kharkov was also planned with Stalin's full support. The attack began on 12 May and at first seemed to promise success. A few days in, however, everything changed. The Germans, who were thought to be focused on capturing Moscow, were in fact planning a decisive offensive in the south. Timoshenko's poorly conceived plans for Kharkov only made their task easier. Despite warnings that the huge Soviet force now risked encirclement, Stalin refused to halt the attack on Kharkov in order to deal with this threat. By the time he decided to suspend the offensive, it was too late. According to General Staff statistics, 277,000 Red Army troops were lost\u2014killed, wounded, or captured\u2014in the Second Battle of Kharkov. The Germans had again been handed a strategic advantage. Hitler's forces were now able to move quickly toward the Caucasus and the Volga.\n\nStalin placed the blame for this defeat squarely at the feet of his commanders, although they were not castigated as harshly as those involved in the Crimean debacle. A few months later, on 24 September 1942, Georgy Malenkov, who had been sent to represent Headquarters at the Stalingrad Front (constituted primarily from the forces of the Southwestern Front), wrote to Stalin: \"While we're on the subject of Timoshenko.... Now that I've been able to see how he's been working here, I can say that Timoshenko looks like a good-for-nothing, indifferent to the fate of the Soviet government and the fate of our motherland.\" Given Malenkov's usual caution, we can assume that he was expressing an opinion with which he knew the _vozhd_ would agree. As with Mekhlis, however, Stalin kept Timoshenko within his inner circle but used him for less critical assignments.\n\nAccusing generals of mistakes and a lack of decisiveness was a leitmotif of Stalin's directives throughout 1942. The generals themselves took a different view. Marshal Konstantin Rokossovsky, for example, wrote in his memoirs that the defeats during the summer of 1942 stemmed from the fact that Headquarters kept repeating the mistakes of the early stages of the war. Commands from the top \"did not correspond to the situation\" and \"only played into the hands of the enemy.\" Instead of gradually pulling troops back to lines prepared in advance (in the summer of 1942, the River Don), Headquarters kept demanding counterattacks. Troops hurriedly moved toward the Germans \"with no time to concentrate, on the fly, went into battle disorganized against an enemy that under these circumstances enjoyed a huge numerical and qualitative advantage.... This was all done in a manner that had nothing to do with the military science we were taught in the colleges and academies, during war games and maneuvers, and it went against all the experience we acquired during the two previous wars.\"\n\nRefusing to recognize any fault on the part of the Supreme Command, Stalin continued to attribute failure solely to the cowardice, treachery, or, at best, incompetence of his subordinates. The ultimate expression of this logic was the notorious Order No. 227, issued on 28 July 1942, just when the German advance in the south seemed unstoppable. Stalin, who undoubtedly wrote the order himself, was exceptionally harsh: \"Panic-mongers and cowards must be exterminated on the spot.\" Commanders \"who retreat from battle positions without an order from above [are] traitors against the Motherland.\" He demanded that commanders be put on trial, starting with army commanders who sanctioned unauthorized retreat. The order provided for the creation of penalty battalions and companies, the ranks of which would be filled by people arrested for violating the Stalinist code of conduct, to be used as cannon fodder at the start of attacks. Anti-retreat units became a regular part of the army and were tasked with \"shooting on the spot panic-mongers and cowards in the case of panic and disorderly retreat by division units.\" These units were not disbanded until October 1944.\n\nThe fight against \"panic-mongers,\" \"cowards,\" and \"saboteurs\" was a centerpiece of Stalin's military policy during the summer of 1942, and fear and panic were indeed a problem. Given the hardships of battle and the long string of defeats, troop morale was inevitably low. But as during the Terror, Stalin's tendency to see saboteurs and wreckers as the root of all failures had no basis in reality. The mental state of Soviet soldiers in the face of the well-organized might of the German Army was just one of many threads in the tangled web of reasons for Red Army retreats. Often orders were disobeyed because they were poorly conceived or simply not realizable. Draconian measures at the front did not guarantee victory. A few weeks after Order No. 227 was issued, the Germans reached the outskirts of Stalingrad.\n\nBeside cowardice and treason, another explanation for Soviet defeats that featured prominently in Stalin's mind was that Hitler was not distracted by a second front in Western Europe. Within the top leadership, the Nazi leader's ability to concentrate his forces on the Soviet front due to inaction by the Allies was a frequent source of anger and frustration. After heavy pressure from Stalin, during a visit by Molotov to Great Britain and the United States in May and June of 1942, Churchill and especially Roosevelt expressed their intention to open up a second front that autumn. These vague promises began to look increasingly chimerical as the situation worsened on all fronts. To soften the blow of his failure to open a European front, Churchill went to see Stalin in Moscow. On 12 August 1942 the two men had their first face-to-face meeting. Stalin found himself in a weakened position due to the numerous defeats suffered by the Soviet side. Meanwhile, the Allies' losses in North Africa and the Mediterranean gave them an excuse for delaying a French landing.\n\nStalin did not hide his irritation at Churchill's explanation. The atmosphere during the first hours of negotiations was extremely tense. The Soviet leader, abandoning diplomacy, disparaged the Allies' wavering and advised them not to fear the Germans. Churchill was just as blunt. He reminded Stalin that Great Britain had been battling the Nazis for a full year, an unmistakable reference to the fact that Britain was already at war with Hitler while Stalin was helping him carve up Poland. With these reproaches out of the way, the allies, who greatly needed one another, settled down to serious discussion. Having given a great deal of thought to his negotiation strategy, Churchill delivered his good news: a landing of American and British forces was planned for the northern coast of French Africa that fall. Stalin took this opportunity for conciliation. He praised the new plan, and subsequent talks went more smoothly. Stalin made the friendly gesture of inviting Churchill to his Kremlin apartment for his last night in Moscow, 15 August, where the evening passed convivially.\n\nThe conclusions to be drawn from Churchill's visit were clear. The USSR would be able to count on its allies mostly for material assistance. Stalin told Churchill that his country particularly needed trucks and aluminum. For now, the Germans could continue fighting on the Eastern Front without worrying about a serious challenge from the West, and the Red Army would continue to suffer defeat and failure. In the south the Germans had entered Stalingrad, had captured the important Don and Kuban agricultural regions, and were drawing near to the petroleum deposits of the North Caucasus and Transcaucasia. According to official Soviet statistics, from January through October 1942 alone, 5.5 million Red Army soldiers had been killed, wounded, or captured. Gradually, however, the formation of new armies and the heroism of the defenders of Stalingrad and the Caucasus allowed the front to stabilize. Hitler's shortage of manpower, as he simultaneously pursued several difficult objectives, also helped shift the momentum. In the ruins of Stalingrad, Soviet troops fought German divisions in pitched battle. By all appearances, this was a replay of late 1941. The battered German armies could advance no farther. Having inflicted huge losses, the Red Army now had an opportunity to seize the initiative. The question was how and when to strike back.\n\n###   **STALINGRAD AND KURSK**\n\nThe counterstrike came outside Stalingrad. This famous Soviet victory was the culmination of heroic efforts and huge sacrifices by the entire country. It showed that Stalin, too, had finally learned from past defeats. The well-prepared Soviet offensive outside Stalin's namesake city began on 19 November 1942. A few days later, Germany's 330,000-man force in Stalingrad, led by General (soon to be Field Marshal) Friedrich Paulus, was surrounded. After thwarting German attempts to break through the encirclement, on 2 February 1943 Soviet forces finally compelled the enemy to capitulate. The protracted battle cost the Germans hundreds of thousands of soldiers and officers. More than 90,000 were taken prisoner, including Paulus himself. The victory marked a major turning point in the war.\n\nDespite this impressive triumph, Stalin continued to act with caution. In planning the new campaign, the Soviet Supreme Command tried not to spread its forces too thinly. The main counterstrike was focused on the Southwestern Direction, where the enemy had already suffered huge losses and was largely disorganized. Hoping to repeat the success of Stalingrad, in January 1943 Stalin ordered the encirclement of the German forces retreating from the North Caucasus. Elsewhere, counteroffensives in the Voronezh and Kharkov Directions made promising beginnings. And on 18 January 1943, at the northern end of the vast Soviet-German Front, the Leningrad Blockade was finally broken and the city again became accessible to Central Russia via land. The liberation of the country's long-suffering historic capital had enormous symbolic and emotional significance.\n\nAmid the rejoicing, Stalin's comrades were eager to crown him with victor's laurels. On 19 January 1943, during a visit to the Voronezh Front, the chief of the General Staff, Vasilevsky, joined the front's leaders in addressing a coded message to Molotov, Beria, and Malenkov. They proposed that following the \"unparalleled successes of our troops at the front,\" Stalin deserved the title \"generalissimo of the Soviet Union.\" The telegram described Stalin as the \"organizer of our victories, a genius and great commander.\" The members of the top leadership, who may very well have inspired this initiative in the first place, greeted the proposal with enthusiasm. On 23 January Molotov, Beria, Malenkov, and Mikoyan signed a motion to that effect and placed it before the Politburo. Nevertheless, it wound up being filed away. Stalin must have felt that his elevation to the rank of generalissimo was premature. Despite hopeful signs, many hard battles lay ahead. Hundreds of thousands of Soviet families were still receiving the dreaded notifications that a loved one had been killed in action. Stalin eventually got the title of generalissimo, but later, after final victory in 1945. For now he settled for the gold-embroidered shoulder boards of a marshal. The resolution elevating him to that rank was published on 7 March 1943. Before Stalin, in January and February respectively, Zhukov and Vasilevsky were also given this honor.\n\nThe rank of marshal was more than sufficient for now. Events at the front soon showed that the Red Army was not safe from further defeats. Significant victories came in the form of the liberation of the North Caucasus and Stavropol and Krasnodar Krais. On the other hand, the Red Army could not carry out its plan to encircle German units in these areas. The enemy managed to maintain its numbers and retreat to the Donets Basin, the lower reaches of the Kuban, and the Taman Peninsula. Soviet forces were successful during early 1943 along the Voronezh, Bryansk, and Southwestern Fronts. Voronezh was liberated in January, and Kursk, Belgorod, and Kharkov in February. But soon the momentum shifted back to the Germans. One reason for this shift was some bad decisions by the Soviet Supreme Command. The Soviet armies were attacking along a broad front, but the enemy, which had stealthily concentrated its forces at strategic points, counterattacked. In March it again occupied Kharkov and Belgorod. The Red Army achieved only modest results in its Western Direction offensive, and its efforts in February and March along the Northwestern Front were not effective.\n\nIn April through June 1943, a strategic lull set in as the two sides prepared their summer campaigns. As the Soviet military leaders' memoirs make clear, nobody doubted that the Germans would strike first at the Kursk salient. By attacking the flanks, the Wehrmacht could encircle and destroy the large number of Soviet forces within the salient and recapture the strategic initiative. The Germans knew that unless they could eliminate the Kursk salient, they would face serious danger. Yet there was some question as to whether the Germans would attack at all. Deciding against an anticipatory offensive, Stalin agreed to meet the enemy from a well-prepared defensive posture, in the hope that it would allow the Red Army to crush the German forces and transition to an offensive posture from a much stronger position.\n\nThe decision to focus on defense shows that Stalin was learning from past mistakes. Whereas earlier he had preferred large-scale lightning attacks before the enemy had time to regroup, he now understood the need to wait, plan, and prepare. Restraint was not easy for him. Twice in May, intelligence suggested that the Germans were about to strike. Soviet forces were put on high alert, but each time proved to be a false alarm. According to General Vasilevsky, in both cases Stalin favored a preemptive attack. \"It took quite an effort by us, by Zhukov, me, and Antonov, to convince him not to do that,\" Vasilevsky wrote. June came, and the Germans still did not attack. Stalin was uneasy and again began pondering a first strike. This time, too, he listened to his generals, who convinced him that it would be advantageous to wait out the enemy.\n\nThe generals were right. The Battle of Kursk began on 5 July 1943 and continued until 23 August. Huge forces, a total of 4 million troops, were arrayed on both sides. This was a major tank battle, and the Soviet side had twice as many as the Germans. The Nazi leaders still hoped that superior organization and up-to-date weaponry\u2014especially the Tiger and Panther tanks\u2014could earn them another victory. It might have turned out that way had they not also faced superior numbers and a more mature and better-prepared force. After wearing down the enemy through a week of fierce fighting from a defensive posture, the Red Army struck back.\n\nAt the height of the counteroffensive, in early August 1943, Stalin visited the front for the first and last time. During the early morning hours of 2 August he boarded a special train disguised to look like a freight carrier that stopped close to his dacha. The part of the front closest to Moscow, the Rzhev-Vyazma salient, the site of preparations for an offensive operation, was chosen for the visit. After arriving at the closest train station, Stalin and his entourage continued by automobile. He spent 3 and 4 August visiting the command posts of each front and meeting with the leaders planning offensives. Here he learned that Soviet troops had retaken Orel and Belgorod. Stalin telephoned Moscow and ordered an artillery salute in honor of this victory. The visitors returned to the train for dinner, and on the evening of 5 August it left for Moscow. Stalin returned to his Kremlin office.\n\nStalin did not like to travel even in peacetime and left Moscow only for vacations. Officially, he was inspecting preparations for the Smolensk offensive operation. In fact, there was no military necessity for this, and his visit did nothing to prevent the operation's failure. The real reason for the trip lay in what we now call \"optics.\" The leader of a country at war has to show solidarity with his army and a willingness to share in its hardships. During the first stage of the war, when Moscow itself was on the front line and Stalin's presence in the besieged capital was of tremendous political significance, solidarity could be demonstrated by his staying in place. Stalin must have understood that even after the tide of war began to turn, such demonstrations were important to sustain his reputation as an involved and compassionate leader.\n\nStalin managed to transform his sole visit to the front lines into a matter of routine. During the summer of 1943 he conducted a heated correspondence with Roosevelt and Churchill. In response to the Allies' refusal to open a second front in northern France in 1943, Stalin refused to participate in summits and grew dilatory in his correspondence. His explanation was that he was too busy rallying the troops. In early August he wrote to his coalition partners: \"I have just returned from the front.... I have had to make more frequent visits to the troops than usual.\" \"I have been compelled to personally spend more time in various sectors of the front and put the interests of the front before all else.\"\n\nAfter returning from the Western Front, Stalin again had to turn his attention to developments in the south, where the Kursk offensive was still raging. The Battle of Kursk put an end to any chance for a German victory, but most of the Nazi forces escaped encirclement and withdrew to prepared defensive lines. Building on Soviet successes, the Supreme Command organized offensives in Ukraine, Crimea, and the Central Direction. The German forces switched to a defensive posture, launching only intermittent counterattacks. The most important developments were taking place at the southern end of the Soviet-German Front. In September the Red Army managed to capture the German bridgehead on the right bank of the Dnieper. At the same time, Hitler's forces were pushed out of the economically important Donets Basin and, to the south, Novorossiisk and the Taman Peninsula. In the predawn hours of 6 November the Red Army liberated the Ukrainian capital of Kiev. By the autumn of 1943, Hitler's forces had been rendered incapable of large-scale offensives. The Red Army advanced six hundred kilometers to the south and three hundred to the west, but these impressive victories came at the expense of heavy losses inflicted by a still capable enemy. Furthermore, many of the objectives assigned by Headquarters were not met. Soviet forces had made little progress in the Western and Northwestern Directions. The attempt to liberate Crimea had failed, and fierce counterattacks by the Wehrmacht made it impossible to build on the ousting of the Nazis from eastern Ukraine. The Germans were managing to evade a decisive blow. The successful approach used in Stalingrad, of encircling and liquidating enemy army groups, could not be repeated. The bloody war would not end any time soon.\n\nBritish and American forces also made progress in 1943. Large deployments of German troops were defeated in North Africa and Sicily, and the southern portion of the Italian peninsula was occupied, bringing down Mussolini's regime and taking Italy out of the war. The Allies were also achieving success against Japan, and Germany's submarine fleet suffered significant losses in the Atlantic, making shipments of supplies and troops from the United States less dangerous. Allied bombing of Germany was causing increasing devastation. The British and Americans no longer worried that the Soviet Union would collapse under the weight of war, and such a realization relieved some of the pressure for major sacrifices by the Western allies. Moreover, the idea of an advance through the Balkans was beginning to look like a viable alternative to the opening of a second front in northern France. Churchill favored the Balkan approach, but Roosevelt, based on American interests, maintained his previous commitment to a landing on the French coast.\n\nFor Stalin, the opening of a second front remained a top priority in relations with his allies. While he of course wanted to relieve the suffering of his battered and exhausted country, he also saw such an opening as a matter of political prestige and a sign of his standing within the Big Three. Not surprisingly, on hearing in June 1943 that Churchill and Roosevelt were planning to postpone the opening of a front in northern France until the next year, his response was icy. \"I must inform you,\" he wrote his partners on 24 June, \"that this is a matter not just of disappointing the Soviet government but of preserving its trust in its allies, trust that has been put to serious tests.\" In August, the Soviet ambassador, who enjoyed good relations with the British establishment, was pointedly recalled from London. But the allies could not afford total alienation, and none wanted to go anywhere near the point of breaking off relations. This was evident in the decision that soon followed, after contentious negotiation, to hold the first face-to-face meeting of the Big Three. In November 1943 the allies gathered in Tehran, the site proposed by Stalin. This concession by Roosevelt and Churchill at least took some of the sting out of their decision to delay an invasion.\n\nThis trip, Stalin's first outside the Soviet Union since coming to power, did not take him far from the Soviet border. After traveling to Baku by train, he took a short flight to the Iranian capital. As far as we know, this was Stalin's first and last flight in an airplane, and he appears to have been anxious about it. According to the memoirs of General Sergei Shtemenko, who accompanied Stalin on the trip, a problem developed at the airport in Baku. Stalin refused to fly in a plane piloted by a high-ranking member of Soviet aviation, General Golovanov (mentioned above), and preferred to be flown by a less eminent pilot. \"General-colonels rarely fly airplanes; we'd be better off flying with a colonel,\" he is quoted as saying. Golovanov categorically denies this account, but he does say that while still in Moscow, Stalin wanted to discuss plans for the flight in detail. Among his instructions, he ordered Golovanov to check the reliability of the pilot. Stalin apparently had a difficult time during the flight. While meeting with UK Ambassador Archibald Kerr and U.S. Ambassador Harriman in September 1944, he told them that his ears hurt for two weeks afterward.\n\nThe Tehran Conference got under way on 28 November 1943. This was Stalin's third meeting with Churchill and his first with Roosevelt. Face-to-face contact with Roosevelt was particularly important. Stalin knew that the American and British leaders did not see eye to eye on everything, and one point of difference was the opening of a second front in northern France. Roosevelt and Stalin, each for his own reason, both advocated this second front. Stalin had two powerful cards in his pocket: the Red Army's victories and a promise to take up arms against Japan after Hitler's defeat. Beside a desire for good long-term relations with the USSR and its help against Japan, Roosevelt was also motivated by his reluctance to have Red Army troops moving deep into Western Europe. The result was a promise in Tehran that the United States and Great Britain would open a second front in the north of France in May 1944. Discussions also covered future Soviet efforts against Japan, the creation of a postwar international security system, the borders of a postwar Poland, and other issues. Stalin had every reason to come away pleased.\n\n###   **VICTORY AND VENGEANCE**\n\nThe Allied successes in 1943 left no doubt that Germany would ultimately lose the war, but when? How many lives would be sacrificed before that happened? Having learned a bitter lesson, Stalin no longer tried to assign a timetable to the fall of the Reich. The Germans put up a desperate fight. Holed up in defensive positions, they launched only occasional counterattacks. Meanwhile, the Red Army pushed forward, sometimes quickening the pace, sometimes slowing it. Both sides endured heavy casualties.\n\nDuring the first five months of 1944 the Red Army achieved impressive victories at both ends of the huge Soviet-German Front, in Ukraine and outside Leningrad. Its forces, fighting fiercely, advanced hundreds of kilometers, in places even going beyond the Soviet border into Romania. But in the center of the Eastern Front, the Germans were unassailable. For the Red Army, the campaign of the summer of 1944 was dedicated to destroying the enemy forces at the front's center. The well and stealthily prepared operation in Belarus was one of the most significant of the entire war. It led to the destruction of a huge Wehrmacht force.\n\nCelebrating his triumph, Stalin ordered up an impressive propaganda spectacle. For several hours, beginning on the morning of 17 July, a column of more than fifty-seven thousand German prisoners of war, with generals and officers at the head of the line, was paraded through central Moscow. That evening they were loaded onto trains and sent to camps. Crowds of Muscovites lined the streets to observe this extraordinary event. \"As the column of prisoners of war passed by,\" Beria reported to Stalin, \"the population behaved in an organized manner.\" He described for the _vozhd_ the shouts that could be heard: \"Numerous enthusiastic exclamations and salutes in honor of the Red Army and our Supreme Commander-in-Chief,\" as well as \"anti-fascist cries of 'Death to Hitler,' 'Death to fascism,' 'Let the scoundrels die,'\" etc. After the columns had passed, crews of water trucks were brought in to pointedly wash the streets clean. On 16 August a similar spectacle took place in Kiev.\n\nThis demeaning procession of German prisoners symbolized the impending collapse of Nazism. On 6 June 1944, British, American, and other Allied troops landed on the beaches of Normandy. Overwhelmed by the Red Army in 1944, Germany's allies Romania and Finland laid down their arms. Red Army troops liberated all Soviet territory, expelled Hitler's forces from a significant portion of Eastern Europe and the Balkans, and moved toward the borders of Germany itself.\n\nThese decisive victories were primarily the result of the Soviet Union's military and economic superiority, attained through the entire country's sacrifices and exertion. By June 1944 the Soviet armed forces exceeded 11 million people. Red Army assets included field forces numbering 6.6 million, approximately 100,000 mortars and artillery, 8,000 tanks and self-propelled artillery, and 13,000 combat aircraft. In terms of personnel, the ratio of forces along the Soviet-German Front was 1.5:1 in favor of the Red Army; for mortars and artillery, 1.7:1; and for combat aircraft, 4.2:1. The two sides were approximately equally matched in tanks. Furthermore, the Soviet side had significant reserves, while the capacity of the Reich and its allies was shrinking by the day. The Red Army and its commanders, led by Stalin, were growing increasingly confident, bolstered by the wealth of their resources and the experience acquired through years of catastrophe and, finally, victory.\n\nFor Stalin, managing the army and continuing to increase its might remained a high priority. Furthermore, liberated areas of the country lay in ruins and desperately needed to be rebuilt. The Nazis had exterminated millions of Soviet civilians, especially Jews. Many towns and villages were completely depopulated. A 1 July 1944 letter to Stalin from the head of Belarus offers a glimpse of the state of territories that had been under German occupation: \"There are 800 people left in Vitebsk; before the war there were 211,000.... Zhlobin has been completely destroyed. There is a small number of wooden buildings and the frames of three stone ones. There is no population in the city.\"\n\nIn addition to repairing physical destruction, the liberation of Soviet territories confronted the leadership with new political problems. For varying durations\u2014from a few weeks to three years\u2014tens of millions of people had lived under Nazi occupation. Many had either been forced to collaborate or had done so out of conviction. Many others had fled to serve with pro-Soviet partisans or had done what they could to help them. Most had simply tried to survive in the new order. Stalin felt no responsibility for the suffering of Soviet citizens who, in Soviet bureaucratic language, \"resided in occupied territory.\" Like soldiers taken prisoner by the Germans, anyone who lived in captured territory was classified as \"suspicious.\" As part of their reintegration into the USSR, liberated areas had to be cleansed of the taint of occupation, and the means of accomplishing this was mass repression. The crime being prosecuted now was abetting the enemy. Stalin was adamant: no mercy could be shown. On 28 December 1943, Beria submitted a memorandum to him about the discovery in Ukraine of so-called \"Volksdeutsche\"\u2014people with German roots. This population, Beria claimed, were privileged supporters of the Nazis during the occupation. Stalin gave the order: \"Arrest them all and keep them in a special camp under special observation and use them for work.\"\n\nAs the war wound down, a new principle shaped Stalinist repression: collective responsibility for collaboration with the occupiers. This principle was expressed in the wholesale internal deportation of a number of Soviet ethnic groups. During late 1943 and the first half of 1944, several peoples were forcibly relocated: Kalmyks, certain North Caucasian ethnic groups (Chechens, Ingush, Karachai, Balkars), and Crimean Tatars, as well as all the Bulgarians, Greeks, and Armenians living in Crimea. Stalin's decision to exile these groups was partially motivated by real evidence of collaboration and noncompliance with government mobilization efforts during the war, mainly evasion of recruitment into the army. But the principle of collective responsibility and punishment had a broader significance. Even before the war, the government had had difficulty integrating many of these peoples into Soviet society. The war only confirmed that this task had never been completed. Moving them to remote areas of the USSR, in Stalin's mind, was a way of solving this problem once and for all. But the job had to be done right. Entire peoples, bound by common ancestry and heritage, had to be relocated. If anyone was left behind to keep the ancestral hearth burning, many others would try to escape exile and return home. In the case of Crimean Tatars, Stalin was probably also worried about their proximity to Turkey, which he regarded as a potentially hostile force. As the ethnic deportations continued in mid-1944, the border regions of Georgia were also targeted. They were purged of Turks, Kurds, and a few other ethnic minorities viewed by the Soviet authorities as fertile ground for Turkish influence and espionage. These expulsions were essentially a continuation of Stalin's long-standing prewar policy of preventative ethnic purging. But the war drove the sweeping nature of the deportations and the decisiveness with which they were carried out. Much of the inhumanity of war stems from the inhuman acts it is used to justify.\n\nThe ethnic deportations of 1943\u20131944 swept up more than a million people. Such a massive endeavor required large numbers of troops and state security personnel. Stalin had the final word in deciding the fates of entire peoples. He was kept constantly informed on the progress of the deportations, and these reports are now available to historians in what is known as \"Stalin's special file\" among NKVD materials. Because of the number of deportees involved (approximately one-half million), the relocation of Chechens and Ingush was particularly complicated and difficult. Beria went personally to the North Caucasus to oversee the effort. On 17 February 1944 he wired Stalin to report that the preliminary stage of the operation had been completed. His telegram made it clear that what the Soviet leadership feared most was \"incidents\"\u2014resistance by the deportees. For this reason, the authorities relied on the element of surprise. Troops assembled under the pretext of training exercises arrested the most active members of the community as a precaution. Stalin, who followed the operation closely, apparently advised Beria not to rely solely on the \"chekist and troop operations\" but to also try to undermine solidarity among the deportees. In a 22 February telegram, Beria reported to Stalin that he had carried out his \"instructions.\" He had summoned top Chechen and Ingush officials and demanded that they help assure that the deportation was carried out without \"excesses.\" To promote calm, Beria informed Stalin, he solicited the help of religious leaders and other local authorities. In exchange, these officials and elders were promised certain privileges in their place of exile, including increased rations and the right to bring property with them. \"I believe that the operation to evict the Chechens and Ingush will be carried out successfully,\" he reported. The following day, 23 February, he proudly described the beginning of the operation, adding: \"There were six attempts to resist by individuals that were suppressed through arrest or use of arms.\" Stalin could rest assured that the task was in good hands.\n\nLike many of Stalin's political tools, reprisals against real and imagined collaborators were a double-edged sword. After the exceptional violence of war, attempts to instill a desire for vengeance against collaborators weakened the army's morale and spawned brutality and abuses. Many incidents served to illustrate the danger of spontaneous eruptions when millions of young men are thrown into a brutal war. Heroism and self-sacrifice coexisted alongside the basest human behaviors and duty, compassion, and decency alongside criminality and rancor. All sorts of people were in the army, including criminals who had been released from the camps early to fight. Documents from 1944 show that Stalin was repeatedly informed of crimes against civilians by soldiers in liberated areas. In late July, Beria wrote him about the arrest of a group of soldiers and junior officers in a tank repair unit in Moldavia after they had gone on a drunken rampage of robbery and rape against the local population. A similar report from Beria in late September informed Stalin of a rape by members of the Red Army in Crimea. This report also recounted instances of robbery and armed encounters with the local police. Summaries of crimes committed by members of the military in September, October, and December also contained descriptions of robberies, rapes, and even murders, both far from the front and close to the fighting. All were committed against Soviet citizens on Soviet territory.\n\nThe situation was much worse when the army entered foreign territory, especially Germany. Feelings of vengeance toward Germans, carefully cultivated by Soviet military propaganda, were not the only reasons for a host of crimes\u2014robbery, murder, and rape\u2014by Soviet soldiers and officers against German civilians. Atrocities by the Nazis within the Soviet Union, the exceptional brutality of the war, the ignorance and criminal pasts of some members of the Red Army, and the weakening of discipline under combat conditions all contributed to, but did not excuse, the firestorm of violence. Stalin was informed of his army's behavior. On 17 March 1945, Beria sent him and Molotov a report on the rapes of German women and their subsequent suicides in eastern Prussia. With the opening of archives from this period, the number of known incidents of this sort will only grow. The history of a dispute with the Yugoslav leadership offers evidence of Stalin's attitude toward such behavior by his military. In late 1944, when the Red Army reached Yugoslav territory and liberated the country together with Yugoslav units, alarming accounts of crimes by members of the Soviet armed forces began to appear. According to the prominent Yugoslav Communist politician and writer Milovan Djilas, there were more than a hundred cases where women were raped and murdered and more than a thousand robberies. The Yugoslav leadership appealed to the Red Army command but was curtly rebuffed. The Yugoslavs were accused of slander. When the matter reached Stalin, he supported his military men and made crude political accusations against the Yugoslavs. Later, when he decided that the conflict needed to be quelled, he had a conciliatory discussion with Djilas during a dinner at his dacha in April 1945:\n\nImagine a man who has fought his way from Stalingrad to Belgrade\u2014thousands of kilometers across his desolated land, seeing the death of his comrades and the people closest to him! Can such a man really react normally? And what's so terrible if he misbehaves with a woman a bit after such horrors? You imagined the Red Army to be ideal. It isn't ideal and wouldn't be ideal even without a certain percentage of criminal elements\u2014we opened up the prisons and took everyone into the army.... You have to understand war. And the Red Army isn't ideal. The important thing was for it to beat the Germans\u2014and it's beating them well. Everything else is secondary.\n\nIf this was Stalin's attitude toward crimes committed on the territory of an allied state, where the government was controlled by Communists loyal to Moscow, it is hardly surprising that he had no desire to take serious measures to prevent abuses in Germany. Stalin's calculations were obvious. All he cared about was the army's military success. If it could be rewarded for its efforts at the expense of the enemy's civilian population, that was fine with him. Nor was he especially worried about reproaches by his Western allies. Remarks made to him by President Roosevelt on 4 February 1945, before the Yalta Conference got under way, probably did not evade his attention: \"Roosevelt states that now that he has seen the senseless destruction perpetrated by the Germans in Crimea, he would like to destroy twice as many Germans as have been destroyed so far. We definitely have to destroy 50,000 German-Prussian officers. He, Roosevelt, remembers how Marshal Stalin proposed a toast in Tehran to the annihilation of 50,000 German-Prussian officers. This was a very good toast.\"\n\nAt some point, however, Stalin had to make a choice. \"Misbehaving with women,\" which he considered a reward for military success, was clearly turning into a problem. Crimes perpetrated by the Soviet military were beginning to serve Nazi propaganda purposes and were feeding German opposition to the Red Army that was not being expressed against the Western Allies. On the eve of a decisive battle for Berlin, Stalin sent the army a clear political signal. On 14 April 1945, _Pravda_ published a scathing critique of a work by the well-known Soviet writer and commentator Ilya Erenburg, hailed for his many furious calls for the killing of Germans. Suddenly these calls, which had been perfectly in harmony with Soviet propaganda, were deemed inappropriate. _Pravda_ explained at length that there is no such thing as a united Germany, that not all Germans behave the same, and that many of them\u2014more and more with time\u2014were turning away from Nazism and even fighting it. Judging by the article's style, it had been touched by Stalin's pen, and certain fragments were probably its product.\n\nPolitical posturing and even the introduction of punishment for crimes by Soviet soldiers improved the situation only slightly. Violence toward civilians within the Soviet zone of occupation continued even after the fighting ended. In the summer of 1945, alarmed by the scale of violence, the supreme commander of the occupying Soviet force, Marshal Zhukov, issued orders demanding an end to \"plunder, violence, and abuses in regard to the local population.\" After these demands had little effect, in early September Zhukov issued a more radical order. Remarking that the \"criminality of military service members has significantly grown,\" he ordered that soldiers be confined to their barracks and obliged officers to move in with their subordinates to maintain order. Stalin, on learning of this order, demanded that it be rescinded. One argument against it was that \"If this order falls in the hands of the leaders of foreign armies, they will not fail to label the Red Army an army of looters.\" In place of Zhukov's strict measures, Stalin proposed more vigorous political work, with the troops bringing guilty officers before so-called officer honor courts. The excesses in Germany continued.\n\n###   **TWEAKING THE MILITARY DICTATORSHIP**\n\nOn 31 July 1943, Stalin signed a directive addressed to the commanders of the Southern Front that stated, among other things, the following: \"I believe it is shameful for Front commanders to allow, through negligence and poor organization, our four infantry regiments to be surrounded by enemy forces. In the third year of war one would think you would have learned how to correctly lead troops.\" This comment reflects how Stalin felt about his two-year experience at the helm of a country at war. His commanders, he believed, were long overdue in mastering skills that had been lacking or poorly developed when the war first broke out. Probably the supreme commander did not feel this assessment fully applied to himself, but his behavior suggests that he knew there were shortcomings in his leadership during the early stages of the war and he was making an effort to correct them. In style and substance, the military \"reforms\" he put in place reflected his preferred approach to any problem. Whether he was industrializing a backward country or waging war, his experiments in leadership had many innocent victims.\n\nOne reason for the Germans' early success against Soviet defenses was the low level of competence up and down the Soviet chain of command. Lacking trust in his generals (sometimes with good reason), Stalin managed using the techniques with which he was most familiar: strong-armed police measures that instilled fear. Commanders were forced to work under the watchful eye of political commissars and NKVD \"special departments.\" Disorganization and panic were addressed through executions in front of the ranks, penalty battalions, and anti-retreat units. Stalin's parallel army of discipline-keepers rushed from crisis to crisis, both at the front and in the rear. As defensive lines collapsed, the enemy advanced, and Stalin lost faith in his commanders, he developed an array of strategies that wound up depriving commanders of flexibility and often increased Red Army casualties.\n\nThese heavy-handed and repressive measures probably do not indicate a conscious choice so much as Stalin's desperation. As strong as his tendency toward violence was, he was certainly aware of the danger inherent in applying it to his own military during a war. He must have grasped that sending troops into battle with guns at their backs was not the ideal way to instill a fighting spirit. He also must have known that on the battlefield it was particularly important to have a single decision maker able to exercise judgment without a political commissar looking over his shoulder. The catastrophes of 1941\u20131942 clearly showed that unsophisticated and rushed maneuvers combined with pressure from political commissars were not the road to success. Fundamental changes were needed in the way the war was being managed. But when could he introduce these changes? Obviously not while the Red Army was fighting with desperate intensity to hold back the German advance. An opportunity may have presented itself in early 1942, after the Red Army's first victories. But Stalin's impatience and his wager on a quick victory only led to further defeat. The lull that set in during the fall of 1942 was used for other purposes, as can be seen in the careful preparations to encircle the Germans in Stalingrad. On the eve of that victory, Stalin finally turned his attention to introducing fundamental changes.\n\nOn 9 October 1942, the Politburo passed a resolution to establish full _edinonachalie_ \u2014an ideological buzzword used during forced industrialization to signify a single responsible decision maker\u2014and abolish the institution of the military commissar within the Red Army. Former commissars would now become deputy commanders. A directive signed by Stalin that same day granted officers additional privileges and assigned orderlies to the commanders of all army units, all the way down to the platoon level. The duties of these orderlies included \"serving the personal everyday needs of commanding officers and carrying out their assignments.\" In January 1943, shoulder boards, which in 1917 had been abolished as a symbol of the tsarist army, were introduced to Red Army uniforms. The title of marshal was given to some senior commanders. The introduction of _edinonachalie,_ along with privileges, medals, and promotions, was intended to empower the Red Army's senior officers. The realities of war forced Stalin to show more trust in his military.\n\nAfter the war's chaotic first stage, there was a change in Stalin's interactions with top military command structures, especially the General Staff. \"I have to admit,\" Vasilevsky later stated, \"that at the beginning of the war the General Staff was thrown into a state of disarray and, strictly speaking, you could not say that it was operating normally.... At the beginning of the war Stalin disbanded the General Staff.\" This disarray meant that a great many decisions were made by Stalin alone, without input from the General Staff. As Vasilevsky described it, things began to change only in September 1942. By fall 1943 a regular schedule was established for consultation between Stalin and the General Staff. At the start of his workday, around ten or eleven in the morning, he heard by telephone the General Staff's first report on the situation at the fronts. Around four or five in the afternoon he heard a report on how things had gone during the first half of the day. Close to midnight, the heads of the General Staff came to him personally with a summary of the day's events. During these meetings, which took place either at Stalin's Kremlin office or his dacha, the group would study the situation at the fronts on maps, and directives were adopted to be sent to the field. Other decisions were made as well. Politburo members often took part in these meetings, as did the heads of various military or civilian bodies. In some cases, the heads of the General Staff visited Stalin several times in a day. The regularity of these meetings led to better management of the war.\n\nIn addition, Stalin had many meetings with other military and civilian leaders. Front commanders were not usually expected to report in person on their assessments and plans, but they were often summoned to Moscow for brief face-to-face meetings. Although Stalin always had the last word, many of these meetings featured a genuine discussion of problems and even debates over large and small questions. A number of memoirs report that as the situation at the fronts improved, meetings grew more businesslike, and the atmosphere became more relaxed and informal. Stalin paced the room as he listened to reports. By remaining on his feet, he lessened the hierarchical divide between him and his military subordinates, who also stood. The _vozhd_ smoked a great deal, but others could also smoke without asking permission. Boxes of _papirosas_ (filterless Russian cigarettes) lay on the table. Members of the top Soviet leadership sat around the table and kept silent until Stalin asked them a question. Less inclined to dictate his own terms or interfere in operational decisions, Stalin became noticeably more respectful toward the military leaders as the war continued.\n\nDuring the second phase of the war, Stalin was not inclined to be hasty in making decisions and usually listened to reports, including upsetting ones, without any sign of irritability, without interrupting, just smoking, pacing, sitting down from time to time, and listening.\n\nLess and less often he imposed his own solutions to individual questions on Front commanders\u2014attack this way and not that way. Earlier he would impose his way, tell them in what direction and in which specific sector it would be more advantageous to attack or to concentrate forces.... By the end of the war there wasn't a hint of this.\n\nStalin's new demeanor was largely a result of his growth as a military leader. As the war progressed, he acquired a huge store of both negative and positive experience. \"After the Battle of Stalingrad and especially Kursk,\" Marshal Vasilevsky wrote, \"he rose to the height of strategic leadership. Now Stalin was thinking in terms of modern warfare and grasped all the issues involved in preparing and conducting operations.\" This view of Stalin's new sophistication was shared by many of the military leaders who worked with him during the war.\n\nStalin's focus on the day-to-day details of operations at the fronts allowed him little time to deal with other matters, particularly the economy. Many spheres of socioeconomic life were removed from the dictator's harsh control as the lines of division among government institutions underwent a spontaneous wartime revision. Under the military dictatorship, at the top of the pyramid was, as always, Stalin, who made decisions either solely or during meetings held either in his Kremlin office or at his dacha. The participants in these meetings included military leaders and the _vozhd_ 's closest comrades. The meetings did not fit into any of the orderly categories of government. Depending on their content, decisions made at these meetings, or solely by Stalin, were drawn up and circulated to those charged with carrying them out in the name of one of the top governmental bodies\u2014the Politburo, the Council of People's Commissars, the State Defense Committee, or the Supreme Command. Meanwhile, many questions having to do with the day-to-day running of the country, including the wartime economy, were being decided without Stalin's direct involvement. Molotov, for example, was in charge of the SNK (the Council of People's Commissars) and regularly presided over the decision-making bodies that basically oversaw all aspects of government not directly tied to military operations. In December 1942 a new body was established to oversee the work by industry and the transportation sector to meet the needs of the front: the State Defense Committee's Operational Bureau. Led at first by Molotov, as the war wound down, it was taken over by Beria. Members of the Politburo and the State Defense Committee also served on these critical managerial bodies, where they had the authority to resolve important issues quickly. Not all of the resolutions produced by these bodies went to Stalin for approval.\n\nIn addition to their duties serving on these top government bodies, each of Stalin's associates had his own individual \"portfolio.\" As the war persisted, this system of putting members of the leadership in charge of particular areas became embedded. For example, in February 1942, the following purviews were assigned to members of the State Defense Committee: Molotov was placed in charge of the production of tanks, Malenkov of aviation, Beria of armaments, Voznesensky of ammunition, and Mikoyan of supplying the army with food and uniforms. These portfolios could change over time. Whatever assignments were given to these top leaders, under the pressures of war and by sheer necessity they operated with significant administrative latitude. What mattered were results. If they met their production targets, they were successful. This system worked, and Stalin had neither the time nor the desire to change it.\n\nThe increased autonomy enjoyed by Stalin's associates inevitably spilled over into the political sphere and affected their interactions with the _vozhd._ As Mikoyan attests, \"During the war there was a certain solidarity among our leadership.... Stalin, who understood that during this difficult time an all-out effort was required, fostered an atmosphere of trust, and every member of the Politburo carried a tremendous load.\" This understanding, of course, did not mean that Stalin's dictatorial dominance over the Politburo was replaced by oligarchic rule. Stalin set the rules of collective leadership. As the situation stabilized at the front and victory over the enemy approached, there were signs that he intended to do away with the slight liberalizations that circumstances had forced upon him. For Mikoyan, the first such sign was a slap on the wrist he received from the _vozhd._ On 17 September 1944 he sent Stalin a draft resolution on advancing grain to a number of oblasts. Although the proposal was rather moderate and did not give the oblasts everything they were asking for, Stalin made a display of his anger, writing onto Mikoyan's resolution: \"I vote against. Mikoyan is behaving in an anti-state manner and is being led around by the oblast committees and is corrupting them. He has completely corrupted Andreev. The procurement commissariat should be taken away from Mikoyan and given to Malenkov, for example.\" The Politburo did so the following day.\n\nAnother sign of coming changes at the top was a shake-up within the military leadership undertaken by Stalin in late 1944. In November the Politburo appointed Nikolai Bulganin to serve as Stalin's deputy at the defense commissariat and made him a member of the State Defense Committee. Bulganin was also given important powers in interacting with the army. His expertise lay in civilian affairs, but during the war he served on the councils of a number of fronts, thus acquiring some military experience and even the rank of general. His assignment to the defense commissariat, and the broad powers he was given, could only mean that Stalin was creating a new counterweight to the military, in particular to the deputy defense commissar and deputy supreme commander, Marshal Zhukov. Evidence can be seen in the demonstrative dressing-down given to Zhukov just two weeks after Bulganin's appointment. In December 1944 Stalin accused Zhukov of exceeding his authority in approving artillery field manuals and issued him a reprimand. The order criticizing Zhukov was circulated to all top military leaders.\n\nAs painful as this lashing out must have been for Stalin's subordinates, his attacks did little to roil the upper echelons of power or change his relative moderation in dealing with the members of the Politburo or the military leadership. Lower down the hierarchy, however, there was no sense of liberalization. The war lent a certain legitimacy to Stalin's brutality, especially given the extreme ruthlessness of the enemy. The intensity of state violence during the war years was comparable to that of the Terror. In addition to the general hardships of war, the front suffered (as noted) from executions, anti-retreat units, and penalty battalions, while members of the civilian population suffered arrest, execution, mass deportations, mobilization, and the mass starvation that resulted from forced grain requisitions by the state and the collapse of agriculture in some of the Soviet Union's most productive areas. While the context of these hardships differed from those experienced in the late 1930s, to those enduring them they must have felt very much the same. As they mounted, just as he had done toward the conclusion of the Terror, Stalin made certain concessions to the populace that cost him little but brought certain tactical advantages.\n\nThe best known concession was a reconciliation with religious institutions and the faithful, most important the country's Orthodox majority. This departure from the anti-religious campaigns of the 1920s and 1930s, from the destruction of churches and the mass executions of clergy members and believers, in favor of the opening of cathedrals and relative freedom of religion, was part of an overall adjustment in official ideology. Russian patriotism was being encouraged before the war, and a revival of images of the heroic past, many placed on a par with the legacy of Bolshevism and the revolution, became more pronounced during the war years. Under Stalin's orders, portraits of the great eighteenth- and nineteenth-century generals Aleksandr Suvorov and Mikhail Kutuzov were placed alongside the photograph of Lenin that hung in his office. To medals based on the symbolism of the revolution were added those commemorating Suvorov, Kutuzov, Prince Aleksandr Nevsky, and Admiral Pavel Nakhimov. At the front, those who had fought in World War I were allowed to wear their tsarist medals along with their Soviet ones.\n\nThe new attitude toward religion received a stunning stamp of approval in September 1943, when a previously unimaginable meeting between Stalin and the leaders of the Russian Orthodox Church was publicly announced. Three metropolitans were brought to Stalin's Kremlin office during the night of 4\u20135 September. They talked with the unusually amiable _vozhd_ for one hour and twenty minutes. After an eighteen-year prohibition, they were granted permission to appoint a patriarch for the Russian Orthodox Church and were even offered the option of using airplanes to bring bishops to Moscow so as to accelerate the selection. Stalin consented to the opening of courses in theology to prepare priests and even proposed organizing theological seminaries and academies. He also supported requests to open new churches and free arrested priests, and he proposed that church leaders improve priests' material well-being by setting up special food stores and assigning them cars. He gave the future patriarch the gift of a three-story house with a garden in the center of Moscow, formerly the home of the German ambassador, including all its furnishings. After discussing a few more items, Stalin escorted the metropolitans to the door of his office. The next day, the meeting with church leaders and the upcoming election of a new patriarch were reported in newspapers.\n\nHistorians have made a rather thorough study of the reasons for Stalin's about-face on religion. Of course the former seminary graduate with the unfinished theological education had no intention of returning to the bosom of the church or asking forgiveness for his sins. Needing to strengthen relations with his allies, he had to respond to the concerns of Western public opinion and influential church circles about the plight of believers in the USSR. Furthermore, the liberation of occupied Soviet territories raised the practical question of what to do about the many churches the Germans had built there. The usual Bolshevik approach of shutting them down was impossible. He needed a reconciliation with the church. Religion had to be put under tight control but not destroyed. Far from the bottom of the list of reasons for this change was Stalin's awareness of the role religion played in uniting the country, in earning the emotional support of the masses, who had endured terrible trials. Soviet values, force-fed into the minds of millions, could not satisfy the spiritual needs of a huge and ancient people. The goal of achieving a universal vision of the path forward turned out to be unattainable. Stalin's grasp of this reality brought him one step closer to victory.\n\n###   **THE STAGES OF VICTORY: CRIMEA, BERLIN, POTSDAM, MANCHURIA**\n\nThe entry of the huge Red Army into Germany was a long-awaited and joyous occasion for the Soviet people and the _vozhd_. The enemy would be finished off in its own den. The time for retribution had come. Such natural and inevitable feelings inspired heroism and self-sacrifice during the war's final battles, when every Soviet soldier could taste victory and was eager for the final assault. Stalin had every reason to be proud of his army.\n\nOne of the Red Army's most successful operations came in January and February 1945. Taking just three weeks to advance five hundred kilometers from the Vistula to the Oder, the Soviet forces shattered critical Nazi lines of defense. Bridgeheads were created for an offensive against Berlin itself, but several months of bloody battles still lay ahead. The German forces defending their country put up a stubborn resistance and even launched counteroffensives, forcing the Red Army to take heavy casualties. Knowing this, Stalin did not hurry to enter Berlin in February. It would take several weeks to eliminate the threat of German counterattacks against the exposed flanks of the advancing Soviet fronts and to bring in reinforcements. Hard-earned experience had taught him prudence.\n\nThe victories of early 1945 had put the Soviet side in a favorable position to negotiate with the Allies on the postwar future. Negotiations first became a practical necessity in late 1944, when the Red Army was advancing through the Balkans and the Western Allies entered France and Italy. In October 1944, Churchill again flew to Moscow to meet with Stalin. The British prime minister raised the question of spheres of influence in Europe, the Balkans in particular. Stalin is unlikely to have been put off by this political cynicism. He agreed that \"England should have the right to a decisive voice in Greece,\" and he was also willing to apportion a Western \"share\" of influence in Romania, Hungary, Bulgaria, and Yugoslavia. The presence of the Red Army in these countries (unlike in Greece) had brought them under Soviet control. For Stalin, this control was decisive. The question of Poland, high on the list of diplomatic issues Churchill brought to Moscow, was much more contentious. By the time of Churchill's visit in late 1944, the USSR had broken off relations with the official Polish government, which had spent the war in exile in Britain, and was promoting a Communist alternative. Britain and the United States did what they could to prevent this outcome. On 1 August 1944, as the Red Army approached, the Polish government in exile organized an uprising in Warsaw with the goal of seizing power in the capital before the arrival of Soviet forces and the pro-Soviet government they were bringing with them. The Red Army, for a variety of reasons, stopped its advance, and the Nazis drowned the uprising in blood. This tragic episode became a source of sharp division between Stalin and his allies, who charged him with intentionally holding back aid to the uprising. This charge was largely just, but Stalin, guided by his own reality, had no intention of relenting. The London Poles had not launched the uprising to help him, so why should he help them?\n\nBurdened by different problems but still united by their common foe, the Big Three met outside the Crimean resort city of Yalta in February 1945. This stunningly beautiful corner of the Soviet Union had only recently been liberated from Nazi occupation and lay in ruins. Sparing no effort or expense, in record time the Soviet authorities created a haven amid the destruction, including residences for the three leaders and their large retinues. Particular attention was paid to security. Camouflage covering was set up to protect against enemy air raids and sturdy shelters were built. Crimea, recently roiled by mass arrests and deportations, was subjected to yet another round of purges. \"Suspicious elements\" were rounded up and taken into custody. A whole army of security personnel was brought to the area. Stalin alone was protected by a force of one hundred operatives and five hundred NKVD troops, plus his usual bodyguards.\n\nWith victory around the corner, the Yalta Conference would have to address a wide range of urgent questions on which the fate of the world hung. At stake were the future of Germany, a redrawing of the map of Europe, and the worldwide balance of power. Generally speaking, the participants' goals were simple. Although their motives and priorities differed, each of the parties wanted to leave Yalta with as many items on his diplomatic wish list as he could. But as long as the war continued, the Allies had to depend on one another and adjust their aspirations to military and political realities. They compromised on many issues. The zones of occupation in Germany were settled. The guiding principles on which a united nations organization would be founded were outlined. The idea was discussed of the Soviet Union annexing new territories at Poland's expense (western Ukraine and Belarus), for which Poland would be compensated with German lands to its west. In exchange for a promise to enter into the war with Japan, Stalin extracted an agreement from the Allies that Soviet borders would be shifted outward to encompass new territory in the Far East and that the country's interests in northern China would be recognized.\n\nBut as the contours of a new world took shape, so did the battle lines of the Cold War. It was not possible to reach a real compromise in regard to Poland. Stalin was determined to put this country under the control of his handpicked government, even if that involved making a few concessions on paper. Another contentious issue was the question of reparations from Germany, a point of particular interest for Stalin.\n\nPerhaps even more indicative of the gulf dividing the Allies was the attitude of Soviet state security personnel in Crimea. The hordes of Westerners who descended on Soviet territory were treated as an enemy penetration. The ships used to bring the Allies' supplies for the conference were surrounded by round-the-clock patrols. Their crews, when given shore leave, were kept under tight NKVD control. \"The entire agent apparatus has been instructed and directed to uncover the nature of ties between foreigners and the port's military personnel and civilians. Female agents who will come into close contact with foreigners have been given particularly careful instructions,\" read one report to the NKVD leadership. One can only imagine what these instructions were.\n\nWith every passing week, Stalin's mistrust of the Western Allies grew, strongly influencing Soviet military plans. Wehrmacht units clearly preferred to surrender in the West, while in the East they fought to the bitter end. Stalin had every reason to fear the possibility, if not of a separate peace, at least that the Allies might make certain separate agreements with the Germans. During the final months of the war, everyone understood what the advances of Allied armies meant for postwar Europe's political landscape. Negotiations in March 1945 in Bern between U.S. intelligence agents and representatives of the Nazis to discuss Germany's capitulation in Italy only heightened Stalin's suspicion.\n\nHad it not unfolded amid other conflicts between the Soviet leadership and the Western Allies, especially in regard to Poland, the Bern incident might not have provoked open confrontation. After lengthy wrangling, on 3 April 1945 Stalin sent Roosevelt a sharply worded letter in which he questioned whether it would be possible to \"preserve and strengthen trust between our countries.\" Now that the archives have been opened, we can see that this letter, unlike many others that went out over his signature, was written entirely by Stalin himself and that he revised it to achieve a sterner tone. Despite the growing friction, Roosevelt, who was committed to cooperating with Stalin, responded with restraint. A letter received by Stalin on 13 April 1945 sought to assure him that \"minor misunderstandings of this character should not arise in the future.\" This letter was one of Roosevelt's final political acts and is part of his testament in regard to relations with the Soviet Union. By the time Stalin received it, Roosevelt was already dead. Stalin appears to have been genuinely saddened by this loss. Nevertheless, he was soon distracted by new and urgent matters.\n\nWorried about his fellow Allies' rapid advance, Stalin decided to speed up the Soviet takeover of the German capital as much as possible. The attack on Berlin began on 16 April 1945, one month earlier than the date Stalin had given his allies. Despite the Soviet forces' overwhelming advantage in manpower and hardware, this key battle was not easy. Out of more than 2 million soldiers of the Red Army and Polish Second Army who took part in the Berlin operation, more than 360,000 were killed, wounded, or went missing in action. German units put up a determined fight in defense of their capital.\n\nThe politically motivated decision to push forward the operation created great hurdles for the Red Army. Although delaying the offensive slightly would have made little difference to its outcome, Stalin required the front commanders to rush the advance of their forces at any cost. This accelerated pace, given the need to break through well-defended enemy positions, meant heavier casualties. The record speed of the operation and the concentration of a huge force directed against Berlin necessitated constant adjustments to the overall plan and field directives. According to the head of the General Staff's Main Operational Directorate, General Sergei Shtemenko, Supreme Command Headquarters was in a state of turmoil throughout the Berlin operation. The General Staff leadership was summoned to Command Headquarters several times a day, sometimes at odd hours; many instructions were drafted under extreme time pressure; and the lightning speed of events made organized operations difficult. But no matter how hurriedly things were done at Headquarters, some historians believe that Stalin could not possibly \"react to the changing situation in time.\" It is unclear whether this lag in the flow of information to and from Headquarters had any real consequences. The performance of the Soviet Supreme Command and Stalin in the Berlin operation has received little scholarly scrutiny.\n\nBut no matter how many obstacles were thrown in the Red Army's path, they were not enough to save the Nazis. On 25 April, Soviet units coming from one direction met U.S. forces coming from the other on the Elbe River. The victors' absolute numerical superiority and high morale sealed the fate of the Third Reich. Early in the morning on 1 May, Stalin learned through an urgent telephone message from Marshal Zhukov that Hitler had committed suicide in his Berlin bunker the day before. On 2 May, the Berlin garrison capitulated. During the night of 8\u20139 May, the final surrender was formulated and signed by Germany. On 24 June, Moscow held a long-awaited and impressive victory parade. Then, on 27 June, Stalin was awarded the title of generalissimo.\n\nNow the leader of a major world power, in July 1945 Stalin set out for a vanquished Berlin for yet another Big Three conference. No firsthand accounts of Stalin's last trip outside the Soviet Union have been preserved. What did he see through the windows of his train? With whom did he meet or spend time during this journey? Undoubtedly he knew the upcoming meeting with his fellow leaders would not be an easy one. With victory, the disagreements among the Allies had only grown more contentious. The Soviet dictator would have his first meeting with the new American president, Harry Truman, among whose advisers advocates of a hard line toward the USSR were gaining ascendancy. The Western Allies were displeased by the sovietization of Romania and Bulgaria, to say nothing of unresolved arguments about the Polish government. Stalin did not trust the Americans and British. This mistrust was fanned when Truman privately informed him of American atom bomb tests. The principles of German demilitarization, de-Nazification, and democratization were unanimously approved, but the Allies argued bitterly about everything else. The search for compromises and mutual concessions was spurred by fears that the war-weary world could be plunged into a new confrontation, by Soviet hopes for economic cooperation with the West, and by Western hopes that the USSR would enter the war against Japan. In the end, Stalin managed to finalize an agreement allowing Poland to expand its territory at the expense of Germany and the Soviet Union to incorporate the Konigsberg area. He did not, however, get his way on reparations or on the creation of Soviet bases on the Turkish Straits and the Mediterranean.\n\nHaving achieved what he could in Europe, Stalin turned his attention to acquiring Japanese lands and gaining footholds in northern China. In Yalta he had agreed to join the war against Japan two or three months after Germany surrendered. Knowing how eager the United States was for Soviet help, he had been able to extract very advantageous terms. The \"status quo\" was preserved in the Mongolian People's Republic, keeping it under de facto Soviet control. The USSR regained the southern portion of Sakhalin, which Russia had lost in the Russo-Japanese War of 1905, and a commercial port and military base in northern China, along with the railroad line leading to it. Of fundamental significance to the USSR was the Allies' agreement to recognize Soviet sovereignty over the strategically important Kuril Islands.\n\nThese agreements all remained in force up to the Berlin Conference, but now, for the first time in history, the nuclear factor came into play. The fact that the Americans had an atom bomb gave them much greater leverage. For one thing, fear of this powerful new technology could lead Japan to surrender even before the Soviet Union entered the war. Stalin preferred not to take the risk. He applied the same strategy in the Far East that he had used in Europe, where actual military possession of territory was more meaningful than agreements at the bargaining table. After the United States used its atom bombs against Japan, Stalin ordered the Red Army to launch an urgent offensive, giving his forces a deadline of 9 August 1945 to turn the Yalta concessions into a reality on the ground. The Soviet numerical advantage coupled with high morale and a seasoned fighting force brought about a quick victory. Even after Japan's capitulation, Soviet forces continued to advance until all territories granted to the USSR at Yalta had been occupied. Then Stalin tried to take a little extra. In the Far East this meant pretentions to jointly occupy Japan proper and share in governing the country using a model similar to the one being applied in Germany. This effort was probably more a test of the new American president's will than an actual demand, but it was accompanied by military preparations. After being decisively rebuffed by the Americans, Stalin quickly backed off, but not without some resentment. Disputes over Japan remained an irritant in Soviet-American relations for months. Japan itself did not recognize the Soviet capture of the Kuril Islands as legitimate.\n\nFor the millions of Soviet people who survived the horrors of war, the disputes and ambitions of politicians were peripheral. The country, finally at peace, could look to the future with hope.\n\n## **FAMILY**\n\n**2 March 1953 at the near dacha. The arrival of the daughter.**\n\nOnce the seriousness of Stalin's condition became clear, his children, Svetlana and Vasily, were called to the dacha. This was largely a symbolic gesture. Over time, Stalin's family had come to play less and less of a role in his life.\n\nStalin met his first wife when he was still a young revolutionary adventurer. Returning to Tiflis in 1905 after escaping from his first exile and traveling through Transcaucasia, he moved in with the Svanidze family. There were five members of this family: Aleksandr Svanidze, who was involved in the revolutionary movement, and his sisters\u2014Sashiko, Kato (Yekaterina), and Masho\u2014as well as Sashiko's husband, whom Stalin had known in the seminary. Sashiko and Kato were well-known dressmakers in the city who had nothing to do with the revolutionary movement. So when he brought Iosif Jughashvili into the household, Aleksandr tried to keep this outsider as far away as possible from his sisters. Nevertheless, an infatuation developed between Iosif and Yekaterina, who were both young and attractive. Kato's sisters could not have been happy about her involvement with an impoverished seminary dropout. Some light is shed on this period by a letter sent to Stalin forty years later, in 1946. An acquaintance of Stalin and the Svanidze family from his Tiflis days asked for help and rather artlessly implied that Stalin owed him a favor. First, Stalin had used the letter writer's room for assignations with Yekaterina. Second, when Stalin proposed to Kato and \"the relatives were opposed,\" \"I told her, if you like him, don't listen to anybody, and she heeded my advice.\"\n\nThe Svanidze family was basically presented with a fait accompli, and in July 1906 the couple was married. This new family member inevitably entangled the Svanidzes in his world. Soon after the wedding, Yekaterina was arrested as an accomplice of revolutionaries. The matter was resolved thanks to her sister Sashiko, who used her ties to wives of police officers. Yekaterina spent about two months under arrest, but instead of being held in a jail cell, she was kept in a local police chief's apartment\u2014apparently at the request of the chief's wife, who was a client of the dressmakers. One important argument for closing Yekaterina's case was that she was pregnant. In March 1907 the future dictator's first child, Yakov, was born. Family life and revolution did not mix. Iosif moved his wife and son with him to Baku, where Yekaterina fell seriously ill. In November 1907 she died. This was a heavy blow to Iosif. Unable to take adequate care of his son, he left Yakov with his wife's family.\n\nThere were other women in Stalin's life. Evidence survives of a relationship with Stefaniia Petrovskaia, a young revolutionary from the landowning class, that began in 1909, when both were exiled to Solvychegodsk in Vologda Province. After serving out her term, Petrovskaia followed Iosif to Baku. When he was arrested in June 1910, the future dictator even asked the police for permission to \"enter into lawful wedlock\" with her. The permission was granted, but the wedding never took place. In September 1910 Jughashvili, still a bachelor, was again sent into exile. During this second exile in Solvychegodsk he registered his place of residence (in the home of M. P. Kuzakova) together with fellow exile Serafima Khoroshenina, suggesting that the two were intimate. Soon, however, Khoroshenina was transferred out of Solvychegodsk. According to rumors now being promoted by some journalists, Stalin then began a relationship with his landlady, Kuzakova, resulting in the birth of a son. There is no hard evidence of this relationship. After finishing his term of exile a few months after the supposed affair with Kuzakova, Jughashvili spent some time living in Vologda. Here he became acquainted with an eighteen-year-old schoolgirl named Pelageia Onufrieva, the fianc\u00e9e of one of his fellow exiles, Petr Chizhikov. The future dictator flirted openly with the girl and gave her a book with the inscription, \"To clever, nasty Polya from the oddball Iosif.\" When Pelageia left Vologda, Jughashvili sent her facetious cards, such as: \"I claim a kiss from you conveyed via Petka [Chizhikov]. I kiss you back, and I don't just kiss you, but passionately (simple kissing isn't worth it). Iosif.\" In his personal files, Stalin kept a photograph of Chizhikov and Onufrieva dating to his time in Vologda: a serious, pretty, round-faced girl in glasses and a serious young man with regular features and a moustache and beard.\n\nThe jocular cards, presents, and photograph attest to the thirty-three-year-old Jughashvili's interest in the young woman but do not prove that he was romantically involved with her. We have only a few vague hints. Around the same time that Stalin left Vologda, in 1912, Chizhikov went to visit his parents in Ukraine, where he fell ill and died suddenly, without marrying Pelageia, as he may or may not have intended. Onufrieva suffered the sort of misfortune that befell many of her compatriots. After Chizhikov's death she married, and as her erstwhile gallant admirer presided over the country, her husband was arrested. It is not known whether she ever tried to appeal to Stalin for help. She died in 1955, having lived her entire life in Vologda.\n\nThe evidence that Iosif Jughashvili had an affair with the even younger Lidiia Pereprygina during his last Turukhansky exile is more solid, although rumors that they had a son together have not been proved. In any case, Stalin never recognized Pereprygina's son or any other illegitimate children attributed to him.\n\nReturning to St. Petersburg after the February 1917 revolution, Stalin was ready to turn a new page. The Alliluev household provided a place of warmth after the upheavals of life underground. The attraction this family held for him is understandable. Stalin had known them since his years in Tiflis and had corresponded with them during his final exile in Kureika. The head of the family, Sergei Alliluev, was a longtime party member who had been arrested many times. The family's two sons and two daughters were often left without adult supervision and led rather freewheeling lifestyles. Iosif was particularly fond of the youngest, the sixteen-year-old schoolgirl Nadezhda, who reciprocated his feelings despite the twenty-three-year difference in their ages. To a young woman from a revolutionary family, he must have seemed like the ideal man: a tried-and-true revolutionary, brave and mysterious but also personable. In 1919 Stalin and Nadezhda tied the knot. As to the nature of their relationship before marriage, we can only guess.\n\nNadezhda, a party member beginning in 1918, was a model Bolshevik wife. She worked in Lenin's secretariat (Lenin knew the Alliluevs and even lived in their apartment in 1917). In 1921 the Stalins had their first child, Vasily. Nadezhda had a hard time keeping up with childrearing, work, and party activism and apparently neglected the last. In late 1921 she was expelled from the party as \"ballast with no interest in the life of the party whatsoever.\" Only through the intercession of top party officials, including Lenin, was her membership restored, although she had to spend a year earning her way back in as a candidate member. Such were the times. Nadezhda herself probably believed in the ideals of equality and party democracy and was not offended by her treatment. In her request to be readmitted she promised to \"prepare herself for party work.\"\n\nIn addition to the birth of Vasily, Nadezhda's life was complicated by the introduction of Stalin's first son, Yakov, into the family. Letters to her mother-in-law, Ekaterine Jughashvili, in 1922 and 1923 included cautious complaints: \"Yasha is going to school, fooling around, and smoking, and does not listen to me\"; \"Yasha is also healthy, but he's not putting much effort into his schoolwork.\" Yakov, fifteen in 1922, was just six years younger than his stepmother. A few years later, in 1926, Nadezhda wrote of Yakov to a female friend: \"I have already lost all hope that he will ever come to his senses. He has absolutely no interests and no goal.\" The boy was also not getting along with his father. Conflict over his intention to marry ended tragically: when he failed to get his father's consent, he tried to commit suicide. On 9 April 1928 Stalin wrote to Nadezhda: \"Tell Yasha from me that he has behaved like a hooligan and a blackmailer with whom I have nothing in common and with whom I can have nothing further to do. Let him live wherever he wants and with whomever he wants.\" For a while Stalin's relationship with his eldest son was in a state of suspension, but on the eve of the war, when Yakov was studying at the Artillery Academy, Stalin was apparently pleased with him. On 5 May 1941, Yakov was present at a large Kremlin reception in honor of military academy graduates. In his remarks to the gathering, Stalin joked that \"I have an acquaintance who studied at the Artillery Academy. I looked over his notes and found that a great deal of time is being spent studying cannons that were decommissioned in 1916.\" This was an obvious reference to Yakov's notes, a sign that the two were spending time together.\n\nIn early 1926 Nadezhda gave birth to a daughter, Svetlana. In sharing the good news with Ordzhonikidze's wife, Zinaida, who was vacationing in the south, Nadezhda wrote, \"In short, we now have a complete family.\" But with Stalin immersed in his official duties and embroiled in a battle for power, this was no usual family. No doubt he loved his wife and children, but for the most part he loved them from a distance. They spent brief stretches of time together at the dacha outside Moscow and while on vacation in the south. Nadezhda, as if emulating her husband, was always busy with work, party activism, and her studies. In a letter to a friend a month before Svetlana's birth, she wrote, \"I very much regret that I've again fettered myself with new family responsibilities,\" obviously referring to the impending arrival of her second child. \"In our time it's not very easy since there are such an awful lot of new prejudices, and if you're not working, then of course you're a _'baba'_ [peasant woman, used derogatively for women in general].... You just have to have an area of expertise that enables you to escape being someone's errand girl, as usually happens in 'secretarial' work, and do everything that has to do with your area of expertise.\" Young and energetic, Nadezhda sincerely and energetically strove to adhere to the new model of the \"Soviet woman.\" This was not easy. Her surviving letters show that to the end of her life her writing was riddled with syntactic errors. In an effort to make up for the shortcomings of her education, she became an assiduous student. In 1929 she enrolled in the Industrial Academy, hoping to receive, in keeping with the ethos of the times, an advanced technical education. Her children were largely handed over to nursemaids, governesses, and tutors. A housekeeper and cook took care of the Stalin Kremlin household. An important part in Vasily and Svetlana's lives was played by relatives, as well as their peers among the children of other Soviet leaders who lived in the Kremlin. Together they formed a boisterous band that spent time together at suburban dachas and each other's Kremlin apartments.\n\nThis manner of family life had its advantages and logic. The infrequency of time spent together could perhaps make \"the heart grow fonder\" and actually strengthen family ties. But the few surviving letters between Stalin and Nadezhda, written during vacations between 1929 and 1931, attest to both love and tension in their relationship. \"I send you a big kiss, like the kiss you gave me when we parted,\" Nadezhda wrote to her husband. She said she missed him and asked doting questions about his health and treatments. Stalin responded in kind. He tenderly called her Tatka and Tatochka (\"Write about everything, my Tatochka\") and even resorted to baby talk. As a loving father, he was always asking about the children: \"How are things with Vaska, with Setanka [his nickname for Svetlana]?\" \"Have Setanka write me something. And Vaska too.\" He sent lemons and peaches home to his family. But this sweetness and consideration could suddenly be darkened by jealousy and irritation. In September 1930, after spending part of her husband's vacation with him and then returning to Moscow, Nadezhda wrote him a letter filled with reproach: \"This summer I didn't feel that delaying my departure would make you happy; quite the opposite. Last summer I could really sense that, but not this time. Of course, there was no point staying with such a mood.\" A few weeks later she wrote: \"For some reason I'm not hearing anything from you.... Probably you're distracted by your quail-hunting trips.... I heard from an interesting young woman that you looked great,... that you were marvelously cheerful and you wouldn't let anyone sit still.... I'm glad to hear it.\" Stalin made a halfhearted effort to dispute her implications: \"As for your assumption that I did not consider it desirable for you to stay in Sochi, your reproaches... are unfair\"; \"You're hinting at some trips. I'm telling you that I have not traveled anywhere (anywhere at all!) and I have no intention of traveling.\"\n\nNadezhda's jealousy was not without grounds. Stalin could be a flagrant philanderer, and his wife was quick to take offense. Many who observed the relationship firsthand commented on Nadezhda's frail mental health. Mental illness apparently ran in the family, afflicting her mother and at least one of her siblings. It is probably here, at the intersection of Stalin's unfaithfulness and Allilueva's mental illness, that the roots of the tragedy should be sought.\n\nOn 8 November 1932, the anniversary of the October Revolution that brought them all to power, Stalin and Allilueva joined other top Soviet leaders and their wives for a celebratory dinner at the Kremlin. The details of what took place at this dinner are unknown. Perhaps Stalin drank too much and started openly flirting with some of the wives. Perhaps Nadezhda was simply in a bad mood or Stalin said something hurtful to her. Or perhaps she was the one who provoked an argument. Whatever the cause, there was an argument, and Nadezhda returned to their Kremlin apartment alone. Sometime during that night she took her own life, using a small pistol that had been a gift from her brother Pavel.\n\nSome have speculated that Allilueva was upset about her husband's policies and felt ardent sympathy for their victims, including those dying from the devastating famine then taking millions of lives. Their daughter, Svetlana, wrote of a suicide note left by her mother that contained, among its grievances, political accusations, although she had no firsthand knowledge of this note and was citing other people's descriptions of it. There is absolutely no hard evidence that Nadezhda objected to her husband's policies. None of her surviving letters mentions the horrific events taking place in the country: violent collectivization, the internal deportations of hundreds of thousands of peasants, and the arrests of countless suspected \"enemies.\" Her letters give the impression that she, like the rest of the Bolshevik elite, was completely isolated from the suffering of tens of millions outside the Kremlin walls. On 10 July 1932, during the famine, when peasant mothers were watching their children starve to death, Nadezhda wrote a note to Stalin's assistant Aleksandr Poskrebyshev complaining that she was not receiving her usual supply of new works of fiction from overseas and asked that the head of the OGPU, Yagoda, do something to fix the problem. Admittedly, we do not know for sure whether Nadezhda ever said anything against her husband's repressive policies in the months before her death, in part because the usual correspondence between Stalin and his wife while he was away on vacation is missing for 1932. Perhaps these letters were destroyed, or perhaps Nadezhda was with her husband during his entire vacation. No evidence has been found to explain the absence of such letters.\n\nHis wife's suicide was apparently a great blow to Stalin. Grief over the loss and pity for his children were combined with anger. Nadezhda had betrayed and humiliated him, cast a cloud over his reputation, and made his personal life a subject of sordid conjecture that endures to this day. \"She did a very bad thing... ; she maimed me for the rest of my life,\" he told relatives some two and a half years later.\n\nOut of habit, Stalin's family led its customary life for a few years after Allilueva's death. Almost every member of the household maintained his or her role within the family routine. Seeking relief from painful memories, Stalin moved to a new apartment in the Kremlin and began construction of the near dacha. The children remained under the care of governesses and nursemaids in Moscow and at the old dacha. Stalin, Vasily, and Svetlana were surrounded by the same relatives, especially the families of Pavel and Anna Alliluev (Nadezhda's brother and sister) and Aleksandr Svanidze (the brother of Stalin's first wife). This was a complicated and often unsavory world. The relatives schemed to outshine one another in Stalin's eyes. Apparently Pavel Alliluev's wife even had a brief affair with the dictator. Stalin appears to have enjoyed the competition among his relatives.\n\nAfter Nadezhda's death, Stalin tried to spend more time with his children. While they were having dinner together in the Kremlin apartment, he asked them how things were going in school, and he sometimes came to the dacha to pick them up and take them to the theater. On occasion, he brought them with him when he vacationed in the south. He was especially fond of Svetlana, who was a promising student and very attached to her father. He began to play a little game with his daughter, calling her _khoziaika_ (which could be translated as \"housekeeper\" or \"the boss\") while he played the role of the _sekretarishka_ (little secretary) who followed her orders: \"Setanka-Housekeeper's wretched Secretary, the poor peasant J. Stalin.\" Svetlana would write out orders for her father: \"I order you to let me go to Zubalovo tomorrow\"; \"I order you to take me to the theater with you\"; \"I order you to let me go to the movies. Ask them to show _Chapaev_ and an American comedy.\" Stalin responded with facetious pomposity. Other members of Stalin's inner circle were appointed Svetlana's _sekretarishkas,_ playing along with the _vozhd_. \"Svetlana the housekeeper will be in Moscow on 27 August. She is demanding permission to leave early for Moscow so that she can check on her secretaries,\" Stalin wrote to Kaganovich from the south on 19 August 1935. Kaganovich replied on 31 August: \"Today I reported to our boss Svetlana on our work, she seemed to deem it satisfactory.\" Until the war began, father and daughter exchanged affectionate letters. \"I give you a big hug, my little sparrow,\" he wrote to her, as he had once written to his wife.\n\nStalin's relationship with his sons was much more fraught. For many years he avoided Yakov and his family, and Vasily gave him a great deal of trouble. The boy understood very early that he was the son of a powerful man. He preferred soccer to studying and often behaved defiantly toward those around him. \"Vasily thinks he's an adult and insists on getting what he wants, which is often foolish,\" the commandant of the Zubalovo dacha reported to Stalin in 1935, when Vasily was fourteen. The situation only grew worse with time. Unable to tolerate the outrageous behavior of his imperious student, in 1938 one of Vasily's teachers complained to the boy's father, telling Stalin that Vasily was getting special treatment from the school administration and that he sometimes used threats of suicide to get his way. Stalin thanked the teacher for his honesty and described his son in extremely negative terms: \"Vasily is a spoiled youth of average ability, a little savage (a real Scythian!) who is not always truthful, loves to blackmail weak authority figures, is often rude, and has a weak, or rather, unfocused will. He has been spoiled by 'kith and kin,' all the while emphasizing that he is 'the son of Stalin.'\" He asked the teacher to be firmer and promised that he would \"take him by the scruff of the neck\" from time to time. As was often the case, the letter was all for show, and the matter was ultimately resolved in typical Stalin manner. A purge of the school was conducted and the directors were fired, along with the teacher who had dared complain to Stalin. Vasily was sent to study at an aviation school in Crimea, where the special treatment continued. He was met at the train station with great fanfare by the school's leadership, quartered away from the other cadets in a hotel, and fed special meals in the officers' mess. Once, obviously pulling a prank, Vasily ordered some special dish. Since the local cook did not know how to make it, someone was sent to a nearby town to find out. Vasily rode all over Crimea in a car and also on a motorcycle. His education was overseen by senior military officials in Moscow. In 1940 he graduated with the rank of lieutenant. He liked to fly, but his character showed no sign of improvement. The system created by the father did irreversible harm to the son.\n\nVasily's departure for Crimea came just as the old Stalin-Alliluev-Svanidze extended family ceased to exist. During the Great Terror, Stalin began to annihilate his own relatives. Between late 1937 and late 1939, Aleksandr Svanidze, his wife, and the husband of Anna Allilueva were arrested and then shot. In late 1938, apparently unable to endure the stress, Pavel Alliluev also died. Stalin had nothing further to do with those relatives who remained at liberty. The war further diminished the family. During its first days, Yakov, who, unlike Vasily, received no special protection, was taken prisoner by the Germans. Stalin ordered the arrest of Yakov's wife but later freed her. Some accounts maintain that Stalin was offered Yakov in exchange for certain German generals (Paulus is most often named) but that he refused. There is no documentary evidence of this claim, and the story lacks credibility since it is hard to understand what would motivate Hitler's leadership to pursue such an exchange. When the war ended, Stalin was given testimony by Yakov's fellow prisoners. After Germany was defeated, Yakov's 1941 interrogation protocol was seized, and testimony was obtained from the guards and commandant of the camp where he died. All this evidence shows that Yakov comported himself honorably as a prisoner. He was shot by a sentry while attempting to leave the prison grounds in 1943. Perhaps this news improved Stalin's opinion of his son, and it may explain why, during his final years, the _vozhd_ took an interest in his young granddaughter, Yakov's daughter.\n\nVasily and Svetlana were disappointments for Stalin during the war. Vasily, who was stationed near Moscow, would host drunken parties at the Zubalovo dacha. At one such gathering, in late 1942, sixteen-year-old Svetlana met the thirty-eight-year-old Soviet filmmaker Aleksei Kapler, who had gained prominence as the screenwriter of popular films about Lenin and the revolution. The two began an affair that ended several months later when Stalin ordered Kapler's arrest. Apparently he was furious over Svetlana's relationship with Kapler, whom she has described as her first love, and considered it all the more inappropriate in wartime. According to Svetlana, his reaction forever destroyed the closeness between them:\n\nI'd never seen my father look that way before.... He was choking with anger and was nearly speechless.... \"Your Kapler is a British spy. He's under arrest!\"...\n\n\"But I love him!\" I protested at last, having found my tongue again.\n\n\"Love!\" screamed my father, with a hatred of the very word. And for the first time in my life he slapped me across the face, twice. \"Just look, nurse, how low she's sunk!\" He could no longer restrain himself. \"Such a war going on, and she's busy the whole time---------!\" Unable to find any other expression, he used the coarse peasant word.\n\nThe next blow came from Vasily. By early 1943 he held the rank of colonel and had been placed in charge of an air regiment. That April he and a group of his subordinates decided to do some fishing. The fish were stunned using explosives. One shell exploded on land, killing one of the regiment's officers and wounding Vasily with shrapnel. He, of course, was treated at the Kremlin hospital in Moscow. Stalin was enraged. Apparently this escapade was one transgression too many, or so one might conclude from an order issued by People's Commissar for Defense I. V. Stalin on 26 May 1943:\n\n(1) Immediately remove V. I. Stalin from his position as commander of an air regiment and do not give him any other command posts in the future until I permit it.\n\n(2) Inform the regiment and former regimental commander Colonel Stalin that he is being removed from his post for drunkenness and debauchery and for spoiling and corrupting the regiment.\n\nBeing long accustomed to his father's empty threats, Vasily was not terribly worried by this reproach. Indeed, he was soon given new, more senior posts, and by war's end he was a twenty-four-year-old general. Stalin's son could get away with almost anything. Around the same time Svetlana, now a university student, married a former schoolmate. She soon gave birth to a son, named Iosif after his grandfather. Nevertheless, Stalin refused to meet with his son-in-law, who was Jewish and had not fought in the war. Perhaps he consented to the marriage only to avoid the acrimony that came with the Kapler affair.\n\nOnce Germany was defeated and wartime pressures receded, Stalin did not return to his family\u2014or rather did not allow his family back into his life. He had grown accustomed to solitude and his nocturnal lifestyle, and he rarely made time for his children. Apparently he never developed grandfatherly feelings. By now he was in his declining years, weary, in poor health, and obsessed with thoughts of treachery and the hunt for enemies. The final blow dealt against his family was the arrest of Pavel Alliluev's wife and Nadezhda's sister Anna. They were released only after his death.\n\nStalin's children, admittedly, were hardly a comfort in his old age. Vasily sank rapidly into alcoholism and dissipation, and by his thirtieth birthday he was already an old man, plagued with a number of chronic diseases. Thanks to his father's indulgence, he nevertheless held increasingly senior army posts and squandered government funds with impunity. The younger Stalin greedily chased the good life: he built and repeatedly renovated his suburban estate, spent lavishly on an elaborate hunting lodge, and established sports teams, luring top athletes with huge salaries and apartments. He had goods shipped in from Germany via airplane, ran through a series of lovers and wives, and drank heavily in the company of sycophantic hangers-on. Toward the end of Stalin's life, after yet another scandalous episode, the father removed the son from the key post of air commander for the Moscow Military District. Vasily was sent to study at a military academy, thereby removing any remaining constraints on his drinking. Meanwhile, Svetlana divorced the husband her father did not like and married one he did\u2014Yuri Zhdanov, son of Stalin's late comrade. This marriage, however, was not happy and did not last long.\n\nAfter his death, Stalin's children suffered deeply symbolic fates. Vasily, after drunkenly insulting his father's successors, was put in prison and died in exile at the age of forty. Svetlana married an Indian Communist. When she was given permission to travel to India for his funeral, she took the opportunity to defect and move to the United States, where she died in 2011. While in emigration, Svetlana published a memoir of life in the Stalin family, _Twenty Letters to a Friend,_ which was both nostalgic and embellished. She placed the blame for her father's pathological cruelty on the scheming and insinuations of Lavrenty Beria. In the end, her attitude toward the system her father created was most eloquently expressed by her defection to the country that he considered socialism's most fearsome enemy.\n\n## **6 THE GENERALISSIMO**\n\nVictory elevated Stalin to unprecedented heights. The exultant show of military might that paraded across Red Square in June 1945 was an important symbol of his new power, now more secure than ever and legitimized with the title of generalissimo. But Stalin was a seasoned enough politician to know that victory, which had transformed the Red Army into one of the most formidable forces on the planet, was just the first step on the long and difficult postwar path toward regaining and holding the country's status as a world power. The Soviet Union was a weakened nation. The extent of suffering and devastation that had befallen it is almost unimaginable. Contemporary demographers speak of 27 million lives lost, and many of those lives were young\u2014the country's future. Thousands of towns and villages lay in ruins, and many people were forced to improvise some form of shelter. Several million wounded veterans needed government support. The demobilization of an army of 11 million and the transition to a peacetime economy also demanded significant resources. The postwar famine\u2014a tragic testament to the devastation wreaked on collectivized agriculture and to the weakness of the Stalinist distribution system\u2014peaked in 1946\u20131947. As many as 1.5 million people died of hunger or disease. Many millions were afflicted by dystrophy and other serious illnesses causing permanent disability. As usual, cannibalism raised its ugly head during the famine years. To all these hardships were added desperate guerrilla wars in western Ukraine and the Baltic states, territories that had been absorbed into the Soviet Union on the eve of the war and given a taste of Stalinist terror.\n\nThere was also a whole new set of international challenges. Relations with the Allies had cooled considerably. Stalin's Soviet Union and the Western democracies, brought together in a marriage of convenience by Nazi aggression, had little in common. Negotiations to resolve the postwar repartition of the world opened up new areas of contention, but the Soviet Union was too weak to put up a decisive fight. It was unnerved by the United States' nuclear monopoly and devoted huge resources to ending it.\n\nA particular danger facing the Stalinist regime was the incongruity between the symbolic triumph of victory for Soviet society and the hard realities of daily life. The war had taken millions of Soviet citizens beyond the country's borders to Europe, an experience that many found shocking. The victors saw that the slaves of capitalism enjoyed a standard of living immeasurably better than theirs. They now knew that for years official Soviet propaganda had been pulling the wool over their eyes. Tens of millions of peasants, many of whom had fought in the war, dreamed of dismantling the kolkhoz system and believed that their sacrifices at the front had earned them this reward. A threatening gulf was opening up between the Soviet people's postwar expectations and their reality. As they struggled to overcome extreme daily hardships, mourned the dead, and listened to stories from returning soldiers, people's conversations inevitably drifted toward ideas and topics that were taboo: the price of war and victory, the questionable privileges enjoyed by party and government officials, and the causes of hunger and deprivation. The system's usual response to such \"incorrect\" thinking was arrest and prosecution for \"anti-Soviet propaganda.\" But would that response work in the new, postwar USSR?\n\nApparently Stalin was not sure how to address these challenges. In the immediate aftermath of victory, he sent mixed messages to the country, including hints at a coming liberalization. Take, for example, the remarks made at a reception honoring Red Army commanders on 24 May 1945:\n\nOur government made more than a few mistakes; there were moments of desperation in 1941\u20131942, when our army was on the retreat, abandoning our native villages and cities.... Another people might have told its government: you have not met our expectations; go away; we will put another government in your place that will sign a truce with Germany and ensure us peace. But the Russian people did not choose to do that since they believed in the correctness of their government's policies and chose to make sacrifices in order to secure the destruction of Germany. And this trust the Russian people placed in the Soviet government proved to be the decisive force that secured a historic victory against an enemy of humanity\u2014against fascism. Thanks to the Russian people for this trust\n\nThis hint of penitence was an effective gesture by a confident, popular, and triumphant leader. But soon Stalin began to sense that such statements could be perilous. They opened the door to discussion of critical questions about the past war, and echoes of these discussions were starting to reach him. In November 1945, he was told about a letter from a propagandist in the Buriat-Mongol republic who was being asked during lectures just what Stalin meant when he mentioned mistakes by the Soviet government: \"I, of course, was not able to answer this question.... I earnestly ask you, Com. Stalin, for your explanation as to what should be the answer to this question.\" More to the point was a letter from N. M. Khmelkov from the village of Maly Uzen in Saratov Oblast that asked, \"How could we allow it to happen that when the war broke out the German Army was better armed than our army?\" Khmelkov recalled prewar promises that the Red Army would soon be fighting \"on the territory from which the enemy comes\" and concluded by asking Stalin a central question, the validity of which Stalinists reject to this day: \"Victors are not judged. But a victorious people is obliged to figure out whether victory was achieved with the least possible expenditure of effort and resources and with the fewest possible casualties, and if it was not, then why: were we given too little time to prepare for war, were the cogs in a complex machine operating poorly... and failing its more complicated parts?\" Stalin instructed that Khmelkov's letter be filed away. He had no intention of responding to such questions or \"figuring out\" what mistakes the government might have made. To forestall undesirable discussion of the price of victory, the performance of the military leadership, and hopes for postwar liberalization, he launched a series of ideological counterattacks.\n\nThe first of these was a reappraisal of the toll taken by the war and the reasons for defeat. In an obvious attempt to downplay the nation's losses, in March 1946 Stalin officially stated that \"as a result of the German invasion, the Soviet Union irretrievably lost approximately 7 million in fighting with the Germans and because of the German occupation and the driving of Soviet people into German hard labor.\" This was a strange number to pick, and it was far from accurate, but it is possible to see how Stalin might have arrived at it. According to General Staff estimates, approximately 7 million Red Army soldiers were killed in the war or died of wounds and disease. He must have known that he was distorting the truth when he included the victims of occupation and those taken to work in Nazi labor camps in this figure. Soviet war losses no longer looked quite so terrible, and the matter was put to rest for many years.\n\nWhile it may have been easy enough to hide the true number of Soviet war dead, the Red Army's catastrophic retreat was another matter. How had the Germans been able to advance all the way to the Volga? At best, discussion of this ignominious episode could be suppressed. The horrible defeats suffered during the war's first eighteen months cast a shameful light on the regime and on Stalin himself, diminishing his stature as the architect of victory. Soviet propaganda had a few stock arguments to explain those early defeats: the might of Hitler's war machine, which enslaved Europe; the fact that the Red Army had not finished rearming; and the Nazis' perfidious surprise attack. Apparently Stalin felt these arguments were not enough. Cautiously and gradually, he tried to introduce another idea into the propaganda arsenal, one that exonerated him as supreme commander: that the Red Army's retreat was a calculated move designed to wear down the enemy. There was a well-known historical precedent that made this argument understandable and familiar: Kutuzov's 1812 strategy of allowing Napoleon's army to enter deep into Russian territory, even relinquishing Moscow, before counterattacking, a strategy that is credited with preserving the army and saving the country.\n\nAn opportunity to promote this new way of explaining the retreat came in the form of a letter Stalin received in early 1946 from Ye. A. Razin, a military academy instructor. Razin was writing the _vozhd_ with general questions about doctrine, but Stalin responded in a letter by offering specific and far-reaching guidelines for understanding Soviet military history. He underscored two central ideas. First, Lenin was not \"an expert in the military sciences\" during the Civil War years or at any other time. Thus Stalin was the only Soviet leader who qualified as a true commander in chief. The second idea offered a more favorable interpretation of the early, catastrophic stage of the war. \"A retreat, under certain disadvantageous conditions,\" Stalin wrote, \"is just as legitimate a form of combat as an offensive.\" He noted the need to take a closer look at the counteroffensive \"after an enemy's successful offensive, [when] the defender gathers strength, switches to a counteroffensive, and hands the enemy a decisive defeat.\" Bolstering this idea with historical parallels, Stalin cited the example of the ancient Parthians, who \"lured\" Roman forces deep inside their country and then \"struck with a counteroffensive and annihilated them.\" He also offered the example of Kutuzov's counteroffensive against the French, calling him a \"brilliant\" commander.\n\nOf course, Stalin did not draw a direct line between these historical precedents and the events of 1941\u20131942, but the implication was obvious. The defeats of the war's first stage were transformed into a manageable phase of preparation for a counteroffensive, a \"legitimate form of combat,\" and not a catastrophe caused by egregious blunders at the top or a broken chain of command. Aware of the questionable validity of this recontextualization, Stalin did not widely disseminate his letter at first. It was written in late February 1946 but not published until a year later.\n\nThe letter to Razin contained another thought that preoccupied Stalin in the first months after the war: the need to avoid \"kowtowing to the West,\" including showing \"unwarranted respect\" for the \"military authorities of Germany.\" The first expression of this sentiment is found in a letter written by Stalin during the autumn of 1945 to his comrades in Moscow while he was vacationing in the south. Denouncing unnamed \"senior officials\" who were \"thrown into fits of childlike glee\" by praise from foreign leaders, he wrote, \"I consider such inclinations to be dangerous since they develop in us kowtowing to foreign figures. A ruthless fight must be waged against obsequiousness toward foreigners.\"\n\nThese loosely formulated ideas were Stalin's response to the \"contamination\" of Soviet society by the ideological influence of the Western allies and to the danger of an inferiority complex on the part of the impoverished victors. Over time, the \"fight against kowtowing\" took the form of specific campaigns and institutions. In August 1946 a Central Committee resolution was published on \"The Magazines _Zvezda_ and _Leningrad\"_ in support of an irate speech to Leningrad writers delivered by Central Committee secretary Andrei Zhdanov. The targets of his ire were the satirist Mikhail Zoshchenko and the poet Anna Akhmatova. The former's writings, according to Zhdanov, were poisoned by the \"venom of a brutish hostility to the Soviet system.\" Akhmatova was labeled a \"whore and a nun, in whom licentiousness is combined with prayer.\" Discussion of the resolution was made mandatory at party meetings across the country\u2014in regional party organizations, factories, and kolkhozes\u2014and marked the beginning of a severe scolding given to the creative intelligentsia.\n\nA leitmotif of the attack on writers was the unmasking of \"kowtowing to the contemporary bourgeois culture of the West\"\u2014a formulation that clearly came from Stalin's own pen. Indeed, archival documents show that Stalin was behind Zhdanov's vitriol and that he read and edited his speech. The archives further reveal that Stalin was the driving force behind other actions designed to promote ideological lockstep, such as the well-known case of the married scientists Nina Kliueva and Grigory Roskin, who were developing a cancer drug in Moscow. In 1947 they were ground-lessly accused of passing secret information to the Americans. The couple was accused of \"kowtowing and servility to anything foreign.\"\n\nThese shrill ideological clich\u00e9s were variations on the canonical themes of Leninism and Stalinism: the USSR, since it was building the most advanced social system, would always and in all respects surpass the rest of the world; the capitalist powers, sensing their inevitable demise, would be ready at any moment to unleash war against the birthplace of socialism. The recent war and the gradual move toward a new \"cold\" war served to confirm this thinking.\n\nMany years of research, especially since the archives of the former USSR and other countries of the socialist bloc have opened up, have provided a wealth of information on the origins of the Cold War. Nevertheless, scholars may never reach agreement about its real causes, which side should take the larger share of blame, and the true motives and calculations of the opposing powers. The Cold War was more a gradual evolution than an event with a clear beginning. The world leaders involved in this process were not simply looking out for their countries' fundamental interests, but were also reacting to specific, often unexpected situations with decisions that were often illogical. Stalin was no exception.\n\nThe intensifying conflict between the World War II Allies was fed by the utter incompatibility of their systems, their competing desires to expand their spheres of influence, mutual grievances dating to the prewar years, and a shared need for a foreign enemy. Specific issues tended to exacerbate the general suspicion and animosity. America's nuclear monopoly and its reluctance to let the Russians take part in the occupation of Japan were among the many frustrations Stalin felt in dealing with the United States. In a meeting with Averell Harriman at the Soviet leader's southern dacha in October 1945, Stalin angrily wondered out loud whether the United States \"needs not an ally but a satellite in Japan? I must say that the Soviet Union is not suited to that role.... It would be more honorable for the USSR to leave Japan entirely rather than remain there like a piece of furniture.\" For his part, Stalin angered Western leaders, already fundamentally opposed to Soviet communism, with his thinly veiled desire to sovietize Eastern Europe using the Red Army and local Communists.\n\nIt is hard to imagine what mutual concessions might have prevented a breakdown in relations between two such different systems. Such a breakdown could only be delayed by tactical calculations and political factors, including the illusion on the part of Western public opinion that an enduring alliance was actually feasible (Soviet public opinion had little say in the matter). Relations also remained civil so long as Stalin harbored hope for Western concessions, particularly in the areas of economic aid and reparations from Germany. The devastation and famine afflicting the USSR after the war made the need for assistance particularly pressing. That Eastern Europe\u2014now within the Soviet sphere of influence\u2014not only suffered its own famine and devastation but was also home to significant anti-Communist sentiment also forced him to act with circumspection.\n\nStalin was restrained in his personal relations with Western leaders. He preferred to let Molotov take hard-line stances during diplomatic negotiations, while he himself would periodically step in and make demonstrative concessions that allowed the Western side to save face or prevented it from breaking off talks. As during the war, Stalin tried to play the Americans and British against one another. In April 1946, after Churchill's \"Iron Curtain\" speech in Fulton, Missouri, Stalin met with the U.S. ambassador in Moscow. After accepting the gifts of a safety razor and transistor radio, Stalin offered a \"friendly\" warning: In pursuing their own interests, \"Churchill and his friends\" might try to push the United States away from the USSR.\n\nSuch face-to-face diplomacy was no match for the powerful forces at play. Truman responded to Soviet attempts to gain footholds in Iran, Turkey, and Greece with a plan to help rebuild Europe, the centerpiece of which became known as the Marshall Plan. Stalin responded by turning down the aid offered under the plan (as did other East European states, under Soviet pressure) and by creating an international Communist organization, the Cominform. During the Cominform's first conference, Zhdanov echoed Stalin's idea that the world was being divided into \"two camps.\" Efforts to sustain the wartime alliance gave way to the traditional call to stand up to \"international imperialism.\"\n\nOn the domestic side, the return to prewar political thinking and practices occurred even earlier. Stalin's conservative inclinations played no small role. Given the array of complex problems facing him, as he approached his seventieth birthday, he neither took an interest in reforms or experiments nor saw any reason to change his country's long-range goals for economic development. He offered a number of production targets in a speech to an election meeting on 9 February 1946: 500 million tons of coal, 60 million tons of steel, 50 million tons of cast iron, 60 million tons of petroleum. Considering the actual figures for 1946\u2014only 13.3 million tons of steel and 9.9 million tons of cast iron were produced, along with 163.8 million tons of coal and 21.7 million tons of petroleum\u2014such targets were obviously wildly ambitious. Furthermore, as the economic historian Eugene Zaleski has noted, a program like Stalin's, purely focused on output targets, reflected a simplistic understanding of economic development.\n\nStalin showed his preference for tried and true methods during the famine of 1946\u20131947, when, as in 1932, draconian laws were enacted against the pilfering of state property. Two 4 June 1947 decrees provided for sentences ranging from five to twenty-five years in a camp for theft. Between 1947 and 1952, more than 2 million people were convicted of this charge. Many if not most were simply ordinary people who committed minor crimes in the face of great material deprivation. Parents who stole a loaf of bread for their hungry children were sentenced to many years in a camp. Mass repression was not limited to the prosecution of theft. Arrests for political crimes continued, and harsh laws were also put in place to combat violations of workplace discipline. Approximately 7 million such sentences, an average of 1 million per year, were handed down between 1946 and 1952. In Stalin's last years, the Gulag grew into a sprawling network that played a central role in the life of the country. On 1 January 1953, more than 2.5 million people were being held in camps, penal colonies, and prisons. \"Special settlements\" in remote regions held another 2.8 million. Some 3 percent of the population was either incarcerated or under internal exile.\n\nMass repression, in the form of large-scale arrests, executions, and internal exile, was now largely focused on the newly absorbed parts of the Soviet Union, where fierce guerrilla campaigns raged. Stalin received regular reports on the pacification of mutinous areas. For the years 1944\u20131952, according to incomplete official statistics, approximately a half million people were killed, arrested, or forcibly exiled from Lithuania, Latvia, and Estonia, along with an equal number in the western provinces of Ukraine. For these small republics and provinces, whose populations totaled just a few million, these were astounding numbers. The Stalinist system had neither changed nor grown less repressive.\n\n###   **KEEPING THE LEADERS IN THEIR PLACE**\n\nAn important aspect of Stalin's postwar consolidation of power was a return to routine shake-ups at the upper echelons of government and the preemptive humiliation of his devoted and obedient comrades. The stable leadership that had governed the country during the war was probably perceived by Stalin as a compromise necessitated by circumstances. Now that they had performed their tasks, he no longer needed influential marshals and members of the State Defense Committee. And as his physical state declined, his tendency toward suspicion grew.\n\nOn 9 October 1945 the Politburo adopted a resolution granting Stalin a vacation so that he could \"rest for a month and a half.\" This was his first trip to the south in nine years, and he may have left reluctantly. The foreign press was full of speculation. On 11 October he received a set of TASS news synopses regarding talk in the West about his poor health and the jockeying for position among potential successors. According to the summary, the _Chicago Tribune's_ London correspondent, citing diplomatic sources, wrote about a bitter behind-the-scenes power struggle between Zhukov and Molotov, both vying to replace Stalin. Zhukov was supposedly supported by the army and Molotov by the party apparat. A week later, the TASS synopsis included a statement by the Soviet ambassador in France: \"Over the past ten months we have been asked fifteen times to confirm reports of Stalin's death.\" An article about Molotov in a Norwegian newspaper stated that \"For public opinion in the U.S.A., England, and other freedom-loving peoples, Molotov represents a new, strong Soviet Union that demands the status of an equal among the world's great powers.\" Stalin was not mentioned. The article spoke only of his successors.\n\nThese foreign press reports reflected the Western view of the postwar configuration of power. The long and horrific war was receding into history, as were the leaders who had achieved victory. Roosevelt was dead. The defeat of the Conservative Party in Great Britain had sent Churchill into retirement. Stalin was aging and rumored to be ill. For the Western observer, these were all elements of the same coherent picture. Stalin, of course, did not share this view. Any hint that the Soviet leader might be replaced only heightened his indignation and suspicion, the brunt of which was borne by his closest comrades\u2014primarily Molotov, as he was first on the list of possible successors. Attacks against Molotov were also a convenient pretext for another shake-up. The ruling Five throughout the war had consisted of Stalin, Molotov, Beria, Malenkov, and Mikoyan. This grouping had been in place uncomfortably long.\n\nStalin's growing irritation with Molotov was on full display during the September 1945 meeting of the Council of Foreign Ministers in London, convened to discuss the new postwar order and peace terms with the vanquished countries. At the outset, Molotov took a liberty in regard to a procedural question. Yielding to a request by the Western Allies, he agreed that in addition to the Soviet Union, United States, and Great Britain, France and China would also be allowed to take part in the drafting of treaties. Under previous agreements, France and China were to be involved in designing terms only with Italy and Japan respectively. Molotov did not see a problem with this change, and strictly speaking there was none. France and China would only offer input on the treaties; they were not given any vote on their approval. Agreeing to this arrangement made perfect sense. Hoping for a productive meeting, Molotov did not want to waste time by provoking conflict over secondary questions.\n\nHis concession would likely have gone unremarked had the negotiations not reached a seemingly insuperable stalemate. Stalin demanded that the Soviet Union be given a real role in deciding the fate of Japan. The Western side would not even place that question on the agenda. Stalin demanded that one of Italy's colonies in North Africa be placed under Soviet trusteeship, thus giving his country a solid foothold on the Mediterranean. The Western side refused. The sides also reached an impasse over Romania and Bulgaria. Considering these countries \"satellites\" (Stalin actually used the cognate in a telegram he sent to Molotov during the meeting), the Soviet authorities had already installed pro-Communist governments there. The United States and Great Britain refused to recognize these governments or sign any accords with them. Stalin decided to increase pressure on his partners, even when it looked as if talks might break down. The question about France and China, whose participation was supported by the United States and Great Britain, offered a convenient pretext. On 21 September Stalin reprimanded Molotov for his procedural concession, and Molotov repented: \"I admit that I committed a grave oversight. I will take immediate measures.\" The following day he withdrew his agreement. The Western Allies were enraged. On the surface it looked as if this simple procedural question had brought the talks to a standstill.\n\nThis incident vividly illustrates Stalin's manipulative personality. While cultivating the image of a moderate and predictable politician in the eyes of his fellow Allies, he forced his comrades to do his dirty work. He was incensed when Molotov revealed that the withdrawal of consent for France's and China's participation came on his orders. For a long time afterward he reminded Molotov of this and similar instances, accusing him of trying to present himself as a reasonable alternative to the inflexibility of \"the Soviet government and Stalin.\"\n\nThese potshots at Molotov were a sign that a more serious attack was on the way. An essential role was played in this drama by the TASS summaries of the foreign press, which Stalin pored over during his vacation. Molotov's troubles began with a 1 December 1945 news item by a correspondent for Britain's _Daily Herald,_ reporting rumors that Stalin might be stepping down as chairman of the Council of People's Commissars and that Molotov might resume that post. The TASS summary quoted the correspondent as saying that the political leadership of the Soviet Union was currently in Molotov's hands, with general directives from the Politburo. For Molotov, nothing could have been more damaging, especially when Stalin was out of Moscow for the first time in years. Furious, on 2 December Stalin telephoned Molotov to demand that more stringent censorship be exercised over the dispatches sent out by foreign correspondents. Molotov gave the foreign affairs commissariat's press office the appropriate orders. The next day, however, there was a bureaucratic snafu. The TASS summary for 3 December included a _New York Times_ piece that had been published on 1 December, before Stalin's order to tighten control. The _Times_ item, like the _Daily Herald_ article, hinted at discord among the Soviet leadership and a weakening of Stalin's position. Stalin read the TASS account of the _Times_ article on 5 December. Apparently that same day he read a 3 December Reuters report that mentioned a relaxing of censorship in regard to foreign correspondents in the USSR. The press agency claimed that after Western journalists collectively had complained to the Soviet authorities, Molotov had said to an American at a 7 November reception, \"I know that you correspondents want to get rid of Russian censorship. What would you say if I agreed to this on condition of reciprocity?\" A few days later, according to Reuters, the Western press corps actually did see signs of relaxed control.\n\nThese reports gave Stalin more than enough ammunition to charge Molotov with scheming against him. On 5 December the _vozhd_ sent Molotov, Beria, Mikoyan, and Malenkov a telegram demanding that the matter be investigated. The following day the four sent Stalin a detailed response. The _New York Times_ article had a simple explanation. It had gone through censorship on 30 November, three days before Stalin asked Molotov to tighten control. The explanation for the Reuters report was just as persuasive. Molotov really had ordered a relaxation of censorship in November since the censors \"often unnecessarily marked out individual words and expressions in the telegrams sent by foreign correspondents.\" As for the conversation at the 7 November reception, Molotov claimed that \"words were attributed to him that he did not say.\"\n\nAfter receiving this response, Stalin went into a rage, either genuine or feigned. That same day, 6 December, he sent a sharply worded telegram to Moscow. Ignoring all the reasonable arguments offered by the four, he stated that Molotov bore the blame for the appearance of \"libels against the Soviet government\" in the foreign press. Furthermore, Molotov's liberal attitude toward foreign correspondents represented an intentional effort to change \"the course of our policies.\" After accusing Malenkov, Beria, and Mikoyan of connivance, Stalin directed extremely harsh words at Molotov. \"I am convinced that Molotov does not care about the interests of our state and the prestige of our government,\" he wrote, \"so long as he gains popularity within certain foreign circles. I can no longer consider such a comrade to be my first deputy.\" To add insult to injury, Stalin sent his telegram only to Malenkov, Beria, and Mikoyan, asking them to summon Molotov and read him its contents but not give him a copy. The reason he gave was extremely insulting to Molotov: \"I did not send [the telegram] to Molotov since I have doubts about some of those close to him.\"\n\nThis telegram contained the strongest accusations Stalin had ever made against a member of his inner circle (unless, of course, we include the Politburo members whom he had executed). The four men were undoubtedly frightened. On 7 December Beria, Malenkov, and Mikoyan sent Stalin a coded telegram in which they reported on the firm approach they had taken in dealing with their associate. \"We summoned Molotov to us and read him the telegram in full. After pausing to think, Molotov said that he had made a lot of mistakes but felt that mistrust toward him was unjust, and then he began to cry.\" There is no way to know whether they were describing this confrontation accurately. This was a drama played out for one spectator who was not even in the theater. What mattered was not the drama itself but the account of how the confrontation was handled, which had to be designed to satisfy Stalin. Molotov played along. That same day he sent Stalin his own telegram: \"Your coded telegram was filled with deep mistrust toward me as a Bolshevik and a man, which I take as the most serious party warning for all my work going forward, wherever that might be. I will try through my deeds to earn your trust, in which every honest Bolshevik sees not simply personal trust, but the trust of the party, which is dearer to me than my life.\" Judging by the correspondence that followed, Stalin felt that he had achieved the desired effect. He clearly knew that Molotov's \"crimes\" had no significance, and his underling had never disobeyed any direct instruction. Molotov had simply used his own discretion on occasions when Stalin's long-distance guidance was intermittent and vague.\n\nThe Molotov scandal was dropped quickly because its true purpose lay elsewhere: Stalin wanted to make changes to the top leadership. He began this reorganization as soon as he returned to Moscow. On 29 December 1945 he brought his old comrade Andrei Zhdanov into the inner circle. The Five were now Six. In October 1946, Nikolai Voznesensky was also admitted to the group, meaning that the country was now governed by the Seven.\n\nThe return of the \"Leningraders\"\u2014Zhdanov and Voznesensky\u2014into Stalin's inner circle provoked competition within the Politburo. Malenkov and Beria, who had pushed the Leningraders aside during the war, were now forced to concede power to them. In May 1946 Stalin removed Malenkov from the post of Central Committee secretary, accusing him of covering up irregularities in the aviation industry, which had been his portfolio during the war. Malenkov's responsibilities overseeing the Central Committee apparat were handed over to Zhdanov. Around the same time, a blow was struck against Beria. Stalin forced Beria's prot\u00e9g\u00e9, Minister for State Security Vsevolod Merkulov, to resign his post in disgrace. A dangerous development was that Stalin appointed the former head of military counterintelligence, Viktor Abakumov, with whom Beria did not get along, to take Merkulov's place. According to the rules of Stalinist shake-ups, the new minister was expected to uncover misconduct or\u2014better yet\u2014crimes by his predecessor. Abakumov was well suited to this role. Both Merkulov and Beria were clearly in danger. As Merkulov attested after Stalin's death, \"The story of my departure from the Ministry of State Security gave Beria a number of unpleasant moments. Beria himself told me that because of me he was in trouble with Comrade Stalin.\"\n\nBeria's and Malenkov's ordeals were relatively painless. Both remained within the top leadership. Presumably they were just being shown who was boss and reminded that they were dispensable. Stalin clearly had no intention of dismantling the system of supreme power that had taken shape. He just wanted to create new counterpoises, new centers of competition.\n\nStalin was just as calculating in dealing with the military leadership. By the war's end, the status of the Soviet Union's marshals and generals was understandably sky-high. For Stalin, who cherished his own reputation as a commander, their popularity was politically undesirable: the victory could be the work of only one genius. Stalin was also concerned about possible conspiracies. The generals, intoxicated by thoughts of their own brilliance, made matters worse. State security, which was always in competition with the military, reported to Stalin on conversations at celebratory dinners where generals lavished one another with praise and made disparaging comments about their _vozhd_. Stalin's natural response was repression. Inevitably his first target was Zhukov, the most famous and influential of the wartime military leaders. Zhukov's life now hung by a thread. Stalin ordered the arrest of a number of generals close to Zhukov and had a case opened against Zhukov himself. A month later, after Malenkov's demotion and Merkulov's firing, Zhukov and other military leaders received a dressing down. A 9 June 1946 order, issued by the minister for the armed forces of the USSR and signed by Stalin, described the wartime commander's transgressions as follows: \"Marshal Zhukov, having lost all modesty and carried away by a sense of personal ambition, felt that his services had not been sufficiently valued and took credit in conversations with subordinates for designing and carrying out all of the Great Patriotic War's major operations, including those operations with which he had nothing to do.\" This condemnation was obviously motivated by Stalin's jealousy and anger at a lack of proper deference from this national hero and other military leaders and his desire to cut them down to size. But he was not prepared to go so far as to physically annihilate Zhukov, who was too symbolic a figure and too closely associated with him. Public discrediting and demotion would suffice. The order relegated Zhukov to a secondary post commanding a military district. Given the fate of some of Stalin's other close associates over the years, such a command might even be considered a reward. Zhukov had lost a great deal but not everything. Toward the end of his life, Stalin agreed to readmit Zhukov to the Central Committee, a sign that he was finally back in the _vozhd_ 's good graces.\n\nBy late 1946 these reshufflings had evened out the balance of power among Stalin's associates. The firings, demotions, and public humiliations more or less restored the structure of top government that had existed before the war. Stalin could now leave his associates in relative peace as he dealt with the country's pressing economic problems.\n\n###   **CURRENCY REFORM AS A REFLECTION OF THE SYSTEM**\n\nMilitarization, physical devastation, famine, an inefficient ration system, crippled agriculture, a degraded social infrastructure, and a reliance on compulsion in mobilizing the labor force\u2014such were the features of the postwar Soviet economy. War's toll was, of course, reflected in the sorry state of the budget. The government had financed the war's huge costs primarily by printing money. The predictable result was spiraling inflation. Something had to be done about the excess currency circulating through the economy. To reduce the amount of money in circulation, the Soviet leadership ordered new rubles printed and old rubles devalued.\n\nIn his memoirs, the wartime finance commissar, Arseny Zverev, states that by late 1943 he had already discussed such measures with Stalin. Evidence that the finance commissariat was planning for currency reform so early can also be found in the archives. Toward the end of 1943 it was decided that the reform would be introduced after the war by reducing the buying power of the ruble through increased prices, exchanging old rubles for new ones, and abolishing the ration system. This is largely the program that went into effect a few years later.\n\nNow that the war was over, the problem of stabilizing the country's finances and doing away with rationing took on tremendous political importance. Doing away with ration cards even more quickly than in capitalist countries would demonstrate the advantages of socialism. The reform measures were planned for 1946, but the famine forced a delay. Throughout that year, Finance Commissar Zverev sent Stalin several memoranda on the upcoming reforms. Judging by Stalin's notations on these documents, he took a great interest in the topic. As preparations reached their final phase, Zverev had frequent face-to-face meetings with the _vozhd_. According to the log of visitors to Stalin's office, during the period leading up to the reform's introduction on 14 December 1947, Zverev was there thirteen times.\n\nFinally, on 13 December 1947, the Politburo voted to approve the main documents instituting the currency reform and abolishing ration cards. It was stipulated that the measures would be announced over the radio at six o'clock in the evening on 14 December and in newspapers the following day. Overnight, between 14 and 15 December, the population was deprived of a significant portion of its savings. For every ten rubles people had in their possession, they would now receive one. There was a more complex system to deal with bank deposits. Accounts with under three thousand rubles were not affected, but those with three to ten thousand rubles would be compensated at a rate of two new rubles for every three old ones. Deposits over ten thousand rubles were compensated at a rate of one to two.\n\nThe Politburo was fully aware that the reform would not be popular. A large part of its resolution, which was intended for publication, was devoted to a detailed explanation of the move's necessity, utility, and fairness. Keenly in tune with widespread prejudices, the text asserted that the reform would hit hardest at \"speculative elements who have amassed large stores of money.\" This assertion was false: the most well-off Soviet citizens were in the best position to convert their cash into other forms of wealth. Nevertheless, the idea that the currency reform was a means of confiscating ill-gotten gains proved extremely popular. As usual, the resolution did not neglect to mention the financial hardships faced by the toiling masses in capitalist countries. Its wording suggests that Stalin played an active role in drafting it. Among the revisions made in his handwriting is the added promise that this would be the Soviet people's \"final sacrifice.\"\n\nMajor reforms are always fraught with difficulty. The new rubles began to be printed in 1946 for introduction at the end of 1947, but at first a high percentage proved defective. To maintain secrecy, the new money was not delivered to Gosbank branches, of which there were many, but to specially set up storage facilities evenly distributed around the country. The new rubles were transported in special, heavily guarded train cars. Finally, when it came time to exchange rubles, in addition to regular Gosbank branches, 46,000 exchange points were set up, for which 170,000 workers were hired.\n\nNo amount of secrecy, of course, could hide such a major operation from public view. Rumors began to spread and became more persistent after salaries and pensions for the second half of November were paid ahead of schedule. Overall, however, the public did not know what the reform would look like. Spurred by contradictory rumors, people scrambled to save their nest eggs. At first the panic affected purchases of durable goods and valuables. On 29 November 1947, Internal Affairs Minister Sergei Kruglov reported to Stalin that customers were flooding stores to buy manufactured goods and crowding into banks to withdraw their savings. Store shelves were emptied, and even items for which there had previously been no demand disappeared. Stores sold out of furniture suites going for tens of thousands of rubles\u2014huge sums, given that the average annual salary for laborers or office workers was approximately 7,000 rubles. One suite costing 101,000 rubles that had languished on the showroom floor for years now had four competing buyers. Customers bought furs, fabrics, watches, jewelry, pianos, and rugs. On 30 November Kruglov reported that hundreds of people had lined up outside Moscow's department stores before opening. People from neighboring oblasts flooded into the city. Huge lines of up to five hundred people formed outside savings banks. After two days of this buying frenzy, the authorities decided to take action. Kruglov informed Stalin that most stores had been closed under the pretext of renovation or taking inventory. The stores that remained open removed valuable items such as gold jewelry from sale. And some were forced to shut their doors because they had nothing left to sell.\n\nKruglov's report of 2 December was not much different. Now that consumer goods were in short supply, people had started to buy up whatever they could find, including musical instruments and phonographs. One store that had been selling six pianos a year sold all eleven it had in stock over two days\u201430 November and 1 December. The shortage of manufactured goods led to a run on non-perishable food items such as smoked sausage, canned goods, candies, tea, and sugar. This hoarding prompted an order to remove these items from sale. Restaurants did a brisk business, and \"drunken individuals would take wads of cash out of their pockets and cry: 'Look at all this paper.'\" Other regions reported similar spending sprees. If Stalin read such reports\u2014and there is every reason to believe he did\u2014he was given an eye-opening lesson on the lives and economic logic of ordinary Soviet citizens.\n\nIt is interesting that the authorities refrained from heavy-handed measures to halt the frenzy. Beginning in early December there was a noticeable increase in small savings bank deposits, an obvious effort to spread savings over multiple small accounts that would counteract the reform's intention of removing rubles from circulation. Even then, no steps were taken. Stalin could see how unpopular the reform was and did not want to further inflame sentiment against it.\n\nBy 15 December it was all over, and the straightforward operation of exchanging old rubles for new and revaluing deposits began. During the eight-day period from 16 to 23 December 1947, Stalin received visitors in his office five times. Each time, Zverev was among them. His visits on 16 and 17 December\u2014the reform's first days\u2014both lasted two hours. Each time, a significant fraction of the Politburo was also present. On 3 January 1948 Zverev sent Stalin a report on the reform's results. It was filled with statistics that must have been encouraging to the government but disheartening to the rest of the population. Before the reform, on 1 December 1947, there were 59 billion rubles in circulation. As a result of the spending spree and ruble exchange, there were now only 4 billion. Deposits in savings accounts had been reduced from 18.6 billion old rubles to 15 billion new ones. The percentage by which prices decreased following the abolition of ration cards was modest in comparison with the number of rubles that had been taken out of people's pockets. The price of bread went down by 20 percent and meat by only 12 percent. Some prices even increased. Woolen fabrics, for example, went up by 27 percent, while clothing in general rose by 11 percent. Overall, the index of state retail prices after the reform went down to 83 percent of what it had been beforehand. Having exchanged ten old rubles for one new one, a consumer's purchasing power was now reduced by a factor of eight. The lion's share of the population's savings had been confiscated.\n\nTo some extent the \"shop window effect\" that followed\u2014the presence of more goods in stores, even if few could afford them\u2014should have softened the blow. But in Stalin's USSR, the shop windows were still not very impressive. Poor output in both the agricultural and consumer goods sectors and the general sluggishness of the state-run economy meant that even relatively weak post-reform demand could not be satisfied. As usual, special measures were taken only in major urban centers, Moscow and Leningrad first and foremost. Generous supplies of food and manufactured goods had been warehoused there in advance. But even in these cities, there were limits placed on purchases: bread\u2014two kilograms per customer; meat and meat products\u2014one kilogram; sausage\u2014half a kilogram; milk\u2014one liter; footwear\u2014one pair; socks\u2014two pairs; soap\u2014one bar; matches\u2014two boxes, etc. In the capitals and in some other major cities, the end of rationing led to supply problems. A few weeks later, Moscow began to receive complaints about empty store shelves, limits on purchases despite the supposed end of rationing, and special shops set up for officials only. One letter from Belgorod read: \"Today is the sixth day in a row that my wife stood in line for bread from 2 in the morning to 10, but, alas, all six days she came home without bread.\" Facing long lines, high prices, and empty stores, people looked back on the days of ration cards with nostalgia.\n\nNot all population segments suffered equally. People in major cities, especially those receiving high salaries or otherwise affluent, were not greatly affected by the reform. Before the devaluation it had been relatively simple for them to convert their old rubles into goods. After the reform they took advantage of the relative availability of goods and the drop in prices in urban _rynoks_ (food markets where peasants could charge a market price for the goods produced on their private plots). But the price drop hit the peasants hard. Deprived of their savings, uncompensated for their labor on kolkhozes, and forced to carry a heavy tax burden, they were desperate for cash. The reduction in state prices, however modest, pushed down food prices in the _rynoks,_ further depressing their income. Once again, the country's rural majority was the main victim of Stalin's policies.\n\nAlthough the government promoted the reform as a tool in combating the illegitimate acquisition of wealth, in fact it had the opposite effect. Corrupt officials and those operating in the shadow economy managed to convert their cash into luxury goods, which they resold at a profit after the devaluation. In Moscow's Tushino District, for example, two store directors (both members of the Communist Party) embarked on a large-scale money-making scheme. Using their own money, they bought up suits, fabrics, hundreds of pairs of shoes, and other items. These goods were stashed away until after the reform, when they were gradually sold at _rynoks_ through a network of sellers, as well as through the directors' stores. The following figures give an idea of how typical such operations were: during the last two weeks of December 1947, approximately 3,000 people working in the retail sector were arrested, of whom 1,100 were store directors and approximately 900 were party members. Such arrests continued at the same rate through January and February. And this was only the tip of the iceberg.\n\nAnother common practice spurred by the devaluation was the backdating of savings account deposits made after the terms of the reform were announced. Many large accounts were broken into smaller ones under the three-thousand-ruble limit. The true scale of such malfeasance is unknown, but records show that this subterfuge was practiced in all regions of the country by a significant proportion of officials. According to incomplete data for March 1948, in just twenty-six oblasts, _krais,_ and republics, more than two thousand officials, including senior party and law enforcement officials, were prosecuted for violating the currency reform law. Party secretaries and the heads of state security and internal affairs branches were found guilty of such operations. Cases were also uncovered where top regional officials tried to subvert justice. Central Committee records show multiple cases where \"certain regional party bodies have dragged out investigations of cases associated with violating the currency law, and in some cases they have even taken under their protection 'major' party and government officials, shifting the full burden of guilt on secondary individuals.\" Another case file stated that \"a significant proportion of senior party and government officials have essentially escaped punishment.\"\n\nResearchers have yet to find evidence of Stalin's reaction to this malfeasance. The absence of major shake-ups in the wake of the monetary reforms suggests that he maintained a fairly condescending attitude toward this blatant corruption. This stance was nothing new. Stalin consistently demonstrated tolerance for the moral failings of his faithful underlings. He cared about political loyalty and administrative competence.\n\nWhile the currency reform cast a spotlight on many of the Stalinist system's flaws, it also had a positive impact on the country's economic development. Ambitious reconstruction plans for 1948 were surpassed. Having taken so much money out of people's pockets, the government could print more without risking inflation, a move that was a great help in making up budgetary shortfalls. The relative financial stability achieved in early 1949 enabled wholesale pricing reform in heavy industry, which in turn created the preconditions for industrial development. Economic indicators for 1948 suggested that the most damaging consequences of the war had been overcome and that the main objectives of postwar recovery had been met. The end of the devastating famine of 1946\u20131947 was especially important. In 1948 the gross grain yield came close to prewar levels, and the production of potatoes (a staple of the Soviet diet) broke all prewar records. In the words of Donald Filtzer, the Soviet Union had entered a period of \"attenuated recovery.\" Nevertheless, Stalin-style industrialization was able to meet only the most basic needs of the population.\n\n###   **CONSOLIDATING THE SOVIET SPACE**\n\nWhile this economic recovery was under way in the USSR, neighboring countries were still roiled by political instability. In early 1948 the liberal democratic government of Czechoslovakia was overthrown in a coup, making Czechoslovakia the last East European country to join the Communist bloc. Establishing Communist control of these countries was, however, just the first step. They had to adopt the Stalinist model of internal development, pledge to be loyal satellites of the USSR, and unquestioningly submit to Stalin as the supreme leader of the bloc. A number of obstacles stood in the way. Despite repression, the presence of the Red Army, the suppression of educated segments of society, and the expansion of state control of the economy, for some time the newly Communist countries retained a degree of socioeconomic, cultural, and political diversity. Furthermore, the majority of East Europeans opposed the Communists, and power struggles within the Communist parties prevented the emergence of the kinds of dictatorial leaders needed to implement Stalinist socialism. Worse, a number of East European leaders showed signs of unacceptable \"liberalism,\" preferring a more flexible model of socialism over the Soviet model.\n\nOne \"bad example\" for any wavering Communists was Yugoslavia's Josip Broz Tito. In the spring of 1948 he became embroiled in a conflict with the Soviet Union that quickly escalated. Stalin was confronted with a worthy adversary. Tito was a born dictator who, unlike some other Communist leaders, had not simply been placed in power by Moscow but had earned it fighting the Nazis. His hand was further strengthened by the absence of Soviet troops in Yugoslavia. Tito pretended to political independence and aspired to be a leader of the Communist bloc, and he translated these pretensions into actions. In short, he ignored one of the key principles of Stalinization: total submission to Moscow.\n\nStalin's hope that severe public accusations would drive a wedge through the Yugoslav leadership and spark mutiny against Tito was disappointed. Tito made quick work of the Kremlin's Yugoslav clients and emerged from the showdown stronger. This defeat was a painful blow for Stalin. For the first time since the struggle with Trotsky, he was being opposed by a major leader within the Communist movement. And unlike Trotsky, Tito had real power and forces capable of protecting him from the ice picks of Stalin's professional killers. Tito's insubordination was not simply a blow to Stalin's self-respect, but also a dangerous precedent and a crack in the monolithic Soviet bloc. Others might follow Tito's lead.\n\nThe dangers of Titoism intensified confrontations with the West. The first serious standoff in Germany between the USSR and its former allies also came in 1948. The Soviet blockade of the Western sectors of Berlin was met with determined resistance. The system used to supply the Western zone by air\u2014the Berlin Airlift\u2014not only demonstrated the effectiveness of the Western bloc, but also promoted its consolidation. In April 1949 the agreement that established NATO was signed. The following month Stalin was forced to lift the blockade, and that autumn, Germany was formally divided into two separate states.\n\nThese foreign policy setbacks ignited Stalin's suspicions and insecurity and strengthened his resolve to force Stalinization in the East European Communist bloc. Moscow intensified its interference in the internal affairs of its satellites, and demands for accelerated sovietization became more implacable and impatient. Using his familiar methods of purges and fabricated political charges, Stalin initiated and oversaw a campaign against \"enemies\" within the leaderships of the socialist countries. In late 1948 he succeeded in getting rid of Poland's unyielding leader, Wladyslaw Gomulka. In Hungary, advisers from Moscow helped orchestrate a case alleging a far-reaching espionage organization, supposedly led by the country's former minister for internal affairs, Laszlo Rajk. In September 1949 Rajk was convicted and given the death sentence. In December, after a lengthy process of fabricated charges (again with the help of Soviet security advisers), the former secretary of the Bulgarian central committee, Traicho Kostov, was put to death. Stalin kept a close watch over all these cases and sanctioned both the falsification of evidence and the death sentences. Rajk's and Kostov's trials prompted arrests in other Communist countries. These tactics brought about a concentration of power in the hands of dictators entirely dependent on Stalin and ready to implement any policy he liked.\n\nWhile overseeing the Stalinization of the Communist bloc, the Soviet dictator still found time to consolidate his power at home\u2014or rather to preempt any possibility that it could be undermined. Setting an example for his satellites, Stalin launched yet another wave of domestic purges. The themes and victims depended to some extent on random developments. One such development was the death of Stalin's close comrade Andrei Zhdanov in August 1948. Zhdanov's duties as Stalin's deputy for party affairs and as head of the Central Committee apparat were taken over by Georgy Malenkov, a shift that upset the balance of power within Stalin's inner circle. Having lost their patron, the Leningrad group, most prominently represented by Gosplan chairman Voznesensky and Central Committee secretary Kuznetsov, found itself weakened, and the group's rivals, Beria and Malenkov, were now stronger. Such shifts prompted a new bout of behind-the-scenes struggle. The combination of these intrigues, international tensions, and Stalin's political calculations spawned the Leningrad Affair, the last purge to roil the upper echelons of power in the USSR. Before it was over blood had been spilled.\n\nThrough the efforts of Malenkov and Beria, who probably did not expect their actions to be as damaging as they proved to be, Stalin received compromising materials against the Leningraders. The infractions these materials exposed were relatively minor. In one instance a decision was made to hold a major trade fair in Leningrad without consulting all of the proper authorities. In another, Voznesensky's agency, Gosplan, made certain errors in putting together plans and misplaced some documents\u2014common occurrences in the highly bureaucratic Soviet system. There were also several instances when regional leaders, mostly Leningraders, attempted to use Voznesensky and Kuznetsov for patronage, but such attempts too were nothing out of the ordinary. They were all the sort of typical rule-bending that Stalin could simply ignore or use as ammunition. He chose to do the latter.\n\nDuring a Politburo meeting presided over by Stalin in February 1949, Kuznetsov, Voznesensky, and other functionaries close to them were charged with attempting to turn the Leningrad party organization into their own fiefdom. Particularly ominous was a resolution comparing their actions to those of Zinoviev in the 1920s, \"when he attempted to turn the Leningrad organization into a power base for his anti-Leninist faction.\" In the months that followed, charges against the beleaguered Leningraders snowballed. They were accused of enemy activity and even espionage. In September 1950, after months of interrogations and torture, Voznesensky, Kuznetsov, and a number of other leaders were sentenced to death in a closed Leningrad courtroom. Several hundred others were given death sentences, imprisoned, or exiled. The purge also affected other regions of the country, where natives of Leningrad held senior posts or had sought support from highly placed Leningraders in Moscow.\n\nThe way the Leningrad Affair unfolded suggests that Stalin was using it to pursue multiple goals. It may have been part of his ongoing pattern of intimidation to consolidate power. The accusations of patronage and the large-scale dismantling of networks of officials who made their careers in Leningrad were typical of the preemptive strikes Stalin liked to launch against informal networks within the nomenklatura. He may also have viewed the Leningrad Affair as part of a larger shake-up at the upper echelons. In any event, the fabrication of evidence against the Leningraders at first unfolded in synchrony with Stalin's attacks against his old comrades Molotov and Mikoyan. These assaults seem all the more likely to be connected as Molotov had maintained close professional ties with Voznesensky and was on friendly terms with him. Furthermore, while the Leningrad Affair was in full swing, Mikoyan's son was preparing to marry Kuznetsov's daughter and, rather surprisingly, proceeded with this plan.\n\nWhatever the reasons for Stalin's displeasure, Molotov and Mikoyan were its most natural targets. They were his oldest and most distinguished comrades, symbols of the collective leadership that might have been, and the presumptive heirs of the aging _vozhd_. The task of bolstering his personal power\u2014Stalin's prime obsession\u2014required him, he felt, to periodically discredit his most influential associates in order to weaken their influence.\n\nFor several years the actions Stalin took against Molotov in late 1945 were known only within the narrow circle of the Politburo. Molotov continued to perform key governmental functions: he chaired a number of Council of Ministers commissions, headed the Ministry of Foreign Affairs, and had a voice on a wide array of questions. This status began to change in 1948. On various pretexts, Stalin used reprimands and limitations on his authority to diminish Molotov's standing. The main means of pressure was the fabrication of evidence against Molotov's ethnically Jewish wife, Polina Zhemchuzhina, showing her to be involved with \"anti-Soviet\" Jewish organizations. Stalin demanded that Molotov divorce her. \"Stalin,\" Molotov later recalled, \"came up to me at the Central Committee: 'You have to divorce your wife!' And she said to me, 'If it's necessary for the party, then we'll get divorced.' In late 1948 we divorced.\"\n\nOn 29 December 1948, \"evidence\" compiled by state security in the Zhemchuzhina case was brought before the Politburo. She was expelled from the party, a move that meant that arrest was imminent. Molotov abstained from voting, an action that put him in direct conflict with Stalin. On 20 January 1949 Molotov sent the _vozhd_ a formulaic expression of remorse:\n\nDuring Central Committee voting on a proposal to expel P. S. Zhemchuzhina from the party I abstained, which I admit to be politically mistaken. I hereby state that having thought over this question, I vote in favor of the Central Committee decision, which corresponds to the interests of the party and the state and teaches a correct understanding of the meaning of Communist Party membership. Furthermore, I admit my grievous guilt in that I did not duly restrain Zhemchuzhina, someone close to me, from false steps and ties with anti-Soviet Jewish nationalists, such as Mikhoels.\n\nIn March 1949, Molotov was dismissed from the post of foreign minister, and Mikoyan was relieved of his duties as minister for foreign trade. These dismissals did not mean that the two men were cast out of the government. Both remained members of the Politburo and deputy chairmen of the government, and in these capacities they fulfilled important administrative functions. But their political authority was damaged, an outcome that undoubtedly was Stalin's true objective.\n\nThe use of Zhemchuzhina's origins in formulating the charges against her reflected a policy of state anti-Semitism that Stalin launched as confrontation with the West intensified. In early 1948 he ordered state security to destroy the prominent Jewish intellectual and theatrical director Solomon Mikhoels. Later that year he ordered the dissolution of the Soviet Jewish Anti-Fascist Committee, which had been founded during the war to mobilize international support for the USSR. The authorities had begun to view the committee as a nest of spies with ties to foreign intelligence agencies. Over the next few years, the Jewish Anti-Fascist Committee Affair gradually engulfed more victims, until it ended with a closed trial held from May through July 1952. All the defendants but one were shot. In 1949, the arrests of Jewish public figures were supplemented with a wide-ranging campaign against \"cosmopolitanism.\" Many Soviet Jews were arrested, fired from their jobs, and made targets of discrimination and contempt.\n\nNewly available documents confirm what most historians have long believed: such campaigns could not have been conducted without Stalin's support and involvement. This fact raises legitimate questions about the motives behind Stalin's anti-Semitism. It is tempting to assume that in the final years of Stalin's life he merely became more open about a Judophobia he had always held as a predictable aspect of his general misanthropy. The evidence, however, suggests that his postwar anti-Semitism was primarily a product of domestic and foreign policy calculations. A complex set of historical factors lay behind his turn toward anti-Semitism as a political tool.\n\nForemost among these factors was the evident growth in anti-Semitism in the USSR. In no small part because of Nazi propaganda, anti-Semitic feelings and beliefs had spread among certain segments of Soviet society. During the war, even highly placed Soviet functionaries did not hesitate to lace their reports with anti-Semitic comments. In January 1944 the deputy commander of Soviet air forces, General Grigory Vorozheikin, wrote to Stalin and other Soviet leaders about the problem of having too many members of the military working in comfortable jobs at headquarters or in commissaries. Regarding those manning the commissaries that sold items to the troops\u2014 _voentorgs_ \u2014he wrote, \"At the fronts they're called not 'voentorgs' but 'abramtorgs.'... All of these 'abramtorgs' should be sent to fight.\" Among the letters Stalin placed in his personal archive during the postwar years we find some expressing anti-Semitic feelings and others complaining about the spread of anti-Semitism. One writer, who accused Jews of shirking physical labor, offered a proposal on how to \"reeducate\" them: \"Separating Jews, as a worthy nation, into a separate republic... and making them work on a justly organized basis would be widely approved by all the other peoples of the Soviet Union.\" Stalin undoubtedly was aware of the prevalence of such feelings and took them into consideration.\n\nLike any totalitarian regime, the Stalinist dictatorship needed to keep society mobilized. This goal was achieved both by provoking anxiety about external threats and by using domestic groups as scapegoats, thereby channeling dissatisfaction away from the country's leaders. The spread of anti-Semitism shows that Jews were the most convenient target for social stigmatization. In the immediate aftermath of the war, however, Stalin was not able to exploit popular anti-Semitism. The complicated games being played in the international arena and the fact that there were advantages still to be derived from his alliance with the West forced him to be circumspect. The ideological campaigns of the first postwar years, designed to combat the rather amorphous idea of \"kowtowing to the West,\" were intended as \"ideological education\" for the intelligentsia and probably had little resonance among the general population.\n\nThe situation changed as tensions spiked with the West, as embodied by the United States with its strong Jewish community. As relations with the new Jewish state of Israel broke down and Israel became allied with the United States, Soviet Jews became more suitable targets. As Yuri Slezkine put it, \"The Jews as a Soviet nationality were now an ethnic diaspora potentially loyal to a hostile foreign state.\" The new ideological paradigm that took shape in 1948\u20131949 brought Stalin's campaign against kowtowing into line with his exploitation of anti-Semitism. The two coalesced in the campaign against \"cosmopolitans,\" appropriately understood by the Soviet masses as targeting Soviet Jews and their foreign patrons. A 1949 letter selected to be shown to Stalin captures the essence of this campaign: \"Just as the entire German people bear responsibility for Hitler's aggression, so too the Jewish people must bear responsibility for the actions of the bourgeois cosmopolitans.\" State anti-Semitism was transformed into a tool of social manipulation.\n\nStalin's personal prejudice undoubtedly played an important role in this new twist in the political line. There are many signs that during the final years of his life, he viewed Jews as a \"counterrevolutionary\" nation, much as he had viewed Poles, Germans, and the peoples of the North Caucasus before and during the war. The repression of the 1930s, the Stalinist regime's failure to protect its citizens from the Holocaust, and postwar anti-Semitism had all dampened the revolutionary fervor many Soviet Jews felt during and after the revolution. Now, Stalin assumed, Jews had turned their gaze westward, toward the United States, and were prepared to serve the West with the enthusiasm they had once shown for the revolution. \"Any Jew-nationalist is an agent of American intelligence,\" Stalin told a meeting of the party's top leadership in late 1952. \"Jew-nationalists believe that their nation was saved by the U.S.A. (there you can become rich, a bourgeois, etc.). They feel they have an obligation to the Americans.\" These suspicions were only intensified by the Jewish wives of some of his closest associates and by his own daughter's Jewish husband. Stalin's political anti-Semitism, taking deep root during his final years, became a key factor in both domestic and foreign policy.\n\n###   **MEETING WITH MAO**\n\nThe setbacks Stalin faced in Europe were partly compensated by the advance of communism in Asia. On 1 October 1949, a Communist victory in the protracted Chinese civil war resulted in the proclamation of the People's Republic of China (PRC) under the leadership of Mao Zedong. The Soviet leadership immediately established diplomatic relations with the new government and severed all ties with the defeated Kuomintang.\n\nThe Communist victory in China no doubt strengthened the Soviet Union's position in the Cold War, but it brought with it a new set of problems associated with the building of Sino-Soviet relations. Despite its dependence on the USSR, Communist China was too imposing a force to remain just another satellite. Stalin had reason to suspect that Mao might confront him with the same assertive intractability he had encountered in Yugoslavia. Considering China's size and its importance within the Third World, such recalcitrance could have much more serious consequences. A major source of friction was economic problems. The need to provide aid to a war-torn friendly power was a heavy burden for the financially strained Soviet Union.\n\nEven before the Chinese Communists had come to power, Stalin had retained personal control over contacts with them. Through Soviet military intelligence he had set up radio communication with Mao, whose army was based in northeastern China. This line of communication was maintained through special Soviet emissaries, who also served as Mao's physicians. Although Mao and Stalin kept up a continuous written correspondence, this was not enough for the Chinese revolutionary leader, who repeatedly expressed a desire to visit the Soviet Union. Probably he saw such a visit in symbolic as well as practical terms: he needed to confirm his status as the leader of the Chinese people and a partner (albeit junior) of Stalin. But Stalin kept finding ways to forestall a visit. At first he felt it inadvisable to demonstrate close ties with the Chinese Communists when they were not the country's official government. The situation in China was extremely fluid, and a Communist victory seemed far from certain.\n\nAfter several postponements by Moscow, Mao began to lose patience. On 4 July 1948 he informed Stalin that he intended to set out for Harbin and fly from there to Moscow. Ten days later he received the following response: \"In view of the commenced grain harvest work, the leading comrades will leave for the provinces in August, where they will remain until November. Therefore the party's Central Committee is asking Com. Mao Zedong to time his visit to Moscow for the end of November so as to have an opportunity to see all the leading comrades.\" Mao had no choice but to comply, but he made his annoyance plain. Stalin's excuse sounded ridiculous, and the Chinese leader did not try to pretend otherwise. The Soviet communications officer attached to Mao even felt compelled to inform Stalin of Mao's reaction:\n\nI have known Mao Zedong for more than 6 years and could tell that his smile and the words \"hao, hao\u2014good, good,\" spoken as he was listening to the translation, did not mean that he was happy with the telegram.... He was sure that he would be going immediately. Probably the trip became necessary for him. He waited for a reply with great eagerness.... Mao Zedong's suitcases were being packed, and even leather shoes were bought (like everybody here, he wears cloth slippers), and a thick wool coat was tailored.... So now he is outwardly calm, polite and attentive, courteous in a purely Chinese manner. But it is hard to see his true soul.\n\nThis visit was becoming a serious headache. From August through December 1948, as the Communists achieved a string of decisive victories, Mao continued to insist on coming. In a telegram dated 28 September 1948 he wrote, \"On a series of questions it is necessary to report personally to the Central Committee and to the _glavny khoziain_ [the boss or chief].\" In early January 1949 he again expressed his desire to come to Moscow to report to the \" _glavny khoziain_.\" Stalin stood firm. In January 1949 the Soviet side again canceled a scheduled visit. Anastas Mikoyan was sent to the Chinese instead. As Mikoyan later recalled, in discussing this matter Stalin had justified the refusal to receive Mao by saying that it would \"be interpreted in the West as a visit to Moscow to receive instructions.... This would lead to a loss of prestige for the Chinese Communist Party and would be used by the imperialists and the Chiang Kai-shek clique against the Chinese Communists.\" This explanation fit nicely with Stalin's policy of caution and demonstrative neutrality.\n\nDuring Mikoyan's visit in February 1949, the Communist march to victory entered a decisive phase. Negotiations were begun on the terms of military and economic assistance from the USSR and what to do about treaties between the Soviets and the Kuomintang. A friendship and cooperation treaty, along with associated accords, had been signed with the Chiang Kai-shek government in August 1945. These documents stemmed from agreements reached with the Allies in Yalta: in exchange for Stalin's promise to enter the war against Japan, the United States and Britain had agreed to give to the USSR lands that the Russian Empire had lost in the 1905 Russo-Japanese War. The Kuomintang government had recognized the independence of the Outer Mongolian Soviet satellite, the People's Republic of Mongolia; the Soviet Union's rights to build a military base in Port Arthur; and its long-term lease of the port of Dalny. The Chinese-Changchun Railway, which connected Port Arthur and Dalny with the USSR proper, had been brought under Soviet administration. There was lingering dissatisfaction over these forced concessions in China. With time, the Soviet presence inside the country began to look increasingly like a politically dangerous anachronism. Both Moscow and the Chinese Communist leadership understood this. Mutual concessions were expected; it was only a question of degree.\n\nAfter the Chinese Communists finally achieved victory, Stalin no longer had grounds to avoid Mao's visit. Furthermore, given the new situation, a face-to-face meeting would be extremely helpful in resolving key questions regarding the Sino-Soviet relationship. Mao left Beijing on 6 December 1949. After a ten-day trip he arrived at Moscow's Yaroslavl Station on 16 December, exactly at noon. Mao's interpreter recalled that the station clock struck twelve just as they pulled up, making the arrival all the more dramatic. A famous photograph capturing the meeting on the station platform shows the head of the honor guard in the front row with his saber drawn, Bulganin in his marshal's uniform, Molotov, and Mao. The Chinese Communist leader, tall and stout next to the slight Molotov and Bulganin, looked imposing in his large fur collar and high fur hat. Later that evening Stalin received Mao in his Kremlin office.\n\nDid the Soviet and Chinese leaders like each other? They certainly had much in common. Both were born in remote provinces to families that were poor but not destitute. Both despised their fathers and loved their mothers. Despite material deprivations, each had obtained an education, joined the revolutionary underground in his youth, and overcome his modest social origins. Each had received much of his education through independent, unguided reading and showed a penchant for abstract, philosophical topics and radical ideas. Both wrote verse and enjoyed literature idealizing rebels and brigands with forceful personalities, physical strength, and indomitable will. Neither had a talent for languages, knew a single foreign language, or even spoke his dominant language very well. Stalin's accent was strongly Georgian, Mao's Xiang (Hunanese). Both were ruthless and decisive. Mao fully shared Stalin's views on attaining sole dictatorial powers and governing and largely borrowed the Soviet leader's methods, carrying out purges, liquidating former comrades, embracing forced rapid industrialization, and presiding over a great famine. The characterization of Mao prepared for the Soviet leadership in 1949 by the doctor and radio communications specialist A. Ya. Orlov describes the Chinese leader as \"Unhurried, even slow.... He moves steadily toward any goal he sets, but not always following a straight path, often with detours.... Is a natural performer. Is able to hide his feelings and can play whatever role is needed.\" This description greatly resembled Stalin. In December 1949, when Stalin was celebrating his seventieth birthday, Mao was about to turn fifty-six. Understandably, Mao looked up to Stalin. Among the Chinese leadership, the Soviet leader was referred to as \"the old man.\"\n\nMao showed his respect for Stalin during the 16 December meeting. He made no demands and did not insist on anything, instead asking for advice and listening to it attentively. Stalin approved of this form of interaction. On hearing Mao's unwelcome but not unexpected question about the fate of the 1945 Sino-Soviet agreement, he launched into a lengthy explanation. The Soviet side wanted to \"formally\" preserve the existing agreement, he stressed, but was prepared to make certain changes that would be advantageous to China. Spelling out the political drawbacks of scrapping the agreement altogether, Stalin explained that it had been part of the Yalta agreements with the United States and Great Britain. Annulling it would \"give America and England the legal grounds to raise questions about modifying also the treaty's provisions concerning the Kurile Islands and South Sakhalin.\" It is unclear whether Mao immediately understood how spurious this argument was; he certainly grasped it later. In any event, he took an understanding tone, and the conversation moved on to pleasanter subjects. Stalin agreed to requests for aid. The talks ended on a high note. Stalin even paid Mao the compliment of proposing to collect and publish his works in Russian.\n\nDespite the atmosphere of goodwill and warmth, the meeting must have left Mao with mixed feelings. Of course the Chinese leader was given many promises and generous displays of respect. In the end, however, Stalin had refused to give him an item near the top of his wish list: an accord that would supersede the 1945 agreement. Politically, such an accord was a high priority for Mao. As subsequent events would show, he decided to bide his time.\n\nThe following days were a bustle of activity not conducive to the discussion of weighty matters. A number of foreign guests arrived for Stalin's seventieth birthday. On 21 December a grand celebration was held at the Bolshoi Theater. Mao was seated in the first row of the presidium with Stalin and was the first foreign guest to give a speech. \"When Mao Zedong stepped up to the podium,\" the Hungarian Communist Party leader Matyas Rakosi later recalled, \"an ovation erupted the likes of which the Bolshoi Theater had probably never seen. I could see that this exultation and such a reception had an effect on Mao Zedong.\"\n\nDespite this show of respect, when the fanfare subsided Mao found himself in an unenviable position. Stalin's refusal to sign a new treaty left a major purpose of his visit unfulfilled. Most historians view the events that unfolded over the rest of his stay in Moscow as a subtle war of nerves. Stalin was clearly showing Mao who was boss. Mao, in response, applied his own form of pressure. After Stalin's death he claimed to have insisted on his demands, but he was probably exaggerating. In fact, claiming illness (he was indeed in a poor physical state), he demonstratively went into seclusion, refusing to take part in various events on his schedule and announcing that he had decided to return to China a month earlier than planned. This tactic was to bear fruit.\n\nScholars have offered a variety of explanations for Stalin's change of position, but probably he had been prepared to strike a deal from the start. Skilled negotiator that he was, Stalin began the talks with a refusal because he was wary that China's strongly nationalistic new leaders might make excessive demands. This was an effective ruse. Mao apparently sensed what Stalin was up to and proved himself a worthy sparring partner. After Stalin agreed to continue negotiations, Mao began to drag his feet. Negotiations were to begin after the arrival of a group of Chinese leaders, but Mao instructed them to take their time. At first they delayed their departure from China, and then they chose a slow means of transportation to the Soviet capital\u2014train.\n\nIt was not until 22 January 1950 that talks resumed among Stalin, Mao, and Mao's associates in Stalin's office. Stalin and Mao both reaffirmed their intention of concluding new agreements and gave instructions on drafting them. After some tough negotiating, on 14 February the Treaty of Friendship, Alliance, and Mutual Assistance was signed in the Kremlin by the USSR and the PRC, along with a number of ancillary treaties. The Soviet side lost almost all of the huge advantages it had gained through the Yalta compromises and the 1945 Sino-Soviet treaty. Under the 1945 agreement, the Chinese-Changchun Railway and Port Arthur were given to the Soviet Union for thirty years, but under the 1950 agreement they were to be returned to China by the end of 1952. China was to take back property leased by the USSR in the port of Dalny almost immediately. As a result, the Soviet Union lost its ice-free port on the Pacific and material resources of significant value. Some authors have described these agreements as \"generosity unprecedented in international treaties.\" The new Chinese leaders did, however, pay a price. They renounced all claims to Outer Mongolia and also signed a secret protocol banning citizens of third-party countries from being given concessions or conducting business in Manchuria and Xinjiang, thereby allowing the USSR to retain exclusive privileges in these border zones.\n\nIt seemed at the time that the USSR, while relinquishing many tactical advantages, was gaining a critical global edge. The country with the planet's largest population now belonged to the Soviet bloc. China had become the gravitational center and a source of real assistance for the many movements throughout Asia opposing Western influence in the region. The idea that the USSR was surrounded by capitalist countries\u2014an enduring theme of Soviet propaganda\u2014had been turned on its head. One could now talk about socialist encirclement of the Western world.\n\nImmediately after signing the treaties, Stalin again showed his respect for the new Chinese leaders by attending a reception held by the Chinese embassy at the Metropol Hotel that same day, 14 February. According to Stalin's interpreter, Nikolai Fedorenko, the choice of where to hold the reception was a source of disagreement between Stalin and Mao. The Soviet leader proposed the Kremlin, but Mao preferred, as a matter of prestige, to hold it elsewhere. \"The Kremlin,\" he explained, \"is a place for state receptions by the Soviet government. Our country, a sovereign state, finds this unsuitable.\" Stalin responded that he could not attend such a reception: \"I never attend receptions at restaurants or foreign embassies. Never.\" Mao insisted. After a conspicuous pause, throughout which Mao kept his intent gaze on the Soviet leader, Stalin relented: \"Fine, Comrade Mao Zedong, I'll come if you want me to so much.\" A standard invitation in the name of the Chinese ambassador to the USSR, handwritten, arrived requesting the presence of Generalissimo Stalin and his wife (the invitation of whom may have reflected diplomatic protocol but more likely showed that the Chinese knew nothing about Stalin's personal life). The attire: dress uniforms with medals.\n\nStalin's appearance was the highlight of the reception. He was late, and as Fedorenko describes it, an aura of anticipation hung over the banquet hall as everyone whispered the same question: Would he show up? He was greeted, Fedorenko wrote, \"with loud applause and noisy exclamations of delight.\" Stalin stopped, paused, and then headed toward Mao. A round of toasts began. \"Everyone who spoke, and not only they, kept their eyes on the two figures standing side by side and occasionally engaging one another in conversation.\" After lengthy and tiresome toasts and ovations, Stalin made a gesture. Once the room settled into silence, he pronounced a toast to Mao and the success of the People's Republic of China. All drained their glasses in synchrony. \"There was another burst of applause, enthusiastic exclamations, and general rejoicing.\"\n\nOn 16 February Stalin hosted a farewell luncheon in honor of the Chinese. The following day the delegation set off for Beijing by train. The heyday of \"Sino-Soviet friendship\" had begun. With the support of the USSR, China repaired its economy and built hundreds of new factories in its most important sectors. The Korean War, which began shortly after Mao Zedong's visit, strengthened the bond between the two regimes, especially its military component. Beneath the surface, however, was the tension that had already manifested itself during Mao's visit. Proclamations of common ideological objectives and unity against a common enemy could not hide differences rooted in diverging national interests. The coming to power of the Chinese Communists was just the beginning of a complicated relationship in which both states pretended to the role of leadership of the international Communist movement. The principles Stalin established to guide his relationship with his vast neighbor to the east would work only so long as the Chinese leadership felt dependent on Soviet aid and support. Like much else that Stalin left to his heirs, these principles would not be viable for long.\n\n###   **THE THREAT OF WORLD WAR III**\n\nThe Communist victory in China coincided with another important development. In late August 1949, having devoted tremendous resources to developing a nuclear capability, the Soviet Union conducted its first test of an atom bomb. With the success of this test, the Stalinist system showed that it was ready to do whatever it took to achieve high-priority military objectives. Lavrenty Beria was put in charge of the atom bomb project, a telling choice given his reputation for ruthlessness and decisiveness. He must have known that failure at this high-priority task could have brought his career\u2014even his life\u2014to a sudden end. Later, after Stalin's death, he recalled that he left for the test site in Kazakhstan \"in a dejected mood.\" Soon, however, he was able to breathe a sigh of relief.\n\nPossession of an atom bomb, despite its tremendous significance for the Soviet Union's stature as a military power, is unlikely to have gone to Stalin's head. He probably took sober account of both the relatively limited options for using such a weapon and the real balance of power in the world. The Western powers had shown decisiveness in opposing the Soviet bloc and building up their already impressive military potential. Stalin could not rely on force alone. In the realm of foreign policy (much more than domestic policy), he exercised caution and pragmatism. Over several years the situation in Korea, the site of the first \"hot\" war between the Western and Communist blocs, offered examples of Stalin's approach.\n\nAfter the defeat of Japan in 1945, Korea was partitioned along the 38th parallel. North of the parallel, the Japanese surrendered to Soviet troops, and in the south, to the United States. As in Europe, a pro-Soviet government was established in the Soviet-occupied zone and a pro-Western one in the U.S.-occupied zone. The starting point for this process was the installation of puppet regimes by each side. The Americans put in power a seventy-year-old professor named Syngman Rhee, who had spent thirty-three years in exile in the United States, where he received his education. In the North, Moscow installed a thirty-three-year-old Red Army officer, Kim Il Sung.\n\nSeveral years after the capitulation of Japan, Korea was far from calm. Small military clashes and saber rattling were a part of everyday life. Both sides were coming to the conclusion that the only path to reuniting Korea was through war\u2014a war kept at bay only by the presence of American and Soviet troops. Fearing a direct confrontation, Stalin and the American leaders preferred to tread with caution. Stalin's approach was summed up in instructions he gave to Soviet representatives in North Korea in May 1947: \"We should not meddle too deeply in Korean affairs.\" In late 1948 Soviet troops left the country, and the United States began to withdraw its contingent the following summer.\n\nThe North Korean leaders saw the American departure as opening the door to military action, but in the fall of 1949 Stalin was still rejecting their insistent requests to sanction an armed offensive against the South. In early 1950, with the victory of Mao Zedong in China and the return home of North Korean units that had fought alongside the Chinese Communists, the situation began to change. Kim Il Sung hoped that the Chinese might offer the Korean Communists reciprocal assistance. He intensified pressure on Moscow, hinting at the possibility of a reorientation toward China. Stalin was confronted with a convoluted web of arguments for and against war that historians are still trying to sort out today.\n\nThe principles of realpolitik that often guided Stalin in the international arena called for caution. Continuing the policy of a divided Korea while strengthening the Communist North as a force to counteract the Americans seemed like the best option. Kim Il Sung's demands, or rather insistent requests, to reunify the country by force were easy for him to continue to turn down. The China factor aside, the North Korean leaders were still Stalin's puppets. Only the USSR could give the North Koreans arms and other vital resources needed for the government to survive. The Chinese themselves relied on Soviet assistance.\n\nTilting the scales in the other direction was the great-power urge for expansion, the natural tendency to fill a void and capture territory that was not clearly spoken for. Many scholars believe that Stalin may have been emboldened by a statement the Americans made in January 1950 about the sphere of the United States' national interests that included no mention of Korea. It sounded like an admission of American weakness after defeat in China. Optimistic assurances by Kim Il Sung and a wager on a pro-Communist uprising in the South's rear offered the prospect of a blitzkrieg that would confront the United States with a fait accompli and leave no time for effective intervention. Also heavily weighing the scales on this side were the pretentions of the USSR, and Stalin personally, to the role of leader of the revolutionary movement in the Third World. Finally, Stalin may have wanted to compensate for setbacks in Europe.\n\nWhatever his thinking was, in early 1950 Stalin decided in favor of action and signaled Kim Il Sung that he could begin preparing an invasion. In April Kim came to Moscow to meet with Stalin and discuss the details. Together they outlined a plan and timeline for the war, and with the help of the USSR, the North Koreans began urgent preparations. By the time combat began, they had acquired a huge advantage over the South. On 25 June 1950, Kim Il Sung's troops began their offensive. Like many other attempts at blitzkrieg, this one met with defeat. The rapid response by the United States, which Stalin had worried about but chosen to discount, dramatically changed the situation. The American leadership saw the aggression in Korea as the start of a broad Soviet offensive that would ultimately include Europe. Having decided to intervene, the Americans quickly outmaneuvered the Soviet bloc diplomatically. A session of the UN Security Council, convened the very day military operations began, condemned the North as the aggressor (Yugoslavia abstained and the Soviet ambassador was absent). Soon afterward, American troops landed in South Korea and were quickly joined by forces from fifteen other states, a fact that was of greater political than military significance.\n\nDespite some initial successes by the North, this start to the war dampened Kim Il Sung's confidence. Stalin demanded that the war go on and encouraged the North Koreans with advice and new deliveries of military hardware. \"In our opinion the attack absolutely must continue and the sooner South Korea is liberated the less chance there is for intervention,\" Stalin wrote the Soviet ambassador in Pyongyang on 1 July 1950. But the wager on a victorious conclusion to the war before serious American forces could reach the peninsula failed. After capturing almost all of South Korea by September, the North Koreans were not able to fully expel its government. The Americans launched a powerful counterstrike. Under the UN flag, coalition forces advanced rapidly and by the end of October had captured most of North Korea and taken Pyongyang. The time had come for the Soviet side to play its final card: the Chinese \"volunteers.\"\n\nNow began the confusing and still little-studied negotiations between Stalin and the Chinese leadership. At one point it appeared they had ended in failure. On 13 October Stalin sent the following directive to Kim Il Sung: \"We feel that continuing resistance is pointless. The Chinese comrades are refusing to take part militarily. Under these circumstances you must prepare to evacuate completely to China and\/or the USSR. It is of the utmost importance to withdraw all troops and military hardware. Draw up a detailed plan of action and follow it rigorously. The potential for fighting the enemy in the future must be preserved.\"\n\nThe Soviet ambassador urgently met with the North Korean leaders and read them Stalin's telegram. As the ambassador reported, \"Kim Il Sung stated that it was very hard for them [to accept Stalin's recommendation], but since there is such advice they will fulfill it.\" How serious was Stalin's directive? Was he truly prepared to lose North Korea? Apparently he was. If the Chinese refused to send troops, Stalin had no other option since he categorically rejected the idea of bringing in Soviet troops. It is also possible, however, that Stalin believed the decision to evacuate forces might lead the Chinese to think twice. The American advance was more threatening to China than to the USSR. Furthermore, having announced his intention to withdraw, Stalin continued to try to engage the Chinese. He made concessions on the question of arms deliveries and offered more specific promises to deploy Soviet air cover. These efforts bore fruit. Mao agreed to enter the war. \"The old man writes to us that we must step up,\" is how he described Stalin's demands to his comrades.\n\nBattered by the Chinese, the South Koreans and their allies withdrew from North Korea. In early 1951 they lost Seoul for the second time. Then came a counterstrike from the South. It was beginning to look like neither side could achieve a decisive victory. The Soviet Union tried to stay in the shadows, although Stalin did keep his promise to provide covert air support for Kim Il Sung's and Mao Zedong's forces. The main victims of this great-power standoff were the Korean people. Millions of lives were lost, and the Koreans were forced to live as a divided nation. Those in the North endured one of history's most brutal dictatorships, a regime that largely followed the Stalinist model.\n\nThe Korean War heightened international tensions and spurred the arms race. While the development of military industries had always been an unquestioned priority for the Soviet leadership, during the final years of Stalin's life the buildup moved to a new level. In January 1951 a meeting was held between the Soviet leadership and top officials from the Eastern bloc. Archival documents relating to this meeting remain classified. The only reason historians know it even took place is that it is mentioned in various memoirs. The most detailed description of what happened there is given in the memoirs of Hungarian Communist Party leader Matyas Rakosi. According to his account, the Soviet side was represented by Stalin and several members of the Politburo and military. The East European countries sent their first party secretaries and defense ministers (only the Polish party secretary was absent). Sergei Shtemenko, chief of the General Staff of the armed forces of the USSR, gave a speech about the growing threat from NATO and the need to counterbalance it with a military buildup by the socialist countries. The Soviet leadership assigned the satellite countries the task of greatly increasing the size of their armies within three years and creating a military-industrial foundation to support this enhanced military might. Shtemenko provided specific numerical targets.\n\nRakosi states that Shtemenko's numbers provoked debate. He quotes the Polish defense minister, Konstantin Rokossovsky, as saying that the army the Poles were being asked to assemble by 1953 was already being planned but would not be attainable until 1956. Other representatives also questioned their countries' abilities to manage such a rapid buildup. The Soviets, however, were adamant. Stalin answered Rokossovsky that the timetable set forth by the Poles could remain in place only if Rokossovsky could guarantee no new wars before 1956. Absent such a guarantee, it was better to adopt Shtemenko's proposal.\n\nWe do not know what plans were on the drawing board for the Soviet military or to what extent they were realized. There is nevertheless sufficient evidence to conclude that Stalin was aiming for a serious military buildup. According to official figures, the army, which had been reduced to 2.9 million soldiers by 1949, had reached 5.8 million by 1953. Investment in the military and naval ministries, as well as production of military arms and hardware, grew by 60 percent in 1951 and 40 percent in 1952. As a comparison, government investment in the non-military sectors of the Soviet economy grew by 6 percent in 1951 and 7 percent in 1952.\n\nDevelopment of nuclear weaponry and delivery systems remained the highest-priority and most expensive military program. In addition to the nuclear project, significant resources were dedicated to rocket technology, jet-propelled aviation, and an air defense system for Moscow. During the final months of his life, Stalin showed his determination to outpace his rivals in the arms race. In February 1953 he approved major programs in aviation and naval ship construction. The first provided for the creation of 106 bomber divisions by the end of 1955, up from 32 as of 1953. In order to outfit new divisions, the plan was to build 10,300 planes during 1953\u20131955 and increase the air and naval forces by 290,000 people. The second program allocated huge resources to the construction of heavy and medium cruisers before 1959. Soviet military bases were established in the Far Eastern regions of Kamchatka and Chukotka, close to the maritime boundary with the United States.\n\nDid this buildup mean that Stalin was planning to launch a preemptive strike and unleash a new world war? There is no evidence to support this line of speculation. It is important to note that the massive arms buildup programs were planned to take place over several years. Historians of Soviet foreign policy also note Stalin's caution and pragmatism in the international arena. During the postwar years he behaved toward the West approximately as he had toward Nazi Germany before the war. He preferred behind-the-scenes maneuvering over direct confrontation. This approach had been on display in the Korean War. While encouraging its continuation, Stalin had consistently avoided direct conflict with the Americans. He had intentionally dragged out the signing of an armistice, seeing the war as a way to let others get their hands dirty weakening the United States. In a private conversation with the Chinese leader Zhou Enlai a few months before his death, Stalin frankly and cynically explained: \"This war is causing the Americans a lot of headaches. The North Koreans have lost nothing, except for the casualties they took during this war.... You have to have self-control, patience. Of course you have to understand the Koreans\u2014they've taken a lot of casualties, but you have to explain to them that this is something big. You have to have patience, you have to have great self-control.\"\n\nIt took Stalin's death to free the Koreans from the obligation of taking casualties to further another country's interests. His heirs pursued a policy of relaxing international tensions and reducing the burden of the arms race. By July 1953 a decision was made to conclude a truce in Korea. Stalin's death brought an end to the USSR's ruinous military buildup, including the creation of armadas of bombers. The country could not endure the strains of the arms race and demanded the reforms that Stalin had refused to give it.\n\n###   **THE INVETERATE CONSERVATIVE**\n\nMilitary spending was not the only reason for a ballooning government budget during Stalin's final years. There is copious evidence of the _vozhd_ 's passion for large-scale, expensive projects toward the end of his life. These projects were often cast by official propaganda as \"the Stalinist building of communism.\" They included huge hydroelectric power plants, canals, and rail lines into the nation's inaccessible polar reaches. To strengthen communication with newly acquired Far Eastern territories, a ferry crossing and a 13.6-kilometer underwater tunnel to the island of Sakhalin were planned, along with a rail line connecting the tunnel with the country's train network. As was usually the case with Stalinism, behind the appealing propagandistic fa\u00e7ade lurked an unsavory reality: communism was largely being built on the backs of prisoners.\n\nExorbitant spending on infrastructure once again plunged the Soviet economy into financial crisis. The chaotic proliferation of projects led to losses on uncompleted construction, which later had to be finished at far greater cost than initially projected. In 1951 and 1952 this extravagance reached its limit. Construction projects fell behind schedule and the launch of new ones was delayed. The picture was completed by stagnation in agriculture and consumer spending\u2014the sectors that funded heavy industry. Undaunted, Stalin devised a plan for a new surge of capital investment in 1953. At the end of his life he stubbornly repeated the mistakes of the First Five-Year Plan's forced industrialization.\n\nAs far as can be determined from available documents, this unfolding crisis was not seriously discussed at the upper echelons of power. Until the very end Stalin demanded the expansion of heavy industry and military buildup at any cost. As in the past, he agreed to limited concessions and policy adjustments only when problems grew so severe that his hand was forced. Clearly unwilling to acknowledge the systemic crisis, he reluctantly addressed only its most obvious manifestations.\n\nAs often happened, the first signs of approaching calamity came from the most disadvantaged sector of the Soviet economy: agriculture. The Soviet countryside bore the brunt of unbalanced economic policies and of the new obligations and taxes that supported growing government expenditures. Under the inefficient kolkhoz system, agriculture was stagnant and incapable of feeding the country. The livestock situation was particularly bad. Even official Soviet statistics showed that there were no more head of cattle in the country in early 1953 than there had been in 1939, and that number was one-third less what it had been in 1928. The number of pigs in 1953 was the same as in 1928. The numerous complaints sent to Moscow from across the country painted a desperate picture. Some of these cries for help reached Stalin.\n\nAmong the letters received in October and November 1952 and selected to be shown to Stalin were a few complaints from various parts of the USSR about the hardships suffered on collective farms. A veterinarian from the Orekhovo-Zuevo District of Moscow Oblast, N. I. Kholodov, called for incentives for work by kolkhozniks, who were essentially forced to labor for no pay. Kholodov wrote:\n\nAccording to our press, we have tremendous achievements in agriculture.... Let us take a look at how matters stand in reality. The rye was poorly harvested, poorly because there is colossal waste in the harvesting process.... The potatoes have been harvested somehow, but what kind of a harvest is this? They were dug up by workers mobilized from plants and factories who were drawing 50% of their salaries for this period, and they do not try to gather all the potatoes because they do not have an interest in this; they try to finish up as quickly as possible and gather only what is on top....\n\nNow let us look at animal husbandry. Even talking about it is embarrassing: annual yield of milk from year to year does not exceed 1,200\u20131,400 liters per forage-fed cow. This is ridiculous\u2014it's what you get from your average goat.\n\nAlongside these tales of dysfunction in the countryside, Stalin's mail in late 1952 contained eloquent accounts of empty store shelves in cities. In early November the _vozhd_ took notice of a letter from V. F. Deikina, the party secretary for a railway station in Riazan Oblast. She wrote:\n\nIt is now October, and here we have to wait in line for black bread, and sometimes you can't get any at all, and workers are saying so many unpleasant words and they don't believe what's written [in newspapers] and say that we're being deceived.... I'll stick to the facts since there's not enough paper to describe it all and send it in a letter.\n\n1. You have to stand in line for black bread.\n\n2. You can't get white bread at all.\n\n3. There's neither butter nor vegetable oil.\n\n4. There's no meat in the stores.\n\n5. There's no sausage.\n\n6. There are no groats of any kind.\n\n7. There's no macaroni or other flour products.\n\n8. There's no sugar.\n\n9. There are no potatoes in the stores.\n\n10. There is no milk or other dairy products.\n\n11. There is no form of animal fat (lard, etc.)....\n\nI'm not a slanderer and I'm not being spiteful; I'm writing the bitter truth, but that's the way it is.... The local leadership gets everything illegally, under the table, so to speak; their underlings deliver everything to their apartments. For them the people can do as they please; that's not their concern.... I am asking for a commission to be sent to bring the guilty to justice, to teach the right people how to plan for needs. Otherwise, those with full bellies don't believe the hungry.\n\nDespite its critical tone, this letter was entirely politically \"correct.\" Deikina was trying to combat the deficiencies and abuses of local officials who did not know how to properly \"plan for needs.\" The letter did not delve into the causes of the lack of food in the country. This was the sort of letter Stalin could like. Averky Aristov, recently appointed as the Central Committee secretary in charge of local party organizations, was sent to investigate. On 17 November 1952 Stalin held a meeting of Central Committee secretaries in his office. As Aristov recounted several years later, Stalin asked him to deliver his findings. Aristov reported that for a long time there had been shortages of bread, cooking oil, and other food items in Riazan Oblast. Stalin grew furious and ordered that the oblast party secretary be removed from his post. Aristov and others present tried to intercede on behalf of the officials from Riazan. Things were no different, they explained, in many other regions, including Ukraine, the country's \"bread basket.\"\n\nFollowing the meeting, Riazan Oblast was allocated food from government supplies. Such measures, of course, did not solve the problem. The country's leadership was again faced with the task of salvaging the agricultural sector. Under the pressure of circumstances, Stalin agreed to review proposals to raise the price paid by the state for livestock produced by kolkhozes. At stake was the fundamental question of whether peasants deserved to be compensated for their labor. The exceptionally low \"purchase price\" paid to kolkhozniks barely masked the fact that everything they produced for the state was basically being confiscated. Growing food was tremendously unprofitable, and those who grew it had no incentive to produce more.\n\nIn December 1952 a commission headed by Nikita Khrushchev was established to draft a resolution raising livestock purchase prices. After working for several weeks, the commission wound up provoking Stalin's displeasure. The _vozhd_ was highly suspicious of attempts to change the existing system for pumping resources out of the countryside. To the dismay of his comrades, who had agreed on an increase in livestock prices, Stalin proposed significantly increasing taxes on the peasantry. Anastas Mikoyan later recalled Stalin's reasoning: \"What is a peasant? He'll turn over his extra hen and that's an end to it.\" Khrushchev and his politically seasoned colleagues on the commission chose the safest course of action. They bided their time. The Soviet leaders would shield themselves from Stalin's anger while they waited for his death. When it finally came, the overdue agricultural reforms were put in place immediately and on a larger scale than initially planned. Stalin's heirs raised procurement prices and lowered taxes on peasants. Although the deep-rooted flaws in the kolkhoz system were preserved, these measures had a positive effect. For the first time in many decades the peasants were given relief, and some improvement in agricultural production was achieved.\n\nReducing the financial burden on the countryside inevitably came with a reduction in extravagant spending on major infrastructure projects. Just a few days after Stalin's death, on 10 March 1953, the chairman of Gosplan presented the new head of the Soviet government, Georgy Malenkov, a report on major construction projects that were \"behind schedule for completion.\" The report stated that it was being presented at Malenkov's request. Members of the top leadership were apparently losing no time in implementing the changes they had been constrained from making while the _vozhd_ was alive. They quickly halted many of Stalin's ambitious projects, including the construction of canals, hydroelectrical systems, and rail lines through difficult terrain. Investment in the military was also reduced. The funds thus freed up could now be put toward dealing with the severe crises in agriculture and social welfare. The Stalinist industrialization system, enabled by the population's low living standard and by the exploitation of the countryside as if it were an internal colony, could now be gradually dismantled.\n\nThese decisions were adopted and realized with unprecedented speed in the months following Stalin's death. The new leaders' decisiveness clearly shows that it was specifically Stalin who was the main obstacle to transformation for long years. Until the very end, the dictator's personal political and economic modus operandi remained extraordinarily conservative and protective. His death opened the door to innovations that were long overdue.\n\n###   **THE DEATH THROES OF THE DICTATORSHIP**\n\nAt the end of his life, Stalin was at the pinnacle of his power. His authority was unassailable and not under threat from any source. But he did not feel that way. Like other dictators, he never stopped fighting for power and never quite trusted his subjects. The methods he used in his never-ending battle for power were universal and simple. They included the elimination of any potential threat from within his inner circle, unrelenting oversight of the secret police, the encouragement of competition and mutual control among the various components of government, and the mobilization of society against perceived enemies both internal and external.\n\nAfter destroying the Leningraders, Stalin began adjusting the balance of power within the Politburo, creating counterweights to the growing influence of Malenkov and Beria. In 1949 he brought Ukrainian party chief Khrushchev to Moscow and made him a Central Committee secretary and head of Moscow's party organization. Soon afterward he began to actively promote Bulganin, who had faithfully served him as defense minister. In April 1950, on Stalin's suggestion, Bulganin was appointed first deputy chairman of the Council of Ministers. For a while this promotion gave Bulganin privileged access to the _vozhd_. Soon, however, Stalin became disenchanted with his prot\u00e9g\u00e9 and stripped him of his authority. This happened without particular acrimony. Bulganin remained a member of the top leadership. A period of relative equilibrium among key Politburo members set in, but it was just the calm before the storm.\n\nAn important factor in Stalin's last battle for power was his declining health. Lightening his workload by relinquishing certain duties or gradually handing over power to subordinates was out of the question. Instead, the weakening _vozhd_ consolidated his dictatorship with enviable energy, compensating for reduced vigor with combativeness. Fierce blows were leveled against the most vulnerable points in the hierarchy of power. The first involved yet another wave of arrests at the Ministry of State Security, over which Stalin never ceased to keep tight control. In July 1951, based on the usual assortment of trumped-up charges and incriminating denunciations, Stalin ordered the arrest of state security minister Viktor Abakumov, who quite recently had been a favorite. The party functionary Semen Ignatiev was appointed in his place. Abakumov's arrest predictably opened the door to a large-scale purge of the ministry.\n\nHaving terrified the chekists, Stalin left for a vacation of more than four months. While in the south, he continued to keep a close eye on state security. The inventory of materials sent to Stalin between 11 August and 21 December 1951 includes more than 160 Ministry of State Security memoranda and reports. He also received an indeterminable number of coded telegrams from the ministry, as well as Politburo and Council of Ministers resolutions having to do with state security. In October Stalin summoned Ignatiev to the south and ordered him to \"kick all the Jews out\" of the ministry. When Ignatiev naively asked, \"Where to?\" Stalin explained to the inexperienced minister: \"I'm not saying you should throw them out onto the street. Lock them up and let them stay in prison.\" Ignatiev turned out to be a quick learner. Mortally terrified, he obediently launched a series of arrests and fabricated cases having to do with a \"Zionist plot\" within his ministry. For Stalin, extending his campaign of state anti-Semitism to state security was a perfectly logical step. Jews, members of a suspect nation and potential henchmen of world imperialism, could not be allowed to work in the regime's most sacred realm. The next targets were just as logical. Immediately after state security, Stalin initiated purges against highly placed functionaries in several branches of the party-state apparat.\n\nThe next round of repression was also orchestrated from his dacha in the south. In September 1951 he received a visit from Georgia's minister for state security, Nikolai Rukhadze. As Rukhadze testified under interrogation after his arrest, Stalin made some general comments at the dinner table about the dominance of Mingrelians (Megrels) in Georgia; he noted that Beria was a Mingrelian and was giving patronage to this group. This comment was the first hint at the target of the next campaign: Georgian officials and their patron. Soon after Rukhadze's visit, the head of Stalin's security team, Nikolai Vlasik, reported to the _vozhd_ that people were complaining about having to pay bribes to enter Georgian colleges and universities. That this information fit perfectly with Stalin's new focus is hardly surprising. Vlasik, who had spent a good portion of his life by Stalin's side, had developed a keen sense of his moods and a talent for telling him what he wanted to hear. He could tell that Stalin was thirsting for blood and sought out the compromising materials that would help satisfy his boss's craving. Rukhadze was assigned to look into Vlasik's allegations.\n\nOn 29 October 1951, Rukhadze reported to Stalin that the bribery charges mostly could not be confirmed. This made no difference. Stalin had decided on a purge in Georgia, and it was only a matter of time before he invented a pretext for it. On 3 November he telephoned Rukhadze and asked him for information about patronage by Georgia's second party secretary, Mikhail Baramiia, the former procurator of the city of Sukhumi, who had been accused of taking bribes. Rukhadze did as he was told, preparing a document suggesting that Baramiia had protected Mingrelian officials guilty of crimes. The case was handled expeditiously. With Stalin's active involvement, sweeping repression was unleashed in Georgia. Many of the republic's leaders, including Baramiia, were arrested. More than eleven thousand people were deported to remote areas of the Soviet Union.\n\nThe Mingrelian and Leningrad Affairs largely followed the same template. Both started with accusations of abuse of power and political protectionism _(shefstvo),_ quickly followed by the arrest and torture of disgraced officials, leading to fabricated evidence of \"anti-Soviet\" and \"espionage\" organizations. As in Leningrad, here too Stalin targeted a specific clan of Soviet officials with ties to influential members of the country's leadership\u2014in this case Beria. Whether to make a mockery of him or simply teach him a lesson in humility, Stalin assigned Beria to hold a plenum of Georgia's Central Committee in 1952, at which he was forced to expose his former clients and feign shock and anger at their behavior. Undoubtedly Beria saw the purge in Georgia as a personal threat. Immediately after Stalin's death he managed to put a stop to the Mingrelian Affair and had its targets freed and returned to senior positions.\n\nBeria weathered the storm. Like many before him, however, he emerged with a renewed sense of the fragility of his political and physical existence. Stalin apparently had his sights on more important targets. The first shot was fired after the Nineteenth Party Congress, which convened in October 1952 after a thirteen-year break. Instead of giving the keynote speech, Stalin limited his appearance at the congress to a brief closing statement. It was as if he was saving his diminishing strength for the main event: the plenum of the newly elected Central Committee, which immediately followed the congress. The plenum would determine the makeup of the party's top governing bodies, most important the Politburo. The election was expected to be a mere formality. Members of the Central Committee usually voted for the candidates proposed from on high without wasting their breath on discussion. But in this case Stalin caught everyone by surprise and introduced some surprising changes.\n\nHis main innovation was the abolition of the Politburo and the creation of two new bodies. The first, which formally replaced the Politburo, was called the Presidium of the Central Committee of the Communist Party of the Soviet Union. Whereas the Politburo had included nine members with full voting rights and two candidate members, the new Presidium was much larger, comprising twenty-five full members and eleven candidate members. The expansion would add younger and relatively unknown party leaders, giving Stalin an even freer hand in regard to his older comrades. The political essence of the reorganization was summed up, probably correctly, by Anastas Mikoyan: \"Since the makeup of the Presidium was so broad, if needed, the disappearance of Presidium members out of favor with Stalin would not be so noticeable. If between congresses five or six people disappeared out of twenty-five, that would look like an insignificant change. If, on the other hand, five or six people out of nine Politburo members disappeared, that would be more noticeable.\"\n\nThis was exactly the sort of apprehension Stalin needed to keep the will of the old guard and potential heirs in check. Not satisfied with the threat implicit in the expanded Central Committee Presidium, Stalin continued his psychological warfare. His next proposal\u2014the creation of a nine-member bureau to serve as the Presidium leadership\u2014was just as unexpected. In principle, the Presidium Bureau made sense. The unwieldy Presidium would hardly be capable of efficient decision making. But Stalin, as he had often done, could of course create a narrow leadership group without formal approval by the Central Committee plenum. The true purpose of this toying with democracy became immediately clear once he disclosed his proposed candidates for the bureau. It turned out that he did not feel it was possible for him to nominate two of his oldest associates\u2014Molotov and Mikoyan\u2014for membership. To add insult to injury, he topped off this announcement by giving the two a public tongue-lashing.\n\nThese two men\u2014Molotov in particular\u2014were seen within the party and among the people as the _vozhd_ 's natural heirs. This perception is specifically why Stalin chose to publicly discredit them by making it known that he did not consider them worthy leaders of the party and the country. Just what charges he brought against Molotov and Mikoyan we do not know, as there is no verbatim transcript of the plenum. Judging by the contradictory recollections of those who took part, Stalin concocted an amalgam of political smears, bending facts and quasi-facts to his purpose. He brought up Molotov's supposed concessions to foreign correspondents and his mistakes at the 1945 foreign ministers' conference and claimed that Molotov, with Mikoyan's support, had proposed raising the procurement prices for grain in order to incentivize work by the peasants. These misdeeds were painted with the brush of \"rightist opportunism.\" Stalin may even have mentioned Molotov's wife and his pro-Jewish sympathies. In the end, the content of the criticism mattered little. The main point was obvious: nobody was worthy of succeeding Stalin. The only hope was that he would live on for many years. Molotov and Mikoyan came to the podium to express their devotion to Stalin. This too only underscored his greatness. Stalin's manner signaled to the gathering that Molotov's and Mikoyan's justifications were not worth listening to. Before Mikoyan could finish what he was saying, according to one eyewitness, Stalin gave a dismissive wave of the hand. \"The hall immediately began to react very emotionally, and people started to yell: 'Enough of your self-justification!'... 'Stop trying to fool the Central Committee!' Mikoyan wanted to say something else, but the hall interrupted him and he sat back down.\" This demonstration of devotion to the _vozhd_ and disdain for apostates brought the plenum to a fitting end.\n\nDespite being anathematized, Molotov and Mikoyan formally held onto most of their official powers\u2014and, most important, their lives\u2014but neither they nor any other member of Stalin's inner circle could feel truly safe. There was also alarming news coming from the country's socialist neighbors. In November 1952, shortly after the conclusion of the Nineteenth Party Congress, the Czechoslovak party leader Rudolf Slansky was put on trial along with other senior party officials. The defendants were found guilty and executed. Recent research has shown that Stalin exercised close personal control over the Slansky trial. Slansky was a Jew, and his trial served as a prelude to Stalin's next act of intimidation: the Doctors' Plot.\n\nThe affair that has come to be known as the Doctors' Plot, to which Stalin devoted a significant portion of his final months, unfolded within a general campaign of state anti-Semitism. The foundation of the case was information \"dug up\" by state security about murderous doctors, mostly Jewish, working in government health care facilities serving the Soviet leadership. Accusations against \"wrecker doctors\" who supposedly killed or plotted to kill Soviet leaders was a leitmotif of the political trials of the 1930s. Toward the end of his life Stalin returned to this theme, possibly because of anxiety about his own mortality or perhaps because he saw in the fabrication of a case against Kremlin doctors a way of putting pressure on their patients. Over many months, Stalin obsessively presided over the fabrication of evidence against Jewish doctors and their supposed patrons within the Ministry of State Security. His eagerness to lash out at this group led him to spew foul threats at Ignatiev, calling state security agents obese \"hippopotamuses\" and promising to drive them \"like sheep\" and \"give it to them in the mug.\"\n\nDuring October and November 1952, when the curtain had closed on the first act of the drama taking place at the upper reaches of government, Stalin approved the arrest of a number of doctors, including Petr Yegorov, head of the body that oversaw Kremlin health services; Vladimir Vinogradov, Stalin's personal physician; and two professors, Miron Vovsi and Vladimir Vasilenko. Stalin met with the heads of state security and instructed them to use torture on the arrestees. On 15 November 1952, Ignatiev reported to him that these instructions had been carried out: \"Means of physical coercion were used on Yegorov, Vinogradov, and Vasilenko and interrogation was intensified, especially in regard to foreign intelligence.... Two workers capable of carrying out special assignments (using physical punishment) in regard to particularly important and particularly dangerous criminals were selected and already used in this case.\"\n\nStalin soon put the \"confessions\" extracted through these brutal techniques to use. On 1 December 1952, during a meeting of the Central Committee Presidium, questions tied to \"wrecking within the field of medicine\" and \"information on the state of the USSR Ministry of State Security\" were placed before the gathering. In keeping with his initial idea of collusion between \"wrecker doctors\" and state security \"conspirators,\" the main targets of Stalin's attack were \"Jewish nationalists\" and chekists. At a subsequent Central Committee Presidium meeting on 4 December, a resolution titled \"On the Situation in the Ministry of State Security\" was adopted, calling for \"active offensive actions\" in intelligence work and intensified party control over the ministry. It defended the use of extreme methods in the fight against \"enemies\" with the idea that \"Many chekists hide behind... rotten and harmful reasoning that the use of diversion and terror against class enemies is supposedly incompatible with Marxism-Leninism. These good-for-nothing chekists have descended from positions of revolutionary Marxism-Leninism to positions of bourgeois liberalism and pacifism.\" Stalin summed up this position more succinctly in a closed-door meeting: \"Communists who take a dim view of intelligence and the work of the cheka, who are afraid of getting their hands dirty, should be thrown down a well head first.\"\n\nAt some point Stalin decided that the Doctors' Plot should be turned into a major campaign. In early January and with his active involvement, two press items were prepared: a TASS report about the arrest of a group of \"wrecker doctors\" and a lead article for _Pravda_ on the same subject. The public was told of the discovery of \"a terrorist group made up of doctors whose goal, using wrecking treatments, was to shorten the lives of the Soviet Union's prominent figures.\" These alleged crimes were being committed on orders from an international Jewish bourgeois-nationalist organization and U.S. and British intelligence services. The Soviet people were urged to exercise vigilance toward enemies receiving support from the imperialist world.\n\nThe publication of these items, on 13 January 1953, launched a large-scale ideological campaign designed to inflame anti-Semitism and bring \"vigilance\" to a fevered pitch. There were widespread rumors of possible pogroms and the internal resettlement of Soviet Jews. In the decades that followed, these rumors evolved into assumptions that Stalin might have been planning show trials against the doctors and the removal of Jews from the European USSR to the Far East, as had been done to Caucasian peoples during the war. Recently opened archives, despite thorough searches, have revealed no direct or indirect evidence to support either assumption. Given that either show trials or a roundup of an entire ethnic group would have required tremendous logistical effort, the absence of any trace of evidence is persuasive.\n\nAnd even the maniacal Stalin, who by now was truly ill, saw no need for a resettlement program or large-scale arrests. The Doctors' Plot campaign was entirely sufficient to his purpose. Remaining within the realm of ideas rather than actions, it manipulated the public mood and fostered a psychology of war-readiness in the absence of any looming war, thereby distracting people from their daily hardships. The arrests of prominent doctors also forced Stalin's comrades to live in a state of anxiety as they tried to guess what testimony would be beaten out of their physicians in the bowels of the Lubyanka. Like other similar acts of demonstrative violence, the Doctors' Plot had a foreign-policy aspect. Some historians believe that Stalin viewed this new campaign of anti-Semitism as a means of putting pressure on his Western opponents, the United States in particular. He was using the implicit threat of anti-Semitic pogroms to extract concessions from Western leaders, who knew no other way to influence him.\n\nHistorians can debate whether calculation or mania played the greater role in Stalin's final campaigns. In either case, his actions attest to a relentless striving to hold onto power until he reached the ultimate impediment: death. The final leg of the journey toward this impediment began on Saturday evening, 28 February 1953, when he invited his four currently closest comrades\u2014Malenkov, Beria, Khrushchev, and Bulganin\u2014to his dacha for the last dinner gathering of his life. The following day his bodyguards found him paralyzed, and the agonizing over whether or not to summon members of the highly suspect medical profession began.\n\n## **THE DICTATORSHIP COLLAPSES**\n\n**A conference in the Kremlin, 2\u20135 March 1953, and the death of Stalin.**\n\nThe arrival of the doctors on the morning of 2 March 1953 fundamentally changed the situation. The very fact that they had been summoned to Stalin's dacha meant that the seriousness of his condition was officially recognized. The doctors confirmed the worst: a stroke had brought the _vozhd_ to death's door. For the first time in many decades, and completely unexpectedly, the USSR was faced with a transfer of power at the highest level.\n\nLike Lenin, Stalin had not anointed a successor or created a legal mechanism for the orderly transfer of power. Instead he did everything he could to hinder the emergence of a successor and to instill a sense of political unworthiness in his associates. By concentrating high-level decision making in his own hands, he ensured that the other members of the Politburo were poorly informed and had little authority even over those areas for which they were immediately responsible. Driven by a thirst for power, political self-centeredness, and senile emotional instability, the Soviet dictator seemed to display an \"Apr\u00e8s moi le d\u00e9luge\" attitude toward the post-Stalinist future.\n\nThus one can only marvel at the ease with which Stalin's heirs got through the critical period of the interregnum. There were a number of reasons why they could do so. One was that even during Stalin's lifetime his comrades had developed a certain independence and the ability to work with one another. Each oversaw a particular component of the party-state apparat. It was not unusual for them to meet without Stalin to work on specific practical matters of government. One set of administrative entities that met quite regularly were the various executive and administrative bodies that came under the Council of Ministers. Officially, Stalin headed these bodies, but he never took part in their day-to-day work. Furthermore, during his lengthy southern vacations the Politburo grew accustomed to deliberating without him. Also, the members of the leadership were united by their common terror of the dictator. Although there was competition to get closer to him, Stalin's comrades were careful not to provoke his fury, and they worked to maintain equilibrium within the leadership group. The Leningrad Affair had shown that no one was safe. There was an elaborate interplay among the instinct for self-preservation, institutional interests, and the need to fend off threats against the system. Dealing as they did with the day-to-day challenge of keeping the country afloat, Stalin's colleagues were keenly aware of the urgent need for change to which he seemed willfully blind. This awareness led to an informal effort to conceive solutions, whose realization was blocked only by Stalin. Gradually and inexorably, under the shadow of dictatorship, the oligarchic system took embryonic form. It was only a matter of days from the first news of Stalin's fatal illness that the oligarchy emerged as a force.\n\nAt 10:40 on the morning of 2 March, an official meeting of the Central Committee Presidium Bureau was convened. It was the first time in many years that a meeting took place in Stalin's Kremlin office without him. In addition to all the members of the Bureau (except for Stalin), the attendees were Molotov, Mikoyan, Nikolai Shvernik (the chairman of the Supreme Soviet), Matvei Shkiriatov (chairman of the Party Control Commission), I. I. Kuperin (head of the Kremlin's health administration), and the neuropathologist R. A. Tkachev. For twenty minutes the group considered one matter: \"The finding of the council of physicians concerning the cerebral hemorrhage of Comrade I. V. Stalin that took place on 2 March and the resulting severe state of his health.\" The Bureau approved the doctors' diagnosis and established a schedule for members of the leadership to keep watch by the _vozhd_ 's bedside. The presence of Molotov and Mikoyan, despite their being out of favor with Stalin and formally expelled from the Bureau, is of central importance. Their inclusion was an act of defiance against the _vozhd_ and an effort to restore the old collective leadership, as well as a natural and sensible step aimed at maintaining unity in a time of crisis. The Soviet leaders, certain that Stalin would not recover, were undertaking to change the system of supreme power that he had established.\n\nAt 8:25 that evening, the same assemblage of newly fledged oligarchs again convened in Stalin's office to consider an official medical update: \"On the state of health of Comrade I. V. Stalin as of the evening of 2 March.\" With every passing hour it became clearer: Stalin had not long to live. The doctor Aleksandr Miasnikov later recalled: \"On the morning of the third the council of physicians had to submit an answer to Malenkov's question about the prognosis. The only answer we could give was a negative one: death was inevitable. Malenkov gave us to understand that he expected such a finding, but then stated that he hoped that medical measures could extend his life for a sufficient time, even if they could not save it. We understood that he was referring to the need to allow time to organize a new government and, at the same time, prepare public opinion.\"\n\nRecords indicate that on the morning of 3 March the Soviet leaders were already assuming that Stalin would not recover and planning accordingly. At noon another meeting was held, this time without any doctors, at which a resolution was adopted to report Stalin's illness in the press and to convene a Central Committee plenum. The decision to convene a plenum signaled preparations to transfer power, even while the exact configuration of the new leadership remained an open question. Malenkov and Beria took upon themselves the task of formulating specific proposals. They had plenty of time to do so. The members of the Presidium kept vigil at Stalin's dacha, two at a time. Malenkov and Beria were teamed for this duty, as were Khrushchev and Bulganin. The shifts lasted many hours, and there was time for far-ranging discussion.\n\nThe fourth of March marked a turning point. That day's newspapers contained the first official announcement of Stalin's illness. With no hope for a recovery, the only option was to accustom the country and world to the news. The same day, Beria and Malenkov prepared proposals for reorganizing the upper echelons of power that were later discussed by the leadership group, including Molotov and Mikoyan. The 4 March document containing these proposals was confiscated from the safe of Malenkov's assistant in 1956. For now we do not know what the initial draft contained, but we do know that it outlined the main decisions that were officially adopted the following day.\n\nStalin's heirs completely dismantled the governmental structure he had put together during his final months of life. The expanded Central Committee Presidium created on Stalin's orders in October 1952 was abolished with the stroke of a pen. The Central Committee's Presidium Bureau was proclaimed to have a new membership: Molotov and Mikoyan were added, and the young prot\u00e9g\u00e9s whom Stalin had made part of the expanded Presidium were expelled from its ranks. In essence this upset meant a return, under a new name, to the collective leadership that had once existed as the Politburo. Stalin's title of chairman of the Council of Ministers was given to Malenkov. This title did not mean, however, that Malenkov was recognized as Stalin's heir or that he possessed Stalin's powers. The new system was designed to include numerous counterpoises that would protect against the appearance of a new tyrant. Malenkov, unlike Stalin, did not simultaneously hold the post of Central Committee secretary; that post was given to Khrushchev. The men designated as Malenkov's first deputies\u2014Beria, Molotov, Bulganin, and Kaganovich\u2014were in no way his juniors within the nomenklatura system. This reshuffling created a balance and satisfied the interests of all the members of the top leadership. Later, none of the participants in the reorganization recalled any controversy or rancor.\n\nThis new arrangement was formally approved by the oligarchs at a joint meeting of the Central Committee plenum, the Council of Ministers, and the Supreme Soviet Presidium on 5 March 1953. Soviet dignitaries gathered in a hall of the Grand Kremlin Palace. One participant, the writer Konstantin Simonov, left the following description of the ceremony's atmosphere:\n\nI arrived long before the appointed time, about forty minutes early, but more than half of the participants had already gathered in the hall, and ten minutes later everyone was there. Maybe two or three people arrived less than a half hour before the start. There were several hundred people there, almost all acquainted with one another... sitting in total silence and waiting for the start. We were sitting side by side, shoulder to shoulder; we saw one another, but nobody said a word to anyone else.... Until the very start it was so quiet in the hall that if I had not sat in that silence for forty minutes myself, I would never have believed that three hundred people sitting so tightly packed could keep quiet like that.\n\nFinally the members of the presidium that was about to be voted into existence appeared. The entire event lasted forty minutes, from 8:00 to 8:40 p.m. The resolutions that the top leadership had already agreed on were, as usual, obediently approved. The Stalin factor was dealt with simply and elegantly. He was deprived of the top posts of chairman of the government and secretary of the Central Committee and then formally included in the Central Committee Presidium. From now on, whatever his physical fate, Stalin's political future and his comrades' liberation from his tyrannical powers were faits accomplis. As Simonov remarked, \"There was a sense that right there, in the Presidium, people were freed from something that had been weighing them down, that had bound them.\"\n\nStalin endured this formal deprivation of power for only one hour. At 9:50 p.m. he died. His death was agonizing, as if in confirmation of the folk wisdom that only the righteous are granted an easy death. His daughter Svetlana, who spent her father's final days by his side, recalled:\n\nThe death agony was horrible. He literally choked to death as we watched. At what seemed like the very last moment he suddenly opened his eyes and cast a glance over everyone in the room. It was a terrible glance, insane or perhaps angry and full of the fear of death and the unfamiliar faces of the doctors bent over him. The glance swept over everyone in a second. Then something incomprehensible and awesome happened that to this day I can't forget and don't understand. He suddenly lifted his left hand as though he were pointing to something above and bringing down a curse on us all. The gesture was incomprehensible and full of menace, and no one could say to whom or at what it might be directed. The next moment, after a final effort, the spirit wrenched itself free of the flesh.\n\nStalin's comrades did not linger at his bedside. A half-hour later, at 10:25 p.m., they were already back in his Kremlin office, several kilometers away. All the main matters of state had been resolved. What remained were the funeral arrangements. The new leaders created a commission to handle these arrangements and appointed Khrushchev to head it. They also adopted a decision to place the sarcophagus with Stalin's embalmed body in Lenin's mausoleum. State security and the propaganda apparat were given their orders. The editor-in-chief of _Pravda,_ Dmitry Shepilov, spent ten minutes at this meeting. One deeply symbolic detail impressed him the most: \"The chair Stalin had occupied as chairman for thirty years was empty; nobody sat in it.\"\n\nFor a while, the Soviet leaders were genuinely equal and united in their determination to prevent the emergence of another tyrant. After what they had endured under Stalin, they were ready to do away with the system of terror, even if that entailed some undesirable political consequences. By 3 April 1953, after the appropriate preparations, the Central Committee Presidium resolved to \"fully rehabilitate and release from custody the doctors and members of their families arrested in association with the so-called Case of the Wrecker-Doctors.\" Thirty-seven people were freed. The state security officers who \"particularly applied themselves in the fabrication of this provocational case\" were to be brought to justice. The next day this resolution was announced in the newspapers, occasioning a variety of responses and a certain consternation among the _vozhd_ 's most ardent supporters. Other political cases in which the collective leadership had a personal interest were quietly subjected to a quick review. Molotov's wife was released from prison. Kaganovich's brother, who had taken his own life on the eve of the war after charges of wrecking, was pronounced innocent. The Mingrelian Affair, which had cast a shadow on Beria's reputation, was also reviewed. Many other prominent victims of political repression were set free or posthumously rehabilitated. After taking care of their own, Stalin's heirs began to grant relative freedom to the rest of the country. They were driven in this direction not only by conscience, but also by the growing crisis that had already been apparent under Stalin. The death of the man who had been unwilling to entertain any talk of change opened the door to reforms to be implemented with amazing speed and decisiveness.\n\nTwo pillars of the dictatorship\u2014state security and the Gulag\u2014were significantly reformed. One symbol of this reform was a Ministry of Internal Affairs order, dated 4 April 1953, banning the use of torture against arrestees. The order recognized the problem of \"arrests of innocent Soviet citizens\" and \"the widespread use of various means of torture: the brutal beating of arrestees; the round-the-clock use of handcuffs behind the back, in isolated cases for several months; long-term sleep deprivation; and locking up unclothed arrestees in cold punishment cells, etc.\" Threatening harsh punishment of anyone who violated the order, the ministry's leadership demanded that torture chambers be closed in prisons and that the implements of torture be destroyed. When the order was read out loud to all state security operatives, it must have made quite an impression. These reforms continued into the spring and summer of 1953, bringing major changes to the camp system. A mass amnesty announced for those convicted of non-political crimes cut the inmate population in half. Many factories and construction projects that were still using a prisoner work force and being overseen by the ministry were shut down or transferred to the economic ministries. A large-scale effort to rehabilitate the victims of Stalinist terror lay in the near future.\n\nSignificant changes to economic policy were made within weeks. Unwieldy construction projects were scaled down, and the rush to \"build communism\" and expand Soviet military capabilities that was putting such a strain on the economy was brought to a halt. The resources thus freed up were directed toward alleviating the crisis in agriculture and meeting the needs of ordinary citizens. The prices paid for agricultural products were raised, and the tax burden on peasants was reduced. A marked improvement in output, especially in the area of livestock, came with amazing speed. Soon there would be ambitious programs to ease the plight of ordinary citizens, including a massive effort to expand housing.\n\nDomestic reforms were accompanied by a moderation of foreign policy. On 19 March 1953 the Council of Ministers passed a resolution calling for \"an end to the war in Korea as soon as possible.\" After tense negotiations, an armistice treaty was signed on 27 July 1953. Moscow gave its blessing to a liberalization of Communist regimes in Eastern Europe. On 2 June 1953 a Council of Ministers directive spelled out Soviet objections to the policies of the East German government and called for measures to improve the republic's political situation.\n\nIn short, Stalin's \"ungrateful\" heirs had little trouble eliminating many of the excesses for which the _vozhd_ bore sole responsibility. Their reforms fundamentally changed the Soviet regime. It was no longer \"Stalinist\"; it was less brutal and more predictable and flexible. Dictatorship, as a form of government in the Soviet Union, had been dealt a blow from which it never recovered. Internal struggles at the upper reaches of government would more than once lead to power changing hands, but never again would a Soviet leader wield the sole power exercised by Stalin.\n\n## **THE FUNERAL**\n\n**The _Vozhd,_ the System, and the People**\n\nFor three days beginning on 6 March 1953, the Soviet Union said its ceremonial farewells to Joseph Stalin. His coffin was put on display in the very center of Moscow, in the House of Unions' Hall of Columns, the traditional site for public mourning of Soviet leaders that had earlier served as the House of Receptions for Moscow's nobility. At four o'clock on the afternoon of the sixth, the public was let in to pay its final respects. The viewing of the body was poorly organized, and the provisions made for the crush of people who headed toward the House of Unions were not conducive to public safety. Those trying to get one last look at the dictator streamed into narrow streets filled with police and trucks meant to serve as barriers. In the chaos and panic many suffered disabling injuries or were crushed to death. The files of investigations into these events have yet to be made accessible to historians. In remarks made to a small gathering in 1962, Khrushchev said that 109 people in the crowd died that day.\n\nNo information about this addition to the long series of Soviet tragedies appeared in newspapers, which were filled with grandiloquent expressions of sorrow and grief for the late _vozhd_. People's true feelings came out in a flood of letters, as eyewitnesses to the tragedy registered their complaints with various government offices:\n\nThis is not the first time that during the movement of a large crowd the police were transformed into a helpless organization, or rather into violators of order. How distressing it was when\u2014in front of a crowd of hundreds and foreigners darting about with their cameras\u2014they began to retrieve the injured and crushed and send them off in ambulances. A simply shocking scene.\n\nFor five hours people were herded all over Moscow, and none of the police knew where the line was! The police were running into columns made up of many thousands of people, with their cars causing casualties, cries, and groans. Hundreds of thousands of people were walking around the blocked-off streets leading to the Hall of Columns and could not find the way in!... Only a wrecker could announce that access would begin at four but announce the route at nine.\n\nIn many ways these letters captured the essence of the Stalin era, both in their lexicon\u2014with references to foreigners \"darting about with their cameras\" and \"wreckers\"\u2014and in the events they describe: the police turning into violators of public order. Relying on brute force, the dictatorship had attained its goals at the expense of countless victims. The boundary between rational order and destructive chaos was blurred. Those charged with maintaining order wound up wreaking havoc.\n\nPerhaps the tragedy in Moscow forced Stalin's heirs to ponder the police state's shortcomings, but for now they had no option but to rely on the institutions and methods bequeathed to them. Stalin's funeral, set for 9 March, was prepared much as the viewing in state had been, but possibly with a bit more care. The top priority was security, ensured by 22,600 secret police agents, policemen, and soldiers. Thirty-five hundred vehicles were commissioned to block streets. The government approved a minute-by-minute schedule of funeral events: the carrying of the coffin from the House of Unions, its placement in front of the Lenin Mausoleum in Red Square, a mourning gathering for the public, the carrying of the coffin into the mausoleum. Several hours before the ceremony, six thousand soldiers and fifteen thousand members of a \"delegation of workers\" were brought to Red Square. This time everything went according to plan.\n\nAlthough incompetent officials bear much responsibility for the casualties in Moscow, another cause of the tragedy was the sheer number of people wanting to catch one last glimpse of the _vozhd_. What drove them? Was it love, curiosity, mass psychosis, or a rare opportunity for a spontaneous display of emotion? Apparently all these elements were present, along with many others. The few available documents that shed light on the public mood reveal a complex range of responses to the _vozhd_ 's illness and death. On 5 March 1953, State Security Minister Ignatiev presented the Soviet leadership with a report on soldiers' reactions to the news that Stalin was ill. The document described a certain pattern in the reaction of the \"faithful.\" One common thread was sympathy toward Stalin the man, who, according to Soviet propaganda, was the embodiment of goodness and benevolence: \"My family takes this news as a terrible sorrow befalling our country\"; \"He worked very hard, and that took a toll on his health.\" \"Positive\" responses often involved expressions of concern over the future of the country and the responder's own future. Two points long emphasized by Soviet propaganda played a part in such positive responses: Stalin was irreplaceable and war was looming: \"It's kind of scary. Who will take his place after his death?\"; \"Maybe this will speed up the onset of a Third World War.\" The chekists also reported on \"negative\" and \"hostile\" statements: \"Serves him right\"; \"That's just fine\"; \"Stalin won't hang on for long, and that's even better. You'll see that everything will immediately change.\" All such letters led to arrests or at least an investigation.\n\nMarch 1953 saw a surge in arrests and convictions of people charged with \"anti-Soviet agitation\" for expressing satisfaction with Stalin's death or otherwise denigrating him. A forty-four-year-old Muscovite named S. M. Telenkov, who worked at a scientific institute, drunkenly proclaimed in a commuter train, \"What a fine day it is today; today we buried Stalin. There'll be one less scoundrel around and now we can get back to living.\" R. S. Rybalko, a twenty-eight-year-old working-class woman from Rostov Oblast, was convicted of using profanity in regard to Stalin. Ya. I. Peit, who had been forcibly resettled in Kazakhstan, was sentenced for destroying and stomping on a portrait of Stalin after an official mourning ceremony. Upon hearing of Stalin's death, P. K. Karpets, a thirty-two-year-old railroad worker from the Ukrainian city of Rovno, swore and exclaimed, \"Smell that? The corpse is already stinking.\" Ye. G. Gridneva, a forty-eight-year-old female railroad worker from Transcaucasia, was not able to contain herself and commented to a coworker, \"A dog dies a dog's death. It's good that he died. There won't be any kolkhozes and life will be a little easier.\"\n\nThe expressions of anti-Stalin sentiment that came to secret police attention were just the tip of the iceberg. Most people had been trained to keep their opinions to themselves. The ubiquity of informants and the habit of fear kept free expression to a minimum, to say nothing of more demonstrative forms of protest. The choice was simple: either accept\u2014or pretend to accept\u2014official values or find yourself in a camp or face to face with an executioner. This circumstance diminishes the value of such normally candid sources as diaries. One must assume that even in the privacy of their own homes, Soviet citizens exercised self-censorship and used their diaries more as potential alibis than vehicles for frankness. Newspaper reports on mass demonstrations, summaries prepared by state security on the public mood, and letters written to the authorities by ordinary citizens provide only part of the picture. Furthermore, many of these documents are still hidden in closed archives. Historians attempting to fathom the public mood during the Stalin era still face major obstacles.\n\nThe 190 million people living in Stalin's Soviet Union on the eve of his death constituted an exceptionally complex community that bore little resemblance to the \"New Man\" featured on the covers of Soviet magazines. Many factors worked to give cohesion to Soviet society and promote support for the regime, and the motives for this support could vary from sincere enthusiasm to reconciliation with the inevitable to ordinary submission in the face of overwhelming power. The huge scale of violence and terror made fear and compulsion the backbone of the Stalinist system, albeit hidden behind a fa\u00e7ade of enthusiasm. At the same time, loyalty and belief in the system and the man were not always feigned. The perpetual fear that was the primary instrument for unifying the people and suppressing independent thought was used alongside \"positive\" mechanisms of social manipulation. Both the carrot and the stick were applied to keep Soviet society moving in the desired direction.\n\nOne by-product of the regime's policies was the creation of a large privileged class of officials. Those holding all but the most junior government or party posts enjoyed many benefits, including high social status and significant material perquisites. After the mass purges of the second half of the 1930s, the ranks of the Soviet nomenklatura stabilized. Repression against officials during the postwar period was more the exception than the rule. Furthermore, there is evidence that on the eve of Stalin's death, officials and their relatives were essentially immune from prosecution. The requirement that any arrest or prosecution of a party member be approved by the leadership of party committees led to a bifurcation of the judicial system. In many cases members of the nomenklatura and their relatives avoided prosecution for administrative or criminal offenses that would bring severe punishment to an ordinary citizen.\n\nAnother category\u2014\"the country's best people\"\u2014approached the status of officials within the huge party-state apparat. These \"best people\" could be found in every social segment and professional group, including workers, peasants, writers, artists, and scientists. The best known examples were the so-called Stakhanovites, real or imagined shock workers at the forefront of production who were held up for admiration as \"beacons\" of the Soviet spirit. Enjoying a stature somewhere between ordinary citizens and officials, the Stakhanovites quickly assimilated the latter's value system, although in theory they kept working away as before. They served as spokespeople, lobbying for the interests of enterprises and regions and enjoying significant material privileges. A typical representative of this category of beneficiary of the Stalinist system was the eponymous miner Aleksei Stakhanov, who earned celebrity and Stalin's favor through his record-breaking productivity. He quickly developed a taste for the nomenklatura lifestyle and bombarded Stalin with requests:\n\nJoseph Vissarionovich! Give me a nice car and I will justify your trust. Soon the Stakhanovite movement will be ten years old, and I'm going to Donbas and will again show people how to work. I keep asking and they keep giving me some broken down war trophy clunker, but if just once I got something nice, I'd stop asking.... Also, about the apartment.... I can't get anywhere with my requests to fix it up. The walls are dirty, the furniture is frayed and broken..., while other people get their walls papered with silk twice a month and get all sorts of furniture. This isn't correct, so I'm asking for a renovation and new furniture so I won't be ashamed to invite people to my apartment.\n\nAnother consequence of the channeling of benefits to the upper crust of Soviet society was the policy of disproportionately allotting resources to cities, especially major ones. Forced industrialization and militarization widened the gulf in living standards and social status between the rural majority and urban minority. Many urbanites, especially in the capitals and major industrial centers, belonged to a relatively privileged and well-remunerated class. During years of famine they may have been hungry, but since they received a government ration, they were not dying of starvation like the peasantry. They had internal passports, unlike the peasants, and relative freedom of movement. Urban populations also enjoyed better health care and a well-developed cultural and educational infrastructure. In the stores of Moscow and Leningrad, where most food and consumer goods were sent, shoppers could find what they needed and even had a degree of choice. The relative accessibility of educational institutions and high-paying jobs gave urbanites much better economic prospects. The monetary reform, which reduced prices in state stores while increasing taxes on peasant production, disproportionately favored the residents of capitals and industrial centers. These measures forced peasants to sell the products of their private plots at lower prices in urban markets. The consequences of these policies apparently escaped Stalin's awareness. Mikoyan, whose duties placed him in charge of certain commercial matters, offers the following account:\n\nI told him [Stalin] that we could not lower the prices on meat and butter, on white bread, first of all because they were in short supply and second because it would affect the procurement prices, which would have a negative effect on the production of these products, and when these goods are in short supply and with this reduction in prices there would be huge lines, which would lead to profiteering; after all, workers cannot go to the store during the day, so the profiteers would buy up all the goods.... But Stalin insisted, saying that this was necessary in the interests of the intelligentsia.\n\nMikoyan here nicely sums up the predictable effect of the politically motivated price reduction: shortages, lines, and a shadow market. But these were of little concern to Stalin. His focus was on the regime's bulwark, the privileged segment of society in major cities. The government's preferential distribution of resources made even the average urbanite many times better off than the rural population. One symptom of this inequality was the number of young rural women streaming into cities to work as housekeepers for urban families for no more than bread and shelter. Clearly, the urban minority and the rural majority had starkly divergent perceptions of reality. It was the urbanite viewpoint that found voice in memoirs and diaries and has disproportionately influenced contemporary understandings of day-to-day life under Stalin.\n\nAnother factor that led Soviet society to tolerate and even support the dictatorship was war. Memories of the horrors of the world and civil wars, the victory over the Nazis (paid for with 27 million lives), and the fear of a third world war all had a huge impact on perceptions\u2014and not only in the Soviet Union. Stalin enjoyed the image of a savior who had delivered the world from a terrible evil. For decades afterward, the 1945 victory lent legitimacy to the Stalinist regime and those of his successors.\n\nThe list of historical circumstances that enabled the Stalinist system to endure could be continued, but even in conjunction with an ever-vigilant apparatus of repression they could not completely hide the contradictions inherent in Soviet society or suppress widespread dissatisfaction. From the moment they came to power as a radical revolutionary party, the Bolsheviks relied on a strategy of dividing society and suppressing the fraction that, for reasons of class origin or societal role, was considered hostile to socialism. This strategy included killing off the members of the hostile groups. The Stalinist revolution devoted tremendous resources to purging society of these \"elements.\" Furthermore, along with the nobility, bourgeoisie, tsarist officers and officials, and anyone else proclaimed persona non grata after 1917, the largest segment of the population was stigmatized: the peasantry. During collectivization, many peasants were branded kulaks and shot, exiled, or driven out of their native villages. Millions of people from every sector were persecuted on a variety of pretexts and put into the camp system or simply killed. Aware that these measures had earned the dictatorship true enemies, Stalin intensified his preemptive purges, most notably during the Great Terror of 1937\u20131938. Repression begat repression. By the end of his rule a significant proportion, if not the majority, of Soviet citizens had at one time or another been arrested, imprisoned in a camp, forcibly relocated, or subjected to some softer form of mistreatment.\n\nThe regime's victims did not necessarily turn into conscious opponents. Terror often had the opposite effect. Intimidation made people more governable and submissive and forced them to demonstrate their loyalty. But it would be wrong to assume that submission was the only possible reaction. The historical record attests to the existence of widespread anti-government feelings or even active forms of resistance. For understandable reasons resistance was most common when the dictatorship was first being consolidated\u2014most notably peasant revolts during collectivization in 1930 and its aftermath. The Terror and the stabilization of the system sharply curtailed opportunities for overt action, especially on a large scale. But it is important to note that access to secret police archives, which would reflect the true state of affairs in the late Stalin era, is extremely limited. We may learn that our image of the 1940s generation as silent and submissive is misinformed.\n\nA root cause of widespread dissatisfaction was the Soviet Union's low standard of living. Agriculture, its productivity severely undermined by collectivization, lurched between crisis and stagnation. Almost every year, the Stalinist government acknowledged that famine or \"food difficulties\" affected either a large swath of the country, as in 1931\u20131933 and 1946\u20131947, or some particular regions. Even in the best years the average diet was meager. Most people lived primarily on grains and potatoes. Budgetary studies conducted on the eve of Stalin's death, during the relatively prosperous year 1952, established the following daily nutritional intake in worker and peasant families: the average Soviet citizen consumed approximately 500 grams of flour products (primarily bread), a small amount of cereals, 400\u2013600 grams of potato, and approximately 200\u2013400 grams of milk or milk products. These items accounted for the bulk of the typical diet. Anything else, especially meat, was a special occasion. The figure for per capita consumption of meat and meat products averaged 40\u201370 grams per day and 15\u201320 grams of fat (animal or plant oils, margarine, or fatback). A few teaspoons of sugar and a bit of fish completed the picture. Average citizens could permit themselves an average of one egg every six days. These rations are approximately equal to the dietary norm for prison camps. The figures were produced by the Central Statistical Directorate, which was under constant political pressure and probably painted an overly rosy picture. Averages could be inflated, for example, by selecting workers at the high end of the pay spectrum or peasants from relatively prosperous kolkhozes in the study. Also, the budgetary studies did not factor in the often poor quality of the food. A resident of Chernigov Oblast wrote to Stalin in November 1952, \"Now they are baking black bread, and even that is of poor quality. It is impossible to eat such bread, especially for people in poor health.\"\n\nThe supply of manufactured goods was just as bad. Prices of factory-made items were traditionally kept exceptionally high. People had to settle for simple, relatively cheap products, but few could afford even these. For example, in 1952 only one out of every four peasants could afford leather footwear. Some lacked even the simplest footwear and clothing. As one resident of a village in Tambov Oblast wrote to Stalin in December 1952, \"In our kolkhoz the kolkhozniks have one article of winter clothing for 3\u20134 family members, and children in 60 percent of the population cannot go to school since they don't have the clothing.\"\n\nFor the majority of the population the housing situation was no better. Under Stalin, housing was the chronically underfunded stepchild that received whatever resources were left after priority items had been taken care of. For years the housing shortage grew continually worse\u2014and then came the devastation of war. As of the beginning of 1953 there was an average of 4.5 square meters of residential housing per urban resident. When temporary residents and those without official registration were taken into account, this ratio grew even worse. The quality of housing was also low. Only 46 percent of state-owned residential space came equipped with running water, 41 percent with sewage hookups, 26 percent with central heating, 3 percent with hot water, and 13 percent with a bathtub. Even these figures reflected the higher standards found in major cities, chiefly the two capitals. A striking indicator of the housing crisis was the prevalence of urban \"barracks\"\u2014flimsy temporary communal housing without plumbing\u2014and the increasing number of people registering such barracks as their residences. In 1945 approximately 2.8 million people lived in urban barracks, but by 1952 the number had grown to 3.8 million. More than 337,000 people in Moscow lived in barracks.\n\nAnother source of hardship for the Soviet people was the exceptionally difficult working conditions in industry and agriculture. The poorly developed system of material incentives led to widespread coercion in the workplace. The use of slave labor was of course most blatant within the Gulag system, but supposedly free industrial and agricultural workers also often toiled under compulsion. The workforce for certain industries, especially the most poorly paid and dangerous, was assembled by pressing young people into service through compulsory mobilization. Evasion was punishable by a term in a labor camp. Beginning in 1940, emergency labor laws were used to bind workers to their places of employment. Peasants, who were essentially not paid for their work in kolkhozes, were prosecuted for failure to fulfill their work quotas. Between 1940 and 1952 approximately 17 million people were convicted of tardiness, leaving their place of employment without permission, or evading mobilization. This huge number, which fails to capture the extent of violations of workplace discipline, belies the propagandists' exultation of Soviet workers' selfless enthusiasm.\n\nBetween the two extremes of devotion and opposition to the regime, the vast majority made empty shows of loyalty but were largely indifferent to politics. Only marginally influenced by propaganda and trying their best to evade the grip of repression, most took comfort in tradition and ritual. Despite state repression of priests and active church members, especially in the 1930s, most Soviet citizens held onto their faith. During the census of January 1937, 57 percent of respondents over the age of sixteen identified themselves as religious\u2014more than 55 million people. Surely many others hid their faith out of fear of persecution.\n\nIn the area of inter-ethnic relations, Stalin left a problematic legacy. The relative liberalism of the early Bolshevik regime, which built what historian Terry Martin calls an \"affirmative action empire,\" came to an end in the early 1930s. Under Stalin, nationalities policy grew increasingly brutal. Mass arrests and executions based on nationality, the internal exile of entire peoples, and the effort to use russification to create a single Soviet nationality laid a minefield under the country's future. Explosions started to go off while Stalin was still alive, when guerrilla wars roiled western Ukraine and the Baltic states. Although a degree of inter-ethnic unity was actually achieved, behind the propaganda fa\u00e7ade extolling the \"friendship of peoples\" seethed many inter-ethnic conflicts. The \"Russian question\" that grew out of the contradictory position of the Russian majority\u2014simultaneously the bulwark of the Soviet empire and one of its chief victims\u2014promoted instability and ultimately destroyed the Soviet Union, an interpretation advanced by Geoffrey Hosking.\n\nWhat did Stalin know about the real life of \"his\" people? The Albanian Communist leader Enver Hoxha visited Moscow in 1947 and later recalled Stalin saying, \"To govern, you have to know the masses, and in order to know them, you have to walk among them.\" Stalin could hardly claim to adhere to his own wisdom. After his famous visit to Siberia in 1928, most of which was spent meeting with functionaries, he almost never walked \"among the masses.\" Official meetings with representatives of the workers were carefully orchestrated propaganda spectacles. During better days, Stalin would occasionally indulge his taste for theatrics and suddenly appear in public. But even these spontaneous meetings inevitably took on the aura of \"Christ appearing to the people.\" In September 1935, accompanied by several Soviet leaders, he toured the outskirts of Sochi and encountered small groups of vacationers. On Stalin's initiative a spontaneous \"fraternization\" was allowed. One vacationer left a striking account of the event:\n\nComrade Stalin... stopped us with the following words: \"Why are you leaving comrades? Why are you so proud that you shun our company? Come here. Where are you from?\" We walked up to him.... \"Well, let's get acquainted,\" Comrade Stalin said, and he introduced us to each of his companions in turn and introduced himself as well. \"This is Comrade Kalinin, this is the wife of Comrade Molotov... and this is I, Stalin,\" he said, shaking everyone's hand. \"Now we'll all have our pictures taken together,\" and Comrade Stalin invited us to stand next to him.... While the photographers were working, Comrade Stalin kept making fun of them: he said they were \"mortal enemies\" and were always trying to interfere with one another. He asked that they photograph not only him but \"all the people.\"... Then Comrade Stalin began to invite the woman selling apples from a kiosk... and a salesman from the food stand to come have their pictures taken. It took a long time before the disconcerted saleswoman could be persuaded to leave her store. Comrade Stalin told her that \"it's not good to be so proud\" and told the photographers not to take the picture until she came. \"The saleswoman,\" Stalin proclaimed, \"should become the most respected woman in our country.\" Finally she came and the photo shoot continued. An empty bus drove up, and Comrade Stalin invited the driver and conductor to have their pictures taken.\n\nObviously such \"walks among the people\" did little to enhance Stalin's understanding of them, and even these mostly stopped after the 1930s. The _vozhd_ never took an interest in seeing the conditions in which the Soviet people were living, what they bought and where, what sort of health care or education they received. His knowledge of \"the masses\" came mostly from what he read in his office. So far we know of two main sources from which he gleaned knowledge of daily life: summary reports from state security about the public mood and letters and complaints from ordinary citizens. A steady stream of such letters arrived in government offices, including some addressed to him personally.\n\nAs far as can be determined from archival studies, state security summaries were a major source of information for the Soviet leadership in the 1920s and 1930s. These reports contained rather candid assessments of the situation in the country, albeit from a chekist perspective, which saw almost all crises and difficulties as the work of enemies. There were a number of types of reports, some providing an overview of sociopolitical processes, others devoted to matters of economics or politics. One problematic aspect of these reports was their length. The leaders for whom they were prepared had to spend hours poring over them. In recent years historians have published a number of informational state security summaries dating to the prewar period. These publications, however, are based on copies found in state security archives\u2014not in Stalin's personal archive. We do not currently know the extent to which, or in what form, they are contained in the Politburo archive, which is part of the Archive of the President of the Russian Federation. Historians therefore cannot be sure to what extent the leadership in general or Stalin in particular read these secret police summaries. There is evidence to suggest that they were mostly unaware of these reports' contents.\n\nWe know more about Stalin's familiarity with letters from Soviet citizens. It would not be an exaggeration to say that most of the country sent complaints, requests, and petitions on a wide array of topics to all sorts of government offices. Such letter writing was an extremely common practice and was even encouraged by the authorities. Within the highly centralized system, letters to the government were one of the few ways of solving everyday problems. The government was virtually the only employer. It also had authority over the allocation or construction of housing. Government stores supplied (or were supposed to supply) all basic needs. Government hospitals were the only places to obtain treatment for serious illnesses. The government determined the rather narrow category of people eligible for pensions or benefits and the size of the payments. Given the flaws of the Soviet judicial system, citizens turned to bureaucrats to resolve conflicts and disputes. Abuses by officials within the huge bureaucratic apparat occasioned countless grievances. Arrests, forcible relocations, imprisonments in camps, or death sentences against tens of millions of people generated millions of complaints and pleas for relief. Arrestees themselves wrote, as did their relatives, and even unrelated people sometimes worked up the courage to intercede on behalf of an acquaintance or colleague. This pursuit of justice was encouraged by the state since it created the illusion of impartial leadership.\n\nAnother practice that was encouraged was denouncing abuses or \"enemy activity.\" Stalin made it no secret that he held denouncers and informers in high regard. All denunciations, including anonymous ones, were investigated. The government's attitude is eloquently illustrated by the fact that even prisoners who were deprived of all other rights had the right to submit denunciations. In February 1936 the NKVD chief signed an order calling for the installation of boxes in all camps, prisons, and penal colonies into which inmates could insert statements addressed to him personally or the head of the Gulag directorate. \"The boxes shall be sealed with the seal of the Directorate of Camps,\" the order read, \"and only the head of the camp or his deputy (in camps) and the head of the Department of Detention Centers or his deputy (in prisons and penal colonies) shall open them.\" All correspondence was to be sent to the NKVD chief personally and \"under no circumstance concealed.\" Inmates were to be informed of \"the purpose of these boxes.\"\n\nTaking advantage of the regime's eagerness to uncover enemies and the almost total impunity enjoyed by slanderers, many Soviet citizens used denunciations to game the system. Informers used the government to attain their own mercenary objectives\u2014to settle scores, get rid of annoying neighbors sharing the same communal apartment, or eliminate those competing for the same job. For the hapless multitudes at the bottom of the societal hierarchy, denunciations were the only means of taking revenge against powerful officials. The state implicitly encouraged people to use this disgraceful means of fighting for their rights.\n\nIn addition to complaints and denunciations, the archives abound with \"helpful\" letters. Some offered ideas for reorganizing government agencies or for various socioeconomic innovations; some offered ideas for renaming cities or creating new holidays or ceremonies; others sought to correct \"errors\" in the press. Writing such letters was one of the few outlets for activism available to ordinary citizens. These letters may have contained an element of self-promotion as their authors tried to draw the top leadership's attention to themselves.\n\nAs the supreme authority, Stalin, of course, was all these correspondents' prime addressee. It is hard to know the precise number of letters addressed to him personally, but it apparently exceeded several hundred thousand per year. Obviously not all of them reached his desk; he was shown a selected sample. The nature of this sample is of interest from a number of perspectives. Primarily, it shows how well informed Stalin was about people's lives and tells us what he expressed an interest in seeing. No doubt the apparat was given criteria for selecting the letters he would be shown.\n\nHandling letters addressed to Stalin was a complicated bureaucratic process. Within the Central Committee's Special Sector, which served as Stalin's personal secretariat, was a division dedicated to processing his mail. After the war this division was called the Special Sector's \"Fifth Section.\" In early 1950 it had a staff of twenty. They received and logged letters addressed to Stalin and immediately forwarded a significant portion of them to various agencies for review. The heads of the Special Sector, especially Stalin's personal assistant, Aleksandr Poskrebyshev, were shown the most important and interesting letters. Poskrebyshev further filtered them, leaving just a few of the most interesting for his boss. As a result of this tiered system, Stalin saw just a tiny percentage of the hundreds of thousands of letters sent to him, and over time this number shrank. In early 1946 Stalin saw about ten letters per month, but by 1952 he was shown just one or two.\n\nThis small sample revealed little about real life in the Soviet Union. Most of the letters reaching Stalin's desk belonged to one of three categories: queries on matters of theory, letters from old acquaintances, and a large number of letters of support. On extremely rare occasions he might be shown correspondence that tiptoed around some unsavory aspect of Soviet reality. Overall, the letters he saw reflected the _vozhd_ 's growing desire to live in the past or savor hopes for the future. Pressing matters of real consequence likely to provoke negative emotions were avoided.\n\nAs ignorant of the life of the people as the _vozhd_ was, the people knew even less what kind of a man he was. Partly due to his personality and partly out of calculation, Stalin, unlike many other dictators, rarely spoke before large audiences. He preferred to express himself in writing. The aggressive propaganda of Stalin's articles, interviews, and theoretical works created the impression that the invisible _vozhd_ was ever-present and all-knowing. His cryptic sententiousness gave him a certain charisma.\n\nTight control over the alchemy of official \"Staliniana\" has created false and doubly majestic images of Stalin and his accomplishments. These images outlive the man himself and have an appeal even in contemporary Russia. The collapse of the Soviet Union, the stresses of the transitional period, corruption, poverty, and glaring social inequality all feed the longing for a social utopia. A significant portion of Russian society seeks recipes for the present by looking to the Stalinist past. Popular images of the greatness of the Stalinist empire\u2014of equality and the fight against corruption, of the joy and purity of this distant life undone by \"enemies\"\u2014are exploited by unscrupulous commentators and politicians. How great is the danger that a blend of historical ignorance, bitterness, and social discontent will provide fertile ground for pro-Stalinist lies and distortions to take root?\n\nCould it really be that Russia in the twenty-first century is in danger of repeating the mistakes of the twentieth?\n\n## **ILLUSTRATIONS**\n\n**Stalin's mother, Ekaterine (Keke) Jughashvili.**\n\n**Russian State Archive of Social and Political History.**\n\n**Stalin as a pupil at the Gori Theological School in the early 1890s.**\n\n**Russian State Archive of Social and Political History.**\n\n**Stalin as a young revolutionary, early 1900s.**\n\n**Russian State Archive of Social and Political History.**\n\n**From party archives, a 1910 arrest record for Stalin from the files of the tsarist political police in Baku. Russian State Archive of Social and Political History.**\n\n**Stalin at the Tsaritsyn front in 1918. Russian State Archive of Social and Political History.**\n\n**Lenin and Stalin at Gorki, Lenin's residence outside Moscow, in 1922, a few months before Lenin's death prompted a fierce power struggle. Russian State Archive of Social and Political History.**\n\n**Stalin with Rykov (left) and Bukharin (right), December 1927. Rykov and Bukharin were shot in 1938. Russian State Archive of Social and Political History.**\n\n**The _vozhd_ with his faithful comrades in 1934. Left to right: Kirov, Kaganovich, Ordzhonikidze, Stalin, and Mikoyan. Kirov was shot later that year by the husband of a staff member, and Ordzhonikidze committed suicide in 1937. Russian State Archive of Social and Political History.**\n\n**Stalin with his wife, Nadezhda Allilueva, Voroshilov, and Voroshilov's wife Yekaterina relaxing in the south in 1932 (with a bodyguard to the right) a few months before Nadezhda's suicide. Russian State Archive of Social and Political History.**\n\n**On vacation in 1933. Stalin's daughter Svetlana sits on the lap of then Georgian party boss Lavrenty Beria. Russian State Archive of Social and Political History.**\n\n**The loving father: Stalin with Svetlana, 1933.**\n\n**Russian State Archive of Social and Political History.**\n\n**A visit in the south, 1933. Left to right: chief of the Red Army General Staff Aleksandr Yegorov; Defense Commissar Kliment Voroshilov; Stalin; Soviet military leader Mikhail Tukhachevsky; Abkhaz leader Nestor Lakoba. Lakoba died in 1936 under mysterious circumstances and was soon proclaimed an \"enemy of the people.\" Tukhachevsky was shot in 1937 and Yegorov in 1938. Russian State Archive of Social and Political History.**\n\n**The near dacha, where Stalin lived (starting after his wife's suicide) and died.**\n\n**Russian State Archive of Social and Political History**\n\n**A rare family gathering in the mid-1930s. Left to right: Stalin's son Vasily, Leningrad party boss Andrei Zhdanov, daughter Svetlana, Stalin, and Stalin's son (by his first wife) Yakov, who was killed in a Nazi POW camp. Russian State Archive of Social and Political History.**\n\n**Stalin inspects new military hardware at the Kremlin, September 1943. Russian State Archive of Social and Political History.**\n\n**The Allies: Churchill, Roosevelt, and Stalin in Crimea, February 1945.**\n\n**Russian State Archive of Social and Political History.**\n\n**Generalissimo Stalin immediately after the war.**\n\n**Russian State Archive of Social and Political History.**\n\n**Stalin and his comrades at a celebration in January 1947. Left to right: Beria, Kaganovich, Malenkov, Molotov, Kuznetsov, Stalin, Kosygin, Voznesensky, Voroshilov, and Shkiriatov. Two years later Kuznetsov and Voznesensky were arrested and shot. Russian State Archive of Social and Political History.**\n\n**Stalin at a party congress in 1952, four months before his death.**\n\n**Unflattering photographs like this one were not published. Russian State Archive of Social and Political History.**\n\n**Millions of copies of Stalin's works were published in all languages. After his death they provided tons of recycled paper pulp. Russian State Archive of Social and Political History.**\n\n## **NOTES**\n\n**Preface**\n\n. Adam. B. Ulam, _Stalin: The Man and His Era_ (New York, 1973); Robert C. Tucker: _Stalin as Revolutionary, 1879\u20131929: A Study in History and Personality_ (New York, 1973), and _Stalin in Power: The Revolution from Above, 1928\u20131941_ (New York, 1990).\n\n. Robert Service, _Stalin: A Biography_ (London, 2004); Hiroaki Kuromiya, _Stalin: Profiles in Power_ (New York, 2005); Sarah Davies and James Harris, eds., _Stalin: A New History_ (New York, 2005); Miklos Kun, _Stalin: An Unknown Portrait_ (Budapest and New York, 2003); Ronald Grigor Suny, _Stalin and the Russian Revolutionary Movement: From Koba to Commissar_ (forthcoming from Oxford University Press). Concerning Stalin the dictator and his relations with the rest of the Soviet leadership, see Oleg V. Khlevniuk, _Master of the House: Stalin and His Inner Circle_ (New Haven and London, 2008), and Yoram Gorlizki and Oleg Khlevniuk, _Cold Peace: Stalin and the Soviet Ruling Circle, 1945\u20131953_ (New York, 2004). Some works have attempted to peer into Stalin's inner world: A. J. Rieber, \"Stalin, Man of the Borderlands,\" _American Historical Review_ 106, no. 4 (2001): 1651\u20131691; Erik van Ree, _The Political Thought of Joseph Stalin: A Study in Twentieth-Century Revolutionary Patriotism_ (London and New York, 2002); B. S. Ilizarov, _Tainaia zhizn' Stalina_ (Moscow, 2002); Donald Rayfield, _Stalin and His Hangmen: The Tyrant and Those Who Killed for Him_ (New York, 2005). Many works on the Terror and the Gulag have added to our understanding of Stalin's personal role in organizing mass repression: Jonathan Brent and Vladimir Naumov, _Stalin's Last Crime: The Plot against the Jewish Doctors, 1948\u20131953_ (New York, 2003); V. N. Khaustov and L. Samuel'son, _Stalin, NKVD i repressii. 1936\u20131938_ (Moscow, 2009); J\u00f6rg Baberowski, _Verbrannte Erde: Stalins Herrschaft der Gewalt_ (Munich, 2012). Despite copious literature on World War II, Stalin's role as supreme commander in chief has yet to be adequately investigated. An analogous lacuna exists in regard to the Cold War and Stalin's handling of foreign policy.\n\n. Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ (New York, 1991); Edvard Radzinsky, _Stalin: The First In-Depth Biography Based on Explosive New Documents from Russia's Secret Archives_ (New York, 1997); Simon Sebag Montefiore: _Stalin: The Court of the Red Tsar_ (London, 2003), and _Young Stalin_ (London, 2007).\n\n. Collections of letters have been published as part of the Annals of Communism Series: Lars T. Lih, Oleg V. Naumov, and Oleg Khlevniuk, eds., _Stalin's Letters to Molotov, 1925\u20131936_ (New Haven, 1995), and R. W. Davies et al., eds., _The Stalin-Kaganovich Correspondence, 1931\u20131936_ (New Haven and London, 2003).\n\n. A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008).\n\n. S. V. Deviatov et al., _Moskovskii Kreml' v gody Velikoi Otechestvennoi Voiny_ (Moscow, 2010), pp. 113\u2013114.\n\n. RGASPI, f. 558, op. 1\u201311. (An _opis'_ [op.] is the equivalent of a drawer in a filing cabinet.) Opis' 11 comprises Stalin's personal archive, brought to RGASPI from the Presidential Archive of the Russian Federation (the former Politburo Archive).\n\n. \"Thematic\" folders _(tematicheskie papki)_ are subject-specific folders of documents that were submitted to the Politburo and Stalin; they comprise the main historical component of the Presidential Archive.\n\n. Sergei Khrushchev, ed.: _Memoirs of Nikita Khrushchev,_ vol. 1: _Commissar_ (University Park, PA, 2004); _Memoirs of Nikita Khrushchev,_ vol. 2: _Reformer_ (University Park, PA, 2006); and _Memoirs of Nikita Khrushchev,_ vol. 3: _Statesman_ (University Park, PA, 2007); A. I. Mikoian, _Tak bylo. Razmyshleniia o minuvshem_ (Moscow, 1999); Anastas Ivanovich Mikoyan, _The Memoirs of Anastas Mikoyan_ (Madison, CT, 1988).\n\n. In a splendid review published soon after Mikoyan's memoirs came out in Russian, Michael Ellman convincingly argued that Mikoyan's text had been altered (Michael Ellman, \"The Road from Il'ich to Il'ich,\" _Slavic Review_ 60, no. 1 [2001]: 141). In a response, Mikoyan's son Sergo categorically stated, \"I did not 'correct' my father's stories\" ( _Slavic Review,_ 60, no. 4 [2001]: 917). This vague formulation came with an important subtext. Sergo Mikoyan was not saying that he did not alter the dictated manuscript, leaving open the possibility that he did supplement the initial transcript of his father's dictation with subsequent accounts by the elder Mikoyan that were not \"correct.\" Clearly, any such additions should have been made explicit or, better yet, placed in a footnote.\n\n. See, for example, Sergo Beria, _Beria, My Father: Inside Stalin's Kremlin_ (London, 2001).\n\n. E. Yu. Zubkova, \"O 'detskoi' literature i drugikh problemakh nashei istoricheskoi pamiati,\" in _Istoricheskie issledovaniia v Rossii. Tendentsii poslednikh let,_ ed. G. A. Bordiugov (Moscow, 1996), pp. 155\u2013178.\n\n**The Seats of Stalin's Power**\n\n. Georgy Maksimilianovich Malenkov (1902\u20131988) was a party bureaucrat who worked for many years in the Central Committee apparat. In the late 1930s, he was elevated by Stalin to the highest echelons of power, buoyed by the waves of mass repression. During the dictator's last years, Malenkov served as his deputy within the government and the Central Committee Secretariat. After Stalin's death he was appointed chairman of the Soviet government, an appointment that seemed to label him as Stalin's unofficial heir. However, Malenkov lost out to Khrushchev in the battle for supreme power and was forced into humiliatingly low-level posts before spending his remaining years in retirement. Other unsuccessful rivals for power had the relative democratization of the USSR to thank for their fates. Under Stalin, disgraced politicians generally paid with their lives.\n\nLavrenty Pavlovich Beria (1899\u20131953) began his career in state security. In the early 1930s, Stalin put Beria in charge of Georgia and in 1938 brought him to Moscow and appointed him people's commissar for internal affairs (head of the NKVD, the main agency of state security); as such, he was assigned to purge the ranks of the secret police and wind down the Great Terror. In subsequent years, Beria became one of Stalin's closest associates. He was his deputy within the government and oversaw the Soviet nuclear project, as well as other important divisions of the Soviet system, including the Gulag. After Stalin's death, Beria brought all \"punitive organs\" under his own control. This move alarmed the other Soviet leaders. They closed ranks and had Beria arrested, accused of countless crimes, and shot. Legends circulated that Beria had had special influence over Stalin and that many of the crimes of the Stalinist regime were his handiwork. In fact, Beria was just one of the men who implemented Stalin's orders and did not play a notably independent role in carrying out the mass repression. See Amy Knight, _Beria: Stalin's First Lieutenant_ (Princeton, NJ, 1993).\n\nNikita Sergeevich Khrushchev (1894\u20131971) came to Moscow from Ukraine to study at the Industrial Academy, where he got to know Stalin's wife, Nadezhda Allilueva. This acquaintance provided the first impetus to his career, and he began to advance through the ranks of the Moscow party committee. In the late 1930s, new opportunities for advancement came as other officials succumbed to the mass repression. He was appointed party secretary for Ukraine, one of the most important Soviet republics. After the war, Stalin placed him in charge of the party organization in Moscow. In the wake of Stalin's death, Khrushchev became head of the Central Committee apparat. This post enabled him to outmaneuver Stalin's other political heirs and become the new Soviet leader. However, Khrushchev was not Stalin. His democratic reforms (the Khrushchev Thaw), his condemnation of Stalin's Cult of Personality, and his advocacy of freedom for Gulag prisoners, along with numerous tactical blunders, led to a plot against him. In late 1964, he was deprived of his post by purely legal means, but not of his life. He lived out his days as a pensioner. While in retirement, he dictated his well-known memoirs. See William Taubman, _Khrushchev: The Man and His Era_ (New York, 2003).\n\nNikolai Aleksandrovich Bulganin (1895\u20131975) was among those who rose through the ranks to fill the vacancies created in the Soviet apparat by the Great Terror. Stalin began to promote Bulganin at the end of the war. As a counterweight to career military men, the civil servant Bulganin was placed in senior posts in the defense commissariat and eventually appointed defense minister. Contemporary accounts portray Bulganin as an expressionless functionary who simply followed orders. After Stalin's death, Bulganin chaired the Council of Ministers, succeeding the disgraced Malenkov. However, he picked the losing side during Khrushchev's rise to power and was sent into retirement.\n\n. Vyacheslav Mikhailovich Molotov (1890\u20131986) was one of Stalin's closest comrades-in-arms, their relationship dating back to prerevolutionary times. From then on, Molotov served as Stalin's faithful supporter and played a key role during Stalin's struggle for supreme power. In return, Molotov was appointed to top government posts. In 1930\u20131941, he chaired the Soviet government (the Council of People's Commissars). When Stalin himself took over this post in 1941, Molotov was made his deputy. For many years Molotov was in charge of foreign affairs. Within the country and the party, he was seen as Stalin's heir. For this reason, toward the end of his life, Stalin began to clamp down on Molotov and, in late 1952, eventually expelled him from the ruling circle. Nevertheless, Molotov remained loyal to Stalin even after his death. This loyalty was one source of tension between Molotov and Khrushchev, who encouraged criticism of the Cult of Personality. Molotov lost out to Khrushchev during the decisive clash of 1957. He held a succession of minor posts before being forced into retirement. See Derek Watson, _Molotov and Soviet Government: Sovnarkom, 1930\u201341_ (Basingstoke, UK, 1996).\n\nAnastas Ivanovich Mikoyan (1895\u20131978) was one of the revolutionary and party activists from Transcaucasia who, thanks to Stalin, wound up making a brilliant career in Moscow. For several decades, Mikoyan was in charge of Soviet trade and the food and consumer-goods industry. In late 1952, Mikoyan fell into disgrace, together with Molotov. After Stalin's death, he restored his position and gave his allegiance to Khrushchev. He played an important role in resolving the 1962 Cuban Missile Crisis. After Khrushchev's removal, Mikoyan's career went into decline. Nevertheless, he is considered a model Soviet political survivor, renowned for his adaptability.\n\nKliment Yefremovich Voroshilov (1881\u20131969) was one of Stalin's closest friends during the 1918\u20131920 Civil War. In the mid-1920s, Stalin placed him in charge of the Red Army, a post for which Voroshilov was clearly not well suited. Shortly before the 1941 German invasion, Stalin was forced to replace him. During World War II, Voroshilov formally remained among the country's top leadership; however, he held secondary posts. After Stalin's death, Voroshilov was appointed to the figurehead post of president of the USSR. He supported Molotov and the other Soviet leaders who opposed Khrushchev in 1957 and soon thereafter went into retirement.\n\n. Yoram Gorlizki, \"Ordinary Stalinism: The Council of Ministers and the Soviet Neo-patrimonial State, 1946\u20131953,\" _Journal of Modern History_ 74, no. 4 (2002): 699\u2013736.\n\n. Interview with Admiral I. S. Isakov in K. Simonov, _Glazami cheloveka moego pokoleniia_ (Moscow, 1989), p. 433.\n\n. A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008), p. 7.\n\n. V. Bogomolova et al., comps., _Moskovskii Kreml' tsitadel' Rossii_ (Moscow, 2009), pp. 310\u2013313.\n\n. After Shumiatsky's arrest in 1938, these records were given to Stalin and placed in his personal archive. They have been published in K. M. Anderson et al., comps., _Kremlevskii kinoteatr. 1928\u20131953_ (Moscow, 2005), pp. 919\u20131053.\n\n. Nadezhda Sergeevna Allilueva (1901\u20131932) grew up in the family of a proletarian revolutionary with whom Stalin had long been acquainted. She and Stalin were married in 1919. Allilueva worked in Lenin's secretariat and in the editorial offices of a Moscow journal before enrolling in the Moscow Industrial Academy. Further details can be found in the section on Stalin's family preceding chapter 6 below.\n\n. This and subsequent information about Stalin's dacha comes from _1953 god. Mezhdu proshlym i budushchim_ (exhibition catalogue) (Moscow, 2003), and S. V. Deviatov, A. Shefov, and Iu. Iur'ev, _Blizhniaia dacha Stalina. Opyt istoricheskogo putevoditelia_ (Moscow, 2011).\n\n. Svetlana Alliluyeva, _Twenty Letters to a Friend,_ trans. Priscilla Johnson McMillian (New York, 1967), p. 21.\n\n. Deviatov, Shefov, and Iur'ev, _Blizhniaia dacha Stalina,_ p. 287. Lozgachev has provided information relating to the postwar years, but there is evidence suggesting that Stalin took an active interest in the productivity of the dacha lands in earlier years as well.\n\n. Lazar Kaganovich mentions the existence of such a notebook in F. I. Chuev, _Kaganovich. Shepilov_ (Moscow, 2001), p. 137.\n\n. Letter to Lazar Kaganovich, 24 September 1931. Cited in R. W. Davies, et al., eds., _The Stalin-Kaganovich Correspondence, 1931\u201336_ (New Haven and London, 2003), p. 98.\n\n. In Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 2: _Reformer_ (University Park, PA, 2006), p. 117.\n\n. In Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 1: _Commissar_ (University Park, PA, 2004), p. 290.\n\n. M. Dzhilas [Milovan Djilas], _Litso totalitarizma_ (Moscow, 1992), p. 108.\n\n. From an account by Hungarian leader M\u00e1ty\u00e1s R\u00e1kosi ( _Istoricheskii arkhiv,_ no. 3 [1997]: 117).\n\n. _1953 god. Mezhdu proshlym i budushchim,_ p. 75.\n\n. Andrei Aleksandrovich Zhdanov (1896\u20131948) joined the Bolshevik party before the revolution and afterward held various provincial party posts. In 1934 Stalin brought him to Moscow and made him a Central Committee secretary. After Kirov's murder, Zhdanov replaced him as Leningrad party boss. Until his death he remained one of Stalin's closest comrades-in-arms and enjoyed good relations with the leader. Zhdanov's son was briefly married to Stalin's daughter.\n\n. In Khrushchev, _Memoirs of Nikita Khrushchev,_ vol. 1, pp. 102\u2013103.\n\n. In Khrushchev, _Memoirs of Nikita Khrushchev,_ vol. 2, p. 68.\n\n. Ibid., p. 117.\n\n. Ibid., pp. 146\u2013147.\n\n. Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ (New York, 1991), p. 571.\n\n. This idea is developed in Erik van Ree, _The Political Thought of Joseph Stalin: A Study in Twentieth-Century Revolutionary Patriotism_ (London and New York, 2002).\n\n. Cited in G. Dmitrov, _Dnevnik_ (Sophia, 1997), p. 128.\n\n. Cited in V. M. Berezhkov, _Riadom so Stalinym_ (Moscow, 1999), p. 371. Berezhkov was Stalin's interpreter.\n\n**Chapter 1. Before the Revolution**\n\n. L. M. Spirin, \"Kogda rodilsia Stalin: Popravki k ofitsial'noi biografii,\" _Izvestiia,_ 25 June 1990; _Izvestiia TsK KPSS,_ no. 11 (1990): 132\u2013134.\n\n. A. Ostrovskii, _Kto stoial za spinoi Stalina?_ (Moscow, 2002), pp. 88\u201389. Ostrovskii's book was the first biography to focus on Stalin's youth and was based on newly discovered documents from Moscow and Georgian archives. Other works appeared later: Miklos Kun, _Stalin: An Unknown Portrait_ (Budapest and New York, 2003); Simon Sebag Montefiore, _Young Stalin_ (London, 2007); Ronald Grigor Suny, _Stalin and the Russian Revolutionary Movement: From Koba to Commissar_ (forthcoming from Oxford University Press). My account of Stalin's early life draws on these books to varying degrees.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 86\u201388, 93, 99.\n\n. RGASPI, f. 558, op. 11, d. 878, l. 73. Unless otherwise noted, translations are by Nora Favorov.\n\n. R. G. Suny, \"Beyond Psychohistory: The Young Stalin in Georgia,\" _Slavic Review_ 46, no. 1 (1991): 52.\n\n. Stalin, _Works,_ vol. 13, p. 115. Interview with the German author Emil Ludwig, 13 December 1931.\n\n. Cited in Iu. G. Murin, comp., _Iosif Stalin v ob\"iatiiakh sem'i. Iz lichnogo arkhiva_ (Moscow, 1993), pp. 6\u201319.\n\n. RGASPI, f. 558, op. 11, d. 1549, l. 83.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 96\u201397, 102\u2013104.\n\n. RGASPI, f. 558, op. 11, d. 876, l. 12.\n\n. RGASPI, f. 558, op. 4, d. 4, l. 1; d. 5, l. 1.\n\n. Cited in Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ (New York, 1991), pp. 7\u20138.\n\n. L. D. Trotsky, _Stalin,_ (Benson, VT, 1985), vol. 1, pp. 32\u201333.\n\nLev Davidovich Trotsky (1879\u20131940) was, for a while, perceived both within the fledgling Soviet state and internationally as second only to Lenin in leading the Bolshevik revolution. The peak of his glory came during the Civil War, in which he led the Red Army to victory. After the war, Trotsky took an active part in the struggle for power and influence that erupted among the Soviet leaders. In 1928, after losing this struggle, Trotsky was sent into exile. He remained politically active in emigration and worked to expose his political nemesis, Stalin, on whose orders he was killed in 1940 in Mexico by a Soviet agent.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 108\u2013111.\n\n. Ibid., pp. 124\u2013125.\n\n. Stalin, _Works,_ vol. 13, pp. 115\u2013116. Interview with the German author Emil Ludwig, 13 December 1931.\n\n. Cited in V. Kaminskii and I. Vereshchagin, \"Detstvo i iunost' vozhdia: Dokumenty, zapiski, rasskazy,\" _Molodaia gvardiia,_ no. 12 (1939): 65.\n\n. Robert C. Tucker, _Stalin as Revolutionary, 1879\u20131929: A Study in History and Personality_ (New York, 1973), pp. 80\u201382.\n\n. RGASPI, f. 558, op. 4, d. 600, ll. 1\u20137; f. 71; op. 10, d. 266, ll. 7\u201311.\n\n. RGASPI, f. 558, op. 4, d. 32, ll. 1\u20132.\n\n. Suny, _Stalin and the Russian Revolutionary Movement,_ ch. 3.\n\n. RGASPI, f. 558, op. 4, d. 53, ll. 1\u201315; Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 148.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 149.\n\n. Kaminskii and Vereshchagin, \"Detstvo i iunost' vozhdia,\" pp. 84\u201385.\n\n. RGASPI, f. 558, op. 4, d. 53, l. 13.\n\n. Ibid., op. 4, d. 60, ll. 1\u20133.\n\n. Ibid., op. 11, d. 879, l. 45.\n\n. Ibid., op. 4, d. 65, ll. 1\u20134.\n\n. Trotsky, _Stalin,_ vol. 1, p. 44.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 154\u2013155.\n\n. A. J. Rieber, \"Stalin, Man of the Borderlands,\" _American Historical Review_ 106, no. 5 (2001): 1651\u20131691; Alfred J. Rieber, \"Stalin as Georgian: The Formative Years,\" in _Stalin: A New History,_ ed. Sarah Davies and James Harris (Cambridge, 2005), pp. 18\u201344.\n\n. I. Baberovski [J. Baberowski], _Vrag est' vezde. Stalinizm na Kavkaze_ (Moscow, 2010), p. 15.\n\n. Documents from Boris Nicolaevsky's archive published by Iu. G. Fel'shtinskii and G. I. Cherniavskii in _Voprosy istorii,_ no. 14 (2012): 16.\n\n. RGASPI, f. 558, op. 4, d. 72, l. 9.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 188\u2013189.\n\n. RGASPI, f. 558, op. 4, d. 619, ll. 175\u2013177.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 212\u2013218.\n\n. Erik van Ree, \"The Stalinist Self: The Case of Ioseb Jughashvili (1898\u20131907),\" _Kritika_ 11, no. 2 (2010): 265\u2013266; Suny, _Stalin and the Russian Revolutionary Movement,_ ch. 4.\n\n. Erik van Ree, \"Reluctant Terrorists? Transcaucasian Social-Democracy, 1901\u20131909,\" _Europe-Asia Studies_ 40, no. 1 (2008); Suny, _Stalin and the Russian Revolutionary Movement,_ ch. 9.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 254.\n\n. RGASPI, f. 558, op. 11, d. 896, l. 115.\n\n. For more details on the heist, see Montefiore, _Young Stalin._ See also Suny, _Stalin and the Russian Revolutionary Movement,_ ch. 11. Miklos Kun has uncovered some evidence that Stalin assisted in the preparations for Kamo's operation ( _Stalin,_ pp. 77\u201379).\n\n. Documents from Boris Nicolaevsky's archive published by Iu. G. Fel'shtinskii and G. I. Cherniavskii in _Voprosy istorii,_ no. 7 (2010): 34, and no. 9 (2010): 11.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 292.\n\n. Z. I. Peregudova, _Politicheskii sysk Rossii (1880\u20131917 gg.)_ (Moscow, 2000), pp. 242\u2013274.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 329\u2013330.\n\n. Cited in Peregudova, _Politicheskii sysk Rossii,_ p. 246.\n\n. Roman Vatslavovich Malinovsky (1876\u20131918) was a metalworker, labor union activist, and member of the Bolshevik party who enjoyed Lenin's special patronage. In 1912 he was elected to the State Duma and in 1913 became chairman of the Duma's Bolshevik faction. Meanwhile, he served many years as a police double agent. Under threat of exposure, he fled Russia in 1914. In 1918 he returned to Soviet Russia hoping to be pardoned. Instead, he was shot.\n\n. These letters were opened by the police and therefore survive in police archives. Copies of them are also in the Stalin Collection (Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 396\u2013398; RGASPI, f. 558, op. 11, d. 1288, ll. 12\u201314, 18, 28, 32\u201335).\n\n. Letter to Roman Malinovsky in late November 1913.\n\n. Letter to T. A. Slavotinskaia, dated 20 November 1913.\n\n. RGASPI, f. 558, op. 1, d. 52, l. 1; Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 402\u2013403.\n\n. RGASPI, f. 558, op. 1, d. 5394, ll. 2\u20133; A. V. Kvashonkin et al., comps., _Bol'shevistskoe rukovodstvo. Perepiska. 1912\u20131927_ (Moscow, 1996), p. 19.\n\n. Ia. M. Sverdlov, _Izbrannye proizvedeniia,_ (Moscow, 1957), vol. 1, p. 227.\n\n. A. S. Allilueva, _Vospominaniia_ (Moscow, 1946), p. 115.\n\n. In Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 2: Reformer (University Park, PA, 2006), p. 132. Translation slightly edited.\n\n. Sverdlov, _Izbrannye proizvedeniia,_ vol. 1, p. 280.\n\n. RGASPI, f. 558, op. 11, d. 1288, ll. 15\u201316; B. S. Ilizarov, _Tainaia zhizn' Stalina_ (Moscow, 2002), pp. 289, 291, 294\u2013297; Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 393.\n\n. RGASPI, f. 558, op. 11, d. 773, ll. 79\u201382; Ilizarov, _Tainaia zhizn' Stalina,_ pp. 297\u2013298.\n\n. In any event, Stalin soon ceased to have anything to do with Pereprygina. After he left exile she married and was later widowed with eight children (Ilizarov, _Tainaia zhizn' Stalina,_ p. 310).\n\n. Letter to O. Ye. Allilueva dated 25 November 1915. RGASPI, f. 558, op. 1, d. 55, l. 2; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 21.\n\n. Trotsky, _Stalin,_ vol. 1, pp. 248\u2013249.\n\n. Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 17\u201320; Ostrovskii, _Kto stoial za spinoi Stalina?,_ pp. 397\u2013401, 412\u2013413, 415.\n\n. RGASPI, f. 558, op. 1, d. 54, l. 1.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 49 (Moscow, 1970), pp. 101, 161.\n\n**The Bulwarks of Stalin's Power**\n\n. There is a tradition that views Stalin's final illness and death as the result of a poisoning organized by Beria. One of the most recent attempts to assess the medical evidence for this view can be found in Jonathan Brent and Vladimir Naumov, _Stalin's Last Crime: The Plot against the Jewish Doctors, 1948\u20131953_ (New York, 2003).\n\nThe basic events of Stalin's last days can be retraced by drawing on multiple sources. In addition to the well-known reminiscences of Khrushchev, who was among the leaders that kept watch over the dying Stalin (Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 1: _Commissar_ [University Park, PA, 2004], pp. 147\u2013149), new sources have appeared, including accounts by Stalin's bodyguards recorded by Dmitri Volkogonov and Edvard Radzinsky (Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ [New York, 1991], pp. 571\u2013572; Edvard Radzinsky, _Stalin: The First In-Depth Biography Based on Explosive New Documents from Russia's Secret Archives_ [New York, 1997], pp. 566\u2013572). Here I make use of all three publications.\n\n. Here and below, on the topic of the Main Guard Directorate, see RGASPI, f. 17, op. 166, d. 858, ll. 2\u201320. It is unclear from the documents in question whether this information applies to all of Stalin's dachas or only to the one in Volynskoe. In any case, the guards and servants were primarily concentrated at the Volynskoe dacha, where Stalin lived.\n\n. S. V. Deviatov et al., _Garazh osobogo naznacheniia. 1921\u20132011_ (Moscow, 2011), pp. 162\u2013163.\n\n. RGASPI, f. 17, op. 162, d. 9, l. 54; V. N. Khaustov et al., comps., _Lubianka. Stalin i VChK-GPU-OGPU-NKVD. Ianvar' 1922\u2013dekabr' 1936_ (Moscow, 2003), pp. 255\u2013256.\n\n. According to a report by senior officials of the Joint State Political Directorate (OGPU) to Stalin, the agent was prevented from making an attempt on Stalin's life by an undercover OGPU agent who had infiltrated the organization and was accompanying the foreign agent. Under interrogation, the foreign agent stated that during an initial attempt he was simply unable to grab his revolver, which was hidden deep under his clothing. The rather large security detail accompanying Stalin prevented him from making a second attempt. (\"Zapiska OGPU Stalinu. 18 noiabria 1931 g.,\" _Istochnik,_ no. 3 [1996]: 161\u2013162; Khaustov et al., _Lubianka. Stalin i VChK-GPU-OGPU-NKVD,_ p. 286.)\n\n. _Gosudarstvennaia okhrana Rossii. 1881\u20132006_ (exhibition catalogue) (Moscow, 2006), pp. 47\u201349.\n\n. Sergei Mironovich Kirov (1886\u20131934) was a Russian revolutionary and Civil War figure. In 1921\u20131926 he served as party chief in Azerbaijan. His career benefited from his years as one of Stalin's clients in Transcaucasia and the personal friendship that developed between the two. In 1926, after the crushing of the opposition, Kirov was appointed to replace Zinoviev as head of the Leningrad party organization, a position that led to his elevation to candidate member of the Politburo. On 1 December 1934 he was killed by a lone gunman. It was long believed that Kirov's murder was arranged by Stalin, but most historians have since rejected this possibility.\n\n. Nikolai Sidorovich Vlasik (1896\u20131967) was born into a peasant family in Belarus, received an elementary-school education, and later supported himself as an unskilled laborer. He fought in the tsarist army during World War I and later joined the Red Army. In 1919 he went to work for state security, where he rose through the ranks. The numerous vacancies created by the mass arrests of 1937\u20131938 accelerated Vlasik's career. In 1952 he was arrested, and two years after Stalin's death he was sentenced to ten years in exile. He was pardoned in 1956.\n\n. After a lengthy investigation, the soldier was shot in 1950.\n\n. S. V. Deviatov et al., _Moskovskii Kreml' v gody Velikoi Otechestvennoi Voiny_ (Moscow, 2010), pp. 161, 164\u2013167.\n\n. Figures are for 1950. E. Iu. Zubkova et al., comps., _Sovetskaia zhizn'. 1945\u20131953_ (Moscow, 2003), p. 501; V. P. Popov, _Rossiiskaia derevnia posle voiny [iiun' 1945\u2013mart 1953]_ (Moscow, 1993), p. 146.\n\n. N. V. Petrov, _Pervyi predsedatel' KGB Ivan Serov_ (Moscow, 2005), pp. 87\u201389.\n\n. From Vlasik's testimony at his 1955 trial; V. M. Loginov, _Teni Stalina. General Vlasik i ego soratniki_ (Moscow, 2000), p. 152.\n\n. RGASPI, f. 17, op. 166, d. 858, ll. 2\u20138.\n\n. Semen Denisovich Ignatiev (1904\u20131983) was born into a peasant family and began his career in the Komsomol (Communist Youth League). After studying at the Industrial Academy in 1935, he landed a job in the Central Committee apparat. For many years he headed various regional party organizations. In 1950 he was placed in charge of the Central Committee department that handled party personnel matters, an important post. In 1951, after a wave of arrests within the leadership of the USSR Ministry of State Security, Stalin appointed Ignatiev to head this institution. Under Stalin's orders, Ignatiev falsified a number of political cases. After Stalin's death this action almost cost him his career or even his life, but Khrushchev's support saved him. Ignatiev was sent to work in the provinces and, in 1960, into retirement.\n\n. Russian State Archive of Contemporary History (RGANI), f. 5, op. 29, d. 3, l. 2; d. 16, ll. 94, 108.\n\n. On the sources of the statistics offered here, see O. Khlevniuk, _Stalin u vlasti. Prioritety i rezul'taty politiki diktatury. Istoriia stalinizma: Itogi i problemy izucheniia_ (Moscow, 2011), pp. 63\u201365.\n\n. In early 1937 the total population of the USSR was 162 million, and in early 1953 it reached 188 million. The adult population was, of course, much lower, totaling in 1937, for example, approximately 100 million.\n\n. Soviet security services have gone through numerous reorganizations and renamings. By tradition, they continued to be called by their initial acronym\u2014ChK ( _chrezvychainaia komissiia,_ or extraordinary commission). This is the origin of the term \"cheka\" or \"chekist.\" Stalin himself often used this designation.\n\n. Grigory Ivanovich Kulik (1890\u20131950) fought alongside Stalin during the Civil War. With Stalin's patronage, he enjoyed a successful military career and in 1940 was elevated to marshal. During the war with Germany, like many other Civil War\u2013era commanders, he did not acquit himself particularly well. In 1942 he was tried and stripped of his rank and given a series of junior command positions. Stalin's lack of trust in Kulik was reciprocated. In 1947 he was arrested along with several other generals who had criticized Stalin in frank discussions with one another. In 1950 he was shot.\n\n. May 1940 letter from the chairman of the Party Control Commission, Andrei Andreev, to Stalin in regard to the Kulik case; K. A. Stoliarov, _Palachi i zhertvy_ (Moscow, 1998), pp. 272\u2013276. RGASPI, f. 73, op. 2, d. 17, ll. 128\u2013148.\n\n. Stoliarov, _Palachi i zhertvy,_ pp. 267\u2013271.\n\n. Solomon Mikhailovich Mikhoels (1890\u20131948) was a stage director, actor, and leader of the Jewish community. During World War II, he headed the Soviet Jewish Anti-Fascist Committee, which mobilized strong support for the Soviet Union in the West. The fact that he was awarded the Stalin Prize (the highest honor granted to cultural figures) immediately after the war testifies to the importance of his services. Nevertheless, soon thereafter Mikhoels became one of the first victims of Stalin's changing foreign policy priorities and the launching of an anti-Semitic campaign in the USSR.\n\n. G. V. Kostyrchenko, _Tainaia Politika Stalina. Vlast' i antisemitizm_ (Moscow, 2001), pp. 388\u2013392.\n\n. N. V. Petrov, _Palachi_ (Moscow, 2011), pp. 66\u201368.\n\n. Ignatiev related this statement during testimony given on 27 March 1953, after Stalin's death (ibid., p. 307).\n\n**Chapter 2. In Lenin's Shadow**\n\n. Lev Borisovich Kamenev (1883\u20131936), the son of an engineer, studied law at Moscow University before being expelled for revolutionary activities. He was one of Lenin's closest associates. Kamenev first met Stalin when they were both engaged in revolutionary work in Transcaucasia. After the 1917 revolution, Kamenev held a number of senior posts within the Soviet government and was among those contending for power after Lenin's death. He became an opposition leader in the 1920s. Once Stalin solidified his victory over the opposition, he dealt brutally with his old friend. In late 1934, Kamenev and his fellow oppositionists were arrested on fabricated charges that they had been involved in Kirov's murder. In August 1936, Kamenev was convicted of espionage and terrorism in the first of a series of major show trials and put to death.\n\n. Lars T. Lih, Oleg V. Naumov, and Oleg Khlevniuk, eds., _Stalin's Letters to Molotov, 1925\u20131936_ (New Haven, 1995), pp. 101\u2013103, 131\u2013132.\n\n. There is a vast body of literature on Bolshevik activities during the Russian revolutionary period, including the following: E. N. Burdzhalov, _Russia's Second Revolution: The February 1917 Uprising in Petrograd,_ trans. and ed. D. J. Raleigh (Bloomington and Indianapolis, 1967); Alexander Rabinowitch, _The Bolsheviks Come to Power_ (Chicago and London, 2004); Richard Pipes, _The Russian Revolution_ (New York, 1990). For Stalin's role in the revolution, see Robert M. Slusser, _Stalin in October: The Man Who Missed the Revolution_ (Baltimore and London, 1987), and Ronald Grigor Suny, _Stalin and the Russian Revolutionary Movement: From Koba to Commissar_ (Oxford University Press, forthcoming), chs. 18 and 19.\n\n. Cited in A. V. Kvashonkin et al., comps., _Bol'shevistskoe rukovodstvo. Perepiska. 1912\u20131927_ (Moscow, 1996), p. 16.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 31 (Moscow, 1969), pp. 11\u201322, 504.\n\n. Ibid., pp. 103\u2013112.\n\n. Cited in N. N. Sukhanov, _Zapiski o revoliutsii,_ vol. 2, bk. 3 (Moscow, 1991), p. 16.\n\n. Cited in _Sed'maia (Aprel'skaia) Vserossiiskaia konferntsiia RSDPR (bol'shevikov). Petrogradskaia obshchegorodskaia konferentsiia RSDPR (bol'shevikov). Protokoly_ (Moscow, 1958), p. 323.\n\n. Grigory Yevseevich Zinoviev (1883\u20131936) was one of Lenin's closest comrades-in-arms. After the revolution, he headed the Leningrad party organization and the Comintern. Failing to take over leadership of the party after Lenin's death, he became an opposition leader and suffered persecution as the opposition was routed. In 1934, Zinoviev, along with Kamenev, was arrested based on fabricated evidence of complicity in Kirov's murder. In August 1936, he and Kamenev were convicted at the first Moscow show trial and shot.\n\n. Speech by Stalin, 3 August 1917, at the Russian Social Democratic Workers' Party (RSDRP) Sixth Party Congress; _Shestoi s\"ezd Rossiiskoi sotsial-demokraticheskoi rabochei partii (bol'shevikov). Avgust 1917 g. Protokoly_ (Moscow, 1958), p. 250.\n\n. RGASPI, f. 558, op. 11, d. 890, l. 8.\n\n. For a detailed investigation of these events, including evidence based on recently discovered documents, see V. T. Loginov, _Neizvestnyi Lenin_ (Moscow, 2010), pp. 261\u2013264.\n\n. Statements by Zinoviev and Kamenev on 11 October 1917; _Protokoly Tsentral'nogo Komiteta RSDRP(b). Avgust 1917\u2013fevral' 1918_ (Moscow, 1958), pp. 87\u201392.\n\n. RGASPI, f. 558, op. 1, d. 66, l. 1.\n\n. _Protokoly Tsentral'nogo Komiteta RSDRP(b). Avgust 1917\u2013fevral' 1918,_ p. 115.\n\n. R. W. Davies, Mark Harrison, and S. G. Wheatcroft, eds., _The Economic Transformation of the Soviet Union, 1913\u20131945_ (Cambridge, 1994), pp. 62\u201364.\n\n. Protocols of Politburo meetings; RGASPI, f. 17, op. 3, dd. 1\u2013125.\n\n. Letter from Stalin to Lenin and Trotsky, 22 June 1918; RGASPI, f. 558, op. 1, d. 5403, l. 1; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 40.\n\n. Letter from Stalin to Lenin, 7 July 1918; RGASPI, f. 558, op. 1, d. 248, l. 1; I. V. Stalin, _Works,_ vol. 4 (Moscow, 1954), pp. 120\u2013121.\n\n. Telegram from Stalin to Trotsky and Lenin, 11 July 1918; RGASPI, f. 558, op. 1, d. 1812, ll. 1\u20132; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 42.\n\n. Letter from Stalin to Lenin, 3 October 1918; RGASPI, f. 558, op. 1, d. 5410, l. 1; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 52.\n\n. RGASPI, f. 558, op. 1, d. 5718, ll. 177, 178, 191, 195, 197.\n\n. Ibid., ll. 196\u2013198.\n\n. Speech by Voroshilov at the Eighth Party Congress in March 1919; _Izvestiia TsK KPSS,_ no. 11 (1989): 160.\n\n. Letter from Stalin to Lenin, 31 August 1918; RGASPI, f. 558, op. 1, d. 5408, l. 4; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 46.\n\n. I. S. Rat'kovskii, _Krasnyi terror i deiatel'nost' VChK v 1918 godu_ (St. Petersburg, 2006), pp. 151, 170.\n\n. _Izvestiia TsK KPSS,_ no. 11 (1989): 157, 168.\n\n. Cited in Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 54.\n\n. Ibid., pp. 52\u201353.\n\n. I. V. Stalin, _Works,_ vol. 4 (Moscow, 1947), p. 271.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 50 (Moscow, 1970), p. 389.\n\n. RGASPI, f. 558, op. 1, d. 1815, ll. 2\u20134; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 142\u2013143.\n\n. RGASPI, f. 558. op. 1, d. 5521, l. 2. Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 148.\n\n. RGASPI, f. 558, op. 1, d. 4137, l. 1; d. 1943, l. 1; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 155.\n\n. RGASPI, f. 558, op. 1, d. 1961, ll. 1\u20132; Stalin, _Works,_ vol. 4, p. 358.\n\n. RGASPI, f. 558, op. 1, d. 4681, l. 1.\n\n. RGASPI, f. 558, op. 1, d. 4458, ll. 1\u20133; Stalin, _Works,_ vol. 4, pp. 360\u2013362.\n\n. RGASPI, f. 558, op. 11, d. 126, l. 4.\n\n. Ibid., op. 1, d. 5213, l. 1; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 156.\n\n. RGASPI, f. 17, op. 3, d. 106, l. 5.\n\n. Ibid., ll. 3, 4.\n\n. _Izvestiia TsK KPSS,_ no. 3 (1991): 167.\n\n. _Deviataia konferentsiia RKP(b). Protokoly_ (Moscow, 1972), pp. 60\u201361, 76\u201377; Iu. N. Amiantov et al., comps., _V. I. Lenin. Neizvestnye dokumenty. 1891\u20131922_ (Moscow, 1999), pp. 382, 390.\n\n. RGASPI, f. 558, op. 1, d. 5541, ll. 1\u20132; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 160\u2013161.\n\n. Stalin's involvement in organizing the so-called Ukrainian Labor Army during the winter and spring of 1920 was an attempt to militarize labor by using the army as a labor force, primarily in the coal mines of Ukraine.\n\n. Meeting of a section of the Twelfth RKP(b) Congress on the nationalities question, 25 April 1923; _Izvestiia TsK KPSS,_ no. 4 (1991): 170. For a detailed account of Stalin's work in the People's Commissariat for Nationalities, see Jeremy Smith, \"Stalin as Commissar of Nationalities,\" in _Stalin: A New History,_ ed. Sarah Davies and James Harris (New York, 2005), pp. 45\u201362, and V. Denningkhaus [Victor D\u00f6nninghaus], _V teni \"bol'shogo brata.\" Zapadnye natsional'nye men'shinstva v SSSR. 1917\u20131938 gg._ (Moscow, 2011), pp. 84\u201391.\n\n. RGASPI, f. 17, op. 3, d. 234, l. 2.\n\n. Ibid., d. 310, l. 2.\n\n. Politburo resolution, 19 October 1922; RGASPI, f. 17, op. 3, d. 318, l. 4.\n\n. Grigory Konstantinovich Ordzhonikidze (1886\u20131937) was one of Stalin's closest friends and comrades-in-arms. In the 1920s he was a top party leader in Transcaucasia before being transferred to Moscow to take up the important post of chairman of the Party Control Commission. In this capacity he helped in Stalin's climb to power. In the 1930s Ordzhonikidze was put in charge of Soviet heavy industry. He tried to oppose Stalin's repression of key personnel, leading to conflict between the two men. In February 1937 Ordzhonikidze committed suicide. How he died became widely known only after Stalin's death. See Oleg V. Khlevniuk, _In Stalin's Shadow: The Career of \"Sergo\" Ordzhonikidze_ (New York, 1995).\n\n. Letters from Nazaretian to Ordzhonikidze, 14 June and after 9 August 1922; RGASPI, f. 85, op. 1c, d. 13, ll. 6, 10; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 256, 257, 262, 263.\n\n. Letters from Nazaretian to Ordzhonikidze, 12 July and after 9 August 1922; RGASPI, f. 85, op. 1c, d. 13, ll. 7, 10; Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 259, 263.\n\n. Letter from Nazaretian to Ordzhonikidze after 9 August 1922; RGASPI, f. 85, op. 1c, d. 13, l. 10. Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ p. 263.\n\n. Reminiscence by N. A. Uglanov, written in January 1925, at a time when Stalin had not yet established his sole power; _Izvestiia TsK KPSS,_ no. 4 (1989): 196.\n\n. Nikolai Ivanovich Bukharin (1888\u20131938) was a Bolshevik leader and theoretician. He took Stalin's side in the confrontation with Trotsky, Zinoviev, and Kamenev, but after Stalin was victorious over these oppositionists, Bukharin himself became Stalin's victim. Bukharin advocated a more moderate course and a gradual transition out of the NEP. Stalin labeled Bukharin and his supporters as \"right deviationists.\" The expulsion of the rightists from the party's leadership helped Stalin solidify his dictatorship. Bukharin was arrested in 1937 and shot the following year. (See Stephen F. Cohen, _Bukharin and the Bolshevik Revolution: A Political Biography, 1888\u20131938_ [New York, 1973]; Paul R. Gregory, _Politics, Murder, and Love in Stalin's Kremlin: The Story of Nikolai Bukharin and Anna Larina_ [Stanford, CA, 2010]).\n\n. Cited in _Izvestiia TsK KPSS,_ no. 12 (1989): 198. For another version of Ulianova's reminiscence, see _Izvestiia TsK KPSS,_ no. 3 (1991): 188.\n\n. _Izvestiia TsK KPSS,_ no. 4 (1989); RGASPI, f. 17, op. 3, d. 303, l. 5.\n\n. Cited in _Izvestiia TsK KPSS,_ no. 12 (1989): 198. Maria Ulianova's memoirs were found among her papers after her death. They were obviously not intended for publication. Their candor and confessional nature add to their credibility as a source.\n\n. _Izvestiia TsK KPSS,_ no. 9 (1989): 191\u2013216.\n\n. Ibid., p. 209.\n\n. Ibid., no. 12 (1989): 191.\n\n. Ibid., pp. 189, 191.\n\n. Cited in ibid., pp. 198\u2013199.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 45 (Moscow, 1970), p. 345.\n\n. Ibid., p. 346.\n\n. Feliks Edmundovich Dzerzhinsky (1877\u20131926) was active in the revolutionary movement in Russia and spent many years in exile, prison, and labor camps. After the revolution he headed the Emergency Commission or Cheka, the Bolsheviks' notorious state security organization. In the 1920s, while still head of the political police, he also ran the commissariats of transport and industry. He was still active at the time of his death from heart failure.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 54 (Moscow, 1975), p. 329.\n\n. Ibid., pp. 329\u2013330.\n\n. Robert C. Tucker, _Stalin as Revolutionary, 1879\u20131929: A Study in History and Personality_ (New York, 1973), p. 277.\n\n. V. I. Lenin, _Polnoe sobranie sochinenii,_ vol. 54, p. 330.\n\n. _Izvestiia TsK KPSS,_ no. 9 (1990): 151; emphasis by Kamenev.\n\n. Ibid., no. 12 (1989): 193.\n\n. Ibid., no. 9 (1990): 151\u2013152.\n\n. V. A. Sakharov, _Politicheskoe zaveshchanie Lenina: Real'nosti istorii i mify politiki_ (Moscow, 2003). See also a critical discussion of this book in _Otechestvennaia istoriia,_ no. 2 (2005): 162\u2013174.\n\n. Moshe Lewin, _Lenin's Last Struggle_ (New York, 1968).\n\n. Cited in V. P. Vilkova, comp., _RKP(b). Vnutripartiinaia bor'ba v dvadtsatye gody. Dokumenty i materialy. 1923_ (Moscow, 2004), p. 129; emphasis by Zinoviev.\n\n. Ibid., pp. 135\u2013136; emphasis by Stalin.\n\n. Transcript of a discussion of the international situation at the 21 August 1923 Politburo meeting. _Istochnik,_ no. 5 (1995): 118, 124.\n\n. Ibid., p. 126.\n\n. Aleksei Ivanovich Rykov (1881\u20131938) was a well-known Bolshevik who served as the Soviet premier after Lenin's death. An economic moderate, he joined forces with Stalin in opposing Trotsky, Zinoviev, and Kamenev. Together with Bukharin, Rykov was accused of \"right deviation\" and removed from the leadership. He was arrested in 1937 and put to death in 1938.\n\n. Ibid.\n\n. Vilkova, _RKP(b). Vnutripartiinaia bor'ba,_ pp. 147\u2013151.\n\n. _Trinadtsatyi s\"ezd PKP(b). Stenograficheskii otchet_ (Moscow, 1963), pp. xxi\u2013xxii.\n\n. RGASPI, f. 558, op. 11, d. 126, l. 68.\n\n. V. Nadtocheev, \"'Triumvirat' ili 'semerka'?\" in _Trudnye voprosy istorii, ed._ V. V. Zhuravlev (Moscow, 1991), pp. 68\u201370.\n\n. _Izvestiia TsK KPSS,_ no. 8 (1991): 182.\n\n. RGASPI, f. 558, op. 11, d. 777, ll. 27\u201328.\n\n. Letters from Kirov to Ordzhonikidze dated 10 and 16 January 1926. Kvashonkin et al., _Bol'shevistskoe rukovodstvo,_ pp. 315, 318.\n\n. Lih, Naumov, and Khlevniuk, _Stalin's Letters to Molotov,_ pp. 115\u2013116.\n\n. A. G. Egorov, ed., _KPSS v rezoliutsiiakh i resheniiakh s\"ezdov, konferentsii i plenumov TsK,_ vol. 4 (Moscow, 1984), pp. 49\u201350.\n\n. See, for example, Stalin's letter to Rykov, Voroshilov, and Molotov dated 20 September 1927; RGASPI, f. 558, op. 11, d. 797, ll. 84\u201385.\n\n. Valerian Valerianovich Osinsky (1887\u20131938) was an Old Bolshevik who took part in various opposition movements and was a follower of Trotsky at one point. Soon after the departure mentioned in the letter to Stalin, Osinsky was removed as head of the Central Statistical Directorate. Nevertheless, in later years he held various senior economic posts. He was shot during the Terror.\n\n. Vladimir Mikhailovich Smirnov (1887\u20131937) was a long-standing party member and an active participant in the revolution and Civil War who became involved in the opposition in the 1920s. In 1928 he was exiled to the Ural region for three years, a term ultimately extended to 1935, at which point he was again arrested. He was shot in 1937.\n\n. Timofei Vladimirovich Sapronov (1887\u20131937) was a long-standing party member and a Moscow Bolshevik leader. After the revolution he held senior government posts. In the 1920s he joined the opposition. In 1928 he was exiled to the Arkhangelsk region for three years. The term of his exile was extended to 1935, as was Smirnov's. In 1935 he was again arrested, and in 1937 he was shot.\n\n. Yuly Osipovich Martov (1873\u20131923) was a leader of the Social Democratic movement in Russia. He collaborated with Lenin during the early stages of his revolutionary career, but in 1903 the two men broke off relations, and later Martov headed the Menshevik party. He participated in the revolutionary movement in Russia but condemned the 1917 Bolshevik overthrow of the Provisional Government. He later tried to work with the Bolsheviks and democratize the Bolshevik dictatorship. In 1920 he was sent abroad and later died of tuberculosis.\n\n. Osinsky's letter and Stalin's following response are in RGASPI, f. 558, op. 11, d. 780, ll. 12\u201314; _Istochnik,_ no. 6 (1994): 88.\n\n. Grigory Yakovlevich Sokolnikov (1888\u20131939), a long-standing party member, escaped abroad after being exiled to Siberia. After the revolution he became a member of the top leadership. His greatest success was the monetary reforms he introduced during the 1920s, which provided Soviet Russia with a stable currency. Sokolnikov was subjected to persecution due to his involvement with the opposition. In 1927 he announced his break with the opposition and for some time held various senior government posts. He was shot during the Stalinist Terror.\n\n. During his speech to the Fifteenth Party Congress in December 1927, Stalin again spoke of an intervention being prepared against the USSR and drew an analogy with the shooting in Sarajevo (I. V. Stalin, _Works,_ vol. 10 [Moscow, 1949], pp. 281, 288).\n\n. RGASPI, f. 558, op. 11, d. 71, ll. 2\u20134ob.\n\n. Yan Ernestovich Rudzutak (1887\u20131938) was a long-standing Bolshevik who spent years in tsarist prisons. After the revolution he held senior party and government posts before being shot during the Stalinist Terror.\n\n. RGASPI, f. 558, op. 11, d. 767, ll. 35\u201339, 45\u201348, 56\u201360.\n\n. Mikhail Ivanovich Kalinin (1875\u20131946) was a long-standing Bolshevik who shortly after the revolution was appointed chairman of the Soviet parliament and held the largely figurehead post of president of the USSR until his death. One of the more moderate members of the Bolshevik leadership, he nevertheless submitted to power. After some wavering, he threw his support behind Stalin. Kalinin's wife was arrested in the 1930s and released shortly before her husband's death.\n\n. RGASPI, f. 558, op. 11, d. 767, ll. 35\u201339, 45\u201348; d. 71. ll. 11, 13\u201314.\n\n. Molotov uses this term since not only Politburo members took part in voting, but also the chairman of the Party Control Commission, Ordzhonikidze, whose post excluded him from Politburo membership.\n\n. RGASPI, f. 558, op. 11, d. 767, ll. 56\u201360.\n\n. Cited in Lih, Naumov, and Khlevniuk, _Stalin's Letters to Molotov,_ p. 139.\n\n. RGASPI, f. 558, op. 11, d. 1110, l. 181.\n\n**A World of Reading and Contemplation**\n\n. RGASPI, f. 558, op. 11, d. 105, ll. 20\u2013126; d. 117, ll. 1\u2013173.\n\n. Ibid., op. 11, d. 70, ll. 85\u2013114.\n\n. B. S. Ilizarov, _Tainaia zhizn' Stalina_ (Moscow, 2002), p. 143.\n\n. M. Ia. Vaiskopf, _Pisatel' Stalin_ (Moscow, 2000), pp. 17\u201322.\n\n. RGASPI, f. 558, op. 3, dd. 1\u2013392. There exists a legal document _(akt)_ instructing that all of Stalin's books with notations be placed in his archive. Books from Stalin's Kremlin and dacha libraries that did not contain any handwritten markings were placed in the library of the Institute of Marxism-Leninism or other research libraries. Whether or not the libraries Stalin left behind at the time of his death were properly catalogued and preserved is an open question. Some books, including those with notations, have disappeared. However, the books that were preserved in the Stalin archival collection appear to be a representative sample.\n\n. Former Soviet transport commissar I. V. Kovalev, in an interview with G. A. Kumanev. Cited in _Novaia i noveishaia istoriia,_ no. 3 (2005): 165.\n\n. Cited in R. W. Davies et al., eds., _The Stalin-Kaganovich Correspondence, 1931\u20131936_ (New Haven and London, 2003), p. 381.\n\n. Cited in A. Artizov and O. Naumov, comps., _Vlast' i khudozhestvennaia intelligentsiia_ (Moscow, 1999), pp. 499, 583, 613. Memorandum from Stalin concerning the script of the film _Ivan the Terrible,_ 13 September 1943; speech by Stalin at a meeting of the Orgburo, 9 August 1946; conversation between Stalin and the creators of the film _Ivan the Terrible,_ 26 February 1947: see Maureen Perrie, _The Cult of Ivan the Terrible in Stalin's Russia_ (Basingstoke and New York, 2001).\n\n. B. S. Ilizarov claims to have found a copy of Fedor Dostoevsky's _The Brothers Karamazov_ with notations by Stalin in a library (Ilizarov, _Tainaia zhizn' Stalina,_ p. 411).\n\n. Mikhail Afanasyevich Bulgakov (1891\u20131940) was a novelist and playwright. Some of his early plays were staged in the 1920s but were harshly criticized for ideological flaws. Gradually, Bulgakov's works were banned and he was deprived of his livelihood. Stalin, who liked Bulgakov's works, gave the writer some support. Bulgakov was given some work, although most of his writing remained prohibited. His best known work, _The Master and Margarita,_ was published many years after Stalin's death.\n\n. Letter from Gorky to the head of the Communist youth organization, 14 April 1936. Cited in L. V. Maksimenkov, comp., _Bol'shaia tsenzura. Pisateli i zhurnalisty v Strane Sovetov. 1917\u20131956_ (Moscow, 2005), p. 413.\n\n. As mentioned above, in 1934\u20131936 the head of the Soviet film industry, Boris Shumiatsky, took notes at several dozen film screenings hosted by Stalin for other top Soviet leaders. K. M. Anderson et al., comps., _Kremlevskii kinoteatr. 1928\u20131953_ (Moscow, 2005), pp. 919\u20131053. The quotes in this paragraph are from this volume.\n\n. Letter from Stalin to members of the Russian Association of Proletarian Writers, 28 February 1929. Cited in Artizov and Naumov, _Vlast' i khudozhestvennaia intelligentsiia,_ p. 110.\n\nVsevolod Emilyevich Meyerhold (1874\u20131940) was a theatrical director and producer and an adherent of revolutionary theatrical experimentation. Meyerhold's works fell out of favor after the proclamation of the Stalinist doctrine of socialist realism. In 1939 Meyerhold was arrested, and he was shot the following year.\n\n. Dmitry Dmitryevich Shostakovich (1906\u20131975) is considered one of the twentieth century's leading composers. On Stalin's instructions, he was branded a \"formalist\" and persecuted in 1936 and 1948. To come to terms with the authorities, Shostakovich was periodically compelled to create \"correct,\" ideologically acceptable works.\n\n. V. A. Nevezhin, _Zastol'ia Iosifa Stalina. Bol'shie kremlevskie priemy 1930-kh\u20131970-kh gg._ (Moscow, 2011), pp. 282\u2013308.\n\n. Stalin's mangling of idioms is difficult to convey in translation. For examples, see Vaiskopf, _Pisatel' Stalin,_ p. 23.\n\n. For an example, see RGASPI, f. 17, op. 163, d. 471, l. 16; d. 494, l. 14.\n\n. Cited in A. Ostrovskii, _Kto stoial za spinoi Stalina?_ (Moscow, 2002), pp. 399, 400\u2013401, 409, 413.\n\n. Iu. G. Murin, comp., _Iosif Stalin v ob\"iatiiakh sem'i. Iz lichnogo arkhiva_ (Moscow, 1993), pp. 30\u201331.\n\n. Ethan Pollock, _Stalin and the Soviet Science Wars_ (Princeton, 2006).\n\n**Chapter 3. His Revolution**\n\n. RGASPI, f. 558, op. 11, d. 767, l. 76.\n\n. Minutes from an 18 January 1928 meeting of the Siberia Krai party leadership attended by Stalin; _Izvestiia TsK KPSS,_ no. 5 (1991): 196\u2013199.\n\n. Ibid., pp. 199\u2013201.\n\n. Stalin's speech at a 20 January 1928 closed meeting of the party leadership of Siberia Krai; RGASPI, f. 558, op. 11, d. 118, ll. 23\u201334; _Izvestiia TsK KPSS,_ no. 6 (1991): 203\u2013212.\n\n. RGASPI, f. 558, op. 11, d. 119, l. 84.\n\n. Ibid., l. 106; _Izvestiia TsK KPSS,_ no. 7 (1991): 178.\n\n. Cited in I. I. Ikonnikova and A. P. Ugrovatov, \"Stalinskaia repetitsiia nastupleniia na krest'ianstvo,\" _Voprosy istorii KPSS,_ no. 1 (1991): 76.\n\n. Mikhail Pavlovich Tomsky (1880\u20131936) was a long-standing member of the Bolshevik party and a Soviet trade union leader after the revolution. In 1922 he took charge of the All-Union Council of Trade Unions and joined the country's top leadership. After Stalin defeated the rightists, Tomsky was relegated to low-level positions. In 1936, under threat of arrest, he took his own life.\n\nNikolai Aleksandrovich Uglanov (1886\u20131937) was a long-standing member of the Bolshevik party who held senior posts in Moscow and the provinces after the revolution. In 1924 he was appointed head of Moscow's party organization, a position that assured him a place at the upper echelons of power. In 1928 he was removed from his post through Stalin's intrigues, given a low-level position, and subjected to persecution. He was arrested and shot during the Terror.\n\n. RGASPI, f. 85. These recent additions to the _fond_ have not yet been assigned an _opis'_ : d. 2, ll. 1\u201311, 28\u201330.\n\n. Cited in A. V. Kvashonkin et al., comps., _Bol'shevistskoe rukovodstvo. Perepiska. 1912\u20131927_ (Moscow, 1996), p. 58.\n\n. New documents pertaining to the conversation between Bukharin and Kamenev and the circumstances under which it came to light have been published. See V. P. Danilov and O. V. Khlevniuk et al., eds., _Kak lomali NEP. Stenogrammy plenumov TsK VKP(b). 1928\u20131929 gg.,_ vol. 4 (Moscow, 2000), pp. 558\u2013567, 685\u2013699.\n\n. Speech delivered at the First All-Union Conference of Leading Personnel of Socialist Industry, 4 February 1931. I. V. Stalin, _Works,_ vol. 13 (Moscow, 1954), p. 43. The translation has been slightly revised.\n\n. RGASPI, f. 558, op. 11, d. 145, ll. 43\u201354.\n\n. I borrow the term \"war on the peasants\" from Andrea Graziosi, _The Great Soviet Peasant War: Bolsheviks and Peasants, 1917\u20131933_ (Cambridge, MA, 1996).\n\n. V. P. Danilov et al., eds., _Tragediia sovetskoi derevni. Kollektivizatsiia i raskulachivanie. 1927\u20131939,_ vol. 2 (Moscow, 2000), pp. 35\u201378.\n\n. Ibid., pp. 75\u201376, 85\u201386.\n\n. Ibid., p. 11.\n\n. Ibid., pp. 703, 789. See also Lynne Viola, _Peasant Rebels under Stalin: Collectivization and the Culture of Peasant Resistance_ (New York and Oxford, 1996).\n\n. In the 1960s V. P. Danilov had an opportunity to acquaint himself with the relevant Politburo archive documents, which have still not been made generally available to historians; Danilov et al., _Tragediia sovetskoi derevni,_ vol. 2, p. 833.\n\n. Ibid., pp. 279, 324. Lynne Viola et al., eds., _Riazanskaia derevniia v 1929\u20131930 gg. Khronika golovokruzheniia_ (Moscow, 1998).\n\n. Danilov et al., _Tragediia sovetskoi derevni,_ vol. 2, p. 270.\n\n. Ibid., pp. 303\u2013305.\n\n. Ibid., p. 804. According to OGPU figures for 1930, 2.5 million people took part in the 10,000 disturbances (out of 13,800) for which an estimate was made. Assuming an average of 245 people per disturbance, we arrive at a figure of 3.4 million people for all 13,800 incidents. It should be borne in mind, however, that the OGPU data were probably not complete.\n\n. Cited in V. Vasil'ev and L. Viola, _Kollektivizatsiia i krest'ianskoe soprotivlenie na Ukraine_ (Vinnitsa, 1997), pp. 213\u2013219, 221.\n\n. RGASPI, f. 85, op. 1c, d. 125, l. 2; Vasil'ev and Viola, _Kollektivizatsiia i krest'ianskoe soprotivlenie,_ p. 233.\n\n. V. N. Zemskov, _Spetsposelentsy v SSSR. 1930\u20131960_ (Moscow, 2003), pp. 16, 20.\n\n. Lynne Viola, _The Unknown Gulag: The Lost World of Stalin's Special Settlements_ (New York, 2007).\n\n. R. W. Davies, Mark Harrison, and S. G. Wheatcroft, eds., _The Economic Transformation of the Soviet Union, 1913\u20131945_ (Cambridge, 1994), p. 289.\n\n. Speech to a Central Committee plenum, 7 January 1933. Stalin, _Works,_ vol. 13, pp. 161\u2013217.\n\n. O. Latsis, \"Problema tempov v sotsialisticheskom stroitel'stve,\" _Kommunist,_ no. 18 (1987): 83.\n\n. R. W. Davies and Stephen G. Wheatcroft, _The Years of Hunger: Soviet Agriculture, 1931\u20131933_ (Basingstoke, 2004), pp. 412\u2013415.\n\n. James C. Scott, _Weapons of the Weak: Everyday Forms of Peasant Resistance_ (New Haven, 1985).\n\n. On proposals submitted to Stalin in 1932 to introduce fixed grain procurement norms, see N. A. Ivnitskii, _Kollektivizatsiia i raskulachivanie (nachalo 30-kh godov)_ (Moscow, 1994), p. 191.\n\n. Politburo resolution, 29 April 1932; RGASPI, f. 17, op. 162, d. 12, l. 115.\n\n. Judging by reports from the head of the Procurement Committee to Stalin, as of 1 July 1933\u2014i.e., before the deliveries of grain from the 1933 harvest\u2014Soviet grain reserves, including all grain cultures, totaled approximately 1.4 million metric tons, including more than 1 million tons of grains for human consumption (APRF [Archive of the President of the Russian Federation], f. 3, op. 40, d. 27, ll. 123, 133). Davies and Wheatcroft found these figures in the archives of the Procurement Committee ( _The Years of Hunger,_ p. 229). It is known that peasant households in Russia annually consumed an average of 262 kilograms of grain per capita. That figure suggests that these reserves would have been sufficient to provide normal rations for approximately 4 million people for an entire year or even more people at below-standard rations. Even more striking is the quantity of grain exported during the famine. Although the government was forced to cut back, grain exports still totaled 1.8 million tons in 1932 and 223,000 tons during the first half of 1933 (Danilov et al., _Tragediia sovetskoi derevni,_ vol. 3, pp. 33\u201334; Davies and Wheat-croft, _The Years of Hunger,_ p. 440).\n\n. Oleg V. Khlevniuk, _The History of the Gulag from Collectivization to the Great Terror_ (New Haven and London, 2004), p. 62; Zemskov, _Spetsposelentsy v SSSR,_ p. 20.\n\n. While formally part of the Russian Federation, the North Caucasus was geographically, economically, and ethnically (due to a significant Ukrainian population) tied to Ukraine.\n\n. Davies and Wheatcroft, _The Years of Hunger,_ pp. 448\u2013449, 470.\n\n. Cited in Iu. Murin, comp., _Pisatel' i vozhd'. Perepiska M. A. Sholokhova s I. V. Stalinym. 1931\u20131950 gody_ (Moscow, 1997), p. 68.\n\nMikhail Aleksandrovich Sholokhov (1905\u20131984) has been called a classic writer of Soviet literature and enjoyed Stalin's particular patronage. Despite his success, Sholokhov continued to live in his native village in the Don region of Russia, a location that exposed him to the realities of collectivization and the Terror. On several occasions Sholokhov appealed directly to Stalin for help.\n\n. R. W. Davies et al., eds., _The Stalin-Kaganovich Correspondence, 1931\u20131936_ (New Haven, 2003), pp. 179\u2013181.\n\n. Hiroaki Kuromiya, _Stalin: Profiles in Power_ (New York, 2005), pp. 111\u2013112. Historians continue to argue about the anti-Ukrainian nature of the famine and whether it represents a case of genocide. See, for example, Andrea Graziosi, _Stalinism, Collectivization and the Great Famine_ (Cambridge, MA, 2009).\n\n. Stalin, _Works,_ vol. 13, pp. 253\u2013254.\n\n. Stalin was referring to a law enacted 7 August 1932 that provided for draconian penalties, including execution, for stealing kolkhoz property.\n\n. RGASPI, f. 558, op. 11, d. 799, ll. 24\u201325, 30\u201331. A transcript of these discussions was first published in 1951: I. V. Stalin, _Sochineniia,_ vol. 13 (Moscow, 1951), pp. 260\u2013273. The published version of the text was redacted and the discussion of the state of the countryside cited here was cut.\n\n. Danilov et al., _Tragediia sovetskoi derevni,_ vol. 3, pp. 527\u2013528, 661\u2013665.\n\n. Cited in Murin, _Pisatel' i vozhd',_ pp. 28\u201358.\n\n. Ibid., pp. 68, 145\u2013147.\n\n. Within the party, many people knew of Trotsky's speeches. They were even quoted at the January 1933 Central Committee plenum, albeit labeled as \"slanderous\" (RGASPI, f. 17, op. 2, d. 514. vyp. 1, l. 55).\n\n. Khlevniuk, _History of the Gulag,_ pp. 56, 57\u201358, 68.\n\n. RGASPI, f. 558, op. 11, d. 779, l. 47.\n\n. RGASPI, f. 17, op. 162, d. 15, ll. 154\u2013155; G. M. Adibekov et al., eds., _Politbiuro TsK RKP(b)-VKP(b) i Evropa. Resheniia 'osoboi papki'_ (Moscow, 2001), pp. 305\u2013306.\n\n. Stalin, _Sochineniia,_ vol. 13, p. 252.\n\n. Davies, Harrison, and Wheatcroft, _The Economic Transformation of the Soviet Union,_ p. 127.\n\n. RGASPI., f. 17, op. 2, d. 530. ll. 78\u201398.\n\n. Khlevniuk. _History of the Gulag,_ p. 63.\n\n. Peter H. Solomon, Jr., _Soviet Criminal Justice under Stalin_ (New York, 1996), pp. 153\u2013195.\n\n. APRF, f. 3, op. 58, d. 71, ll. 11\u201331.\n\n. RGASPI, f. 17, op. 162, d. 16, ll. 88\u201389. Subsequently, Aleksei Seliavkin fared relatively well. He survived the repression of 1937\u20131938 and fought in World War II, earning the rank of colonel. He even managed to publish his memoirs in the early 1980s (A. I. Seliavkin, _V trekh voinakh na bronevikakh i tankakh_ [Kharkov, 1981]), a testament to the position of respect he held in Soviet society.\n\n. Khlevniuk, _History of the Gulag,_ pp. 121\u2013123.\n\n. RGASPI, f. 17, op. 162, d. 17, l. 31; V. N. Khaustov et al., comps., _Lubianka. Stalin i VChK-GPU-OGPU-NKVD. Ianvar' 1922\u2013dekabr' 1936_ (Moscow, 2003), p. 566; V. N. Khaustov and L. Samuel'son, _Stalin, NKVD i repressii. 1936\u20131938_ (Moscow, 2009), p. 70.\n\n. A major part in promoting such accounts was played by the works of Roy Medvedev. See, for example, Roy Medvedev, _Let History Judge: The Origin and Consequences of Stalinism_ (New York, 1972).\n\n. For more details, see Oleg V. Khlevniuk, _Master of the House: Stalin and His Inner Circle_ (New Haven and London, 2008), pp. 108\u2013116.\n\n. An examination of the most important evidence is offered in Matthew E. Lenoe, _The Kirov Murder and Soviet History_ (New Haven and London, 2010). My discussion of this event relies heavily on this highly professional and detailed study and on A. Kirilina, _Neizvestnyi Kirov_ (St. Petersburg and Moscow, 2001).\n\n. One of the most recent publications on this subject is based on documents from the archives of the RF Federal Protection Service, the agency responsible for protecting senior officials. See S. Deviatov et al., \"Gibel' Kirova. Fakty i versii,\" _Rodina,_ no. 3 (2005): 64.\n\n. Cited in F. Chuev, _Sto sorok besed s Molotovym_ (Moscow, 1991), p. 310.\n\n. Cited in _Voprosy istorii,_ no. 2 (1995): 16\u201317.\n\nNikolai Ivanovich Yezhov (1895\u20131940) played a central role in carrying out Stalin's plans for the mass purges and repression in 1935\u20131938. Yezhov initially oversaw this campaign in his capacity as the Central Committee secretary charged with monitoring the NKVD. In late 1936 he was placed directly in charge of the organization. Under Stalin's guidance, Yezhov conducted the large-scale repressive operations of 1937\u20131938 that constituted the core of the Great Terror. After carrying out the duties that had been placed on his shoulders, Yezhov was arrested and shot.\n\nAleksandr Vasilyevich Kosarev (1903\u20131939) was head of the Komsomol, the Soviet youth organization. He was arrested in 1938 and shot in 1939.\n\n. A. N. Artizov et al., comps., _Reabilitatsiia: Kak eto bylo,_ vol. 2 (Moscow, 2003), pp. 546, 548\u2013549, and vol. 3 (Moscow, 2004), pp. 491\u2013492.\n\n. Nikolaev's relatives also met tragic fates. Almost all of them\u2014his mother, two sisters, his younger sister's husband, his brother's wife, and, in addition to Milda Draule herself, her sister, her sister's husband, and even Nikolaev's neighbor\u2014were shot or perished in prison (Kirilina, _Neizvestnyi Kirov,_ p. 367).\n\n. Genrikh Grigoryevich Yagoda (1891\u20131938) served as deputy chairman of the OGPU beginning in 1923 and as people's commissar for internal affairs (NKVD chief) from 1934 to 1936. He was arrested in 1937 and shot in 1938.\n\n. Artizov et al., _Reabilitatsiia,_ vol. 3, pp. 466\u2013467. Nikolaev officially registered his revolver in 1924 and 1930.\n\n. Ibid., pp. 490, 499.\n\n. Ibid., p. 493.\n\n. Kirilina, _Neizvestnyi Kirov,_ pp. 344\u2013347; Artizov et al., _Reabilitatsiia,_ vol. 3, pp. 494\u2013498.\n\n. Cited in Iu. G. Murin, comp., _Iosif Stalin v ob\"iatiiakh sem'i. Iz lichnogo arkhiva_ (Moscow, 1993), p. 168.\n\n. RGASPI, f. 17, op. 163, d. 1052, l. 152.\n\n. Ibid., ll. 152, 153. For the complete text of Stalin's memorandum, see ibid., f. 71, op. 10, d. 130, ll. 13\u201315.\n\n. Cited in _Pravda,_ 2 December 1935.\n\n. From the diary of Maria Svanidze; cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ pp. 173\u2013175.\n\n. Speech at the March 1937 Central Committee plenum; cited in _Voprosy istorii,_ no. 3 (1995): 14.\n\n. D. A. Volkogonov, _Triumf i tragediia,_ vol. 2, pt. 2 (Moscow, 1989), p. 249.\n\n. _Izvestiia TsK KPSS,_ no. 7 (1989): 86\u201393.\n\n. Avel Safronovich Yenukidze (1877\u20131937) was a long-standing member of the Bolshevik party who became friends with Stalin when they were both working in the revolutionary underground in Transcaucasia. After the revolution Yenukidze held a senior post in the Soviet parliament. Among his duties was accommodating the material needs of the top Soviet leadership. In that post he developed a reputation as someone who enjoyed a lavish lifestyle, and it probably contributed to his fall from favor. In 1935 he was removed from his senior post based on fabricated charges and in 1937 he was shot.\n\n. Khaustov et al., _Lubianka. Stalin i VChK-GPU-OGPU-NKVD,_ pp. 599, 601\u2013612, 618\u2013619, 626\u2013637, 638\u2013650, 663\u2013669.\n\n. An account of the relationship between these two men is offered in Oleg V. Khlevniuk, _In Stalin's Shadow: The Career of \"Sergo\" Ordzhonikidze_ (New York, 1995).\n\n. Mikhail Nikolaevich Tukhachevsky (1893\u20131937) was a Bolshevik hero of the Civil War who had held senior posts in the Red Army before being appointed deputy to the people's commissar for defense, Kliment Voroshilov, with whom he had numerous run-ins. Stalin and many other Soviet military leaders were suspicious of Tukhachevsky as a potential conspirator because of his long years serving under Trotsky. Tukhachevsky and many of his fellow military leaders were shot based on fabricated political charges.\n\n. Khaustov and Samuel'son, _Stalin, NKVD i represii,_ pp. 106\u2013121.\n\n**Trepidation in the Inner Circle**\n\n. On this point the bodyguards' accounts are fully consistent with Khrushchev's. See Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 2: _Reformer_ (University Park, PA, 2006), p. 147; Edvard Radzinsky, _Stalin: The First In-Depth Biography Based on Explosive New Documents from Russia's Secret Archives_ (New York, 1997), p. 573.\n\n. Khrushchev, _Memoirs of Nikita Khrushchev,_ vol. 2, p. 147.\n\n. Radzinsky, _Stalin,_ p. 573.\n\n. A. L. Miasnikov, _Ia lechil Stalina_ (Moscow, 2011), pp. 302, 304\u2013305.\n\n. Lazar Moiseevich Kaganovich (1893\u20131991) was one of Stalin's closest associates in the 1930s. Beginning in 1931 he essentially acted as Stalin's deputy in party matters. Before the war his political influence was somewhat diminished, and he was sent to work in economic posts, but because of his boundless devotion to Stalin, he continued to be a part of the inner circle. In 1957 he opposed Khrushchev's ascent and was forced into retirement. He lived to be almost one hundred and remained a confirmed Stalinist until his death. See E. A. Rees, _Iron Lazar: A Political Biography of Lazar Kaganovich_ (London and New York, 2012).\n\n. Nikolai Alekseevich Voznesensky (1903\u20131950) was a member of the post-revolutionary generation of Stalinist functionaries. He joined the party after the Civil War, studied at Moscow's Institute of the Red Professoriat, and went on to hold several government posts. Voznesensky's career benefited from his time working directly under Andrei Zhdanov in Leningrad. When Zhdanov was promoted to the top leadership, he took his clients with him. Voznesensky also benefited from all the job openings created by mass repression. In 1938 he was appointed to head the State Planning Commission, and in 1941 he became Stalin's first deputy chairman at the Council of People's Commissars. After the war he became a member of the country's top leadership, but after Zhdanov's death in 1948 he, along with Zhdanov's other prot\u00e9g\u00e9s, began to lose influence. In 1949 Stalin arranged the series of fabricated cases that constituted the Leningrad Affair. Voznesensky was arrested and shot.\n\nAleksei Aleksandrovich Kuznetsov (1905\u20131950) also rose to prominence under Zhdanov's patronage. He held many party posts in Leningrad and was transferred to Moscow after the war. There he became a Central Committee secretary and was placed in charge of CC personnel matters. He was arrested and shot in association with the Leningrad Affair.\n\n. M. A. Men'shikov, _S vintovkoi i vo frake_ (Moscow, 1996), p. 138.\n\n. Note from Ignatiev to Beria dated 27 March 1953; cited in N. V. Petrov, _Palachi_ (Moscow, 2011), p. 299.\n\n. K. M. Simonov, _Glazami cheloveka moego pokoleniia_ (Moscow, 1989), pp. 341\u2013343.\n\n. Pavel Sudoplatov claims that in 1950 Stalin ordered that listening devices be installed to spy on Molotov and Mikoyan (Pavel Sudoplatov et al., _Special Tasks: The Memoirs of an Unwanted Witness\u2014A Soviet Spymaster_ [New York, 1994], p. 332). Even if Sudoplatov is mistaken about the time and target of this eavesdropping, the very mention of such orders by Stalin reflects an actual phenomenon recalled by this highly placed security official.\n\n. Simonov, _Glazami cheloveka moego pokoleniia,_ pp. 160\u2013161. Quoted from Yoram Gorlizki and Oleg Khlevniuk, _Cold Peace: Stalin and the Soviet Ruling Circle, 1945\u20131953_ (New York, 2004), p. 83.\n\n. Interview with V. G. Trukhanovsky in _Novaia i noveishaia istoriia,_ no. 6 (1994): 78\u201379.\n\n. Cited in O. V. Khlevniuk et al., comps., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR. 1945\u20131953_ (Moscow, 2002), p. 399. See also Yoram Gorlizki, \"Ordinary Stalinism: The Council of Ministers and the Soviet Neopatrimonial State, 1946\u20131953,\" _Journal of Modern History_ 74, no. 4 (2002): 723\u2013725.\n\n. Cited in Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ p. 409.\n\n. O. V. Khlevniuk et al., comps., _Regional'naia politika Khrushcheva. TsK VKP(b) i mestnye partinye komitety. 1953\u20131964_ (Moscow, 2009), p. 161.\n\n. In early 1951 Soviet ministers were paid a monthly salary of twenty thousand rubles, and their deputies received ten thousand (RGANI, f. 5, op. 25, d. 279, l. 17). Other senior officials in Moscow and around the country received salaries totaling several thousand rubles, as well as significant perquisites. L. V. Maksimenkov, comp., _Bol'shaia tsenzura. Pisateli i zhurnalisty v Strane Sovetov. 1917\u20131956_ (Moscow, 2005), p. 627, describes fees totaling in the millions of rubles paid to writers. For comparison, the average per capita income of a peasant household in 1950 was less than one hundred rubles per month (V. P. Popov, _Rossiiskaia derevnia posle voiny [iiun' 1945\u2013mart 1953]_ [Moscow, 1993], p. 146). Meanwhile, many top officials were not subject to taxes, while the tax burden on the population at large was constantly growing.\n\n. Cited in Khrushchev, _Memoirs of Nikita Khrushchev,_ vol. 2, p. 89.\n\n. N. Fedorenko, \"Nochnye besedy,\" _Pravda,_ 23 October 1988, p. 4.\n\n. _Izvestiia TsK KPSS,_ no. 9 (1990): 113, 118.\n\n**Chapter 4. Terror and Impending War**\n\n. Robert Conquest, _The Great Terror: Stalin's Purges of the 1930s_ (New York, 1968). The orders and other documents associated with the large-scale operations of 1937\u20131938 have been published in English translation (see Oleg V. Khlevniuk, _The History of the Gulag from Collectivization to the Great Terror_ [New Haven and London, 2004], pp. 140\u2013165). By now there is a vast literature outlining the mechanism by which the Terror was carried out. Among general works on the subject available in English are J. Arch Getty and Oleg V. Naumov, eds., _The Road to Terror: Stalin and the Self-Destruction of the Bolsheviks, 1932\u20131939,_ updated and abridged edition (New Haven, 2010); David R. Shearer, _Policing Stalin's Socialism: Repression and Social Order in the Soviet Union, 1924\u20131953_ (New Haven and London, 2009); Paul Hagenloh, _Stalin's Police: Public Order and Mass Repression in the USSR, 1926\u20131941_ (Washington, D.C. and Baltimore, 2009).\n\n. State Archive of the Russian Federation (GARF), f. R-9401, op. 1, d. 4157, ll. 201\u2013205. These figures have appeared in numerous publications. See, for example, Khlevniuk, _History of the Gulag,_ pp. 165\u2013170, 289\u2013290.\n\n. Note written by Stalin on a telegram from the NKVD chief for Sverdlovsk Oblast; dated 10 September 1937; cited in V. N. Khaustov et al., comps., _Lubianka. Stalin i Glavnoe upravlenie gosbezopasnosti NKVD. 1937\u20131938_ (Moscow, 2004), pp. 348\u2013351.\n\n. Instructions to Yezhov (most likely), dated 13 September 1937; ibid., p. 352.\n\n. Note written by Stalin on a progress report from Yezhov concerning the \"operation to liquidate Polish espionage cadres\"; dated 14 September 1937; ibid., pp. 352\u2013359.\n\n. Stalin's instructions written in response to an NKVD summary of testimony by arrestees; dated 30 April 1938; ibid., pp. 527\u2013537.\n\n. Stalin's instructions written in response to an NKVD report on a \"terrorist group\" within the rubber industry; dated 2 September 1938; ibid., pp. 546\u2013547.\n\n. Cited in N. S. Tarkhova et al., comps., _Voennyi sovet pri narodnom komissare oborony SSSR. 1\u20134 iiunia 1937 g._ (Moscow, 2008), p. 137.\n\n. Cited in V. A. Nevezhin, comp., _Zastol'nye rechi Stalina. Dokumenty i materialy_ (Moscow, 2003), pp. 132\u2013135.\n\n. Rozengolts was arrested on 7 October 1937; V. N. Khaustov and L. Samuel'son, _Stalin, NKVD i repressii. 1936\u20131938_ (Moscow, 2009), pp. 138\u2013139.\n\n. RGASPI, f. 17, op. 162, d. 20, l. 87.\n\n. APRF, f. 3, op. 65, d. 223, l. 90; Oleg Khlevniuk, \"The Reasons for the 'Great Terror': The Foreign-Political Aspect,\" in _Russia in the Age of Wars 1914\u20131945, ed._ Silvio Pons and Andrea Romano (Milan, 2000), pp. 165\u2013166.\n\n. APRF, f. 3, op. 65, d. 223, l. 142.\n\n. RGASPI, f. 558, op. 11, d. 772, l. 14.\n\n. Ibid., l. 88.\n\n. \"Stenogramma zasedanii fevral'sko-martovskogo plenuma 1937 g.,\" _Voprosy istorii,_ no. 3 (1995): 13\u201314.\n\n. RGASPI, f. 558, op. 11, d. 203, ll. 62, 77\u201378.\n\n. Cited in Tarkhova et al., _Voennyi sovet pri narodnom komissare oborony SSSR,_ p. 133.\n\n. Edward Hallet Carr, _The Comintern and the Spanish Civil War_ (London and Basingstoke, 1984), p. 44.\n\n. RGASPI, f. 17, op. 162, d. 21, l. 157; N. F. Bugai, \"Vyselenie sovetskikh koreitsev s Dal'nego Vostoka,\" _Voprosy istorii,_ no. 5 (1994): 144.\n\n. F. Chuev, _Sto sorok besed s Molotovym_ (Moscow, 1991), pp. 390, 391, 416.\n\n. L. M. Kaganovich, _Pamiatnye zapiski_ (Moscow, 1996), pp. 549, 558.\n\n. Cited in A. S. Iakovlev, _Tsel' zhizni_ (Moscow, 1987), p. 212.\n\n. The discovery of \"counterrevolutionary groups\" (rather than lone enemies) was one of the primary goals of the process of extracting confessions from arrestees.\n\n. Cited in Khlevniuk, _History of the Gulag,_ p. 163.\n\n. A detailed study of Stalin's role in organizing the Terror has been done using a vast body of archival documents. See Khaustov and Samuel'son, _Stalin, NKVD i repressii._\n\n. These calculations were made based on the clerical numbering of Yezhov's reports published in Khaustov et al., _Lubianka. Stalin i Glavnoe upravlenie gosbezopasnosti NKVD._ I am grateful to N. V. Petrov, who pointed out the possibility of using this source.\n\n. Oleg V. Khlevniuk. _Master of the House: Stalin and His Inner Circle_ (New Haven and London, 2008), p. 270.\n\n. RGASPI, f. 671, op. 1, d. 265, l. 22.\n\n. _Istoricheskii arkhiv,_ no. 1 (1992): 125\u2013128.\n\n. Official figures for industrial growth gave 28.7 percent for 1936, 11.2 percent for 1937, and 11.8 percent for 1938. Economists have calculated that using modern methods, these figures would correspond to 10.4, 2.3, and 1.1 percent growth respectively. See R. W. Davies, Mark Harrison, and S. G. Wheatcroft, eds., _The Economic Transformation of the Soviet Union, 1913\u20131945_ (Cambridge, 1994), pp. 302\u2013303.\n\n. During 1937\u20131938 a total of thirty-five thousand commanders were discharged from the Red Army (not including the air force and navy). Many of them were arrested. As of early 1940, approximately eleven thousand of them had been returned to the army, so approximately twenty-four thousand were lost. A sense of the scale of this attenuation can be gained by comparing these figures to the number of graduates of military colleges and academies during the three-year period of 1935\u20131937: slightly more than twenty-seven thousand ( _Izvestiia TsK KPSS,_ no. 1 [1990]: 186\u2013189). It goes without saying that the officers who had been discharged, arrested, and then returned to duty suffered serious emotional trauma that affected their performance. Furthermore, the fear that they too could be arrested surely also had an effect on those who were not.\n\n. GARF, R-8131, op. 37, d. 112, l. 16.\n\n. Cited in A. I. Kartunova, \"1938-i. Poslednii god zhizni i deiatel'nosti marshala V. K. Bliukhera,\" _Novaia i noveishaia istoriia,_ no. 1 (2004): 175.\n\n. RGASPI, f. 17, op. 162, d. 24, l. 17.\n\n. Stenographic record of the Eighteenth Party Congress; _XVIII s\"ezd Vsesoiuznoi Kommunisticheskoi Partii (b). 10\u201321 marta 1939 g._ (Moscow, 1939), pp. 12\u201315.\n\n. Maksim Maksimovich Litvinov (1876\u20131951), who joined what would become the Bolshevik party long before the revolution, served the cause of Soviet foreign affairs in one capacity or another most of his adult life. After years as deputy commissar and then commissar, he fell into disgrace in the late 1930s. During the war, Stalin decided to take advantage of the ties Litvinov had developed in the West and the reputation he enjoyed there and appointed him Soviet ambassador to the United States. Toward the war's end Litvinov was dismissed for the final time, but he was never arrested and was allowed to live out his life.\n\n. A. I. Mikoian, _Tak bylo. Razmyshleniia o minuvshem_ (Moscow, 1999), p. 534.\n\n. S. Z. Sluch, \"Stalin i Gitler, 1933\u20131941: Raschety i proschety Kremlia,\" _Otechestvennaia istoriia,_ no. 1 (2005): 98\u2013119.\n\n. RGASPI, f. 17, op. 166, d. 592, l. 107.\n\n. Cited in G. Ia. Rudoi, comp., _Otkroveniia i priznaniia. Natsistskaia verkhushka o voine \"tret'ego reikha\" protiv SSSR_ (Moscow, 1996), p. 65.\n\n. V. G. Komplektov et al., eds., _Dokumenty vneshnei politiki SSSR. 1939,_ vol. 22 (Moscow, 1992), vol. 1, p. 624; vol. 2, p. 585. This correspondence was also preserved in Stalin's personal archive: RGASPI, f. 558, op. 11, d. 296, ll. 1\u20133.\n\n. Cited in S. Z. Sluch, \"Rech' Stalina, kotoroi ne bylo,\" _Otechestvennaia istoriia,_ no. 1 (2004): 114. In this article Sluch provides a detailed history of this alleged speech and persuasively argues that it was a fake.\n\n. Cited in G. M. Adibekov et al., eds., _Politbiuro TsK RKP(b) VKP(b) i Komintern. 1919\u20131943. Dokumenty_ (Moscow, 2004), pp. 780\u2013781.\n\n. Anna M. Cienciala, Natalia S. Lebedeva, and Wojciech Materski, eds., _Katyn: A Crime without Punishment_ (New Haven and London, 2007).\n\n. Alfred Bilmanis, comp., _Latvian-Russian Relations: Documents_ (Washington, D.C., 1944), pp. 196\u2013197.\n\n. Cited in L. E. Reshin et al., comps., _1941 god,_ vol. 2 (Moscow, 1998), pp. 595\u2013596.\n\n. Notation by Stalin on a coded message from Belarusian Central Committee secretary Ponomarenko to Stalin; dated 13 November 1939; RGASPI, f. 558, op. 11, d. 66, l. 13.\n\n. O. A. Rzheshevskii and O. Vekhviliainen, eds., _Zimniaia voina 1939\u20131940_ (Moscow, 1999), vol. 1, pp. 324\u2013325.\n\n. Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 1: _Commissar_ (University Park, PA, 2004), p. 266.\n\n. Khlevniuk, _History of the Gulag,_ p. 236.\n\n. Soviet transcripts of Molotov's conversations with Hitler and von Ribbentrop on 13 November 1940 have been published in G. E. Mamedov et al., eds., _Dokumenty vneshnei politiki_ (Moscow, 1998), vol. 23, bk. 2, pt. 1, pp. 63\u201378.\n\n. Ibid., pp. 135\u2013137.\n\n. G. A. Kumanev, _Riadom so Stalinym_ (Smolensk, 2001), pp. 463\u2013470.\n\n. According to Chadaev in ibid., the chairman of Gosplan, Nikolai Voznesensky, was also at the meeting. At the time, Voznesensky was not yet a Politburo member.\n\n. A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008), pp. 317\u2013318.\n\n. Aleksandr Sergeevich Shcherbakov (1901\u20131945) was a member of the post-revolutionary generation that Stalin placed in charge of propaganda within the Central Committee apparat. In 1938 he was made first secretary of Moscow's party organization as well as a Central Committee secretary. Shcherbakov died at an early age.\n\n. Remarks by Stalin at a meeting on 17 January 1941 as recorded by V. A. Malyshev in his diary; cited in _Istochnik,_ no. 5 (1997): 114.\n\n. Mikoian, _Tak bylo,_ p. 346.\n\n. RGASPI, f. 558, op. 11, d. 769, ll. 176\u2013176ob.\n\n. _Istoricheskii arkhiv,_ no. 5 (1994): 222.\n\n. Cited in Yoram Gorlizki and Oleg Khlevniuk, \"Stalin and His Circle,\" in _The Cambridge History of Russia,_ ed. Ronald Grigor Suny, vol. 3 (Cambridge, 2006), p. 248.\n\n. Stalin actually spoke at this reception several times, but for simplicity's sake, I will treat these remarks as a single speech. The stenographic record of Stalin's remarks has not been preserved, but several witnesses describe him as saying essentially the same thing. See Nevezhin, _Zastol'nye rechi Stalina,_ pp. 273\u2013296.\n\n. Speech by Stalin at a meeting of Moscow and Leningrad propagandists; _Istoricheskii arkhiv,_ no. 5 (1994): 13.\n\n. E. N. Kul'kov and O. A. Rzheshevskii, eds., _Zimniaia voina 1939\u20131940_ (Moscow, 1999), vol. 2, pp. 281\u2013282.\n\n. Debate around this topic has become particularly active over the past twenty years. Overall, the numerous arguments in favor of the idea that Stalin was planning a preventive strike\u2014some of which appear to be politically motivated\u2014do not seem to warrant serious attention, but this theory has generated a number of works presenting interesting evidence and arguments. I make use of statistical data offered in a study by Mikhail Meltiukhov, although I am not convinced by his overall argument. See M. Mel'tiukhov, _Upushchennyi shans Stalina. Sovetskii Soiuz i bor'ba za Evropu. 1939\u20131941_ (Moscow, 2002).\n\n. Ibid., pp. 360, 392\u2013393.\n\n. Davies, Harrison, and Wheatcroft, _The Economic Transformation of the Soviet Union,_ p. 321.\n\n. Mel'tiukhov, _Upushennyi shans Stalina,_ pp. 392, 393.\n\n. Cited in E. A. Osokina, _Za fasadom \"stalinskogo izobiliia\"_ (Moscow, 2008), pp. 272\u2013277.\n\n. _Voenno-istoricheskii zhurnal,_ no. 1 (1991): 17.\n\n. In September 1940, the government permitted such convicts to be sent to the Gulag to serve their prison terms, a violation of its own law (GARF, f. R-5446, op. 57, d. 79, l. 31). These prisoners suffered a terrible fate, and they were not always released after serving the short terms handed down by the courts.\n\n. From a 15 April 1942 conversation between Stalin and General Nikolai Biriukov, one of the heads of the Main Mechanized Directorate; N. Biriukov, _Tanki\u2013frontu. Zapiski sovetskogo generala_ (Smolensk, 2005), pp. 143\u2013144.\n\n. Reshin et al., _1941 god,_ pp. 54\u201355.\n\n. Mel'tiukov, _Upushchennyi shans Stalina,_ p. 246; M. Iu. Mukhin, _Aviapromyshlennost' SSSR v 1921\u20131941 godakh_ (Moscow, 2006), pp. 154\u2013155, 291\u2013299.\n\n. David Murphy, who has made a careful study of all available Soviet intelligence reports on the eve of the war, gives Soviet espionage rather high marks. However, he notes an effort on the part of the leaders of Soviet intelligence to adapt their findings to Stalin's preconceptions. In this regard, Murphy draws historical parallels: the reluctance of the conservative government of Great Britain in the 1930s to properly assess the Nazi threat and the myopic focus of U.S. intelligence on hunting down weapons of mass destruction in Iraq, while earlier administrations missed clues of an impending terrorist attack on U.S. soil. See David E. Murphy, _What Stalin Knew: The Enigma of Barbarossa_ (New Haven and London, 2005), pp. xviii\u2013xix.\n\n. Cited in Reshin et al., _1941 god,_ pp. 382\u2013383.\n\n**Patient Number 1**\n\n. Sergei Khrushchev, ed., _Memoirs of Nikita Khrushchev,_ vol. 2: _Reformer_ (University Park, PA, 2006), p. 148.\n\n. A. L. Miasnikov, _Ia lechil Stalina_ (Moscow, 2011), pp. 294\u2013295.\n\n. Ibid., p. 302.\n\n. B. S. Ilizarov, _Tainaia zhizn' Stalina_ (Moscow, 2002), p. 110.\n\n. RGASPI, f. 558, op. 1, d. 4327, l. 1.\n\n. Ibid., op. 4, d. 619, ll. 172, 173.\n\n. Ilizarov, _Tainaia zhizn' Stalina,_ p. 110.\n\n. Letter from Stalin to Malinovsky, November 1913; cited in A. Ostrovskii, _Kto stoial za spinoi Stalina?_ (Moscow, 2002), pp. 397\u2013398.\n\n. Ilizarov, _Tainaia zhizn' Stalina,_ p. 110.\n\n. RGASPI, f. 17, op. 3, d. 154, l. 2.\n\n. Ibid., d. 303, l. 5.\n\n. Svetlana Alliluyeva, _Twenty Letters to a Friend_ trans. Priscilla Johnson McMillan (New York, 1967), p. 33.\n\n. No information has been found about Stalin's travels in the south in 1924, although an August 1924 Politburo decision granted him a two-month vacation; RGASPI, f. 17, op. 3, d. 459, l. 2.\n\n. Ilizarov, _Tainaia zhizn' Stalin,_ pp. 112\u2013113, 118\u2013119.\n\n. Cited in Lars T. Lih, Oleg V. Naumov, and Oleg Khlevniuk, eds., _Stalin's Letters to Molotov, 1925\u20131936_ (New Haven, 1995), p. 91.\n\n. RGASPI, f. 558, op. 11, d. 69, ll. 53\u201354.\n\n. Cited in Lih, Naumov, and Khlevniuk, _Stalin's Letters to Molotov,_ p. 113.\n\n. RGASPI, f. 558, op. 11, d. 69, l. 67ob.\n\n. Ibid., l. 68.\n\n. From Valedinsky's memoirs; cited in _Istochnik,_ no. 2 (1998): 68.\n\n. Cited in Lih, Naumov, and Khlevniuk, _Stalin's Letters to Molotov,_ p. 138.\n\n. Cited in _Istochnik,_ no. 2 (1998): 69.\n\n. Ibid., p. 69; Ilizarov, _Tainaia zhizn' Stalin,_ pp. 112\u2013113.\n\n. Cited in Lih, Naumov, and Khlevniuk, _Stalin's Letters to Molotov,_ p. 175.\n\n. Iu. G. Murin, comp., _Iosif Stalin v ob\"iatiiakh sem'i. Iz lichnogo arkhiva_ (Moscow, 1993), p. 32.\n\n. RGASPI, f. 558, op. 11, d. 728, l. 29.\n\n. Cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ p. 37.\n\n. I. V. Stalin, _Works,_ vol. 13 (Moscow, 1954), p. 136. Translation slightly revised.\n\n. Cited in O. V. Khlevniuk et al., comps., _Stalin i Kaganovich. Perepiska. 1931\u20131936_ (Moscow, 2001), p. 180.\n\n. S. V. Deviatov et al., _Garazh osobogo naznacheniia. 1921\u20132011_ (Moscow, 2011), p. 157.\n\n. Letters from Stalin to Yenukidze, dated 16 August and 13 September 1933; RGASPI, f. 558, op. 11, d. 728, ll. 38, 40.\n\n. Letter dated 7 September 1933; cited in A. V. Kvashonkin et al., comps., _Bol'shevistskoe rukovodstvo. Perepiska. 1912\u20131927_ (Moscow, 1996), p. 254.\n\n. From the diary of Maria Svanidze; cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ p. 158.\n\n. Letter to A. I. Ugarov, dated 16 August 1934; cited in A. Kirilina, _Neizvestnyi Kirov_ (St. Petersburg and Moscow, 2001), p. 141.\n\n. From the diary of Maria Svanidze; cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ p. 183.\n\n. From the memoirs of Dr. Valedinsky; cited in _Istochnik,_ no. 2 (1998): 70.\n\n. Ibid., p. 70.\n\n. RGASPI, f. 558, op. 11, d. 377, l. 60.\n\n. Stalin left Moscow on 9 October 1945 and returned 17 December; O. V. Khlevniuk et al., comps., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR. 1945\u20131953_ (Moscow, 2002), p. 398.\n\n. Ibid.\n\n. Deviatov et al., _Garazh osobogo naznacheniia,_ p. 201.\n\n. Descriptions of Stalin's lifestyle at his southern dachas can be found in the memoirs of the Georgian party boss Akaky Mgeladze, a young prot\u00e9g\u00e9 of Stalin who enjoyed his particular favor; A. I. Mgeladze, _Stalin. Kakim ia ego znal,_ (n.p., 2001).\n\n. Stalin's medical records; RGASPI, f. 558, op. 11, d. 1483, ll. 1\u2013101; Ilizarov, _Tainaia zhizn' Stalina,_ pp. 126, 129.\n\n. M. Dzhilas [Milovan Djilas], _Litso totalitarizma_ (Moscow, 1992), p. 60.\n\n. Cited in _Istoricheskii arkhiv,_ no. 3 (1997): 117.\n\n. Mgeladze, _Stalin,_ p. 125.\n\n. Cited in E. Khodzha [Enver Hoxha], _So Stalinym. Vospominaniia_ (Tirana, 1984), p. 137.\n\n. Alliluyeva, _Twenty Letters,_ p. 22.\n\n. Ibid., pp. 206\u2013207.\n\n. Miasnikov, _Ia lechil Stalina,_ p. 302.\n\n. Alliluyeva, _Twenty Letters,_ p. 207.\n\n. Miasnikov, _Ia lechil Stalina,_ pp. 304\u2013305.\n\n. Transcript of a conversation in March 1978 published in F. Chuev, _Sto sorok besed s Molotovym_ (Moscow, 1991), p. 324.\n\n**Chapter 5. Stalin at War**\n\n. The following descriptions of meetings in Stalin's office on 21 and 22 June 1941 are based on G. K. Zhukov, _Vospominaniia i razmyshleniia_ (Moscow, 2002), vol. 1, pp. 260\u2013269; A. I. Mikoian, _Tak bylo. Razmyshleniia o minuvshem_ (Moscow, 1999), p. 388; and A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008), pp. 337\u2013338.\n\n. Semen Konstantinovich Timoshenko (1895\u20131970) was a commander of the First Cavalry Army during the Civil War, in which capacity he worked closely with Stalin. He went on to make a successful military career and, after the debacle in Finland, replaced Voroshilov as defense commissar and was elevated to marshal. However, during the war with Germany, Timoshenko did not prove to be particularly able and was forced into the background. After the war and until his retirement in 1960 he was given secondary posts commanding various military districts.\n\nGeorgy Konstantinovich Zhukov (1896\u20131974) made a military career after serving with the Red Army during the Civil War. He advanced rapidly through the ranks during the late 1930s, when purges among the officer corps created opportunities. Zhukov proved an able commander during military conflicts with Japan in 1939. Before the war with Germany he was appointed chief of the General Staff. The war proved to be his finest hour. He rose to be one of the Soviet Union's leading marshals and served as deputy to the commander in chief (Stalin). When it was over, Zhukov fell into disfavor but enjoyed a brief return to prominence after Stalin's death, serving as defense minister from 1955 to 1957. Khrushchev, however, was wary of the ambitious marshal and forced him into retirement. After Khrushchev was expelled as Soviet leader, Zhukov was allowed to publish his memoirs (the first edition of which came out in 1969). Although they were heavily censored, they remain an important source for historians of the Great Patriotic War (as the war with Germany is known in Russia). Recent editions of his memoirs restore materials excised by the censors, but we will never know to what extent Zhukov self-censored his original manuscript.\n\n. Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, p. 260.\n\n. Ibid, p. 264\n\n. Cited in a speech written by Zhukov in May 1956 to be given at a Central Committee plenum that was to be devoted to the Cult of Personality but never took place; cited in _Istochnik,_ no. 2 (1995): 147.\n\n. Chernobaev, _Na prieme u Stalina,_ p. 337.\n\n. Cited in Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, p. 265.\n\n. Cited in L. E. Reshin et al., comps., _1941 god_ (Moscow, 1998), vol. 2, p. 432.\n\n. Chernobaev, _Na prieme u Stalina,_ p. 337.\n\nLev Zakharovich Mekhlis (1889\u20131953) was one of Stalin's assistants in the 1920s, after which he held a number of senior posts and enjoyed Stalin's wholehearted trust. After war with Germany broke out, Stalin put Mekhlis in charge of the political offices within the Red Army that were supposed to exercise political control over commanders. Mekhlis's bungling at the front infuriated Stalin but did not undermine his trust in his faithful helper. Mekhlis went on to hold a number of senior posts on various fronts. After the war, he was put in charge of the Ministry of State Control. Poor health forced him into retirement. He died several weeks before Stalin and was buried at the foot of the Kremlin walls, alongside other Soviet leaders and heroes.\n\n. Cited in Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, p. 265.\n\n. Reshin et al., _1941 god,_ p. 431.\n\n. For versions of Molotov's speech, see _Istoricheskii arkhiv,_ no. 2 (1995): 34\u201339.\n\n. John Erickson, _The Road to Stalingrad_ (London, 2003), p. 177.\n\n. Boris Mikhailovich Shaposhnikov (1882\u20131945) was one of the few Red Army officers to retain Stalin's trust even as he reached a position of seniority. On the eve of the war with Germany and during its initial phase Shaposhnikov was head of the army's General Staff and deputy defense commissar, but he had to resign due to illness. He died a few weeks before the fall of Berlin.\n\n. Reshin et al., _1941 god,_ pp. 439\u2013440.\n\n. Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, p. 268.\n\n. _Rodina,_ no. 4 (2005): 4.\n\n. M. I. Mel'tiukhov, _Upushchennyi shans Stalina. Sovetskii Soiuz i bor'ba za Evropu. 1939\u20131941_ (Moscow, 2002), p. 413.\n\n. Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, p. 340.\n\n. Memoirs of Chadaev published in _Otechestvennaia istoriia,_ no. 2 (2005): 7.\n\n. Semen Mikhailovich Budenny (1883\u20131973) commanded the First Cavalry Army during the Civil War and was a supporter of Stalin during this period. He went on to become a marshal and held top military posts, including first deputy people's commissar of defense.\n\n. Admiral Nikolai Gerasimovich Kuznetsov (1902\u20131974) was head of the naval commissariat and commander in chief of the navy from 1939 to 1946. After the war he fell into disfavor and was demoted, but in 1951\u20131953 he was again placed in charge of the naval ministry. He lost command of the navy for good in 1955 after the loss of a battleship.\n\n. RGASPI, f. 17, op. 162, d. 36, l. 22; _Izvestiia TsK KPSS,_ no. 6 (1990): 196\u2013197.\n\n. N. G. Kuznetsov, _Nakanune_ (Moscow, 1989), p. 327.\n\n. N. V. Petrov, _Palachi_ (Moscow, 2011), p. 85\u201393.\n\n. _Otechestvennye arkhivy,_ no. 2 (1995): 29\u201332; _Izvestiia TsK KPSS,_ no. 6 (1990): 208\u2013209, 212\u2013214.\n\n. Interview by Georgy Kumanev of I. V. Kovalev, who was serving as deputy commissar for state control when the war broke out and was in charge of rail transport. _Novaia i noveishaia istoriia,_ no. 3 (2005): 149\u2013150.\n\n. Reshin et al., _1941 god,_ p. 497; F. Chuev, _Sto sorok besed s Molotovym_ (Moscow, 1991), p. 52.\n\n. In his memoirs Zhukov states that Stalin came to the defense commissariat twice ( _Vospominaniia i razmyshleniia,_ vol. 1, p. 287); however, there are no other sources to corroborate this assertion.\n\n. Mikoyan's recollections are reported in Reshin et al., _1941 god,_ pp. 497\u2013498.\n\n. Letter to the Soviet leadership from Lavrenty Beria after his arrest in 1953; published in _Istochnik,_ no. 4 (1994): 7; recollections of Anastas Mikoyan published in Reshin et al., _1941 god,_ pp. 498\u2013499.\n\n. Cited in Reshin et al., _1941 god,_ p. 498.\n\n. Cited in Chuev, _Sto sorok besed s Molotovym,_ p. 330.\n\n. Recollections of Anastas Mikoyan published in Reshin et al., _1941 god,_ pp. 498\u2013499.\n\n. The original text is among Mikoyan's personal papers, which are held by RGASPI.\n\n. Mikoian, _Tak bylo,_ p. 391.\n\n. Iu. A. Gor'kov, _Gosudarstvennyi Komitet Oborony postanovliaet (1941\u20131945)_ (Moscow, 2002), pp. 30\u201331.\n\n. _Izvestiia TsK KPSS,_ no. 7 (1990): 208, and no. 8 (1990): 208; RGASPI, f. 17, op. 163, d. 1319, l. 93.\n\n. Interview of Ivan Peresypkin, wartime communications commissar, by Georgy Kumanev; _Otechestvennaia istoriia,_ no. 3 (2003): 65.\n\n. The speech is cited in _Pravda,_ 3 July 1941.\n\n. Order by People's Commissar for Defense Stalin, 28 June 1941. In V. A. Zolotarev, ed., _Russkii arkhiv. Velikaia Otechestvennaia. Prikazy narodnogo komissara oborony SSSR. 22 iunia 1941 g.\u20131942 g.,_ vol. 13 (2\u20132) (Moscow, 1997), pp. 37\u201338.\n\n. G. F. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti. Kniga poter'_ (Moscow, 2009), pp. 60\u201361.\n\n. Conversations between Stalin and commanders in the Western Direction, 26 July 1941; cited in V. A. Zolotarev, ed., _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1941 g.,_ vol. 16 (5\u20131) (Moscow, 1996), pp. 92\u201393.\n\n. RGASPI, f. 17, op. 167, d. 60, l. 49.\n\n. D. A. Volkogonov, _Triumf i tragediia,_ vol. 2, pt. 1 (Moscow, 1989), p. 167.\n\n. Cited in Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1941 g.,_ vol. 16 (5\u20131), p. 361.\n\n. RGASPI, f. 558, op. 11, d. 492, l. 35; _Izvestiia TsK KPSS,_ no. 9 (1990): 213.\n\n. V. N. Khaustov et al., comps., _Lubianka. Stalin i NKVD-NKGB-GUKR \"Smersh.\" 1939\u20131946_ (Moscow, 2006), pp. 317\u2013318.\n\n. For the texts of these decisions with changes entered by Stalin, see _Vestnik arkhiva Preszidenta Rossiiskoi Federatsii. Voina. 1941\u20131945_ (Moscow, 2010), pp. 37\u201340.\n\n. Khaustov et al., _Lubianka. Stalin i NKVD-NKGB-GUKR \"Smersh,\"_ pp. 317\u2013318.\n\n. Reshin et al., _1941 god,_ pp. 476\u2013479.\n\n. Cited in _Izvestiia TsK KPSS,_ no. 7 (1990): 209.\n\n. _Istoricheskii arkhiv,_ no. 1 (1993): 45\u201346.\n\n. Interview of I. V. Kovalev by Georgy Kumanev published in _Novaia i noveishaia istoriia,_ no. 3 (2005): 160\u2013161.\n\n. RGASPI, f. 558, op. 11, d. 235, l. 123.\n\n. Zhukov, _Vospominaniia i razmyshleniia,_ vol. 1, pp. 350\u2013353.\n\n. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti,_ p. 84.\n\n. Cited in Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1941 g.,_ vol. 16 (5\u20131), pp. 108\u2013109; _Izvestiia TsK KPSS,_ no. 9 (1990): 199\u2013200.\n\n. A. M. Vasilevskii, _Delo vsei zhizni_ (Moscow, 1978), p. 132.\n\nAleksandr Mikhailovich Vasilevsky (1895\u20131977) was a renowned Soviet marshal and leading figure in the Great Patriotic War who served as deputy chief and then chief of the General Staff and commanded Soviet troops in the Far East during the war with Japan. After the war he served as minister of defense.\n\n. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti,_ p. 85.\n\n. Interview with Zhukov published in K. Simonov, _Glazami cheloveka moego pokoleniia_ (Moscow, 1989), p. 361.\n\n. Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1941 g.,_ vol. 16 (5\u20131), pp. 175.\n\n. Simonov, _Glazami cheloveka moego pokoleniia,_ pp. 361\u2013363.\n\n. _Voenno-istoricheskii zhurnal,_ no. 1 (1992): 77, and nos. 6\u20137 (1992): 17; Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1941 g.,_ vol. 16 (5\u20131), pp. 378\u2013379.\n\n. A. E. Golovanov, _Dal'niaia bombardirovochnaia_... (Moscow, 2004), p. 78.\n\n. _Izvestiia TsK KPSS,_ no. 12 (1990): 210\u2013211.\n\n. Mikoian, _Tak bylo,_ p. 417. Mikoyan writes that this was on 16 October, but he definitely refers to the discussion of the order to evacuate Moscow that was actually adopted on 15 October. This meeting evidently took place in Stalin's apartment.\n\n. _Izvestiia TsK KPSS,_ no. 12 (1990): 217.\n\n. Interview with Aleksandr Vasilevsky in Simonov, _Glazami cheloveka moego pokoleniia,_ p. 446.\n\n. NKVD report dated 21 October 1941; published in _Istochnik,_ no. 5 (1995): 152.\n\n. M. M. Gorinov et al., comps., _Moskva voennaia. 1941\u20131945. Memuary i arkhivnye dokumenty_ (Moscow, 1995), p. 550; _Izvestiia TsK KPSS,_ no. 1 (1991): 217.\n\n. Gorinov et al., _Moskva voennaia,_ pp. 111, 116\u2013119; Mikoian, _Tak bylo,_ pp. 419\u2013420.\n\n. A. I. Shakhurin, _Kryl'ia pobedy_ (Moscow, 1990), pp. 156\u2013157.\n\n. _Izvestiia TsK KPSS,_ no. 1 (1991): 215\u2013216, and no. 4 (1991): 210\u2013214; _Istoricheskii arkhiv,_ no. 3 (1997): 92.\n\n. S. V. Tochenov, \"Volneniia i zabastovki na tekstil'nykh predpriiatiiakh Ivanovskoi oblast osen'iu 1941 goda,\" _Otechestvennaia istoriia,_ no. 3 (2004): pp. 42\u201347; _Istoricheskii arkhiv,_ no. 2 (1994): 111\u2013136.\n\n. Cited in G. K. Zhukov, _Vospominaniia i razmyshleniia_ (Moscow, 2002), vol. 2, pp. 26\u201327.\n\n. Cited in _Pravda,_ 7 November 1941.\n\n. S. V. Deviatov et al., _Moskovskii Kreml' v gody Velikoi Otechestvennoi Voiny_ (Moscow, 2010), p. 87.\n\n. Ibid., pp. 57\u201361, 64.\n\n. _Pravda,_ 8 November 1941.\n\n. Deviatov et al., _Moskovskii Kreml',_ p. 61.\n\n. K. M. Anderson et al., comps., _Kremlevskii kinoteatr. 1928\u20131953_ (Moscow, 2005), p. 639.\n\n. Cited in V. A. Zolotarev, ed., _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1942 g.,_ vol. 16 (5\u20132) (Moscow, 1996), pp. 33\u201335.\n\n. R. V. Mazurkevich, \"Plany i real'nost',\" _Voenno-istoricheskii zhurnal,_ no. 2 (1992): 24\u201325.\n\n. Vasilevskii, _Delo vsei zhizni,_ p. 189.\n\n. Cited in _Istochnik,_ no. 5 (1995): 41.\n\n. GARF (not yet catalogued). From the memoirs of Yakov Chadaev.\n\n. Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1942 g.,_ vol. 16 (5\u20132), pp. 236\u2013239.\n\n. Interview with Georgy Zhukov in Simonov, _Glazami cheloveka moego pokoleniia,_ p. 366.\n\n. Vasilevskii, _Delo vsei zhizni,_ pp. 195\u2013196.\n\n. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti,_ p. 179.\n\n. Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1942 g.,_ vol. 16 (5\u20132), pp. 263\u2013264.\n\n. After reading this letter, Stalin placed it in his personal archive. RGASPI, f. 558, op. 11, d. 762, ll. 6\u20138.\n\n. K. K. Rokossovskii, _Soldatskii dolg_ (Moscow, 2013), p. 211.\n\nKonstantin Konstantinovich Rokossovsky (1896\u20131968) was a much-acclaimed marshal of the Great Patriotic War. He was arrested during the purges and spent 1937\u20131940 in prison. During the war he was placed in charge of armies and fronts. In 1949\u20131956 he served as Poland's minister of defense before holding senior posts in the Soviet ministry. His rather candid memoirs, _Soldatskii dolg_ (A soldier's duty), were published in 1968 with major excisions. An uncensored edition came out in 1997.\n\n. Cited in Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1942 g.,_ vol. 16 (5\u20132), pp. 276\u2013279.\n\n. O. A. Rzheshevskii, _Stalin i Cherchill'. Vstrechi. Besedy. Diskussii_ (Moscow, 2004), pp. 348\u2013383.\n\n. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti,_ pp. 60\u201361.\n\n. _Rodina,_ no. 4 (2005): 65.\n\n. Aleksei Innokentievich Antonov (1896\u20131962) was a senior Soviet military officer who served as deputy head of the General Staff during the war years and often reported directly to Stalin.\n\n. Vasilevskii, _Delo vsei zhizni,_ pp. 311.\n\n. A. Eremenko, _Gody vozmezdiia_ (Moscow, 1986), pp. 36, 38; Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ (New York, 1991), p. 481; Deviatov et al., _Moskovskii Kreml',_ pp. 184, 186.\n\n. _Perepiska Predsedatelia Soveta Ministrov SSSR s prezidentami SShA i prem'er-ministrami Velikobritanii vo vremia Velikoi Otechestvennoi Voiny 1941\u20131945 gg._ (Moscow, 1957). Stalin's letter to Churchill is dated 9 August 1943 (vol. 1, pp. 141\u2013142). Stalin's letter to Roosevelt is dated 8 August 1943 (vol. 2, p. 77).\n\n. Stalin's letter to Churchill, 24 June 1943, which he sent that same day to Roosevelt for his information; _Perepiska,_ vol. 2, pp. 72\u201375.\n\n. Cited in S. M. Shtemenko, _General'nyi shtab v gody voiny_ (Moscow, 1989), p. 148.\n\n. Golovanov, _Dal'niaia bombardirovochnaia,_ pp. 351\u2013356.\n\n. RGASPI, f. 558, op. 11, d. 377, l. 61.\n\n. Cited in Gorinov et al., _Moskva voennaia,_ pp. 694\u2013695; _Istochnik,_ no. 2 (1995): 138\u2013139.\n\n. _Istoricheskii arkhiv,_ no. 1 (1997): 66\u201368.\n\n. V. A. Zolotarev, ed., _Russkii arkhiv, Velikaia Otechestvennaia. Stavka VGK. 1944\u20131945,_ vol. 16 (5\u20134) (Moscow, 1999), p. 12.\n\n. For an examination of the populations affected by mass killings in the geographic region between Germany and the Soviet Union by both Stalin and Hitler, see Timothy Snyder, _Bloodlands: Europe between Hitler and Stalin_ (New York, 2010).\n\n. _Vestnik arkhiva Preszidenta Rossiiskoi Federatsii. Voina. 1941\u20131945,_ pp. 346\u2013348.\n\n. Khaustov et al., _Lubianka. Stalin i NKVD-NKGB-GUKR \"Smersh,\"_ p. 405.\n\n. Alexander Statiev, \"The Nature of Anti-Soviet Armed Resistance, 1942\u20131944: The North Caucasus, the Kalmyk Autonomous Republic, and Crimea,\" _Kritika: Explorations in Russian and Eurasian History_ 6, no. 2 (Spring 2005): 285\u2013318.\n\n. V. A. Kozlov and S. V. Mironenko, eds., \" _Osobaia papka\" Stalina. Iz materialov Sekretariata NKVD-MVD SSSR. 1944\u20131953_ (Moscow, 1994).\n\n. GARF, f. R-9401, op. 2, d. 64, l. 167.\n\n. Ibid., l. 166.\n\n. Ibid., l. 165\n\n. Ibid., d. 66, ll. 9\u201310, 40\u201346.\n\n. Ibid., ll. 334\u2013340.\n\n. Ibid., d. 67, ll. 319\u2013324; d. 68, ll. 268\u2013273.\n\n. N. M. Naimark, _Russians in Germany: A History of the Soviet Zone of Occupation, 1945\u20131949_ (Cambridge, MA, 1995).\n\n. Khaustov et al., _Lubianka. Stalin i NKVD-NKGB-GUKR \"Smersh,\"_ pp. 502\u2013504.\n\n. Cited in M. Dzhilas [Milovan Djilas], _Litso totalitarizma_ (Moscow, 1992), p. 82.\n\n. Transcript of conversations between Stalin and Roosevelt; RGASPI, f. 558, op. 11, d. 235, l. 8.\n\n. N. V. Petrov, _Po stsenariiu Stalina: Rol' organov NKVD-MGB SSSR v sovetizatsii stran Tsentral'noi i Vostochnoi Evropy. 1945\u20131953_ (Moscow, 2011), pp. 44\u201352.\n\n. Cited in V. A. Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1943 g.,_ vol. 16 (5\u20133) (Moscow, 1996), p. 185.\n\n. RGASPI, f. 17, op. 3, d. 1045, l. 55.\n\n. Zolotarev, _Russkii arkhiv. Velikaia Otechestvennaia. Stavka VGK. 1942,_ vol. 16 (5\u20132), p. 420.\n\n. Interview with Aleksandr Vasilevsky in Simonov, _Glazami cheloveka moego pokoleniia,_ p. 446.\n\n. Vasilevskii, _Delo vsei zhizni,_ pp. 496\u2013497.\n\n. Shtemenko, _General'nyi shtab v gody voiny,_ pp. 102\u2013104. V. A. Zolotarev, ed., _Russkii arkhiv, Velikaia Otechestvennaia. General'nyi shtab v gody Velikoi Otechestvennoi voiny. 1941,_ vol. 23 (12\u20131) (Moscow, 1997), pp. 11\u201312.\n\n. From a 2 April 1965 memorandum from Marshal Konev to the Central Committee Presidium; Shtemenko, _General'nyi shtab v gody voiny,_ pp. 104, 192; I. S. Konev, _Zapiski komanduiushchego frontom_ (Moscow, 2000), p. 498.\n\n. Interview with Zhukov in Simonov, _Glazami cheloveka moego pokoleniia,_ p. 377.\n\n. Konev, _Zapiski komanduiushchego frontom,_ p. 498.\n\n. Vasilevskii, _Delo vsei zhizni,_ p. 497.\n\n. The SNK Bureau's Commission on Current Issues existed from June 1941 to December 1942. The SNK Bureau met in regular session from December 1942 through August 1945. Background on the establishment of these bodies can be found in RGASPI, f. 17, op. 163, d. 1326, l. 233; d. 1350, l. 40; d. 1356, ll. 120\u2013121; d. 1406, l. 27.\n\n. Ibid., d. 1356, ll. 120\u2013121.\n\n. Ibid., d. 1406, l. 27.\n\n. State Defense Committee resolution dated 4 February 1942; RGASPI, f. 644, op. 2, d. 36, ll. 32\u201335.\n\n. Mikoian, _Tak bylo,_ p. 465.\n\n. APRF, f. 3, op. 52, d. 251, l. 93.\n\n. In December 1943, Andrei Andreev, a Central Committee secretary and Politburo member, was appointed people's commissar for agriculture.\n\n. APRF, f. 3, op. 52, d. 251, l. 93; Mikoian, _Tak bylo,_ p. 466.\n\n. RGASPI, f. 17, op. 163, d. 1420, l. 136.\n\n. Bulganin was made a member of the State Defense Committee in place of Voroshilov, with whose performance Stalin was displeased. Ibid., op. 3, d. 1051, l. 44, 46.\n\n. V. A. Zolotarev, ed., _Russkii arkhiv. Prikazy narodnogo komissara oborony SSSR. 1943\u20131945 gg.,_ vol. 13 (2\u20133) (Moscow, 1997), p. 332.\n\n. Ibid., p. 337\u2013338.\n\n. David Brandenberger, _National Bolshevism: Stalinist Mass Culture and Formation of Modern Russian National Identity, 1931\u20131956_ (Cambridge, MA, and London, 2002).\n\n. Chernobaev, _Na prieme u Stalina,_ p. 417.\n\n. From notes taken by the head of the Council on the Affairs of the Russian Orthodox Church, G. G. Karpov, on the meeting between Stalin and the church leaders. GARF, f. R-6991, op. 1, d. 1, ll. 1\u201310; M. I. Odintsov, _Russkie patriarkhi XX veka_ (Moscow, 1994), pp. 283\u2013291.\n\n. Rzheshevskii, _Stalin i Cherchill',_ p. 420; Michael Ellman, \"Churchill on Stalin: A Note,\" _Europe-Asia Studies_ 58, no. 6 (September, 2006): 969\u2013970.\n\n. GARF, f. R-9401, op. 2, d. 94, ll. 15\u201327. _Istoricheskii arkhiv,_ no. 5 (1993): 123\u2013128.\n\n. Cited in D. Omel'chuk and S. Iurchenko, \"Krymskaia konferentsiia: Neizvestnye stranitsy,\" _Svododnaia mysl,_ no. 2 (2001): 122\u2013123.\n\n. _Perepiska,_ vol. 2, pp. 204, 205; V. Pechatnov, _Stalin, Ruzvel't, Trumen: SSSR i SShA v 1940-kh gg._ (Moscow, 2006), pp. 305\u2013306.\n\n. _Perepiska,_ vol. 2, pp. 211, 212; Commission for the Publication of Diplomatic Documents under the Ministry of Foreign Affairs of the U.S.S.R., comp., _Correspondence between Stalin, Roosevelt, Truman, Churchill and Atlee during World War II_ (Honolulu, 2001), p. 214.\n\n. Secret telegram from Joseph Stalin to Dwight D. Eisenhower on the eve of the Battle of Berlin; _Novaia i noveishaia istoriia,_ no. 3 (2000): 180\u2013181.\n\n. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti,_ p. 171.\n\n. Shtemenko, _General'nyi shtab v gody voiny,_ p. 265.\n\n. V. A. Zolotarev and G. N. Sevast'ianov, eds., _Velikaia Otechestvennaia Voina. 1941\u20131945. Voenno-istoricheskie ocherki,_ vol. 3 (Moscow, 1999), p. 279.\n\n. _Rodina,_ no. 4 (2005): 99.\n\n**Family**\n\n. A. Ostrovskii, _Kto stoial za spinoi Stalina?_ (Moscow, 2002), pp. 235\u2013236.\n\n. This letter was included in a summary of incoming correspondence prepared for Stalin and then sent to Bulganin, evidently so he could look into granting the requests for assistance; RGASPI, f. 558, op. 11, d. 895, l. 59.\n\n. Ostrovskii, _Kto stoial za spinoi Stalina?,_ p. 249.\n\n. Ibid., pp. 251\u2013252.\n\n. Ibid., pp. 308\u2013309, 329, 332\u2013334.\n\n. Ibid., pp. 340\u2013341.\n\n. Cited in ibid., pp. 349, 357.\n\n. _Izvestiia TsK KPSS,_ no. 10 (1989): 190.\n\n. _Izvestiia TsK KPSS,_ no. 8 (1991): 150.\n\n. Cited in Iu. G. Murin, comp., _Iosif Stalin v ob\"iatiiakh sem'i. Iz lichnogo arkhiva_ (Moscow, 1993), pp. 7\u20138.\n\n. Ibid., p. 154.\n\n. Ibid., p. 22.\n\n. Cited in V. A. Nevezhin, _Zastol'ia Iosifa Stalina. Bol'shie kremlevskie priemy 1930-kh\u20131970-kh gg._ (Moscow, 2011), p. 279.\n\n. \"Pis'ma N. S. Alliluevoi Z. G. Ordzhonikidze,\" _Svobodnaia mysl',_ no. 5 (1993): 74.\n\n. Letter from Nadezhda Allilueva to Maria Svanidze, 11 January 1926; cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ p. 154.\n\n. Ibid., pp. 22\u201340.\n\n. Simon Sebag Montefiore explores possible scenarios of what took place that evening in the prologue to his book _Stalin: The Court of the Red Tsar_ (London, 2003).\n\n. RGASPI, f. 558, op. 11, d. 786, ll. 123\u2013124.\n\n. According to the diary of Maria Svanidze; cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ p. 177.\n\n. Ibid., pp. 157\u2013158.\n\n. Svetlana Alliluyeva, _Twenty Letters to a Friend,_ trans. Priscilla Johnson McMillan (New York, 1967), pp. 151\u2013152.\n\n. Cited in R. W. Davies et al., eds., _The Stalin-Kaganovich Correspondence, 1931\u20131936_ (New Haven, 2003), pp. 297, 304.\n\n. Alliluyeva, _Twenty Letters,_ p. 151.\n\n. The discussion of Vasily's relationship with Stalin is based on Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ pp. 54\u201365, 68\u201369.\n\n. GARF, f. R-9401, op. 2, d. 93, ll. 276\u2013278; V. N. Khaustov et al., comps., _Lubianka. Stalin i NKVD-NKGB-GUKR \"Smersh.\" 1939\u20131946_ (Moscow, 2006), pp. 493\u2013494; Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ pp. 92\u201393.\n\n. Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ pp. 69\u201389, 96\u2013100.\n\n. Alliluyeva, _Twenty Letters,_ p. 180.\n\n. Cited in Murin, _Iosif Stalin v ob\"iatiiakh sem'i,_ pp. 91\u201392.\n\n**Chapter 6. The Generalissimo**\n\n. Cited in _Pravda,_ 25 May 1945.\n\n. Letter from G. Tsydenov, 23 October 1945; RGASPI, f. 558, op. 11, d. 865, l. 6.\n\n. Letter dated 16 February 1946; RGASPI, f. 558, op. 11, d. 867, ll. 14\u201315; E. Iu. Zubkova et al., comps., _Sovetskaia zhizn'. 1945\u20131953_ (Moscow, 2003), pp. 612\u2013613.\n\n. The summary of incoming correspondence that included quotes from this letter features a notation by Poskrebyshev: \"Archive.\" It could only have been made on Stalin's instructions since a number of other letters from the summary were sent to be taken care of by the appropriate official; RGASPI, f. 558, op. 11, d. 867, ll. 1\u20132.\n\n. _Pravda,_ 14 March 1946.\n\n. G. F. Krivosheev et al., _Velikaia Otechestvennaia bez grifa sekretnosti. Kniga poter'_ (Moscow, 2009), p. 42. Without citing a source, Dmitri Volkogonov states that in January 1946 Stalin was given a figure of 15 million dead, including 7.5 million soldiers killed, dying of wounds, or missing in action; Dmitri Volkogonov, _Stalin: Triumph and Tragedy_ (New York, 1991), p. 505. It has not been possible to verify this information.\n\n. For the original of Stalin's letter, edited in his own hand, see RGASPI, f. 558, op. 11, d. 794, ll. 85\u201389. The letter was published in the magazine _Bol'shevik_ in 1947 (no. 3, pp. 6\u20138).\n\n. From a coded telegram from Stalin to Molotov, Beria, Malenkov, and Mikoyan dated 10 November 1945; cited in L. V. Maksimenkov, comp., _Bol'shaia tsenzura. Pisateli i zhurnalisty v Strane Sovetov. 1917\u20131956_ (Moscow, 2005), pp. 556\u2013557.\n\n. _Pravda,_ 23 September 1946.\n\nMikhail Mikhailovich Zoshchenko (1895\u20131958) was a popular satirical writer and playwright. The scathing criticism to which he was subjected in 1946 led to his being deprived of the right to publish. After Stalin's death he was given work writing for magazines but was still a target of discrimination. The 1946 decree criticizing Zoshchenko and Akhmatova was rescinded only in the late 1980s during Gorbachev's perestroika.\n\nAnna Andreevna Akhmatova (1889\u20131966) was among Russia's most important poets. Under Stalin, she was subjected to ongoing persecution. Her first husband was shot and the second died in a labor camp, and her only son spent many years in a camp. A number of anti-Stalinist works by Akhmatova are famous, her poetic cycle _Requiem_ first and foremost.\n\n. RGASPI, f. 558, op. 11, d. 732, ll. 1\u201319.\n\n. Nikolai Krementsov, _Stalinist Science_ (Princeton, 1997); V. D. Esakov and E. S. Levina, _Stalinskie \"sudy chesti\": \"Delo KR\"_ (Moscow, 2005).\n\n. Cited in V. Pechatnov, _Stalin, Ruzvel't, Trumen: SSSR i SShA v 1940-kh gg._ (Moscow, 2006), pp. 392\u2013393.\n\n. RGASPI, f. 558, op. 11, d. 382, l. 45; Pechatnov, _Stalin, Ruzvel't, Trumen,_ p. 421.\n\n. G. Procacci and G. Adibekov et al., eds., _The Cominform: Minutes of the Three Conferences, 1947\/1948\/1949_ (Milan, 1994), pp. 225\u2013226.\n\n. Eugene Zaleski, _Stalinist Planning for Economic Growth, 1933\u20131952_ (Chapel Hill, 1980), pp. 347\u2013348.\n\n. N. Vert and S. V. Mironenko, eds., _Istoriia stalinskogo Gulaga. Konets 1920-kh\u2013pervaia polovina 1950-kh godov,_ vol. 1: _Massovye repressii v SSSR_ (Moscow, 2004), p. 610.\n\n. A. I. Kokurin and N. V. Petrov, comps., _GULAG. 1917\u20131960_ (Moscow, 2000), pp. 435, 447; V. N. Zemskov, _Spetsposelentsy v SSSR, 1930\u20131960_ (Moscow, 2003), p. 225.\n\n. The total population of the USSR at the beginning of 1953 was 188 million; V. P. Popov, _Ekonomicheskaia politika sovetskogo gosudarstva. 1946\u20131953 gg._ (Moscow and Tambov, 2000), p. 16.\n\n. V. A. Kozlov and S. V. Mironenko, eds., \" _Osobaia papka\" Stalina. Iz materialov Sekretariata NKVD-MVD SSSR. 1944\u20131953_ (Moscow, 1994).\n\n. E. Iu. Zubkova, _Pribaltika i Kreml'_ (Moscow, 2008), p. 256; V. Naumov and Iu. Sigachev, comps., _Lavrentii Beriia. 1953. Stenogramma iul'skogo plenuma TsK KPSS i drugie dokumenty_ (Moscow, 1999), p. 47.\n\n. RGASPI, f. 558, op. 11, d. 1481, l. 45.\n\n. Ibid., d. 97, ll. 35\u201336.\n\n. Ibid., ll. 96\u201399.\n\n. V. O. Pechatnov, \"'The Allies Are Pressing on You to Break Your Will....' Foreign Policy Correspondence between Stalin and Molotov and Other Politburo Members, September 1945\u2013December 1946,\" Cold War International History Project, Working Paper No. 26 (September 1999).\n\n. Ibid., p. 2.\n\n. Cited in ibid., p. 4.\n\n. O. V. Khlevniuk et al., comps., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR. 1945\u20131953_ (Moscow, 2002), pp. 198\u2013199.\n\n. RGASPI, f. 558, op. 11, d. 771, ll. 9\u201310.\n\n. Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ pp. 195, 196.\n\n. RGASPI, f. 558, op. 11, d. 771, l. 11.\n\n. Ibid., ll. 7\u20138.\n\n. Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ p. 195. This conflict is also described in Pechatnov, \"'The Allies Are Pressing on You to Break Your Will,'\" pp. 8\u201315.\n\n. Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ pp. 196\u2013197.\n\n. Cited in ibid., pp. 197\u2013198.\n\n. Ibid., pp. 198\u2013199.\n\n. Ibid., p. 200.\n\n. Ibid., pp. 24\u201325, 38.\n\n. Vsevolod Nikolaevich Merkulov (1895\u20131953) was a longtime aid to Beria who had come with him to Moscow in 1938 and was appointed his first deputy at the NKVD. In 1943 Merkulov was in charge of the State Security Commissariat, which had been made into a separate agency outside of the internal affairs commissariat (the NKVD). After being removed from this post amid scandal, he still held high-level positions and during Stalin's final years headed the State Control Ministry. As a client of Beria, he was arrested and shot in late 1953 after Beria himself.\n\n. Viktor Semenovich Abakumov (1908\u20131954) rose through the state security ranks and during the war served as Stalin's deputy at the defense commissariat in charge of military counterintelligence. In 1946\u20131951 he served as state security minister before being arrested in 1951. Even after Stalin's death he was shot rather than being released from prison.\n\n. Memorandum from Merkulov dated 23 July 1953; cited in V. A. Kozlov, ed., _Neizvestnaia Rossiia XX vek,_ vol. 3 (Moscow, 1993), p. 73.\n\n. RGASPI, f. 558, op. 11, d. 442, ll. 202\u2013206; V. Naumov et al., comps., _Georgii Zhukov. Stenogramma oktiabr'skogo (1957 g.) plenuma TsK KPSS i drugie dokumenty_ (Moscow, 2001), pp. 16\u201317.\n\n. A. G. Zverev, _Zapiski ministra_ (Moscow, 1973), pp. 231\u2013234.\n\n. Iu. I. Kashin, comp., _Po stranitsam arkhivnykh fondov Tsentral'nogo banka Rossiiskoi Federatsii,_ vol. 3 (Moscow, 2007), pp. 31\u201332.\n\n. Popov, _Ekonomicheskaia politika sovetskogo gosudarstva,_ pp. 83\u201388. A key memorandum by Zverev dated 8 October 1946 and providing an overview of the experience of the 1922\u20131924 Soviet currency reform, including notations by Stalin, has been published in _Istochnik,_ no. 5 (2001): 21\u201347. The memorandum is held in the APRF.\n\n. A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008), p. 617.\n\n. RGASPI, f. 17, op. 163, d. 1506, l. 22.\n\n. Kashin, _Po stranitsam arkhivnykh fondov Tsentral'nogo banka,_ vol. 3, pp. 96\u201397.\n\n. E. Iu. Zavadskaia and T. V. Tsarevskaia, \"Denezhnaia reforma 1947 goda: Reaktsiia naseleniia. Po dokumentam iz 'osobykh papok' Stalina,\" _Otechestvennaia istoriia,_ no. 6 (1997): 135\u2013137.\n\n. Zubkova et al., _Sovetskaia zhizn',_ pp. 561\u2013564.\n\n. Ibid., p. 564\u2013567.\n\n. Iu. Aksenov and A. Uliukaev, \"O prostykh resheniiakh neprostykh problem. Denezhnaia reforma 1947 goda,\" _Kommunist,_ no. 6 (1990): 83.\n\n. Chernobaev, _Na prieme u Stalina,_ pp. 495\u2013496.\n\n. _Istochnik,_ no. 5 (2001): 51.\n\n. Zubkova et al., _Sovetskaia zhizn',_ p. 529.\n\n. \"On Per Person Norms for Sales of Food and Manufactured Goods\"; USSR Council of Ministers Resolution No. 3867, dated 14 December 1947; GARF, f. R-5446, op. 1, d. 316, ll. 288\u2013289. These limits remained in effect until 1958.\n\n. Aksenov and Uliukaev, \"O prostykh resheniiakh neprostykh problem,\" pp. 84\u201385.\n\n. Julie Hessler, _A Social History of Soviet Trade: Trade Policy, Retail Practices, and Consumption, 1917\u20131953_ (Princeton, 2004), p. 314.\n\n. Zubkova et al., _Sovetskaia zhizn',_ p. 578.\n\n. RGASPI, f. 17, op. 122, d. 308, l. 183.\n\n. Ibid., op. 88, d. 900, l. 178.\n\n. Donald Filtzer, _Soviet Workers and Late Stalinism: Labour and the Restoration of the Stalinist System after World War II_ (Cambridge, 2002), pp. 77\u2013116.\n\n. In recent years a huge number of documents pertaining to the sovietization of Eastern Europe and Stalin's role in this process has been published. For a multifaceted study of these questions, see T. V. Volokitina et al., _Moskva i Vostochnaia Evropa. Stanovlenie politicheskikh rezhimov sovetskogo tipa (1949\u20131953)_ (Moscow, 2008).\n\n. Ibid., pp. 430\u2013550.\n\n. For a more in-depth account, see Yoram Gorlizki and Oleg Khlevniuk, _Cold Peace: Stalin and the Soviet Ruling Circle, 1945\u20131953_ (New York, 2004), pp. 79\u201389.\n\n. Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ p. 67.\n\n. For an interpretation of the Leningrad Affair as Stalin's response to the spread and strengthening of patron-client relations within the Soviet nomenklatura, see Benjamin Tromly, \"The Leningrad Affair and Soviet Patronage Politics, 1949\u20131950,\" _Europe-Asia Studies_ 56, no. 5 (July 2004): 707\u2013729.\n\n. Cited in F. Chuev, _Sto sorok besed s Molotovym_ (Moscow, 1991), p. 475.\n\n. Voting was carried out by _opros_ (polling); in other words, members voted remotely, not while they were seated together in a Politburo meeting. According to the tally compiled by Poskrebyshev, who handled most of the clerical aspects of Politburo resolutions, Stalin, Bulganin, Voroshilov, Voznesensky, Shvernik, Kaganovich, Mikoyan, Andreev, Beria, Malenkov, and Kosygin voted in favor of expelling Zhemchuzhina from the party. \"Com. Molotov abstained\"; RGASPI, f. 17, op. 163, d. 1518, l. 162.\n\n. Ibid., l. 164; Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ p. 313.\n\n. Joshua Rubenstein and Vladimir P. Naumov, eds., _Stalin's Secret Pogrom: The Postwar Inquisition of the Jewish Anti-Fascist Committee_ (New Haven, 2001).\n\n. Cited in _Vestnik arkhiva prezidenta Rossiiskoi Federatsii. Voina 1941\u20131945_ (Moscow, 2010), p. 333.\n\n. Letter from Gorbenko, a member of the military, dated 15 July 1945; RGASPI, f. 558, op. 11, d. 863, ll. 79\u201386.\n\n. Yuri Slezkine, _The Jewish Century_ (Princeton, 2004), p. 297.\n\n. Letter from military journalist S. A. Lifshits dated March 1949; RGASPI, f. 558, op. 11, d. 876, l. 15; f. 17, op. 132, d. 118, ll. 1\u20133.\n\n. From the diary of People's Commissar V. A. Malyshev, who was present at the meeting; cited in _Istochnik,_ no. 5 (1997): 140\u2013141.\n\n. This and subsequent excerpts from documents pertaining to preparations for Mao's visit are quoted, with minor modifications, from Sergey Radchenko and David Wolff, \"To the Summit via Proxy-Summits: New Evidence from Soviet and Chinese Archives on Mao's Long March to Moscow, 1949,\" _Cold War International History Project Bulletin,_ no 16. (Spring 2008): 118\u2013129.\n\n. A. M. Ledovskii, _SSSR i Stalin v sud'bakh Kitaia. Dokumenty i svidetel'stva uchastnika sobytii. 1937\u20131952_ (Moscow, 1999), p. 55.\n\n. Chen Jian, \"The Sino-Soviet Alliance and China's Entry into the Korean War,\" Cold War International History Project, Working Paper No. 1 (June 1992), p. 19.\n\n. A. V. Pantsov, _Mao Tzedun_ (Moscow, 2007), p. 47.\n\n. Cited in A. V. Pantsov, comp., _Mao Tzedun. Avtobiografiia. Stikhi_ (Moscow, 2008), p. 166.\n\n. A. M. Ledovskii, \"Stalin, Mao Tzedun i koreiskaia voina 1950\u20131953,\" _Novaia i noveishaia istoriia,_ no. 5 (2005): 106.\n\n. RGASPI, f. 558, op. 11, d. 329, ll. 10\u201317; Chen Jian et al., eds., \"Stalin's Conversations: Talks with Mao Zedong, December 1949\u2013January 1950, and with Zhou Enlai, August\u2013September 1952,\" _Cold War International History Project Bulletin,_ nos. 6\u20137 (Winter 1995\u20131996): 5\u20137.\n\n. From the memoirs of Matyas Rakosi; cited in _Istoricheskii arkhiv,_ no. 3 (1997): 142\u2013143.\n\n. Odd Arne Westad, \"Fighting for Friendship: Mao, Stalin, and the Sino-Soviet Treaty of 1950,\" _Cold War International History Project Bulletin,_ nos. 8\u20139 (Winter 1996\u20131997): 227\u2013228; Dieter Heinzig, _The Soviet Union and Communist China, 1945\u20131950: The Arduous Road to the Alliance_ (London, 2003), pp. 281\u2013282, 286\u2013289.\n\n. Ledovskii, _SSSR i Stalin v sud'bakh Kitaia,_ p. 143.\n\n. Cited in N. Fedorenko, \"Nochnye besedy,\" _Pravda,_ 23 October 1988, p. 4.\n\n. RGASPI, f. 558, op. 11, d. 329, l. 51.\n\n. Fedorenko, \"Nochnye besedy.\"\n\n. For a detailed examination of Stalin's role in the Soviet nuclear project, see David Holloway, _Stalin and the Bomb: The Soviet Union and Atomic Energy, 1939\u20131956_ (New Haven, 1996).\n\n. Letter written to the members of the Soviet leadership by Beria from prison, 1 July 1953; cited in Naumov and Sigachev, _Lavrentii Beriia,_ p. 75.\n\n. Cited in A. V. Torkunov, _Zagadochnaia voina: Koreiskii konflikt 1950\u20131953_ (Moscow, 2000), pp. 6\u20138.\n\n. Kathryn Weathersby, \"To Attack, or Not to Attack? Stalin, Kim Il Sung, and the Prelude to War,\" _International History Project Bulletin,_ no 5. (Spring 1995): 7\u20138.\n\n. Ibid., p. 9; Chernobaev, _Na prieme u Stalina,_ p. 533.\n\n. K. Vezersbi [Weathersby], \"Sovetskie tseli v Koree, 1945\u20131950 gg.,\" in _Kholodnaia voina. Novye podkhody, novye dokumenty,_ ed. M. M. Narinskii (Moscow, 1995), p. 316.\n\n. In January 1950 the USSR was boycotting the United Nations, demanding that the new Communist government of China be allowed representation. Starting the war in Korea at a time when the Soviet representative to the Security Council was absent was a clear blunder by Stalin, one of which the United States took full advantage.\n\n. Cited in Kathryn Weathersby (introduction and translations), \"New Russian Documents on the Korean War,\" _Cold War International History Project Bulletin,_ nos. 6\u20137 (Winter 1995\u20131996): 40.\n\n. Cited in Torkunov, _Zagadochnaia voina,_ p. 97.\n\n. Cited in Alexandre Y. Mansourov, \"Stalin, Mao, Kim, and China's Decision to Enter the Korean War,\" _Cold War International History Project Bulletin,_ nos. 6\u20137 (Winter 1995\u20131996): 118. (Bracketed insertion is Mansourov's.)\n\n. Cited in Ledovskii, \"Stalin, Mao Tzedun i koreiskaia voina,\" p. 106.\n\n. From the memoirs of Matyas Rakosi; cited in _Istoricheskii arkhiv,_ nos. 5\u20136 (1997): 7\u20138. The fact that this meeting took place is also confirmed by the defense minister of Czechoslovakia, Alexeje \u010cepi\u010dka; _Voprosy istorii,_ no. 10 (1999): 85\u201386.\n\n. Zaleski, _Stalinist Planning for Economic Growth,_ pp. 668\u2013669.\n\n. Russian State Archive of the Economy (RGAE), f. 4372, op. 11, d. 677, ll. 9\u201310. Figures for military expenditures are for four ministries created after Stalin's death: defense (which brought together the former defense and naval ministries), defense industry (an updated version of the former armaments ministry), the aviation industry, and medium-machine building. These ministries accounted for the lion's share (although not all) of military spending.\n\n. N. S. Simonov, _Voenno-promyshlennyi kompleks SSSR v 1920\u20131950-e gody_ (Moscow, 1996), pp. 210\u2013266.\n\n. Council of Ministers resolutions dated 9 and 19 February 1953; A. A. Danilov and A. V. Pryzhikov, _Rozhdenie sverkhderzhavy. SSSR v pervye poslevoennye gody_ (Moscow, 2001), pp. 92\u201393.\n\n. RGASPI, f. 558, op. 11, d. 329, ll. 66; Ledovskii, _SSSR i Stalin v sud'bakh Kitaia,_ p. 160.\n\n. A. I. Kokurin and Iu. N. Morukov, _Stalinskie stroiki GULAGA. 1930\u20131953_ (Moscow, 2005).\n\n. RGAE, f. 4372, op. 11, d. 282, l. 66.\n\n. _Narodnoe khoziastvo SSSR. Statisticheskii sbornik_ (Moscow, 1956), p. 118.\n\n. RGASPI, f. 558, op. 11, d. 882, ll. 57\u201358.\n\n. Letter dated 1 November 1952; RGASPI, f. 558, op. 11, d. 903, ll. 42\u201346.\n\n. This undated letter was sent from Stalin's secretariat for Malenkov to deal with on 4 November 1952; RGASPI, f. 558, op. 11, d. 901, ll. 39\u201340.\n\n. Chernobaev, _Na prieme u Stalina,_ p. 551; N. Kovaleva et al., comps., _Molotov, Malenkov, Kaganovich. 1957. Stenogramma iiun'skogo plenuma TsK KPSS i drugie dokumenty_ (Moscow, 1998), pp. 193\u2013194.\n\n. On the works of this commission and Stalin's position on the subject, see Gorlizki and Khlevniuk, _Cold Peace,_ pp. 139\u2013140.\n\n. A. I. Mikoian, _Tak bylo. Razmyshleniia o minuvshem_ (Moscow, 1999), p. 578.\n\n. RGAE, f. 4372, op. 11, d. 459, ll. 164\u2013170.\n\n. Kokurin and Petrov, _GULAG. 1917\u20131960,_ pp. 788\u2013791; RGAE, f. 4372, op. 11, d. 677, l. 9.\n\n. The inventories did not specify the agency originating the coded telegrams. RGASPI, f. 558, op. 11, d. 117, ll. 1\u2013173.\n\n. Ignatiev told this story in testimony given 27 March 1953; N. V. Petrov, _Palachi_ (Moscow, 2011), p. 307.\n\n. K. A. Stoliarov, _Palachi i zhertvy_ (Moscow, 1998), p. 163.\n\n. Ibid., pp. 225\u2013226.\n\n. Ibid., pp. 167\u2013168.\n\n. Naumov and Sigachev, _Lavrentii Beriia,_ pp. 34\u201335.\n\n. For more details, see Timothy Blauvelt, \"Abkhazia: Patronage and Power in the Stalin Era,\" _Nationalities Papers_ 35, no. 2 (2007): 220, 222\u2013223.\n\n. Naumov and Sigachev, _Lavrentii Beriia,_ pp. 29\u201340.\n\n. It was at the Nineteenth Party Congress that the party's name was officially changed from the All-Union Communist Party of Bolsheviks, commonly referred to by the acronym VKP(b), to simply the Communist Party of the Soviet Union or KPSS. This name endured until the party and country were abolished in 1991.\n\n. Mikoian, _Tak bylo,_ p. 573.\n\n. Ibid., pp. 574\u2013576; Chuev, _Sto sorok besed s Molotovym,_ p. 469; L. N. Efremov, _Dorogami bor'by i truda_ (Stavropol, 1998), pp. 12\u201316.\n\n. N. Mukhitdinov, _Reka vremeni. Ot Stalina do Gorbacheva. Vospominaniia_ (Moscow, 1995), pp. 88\u201389.\n\n. Volokitina et al., _Moskva i Vostochnaia Evropa,_ pp. 558\u2013566.\n\n. Explanatory memorandum from Ignatiev to Beria dated 27 March 1953; cited in Petrov, _Palachi,_ p. 297.\n\n. Ibid., pp. 287, 299\u2013300.\n\n. Cited in V. N. Khaustov et al., comps. _Lubianka. Stalin i MGB SSSR. Mart 1946\u2013mart 1953_ (Moscow, 2007), pp. 522\u2013523.\n\n. Cited in N. V. Petrov, _Pervyi predsedatel' KGB Ivan Serov_ (Moscow, 2005), p. 124.\n\n. From a transcript of remarks by Stalin to a commission on reorganizing the Ministry of State Security's intelligence service, November\u2013December 1952; cited in _Istochnik,_ no. 5 (2001): 132.\n\n. These press items were edited by Stalin. RGASPI, f. 558, op. 11, d. 157, ll. 9\u201314, 29\u201333; Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ pp. 392\u2013397.\n\n. For a detailed examination of this theory about the deportation of Jews, see G. V. Kostyrchenko, _Stalin protiv \"kosmopolitov\". Vlast' i evreiskaia intelligentsiia v SSSR_ (Moscow, 2009), pp. 329\u2013380.\n\n. B. S. Klein, \"Politika SShA i 'delo vrachei,'\" _Voprosy istorii,_ no. 6 (2006): 35\u201347.\n\n**The Dictatorship Collapses**\n\n. A. A. Chernobaev, ed., _Na prieme u Stalina. Tetradi (zhurnaly) zapisei lits, priniatykh I. V. Stalinym (1924\u20131953 gg.)_ (Moscow, 2008), p. 553; O. V. Khlevniuk et al., comps., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR. 1945\u20131953_ (Moscow, 2002), p. 436. When the log of visitors to Stalin's office was published, Tkachev's name was mistakenly given as Tolkachev.\n\n. Chernobaev, _Na prieme u Stalina,_ p. 553; Khlevniuk et al., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR,_ p. 436.\n\n. A. L. Miasnikov, _Ia lechil Stalina_ (Moscow, 2011), p. 295.\n\n. Khlevniuk et al., _TsK VKP(b) i Sovet Ministrov SSSR,_ pp. 436\u2013437.\n\n. N. Kovaleva et al., comps., _Molotov, Malenkov, Kaganovich. 1957. Stenogramma iiun'skogo plenuma TsK KPSS i drugie dokumenty_ (Moscow, 1998), pp. 42, 45. The papers were removed when Malenkov's assistant was arrested.\n\n. The decisions were recorded in the minutes of the 5 March 1953 joint meeting of the Central Committee plenum, the Council of Ministers, and the Presidium of the Supreme Soviet. _Istochnik,_ no. 1 (1994): 107\u2013111.\n\n. K. M. Simonov, _Glazami cheloveka moego pokoleniia_ (Moscow, 1989), pp. 257\u2013258.\n\n. Ibid., p. 260.\n\n. Svetlana Alliluyeva, _Twenty Letters to a Friend,_ trans. Priscilla Johnson McMillan (New York, 1967), p. 10.\n\n. Chernobaev, _Na prieme u Stalina,_ p. 553.\n\n. From Shepilov's memoirs; cited in _Voprosy istorii,_ no. 3 (1998): 15.\n\n. A. N. Artizov et al., comps., _Reabilitatsiia: Kak eto bylo,_ vol. 1 (Moscow, 2000), p. 19.\n\n. V. Naumov and Iu. Sigachev, comps., _Lavrentii Beriia. 1953. Stenogramma iul'skogo plenuma TsK KPSS i drugie dokumenty_ (Moscow, 1999), pp. 28\u201329.\n\n. Oleg Khlevniuk, \"The Economy of the OGPU, NKVD and MVD of the USSR, 1930\u20131953: The Scale, Structure and Trends of Development,\" in _The Economics of Forced Labor: The Soviet Gulag,_ ed. Paul R. Gregory and Valery Lazarev (Stanford, CA, 2003), pp. 54\u201355.\n\n. According to official statistics, between 1 January and 1 October 1953 the number of cows increased from 24.3 million to 26 million, and almost 1 million of that increase took place outside of the collective and state farm system. During that same period the number of pigs increased from 28.5 to 47.6 million, including an increase of 12 million in private herds; _Narodnoe khoziastvo SSSR. Statisticheskii sbornik_ (Moscow, 1956), pp. 119\u2013120. Even with the consideration of possible seasonal fluctuations, these numbers are significant and surely attributable to lower taxes and higher procurement prices.\n\n. A. V. Torkunov, _Zagadochnaia voina: Koreiskii konflikt 1950\u20131953_ (Moscow, 2000), pp. 272\u2013279.\n\n. This directive was largely in response to the large number of defections from East Germany to the West. See Naumov and Sigachev, _Lavrentii Beriia,_ pp. 55\u201359.\n\n**The Funeral**\n\n. Speech by Khrushchev at a dinner in the Bulgarian city of Varna during an official visit on 16 May 1962; cited in _Istochnik,_ no. 6 (2003): 130.\n\n. Letter dated 10 March 1953 from a group of citizens to the Central Committee and the Supreme Soviet; GARF, f. R-7523, op. 52, d. 18, ll. 94\u201395.\n\n. Anonymous letter addressed to Georgy Malenkov, dated 6 March 1953; RGASPI, f. 558, op. 11, d. 1486, l. 157.\n\n. Ibid., d. 1487, l. 55.\n\n. Ibid., ll. 66\u201371.\n\n. Cited in V. A. Kozlov, _Neizvestnaia Rossiia XX vek,_ vol. 2 (Moscow, 1992), pp. 254\u2013258.\n\n. Cited in V. A. Kozlov and S. V. Mironenko, _58\u201310. Nadzornye proizvodstva Prokuratury SSSR po delam ob antisovetskoi agitatsii i propagande. Annotirovannyi katalog. Mart 1953\u20131991_ (Moscow, 1999), pp. 13, 21, 23, 32.\n\n. There is a long list of published documents and studies on the public mood and mechanisms used to shape it and on social adaptation and the particular mindset that Stalinism strove to shape. Studies vary in terms of their authors' viewpoints and the aspect of reality they emphasize. See, for example, the following: Sheila Fitzpatrick: _The Cultural Front: Power and Culture in Revolutionary Russia_ (Ithaca, NY, 1992), and _Tear off the Masks! Identity and Imposture in Twentieth-Century Russia_ (Princeton, 2005); Stephen Kotkin, _Magnetic Mountain: Stalinism as a Civilization_ (Berkeley and Los Angeles, 1995); Sarah Davies, _Popular Opinion in Stalin's Russia: Terror, Propaganda and Dissent, 1934\u20131941_ (Cambridge, 1997); Elena Zubkova, _Russia after the War: Hopes, Illusions, and Disappointments, 1945\u20131957_ (New York, 1998); Jochen Hellbeck, _Revolution on My Mind: Writing a Diary under Stalin_ (Cambridge, MA, 2006).\n\n. Yoram Gorlizki, \"Political Reform and Local Party Interventions under Khrushchev,\" in _Reforming Justice in Russia, 1864\u20131996,_ ed. Peter H. Solomon (New York and London, 1997), pp. 259\u2013260.\n\n. Letter from Stakhanov to Stalin in May 1945; RGASPI, f. 558, op. 11, d. 891, l. 128. For a similar letter sent to Molotov before the war, see GARF, f. R-5446, op. 82, d. 108, l. 145; d. 120, l. 74.\n\n. According to official statistics, at the start of 1953 more than 40 percent of the country's population lived in cities. It should be kept in mind, however, that this figure included residents of small cities and settlements where the standard of living was close to that of the peasants.\n\n. In 1952, out of the 443,000 tons of meat sold through state and cooperative outlets across the USSR, 110,000 were sent to Moscow and 57,400 were sent to Leningrad; GARF, f. R-5446, op. 87, d. 1162, l. 171.\n\n. A. I. Mikoian, _Tak bylo. Razmyshleniia o minuvshem_ (Moscow, 1999), p. 355.\n\n. Amir Weiner, _Making Sense of War: The Second World War and the Fate of the Bolshevik Revolution_ (Princeton, 2000).\n\n. Golfo Alexopoulos, _Stalin's Outcasts: Aliens, Citizens, and the Soviet State, 1926\u20131936_ (Ithaca, NY, and London, 2003).\n\n. Lynne Viola, _Peasant Rebels under Stalin: Collectivization and the Culture of Peasant Resistance_ (New York and Oxford, 1996); Lynne Viola, ed., _Contending with Stalinism: Soviet Power and Popular Resistance in the 1930s_ (Ithaca, NY, 2002); Jeffrey J. Rossman, _Worker Resistance under Stalin: Class and Revolution on the Shop Floor_ (Cambridge, MA, and London, 2005).\n\n. In recent years historians have produced several valuable studies on this problem. See, for example, the following: Sheila Fitzpatrick, _Everyday Stalinism: Ordinary Life in Extraordinary Times: Soviet Russia in the 1930s_ (New York, 1999); Elena Osokina, _Our Daily Bread: Socialist Distribution and the Art of Survival in Stalin's Russia, 1927\u20131941_ (New York and London, 2001); Donald Filtzer, _The Hazards of Urban Life in Late Stalinist Russia: Health, Hygiene, and Living Standards, 1943\u20131953_ (Cambridge, 2010).\n\n. Calculations based on E. Iu. Zubkova et al., comps., _Sovetskaia zhizn'. 1945\u20131953_ (Moscow, 2003), pp. 102\u2013103; O. V. Khlevniuk et al., comps., _Politbiuro TsK VKP(b) i Sovet Ministrov SSSR, 1945\u20131953_ (Moscow, 2002), pp. 388\u2013389. For comparison, see A. I. Kokurin and N. V. Petrov, _GULAG. 1917\u20131960_ (Moscow, 2000), pp. 543\u2013551.\n\n. This letter was given to Malenkov to read; RGASPI, f. 558, op. 11, d. 901, l. 37.\n\n. Zubkova et al., _Sovetskaia zhizn',_ p. 107.\n\n. Cited in ibid., p. 263.\n\n. Figures for state and private urban housing are from RGAE, f. 1562, op. 41, d. 56, ll. 30\u201333. Figures for the urban population as of early 1953 are from V. P. Popov, _Ekonomicheskaia politika Sovetskogo gosudarstva. 1946\u20131953 gg._ (Moscow and Tambov, 2000), p. 16.\n\n. RGAE, f. 1562, op. 41, d. 56, ll. 30\u201333. The inventory of publicly owned residential buildings included the best-built ones, which belonged to local government councils (soviets) and agencies. A significant proportion of urban housing was in private hands. These buildings were in much worse shape.\n\n. Zubkova et al., _Sovetskaia zhizn',_ p. 179.\n\n. N. Vert and S. V. Mironenko, eds., _Istoriia stalinskogo Gulaga. Konets 1920-kh\u2013pervaia polovina 1950-kh godov,_ vol. 1: _Massovye repressii v SSSR_ (Moscow, 2004), pp. 623\u2013624.\n\n. B. V. Zhiromskaia, I. N. Kiselev, and Iu. A. Poliakov, _Polveka pod grifom \"sekretno\": Vsesoiuznaia perepis' naseleniia 1937 goda_ (Moscow, 1996), pp. 98, 100.\n\n. Terry Martin, _The Affirmative Action Empire: Nations and Nationalism in the Soviet Union, 1923\u20131939_ (Ithaca, NY, and London, 2001).\n\n. See one recent study: Timothy Snyder, _Bloodlands: Europe between Hitler and Stalin_ (New York, 2010).\n\n. For documents and letters characterizing inter-ethnic conflicts during the final period of Stalin's rule, see L. P. Kosheleva et al., comps., _Sovetskaia natsional'naia politika. Ideologiia i praktiki realizatsii_ (Moscow, 2013).\n\n. Geoffrey Hosking, _Rulers and Victims: The Russians in the Soviet Union_ (Cambridge, MA, and London, 2006).\n\n. E. Khodzha [Enver Hoxha], _So Stalinym. Vospominaniia_ (Tirana, 1984), p. 90.\n\n. RGASPI, f. 558, op. 11, d. 1479, ll. 14\u201318.\n\n. A. Berelovich and V. Danilov, eds., _Sovetskaia derevnia glazami VChK-OGPU-NKVD: 1918\u20131939 gg.,_ vols. 1\u20134 (Moscow, 1998\u20132012); G. N. Sevost'ianov et al., eds., _\"Sovershenno sekretno\": Lubianka\u2013Stalinu o polozhenii v strane (1922\u20131934),_ vols. 1\u20139 (Moscow, 2001\u20132013).\n\n. GARF, f. R-9401, op. 12, d. 100, ll. 91\u201392.\n\n. When the apparat of the Special Sector was being reorganized in 1939, provisions were made for the creation of fifteen staff positions for people reading letters addressed to Stalin. Their duties included familiarizing themselves with the letters and sorting them (APRF, f. 3, op. 22, d. 65, l. 37). If we assume that each reader spent an average of ten minutes per letter, in working an eight-hour day, all fifteen readers would be able to review 720 letters per day or approximately 260,000 per year. Probably the number was higher. Experienced readers would process letters quickly, especially as many letters were short. Furthermore, using a shift system, the apparat worked essentially around the clock, and shifts were not strictly limited to eight hours.\n\n. APRF, f. 3, op. 22, d. 65, l. 51. The Special Sector's Fifth Section also took care of Stalin's library.\n\n. The letters shown to the Special Sector leadership during 1945\u20131953 have been preserved. See RGASPI, f. 558, op. 11, dd. 888\u2013904.\n\n. Letters selected to be shown to Stalin were accompanied by a list entitled \"Letters and Petitions Received Addressed to Com. Stalin.\" In addition to the letters presented to Stalin, this list included certain letters sent for review by other Soviet leaders. Apparently these were letters it was felt Stalin did not need to see but about which he would be interested in knowing. Stalin's personal archive contains a rather complete set of such lists of letters only for 1945\u20131952 (but lacks those received while he was vacationing in the south); RGASPI, f. 558, op. 11, dd. 862\u2013882.\n\n. Jan Plamper, _The Stalin Cult: A Study in the Alchemy of Power_ (New Haven, 2012).\n\n## **ACKNOWLEDGMENTS**\n\nWhen Jonathan Brent and Vadim Staklo\u2014then the editorial director and project manager for Yale University Press's Annals of Communism series respectively\u2014suggested that I write a biography of Stalin, I was more puzzled than glad. But now that the book has been completed, I am truly thankful to them.\n\nFew know more about the Stalin era than my friends Yoram Gorlizki, Andrea Graziosi, Jan Plamper, and David Shearer, and I am grateful to them for reading the manuscript and making valuable comments. This work also greatly benefited from the skillful editing of William Frucht, the press's executive editor, the keen eye and remarkable memory of the manuscript's copy editor, Bojana Ristich, and the expertise of production editor Margaret Otzel. A critical role was played by the book's translator, Nora Favorov, my most attentive and demanding reader.\n\nThis biography is the culmination of long years of studying Soviet history. These years brought collaboration and friendship with many knowledgeable colleagues. My interactions with all of them have helped prepare me to produce this work.\n\nTo start with those no longer living, I learned a great deal from Moshe Lewin, Viktor Petrovich Danilov, Victor Zaslavsky, and Derek Watson\u2014all prominent historians and wonderful human beings.\n\nNext, this book would not have been possible without decades of work alongside my friends and fellow archival researchers. At RGASPI, I have been fortunate to work with Andrei Sorokin, Lyudmila Kosheleva, Marina Astakhova, Galina Gorskaia, and Elena Kirillova. My work at GARF would have been impossible without the constant support of Sergei Mironenko, Larisa Rogovaya, Larisa Malashenko, Dina Nokhotovich, Sofia Somonova, Galina Kuznetsova, and Tatiana Zhukova. Together we compiled a number of collections of historical documents.\n\nFor twenty-five years now I have been a proud member of the Robert Davies team. His dedication to the study of history and his amazing productivity are an example for us all.\n\nCollaboration, interaction, and friendship with a number of historians have greatly contributed to my work. I would like to express my sincere gratitude to Golfo Alexopoulos, J\u00f6rg Baberowski, Alain Blum, Yves Cohen, Marta Craveri, Victor D\u00f6nninghaus, Michael David-Fox, Mark Elie, Benno Ennker, Klaus Gestwa, Mark Harrison, Jana Howlett, Melanie Ilic, Nicolaus Katzer, Vladimir Kozlov, Sergei Kudryashov, Hiroaki Kuromiya, Terry Martin, Silvio Pons, Valeri Pozner, Arfon Rees, Andrea Romano, Ingrid Schierle, Robert Service, Jeremy Smith, Takeshi Tomita, Aleksandr Vatlin, Lynne Viola, Amir Weiner, Nicolas Werth, Stephen Wheatcroft, and Elena Zubkova.\n\nPaul Gregory, Ron Suny, Sheila Fitzpatrick, Piter Solomon, and Dietrich Beyrau have been attentive and patient conversation partners for many years.\n\nI would also like to express gratitude to the German Historical Institute in Moscow, Fondation Maison des Sciences de l'Homme, and the Ukrainian Studies Fund for their support.\n\nAs always, I would not miss this opportunity to wish success to my daughter Dasha.\n\nAs fate would have it, when I began work on this project my wife Katya fell ill. By the time I completed the manuscript, she was no longer with us.\n\nTo her, I dedicate this book.\n\n## **INDEX**\n\nIllustrations are indicated by Gallery number\n\nAbakumov, Viktor Semenovich, , , n39\n\nAgriculture: collectivization, \u2013113, \u2013116, , ,\n\nand famine policy, ,\n\nand food shortages, \u2013185, , \u2013301, \u2013324\n\ngrain expropriation, , \u2013104, , , \u2013120\n\ngrain reserves, n35\n\ninterregnum period reforms, ,\n\nlivestock production, \u2013300, n15\n\nand New Economic Policy (NEP), ,\n\nprivate plots, , , , ,\n\nquotas for grain deliveries, \u2013124\n\ntsarist reforms, . _See also_ Peasants\n\nAir Force: accident rate in,\n\ncapacity of, ,\n\nmodernization of,\n\ndestruction of at start of war, \u2013202\n\nAkhmatova, Anna, , n9\n\nAlcohol consumption, at Stalin's dacha, \u20136\n\nAlekseev, N. P., \u201357\n\nAlexander III,\n\nAlliluev, Pavel, , , ,\n\nAlliluev, Sergei,\n\nAllilueva, Anna, , , ,\n\nAllilueva, Nadezhda. _See_ Stalin, Nadezhda Allilueva\n\nAllilueva (Alliluyeva), Svetlana: birth of,\n\nchildhood of, , , Gallery , ,\n\non dacha lifestyle, ,\n\ndefection of,\n\nJewish husband of, ,\n\nin public appearance, ,\n\nrelationship with father, \u2013257, ,\n\nat Stalin's deathbed, , \u2013314\n\non Stalin's health, \u2013197\n\non suicide of mother,\n\nAlliluev family, Stalin's friendship with, \u201349,\n\nAndreev, Andrei, n142\n\nAnti-Comintern Pact,\n\nAnti-retreat units, , ,\n\nAnti-Semitism, as state policy, \u2013285, ,\n\nAntonov, Aleksei Innokentievich, , n99\n\nAppeasement policy, , ,\n\nArchival sources: citizens' letters, \u2013301, \u2013329, n35, n38\n\nnewly accessible, ix, xii, xv\n\nStalin Collection, xiii\u2013xiv\n\ntypes of, xii\n\nvisitor logs, xiii\n\nAristov, Averky,\n\nAtom bomb: Soviet,\n\nU.S.,\n\nBaberowski (Baberovski), J\u00f6rg,\n\nBaku, Social Democratic movement in, \u201327\n\nBalitsky, Vsevolod,\n\nBank deposits, and currency reform, , ,\n\nBaramiia, Mikhail, \u2013305\n\nBatum, revolutionary violence in, \u201323\n\nBeaverbrook, Lord,\n\nBelarus: German invasion of,\n\nand German-Soviet non-aggression pact,\n\nRed Army campaign of 1944,\n\nforced resettlement campaign in,\n\nSoviet occupation of,\n\nBeria, Lavrenty Pavlovich: , , , , Gallery ,\n\nbiography of, \u2013333n1\n\nbribery charge against, \u2013305\n\non collaborators,\n\nand ethnic deportations, \u2013234\n\nfall of, n1\n\non German prisoners of war,\n\nin interregnum period,\n\nin leadership reorganizations, ,\n\nand Molotov scandal, ,\n\npoisoning charge against, n1\n\nPolitburo appointment of,\n\non Red Army crimes against civilians,\n\nat Stalin's deathbed, , , ,\n\nand Stalin's wartime safety,\n\nstate security under,\n\nin wartime leadership, , , , , , ,\n\nas Yezhov's deputy, \u2013160\n\nBerlin blockade,\n\nBerlin Conference, \u2013248\n\nBerlin operation, \u2013247\n\nBessarabia, , ,\n\nBiography: archival expos\u00e9s, ix\u2013x\n\nmemoirs as source for, xiv\u2013xv\n\npitfalls of genre, xi\u2013xii\n\nBliukher, Vasily,\n\nBogdanov, Aleksandr,\n\nBolsheviks: armed robbery by, \u201326\n\ncentralized party organization of, \u201338\n\nat European party congresses,\n\nfalsification of history of,\n\nrevolutionary agenda of,\n\nStalin's conflicts with colleagues,\n\nStalin's purge of old guard, \u2013139,\n\nStalin's rise in leadership ranks, , , \u201366\n\nwar readiness of, . _See also_ Civil War\n\nCollective leadership\n\nLenin\n\nRevolution of 1917\n\nBorisov, Mikhail, , ,\n\nBritain: appeasement of Hitler, ,\n\nChurchill-Stalin meetings, , ,\n\nand Polish invasion, \u2013170\n\nin postwar settlement,\n\nsecond front plan, \u2013224, ,\n\nwartime aid to Stalin, ,\n\nBudenny, Semen, , n21\n\nBukharin, Nikolai Ivanovich: , , , Gallery\n\non anti-Trotsky coalition, \u201382\n\nbiography of, n55\n\nexecution of,\n\n-Kamenev secret meeting,\n\nas oppositionist, , ,\n\nin power struggle, \u201376, , ,\n\nBukovina,\n\nBulgakov, Mikhail Afanasyevich, , n10\n\nBulganin, Nikolai Aleksandrovich,\n\nbiography of, n1\n\nin interregnum period,\n\nin Mao's Moscow visit,\n\nat Stalin's deathbed, , ,\n\nStalin's promotion of,\n\nin wartime leadership,\n\nBulgaria, in Communist bloc,\n\nCannibalism, in famine of 1932\u20131933,\n\nCapital investment, postwar, ,\n\nCapture by enemy provisions (Order No. 270),\n\nCaucasus region, culture of violence in, \u201322\n\nCentral Committee: February-March 1937 plenum, \u2013155\n\nmajority faction,\n\nPresidium, \u2013306,\n\nPresidium Bureau, , ,\n\nSpecial Sector,\n\nStalin's reorganization of,\n\nChadaev, Yakov, , ,\n\nChamberlain, Austen,\n\nChamberlain, Neville,\n\n_Chapaev_ (film),\n\nCharkviani, Khristofor,\n\nChechens, forced relocation of,\n\nChiang Kai-shek,\n\n\"Children's literature,\" xiv\u2013xv\n\nChina-Soviet relations: collective leadership debate on,\n\nin Korean War, ,\n\nMao's Soviet visit, , \u2013288, \u2013293\n\nnon-aggression pact of 1937,\n\ntreaty of 1945, , ,\n\ntreaty of 1950, \u2013292\n\nChinese civil war, ,\n\nChizhikov, Petr,\n\nChochia, Grigory,\n\nChubar, Vlas,\n\nChuikov, Vasily, \u2013172\n\nChurchill, Winston,\n\nIron Curtain speech of,\n\nand second front plan, \u2013224, ,\n\n-Stalin meetings, , , , Gallery\n\nCivil War: casualties of,\n\nopponents of Bolsheviks in,\n\nPolish front, \u201361\n\nStalin's missions during, \u201356\n\nStalin-Trotsky conflict, \u201363\n\nTsaritsyn terror campaign during, \u201359\n\nClausewitz, Carl von,\n\nCold War, origins of, \u2013267\n\nCollaborators with Nazi occupation, prosecution of,\n\nCollective leadership, Gallery, ,\n\nanti-Trotsky coalition in, , \u201382\n\ndivision of functions, \u201381\n\nexpulsion of leftist opposition, \u201386, ,\n\nand foreign threats campaign, \u201389\n\nand New Economic Policy (NEP), , , , ,\n\nopponents of Stalin's rise to power, \u201383, \u2013105, ,\n\npolicy debates within, \u201391\n\npurge of oppositionists, \u2013139\n\nStalin faction in, ,\n\nafter Stalin's death, \u2013313\n\nStalin's resignation offer, \u201380,\n\nStalin's restructuring of NEP, \u2013106\n\nvictory of Stalin faction, \u2013108\n\nCollectivization, agricultural, \u2013113, \u2013116, , ,\n\nCominform, creation of,\n\nConquest, Robert,\n\nConstitution, liberalization of, \u2013135\n\nConsumer goods: price of, \u2013322,\n\nshortages of, \u2013278\n\nCorruption, and currency reform, \u2013279\n\nCosmopolitanism campaign, ,\n\nCrimea: ethnic deportations from, ,\n\nSoviet-German Front in, , ,\n\nYalta Conference, \u2013246, , Gallery\n\nCurrency reform, \u2013280,\n\nCzechoslovakia: in Communist bloc,\n\nGerman invasion of,\n\nMunich Agreement on, \u2013164\n\nSlansky trial,\n\n\u2013Soviet mutual assistance treaty,\n\nDacha (\"near dacha\"): Gallery\n\nlandscaping of, \u20135\n\nlibrary at, \u201396, n5\n\nrenovations of, \u20134\n\nsecurity personnel at, , , , \u2013143, , n2\n\nsocial gatherings at, \u20137, ,\n\nStalin's death at, , , \u2013144, n1\n\nDachas, southern, , , n42\n\n_Daily Herald_ , \u2013271\n\nDaladier, \u00c9douard,\n\nDanilov, V. P., n19\n\nDeikina, V. F., \u2013301\n\nDenunciations,\n\nDimitrov, Georgy,\n\nDisease epidemics,\n\n\"Dizzy with Success\" (Stalin),\n\nDjilas, Milovan, , ,\n\nDoctors' Plot, , \u2013309,\n\nDraule, Milda,\n\nDureiko, N. M., \u2013133\n\nDzerzhinsky, Feliks Edmundovich, , , n66\n\nEastern Europe: Berlin blockade,\n\nliberalization in interregnum period,\n\nmilitary buildup in,\n\nsovietization of, , , , , n62\n\nStalin's enemies campaign in,\n\nand Titoism, \u2013281\n\nEastern Front. _See_ Soviet\u2013German Front\n\nEconomy: and capital investment, ,\n\nand currency reform, \u2013280,\n\ninterregnum period reforms, ,\n\npostwar recovery, \u2013280\n\nand price reduction, \u2013322\n\nSovnarkom Bureau oversight of, \u2013179\n\nstandard of living,\n\nTerror's impact on,\n\nurbanite advantages in,\n\nand war mobilization, \u2013185,\n\nwartime leadership of, \u2013241. _See also_ Agriculture\n\nFamine\n\nIndustrialization\n\nNew Economic Policy (NEP)\n\n_Edinonachalie_ , in Red Army,\n\nEikhe, Robert,\n\nEllman, Michael, n10\n\nEngels, Friedrich,\n\nErenburg, Ilya,\n\nErickson, John,\n\nEspionage, Soviet, \u2013358n76\n\nEstonia: and German-Soviet non-aggression pact,\n\npostwar repression in,\n\nforced resettlement campaign in,\n\nsovietization of, \u2013171,\n\nEthnic groups: forced resettlement of, \u2013234\n\nand russification policy, . _See also_ Jews, Soviet\n\nFamine: of 1921\u20131922, ,\n\nof 1931\u20131933, , , , , \u2013122\n\nof 1936,\n\nas political weapon,\n\npostwar, ,\n\npreferential treatment of urbanites,\n\nFedorenko, Nikolai,\n\nFeuchtwanger, Lion,\n\nFifth column, Stalin's suspicions of, \u2013157,\n\nFilm screenings, \u20133, , \u201397, n12\n\nFiltzer, Donald,\n\nFinland: and German-Soviet non-aggression pact,\n\nSoviet invasion of (Winter War), \u2013173,\n\n\"Five\" (ruling group),\n\nFive-Year Plan: First, , , ,\n\nSecond,\n\nFood shortages, \u2013185, , \u2013301, \u2013324\n\nForeign intelligence, Stalin's suspicions of, \u2013157,\n\nForeign policy: arms race, \u2013298\n\nBaltic states occupation, \u2013171\n\nbreakdown in relations with West, \u2013267\n\nChina ( _See_ China-Soviet relations)\n\nunder collective leadership, \u201389\n\nDoctors' Plot as tool of,\n\nand European alliances, , ,\n\nFinland invasion (Winter War), \u2013173,\n\nGerman-Soviet non-aggression pact, \u2013169, ,\n\nHitler-Molotov four-way alliance negotiations, \u2013176\n\nin interregnum period, \u2013316\n\nand Japanese border clashes, , \u2013169\n\nand Japanese postwar settlement, \u2013249,\n\nand Japanese threat, , ,\n\nand Korean War, \u2013296, ,\n\n\"kowtowing to the West\" campaign, \u2013266, ,\n\nand Munich Agreement, \u2013164\n\nand nuclear capability,\n\nPolish occupation,\n\npostwar challenges for, \u2013262\n\npreemptive strike plan, \u2013183\n\nsovietization of postwar Eastern Europe, , , , , n62\n\nsovietization under Molotov\u2013Ribbentrop pact, , \u2013174\n\nand Spanish civil war, \u2013154\n\nand spheres of influence, ,\n\nStalin's caution and pragmatism in,\n\nTerror's consequences for,\n\nand Titoism, \u2013281\n\nUnited Nations boycott, n95\n\nwar readiness in, , , \u2013188\n\nForeign press, rumors of power struggle in, , \u2013271\n\nFrance, Anatole,\n\nFrance: appeasement of Hitler, ,\n\nfall of,\n\nand Polish invasion, \u2013170\n\n-Soviet mutual assistance treaty,\n\nFranco, Francisco,\n\nGenghis Khan,\n\nGeorgia: ethnic deportations from,\n\nMingrelian Affair, \u2013305, \u2013315\n\nin Transcaucasian Federation dispute, \u201372, , ,\n\nGeorgian language, Stalin's use of,\n\nGerman\u2013Soviet Front. _See_ Soviet-German Front Germany: Berlin blockade,\n\nPolitburo plan for revolution in, \u201378\n\nRed Army crimes against civilians, \u2013235\n\nWeimar, . _See also_ Nazi Germany\n\nGoebbels, Joseph, xiv\n\nGolovanov, Aleksandr, ,\n\nGomulka, Wladyslaw,\n\nGorbachev, M.,\n\nGori, Stalin's birth in,\n\nGori Theological School, , \u201315, Gallery\n\nGorky, Maxim, ,\n\nGorlizki, Yoram,\n\nGrain requisitions, forced, , \u2013104, , , \u2013120\n\nGrain reserves, n35\n\nGreat Leap policy, \u2013109,\n\nGreat Terror. _See_ Terror\n\nGridneva, Ye. G.,\n\nHarriman, Averell, \u20139, , ,\n\n_History of the All-Union Communist Party (Bolsheviks): Short Course_ ,\n\nHitler, Adolf, ,\n\nappeasement of, , ,\n\n-Molotov four-way alliance negotiations, \u2013176\n\npurges of,\n\n-Stalin non-aggression pact, \u2013169,\n\nsuicide of, . _See also_ Nazi Germany\n\nHoarding, \u2013277\n\nHopkins, Harry,\n\nHosking, Geoffrey,\n\nHousing, , n23\n\nHungary, in Communist bloc,\n\nIgnatiev, Semen Denisovich, , , , , , , , , n15\n\nIndustrialization: absenteeism and employee turnover,\n\nconsumer goods production, \u2013278, \u2013322,\n\nFirst Five-Year Plan growth targets, , \u2013117\n\nGreat Leap policy, \u2013109\n\ninterregnum period reforms, ,\n\nand liberalization policy, \u2013125\n\nmilitary production, \u2013184, \u2013298\n\nNew Economic Policy (NEP), ,\n\npostwar production,\n\nand Stakhanovites, \u2013321\n\nTerror's impact on,\n\nwartime management of,\n\nworking conditions under, \u2013325\n\nIngush, forced relocation of,\n\nInner circle, Gallery\n\ncomposition of, , \u2013334n1\n\ndachas and apartments of, ,\n\nface-to-face communications with, \u20132\n\nmemoirs of, xiv\n\nMolotov scandal, \u2013272\n\nmovie screenings for, \u20133, n12\n\nshifts in balance of power, \u2013179, \u2013274, , , \u2013307\n\nat Stalin's deathbed, \u2013144,\n\nStalin's power over, , \u2013147\n\nat Stalin's social gatherings, \u20137,\n\nwartime domestic duties of, \u2013241\n\nIntelligentsia, attacks on,\n\nIsrael, \u2013286\n\nItaly: Allied victories in,\n\npostwar settlement,\n\nin Tripartite Pact,\n\nIvan the Terrible, Stalin's view of, ,\n\nJapan: in Anti-Comintern Pact,\n\ninvasion of China,\n\nmilitary threat from, , ,\n\nin postwar settlement, \u2013249,\n\n-Soviet border clashes, , \u2013169\n\nin Tripartite Pact,\n\nJewish Anti-Fascist Committee,\n\nJews, Soviet: cosmopolitanism campaign against, ,\n\ndiscrimination against, ,\n\nin Doctors' Plot, \u2013309\n\nand Israel, \u2013286\n\npurge of Molotov's wife, \u2013284\n\nand Stalin's anti-Semitism, \u2013285,\n\n_Jolly Fellows_ (film),\n\nJughashvili, Besarionis (Stalin's father), , ,\n\nJughashvili (Geladze), Ekaterine (Stalin's mother), , , \u201314, , , , , Gallery\n\nJughashvili, Iosif. _See_ Stalin, Joseph\n\nKaganovich, Lazar Moiseevich, Gallery ,\n\nbiography of, n5\n\nin interregnum period,\n\npurge of family member,\n\nand Stalin's daughter, \u2013257\n\nStalin's power over, , ,\n\nin wartime leadership, ,\n\nKalinin, Mikhail Ivanovich: biography of, \u2013346n102\n\ncompromising evidence against,\n\nand expulsion of left opposition,\n\nas oppositionist,\n\npurge of wife,\n\nKamenev, Lev Borisovich, , , ,\n\nand abdication of tsar,\n\nbiography of, \u2013341n1\n\nBukharin's secret meeting with,\n\nexecution of,\n\nPolitburo expulsion of, \u201385\n\nand Kirov's murder charge,\n\nand Kremlin Affair,\n\non Lenin-Stalin conflict,\n\nin moderate (rightist) faction, \u201344,\n\nopposition to Bolshevik seizure of power, , ,\n\nas oppositionist, \u201384,\n\nin power struggle, ,\n\nProvisional Government crackdown on,\n\nKapler, Aleksei,\n\nKarpets, P. K.,\n\nKatyn massacre,\n\nKautsky, Karl,\n\nKazbegi, Alexandre,\n\nKelbakiani, \u201320\n\nKerensky, Aleksandr,\n\nKerr, Archibald,\n\nKetskhoveli, Lado, ,\n\nKharkov, Battle of, ,\n\nKhmelkov, N. M.,\n\nKholodov, N. I.,\n\nKhoroshenina, Serafima,\n\nKhrushchev, Nikita: on agricultural commission, \u2013302\n\nbiography of, n1\n\non fall of France,\n\nin interregnum period, ,\n\nin Kirov murder investigation,\n\nmemoirs of, xiv, n1\n\nPolitburo appointment of,\n\nStalin characterized by,\n\non Stalin's ceremonial farewell,\n\nat Stalin's deathbed, , , , , n1\n\non Stalin's exile,\n\nStalin's promotion of,\n\nat Stalin's social gatherings, , ,\n\nin wartime leadership,\n\nand Zhukov, n2\n\nKibirov, I. I.,\n\nKiev: fall of, \u2013213\n\nliberation of,\n\nKiev Theological Seminary,\n\nKim Il Sung, , ,\n\nKirov, Sergei Mironovich, Gallery\n\nbiography of, n7\n\nbodyguards of,\n\nand collective leadership,\n\nmotive for murder of, \u2013131\n\nmurder of, , \u2013129\n\nand plot against Stalin,\n\nStalin's involvement in murder of, \u2013134\n\nvacation with Stalin, ,\n\nKirponos, Mikhail, ,\n\nKliueva, Nina,\n\nKolkhozes (collective farms), \u2013113, ,\n\nKorean War, \u2013296, ,\n\nKornilov, Lavr,\n\nKosarev, Aleksandr Vasilyevich, , n66\n\nKosior, Stanislav,\n\nKostov, Traicho,\n\nKosygin, Gallery\n\nKovalev, I. V., n27\n\n\"Kowtowing to the West\" campaign, \u2013266, ,\n\nKremlin: German bombing of,\n\nmovie theater in, \u20133, \u201397\n\npurge of staff,\n\nStalin's office in, , ,\n\nKremlin Affair,\n\nKronstadt rebellion,\n\nKruglov, Sergei,\n\nKrupskaia, Nadezhda, , ,\n\nKulaks: collectivization campaign against, \u2013113\n\nexecutions of, ,\n\ngrain requisitions from, , \u2013104, ,\n\nforced resettlement of, , ,\n\nKulik, Grigory Ivanovich: biography of, n20\n\nmurder of wife,\n\nas Stalin's emissary to front,\n\nKulik-Simonich, Kira,\n\nKuomintang, ,\n\nKuperin, I. I.,\n\nKurile Islands, , ,\n\nKuromiya, Hiroaki,\n\nKursk, Battle of, \u2013227,\n\nKutuzov, Mikhail, , ,\n\nKuzakova, M. P.,\n\nKuznetsov, Aleksi Aleksandrovich, Gallery\n\nbiography of, n6\n\nin leadership reorganization,\n\nin Leningrad Affair, , \u2013283\n\nKuznetsov, Nikolai Gerasimovich, , , n22\n\nLabor camps, , ,\n\nLake Khasan, Battle of,\n\nLakoba, Nestor, Gallery\n\nLashevich, Mikhail,\n\nLatvia: and German-Soviet non-aggression pact,\n\npostwar repression in,\n\nforced resettlement campaign in,\n\nsovietization of, \u2013171,\n\nLeague of Nations, ,\n\nLend-Lease aid,\n\nLenin, Vladimir Ilyich: assassination attempt on, ,\n\ndeath of,\n\ndeath mask of,\n\nevacuation of sarcophagus of,\n\nfederation proposal of, \u201369\n\nhealth of, , , ,\n\nas military leader,\n\nNew Economic Policy (NEP) of, , \u201365\n\nand invasion of Poland, \u201360\n\nin Provisional Government crackdown, ,\n\nreturn from Switzerland,\n\nrevolutionary action plan of, \u201347\n\nrevolutionary teachings of,\n\nand seizure of power, \u201351,\n\nStalin attacked by, \u201374,\n\n\u2013Stalin relationship, , , , , \u201365, \u201368, \u201375\n\nStalin's correspondence with, ,\n\nStalin's support for revolutionary agenda of, \u201349, \u201353\n\nStalin as student of, , , \u201394\n\nand Stalin\u2013Trotsky conflict, \u201363\n\n\"testament\" of, , , ,\n\n-Trotsky relationship, \u201352, \u201366,\n\non Tsaritsyn terror campaign,\n\nLeningrad Affair, , \u2013283,\n\nLeningrad Blockade, ,\n\n_Leningrad_ magazine,\n\nLetter writing, citizen, \u2013301, \u2013329, n35, n38\n\nLewin, Moshe,\n\nLiberalization policy, \u2013125\n\nLiterature: censorship of,\n\nin Stalin's library,\n\nStalin's taste in, , , \u201396\n\nLithuania: and German-Soviet non-aggression pact,\n\npostwar repression in,\n\nforced resettlement campaign from,\n\nsovietization of, \u2013171,\n\nLitvinov, Maksim Maksimovich, , n37\n\nLiushkov, Genrikh,\n\nLivestock production, postwar, \u2013300, n15\n\nLozgachev, P. V.,\n\nLudwig, Emil,\n\nMain Guard Directorate, \u201334, \u201336\n\nMakhrovsky, Konstantin, \u201357\n\nMalenkov, Georgy Maksimillianovich: , Gallery\n\nbiography of, n1\n\ncapital investment reduction under,\n\nin interregnum period,\n\nin leadership reorganizations, , , \u2013273, ,\n\nand Molotov scandal, ,\n\nPolitburo appointment of,\n\nat Stalin's deathbed, , , ,\n\non Timoshenko,\n\nin wartime leadership, , , , , , , , , ,\n\nMalinovsky, Roman, , , , n48\n\nMalinovsky, S. V., \u201315\n\nMao Zedong,\n\nand Korean War,\n\nMoscow visit of, , \u2013288, \u2013293\n\nMarshall Plan,\n\nMartin, Terry,\n\nMartov, Yuly Osipovich, , , n95\n\nMekhlis, Lev Zakharovich, , , , n9\n\nMensheviks: opposition to armed robberies,\n\nin party congresses,\n\nin Petrograd Soviet, ,\n\nand Provisional Government,\n\nrevolutionary agenda of,\n\nMenshikov, Mikhail,\n\nMerkulov, Vsevolod Nikolaevich, , n38\n\nMerzliakov, M. A.,\n\nMeyerhold, Vsevolod Emilyevich, , n13\n\nMgeladze, Akaky, n42\n\nMiasnikov, Aleksandr, , ,\n\nMikhail, Grand Duke,\n\nMikhoels, Solomon Mikhailovich, , , n23\n\nMikoyan (Mikoian), Anastas Ivanovich, Gallery\n\non agricultural policy,\n\nassassination attempt on,\n\nbiography of, n2\n\nChina visit of,\n\ndismissal of, ,\n\nin interregnum period,\n\nin leadership reorganizations, \u2013179, \u2013307\n\nand Leningrad Affair,\n\nmemoirs of, xiv, n10\n\nand Molotov scandal, ,\n\non Moscow evacuation,\n\non Presidium makeup,\n\non price reduction, \u2013322\n\nin Stalin faction,\n\nat Stalin's social gatherings,\n\nStalin's threat to,\n\nStalin's vacation with,\n\nin wartime leadership, , , , , , , ,\n\nMikoyan (Mikoian), Sergo, , n10\n\nMilitary. _See_ Air Force\n\nSoviet-German Front\n\nRed Army\n\nMilitary production: German, \u2013187\n\nSoviet, \u2013184, \u2013298\n\nMingrelian Affair, \u2013305, \u2013315\n\nMola, Emilio,\n\nMolotov, Vyacheslav Mikhailovich: , Gallery\n\nin Baltic states negotiations,\n\nbiography of, \u2013334n2\n\non collective leadership, \u201390\n\ndismissal of, ,\n\nin foreign affairs post, \u2013165,\n\nforeign press rumors blamed on, , \u2013271\n\nand German invasion, , \u2013201\n\non grain expropriation,\n\n-Hitler four-way alliance negotiations, \u2013176\n\nand Hitler-Stalin non-aggression pact,\n\nin interregnum period, ,\n\non Kaganovich,\n\nin leadership reorganizations, , \u2013180, , \u2013307\n\nin Mao's Moscow visit,\n\nnegotiations with West,\n\nand postwar settlement, \u2013270\n\npurge of wife, \u2013284,\n\nin Stalin faction,\n\nStalin's attacks on, \u2013182, \u2013272,\n\non Stalin's mental state, , \u2013205\n\non Stalin's security,\n\nStalin's threat to, ,\n\non Trotsky,\n\nin wartime leadership, , , , , , , , ,\n\nWestern allies visited by,\n\nMonetary reform, \u2013280,\n\nMontefiore, Simon Sebag, ix Moscow Conference,\n\nMoscow, Siege of: defenses in,\n\nand evacuation plans, \u2013216\n\nOctober Revolution anniversary celebrations in, \u2013219\n\npolitical stability during,\n\npopular uprisings during, \u2013217\n\nStalin's presence during, \u2013228\n\nMovie screenings, \u20133, , \u201397, n12\n\nMunich Agreement, \u2013164,\n\nMunitions industry, \u2013184, \u2013186\n\nMurphy, David, \u2013358n76\n\nMusic, Stalin's taste in,\n\nNakhimov, Pavel,\n\nNationalities. _See_ Ethnic groups Nationalities commission,\n\nNATO, ,\n\nNazaretian, Amaiak, \u201367\n\nNazi Germany: in Anti-Comintern Pact,\n\ncollaborators in occupied territories,\n\ncollapse of, ,\n\ninvasion of Czechoslovakia,\n\ninvasion of Poland, \u2013170\n\ninvasion of Soviet Union, , , \u2013188, \u2013202\n\nmilitary production of, \u2013187\n\nin Munich Agreement, \u2013164,\n\noccupation of Europe, ,\n\n-Soviet non-aggression pact, \u2013169,\n\nStalin's overtures to, \u2013166\n\nin Tripartite Pact, . _See also_ Soviet-German Front\n\nNevsky, Aleksandr,\n\nNew Economic Policy (NEP): under collective leadership, , ,\n\nunder Lenin, , \u201365\n\nparty infighting over, ,\n\nStalin's restructuring of, \u2013104\n\n_New York Times_ ,\n\nNicholas II, abdication of,\n\nNicolaevsky, Boris,\n\nNikolaev, Leonid: background of, \u2013130\n\nexecution of relatives, n68\n\nmotives for Kirov's murder, \u2013131\n\nmurder of Kirov, \u2013129, \u2013132, n68\n\nprior detentions by NKVD,\n\nNKVD: fabricated evidence of,\n\nformation of,\n\nin Kirov murder investigation, \u2013133\n\nin Kremlin Affair,\n\nOrder No.00447,\n\npurge of, ,\n\nin Trotsky murder,\n\nYezhov's direction of Terror, , \u2013159\n\nNomenklatura: new stock,\n\nprivileged lifestyle of,\n\npurge of, \u2013140, , , \u2013283, ,\n\nStalin's control over, \u201337, \u2013148\n\nStalin's reorganization of,\n\nNorway, German occupation of,\n\nNuclear development, ,\n\nOctober Revolution, , \u201353\n\nanniversary of, \u2013219\n\nOnufrieva, Pelageia, \u2013252\n\nOrder No. 227,\n\nOrder No. 270,\n\nOrdzhonikidze, Grigory Konstantinovich: , , , Gallery\n\nbiography of, n55\n\non Central Committee expulsion of left opposition,\n\non factional disputes, \u2013107\n\nand \"Georgian Affair,\" ,\n\nhealth of,\n\non industrialization,\n\nand Lenin-Stalin conflict,\n\non peasant uprisings,\n\nin power struggle,\n\nin Stalin faction,\n\nsuicide of, ,\n\nOrdzhonikidze, Zinaida,\n\nOrlov, A. Ya.,\n\nOsinsky, Valerian Valerianovich, \u201386, n92\n\nOstrovskii, Aleksandr,\n\nPanic-mongers and cowards, campaign against,\n\n_Parade of Our Troops on Moscow's Red Square on 7 November 1941_ (film), \u2013219\n\nParty apparat. _See_ Nomenklatura\n\n_Patricide, The_ (Kazbegi),\n\nPaulus, Friedrich, ,\n\nPavlov, Dmitry,\n\nPeasants: currency reform impact on,\n\nand famines, , , \u2013122\n\nincome of, n16\n\npostwar discontent of,\n\nforced resettlement of, ,\n\nrevolts of, , \u2013115,\n\ntaxation of,\n\nworking conditions of, \u2013325. _See also_ Agriculture\n\nKulaks\n\nPenal colonies,\n\nPenalty battalions, , ,\n\nPereprygina, Lidiia, , , , n60\n\nPeter the Great, Stalin's view of, ,\n\nPetrograd Soviet, , ,\n\nPetrovskaia, Stefaniia,\n\nPetrovsky, Grigory,\n\nPlato,\n\nPlekhanov, Georgy,\n\nPoland: Civil War campaign in, \u201361\n\nin Communist bloc, ,\n\nGerman invasion of, \u2013170\n\nand German-Soviet non-aggression pact,\n\nKatyn massacre,\n\npostwar settlement in, ,\n\nWarsaw uprising,\n\nPolitburo: abolition of, \u2013306\n\neavesdropping on members, , n10\n\nexecution of members, ,\n\nexpulsion of leftist opposition, \u201386\n\nindependence in day-to-day work, \u2013311\n\npower struggle in, \u201381, \u201387\n\nreorganizations of leadership, , \u2013179,\n\nreplaced during Great Terror, \u2013178\n\nStalin's appointment to,\n\nStalin's dictatorial powers approved by, \u2013181\n\nStalin's domination of, \u2013147, \u2013177,\n\nvoting process in, n66\n\nwartime domestic duties of, \u2013241\n\nand wartime leadership, , , , , , ,\n\nand world revolution, \u201378\n\nyoung generation in, , . _See also_ Collective leadership\n\nPopular front against fascism,\n\nPoskrebyshev, Aleksandr, ,\n\nPostyshev, Pavel,\n\n_Pravda_ , , , , , , , , , , ,\n\nPresidential Archive of the Russian Federation (APRF), xiv\n\nPresidium, of Central Committee, \u2013306,\n\nPresidium Bureau, of Central Committee, , ,\n\nPreventive war theory, ,\n\nPrisoners of war: German,\n\nSoviet,\n\nPrivate peasant plots, , , , ,\n\n_Prosveshchenie_ ( _Enlightenment_ ) (magazine),\n\nProvisional Government: crackdown on Bolsheviks, \u201349\n\nformation of,\n\nand Kornilov mutiny, \u201350\n\noverthrow of, , \u201353\n\nsocialist support for,\n\nPurges: during Civil War, \u201359\n\nunder collective leadership,\n\nDoctors' Plot, , \u2013309,\n\nLeningrad Affair, , \u2013283,\n\nMingrelian Affair, \u2013305, \u2013315\n\nof Molotov's wife, \u2013284,\n\nof nomenklatura, \u2013140, , , \u2013283, ,\n\nrehabilitation of victims, \u2013315\n\nShakhty Affair,\n\nof state security, \u201335, , , \u2013304. _See also_ Terror\n\nRadzinsky, Edvard, ix, n1\n\nRajk, Laszlo,\n\nRakosi, Matyas, , ,\n\nRape, by Red Army, \u2013235\n\nRationing, abolition of, ,\n\nRazin, Ye. A., ,\n\nReconciliation campaign, \u2013136\n\nRed Army: anti\u2013government sentiment of peasant recruits,\n\nin Baltic states, \u2013171\n\nBerlin operation of, \u2013247\n\nin Civil War, \u201356, \u201361\n\ncrimes against civilians, \u2013236\n\nin Eastern Europe,\n\nin Finland (Winter War), \u2013173,\n\ngrowth of, \u2013186\n\nin Japanese border clashes,\n\nmodernization of, \u2013184\n\nin Poland,\n\npostwar buildup, \u2013298\n\npurge of, , , \u2013163, , n31\n\nStalin's critique of command, , , \u2013237\n\nStalin's reorganizations of command, \u2013239, \u2013274\n\nStalin-Trotsky conflict over, \u201363. _See also_ Soviet-German Front\n\nReligion, \u2013243,\n\nRetail prices, and currency reform, ,\n\nRevolution of 1905, \u201325\n\nRevolution of 1917: Bolshevik seizure of power, \u201353\n\nescalation of,\n\nLenin's radical action plan, \u201346\n\nLenin\u2013Stalin collaboration during, , \u201349,\n\nmoderate (rightist) Bolshevik faction in, \u201344, \u201347, ,\n\noutbreak of, . _See also_ Provisional Government\n\nRhee, Syngman,\n\nRibbentrop, Joachim von, , ,\n\nRieber, Alfred,\n\nRobins, Raymond, \u2013121\n\nRokossovsky, Konstantin Konstantinovich, \u2013223, , n94\n\nRoosevelt, Franklin: , Gallery\n\ncorrespondence with Stalin, ,\n\nand Hopkins mission to Moscow,\n\nand Red Army criminal behavior, \u2013236\n\nrepresentatives at Moscow Conference,\n\nand second front plan, , , ,\n\nRoskin, Grigory,\n\nRozengolts, Arkady,\n\nRudzutak, Yan Ernestovich: biography of, n100\n\ncompromising evidence against,\n\nexecution of,\n\nas oppositionist,\n\nRukhadze, Nikolai, \u2013305\n\nRussian Federation, proposals for, \u201369\n\nRussian language proficiency, Stalin's, \u201398\n\nRussian Orthodox Church, reconciliation with, \u2013243\n\nRussification policy,\n\nRusso\u2013Japanese War of 1905,\n\nRybalko, R. S.,\n\nRykov, Aleksei Ivanovich: , , Gallery\n\nbiography of, n80\n\nexecution of,\n\nexpulsion of,\n\nand left opposition,\n\nas oppositionist, , ,\n\nSakhalin: oil pipeline,\n\nand postwar settlement, ,\n\nSalaries, , n16\n\nSaltykov-Shchedrin, Mikhail,\n\nSapronov, Timofei Vladimirovich, , n94\n\nSavings bank deposits, and currency reform, , ,\n\nSchulenburg, Friedrich von der,\n\nSecond front plan, \u2013224, , ,\n\n\"Secret Five\" group, \u2013178\n\nSecurity system. _See_ NKVD\n\nState security Seliavkin, Aleksei, , n58\n\nSevastopol, siege of,\n\nShakhty Affair,\n\nShakhurin, Aleksei, , \u2013216\n\nShaposhnikov, Boris Mikhailovich, , n14\n\nShcherbakov, Aleksandr Sergeevich: biography of, \u2013357n57\n\nPolitburo appointment of,\n\nin wartime leadership, ,\n\nShepilov, Dmitry,\n\nShkiriatov, Matvei, Gallery ,\n\nSholokhov, Mikhail Aleksandrovich, , \u2013122, n39\n\nShostakovich, Dmitry Dmitryevich, , n14\n\nShtemenko, Sergei, \u2013230, ,\n\nShumiatsky, Boris, , , n12\n\nShvernik, Nikolai,\n\nSiberia: grain expropriation in, , \u2013104\n\nStalin's exile in, , \u201332,\n\nSimonov, Konstantin, \u2013145, ,\n\nSlansky, Rudolf,\n\nSmirnov, Vladimir Mikhailovich, , n93\n\nSmolensk, Battle of, \u2013211\n\nSmolny Institute, Kirov's murder at, \u2013129\n\nSocial Democratic movement: in Baku, \u201327\n\nEuropean congresses of,\n\nfactions of,\n\nradical wing of,\n\nin revolution of 1905, \u201325\n\nStalin's involvement in, , , , \u201324. _See also_ Bolsheviks\n\nMensheviks\n\nSocialist Revolutionaries (SRs), , ,\n\nSokolnikov, Grigory Yakovlevich, , n97\n\nSoviet-German Front: capture by enemy provisions (Order No. 270),\n\ncasualties of, , , , , , ,\n\nin Crimea, , ,\n\ncrimes against civilians, \u2013236\n\ndefeats in early stages of war, , \u2013210, , , \u2013265\n\nfinal battles, \u2013244\n\nfirst days of combat, \u2013205\n\nKharkov, Battle of, ,\n\nKursk, Battle of, \u2013227,\n\nLend-Lease aid,\n\nLeningrad Blockade, ,\n\nMoscow Siege, \u2013219, \u2013228\n\npenalty battalions\/anti-retreat units, , ,\n\nphysical destruction in, ,\n\npopular uprisings\/disturbances in, \u2013217\n\nratio of forces,\n\nrepressive measures in, \u2013210, , ,\n\nSmolensk, Battle of, \u2013211\n\nStalingrad, Battle of, \u2013225, ,\n\nStalin's emissaries at, , \u2013209\n\nStalin's strategic directives in, , , \u2013213, , \u2013221, , \u2013227, \u2013240\n\nStalin's visits to, \u2013228\n\nin Ukraine, \u2013213, , ,\n\nvictories of 1944, \u2013231\n\nZhukov's commands, , , , n2. _See also_ Stalin, Joseph, wartime leadership of\n\nSovnarkom Bureau of the USSR: Commission on Current Issues, n136\n\nformation of, \u2013179\n\nfunction of,\n\nSpain, repression of foreign espionage,\n\nSpanish Civil War, \u2013154,\n\nSpecial Sector, , nn35\u201337\n\nSpheres of influence in Europe,\n\nStakhanov, Aleksei, \u2013321\n\nStakhanovites, \u2013321\n\nStalin, Joseph, Gallery , ,\n\n\"agree and ignore\" approach of, \u2013106\n\nin airplane flight, \u2013230\n\nanti-Semitism of, \u2013285,\n\napologists for, ix, x\u2013xi\n\narrests of, , , , Gallery\n\nbiographies of, ix\u2013x, , n2\n\nbirth of,\n\nbodyguards of, \u201336, , , \u2013339n5\n\nand Bolshevik armed robbery,\n\nin car crash,\n\nand Caucasian culture of violence, \u201322\n\nchildhood and youth of, \u201314\n\nin Civil War Southwestern Front, \u201361\n\nin Civil War Tsaritsyn command, \u201356, Gallery\n\nCivil War Tsaritsyn terror campaign of, \u201359\n\nin collective leadership ( _See_ Collective leadership)\n\ncollectivization policy of, \u2013113, \u2013116, ,\n\nconflict with Bolshevik colleagues,\n\nconsolidation of power, \u2013181\n\nconstitutional liberalization by, \u2013135\n\ndachas of, \u20137, , , Gallery\n\ndeath of, \u2013314\n\ndictatorial powers of, \u201339, \u2013181,\n\n\"Dizzy with Success,\"\n\ndouble agent rumors about,\n\neconomy under ( _See_ Economy)\n\neditor of _Pravda_ , ,\n\neducation of, , , \u201320, Gallery\n\non espionage threat, \u2013155\n\nat European party congresses,\n\nexile of, , , \u201332, , , ,\n\nfamily life\/relations with children, , , \u2013260, Gallery ,\n\nfamine explanation of, \u2013122\n\nfederation proposal of,\n\nfilm favorites of, \u201397\n\nfinal days of, , , \u2013144, , \u2013313, n1\n\non foreign intelligence threat, \u2013157, ,\n\nforeign policy of ( _See_ Foreign policy)\n\nand foreign press rumors of leadership struggle, , \u2013271\n\nfuneral and ceremonial farewells, , \u2013318\n\ngeneralissimo rank of, \u2013226,\n\nas general secretary of party, \u201367,\n\ngrain expropriation policy of, , \u2013104, , , \u2013120\n\nhealth of, , \u2013197,\n\nhistorical interests of, \u201395\n\n-Hitler non-aggression pact, \u2013169,\n\nideological influences on, , \u201394\n\nindustrialization policy of, \u2013109, , \u2013125, , ,\n\ninner circle of ( _See_ Inner circle)\n\nand Kirov's murder, , , \u2013134\n\nKremlin office of, , ,\n\nlanguages, knowledge of, \u201398,\n\nLenin cited in speeches of,\n\n-Lenin relationship, , , , , \u201365, \u201368, \u201375, Gallery\n\nLenin's attack on, \u201374,\n\nLenin's ideological influence on, , , \u201394\n\nLenin's revolutionary agenda supported by, \u201349, , \u201353\n\nletters from public sent to, \u2013301, \u2013329, n35, n38\n\nlibraries of, \u201396, , n5\n\nliterary tastes of, , , \u201396\n\nMao's visit to, , \u2013288, \u2013293\n\nmarriage to Nadezhda Allilueva, , \u2013256, Gallery\n\nmarriage to Yekaterina Svanidze, , , \u2013251\n\nmarshal rank of,\n\nmedical care of, , , \u2013197,\n\nmental state of, \u2013153, , \u2013205\n\non military professionals,\n\nmodest facade of, \u2013149\n\n-Molotov relationship, \u2013182, , \u2013272\n\nmusical tastes of,\n\nname change of,\n\nofficial trips of,\n\noratory of,\n\nparents of, , , Gallery\n\npersonality of, , \u201322, \u201341, , , , ,\n\nphysical defects of, \u201314,\n\nplot against,\n\npoetry of,\n\npostwar challenges to, \u2013263\n\npower over subordinates, , \u2013149\n\nin power struggle for leadership, \u201381, \u201387\n\non preemptive strike, \u2013183\n\nand private agriculture, , ,\n\npromotion of younger generation leaders, , ,\n\npublic appearances of, \u2013137,\n\npublic image of, \u2013330\n\npublic sentiment at death of, \u2013319\n\nradicalization during seminary years, \u201320,\n\nreading materials\/information sources of, \u201396,\n\nreconciliation campaign of, \u2013136\n\nreorganization of leadership, \u2013179, \u2013274, , \u2013307\n\nreorganization of military command, \u2013239, \u2013274\n\nreorganization of political structure, \u2013306\n\nresearch on ( _See_ Archival sources)\n\nin revolution of 1905, \u201325\n\nin revolutionary movement, , \u201344, \u201347,\n\nrise in Bolshevik leadership ranks, , , \u201366\n\nromantic relationships of, , \u2013252,\n\nseats of power, \u20132\n\nand \"Secret Five\" leadership group, \u2013178\n\nseventieth birthday celebration,\n\nSiberian visit of 1928, \u2013104,\n\nin Social Democratic movement, , , , \u201324\n\nsocial gatherings of, \u20137,\n\nsovietization goal of, \u2013172\n\nand Spanish Civil War,\n\nstate security controlled by, \u201340, ,\n\nsuccession to, \u2013316\n\ntheatrical tastes of,\n\nin Tiflis weather station job,\n\n-Trotsky conflict over Polish Front, \u201363\n\nas tsarist heir, \u20139\n\nvacations in south, \u2013194, \u2013269, \u2013304, Gallery ,\n\nunder Vlasik,\n\nwar preparations of, \u2013187\n\nworld-view of, \u20138, \u201399\n\nwritings of, , Gallery\n\nwriting style of,\n\n_See also_ Purges\n\nStalinist system\n\nTerror\n\nStalin, Joseph, wartime leadership of: Gallery\n\nat Berlin Conference, \u2013248\n\nin Churchill meetings, ,\n\nand collaborators with Nazi occupation,\n\ncombat operation directives in, , , \u2013213, , \u2013221, , \u2013227, \u2013240\n\nCommand Headquarters in,\n\ndefeats blamed on subordinates, \u2013223\n\nand domestic policy, \u2013241\n\nemissaries to front, , \u2013209\n\nethnic deportations policy, \u2013234\n\nexplanation of Red Army retreat, \u2013265\n\nin first days of war, \u2013205\n\nfront line visit of, \u2013228\n\nand German invasion, reaction to, \u2013188, \u2013202\n\nand health problems, \u2013196,\n\nin Japanese war, \u2013249\n\nand Lend\u2013Lease aid, \u2013212\n\nand mental health problem, \u2013205\n\nand military command structure, \u2013239\n\nand military leadership reorganization, \u2013242\n\nand military staffing decisions, \u2013221\n\nmistrust of Western Allies, , ,\n\nand Moscow evacuation, \u2013216\n\nand prisoners of war procession,\n\npropaganda campaign, \u2013219\n\nradio address (3 July 1942),\n\nand Red Army crimes against civilians, \u2013236\n\nRed Army criticized by, , , \u2013237\n\nand religious reconciliation, \u2013243\n\nrepressive measures in, \u2013210, , ,\n\nand second front plan, \u2013224, ,\n\nState Defense Committee in, \u2013207\n\nat Tehran Conference, \u2013230\n\nat Yalta Conference, \u2013246, Gallery\n\nStalin, Nadezhda Allilueva (second wife): Gallery\n\nbiography of, n8\n\ncareer of, \u2013253\n\nand Khrushchev, n1\n\nmarriage of, , , \u2013255\n\nsuicide of, , \u2013256\n\nStalin, Svetlana (daughter). _See_ Allilueva (Alliluyeva), Svetlana Stalin, Vasily (son): Gallery\n\nbirth of,\n\nchildhood of, ,\n\ndeath of,\n\npublic appearance of, \u2013137\n\nrelationship with father, , , \u2013260\n\nat Stalin's deathbed,\n\nStalin, Yakov (son): Gallery\n\nbirth of, \u2013251\n\nchildhood and youth of, , , \u2013253\n\nas prisoner of war,\n\nrelationship with father, ,\n\nStalin, Yekaterina Svanidze (first wife), , , \u2013251\n\nStalin Collection of the Russian State Archive of Social and Political History (RGASPI), xiii\u2013xiv\n\nStalingrad, Battle of, \u2013225, ,\n\nStalingrad (Tsaritsyn), in Civil War, \u201359\n\nStalinist system: beneficiaries of, \u2013322\n\ncampaigns and counter-campaigns of,\n\n\"inevitable Stalinism\" theory, xi\n\nmodernization as justification for, x\u2013xi\n\nnostalgia for,\n\npublic mood under, \u2013330\n\nvictims of, \u201338, \u2013323\n\nStandard of living, , \u2013324\n\nState Defense Committee, \u2013207,\n\nState security: under Beria, , \u2013333n1\n\nformation of NKVD,\n\ninterregnum period reform of,\n\ninvestigation of secret police abuses, \u2013126\n\nparty opposition suppressed by, \u201385\n\npurges of, \u201335, , , \u2013304\n\nreorganization of, n19\n\nreports\/summaries as information source,\n\nStalin's control of, \u201340, ,\n\nStalin's guards, \u201334, \u201336\n\ntorture by,\n\nin wartime,\n\nat Yalta Conference, \u2013246. _See also_ NKVD\n\nSudoplatov, Pavel, n10\n\nSuny, Ronald Grigor, \u201313\n\nSuvorov, Aleksandr, ,\n\nSvanidze, Aleksandr, , ,\n\nSvanidze, Maria, ,\n\nSvanidze, Yekaterina, , , \u2013251\n\nSvanidze family,\n\nSvechin, Aleksandr,\n\nSverdlov, Yakov, ,\n\nTehran Conference, \u2013230\n\nTelenkov, S. M.,\n\nTer-Petrosian, Simon (Kamo),\n\nTerror: anti-Semitic campaign, n23\n\n\"anti-Soviet elements\" operations, \u2013151\n\napologists' view of, x\n\ncasualties of, ,\n\nand class war,\n\nconspiracy theories as pretext for,\n\neconomic consequences of,\n\nended by Stalin's successors, \u2013315\n\nexpansion of,\n\nforeign intelligence threat as pretext for, \u2013157\n\nforeign relations consequences of,\n\ngrievances and complaints about,\n\nagainst Kremlin staff,\n\nagainst Kulaks, ,\n\nKirov's murder as pretext for, , , \u2013132,\n\nliterature on, n1\n\nmilitary consequences of, \u2013163,\n\n\"nationalities\" operations,\n\nagainst old guard Bolsheviks, \u2013139, ,\n\novercrowding of camps and prisons in, \u2013123,\n\nagainst party and state apparats, \u2013140,\n\nagainst Politburo members,\n\npolitical consequences of, \u2013177\n\npostwar repression,\n\npullback on, \u2013127\n\nagainst Red Army, , , \u2013163, n32\n\nand reconciliation campaign, \u2013136\n\nStalin's instigation and orders in, \u201341, \u2013153, \u2013159\n\nStalin's relatives as victims of, \u2013258\n\nagainst state security, \u201335, , ,\n\ntargeted killings,\n\nas tool of Stalinist system, \u2013323\n\nwinding down of, , \u2013162,\n\nYezhov's direction of, , , , , \u2013159, n66\n\nTheater, Stalin's taste in,\n\nTiflis: Bolshevik armed robbery at, \u201326\n\nSocial Democratic movement in, , ,\n\nTiflis Meteorological Station,\n\nTiflis Theological Seminary, \u201320,\n\nTimoshenko, Semen Konstantinovich: biography of, \u2013360n2\n\nand German invasion, , , , ,\n\nin wartime leadership, , ,\n\nTito, Josip Broz, \u2013281\n\nTkachev, R. A.,\n\nTolstoy, Leo,\n\nTomsky, Mikhail Pavlovich: biography of, \u2013348n8\n\nPolitburo expulsion of,\n\nas oppositionist, ,\n\nsuicide of,\n\nTorture,\n\nban on,\n\nTovstukha, Ivan,\n\nTripartite Pact, \u2013176\n\nTrotsky, Lev Davidovich, , ,\n\nanti-Trotsky coalition, , \u201382\n\nbiography of, n13\n\nand Bukharin\u2013Kamenev meeting,\n\nexpulsion of, \u201385,\n\n-Lenin relationship, \u201352, \u201366,\n\nmurder of, , ,\n\nNEP strategy criticized by,\n\noratory of,\n\npower struggle with Stalin, ,\n\nRed Army command of,\n\n-Stalin conflict over Polish Front, \u201363\n\nTsaritsyn terror campaign opposed by, \u201359\n\nworks of, ,\n\nTruman, Harry, ,\n\nTsaritsyn (Stalingrad), in Civil War, \u201359, Gallery\n\nTucker, Robert, ix,\n\nTukhachevsky, Mikhail Nikolaevich, , n85, Gallery\n\n_Twenty Letters to a Friend_ (Allilueva),\n\nUglanov, Nikolai Aleksandrovich, , , , n8\n\nUkraine: collaboration policy in,\n\nfamine of 1932\u20131933,\n\nand German-Soviet non-aggression pact,\n\npeasant uprisings in, \u2013115\n\nforced resettlement campaign in,\n\nSoviet-German Front in, \u2013213, ,\n\nSoviet occupation of,\n\nUkrainian Labor Army, n45\n\nUlam, Adam, ix\n\nUlianova, Maria Ilinichna, , , , n58\n\nUnited Nations: and Korean War,\n\nSoviet boycott of, n95\n\nUnited States: at Berlin Conference,\n\n\u2013Israel relations, \u2013286\n\nin Korean War, ,\n\nLend-Lease aid of, ,\n\nMarshall Plan,\n\nin postwar settlement,\n\nsecond front plan of, \u2013224, , ,\n\nat Tehran Conference,\n\nUrban population, , n11\n\npreferential treatment of, \u2013322\n\nValedinsky, L. A.,\n\nVasilenko, Vladimir, \u2013308\n\nVasilevsky, Aleksandr Mikhailovich: biography of, n59\n\non command structure,\n\non Kiev defeat,\n\nmarshal rank of,\n\non Moscow evacuation,\n\non Stalin's strategic decisions, , ,\n\nVinogradov, Vladimir, , ,\n\nViolence: Caucasian culture of, \u201322\n\nStalin's pathology of, \u201341\n\nVisitor logs, in archives, xiii Vlasik, Nikolai Sidorovich, , , n8\n\nVoikov, Petr, murder of, \u201389\n\nVolkogonov, Dmitri, ix, , n1, n6\n\nVoroshilov, Kliment Yefremovich: , , , Gallery , ,\n\nbiography of, n2\n\nin Civil War command, ,\n\nin collective leadership,\n\nin Stalin faction,\n\nvacation with Stalin,\n\nin wartime leadership, , , , \u2013221\n\nVorozheikin, Grigory,\n\nVovsi, Miron, \u2013308\n\nVoznesensky, Nikolai Alekseevich: Gallery\n\nbiography of, \u2013353n6\n\nin leadership reorganization, ,\n\nin Leningrad Affair, , \u2013283\n\nPolitburo appointment of,\n\nSovnarkom Bureau appointment of, \u2013179,\n\nin wartime leadership, , , , ,\n\nVyshinsky, Andrei,\n\nWarsaw uprising,\n\nWhite movement,\n\nWinter War, \u2013173\n\nWorkers' and Peasants' Inspectorate,\n\nWorking conditions, \u2013325\n\nWorld revolution, Politburo plans for, \u201378\n\nWorld War II: Allied victories in 1943,\n\nBerlin Conference, \u2013248\n\nBerlin operation, \u2013247\n\ncollapse of Nazi Germany,\n\nfinal battles of, \u2013244,\n\nGerman invasion of Poland, \u2013170\n\nGerman invasion of Soviet Union, , \u2013188, \u2013202\n\nGerman occupation of Europe, ,\n\npostwar settlement, \u2013270\n\nsecond front plan, \u2013224, , ,\n\nTehran Conference, \u2013230\n\nYalta Conference, \u2013246, , Gallery . _See also_ Soviet-German Front\n\nStalin, Joseph, wartime leadership of\n\nWrangel, Petr, , ,\n\nYagoda, Genrikh Grigoryevich,\n\nbiography of, n69\n\nand Kirov murder investigation,\n\npurge of, ,\n\nYakovlev, Aleksandr,\n\nYalta Conference, \u2013246, , Gallery\n\nYegorov, Alexandr, Gallery\n\nYegorov, Petr, , Yemelianov, Nikolai,\n\nYenukidze, Avel Safronovich, , , n82\n\nYezhov, Nikolai Ivanovich,\n\nbiography of, n66\n\nexecution of,\n\non Kirov's murder,\n\nremoval of, , \u2013160\n\nTerror directed by, , , , , \u2013159\n\nYugoslavia, , \u2013281\n\nZagumenny, Sergei,\n\nZaleski, Eugene,\n\nZhdanov, Andrei Aleksandrovich: Gallery\n\nin anti-intelligentsia campaign,\n\nbiography of, n19\n\nand breakdown in relations with West,\n\nand Central Committee,\n\ndeath of,\n\nin leadership reorganizations, , , ,\n\nPolitburo appointment of,\n\nprot\u00e9g\u00e9s of, \u2013353n6\n\nat Stalin's social gatherings,\n\nin wartime leadership,\n\nZhdanov, Yuri,\n\nZhemchuzhina, Polina (Molotov's wife), \u2013284,\n\nZhou Enlai,\n\nZhukov, Georgy Konstantinovich: biography of, n2\n\nin Command Headquarters, ,\n\non Crimean defeat,\n\non crimes against civilians,\n\nand foreign press rumors,\n\non German intelligence,\n\nand German invasion, , \u2013200, ,\n\nmarshal rank of,\n\nStalin's criticism of, , \u2013274\n\nas Stalin's emissary,\n\nand Stalin's strategic decisions, ,\n\nwartime commands of, , , , n2\n\nZinoviev, Grigory Yevseevich, , , ,\n\nand anti-Trotsky coalition, \u201382\n\nbiography of, n9\n\nBolshevik seizure of power opposed by, , ,\n\nexecution of,\n\nexpulsion of, \u201385,\n\nand Kirov's murder charge,\n\nas oppositionist, \u201384,\n\nin power struggle, \u201376, , , ,\n\nin Provisional Government crackdown, ,\n\nZoshchenko, Mikhail, , n9\n\nZubkova, Elena, xiv\u2013xv\n\nZverev, Arseny, , ,\n\n_Zvezda_ (magazine), \n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\n\n\nProduced by Chuck Greif and the Online Distributed\nProofreading Team at http:\/\/www.pgdp.net (This file was\nproduced from images available at The Internet Archive)\n\n\n\n\n\n\n\n\n\n                         KATHARINE LAUDERDALE\n\n                       [Illustration: colophone]\n\n[Illustration: \"She was very white as she turned her face to him.\"--Vol.\n                             II., p. 314.]\n\n\n\n\n                         KATHARINE LAUDERDALE\n\n                                  BY\n\n                          F. MARION CRAWFORD\n           AUTHOR OF \"SARACINESCA,\" \"PIETRO GHISLERI,\" ETC.\n\n                                VOL. II\n\n                 WITH ILLUSTRATIONS BY ALFRED BRENNAN\n\n                               New York\n                           MACMILLAN AND CO.\n                              AND LONDON\n                                 1894\n\n                         _All rights reserved_\n\n\n                           COPYRIGHT, 1893,\n                        BY F. MARION CRAWFORD.\n\n\n                            Norwood Press:\n                 J. S. Cushing & Co.--Berwick & Smith.\n                         Boston, Mass., U.S.A.\n\n\n\n\nCONTENTS.\n\n\n                       PAGE\n\nCHAPTER XVI.              1\n\nCHAPTER XVII.            23\n\nCHAPTER XVIII.           45\n\nCHAPTER XIX.             67\n\nCHAPTER XX.              89\n\nCHAPTER XXI.            114\n\nCHAPTER XXII.           135\n\nCHAPTER XXIII.          157\n\nCHAPTER XXIV.           178\n\nCHAPTER XXV.            202\n\nCHAPTER XXVI.           225\n\nCHAPTER XXVII.          247\n\nCHAPTER XXVIII.         269\n\nCHAPTER XXIX.           291\n\nCHAPTER XXX.            315\n\n\n\n\nLIST OF ILLUSTRATIONS.\n\nVOL. II.\n\n\nPAGE\n\n\"'I'm glad to see you, my dear child!' he said warmly\"      3\n\n\"Before he could even raise his head, Ralston was out of\nthe door and in the street\"                                 57\n\n\"She knew that life could never be the same again, if she\ncould not believe her son\"                                 142\n\n\"'That's good, Crowdie,' he said thoughtfully. 'It's\ndistinctly good'\"                                          189\n\n\"She was very white as she turned her face to him\"         314\n\n\n\n\nKATHARINE LAUDERDALE.\n\n\n\n\nCHAPTER XVI.\n\n\nKatharine let Ralston accompany her within a block of Robert\nLauderdale's house and then sent him away.\n\n\"It's getting late,\" she said. \"It must be nearly ten o'clock, isn't it?\nYes. People are all going out at this hour in the morning, and it's of\nno especial use to be seen about together. There's the Assembly ball\nto-night, and of course you'll come and talk to me, but I shall see\nyou--or no--I'll write you a note, with a special delivery stamp, and\npost it at the District Post-Office. You'll get it in less than an hour,\nand then you'll know what uncle Robert says.\"\n\n\"I know already what he'll say,\" answered Ralston. \"But why mayn't I\nwait for you here?\"\n\n\"Now, Jack! Don't be so ridiculously hopeless about things. And I don't\nwant you to wait, for I haven't the least idea how long it may last, and\nas I said, there's no object in our being seen to meet, away up here by\nthe Park, at this hour. Good-bye.\n\n\"I hate to leave you,\" said Ralston, holding out one hand, with a\nresigned air, and raising his hat with the other.\n\n\"I like that in you!\" exclaimed Katharine, noticing the action. \"I like\nyou to take off your hat to me just the same--though you are my\nhusband.\" She looked at him a moment. \"I'm so glad we've done it!\" she\nadded with much emphasis, and a faint colour rose in her face.\n\nThen she turned away and walked quickly in the direction of Robert\nLauderdale's house, which was at the next corner. As she went she\nglanced at the big polished windows which face the Park, to see whether\nany one had noticed her. She knew the people who lived in one of the\nhouses, and she had an idea that others might know her by sight, as the\nniece of the great man who had built the whole block. But there were\nonly two children at one of the windows, flattening their rosy faces\nagainst the pane and drumming on it with fat hands; very smartly dressed\nchildren, with bright eyes and gayly-coloured ribbons.\n\nAs Katharine had expected, Robert Lauderdale was at home, had finished\nhis breakfast and was in his library attending to his morning letters.\nShe was ushered in almost immediately, and as she entered the room the\nrich man's secretary stood aside\n\n[Illustration: \"'I'm glad to see you, my dear child!' he said\nwarmly.\"--Vol. II., p. 3.]\n\nto let her pass through the door and then went out--a quiet, faultlessly\ndressed young man who had the air of a gentleman. He wore gold-rimmed\nspectacles, which looked oddly on his young face.\n\nRobert Lauderdale did not rise to meet Katharine, as he sat sideways by\na broad table, in an easy position, with one leg crossed over the other\nand leaning back in his deep chair. But a bright smile came into his\ncheerful old face, and stretching out one long arm he took her hand and\ndrew her down and gave her a hearty kiss. Still holding her by the hand,\nhe made her sit in the chair beside him, left vacant by the secretary.\n\n\"I'm glad to see you, my dear child!\" he said warmly. \"What brings you\nso early?\"\n\nHe was a big old man and was dressed in a rough tweed of a light colour,\nwhich was very becoming to his fresh complexion. His thick hair had once\nbeen red, but had turned to a bright sandy grey, something like the\nsands at Newport. His face was laid out in broad surfaces, rich in\nhealthy colour and deeply freckled where the skin was white. His keen\nblue eyes were small, but very clear and honest, and the eyebrows were\nred still, and bushy, with a few white hairs. Two deep, clean furrows\nextended from beside the nostrils into the carefully brushed beard, and\nthere were four wrinkles, and no more, across the broad forehead. No one\nwould have supposed that Robert Lauderdale was much over sixty, but in\nreality he was ten years older. His elder brother, the philanthropist,\nlooked almost as though he might have been his father. It was clear\nthat, like many of the Lauderdales, the old man had possessed great\nphysical strength, and that he had preserved his splendid constitutional\nvitality even in his old age.\n\nKatharine did not answer his question immediately. She was by no means\ntimid, as has been seen, but she felt a little less brave and sure of\nherself in the presence of the head of her family than when she had been\nwith Ralston a few minutes earlier. She was not aware of the fact that\nin many ways she dominated the man who was now her husband, and she\nwould very probably not have wished to believe she did; but she was very\ndistinctly conscious that she could never, under any imaginable\ncircumstances, exert any direct influence over her uncle Robert, though\nshe might persuade him to do much for her. He was by nature himself of\nthe dominant tribe, and during forty years he had been accustomed to\ncommand with that absolute certainty of being obeyed which few positions\ninsure as completely as very great wealth does. As she looked at him for\na moment before speaking, the little opening speech she had framed began\nto seem absolutely inadequate, and she could not find words wherewith to\ncompose another at such short notice. Being courageous, however, she\ndid not hesitate long, but characteristically plunged into the very\nheart of the matter by telling him just what she felt.\n\n\"I've done something very unusual, uncle Robert,\" she began. \"And I've\ncome to tell you all about it, and I prepared a speech for you. But it\nwon't do. Somehow, though I'm not a bit afraid of you--\" she smiled as\nshe met his eyes--\"you seem ever so much bigger and stronger than I\nthought you were, now that I've got here.\"\n\nUncle Robert laughed and patted her hand as it lay on the desk.\n\n\"Out with it, child!\" he exclaimed. \"I suppose you're in trouble, in\nsome way or other, and you want me to help you. Is that it?\"\n\n\"You must help me,\" answered Katharine. \"Nobody else can. Uncle\nRobert--\" She paused, though a pause was certainly not necessary in\norder to give the plain statement more force. \"I've just been married to\nJack Ralston.\"\n\n\"Good--gracious--heavens!\"\n\nThe old man half rose from his seat as he uttered the words, one by one,\nin his deep voice. Then he dropped into his chair again and stared at\nthe young girl in downright amazement.\n\n\"What in the name of common sense induced you to do such a mad thing?\"\nhe asked very quietly, as soon as he had drawn breath.\n\nKatharine had expected that he would be surprised, as was rather\nnatural, and regained her coolness and decision at once.\n\n\"We've loved each other ever since we were children,\" she said, speaking\ncalmly and distinctly. \"You know all about it, for I've told you before\nnow just how I felt. Everybody opposed it--even my mother, at\nlast--except you, and you certainly never gave us any encouragement.\"\n\n\"I should think not, indeed!\" exclaimed old Lauderdale, shaking his\ngreat head and beating a tattoo on the table with his heavy fingers.\n\n\"I don't know why not, I'm sure,\" Katharine answered, with rising\nenergy. \"There's no reason in the world why we shouldn't love each\nother, and it wouldn't make the slightest difference to me if there\nwere. I should love him just the same, and he would love me. He went to\nmy father last year, as you know, and papa treated him\noutrageously--wanted to forbid him to come to the house, but of course\nthat was absurd. Jack behaved splendidly through it all--even papa had\nto acknowledge that, though he didn't wish to in the least. And I hoped\nand hoped, and waited and waited, but things went no better. You know\nwhen papa makes up his mind to a thing, no matter how unreasonable it\nis, one might just as well talk to a stone wall. But I hadn't the\nsmallest intention of being made miserable for the rest of my life, so I\npersuaded Jack to marry me--\"\n\n\"I suppose he didn't need much persuasion,\" observed the old gentleman,\nangrily.\n\n\"You're quite wrong, uncle Robert! He didn't want to do it at all. He\nhad an idea that it wasn't all right--\"\n\n\"Then why in the world did he do it? Oh, I hate that sort of young\nfellow, who pretends that he doesn't want to do a thing because he means\nto do it all the time--and knows perfectly well that it's a low thing to\ndo!\"\n\n\"I won't let you say that of Jack!\" Katharine's grey eyes began to\nflash. \"If you knew how hard it was to persuade him! He only consented\nat last--and so did the clergyman--because I promised to come and tell\nyou at once--\"\n\n\"That's just like the young good-for-nothing, too!\" muttered the old\nman. \"Besides--how do I know that you're really married? How do I know\nthat you're not--\"\n\n\"Stop, please! There's the certificate. Please persuade yourself, before\nyou accuse me of telling falsehoods.\"\n\nKatharine was suddenly very angry, and Robert Lauderdale realized that\nhe had gone too far in his excitement. But he looked at the certificate\ncarefully, then took out his note-book and wrote down the main facts\nwith great care.\n\n\"I didn't mean to doubt what you told me, child,\" he said, while he was\nwriting. \"You've rather startled me with this piece of news. Human life\nis very uncertain,\" he added, using the clergyman's own words, \"and it\nmay be just as well that there should be a note made of this. Hadn't you\nbetter let me keep the certificate itself? It will be quite safe with my\npapers.\"\n\n\"I wish you would,\" answered Katharine, after a moment's thought.\n\nThe production of the certificate had produced a momentary cessation of\nhostilities, so to speak, but the old gentleman had by no means said his\nlast word yet, nor Katharine either.\n\n\"Go on, my dear,\" he resumed gravely. \"If I'm to know anything, I should\nknow everything, I suppose.\"\n\n\"There's not very much more to tell,\" Katharine replied. \"I repeat that\nit was all I could do to persuade Jack to take the step. He resisted to\nthe very last--\"\n\n\"Hm! He seems to have taken an active part in the proceedings in spite\nof his resistance--\"\n\n\"Of course he did, after I had persuaded him to. It was up to that point\nthat he resisted--and even after everything was ready--even this\nmorning, when I met him, he told me that I ought not to have come.\"\n\n\"His spirit seems to have been willing to have some sense--but the flesh\nwas weak,\" observed the old gentleman, without a smile.\n\n\"I insist upon taking the whole responsibility,\" said Katharine. \"It was\nI who proposed it, and it was I who made him do it.\"\n\n\"You're evidently the strong-minded member, my dear.\"\n\n\"In this--yes. I love him, and I made up my mind that it was right to\nlove him and that I would marry him. Now I have.\"\n\n\"It is impossible to make a more direct statement of an unpleasant\ntruth. And now that you've done it, you mean that your family shall take\nthe consequences--which shows a strong sense of that responsibility you\nmentioned--and so you've come to me. Why didn't you come to me\nyesterday? It would have been far more sensible.\"\n\n\"I did think of coming yesterday afternoon--and then it rained, and\nCharlotte came--\"\n\n\"Yes--it rained--I remember.\" Robert Lauderdale's mouth quivered, as\nthough he should have liked to smile at the utter insignificance of the\nshower as compared with the importance of Katharine's action. \"You might\nhave taken a cab. There's a stand close by your house, at the Brevoort.\"\n\n\"Oh, yes--of course--though I should have had to ask mamma for some\nmoney, and that would have been very awkward, you know. And if I had\nreally and truly meant to come, I suppose I shouldn't have minded the\nrain.\"\n\n\"Well--never mind the rain now!\" Uncle Robert spoke a little\nimpatiently. \"You didn't come--and you've come to-day, when it's too\nlate to do anything--except regret what you've done.\"\n\n\"I don't regret it at all--and I don't intend to,\" Katharine answered\nfirmly.\n\n\"And what do you mean to do in the future? Live with Ralston's mother?\nIs that your idea?\"\n\n\"Certainly not. I want you to give Jack something to do, and we'll live\ntogether, wherever you make him go--if it's to Alaska.\"\n\n\"Oh--that's it, is it? I begin to understand. I suppose Jack would think\nit would simplify matters very much if I gave him a hundred thousand\ndollars, wouldn't he? That would be an even shorter way of giving him\nthe means to support his family.\"\n\n\"Jack wouldn't take money from you,\" answered Katharine, quickly.\n\n\"Wouldn't he? If it were not such a risk, I'd try it, just to convince\nyou. You seem to have a very exalted idea of Jack Ralston, altogether.\nI've not. Do you know anything about his life?\"\n\n\"Of course I do. I know how you all talk about the chances you've given\nhim--between you. And I know just what they were--to try his hand at\nbeing a lawyer's clerk first, and a banker's clerk afterwards, with no\nsalary and--\"\n\n\"If he had stuck to either for a year he would have had a very\ndifferent sort of chance,\" interrupted the old gentleman. \"I told him\nso. There was little enough expected of him, I'm sure--just to go to an\noffice every day, as most people do, and write what he was told to\nwrite. It wasn't much to ask. Take the whole thing to pieces and look at\nit. What can he do? What do most men do who must make their way in the\nworld? He has no exceptional talent, so he can't go in for art or\nliterature or that sort of thing. His father wouldn't educate him for\nthe navy, where he would have found his level, or where the Admiral's\nname would have helped him. He didn't get a technical education, which\nwould have given him a chance to try engineering. There were only two\nthings left--the law or business. I explained all that to him at the\ntime. He shook his head and said he wanted something active. That's just\nthe way all young men talk who merely don't want to stay in-doors and\nwork decently hard, like other people. An active life! What is an active\nlife? Ranching, I suppose he means, and he thinks he should do well on a\nranch merely because he can ride fairly well. Riding fairly well doesn't\nmean much on a ranch. The men out there can all ride better than he ever\ncould, and he knows nothing about horses, nor cattle, nor about anything\nuseful. Besides, with his temper, he'd be shot before he'd been out\nthere a year--\"\n\n\"But there are all sorts of other things, and you forget Hamilton\nBright, who began on a ranch--\"\n\n\"Ham Bright is made of different stuff. He had been brought up in the\ncountry, too, and his father was a Western man--from Cincinnati, at all\nevents, though that isn't West nowadays. No. Jack Ralston could never\nsucceed at that--and I haven't a ranch to give him, and I certainly\nwon't go and buy land out there now. I repeat that his only chance lay\nin law or business. Law would have done better. He had the advantage of\nhaving a degree to begin with, and I would have found him a partner, and\nthere's a lot of law connected with real estate which doesn't need a\ngenius to work it, and which is fairly profitable. But no! He wanted\nsomething active! That's exactly what a kitten wants when it runs round\nafter its own tail--and there's about as much sense in it. Upon my word,\nthere is!\"\n\n\"You're very hard on him, uncle Robert. And I don't think you're quite\nreasonable. It was a good deal the old Admiral's fault--\"\n\n\"I'm not examining the cause, I'm going over the facts,\" said old\nLauderdale, impatiently. \"I tried him, and I very soon got to the end of\nhim. He meant to do nothing. It was quite clear from the first. If he'd\nbeen a starving relation it would have been different. I should have\nmade him work whether he liked it or not. As it was, I gave it up as a\nbad job. He wants to be idle, and he has the means to be idle if he's\nwilling to live on his mother. She has ten thousand dollars a year, and\na house of her own, and they can live very well on that--just as well as\nthey want to. When his mother dies that's what Jack will have, and if he\nchooses to marry on it--\"\n\n\"You seem to forget that he's married already--\"\n\n\"By Jove! I did! But it doesn't change things in the least. My position\nis just the same as it was before. With ten thousand a year Katharine\nRalston couldn't support a family--\"\n\n\"Indeed, I could! I'm Katharine Ralston, and I should be--\"\n\n\"Nonsense! You're Katharine Lauderdale. I'm speaking of Jack's mother. I\nsuppose you'll admit that she's not able to support her son's wife out\nof what she has. It would mean a great change in her way of living. At\npresent she doesn't need more. She's often told me so. If she wanted\nmoney for herself, just to spend on herself, mind you--I'd give\nher--well, I won't say how much. But she doesn't. It's for Jack that she\nwants it. She's perfectly honest. She's just like a man in her way of\ntalking, anyhow. And I don't want Jack to be throwing my money into the\nstreets. I can do more good with it in other ways, and she gives him\nmore than is good for him, as it is. People seem to think that if a man\nhas more than a certain amount of money, he's under a sort of moral\nobligation to society to throw it out of the window. That's a point of\nview I never could understand, though it comes quite naturally to Jack,\nI daresay. But I go back. I want to insist on that circumstance, and I\nwant you to see the facts just as they are. If I were to settle another\nhundred thousand dollars on Jack's mother, it would be precisely the\nsame thing, at present, as though I'd settled it on him, or on you. Now\nyou say he wouldn't take any money if I offered it to him.\"\n\n\"No. He wouldn't, and I wouldn't let him if he wanted to.\"\n\n\"You needn't be afraid, my dear. I've no intention of doing anything so\ngood-natured and foolish. If anything could complete Jack's ruin for all\npractical purposes, that would. No, no! I won't do it. I've given Kate\nRalston a good many valuable jewels at one time and another since she\nmarried the Admiral--she's fond of good stones, you know. If Jack\nchooses to go to her and tell her the truth, and if she chooses to sell\nthem and give him the money, it will keep you very comfortably for a\nlong time--\"\n\n\"How can you suggest such a thing!\" cried Katharine, indignantly. \"As\nthough he would ever stoop to think of it!\"\n\n\"Well--I hope he wouldn't. It wouldn't be pretty, if he did. But I'm a\npractical man, my dear, and I'm an old fellow and I've seen the world\non both sides of the Atlantic Ocean for over seventy years. So I look at\nthe case from all possible points of view, fair and unfair, as most\npeople would. But I don't mean to be unfair to Jack.\"\n\n\"I think you are, uncle Robert. If you've proved anything, you've proved\nthat he isn't fit for a ranch--and so you say there's nothing left but\nthe law or business. It seems to me that there are ever so many\nthings--\"\n\n\"If you'll name them, you'll help me,\" said old Lauderdale, seriously.\n\n\"I mean active things--to do with railroads, and all that--\" Katharine\nstopped, feeling that her knowledge was rather vague.\n\n\"Oh! You mean to talk about railroading. I don't own any railroads\nmyself, as I daresay you know, but I've picked up some information about\nthem. Apart from the financing of them--and that's banking, which Jack\nobjects to--there's the law part, which he doesn't like either, and the\nbuilding of them, which he's too old to learn, and the mechanical part\nof them, such as locomotives and rolling stock, which he can't learn\neither--and then there are two places which men covet and for which\nthere's an enormous competition amongst the best men for such matters in\nthe country--I mean the freight agent's place and the passenger agent's.\nThey are two big men, and they understand their business practically,\nbecause they've learned it practically. To understand freight, a man\nmust begin by putting on rough clothes and going down to the shed and\nhandling freight himself, with the common freight men. There are\ngentlemen who have done that sort of thing--just as fine gentlemen as\nJack Ralston, but made of quite different stuff. And it takes a very\nlong time to reach a high position in that way, though it's worth having\nwhen you get it. Do you understand?\"\n\n\"Yes--I suppose I do. But one always hears of men going off and\nsucceeding in some out-of-the-way place--\"\n\n\"But you hear very little about the ones who fail, and they're the\nmajority. And you hear, still more often, people saying, as they do of\nJack Ralston, that he ought to go away, and show some enterprise, and\nget something to do in the West. It's always the West, because most of\nthe people who talk know nothing whatever about it. I tell you,\nKatharine, my dear, it's just as hard to start in this country as it is\nanywhere else, though men get on faster after they're once started--and\nall this talk about something active and an out-of-door existence is\npure nonsense. It's nothing else. A man may have luck soon or late or\nnever, but the safest plan for city-bred men is to begin at a bank. I\ndid, and I've not regretted it. Just as soon as a fellow shows that he\nhas something in him, he's wanted, and if he has friends, as Jack has,\nthey'll help him. But as long as a man hangs about the clubs all day\nwith a cigarette in his mouth, sensible people, who want workers, will\nfight shy of him. Just tell Jack that, the next time you see him. It's\nall I've got to say, and if it doesn't satisfy him nothing can.\"\n\nThe old gentleman's anger had quite disappeared while he was speaking,\nthough it was ready to burst out again on very small provocation. He\nspoke so earnestly, and put matters so plainly, that Katharine began to\nfeel a blank disappointment closing in between her and her visions of\nthe future in regard to an occupation for John. For the rest, she would\nhave been just as determined to marry him after hearing all that her\nuncle had to say as she had been before. But she could not help showing\nwhat she felt, in her face and in the tone of her voice.\n\n\"Still--men do succeed, uncle Robert,\" she said, clinging rather\ndesperately to the hope that he had only been lecturing her and had some\npleasant surprise in store.\n\n\"Of course they do, my dear,\" he answered. \"And it's possible for Jack\nto succeed, too, if he'll go about it in the right way.\"\n\n\"How?\" asked Katharine, eagerly, and immediately her face brightened\nagain.\n\n\"Just as I said. If he'll show that he can stick to any sort of\noccupation for a year, I'll see what can be done.\"\n\n\"But that sticking, as you call it--all day at a desk--is just what he\ncan't do. He wasn't made for it, he--\"\n\n\"Well then, what is he made for? I wish you would get him to make a\nstatement explaining his peculiar gifts--\"\n\n\"Now don't be angry again, uncle Robert! This is rather a serious matter\nfor Jack and me. Do you tell me, in real earnest, quite, quite honestly,\nthat as far as you know the only way for Jack to earn his living is to\ngo into an office for a year, to begin with? Is that what you mean?\"\n\n\"Yes, child. Upon my word--there, you'll believe me now, won't you?\nThat's the only way I can see, if he really means to work. My dear--I'm\nnot a boy, and I'm very fond of you--I've no reason for deceiving you,\nhave I?\"\n\n\"No, uncle dear--but you were angry at first, you know.\"\n\n\"No doubt. But I'm not angry now, nor are you. We've discussed the\nmatter calmly. And we're putting out of the question the fact that if I\nchose to give Jack anything in the way of money, my cheque-book is in\nthis drawer, and I have the power to do it--without any inconvenience,\"\nadded the very rich man, thoughtfully. \"But you tell me that he would\nnot accept it. It's hard to believe, but you know him better than I do,\nand I accept your statement. I may as well tell you that for the honour\nof the family and to get rid of all this nonsense about a secret\nmarriage I'm perfectly willing to do this. Listen. I'll invite you\nall--the whole family--to my place on the river, and I'll tell them all\nwhat has happened and we'll have a sort of 'post facto' wedding there,\nvery quietly, and then announce it to the world. And I'll settle enough\non you, personally--not on your husband--to give you an income you can\nmanage to live on comfortably--\"\n\n\"Oh!\" cried Katharine. \"You're too kind, uncle Robert--and I thank you\nwith all my heart--just as though we could take it from you--I do,\nindeed--\"\n\n\"Never mind that, child. But you say you can't take it. You mean, I\nsuppose, that if it were your money--if I made it so--Jack would refuse\nto live on it. Let's be quite clear.\"\n\n\"That's exactly it. He would never consent to live on it. He would\nfeel--he'd be quite right, too--that we had got married first in order\nto force money out of you, for the honour of the family, as you said\nyourself.\"\n\n\"Yes. And it's particularly hard to force money out of me, too, though\nI'm not stingy, my dear. But I must say, if you had meant to do it, you\ncouldn't have invented anything more ingenious, or more successful. I\ncouldn't allow a couple of young Lauderdales to go begging. They'd have\npictures of me in the evening papers, you know. And apart from that, I'm\ndevilish fond of you--I mean I'm very fond of you--you must excuse an\nold bachelor's English, sometimes. But you won't take the money, so that\nsettles it. Then there's no other way but for Jack to go to work like a\nman and stick to it. To give him a salary for doing no work would be\njust the same as to give him money without making any pretence about it.\nHe can have a desk at my lawyer's, or he can go back to Beman\nBrothers',--just as he prefers. If he'll do that, and honestly try to\nunderstand what he's doing, he shan't regret it. If he'll do what there\nis to be done, I'll make him succeed. I could make him succeed if he had\n'failure' written all over him in letters a foot high--because it's\nwithin the bounds of possibility. But it's of no use to ask me to do\nwhat's not possible. I can't make this country over again. I can't\ncreate a convenient, active, out-of-door career at a good salary, when\nthe thing doesn't exist. In other words, I can't work miracles, and he\nwon't take money, so he must content himself to run on lines of\npossibility. My lawyer would do most things for me, and so would Beman\nBrothers. Beman, to please me, would make Jack a partner, as he has done\nfor Ham Bright. But Jack must either work or put in capital, and he has\nno capital to put in, and won't take any from me. And to be a partner in\na law firm, a man must have some little experience--something beyond his\nbare degree. Do you see it all now, Katharine?\"\n\n\"Indeed, I do,\" she answered, with a little sigh. \"And meanwhile--uncle\nRobert--meanwhile--\"\n\n\"Yes--I know--you're married. That's the very devil, that marriage\nbusiness.\"\n\nHe seemed to be thinking it over. There was something so innocently\nsincere in his strong way of putting it that Katharine could not help\nsmiling, even in her distress. But she waited for him to speak,\nforeseeing what he would say, and did.\n\n\"There's nothing for it,\" he said, at last. \"You won't take money, and\nyou can't live with your mother, and as for telling your father at this\nstage--well, you know him! It really wouldn't be safe. So there's\nnothing for it but--I hate to say it, my dear,\" he added kindly.\n\n\"But to keep it a secret, you mean,\" she said sadly.\n\n\"You see,\" he answered, in a tone that was almost apologetic, \"it would\nbe a mistake, socially, to say you were married, and to go on living\neach with your own family--besides, your father would know it like\neverybody else. He'd make your life very--unbearable, I should think.\"\n\n\"Yes--he would. I know that.\"\n\n\"Well--come and see me again soon, and we'll talk it over. You'll have\nto consider it just as a--I don't know exactly how to put it--a sort of\nformal betrothal between yourselves, such as they used to have in old\ntimes. And I suppose I'm the head of the family, though your grandfather\nis older than I am. Anyhow, you must consider it as though you were\nsolemnly engaged, with the approval of the head of the family, and as\nthough you were to be married, say, next year. Can you do that? Can you\nmake him look at it in that light, child?\"\n\n\"I'll try, since there's really nothing else to be done. But oh, uncle\nRobert, I wish I'd come before. You've been so kind! Why did it rain\nyesterday--oh, why did it rain?\"\n\n\n\n\nCHAPTER XVII.\n\n\nWhen Katharine left Robert Lauderdale's house that morning, she felt\nthat trouble had begun and was not to cease for a long time. She had\nentered her uncle's library full of hope, sure of success and believing\nthat John Ralston's future depended only upon the rich man's good will\nand good word. She went out fully convinced at last that he must take\none or the other of the much-despised chances he had neglected and\nforthwith do the best he could with it. She thought it was very hard,\nbut she understood old Lauderdale's clear statement and she saw that\nthere was no other way.\n\nShe sympathized deeply with John in his dislike of the daily drudgery,\nfor which it was quite true that he was little fitted by nature or\ntraining. But she did her best to analyze that unfitness, so as to try\nand discover some gift or quality to balance it and neutralize it. And\nher first impulse was not to find him at once and tell him what had\nhappened, but rather to put off the evil moment in which she must tell\nhim the truth. This was the first sign of weakness which she had\nexhibited since that Monday afternoon on which she had persuaded him to\ntake the decisive step.\n\nShe turned into Madison Avenue as soon as she could, for the sake of the\nquiet. The morning sun shone full in her eyes as she began to make her\nway southwards, and she was glad of the warmth, for she felt cold and\ninwardly chilled in mind and body. She had walked far, but she still\nwalked on, disliking the thought of being penned in with a dozen or more\nof unsympathizing individuals for twenty minutes in a horse-car.\nMoreover, she instinctively wished to tire herself, as though to bring\ndown her bodily energy to the low ebb at which her mental activity\nseemed to be stagnating. Strong people will understand that desire to\nbalance mind and body.\n\nShe was quite convinced that her uncle was right. The more she turned\nthe whole situation over, the clearer what he had said became to her.\nThe only escape was to accept the money which he was willing to give\nher--for the honour of the family. But if neither she nor John would\ntake that, there was no alternative but for John to go to work in the\nordinary way, and show that he could be steady for at least a year. That\nseemed a very long time--as long as a year can seem to a girl of\nnineteen, which is saying much.\n\nKatharine had seen such glorious visions for that year, too, that the\ndarkness of the future was a tangible horror now that they were fading\naway. The memory of a dream can be as vivid as the recollection of a\nreality. The something which John was to find to do had presented itself\nto her mind as a sort of idyllic existence somewhere out of the world,\nin which there should be woods and brooks and breezes, and a convenient\ntown not far away, where things could be got, and a cottage quite unlike\nother cottages, and a good deal of shooting and fishing and riding, with\nan amount of responsibility for all these things equal in money to six\nor seven thousand dollars a year, out of which Katharine was sure that\nshe could save a small fortune in a few years. It had not been quite\nclear to her why the responsibility was to be worth so much in actual\ncoin of the Republic, but people certainly succeeded very quickly in the\nWest. Besides, she was quite ready to give up all the luxuries and\namusements of social existence--much more ready to do so than John\nRalston, if she had known the truth.\n\nIt must not be believed that she was utterly visionary and unpractical,\nbecause she had taken this rose-coloured view of the life uncle Robert\nwas to provide for her and her husband. There are probably a great many\nyoung women in the Eastern cities who imagine just such things to be\nquite possible, and quite within the power and gift of a millionaire, in\nthe American sense of that word, which implies the possession of more\nthan one million, and more often refers in actual use to income than\nmerely to capital. In Paris, a man who has twenty thousand dollars a\nyear is called a millionaire. In New York a man with that income is but\njust beyond the level of the estimable society poor, and within the\nranks of the 'fairly well-off.' The great fortunes being really as\nfabulous as those in fairy tales, it is not surprising that the\npossession of them should be supposed to bring with it an almost\nfabulous power in all directions. Men like Robert Lauderdale, the\nadministration of whose estates requires a machinery not unlike that of\na small nation's treasury, are thought to have in their gift all sorts\nof remunerative positions, for which the principal qualifications are an\nunlimited capacity for enjoying the fresh air and some talent for\nfishing. As a matter of fact, though so much richer than ordinary men,\nthey are so much poorer than all except the very small nations that they\ncannot support so many idlers.\n\nKatharine knew a good deal about life in New York and its possibilities,\nbut very little of what could be done elsewhere. She was perfectly well\naware of the truth of all that her uncle had told her concerning the\nrequirements for business or the law, for she had heard such matters\ndiscussed often enough. In her own city she was practical, for she\nunderstood her surroundings as well as any young girl could. It was\nbecause she understood them that she dreamed of getting out of them as\nsoon as practicable, and of beginning that vaguely active and\nremunerative existence which, for her, lay west of Illinois and anywhere\nbeyond that, even to the shores of the Pacific Ocean. John Ralston\nhimself knew very little about it, but he had rightly judged its\nmythical nature when he had told her that Robert Lauderdale would do\nnothing for him.\n\nThe sun warmed Katharine as she walked down Madison Avenue, but\neverything was black--felt black, she would have said, had she thought\naloud. Ralston would not turn upon her and say, 'I told you so,' because\nhe loved her, but she could see the expression of his face as she looked\nforward to the interview. He would nod his head slowly and say nothing.\nThe corners of his mouth would be drawn down for a moment and his\neyelids would contract a little while he looked away from her. He would\nthink the matter over during about half a minute, and then, with a look\nof determination, he would say that he would try what uncle Robert\nproposed. He would not say anything against the plan of keeping the\nmarriage a secret, now that old Lauderdale knew of it, for he would see\nat once that there was absolutely nothing else to be done. They had gone\nover the possibilities so often--there was not one which they had not\ncarefully considered. It was all so hopelessly against them still, in\nspite of the one great effort Katharine had made that morning.\n\nShe walked more slowly after she had passed the high level above the\nrailway, where it runs out of the city under ground from the central\nstation. As she came nearer to the neighbourhood in which John lived,\nshe felt for the first time in her life that she did not wish to meet\nhim. Though she did not admit to herself that she feared to tell him the\nresult of her conversation with her uncle, and though she had no\nintention of going to his mother's house and asking for him, her pace\nslackened at the mere idea of being nearer to him.\n\nThen she realized what she was doing, and with a bitter little smile of\ncontempt at her own weakness she walked on more briskly. She had often\nread in books of that sudden change in the aspect of the outer world\nwhich disappointment brings, but she had never quite believed in it\nbefore. She realized it now. There was no light in anything. The faces\nof the people who passed her looked dead and uninteresting. Every house\nlooked as though a funeral procession might at any moment file out of\nits door. The very pavement, drying in patches in the sunshine, felt\ncold and unsympathetic under her feet.\n\nShe began to wonder what she had better do,--whether she should write\nJohn Ralston a long letter, explaining everything, or whether she\nshould write him a short one, merely saying that the news was\nunfavourable--'unfavourable' sounded better than 'bad' or\n'disappointing,' she thought--and asking him to come and see her in the\nafternoon. The latter course seemed preferable, and had, moreover, the\nadvantage of involving fewer practical difficulties, for her command\nover her mother tongue was by no means very great when subjected to the\ntest of black and white, though in conversation it was quite equal to\nher requirements on most occasions. She could even entirely avoid the\nuse of slang, by making a determined effort, for her father detested it,\nand her mother's conversational weaknesses were Southern and of a\ndifferent type. But on paper she was never sure of being quite right.\nPunctuation was a department which she affected to despise, but which\nshe inwardly feared, and when alone she admitted that there were words\nwhich she seemed to spell not as they were spelled in books--'parallel,'\nfor instance, 'psychology' and 'responsibility.' She avoided those\nwords, which were not very necessary to her, but with a disagreeable\nsuspicion that there might be others. Had 'develop' an 'e' at the end of\nit, or had it not? She could never remember, and the dictionary lived in\nher grandfather's den, at some distance from her own room. The\ndifficulties of writing a long letter to John Ralston, whose mother had\ntaught him his English before it could be taught him all wrong at a\nfashionable school, rose before her eyes with absurd force, and she\ndecided forthwith to send for Ralston in the afternoon.\n\nHaving come to a preliminary conclusion, life seemed momentarily a\nlittle easier. She turned out of her way into Fourth Avenue, took a\nhorse-car, got transferred to a Christopher Street one, and in the\ncourse of time got out at the corner of Clinton Place. She wrote the\nshortest possible note to John Ralston, went out again, bought a special\ndelivery stamp and took the letter up to the Thirteenth Street\nPost-Office--instead of dropping it into an ordinary letter-box. She did\neverything, in short, to make the message reach its destination as\nquickly as possible without employing a messenger.\n\nCharlotte Slayback appeared at luncheon. She preferred that meal when\nshe invited herself, because her father was never present, and a certain\namount of peaceful conversation was possible in his absence. It was some\ntime since she had been in New York, and the glimpse of her old room on\nthe previous afternoon irresistibly attracted her again. Katharine\nhoped, however, that she would not stay long, as Ralston was to come at\nthree o'clock, this being usually the safest hour for his visits. Mrs.\nLauderdale would then be either at work or out of the house, the\nphilanthropist would be dozing upstairs in a cloud of smoke before a\ntable covered with reports, and Alexander Junior would be still down\ntown. In consideration of the importance of getting Charlotte out of the\nway, Katharine was more than usually cordial to her--a mistake often\nmade by young people, who do not seem to understand the very simple fact\nthat the best way to make people go away is generally to be as\ndisagreeable as possible.\n\nThe consequence was that Charlotte enjoyed herself immensely, and it\nrequired the sight of her father's photograph, which stood upon Mrs.\nLauderdale's writing-table in the library, to keep her from proposing to\nspend two or three days in the house after her husband should have gone\nback to Washington. But the photograph was there, and it was one taken\nby the platinum process, which made the handsome, steely face look more\nmetallic than ever. Charlotte gazed at it thoughtfully, and could almost\nhear the maxims of virtue and economy with which those even lips had\npreached her down since she had been a child, and she decided that she\nwould not stay. Her husband was not to her taste, but he never preached.\n\nMrs. Lauderdale had for her eldest daughter that sentiment which is\ngenerally described as a mother's love, and which, as Frank Miner had\nonce rather coarsely put it, will stand more knocking about than old\nboots. Charlotte was spoiled, capricious, frivolous in the extreme,\nungrateful beyond description, weak where she should have been strong\nand strong where she should have been tender. And Mrs. Lauderdale knew\nit all, and loved her in spite of it all, though she disapproved of her\nalmost at every point. Charlotte had one of those characters of which\npeople are apt to say that they might have turned out splendidly, if\nproperly trained, than which no more foolish expression falls from the\nlips of commonplace, virtuous humanity. Charlotte, like many women who\nresemble her, had received an excellent training. The proof was that,\nwhen she chose to behave herself, no one could seem to be more docile,\nmore thoughtful and considerate of others or more charming in\nconversation. She had only to wish to appear well, as the phrase goes,\nand the minutest details necessary to success were absolutely under her\ncontrol. What people meant when they said that she might have turned out\nsplendidly--though they did not at all understand the fact--was that a\nwoman possessing Charlotte Slayback's natural gifts and acquired\naccomplishments might have been a different person if she had been born\nwith a very different character--a statement quite startling in its\ngreat simplicity. As it was, there was nothing to be done. Charlotte had\nbeen admirably 'trained' in every way--so well that she could exhibit\nthe finest qualities, on occasion, without any perceptible effort, even\nwhen she felt the utmost reluctance to do so. But the occasions were\nfew, and were determined by questions of personal advantage, and even\nmore often by mere caprice.\n\nOn that particular day, when she lunched quietly in her old home, her\nconduct was little short of angelic, and Katharine found it hard to\nrealize that she was the same woman who on the previous afternoon had\nmade such an exhibition of contemptible pettiness and unreasoning\ndiscontent. Katharine, had she known her sister less well, would almost\nhave been inclined to believe that Benjamin Slayback of Nevada was a\nperson with whom no wife of ordinary sensibility would possibly live.\nBut she knew Charlotte very well indeed.\n\nAnd as the hands of the clock went round towards three, Charlotte showed\nno intention of going away, to Katharine's infinite annoyance, for she\nknew that Ralston would be punctual, and would probably come even a\nlittle before the time she had named. It would not do to let him walk\ninto the library, after the late scene between him and her mother. The\nlatter had said nothing more about the matter, but only one day had\nintervened since Mrs. Lauderdale had so unexpectedly expressed her total\ndisapproval of Katharine's relations with John. It was not probable\nthat Mrs. Lauderdale, who was not a changeable woman, would go back to\nher original position in the course of a few hours, and there would\ncertainly be trouble if John appeared with no particular excuse.\n\nKatharine, as may be imagined, was by no means in a normal mood, and if\nshe made herself agreeable to her sister, it was not at first without a\ncertain effort, which did not decrease, in spite of Charlotte's own\nexceptionally good temper, because as the latter grew more and more\namiable, she also seemed more and more inclined to spend the whole\nafternoon where she was.\n\nHints about going out, about going upstairs to the room in which Mrs.\nLauderdale painted, about possible visitors, had no effect whatever.\nCharlotte was enjoying herself and her mother was delighted to keep her\nand listen to her conversation. Katharine thought at last that she\nshould be reduced to the necessity of waiting in the entry until Ralston\ncame, in order to send him away again before he could get into the\nlibrary by mistake. She hated the plan, which certainly lacked dignity,\nand she watched the hands of the clock, growing nervous and absent in\nwhat she said, as she saw that the fatal hour was approaching.\n\nAt twenty minutes to three Charlotte was describing to her mother the\ngown worn by the English ambassadress at the last official dinner at\nthe White House. At a quarter to three she was giving an amusing account\nof the last filibustering affray in the House, which she had\nwitnessed--it having been arranged beforehand to take place at a given\npoint in the proceedings--from the gallery reserved for members'\nfamilies. Five minutes later she was telling anecdotes about a\ndeputation from the South Sea Islands. Katharine could hardly sit still\nas she watched the inexorable hands. At five minutes to three Charlotte\nstruck the subject of painting, and Katharine felt that it was all over.\nSuddenly Charlotte herself glanced at the clock and sprang up.\n\n\"I had forgotten all about poor little Crowdie!\" she exclaimed. \"He was\ncoming at three to take me to the Loan Exhibition,\" she added, looking\nabout her for her hat and gloves.\n\n\"Here?\" asked Katharine, aghast.\n\n\"Oh, no--at the hotel, of course. I must run as fast as I can. There are\nstill cabs at the Brevoort House corner, aren't there? Thank you, my\ndear--\" Katharine had found all her things and was already tying on the\nlittle veil. \"I do hope he'll wait.\"\n\n\"Of course he will!\" answered Katharine, with amazing certainty. \"You're\nall right, dear--now run!\" she added, pushing her sister towards the\ndoor.\n\n\"Do come to dinner, Charlie!\" cried Mrs. Lauderdale, following her.\n\"It's so nice to see something of you!\"\n\n\"Oh, yes--she'll come--but you mustn't keep her, mamma--she's awfully\nlate as it is!\"\n\nFrom a condition of apparently hopeless apathy, Katharine was suddenly\nroused to exert all her energies. It was two minutes to three as she\nclosed the glass door behind her sister. Fortunately Ralston had not\ncome before his time.\n\n\"I suppose you're going to work now, mamma?\" Katharine suggested, doing\nher best to speak calmly, as she turned to her mother, who was standing\nin the door of the library.\n\nShe had never before wished that Ralston were an unpunctual man, nor\nthat her mother, to whom she was devotedly attached, were at the bottom\nof the sea.\n\n\"Oh, yes! I suppose so,\" answered Mrs. Lauderdale. \"How delightful\nCharlotte was to-day, wasn't she?\"\n\nHer face was fresh and rested. She leaned against the doorpost as though\ndeciding whether to go upstairs at once or to go back into the library.\nWith a movement natural to her she raised her graceful arms, folding her\nhands together behind her head, and leaning back against the woodwork,\nlooking lazily at Katharine as she did so. She felt that small\ndifficulty, at the moment, of going back to the daily occupation after\nspending an exceptionally pleasant hour in some one's company, which is\nfamiliar to all hard workers. Katharine stood still, trying to hide her\nanxiety. The clock must be just going to strike, she thought.\n\n\"What's the matter, child? You seem nervous and worried about\nsomething.\" She asked the question with a certain curiosity.\n\n\"Do I?\" asked Katharine, trying to affect indifference.\n\nMrs. Lauderdale did not move. In the half light of the doorway she was\nstill very beautiful, as she stood there trying to make up her mind to\ngo to her work. Katharine was in despair, and turned over the cards that\nlay in a deep dish on the table, reading the names mechanically.\n\n\"Yes,\" continued her mother. \"You look as though you were expecting\nsomething--or somebody.\"\n\nThe clock struck, and almost at the same instant Katharine heard\nRalston's quick, light tread on the stone steps outside the house. She\nhad a sudden inspiration.\n\n\"There's a visitor coming, mother!\" she whispered quickly. \"Run away,\nand I'll tell Annie not to let him in.\"\n\nMrs. Lauderdale, fortunately, did not care to receive any one, but\ninstead of going upstairs she merely nodded, just as the bell rang, and\nretired into the library again, shutting the door behind her. Katharine\nwas left alone in the entry, and she could see the dark, indistinct\nshape of John Ralston through the ground-glass pane of the front door.\nShe hesitated an instant, doubting whether it would not be wisest to\nopen the door herself, send him away, and then, slipping on her things,\nto follow him a moment later into the street. But in the same instant\nshe reflected that her mother had very possibly gone to the window to\nsee who the visitor had been when he should descend the steps again.\nMost women do that in houses where it is possible. Then, too, her mother\nwould expect to hear Annie's footsteps passing the library, as the girl\nwent to the front door.\n\nThere was the dining-room, and it could be reached from the entry by\npassing through the pantry. Annie was devoted to Katharine, and at a\nwhispered word would lead Ralston silently thither. The closed room\nbetween the dining-room and the library would effectually cut off the\nsound of voices. But that, too, struck Katharine as being beneath\nher--to confide in a servant! She could not do it, and was further\njustified by the reflection that even if she followed that course, her\nmother, who was doubtless at the window, would not see Ralston go away,\nand would naturally conclude that the visitor had remained in the house,\nwhoever he might be.\n\nKatharine stood irresolute, watching Ralston's shadow on the pane, and\nlistening to Annie's rapidly approaching tread from the regions of the\npantry at the end of the entry. A moment later and the girl was by her\nside.\n\n\"If it's Mr. Ralston, don't shut the door again till I've spoken to\nhim,\" she said, in a low voice. \"My mother isn't receiving, if it's a\nvisitor.\"\n\nShe stood behind Annie as the latter opened the door. John was there, as\nshe had expected, and Annie stepped back. Katharine raised her finger to\nher lips, warning him not to speak. He looked surprised, but stood\nbareheaded on the threshold.\n\n\"You must go away at once, Jack,\" she whispered. \"My mother is in the\nlibrary, looking out of the window, and I can't possibly see you alone.\nWait for me near the door at the Assembly to-night. Go, dear--it's\nimpossible now. I'll tell you afterwards.\"\n\nIn her anxiety not to rouse her mother's suspicions, she shut the door\nalmost before he had nodded his assent. She scarcely saw the blank look\nthat came into his face, and the utter disappointment in his eyes.\n\nSeeing that the door was shut, Annie turned and went away. Katharine\nhesitated a moment, passed her hand over her brow, glanced mechanically\nonce more at the cards in the china dish on the table and then went into\nthe library. To her surprise her mother was not there, but the folding\ndoor which led to the dark drawing-room was half rolled back, and it\nwas clear that Mrs. Lauderdale had gone through the dining-room, and had\nprobably reached her own apartment by the back staircase of the house.\nKatharine was on the point of running into the street and calling\nRalston back. She hesitated a moment, and then going hastily to the\nwindow threw up the sash and looked out, hoping that he might be still\nwithin hearing. But looking eastward, towards Fifth Avenue, he was not\nto be seen amongst the moving pedestrians, of whom there were many just\nthen. She turned to see whether he had taken the other direction, and\nsaw him at once, but already far down the street, walking fast, with his\nhead bent low and his hands in the pockets of his overcoat. He was\nevidently going to take the elevated road up town.\n\n\"Oh, Jack--I'm so sorry!\" she exclaimed softly to herself, still looking\nafter him as he disappeared in the distance.\n\nThen she drew down the window again, and went and sat in her accustomed\nplace in the small armchair opposite to her mother's sofa. She thought\nvery uncharitably of Charlotte during the next quarter of an hour, but\nshe promised herself to get into a corner with Ralston that evening, at\nthe great ball, and to explain all the circumstances to him as minutely\nas they have been explained here. She was angry with her mother, too,\nfor not having gone up the front staircase, as she might just as well\nhave done, but she was very glad she had not condescended to the\nman\u0153uvre of introducing John into the dining-room by the back way, as\nshe would have probably just met Mrs. Lauderdale as the latter passed\nthrough. On the whole, it seemed to Katharine that she had done as\nwisely as the peculiarly difficult circumstances had allowed, and that\nalthough there was much to regret, she had done nothing of which she\nneeded to repent.\n\nIt seemed to her, too, as she began to recover from the immediate\nannoyance of failure, that she had gained several hours more than she\nhad expected, in which to think over what she should say to Ralston when\nthey met. And she at once set herself the task of recalling everything\nthat Robert Lauderdale had said to her, with the intention of repeating\nit as accurately as possible, since she could not expect to say it any\nbetter than he had said it himself. It was necessary that Ralston should\nunderstand it, as she had understood it, and should see that although\nuncle Robert was quite ready to be generous he could not undertake to\nperform miracles. Those had been the old gentleman's own words.\n\nThen she began to wonder whether, after all, it would not be better to\naccept what he offered--the small, settled income which was so good to\nthink of--and to get rid of all this secrecy, which oppressed her much\nmore since she had been told that it must last, than when she had\nexpected that it would involve at most the delay of a week. The deep\ndepression which she began to feel at her heart, now that she was alone\nagain, made the simple means of escape from all her anxieties look very\ntempting to her, and she dwelt on it. If she begged Ralston to forget\nhis pride for her sake, as she was willing to forget her own for his,\nand to let her take the money, he would surely yield. Once together,\nopenly married before the world, things would be so much easier. He and\nshe could talk all day, unhindered and unobserved, and plan the future\nat their leisure, and it was not possible that with all the joint\nintelligence they could bring to bear upon the problem, it should still\nremain unsolved.\n\nMeanwhile, Ralston had gone up town, very much more disappointed than\nKatharine knew. Strange to say, their marriage seemed far more important\nin his eyes than in hers, and he had lived all day, since they had\nparted at ten o'clock in the morning, in nervous anticipation of seeing\nher again before night. He had gone home at once, and had spent the\nhours alone, for his mother had gone out to luncheon. Until the\nmessenger with Katharine's specially stamped letter rang at the door, he\nwould not have gone out of the house for any consideration, and after\nhe had read it he sat counting the minutes until he could reasonably\nexpect to use up the remaining time in walking to Clinton Place. As it\nwas, he had reached the corner a quarter of an hour before the time, and\nhis extreme punctuality was to be accounted for by the fact that he had\nset his watch with the Lauderdales' library clock,--as he always did\nnowadays,--and that he looked at it every thirty seconds, as he walked\nup and down the street, timing himself so exactly that the hands were\nprecisely at the hour of three when he took hold of the bell.\n\nThere are few small disappointments in the world comparable with that of\na man who has been told by the woman he loves to come at a certain hour,\nwho appears at her door with military punctuality and who is told to go\naway again instantly, no adequate excuse being given for the summary\ndismissal. Men all know that, but few women realize it.\n\n\"Considering the rather unusual situation,\" thought Ralston, angrily,\n\"she might have managed to get her mother out of the way for half an\nhour. Besides, her mother wouldn't have stoned me to death, if she had\nlet me come in--and, after last night, I shouldn't think she would care\nvery much for the sort of privacy one has in a ball-room.\"\n\nHe had waited all day to see her, and he had nothing to do until the\nevening, when he had to go to a dinner-party before the Assembly ball.\nHe naturally thought of his club, as a quiet place where he could be\nalone with his annoyances and disappointments between three and four\no'clock, and he took the elevated road as the shortest way of getting\nthere.\n\n\n\n\nCHAPTER XVIII.\n\n\nRalston was in a thoroughly bad humour when he reached his club. The\nabsurdity of a marriage, which was practically no marriage at all, had\nbeen thrust upon him on the very first day, and he felt that he had been\nled into a romantic piece of folly, which could not possibly produce any\ngood results, either at the present time or afterwards. He was as\nproperly and legally the husband of Katharine as the law and the church\ncould make him, and yet he could not even get an interview of a quarter\nof an hour with his wife. He could not count, with certainty, upon\nseeing her anywhere, except at such a public place as the ball they were\nboth going to that night, under the eyes of all New York society, so far\nas it existed for them. The position was ludicrous, or would have been,\nhad he not been the principal actor in the comedy.\n\nHe was sure, too, that if Katharine had got any favourable answer from\ntheir uncle Robert, she would have said at least a word to this effect,\neven while she was in the act of thrusting him from the door. Two words,\n'all right,' would have been enough. But she had only seemed anxious to\nget rid of him as quickly as possible, and he felt that he was not to\nbe blamed for being angry. The details of the situation, as she had seen\nit, were quite unknown to him. He was not aware that Charlotte Slayback\nhad been at luncheon, and had stayed until the last minute, nor that\nKatharine had really done everything in her power to make her mother go\nupstairs. The details, indeed, taken separately, were laughable in their\ninsignificance, and it would hardly be possible for Katharine to explain\nthem to him, so as to make him see their importance when taken all\ntogether. He was ignorant of them all, except of the fancied fact that\nMrs. Lauderdale had been at the window of the library. Katharine had\ntold him so, and had believed it herself, as was natural. She had not\nhad time to explain why she believed it, and he would be more angry than\never if she ever told him that she had been mistaken, and that he might\njust as well have come and stayed as long as he pleased. He knew that a\nconsiderable time must have elapsed between the end of luncheon and his\narrival at the door of the house; he supposed that Katharine had been\nalone with her mother and grandfather, as usual, and he blamed her for\nnot exerting a little tact in getting her mother out of the way, when\nshe must have had nearly an hour in which to do so. He went over and\nover all that he knew of the facts, and reached always the same\nconclusion--Katharine had not taken the trouble, and had probably only\nremembered when it was too late that he was to come at three o'clock.\n\nIt must not be supposed that Ralston belonged to the class of hasty and\ncapricious men, who hate the object of their affections as soon as they\nare in the least annoyed with anything she has done--or who, at all\nevents, act as though they did. Ralston was merely in an excessively bad\ntemper with himself, with everything he had done and with the world at\nlarge. Had he received a note from Katharine at any time later in the\nafternoon, telling him to come back, he would have gone instantly, with\njust as much impatience as he had shown at three o'clock, when he had\nreached Clinton Place a quarter of an hour before the appointed time. He\nwould probably not have alluded, nor even have wished to allude, to his\nsummary dismissal at his first attempt. But he would come. He satisfied\nhimself of that, for he sent a message from his club to his home,\ndirecting the servant to send on any note which might come for him; and,\non repeating the message an hour later, he was told that there was\nnothing to send.\n\nSo he sat in the general room at the club, downstairs, and turned over a\nnewspaper half a dozen times without understanding a word of its\ncontents, and smoked discontentedly, but without ceasing. At last, by a\nmere accident, his eye fell upon the column of situations offered and\nwanted, and, with a sour smile, he began to read the advertisements.\nThat sort of thing suited his case, at all events, he thought. He was\nvery soon struck by the balance of numbers in favour of the unemployed,\nand by the severe manner in which those who offered situations spoke of\nthorough knowledge and of certificates of service.\n\nIt did not take him long to convince himself that he was fit for nothing\nbut a shoeblack or a messenger boy, and he fancied that his age would be\na drawback in either profession. He dropped the paper in disgust at\nlast, and was suddenly aware that Frank Miner was seated at a small\ntable opposite to him, but on the other side of the room. Miner looked\nup at the same moment, from a letter he was writing, his attention being\nattracted by the rustling of the paper.\n\n\"Hallo, Jack!\" he cried, cheerily. \"I knew those were your legs all the\ntime.\"\n\n\"Why didn't you speak, then?\" asked Ralston, rather coldly, and looking\nup and down the columns of the paper he had dropped upon his knee.\n\n\"I don't know. Why should I?\" Miner went on with his letter, having\nevidently interrupted himself in the midst of a sentence.\n\nRalston wished something would happen. He felt suddenly inclined to\nthrow something at Miner, who generally amused him when he talked, but\nwas clearly very busy, and went on writing as though his cheerful little\nlife depended on it. But it was not probable that anything should happen\njust at that hour. There were three or four other men in different parts\nof the big room, writing or reading letters. There were doubtless a few\nothers somewhere in the house, playing cards or drinking a quiet\nafternoon cocktail. It was a big club, having many rooms. But Ralston\ndid not feel inclined to play poker, and he wished not to drink, if he\ncould help it, and Miner went on writing, so he stayed where he was, and\nbrooded over his annoyances. Suddenly Miner's pen ceased with a scratch\nand a dash, audible all over the room, and he began to fold his letter.\n\n\"Come and have a drink, Jack!\" he called out to Ralston, as he took up\nan envelope. \"I've earned it, if you haven't.\"\n\n\"I don't want to drink,\" answered Ralston, gloomily, and, out of pure\ncontrariety, he took up his paper again.\n\nMiner looked long and steadily at him, closed his letter, put it into\nhis pocket and crossed the room.\n\n\"I say, Jack,\" he said, in an absurdly solemn tone, \"are you ill, old\nman?\"\n\n\"Ill? No. Why? Never was better in my life. Don't be an idiot, Frank.\"\nAnd he kept his paper at the level of his eyes.\n\n\"There's something wrong, anyhow,\" said Miner, thoughtfully. \"Never knew\nyou to refuse to drink before. I'll be damned, you know!\"\n\n\"I haven't a doubt of it, my dear fellow. I always told you so.\"\n\n\"For a gentle and unassuming manner, I think you take the cake, Jack,\"\nanswered Miner, without a smile. \"What on earth is the matter with you?\nLet me see--you've either lost money, or you're in love, or your liver's\nout of order, or all three, and if that's it, I pity you.\"\n\n\"I tell you there's nothing the matter with me!\" cried Ralston, with\nsome temper. \"Why do you keep bothering me? I merely said I didn't want\nto drink. Can't a man not be thirsty? Confound it all, I'm not obliged\nto drink if I don't want to!\"\n\n\"Oh, well, don't get into a fiery green rage about it, Jack. I'm thirsty\nmyself, and I didn't want to drink alone. Only, don't go west of Maine\nso long as this lasts. They're prohibition there, you know. Don't try\nit, Jack; you'd come back on ice by the next train.\"\n\n\"I'm going to stay here,\" answered Ralston, without a smile. \"Go ahead\nand get your drink.\"\n\n\"All right! If you won't, you won't, I know. But when you're scratching\nround and trying to get some sympathetic person, like Abraham and\nLazarus, to give you a glass of water, think of what you've missed this\nafternoon!\"\n\n\"Dives,\" said Ralston, savagely, \"is the only man ever mentioned in the\nBible as having asked for a glass of water, and he's--where he ought to\nbe.\"\n\n\"That's an old, cold chestnut,\" retorted Miner, turning to go, but not\nreally in the least annoyed.\n\nAt that moment a servant crossed the room and stood before Ralston.\nMiner waited to see what would happen, half believing that Ralston was\nnot in earnest, but had surreptitiously touched the electric bell on the\ntable at his elbow, with the intention of ordering something.\n\n\"Mr. Lauderdale wishes to speak to you at the telephone, sir,\" said the\nservant.\n\nThe man's expression betrayed his respect for the name, and for a person\nwho had a telephone in his house--an unusual thing in New York. It was\nthe sort of expression which the waiters at restaurants put on when they\npresent to the diner a dish of terrapin or a canvas-back duck, or open a\nvery particularly old bottle of very particularly fine wine--quite\ndifferent from the stolid look they wear for beef and table-claret.\n\n\"Which Mr. Lauderdale?\" asked Ralston, with a sudden frown. \"Mr.\nAlexander Lauderdale Junior?\"\n\n\"I don't know, sir. The gentleman's at the telephone, sir.\"\n\nThis seemed to be added as a gentle hint not to keep any one of the\nname of Lauderdale waiting too long.\n\nRalston rose quickly, and Miner watched him as he passed out with long\nstrides and a rather anxious face, wondering what could be the matter\nwith his friend, and somehow connecting his refusal to drink with the\nsummons to the instrument. Then Miner followed slowly in the same\ndirection, with his hands in his pockets and his lips pursed as though\nhe were about to whistle. He knew the man well enough to be aware that\nhis refusal to drink might proceed from his having taken all he could\nstand for the present, and Ralston's ill temper inclined Miner to\nbelieve that this might be the case. Ralston rarely betrayed himself at\nall, until he suddenly became viciously unmanageable, a fact which made\nhim always the function of a doubtful quantity, as Miner, who had once\nlearned a little mathematics, was fond of expressing it.\n\nThe little man was essentially sociable, and though he might want the\nvery small and mild drink he was fond of ever so much, he preferred, if\npossible, to swallow it in company. Instead of ringing, therefore, he\nstrolled away in search of another friend. As luck would have it, he\nalmost ran against Walter Crowdie, who was coming towards him, but\nlooking after Ralston, as the latter disappeared at the other end of the\nhall. Crowdie seemed excessively irritated about something.\n\n\"Confound that fellow!\" he exclaimed, giving vent to his feelings as he\nturned and saw Miner close upon him.\n\n\"Who? Me?\" enquired the little man, with a laugh. \"Everybody's purple\nwith rage in this club to-day--I'm going home.\"\n\n\"You? No--is that you, Frank? No--I mean that everlasting Ralston.\"\n\n\"Oh! What's he done to you? What's the matter with Ralston?\"\n\n\"Drunk again, I suppose,\" answered Crowdie. \"But I wish he'd keep out of\nmy way when he is--runs into me, treads on both my feet--with his heels,\nI believe, though I don't understand how that's possible--pushes me out\nof the way and goes straight on without a word. Confound him, I say! You\nused to be able to swear beautifully, Frank--can't you manage to say\nsomething?\"\n\n\"At any other time--oh, yes! But you'd better get Ralston himself to do\nit for you. I'm not in it with him to-day. He's been giving me the life\nto come--hot--and Abraham and Isaac and Lazarus and the rich man, and\nthe glass of water, all in a breath. Go and ask him for what you want.\"\n\n\"Oh--then he is drunk, is he?\" asked Crowdie, with a disagreeable sneer\non his red lips.\n\n\"I suppose so,\" answered Miner, quite carelessly. \"At all events, he\nrefused to drink--that's always a bad sign with him.\"\n\n\"Of course--that makes it a certainty. Gad, though! It doesn't make him\nlight on his feet, if he happens to tread on yours. It serves me right\nfor coming to the club at this time of day! Perdition on the fellow!\nI've got on new shoes, too!\"\n\n\"What are you two squabbling about?\" enquired Hamilton Bright, coming\nsuddenly upon them out of the cloak-room.\n\n\"We're not squabbling--we're cursing Ralston,\" answered Miner.\n\n\"I wish you'd go and look after him, Ham,\" said Crowdie to his\nbrother-in-law. \"He's just gone off there. He's as drunk as the dickens,\nand swearing against everybody and treading on their toes in the most\ninsolent way imaginable. Get him out of this, can't you? Take him\nhome--you're his friend. If you don't he'll be smashing things before\nlong.\"\n\n\"Is he as bad as that, Frank?\" asked Bright, gravely. \"Where is he?\"\n\n\"At the telephone--I don't know--he trod on Crowdie's feet and Crowdie's\nperfectly wild and exaggerates. But there's something wrong, I know. I\nthink he's not exactly screwed--but he's screwed up--well, several pegs,\nby the way he acts. They call drinks 'pegs' somewhere, don't they? I\nwanted to make a joke. I thought it might do Crowdie good--\"\n\n\"Well, it's a very bad one,\" said Bright. \"He's at the telephone, you\nsay?\"\n\n\"Yes. The man said Mr. Lauderdale wanted to speak to him--he didn't know\nwhich Mr. Lauderdale--but it's probably Alexander the Safe, and if it\nis, there's going to be a row over the wires. When Jack's shut up there\nalone in the dark in the sound-proof box with the receiver under his\nnose and Alexander at the other end--if the wires don't melt--that's\nall! And Alexander's a metallic sort of man--I should think he'd draw\nthe lightning right down to his toes.\"\n\nAt that moment Ralston came swinging down the hall at a great pace, pale\nand evidently under some sort of powerful excitement. He nodded\ncarelessly to the three men as they stood together and disappeared into\nthe cloak-room. Bright followed him, but Ralston, with his hat on, his\nhead down and struggling into his overcoat, rushed out as Bright reached\nthe door, and ran into the latter, precisely as he had run into Crowdie.\nBright was by far the heavier man, however, and Ralston stumbled at the\nshock. Bright caught him by one arm and held him a moment.\n\n\"All right, Ham!\" he exclaimed. \"Everybody gets into my way to-day. Let\ngo, man! I'm in a hurry!\"\n\n\"Wait a bit,\" said Bright. \"I'll come with you--\"\n\n\"No--you can't. Let me go, Ham! What the deuce are you holding me for?\"\n\nHe shook Bright's arm angrily, for between the message he had received\nand the obstacles he seemed to meet at every step, he was, by this time,\nvery much excited. Bright thought he read certain well-known signs in\nhis face, and believed that he had been drinking hard and might get into\ntrouble if he went out alone, for Ralston was extremely quarrelsome at\nsuch times, and was quite capable of hitting out on the slightest\nprovocation, and had been in trouble more than once for doing so, as\nBright was well aware.\n\n\"I'm going with you, Jack, whether you like it or not,\" said the latter,\nwith mistaken firmness in his good intentions.\n\n\"You're not, I can tell you!\" answered Ralston, in a lower tone. \"Just\nlet me go--or there'll be trouble here.\"\n\nHe was furious at the delay, but Bright's powerful hand did not relax\nits grasp on his arm.\n\n\"Jack, old man,\" said Bright, in a coaxing tone, \"just come upstairs for\na quarter of an hour, and get quiet--\"\n\n\"Oh--that's it, is it? You think I'm screwed. I'm not. Let me\ngo--once--twice--\"\n\nRalston's face was now white with anger. The\n\n[Illustration: \"Before he could even raise his head, Ralston was out of\nthe door and in the street.\"--Vol. II., p. 57.]\n\nunjust accusation was the last drop. He was growing dangerous, but\nBright, in the pride of his superior strength, still held him firmly.\n\n\"Take care!\" said Ralston, almost in a whisper. \"I've counted two.\" He\npaused a full two seconds. \"Three! There you go!\"\n\nThe other men saw his foot glide forward like lightning over the marble\npavement. Instantly Bright was thrown heavily on his back, and before he\ncould even raise his head, Ralston was out of the door and in the\nstreet. Crowdie and Miner ran forward to help the fallen man, as they\nhad not moved from where they had stood, a dozen paces away. But Bright\nwas on his feet in an instant, pale with anger and with the severe shock\nof his fall. He turned his back on his companions at once, pretending to\nbrush the dust from his coat by the bright light which fell through the\nglass door. Frank Miner stood near him, very quiet, his hands in his\npockets, as usual, and a puzzled look in his face.\n\n\"Look here, Bright,\" he said gravely, watching Bright's back. \"This sort\nof thing can't go on, you know.\"\n\nBright said nothing, but continued to dust himself, though there was not\nthe least mark on his clothes.\n\n\"Upon my word,\" observed Crowdie, walking slowly up and down in his\nungraceful way, \"I think we'd better call a meeting at once and have\nhim requested to take his name off. If that isn't conduct unbecoming a\ngentleman, I don't know what is.\"\n\n\"No,\" said Miner. \"That wouldn't do. It would stick to him for life. All\nthe same, Bright, this is a club--it isn't a circus--and this sort of\nhorse-play is just a little too much. Why don't you turn round? There's\nno dust on you--they keep the floor of the arena swept on purpose when\nRalston's about. But it's got to stop--it's got to stop right here.\"\n\nBright's big shoulders squared themselves all at once and he faced\nabout, apparently quite cool again.\n\n\"I say,\" he began, \"did anybody see that but you two?\" He looked up and\ndown the deserted hall.\n\n\"No--wait a bit, though--halloa! Where are the hall servants? There\nought to be two of them. They must have just gone off. There they are,\non the other side of the staircase. Robert! And you--whatever your name\nis--come here!\"\n\nThe two servants came forward at once. They had retired to show their\ndiscretion and at the same time to observe what happened, the moment\nthey had seen Bright catch Ralston's arm.\n\n\"Look here,\" said Bright to them. \"If you say anything about what you\nsaw just now, you'll have to go. Do you understand? As we shan't speak\nof it, we shall know that you have, if it's talked about. That's all\nright--you can go now. I just wanted you to understand.\"\n\nThe two servants bowed gravely. They respected Bright, and, like all\nservants, they worshiped Ralston. There was little fear of their\nindiscretion. Bright turned to Crowdie and Miner.\n\n\"If anybody has anything to say about this, I have,\" he said. \"I'm the\ninjured person if any one is. And of course I shall say nothing, and\nI'll beg you to say nothing either. Of course, if he ever falls foul of\nyou, you're free to do as you please, and of course you might, if you\nchose, bring this thing before the committee. But I know you won't speak\nof it--either of you. We've all been screwed once or twice in our lives,\nI suppose. As for me, I'm his friend, and he didn't know what he was\ndoing. He's a deuced good fellow at heart, but he's infernally hasty\nwhen he's had too much. That's all right, isn't it? I can trust you,\ncan't I?\"\n\n\"Oh, yes, as far as I'm concerned,\" said Crowdie, speaking first. \"If\nyou like that sort of thing, I've nothing to say. You're quite big\nenough to take care of yourself. I hope Hester won't hear it. She\nwouldn't like the idea of her brother being knocked about without\ndefending himself. I don't particularly like it myself.\"\n\n\"That's nonsense, Walter, and you know it is,\" answered Bright, curtly,\nand he turned to Miner with a look of enquiry.\n\n\"All right, Ham!\" said the little man. \"I'm not going to tell tales, if\nyou aren't. All the same--I don't want to seem squeamish, and\nold-maid-ish, and a frump generally--but I don't think I do remember\njust such a thing happening in any club I ever belonged to. Oh, well!\nDon't let's stand here talking ourselves black in the face. He's gone,\nthis time, and he'll never find his way back if he once gets round the\ncorner. You'll hear to-morrow that he's been polishing Tiffany's best\nwindow with a policeman. That's about his pressure when he gets a\nregular jag on. As for me, I've been trying to get somebody to have a\ndrink with me for just three quarters of an hour, and so far my\ninvitations have come back unopened. I suppose you won't refuse a\npilot's two fingers after the battle, Ham?\"\n\n\"What's a pilot's two fingers?\" asked Bright. \"I'll accept your\nhospitality to that modest extent, anyhow. Show us.\"\n\n\"It's this,\" said Miner, holding up his hand with the forefinger and\nlittle finger extended and the others turned in. \"The little finger is\nthe bottom,\" he explained, \"and you don't count the others till you get\nto the forefinger, and just a little above the top of that you can see\nthe whiskey. Understand? What will you have, Crowdie?\"\n\n\"A drop of maraschino, thanks,\" said the painter.\n\n\"Maraschino!\" Miner made a wry face at the thought of the sugary stuff.\n\"All right then, come in!\"\n\nThey all went back together into the room in which Ralston and Miner had\nbeen sitting before the trouble began. Crowdie and his brother-in-law\nwere not on very good terms. The former behaved well enough when they\nmet, but Bright's dislike for him was not to be concealed--which was\nstrange, considering that Bright was a sensible and particularly\nself-possessed man, who was generally said to be of a gentle\ndisposition, inclined to live harmoniously with his surroundings. He\nsoon went away, leaving the artist and the man of letters to themselves.\nMiner did not like Crowdie very much either, but he admired him as an\nartist and had the faculty of making him talk.\n\nIf Ralston had really been drinking, he could not have been in a more\nexcited state than when he left the club, leaving his best friend\nstretched on his back in the hall. He was half conscious of having done\nsomething which would be considered wholly outrageous among his\nassociates, and among gentlemen at large. The fact that Bright was his\ndistant cousin was hardly an excuse for tripping him up even in jest,\nand if the matter were to be taken in earnest, Bright's superior\nstrength would not excuse Ralston for using his own far superior skill\nand quickness, in the most brutal way, and on rather slender\nprovocation. No one but he himself, however, even knew that he had been\nmaking a great effort to cure himself of a bad habit, and that although\nit was now Thursday, he had taken nothing stronger than a little weak\nwine and water and an occasional cup of coffee since Monday afternoon.\nBright could therefore have no idea of the extent to which his\naccusation had wounded and exasperated the sensitive man--rendered ten\ntimes more sensitive than usual by his unwonted abstention.\n\nRalston, however, did not enter into any such elaborate consideration of\nthe matter as he hurried along, too much excited just then to stop and\nlook for a cab. He was still whole-heartedly angry with Bright, and was\nglad that he had thrown him, be the consequences what they might. If\nBright would apologize for having laid rough hands on him, Ralston would\ndo as much--not otherwise. If the thing were mentioned, he would leave\nthe club and frequent another to which he belonged. Nothing could be\nsimpler.\n\nBut he had received a much more violent impression than he fancied, and\nhe forgot many things--forgetting even for a moment where he was going.\nPassing an up-town hotel on his way, he entered the bar by sheer force\nof habit--the habit of drinking something whenever his nerves were not\nquite steady. He ordered some whiskey, still thinking of Bright, and it\nwas not until he had swallowed half of it that he realized what he was\ndoing. With a half-suppressed oath he set down the liquor unfinished,\ndropped his money on the metal table and went out, more angry than ever.\n\nRealizing that he was not exactly in a condition to talk quietly to any\none, he turned into a side street, lit a strong cigar and walked more\nslowly for a few minutes, trying to collect his thoughts, and at last\nsucceeding to a certain extent, aided perhaps by the tonic effect of the\nspoonful of alcohol he had swallowed.\n\nThe whole thing had begun in a very simple way--the gradual increase of\ntension from the early morning until towards evening had been produced\nby small incidents following upon the hasty marriage ceremony, which, as\nhas been said, had produced a far deeper impression upon him than upon\nKatharine herself. The endless hours of waiting, the solitary luncheon,\nthe waiting again, Katharine's summary dismissal of him, almost without\na word of explanation--then more waiting, and Miner's tiresome\nquestions, and the sudden call to the telephone, and stumbling against\nCrowdie--and all the rest of it. Small things, all of them, after the\nmarriage itself, but able to produce at least a fit of extremely bad\ntemper by their cumulative action upon such a character. Ralston was\nundoubtedly a dangerous man to exasperate at five o'clock on that\nThursday afternoon.\n\nHe had been summoned by Robert Lauderdale himself, and this had\ncontributed not a little to the haste which had brought him into\ncollision with Bright. The old gentleman had asked him to come up to his\nhouse at once; John had said that he would come immediately, but on\nasking a further question he found the communication closed.\n\nIt immediately struck him that Katharine had not found uncle Robert at\nhome in the morning, that she had very possibly gone to him again in the\nafternoon, and that they were perhaps together at that very moment, and\nhad agreed to send for Ralston in order to talk matters over. It was\nnatural enough, considering his strong desire to see Katharine before\nthe ball, and his anxiety to hear Robert Lauderdale's definite answer,\nupon which depended everything in the immediate present and future, that\nhe should not have cared to waste time in exchanging civilities in the\nhall of the club with Bright, whom he saw almost every day, or with\nCrowdie, whom he detested. The rest has been explained.\n\nNor was it at all unnatural that the three men should all have been\nsimultaneously deceived into believing that he had been drinking more\nthan was good for him. A man who is known to drink habitually can\nhardly get credit for being sober when he is perfectly quiet--never,\nwhen he is in the least excited. Ralston had been more than excited. He\nhad been violent. He had disgraced himself and the club by a piece of\noutrageous brutality. If any one but Bright had suffered by it, there\nwould have been a meeting of the committee within twenty-four hours, and\nJohn Ralston's name would have disappeared from the list of members\nforever. It was fortunate for him that Bright chanced to be his best\nfriend.\n\nRalston scarcely realized how strongly the man was attached to him.\nEmbittered as he was by being constantly regarded as the failure of the\nfamily, he could hardly believe that any one but his mother and\nKatharine cared what became of him. A young man who has wasted three or\nfour years in fruitless, if not very terrible, dissipation, whose nerves\nare a trifle affected by habits as yet by no means incurable, and who\nhas had the word 'failure' daily branded upon him by his discriminating\nrelatives, easily believes that for him life is over, and that he can\nnever redeem the time lost--for he is constantly reminded of this by\npersons who should know better. And if he is somewhat melancholic by\nnature, he is very ready to think that the future holds but two\npossibilities,--the love of woman so long as it may last, and an easy\ndeath of some sort when there is no more love. That was approximately\nJohn Ralston's state of mind as he ascended the steps of Robert\nLauderdale's house on that Thursday afternoon.\n\n\n\n\nCHAPTER XIX.\n\n\nRalston shook himself and stamped his feet softly upon the rug as he\ntook off his overcoat in the hall of Robert Lauderdale's house. He was\nconscious that he was nervous and tried to restore the balance of forces\nby a physical effort, but he was not very successful. The man went\nbefore him and ushered him into the same room in which Katharine had\nbeen received that morning. The windows were already shut, and several\nshaded lamps shed a soft light upon the bookcases, the great desk and\nthe solid central figure of the great man. Ralston had not passed the\nthreshold before he was conscious that Katharine was not present, as he\nhad hoped that she might be. His excitement gave place once more to the\ncold sensation of something infinitely disappointing, as he took the old\ngentleman's hand and then sat down in a stiff, high-backed chair\nopposite to him--to be 'looked over,' he said to himself.\n\n\"So you're married,\" said Robert Lauderdale, abruptly opening the\nconversation.\n\n\"Then you've seen Katharine,\" answered the young man. \"I wasn't sure you\nhad.\"\n\n\"Hasn't she told you?\"\n\n\"No. I was to have seen her this afternoon, but--she couldn't do more\nthan tell me that she would talk it all over this evening.\"\n\n\"Oh!\" ejaculated the old man. \"That rather alters the case.\"\n\n\"How?\" enquired Ralston, whose bad temper made him instinctively choose\nto understand as little as possible of what was said.\n\n\"Well, in this way, my dear boy. Katharine and I had a long interview\nthis morning, and as I supposed you must have met before now, I\nnaturally thought she had explained things to you.\"\n\n\"What things?\" asked Ralston, doggedly.\n\n\"Oh, well! If I've got to go through the whole affair again--\" The old\nman stopped abruptly and tapped the table with his big fingers, looking\nacross the room at one of the lamps.\n\n\"I don't think that will be necessary,\" said Ralston. \"If you'll tell me\nwhy you sent for me that will be quite enough.\"\n\nRobert Lauderdale looked at him in some surprise, for the tone of his\nvoice sounded unaccountably hostile.\n\n\"I didn't ask you to come for the sake of quarrelling with you, Jack,\"\nhe replied.\n\n\"No. I didn't suppose so.\"\n\n\"But you seem to be in a confoundedly bad temper all the same,\" observed\nthe old gentleman, and his bushy eyebrows moved oddly above his bright\nold eyes.\n\n\"Am I? I didn't know it.\" Ralston sat very quietly in his chair, holding\nhis hat on his knees, but looking steadily at Mr. Lauderdale.\n\nThe latter suddenly sniffed the air discontentedly, and frowned.\n\n\"It's those abominable cocktails you're always drinking, Jack,\" he said.\n\n\"I've not been drinking any,\" answered Ralston, momentarily forgetting\nthe forgetfulness which had so angered him ten minutes earlier.\n\n\"Nonsense!\" cried the old man, angrily. \"Do you think that I'm in my\ndotage, Jack? It's whiskey. I can smell it!\"\n\n\"Oh!\" Ralston paused. \"It's true--on my way here, I began to drink\nsomething and then put it down.\"\n\n\"Hm!\" Robert Lauderdale snorted and looked at him. \"It's none of my\nbusiness how many cocktails you drink, I suppose--and it's natural that\nyou should wish to celebrate the wedding day. Might drink wine, though,\nlike a gentleman,\" he added audibly.\n\nAgain Ralston felt that sharp thrust of pain which a man feels under a\nwholly unjust accusation brought against him when he has been doing his\nbest and has more than partially succeeded. The fiery temper--barely\nunder control when he had entered the house--broke out again.\n\n\"If you've sent for me to lecture me on my habits, I shall go,\" he said,\nmoving as though about to rise.\n\n\"I didn't,\" answered the old gentleman, with flashing eyes. \"I asked you\nto come here on a matter of business--and you've come smelling of\nwhiskey and flying into a passion at everything I say--and I tell\nyou--pah! I can smell it here!\"\n\nHe took a cigar from the table and lit it hastily. Meanwhile Ralston\nrose to his feet. He evidently had no intention of quarrelling with his\nuncle unnecessarily, but the repeated insult stung him past endurance.\nThe old man looked up, with the cigar between his teeth, and still\nholding the match at the end of it. With the other hand he took a bit of\npaper from the table and held it out towards Ralston.\n\n\"That's what I sent for you about,\" he said.\n\nRalston turned suddenly and faced him.\n\n\"What is it?\" he asked sharply.\n\n\"Take it, and see.\"\n\n\"If it's money, I won't touch it,\" Ralston answered, beginning to grow\npale, for he saw that it was a cheque, and it seemed just then like a\nworse insult than the first.\n\n\"It's not for you. It's a matter of business. Take it!\"\n\nRalston shifted his hat into his left hand and took the cheque in his\nright, and glanced at it. It was drawn in favour of Katharine Lauderdale\nfor one hundred thousand dollars. He laughed in the old man's face,\nbeing very angry.\n\n\"It's a curiosity, at all events,\" he said with contempt, laying it on\nthe table.\n\n\"What do you mean?\" cried his uncle, growing redder as Ralston turned\nwhite.\n\n\"There is no Katharine Lauderdale, in the first place,\" answered the\nyoung man. \"The thing isn't worth the paper it's written on. If it were\nworth money, I'd tear it up--if it were for a million.\"\n\n\"Oh--would you?\" The old gentleman looked at Ralston with a sort of\nfierce, contemptuous unbelief.\n\n\"Yes--I would. So would Katharine. I daresay she told you so.\"\n\nRobert Lauderdale bit his cigar savagely. It was a little too much to be\nbrowbeaten by a mere boy, when he had been used to commanding all his\nlife. Whether he understood Ralston, or whether he completely lost his\nhead, was never clear to either of them, then, or afterwards. He took a\nfresh cheque and filled it in carefully. His face was scarlet now, and\nhis sandy eyebrows were knitted angrily together. When he had done, he\nscrutinized the order closely, and then laid it upon the end of the desk\nunder Ralston's eyes.\n\n'Pay to the order of John Ralston one million dollars, Robert\nLauderdale.'\n\nRalston glanced at the writing without touching the paper, and\ninvoluntarily his eyes were fascinated by it for a moment. There was\nnothing wrong about the cheque this time.\n\nIn the instant during which he looked at it, as it lay there, the\ntemptation to take it was hardly perceptible to him. He knew it was\nreal, and yet it did not look real. In the progress of his increasing\nanger there was a momentary pause. The exceeding magnitude of the figure\narrested his attention and diverted his thoughts. He had never seen a\ncheque for a million of dollars before, and he could not help looking at\nit, for its own sake.\n\n\"That's a curiosity, too,\" he said, almost unconsciously. \"I never saw\none.\"\n\nA moment later he set down his hat, took the slip of paper and tore it\nacross, doubled it and tore it again, and mechanically looked for the\nwaste-paper basket. Robert Lauderdale watched him, not without an\nanxiety of which he was ashamed, for he had realized the stupendous risk\ninto which his anger had led him as soon as he had laid the cheque on\nthe desk, but had been too proud to take it back. He would not have been\nRobert the Rich if he had often been tempted to such folly, but the\nyoung man's manner had exasperated him beyond measure.\n\n\"That was a million of dollars,\" he said, in an odd voice, as the shreds\nfell into the basket.\n\n\"I suppose so,\" answered Ralston, with a sneer, as he took his hat\nagain. \"You could have drawn it for fifty millions, I daresay, if you\nhad chosen. It's lucky you do that sort of thing in the family.\"\n\n\"You're either tipsy--or you're a better man than I took you for,\" said\nRobert Lauderdale, slowly regaining his composure.\n\n\"You've suggested already that I am probably drunk,\" answered Ralston,\nbrutally. \"I'll leave you to consider the matter. Good evening.\"\n\nHe went towards the door. Old Lauderdale looked after him a moment and\nthen rose, heavily, as big old men do.\n\n\"Jack! Come back! Don't be a fool, my boy!\"\n\n\"I'm not,\" replied the young man. \"The wisest thing I can do is to\ngo--and I'm going.\" He laid hold of the handle of the door. \"It's of no\nuse for me to stay,\" he said. \"We shall come to blows if this goes on.\"\n\nHis uncle came towards him as he stood there. Hamilton Bright was more\nlike him in size and figure than any of the other Lauderdales.\n\n\"I don't want you to go just yet, Jack,\" he said, more kindly than he\nhad spoken yet, and laying his hand on Ralston's arm very much as Bright\nhad done in the club.\n\nRalston shrank from his touch, not because he was in the least afraid\nof being violent with an old man, but because the mere thought of such a\nthing offended his sense of honour, and the position in which the two\nwere standing reminded him of what had happened but a short time\npreviously.\n\n\"Just tell me one thing, my dear boy,\" began Robert Lauderdale, whose\nshort fits of anger were always succeeded immediately by a burst of\nsunshiny good humour. \"I want to know what induced you to go and marry\nKatharine in that way?\"\n\nRalston drew back still further, trying to avoid his touch. It was\nutterly impossible for him to answer that he had very reluctantly\nyielded to Katharine's own entreaties. Nor was his anger by any means as\ntransient as the old man's.\n\n\"I entirely refuse to discuss the matter,\" he said, and paused. \"Do you\nwant a plain statement?\" he asked, a moment later. \"Very well. It was\nunderstood that Katharine was to tell you about the marriage, and she\nhas done so. You're the head of the family, and you have a right to\nknow. If I ever had any intention of asking anything of you, it\ncertainly wasn't money. And I've asked nothing. Possibly, just now, you\nmeant to be generous. It struck me in rather a different light. I\nthought it was pretty clear, in the first place, that you took me for\nthe sort of man who would be willing to live on his wife's money, if\nshe had any. If you meant to give her the money, there was no reason for\nputting the cheque into my hands--nor for writing a cheque at all. You\ncould, and you naturally should, have written a note to Beman to place\nthe sum to her credit. That was a mere comedy, to see what I would\ndo--to try me, as I suppose you said to yourself. Thank you. I never\noffered myself to be a subject for your experiments. As for the cheque\nfor a million--that was pure farce. You were so angry that you didn't\nknow what you were doing, and then your fright--yes, your fright--calmed\nyou again. But there's no harm done. You saw me throw it into the\nwaste-paper basket. That's all, I think. As you seem to think I'm not\nsober, you may as well let me take myself off. But if I'm drunk--well,\ndon't try any of those silly experiments on men who aren't. You'll get\ncaught, and a million is rather a high price to pay for seeing a man's\nexpression of face change. Good night--let me go, please.\"\n\nDuring this long tirade Robert Lauderdale had walked up and down before\nhim with short, heavy steps, uttering occasional ejaculations, but at\nthe last words he took hold of Ralston's arm again--rather roughly this\ntime.\n\n\"You're an insolent young vagabond!\" he cried, breaking into a fresh fit\nof anger. \"You're insulting me in my own house.\"\n\n\"You've been insulting me in your own house for the last quarter of an\nhour,\" retorted Ralston.\n\n\"And you're throwing away the last chance you'll ever get from me--\"\n\n\"It wasn't much of a chance--for a gentleman,\" sneered the young man,\ninterrupting him.\n\n\"Confound it! Can't you let me speak? I say--\" He hesitated, losing the\nthread of his intended speech in his anger.\n\n\"You don't seem to have anything especial to say, except in the way of\nabuse, and there's no reason at all why I should listen to that sort of\nthing. I'm not your son, and I'm not your butler--I'm thankful I'm not\nyour dog!\"\n\n\"John!\" roared the old man, shaking him by the arm. \"Be silent, sir! I\nwon't submit to such language!\"\n\n\"What right have you to tell me what I shall submit to, or not submit\nto? Because you're a sort of distant relation, I suppose, and have got\ninto the habit of lording it over the whole tribe--who would lick the\nheels of your boots for your money--every one of them, except my mother\nand Katharine and me. Don't tell me what I'm to submit to--\"\n\n\"I didn't say you!\" shouted old Lauderdale. \"I said that I wouldn't hear\nsuch language from you--you're drunk, John Ralston--you're mad drunk.\"\n\n\"Then you'll have to listen to my ravings just as long as you force me\nto stay under your roof,\" answered Ralston, almost trembling with rage.\n\"If you keep me here, I shall tell you just what I think of you--\"\n\n\"By the Eternal--this is too much--you young--puppy! You graceless,\nungrateful--\"\n\n\"I should really like to know what I'm to be grateful to you for,\" said\nRalston, feeling that his hands were growing icy cold. \"You've never\ndone anything for me or mine in your life--as you know. You'd much\nbetter let me go. You'll regret it if you don't.\"\n\n\"And you dare to threaten me, too--I tell you--I'll make you--\" His\nwords choked him, and again he shook Ralston's arm violently.\n\n\"You won't make me forget that you're three times my age, at all\nevents,\" answered the young man. \"But unless you're very careful during\nthe next ten minutes you'll have a fit of apoplexy. You'd much better\nlet me go away. This sort of thing isn't good for a man of your age--and\nit's not particularly dignified either. You'd realize it if you could\nsee yourself and hear yourself--oh! take care, please! That's my hat.\"\n\nRobert Lauderdale's fury had boiled over at last and expressed itself in\na very violent gesture, not intended for a blow, but very like one, and\nutterly destructive to Ralston's hat, which rolled shapeless upon the\npolished wooden floor. The young man stooped as he spoke the last words,\nand picked it up.\n\n\"Oh, I say, Jack! I didn't mean to do that, my boy!\" said the old\ngentleman, with that absurdly foolish change of tone which generally\ncomes into the voice when one in anger has accidentally broken\nsomething.\n\n\"No--I daresay not,\" answered Ralston, coldly.\n\nWithout so much as a glance at old Lauderdale, he quickly opened the\ndoor and left the room, as he would have done some minutes earlier if\nhis uncle had not held him by the arm. The library was downstairs, and\nhe was out of the house before Lauderdale had sufficiently recovered\nfrom his surprise to call him back.\n\nThat, indeed, would have been quite useless, for Ralston would not have\nturned his head. He had never been able to understand how a man could be\nin a passion at one moment and brimming with good nature at the next,\nfor his own moods were enduring, passionate and brooding.\n\nIt had all been very serious to him, much more so than to the old\ngentleman, though the latter had been by far the more noisy of the two\nin his anger. If he had been able to reflect, he might have soon come to\nthe conclusion that the violent scene had been the result of a\nmisunderstanding, in the first instance, and secondly, of Robert\nLauderdale's lack of wisdom in trying to make him take money for\nKatharine. In the course of time he would have condoned the latter\noffence and forgiven the former, but just now both seemed very hard to\nbear.\n\nAfter being exceptionally abstemious,--and he alone knew at what a cost\nin the way of constant self-control,--he had been accused twice within\nan hour of being drunk. And as though that were not enough, with all the\nother matters which had combined to affect his temper on that day,\nRobert Lauderdale had first tried to make him act dishonourably, as\nRalston thought, or at least in an unmanly way, and had then tried to\nmake a fool of him with the cheque for a million. He almost wished that\nhe could have kept the latter twenty-four hours for the sake of\nfrightening the old man into his senses. It would have been a fair act\nof retaliation, he thought, though he would not in reality have stooped\nto do it.\n\nIt was quite dark when he came out upon Fifty-ninth Street, and the\nweather was foggy and threatening, though it was not cold. He had\nforgotten his overcoat in his hurry to get away, and did not notice even\nnow that he was without it. Half mechanically he had pushed his high hat\ninto some sort of shape and put it on, and had already forgotten that it\nwas not in its normal condition. His face was very pale, and his eyes\nwere bright. Without thinking of the direction he was taking, he turned\ninto Fifth Avenue by force of habit. As he walked along, several men who\nknew him passed him, walking up from their clubs to dress for dinner.\nThey most of them nodded, smiled rather oddly and went on. He noticed\nnothing strange in their behaviour, being very much absorbed in his own\nunpleasant reflections, but most of them were under the impression, from\nthe glimpse they had of him under the vivid electric light, that he was\nvery much the worse for drink, and that he had lost his overcoat and had\nhis hat smashed in some encounter with a rough or roughs unknown. One or\ntwo of his rows had remained famous. But he was well known, too, for his\npower of walking straight and of taking care of himself, even when he\nwas very far gone, and nobody who met him ventured to offer him any\nassistance. On the other hand, no one would have believed that he was\nperfectly sober, and that his hat had been destroyed by no less a person\nthan the great Robert Lauderdale himself.\n\nHe certainly deserved much more pity than he got that day. But good and\nbad luck run in streaks, as the winds blow across land-locked waters,\nand it is not easy to get across from one to the other. Ralston was\ndrifting in a current of circumstances from which he could not escape,\nbeing what he was, a man with an irritable temper, more inclined to\nresent the present than to prepare the future. Presently he turned\neastwards out of Fifth Avenue. He remembered afterwards that it must\nhave been somewhere near Forty-second Street, for he had a definite\nimpression of having lately passed the great black wall of the old\nreservoir. He did not know why he turned just there, and he was probably\nimpelled to do so by some slight hindrance at the crossing he had\nreached. At all events, he was sure of having walked at least a mile\nsince he had left Robert Lauderdale's house.\n\nThe cross street was very dark compared with the Avenue he had left. He\nstopped to light a cigar, in the vague hope that it might help him to\nthink, for he knew very well that he must go home before long and dress\nfor a dinner party, and then go on to the great Assembly ball at which\nhe was to meet Katharine. It struck him as he thought of the meeting\nthat he would have much more to tell her about their uncle Robert than\nshe could possibly have to relate of her own experience. He lit his\ncigar very carefully. Anger had to some extent the effect of making him\ndeliberate and precise in his small actions. He held the lighted taper\nto the end of his cigar several seconds, and then dropped it. It had\ndazzled him, so that for the moment the street seemed to be quite black\nin front of him. He walked on boldly, suspecting nothing, and a moment\nlater he fell to his full length upon a heap of building material piled\nupon the pavement.\n\nIt is worth remarking, for the sake of those who take an interest in\ntracing the relations of cause and effect, that this was the first, the\nlast and the only real accident which happened to John Ralston on that\nday, and it was not a very serious one, nor, unfortunately, a very\nunfrequent one in the streets of New York. But it happened to him, as\nsmall accidents so often do, at an hour which gave it an especial\nimportance.\n\nHe lay stunned as he had fallen for more than a minute, and when he came\nto himself he discovered that he had struck his head. The brim of his\nalready much injured hat had saved him from a wound; but the blow had\nbeen a violent one, and though he got upon his feet almost immediately\nand assured himself that he was not really injured, yet, when he had got\nbeyond the obstacle over which he had stumbled, he found it impossible\nto recollect which way he should go in order to get home. The slight\nconcussion of the brain had temporarily disturbed the sense of\ndirection, a phenomenon not at all uncommon after receiving a violent\nblow on the head, as many hard riders and hunting men are well aware.\nBut it was new to Ralston, and he began to think that he was losing his\nmind. He stopped under a gas-lamp and looked at his watch, by way of\ntesting his sanity. It was half past six, and the watch was going. He\nimmediately began a mental calculation to ascertain whether he had been\nunconscious for any length of time. He remembered that it had been after\nfive o'clock when he had been called to the telephone at the club. His\nstruggle with Bright had kept him some minutes longer, he had walked to\nRobert Lauderdale's, and his interview had lasted nearly half an hour,\nand on recalling what he had done since then he had that distinct\nimpression of having lately seen the reservoir, of which mention has\nalready been made.\n\nHe walked on like a man in a dream, and more than half believing that he\nwas really dreaming. He was going eastwards, as he had been going when\nhe had entered the street, but he found it impossible to understand\nwhich way his face was turned. He came to Madison Avenue, and knew it at\nonce, recognizing the houses, but though he stood still several minutes\nat the corner, he could not distinguish which was up town and which down\ntown. He believed that if he could have seen the stars he could have\nfound his way, but the familiar buildings, recognizable in all their\nfeatures to his practised eye even in the uncertain gaslight, conveyed\nto him no idea of direction, and the sky was overcast. In despair, at\nlast, he continued in the direction in which he had been going. If he\nwas crossing the avenue he must surely strike the water, whether he\nwent forwards or backwards, and he was positive that he should know the\nEast River from the North River, even on the darkest night, by the look\nof the piers. But to all intents and purposes, though he knew where he\nwas, he was lost, being deprived of the sense of direction.\n\nThe confusion increased with the darkness of the next street he\ntraversed, and to his surprise the avenue beyond that did not seem\nfamiliar. It was Park Avenue where it is tunnelled along its length for\nthe horse-cars which go to the Central Station. It was very dark, but in\na moment he again recognized the houses. By sheer instinct he turned to\nthe right, trusting to luck and giving up all hope of finding his way by\nany process of reasoning. The darkness, the blow he had received when he\nhad fallen and all that had gone before, combined with the cold he felt,\ndeadened his senses still more.\n\nHe noticed for the first time that his overcoat was gone, and he\nwondered vaguely whether it had been stolen from him when he had fallen.\nIn that case he must have been unconscious longer than he had imagined.\nHe felt for his watch, though he had looked at it a few moments\npreviously. It was in his pocket as well as his pocket-book and some\nsmall change. He felt comforted at finding that he had money about him,\nand wished he might come across a stray cab. Several passed him, but he\ncould see by the lamplight that there were people in them, dressed for\ndinner. It was growing late, since they were already going to their\ndinner-parties. He felt very cold, and suddenly the flakes of snow began\nto fall thick and fast in his face. The weather had changed in half an\nhour, and a blizzard was coming. He shivered and trudged on, not knowing\nwhither. He walked faster and faster, as men generally do when they have\nlost their way, and he turned in many directions, losing himself more\ncompletely at every new attempt, yet walking ever more rapidly, pursued\nby the nervous consciousness that he should be dressing for dinner and\nthat there was no time to be lost. He did not feel dizzy nor weak, but\nhe was utterly confused, and began to be unconscious of the distance he\nwas traversing and of the time as it passed.\n\nAll at once he came upon a vast, dim square full of small trees. At\nfirst he thought he was in Gramercy Park, but the size of the place soon\ntold him that he was mistaken. By this time it was snowing heavily and\nthe pavements were already white. He pulled up the collar of his frock\ncoat and hid his right hand in the front of it, between the buttons,\nblowing into his left at the same time, for both were freezing. He\nstared up at the first corner gas-lamp he came to, and read without\ndifficulty the name in black letters. He was in Tompkins Square.\n\nHe had been there once or twice in his life, and had been struck by the\ngreat, quiet, open place, and he understood once more where he was, and\nlooked at his watch. It was nearly ten o'clock. He rubbed his eyes, and\nthen rubbed the snow-flakes off the glass, for they fell so fast that he\ncould not hold it to the light a moment before one of them fell into the\nopen case. He had been wandering for nearly three hours, dinnerless, in\nthe snow, and he suddenly felt numb and hungry and thirsty all at once.\nBut at the same time, as though by magic, the sense of locality and\ndirection returned. He put his watch into his pocket again, stamped the\nwet snow from his shoes and struck resolutely westward. He knew how\nhopeless it was to expect to find a carriage of any sort in that poor\nquarter of the city. Oddly enough, the first thing that struck him was\nthe absurdity of his own conduct in not once asking his way, for he was\ncertain that he had met many hundreds of people during those hours of\nwandering. He marched on through the snow, perfectly satisfied at having\nrecovered his senses, though he now for the first time felt a severe\npain in his head.\n\nBefore long he reached a horse-car track and waited for the car to come\nup, without the least hesitation as to its direction. He got on without\ndifficulty, though he noticed that the conductor looked at him keenly\nand seemed inclined to help him. He paid his five cents and sat down in\nthe corner away from the door. It was pleasantly warm by contrast with\nthe weather he had been facing for hours, and the straw under his feet\nseemed deliciously comfortable. He remembered being surprised at finding\nhimself so tired, and at the pain in his head. There was one other man\nin the car, who stood near the door talking with the conductor. He was a\nshort man, very broad in the shoulders and thick about the neck, but not\nat all fat, as Ralston noticed, being a judge of athletes. This man wore\nan overcoat with a superb sable collar, and a gorgeous gold chain was\nstretched across the broad expanse of his waistcoat. He was perfectly\nclean shaven, and looked as though he might be a successful prize\nfighter. At this point in his observation John Ralston fell asleep.\n\nHe had two more intervals of consciousness.\n\nHe had gone to sleep in the horse-car. He woke to find himself fighting\nthe man with the fur coat and the chain, out under the falling snow,\nwith half a dozen horse-car drivers and conductors making a ring, each\nwith a lantern. He thought he remembered seeing a red streak on the face\nof his adversary. A moment later he saw a vivid flash of light, and\nthen he was unconscious again.\n\nWhen he opened his eyes once more he looked into his mother's face, and\nhe saw an expression there which he never forgot as long as he lived.\n\n\n\n\nCHAPTER XX.\n\n\nKatharine looked in vain for Ralston near the door of the ball-room that\nnight, as she entered with her mother, passed up to curtsy to one of the\nladies whose turn it was to receive and slowly crossed the polished\nfloor to the other side. He was nowhere to be seen, and immediately she\nfelt a little chill of apprehension, as though something had warned her\nthat he was in trouble. The sensation was merely the result of her\ndisappointment. Hitherto, even to that very afternoon, he had always\nshown himself to be the most scrupulously exact and punctual man of her\nacquaintance, and it was natural enough that the fact of his not\nappearing at such an important juncture as the present should seem very\nstrange. Katharine, however, attributed what she felt to a presentiment\nof evil, and afterwards remembered it as though it had been something\nlike a supernatural warning.\n\nWhen she had assured herself that he was really not at the ball, her\nfirst impulse was to ask every one she met if he had been seen, and as\nthat was impossible, she looked about for some member of the family who\nmight enlighten her and of whom she might ask questions without\nexciting curiosity. It was not an easy matter, however, to find just\nsuch a person as should fulfil the requirements of the case. Hamilton\nBright or Frank Miner would have answered her purpose, and it was just\npossible that one or both of them might appear at a later hour, though\nneither of them were men who danced. Crowdie would come, of course, with\nhis wife, but she felt that she could not ask him questions about\nRalston, and Hester would hardly be likely to know anything of the\nlatter's movements.\n\nIt was quite out of the question for Katharine to sit in a quiet corner\nunder one of the galleries, and watch the door, as a cat watches the\nhole from which she expects a mouse to appear. She was too much\nsurrounded by the tribe of high-collared, broad-tied, smooth-faced,\nempty-headed, and very young men who, in an American ball-room, make it\nmore or less their business to inflict their company upon the most\nbeautiful young girl present at any one time. Older men would often be\nonly too glad to talk with her, and she would prefer them to her bevy of\nhalf-fledged admirers, but the older man naturally shrinks from\nintruding himself amongst a circle of very young people, and\nsystematically keeps away. On the whole, too, the young girls enjoy\nthemselves exceedingly well and do not complain of their following.\n\nAt last, however, Katharine determined to speak to her mother. She had\nseen the latter in close conversation with Crowdie. That was natural\nenough. Crowdie thought more of beauty than of any other gift, and if\nMrs. Lauderdale had been a doll, which she was not, he would always have\nspent half an hour with her if he could, merely for the sake of studying\nher face. She was very beautiful to-night, and there was no fear of a\nrepetition of the scene which had occurred by the fireplace in Clinton\nPlace on Monday night. It seemed as though she had recalled the dazzling\nfreshness of other days--not long past, it is true--by an act of will,\ndetermined to be supreme to the very end. She knew it, too. She was\nconscious that the lights were exactly what they should be, that the\ntemperature was perfect, that her gown could not fit her better and that\nshe had arrived feeling fresh and rested. Charlotte's visit had done her\ngood, also, for Charlotte had made herself very charming on that\nafternoon, as will be remembered by those who have had the patience to\nfollow the minor events of the long day. Even her husband had been more\nthan usually unbending and agreeable at dinner, and it was probably her\nappearance which had produced that effect on him. Like most very strong\nand masculine men, whatever be their characters, he was very really\naffected by woman's beauty. For some time he had silently regretted the\nchange in his wife's appearance, and this evening he had noticed the\nreturn of that brilliancy which had attracted him long ago. He had even\nkissed her before his daughter, when he had put on her cloak for her,\nwhich was a very rare occurrence. Crowdie had seen Mrs. Lauderdale as\nsoon as he had left Hester to her first partner and had been at liberty\nto wander after his own devices, and had immediately gone to her.\nKatharine had observed this, for she had good eyes and few things within\nher range of vision escaped her. Naturally enough, too, she had glanced\nat her mother more than once and had seen that the latter was evidently\nmuch interested by some story which Crowdie was telling. Her own mind\nbeing entirely occupied with Ralston, it was not surprising that she\nshould imagine that they were talking of him.\n\nShe watched her opportunity, and when Crowdie at last left her mother's\nside, went to her immediately. They were a wonderful pair as they stood\ntogether for a few moments, and many people watched them. Mrs.\nLauderdale, who was especially conscious of the admiration she was\nreceiving that night, felt so vain of herself that she did not attempt\nto avoid the comparison, but drew herself up proudly to her great height\nin the full view of every one, and as though remembering and repenting\nof the bitter envy she had felt of Katharine's youth even as lately as\nthe previous day, she looked down calmly and lovingly into the girl's\nface. Katharine was not in the least aware that any one was looking at\nthem, nor did she imagine any comparison possible between her mother and\nherself. Her faults of character certainly did not lie in the direction\nof personal vanity. Many people, too, thought that she was not looking\nher best, as the phrase goes, on that evening, while others said that\nshe had never looked as well before. She was transparently pale, with\nthat fresh pallor which is not unbecoming in youth and health when it is\nnatural, or the result of an emotion. The whiteness of her face made her\ndeep grey eyes seem larger and deeper than ever, and the broad, dark\neyebrows gave a look of power to the features, which was striking in one\nso young. Passion, anxiety, the alternations of hope and fear, even the\nsense of unwonted responsibility, may all enhance beauty when they are\nof short duration, though in time they must destroy it, or modify its\nnature, spiritualizing or materializing it, according to the objects and\nreasons from which they proceed. The beauty of Napoleon's death mask is\nvery different from that of Goethe's, yet both, perhaps, at widely\ndifferent ages, approached as nearly to perfection of feature as\nhumanity ever can.\n\n\"Well, child, have you come back to me?\" asked Mrs. Lauderdale, with a\nsmile.\n\nThere was nothing affected in her manner, for she had too long been\nfirst, yet she knew that her smile was not lost on others--she could\nfeel that the eyes of many were on her, and she had a right to be as\nhandsome as she could. Even Katharine was struck by the wonderful return\nof youth.\n\n\"You're perfectly beautiful to-night, mother!\" she exclaimed, in genuine\nadmiration.\n\nThere was something in the whole-hearted, spontaneous expression of\napproval from her own daughter which did more to assure the elder woman\nof her appearance than all Crowdie's compliments could have done.\nKatharine rarely said such things.\n\n\"You're not at all ugly yourself to-night, my dear!\" laughed Mrs.\nLauderdale. \"You're a little pale--but it's very becoming. What's the\nmatter? Are you out of breath? Have you been dancing too long?\"\n\n\"I didn't know that I was pale,\" answered Katharine. \"No, I'm not out of\nbreath--nor anything. I just came over to you because I saw you were\nalone for a moment. By the bye, mother, have you seen Jack anywhere?\"\n\nIt was not very well done, and it was quite clear that she had crossed\nthe big ball-room solely for the purpose of asking the question. Mrs.\nLauderdale hesitated an instant before giving any answer, and she had a\npuzzled expression.\n\n\"No,\" she said, at last. \"I've not seen him. I don't believe he's here.\nIn fact--\" she was a truthful woman--\"in fact, I'm quite sure he's not.\nDid you expect him?\"\n\n\"Of course,\" answered Katharine, in a low voice. \"He always comes.\"\n\nShe knew her mother's face very well, and was at once convinced that she\nhad been right in supposing that Crowdie had been speaking of Ralston.\nShe saw the painter at some distance, and tried to catch his glance and\nbring him to her, but he suddenly turned away and went off in the\nopposite direction. She reflected that Crowdie did not pass for a\ndiscreet or reticent person, and that if there were anything especial to\nbe told he had doubtless confided it to his wife before coming to the\nball. She looked about for Hester, but could not see her at first,\nneither could she discover Bright or Miner in the moving crowd. She\nstood quietly by her mother for a time, glad to escape momentarily from\nher usual retinue of beardless young dandies. Mrs. Lauderdale still\nseemed to hesitate as to whether she should say any more. The story\nCrowdie had told her was a very strange one, she thought, and she\nherself doubted the accuracy of the details. And he had exacted a sort\nof promise of secrecy from her, which, in her experience, very generally\nmeant that a part, or the whole of what was told, might be untrue.\nNevertheless, she had never thought that the painter was a spiteful\nperson. She was puzzled, therefore, but she very soon resolved that she\nshould tell Katharine nothing, which was, after all, the wisest plan.\n\nJust then a tall, lean man made his way up to her and bowed rather\nstiffly. He was powerfully made, and moved like a person more accustomed\nto motion than to rest. He had a weather-beaten, kindly face, clean\nshaven, thin and bony. His features were decidedly ugly, though by no\nmeans repulsive. His hair was thick and iron grey, and he was about\nfifty years of age. Mrs. Lauderdale gave him her hand, and seemed glad\nto see him.\n\n\"Mr. Griggs--my daughter,\" she said, introducing him to Katharine, who\nhad immediately recognized him, for she had seen him at a distance on\nthe previous evening at the Thirlwalls' dance.\n\nPaul Griggs bowed again in his stiff, rather foreign way, and Katharine\nsmiled and bent her head a little. She had always wished she might meet\nhim, for she had read some of his books and liked them, and he was\nreported to have led a very strange life, and to have been everywhere.\n\n\"I saw you talking to Mrs. Crowdie,\" said Mrs. Lauderdale. \"She's\ncharming, isn't she?\"\n\n\"Very,\" answered Mr. Griggs, in a deep, manly voice, but without any\nspecial emphasis. \"Very,\" he repeated vaguely. \"She was a mere girl--not\nout yet--when I was last at home,\" he added, suddenly showing some\ninterest.\n\n\"By the bye, where is she?\" asked Katharine, in the momentary pause\nwhich followed. \"I was looking for her.\"\n\n\"Over there,\" replied Mr. Griggs, nodding almost imperceptibly in the\ndirection he meant to indicate. As he was over six feet in height, and\ncould see over the heads of most of the people, Katharine had not gained\nany very accurate information.\n\n\"You can see her,\" he continued in explanation. \"She's sitting up among\nthe frumps; she's looking for her husband, and there's a man with yellow\nhair talking to her--it's her brother--over there between the first and\nsecond windows from the end where the music is. Do you make her out?\"\n\n\"Yes. How can you tell that she is looking for her husband at this\ndistance?\" Katharine laughed.\n\n\"By her eyes,\" answered Mr. Griggs. \"She's in love with him, you\nknow--and she's anxious about him for some reason or other. But I\nbelieve he's all right now. I used to know him very well in Paris once\nupon a time. Clever fellow, but he had--oh, well, it's nobody's\nbusiness. What a beautiful ball it is, Mrs. Lauderdale--\"\n\n\"What did Mr. Crowdie have in Paris?\" asked Katharine, with sudden\ninterest, and interrupting him.\n\n\"Oh--he was subject to bad colds in winter,\" answered Mr. Griggs,\ncoolly. \"Lungs affected, I believe--or something of that sort. As I was\nsaying, Mrs. Lauderdale, this is a vast improvement on the dances they\nused to have in New York when I was young. That was long before your\ntime, though I daresay your husband can remember them.\"\n\nAnd he went on speaking, evidently making conversation of a most\nunprofitable kind in the most cold-blooded and cynical manner, by sheer\nforce of habit, as people who have the manners of the world without its\ninterests often do, until something strikes them.\n\nA young man, whose small head seemed to have just been squeezed through\nthe cylinder of enamelled linen on which it rested as on a pedestal,\ncame up to Katharine and asked her for a dance. She went away on his\narm. After a couple of turns, she made him stop close to Hester Crowdie.\n\n\"Thanks,\" she said, nodding to her partner. \"I want to speak to my\ncousin. You don't mind--do you? I'll give you the rest of the dance some\nother time.\"\n\nAnd without waiting for his answer, she stepped upon the low platform\nwhich ran round the ball-room, and took the vacant seat by Hester's\nside. Hamilton Bright, who had only been exchanging a word with his\nsister when Griggs had caught sight of him, was gone, and she was\nmomentarily alone.\n\n\"Hester,\" began Katharine, \"where is Jack Ralston? I'm perfectly sure\nyour husband knows, and has told you, and I know that he has told my\nmother, from the way she spoke--\"\n\n\"How did you guess that?\" asked Mrs. Crowdie, starting a little at the\nfirst words. \"But I'm sorry if he has spoken to your mother about it--\"\nShe stopped suddenly, feeling that she had made a mistake.\n\nShe was very nervous herself that evening, and as Griggs had said, she\nwas anxious about her husband. There was no real foundation for her\nanxiety, but since her recent experience, she was very easily\nfrightened. Crowdie had spoken excitedly to her about Ralston's conduct\nat the club that afternoon, and she had fancied that there was something\nunusual in his look.\n\n\"Oh, Hester, what is it?\" asked Katharine, bending nearer to her and\nlaying a hand on hers.\n\n\"Don't look so awfully frightened, dear!\" Hester smiled, but not very\nnaturally. \"It's nothing very serious. In fact, I believe it's only that\nWalter saw him at the club late this afternoon and got the idea that he\nwasn't--quite well.\"\n\n\"Not well? Is he ill? Where is he? At home?\" Katharine asked the\nquestions all in a breath, with no suspicion that Hester had softened\nthe truth almost altogether into something else.\n\n\"I suppose he's at home, since he's not here,\" answered Mrs. Crowdie,\nwishing that she had said so at first and had said nothing more.\n\n\"Oh, Hester! What is it? I know it's something dreadful!\" cried\nKatharine. \"I shall go and ask Mr. Crowdie if you won't tell me.\"\n\n\"Don't!\" exclaimed Mrs. Crowdie, so quickly and so loudly that the\npeople near her turned to see what was the matter.\n\n\"You've told me, now--he must be very ill, or you wouldn't speak like\nthat!\" Katharine's lips began to turn white, and she half rose from her\nseat.\n\nMrs. Crowdie drew her back again very gently.\n\n\"No, dear--no, I assure--I give you my word it's not that, dear--oh, I'm\nso sorry I said anything!\" Katharine yielded, and resumed her seat.\n\n\"Hester, what is it?\" she asked very gravely for the third time. \"You're\nmy best friend--the only friend I have besides him. If it's anything\nbad, I'd much rather hear it from you. But I can't stand this suspense.\nI shall ask everybody until somebody tells me the truth.\"\n\nMrs. Crowdie seemed to reflect for a moment before answering, but even\nwhile she was thinking of what she should say, her passionate eyes\nsought for her husband's pale face in the crowd--the pale face and the\nred lips that so many women thought repulsive.\n\n\"Dear,\" she said at last, \"it's foolish to make such a fuss and to\nfrighten you. That sort of thing has happened to almost all men at one\ntime or another--really, you know! You mustn't blame Jack too much--\"\n\n\"For what? For what? Speak, Hester! Don't try to--\"\n\n\"Katharine darling, Walter says that Jack was--well--you know--just a\nlittle far gone--and they had some trouble with him at the club. I don't\nknow--it seems that my brother tried to hold him for some reason or\nother--it's not quite clear--and Jack threw Ham down, there in the hall\nof the club, before a lot of people--Katharine dearest, I'm so sorry I\nspoke!\"\n\nKatharine was leaning back against the cushion, her hands folded\ntogether, and her face set like a mask; but she said nothing, and\nscarcely seemed to be listening, though she heard every word.\n\n\"Of course, dear,\" continued Mrs. Crowdie, \"I know how you love him--but\nyou mustn't think any the worse of him for this. Ham just told me it\nwasn't--well--it wasn't as bad as Walter made out, and he was very angry\nwith Walter for telling me--as though he would keep anything from me!\"\n\nShe stopped again, being much more inclined to talk of Crowdie than of\nRalston, and to defend his indiscretion. Katharine did not move nor\nchange her position, and her eyes looked straight before her, though it\nwas clear that they saw nothing.\n\n\"I'm glad it was you who told me,\" she said in a low, monotonous tone.\n\n\"So am I,\" answered her friend, sympathetically. \"And I'm sure it's not\nhalf as bad as they--\"\n\n\"They all know it,\" continued Katharine, not heeding her. \"I can see it\nin their eyes when they look at me.\"\n\n\"Nonsense, Katharine--nobody but Walter and Ham--\"\n\n\"Your husband told my mother, too. She spoke very oddly. He's been\ntelling every one. Why does he want to make trouble? Does he hate Jack\nso?\"\n\n\"Hate him? No, indeed! I think he's rather fond of him--\"\n\n\"It's a very treacherous sort of fondness, then,\" answered Katharine,\nwith a bitter little laugh, and changing her position at last, so that\nshe looked into her friend's face.\n\n\"Katharine!\" exclaimed Hester. \"How can you talk like that--telling me\nthat Walter is treacherous--\"\n\n\"Oh--you mustn't mind what I say--I'm a little upset--I didn't mean to\nhurt you, dear.\"\n\nKatharine rose, and without another word she left her friend and began\nto go up the side of the room alone, looking for some one as she went.\nIn a moment one of her numerous young adorers was by her side. He had\nseen her talking to Mrs. Crowdie, and had watched his opportunity.\n\n\"No,\" said Katharine, absently, and without looking at him. \"I don't\nwant to dance, thanks. I want to find my cousin, Hamilton Bright. Have\nyou seen him?\"\n\n\"Oh--ah--yes!\" answered the young man, with an imitation of the advanced\nEnglish manner of twenty years ago, which seems to have become the ideal\nof our gilded youth of to-day. \"He's in the corner under the\nbalcony--he's been--er--rather leathering into Crowdie--you\nknow--er--for talking about Jack Ralston's last, all over the place--I\ndaresay you've heard of it, Miss Lauderdale--being--er--a cousin of your\nown, too. No end game, that Ralston chap!\"\n\nKatharine lost her temper suddenly. She stopped and looked the young\ndandy in the eyes. He never forgot the look of hers, nor the paleness of\nher lips as she spoke.\n\n\"You're rather young to speak like that of older men, Mr. Van De Water,\"\nshe said.\n\nShe coolly turned her back on the annihilated youth and walked away from\nhim alone, almost as surprised at what she had done as he was. He, poor\nboy, got very red in the face, stood still, helped himself into\ncountenance by sticking a single glass in his eye and then went in\nsearch of his dearest friend, the man who had just discovered that\nextraordinary tailor in New Burlington Street, you know.\n\nKatharine had been half stunned by what Hester Crowdie had told her,\nwhich she felt instinctively was not more than a moiety of the truth.\nShe had barely recovered her self-possession when she was met by what\nrang like an insult in her ears. It was no wonder that her blood boiled.\nWithout looking to the right or to the left, she went forward till she\nwas under the great balcony, and there, by one of the pillars, she came\nupon Bright and Crowdie talking together in low, excited tones.\n\nBright's big shoulders slowly heaved as in his anger he took about twice\nas much breath as he needed into his lungs at every sentence. His fresh,\npink face was red, and his bright blue eyes flashed visibly. What the\nyoung dandy had said was evidently true. He was still 'leathering into'\nCrowdie with all his might, which was considerable.\n\nCrowdie, perfectly cool and collected, leaned against the wooden pillar\nwith a disagreeable sneer on his red mouth. One hand was in his pocket;\nthe other hung by his side, and his fingers quietly tapped a little\nmeasure upon the fluted column. Almost every one has that trick of\ntapping upon something in moments of anxiety or uncertainty, but the\nway in which it is done is very characteristic of the individual.\nCrowdie's pointed white fingers did it delicately, drawing back lightly\nfrom contact with the wood, as a woman's might, or as though he were\nplaying upon a fine instrument.\n\n\"It's just like you, Walter,\" Bright was saying, to go about telling the\nthing to all the women. Didn't I tell you this afternoon that I was the\nprincipal person concerned, that it was my business and not yours and\nthat if I wished it kept quiet, nobody need tell? And you said yourself\nthat you hoped Hester might not hear it, and then the very first thing I\nfind is that you've told her and cousin Emma and probably Katharine\nherself--\"\n\n\"No, I've not told Katharine,\" said Crowdie, calmly. \"I shan't, because\nshe loves him. The Lord knows why! Drunken beast! I shall leave the club\nmyself, since he's not to be turned out--\"\n\nCrowdie stopped suddenly, for he was more timid than most men, and his\nface plainly expressed fear at that moment--but not of Hamilton Bright.\nKatharine Lauderdale was looking at him over Bright's shoulder and had\nplainly heard what he had said. A man's fear of woman under certain\ncircumstances exceeds his utmost possible fear of man. The painter knew\nat once that he had accidentally done Katharine something like a mortal\ninjury. He felt as a man must feel who has accidentally shot some one\nwhile playing with a loaded pistol.\n\nAs for Katharine, this was the third blow she had received within five\nminutes. The fact that she was in a measure prepared for it had not\ndiminished its force. It had the effect, however, of quenching her\nrising anger instead of further inflaming it, as young Van De Water's\nfoolish remarks had done. She begun to feel that she had a real calamity\nto face--something against which mere anger would have no effect. She\nheard every word Crowdie said, and each struck her with cruel precision\nin the same aching spot. But she drew herself up proudly as she came\nbetween the two men. There was something almost queenly in the quiet\ndignity with which she affected to ignore what she had heard, even\ntrying to give her white lips the shadow of a civil smile as she spoke.\n\n\"Mr. Crowdie, I wish to speak to Hamilton a moment--you don't mind, do\nyou?\"\n\nCrowdie looked at her with undisguised amazement and admiration. He\nuttered some polite but half inaudible words and moved away, glad,\nperhaps, to get out of the sphere of Bright's invective. Bright\nunderstood very well that Katharine had heard, and admired her calmness\nalmost as much as Crowdie did, though he did not know as much as the\nlatter concerning Katharine's relations with Ralston. Hester Crowdie,\nwho told her husband everything, had told him most of what Katharine had\nconfided to her, not considering it a betrayal of confidence, because\nshe trusted him implicitly. No day of disenchantment had yet come for\nher.\n\n\"Won't you come and sit down?\" asked Bright, rather anxiously. \"There's\na corner there.\"\n\n\"Yes,\" said Katharine, moving in the direction of the vacant seats.\n\n\"I'm afraid you heard what that brute said,\" Bright remarked before they\nhad reached the place. \"If I'd seen you coming--\"\n\n\"It wouldn't have made any difference,\" Katharine answered. Then they\nsat down side by side. \"It's much too serious a matter to be angry\nabout,\" she continued, settling herself and looking at his face, and\nfeeling that it was a relief to see a pair of honest blue eyes at last.\n\"That's why I come to you. It happened to you, it seems. Everybody is\ntalking about it, and I have some right to know--\" She hesitated and\nthen continued. \"He's a near relation and all that, of course, and\nwhatever he does makes a difference to us all--my mother has heard,\ntoo--I'm sure Mr. Crowdie told her. Didn't he?\"\n\n\"I believe so,\" answered Bright. \"He's just like a--oh, well! I'll swear\nat him when I'm alone.\"\n\n\"I'm glad you're angry with him,\" said Katharine, and her eyes flashed\na little. \"It's so mean! But that's not the question. I want to know\nfrom your own lips what happened--and why he's not here. I have a right\nto know because--because we were going to dance the cotillion\ntogether--and besides--\"\n\nShe hesitated again, and stopped altogether this time.\n\n\"It's very natural, I'm sure,\" said Bright, who was not the type of men\nwho seek confidences. \"Crowdie has made it all out much worse than it\nwas. He's a--I mean--I wish I'd met him when I was driving cattle in the\nNacimiento Valley!\"\n\nKatharine had never seen Bright so angry before, and the sight was very\nsoothing and comforting to her. She fully concurred in Bright's\nlast-expressed wish.\n\n\"You're Jack's best friend, aren't you?\" she asked.\n\n\"Oh, well--a friend--he always says he hasn't any. But I daresay I'd do\nas much for him as most of them, though, if I had to. I always liked the\nfellow for his dash, and we generally get on very well together. He's\njust a trifle lively sometimes, and he doesn't go well on the curb when\nhe's had--when he's too lively--\"\n\n\"Why don't you say when he drinks?\" asked Katharine, biting on the\nwords, as it were, though she forced herself to say them.\n\n\"Well, he doesn't drink exactly,\" said Bright. \"He's got an awfully\nstrong head and a cast-iron constitution, but he's a queer chap. He gets\nmelancholy, and thinks he's a failure and tries to cheer himself with\ncocktails. And then, you see, having such a nerve, he doesn't know\nexactly how many he takes; and there's a limit, of course--and the last\none does the trick. Then he won't take anything to speak of for days\ntogether. He got a little too much on board last Monday--but that was\nexcusable, and I hadn't seen him that way for a long time. I daresay you\nheard of it? He saved a boy's life between a lot of carts and\nhorse-cars, and got a bad fall; and then, quite naturally--just as I\nshould have done myself--he swallowed a big dose of something, and it\nwent to his head. But he went straight home in a cab, so I suppose it\nwas all right. It was a pretty brave thing he did--talk of baseball! It\nwas one of the smartest bits of fielding I ever saw--the way he caught\nup the little chap, and the dog and the perambulator--forgot nothing,\nthough it was a close shave. Oh--he's brave enough! It's a pity he can't\nfind anything to do.\"\n\n\"Monday,\" repeated Katharine, thoughtfully. \"Yes--I heard about it. Go\non, please, Ham--about to-day. I want to hear everything there is.\"\n\n\"Oh--Crowdie talks like a fool about it. I suppose Jack was a little\ndepressed, or something, and had been trying to screw himself up a bit.\nAnyway, he looked rather wild, and I tried to persuade him to stay a\nlittle while before going out of the club--it was in the hall, you know.\nI behaved like an ass myself--you know I'm awfully obstinate. He really\ndid look a little wild, though! I held his arm--just like that, you\nknow--\" he laid his broad hand upon Katharine's glove--\"and then,\nsomehow, we got fooling together--there in the hall--and he tripped me\nup on my back, and ran out. It was all over in a minute; and I was\nrather angry at the time, because Crowdie and little Frank Miner were\nthere, and a couple of servants. But I give you my word, I didn't say\nanything beyond making them all four swear that they wouldn't tell--\"\n\n\"And this is the result!\" said Katharine, with a sigh. \"What was that he\nsaid about being turned out of the club?\"\n\n\"Crowdie? Oh--some nonsense or other! He felt his ladyship offended\nbecause there had been a bit of a wrestling match in the hall of his\nclub, that's all, and said he meant to leave it--\"\n\n\"No--but about Jack being turned out--\"\n\n\"It's all nonsense of Crowdie's. Men are turned out of a club for\ncheating at cards, and that sort of thing. Besides, Jack's popular with\nmost of the men. I don't believe you could get a committee to sit on his\noffences--not if he locked the oldest member up in the ice-chest, and\nthrew the billiard-table out of the window. He says he has no\nfriends--but it's all bosh, you know--everybody likes him, except that\ndoughy brother-in-law of mine!\"\n\nKatharine was momentarily comforted by Bright's account of the matter,\ndelivered in his familiar, uncompromising fashion. But she was very far\nfrom regaining her composure. She saw that Bright was purposely making\nlight of the matter; and in the course of the silence, which lasted\nseveral minutes after he had finished speaking, it all looked worse than\nit had looked before she had known the exact truth.\n\nShe felt, too, an instinct of repulsion from Ralston, which she had\nnever known, nor dreamed possible. Could he not have controlled himself\na few hours longer? It was their wedding day. Twelve hours had not\npassed from the time when they had left the church together until he had\nbeen drunk--positively drunk, to the point of knocking down his best\nfriend in such a place as a club. She could not deny the facts. Even\nHamilton Bright, kind--more than kind, devoted--did not attempt to\nconceal the fact that Ralston had been what he called 'lively.' And if\nBright could not try to make him out to have been sober, who could?\n\nAnd they had been married that morning! If he had been sober--the word\ncut her like a whip--if he had been sober, they would at that very\nmoment have been sitting together--planning their future--perhaps in\nthat very corner.\n\nShe did not know all yet, either. The clock was striking twelve. It was\nabout at that time that John Ralston was brought into his mother's house\nby a couple of policemen, who had found his card-case in his pocket, and\nhad the sense--with the hope of a handsome fee--to bring him home,\ninsensible, stunned almost to death with the blow he had received.\n\nThey had waked him roughly, the conductor and the other man, who was\nreally a prize fighter, at the end of the run, in front of the horse-car\nstables, and John had struck out before he was awake, as some excitable\nmen do. The fight had followed as a matter of course, out in the snow.\nThe professional had not meant to hurt him, but had lost his temper when\nJohn had reached him and cut his lip, and a right-handed counter had\nsettled the matter--a heavy right-hander just under John's left ear.\n\nThe policemen said they had picked him up out of a drunken brawl.\nAccording to them, everybody was drunk--Ralston, the prize fighter,--who\nhad paid five dollars to be left in peace after the adventure,--the\nconductor, the driver and every living thing on the scene of action,\nincluding the wretched horses of the car.\n\nThere was a short account of the affair in the morning papers, but only\none or two of them mentioned Ralston's name.\n\nKatharine had yet much to learn about the doings on her wedding day,\nwhen she suddenly announced her intention of going home before the ball\nwas half over. Hester Crowdie took her, in her own carriage; and Mrs.\nLauderdale and Crowdie stayed till the end.\n\nNow against all this chain of evidence, including that of several men\nwho had met John in Fifth Avenue about six o'clock, with no overcoat and\nhis hat badly smashed, against evidence that would have hanged a man ten\ntimes over in a murder case, stood the plain fact, which nobody but\nRalston knew, and which no one would ever believe--the plain fact that\nhe had drunk nothing at all.\n\n\n\n\nCHAPTER XXI.\n\n\nIn the grey dawn of Friday morning Katharine woke from broken sleep to\nface the reality of what she had done twenty-four hours earlier. It had\nsnowed very heavily during the night, and her first conscious perception\nwas of that strange, cold glare which the snow reflects, and which makes\neven a bedroom feel like a chilly outer hall into which the daylight\npenetrates through thick panes of ground glass.\n\nShe had slept very little, and against her will, losing consciousness\nfrom time to time out of sheer exhaustion, and roused again by the cruel\nreuniting of the train of thought. Those who have received a wound by\nwhich a principal nerve has been divided, know how intense is the\nsuffering when the severed cords begin to grow together, with agonizing\nslowness, day by day and week by week, convulsing the whole frame of the\nman in their meeting. Katharine felt something like that each time that\nthe merciful curtains of sleep were suddenly torn asunder between\nherself and the truth of the present.\n\nThe pain was combined of many elements, too, and each hurt her in its\nown way. There was the shame of the thing, first, the burning, scarlet\nshame--the thought of it had a colour for her. John Ralston was\ndisgraced in the eyes of all the world. Even the smooth-faced dandy,\nfresh from college, young Van De Water, might sneer at him and welcome,\nand feel superior to him, for never having gone so far in folly. Now if\nsuch men as Van De Water knew the story, it was but a question of hours,\nand all society must know it, too. Society would set down John Ralston\nas a hopeless case. Katharine wondered, with a sickening chill, whether\nthe virtuous--like her father--would turn their backs on Ralston and\nrefuse to know him. She did not know. But Ralston was her husband.\n\nThe thought almost drove her mad. There was that condition of the\ninevitable in her position which gives fate its hold over men's minds.\nShe could not escape. She could not go back to the point where she had\nbeen yesterday morning, and begin her life again. As she had begun it,\nso it must go on to the very end, 'until death them should part'--the\nlife of a spotless girl married to a man who was the very incarnation of\na disgusting vice. In those first moments it would have been a human\nsatisfaction to have been free to blame some one besides herself for\nwhat she had done.\n\nBut even now, when every bitter thought seemed to rise up against John\nRalston, she could not say that the fault had been his if she had bound\nherself to him. To the very last he had resisted. This was Friday\nmorning, and on the Wednesday night at the Thirlwalls' he had told her\nthat he could not be sure of himself. By and by, perhaps, that brave act\nof his might begin to tell in his favour with her, but not yet. The\nfaces, the expressions, the words, of those from whom she had learned\nthe story of his doings were before her eyes and present in her hearing\nnow, as she lay wide awake in the early morning, staring with hot eyes\nat the cold grey ceiling of her room.\n\nIt was only yesterday that her sister Charlotte had sat there, lamenting\nher imaginary woes. How Katharine had despised her! Had she not\ndeliberately chosen, of her own free will, and was she not bound to\nstand by her choice, out of mere self-respect? And Katharine had felt\nthen that, come what might, for good or ill, better or worse, honour or\ndishonour, she was glad that she had married John Ralston and that she\nwould face all imaginable deaths to help him, even a little. But\nnow--now, it was different. He had failed her at the very outset. It was\nnot that others had turned upon him, despising him wholly for a partial\nfault. The public disgrace made it all worse than it might have been,\nbut it was only secondary, after all. The keenest pain was from the\nthrust that had entered Katharine's own heart. It had been with him as\nthough she had not existed. He had not been strong enough, for her sake,\non their wedding day--the day of days to her--to keep himself sober from\nthree o'clock in the afternoon until ten o'clock at night. Only seven\nhours, Katharine repeated to herself in the cold snow-glare of the early\nmorning--seven little hours; her lips were hot and dry with anger, and\nher hands were cold, as she thought of it. It was not only the weakness\nof him, contemptible as that was--if it had at least been weakness for\nsomething less brutal, less beastly, less degrading. Katharine chose the\nstrongest words she could think of, and smote him with them in her\nheart. Was he not her husband, and had she not the right to hate and\ndespise what he had done? It was bad enough, as she said it, and as it\nappeared to most people that morning. There was not a link missing in\nthe evidence, from the moment when John had begun to lose his temper\nwith Miner at the club, until he had been brought home insensible to his\nmother's house by a couple of policemen. His relations and his best\nfriends were all convinced that he had been very drunk, and there was no\nreason why society in general should be more merciful than his own\npeople. Robert Lauderdale said nothing, but when he saw the paragraph\nin a morning paper describing 'Mr. John R----'s drunken encounter with\na professional pugilist,' he regarded the statement as an elucidatory\ncomment on his interview with his great-nephew. No one spoke of the\nmatter in Robert Lauderdale's presence, but the old gentleman felt that\nit was a distinct shame to the whole family, and he inwardly expressed\nhimself strongly. The only one who tried to make matters look a little\nbetter than every one believed they were, was Hamilton Bright. He could\nnot deny the facts, but he put on a cheerful countenance and made the\nbest of them, laughing good-humouredly at John's misfortune, and asking\nevery one who ventured an unfavourable comment whether John was the only\nman alive on that day in the city of New York who had once been a little\nlively, recommending the beardless critics of his friend's conduct to go\nout and drive cattle in the Nacimiento Valley if they wished to\nunderstand the real properties of alcohol, and making the older ones\nfeel uncomfortable by reminding them vividly of the errors of their\nyouth. But no one else said anything in Ralston's favour. He was down\njust then, and it was as well to hit him when everybody was doing the\nsame thing.\n\nKatharine tried to make up her mind as to what she should do, and she\ndid not find it an easy matter. It would be useless to deny the fact\nthat what she felt for Ralston on that morning bore little resemblance\nto love. She remembered vaguely, and with wonder, how she had promised\nto stand by him and help him to her utmost to overcome his weakness. How\nwas she to help him now? How could she play a part and conceal the\nanger, the pain, the shame that boiled and burned in her? If he should\ncome to her, what should she say? She had promised that she would never\nrefer to the matter in any way, when it had seemed but the shadow of a\npossibility. But it had turned into the reality so soon, and into such a\nreality--far more repulsive than anything of which she had dreamed.\nBesides, she added in her heart, it was unpardonable on that day of all\ndays. Married she was, but forgive she could not and would not. Wounded\nlove is less merciful than any hatred, and Katharine could not help\ndeepening the wound by recalling every circumstance of the previous\nevening, from the moment when she had looked in vain for John's face in\nthe crowded room, until she had broken down and asked Hester Crowdie to\nbring her home.\n\nShe rose at last to face the day, undecided, worn out with fatigue, and\nscared, had she been willing to admit the fact, by the possibilities of\nthe next twelve hours. Half dressed, she paused and sat down to think it\nall over again--all she knew, for she had yet to learn the end of the\nstory.\n\nShe had been married just four and twenty hours. Yesterday, at that\nvery time, life had been before her, joyous, hopeful, merry. All that\nwas to be had glistened with gold and gleamed with silver, with the\nsilver of dreamland and the gold of hope, having love set as a jewel in\nthe midst. To-day the precious things were but dross and tinsel and\ncheap glass. For it was all over, and there was no returning. Real life\nwas beginning, began, had begun--the reality of an existence not defined\nexcept in the extent of its suffering, but desperately limited in the\npossibilities of its happiness.\n\nKatharine tried to think it over in some other way. The snow-glare was\nmore grey than ever, and her eyes ached with it, whichever way she\nturned. The room was cold, and her teeth chattered as she sat there,\nhalf dressed. Then, when she let in the hot air from the furnace, it was\ndry and unbearable. And she tried hard to find some other way in which\nto save her breaking heart--if so be that she might look at it so as not\nto see the break, and so, perhaps--if there were mercy in heaven, beyond\nthat aching snow-glare--that by not seeing she might feel a little less,\nonly a little less. It was hard that she should have to feel so much and\nso very bitterly, and all at once. But there was no other way. Instead\nof facing life with John Ralston, she had now to face life and John\nRalston. How could she guess what he might do next? A drunken man has\nlittle control of his faculties--John might suddenly publish in the club\nthe fact that he was her husband.\n\nHe was not the same John Ralston whom she had married yesterday morning,\nand whom she had seen yesterday afternoon for one moment at her door.\nThe hours had changed him. Instead of his face there was a horrible\nmask; instead of his straight, elastic figure there was the reeling,\ndelapidated body of the drunken wretch her father had once shown her in\nthe streets. How could she love that thing? It was not even a man. She\nloathed it and hated it, for it had broken her life. She remembered\nhaving once broken a thermometer when she had been a little girl. She\nremembered the jagged edge of glass, and how the bright mercury had all\nrun out and lost itself in tiny drops in the carpet. She recalled it\nvividly, and she felt that she was like the broken thermometer, and the\nidea was not ridiculous to her, as it must be to any one else, because\nshe was badly hurt.\n\nVague ideas of a long and painful sacrifice rose before her--of\nsomething which must inevitably be begun and ended, like an execution.\nShe had never understood what the inevitable meant until to-day.\n\nThen, all at once, the great question presented itself clearly, the\ngreat query, the enormous interrogation of which we are all aware, more\nor less dimly, more or less clearly--the question which is like the\ndeath-rattle in the throat of the dying nineteenth century,--'What is it\nall for?'\n\nIt came in a rush of passionate disappointment and anger and pain. It\nhad come to Katharine before then, and she had faced it with the easy\nanswer, that it was for love--that it was all for love of John\nRalston--life, its thoughts, its deeds, its hopes, its many fears--all\nfor him, so far as Katharine Lauderdale was concerned. Love made God\ntrue, and heaven a fact, the angels her guardians now and her companions\nhereafter. And her love had been so great that it had seemed to demand a\nwider wealth of heavenly things wherewith to frame it. God was hardly\ngood enough nor heaven broad enough.\n\nBut if this were to be the end, what had it all meant? She stood before\nthe window and looked at the grey sky till the reflection from the dead\nwhite snow beneath her window and on the opposite roof was painful. Yet\nthe little physical pain was a relief. She turned, quite suddenly, and\nfell upon her knees beside the corner of the toilet table, and buried\nher face in her hands and became conscious of prayer.\n\nThat seems to be the only way of describing what she felt. The wave of\npain beat upon her agonized heart, and though the wave could not speak\nwords, yet the surging and the moaning, and the forward rushing, and\nthe backward, whispering ebb, were as the sounds of many prayers.\n\nWas God good? How could she tell? Was He kind? She did not know.\nMerciful? What would be mercy to her? God was there--somewhere beyond\nthe snow-glare that hurt so, and the girl's breaking heart cried to Him,\nquite incoherently, and expecting nothing, but consciously, though it\nknew more of its own bitterness than of God's goodness, just then.\n\nMomentarily the great question sank back into the outer darkness with\nwhich it was concerned, and little by little the religious idea of a\nsacrifice to be made was restored with greater stability than before.\nShe had chosen her own burden, her own way of suffering, and she must\nbear all as well as she could. The waves of pain beat and crashed\nagainst her heart--she wondered, childishly, whether it were broken yet.\nShe knew it was breaking, because it hurt her so.\n\nThere was no connected thread of thought in the torn tissue of her mind,\nany more than there was any coherence in the few words which from time\nto time tried to form themselves on her lips without her knowledge. So\nlong as she had been lying still and staring at the grey ceiling, the\nstorm had been brooding. It had burst now, and she was as helpless in it\nas though it had been a real storm on a real sea, and she alone on a\ndriving wreck.\n\nShe lifted her face and wrung her hands together. It was as though some\none from behind had taken a turn of rough rope round her breast--some\none who was very strong--and as though the rope were tightening fast.\nSoon she should not be able to draw breath against it. As she felt it\ncrushing her, she knew that the hideous picture her mind had made of\nJohn was coming before her eyes again. In a moment it must be there.\nThis time she felt as though she must scream when she saw it. But when\nit came she made no sound. She only dropped her head again, and her\nforehead beat upon the back of her hands and her fingers scratched and\ndrew the cover of the toilet table. Then the picture was drowned in the\ntide of pain--as though it had fallen flat upon the dark sands between\nher and the cruel surf of her immense suffering that roared up to crash\nagainst her heart again. It must break this time, she thought. It could\nnot last forever--nor even all day long. God was there--somewhere.\n\nA lull came, and she said something aloud. It seemed to her that she had\nforgotten words and had to make new ones--although those she spoke were\nold and good. With the sound of her own voice came a little courage, and\nenough determination to make her rise from her knees and face daylight\nagain.\n\nMechanically, as she continued to dress, she looked at herself in the\nmirror. Her features did not seem to be her own. She remembered to have\nseen a plaster cast from a death mask, in a museum, and her face made\nher think of that. There were no lines in it, but there were shadows\nwhere the lines would be some day. The grey eyes had no light in them,\nand scarcely seemed alive. Her colour was that of wax, and there was\nsomething unnatural in the strong black brows and lashes.\n\nThe door opened at that moment, and Mrs. Lauderdale entered the room.\nShe seemed none the worse for having danced till morning, and the\nfreshness which had come back to her had not disappeared again. She\nstood still for a moment, looking at Katharine's face as the latter\nturned towards her with an enquiring glance, in which there was\nsomething of fear and something of shyness. A nervous thoroughbred has\nthe same look, if some one unexpectedly enters its box. Mrs. Lauderdale\nhad a newspaper in her hand.\n\n\"How you look, child!\" she exclaimed, as she came forward. \"Haven't you\nslept? Or what is the matter?\"\n\nShe kissed Katharine affectionately, without waiting for an answer.\n\n\"Well, I don't wonder,\" she added, a moment later, as though speaking to\nherself. \"I've been reading this--\"\n\nShe paused and hesitated, as though not sure whether she should give\nKatharine the paper or not, and she glanced once more at the paragraph\nbefore deciding.\n\n\"What is it about?\" Katharine asked, in a tired voice. \"Read it.\"\n\n\"Yes--but I ought to tell you first. You know, last night--you asked me\nabout Jack Ralston, and I wouldn't tell you what I had heard. Then I saw\nthat somebody else had told you--you really ought to be more careful,\ndear! Everybody was noticing it.\"\n\n\"What?\"\n\n\"Why--your face! It's of no use to advertise the fact that you are\ninterested in Jack's doings. They don't seem to have been very\ncreditable--it's just as well that he didn't try to come to the ball in\nhis condition. Do you know what he was doing, late last night, just\nabout supper-time? I'm so glad I spoke to you both the other day.\nImagine the mere idea of marrying a man who gets into drunken brawls\nwith prize fighters and is taken home by the police--\"\n\n\"Stop--please! Don't talk like that!\" Katharine was trembling visibly.\n\n\"My dear child! It's far better that I should tell you--it's in the\npapers this morning. That sort of thing can't be concealed, you know.\nThe first person you meet will talk to you about it.\"\n\nKatharine had turned from her and was facing the mirror, steadying\nherself with her hands upon the dressing table.\n\n\"And as for behaving as you did last night--he's not worth it. One might\nforgive him for being idle and all that--but men who get tipsy in the\nstreets and fight horse-car conductors and pugilists are not exactly the\nkind of people one wants to meet in society--to dance with, for\ninstance. Just listen to this--\"\n\n\"Mother!\"\n\n\"No--I want you to hear it. You can judge for yourself. 'Mr. John R----,\na well-known young gentleman about town and a near relation of--'\"\n\n\"Mother--please don't!\" cried Katharine, bending over the table as\nthough she could not hold up her head.\n\n\"'--one of our financial magnates,'\" continued Mrs. Lauderdale,\ninexorably, \"and the hero of more than one midnight adventure, has at\nlast met his match in the person of Tam Shelton, the famous light-weight\npugilist. An entirety unadvertised and scantily attended encounter took\nplace between these two gentlemen last night between eleven and twelve\no'clock, in consequence of a dispute which had arisen in a horse-car. It\nappears that the representative of the four hundred had mistaken the\npublic conveyance for his own comfortable quarters, and suddenly feeling\nvery tired had naturally proceeded to go to bed--'\"\n\nWith a very quick motion Katharine turned, took the paper from her\nmother's hands and tore the doubled fourfold sheet through twice, almost\nwithout any apparent effort, before Mrs. Lauderdale could interfere. She\nsaid nothing as she tossed the torn bits under the table, but her eyes\nhad suddenly got life in them again.\n\n\"Katharine!\" exclaimed Mrs. Lauderdale, in great annoyance. \"How can you\nbe so rude?\"\n\n\"And how can you be so unkind, mother?\" asked Katharine, facing her.\n\"Don't you know what I'm suffering?\"\n\n\"It's better to know everything, and have it over,\" answered Mrs.\nLauderdale, with astonishing indifference. \"It only seemed to me that as\nevery one would be discussing this abominable affair, you should know\nbeforehand just what the facts were. I don't in the least wish to hurt\nyour feelings--but now that it's all over with Jack, you may as well\nknow.\"\n\n\"What may I as well know? That you hate him? That you have suddenly\nchanged your mind--\"\n\n\"My dear, I'll merely ask you whether a man who does such things is\nrespectable. Yes, or no?\"\n\n\"That's not the question,\" answered Katharine, with rising anger.\n\"Something strange has happened to you. Until last Tuesday you never\nsaid anything against him. Then you changed, all in a moment--just as\nyou would take off one pair of gloves and put on another. You used to\nunderstand me--and now--oh, mother!\"\n\nHer voice shook, and she turned away again. The little momentary flame\nof her anger was swept out of existence by the returning tide of pain.\n\nMrs. Lauderdale's whole character seemed to have changed, as her\ndaughter said that it had, between one day and the next. A strong new\npassion had risen up in the very midst of it and had torn it to shreds,\nas it were. Even now, as she gazed at Katharine, she was conscious that\nshe envied the girl for being able to suffer without looking old. She\nhated herself for it, but she could not resist it, any more than she\ncould help glancing at her own reflection in the mirror that morning to\nsee whether her face showed any fatigue after the long ball. This at\nleast was satisfactory, for she was as brilliantly fresh as ever. She\ncould hardly understand how she could have seemed so utterly broken down\nand weary on Monday night and all day on Tuesday, but she could never\nforget how she had then looked, and the fear of it was continually upon\nher. Nevertheless she loved Katharine still. The conflict between her\nlove and her envy made her seem oddly inconsequent and almost frivolous.\nKatharine fancied that her mother was growing to be like Charlotte. The\nappealing tone of the girl's last words rang in Mrs. Lauderdale's ears\nand accused her. She stretched out her hand and tried to draw Katharine\ntowards her, affectionately, as she often did when she was seated and\nthe girl was standing.\n\n\"Katharine, dear child,\" she began, \"I'm not changed to you--it's\nonly--\"\n\n\"Yes--it's only Jack!\" answered Katharine, bitterly.\n\n\"We won't talk of him, darling,\" said Mrs. Lauderdale, softly, and\ntrying to soothe her. \"You see, I didn't know how badly you felt about\nit--\"\n\n\"You might have guessed. You know that I love him--you never knew how\nmuch!\"\n\n\"Yes, sweetheart, but now--\"\n\n\"There is no 'but'--it's the passion of my life--the first, the last,\nand the only one!\"\n\n\"You're so young, my darling, that it seems to you as though there could\nnever be anything else--\"\n\n\"Seems! I know.\"\n\nThough Mrs. Lauderdale had already repented of what she had done and\nreally wished to be sympathetic, she could not help smiling faintly at\nthe absolute conviction with which Katharine spoke. There was something\nso young and whole-hearted in the tone as well as in those words that\nonly found an echo far back in the forgotten fields of the older\nwoman's understanding. She hardly knew what to answer, and patted\nKatharine's head gently while she sought for something to say. But\nKatharine resented the affectionate manner, being in no humour to\nappreciate anything which had a savour of artificiality about it. She\nwithdrew her hand and faced her mother again.\n\n\"I know all that you can tell me,\" she said. \"I know all there is to be\nknown, without reading that vile thing. But I don't know what I shall\ndo--I shall decide. And, please--mother--if you care for me at\nall--don't talk about it. It's hard enough, as it is--just the thing,\nwithout any words.\"\n\nShe spoke with an effort, almost forcing the syllables from her lips,\nfor she was suffering terribly just then. She wished that her mother\nwould go away, and leave her to herself, if only for half an hour. She\nhad so much more to think of than any one could know, or guess--except\nold Robert Lauderdale and Jack himself.\n\n\"Well, child--as you like,\" said Mrs. Lauderdale, feeling that she had\nmade a series of mistakes. \"I'm sure I don't care to talk about it in\nthe least, but I can't prevent your father from saying what he pleases.\nOf course he began to make remarks about your not coming to breakfast\nthis morning. I didn't go down myself until he had nearly finished, and\nhe seemed hurt at our neglecting him. And then, he had been reading the\npaper, and so the question came up. But, dearest, don't think I'm unkind\nand heartless and all that sort of thing. I love you dearly, child.\nDon't you believe me?\"\n\nShe put her arm round Katharine's neck and kissed her.\n\n\"Oh, yes!\" Katharine answered wearily. \"I'm sure you do.\"\n\nMrs. Lauderdale looked into her face long and earnestly.\n\n\"It's quite wonderful!\" she exclaimed at last. \"You're a little\npale--but, after all, you're just as pretty as ever this morning.\"\n\n\"Am I?\" asked Katharine, indifferently. \"I don't feel pretty.\"\n\n\"Oh, well--that will all go away,\" answered Mrs. Lauderdale, withdrawing\nher arm and turning towards the door. \"Yes,\" she repeated thoughtfully,\nas though to herself, \"that will all go away. You're so young--still--so\nyoung!\"\n\nHer head sank forward a little as she went out and she did not look back\nat her daughter.\n\nKatharine drew a long breath of relief when she found herself alone. The\ninterview had not lasted many minutes, but it had seemed endless. She\nlooked at the torn pieces of the newspaper which lay on the floor, and\nshe shuddered a little and turned from them uneasily, half afraid that\nsome supernatural power might force her to stoop down and pick them up,\nand fit them together and read the paragraph to the end. She sat down to\ntry and collect her thoughts.\n\nBut she grew more and more confused as she reviewed the past and tried\nto call up the future. For instance, if John Ralston came to the house\nthat afternoon, to explain, to defend himself, to ask forgiveness of\nher, what should she say to him? Could she send him away without a word\nof hope? And if not, what hope should she give him? And hope of what? He\nwas her husband. He had a right to claim her if he pleased--before every\none.\n\nThe words all seemed to be gradually losing their meaning for her. The\nbells of the horse-cars as they passed through Clinton Place sang queer\nlittle songs to her, and the snow-glare made her eyes ache. There was no\nlonger any apparent reason why the day should go on, nor why it should\nend. She did not know what time it was, and she did not care to look.\nWhat difference did it make?\n\nHer ball gown was lying on the sofa, as she had laid it when she had\ncome home. She looked at it and wondered vaguely whether she should ever\nagain take the trouble to put on such a thing, and to go and show\nherself amongst a crowd of people who were perfectly indifferent to her.\n\nOn reflection, for she seriously tried to reflect, it seemed more\nprobable that John would write before coming, and this would give her\nan opportunity of answering. It would be easier to write than to speak.\nBut if she wrote, what should she say? It was just as hard to decide,\nand the words would look more unkind on paper, perhaps, than she could\npossibly make them sound.\n\nWas it her duty to speak harshly? She asked herself the question quite\nsuddenly, and it startled her. If her heart were really broken, she\nthought, there could be nothing for her to do but to say once what she\nthought and then begin the weary life that lay before her--an endless\nstretch of glaring snow, and endless jingling of horse-car bells.\n\nShe rose suddenly and roused herself, conscious that she was almost\nlosing her senses. The monstrous incongruity of the thoughts that\ncrossed her brain frightened her. She pressed her hand to her forehead\nand with characteristic strength determined there and then to occupy\nherself in some way or other during the day. To sit there in her room\nmuch longer would either drive her mad or make her break down\ncompletely. She feared the mere thought of those tears in which some\nwomen find relief, almost as much as the idea of becoming insane, which\npresented itself vividly as a possibility just then. Whatever was to\nhappen during the day, she must at any cost have control over her\noutward actions. She stood for one moment with her hands clasped to her\nbrows, and then turned and left the room.\n\n\n\n\nCHAPTER XXII.\n\n\nOn the present occasion John Ralston deserved very much more sympathy\nthan he got from the world at large, which would have found it very hard\nto believe the truth about his doings on the afternoon and night of\nThursday. He was still unconscious when he was carried into the house by\nthe two policemen and deposited upon his own bed. When he opened his\neyes, they met his mother's, staring down upon him with an expression in\nwhich grief, fear and disgust were all struggling for the mastery. She\nwas standing by his bedside, bending over him, and rubbing something on\nhis temples from time to time. He was but just conscious that he was at\nhome at last, and that she was with him, and he smiled faintly at her\nand closed his eyes again.\n\nHe had hardly done so, however, when he realized what a look was in her\nface. He was not really injured in any way, he was perfectly sober, and\nhe was very hungry. As soon as the effect of the last blow began to wear\noff, his brain worked clearly enough. He understood at once that his\nmother must suppose him to be intoxicated. It was no wonder if she did,\nas he knew. He was in a far worse plight now than he had been on Monday\nafternoon, as far as appearances were concerned. His clothes were\ndrenched with the wet snow, his hat had altogether disappeared in the\nfight, his head was bruised, and his face was ghastly pale. He kept his\neyes shut for a while and tried to recall what had happened last. But it\nwas not at all clear to him why he had been fighting with the man who\nwore the fur collar and the chain, nor why he had wandered to Tompkins\nSquare. Those were the two facts which recalled themselves most vividly\nat first, in a quite disconnected fashion. Next came the vision of\nRobert Lauderdale and the recollection of the violent gesture with which\nthe latter had accidentally knocked John's hat out of his hand; and\nafter that he recalled the scene at the club. It seemed to him that he\nhad been through a series of violent struggles which had no connection\nwith each other. His head ached terribly and he should have liked to be\nleft in the dark to try and go to sleep. Then, as he lay there, he knew\nthat his mother was still looking at him with that expression in which\ndisgust seemed to him to be uppermost. It flashed across his mind\ninstantly that she must naturally think he had been drinking. But though\nhis memory of what had happened was very imperfect, and though he was\ndizzy and faint, he knew very well that he was sober, and he realized\nthat he must impress the fact upon his mother at any cost, immediately,\nboth for his own sake and for hers. He opened his eyes once more and\nlooked at her, wondering how his voice would sound when he should speak.\n\n\"Mother dear--\" he began. Then he paused, watching her face.\n\nBut her expression did not unbend. It was quite clear now that she\nbelieved the very worst of him, and he wondered whether the mere fact of\nhis speaking connectedly would persuade her that he was telling the\ntruth.\n\n\"Don't try to talk,\" she said in a low, hard voice. \"I don't want to\nknow anything about your doings.\"\n\n\"Mother--I'm perfectly sober,\" said John Ralston, quietly. \"I want you\nto listen to me, please, and persuade yourself.\"\n\nMrs. Ralston drew herself up to her full height as she stood beside him.\nHer even lips curled scornfully, and the lines of temper deepened into\nsoft, straight furrows in her keen face.\n\n\"You may be half sober now,\" she answered with profound contempt.\n\"You're so strong--it's impossible to tell.\"\n\n\"So you don't believe me,\" said John, who was prepared for her\nincredulity. \"But you must--somehow. My head aches badly, and I can't\ntalk very well, but I must make you believe me. It's--it's very\nimportant that you should, mother.\"\n\nThis time she said nothing. She left the bedside and moved about the\nroom, stopping before the dressing table and mechanically putting the\nbrushes and other small objects quite straight. If she had felt that it\nwere safe to leave him alone she would have left him at once and would\nhave locked herself into her own room. For she was very angry, and she\nbelieved that her anger was justified. So long as he had been\nunconscious, she had felt a certain fear for his safety which made a\nlink with the love she bore him. But, as usual, his iron constitution\nseemed to have triumphed. She remembered clearly how, on Monday\nafternoon, he had evidently been the worse for drink when he had entered\nher room, and yet how, in less than an hour, he had reappeared\napparently quite sober. He was very strong, and there was no knowing\nwhat he could do. She had forgiven him that once, but it was not in her\nnature to forgive easily, and she told herself that this time it would\nbe impossible. He had disgraced himself and her.\n\nShe continued to turn away from him. He watched her, and saw how\ndesperate the situation was growing. He knew well enough that there\nwould be some talk about him on the morrow and that it would come to\nKatharine's ears, in explanation of his absence from the Assembly ball.\nHis mind worked rapidly and energetically now, for it was quite clear to\nhim that he had no time to lose. If he should fall asleep without having\npersuaded his mother that he was quite himself, he could never, in all\nhis life, succeed in destroying the fatal impression she must carry with\nher. While she was turning from him he made a great effort, and putting\nhis feet to the ground, sat upon the edge of his bed. His head swam for\na moment, but he steadied himself with both hands and faced the light,\nthinking that the brilliant glare might help him.\n\n\"You must believe me, now,\" he said, \"or you never will. I've had rather\na bad day of it, and another accident, and a fight with a better man\nthan myself, so that I'm rather battered. But I haven't been drinking.\"\n\n\"Look at yourself!\" answered Mrs. Ralston, scornfully. \"Look at yourself\nin the glass and see whether you have any chance of convincing me of\nthat. Since you're not killed, and not injured, I shall leave you to\nyourself. I hope you won't talk about it to-morrow. This is the second\ntime within four days. It's just a little more than I can bear. If you\ncan't live like a gentleman, you had better go away and live in the way\nyou prefer--somewhere else.\"\n\nAs she spoke, her anger began to take hold of her, and her voice fell\nto a lower pitch, growing concentrated and cruel.\n\n\"You're unjust, though you don't mean to be,\" said John. \"But, as I\nsaid, it's very important that you should recognize the truth. All sorts\nof things have happened to me, and many people will say that I had been\ndrinking. And now that it's over I want you to establish the fact that I\nhave not. It's quite natural that you should think as you do, of course.\nBut--\"\n\n\"I'm glad you admit that, at least,\" interrupted Mrs. Ralston. \"Nothing\nyou can possibly say or do can convince me that you've been sober. You\nmay be now--you're such a curiously organized man. But you've not been\nall day.\"\n\n\"Mother, I swear to you that I have!\"\n\n\"Stop, John!\" cried Mrs. Ralston, crossing the room suddenly and\nstanding before him. \"I won't let you--you shan't! We've not all been\ngood in the family, but we've told the truth. If you were sober you\nwouldn't--\"\n\nJohn Ralston was accustomed to be believed when he made a statement,\neven if he did not swear to it. His virtues were not many, and were not\nvery serviceable, on the whole; but he was a truthful man, and his anger\nrose, even against his own mother, when he saw that she refused to\nbelieve him. He forgot his bruises and his mortal weariness, and sprang\nto his feet before her. Their eyes met steadily, as he spoke.\n\n\"I give you my sacred word of honour, mother.\"\n\nHe saw a startled look come into his mother's eyes, and they seemed to\nwaver for a moment and then grow steady again. Then, without warning,\nshe turned from him once more, and went and seated herself in a small\narm-chair by the fire. She sat with her elbow resting on her knee, while\nher hand supported her chin, and she stared at the smouldering embers as\nthough in deep thought.\n\nHer principal belief was in the code of honour, and in the absolute\nsanctity of everything connected with it, and she had brought up her son\nin that belief, and in the practice of what it meant. He did not give\nhis word lightly. She did not at that moment recall any occasion upon\nwhich he had given it in her hearing, and she knew what value he set\nupon it.\n\nThe evidence of her senses, on the other hand, was strong, and that of\nher reason was stronger still. It did not seem conceivable that he could\nbe telling the truth. It was not possible that as his sober, natural\nself he should have got into the condition in which he had been brought\nhome to her. But it was quite within the bounds of possibility, she\nthought, that he should have succeeded in steadying himself so far as to\nbe able to speak connectedly. In that case he had lied to her, when he\nhad given his word of honour, a moment ago.\n\nShe tried to look at it fairly, for it was a question quite as grave in\nher estimation as one of life or death. She would far rather have known\nhim dead than dishonourable, and his honour was arraigned at her\ntribunal in that moment. Her impulse was to believe him, to go back to\nhim, and kiss him, and ask his forgiveness for having accused him\nwrongly. But the evidence stood between him and her as a wall of ice.\nThe physical impression of horror and disgust was too strong. The\noutward tokens were too clear. Even the honesty of his whole life from\nhis childhood could not face and overcome them.\n\nAnd so he must have lied to her. It was a conviction, and she could not\nhelp it. And then she, too, felt that iron hands were tightening a band\nround her breast, and that she could not bear much more. There was but\none small, pitiful excuse for him. In spite of his quiet tones, he might\nbe so far gone as not to know what he was saying when he spoke. It was a\nforlorn hope, a mere straw, a poor little chance of life for her\nmother's love. She knew that life could never be the same again, if she\ncould not believe her son.\n\nThe struggle went on in silence. She did not move from her seat nor\nchange her position. Her eyelids scarcely quivered as she gazed steadily\nat the coals of the dying wood fire. Behind her,\n\n[Illustration: \"She knew that life could never be the same again, if she\ncould not believe her son.\"--Vol. II., p. 142.]\n\nJohn Ralston slowly paced the room, following the pattern of the carpet,\nand glancing at her from time to time, unconscious of pain or fatigue,\nfor he knew as well as she herself that his soul was in the balance of\nher soul's justice. But the silence was becoming intolerable to him. As\nfor her, she could not have told whether minutes or hours had passed\nsince he had spoken. The trial was going against him, and she almost\nwished that she might never hear his voice again.\n\nThe questions and the arguments and the evidence chased each other\nthrough her brain faster and faster, and ever in the same vicious\ncircle, till she was almost distracted, though she sat there quite\nmotionless and outwardly calm. At last she dropped both hands upon her\nknees; her head fell forward upon her breast, and a short, quick sound,\nneither a sigh nor a groan, escaped her lips. It was finished. The last\nargument had failed; the last hope was gone. Her son had disgraced\nhimself--that was little; he had lied on his word of honour--that was\ngreater and worse than death.\n\n\"Mother, you've always believed me,\" said John, standing still behind\nher and looking down at her bent head.\n\n\"Until now,\" she answered, in a low, heart-broken voice.\n\nJohn turned away sharply, and began to pace the floor again with\nquickening steps. He knew as well as she what it must mean if he did\nnot convince her then and there. In a few hours it would be too late.\nAll sorts of mad and foolish ideas crossed his mind, but he rejected\nthem one after the other. They were all ridiculous before the magnitude\nof her conviction. He had never seen her as she was now, not even when\nhis father had died. He grew more and more desperate as the minutes\npassed. If his voice, his manner, his calm asseveration of the truth\ncould not convince her, he asked himself if anything could. And if not,\nwhat could convince Katharine to-morrow? His recollections were all\ncoming back vividly to him now. He remembered everything that had\nhappened since the early morning. Strange to say,--and it is a\nwell-known peculiarity of such cases,--he recalled distinctly the\ncircumstances of his fall in the dark, and the absence of all knowledge\nof the direction he was taking afterwards. He knew, now, how he had\nwandered for hours in the great city, and he remembered many things he\nhad seen, all of which were perfectly familiar, and each of which, at\nany other time, would have told him well enough whither he was going. He\nreconstructed every detail without effort. He even knew that when he had\nfallen over the heap of building material he had hurt one of his\nfingers, a fact which he had not noticed at the time. He looked at his\nhand now to convince himself. The finger was badly scratched, and the\nnail was torn to the quick.\n\n\"Will nothing make you change your mind?\" he asked, stopping in the\nmiddle of the room. \"Will nothing I can do convince you?\"\n\n\"It would be hard,\" answered Mrs. Ralston, shaking her head.\n\n\"I've done all I can, then,\" said John. \"There's nothing more to be\nsaid. You believe that I can lie to you and give you my word for a lie.\nIs that it?\"\n\n\"Don't say it, please--it's bad enough without any more words.\" She\nrested her chin upon her hand once more and stared at the fire.\n\n\"There is one thing more,\" answered John, suddenly. \"I think I can make\nyou believe me still.\"\n\nA bitter smile twisted Mrs. Ralston's even lips, but she did not move\nnor speak.\n\n\"Will you believe the statement of a good doctor on his oath?\" asked\nJohn, quietly.\n\nMrs. Ralston looked up at him suddenly. There was a strange expression\nin her eyes, something like hope, but with a little distrust.\n\n\"Yes,\" she said, after a moment's thought. \"I would believe that.\"\n\n\"Most people would,\" answered John, with sudden coldness. \"Will you send\nfor a doctor? Or shall I go myself?\"\n\n\"Are you in earnest?\" asked Mrs. Ralston, rising slowly from her seat\nand looking at him.\n\n\"I'm in earnest--yes. You seem to be. It's rather a serious matter to\ndoubt my word of honour--even for my mother.\"\n\nBeing quite sure of himself, he spoke very bitterly and coldly. The time\nfor appealing to her kindness, her love, or her belief in him was over,\nand the sense of approaching triumph was thrilling, after the\nhumiliation he had suffered in silence. Mrs. Ralston, strange to say,\nhesitated.\n\n\"It's very late to send for any one now,\" she said.\n\n\"Very well; I'll go myself,\" answered John. \"The man should come, if it\nwere within five minutes of the Last Judgment. Will you go to your room\nfor a moment, mother, while I dress? I can't go as I am.\"\n\n\"No. I'll send some one.\" She stood still, watching his face. \"I'll ring\nfor a messenger,\" she said, and left the room.\n\nBy this time her conviction was so deep seated that she had many reasons\nfor not letting him leave the house, nor even change his clothes. He was\nvery strong. It was evident, too, that he had completely regained\npossession of his faculties, and she believed that he was capable, at\nshort notice, of so restoring his appearance as to deceive the keenest\ndoctor. She remembered what had happened on Monday, and resolved that\nthe physician should see him just as he was. It did not strike her, in\nher experience, that a doctor does not judge such matters as a woman\ndoes.\n\nDuring her brief absence from the room, John was thinking of very\ndifferent matters. It did not even strike him that he might smooth his\nhair or wash his soiled and blood-stained hands, and he continued to\npace the room under strong excitement.\n\n\"Doctor Routh will come, I think,\" said Mrs. Ralston, as she came in.\n\nShe sat down where she had been sitting before, in the small easy chair\nbefore the fire. She leaned back and folded her hands, in the attitude\nof a person resigned to await events. John merely nodded as she spoke,\nand did not stop walking up and down. He was thinking of the future now,\nfor he knew that he had made sure of the present. He was weighing the\nchances of discretion on the part of the two men who had been witnesses\nof his struggle with Bright in the hall of the club. As for Bright\nhimself, though he was the injured party, John knew that he could be\ntrusted to be silent. He might never forgive John, but he could not\ngossip about what had happened. Frank Miner would probably follow\nBright's lead. The dangerous man was Crowdie, who would tell what he had\nseen, most probably to Katharine herself, and that very night. He might\naccount for his absence from the dinner-party to which he had been\nengaged, and from the ball, on the ground of an accident. People might\nsay what they pleased about that, but it would be hard to make any one\nbelieve that he had been sober when he had so suddenly lost his temper\nand tripped up the pacific Hamilton Bright in the afternoon.\n\nHe knew, of course, that his mother's testimony would have counted for\nnothing, even if she had believed him, and bitterly as he resented her\nunbelief, he recognized that it was bringing about a good result. No one\ncould doubt the evidence of such a man as Doctor Routh, and the latter\nwould of course be ready at any time to repeat his statement, if it were\nnecessary to clear John's reputation.\n\nBut when he thought of Katharine, his instinct told him that matters\ncould not be so easily settled. It was quite true that he was in no way\nto blame for having fallen over a heap of stones in a dark street, but\nhe knew how anxiously she must have waited for him at the ball, and what\nshe must have felt if, as he suspected, Crowdie had given her his own\nversion of what had taken place in the afternoon. It was not yet so late\nbut that he might have found her still at the Assembly rooms, and so far\nas his strength was concerned, he would have gone there even at that\nhour. Tough as he was, a few hours, more or less, of fatigue and effort\nwould make little difference to him, though he had scarcely touched food\nthat day. He was one of those men who are not dependent for their\nstrength on the last meal they happen to have eaten, as the majority\nare, and who break down under a fast of twenty-four hours. In spite of\nall he had been through, moreover, his determined abstinence during the\nlast days was beginning to tell favourably on him, for he was young, and\nhis nerves had a boundless recuperative elasticity. Hungry and tired and\nbruised as he was, and accustomed as he had always been to swallow a\nstimulant when the machinery was slackened, he did not now feel that\ncraving at all as he had felt it on the previous night, when he had\nstood in the corner at the Thirlwalls' dance. That seemed to have been a\nturning-point with him. He had thought so at the time, and he was sure\nof it now. He felt that just as he was he could dress himself, and go to\nthe Assembly if he pleased, and that he should not break down.\n\nBut his appearance was against him, as he was obliged to admit when he\nlooked at himself in the mirror. His face was swollen and bruised, his\neyes were sunken and haggard, and his skin was almost livid in its\nsallow whiteness. Others would judge him as his mother had judged, and\nKatharine might be the first to do so. On the whole, it seemed wisest to\nwrite to her early in the morning, and to explain exactly what had\nhappened. In the course of the day he could go and see her.\n\nHe had reached this conclusion, when the sound of wheels, grating out of\nthe snow against the curb-stone of the pavement, interrupted his\nmeditations, and he stopped in his walk. At the same moment Mrs. Ralston\nrose from her seat.\n\n\"I'll let him in,\" she said briefly, as John advanced towards the door.\n\n\"Let me go,\" he said. \"Why not?\" he asked, as she pushed past him.\n\n\"Because--I'd rather not. Stay here!\" In a moment she was descending the\nstairs.\n\nJohn listened at the open door, and heard the latch turned, and\nimmediately afterwards the sound of a man's voice, which he recognized\nas that of Doctor Routh. The doctor had been one of the Admiral's\nfirmest friends, and was, moreover, a man of very great reputation in\nNew York. It was improbable that, except for some matter of life and\ndeath, any one but Mrs. Ralston could have got him to leave his fireside\nat midnight and in such weather.\n\n\"It's an awful night, Mrs. Ralston,\" John heard him say, and the words\nwere accompanied by a stamping of feet, followed by the unmistakable\nsoft noise of india-rubber overshoes kicked off, one after the other,\nupon the marble floor of the entry.\n\nJohn retired into his room again, leaving the door open, and waited\nbefore the fireplace. Far down below he could hear the voices of his\nmother and Doctor Routh. They were evidently talking the matter over\nbefore coming up. Then their soft tread upon the carpeted stairs told\nhim that they were on their way to his room.\n\nMrs. Ralston entered first, and stood aside to let the doctor pass her\nbefore she closed the door. Doctor Routh was enormously tall. He wore a\nlong white beard, and carried his head very much bent forward. His eyes\nwere of the very dark blue which is sometimes called violet, and when he\nwas looking directly in front of him, the white was visible below the\niris. He had delicate hands, but was otherwise rough in appearance, and\nwalked with a heavy tread and a long stride, as a strong man marches\nwith a load on his back.\n\nHe stopped before John, looked keenly at him, and smiled. He had known\nhim since he had been a boy.\n\n\"Well, young man,\" he said, \"you look pretty badly used up. What's the\nmatter with you?\"\n\n\"Have I been drinking, doctor? That's the question.\" John did not smile\nas he shook hands.\n\n\"I don't know,\" answered the physician. \"Let me look at you.\"\n\nHe was holding the young man's hand, and pressing it gently, as though\nto judge of its temperature. He made him sit down under the bright\ngas-light by the dressing table, and began to examine him carefully.\n\nMrs. Ralston turned her back to them both, and leaned against the\nmantelpiece. There was something horrible to her in the idea of such an\nexamination for such a purpose. There was something far more horrible\nstill in the verdict which she knew must fall from the doctor's lips\nwithin the next five minutes--the words which must assure her that John\nhad lied to her on his word of honour. She had no hope now. She had\nwatched the doctor nervously when he had entered the room, and when he\nhad spoken to John she had seen the smile on his face. There had been no\ndoubt in his mind from the first, and he was amused--probably at the\nbare idea that any one could look as John looked who had not been very\ndrunk indeed within the last few hours. Presently he would look grave\nand shake his head, and probably give John a bit of good advice about\nhis habits. She turned her face to the wall above the mantelpiece and\nwaited. It could not take long, she thought. Then it came.\n\n\"If you're not careful, my boy--\" the doctor began, and stopped.\n\n\"What?\" asked John, rather anxiously.\n\nMrs. Ralston felt as though she must stop her ears to keep out the sound\nof the next words. Yet she knew that she must hear them before it was\nall over.\n\n\"You'll injure yourself,\" said Doctor Routh, completing his sentence\nvery slowly and thoughtfully.\n\n\"That's of no consequence,\" answered John. \"What I want to know is,\nwhether I have been drinking or not. Yes or no?\"\n\n\"Drinking?\" Doctor Routh laughed contemptuously. \"You know as well as I\ndo that you haven't had a drop of anything like drink all day. But\nyou've had nothing to eat, either, for some reason or other--and\nstarvation's a precious deal worse than drinking any day. Drinking be\ndamned! You're starving--that's what's the matter with you. Excuse me,\nMrs. Ralston, forgot you were there--\"\n\nMrs. Ralston had heard every word. Her hands dropped together inertly\nupon the mantelpiece, and she turned her head slowly toward the two men.\nHer face had a dazed expression, as though she were waking from a dream.\n\n\"Never mind the starvation, doctor,\" said John, with a hard laugh.\n\"There's a Bible somewhere in the room. Perhaps you won't mind swearing\non it that I'm sober--before my mother, please.\"\n\n\"I shouldn't think any sane person would need any swearing to convince\nthem!\" Doctor Routh seemed to be growing suddenly angry. \"You've been\nbadly knocked about, and you've been starving yourself for days--or\nweeks, very likely. You've had a concussion of the brain that would\nhave laid up most people for a week, and would have killed some that I\nknow. You're as thin as razor edges all over--there's nothing to you but\nbone and muscle and nerve. You ought to be fed and put to bed and looked\nafter, and then you ought to be sent out West to drive cattle, or go to\nsea before the mast for two or three years. Your lungs are your weak\npoint. That's apt to be the trouble with thoroughbreds in this country.\nOh--they're sound enough--enough for the present, but you can't go on\nlike this. You'll give out when you don't expect it. Drinking? No! I\nshould think a little whiskey and water would do you good!\"\n\nWhile he was speaking, Mrs. Ralston came slowly forward, listening to\nevery word he said, in wide-eyed wonder. At last she laid her hand upon\nhis arm. He felt the slight pressure and looked down into her eyes.\n\n\"Doctor Routh--on your word of honour?\" she asked in a low voice.\n\nJohn laughed very bitterly, rose from his chair, and crossed the room.\nThe old man's eyes flashed suddenly, and he drew himself up.\n\n\"My dear Mrs. Ralston, I don't know what has happened to you, nor what\nyou have got into your head. But if you're not satisfied that I'm enough\nof a doctor to tell whether a man is drunk or sober, send for some one\nin whom you've more confidence. I'm not used to going about swearing my\nprofessional opinion on Bibles and things, nor to giving my word of\nhonour that I'm in earnest when I've said what I think about a patient.\nBut I'll tell you--if I had fifty words of honour and the whole Bible\nHouse to swear on--well, I'll say more--if it were a case of a trial,\nI'd give my solemn evidence in court that Master John Ralston has had\nnothing to drink. Upon my word, Mrs. Ralston! Talk of making mountains\nof mole-hills! You're making a dozen Himalayas out of nothing at all, it\nseems to me. Your boy's starving, Mrs. Ralston, and I daresay he takes\ntoo much champagne and too many cocktails occasionally. But he's not\nbeen doing it to-day, nor yesterday, nor the day before. That is my\nopinion as a doctor. Want my word of honour and the Bible again? Go to\nbed! Getting your old friend away from his books and his pipe and his\nfire at this hour, on such a night as this! You ought to be ashamed of\nyourself, young lady! Well--if I've done you any good, I'm not\nsorry--but don't do it again. Good night--and get that young fellow out\nof this as soon as you can. He's not fit for this sort of life, anyhow.\nDon't take thoroughbreds for cart horses--they stand it for a bit, and\nthen they go crack! Good night--no, I know my way all right--don't come\ndown.\"\n\nJohn followed him, however, but before he left the room he glanced at\nhis mother's face. Her eyes were cast down, and her lips seemed to\ntremble a little. She did not even say good night to Doctor Routh.\n\n\n\n\nCHAPTER XXIII.\n\n\nIt was nearly one o'clock when John Ralston let Doctor Routh out of the\nhouse and returned to his own room. He found his mother standing there,\nopposite the door, as he entered, and her eyes had met his even before\nhe had passed the threshold. She came forward to meet him, and without a\nword laid her two hands upon his shoulders and hid her face against his\ntorn coat. He put one arm around her and gently stroked her head with\nthe other hand, but he looked straight before him at the bright globe of\nthe gas-light, and said nothing.\n\nThere was an unsettled expression on his pale face. He did not wish to\nseem triumphant, and he did wish that his anger against her might\nsubside immediately and be altogether forgotten. But although he had\nenough control of his outward self to say nothing and to touch her\ntenderly, the part of him that had been so deeply wounded was not to be\nhealed in a moment. Her doubt--more, her openly and scornfully outspoken\ndisbelief had been the very last straw that day. It had been hard, just\nwhen he had been doing his best to reform, to be accused by every one,\nfrom Hamilton Bright, his friend, to the people on the horse-car; but\nit had been hardest of all to be accused by his mother, and not to be\nbelieved even on his pledged word. That was a very different matter.\n\nTo a man of a naturally melancholic and brooding temper, as John Ralston\nwas, illusions have a very great value. Such men have few of them, as a\nrule, and regard them as possessions with which no one has any right to\ninterfere. They ask little or nothing of the world at large, except to\nbe allowed to follow their own inclinations and worship their own idols\nin their own way. But of their idols they ask much, and often give them\nlittle in return except acts of idolatry. And the first thing they ask,\nwhether they express the demand openly or not, is that their idols\nshould believe in them in spite of every one and everything. They are\nnot, as a rule, capricious men. They cannot replace one object of\nadoration by another, at short notice. Perhaps the foundation of such\ncharacters is a sort of honourable selfishness, a desire to keep what\nthey care for to themselves, beyond the reach of every one else,\ntogether with an inward conviction that their love is eminently worth\nhaving from the mere fact that they do not bestow it lightly. When the\nidol expresses a human and pardonable doubt in their sincerity, an\nillusion is injured, if not destroyed--even when that doubt is well\nfounded. But when the doubt is groundless, it makes a bad wound which\nleaves an ugly scar, if it ever heals at all.\n\nJohn Ralston was very like his mother, and she knew it and understood\ninstinctively that words could be of no use. There was nothing to be\ndone but to throw herself upon his mercy, as it were, and to trust that\nhe would forgive an injury which nothing could repair. And John\nunderstood this, and did his best to meet her half way, for he loved her\nvery much. But he could not help the expression on his face, not being\ngood at masking nor at playing any part. She, womanly, could have done\nthat better than he.\n\nShe wished to act no comedy, however. The thing was real and true, and\nshe was distressed beyond measure. She looked up at his face and saw\nwhat was in his mind, and she knew that for the present she could do\nnothing. Then she gently kissed the sleeve of his coat, and withdrew her\nhands from him.\n\n\"You're wet, Jack,\" she said, trying to speak naturally. \"Go to bed, and\nI'll bring you something to eat and something hot to drink.\"\n\n\"No, mother--thank you. I don't want anything. But I think I'll go to\nbed. Good night.\"\n\n\"Let me bring you something--\"\n\n\"No, thank you. I'd rather not. It's all right, mother. Don't worry.\"\n\nIt was hard to say even that little, just then, but he did as well as he\ncould. Then he kissed her on the forehead and opened the door for her.\nShe bent her head low as she passed him, but she did not look up.\n\nHalf an hour later, when John was about to put out his light, he heard\nthe little clinking of glasses and silver on a tray outside his door.\nThen there was a knock.\n\n\"I've brought you something to eat, Jack,\" said his mother's voice.\n\"Just what I could find--\"\n\nJohn turned as he was crossing the room--a gaunt figure in his loose,\nstriped flannels--and hesitated a moment before he spoke.\n\n\"Oh--thank you, very much,\" he answered. \"Would you kindly set it down?\nI'll take it in presently. It's very good of you, mother--thank\nyou--good night again.\"\n\nHe heard her set down the tray, and the things rattled and clinked.\n\n\"It's here, when you want it,\" said the voice.\n\nHe fancied there was a sigh after the words, and two or three seconds\npassed before the sound of softly departing footsteps followed. He\nlistened, with a weary look in his eyes, then went to the fireplace and\nleaned against the mantelpiece for a moment. As though making an effort,\nhe turned again and went to the door and opened it and brought in the\ntray. There were dainty things on it, daintily arranged. There was also\na small decanter of whiskey, a pint of claret and a little jug of hot\nwater. John set the tray upon one end of his writing table and looked at\nit, with an odd, sour smile. He was really so tired that he wanted\nneither food nor drink, and the sight of both in abundance was almost\nnauseous to him. He reflected that the servant would take away the\nthings in the morning, and that his mother would never know whether he\nhad taken what she had brought him or not, unless she asked him, which\nwas impossible. He took up the tray again, set it down on the floor, in\na corner, and instead of going to bed seated himself at his writing\ntable.\n\nIt seemed best to write to Katharine and send his letter early in the\nmorning. It was hard work, and he could scarcely see the words he wrote,\nfor the pain in his head was becoming excruciating. It was necessarily a\nlong letter, too, and a complicated one, and his command of the English\nlanguage seemed gone from him. Nevertheless, he plodded on diligently,\ntelling as nearly as he could remember what had happened to him since he\nhad left Katharine's door at three o'clock in the afternoon, up to the\nmoment when Doctor Routh had pronounced his verdict. It was not well\nwritten, but on the whole it was a thoroughly clear account of events,\nso far as he himself could be said to know what had happened to him. He\naddressed the letter and put a special delivery stamp upon it, thinking\nthat this would be a means of sending it to its destination quickly\nwithout attracting so much attention to it as though he should send a\nmessenger himself. Then he put out the gas, drew up the shades, so that\nthe morning light should wake him early, in spite of his exhaustion, and\nat last went to bed.\n\nIt was unfortunate that the messenger who took the specially stamped\nletter to Clinton Place on the following morning should have rung the\nbell exactly when he did, that is to say, at the precise moment when\nAlexander Junior was putting on his overcoat and overshoes in the entry.\nIt was natural enough that Mr. Lauderdale should open the door himself\nand confront the boy, who held up the letter to him with the little book\nin which the receipt was to be signed. It was the worse for the boy,\nbecause Katharine would have given him five or ten cents for himself,\nwhereas Alexander Junior signed the receipt, handed it back and shut the\ndoor in the boy's face. And it was very much the worse for John Ralston,\nsince Mr. Lauderdale, having looked at the handwriting and recognized\nit, put the letter into his pocket without a word to any one and went\ndown town for the day.\n\nNow it was his intention to do the thing which was right according to\nhis point of view. He was as honourable a man, in his own unprejudiced\nopinion, as any living, and he would no more have forfeited his right\nto congratulate himself upon his uprightness than he would have given\nten cents to the messenger boy, or a holiday to a clerk, or a\nsubscription for anything except his pew in church. The latter was\nreally a subscription to his own character, and therefore not an\nextravagance. It would never have entered into his mind that he could\npossibly break the seal of Ralston's specially stamped envelope. The\nletter was as safe in his pocket as though it had been put away in his\nown box at the Safe Deposit--where there were so many curious things of\nwhich no one but Alexander Junior knew anything. But he did not intend\nthat his daughter should ever read it either. He disapproved of John\nfrom the very bottom of his heart, partly because he did, which was an\nexcellent reason, partly because there could be no question as to John's\nmode of life, and partly because he had once lost his temper when John\nhad managed to keep his own. So far as he allowed himself to swear, he\nhad sworn that John should never marry Katharine--unless, indeed, John\nshould inherit a much larger share of Robert Lauderdale's money than was\njust, in which case justice itself would make it right to enter into a\nmatrimonial alliance with the millions. Meanwhile, however, Robert the\nRich was an exceedingly healthy old man.\n\nUnder present circumstances, therefore, if accident threw into his\nhands one of Ralston's letters to Katharine, it was clearly the duty of\nsuch a perfectly upright and well-conducted father as Alexander Junior\nto hinder it from reaching its destination. Only one question as to his\nconduct presented itself to his mind, and he occupied the day in solving\nit. Should he quietly destroy the letter and say nothing about it to any\none, or should he tell Katharine that he had it, and burn it in her\npresence after showing her that it was unopened? His conscience played\nan important part in his life, though Robert Lauderdale secretly\nbelieved that he had none at all; and his conscience bade him be quite\nfrank about what he had done, and destroy the letter under Katharine's\nown eyes. He took it from his pocket as he sat in his brilliantly\npolished chair before his shiny table, under the vivid snow-glare which\nfell upon him through his magnificent plate-glass windows. He looked at\nit again, turned it over thoughtfully, and returned it at last to his\npocket, where it remained until he came home late in the afternoon.\nWhile he sipped his glass of iced water at luncheon time, he prepared a\nlittle speech, which he repeated to himself several times in the course\nof the day.\n\nIn the meantime Katharine, not suspecting that John had written to her,\nand of course utterly ignorant of the truth about his doings on the\npreceding day, felt that she must find some occupation, no matter how\ntrivial, to take her mind out of the strong current of painful thought\nwhich must at last draw her down into the very vortex of despair's own\nwhirlpool. It seemed to her that she had never before even faintly\nguessed the meaning of pain nor the unknown extent of possible mental\nsuffering. As for forming any resolution, or even distinguishing the\ndirection of her probable course in the immediate future, she was\nutterly incapable of any such effort or thought. The longing for total\nannihilation was perhaps uppermost among her instincts just then, as it\noften is with men and women who have been at once bitterly disappointed\nand deeply wounded, and who find themselves in a position from which no\nescape seems possible. Katharine wished with all her young heart that\nthe world were a lighted candle and that she could blow it out.\n\nIt must not be believed, however, that her love for John Ralston had\ndisappeared as suddenly and totally as she should have liked to\nextinguish the universe. It had not been of sudden growth nor of\ncapricious blooming. Its roots were deep, its stem was strong, its\nflowers were sweet--and the blight which had fallen upon it was the more\ncruel. A frostbitten rose-tree is a sadder sight than a withered\nmushroom or a blade of dried grass. It was real, honest, unsuspecting,\nstrong, maidenly love, and it stood there still in the midst of her\nheart, hanging its head in the cold, while she gazed at it and\nwondered, and choked with anguish. But she could not lift her hand to\nprop it, nor to cover it and warm it again, still less to root it up and\nburn it.\n\nShe could only try to escape from seeing it, and she resolutely set\nabout making the attempt. She left her room and went downstairs,\ntreading more softly as she passed the door of the room in which her\nmother worked during the morning hours. She did not wish to see her\nagain at present, and as she descended she could not help thinking with\nwonder of the sudden and unaccountable change in their relations.\n\nShe entered the library, but though it was warm, it had that chilly look\nabout it which rooms principally used in the evening generally have when\nthere is no fire in them. The snow-glare was on everything, too, and\nmade it worse. She stood a moment in hesitation before the writing\ntable, and laid her hand uncertainly upon a sheet of writing paper. But\nshe realized that she could not write to John, and she turned away\nalmost immediately.\n\nWhat could she have written? It was easy to talk to herself of a letter;\nit was quite another matter to find words, or even to discover the\nmeaning of her own thoughts. She did not wish to see him. If she wished\nanything, it was that she might never see him again. Nothing could have\nbeen much worse than to meet him just then, and talking on paper was\nnext to talking in fact. It all rushed back upon her as she moved away,\nand she paused a moment and steadied herself against her favourite chair\nby the empty fireplace. Then she raised her head again, proudly, and\nleft the room, looking straight before her.\n\nThere was nothing to be done but to go out. The loneliness of the house\nwas absolutely intolerable, and she could not wander about in such an\naimless fashion all day long. Again she went upstairs to her room to put\non her hat and things. Mechanically she took the hat she had worn on the\nprevious day, but as she stood before the mirror and caught sight of it,\nshe suddenly took it from her head again and threw it behind her with a\npassionate gesture, stared at herself a moment and then buried her face\nin her hands. She had unconsciously put on the same frock as\nyesterday--the frock in which she had been married--it was the rough\ngrey woollen one she had been wearing every day. And there were the same\nsimple little ornaments, the small silver pin at her throat, the tiny\ngold bar of her thin watch chain at the third button from the top--the\nhat had made it complete--just as she had been married. She could not\nbear that.\n\nA few moments later she rose, and without looking at herself in the\nglass, began to change her clothes. She dressed herself entirely in\nblack, put on a black hat and a gold pin, and took a new pair of brown\ngloves from a drawer. There was a relief, now, in her altered\nappearance, as she fastened her veil. She felt that she could behave\ndifferently if she could get rid of the outward things which reminded\nher of yesterday. It is not wise to reflect contemptuously upon the\nsmallness of things which influence passionate people at great moments\nin their lives. It needs less to send a fast express off the track, if\nthe obstacle be just so placed as to cause an accident, than it does to\nupset a freight train going at twelve miles an hour.\n\nKatharine descended the stairs again with a firm step, holding her head\nhigher than before, and with quite a different look in her eyes. She had\nput on a sort of shell with her black clothes. It seemed to conceal her\nreal self from the outer world, the self that had worn rough grey\nwoollen and a silver pin and had been married to John Ralston yesterday\nmorning. She did not even take the trouble to tread softly as she passed\nher mother's studio, for she felt able to face any one, all at once. If\nJohn himself had been standing in the entry below, and if she had come\nupon him suddenly, she should have known how to meet him, and what to\nsay. She would have hurt him, and she would have been glad of it, with\nall of her. What right had John Ralston to ruin her life?\n\nBut John was not there, nor was there any possibility of her meeting\nhim that morning. He had shut himself up in his room and was waiting for\nher answer to the letter which Alexander Lauderdale had taken down town\nin his pocket, and which he meant to burn before her eyes that evening\nafter delivering his little speech. It was not probable that John would\ngo out of the house until he was convinced that no answer was to be\nexpected.\n\nKatharine went out into the street and paused on the last step. The snow\nwas deep everywhere, and wet and clinging. No attempt had as yet been\nmade to clear it away, though the horse-cars had ploughed their black\nchannel through, and it had been shovelled off the pavements before some\nof the houses. There was a slushy muddiness about it where it was not\nstill white, which promised ill for a walk. Katharine knew exactly what\nWashington Square would be like on such a morning. The little birds\nwould all be draggled and cold, the leafless twigs would be dripping,\nthe paths would be impracticable, and all the American boys would be\nsnowballing the Italian and French boys from South Fifth Avenue. The\nUniversity Building would look more than usual like a sepulchre to let,\nand Waverley Place would be more savagely respectable than ever, as its\nquiet red brick houses fronted the snow. Overhead the sky was of a\nuniform grey. It was impossible to tell from any increase of light where\nthe sun ought to be. The air was damp and cold, and all the noises of\nthe street were muffled. Far away and out of sight, a hand-organ was\nplaying 'Ah quell' amore ond'ardo'--an air which Katharine most\nespecially and heartily detested. There was something ghostly in the\nsound, as though the wretched instrument were grinding itself to death\nout of sheer weariness. Katharine thought that if the world were making\nmusic in its orbit that morning, the noise must be as melancholy and as\njarring as that of the miserable hurdy-gurdy. She thought vaguely, too,\nof the poor old man who has stood every day for years with his back to\nthe railings on the south side of West Fourteenth Street, before you\ncome to Sixth Avenue, feebly turning the handle of a little box which\nseems to be full of broken strings, which something stirs up into a\nscarcely audible jangle at every sixth or seventh revolution. He has\nyellowish grey hair, long and thick, and is generally bareheaded. She\nfelt inclined to go and see whether he were there now, in the wet snow,\nwith his torn shoes and his blind eyes, that could not feel the glare.\nShe found herself thinking of all the many familiar figures of distress,\njust below the surface of the golden stream as it were, looking up out\nof it with pitiful appealing faces, and without which New York could not\nbe itself. Her father said they made a good living out of their starving\nappearance, and firmly refused to encourage what he called pauperism by\nwhat other people called charity. Even if they were really poor, he\nsaid, they probably deserved to be, and were only reaping the fruit of\ntheir own improvidence, a deduction which did not appeal to Katharine.\n\nShe turned eastwards and would have walked up to Fourteenth Street in\norder to give the hurdy-gurdy beggar something, had she not remembered\nalmost immediately that she had no money with her. She never had any\nexcept what her mother gave her for her small expenses, and during the\nlast few days she had not cared to ask for any. In very economically\nconducted families the reluctance to ask for small sums is generally\neither the sign of a quarrel or the highest expression of sympathetic\nconsideration. Every family has its private barometer in which money\ntakes the place of mercury.\n\nKatharine suddenly remembered that she had promised Crowdie another\nsitting at eleven o'clock on Friday. It was the day and it was the hour,\nand though by no means sure that she would enter the house when she\nreached Lafayette Place, she turned in that direction and walked on,\npicking her way across the streets as well as she could. The last time\nshe had gone to Crowdie's she had gone with John, who had left her at\nthe door in order to go in search of a clergyman. She remembered that,\nas she went along, and she chose the side of the street opposite to the\none on which she had gone with Ralston.\n\nAt the door of Crowdie's house, she hesitated again. Crowdie was one of\nthe gossips. It was he who had told the story of John's quarrel with\nBright. It seemed as though he must be more repulsive to her than ever.\nOn the other hand, she realized that if she failed to appear as she had\npromised, he would naturally connect her absence with what had happened\nto Ralston. He could hardly be blamed for that, she thought, but she\nwould not have such a story repeated if she could help it. She felt very\nbrave, and very unlike the Katharine Lauderdale of two hours earlier,\nand after a moment's thought, she rang the bell and was admitted\nimmediately.\n\nHester Crowdie was just coming down the stairs, and greeted Katharine\nbefore reaching her. She seemed annoyed about something, Katharine\nthought. There was a little bright colour in her pale cheeks, and her\ndark eyes gleamed angrily.\n\n\"I'm so glad you've come!\" she exclaimed, helping her friend to take off\nher heavy coat. \"Come in with me for a minute, won't you?\"\n\n\"What's the matter?\" asked Katharine, going with her into the little\nfront room. \"You look angry.\"\n\n\"Oh--it's nothing! I'm so foolish, you know. It's silly of me. Sit\ndown.\"\n\n\"What is it, dear?\" asked Katharine, affectionately, as she sat down\nbeside Hester upon a little sofa. \"Have you and he been quarrelling?\"\n\n\"Quarrelling!\" Hester laughed gaily. \"No, indeed. That's impossible!\nNo--we were all by ourselves--Walter was singing over his work, and I\nwas just lying amongst the cushions and listening and thinking how\nheavenly it was--and that stupid Mr. Griggs came in and spoiled it all.\nSo I came away in disgust. I was so angry, just for a minute--I could\nhave killed him!\"\n\n\"Poor dear!\" Katharine could not help smiling at the story.\n\n\"Oh, of course, you laugh at me. Everybody does. But what do I care? I\nlove him--and I love his voice, and I love to be all alone with him up\nthere under the sky--and at night, too, when there's a full moon--you\nhave no idea how beautiful it is. And then I always think that the snowy\ndays, when I can't go out on foot, belong especially to me. You're\ndifferent--I knew you were coming at eleven--but that horrid Mr.\nGriggs!\"\n\n\"Poor Mr. Griggs! If he could only hear you!\"\n\n\"Walter pretends to like him. That's one of the few points on which we\nshall never agree. There's nothing against him, I know, and he's rather\nmodest, considering how he has been talked about--and all that. But one\ndoesn't like one's husband's old friends to come--bothering--you know,\nand getting in the way when one wants to be alone with him. Oh, no! I've\nnothing against the poor man--only that I hate him! How are you,\ndearest, after the ball, last night? You seemed awfully tired when I\nbrought you home. As for me, I'm worn out. I never closed my eyes till\nWalter came home--he danced the cotillion with your mother. Didn't you\nthink he was looking ill? I did. There was one moment when I was just a\nlittle afraid that--you know--that something might happen to him--as it\ndid the other day--did you notice anything?\"\n\n\"No,\" answered Katharine, thoughtfully. \"He's naturally pale. Don't you\nthink that just happened once, and isn't likely to occur again? He's\nbeen perfectly well ever since Monday, hasn't he?\"\n\n\"Oh, yes--perfectly. But you know it's always on my mind, now. I want to\nbe with him more than ever. I suppose that accounts for my being so\nangry with poor Mr. Griggs. I think I'd ask him to stay to luncheon if I\nwere sure he'd go away the minute it's over. Shouldn't you like to stay,\ndear? Shall I ask him? That will just make four. Do! I shall feel that\nI've atoned for being so horrid about him. I wish you would!\"\n\nKatharine did not answer at once. The vision of her luncheon at home\nrose disagreeably before her--there would be her mother and her\ngrandfather, and probably Charlotte. The latter was quite sure to have\nheard something about John, and would, of course, seize the occasion to\nmake unpleasant remarks. This consideration was a decisive argument.\n\n\"Dear,\" she said at last, \"if you really want me, I think I will stay.\nOnly--I don't want to be in the way, like Mr. Griggs. You must send me\naway when you've had enough of me.\"\n\n\"Katharine! What an idea! I only wish you would stay forever.\"\n\n\"Oh, no, you don't!\" answered Katharine, with a smile.\n\nHester rang the bell, and the immaculate and magnificent Fletcher\nappeared to receive her orders about the luncheon. Katharine meanwhile\nbegan to wonder at herself. She was so unlike what she had been a few\nhours earlier, in the early morning, alone in her room. She wondered\nwhether, after all, she were not heartless, or whether the memory of all\nthat had lately happened to her might not be softened, like that of a\nbad dream, which is horrible while it lasts, and at which one laughs at\nbreakfast, knowing that it has had no reality. Had her marriage any\nreality? Last night, before the ball, the question would have seemed\nblasphemous. It presented itself quite naturally just now. What value\nhad that contract? What power had the words of any man, priest or\nlayman, to tie her forever to one who had not the common decency to\nbehave like a gentleman, and to keep his appointment with her on the\nsame evening--on the evening of their wedding day? Was there a\nmysterious magic in the mere words, which made them like a witch's spell\nin a fairy story? She had not seen him since. What was he doing? Had he\nnot even enough respect for her to send her a line of apology? Merely\nwhat any man would have sent who had missed an appointment? Had she sold\nher soul into bondage for the term of her natural life by uttering two\nwords--'I will'? It was only her soul, after all. She had not seen his\nface save for a moment at her own door in the afternoon. Did he think\nthat since they had been married he need not have even the most common\nconsideration for her? It seemed so. What had she dreamed, what had she\nimagined during all those weeks and months before last Monday, while she\nhad been making up her mind that she would sacrifice anything and\neverything for the sake of making him happy? She could not be mistaken,\nnow, for she was thinking it all over quite coldly during these two\nminutes, while Hester was speaking to the butler. She was more than\ncold. She was indifferent. She could have gone back to her room and put\non her grey frock, and the little silver pin again, and could have\nlooked at herself in the mirror for an hour without any sensation but\nthat of wonder--amazement at her own folly.\n\nTalk of love! There was love between Walter Crowdie and his wife. Hester\ncould not be with any one for five minutes without speaking of him, and\nas for Crowdie himself, he was infatuated. Everybody said so. Katharine\npardoned him his pale face, his red lips, and the incomprehensible\nrepulsion she felt for him, because he loved his wife.\n\n\n\n\nCHAPTER XXIV.\n\n\nKatharine and Hester went up to the studio together, and Hester opened\nthe door.\n\n\"I've brought your sitter, Walter,\" she said, announcing Katharine.\n\"I've come back with a reinforcement.\"\n\n\"Oh, Miss Lauderdale, how do you do?\" Crowdie came forward. \"Do you know\nMr. Griggs?\" he asked in a low voice.\n\n\"Yes, he was introduced to me last night,\" explained Katharine in an\nundertone, and bending her head graciously as the elderly man bowed from\na distance.\n\n\"Oh! that's very nice,\" observed Crowdie. \"I didn't know whether you had\nmet. I hate introducing people. They're apt to remember it against one.\nGriggs is an old friend, Miss Lauderdale.\"\n\nKatharine looked at the painter and thought he was less repulsive than\nusual.\n\n\"I know,\" she answered. \"Do you really want me to sit this morning, Mr.\nCrowdie? You know, we said Friday--\"\n\n\"Of course I do! There's your chair, all ready for you--just where it\nwas last time. And the thing--it isn't a picture yet--is in the corner\nhere. Hester, dear, just help Miss Lauderdale to take off her hat, won't\nyou?\"\n\nHe crossed the room as he spoke, and began to wheel up the easel on\nwhich Katharine's portrait stood. Griggs said nothing, but watched the\ntwo women as they stood together, trying to understand the very opposite\nimpressions they made upon him, and wondering with an excess of cynicism\nwhich Crowdie thought the more beautiful. For his own part, he fancied\nthat he should prefer Hester's face and Katharine's character, as he\njudged it from her appearance.\n\nPresently Katharine seated herself, trying to assume the pose she had\ntaken at the first sitting. Crowdie disappeared behind the curtain in\nsearch of paint and brushes, and Hester sat down on the edge of a huge\ndivan. As there was no chair except Katharine's, Griggs seated himself\non the divan beside Mrs. Crowdie.\n\n\"There's never more than one chair here,\" she explained. \"It's for the\nsitter, or the buyer, or the lion-hunter, according to the time of day.\nOther people must sit on the divan or on the floor.\"\n\n\"Yes,\" answered Griggs. \"I see.\"\n\nKatharine did not think the answer a very brilliant one for a man of\nsuch reputation. Hitherto she had not had much experience of lions.\nCrowdie came back with his palette and paints.\n\n\"That's almost it,\" he said, looking at Katharine. \"A little more to the\nleft, I think--just the shade of a shadow!\"\n\n\"So?\" asked Katharine, turning her head a very little.\n\n\"Yes--only for a moment--while I look at you. Afterwards you needn't\nkeep so very still.\"\n\n\"Yes--I know. The same as last time.\"\n\nMeanwhile, Hester remembered that she had not yet asked Griggs to stay\nto luncheon, though she had taken it for granted that he would.\n\n\"Won't you stay and lunch with us?\" she asked. \"Miss Lauderdale says she\nwill, and I've told them to set a place for you. We shall be four. Do,\nif you can!\"\n\n\"You're awfully kind, Mrs. Crowdie,\" answered Griggs. \"I wish I could. I\nbelieve I have an engagement.\"\n\n\"Oh, of course you have. But that's no reason.\" Hester spoke with great\nconviction. \"I daresay you made that particular engagement very much\nagainst your will. At all events, you mean to stay, because you only say\nyou 'believe' you're engaged. If you didn't mean to stay, you would say\nat once that you 'had' an engagement which you couldn't break. Wouldn't\nyou? Therefore you will.\"\n\n\"That's a remarkable piece of logic,\" observed Griggs, smiling.\n\n\"Besides, you're a lion just now, because you've been away so long. So\nyou can break as many engagements as you please--it won't make any\ndifference.\"\n\n\"There's a plain and unadorned contempt for social rules in that, which\nappeals to me. Thanks; if you'll let me, I'll stay.\"\n\n\"Of course!\" Hester laughed. \"You see I'm married to a lion, so I know\njust what lions do. Walter, Katharine and Mr. Griggs are going to stay\nto luncheon.\"\n\n\"I'm delighted,\" answered Crowdie, from behind his easel. He was putting\nin background with an enormous brush. \"I say, Griggs--\" he began again.\n\n\"Well?\"\n\n\"Do you like Rockaways or Blue Points? I'm sure Hester has forgotten.\"\n\n\"'When love was the pearl of' my 'oyster,' I used to prefer Blue\nPoints,\" answered Griggs, meditatively.\n\n\"So does Walter,\" said Mrs. Crowdie.\n\n\"Was that a quotation--or what?\" asked Katharine, speaking to Crowdie in\nan undertone.\n\n\"Swinburne,\" answered the painter, indistinctly, for he had one of his\nbrushes between his teeth.\n\n\"Not that it makes any difference what a man eats,\" observed Griggs in\nthe same thoughtful tone. \"I once lived for five weeks on ship biscuit\nand raw apples.\"\n\n\"Good heavens!\" laughed Hester. \"Where was that? In a shipwreck?\"\n\n\"No; in New York. It wasn't bad. I used to eat a pound a day--there were\ntwelve to a pound of the white pilot-bread, and four apples.\"\n\n\"Do you mean to say that you were deliberately starving yourself? What\nfor?\"\n\n\"Oh, no! I had no money, and I wanted to write a book, so that I\ncouldn't get anything for my work till it was done. It wasn't like\nlittle jobs that one's paid for at once.\"\n\n\"How funny!\" exclaimed Hester. \"Did you hear that, Walter?\" she asked.\n\n\"Yes; but he's done all sorts of things.\"\n\n\"Were you ever as hard up as that, Walter?\"\n\n\"Not for so long; but I've had my days. Haven't I, Griggs? Do you\nremember--in Paris--when we tried to make an omelet without eggs, by the\nrecipe out of the 'Noble Booke of Cookerie,' and I wanted to colour it\nwith yellow ochre, and you said it was poisonous? I've often thought\nthat if we'd had some saffron, it would have turned out better.\"\n\n\"You cooked it too much,\" answered Griggs, gravely. \"It tasted like an\nold binding of a book--all parchment and leathery. There's nothing in\nthat recipe anyhow. You can't make an omelet without eggs. I got hold of\nthe book again, and copied it out and persuaded the great man at\nVoisin's to try it. But he couldn't do anything with it. It wasn't much\nbetter than ours.\"\n\n\"I'm glad to know that,\" said Crowdie. \"I've often thought of it and\nwondered whether we hadn't made some mistake.\"\n\nKatharine was amused by what the two men said. She had supposed that a\nfamous painter and a well-known writer, who probably did not spend a\nmorning together more than two or three times a year, would talk\nprofoundly of literature and art. But it was interesting, nevertheless,\nto hear them speak of little incidents which threw a side-light on their\nformer lives.\n\n\"Do people who succeed always have such a dreadfully hard time of it?\"\nshe asked, addressing the question to both men.\n\n\"Oh, I suppose most of them do,\" answered Crowdie, indifferently.\n\n\"'Jordan's a hard road to travel,'\" observed Griggs, mechanically.\n\n\"Sing it, Walter--it is so funny!\" suggested Hester.\n\n\"What?\" asked the painter.\n\n\"'Jordan's a hard road'--\"\n\n\"Oh, I can't sing and paint. Besides, we're driving Miss Lauderdale\ndistracted. Aren't we, Miss Lauderdale?\"\n\n\"Not at all. I like to hear you two talk--as you wouldn't to a reporter,\nfor instance. Tell me something more about what you did in Paris. Did\nyou live together?\"\n\n\"Oh, dear, no! Griggs was a sort of little great man already in those\ndays, and he used to stay at Meurice's--except when he had no money, and\nthen he used to sleep in the Calais train--he got nearly ten hours in\nthat way--and he had a free pass--coming back to Paris in time for\nbreakfast. He got smashed once, and then he gave it up.\"\n\n\"That's pure invention, Crowdie,\" said Griggs.\n\n\"Oh, I know it is. But it sounds well, and we always used to say it was\ntrue because you were perpetually rushing backwards and forwards. Oh,\nno, Miss Lauderdale--Griggs had begun to 'arrive' then, but I was only a\nstudent. You don't suppose we're the same age, do you?\"\n\n\"Oh, Walter!\" exclaimed Hester, as though the suggestion were an insult.\n\n\"Yes, Griggs is--how old are you, Griggs? I've forgotten. About fifty,\naren't you?\"\n\n\"About fifty thousand, or thereabouts,\" answered the literary man, with\na good-humoured smile.\n\nKatharine looked at him, turning completely round, for he and Mrs.\nCrowdie were sitting on the divan behind her. She thought his face was\nold, especially the eyes and the upper part, but his figure had the\nsinewy elasticity of youth even as he sat there, bending forward, with\nhis hands folded on his knees. She wished she might be with him alone\nfor a while, for she longed to make him talk about himself.\n\n\"You always seemed the same age, to me, even then,\" said Crowdie.\n\n\"Does Mr. Crowdie mean that you were never young, Mr. Griggs?\" asked\nKatharine, who had resumed her pose and was facing the artist.\n\n\"We neither of us mean anything,\" said Crowdie, with a soft laugh.\n\n\"That's reassuring!\" exclaimed Katharine, a little annoyed, for Crowdie\nlaughed as though he knew more about Griggs than he could or would tell.\n\n\"I believe it's the truth,\" said Griggs himself. \"We don't mean anything\nespecial, except a little chaff. It's so nice to be idiotic and not to\nhave to make speeches.\"\n\n\"I hate speeches,\" said Katharine. \"But what I began by asking was this.\nMust people necessarily have a very hard time in order to succeed at\nanything? You're both successful men--you ought to know.\"\n\n\"They say that the wives of great men have the hardest time,\" said\nGriggs. \"What do you think, Mrs. Crowdie?\"\n\n\"Be reasonable!\" exclaimed Hester. \"Answer Miss Lauderdale's\nquestion--if any one can, you can.\"\n\n\"It depends--\" answered Griggs, thoughtfully. \"Christopher Columbus--\"\n\n\"Oh, I don't mean Christopher Columbus, nor any one like him!\" Katharine\nlaughed, but a little impatiently. \"I mean modern people, like you two.\"\n\n\"Oh--modern people. I see.\" Mr. Griggs spoke in a very absent tone.\n\n\"Don't be so hopelessly dull, Griggs!\" protested Crowdie. \"You're here\nto amuse Miss Lauderdale.\"\n\n\"Yes--I know I am. I was thinking just then. Please don't think me rude,\nMiss Lauderdale. You asked rather a big question.\"\n\n\"Oh--I didn't mean to put you to the trouble of thinking--\"\n\n\"By the bye, Miss Lauderdale,\" interrupted Crowdie, \"you're all in black\nto-day, and on Wednesday you were in grey. It makes a good deal of\ndifference, you know, if we are to go on. Which is to be in the picture?\nWe must decide now, if you don't mind.\"\n\n\"What a fellow you are, Crowdie!\" exclaimed Griggs.\n\n\"I'll have it black, if it's the same to you,\" said Katharine, answering\nthe painter's question.\n\n\"What are you abusing me for, Griggs?\" asked Crowdie, looking round his\neasel.\n\n\"For interrupting. You always do. Miss Lauderdale asked me a question,\nand you sprang at me like a fiery and untamed wild-cat because I didn't\nanswer it--and then you interrupt and begin to talk about dress.\"\n\n\"I didn't suppose you had finished thinking already,\" answered Crowdie,\ncalmly. \"It generally takes you longer. All right. Go ahead. The\ncurtain's up! The anchor's weighed--all sorts of things! I'm listening.\nMiss Lauderdale, if you could look at me for one moment--\"\n\n\"There you go again!\" exclaimed Griggs.\n\n\"Bless your old heart, man--I'm working, and you're doing nothing. I\nhave the right of way. Haven't I, Miss Lauderdale?\"\n\n\"Of course,\" answered Katharine. \"But I want to hear Mr. Griggs--\"\n\n\"'Griggs on Struggles'--it sounds like the title of a law book,\"\nobserved Crowdie.\n\n\"You seem playful this morning,\" said Griggs. \"What makes you so\nterribly pleasant?\"\n\n\"The sight of you, my dear fellow, writhing under Miss Lauderdale's\nquestions.\"\n\n\"Doesn't Mr. Griggs like to be asked general questions?\" enquired\nKatharine, innocently.\n\n\"It's not that, Miss Lauderdale,\" said Griggs, answering her question.\n\"It's not that. I'm a fidgety old person, I suppose, and I don't like to\nanswer at random, and your question is a very big one. Not as a matter\nof fact. It's perfectly easy to say yes, or no, just as one feels about\nit, or according to one's own experience. In that way, I should be\ninclined to say that it's a matter of accident and\ncircumstances--whether men who succeed have to go through many material\ndifficulties or not. You don't hear much of all those who struggle and\nnever succeed, or who are heard of for a moment and then sink. They're\nby far the most numerous. Lots of successful men have never been poor,\nif that's what you mean by hard times--even in art and literature.\nMichael Angelo, Raphael, Leonardo da Vinci, Chaucer, Montaigne, Goethe,\nByron--you can name any number who never went through anything like what\nnine students out of ten in Paris, for instance, suffer cheerfully. It\ncertainly does not follow that because a man is great he must have\nstarved at one time or another. The very greatest seem, as a rule, to\nhave had fairly comfortable homes with everything they could need,\nunless they had extravagant tastes. That's the material view of the\nquestion. The answer is reasonable enough. It's a disadvantage to begin\nvery poor, because energy is used up in fighting poverty which might be\nused in attacking intellectual difficulties. No doubt the average man,\nwhose faculties are not extraordinary to begin with, may develop them\nwonderfully, and even be very successful--from sheer necessity, sheer\nhunger; when, if he were comfortably off, he would do nothing in the\nworld but lie on his back in the sunshine, and smoke a pipe, and\ncriticise other people. But to a man who\n\n[Illustration: \"'That's good, Crowdie,' he said thoughtfully. 'It's\ndistinctly good.'\"--Vol. II., p. 189.]\n\nis naturally so highly gifted that he would produce good work under any\ncircumstances, poverty is a drawback.\"\n\n\"You didn't know what you were going to get, Miss Lauderdale, when you\nprevailed on Griggs to answer a serious question,\" said Crowdie, as\nGriggs paused a moment. \"He's a didactic old bird, when he mounts his\nhobby.\"\n\n\"There's something wrong about that metaphor, Crowdie,\" observed Griggs.\n\"Bird mounting hobby--you know.\"\n\n\"Did you never see a crow on a cow's back?\" enquired Crowdie, unmoved.\n\"Or on a sheep? It's funny when he gets his claws caught in the wool.\"\n\n\"Go on, please, Mr. Griggs,\" said Katharine. \"It's very interesting.\nWhat's the other side of the question?\"\n\n\"Oh--I don't know!\" Griggs rose abruptly from his seat and began to pace\nthe room. \"It's lots of things, I suppose. Things we don't understand\nand never shall--in this world.\"\n\n\"But in the other world, perhaps,\" suggested Crowdie, with a smile which\nKatharine did not like.\n\n\"The other world is the inside of this one,\" said Griggs, coming up to\nthe easel and looking at the painting. \"That's good, Crowdie,\" he said,\nthoughtfully. \"It's distinctly good. I mean that it's like, that's all.\nOf course, I don't know anything about painting--that's your business.\"\n\n\"Of course it is,\" answered Crowdie; \"I didn't ask you to criticise. But\nI'm glad if you think it's like.\"\n\n\"Yes. Don't mind my telling you, Crowdie--Miss Lauderdale, I hope you'll\nforgive me--there's a slight irregularity in the pupil of Miss\nLauderdale's right eye--it isn't exactly round. It affects the\nexpression. Do you see?\"\n\n\"I never noticed it,\" said Katharine in surprise.\n\n\"By Jove--you're right!\" exclaimed Crowdie. \"What eyes you have,\nGriggs!\"\n\n\"It doesn't affect your sight in the least,\" said Griggs, \"and nobody\nwould notice it, but it affects the expression all the same.\"\n\n\"You saw it at once,\" remarked Katharine.\n\n\"Oh--Griggs sees everything,\" answered Crowdie. \"He probably observed\nthe fact last night when he was introduced to you, and has been thinking\nabout it ever since.\"\n\n\"Now you've interrupted him again,\" said Katharine. \"Do sit down again,\nMr. Griggs, and go on with what you were saying--about the other side of\nthe question.\"\n\n\"The question of success?\"\n\n\"Yes--and difficulties--and all that.\"\n\n\"Delightfully vague--'all that'! I can only give you an idea of what I\nmean. The question of success involves its own value, and the ultimate\nhappiness of mankind. Do you see how big it is? It goes through\neverything, and it has no end. What is success? Getting ahead of other\npeople, I suppose. But in what direction? In the direction of one's own\nhappiness, presumably. Every one has a prime and innate right to be\nhappy. Ideas about happiness differ. With most people it's a matter of\ntaste and inherited proclivities. All schemes for making all mankind\nhappy in one direction must fail. A man is happy when he feels that he\nhas succeeded--the sportsman when he has killed his game, the parson\nwhen he believes he has saved a soul. We can't all be parsons, nor all\ngood shots. There must be variety. Happiness is success, in each\nvariety, and nothing else. I mean, of course, belief in one's own\nsuccess, with a reasonable amount of acknowledgment. It's of much less\nconsequence to Crowdie, for instance, what you think, or I think, or\nMrs. Crowdie thinks about that picture, than it is to himself. But our\nopinion has a certain value for him. With an amateur, public opinion is\neverything, or nearly everything. With a good professional it is quite\nsecondary, because he knows much better than the public can, whether his\nwork is good or bad. He himself is his world--the public is only his\nweather, fine one day and rainy the next. He prefers his world in fine\nweather, but even when it rains he would not exchange it for any other.\nHe's his own king, kingdom and court. He's his own enemy, his own\nconqueror, and his own captive--slave is a better word. In the course of\ntime he may even become perfectly indifferent to the weather in his\nworld--that is, to the public. And if he can believe that he is doing a\ngood work, and if he can keep inside his own world, he will probably be\nhappy.\"\n\n\"But if he goes beyond it?\" asked Katharine.\n\n\"He will probably be killed--body or soul, or both,\" said Griggs, with a\nqueer change of tone.\n\n\"It seems to me, that you exclude women altogether from your paradise,\"\nobserved Mrs. Crowdie, with a laugh.\n\n\"And amateurs,\" said her husband. \"It's to be a professional paradise\nfor men--no admittance except on business. No one who hasn't had a\npicture on the line need apply. Special hell for minor poets. Crowns of\nglory may be had on application at the desk--fit not guaranteed in cases\nof swelled head--\"\n\n\"Don't be vulgar, Crowdie,\" interrupted Griggs.\n\n\"Is 'swelled head' vulgar, Miss Lauderdale?\" enquired the painter.\n\n\"It sounds like something horrid--mumps, or that sort of thing. What\ndoes it mean?\"\n\n\"It means a bad case of conceit. It's a good New York expression. I\nwonder you haven't heard it. Go on about the professional persons,\nGriggs. I'm not half good enough to chaff you. I wish Frank Miner were\nhere. He's the literary man in the family.\"\n\n\"Little Frank Miner--the brother of the three Miss Miners?\" asked\nGriggs.\n\n\"Yes--looks a well-dressed cock sparrow--always in a good humour--don't\nyou know him?\"\n\n\"Of course I do--the brother of the three Miss Miners,\" said Griggs,\nmeditatively. \"Does he write? I didn't know.\" Crowdie laughed, and\nHester smiled.\n\n\"Such is fame!\" exclaimed Crowdie. \"But then, literary men never seem to\nhave heard of each other.\"\n\n\"No,\" answered Griggs. \"By the bye, Crowdie, have you heard anything of\nChang-Li-Ho lately?\"\n\n\"Chang-Li-Ho? Who on earth is he? A Chinese laundryman?\"\n\n\"No,\" replied Griggs, unmoved. \"He's the greatest painter in the Chinese\nEmpire. But then, you painters never seem to have heard of one another.\"\n\n\"By Jove! that's not fair, Griggs! Is he to be in the professional\nheaven, too?\"\n\n\"I suppose so. There'll probably be more Chinamen than New Yorkers\nthere. They know a great deal more about art.\"\n\n\"You're getting deucedly sarcastic, Griggs,\" observed Crowdie. \"You'd\nbetter tell Miss Lauderdale more about the life to come. Your hobby\ncan't be tired yet, and if you ride him industriously, it will soon be\ntime for luncheon.\"\n\n\"We'd better have it at once if you two are going to quarrel,\" suggested\nHester, with a laugh.\n\n\"Oh, we never quarrel,\" answered Crowdie. \"Besides, I've got no soul,\nGriggs says, and he sold his own to the printer's devil ages ago--so\nthat the life to come is a perfectly safe subject.\"\n\n\"What do you mean by saying that Walter has no soul?\" asked Hester,\nlooking up quickly at Griggs.\n\n\"My dear lady,\" he answered, \"please don't be so terribly angry with me.\nIn the first place, I said it in fun; and secondly, it's quite true; and\nthirdly, it's very lucky for him that he has none.\"\n\n\"Are you joking now, or are you unintentionally funny?\" asked Crowdie.\n\n\"I don't think it's very funny to be talking about people having no\nsouls,\" said Katharine.\n\n\"Do you think every one has a soul, Miss Lauderdale?\" asked Griggs,\nbeginning to walk about again.\n\n\"Yes--of course. Don't you?\"\n\nGriggs looked at her a moment in silence, as though he were hesitating\nas to what he should say.\n\n\"Can you see the soul, as you did the defect in my eyes?\" asked\nKatharine, smiling.\n\n\"Sometimes--sometimes one almost fancies that one might.\"\n\n\"And what do you see in mine, may I ask? A defect?\"\n\nHe was quite near to her. She looked up at him earnestly with her pure\ngirl's eyes, wide, grey and honest. The fresh pallor of her skin was\nthrown into relief by the black she wore, and her features by the rich\nstuff which covered the high back of the chair. There was a deeper\ninterest in her expression than Griggs often saw in the faces of those\nwith whom he talked, but it was not that which fascinated him. There was\nsomething suggestive of holy things, of innocent suffering, of the\nromance of a virgin martyr--something which, perhaps, took him back to\nstrange sights he had seen in his youth.\n\nHe stood looking down into her eyes, a gaunt, world-worn fighter of\nfifty years, with a strong, ugly, determined but yet kindly face--the\nface of a man who has passed beyond a certain barrier which few men ever\nreach at all.\n\nCrowdie dropped his hand, holding his brush, and gazing at the two in\nsilent and genuine delight. The contrast was wonderful, he thought. He\nwould have given much to paint them as they were before him, with their\nexpressions--with the very thoughts of which the look in each face was\nborn. Whatever Crowdie might be at heart, he was an artist first.\n\nAnd Hester watched them, too, accustomed to notice whatever struck her\nhusband's attention. A very different nature was hers from any of the\nthree--one reserved for an unusual destiny, and with something of fate's\nshadowy painting already in all her outward self--passionate, first, and\nhaving, also, many qualities of mercy and cruelty at passion's command,\nbut not having anything of the keen insight into the world spiritual,\nand material, which in varied measure belonged to each of the others.\n\n\"And what defect do you see in my soul?\" asked Katharine, her exquisite\nlips just parting in a smile.\n\n\"Forgive me!\" exclaimed Griggs, as though roused from a reverie. \"I\ndidn't realize that I was staring at you.\" He was an oddly natural man\nat certain times. Katharine almost laughed.\n\n\"I didn't realize it either,\" she answered. \"I was too much interested\nin what I thought you were going to say.\"\n\n\"He's a very clever fellow, Miss Lauderdale,\" said Crowdie, going on\nwith his painting. \"But you'll turn his head completely. To be so much\ninterested--not in what he has said, or is saying, or even is going to\nsay, but just in what you think he possibly may say--it's amazing!\nGriggs, you're not half enough nattered! But then, you're so spoilt!\"\n\n\"Yes--in my old age, people are spoiling me.\" Griggs smiled rather\nsourly. \"I can't read souls, Miss Lauderdale,\" he continued. \"But if I\ncould, I should rather read yours than most books. It has something to\nsay.\"\n\n\"It's impossible to be more vague, I'm sure,\" observed Crowdie.\n\n\"It's impossible to be more flattering,\" said Katharine, quietly. \"Thank\nyou, Mr. Griggs.\"\n\nShe was beginning to be tired of Crowdie's observations upon what Griggs\nsaid--possibly because she was beginning to like Griggs himself more\nthan she had expected.\n\n\"I didn't mean to be either vague or flattering. It's servile to be the\none and weak to be the other. I said what I thought. Do you call it\nflattery to paint a beautiful portrait of Miss Lauderdale?\"\n\n\"Not unless I make it more beautiful than she is,\" answered the painter.\n\n\"You can't.\"\n\n\"That's decisive, at all events,\" laughed Crowdie. \"Not but that I agree\nwith you, entirely.\"\n\n\"Oh, I don't mean it as you do,\" answered Griggs. \"That would be\nflattery--exactly what I don't mean. Miss Lauderdale is perfectly well\naware that you're a great portrait painter and that she is not\naltogether the most beautiful young lady living at the present moment.\nYou mean flesh and blood and eyes and hair. I don't. I mean all that\nflesh and blood and eyes and hair don't mean, and never can mean.\"\n\n\"Soul,\" suggested Crowdie. \"I was talking about that to Miss Lauderdale\nthe last time she sat for me--that was on Wednesday, wasn't it--the day\nbefore yesterday? It seems like last year, for some reason or other.\nYes, I know what you mean. You needn't get into such a state of frenzied\nexcitement.\"\n\n\"I appeal to you, Mrs. Crowdie--was I talking excitedly?\"\n\n\"A little,\" answered Hester, who was incapable of disagreeing with her\nhusband.\n\n\"Oh--well--I daresay,\" said Griggs. \"It hasn't been my weakness in life\nto get excited, though.\" He laughed.\n\n\"Walter always makes you talk, Mr. Griggs,\" answered Mrs. Crowdie.\n\n\"A great deal too much. I think I shall be rude, and not stay to\nluncheon, after all.\"\n\n\"Nonsense!\" exclaimed Crowdie. \"Don't go in for being young and\neccentric--the 'man of genius' style, who runs in and out like a hen in\na thunder-storm, and is in everybody's way when he's not wanted and\ncan't be found when people want him. You've outgrown that sort of\nabsurdity long ago.\"\n\nKatharine would have liked to see Griggs' face at that moment, but he\nwas behind her again. There was something in the relation of the two\nmen which she found it hard to understand. Crowdie was much younger than\nGriggs--fourteen or fifteen years, she fancied, and Griggs did not seem\nto be at all the kind of man with whom people would naturally be\nfamiliar or take liberties, to use the common phrase. Yet they talked\ntogether like a couple of schoolboys. She should not have thought,\neither, that they could be mutually attracted. Yet they appeared to have\nmany ideas in common, and to understand each other wonderfully well.\nCrowdie was evidently not repulsive to Griggs as he was to many men she\nknew--to Bright and Miner, for instance--and the two had undoubtedly\nbeen very intimate in former days. Nevertheless, it was strange to hear\nthe younger man, who was little more than a youth in appearance,\ncomparing the celebrated Paul Griggs to a hen in a thunder-storm, and\nstill stranger to see that Griggs did not resent it at all. An older\nwoman might have unjustly suspected that the elderly man of letters was\nin love with Hester Crowdie, but such an idea could never have crossed\nKatharine's mind. In that respect she was singularly unsophisticated.\nShe had been accustomed to see her beautiful mother surrounded and\ncourted by men of all ages, and she knew that her mother was utterly\nindifferent to them except in so far as she liked to be admired. In some\nbooks, men fall in love with married women, and Katharine had always\nbeen told that those were bad books, and had accepted the fact without\nquestion and without interest.\n\nBut in ordinary matters she was keen of perception. It struck her that\nthere was some bond or link between the two men, and it seemed strange\nto her that there should be--as strange as though she had seen an old\nwolf playing amicably with a little rabbit. She thought of the two\nanimals in connection with the two men.\n\nWhile she had been thinking, Hester and Griggs had been talking together\nin lower tones, on the divan, and Crowdie had been painting\nindustriously.\n\n\"It's time for luncheon,\" said Mrs. Crowdie. \"Mr. Griggs says he really\nmust go away very early, and perhaps, if Katharine will stay, she will\nlet you paint for another quarter of an hour afterward.\"\n\n\"I wish you would!\" answered Crowdie, with alacrity. \"The snow-light is\nso soft--you see the snow lies on the skylight like a blanket.\"\n\nKatharine looked up at the glass roof, turning her head far back, for it\nwas immediately overhead. When she dropped her eyes she saw that Griggs\nwas looking at her again, but he turned away instantly. She had no\nsensation of unpleasantness, as she always had when she met Crowdie's\nwomanish glance; but she wondered about the man and his past.\n\nHester was just leaving the studio, going downstairs to be sure that\nluncheon was ready, and Crowdie had disappeared behind his curtain to\nput his palette and brushes out of sight, as usual. Katharine was alone\nwith Griggs for a few moments. They stood together, looking at the\nportrait.\n\n\"How long have you known Mr. Crowdie?\" she asked, yielding to an\nirresistible impulse.\n\n\"Crowdie?\" repeated Griggs. \"Oh--a long time--fifteen or sixteen years,\nI should think. That's going to be a very good portrait, Miss\nLauderdale--one of his best. And Crowdie, at his best, is first rate.\"\n\n\n\n\nCHAPTER XXV.\n\n\nKatharine was conscious that during the time she had spent in the studio\nshe had been taken out of herself. She had listened to what the others\nhad said, she had been interested in Griggs, she had speculated upon the\nprobable origin of his apparent friendship with Crowdie; in a word, she\nhad temporarily lulled the tempest which had threatened to overwhelm her\naltogether in the earlier part of the morning. She was not much given to\nanalyzing herself and her feelings, but as she descended the stairs,\nfollowed by Crowdie and Griggs, she was inclined to doubt whether she\nwere awake, or dreaming. She told herself that it was all true; that she\nhad been married to John Ralston on the previous morning in the quiet,\nremote church, that she had seen John for one moment in the afternoon,\nat her own door, that he had failed her in the evening, and that she\nknew only too certainly how he had disgraced himself in the eyes of\ndecent people during the remainder of the day. It was all true, and yet\nthere was something misty about it all, as though it were a dream. She\ndid not feel angry or hurt any more. It only seemed to her that John,\nand everything connected with him, had all at once passed out of her\nlife, beyond the possibility of recall. And she did not wish to recall\nit, for she had reached something like peace, very unexpectedly.\n\nIt was, of course, only temporary. Physically speaking, it might be\nexplained as the reaction from violent emotions, which had left her\nnerves weary and deadened. And speaking not merely of the material side,\nit is true that the life of love has moments of suspended animation,\nduring which it is hard to believe that love was ever alive at\nall--times when love has a past and a future, but no present.\n\nIf she had met John at that moment, on the stairs, she would very\nprobably have put out her hand quite naturally, and would have greeted\nhim with a smile, before the reality of all that had happened could come\nback to her. Many of us have dreamed that those dearest to us have done\nus some cruel and bitter wrong, struck us, insulted us, trampled on our\nlife-long devotion to them; and in the morning, awaking, we have met\nthem, and smiled, and loved them just the same. For it was only a dream.\nAnd there are those who have known the reality; who, after much time,\nhave very suddenly found out that they have been betrayed and wickedly\ndeceived, and used ill, by their most dear--and who, in the first\nmoment, have met them, and smiled, and loved them just the same. For it\nwas only a dream, they thought indeed. And then comes the waking, which\nis as though one fell asleep upon his beloved's bosom and awoke among\nthorns, and having a crown of thorns about his brows--very hard to bear\nwithout crying aloud.\n\nKatharine pressed the polished banister of the staircase with her hand,\nand with the other she found the point of the little gold pin she wore\nat her throat and made it prick her a little. It was a foolish idea and\na childish thought. She knew that she was not really dreaming, and yet,\nas though she might have been, she wanted a physical sensation to assure\nher that she was awake. Griggs was close behind her. Crowdie had stopped\na moment to pull the cord of a curtain which covered the skylight of the\nstaircase.\n\n\"I wonder where real things end, and dreams begin!\" said Katharine, half\nturning her head, and then immediately looking before her again.\n\n\"At every minute of every hour,\" answered Griggs, as quickly as though\nthe thought had been in his own mind.\n\nFrom higher up came Crowdie's golden voice, singing very softly to\nhimself. He had heard the question and the answer.\n\n\"'La vie est un songe,'\" he sang, and then, breaking off suddenly,\nlaughed a little and began to descend.\n\nAt the first note, Katharine stood still and turned her face upwards.\nGriggs stopped, too, and looked down at her. Even after Crowdie had\nlaughed Katharine did not move.\n\n\"I wish you'd go on, Mr. Crowdie!\" she cried, speaking so that he could\nhear her.\n\n\"Griggs is anxious for the Blue Points,\" he answered, coming down.\n\"Besides, he hates music, and makes no secret of the fact.\"\n\n\"Is it true? Do you really hate music?\" asked Katharine, turning and\nbeginning to descend again.\n\n\"Quite true,\" answered Griggs, quietly. \"I detest it. Crowdie's a\nnuisance with his perpetual yapping.\"\n\nCrowdie laughed good naturedly, and Katharine said nothing. As they\nreached the lower landing she turned and paused an instant, so that\nGriggs came beside her.\n\n\"Did you always hate music?\" she asked, looking up into his\nweather-beaten face with some curiosity.\n\n\"Hm!\" Griggs uttered a doubtful sound. \"It's a long time since I heard\nany that pleased me, at all events.\"\n\n\"There are certain subjects, Miss Lauderdale, upon which Griggs is\nunapproachable, because he won't say anything. And there are others upon\nwhich it is dangerous to approach him, because he is likely to say too\nmuch. Hester! Where are you?\"\n\nHe disappeared into the little room at the front of the house in search\nof his wife, and Katharine stood alone with Griggs in the entry. Again\nshe looked at him with curiosity.\n\n\"You're a very good-humoured person, Mr. Griggs,\" she said, with a\nsmile.\n\n\"You mean about Crowdie? Oh, I can stand a lot of his chaff--and he has\nto stand mine, too.\"\n\n\"That was a very interesting answer you gave to my question about\ndreams,\" said Katharine, leaning against the pillar of the banister.\n\n\"Was it? Let me see--what did I say?\" He seemed to be absent-minded\nagain.\n\n\"Come to luncheon!\" cried Crowdie, reappearing with Hester at that\nmoment. \"You can talk metaphysics over the oysters.\"\n\n\"Metaphysics!\" exclaimed Griggs, with a smile.\n\n\"Oh, I know,\" answered Crowdie. \"I can't tell the difference between\nmetaphysics and psychics, and geography and Totem. It is all precisely\nthe same to me--and it is to Griggs, if he'd only acknowledge it. Come\nalong, Miss Lauderdale--to oysters and culture!\"\n\nHester laughed at Crowdie's good spirits, and Griggs smiled. He had\nlarge, sharp teeth, and Katharine thought of the wolf and the rabbit\nagain. It was strange that they should be on such good terms.\n\nThey sat down to luncheon. The dining-room, like every other part of\nthe small house, had been beautified as much as its position and\ndimensions would allow. It had originally been small, but an extension\nof glass had been built out into the yard, which Hester had turned into\na fernery. There were a great number of plants of many varieties, some\nof which had been obtained with great difficulty from immense distances.\nHester had been told that it would be impossible to make them grow in an\ninhabited room, but she had succeeded, and the result was something\naltogether out of the common.\n\nShe admitted that, besides the attention she bestowed upon the plants\nherself, they occupied the whole time of a specially trained gardener.\nThey were her only hobby, and where they were concerned, time and money\nhad no value for her. The dining-room itself was simple, but exquisite\nin its way. There were a few pieces of wonderfully chiselled silver on\nthe sideboard, and the glasses on the table were Venetian and Bohemian,\nand very old. The linen was as fine as fine writing paper, the porcelain\nwas plain white S\u00e8vres. There was nothing superfluous, but there were\nall the little, unobtrusive, almost priceless details which are the\nhighest expression \"of intimate luxury--in which the eye alone receives\nrest, while the other senses are flattered to the utmost. Colour and the\nprecious metals are terribly cheap things nowadays compared with what\nappeals to touch and taste. There are times when certain dainties, like\nterrapin, for instance, are certainly worth much more than their weight\nin silver, if not quite their weight in gold. But as for that, to say\nthat a man is worth his weight in gold has ceased to mean very much.\nSome ingenious persons have lately calculated that the average man's\nweight in gold would be worth about forty thousand dollars, and that a\nfew minutes' worth of the income of some men living would pay for a\nlife-sized golden calf. The further development of luxury will be an\ninteresting thing to watch during the next century. A poor woman in New\nYork recently returned a roast turkey to a charitable lady who had sent\nit to her, with the remark that she was accustomed to eat roast beef at\nChristmas, though she 'did not mind turkey on Thanksgiving Day.'\n\nKatharine wondered how far such a man as Griggs, who said that he hated\nmusic, could appreciate the excessive refinement of a luxury which could\nbe felt rather than seen. It was all familiar to Katharine, and there\nwere little things at the Crowdies which she longed to have at home.\nGriggs ate his oysters in silence. Fletcher came to his elbow with a\ndecanter.\n\n\"Vin de Grave, sir?\" enquired the old butler in a low voice.\n\n\"No wine, thank you,\" said Griggs.\n\n\"There's Sauterne, isn't there, Walter?\" asked Hester. \"Perhaps Mr.\nGriggs--\"\n\n\"Griggs is a cold water man, like me,\" answered Crowdie. \"His secret\nvice is to drink a bucket of it, when nobody is looking.\"\n\nFletcher looked disappointed, and replaced the decanter on the\nsideboard.\n\n\"It's uncommon to see two men who drink nothing,\" observed Hester. \"But\nI remember that Mr. Griggs never did.\"\n\n\"Never--since you knew me, Mrs. Crowdie. I did when I was younger.\"\n\n\"Did you? What made you give it up?\"\n\nKatharine felt a strange pain in her heart, as they began to talk of the\nsubject. The reality was suddenly coming back out of dreamland.\n\n\"I lost my taste for it,\" answered Griggs, indifferently.\n\n\"About the same time as when you began to hate music, wasn't it?\" asked\nCrowdie, gravely.\n\n\"Yes, I daresay.\"\n\nThe elder man spoke quietly enough, and there was not a shade of\ninterest in his voice as he answered the question. But Katharine, who\nwas watching him unconsciously, saw a momentary change pass over his\nface. He glanced at Crowdie with an expression that was almost savage.\nThe dark, weary eyes gleamed fiercely for an instant, the great veins\nswelled at the lean temples, the lips parted and just showed the big,\nsharp teeth. Then it was all over again and the kindly look came back.\nCrowdie was not smiling, and the tone in which he had asked the question\nshowed plainly enough that it was not meant as a jest. Indeed, the\npainter himself seemed unusually serious. But he had not been looking at\nGriggs, nor had Hester seen the sudden flash of what was very like\nhalf-suppressed anger. Katharine wondered more and more, and the little\nincident diverted her thoughts again from the suggestion which had given\nher pain.\n\n\"Lots of men drink water altogether, nowadays,\" observed Crowdie. \"It's\na mistake, of course, but it's much more agreeable.\"\n\n\"A mistake!\" exclaimed Katharine, very much astonished.\n\n\"Oh, yes--it's an awful mistake,\" echoed Griggs, in the most natural way\npossible.\n\n\"I'm not so sure,\" said Hester Crowdie, in a tone of voice which showed\nplainly that the idea was not new to her.\n\n\"I don't understand,\" said Katharine, unable to recover from her\nsurprise. \"I always thought that--\" she checked herself and looked\nacross at the ferns, for her heart was hurting her again.\n\nShe suddenly realized, also, that considering what had happened on the\nprevious night, it was very tactless of Crowdie not to change the\nsubject. But he seemed not at all inclined to drop it yet.\n\n\"Yes,\" he said. \"In the first place, total abstinence shortens life.\nStatistics show that moderate consumers of alcoholic drinks live\nconsiderably longer than drunkards and total abstainers.\"\n\n\"Of course,\" assented Griggs. \"A certain amount of wine makes a man lazy\nfor a time, and that rests his nerves. We who drink water accomplish\nmore in a given time, but we don't live so long. We wear ourselves out.\nIf we were not the strongest generation there has been for centuries, we\nshould all be in our graves by this time.\"\n\n\"Do you think we are a very strong generation?\" asked Crowdie, who\nlooked as weak as a girl.\n\n\"Yes, I do,\" answered Griggs. \"Look at yourself and at me. You're not an\nathlete, and an average street boy of fifteen or sixteen might kill you\nin a fight. That has nothing to do with it. The amount of actual hard\nwork, in your profession, which you've done--ever since you were a mere\nlad--is amazing, and you're none the worse for it, either. You go on,\njust as though you had begun yesterday. Heaving weights and rowing races\nis no test of what a man's strength will bear in everyday life. You\ndon't need big muscles and strong joints. But you need good nerves and\nenormous endurance. I consider you a very strong man--in most ways that\nare of any use.\"\n\n\"That's true,\" said Mrs. Crowdie. \"It's what I've always been trying to\nput into words.\"\n\n\"All the same,\" continued Griggs, \"one reason why you do more than other\npeople is that you drink water. If we are strong, it's because the last\ngeneration and the one before it lived too well. The next generation\nwill be ruined by the advance of science.\"\n\n\"The advance of science!\" exclaimed Katharine. \"But, Mr. Griggs--what\nextraordinary ideas you have!\"\n\n\"Have I? It's very simple, and it's absolutely true. We've had the\nsurvival of the fittest, and now we're to have the survival of the\nweakest, because medical science is learning how to keep all the\nweaklings alive. If they were puppies, they'd all be drowned, for fear\nof spoiling the breed. That's rather a brutal way of putting it, but\nit's true. As for the question of drink, the races that produce the most\neffect on the world are those that consume the most meat and the most\nalcohol. I don't suppose any one will try to deny that. Of course, the\nconsequences of drinking last for many generations after alcohol has\ngone out of use. It's pretty certain that before Mohammed's time the\nnational vice of the Arabs was drunkenness. So long as the effects\nlasted--for a good many generations--they swept everything before them.\nThe most terrible nation is the one that has alcohol in its veins but\nnot in its head. But when the effects wore out, the Arabs retired from\nthe field before nations that drank--and drank hard. They had no\nchance.\"\n\n\"What a horrible view to take!\" Katharine was really shocked by the\nman's cool statements, and most of all by the appearance of indisputable\ntruth which he undoubtedly gave to them.\n\n\"And as for saying that drink is the principal cause of crime,\" he\ncontinued, quietly finishing a piece of shad on his plate, \"it's the\nmost arrant nonsense that ever was invented. The Hindus are total\nabstainers and always have been, so far as we know. The vast majority of\nthem take no stimulant whatever, no tea, no coffee. They smoke a little.\nThere are, I believe, about two hundred millions of them alive now, and\ntheir capacity for most kinds of wickedness is quite as great as ours.\nAny Indian official will tell you that. It's pure nonsense to lay all\nthe blame on whiskey. There would be just as many crimes committed\nwithout it, and it would be much harder to detect them, because the\ncriminals would keep their heads better under difficulties. Crime is in\nhuman nature, like virtue--like most things, if you know how to find\nthem.\"\n\n\"That's perfectly true,\" said Crowdie. \"I believe every word of it. And\nI know that if I drank a certain amount of wine I should have a better\nchance of long life, but I don't like the taste of it--couldn't bear it\nwhen I was a boy. I like to see men get mellow and good-natured over a\nbottle of claret, too. All the same, there's nothing so positively\ndisgusting as a man who has had too much.\"\n\nHester looked at him quickly, warning him to drop the subject. But\nGriggs knew nothing of the circumstances, and went on discussing the\nmatter from his original point of view.\n\n\"There's a beast somewhere, in every human being,\" he said,\nthoughtfully. \"If you grant the fact that it is a beast, it's no worse\nto look at than other beasts. But it's quite proper to call a drunkard a\nbeast, because almost all animals will drink anything alcoholic which\nhasn't a bad taste, until they're blind drunk. It's a natural instinct.\nDid you ever see a goat drink rum, or a Western pony drink a pint of\nwhiskey? All animals like it. I've tried it on lots of them. It's an old\nsailors' trick.\"\n\n\"I think it's horrid!\" exclaimed Hester. \"Altogether, it's a most\nunpleasant subject. Can't we talk of something else?\"\n\n\"Griggs can talk about anything except botany, my dear,\" said Crowdie.\n\"Don't ask him about ferns, unless you want an exhibition of ignorance\nwhich will startle you.\"\n\nKatharine sat still in silence, though it would have been easy for her\nat that moment to turn the conversation into a new channel, by asking\nGriggs the first question which chanced to present itself. But she could\nnot have spoken just then. She could not eat, either, though she made a\npretence of using her fork. The reality had come back out of dreamland\naltogether this time, and would not be banished again. The long\ndiscussion about the subject which of all others was most painful to\nher, and the cynical indifference with which the two men had discussed\nit, had goaded her memory back through all the details of the last\ntwenty-four hours. She was scarcely conscious that Hester had\ninterfered, as she became more and more absorbed by her own suffering.\n\n\"Shall we talk of roses and green fields and angels' loves?\" asked\nGriggs. \"How many portraits have you painted since last summer,\nCrowdie?\"\n\n\"By way of reminding me of roses you stick the thorns into me--four, I\nthink--and two I'm doing now, besides Miss Lauderdale's. There's been a\ndepression down town. That accounts for the small number. Portrait\npainters suffer first. In hard times people don't want them.\"\n\n\"Yes,\" answered Griggs, thoughtfully. \"Portrait painters and hatters.\nDid you know that, Crowdie? When money is tight in Wall Street, people\ndon't bet hats, and the hatters say it makes a great difference.\"\n\n\"That's queer. And you--how many books have you written?\"\n\n\"Since last summer? Only one--a boshy little thing of sixty thousand.\"\n\n\"Sixty thousand what?\" asked Hester. \"Dollars?\"\n\n\"Dollars!\" Griggs laughed. \"No--only words. Sixty thousand words. That's\nthe way we count what we do. No--it's a tiresome little thing. I had an\nidea,--or thought I had,--and just when I got to the end of it I found\nit was trash. That's generally the way with me, unless I have a stroke\nof luck. Haven't you got an idea for me, Mrs. Crowdie? I'm getting old\nand people won't give me any, as they used to.\"\n\n\"I wish I had! What do you want? A love story?\"\n\n\"Of course. But what I want is a character. There are no new plots, nor\nincidents, nor things of that sort, you know. Everything that's ever\nhappened has happened so often. But there are new characters. The end of\nthe century, the sharp end of the century, is digging them up out of the\nsands of life--as you might dig up clams with a pointed stick.\"\n\n\"That's bathos!\" laughed Crowdie. \"The sands of life--and clams!\"\n\n\"I wish you'd stick to your daubs, Crowdie, and leave my English alone!\"\nsaid Griggs. \"It sells just as well as your portraits. No--what I mean\nis that just when fate is twisting the tail of the century--\"\n\n\"Really, my dear fellow--that's a little too bad, you know! To compare\nthe century to a refractory cow!\"\n\n\"Crowdie,\" said Griggs, gravely, \"in a former state I was a wolf, and\nyou were a rabbit, and I gobbled you up. If you go on interrupting me,\nI'll do it again and destroy your Totem.\"\n\nKatharine started suddenly and stared at Griggs. It seemed so strange\nthat he should have used the very words--wolf and rabbit--which had been\nin her mind more than once during the morning.\n\n\"What is it, Miss Lauderdale?\" he asked, in some surprise. \"You look\nstartled.\"\n\n\"Oh--nothing!\" Katharine hastened to say. \"I happened to have thought of\nwolves and rabbits, and it seemed odd that you should mention them.\"\n\n\"Write to the Psychical Research people,\" suggested Crowdie. \"It's a\ndistinct case of thought-transference.\"\n\n\"I daresay it is,\" said Griggs, indifferently. \"Everything is\ntransferable--why shouldn't thoughts be?\"\n\n\"Everything?\" repeated Crowdie. \"Even the affections?\"\n\n\"Oh, yes--even the affections--but punched, like a railway ticket,\"\nanswered Griggs, promptly. Everybody laughed a little, except Griggs\nhimself.\n\n\"Of course the affections are transferable,\" he continued, meditatively.\n\"The affections are the hat--the object is only the peg on which it's\nhung. One peg is almost as good as another--if it's within reach; but\nthe best place for the hat is on the man's own head. Nothing shields a\nman like devoting all his affections to himself.\"\n\n\"That's perfectly outrageous!\" exclaimed Hester Crowdie. \"You make one\nthink that you don't believe in anything! Oh, it's too bad--really it\nis!\"\n\n\"I believe in ever so many things, my dear lady,\" answered Griggs,\nlooking at her with a singularly gentle expression on his weather-beaten\nface. \"I believe in lots of good things--more than Crowdie does, as he\nknows. I believe in roses, and green fields, and love, as much as you\ndo. Only--the things one believes in are not always good for one--it\ndepends--love's path may lie among roses or among thorns; yet the path\nalways has two ends--the one end is life, if the love is true.\"\n\n\"And the other?\" asked Katharine, meeting his far-away glance.\n\n\"The other is death,\" he answered, almost solemnly.\n\nA momentary silence followed the words. Even Crowdie made no remark,\nwhile both Hester and Katharine watched the elder man's face, as women\ndo when a man who has known the world well speaks seriously of love.\n\n\"But then,\" added Griggs himself, more lightly, and as though to destroy\nthe impression he had made, \"most people never go to either end of the\npath. They enter at one side, look up and down it, cross it, and go out\nat the other. Something frightens them, or they don't like the colour of\nthe roses, or they're afraid of the thorns--in nine cases out of ten,\nsomething drives them out of it.\"\n\n\"How can one be driven out of love?\" asked Katharine, gravely.\n\n\"I put the thing generally, and adorned it with nice similes and\nthings--and now you want me to explain all the details!\" protested\nGriggs, with a little rough laugh. \"How can one be driven out of love?\nIn many ways, I fancy. By a real or imaginary fault of the other person\nin the path, I suppose, as much as by anything. It won't do to stand at\ntrifles when one loves. There's a meaning in the words of the marriage\nservice--'for better, for worse.'\"\n\n\"I know there is,\" said Katharine, growing pale, and choking herself\nwith the words in the determination to be brave.\n\n\"Of course there is. People don't know much about one another when they\nget married. At least, not as a rule. They've met on the stage like\nactors in a play--and then, suddenly, they meet in private life, and are\nquite different people. Very probably the woman is jealous and\nextravagant, and has a temper, and has been playing the ingenuous young\ngirl's parts on the stage. And the man, who has been doing the\nself-sacrificing hero, who proposes to go without butter in order to\nsupport his starving mother-in-law, turns out to be a gambler--or\ndrinks, or otherwise plays the fool. Of course that's all very\ndistressing to the bride or the bridegroom, as the case may be. But it\ncan't be helped. They've taken one another 'for better, for worse,' and\nit's turned out to be for worse. They can go to Sioux City and get a\ndivorce, but then that's troublesome and scandalous, and one thing and\nanother. So they just put up with it. Besides, they may love each other\nso much that the defects don't drive them out of it. Then the bad one\ndrags down the good one--or, in rare cases, the good one raises the bad\none. Oh, yes--I'm not a cynic--that happens, too, from time to time.\"\n\nCrowdie looked at his wife with his soft, languishing glance, and if\nKatharine had been watching him, she might have seen on his red lips\nthe smile she especially detested. But she was looking down and pressing\nher hands together under the table. Hester Crowdie's eyes were fixed on\nher face, for she was very pale and was evidently suffering. Griggs also\nlooked at her, and saw that something unusual was happening.\n\n\"Mrs. Crowdie,\" he said, vigorously changing the subject, as a man can\nwho has been leading the conversation, \"if it isn't a very rude\nquestion, may I ask where you get the extraordinary ham you always have\nwhenever I lunch with you? I've been all over the world, and I've never\neaten anything like it. I'm not sure whether it's the ham itself, or\nsome secret in the cooking.\"\n\nMrs. Crowdie glanced at Katharine's face once more, and then looked at\nhim. Crowdie also turned towards him, and Katharine slowly unclasped her\nhands beneath the table, as though the bitterness of death were passed.\n\n\"Oh--the ham?\" repeated Mrs. Crowdie. \"They're Yorkshire hams, aren't\nthey, Walter? You always order them.\"\n\n\"No, my dear,\" answered Crowdie. \"They're American. We've not had any\nEnglish ones for two or three years. Fletcher gets them. He's a better\njudge than the cook. Griggs is quite right--there's a trick about\nboiling them--something to do with changing the water a certain number\nof times before you put in the wine. Are you going to set up\nhousekeeping, Griggs? I should think that oatmeal and water and dried\nherrings would be your sort of fare, from what I remember.\"\n\n\"Something of that kind,\" answered Griggs. \"Anything's good enough that\nwill support life.\"\n\nThe luncheon came to an end without any further incident, and the\nconversation ran on in the very smallest of small talk. Then Griggs, who\nwas a very busy man, lighted a cigarette and took his departure. As he\nshook hands with Katharine, and bowed in his rather foreign way, he\nlooked at her once more, as though she interested him very much.\n\n\"I hope I shall see you again,\" said Katharine, quietly.\n\n\"I hope so, indeed,\" answered Griggs. \"You're very kind to say so.\"\n\nWhen he was gone the other three remained together in the little front\nroom, which has been so often mentioned.\n\n\"Will you sit for me a little longer, Miss Lauderdale?\" asked Crowdie.\n\n\"Oh, don't work any more just yet, Walter!\" cried Hester, with sudden\nanxiety.\n\n\"Why? What's the matter?\" enquired Crowdie in some surprise.\n\n\"You know what Mr. Griggs was just saying at luncheon. You work so\nhard! You'll overdo it some day. It's perfectly true, you know. You\nnever give yourself any rest!\"\n\n\"Except during about one-half of the year, my dear, when you and I do\nabsolutely nothing together in the most beautiful places in the\nworld--in the most perfect climates, and without one solitary little\nshadow of a care for anything on earth but our two selves.\"\n\n\"Yes--I know. But you work all the harder the rest of the time. Besides,\nwe haven't been abroad this year, and you say we can't get away for at\nleast two months. Do give yourself time to breathe--just after luncheon,\ntoo. I'm sure it's not good for him, is it Katharine?\" she asked,\nappealing to her friend.\n\n\"Of course not!\" answered Katharine. \"And besides, I must run home. My\ndear, just fancy! I forgot to ask you to send word to say that I wasn't\ncoming, and they won't know where I am. But we lunch later than you\ndo--if I go directly, I shall find them still at table.\"\n\n\"Nonsense!\" exclaimed Hester. \"You don't want to go really? Do you? You\nknow, I could send word still--it wouldn't be too late.\" She glanced at\nher husband, who shook his head, and smiled--he was standing behind\nKatharine. \"Well--if you must, then,\" continued Hester, \"I won't keep\nyou. But come back soon. It seems to me that I never see you now--and I\nhave lots of things to tell you.\"\n\nKatharine shook hands with Crowdie, whose soft, white fingers felt cold\nin hers. Hester went out with her into the entry, and helped her to put\non her thick coat.\n\n\"Take courage, dear!\" said Mrs. Crowdie in a low voice, as she kissed\nher. \"It will come right in the end.\"\n\nKatharine looked fixedly at her for a few seconds, buttoning her coat.\n\n\"It's not courage that I need,\" she said slowly, at last. \"I think I\nhave enough--good-bye--Hester, darling--good-bye!\"\n\nShe put her arms round her friend and kissed her three times, and then\nturned quickly and let herself out, leaving Hester standing in the\nentry, wondering at the solemn way in which she had taken leave of her.\n\n\n\n\nCHAPTER XXVI.\n\n\nKatharine's mood had changed very much since she had entered the\nCrowdies' house. She had felt then a certain sense of strength which had\nbeen familiar to her all her life, but which had never before seemed so\nreal and serviceable. She had been sure that she could defy the\nworld--in that black frock she wore--and that her face would be of\nmarble and her heart of steel under all imaginable circumstances. She\nhad carried her head high and had walked with a firm tread. She had felt\nthat if she met John Ralston she could tell him what she thought of him,\nand hurt him, so that in his suffering, at least, he should repent of\nwhat he had done.\n\nIt was different now. She did not attempt to find reasons for the\ndifference, and they would have been hard to discover. But she knew that\nshe had been exposed to a sort of test of her strength, and had broken\ndown, and that Hester Crowdie had seen her defeat. Possibly it was the\nknowledge that Hester had seen and understood which was the most\nimmediately painful circumstance at the present moment; but it was not\nthe most important one, for she was really quite as brave as she had\nbelieved herself, and what suffered most in her was not her vanity.\n\nThe conversation at table had somehow brought the whole truth more\nclearly before her, as the developer brings out the picture on a\nphotographer's plate. The facts were fixed now, and she could not hide\nthem nor turn from them at will.\n\nWhether she were mistaken or not, the position was bad enough. As she\nsaw it, it was intolerable. By her own act, by the exercise of her own\nwill, and by nothing else, she had been secretly married to John\nRalston. She had counted with certainty upon old Robert Lauderdale to\nprovide her husband with some occupation immediately, feeling sure that\nwithin a few days she should be able to acknowledge the marriage and\nassume her position before the world as a married woman. But Robert\nLauderdale had demonstrated to her that this was impossible under the\nconditions she required, namely, that John should support himself. He\nhad indeed offered to make her independent, but that solution of the\ndifficulty was not acceptable. To obtain what she and Ralston had both\ndesired, it was necessary, and she admitted the fact, that John should\nwork regularly in some office for a certain time. Robert Lauderdale\nhimself could not take an idle man from a fashionable club and suddenly\nturn him into a partner in a house of business or a firm of lawyers, if\nthe idle man himself refused to accept money in any shape. Even if he\nhad accepted it, such a proceeding would have been criticised and\nlaughed at as a piece of plutocratic juggling. It would have made John\ncontemptible. Therefore it was impossible that John and Katharine should\nhave a house of their own and appear as a married couple for some time,\nfor at least a year, and probably for a longer period. Under such\ncircumstances to declare the marriage would have been to make themselves\nthe laughing-stock of society, so long as John continued to live under\nhis mother's roof, and Katharine with her father. The secret marriage\nwould have to be kept a secret, except, perhaps, from the more discreet\nmembers of the family. Alexander Lauderdale would have to be told, and\nlife would not be very pleasant for Katharine until she could leave the\npaternal dwelling. She knew that, but she would have been able to bear\nit, to look upon the next year or two as years of betrothal, and to give\nher whole heart and soul to help John in his work. It was the worst\ncontingency which she had foreseen when she had persuaded him to take\nthe step with her, and she had certainly not expected that it could\narise; but since it had arisen, she was ready to meet it. There was\nnothing within the limits of reason which she would not have done for\nJohn, and she had driven those limits as far from ordinary common sense\nas was possible, to rashness, even to the verge of things desperate in\ntheir folly.\n\nShe knew that. But she had counted on John Ralston with that singularly\nwhole-hearted faith which characterizes very refined women. Many years\nago, when analytical fiction was in its infancy, Charles de Bernard made\nthe very wise and true observation that no women abandon themselves more\ncompletely in thought and deed to the men they love, or make such real\nslaves of themselves, as those whom he calls 'great ladies,'--that is,\nas we should say, women of the highest refinement, the most unassailable\nsocial position, and the most rigid traditions. The remark is a very\nprofound one. The explanation of the fact is very simple. Women who have\ngrown up in surroundings wherein the letter of honour is rigidly\nobserved, and in which the spirit of virtue prevails for honour's sake,\nreadily believe that the men they love are as honourable as they seem,\nand more virtuous in all ways than sinful man is likely to be. The man\nwhom such a woman loves with all her heart, before she has met truth\nface to face, cannot possibly be as worthy as she imagines that he is;\nand if he be an honest man, he must be aware of the fact, and must\nconstantly suffer by the ever present knowledge that he is casting a\nshadow greater than himself, so to say--and to push the simile further,\nit is true that in attempting to overtake that shadow of himself, he\noften deliberately walks away from the light which makes him cast it.\n\nJohn Ralston could never, under any circumstances, have done all that\nKatharine had expected of him, although she had professed to expect so\nlittle. Woman fills the hours of her lover's absence with scenes from\nher own sweet dreamland. In nine cases out of ten, when she has the\nchance of comparing what she has learned with what she has imagined, she\nhas a moment of sickening disappointment. Later in life there is an\nadjustment, and at forty years of age she merely warns her daughter\nvaguely that she must not believe too much in men. That is the usual\nsequence of events.\n\nBut Katharine's case just now was very much worse than the common. It is\nnot necessary to recapitulate the evidence against John's soberness on\nthat memorable Thursday. It might have ruined the reputation of a Father\nof the Church. Up to one o'clock on the following day no one but Mrs.\nRalston and Doctor Routh were aware that there was anything whatsoever\nto be said on the other side of the question. So far as Katharine or any\none else could fairly judge, John had been through one of the most\noutrageous and complete sprees of which New York society had heard for\na long time. A certain number of people knew that he had practically\nfought Hamilton Bright in the hall of his club, and had undoubtedly\ntripped him up and thrown him. Katharine, naturally enough, supposed\nthat every one knew it, and in spite of Bright's reassuring words on the\nprevious night, she fully expected that John would have to withdraw from\nthe club in question. Even she, girl as she was, knew that this was a\nsort of public disgrace.\n\nThere was no other word for it. The man she loved, and to whom she had\nbeen secretly married, had publicly disgraced himself on the very day of\nthe marriage, had been tipsy in the club, had been seen drunk in the\nstreets, had been in a light with a professional boxer, and had been\nincapable of getting home alone--much more of going to meet his wife at\nthe Assembly ball.\n\nIf he had done such things on their wedding day, what might he not do\nhereafter? The question was a natural one. Katharine had bound herself\nto a hopeless drunkard. She had heard of such cases, unfortunately,\nthough they have become rare enough in society, and she knew what it all\nmeant. There would be years of a wretched existence, of a perfectly\nhopeless attempt to cure him. She had heard her father tell such\nstories, for Alexander Junior was not a peaceable abstainer like Griggs\nand Crowdie. He was not an abstainer at all--he was a man of ferocious\nmoderation. She remembered painful details about the drunkard's\nchildren. Then there was a story of a blow--and then a separation--a\nwife who, for her child's sake, would not go to another State and be\ndivorced--and the going back to the father's house to live, while the\nhusband sank from bad to worse, and his acquaintances avoided him in the\nstreet, till he had been seen hanging about low liquor saloons and\ntelling drunken loafers the story of his married life--speaking to them\nof the pure and suffering woman who was still his lawful wife--and\nlaughing about it. Alexander had told it all, as a wholesome lesson to\nhis household, which, by the way, consisted of his aged father, his\nwife, and his two daughters, none of whom, one might have thought, could\never stand in need of such lessons. Charlotte had laughed then, and\nKatharine had been disgusted. Mrs. Lauderdale's perfectly classical face\nhad expressed nothing, for she had been thinking of something else, and\nthe old philanthropist had made some remarks about the close connection\nbetween intemperance and idiocy. But the so-called lesson was telling\nheavily against John Ralston now, two or three years after it had been\ndelivered.\n\nIt was clear to Katharine that her life was ruined before it had begun.\nIn those first hours after the shock it did not occur to her that she\ncould ever forgive John. She was therefore doubly sure that the ruin he\nhad wrought was irretrievable. She could not naturally think now of the\npossibility of ever acknowledging her marriage. To proclaim it meant to\nattempt just such a life as she had heard her father describe.\nUnfortunately, too, in that very case, she knew the people, and knew\nthat Alexander Junior, who never exaggerated anything but the terrors of\nthe life to come, had kept within the truth rather than gone beyond it.\n\nShe did not even tell herself that matters would have been still worse\nif she had been made publicly John Ralston's wife on the previous day.\nAt that moment she did not seek to make things look more bearable, if\nthey might. She had faced the situation and it was terrible--it\njustified anything she might choose to do. If she chose to do something\ndesperate to free herself, she wished to be fully justified, and that\ndesired justification would be weakened by anything which should make\nher position seem more easy to bear.\n\nIndeed, she could hardly have been blamed, whatever she had done. She\nwas bound without being united, married and yet not married, but\nnecessarily shut off from all future thought of marriage, so long as\nJohn Ralston lived.\n\nShe had assumed duties, too, which she was far from wishing to avoid. In\nher girlish view, the difference between the married and the single\nstate lay mainly in the loss of the individual liberty which seemed to\nbelong to the latter. She had been brought up, as most American girls\nare, in old-fashioned ideas on the subject, which are good,--much better\nthan European ideas,--though in extended practice they occasionally lead\nto some odd results, and are not always carried out in after life. In\ntwo words, our American idea is that, on being married, woman assumes\ncertain responsibilities, and ceases, so to say, to be a free dancer in\na ball-room. The general idea in Europe is that, at marriage, a woman\ngets rid of as many responsibilities as she can, and acquires the\nliberty to do as she pleases, which has been withheld from her before.\n\nKatharine felt, therefore, even at that crisis, that she had forfeited\nher freedom, and, amidst all she felt, there was room for that bitter\nregret. A French girl could hardly understand her point of view; a\ncertain number of English girls might appreciate it, and some might\npossibly feel as she did; to an American girl it will seem natural\nenough. It was not merely out of a feeling of self-respect that she\nlooked upon a change as necessary, nor out of a blind reverence for the\nreligious ceremony which had taken place. Every inborn and cultivated\ninstinct and tradition told her that as a married woman, though the\nwhole world should believe her to be a young girl, she could not behave\nas she had behaved formerly; that a certain form of perfectly innocent\namusement would no longer be at all innocent now; that she had forfeited\nthe right to look upon every man she met as a possible admirer--she went\nno further than that in her idea of flirtation--and finally that,\nsomehow, she should feel out of place in the parties of very young\npeople to which she was naturally invited.\n\nShe was a married woman, and she must behave as one, for the rest of her\nnatural life, though no one was ever to know that she was married. It\nwas a very general idea, with her, but it was a very strong one, and\nnone the less so for its ingenuous simplicity.\n\nBut the fact that she regretted her liberty did not even distantly\nsuggest that she might ever fall in love with any one but John Ralston.\nHer only wish was to make him feel bitterly what he had done, that he\nmight regret it as long as he lived, just as she must regret her\nliberty. The offence was so monstrous that the possibility of forgiving\nit did not cross her mind. She did not, however, ask herself whether the\nlove that still remained was making the injury he had done it seem yet\nmore atrocious. Love was still in a state of suspended animation--there\nwas no telling what he might do when he came to life again. For the time\nbeing he was not to be taken into consideration at all. If she were to\nlove him during the coming years, that would only make matters much\nworse.\n\nThere is not, perhaps, in the yet comparatively passionless nature of\nmost young girls so great a capacity for real suffering as there is in\nolder women. But there is something else instead. There is a\nsensitiveness which most women lose by degrees to a certain point,\nthough never altogether, the sensitiveness of the very young animal when\nit is roughly exposed to the first storm of its first winter, if it has\nbeen born under the spring breezes and reared amongst the flowers of\nsummer.\n\nIt will suffer much more acutely later,--lash and spur, or shears and\nknife, sharper than wind and snow,--but it will never be so sensitive\nagain. It will never forget how the cruel cold bit its young skin, and\ngot into its delicate throat, and made its slender limbs tremble like\nthe tendrils of a creeper.\n\nIt was snowing again, but Katharine walked slowly, and went out of her\nway in her unformulated wish to lose time, and to put off the moment at\nwhich she must meet the familiar faces and hear the well-known voices at\nhome. Until Griggs had broached the fatal subject at table, she had been\ntaken out of herself at the Crowdies'. She must go back to herself now,\nand she hated the thought as she hated all her own existence. But the\nregions between Clinton Place and Fourth Avenue are not the part of New\nYork in which it is best for a young girl to walk about alone. She did\nnot like to be stared at by the loafers at the corners, nor to be\ntreated with too much familiarity by the patronizing policeman who saw\nher over the Broadway crossing. Then, too, she remembered that she had\ngiven no notice of her absence from luncheon, and that her mother might\nperhaps be anxious about her. There was nothing for it but to take\ncourage and go home. She only hoped that Charlotte might not be there.\n\nBut Charlotte had come, in the hope of enjoying herself as she had done\non the previous day. Katharine ascertained the fact from the girl who\nlet her in, and went straight to her room, sending word to her mother\nthat she had lunched with the Crowdies and would come down presently.\nEven as she went up the stairs she felt a sharp pain at the thought that\nher mother and sister were probably at that very moment discussing\nJohn's mishaps, and comparing notes about the stories they had\nheard--and perhaps reading more paragraphs from the papers. The shame of\nthe horrible publicity of it all overcame her, and she locked her door,\nand tried the handle to be sure that it was fast--with a woman's\ndistrust of all mechanical contrivances when she wishes to be quite sure\nof a situation. It was instinctive, and she had no second thought which\nshe tried to hide from herself.\n\nAs she took off her hat and coat she grew very pale, and the deep\nshadows came under her eyes--so dark that she wondered at them vaguely\nas she glanced at herself in the mirror. She felt faint and sick. She\ndrank a little water, and then, with a sudden impulse, threw herself\nupon her bed, and lay staring at the ceiling, as she had lain at dawn.\nThe same glare still came in from the street and penetrated every\ncorner, but not so vividly as before, for the snow was falling fast, and\nthe mist of the whirling flakes softened the light.\n\nIt was a forlorn little room. Robert the Rich would have been very much\nsurprised if he could have seen it. He was a generous man, and was very\nfond of his grand-niece, and if he had known exactly how she lived under\nher father's roof it would have been like him to have interfered. All\nthat he ever saw of the house was very different. There was great\nsimplicity downstairs, and his practised eye detected the signs of a\nrigid economy--far too rigid, he thought, when he calculated what\nAlexander Junior must be worth; a ridiculously exaggerated economy, he\nconsidered, when he thought of his own wealth, and that his only\nsurviving brother lived in the house in Clinton Place. But there was\nnothing squalid or mean about it all. The meanness was relative. It was\nlike an aspersion upon the solidity of Robert's fortune, and upon his\nintention of providing suitably for all his relations.\n\nUpstairs, however, and notably in Katharine's room, things had a\ndifferent aspect. Nothing had been done there since long before\nCharlotte had been married. The wall-paper was old-fashioned, faded, and\nbadly damaged by generations of tacks and pins. The carpet was\nthreadbare and patched, and there were holes where even a patch had not\nbeen attempted. The furniture was in the style of fifty years ago or\nmore, veneered with dark mahogany, but the veneering was coming off in\nplaces, leaving bare little surfaces of dusty pine wood smeared with\nyellowish, hardened glue. Few objects can look more desperately shabby\nthan veneered furniture which is coming to pieces. There was nothing in\nthe room which Katharine could distinctly remember to have seen in good\ncondition, except the old carpet, which had been put down when she and\nCharlotte had been little girls. To Charlotte herself, when she had come\nin on Wednesday afternoon, there had been something delightful in the\nrenewal of acquaintance with all her old dinginess of intimate\nsurroundings. Charlotte's own life was almost oppressed with luxury, so\nthat it destroyed her independence. But to Katharine, worn out and\nheartsore with the troubles of her darkening life, it was all\ninexpressibly depressing. She stared at the ceiling as she lay there, in\norder not to look at the room itself. She was very tired, too, and she\nwould have given anything to go to sleep.\n\nIt was not merely sleep for which she longed. It was a going out. Again\nthe thought crossed her mind, as it had that morning, that if the whole\nworld were a single taper, she would extinguish the flame with one short\nbreath, and everything would be over. And now, too, in her exhaustion,\ncame the idea that something less complete, but quite as effectual, was\nin her power. It had passed through her brain half an hour previously,\nwhen she had bidden Hester Crowdie good-bye--with a sort of intuitive\ncertainty that she was never to see her friend again. She had left\nHester with a vague and sudden presentiment of darkness. She had\nassuredly not any intention of seeking death in any definite form, but\nit had seemed to be close to her as she had said those few words of\nfarewell. It came nearer still as she lay alone in her own room. It came\nnearer, and hovered over her, and spoke to her.\n\nIt would be the instant solution of all difficulties, the end of all\ntroubles. The deep calm against which no storm would have power any\nmore. On the one hand, there was life in two aspects. Either to live an\nexistence of misery and daily torture with the victim of a most\ndegrading vice, a man openly disgraced, and at whom every one she\nrespected would forever look askance. Or else to live out that other\nlife of secret bondage, neither girl nor wife, so long as John Ralston\nwas alive, suffering each time he was dragged lower, as she was\nsuffering to-day, bound, tied in every way, beyond possibility of\nescaping. Why should she suffer less to-morrow than now? It would be the\nsame, since all the conditions must remain unchanged. It would be the\nsame always. Those were the two aspects of living on in the future which\npresented themselves. The torn carpet and the broken veneering of the\nfurniture made them seem even more terrible. There may be a point at\nwhich the trivial has the power to push the tragic to the last\nextremity.\n\nAnd on the other side stood death, the liberator, with his white smile\nand far-away eyes. The snow-glare was in his face, and he did not seem\nto feel it, but looked quietly into it, as though he saw something very\npeaceful beyond. It was a mere passing fancy that evoked the picture in\nthe weary, restless mind, but it was pleasant to gaze at it, so long as\nit lasted. It was gone in a moment again, leaving, however, a new\nimpression--that of light, rather than of darkness. She wished it would\ncome back.\n\nPossibly she had been almost or quite dozing, seeing that she was so\nmuch exhausted. But she was wide awake again now. She turned upon her\nside with a long-drawn sigh, and stared at the hideous furniture, the\nragged carpet, and the dilapidated wall-paper. It was not that they\nmeant anything of themselves--certainly not poverty, as they might have\nseemed to mean to any one else. They were the result of a curious\ncombination of contradictory characters in one family, which ultimately\nproduced stranger results than Katharine Lauderdale's secret marriage,\nsome of which shall be chronicled hereafter. The idea of poverty was not\nassociated with the absence of money in Katharine's mind. She might be\nin need of a pair of new gloves, and she and her mother might go to the\nopera upstairs, because the stalls were too dear. But poverty! How could\nit enter under the roof of any who bore the name of Lauderdale? If,\nyesterday, she had begged uncle Robert to give her half a million,\ninstead of refusing a hundred thousand, it was quite within the bounds\nof possibility that he might have written the cheque there and then. No.\nThe shabby furniture in Katharine's room had nothing to do with poverty,\nnor with the absence of money, either. It was the fatal result of\ncertain family peculiarities concerning which the public knew nothing,\nand it was there, and at that moment it had a strong effect upon\nKatharine's mind. It represented the dilapidation of her life, the\nliteral dilapidation, the tearing down of one stone after another from\nthe crowning point she had reached yesterday to the deep foundation\nwhich was laid bare as an open tomb to-day. She dwelt on the idea now,\nand she stared at the forlorn objects, as she had at first avoided both.\n\nDeath has a strange fascination, sometimes, both for young and old\npeople. Men and women in the prime and strength of life rarely fall\nunder its influence. It is the refuge of those who, having seen little,\nbelieve that there is little to see, and of the others who, having seen\nall, have died of the sight, inwardly, and desire bodily death as the\ncompletion of experience. Let one, or both, be wrong or right; it\nmatters little, since the facts are there. But the fascination aforesaid\nis stronger upon the young than upon the old. They have fewer ties, and\nless to keep them with the living. For the ascendant bond is weaker than\nthe descendant in humanity, and the love of the child for its mother is\nnot as her love for the child. It is right that it should be so. In\nspite of many proverbs, we know that what the child owes the parent is\nas nothing compared with the parent's debt to it. Have we all found it\nso easy to live that we should cast stones upon heart-broken youths and\nmaidens who would fain give back the life thrust upon them without their\nconsent?\n\nKatharine clasped her hands together, as she lay on her side, and prayed\nfervently that she might die that day--at that very hour, if possible.\nIt would be so very easy for God to let her die, she thought, since she\nwas already so tired. Her heart had almost stopped beating, her hands\nwere cold, and she felt numb, and weary, and miserable. The step was so\nshort. She wondered whether it would hurt much if she took it herself,\nwithout waiting. There were things which made one go to sleep--without\nwaking again. That must be very easy and quite, quite painless, she\nthought. She felt dizzy, and she closed her eyes again.\n\nHow good it would be! All alone, in the old room, while the snow was\nfalling softly outside. She should not mind the snow-glare any more\nthen. It would not tire her eyes. That white smile--it came back to her\nat last, and she felt it on her own face. It was very strange that she\nshould be smiling now--for she was so near crying--nearer than she\nthought, indeed, for as the delicate lips parted with the slow, sighing\nbreath, the heavy lids--darkened as though they had been hurt--were\nsoftly swelling a little, and then very suddenly and quickly two great\ntears gathered and dropped and ran and lost themselves upon the pillow.\n\nAh, how peaceful it would be--never to wake again, when the little step\nwas passed! Perhaps, if she lay quite still, it would come. She had\nheard strange stories of people in the East, who let themselves die when\nthey were weary. Surely, none of them had ever been as weary as she.\nStrange--she was always so strong! Every one used to say, 'as strong as\nKatharine Lauderdale.' If they could see her now!\n\nShe wanted to open her eyes, but the snow-glare must be still in the\nroom, and she could not bear it--and the shabby furniture. She would\nbreathe more slowly. It seemed as though with each quiet sigh the\nlingering life might float away into that dear, peaceful beyond--where\nthere would be no snow-glare and the furniture would not be shabby--if\nthere were any furniture at all--beyond--or any John Ralston--no\n'marriage nor giving in marriage'--all alone in the old room--\n\nTwo more tears gathered, more slowly this time, though they dropped and\nlost themselves just where the first had fallen, and then, somehow, it\nall stopped, for what seemed like one blessed instant, and then there\ncame a loud knocking, with a strange, involved dream of carpenters and\nboxes and a journey and being late for something, and more knocking, and\nher mother's voice calling to her through the door.\n\n\"Katharine, child! Wake up! Don't forget that you're to dine at the Van\nDe Waters' at eight! It's half past six now!\"\n\nIt was quite dark, save for the flickering light thrown upon the ceiling\nfrom the gas-lamp below. Katharine started up from her long sleep,\nhardly realizing where she was.\n\n\"All right, mother--I'm awake!\" she answered sleepily.\n\nAs she listened to her mother's departing footsteps, it all came back to\nher, and she felt faint again. She struggled to her feet in the gloom\nand groped about till she had found a match, and lit the gas and drew\ndown the old brown shades of the window. The light hurt her eyes for a\nmoment, and as she pressed her hands to them she felt that they were\nwet.\n\n\"I suppose I've been crying in my sleep!\" she exclaimed aloud. \"What a\nbaby I am!\"\n\nShe looked at herself in the mirror with some curiosity, before\nbeginning to dress.\n\n\"I'm an object for men and angels to stare at!\" she said, and tried to\nlaugh at her dejected appearance. \"However,\" she added, \"I suppose I\nmust go. I'm Katharine Lauderdale--'that nice girl who never has\nheadaches and things'--so I have no excuse.\"\n\nShe stopped for a moment, still looking at herself.\n\n\"But I'm not Katharine Lauderdale!\" she said presently, whispering the\nwords to herself. \"I'm Katharine Ralston--if not, what am I? Ah, dear\nme!\" she sighed. \"I wonder how it will all end!\"\n\nAt all events, Katharine Lauderdale, or Katharine Ralston, she was\nherself again, as she turned from the mirror and began to think of what\nshe must wear at the Van De Waters' dinner-party.\n\n\n\n\nCHAPTER XXVII.\n\n\nEven John Ralston's tough constitution could not have been expected to\nshake off in a few hours the fatigue and soreness of such an experience\nas he had undergone. Even if he had been perfectly well, he would have\nstayed at home that day in the expectation of receiving an answer from\nKatharine; and as it was, he needed as much rest as he could get. He had\nnot often been at the trouble of taking care of himself, and the\nsensation was not altogether disagreeable, as he sat by his own\nfireside, in the small room which went by the name of 'Mr. Ralston's\nstudy.' He stretched out his feet to the fire, drank a little tea from\ntime to time, stared at the logs, smoked, turned over the pages of a\nmagazine without reading half a dozen sentences, and revolved the\npossibilities of his life without coming to any conclusion.\n\nHe was stiff and bruised. When he moved his head, it ached, and when he\ntried to lean to the right, his neck hurt him on the left side. But if\nhe did not move at all, he felt no pain. There was a sort of perpetual\ndrowsy hum in his ears, partly attributable, he thought, to the singing\nof a damp log in the fire, and partly to his own imagination. When he\ntried to think of anything but his own rather complicated affairs, he\nalmost fell asleep. But when his attention was fixed on his present\nsituation, it seemed to him that his life had all at once come to a\nstandstill just as events had been moving most quickly. As for really\nsleeping in the intervals of thought, his constant anxiety for\nKatharine's reply to his letter kept his faculties awake. He knew,\nhowever, that it would be quite unreasonable to expect anything from her\nbefore twelve o'clock. He tried to be patient.\n\nBetween ten and eleven, when he had been sitting before his fire for\nabout an hour, the door opened softly and Mrs. Ralston entered the room.\nShe did not speak, but as John rose to meet her she smiled quietly and\nmade him sit down again. Then she kneeled before the hearth and began to\narrange the fire, an operation which she had always liked, and in which\nshe displayed a singular talent. Moreover, at more than one critical\nmoment in her life, she had found it a very good resource in\nembarrassment. A woman on her knees, making up a fire, has a distinct\nadvantage. She may take as long as she pleases about it, for any amount\nof worrying about the position of a particular log is admissible. She\nmay change colour twenty times in a minute, and the heat of the flame\nas well as the effort she makes in moving the wood will account\nsatisfactorily for her blushes or her pallor. She may interrupt herself\nin speaking, and make effective pauses, which will be attributed to the\nconcentration of her thoughts upon the occupation of her hands. If a man\ncomes too near, she may tell him sharply to keep away, either saying\nthat she can manage what she is doing far better if he leaves her alone,\nor alleging that the proximity of a second person will keep the air from\nthe chimney and make it smoke. Or if the gods be favourable and she\nwilling, she may at any moment make him kneel beside her and help her to\nlift a particularly heavy log. And when two young people are kneeling\nside by side before a pile of roaring logs in winter, the flames have a\nstrange bright magic of their own; and sometimes love that has\nsmouldered long blazes up suddenly and takes the two hearts with it--out\nof sheer sympathy for the burning oak and hickory and pine.\n\nBut Mrs. Ralston really enjoyed making up a fire, and she went to the\nhearth quite naturally and without reflecting that after what had\noccurred she felt a little timid in her son's presence. He obeyed her\nand resumed his seat, and sat leaning forward, his arms resting on his\nknees and his hands hanging down idly, while he watched his mother's\nskilful hands at work.\n\n\"Jack dear--\" she paused in her occupation, having the tongs in one hand\nand a little piece of kindling-wood in the other, but did not turn\nround--\"Jack, I can't make up to you for what I did last night, can I?\"\n\nShe was motionless for a moment, listening for his reply. It came\nquietly enough after a second or two.\n\n\"No, mother, you can't. But I don't want to remember it, any more than\nyou do.\"\n\nMrs. Ralston did not move for an instant after he had spoken. Then she\noccupied herself with the fire again.\n\n\"You're quite right,\" she said presently. \"You wouldn't be my son, if\nyou said anything else. If I were a man, one of us would be dead by this\ntime.\"\n\nShe spoke rather intensely, so to say, but she used her hands as gently\nas ever in what she was doing. John said nothing.\n\n\"Men don't forgive that sort of thing from men,\" she continued\npresently. \"There's no reason why a woman should be forgiven, I suppose,\neven if the man she has insulted is her own son.\"\n\n\"No,\" John answered thoughtfully. \"There is no more reason for forgiving\nit. But there's every reason to forget it, if you can.\"\n\n\"If you can. I don't wish to forget it.\"\n\n\"You should, mother. Of course, you brought me up to believe--you and my\nfather--that to doubt a man's word is an unpardonable offence, because\nlying is a part of being afraid, which is the only unpardonable sin. I\nbelieve it. I can't help it.\"\n\n\"I don't expect you to. We've always--in a way--been more like two men,\nyou and I, than like a mother and her son. I don't want the allowances\nthat are made for women. I despise them. I've done you wrong, and I'll\ntake the consequences. What are they? It's a bad business, Jack. I've\nrun against a rock. I'll do anything you ask. I'll give you half my\nincome, and we can live apart. Will you do that?\"\n\n\"Mother!\" John Ralston fairly started in his surprise. \"Don't talk like\nthat!\"\n\n\"There!\" exclaimed Mrs. Ralston, hanging up the hearthbrush on her left,\nafter sweeping the feathery ashes from the shining tiles within the\nfender. \"It will burn now. Nobody understands making a fire as I do.\"\n\nShe rose to her feet swiftly, drew back from John, and sat down in the\nother of the two easy chairs which stood before the fireplace. She\nglanced at John and then looked at the fire she had made, clasping her\nhands over one knee.\n\n\"Smoke, won't you?\" she said presently. \"It seems more natural.\"\n\n\"All right--if you like.\"\n\nJohn lit a cigarette and blew two or three puffs into the air, high\nabove his head, very thoughtfully.\n\n\"I'm waiting for your answer, Jack,\" said Mrs. Ralston, at last.\n\n\"I don't see what I'm to say,\" replied John. \"Why do you talk about it?\"\n\n\"For this reason--or for these reasons,\" said Mrs. Ralston, promptly, as\nthough she had prepared a speech beforehand, which was, in a measure,\nthe truth. \"I've done you a mortal injury, Jack. I know that sounds\ndramatic, but it's not. I'll tell you why. If any one else, man or\nwoman, had deliberately doubted your statement on your word of honour,\nyou would never have spoken to him or her again. Of course, in our\ncountry, duelling isn't fashionable--but if it had been a man--I don't\nknow, but I think you would have done something to him with your hands.\nYes, you can't deny it. Well, the case isn't any better because\nsatisfaction is impossible, is it? I'm trying to look at it logically,\nbecause I know what you must feel. Don't you see, dear?\"\n\n\"Yes. But--\"\n\n\"No! Let me say all I've got to say first, and then you can answer me.\nI've been thinking about it all night, and I know just what I ought to\ndo. I know very well, too, that most women would just make you forgive\nas much as you could and then pretend to you and to themselves that\nnothing had ever happened. But we're not like that, you and I. We're\nlike two men, and since we've begun in that way, it's not possible to\nturn round and be different now, in the face of a difficulty. There are\npeople who would think me foolish, and call me quixotic, and say, 'But\nit's your own son--what a fuss you're making about nothing.' Wouldn't\nthey? I know they would. It seems to me that, if anything, it's much\nworse to insult one's own son, as I did you, than somebody else's son,\nto whom one owes nothing. I'm not going to put on sackcloth and sit in\nthe ashes and cry. That wouldn't help me a bit, nor you either. Besides,\nother people, as a rule, couldn't understand the thing. You never told\nme a lie in your life. Last Monday when you came home after that\naccident, and weren't quite yourself, you told me the exact truth about\neverything that had happened. You never even tried to deceive me. Of\ncourse you have your life, and I have mine. I have always respected your\nsecrets, haven't I, Jack?\"\n\n\"Indeed you have, mother.\"\n\n\"I know I have, and if I take credit for it, that only makes all this\nworse. I've never asked you questions which I thought you wouldn't care\nto answer. I've never been inquisitive about all this affair with\nKatharine. I don't even know at the present moment whether you're\nengaged to her still, or not. I don't want to know--but I hope you'll\nmarry her some day, for I'm very fond of her. No--I've never interfered\nwith your liberty, and I've never been willing to listen to what people\nwished to tell me about you. I shouldn't think it honest. And in that\nway we've lived very harmoniously, haven't we?\"\n\n\"Mother, you know we have,\" answered John, earnestly.\n\n\"All that makes this very much worse. One drop of blood will turn a\nwhole bowl of clean water red. It wouldn't show at all if the water were\nmuddy. If you and I lived together all our lives, we should never forget\nlast night.\"\n\n\"We could try to,\" said John. \"I'm willing.\"\n\nMrs. Ralston paused and looked at him a full minute in silence. Then she\nput out her hand and touched his arm.\n\n\"Thank you, Jack,\" she said gravely.\n\nJohn tried to press her hand, but she withdrew it.\n\n\"But I'm not willing,\" she resumed, after another short pause. \"I've\ntold you--I don't want a woman's privilege to act like a brute and be\ntreated like a spoiled child afterwards. Besides, there are many other\nthings. If what I thought had been true, I should never have allowed\nmyself to act as I did. I ought to have been kind to you, even if you\nhad been perfectly helpless. I know you're wild, and drink too much\nsometimes. You have the strength to stop it if you choose, and you've\nbeen trying to since Monday. You've said nothing, and I've not watched\nyou, but I've been conscious of it. But it's not your fault if you have\nthe tendency to it. Your father drank very hard sometimes, but he had a\ndifferent constitution. It shortened his life, but it never seemed to\naffect him outwardly. I'm conscious--to my shame--that I didn't\ndiscourage him, and that when I was young and foolish I was proud of him\nbecause he could take more than all the other officers and never show\nit. Men drank more in those days. It was not so long after the war. But\nyou're a nervous man, and your father wasn't, and you have his taste for\nit without that sort of quiet, phlegmatic, strong, sailor's nature that\nhe had. So it's not your fault. Perhaps I should have frightened you\nabout it when you were a boy. I don't know. I've made mistakes in my\nlife.\"\n\n\"Not many, mother dear.\"\n\n\"Well--I've made a great one now, at all events. I'm not going back over\nanything I've said already. It's the future I'm thinking of. I can't do\nmuch, but I can manage a 'modus vivendi' for us--\"\n\n\"But why--\"\n\n\"Don't interrupt me, dear! I've made up my mind what to do. All I want\nof you now, is your advice as a man, about the way of doing it. Listen\nto me, Jack. After what has happened between us--no matter how it turns\nout afterwards, for we can't foresee that--it's impossible that we\nshould go on living as we've lived since your father died. I don't mean\nthat we must part, unless you want to leave me, as you would have a\nperfect right to do.\"\n\n\"Mother!\"\n\n\"Jack--if I were your brother, instead of your mother--still more, if I\nwere any other relation--would you be willing to depend for the rest of\nyour life on him, or on any one who had treated you as I treated you\nlast night?\"\n\nShe paused for an answer, but John Ralston was silent. With his\ncharacter, he knew that she was quite right, and that nothing in the\nworld could have induced him to accept such a situation.\n\n\"Answer me, please, dear,\" she said, and waited again.\n\n\"Mother--you know! Why should I say it?\"\n\n\"You would refuse to be dependent any longer on such a person?\"\n\n\"Well--yes--since you insist upon my saying it,\" answered John,\nreluctantly. \"But with you, it's--\"\n\n\"With me, it's just the same--more so. I have had a longer experience of\nyou than any one else could have had, and you've never deceived me.\nConsequently, it was more unpardonable to doubt you. I don't wish you to\nbe dependent on me any longer, Jack. It's an undignified position for\nyou, after this.\"\n\n\"Mother--I've tried--\"\n\n\"Hush, dear! I'm not talking about that. If there had been any\nnecessity, if you had ever had reason to suppose that it wasn't my\ngreatest happiness to have you with me--or that there wasn't quite\nenough for us both--you'd have just gone to sea before the mast, or done\nsomething of the same kind, as all brave boys do who feel that they're a\nburden on their mothers. But there's always been enough for us both, and\nthere is now. I mean to give you your share, and keep what I need\nmyself. That will be yours some day, too, when I'm dead and gone.\"\n\n\"Please don't speak of that,\" said John, quickly and earnestly. \"And as\nfor this idea of your--\"\n\n\"Oh, I'm in no danger of dying young,\" interrupted Mrs. Ralston, with a\nlittle dry laugh. \"I'm very strong. All the Lauderdales are, you\nknow--we live forever. My father would have been seventy-one this year\nif he hadn't been killed. And as long as I live, of course, I must have\nsomething to live on. I don't mean to go begging to uncle Robert for\nmyself, and I shouldn't care to do it for you, though I would if it were\nnecessary. Now, we've got just twelve thousand dollars a year between\nus, and the house, which is mine, you know. That will give us each six\nthousand dollars a year. I shall see my lawyer this morning and it can\nbe settled at once. Whenever the house is let, if we're both abroad, you\nshall have half of the rent. When we're both here, half of it is yours\nto live in--or pull down, if you like. If you marry, you can bring your\nwife here, and I'll go away. Now, I think that's fair. If it isn't, say\nso before it's too late.\"\n\n\"I won't listen to anything of the kind,\" answered John, calmly.\n\n\"You must,\" answered his mother.\n\n\"I don't think so, mother.\"\n\n\"I do. You can't prevent me from making over half the estate to you, if\nI choose, and when that's done, it's yours. If you don't like to draw\nthe rents, you needn't. The money will accumulate, for I won't touch it.\nYou shall not be in this position of dependence on me--and at your\nage--after what has happened.\"\n\n\"It seems to me, mother dear, that it's very much the same, whether you\ngive me a part of your income, or whether you make over to me the\ncapital it represents. It's the same transaction in another shape,\nthat's all.\"\n\n\"No, it's not, Jack! I've thought of that, because I knew you'd say it.\nIt's so like you. It's not at all the same. You might as well say that\nit was originally intended that you should never have the money at all,\neven after I died. It was and is mine, for me and my children. As I have\nonly one child, it's yours and mine jointly. As long as you were a boy,\nit was my business to look after your share of it for you. As soon as\nyou were a man, I should have given you your share of it. It would have\nbeen much better, though there was no provision in either of the wills.\nIf it had been a fortune, I should have done it anyhow, but as it was\nonly enough for us two to live on, I kept it together and was as careful\nof it as I could be.\"\n\n\"Mother--I don't want you to do this,\" said John. \"I don't like this\nsordid financial way of looking at it--I tell you so quite frankly.\"\n\nMrs. Ralston was silent for a few moments, and seemed to be thinking the\nmatter over.\n\n\"I don't like it either, Jack,\" she said at last. \"It isn't like us. So\nI won't say anything more about it. I'll just go and do it, and then it\nwill be off my mind.\"\n\n\"Please don't!\" cried Ralston, bending forward, for she made as though\nshe would rise from her seat.\n\n\"I must,\" she answered. \"It's the only possible basis of any future\nexistence for us. You shall live with me from choice, if you like. It\nwill--well, never mind--my happiness is not the question! But you shall\nnot live with me as a matter of necessity in a position of dependence.\nThe money is just as much yours as it's mine. You shall have your share,\nand--\"\n\n\"I'd rather go to sea--as you said,\" interrupted John.\n\n\"And let your income accumulate. Very well. But I--I hope you won't,\ndear. It would be lonely. It wouldn't make any difference so far as this\nis concerned. I should do it, whatever you did. As long as you like,\nlive here, and pay your half of the expenses. I shall get on very well\non my share if I'm all alone. Now I'm going, because there's nothing\nmore to be said.\"\n\nMrs. Ralston rose this time. John got up and stood beside her, and they\nboth looked at the fire thoughtfully.\n\n\"Mother--please--I entreat you not to do this thing!\" said John,\nsuddenly. \"I'm a brute even to have thought twice of that silly affair\nlast night--and to have said what I said just now, that I couldn't\nexactly feel as though anything could undo what had been done.\nIndeed--if there's anything to forgive, it's forgiven with all my heart,\nand we'll forget it and live just as we always have. We can, if we\nchoose. How could you help it--the way I looked! I saw myself in the\nglass. Upon my word, if I'd drunk ever so little, I should have been\nquite ready to believe that I was tipsy, from my own appearance--it was\nnatural, I'm sure, and--\"\n\n\"Hush, Jack!\" exclaimed Mrs. Ralston. \"I don't want you to find excuses\nfor me. I was blind with anger, if that's an excuse--but it's not. And\nmost of all--I don't want you to imagine for one moment that I'm going\nto make this settlement of our affairs with the least idea that it is a\nreparation to you, or anything at all of that sort. Not that you'd ever\nmisunderstand me to that extent. Would you?\"\n\n\"No. Certainly not. You're too much like me.\"\n\n\"Yes. There's no reparation about it, because that's more possible. As\nit is, no particular result will follow unless you wish it. You'll be\nfree to go away, if you please, that's all. And if you choose to marry\nKatharine, and if she is willing to marry you on six thousand a year,\nyou'll feel that you can, though it's not much. And for the matter of\nthat, Jack dear--you know, don't you? If it would make you happy, and if\nshe would--I don't think I should be any worse than most\nmothers-in-law--and all I have is yours, Jack, besides your share. But\nthose are your secrets--no, it's quite natural.\"\n\nJohn had taken her hand gently and kissed it.\n\n\"I don't want any gratitude for that,\" she continued. \"It's perfectly\nnatural. Besides, there's no question of gratitude between you and me.\nIt's always been share and share alike--of everything that was good. Now\nI'm going. You'll be in for luncheon? Do take care of yourself to-day.\nSee what weather we're having! And--well--it's not for me to lecture you\nabout your health, dear. But what Doctor Routh said is true. You've\ngrown thinner again, Jack--you grow thinner every year, though you are\nso strong.\"\n\n\"Don't worry about me, mother dear. I'm all right. And I shan't go out\nto-day. But I have a dinner-party this evening, and I shall go to it. I\nthink I told you--the Van De Waters'--didn't I? Yes. I shall go to that\nand show myself. I'm sure people have been talking about me, and it was\nprobably in the papers this morning. Wasn't it?\"\n\n\"Dear--to tell you the truth, I wouldn't look to see. It wasn't very\nbrave of me--but--you understand.\"\n\n\"I certainly shan't look for the report of my encounter with the\nprize-fighter. I'm sure he was one. I shall probably be stared at\nto-night, and some of them will be rather cold. But I'll face it\nout--since I'm in the right for once.\"\n\n\"Yes. I wouldn't have you stay at home. People would say you were afraid\nand were waiting for it to blow over. Is it a big dinner?\"\n\n\"I don't know. I got the invitation a week ago, at least, so it isn't an\ninformal affair. It's probably to announce Ruth Van De Water's\nengagement to that foreigner--you know--I've forgotten his name. I know\nBright's going--because they said he wanted to marry her last year--it\nisn't true. And there'll probably be some of the Thirlwalls, and the\nyoung Trehearns, and Vanbrugh and his wife--you know, all the Van De\nWater young set. Katharine's going, too. She told me when she got the\ninvitation, some time last week. There'll be sixteen or eighteen at\ntable, and I suppose they'll amuse themselves somehow or other\nafterwards. Nobody wants to dance to-night, I fancy--at least none of\nour set, after the Thirlwalls', and the Assembly, and I don't know how\nmany others last week.\"\n\n\"They'll probably put you next to Katharine,\" said Mrs. Ralston.\n\n\"Probably--especially there, for they always do--with Frank Miner on her\nother side to relieve my gloom. Second cousins don't count as relations\nat a dinner-party, and can be put together. Half of the others are own\ncousins, too.\"\n\n\"Well, if it's a big dinner it won't be so disagreeable for you. But if\nyou'd take my advice, Jack--however--\" She stopped.\n\n\"What is it, mother?\" he asked. \"Say it.\"\n\n\"Well--I was going to say that if any one made any disagreeable\nremarks, or asked you why you weren't at the Assembly last night, I\nshould just tell the whole story as it happened. And you can end by\nsaying that I was anxious about you and sent for Doctor Routh, and refer\nthem to him. That ought to silence everybody.\"\n\n\"Yes.\" John paused a moment. \"Yes,\" he repeated. \"I think you're right.\nI wish old Routh were going to be there himself.\"\n\n\"He'd go in a minute if he were asked,\" said Mrs. Ralston.\n\n\"Would he? With all those young people?\"\n\n\"Of course he would--only too delighted! Dear old man, it's just the\nsort of thing he'd like. But I'm going, Jack, or I shall stay here\nchattering with you all the morning.\"\n\n\"That other thing, mother--about the money--don't do it!\" Jack held her\na moment by the hand.\n\n\"Don't try to hinder me, dear,\" she answered. \"It's the only thing I can\ndo--to please my own conscience a little. Good-bye. I'll see you at\nluncheon.\"\n\nShe left the room quickly, and John found himself alone with his own\nthoughts again.\n\n\"It's just like her,\" he said to himself, as he lighted a cigar and sat\ndown to think over the situation. \"She's just like a man about those\nthings.\"\n\nHe had perhaps never admired and loved his mother as he did then; not\nfor what she was going to do, but for the spirit in which she was doing\nit. He was honest in trying to hinder her, because he vaguely feared\nthat the step might cause her some inconvenience hereafter--he did not\nexactly know how, and he was firmly resolved that he would not under any\ncircumstances take advantage of the arrangement to change his mode of\nlife. Everything was to go on just as before. As a matter of theory, he\nwas to have a fixed, settled income of his own; but as a matter of fact,\nhe would not regard it as his. What he liked about it, and what really\nappealed to him in it all, was his mother's man-like respect for his\nhonour, and her frank admission that nothing she could do could possibly\nwipe out the slight she had put upon him. Then, too, the fact and the\ntheory were at variance and in direct opposition to one another. As a\nmatter of theory, nothing could ever give him back the sensation he had\nalways felt since he had been a boy--that his mother would believe him\non his word in the face of any evidence whatsoever which there might be\nagainst him. But as a matter of fact, the evil was not only completely\nundone, but there was a stronger bond between them than there had ever\nbeen before.\n\nThat certainly was the first good thing which had come to him during the\nlast four and twenty hours, and it had an effect upon his spirits.\n\nHe thought over what his mother had said about the evening, too, and was\nconvinced that she was right in advising him to tell the story frankly\nas it had happened. But he was conscious all the time that his anxiety\nabout Katharine's silence was increasing. He had roused himself at dawn,\nin spite of his fatigue, and had sent a servant out to post the letter\nwith the special delivery stamp on it. Katharine must have received it\nlong ago, and her answer might have been in his hands before now.\nNevertheless, he told himself that he should not be impatient, that she\nhad doubtless slept late after the ball, and that she would send him an\nanswer as soon as she could. By no process of reasoning or exaggeration\nof doubting could he have reached the conclusion that she had never\nreceived his letter. She had always got everything he sent her, and\nthere had never been any difficulty about their correspondence in all\nthe years during which they had exchanged little notes. He took up the\nmagazine again, and turned over the pages idly. Suddenly Frank Miner's\nname caught his eye. The little man had really got a story into one of\nthe great magazines, a genuine novel, it seemed, for this was only a\npart, and there were the little words at the end of it, in italics and\nin parenthesis, 'to be continued,' which promised at least two more\nnumbers, for as John reflected, when the succeeding number was to be the\nlast, the words were 'to be concluded.' He was glad, for Miner's sake,\nof this first sign of something like success, and began to read the\nstory with interest.\n\nIt began well, in a dashing, amusing style, as fresh as Miner's\nconversation, but with more in it, and John was beginning to\ncongratulate himself upon having found something to distract his\nattention from his bodily ills and his mental embarrassments, when the\ndoor opened, and Miner himself appeared.\n\n\"May I come in, Ralston?\" he enquired, speaking softly, as though he\nbelieved that his friend had a headache.\n\n\"Oh--hello, Frank! Is that you? Come in! I'm reading your novel. I'd\njust found it.\"\n\nLittle Frank Miner beamed with pleasure as he saw that the magazine was\nreally open at his own story, for he recognized that this, at least,\ncould not be a case of premeditated appreciation.\n\n\"Why--Jack--\" he stammered a moment later, in evident surprise. \"You\ndon't look badly at all!\"\n\n\"Did they say I was dead?\" enquired Ralston, with a grim smile. \"Take a\ncigar. Sit down. Tell me all about my funeral.\"\n\nMiner laughed as he carefully cut off the end of the cigar and lit it--a\nsort of continuous little gurgling laugh, like the purling of a brook.\n\n\"My dear boy,\" he said, blowing out a quantity of smoke, and curling\nhimself up in the easy chair, \"you're the special edition of the day.\nThe papers are full of you--they're selling like hot cakes\neverywhere--your fight with Tom Shelton, the champion light weight--and\nyour turning up in the arms of two policemen--talk of a 'jag!' Lord!\"\n\n\n\n\nCHAPTER XXVIII.\n\n\nJohn looked at Miner quietly for a few seconds, without saying anything.\nThe little man was evidently lost in admiration of the magnitude of his\nfriend's 'jag,' as he called it.\n\n\"I say, Frank,\" said Ralston, at last, \"it's all a mistake, you know. It\nwas a series of accidents from beginning to end.\"\n\n\"Oh--yes--I suppose so. You managed to accumulate quite a number of\naccidents, as you say.\"\n\nRalston was silent again. He was well aware of the weight of the\nevidence against him, and he wished to enter upon his explanation by\ndegrees, in order that it might be quite clear to Miner.\n\n\"Look here,\" he began, after a while. \"I'm not the sort of man who tries\nto wriggle out of things, when he's done them, am I? Heaven knows--I've\nbeen in scrapes enough! But you never knew me to deny it, nor to try and\nmake out that I was steady when I wasn't. Did you, Frank?\"\n\n\"No,\" answered Miner, thoughtfully. \"I never did. That's a fact. It's\nquite true.\"\n\nThe threefold assent seemed to satisfy Ralston.\n\n\"All right,\" he said. \"Now I want you to listen to me, because this is\nrather an extraordinary tale. I'll tell it all, as nearly as I can, but\nthere are one or two gaps, and there's a matter connected with it about\nwhich I don't want to talk to you.\"\n\n\"Go ahead,\" answered Miner. \"I've got some perfectly new faith out--and\nI'm just waiting for you. Produce the mountain, and I'll take its\nmeasure and remove it at a valuation.\"\n\nRalston laughed a little and then began to tell his story. It was, of\ncourse, easy for him to omit all mention of Katharine, and he spoke of\nhis interview with Robert Lauderdale as having taken place in connection\nwith an idea he had of trying to get something to do in the West, which\nwas quite true. He omitted also to mention the old gentleman's amazing\nmanifestation of eccentricity--or folly--in writing the cheque which\nJohn had destroyed. For the rest, he gave Miner every detail as well as\nhe could remember it. Miner listened thoughtfully and never interrupted\nhim once.\n\n\"This isn't a joke, is it, Jack?\" he asked, when John had finished with\na description of Doctor Routh's midnight visit.\n\n\"No,\" answered Ralston, emphatically. \"It's the truth. I should be glad\nif you would tell any one who cares to know.\"\n\n\"They wouldn't believe me,\" answered Miner, quietly.\n\n\"I say, Frank--\" John's quick temper was stirred already, but he checked\nhimself.\n\n\"It's all right, Jack,\" answered Miner. \"I believe every word you've\ntold me, because I know you don't invent--except about leaving cards on\nstray acquaintances at the Imperial, when you happen to be thirsty.\"\n\nHe laughed good-naturedly.\n\n\"That's another of your mistakes,\" said Ralston. \"I know--you mean last\nMonday. I did leave cards at the hotel. I also had a cocktail. I didn't\nsay I wasn't going to, and I wasn't obliged to say so, was I? It wasn't\nyour business, my dear boy, nor Ham Bright's, either.\"\n\n\"Well--I'm glad you did, then. I'm glad the cards were real, though it\nstruck me as thin at the time. I apologize, and eat humble pie. You know\nyou're one of my illusions, Jack. There are two or three to which I\ncling. You're a truthful beggar, somehow. You ought to have a little\nhatchet, like George Washington--but I daresay you'd rather have a\nlittle cocktail. It illustrates your nature just as well. Bury the\nhatchet and pour the cocktail over it as a libation--where was I?\nOh--this is what I meant, Jack. Other people won't believe the story, if\nI tell it, you know.\"\n\n\"Well--but there's old Routh, after all. People will believe him.\"\n\n\"Yes--if he takes the trouble to write a letter to the papers, over his\nname, degrees and qualifications. Of course they'll believe him. And the\neditors will do something handsome. They won't apologize, but they'll\nsay that a zebra got loose in the office and upset the type while they\nwere in Albany attending to the affairs of the Empire State--and that's\njust the same as an apology, you know, which is all you care for. You\ncan't storm Park Row with the gallant Four Hundred at your back. In the\nfirst place, Park Row's insured, and secondly, the Four Hundred would\nsee you--further--before they'd lift one of their four thousand fingers\nto help you out of a scrape which doesn't concern them. You'd have to be\na parson or a pianist, before they'd do anything for you. It's 'meat,\ndrink and pantaloons' to be one of them, anyhow--and you needn't expect\nanything more.\"\n\n\"Where do you get your similes from, Frank!\" laughed Ralston.\n\n\"I don't know. But they're good ones, anyway. Why don't you get Routh to\nwrite a letter, before the thing cools down? It could be in the evening\nedition, you know. There have been horrid things this\nmorning--allusions--that sort of thing.\"\n\n\"Allusions to what?\" asked Ralston, quickly and sharply.\n\n\"To you, of course--what did you suppose?\"\n\n\"Oh--to me! As though I cared! All the same, if old Routh would write,\nit would be a good thing. I wish he were going to be at the Van De\nWaters' dinner to-night.\"\n\n\"Why? Are you going there? So am I.\"\n\n\"It seems to be a sort of family tea-party,\" said Ralston. \"Bright's\ngoing, and cousin Katharine, and you and I. It only needs the Crowdies\nand a few others to make it complete.\"\n\n\"Well--you see, they're cousins of mine, and so are you, and that sort\nof makes us all cousins,\" observed Miner, absently. \"I say, Jack--tell\nthe story at table, just as you've told it to me. Will you? I'll set you\non by asking you questions. Stunning effect--especially if we can get\nRouth to write the letter. I'll cut it out of a paper and bring it with\nme.\"\n\n\"You know him, don't you?\" asked Ralston.\n\n\"Know him? I should think so. Ever since I was a baby. Why?\"\n\n\"I wish you'd go to him this morning, Frank, and get him to write the\nletter. Then you could take it to one of the evening papers and get them\nto put it in. You know all those men in Park Row, don't you?\"\n\n\"Much better than some of them want to know me,\" sighed the little man.\n\"However,\" he added, his bright smile coming back at once, \"I ought not\nto complain. I'm getting on, now. Let me see. You want me to go to Routh\nand get him to write a formal letter over his name, denying all the\nstatements made about you this morning. Isn't that taking too much\nnotice of the thing, after all, Jack?\"\n\n\"It's going to make a good deal of difference to me in the end,\"\nanswered Ralston. \"It's worth taking some trouble for.\"\n\n\"I'm quite willing,\" said Miner. \"But--I say! What an extraordinary\nstory it is!\"\n\n\"Oh, no. It's only real life. I told you--I only had one accident, which\nwas quite an accident--when I tumbled down in that dark street.\nEverything else happened just as naturally as unnatural things always\ndo. As for upsetting Ham Bright at the club, I was awfully sorry about\nthat. It seemed such a low thing to do. But then--just remember that I'd\nbeen making a point of drinking nothing for several days, just by way of\nan experiment, and it was irritating, to say the least of it, to be\ngrabbed by the arm and told that I was screwed. Wasn't it, Frank? And\njust at that moment, uncle Robert had telephoned for me to come up, and\nI was in a tremendous hurry. Just look at in that way, and you'll\nunderstand why I did it. It doesn't excuse it--I shall tell Ham that\nI'm sorry--but it explains it. Doesn't it?\"\n\n\"Rather!\" exclaimed Miner, heartily.\n\n\"By the bye,\" said Ralston, \"I wanted to ask you something. Did that\nfellow Crowdie hold his tongue? I suppose he was at the Assembly last\nnight.\"\n\n\"Well--since you ask me--\" Miner hesitated. \"No--he didn't. Bright gave\nit to him, though, for telling cousin Emma.\"\n\n\"Brute! How I hate that man! So he told cousin Emma, did he? And the\nrest of the family, too, I suppose.\"\n\n\"I suppose so,\" answered Miner, knowing that Ralston meant Katharine.\n\"Everybody knew about the row at the club, before the evening was half\nover. Teddy Van De Water said he supposed you'd back out of the dinner\nto-night and keep quiet till this blew over. I told Teddy that perhaps\nhe'd better come round and suggest that to you himself this morning, if\nhe wanted to understand things quickly. He grinned--you know how he\ngrins--like an organ pipe in a white tie. But he said he'd heard Bright\nleathering into Crowdie--that's one of Teddy's expressions--so he\nsupposed that things weren't as bad as people said--and that Crowdie was\nonly a 'painter chap,' anyhow. I didn't know what that meant, but feebly\npointed out that Crowdie was a great man, and that his wife was a sort\nof cousin of mine, and that she, at least, had a good chance of having\nsome of cousin Robert's money one of these days. Not that I wanted to\ndefend Crowdie, or that I don't like Teddy much better--but then, you\nknow what I mean! He'll be calling me 'one of those literary chaps,'\nnext, with just the same air. One's bound to stand up for art and\nliterature when one's a professional, you know, Jack. Wasn't I right?\"\n\n\"Oh, perfectly!\" answered Ralston, with a smile. \"But will you do that\nfor me, Frank?\"\n\n\"Of course I will. You're one of my illusions, as I told you. I'm\nwilling to do lots of things for my illusions. I'll go now, and then\nI'll come back and tell you what the old chap says. If by any chance he\ngets into a rage, I'll tell him that I didn't come so much to talk about\nyou as to consult him about certain symptoms of nervous prostration I'm\nbeginning to feel. He's death on nervous prostration--he's a perfect\nterror at it--he'll hypnotise me, and put me into a jar of spirits, and\npaint my nose with nervine and pickled electricity and things, and sort\nof wake me up generally.\"\n\n\"All right--if you can stand it, I can,\" said Ralston. \"I'd go\nmyself--only only--\"\n\n\"You're pretty badly used up,\" interrupted Miner, completing the\nsentence in his own way. \"I know. I remember trying to play football\nonce. Those little games aren't much in my line. Nature meant me for\nhigher things. I tried football, though, and then I said, like\nNapoleon--you remember?--'Ces balles ne sont pas pour moi.' I couldn't\ntell where I began and the football ended--I felt that I was a safe\nunder-study for a shuttlecock afterwards. That's just the way you feel,\nisn't it? As though it were Sunday, and you were the frog--and the boys\nhad gone back to afternoon church? I know! Well--I'll come back as soon\nas I've seen Routh. Good-bye, old man--don't smoke too much. I do--but\nthat's no reason.\"\n\nThe little man nodded cheerfully, knocked the ashes carefully from the\nend of his cigar--he was neat in everything he did--and returned it to\nhis lips as he left the room. Ralston leaned back in his arm-chair again\nand rested his feet on the fender. The fire his mother had made so\ncarefully was burning in broad, smoking flames. He felt cold and\nunderfed and weary, so that the warmth was very pleasant; and with all\nthat came to his heart now, as he thought of his mother, there mingled\nalso a little simple, childlike gratitude to her for having made up such\na good fire.\n\nThe time passed, and still no word came from Katharine. He was willing\nto find reasons or, at least, excuses, for her silence, but he was\nconscious that they were of little value. He knew, now, that there had\nreally been paragraphs in the papers about him, as he had expected, and\nthat they had been of a very disagreeable nature. Katharine had probably\nseen them, or one of them, besides having heard the stories that had\nbeen circulated by Crowdie and others during the previous evening. He\nfancied that he could feel her unbelief, hurting him from a distance, as\nit were. Her face, cold and contemptuous, rose before him out of the\nfire, and he took up the magazine again, and tried to hide it. But it\ncould not be hidden.\n\nSurely by this time she must have got his letter. There could be no\nreasonable doubt of that. He looked at his watch again, as he had done\nonce in every quarter of an hour for some time. It was twelve o'clock.\nMiner had not stayed long.\n\nJohn went over the scene on Wednesday evening, at the Thirlwalls'.\nKatharine had been very sure of herself, at the last--sure that,\nwhatever he did, she should always stand by him. Events had put her to\nthe test soon enough, and this was the result. They had been married\ntwenty-four hours, and she would not even answer his note, because\nappearances were against him.\n\nAnd the great, strong sense of real innocence rose in him and defied and\ndespised the woman who could not trust him even a little. If the very\nleast of the accusations had been true, he would have humbled himself\nhonestly and said that she was right, and that she had promised too\nmuch, in saying all she had said. At all times he was a man ready to\ntake the full blame of all he had done, to make himself out worse than\nhe really was, to assume at once that he was a failure and could do\nnothing right. On the slightest ground, he was ready to admit everything\nthat people brought against him. Katharine, if he had even been living\nas usual, would have been at liberty to reproach him as bitterly as she\npleased with his weakness, to turn her back on him and condemn him\nunheard, if she chose. He would have been patient and would have\nadmitted that he deserved it all, and more also. He was melancholy, he\nwas discouraged with himself, and he was neither vain nor untruthful.\n\nBut he had made an effort, and a great one. There was in him something\nof the ascetic, with all his faults, and something of the enthusiasm\nwhich is capable of sudden and great self-denial if once roused. He knew\nwhat he had done, for he knew what it had cost him, mentally and\nphysically. Lean as he had been before, he had grown perceptibly thinner\nsince Monday. He knew that, so far, he had succeeded. For the first\ntime, perhaps, he had every point of justice on his side. If he had\nbeen inclined to be merciful and humble and submissive towards those who\ndoubted him now, he would not have been human. The two beings whom he\nloved in the world, his mother and Katharine, were the very two who had\ndoubted him most. As for his mother, he had not persuaded her, for she\nhad persuaded herself--by means of such demonstration as no sane being\ncould have rejected, namely, the authoritative statement of a great\ndoctor, personally known to her. What had followed had produced a\nstrange result, for he felt that he was more closely bound to her than\never before, a fact which showed, at least, that he did not bear malice,\nhowever deeply he had been hurt. But he could not go about everywhere\nfor a week with Doctor Routh at his heels to swear to his sobriety. He\ntold himself so with some contempt, and then he thought of Katharine,\nand his face grew harder as the minutes went by and no answer came to\nhis letter.\n\nIt was far more cruel of her than it had been in his mother's case.\nKatharine had only heard stories and reports of his doings, and she\nshould be willing to accept his denial of them on her faith in him. He\nhad never lied to her. On Wednesday night, he had gratuitously told her\nthe truth about himself--a truth which she had never suspected--and had\ninsisted upon making it out to be even worse than it was. His wisdom\ntold him that he had made a mistake then, in wilfully lowering himself\nin her estimation, and that this was the consequence of that; if he had\nnot forced upon her an unnecessary confession of his weakness, she would\nnow have believed in his strength. But his sense of honour rose and\nshamed his wisdom, and told him that he had done right. It would have\nbeen a cowardly thing to accept what Katharine had then been forcing\nupon him, and had actually made him accept, without telling her all the\ntruth about himself.\n\nHe had done wrong to yield at all. That he admitted, and repeated,\nreadily enough. He made no pretence of having a strong character, and he\nhad been wretchedly weak in allowing her to persuade him to the secret\nmarriage. He should have folded his arms and refused, from the first. He\nhad foreseen trouble, though not of the kind which had actually\novertaken him, and he should have been firm. Unfortunately, he was not\nfirm, by nature, as he told himself, with a sneer. Not that Katharine\nhad been to blame, either. She had made her reasons seem good, and he\nshould not have blamed her had she been ever so much in the wrong. There\nhis honour spoke again, and loudly.\n\nBut for what she was doing now, in keeping silence, leaving him without\na word when she must know that he was most in need of her faith and\nbelief--for abandoning him when it seemed as though every man's hand\nwere turned against him--he could not help despising her. It was so\ncowardly. Had it all been ten times true, she should have stood by him\nwhen every one was abusing him.\n\nIt was far more cruel of her than of his mother, for all she knew of the\nstory had reached her by hearsay, whereas his mother had seen him, as he\nhad seen himself, and his appearance might well have deceived any one\nbut Doctor Routh. He did not ask himself whether he could ever forgive\nher, for he did not wish to hear in his heart the answer which seemed\ninevitable. As for loving her, or not loving her, he thought nothing\nabout it at that moment. With him, too, as with her, love was in a state\nof suspended animation. It would have been sufficiently clear to any\noutsider acquainted with the circumstances that when the two met that\nevening, something unusual would probably occur. Katharine, indeed,\nbelieved that John would not appear at the dinner-party; but John, who\nfirmly intended to be present, knew that Katharine was going, and he\nexpected to be placed beside her. It was perfectly well known that they\nwere in love with one another, and the least their greatest friends\ncould do was to let them enjoy one another's society. This may have\nbeen done partly as a matter of policy, for both were young enough and\ntactless enough to show their annoyance if they were separated when they\nchanced to be asked to sit at the same table. John looked forward to the\ncoming evening with some curiosity, and without any timidity, but also\nwithout any anticipation of enjoyment.\n\nHe was trying to imagine what the conversation would be like, when Frank\nMiner returned, beaming with enthusiasm, and glowing from his walk in\nthe cold, wet air. He had been gone a long time.\n\n\"Well?\" asked John, as his friend came up to the fire, and held out his\nhands to it.\n\n\"Very well--very well, indeed, thank you,\" answered the latter, with a\ncheerful laugh. \"I'll bet you twenty-five cents to a gold watch that you\ncan't guess what's happened--at Routh's.\"\n\n\"Twenty-five cents--to a gold watch? Oh--I see. Thank you--the odds\ndon't tempt me. What did happen?\"\n\n\"I say--those were awfully good cigars of yours, Jack!\" exclaimed Miner,\nby way of answer. \"Haven't you got another?\"\n\n\"There's the box. Take them all. What happened?\"\n\n\"No--I'll only take one--it would look like borrowing if I took two, and\nI can't return them. Jack, there's a lot of good blood knocking about\nin this family, do you know? I don't mean about the cigars--I'm\nnaturally a generous man when it comes to taking things I like. But the\nother thing. Do you know that somebody had been to Routh about making\nhim write the letter, before I got there?\"\n\n\"What? To make him write it? Not Ham Bright? It would be like him--but\nhow should he have known about Routh?\"\n\n\"No. It wasn't Bright. Want to guess? Well--I'll tell you. It was your\nmother, Jack. Nice of her, wasn't it?\"\n\n\"My mother!\"\n\nRalston leaned forward and began to poke the logs about. He felt a\ncurious sensation of gladness in the eyes, and weakness in the throat.\n\n\"Tell me about it, Frank,\" he added, in a rather thick voice.\n\n\"There's not much to tell. I marched in and stated my case. He's between\nseven and eight feet high, I believe, and he stood up all the time--felt\nas though I were talking to scaffold poles. He listened in the calmest\nway till I'd finished, and then took up a letter from his desk and\nhanded it to me to read and to see whether I thought it would do. I\nasked what it meant, and he said he'd just written it at the request of\nMrs. Ralston, who had left him a quarter of an hour ago, and that if I\nwould take it to the proper quarter--as he expressed it--he should be\nmuch obliged. He's a brick--a tower of strength--a tower of bricks--a\nperfect Babel of a man. You'll see, when the evening papers come out--\"\n\n\"Did you take it down town?\"\n\n\"Of course. And I got hold of one of the big editors. I sent in word\nthat I had a letter from Doctor Routh which must be published in the\nfront page this evening unless the paper wanted Mr. Robert Lauderdale to\nbring an action against them for libel to-morrow morning. You should\nhave seen things move. What a power cousin Robert is! I suppose I took\nhis name in vain--but I don't care. Old Routh is not to be sneezed at,\neither. You'll see the letter. There's some good old English in it. Oh,\nit's just prickly with epithets--'unwarrantable liberty,' 'impertinent\nscurrility'--I don't know what the old doctor had for breakfast. It's\nnot like him to come out like that, not a bit. He's a cautious old bird,\nas a rule, and not given to slinging English all over the ten-acre lot,\nlike that. You see, he takes the ground that you're his patient, that\nyou had some sort of confustication of the back of your head, and that\nto say that you were screwed when you were ill was a libel, that the\nterms in which the editor had allowed the thing to appear proved that it\nwas malicious, and that as the editor was supposed to exercise some\ncontrol, and to use his own will in the matter of what he published and\ncirculated, it was wilfully published, since the city paid for places in\nwhich people who had no control over their wills were kept for the\npublic safety, and that therefore the paragraph in question was a\nwilfully malicious libel evidently published with the intention of doing\nharm--and much more of the same kind of thing--all of which the editor\nwould have put into the waste paper basket if it had not been signed,\nMartin Routh, M.D., with the old gentleman's address. Moreover, the\neditor asked me why, in sending in a message, I had made use of\nthreatening language purporting to come from Mr. Robert Lauderdale. But\nas you had told me the whole story, I knew what to say. I just told him\nthat you had left the house of your uncle, Mr. Robert Lauderdale, after\nspending some time with him, when you met with the accident in the\nstreet which led to all your subsequent adventures. That seemed to\nsettle him. He said the whole thing had been a mistake, and that he\nshould be very sorry to have given Mr. Lauderdale any annoyance,\nespecially at this time. I don't know what he meant by that, I'm\nsure--unless uncle Robert is going to buy the paper for a day or two to\nsee what it's like--you know the proprietor's dead, and they say the\nheirs are going to sell. Well--that's all. Confound it, my cigar's out.\nI'm a great deal too good to you, Jack!\"\n\nRalston had listened without comment while the little man told his\nstory, satisfied, as he proceeded from point to point, that everything\nwas going well for him, at last, and mentally reducing Miner's strong\nexpressions to the lowest key of probability.\n\n\"So it was my mother who went first to Doctor Routh,\" he said, as though\ntalking with himself, while Miner relighted his cigar.\n\n\"Yes,\" answered Miner, between two puffs. \"I confess to having been\nimpressed.\"\n\n\"It's like her,\" said John. \"It's just like her. You didn't happen to\nsee any note for me lying on the hall table, did you?\" he asked, rather\nirrelevantly.\n\n\"No--but I'll go and look, if you like.\"\n\n\"Oh--it's no matter. Besides, they know I haven't been out this morning,\nand they'd bring anything up. I'm very much obliged to you, Frank, for\nall this. And I know that you'll tell anybody who talks about it just\nwhat I've told you. I should like to feel that there's a chance of some\none's knowing the truth when I come into the room this evening.\"\n\n\"Oh, they'll all know it by that time. Routh's letter will run along the\nground like fire mingled with hail. As for Teddy Van De Water, he lives\non the papers. Of course they won't fly at you and congratulate you all\nover, and that sort of thing. They'll just behave as though nothing at\nall had happened, and afterwards, when we men are by ourselves, smoking,\nthey'll all begin to ask you how it happened. That is, unless you want\nto tell the story yourself at table, and in that case I'll set you on,\nas I said.\"\n\n\"I don't care to talk about it,\" answered John. \"But--look here,\nFrank--listen! You're as quick as anybody to see things. If you notice\nthat a number of the set don't know about Routh's letter--that there's a\nsort of hostile feeling against me at table--why, then just set me on,\nas you call it, and I'll defend myself. You see, I've such a bad temper,\nand my bones ache, and I'm altogether so generally knocked out, that it\nwill be much better to give me my head with for a clear run, than to let\npeople look as though they should like to turn their backs on me, but\ndidn't dare to. Do you understand?\"\n\n\"All right, Jack. I won't make any mistake about it.\"\n\n\"Very well, then. It's a bargain. We won't say anything more about it.\"\n\nMiner presently took his departure, and John was left alone again. In\nthe course of time he gave up looking at his watch, and relinquished all\nhope of hearing from Katharine. Little by little, the certainty formed\nitself in his mind that the meeting that evening was to be a hostile\none.\n\nNot very long after Miner had gone, another hand opened the door, and\nJohn sprang to his feet, for even in the slight sound he recognized the\ntouch. Mrs. Ralston entered the room. With more impulsiveness than was\nusual in him he went quickly to meet her, and threw his arms round her,\nkissing her through her veil, damp and cold from the snowy air.\n\n\"Mother, darling--how good you are!\" he exclaimed softly. \"There isn't\nanybody like you--really.\"\n\n\"Why--Jack? What is it?\" asked Mrs. Ralston, happy, but not\nunderstanding.\n\n\"Miner was here--he told me about your having been to old Routh to make\nhim write--\"\n\n\"That? Oh--that's nothing. Of course I went--the first thing. Didn't he\nsay last night that he'd give his evidence in a court of law? I thought\nhe might just as well do it. The business is all settled, dear boy. I've\nseen the lawyer, and he's making out the deed. He'll bring it here for\nme to sign when he comes up from his office, and the transfers of the\ntitles will be registered to-morrow morning--just in time before\nSunday.\"\n\n\"Don't talk about that, mother!\" answered John. \"I didn't want you to do\nit, and it's never going to make the slightest difference between us.\"\n\n\"Well--perhaps not. But it makes all the difference to me. Promise me\none thing, Jack.\"\n\n\"Yes, mother--anything you like.\"\n\n\"Promise me to remember that if you and Katharine choose to get married,\nin spite of her father and all the Lauderdales, this is your house, and\nthat you have a right to it. You won't have much to live on, but you\nwon't starve. Promise me to remember that, Jack. Will you?\"\n\n\"I'll promise to remember it, mother. But I'll not promise to act on\nit.\"\n\n\"Well--that's a matter for your judgment. Go and get ready for luncheon.\nIt must be time.\"\n\nOnce more John put his arms round her neck, and drew her close to him.\n\n\"You're very good to me, mother--thank you!\"\n\n\n\n\nCHAPTER XXIX.\n\n\nKatharine spent more time than necessary over dressing for dinner on\nthat evening, not because she bestowed more attention than usual upon\nher appearance, but because there were long pauses of which she was\nscarcely conscious, although the maid reminded her from time to time\nthat it was growing late. The result, however, was satisfactory in the\nopinion of her assistant, a sober-minded Scotch person of severe tastes,\nwho preferred black and white to any colours whatsoever, and thought\nthat the trees showed decided frivolity in being green, and that the\nwoods in autumn were positively improper.\n\nIt was undoubtedly true that the simple black gown, without ornament and\nwith very little to break its sweeping line, was as becoming to\nKatharine's strong beauty as it was appropriate to her frame of mind. It\nmade her look older than she was, perhaps, but being so young, the loss\nwas almost gain. It gave her dignity a background and a reason, as it\nwere. Her face was pale still, but not noticeably so, and her eyes were\nquiet if not soft. Only a person who knew her very well would have\nobserved the slight but steady contraction of the broad eyebrows, which\nwas unusual. As a rule, if it came at all, it disappeared almost\ninstantly again. She remembered afterwards--as one remembers the absurd\ndetails of one's own thoughts--that when she had looked into the mirror\nfor the last time, she had been glad that her front hair did not curl,\nand that she had never yielded to the temptation to make it curl, as\nmost girls did. She had been pleased by the simplicity of the two thick,\nblack waves which lay across the clear paleness of her forehead, like\ndark velvet on cream-white silk. She forgot the thought instantly, but,\nlater, she remembered how severe and straight it had looked, and the\nconsciousness was of some value to her--as the least vain man, taken\nunexpectedly to meet and address a great assembly, may be momentarily\nglad if he chances to be wearing a particularly good coat. The gravest\nof us have some consciousness of our own appearance, and be our strength\nwhat it may, when it is appropriate to appear in the wedding garment, it\nis good for us to be wearing one.\n\nKatharine stopped at her mother's door as she descended the stairs. Mrs.\nLauderdale was dining at home, and the Lauderdales dined at eight\no'clock, so that she was still in her room at ten minutes before the\nhour. Katharine knocked and entered. Her mother was standing before the\nmirror. The door which led to her father's dressing-room, by a short\npassage between two wardrobes built into the house, was wide open.\nKatharine heard him moving some small objects on his dressing-table.\n\n\"You're late, child,\" said Mrs. Lauderdale, not turning, for as\nKatharine entered, she could see her reflection in the mirror. \"Are you\ngoing to take Jane with you? If not, I wish you'd tell her to come here,\nas you go down--I let you have her because I knew you'd be late.\"\n\n\"No,\" answered Katharine, \"I don't want her--she's only in the way. It's\nthe Van De Waters', you know. Good night, mother.\"\n\n\"Good night, darling--enjoy yourself--you'll be late, of course--they'll\ndance, or something.\"\n\n\"Yes--but I shan't stay. I'm tired. Good night again.\"\n\nKatharine was going to the door, when her father appeared from his\ndressing-room, serenely correct, as usual, but wearing his black tie\nbecause no one was coming to dinner.\n\n\"I want to speak to you, Katharine,\" he said.\n\nShe turned and stood still in the middle of the room, facing him. He had\na letter in his hand.\n\n\"Yes, papa,\" she answered quietly, not anticipating trouble.\n\n\"I'm sorry I could not see you earlier,\" said Alexander Junior, coming\nforward and fixing his steely eyes on his daughter's face. \"But I\nhadn't an opportunity, because I was told that you were asleep when I\ncame home. This morning, as I was leaving the house as usual, a\nmessenger put this letter into my hands. It has a special delivery stamp\non it, and you will see that the mark on the dial edge stands at eight\nforty-five A.M. Consequently, the boy who brought it was dilatory in\ndoing his duty. It is addressed to you in John Ralston's handwriting.\"\n\n\"Why didn't you send it up to me, instead of keeping it all day?\"\nenquired Katharine, with cold surprise.\n\n\"Because I do not intend that you shall read it,\" answered her father,\nhis lips opening and shutting on the words like the shears of a\ncutting-machine.\n\nMrs. Lauderdale turned round from the mirror and looked at her husband\nand daughter. It would have been impossible to tell from her face\nwhether she had been warned of what was to be done or not, but there was\nan odd little gleam in her eyes, of something which might have been\nannoyance or satisfaction.\n\n\"Why don't you intend me to read my letters?\" asked Katharine in a lower\ntone.\n\n\"I don't wish you to correspond with John Ralston,\" answered Alexander\nJunior. \"You shall never marry him with my consent, especially since he\nhas disgraced himself publicly as he did yesterday. There was an account\nof his doings in the morning papers. I daresay you've not seen it. He\nwas taken home last night in a state of beastly intoxication by two\npolicemen, having been picked up by them out of a drunken brawl with a\nprize-fighter. To judge from the handwriting of the address on this\nletter, it appears to have been written while he was still under the\ninfluence of liquor. I don't mean that my daughter shall receive letters\nwritten by drunken men, if I can help it.\"\n\n\"Show me the letter,\" said Katharine, quietly.\n\n\"I'll show it to you because, though you've never had any reason to\ndoubt my statements, I wish you to have actually seen that it has not\nbeen opened by me, nor by any one. My judgment is formed from the\nhandwriting solely, but I may add that it is impossible that a man who\nwas admittedly in a state of unconsciousness from liquor at one o'clock\nin the morning, should be fit to write a letter to Katharine Lauderdale,\nor to any lady, within six hours. The postmark on the envelope is\nseven-thirty. Am I right?\" He turned deliberately to Mrs. Lauderdale.\n\n\"Perfectly,\" she answered, with sincere conviction.\n\nAnd it must be allowed that, from his point of view, he was not wrong.\nHe beckoned Katharine to the gas-light beside the mirror and held up\nthe letter, holding it at the two sides of the square envelope in the\nfirm grip of his big, thin fingers, as though he feared lest she should\ntry to take it. But Katharine did not raise her hands, as she bent\nforward and inspected the address. It was assuredly not written in\nJohn's ordinary hand, though the writing was recognizable as his, beyond\ndoubt. There was an evident attempt at regularity, but a too evident\nfailure. It looked a little as though he had attempted to write with his\nleft hand. At one corner there was a very small stain of blood, which,\nas every one knows, retains its colour on writing paper, even under\ngas-light, for a considerable time. It will be remembered that John had\nhurt his right hand.\n\nKatharine's brows contracted more heavily. She was disgusted, but she\nwas also pained. She looked long and steadily at the writing, and her\nlips curled slightly. Alexander Lauderdale turned the letter over to\nshow her that it was sealed. Again, where the finger had hurriedly\npressed the gummed edge of the envelope, there was a little mark of\nblood. Katharine drew back very proudly, as from something at once\nrepulsive and beneath her woman's dignity. Her father looked at her\nkeenly and coldly.\n\n\"Have you satisfied yourself?\" he enquired. \"You see that it has not\nbeen opened, do you?\"\n\n\"Yes.\"\n\n\"I will burn it,\" said Alexander Lauderdale, still watching her.\n\n\"Yes.\"\n\nHe seemed surprised, for he had expected resistance, and perhaps some\nattempt on her part to get possession of the letter and read it. But she\nstood upright, silent, and evidently disgusted. He lifted his hand and\nheld the letter over the flame of the gas-light until it had caught fire\nthoroughly. Then he laid it in the fireless grate--the room, like all\nthe rest of the house, was heated by the furnace,--and with his usual\nprecise interpretation of his own conscience's promptings, he turned his\nback on it, lest by any chance he should see and accidentally read any\nword of the contents as the paper curled and flared and blackened and\nfell to ashes. Katharine, however, was well aware that a folded letter\nwithin its envelope will rarely burn through and through if left to\nitself. She went to the hearth and watched it. It had fallen flat upon\nthe tiles, and one thickness after another flamed, rose from one end and\ncurled away as the one beneath it took fire. She would not attempt to\nread one of the indistinct words, but she could not help seeing that it\nhad been a long letter, scrawled in a handwriting even more irregular\nthan that on the envelope. The leaves turned black, one by one, rising\nand remaining upright like black funeral feathers, till at last there\nwas only a little blue light far down in the heart of them. That, too,\nwent out, and a small, final puff of smoke rose and vanished. Katharine\nturned the heap over with the tongs. Only one little yellow bit of paper\nremained unconsumed at the bottom. It was almost round, and as she\nturned it over, she read on it the number of the house. That was all\nthat had not been burned.\n\n\"I'm glad to see that you look at the matter in its true light,\" said\nher father, as she stood up again.\n\n\"How should I look at it?\" asked Katharine, coldly. \"Good night,\nmother--good night,\" she repeated, nodding to her father.\n\nShe turned and left the room. A moment later she was on her way to the\nVan De Waters' house, leaning back in the dark, comfortable brougham,\nher feet toasting on the foot-warmer, and the furs drawn up closely\nround her. It was a bitterly cold night, for a sharp frost had succeeded\nthe snow-storm after sunset. Even inside the carriage Katharine could\nfeel that there was something hard and ringing in the quality of the air\nwhich was in harmony with her own temper. She had plenty of time to go\nover the scene which had taken place in her mother's room, but she felt\nno inclination to analyze her feelings. She only knew that this letter\nof John's, written when he was still half senseless with drink, was\nanother insult, and one deeper than any she had felt before. It was a\ndirect insult--a sin of commission, and not merely of omission, like his\nabsence from the ball on the previous night.\n\nShe supposed, naturally enough, that he would not appear at the\ndinner-party, but at that moment she was almost indifferent as to\nwhether he should come or not. She was certainly not afraid to meet him.\nIt would be far more probable, she thought, that he should be afraid to\nmeet her.\n\nIt was a quarter past eight when she reached the Van De Waters', and she\nwas the last to arrive. It was a party of sixteen, almost all very\nyoung, and most of them unmarried--a party very carefully selected with\na view to enjoyment--an intimate party, because many out of the number\nwere more or less closely connected and related, and it was indicative\nof the popularity of the Lauderdales, that amongst sixteen young persons\nthere should be four who belonged more or less to the Lauderdale tribe.\nThere was Katharine, there was Hamilton Bright,--the Crowdies had been\nomitted because so many disliked Crowdie himself,--there was little\nFrank Miner, who was a near relation of the Van De Waters, and there\nstood John Ralston, talking to Ruth Van De Water, before Crowdie's new\nportrait of her, as though nothing had happened.\n\nKatharine saw him the moment she entered the room, and he knew, as he\nheard the door opened, that she must be the last comer, since every one\nelse had arrived. Without interrupting his conversation with Miss Van De\nWater, he turned his head a little and met Katharine's eyes. He bowed\njust perceptibly, but she gave him no sign of recognition, which was\npardonable, however, as he knew, since there were people between them,\nand she had not yet spoken to Ruth herself, who, with her brother, had\ninvited the party. The elder Van De Waters had left the house to the\nyoung people, and had betaken themselves elsewhere for the evening.\n\nJohn continued to talk quietly, as Katharine came forward. As he had\nexpected, he had found her name on the card in the little envelope which\nhad been handed to him when he arrived, and he was to take her in to\ndinner. Until late in the afternoon the brother and sister had hoped\nthat John would not come, and had already decided to ask in his place\nthat excellent man, Mr. Brown, who was always so kind about coming when\nasked at the last minute. Then Frank Miner had appeared, with an evening\npaper containing Doctor Routh's letter, and had explained the whole\nmatter, so that they felt sympathy for John rather than otherwise,\nthough no one had as yet broached to him the subject of his adventures.\nNaturally enough, the Van De Waters both supposed that Katharine should\nhave been among the first to hear the true version of the story, and\nthey would not disarrange their table in order to separate two young\npeople who were generally thought to be engaged to be married. There\nwere, of course, a few present who had not heard of Doctor Routh's\njustification of John.\n\nKatharine came across directly, towards Ruth Van De Water, and greeted\nher affectionately. John came forward a little, waiting to be noticed\nand to shake hands in his turn. Katharine prolonged the first exchange\nof words with her young hostess rather unnecessarily, and then, since\nshe could not avoid the meeting, held out her hand to John, looking\nstraight and coldly into his eyes.\n\n\"You're to take Miss Lauderdale in, you know, Mr. Ralston,\" said Miss\nVan De Water, who knew that dinner would be announced almost\nimmediately, and that Katharine would wish to speak to the other guests\nbefore sitting down.\n\n\"Yes--I found my card,\" answered John, as Katharine withdrew her hand\nwithout having given his the slightest pressure.\n\nIt was a strange meeting, considering that they had been man and wife\nsince the previous morning, and could hardly be said to have met since\nthey had parted after the wedding. Katharine, who was cold and angry,\nwondered what all those young people would say if she suddenly announced\nto them, at table, that John Ralston was her husband. But just then she\nhad no definite intention of ever announcing the fact at all.\n\nJohn only partly understood, for he was sure that she must have received\nhis letter. But what he saw was enough to convince him that she had not\nin the least believed what he had written, and had not meant to answer\nhim. He was pale and haggard already, but during the few minutes that\nfollowed, while Katharine moved about the room, greeting her friends,\nthe strong lines deepened about his mouth and the shadows under his eyes\ngrew perceptibly darker.\n\nA few minutes later the wide doors were thrown back and dinner was\nannounced. Without hesitation he went to Katharine's side, and waited\nwhile she finished speaking with young Mrs. Vanbrugh, his right arm\nslightly raised as he silently offered it.\n\nKatharine deliberately finished her sentence, nodded and smiled to Dolly\nVanbrugh, who was a friend of hers, and had been in some way concerned\nin the famous Darche affair three or four years ago, as Mrs. Darche's\nintimate and confidante. Then she allowed her expression to harden\nagain, and she laid her hand on John's arm and they all moved in to\ndinner.\n\n\"I'm sorry,\" said John, in a low, cold voice. \"I suppose they couldn't\nupset their table.\"\n\nKatharine said nothing, but looked straight before her as they traversed\none beautiful room after another, going through the great house to the\ndining-room at the back.\n\n\"You got my letter, I suppose,\" said John, speaking again as they\ncrossed the threshold of the last door but one, and came in sight of the\ntable, gleaming in the distance under soft lights.\n\nKatharine made a slight inclination of the head by way of answer, but\nstill said nothing. John thought that she moved her hand, as though she\nwould have liked to withdraw it from his arm, and he, for his part,\nwould gladly have let it go at that moment.\n\nIt was a very brilliant party, of the sort which could hardly be\ngathered anywhere except in America, where young people are not\nunfrequently allowed to amuse themselves together in their own way\nwithout the interference or even the presence of elders--young people\nborn to the possession, in abundance, of most things which the world\nthinks good, and as often as anywhere, too, to the inheritance of things\ngood in themselves, besides great wealth--such as beauty, health, a fair\nshare of wit, and the cheerful heart, without which all else is as\nashes.\n\nNear one end of the table sat Frank Miner, who had taken in Mrs.\nVanbrugh, and who was amusing every one with absurd stories and\njokes--the small change of wit, but small change that was bright and\nnew, ringing from his busy little mint.\n\nAt the other end sat Teddy Van De Water, a good fellow at heart in spite\nof his eyeglass and his affectations, discussing yachts and centreboards\nand fin-keels with Fanny Trehearne, a girl who sailed her own boats at\nNewport and Bar Harbor, and who cared for little else except music,\nstrange to say. Nearly opposite to Katharine and John was Hamilton\nBright, between two young girls, talking steadily and quietly about\nsociety, but evidently much preoccupied, and far more inclined to look\nat Katharine than at his pretty neighbours. He had seen Routh's letter,\nand had, moreover, exchanged a few words with Ralston in the hall,\nhaving arrived almost at the same instant, and he saw that Katharine did\nnot understand the truth. Ralston had begun by apologizing to his friend\nfor what had happened at the club, but Bright, who bore no malice, had\nstopped him with a hearty shake of the hand, and a challenge to wrestle\nwith him any day, for the honour of the thing, in the hall of the club\nor anywhere else.\n\nFrank Miner, too, from a distance, watched John and Katharine, and saw\nthat the trouble was great, though he laughed and chatted and told\nstories, as though he were thoroughly enjoying himself. In reality he\nwas debating whether he should not bring up the subject which must be\nnear to every one's thoughts, and give John a chance of telling his own\nstory. Seeing how the rest of the people were taking the affair, he\nwould not have done so, since all was pleasant and easy, but he saw also\nthat John could not possibly have an explanation with Katharine at\ntable, and that both were suffering. His kindly heart decided the\nquestion. It would be a very easy matter to accomplish, and he waited\nfor a convenient opportunity of attracting attention to himself, so as\nto obtain the ear of the whole large table, before he began. He was\nperfectly conscious of his own extreme popularity, and knew that, for\nonce, he could presume upon it, though he was quite unspoiled by a long\ncareer of little social successes.\n\nJohn and Katharine exchanged a few words from time to time, for the sake\nof appearances, in a coldly civil tone, and without the slightest\nexpression of interest in one another. John spoke of the weather, and\nKatharine admitted that it had been very bad of late. She observed that\nMiss Van De Water was looking very well, and that a greenish blue was\nbecoming to fair people. John answered that he had expected to hear of\nMiss Van De Water's engagement to that foreigner whose name he had\nforgotten, and Katharine replied that he was not a foreigner but an\nEnglishman, and that his name was Northallerton, or something like that.\nJohn said he had heard that they had first met in Paris, and Katharine\ntook some salt upon her plate and admitted that it was quite possible.\nShe grew more coldly wrathful with every minute, and the iron entered\ninto John's soul, and he gave up trying to talk to her--of which she was\nvery glad.\n\nIt was some time before the occasion which Miner sought presented\nitself, and the dinner proceeded brilliantly enough amid the laughter of\nyoung voices and the gladness of young eyes. For young eyes see flowers\nwhere old ones see but botany, so to speak.\n\nKatharine had not believed that it would hurt her as it did, nor Ralston\nthat love could seem so far away. They turned from each other and talked\nwith their neighbours. John almost thought that Katharine once or twice\ngathered her black skirts nearer to her, as though to keep them from a\nsort of contamination. He was on her left, and he was conscious that in\npretending to eat he used his right arm very cautiously because he did\nnot wish even to run the risk of touching hers by accident.\n\nNow, in the course of events, it happened that the subject of yachts\ntravelled from neighbour to neighbour, as subjects sometimes do at big\ndinners, until, having been started by Teddy Van De Water and Fanny\nTrehearne, it came up the table to Frank Miner. He immediately saw his\nchance, and plunged into his subject.\n\n\"Oh, I don't take any interest in yacht races, compared with prize\nfights, since Jack Ralston has gone into the ring!\" he said, and his\nhigh, clear voice made the words ring down the table with the cheery,\nlaughing cadence after them.\n\n\"What's that about me, Frank?\" asked John, speaking over Katharine's\nhead as she bent away from him towards Russell Vanbrugh, who was next to\nher on the other side.\n\n\"Oh, nothing--talking about your round with Tom Shelton. Tell us all\nabout it, Jack. Don't be modest. You're the only man here who's ever\nstood up to a champion prize-fighter without the gloves on, and it seems\nyou hit him, too. You needn't be ashamed of it.\"\n\n\"I'm not in the least ashamed of it,\" answered Ralston, unbending a\nlittle.\n\nHe spoke in a dead silence, and all eyes were turned upon him. But he\nsaid nothing more. Even the butler and the footmen, every one of whom\nhad read both the morning and the evening papers, paused and held their\nbreath, and looked at John with admiration.\n\n\"Go ahead, Ralston!\" cried Teddy Van De Water, from his end. \"Some of us\nhaven't heard the story, though everybody saw those horrid things in the\npapers this morning. It was too bad!\"\n\nKatharine had attempted to continue her conversation with Russell\nVanbrugh, but it had proved impossible. Moreover, she was herself\nalmost breathless with surprise at the sudden appeal to Ralston himself,\nwhen she had been taking it for granted that every one present,\nincluding his hosts, despised him, and secretly wished that he had not\ncome.\n\nVan De Water had spoken from the end of the table. Frank Miner responded\nagain from the other, looking hard at Katharine's blank face, as he\naddressed John.\n\n\"Tell it, Jack!\" he cried. \"Don't be foolish. Everybody wants to know\nhow it happened.\"\n\nRalston looked round the table once more, and saw that every one was\nexpecting him to speak, all with curiosity, and some of the men with\nadmiration. His eyes rested on Katharine for a moment, but she turned\nfrom him instantly--not coldly, as before, but as though she did not\nwish to meet his glance.\n\n\"I can't tell a story by halves,\" said he. \"If you really want to have\nit, you must hear it from the beginning. But I told Frank Miner this\nmorning--he can tell it better than I.\"\n\n\"Go on, Jack--you're only keeping everybody waiting!\" said Hamilton\nBright, from across the table. \"Tell it all--about me, too--it will make\nthem laugh.\"\n\nJohn saw the honest friendship in the strong Saxon face, and knew that\nto tell the whole story was his best plan.\n\n\"All right,\" he said. \"I'll do my best. It won't take long. In the first\nplace--you won't mind my going into details, Miss Van De Water?\"\n\n\"Oh, no--we should rather prefer it,\" laughed the young girl, from her\ndistant place.\n\n\"Then I'll go on. I've been going in for reform lately--I began last\nMonday morning. Yes--of course you all laugh, because I've not much of a\nreputation for reform, or anything else. But the statement is necessary\nbecause it's true, and bears on the subject. Reform means claret and\nsoda, and very little of that. It had rather affected my temper, as I\nwasn't used to it, and I was sitting in the club yesterday afternoon,\ntrying to read a paper and worrying about things generally, when Frank,\nthere, wanted me to drink with him, and I wouldn't, and I didn't choose\nto tell him I was trying to be good, because I wasn't sure that I was\ngoing to be. Anyhow, he wouldn't take 'no,' and I wouldn't say\n'yes'--and so I suppose I behaved rather rudely to him.\"\n\n\"Like a fiend!\" observed Miner, from a distance.\n\n\"Exactly. Then I was called to the telephone, and found that my uncle\nRobert wanted me at once, that very moment, and wouldn't say why. So I\ncame back in a hurry, and as I was coming out of the cloak-room with my\nhat and coat on I ran into Bright, who generally saves my life when the\nthing is to be done promptly. I suppose I looked rather wild, didn't I,\nHam?\"\n\n\"Rather. You were white--and queer altogether. I thought you 'had it\nbad.'\"\n\nThere was a titter and a laugh, as the two men looked at one another and\nsmiled.\n\n\"Well, you've not often been wrong, Ham,\" said Ralston, laughing too. \"I\ndon't propose to let my guardian angel lead a life of happy idleness--\"\n\n\"Keep an angel, and save yourself,\" suggested Miner.\n\n\"Don't make them laugh till I've finished,\" said Ralston, \"or they won't\nunderstand. Well--Ham tried to hold me, and I wouldn't be held. He's\nabout twenty times stronger than I am, anyhow, and he'd got hold of my\narm--wanted to calm me before I went out, as he thought. I lost my\ntemper--\"\n\n\"Your family's been advertising a reward if it's found, ever since you\nwere born,\" observed Miner.\n\n\"Suppress that man, can't you--somebody?\" cried Ralston, good-naturedly.\n\"So I tripped Bright up under Miner's nose--and there was Crowdie there,\nand a couple of servants, so it was rather a public affair. I got out of\nthe door, and made for the park--uncle Robert's, you know. Being in a\nrage, I walked, and passing the Murray Hill Hotel, I went in, from sheer\nforce of habit, and ordered a cocktail. I hadn't more than tasted it\nwhen I remembered what I was about, and promptly did the Spartan\ndodge--to the surprise of the bar-tender--and put it down and went out.\nThen uncle Robert and I had rather a warm discussion. Unfortunately,\ntoo, just that drop of whiskey--forgive the details, Miss Van De\nWater--you know I warned you--just that drop of whiskey I had touched\nwas distinctly perceptible to the old gentleman's nostrils, and he began\nto call me names, and I got angry, and being excited already, I daresay\nhe really thought I wasn't sober. Anyhow, he managed to knock my hat out\nof my hand and smash it--ask him the first time you see him, if any of\nyou doubt it.\"\n\n\"Oh, nobody doubts you, Jack,\" said Teddy Van De Water, vehemently.\n\"Don't be an idiot!\"\n\n\"Thank you, Teddy,\" laughed Ralston. \"Well, the next thing was that I\nbolted out of the house with a smashed hat, and forgot my overcoat in my\nrage. It's there still, hanging in uncle Robert's hall. And, of course,\nbeing so angry, I never thought of my hat. It must have looked oddly\nenough. I went down Fifth Avenue, past the reservoir--nearly a mile in\nthat state.\"\n\n\"I met you,\" observed Russell Vanbrugh. \"I was just coming home--been\nlate down town. I thought you looked rather seedy, but you walked\nstraight enough.\"\n\n\"Of course I did--being perfectly sober, and only angry. I must have\nturned into East Fortieth or Thirty-ninth, when I stopped to light a\ncigar. The waxlight dazzled me, I suppose, for when I went on I fell\nover something--that street is awfully dark after the avenue--and I hurt\nmy head and my hand. This finger--\"\n\nHe held up his right hand of which one finger was encased in black silk.\nKatharine remembered the spot of blood on the letter.\n\n\"Then I don't know what happened to me. Doctor Routh said I had a\nconcussion of the brain and lost the sense of direction, but I lost my\nsenses, anyhow. Have any of you fellows ever had that happen to you?\nIt's awfully queer?\"\n\n\"I have,\" said Bright. \"I know--you're all right, but you can't tell\nwhere you're going.\"\n\n\"Exactly--you can't tell which is right and which is left. You recognize\nhouses, but don't know which way to turn to get to your own. I lost\nmyself in New York. I'm glad I've had the experience, but I don't want\nit again. Do you know where I found myself and got my direction again?\nAway down in Tompkins Square. It was ten o'clock, and I'd missed a\ndinner-party, and thought I should just have time to get home and dress,\nand go to the Assembly. But I wasn't meant to. I was dazed and queer\nstill, and it had been snowing for hours and I had no overcoat. I found\na horse-car going up town and got on. There was nobody else on it but\nthat prize-fighter chap, who turns out to have been Tom Shelton. It was\nnice and warm in the car, and I must have been pretty well fagged out,\nfor I sat down at the upper end and dropped asleep without telling the\nconductor to wake me at my street. I never fell asleep in a horse-car\nbefore in my life, and didn't expect to then. I don't know what happened\nafter that--at least not distinctly. They must have tried to wake me\nwith kicks and screams, or something, for I remember hitting out, and\nthen a struggle, and I was pitched out into the snow by the conductor\nand the prize-fighter. Of course I jumped up and made for the fighting\nman, and I remember hearing something about a fair fight, and then a lot\nof men came running up with lanterns, and I was squaring up to Tom\nShelton. I caught him one on the mouth, and I suppose that roused him. I\ncan see that right-hand counter of his coming at me now, but I couldn't\nstop it for the life of me--and that was the last I saw, until I opened\nmy eyes in my own room and saw my mother looking at me. She sent for\nDoctor Routh, and he saw that I wasn't going to die and went home,\nleaving everybody considerably relieved. But he wasn't at all sure that\nI hadn't been larking, when he first came, so he took the trouble to\nmake a thorough examination. I wasn't really hurt much, and though I'd\nhad such a crack from Shelton, and the other one when I tumbled in the\ndark, I had pretty nearly an hour's sleep in the horse-car as a\nset-off. Then my mother brought me things to eat--of course all the\nservants were in bed, and she'd rung for a messenger in order to send\nfor Routh. And I sat up and wrote a long letter before I went to bed,\nthough it wasn't easy work, with my hand hurt and my head rather queer.\nI wish I hadn't, though--it was more to show that I could, than anything\nelse. There--I think I've told you the whole story. I'm sorry I couldn't\nmake it shorter.\"\n\n\"It wasn't at all too long, Jack,\" said Katharine, in clear and gentle\ntones.\n\nShe was very white as she turned her face to him. Every one agreed with\nher, and every one began talking at once. But John did not look at her.\nHe answered some question put to him by the young girl on his left, and\nat once entered into conversation at that side, without taking any more\nnotice of Katharine than she had taken of him before.\n\n\n\n\nCHAPTER XXX.\n\n\nThe dinner was almost at an end, when John spoke to Katharine again.\nEvery one was laughing and talking at once. The point had been reached\nat which young people laugh at anything out of sheer good spirits, and\nFrank Miner had only to open his lips, at his end of the table, to set\nthe clear voices ringing; while at the other, Teddy Van De Water, whose\nconversational powers were not brilliant, but who possessed considerable\npower over his fresh, thin, plain young face, excited undeserved\napplause by putting up his eyeglass every other minute, staring solemnly\nat John as the hero of the evening, and then dropping it with a\nridiculous little smirk, supposed to be expressive of admiration and\nrespect.\n\nJohn saw him do it two or three times, while turning towards him in the\nact of talking to his neighbour on his left, and smiled good-naturedly\nat each repetition of the trick. To tell the truth, the evident turn of\nfeeling in his favour had so far influenced his depressed spirits that\nhe smiled almost naturally, out of sympathy, because every one was so\nhappy and so gay. But he was soon tired of young Van De Water's joke,\nbefore the others were, and looking away in order not to see the\neyeglass fall again, he caught sight of Katharine's face.\n\nHer eyes were not upon him, and she might have been supposed to be\nlooking past him at some one seated farther down the table, but she saw\nhim and watched him, nevertheless. She was quite silent now, and her\nface was pale. He only glanced at her, and was already turning his head\naway once more when her lips moved.\n\n\"Jack!\" she said, in a low voice, that trembled but reached his ear,\neven amidst the peals of laughter which filled the room.\n\nHe looked at her again, and his features hardened a little in spite of\nhim. But he knew that Bright, who sat opposite, was watching both\nKatharine and himself, and he did his best to seem natural and\nunconcerned.\n\n\"What is it?\" he asked.\n\nShe did not find words immediately with which to answer the simple\nquestion, but her face told all that her voice should have said, and\nmore. The contraction of the broad brows was gone at last, and the great\ngrey eyes were soft and pleading.\n\n\"You know,\" she said, at last.\n\nJohn felt that his lips would have curled rather scornfully, if he had\nallowed them. He set his mouth, by an effort, in a hard, civil smile.\nIt was the best he could do, for he had been badly hurt. Repentance\nsometimes satisfies the offender, but he who has been offended demands\nblood money. John deserved some credit for saying nothing, and even for\nhis cold, conventional smile.\n\n\"Jack--dear--aren't you going to forgive me?\" she asked, in a still\nlower tone than before.\n\nRalston glanced up and down the table, man-like, to see whether they\nwere watched. But no one was paying any attention to them. Hamilton\nBright was looking away, just then.\n\n\"Why didn't you answer my letter?\" asked John, at last, but he could not\ndisguise the bitterness of his voice.\n\n\"I only--it only came--that is--it was this evening, when I was all\ndressed to come here.\"\n\nJohn could not control his expression any longer, and his lip bent\ncontemptuously, in spite of himself.\n\n\"It was mailed very early this morning, with a special delivery stamp,\"\nhe said, coldly.\n\n\"Yes, it reached the house--but--oh, Jack! How can I explain, with all\nthese people?\"\n\n\"It wouldn't be easy without the people,\" he answered. \"Nobody hears\nwhat we're saying.\"\n\nKatharine was silent for a moment, and looked at her plate. In a lover's\nquarrel, the man has the advantage, if it takes place in the midst of\nacquaintances who may see what is happening. He is stronger and, as a\nrule, cooler, though rarely, at heart, so cold. A woman, to be\npersuasive, must be more or less demonstrative, and demonstrativeness is\nvisible to others, even from a great distance. Katharine did not\nbelittle the hardness of what she had to do in so far as she reckoned\nthe odds at all. She loved John too well, and knew again that she loved\nhim; and she understood fully how she had injured him, if not how much\nshe had hurt him. She was suffering herself, too, and greatly--much more\nthan she had suffered so long as her anger had lasted, for she knew, too\nlate, that she should have believed in him when others did not, rather\nthan when all were for him and with him, so that she was the very last\nto take his part. But it was hard, and she tried to think that she had\nsome justification.\n\nAfter Ralston had finished telling his story, Russell Vanbrugh, who was\nan eminent criminal lawyer, had commented to her upon the adventure,\ntelling her how men had been hanged upon just such circumstantial\nevidence, when it had not chanced that such a man as Doctor Routh, at\nthe head of his profession and above all possible suspicion, had\nintervened in time. She tried to argue that she might be pardoned for\nbeing misled, as she had been. But her conscience told her flatly that\nshe was deceiving herself, that she had really known far less than most\nof the others about the events of the previous day, some of which were\nnow altogether new to her, that she had judged John in the worst light\nfrom the first words she had heard about him at the Assembly ball, and\nhad not even been at pains to examine the circumstances so far as she\nmight have known them. And she remembered how, but a short time previous\nto the present moment, she had looked at the sealed envelope with\ndisgust--almost with loathing, and had turned over its ashes with the\ntongs. Yet that letter had cost him a supreme effort of strength and\nwill, made for her sake, when he was bruised and wounded and exhausted\nwith fatigue.\n\n\"Jack,\" she said at last, turning to him again, \"I must talk to you.\nPlease come to me right after dinner--when you come back with the\nmen--will you?\"\n\n\"Certainly,\" answered John.\n\nHe knew that an explanation was inevitable. Oddly enough, though he now\nhad by far the best of the situation, he did not wish that the\nexplanatory interview might come so soon. Perhaps he did not wish for it\nat all. With Katharine love was alive again, working and suffering. With\nhim there was no response, where love had been. In its place there was\nan unformulated longing to be left alone for a time, not to be forced to\nrealize how utterly he had been distrusted and abandoned when he had\nmost needed faith and support. There was an unwilling and unjust\ncomparison of Katharine with his mother, too, which presented itself\nconstantly. Losing the sense of values and forgetting how his mother had\ndenied his word of honour, he remembered only that her disbelief had\nlasted but an hour, and that hour seemed now but an insignificant\nmoment. She had done so much, too, and at once. He recalled, amid the\nnoise and laughter, the clinking of the things on the little tray she\nhad brought up for him and set down outside his door--a foolish detail,\nbut one of those which strike fast little roots as soon as the seed has\nfallen. The reaction, too, after all he had gone through, was coming at\nlast and was telling even on his wiry organization. Most men would have\nbroken down already. He wished that he might be spared the necessity of\nKatharine's explanation--that she would write to him, and that he might\nread in peace and ponder at his leisure--and answer at his discretion.\nYet he knew very well that the situation must be cleared up at once. He\nregretted having given Katharine but that one word in answer to her\nappeal--for he did not wish to seem even more unforgiving than he felt.\n\n\"I'll come as soon as possible,\" he said, turning to her. \"I'll come\nnow, if you like.\"\n\nIt would have been a satisfaction to have it over at once. But Katharine\nshook her head.\n\n\"You must stay with the men--but--thank you, Jack.\"\n\nHer voice was very sweet and low. At that moment Ruth Van De Water\nnodded to her brother, and in an instant all the sixteen chairs were\npushed back simultaneously, and the laughter died away in the rustle of\nsoft skirts and the moving of two and thirty slippered feet on the thick\ncarpet.\n\n\"No!\" cried Miss Van De Water, looking over her shoulder with a little\nlaugh at the man next to her, who offered his arm in the European\nfashion. \"We don't want you--we're not in Washington--we're going to\ntalk about you, and we want to be by ourselves. Stay and smoke your\ncigars--but not forever, you know,\" she added, and laughed again, a\nsilvery, girlish laugh.\n\nRalston stood back and watched the fair young girls and women as they\nfiled out. After all, there was not one that could compare with\nKatharine--whether he loved her, or not, he added mentally.\n\nWhen the men were alone, they gathered round him under a great cloud of\nsmoke over their little cups of coffee and their tiny liqueur glasses\nof many colours. He had always been more popular than he had been\nwilling to think, which was the reason why so much had been forgiven\nhim. He had assuredly done nothing heroic on the present occasion,\nunless his manly effort to fight against his taste for drinking was\nheroic. If it was, the majority of the seven other men did not think of\nit nor care. But he did not deserve such very great credit even for\nthat, perhaps, for there was that strain of asceticism in him which\nmakes such things easier for some people than for others. Most of them,\nbeing young, envied and admired him for having stood up to a champion\nprize-fighter in fair combat, heavily handicapped as he had been, and\nfor having reached his antagonist once, at least, before he went down. A\ngood deal of the enthusiasm young men occasionally express for one of\nthemselves rests on a similar basis, and yet is not to be altogether\ndespised on that account.\n\nJohn warmed to something almost approaching to geniality, in the midst\nof so much good-will, in spite of his many troubles and of the painful\ninterview which was imminent. When Van De Water dropped the end of his\ncigar and suggested that they should go into the drawing-room and not\nwaste the evening in doing badly what they could do well at their clubs\nfrom morning till night, John would have been willing to stay a little\nlonger. He was very tired. Three or four glasses of wine would have\nwarmed him and revived him earlier, but he had not broken down in his\nresolution yet--and coffee and cigars were not bad substitutes, after\nall. The chair was comfortable, it was warm and the lights were soft. He\nrose rather regretfully and followed the other men through the house to\njoin the ladies.\n\nWithout hesitation, since it had to be done, he went up to Katharine at\nonce. She had managed to keep a little apart from the rest, and in the\nchanging of places and positions which followed the entrance of the men,\nshe backed by degrees towards a corner in which there were two vacant\neasy chairs, one on each side of a little table covered with bits of\nrare old silver-work, and half shielded from the rest of the room by the\nend of a grand piano. It would have been too remote a seat for two\npersons who wished to flirt unnoticed, but Katharine knew perfectly well\nthat most of her friends believed her to be engaged to marry John\nRalston, and was quite sure of being left to talk with him in peace if\nshe chose to sit down with him in a corner.\n\nGravely, now, and with no inclination to let his lips twist\ncontemptuously, John sat down beside her, drawing his chair in front of\nthe small table, and waiting patiently while she settled herself.\n\n\"It was impossible to talk at table,\" she said nervously, and with a\nslight tremor in her voice.\n\n\"Yes--with all those people,\" assented John.\n\nA short silence followed. Katharine seemed to be choosing her words. She\nlooked calm enough, he thought, and he expected that she would begin to\nmake a deliberate explanation. All at once she put out her hand\nspasmodically, drew it back again, and began to turn over and handle a\ntiny fish of Norwegian silver which lay among the other things on the\ntable.\n\n\"It's all been a terrible misunderstanding--I don't know where to\nbegin,\" she said, rather helplessly.\n\n\"Tell me what became of my letter,\" answered John, quietly. \"That's the\nimportant thing for me to know.\"\n\n\"Yes--of course--well, in the first place, it was put into papa's hands\nthis morning just as he was going down town.\"\n\n\"Did he keep it?\" asked Ralston, his anger rising suddenly in his eyes.\n\n\"No--that is--he didn't mean to. He thought I was asleep--you see he had\nread those things in the papers, and was angry and recognized your\nhandwriting--and he thought--you know the handwriting really was rather\nshaky, Jack.\"\n\n\"I've no doubt. It wasn't easy to write at all, just then.\"\n\n\"Oh, Jack dear! If I'd only known, or guessed--\"\n\n\"Then you wouldn't have needed to believe a little,\" answered John.\n\"What did your father do with the letter?\"\n\n\"He had it in his pocket all day, and brought it home with him in the\nevening. You see--I'd been out--at the Crowdies'--and then I came home\nand shut myself up. I was so miserable--and then I fell asleep.\"\n\n\"You were so miserable that you fell asleep,\" repeated Ralston, cruelly.\n\"I see.\"\n\n\"Jack! Please--please listen to me--\"\n\n\"Yes. I beg your pardon, Katharine. I'm out of temper. I didn't mean to\nbe rude.\"\n\n\"No, dear. Please don't. I can't bear it.\" Her lip quivered. \"Jack,\" she\nbegan again, after a moment, \"please don't say anything till I've told\nyou all I have to say. If you do--no--I can't help it--I'm crying now.\"\n\nHer eyes were full of tears, and she turned her face away quickly to\nrecover her self-control. John was pained, but just then he could find\nnothing to say. He bent his head and looked at his hand, affecting not\nto see how much moved she was.\n\nA moment later she turned to him, and the tears seemed to be gone again,\nthough they were, perhaps, not far away. Strong women can make such\nefforts in great need.\n\n\"I went into my mother's room on my way down to the carriage to come\nhere,\" she continued. \"Papa came in, bringing your letter. He had not\nopened it, of course--he only wanted to show me that he had received it,\nand he said he would destroy it after showing it to me. I looked at\nit--and oh, the handwriting was so shaky, and there were spots on the\nenvelope--Jack--I didn't want to read it. That's the truth. I let him\nburn it. I turned over the ashes to see that there was nothing left.\nThere--I've told you the truth. How could I know--oh, how could I know?\"\n\nJohn glanced at her and then looked down again, not trusting himself to\nspeak yet. The thought that she had not even wished to read that letter,\nand that she had stood calmly by while her father destroyed it,\ndeliberately turning over the ashes afterwards, was almost too much to\nbe borne with equanimity. Again he remembered what it had cost him to\nwrite it, and how he had felt that, having written it, Katharine, at\nleast, would be loyal to him, whatever the world might say. He would\nhave been a little more than human if he could have then and there\nsmiled, held out his hand, and freely forgiven and promised to forget.\n\nAnd yet she, too, had some justice on her side, though she was ready and\nwilling to forget it all, and to bear far more of blame than she\ndeserved. Russell Vanbrugh had told her that a man might easily be\nconvicted on such evidence. Yet in her heart she knew that her disbelief\nhad waited for no proofs last night, but had established itself supreme\nas her disappointment at John's absence from the ball.\n\n\"Jack,\" she began again, seeing that he did not speak, \"say\nsomething--say that you'll try to forgive me. It's breaking my heart.\"\n\n\"I'll try,\" answered John, in a voice without meaning.\n\n\"Ah--not that way, dear!\" answered Katharine, with a breaking sigh. \"Be\nkind--for the sake of all that has been!\"\n\nThere was a deep and touching quaver in the words. He could say nothing\nyet.\n\n\"Of all that might have been, Jack--it was only yesterday morning that\nwe were married--dear--and now--\"\n\nHe lifted his face and looked long into her eyes--she saw nothing but\nregret, coldness, interrogation in his. And still he was silent, and\nstill she pleaded for forgiveness.\n\n\"But it can't be undone, now. It can never be undone--and I'm your wife,\nthough I have distrusted you, and been cruel and heartless and unkind.\nDon't you see how it all was, dear? Can't you be weak for a moment, just\nto understand me a little bit? Won't you believe me when I tell you how\nI hate myself and despise myself and wish that I could--oh, I don't\nknow!--I wish I could wash it all away, if it were with my heart's\nblood! I'd give it, every drop, for you, now--dear\none--sweetheart--forgive me! forgive me!\"\n\n\"Don't, Katharine--please don't,\" said John, in an uncertain tone, and\nlooking away from her again.\n\n\"But you must,\" she cried in her low and pleading voice, leaning far\nforward, so that she spoke very close to his averted face. \"It's my\nlife--it's all I have! Jack--haven't women done as bad things and been\nforgiven and been loved, too, after all was over? No--I know--oh, God!\nIf I had but known before!\"\n\n\"Don't talk like that, Katharine!\" said Ralston, distressed, if not\nmoved. \"What's done is done, and we can't undo it. I made a bad mistake\nmyself--\"\n\n\"You, Jack? What? Yesterday?\" She thought he spoke of their marriage.\n\n\"No--the night before--at the Thirlwalls', when I told you that I\nsometimes drank--and all that--\"\n\n\"Oh, no!\" exclaimed Katharine. \"You were so right. It was the bravest\nthing you ever did!\"\n\n\"And this is the result,\" said John, bitterly. \"I put it all into your\nhead then. You'd never thought about it before. And of course things\nlooked badly--about yesterday--and you took it for granted. Isn't that\nthe truth?\"\n\n\"No, dear. It's not--you're mistaken. Because I thought you brave, night\nbefore last, was no reason why I should have thought you a coward\nyesterday. No--don't make excuses for me, even in that way. There are\nnone--I want none--I ask for none. Only say that you'll try to forgive\nme--but not as you said it just now. Mean it, Jack! Oh, try to mean it,\nif you ever loved me!\"\n\nRalston had not doubted her sincerity for a moment, after he had caught\nsight of her face when he had finished telling his story at the\ndinner-table. She loved him with all her heart, and her grief for what\nshe had done was real and deep. But he had been badly hurt. Love was\nhalf numb, and would not wake, though his tears were in her voice.\n\nNevertheless, she had moved John so far that he made an effort to meet\nher, as it were, and to stretch out his hand to hers across the gulf\nthat divided them.\n\n\"Katharine,\" he said, at last, \"don't think me hard and unfeeling. You\nmanaged to hurt me pretty badly, that's all. Just when I was down, you\nturned your back on me, and I cared. I suppose that if I didn't love\nyou, I shouldn't have cared at all, or not so much. Shouldn't you think\nit strange if I'd been perfectly indifferent, and if I were to say to\nyou now--'Oh, never mind--it's all right--it wasn't anything'? It seems\nto me that would just show that I'd never loved you, and that I had\nacted like a blackguard in marrying you yesterday morning. Wouldn't it?\"\n\nKatharine looked at him, and a gleam of hope came into her eyes. She\nnodded twice in silence, with close-set lips, waiting to hear what more\nhe would say.\n\n\"I don't like to talk of forgiveness and that sort of thing between you\nand me, either,\" he continued. \"I don't think it's a question of\nforgiveness. You're not a child, and I'm not your father. I can't\nexactly forgive--in that sense. I never knew precisely what the word\nmeant, anyhow. They say 'forgive and forget'--but if forgiving an injury\nisn't forgetting it, what is it? Love bears, but doesn't need to\nforgive, it seems to me. The forgiveness consists in the bearing. Well,\nyou don't mean to make me bear anything more, do you?\"\n\nA smile came into his face, not a very gentle one, but nevertheless a\nsmile. Katharine's hand went out quickly and touched his own.\n\n\"No, dear, never,\" she said simply.\n\n\"Well--don't. Perhaps I couldn't bear much more just now. You see, I've\nloved you very much.\"\n\n\"Don't say it as though it were past, Jack,\" said Katharine, softly.\n\n\"No--I was thinking of the past, that's all.\"\n\nHe paused a moment. His heart was beating a little faster now, and\ntender words were not so far from his lips as they had been five minutes\nearlier. He could be silent and still be cold. But she had made him feel\nthat she loved him dearly, and her voice waked the music in his own as\nhe spoke.\n\n\"It was because I loved you so, that I felt it all,\" he said. \"A little\nmore than you thought I could--dear.\"\n\nIt was he, now, who put out his hand and touched a fold of her gown\nwhich was near him, as she had touched his arm. The tears came back to\nKatharine's eyes suddenly and unexpectedly, but they did not burn as\nthey had burned before.\n\n\"I've never loved any one else,\" he continued presently. \"Yes--and I\nknow you've not. But I'm older, and I know men who have been in\nlove--what they call being in love--twice and three times at my age.\nI've not. I've never cared for any one but you, and I don't want to.\nI've been a failure in a good many ways, but I shan't be in that one\nway. I shall always love you--just the same.\"\n\nKatharine caught happily at the three little words.\n\n\"Just the same--as though all this had never happened, Jack?\" she asked,\nbending towards him, and looking into his brown eyes. \"If you'll say\nthat again, dear, I shall be quite happy.\"\n\n\"Yes--in a way--just the same,\" answered Ralston, as though weighing his\nwords.\n\nKatharine's face fell.\n\n\"There's a reservation, dear--I knew there would be,\" she said, with a\nsigh.\n\n\"No,\" answered Ralston. \"Only I didn't want to say more than just what I\nmeant. I've been angry myself--I was angry at dinner--perhaps I was\nangry still when I sat down here. I don't know. I didn't mean to be.\nIt's hard to say exactly what I do mean. I love you--just the same as\never. Only we've both been very angry and shall never forget that we\nhave been, though we may wonder some day why we were. Do you understand?\nIt's not very clear, but I'm not good at talking.\"\n\n\"Yes.\" Katharine's face grew brighter again. \"Yes,\" she repeated, a\nmoment later; \"it's what I feel--only I wish that you might not feel it,\nbecause it's all my fault--all of it. And yet--oh, Jack! It seems to me\nthat I never loved you as I do now--somehow, you seem dearer to me since\nI've hurt you, and you've forgiven me--but I wasn't to say that!\"\n\n\"No, dear--don't talk of forgiveness. Tell me you love me--I'd rather\nhear it.\"\n\n\"So would I--from you, Jack!\"\n\nSome one had sat down at the piano. The keyboard was away from them, so\nthat they could not see who it was, but as Katharine spoke a chord was\nstruck, then two or three more followed, and the first bars of a waltz\nrang through the room. It was the same which the orchestra had been\nplaying on the previous evening, just when Katharine had left the\nAssembly rooms with Hester Crowdie.\n\n\"They were playing that last night,\" she said, leaning toward him once\nmore in the shadow of the piano. \"I was so unhappy--last night--\"\n\nNo one was looking at them in their corner. John Ralston caught her hand\nin his, pressed it almost sharply, and then held it a moment.\n\n\"I love you with all my heart,\" he said.\n\nThe deep grey eyes melted as they met his, and the beautiful mouth\nquivered.\n\n\"I want to kiss you, dear,\" said Katharine. \"Then I shall know. Do you\nthink anybody will see?\"\n\n       *       *       *       *       *\n\nThat is the story of those five days, from Monday afternoon to Friday\nevening, in reality little more than four times twenty-four hours. It\nhas been a long story, and if it has not been well told, the fault lies\nwith him who has told it, and may or may not be pardoned, according to\nthe kindness of those whose patience has brought them thus far. And if\nthere be any whose patience will carry them further, they shall be\nsatisfied before long, unless the writer be meanwhile gathered among\nthose who tell no tales.\n\nFor there is much more to be said about John Ralston and Katharine, and\nabout all the other people who have entered into their lives. For\ninstance, it may occur to some one to wonder whether, after this last\nevening, John and Katharine declared their marriage at once, or whether\nthey were obliged to keep the secret much longer, and some may ask\nwhether John Ralston's resolution held good against more of such\ntemptations as he had resisted on Wednesday night at the Thirlwalls'\ndance. Some may like to know whether old Robert Lauderdale lived many\nyears longer, and, if he died, what became of the vast Lauderdale\nfortune; whether it turned out to be true that Alexander Junior was\nrich, or, at least, not nearly so poor as he represented himself to be;\nwhether Walter Crowdie had another of those strange attacks which had so\nterrified his wife on Monday night; whether he and Paul Griggs, the\nveteran man of letters, were really bound by some common tie of a former\nhistory or not, and, finally, perhaps, whether Charlotte Slayback got\ndivorced from Benjamin Slayback of Nevada, or not. There is also a\npretty little tale to be told about the three Misses Miner, Frank's\nold-maid sisters. And some few there may be who will care to know what\nKatharine's convictions ultimately became and remained, when, after\npassing through this five days' storm, she found time once more for\nthought and meditation. All these things may interest a few patient\nreaders, but the main question here raised and not yet answered is\nwhether that hasty, secret marriage between Katharine and John turned\nout to have been really such a piece of folly as it seemed, or whether\nthe lovers were ultimately glad that they had done as they did. It is\nassuredly very rash to be married secretly, and some of the reasons\ngiven by Katharine when she persuaded John to take the step were not\nvery valid ones, as he, at least, was well aware at the time. But, on\nthe other hand, such true love as they really bore one another is good,\nand a rare thing in the world, and when men and women feel such love,\nhaving felt it long, and knowing it, they may be right to do such things\nto make sure of not being parted; and they may live to look each into\nthe other's eyes and say, long afterwards, 'Thank God that we were not\nafraid.' But this must not be asserted of them positively by others\nwithout proof.\n\nFor better, or for worse, Katharine Lauderdale is Katharine Ralston, and\nmust be left sitting behind the piano with her husband after the Van De\nWaters' dinner-party. And if she is the centre of any interest, or even\nof any idle speculation for such as have read these pages of her\nhistory, they have not been written in vain. At all events, she has\nmade a strange beginning in life, and almost unawares she has been near\nsome of the evil things which lie so close to the good, at the root of\nall that is human. But youth does not see the bad sights in its path.\nIts young eyes look onward, and sometimes upward, and it passes by on\nthe other side.\n\n\nTHE END.\n\n       *       *       *       *       *\n\n\nF. Marion Crawford's Novels.\n\nNEW UNIFORM AND COMPLETE EDITION.\n\n12mo. Cloth. $1.00 each.\n\n\nSARACINESCA.\n\n\"The work has two distinct merits, either of which would serve to make\nit great,--that of telling a perfect story in a perfect way, and of\ngiving a graphic picture of Roman society in the last days of the Pope's\ntemporal power.... The story is exquisitely told.\"--_Boston Traveller._\n\n\nSANT' ILARIO.\n\nA Sequel to _SARACINESCA_.\n\n\"A singularly powerful and beautiful story.... It fulfils every\nrequirement of artistic fiction. It brings out what is most impressive\nin human action, without owing any of its effectiveness to\nsensationalism or artifice. It is natural, fluent in evolution,\naccordant with experience, graphic in description, penetrating in\nanalysis, and absorbing in interest.\"--_New York Tribune._\n\n\nDON ORSINO.\n\nA Sequel to _SARACINESCA_ and _SANT' ILARIO_.\n\n\"Perhaps the cleverest novel of the year.... There is not a dull\nparagraph in the book, and the reader may be assured that once begun,\nthe story of _Don Orsino_ will fascinate him until its close.\"--_The\nCritic._\n\n\nPIETRO CHISLERI.\n\n\"The imaginative richness, the marvellous ingenuity of plot, the power\nand subtlety of the portrayal of character, the charm of the romantic\nenvironment,--the entire atmosphere, indeed,--rank this novel at once\namong the great creations.\"--_The Boston Budget._\n\n\nA TALE OF A LONELY PARISH.\n\n\"It is a pleasure to have anything so perfect of its kind as this brief\nand vivid story.... It is doubly a success, being full of human\nsympathy, as well as thoroughly artistic in its nice balancing of the\nunusual with the commonplace, the clever juxtaposition of innocence and\nguilt, comedy and tragedy, simplicity and intrigue.\"--_Critic._\n\n\nMACMILLAN & CO.,\n\n66 FIFTH AVENUE, NEW YORK.\n\n       *       *       *       *       *\n\nMR. ISAACS.\n\nA Tale of Modern India.\n\n\"Under an unpretentious title we have here the most brilliant novel, or\nrather romance, that has been given to the world for a very long\ntime.\"--_The American._\n\n\nDR. CLAUDIUS.\n\nA True Story.\n\n\"It by no means belies the promises of its predecessor. The story, an\nexceedingly improbable and romantic one, is told with much skill; the\ncharacters are strongly marked without any suspicion of caricature, and\nthe author's ideas on social and political subjects are often brilliant\nand always striking. It is no exaggeration to say that there is not a\ndull page in the book, which is peculiarly adapted for the recreation of\nstudent or thinker.\"--_Living Church._\n\n\nTO LEEWARD.\n\n\"A story of remarkable power.\"--_The Review of Reviews._\n\n\"The four characters with whose fortunes this novel deals, are, perhaps,\nthe most brilliantly executed portraits in the whole of Mr. Crawford's\nlong picture gallery, while for subtle insight into the springs of human\npassion and for swift dramatic action none of the novels surpasses this\none.\"--_The News and Courier._\n\n\nTHE THREE FATES.\n\n\"Mr. Crawford has manifestly brought his best qualities as a student of\nhuman nature and his finest resources as a master of an original and\npicturesque style to bear upon this story. Taken for all in all it is\none of the most pleasing of all his productions in fiction, and it\naffords a view of certain phases of American, or perhaps we should say\nof New York, life that have not hitherto been treated with anything like\nthe same adequacy and felicity.\"--_Boston Beacon._\n\n\nA CIGARETTE-MAKER'S ROMANCE.\n\n\"The interest is unflagging throughout. Never has Mr. Crawford done more\nbrilliant realistic work than here. But his realism is only the case and\ncover for those intense feelings which, placed under no matter what\nhumble conditions, produce the most dramatic and the most tragic\nsituations.... This is a secret of genius, to take the most coarse and\ncommon material, the meanest surroundings, the most sordid material\nprospects, and out of the vehement passions which sometimes dominate all\nhuman beings to build up with these poor elements, scenes, and passages,\nthe dramatic and emotional power of which at once enforce attention and\nawaken the profoundest interest.\"--_New York Tribune._\n\n\nAN AMERICAN POLITICIAN.\n\n\nMACMILLAN & CO.,\n\n66 FIFTH AVENUE, NEW YORK.\n\n       *       *       *       *       *\n\nTHE WITCH OF PRAGUE.\n\nA Fantastic Tale.\n\nIllustrated by W. J. HENNESSY.\n\n\"The artistic skill with which this extraordinary story is constructed\nand carried out is admirable and delightful.... Mr. Crawford has scored\na decided triumph, for the interest of the tale is sustained\nthroughout.... A very remarkable, powerful, and interesting\nstory.\"--_New York Tribune._\n\n\nGREIFENSTEIN.\n\n\" ...Another notable contribution to the literature of the day. It\npossesses originality in its conception and is a work of unusual\nability. Its interest is sustained to the close, and it is an advance\neven on the previous work of this talented author. Like all Mr.\nCrawford's work this novel is crisp, clear, and vigorous, and will be\nread with a great deal of interest.\"--_New York Evening Telegram._\n\n\nWITH THE IMMORTALS.\n\n\"The strange central idea of the story could have occurred only to a\nwriter whose mind was very sensitive to the current of modern thought\nand progress, while its execution, the setting it forth in proper\nliterary clothing, could be successfully attempted only by one whose\nactive literary ability should be fully equalled by his power of\nassimilative knowledge both literary and scientific, and no less by his\ncourage and capacity for hard work. The book will be found to have a\nfascination entirely new for the habitual reader of novels. Indeed, Mr.\nCrawford has succeeded in taking his readers quite above the ordinary\nplane of novel interest.\"--_Boston Advertiser._\n\n\nZOROASTER.\n\n\"It is a drama in the force of its situations and in the poetry and\ndignity of its language; but its men and women are not men and women of\na play. By the naturalness of their conversation and behavior they seem\nto live and lay hold of our human sympathy more than the same characters\non a stage could possibly do.\"--_The New York Times._\n\n\nA ROMAN SINGER.\n\n\"One of the earliest and best works of this famous novelist.... None but\na genuine artist could have made so true a picture of human life,\ncrossed by human passions and interwoven with human weakness. It is a\nperfect specimen of literary art.\"--_The Newark Advertiser._\n\n\nPAUL PATOFF.\n\n\nMACMILLAN & CO.,\n\n66 FIFTH AVENUE, NEW YORK.\n\n       *       *       *       *       *\n\nKHALED.\n\nA Story of Arabia.\n\n\"Throughout the fascinating story runs the subtlest analysis, suggested\nrather than elaborately worked out, of human passion and motive, the\nbuilding out and development of the character of the woman who becomes\nthe hero's wife and whose love he finally wins being an especially acute\nand highly finished example of the story-teller's art.... That it is\nbeautifully written and holds the interest of the reader, fanciful as it\nall is, to the very end, none who know the depth and artistic finish of\nMr. Crawford's work need be told.\"--_The Chicago Times._\n\n\nCHILDREN OF THE KING.\n\n\"One of the most artistic and exquisitely finished pieces of work that\nCrawford has produced. The picturesque setting, Calabria and its\nsurroundings, the beautiful Sorrento and the Gulf of Salermo, with the\nbewitching accessories that climate, sea, and sky afford, give Mr.\nCrawford rich opportunities to show his rare descriptive powers. As a\nwhole the book is strong and beautiful through its simplicity, and ranks\namong the choicest of the author's many fine productions.\"--_Public\nOpinion._\n\n\nMARZIO'S CRUCIFIX.\n\n\"This work belongs to the highest department of character-painting in\nwords.\"--_The Churchman._\n\n\"We have repeatedly had occasion to say that Mr. Crawford possesses in\nan extraordinary degree the art of constructing a story. His sense of\nproportion is just, and his narrative flows along with ease and\nperspicuity. It is as if it could not have been written otherwise, so\nnaturally does the story unfold itself, and so logical and consistent is\nthe sequence of incident after incident. As a story _Marzio's Crucifix_\nis perfectly constructed.\"--_New York Commercial Advertiser._\n\n\nMARION DARCHE.\n\n\"Full enough of incident to have furnished material for three or four\nstories.... A most interesting and engrossing book. Every page unfolds\nnew possibilities, and the incidents multiply rapidly.\"--_Detroit Free\nPress._\n\n\"We are disposed to rank _Marion Darche_ as the best of Mr. Crawford's\nAmerican stories.\"--_The Literary World._\n\n\nTHE NOVEL: What It Is.\n\n18mo. Cloth. 75 cents.\n\n\"When a master of his craft speaks, the public may well listen with\ncareful attention, and since no fiction-writer of the day enjoys in this\ncountry a broader or more enlightened popularity than Marion Crawford,\nhis explanation of _The Novel: What It Is_, will be received with\nflattering interest.\"--_The Boston Beacon._\n\n\nMACMILLAN & CO.,\n\n66 FIFTH AVENUE, NEW YORK.\n\n\n\n\n\n\nEnd of the Project Gutenberg EBook of Katherine Lauderdale; Vol. 2 of 2, by \nF. Marion Crawford\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \nIMAGES OF WAR SPECIAL\n\nTHE PANZER IV\n\nHITLER'S ROCK\n\nSmiling and confident-looking black-clad panzertruppen with their Panzer IV Ausf B in Poland in 1939.\nIMAGES OF WAR SPECIAL\n\n# THE PANZER IV\n\nHITLER'S ROCK\n\nANTHONY TUCKER-JONES\n\nIllustrated by\n\nDavid Lee Hemingway\n\nFirst published in Great Britain in 2017 by\n\nPEN & SWORD MILITARY\n\nan imprint of\n\nPen & Sword Books Ltd,\n\n47 Church Street,\n\nBarnsley,\n\nSouth Yorkshire.\n\nS70 2AS\n\nCopyright \u00a9 Anthony Tucker-Jones, 2017\n\nA CIP record for this book is available from the British Library.\n\nISBN 978 1 52670 225 8\n\neISBN 978 1 47385 676 9\n\nMobi ISBN 978 1 47385 677 6\n\nThe right of Anthony Tucker-Jones to be identified as Author of this Work has been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.\n\nAll rights reserved. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical including photocopying, recording or by any information storage and retrieval system, without permission from the Publisher in writing.\n\nPen & Sword Books Ltd incorporates the Imprints of Pen & Sword Aviation, Pen & Sword Maritime, Pen & Sword Military, Wharncliffe Local History, Pen & Sword Select, Pen & Sword Military Classics and Leo Cooper.\n\nFor a complete list of Pen & Sword titles please contact\n\nPen & Sword Books Limited\n\n47 Church Street, Barnsley, South Yorkshire, S70 2AS, England\n\nE-mail: enquiries@pen-and-sword.co.uk\n\nWebsite: www.pen-and-sword.co.uk\n\n# Contents\n\n## Introduction\n\n## Photograph Sources\n\n## Chapter One\n\n## Early Days \u2013 Ausf A\u2013C\n\n## Chapter Two\n\n## Ramping up Production \u2013Ausf D\u2013F\n\n## Chapter Three\n\n## Something 'Special' \u2013 Ausf F2\n\n## Chapter Four\n\n## Panzerwaffe Backbone \u2013 Ausf G\u2013H\n\n## Chapter Five\n\n## No Frills \u2013 Ausf J\n\n## Chapter Six\n\n## The Rhino and other Beasts\n\n## Chapter Seven\n\n## In the Desert and Mountains\n\n## Chapter Eight\n\n## From Barbarossa to Berlin\n\n## Chapter Nine\n\n## Beyond the Seine\n\n## Chapter Ten\n\n## Hitler's Rock\n\n## Further Reading\n\n## Introduction\n\nThe more powerful Panther and Tiger tanks have always eclipsed the Panzer IV and yet it played a far more pivotal role throughout the entire course of the Second World War. Likewise it outshone its stablemate the Panzer III, which was eventually abandoned as a gun tank. As well as providing a well-rounded tank, the Panzer IV also supplied a highly versatile platform for a whole family of self-propelled guns and tank destroyers.\n\nThese different types of Panzer IV saw combat with both the German Army and the Waffen-SS in all the major theatres of operation. In addition the Panzer IV was widely deployed by Hitler's Axis allies. It was supplied in limited quantities to Bulgaria, Croatia, Finland, Hungary, Italy and Romania, as well as Spain and Turkey. In some cases it was subsequently turned against the Germans.\n\nThe British Army first encountered the Panzer IV in France in 1940, where its lowvelocity gun was not a great threat to the very heavily armoured but cripplingly slow Matilda II tank. Rommel lost a number of Panzer IV at Arras in the face of a valiant but ill-fated British counter-attack. However, in the deserts of North Africa the greater range of the early Panzer IV's high explosive shells gave it a distinct advantage before British tanks could close.\n\nOnce it was up-gunned it enjoyed an even greater advantage and came into its own as a gun tank. Luckily for the British the latter were never deployed in great numbers in Libya and Tunisia. In Russia and Normandy it was a different matter and the Panzer IV became the workhorse of the panzer divisions. On the Eastern Front the Panzer IV ultimately proved a worthy adversary of the highly-respected T-34 medium tank.\n\nAlthough the Panzer IV went into production in the late 1930s, there were 50 per cent less than Russia's brand-new T-34 by 1941. Nor was it produced in significant quantities until late 1942. Despite going on to form the backbone of Hitler's panzer divisions, the Panzer IV actually started life as an infantry support tank and was not intended as a battle tank. Its low-velocity gun, also fitted to the Sturmgesch\u00fctz III, was designed to fire high explosive rounds and not armour piercing. Nonetheless, Hitler's early Mk IVs initially proved useful during his Blitzkrieg into the West and Russia.\n\nWhile the Panzer III was a good design at the start of the war, it rapidly became clear that its 50mm anti-tank gun was inadequate against the newer Soviet tanks armed with a 76.2mm gun. This 50mm gun had a much higher velocity than the weapon on the Panzer IV. Therefore it was obvious soon after the invasion of the Soviet Union that the T-34 outmatched both the Panzer III and the Panzer IV. This inevitably led to an arms race as both sides sought to produce a more effective tank killer. In addition to up-gunning the Panzer IV, to counter the T-34 the Germans came up with the Tiger and the Panther but these were never built in decisive numbers.\n\nWhile the Panzer III was completely given over to assault gun production, the upgunned Panzer IV was kept in production as a gun tank, seeing combat in one form or another throughout the whole of the Second World War. The Panzer IV had an ardent supporter in General Heinz Guderian in his role as Inspector-General of Armoured Troops. It was he who lobbied to keep it in service even when newer types of panzer were being put into the field. Guderian never saw the over-engineered Panther and Tiger as viable solutions to the panzer divisions' problem warding off ever-growing numbers of enemy tanks.\n\nIn fact, despite Nazi engineering ingenuity, they were never able to dispense with the Mk IV's services, which became the equivalent of the Russian T-34 and American M4 Sherman. However, although it appeared in a number of successful guises, the Mk IV was never up-gunned to the extent that it became a decisive weapon in the same manner that the later T-34\/85 tank did.\n\nThere were ten different production models of the Panzer IV \u2013 essentially they all looked the same, though the last four types sported a very distinctive long-barrelled anti-tank gun. It was these later models that were the real tank killers. Otherwise ongoing changes mainly related to improving the powerpack and enhancing the armour. In all some 8,500 Panzer IV were built from 1937 to 1945.\n\nThroughout the war older models of the Panzer IV were progressively upgraded. This meant that they often had a mix-and-match of different parts, making it on occasions difficult to precisely identify photographs of retrofitted models. In addition there were three types of Panzer IV tank destroyer and a self-propelled anti-tank gun. These and other various specialized tracked armoured fighting vehicles, including self-propelled anti-aircraft guns, accounted for another 4,900 Panzer IV chassis.\n\nAs a good all-rounder the Panzer IV was a greatly respected adversary on both the Eastern and Western Fronts. It was on the battlefields of North Africa's deserts, Russia's steppe and Normandy's hedgerows that the Panzer IV gained its tough reputation for reliability and hitting power. Such was the bravery and tenacity of the crews that Panzer IVs were even known to tackle the mighty Joseph Stalin heavy tank. Ultimately the Panzer IV proved vastly more reliable and numerous than the Tiger and the Panther. In this respect it proved to be the better of the three and therefore was arguably the best panzer of the Second World War.\n\n## Photograph Sources\n\nAll photographs in this book are sourced via the author.\n\n## Chapter One\n\n## Early Days \u2013 Ausf A\u2013C\n\nDevelopment of Hitler's Panzer Mk IV medium support tank commenced in the mid-1930s at the same time as that of the Mk III medium tank. The latter was codenamed the Zugf\u00fchrerwagen (platoon commander's vehicle) and the former the Bataillonsf\u00fchrerwagen (battalion commander's vehicle), so were dubbed the ZW and BW series respectively. Each was assigned differing roles. As early as 1930 the German Army High Command, through the Heereswaffenamt Wa Pr\u00fcf VI (Army Ordnance Department 6), had requested that Krupp and Rheinmetall-Borsig each produce a support tank prototype. This had to be done secretly, along with all other tank development, because of the restrictions on German rearmament following the First World War.\n\nThe first prototype was dubbed the Vollkettenkraftfahrzeug 2001, which translates as fully-tracked experimental vehicle. Built in 1935 the Rheinmetall-Borsig Rh-B BW VK2001, with four pairs of road wheels, three return rollers, front drive sprocket and rear idler, was powered by a 300PS motor giving a speed of 35km\/hr. This 18-ton prototype utilized the Wilson type steering and many of the design features were later incorporated into the development of the Krupp BW.\n\nThat same year Krupp-Gruson and Maschinenfabrik & Augsburg-Nurnberg also produced competing prototypes. A Daimler design never got much further than the drawing board. Interestingly all the initial drawings for the Daimler, Krupp and MAN plans had large inter-leaved road wheels, a feature that was later incorporated in the Panther and Tiger tanks. Following trials at the K\u00fcmmersdorf and Ulm testing grounds in southern Germany, the Krupp model was selected as the most promising. The MAN prototype was too high and the Rheinmetall-Borsig pilot model had a vulnerable external suspension mechanism for the road wheels.\n\nAusf A\n\nWhile the first Panzer III built by Daimler-Benz, armed with a 37mm gun, was initially intended to be in the 15-ton category, the Panzer IV, armed with a 75mm gun, was supposed to be five tons heavier. In reality wartime requirements meant that both tanks ended up over 20 tons with the final production models being 23 tons and 25 tons respectively. Both looked very similar and required a five-man crew, but the Panzer III ultimately proved problematic as it was very difficult to up-gun the turret.\n\nKrupp was issued the development contract for the 7.5cm Gesch\u00fctz-Panzerwagen (Vs Kfz 618) or the experimental 75mm gun armoured vehicle No.618. Rather confusingly the Panzer III was designated the 37mm gun armoured vehicle N.619. Gesch\u00fctz-Panzerwagen was then altered to Panzerkampfwagen (tank armoured fighting vehicle) or PzKpfw for short in 1937. The Panzer IV was re-designated the Vs Kfz 622 which was originally assigned to the Panzer II armed with a 20mm gun.\n\nWhat gave the Panzer IV a better punch from the infantry's point of view than the Mk III was its short-barrelled low-velocity 75mm KwK37 L\/24 gun (KwK \u2013 Kampfwagenkanone or tank gun). This was primarily a close-support weapon designed mainly for firing high explosive shells. Therefore ammunition comprised 65 per cent HE, 25 per cent armour-piercing and 10 per cent smoke rounds. Its armour-piercing capability was relatively poor due to the low muzzle velocity. Nonetheless, it had a much greater range. As a result the Panzer III was later up-gunned with a more effective 50mm weapon.\n\nThe 37mm KwK L\/46.5 on the first five models of Panzer III depended on ammunition having a muzzle velocity of 745m\/s and at 100m could penetrate 34mm of armour or at 500m some 29mm. The 75mm L\/24 could manage less than 400m\/s. It was electrically fired with a semi-automatic breech action. The inner main gun mantlet on the Panzer IV featured a right-hand coaxial MG34 machine gun (fitted on the A, B and C models), with a second one in a ball mount on the right-hand side of the superstructure on the Ausf A. The KwK37 was also used on the early model Sturmgesch\u00fctz III which had the same role as the initial Panzer IV.\n\nThe Panzer IV consisted of four major sub-assemblies \u2013 the hull, front and rear superstructure and the turret. These were all bolted together in the final assembly stage. The hull itself was divided into three by two bulkheads. The engine was positioned in the rear with the drive shaft powering the front sprockets running forward to the driving compartment under the fighting compartment floor. The gearbox was located in the middle of the front compartment, with the driver to the left and the radio operator to the right. The superstructure overhung the hull sides, allowing good internal storage.\n\nVisibility was provided for the tank commander by a prominent vertical drum cupola with a total of eight vision slits. This was the same type as that used on the Panzer III Ausf B and had a similar overhang on the rear turret plate. The turret's power traverse was driven by a 500cc two-stroke petrol engine located to the left of the main engine. A number of pistol ports and vision slits were installed throughout the turret and hull.\n\nWhereas the driver and hull gunner's front plate was flat on the Panzer III, on the early IVs the driver's section was stepped forward of the rest of the superstructure. This feature permitted the driver to see to his right as well as giving more ammunition storage space. The driver also had a forward-facing vision port and binocular episcope. Roof hatches served both the driver and radio operator. Hinged flaps on the glacis plate gave access to the steering mechanism and gearbox.\n\nThe initial Panzer IV was powered by a V-12 cylinder 230hp Maybach engine (first the HL 108TR and then the HL 120TRM \u2013 also used in the Panzer III), which gave a speed of 31km\/hr and a range of 150km. Subsequent improvements to the engine would provide later models a speed of 40km\/hr and a range of 200km. The drive was powered by the gasoline engine via a five speed transmission with an epicyclic clutch and brake steering system.\n\nEach side of the hull had four pairs of rubber-tyred road wheels with a front drive sprocket and adjustable rear idler, plus four upper return rollers. This immediately made the Panzer IV easy to distinguish from the Panzer III, which only had three pairs of road wheels and three return rollers. On the Panzer IV the rear roller was set slightly lower than the others in order to run the track down onto the idler which was set lower than the sprocket.\n\nThe suspension was of leaf spring design rather than the much newer torsion bar system. This was in part as a result of Krupp-Gruson drawing on its experience with the Panzer I. The mounting bracket for the bogies and suspension was bolted to the side and base of the hull. Under the leading axle of each bogie were quarter-elliptic leaf springs, with the tail of the spring resting under a trailing axle on a roller.\n\nUp to the Ausf E the Panzer IV idler wheel was fabricated from steel plate. The metal tracks were the single pin skeleton type utilising a central triple guide horn to slot the links together. This meant they could be manufactured lighter than other tracks and were suitable for almost all terrain types. The first tracks consisted of 101 links either side, were 38cm wide and cast from manganese steel. From the Ausf F onwards the tracks were widened by a modest 2cm.\n\nProduction of the Panzer IV Ausf (Model) A commenced in the autumn of 1937, with a total of just thirty-five completed by Krupp-Gruson by March 1938. This small number rather suggested that it was just another developmental prototype undergoing extensive trials. However, all of them were accepted for service by the army with the first three being issued to the panzertruppen in January 1938. The numbers actually reflected their support role and there was a greater requirement for the Panzer III. By April that year thirty were in service and the Ausf A went on to see combat in Poland, Norway and France.\n\nThe main drawback with the Ausf A was that its very thin armour \u2013 just 15mm \u2013 was no better than the initial Panzer III's and it was slower than the latter. Access for the driver and radio operator was not ideal as they both had two-piece hatches that opened backward and forward. The rear section could catch on the main gun barrel and the mantlet. As a result it was withdrawn before the spring campaigns of 1941.\n\nAusf B\n\nThe frontal armour on the next three models, Ausf B\u2013D, was doubled to 30mm. Krupp-Gruson were instructed to produce forty-five improved Ausf B in April 1938 but only forty-two were finished due to problems with parts. Key changes from the A model included the doubling of the frontal armour and a more powerful 300hp Maybach HL120 TR engine and a six-speed SSG 76 transmission. This boosted the speed from 31km\/hr to 40km\/hr.\n\nThe Ausf B also had a new type of stepped cupola, offering better protection for the commander. Other differences involved the installation of single-piece hatches over the driver and radio operator that only opened forward. The superstructure front was also one straight piece, thereby losing the A model's step, with the hull MG 34 replaced by a visor and pistol port. Limited numbers of Ausf B saw combat in Poland, France, the Balkans and Russia. They were phased out through attrition by late 1943.\n\nAusf C\n\nJust as the limited production run of the Ausf B was coming to an end, work started on the 3rd series BW or Ausf C. This proved to be the most numerous of the first three models with 134 built between September 1938 and August 1939. The Inspectorate for Engineers was also provided with six chassis for bridge-laying tanks. The initial Ausf C order had been for 300 tanks, but this was cut by 160 before production even started.\n\nChanges on the Ausf C were largely internal with an improved turret race (or fitting), engine mount, redesigned gun mantlet housing and armoured sleeve to protect the coaxial machine gun. The engine was uprated to the Maybach HL 120TRM. Like its predecessor the Ausf C was progressively up-armoured with bolt on armoured plates and remained in service until 1943.\n\nA few of the early model Panzer IVs continued in service well into 1944, which were not upgraded. For example, the 21st Panzer Division had half a dozen Ausf B or C still with the short 75mm gun and the 116th Panzer Division had three on its books. They should have been brought up to G and H standard by this stage of the war. These outdated models were presumably employed for training or as observations tanks (in the case of one its crew were photographed shopping for cheese!). Nonetheless, they ended up being sent into action in France.\n\nAt the end of August 1944 one of the Ausf B\/C belonging to 21st Panzer was photographed abandoned in Normandy that had clearly not been up-gunned or uparmoured. It was still armed with the 24-calibre 75mm gun, had a pistol port instead of a hull machine gun and featured the narrow 36cm tracks. It looked undamaged and one of the glacis access hatches was missing, indicating it had broken down rather than been knocked out.\n\nThe Panzer IV Ausf A is immediately identifiable by its distinctive drum cupola and forward driver's position. All Panzer IV had eight road wheels and four return rollers (except for the final Ausf J, some of which only had three return rollers). Krupp-Gruson built just thirty-five of the A model from October 1937 to March 1938.\n\nThis Ausf A was photographed during the invasion of Poland in 1939. The solid white cross or balkenkruez was specific to this particular campaign. The design was soon abandoned as it made the tank far too conspicuous. The white tactical number painted over the turret vision port indicates it is the third tank of the third troop in the fourth company.\n\nThe crew have removed the right-hand drive sprocket and tracks on this Ausf A. Limited numbers of the Ausf A saw combat not only in Poland but also Norway and France before being withdrawn from the panzer regiments prior the spring campaigns of 1941.\n\nAusf B or C being given a warm welcome in Poland in 1939.\n\nThe burnt and smashed remains of a Panzer IV \u2013 the armoured sleeve protecting the coaxial machine gun identifies it as a C model. It has lost its glacis plate after taking a direct hit.\n\nThe Ausf C was almost identical to the Ausf B, having minor modifications to its armour and internal fittings.\n\nNote the drive sprocket on this early Panzer IV: from the Ausf E it was a much more simplified design. This tank bears the emblem of the 11th Panzer Division.\n\nThe drive assembly to the front sprocket has been removed from this Ausf C, showing the gearing from the brake\/steering unit. Also clearly visible is the new type of turret cupola fitted to the Ausf B and C that replaced the early drum cupola on the Ausf A.\n\n## Chapter Two\n\n## Ramping up Production \u2013 Ausf D\u2013F\n\nImprovements to the Panzer IV were largely driven by the regular need to enhance the armour. The problem with this was that the Panzerwaffe were issued with a constantlychanging tank, which was a mixed blessing as they did not have the same capabilities \u2013 the Panzerwaffe faced exactly the same problem with the similarly-evolving Panzer III.\n\nAusf D\n\nConfusingly the Ausf D was known as the 4th and 5th series Panzer IV. Krupp was instructed to build 200 in the 4th Series BW and 48 in the 5th Series BW in January 1938. In the event just 229 were finished as gun tanks, the remaining nineteen chassis being used for sixteen bridge-laying tanks, two self-propelled gun mounts and an ammunition carrier for the super-heavy Karl mortar. Also, as part of the experiments to up-gun the Panzer IV, an Ausf D was fitted with a 50mm KwK39 L\/60, which had double the muzzle velocity of the 75mm KwK37 L\/24 and therefore vastly better penetration.\n\nThe Ausf D had its side and rear armour enhanced from the 15mm of its predecessors to 20mm, and the main gun was fitted with an external mantlet. The front of the superstructure was once more stepped with the driver forward of the radio operator. The driver was provided with a pistol port to his right and the hull machine gun was reinstated for the radio operator. Ausf Ds at the tail-end of the production run had 30mm plates bolted and welded to the hull front and superstructure front as well as an extra 20mm on the sides in 1943. A few Ausf D were armed with the long-barrelled 75mm KwK L\/48 gun and deployed with training and replacement units. This gun was installed in late Ausf G and the subsequent H and J models.\n\nProduction of the Panzer IV support tank was such that by May 1940 every tank unit with a medium tank company could deploy six to eleven Mk IVs. At the start of the invasion of France on 10 May 1940 Hitler's panzer divisions were able to field a total of 280 Ausf A, B, C and D. Before it was phased out in 1944 the Ausf D saw action in France, the Balkans, Africa and Russia.\n\nAusf E\n\nThe 6th Series Ausf E went into production in September 1940. It had a new design of cupola, other modifications to the turret and increased armour. The turret rear now had a single bent plate, which eliminated the cupola overhang. In addition the turret roof was fitted with an exhaust fan to remove noxious gun fumes from the fighting compartment. The front of the Ausf E hull had 50mm of armour, with 20mm plate bolted on the hull and superstructure sides. Other modifications included countersinking the glacis hatches level with the surface of the glacis, a new driver's visor that pivoted and a simplified drive sprocket. In total forty Ausf D and E were shipped to North Africa while some 438 Ausf B to F were committed to the assault on the Soviet Union. Like its predecessor, the Ausf E was phased out in 1944.\n\nAusf F\n\nWhereas the Ausf A, B, C, D and E Panzer IV had all been produced by Krupp-Gruson, construction of the Ausf F, or Ausf F1 as it became known, was extended to the manufacturers Nibelungenwerke and Vomag. Krupp was initially instructed to build 500 of the Panzer IV 7th series, but Vomag then received an order for 100 and Nibelungenwerke for 25. However, the German Army wanted the Panzer IV upgunned with the long-barrel 75mm KwK40 L\/43 anti-tank gun as soon as possible to produce the F2. The result was that twenty-five of the F1 were converted before they were ever issued to the panzertruppen.\n\nOnce again the main improvement on the Ausf F1 was an increase in armour, which meant that it was a ton heavier than the Ausf E. The change in the armour also required modifications to the driver's visor (with installation of the Fahrersehklappe 50), vision ports, pistol ports, turret doors and the hull machine-gun fitting (the Kugelblende 50 ball mount replacing the Kugelblende 30 gimbal mount). The turret side door in previous models comprised a single hatch either side, but on the F double doors replaced these to make access easier. The glacis hatches also featured raised air intake cowls to cool the steering brakes.\n\nFrom the Ausf F1 onward the rear idler wheel was simplified and fabricated from tubular steel with plate reinforcing webs. This had seven rather than the previous eight spokes. Likewise the tracks were modified on the F1 with a reduction of links from 101 to 99. Also the track width increased from 38cm to 40cm to help spread the ground pressure of the tank.\n\nThe F1 was mainly used as combat replacements, but several new units were equipped with it and it refitted the 2nd and 5th Panzer Divisions. Some 208 Ausf B\u2013F1 were in the field when Hitler commenced Operation Blue his summer offensive on the Eastern Front in June 1942. By the time of the Kursk offensive the following summer these had been reduced to just sixty tanks.\n\nThe 4th and 5th Series Panzer IV was known as the Ausf D. Like the initial Ausf A, it also had the driver's position set forward of the radio operator's position, creating a step in the superstructure front plate. However, it had an external mantlet for the main gun.\n\nThe crew of this Ausf D are making sure they do not run out of fuel while on operations. The turret has been stacked with 'Jerricans'. The panzers also used towed trailers during operational deployment.\n\nThis D model was photographed while on exercise in early 1940. Like the earlier versions the cupola overhung the rear of the turret. An equipment bustle or storage bin was normally attached to the rear of the Panzer IV turret.\n\nThis Ausf D lost its turret and part of the engine during the fighting in France. Note the pistol port to the right-hand side of the driver's visor. A direct hit or the ammunition 'cooking off' caused the damage.\n\nAnother knocked-out Ausf D.\n\nAmongst an assortment of Panzer IIIs and IIs, this Panzer IV Ausf D was a casualty of the fighting in Libya. The crew attempted to increase the armour by adding track links to the front and the driver's compartment and the turret. The road wheel on the turret was also intended to serve the same purpose.\n\nThe drive sprocket on the Ausf E was simplified, making it easier to manufacturer and stronger. Likewise, it had a different cupola to the Ausf B and C.\n\nImprovements on the Ausf E included installing an extractor fan in the turret to remove gun fumes. The circular vent can be seen on the right side of the turret roof. In addition the glacis hatches were countersunk level with the glacis surface and it had a pivoting driver's visor.\n\nPanzertruppen examine a badlydamaged Ausf D in Libya. This model, along with the E and F1, were the most common in North Africa, making up 25 per cent of Rommel's armoured formations. It had the distinct advantage of being able to fire both high explosive and armour piercing rounds \u2013 British tanks could only fire armour piercing at close range.\n\nWhitewashed for winter operations, these Auf E are being shipped to Russia. The large barrelshaped object at the back of the tank is the engine silencer and exhaust outlet \u2013 the design of this changed as development of the Panzer IV progressed.\n\nThe Ausf F1 went into production in April 1941. New features included raised air-intake cowls on the glacis hatches. The front of the superstructure had a single 50mm plate with a new machinegun ball mount and an improved visor for the driver.\n\nThe sprocket was now dish-shaped on the Ausf F1 and the idler had seven tubular steel spokes rather than the previous design with eight made from steel plate. The turret doors were also replaced by double hatches either side.\n\nThe Kugelblende 50 ball mount for the hull machine gun on this F1 had been covered to prevent fouling by the mud.\n\nTo combat the dust being thrown up in Russia this F1 crew are wearing goggles.\n\nThis F1 has some sort of large storage box attached to the glacis plate.\n\n## Chapter Three\n\n## Something 'Special' \u2013 Ausf F2\n\nBy late 1941 it was very apparent that the 76.2mm gun mounted in the Soviet KV-1 heavy and T-34 medium tanks was superior to that of the Panzer IV's short 75mm gun. This meant it was imperative to install the much more powerful long-barrelled 75mm KwK40 L\/43 anti-tank gun as quickly as possible. This process had started in the winter of 1941 and the intention was that this would commence with the Ausf G in May 1942. However, to speed things up it was decided to co-opt Ausf F production first.\n\nAusf F2\n\nA month's worth of production of the Ausf F was disrupted in March 1942 in order to create the F2. The introduction of this model saw the length of the 75mm gun increased from 24 to 43 calibres. To compensate for the long barrel a coil-spring counter-balance was installed. It was designed to act as both an anti-tank and a high explosive firing weapon. None of its parts were interchangeable with the earlier gun, but like the 75mm KwK37 L\/24 it was electrically fired and had a semi-automatic breech action. The barrel was fitted with a muzzle brake of which there were about four different types.\n\nInitially the KwK40 L\/43 was fitted with a very distinctive globular single-baffle muzzle brake, which was followed by a series of double baffle designs. As there was a two or three-month overlap in production of both the F2 and the subsequent Ausf G, which came out of the same Krupp, Nibelungenwerke and Vomag factories, there was inevitably some overlap in appearance and design modifications. Essentially they were the same tank, with the F2 being the interim version.\n\nIn the summer of 1942 the Ausf G went over to the double-baffle muzzle brake and it is quite possible some of the later F2 featured this. Similarly some up-gunned Panzer IVs sent to the Eastern Front, while they had the single baffle muzzle brake, also had the newer-style Ausf G turret that eliminated the side vision ports.\n\nThe L\/43 was a fundamental upgrade because it changed the Panzer IV's primary role from being a support weapon to a tank-to-tank weapon. It also meant it firmly superseded the Panzer III on the battlefield. From the F2 onwards the Panzer IV went from being a medium support tank to a medium battle tank. The StuG III assault gun went through exactly the same evolution when its close support weapon was upgraded to create the StuG Ausf F.\n\nThe earlier L\/24 gun had a muzzle velocity of 385m\/s and could penetrate 41mm of armour at 100m, 39mm at 400mm and 35mm at 1,000m. This degraded to 33mm and 30mm at 1,500m and 2,000m respectively but at such ranges accuracy was a problem. In contrast the much more powerful long-barrel L\/43 had a muzzle velocity of 740m\/s and could cut through almost 100mm of armour at 100m, 90mm at 500m and 80mm at 1,000m.\n\nThis was exactly what the Panzerwaffe needed at this crucial moment in the war. It was only now that the Panzer IV truly came into its own and was to remain the backbone of the Panzerwaffe for the rest of the war despite the appearance of the Tiger and Panther. From mid-1942 onwards the Panzer IV began to take over the role of the Panzer III. Shortly after the chassis of the latter was given over to assault gun production.\n\nProvision for full indirect fire was never made on German tanks. On the Panzer IV targeting of the main gun was achieved using a fixed-eyepiece telescope-type sight, known as the TZF5b manufactured by E. Leitz of Wetzlar. This had compound object glass and moving graticules giving magnification of about 2.5 times for the main gun and the coaxial machine gun. In the Ausf F2 and G a clinometer was fitted to measure the angle of elevation.\n\nThe main differences between the F1 and F2 other than the barrel length, involved ammunition storage design which had to be modified to allow for the larger rounds. The commander and gunner's seats were altered to allow more ammunition to be carried. An auxiliary hand traverse was fitted for use by the loader and the elevation mechanism was modified. The commander's cupola was also moved slightly forward.\n\nWhen the F2 pitched up in North Africa the British dubbed it the 'Mk IV Special' with good reason. The heaviest British anti-tank gun could not compete. The 75mm gun on the American-supplied M3 Grant tank was inferior to the F2's because it could only penetrate 45mm of armour at 1,000 yards.\n\nGerman designers looked at ways of up-gunning the Ausf F even further. An experimental mock up was produced of a Panzer IV F2 armed with a 75mm KwK 42 L\/70. Although this weapon did not feature a muzzle brake the barrel length made the tank very nose-heavy and unwieldy. The L\/70 was at least twice as long as the L\/43 and on the Ausf F was not really a practical design for a gun tank. While this up-gunned variant of the F2 did not go into production, the gun with muzzle brake was later employed on the Panther, which appeared in 1943. Similarly it was used the following summer to arm the Panzer IV\/70 tank destroyer but without a muzzle brake. This Panzer IV variant again proved nose-heavy, which resulted in excessive wear on the front road wheels.*\n\nThe Ausf F was similarly used as an experimental mount for a weapon using the Gerlich principle. The Panzer IV F with Waffe 0725 featured a gun using a taperedbore barrel of 75\/55mm. This fired a skirted round that was compressed as it travelled along the bore of the barrel, which greatly increased the velocity of the projectile. The round had added punch thanks to a core of tungsten-carbide, but a shortage of the latter meant that the project was not viable and this F2 variant was also abandoned. Instead it was decided the best way to improve the armament of the Ausf F was to install the larger-calibre KwK40 L\/48, which led to the Ausf G.\n\nBetween March and July 1942 some 200 interim Ausf F2 were produced including the 25 converted from F1. By the summer of 1942 there were 135 Panzer IVs with the L\/43 gun on the Eastern Front. This could penetrate the T-34 out to a range of 1,600m. Although the Tiger, armed with an 88mm gun, appeared late that year it was not available in any great numbers, leaving the Panzer IV to do all the work. The Panther armed with a 75mm gun that was more powerful than the Ausf F2\/G\/H did not appear until the summer of 1943.\n\nWith the advent of the Ausf F2 the Panzer IV became a true gun tank. German officers examine a new F2 which is easily recognizable by the early single-baffle globular muzzle brake fitted to the Kwk40 L\/43 75mm gun. Later models were equipped with at least four different types of larger double-baffle muzzle brake.\n\nThe F2 began to arrive in North Africa in the summer of 1942 where the British dubbed it the Panzer IV 'Special'. It posed a threat to the under-gunned and underarmoured tanks of the 8th Army but was never available in sufficient numbers to tilt the balance.\n\nThe cupola on this tank has been blasted off. Rommel received just twenty-seven F2s by August 1942. The Panzer III remained the main German tank type in North Africa.\n\nDeployment of the F2 on the Eastern Front provided much-needed parity with the ever-growing number of T-34s.\n\nThis F2 was photographed while on garrison duty in occupied France.\n\nPanzer IV with the KwK40 early muzzle brake lost on the Eastern Front \u2013 interestingly it has no vision ports forward of the doors on the sides of the turret, suggesting that this is an early Ausf G or late-production F2.\n\nThis could be a late-production F2 with the new muzzle brake fighting in Tunisia. The lack of visible turret vision ports forward of the side hatch indicated it is an Ausf G which was quite rare in North Africa.\n\nThis Panzer IV was photographed on 10 May 1943 in Tunisia shortly before the German surrender. Judging by the muzzle brake and the lack of turret vision ports this is an early Ausf G.\n\nThe more numerous Ausf G first appeared on the Eastern Front during the summer of 1942, shortly after the F2.\n\n* For further information on the Panzer IV\/70 see the author's German Assault Guns and Tank Destroyers 1940-1945 also published by Pen & Sword.\n\n## Chapter Four\n\n## Panzerwaffe Backbone \u2013 Ausf G\u2013H\n\nThe next two production models, the Ausf G and H also armed with the KwK40, formed the bulk of the Mk IV panzer force, with 1,687 Gs and 3,774 Hs built during 1942 to 1944. From 1943 panzer divisions were meant to have one battalion equipped with Mk IVs and one with Panthers, but due to problems with the latter this often did not happen. The net result was that there were always more Panzer IVs than Panthers. Most of the Ausf G went to Russia while in 1944 most of the Mk IVs in France were Ausf H supplemented by Ausf J.\n\nAusf G\n\nThe Panzer IV Ausf G first went into production in May 1942 and was initially armed with the same main gun as the interim F2. Visually there was very little to distinguish the two. Notably, on the G model the vision ports on the sides of the turret were dispensed with as well as the one serving the loader's position on the front of the turret. By the summer of 1942 other alterations included a new muzzle brake and a system that permitted siphoning coolant from one panzer to another to help coldweather starting. The smoke dispensers were also moved from the rear of the hull to the turret sides.\n\nIt was very quickly decided to up-armour some of the G model. From 20 June 1942 deliveries included additional 30mm armour plate bolted or welded to the front of the hull and superstructure. This gave the tank a total of 80mm on the front. It had been hoped to boost it to 100mm, but trials showed the Ausf G became too nose heavy and difficult to steer. From July to November 1942 every month some sixteen production tanks were fitted with the additional protection. From December 1942 the number went up to 50 per cent of all production tanks which resulted in about 700 Ausf G having the extra armour.\n\nAt the beginning of 1943 the driver's KFF2 episcope on the Ausf G was removed. Also that year spaced armour consisting of thin steel plates or panels hung from steel brackets, known as Sch\u00fcrzen or skirts, were added to the sides of the hull and the sides and rear of the turret in response to a fresh Soviet threat. Hence the removal of the turret's side vision ports. To allow the turret side access doors to open hinged double doors were fitted to either side of the 8mm turret skirts. Despite these it was possible to escape via the turret without using the doors in the spaced plates.\n\nThe 5mm hull Sch\u00fcrzen consisted of six plates either side with the front two and the rear one tapered. The foremost was the smallest and created a greater taper than at the back, but this was easily damaged and often absent. These skirts covered the hull and superstructure, the return rollers and a quarter of the drive sprocket and idler. The eight road wheels were left exposed.\n\nMade of mild steel boiler plate, the Sch\u00fcrzen were intended to help defend the tank against attack from the new Soviet RPG-43 shaped-charge high explosive anti-tank (HEAT) hand grenade and later hollow-charge weapons such as the American Bazooka and the British PIAT (Portable Infantry Anti-Tank). The RPG-43 entered service in 1943 and could penetrated 75mm of armour at a 90\u00ba angle. Although it had to be thrown at very close range, it gave no warning like other anti-tank weapons. The skirts would prematurely detonate an incoming projectile inches in front of the main armour and dissipate the force of the HEAT round. Although the plate would be destroyed, it saved the tank from direct impact. The hull Sch\u00fcrzen were fragile and easily damaged in combat or torn off whilst the tank was on the move. As a result the additional turret skirts tended to last a lot longer. Panzer IV were regularly photographed with the hull Sch\u00fcrzen in varying states of disrepair with sections of plate missing.\n\nAnother form of protection was added from early 1943. This was a light grey paste known as Zimmerit, which was applied to the vertical surfaces of the hulls and turrets of most tanks and assault guns. Panzer IVs were coated in the factory before painting and the paste was raked with a spreader, creating ridged and criss-cross patterns to increase the depth. The paste was then hardened off and spray-painted. This prevented the attaching of magnetic anti-tank hollow charges. On occasion the Sch\u00fcrzen were also coated in it. The Zimmerit patterns varied from factory to factory.\n\nFinal production Ausf G were fitted with a new type of drive sprocket and the radio antenna was moved from the right of the hull rear to the left. The later alteration made it impossible to distinguish late Ausf G from early Ausf H. The new design sprocket wheel was intended to improve traction with the tracks. Mid-model Ausf G were fitted with a double-baffle muzzle brake on the main gun used on the subsequent H and J models.\n\nA total of 1,275 Ausf G were armed with the KwK40 L\/43 gun. Then from late March 1943 the Ausf G followed by the H and J were armed with a gun with an even longer barrel, the 75mm KwK40 L\/48. Although this fired the same ammunition as the 43-calibre gun, it had a barrel five calibres longer. The breech mechanism was also simplified to help with production. Nonetheless, apart from a few components, the barrel and breech rings of the 43- and 48-calibre guns were interchangeable.\n\nProduction of the Ausf G ended in June 1943. In total ten separate orders with Krupp-Gruson, Nibelungenwerke and Vomag amounted to 1,750, but only 1,687 were built as Ausf G. The rest were employed as prototypes for the Hummel (Bumble Bee) and Brummb\u00e4r (Grizzly Bear) consisting of ten and fifty-three chassis respectively. When Hitler's summer offensive commenced on the Eastern Front in June 1942 he had about 170 F2 and G available. By the start of the Kursk offensive the following summer Army Groups Centre and South had 841 long-barrel Panzer IVs.\n\nAusf H\n\nIn early 1943 General Heinz Guderian was appointed Inspector-General of Armoured Troops. He was not impressed with the new Panzer V dubbed the Panther, which was having unending 'teething' problems. He recalled, 'In the tank production field it was decided during April, in accordance with my suggestion, that the Panzer IV should continue to be built until such time as a high level of mass-production was absolutely assured for the Panther.' Later in the year Guderian was very displeased to learn that Panzer IV production was to be partially diverted to the construction of assault guns and tank destroyers.\n\nThe Ausf H went into production in April 1943 and was by far the most numerous type of Panzer IV ever built. Once more changes from its predecessor were fairly minimal and were alterations one would expect. On the front of the H model the armour evolved from the 50mm basic plus additional 30mm to 80mm basic \u2013 inevitably this meant a sizeable weight increase by one and a half tons bringing the tank's overall weight to 25 tons. It also meant that the tank's armour had increased fourfold over that of the original version \u2013 the Ausf A.\n\nThe turret featured a better-armoured cupola, a single-piece cupola hatch hinged to the left (all the previous models had a two-piece hatch that opened in opposite directions) and a cupola mount for an anti-aircraft machine gun. On the hull, other changes involved deletion of the driver and radio operator's side vision ports to give the armour better integrity, new-style drive sprockets and idler wheels, all-steel return rollers and external air filters.\n\nIn order to give the Panzer IV better ground clearance, in 1943 designers and engineers tried to alter the suspension, which had remained largely unchanged since 1937. These experiments were not fruitful and the Panzer IV retained the same basic suspension until the end of the war.\n\nA total of 3,935 Ausf H chassis were built, of which 3,744 were gun tanks, while 130 were diverted to Brummb\u00e4r construction by Deutsche Eisenwerke. Nibelungenwerke diverted another thirty to Krupp for the initial Sturmgesch\u00fctz IV production.* By June 1944 some two-thirds of the Panzer IV deployed in France were the H model, the rest being the new Ausf J.\n\nAs the war progressed most of the older models of Panzer IV were recalled for upgrading. For example, the Tank Museum at Bovington has an Ausf D that was factory-modified to Ausf H specifications. This principally included installing the longbarrel KwK40 L\/43 gun, while in addition the driver and hull machine gunner's positions were up-armoured using bolt on armour plate. The bow plate armour was increased and extra armour was bolted to the front plate. The turret was fitted with spaced armour and wider tracks were fitted. From a distance it looked like any other latemodel Panzer IV but up close the forward driver's compartment of the Ausf D is clearly visible.\n\nThis seems to be an Ausf G fitted with the new muzzle brake, which went into production from May 1942. The Sch\u00fcrzen spaced armour on the turret was introduced in early 1943 along with spaced side skirts in response to a new type of Soviet anti-tank grenade.\n\nA disabled Ausf H captured during the British advance on Villa Grande, Italy in 1943. Note the new design of single-piece hatch on the cupola. The engine silencer around the exhaust has been badly damaged. The tank has also thrown its left-hand track.\n\nThis abandoned late-production Ausf G or early Ausf H '725' was found hiding in a haystack near Szee, Italy. Modifications during the H production run included deleting the side vision ports for the driver and radio operator \u2013 this vehicle still has them fitted.\n\nThe crew of this Ausf H are repairing the tracks during the fighting at Monte Cassino in early 1944. They have left the doors open on the turret Sch\u00fcrzen to enable them to jump back in quickly if they come under fire.\n\nAusf H '802' serving with the 2nd Panzer Division was destroyed at Pontfarcy in Normandy. It has no Zimmerit and although the rail is fitted, the hull Sch\u00fcrzen is absent. At the beginning of June 1944 the division had ninety-four Panzer IVs.\n\nBelonging to the 21st Panzer Division, this Ausf H was knocked out on the plain north-east of Caen near Lebisey in June 1944. This tank was fighting 'hull-down' desert-style and appears to have been further concealed by the surrounding boards. Good shooting penetrated the turret's 50mm armour. Zimmerit can be seen on the front of the superstructure and the turret.\n\nRemains of a burnt-out Panzer IV along with other vehicles caught in the open by Allied fighterbombers in Normandy.\n\nThis derelict Ausf H, most likely from the 13th Panzer Division, was lost during the siege of Budapest in early 1945. The gun lacks a muzzle brake.\n\nTwo Ausf H burning on the Eastern Front: the second one is just visible through the smoke on the far right. Smoke from the nearest is pouring from the side of the turret, indicating a fighting compartment fire.\n\n* See German Assault Guns and Tank Destroyers 1940\u20131945.\n\n## Chapter Five\n\n## No Frills \u2013 Ausf J\n\nGuderian knew that the German armed forces could not dispense with the Panzer IV. Upon taking up his post as Inspector-General of Armoured Troops he had assessed that 'the production of the Panzer IV must be increased during the year 1944\u201345, so far as this can be done without damaging the production of Panthers and Tigers.'\n\nThe last model of the Mk IV had many features dropped during production in order to simplify and speed up construction. Notably one of the four return rollers was dispensed with as well as the pistol ports; some of the vision ports and the electric turret traverse plus the exhaust system were simplified. In the name of further expediency wire mesh screens were mounted on the sides of the hull instead of the normal Sch\u00fcrzen. It also had steel-tyred road wheels due to the shortage of rubber. Some 1,758 of this type were built from June 1944 to March 1945. The Ausf J saw combat in the battles for France, the Ardennes, the Rhineland and on the Eastern Front.\n\nAusf J\n\nIn light of problems refuelling and re-equipping in the field, German tank designers in their infinite wisdom decided to increase the Panzer IV's range by 110km. For Operation Barbarossa, tanks, including the Panzer IV, had been furnished with a fieldmanufactured fuel trailer. This could carry two extra 200-litre petrol tanks. The 20-litre 'Jerricans' (or jerrycan) were also strapped to the turret. However, both measures were largely temporary fixes, especially as the trailers were easily damaged.\n\nThe designers boosted the 210km range of the Ausf G and H to 320km. This was achieved by increasing the 470-litre fuel capacity of the Ausf H to 680 litres. This came at a price because the extra space needed meant removing the auxiliary engine that powered the electric turret traverse on the Ausf J. On previous models this was positioned at the rear of the hull to the left of the large exhaust outlet. A dual gearratio hand traverse was installed instead. This was clearly a retrograde step and slowed firing response times. On late Ausf J the bulky engine silencer and exhaust outlet was replaced by two simple vertical exhaust mufflers.\n\nThere had been some hope of installing the Panther tank turret armed with the KwK42 L\/70 gun on the Panzer IV chassis. The turret proved much too heavy and overloaded the chassis and transmission. Instead, the thickness of the turret roof armour on the Ausf J was increased and for close defence a N\u00e4hverteidigungswaffe smoke projector was fitted. This final model also used at least two different types of the modified East Tracks, which gave a better overall performance. As the name suggests these were developed as a result of battlefield experience on the Eastern Front.\n\nIn North Africa the Sherman had been able to hold its own against the early-model Panzer IVs, but in Normandy the frontal 80mm hull and superstructure armour of the Ausf H and J easily resisted its 75mm gun. Only the turret remained vulnerable. Limited numbers of Shermans were up-gunned with 17-pounder or 76mm guns but these were not up-armoured, leaving them at a real disadvantage.\n\nThe first Ausf J deployed to Normandy were fitted with standard Sch\u00fcrzen and coated in Zimmerit. While the hull Sch\u00fcrzen on the Ausf G and initial Ausf J consisted of six adjoining steel plates, on later production Ausf J they were replaced by simple wire mesh. These screens consisted of only three sections but covered the same area. Also by this stage the use of Zimmerit had been discontinued and it was not applied to the final batches of Ausf J. Both types of Sch\u00fcrzen were fitted to the Panzer IV\/70 tank destroyer \u2013 photographic evidence shows a Panzer IV\/70(A) with wire-mesh skirts.\n\nBy late 1943 production of the Panzer IV was increasingly under pressure. Although Guderian supported the factories being switched solely to the Panther, retooling would simply have been too disruptive. Panther manufacture was lagging behind and by this stage Germany was increasingly on the defensive on the Eastern Front. The result of this was that the Panzer IV was kept in production and was increasingly diverted to making assault guns and tank destroyers. This was exactly the same fate that had befallen the Panzer III.\n\nIn December 1943 Krupp-Gruson finally stopped building the Panzer IV gun tank and in the New Year switched production over to the StuG IV. Likewise in January 1944 Vomag began producing the Jagdpanzer IV alongside its Panzer IV. This continued until May when tank production was abandoned in favour of the Jagdpanzer IV and then the Panzer IV\/70. In August 1944 Nibelungenwerke started building its own version of the Panzer IV\/70. The result of this was that Nibelungenwerke was the only company to produce the Ausf J. As well as the 1,758 gun tanks, some 278 Ausf J chassis were diverted to Panzer IV\/70 production from August 1944. Another 142 chassis were converted to Brummb\u00e4rs.\n\nProduction was concentrated with Nibelungenwerke for a reason \u2013 the Allied bombing offensive. The original Panzer IV prototypes had been built by Krupp-Gruson at Magdeburg. Upon series production the Krupp factories in Essen and Eisen un Huettenwerke had built the hulls and turrets. Subcomponents had come from other companies. It was only with the Ausf F\/G that production needed to be shared. At this point Vomag and Nibelungenwerke became involved. In the face of escalating Allied bombing of Germany's industrial centres, the main production was then relocated to Nibelungenwerke which was out of range of the bombers.\n\nPanzer IV Ausf H on manoeuvres in France in 1944. They have a new type of drive sprocket and idler wheel. Also the turret is fitted with the new cupola with thicker armour and a single-piece cupola hatch. The haphazard three-tone camouflage has been sprayed on with little precision.\n\nAn Ausf J with the 2nd SS Panzer Division in France: half the nearside Sch\u00fcrzen is missing. This was added to afford some protection from the Soviet RPG-43 anti-tank grenade.\n\nThis late-production Ausf H lacks Zimmerit \u2013 this stopped being applied in September 1944 because it was time-consuming and due to unfounded fears it might be flammable. The tank has a very crudely painted three-tone camouflage scheme.\n\nThe Sch\u00fcrzen are intact on this Ausf J serving in Poland in 1944. The small front plate has a distinctive notch above the sprocket, designed to prevent snagging on the tracks.\n\nRemains of an Ausf J from the 26th Panzer Division, which fought in Italy in the Salerno and Cassino areas from January to May 1944. The hull has been shattered and the muzzle brake is clogged with dirt.\n\nTwo Ausf J belonging to the 2nd SS Panzer Division at St Fromond in Normandy in July 1944. Although the nearest tank has the standard engine muffler, the auxiliary engine for the electricpowered turret traverse is clearly missing.\n\nPhotographed near Li\u00e8ge in Belgium, this Ausf J with its turret reversed has a simplified twin exhaust muffler. There is evidence of Zimmerit on the rear of the hull and the front of the turret, showing it was built before September 1944.\n\nThis burnt-out Ausf J pictured in January 1945 still has the remains of its mesh-screen Sch\u00fcrzen. This was from the early production run as it has four return rollers: on the later Js one roller was dropped.\n\n## Chapter Six\n\n## The Rhino and Other Beasts\n\nAs well as assault guns and tank destroyers, the Panzer IV chassis was additionally diverted to create a wide variety of self-propelled weapon mounts. These either had anti-tank, anti-aircraft or artillery roles. Key amongst them were the Hornisse (Hornet), Hummel (Bumble Bee), Brummb\u00e4r (Grizzly Bear) and M\u00f6belwagen (Furniture Van).\n\nSelf-Propelled and Support Guns\n\nHornisse\n\nWork on creating a self-propelled platform for the formidable 88mm Pak43 anti-tank gun started in 1942. This led to the powerful Hornisse (Hornet) that was later called the Nashorn (Rhinoceros). It was decided that this and the Hummel, armed with a 150mm howitzer, would utilize a hybrid Panzer III and IV chassis as a zwischenl\u00f6sung or interim solution until a dedicated self-propelled gun platform could be designed.\n\nA hybrid prototype was presented to Hitler in October 1942. The Panzer III\/IV platform utilized a lengthened Panzer IV hull as the centrepiece of the design. This kept the same basic suspension of four pairs of road wheels and four return rollers except for spacing between the components, while the drive sprocket was that designed for the Panzer III. The glacis plate was extended and on the left-hand side was fitted a small raised armoured compartment for the driver. A circular forward-opening hatch that was set back from the driver's position served the radio operator on the right. This self-propelled mount was known as the Gesch\u00fctzwagen III\/IV.\n\nThe engine was the same Maybach HL120 used in the Panzer III and IV, giving a 42km\/hr speed and a range of 210km. To allow for the open top-fighting compartment at the rear for the 88mm gun crew, the engine was moved to a central position. The compartment was protected on all four sides by slanted armour plates bolted to the hull. These were fairly thin and on the front of the hull amounted to 30mm with just 10mm on the gunshield. The side armour was 10mm on the superstructure and 20mm on the hull.\n\nThe initial order with Deutsche-Eisenwerke was for 500 Hornisse, of which 494 were completed between February 1943 and March 1945. After Hitler approved the prototype it was initially agreed to have 100 Hornisse ready by 12 May 1943 in time for his summer offensive and the attack at Kursk. They were first deployed on the Eastern Front with the 655th Heavy Panzerj\u00e4ger Battalion. Other units equipped with the Hornisse also saw combat on the Western Front and in Italy. Hitler renamed it the Nashorn in 1944.\n\nHummel\n\nWhile the Panzer II provided a useful platform for the 105mm field gun it was decided to mount the heavier 150mm howitzer on the Gesch\u00fctzwagen III\/IV hybrid chassis. This was to equip the heavy batteries of the self-propelled armoured artillery detachments serving with the panzer and panzergrenadier divisions. At least four different selfpropelled howitzer designs utilizing a 105mm gun mounted on the Panzer IV never got much beyond the prototype stage and did not go into full series production.\n\nThe 150mm sFH18\/1 L\/30 was positioned in the middle of the hull over the engine, which resulted in the Hummel having a higher silhouette than the Hornisse. The prototype featured a muzzle brake but this was dispensed with on the production model. Whereas the Hornisse required a four-man crew the Hummel needed six.\n\nLike the Hornisse an order for 100 Hummel was placed with a view to having them ready for the Kursk offensive. Ammunition carriers were also ordered to support the Hummel batteries. To start with, each panzer division only had a single heavy battery equipped with six Hummel plus two ammunition carriers. Later as production increased some had a second heavy battery. Like the Hornisse, the Hummel proved a useful and popular weapon.\n\nThose Hummel built from early 1944 had a crew compartment that extended the initial forward driver's position right across the full width of the hull. In total over 600 had been built by late 1944 with 150 converted into ammunition carriers, as lorries were not up to the job. A variant of the Hummel, known as the Oskette, was produced with wider tracks for the winter fighting on the Eastern Front. German self-propelled guns were not coated in Zimmerit.\n\nBrummb\u00e4r\n\nAlso produced in time for Kursk was the mighty Brummb\u00e4r assault infantry gun. Alkett developed the Sturmpanzer concept while Krupp modified the Panzer IV chassis to accommodate the 150mm StuH43 L\/12 gun (StuH \u2013 Sturmhaubitze or assault howitzer). This required a fully enclosed box superstructure for the gunners. Initially the Brummb\u00e4r utilized fifty-two new Ausf G chassis and eight rebuilt Ausf E and F chassis. Rather than the basic 80mm of armour on the hull front, the first sixty had 50mm plus an additional 50mm armour plate bolted on. The superstructure was also protected by 100mm of armour. Although weighing over 28 tons, the overloaded chassis could still achieve 40km\/hr.\n\nThe initial production models featured a crude sliding-shutter visor for the driver, similar to that on the Tiger I. The middle production version included a periscope for the driver. Both lacked a close-defence weapon, which left the Brummb\u00e4r at risk from enemy infantry. Built from April 1944 to the end of the war the final version had a redesigned superstructure armed with a ball-mounted machine gun in the top lefthand corner of the front plate. A cupola was also installed for the commander.\n\nAfter seeing Alkett's plans in October 1942 Hitler demanded up to sixty Brummb\u00e4r be ready as soon as possible. On 7 February 1943 it was agreed that forty should be completed in time for Kursk with a follow on order for twenty. The initial production run was conducted during April to May 1943; long-term production was then instigated in November 1943 and lasted until the end of the war. In total 298 Brummb\u00e4rs were built at Duisburg with another eight converted from Panzer IV gun tanks.\n\nBefore leaving the factory all vertical surfaces on the mid-production Brummb\u00e4r were coated in Zimmerit. Sch\u00fcrzen side skirt armour was also fitted. They were factorysprayed in dunkel gelb, and areas were then oversprayed with small patches of diluted oliv grun. The first unit to deploy the Brummb\u00e4r was the 216th Sturmpanzer Battalion, which fought at Kursk and was involved in the defensive battles near Zaparozhye ending in October 1943. Three other battalions were formed fighting on the Eastern and Western Fronts as well as in Italy.\n\nAssault Guns and Tank Destroyers\n\nFrom late 1943 the Panzer IV chassis was also used as the basis for a series of turretless assault guns and tank destroyers. These consisted of the Sturmgesch\u00fctz IV, Jagdpanzer IV and the Panzer IV\/70. Combined these represented some 3,120 vehicles with roughly a thousand of each type. These are noteworthy because they diverted valuable tank production just as the tide was turning to what were essentially defensive rather than offensive operations. Their lack of turret and very limited traverse with the hullmounted main gun naturally limited their tactical flexibility.\n\nSturmgesch\u00fctz IV, Jagdpanzer IV and Panzer IV\/70*\n\nThe StuG IV first went into production in December 1943 and continued until the end of the war. As a result Krupp-Gruson completely abandoned building Panzer IV gun tanks. The Vomag Jagdpanzer IV was only built during 1944 and overlapped with the Panzer IV\/70(V), which remained in production until March 1945.\n\nWhile Vomag also discontinued Panzer IV gun tank manufacture, Nibelungenwerke built both Ausf J gun tanks and chassis for the Panzer IV\/70 until the very end. Notably, the Panzer IV\/70(A) built by Nibelungenwerke was a stopgap model using chassis intended for gun tank production. As a result the chassis did not have a modified nose, which meant mounting the superstructure half a metre higher than that on the Jagdpanzer IV and the Panzer IV\/70(V).\n\nSelf-Propelled Flak Guns\n\nGrowing Allied air superiority by 1943 made it increasingly necessary for the Germans to divert a greater proportion of their armoured fighting vehicle production to the output of anti-aircraft tanks or flakpanzers. By 1944 the panzer divisions were in need of fully-tracked armoured anti-aircraft guns that could accompany them into combat. This requirement resulted in the M\u00f6belwagen (Furniture Van), Ostwind (East Wind) and Wirbelwind (Whirlwind), all employing the Panzer IV chassis. These were issued to the anti-aircraft platoons of the panzer regiments. The M\u00f6belwagen was by far the most numerous.\n\nM\u00f6belwagen\n\nThe M\u00f6belwagen was armed with a 37mm FlaK43 L\/60 anti-aircraft gun, with the chassis supplied by Krupp-Gruson, and it proved quite successful. The FlaK43 was designed to replace the earlier FlaK37, which used a two-wheeled trailer similar to the 20mm FlaK30. This newer gun featured a FlaK37 body but also had a gas-actuated mechanism from an aircraft cannon replacing the original recoil-actuated mechanism. This almost doubled the rate of fire. Its effective ceiling was 4,200m. The gun was also mounted in a twin configuration with one gun above the other, rather than the more usual side-by-side. This double configuration was not tried on the M\u00f6belwagen as it would have left the crew permanently exposed apart from the gun shield.\n\nOn the M\u00f6belwagen a fighting compartment for the gunners was created by a simple four-sided 50mm plate superstructure, which could be lowered to a horizontal position to permit the gun to traverse 360\u00ba at low elevation. The hull had 80mm of armour and the superstructure 50mm. Between March 1944 and March the following year some 240 M\u00f6belwagen were produced by Deutsche-Eisenwerke. It does not appear that the M\u00f6belwagen hull was given an application of Zimmerit on leaving the factory.\n\nWirbelwind\n\nWhereas the M\u00f6belwagen only had a flat, largely exposed platform for the gun, the Wirbelwind was equipped with an open-topped turret into which was mounted a 20mm Flakvierling 38 quad. The turret offered just 16mm of armour. This anti-aircraft gun was an improvement on the 20mm FlaK30, which had seen action in Spain and was deployed on a two-wheeled trailer. The rate of fire was increased from 280 to 420 rounds per minute. The Flakvierling 38 was then tried as twin guns and finally a quad mounting was adopted, capable of firing 1,680 shells per minute. The gun's ceiling was 2,000m.\n\nThe Wirbelwind was designed to supplement M\u00f6belwagen production by using existing Panzer IV chassis that had been returned from the front for factory repair. From July to November 1944 Ostbau converted just eighty-six Wirbelwind before the use of the inadequate 20mm gun was abandoned in favour of the 37mm gun.\n\nOstwind\n\nTrials were held in July 1944 mounting a 37mm FlaK43 gun in a turret similar to that used on the Wirbelwind. This resulted in the Ostwind, comprising a six-sided open-top turret on a converted Panzer IV chassis. An order for 100 was placed with Ostbau on 18 August 1944 but by the end of the war they had produced just thirty-six conversions.\n\nThe majority of the Flakpanzer IV were deployed to the Eastern Front, though some M\u00f6belwagen saw action in France in 1944. Wirbelwind and Ostwind flak panzers saw combat during Hitler's Ardennes offensive in December 1944 and were also used in a ground support role. Both these type of Flakpanzer IV had their hulls finished with Zimmerit.\n\nIt had been planned to replace the Ostwind and M\u00f6belwagen with the Panzer IVbased Kugelblitz, armed with twin 30mm guns. Just two Kugelblitz were ever completed, so seven new chassis earmarked for the project were also used to produce Ostwind. Ostbau was looking to produce an Ostwind II armed with two 37mm guns and a Flakpanzer IV armed with a 30mm quad, but loss of their facilities ensured this did not happen.\n\nSupport Vehicles\n\nAs well as being diverted for self-propelled guns and tanks destroyers, the Panzer IV chassis was employed in a wide range of command and support roles.\n\nCommand Tank\n\nThe Panzer III was likewise deployed in a very large variety of roles, including the Panzerbefehlswagen or command vehicle and the Panzerbeobachtungswagen or artillery observation post vehicle. Hundreds of these were produced but by 1944 they were increasingly in short supply due to the end of Panzer III gun tank production. To make up the shortfall the German Army had little choice but to convert existing Panzer IVs to Panzerbefehlswagen mit 7.5cm KwK L\/48.\n\nAbout half the Panzer III command vehicles did not have a main gun in order to create extra workspace. None of the Panzer III observation vehicles had their main gun. This meant they were reliant on a single machine gun for protection and were at risk if they got too close to the front. In contrast the Panzer IV command vehicle retained its 75mm gun.\n\nFrom March to September 1944 Nibelungenwerke converted some ninety-seven Mk IVs to a command role. This primarily involved the installation of the FuG5 (10-watt transmitter) and FuG7 or the FuG8 radios (20- and 30-watt transmitters respectively). These were the standard tank, ground-to-air and divisional link sets. The FuG5 aerial along with a TSR1 1.4m high periscope was fixed to the turret top. The distinctive star antenna for the FuG7\/8 was attached to the rear of the hull on the right-hand edge of the tail plate. These command tanks were issued to Panzer IV battalions.\n\nObservation Tank\n\nThe artillery observation post vehicle was essentially the same but with different radio equipment, comprising the FuG4 and FuG8. This provided an artillery-control set and a divisional link. This conversion work was conducted by Nibelungenwerke but started slightly later, although it involved similar quantities of tanks. A total of ninety Panzer IVs were converted into Panzerbeobachtungswagen IV from July 1944 to March 1945. These were issued to the Hummel batteries.\n\nSubmersible Tank\n\nLike the Panzer III, a number of Mk IVs were converted into submersible tanks known as Tauchpanzer in mid-1940. This was done in preparation for Hitler's Operation Sealion, the proposed invasion of England. In theory it allowed the panzers to operate in water up to 15m deep. Whilst almost 170 Tauchpanzer III were produced there were just forty-two Tauchpanzer IV conversions.\n\nThis comprised fitting the air-intakes with locking covers and the exhausts with non-return valves replacing the normal mufflers. Also waterproof fabric covers were applied to the hull machine gun, the gun mantlet and the cupola. An inflatable rubber tube encompassed the turret ring. A specially-designed metal cover with a vision block made the driver's visor watertight. A pipe suspended from a floating snorkel connected to the back of the tank allowed it to draw air when submerged and a gyrocompass facilitated underwater navigation.\n\nTrials and training showed that the Tauchpanzer was a reasonable successful innovation. When Hitler finally abandoned Sealion they were made redundant. Nevertheless, the Tauchpanzers did not go to waste as they were shipped to Milowitz near Prague in the spring of 1941 and prepared for another mission. Most of them had a long rigid snorkel attached to the commander's cupola, making them capable of deep river crossings. These Tauchpanzers took part in Operation Barbarossa with the 18th Panzer Division in June 1941.\n\nBridge-layer\n\nDuring the 1930s the emphasis on fixed fortifications in Europe resulted in the German Inspectorate of Engineers concluding that there was a strong requirement for an armoured bridge layer. Early experiments utilizing the Panzer I and II were soon abandoned as they could not carry a large-enough tank bridge. Instead they came up with the Br\u00fcckenleger IV.\n\nFour Panzer IV Ausf C chassis were assigned for conversion in August 1939. The following month an additional 16 Ausf D chassis were requisitioned. By January 1940 it was anticipated that twelve BL IV would be delivered with two employing a Kruppdesigned bridge and the rest using a Magirus development. One had a forward-pivoting gantry while the other slid the bridge into position horizontally.\n\nThe original order had been for fifty 9m bridges, but this was cut by thirty so a new design could be fitted to sixty Panzer IVs. Experience in France and the Low Countries showed there was actually little call for the services of the BL IV and the programme was cancelled. During 1940\u201341 it was decided to restore fifteen BL IV to a gun tank role. In the end just twenty bridgelayers were issued to five panzer divisions.\n\nInfantry Assault Bridge\n\nThere was another design that employed a 50m extending bridge using Panzer IV Ausf C chassis. Known as the Infanterie Sturmsteg or infantry assault bridge, this Magirus construction comprised two bridges side-by-side that created beams for a walkway very similar to a fire-fighting ladder. Just two Sturmsteg were completed in February 1940 and saw use in the invasion of France. One was disabled, losing its tracks and several road wheels. No further conversions were carried out.\n\nArmoured Recovery Vehicle\n\nLate in the war a very limited number of Bergepanzer IV or recovery vehicles were produced to support the panzer divisions. This was a fairly crude affair and consisted of replacing the turret with a wooden box-body on top of the fighting compartment for the mechanics. Mountings were also installed on the rear engine deck to take a derrick crane for engine lifting. Just thirty-six Bergepanzer IV were converted from October 1944 to December 1944.\n\nIn contrast 150 Bergepanzer III were produced during that same year as the Panzer III had outlived its usefulness by this stage. Instructions were that all Mk IIIs being returned from the front for maintenance were to be converted to a recovery role. It is likely that others were produced unofficially as field conversions from damaged tanks or those under repair. Both types of Bergepanzer were issued to Panzer IV and StuG III\/IV workshop companies.\n\nHummel Ammunition Carrier\n\nExternally the Hummel ammunition carrier differed from the Hummel, simply by having its gun removed and the resulting gap plated over. Internally the fighting compartment was converted to allow for extra ammunition stowage. These vehicles were produced both in the factory and frequently as field conversions.\n\nKarl Mortar Ammunition Carrier\n\nDuring the winter of 1940 and the summer of 1941 the Germans produced six Karlger\u00e4t, which were enormous 600mm self-propelled super-heavy siege mortars. These required tracked ammunition carriers and in October 1939 a brand-new Panzer Ausf D chassis was used as a prototype. A new superstructure was constructed, while over the engine compartment were fitted racks that could take four massive 600mm rounds. To get these rounds to the mortar a crane was installed on the front righthand side of the superstructure.\n\nOnce green-lighted it was decided in 1941 to use a newer model Panzer IV and 13 Ausf F1 chassis were converted to Munitionsschlepper f\u00fcr Karlger\u00e4t. A number of rebuilt Panzer IV chassis were also pressed into service. These carriers saw action in support of the mortars on the Eastern Front, most memorably during the siege of Sevastopol in 1942.\n\nThe Hornisse self-propelled heavy anti-tank gun was a hybrid using components from the Panzer III and IV. Armament comprised the powerful 88mm PaK43\/1 L\/71 gun which was far too large to mount in a conventional tank turret.\n\nThis shot shows how the main armament on the Hornisse made it nose-heavy and difficult to manoeuvre in anything other than open spaces. The gun's long range made it ideal for the open Russian steppe.\n\nThe armour protecting the gunners' fighting compartment on the Hornisse was very poor, amounting to just 10mm. Nonetheless, this tank killer proved very successful and almost 500 had been built by the end of the war.\n\nThe Hummel, like the Hornisse, first appeared in the summer of 1943. This self-propelled gun carried the 150mm sFH18\/1 l\/30 howitzer. The early production models built in 1943 had a raised armoured forward driving compartment.\n\nA direct hit destroyed this final-model Hummel in Berlin in April 1945. The two blurred bands round the barrel near the muzzle suggest it was being using in an anti-tank role.\n\nA crew of five gunners and a driver were required to operate the Hummel. This is a 1944 model as the superstructure front plate for the driver and radio operator are flush.\n\nA 1944\u20135 model Hummel captured by the Red Army in Budapest, Hungary in 1945. The superstructure and gun shield on the Hummel only had 10mm of armour. The hull front had 30mmm with just 20mm on the sides and rear.\n\nThe remains of two Brummb\u00e4r numbered '222' and '110' captured at the end of the war. Both appear to have been cannibalized or were incomplete as their drive sprockets are missing. Almost 300 Sturmpanzer IV were built.\n\nBoth Brummb\u00e4r were fitted with rails to hold the Sch\u00fcrzen. The commander's cupola and machine-gun ball mount at the front identifies them as late-production models.\n\nThe requirement for tracked anti-aircraft guns to support the panzers resulted in the M\u00f6belwagen. This consisted of a 37mm FlaK43 L\/60 anti-aircraft gun mounted on a turretless Panzer IV chassis.\n\nThe sides of the fighting compartment folded down on the M\u00f6belwagen to allow full 360\u00ba traverse.\n\nA M\u00f6belwagen getting the once-over from curious American troops. The gun shield offered just 25mm of protection but the fighting compartment front plate that is folded down was 50mm thick.\n\nThe Wirbelwind was produced to supplement the M\u00f6belwagen. Its armament consisted of a 20mm Flakvierling 38 quad, but this was not as effective against low-flying fighter-bombers as the 37mm gun. These well-camouflaged examples were keen not to attract attention to themselves while on the move.\n\nThe hull and superstructure of this Wirbelwind has been coated in the anti-magnetic Zimmerit paste.\n\nTwo Wirbelwind captured on the Western Front during the winter of 1944\u20135. Less than 100 were built during the second half of 1944 as it was intended to replace them with the Ostwind armed with the FlaK43.\n\nTo support the invasion of England in 1940 the Germans produced the Tauchpanzer or submersible tank. Only forty-two Tauchpanzer IV conversions were ever produced and these were involved in the attack on the Soviet Union.\n\nA number of Panzer IV bridgelayer variants were produced including twenty Br\u00fcckenleger IV and two Infanterie Sturmsteg, one of which is seen here.\n\nThe Infanterie Sturmsteg took part in the invasion of France in 1940 where this one was disabled. The Germans abandoned their Panzer IV bridgelayer concept shortly after.\n\n* These are covered in greater detail in German Assault Guns and Tank Destroyers 1940\u20131945.\n\n## Chapter Seven\n\n## In the Desert and Mountains\n\nOn the eve of the Second World War Hitler's Wehrmacht had just over 3,300 tanks, of which only 629 were Panzer III and IV. By May 1940 every tank detachment had a medium tank company of six to eleven Panzer IVs. When the invasion of France and the Low Countries commenced there were 280 Ausf A, B, C and D Mk IVs equipping the panzer divisions. After the defeat of France some of these were sent to North Africa to help the Italians.\n\nWhen the armoured units of Rommel's Deutsches Afrika Korps (DAK) arrived in Libya they were equipped with the Panzer IV Ausf C and D and then later the Ausf E and F1, armed with the 75mm KwK L\/24 gun that fired the same high-explosive projectile as the Ausf F2 (dubbed the Panzer IV 'Special' by the British) equipped with the L\/43.\n\nThe first elements of DAK landed in Tripoli on 14 February 1941 with the disembarkation of the 3rd Reconnaissance Battalion, 5th Light Division; other units, most notably two battalions of the 5th Panzer Regiment, followed. The 5th and 8th Panzer Regiments initially took just forty Ausf D and E Mk IVs to North Africa. These regiments formed the spearhead of the 21st and 15th Panzer Divisions respectively.\n\nSignificantly, the most common Panzer IV were the Ausf D, E and F1 which made up around 25 per cent of Rommel's armoured formations. Armed with the short 75mm KwK37 L\/24 gun, they were inferior to the later 50mm gun of the Panzer III. Prior to the Crusader battles in November 1941 the British were able to field 748 tanks armed with 40mm or 37mm high-velocity guns against 248 panzers, of which 174 were Panzer III and IVs, the rest being Panzer IIs with a 20mm gun. The Italians supported Rommel with 146 inferior tanks that were armed with a low-velocity 47mm gun. What really gave Rommel an advantage was his superior tactics.\n\nIn the summer of 1942 he began to receive the up-gunned F2 armed with the longbarrelled 75mm KwK40 L\/43 gun. This could punch through 85mm of armour at 1,000 yards and was superior to the British 2-pounder (40mm) and 6-pounder (57mm) and the American-supplied M3 Grant's 75mm gun. Luckily for the British the Germans had only received twenty-seven Panzer IV F2s by August 1942, which they employed to spearhead Rommel's counteroffensive. Despite their presence Rommel's attack at Alam Halfa was held.\n\nCrucially the F2 was never available in sufficient numbers, with around thirty with each of DAK's panzer divisions at any one time compared to 100 Mk IIIs. While its gun could penetrate all British and American armour at a distance and more arrived between August and October 1942, they were nothing like the quantities of tanks reaching the British 8th Army prior to El Alamein. Some Ausf G also fought with the Germans in Tunisia.\n\nBritish armour could cope with the Panzer I and II but not the subsequent two models. The Panzer III armed with a 50mm gun was superior to any Allied armour until 1942 and the arrival of the M3 Grant. The early Mk IV with its short 75mm gun was able to fire armour piercing, high explosive and smoke so could outshoot very vulnerable British cruiser tanks and shell exposed 25-pounder gun crews.\n\nBritish cruiser tanks, as well as the Matilda and Valentine infantry support tanks armed with the 2-pounder gun, could only fire armour piercing rounds. Initially the 6-pounder installed on the Crusader and the Churchill was only intended to fire armour piercing. While the Panzer IV could fire from 3,000 yards with HE, British tanks had to wait for them to close to within 1,000\u2013500 yards to engage with their solid AP shot: in the meantime their exposed artillery would have to retreat.\n\nThe Panzer III and Panzer IV F2 therefore had little trouble dealing with their opponents. British cruiser tanks such as the Mk II and IV had a maximum of 30mm of armour, while the Mk I and III were even more lightly armoured at just 14mm. The Mk VI Crusader was little better \u2013 with the final production version the Crusader III sporting 51mm of armour. It was also up-gunned to the British 6-pounder. The Matilda and Valentine had a respectable maximum of 78mm and 65mm respectively but like all British tanks were slower than the panzers and under-gunned. The Grant and Sherman had 37mm and 75mm armour respectively.\n\nAgainst the F2 the Grant was not only disadvantaged by its gun, but also by the mounting the weapon in the hull instead of a fully revolving turret. This meant it could not fight dug-in from a 'hull-down' position when on the defensive. Likewise the Grant's high silhouette exposed the top of the tank to enemy fire when on the offensive. The Panzer IV also proved more than a match for the Sherman in North Africa.\n\nUltimately though, the Panzer III and IV were overwhelmed by superior numbers and air power. Hitler never adequately reinforced his troops in North Africa until it was far too late and they were trapped. Despite the effectiveness of the Panzer IV F2 and the introduction of a few Tigers, particularly against the inexperienced American army, capitulation was inevitable. The very last of the Panzer IVs were disabled by Allied bombers \u2013 one of which was photographed on 10 May 1943 just two days before the Axis surrendered in Tunisia.\n\nSimilarly, significant numbers of Panzer IVs were never available in Italy. German panzer divisions were always thin on the ground during the Italian campaign. Generally German infantry divisions relied on the support of panzergrenadier units, which had fewer armoured fighting vehicles than the regular panzer divisions and relied on assault guns, not tanks.\n\nThe key tank unit was the 26th Panzer Division, which transferred to Italy in 1943 and remained there for the rest of the war until its surrender near Bologna in May 1945. The 16th Panzer Division only fought in Italy for six months between June and November 1943 seeing action at Salerno and Naples before being sent to the Eastern Front. These divisions were equipped with the Panzer IV Ausf G, H and J.\n\nThe Luftwaffe also fielded the Panzer IV in Italy. The Hermann G\u00f6ring Panzer Division, which had been destroyed in Tunisia, was reformed in southern Italy and Sicily and played a key role in the Sicilian campaign in July and August 1943. Escaping to the Italian mainland following the Allied landings on the island it was given the title Fallschirm Panzer Division Hermann G\u00f6ring, although the Fallschirm (Parachute) designation was purely honorary. The Parachute Panzer Regiment Hermann G\u00f6ring included a panzer and assault gun battalion.\n\nA British Crusader passes a burning Panzer IV during Operation Crusader on 27 November 1941. British tank armament, comprising the 2-pounder gun, was wholly inadequate against the Panzer III and IV.\n\nAnother shot of the burning Panzer IV. The most common models in North Africa were the Ausf D, E and F1, all armed with the close-support short 75mm gun.\n\nPanzer IV '618' belonging to the 1st SS Panzer Division in Milan during September 1943 following the German occupation of Italy.\n\nBritish troops sheltering by two burning Panzer IVs caught on the road near Salerno on 22 September 1943, following the landings on the Italian mainland.\n\nMen hurry past the same tanks. Both have the turret Sch\u00fcrzen but the side plates are missing.\n\nA battery of Hornisse self-propelled anti-tank guns near Anzio, Italy in 1944.\n\nBr\u00fcmmbar Sturmpanzer IV during the battle for Anzio. The Br\u00fcmmbar also saw action on the Western and Eastern Fronts. This particular example has lost the central plate on the left-hand skirt.\n\nThis Ausf G was photographed near Lucera in Italy on 27 October 1943. It was being used for weapons tests by the British 8th Army.\n\nThis Panzer IV Ausf H was destroyed near Salerno, Italy. The engine compartment and hull have been badly damaged by an explosion.\n\n## Chapter Eight\n\n## From Barbarossa to Berlin\n\nThose short-barrel Panzer IVs lost during the invasion of the Soviet Union in the summer of 1941 were replaced by the spring of 1942, but crucially there was no increase in overall numbers. However, the Red Army's T-34 tank, with a maximum of 70mm frontal armour, was now vulnerable to the recently-introduced Panzer IV Ausf F2. Even at 1,000 yards its 75mm Kwk40 L\/43 could penetrate 87mm of armour and the subsequent Panzer IV Ausf G and H were both armed with the more powerful Kwk40 L\/48. They finally gave the Panzerwaffe parity.\n\nNonetheless in terms of tank production the Germans could not compete, in November 1942 they managed just 100 Panzer IVs compared to 1,000 Soviet T-34s. The following month, rather than retool for the Panzer IV, the Panzer III assembly lines were turned over to StuG III assault gun production. Following Hitler's major summer offensive in 1942, by early 1943 the eighteen panzer divisions on the Eastern Front had less than 600 tanks. This did not bode well for his Kursk offensive. However, Hitler managed to gather 1,850 panzers supported by 200 obsolete tanks and over 530 assault guns. Along with the Panzer IV, the Panzer III provided the bulk of Hitler's armour employed during Operation Citadel. The Panther, making its debut, was in short supply, as was the Tiger.\n\nAlso making their debut at Kursk in July 1943 were the Panzer IV-based Hummel and Hornisse. While the latter had a very effective anti-tank gun it was really a defensive rather than offensive armoured fighting vehicle. The numbers available though were once again a problem. Both had only gone into production in early 1943, with about 100 of each type ready for the summer. The Hummel units were supported by obsolete Panzer IIs equipped with radios to act as command and observation tanks. Also to help smash Soviet fortifications were sixty-six newly-built Brummb\u00e4r comprising a short 150mm howitzer mounted on a Panzer IV chassis.\n\nDespite their best efforts the Panzer IV, and the newly-introduced Panther and Tiger, were simply swamped at Kursk by a well-prepared and more numerous enemy. Hitler had been warned he was outnumbered but refused to give ground or let the initiative pass to the Red Army so had chosen to attack first with dire results. The panzers were crushed in a massive tank battle at Prokhorovka. Hitler lost around 1,600 panzers of which 300 were irreparable losses.\n\nAlthough the Panzer IV remained the German workhorse on the Eastern Front, the earlier Ausf D, E and F armed with the short 75mm gun were completely phased out through attrition in early 1944. The panzer divisions were equipped with the newer F2, G and H variants, although they were in danger of being outclassed by some of the newer heavily-armed Soviet armour.\n\nIn the summer of 1944 Hitler had too few Panzer IVs supporting Army Group Centre that was holding the exposed frontline in Byelorussia. The army group under Field Marshal Ernst Busch was predominantly equipped with assault guns. Busch's only unit with Panzer IVs, that constituted his sole reserve, was the 20th Panzer Division. Army Group North was little better off with just the 12th Panzer Division equipped with Mk IVs. The two Army Groups in Ukraine were much stronger with fourteen panzer divisions but this was where Hitler was anticipating Stalin would strike.\n\nStalin conducted a major offensive to liberate Minsk and Byelorussia in late June 1944. General Jordan, commanding the German 9th Army, sought and received Busch's permission to commit the 20th Panzer Division to try and stem the Soviet tide. The division could muster seventy Panzer IVs, its other panzer battalion being busy re-equipping with Panthers. At that very moment General Batov's 65th Army broke through on the southern approaches to Bobruisk and General Rokossovsky committed the 1st Guards Tank Corps to exploit the breach. A dithering Jordan ordered 20th Panzer to retrace its tracks and head south, bumping into the Soviets near Slobodka south of Bobruisk.\n\nIt rapidly became apparent that not only was German-held Bobruisk under threat but also those German divisions still east of the Berezina River. By 26 June the battered 20th Panzer had been driven back to the city with the Soviet 9th Tank Corps bearing down on it from the east and the Soviet 1st Guards Tank Corps from the south. The 1st Guards Tank Corps cut the roads from Bobruisk to the north and north-west on the night of 26\/27 June, closing the trap.\n\nThe 5th Panzer Division was moved up from Kovel to defensive positions east of Borisov with a view to covering those troops withdrawing from Mogilev, which had been overrun late on the 27th. Under General Karl Decker, 5th Panzer began to arrive in Minsk on 26 June with the all-but-impossible task of holding the Moscow-Minsk Highway. The division was equipped with just fifty-five Panzer IVs and seventy Panthers, supported by Captain von Beschwitz's twenty-nine Tiger Is of Heavy Panzer Battalion 505. Their first mission was to put a stop line in place north-east of Borisov. Like 20th Panzer, they too failed to stem the Red Army's tidal wave.\n\nThe year was a complete disaster for Hitler and those regiments equipped with the Panzer IV. Throughout 1944 Hitler lost 2,640 Panzer IVs on the Eastern Front. This was more than could be replaced. The previous year he had suffered losses of 2,350 Panzer IVs, with some divisions down to as few as twelve tanks. The no-frills Panzer IV Ausf J which appeared in mid-1944 was the final production model and was greatly simplified to speed construction. However, the manual turret traverse was not greatly liked by the crews. Production ran until March 1945 by which time less than 2,000 had been produced. It was not enough\n\nReportedly 287 Panzer IVs were lost on the Eastern Front during January 1945. Within months the Red Army was poised before the very gates of Berlin and victory. By this stage the German armies tasked with protecting the capital had very few panzers left. In total it is estimated that the Red Army accounted for 6,150 Panzer IVs or about 75 per cent of all Panzer IV losses during the war.\n\nWhitewashed shortbarrelled Panzer IVs on the road to Moscow. The bad weather and Red Army counterattacks ensured they did not capture the city.\n\nThose early-model Ausf C, D and E lost during the invasion of Russia were replaced by the up-gunned Ausf F2 and G during the spring of 1942.\n\nA Soviet soldier examines a knocked-out Panzer IV Ausf H in the summer of 1943.\n\nThis illustrates the extent to which the Sch\u00fcrzen greatly enhanced the protection of the Panzer IV's hull and turret. The side plates, though, were invariably lost and once damaged were often not replaced. These show signs of impact marks.\n\nAusf G\/H serving in Russia. Although both tanks have the brackets for the side plates only the turret Sch\u00fcrzen remains. The nearest tank bears the tactical number '325'.\n\nHummels being shipped to the front. The self-propelled mounting known as the Gesch\u00fctzwagen III\/IV proved highly successful with the Hummel and Hornisse.\n\nThe Hummel, like the Hornisse, made its debut in the summer of 1943 at Kursk, where it served in the heavy batteries of the armoured artillery detachments with the panzer divisions. The exposed crew had no protection from indirect fire.\n\nA Hornisse camouflaged for operations during the Russian winter. Renamed the Nashorn in 1944 it stayed in production until the end of the war, although the Jagdpanzer IV and Panzer IV\/70 took priority. The Hornisse made its first appearance at Kursk. Its powerful and long-ranged 88mm gun meant it could kill Soviet tanks at great distances \u2013 this helped cancel out the risks caused by its high profile and light armour.\n\nThese Hornisse belong to the 519th Heavy Panzerj\u00e4ger Battalion serving on the Eastern Front in the winter of 1944. This was one of six such battalions equipped with this Panzer IV-based tank destroyer.\n\nA Panzer IV Ausf H on the Eastern Front with its Sch\u00fcrzen intact. It bears the tactical number '505' and has a two-tone camouflage scheme.\n\nHummel belonging to the 1st SS Panzer Division being transported by rail. Tarpaulins have been fitted to keep the rain out of the fighting compartment. The key and shield divisional insignia can just be seen on the front of the superstructure on the right-hand side.\n\n## Chapter Nine\n\n## Beyond the Seine\n\nThe Panzer IV played a significant part in the Battle for Normandy. The most common type of panzer resisting the Allies in occupied France in 1944 was the Panzer IV Ausf H and Ausf J totalling around 750 tanks. Around two-thirds of the Panzer IV battalions were armed with the Ausf H and the rest with the newer Ausf J. With frontal armour of 80mm and the 75mm KwK 40 L\/48 anti-tank gun, the Panzer IV provided the fighting backbone of the ten panzer divisions deployed to France (in all five army and five Waffen-SS divisions fielded it in Normandy).\n\nIt, along with the Panther and Tiger tanks, gave the Germans a distinct advantage in tank-to-tank engagements during the battles in the Normandy bocage. Notably its gun had a 20 per cent higher muzzle velocity than that of the ubiquitous American-built M4 Sherman's 75mm gun, meaning it could punch through 92mm of armour at 500 yards, while the Sherman could only manage 68mm. In addition the Panzer IV proved remarkably reliable, maintaining consistently good operational rates.\n\nNormally the Panzer IV was allocated to the 2nd Battalion or II Abteilung of a panzer regiment, while the 1st was equipped with the Panther, although there were a number of exceptions in France. In the case of the 9th Panzer Division the 1st Battalion of its Panzer Regiment 33 was equipped with Panzer IVs and both battalions of 21st Panzer's Panzer Regiment 22 were equipped with it.\n\nAt the beginning of June 1944 the 2nd Panzer Division, deployed east of the Seine at the start of the campaign, had a total of ninety-eight Panzer IVs on its strength. By early July the division still had eighty-five of them in the field with another eleven in the workshop. The 9th Panzer Division, based in the south of France, arrived in the Normandy in early August with a total of eighty-two Panzer IVs.\n\nThe 21st Panzer Division, already in Normandy from the start, had a total of 104 Panzer IVs, including six old Ausf G and six even older Ausf B or C with the short 75mm gun. None of the latter seems to have been upgraded and lacked Zimmerit and Sch\u00fcrzen. Before D-Day one of the Ausf B\/C dubbed 'Heidi' was photographed in St Martin de Fresnay, south-east of St Pierre-sur-Dives, where its crew were happily shopping for Camembert cheese. It is not clear if this was an authorized use of official military equipment or simply a PR exercise.\n\nOn 24 May 1944 another fourteen Panzer IVs were despatched to 21st Panzer, but it is unlikely that they had arrived by early June. During the battle for Normandy another thirty were sent to the division including three command tanks. The final ten gun tanks probably did not arrive in time to take part in the fighting.\n\nPanzer Lehr Division, initially north-west of Orleans, had ninety-nine Panzer IVs. It was sent another eleven Mk IVs as replacements on 8 July 1944. Following heavy fighting with the Americans, by 22 August the division had just ten left. When the 116th Panzer Division Windhund, north of Paris, was committed to the battle for Normandy in late July it had 86 Panzer IVs. Its other armoured fighting vehicles include three older Panzer IVs with the short 75mm gun. Following the German retreat and the fighting at St Lambert the division was only able to retrieve four Panzer IVs from the battlefield.\n\nThe Waffen-SS panzer divisions committed to Normandy were also equipped with the Panzer IV. The 1st SS Panzer Division Leibstandarte Adolf Hitler, east of Brussels, mustered forty-two Panzer IVs in its 1st Tank Battalion. A further eight were in for repair. Deliveries during June amounted to another fifty-three tanks. After being sent to Normandy by late July the division was able to field sixty-one Panzer IVs.\n\nThe 2nd SS Panzer Division Das Reich by the beginning of June had fifty-four Panzer IVs, of which ten were in the workshop. Further deliveries of armour meant that the 2nd SS was to field a total of eighty-three Panzer IVs, which included the newer Ausf J. Many of these were lost in the Roncey pocket at the end of July, though some of the Mk IVs from 2nd SS managed to escape.\n\nThe 9th SS Panzer Division Hohenstaufen had forty-eight Panzer IVs at the start of the campaign, but seven of these were in the shop. Its sister division 10th SS Frundsberg was able to deploy thirty-nine and by mid-August had only lost twelve. However, just eight Panzer IVs were available to hold the Americans at bay. The formidable 12th SS Hitlerjugend had an authorized strength of 101 Panzer IVs though only ninety-one were combat ready. By 10 August after much heavy fighting the division had just fifteen left.\n\nBy 1944 the panzer divisions were largely relying on flak guns mounted on half-tracks for air defence, including the SdKfz 10\/4 and SdKfz 7\/1. In Normandy the Flakpanzer 38(t) using Czech-built chassis was in service instead of the Flakpanzer IV with the 2nd, 9th, Panzer Lehr and 1st SS Panzer Divisions. In theory eight Flakpanzer IV M\u00f6belwagen were allocated to the anti-aircraft section of a panzer regiments headquarters' company. In July 1944 (on paper at least) M\u00f6belwagens were assigned to the 12th SS and 116th Panzer Divisions. Very limited numbers of Wirbelwind also saw combat in Normandy.\n\nThe lack of infantry divisions, which were held north of the Seine, meant the panzers bore the brunt of the fighting. By the beginning of July 1944 the unrelenting operational commitment of the panzers had taken its toll: 42 per cent of the Panzer IVs and 58 per cent of the Panthers were in the maintenance depots. Prior to Montgomery's Goodwood offensive on 18 July Panzer IVs of 21st Panzer along with Tiger tanks from Heavy Panzer Battalion 503 were caught in the Allied saturation bombing near Ch\u00e2teau de Manneville 10km east of Caen. The effects were devastating, with tanks simply tossed upside-down like they were toys. From a force of about fifty panzers over half were lost, many others suffering mechanical failures.\n\nAlthough Goodwood failed to breakthrough the incredibly tough German defences it contributed further to the attrition of the Panzer IVs in Normandy and helped pin them in the British sector. In August for the renewed counter-attack at Mortain against the Americans the Germans could only gather seventy-seven Panzer Mark IVs and forty-seven Panthers, roughly the same inadequate number that had been launched in the initial attack, which involved fifty-seven Panzer IVs and seventy Panthers.\n\nA few weeks later, unable to contain the American breakout, the German Army was in headlong flight and then trapped at Falaise and defeated. After the Battle for Normandy despite the presence of surviving panzers belonging to the SS at Arnhem, Montgomery was not put off launching Operation Market Garden that resulted in the disastrous battle for the Rhine crossing.\n\nFollowing the Panzerwaffe's mauling in Normandy by late 1944 the Panzer IV was in very short supply. As a result of continual battlefield attrition and the disruption caused by Allied bombing the panzer divisions struggled to field both a battalion of Panzer IVs and a battalion of Panthers. The 2nd and 11th Panzer and the 2nd SS and 9th SS Panzer Divisions each had two companies of StuG assault guns instead of Panzer IVs; the 9th and 116th Panzer Divisions had no Panzer IV at all, fielding instead three companies of StuGs. Other divisions had a single battalion of mixed Panzer IVs and Panthers. The second battalion had to be substituted by panzerj\u00e4ger and sturmgesch\u00fctz units.\n\nTo make up for such shortfalls, prior to Hitler's winter Ardennes offensive the 1st SS was supplemented by a battalion of heavy Tiger IIs, while the other divisions had to be brought up to some semblance of strength using StuGs and Jagdpanthers. The 1st SS ended up with a panzer regiment consisting of one battalion of mixed Panzer IVs and Panthers and one of Tigers. The division should also have had twenty-one Panzer IV\/70 tank destroyers but its SS panzerj\u00e4ger battalion only had ten on the eve of the battle. Nonetheless, a total of almost 140 Panzer IV\/70 were available for the Ardennes offensive. The panzer divisions had Jagdpanzer IV and Panzer IV\/70 tank destroyers equipping two companies of their panzerj\u00e4ger battalion, the third company being equipped with towed anti-tank guns.\n\nThe 1st SS and 12th SS Panzer Divisions' Ardennes spearhead included 100 Panzer IVs. Two tank companies from 1st SS Panzer consisting of thirty-four Panzer IVs formed Kampfgruppe Peiper that led the daring attack toward Antwerp. The 1st SS also had a number of Flakpanzer IVs, which included Ostwind and Wirbelwind. Some of the latter were involved in the fighting at Stoumont where they acted in support of the infantry.\n\nOne knocked-out Panzer IV was photographed north-west of Bastogne. A massive explosion had shattered the glacis plate, tearing out the gearbox and differential assembly. The turret had also been blown off and rested on its side vertical to the hull. After the war this much-published image came to epitomize the Battle of the Bulge and the defeat of the panzers in North-west Europe.\n\nPanzer IVs with the 12th SS Panzer Division on parade in France in early 1944. The majority of the Panzer IV deployed on occupation duties were Ausf H or J.\n\nThis 12th SS Ausf H has the side vision ports for the radio operator and the driver, indicating it is an early production version. It has the anti-aircraft machine gun fitted to the cupola and the onepiece hatch. It also has the latest drive sprocket but the mid-model idler. There is Zimmerit on the front of the turret.\n\nThe Sch\u00fcrzen on Ausf J '625' is in pristine condition and has been painted, along with the rest of the tank, in three-tone camouflage. The basic dunkel gelb has been oversprayed with irregular patches of oliv grun and rot brun. The tactical number was hand painted afterwards. The hull front and the gun mantlet have the zimmerit finish.\n\nThe crew of this tank in Normandy used track links to up-armour the glacis plate, the superstructure and even the front of the turret. It has the type three muzzle brake. What appear to be scorch marks around the driver and radio operator hatches suggests that it caught fire. It is possible the tank was deliberately torched by its crew after breaking down.\n\nGIs examine a Hummel captured during the fighting in Normandy. It has a three-tone camouflage enhanced by foliage. It bears the national identification symbol and the tactical number '110'.\n\nA StuG IV formerly serving with the 2nd SS Panzer Division captured during the Battle for Normandy. Although quicker to build, assault guns and tank destroyers were a distraction from Panzer IV gun tank production.\n\nA sea of military debris. This Normandy scrapyard is full of discarded Panzer IVs. Around 750 saw combat in northern France.\n\nThis Ausf J was knocked out by Allied fighter-bombers in front of Patton's US 3rd Army during the Battle of the Bulge. Ironically the frame protruding from the cupola is the mount for an anti-aircraft machine gun.\n\nThis late-war Hummel was captured in April 1945 near Wurzen, Germany. An oliv grun pattern has been applied over the dunkel gelb, and the remains of foliage camouflage can be seen. The vehicle is undamaged and the gun is locked in the travelling position.\n\nThe victim of this concealed American M10 tank destroyer is a Panzer IV Ausf J.\n\n## Chapter Ten\n\n## Hitler's Rock\n\nThe Panzer IV proved to be the one constant in the Panzerwaffe's armoury during the Second World War. Like the T-34 it served throughout the conflict, although it was not built in sufficient numbers and up-gunned until late 1942 and early 1943, by which time it was too late. Hitler's hopes that he could step up production of the technically superior Panther and Tiger tanks never came to fruition because they were overengineered and therefore difficult to mass-produce.\n\nThe Panzer IV appeared in ten different models (with three different guns) as well as a dozen different armoured fighting vehicles, whereas the T-34 essentially comprised two \u2013 the T-34\/76 and T-34\/85 (though to be fair there were five and three production models respectively of each, as well as half a dozen AFV variants). To put the Panzer IV's contribution in perspective, 8,500 were built compared with 55,000 T-34s. Nonetheless it proved to be one of the key if not best panzers of the Second World War and was Hitler's rock.\n\nThe Panzer IV proved to be quite a remarkable weapon. The Panzerwaffe recognized it for what it was \u2013 a robust and durable design. As a result it was the only German tank to remain in continuous production throughout the war, and was in production longer than any Allied tank. It was only rivalled by the Soviet T-34, which also stayed in production throughout. However, T-34 mass production did not commence until mid-1940, giving the Panzer IV a slight lead. In its favour the T-34 was almost impervious to the standard German 37mm and 50mm anti-tank guns, and of the Panzer III and IV, it outclassed the former in all respects and the latter in all except gun power where the up-gunned variants equalled it.\n\nDue to the urgent need to find a counterpart to Soviet tank designs, production of the Panzer IV was in part marred by wasteful experimentation. This was because of Hitler's inability to standardize his tank force and settle on a single utilitarian design. He struggled with quality over quantity. As a result for example in October 1942 just 100 Mk IVs were built against about 900 T-34s.\n\nFrom spring 1942 to the summer of 1943 just over 1,800 of the vastly improved Ausf F2 and G were built. It was not enough and many of them were swallowed up as combat replacements. In a similar timeframe, from spring 1943 to the summer of 1944, production was finally ramped up with almost 3,800 Ausf H, but after that production declined as the chassis was diverted to tank destroyers. The Panzer IV and indeed the Panther and Tiger could simply not compete with the enormous numbers of T-34s coming out of Stalin's factories.\n\nNonetheless, the Panzer IV accounted for over a third of German wartime tank production making it the most widely used tank of the war. It saw combat with the panzer divisions in all theatres of operation including Poland, France, the Balkans, North Africa, Russia, Italy and North-west Europe.\n\nManufacture of the Panzer IV easily outstripped that of the overrated Tiger and Panther combined. These totalled around 1,800 and 6,000 respectively. Panzer IV gun tank production amounted to over 8,500 \u2013 some 2,400 more than the Panzer III. When all the other variants using the Panzer IV chassis are taken into account, the overall total is 13,400. In contrast just over 6,100 Panzer III were built between 1937 and 1943. It proved far more useful as an assault gun platform with 8,600 StuG IIIs \u2013 the latter though were not tanks.\n\nHitler had to maintain Panzer IV production because of the German Army's constant urgent need for serviceable tanks. This prevented him from switching solely to the Panther and Tiger; however, this is certainly no reflection on the excellent basic design of the Panzer IV. Both the Panther and Tiger were labour intensive while they had exceedingly good anti-tank guns were often mechanically unreliable. Their weight strained the transmission and drive train and made them very difficult to recover. Once swamped by enemy tanks the Panther and Tiger's long-range stand-off kill capability was completely nullified.\n\nCrucially the Panzer IV was much more reliable than both of them and at half their weight was easier to recover and repair. Like its key adversary, the T-34, the Panzer IV's design ensured that it could be regularly upgraded. American and British tanks did not offer such flexibility \u2013 the Sherman and Cromwell are prime examples of this very serious shortcoming.\n\nLikewise, the size of the Panther and the Tiger proved a problem, making them easier targets. Panzer General von Manteuffel said the Panther 'would have been close to ideal' but for its high and bulky silhouette. The Tiger was almost 4m wide and 3m high, the Panther 3.5m wide and 3m high. The Panzer IV measured in at 2.8m and 2.7m (which included a bulkier cupola) respectively.\n\nAll these faults, it might be argued, permit the Panzer IV to steal the laurels as the best German tank of the Second World War. After the war such was its reputation that some surviving Panzer IVs even had a second career, seeing combat with the Syrian Army in the late 1960s. In contrast the Tiger and Panther never saw action again.\n\nThe early-model Panzer IVs were never intended as tank-to-tank weapons and suffered as a result. Once up-gunned the Panzer IV became the backbone of Hitler's panzer forces.\n\nLate-model Panzer IVs being shipped to the front \u2013 in total over 8,500 Panzer IV were built along with almost 5,000 specialized AFV variants.\n\nA Panzer IV lost during the fighting for Kharkov. The Panzer IV, along with the Panther and Tiger, proved a match for the T-34, but were never available in sufficient numbers in the face of growing attrition rates.\n\nThe snow-covered remains of an Ausf J, the very last model Panzer IV. Nearly 1,800 of these were built by the end of the war\n\nFrozen Panzer IVs lost on the Eastern Front \u2013 some 6,100 Panzer IVs were lost to the Red Army.\n\nThe successful Panzer IV-based Hummel made its debut at Kursk and fought to the very end on the streets of Berlin. The gunners of this early model have been distracted from their target by something going on behind them\n\nGIs souvenir-hunt amongst the wreckage of a Panzer IV.\n\nOne of the very last Hummels, destroyed during the battle for Berlin in April 1945.\n\nAn abandoned late-model Hummel self-propelled gun left in its firing position.\n\n## Further Reading\n\nTucker-Jones, Anthony, illustrated by Hemingway, David Lee, Images of War Special: The Panther Tank, Hitler's T-34 Killer (Pen & Sword Military, 2016)\n\nTucker-Jones, Anthony, Images of War: German Assault Guns and Tank Destroyers 1940-1945 (Pen & Sword Military, 2016)\n\nTucker-Jones, Anthony, illustrated by Hemingway, David Lee, Images of War Special: T-34. The Red Army's Legendary Medium Tank (Pen & Sword Military, 2015)\n\nTucker-Jones, Anthony, illustrated by Delf, Brian, Images of War Special: Tiger I & Tiger II (Pen & Sword Military, 2013)\nColour Plates\n\nAt the start of the Second World War German military vehicles were painted in a twotone scheme using dark grey (dunkel grau) and dark brown (dunkel braun). Irregular patches of the brown were added to break up the factory grey base coat. In mid-1940 the brown was discontinued. In Europe after the Polish campaign most panzers were grey with a white Balkenkreuz (Balkan Cross) with a black centre. Vehicles shipped to North Africa were painted dark yellow or salmon sand, over which was sometimes added a vague hint of camouflage using darker colours.\n\nGerman Army Memorandum No.181, issued in February 1943, decreed that the basic overall colour for all military vehicles be a deep sand-yellow (dunkel gelb). From this point all vehicles were sprayed this colour, which superseded the previous panzer grey, before leaving the factory. This led to an enormous range of camouflage patterns being employed on the Eastern and Western Fronts as well as in Italy, using two other colours, olive green (oliv grun) and a red-brown (rotbraun) either together or singularly. During the winter on the Eastern and Western Fronts Panzer IVs were snow-camouflaged with varying degrees of success using a whitewash finish.\n\nPanzer IVs, in common with other German military vehicles, normally had up to four types of marking that identified their unit of origin and role. These consisted of national, divisional, tactical and personal markings. The most readily recognizable was the Balkenkreuz, which went through a number of variations as the war progressed.\n\nThe cross was usually located on the sides of the superstructure and the tactical number painted onto the turret sides or on the sides of the turret spaced armour (which formed part of the Sch\u00fcrzen: it always lasted longer than the skirt plates which were damaged or torn off whilst the tank was manoeuvring). In Normandy Panzer IVs were photographed with the cross on the forward sides and rear of the turret spaced armour. The three-digit number painted on the turret, usually in black or red lined in white, indicated, left to right, the company number, troop number and lastly the vehicle number.\n\nAUSF D\n\nThis Ausf D is finished in the early war standard panzer grey, which in many black and white and colour photographs is very dark and looks almost black. Panzer IVs fought in the initial campaigns on the Western and Eastern Fronts in this colour.\n\nAUSF D\n\nThis tank has the Balkenkreuz in black with white outline and the white tactical number 301 on the hull. The side and rear armour was increased from 15mm on the earlier models to 20mm.\n\nThe Ausf D, like the earlier A model, had a distinctive forward-set driver's position visible here from the top view.\n\nAUSF E\n\nThe similar-looking Ausf E turret had a redesigned cupola and an exhaust fan vent forward to the right to expel gun fumes. Also the glacis hatches were countersunk level with the glacis plate.\n\nAUSF E\n\nThe tank bears the insignia of the 11th Panzer Division which was formed in August 1940. It fought in the Balkans, Russia and France.\n\nThis particular example is finished in dunkel gelb, indicating it had been sent back to the factory for refurbishment.\n\nAUSF F2\n\nThis is the upgunned version of the Mk IV armed with the powerful 75mm KwK40 L\/43 anti-tank gun, which the British dubbed the 'Pz IV Special.' Bearing the tactical number '444' this F2 belonging to the 21st Panzer Division was photographed in North Africa in 1942. The original image indicates a quite dark base colour. The Afrika Korps swastika and palm tree insignia appeared in a number of slightly different variations.\n\nAUSF F2\n\nSide-on, the F2's gun had a very distinctive circular muzzle brake that made it easy to distinguish from subsequent versions. Also note the ubiquitous Jerry cans that were carried in racks on the sides and at the rear. On some models these side racks were replaced by a bracket for two spare wheels, that also served to up-armour the sides of the fighting compartment.\n\nThe dunkel gelb has either faded with the weather, dirt and dust or was toned down by the crew for operations in Tunisia during the winter months. Some German vehicles in Libya and Tunisia had a two-tone camouflage pattern. When applied in North Africa the Balkenkreuz was solid white with a black outline or was simply just a white outline.\n\nAUSF G\n\nThis formidable-looking Ausf G is fitted with Sch\u00fcrzen on both the turret and hull sides. It is finished in the three-tone camouflage used by the panzers in Normandy during the summer of 1944. The basic dunkel gelb coating has been oversprayed with very irregular patches of oliv grun and rotbraun. Mud and foliage also served to disrupt the outline of the panzers.\n\nThis Ausf G sports the German national cross but lacks a tactical number which was normally applied to the forward edge of the turret Sch\u00fcrzen. Photographic evidence suggests this omission was quite common in the later stages of the war.\n\nAUSF G\n\nThis top view shows just how effective the Normandy camouflage was from the air: this was vital in light of the threat posed by Allied fighter-bombers.\n  1. Cover\n  2. Title\n  3. Copyright\n  4. Contents\n  5. Introduction\n  6. Photograph Sources\n  7. Chapter One Early Days - Ausf A-C\n  8. Chapter Two Ramping up Production -Ausf D-F\n  9. Chapter Three Something 'Special' - Ausf F2\n  10. Chapter Four Panzerwaffe Backbone - Ausf G-H\n  11. Chapter Five No Frills - Ausf J\n  12. Chapter Six The Rhino and other Beasts\n  13. Chapter Seven In the Desert and Mountains\n  14. Chapter Eight From Barbarossa to Berlin\n  15. Chapter Nine Beyond the Seine\n  16. Chapter Ten Hitler's Rock\n  17. Further Reading\n  18. Plate section\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"\n\n\n\nProduced by Sandra Laythorpe\n\n\n\n\n\n\nTHE HERD BOY AND HIS HERMIT\n\nBy Charlotte M. Yonge\n\n\n\n\n     Henry, thou of holy birth,\n     Thou, to whom thy Windsor gave\n     Nativity and name and grave\n     Heavily upon his head\n     Ancestral crimes were visited.\n     Meek in heart and undefiled,\n     Patiently his soul resigned,\n     Blessing, while he kissed the rod,\n     His Redeemer and his God.\n                              SOUTHEY\n\n\n\n\nLIST OF CONTENTS\n\n\nI. IN THE MOSS\n\nII. THE SNOW-STORM\n\nIII. OVER THE MOOR\n\nIV. A SPORTING PRIORESS\n\nV. MOTHER AND SON\n\nVI. A CAUTIOUS STEPFATHER\n\nVII. ON DERWENT BANKS\n\nVIII. THE HERMIT\n\nIX. HENRY OF WINDSOR\n\nX. THE SCHOLAR OF THE MOUNTAINS\n\nXI. THE RED ROSE\n\nXII. A PRUDENT RECEPTION\n\nXIII. FELLOW TRAVELLERS\n\nXIV. THE JOURNEY\n\nXV. BLETSO\n\nXVI. THE HERMIT IN THE TOWER\n\nXVII. A CAPTIVE KING\n\nXVIII. AT THE MINORESSES\n\nXIX. A STRANGE EASTER EVE\n\nXX. BARNET\n\nXXI. TEWKESBURY\n\nXXII. THE NUT BROWN MAID\n\nXXIII. BROUGHAM CASTLE\n\n\n\n\n\nTHE HERD BOY AND HIS HERMIT\n\n\n\n\nCHAPTER I. -- IN THE MOSS\n\n\n\n     I can conduct you, lady, to a low\n     But loyal cottage where you may be safe\n     Till further quest.--MILTON.\n\n\nOn a moorland <DW72> where sheep and goats were dispersed among the\nrocks, there lay a young lad on his back, in a stout canvas cassock\nover his leathern coat, and stout leathern leggings over wooden shoes.\nTwilight was fast coming on; only a gleam of purple light rested on the\ntop of the eastern hills, but was gradually fading away, though the sky\nto the westward still preserved a little pale golden light by the help\nof the descending crescent moon.\n\n'Go away, horned moon,' murmured the boy. 'I want to see my stars come\nout before Hob comes to call me home, and the goats are getting up\nalready. Moon, moon, thou mayst go quicker. Thou wilt have longer time\nto-morrow--and be higher in the sky, as well as bigger, and thou mightst\nlet me see my star to-night! Ah! there is one high in the sunset, pale\nand fair, but not mine! That's the evening star--one of the wanderers.\nIs it the same as comes in the morning betimes, when we do not have\nit at night? Like that it shines with steady light and twinkles not. I\nwould that I knew! There! there's mine, my own star, far up, only paling\nwhile the sun glaring blazes in the sky; mine own, he that from afar\ndrives the stars in Charles's Wain. There they come, the good old\ntwinkling team of three, and the four of the Wain! Old Billy Goat knows\nthem too! Up he gets, and all in his wake \"Ha-ha-ha\" he calls, and the\nNannies answer. Ay, and the sheep are rising up too! How white they look\nin the moonshine! Piers--deaf as he is--waking at their music. Ba, they\ncall the lambs! Nay, that's no call of sheep or goat! 'Tis some child\ncrying, all astray! Ha! Hilloa, where beest thou? Tarry till I come!\nMove not, or thou mayst be in the bogs and mosses! Come, Watch'--to a\ngreat unwieldy collie puppy--'let us find her.'\n\nA feeble piteous sound answered him, and following the direction of the\nreply, he strode along, between the rocks and thorn-bushes that guarded\nthe <DW72> of the hill, to a valley covered with thick moss, veiling\ntreacherously marshy ground in which it was easy to sink.\n\nThe cry came from the further side, where a mountain stream had force\nenough to struggle through the swamp. There were stepping-stones across\nthe brook, which the boy knew, and he made his way from one to the\nother, calling out cheerily to the little figure that he began to\ndiscern in the fading light, and who answered him with tones evidently\ngirlish, 'O come, come, shepherd! Here I am! I am lost and lorn! They\nwill reward thee! Oh, come fast!'\n\n'All in good time, lassie! Haste is no good here! I must look to my\nfooting.'\n\nPresently he was by the side of the wanderer, and could see that it was\na maiden of ten or twelve years old, who somehow, even in the darkness,\nhad not the air of one of the few inhabitants of that wild mountain\ndistrict.\n\n'Lost art thou, maiden,' he said, as he stood beside her; 'where is\nthine home?'\n\n'I am at Greystone Priory,' replied the girl. 'I went out hawking to-day\nwith the Mother Prioress and the rest. My pony fell with me when we were\nriding after a heron. No one saw me or heard me, and my pony galloped\nhome. I saw none of them, and I have been wandering miles and miles! Oh\ntake me back, good lad; the Mother Prioress will give thee--'\n\n''Tis too far to take thee back to-night,' he said. 'Thou must come with\nme to Hob Hogward, where Doll will give thee supper and bed, and we will\nhave thee home in the morning.'\n\n'I never lay in a hogward's house,' she said primly.\n\n'Belike, but there be worse spots to be harboured in. Here, I must carry\nthee over the burn, it gets wider below! Nay, 'tis no use trying to leap\nit in the dark, thou wouldst only sink in. There!'\n\nAnd as he raised her in his arms, the touch of her garment was delicate,\nand she on her side felt that his speech, gestures and touch were not\nthose of a rustic shepherd boy; but nothing was said till he had waded\nthrough the little narrow stream, and set her down on a fairly firm\nclump of grass on the other side. Then she asked, 'What art thou,\nlad?--Who art thou?'\n\n'They call me Hal,' was the answer; 'but this is no time for questions.\nLook to thy feet, maid, or thou wilt be in a swamp-hole whence I may\nhardly drag thee out.'\n\nHe held her hand, for he could hardly carry her farther, since she\nwas almost as tall as himself, and more plump; and the rest of the\nconversation for some little time consisted of, 'There!' 'Where?' 'Oh,\nI was almost down!' 'Take heed; give me thy other hand! Thou must leap\nthis!' 'Oh! what a place! Is there much more of it?' 'Not much! Come\nbravely on! There's a good maid.' 'Oh, I must get my breath.' 'Don't\nstand still. That means sinking. Leap! Leap! That's right. No, not that\nway, turn to the big stair.' 'Oh--h!' 'That's my brave wench! Not far\nnow.' 'I'm down, I'm down!' 'Up! Here, this is safe! On that white\nstone! Now, here's sound ground! Hark!' Wherewith he emitted a strange\nwild whoop, and added, 'That's Hob come out to call me!' He holloaed\nagain. 'We shall soon be at home now. There's Mother Doll's light! Her\nlight below, the star above,' he added to himself.\n\nBy this time it was too dark for the two young people to see more than\ndim shapes of one another, but the boy knew that the hand he still held\nwas a soft and delicate one, and the girl that those which had grasped\nand lifted her were rough with country labours. She began to assert her\ndignity and say again, 'Who art thou, lad? We will guerdon thee well for\naiding me. The Lord St. John is my father. And who art thou?'\n\n'I? Oh, I am Hob Hogward's lad,' he answered in an odd off-hand tone,\nbefore whooping again his answer to the shouts of Hob, which were coming\nnearer.\n\n'I am so hungry!' said the little lady, in a weak, famished tone. 'Hast\naught to eat?'\n\n'I have finished my wallet, more's the pity!' said the boy, 'but never\nfear! Hold out but a few steps more, and Mother Doll will give thee bite\nand sup and bed.'\n\n'Alack! Is it much further! My feet! they are so sore and weary--'\n\n'Poor maiden, let me bear thee on!'\n\nHal took her up again, but they went more slowly, and were glad to see a\ntall figure before them, and hear the cry, 'How now, Hal boy, where hast\nbeen? What hast thou there?'\n\n'A sorely weary little lady, Daddy Hob, lost from the hawking folk from\nthe Priory,' responded Hal, panting a little as he set his burthen down,\nand Hob's stronger arms received her.\n\nHal next asked whether the flock had come back under charge of Piers,\nand was answered that all were safely at home, and after 'telling the\ntale' Hob had set out to find him. 'Thou shouldst not stray so far,' he\nsaid.\n\n'I heard the maid cry, and went after her,' said Hal, 'all the way to\nthe Blackreed Moss, and the springs, and 'twas hard getting over the\nswamp.'\n\n'Well indeed ye were not both swallowed in it,' said Hob; 'God be\npraised for bringing you through! Poor wee bairn! Thou hast come far!\nFrom whence didst say?'\n\n'From Greystone Priory,' wearily said the girl, who had her head down on\nHob's shoulder, and seemed ready to fall asleep there.\n\n'Her horse fell with her, and they were too bent on their sport to heed\nher,' explained the boy, as he trudged along beside Hob and his charge,\n'so she wandered on foot till by good hap I heard her moan.'\n\n'Ay, there will be a rare coil to-night for having missed her,' said\nHob; 'but I've heard tell, my Lady Prioress heeds her hawks more than\nher nuns! But be she who she may, we'll have her home, and Mother Doll\nshall see to her, for she needs it sure, poor bairn. She is asleep\nalready.'\n\nSo she was, with her head nestled into the shepherd's neck, nor did she\nwaken when after a tramp of more than a mile the bleatings of the folded\nsheep announced that they were nearly arrived, and in the low doorway\nthere shone a light, and in the light stood a motherly form, in a white\nwoollen hood and dark serge dress. Tired as he was, Hal ran on to her,\nexclaiming 'All well, Mammy Doll?'\n\n'Ah well!' she answered, 'thank the good God! I was in fear for thee, my\nboy! What's that Daddy hath? A strayed lamb?'\n\n'Nay, Mammy, but a strayed maiden! 'Twas that kept me so long. I had to\nbear her through the burn at Blackreed, and drag her on as best I might,\nand she is worn out and weary.'\n\n'Ay,' said Hob, as he came up. 'How now, my bit lassie?' as he put her\ninto the outstretched arms of his wife, who sat down on the settle to\nreceive her, still not half awake.\n\n'She is well-nigh clemmed,' said Hal. 'She has had no bite nor sup all\nday, since her pony fell with her out a-hawking, and all were so hot on\nthe chase that none heeded her.'\n\nMother Doll's exclamations of pity were profuse. There was a kettle of\nbroth on the peat fire, and after placing the girl in a corner of the\nsettle, she filled three wooden bowls, two of which she placed before\nHal and the shepherd, making signs to the heavy-browed Piers to wait;\nand getting no reply from her worn-out guest, she took her in her arms,\nand fed her from a wooden spoon. Though without clear waking, mouthfuls\nwere swallowed down, till the bowl was filled again and set before\nPiers.\n\n'There, that will be enough this day!' said the good dame. 'Poor bairn!\n'Twas scurvy treatment. Now will we put her to bed, and in the morn we\nwill see how to deal with her.'\n\nHal insisted that the little lady should have his own bed--a\nchaff-stuffed mattress, covered with a woollen rug, in the recess behind\nthe projecting hearth--a strange luxury for a farm boy; and Doll yielded\nvery unwillingly when he spoke in a tone that savoured of command.\nThe shaggy Piers had already curled himself up in a corner and gone to\nsleep.\n\n\n\n\nCHAPTER II. -- THE SNOW-STORM\n\n\n\n     Yet stay, fair lady, rest awhile\n     Beneath the cottage wall;\n     See, through the hawthorns blows the cold wind,\n     And drizzling rain doth fall.--OLD BALLAD.\n\n\nThough Hal had gone to sleep very tired the night before, and only on\na pile of hay, curled up with Watch, having yielded his own bed to the\nstrange guest, he was awake before the sun, for it was the decline of\nthe year, and the dawn was not early.\n\nHe was not the first awake--Hob and Piers were already busy on the\noutside, and Mother Doll had emerged from the box bed which made almost\na separate apartment, and was raking together the peat, so as to revive\nthe slumbering fire. The hovel, for it was hardly more, was built of\nrough stone and thatched with reeds, with large stones to keep the\nroof down in the high mountain blasts. There was only one room, earthen\nfloored, and with no furniture save a big chest, a rude table, a settle\nand a few stools, besides the big kettle and a few crocks and wooden\nbowls. Yet whereas all was clean, it had an air of comfort and\ncivilisation beyond any of the cabins in the neighbourhood, more\nespecially as there was even a rude chimney-piece projecting far into\nthe room, and in the niche behind this lay the little girl in her\nclothes, fast asleep.\n\nVery young and childish she looked as she lay, her lips partly unclosed,\nher dark hair straying beyond her hand, and her black lashes resting on\nher delicate brunette cheeks, slightly flushed with sleep. Hal could\nnot help standing for a minute gazing at her in a sort of wondering\ncuriosity, till roused by the voice of Mother Doll.\n\n'Go thy ways, my bairn, to wash in the burn. Here's thy comb. I must\nhave the lassie up before the shepherd comes back, though 'tis amost\na pity to wake her! There, she is stirring! Best be off with thee, my\nbonnie lad.'\n\nIt was spoken more in the tone of nurse to nursling than of mother\nto son, still less that of mistress to farm boy; but Hal obeyed, only\nobserving, 'Take care of her.'\n\n'Ay, my pretty, will not I,' murmured the old woman, as the child turned\nround on her pillow, put up a hand, rubbed her eyes, and disclosed a\npair of sleepy brown orbs, gazed about, and demanded, 'What's this?\nWho's this?'\n\n''Tis Hob Hogward's hut, my bonnie lamb, where you are full welcome!\nHere, take a sup of warm milk.'\n\n'I mind me now,' said the girl, sitting up, and holding out her hands\nfor the bowl. 'They all left me, and the lad brought me--a great lubber\nlout--'\n\n'Nay, nay, mistress, you'll scarce say so when you see him by day--a\nwell-grown youth as can bear himself with any.'\n\n'Where is he?' asked the girl, gazing round; 'I want him to take me\nback. This place is not one for me. The Sisters will be seeking me! Oh,\nwhat a coil they must be in!'\n\n'We will have you back, my bairn, so soon as my goodman can go with you,\nbut now I would have you up and dressed, ay, and washed, ere he and Hal\ncome in. Then after meat and prayer you will be ready to go.'\n\n'To Greystone Priory,' returned the girl. 'Yea, I would have thee to\nknow,' she added, with a little dignity that sat drolly on her bare feet\nand disordered hair and cap as she rose out of bed, 'that the Sisters\nare accountable for me. I am the Lady Anne St. John. My father is a lord\nin Bedfordshire, but he is gone to the wars in Burgundy, and bestowed\nme in a convent at York while he was abroad, but the Mother thought her\nhouse would be safer if I were away at the cell at Greystone when Queen\nMargaret and the Red Rose came north.'\n\n'And is that the way they keep you safe?' asked the hostess, who\nmeanwhile was attending to her in a way that, if the Lady Anne had known\nit, was like the tendance of her own nurse at home, instead of that of a\nrough peasant woman.\n\n'Oh, we all like the chase, and the Mother had a new cast of hawks that\nshe wanted to fly. There came out a heron, and she threw off the new\none, and it went careering up--and up--and we all rode after, and just\nas the bird was about to pounce down, into a <DW18> went my pony, Imp, and\nnot one of them saw! Not Bertram Selby, the Sisters, nor the groom, nor\nthe rabble rout that had come out of Greystone; and before I could get\nfree they were off; and the pony, Imp of Evil that he is, has not learnt\nto know me or my voice, and would not let me catch him, but cantered\noff--either after the other horses or to the Priory. I knew not where I\nwas, and halloaed myself hoarse, but no one heard, and I went on and on,\nand lost my way!'\n\n'I did hear tell that the Lady Prioress minded her hawks more than her\nHours,' said Mother Doll.\n\n'And that's sooth,' said the Lady Anne, beginning to prove herself a\nchatterbox. 'The merlins have better hoods than the Sisters; and as\nto the Hours, no one ever gets up in the night to say Nocturns or even\nMatins but old Sister Scholastica, and she is as strict and cross as may\nbe.'\n\nHere the flow of confidence was interrupted by the return of Hal, who\ngazed eagerly, though in a shamefaced way, at the guest as he set down a\nbowl of ewe milk. She was a well-grown girl of ten, slender, and bearing\nherself like one high bred and well trained in deportment; and her face\nwas delicately tinted on an olive skin, with fine marked eyebrows, and\ndark bright eyes, and her little hunting dress of green, and the hood,\nset on far back, became the dark locks that curled in rings beneath.\n\nShe saw a slender lad, dark-haired and dark-eyed, ruddy and embrowned\nby mountain sun and air; and the bow with which he bent before her had\nsomething of the rustic lout, and there was a certain shyness over him\nthat hindered him from addressing her.\n\n'So, shepherd,' she said, 'when wilt thou take me back to Greystone?'\n\n'Father will fix that,' interposed the housewife; 'meanwhile, ye had\nbest eat your porridge. Here is Father, in good time with the cows'\nmilk.'\n\nThe rugged broad-shouldered shepherd made his salutation duly to the\nyoung lady, and uttered the information that there was a black cloud,\nlike snow, coming up over the fells to the south-west.\n\n'But I must fare back to Greystone!' said the damsel. 'They will be in a\nmighty coil what has become of me.'\n\n'They would be in a worse coil if they found your bones under a snow\nwreath.'\n\nHal went to the door and spied out, as if the tidings were rather\npleasant to him than otherwise. The goodwife shivered, and reached out\nto close the shutter, and there being no glass to the windows, all the\nlight that came in was through the chinks.\n\n'It would serve them right for not minding me better,' said the maiden\ncomposedly. 'Nay, it is as merry here as at Greystone, with Sister\nMargaret picking out one's broidery, and Father Cuthbert making one pore\nover his crabbed parchments.'\n\n'Oh, does this Father teach Latin?' exclaimed Hal with eager interest.\n\n'Of course he doth! The Mother at York promised I should learn whatever\nbecame a damsel of high degree,' said the girl, drawing herself up.\n\n'I would he would teach me!' sighed the boy.\n\n'Better break thy fast and mind thy sheep,' said the old woman, as if\nshe feared his getting on dangerous ground; and placing the bowl of\nporridge on the rough table, she added, 'Say the Benedicite, lad, and\nfall to.' Then, as he uttered the blessing, she asked the guest whether\nshe preferred ewes' milk or cows' milk, a luxury no one else was\nallowed, all eating their porridge contentedly with a pinch of salt, Hob\nshowing scant courtesy, the less since his guest's rank had been made\nknown.\n\nBy the time they had finished, snowflakes--an early autumn storm--were\ndrifting against the shutter, and a black cloud was lowering over the\nhills. Hob foretold a heavy fall of snow, and called on Hal to help\nhim and Piers fold the flock more securely, sleepy Watch and his old\nlong-haired collie mother rising at the same call. Lady Anne sprang up\nat the same time, insisting that she must go and help to feed the poor\nsheep, but she was withheld, much against her will, by Mother Dolly,\nthough she persisted that snow was nothing to her, and it was a fine\njest to be out of the reach of the Sisters, who mewed her up in a\ncell, like a messan dog. However, she was much amused by watching,\nand thinking she assisted in, Mother Dolly's preparations for ewe milk\ncheese-making; and by-and-by Hal came in, shaking the snow off the\nsheepskin he had worn over his leathern coat. Hob had sent him in, as\nthe weather was too bad for him, and he and Anne crouched on opposite\nsides of the wide hearth as he dried and warmed himself, and cosseted\nthe cat which Anne had tried to caress, but which showed a decided\npreference for the older friend.\n\n'Our Baudrons at Greystone loves me better than that,' said Anne. 'She\nwill come to me sooner than even to Sister Scholastica!'\n\n'My Tib came with us when we came here. Ay, Tib! purr thy best!' as he\nheld his fingers over her, and she rubbed her smooth head against him.\n\n'Can she leap? Baudrons leaps like a horse in the tilt-yard.'\n\n'Cannot she! There, my lady pussy, show what thou canst do to please the\ndemoiselle,' and he held his arms forward with clasped hands, so that\nthe grey cat might spring over them, and Lady Anne cried out with\ndelight.\n\nAgain and again the performance was repeated, and pussy was induced\nto dance after a string dangled before her, to roll over and play in\napparent ecstasy with a flake of wool, as if it were a mouse, and Watch\njoined in the game in full amity. Mother Dolly, busy with her distaff,\nlooked on, not displeased, except when she had to guard her spindle from\nthe kitten's pranks, but she was less happy when the children began to\ntalk.\n\n'You have seen a tilt-yard?'\n\n'Yea, indeed,' he answered dreamily. 'The poor squire was hurt--I did\nnot like it! It is gruesome.'\n\n'Oh, no! It is a noble sport! I loved our tilt-yard at Bletso. Two\nknights could gallop at one another in the lists, as if they were out\nhunting. Oh! to hear the lances ring against the shields made one's\nheart leap up! Where was yours?'\n\nHere Dolly interrupted hastily, 'Hal, lad, gang out to the shed and\nbring in some more sods of turf. The fire is getting low.'\n\n'Here's a store, mother--I need not go out,' said Hal, passing to a pile\nin the corner. 'It is too dark for thee to see it.'\n\n'But where was your castle?' continued the girl. 'I am sure you have\nlived in a castle.'\n\nInsensibly the two children had in addressing one another changed the\nhomely singular pronoun to the more polite, if less grammatical, second\nperson plural. The boy laughed, nodded his head, and said, 'You are a\nlittle witch.'\n\n'No great witchcraft to hear that you speak as we do at home in\nBedfordshire, not like these northern boors, that might as well be\nScots!'\n\n'I am not from Bedfordshire,' said the lad, looking much amused at her\nperplexity.\n\n'Who art thou then?' she cried peremptorily.\n\n'I? I am Hal the shepherd boy, as I told thee before.'\n\n'No shepherd boy are you! Come, tell me true.'\n\nDolly thought it time to interfere. She heard an imaginary bleat, and\nordered Hal out to see what was the matter, hindering the girl by force\nfrom running after him, for the snow was coming down in larger flakes\nthan ever. Nevertheless, when her husband was heard outside she threw a\ncloak over her head and hurried out to speak with him. 'That maid will\nmake our lad betray himself ere another hour is over their heads!'\n\n'Doth she do it wittingly?' asked the shepherd gravely.\n\n'Nay, 'tis no guile, but each child sees that the other is of gentle\nblood, and women's wits be sharp and prying, and the maid will never\nrest till she has wormed out who he is.'\n\n'He promised me never to say, nor doth he know.'\n\n'Thee! Much do the hests of an old hogherd weigh against the wiles of a\nyoung maid!'\n\n'Lord Hal is a lad of his word. Peace with thy lords and ladies, woman,\nthou'lt have the archers after him at once.'\n\n'She makes no secret of being of gentle blood--a St. John of Bletso.'\n\n'A pestilent White Rose lot! We shall have them on the scent ere many\ndays are over our head! An unlucky chance this same snow, or I should\nhave had the wench off to Greystone ere they could exchange a word.'\n\n'Thou wouldst have been caught in the storm. Ill for the maid to have\nfallen into a drift!'\n\n'Well for the lad if she never came out of it!' muttered the gruff\nold shepherd. 'Then were her tongue stilled, and those of the clacking\nwenches at York--Yorkists every one of them.'\n\nMother Dolly's eyes grew round. 'Mind thee, Hob!' she said; 'I ken thy\nbark is worse than thy bite, but I would have thee to know that if aught\nbefall the maid between this and Greystone, I shall hold thee--and so\nwill my Lady--guilty of a foul deed.'\n\n'No fouler than was done on the stripling's father,' muttered the\nshepherd. 'Get thee in, wife! Who knows what folly those two may be\nafter while thou art away? Mind thee, if the maid gets an inkling of who\nthe boy is, it will be the worse for her.'\n\n'Oh!' murmured the goodwife, 'I moaned once that our Piers there should\nbe deaf and well-nigh dumb, but I thank God for it now! No fear of\nperilous word going out through him, or I durst not have kept my poor\nsister's son!'\n\nMother Doll trusted that her husband would never have the heart to leave\nthe pretty dark-haired girl in the snow, but she was relieved to find\nHal marking down on the wide flat hearth-stone, with a bit of charcoal,\nall the stars he had observed. 'Hob calls that the Plough--those seven!'\nhe said; 'I call it Charles's Wain!'\n\n'Methinks I have seen that!' she said, 'winter and summer both.'\n\n'Ay, he is a meuseful husbandman, that Charles! And see here! This\nmiddle mare of the team has a little foal running beside her'--he made\na small spot beside the mark that stood for the central star of what we\ncall the Bear's Tail.\n\n'I never saw that!'\n\n'No, 'tis only to be seen on a clear bright night. I have seen it, but\nHob mocks at it. He thinks the only use of the Wain is to find the North\nStar, up beyond there, pointing by the back of the Plough, and go by it\nwhen you are lost.'\n\n'What good would finding the North Star do? It would not have helped me\nhome if you had not found me!'\n\n'Look here, Lady Anne! Which way does Greystone lie?'\n\n'How should I tell?'\n\n'Which way did the sun lie when you crossed the moor?'\n\nAnne could not remember at first, but by-and-by recollected that it\ndazzled her eyes just as she was looking for the runaway pony; and Hal\ndeclared that it proved that the convent must have been to the south of\nthe spot of her fall; but his astronomy, though eagerly demonstrated,\nwas not likely to have brought her back to Greystone. Still Doll\nwas thankful for the safe subject, as he went on to mark out what he\npromised that she should see in the winter--the swarm of glow-worms,\nas he called the Pleiades; and 'Our Lady's Rock,' namely, distaff,\nthe northern name for Orion; and then he talked of the stars that so\nperplexed him, namely, the planets, that never stayed in their places.\n\nBy-and-by, when Mother Dolly's work was over the kettle was on the fire,\nand she was able to take out her own spinning, she essayed to fill up\nthe time by telling them lengthily the old stories and ballads handed\ndown from minstrel to minstrel, from nurse to nurse, and they sat\nentranced, listening to the stories, more than even Hal knew she\npossessed, and holding one another by the hand as they listened.\n\nMeantime the snow had ceased--it was but a scud of early autumn on\nthe mountains--the sun came out with bright slanting beams before his\nsetting, there was a soft south wind; and Hob, when he came in, growled\nout that the thaw had set in, and he should be able to take the maid\nback in the morning. He sat scowling and silent during supper, and\nordered Hal about with sharp sternness, sending him out to attend to the\nlitter of the cattle, before all had finished, and manifestly treated\nhim as the shepherd's boy, the drudge of the house, and threatening\nhim with a staff if he lingered, soon following himself. Mother Dolly\ninsisted on putting the little lady to bed before they should return,\nand convent-bred Anne had sufficient respect for proprieties to see that\nit was becoming. She heard no more that night.\n\n\n\n\nCHAPTER III. -- OVER THE MOOR\n\n\n\n     In humblest, simplest habit clad,\n     But these were all to me.--GOLDSMITH.\n\n\n'Hal! What is your name?'\n\nShe stood at the door of the hovel, the rising sun lighting up her\nbright dark eyes, and smiling in the curly rings of her hair while Hal\nstood by, and Watch bounded round them.\n\n'You have heard,' he said, half smiling, and half embarrassed.\n\n'Hal! That's no name.'\n\n'Harry, an it like you better.'\n\n'Harry what?' with a little stamp of her foot.\n\n'Harry Hogward, as you see, or Shepherd, so please you.'\n\n'You are no Hogward, nor shepherd! These folk be no kin to you, I can\nsee. Come, an you love me, tell me true! I told you true who I am, Red\nRose though I see you be! Why not trust me the same?'\n\n'Lady, I verily ken no name save Harry. I would trust you, verily I\nwould, but I know not myself.'\n\n'I guess! I guess!' she cried, clapping her hands, but at the moment\nDolly laid a hand on her shoulder.\n\n'Do not guess, maiden,' she said. 'If thou wouldst not bring evil on the\nlad that found thee, and the roof that sheltered thee, guess not, yea,\nand utter not a word save that thou hast lain in a shepherd's hut.\nForget all, as though thou hadst slept in the castle on the hill that\nfades away with the day.'\n\nShe ended hastily, for her husband was coming up with a rough pony's\nhalter in his hand. He was in haste to be off, lest a search for the\nlost child might extend to his abode, and his gloomy displeasure and\nill-masked uneasiness reduced every-one to silence in his presence.\n\n'Up and away, lady wench!' he said. 'No time to lose if you are to be at\nGreystone ere night! Thou Hal, thou lazy lubber, go with Piers and the\nsheep--'\n\n'I shall go with you,' replied Hal, in a grave tone of resolution. 'I\nwill only go within view of the convent, but go with you I will.'\n\nHe spoke with a decided tone of authority, and Hob Hogward muttered a\nlittle to himself, but yielded.\n\nHal assisted the young lady to mount, and they set off along the track\nof the moss, driving the cows, sheep, and goats before them--not a very\nconsiderable number--till they came to another hut, much smaller and\nmore rude than that where they had left Mother Doll.\n\nPiers was a wild, shaggy-haired lad, with a sheepskin over his\nshoulders, and legs bare below the knee, and to him the charge of\nthe flock was committed, with signs which he evidently understood and\nreplied to with a gruff 'Ay, ay!' The three went on the way, over the\n<DW72> of a hill, partly clothed with heather, holly and birch trees, as\nit rose above the moss. Hob led the pony, and there was something in his\ngrim air and manner that hindered any conversation between the two young\npeople. Only Hal from time to time gathered a flower for the young lady,\nscabious and globe flowers, and once a very pink wild rose, mingled\nwith white ones. Lady Anne took them with a meaning smile, and a merry\ngesture, as though she were going to brush Hal's face with the petals.\nHal laughed, and said, 'You will make them shed.'\n\n'Well and good, so the disputes be shed,' said Anne, with more meaning\nthan perhaps Hal understood. 'And the white overcomes the red.'\n\n'May be the red will have its way with spring--'\n\nBut there Hob looked round on them, and growled out, 'Have done with\nthat folly! What has a herd boy like thee to do with roses and frippery?\nCome away from the lady's rein. Thou art over-held to thrust thyself\nupon her.'\n\nNevertheless, as Hal fell back, the dark eyes shot a meaning glance\nat him, and the party went on in silence, except that now and then\nHob launched at Hal an order that he endeavoured to render savagely\ncontemptuous and harsh, so that Lady Anne interfered to say, 'Nay, the\npoor lad is doing no harm.'\n\n'Scathe enough,' answered Hob. 'He always will be doing ill if he can.\nHeed him not, lady, it only makes him the more malapert.'\n\n'Malapert,' repeated Anne, not able to resist a little teasing of\nthe grim escort; 'that's scarce a word of the dales. 'Tis more like a\nman-at-arms.'\n\nThis Hob would not hear, and if he did, it produced a rough imprecation\non the pony, and a sharp cut with his switch.\n\nThey had crossed another burn, travelled through the moss, and mounted\nto the brow of another hill, when, far away against the sky, on the top\nof yet another height, were to be seen moving figures, not cattle, but\nAnne recognised them at once. 'Men-at-arms! archers! lances! A search\nparty for me! The Prioress must have sent to the Warden's tower.'\n\n'Off with thee, lad!' said Hob, at once turning round upon Hal. 'I'll\nnot have thee lingering to gape at the men-at-arms! Off I say, or--'\n\nHe raised his stout staff as though to beat the boy, who looked up in\nhis face with a laugh, as if in very little alarm at his threat,\nsmiled up in the young lady's face, and as she held out her hand with\n'Farewell, Hal; I'll keep your rose-leaves in my breviary,' he bent over\nand kissed the fingers.\n\n'How now! This impudence passes! As if thou wert of the same blood as\nthe damsel!' exclaimed Hob in considerable anger, bringing down his\nstick. 'Away with thee, ill-bred lubber! Back to thy sheep, thou lazy\nloiterer! Get thee gone and thy whelp with thee!'\n\nHal obeyed, though not without a parting grin at Anne, and had sped away\ndown the side of the hill, among the hollies and birches, which entirely\nconcealed him and the bounding puppy.\n\nHob went on in a gruff tone: 'The insolence of these loutish lads! See\nyou, lady, he is a stripling that I took up off the roadside out of mere\ncharity, and for the love of Heaven--a mere foundling as you may say,\nand this is the way he presumes!'\n\n'A foundling, sayest thou?' said Anne, unable to resist teasing him a\nlittle, and trying to gratify her own curiosity.\n\n'Ay, you may say so! There's a whole sort of these orphans, after all\nthe bad luck to the land, to be picked up on every wayside.'\n\n'On Towton Moor, mayhap,' said Anne demurely, as she saw her surly guide\nstart. But he was equal to the occasion, and answered:\n\n'Ay, ay, Towton Moor; 'twas shame to see such bloody work; and there\nwere motherless and fatherless children, stray lambs, to be met with,\nweeping their little hearts out, and starving all around unless some\ngood Christian took pity on them.'\n\n'Was Hal one of these?' asked Lady Anne.\n\n'I tell you, lady, I looked into a church that was full of weeping\nand wailing folk, women and children in deadly fear of the cruel,\nbloody-minded York folk, and the Lord of March that is himself King\nEdward now, a murrain on him!'\n\n'Don't let those folk hear you say so!' laughed Lady Anne. 'They would\nthink nothing of hauling thee off for a black traitor, or hanging thee\nup on the first tree stout enough to bear thee.'\n\nShe said it half mischievously, but the only effect was a grunt, and a\nstolid shrug of his shoulders, nor did he vouchsafe another word for the\nrest of the way before they came through the valley, and through the low\nbrushwood on the bank, and were in sight of the search party, who set up\na joyful halloo of welcome on perceiving her.\n\nA young man, the best mounted and armed, evidently an esquire, rode\nforward, exclaiming, 'Well met, fair Lady Anne! Great have been the\nMother Prioress's fears for you, and she has called up half the country\nside, lest you should be fallen into the hands of Robin of Redesdale, or\nsome other Lancastrian rogue.'\n\n'Much she heeded me in comparison with hawk and heron!' responded Anne.\n'Thanks for your heed, Master Bertram.'\n\n'I must part from thee and thy sturdy pony. Thanks for the use of it,'\nadded she, as the squire proceeded to take her from the pony. He would\nhave lifted her down, but she only touched his hand lightly and sprang\nto the ground, then stood patting its neck. 'Thanks again, good pony. I\nam much beholden to thee, Gaffer Hob! Stay a moment.'\n\n'Nay, lady, it would be well to mount you behind Archie. His beast is\nbest to carry a lady.'\n\nArchie was an elderly man, stout but active, attached to the service of\nthe convent. He had leapt down, and was putting on a belt, and arranging\na pad for the damsel, observing, 'Ill hap we lost you, damsel! I saw you\nnot fall.'\n\n'Ay,' returned Anne, 'your merlin charmed you far more. Master Bertram,\nthe loan of your purse. I would reward the honest man who housed me.'\n\nBertram laughed and said, tossing up the little bag that hung to his\ngirdle, 'Do you think, fair damsel, that a poor Border squire carries\nabout largesse in gold and silver? Let your clown come with us to\nGreystone, and thence have what meed the Prioress may bestow on him, for\na find that your poor servant would have given worlds to make.'\n\n'Hearest thou, Hob?' said Anne. 'Come with us to the convent, and thou\nshalt have thy guerdon.'\n\nHob, however, scratched his head, with a more boorish air than he had\nbefore manifested, and muttered something about a cow that needed his\nattention, and that he could not spare the time from his herd for all\nthat the Prioress was like to give him.\n\n'Take this, then,' said Anne, disengaging a gold clasp from her neck,\nand giving it to him. 'Bear it to the goodwife and bid her recollect me\nin her prayers.'\n\n'I shall come and redeem it from thee, sulky carle as thou art,' said\nBertram. 'Such jewels are not for greasy porridge-fed housewives. Hark\nthee, have it ready for me! I shall be at thy hovel ere long'--as Anne\nwaved to Hob when she was lifted to her seat.\n\nBut Hob had already turned away, and Anne, as she held on by Archie's\nleathern belt, in her gay tone was beginning to defend him by declaring\nthat porridge and grease did not go together, so the nickname was not\nrightly bestowed on the kindly goodwife.\n\n'Ay! Greasy from his lord's red deer,' said Bertram, 'or his tainted\nmutton. Trust one of these herds, and a sheep is tainted whenever he\nwants a good supper. Beshrew me but that stout fellow looks lusty and\nhearty enough, as if he lived well.'\n\n'They were good and kind, and treated me well,' said Anne. 'I should be\ndead if they had not succoured me.'\n\n'The marvel is you are not dead with the stench of their hovel, and the\nfoulness of their food.'\n\n'It was very good food--milk, meat, and oaten porridge,' replied Anne.\n\n'Marvellous, I say!' cried Bertram with a sudden thought. 'Was it not\nsaid that there were some of those traitorous Lancastrian folk\nlurking about the mountains and fells? That rogue had the bearing of\na man-at-arms, far more than of a mere herd. Deemedst thou not so,\nArchie?' to the elderly man who rode before the young damsel.\n\n'Herdsmen here are good with the quarter-staff. They know how to stand\nagainst the Scots, and do not get bowed like our Midland serfs,' put\nin Anne, before Archie could answer, which he did with something of a\nsnarl, as Bertram laughed somewhat jeeringly, and declared that the Lady\nAnne had become soft-hearted. She looked down at her roses, but in the\ndismounting and mounting again the petals of the red rose had floated\naway, and nothing was left of it save a slender pink bud enclosed within\na dark calyx.\n\nArchie, hard pressed, declared, 'There are poor fellows lurking about\nhere and there, but bad blood is over among us. No need to ferret about\nfor them.'\n\n'Eh! Not when there may be a lad among them for whose head the king and\nhis brothers would give the weight of it in gold nobles?'\n\nAnne shivered a little at this, but she cried out, 'Shame on you, Master\nBertram Selby, if you would take a price for the head of a brave foe!\nYou, to aspire to be a knight!'\n\n'Nay, lady, I was but pointing out to Archie and the other grooms here,\nhow they might fill their pouches if they would. I verily believe thou\nknowst of some lurking-place, thou art so prompt to argue! Did I not\nsee another with thee, who made off when we came in view? Say! Was he\na blood-stained Clifford? I heard of the mother having married in these\nparts.'\n\n'He was Hob Hogward's herd boy,' answered Anne, as composedly as she\ncould. 'He hied him back to mind his sheep.'\n\nNor would Anne allow another word to be extracted from her ere the grey\nwalls of the Priory of Greystone rose before her, and the lay Sister at\nthe gate shrieked for joy at seeing her riding behind Archie.\n\n\n\n\nCHAPTER IV. -- A SPORTING PRIORESS\n\n\n\n     Yet nothing stern was she in cell,\n     And the nuns loved their abbess well.--SCOTT.\n\n\nThe days of the Wars of the Roses were evil times for the discipline of\nconvents, which, together with the entire Western Church, suffered from\nthe feuds of the Popes with the Italian princes.\n\nSmall remote houses, used as daughters or auxiliaries to the large\nconvents, were especially apt to fall into a lax state, and in truth\nthe little priory of Greystone, with its half-dozen of Sisters, had been\nplaced under the care of the Lady Agnes Selby because she was too highly\nconnected to be dealt with sharply, and too turbulent and unmanageable\nfor the soberminded house at York. So there she was sent, with the\ndeeply devout and strict Sister Scholastica, to keep the establishment\nin order, and deal with the younger nuns and lay Sisters. Being not\nentirely out of reach of a raid from the Scottish border, it was\nhardly a place for the timid, although the better sort of moss troopers\ngenerally spared monastic houses. Anne St. John had been sent thither at\nthe time when Queen Margaret was making her attempt in the north, where\nthe city of York was Lancastrian, as the Mother Abbess feared that her\npresence might bring vengeance upon the Sisterhood.\n\nThere was no great harm in the Mother Agnes, only she was a maiden\nwhom nothing but family difficulties could have forced into a monastic\nlife--a lively, high-spirited, out-of-door creature, whom the close\nconventionalities of castle life and even whipping could not tame, and\nwho had been the despair of her mother and of the discreet dames to whom\nher first childhood had been committed, to say nothing of a Lady Abbess\nor two. Indeed, from the Mother of Sopwell, Dame Julian Berners, she\nhad imbibed nothing but a vehement taste for hawk, horse, and hound.\nThe recluses of St. Mary, York, after being heartily scandalised by her\nhabits, were far from sorry to have a good excuse for despatching her to\ntheir outlying cell, where, as they observed, she would know how to show\na good face in case the Armstrongs came over the Border.\n\nShe came flying down on the first rumour of Lady Anne's return, her veil\nturned back, her pace not at all accordant with the solemn gait of a\nPrioress, her arms outstretched, her face, not young nor handsome, but\nsunburnt, weather-beaten and healthy, and full of delight. 'My child,\nmy Nan, here thou art! I was just mounting to seek for thee to the west,\nwhile Bertram sought again over the mosses where we sent yester morn.\nWhere hast thou been in the snow?'\n\n'A shepherd took me to his hut, Lady Mother,' answered Anne rather\ncoldly.\n\n'Little didst thou think of our woe and grief when thy palfrey was found\nstanding riderless at the stable door, and Sister Scholastica told us\nthat there he had been since nones! And she had none to send in quest\nbut Cuddie, the neatherd.'\n\n'My palfrey fell with me when you were in full chase of hawk and heron,\n'and none ever turned a head towards me nor heard me call.'\n\n'Poor maid! But it was such a chase as never you did watch. On and on\nwent the heron, the falcon ever mounting higher and higher, till she was\nbut a speck in the clouds, and Tam Falconer shouting and galloping, mad\nlest she should go down the wind. Methought she would have been back to\nNorroway, the foul jade!'\n\n'Did you capture her, Mother?' asked Anne.\n\n'Ay, she pounced at last, and well-nigh staked herself on the heron's\nbeak! But we had a long ride, and were well-nigh at the Tyne before we\nhad caught her. Full of pranks, but a noble hawk, as I shall write to my\nbrother by the next messenger that comes our way. I call it a hawk worth\nher meat that leads one such a gallop.'\n\n'What would you have done, reverend Mother, if she had crossed the\nBorder?' asked Bertram.\n\n'Ridden after her. No Scot would touch a Lady Prioress on the chase,'\nresponded Mother Agnes, looking not at all like a reverend Mother. 'Now,\npoor Anne, thou must be hungered. Thou shalt eat with Master Bertram and\nme in the refectory anon. Take her, Sister Joan, and make her ready to\nbreak her fast with us.'\n\nAnne quickly went to her chamber. It was not quite a cell, the bare\nstone walls being hung with faded woollen tapestry, the floor covered\nwith a deerskin, the small window filled with dark green glass, a chest\nserving the double purpose of seat and wardrobe, and further, a bed hung\nwith thick curtains, in which she slept with the lay Sister, Joan, who\nfurther fetched a wooden bowl of water from the fountain in the\ncourt that she might wash her face and hands. She changed her soiled\nriding-dress for a tight-fitting serge garment of dark green with long\nhanging sleeves, assisted by Joan, who also arranged her dark hair in\ntwo plaits, and put over it a white veil, fastened over a framework to\nkeep it from hanging too closely.\n\nAll the time Joan talked, telling of the fright the Mother had been\nin when the loss of the Lady Anne had been discovered, and how it was\nfeared that she had been seized by Scottish reivers, or lost in the snow\non the hills, or captured by the Lancastrians.\n\n'For there be many of the Red Rose rogues about on the mosses--comrades,\n'tis said, of that noted thief Robin of Redesdale.'\n\n'I was with good folk, in a shepherd's sheiling,' replied Anne.\n\n'Ay, ay. Out on the north hill, methinks.'\n\n'Nay. Beyond Deadman's Pool,' said Anne. 'By Blackreed Moss. That was\nwhere the pony fell.'\n\n'Blackreed Moss! That moor belongs to the De Vescis, the blackest\nLancaster fellow of all! His daughter is the widow of the red-handed\nClifford, who slew young Earl Edmund on Wakefield Bridge. They say her\nyoung son is in hiding in some moss in his lands, for the King holds him\nin deadly feud for his brother's death.'\n\n'He was a babe, and had nought to do with it,' said Anne.\n\n'He is of his father's blood,' returned Sister Joan, who in her convent\nwas still a true north country woman. 'Ay, Lady Anne, you from your\nshires know nought of how deep goes the blood feud in us of the\nBorderland! Ay, lady, was not mine own grandfather slain by the Musgrave\nof Leit Hill, and did not my father have his revenge on his son by\nSolway Firth? Yea, and now not a Graeme can meet a Musgrave but they\ncome to blows.'\n\n'Nay, but that is not what the good Fathers teach,' Anne interposed.\n\n'The Fathers have neither chick nor child to take up their quarrel. They\nknow nought about blood crying for blood! If King Edward caught that\nbrat of Clifford he would make him know what 'tis to be born of a bloody\nhouse.'\n\nAnne tried to say something, but the lay Sister pushed her along.\n'There, there, go you down--you know nothing about what honour requires\nof you! You are but a south country maid, and have no notion of what is\ndue to them one came from.'\n\nJoan Graeme was only a lay Sister, her father a small farmer when not a\nmoss trooper; but all the Border, on both sides, had the strongest\nideas of persistent vendetta, such as happily had never been held in the\nmidland and southern counties, where there was less infusion of Celtic\nblood. Anne was a good deal shocked at the doctrine propounded by the\nattendant Sister, a mild, good-natured woman in daily life, but the\nconversation confirmed her suspicions, and put her on her guard as she\nremembered Hob's warning. She had liked the shepherd lad far too much,\nand was far too grateful to him, to utter a word that might give him up\nto the revengers of blood.\n\nAt the foot of the stone stairs that led into the quadrangle she met the\nblack-robed, heavily hooded Sister Scholastica on her way to the chapel.\nThe old nun held out her arms. 'Safely returned, my child! God be\nthanked! Art thou come to join thy thanksgiving with ours at this hour\nof nones?'\n\n'Nay, I am bound to break my fast with the Mother and Master Bertram.'\n\n'Ah! thou must needs be hungered! It is well! But do but utter thy\nthanks to Him Who kept thee safe from the storm and from foul doers.'\n\nAnne did not break away from the good Sister, but went as far as the\nchapel porch, was touched with holy water, and bending her knee, uttered\nin a low voice her 'Gratias ago,' then hastened across the court to the\nrefectory, where the Prioress received her with a laugh and, 'So Sister\nScholastica laid hands on thee; I thought I should have to come and\nrescue thee ere the grouse grew cold.'\n\nBertram, as a courteous squire of dames, came forward bowing low, and\nthe party were soon seated at the board--literally a board, supported\nupon trestles, only large enough to receive the Prioress, the squire and\nthe recovered girl, but daintily veiled in delicate white napery.\n\nIt was screened off from the rest of the refectory, where the few\nSisters had already had their morning's meal after Holy Communion; and\nfrom it there was a slight barrier, on the other side of which Bertram\nSelby ought to have been, but rules sat very lightly on the Prioress\nSelby. Bertram was of kin to her, and she had no demur as to admitting\nhim to her private table. He was, in fact, a squire of the household\nof the Marquess of Montagu, brother of the Kingmaker and had been\ndespatched with letters to the south. He had made a halt at his cousin's\npriory, had been persuaded to join in flying the new hawks, and then had\nfirst been detained by the snow-storm, and then joined in the quest for\nthe lost Lady Anne St. John.\n\nNo doubt had then arisen that the Nevils were firm in their attachment\nto Edward IV., and, as a consequence, in enmity to the House of\nClifford, and both these scions of Selby had been excited at a rumour\nthat the widow of the Baron who had slain young Edmund of York had\nmarried Sir Lancelot Threlkeld of Threlkeld, and that her eldest son,\nthe heir of the line, might be hidden somewhere on the De Vesci estates.\n\nBertram had already told the Prioress that his men had spied a lad\naccompanying the shepherd who escorted the lady, and who, he thought,\nhad a certain twang of south country speech; and no sooner had he carved\nfor the ladies, according to the courtly duty of an esquire, than the\ninquiry began as to who had found the maiden and where she had been\nlodged. Prioress Agnes, who had already broken her fast, sat meantime\nwith the favourite hawk on her wrist and a large dog beside her, feeding\nthem alternately with the bones of the grouse.\n\n'Come, tell us all, sweet Nan! Where wast thou in that untimely\nsnow-storm? In a cave, starved with cold, eh?'\n\n'I was safe in a cabin with a kind old gammer.'\n\n'Eh! And how cam'st thou there? Wandering thither?'\n\n'Nay, the shepherd heard me call.'\n\n'The shepherd! What, the churl that came with thee?'\n\n'He carried me to the hut.'\n\nAnne was on her guard, though Bertram probed her well. Was there only\none shepherd? Was there not a boy with her on the hill-side where\nBertram met her? The shepherd lad in sooth! What became of him? The\nshepherd sent him back, he had been too long away from his flock. What\nwas his name? What was the shepherd's name? Who was his master? Anne did\nnot know--she had heard no names save Hob and Hal, she had seen no arms,\nshe had heard nothing southland. The lad was a mere herd-boy, ordered\nout to milk ewes and tend the sheep. She answered briefly, and with a\ncertain sullenness, and young Selby at last turned on her. 'Look thee\nhere, fair lady, there's a saying abroad that the heir of the red-handed\nHouse of Clifford is lurking here, on the look-out to favour Queen\nMargaret and her son. Couldst thou put us on the scent, King Edward\nwould favour thee and make thee a great dame, and have thee to his\nCourt--nay, maybe give thee what is left of the barony of Clifford.'\n\n'I know nothing of young lords,' sulkily growled Anne, who had been\nhitherto busy with her pets, striking her hand on the table.\n\n'And I tell thee, Bertram Selby,' exclaimed the Prioress, 'that if thou\nart ware of a poor fatherless lad lurking in hiding in these parts, it\nis not the part of an honest man to seek him out for his destruction,\nand still less to try to make the maid he rescued betray him. Well done,\nlittle Anne, thou knowest how to hold thy tongue.'\n\n'Reverend Mother,' expostulated Bertram, 'if you knew what some would\ngive to be on the scent of the wolf-cub!'\n\n'I know not, nor do I wish to know, for what price a Selby would sell\nhis honour and his bowels of mercy,' said Mother Agnes. 'Come away, Nan;\nthou hast done well.'\n\nBertram muttered something about having thought her a better Yorkist,\nwomen not understanding, and mischief that might be brewing; but\nthe Prioress, taking Anne by the hand, went her way, leaving Bertram\nstanding confused.\n\n'Oh, mother,' sighed Anne, 'do you think he will go after him? He will\nthink I was treacherous!'\n\n'I doubt me whether he will dare,' said the Prioress. 'Moreover, it is\ntoo late in the day for a search, and another snow-shower seems coming\nup again. I cannot turn the youth, my kinsman, from my door, and he is\nsafer here than on his quest, but he shall see no more of thee or me\nto-night. I may hold that Edward of March has the right, but that does\nnot mean hunting down an orphan child.'\n\n'Mother, mother, you are good indeed!' cried Anne, almost weeping for\njoy.\n\nBertram, though hurt and offended, was obliged by advance of evening to\nremain all night in the hospitium, with only the chaplain to bear him\ncompany, and it was reported that though he rode past Blackpool, no\ntrace of shepherd or hovel was found.\n\n\n\n\nCHAPTER V. -- MOTHER AND SON\n\n\n\n     My own, my own, thy fellow-guest\n     I may not be, but rest thee, rest--\n     The lowly shepherd's life is best.\n     --WORDSWORTH.\n\n\nThe Lady Threlkeld stood in the lower storey of her castle, a sort of\nrough-built hall or crypt, with a stone stair leading upward to the\nreal castle hall above, while this served as a place where she met her\nhusband's retainers and the poor around, and administered to their wants\nwith her own hands, assisted by the maidens of her household.\n\nAmong the various hungry and diseased there limped in a sturdy\nbeggar with a wallet on his back, and a broad shady hat, as though on\npilgrimage. He was evidently a stranger among the rest, and had his leg\nand foot bound up, leaning heavily on a stout staff.\n\n'Italy pilgrim, what ails thee?' demanded the lady, as he approached\nher.\n\n'Alack, noble dame! we poor pilgrims must ever be moving on, however\nmuch it irks foot and limb, over these northern stones,' he answered,\nand his accent and tone were such that a thrill seemed to pass over the\nlady's whole person, but she controlled it, and only said, 'Tarry till\nthese have received their alms, then will I see to thee and thy maimed\nfoot. Give him a stool, Alice, while he waits.'\n\nThe various patients who claimed the lady's assistance were attended\nto, those who needed food were relieved, and in due time the hall was\ncleared, excepting of the lady, an old female servant, and Hob, who\nhad sat all the time with his foot on a stool, and his back against\nthe wall, more than half asleep after the toils and long journey of the\nnight.\n\nThen the Lady Threlkeld came to him, and making him a sign not to rise,\nsaid aloud, 'Good Gaffer, let me see what ails thy leg.' Then kneeling\ndown and busying herself with the bandages, she looked up piteously in\nhis face, with the partly breathed inquiry, 'My son?'\n\n'Well, my lady, and grown into a stalwart lad,' was Hob's answer, with\nan eye on the door, and in a voice as low as his gruff tones would\npermit.\n\n'And wherefore? What is it?' she asked anxiously. 'Be they on the track\nof my poor boy?'\n\n'They may be,' answered Hob, 'wherefore I deemed it well to shift our\nquarters. As hap would have it, the lad fell upon a little wench lost in\nthe mosses, and there was nothing for it but to bring her home for the\nnight. I would have had her away as soon as day dawned, and no questions\nasked, but the witches, or the foul fiend himself, must needs bring up a\nsnow-storm, and there was nothing for it but to let her bide in the cot\nall day, giving tongue as none but womenfolk can do; and behold she is\nthe child of the Lord St. John of Bletso.'\n\n'Nay, what should bring her north?'\n\n'She wonnes at Greystone with the wild Prioress Selby, who lost her out\nhawking. Her father is a black Yorkist. I saw him up to his stirrups in\nblood at St. Albans!'\n\n'But sure my boy did not make himself known to her?' exclaimed the lady.\n\n'I trow not. He has been well warned, and is a lad of his word; but the\ntwo bairns, left to themselves, could scarce help finding out that each\nwas of gentle blood and breeding, and how much more my goodwife cannot\ntell. I took the maid back so soon as it was safe yester morn, and sent\nback my young lord, much against his will, half-way to Greystone. And\nwell was it I did so, for he was scarce over the ridge when a plump of\nspears came in sight on the search for him, and led by the young squire\nof Selby.'\n\n'Ah! and if the damsel does but talk, even if she knows nought, the foe\nwill draw their conclusions!' said the lady, clasping her hands. 'Oh,\nwould that I had sent him abroad with his little brothers!'\n\n'Nay, then might he have fallen into the hands of Bletso himself, and\nthey say Burgundy is all for the Yorkists now,' said Hob. 'This is what\nI have done, gracious lady. I bade my good woman carry off all she could\nfrom the homestead and burn the rest; and for him we wot on, I sent\nhim and his flock off westward, appointing each of them the same\ntrysting-place--on the <DW72> beneath Derwent Hill, my lady--whence I\nthought, if it were your will and the good knight Sir Lancelot's, we\nmight go nigher to the sea and the firth, where the Selby clan have no\ncall, being at deadly feud with the Ridleys. So if the maiden's tongue\ngoes fast, and the Prioress follows up the quest with young Selby, they\nwill find nought for their pains.'\n\n'Thou art a good guardian, Hob! Ah! where would my boy be save for thee?\nAnd thou sayest he is even now at the very border of the forest ground!\nSure, there can be no cause that I should not go and see him. My heart\nhungers for my children. Oh, let me go with thee!'\n\n'Sir Lancelot--' began Hob.\n\n'He is away at the Warden's summons. He will scarce be back for a week\nor more. I will, I must go with thee, good Hob.'\n\n'Not in your own person, good madam,' stipulated Hob. 'As thou knowest,\nthere are those in Sir Lancelot's following who might be too apt to\nreport of secret visits, and that were as ill as the Priory folk.'\n\nIt was then decided that the lady should put on the disguise of a\ncountrywoman bringing eggs and meat to sell at the castle, and meet Hob\nnear the postern, whence a path led to Penrith.\n\nHob, having received a lump of oatcake and a draught of very small ale,\nlimped out of the court, and, so soon as he could find a convenient spot\nbehind the gorse bushes, divested himself of his bandages, and\nchanged the side of his shepherd's plaid to one much older and more\nweather-beaten; also his pilgrim's hat for one in his pouch--a blue\nbonnet, more like the national Scottish head-gear, hiding the hat in the\ngorse.\n\nThen he lay down and waited, where he could see a window, whence a red\nkerchief was to be fluttered to show when the lady would be ready for\nhim to attend her. He waited long, for she had first to disarm suspicion\nby presiding at the general meal of the household, and showing no undue\nhaste.\n\nAt last, though not till after he had more than once fallen asleep and\nfeared that he had missed the signal, or that his wife and 'Hal' might\nbe tempted to some imprudence while waiting, he beheld the kerchief\nwaving in the sunset light of the afternoon, and presently, shrouded in\nsuch a black and white shepherd's maud as his own, and in a russet gown\nwith a basket on her arm, his lady came forth and joined him.\n\nHis first thought was how would she return again, when the darkness was\nbegun, but her only answer was, 'Heed not that! My child, I must see.'\n\nIndeed, she was almost too breathless and eager with haste, as he guided\nher over the rough and difficult path, or rather track, to answer his\ninquiries as to what was to be done next. Her view, however, agreed with\nhis, that they must lurk in the borders of the woodland for a day or two\ntill Sir Lancelot's return, when he would direct them to a place where\nhe could put them under the protection of one of the tenants of his\nmanor. It was a long walk, longer than Hob had perhaps felt when he had\nundertaken to conduct the lady through it, for ladies, though inured to\nmany dangers in those days, were unaccustomed to travelling on their own\nfeet; but the mother's heart seemed to heed no obstacle, though moments\ncame when she had to lean heavily on her companion, and he even had to\nlift her over brooks or pools; but happily the sun had not set when they\nmade their way through the tangles of the wood, and at last saw before\nthem the fitful glow of a fire of dead leaves, branches and twigs, while\nthe bark of a dog greeted the rustling, they made.\n\n'Sweetheart, my faithful!' then shouted Hob, and in another moment there\nwas a cry, 'Ha! Halloa! Master Hob--beest there?'\n\n'His voice!--my son's!' gasped the lady, and sank for a moment of\noverwhelming joy against the faithful retainer, while the shaggy dog\nleapt upon them both.\n\n'Ay, lad, here--and some one else.'\n\nThe boy crashed through the underwood, and stood on the path in a\nmoment's hesitation. Mother and son were face to face!\n\nThe years that had passed had changed the lad from almost a babe into a\nwell-grown strong boy but the mother was little altered, and as she held\nout her arms no word was wasted ere he sprang into them, and his face\nwas hidden on her neck as when he knew his way into her embrace of old!\n\nWhen the intense rapturous hold was loosed they were aware of Goodwife\nDolly looking on with clasped hands and streaming eyes, giving thanks\nfor the meeting of her dear lady and the charge whom she and her husband\nhad so faithfully kept.\n\nWhen the mother and son had leisure to look round, and there was a\npleased survey of the boy's height and strength, Goodwife Dolly came\nforward to beg the lady to come to her fire, and rest under the gipsy\ntent which she and nephew Piers--her _real_ herd-boy, a rough, shaggy,\nalmost dumb and imbecile lad--had raised with branches, skins and\ncanvas, to protect their few articles of property. There was a\nsmouldering fire, over which Doll had prepared a rabbit which the dog\nhad caught, and which she had intended for Hal's supper and that of her\nhusband if he came home in time. While the lady lavished thanks upon her\nfor all she had done for the boy she was intent on improving the rude\nmeal, so as to strengthen her mistress after her long walk, and for the\nreturn. The lady, however, could see and think of nothing but her son,\nwhile he returned her tearful gaze with open eyes, gathering up his old\nrecollections of her.\n\n'Mother!' he said--with a half-wondering tone, as the recollections of\nsix years old came back to him more fully, and then he nestled again in\nher arms as if she were far more real to him than at first--'Mother!'\nAnd then, as she sobbed over him, 'The little one?'\n\n'The babe is well, when last I heard of her, in a convent at York. Thou\nrememberest her?'\n\n'Ay--my little sister! Ay,' he said, with a considering interrogative\nsound, 'I mind her well, and old Bunce too, that taught me to ride.'\n\nBut Hob interrupted the reminiscences by bringing up the pony on which\nAnne had ridden, and insisting that the lady should not tarry longer.\n'He,' indicating Hal, might walk beside her through the wood, and thus\nprolong their interview, but, as she well knew, it was entirely unsafe\nto remain any longer away from the castle.\n\nThere were embraces and sobbing thanks exchanged between the lady and\nher son's old nurse, and then Hal, at a growling hint from Hob, came\nforward, and awkwardly helped her to her saddle. He walked by her side\nthrough the wood, holding her rein, while Hob, going before, did his\nbest in the twilight to clear away the tangled branches and brambles\nthat fell across the path, and were near of striking the lady across the\nface as she rode.\n\nOn the way she talked to her son about his remembrances, anxious to\nknow how far his dim recollections went of the old paternal castle in\nBedfordshire, of his infant sister and brother, and his father. Of him\nhe had little recollection, only of being lifted in his arms, kissed\nand blessed, and seeing him ride away with his troop, clanking in their\narmour. After that he remembered nothing, save the being put into a\nhomelier dress, and travelling on Nurse Dolly's lap in a wain, up and\ndown, it seemed to him, for ever, till at last clearer recollections\nawoke in him, and he knew himself as Hal the shepherd's boy, with the\nsheep around him, and the blue starry sky above him.\n\n'Dost thou remember what thou wast called in those times?' asked his\nmother.\n\n'I was always Hal. The little one was Meg,' he said.\n\n'Even so, my boy, my dear boy! But knowst thou no more than this?'\n\n'Methinks, methinks there were serving-men that called me the young\nLord. Ay, so! But nurse said I must forget all that. Mother dear,\nwhen that maiden came and talked of tilts and lances, meseemed that I\nrecollected somewhat. Was then my father a knight?'\n\n'Alack! alack! my child, that thou shouldst not know!'\n\n'Memories came back with that maiden's voice and thine,' said Hal, in a\nbewildered tone. 'My father! Was he then slain when he rode farther?'\n\n'Ah! I may tell thee now thou art old enough to guard thyself,' she\nsaid. 'Thy father, whom our blessed Lord assoilzie, was the Lord\nClifford, slain by savage hands on Towton field for his faith to King\nHarry! Thou, my poor boy, art the Baron of Clifford, though while this\ncruel House of York be in power thou must keep in hiding from them in\nthis mean disguise. Woe worth the day!'\n\n'And am I then a baron--a lord?' said the boy. 'Great lords have books.\nWere there not some big ones on the hall window seats? Did not Brother\nEldred begin to teach me my letters? I would that I could go on to learn\nmore!'\n\n'Oh, I would that thou couldst have all knightly training, and learn to\nuse sword and lance like thy gallant father!'\n\n'Nay, but I saw a poor man fall off his horse and lie hurt, I do not\nwant those hard, cruel ways. And my father was slain. Must a lord go to\nbattle?'\n\n'Boy, boy, thou wilt not belie thy Clifford blood,' cried the lady in\nconsternation, which was increased when he said, 'I have no mind to go\nout and kill folks or be killed. I had rather mark the stars and tend my\nsheep.'\n\n'Alack! alack! This comes of keeping company with the sheep. That my\nson, and my lord's son, should be infected with their sheepish nature!'\n\n'Never fear, madam,' said Hob. 'When occasion comes, and strength is\ngrown, his blood will show itself.'\n\n'If I could only give him knightly breeding!' sighed the lady. 'Sir\nLancelot may find the way. I cannot see him grow up a mere shepherd\nboy.'\n\n'Content you, madam,' said Hob. 'Never did I see a shepherd boy with the\nwisdom and the thought there is in that curly pate!'\n\n'Wisdom! thought!' muttered the lady. 'Those did not save our good King,\nonly made him a saint. I had rather hear the boy talk of sword and lance\nthan prate of books and stars! And that wench, whom to our misfortune\nthou didst find! What didst tell her?'\n\n'I told her nought, mother, for I had nought to tell.'\n\n'She scented mystery, though,' said Hob. 'She saw he was no herd boy.'\n\n'Nay? Though he holds himself like a lout untrained! Would that I could\nhave thee in hand, my son, to make thee meet to tread in thy brave\nfather's steps! But now, comrade of sheep thou art, and I fear me thou\nwilt ever be! But that maid, I trust that she perceived nothing in thy\nbearing or speech?'\n\n'She will not betray whatever she perceived,' said Hal stoutly.\n\nThe wood was by this time nearly past, and the moment of parting had\ncome. The lady had decided on going on foot to the little grey stone\nchurch whose low square tower could be seen rising like another rock.\nThither she could repair in her plaid, and by-and-by throw it off, and\nreturn in her own character to the castle, as though she had gone forth\nto worship there. When lifted off the shaggy pony she threw her arms\nround Hal, kissed him passionately, and bade him never breathe a word\nof it, but never to forget that a baron he was, and bound to be a good\nbrave knight, fit to avenge his father's death!\n\nHal came to understand from Dolly's explanations that his recent\nabode had been on the estate of his grandfather, Baron de Vesci, at\nLondesborough, but his mother had since married Sir Lancelot Threlkeld,\nand had intimated that her boy should be removed thither as soon as\nmight be expedient, and therefore the house on the Yorkshire moor had\nbeen broken up.\n\n\n\n\nCHAPTER VI. -- A CAUTIOUS STEPFATHER\n\n\n\n     Thou tree of covert and of rest\n     For this young bird that was distrest.\n     --WORDSWORTH.\n\n\nA baron--bound to be a good knight, and to avenge my father's death!\nWhat does it all mean?' murmured Hal to himself as he lay on his back in\nthe morning sunshine, on the hill-side, the wood behind him, and before\nhim a distance of undulating ground, ending in the straight mysterious\nblue-grey line that Hob Hogward had told him was the sea.\n\n'Baron! Lord Clifford, like my father! He was a man in steel armour; I\nremember how it rang, and how his gorget--yes, that was the thing round\nhis throat--how it hurt me when he lifted me up to kiss me, and how they\nblamed me for crying out. Ay, and he lived in a castle with dark, dull,\nnarrow chambers, all save the hall, where there was ever a tramping and\na clamouring, and smells of hot burning meat, and horses, and all sorts\nof things, and they sat and sat over their meat and wine, and drank\nhealth to King Harry and the Red Rose. I mind now how they shouted and\nroared, and how I wanted to go and hide on the stairs, and my father\nwould have me shout with them, and drink confusion to York out of his\ncup, and shook me and cuffed me when I cried. Oh! must one be like that\nto be a knight? I had rather live on these free green hills with the\nclear blue sky above me, and my good old ewe for my comrade'--and he\nfell to caressing the face of an old sheep which had come up to him,\na white, mountain-bleached sheep with fine and delicate limbs. 'Yes,\nI love thee, good, gentle, little ewe, and thee, faithful Watch,' as\na young collie pressed up to him, thrusting a long nose into his hand,\n'far better than those great baying hounds, or the fierce-eyed hawks\nthat only want to kill. If I be a baron, must it be in that sort?\nAvenge! avenge! what does that mean? Is it, as in Goodwife Dolly's\nballads, going forth to kill? Why should I? I had rather let them be!\nHark! Yea, Watch,' as the dog pricked his ears and raised his graceful\nhead, then sprang up and uttered a deep-mouthed bark. The sheep darted\naway to her companions, and Hal rose to his feet, as the dog began to\nwave his tail, and Hob came forward accompanied by a tall, grave-looking\ngentleman. 'Here he be, sir. Hal, come thou and ask the blessing of thy\nknightly stepfather.'\n\nHal obeyed the summons, and coming forward put a knee to the ground,\nwhile Sir Lancelot Threlkeld uttered the conventional blessing,\nadding, 'Fair son, I am glad to see thee. Would that we might be better\nacquainted, but I fear it is not safe for thee to come and be trained\nfor knighthood in my poor house. Thou art a well grown lad, I rejoice to\nsee, and strong and hearty I have no doubt.'\n\n'Ay, sir, he is strong enow, I wis; we have done our best for him,'\nresponded Hob, while Hal stood shy and shamefaced; but there was\nsomething about his bearing that made Sir Lancelot observe, 'Ay, ay, he\nshows what he comes of more than his mother made me fear. Only thou must\nnot slouch, my fair son. Raise thy head more. Put thy shoulders back.\nSo! so! Nay.'\n\nPoor Hal tried to obey, the colour mounting in his face, but he\nonly became more and more stiff when he tried to be upright, and his\nexpression was such that Sir Lancelot cried out, 'Put not on the visage\nof one of thine own sheep! Ah! how shalt thou be trained to be a worthy\nknight? I cannot take thee to mine house, for I have men there who might\ninform King Edward that thy mother harboured thee. And unless I could\nfirst make interest with Montagu or Salisbury, that would be thy death,\nif not mine.'\n\nThe boy had nothing to say to this, and stood shy by, while his\nstepfather explained his designs to Hal. It was needful to remove the\nyoung Baron as far as possible from the suspicion of the greater part\nof Sir Lancelot Threlkeld's household, and the present resting-place,\nwithin a walk of his castle, was therefore unsafe; besides that,\nfreebooters might be another danger, so near the outskirts of the wood,\nsince the northern districts of moor and wood were by no means clear of\nthe remnants of the contending armies, people who were generally of the\nparty opposite to that which they intended to rob.\n\nBut on the banks of the Derwent, not far from its fall into the sea, Sir\nLancelot had granted a tenure to an old retainer of the De Vescis,\nwho had followed his mistress in her misfortunes; and on his lands Hob\nHogward might be established as a guardian of the herds with his family,\nwhich would excite no suspicion. Moreover, he could train the young\nBaron in martial exercises, the only other way of fitting him for his\nstation unless he could be sent to France or Burgundy like his brother;\nbut besides that the journey was a difficulty, it was always uncertain\nwhether there would be revengeful exiles of one or other side in the\nservice of their King, who might wreak the wrongs of their party on\nClifford's eldest son. There was reported to be a hermit on the coast,\nwho, if he was a scholar, might teach the young gentleman. To Sir\nLancelot's surprise, his stepson's face lighted up more at this\nsuggestion than at that of being trained in arms.\n\nHob had done nothing in that way, not even begun to teach him the\nquarterstaff, though he avouched that when there was cause the young\nlord was no craven, no more than any Clifford ever was--witness when he\ndrove off the great hound, which some said was a wolf, when it fell upon\nthe flock, or when none could hold him from climbing down the Giant's\nCliff after the lamb that had fallen. No fear but he had heart enough to\nmake his hand keep his own or other folks' heads.\n\n'That is well,' said Sir Lancelot, looking at the lad, who stood\ntwisting his hands in the speechless silence induced by being the\nsubject of discussion; 'but it would be better, as my lady saith, if he\ncould only learn not to bear himself so like a clown.'\n\nHowever, there was no more time, for Simon Bunce, the old man-at-arms\nwhom Sir Lancelot had appointed to meet him there, came in sight through\nthe trees, riding an old grey war-horse, much resembling himself in the\nbattered and yet strong and effective air of both. Springing down, the\nold man bent very low before the young Baron, raising his cap as he gave\nthanks to Heaven for permitting him to see his master's son. Then, after\nobeisance to his present master, he and Hob eagerly shook hands as old\ncomrades and fellow-soldiers who had thought never to meet again.\n\nThen turning again to the young noble, he poured out his love, devotion\nand gratitude for being able to serve his beloved lord's noble son;\nwhile poor Hal stood under the discomfort of being surrounded with\nfriends who knew exactly what to say and do to him, their superior,\nwhile he himself was entirely at a loss how to show himself gracious or\ngrateful as he knew he ought to do. It was a relief when Sir Lancelot\nsaid 'Enough, good Simon! Forget his nobility for the present while he\ngoes with thee to Derwentside as herd boy to Halbert Halstead here; only\nthou must forget both their names, and know them only as Hal and Hob.'\n\nWith a gesture of obedience, Simon listened to the further directions,\nand how he was to explain that these south country folks had been sent\nup in charge of an especial flock of my lady's which she wished to have\non the comparatively sheltered valley of the Derwent. Perhaps further\ndirections as to the training of the young Baron were added later, but\nHal did not hear them. He was glad to be dismissed to find Piers and\ngather the sheep together in preparation for the journey to their new\nquarters. Yet he did not fail to hear the sigh with which his stepfather\nnoted that his parting salutation was far too much in the character of\nthe herd boy.\n\n\n\n\nCHAPTER VII. -- ON DERWENT BANKS\n\n\n\n     When under cloud of fear he lay\n     A shepherd clad in homely grey.\n     --WORDSWORTH.\n\n\nSimon Bunce came himself to conduct his new tenants to their abode. It\nwas a pleasant spot, a ravine, down which the clear stream rushed on\nits course to mingle its waters with those of the ocean. The rocks and\nbrushwood veiled the approach to an open glade where stood a rude stone\nhovel, rough enough, but possessing two rooms, a hearth and a chimney,\nand thus superior to the hut that had been left on the moor. There were\nsheds for the cattle around, and the grass was fresh and green so that\nthe sheep, the goat and the cow began eagerly feeding, as did the pony\nwhich Hal and Piers were unloading.\n\nOn one side stretched the open moor rising into the purple hills, just\ntouched with snow. On the other was the wooded valley of the Derwent,\ngrowing wider ever before it debouched amid rocks into the sea. The\ngoodwife at once discovered that there had been recent habitation, and\nasked what had become of the former dwellers there.\n\n'The woman fretted for company,' said Simon, 'and vowed she was in fear\nof the Scots, so I even let her have her way and go down to the town.'\n\nThe town in north country parlance only meant a small village, and Hob\nasked where it lay.\n\nIt was near the junction of the two streams, where Simon lived himself\nin a slightly fortified farmhouse, just high up enough to be fairly safe\nfrom flood tides. He did not advise his newly arrived tenants to be much\nseen at this place, where there were people who might talk. They were\nalmost able to provide for their daily needs themselves, excepting for\nmeal and for ale, and he would himself see to this being supplied from\na more distant farm on the coast, which Hob and Piers might visit from\ntime to time with the pony.\n\nGoodwife Dolly inquired whether they might safely go to church, from\nwhich she had been debarred all the time they had been on the move. 'So\nill for both us and the lad,' she said.\n\nSimon looked doubtful. 'If thou canst not save thy soul without,' he\nsaid, 'thou mightst go on some feast day, when there is such a concourse\nof folk that thou mightst not be noticed, and come away at once without\nhalting for idle clavers, as they call them here.'\n\n'That's what the women folk are keen for with their church-going,' said\nHob with a grin.\n\n'Now, husband, thou knowst,' said Dolly, injured, though she was more\nthan aware he spoke with intent to tease her. 'Have I not lived all this\nwhile with none to speak to save thee and the blessed lads, and never\nmurmured.'\n\n'Though thy tongue be sore for want of speech!' laughed Hob, 'thou beest\na good wife, Dolly, and maybe thy faithfulness will tell as much in the\nsaving of thy soul as going to church.'\n\n'Nay, but,' said Hal with eagerness, 'is there not a priest?'\n\n'The priest comes of a White Rose house--I trust not him. Ay, goodwife,\nbeware of showing thyself to him. I give him my dues, that he may have\nno occasion against me or Sir Lancelot, but I would not have him pry\ninto knowledge that concerns him not.'\n\n'Did not Sir Lancelot say somewhat of a scholarly hermit who might learn\nme in what I ought to know?' asked the boy.\n\n'Never you fear, sir! Here are Hob Halstead and I, able to train any\nyoung noble in what behoves him most to know.'\n\n'Yea, in arms and sports. They must be learnt I know, but a noble needs\nbooklore too,' said the boy. 'Cannot this same hermit help me? Sir\nLancelot--'\n\nSimon Bunce interrupted sharply. 'Sir Lancelot knows nought of the\nhermit! He is--he is--a holy man.'\n\n'A priest,' broke in Dolly, 'a priest!'\n\n'No such thing, dame, no clerk at all, I tell thee. And ye lads had best\nnot molest him! He is for ever busy with his prayers, and wants none\nnear him.'\n\nHal was disappointed, for his mind was far less set on the exercises of\na young knight than on the desire to acquire knowledge, that study which\nseemed to be thrown away on the unwilling ears of Anne St. John.\n\nHob had been awakened by contact with his lady and her husband, as well\nas with the old comrade, Simon Bunce, to perceive that if there were any\nchance of the young Lord Clifford's recovering his true position he\nmust not be allowed to lounge and slouch about like Piers, and he was\ncontinually calling him to order, making him sit and stand upright, as\nhe had seen the young pages forced to do at the castle, learn how to\nhandle a sword, and use the long stick which was the substitute for a\nlance, and to mount and sit on the old pony as a knight should do, till\npoor Hal had no peace, and was glad to get away upon the moor with Piers\nand the sheep, where there was no one to criticise him, or predict that\nnothing would ever make him do honour to his name if he were proved ten\ntimes a baron.\n\nIt was still worse when Bunce came over, and brought a taller horse, and\nsuch real weapons as he deemed that the young lord might be taught to\nuse, and there were doleful auguries and sharp reproofs, designed in\ncomically respectful phrases, till he was almost beside himself with\nbeing thus tormented, and ready to wish never to hear of being a baron.\n\nHis relief was to wander away upon the moors, watch the lights and\nshadows on the wondrous mountains, or dream on the banks of the river,\nby which he could make his way to the seashore, a place of endless\nwonder and contemplation, as he marvelled why the waters flowed in and\nretreated again, watched the white crests, and the glassy rolls of\nthe waves, felt his mind and aspiration stretched as by something\nillimitable, even as when he looked up to the sky, and saw star beyond\nstar, differing from one another in brightness. There were those white\nbirds too, differing from all the night-jars and plovers he had seen on\nthe moor, floating now over the waves, now up aloft and away, as if they\nwere soaring into the very skies. Oh, would that he could follow them,\nand rise with them to know what were those great grey or white clouds,\nand what was above or below in those blue vastnesses! And whence came\nall those strange things that the water spread at his feet the long,\nbrown, wet streamers, or the delicate red tracery that could be seen in\nthe clear pools, where were sometimes those lumps like raw flesh when\nclosed, but which opened into flowers? Or the things like the snails on\nthe heath, yet not snails, and all the strange creatures that hopped and\ndanced in the water?\n\nWhy would no one explain such things to him? Nay, what a pity everyone\ntreated it as mere childish folly in him to be thus interested! They did\nnot quite dare to beat him for it--that was one use of being a baron.\nIndeed, one day when Simon Bunce struck him sharply and hard over the\nshoulders for dragging home a great piece of sea-weed with numerous\ncurious creatures upon it, Goodwife Dolly rushed out and made such an\noutcry that the esquire was fain to excuse himself by declaring that it\nwas time that my lord should know how to bide a buffet, and answer it.\nHe was ready and glad to meet the stroke in return! 'Come on, sir!'\n\nAnd Hob put a stout headless lance in the boy's hand, while Simon stood\nup straight before him. Hob adjusted the weapon in his inert hand, and\ntold him how and where to strike. But 'It is not in sooth. I don't want\nto hurt Master Simon,' said the child, as they laughed, and yet with\ndispleasure as his blow fell weak and uncertain.\n\n'Is it a mouse's tail?' cried Simon in derision.\n\n'Come, sir, try again,' said Hob. 'Strike as you did when the black bull\ncame down. Why cannot you do the like now, when you are tingling from\nBunce's stroke?'\n\n'Ah! then I thought the bull would fall on Piers,' said Hal.\n\n'Come on, think so now, sir. One blow to do my heart good, and show you\nhave the arm of your forebears.'\n\nThus incited, with Hob calling out to him to take heart of grace, while\nSimon made a feint of trying to beat Mother Dolly, Hal started forward\nand dealt a blow sufficient to make Simon cry out, 'Ha, well struck,\nsir, if you had had a better grip of your lance! I even feel it through\nmy buff coat.'\n\nHe spoke as though it had been a kiss; but oh! and alack! why were these\nrough and dreary exercises all that these guardians--yea, and even Sir\nLancelot and his mother--thought worth his learning, when there was so\nmuch more that awoke his delight and interest? Was it really childish to\nheed these things? Yet even to his young, undeveloped brain it seemed\nas if there must be mysteries in sky and sea, the unravelling of which\nwould make life more worth having than the giving and taking of blows,\nwhich was all they heeded.\n\n\n\n\nCHAPTER VIII. -- THE HERMIT\n\n\n\n     No hermit e'er so welcome crost\n     A child's lone path in woodland lost.\n     --KEBLE.\n\n\nHal had wandered farther than his wont, rather hoping to be out of call\nif Simon arrived to give him a lesson in chivalrous sports. He found\nhimself on the <DW72> of one of the gorges down which smaller streams\nrushed in wet weather to join the Derwent. There was a sound of tinkling\nwater, and leaning forward, Hal saw that a tiny thread of water dropped\nbetween the ferns and the stones. Therewith a low, soft chant in a manly\nvoice, mingling with the drip of the water.\n\nThe words were strange to him&&\n\n\n          Lucis Creator optime,\n          Lucem dierum proferens&&\n\n\nbut they were very sweet, and in leaning forward to look between the\nrowan branches and hear and see more, his foot slipped, and with Watch\nbarking round him, he rolled helplessly down the rock, and found himself\nbefore a tall light-haired man, in a dark dress, who gave a hand to\nraise him, asking kindly, 'Art hurt, my child?'\n\n'Oh, no, sir! Off, off, Watch!' as the dog was about to resent anyone's\ntouching his master. 'Holy sir, thanks, great thanks,' as a long fair\nhand helped him to his feet, and brushed his soiled garment.\n\n'Unhurt, I see,' said that sweet voice. 'Hast thou lost thy way? Good\ndog, thou lovest thy master! Art thou astray?'\n\n'No, sir, thank you, I know my way home.'\n\n'Thou art the boy who lives with the shepherd at Derwentside, on Bunce's\nground?'\n\n'Ay, Hob Hogward's herd boy,' said Hal. 'Oh, sir, are you the holy\nhermit of the Derwent vale?'\n\n'A hermit for the nonce I am,' was the answer, with something of a smile\nresponsive to the eager face.\n\n'Oh, sir, if you be not too holy to look at me or speak to me! If\nyou would help me to some better knowledge--not only of sword and\nsingle-stick!'\n\n'Better knowledge, my child! Of thy God?' said the hermit, a sweet look\nof joy spreading over his face.\n\n'Goodwife Dolly has told me of Him, and taught me my Pater and Credo,\nbut we have lived far off, and she has not been able to go to church\nfor weeks and years. But what I long after is to tell me what means all\nthis--yonder sea, and all the stars up above. And they will call me a\nsimpleton for marking such as these, and only want me to heed how to\nshoot an arrow, or give a stroke hard enough to hurt another. Do such\nrude doings alone, fit for a bull or a ram as meseems, go to the making\nof a knight, fair sir?'\n\n'They go to the knight's keeping of his own, for others whom he ought\nto defend,' said the hermit sadly; 'I would have thee learn and practise\nthem. But for the rest, thou knowest, sure, who made the stars?'\n\n'Oh yes! Nurse Dolly told me. She saw it all in a mystery play long long\nago--when a Hand came out, and put in the stars and sun and moon.'\n\n'Knowest thou whose Hand was figured there, my child?'\n\n'The Hand of God,' said Hal, removing his cap. 'They be sparks to show\nHis glory! But why do some move about among the others--one big one\nmoves from the Bull's face one winter to half-way beyond it. And is the\nmorning star the evening one?'\n\n'Ah! thou shouldst know Ptolemy and the Almagest,' said the hermit\nsmiling, 'to understand the circuits of those wandering stars--Coeli\nenarrant gloriam Dei.'\n\n'That is Latin,' said the boy, startled. 'Are you a priest, sir?'\n\n'No, not I--I am not worthy,' was the answer, 'but in some things I may\naid thee, and I shall be blessed in so doing. Canst say thy prayers?'\n\n'Oh, yes! nurse makes me say them when I lie down and when I get\nup--Credo and Pater. She says the old parson used to teach them our own\ntongue for them, but she has well-nigh forgot. Can you tell me, holy\nman?'\n\n'That will I, with all my heart,' responded the hermit, laying his long\ndelicate hand on Hal's head. 'Blessed be He who has sent thee to me!'\n\nThe boy sat at the hermit's feet, listening with the eagerness of one\nwhose soul and mind had alike been under starvation, and how time went\nneither knew till there was a rustling and a step. Watch sprang up,\nbut in another moment Simon Bunce, cap in hand, stood before the hut,\nbeginning with 'How now, sir?'\n\nThe hermit raised his hand, as if to make a sign, saying, 'Thou seest I\nhave a guest, good friend.'\n\nBunce started back with 'Oh! the young Lord! Sworn to silence, I trust!\nI bade him not meddle with you, sir.'\n\n'It was against his will, I trow,' said the hermit. 'He fell over the\nrock by the waterfall, but since he is here, I will answer for him that\nhe does no hurt by word or deed!'\n\n'Never, holy sir!' eagerly exclaimed Hal. 'Hob Hogward knows that I can\nkeep my mouth shut. And may I come again?'\n\nSimon was shaking his head, but the hermit took on him to say, 'Gladly\nwill I welcome thee, my fair child, whensoever thou canst find thy way\nto the weary old anchoret! Go thy way now! Or hast thou lost it?'\n\n'No, sir; I ken the woodland and can soon be at home,' replied Hal;\nthen, putting a knee to the ground, 'May I have your blessing, holy\nman?'\n\n'Alack, I told thee I am no priest,' said the hermit; 'but for such as I\nam, I bless thee with all my soul, thou fatherless lad,' and he laid\nhis hand on the young lad's wondering brow, then bade him begone, since\nSimon and himself had much to say to one another.\n\nHal summoned Watch, and turned to a path through the wood, leading\ntowards the coast, wondering as he walked how the hermit seemed to know\nhim--him whose presence had been so sedulously concealed. Could it be\nthat so very holy a man had something of the spirit of prophecy?\n\nHe kept his promise of silence, and indeed his guardians were so much\naccustomed to his long wanderings that he encountered no questions, only\none of Hob's growls that he should always steal away whenever there was\na chance of Master Bunce's coming to try to make a man of him.\n\nHowever, Bunce himself arrived shortly after, and informed Hob that\nsince young folks always pried where they were least wanted, and my lord\nhad stumbled incontinently on the anchoret's den, it was the holy man's\nwill that he might come there whenever he chose. A pity and shame\nit was, but it would make him more than ever a mere priestling, ever\nhankering after books and trash!\n\n'Were it not better to ask my lady and Sir Lancelot if they would have\nit so? I could walk over to Threlkeld!'\n\n'No, no, no, on your life not,' exclaimed Simon, striking his staff on\nthe ground in his vehemence. 'Never a word to the Threlkeld or any of\nhis kin! Let well alone! I only wish the lad had never gone a-roaming\nthere! But holy men must not be gainsaid, even if it does make a poor\ncraven scholar out of his father's son.'\n\nAnd thus began a time of great contentment to the Lord Clifford. There\nwere few days on which he did not visit the hermitage. It was a small\nlog hut, but raised with some care, and made weatherproof with moss and\nclay in the crevices, and there was an inner apartment, with a little\noil lamp burning before a rough wooden cross, where Hal, if the hermit\nwere not outside, was certain to find him saying his prayers. Food was\nsupplied by Simon himself, and, since Hal's admission, was often carried\nby him, and the hermit seemed to spend his time either in prayer or in\na gentle dreamy state of meditation, though he always lighted up into\nanimation at the arrival of the boy whom he had made his friend. Hal had\nthought him old at first, on the presumption that all hermits must be\naged, nor was it likely that age should be estimated by one living such\na life, but the light hair, untouched with grey, the smooth cheeks and\nthe graceful figure did not belong to more than a year or two above\nforty. And he had no air of ill health, yet this calm solitary residence\nin the wooded valley seemed to be infinite rest to him.\n\nHal had no knowledge nor experience to make him wonder, and accepted the\ngreat quiet and calm of the hermit as the token of his extreme holiness\nand power of meditation. He himself was always made welcome with Watch\nby his side, and encouraged to talk and ask questions, which the hermit\nanswered with what seemed to the boy the utmost wisdom, but older heads\nwould have seen not to be that of a clever man, but of one who had been\nfairly educated for the time, had had experience of courts and camps,\nand referred all the inquiries and wonderments which were far beyond him\ndirect to Almighty Power.\n\nThe mind of the boy advanced much in this intercourse with the first\ncultivated person he had encountered, and who made a point of actually\nteaching and explaining to him all those mysteries of religion which\npoor old Dolly only blindly accepted and imparted as blindly to her\nnursling. Of actual instruction, nothing was attempted. A little\nportuary, or abbreviated manual of the service, was all that the hermit\npossessed, treasured with his small crucifix in his bosom, and of course\nit was in Latin. The Hours of the Church he knew by heart, and never\nfailed to observe them, training his young pupil in the repetition and\nEnglish meaning of such as occurred during his visits. He also told much\nof the history of the world, as he knew it, and of the Church and the\nsaints, to the eager mind that absorbed everything and reflected on it,\ncoming with fresh questions that would have been too deep and perplexing\nfor his friend if he had not always determined everything with 'Such is\nthe will of God.'\n\nSomewhat to the surprise of Simon Bunce and Hob Hogward, Hal improved\ngreatly, not only in speech but in bearing; he showed no such dislike\nor backwardness in chivalrous exercises as previously; and when once Sir\nLancelot Threlkeld came over to see him, he was absolutely congratulated\non looking so much more like a young knight.\n\n'Ay,' said Bunce, taking all the merit to himself, 'there's nought like\nhaving an old squire trained in the wars in France to show a stripling\nhow to hold a lance.'\n\nHal had been too well tutored to utter a word of him to whom his\nimprovement was really due, not by actual training, but partly by\nunconscious example in dignified grace and courtesy of demeanour, and\npartly by the rather sad assurances that it was well that a man born to\nhis station, if he ever regained it, should be able to defend himself\nand others, and not be a helpless burthen on their hands. Tales of\nthe Seven Champions of Christendom and of King Arthur and his Knights\nlikewise had their share in the moulding of the youthful Lord Clifford.\n\nHis great desire was to learn to read, but it was not encouraged by the\nhermit, nor was there any book available save the portuary, crookedly\nand contractedly written on vellum, so as to be illegible to anyone\nunfamiliar with writing, with Latin, or the service. However, the\nanchoret yielded to his importunity so far as to let him learn the\nalphabet, traced on the door in charcoal, and identify the more sacred\nwords in the book--which, indeed, were all in gold, red and blue.\n\nHe did not advance more than this, for his teacher was apt to go off in\na musing dream of meditation, repeating over and over in low sweet tones\nthe holy phrases, and not always rousing himself when his pupil made\na remark or asked a question. Yet he was always concerned at his own\ninattention when awakened, and would apologise in a tone of humility\nthat always made Hal feel grieved and ashamed of having been\nimportunate. For there was a dignity and gentleness about the hermit\nthat always made the boy feel the contrast with his own roughness and\nuncouthness, and reverence him as something from a holier world.\n\n'Nurse, I do think he is a saint,' one day said Hal.\n\n'Nay, nay, my laddie, saints don't come down from heaven in these days\nof evil.'\n\n'I would thou could see him when one comes upon him at his prayers.\nHis face is like the angel at the cross I saw so long ago in the castle\nchapel.'\n\n'Dost thou remember that chapel? Thou wert a babe when we quitted it.'\n\n'I had well nigh forgotten it, but the good hermit's face brought all\nback again, and the voice of the father when he said the Service.'\n\n'That thou shouldst mind so long! This hermit is no priest, thou sayst?'\n\n'No, he said he was not worthy; but sure all saints were not priests,\nnurse.'\n\n'Nay, it is easy to be more worthy than the Jack Priests I have known.\nThough I would they would let me go to church. But look thee here,\nHal, if he be such a saint as thou sayst, maybe thou couldst get him to\nbestow a blessing on poor Piers, and give him his hearing and voice.'\n\nHal was sure that his own special saint was holy enough for anything,\nand accordingly asked permission of him to bring his silent companion\nfor blessing and healing.\n\nThe mild blue eye lighted for a moment. 'Is the poor child then\nafflicted with the King's Evil?' the hermit asked.\n\n'Nay, he is sound enough in skin and limb. It is that he can neither\nhear nor speak, and if you, holy sir, would lay thine hand on him, and\nsign him with the rood, and pray, mayhap your holiness--'\n\n'Peace, peace,' cried the hermit impetuously, lifting up his hand. 'Dost\nnot know that I am a sinner like unto the rest--nay, a greater sinner,\nin that a burthen was laid on me that I had not the soul to rise to, so\nthat the sin and wickedness of thousands have been caused by my craven\nfaint heart for well nigh two score years? O miserere Domine.'\n\nHe threw himself on the ground with clasped hands, and Hal, standing\nby in awestruck amazement, heard no more save sobs, mingled with the\nsupplications of the fifty-first Psalm.\n\nHe was obliged at last to go away without having been able to recall\nthe attention of his friend from his agony of prayer. With the reticence\nthat had grown upon him, he did not mention at home the full effect of\nhis request, but when he thought it over he was all the more convinced\nthat his friend was a great saint. Had he not always heard that saints\nbelieved themselves great sinners, and went through many penances? And\nwhy did he speak as if he could have cured the King's Evil? He asked\nDolly what it was, and she replied that it was the sickness that only\nthe King's touch could heal.\n\n\n\n\nCHAPTER IX. -- HENRY OF WINDSOR\n\n\n\n     My crown is in my heart, not on my head;\n     Not deck'd with diamonds, and Indian stones,\n     Nor to be seen.  My crown is call'd Content.\n     --SHAKESPEARE.\n\n\nSummer had faded, and an early frost had tinted the fern-leaves with\ngold here and there, and made the hermit wrap himself close in a cloak\nlined with thick brown fur.\n\nSimon, who was accustomed very respectfully to take the command of him,\ninsisted that he should have a fire always burning on a rock close to\nhis door, and that Piers, if not Hal, should always take care that it\nnever went out, smothering it with peat, as every shepherd boy knew how\nto do, so as to keep it alight, or, in case of need, to conceal it with\nturf.\n\nOne afternoon, as Hal lay on the grass, whiling away the time by\nalternately playing with Watch and trying to unravel the mysteries of a\nflower of golden-rod, until the hermit should have finished his prayers\nand be ready to attend to him, Piers came through the wood, evidently\nsent on a message, and made him understand that he was immediately\nwanted at home.\n\nHal turned to take leave of his host, but the hermit's eyes were raised\nin such rapt contemplation as to see nought, and, indeed, it might\nbe matter of doubt whether he had ever perceived the presence of his\nvisitor.\n\nHal directed Piers to arrange the fire, and hurried away, becoming\nconscious as he came in sight of the cottage that there were horses\nstanding before it, and guessing at once that it must be a visit from\nSir Lancelot Threlkeld.\n\nIt was Simon Bunce, however, who, with demonstrations of looking for\nhim, came out to meet him as he emerged from the brushwood, and said\nin a gruff whisper, clutching his shoulder hard, 'Not a word to give a\nclue! Mum! More than your life hangs on it.'\n\nNo more could pass, to explain the clue intended, whether to the\npresence of the young Lord Clifford himself, which was his first\nthought, or to the inhabitant of the hermitage. For Sir Lancelot's\ncheerful voice was exclaiming, 'Here he is, my lady! Here's your son!\nHow now, my young lord? Thou hast learnt to hold up thy head! Ay, and to\nbow in better sort,' as, bending with due grace, Hal paused for a second\nere hurrying forward to kneel before his mother, who raised him in her\narms and kissed him with fervent affection. 'My son! mine own dear\nboy, how art thou grown! Thou hast well nigh a knightly bearing!' she\nexclaimed. 'Master Bunce hath done well by thee.'\n\n'Good blood will out, my lady,' quoth Simon, well pleased at her praise.\n\n'He hath had no training but thine?' said Sir Lancelot, looking full at\nSimon.\n\n'None, Sir Knight, unless it be honest Halstead's here.'\n\n'Methought I heard somewhat of the hermit in the glen,' put in the lady.\n\n'He is a saint!' declared two or three voices, as if this precluded his\nbeing anything more.\n\n'A saint,' repeated the lady. 'Anchorets are always saints. What doth\nhe?'\n\n'Prayeth,' answered Simon. 'Never doth a man come in but he is at his\nprayers. 'Tis always one hour or another!'\n\n'Ay?' said Sir Lancelot, interrogatively. 'Sayest thou so? Is he an old\nman?'\n\nSimon put in his word before Hal could speak: 'Men get so knocked about\nin these wars that there's no guessing their age. I myself should deem\nthat the poor rogue had had some clouts on the head that dazed him and\nmade him fit for nought save saying his prayers.'\n\nHere Sir Lancelot beckoned Simon aside, and walked him away, so as to\nleave the mother and son alone together.\n\nLady Threlkeld questioned closely as to the colour of the eyes and\nhair, and the general appearance of the hermit, and Hal replied, without\nsuspicion, that the eyes were blue, the hair, he thought, of a light\ncolour, the frame tall and slight, graceful though stooping; he had\nthought at first that the hermit must be old, very old, but had since\ncome to a different conclusion. His dress was a plain brown gown like\na countryman's. There was nobody like him, no one whom Hal so loved and\nvenerated, and he could not help, as he stood by his mother, pouring out\nto her all his feeling for the hermit, and the wise patient words that\nnow and then dropped from him, such as 'Patience is the armour and\nconquest of the godly;' or, 'Shall a man complain for the punishment of\nhis sins?' 'Yet,' said Hal, 'what sins could the anchoret have? Never\ndid I know that a man could be so holy here on earth. I deemed that was\nonly for the saints in heaven.'\n\nThe lady kissed the boy and said, 'I trow thou hast enjoyed a great\nhonour, my child.'\n\nBut she did not say what it was, and when her husband summoned her,\nshe joined him to repair to Penrith, where they were keeping an autumn\nretirement at a monastery, and had contrived to leave their escort and\nmake this expedition on their way.\n\nSimon examined Hal closely on what he had said to his mother, sighed\nheavily, and chided him for prating when he had been warned against it,\nbut that was what came of dealing with children and womenfolk.\n\n'What can be the hurt?' asked Hal. 'Sir Lancelot knows well who I am! No\nlack of prudence in him would put men on my track.'\n\n'Hear him!' cried Simon; 'he thinks there is no nobler quarry in the\nwoods than his lordship!'\n\n'The hermit! Oh, Simon, who is he?'\n\nBut Simon began to shout for Hob Hogward, and would not hear any further\nquestions before he rode away, as far as Hal could see, in the opposite\ndirection to the hermitage. But when he repaired thither the next day\nhe was startled by hearing voices and the stamp of horses, and as he\nreconnoitred through the trees he saw half a dozen rough-looking men,\nwith bows and arrows, buff coats, and steel-guarded caps--outlaws and\nrobbers as he believed.\n\nHis first thought was that they meant harm to the gentle hermit, and his\nimpulse was to start forward to his protection or assistance, but as\nhe sprang into sight one of the strangers cried out: 'How now! Here's\na shepherd thrusting himself in. Back, lad, or 'twill be the worse for\nyou.'\n\n'The hermit! the hermit! Do not meddle with him! He's a saint,' shouted\nHal.\n\nBut even as he spoke he became aware of Simon, who called out: 'Hold,\nsir; back, Giles; this is one well nigh in as much need of hiding as him\nyonder. Well come, since you be come, my lord, for we cannot get _him_\nthere away without a message to you, and 'tis well he should be off ere\nthe sleuth-hounds can get on the scent.'\n\n'What! Where! Who?' demanded the bewildered boy, breaking off, as at\nthat moment his friend appeared at the door of the hovel, no longer\nin the brown anchoret's gown but in riding gear, partially defended\nby slight armour, and with a cap on his head, which made him look much\nyounger than he had before done.\n\n'Child, art thou there? It is well; I could scarce have gone without\nbidding thee farewell,' he said in his sweet voice; 'thou, the dear\ncompanion of my loneliness.'\n\n'O sir, sir, and are you going away?'\n\n'Yea, so they will have it! These good fellows are come to guard me.'\n\n'Oh! may I not go with thee?'\n\n'Nay, my fair son. Thou art beneath thy mother's wing, while I am like\none who was hunted as a partridge on the mountains.'\n\n'Whither, oh whither?' gasped Hal.\n\n'That I know not! It is in the breasts of these good men, who are\ncharged by my brave wife to have me in their care.'\n\n'Oh! sir, sir, what shall I do without you? You that have helped me, and\ntaught me, and opened mine eyes to all I need to know.'\n\n'Hush, hush; it is a better master than I could ever be that thou\nneedest. But,' as tokens of impatience manifested themselves among the\nrude escort, 'take thou this,' giving him the little service-book, as he\nknelt to receive it, scarce knowing why. 'One day thou wilt be able to\nread it. Poor child! whose lot it is to be fatherless and landless for\nme and mine, I would I could do more for thee.'\n\n'Oh! you have done all,' sobbed Hal.\n\n'Nay, now, but this be our covenant, my boy! If thou, and if mine own\nson both come to your own, thou wilt be a true and loyal man to him,\neven as thy father was to me, and may God Almighty make it go better\nwith you both.'\n\n'I will, I will! I swear by all that is holy!' gasped Hal Clifford, with\na flash of perception, as he knelt.\n\n'Come, my liege, we have far to go ere night. No time for more parting\nwords and sighs.'\n\nHal scarcely knew more except that the hands were laid on his head, and\nthe voice he had learnt to love so well said: 'The blessing of God\nthe Father be upon thee, thou fatherless boy, and may He reward thee\nsevenfold for what thy father was, who died for his faithfulness to me,\na sinner! Fare thee well, my boy.'\n\nAs the hand that Hal was fervently kissing was withdrawn from him he\nsank upon his face, weeping as one heartbroken. He scarce heard the\nsounds of mounting and the trampling of feet, and when he raised his\nhead he was alone, the woods and rocks were forsaken.\n\nHe sprang up and ran along at his utmost speed on the trampled path,\nbut when he emerged from it he could only see a dark party, containing\na horseman or two, so far on the way that it was hopeless to overtake\nthem.\n\nHe turned back slowly to the deserted hut, and again threw himself on\nthe ground, weeping bitterly. He knew now that his friend and master had\nbeen none other than the fugitive King, Henry of Windsor.\n\n\n\n\nCHAPTER X. -- THE SCHOLAR OF THE MOUNTAINS\n\n\n     Not in proud pomp nor courtly state;\n     Him his own thoughts did elevate,\n     Most happy in the shy recess.\n     --WORDSWORTH.\n\n\nThe departure of King Henry was the closing of the whole intellectual\nand religious world that had been opened to the young Lord Clifford. To\nthe men of his own court, practical men of the world, there were times\nwhen poor Henry seemed almost imbecile, and no doubt his attack of\nmelancholy insanity, the saddest of his ancestral inheritances, had\nshattered his powers of decision and action; but he was one who 'saw far\non holy ground,' and he was a well-read man in human learning, besides\nhaving the ordinary experience of having lived in the outer world, so\nthat in every way his companionship was delightful to a thoughtful boy,\nwakening to the instincts of his race.\n\nTo think of being left to the society of the sheep, of dumb Piers and\nhis peasant parents was dreariness in the extreme to one who had begun\nto know something like conversation, and to have his countless questions\nanswered, or at any rate attended to. Add to this, he had a deep\npersonal love and reverence for his saint, long before the knowing him\nas his persecuted King, and thus his sorrow might well be profound,\nas well as rendered more acute by the terror lest his even unconscious\ndescription to his mother might have been treason!\n\nHe wept till he could weep no longer, and lay on the ground in his\ndespair till darkness was coming on, and Piers came and pulled him up,\nindicating by gestures and uncouth sounds that he must go home. Goodwife\nDolly was anxiously looking out for him.\n\n'Laddie, there thou beest at last! I had begun to fear me whether the\nrobber gang had got a hold of thee. Only Hob said he saw Master Simon\nwith them. Have they mishandled thee, mine own lad nurse's darling? Thou\nlookest quite distraught.'\n\nAll Hal's answer was to hide his head in her lap and weep like a babe,\nthough she could, with all her caresses, elicit nothing from him but\nthat his hermit was gone. No, no, the outlaws had not hurt him, but they\nhad taken him away, and he would never come back.\n\n'Ay, ay, thou didst love him and he was a holy man, no doubt, but one of\nthese days thou shalt have a true knight, and that is better for a young\nbaron to look to than a saint fitter for Heaven than for earth! Come\nnow, stand up and eat thy supper. Don't let Hob come in and find thee\ncrying like a swaddled babe.'\n\nWith which worldly consolations and exhortations Goodwife Dolly brought\nhim to rise and accept his bowl of pottage, though he could not swallow\nmuch, and soon put it aside and sought his bed.\n\nIt was not till late the next day that Simon Bunce was seen riding\nhis rough pony over the moor. Hal repaired to him at once, with the\nbreathless inquiry, 'Where is he?'\n\n'In safe hands! Never you fear, sir! But best know nought.'\n\n'O Simon, was I--? Did I do him any scathe?--I--I never knew--I only\ntold my lady mother it was a saint.'\n\n'Ay, ay, lad, more's the pity that he is more saint than king! If my\nlady guessed aught, she would be loyal as became your father's wife, and\nmethinks she would not press you hard for fear she should be forced to\nbe aware of the truth.'\n\n'But Sir Lancelot?'\n\n'As far as I can gather,' explained Simon, 'Sir Lancelot is one that\nhath kept well with both sides, and so is able to be a protector. But\ndown came orders from York and his crew that King Harry is reported to\nbe lurking in some of these moors, and the Countess Clifford being his\nwife, he fell under suspicion of harbouring him. Nay, there was some\nperilous talk in his own household, so that, as I understand the matter,\nhe saw the need of being able to show that he knew nothing; or, if he\nfound that the King was living within these lands, of sending him a\nwarning ere avowing that he had been there. So I read what was said to\nme.'\n\n'He knew nothing from me! Neither he nor my lady mother,' eagerly said\nHal. 'When I mind me I am sure my mother cut me short when I described\nthe hermit too closely, lest no doubt she should guess who he was.'\n\n'Belike! It would be like my lady, who is a loyal Lancastrian at heart,\nthough much bent on not offending her husband lest his protection should\nbe withdrawn from you.'\n\n'Better--O, a thousand times better!--he gave me up than the King!'\n\n'Hush! What good would that do? A boy like you? Unless they took you\nin hand to make you a traitor, and offered you your lands if you would\nswear allegiance to King Edward, as he calls himself.'\n\n'Never, though I were cut into quarters!' averred Hal, with a fierce\ngesture, clasping his staff. 'But the King? Where and what have they\ndone with him?'\n\n'Best not to know, my lord,' said Simon. 'In sooth, I myself do not know\nwhither he is gone, only that he is with friends.'\n\n'But who--what were they? They looked like outlaws!'\n\n'So they were; many a good fellow is of Robin of Redesdale's train.\nThere are scores of them haunting the fells and woods, all Red Rose men,\nkeeping a watch on the King,' replied Simon. 'We had made up our minds\nthat he had been long enough in one place, and that he must have taken\nshelter the winter through, when I got notice of these notions of Sir\nLancelot, and forthwith sent word to them to have him away before worse\ncame of it.'\n\n'Oh! why did you not let me go with him? I would have saved him, waited\non him, fought for him.'\n\n'Fine fighting--when there's no getting you to handle a lance, except\nas if you wanted to drive a puddock with a reed! Though you have been\nbetter of late, little as your hermit seemed the man to teach you.'\n\n'He said it was right and became a man! Would I were with him! He, my\ntrue King! Let me go to him when you know where, good Simon. I, that am\nhis true and loving liegeman, should be with him.'\n\n'Ay! when you are a man to keep his head and your own.'\n\n'But I could wait on him.'\n\n'Would you have us bested to take care of two instead of one, and my\nlady, moreover, in a pother about her son, and Sir Lancelot stirred to\nmake a hue and cry all the more? No, no, sir, bide in peace in the safe\nhomestead where you are sheltered, and learn to be a man, minding your\nexercises as well as may be till the time shall come.'\n\n'When I shall be a man and a knight, and do deeds of derring-do in his\ncause,' cried Hal.\n\nAnd the stimulus drove him on to continual calls to Hob, in Simon's\ndefault, to jousts with sword or spear, represented generally by staves;\nand when these could not be had, he was making arrows and practising\nwith them, so as to become a terror to the wild ducks and other\nneighbours on the wolds, the great geese and strange birds that came\nin from the sea in the cold weather. When it was not possible to go far\nafield in the frosts and snows, he conned King Henry's portuary, trying\nto identify the written words with those he knew by heart, and sometimes\ntrying to trace the shapes of the letters on the snow with a stick;\nvisiting, too, the mountains and looking into the limpid grey waters of\nthe lakes, striving hard to guess why, when the sea rose in tides, they\nwere still. More than ever, too, did the starry skies fill him with\ncontemplation and wonder, as he dwelt on the scraps alike of astronomy,\nastrology, and devotion which he had gathered from his oracle in the\nhermitage, and longed more and more for the time to return when he\nshould again meet his teacher, his saint, and his King.\n\nAlas! that time was never to come. The outlawed partisans of the\nRed Rose had secret communications which spread intelligence rapidly\nthroughout the country, and long before Sir Lancelot and his lady knew,\nand thus it was that Simon Bunce learnt, through the outlaws, that poor\nKing Henry had been betrayed by treachery, and seized by John Talbot\nat Waddington Hall in Lancashire. Deep were the curses that the outlaws\nuttered, and fierce were the threats against the Talbot if ever he\nshould venture himself on the Cumbrian moors; and still hotter was their\nwrath, more bitter the tears of the shepherd lord, when the further\ntidings were received that the Earl of Warwick had brought the gentle,\nharmless prince, to whom he had repeatedly sworn fealty, into London\nwith his feet tied to the stirrups of a sorry jade, and men crying\nbefore him, 'Behold the traitor!'\n\nThe very certainty that the meek and patient King would bear all with\nrejoicing in the shame and reproach that led him in the steps of his\nMaster, only added to the misery of Hal as he heard the tale; and he lay\non the ground before his hut, grinding his teeth with rage and longing\nto take revenge on Warwick, Edward, Talbot--he knew not whom--and\ngrasping at the rocks as if they were the stones of the Tower which he\nlonged to tear down and liberate his beloved saint.\n\nNor, from that time, was there any slackness in acquiring or practising\nall skill in chivalrous exercises.\n\n\n\n\nCHAPTER XI. -- THE RED ROSE\n\n\n\n     That Edward is escaped from your brother\n     And fled, as he hears since, to Burgundy.\n     --SHAKESPEARE.\n\n\nYears passed on, and still Henry Clifford continued to be the shepherd.\nMatters were still too unsettled, and there were too many Yorkists in\nthe north, keeping up the deadly hatred of the family against that of\nClifford, for it to be safe for him to show himself openly. He was a\ntall, well-made, strong youth, and his stepfather spoke of his going to\nlearn war in Burgundy; but not only was his mother afraid to venture him\nthere, but he could not bear to leave England while there was a hope\nof working in the cause of the captive King, though the Red Rose hung\nwithered on the branches.\n\nReports of misunderstandings between King Edward and the Earl of Warwick\ncame from time to time, and that Queen Margaret and her son were busy\nbeyond seas, which kept up hope; and in the meantime Hal grew in the\nknowledge of all country lore, of herd and wood, and added to it all his\nown earnest love of the out-of-door world, of sun, moon, and stars,\nsea and hills, beast and bird. The hermit King, who had been a\nwell-educated, well-read man in his earlier days, had given him the\nframework of such natural science as had come down to the fifteenth\ncentury, backed by the deepest faith in scriptural descriptions; and\nthese inferences and this philosophy were enough to lead a far acuter\nand more able intellect, with greater opportunities of observation, much\nfurther into the fields of the mystery of nature than ever the King had\ngone.\n\nHe said nothing, for never had he met one who understood a word he said\napart from fortune telling, excepting the royal teacher after whom\nhe longed; but he watched, he observed, and he dreamt, and came to\nconclusions that his King's namesake cousin, Enrique of Portugal, the\ndiscoverer, in his observatory at St. Vincent, might have profited by.\nBrother Brian, a friar, for whose fidelity Simon Bunce's outlaw could\nabsolutely answer, and who was no Friar Tuck, in spite of his rough\nlife, gave Dolly much comfort religiously, carried on some of the\neducation for which Hal longed, and tried to teach him astrology. Some\nof the yearnings of his young soul were thus gratified, but they were\nthe more extended as he grew nearer manhood, and many a day he stood\nwith eyes stretched over the sea to the dim line of the horizon, with\narms spread for a moment as if he would join the flight of the sea-gulls\nfloating far, far away, then clasped over his breast in a sort of\ndespair at being bound to one spot, then pressed the tighter in the\nstrong purpose of fighting for his imprisoned King when the time should\ncome.\n\nFor this he diligently practised with bow and arrow when alone, or only\nwith Piers, and learnt all the feats of arms that Simon Runce or Giles\nSpearman could teach him. Spearman was evidently an accomplished knight\nor esquire; he had fought in France as well as in the home wars, and\nknew all the refinements of warfare in an age when the extreme weight\nof the armour rendered training and skill doubly necessary. Spearman\nwas evidently not his real name, and it was evident that he had some\nknowledge of Hal's real rank, though he never hazarded mention of other\nname or title. The great drawback was the want of horses. The little\nmountain ponies did not adequately represent the warhorses trained\nto charge under an enormous load, and the buff jerkins and steel\nbreast-plates of the outlaws were equally far from showing how to move\nunder 'mail and plates of Milan steel.' Nor would Sir Lancelot Threlkeld\nlend or give what was needful. Indeed, he was more cautious than ever,\nand seemed really alarmed as well as surprised to see how tall and manly\nhis step-son was growing, and how like his father. He would not hear\nof a visit to Threlkeld under any disguise, though Lady Clifford was\nin failing health, nor would he do anything to forward the young lord's\nknightly training. In effect, he only wanted to keep as quiet and\nunobserved as possible, for everything was in a most unsettled and\ndangerous condition, and there was no knowing what course was the safest\nfor one by no means prepared to lose life or lands in any cause.\n\nThe great Earl of Warwick, on whom the fate of England had hitherto\nhinged, was reported to have never forgiven King Edward for his marriage\nwith Dame Elizabeth Grey, and to be meditating insurrection. Encouraged\nby this there was a great rising in Yorkshire of the peasants under\nRobin of Redesdale, and a message was brought to Giles Spearman and his\nfollowers to join them, but he and Brother Brian demurred, and news soon\ncame that the Marquess of Montagu had defeated the rising and beheaded\nRedesdale.\n\nSir Lancelot congratulated his step-son on having been too late to take\nup arms, and maintained that the only safe policy was to do nothing, a\nplan which suited age much better than youth.\n\nHe still lived with Hob and Piers, and slept at the hut, but he went\nfurther and further afield among the hills and mosses, often with no\ncompanion save Watch, so that he might without interruption watch the\nclear streams and wonder what filled their fountains, and why the sea\nwas never full, or stand on the sea-shore studying the tides, and\ntrying to construct a theory about them. King Henry was satisfied with\n'Hitherto shalt thou come and no farther,' but He who gave that decree\nmust have placed some cause or rule in nature thus to affect them. Could\nit be the moon? The waves assuredly obeyed the changes of the moon, and\nHal was striving to keep a record in strokes marked by a stick on soft\nearth or rows of pebbles, so as to establish a rule. 'Aye, aye,' quoth\nHob. 'Poor fellow, he is not much wiser than the hermit. See how he\nplays with pebbles and stones. You'll make nought of him, fine grown lad\nas he is. Why, he'll sit dazed and moonstruck half a day, and all the\nnight, staring up at the stars as if he would count them!'\n\nSo spoke the stout shepherd to Simon Bunce, pointing to the young man,\nwho lay at his length upon the grass calculating the proportions of the\nstones that marked the relations of hours of the flood tide and those\nof the height of the moon. Above and beyond was a sundial cut out in the\nturf, from his own observations after the hints that the hermit and the\nfriar had given him.\n\n'Ha now, my lord, I have rare news for you.'\n\nThe unwonted title did not strike Hal's unaccustomed ears, and he\ncontinued moving his lips, 'High noon, spring tide.'\n\n'There, d'ye see?' said Hob, 'he heeds nothing. 'That I and my goodwife\nshould have bred up a mooncalf! Here, Hal, don't you know Simon? Hear\nhis tidings!'\n\n'Tidings enow! King Henry is freed, King Edward is fled. My Lord\nof Warwick has turned against him for good and all. King Henry is\nproclaimed in all the market-places! I heard it with my own ears at\nPenrith!' And throwing up his cap into the air, while the example\nwas followed by Hob, with 'God save King Henry, and you my Lord of\nClifford.'\n\nThe sound was echoed by a burst of voices, and out of the brake suddenly\nstood the whole band of outlaws, headed by Giles Spearman, but Hal still\nstood like one dazed. 'King Harry, the hermit, free and on his throne,'\nhe murmured, as one in a dream.\n\n'Ay, all things be upset and reversed,' said Spearman, with a hand on\nhis shoulder. 'No herd boy now, but my Lord of Clifford.'\n\n'Come to his kingdom,' repeated Hal. 'My own King Harry the hermit! I\nwould fain go and see him.'\n\n'So you shall, my brave youth, and carry him your homage and mine,'\nsaid Spearman. 'He will know me for poor Giles Musgrave, who upheld\nhis standard in many a bloody field. We will off to Sir Lancelot at\nThrelkeld now! Spite of his policy of holes and corners, he will not now\nrefuse to own you for what you are, aye, and fit you out as becomes a\nknight.'\n\n'God grant he may!' muttered Bunce, 'without his hum and ha, and swaying\nthis way and that, till he never moves at all! Betwixt his caution,\nand this lad's moonstruck ways, you have a fair course before you, Sir\nGiles! See, what's the lad doing now?'\n\nThe lad was putting into his pouch the larger white pebbles that had\nrepresented tens in his calculation, and murmuring the numbers they\nstood for. 'He will understand,' he said almost to himself, but he\nshowed himself ready to go with the party to Threlkeld, merely pausing\nat Hob's cottage to pick up a few needful equipments. In the skin of a\nrabbit, carefully prepared, and next wrapped in a silken kerchief,\nand kept under his chaff pillow, was the hermit's portuary, which was\ncarefully and silently transferred by Hal to his own bosom. Sir Giles\nMusgrave objected to Watch, in city or camp, and Hal was obliged to\nleave him to Goodwife Dolly and to Piers.\n\nWith each it was a piteous parting, for Dolly had been as a mother to\nhim for almost all his boyhood, and had supplied the tenderness that\nhis mother's fears and Sir Lancelot's precautions had prevented his\nreceiving at Threlkeld. He was truly as a son to her, and she sobbed\nover him, declaring that she never would see him again, even if he came\nto his own, which she did not believe was possible, and who would see to\nhis clean shirts?\n\n'Never fear, goodwife,' said Giles Musgrave; 'he shall be looked to as\nmine own son.'\n\n'And what's that to a gentle lad that has always been tended as becomes\nhim?'\n\n'Heed not, mother! Be comforted! I must have gone to the wars, anyway.\nIf so be I thrive, I'll send for thee to mine own castle, to reign there\nas I remember of old. Here now! Comfort Piers as thou only canst do.'\n\nPiers, poor fellow, wept bitterly, only able to understand that\nsomething had befallen his comrade of seven years, which would take him\naway from field and moor. He clung to Hal, and both lads shed tears,\ntill Hob roughly snatched Piers away and threw him to his aunt, with\nthreats that drew indignant, though useless, interference from Hal,\nthough Simon Bunce was muttering, 'As lief take one lad as the other!'\nwhile Dolly's angry defence of her nursling's wisdom broke the sadness\nof the parting.\n\n\n\n\nCHAPTER XII. -- A PRUDENT RECEPTION\n\n\n\n     So doth my heart misgive me in these conflicts,\n     What may befall him to his harm and ours.\n     --SHAKESPEARE.\n\n\nThrough the woods the party went to the fortified house of Threlkeld,\nwhere the gateway was evidently prepared to resist any passing attack,\nby stout gates and a little watch-tower.\n\nSir Giles blew a long blast on his bugle-horn, and had to repeat it\ntwice before a porter looked cautiously out at a wicket opening in the\nheavy door, and demanded 'Who comes?'\n\n'Open, porter, open in the name of King Harry, to the Lords of Clifford\nand of Peelholm.'\n\nThe porter fell back, observing, 'Sir, pardon, while I have speech with\nmy master, Sir Lancelot Threlkeld.'\n\nSome delay and some sounds of conversation were heard, then, on a\nrenewed and impatient blast on Sir Giles's horn, Sir Lancelot Threlkeld\nhimself came to the wicket, and his thin anxious voice might be heard\ndemanding, 'What madness is this?'\n\n'The madness is past, soundness is come,' responded Sir Giles. 'King\nHarry is on his throne, the traitors are fled, and your own fair son\ncomes forth in his proper person to uphold the lawful sovereign; but he\nwould fain first see his lady mother, and take her blessing with him.'\n\n'And by his impatience destroy himself, after all the burthen of care\nand peril he hath been to me all these years,' lamented Sir Lancelot.\n'But come in, fair lad. Open the gates, porter. I give you welcome, Lord\nMusgrave of Peelholm. But who are these?' he added, looking at the troop\nof buff-coated archers in the rear.\n\n'They are bold champions of the Red Rose, returned Sir Giles, 'who\nhave lived with me in the wolds, and now are on the way to maintain our\nKing's quarrel.''\n\nSir Lancelot, however, would not hear of admitting the outlaws. Young\nClifford and the Lord of Peelholm should be welcome, or more truly he\ncould not help receiving them, but the archers must stay outside, their\nentertainment in beef and ale being committed to Bunce and the chief\nwarder, while the two noblemen were conducted to the castle hall. For\nthe first time in his life Clifford was received in his mother's home,\nand accepted openly, as he knelt before her to ask her blessing. A fine,\nactive, handsome youth was he, with bright, keen eyes, close-curled\nblack locks and hardy complexion, telling of his out-of-door life, and\na free use of his limbs, and upright carriage, though still with more\nof the grace of the free mountain than of the training of pagedom and\nsquiredom.\n\nNor could he speak openly and freely to her, not knowing how much he\nmight say of his past intercourse with King Henry, and of her endeavour\nto discover it; and he sat beside her, neither of them greatly at ease,\nat the long table, which, by the array of silver cups, of glasses\nand the tall salt cellar separating the nobility and their followers,\nrecalled to him dim recollections of the scenes of his youth.\n\nHe asked for his sister--he knew his little brother had died in the\nNetherlands--and he heard that she had been in the Priory of St.\nHelen's, and was now in the household of my Lady of Hungerford, who\nhad promised to find a good match for her. There was but one son of the\nunion with the knight of Threlkeld, and him Hal had never seen; nor was\nhe at home, being a page in the household of the Earl of Westmoreland,\naccording to the prevailing fashion of the castles of the great feudal\nnobles becoming schools of arms, courtesy and learning for the young\ngentlemen around. Indeed, Lady Clifford surveyed her eldest son with\na sigh that such breeding was denied him, as she observed one or two\nlittle deficiencies in what would be called his table manners--not very\nimportant, but revealing that he had grown up in the byre instead of\nthe castle, where there was a very strict and punctilious code, which\nfigured in catechisms for the young.\n\nShe longed to keep him, and train him for his station, but in the first\nplace, Sir Lancelot still held that it could not safely be permitted,\nsince he had little confidence in the adherence of the House of Nevil\nto the Red Rose; and moreover Hal himself utterly refused to remain\nconcealed in Cumberland instead of carrying his service to the King he\nloved.\n\nIn fact, when he heard the proposal of leaving him in the north, he\nstood up, and, with far more energy than had been expected from him,\nsaid, 'Go I must, to my lawful King's banner, and my father's cause. To\nKing Harry I carry my homage and whatever my hand can do!'\n\nSuch an expression of energy lighted his hitherto dreamy eyes, that all\nbeholders turned their glances on his face with a look of wonder. Sir\nLancelot again objected that he would be rushing to his ruin.\n\n'Be it so,' replied Hal. 'It is my duty.'\n\n'The time seems to me to be come,' added Musgrave, 'that my young lord\nshould put himself forward, though it may be only in a losing cause. Not\nso much for the sake of success, as to make himself a man and a noble.'\n\n'But what can he do?' persisted Threlkeld; 'he has none of the training\nof a knight. How can you tilt in plate armour, you who have never\nbestridden a charger? These are not the days of Du Guesclin, when a lad\ncame in from the byre and bore down all foes before him.'\n\nThe objection was of force, for the defensive armour of the fifteenth\ncentury had reached a pitch of cumbrousness that required long practice\nfor a man to be capable of moving under it.\n\n'So please you, sir,' said Hal, 'I am not wholly unskilled. The good Sir\nGiles and Simon Bunce have taught me enough to strike a blow with a good\nwill for a good cause.'\n\n'With horse and arms as befits him,' began Musgrave.\n\n'I know not that a horse is here that could be depended on,' began\nThrelkeld. 'Armour too requires to be fitted and proved.'\n\nHe spoke in a hesitating voice that showed his unwillingness, and Hal\nexclaimed, 'My longbow is mine own, and so are my feet. Sir Giles,\nwill you own me as an archer in your troop, where I will strive not to\ndisgrace you or my name?'\n\n'Bravely spoken, young lord,' said Sir Giles heartily; 'right willingly\nwill I be your godfather in chivalry, since you find not one nigher\nhome.'\n\n'So may it best be,' observed his mother, 'since he is bent on going.\nThus his name and rank may be kept back till it be plain whether the\nenmity of my Lords of Warwick and Montagu still remain against our poor\nhouse.'\n\nThere was no desire on either side to object when the Lord Musgrave\nof Peelholm decided on departing early on the morrow. Their host was\nevidently not sorry to speed them on their way, and his reluctant\nhospitality made them anxious to cumber him no longer than needful; and\nhis mind was relieved when it was decided that the heir of the De Vescis\nand Cliffords should be known as Harry of Derwentdale.\n\nOnly, when all was preparation in the morning, and a hearty service had\nbeen said in the chapel, the lady called her son aside, and looking up\ninto his dark eyes, said in a low voice, 'Be not angered with my lord\nhusband's prudence, my son. Remember it is only by caution that he has\nsaved thine head, or mine, or thy sister's!'\n\n'Ay, ay, mother, I know,' he said, more impatiently than perhaps he\nknew.\n\n'It was by the same care that he preserved us all when Edgecotefield was\nfought. Chafe not at him. Thou mayst be thankful even now, mayhap, to\nfind a shelter preserved, while that rogue and robber Nevil holds our\nlands.'\n\n'I am more like to have to protect thee, lady mother, and bring thee to\nthy true home again!' said Hal.\n\n'Meantime, my child, take this purse and equip thyself at York or\nwhenever thou canst. Nay, thou needst not shrug and refuse! How like thy\nfather the gesture, though I would it were more gracious and seemly.\nBut this is mine, mine own, none of my husband's, though he would be\nwilling. It comes from the De Vesci lands, and those will be thine after\nme, and thine if thou winnest not back thy Clifford inheritance. And oh!\nmy son, crave of Sir Giles to teach thee how to demean thyself that they\nmay not say thou art but a churl.'\n\n'I trust to be no churl in heart, if I be in manners,' said Hal, looking\ndown on his small clinging mother.\n\n'Only be cautious, my son. Remember that you are the last of the name,\nand it is your part to bring it to honour.'\n\n'Which I shall scarce do by being cautious,' he said, with something of\na smile. 'That was not my father's way.'\n\n'Ah me! You have his spirit in you, and how did it end?'\n\n'My Lord of Clifford,' said a voice from the court, 'you are waited\nfor!'\n\n'And remember,' cried his mother, with a last embrace, 'there will be\nsafety here whenever thou shalt need it.'\n\n'With God's grace, I am more like to protect you and your husband,' said\nthe lad, bending for another kiss and hurrying away.\n\n\n\n\nCHAPTER XIII. -- FELLOW TRAVELLERS\n\n\n\n     And sickerlie she was of great disport,\n     And full pleasant and amiable of port;\n     Of small hounds had she that she fed\n     With roasted flesh and milk and wastel bread.\n     --CHAUCER.\n\n\nSir Giles Musgrave of Peelholm was an old campaigner, and when Hal came\nout beyond the gate of the Threlkeld fortalice, he found him reviewing\nhis troop; a very disorderly collection, as Sir Lancelot pronounced with\na sneer, looking out on them, and strongly advising his step-son not to\ncast in his lot with them, but to wait and see what would befall, and\nwhether the Nevils were in earnest in their desertion of the House of\nYork.\n\nHal restrained himself with difficulty enough to take a courteous leave\nof his mother's husband, to whose prudence and forbearance he was really\nmuch beholden; though, with his spirit newly raised and burning for his\nKing, it was hard to have patience with neutrality.\n\nHe found Sir Giles employed in examining his followers, and rigidly\nsending home all not properly equipped with bow, sheaf of arrows, strong\nknife or pike, buff coat, head-piece and stout shoes; also a wallet of\nprovisions for three days, or a certain amount of coin. He would have\nno marauding on the way, and refused to take any mere lawless camp\nfollower, thus disposing of a good many disreputable-looking fellows who\nhad flocked in his wake. Sir Lancelot's steward seconded him heartily\nby hunting back his master's retainers; and there remained only about\nfive-and-twenty--mostly, in fact, yeomen or their sons--men who had\nbeen in arms for Queen Margaret and had never made their submission,\nbut lived on unmolested in the hills, really outlawed, but not coming in\ncollision with the authorities enough to have their condition inquired\ninto. They had sometimes attacked Yorkist parties, sometimes resisted\nScottish raids, or even made a foray in return, and they were well used\nto arms. These all had full equipments, and some more coin in their\npouches than they cared to avow. Three or four of them brought an ox,\ncalf or sheep, or a rough pony loaded with provisions, and driven by a\nherd boy or a son eager to see life and 'the wars.' Simon Bunce, well\narmed, was of this party. Hob Hogward, though he had come to see what\nbecame of his young lord, was pronounced too stiff and aged to join the\nband, which might now really be called a troop, not a mere lawless\ncrowd of rough lads. There were three trained men-at-arms, the regular\nretainers of Sir Giles, who held a little peel tower on the borders\nwhere nobody durst molest him, and these marshalled the little band in\nfair order.\n\nIt was no season for roses, but a feather was also the cognisance of\nHenry VI., and every one's barret-cap mounted a feather, generally\nborrowed from the goodwife's poultry yard at home, but sometimes picked\nup on the moors, and showing the barred black and brown patterns of the\nhawk's or the owl's plumage. It was a heron's feather that Hal assumed,\non the counsel of Sir Giles, who told him it was an old badge of the\nCliffords, and it became well his bright dark hair and brown face.\n\nOn they went, a new and wonderful march to Hal, who had only looked with\ninfant eyes on anything beyond the fells, and had very rarely been into\na little moorland church, or seen enough people together for a market\nday in Penrith. Sir Giles directed their course along the sides of the\nhills till he should gain further intelligence, and know how they would\nbe received. For the most part the people were well inclined to King\nHenry, though unwilling to stir on his behalf in fear of Edward's\ncruelty.\n\nHowever, it was as they had come down from the hills intending to\nobtain fresh provisions at one of the villages, and Hal was beginning\nto recognise the moors he had known in earlier childhood, that they\nperceived a party on the old Roman road before them, which the outlaws'\nkeen eyes at once discovered to be somewhat of their own imputed trade.\nThere seemed to be a waggon upset, persons bound, and a buzz of men,\nlike wasps around a honeycomb preying on it. Something like women's\nveiled forms could be seen. 'Ha! Mere robbery. This must not be. Upon\nthem! Form! Charge!' were the brief commands of the leader, and the\ncompact body ran at a rapid but a regulated pace down the little <DW72>\nthat gave them an advantage of ground with some concealment by a brake\nof gorse. 'Halt! Pikes forward!' was the next order. The little band\nwere already close upon the robbers, in whom they began to recognise\nsome of those whom Sir Giles had dismissed as mere ruffians unequipped\na few days before. It was with a yell of indignation that the troop fell\non them, Sir Giles with a sharp blow severing the bridle of a horse that\na man was leading, but there was a cry back, 'We are for King Harry!\nThese be Yorkists!'\n\n'Nay! nay!' came back the voices of the overthrown. 'Help! help! for\nKing Harry and Queen Margaret! These be rank thieves who have set on us!\nHoly women are here!'\n\nThese exclamations came broken and in utter confusion, mingled with\ncries for mercy and asseverations on the part of the thieves, and fierce\nshouts from Sir Giles's men. All was hubbub, barking dogs, shouting\nmen, and Hal scarcely knew anything till he was aware of two or three\nshrouded nuns, as it seemed, standing by their ponies, of merchantmen\nor carters trying to quiet and harness frightened mules, of waggons\noverturned, of a general confusion over which arose Lord Musgrave's\npowerful authoritative voice.\n\n'Kit of Clumber! Why should I not hang you for thieving on yonder tree,\nwith your fellow thieves?'\n\n'Yorkists, sir! It was all in the good cause,' responded a sullen voice,\nas a grim red and scarred face was seen on a ruffian held by two of the\narchers.\n\n'No Yorkists we, sir!' began a stout figure, coming forward from the\nwaggon. 'We be peaceable merchants and this is a holy dame, the--'\n\n'The Prioress Selby of Greystone,' interrupted one of the nuns, coming\nforward with a hawk on her wrist. 'Sir Giles of Musgrave, I am beholden\nto you! I was on my way to take the young damsel of Bletso to her\nfather, the Lord St. John, with Earl Warwick in London. He sent us an\nescort, but they being arrant cravens, as it seems, we thought it well\nto join company with these same merchants, and thus we became a bait for\nthe outlaws of the Border.'\n\n'Lady, lady,' burst from one of the prisoners, 'I swear that we kenned\nnot holy dames to be of the company! Sir, my lord, we thought to serve\nthe cause of King Harry, and how any man is to guess which side is Earl\nWarwick's is past an honest man.'\n\n'An honest man whose cause is his own pouch!' returned Sir Giles.\n'Miscreants all! But I trow we are scarce yet out of the land of\nmisrule! So if the Lady Prioress will say a word for such a sort of\nsorners, I'll e'en let you go on your way.'\n\n'They have had a warning, the poor rogues, and that will suffice for\nthis time! Nay, now, fellows, let my wimple alone! You'll not find\nanother lord to let you off so easy, nor another Prioress to stand your\nfriend. Get off, I say.'\n\nAn archer enforced her words with a blow, and by some means, rough or\notherwise, a certain amount of order was restored, the ruffians slinking\noff among the gorse bushes, their flight hastened by the pointing of\npikes and levelling of arrows at them. While the merchants, diving into\ntheir packages, produced horns of ale which a younger man offered to\ntheir defenders, the chief of the party, a portly fellow, interrupted\ncertain civilities between the Prioress and Sir Giles by praying them to\npartake of a cup of malmsey, and adding an entreaty that they might be\nallowed to join company with so brave an escort, explaining that he was\na poor merchant of London and the Hans towns who had been beguiled into\nan expedition to Scotland to the young King James, who was said to have\na fair taste. He waved his hands as if his sufferings had been beyond\ndescription.\n\n'Went for wool and came back shorn!' said the Prioress, laughing. 'Well,\nmy Lord Musgrave, what say you to letting us join company?--as I see\nyour band is afoot it will be no great delay, and the more the safer as\nwell as the merrier! Here, let me present to you my young maid, the Lady\nAnne of Bletso, whom I in person am about to deliver to her father.'\n\n'And let me present privately to both ladies,' said Sir Giles, 'the\nyoung squire Harry of Derwentdale, who hath been living as a shepherd in\nthe hills during the York rule.'\n\n'Ha! my lord, methinks this may not be the first meeting between Lady\nAnne and you, though she would not know who the herd boy was who found\nher, a stray lambkin on the moor.'\n\nThe young people looked at each other with eyes of recognition, and as\nHal made his best bow, he said, 'Forsooth, lady, I did not know myself\ntill afterwards.'\n\n'Your shepherd and his wife gave me to understand that I should do hurt\nby inquiring too much,' said the young lady smiling, and holding out her\nhand, which Hal did not know whether to kiss or to shake. 'I hope the\nkind old goodwife is well, who cosseted me so lovingly.'\n\n'She fares well, indeed, lady, only grieved at parting with me.'\n\n'There now,' said the Prioress, 'since we are quit of the robbers,\nmethinks we cannot do better than halt awhile for Master Lorimer's folk\nto mend the tackling of their gear, while we make our noonday meal and\nprovide for our further journey. Allow me to be your hostess for the\nnonce, my lords.'\n\nAnd between the lady's sumpter mules and the merchant's stores a far\nmore sumptuous meal was produced than would have otherwise been the\nshare of the Lancastrian party.\n\n\n\n\nCHAPTER XIV. -- THE JOURNEY\n\n\n\n     'Twas sweet to see these holy maids,\n      Like birds escaped to greenwood shades,\n     --SCOTT.\n\n\nThe Prioress Agnes Selby of Greystone was a person who would have made\na much fitter lady of a castle than head of a nunnery. She would have\nworked for and with her lord, defended his lands for him, governed his\nhouse and managed her sons with untiring zest and energy. But a vow\nof her parents had consigned her to a monastic life at York, where she\ncould only work off her vigour by teasing the more devout and grave\nsisters, and when honourably banished to the more remote Greystone,\nin field sports, and in fortifying her convent against Scots or\nLancastrians who, somewhat to her disappointment, never did attack her.\nNo complaint or scandal had ever attached itself to her name, and she\nlet Mother Scholastica manage the nuns, and regulate the devotions,\nwhile Greystone was known as a place where a thirsty warrior might be\nrefreshed, where tales and ballads of Border raids were welcome, and\nwhere good hawk or hound was not despised.\n\nIt had occurred to the Lord St. John of Bletso that the little daughter\nwhom he had left at York might be come to a marriageable age, and he had\nlistened to the proposal of one of the cousins of the house of Nevil\nfor a contract between her and his son, sending an escort northwards to\nfetch her, properly accompanied.\n\nShe had been all these years at Greystone, and the Prioress immediately\ndecided that this would be an excellent opportunity of seeing the\nsouthern world, and going on a round of pilgrimages which would make the\nexpedition highly decorous. The ever restless spirit within her rose\nin delight, and the Sisterhood of York were ready to acquiesce, having\nfaith in Mother Agnes' good sense to guide her and her pupil to his\ncastle in Bedfordshire by the help of Father Martin through any tangles\nof the White and Red Roses that might await her, as well to her real\nprinciple for avoiding actual evil, though she might startle monastic\nproprieties.\n\nThere was no doubt but that conversation, when she could have it, was as\ngreat a joy to her as ever was galloping after a deer; and there she sat\nwith her beautiful hound by her side, and her hawk on a pole, exchanging\nsentiments of speculation as to Warwick's change of front with Sir Giles\nMusgrave, Father Martin, and Master Ralph Lorimer, while discussing\na pasty certainly very superior to anything that had come out of the\nPenrith stores.\n\nYoung Clifford and Lady Anne sat on the grass near, too shy for the\npresent to renew their acquaintance, but looking up at one another under\ntheir eyelashes, and the first time their eyes met, the girl breaking\ninto a laugh, but it was not till towards the end of the refection that\nthey were startled into intercourse by a general growling and leaping\nup of the great hound, and of the two big ungainly dogs chained to the\nwaggon, as wet, lean, bristling but ecstatic, Watch dashed in among\nthem, and fell on his master.\n\nFor four days (unless he was tied up at first) the good dog must have\nbeen tracking him. 'Off! off!' cried the Prioress, holding back her\ndeer-hound by main strength. 'Off, Florimond! he sets thee a pattern of\nfaithfulness! Be quiet and learn thy devoir!'\n\n'O sir, I cannot send him back!' entreated Hal, also embracing and\ncaressing the shaggy neck.\n\n'Send him back! Nay, indeed. As saith the Reverend Mother, it were well\nif some earls and lords minded his example,' said Sir Giles.\n\n'Here! Watch, I mind thee well,' added Anne. 'Here's a slice of pasty\nto reward thee. Oh! thou art very hungry,' as the big mouth bolted it\nwhole.\n\n'Nearly famished, poor rogue!' said Hal, administering a bone. 'How far\nhast thou run, mine own lad! Art fain to come with thy master and see\nthe hermit?'\n\n'Thou must e'en go,' growled Simon Bunce, 'unless the lady's dog make an\nend of thee! 'Tis ever the worthless that turn up.'\n\n'I would Florimond would show himself as true,' said the Prioress.\n'Don't show thy teeth, sir! I can honour Watch, yet love thee.'\n\n''Tis jealousy as upsets faith,' said the merchant. 'The hound is a\nknightly beast with his proud head, but he brooks not to see a Woodville\ncreep in.'\n\n'Nay, or a Beaufort!' suggested Sir Giles.\n\n'No treason, Lord Musgrave!' said the Prioress, laughing.\n\n'Ah, madam,' responded Sir Giles, 'what is treason?'\n\n'Whatever is against him that has the best of it,' observed Master\nLorimer. 'Well that it is not the business of a poor dealer in\nhorse-gear and leather-work. He asks not which way his bridles are to\nturn! How now, Tray and Blackchaps? Never growl and gird. You have no\npart in the fray!'\n\nFor they were chained, and could only champ, bark and howl, while\nFlorimond and Watch turned one another over, and had to be pulled\nforcibly back, by Hal on the one hand and on the other by the Mother\nAgnes, who would let nobody touch Florimond except herself. After\nthis, the two dogs subsided into armed neutrality, and gradually became\ndevoted friends.\n\nThe curiously composed cavalcade moved on their way southward. The\nPrioress was mounted on the fine chestnut horse that Sir Giles had\nrescued. She was attended by a nun, Sister Mabel, and a lay Sister,\nboth as hardy as herself, and riding sturdy mountain ponies; but her\nchaplain, a thin delicate-looking man with a bad cough, only ventured\nupon a sturdy ass; Anne St. John had a pretty little white palfrey and\ntwo men-at-arms. There were two grooms, countrymen, who had run away on\nthe onset of the thieves, but came sneaking back again, to be soundly\nrated by the Prioress, who threatened to send them home again or have\nthem well scourged, but finally laughed and forgave them.\n\nThe merchant, Master Lorimer--who dealt primarily in all sorts of horse\nfurniture, but added thereto leather-work for knights and men-at-arms,\nand all that did not too closely touch the armourer's trade--had\nthree sturdy attendants, having lost one in an attack by the Scottish\nBorderers, and he had four huge Flemish horses, who sped along the\nbetter for their loads having been lightened by sales in Edinburgh,\nwhere he had hardly obtained skins enough to make up for the weight.\nHis headquarters, he said, were at Barnet, since tanning and\nleather-dressing, necessary to his work, though a separate guild,\nliterally stank in the nostrils of the citizens of London.\n\nTo these were added Sir Giles Musgrave's twenty archers, making a very\nfair troop, wherewith to proceed, and the Prioress decided on not going\nto York. She was not particularly anxious for an interview with the\nAbbess of her Order, and it would have considerably lengthened the\njourney, which both Musgrave and Lorimer were anxious to make as short\nas possible. They preferred likewise to keep to the country, that was\nstill chiefly open and wild, with all its destiny in manufactories\nyet to come, though there were occasionally such towns, villages and\nconvents on the way where provisions and lodging could be obtained.\n\nEvery fresh scene of civilisation was a new wonder to Hal Clifford,\nand scarcely less so to Anne St. John, though her life in the moorland\nconvent had begun when she was not quite so young as he had been when\ntaken to the hills of Londesborough. He had only been two or three times\nin the church at Threlkeld, which was simple and bare, and the full\ndisplay of a monastic church was an absolute amazement, making him kneel\nalmost breathless with awe, recollecting what the royal hermit had told\nhim. He was too illiterate to follow the service, but the music and the\nmajestic flow of the chants overwhelmed him, and he listened with hands\nclasped over his face, not daring to raise his eyes to the dazzling gold\nof the altar, lighted by innumerable wax tapers.\n\nThe Prioress was amused. 'Art dazed, my friend? This is but a poor\ncountry cell; we will show you something much finer when we get to\nDerby.'\n\nHal drew a long breath. 'Is that meant to be like the saints in Heaven?'\nhe said. 'Is that the way they sing there?'\n\n'I should hope they pronounce their Latin better,' responded the\nPrioress, who, it may be feared, was rather a light-minded woman. At any\nrate there was a chill upon Hal which prevented him from directing any\nof his remarks or questions to her for the future. The chaplain told him\nsomething of what he wanted to know, but he met with the most sympathy\nfrom the Lady Anne.\n\n'Which, think you, is the fittest temple and worship?' he said; as they\nrode out together, after hearing an early morning service, gone through\nin haste, and partaking of a hurried meal. The sun was rising over the\nhills of Derbyshire, dyeing them of a red purple, standing out sharply\nagainst a flaming sky, flecked here and there with rosy clouds, and\nfading into blue that deepened as it rose higher. The elms and beeches\nthat bordered the monastic fields had begun to put on their autumn\nlivery, and yellow leaves here and there were like sparks caught from\nthe golden light.\n\nHal drew off his cap as in homage to the glorious sight.\n\n'Ah, it is fine!' said Anne, 'it is like the sunrise upon our own moors,\nwhen one breathes freely, and the clouds grow white instead of grey.'\n\n'Ah!' said Hal, 'I used to go out to the high ground and say the prayer\nthe hermit taught me--\"Jam Lucis,\" it began. He said it was about the\nmorning light.'\n\n'I know that \"Jam Lucis,\"' said Anne; 'the Sisters sing it at prime, and\nSister Scholastica makes us think how it means about light coming and\nour being kept from ill,' and she hummed the chant of the first verse.\n\n'I think this blue sky and royal sun, and the moon and stars at night,\nare God's great hall of praise,' said Hal, still keeping his cap off, as\nhe had done through Anne's chant of praise.\n\n'Verily it is! It is the temple of God Almighty, Creator of Heaven and\nearth, as the Credo says,' replied Anne, 'but, maybe, we come nearer\nstill to Him in God the Son when we are in church.'\n\n'I do not know. The dark vaulted roof and the dimness seem to crush me\ndown,' said the mountain lad, 'though the singing lifts me sometimes,\nthough at others it comes like a wailing gust, all mournful and sad! If\nI could only understand! My royal hermit would tell me when I can come\nto him.'\n\n'Do you think, now he is a king again, he will be able to take heed to\nyou?'\n\n'I know he cares for me,' said Hal with confidence.\n\n'Ah yea, but will the folk about him care to let him talk to you? I have\nheard say that he was but a puppet in their hands. Yea, you are a great\nlord, that is true, but will that great masterful Earl Warwick let you\nto him, or say all these thoughts of his and yours are but fancies for\nbabes?'\n\n'Simon Bunce did mutter such things, and that one of us was as great an\ninnocent as the other,' said Hal, 'but I trust my hermit's love.'\n\n'Ay, you know you are going to someone you love, and who loves you,'\nsighed Anne, 'but how will it be with me?'\n\n'Your father?' suggested Hal.\n\n'My father! What knows he of me or I of him? I tell thee, Harry\nClifford, he left me at York when I was not eight years old, and I have\nnever seen him since. He gave a charge on his lands to a goldsmith at\nYork to pay for my up-bringing, and I verily believe thought no more of\nme than if I had been a messan dog. He wedded a lady in Flanders and\nhad a son or twain, but I have never seen them nor my stepdame; and now\nGilbert there, who brought the letter to the Mother Prioress, says\nshe is dead, and the little heir, whose birth makes me nobody, is at\na monastery school at Ghent. But my Lord of Redgrave must needs make\novertures to my father for me, whether for his son or himself Gilbert\ncannot say. So my father sends to bring me back for a betrothal. The\ngood Prioress goes with me. She saith that if it be the old Lord, who is\na fierce old rogue with as ill a name as Tiptoft himself, the butcher,\nshe will make my Lord St. John know the reason why! But what will he\ncare?'\n\n'It would be hard not to hear my Lady Prioress!' said Hal, looking\nback at the determined black figure, gesticulating as she talked to Sir\nGiles.\n\nAnne laughed, half sadly, 'So you think! But you have never seen the\ngrim faces at Bletso! They will say she is but a woman and a nun, and\nwhat are her words to alliance with a friend of the Lord of Warwick? Ah!\nit is a heartless hope, when I come to that castle!'\n\n'Nay, Anne, if my King gives me my place then&&\n\n'Lady Anne! Lady Anne!' called Sir Giles Musgrave, 'the Mother Prioress\nthinks it not safe for you to keep so much in the front. There might be\nill-doers in the thickets.'\n\nAnne perforce reined in, but Hal fed on the idea that had suddenly\nflashed on him.\n\n\n\n\nCHAPTER XV. -- BLETSO\n\n\n\n     Matter of marriage was the charge he gave me.\n     --SHAKESPEARE,\n\n\nThe cavalcade journeyed on not very quickly, as the riders accommodated\nthemselves to those on foot. They avoided the towns when they came into\nthe more inhabited country, the Prioress preferring the smaller hostels\nfor pilgrims and travellers, and, it may be suspected, monasteries to\nthe nunneries, where she said the ladies had nothing to talk about but\nwonder at her journey, and advice to stay in shelter till after the\nwinter weather. Meantime it was a fine autumn still, and with bright\ncolours on the woods, where deer, hare, rabbit, or partridge tempted the\nhounds, not to say their mistress, but she kept them well in leash, and\nher falcon with hood and jesses, she being too well nurtured not to\nbe well aware of the strict laws of the chase, except when some\ngood-natured monk gave her leave and accompanied her--generally\nAugustinians, who were more of country squires than ecclesiastics. Watch\nneeded no leash--he kept close to his master, except when occasionally\ntempted to a little amateur shepherding, from which Hal could easily\ncall him off. The great stag-hounds evidently despised him, and the curs\nof the waggon hated him, and snarled whenever he came near them, but the\nPrioress respected him, and could well believe that the hermit King had\nloved him. 'He had just the virtues to suit the good King Harry,' she\nsaid, 'dutifulness and harmlessness.'\n\nThe Prioress was the life of the party, with her droll descriptions of\nthe ways of the nuns who received her, while the males of the party had\nto be content with the hostel outside. Sir Giles and Master Lorimer,\nriding on each side of her, might often be heard laughing with her. The\nyoung people were much graver, especially as there were fewer and fewer\ndays' journeys to Bletso, and Anne's unknown future would begin with\nseparation from all she had ever known, unless the Mother Prioress\nshould be able to remain with her.\n\nAnd to Harry Clifford the loss of her presence grew more and more to\nbe dreaded as each day's companionship drew them nearer together in\nsympathy, and he began to build fanciful hopes of the King's influence\nupon the plans of Lord St. John, unless the contract of betrothal had\nbeen actually made, and therewith came a certain zest in looking to his\nprobable dignity such as he had never felt before.\n\nThe last day's journey had come. The escort who had acted as guides were\nin familiar fields and lanes, and one, the leader, rode up to Lady Anne\nand pointed to the grey outline among the trees of her home, while he\nsent the other to hurry forward and announce her.\n\nAnne shivered a little, and Hal kept close to her. He had made the\njourney on foot, because he had chosen to be reckoned among Musgrave's\narchers till he had received full knightly training; and, besides, he\nhad more freedom to attach himself to Anne's bridle rein, and be at hand\nto help through difficult passages. Now he came up close to her, and she\nheld out her hand. He pressed it warmly.\n\n'You will not forget?'\n\n'Never, never! That red rose in the snow--I have the leaf in my\nbreviary. And Goodwife Dolly, tell her I'll never forget how she\ncosseted the wildered lamb.'\n\n'Poor Mother Dolly, when shall I see her?'\n\n'Oh! you will be able to have her to share your state, and Watch too! I\ntake none with me.'\n\n'If we are all in King Harry's cause, there will be hope of meeting, and\nthen if--'\n\n'Ah! I see a horseman coming! Is it my father?'\n\nIt was a horseman who met them, taking off his cap of maintenance and\nbowing low to the Prioress and the young lady, but it was the seneschal\nof the castle, not the father whom Anne so dreaded, but an old\ngentleman, Walter Wenlock, with whom there was a greeting as of an old\nfriend. My lord had gone with the Earl of Warwick to Queen Margaret in\nFrance, and had sent a messenger with a letter to meet his daughter\nat York, and tell her to go to the house of the Poor Clares in London\ninstead of coming home, 'and there await him.'\n\nThe route that had been taken by the party accounted for their not\nhaving met the messenger and it was plain that they must go on to\nLondon. The evening was beginning to draw in, and a night's lodging was\nnecessary. Anne assumed a little dignity.\n\n'My good friends who have guarded me, I hope you will do me the honour\nto rest for the night in my father's castle.'\n\nThe seneschal bowed acquiescence, but the poor man was evidently sorely\nperplexed by such an extensive invitation on the part of his young lady\non his peace establishment, though the Prioress did her best to assist\nAnne to set him at ease. 'Here is Sir Giles Musgrave, the Lord of\nPeelholm on the Borders, a staunch friend of King Harry, with a band of\nstout archers, and this gentleman from the north is with him.' (It had\nbeen agreed that the Clifford name should not be mentioned till the way\nhad been felt with Warwick, one of whose cousins had been granted the\nlands of the Black Lord Clifford.)\n\nThe seneschal bent before Musgrave courteously, saying he was happy\nto welcome so good and brave a knight, and he prayed his followers to\nexcuse if their fare was scant and homely, being that he was unprovided\nfor the honour.\n\n'No matter, sir,' returned Musgrave; 'we are used to soldiers' fare.'\n\n'And,' proceeded Anne, 'Master Lorimer must lie here, and his wains.'\n\n'Master Lorimer,' said the Prioress, 'with whom belike--Lorimer of\nBarnet--Sir Seneschal has had dealings,' and she put forward the\nmerchant, who had been falling back to his waggon.\n\n'Yea,' said Walter Wenlock frankly, holding out his hand. 'We have\nbought your wares and made proof of them, good sir. I am glad to welcome\nyou, though I never saw you to the face before.'\n\n'Great thanks, good seneschal. All that I would ask would be licence for\nmy wains to stand in your court to-night while my fellows and I sup and\nlodge at the hostel.'\n\nThe hospitality of Bletso could not suffer this, and both Anne and the\nseneschal were urgent that all should remain, Wenlock reflecting that if\nthe store for winter consumption were devoured, even to the hog waiting\nto be killed, he could obtain fresh supplies from the tenants, so he\nushered all into the court, and summoned steward, cooks, and scullions\nto do their best. It was not a castle, only a castellated house, which\nwould not have been capable of long resistance in time of danger, but\nthe court and stables gave ample accommodation for the animals and the\nwaggons, and the men were bestowed in the great open hall, reaching to\nthe top of the house, where all would presently sup.\n\nIn the meantime the seneschal conducted the ladies and their two\nattendants to a tiny chamber, where an enormous bed was being made ready\nby the steward's wife and her son, and in which all four ladies would\nsleep, the Prioress and Anne one way, the other two foot to foot with\nthem! They had done so before, so were not surprised, and the lack of\nfurniture was a matter of course. Their mails were brought up, a pitcher\nof water and a bowl, and they made their preparations for supper. Anne\nwas in high spirits at the dreaded meeting, and still more dreaded\nparting, having been deferred, and she skipped about the room, trying to\ngather up her old recollections. 'Yes, I remember that bit of tapestry,\nand the man that stands there among the sheep. Is it King David, think\nyou, Mother, about to throw his stone at the lion and the bear?'\n\n'Lion and bear, child! 'Tis the three goddesses and Paris choosing the\nfairest to give the golden apple.'\n\n'Methought that was the lion's mane, but I see a face.'\n\n'What would the Lady Venus say to have her golden locks taken for a\nlion's mane?'\n\n'I like black hair,' said Anne.\n\n'Better not fix thy mind on any hue! We poor women have no choice save\nwhat fathers make for us.'\n\n'O good my mother, peace! They are all in France, and there's no need\nto spoil this breathing time with thinking of what is coming! Good\nold Wenlock! I used to ride on his shoulder! I'm right glad to see him\nagain! I must tell him in his ear to put Hal well above the salt! May\nnot I tell him in his ear who he is?'\n\n'Safer not, my maid, till we know what King Harry can do for him. Better\nthat his name should not get abroad till he can have his own.'\n\nA great bell brought all down, and Anne was pleased to see that her\nseneschal made no question about placing Harry Clifford beside the\nPrioress, who sat next to the Lord of Peelholm, who sat next to the\nyoung daughter of the house in the seat of honour.\n\nThe nuns, Master Lorimer, and one of the archers, who was a Border\nsquire, besides Master Wenlock, occupied the high table on the dais, and\nthe archers, grooms, and the rest of the household were below.\n\nThe fare was not scanty nor unsubstantial, but evidently hastily\nprepared, being chiefly broiled slices of beef, on which salting had\nbegun; but there was a lack of bread, even of barley, though there was\nno want of drink.\n\nHowever, the Prioress was good-humoured, and forestalled all excuses by\njests about travellers' meals and surprises in the way of guests, and\nboth she and Sir Giles were anxious for Wenlock's news of the state of\nthings.\n\nHe knew much more of the course of affairs than they in their northern\nhomes and on their journey.\n\n'The realm is divided,' he said. 'Those who hold to King Harry, as you\ngentles do, are in high joy, but there be many, spoken with respect, who\ncannot face about so fast, and hold still for York, though they mislike\nthe Queen's kindred. Of such are the merchantmen of London.'\n\n'Is it so?' asked Lorimer. 'If King Edward be as deep in debt to them\nas to me for housings and bridle reins methinks he should not be in good\nodour in their nostrils.'\n\n'Yea,' said Wenlock, 'but if he be gone a beggar to Burgundy what\nbecomes of their debt?'\n\n'I would not give much for it were he restored a score of times,' said\nthe Prioress. 'What would he do but plunge deeper?'\n\n'There would be hope, though, of getting an order on the royal demesne,\nor the crown jewels, or the taxes,' said Lorimer. 'Nay, I hold one even\nnow that will be but waste if he come not back.'\n\n'And this poor King spendeth nothing save on priests and masses,' said\nWenlock.\n\nHal started forward, eager to hear of his King, and Musgrave said, 'A\nholy man is he.'\n\n'Too holy for a King,' said the seneschal. 'He looked like a woolsack\nacross a horse when my Lord of Warwick led him down Cheapside; and only\nthe rabble cried out \"Long live King Harry!\" but some scoffed and said\nthey saw a mere gross monk with a baby face where they had been wont to\nsee a comely prince full of manhood, with a sword instead of beads.'\n\n'His son will please them,' said Musgrave. 'He was a goodly child, full\nof spirit, when last I saw him.'\n\n'If so be he have not too much of the Frenchwoman, his mother, in him,'\nsaid Wenlock. 'A losing lot, as poor as any rats, and as proud as very\npeacocks.'\n\n'She was gracious enough and won all hearts on the Border,' replied\nMusgrave.\n\n'Come, come!' put in the Prioress, 'you may have the chance yet to break\na lance on her behalf. No fear but she is royal enough to shine down\nKing Edward's low-born love, the Widow Grey!'\n\n'Ay, there lay the cause of discontent,' said Lorimer; 'the upstart ways\nof her kin were not to be borne. To hear Dick Woodville chaffer\nabout the blazoning of his horse-gear when he was wedding the\nfourscore-year-old Duchess of Norfolk, one would have thought he was an\nemperor at the very least.'\n\n'Widow Grey has done something for her husband's cause,' said the\nseneschal, 'in bringing him at last a fair son, all in his exile, and\nshe in sanctuary at Westminster. The London citizens are ever touched\nthrough all the fat about their hearts by whatever would sound well in\nthe mouth of a ballad-monger.'\n\n'My King, my King, what of him?' sighed Hal in the Prioress's ear,\nand she made the inquiry for him: 'What said you of King Henry, Sir\nSeneschal? How did he fare in his captivity?'\n\n'Not so ill, methinks,' said the seneschal. 'He had the range of the\nTower, and St. Peter's in the Fetters to pray in, which was what he\nheeded most; also he had a messan dog, and a tame bird. Indeed, men said\nhe had laid on much flesh since he had been mewed up there; and my lord,\nwho went with my Lord of Warwick to fetch him, said his garments were\nscarce so cleanly as befitted. 'Twas hard to make him understand. First\nhe clasped his hands, and bowed his head, crying out that he forgave\nthose who came to slay him, and when he found it was all the other way,\nhe stood like one dazed, let his hand be kissed, and they say is still\nin the hands of my Lord Archbishop of York just as if he were the waxen\nimage of St. John in a procession.'\n\n'The Earl and the Queen will have to do the work,' said the Prioress,\n'and they will no more hold together than a couple of wild hawks will\nhunt in company. How long do you give them to tear out one another's\neyes?'\n\n'Son and daughter may keep them together,' said Musgrave,\n\n'Hatred of the Woodvilles is more like, a poor band though it be,'\nsaid the Prioress. 'These are stirring times! I'll not go back to\nmy anchoress lodge in the north till I see what works out of them!\nMeantime, to our beds, sweet Anne, since 'tis an early start tomorrow.'\n\nThe Prioress, who had become warmly interested in Hal, and had divined\nthe feeling between him and Anne, thought that if she could obtain\naccess to the Archbishop of York, Warwick's brother George, she could\ndeal with him to procure Clifford's restitution in name and in blood,\nand at least his De Vesci inheritance, if Dick Nevil, who had grasped\nthe Clifford lands, could not be induced to give them up.\n\n'I have seen George Nevil,' she said, 'when I was instituted to\nGreystone. He is of kindlier mood than his brothers, and more a valiant\ntrencherman and hunter than aught else. If I had him on the moors and\ncould show him some sport with a red deer, I could turn him round my\nfinger.'\n\n\n\n\nCHAPTER XVI. -- THE HERMIT IN THE TOWER\n\n\n\n     Thy pity hath been balm to heal their wounds,\n     Thy mildness hath allayed their swelling griefs,\n     Thy mercy dried their ever flowing tears.\n     --SHAKESPEARE.\n\n\nEarly in the morning, while the wintry sun was struggling with mists,\nand grass and leaves were dark with frost, the Prioress was in her\nsaddle. Perhaps the weather might have constrained a longer stay, but\nthat it was clear to her keen eyes that, however welcome Wenlock might\nmake his young lady, there was little provision and no welcome for\nthorough-going Lancastrians like Sir Giles's troop, who had besides a\ndoubtful Robin Hood-like reputation; and as neither she nor Anne wished\nto ride forward without them, they decided to go on all together as\nbefore.\n\nAnd a very wet and slightly snowy journey they had, 'meeting in snow\nand parting in snow,' as Hal said, as he marched by Anne's bridle-rein,\nleading her pony, so as to leave her hands free to hold cloak and hood\nclose about her.\n\nShe sighed, and put one hand on his, but a gust of wind took that\nopportunity of getting under her cloak and sending it fluttering over\nher back, so that he had to catch it and return it to her grasp.\n\n'Let us take that as a prophecy that storms shall not hinder our further\nmeeting! It may be! It may be! Who knows what my King may do for us?'\n\n'Only a storm can bring us together! But that may--'\n\nHer breath was blown away again before the sentence was finished, if\nit was meant to be finished, and Master Lorimer came to insist on the\nladies taking shelter in his covered waggon, where the Prioress was\nalready installed.\n\nThrough rain and sleet they reached Chipping Barnet in due time on the\nthird day's journey, and here they were to part from the merchant's\nwains. He had sent forward, and ample cheer was provided at the handsome\ntimbered and gabled house at the porch of which stood his portly wife,\nwith son, daughter, and son-in-law, ready to welcome the party, bringing\nthem in to be warmed and dried before sitting down to the excellent\nmeal which it had been Mistress Lorimer's pride and pleasure to provide.\nThere was a small nunnery at Barnet, but not very near, and the Prioress\nAgnes did not think herself bound to make her way thither in the dark\nand snow, so she remained, most devoutly waited on by her hostess, and\ndiscussed the very last tidings, which had been brought that morning\nby the foreman whom Mistress Lorimer had sent to bring the news to her\nhusband.\n\nIt was probable that the Lord of Bletso was with Warwick and the Queen,\nas he had not been heard of at his home. The King was in the royal\napartments of the Tower, under the charge of the Chancellor. The Earl of\nOxford, a steady partisan of the Red Rose, was Constable of the Kingdom,\nand was guarding the Tower.\n\nOn hearing this, Musgrave decided to repair at once to the Earl, one of\nthe few men in whom there was confidence, since he had never changed\nhis allegiance, and to take his counsel as to the recognition of young\nClifford. On the way to the Tower they would leave the Prioress and her\nsuite at the Sister Minoresses', till news could be heard of the Baron\nSt. John.\n\nSo for the last time the travellers rode forth in slightly improved\nweather. Harry's heart beat high with the longing soon to be in the\npresence of him who had opened so many doors of life to his young mind,\nwhom he so heartily loved, and who, it might be, could give him that\nwhich he began to feel would be the joy of his life.\n\nThe archers, who had been lodged in the warehouses, were drawn up in a\ncompact body, and Master Lorimer, who had a shop in Cheapside, decided\non accompanying them, partly to be at the scene of action and partly to\nfacilitate their entrance.\n\nSo Hal walked by the side of Anne St. John's bridle-rein, with a very\nfull heart, swelling with sensations he did not understand, and which\nkept him absolutely silent, untrained as he was in the conventionalities\nwhich would have made speech easier to him. Nor had Anne much more\ncommand of tongue, and all she did was to keep her hand upon the\nshoulder of her squire; but there was much involuntary meaning in the\nyearning grasp of those fingers, and both fed on the hopes the Prioress\nhad given them.\n\nChristmas was close at hand, and fatted cattle on their way to market\nimpeded the way, so that Hal's time was a good deal taken up in steering\nthe pony along, and in preventing Watch from getting into a battle with\nthe savage dogs that guarded them. Penrith market, where once he had\nbeen, had never shown him anything like such a concourse, and he could\nhear muttered exclamations from the archers, who walked by Sir Giles's\norders in a double line on each side the horses, their pikes keeping off\nthe blundering approach of bullocks or sheep. 'By the halidome, if\nthe Scots were among them, they might victual their whole kingdom till\nDomesday!'\n\nThe tall spire of old St. Paul's and the four turrets of the Tower began\nto rise on them, and were pointed out by Master Lorimer, for even Sir\nGiles had only once in his life visited the City, and no one else of the\nwhole band from the north had ever been there. The road was bordered by\nthe high walls of monasteries, overshadowed by trees, and at the deep\ngateway of one of these Lorimer called a halt. It was the house of the\nMinoresses or Poor Clares, where the ladies were to remain. The six\nweeks' companionship would come to an end, and the Prioress was heartily\nsorry for it. 'I shall scarce meet such good company at the Clares','\nshe said, laughing, as she took leave of Lord Musgrave, 'Mayhap when\nI go back to my hills I shall remember your goodwife's offer of\nhospitality, Master Lorimer.'\n\nMaster Lorimer bowed low, expressed his delight in the prospect, and\nkissed the Prioress's hand, but the heavy door was already being opened,\nand with an expressive look of drollery and resignation, the good lady\nwithdrew her hand, hastily brought her Benedictine hood and veil closely\nover her face, and rode into the court, followed by her suite. Anne had\ntime to let her hand be kissed by Sir Giles and Hal, who felt as if a\nworld had closed on him as the heavy doors clanged together behind the\nSisters. But the previous affection of his young life lay before him as\nSir Giles rode on to the fortified Aldgate, and after a challenge from\nthe guard, answered by a watchword from Lorimer, and an inquiry for whom\nthe knight held, they were admitted, and went on through an increasing\ncrowd trailing boughs of holly and mistletoe, to the north gateway of\nthe Tower. Here they parted with Lorimer, with friendly greetings and\npromises to come and see his stall at Cheapside.\n\nThere was a man-at-arms with the star of the De Veres emblazoned on his\nbreast, and a red rosette on his steel cap, but he would not admit the\nnew-comers till Sir Giles had given his name, and it had been sent in by\nanother of the garrison to the Earl of Oxford.\n\nPresently, after some waiting in the rain, and looking up with awe at\nthe massive defences, two knights appeared with outstretched hands of\nwelcome. Down went the drawbridge, up went the portcullis, the horses\nclattered over the moat, and the reception was hearty indeed. 'Well met,\nmy Lord of Musgrave! I knew you would soon be where Red Roses grew.'\n\n'Welcome, Sir Giles! Methought you had escaped after the fight at\nHexham.'\n\n'Glad indeed to meet you, brave Sir John, and you, good Lord of\nHolmdale! Is all well with the King?'\n\n'As well as ever it will be. The Constable is nigh at hand! You have\nbrought us a stout band of archers, I see! We will find a use for them\nif March chooses to show his presumptuous nose here again!'\n\n'And hither comes my Lord Constable! It rejoices his heart to hear of\nsuch staunch following.'\n\nThe Earl of Oxford, a stern, grave man of early middle age, was coming\nacross the court-yard, and received Sir Giles with the heartiness that\nbecame the welcome of a proved and trustworthy ally. After a few words,\nMusgrave turned and beckoned to Hal, who advanced, shy and colouring.\n\n'Ha! young Lord Clifford! I am glad to see you! I knew your father well,\nrest his soul! The King spoke to me of the son of a loyal house living\namong the moors.'\n\n'The King was very good to me,' faltered Hal, crimson with eagerness.\n\n'Ay, ay! I sent not after you, having enough to do here; and besides,\ntill we have the strong hand, and can do without that heady kinsman\nof Warwick, it will be ill for you to disturb the rogue--what's his\nname--to whom your lands have been granted, and who might turn against\nthe cause and maybe make a speedy end of you if he knew you present.\nBe known for the present as Sir Giles counsels. Better not put his name\nforward,' he added to Musgrave.\n\n'I care not for lands,' said Hal, 'only to see the King.'\n\n'See him you shall, my young lord, and if he be not in one of his\ntrances, he will be right glad to see you and remember you. But he is\nscarce half a man,' added Oxford, turning to Musgrave. 'Cares for nought\nbut his prayers! Keeps his Hours like a monk! We can hardly bring him to\nsit in the Council, and when he is there he sits scarce knowing what we\nsay. 'Tis my belief, when the Queen and Prince come, that we shall have\nto make the Prince rule in his name, and let him alone to his prayers!\nHe will be in the church. 'Tis nones, or some hour as they call it, and\nhe makes one stretch out to another.'\n\nThey entered the low archway of St. Peter ad Vincula, and there Hal\nperceived a figure in a dark mantle just touched with gold, kneeling\nnear the chancel step, almost crouching. Did he not know the attitude,\nthough the back was broader than of old? He paused, as did his\ncompanions; but there was one who did not pause, and would not be left\noutside. Watch unseen had pattered up, and was rearing up, jumping and\nfawning. There was a call of 'Watch! here sirrah!' but 'Watch! Watch!\nGood dog! Is it thou indeed?' was exclaimed at the same moment, and with\nWatch springing up, King Henry stood on his feet looking round with his\ndazed glance.\n\n'My King! my hermit father! Forgive! Down, Watch!' cried Hal, falling\ndown at his feet, with one arm holding down Watch, who tried to lick his\nface and the King's hand by turns.\n\n'Is it thou, my child, my shepherd?' said Henry, his hands on the lad's\nhead. 'Bless thee! Oh, bless thee, much loved child of my wanderings! I\nhave longed after thee, and prayed for thee, and now God hath given thee\nto me at this shrine! Kneel and give the Lord thy best thanks, my\nlad! Ah! how tall thou art! I should not have known thee, Hal, but for\nWatch.'\n\n'It is well,' muttered Oxford to Musgrave. 'I have not seen him so well\nnor so cheery all this day. The lad will waken him up and do him good.'\n\n\n\n\nCHAPTER XVII. -- A CAPTIVE KING\n\n\n\n     And we see far on holy ground,\n     If duly purged our mental view.--KEBLE.\n\n\nThe King held Harry Clifford by the hand as he left St. Peter's Church.\n'My child, my shepherd boy,' he said, and he called Watch after him, and\ninterested himself in establishing a kind of suspicious peace between\nthe shaggy collie and his own 'Minion,' a small white curly-haired dog,\nwhich belonged to a family that had been brought by Queen Margaret from\nProvence.\n\nHis attendant knight, Sir Nicolas Romford, told Sir Giles Musgrave\nthat he had really never seemed so happy since his deliverance, and Sir\nNicolas had waited on him ever since his capture, six years previously.\nHe led the youth along to the royal rooms, asking on the way after his\nsheep and the goodwife who had sent him presents of eggs, then showing\nhim the bullfinch, that greeted his return with loving chirps, and when\nreleased from its cage came and sat upon his shoulder and played with\nhis hair, 'A better pet than a fierce hawk, eh, Hal?' he said.\n\nHe laughed when he found that Harry thought he had spent all this time\nin a dark underground dungeon with fetters on his feet.\n\n'Oh no!' he said; 'they were kindly jailors. They dealt better with me\nthan with my Master.'\n\n'Sir, sir, that terrible ride through Cheapside!' said Harry. 'We heard\nof it at Derwent-side, and we longed to have our pikes at the throats of\nthe villain traitors.'\n\nThe King looked as if he hardly remembered that cruel procession, when\nhe was set upon a sorry jade with his feet tied to the stirrups, and\nshouts of 'Behold the traitor!' around him. Then with a sweet smile of\nsudden recollection, he said, 'Ah! I recall it, and how I rejoiced to\nbe led in the steps of my Lord, and how the cries sounded, \"We will not\nhave this man to reign over us!\" Gratias ago, unworthy me, who by my own\nfault could not reign.'\n\nHarry was silenced, awe-struck, and by-and-by the King took him to see\nhis old chamber in the White Tower, up a winding stone stair. It was\nnot much inferior to the royal lodgings, except in the matter of dais,\ncanopy, and tapestry, and the window looked out into the country, so\nthat the King said he had loved it, and it had many a happy thought\nconnected with it.\n\nHal followed him in a sort of silent wonder, if not awe, not daring\nto answer him in monosyllables. This was not quite the hermit of\nDerwentdale. It was a broader man--not with the breadth of full\nstrength, but of inactivity and advance of years, though the fiftieth\nyear was only lately completed--and the royal robe of crimson, touched\nwith gold, suited him far less than the brown serge of the anchoret.\nThe face was no longer thin, sunburnt, and worn, but pale, and his\nchecks slightly puffed, and the eyes and smile, with more of the strange\nlook of innocent happiness than of old, and of that which seemed to\nbring back to his young visitor the sense of peace and well-being that\nthe saintly hermit had always given him.\n\nThere was consultation that evening between Lord Oxford and Sir Giles\nMusgrave. It was better, they agreed, to let young Clifford remain with\nthe King as much as possible, but without divulging his name. The\nKing knew it, and indeed had known it, when he received the boy at his\nhermitage, but he seemed to have forgotten it, as he had much besides.\nOxford said that though he could be roused into actual fulfilment of\nsuch forms as were required of him, and understood what was set before\nhim, his memory and other powers seemed to have been much impaired, and\nit was held wiser not to call on him more than could be helped, till\nthe Queen and her son should come to supply the energy that was wanting.\nThey would make the gay and brilliant appearance that the Londoners had\nadmired in Edward of York, and which could not be obtained from poor\nHenry.\n\nHis memory for actual matters was much impaired. Never for two days\ntogether could he recollect that his son and Warwick's daughter were\nmarried, and it was always by an effort that he remembered that the\nPrince of Wales was not the eight-years-old child whom he had last\nseen. As to young Clifford, he sometimes seemed to think the tall\nnineteen-years-old stripling was just where he had left the child of\ntwelve or thirteen, and if he perceived the age, was so far confused\nthat it was not quite certain that he might not mix him up with his own\nson, though the knight in constant attendance was sure that he was clear\non that point, and only looked on 'Hal' as the child of his teaching and\nprayers.\n\nBut Harry Clifford could not persuade him to enter into that which more\nand more lay near the youthful heart, the rescuing Anne St. John from\nthe suitor of whom little that was hopeful was heard; and the obtaining\nher from his father. Of course this could not be unless Harry could win\nhis father's property, and no longer be under the attaint in blood, so\nas to be able to lay claim to the lands of the De Vescis through his\nmother; but though the King listened with kindly interest to the\nstory of the children's adventure on the Londesborough moor, and the\nsubsequent meeting in Westmorland, the rescue from the outlaws, and the\njourney together, it was all like a romance to him--he would nod\nhis head and promise to do what he could, if he could, but he never\nremembered it for two days together, and if Hal ventured on anything\nlike pressure, the only answer was, 'Patience, my son, patience must\nhave her work! It is the will of God, it will be right.'\n\nAnd when Hal began to despair and work himself up and seek to do more\nwith one so impracticable, Lord Oxford and Sir Giles warned him not to\nforce his real name and claims too much, for he did not need too many\nenemies nor to have Lord St. John and the Nevil who held his lands both\nanxious to sweep him from their path.\n\nNor was anything heard from or of the Prioress of Greystone, and\nwhenever the name of George Nevil, the Chancellor and Archbishop of\nYork, was heard, Hal's heart burnt with anxiety, and fear that the lady\nhad forgotten him, though as Dick Nevil, who held the lands of Clifford,\nwas known to be in his suite, it was probable that she was acting out of\nprudence.\n\nThe turmoil of anxious impatience seemed to be quelled when Hal sat on\na stool before the King, with Watch leaning against his knee. The\ninstruction or meditation seemed to be taken up much where it had been\nleft six years before, with the same unanswerable questions, only the\nyouth had thought out a great deal more, and the hermit had advanced in\na wisdom which was not that of the rough, practical world.\n\nPart of Clifford's day was spent in the tilt-yard, where his two\nfriends, as well as himself, were anxious that he should acquire\nproficiency and ease such as would become his station, when he recovered\nit; and a martinet old squire of Oxford proved himself nearly as hard a\nmaster as ever Simon Bunce had been.\n\nOne very joyous day came to Henry in his regal capacity. Christmas Day\nhad been quietly spent. There was much noisy revelling in the city,\nand the guards in the castle had their feastings, but Warwick was\ndaily expected to return from France, and neither his brother nor\nthe Archbishop thought that there was much policy in making a public\nspectacle of a puppet King.\n\nBut there was one ceremony from which Henry would not be debarred. He\nwould make the public offering on the Epiphany in Westminster Abbey. He\nhad done so ever since he was old enough to totter up to the altar and\nhold the offerings; and his heart was set on doing so once more. So a\nlarge and quiet cream- Flemish horse was brought for him, he was\nrobed in purple and ermine, with a coronal around the cap that covered\nhis hair, fast becoming white. His train in full array followed him, and\nthe streets were thronged, but there was an ominous lack of applause,\nand even a few audible jeers at the monk dressed up like the jackdaw\nin peacock's plumes, and comparisons with Edward, in sooth a king worth\nlooking at.\n\nHenry seemed not to heed or hear. His blue eyes looked upward, his face\nwas set in peaceful contemplation, his lips were moving, and those who\nwere near enough caught murmurs of 'Vidimus enim stellam Ejus in Oriente\net venimus adorare Eum.' Truly the one might be a king to suit the\nkingdoms of this world, the other had a soul near the Kingdom of Heaven.\n\nThe Dean and choir received him at the west door, and with the same rapt\ncountenance he paced up to the sanctuary, and knelt before the chair\nappropriated to him, while the grand Epiphany Celebration was gone\nthrough, in all its glory and beauty of sound and sight, and with the\nKing kneeling with clasped hands, and a radiant look of happiness almost\ntransfiguring that worn face.\n\nWhen the offertory anthem was sung, he rose up, and advanced to the\naltar. A salver of gold coins was presented to him, which he took and\nsolemnly laid on the altar, but paused for a moment, and removed his\ncrown with both hands, placing it likewise on the altar, and kneeling\nfor a moment ere he turned to take the vase whence breathed the fragrant\nodour of frankincense; and presenting this, and afterwards kneeling and\nbowing low with clasped hands, he again took the salver in which the\nmyrrh was laid. This again he placed on the altar, and remained kneeling\nin intense devotion through the remainder of the service, only looking\nup at the 'Sursum Corda,' when those near enough to see his countenance\nsaid that they never knew before the full import of those words, nor how\nthe heart could be uplifted.\n\nIt was the first time that Hal Clifford had ever joined in the full\nceremonial of the Church, or in such splendid accompaniment, for though\nthere had been the rightful ritual at St. Peter's in the Tower, the\nspace had been confined, and the clergy few, and the whole, even on\nChristmas Day, had been more or less a training to him to enter into\nwhat he now saw and heard. He had in these last weeks gathered much\nof the meaning of all this from the King, who perhaps never fully\ndisentangled the full-grown youth from the boy he had taught at\nDerwentdale, but who, perhaps for that very cause, really suited better\nthe strange mixture of ignorance, simplicity, observation and aspiration\nof the shepherd lord.\n\nThe King did not help more but less than he had done before in Hal's\nresearches and wonderings about natural objects; he had forgotten\nthe philosophies he had once read, and the supposed circuits of moon,\nplanets and stars only perplexed and worried his brain. It was much more\nsatisfactory to refer all to 'He hath made them fast for ever and ever,\nHe hath given them a law which shall not be broken,' and he could not\nunderstand Hal's desire to find out what that law was, and far less his\ncalculations about the tides. He had scarcely ever seen the sea, and as\nto its motions, 'Hitherto shalt thou come and no farther' was sufficient\nexplanation, and when Hal tried to show him the correspondence between\nspring tides and full moons he either waved him away or fell asleep.\n\nBut on the spiritual side of his mind there was no torpor. He loved to\nexplain the sense of the prayers to his willing pupil, and to tell\nhim the Gospel story, dwelling on whatever could waken or carry on the\nChristian life; and between the tiltyard and the oratory Hal spent a\nstrange life.\n\nThat question which had occurred to him on the journey Hal ventured to\nlay before his King--'Was it really and truly better and more acceptable\nworship that came to breathe through him when alone with God under the\nopen vault of Heaven, with endless stars above and beyond, or was the\nbest that which was beautified and guided by priests, with all that\nman's devices could lavish upon its embellishment?' Such, though in more\nbroken and hesitating words, was the herd boy's difficulty, and Henry\nput his head back, and after having once said, 'Adam had the one, God\ndirected the other,' he shut his eyes, and Hal feared he would put it\naside as he had with the moon and the tides, but after some delay, he\nleant forward and said, 'My son, if man had always been innocent, that\nworship as Adam and Eve had it might--nay, would--have sufficed them.\nThe more innocent man is, the better his heart rises. But sin came into\nthe world, and expiation was needed, not only here on earth, but before\nthe just God in Heaven above. Therefore doth He, who hath once offered\nHimself in sacrifice for us, eternally present His offering in Heaven\nbefore the Mercy-Seat, and we endeavour as much as our poor feeble\nefforts can, to take part in what He does above, and bring it home to\nour senses by all that can represent to us the glories of Heaven.'\n\nThere was much in this that went beyond Hal, who knitted his brow,\nand would have asked further, but the King fell into a state of\ncontemplation, and noticed nothing, until presently he broke out into\na thanksgiving: 'Blessed be my Lord, who hath granted me once more to\nfollow in the steps of the kings of the East, though but as in a dream,\nand lay my crown and my prayer before Him. Once more I thank Thee, O my\ntrue King of kings, and Lord of lords.'\n\n'Oh, do not say once more!' exclaimed Hal. 'Again and again, I trust,\nsir. It is no dream. It is real.'\n\nThe King smiled and shook his head. 'It is all a dream to me,' he said,\n'the pageants and the whole. They will not last! Oh, no! It is all but\nan empty show.'\n\nHal looked up anxiously, and the King went on: 'Well do I remember the\nday when, scarce able to walk, and weighed down by my robes, I tottered\nup to the altar and was well pleased to make my offering, and how my\nLord of Warwick, who was then, took me in his arms, and showed me my\ngreat father's figure on his grave, and told me I was bound to be such a\nking as he! Alas! was it mine own error that I so failed?&&\n\n\n     Henry born at Monmouth shall short live and gain all,\n     Henry born at Windsor shall long live and lose all.'\n\n\n'Oh, sir, sir, do not speak of that old saw!'\n\nStill the King smiled. 'It has come true, my child. All is lost, and\nit may be well for my soul that thus it should be, and that I should\ngo into the presence of my God freed from the load of what was gained\nunjustly. I know not whether, if my hand had been stronger, I should\nhave striven to have borne up the burthen of these two realms, but they\nnever ought to have been mine, and if the sins of the forefathers be\nvisited on the children to the third and fourth generation, no marvel\nthat my brain and mine arm could but sink under the weight. Would that\nI had yielded at once, and spared the bloodshed and sacrilege! Miserere\nmei! My son was a temptation. Oh, my poor boy! is he to be the heir to\nall that has come on me? Have pity on him, good Lord!'\n\n'Nay, sir, your brave son will come home to comfort you, and help you\nand make all well.'\n\n'I know not! I know not! I cannot believe that I shall see him again,\nor that the visitation of these crimes is not still to come! My son, my\nsweet son, I can only pray that he might give up his soul sackless and\nfreer of guilt than his father can be, when I remember all that I ought\nto have hindered when I could think and use my will! Now, now all is but\nconfusion! God has taken away my judgment, even as He did with my French\ngrandsire, and I can only let others act as they will, and pray for them\nand for myself.'\n\nHe had never spoken at such length, nor so clearly, and whenever he was\nrequired to come forward, he merely walked, rode, sat or signed rolls\nas he was told to do, and continually made mistakes as to the persons\nbrought to him, generally calling them by their fathers' names, if\nhe recognised them at all, but still to his nearest attendants, and\nespecially to his beloved herd boy, he was the same gentle, affectionate\nbeing, never so happy as at his prayers, and sometimes speaking of holy\nthings as one almost inspired.\n\n\n\n\nCHAPTER XVIII. -- AT THE MINORESSES'\n\n\n\n     The bird that hath been limed in a bush,\n     With trembling wings misdoubteth every bush.\n     --SHAKESPEARE.\n\n\nOne day, soon after that Twelfth Day, Hal accompanied Sir Giles Musgrave\nto the shop or stall of Master Lorimer in Cheapside, a wide space, open\nby day but closed by shutters at night, where all sorts of gilded and\nemblazoned leather-works for man or horse were displayed, and young\n'prentices called, 'What d'ye lack?' 'Saddle of the newest make?' 'Buff\ncoat fit to keep out the spear of Black Douglas himself?'\n\n''Tis Master Lorimer himself I lack,' said Musgrave with a good-humoured\nsmile, and the merchant appeared from a room in the rear, something\nbetween a counting-house and a bedroom, where he welcomed his former\ncompanions, and insisted on their tasting the good sherris sack that had\nbeen sent with his last cargo of Spanish leather.\n\n'I would I could send a flask to our good Prioress,' he said, 'to cheer\nher heart. I went to the Minoresses' as she bade me, to settle some\nmatters of account with her, and after some ado, Sister Mabel came down\nto the parlour and told me the Prioress is very sick with a tertian\nfever, and they misdoubt her recovering.'\n\n'And the young Lady of St. John.'\n\n'She is well enough, but sadly woeful as to the Mother Prioress, and\nlikewise as to what they hear of the Lord Redgrave. It is the old man,\nnot his son, a hard and stark old man, as I remember. He would have\nbargained with me for the coats of the poor rogues slain at St. Albans,\nand right evil was his face as he spoke thereof, he being then for Queen\nMargaret; but then he went over to King Edward, and glutted himself with\nslaughter at Towton, and here he calls himself Red Rose again. Ill-luck\nto the poor young maid if she falls to him!'\n\nIt was terrible news for Hal, and Musgrave could not but gratify him\nby riding by the Minories to endeavour to hear further tidings of the\nPrioress.\n\nIt was a grand building in fine pointed architecture, for the Clares,\nthough once poor, in imitation of St. Clara and St. Francis, had been\ndispensed collectively from their vow of poverty, and though singly\nincapable of holding property, had a considerable accumulation en masse.\nThey were themselves a strict Order, but they often gave lodgings to\nladies either in retreat or for any cause detained near London.\n\nSir Giles and Harry were only admitted to the outer court, whence the\nportress went with their message of inquiry. They waited a long time,\nand then the Greystone lay Sister who had been the companion of their\njourney came back in company with the portress.\n\n'Benedicite, dear gentles,' she said; 'oh, you are a sight for sair\neen.'\n\n'And how fares the good Mother Prioress?' asked the Lord of Peelholm.\n\n'Alack! she is woefully ill when the fever takes her, and she is wasted\naway so that you would scarce know her; but this is one of the better\ndays, and if you, sir, will come into the parlour, she will see you. She\nwas arraying herself as I came down. She was neither to have nor to hold\nwhen she heard you were there, and said a north country face would be\nbetter to her than all the Sisters' potions!'\n\nThey were accordingly conducted through a graceful cloister, overgrown\nwith trailing ivy, to a bare room, with mullioned windows, and frescoes\non the Walls with the history of St. Francis relieving beggars,\npreaching to the birds, &c., and with a stout open work barrier cutting\noff half the room.\n\nPresently the Prioress tottered in, leaning heavily on the arms of\nSister Mabel and of Anne St. John, while her own lay Sister and another\nplaced a seat for her; but before she would sit down, she would go up\nto the opening, and turning back her veil, put out a hand to be grasped.\n'Right glad am I to see you, good Sir Giles and young Harry. Are you\ngoing back to the wholesome winds of our moors?'\n\n'Not yet, holy Mother. It grieves me to see you faring so ill.'\n\n'Ah! a breeze from the north would bring life back to my old bones. Aye,\nGiles, this place has made an old woman of me.' And truly her bright\nruddy face was faded to a purple hue, and her cheeks hung haggard and\nalmost withered, but as her visitors expressed their grief and sympathy,\nshe went on in her own tone. 'And tell me somewhat of how things are\ngoing. How doth Richard of Warwick comport himself to the King? Hath\nyour King zest enough to reign? Is my White Rose King still abroad in\nBurgundy?' And as Sir Giles replied to each inquiry in turn, and told\nall he could of political matters, she exclaimed: 'Ah! that is better\nthan the hearing whether the black hen hath laid an egg, or the skein of\nyellow silk matches. I am weary, O! I am weary. Moreover, young Hal, I\nknow as matters are that could I see George Nevil face to face I could\ndo somewhat with him, and I laid my plans to obtain a meeting, but\ntherewith, what with vexation and weariness and lack of air, comes this\nsickness, and I am laid aside and can do nought but pray, and lay my\nplans to meet him some day in the fields, and show him what a hawk can\ndo, then shame him into listening to my tale. But I must be a sound\nwoman first! And maybe his brother Warwick, being a sturdy gentleman who\nloves a brave man, will be better to deal with. I am a sinful woman,\nand maybe my devotions here will help me to be more worthy to be heard.\nMoreover, I hoped you had done somewhat in thine own cause with thy King\nand Earl Oxford,' she proceeded. 'Thou hast an esquire's coat; hast thou\nany hope of thy lands?'\n\n'I must strive to earn them by deeds,' said Hal. 'And--'\n\n'Well spoken, lad! 'Tis the manly way; but methought you hadst interest\nwith this King of thine, or hath he only a royal memory for services?'\n\n'He is good to me. Yea, most good,' began Harry.\n\n'Ay, he loves the boy,' said Sir Giles, 'no question about that; but his\nmemory for all that is about him hath failed, and there is nothing for\nit save to wait for the Queen and the Prince, who will bear the boy's\nfather's services in mind.'\n\n'And wherefore tarries the French woman? This maid's father is to come\nover with her. He is forming her English court, I trow; she can have few\nbeside from England.'\n\n'When he comes,' said Harry, with a look into Anne's eyes that made\nthem droop and her cheeks burn, 'then shall we put it to the touch. Then\nshall I know whether I have mine own, and what is more than mine own.'\n\n'Thine own,' whispered Anne. 'Oh, better live in the sheepfolds with\nthee than with this Baron! I shudder at the thought.'\n\nThis, and a few more such words were an aside, while the Prioress\ncontinued her conversation with Sir Giles, and went on to say that she\nwas sure she should never recover till she was out of these walls, and\naway from London smoke and London smells, and she naughtily added in a\nwhisper the weary talk of these good nuns, who had never flown a hawk or\nchased a deer in their lives, and thought Florimond a mere wolf, if\nnot the evil one himself, and kept the poor hound chained up like a\nmalefactor in gyves, till she was fain to send him away with Master\nLorimer to keep for her.\n\nShe would not go back to her Priory till Anne's fate was settled, being\nin hopes of doing something yet for the poor wench; but meantime she\nshould die if she stayed there much longer, and she meant to set forth\non pilgrimage in good time, before she had scandalised the good ladies\nenough to make them gossip to the dames of St. Helen's, who would be\nonly too glad to have a story against the Benedictines. A ride over the\nKentish downs was the only cure for her or for Anne, who had been pining\never since they had been mewed up here, though, looking across at the\ngirl, whose head was leaning against the bars, Sir Giles seemed to have\nbrought a remedy to judge by those cheeks.\n\n'Would that we could hope it would be an effectual and lasting remedy,'\nsighed Sir Giles; 'but unless this poor King could be roused to insist,\nor the Earl of Warwick fell out with his cousin, I do not see much\nchance for the lad.'\n\n'Is it Warwick who is his chief foe or King Edward?' asked the Prioress.\n\n'King Edward, doubtless, for his father's slaughter of young Rutland at\nWakefield.'\n\n'That bodes ill,' said the lady. 'By all I gather, King Edward is a\ntiger when once roused, but at other times is like that same tiger,\npurring and slow to move. But there's a bell that warns us to vespers.\nThey are mightily more strict here than ever we are at Greystone. Ah!\nyou won't tell tales, Sir Giles! You'll soon hear of me at St. Thomas's\nshrine at Canterbury.'\n\nThe knight took his leave. It was impossible not to like and pity the\nPrioress, though the life among devout nuns was clearly beyond her\npowers.\n\nThe dreamy peaceful days of the Tower of London were stirred by the\narrival of the great Earl of Warwick, the Kingmaker, as people already\ncalled him. He took up his residence in his own mighty establishment at\nWarwick House near St. Paul's; and the day after his arrival, he came\nclanking over London Bridge with a great following of knights and\nsquires to pay his respects to King Henry.\n\nHenry Clifford was not disposed to meet him, and only watched from\na window when the drawbridge was lowered, and the sturdy man, with\ngrizzled hair and marked, determined features, rode into the gateway,\nwhere he was received by the Earl of Oxford.\n\nThe interview was long, and when it was finished, the two Earls made\nthe round of the defences, and Oxford drew up his garrison on the Tower\nGreen to be inspected.\n\nWhen Warwick had taken his leave, Hal was summoned to the Constable's\nhall. 'We must be jogging, my young master,' he said. 'There are rumours\nof King Edward making another attempt for his crown, and my Lord of\nWarwick would have me go and watch the eastern seaboard. And you had\nbest go with me.'\n\n'The King--' began Hal.\n\n'You will come back to the King by-and-by if so be he misses you, but\nhe was more dazed than ever to-day, and perhaps it was well, for Warwick\nbrought with him Dick Nevil, who has got your lands of Clifford, and\nmight be tempted to put you out of the way in one of the dungeons that\nlie so handy.'\n\n'No one save the King knows who I am,' said Hal, 'and he forgets from\nday to day all save that I am the herd boy, and I think it cheers him to\nhave me with him. I will stay beside him even as a varlet.'\n\n'Nay, my lord, that may not be. 'Tis true he loves thee, but he will\nforget anon, and I may not suffer the risk. Too many know or guess.'\n\nHarry Clifford repeated that he recked not of the risk when he could\nserve and comfort his beloved King, and, indeed, his mind was made up\non the subject. He had taken measures for remaining as one of the\nmen-at-arms of the garrison; but King Henry himself surprised him by\nsaying, 'My young Lord of Clifford, fare thee well. Thou goest forth\nto-morrow with the Constable of Oxford. Take my blessing with thee, my\nchild. Thou hast been granted to me to make life very sweet to me of\nlate, and I thank God for it, but the time is come that thou must part\nfrom me.'\n\n'Oh, sir, never! None was ever so dear to me! For weal or woe I will\nbe with you! Suffer me to be your meanest varlet, and serve you as none\nother can do.'\n\nHenry shook his head. 'It may not be, my child, let not thy blood also\nbe on my head! Go with Oxford and his men. Thou hast learnt to draw\nsword and use lance. Thou wilt be serving me still if again there be,\nwhich Heaven forefend, stricken fields in my cause or my son's.'\n\n'Sir, if I must fight, let no less holy hand than thine lay knighthood\non my shoulder,' sobbed Hal, kneeling.\n\nHenry smiled. 'I have well-nigh forgotten the fashion. But if it will\nplease thee, my son, give me thy sword, Oxford. In the name of God and\nSt. George of England I dub thee knight. For the Church, for the honour\nof God, for a good cause, fight. Arise, Sir Henry Clifford!'\n\n\n\n\n\nCHAPTER XIX. -- A STRANGE EASTER EVE\n\n\n\n     And spare, O spare\n     The meek usurper's holy head.\n     --GRAY.\n\n\nOnce more, at the close of morning service, while it was still dark, did\nHarry Clifford, the new-made knight, kneel before King Henry and feel\nhis hand in blessing on his head. Then he went forth to join Musgrave\nand the troop that the Earl of Oxford was leading from the Tower to\nraise the counties of East Anglia and watch the coast against a descent\nof King Edward from the Low Countries.\n\nAs they passed the walls enclosing the Minories Convent, and Hal gazed\nat it wistfully, the wide gateway was opened and out came a party of\nblack-hooded nuns, mounted on ponies and mules, evidently waiting till\nOxford's band had gone by. Harry drew Sir Giles's attention, and they\nlingered, as they became certain that they beheld the Prioress Selby of\nGreystone, hawk, hound and all, riding forth, nearly smothered in her\nhood, and not so upright as of old.\n\n'Ay, here I am!' she said, as he reined up and bowed his greeting. 'Here\nI am on my pilgrimage! I got Father Ridley, the Benedictine head, to\norder me forth. Methinks he was glad, being a north countryman, to send\nme out before I either died on the Poor Clares' hands, or gave them a\nfuller store of tales against us of St. Bennet's! Not but that they are\ngood women, too godly and devout for a poor wild north country Selby\nlike me, who cannot live without air.\n\n\n          O the oak and the ash and the bonny ivy tree,\n          They flourish best at home in the north countree.\n\n\nFlori, Flori, whither away? Ah! thou hast found thine old friend. Birds\nof a feather. Eh? the young folk have foregathered likewise. Watch! And\nthou, sir knight, whither are you away?'\n\n'On our way to Norfolk in case the Duke of York should show himself on\nthe coast. And yours, reverend Mother?'\n\n'To Canterbury first by easy journeys. We sleep to-night at the Tabard,\nwhere we shall meet other pilgrims.'\n\n'Here, alack! our way severs from yours. Farewell, holy Mother, may you\nfind health on your pilgrimage.'\n\n'Every breath I take in is health,' said the Mother, who had already\nmanoeuvred an opening in her veil, and gasped to throw it back as soon\nas she should attain an unfrequented place. 'There are so many coming\nand going here that all the air is used up by their greasy nostrils!\nWell! good luck, and God's blessing go with you, and you, young Hal, I\nmay say so far, whichever side ye be, but still I hold that York has the\nright, and yours may be a saint, but not a king.'\n\nHal had meantime 'forgathered' as the Prioress said with Anne, marching,\nin spite of his new honours, close to her stirrup, and venturing to\nwhisper to her that he was now her knight, and 'her colours,' which he\nwas to wear for her, were only a tiny scrap of ribbon from her glove,\nwhich he cut off with his dagger, and kissed, saying he should wear it\nnext his heart, though he might not do so openly.\n\nTheir love was more implied than ever it had been before, and she\nrepeated her confidence that the kind Prioress would never leave her\ntill she had done her utmost for them both.\n\n'But you, my good stripling, I am ashamed to see you. I have done\nnothing for you. I sent a humble message to ask to see the Archbishop,\nbut had no answer, and by-and-by, when I stirred again, who should come\nto sec me but young Bertram Selby, and \"Kinswoman,\" said he, \"you had\nbest keep quiet. The Archbishop hath asked me whether rumours were sooth\nthat yours was scarce a regular Priory.\" The squire stood up for me and\nsaid, as became one of the family, that an outlying cell, where there\nwere ill neighbours of Scots, thieves, borderers, and the like, could\nscarce look to be as trim as a city nunnery, and that none had ever\nheard harm of Mother Agnes. But then one of his priests took on him to\nwhisper in his ear, and he demanded whether we had not gone so far as to\nhide traitors from justice, to which Bertram returned a stout denial as\nwell he might, though he thought it well to give me warning, but for the\npresent there was no use in attempting anything more. The Archbishop was\nexceedingly busy with the work of his office and the defence of London\nin case of Edward's threatened return; but he had not yet come, and no\none thought there was a reasonable doubt that Warwick, the Kingmaker,\nwould not be victorious, and he had carried his son-in-law, the Duke of\nClarence, with him.' After the cause of the Red Rose was won, there was\nno fear but that the services of Clifford would be remembered. So Harry\nClifford parted with Anne, promising himself and her that there should\nbe fresh Clifford services, winning a recognition of the De Vesci\ninheritance if of no more.\n\nThe ladies went on their way in the track which Chaucer has made\nmemorable, laying their count to meet Queen Margaret and her son, and\nwin their ears beforehand, and wondering that they came not. Kentish\nbreezes soon revived the Prioress, and she went through many strange\ndevotions at the shrine of Becket, which, it might be feared, did not\nimprove her spiritual, so much as her bodily, health, while Anne's\nchiefly resolved themselves into prayers that Harry Clifford might\nbe guarded and restored, and that she herself might be saved from the\ndreaded Lord Redgrave.\n\nThey did not set out on the return to London till they had inhaled\nplenty of sea breezes by visiting the shrine of St. Mildred in the isle\nof Thanet, and St. Eanswith at Folkestone, till Lent had begun, and\nthe first fresh tidings that they met were that Edward had landed in\nYorkshire, but his fleet had been dispersed by storms, and the people\ndid not rise to join him, so that he was fain to proclaim that he only\ncame to assert his right to his father's inheritance of the Dukedom of\nYork.\n\nAt the Minoresses' Convent they found that a messenger had arrived,\nbidding Anne go to meet her father at his castle in Bedfordshire. He was\ncoming over with the Queen whenever she could obtain a convoy from King\nLouis of France. Lord Redgrave was with him, and the marriage should\ntake place as soon as they arrived.\n\n'Never fear, child,' said the Prioress; 'many is the slip between the\ncup and the lip.'\n\nFurther tidings came that Edward had thrown off his first plea, that he\nhad passed Warwick's brother Montagu at Pontefract, and that men from\nhis own hereditary estates were flocking to his royal banner. Warwick\nwas calling up his men in all directions, and both armies were advancing\non London. Then it was known that 'false, fleeting, perjured Clarence'\nhad deserted his father-in-law, and returned to his brother; and\nworthless as he individually was, it boded ill for Lancaster, though\nstill hope continued in the uniform success of the Kingmaker. Warwick\nwas about twenty miles in advance of Edward, till that King actually\npassed him and reached the town of Warwick itself. Still the Earl wrote\nto his brother that if he could only hold out London for forty-eight\nhours all would be well.\n\nOnce more poor King Henry was set on horseback and paraded through the\nstreets. Brother Martin went out with the chaplain of the Poor Clares to\ngaze upon him, and they came back declaring that he was more than ever\nlike the image carried in a procession, seeming quite as helpless and\nindifferent, except, said Brother Martin, when he passed a church, and\nthen a heavenly look came over his still features as he bowed his head;\nbut none of the crowd who came out to gaze cried 'Save King Harry!' or\n'God bless him!'\n\nThere were two or three thousand Yorkists in the various sanctuaries of\nLondon, and they were preparing to rise in favour of their King Edward,\nand only a few hundred were mustering in St. Paul's Churchyard for the\nRed Rose.\n\nThe Poor Clares were in much terror, though nunneries and religious\nhouses, and indeed non-combatants in general, were usually respected\nby each side in these wars; but the Prioress of Greystone was not sorry\nthat the summons to her protegee called her party off on the way to\nBedfordshire, and they all set forward together, intending to make\nMaster Lorimer's household at Chipping Barnet their first stage, as they\nhad engaged to do.\n\nTheir intention had been notified to Lorimer's people in his London\nshop, who had sent on word to their master, and the good man came out\nto meet them, full of surprise at the valour of the ladies in attempting\nthe journey. But they could not possibly go further. King Edward was at\nSt. Albans, and was on his way to London, and the Earl of Warwick was\ncoming up from Dunstable with the Earls of Somerset and Oxford. For\nladies, even of religious orders, to ride on between the two hosts was\nmanifestly impossible, and he and his wife were delighted to entertain\nthe Lady Prioress till the roads should be safe.\n\nThe Prioress was nothing loth. She always enjoyed the freedom of a\nsecular household, and she was glad to remain within hearing of the last\nnews in this great crisis of York and Lancaster.\n\n'I marvel if there will be a battle,' she said. 'Never have I had the\ngood luck to see or hear one.'\n\n'Oh! Mother, are you not afraid?' cried Sister Mabel.\n\n'Afraid! What should I be afraid of, silly maid? Do you think the\nmen-at-arms are wolves to snap you up?'\n\n'And,' murmured Anne, 'we shall know how it goes with my Lord of\nOxford's people.'\n\nThese were the last days of Lent, and were carefully kept in the matter\nof food by the household, but the religious observances were much\ndisturbed by the tidings that poured in. King Henry and Archbishop Nevil\nhad taken refuge in the house of Bishop Kemp of London, Urswick the\nRecorder, with the consent of the Aldermen, had opened the gates to\nEdward, and the Good Friday Services at Barnet, the Psalms and prayers\nin the church, were disturbed by men-at-arms galloping to and fro, and\nreports coming in continually.\n\nThere could be no going out to gather flowers to deck the Church the\nnext day, for King Edward was on the London side, and Warwick with\nhis army had reached the low hills of Hadley, and their tents, their\nbanners, and the glint of their armour might be seen over the heathy\n<DW72> between them and the lanes and fields, surrounded by hedges, that\nfenced in the valley of Barnet. The little town itself, though lying\nbetween the two armies, remained unoccupied by either party, and only\nmen-at-arms came down into it, not as plunderers, but to buy food.\n\nWarwick's cannon, however, thundered all night, a very awful sound to\nsuch unaccustomed ears, but they were so directed that the charges flew\nfar away from Barnet, under a false impression as to the situation of\nthe Yorkist forces.\n\nMistress Lorimer had heard them before, but accompanied every report\nwith a pious prayer; Sister Mabel screamed at each, then joined in; the\nPrioress was greatly excited, and walked about with Master Lorimer,\nnow on the roof, trying to see, now at the gate, trying to hear. Anne\nfancied it meant victory to Hal's party, but knelt, tried to pray while\nshe listened, and the dogs barked incessantly. And that Hal must be in\nthe army above the little town they guessed, for in the evening Watch\ncame floundering into the courtyard, hungry and muddy, but full of\naffectionate recognition of his old friends and the quarters he had\nlearnt to know. Florimond, who happened to be loose, had a romp with\nhim in their old fashion, and to the vexation and alarm of his mistress,\nthey both ran off together, and must have gone hunting on the heath, for\nthere was no response to her silver whistle.\n\n\n\n\nCHAPTER XX. -- BARNET\n\n\n\n     A dead hush fell; but when the dolorous day\n     Grew drearier toward twilight falling, came\n     A bitter wind, clear from the North, and blew\n     The mist aside.\n     --TENNYSON.\n\n\nAnd Sir Henry Clifford? Still he was Hal of Derwentdale, for the\nperilous usurper, Sir Richard Nevil, was known to be continually with\nWarwick, and Musgrave was convinced that the concealment was safest.\n\nThe youth then remained with the Peelholm men, and became a good deal\nmore practised in warlike affairs, and accustomed to campaigning, during\nthe three months when Oxford was watching the eastern coast. On this\nEaster night he lay down on the hill-side with Watch beside him, his\nshepherd's plaid round him, his heart rising as he thought himself\nnear upon gaining fame and honour wherewith to win his early love, and\nwinning victory and safety for his beloved King, or rather his hermit.\nFor as his hermit did that mild unearthly face always come before him.\nHe could not think of it wearing that golden crown, which seemed alien\nto it, but rather, as he lay on his back, after his old habit looking\nup at the stars, either he saw and recognised the Northern Crown, or his\ndazed and sleepy fancy wove a radiant coronet of stars above that meek\ncountenance that he knew and loved so well; and as at intervals the\ncannon boomed and wakened him, he looked on at the bright Northern Cross\nand dreamily linked together the cross and crown.\n\nEaster Sunday morning came dawning, but no one looked to see the sun\ndance, even if the morning had not been dull and grey, a thick fog\ncovering everything; but through it came a dull and heavy sound, and\nthe clang of armour. Even by their own force the radiant star of the De\nVeres could hardly be seen on the banner, as the Earl of Oxford rode up\nand down, putting his men in battle array. Hal was on foot as an archer,\nmeaning to deserve the spurs that he had not yet worn. The hosts were\nclose to one another, and at first only the continual rain of arrows\ndarkened the air; but as the sun rose and the two armies saw one\nanother, Oxford's star was to be seen carried into the very midst of the\nopposing force under Lord Hastings. On, on, with cries of victory, the\nknights rode, the archers ran across the heath carrying all before them,\nnever doubting that the day was theirs, but not knowing where they were\ntill trumpets sounded, halt was called, and they were drawn up together,\nas best they might, round their leading star. But as they advanced,\nbehold there was an unexpected shout of treason. Arrows came thickly\non them, men-at-arms bearing Warwick's ragged staff came thundering\nheadlong upon them. 'Treason, treason,' echoed on all sides, and with\nthat sound in his ears Harry Clifford was cut down, and fell under a\nhuge horse and man, and lay senseless under a gorse-bush.\n\nHe knew no more but that horses and men seemed for ever trampling over\nhim and treading him down, and then all was lost to him--for how long he\nknew not, but for one second he was roused so far as to hear a furious\ngrowling and barking of Watch, but with dazed senses he thought it\nwas over the sheep, tried to raise himself, could not, thought himself\ndying, and sank back again.\n\nThe next thing he knew was 'Here, Master Lorimer, you know this gear\nbetter than I; unfasten this buff coat. There, he can breathe. Drink\nthis, my lad.'\n\nIt was the Prioress's voice! He felt a jolt as of a waggon, and opened\nhis eyes. It was dark, but he knew he was under the tilt of Lorimer's\nwaggon, which was moving on. The Prioress was kneeling over him on one\nside, Lorimer on the other, and his head was on a soft lap--nay, a warm\ntear dropped on his face, a sweet though stifled voice said, 'Is he\ntruly better?'\n\nThen came sounds of 'hushing,' yet of reassurance; and when there was a\nhalt, and clearer consciousness began to revive, while kind hands were\nbusy about him, and a cordial was poured down his throat, by the light\nof a lantern cautiously shown, Hal found speech to say, as he felt a\nlong soft tongue on his face, 'Watch, Watch, is it thou, man?'\n\n'Ay, Watch it is,' said the Prioress. 'Well may you thank him! It is to\nhim you owe all, and to my good Florimond.'\n\n'But what--how--where am I?' asked Hal, trying to look round, but\nfeeling sharp thrills and shoots of pain at every motion.\n\n'Lie still till they bring their bandages, and I will tell you. Gently,\nNan, gently--thy sobs shake him!' But, as he managed to hold and press\nAnne's hand, the Prioress went on, 'You are in good Lorimer's warehouse.\nSafer thus, though it is too odorous, for the men of York do not respect\nsanctuary in the hour of victory.'\n\nThe word roused Hal further. 'The victory was ours!' he said. 'We had\ndriven Hastings' banner off the field! Say, was there a cry of treason?'\n\n'Even so, my son. So far as Master Lorimer understands, Lord Oxford's\nbanner of the beaming star was mistaken for the sun of York, and the men\nof Warwick turned on you as you came back from the chase, but all was\nutter confusion. No one knows who was staunch and who not, and the\nfields and lanes are full of blood and slaughtered men; and Edward's\nroyal banner is set up on the market cross, and trumpets were sounding\nround it. And here come Master Lorimer and the goodwife to bind these\nwounds.'\n\n'But Sir Giles Musgrave?' still asked Hal.\n\n'Belike fled with Lord Oxford and his men, who all made off at the cry\nof treason,' was the answer.\n\nLorimer returned with his wife and various appliances, and likewise with\nfresh tidings. There was no doubt that the brothers Warwick and Montagu\nhad been slain. They had been found--Warwick under a hedge impeded by\nhis heavy armour, and Montagu on the field itself. Each body had been\nthrown over a horse, and shown at the market cross; and they would be\ncarried to London on the morrow. 'And so end,' said Lorimer, 'two brave\nand open-handed gentlemen as ever lived, with whom I have had many\nfriendly dealings.'\n\nOne thing more Hal longed to hear--namely, how he had been saved. He\nremembered that Watch had come back to him with Florimond the evening\nbefore. They had probably been hunting together, and the hound, who had\nalways been very fond of him on the journey, had accompanied Watch to\nhis side before going back to his chain in Barnet; but he had lost sight\nof them in the morning, and regretted that he could not find Watch to\nprovide for his safety. He knew, he said, by the presence of Florimond,\nwho must be in Barnet. And he also had a dim recollection of being\nlicked by Watch's tongue as he lay, and likewise of hearing a furious\nbarking, yelling and growling, whether of one or both dogs he was not\nsure.\n\nIt seemed that towards the evening, when the battle-cries had grown\nfainter, and the sun was going down, Florimond had burst in on his\nmistress, panting and blood-stained--but not with his own blood, as was\nsoon ascertained--and made vehement demonstrations by which, as a true\ndog-lover, the Prioress perceived that he wanted her to follow him. And\nAnne, who thought she saw a piece of Hal's plaid caught in his collar,\nwas 'neither to have nor to hold,' as the Mother said, till Master\nLorimer was found, and entreated to follow the hound, ay, and to take\nthem with him. He demurred much as to their safety, but the Prioress\ndeclared that it was the part of the religious to take care of the\nwounded, and not inconsistent with her vow. See the Sisters of St.\nKatharine's of the Tower! And though her interpretation was a broad one,\nand would have shocked alike her own Abbess and her of the Minoresses,\nhe was fain to accept it in such a cause; but he commanded his waggoners\nto bring the wain in the rear, both as an excuse, and a possible\nprotection for the ladies, and, it might be, a conveyance for the\nwounded.\n\nFlorimond, who had sprung about, barked, fawned and made entreating\nsounds all this time (longer in narrative than in reality) led them, not\nthrough the central field of slaughter, but somewhat to the left, among\nthe heath--where, in fact, Oxford had lost his way in the fog, and his\nown allies had charged him, but had not followed far beyond the place\nof Hal's fall, discovering the fatal error that spread confusion through\ntheir ranks, where everyone distrusted his fellow leader.\n\nThere, after a weary and perilous way, diversified by the horrid shouts\nof plunderers of the slain, happily not near at hand, and when Lorimer,\nbut for the ladies, would have given up the quest as useless, they were\ngreeted by Watch's bark, and found him lying with his fine head alert\nand ready over his senseless master.\n\nThere was no doubt but that the two good creatures, both powerful and\nformidable animals, must have saved him from the spoilers, and then been\nsagacious enough to let the hound go down to fetch assistance while the\nsheep-dog remained as his master's faithful guardian. How honoured and\ncaressed they were can hardly be described, but all will know.\n\nThe joy and gratitude of knowing of Anne's devotion, and the pleasure of\nhis good dog's faithfulness, helped Hal through the painful process\nof having his hurts dealt with. Surgeons, even barbers, were fully\noccupied, and Lorimer did not wish to have it known that a Lancastrian\nwas in his house. His wife and her old nurse, as well as the Prioress,\nhad some knowledge of simple practical surgery; and Hal's disasters\nproved to be a severe cut on the head, a slash on the shoulder, various\nbruises, and a broken rib and thigh-bone, all which were within their\ncapabilities, with assistance from the master's stronger hand. No one\ncould tell whether the savage nature of the York brothers might not\nslake their revenge in a general massacre of their antagonists; so\nLorimer caused Hal's bed to be made in the waggon in the warehouse,\nwhere he was safe from detection until the victorious army should have\nquitted Barnet.\n\n\n\n\nCHAPTER XXI. -- TEWKESBURY\n\n\n\n     The last shoot of that ancient tree\n     Was budding fair as fair might be;\n       Its buds they crop\n       Its branches lop\n     Then leave the sapless stem to die.\n     --SOPHOCLES (Anstice).\n\n\nHarry Clifford lay fevered, and knowing little of what passed, for\nseveral days, only murmuring sometimes of his flock at home, sometimes\nof the royal hermit, and sometimes in distress of the men-at-arms with\nwhom he had been thrown, and whose habits and language had plainly been\na great shock to his innocent mind, trained by the company of the sheep,\nand the hermit. He took the Prioress's hand for Good-wife Dolly's, but\nhe generally knew Anne, who could soothe him better than any other.\n\nMaster Lorimer was fully occupied by combatants who came to have their\nequipments renewed or repaired, and he spent the days in his shop in\nLondon, but rode home in the long evenings with his budget of news. King\nHenry was in the Tower again, as passive as ever, but on the very day of\nthe battle of Barnet Queen Margaret had landed at Weymouth with her son,\nand the war would be renewed in Somersetshire.\n\nSearch for prisoners being over at Barnet, Hal was removed to the guest\nchamber of his hosts, where he lay in a huge square bed, and in the\nbetter air began to recover, understand what was going on round him,\nand be anxious for his friends, especially Sir Giles Musgrave and Simon\nBunce. The ladies still attended to him, as Lorimer pronounced the\njourney to be absolutely unsafe, while so many soldiers disbanded, or on\ntheir way to the Queen's army, were roaming about, and the Burgundians\nbrought by Edward might not be respectful to an English Prioress. It was\nsafer to wait for tidings from Lord St. John, which were certain to come\neither from Bletso or the Minoresses'.\n\nSo May had begun when Lorimer hurried home with the tidings that a\nmessenger had come in haste from King Edward from the battlefield of\nTewkesbury, with the tidings of a complete victory. Prince Edward, the\nfair and spirited hope of Lancaster, was slain, Somerset and his friends\nhad taken sanctuary in the Abbey Church, Queen Margaret and the young\nwife of the prince in a small convent, and beyond all had been flight\nand slaughter.\n\nFor a few days no more was known, but then came fuller and sadder\ntidings. The young prince had been brutally slain by his cousins,\nEdward, George, and Richard, excited as they were to tiger-like ferocity\nby the late revolt. The nobles in the sanctuary, who had for one night\nbeen protected by a cord drawn in front of them by a priest, had in the\nmorning been dragged out and beheaded. Among them was Anne's father,\nLord St. John of Bletso, and on the field the heralds had recognised the\ncorpse of her suitor, Lord Redgrave. To expect that Anne felt any acute\nsorrow for a father whom she had never seen since she was six years old,\nand who then had never seemed to care for her, was not possible.\n\nAnd what was to be her fate? Her young brother, the heir of Bletso, was\nin Flanders with his foreign mother, and she knew not what might be\nher own claims through her own mother, though the Prioress and Master\nLorimer knew that it could be ascertained through the seneschal at\nBletso, if he had not perished with his lord, or the agents at York\nthrough whom Anne's pension had been paid. If she were an heiress, she\nwould become a ward of the Crown, a dreary prospect, for it meant to be\ndisposed of to some unknown minion of the Court.\n\n\n\n\nCHAPTER XXII. -- THE NUT-BROWN MAID\n\n\n\n     All my wellfare to trouble and care\n       Should change if you were gone,\n     For in my mynde, of all mankind\n       I love but you alone.\n     --NUT-BROWN MAID.\n\n\nAnne St. John, in her 'doul' or deep mourning, sat by Hal's couch or\ndaybed in tears, as he lay in the deep bay of the mullioned window, and\ntold him of the consultation that had been held.\n\n'Ah, dear lady!' he said, 'now am I grieved that I have not mine own to\nendow you with! Well would I remain the landless shepherd were it not\nfor you.'\n\n'Nay,' she said, looking up through her tears, 'and wherefore should I\nnot share your shepherd's lot?'\n\n'You! Nan, sweet Nan, tenderly nurtured in the convent while I have ever\nlived as a rough hardy shepherd!'\n\n'And I have ever been a moorland maid,' she answered, 'bred to no soft\nways. I know not how to be the lady of a castle--I shall be a much\nbetter herdsman's wife, like your good old Dolly, whom I have always\nloved and envied.'\n\n'You never saw us snowed up in winter with all things scarce, and hardly\nable to milk a goat.'\n\n'Have not we been snowed up at Greystone for five weeks at a time?'\n\n'Ay, but with thick walls round and a stack of peat at hand,' said Hal,\nhis heart beating violently as more and more he felt that the maiden did\nnot speak in jest, but in full earnestness of love.\n\n'Verily one would deem you took me for a fine dainty dame, such as I saw\nat the Minoresses', shivering at the least gust of fresh wind, and not\ndaring to wet their satin shoes if there had been a shower of rain\nin the cloisters. Were we not all stifled within the walls, and never\nbreathed till we were out of them? Nay, Hal, there is none to come\nbetween us now. Take me to your moors and hills! I will be your good\nhousewife and shepherdess, and make you such a home! And you will teach\nme of the stars and of the flowers and all the holy lore of your good\nroyal hermit.'\n\n'Ah! my hermit, my master, how fares it with him? Would that I could go\nand see!'\n\n'Which do you love best--me or the hermit?' asked Anne archly, lifting\nup her head, which was lying on his shoulder.\n\n'I love you, mine own love and sweetheart, with all my heart,' he said,\nregaining her hand, 'but my King and master with my soul; and oh! that\nI had any strength to give him! I love him as my master in holy things,\nand as my true prince, and what would I not give to know how it is with\nhim and how he bears these dreadful tidings!'\n\nHe bent his head, choking with sobs as he spoke, and Anne wept with him,\nher momentary jealousy subdued by the picture of the lonely prisoner,\nhis friends slain in his cause, and his only child cut off in early\nprime; but she tried the comfort of hoping that his Queen would be with\nhim. Thus talking now of love, now of grief, now of the future, now of\nthe past, the Prioress found them, and as she was inclined to blame\nAnne for letting her patient weep, the maiden looked up to her and said,\n'Dear Mother, we are disputing--I want this same Hal to wed me so soon\nas he can stand and walk. Then I would go home with him to Derwentside,\nand take care of him.'\n\nThe Prioress burst out laughing. 'Make porridge, milk the ewes and spin\ntheir wool? Eh? Meet work for a baron's daughter!'\n\n'So I tell her,' said Harry. 'She knows not how hard the life is.'\n\n'Do I not?' said Anne. 'Have I not spent a night and day, the happiest\nmy childhood knew, in your hut? Has it not been a dream of joy ever\nsince?'\n\n'Ay, a summer's dream!' said Hal. 'Tell her the folly of it.'\n\n'I verily believe he does not want me. If he had not a lame leg, I trow\nhe would be trying to be mewed up with his King!'\n\n'It would be my duty,' murmured Hal, 'nor should I love thee the less.'\n\n''Tis a duty beyond your reach,' said the Prioress. 'Master Lorimer\nhears that none have access to King Henry, God help him! and he sits as\nin a trance, as though he understood and took heed of nothing--not even\nof this last sore battle.'\n\n'God aid him! Aye, and his converse is with Him,' said Hal, with a gush\nof tears. 'He minds nought of earth, not even earthly griefs.'\n\n'But we, we are of earth still, and have our years before us,' said\nAnne, 'and I will not spend mine the dreary lady of a dull castle.\nEither I will back and take my vows in your Priory, reverend Mother, if\nHal there disdains to have me.'\n\n'Nan, Nan! when you know that all I dread is to have you mewed behind\na wall of snow as thick as the walls of the Tower and freezing to the\nbone!'\n\n'With you behind it telling all the tales. Mother, prithee prove to him\nthat I am not made of sugar like the Clares, but that I love a fresh\nwind and the open moorlands.'\n\nThe Prioress laughed and took her away, but in private the maiden\nconvinced her that the proposal, however wild, was in full earnest, and\nnot in utter ignorance of the way of life that was preferred.\n\nAfterwards the good lady discussed it with the Lorimers. 'For my part,'\nshe said, 'I see nought to gainsay the children having their way. They\nare equal in birth and breeding, and love one another heartily, and the\ntimes may turn about to bring them to their own proper station.'\n\n'But the hardness and the roughness of the life,' objected Mistress\nLorimer, 'for a dainty, convent-bred lady.'\n\n'My convent--God, forgive me!--is not like the Poor Clares. We knew\nthere what cold and hunger mean, as well as what free air and mountains\nare. Moreover, though the maid thinks not of it, I do not believe the\nlife will be so bare and comfortless. The lad's mother hath not let him\nwant, and there is a heritage through the Vescis that must come to him,\neven if he never can claim the lands of Clifford.'\n\n'And now that all Lancaster is gone, King Edward may be less vindictive\nagainst the Red Rose,' said Lorimer.\n\n'There must be a dowry secured to the maid,' said the Prioress. 'Let\nthem only lie quiet for a time till the remains of the late tempest have\nblown over, and all will be well with them. Ay, and Master Lorimer, the\nLady Threlkeld, as well as myself, will fully acquit ourselves of the\nheavy charges you have been put to for your hospitality to us.'\n\nMaster Lorimer disclaimed all save his delight in the honour paid to\nhis poor house, and appealed to his wife, who seconded him courteously,\nthough perhaps the expenses of a wounded knight, three nuns, a noble\ndamsel and their horses, were felt by her enough to make the promise\ngratifying.\n\nWhile the elders talked, a horseman was heard in the court, asking\nwhether the young demoiselle of Bletso were lodged there. It was the\nseneschal Wenlock, who had come with what might be called the official\nreport of his lord's death, and to consider of the disposal of the young\nlady, being glad to find the Prioress of Greystone, to whom she had\noriginally been committed by her father.\n\nBefore summoning her, he explained to the Prioress that a small estate\nwhich had belonged to her mother devolved upon her. The proceeds of the\nproperty were not large, but they had been sufficient to keep her at the\nconvent, on the moderate charges of the time. Anne was only eighteen,\nand at no time of their lives were women, even widows, reckoned able to\ndispose of themselves. She would naturally become a ward of the Crown,\nand Lord Redgrave having been killed, the seneschal was about to go and\ninform King Edward of the situation.\n\n'But,' said the Prioress, 'suppose you found her already betrothed to\na gentleman of equal birth, and with claims to an even greater\ninheritance? Would you not be silent till the match was concluded, and\nthe King had no chance of breaking it?'\n\n'If it were well for the maid's honour and fortune,' said the seneschal.\n'If you, reverend Mother, have found a fair marriage for her, it might\nbe better to let well alone.'\n\nThen the Prioress set forth the situation and claims of young Clifford,\nand the certainty, that even if it were more prudent not to advance\nthem at present, yet the ruin of the house of Nevil removed one great\nbarrier, and at least the Vesci inheritance held by his mother must come\nto him, and she was the more likely to make a portion over to him when\nshe found that he had married nobly.\n\nThe seneschal acquiesced, even though the Prioress confessed that the\nbetrothal had not actually taken place. In fact he was relieved that the\nmaiden, whom he had known as a fair child, should be off his hands, and\nsecured from the greed of some Yorkist partisan needing a reward.\n\nWhen Anne, her dark eyes and hair shaded by her mourning veil, came\ndown, and had heard his greeting, with such details of her father's\ndeath and the state of the family as he could give her, she rose and\nsaid: 'Sir, there have been passages between Sir Harry Clifford and\nmyself, and I would wed none other than him.'\n\nNor did the seneschal gainsay her.\n\nAll that he desired was that what was decided upon should be done\nquickly, before heralds or lawyers brought to the knowledge of the\nWoodvilles that there was any sort of prize to be had in the damsel of\nSt. John, and he went off, early the next morning, back to Bletso, that\nhe might seem to know nothing of the matter.\n\nThe Prioress laughed at men being so much more afraid than women. She\nwas willing to bear all the consequences, but then the Plantagenets were\nnot in the habit of treating ladies as traitors. However, all agreed\nthat it would be wiser to be out of reach of London as soon as possible,\nand Master Lorimer, who had become deeply interested in this romance of\ntrue love, arranged to send one of his wains to York, in which the bride\nand bridegroom might travel unsuspected, until the latter should be able\nto ride and all were out of reach of pursuit. The Prioress would go thus\nfar with them, 'And then! And then,' she said sighing, 'I shall have to\ndree my penance for all my friskings!'\n\n'But, oh, what kindly friskings!' cried Anne, throwing herself into\nthose tender arms.\n\n'Little they will reck of kindness out of rule,' sighed the Prioress.\n'If only they will send me back to Greystone, then shall I hear of thee,\nand thou hadst better take Florimond, poor hound, or the Sisters at York\nmay put him to penance too!'\n\nHenry Clifford was able to walk again, though still lame, when, in the\nearly morning of Ascension Day, he and Anne St. John were married in the\nhall of Master Lorimer's house by a trusty priest of Barnet, and in the\nafternoon, when the thanksgiving worship at the church had been gone\nthrough, they started in the waggon for the first stage of the journey,\nto be overtaken at the halting-place by the Prioress and Master Lorimer,\nwho had had to ride into London to finish some business.\n\nAnd he brought tidings that rendered that wedding-day one of mournful,\nif peaceful, remembrances.\n\nFor he had seen, borne from the Tower, along Cheapside, the bier on\nwhich lay the body of King Henry, his hands clasped on his breast, his\nwhite face upturned with that heavenly expression which Hal knew so\nwell, enhanced into perfect peace, every toil, every grief at an end.\n\nWhether blood dropped as the procession moved along, Lorimer could not\ncertainly tell. Whether so it was, or whoever shed it, there was no\nmarring the absolute rest and joy that had crowned the 'meek usurper's\nholy head,' after his dreary half-century of suffering under the\nretribution of the ancestral sins of two lines of forefathers. All had\nbeen undergone in a deep and holy trust and faith such as could render\neven his hereditary insanity an actual shield from the poignancy of\ngrief.\n\nTears were shed, not bitter nor vengeful. Such thoughts would have\nseemed out of place with the memory of the gentle countenance of love,\ngood-will and peace, and as Harry and Anne joined in the service\nthat the Prioress had requested to have in the early daylight before\nstarting, Hal felt that to the hermit saint of his boyhood he verily\nowed his own self.\n\n\n\n\nCHAPTER XXIII. -- BROUGHAM CASTLE\n\n\n\n     And now am I an Earlis son,\n     And not a banished man.\n     --NUT-BROWN MAID.\n\n\nThat journey northward in the long summer days was a honeymoon to the\nyoung couple. The Prioress left them as much to themselves as possible,\ntrying to rejoice fully in their gladness, and not to think what might\nhave been hers but for that vow of her parents, keeping her hours\ndiligently in preparation for the stricter rule awaiting her.\n\nWhen they parted she sent Florimond with them, to be restored if she\nwere allowed to return to Greystone, and Anne parted with her with many\ntears as the truest mother and friend she had ever known.\n\nBy this time Harry was able to ride, and the two, with a couple of\nmen-at-arms hired as escort, made their way over the moors, Harry's\nhead throbbing with gladness, as, with a shout of joy, he hailed his own\nmountain-heads, Helvellyn and Saddleback, in all their purple cloud-like\nmajesty.\n\nThey agreed first to go to Dolly's homestead, drawn as much by affection\nas by prudence. Delight it was to Hal to point out the rocks and bushes\nof his home; but when he came in sight of Piers and the sheep, the dumb\nboy broke out into a cry of terror, and rushed away headlong, nor did\nhe turn till he felt Watch's very substantial paws bounding on him in\necstasy.\n\nWatch was indeed a forerunner, for Dolly and her husband could scarcely\nbe induced by his solid presence and caresses to come out and see for\nthemselves that the tall knight and lady were no ghostly shades, nor\nbewildered travellers, but that this was their own nursling Hal, whom\nSimon Bunce had reported to be lying dead under a gorse-bush at Barnet,\nand further that the lovely brunette lady was the little lost child whom\nDolly had mothered for a night.\n\nWhile the happy goodwife was regaling them with the best she had to\noffer, Hob set forth to announce their arrival at Threlkeld, being not\ncertain what the cautious Sir Lancelot would deem advisable, since the\nLancaster race had perished, and York was in the ascendant.\n\nThere was a long time to wait, but finally Sir Lancelot himself came\nriding through the wood, no longer afraid to welcome his stepson at the\ncastle, and the more willing since the bride newly arrived was no maiden\nof low degree, but a damsel of equal birth and with unquestioned rights.\n\nSo all was well, and the lady no longer had to embrace her son in fear\nand trembling, but to see him a handsome and thoughtful young man, well\nable to take his place in her halls.\n\nSince he had been actually in arms against King Edward it was not\nthought safe to assert his claims to his father's domains, but the lady\ngave up to him a portion of her own inheritance from the Vescis, where\nhe and Anne were able to live in Barden Tower in Yorkshire, not far from\nBolton Abbey. So Hal's shepherd days were over, though he still loved\ncountry habits and ways. Hob came to be once more his attendant, Dolly\nwas Anne's bower-woman, and Simon Bunce Sir Harry's squire, though he\nnever ceased blaming himself for having left his master, dead as he\nthought, when even a poor hound was more trusty.\n\nFlorimond was restored to the Prioress, who was reinstated at Greystone,\na graver woman than before she had set forth, the better for having\nwatched deeper devotion at the Minoresses', and still more for the\nterrible realities of the battle of Barnet. At Bolton Abbey Harry found\nmonks who encouraged his craving for information on natural science,\nand could carry him on much farther in these researches than his hermit,\nthough he always maintained that the royal anchorite and prisoner saw\nfarther into heavenly things than any other whom he had known, and\nthat his soul and insight rose the higher with his outward troubles and\nbodily decay.\n\nSo peacefully went the world with them till Henry was one-and-thirty,\nand then the tidings of Bosworth Field came north. The great tragedy of\nPlantagenet was complete, and the ambitious and blood-stained house\nof York, who had avenged the usurpation of Henry of Lancaster, had\nperished, chiefly by the hands of each other, and the distantly related\ndescendant of John of Gaunt, Henry Tudor, triumphed.\n\nThe Threlkelds were not slow to recollect that it was time for the\nCliffords to show their heads; moreover, that the St. Johns of Bletso\nwere related to the Tudors. Though now an aged woman, she descended\nfrom her hills, called upon her son and his wife with their little\nnine-year-old son to come with her, and pay homage to the new sovereign\nin their own names, and rode with them to Westminster.\n\nThere a very different monarch from the saint of Harry's memory received\nand favoured him. The lands of Westmoreland were granted to him as his\nright, and on their return, Master Lorimer coming by special invitation,\nthe family were welcomed at Brougham Castle, the cradle of their\nrace, where Harry Clifford, no longer an outlaw, began the career thus\ndescribed:\n\n\n          Love had he found in huts where poor men lie,\n            His daily teachers had been woods and rills,\n          The silence that is in the starry sky,\n            The sleep that is among the lonely hills.\n\n          In him the savage virtue of the race,\n            Revenge, and all ferocious thoughts were dead,\n          Nor did he change, but kept in lofty place\n            The wisdom that adversity had bred.\n\n          Glad were the vales, and every cottage hearth,\n            The Shepherd Lord was honoured more and more,\n          And ages after he was laid in earth\n            The Good Lord Clifford was the name he bore.\n\n\n\nFINIS\n\n\n\n\n\n\nEnd of Project Gutenberg's The Herd Boy and His Hermit, by Charlotte M. Yonge\n\n*** ","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n\nIt had been the Pagan Stone for hundreds of years, long before three boys stood around it and spilled their blood in a bond of brotherhood, unwittingly releasing a force bent on destruction...\n\nEvery seven years, there comes a week in July when the locals do unspeakable things\u2014and then don't seem to remember them. The collective madness has made itself known beyond the town borders and has given Hawkins Hollow the reputation of a village possessed.\n\nThis modern-day legend draws reporter and author Quinn Black to Hawkins Hollow with the hope of making the eerie happening the subject of her new book. It is only February, but Caleb Hawkins, descendent of the town founders, has already seen and felt the stirrings of evil. Though he can never forget the beginning of the terror in the woods twenty-one years ago, the signs have never been this strong before. Cal will need the help of his best friends, Fox and Gage, but surprisingly he must rely on Quinn as well. She, too, can see the evil that the locals cannot, somehow connecting her to the town\u2014and to Cal. As winter turns to spring, Cal and Quinn will shed their inhibitions, surrendering to a growing desire. They will form the cornerstone of a group of men and women bound by fate, passion, and the fight against what is to come from out of the darkness...\n\nTurn the page for a complete list of titles by Nora Roberts and J. D. Robb from The Berkley Publishing Group...\n\n## Nora Roberts & J. D. Robb\n\nREMEMBER WHEN\n\nNora Roberts\n\nHOT ICE\n\nSACRED SINS\n\nBRAZEN VIRTUE\n\nSWEET REVENGE\n\nPUBLIC SECRETS\n\nGENUINE LIES\n\nCARNAL INNOCENCE\n\nDIVINE EVIL\n\nHONEST ILLUSIONS\n\nPRIVATE SCANDALS\n\nHIDDEN RICHES\n\nTRUE BETRAYALS\n\nMONTANA SKY\n\nSANCTUARY\n\nHOMEPORT\n\nTHE REEF\n\nRIVER'S END\n\nCAROLINA MOON\n\nTHE VILLA\n\nMIDNIGHT BAYOU\n\nTHREE FATES\n\nBIRTHRIGHT\n\nNORTHERN LIGHTS\n\nBLUE SMOKE\n\nANGELS FALL\n\nHIGH NOON\n\nSeries\n\nBorn In Trilogy\n\nBORN IN FIRE\n\nBORN IN ICE\n\nBORN IN SHAME\n\nDream Trilogy\n\nDARING TO DREAM\n\nHOLDING THE DREAM\n\nFINDING THE DREAM\n\nChesapeake Bay Saga\n\nSEA SWEPT\n\nRISING TIDES\n\nINNER HARBOR\n\nCHESAPEAKE BLUE\n\nGallaghers of Ardmore Trilogy\n\nJEWELS OF THE SUN\n\nTEARS OF THE MOON\n\nHEART OF THE SEA\n\nThree Sisters Island Trilogy\n\nDANCE UPON THE AIR\n\nHEAVEN AND EARTH\n\nFACE THE FIRE\n\nKey Trilogy\n\nKEY OF LIGHT\n\nKEY OF KNOWLEDGE\n\nKEY OF VALOR\n\nIn the Garden Trilogy\n\nBLUE DAHLIA\n\nBLACK ROSE\n\nRED LILY\n\nCircle Trilogy\n\nMORRIGAN'S CROSS\n\nDANCE OF THE GODS\n\nVALLEY OF SILENCE\n\nSign of Seven Trilogy\n\nBLOOD BROTHERS\n\nAnthologies\n\nFROM THE HEART\n\nA LITTLE MAGIC\n\nA LITTLE FATE\n\nMOON SHADOWS (with Jill Gregory, Ruth Ryan Langan, and Marianne Willman)\n\nThe Once Upon Series (with Jill Gregory, Ruth Ryan Langan, and Marianne Willman)\n\nONCE UPON A CASTLE\n\nONCE UPON A STAR\n\nONCE UPON A DREAM\n\nONCE UPON A ROSE\n\nONCE UPON A KISS\n\nONCE UPON A MIDNIGHT\n\nJ. D. Robb\n\nNAKED IN DEATH\n\nGLORY IN DEATH\n\nIMMORTAL IN DEATH\n\nRAPTURE IN DEATH\n\nCEREMONY IN DEATH\n\nVENGEANCE IN DEATH\n\nHOLIDAY IN DEATH\n\nCONSPIRACY IN DEATH\n\nLOYALTY IN DEATH\n\nWITNESS IN DEATH\n\nJUDGMENT IN DEATH\n\nBETRAYAL IN DEATH\n\nSEDUCTION IN DEATH\n\nREUNION IN DEATH\n\nPURITY IN DEATH\n\nPORTRAIT IN DEATH\n\nIMITATION IN DEATH\n\nDIVIDED IN DEATH\n\nVISIONS IN DEATH\n\nSURVIVOR IN DEATH\n\nORIGIN IN DEATH\n\nMEMORY IN DEATH\n\nBORN IN DEATH\n\nINNOCENT IN DEATH\n\nCREATION IN DEATH\n\nAnthologies\n\nSILENT NIGHT (with Susan Plunkett, Dee Holmes, and Claire Cross)\n\nOUT OF THIS WORLD (with Laurell K. Hamilton, Susan Krinard, and Maggie Shayne)\n\nBUMP IN THE NIGHT (with Mary Blayney, Ruth Ryan Langan, and Mary Kay McComas)\n\nDEAD OF NIGHT (with Mary Blayney, Ruth Ryan Langan, and Mary Kay McComas)\n\nAlso available...\n\nTHE OFFICIAL NORA ROBERTS COMPANION (edited by Denise Little and Laura Hayden)\n\n## NORA ROBERTS\n\n## BLOOD BROTHERS\n\nJOVE BOOKS, NEW YORK\nTHE BERKLEY PUBLISHING GROUP  \nPublished by the Penguin Group  \nPenguin Group (USA) Inc. 375 Hudson Street, New York, New York 10014, USA  \nPenguin Group (Canada), 90 Eglinton Avenue East, Suite 700, Toronto, Ontario M4P 2Y3, Canada (a division of Pearson Penguin Canada Inc.)  \nPenguin Books Ltd., 80 Strand, London WC2R 0RL, England  \nPenguin Group Ireland, 25 St. Stephen's Green, Dublin 2, Ireland (a division of Penguin Books Ltd.) Penguin Group (Australia), 250 Camberwell Road, Camberwell, Victoria 3124, Australia (a division of Pearson Australia Group Pty. Ltd.)  \nPenguin Books India Pvt. Ltd., 11 Community Centre, Panchsheel Park, New Delhi\u2014110 017, India Penguin Group (NZ), 67 Apollo Drive, Rosedale, North Shore 0632, New Zealand (a division of Pearson New Zealand Ltd.)  \nPenguin Books (South Africa) (Pty.) Ltd., 24 Sturdee Avenue, Rosebank, Johannesburg 2196, South Africa\n\nPenguin Books Ltd., Registered Offices: 80 Strand, London WC2R 0RL, England\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, business establishments, events, or locales is entirely coincidental. The publisher does not have any control over and does not assume any responsibility for author or third-party websites or their content.\n\nBLOOD BROTHERS\n\nA Jove Book \/ published by arrangement with the author\n\nCopyright \u00a9 2007 by Nora Roberts.  \nExcerpt from The Hollow copyright \u00a9 2007 by Nora Roberts.\n\nAll rights reserved.  \nNo part of this book may be reproduced, scanned, or distributed in any printed or electronic form without permission. Please do not participate in or encourage piracy of copyrighted materials in violation of the author's rights. Purchase only authorized editions.  \nFor information, address: The Berkley Publishing Group,  \na division of Penguin Group (USA) Inc.,  \n375 Hudson Street, New York, New York 10014.\n\nISBN: 1-101-14733-4\n\nJOVE\u00ae  \nJove Books are published by The Berkley Publishing Group, a division of Penguin Group (USA) Inc., 375 Hudson Street, New York, New York 10014.  \nJOVE\u00ae is a registered trademark of Penguin Group (USA) Inc.  \nThe \"J\" design is a trademark belonging to Penguin Group (USA) Inc.\nTo my boys,  \nwho roamed the woods,  \neven when they weren't supposed to.\nWhere God hath a temple,  \nthe Devil will have a chapel.\n\n\u2014ROBERT BURTON\n\nThe childhood shows the man  \nAs morning shows the day.\n\n\u2014JOHN MILTON\n\n## Contents\n\nPrologue\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\n## Prologue\n\nHawkins Hollow  \nMaryland Province  \n1652\n\nIT CRAWLED ALONG THE AIR THAT HUNG HEAVY as wet wool over the glade. Through the snakes of fog that slid silent over the ground, its hate crept. It came for him through the heat-smothered night.\n\nIt wanted his death.\n\nSo he waited as it pushed its way through the woods, its torch raised toward the empty sky, as it waded across the streams, around the thickets where small animals huddled in fear of the scent it bore with it.\n\nHellsmoke.\n\nHe had sent Ann and the lives she carried in her womb away, to safety. She had not wept, he thought now as he sprinkled the herbs he'd selected over water. Not his Ann. But he had seen the grief on her face, in the deep, dark eyes he had loved through this lifetime, and all the others before.\n\nThe three would be born from her, raised by her, and taught by her. And from them, when the time came, there would be three more.\n\nWhat power he had would be theirs, these sons, who would loose their first cries long, long after this night's work was done. To leave them what tools they would need, the weapons they would wield, he risked all he had, all he was.\n\nHis legacy to them was in blood, in heart, in vision.\n\nIn this last hour he would do all he could to provide them with what was needed to carry the burden, to remain true, to see their destiny.\n\nHis voice was strong and clear as he called to wind and water, to earth and fire. In the hearth the flames snapped. In the bowl the water trembled.\n\nHe laid the bloodstone on the cloth. Its deep green was generously spotted with red. He had treasured this stone, as had those who'd come before him. He had honored it. And now he poured power into it as one would pour water into a cup.\n\nSo his body shook and sweat and weakened as light hovered in a halo around the stone.\n\n\"For you now,\" he murmured, \"sons of sons. Three parts of one. In faith, in hope, in truth. One light, united, to strike back dark. And here, my vow. I will not rest until destiny is met.\"\n\nWith the athame, he scored his palm so his blood fell onto the stone, into the water, and into the flame.\n\n\"Blood of my blood. Here I will hold until you come for me, until you loose what must be loosed again on the world. May the gods keep you.\"\n\nFor a moment there was grief. Even through his purpose, there was grief. Not for his life, as the sands of it were dripping down the glass. He had no fear of death. No fear of what he would soon embrace that was not death. But he grieved that he would never lay his lips on Ann's again in this life. He would not see his children born, nor the children of his children. He grieved that he would not be able to stop the suffering to come, as he had been unable to stop the suffering that had come before, in so many other lifetimes.\n\nHe understood that he was not the instrument, but only the vessel to be filled and emptied at the needs of the gods.\n\nSo, weary from the work, saddened by the loss, he stood outside the little hut, beside the great stone, to meet his fate.\n\nIt came in the body of a man, but that was a shell. As his own body was a shell. It called itself Lazarus Twisse, an elder of \"the godly.\" He and those who followed had settled in the wilderness of this province when they broke with the Puritans of New England.\n\nHe studied them now in their torchlight, these men and the one who was not a man. These, he thought, who had come to the New World for religious freedom, and then persecuted and destroyed any who did not follow their single, narrow path.\n\n\"You are Giles Dent.\"\n\n\"I am,\" he said, \"in this time and this place.\"\n\nLazarus Twisse stepped forward. He wore the unrelieved formal black of an elder. His high-crowned, wide-brimmed hat shadowed his face. But Giles could see his eyes, and in his eyes, he saw the demon.\n\n\"Giles Dent, you and the female known as Ann Hawkins have been accused and found guilty of witchcraft and demonic practices.\"\n\n\"Who accuses?\"\n\n\"Bring the girl forward!\" Lazarus ordered.\n\nThey pulled her, a man on each arm. She was a slight girl, barely six and ten by Giles's calculation. Her face was wax white with fear, her eyes drenched with it. Her hair had been shorn.\n\n\"Hester Deale, is this the witch who seduced you?\"\n\n\"He and the one he calls wife laid hands on me.\" She spoke as if in a trance. \"They performed ungodly acts upon my body. They came to my window as ravens, flew into my room in the night. They stilled my throat so I could not speak or call for help.\"\n\n\"Child,\" Giles said gently, \"what has been done to you?\"\n\nThose fear-swamped eyes stared through him. \"They called to Satan as their god, and cut the throat of a cock in sacrifice. And drank its blood. They forced its blood on me. I could not stop them.\"\n\n\"Hester Deale, do you renounce Satan?\"\n\n\"I do renounce him.\"\n\n\"Hester Deale, do you renounce Giles Dent and the woman Ann Hawkins as witches and heretics?\"\n\n\"I do.\" Tears spilled down her cheeks. \"I do renounce them, and pray to God to save me. Pray to God to forgive me.\"\n\n\"He will,\" Giles whispered. \"You are not to blame.\"\n\n\"Where is the woman Ann Hawkins?\" Lazarus demanded, and Giles turned his clear gray eyes to him.\n\n\"You will not find her.\"\n\n\"Stand aside. I will enter this house of the devil.\"\n\n\"You will not find her,\" Giles repeated. For a moment he looked beyond Lazarus to the men and the handful of women who stood in his glade.\n\nHe saw death in their eyes, and more, the hunger for it. This was the demon's power, and his work.\n\nOnly in Hester's did Giles see fear or sorrow. So he used what he had to give, pushed his mind toward hers. Run!\n\nHe saw her jolt, stumble back, then he turned to Lazarus.\n\n\"We know each other, you and I. Dispatch them, release them, and it will be between us alone.\"\n\nFor an instant he saw the gleam of red in Lazarus's eyes. \"You are done. Burn the witch!\" he shouted. \"Burn the devil house and all within it!\"\n\nThey came with torches, and with clubs. Giles felt the blows rain on him, and the fury of the hate that was the demon's sharpest weapon.\n\nThey drove him to his knees, and the wood of the hut began to flame and smoke. Screams rang in his head, the madness of them.\n\nWith the last of his power he reached out toward the demon inside the man, with red rimming its dark eyes as it fed on the hate, the fear, the violence. He felt it gloat, he felt it rising, so sure of its victory, and the feast to follow.\n\nAnd he ripped it to him, through the smoking air. He heard it scream in fury and pain as the flames bit into flesh. And he held it to him, close as a lover as the fire consumed them.\n\nAnd with that union the fire burst, spread, destroyed every living thing in the glade.\n\nIt burned for a day and a night, like the belly of hell.\n\n## One\n\nHawkins Hollow  \nMaryland  \nJuly 6, 1987\n\nINSIDE THE PRETTY KITCHEN OF THE PRETTY house on Pleasant Avenue, Caleb Hawkins struggled not to squirm as his mother packed her version of campout provisions.\n\nIn his mother's world, ten-year-old boys required fresh fruit, homemade oatmeal cookies (they weren't so bad), half a dozen hard-boiled eggs, a bag of Ritz crackers made into sandwiches with Jif peanut butter for filling, some celery and carrot sticks (yuck!), and hearty ham-and-cheese sandwiches.\n\nThen there was the thermos of lemonade, the stack of paper napkins, and the two boxes of Pop-Tarts she wedged into the basket for breakfast.\n\n\"Mom, we're not going to starve to death,\" he complained as she stood deliberating in front of an open cupboard. \"We're going to be right in Fox's backyard.\"\n\nWhich was a lie, and kinda hurt his tongue. But she'd never let him go if he told her the truth. And, sheesh, he was ten. Or would be the very next day.\n\nFrannie Hawkins put her hands on her hips. She was a pert, attractive blonde with summer blue eyes and a stylish curly perm. She was the mother of three, and Cal was her baby and only boy. \"Now, let me check that backpack.\"\n\n\"Mom!\"\n\n\"Honey, I just want to be sure you didn't forget anything.\" Ruthless in her own sunny way, Frannie unzipped Cal's navy blue pack. \"Change of underwear, clean shirt, socks, good, good, shorts, toothbrush. Cal, where are the Band-Aids I told you to put in, and the Bactine, the bug repellant.\"\n\n\"Sheesh, we're not going to Africa.\"\n\n\"All the same,\" Frannie said, and did her signature finger wave to send him along to gather up the supplies. While he did, she slipped a card out of her pocket and tucked it into the pack.\n\nHe'd been born\u2014after eight hours and twelve minutes of vicious labor\u2014at one minute past midnight. Every year she stepped up to his bed at twelve, watched him sleep for that minute, then kissed him on the cheek.\n\nNow he'd be ten, and she wouldn't be able to perform the ritual. Because it made her eyes sting, she turned away to wipe at her spotless counter as she heard his tromping footsteps.\n\n\"I got it all, okay?\"\n\nSmiling brightly, she turned back. \"Okay.\" She stepped over to rub a hand over his short, soft hair. He'd been her towheaded baby boy, she mused, but his hair was darkening, and she suspected it would be a light brown eventually.\n\nJust as hers would be without the aid of Born Blonde.\n\nIn a habitual gesture, Frannie tapped his dark-framed glasses back up his nose. \"You make sure you thank Miss Barry and Mr. O'Dell when you get there.\"\n\n\"I will.\"\n\n\"And when you leave to come home tomorrow.\"\n\n\"Yes, ma'am.\"\n\nShe took his face in her hands, looked through the thick lenses into eyes the same color as his father's calm gray ones. \"Behave,\" she said and kissed his cheek. \"Have fun.\" Then the other. \"Happy birthday, my baby.\"\n\nUsually it mortified him to be called her baby, but for some reason, just then, it made him feel sort of gooey and good.\n\n\"Thanks, Mom.\"\n\nHe shrugged on the backpack, then hefted the loaded picnic basket. How the hell was he going to ride all the way out to Hawkins Wood with half the darn grocery store on his bike?\n\nThe guys were going to razz him something fierce.\n\nSince he was stuck, he carted it into the garage where his bike hung tidily\u2014by Mom decree\u2014on a rack on the wall. Thinking it through, he borrowed two of his father's bungee cords and secured the picnic basket to the wire basket of his bike.\n\nThen he hopped on his bike and pedaled down the short drive.\n\nFOX FINISHED WEEDING HIS SECTION OF THE vegetable garden before hefting the spray his mother mixed up weekly to discourage the deer and rabbits from invading for an all-you-can-eat buffet. The garlic, raw egg, and cayenne pepper combination stank so bad he held his breath as he squirted it on the rows of snap beans and limas, the potato greens, the carrot and radish tops.\n\nHe stepped back, took a clear breath, and studied his work. His mother was pretty damn strict about the gardening. It was all about respecting the earth, harmonizing with Nature, and that stuff.\n\nIt was also, Fox knew, about eating, and making enough food and money to feed a family of six\u2014and whoever dropped by. Which was why his dad and his older sister, Sage, were down at their stand selling fresh eggs, goat's milk, honey, and his mother's homemade jams.\n\nHe glanced over to where his younger brother, Ridge, was stretched out between the rows playing with the weeds instead of yanking them. And because his mother was inside putting their baby sister, Sparrow, down for her nap, he was on Ridge duty.\n\n\"Come on, Ridge, pull the stupid things. I wanna go.\"\n\nRidge lifted his face, turned his I'm-dreaming eyes on his brother. \"Why can't I go with you?\"\n\n\"Because you're eight and you can't even weed the dumb tomatoes.\" Annoyed, Fox stepped over the rows to Ridge's section and, crouching, began to yank.\n\n\"Can, too.\"\n\nAs Fox hoped, the insult had Ridge weeding with a vengeance. Fox straightened, rubbed his hands on his jeans. He was a tall boy with a skinny build, a mass of bark brown hair worn in a waving tangle around a sharp-boned face. His eyes were tawny and reflected his satisfaction now as he trooped over for the sprayer.\n\nHe dumped it beside Ridge. \"Don't forget to spray this shit.\"\n\nHe crossed the yard, circling what was left\u2014three short walls and part of a chimney\u2014of the old stone hut on the edge of the vegetable garden. It was buried, as his mother liked it best, in honeysuckle and wild morning glory.\n\nHe skirted past the chicken coop and the cluckers that were pecking around, by the goat yard where the two nannies stood slack-hipped and bored, edged around his mother's herb garden. He headed toward the kitchen door of the house his parents had mostly built. The kitchen was big, and the counters loaded with projects\u2014canning jars, lids, tubs of candle wax, bowls of wicks.\n\nHe knew most of the people in and around the Hollow thought of his family as the weird hippies. It didn't bother him. For the most part they got along, and people were happy to buy their eggs and produce, his mother's needlework and handmade candles and crafts, or hire his dad to build stuff.\n\nFox washed up at the sink before rooting through the cupboards, poking in the big pantry searching for something that wasn't health food.\n\nFat chance.\n\nHe'd bike over to the market\u2014the one right outside of town just in case\u2014and use some of his savings to buy Little Debbies and Nutter Butters.\n\nHis mother came in, tossing her long brown braid off the shoulder bared by her cotton sundress. \"Finished?\"\n\n\"I am. Ridge is almost.\"\n\nJoanne walked to the window, her hand automatically lifting to brush down Fox's hair, staying to rest on his neck as she studied her young son.\n\n\"There's some carob brownies and some veggie dogs, if you want to take any.\"\n\n\"Ah.\" Barf. \"No, thanks. I'm good.\"\n\nHe knew that she knew he'd be chowing down on meat products and refined sugar. And he knew she knew he knew. But she wouldn't rag him about it. Choices were big with Mom.\n\n\"Have a good time.\"\n\n\"I will.\"\n\n\"Fox?\" She stood where she was, by the sink with the light coming in the window and haloing her hair. \"Happy birthday.\"\n\n\"Thanks, Mom.\" And with Little Debbies on his mind, he bolted out to grab his bike and start the adventure.\n\nTHE OLD MAN WAS STILL SLEEPING WHEN GAGE shoved some supplies into his pack. Gage could hear the snoring through the thin, crappy walls of the cramped, crappy apartment over the Bowl-a-Rama. The old man worked there cleaning the floors, the johns, and whatever else Cal's father found for him to do.\n\nHe might've been a day shy of his tenth birthday, but Gage knew why Mr. Hawkins kept the old man on, why they had the apartment rent-free with the old man supposedly being the maintenance guy for the building. Mr. Hawkins felt sorry for them\u2014and mostly sorry for Gage because he was stuck as the motherless son of a mean drunk.\n\nOther people felt sorry for him, too, and that put Gage's back up. Not Mr. Hawkins though. He never let the pity show. And whenever Gage did any chores for the bowling alley, Mr. Hawkins paid him in cash, on the side. And with a conspirator's wink.\n\nHe knew, hell, everybody knew, that Bill Turner knocked his kid around from time to time. But Mr. Hawkins was the only one who'd ever sat down with Gage and asked him what he wanted. Did he want the cops, Social Services, did he want to come stay with him and his family for a while?\n\nHe hadn't wanted the cops or the do-gooders. They only made it worse. And though he'd have given anything to live in that nice house with people who lived decent lives, he'd only asked if Mr. Hawkins would please, please, not fire his old man.\n\nHe got knocked around less whenever Mr. Hawkins kept his father busy and employed. Unless, of course, good old Bill went on a toot and decided to whale in.\n\nIf Mr. Hawkins knew how bad it could get during those times, he would call the cops.\n\nSo he didn't tell, and he learned to be very good at hiding beatings like the one he'd taken the night before.\n\nGage moved carefully as he snagged three cold ones out of his father's beer supply. The welts on his back and butt were still raw and angry and they stung like fire. He'd expected the beating. He always got one around his birthday. He always got another one around the date of his mother's death.\n\nThose were the big, traditional two. Other times, the whippings came as a surprise. But mostly, when the old man was working steady, the hits were just a careless cuff or shove.\n\nHe didn't bother to be quiet when he turned toward his father's bedroom. Nothing short of a raid by the A-Team would wake Bill Turner when he was in a drunken sleep.\n\nThe room stank of beer sweat and stale smoke, causing Gage to wrinkle his handsome face. He took the half pack of Marlboros off the dresser. The old man wouldn't remember if he'd had any, so no problem there.\n\nWithout a qualm, he opened his father's wallet and helped himself to three singles and a five.\n\nHe looked at his father as he stuffed the bills in his pocket. Bill sprawled on the bed, stripped down to his boxers, his mouth open as the snores pumped out.\n\nThe belt he'd used on his son the night before lay on the floor along with dirty shirts, socks, jeans.\n\nFor a moment, just a moment, it rippled through Gage with a kind of mad glee\u2014the image of himself picking up that belt, swinging it high, laying it snapping hard over his father's bare, sagging belly.\n\nSee how you like it.\n\nBut there on the table with its overflowing ashtray, the empty bottle, was the picture of Gage's mother, smiling out.\n\nPeople said he looked like her\u2014the dark hair, the hazy green eyes, the strong mouth. It had embarrassed him once, being compared to a woman. But lately, since everything but that one photograph was so faded in his head, when he couldn't hear her voice in his head or remember how she'd smelled, it steadied him.\n\nHe looked like his mother.\n\nSometimes he imagined the man who drank himself into a stupor most nights wasn't his father.\n\nHis father was smart and brave and sort of reckless.\n\nAnd then he'd look at the old man and know that was all bullshit.\n\nHe shot the old bastard the finger as he left the room. He had to carry his backpack. No way he could put it on with the welts riding his back.\n\nHe took the outside steps down, went around the back where he chained up his thirdhand bike.\n\nDespite the pain, he grinned as he got on.\n\nFor the next twenty-four hours, he was free.\n\nTHEY'D AGREED TO MEET ON THE WEST EDGE OF town where the woods crept toward the curve of the road. The middle-class boy, the hippie kid, and the drunk's son.\n\nThey shared the same birthday, July seventh. Cal had let out his first shocked cry in the delivery room of Washington County Hospital while his mother panted and his father wept. Fox had shoved his way into the world and into his laughing father's waiting hands in the bedroom of the odd little farmhouse while Bob Dylan sang \"Lay, Lady, Lay\" on the record player, and lavender-scented candles burned. And Gage had struggled out of his terrified mother in an ambulance racing up Maryland Route 65.\n\nNow, Gage arrived first, sliding off his bike to walk it into the trees where nobody cruising the road could spot it, or him.\n\nThen he sat on the ground and lit his first cigarette of the afternoon. They always made him a little sick to his stomach, but the defiant act of lighting up made up for the queasiness.\n\nHe sat and smoked in the shady woods, and imagined himself on a mountain path in Colorado or in a steamy South American jungle.\n\nAnywhere but here.\n\nHe'd taken his third puff, and his first cautious inhale, when he heard the bumps of tires over dirt and rock.\n\nFox pushed through the trees on Lightning, his bike so named because Fox's father had painted lightning bolts on the bars.\n\nHis dad was cool that way.\n\n\"Hey, Turner.\"\n\n\"O'Dell.\" Gage held out the cigarette.\n\nThey both knew Fox took it only because to do otherwise made him a dweeb. So he took a quick drag, passed it back. Gage nodded to the bag tied to Lightning's handlebars. \"What'd you get?\"\n\n\"Little Debbies, Nutter Butters, some Tasty Kake pies. Apple and cherry.\"\n\n\"Righteous. I got three cans of Bud for tonight.\"\n\nFox's eyes didn't pop out of his head, but they were close. \"No shit?\"\n\n\"No shit. Old man was trashed. He'll never know the difference. I got something else, too. Last month's Penthouse magazine.\"\n\n\"No way.\"\n\n\"He keeps them buried under a bunch of crap in the bathroom.\"\n\n\"Lemme see.\"\n\n\"Later. With the beer.\"\n\nThey both looked over as Cal dragged his bike down the rough path. \"Hey, jerkwad,\" Fox greeted him.\n\n\"Hey, dickheads.\"\n\nThat said with the affection of brothers, they walked their bikes deeper into the trees, then off the narrow path.\n\nOnce the bikes were deemed secure, supplies were untied and divvied up.\n\n\"Jesus, Hawkins, what'd your mom put in here?\"\n\n\"You won't complain when you're eating it.\" Cal's arms were already protesting the weight as he scowled at Gage. \"Why don't you put your pack on, and give me a hand?\"\n\n\"Because I'm carrying it.\" But he flipped the top on the basket and after hooting at the Tupperware, shoved a couple of the containers into his pack. \"Put something in yours, O'Dell, or it'll take us all day just to get to Hester's Pool.\"\n\n\"Shit.\" Fox pulled out a thermos, wedged it in his pack. \"Light enough now, Sally?\"\n\n\"Screw you. I got the basket and my pack.\"\n\n\"I got the supplies from the market and my pack.\" Fox pulled his prized possession from his bike. \"You carry the boom box, Turner.\"\n\nGage shrugged, took the radio. \"Then I pick the tunes.\"\n\n\"No rap,\" Cal and Fox said together, but Gage only grinned as he walked and tuned until he found some Run-DMC.\n\nWith a lot of bitching and moaning, they started the hike.\n\nThe leaves, thick and green, cut the sun's glare and summer heat. Through the thick poplars and towering oaks, slices and dabs of milky blue sky peeked. They aimed for the wind of the creek while the rapper and Aero-smith urged them to walk this way.\n\n\"Gage has a Penthouse,\" Fox announced. \"The skin magazine, numbnut,\" he said at Cal's blank stare.\n\n\"Uh-uh.\"\n\n\"Uh-huh. Come on, Turner, break it out.\"\n\n\"Not until we're camped and pop the beer.\"\n\n\"Beer!\" Instinctively, Cal sent a look over his shoulder, just in case his mother had magically appeared. \"You got beer?\"\n\n\"Three cans of suds,\" Gage confirmed, strutting. \"Smokes, too.\"\n\n\"Is this far-out or what?\" Fox gave Cal a punch in the arm. \"It's the best birthday ever.\"\n\n\"Ever,\" Cal agreed, secretly terrified. Beer, cigarettes, and pictures of naked women. If his mother ever found out, he'd be grounded until he was thirty. That didn't even count the fact he'd lied. Or that he was hiking his way through Hawkins Wood to camp out at the expressly forbidden Pagan Stone.\n\nHe'd be grounded until he died of old age.\n\n\"Stop worrying.\" Gage shifted his pack from one arm to the other, with a wicked glint of what-the-hell in his eyes. \"It's all cool.\"\n\n\"I'm not worried.\" Still, Cal jolted when a fat jay zoomed out of the trees and let out an irritated call.\n\n## Two\n\nHESTER'S POOL WAS ALSO FORBIDDEN IN CAL'S world, which was only one of the reasons it was irresistible.\n\nThe scoop of brown water, fed by the winding Antietam Creek and hidden in the thick woods, was supposed to be haunted by some weird Pilgrim girl who'd drowned in it way back whenever.\n\nHe'd heard his mother talk about a boy who'd drowned there when she'd been a kid, which in Mom Logic was the number one reason Cal was never allowed to swim there. The kid's ghost was supposed to be there, too, lurking under the water, just waiting to grab another kid's ankle and drag him down to the bottom so he'd have somebody to hang out with.\n\nCal had swum there twice that summer, giddy with fear and excitement. And both times he'd sworn he'd felt bony fingers brush over his ankle.\n\nA dense army of cattails trooped along the edges, and around the slippery bank grew bunches of the wild orange lilies his mother liked. Fans of ferns climbed up the rocky slope, along with brambles of wild berries, which when ripe would stain the fingers a kind of reddish purple that looked a little like blood.\n\nThe last time they'd come, he'd seen a black snake slither its way up the slope, barely stirring the ferns.\n\nFox let out a shout, dumped his pack. In seconds he'd dragged off his shoes, his shirt, his jeans and was sailing over the water in a cannonball without a thought for snakes or ghosts or whatever else might be under that murky brown surface.\n\n\"Come on, you pussies!\" After a slick surface dive, Fox bobbed around the pool like a seal.\n\nCal sat, untied his Converse All Stars, carefully tucked his socks inside them. While Fox continued to whoop and splash, he glanced over where Gage simply stood looking out over the water.\n\n\"You going in?\"\n\n\"I dunno.\"\n\nCal pulled off his shirt, folded it out of habit. \"It's on the agenda. We can't cross it off unless we all do it.\"\n\n\"Yeah, yeah.\" But Gage only stood as Cal stripped down to his Fruit of the Looms.\n\n\"We have to all go in, dare the gods and stuff.\"\n\nWith a shrug, Gage toed off his shoes. \"Go on, what are you, a homo? Want to watch me take my clothes off?\"\n\n\"Gross.\" And slipping his glasses inside his left shoe, Cal sucked in breath, gave thanks his vision blurred, and jumped.\n\nThe water was a quick, cold shock.\n\nFox immediately spewed water in his face, fully blinding him, then stroked off toward the cattails before retaliation. Just when he'd managed to clear his myopic eyes, Gage jumped in and blinded him all over again.\n\n\"Sheesh, you guys!\"\n\nGage's choppy dog paddle worked up the water, so Cal swam clear of the storm. Of the three, he was the best swimmer. Fox was fast, but he ran out of steam. And Gage, well, Gage sort of attacked the water like he was in a fight with it.\n\nCal worried\u2014even as part of him thrilled at the idea\u2014that he'd one day have to use the lifesaving techniques his dad had taught him in their aboveground pool to save Gage from drowning.\n\nHe was picturing it, and how Gage and Fox would stare at him with gratitude and admiration, when a hand grabbed his ankle and yanked him underwater.\n\nEven though he knew it was Fox who pulled him down, Cal's heart slammed into his throat as the water closed over his head. He floundered, forgetting all his training in that first instant of panic. Even as he managed to kick off the hold on his ankle and gather himself to push to the surface, he saw a movement to the left.\n\nIt\u2014she\u2014seemed to glide through the water toward him. Her hair streamed back from her white face, and her eyes were cave black. As her hand reached out, Cal opened his mouth to scream. Gulping in water, he clawed his way to the surface.\n\nHe could hear laughter all around him, tinny and echoing like the music out of the old transistor radio his father sometimes used. With terror biting inside his throat, he slapped and clawed his way to the edge of the pool.\n\n\"I saw her, I saw her, in the water, I saw her.\" He choked out the words while fighting to climb out.\n\nShe was coming for him, fast as a shark in his mind, and in his mind he saw her mouth open, and the teeth gleam sharp as knives.\n\n\"Get out! Get out of the water!\" Panting, he crawled through the slippery weeds and rolling, saw his friends treading water. \"She's in the water.\" He almost sobbed it, bellying over to fumble his glasses out of his shoe. \"I saw her. Get out. Hurry up!\"\n\n\"Oooh, the ghost! Help me, help me!\" With a mock gurgle, Fox sank underwater.\n\nCal lurched to his feet, balled his hands into fists at his sides. Fury tangled with terror to have his voice lashing through the still summer air. \"Get the fuck out.\"\n\nThe grin on Gage's face faded. Eyes narrowed on Cal, he gripped Fox by the arm when Fox surfaced laughing.\n\n\"We're getting out.\"\n\n\"Come on. He's just being spaz because I dunked him.\"\n\n\"He's not bullshitting.\"\n\nThe tone got through, or when he bothered to look, the expression on Cal's face tripped a chord. Fox shot off toward the edge, spooked enough to send a couple of wary looks over his shoulder.\n\nGage followed, a careless dog paddle that made Cal think he was daring something to happen.\n\nWhen his friends hauled themselves out, Cal sank back down to the ground. Drawing his knees up, he pressed his forehead to them and began to shake.\n\n\"Man.\" Dripping in his underwear, Fox shifted from foot to foot. \"I just gave you a tug, and you freak out. We were just fooling around.\"\n\n\"I saw her.\"\n\nCrouching, Fox shoved his sopping hair back from his face. \"Dude, you can't see squat without those Coke bottles.\"\n\n\"Shut up, O'Dell.\" Gage squatted down. \"What did you see, Cal?\"\n\n\"Her. She had all this hair swimming around her, and her eyes, oh man, her eyes were black like the shark in Jaws. She had this long dress on, long sleeves and all, and she reached out like she was going to grab me\u2014\"\n\n\"With her bony fingers,\" Fox put in, falling well short of his target of disdain.\n\n\"They weren't bony.\" Cal lifted his head now, and behind the lenses his eyes were fierce and frightened. \"I thought they would be, but she looked, all of her, looked just...real. Not like a ghost or a skeleton. Oh man, oh God, I saw her. I'm not making it up.\"\n\n\"Well Jee-sus.\" Fox crab-walked another foot away from the pond, then cursed breathlessly when he tore his forearm on berry thorns. \"Shit, now I'm bleeding.\" Fox yanked a handful of weedy grass, swiped at the blood seeping from the scratches.\n\n\"Don't even think about it.\" Cal saw the way Gage was studying the water\u2014that thoughtful, wonder-what'll-happen gleam in his eye. \"Nobody's going in there. You don't swim well enough to try it anyway.\"\n\n\"How come you're the only one who saw her?\"\n\n\"I don't know and I don't care. I just want to get away from here.\"\n\nCal leaped up, grabbed his pants. Before he could wiggle into them, he saw Gage from behind. \"Holy cow. Your back is messed up bad.\"\n\n\"The old man got wasted last night. It's no big deal.\"\n\n\"Dude.\" Fox walked around to get a look. \"That's gotta hurt.\"\n\n\"The water cooled it off.\"\n\n\"I've got my first aid kit\u2014\" Cal began, but Gage cut him off.\n\n\"I said no big deal.\" He grabbed his shirt, pulled it on. \"If you two don't have the balls to go back in and see what happens, we might as well move on.\"\n\n\"I don't have the balls,\" Cal said in such a deadpan, Gage snorted out a laugh.\n\n\"Then put your pants on so I don't have to wonder what that is hanging between your legs.\"\n\nFox broke out the Little Debbies, and one of the six-pack of Coke he'd bought at the market. Because the incident in the pond and the welts on Gage's back were too important, they didn't speak of them. Instead, hair still dripping, they resumed the hike, gobbling snack cakes and sharing a can of warm soda.\n\nBut with Bon Jovi claiming they were halfway there, Cal thought of what he'd seen. Why had he been the only one? How had her face been so clear in the murky water, and with his glasses tucked in his shoe? How could he have seen her? With every step he took away from the pond, it was easier to convince himself he'd just imagined it.\n\nNot that he'd ever, ever admit that maybe he'd just freaked out.\n\nThe heat dried his damp skin and brought on the sweat. It made him wonder how Gage could stand having his shirt clinging to his sore back. Because, man, those marks were all red and bumpy, and really had to hurt. He'd seen Gage after Old Man Turner had gone after him before, and it hadn't ever, ever been as bad as this. He wished Gage had let him put some salve on his back.\n\nWhat if it got infected? What if he got blood poisoning, got all delirious or something when they were all the way to the Pagan Stone?\n\nHe'd have to send Fox for help, yeah, that's what he'd do\u2014send Fox for help while he stayed with Gage and treated the wounds, got him to drink something so he didn't\u2014what was it?\u2014dehydrate.\n\nOf course, all their butts would be in the sling when his dad had to come get them, but Gage would get better.\n\nMaybe they'd put Gage's father in jail. Then what would happen? Would Gage have to go to an orphanage?\n\nIt was almost as scary to think about as the woman in the pond.\n\nThey stopped to rest, then sat in the shade to share one of Gage's stolen Marlboros. They always made Cal dizzy, but it was kind of nice to sit there in the trees with the water sliding over rocks behind them and a bunch of crazy birds calling out to each other.\n\n\"We could camp right here,\" Cal said half to himself.\n\n\"No way.\" Fox punched his shoulder. \"We're turning ten at the Pagan Stone. No changing the plan. We'll be there in under an hour. Right, Gage?\"\n\nGage stared up through the trees. \"Yeah. We'd be moving faster if you guys hadn't brought so much shit with you.\"\n\n\"Didn't see you turn down a Little Debbie,\" Fox reminded him.\n\n\"Nobody turns down Little Debbies. Well...\" He crushed out the cigarette, then planted a rock over the butt. \"Saddle up, troops.\"\n\nNobody came here. Cal knew it wasn't true, knew when deer was in season these woods were hunted.\n\nBut it felt like nobody came here. The two other times he'd been talked into hiking all the way to the Pagan Stone he'd felt exactly the same. And both those times they'd started out early in the morning instead of afternoon. They'd been back out before two.\n\nNow, according to his Timex, it was nearly four. Despite the snack cake, his stomach wanted to rumble. He wanted to stop again, to dig into what his mother had packed in the stupid basket.\n\nBut Gage was pushing on, anxious to get to the Pagan Stone.\n\nThe earth in the clearing had a scorched look about it, as if a fire had blown through the trees there and turned them all to ash. It was almost a perfect circle, ringed by oaks and locus and the bramble of wild berries. In its center was a single rock that jutted two feet out of the burned earth and flattened at the top like a small table.\n\nSome said altar.\n\nPeople, when they spoke of it at all, said the Pagan Stone was just a big rock that pushed out of the ground. Ground so colored because of minerals, or an underground stream, or maybe caves.\n\nBut others, who were usually more happy to talk about it, pointed to the original settlement of Hawkins Hollow and the night thirteen people met their doom, burned alive in that very clearing.\n\nWitchcraft, some said, and others devil worship.\n\nAnother theory was that an inhospitable band of Indians had killed them, then burned the bodies.\n\nBut whatever the theory, the pale gray stone rose out of the soot-colored earth like a monument.\n\n\"We made it!\" Fox dumped his pack and his bag to dash forward and do a dancing run around the rock. \"Is this cool? Is this cool? Nobody knows where we are. And we've got all night to do anything we want.\"\n\n\"Anything we want in the middle of the woods,\" Cal added. Without a TV, or a refrigerator.\n\nFox threw back his head and let out a shout that echoed away. \"See that? Nobody can hear us. We could be attacked by mutants or ninjas or space aliens, and nobody would hear us.\"\n\nThat, Cal realized, didn't make his stomach feel any steadier. \"We need to get wood for a campfire.\"\n\n\"The Boy Scout's right,\" Gage decided. \"You guys find some wood. I'll go put the beer and the Coke in the stream. Cool off the cans.\"\n\nIn his tidy way, Cal organized the campsite first. Food in one area, clothes in another, tools in another still. With his Scout knife and compass in his pocket, he set off to gather twigs and small branches. The brambles nipped and scratched as he picked his way through them. With his arms loaded, he didn't notice a few drops of his blood drip onto the ground at the edge of the circle.\n\nOr the way the blood sizzled, smoked, then was sucked into that scarred earth.\n\nFox set the boom box on the rock, so they set up camp with Madonna and U2 and the Boss. Following Cal's advice, they built the fire, but didn't set it to light while they had the sun.\n\nSweaty and filthy, they sat on the ground and tore into the picnic basket with grubby hands and huge appetites. As the food, the familiar flavors filled his belly and soothed his system, Cal decided it had been worth hauling the basket for a couple of hours.\n\nReplete, they stretched out on their backs, faces to the sky.\n\n\"Do you really think all those people died right here?\" Gage wondered.\n\n\"There are books about it in the library,\" Cal told him. \"About a fire of, like, 'unknown origin' breaking out and these people burned up.\"\n\n\"Kind of a weird place for them to be.\"\n\n\"We're here.\"\n\nGage only grunted at that.\n\n\"My mom said how the first white people to settle here were Puritans.\" Fox blew a huge pink bubble with the Bazooka he'd bought at the market. \"A sort of radical Puritan or something. How they came over here looking for religious freedom, but really only meant it was free if it was, you know, their way. Mom says lots of people are like that about religion. I don't get it.\"\n\nGage thought he knew, or knew part. \"A lot of people are mean, and even if they're not, a lot more people think they're better than you.\" He saw it all the time, in the way people looked at him.\n\n\"But do you think they were witches, and the people from the Hollow back then burned them at the stake or something?\" Fox rolled over on his belly. \"My mom says that being a witch is like a religion, too.\"\n\n\"Your mom's whacked.\"\n\nBecause it was Gage, and because it was said jokingly, Fox grinned. \"We're all whacked.\"\n\n\"I say this calls for a beer.\" Gage pushed up. \"We'll share one, let the others get colder.\" As Gage walked off to the stream, Cal and Fox exchanged looks.\n\n\"You ever had beer before?\" Cal wanted to know.\n\n\"No. You?\"\n\n\"Are you kidding? I can only have Coke on special occasions. What if we get drunk and pass out or something?\"\n\n\"My dad drinks beer sometimes. He doesn't, I don't think.\"\n\nThey went quiet when Gage walked back with the dripping can. \"Okay. This is to, you know, celebrate that we're going to stop being kids at midnight.\"\n\n\"Maybe we shouldn't drink it until midnight,\" Cal supposed.\n\n\"We'll have the second one after. It's like...it's like a ritual.\"\n\nThe sound of the top popping was loud in the quiet woods, a quick crack, almost as shocking to Cal as a gunshot might have been. He smelled the beer immediately, and it struck him as a sour smell. He wondered if it tasted the same.\n\nGage held the beer up in one hand, high, as if he gripped the hilt of a sword. Then he lowered it, took a long, deep gulp from the can.\n\nHe didn't quite mask the reaction, a closing in of his face as if he'd swallowed something strange and unpleasant. His cheeks flushed as he let out a short, gasping breath.\n\n\"It's still pretty warm but it...\" He coughed once. \"It hits the spot. Now you.\"\n\nHe passed the can to Fox. With a shrug, Fox took the can, mirrored Gage's move. Everyone knew if there was anything close to a dare, Fox would jump at it. \"Ugh. It tastes like piss.\"\n\n\"You been drinking piss lately?\"\n\nFox snorted at Gage's question and passed the can to Cal. \"Your turn.\"\n\nCal studied the can. It wasn't like a sip of beer would kill him or anything. So he sucked in a breath and swallowed some down.\n\nIt made his stomach curl and his eyes water. He shoved the can back at Gage. \"It does taste like piss.\"\n\n\"I guess people don't drink it for how it tastes. It's how it makes you feel.\" Gage took another sip, because he wanted to know how it made him feel.\n\nThey sat cross-legged in the circular clearing, knees bumping, passing the can from hand to hand.\n\nCal's stomach pitched, but it didn't feel sick, not exactly. His head pitched, too, but it felt sort of goofy and fun. And the beer made his bladder full. When he stood, the whole world pitched and made him laugh helplessly as he staggered toward a tree.\n\nHe unzipped, aimed toward the tree but the tree kept moving.\n\nFox was struggling to light one of the cigarettes when Cal stumbled back. They passed that around the circle as well until Cal's almost ten-year-old stomach revolted. He crawled off to sick it all up, crawled back, and just lay flat, closing his eyes and willing the world to go still again.\n\nHe felt as if he were once again swimming in the pond, and being slowly pulled under.\n\nWhen he surfaced again it was nearly dusk.\n\nHe eased up, hoping he wouldn't be sick again. He felt a little hollow inside\u2014belly and head\u2014but not like he was going to puke. He saw Fox curled against the stone, sleeping. He crawled over on all fours for the thermos and as he washed the sick and beer out of his throat, he was never so grateful for his mother and her lemonade.\n\nSteadier, he rubbed his fingers on his eyes under his glasses, then spotted Gage sitting, staring at the tented wood of the campfire they'd yet to light.\n\n\"'Morning, Sally.\"\n\nWith a wan smile, Cal scooted over.\n\n\"I don't know how to light this thing. I figured it was about time to, but I needed a Boy Scout.\"\n\nCal took the book of matches Gage handed him, and set fire to several spots on the pile of dried leaves he'd arranged under the wood. \"That should do it. Wind's pretty still, and there's nothing to catch in the clearing. We can keep feeding it when we need to, and just make sure we bury it before we go tomorrow.\"\n\n\"Smokey the Bear. You all right?\"\n\n\"Yeah. I guess I threw most everything up.\"\n\n\"I shouldn't have brought the beer.\"\n\nCal lifted a shoulder, glanced toward Fox. \"We're okay, and now we won't have to wonder what it tastes like. We know it tastes like piss.\"\n\nGage laughed a little. \"It didn't make me feel mean.\" He picked up a stick, poked at the little flames. \"I wanted to know if it would, and I figured I could try it with you and Fox. You're my best friends, so I could try it with you and see if it made me feel mean.\"\n\n\"How did it make you feel?\"\n\n\"It made my head hurt. It still does a little. I didn't get sick like you, but I sorta wanted to. I went and got one of the Cokes and drank that. It felt better then. Why does he drink so goddamn much if it makes him feel like that?\"\n\n\"I don't know.\"\n\nGage dropped his head on his knees. \"He was crying when he went after me last night. Blubbering and crying the whole time he used the belt on me. Why would anybody want to feel like that?\"\n\nCareful to avoid the welts on Gage's back, Cal draped an arm over his shoulders. He wished he knew what to say.\n\n\"Soon as I'm old enough I'm getting out. Join the army maybe, or get a job on a freighter, maybe an oil rig.\"\n\nGage's eyes gleamed when he lifted his head, and Cal looked away because he knew the shine was tears. \"You can come stay with us when you need to.\"\n\n\"It'd just be worse when I went back. But I'm going to be ten in a few hours. And in a few years I'll be as big as he is. Bigger maybe. I won't let him come after me then. I won't let him hit me. Screw it.\" Gage rubbed his face. \"Let's wake Fox up. Nobody sleeps tonight.\"\n\nFox moaned and grumbled, and he got himself up to pee and fetch a cool Coke from the stream. They shared it with another round of Little Debbies. And, at last, the copy of Penthouse.\n\nCal had seen naked breasts before. You could see them in the National Geographic in the library, if you knew where to look.\n\nBut these were different.\n\n\"Hey guys, did you ever think about doing it?\" Cal asked.\n\n\"Who doesn't?\" they both replied.\n\n\"Whoever does it first has to tell the other two everything. All about how it feels,\" Cal continued. \"And how you did it, and what she does. Everything. I call for an oath.\"\n\nA call for an oath was sacred. Gage spat on the back of his hand, held it out. Fox slapped his palm on, spat on the back of his hand, and Cal completed the contact.\n\n\"And so we swear,\" they said together.\n\nThey sat around the fire as the stars came out, and deep in the woods an owl hooted its night call.\n\nThe long, sweaty hike, ghostly apparitions, and beer puke were forgotten.\n\n\"We should do this every year on our birthday,\" Cal decided. \"Even when we're old. Like thirty or something. The three of us should come here.\"\n\n\"Drink beer and look at pictures of naked girls,\" Fox added. \"I call for\u2014\"\n\n\"Don't.\" Gage spoke sharply. \"I can't swear. I don't know where I'm going to go, but it'll be somewhere else. I don't know if I'll ever come back.\"\n\n\"Then we'll go where you are, when we can. We're always going to be best friends.\" Nothing would change that, Cal thought, and took his own, personal oath on it. Nothing ever could. He looked at his watch. \"It's going to be midnight soon. I have an idea.\"\n\nHe took out his Boy Scout knife and, opening the blade, held it in the fire.\n\n\"What's up?\" Fox demanded.\n\n\"I'm sterilizing it. Like, ah, purifying it.\" It got so hot he had to pull back, blow on his fingers. \"It's like Gage said about ritual and stuff. Ten years is a decade. We've known each other almost the whole time. We were born on the same day. It makes us...different,\" he said, searching for words he wasn't quite sure of. \"Like special, I guess. We're best friends. We're like brothers.\"\n\nGage looked at the knife, then into Cal's face. \"Blood brothers.\"\n\n\"Yeah.\"\n\n\"Cool.\" Already committed, Fox held out his hand.\n\n\"At midnight,\" Cal said. \"We should do it at midnight, and we should have some words to say.\"\n\n\"We'll swear an oath,\" Gage said. \"That we mix our blood, um, three into one? Something like that. In loyalty.\"\n\n\"That's good. Write it down, Cal.\"\n\nCal dug pencil and paper out of his pack. \"We'll write words down, and say them together. Then we'll do the cut and put our wrists together. I've got Band-Aids for after if we need them.\"\n\nCal wrote the words with his Number Two pencil on the blue lined paper, crossing out when they changed their minds.\n\nFox added more wood to the fire so that the flames crackled as they stood by the Pagan Stone.\n\nAt moments to midnight, they stood, three young boys with faces lit by fire and starlight. At Gage's nod, they spoke together in voices solemn and achingly young.\n\n\"We were born ten years ago, on the same night, at the same time, in the same year. We are brothers. At the Pagan Stone we swear an oath of loyalty and truth and brotherhood. We mix our blood.\"\n\nCal sucked in a breath and geared up the courage to run the knife across his wrist first. \"Ouch.\"\n\n\"We mix our blood.\" Fox gritted his teeth as Cal cut his wrist.\n\n\"We mix our blood.\" And Gage stood unflinching as the knife drew over his flesh.\n\n\"Three into one, and one for the three.\"\n\nCal held his arm out. Fox, then Gage pressed their scored wrists down to his. \"Brothers in spirit, in mind. Brothers in blood for all time.\"\n\nAs they stood, clouds shivered over the fat moon, misted over the bright stars. Their mixed blood dripped and fell onto the burnt ground.\n\nThe wind exploded with a voice like a raging scream. The little campfire spewed up flame in a spearing tower. The three of them were lifted off their feet as if a hand gripped them, tossed them. Light burst as if the stars had shattered.\n\nAs he opened his mouth to shout, Cal felt something shove inside him, hot and strong, to smother his lungs, to squeeze his heart in a stunning agony of pain.\n\nThe light shut off. In the thick dark blew an icy cold that numbed his skin. The sound the wind made now was like an animal, like a monster that only lived inside books. Beneath him the ground shook, heaving him back as he tried to crawl away.\n\nAnd something came out of that icy dark, out of that quaking ground. Something huge and horrible.\n\nEyes bloodred and full of...hunger. It looked at him. And when it smiled, its teeth glittered like silver swords.\n\nHe thought he died, and that it took him in, in one gulp.\n\nBut when he came to himself again, he could hear his own heart. He could hear the shouts and calls of his friends.\n\nBlood brothers.\n\n\"Jesus, Jesus, what was that? Did you see?\" Fox called out in a voice thin as a reed. \"Gage, God, your nose is bleeding.\"\n\n\"So's yours. Something...Cal. God, Cal.\"\n\nCal lay where he was, flat on his back. He felt the wet warmth of blood on his face. He was too numb to be frightened by it. \"I can't see.\" He croaked out a weak whisper. \"I can't see.\"\n\n\"Your glasses are broken.\" Face filthy with soot and blood, Fox crawled to him. \"One of the lenses is cracked. Dude, your mom's going to kill you.\"\n\n\"Broken.\" Shaking, Cal reached up to pull off his glasses.\n\n\"Something. Something was here.\" Gage gripped Cal's shoulder. \"I felt something happen, after everything went crazy, I felt something happen inside me. Then...did you see it? Did you see that thing?\"\n\n\"I saw its eyes,\" Fox said, and his teeth chattered. \"We need to get out of here. We need to get out.\"\n\n\"Where?\" Gage demanded. Though his breath still wheezed, he grabbed Cal's knife from the ground, gripped it. \"We don't know where it went. Was it some kind of bear? Was it\u2014\"\n\n\"It wasn't a bear.\" Cal spoke calmly now. \"It was what's been here, in this place, a long time. I can see...I can see it. It looked like a man once, when it wanted. But it wasn't.\"\n\n\"Man, you hit your head.\"\n\nCal turned his eyes on Fox, and the irises were nearly black. \"I can see it, and the other.\" He opened the hand of the wrist he'd cut. In the palm was a chunk of a green stone spotted with red. \"His.\"\n\nFox opened his hand, and Gage his. In each was an identical third of the stone. \"What is it?\" Gage whispered. \"Where the hell did it come from?\"\n\n\"I don't know, but it's ours now. Uh, one into three, three into one. I think we let something out. And something came with it. Something bad. I can see.\"\n\nHe closed his eyes a moment, then opened them to look at his friends. \"I can see, but not with my glasses. I can see without them. It's not blurry. I can see without my glasses.\"\n\n\"Wait.\" Trembling, Gage pulled up his shirt, turned his back.\n\n\"Man, they're gone.\" Fox reached out to touch his fingers to Gage's unmarred back. \"The welts. They're gone. And...\" He held out his wrist where the shallow cut was already healing. \"Holy cow, are we like superheroes now?\"\n\n\"It's a demon,\" Cal said. \"And we let it out.\"\n\n\"Shit.\" Gage stared off into the dark woods. \"Happy goddamn birthday to us.\"\n\n## Three\n\nHawkins Hollow  \nFebruary 2008\n\nIT WAS COLDER IN HAWKINS HOLLOW, MARYLAND, than it was in Juno, Alaska. Cal liked to know little bits like that, even though at the moment he was in the Hollow where the damp, cold wind blew like a mother and froze his eyeballs.\n\nHis eyeballs were about the only things exposed as he zipped across Main Street from Coffee Talk, with a to-go cup of mochaccino in one gloved hand, to the Bowl-a-Rama.\n\nThree days a week, he tried for a counter breakfast at Ma's Pantry a couple doors down, and at least once a week he hit Gino's for dinner.\n\nHis father believed in supporting the community, the other merchants. Now that his dad was semiretired and Cal oversaw most of the businesses, he tried to follow that Hawkins tradition.\n\nHe shopped the local market even though the chain supermarket a couple miles outside town was cheaper. If he wanted to send a woman flowers, he resisted doing so with a couple of clicks on his computer and hauled himself down to the Flower Pot.\n\nHe had relationships with the local plumber, electrician, painter, the area craftsmen. Whenever possible, he hired for the town from the town.\n\nExcept for his years away at college, he'd always lived in the Hollow. It was his place.\n\nEvery seven years since his tenth birthday, he lived through the nightmare that visited his place. And every seven years, he helped clean up the aftermath.\n\nHe unlocked the front door of the Bowl-a-Rama, relocked it behind him. People tended to walk right in, whatever the posted hours, if the door wasn't locked.\n\nHe'd once been a little more casual about that, until one fine night while he'd been enjoying some after-hours Strip Bowling with Allysa Kramer, three teenage boys had wandered in, hoping the video arcade was still open.\n\nLesson learned.\n\nHe walked by the front desk, the six lanes and ball returns, the shoe rental counter and the grill, turned and jogged up the stairs to the squat second floor that held his (or his father's if his father was in the mood) office, a closet-sized john, and a mammoth storage area.\n\nHe set the coffee on the desk, stripped off gloves, scarf, watch cap, coat, insulated vest.\n\nHe booted up his computer, put on the satellite radio, then sat down to fuel up on caffeine and get to work.\n\nThe bowling center Cal's grandfather had opened in the postwar forties had been a tiny, three-lane gathering spot with a couple of pinball machines and counter Cokes. It expanded in the sixties, and again, when Cal's father took the reins, in the early eighties.\n\nNow, with its six lanes, its video arcade, and its private party room, it was the place to gather in the Hollow.\n\nCredit to Grandpa, Cal thought as he looked over the party reservations for the next month. But the biggest chunk of credit went to Cal's father, who'd morphed the lanes into a family center, and had used its success to dip into other areas of business.\n\nThe town bears our name, Jim Hawkins liked to say. Respect the name, respect the town.\n\nCal did both. He'd have left long ago otherwise.\n\nAn hour into the work, Cal glanced up at the rap on his doorjamb.\n\n\"Sorry, Cal. Just wanted you to know I was here. Thought I'd go ahead and get that painting done in the rest rooms since you're not open this morning.\"\n\n\"Okay, Bill. Got everything you need?\"\n\n\"Sure do.\" Bill Turner, five years, two months, and six days sober, cleared his throat. \"Wonder if maybe you'd heard anything from Gage.\"\n\n\"Not in a couple months now.\"\n\nTender area, Cal thought when Bill just nodded. Boggy ground.\n\n\"I'll just get started then.\"\n\nCal watched as Bill moved away from the doorway. Nothing he could do about it, he told himself. Nothing he was sure he should do.\n\nDid five years clean and sober make up for all those whacks with a belt, for all those shoves and slaps, all those curses? It wasn't for him to judge.\n\nHe glanced down at the thin scar that ran diagonally across his wrist. Odd how quickly that small wound had healed, and yet the mark of it remained\u2014the only scar he carried. Odd how so small a thing had catapulted the town and people he knew into seven days of hell every seven years.\n\nWould Gage come back this summer, as he had every seventh year? Cal couldn't see ahead, that wasn't his gift or his burden. But he knew when he, Gage, and Fox turned thirty-one, they would all be together in the Hollow.\n\nThey'd sworn an oath.\n\nHe finished up the morning's work, and because he couldn't get his mind off it, composed a quick e-mail to Gage.\n\nHey. Where the hell are you? Vegas? Mozambique? Duluth? Heading out to see Fox. There's a writer coming into the Hollow to do research on the history, the legend, and what they're calling the anomalies. Probably got it handled, but thought you should know.\n\nIt's twenty-two degrees with a windchill factor of fifteen. Wish you were here and I wasn't.\n\nCal\n\nHe'd answer eventually, Cal thought as he sent the e-mail, then shut down the computer. Could be in five minutes or in five weeks, but Gage would answer.\n\nHe began to layer on the outer gear again over a long and lanky frame passed down by his father. He'd gotten his outsized feet from dear old Dad, too.\n\nThe dark blond hair that tended to go as it chose was from his mother. He knew that only due to early photos of her, as she'd been a soft, sunny blonde, perfectly groomed, throughout his memory.\n\nHis eyes, a sharp, occasionally stormy gray, had been twenty-twenty since his tenth birthday.\n\nEven as he zipped up his parka to head outside, he thought that the coat was for comfort only. He hadn't had so much as a sniffle in over twenty years. No flu, no virus, no hay fever.\n\nHe'd fallen out of an apple tree when he'd been twelve. He'd heard the bone in his arm snap, had felt the breathless pain.\n\nAnd he'd felt it knit together again\u2014with more pain\u2014before he'd made it across the lawn to the house to tell his mother.\n\nSo he'd never told her, he thought as he stepped outside into the ugly slap of cold. Why upset her?\n\nHe covered the three blocks to Fox's office quickly, shooting out waves or calling back greetings to neighbors and friends. But he didn't stop for conversation. He might not get pneumonia or postnasal drip, but he was freaking tired of winter.\n\nGray, ice-crusted snow lay in a dirty ribbon along the curbs, and above, the sky mirrored the brooding color. Some of the houses or businesses had hearts and Valentine wreaths on doors and windows, but they didn't add a lot of cheer with the bare trees and winter-stripped gardens.\n\nThe Hollow didn't show to advantage, to Cal's way of thinking, in February.\n\nHe walked up the short steps to the little covered porch of the old stone townhouse. The plaque beside the door read: FOX B. O'DELL, ATTORNEY AT LAW.\n\nIt was something that always gave Cal a quick jolt and a quick flash of amusement. Even after nearly six years, he couldn't quite get used to it.\n\nThe long-haired hippie freak was a goddamn lawyer.\n\nHe stepped into the tidy reception area, and there was Alice Hawbaker at the desk. Trim, tidy in her navy suit with its bowed white blouse, her snowcap of hair and no-nonsense bifocals, Mrs. Hawbaker ran the office like a Border collie ran a herd.\n\nShe looked sweet and pretty, and she'd bite your ankle if you didn't fall in line.\n\n\"Hey, Mrs. Hawbaker. Boy, it is cold out there. Looks like we might get some more snow.\" He unwrapped his scarf. \"Hope you and Mr. Hawbaker are keeping warm.\"\n\n\"Warm enough.\"\n\nHe heard something in her voice that had him looking more closely as he pulled off his gloves. When he realized she'd been crying he instinctively stepped to the desk. \"Is everything okay? Is\u2014\"\n\n\"Everything's fine. Just fine. Fox is between appointments. He's in there sulking, so you go right on back.\"\n\n\"Yes, ma'am. Mrs. Hawbaker, if there's anything\u2014\"\n\n\"Just go right on back,\" she repeated, then made herself busy with her keyboard.\n\nBeyond the reception area a hallway held a powder room on one side and a library on the other. Straight back, Fox's office was closed off by a pair of pocket doors. Cal didn't bother to knock.\n\nFox looked up when the doors slid open. He did appear to be sulking as his gilded eyes were broody and his mouth was in full scowl.\n\nHe sat behind his desk, his feet, clad in hiking boots, propped on it. He wore jeans and a flannel shirt open over a white insulated tee. His hair, densely brown, waved around his sharp-featured face.\n\n\"What's going on?\"\n\n\"I'll tell you what's going on. My administrative assistant just gave me her notice.\"\n\n\"What did you do?\"\n\n\"Me?\" Fox shoved back from the desk and opened the minifridge for a can of Coke. He'd never developed a taste for coffee. \"Try we, brother. We camped out at the Pagan Stone one fateful night, and screwed the monkey.\"\n\nCal dropped into a chair. \"She's quitting because\u2014\"\n\n\"Not just quitting. They're leaving the Hollow, she and Mr. Hawbaker. And yeah, because.\" He took a long, greedy drink the way some men might take a pull on a bottle of whiskey. \"That's not the reason she gave me, but that's the reason. She said they decided to move to Minneapolis to be close to their daughter and grandchildren, and that's bogus. Why does a woman heading toward seventy, married to a guy older than dirt, pick up and move north? They've got another kid lives outside of D.C., and they've got strong ties here. I could tell it was bull.\"\n\n\"Because of what she said, or because you took a cruise through her head?\"\n\n\"First the one, then the other. Don't start on me.\" Fox gestured with the Coke, then slammed it down on his desk. \"I don't poke around for the fun of it. Son of a bitch.\"\n\n\"Maybe they'll change their minds.\"\n\n\"They don't want to go, but they're afraid to stay. They're afraid it'll happen again\u2014which I could tell her it will\u2014and they just don't want to go through it again. I offered her a raise\u2014like I could afford it\u2014offered her the whole month of July off, letting her know that I knew what was at the bottom of it. But they're going. She'll give me until April first. April frickin' Fools,\" he ranted. \"To find somebody else, for her to show them the ropes. I don't know where the damn ropes are, Cal. I don't know half the stuff she does. She just does it. Anyway.\"\n\n\"You've got until April, maybe we'll think of something.\"\n\n\"We haven't thought of the solution to this in twenty years plus.\"\n\n\"I meant your office problem. But yeah, I've been thinking a lot about the other.\" Rising, he walked to Fox's window, looked out on the quiet side street. \"We've got to end it. This time we've got to end it. Maybe talking to this writer will help. Laying it out to someone objective, someone not involved.\"\n\n\"Asking for trouble.\"\n\n\"Maybe it is, but trouble's coming anyway. Five months to go. We're supposed to meet her at the house.\" Cal glanced at his watch. \"Forty minutes.\"\n\n\"We?\" Fox looked blank for a moment. \"That's today? See, see, I didn't tell Mrs. H, so it didn't get written down somewhere. I've got a deposition in an hour.\"\n\n\"Why don't you use your damn BlackBerry?\"\n\n\"Because it doesn't follow my simple Earth logic. Reschedule the writer. I'm clear after four.\"\n\n\"It's okay, I can handle it. If she wants more, I'll see about setting up a dinner, so keep tonight open.\"\n\n\"Be careful what you say.\"\n\n\"Yeah, yeah, I'm going to. But I've been thinking. We've been careful about that for a long time. Maybe it's time to be a little reckless.\"\n\n\"You sound like Gage.\"\n\n\"Fox...I've already started having the dreams again.\"\n\nFox blew out a breath. \"I was hoping that was just me.\"\n\n\"When we were seventeen they started about a week before our birthday, then when we were twenty-four, over a month. Now, five months out. Every time it gets stronger. I'm afraid if we don't find the way, this time could be the last for us, and the town.\"\n\n\"Have you talked to Gage?\"\n\n\"I just e-mailed him. I didn't tell him about the dreams. You do it. Find out if he's having them, too, wherever the hell he is. Get him home, Fox. I think we need him back. I don't think we can wait until summer this time. I gotta go.\"\n\n\"Watch your step with the writer,\" Fox called out as Cal started for the door. \"Get more than you give.\"\n\n\"I can handle it,\" Cal repeated.\n\nQUINN BLACK EASED HER MINI COOPER OFF THE exit ramp and hit the usual barrage at the interchange. Pancake House, Wendy's, McDonald's, KFC.\n\nWith great affection, she thought of a Quarter Pounder, with a side of really salty fries, and\u2014natch\u2014a Diet Coke to ease the guilt. But since that would be breaking her vow to eat fast food no more than once a month, she wasn't going to indulge.\n\n\"There now, don't you feel righteous?\" she asked herself with only one wistful glance in the rearview at the lovely Golden Arches.\n\nHer love of the quick and the greasy had sent her on an odyssey of fad diets, unsatisfying supplements, and miracle workout tapes through her late teens and early twenties. Until she'd finally slapped herself silly, tossed out all her diet books, her diet articles, her I LOST TWENTY POUNDS IN TWO WEEKS\u2014AND YOU CAN, TOO! ads, and put herself on the path to sensible eating and exercising.\n\nLifestyle change, she reminded herself. She'd made a lifestyle change.\n\nBut boy, she missed those Quarter Pounders more than she missed her ex-fianc\u00e9.\n\nThen again, who wouldn't?\n\nShe glanced at the GPS hooked to her dashboard, then over at the directions she'd printed out from Caleb Hawkins's e-mail. So far, they were in tandem.\n\nShe reached down for the apple serving as her midmorning snack. Apples were filling, Quinn thought as she bit in. They were good for you, and they were tasty.\n\nAnd they were no Quarter Pounder.\n\nIn order to keep her mind off the devil, she considered what she hoped to accomplish on this first face-to-face interview with one of the main players in the odd little town of Hawkins Hollow.\n\nNo, not fair to call it odd, she reminded herself. Objectivity first. Maybe her research leaned her toward the odd label, but there would be no making up her mind until she'd seen for herself, done her interviews, taken her notes, scoped out the local library. And, maybe most important, seen the Pagan Stone in person.\n\nShe loved poking at all the corners and cobwebs of small towns, digging down under the floorboards for secrets and surprises, listening to the gossip, the local lore and legend.\n\nShe'd made a tiny name for herself doing a series of articles on quirky, off-the-mainstream towns for a small press magazine called Detours. And since her professional appetite was as well-developed as her bodily one, she'd taken a risky leap and written a book, following the same theme, but focusing on a single town in Maine reputed to be haunted by the ghosts of twin sisters who'd been murdered in a boardinghouse in 1843.\n\nThe critics had called the result \"engaging\" and \"good, spooky fun,\" except for the ones who'd deemed it \"preposterous\" and \"convoluted.\"\n\nShe'd followed it up with a book highlighting a small town in Louisiana where the descendent of a voodoo priestess served as mayor and faith healer. And, Quinn had discovered, had been running a very successful prostitution ring.\n\nBut Hawkins Hollow\u2014she could just feel it\u2014was going to be bigger, better, meatier.\n\nShe couldn't wait to sink her teeth in.\n\nThe fast-food joints, the businesses, the ass-to-elbow houses gave way to bigger lawns, bigger homes, and to fields sleeping under the dreary sky.\n\nThe road wound, dipped and lifted, then veered straight again. She saw a sign for the Antietam Battlefield, something else she meant to investigate and research firsthand. She'd found little snippets about incidents during the Civil War in and around Hawkins Hollow.\n\nShe wanted to know more.\n\nWhen her GPS and Caleb's directions told her to turn, she turned, following the next road past a grove of naked trees, a scatter of houses, and the farms that always made her smile with their barns and silos and fenced paddocks.\n\nShe'd have to find a small town to explore in the Midwest next time. A haunted farm, or the weeping spirit of a milkmaid.\n\nShe nearly ignored the directions to turn when she saw the sign for Hawkins Hollow (est. 1648). As with the Quarter Pounder, her heart longed to indulge, to drive into town rather than turn off toward Caleb Hawkins's place. But she hated to be late, and if she got caught up exploring the streets, the corners, the look of the town, she certainly would be late for her first appointment.\n\n\"Soon,\" she promised, and turned to take the road winding by the woods she knew held the Pagan Stone at their heart.\n\nIt gave her a quick shiver, and that was strange. Strange to realize that shiver had been fear and not the anticipation she always felt with a new project.\n\nAs she followed the twists of the road, she glanced with some unease toward the dark and denuded trees. And hit the brakes hard when she shifted her eyes back to the road and saw something rush out in front of her.\n\nShe thought she saw a child\u2014oh God, oh God\u2014then thought it was a dog. And then...it was nothing. Nothing at all on the road, nothing rushing to the field beyond. Nothing there but herself and her wildly beating heart in the little red car.\n\n\"Trick of the eye,\" she told herself, and didn't believe it. \"Just one of those things.\"\n\nBut she restarted the car that had stalled when she'd slammed the brakes, then eased to the strip of dirt that served as the shoulder of the road. She pulled out her notebook, noted the time, and wrote down exactly what she thought she'd seen.\n\nYoung boy, abt ten. Lng blck hair, red eyes. He LOOKED right at me. Did I blink? Shut my eyes? Opened, & saw lrg blck dog, not boy. Then poof. Nothing there.\n\nCars passed her without incident as she sat a few moments more, waited for the trembling to stop.\n\nIntrepid writer balks at first possible phenom, she thought, turns around, and drives her adorable red car to the nearest Mickey D's for a fat-filled antidote to nerves.\n\nShe could do that, she considered. Nobody could charge her with a felony and throw her into prison. And if she did that, she wouldn't have her next book, or any self-respect.\n\n\"Man up, Quinn,\" she ordered. \"You've seen spooks before.\"\n\nSteadier, she swung back out on the road, and made the next turn. The road was narrow and twisty with trees looming on both sides. She imagined it would be lovely in the spring and summer, with the green dappling, or after a snowfall with all those trees ermine drenched. But under a dull gray sky the woods seemed to crowd the road, bare branches just waiting to reach out and strike, as if they and only they were allowed to live there.\n\nAs if to enforce the sensation, no other car passed, and when she turned off her radio as the music seemed too loud, the only sound was the keening curse of the wind.\n\nShould've called it Spooky Hollow, she decided, and nearly missed the turn into the gravel lane.\n\nWhy, she wondered, would anyone choose to live here? Amid all those dense, thrusting trees where bleak pools of snow huddled to hide from the sun? Where the only sound was the warning growl of Nature. Everything was brown and gray and moody.\n\nShe bumped over a little bridge spanning a curve of a creek, followed the slight rise of the stingy lane.\n\nThere was the house, exactly as advertised.\n\nIt sat on what she would have termed a knoll rather than a hill, with the front slope tamed into step-down terraces decked with shrubs she imagined put on a hell of a show in the spring and summer.\n\nThere wasn't a lawn, so to speak, and she thought Hawkins had been smart to go with the thick mulch and shrubs and trees skirting the front instead of the traditional grass that would probably be a pain in the ass to mow and keep clear of weeds.\n\nShe approved of the deck that wrapped around the front and sides, and she'd bet the rear as well. She liked the earthy tones of the stone and the generous windows.\n\nIt sat like it belonged there, content and well-settled in the woods.\n\nShe pulled up beside an aging Chevy pickup, got out of her car to stand and take a long view.\n\nAnd understood why someone would choose this spot. There was, unquestionably, an aura of spookiness, especially for one who was inclined to see and feel such things. But there was considerable charm as well, and a sense of solitude that was far from lonely. She could imagine very well sitting on that front deck some summer evening, drinking a cold one, and wallowing in the silence.\n\nBefore she could move toward the house, the front door opened.\n\nThe sense of d\u00e9j\u00e0 vu was vivid, almost dizzying. He stood there at the door of the cabin, the blood like red flowers on his shirt.\n\nWe can stay no longer.\n\nThe words sounded in her head, clear, and in a voice she somehow knew.\n\n\"Miss Black?\"\n\nShe snapped back. There was no cabin, and the man standing on the lovely deck of his charming house had no blood blooming on him. There was no force of great love and great grief shining in his eyes.\n\nAnd still, she had to lean back against her car for a minute and catch her breath. \"Yeah, hi. I was just...admiring the house. Great spot.\"\n\n\"Thanks. Any trouble finding it?\"\n\n\"No, no. Your directions were perfect.\" And, of course, it was ridiculous to be having this conversation outside in the freezing wind. From the quizzical look on his face, he obviously felt the same.\n\nShe pushed off the car, worked up what she hoped was a sane and pleasant expression as she walked to the trio of wooden steps.\n\nAnd wasn't he a serious cutie? she realized as she finally focused on the reality. All that windblown hair and those strong gray eyes. Add the crooked smile, the long, lean body in jeans and flannel, and a woman might be tempted to hang a SOLD! sign around his neck.\n\nShe stepped up, held out a hand. \"Quinn Black, thanks for meeting with me, Mr. Hawkins.\"\n\n\"Cal.\" He took her hand, shook it, then held it as he gestured to the door. \"Let's get you out of the wind.\"\n\nThey stepped directly into a living room that managed to be cozy and male at the same time. The generous sofa faced the big front windows, and the chairs looked as though they'd allow an ass to sink right in. Tables and lamps probably weren't antiques, but looked to be something a grandmother might have passed down when she got the urge to redecorate her own place.\n\nThere was even a little stone fireplace with the requisite large mutt sprawled sleeping in front of it.\n\n\"Let me take your coat.\"\n\n\"Is your dog in a coma?\" Quinn asked when the dog didn't move a muscle.\n\n\"No. Lump leads an active and demanding internal life that requires long periods of rest.\"\n\n\"I see.\"\n\n\"Want some coffee?\"\n\n\"That'd be great. So would the bathroom. Long drive.\"\n\n\"First right.\"\n\n\"Thanks.\"\n\nShe closed herself into a small, spotlessly clean powder room as much to pull herself back together from a couple of psychic shocks as to pee.\n\n\"Okay, Quinn,\" she whispered. \"Here we go.\"\n\n## Four\n\nHE'D READ HER WORK; HE'D STUDIED HER AUTHOR photos and used Google to get some background, to read her interviews. Cal wasn't one to agree to talk to any sort of writer, journalist, reporter, Internet blogger about the Hollow, himself, or much of anything else without doing a thorough check.\n\nHe'd found her books and articles entertaining. He'd enjoyed her obvious affection for small towns, had been intrigued by her interest and treatment of lore, legend, and things that went bump in the night.\n\nHe liked the fact that she still wrote the occasional article for the magazine that had given her a break when she'd still been in college. It spoke of loyalty.\n\nHe hadn't been disappointed that her author photo had shown her to be a looker, with a sexy tumble of honey blond hair, bright blue eyes, and the hint of a fairly adorable overbite.\n\nThe photo hadn't come close.\n\nShe probably wasn't beautiful, he thought as he poured coffee. He'd have to get another look when, hopefully, his brain wouldn't go to fuzz, then decide about that.\n\nWhat he did know, unquestionably, was she just plain radiated energy and\u2014to his fuzzed brain\u2014sex.\n\nBut maybe that was because she was built, another thing the photo hadn't gotten across. The lady had some truly excellent curves.\n\nAnd it wasn't as if he hadn't seen curves on a woman before or, in fact, seen his share of naked female curves alive and in person. So why was he standing in his own kitchen frazzled because an attractive, fully dressed woman was in his house? For professional purposes.\n\n\"Jesus, grow up, Hawkins.\"\n\n\"Sorry?\"\n\nHe actually jumped. She was in the kitchen, a few steps behind him, smiling that million-watt smile.\n\n\"Were you talking to yourself? I do that, too. Why do people think we're crazy?\"\n\n\"Because they want to suck us into talking to them.\"\n\n\"You're probably right.\" Quinn shoved back that long spill of blond.\n\nCal saw he was right. She wasn't beautiful. The top-heavy mouth, the slightly crooked nose, the oversized eyes weren't elements of traditional beauty. He couldn't label her pretty, either. It was too simple and sweet a word. Cute didn't do it.\n\nAll he could think of was hot, but that might have been his brain blurring again.\n\n\"I didn't ask how you take your coffee.\"\n\n\"Oh. I don't suppose you have two percent milk.\"\n\n\"I often wonder why anybody does.\"\n\nWith an easy laugh that shot straight to his bloodstream, she wandered over to study the view outside the glass doors that led\u2014as she'd suspected\u2014to the rear portion of the circling deck. \"Which also means you probably don't have any fake sugar. Those little pink, blue, or yellow packets?\"\n\n\"Fresh out. I could offer you actual milk and actual sugar.\"\n\n\"You could.\" And hadn't she eaten an apple like a good girl? \"And I could accept. Let me ask you something else, just to satisfy my curiosity. Is your house always so clean and tidy, or did you do all this just for me?\"\n\nHe got out the milk. \"Tidy's a girlie word. I prefer the term organized. I like organization. Besides.\" He offered her a spoon for the sugar bowl. \"My mother could\u2014and does\u2014drop by unexpectedly. If my house wasn't clean, she'd ground me.\"\n\n\"If I don't call my mother once a week, she assumes I've been hacked to death by an ax murderer.\" Quinn held herself to one scant spoon of sugar. \"It's nice, isn't it? Those long and elastic family ties.\"\n\n\"I like them. Why don't we go sit in the living room by the fire?\"\n\n\"Perfect. So, how long have you lived here? In this particular house,\" she added as they carried their mugs out of the kitchen.\n\n\"A couple of years.\"\n\n\"Not much for neighbors?\"\n\n\"Neighbors are fine, and I spend a lot of time in town. I like the quiet now and then.\"\n\n\"People do. I do myself, now and again.\" She took one of the living room chairs, settled back. \"I guess I'm surprised other people haven't had the same idea as you, and plugged in a few more houses around here.\"\n\n\"There was talk of it a couple of times. Never panned out.\"\n\nHe's being cagey, Quinn decided. \"Because?\"\n\n\"Didn't turn out to be financially attractive, I guess.\"\n\n\"Yet here you are.\"\n\n\"My grandfather owned the property, some acres of Hawkins Wood. He left it to me.\"\n\n\"So you had this house built.\"\n\n\"More or less. I'd liked the spot.\" Private when he needed to be private. Close to the woods where everything had changed. \"I know some people in the trade, and we put the house up. How's the coffee?\"\n\n\"It's terrific. You cook, too?\"\n\n\"Coffee's my specialty. I read your books.\"\n\n\"How were they?\"\n\n\"I liked them. You probably know you wouldn't be here if I hadn't.\"\n\n\"Which would've made it a lot tougher to write the book I want to write. You're a Hawkins, a descendent of the founder of the settlement that became the village that became the town. And one of the main players in the more recent unexplained incidents related to the town. I've done a lot of research on the history, the lore, the legends, and the various explanations,\" she said, and reached in the bag that served as her purse and her briefcase. Taking out a minirecorder, she switched it on, set it on the table between them.\n\nHer smile was full of energy and interest when she set her notebook on her lap, flipped pages to a clear one. \"So, tell me, Cal, about what happened the week of July seventh, nineteen eighty-seven, ninety-four, and two thousand one.\"\n\nThe tape recorder made him...itchy. \"Dive right in, don't you?\"\n\n\"I love knowing things. July seventh is your birthday. It's also the birthday of Fox O'Dell and Gage Turner\u2014born the same year as you, who grew up in Hawkins Hollow with you. I read articles that reported you, O'Dell, and Turner were responsible for alerting the fire department on July eleventh, nineteen eighty-seven, when the elementary school was set on fire, and also responsible for saving the life of one Marian Lister who was inside the school at the time.\"\n\nShe continued to look straight into his eyes as she spoke. He found it interesting she didn't need to refer to notes, and that she didn't appear to need the little breaks from direct eye contact.\n\n\"Initial reports indicated the three of you were originally suspected of starting the fire, but it was proven Miss Lister herself was responsible. She suffered second-degree burns on nearly thirty percent of her body as well as a concussion. You and your friends, three ten-year-old boys, dragged her out and called the fire department. Miss Lister was, at that time, a twenty-five-year-old fourth-grade teacher with no history of criminal behavior or mental illness. Is that all correct information?\"\n\nShe got her facts in order, Cal noted. Such as the facts were known. They fell far short of the abject terror of entering that burning school, of finding the pretty Miss Lister cackling madly as she ran through the flames. Of how it felt to chase her through those hallways as her clothes burned.\n\n\"She had a breakdown.\"\n\n\"Obviously.\" Smile in place, Quinn lifted her eyebrows. \"There were also over a dozen nine-one-one calls on domestic abuse during that single week, more than previously had been reported in Hawkins Hollow in the six preceding months. There were two suicides and four attempted suicides, numerous accounts of assault, three reported rapes, and a hit-and-run. Several homes and businesses were vandalized. None\u2014virtually none\u2014of the people involved in any of the reported crimes or incidents has a clear memory of the events. Some speculate the town suffered from mass hysteria or hallucinations or an unknown infection taken through food or water. What do you think?\"\n\n\"I think I was ten years old and pretty much scared shitless.\"\n\nShe offered that brief, sunny smile. \"I bet.\" Then it was gone. \"You were seventeen in nineteen ninety-four when during the week of July seventh another\u2014let's say outbreak\u2014occurred. Three people were murdered, one of them apparently hanged in the town park, but no one came forward as a witness or to admit participation. There were more rapes, more beatings, more suicides, two houses burned to the ground. There were reports that you, O'Dell, and Turner were able to get some of the wounded and traumatized onto a school bus and transport them to the hospital. Is that accurate?\"\n\n\"As far as it goes.\"\n\n\"I'm looking to go further. In two thousand one\u2014\"\n\n\"I know the pattern,\" Cal interrupted.\n\n\"Every seven years,\" Quinn said with a nod. \"For seven nights. Days\u2014according again to what I can ascertain\u2014little happens. But from sundown to sunset, all hell breaks loose. It's hard to believe that it's a coincidence this anomaly happens every seven years, with its start on your birthday. Seven's considered a magickal number by those who profess to magicks, black and white. You were born on the seventh day of the seventh month of nineteen seventy-seven.\"\n\n\"If I knew the answers, I'd stop it from happening. If I knew the answers, I wouldn't be talking to you. I'm talking to you because maybe, just maybe, you'll find them, or help find them.\"\n\n\"Then tell me what happened, tell me what you do know, even what you think or sense.\"\n\nCal set his coffee aside, leaned forward to look deep into her eyes. \"Not on a first date.\"\n\nSmart-ass, she thought with considerable approval. \"Fine. Next time I'll buy you dinner first. But now, how about playing guide and taking me to the Pagan Stone.\"\n\n\"It's too late in the day. It's a two-hour hike from here. We wouldn't make it there and back before dark.\"\n\n\"I'm not afraid of the dark.\"\n\nHis eyes went very cool. \"You would be. I'll tell you this, there are places in these woods no one goes after dark, not any time of the year.\"\n\nShe felt the prick of ice at the base of her spine. \"Have you ever seen a boy, about the age you'd have been in eighty-seven. A boy with dark hair. And red eyes.\" She saw by the way Cal paled she'd flicked a switch. \"You have seen him.\"\n\n\"Why do you ask about that?\"\n\n\"Because I saw him.\"\n\nNow Cal pushed to his feet, paced to the window, stared out at the woods. The light was dimmer, duller already than it had been an hour before.\n\nThey'd never told anyone about the boy\u2014or the man\u2014whatever form the thing chose to take. Yes, he'd seen him, and not only during that one hellish week every seven years.\n\nHe'd seen it in dreams. He'd seen it out of the corner of his eye, or loping through the woods. Or with its face pressed to the dark glass of his bedroom window...and its mouth grinning.\n\nBut no one, no one but he, Fox, and Gage had ever seen it in the between times.\n\nWhy had she?\n\n\"When and where did you see him?\"\n\n\"Today, just before I turned off onto Pagan Road. He ran in front of my car. Came out of nowhere. That's what people always say, but this time it's true. A boy, then it wasn't a boy but a dog. Then it wasn't anything. There was nothing there.\"\n\nHe heard her rise, and when he turned was simply stunned to see that brilliant smile on her face. \"And this kind of thing makes you happy?\"\n\n\"It makes me thrilled. Excited. I'm saying wow! I had myself what we could call a close encounter with an unspecified phenomenon. Scary, I grant you, but again, wow. This sort of thing completely winds me up.\"\n\n\"I can see that.\"\n\n\"I knew there was something here, and I thought it was big. But to have it confirmed, the first day out, that's hitting the mother lode with the first whack of the pick.\"\n\n\"I haven't confirmed anything.\"\n\n\"Your face did.\" She picked up her recorder, turned it off. He wasn't going to tell her anything today. Cautious man, Caleb Hawkins. \"I need to get into town, check into the hotel, get a lay of the land. Why don't I buy you that dinner tonight?\"\n\nShe moved fast, and he made a habit of taking his time. \"Why don't you take some time to settle in? We can talk about dinner and so on in a couple days.\"\n\n\"I love a man who's hard to get.\" She slipped her recorder, her notepad back in her bag. \"I guess I'll need my coat.\"\n\nAfter he'd brought it to her, she studied him as she shrugged it on. \"You know, when you first came outside, I had the strangest sensation. I thought I recognized you, that I'd known you before. That you'd waited for me before. It was very strong. Did you feel anything like that?\"\n\n\"No. But maybe I was too busy thinking, she looks better than her picture.\"\n\n\"Really? Nice, because I looked terrific in that picture. Thanks for the coffee.\" She glanced back to the dog who'd snored lightly the entire time they'd talked. \"See you later, Lump. Don't work so hard.\"\n\nHe walked her out. \"Quinn,\" he said as she started down the stairs. \"Don't get any ideas about Lois Laning it and trying to find the Pagan Stone on your own. You don't know the woods. I'll take you there myself, sometime this week.\"\n\n\"Tomorrow?\"\n\n\"I can't, I've got a full plate. Day after if you're in a hurry.\"\n\n\"I almost always am.\" She walked backward toward her car so she could keep him in view. \"What time?\"\n\n\"Let's say we'll meet here at nine, weather permitting.\"\n\n\"That's a date.\" She opened her car door. \"The house suits you, by the way. Country boy with more style than pretention. I like it.\"\n\nHe watched her drive off\u2014strange and sexy Quinn Black.\n\nAnd he stood for a long time watching the light go dimmer in the woods where he'd made his home.\n\nCAL HEADED FOX OFF WITH A PHONE CALL AND arranged to meet him at the bowling alley. Since the Pin Boys and the Alley Cats were having a league game on lanes one and two, he and Fox could have dinner and a show at the grill.\n\nAdded to it, there was little as noisy as a bowling alley, so their conversation would be covered by the crash of balls against pins, the hoots and hollers.\n\n\"First, let's backtrack into the land of logic for a minute.\" Fox took a swig of his beer. \"She could've made it up to get a reaction.\"\n\n\"How did she know what to make up?\"\n\n\"During the Seven, there are people who see it\u2014who've said they did before it starts to fade on them. She got wind.\"\n\n\"I don't think so, Fox. Some talked about seeing something\u2014boy, man, woman, dog, wolf\u2014\"\n\n\"The rat the size of a Doberman,\" Fox remembered.\n\n\"Thanks for bringing that one back. But no one ever claimed they'd seen it before or after the Seven. No one but us, and we've never told anyone.\" Cal arched his brow in question.\n\n\"No. You think I'm going to spread it around that I see red-eyed demons? I'd just rake in the clients that way.\"\n\n\"She's smart. I don't see why she'd claim to have seen it, outside the norm\u2014ha-ha\u2014if she hadn't. Plus she was psyched about it. Juiced up. So, let's accept she did and continue to dwell in the land of logic. One logical assumption is that the bastard's stronger, we know he will be. But strong enough to push out of the Seven into the between time.\"\n\nFox brooded over his beer. \"I don't like that logic.\"\n\n\"Second option could be she's somehow connected. To one of us, the town, the incident at the Pagan Stone.\"\n\n\"I like that better. Everyone's connected. It's not just Kevin Bacon. If you work at it, you can put a handful of degrees between almost any two people.\" Thoughtful, Fox picked up his second slice of pizza. \"Maybe she's a distant cousin. I've got cousins up the wazoo and so do you. Gage, not so much, but there's some out there.\"\n\n\"Possible. But why would a distant cousin see something none of our immediate family has? They'd tell us, Fox. They all know what's coming better and clearer than anyone else.\"\n\n\"Reincarnation. That's not off the Planet Logic, considering. Besides, reincarnation's big in the family O'Dell. Maybe she was there when it all happened. Another life.\"\n\n\"I don't discount anything. But more to the point, why is she here now? And will it help us put a goddamn end to this?\"\n\n\"It's going to take more than an hour's chat in front of the fire to figure that out. I don't guess you heard from Gage.\"\n\n\"Not yet. He'll be in touch. I'm going to take her out to the stone day after tomorrow.\"\n\n\"Leaping forward fast, Cal.\"\n\nCal shook his head. \"If I don't take her soon enough, she'll try it on her own. If something happened...We can't be responsible for that.\"\n\n\"We are responsible\u2014isn't that the point? On some level it's on us.\" Frowning now, he watched Don Myers, of Myers Plumbing, make a seven-ten split to appropriate hoots and shouts. All three hundred twenty pounds of Myers did a flab-wriggling victory dance that was not a pretty sight.\n\n\"You go on,\" Fox said quietly, \"day after day, doing what you do, living your life, making your life. Eating pizza, scratching your ass, getting laid if you're lucky. But you know, on some level you try to keep buried just to get through, that it's coming back. That some of the people you see on the street every day, maybe they won't make it through the next round. Maybe we won't. What the hell.\" He rapped his beer against Cal's. \"We've got the now, plus five months to figure this out.\"\n\n\"I can try to go back again.\"\n\n\"Not unless Gage is here. We can't risk it unless we're together. It's not worth it, Cal. The other times you only got bits and pieces, and took a hell of a beating for it.\"\n\n\"Older and wiser now. And I'm thinking, if it's showing itself now\u2014our dreams, what happened to Quinn\u2014it's expending energy. I might get more than I have before.\"\n\n\"Not without Gage. That's...Hmm,\" he said as his attention wandered over his friend's shoulder. \"Fresh flowers.\"\n\nGlancing back, Cal saw Quinn standing behind lane one, her coat open and a bemused expression on her face as she watched Myers, graceful as a hippo in toe shoes, make his approach and release his lucky red ball.\n\n\"That's Quinn.\"\n\n\"Yeah, I recognized her. I read the books, too. She's hotter than her picture, and that was pretty hot.\"\n\n\"I saw her first.\"\n\nFox snorted, shifted his eyes to sneer at Cal. \"Dude, it's not about who saw her first, it's who she sees. I pull out the full power of my sexual charm, and you'll be the Invisible Man.\"\n\n\"Shit. The full power of your sexual charm wouldn't light up a forty-watt bulb.\"\n\nCal pushed off the stool when Quinn walked toward him.\n\n\"So this is why I got the brush-off tonight,\" she said. \"Pizza, beer, and bowling.\"\n\n\"The Hawkins Hollow hat trick. I'm on manager duty tonight. Quinn, this is Fox O'Dell.\"\n\n\"The second part of the triad.\" She shook Fox's hand. \"Now I'm doubly glad I decided to check out what seems to be the town's hot spot. Mind if I join you?\"\n\n\"Wouldn't have it any other way. Buy you a beer?\" Fox asked.\n\n\"Boy, could you, but...make it a light one.\"\n\nCal stepped back to swing around the counter. \"I'll take care of it. Anything to go with it? Pizza?\"\n\n\"Oh.\" She looked at the pizza on the counter with eyes that went suddenly dewy. \"Um, I don't suppose you have any with whole-wheat crust and low-fat mozzarella?\"\n\n\"Health nut?\" Fox asked.\n\n\"Just the opposite.\" Quinn bit her bottom lip. \"I'm in a lifestyle change. Damn it, that really looks good. How about if we cut one of those slices in half.\" She sawed the side of her hand over the plate.\n\n\"No problem.\"\n\nCal got a pizza cutter and slid it down a slice.\n\n\"I love fat and sugar like a mother loves her child,\" Quinn told Fox. \"I'm trying to eat more sensibly.\"\n\n\"My parents are vegetarians,\" Fox said as they each picked up a half slice. \"I grew up on tofu and alfalfa.\"\n\n\"God. That's so sad.\"\n\n\"Which is why he ate at my house whenever he could manage it, and spent all his money on Little Debbies and Slim Jims.\"\n\n\"Little Debbies are food for the gods.\" She smiled at Cal when he set her beer on the counter. \"I like your town. I took a walk up and down several blocks of Main Street. And since I was freezing my ass off, went back to the really charming Hotel Hollow, sat on my windowsill, and watched the world go by.\"\n\n\"Nice world,\" Cal said, \"that moves a little slow this time of year.\"\n\n\"Umm,\" was her agreement as she took a minute bite of the point of her narrow triangle of pizza. She closed her eyes on a sigh. \"It is good. I was hoping, being bowling-alley pizza, it wouldn't be.\"\n\n\"We do okay. Gino's across the street is better, and has more selections.\"\n\nShe opened her eyes to find him smiling at her. \"That's a lousy thing to tell a woman in the middle of a lifestyle change.\"\n\nCal leaned on the counter, bringing that smile a little closer, and Quinn found herself losing her train of thought. He had the best quick and crooked grin, the kind a woman wanted to take a testing nibble of.\n\nBefore he could speak, someone hailed him, and those eyes of quiet gray glanced away from hers toward the end of the counter. \"Be right back.\"\n\n\"Well.\" Jeez, her pulse had actually tripped. \"Alone at last,\" she said to Fox. \"So you and Cal and the as-yet-absent Gage Turner have been friends since you were kids.\"\n\n\"Babies, actually. In utero, technically. Cal's and Gage's mother got together with mine when my mother was teaching a Lamaze class. They had a kind of roundup with the class a couple months after everyone delivered the packages, and the deal about the three of us being born on the same day, same time came out.\"\n\n\"Instant mommy bonding.\"\n\n\"I don't know. They always got along, even though you could say they all came from different planets. They were friendly without being friends. My parents and Cal's still get along fine, and Cal's dad kept Gage's employed when nobody else in town would've hired him.\"\n\n\"Why wouldn't anyone have hired him?\"\n\nFox debated for a minute, drank some of his beer. \"It's no secret,\" he decided. \"He drank. He's been sober for a while now. About five years, I guess. I always figured Mr. Hawkins gave him work because that's just the way he is, and, in a big part, he did it for Gage. Anyway, I don't remember the three of us not being friends.\"\n\n\"No 'you like him better than me,' major falling-outs or your basic and usual drifting apart?\"\n\n\"We fought\u2014fight still\u2014now and then.\" Didn't all brothers? Fox thought. \"Had your expected pissy periods, but no. We're connected. Nothing can snap that connection. And the 'you like him better than me'? Mostly a girl thing.\"\n\n\"But Gage doesn't live here anymore.\"\n\n\"Gage doesn't live anywhere, really. He's the original footloose guy.\"\n\n\"And you? The hometown boy.\"\n\n\"I thought about the bright lights, big city routine, even gave it a short try.\" He glanced over in the direction of the moans coming from one of the Alley Cats who had failed to pick up a spare. \"I like the Hollow. I even like my family, most of the time. And I like, as it turns out, practicing small-town law.\"\n\nTruth, Quinn decided, but not the whole truth of it. \"Have you seen the kid with the red eyes?\"\n\nOff balance, Fox set down the beer he'd lifted to drink. \"That's a hell of a segue.\"\n\n\"Maybe. But that wasn't an answer.\"\n\n\"I'm going to postpone my answer until further deliberation. Cal's taking point on this.\"\n\n\"And you're not sure you like the idea of him, or anyone, talking to me about what may or may not go on here.\"\n\n\"I'm not sure what purpose it serves. So I'm weighing the information as it comes in.\"\n\n\"Fair enough.\" She glanced over as Cal came back. \"Well, boys, thanks for the beer and the slice. I should get back to my adorable room.\"\n\n\"You bowl?\" Cal asked her, and she laughed.\n\n\"Absolutely not.\"\n\n\"Oh-oh,\" Fox said under his breath.\n\nCal walked around the counter, blocking Quinn before she could slide off the stool. He took a long, considering look at her boots. \"Seven and a half, right?\"\n\n\"Ah...\" She looked down at her boots herself. \"On the money. Good eye.\"\n\n\"Stay.\" He tapped her on the shoulder. \"I'll be right back.\"\n\nQuinn frowned after him, then looked at Fox. \"He is not going to get me a pair of bowling shoes.\"\n\n\"Oh yeah, he is. You mocked the tradition, which\u2014if you give him any tiny opening\u2014he'll tell you started five thousand years ago. Then he'll explain its evolution and so on and so on.\"\n\n\"Well, Christ,\" was all Quinn could think to say.\n\nCal brought back a pair of maroon and cream bowling shoes, and another, larger pair of dark brown ones, which were obviously his. \"Lane five's open. You want in, Fox?\"\n\n\"Sadly, I have a brief to finish writing. I'll rain-check it. See you later, Quinn.\"\n\nCal tucked the shoes under his arm, then, taking Quinn's hand, pulled her off the stool. \"When's the last time you bowled?\" he asked as he led her across the alley to an open lane.\n\n\"I think I was fourteen. Group date, which didn't go well, as the object of my affection, Nathan Hobbs, only had eyes for the incessantly giggly and already well-developed Missy Dover.\"\n\n\"You can't let previous heartbreak spoil your enjoyment.\"\n\n\"But I didn't like the bowling part either.\"\n\n\"That was then.\" Cal sat her down on the smooth wooden bench, slid on beside her. \"You'll have a better time with it tonight. Ever make a strike?\"\n\n\"Still talking bowling? No.\"\n\n\"You will, and there's nothing much that beats the feeling of that first strike.\"\n\n\"How about sex with Hugh Jackman?\"\n\nHe stopped tying his bowling shoe to stare over at her. \"You had sex with Hugh Jackman?\"\n\n\"No, but I'm willing to bet any amount of money that having sex with Hugh Jackman would, for me, beat out the feeling of knocking down ten pins with one ball.\"\n\n\"Okay. But I'm willing to bet\u2014let's make it ten bucks\u2014that when you throw a strike, you'll admit it's up there on the Thrill-O-Meter.\"\n\n\"First, it's highly unlikely I'll throw anything resembling a strike. Second, I could lie.\"\n\n\"You will. And you won't. Change your shoes, Blondie.\"\n\n## Five\n\nIT WASN'T AS RIDICULOUS AS SHE'D ASSUMED IT would be. Silly, yes, but she had plenty of room for silly.\n\nThe balls were mottled black\u2014the small ones without the three holes. The job was to heave it down the long polished alley toward the red-necked pins he called Duck Pins.\n\nHe watched as she walked up to the foul line, swung back, and did the heave.\n\nThe ball bounced a couple of times before it toppled into the gutter.\n\n\"Okay.\" She turned, tossed back her hair. \"Your turn.\"\n\n\"You get two more balls per frame.\"\n\n\"Woo-hoo.\"\n\nHe shot her the quick grin. \"Let's work on your delivery and follow-through, then we'll tackle approach.\" He walked toward her with another ball as he spoke. He handed her the ball. \"Hold it with both hands,\" he instructed as he turned her around to face the pins. \"Now you want to take a step forward with your left foot, bend your knees like you were doing a squat, but bend over from the waist.\"\n\nHe was snuggled up right behind her now, his front sort of bowing over her back. She tipped her face around to meet his eyes.\n\n\"You use this routine to hit on women, right?\"\n\n\"Absolutely. Eighty-five percent success ratio. You're going to want to aim for the front pin. You can worry about the pockets and the sweet spot later. Now you're just going to bring your right arm back, then sweep it forward with your fingers aimed at the front pin. Let the ball go, following your fingers.\"\n\n\"Hmm.\" But she tried it. This time the ball didn't bounce straight into the gutter, but actually stayed on the lane long enough to bump down the two pins on the far right.\n\nSince the woman in the next lane, who had to be sixty if she was a day, slid gracefully to the foul line, released, and knocked down seven pins, Quinn didn't feel like celebrating.\n\n\"Better.\"\n\n\"Two balls, two pins. I don't think that earns my bootie dance.\"\n\n\"Since I'm looking forward to your bootie dance, I'll help you do better yet. More from your shoulder down this time. Nice perfume,\" he added before he walked back to get her another ball.\n\n\"Thanks.\" Stride, bend, swing, release, she thought. And actually managed to knock down the end pin on the other side of the alley.\n\n\"Overcompensated.\" He hit the reset button. The grate came down, pins were swept off with a lot of clattering, and another full triangle thudded into place.\n\n\"She knocked them all down.\" Quinn gave a head nod toward the woman in the next lane who'd taken her seat. \"She didn't seem all that excited.\"\n\n\"Mrs. Keefafer? Bowls twice a week, and has become jaded. On the outside. Inside, believe me, she's doing her bootie dance.\"\n\n\"If you say so.\"\n\nHe adjusted Quinn's shoulders, shifted her hips. And yeah, she could see why he had such a high success rate with this routine. Eventually, after countless attempts, she was able to take down multiple pins that took odd bites out of the triangle.\n\nThere was a wall of noise, the low thunder of balls rolling, the sharp clatter of pins, hoots and cheers from bowlers and onlookers, the bright bells of a pinball machine.\n\nShe smelled beer and wax, and the gooey orange cheese\u2014a personal favorite\u2014from the nachos someone munched on in the next lane.\n\nTimeless, all-American, she mused, absently drafting an article on the experience. Centuries-old sport\u2014she'd need to research that part\u2014to good, clean, family fun.\n\nShe thought she had the hang of it, more or less, though she was shallow enough to throw a deliberate gutter ball here and there so Cal would adjust her stance.\n\nAs he did, she considered changing the angle of the article from family fun to the sexiness of bowling. The idea made her grin as she took her position.\n\nThen it happened. She released the ball and it rolled down the center of the alley. Surprised, she took a step back. Then another with her arms going up to clamp on the sides of her head.\n\nSomething tingled in her belly as her heartbeat sped up.\n\n\"Oh. Oh. Look! It's going to\u2014\"\n\nThere was a satisfying crack and crash as ball slapped pins and pins tumbled in all directions. Bumping into each other, rolling, spinning, until the last fell with a slow, drunken sway.\n\n\"Well, my God!\" She actually bounced on the toes of her rented shoes. \"Did you see that? Did you\u2014\" And when she spun around, a look of stunned delight on her face, he was grinning at her.\n\n\"Son of a bitch,\" she muttered. \"I owe you ten bucks.\"\n\n\"You learn fast. Want to try an approach?\"\n\nShe wandered back toward him. \"I believe I'm...spent. But I may come by some evening for lesson number two.\"\n\n\"Happy to oblige.\" Sitting hip-to-hip, they changed shoes. \"I'll walk you back to the hotel.\"\n\n\"All right.\"\n\nHe got his coat, and on the way out shot a wave at the skinny young guy behind the shoe rental counter. \"Back in ten.\"\n\n\"Quiet,\" she said the minute they stepped outside. \"Just listen to all that quiet.\"\n\n\"The noise is part of the fun and the quiet after part of the reward.\"\n\n\"Did you ever want to do anything else, or did you grow up with a burning desire to manage a bowling alley?\"\n\n\"Family fun center,\" he corrected. \"We have an arcade\u2014pinball, skee-ball, video games, and a section for kids under six. We do private parties\u2014birthday parties, bachelor parties, wedding receptions\u2014\"\n\n\"Wedding receptions?\"\n\n\"Sure. Bar mitzvahs, bat mitzvahs, anniversaries, corporate parties.\"\n\nDefinitely meat for an article, she realized. \"A lot of arms on one body.\"\n\n\"You could say that.\"\n\n\"So why aren't you married and raising the next generation of Bowl-a-Rama kingpins, pun intended.\"\n\n\"Love has eluded me.\"\n\n\"Aw.\"\n\nDespite the biting cold, it was pleasant to walk beside a man who naturally fit his stride to hers, to watch the clouds of their breath puff out, then merge together before the wind tore them to nothing.\n\nHe had an easy way about him and killer eyes, so there were worse things than feeling her toes go numb with cold in boots she knew were more stylish than practical.\n\n\"Are you going to be around if I think of some pertinent question to ask you tomorrow?\"\n\n\"'Round and about,\" he told her. \"I can give you my cell phone number if\u2014\"\n\n\"Wait.\" She dug into her bag and came out with her own phone. Still walking, she punched a few keys. \"Shoot.\"\n\nHe rattled it off. \"I'm aroused by a woman who not only immediately finds what she's looking for in the mysterious depths of her purse, but who can skillfully operate electronic devices.\"\n\n\"Is that a sexist remark?\"\n\n\"No. My mother always knows where everything is, but is still defeated by the universal remote. My sister Jen can operate anything from a six-speed to a wireless mouse, but can never find anything without a twenty-minute hunt, and my other sister, Marly, can't find anything, ever, and gets intimidated by her electric can opener. And here you are, stirring me up by being able to do both.\"\n\n\"I've always been a siren.\" She tucked her phone back in her bag as they turned to the steps leading to the long front porch of the hotel. \"Thanks for the escort.\"\n\n\"No problem.\"\n\nThere was one of those beats; she recognized it. Both of them wondering, did they shake hands, just turn and go, or give in to curiosity and lean into a kiss.\n\n\"Let's stay to the safe road for now,\" she decided. \"I admit, I like the look of your mouth, but moving on that's bound to tangle things up before I really get started on what brought me here.\"\n\n\"It's a damn shame you're right about that.\" He dipped his hands into his pockets. \"So I'll just say good night. I'll wait, make sure you get inside.\"\n\n\"Good night.\" She walked up the steps to the door, eased it open. Then glanced back to see him standing, hands still in his pockets, with the old-fashioned streetlight spotlighting him.\n\nOh, yeah, she thought, it was a damn shame.\n\n\"See you soon.\"\n\nHe waited until the door shut behind her, then taking a couple of steps, studied the windows of the second and third floor. She'd said her window faced Main Street, but he wasn't sure what level she was on.\n\nAfter a few moments, a light flashed on in a second-floor window, telling him Quinn was safe in her room.\n\nHe turned and had taken two steps when he saw the boy. He stood on the sidewalk half a block down. He wore no coat, no hat, no protection against the bite of wind. The long stream of his hair didn't stir in it.\n\nHis eyes gleamed, eerily red, as his lips peeled back in a snarl.\n\nCal heard the sound inside his head while ice balled in his belly.\n\nNot real, he told himself. Not yet. A projection only, like in the dreams. But even in the dreams, it could hurt you or make you think you were hurt.\n\n\"Go back where you came from, you bastard.\" Cal spoke clearly, and as calmly as his shaken nerves would allow. \"It's not your time yet.\"\n\nWhen it is, I'll devour you, all of you, and everything you hold precious.\n\nThe lips didn't move with the words, but stayed frozen in that feral snarl.\n\n\"We'll see who feels the bite this round.\" Cal took another step forward.\n\nAnd the fire erupted. It spewed out of the wide brick sidewalk, fumed across the street in a wall of wild red. Before he could register that there was no heat, no burn, Cal had already stumbled back, thrown up his hands.\n\nThe laughter rang in his head, as wild as the flames. Then both snapped off.\n\nThe street was quiet, the brick and buildings unmarred. Tricks up his sleeve, Cal reminded himself. Lots of tricks up his sleeve.\n\nHe made himself stride forward, through where the false fire had run. There was a strong acrid odor that puffed then vanished like the vapor of his own breath. In that instant he recognized it.\n\nBrimstone.\n\nUPSTAIRS IN THE ROOM THAT MADE HER BLISSFULLY happy with its four-poster bed and fluffy white duvet, Quinn sat at the pretty desk with its curved legs and polished surface writing up the day's notes, data, and impressions on her laptop.\n\nShe loved that there were fresh flowers in the room, and a little blue bowl of artfully arranged fresh fruit. The bath held a deep and delightful claw-foot tub and a snowy white pedestal sink. There were thick, generous towels, two bars of soap, and rather stylish minibottles of shampoo, body cream, and bath gel.\n\nInstead of boring, mass-produced posters, the art on the walls were original paintings and photos, which the discreet note on the desk identified as works by local artists available at Artful, a shop on South Main.\n\nThe room was full of homey welcoming touches, and provided high-speed Internet access. She made a note to reserve the same room after her initial week was up, for the return trips she planned in April, then again in July.\n\nShe'd accomplished quite a bit on her first day, which was a travel day on top of it. She'd met two of the three focal players, had an appointment to hike to the Pagan Stone. She'd gotten a feel for the town, on the surface in any case. And had, she believed, a personal experience with the manifestation of an unidentified (as yet) force.\n\nAnd she had the bare bones for a bowling article that should work for her friends at Detour.\n\nNot bad, especially when you added in she'd dined sensibly on the grilled chicken salad in the hotel dining room, had not given in to temptation and inhaled an entire pizza but had limited herself to half a slice. And she'd bowled a strike.\n\nOn the personal downside, she supposed, as she shut down to prepare for bed, she'd also resisted the temptation to lock lips with the very appealing Caleb Hawkins.\n\nWasn't she all professional and unsatisfied?\n\nOnce she'd changed into her bedtime flannel pants and T-shirt, she nagged herself into doing fifteen minutes of pilates (okay, ten), then fifteen of yoga, before burrowing under the fabulous duvet with her small forest of down pillows.\n\nShe took her current book off the nightstand, burrowed into that as well until her eyes began to droop.\n\nJust past midnight, she marked the novel, switched off the lamp, and snuggled into her happy nest.\n\nAs was her habit, she was asleep in a finger snap.\n\nQuinn recognized the dream as a dream. Always, she enjoyed the sensation of the disjointed, carnival world of dreamscapes. It was, for her, like having some crazy adventure without any physical exertion. So when she found herself on a crooked path through a thick wood where the moonshine silvered the leaves and the curling fog rippled along the ground, a part of her mind thought: Oh boy! Here we go.\n\nShe thought she heard chanting, a kind of hoarse and desperate whisper, but the words themselves were indiscernible.\n\nThe air felt like silk, so soft, as she waded through the pools of fog. The chanting continued, drawing her toward it. A single word seemed to fly out of that moonstruck night, and the word was bestia.\n\nShe heard it over and over as she followed the crooked path through the silken air and the silver-laced trees. She felt a sexual pull, a heat and reaching in the belly toward whatever, whoever called out in the night.\n\nTwice, then three times, the air seemed to whisper. Beatus. The murmur of that warmed her skin. In the dream, she quickened her steps.\n\nOut of the moon-drenched trees swam a black owl, its great wings stirring a storm in that soft air, chilling it until she shivered. And was, even in the dream, afraid.\n\nWith that cold wind stirring, she saw, stretched across the path, a golden fawn. The blood from its slit throat drenched the ground so it gleamed wet and black in the night.\n\nHer heart squeezed with pity. So young, so sweet, she thought as she made herself approach it. Who could have done such a thing?\n\nFor a moment, the dead, staring eyes of the fawn cleared, shone as gold as its hind. It looked at her with such sorrow, such wisdom, tears gathered in her throat.\n\nThe voice came now, not through the whipped air, but in her mind. The single word: devoveo.\n\nThen the trees were bare but for the ice that sheathed trunk and branch, and the silver moonlight turned gray. The path had turned, or she had, so now she faced a small pond. The water was black as ink, as if any light the sky pushed down was sucked into its depths and smothered there.\n\nBeside the pond was a young woman in a long brown dress. Her hair was chopped short, with the strings and tufts of it sticking out wildly. Beside the black pond she bent to fill the pockets of her brown dress with stones.\n\nHello! Quinn called out. What are you doing?\n\nThe girl only continued to fill her pockets. As Quinn walked closer, she saw the girl's eyes were full of tears, and of madness.\n\nCrap. You don't want to do that. You don't want to go Virginia Woolf. Wait. Just wait. Talk to me.\n\nThe girl turned her head, and for one shocked moment, Quinn saw the face as her own. He doesn't know everything, the mad girl said. He didn't know you.\n\nShe threw out her arms, and her slight body, weighed heavy with her cache of stones, tipped, tipped, tipped until it met the black water. The pond swallowed it like a waiting mouth.\n\nQuinn leaped\u2014what else could she do? Her body braced for the shock of cold as she filled her lungs with air.\n\nThere was a flash of light, a roar that might have been thunder or something alive and hungry. She was on her knees in a clearing where a stone rose out of the earth like an altar. Fire spewed around her, above her, through her, but she felt none of its heat.\n\nThrough the flames she saw two shapes, one black, one white, grappling like mad animals. With a terrible rending sound, the earth opened up, and like the waiting mouth of the pond, swallowed everything.\n\nThe scream ripped from her throat as that maw widened to take her. Clawing, she dragged herself toward the stone, fought to wrap her arms around it.\n\nIt broke into three equal parts, sending her tumbling, tumbling into that open, avid mouth.\n\nShe woke, huddled on the lovely bed, the linens tangled around her legs as she gripped one of the bedposts as if her life depended on it.\n\nHer breath was an asthmatic's wheeze, and her heart beat so fast and hard it had her head spinning.\n\nA dream, just a dream, she reminded herself, but couldn't force herself\u2014not quite yet\u2014to release her hold on the bedpost.\n\nClinging to it, she let her cheek rest on the wood, closed her eyes until the shaking had lessened to an occasional quiver.\n\n\"Hell of a ride,\" she mumbled.\n\nThe Pagan Stone. That's where she'd been at the end of the dream, she was certain of it. She recognized it from pictures she'd seen. Small wonder she'd have a scary dream about it, about the woods. And the pond...Wasn't there something in her research about a woman drowning in the pond? They'd named it after her. Hester's Pond. No, pool. Hester's Pool.\n\nIt all made sense, in dream logic.\n\nYeah, a hell of a ride, and she'd die happy if she never took another like it.\n\nShe glanced at her travel alarm, and saw by its luminous dial it was twenty after three. Three in the morning, she thought, was the dead time, the worst time to be wakeful. So she'd go back to sleep, like a sensible woman. She'd straighten the bed, get herself a nice cool drink of water, then tune out.\n\nShe'd had enough jolts and jumps for her first day.\n\nShe slid out of bed to tug the sheets and duvet back into some semblance of order, then turned, intending to go to the adjoining bath for a glass of water.\n\nThe scream wouldn't sound. It tore through her head like scrabbling claws, but nothing could tear its way out of the hot lock of her throat.\n\nThe boy grinned obscenely through the dark window. His face, his hands pressed against the glass bare inches away from her own. She saw its tongue flick out to roll across those sharp, white teeth, and those eyes, gleaming red, seemed as bottomless and hungry as the mouth of earth that had tried to swallow her in her dream.\n\nHer knees wanted to buckle, but she feared if she dropped to the ground it would come crashing through the glass to latch those teeth on her throat like a wild dog.\n\nInstead, she lifted her hand in the ancient sign against evil. \"Get away from here,\" she whispered. \"Stay away from me.\"\n\nIt laughed. She heard the horrible, giddy sound of it, saw its shoulders shake with mirth. Then it pushed off the glass into a slow, sinuous somersault. It hung suspended for a moment above the sleeping street. Then it...condensed, was all she could think. It shrank into itself, into a pinpoint of black, and vanished.\n\nQuinn launched herself at the window, yanked the shade down to cover every inch of glass. And lowering to the floor at last, she leaned back against the wall, trembling.\n\nWhen she thought she could stand, she used the wall as a brace, quick-stepping to the other windows. She was out of breath again by the time all the shades were pulled, and tried to tell herself the room didn't feel like a closed box.\n\nShe got the water\u2014she needed it\u2014and gulped down two full glasses. Steadier, she stared at the covered windows.\n\n\"Okay, screw you, you little bastard.\"\n\nPicking up her laptop, she went back to her position on the floor\u2014it just felt safer under the line of the windowsills\u2014and began to type up every detail she remembered from the dream, and from the thing that pressed itself to the night glass.\n\nWHEN SHE WOKE, THE LIGHT WAS A HARD YELLOW line around the cream linen of the shades. And the battery of her laptop was stone dead. Congratulating herself on remembering to back up before she'd curled onto the floor to sleep, she got her creaky self up.\n\nStupid, of course, she told herself as she tried to stretch out the worst of the stiffness. Stupid not to turn off her machine, then crawl back into that big, cozy bed. But she'd forgotten the first and hadn't even considered the second.\n\nNow, she put the computer back on the pretty desk, plugged it in to recharge the batteries. With some caution\u2014after all, it had been broad daylight when she'd seen the boy the first time\u2014she approached the first window. Eased up the shade.\n\nThe sun was lancing down out of a boiled blue sky. On the pavement, on awnings and roofs, a fresh white carpet of snow shimmered.\n\nShe spotted a few merchants or their employees busily shoveling sidewalks or porches and steps. Cars putted along the plowed street. She wondered if school had been called or delayed due to the snow.\n\nShe wondered if the boy had demon classes that day.\n\nFor herself, Quinn decided she was going to treat her abused body to a long soak in the charming tub. Then she'd try Ma's Pantry for breakfast, and see who she could get to talk to her over her fruit and granola about the legends of Hawkins Hollow.\n\n## Six\n\nCAL SAW HER COME IN WHILE HE CUT INTO HIS short stack at the counter. She had on those high, sharp-heeled boots, faded jeans, and a watch cap, bright as a cardinal, pulled over her hair.\n\nShe'd wound on a scarf that made him think of Joseph's coat of many colors, which added a jauntiness with her coat opened. Under it was a sweater the color of ripe blueberries.\n\nThere was something about her, he mused, that would have been bright and eye-catching even in mud brown.\n\nHe watched her eyes track around the diner area, and decided she was weighing where to sit, whom to approach. Already working, he concluded. Maybe she always was. He was damn sure, even on short acquaintance, that her mind was always working.\n\nShe spotted him. She aimed that sunbeam smile of hers, started over. He felt a little like the kid in the pickup game of ball, who got plucked from all the others waving their arms and shouting: Me! Me! Pick me!\n\n\"Morning, Caleb.\"\n\n\"Morning, Quinn. Buy you breakfast?\"\n\n\"Absolutely.\" She leaned over his plate, took a long, dramatic sniff of his butter-and-syrup-loaded pancakes. \"I bet those are fabulous.\"\n\n\"Best in town.\" He stabbed a thick bite with his fork, held it out. \"Want a sample?\"\n\n\"I can never stop at a taste. It's a sickness.\" She slid onto the stool, swiveled around to beam at the waitress as she unwound her scarf. \"Morning. I'd love some coffee, and do you have any granola-type substance that could possibly be topped with any sort of fruit?\"\n\n\"Well, we got Special K, and I could slice you up some bananas with it.\"\n\n\"Perfect.\" She reached over the counter. \"I'm Quinn.\"\n\n\"The writer from up in PA.\" The waitress nodded, took Quinn's hand in a firm grip. \"Meg Stanley. You watch this one here, Quinn,\" Meg said with a poke at Cal. \"Some of those quiet types are sneaky.\"\n\n\"Some of us mouthy types are fast.\"\n\nThat got a laugh out of Meg as she poured Quinn's coffee. \"Being quick on your feet's a strong advantage. I'll get that cereal for you.\"\n\n\"Why,\" Cal wondered aloud as he forked up another dripping bite of pancake, \"would anyone willingly choose to eat trail mix for breakfast?\"\n\n\"It's an acquired taste. I'm still acquiring it. But knowing myself, and I do, if I keep coming in here for breakfast, I'll eventually succumb to the allure of the pancake. Does the town have a gym, a health club, a burly guy who rents out his Bowflex?\"\n\n\"There's a little gym down in the basement of the community center. You need a membership, but I can get you a pass on that.\"\n\n\"Really? You're a handy guy to know, Cal.\"\n\n\"I am. You want to change your order? Go for the gold, then the treadmill?\"\n\n\"Not today, but thanks. So.\" After she'd doctored her coffee, she picked up the cup with both hands, sipping as she studied him through the faint rise of steam. \"Now that we're having our second date\u2014\"\n\n\"How'd I miss the first one?\"\n\n\"You bought me pizza and a beer and took me bowling. In my dictionary, that falls under the definition of date. Now you're buying me breakfast.\"\n\n\"Cereal and bananas. I do appreciate a cheap date.\"\n\n\"Who doesn't? But since we're dating and all...\" She took another sip as he laughed. \"I'd like to share an experience with you.\"\n\nShe glanced over as Meg brought her a white stoneware bowl heaped with cereal and sliced bananas. \"Figured you'd be going for the two percent milk with this.\"\n\n\"Perceptive and correct, thanks.\"\n\n\"Get you anything else?\"\n\n\"We're good for now, Meg,\" Cal told her. \"Thanks.\"\n\n\"Just give a holler.\"\n\n\"An experience,\" Cal prompted, as Meg moved down the counter.\n\n\"I had a dream.\"\n\nHis insides tensed even before she began to tell him, in a quiet voice and in careful detail of the dream she'd had during the night.\n\n\"I knew it was a dream,\" she concluded. \"I always do, even during them. Usually I get a kick out of them, even the spooky ones. Because, you know, not really happening. I haven't actually grown a second head so I can argue with myself, nor am I jumping out of a plane with a handful of red balloons. But this...I can't say I got a charge out of it. I didn't just think I felt cold, for instance. I was cold. I didn't just think I felt myself hit and roll on the ground. I found bruises this morning that weren't there when I went to bed. Fresh bruises on my hip. How do you get hurt in a dream, if it's just a dream?\"\n\nYou could, he thought, in Hawkins Hollow. \"Did you fall out of bed, Quinn?\"\n\n\"No, I didn't fall out of bed.\" For the first time, there was a whiff of irritation in her voice. \"I woke up with my arms locked around the bedpost like it was my long-lost lover. And all this was before I saw that red-eyed little bastard again.\"\n\n\"Where?\"\n\nShe paused long enough to spoon up some cereal. He wasn't sure if the expression of displeasure that crossed her face was due to the taste, or her thoughts. \"Did you ever read King's Salem's Lot?\"\n\n\"Sure. Small town, vampires. Great stuff.\"\n\n\"Remember that scene? The little boys, brothers. One's been changed after they snatched him off the path in the woods. He comes to visit his brother one night.\"\n\n\"Nothing scarier than kiddie vampires.\"\n\n\"Not much, anyway. And the vampire kid's just hanging outside the window. Just floating out there, scratching on the glass. It was like that. He was pressed to the glass, and I'll point out I'm on the second floor. Then he did a stylish back flip in the air, and poofed.\"\n\nHe laid a hand over hers, found it cold, rubbed some warmth into it. \"You have my home and cell numbers, Quinn. Why didn't you call me?\"\n\nShe ate a little more, then, smiling at Meg, held up her cup for a top-off. \"I realize we're dating, Cal, but I don't call all the guys I go bowling with at three thirty in the morning to go: eek! I slogged through swamps in Louisiana on the trail of the ghost of a voodoo queen\u2014and don't think I don't know how that sounds. I spent the night, alone, in a reputedly haunted house on the coast of Maine, and interviewed a guy who was reported to be possessed by no less than thirteen demons. Then there was the family of werewolves in Tallahassee. But this kid...\"\n\n\"You don't believe in werewolves and vampires, Quinn.\"\n\nShe turned on the stool to face him directly. \"My mind's as open as a twenty-four-hour deli, and considering the circumstances, yours should be, too. But no, I don't think this thing is a vampire. I saw him in broad daylight, after all. But he's not human, and just because he's not human doesn't make him less than real. He's part of the Pagan Stone. He's part of what happens here every seven years. And he's early, isn't he?\"\n\nYeah, he thought, her mind was always working and it was sharp as a switchblade. \"This isn't the best place to go into this any deeper.\"\n\n\"Say where.\"\n\n\"I said I'd take you to the stone tomorrow, and I will. We'll get into more detail then. Can't do it today,\" he said, anticipating her. \"I've got a full plate, and tomorrow's better anyway. They're calling for sun and forties today and tomorrow.\" He hitched up a hip to take out his wallet. \"Most of this last snow'll be melted.\" He glanced down at her boots as he laid bills on the counter to cover both their tabs. \"If you don't have anything more suitable to hike in than those, you'd better buy something. You won't last a half mile otherwise.\"\n\n\"You'd be surprised how long I can last.\"\n\n\"Don't know as I would. I'll see you tomorrow if not before.\"\n\nQuinn frowned at him as he walked out, then turned back as Meg slid her rag down the counter. \"Sneaky. You were right about that.\"\n\n\"Known the boy since before he was born, haven't I?\"\n\nAmused, Quinn propped an elbow back on the bar as she toyed with the rest of her cereal. Apparently a serious scare in the night and mild irritation with a man in the morning was a more effective diet aid than any bathroom scale. Meg struck her as a comfortable woman, wide-hipped in her brown cords and flannel shirt, sleeves rolled up at the elbows. Her hair curled tight as a poodle's fur in a brown ball around a soft and lined face. And there was a quick spark in her hazel eyes that told Quinn she'd be inclined to talk.\n\n\"So, Meg, what else do you know? Say about the Pagan Stone.\"\n\n\"Buncha nonsense, you ask me.\"\n\n\"Really?\"\n\n\"People just get a little\"\u2014she circled her finger at her ear\u2014\"now and again. Tip too much at the bottle, get all het up. One thing leads to another. Good for business though, the speculation, if you follow me. Get plenty of flatlanders in here wondering about it, asking about it, taking pictures, buying souvenirs.\"\n\n\"You never had any experiences?\"\n\n\"Saw some people usually have good sense acting like fools, and some who got a mean streak in them acting meaner for a spell of time.\" She shrugged. \"People are what people are, and sometimes they're more so.\"\n\n\"I guess that's true.\"\n\n\"If you want more about it, you should go on out to the library. There's some books there written about the town, the history and whatnot. And Sally Keefafer\u2014\"\n\n\"Bowling Sally?\"\n\nMeg snorted a laugh. \"She does like to bowl. Library director. She'll bend your ear plenty if you ask her questions. She loves to talk, and never found a subject she couldn't expound on till you wanted to slap some duct tape over her mouth.\"\n\n\"I'll do that. You sell duct tape here?\"\n\nMeg hooted out another laugh, shook her head. \"If you really want to talk, and get some sense out of it, you want Mrs. Abbott. She ran the old library, and she's at the new one for a spell most every day.\"\n\nThen scooping up the bills Cal left, she went to refill waiting cups at the other end of the counter.\n\nCAL HEADED STRAIGHT TO HIS OFFICE. HE HAD the usual morning's paperwork, phone calls, e-mails. And he had a morning meeting scheduled with his father and the arcade guy before the center opened for the afternoon leagues.\n\nHe thought of the wall of fire across Main Street the night before. Add that to two sightings by Quinn\u2014an outsider\u2014and it sure as hell seemed the entity that plagued the town was starting its jollies early.\n\nHer dream troubled him as well. The details\u2014he'd recognized where she'd been, what she'd seen. For her to have dreamed so lucidly about the pond, about the clearing, to have bruises from it, meant, in his opinion, she had to be connected in some way.\n\nA distant relation wasn't out of the question, and there should be a way to do a search. But he had other relations, and none but his immediate family had ever spoken of any effects, even during the Seven.\n\nAs he passed through the bowling center, he sent a wave toward Bill Turner, who was buffing the lanes. The big, burly machine's throaty hum echoed through the empty building.\n\nThe first thing he checked in his office was his e-mail, and he let out a breath of relief when he saw one from Gage.\n\nPrague. Got some business to clear up. Should be back in the U.S. of A. inside a couple weeks. Don't do anything stupider than usual without me.\n\nNo salutation, no signature. Very Gage, Cal thought. And it would have to do, for now.\n\nContact me as soon as you're Stateside, Cal wrote back. Things are already rumbling. Will always wait for you to do the stupid, because you're better at it.\n\nAfter clicking Send, he dashed another off to Fox.\n\nNeed to talk. My place, six o'clock. Got beer. Bring food that's not pizza.\n\nBest he could do, for now, Cal thought. Because life just had to keep rolling on.\n\nQUINN WALKED BACK TO THE HOTEL TO RETRIEVE her laptop. If she was going to the library, she might as well use it for a couple hours' work. And while she expected she had most, if not all, of the books tucked into the town's library already, maybe this Mrs. Abbott would prove to be a valuable source.\n\nCaleb Hawkins, it appeared, was going to be a clam until the following day.\n\nAs she stepped into the hotel lobby she saw the pert blond clerk behind the desk\u2014Mandy, Quinn thought after a quick scroll through her mental PDA\u2014and a brunette in the curvy chair being checked in.\n\nQuinn's quick once-over registered the brunette with the short, sassy do as mid to late twenties, with a travel-weary look about her that didn't do anything much to diminish the seriously pretty face. Jeans and a black sweater fit well over an athletic build. Pooled at her feet were a suitcase, a laptop case, a smaller bag probably for cosmetics and other female necessities, and an excellent and roomy hobo in slick red leather.\n\nQuinn had a moment of purse envy as she aimed a smile.\n\n\"Welcome back, Miss Black. If you need anything, I'll be with you in just a minute.\"\n\n\"I'm fine, thanks.\"\n\nQuinn turned to the stairs and, starting up, heard Mandy's cheerful, \"You're all checked in, Miss Darnell. I'm just going to call Harry to help with your bags.\"\n\nAs was her way, Quinn speculated on Miss Gorgeous Red Bag Darnell as she climbed up to her room. Passing through on her way to New York. No, too odd a place to stop over, and too early in the day to stop a road trip.\n\nVisiting relatives or friends, but why wouldn't she just bunk with said relatives or friends? Then again, she had some of both she'd rather not bunk with.\n\nMaybe a business trip, Quinn mused as she let herself into her room.\n\nWell, if Red Bag I Want for My Very Own stayed more than a few hours, Quinn would find out just who and what and why. It was, after all, what she was best at.\n\nQuinn packed up her laptop, added a spare notebook and extra pencils in case she got lucky. Digging out her phone, she set it on vibrate. Little was more annoying, to her mind, than ringing cell phones in libraries and theaters.\n\nShe slipped a county map into her case in the event she decided to explore.\n\nArmed, she headed down for the drive to the other end of town and the Hawkins Hollow Library.\n\nFrom her own research, Quinn knew that the original stone building tucked on Main Street now housed the community center, and the gym she intended to make use of. At the turn of the current century the new library had been built on a pretty rise of land on the south end of town. It, too, was stone, though Quinn was pretty sure it was the facing used on concrete and such rather than quarried. It was two levels with short wings on either side and a portico-style entrance. The style, she thought, was attractively old-fashioned. One, she guessed, the local historic society had likely fought a war to win.\n\nShe admired the benches, and the trees she imagined made shady reading nooks in season as she pulled up to park in the side lot.\n\nIt smelled like a library, she thought. Of books and a little dust, of silence.\n\nShe saw a brightly lettered sign announcing a Story Hour in the Children's section at ten thirty.\n\nShe wound her way through. Computers, long tables, carts, a few people wandering the stacks, a couple of old men paging through newspapers. She heard the soft hum-chuck of a copier and the muted ringing of a phone from the Information Desk.\n\nReminding herself to focus because if she wandered she'd be entranced by the spell she believed all libraries wove, she aimed straight for Information. And in the hushed tone reserved for libraries and churches, addressed the stringy man on duty. \"Good morning, I'm looking for books on local history.\"\n\n\"That would be on the second floor, west wing. Steps over to the left, elevator straight back. Anything in particular you're after?\"\n\n\"Thanks, but I'm just going to poke. Is Mrs. Abbott in today?\"\n\n\"Mrs. Abbott is retired, but she's in most every day by eleven. In a volunteer capacity.\"\n\n\"Thanks again.\"\n\nQuinn used the stairs. They had a nice curve to them, she thought, almost a Gone With the Wind sort of swish. She put on mental blinders so as not to be tempted by stacks and reading areas until she found herself in Local Interest.\n\nIt was more a room\u2014a mini-library\u2014than a section. Nice cozy chairs, tables, amber-shaded lamps, even footrests. And it was larger than she'd expected.\n\nThen again, she should have accounted for the fact that there had been battles fought in and around the Hollow in both the Revolutionary and Civil Wars.\n\nBooks pertaining to those were arranged in separate areas, as were books on the county, the state, and the town.\n\nIn addition there was a very healthy section for local authors.\n\nShe tried that section first and saw she'd hit a treasure trove. There had to be more than a dozen she hadn't come across on her own hunt before coming to town. They were self-published, vanity-pressed, small local publishers.\n\nTitles like Nightmare Hollow and The Hollow, The Truth had her giddy with anticipation. She set up her laptop, her notebook, her recorder, then pulled out five books. It was then she noticed the discreet bronze plaque.\n\nThe Hawkins Hollow Library gratefully acknowledges the generosity of the Franklin and Maybelle Hawkins Family\n\nFranklin and Maybelle. Very probably Cal's ancestors. It struck Quinn as both suitable and generous that they would have donated the funds to sponsor this room. This particular room.\n\nShe settled at the table, chose one of the books at random, then began to read.\n\nShe'd covered pages of her notebook with names, locations, dates, reputed incidents, and any number of theories when she scented lavender and baby powder.\n\nSurfacing, she saw a trim and tidy old woman standing in black, sensible shoes with her hands folded neatly at the waist of her purple suit.\n\nHer hair was a thinning snowball; her clear framed glasses so thickly lensed Quinn wondered how the tiny nose and ears supported their weight.\n\nShe wore pearls around her neck, a gold wedding band on her finger, and a leather-banded watch with a huge face that looked to be as practical as her thick-soled shoes.\n\n\"I'm Estelle Abbott,\" she said in her creaky voice. \"Young Dennis said you asked after me.\"\n\nAs Quinn had gauged Dennis at Information as tumbling down the back end of his sixties, she imagined the woman who termed him young must have him by a good two decades.\n\n\"Yes.\" Quinn got to her feet, crossed over to offer her hand. \"I'm Quinn Black, Mrs. Abbott. I'm\u2014\"\n\n\"Yes, I know. The writer. I've enjoyed your books.\"\n\n\"Thank you very much.\"\n\n\"No need. If I hadn't liked them I'd've told you straight-out. You're researching for a book on the Hollow.\"\n\n\"Yes, ma'am, I am.\"\n\n\"You'll find quite a bit of information here. Some of it useful.\" She peered at the books on the table. \"Some of it nonsense.\"\n\n\"Then in the interest of separating the wheat from the chaff, maybe you could find some time to talk to me at some point. I'd be happy to take you to lunch or dinner whenever you\u2014\"\n\n\"That's very nice of you, but unnecessary. Why don't we sit down for a while, and we'll see how things go?\"\n\n\"That would be great.\"\n\nEstelle crossed to a chair, sat, then with her back ruler-straight and her knees glued together, folded her hands in her lap. \"I was born in the Hollow,\" she began, \"lived here all of my ninety-seven years.\"\n\n\"Ninety-seven?\" Quinn didn't have to feign the surprise. \"I'm usually pretty good at gauging age, and I'd put you a solid decade under that.\"\n\n\"Good bones,\" Estelle said with an easy smile. \"I lost my husband, John, also born and raised here, eight years back come the fifth of next month. We were married seventy-one years.\"\n\n\"What was your secret?\"\n\nThat brought on another smile. \"Learn to laugh, otherwise, you'll beat them to death with a hammer first chance.\"\n\n\"Just let me write that down.\"\n\n\"We had six children\u2014four boys, two girls\u2014and all of them living still and not in jail, thank the Lord. Out of them, we had ourselves nineteen grandchildren, and out of them got ourselves twenty-eight greats\u2014last count, and five of the next generation with two on the way.\"\n\nQuinn simply goggled. \"Christmas must be insane in a good way.\"\n\n\"We're scattered all over, but we've managed to get most everybody in one place at one time a few times.\"\n\n\"Dennis said you were retired. You were a librarian?\"\n\n\"I started working in the library when my youngest started school. That would be the old library on Main Street. I worked there more than fifty years. Went back to school myself and got my degree. Johnnie and I traveled, saw a lot of the world together. For a time we thought about moving on down to Florida. But our roots here were too deep for that. I went to part-time work, then I retired when my Johnnie got sick. When he passed, I came back\u2014still the old one while this was being built\u2014as a volunteer or as an artifact, however you look at it. I tell you this so you'll have some idea about me.\"\n\n\"You love your husband and your children, and the children who've come from them. You love books, and you're proud of the work you've done. You love this town, and respect the life you've lived here.\"\n\nEstelle gave her a look of approval. \"You have an efficient and insightful way of summing up. You didn't say I loved my husband, but used the present tense. That tells me you're an observant and sensitive young woman. I sensed from your books that you have an open and seeking mind. Tell me, Miss Black, do you also have courage?\"\n\nQuinn thought of the thing outside the window, the way its tongue had flicked over its teeth. She'd been afraid, but she hadn't run. \"I like to think so. Please call me Quinn.\"\n\n\"Quinn. A family name.\"\n\n\"Yes, my mother's maiden.\"\n\n\"Irish Gaelic. I believe it means 'counselor.'\"\n\n\"It does, yes.\"\n\n\"I have a well of trivial information,\" Estelle said with a tap of her finger to her temple. \"But I wonder if your name isn't relevant. You'll need to have the objectivity, and the sensitivity of a counselor to write the book that should be written on Hawkins Hollow.\"\n\n\"Why haven't you written it?\"\n\n\"Not everyone who loves music can play the tune. Let me tell you a few things, some of which you may already know. There is a place in the woods that borders the west of this town, and that place was sacred ground, sacred and volatile ground long before Lazarus Twisse sought it out.\"\n\n\"Lazarus Twisse, the leader of the Puritan sect\u2014the radical sect\u2014which broke off or, more accurately, was cut off, from the godly in Massachusetts.\"\n\n\"According to the history of the time, yes. The Native Americans held that ground as sacred. And before them, it's said, powers battled for that circle of ground, both\u2014the dark and the light, good and evil, whatever terms you prefer\u2014left some seeds of that power there. They lay dormant, century by century, with only the stone to mark what had passed there. Over time the memories of the battle were forgotten or bastardized in folklore, and only the sense many felt that this ground and its stone were not ordinary dirt and rock remained.\"\n\nEstelle paused, fell into silence so that Quinn heard the click and hum of the heater, and the light slap of leather shoes on the floor as someone passed by the room toward other business.\n\n\"Twisse came to the Hollow, already named for Richard Hawkins, who, with his wife and children, had carved a small settlement in 1648. You should remark that Richard's eldest daughter was Ann. When Twisse came, Hawkins, his family, and a handful of others\u2014some who'd fled Europe as criminals, political or otherwise\u2014had made their life here. As had a man calling himself Giles Dent. And Dent built a cabin in the woods where the stone rose out of the ground.\"\n\n\"What's called the Pagan Stone.\"\n\n\"Yes. He troubled no one, and as he had some skill and knowledge of healing, was often sought out for sickness or injury. There are some accounts that claim he was known as the Pagan, and that this was the basis of the name the Pagan Stone.\"\n\n\"You're not convinced those accounts are accurate.\"\n\n\"It may be that the term stuck, entered the language and the lexicon at that time. But it was the Pagan Stone long before the arrival of Giles Dent or Lazarus Twisse. There are other accounts that claim Dent dabbled in witchcraft, that he enspelled Ann Hawkins, seduced and impregnated her. Others state that Ann and Dent were indeed lovers, but that she went to his bed of her own free will, and left her family home to live with him in the little cabin with the Pagan Stone.\"\n\n\"It would've been difficult for her\u2014for Ann Hawkins\u2014either way,\" Quinn speculated. \"Enspelled or free will, to live with a man, unmarried. If it was free will, if it was love, she must have been very strong.\"\n\n\"The Hawkinses have always been strong. Ann had to be strong to go to Dent, to stay with him. Then she had to be strong enough to leave him.\"\n\n\"There are a lot of conflicting stories,\" Quinn began. \"Why do you believe Ann Hawkins left Giles Dent?\"\n\n\"I believe she left to protect the lives growing inside her.\"\n\n\"From?\"\n\n\"Lazarus Twisse. Twisse and those who followed him came to Hawkins Hollow in sixteen fifty-one. He was a powerful force, and soon the settlement was under his rule. His rule decreed there would be no dancing, no singing, no music, no books but the Bible. No church but his church, no god but his god.\"\n\n\"So much for freedom of religion.\"\n\n\"Freedom was never Twisse's goal. In the way of those thirsty for power above all else, he intimidated, terrorized, punished, banished, and used as his visible weapon, the wrath of his chosen god. As Twisse's power grew, so did his punishments and penalties. Stocks, lashings, the shearing of a woman's hair if she was deemed ungodly, the branding of a man should he be accused of a crime. And finally, the burning of those he judged to be witches. On the night of July the seventh, sixteen fifty-two, on the accusation of a young woman, Hester Deale, Twisse led a mob from the settlement to the Pagan Stone, and to Giles Dent. What happened there...\"\n\nQuinn leaned forward. But Estelle sighed and shook her head. \"Well, there are many accounts. As there were many deaths. Seeds planted long before stirred in the ground. Some may have sprouted, only to die in the blaze that scorched the clearing.\n\n\"There are...fewer reports of what immediately followed, or followed over the next days and weeks. But in time, Ann Hawkins returned to the settlement with her three sons. And Hester Deale gave birth to a daughter eight months after the killing blaze at the Pagan Stone. Shortly, very shortly after her child, whom she claimed was sired by the devil, was born, Hester drowned herself in a small pond in Hawkins Wood.\"\n\nLoading her pockets with stones, Quinn thought with a suppressed shudder. \"Do you know what happened to her child? Or the children of Ann Hawkins?\"\n\n\"There are some letters, some journals, family Bibles. But most concrete information has been lost, or has never come to light. It will take considerable time and effort to dig out the truth. I can tell you this, those seeds stayed dormant until a night twenty-one years ago this July. They were awakened, and what sowed them awakened. They bloom for seven nights every seven years, and they strangle Hawkins Hollow. I'm sorry, I tire so quickly these days. It's irritating.\"\n\n\"Can I get you something? Or drive you home?\"\n\n\"You're a good girl. My grandson will be coming along to pick me up. You'll have spoken, I imagine, to his son by now. To Caleb.\"\n\nSomething in the smile turned a switch in Quinn's brain. \"Caleb would be your\u2014\"\n\n\"Great-grandson. Honorary, you could say. My brother Franklin and his wife, my dearest friend, Maybelle, were killed in an accident just before Jim\u2014Caleb's father was born. My Johnnie and I stood as grandparents to my brother's grandchildren. I'd have counted them and theirs in that long list of progeny before.\"\n\n\"You're a Hawkins by birth then.\"\n\n\"I am, and our line goes back, in the Hollow, to Richard Hawkins, the founder\u2014and through him to Ann.\" She paused a moment as if to let Quinn absorb, analyze. \"He's a good boy, my Caleb, and he carries more than his share of weight on his shoulders.\"\n\n\"From what I've seen, he carries it well.\"\n\n\"He's a good boy,\" Estelle repeated, then rose. \"We'll talk again, soon.\"\n\n\"I'll walk you downstairs.\"\n\n\"Don't trouble. They'll have tea and cookies for me in the staff lounge. I'm a pet here\u2014in the nicest sense of the word. Tell Caleb we spoke, and that I'd like to speak with you again. Don't spend all this pretty day inside a book. As much as I love them, there's life to be lived.\"\n\n\"Mrs. Abbott?\"\n\n\"Yes?\"\n\n\"Who do you think planted the seeds at the Pagan Stone?\"\n\n\"Gods and demons.\" Estelle's eyes were tired, but clear. \"Gods and demons, and there's such a thin line between the two, isn't there?\"\n\nAlone, Quinn sat again. Gods and demons. Those were a big, giant step up from ghosts and spirits, and other bump-in-the-night residents. But didn't it fit, didn't it click right together with the words she remembered from her dreams?\n\nWords she'd looked up that morning.\n\nBestia, Latin for beast.\n\nBeatus, Latin for blessed.\n\nDevoveo, Latin for sacrifice.\n\nOkay, okay, she thought, if we're heading down that track, it might be a good time to call in the reserves.\n\nShe pulled out her phone. When she was greeted by voice mail, Quinn pushed down impatience and waited for her cue to leave a message.\n\n\"Cyb, it's Q. I'm in Hawkins Hollow, Maryland. And, wow, I've hooked a big one. Can you come? Let me know if you can come. Let me know if you can't come so I can talk you into it.\"\n\nShe closed the phone, and for the moment she ignored the stack of books she'd selected. Instead, she began to busily type up notes from Estelle Hawkins Abbott's recitation.\n\n## Seven\n\nCAL DID WHAT HE THOUGHT OF AS THE PASS-OFF to his father. Since the meetings and the morning and afternoon league games were over and there was no party or event scheduled, the lanes were empty but for a couple of old-timers having a practice game on lane one.\n\nThe arcade was buzzing, as it tended to between the last school bell and the dinner hour. But Cy Hudson was running herd there, and Holly Lappins manned the front desk. Jake and Sara worked the grill and fountain, which would start hopping in another hour.\n\nEverything, everyone was in its place, so Cal could sit with his father at the end of the counter over a cup of coffee before he headed for home, and his dad took over the center for the night.\n\nThey could sit quietly for a while, too. Quiet was his father's way. Not that Jim Hawkins didn't like to socialize. He seemed to like crowds as much as his alone time, remembered names, faces, and could and would converse on any subject, including politics and religion. The fact that he could do so without pissing anyone off was, in Cal's opinion, one of his finest skills.\n\nHis sandy-colored hair had gone a pure and bright silver over the last few years, and was trimmed every two weeks at the local barbershop. He rarely altered his uniform of khakis, Rockports, and oxford shirts on workdays.\n\nSome would have called Jim Hawkins habitual, even boring. Cal called him reliable.\n\n\"Having a good month so far,\" Jim said in his take-your-time drawl. He took his coffee sweet and light, and by his wife's decree, cut off the caffeine at six p.m. sharp. \"Kind of weather we've been having, you never know if people are going to burrow in, or get cabin fever so bad they want to be anywhere but home.\"\n\n\"It was a good idea, running the three-game special for February.\"\n\n\"I get one now and again.\" Jim smiled, lines fanning out and deepening around his eyes. \"So do you. Your mom's wishing you'd come by, have dinner some night soon.\"\n\n\"Sure. I'll give her a call.\"\n\n\"Heard from Jen yesterday.\"\n\n\"How's she doing?\"\n\n\"Fine enough to flaunt that it was seventy-four in San Diego. Rosie's learning to write her letters, and the baby's getting another tooth. Jen said she'd send us pictures.\"\n\nCal heard the wistfulness. \"You and Mom should take another trip out there.\"\n\n\"Maybe, maybe in a month or two. We're heading to Baltimore on Sunday to see Marly and her brood. I saw your great-gran today. She told me she had a nice chat with that writer who's in town.\"\n\n\"Gran talked with Quinn?\"\n\n\"In the library. She liked the girl. Likes the idea of this book, too.\"\n\n\"And how about you?\"\n\nJim shook his head, contemplated as Sara drew off Cokes for a couple of teenagers taking a break from the arcade. \"I don't know what I think, Cal, that's the plain truth. I ask myself what good's it going to do to have somebody\u2014and an outsider at that\u2014write all this down so people can read about it. I tell myself that what happened before won't happen again\u2014\"\n\n\"Dad.\"\n\n\"I know that's not true, or most likely not true.\"\n\nFor a moment Jim just listened to the voices from the boys at the other end of the counter, the way they joked and poked at each other. He knew those boys, he thought. He knew their parents. If life worked as it ought to work, he'd know their wives and kids one day.\n\nHadn't he joked and poked at his own friends here once upon a time, over fountain Cokes and fries? Hadn't his own children run tame through this place? Now his girls were married and gone, with families of their own. And his boy was a man, sitting with worry in his eyes over problems too big to be understood.\n\n\"You have to prepare for it to happen again,\" Jim continued. \"But for most of us, it all hazes up, it just hazes up so you can barely remember what did happen. Not you, I know. It's clear for you, and I wish that wasn't so. I guess if you believe this writer can help find the answers, I'm behind you on that.\"\n\n\"I don't know what I believe. I haven't worked it out yet.\"\n\n\"You will. Well. I'm going to go check on Cy. Some of the evening rollers'll be coming in before long, wanting a bite before they suit up.\"\n\nHe pushed away from the counter, took a long look around. He heard the echoes of his boyhood, and the shouts of his children. He saw his son, gangly with youth, sitting at the counter with the two boys Jim knew were the same as brothers to him.\n\n\"We've got a good place here, Cal. It's worth working for. Worth fighting to hold it steady.\"\n\nJim gave Cal a pat on the shoulder, then strolled away.\n\nNot just the center, Cal thought. His father had meant the town. And Cal was afraid that holding it steady this time was going to be one hell of a battle.\n\nHe went straight home where most of the snow had melted off the shrubs and stones. Part of him had wanted to hunt Quinn down, pump out of her what she and his great-grandmother had talked about. Better to wait, he thought as he jingled his keys, better to wait then ease it out of her the next day. When they went to the Pagan Stone.\n\nHe glanced toward the woods where trees and shadows held pockets and rivers of snow, where he knew the path would be muddy from the melt.\n\nWas it in there now, gathering itself? Had it somehow found a way to strike outside the Seven? Maybe, maybe, but not tonight. He didn't feel it tonight. And he always did.\n\nStill, he couldn't deny he felt less exposed when he was inside the house, after he'd put on lights to push away the gloom.\n\nHe went through to the back door, opened it, and gave a whistle.\n\nLump took his time as Lump was wont to do. But the dog eased his way out of the doghouse and even stirred up the energy for a couple of tail wags before he moseyed across the backyard to the bottom of the deck stairs.\n\nHe gave a doggie sigh before clumping up the short flight. Then he leaned his whole body against Cal.\n\nAnd that, Cal thought, was love. That was welcome home, how ya doing, in Lump's world.\n\nHe crouched down to stroke and ruffle the fur, to scratch between the floppy ears while Lump gazed at him soulfully. \"How's it going? Get all your work done? What do you say we have a beer?\"\n\nThey went inside together. Cal filled the dog bowl from the bin of chow while Lump sat politely, though Cal assumed a large portion of his dog's manners was sheer laziness. When the bowl was set in front of him, Lump ate slowly, and with absolute focus on the task at hand.\n\nCal pulled a beer out of the fridge and popped the top. Leaning back on the counter he took that first long swallow that signaled the end of the workday.\n\n\"Got some serious shit on my mind, Lump. Don't know what to do about it, think about it. Should I have found a way to stop Quinn from coming here? Not sure that would've worked since she seems to go where the hell she wants, but I could've played it different. Laughed it off, or pushed it higher, so the whole thing came off as bogus. Played it straight, so far, and I don't know where that's going to lead.\"\n\nHe heard the front door open, then Fox shouted, \"Yo!\" Fox came in carrying a bucket of chicken and a large white takeout bag. \"Got tub-o-cluck, got fries. Want beer.\"\n\nAfter dumping the food on the table, Fox pulled out a beer. \"Your summons was pretty abrupt, son. I might've had a hot date tonight.\"\n\n\"You haven't had a hot date in two months.\"\n\n\"I'm storing it up.\" After the first swig, Fox shrugged off his coat, tossed it over a chair. \"What's the deal?\"\n\n\"Tell you while we eat.\"\n\nAs he'd been too brainwashed by his mother to fall back on the single-man's friend of paper plates, Cal set out two of stoneware in dull blue. They sat down to fried chicken and potatoes with Lump\u2014as the only thing that lured the dog from food was more food\u2014caging fries by leaning against Cal's knee or Fox's.\n\nHe told Fox everything, from the wall of fire, through Quinn's dream, and up to the conversation she'd had with his great-grandmother.\n\n\"Seeing an awful lot of the fucker for February,\" Fox mused. \"That's never happened before. Did you dream last night?\"\n\n\"Yeah.\"\n\n\"Me, too. Mine was a replay of the first time, the first summer. Only we didn't get to the school in time, and it wasn't just Miss Lister inside. It was everybody.\" He scrubbed a hand over his face before taking a long pull of beer. \"Everybody in town, my family, yours, all inside. Trapped, beating on the windows, screaming, their faces at the windows while the place burned.\" He offered Lump another fry, and his eyes were as dark and soulful as the dog's. \"Didn't happen that way, thank Christ. But it felt like it did. You know how that goes.\"\n\n\"Yeah.\" Cal let out a breath. \"Yeah, I know how that goes. Mine was from that same summer, and we were all riding our bikes through town the way we did. Buildings were burned out, windows broken, cars wrecked and smoking. Bodies everywhere.\"\n\n\"It didn't happen that way,\" Fox repeated. \"We're not ten anymore, and we're not going to let it happen that way.\"\n\n\"I've been asking myself how long we can do this, Fox. How long can we hold it back as much as we do? This time, the next. Three more times? How many more times are we going to watch people we know, people we see most every day turn? Go crazy, go mean. Hurt each other, hurt themselves?\"\n\n\"As long as it takes.\"\n\nCal shoved his plate aside. \"Not good enough.\"\n\n\"It's all we've got, for now.\"\n\n\"It's like a virus, an infection, passing from one person to another. Where's the goddamn antidote?\"\n\n\"Not everyone's affected,\" Fox reminded him. \"There has to be a reason for that.\"\n\n\"We've never found it.\"\n\n\"No, so maybe you were right. Maybe we do need fresh eyes, an outsider, objectivity we just don't have. Are you still planning to take Quinn to the stone tomorrow?\"\n\n\"If I don't, she'll go anyway. So yeah, it's better I'm there.\"\n\n\"You want me? I can cancel some stuff.\"\n\n\"I can handle it.\" Had to handle it.\n\nQUINN STUDIED THE MENU IN THE HOTEL'S ALMOST empty dining room. She'd considered getting some takeout and eating in her room over her laptop, but she fell too easily into that habit, she knew. And to write about a town, she had to experience the town, and couldn't do that closed up in her pretty room eating a cold-cut sub.\n\nShe wanted a glass of wine, something chilly with a subtle zip. The hotel's cellar was more extensive than she'd expected, but she didn't want a whole bottle. She was frowning over the selections offered by the glass when Miss Fabulous Red Bag stepped in.\n\nShe'd changed into black pants, Quinn noted, and a cashmere sweater in two tissue-thin layers of deep blue under pale. The hair was great, she decided, pin straight with those jagged ends just past chin length. What Quinn knew would look messy on her came off fresh and stylish on the brunette.\n\nQuinn debated catching her eye, trying a wave. She could ask Red Purse to join her for dinner. After all, who didn't hate to eat alone? Then she could pump her dinner companion for the really important details. Like where she got that bag.\n\nEven as she charged up her smile, Quinn saw it.\n\nIt slithered across the glossy planks of the oak floor, leaving a hideous trail of bloody ooze behind it. At first she thought snake, then slug, then could barely think at all as she watched it slide up the legs of a table where an attractive young couple were enjoying cocktails by candlelight.\n\nIts body, thick as a truck tire, mottled red over black, wound its way over the table, leaving that ugly smear on the snowy linen while the couple laughed and flirted.\n\nA waitress walked briskly in, stepped in and through the sludge on the floor, to serve the couple their appetizers.\n\nQuinn swore she could hear the table creak under its weight.\n\nAnd its eyes when they met hers were the eyes of the boy, the red gleam in them bright and somehow amused. Then it began to wiggle wetly down the skirt of the tablecloth, and toward the brunette.\n\nThe woman stood frozen in place, her face bone white. Quinn pushed to her feet and, ignoring the surprised look from the waitress, leaped over the ugly path. She gripped the brunette's arm, pulled her out of the dining room.\n\n\"You saw it, too,\" Quinn said in a whisper. \"You saw that thing. Let's get out of here.\"\n\n\"What? What?\" The brunette cast shocked glances over her shoulder as she and Quinn stumbled for the door. \"You saw it?\"\n\n\"Sluggy, red-eyed, very nasty wake. Jesus. Jesus.\" She gulped in the raw February air on the hotel's porch. \"They didn't see it, but you did. I did. Why is that? Fuck if I know, but I have an idea who might. That's my car right there. Let's go. Let's just go.\"\n\nThe brunette didn't say another word until they were in the car and Quinn was squealing away from the curb. \"Who the hell are you?\"\n\n\"Quinn. Quinn Black. I'm a writer, mostly on the spooky. Of which there is a surplus in this town. Who are you?\"\n\n\"Layla Darnell. What is this place?\"\n\n\"That's what I want to find out. I don't know if it's nice to meet you or not, Layla, under the circumstances.\"\n\n\"Same here. Where are we going?\"\n\n\"To the source, or one of them.\" Quinn glanced over, saw Layla was still pale, still shaky. Who could blame her? \"What are you doing in Hawkins Hollow?\"\n\n\"I'm damned if I know, but I think I've decided to cut my visit short.\"\n\n\"Understandable. Nice bag, by the way.\"\n\nLayla worked up a wan smile. \"Thanks.\"\n\n\"Nearly there. Okay, you don't know why you're here, so where did you come from?\"\n\n\"New York.\"\n\n\"I knew it. It's the polish. Do you love it?\"\n\n\"Ah.\" Layla combed her fingers through her hair as she swiveled to look back. \"Most of the time. I manage a boutique in SoHo. Did. Do. I don't know that anymore either.\"\n\nNearly there, Quinn thought again. Let's keep calm. \"I bet you get great discounts.\"\n\n\"Yeah, part of the perks. Have you seen anything like that before. Like that thing?\"\n\n\"Yeah. Have you?\"\n\n\"Not when I was awake. I'm not crazy,\" Layla stated. \"Or I am, and so are you.\"\n\n\"We're not crazy, which is what crazy people tend to say, so you'll just have to take my word.\" She swung onto Cal's lane, and aimed the car over the little bridge toward the house where lights\u2014thank God\u2014glowed in the windows.\n\n\"Whose house is this?\" Layla gripped the front edge of her seat. \"Who lives here?\"\n\n\"Caleb Hawkins. His ancestors founded the town. He's okay. He knows about what we saw.\"\n\n\"How?\"\n\n\"It's a long story, with a lot of holes in it. And now you're thinking, what am I doing in this car with a complete stranger who's telling me to go into this house pretty much in the middle of nowhere.\"\n\nLayla took firm hold on the short strap of her bag, as if she might use it as a weapon. \"The thought's crossed.\"\n\n\"Your instinct put you in the car with me, Layla. Maybe you could follow along with that for the next step. Plus, it's cold. We didn't bring our coats.\"\n\n\"All right. Yes, all right.\" With a bracing breath, Layla opened the door, and with Quinn walked toward the house. \"Nice place. If you like isolated houses in the woods.\"\n\n\"Culture shock for the New Yorker.\"\n\n\"I grew up in Altoona, Pennsylvania.\"\n\n\"No kidding. Philadelphia. We're practically neighbors.\" Quinn knocked briskly on the door, then just opened the door and called in, \"Cal!\"\n\nShe was halfway across the living room when he hurried in. \"Quinn? What?\" Spotted Layla. \"Hello. What?\"\n\n\"Who's here?\" Quinn demanded. \"I saw another car in the drive.\"\n\n\"Fox. What's going on?\"\n\n\"The bonus-round question.\" She sniffed. \"Do I smell fried chicken? Is there food? Layla\u2014this is Layla Darnell; Layla, Cal Hawkins\u2014Layla and I haven't had dinner.\"\n\nShe moved right by him, and walked toward the kitchen.\n\n\"I'm sorry, I think, to bust in on you,\" Layla began. It passed through her mind that he didn't look like a serial killer. But then again, how would she know? \"I don't know what's happening, or why I'm here. I've had a confusing few days.\"\n\n\"Okay. Well, come on back.\"\n\nQuinn already had a drumstick in her hand, and was taking a swig of Cal's beer. \"Layla Darnell, Fox O'Dell. I'm not really in the mood for beer,\" she said to Cal. \"I was about to order some wine when Layla and I were disgustingly interrupted. Got any?\"\n\n\"Yeah. Yeah.\"\n\n\"Is it decent? If you run to jug or twist caps, I'll stick with beer.\"\n\n\"I've got some damn decent wine.\" He yanked a plate out, pushed it at her. \"Use a plate.\"\n\n\"He's completely Sally about things like that,\" Fox told her. He'd risen, and pulled out a chair. \"You look a little shaken up\u2014Layla, right? Why don't you sit down?\"\n\nShe just couldn't believe psycho killers sat around a pretty kitchen eating bucket chicken and debating wine over beer. \"Why don't I? I'm probably not really here.\" She sat, dropped her head in her hands. \"I'm probably in some padded room imagining all this.\"\n\n\"Imagining all what?\" Fox asked.\n\n\"Why don't I take it?\" Quinn glanced at Layla as Cal got out wineglasses. \"Then you can fill in as much of your own backstory as you want.\"\n\n\"Fine. That's fine.\"\n\n\"Layla checked into the hotel this morning. She's from New York. Just a bit ago, I was in the hotel dining room, considering ordering the green salad and the haddock, along with a nice glass of white. Layla was just coming in, I assume, to have her own dinner. I was going to ask you to join me, by the way.\"\n\n\"Oh. Ah, that's nice.\"\n\n\"Before I could issue the invite, what I'd describe as a sluglike creature thicker than my aunt Christine's thigh and about four feet in length oozed its way across the dining room, up over the table where a couple happily continued their dining foreplay, then oozed down again, leaving a revolting smear of God-knows-what behind it. She saw it.\"\n\n\"It looked at me. It looked right at me,\" Layla whispered.\n\n\"Don't be stingy with the wine, Cal.\" Quinn stepped over to rub a hand on Layla's shoulder. \"We were the only ones who saw it, and no longer wishing to dine at the hotel, and believing Layla felt the same, we booked. And I'm now screwing my caloric intake for the day with this drumstick.\"\n\n\"You're awfully...blithe. Thanks.\" Layla accepted the wineglass Cal offered, then drank half the contents at one go.\n\n\"Not really. Defense mechanism. So here we are, and I want to know if either of you have ever seen anything like I just described.\"\n\nThere was a moment of silence, then Cal picked up his beer, drank. \"We've seen a lot of things. The bigger question for me is, why are you seeing them, and part two, why are you seeing them now?\"\n\n\"Got a theory.\"\n\nCal turned to Fox. \"Such as?\"\n\n\"Connections. You said yourself there had to be some connection for Quinn to see it, to have the dream\u2014\"\n\n\"Dreams.\" Layla's head came up. \"You've had dreams?\"\n\n\"And so, apparently, have you,\" Fox continued. \"So we'll connect Layla. Figuring out how they're connected may take a while, but let's just go with the hypothesis that they are, and say, what if. What if, due to this connection, due to Quinn, then Layla being in the Hollow, particularly during the seventh year, gives it some kind of psychic boost? Gives it the juice to manifest?\"\n\n\"That's not bad,\" Cal replied.\n\n\"I'd say it's damn good.\" Quinn cocked her head as she considered. \"Energy. Most paranormal activity stems from energy. The energy the...well, entity or entities, the actions, the emotions thereof, leave behind, and the energy of the people within its sphere, let's say. And we could speculate that this psychic energy has built over time, strengthened, so that now, with the addition of other connected energies, it's able to push out into our reality, to some extent, outside of its traditional time frame.\"\n\n\"What in God's name are you people talking about?\" Layla demanded.\n\n\"We'll get to that, I promise.\" Quinn offered her a bolstering smile. \"Why don't you eat something, settle the nerves?\"\n\n\"I think it's going to be a while before food holds any appeal for me.\"\n\n\"Mr. Slug slimed right over the bread bowl,\" Quinn explained. \"It was pretty damn gross. Sadly, nothing puts me off food.\" She snagged a couple of cold fries. \"So, if we run with Fox's theory, where is its counterpoint? The good to its bad, the white to its dark. All my research on this points to both sides.\"\n\n\"Maybe it can't pull out yet, or it's hanging back.\"\n\n\"Or the two of you connect to the dark, and not the light,\" Cal added.\n\nQuinn narrowed her eyes at him, with something glinting between her lashes. Then she shrugged. \"Insulting, but unarguable at this time. Except for the fact that, logically, if we were more a weight on the bad side, why is said bad side trying to scare the living daylights out of us?\"\n\n\"Good point,\" Cal conceded.\n\n\"I want some answers.\"\n\nQuinn nodded at Layla. \"I bet you do.\"\n\n\"I want some serious, sensible answers.\"\n\n\"Thumbnail: The town includes an area in the woods known as the Pagan Stone. Bad stuff happened there. Gods, demons, blood, death, fire. I'm going to lend you a couple of books on the subject. Centuries pass, then something opened it up again. Since nineteen eighty-seven, for seven nights in July, every seventh year, it comes out to play. It's mean, it's ugly, and it's powerful. We're getting a preview.\"\n\nGratefully, Layla held out her glass for more wine as she studied Quinn. \"Why haven't I ever heard of this? Or this place?\"\n\n\"There have been some books, some articles, some reports\u2014but most of them hit somewhere between alien abductions and sightings of Bigfoot,\" Quinn explained. \"There's never been a serious, thorough, fully researched account published. That's going to be my job.\"\n\n\"All right. Say I believe all this, and I'm not sure I'm not just having the mother of all hallucinations, why you, and you?\" she said to Fox and Cal. \"Where do you come in?\"\n\n\"Because we're the ones who opened it,\" Fox told her. \"Cal, me, and a friend who's currently absent. Twenty-one years ago this July.\"\n\n\"But you'd have been kids. You'd have had to have been\u2014\"\n\n\"Ten,\" Cal confirmed. \"We share a birthday. It was our tenth birthday. Now, we showed some of ours. How about seeing some of yours. Why did you come here?\"\n\n\"Fair enough.\" Layla took another slow sip of her wine. Whether it was that or the brightly lit kitchen with a dog snoring under the table or just having a group of strangers who were likely to believe what she was about to tell them, her nerves were steadier.\n\n\"I've been having dreams for the last several nights. Nightmares or night terrors. Sometimes I'd wake up in my bed, sometimes I'd wake up trying to get out the door of my apartment. You said blood and fire. There was both in the dreams, and a kind of altar in a clearing in the woods. I think it was stone. And there was water, too. Black water. I was drowning in it. I was captain of the swim team in high school, and I was drowning.\"\n\nShe shuddered, took another breath. \"I was afraid to sleep. I thought I heard voices even when I wasn't asleep. I couldn't understand them, but I'd be at work, doing my job, or stopping by the dry cleaners on the way home, and these voices would just fill my head. I thought I was having a breakdown. But why? Then I thought maybe I had a brain tumor. I even thought about making an appointment with a neurologist. Then last night, I took a sleeping pill. Maybe I could just drug my way out of it. But it came, and in the dream something was in bed with me.\"\n\nHer breath trembled out this time. \"Not my bed, but somewhere else. A small room, a small hot room with a tiny window. I was someone else. I can't explain it, really.\"\n\n\"You're doing fine,\" Quinn assured her.\n\n\"It was happening to me, but I wasn't me. I had long hair, and the shape of my body, it was different. I was wearing a long nightgown. I know because it...it pulled it up. It was touching me. It was cold, it was so cold. I couldn't scream, I couldn't fight, even when it raped me. It was inside me, but I couldn't see, I couldn't move. I felt it, all of it, as if it were happening, but I couldn't stop it.\"\n\nShe wasn't aware of the tears until Fox pressed a napkin into her hand. \"Thanks. When it was over, when it was gone, there was a voice in my head. Just one voice this time, and it calmed me, it made me warm again and took away the pain. It said: 'Hawkins Hollow.'\"\n\n\"Layla, were you raped?\" Fox spoke very quietly. \"When you came out of the dream, was there any sign you'd been raped?\"\n\n\"No.\" She pressed her lips together, kept her gaze on his face. His eyes were golden brown, and full of compassion. \"I woke up in my own bed, and I made myself go...check. There was nothing. It hurt me, so there would've been bruises, there would've been marks, but there was nothing. It was early in the morning, not quite four in the morning, and I kept thinking Hawkins Hollow. So I packed, and I took a cab out to the airport to rent a car. Then I drove here. I've never been here.\"\n\nShe paused to look at Quinn now, at Cal. \"I've never heard of Hawkins Hollow that I can remember, but I knew what roads to take. I knew how to get here, and how to get to the hotel. I checked in this morning, went up to the room they gave me, and I slept like the dead until nearly six. When I walked into the dining room and saw that thing, I thought I was still asleep. Dreaming again.\"\n\n\"It's a wonder you didn't bolt,\" Quinn commented.\n\nLayla sent her an exhausted look. \"To where?\"\n\n\"There's that.\" Quinn put a hand on Layla's shoulder, rubbing gently as she spoke. \"I think we all need as much information as there is to be had, from every source there is. I think, from this point, it's share and share alike, one for goddamn all and all for goddamn one. You don't like that,\" she said with a nod toward Cal, \"but I think you're going to have to get used to it.\"\n\n\"You've been in this for days. Fox and I have lived with it for years. Lived in it. So, don't put on your badge and call yourself captain yet, Blondie.\"\n\n\"Living in it for twenty-one years gives you certain advantages. But you haven't figured it out, you haven't stopped it or even identified it, as far as I can tell, in your twenty-one-year experience. So loosen up.\"\n\n\"You poked at my ninety-seven-year-old great-grandmother today.\"\n\n\"Oh, bull. Your remarkable and fascinating ninety-seven-year-old great-grandmother came up to where I was researching in the library, sat down, and had a conversation with me of her own free will. There was no poking. My keen observation skills tell me you didn't inherit your tight-ass tendencies from her.\"\n\n\"Kids, kids.\" Fox held up a hand. \"Tense situation, agreed, but we're all on the same side, or are on the same side potentially. So chill. Cal, Quinn makes a good point, and it bears consideration. At the same time, Quinn, you've been in the Hollow a couple of days, and Layla less than that. You're going to have to be patient, and accept the fact that some areas of information are more sensitive than others, and may take time to be offered. Even if we start with what can and has been corroborated or documented\u2014\"\n\n\"What are you, a lawyer?\" Layla asked.\n\n\"Yeah.\"\n\n\"Figures,\" she said under her breath.\n\n\"Let's just table this,\" Cal suggested. \"Let's let it sit, so we can all think about it for the night. I said I'd take you to the Pagan Stone tomorrow, and I will. Let's see how it goes.\"\n\n\"Accepted.\"\n\n\"Are you two all right at the hotel? You can stay here if you're not easy about going back.\"\n\nThe fact that he'd offered had Quinn's hackles smoothing down again. \"We're not wimps, are we, Layla?\"\n\n\"I wouldn't have said I was a few days ago. Now, I'm not so sure. But I'll be all right at the hotel.\" In fact, she wanted to go back, crawl into that big, soft bed and pull the covers over her head. \"I slept better there than I have all week, so that's something.\"\n\nQuinn decided she'd wait until they were back before she advised Layla to lower all the shades, and maybe leave a light burning.\n\n## Eight\n\nIN THE MORNING, QUINN PRESSED AN EAR AGAINST the door to Layla's room. Since she heard the muted sounds of the Today show, she gave the door a knuckle rap. \"It's Quinn,\" she added, in case Layla was still jumpy.\n\nLayla opened the door in a pretty damn cute pair of purple-and-white-striped pajama pants and a purple sleep tank. There was color in her cheeks, and her quiet green eyes had the clarity that told Quinn she'd been awake awhile.\n\n\"I'm about to head out to Cal's. Mind if I come in a minute?\"\n\n\"No.\" She stepped back. \"I was trying to figure out what I'm supposed to do with myself today.\"\n\n\"You can come with me if you want.\"\n\n\"Into the woods? Not quite ready for that, thanks. You know...\" Layla switched off the TV before dropping into a chair. \"I was thinking about the wimp statement you made last night. I've never been a wimp, but it occurred to me as I was huddled in bed with the shades drawn and this stupid chair under the doorknob that I've never had anything happen that tested that before. My life's been pretty normal.\"\n\n\"You came here, and you're still here. So I'm thinking that puts you pretty low on the wimp scale. How'd you sleep?\"\n\n\"Good. Once I got there, good. No dreams, no visitations, no bumps in the night. So, of course, now I'm wondering why.\"\n\n\"No dreams for me either.\" Quinn glanced around the room. Layla's bed was a sleigh style and the color scheme was muted greens and creams. \"We could theorize that your room here's a safe zone, but that's off because mine isn't, and it's two doors down. It could be that whatever it is just took the night off. Maybe needed to recharge some expended energy.\"\n\n\"Happy thought.\"\n\n\"You've got my cell number, Cal's, Fox's. We've got yours. We're\u2014connected. I wanted to let you know that the diner across the street, figuring you're not going to try the dining room here again, has a nice breakfast.\"\n\n\"I'm thinking I might try room service, and start on the books that you gave me last night. I didn't want to try them for bedtime reading.\"\n\n\"Wise. Okay. If you head out, it's a nice town. Some cute little shops, a little museum I haven't had time to explore so can't give you a rating, and there's always the Bowl-a-Rama.\"\n\nA hint of a smile appeared around Layla's mouth. \"Is there?\"\n\n\"It's Cal's family's place. Interesting, and it feels like the hub of the town. So, I'll look you up when I get back?\"\n\n\"Okay. Quinn?\" Layla added as Quinn reached for the door. \"Wimp scale or not, I'm not sure I'd still be here if I hadn't run into you.\"\n\n\"I know how you feel. I'll see you later.\"\n\nCAL WAS WAITING FOR HER WHEN SHE DROVE UP. He stepped out, started down the steps, the dog wandering behind him, as she got out of the car. He took a scan, starting with her feet. Good, sturdy hiking boots that showed some scars and wear, faded jeans, tough jacket in I'm-Not-a-Deer red, and a multistriped scarf that matched the cloche-style cap on her head. Silly hat, he mused, that was unaccountably appealing on her.\n\nIn any case, he decided she knew what to wear on a hike through the winter woods.\n\n\"Do I pass muster, Sergeant?\"\n\n\"Yeah.\" He came down the rest of the steps. \"Let's start this off with me saying I was off base by a couple inches last night. I haven't completely resolved dealing with you, and now there's another person in the mix, another unknown. When you live with this as long as I have, part of you gets used to it, and other parts just get edgier. Especially when you're into the seventh year. So, I'll apologize, if you need it.\"\n\n\"Well. Wind, sails sucked out. Okay, I can't be pissed off after that or it's just bitchy instead of righteous. So let me say this. Before I came here, this was an idea for a book, a job I enjoy on a level some might consider twisted, and that I consider vastly fascinating. Now, it's more personal. While I can appreciate you being somewhat edgy, and somewhat proprietary, I'm bringing something important to the table. Experience and objectivity. And guts. I've got some impressive guts.\"\n\n\"I've noticed.\"\n\n\"So, we're going to do this thing?\"\n\n\"Yeah, we're going to do it.\"\n\nShe gave the dog who came over to lean on her a rub. \"Is Lump seeing us off on our adventure?\"\n\n\"He's coming. He likes to walk in the woods when the mood strikes. And if he's had enough, he'll just lie down and sleep until he's in the mood to walk back home again.\"\n\n\"Strikes me as a sensible attitude.\" She picked up a small pack, hitched it on, then drew her tape recorder out of her pocket. It was attached to the pocket with a small clamp. \"I'm going to want to record observations, and whatever you tell me. Okay with that?\"\n\n\"Yeah.\" He'd given it a lot of thought overnight. \"I'm okay with that.\"\n\n\"Then I'm ready when you are, Tonto.\"\n\n\"Trail's going to be sloppy,\" he said as they started toward the woods. \"Given that, from this point it'll take about two hours\u2014a little more depending\u2014to reach the clearing.\"\n\n\"I'm in no hurry.\"\n\nCal glanced up at the sky. \"You will be if the weather turns, or anything holds us up after sundown.\"\n\nShe clicked on her recorder, and hoped she'd been generous enough with her cache of extra tapes and batteries. \"Why?\"\n\n\"Years back people hiked or hunted in this section of the woods routinely. Now they don't. People got lost, turned around, spooked. Some reported hearing what they thought were bear or wolves. We don't have wolves and it's rare for bear to come this far down the mountains. Kids, teens mostly, used to sneak in to swim in Hester's Pool in the summer, or to screw around. Now they don't. People used to say the pool was haunted, it was kind of a local legend. Now, people don't like to talk about it.\"\n\n\"Do you think it's haunted?\"\n\n\"I know there's something in it. I saw it myself. We'll talk about that once we get to the pool. No point in going into it now.\"\n\n\"All right. Is this the way the three of you came in on your birthday twenty-one years ago?\"\n\n\"We came in from the east.\" He gestured. \"Track closest to town. This way's shorter, but it would've been a longer ride around for us from town. There wasn't anything...off about it, until we got to the pool.\"\n\n\"Have the three of you been back together since that night?\"\n\n\"Yeah, we went back. More than once.\" He glanced toward her. \"I can tell you that going back anytime near the Seven isn't an experience I look forward to repeating.\"\n\n\"The Seven?\"\n\n\"That's what we call the week in July.\"\n\n\"Tell me more about what happens during the Seven.\"\n\nIt was time to do just that, he thought. To say it straight-out to someone who wanted to know. To someone, maybe, who was part of the answer.\n\n\"People in the Hollow get mean, violent, even murderous. They do things they'd never do at any other time. Destroy property, beat the hell out of each other, start fires. Worse.\"\n\n\"Murders, suicides.\"\n\n\"Yeah. After the week's up, they don't remember clearly. It's like watching someone come out of a trance, or a long illness. Some of them are never the same. Some of them leave town. And some fix up their shop or their house, and just go on. It doesn't hit everyone, and it doesn't hit those it does all in the same way. The best I can explain is it's like a mass psychotic episode, and it gets stronger each time.\"\n\n\"What about the police?\"\n\nOut of habit, Cal reached down, picked up a stick. There was no point in tossing it for Lump, that would only embarrass them both. So he held it down so Lump could take it into his mouth and plod contentedly along.\n\n\"Chief Larson was in charge last time. He was a good man, went to school with my father. They were friends. The third night, he locked himself in his office. I think he, some part of him anyway, knew what was happening to him, and didn't want to risk going home to his wife and kids. One of the deputies, guy named Wayne Hawbaker, nephew to Fox's secretary, came in looking for him, needed help. He heard Larson crying in the office. Couldn't get him to come out. By the time Wayne knocked down the door, Larson had shot himself. Wayne's chief of police now. He's a good man, too.\"\n\nHow much loss had he seen? Quinn wondered. How many losses had he suffered since his tenth birthday? And yet he was walking back into these woods, back where it all began for him. She didn't think she'd ever known a braver stand.\n\n\"What about the county cops, the state cops?\"\n\n\"It's like we're cut off for that week.\" A cardinal winged by, boldly red, carelessly free. \"Sometimes people get out, sometimes they get in, but by and large, we're on our own. It's like...\" He groped for words. \"It's like this veil comes down, and nobody sees, not clearly. Help doesn't come, and after, nobody questions it too closely. Nobody looks straight on at what happened, or why. So it ends up being lore, or Blair Witch stuff. Then it fades off until it happens again.\"\n\n\"You stay, and you look at it straight on.\"\n\n\"It's my town,\" he said simply.\n\nNo, Quinn thought, that was the bravest stand she'd ever known.\n\n\"How'd you sleep last night?\" he asked her.\n\n\"Dreamlessly. So did Layla. You?\"\n\n\"The same. Always before, once it started, it didn't stop. But then, things are different this time around.\"\n\n\"Because I saw something, and so did Layla.\"\n\n\"That's the big one. And it's never started this early, or this strong.\" As they walked, he studied her face. \"Have you ever had a genealogy done?\"\n\n\"No. You think we're related back when, or I'm related to someone who was involved in whatever happened at the Pagan Stone way back when?\"\n\n\"I think, we've always thought, this was about blood.\" Absently, he glanced at the scar on his wrist. \"So far, knowing or sensing that hasn't done any good. Where are your ancestors from?\"\n\n\"England primarily, some Irish tossed in.\"\n\n\"Mine, too. But then a lot of Americans have English ancestry.\"\n\n\"Maybe I should start researching and find out if there are any Dents or Twisses in my lineage?\" She shrugged when he frowned at her. \"Your great-grandmother sent me down that path. Have you tried to trace them? Giles Dent and Lazarus Twisse?\"\n\n\"Yeah. Dent may be an ancestor, if he did indeed father the three sons of Ann Hawkins. There's no record of him. And other than accounts from the time, some old family letters and diaries, no Giles Dent on anything we've dug up. No record of birth, death. Same for Twisse. They could've dropped down from Pluto as far as we've been able to prove.\"\n\n\"I have a friend who's a whiz on research. I sent her a heads-up. And don't get that look on your face again. I've known her for years, and we've worked together on other projects. I don't know as yet if she can or will come in on this, but trust me, if she does, you'll be grateful. She's brilliant.\"\n\nRather than respond, he chewed on it. How much of his resistance was due to this feeling of losing control over the situation? And had he ever had any control to begin with? Some, he knew, was due to the fact that the more people who became involved, the more people he felt responsible for.\n\nAnd maybe most of all, how much was all this exposure going to affect the town?\n\n\"The Hollow's gotten some publicity over the years, focused on this whole thing. That's how you found out about us to begin with. But it's been mild, and for the most part, hasn't done much more than bring interested tourists through. With your involvement, and now potentially two others, it could turn the Hollow into some sort of lurid or ridiculous caption in the tourist guides.\"\n\n\"You knew that was a risk when you agreed to talk to me.\"\n\nShe was keeping pace with him, stride-by-stride on the sloppy ground. And, she was striding into the unknown without a quake or a quiver. \"You'd have come whether or not I agreed.\"\n\n\"So part of your cooperation is damage control.\" She nodded. \"Can't blame you. But maybe you should be thinking bigger picture, Cal. More people invested means more brains and more chance of figuring out how to stop what's been happening. Do you want to stop it?\"\n\n\"More than I can possibly tell you.\"\n\n\"I want a story. There's no point in bullshitting you about that. But I want to stop it, too. Because despite my famous guts, this thing scares me. Better shot at that, it seems to me, if we work together and utilize all our resources. Cybil's one of mine, and she's a damn good one.\"\n\n\"I'll think about it.\" For now, he thought, he'd given her enough. \"Why don't you tell me what made you head down the woo-woo trail, writing-wise.\"\n\n\"That's easy. I always liked spooky stuff. When I was a kid and had a choice between, say, Sweet Valley High or Stephen King, King was always going to win. I used to write my own horror stories and give my friends nightmares. Good times,\" she said and made him laugh. \"Then, the turning point, I suppose, was when I went into this reputed haunted house with a group of friends. Halloween. I was twelve. Big dare. Place was falling down and due to be demolished. We were probably lucky we didn't fall through floorboards. So we poked around, squealed, scared ourselves, and had some laughs. Then I saw her.\"\n\n\"Who?\"\n\n\"The ghost, of course.\" She gave him a friendly elbow poke. \"Keep up. None of the others did. But I saw her, walking down the stairs. There was blood all over her. She looked at me,\" Quinn said quietly now. \"It seemed like she looked right at me, and walked right by. I felt the cold she carried with her.\"\n\n\"What did you do? And if I get a guess, I'm guessing you followed her.\"\n\n\"Of course, I followed her. My friends were running around, making spooky noises, but I followed her into the falling-down kitchen, down the broken steps to the basement by the beam of my Princess Leia flashlight. No cracks.\"\n\n\"How can I crack when I had a Luke Skywalker flashlight?\"\n\n\"Good. What I found were a lot of spiderwebs, mouse droppings, dead bugs, and a filthy floor of concrete. Then the concrete was gone and it was just a dirt floor with a hole\u2014a grave\u2014dug in it. A black-handled shovel beside it. She went to it, looked at me again, then slid down, hell, like a woman might slide into a nice bubble bath. Then I was standing on the concrete floor again.\"\n\n\"What did you do?\"\n\n\"Your guess?\"\n\n\"I'd guess you and Leia got the hell out of there.\"\n\n\"Right again. I came out of the basement like a rocket. I told my friends, who didn't believe me. Just trying to spook them out as usual. I didn't tell anyone else, because if I had, our parents would have known we were in the house and we'd have been grounded till our Social Security kicked in. But when they demolished the house, started jackhammering the concrete floor, they found her. She'd been in there since the thirties. The wife of the guy who'd owned the house had claimed she'd run off. He was dead by then, so nobody could ask him how or why he'd done it. But I knew. From the time I saw her until they found her bones, I dreamed about her murder, I saw it happen.\n\n\"I didn't tell anyone. I was too afraid. Ever since, I've told what I find, confirming or debunking. Maybe partly to make it up to Mary Bines\u2014that was her name. And partly because I'm not twelve anymore, and nobody's going to ground me.\"\n\nHe said nothing for a long time. \"Do you always see what happened?\"\n\n\"I don't know if it's seeing or just intuiting, or just my imagination, which is even more far-famed than my guts. But I've learned to trust what I feel, and go with it.\"\n\nHe stopped, gestured. \"This is where the tracks cross. We came in from that direction, picked up the cross trail here. We were loaded down. My mother had packed a picnic basket, thinking we were camping out on Fox's family farm. We had his boom box, his load from the market, our backpacks full of the stuff we figured we couldn't live without. We were still nine years old. Kids, pretty much fearless. That all changed before we came out of the woods again.\"\n\nWhen he started to walk once more, she put a hand on his arm, squeezed. \"Is that tree bleeding, or do you just have really strange sap in this part of the world?\"\n\nHe turned, looked. Blood seeped from the bark of the old oak, and seeped into the soggy ground at its trunk.\n\n\"That kind of thing happens now and again. It puts off the hikers.\"\n\n\"I bet.\" She watched Lump plod by the tree after only a cursory sniff. \"Why doesn't he care?\"\n\n\"Old hat to him.\"\n\nShe started to give the tree a wide berth, then stopped. \"Wait, wait. This is the spot. This is the spot where I saw the deer across the path. I'm sure of it.\"\n\n\"He called it, with magick. The innocent and pure.\"\n\nShe started to speak, then looking at Cal's face, held her tongue. His eyes had darkened; his cheeks had paled.\n\n\"Its blood for the binding. Its blood, his blood, the blood of the dark thing. He grieved when he drew the blade across its neck, and its life poured onto his hands and into the cup.\"\n\nAs his head swam, Cal bent over from the waist. Prayed he wouldn't be sick. \"Need a second to get my breath.\"\n\n\"Take it easy.\" Quickly, Quinn pulled off her pack and pulled out her water bottle. \"Drink a little.\"\n\nMost of the queasiness passed when she took his hand, pressed the bottle into it. \"I could see it, feel it. I've gone by this tree before, even when it's bled, and I never saw that. Or felt that.\"\n\n\"Two of us this time. Maybe that's what opened it up.\"\n\nHe drank slowly. Not just two, he thought. He'd walked this path with Fox and Gage. We two, he decided. Something about being here with her. \"The deer was a sacrifice.\"\n\n\"I get that. Devoveo. He said it in Latin. Blood sacrifice. White witchery doesn't ascribe to that. He had to cross over the line, smear on some of the black to do what he felt he needed to do. Was it Dent? Or someone who came long before him?\"\n\n\"I don't know.\"\n\nBecause she could see his color was eking back, her own heart rate settled. \"Do you see what came before?\"\n\n\"Bits, pieces, flashes. Not all of it. I generally come back a little sick. If I push for more, it's a hell of a lot worse.\"\n\n\"Let's not push then. Are you okay to go on?\"\n\n\"Yeah. Yeah.\" His stomach was still mildly uneasy, but the light-headedness had passed. \"We'll be coming to Hester's Pool soon.\"\n\n\"I know. I'm going to tell you what it looks like before we get there. I'm telling you I've never been there before, not in reality, but I've seen it, and I stood there night before last. There are cattails and wild grass. It's off the path, through some brush and thorny stuff. It was night, so the water looked black. Opaque. Its shape isn't quite round, not really oval. It's more of a fat crescent. There were a lot of rocks. Some more like boulders, some no more than pebbles. She filled her pockets with them\u2014they looked to be about hand-sized or smaller\u2014until her pockets were sagging with the weight. Her hair was cut short, like it'd been hacked at, and her eyes looked mad.\"\n\n\"Her body didn't stay down, not according to reports.\"\n\n\"I've read them,\" Quinn acknowledged. \"She was found floating in the pool, which came to bear her name, and because it was suicide, they buried her in unconsecrated ground. Records I've dug up so far don't indicate what happened to the infant daughter she left behind.\"\n\nBefore replacing the pack, she took out a bag of trail mix. Opened it, offered. Cal shook his head. \"There's plenty of bark and twigs around if I get that desperate.\"\n\n\"This isn't bad. What did your mother pack for you that day?\"\n\n\"Ham-and-cheese sandwiches, hard-boiled eggs, apple slices, celery and carrot sticks, oatmeal cookies, lemonade.\" Remembering made him smile. \"Pop-Tarts, snack pack cereal for breakfast.\"\n\n\"Uppercase M Mom.\"\n\n\"Yeah, always has been.\"\n\n\"How long do we date before I meet the parents?\"\n\nHe considered. \"They want me to come for dinner some night soon if you want in.\"\n\n\"A home-cooked meal by Mom? I'm there. How does she feel about all this?\"\n\n\"It's hard for them, all of this is hard. And they've never let me down in my life.\"\n\n\"You're a lucky man, Cal.\"\n\nHe broke trail, skirting the tangles of blackberry bushes, and following the more narrow and less-trod path. Lump moved on ahead, as if he understood where they were headed. The first glint of the pool brought a chill down his spine. But then, it always did.\n\nBirds still called, and Lump\u2014more by accident than design, flushed a rabbit that ran across the path and into another thicket. Sunlight streamed through the empty branches onto the leaf-carpeted ground. And glinted dully on the brown water of Hester's Pool.\n\n\"It looks different during the day,\" Quinn noted. \"Not nearly as ominous. But I'd have to be very young and very hot to want to go splashing around in that.\"\n\n\"We were both. Fox went in first. We'd snuck out here before to swim, but I'd never much liked it. Who knew what was swimming under there? I always thought Hester's bony hand was going to grab my ankle and pull me under. Then it did.\"\n\nQuinn's eyebrows shot up, and when he didn't continue, she sat on one of the rocks. \"I'm listening.\"\n\n\"Fox was messing with me. I was a better swimmer, but he was sneaky. Gage couldn't swim for crap, but he was game. I thought it was Fox again, dunking me, but it was her. I saw her when I went under. Her hair wasn't short the way you saw her. I remember how her hair streamed out. She didn't look like a ghost. She looked like a woman. Girl,\" he corrected. \"I realized when I got older she was just a girl. I couldn't get out fast enough, and I made Fox and Gage get out. They hadn't seen anything.\"\n\n\"But they believed you.\"\n\n\"That's what friends do.\"\n\n\"Did you ever go back in?\"\n\n\"Twice. But I never saw her again.\"\n\nQuinn gave Lump, who wasn't as particular as his master, a handful of trail mix. \"It's too damn cold to try now, but come June, I'd like to take a dip and see what happens.\" She munched some mix as she looked around. \"It's a nice spot, considering. Primitive, but still picturesque. Seems like a great place for three boys to run a little wild.\"\n\nShe cocked her head. \"So do you usually bring your women here on dates?\"\n\n\"You'd be the first.\"\n\n\"Really? Is that because they haven't been interested, or you haven't wanted to answer questions pertaining.\"\n\n\"Both.\"\n\n\"So I'm breaking molds here, which is one of my favorite hobbies.\" Quinn stared out over the water. \"She must've been so sad, so horribly sad to believe there was no other way for her. Crazy's a factor, too, but I think she must've been weighed down by sadness and despair before she weighed herself down with rocks. That's what I felt in the dream, and it's what I feel now, sitting here. Her horrible, heavy sadness. Even more than the fear when it raped her.\"\n\nShe shuddered, rose. \"Can we move on? It's too much, sitting here. It's too much.\"\n\nIt would be worse, he thought. If she felt already, sensed or understood this already, it would be worse. He took her hand to lead her back to the path. Since, at least for the moment, it was wide enough to walk abreast, he kept ahold of her hand. It almost seemed as if they were taking a simple walk in the winter woods.\n\n\"Tell me something surprising about you. Something I'd never guess.\"\n\nHe cocked his head. \"Why would I tell you something about me you'd never guess?\"\n\n\"It doesn't have to be some dark secret.\" She bumped her hip against his. \"Just something unexpected.\"\n\n\"I lettered in track and field.\"\n\nQuinn shook her head. \"Impressive, but not surprising. I might've guessed that. You've got a yard or so of leg.\"\n\n\"All right, all right.\" He thought it over. \"I grew a pumpkin that broke the county record for weight.\"\n\n\"The fattest pumpkin in the history of the county?\"\n\n\"It missed the state record by ounces. It got written up in the paper.\"\n\n\"Well, that is surprising. I was hoping for something a bit more salacious, but am forced to admit, I'd never have guessed you held the county record for fattest pumpkin.\"\n\n\"How about you?\"\n\n\"I'm afraid I've never grown a pumpkin of any size or weight.\"\n\n\"Surprise me.\"\n\n\"I can walk on my hands. I'd demonstrate, but the ground's not conducive to hand-walking. Come on. You wouldn't have guessed that.\"\n\n\"You're right. I will, however, insist on a demo later. I, after all, have documentation of the pumpkin.\"\n\n\"Fair enough.\"\n\nShe kept up the chatter, light and silly enough to make him laugh. He wasn't sure he'd laughed along this path since that fateful hike with his friends. But it seemed natural enough now, with the sun beaming down through the trees, the birds singing.\n\nUntil he heard the growl.\n\nShe'd heard it, too. He couldn't think of another reason her voice would have stopped so short, or her hand would have gripped his arm like a vise. \"Cal\u2014\"\n\n\"Yeah, I hear it. We're nearly there. Sometimes it makes noise, sometimes it makes an appearance.\" Never this time of year, he thought, as he hitched up the back of his jacket. But these, apparently, were different times. \"Just stay close.\"\n\n\"Believe me, I...\" Her voice trailed off this time as he drew the large, jagged-edged hunting knife. \"Okay. Okay. Now that would have been one of those unexpected things about you. That you, ah, carry a Crocodile Dundee around.\"\n\n\"I don't come here unarmed.\"\n\nShe moistened her lips. \"And you probably know how to use it, if necessary.\"\n\nHe shot her a look. \"I probably do. Do you want to keep going, or do you want to turn around and go back?\"\n\n\"I'm not turning tail.\"\n\nHe could hear it rustling in the brush, could hear the slide of mud underfoot. Stalking them, he thought. He imagined the knife was as useless as a few harsh words if the thing meant business, but he felt better with it in his hand.\n\n\"Lump doesn't hear it,\" Quinn murmured, lifting her chin to where the dog slopped along the path a few feet ahead. \"Even he can't be that lazy. If he heard it, scented it, he'd show some concern. So it's not real.\" She took a slow breath. \"It's just show.\"\n\n\"Not real to him, anyway.\"\n\nWhen the thing howled, Cal took her firmly by the arm and pulled her through the edge of the trees into the clearing where the Pagan Stone speared up out of the muddy earth.\n\n\"I guess, all things considered, I was half expecting something along the lines of the king stone from Stonehenge.\" Quinn stepped away from Cal to circle the stone. \"It's amazing enough though, when you take a good look, the way it forms a table, or altar. How flat and smooth the top is.\" She laid her hand on it. \"It's warm,\" she added. \"Warmer than stone should be in a February wood.\"\n\nHe put his hand beside hers. \"Sometimes it's cold.\" He fit the knife back into its sheath. \"Nothing to worry about when it's warm. So far.\" He shoved his sleeve back, examined the scar on his wrist. \"So far,\" he repeated.\n\nWithout thinking, he laid his hand over hers. \"As long as\u2014\"\n\n\"It's heating up! Feel that? Do you feel that?\"\n\nShe shifted, started to place her other hand on the stone. He moved, felt himself move as he might have through that wall of fire. Madly.\n\nHe gripped her shoulders, spinning her around until her back was pressed to the stone. Then sated the sudden, desperate appetite by taking her mouth.\n\nFor an instant, he was someone else, as was she, and the moment was full of grieving desperation. Her taste, her skin, the beat of her heart.\n\nThen he was himself, feeling Quinn's lips heat under his as the stone had heated under their hands. It was her body quivering against his, and her fingers digging into his hips.\n\nHe wanted more, wanted to shove her onto the table of rock, to cover her with his body, to surround himself with all she was.\n\nNot him, he thought dimly, or not entirely him. And so he made himself pull back, forced himself to break that connection.\n\nThe air wavered a moment. \"Sorry,\" he managed. \"Not altogether sorry, but\u2014\"\n\n\"Surprised.\" Her voice was hoarse. \"Me, too. That was definitely unexpected. Made me dizzy,\" she whispered. \"That's not a complaint. It wasn't us, then it was.\" She took another steadying breath. \"Call me a slut, but I liked it both ways.\" With her eyes on his, she placed her hand on the stone again. \"Want to try it again?\"\n\n\"I think I'm still a man, so damn right I do. But I don't think it'd be smart, or particularly safe. Plus, I don't care for someone\u2014something\u2014else yanking on my hormones. Next time I kiss you, it's just you and me.\"\n\n\"All right. Connections.\" She nodded. \"I'm more in favor than ever about the theory regarding connections. Could be blood, could be a reincarnation thing. It's worth exploring.\"\n\nShe sidestepped away from the stone, and him. \"So, no more contact with each other and that thing for the time being. And let's take it back to the purpose at hand.\"\n\n\"Are you okay?\"\n\n\"Stirred me up, I'll admit. But no harm, no foul.\" She took out her water bottle, and this time drank deep.\n\n\"I wanted you. Both ways.\"\n\nLowering the bottle, she met those calm gray eyes. She'd just gulped down water, she thought, but now her throat was dry again. \"I know. What I don't know is if that's going to be a problem.\"\n\n\"It's going to be a problem. I'm not going to care about that.\"\n\nHer pulse gave a couple of quick jumps. \"Ah...This probably isn't the place to\u2014\"\n\n\"No, it's not.\" He took a step forward, but didn't touch her. And still her skin went hot. \"There's going to be another place.\"\n\n\"Okay.\" She cleared her throat. \"All right. To work.\"\n\nShe did another circle while he watched her. He'd made her a little jumpy. He didn't mind that. In fact, he considered it a point for his side. Something might have pushed him to kiss her that way, but he knew what he'd felt as that something released its grip. He knew what he'd been feeling since she'd stepped out of her car at the top of his lane.\n\nPlain and simple lust. Caleb Hawkins for Quinn Black.\n\n\"You camped here, the three of you, that night.\" Apparently taking Cal at his word about the safety of the area, Quinn moved easily around the clearing. \"You\u2014if I have any understanding of young boys\u2014ate junk food, ragged on each other, maybe told ghost stories.\"\n\n\"Some. We also drank the beer Gage stole from his father, and looked at the skin mags he'd swiped.\"\n\n\"Of course, though I'd have pegged those activities for more like twelve-year-olds.\"\n\n\"Precocious.\" He ordered himself to stop thinking about her, to take himself back. \"We built a fire. We had the boom box on. It was a pretty night, still hot, but not oppressive. And it was our night. It was, we thought, our place. Sacred ground.\"\n\n\"So your great-grandmother said.\"\n\n\"It called for ritual.\" He waited for her to turn to him. \"We wrote down words. Words we made. We swore an oath, and at midnight, I used my Boy Scout knife to cut our wrists. We said the words we'd made and pressed our wrists together to mix the blood. To make us blood brothers. And hell opened up.\"\n\n\"What happened?\"\n\n\"I don't know, not exactly. None of us do, not that we can remember. There was a kind of explosion. It seemed like one. The light was blinding, and the force of it knocked me back. Lifted me right off my feet. Screams, but I've never known if they were mine, Fox's, Gage's, or something else. The fire shot straight up, there seemed to be fire everywhere, but we weren't burned. Something pushed out, pushed into me. Pain, I remember pain. Then I saw some kind of dark mass rising out, and felt the cold it brought with it. Then it was over, and we were alone, scared, and the ground was scorched black.\"\n\nTen years old, she thought. Just a little boy. \"How did you get out?\"\n\n\"We hiked out the next morning pretty much as we'd hiked in. Except for a few changes. I came into this clearing when I was nine. I was wearing glasses. I was nearsighted.\"\n\nHer brows rose. \"Was?\"\n\n\"Twenty\u2013one hundred in my left eye, twenty-ninety in my right. I walked out ten, and twenty-twenty. None of us had a mark on him when we left, though Gage especially had some wounds he brought in with him. Not one of us has been sick a day since that night. If we're injured, it heals on its own.\"\n\nThere was no doubt on her face, only interest with a touch, he thought, of fascination. It struck him that other than his family she was the only one who knew. Who believed.\n\n\"You were given some sort of immunity.\"\n\n\"You could call it that.\"\n\n\"Do you feel pain?\"\n\n\"Damn right. I came out with perfect vision, not X-ray. And the healing can hurt like a mother, but it's pretty quick. I can see things that happened before, like out on the trail. Not all the time, not every time, but I can see events of the past.\"\n\n\"A reverse clairvoyance.\"\n\n\"When it's on. I've seen what happened here on July seventh, sixteen fifty-two.\"\n\n\"What happened here, Cal?\"\n\n\"The demon was bound under the stone. And Fox, Gage, and I, we cut the bastard loose.\"\n\nShe moved to him. She wanted to touch him, to soothe that worry from his face, but was afraid to. \"If you did, you weren't to blame.\"\n\n\"Blame and responsibility aren't much different.\"\n\nThe hell with it. She laid her hands on his cheeks even when he flinched. Then touched her lips gently to his. \"That was normal. You're responsible because, to my mind, you're willing to take responsibility. You've stayed when a lot of other men would've walked, if not run, away from here. So I say there's a way to beat it back where it belongs. And I'm going to do whatever I can to help you do just that.\"\n\nShe opened her pack. \"I'm going to take photos, some measurements, some notes, and ask a lot of annoying questions.\"\n\nShe'd shaken him. The touch, the words, the faith. He wanted to draw her in, hold on to her. Just hold on. Normal, she'd said, and looking at her now, he craved the bliss of normality.\n\nNot the place, he reminded himself, and stepped back. \"You've got an hour. We start back in an hour. We're going to be well out of the woods before twilight.\"\n\n\"No argument.\" This time, she thought, and went to work.\n\n## Nine\n\nSHE SPENT A LOT OF TIME, TO CAL'S MIND, WANDERING around, taking what appeared to be copious notes and a mammoth number of photographs with her tiny little digital, and muttering to herself.\n\nHe didn't see how any of that was particularly helpful, but since she seemed to be absorbed in it all, he sat under a tree with the snoring Lump and let her work.\n\nThere was no more howling, no more sense of anything stalking the clearing, or them. Maybe the demon had something else to do, Cal thought. Or maybe it was just hanging back, watching. Waiting.\n\nWell, he was doing the same, he supposed. He didn't mind waiting, especially when the view was good.\n\nIt was interesting to watch her, to watch the way she moved. Brisk and direct one minute, slow and wandering the next. As if she couldn't quite make up her mind which approach to take.\n\n\"Have you ever had this analyzed?\" she called out. \"The stone itself? A scientific analysis?\"\n\n\"Yeah. We took scrapings when we were teenagers, and took them to the geology teacher at the high school. It's limestone. Common limestone. And,\" he continued, anticipating her, \"we took another sample a few years later, that Gage took to a lab in New York. Same results.\"\n\n\"Okay. Any objection if I take a sample, send it to a lab I've used, just for one more confirmation?\"\n\n\"Help yourself.\" He started to hitch up a hip for his knife, but she was already taking a Swiss Army out of her pocket. He should've figured her for it. Still, it made him smile.\n\nMost of the women he knew might have lipstick in their pocket, but wouldn't consider a Swiss Army. He was betting Quinn had both.\n\nHe watched her hands as she scraped stone dust into a Baggie she pulled out of her pack. A trio of rings circled two fingers and the thumb of her right hand to catch quick glints of the sun with the movement.\n\nThe glints brightened, beamed into his eyes.\n\nThe light changed, softened like a summer morning even as the air warmed and took on a weight of humidity. Leaves budded, unfurled, then burst into thick green on the trees, casting shade and light in patterns on the ground, on the stone.\n\nOn the woman.\n\nHer hair was long and loose, the color of raw honey. Her face was sharp-featured with eyes long and tipped up slightly. She wore a long dress of dusky blue under a white apron. She moved with care, and still with grace, though her body was heavily pregnant. And she carried two pails across the clearing toward a little shed behind the stone.\n\nAs she walked she sang in a voice clear and bright as the summer morning.\n\nAll in a garden green where late I laid me down upon a bank of chamomile where I saw upon a style sitting, a country clown...\n\nHearing her, seeing her, Cal was filled with love so urgent, so ripe, he thought his heart might burst from it.\n\nThe man stepped through the door of the shed, and that love was illuminated on his face. The woman stopped, gave a knowing, flirtatious toss of her head, and sang as the man walked toward her.\n\n...holding in his arms a comely country maid. Courting her with all his skill, working her unto his will. Thus to her he said, Kiss me in kindness, sweetheart.\n\nShe lifted her face, offered her lips. The man brushed them with his, and as her laugh burst like a shooting star, he took the pails from her, setting them on the ground before wrapping her in an embrace.\n\nHave I not told you, you are not to carry water or wood? You carry enough.\n\nHis hands stroked over the mound of her belly, held there when hers covered them. Our sons are strong and well. I will give you sons, my love, as bright and brave as their father. My love, my heart. Now Cal saw the tears glimmer in those almond-shaped eyes. Must I leave you?\n\nYou will never leave me, not truly, nor I you. No tears. He kissed them away, and Cal felt the wrench of his own heart. No tears.\n\nNo. I swore an oath against them. So she smiled. There is time yet. Soft mornings and long summer days. It is not death. You swear to me?\n\nIt is not death. Come now. I will carry the water.\n\nWhen they faded, he saw Quinn crouched in front of him, heard her saying his name sharply, repeatedly.\n\n\"You're back. You went somewhere. Your eyes...Your eyes go black and...deep is the only word I can think of when you go somewhere else. Where did you go, Cal?\"\n\n\"She's not you.\"\n\n\"Okay.\" She'd been afraid to touch him before, afraid if she did she'd push them both into that somewhere else, or yank him back before he was done. Now she reached out to rest her hand on his knee. \"I'm not who?\"\n\n\"Whoever I was kissing. Started to, then it was you, but before, at first...Jesus.\" He clamped the heels of his hands at his temple. \"Headache. Bitch of a headache.\"\n\n\"Lean back, close your eyes. I'll\u2014\"\n\n\"It'll pass in a minute. They always do. We're not them. It's not a reincarnation deal. It doesn't feel right. Sporadic possession maybe, which is bad enough.\"\n\n\"Who?\"\n\n\"How the hell do I know?\" His head screamed until he had to lower his head between his knees to fight off the sudden, acute nausea. \"I'd draw you a damn picture if I could draw. Give me a minute.\"\n\nRising, Quinn went behind him and, kneeling, began to massage his neck, his shoulders.\n\n\"Okay, all right. Sorry. Christ. It's like having an electric drill inside my head, biting its way out through my temples. It's better. I don't know who they were. They didn't call each other by name. But best guess is Giles Dent and Ann Hawkins. They were obviously living here, and she was really, really pregnant. She was singing,\" he said and told her what he'd seen.\n\nQuinn continued to rub his shoulders while she listened. \"So they knew it was coming, and from what you say, he was sending her away before it did. 'Not death.' That's interesting, and something to look into. But for now, I think you've had enough of this place. And so have I.\"\n\nShe sat on the ground then, hissed a breath out, sucked one in. \"While you were out, let's say, it came back.\"\n\n\"Jesus Christ.\" He started to spring up, but she gripped his arm.\n\n\"It's gone. Let's just sit here until we both get our legs back under us. I heard it growling, and I spun around. You were taking a trip, and I quashed my first instinct to grab you, shake you out of it, in case doing that pulled me in with you.\"\n\n\"And we'd both be defenseless,\" he said in disgust.\n\n\"And now Mr. Responsibility is beating himself up because he didn't somehow see this coming, fight off the magickal forces so he could stay in the here and now and protect the girl.\"\n\nEven with the headache, he could manage a cool, steely stare. \"Something like that.\"\n\n\"Something like that is appreciated, even if it is annoying. I had my handy Swiss Army knife, which, while it isn't up to Jim Bowie standards, does include a nice corkscrew and tweezers, both of which you never know when you may need.\"\n\n\"Is that spunk? Are you being spunky?\"\n\n\"I'm babbling until I level out and I'm nearly there. The thing is, it just circled, making its nasty 'I'll eat you, my pretty and your big, lazy dog, too.' Rustling, growling, snarling. But it didn't show itself. Then it stopped, and you came back.\"\n\n\"How long?\"\n\n\"I don't know. I think just a couple minutes, though it seemed longer at the time. However long, I'm so ready to get gone. I hope to hell you can walk back, Cal, because strong and resilient as I am, there's no way I can carry you piggyback.\"\n\n\"I can walk.\"\n\n\"Good, then let's get the hell out of here, and when we get to civilization, Hawkins, you're buying me a really big drink.\"\n\nThey gathered their packs; Cal whistled Lump awake. As they started back he wondered why he hadn't told her of the bloodstone\u2014the three pieces he, Fox, and Gage held. The three pieces that he now knew formed the stone in the amulet Giles Dent had worn when he'd lived at the Pagan Stone.\n\nWHILE CAL AND QUINN WERE HIKING OUT OF Hawkins Wood, Layla was taking herself out for an aimless walk around town. It was odd to just let her feet choose any direction. During her years in New York she'd always had a specific destination, always had a specific task, or several specific tasks to accomplish within a particular time frame.\n\nNow, she'd let the morning stretch out, and had accomplished no more than reading sections of a few of the odd books Quinn had left with her. She might have stayed right there, inside her lovely room, inside that safe zone as Quinn had termed it.\n\nBut she'd needed to get away from the books. In any case, it gave the housekeeper an opportunity to set the room to rights, she supposed. And gave herself an opportunity to take a real look at the town she'd been compelled to visit.\n\nShe didn't have the urge to wander into any of the shops, though she thought Quinn's assessment was on the mark. There were some very interesting possibilities.\n\nBut even window shopping made her feel guilty for leaving the staff of the boutique in the lurch. Taking off the way she had, barely taking the time to call in from the road to tell the owner she'd had a personal emergency and wouldn't be in for the next several days.\n\nPersonal emergency covered it, Layla decided.\n\nAnd it could very well get her fired. Still, even knowing that, she couldn't go back, pick things up, forget what had happened.\n\nShe'd get another job if she had to. When and if, she'd find another. She had some savings, she had a cushion. If her boss couldn't cut her some slack, she didn't want that stupid job anyway.\n\nAnd, oh God, she was already justifying being unemployed.\n\nDon't think about it, she ordered herself. Don't think about that right this minute.\n\nShe didn't think about it, and didn't think twice when her feet decided to continue on beyond the shops. She couldn't have said why they wanted to stop at the base of the building. LIBRARY was carved into the stone lintel over the door, but the glossy sign read HAWKINS HOLLOW COMMUNITY CENTER.\n\nInnocuous enough, she told herself. But when a chill danced over her skin she ordered her feet to keep traveling.\n\nShe considered going into the museum, but couldn't work up the interest. She thought about crossing the street to Salon A and whiling away some time with a manicure, but simply didn't care about the state of her nails.\n\nTired and annoyed with herself, she nearly turned around and headed back. But the sign that caught her eye this time drew her forward.\n\nFOX O'DELL, ATTORNEY AT LAW.\n\nAt least he was someone she knew\u2014more or less. The hot lawyer with the compassionate eyes. He was probably busy with a client or out of the office, but she didn't care. Going in was something to do other than wander around feeling sorry for herself.\n\nShe stepped into the attractive, homespun reception area. The woman behind the gorgeous old desk offered a polite smile.\n\n\"Good morning\u2014well, afternoon now. Can I help you?\"\n\n\"I'm actually...\" What? Layla wondered. What exactly was she? \"I was hoping to speak to Mr. O'Dell for a minute if he's free.\"\n\n\"Actually, he's with a client, but they shouldn't be much longer if you'd like to...\"\n\nA woman in tight jeans, a snug pink sweater, and an explosion of hair in an improbable shade of red marched out on heeled boots. She dragged on a short leather jacket. \"I want him skinned, Fox, you hear? I gave that son of a bitch the best two years and three months of my life, and I want him skinned like a rabbit.\"\n\n\"So noted, Shelley.\"\n\n\"How could he do that to me?\" On a wail she collapsed into Fox's arms.\n\nHe wore jeans as well, and an untucked pinstriped shirt, along with an expression of resignation as he glanced over at Layla. \"There, there,\" he said, patting the sobbing Shelley's back. \"There, there.\"\n\n\"I just bought him new tires for his truck! I'm going to go slash every one of them.\"\n\n\"Don't.\" Fox took a good hold of her before Shelley, tears streaming away in fresh rage, started to yank back. \"I don't want you to do that. You don't go near his truck, and for now, honey, try to stay away from him, too. And Sami.\"\n\n\"That turncoat slut of a bitch.\"\n\n\"That's the one. Leave this to me for now, okay? You go on back to work and let me handle this. That's why you hired me, right?\"\n\n\"I guess. But you skin him raw, Fox. You crack that bastard's nuts like pecans.\"\n\n\"I'm going to get right on that,\" he assured her as he led her to the door. \"You just stay above it all, that's the way. I'll be in touch.\"\n\nAfter he'd closed the door, leaned back on it, he heaved out a breath. \"Holy Mother of God.\"\n\n\"You should've referred that one,\" Alice told him.\n\n\"You can't refer off the first girl you got to second base with when she's filing for divorce. It's against the laws of God and Man. Hello, Layla, need a lawyer?\"\n\n\"I hope not.\" He was better looking than she remembered, which just went to show the shape she'd been in the night before. Plus he didn't look anything like a lawyer. \"No offense.\"\n\n\"None taken. Layla...It's Darnell, right?\"\n\n\"Yes.\"\n\n\"Layla Darnell, Alice Hawbaker. Mrs. H, I'm clear for a while?\"\n\n\"You are.\"\n\n\"Come on back, Layla.\" He gestured. \"We don't usually put a show on this early in the day, but my old pal Shelley walked into the back room over at the diner to visit her twin sister, Sami, and found her husband\u2014that would be Shelley's husband, Block\u2014holding Sami's tip money.\"\n\n\"I'm sorry, she's filing for divorce because her husband was holding her sister's tip money?\"\n\n\"It was in Sami's Victoria's Secret Miracle Bra at the time.\"\n\n\"Oh. Well.\"\n\n\"That's not privileged information as Shelley chased them both out of the back room and straight out onto Main Street\u2014with Sami's miraculous bra in full view\u2014with a rag mop. Want a Coke?\"\n\n\"No, I really don't. I don't think I need anything to give me an edge.\"\n\nSince she looked inclined to pace, he didn't offer her a chair. Instead, he leaned back against his desk. \"Rough night?\"\n\n\"No, the opposite. I just can't figure out what I'm doing here. I don't understand any of this, and I certainly don't understand my place in it. A couple hours ago I told myself I was going to pack and drive back to New York like a sane person. But I didn't.\" She turned to him. \"I couldn't. And I don't understand that either.\"\n\n\"You're where you're supposed to be. That's the simplest answer.\"\n\n\"Are you afraid?\"\n\n\"A lot of the time.\"\n\n\"I don't think I've ever been really afraid. I wonder if I'd be so damned edgy if I had something to do. An assignment, a task.\"\n\n\"Listen, I've got to drive to a client a few miles out of town, take her some papers.\"\n\n\"Oh, sorry. I'm in the way.\"\n\n\"No, and when I start thinking beautiful women are in my way, please notify my next of kin so they can gather to say their final good-byes before my death. I was going to suggest you ride out with me, which is something to do. And you can have chamomile tea and stale lemon snaps with Mrs. Oldinger, which is a task. She likes company, which is the real reason she had me draw up the fifteenth codicil to her will.\"\n\nHe kept talking, knowing that was one way to help calm someone down when she looked ready to bolt. \"By the time that's done, I can swing by another client who's not far out of the way and save him a trip into town. By my way of thinking, Cal and Quinn should be just about back home by the time we're done with all that. We'll go by, see what's what.\"\n\n\"Can you be out of the office all that time?\"\n\n\"Believe me.\" He grabbed his coat, his briefcase. \"Mrs. H will holler me back if I'm needed here. But unless you've got something better to do, I'll have her pull out the files I need and we'll take a drive.\"\n\nIt was better than brooding, Layla decided. Maybe she thought it was odd for a lawyer, even a small-town lawyer to drive an old Dodge pickup with a couple of Ring Ding wrappers littering the floorboards.\n\n\"What are you doing for the second client?\"\n\n\"That's Charlie Deen. Charlie got clipped by a DUI when he was driving home from work. Insurance company's trying to dance around some of the medical bills. Not going to happen.\"\n\n\"Divorce, wills, personal injury. So you don't specialize?\"\n\n\"All law, all the time,\" he said and sent her a smile that was a combination of sweet and cocky. \"Well, except for tax law if I can avoid it. I leave that to my sister. She's tax and business law.\"\n\n\"But you don't have a practice together.\"\n\n\"That'd be tough. Sage went to Seattle to be a lesbian.\"\n\n\"I beg your pardon?\"\n\n\"Sorry.\" He boosted the gas as they passed the town limits. \"Family joke. What I mean is my sister Sage is gay, and she lives in Seattle. She's an activist, and she and her partner of, hmm, I guess about eight years now run a firm they call Girl on Girl. Seriously,\" he added when Layla said nothing. \"They specialize in tax and business law for gays.\"\n\n\"Your family doesn't approve?\"\n\n\"Are you kidding? My parents eat it up like tofu. When Sage and Paula\u2014that's her partner\u2014got married. Or had their life-partner affirmation, whatever\u2014we all went out there and celebrated like mental patients. She's happy and that's what counts. The alternate lifestyle choice is just kind of a bonus for my parents. Speaking of family, that's my little brother's place.\"\n\nLayla saw a log house all but buried in the trees, with a sign near the curve of the road reading HAWKINS CREEK POTTERY.\n\n\"Your brother's a potter.\"\n\n\"Yeah, a good one. So's my mother when she's in the mood. Want to stop in?\"\n\n\"Oh, I...\"\n\n\"Better not,\" he decided. \"Ridge'll get going and Mrs. H has called Mrs. Oldinger by now to tell her to expect us. Another time.\"\n\n\"Okay.\" Conversation, she thought. Small talk. Relative sanity. \"So you have a brother and sister.\"\n\n\"Two sisters. My baby sister owns the little vegetarian restaurant in town. It's pretty good, considering. Of the four of us I veered the farthest off the flower-strewn path my counterculture parents forged. But they love me anyway. That's about it for me. How about you?\"\n\n\"Well...I don't have any relatives nearly as interesting as yours sound, but I'm pretty sure my mother has some old Joan Baez albums.\"\n\n\"There, that strange and fateful crossroads again.\"\n\nShe started to laugh, then gasped with pleasure as she spotted the deer. \"Look! Oh, look. Aren't they gorgeous, just grazing there along the edge of the trees?\"\n\nTo accommodate her, Fox pulled over to the narrow shoulder so she could watch. \"You're used to seeing deer, I suppose,\" she said.\n\n\"Doesn't mean I don't get a kick out of it. We had to run herds off the farm when I was a kid.\"\n\n\"You grew up on a farm.\"\n\nThere was that urban-dweller wistfulness in her voice. The kind that said she saw the pretty deer, the bunnies, the sunflowers, and happy chickens. And not the plowing, the hoeing, weeding, harvesting. \"Small, family farm. We grew our own vegetables, kept chickens and goats, bees. Sold some of the surplus, some of my mother's crafts, my father's woodwork.\"\n\n\"Do they still have it?\"\n\n\"Yeah.\"\n\n\"My parents owned a little dress shop when I was a kid. They sold out about fifteen years ago. I always wished\u2014Oh God, oh my God!\"\n\nHer hand whipped over to clamp on his arm.\n\nThe wolf leaped out of the trees, onto the back of a young deer. It bucked, it screamed\u2014she could hear its high-pitched screams of fear and pain\u2014it bled while the others in the small herd continued to crop at grass.\n\n\"It's not real.\"\n\nHis voice sounded tinny and distant. In front of her horrified eyes the wolf took the deer down, then began to tear and rip.\n\n\"It's not real,\" he repeated. He put his hands on her shoulders, and she felt something click. Something inside her pushed toward him and away from the horror at the edge of the trees. \"Look at it, straight on,\" he told her. \"Look at it and know it's not real.\"\n\nThe blood was so red, so wet. It flew in ugly rain, smearing the winter grass of the narrow field. \"It's not real.\"\n\n\"Don't just say it. Know it. It lies, Layla. It lives in lies. It's not real.\"\n\nShe breathed in, breathed out. \"It's not real. It's a lie. It's an ugly lie. A small, cruel lie. It's not real.\"\n\nThe field was empty; the winter grass ragged and unstained.\n\n\"How do you live with this?\" Shoving around in her seat, Layla stared at him. \"How do you stand this?\"\n\n\"By knowing\u2014the way I knew that was a lie\u2014that some day, some way, we're going to kick its ass.\"\n\nHer throat burned dry. \"You did something to me. When you took my shoulders, when you were talking to me, you did something to me.\"\n\n\"No.\" He denied it without a qualm. He'd done something for her, Fox told himself. \"I just helped you remember it wasn't real. We're going on to Mrs. Oldinger. I bet you could use that chamomile tea about now.\"\n\n\"Does she have any whiskey to go with it?\"\n\n\"Wouldn't surprise me.\"\n\nQUINN COULD SEE CAL'S HOUSE THROUGH THE trees when her phone signaled a waiting text-message. \"Crap, why didn't she just call me?\"\n\n\"Might've tried. There are lots of pockets in the woods where calls drop out.\"\n\n\"Color me virtually unsurprised.\" She brought up the message, smiling a little as she recognized Cybil's shorthand.\n\nBzy, but intrig'd. Tell u more when. Cn B there in a wk, 2 latest. Tlk whn cn. Q? B-ware. Serious. C.\n\n\"All right.\" Quinn replaced the phone and made the decision she'd been weighing during the hike back. \"I guess we'll call Fox and Layla when I'm having that really big drink by the fire you're going to build.\"\n\n\"I can live with that.\"\n\n\"Then, seeing as you're a town honcho, you'd be the one to ask about finding a nice, attractive, convenient, and somewhat roomy house to rent for the next, oh, six months.\"\n\n\"And the tenant would be?\"\n\n\"Tenants. They would be me, my delightful friend Cybil, whom I will talk into digging in, and most likely Layla, whom\u2014I believe\u2014will take a bit more convincing. But I'm very persuasive.\"\n\n\"What happened to staying a week for initial research, then coming back in April for a follow-up?\"\n\n\"Plans change,\" she said airily, and smiled at him as they stepped onto the gravel of his driveway. \"Don't you just love when that happens?\"\n\n\"Not really.\" But he walked with her onto the deck and opened the door so she could breeze into his quiet home ahead of him.\n\n## Ten\n\nTHE HOUSE WHERE CAL HAD GROWN UP WAS, IN his opinion, in a constant state of evolution. Every few years his mother would decide the walls needed \"freshening,\" which meant painting\u2014or often in his mother's vocabulary a new \"paint treatment.\"\n\nThere was ragging, there was sponging, there was combing, and a variety of other terms he did his best to tune out.\n\nNaturally, new paint led to new upholstery or window treatments, certainly to new bed linens when she worked her way to bedrooms. Which invariably led to new \"arrangements.\"\n\nHe couldn't count the number of times he'd hauled furniture around to match the grafts his mother routinely generated.\n\nHis father liked to say that as soon as Frannie had the house the way she wanted, it was time for her to shake it all up again.\n\nAt one time, Cal had assumed his mother had fiddled, fooled, painted, sewed, arranged, and re-arranged out of boredom. Although she volunteered, served on various committees, or stuck her oar in countless organizations, she'd never worked outside the home. He'd gone through a period in his late teens and early twenties where he'd imagined her (pitied her) as an unfulfilled, semidesperate housewife.\n\nAt one point he, in his worldliness of two college semesters, got her alone and explained his understanding of her sense of repression. She'd laughed so hard she'd had to set down her upholstery tacks and wipe her eyes.\n\n\"Honey,\" she'd said, \"there's not a single bone of repression in my entire body. I love color and texture and patterns and flavors. And oh, just all sorts of things. I get to use this house as my studio, my science project, my laboratory, and my showroom. I get to be the director, the designer, the set builder, and the star of the whole show. Now, why would I want to go out and get a job or a career\u2014since we don't need the money\u2014and have somebody else tell me what to do and when to do it?\"\n\nShe'd crooked her finger so he leaned down to her. And she'd laid a hand on his cheek. \"You're such a sweetheart, Caleb. You're going to find out that not everybody wants what society\u2014in whatever its current mood or mode might be\u2014tells them they should want. I consider myself lucky, even privileged, that I was able to make the choice to stay home and raise my children. And I'm lucky to be able to be married to a man who doesn't mind if I use my talents\u2014and I'm damned talented\u2014to disrupt his quiet home with paint samples and fabric swatches every time he turns around. I'm happy. And I love knowing you worried I might not be.\"\n\nHe'd come to see she was exactly right. She did just as she liked, and was terrific at what she did. And, he'd come to see that when it came down to the core, she was the power in the house. His father brought in the money, but his mother handled the finances. His father ran his business, his mother ran the home.\n\nAnd that was exactly the way they liked it.\n\nSo he didn't bother telling her not to fuss over Sunday dinner\u2014just as he hadn't attempted to talk her out of extending the invitation to Quinn, Layla, and Fox. She lived to fuss, and enjoyed putting on elaborate meals for people, even if she didn't know them.\n\nSince Fox volunteered to swing into town and pick up the women, Cal went directly to his parents' house, and went early. It seemed wise to give them some sort of groundwork\u2014and hopefully a few basic tips on how to deal with a woman who intended to write a book on the Hollow, since the town included people, and those people included his family.\n\nFrannie stood at the stove, checking the temperature of her pork tenderloin. Obviously satisfied with that, she crossed to the counter to continue the layers of her famous antipasto squares.\n\n\"So, Mom,\" Cal began as he opened the refrigerator.\n\n\"I'm serving wine with dinner, so don't go hunting up any beer.\"\n\nChastised, he shut the refrigerator door. \"Okay. I just wanted to mention that you shouldn't forget that Quinn's writing a book.\"\n\n\"Have you noticed me forgetting things?\"\n\n\"No.\" The woman forgot nothing, which could be a little daunting. \"What I mean is, we should all be aware that things we say and do may end up in a book.\"\n\n\"Hmm.\" Frannie layered pepperoni over provolone. \"Do you expect me or your father to say or do something embarrassing over appetizers? Or maybe we'll wait until dessert. Which is apple pie, by the way.\"\n\n\"No, I\u2014You made apple pie?\"\n\nShe spared him a glance, and a knowing smile. \"It's your favorite, isn't it, my baby?\"\n\n\"Yeah, but maybe you've lost your knack. I should sample a piece before company gets here. Save you any embarrassment if it's lousy pie.\"\n\n\"That didn't work when you were twelve.\"\n\n\"I know, but you always pounded the whole if-you don't-succeed chestnut into my head.\"\n\n\"You just keep trying, sweetie. Now, why are you worried about this girl, who I'm told you've been seen out and about with a few times, coming around for dinner?\"\n\n\"It's not like that.\" He wasn't sure what it was like. \"It's about why she's here at all. We can't forget that, that's all I'm saying.\"\n\n\"I never forget. How could I? We have to live our lives, peel potatoes, get the mail, sneeze, buy new shoes, in spite of it all, maybe because of it all.\" There was a hint of fierceness in her voice he recognized as sorrow. \"And that living includes being able to have a nice company meal on a Sunday.\"\n\n\"I wish it were different.\"\n\n\"I know you do, but it's not.\" She kept layering, but her eyes lifted to his. \"And, Cal, my handsome boy, you can't do more than you do. If anything, there are times I wish you could do less. But...Tell me, do you like this girl? Quinn Black?\"\n\n\"Sure.\" Like to get a taste of that top-heavy mouth again, he mused. Then broke off that train of thought quickly since he knew his mother's skill at reading her children's minds.\n\n\"Then I intend to give her and the others a comfortable evening and an excellent meal. And, Cal, if you didn't want her here, didn't want her to speak with me or your dad, you wouldn't let her in the door. I wouldn't be able, though my powers are fierce, to shove you aside and open it myself.\"\n\nHe looked at her. Sometimes when he did, it surprised him that this pretty woman with her short, streaked blond hair, her slim build and creative mind could have given birth to him, could have raised him to be a man. He could look and think she was delicate, and then remember she was almost terrifyingly strong.\n\n\"I'm not going to let anything hurt you.\"\n\n\"Back at you, doubled. Now get out of my kitchen. I need to finish up the appetizers.\"\n\nHe'd have offered to lend her a hand, but would have earned one of her pitying stares. Not that she didn't allow kitchen help. His father was not only allowed to grill, but encouraged to. And any and all could and were called in as line chefs from time to time.\n\nBut when his mother was in full-out company-coming mode, she wanted the kitchen to herself.\n\nHe passed through the dining room where, naturally, the table was already set. She'd used festive plates, which meant she wasn't going for elegant or drop-in casual. Tented linen napkins, tea lights in cobalt rounds, inside a centerpiece of winter berries.\n\nEven during the worst time, even during the Seven, he could come here and there would be fresh flowers artfully arranged, furniture free of dust and gleaming with polish, and intriguing little soaps in the dish in the downstairs powder room.\n\nEven hell didn't cause Frannie Hawkins to break stride.\n\nMaybe, Cal thought as he wandered into the living room, that was part of the reason\u2014even the most important reason\u2014he got through it himself. Because whatever else happened, his mother would be maintaining her own brand of order and sanity.\n\nJust as his father would be. They'd given him that, Cal thought. That rock-solid foundation. Nothing, not even a demon from hell had ever shaken it.\n\nHe started to go upstairs, hunt down his father who, he suspected, would be in his home office. But saw Fox's truck pull in when he glanced out the window.\n\nHe stood where he was, watched Quinn jump out first, cradling a bouquet wrapped in green florist paper. Layla slid out next, holding what looked to be a wine gift bag. His mother, Cal thought, would approve of the offerings. She herself had shelves and bins in her ruthlessly organized workroom that held carefully selected emergency hostess gifts, gift bags, colored tissue paper, and an assortment of bows and ribbons.\n\nWhen Cal opened the door, Quinn strode straight in. \"Hi. I love the house and the yard! Shows where you came by your eye for landscaping. What a great space. Layla, look at these walls. Like an Italian villa.\"\n\n\"It's their latest incarnation,\" Cal commented.\n\n\"It looks like home, but with a kick of style. Like you could curl up on that fabulous sofa and take a snooze, but you'd probably read Southern Homes first.\"\n\n\"Thank you.\" Frannie stepped out. \"That's a lovely compliment. Cal, take everyone's coats, will you? I'm Frannie Hawkins.\"\n\n\"It's so nice to meet you. I'm Quinn. Thanks so much for having us. I hope you like mixed bouquets. I have a hard time deciding on one type of mostly anything.\"\n\n\"They're wonderful, thank you.\" Frannie accepted the flowers, smiled expectantly at Layla.\n\n\"I'm Layla Darnell, thank you for having us in your home. I hope the wine's appropriate.\"\n\n\"I'm sure it is.\" Frannie took a peek inside the gift bag. \"Jim's favorite cabernet. Aren't you clever girls? Cal, go up and tell your father we have company. Hello, Fox.\"\n\n\"I brought you something, too.\" He grabbed her, lowered her into a stylish dip, and kissed both her cheeks. \"What's cooking, sweetheart?\"\n\nAs she had since he'd been a boy, Frannie ruffled his hair. \"You won't have long to wait to find out. Quinn and Layla, you make yourselves comfortable. Fox, you come with me. I want to put these flowers in water.\"\n\n\"Is there anything we can do to help?\"\n\n\"Not a thing.\"\n\nWhen Cal came down with his father, Fox was doing his version of snooty French waiter as he served appetizers. The women were laughing, candles were lit, and his mother carried in her grandmother's best crystal vase with Quinn's flowers a colorful filling.\n\nSometimes, Cal mused, all really was right with the world.\n\nHALFWAY THROUGH THE MEAL, WHERE THE CONVERSATION stayed in what Cal considered safe territories, Quinn set down her fork, shook her head. \"Mrs. Hawkins, this is the most amazing meal, and I have to ask. Did you study? Did you have a career as a gourmet chef at some point or did we just hit you on a really lucky day?\"\n\n\"I took a few classes.\"\n\n\"Frannie's taken a lot of 'a few classes,'\" Jim said. \"In all kinds of things. But she's just got a natural talent for cooking and gardening and decorating. What you see around here, it's all her doing. Painted the walls, made the curtains\u2014sorry, window treatments,\" he corrected with a twinkle at his wife.\n\n\"Get out. You did all the faux and fancy paintwork? Yourself?\"\n\n\"I enjoy it.\"\n\n\"Found that sideboard there years back at some flea market, had me haul it home.\" Jim gestured toward the gleaming mahogany sideboard. \"A few weeks later, she has me haul it in here. Thought she was pulling a fast one, had snuck out and bought something from an antique store.\"\n\n\"Martha Stewart eats your dust,\" Quinn decided. \"I mean that as a compliment.\"\n\n\"I'll take it.\"\n\n\"I'm useless at all of that. I can barely paint my own nails. How about you?\" Quinn asked Layla.\n\n\"I can't sew, but I like to paint. Walls. I've done some ragging that turned out pretty well.\"\n\n\"The only ragging I've done successfully was on my ex-fianc\u00e9.\"\n\n\"You were engaged?\" Frannie asked.\n\n\"I thought I was. But our definition of same differed widely.\"\n\n\"It can be difficult to blend careers and personal lives.\"\n\n\"Oh, I don't know. People do it all the time\u2014with varying degrees of success, sure, but they do. I think it just has to be the right people. The trick, or the first of probably many tricks, is recognizing the right person. Wasn't it like that for you? Didn't you have to recognize each other?\"\n\n\"I knew the first time I saw Frannie. There she is.\" Jim beamed down the table at his wife. \"Frannie now, she was a little more shortsighted.\"\n\n\"A little more practical,\" Frannie corrected, \"seeing as we were eight and ten at the time. Plus I enjoyed having you moon over and chase after me. Yes, you're right.\" Frannie looked back at Quinn. \"You have to see each other, and see in each other something that makes you want to take the chance, that makes you believe you can dig down for the long haul.\"\n\n\"And sometimes you think you see something,\" Quinn commented, \"but it was just a\u2014let's say\u2014trompe l'oeil.\"\n\nONE THING QUINN KNEW HOW TO DO WAS FINAGLE. Frannie Hawkins wasn't an easy mark, but Quinn managed to charm her way into the kitchen to help put together dessert and coffee.\n\n\"I love kitchens. I'm kind of a pathetic cook, but I love all the gadgets and tools, all the shiny surfaces.\"\n\n\"I imagine with your work, you eat out a lot.\"\n\n\"Actually, I eat in most of the time or call for takeout. I implemented a lifestyle change\u2014nutrition-wise\u2014a couple of years ago. Determined to eat healthier, depend less on fast or nuke-it-out-of-a-box food. I make a really good salad these days. That's a start. Oh God, oh God, that's apple pie. Homemade apple pie. I'm going to have to do double duty in the gym as penance for the huge piece I'm going to ask for.\"\n\nHer enjoyment obvious, Frannie shot her a wicked smile. \"\u00c0 la mode, with vanilla bean ice cream?\"\n\n\"Yes, but only to show my impeccable manners.\" Quinn hesitated a moment, then jumped in. \"I'm going to ask you, and if you want this off-limits while I'm enjoying your hospitality, just tell me to back off. Is it hard for you to nurture this normal life, to hold your family, yourself, your home together when you know all of it will be threatened?\"\n\n\"It's very hard.\" Frannie turned to her pies while the coffee brewed. \"Just as it's very necessary. I wanted Cal to go, and if he had I would have convinced Jim to leave. I could do that, I could turn my back on it all. But Cal couldn't. And I'm so proud of him for staying, for not giving up.\"\n\n\"Will you tell me what happened when he came home that morning, the morning of his tenth birthday?\"\n\n\"I was in the yard.\" Frannie walked over to the window that faced the back. She could see it all, every detail. How green the grass was, how blue the sky. Her hydrangeas were headed up and beginning to pop, her delphiniums towering spears of exotic blue.\n\nDeadheading her roses, and some of the coreopsis that had bloomed off. She could even hear the busy snip, snip of her shears, and the hum of the neighbor's\u2014it had been the Petersons, Jack and Lois, then\u2014lawn mower. She remembered, too, she'd been thinking about Cal, and his birthday party. She'd had his cake in the oven.\n\nA double-chocolate sour cream cake, she remembered. She'd intended to do a white frosting to simulate the ice planet from one of the Star Wars movies. Cal had loved Star Wars for years and years. She'd had the little action figures to arrange on it, the ten candles all ready in the kitchen.\n\nHad she heard him or sensed him\u2014probably some of both\u2014but she'd looked around as he'd come barreling up on his bike, pale, filthy, sweaty. Her first thought had been accident, there'd been an accident. And she'd been on her feet and rushing to him before she'd noticed he wasn't wearing his glasses.\n\n\"The part of me that registered that was ready to give him a good tongue-lashing. But the rest of me was still running when he climbed off his bike, and ran to me. He ran to me and he grabbed on so tight. He was shaking\u2014my little boy\u2014shaking like a leaf. I went down on my knees, pulling him back so I could check for blood or broken bones.\"\n\nWhat is it, what happened, are you hurt? All of that, Frannie remembered had flooded out of her, so fast it was like one word. In the woods, he'd said. Mom. Mom. In the woods.\n\n\"There was that part of me again, the part that thought what were you doing in the woods, Caleb Hawkins? It all came pouring out of him, how he and Fox and Gage planned this adventure, what they'd done, where they'd gone. And that same part was coldly devising the punishment to fit the crime, even while the rest of me was terrified, and relieved, so pitifully relieved I was holding my dirty, sweaty boy. Then he told me the rest.\"\n\n\"You believed him?\"\n\n\"I didn't want to. I wanted to believe he'd had a nightmare, which he richly deserved, that he'd stuffed himself on sweets and junk food and had a nightmare. Even, that someone had gone after them in the woods. But I couldn't look at his face and believe that. I couldn't believe the easy that, the fixable that. And then, of course, there were his eyes. He could see a bee hovering over the delphiniums across the yard. And under the dirt and sweat, there wasn't a bruise on him. The nine-year-old I'd sent off the day before had scraped knees and bruised shins. The one who came back to me hadn't a mark on him, but for the thin white scar across his wrist he hadn't had when he left.\"\n\n\"Even with that, a lot of adults, even mothers, wouldn't have believed a kid who came home with a story like that.\"\n\n\"I won't say Cal never lied to me, because obviously he did. He had. But I knew he wasn't lying. I knew he was telling me the truth, all the truth he knew.\"\n\n\"What did you do?\"\n\n\"I took him inside, told him to clean up, change his clothes. I called his father, and got his sisters home. I burned his birthday cake\u2014completely forgot about it, never heard the timer. Might've burned the house down if Cal himself hadn't smelled the burning. So he never got his ice planet or his ten candles. I hate remembering that. I burned his cake and he never got to blow out his birthday candles. Isn't that silly?\"\n\n\"No, ma'am. No,\" Quinn said with feeling when Frannie looked at her, \"it's not.\"\n\n\"He was never really, not wholly, a little boy again.\" Frannie sighed. \"We went straight over to the O'Dells, because Fox and Gage were already there. We had what I guess you could call our first summit meeting.\"\n\n\"What did\u2014\"\n\n\"We need to take in the dessert and coffee. Can you handle that tray?\"\n\nUnderstanding the subject was closed for now, Quinn stepped over. \"Sure. It looks terrific, Mrs. Hawkins.\"\n\nIn between moans and tears of joy over the pie, Quinn aimed her charm at Jim Hawkins. Cal, she was sure, had been dodging and weaving, avoiding and evading her since their hike to the Pagan Stone.\n\n\"Mr. Hawkins, you've lived in the Hollow all your life.\"\n\n\"Born and raised. Hawkinses have been here since the town was a couple of stone cabins.\"\n\n\"I met your grandmother, and she seems to know town history.\"\n\n\"Nobody knows more.\"\n\n\"People say you're the one who knows real estate, business, local politics.\"\n\n\"I guess I do.\"\n\n\"Then you may be able to point me in the right direction.\" She slid a look at Cal, then beamed back at his father. \"I'm looking to rent a house, something in town or close to it. Nothing fancy, but I'd like room. I have a friend coming in soon, and I've nearly talked Layla into staying longer. I think we'd be more comfortable, and it would be more efficient, for the three of us to have a house instead of using the hotel.\"\n\n\"How long are you looking for?\"\n\n\"Six months.\" She saw it register on his face, just as she noticed the frown form on Cal's. \"I'm going to stay through July, Mr. Hawkins, and I'm hoping to find a house that would accommodate three women\u2014potentially three\u2014\" she said with a glance at Layla.\n\n\"I guess you've thought that over.\"\n\n\"I have. I'm going to write this book, and part of the angle I'm after is the fact that the town remains, the people\u2014a lot of them\u2014stay. They stay and they make apple pie and have people over to Sunday dinner. They bowl, and they shop. They fight and they make love. They live. If I'm going to do this right, I want to be here, before, during, and after. So I'd like to rent a house.\"\n\nJim scooped up some pie, chased it with coffee. \"It happens I know a place on High Street, just a block off Main. It's old, main part went up before the Civil War. It's got four bedrooms, three baths. Nice porches, front and back. Had a new roof on her two years ago. Kitchen's eat-in size, though there's a little dining room off it. Appliances aren't fancy, but they've only got five years on them. Just been painted. Tenants moved out just a month ago.\"\n\n\"It sounds perfect. You seem to know it well.\"\n\n\"Should. We own it. Cal, you should take Quinn by. Maybe run her and Layla over there on the way home. You know where the keys are.\"\n\n\"Yeah,\" he said when Quinn gave him a big, bright smile. \"I know where the keys are.\"\n\nAS IT MADE THE MOST SENSE, QUINN HITCHED A ride with Cal, and left Fox and Layla to follow. She stretched out her legs, let out a sigh.\n\n\"Let me start off by saying your parents are terrific, and you're lucky to have grown up in such a warm, inviting home.\"\n\n\"I agree.\"\n\n\"Your dad's got that Ward Cleaver meets Jimmy Stewart thing going. I could've eaten him up like your mother's\u2014Martha Stewart meets Grace Kelly by way of Julia Child\u2014apple pie.\"\n\nHis lips twitched. \"They'd both like those descriptions.\"\n\n\"You knew about the High Street house.\"\n\n\"Yeah, I did.\"\n\n\"You knew about the High Street house, and avoided telling me about it.\"\n\n\"That's right. You found out about it, too, before dinner, which is why you did the end-run around me to my father.\"\n\n\"Correct.\" She tapped her finger on his shoulder. \"I figured he'd point me there. He likes me. Did you avoid telling me because you're not comfortable with what I might write about Hawkins Hollow?\"\n\n\"Some of that. More, I was hoping you'd change your mind and leave. Because I like you, too.\"\n\n\"You like me, so you want me gone?\"\n\n\"I like you, Quinn, so I want you safe.\" He looked at her again, longer. \"But some of the things you said about the Hollow over apple pie echoed pretty closely some of the things my mother said to me today. It all but eliminates any discomfort with what you may decide to write. But it makes me like you more, and that's a problem.\"\n\n\"You had to know, after what happened to us in the woods, I wouldn't be leaving.\"\n\n\"I guess I did.\" He pulled off into a short, steep driveway.\n\n\"Is this the house? It is perfect! Look at the stonework, and the big porch, the windows have shutters.\"\n\nThey were painted a deep blue that stood out well against the gray stone. The little front yard was bisected by a trio of concrete steps and the narrow walkway. A trim tree Quinn thought might be a dogwood highlighted the left square of front yard.\n\nAs Fox's truck pulled in behind, Quinn popped out to stand, hands on hips. \"Pretty damned adorable. Don't you think, Layla?\"\n\n\"Yes, but\u2014\"\n\n\"No buts, not yet. Let's take a look inside.\" She cocked her head at Cal. \"Okay, landlord?\"\n\nAs they trooped up to the porch, Cal took out the keys he'd grabbed off their hook from his father's home office. The ring was clearly labeled with the High Street address.\n\nThe fact that the door opened without a creak told Quinn the landlords were vigilant in the maintenance department.\n\nThe door opened straight into the living area that stood twice as long as it was wide, with the steps to the second floor a couple of strides in on the left. The wood floors showed wear, but were spotlessly clean. The air was chilly and carried the light sting of fresh paint.\n\nThe small brick fireplace delighted her.\n\n\"Could use your mother's eye in the paint department,\" Quinn commented.\n\n\"Rental properties get eggshell, through and through. It's the Hawkins's way. Tenants want to play around with that, it's their deal.\"\n\n\"Reasonable. I want to start at the top, work down. Layla, do you want to go up and fight over who gets which bedroom?\"\n\n\"No.\" Cal thought there was mutiny, as well as frustration on her face. \"I have a bedroom. In New York.\"\n\n\"You're not in New York,\" Quinn said simply, then dashed up the steps.\n\n\"She's not listening to me,\" Layla muttered. \"I don't seem to be listening to me either about going back.\"\n\n\"We're here.\" Fox gave a shrug. \"Might as well poke around. I really dig empty houses.\"\n\n\"I'll be up.\" Cal started up the stairs.\n\nHe found her in one of the bedrooms, one that faced the tiny backyard. She stood at the long, narrow window, the fingertips of her right hand pressed to the glass. \"I thought I'd go for one of the rooms facing the street, catch the who's going where when and with who. I usually go for that. Just have to know what's going on. But this is the one for me. I bet, in the daylight, you can stand here, see backyards, other houses, and wow, right on to the mountains.\"\n\n\"Do you always make up your mind so fast?\"\n\n\"Yeah, usually. Even when I surprise myself like now. Bathroom's nice, too.\" She turned enough to gesture to the door on the side of the room. \"And since it's girls, if any of us share that one, it won't be too weird having it link up the two bedrooms on this side.\"\n\n\"You're sure everyone will fall in line.\"\n\nNow she turned to him, fully. \"Confidence is the first step to getting what you want, or need. But we'll say I'm hoping Layla and Cyb will agree it's efficient, practical, and would be more comfortable to share the house for a few months than to bunk at the hotel. Especially considering the fact that both Layla and I are pretty well put off of the dining room there after Slugfest.\"\n\n\"You don't have any furniture.\"\n\n\"Flea markets. We'll pick up the essentials. Cal, I've stayed in less stellar accommodations and done it for one thing. A story. This is more. Somehow or other I'm connected to this story, this place. I can't turn that off and walk away.\"\n\nHe wished she could, and knew if she could his feelings for her wouldn't be as strong or as complex. \"Okay, but let's agree, here and now, that if you change your mind and do just that, no explanations needed.\"\n\n\"That's a deal. Now, let's talk rent. What's this place going to run us?\"\n\n\"You pay the utilities\u2014heat, electric, phone, cable.\"\n\n\"Naturally. And?\"\n\n\"That's it.\"\n\n\"What do you mean, that's it?\"\n\n\"I'm not going to charge you rent, not when you're staying here, at least in part, because of me. My family, my friends, my town. We're not going to make a profit off that.\"\n\n\"Straight arrow, aren't you, Caleb?\"\n\n\"About most.\"\n\n\"I'll make a profit\u2014she says optimistically\u2014from the book I intend to write.\"\n\n\"If we get through July and you write a book, you'll have earned it.\"\n\n\"Well, you drive a hard bargain, but it looks like we have a deal.\" She stepped forward, offered a hand.\n\nHe took it, then cupped his other at the back of her neck. Surprise danced in her eyes, but she didn't resist as he eased her toward him.\n\nHe moved slow, the closing together of bodies, the meeting of lips, the testing slide of tongues. There was no explosion of need as there had been in that moment in the clearing. No sudden, almost painful shock of desire. Instead, it was a long and gradual glide from interest to pleasure to ache while her head went light and her blood warmed. It seemed everything inside her went quiet so that she heard, very clearly, the low hum in her own throat as he changed the angle of the kiss.\n\nHe felt her give, degree by degree, even as he felt the hand he held in his go lax. The tension that had dogged him throughout the day drained away, so there was only the moment, the quiet, endless moment.\n\nEven when he drew back, that inner stillness held. And she opened her eyes, met his.\n\n\"That was just you and me.\"\n\n\"Yeah.\" He stroked his fingers over the back of her neck. \"Just you and me.\"\n\n\"I want to say that I have a policy against becoming romantically, intimately, or sexually\u2014just to cover all my bases\u2014involved with anyone directly associated with a story I'm researching.\"\n\n\"That's probably smart.\"\n\n\"I am smart. I also want to say I'm going to negate that policy in this particular case.\"\n\nHe smiled. \"Damn right you are.\"\n\n\"Cocky. Well, mixed with the straight arrow, I have to like it. Unfortunately, I should get back to the hotel. I have a lot of...things. Details to see to before I can move in here.\"\n\n\"Sure. I can wait.\"\n\nHe kept her hand in his, switching off the light as he led her out.\n\n## Eleven\n\nCAL SENT A DOZEN PINK ROSES TO HIS MOTHER. She liked the traditional flower for Valentine's Day, and he knew his father always went for the red. If he hadn't known, Amy Yost in the flower shop would have reminded him, as she did every blessed year.\n\n\"Your dad ordered a dozen red last week, for delivery today, potted geranium to his grandma, and he sent the Valentine's Day Sweetheart Special to your sisters.\"\n\n\"That suck-up,\" Cal said, knowing it would make Amy gasp and giggle. \"How about a dozen yellow for my gran. In a vase, Amy. I don't want her to have to fool with them.\"\n\n\"Aw, that's sweet. I've got Essie's address on file, you just fill out the card.\"\n\nHe picked one out of the slot, gave it a minute's thought before writing: Hearts are red, these roses are yellow. Happy Valentine's Day from your best fellow.\n\nCorny, sure, he decided, but Gran would love it.\n\nHe reached for his wallet to pay when he noticed the red-and-white-striped tulips behind the glass doors of the refrigerated display. \"Ah, those tulips are...interesting.\"\n\n\"Aren't they pretty? And they just make me feel like spring. It's no problem if you want to change either of the roses for them. I can just\u2014\"\n\n\"No, no, maybe...I'll take a dozen of them, too. Another delivery in a vase, Amy.\"\n\n\"Sure.\" Her cheerful round face lit up with curiosity and the anticipation of good gossip. \"Who's your valentine, Cal?\"\n\n\"It's more a housewarming kind of thing.\" He couldn't think of any reason why not to send Quinn flowers. Women liked flowers, he thought as he filled out the delivery form. It was Valentine's Day, and she was moving into the High Street house. It wasn't like he was buying her a ring and picking out a band for the wedding.\n\nIt was just a nice gesture.\n\n\"Quinn Black.\" Amy wiggled her eyebrows as she read the name on the form. \"Meg Stanley ran into her at the flea market yesterday, along with that friend of hers from New York. They bought a bunch of stuff, according to Meg. I heard you were going around with her.\"\n\n\"We're not...\" Were they? Either way, it was best to leave it alone. \"Well, what's the damage, Amy?\"\n\nWith his credit card still humming, he stepped outside, hunched his shoulders against the cold. There might be candy-striped tulips, but it didn't feel as if Mother Nature was giving so much as a passing thought to spring. The sky spat out a thin and bitter sleet that lay slick as grease on the streets and sidewalks.\n\nHe'd walked down from the bowling center as was his habit, timing his arrival at the florist to their ten o'clock opening. It was the best way to avoid the panicked rush of others who had waited until the last minute to do the Valentine's thing.\n\nIt didn't appear he'd needed to worry. Not only had no other customers come in while he'd been buying his roses and impulsive tulips, but there was no one on the sidewalks, no cars creeping cautiously toward the curb in front of the Flower Pot.\n\n\"Strange.\" His voice sounded hollow against the sizzle of sleet striking asphalt. Even on the crappiest day, he'd pass any number of people on his walks around town. He shoved his gloveless hands into his pockets and cursed himself for not breaking his routine and driving.\n\n\"Creatures of habit freeze their asses off,\" he muttered. He wanted to be inside in his office, drinking a cup of coffee, even preparing to start the cancellation process on the evening's scheduled Sweetheart Dance if the sleet worsened. If he'd just taken the damn truck, he'd already be there.\n\nSo thinking, he looked up toward the center, and saw the stoplight at the Town Square was out.\n\nPower down, Cal thought, and that was a problem. He quickened his steps. He knew Bill Turner would make certain the generator kicked on for the emergency power, but he needed to be there. School was out, and that meant kids were bound to be scattered around in the arcade.\n\nThe hissing of the sleet increased until it sounded like the forced march of an army of giant insects. Despite the slick sidewalk, Cal found himself breaking into a jog when it struck him.\n\nWhy weren't there any cars at the Square, or parked at the curbs? Why weren't there any cars anywhere?\n\nHe stopped, and so did the hiss of the sleet. In the ensuing silence, he heard his own heart thumping like a fist against steel.\n\nShe stood so close he might have reached out to touch her, and knew if he tried, his hand would pass through her as it would through water.\n\nHer hair was deep blond, worn long and loose as it had been when she'd carried the pails toward the little cabin in Hawkins Wood. When she'd sung about a garden green. But her body was slim and straight in a long gray dress.\n\nHe had the ridiculous thought that if he had to see a ghost, at least it wasn't a pregnant one.\n\nAs if she heard his thoughts, she smiled. \"I am not your fear, but you are my hope. You and those who make up the whole of you. What makes you, Caleb Hawkins, is of the past, the now, and the yet to come.\"\n\n\"Who are you? Are you Ann?\"\n\n\"I am what came before you, and you are formed through love. Know that, know that long, long before you came into the world, you were loved.\"\n\n\"Love isn't enough.\"\n\n\"No, but it is the rock on which all else stands. You have to look; you have to see. This is the time, Caleb. This was always to be the time.\"\n\n\"The time for what?\"\n\n\"The end of it. Seven times three. Death or life. He holds it, prevents it. Without his endless struggle, his sacrifice, his courage, all this...\" She held out her arms. \"All would be destroyed. Now it is for you.\"\n\n\"Just tell me what I need to do. Goddamn it.\"\n\n\"If I could. If I could spare you.\" She lifted a hand, let it fall again. \"There must be struggle, and sacrifice, and great courage. There must be faith. There must be love. It is courage, faith, love that holds it so long, that prevents it from taking all who live and breathe within this place. Now it is for you.\"\n\n\"We don't know how. We've tried.\"\n\n\"This is the time,\" she repeated. \"It is stronger, but so are you, and so are we. Use what you were given, take what it sowed but could never own. You cannot fail.\"\n\n\"Easy for you to say. You're dead.\"\n\n\"But you are not. They are not. Remember that.\"\n\nWhen she started to fade, he did reach out, uselessly. \"Wait, damn it. Wait. Who are you?\"\n\n\"Yours,\" she said. \"Yours as I am and always will be his.\"\n\nShe was gone, and the sleet sizzled on the pavement again. Cars rumbled by as the traffic light on the Square glowed green.\n\n\"Not the spot for daydreaming.\" Meg Stanley skidded by, giving him a wink as she pulled open the door of Ma's Pantry.\n\n\"No,\" Cal muttered. \"It's not.\"\n\nHe started toward the center again, then veered off to take a detour to High Street.\n\nQuinn's car was in the drive, and through the windows he could see the lights she must've turned on to chase back the gloom. He knocked, heard a muffled call to come in.\n\nWhen he did, he saw Quinn and Layla trying to muscle something that resembled a desk up the stairs.\n\n\"What are you doing? Jesus.\" He stepped over to grip the side of the desk beside Quinn. \"You're going to hurt yourselves.\"\n\nIn an annoyed move, she tossed her head to flip the hair away from her face. \"We're managing.\"\n\n\"You'll be managing a trip to the ER. Go on up, take that end with Layla.\"\n\n\"Then we'll both be walking backward. Why don't you take that end?\"\n\n\"Because I'm going to be taking the bulk of the weight this way.\"\n\n\"Oh.\" She let go, squeezed between the wall and the desk.\n\nHe didn't bother to ask why it had to go up. He'd lived with his mother too long to waste his breath. Instead he grunted out orders to prevent the edge of the desk from bashing into the wall as they angled left at the top of the stairs. Then followed Quinn as she directed the process to the window in the smallest bedroom.\n\n\"See, we were right.\" Quinn panted, and tugged down a Penn State sweatshirt. \"This is the spot for it.\"\n\nThere was a seventies chair that had seen better days, a pole lamp with a rosy glass shade that dripped long crystals, and a low bookshelf varnished black over decades that wobbled when he set a hand on it.\n\n\"I know, I know.\" Quinn waved away his baleful look. \"But it just needs a little hammering or something, and it's really just to fill things out. We were thinking about making it a little sitting room, then decided it would be better as a little office. Hence the desk we originally thought should be in the dining room.\"\n\n\"Okay.\"\n\n\"The lamp looks like something out of The Best Little Whorehouse in Texas.\" Layla gave one of the crystals a flick with her fingers. \"But that's what we like about it. The chair is hideous.\"\n\n\"But comfortable,\" Quinn inserted.\n\n\"But comfortable, and that's what throws are for.\"\n\nCal waited a beat as both of them looked at him expectantly. \"Okay,\" he repeated, which was generally how he handled his mother's decorating explanations.\n\n\"We've been busy. We turned in Layla's rental car, then hit the flea market just out of town. Bonanza. Plus we agreed no secondhand mattresses. The ones we ordered should be here this afternoon. Anyway, come see what we've got going so far.\"\n\nQuinn grabbed his hand, pulled him across the hall to the room she'd chosen. There was a long bureau desperately in need of refinishing, topped by a spotted mirror. Across the room was a boxy chest someone had painted a murderous and shiny red. On it stood a Wonder Woman lamp.\n\n\"Homey.\"\n\n\"It'll be very livable when we're done.\"\n\n\"Yeah. You know I think that lamp might've been my sister Jen's twenty, twenty-five years ago.\"\n\n\"It's classic,\" Quinn claimed. \"It's kitschy.\"\n\nHe fell back on the standard. \"Okay.\"\n\n\"I think I have Danish modern,\" Layla commented from the doorway. \"Or possibly Flemish. It's absolutely horrible. I have no idea why I bought it.\"\n\n\"Did you two haul this stuff up here?\"\n\n\"Please.\" Quinn tossed her head.\n\n\"We opted for brain over brawn.\"\n\n\"Every time. That and a small investment. Do you know how much a couple of teenage boys will cart and carry for twenty bucks each and the opportunity to ogle a couple of hot chicks such as we?\" Quinn fisted a hand on her hip, struck a pose.\n\n\"I'd've done it for ten. You could have called.\"\n\n\"Which was our intention, actually. But the boys were handy. Why don't we go down and sit on our new third-or fourthhand sofa?\"\n\n\"We did splurge,\" Layla added. \"We have an actual new coffeemaker and a very eclectic selection of coffee mugs.\"\n\n\"Coffee'd be good.\"\n\n\"I'll get it started.\"\n\nCal glanced after Layla. \"She seems to have done a one-eighty on all this.\"\n\n\"I'm persuasive. And you're generous. I think I should plant one on you for that.\"\n\n\"Go ahead. I can take it.\"\n\nLaughing, she braced her hands on his shoulders, gave him a firm, noisy kiss.\n\n\"Does that mean I don't get ten bucks?\"\n\nHer smile beamed as she poked him in the belly. \"You'll take the kiss and like it. Anyway, part of the reason for Layla hanging back was the money. The idea of staying was\u2014is\u2014difficult for her. But the idea of taking a long leave, unpaid, from her job, coming up with rent money here, keeping her place in New York, that was pretty much off the table.\"\n\nShe stepped up to the bright red chest to turn her Wonder Woman lamp on and off. From the look on her face, Cal could see the act pleased her.\n\n\"So, the rent-free aspect checked one problem off her list,\" Quinn went on. \"She hasn't completely committed. Right now, it's a day at a time for her.\"\n\n\"I've got something to tell you, both of you, that may make this her last day.\"\n\n\"Something happened.\" She dropped her hand, turned. \"What happened?\"\n\n\"I'll tell you both. I want to call Fox first, see if he can swing by. Then I can tell it once.\"\n\nHE HAD TO DO IT WITHOUT FOX, WHO, ACCORDING to Mrs. Hawbaker, was at the courthouse being a lawyer. So he sat in the oddly furnished living room on a couch so soft and saggy he was already wishing for the opportunity to get Quinn naked on it, and told them about the visitation on Main Street.\n\n\"An OOB,\" Quinn decided.\n\n\"An oob?\"\n\n\"No, no. Initials, like CYA. Out of body\u2014experience. It sounds like that might be what you had, or maybe there was a slight shift in dimensions and you were in an alternate Hawkins Hollow.\"\n\nHe might have spent two-thirds of his life caught up in something beyond rational belief, but he'd never heard another woman talk like Quinn Black. \"I was not in an alternate anything, and I was right inside my body where I belong.\"\n\n\"I've been studying, researching, and writing about the paranormal for some time now.\" Quinn drank some coffee and brooded over it.\n\n\"It could be he was talking to a ghost who caused the illusion that they were alone on the street, and caused everyone else out there to\u2014I don't know\u2014blip out for a few minutes.\" Layla shrugged at Quinn's narrowed look. \"I'm new at this, and I'm still working really hard not to hide under the covers until somebody wakes me up and tells me this was all a dream.\"\n\n\"For the new kid, your theory's pretty good,\" Quinn told her.\n\n\"How about mine? Which is what she said is a hell of a lot more important right now than how she said it.\"\n\n\"Point taken.\" Quinn nodded at Cal. \"This is the time, she said. Three times seven. That one's easy enough to figure.\"\n\n\"Twenty-one years.\" Cal pushed up to pace. \"This July makes twenty-one years.\"\n\n\"Three, like seven, is considered a magickal number. It sounds like she was telling you it was always going to come now, this July, this year. It's stronger, you're stronger, they're stronger.\" Quinn squeezed her eyes shut.\n\n\"So, it and this woman\u2014this spirit\u2014have both been able to...\"\n\n\"Manifest.\" Quinn finished Layla's thought. \"That follows the logic.\"\n\n\"Nothing about this is logical.\"\n\n\"It is, really.\" Opening her eyes again, Quinn gave Layla a sympathetic look. \"Inside this sphere, there's logic. It's just not the kind we deal with, or most of us deal with, every day. The past, the now, the yet to be. Things that happened, that are happening, and that will or may are all part of the solution, the way to end it.\"\n\n\"I think there's more to that part.\" Cal turned back from the window. \"After that night in the clearing, the three of us were different.\"\n\n\"You don't get sick, and you heal almost as soon as you're hurt. Quinn told me.\"\n\n\"Yeah. And I could see.\"\n\n\"Without your glasses.\"\n\n\"I could also see before. I started\u2014right there minutes afterward\u2014to have flashes of the past.\"\n\n\"The way you did\u2014both of us did,\" Quinn corrected, \"when we touched the stone together. And later, when we\u2014\"\n\n\"Like that, not always that clear, not always so intense. Sometimes awake, sometimes like a dream. Sometimes completely irrelevant. And Fox...It took him a while to understand. Jesus, we were ten. He can see now.\" Annoyed with himself, Cal shook his head. \"He can see, or sense what you're thinking, or feeling.\"\n\n\"Fox is psychic?\" Layla demanded.\n\n\"Psychic lawyer. He's so hired.\"\n\nDespite everything, Quinn's announcement made Cal's lips twitch. \"Not like that, not exactly. It's never been something we can completely control. Fox has to deliberately push it, and it doesn't always work then. But since then he has an instinct about people. And Gage\u2014\"\n\n\"He sees what could happen,\" Quinn added. \"He's the soothsayer.\"\n\n\"It's hardest for him. That's why\u2014one of the reasons why\u2014he doesn't spend much time here. It's harder here. He's had some pretty damn vicious dreams, visions, nightmares, whatever the hell you want to call them.\"\n\nAnd it hurts you when he hurts, Quinn thought. \"But he hasn't seen what you're meant to do?\"\n\n\"No. That would be too easy, wouldn't it?\" Cal said bitterly. \"Has to be more fun to mess up the lives of three kids, to let innocent people die or kill and maim each other. Stretch that out for a couple of decades, then say: Okay, boys, now's the time.\"\n\n\"Maybe there was no choice.\" Quinn held up a hand when Cal's eyes fired. \"I'm not saying it's fair. In fact, it sucks. Inside and out, it sucks. I'm saying maybe it couldn't be another way. Whether it was something Giles Dent did, or something set in motion centuries before that, there may have been no other choice. She said he was holding it, that he was preventing it from destroying the Hollow. If it was Ann, and she meant Giles Dent, does that mean he trapped this thing, this bestia, and in some form\u2014beatus\u2014has been trapped with it, battling it, all this time? Three hundred and fifty years and change. That sucks, too.\"\n\nLayla jumped at the brisk knock on the door, then popped up. \"I'll get it. Maybe it's the delivery.\"\n\n\"You're not wrong,\" Cal said quietly. \"But it doesn't make it easier to live through it. It doesn't make it easier to know, in my gut, that we're coming up to our last chance.\"\n\nQuinn got to her feet. \"I wish\u2014\"\n\n\"It's flowers!\" Layla's voice was giddy with delight as she came in carrying the vase of tulips. \"For you, Quinn.\"\n\n\"Jesus, talk about weird timing,\" Cal muttered.\n\n\"For me? Oh God, they look like lollipop cups. They're gorgeous!\" Quinn set them on the ancient coffee table. \"Must be a bribe from my editor so I'll finish that article on\u2014\" She broke off as she ripped open the card. Her face was blank with shock as she lifted her eyes to Cal. \"You sent me flowers?\"\n\n\"I was in the florist before\u2014\"\n\n\"You sent me flowers on Valentine's Day.\"\n\n\"I hear my mother calling,\" Layla announced. \"Coming, Mom!\" She made a fast exit.\n\n\"You sent me tulips that look like blooming candy canes on Valentine's Day.\"\n\n\"They looked like fun.\"\n\n\"That's what you wrote on the card. 'These look like fun.' Wow.\" She scooped a hand through her hair. \"I have to say that I'm a sensible woman, who knows very well Valentine's Day is a commercially generated holiday designed to sell greeting cards, flowers, and candy.\"\n\n\"Yeah, well.\" He slid his hands into his pockets. \"Works.\"\n\n\"And I'm not the type of woman who goes all mushy and gooey over flowers, or sees them as an apology for an argument, a prelude to sex, or any of the other oft-perceived uses.\"\n\n\"I just saw them, thought you'd get a kick out of them. Period. I've got to get to work.\"\n\n\"But,\" she continued and moved toward him, \"strangely, I find none of that applies in the least in this particular case. They are fun.\" She rose up on her toes, kissed his cheek. \"And they're beautiful.\" Then his other cheek. \"And thoughtful.\" Now his lips. \"Thank you.\"\n\n\"You're welcome.\"\n\n\"I'd like to add that...\" She trailed her hands down his shirt, up again. \"If you'll tell me what time you finish up tonight, I'll have a bottle of wine waiting in my bedroom upstairs, where I can promise you, you're going to get really, really lucky.\"\n\n\"Eleven,\" he said immediately. \"I can be here at eleven-oh-five. I\u2014Oh shit. Sweetheart Dance, that's midnight. Special event. No problem. You'll come.\"\n\n\"That's my plan.\" When he grinned, she rolled her eyes. \"You mean to this dance. At the Bowl-a-Rama. A Sweetheart Dance at the Bowl-a-Rama. God, I'd love that. But, I can't leave Layla here, not at night. Not alone.\"\n\n\"She can come, too\u2014to the dance.\"\n\nNow her eyeroll was absolutely sincere. \"Cal, no woman wants to tag along with a couple to a dance on Valentine's Day. It paints a big L for loser in the middle of her forehead, and they're so damn hard to wash off.\"\n\n\"Fox can take her. Probably. I'll check.\"\n\n\"That's a possibility, especially if we make it all for fun. You check, then I'll check, then we'll see. But either way.\" She grabbed a fistful of his shirt, and this time brought him to her for a long, long kiss. \"My bedroom, twelve-oh-five.\"\n\nLAYLA SAT ON HER BRAND-NEW DISCOUNT MATTRESS while Quinn busily checked out the clothes she'd recently hung in her closet.\n\n\"Quinn, I appreciate the thought, I really do, but put yourself in my place. The third-wheel position.\"\n\n\"It's perfectly acceptable to be the third wheel when there're four wheels altogether. Fox is going.\"\n\n\"Because Cal asked him to take pity on the poor dateless V-Day loser. Probably told him or bribed him or\u2014\"\n\n\"You're right. Fox certainly had to have his arm twisted to go out with such an ugly hag like yourself. I admit every time I look at you, I'm tempted to go: woof, woof, what a dog. Besides...Oh, I love this jacket! You have the best clothes. But this jacket is seriously awesome. Mmm.\" Quinn stroked it like a cat. \"Cashmere.\"\n\n\"I don't know why I packed it. I don't know why I packed half the stuff I did. I just started grabbing things. And you're trying to distract me.\"\n\n\"Not really, but it's a nice side benefit. What was I saying? Oh, yeah. Besides, it's not a date. It's a gang bang,\" she said and made Layla laugh. \"It's just the four of us going to a bowling alley, for God's sake, to hear some local band play and dance a little.\"\n\n\"Sure. After which, you'll be hanging a scarf over the doorknob of your bedroom. I went to college, Quinn. I had a roommate. Actually, I had a nympho of a roommate who had an endless supply of scarves.\"\n\n\"Is it a problem?\" Quinn stopped poking in the closet long enough to look over her shoulder. \"Cal and me, across the hall?\"\n\n\"No. No.\" And now didn't she feel stupid and petty? \"I think it's great. Really, I do. Anybody can tell the two of you rev like engines when you're within three feet of each other.\"\n\n\"They can?\" Quinn turned all the way around now. \"We do.\"\n\n\"Vroom, vroom. He's great, it's great. I just feel...\" Layla rolled her shoulders broadly. \"In the way.\"\n\n\"You're not. I couldn't stay here without you. I'm pretty steady, but I couldn't stay in this house alone. The dance isn't a big deal. We don't have to go, but I think it'd be fun, for all of us. And a chance to do something absolutely normal to take our minds off everything that isn't.\"\n\n\"That's a good point.\"\n\n\"So get dressed. Put on something fun, maybe a little sexy, and let's hit the Bowl-a-Rama.\"\n\nTHE BAND, A LOCAL GROUP NAMED HOLLOWED Out, was into its first set. They were popular at weddings and corporate functions, and regularly booked at the center's events because their playlist ran the gamut from old standards to hip-hop. The something-for-everybody kept the dance floor lively while those sitting one out could chat at one of the tables circling the room, sip drinks, or nibble from the light buffet set up along one of the side walls.\n\nCal figured it was one of the center's most popular annual events for good reason. His mother headed up the decorating committee, so there were flowers and candles, red and white streamers, glittering red hearts. It gave people a chance to get a little dressed up in the dullness that was February, get out and socialize, hear some music, show off their moves if they had them. Or like Cy Hudson, even if they didn't.\n\nIt was a little bright spot toward the end of a long winter, and they never failed to have a full house.\n\nCal danced with Essie to \"Fly Me to the Moon.\"\n\n\"Your mother was right to make you take those dance lessons.\"\n\n\"I was humiliated among my peers,\" Cal said. \"But light on my feet.\"\n\n\"Women tend to lose their heads over a good dancer.\"\n\n\"A fact I've exploited whenever possible.\" He smiled down at her. \"You look so pretty, Gran.\"\n\n\"I look dignified. Now, there was a day when I turned plenty of heads.\"\n\n\"You still turn mine.\"\n\n\"And you're still the sweetest of my sweethearts. When are you going to bring that pretty writer to see me?\"\n\n\"Soon, if that's what you want.\"\n\n\"It feels like time. I don't know why. And speaking of\u2014\" She nodded toward the open double doors. \"Those two turn heads.\"\n\nHe looked. He noticed Layla, in that she was there. But his focus was all for Quinn. She'd wound that mass of blond hair up, a touch of elegance, and wore an open black jacket over some kind of lacey top\u2014camisole, he remembered. They called them camisoles, and God bless whoever invented them.\n\nThings glittered at her ears, at her wrists, but all he could think was she had the sexiest collarbone in the history of collarbones, and he couldn't wait to get his mouth on it.\n\n\"You're about to drool, Caleb.\"\n\n\"What?\" He blinked his attention back to Essie. \"Oh. Jeez.\"\n\n\"She does look a picture. You take me on back to my table now and go get her. Bring her and her friend around to say hello before I leave.\"\n\nBy the time he got to them, Fox had already scooped them up to one of the portable bars and sprung for champagne. Quinn turned to Cal, glass in hand, and pitched her voice over the music. \"This is great! The band's hot, the bubbly's cold, and the room looks like a love affair.\"\n\n\"You were expecting a couple of toothless guys with a washboard and a jug, some hard cider, and a few plastic hearts.\"\n\n\"No.\" She laughed, jabbed him with her finger. \"But something between that and this. It's my first bowling alley dance, and I'm impressed. And look! Isn't that His Honor, the mayor, getting down?\"\n\n\"With his wife's cousin, who is the choir director for the First Methodist Church.\"\n\n\"Isn't that your assistant, Fox?\" Layla gestured to a table.\n\n\"Yeah. Fortunately, the guy she's kissing is her husband.\"\n\n\"They look completely in love.\"\n\n\"Guess they are. I don't know what I'm going to do without her. They're moving to Minneapolis in a couple months. I wish they'd just take off for a few weeks in July instead of\u2014\" He caught himself. \"No shop talk tonight. Do you want to scare up a table?\"\n\n\"Perfect for people-watching,\" Quinn agreed, then spun toward the band. \"'In the Mood'!\"\n\n\"Signature piece for them. Do you swing?\" Cal asked her.\n\n\"Damn right.\" She glanced at him, considered. \"Do you?\"\n\n\"Let's go see what you've got, Blondie.\" He grabbed her hand, pulled her out to the dance floor.\n\nFox watched the spins and footwork. \"I absolutely can't do that.\"\n\n\"Neither can I. Wow.\" Layla's eyes widened. \"They're really good.\"\n\nOn the dance floor, Cal set Quinn up for a double spin, whipped her back. \"Lessons?\"\n\n\"Four years. You?\"\n\n\"Three.\" When the song ended and bled into a slow number, he fit Quinn's body to his and blessed his mother. \"I'm glad you're here.\"\n\n\"Me, too.\" She nuzzled her cheek to his. \"Everything feels good tonight. Sweet and shiny. And mmm,\" she murmured when he led her into a stylish turn. \"Sexy.\" Tipping back her head, she smiled at him. \"I've completely reversed my cynical take on Valentine's Day. I now consider it the perfect holiday.\"\n\nHe brushed his lips over hers. \"After this dance, why don't we sneak off to the storeroom upstairs and neck?\"\n\n\"Why wait?\"\n\nWith a laugh, he started to bring her close again. And froze.\n\nThe hearts bled. The glittery art board dripped, and splattered red on the dance floor, plopped on tables, slid down the hair and faces of people while they laughed, or chatted, strolled or swayed.\n\n\"Quinn.\"\n\n\"I see it. Oh God.\"\n\nThe vocalist continued to sing of love and longing as the red and silver balloons overhead popped like gunshots. And from them rained spiders.\n\n## Twelve\n\nQUINN BARELY MANAGED TO MUFFLE A SCREAM, and would have danced back as the spiders skittered over the floor if Cal hadn't gripped her.\n\n\"Not real.\" He said it with absolute and icy calm. \"It's not real.\"\n\nSomeone laughed, and the sound spiked wildly. There were shouts of approval as the music changed tempo to hip-grinding rock.\n\n\"Great party, Cal!\" Amy from the flower shop danced by with a wide, blood-splattered grin.\n\nWith his arm still tight around Quinn, Cal began to back off the floor. He needed to see his family, needed to see...And there was Fox, gripping Layla's hand as he wound his way through the oblivious crowd.\n\n\"We need to go,\" Fox shouted.\n\n\"My parents\u2014\"\n\nFox shook his head. \"It's only happening because we're here. I think it only can happen because we're here. Let's move out. Let's move.\"\n\nAs they pushed between tables, the tiny tea lights in the centerpieces flashed like torches, belching a volcanic spew of smoke. Cal felt it in his throat, stinging, even as his foot crunched down on a fist-sized spider. On the little stage, the drummer swung into a wild solo with bloodied sticks. When they reached the doors, Cal glanced back.\n\nHe saw the boy floating above the dancers. Laughing.\n\n\"Straight out.\" Following Fox's line of thought, Cal pulled Quinn toward the exit. \"Straight out of the building. Then we'll see. Then we'll damn well see.\"\n\n\"They didn't see.\" Out of breath, Layla stumbled outside. \"Or feel. It wasn't happening for them.\"\n\n\"It's outside the box, okay, it's pushed outside the lines. But only for us.\" Fox stripped off his jacket and tossed it over Layla's shaking shoulders. \"Giving us a preview of coming attractions. Arrogant bastard.\"\n\n\"Yes.\" Quinn nodded, even as her stomach rolled. \"I think you're right, because every time it puts on a show, it costs energy. So we get that lull between production numbers.\"\n\n\"I have to go back.\" He'd left his family. Even if retreat was to defend, Cal couldn't stand and do nothing while his family was inside. \"I need to be in there, need to close down when the event's over.\"\n\n\"We'll all go back,\" Quinn linked her cold fingers with Cal's. \"These performances are always of pretty short duration. It lost its audience, and unless it's got enough for a second act, it's done for tonight. Let's go back. It's freezing out here.\"\n\nInside, the tea lights glimmered softly, and the hearts glittered. The polished dance floor was unstained. Cal saw his parents dancing, his mother's head resting on his father's shoulder. When she caught his eye and smiled at him, Cal felt the fist twisting in his belly relax.\n\n\"I don't know about you, but I'd really like another glass of champagne.\" Quinn blew out a breath, as her eyes went sharp and hard. \"Then you know what? Let's dance.\"\n\nFOX WAS SPRAWLED ON THE COUCH WATCHING some drowsy black-and-white movie on TV when Cal and Quinn came into the rental house after midnight. \"Layla went up,\" he said as he shoved himself to sitting. \"She was beat.\"\n\nThe subtext, that she'd wanted to be well tucked away before her housemate and Cal came up, was perfectly clear.\n\n\"Is she all right?\" Quinn asked.\n\n\"Yeah. Yeah, she handles herself. Anything else happen after we left?\"\n\nCal shook his head as his gaze tracked over to the window, and the dark. \"Just a big, happy party momentarily interrupted for some of us by supernatural blood and spiders. Everything okay here?\"\n\n\"Yeah, except for the fact these women buy Diet Pepsi. Classic Coke,\" he said to Quinn. \"A guy has to have some standards.\"\n\n\"We'll look right into that. Thanks, Fox.\" She stepped up and kissed his cheek. \"For hanging out until we got back.\"\n\n\"No big. It got me out of cleanup duty and let me watch...\" He looked back at the little TV screen. \"I have no idea. You ought to think about getting cable. ESPN.\"\n\n\"I don't know how I've lived without it these last few days.\"\n\nHe grinned as he pulled on his coat. \"Humankind shouldn't live by network alone. Call me if you need anything,\" he added as he headed for the door.\n\n\"Fox.\" Cal trailed behind him. After a murmured conversation, Fox sent Quinn a quick wave and left.\n\n\"What was that?\"\n\n\"I asked if he'd bunk at my place tonight, check on Lump. It's no problem. I've got Coke and ESPN.\"\n\n\"You've got worry all over you, Cal.\"\n\n\"I'm having a hard time taking it off.\"\n\n\"It can't hurt us, not yet. It's all head games. Mean, disgusting, but just psychological warfare.\"\n\n\"It means something, Quinn.\" He gave her arms a quick, almost absent rub before turning to check the dark, again. \"That it can do it now, with us. That I had that episode with Ann. It means something.\"\n\n\"And you have to think about it. You think a lot, have all sorts of stores up here.\" She tapped her temple. \"The fact that you do is, well, it's comforting to me and oddly attractive. But you know what? After this really long, strange day, it might be good for us not to think at all.\"\n\n\"That's a good idea.\" Take a break, he told himself. Take some normal. Walking back to her, he skimmed his fingers over her cheek, then let them trail down her arm until they linked with hers. \"Why don't we try that?\"\n\nHe drew her toward the steps, started up. There were a few homey creaks, the click and hum of the furnace, and nothing else.\n\n\"Do you\u2014\"\n\nHe cut her off by cupping a hand on her cheek, then laying his lips on hers. Soft and easy as a sigh. \"No questions either. Then we'd have to think of the answers.\"\n\n\"Good point.\"\n\nJust the room, the dark, the woman. That was all there would be, all he wanted for the night. Her scent, her skin, the fall of her hair, the sounds two people made when they discovered each other.\n\nIt was enough. It was more than enough.\n\nHe closed the door behind him.\n\n\"I like candles.\" She drew away to pick up a long, slim lighter to set the candles she'd scattered around the room to flame.\n\nIn their light she looked delicate, more delicate than she was. He enjoyed the contrast of reality and illusion. The mattress and box spring sat on the floor, covered by sheets that looked crisp and pearly against a blanket of deep, rich purple. His tulips sat like a cheerful carnival on the scarred wood of her flea market dresser.\n\nShe'd hung fabric in a blurry blend of colors over the windows to close out the night. And when she turned from them, she smiled.\n\nIt was, for him, perfect.\n\n\"Maybe I should tell you\u2014\"\n\nHe shook his head, stepped toward her.\n\n\"Later.\" He did the first thing that came to mind, lifting his hands to her hair. He drew the pins out, let them fall. When the weight of it tumbled free, over her shoulders, down her back, he combed his fingers through it. With his eyes on hers, he wrapped her hair around his fist like a rope, gave a tug.\n\n\"There's still a lot of later,\" he said, and took her mouth with his.\n\nHer lips, for him, were perfect. Soft and full, warm and generous. He felt a quick tremble from her as her arms wound around him, as she pressed her body to his. She didn't yield, didn't soften\u2014not yet. Instead she met his slow, patient assault with one of her own.\n\nHe slid the jacket from her shoulders, let it fall like the pins so his hands, his fingertips could explore silk and lace and flesh. While their lips brushed, rubbed, pressed, her hands came to his shoulders, then shoved at his jacket until it dropped away.\n\nHe tasted her throat, heard her purr of approval. As he eased back, he danced his fingers over the alluring line of her collarbone. Her eyes were vivid, alight with anticipation. He wanted to see them heavy. He wanted to see them go blind. Watching them, watching her, he let his fingers trail down to the swell of her breast where the lace flirted. And watching her still, glided them over the lace, over the silk to cup her while his thumb lightly rubbed, rubbed to tease her nipple.\n\nHe heard her breath catch, release, felt her shiver even as she reached to him to unbutton his shirt. Her hands slid up his torso, spread. He knew his heartbeat skipped, but his own hand made the journey almost lazily to the waistband of her pants. The flesh there was warm, and her muscles quivered as his fingers did a testing sweep. Then with a flick and a tug, her pants floated down her legs.\n\nThe move was so sudden, so unexpected, she couldn't anticipate or prepare. Everything had been so slow, so dreamy, then his hands hooked under her arms, lifted her straight off her feet. The quick, careless show of strength shocked her system, made her head swim. Even when he set her back down, her knees stayed weak.\n\nHis gaze skimmed down, over the camisole, over the frothy underwear she'd donned with the idea of making him crazy. His lips curved as his eyes came back to hers.\n\n\"Nice.\"\n\nIt was all he said, and her mouth went dry. It was ridiculous. She'd had other men look at her, touch her, want her. But he did, and her throat went dry. She tried to find something clever and careless to say back, but could barely find the wit to breathe.\n\nThen he hooked his finger in the waist of her panties, gave one easy tug. She stepped toward him like a woman under a spell.\n\n\"Let's see what's under here,\" he murmured, and lifted the camisole over her head. \"Very nice,\" was his comment as he traced his fingertip along the edge of her bra.\n\nShe couldn't remember her moves, had to remind herself she was good at this\u2014actively good, not just the type who went limp and let a guy do all the work. She reached for the hook of his trousers, fumbled.\n\n\"You're shaking.\"\n\n\"Shut up. I feel like an idiot.\"\n\nHe took her hands, brought them both to his lips and she knew she was as sunk as the Titanic. \"Sexy,\" he corrected. \"What you are is stupendously sexy.\"\n\n\"Cal.\" She had to concentrate to form the words. \"I really need to lie down.\"\n\nThere was that smile again, and though it might have transmitted self-satisfied male, she really didn't give a damn.\n\nThen they were on the bed, aroused bodies on cool, crisp sheets, candlelight flickering like magic in the dark. And his hands, his mouth, went to work on her.\n\nHe runs a bowling alley, she thought as he simply saturated her with pleasure. How did he get hands like this? Where did he learn to...Oh my God.\n\nShe came in a long, rolling wave that seemed to curl up from her toes, ride over her legs, burst in her center then wash over heart and mind. She clung to it, greedily wringing every drop of shock and delight until she was both limp and breathless.\n\nOkay, okay, was all her brain could manage. Okay, wow.\n\nHer body was a feast of curves and quivers. He could have lingered over those lovely breasts, the strong line of torso, that feminine flare of hip for days. Then there were her legs, smooth and strong and...sensitive. So many places to touch, so much to taste, and all the endless night to savor.\n\nShe rose to him, wrapped around him, arched and flowed and answered. He felt her heart thundering under his lips, heard her moan as he used his tongue to torment. Her fingers dug into his shoulders, his hips, her hands squeezing then gliding to fray the taut line of his control.\n\nKisses became more urgent. The cool air of the room went hot, went thick as smoke. When the need became a blur, he slipped inside her. And yes, watched her eyes go blind.\n\nHe gripped her hands to anchor himself, to stop himself from simply plunging, from bulleting by the aching pleasure to release. Her fingers tightened on his, and that pleasure glowed on her face with each long, slow thrust. Stay with me, he thought, and she did, beat for beat. Until it built and built in her ragged breaths, in the shivering of her body. She made a helpless sound as she closed her eyes, turned her head on the pillow. When her body melted under him, he pressed his face to that exposed curve of her neck. And let himself go.\n\nHE LAY QUIET, THINKING SHE MIGHT HAVE FALLEN asleep. She'd rolled so that her head was on his shoulder, her arm tossed across his chest, and her leg hooked around his. It was, he thought, a little like being tied up with a Quinn bow. And he couldn't find anything not to like about it.\n\n\"I was going to say something.\"\n\nNot asleep, he realized, though her words were drunk and slurry.\n\n\"About what?\"\n\n\"Mmm. I was going to say, when we first came into the room. I was going to say something.\" She curled closer, and he realized the heat sex had generated had ebbed, and she was cold.\n\n\"Hold on.\" He had to unwind her, to which she gave a couple of halfhearted mutters of protest. But when he pulled up the blanket, she snuggled right in. \"Better?\"\n\n\"Couldn't be any. I was going to say that I've been\u2014more or less\u2014thinking about getting you naked since I met you.\"\n\n\"That's funny. I've been more or less thinking the same about you. You've got an amazing body there, Quinn.\"\n\n\"Lifestyle change, for which I could now preach like an evangelist. However.\" She levered up so she could look down into his face. \"Had I known what it would be like, I would've had you naked in five minutes flat.\"\n\nHe grinned. \"Once again, our thoughts run on parallel lines. Do that thing again. No,\" he said with a laugh when her eyebrows wiggled. \"This thing.\"\n\nHe tugged her head down again until it rested on his shoulder, then drew her arm over his chest. \"And the leg. That's it,\" he said when she obliged. \"That's perfect.\"\n\nThe fact that it was gave her a nice warm glow under her heart. Quinn closed her eyes, and without a worry in the world, drifted off to sleep.\n\nIN THE DARK, SHE WOKE WHEN SOMETHING FELL on her. She managed a breathless squeal, shoved herself to sitting, balled her hands into fists.\n\n\"Sorry, sorry.\"\n\nShe recognized Cal's whisper, but it was too late to stop the punch. Her fist jabbed into something hard enough to sting her knuckles. \"Ow! Ow! Shit.\"\n\n\"I'll say,\" Cal muttered.\n\n\"What the hell are you doing?\"\n\n\"Tripping, falling down, and getting punched in the head.\"\n\n\"Why?\"\n\n\"Because it's pitch-dark.\" He shifted, rubbed his sore temple. \"And I was trying not to wake you up, and you hit me. In the head.\"\n\n\"Well, I'm sorry,\" she hissed right back. \"For all I knew you could've been a mad rapist, or more likely, given the location, a demon from hell. What are you doing milling around in the dark?\"\n\n\"Trying to find my shoes, which I think is what I tripped over.\"\n\n\"You're leaving?\"\n\n\"It's morning, and I've got a breakfast meeting in a couple hours.\"\n\n\"It's dark.\"\n\n\"It's February, and you've got those curtain deals over the windows. It's about six thirty.\"\n\n\"Oh God.\" She plopped back down. \"Six thirty isn't morning, even in February. Or maybe especially.\"\n\n\"Which is why I was trying not to wake you up.\"\n\nShe shifted. She could make him out now, a little, as her eyes adjusted. \"Well, I'm awake, so why are you still whispering?\"\n\n\"I don't know. Maybe I have brain damage from getting punched in the head.\"\n\nSomething about the baffled irritation in his voice stirred her juices. \"Aw. Why don't you crawl back in here with me where it's all nice and warm? I'll kiss it and make it better.\"\n\n\"That's a cruel thing to suggest when I have a breakfast meeting with the mayor, the town manager, and the town council.\"\n\n\"Sex and politics go together like peanut butter and jelly.\"\n\n\"That may be, but I've got to go home, feed Lump, drag Fox out of bed as he's in on this meeting. Shower, shave, and change so it doesn't look like I've been having hot sex.\"\n\nAs he dragged on his shoes, she roused herself to push up again, then slither around him. \"You could do all that after.\"\n\nHer breasts, warm and full, pressed against his back as she nibbled on the side of his throat. And her hand snuck down to where he'd already gone rock hard.\n\n\"You've got a mean streak, Blondie.\"\n\n\"Maybe you ought to teach me a lesson.\" She let out a choked laugh when he swiveled and grabbed her.\n\nThis time when he fell on her, it was on purpose.\n\nHE WAS LATE FOR THE MEETING, BUT HE WAS feeling too damn good to care. He ordered an enormous breakfast\u2014eggs, bacon, hash browns, two biscuits. He worked his way through it while Fox gulped down Coke as if it were the antidote to some rare and fatal poison in his bloodstream, and the others engaged in small talk.\n\nSmall talk edged into town business. It may have been February, but plans for the annual Memorial Day parade had to be finalized. Then there was the debate about installing new benches in the park. Most of it washed over Cal as he ate, as he thought about Quinn.\n\nHe tuned back in, primarily because Fox kicked him under the table.\n\n\"The Branson place is only a couple doors down from the Bowl-a-Rama,\" Mayor Watson continued. \"Misty said it looked like the house on either side went dark, too, but across the street, the lights were on. Phones went out, too. Spooked her pretty good, she said when Wendy and I picked her up after the dance. Only lasted a few minutes.\"\n\n\"Maybe a breaker,\" Jim Hawkins suggested, but he looked at his son.\n\n\"Maybe, but Misty said it all flickered and snapped for a few seconds. Power surge maybe. But I think I'm going to urge Mike Branson to get his wiring checked out. Could be something's shorting out. We don't want an electrical fire.\"\n\nHow did they manage to forget? Cal wondered. Was it a defense mechanism, amnesia, or simply part of the whole ugly situation?\n\nNot all of them. He could see the question, the concern in his father's eyes, in one or two of the others. But the mayor and most of the council were moving on to a discussion of painting the bleachers in the ballpark before Little League season began.\n\nThere had been other odd power surges, other strange power outages. But never until June, never before that final countdown to the Seven.\n\nWhen the meeting was over, Fox walked to the bowling center with Cal and his father. They didn't speak until they were inside, and the door closed behind them.\n\n\"It's too early for this to happen,\" Jim said immediately. \"It's more likely a power surge, or faulty wiring.\"\n\n\"It's not. Things have been happening already,\" Cal told him. \"And it's not just Fox and I who've seen them. Not this time.\"\n\n\"Well.\" Jim sat down heavily at one of the tables in the grill section. \"What can I do?\"\n\nTake care of yourself, Cal thought. Take care of Mom. But it would never be enough. \"Anything feels off, you tell me. Tell Fox, or Gage when he gets here. There are more of us this time. Quinn and Layla, they're part of it. We need to figure out how and why.\"\n\nHis great-grandmother had known Quinn was connected, Cal thought. She'd sensed something. \"I need to talk to Gran.\"\n\n\"Cal, she's ninety-seven. I don't care how spry she is, she's still ninety-seven.\"\n\n\"I'll be careful.\"\n\n\"You know, I'm going to talk to Mrs. H again.\" Fox shook his head. \"She's jumpy, nervous. Making noises about leaving next month instead of April. I figured it was just restlessness now that she's decided to move. Maybe it's more.\"\n\n\"All right.\" Jim blew out a breath. \"You two go do what you need to do. I'll handle things here. I know how to run the center,\" he said before Cal could protest. \"Been doing it awhile now.\"\n\n\"Okay. I'll run Gran to the library if she wants to go today. I'll be back after, and we can switch off. You can pick her up, take her home.\"\n\nCAL WALKED TO ESSIE'S HOUSE. SHE ONLY LIVED a block away in the pretty little house she shared with his cousin Ginger. Essie's concession to her age was to have Ginger live in, take care of the house, the grocery shopping, most of the cooking, and be her chauffeur for duties like doctor and dentist appointments.\n\nCal knew Ginger to be a sturdy, practical sort who stayed out of his gran's way\u2014and her business\u2014unless she needed to do otherwise. Ginger preferred TV to books, and lived for a trio of afternoon soaps. Her disastrous and childless marriage had turned her off men, except television beefcake or those within the covers of People magazine.\n\nAs far as Cal could tell, his gran and his cousin bumped along well enough in the little dollhouse with its trim front yard and cheerful blue porch.\n\nWhen he arrived he didn't see Ginger's car at the curb, and wondered if his gran had an early medical appointment. His father kept Essie's schedule in his head, as he kept so much else, but he'd been upset that morning.\n\nStill, it was more likely that Ginger had taken a run to the grocery store.\n\nHe crossed the porch and knocked. It didn't surprise him when the door opened. Even upset, his father rarely forgot anything.\n\nBut it did surprise him to see Quinn at the threshold.\n\n\"Hi. Come on in. Essie and I are just having some tea in the parlor.\"\n\nHe gripped her arm. \"Why are you here?\"\n\nThe greeting smile faded at the sharp tone. \"I have a job to do. And Essie called me.\"\n\n\"Why?\"\n\n\"Maybe if you come in instead of scowling at me, we'll both find out.\"\n\nSeeing no other choice, Cal walked into his great-grandmother's lovely living room where African violets bloomed in purple profusion in the windows, where built-in shelves Fox's father had crafted were filled with books, family pictures, little bits and bobs of memories. Where the company tea set was laid out on the low table in front of the high-backed sofa his mother had reupholstered only the previous spring.\n\nWhere his beloved gran sat like a queen in her favored wingback chair. \"Cal.\" She lifted her hand for his, and her cheek for his kiss. \"I thought you'd be tied up all morning between the meeting and center business.\"\n\n\"Meeting's over, and Dad's at the center. I didn't see Ginger's car.\"\n\n\"She's off running some errands since I had company. Quinn's just pouring the tea. Go get yourself a cup out of the cupboard.\"\n\n\"No, thanks. I'm fine. Just had breakfast.\"\n\n\"I would've called you, too, if I'd realized you'd have time this morning.\"\n\n\"I've always got time for you, Gran.\"\n\n\"He's my boy,\" she said to Quinn, squeezing Cal's hand before she released it to take the tea Quinn offered. \"Thank you. Please, sit down, both of you. I might as well get right to it. I need to ask you if there was an incident last night, during the dance. An incident just before ten.\"\n\nShe looked hard at Cal's face as she asked, and what she saw had her closing her eyes. \"So there was.\" Her thin voice quivered. \"I don't know whether to be relieved or afraid. Relieved because I thought I might be losing my mind. Afraid because I'm not. It was real then,\" she said quietly. \"What I saw.\"\n\n\"What did you see?\"\n\n\"It was as if I were behind a curtain. As if a curtain had dropped, or a shroud, and I had to look through it. I thought it was blood, but no one seemed to notice. No one noticed all the blood, or the things that crawled and clattered over the floor, over the tables.\" Her hand lifted to rub at her throat. \"I couldn't see clearly, but I saw a shape, a black shape. It seemed to float in the air on the other side of the curtain. I thought it was death.\"\n\nShe smiled a little as she lifted her tea with a steady hand. \"You prepare for death at my age, or you damn well should. But I was afraid of that shape. Then it was gone, the curtain lifted again, and everything was exactly as it should be.\"\n\n\"Gran\u2014\"\n\n\"Why didn't I tell you last night?\" she interrupted. \"I can read your face like a book, Caleb. Pride, fear. I simply wanted to get out, to be home, and your father drove me. I needed to sleep, and I did. This morning, I needed to know if it was true.\"\n\n\"Mrs. Hawkins\u2014\"\n\n\"You'll call me Essie now,\" she said to Quinn.\n\n\"Essie, have you ever had an experience like this before?\"\n\n\"Yes. I didn't tell you,\" she said when Cal cursed. \"Or anyone. It was the summer you were ten. That first summer. I saw terrible things outside the house, things that couldn't be. That black shape that was sometimes a man, sometimes a dog. Or a hideous combination of both. Your grandfather didn't see, or wouldn't. I always thought he simply wouldn't see. There were horrible things that week.\"\n\nShe closed her eyes a moment, then took another soothing sip of tea. \"Neighbors, friends. Things they did to themselves and each other. After the second night, you came to the door. Do you remember, Cal?\"\n\n\"Yes, ma'am, I remember.\"\n\n\"Ten years old.\" She smiled at Quinn. \"He was only a little boy, with his two young friends. They were so afraid. You could see and feel the fear and the, valor, I want to say, coming off them like light. You told me we had to pack up, your grandfather and I. We had to come stay at your house. That it wasn't safe in town. Didn't you ever wonder why I didn't argue, or pat you on the head and shoo you on home?\"\n\n\"No. I guess there was too much else going on. I just wanted you and Pop safe.\"\n\n\"And every seven years, I packed for your grandfather and me, then when he died, just for me, now this year it'll be Ginger and me. But it's coming sooner and stronger this time.\"\n\n\"I'll pack for you, Gran, for you and Ginger right now.\"\n\n\"Oh, I think we're safe enough for now,\" she said to Cal. \"When it's time, Ginger and I can put what we'll need together. I want you to take the books. I know I've read them, you've read them. It seems countless times. But we've missed something, somehow. And now, we have fresh eyes.\"\n\nQuinn turned toward Cal, narrowed her eyes. \"Books?\"\n\n## Thirteen\n\nFOX MADE A RUN TO THE BANK. IT WAS COMPLETELY unnecessary since the papers in his briefcase could have been dropped off at any time\u2014or more efficiently, the client could have come into his office to ink them.\n\nBut he'd wanted to get out, get some air, walk off his frustration.\n\nIt was time to admit that he'd still held on to the hope that Alice Hawbaker would change her mind, or that he could change it for her. Maybe it was selfish, and so what? He depended on her, he was used to her. And he loved her.\n\nThe love meant he had no choice but to let her go. The love meant if he could take back the last twenty minutes he'd spent with her, he would.\n\nShe'd nearly broken down, he remembered as he strode along in his worn-down hiking boots (no court today). She never broke. She never even cracked, but he'd pushed her hard enough to cause fissures. He'd always regret it.\n\nIf we stay, we'll die. She'd said that with tears in her voice, with tears glimmering in her eyes.\n\nHe'd only wanted to know why she was so set to leave, why she was jumpier every day to the point she wanted to go sooner than originally planned.\n\nSo he'd pushed. And finally, she'd told him.\n\nShe'd seen their deaths, over and over, every time she closed her eyes. She'd seen herself getting her husband's deer rifle out of the locked case in his basement workroom. Seen herself calmly loading it. She'd watched herself walk upstairs, through the kitchen where the dinner dishes were loaded into the dishwasher, the counters wiped clean. Into the den where the man she'd loved for thirty-six years, had made three children with, was watching the Orioles battle the Red Sox. The O's were up two-zip, but the Sox were at the plate, with a man on second, one out. Top of the sixth. The count was one and two.\n\nWhen the pitcher wound up, she pumped a bullet through the back of her husband's head as he sat in his favorite recliner.\n\nThen she'd put the barrel under her own chin.\n\nSo, yes, he had to let her go, just as he'd had to make an excuse to leave the office because he knew her well enough to understand she didn't want him around until she was composed again.\n\nKnowing he'd given her what she wanted and needed didn't stop him from feeling guilty, frustrated, and inadequate.\n\nHe ducked in to buy flowers. She'd accept them as a peace offering, he knew. She liked flowers in the office, and often picked them up herself as he tended to forget.\n\nHe came out with an armload of mixed blooms, and nearly ran over Layla.\n\nShe stumbled back, even took a couple extra steps in retreat. He saw upset and unhappiness on her face, and wondered if it was his current lot to make women nervous and miserable.\n\n\"Sorry. Wasn't looking.\"\n\nShe didn't smile, just started fiddling with the buttons of her coat. \"It's okay. Neither was I.\"\n\nHe should just go. He didn't have to tap in to her mind to feel the jangle of nerves and misery surrounding her. It seemed to him she never relaxed around him, was always making that little move away. Or maybe she never relaxed ever. Could be a New York thing, he mused. He sure as hell hadn't been able to relax there.\n\nBut there was too much of the how-can-I-fix-this in him. \"Problem?\"\n\nNow her eyes glimmered with tears, and Fox quite simply wanted to step into the street into the path of a passing truck.\n\n\"Problem? How could there possibly be a problem? I'm living in a strange house in a strange town, seeing things that aren't there\u2014or worse, are there and want me dead. Nearly everything I own is sitting in my apartment in New York. An apartment I have to pay for, and my very understanding and patient boss called this morning to tell me, regretfully, that if I couldn't come back to work next week, she'll have to replace me. So do you know what I did?\"\n\n\"No.\"\n\n\"I started to pack. Sorry, really, sorry, but I've got a life here. I have responsibilities and bills and a goddamn routine.\" She gripped her elbows in opposite hands as if to hold herself in place. \"I need to get back to them. And I couldn't. I just couldn't do it. I don't even know why, not on any reasonable level, but I couldn't. So now I'm going to be out of a job, which means I won't be able to afford my apartment. And I'm probably going to end up dead or institutionalized, and that's after my landlord sues me for back rent. So problems? No, not me.\"\n\nHe listened all the way through without interruption, then just nodded. \"Stupid question. Here.\" He shoved the flowers at her.\n\n\"What?\"\n\n\"You look like you could use them.\"\n\nFlummoxed, she stared at him, stared at the colorful blooms in her arms. And felt the sharpest edge of what might have been hysteria dulling into perplexity. \"But...you bought them for someone.\"\n\n\"I can buy more.\" He waved a thumb at the door of the flower shop. \"And I can help with the landlord if you get me the information. The rest, well, we're working on it. Maybe something pushed you to come here, and maybe something's pushing you to stay, but at the bottom of it, Layla, it's your choice. If you decide you have to leave...\" He thought of Alice again, and some of his own frustration ebbed. \"Nobody's going to blame you for it. But if you stay, you need to commit.\"\n\n\"I've\u2014\"\n\n\"No, you haven't.\" Absently, he reached out to secure the strap of her bag, which had slipped down to the crook of her elbow, back on her shoulder. \"You're still looking for the way out, the loophole in the deal that means you can pack your bags and go without consequences. Just go back to the way things were. Can't blame you for it. But choose, then stick. That's all. I've got to finish up and get back. Talk to you later.\"\n\nHe stepped back into the florist and left her standing speechless on the sidewalk.\n\nQUINN SHOUTED DOWN FROM THE SECOND FLOOR when Layla came in.\n\n\"It's me,\" Layla called up, and still conflicted, walked back to the kitchen with the flowers and the bottles and pots she'd bought in a gift shop on the walk home.\n\n\"Coffee.\" Quinn bustled in a few moments later. \"Going to need lots and lots of...Hey, pretty,\" she said when she saw the flowers Layla was clipping to size and arranging in various bottles.\n\n\"They really are. Quinn, I need to talk to you.\"\n\n\"Need to talk to you, too. You go first.\"\n\n\"I was going to leave this morning.\"\n\nQuinn stopped on the way to fill the coffeepot. \"Oh.\"\n\n\"And I was going to do my best to get out before you came back, and talked me out of it. I'm sorry.\"\n\n\"Okay. It's okay.\" Quinn busied herself making the coffee. \"I'd avoid me, too, if I wanted to do something I didn't want me to do. If you get me.\"\n\n\"Oddly enough, I do.\"\n\n\"Why aren't you gone?\"\n\n\"Let me backtrack.\" While she finished fussing with the flowers, Layla related the telephone conversation she'd had with her boss.\n\n\"I'm sorry. It's so unfair. I don't mean your boss is unfair. She's got a business to run. But that this whole thing is unfair.\" Quinn watched Layla arrange multicolored daisies in an oversized teacup. \"On a practical level I'm okay, because this is my job, or the job I picked. I can afford to take the time to be here and supplement that with articles. I could help\u2014\"\n\n\"That's not what I'm looking for. I don't want you to loan me money, or to carry my share of the expenses. If I stay, it's because I've chosen to stay.\" Layla looked at the flowers, thought of what Fox had said. \"I think, until today, I didn't accept that, or want to accept it. Easier to think I'd been driven to come here, and that I was being pressured to stay. I wanted to go because I didn't want any of this to be happening. But it is. So I'm staying because I've decided to stay. I'll just have to figure out the practicalities.\"\n\n\"I've got a couple of ideas on that, maybe just a thumb in the dike. Let me think about them. The flowers were a nice idea. Cheer up a bad news day.\"\n\n\"Not my idea. Fox gave them to me when I ran into him outside the florist. I cut loose on him.\" Layla shrugged, then gathered up the bits of stems she'd cut off, the florist wrappings. \"He's basically, 'How are you doing,' and I'm 'How am I doing? I'll tell you how I'm doing.'\" She tossed the leavings in the trash, then leaned back and laughed. \"God, I just blasted him. So he gives me the flowers he'd just bought, thrust them at me, really, and gave me a short, pithy lecture. I guess I deserved it.\"\n\n\"Hmm.\" Quinn added the information to the think-pot she was stirring. \"And you feel better?\"\n\n\"Better?\" Layla walked into the little dining room to arrange a trio of flowers on the old, drop-leaf table they'd picked up at the flea market. \"I feel more resolved. I don't know if that's better.\"\n\n\"I've got something to keep you busy.\"\n\n\"Thank God. I'm used to working, and all this time on my hands makes me bitchy.\"\n\n\"Come with me. Don't leave all the flowers; you should have some of them in your room.\"\n\n\"I thought they'd be for the house. He didn't buy them for me or\u2014\"\n\n\"He gave them to you. Take some of them up. You made me take the tulips up to mine.\" To solve the matter, Quinn picked up one of the little pots and a slender bottle herself. \"Oh, coffee.\"\n\n\"I'll get it.\" Layla poured one of the mugs for Quinn, doctored it, then got a bottle of water for herself. \"What's the project that's going to keep me busy?\"\n\n\"Books.\"\n\n\"We already have the books from the library.\"\n\n\"Now we have some from Estelle Hawkins's personal store. Some of them are journals. I haven't really scratched the surface yet,\" Quinn explained as they headed up. \"I'd barely gotten home ahead of you. But there are three of them written by Ann Hawkins. After her children were born. Her children with Giles Dent.\"\n\n\"But Mrs. Hawkins must have read them before, shown them to Cal.\"\n\n\"Right, and right. They've all been read, studied, pondered over. But not by us, Layla. Fresh eyes, different angle.\" She detoured to Layla's room to set the flowers down, then took the coffee mug on her way to the office. \"And I've already got the first question on my notes: Where are the others?\"\n\n\"Other journals?\"\n\n\"Ann's other journals, because I'm betting there are more, or were. Where's the journal she kept when she lived with Dent, when she was carrying her triplets? That's one of the new angles I hope our fresh eyes can find. Where would they be, and why aren't they with the others?\"\n\n\"If she did write others, they might have been lost or destroyed.\"\n\n\"Let's hope not.\" Quinn's eyes were sharp as she sat, lifted a small book bound in brown leather. \"Because I think she had some of the answers we need.\"\n\nCAL COULDN'T REASONABLY BREAK AWAY FROM the center until after seven. Even then he felt guilty leaving his father to handle the rest of the night. He'd called Quinn in the late afternoon to let her know he'd be by when he could. And her absent response had been for him to bring food with him.\n\nShe'd have to settle for pizza, he thought as he carried the takeout boxes up the steps. He hadn't had the time or inclination to figure out what her lifestyle-change option might be.\n\nAs he knocked, the wind whistled across the back of his neck, had him glancing uneasily behind him. Something coming, he thought. Something's in the wind.\n\nFox answered the door. \"Thank God, pizza and a testosterone carrier. I'm outnumbered here, buddy.\"\n\n\"Where's the estrogen?\"\n\n\"Up. Buried in books and notes. Charts. Layla makes charts. I made the mistake of telling them I had a dry-wipe board down at the office. They made me go get it, haul it in here, haul it upstairs.\" The minute Cal set the pizza down on the kitchen counter, Fox shoved up the lid and took out a slice. \"There's been talk of index cards. Colored index cards. Don't leave me here alone again.\"\n\nCal grunted, opened the fridge, and found, as he'd hoped and dreamed, Fox had stocked beer. \"Maybe we were never organized enough, so we missed some detail. Maybe\u2014\"\n\nHe broke off as Quinn rushed in. \"Hi! Pizza. Oh-oh. Well, I'll work it off with the power of my mind and with a session in the gym tomorrow morning.\"\n\nShe got down plates, passed one to Fox, who was already halfway through with his first slice. Then she smiled that smile at Cal. \"Got anything else for me?\"\n\nHe leaned right in, laid his mouth on hers. \"Got that.\"\n\n\"Coincidentally, exactly what I wanted. So how about some more.\" She got a fistful of his shirt and tugged him down for another, longer kiss.\n\n\"You guys want me to leave? Can I take the pizza with me?\"\n\n\"As a matter of fact,\" Cal began.\n\n\"Now, now.\" Quinn patted Cal's chest to ease him back. \"Mommy and Daddy were just saying hello,\" she told Fox. \"Why don't we eat in the dining room like the civilized. Layla's coming right down.\"\n\n\"How come I can't say hello to Mommy?\" Fox complained as Quinn sailed off with the plates.\n\n\"Because then I'd have to beat you unconscious.\"\n\n\"As if.\" Amused, Fox grabbed the pizza boxes and started after Quinn. \"Beverages on you, bro.\"\n\nShortly after they were seated, drinks, plates, napkins, pizza passed around, Layla came in with a large bowl and a stack of smaller ones. \"I put this together earlier. I wasn't sure what you might bring,\" she said to Cal.\n\n\"You made salad?\" Quinn asked.\n\n\"My specialty. Chop, shred, mix. No cooking.\"\n\n\"Now, I'm forced to be good.\" Quinn gave up the dream of two slices of pizza, settled on one and a bowl of Layla's salad. \"We made progress,\" she began as she forked up the first bite.\n\n\"Yeah, ask the ladies here how to make tallow candles or black raspberry preserves,\" Fox suggested. \"They've got it down.\"\n\n\"So, some of the information contained in the books we're going through may not currently apply to our situation.\" Quinn raised her eyebrows at Fox. \"But one day I may be called on in some blackout emergency to make a tallow candle. By progress, however, I mean that there's a lot of interesting information in Ann's journals.\"\n\n\"We've read them,\" Cal pointed out. \"Multiple times.\"\n\n\"You're not women.\" She held up a finger. \"And, yes, Essie is. But Essie's a woman who's a descendent, who's part of this town and its history. And however objective she might try to be, she may have missed some nuances. First question, where are the others?\"\n\n\"There aren't any others.\"\n\n\"I disagree. There aren't any others that were found. Essie said these books were passed to her by her father, because she loved books. I called her to be sure, but he never said if there were more.\"\n\n\"If there'd been more,\" Cal insisted, \"he'd have given them to her.\"\n\n\"If he had them. There's a long span between the sixteen hundreds and the nineteen hundreds,\" Quinn pointed out. \"Things get misplaced, lost, tossed out. According to the records and your own family's oral history, Ann Hawkins lived most of her life in what's now the community center on Main Street, which was previously the library. Books, library. Interesting.\"\n\n\"A library Gran knew inside and out,\" Cal returned. \"There couldn't have been a book in there she didn't know about. And something like this?\" He shook his head. \"She'd have it if it was to be had.\"\n\n\"Unless she never saw it. Maybe it was hidden, or maybe, for the sake of argument, she wasn't meant to find it. It wasn't meant to be found, not by her, not then.\"\n\n\"Debatable,\" Fox commented.\n\n\"And something to look into. Meanwhile, she didn't date her journals, so Layla and I are dating them, more or less, by how she writes about her sons. In what we're judging to be the first, her sons are about two to three. In the next they're five because she writes about their fifth birthday very specifically, and about seven, we think, when that one ends. The third it seems that they're young men. We think about sixteen.\"\n\n\"A lot of years between,\" Layla said.\n\n\"Maybe she didn't have anything worth writing about during those years.\"\n\n\"Could be,\" Quinn said to Cal. \"But I'm betting she did, even if it was just about blackberry jam and a trio of active sons. More important now, at least I think so, is where is the journal or journals that cover her time with Dent, to the birth of her sons through to the first two years of their lives? Because you can just bet your ass those were interesting times.\"\n\n\"She writes of him,\" Layla said quietly. \"Of Giles Dent. Again and again, in all the journals we have. She writes about him, of her feelings for him, her dreams about him.\"\n\n\"And always in the present tense,\" Quinn added.\n\n\"It's hard to lose someone you love.\" Fox turned his beer bottle in his hand.\n\n\"It is, but she writes of him, consistently, as if he were alive.\" Quinn looked at Cal. \"It is not death. We talked about this, how Dent found the way to exist, with this thing. To hold it down or through or inside. Whatever the term. Obviously he couldn't\u2014or didn't\u2014kill or destroy it, but neither could it kill or destroy him. He found a way to keep it under, and to continue to exist. Maybe only for that single purpose. She knew it. Ann knew what he did, and I'm betting she knew how he did it.\"\n\n\"You're not taking into account love and grief,\" Cal pointed out.\n\n\"I'm not discounting them, but when I read her journals, I get the sense of a strong-minded woman. And one who shared a very deep love with a strong-minded man. She defied convention for him, risked shunning and censure. Shared his bed, but I believe, shared his obligations, too. Whatever he planned to do, attempted to do, felt bound to do, he would have shared it with her. They were a unit. Isn't that what you felt, what we both felt, when we were in the clearing?\"\n\n\"Yeah.\" He couldn't deny it, Cal thought. \"That's what I felt.\"\n\n\"Going off that, Ann knew, and while she may have told her sons when they were old enough, that part of the Hawkins's oral history could have been lost or bastardized. It happens. I think she would have written it out, too. And put the record somewhere she believed would be safe and protected, until it was needed.\"\n\n\"It's been needed for twenty-one years.\"\n\n\"Cal, that's your responsibility talking, not logic. At least not the line of logic that follows this route. She told you this was the time. That it was always to be this time. Nothing you had, nothing you could have done would have stopped it before this time.\"\n\n\"We let it out,\" Fox said. \"Nothing would have been needed if we hadn't let it out.\"\n\n\"I don't think that's true.\" Layla shifted toward him, just a little. \"And maybe, if we find the other journals, we'll understand. But, we noticed something else.\"\n\n\"Layla caught it right off the bat,\" Quinn put in.\n\n\"Because it was in front of me first. But in any case, it's the names. The names of Ann's sons. Caleb, Fletcher, and Gideon.\"\n\n\"Pretty common for back then.\" Cal gave a shrug as he pushed his plate away. \"Caleb stuck in the Hawkins line more than the other two did. But I've got a cousin Fletch and an uncle Gideon.\"\n\n\"No, first initials,\" Quinn said impatiently. \"I told you they'd missed it,\" she added to Layla. \"C, F, G. Caleb, Fox, Gage.\"\n\n\"Reaching,\" Fox decided. \"Especially when you consider I'm Fox because my mother saw a pack of red foxes running across the field and into the woods about the time she was going into labor with me. My sister Sage? Mom smelled the sage from her herb garden right after Sage was born. It was like that with all four of us.\"\n\n\"You were named after an actual fox? Like a...release-the-hounds fox?\" Layla wanted to know.\n\n\"Well, not a specific one. It was more a...You have to meet my mother.\"\n\n\"However Fox got his famous name, I don't think we discount coincidences.\" Quinn studied Cal's face, saw he was considering it. \"And I think there's more than one of Ann Hawkins's descendents at this table.\"\n\n\"Quinn, my father's people came over from Ireland, four generations back,\" Fox told her. \"They weren't here in Ann Hawkins's time because they were plowing fields in Kerry.\"\n\n\"What about your mother's?\" Layla asked.\n\n\"Wider mix. English, Irish. I think some French. Nobody ever bothered with a genealogy, but I've never heard of any Hawkins on the family tree.\"\n\n\"You may want to take a closer look. How about Gage?\" Quinn wondered.\n\n\"No idea.\" And Cal was more than considering it now. \"I doubt he does either. I can ask Bill, Gage's father. If it's true, if we're direct descendents, it could explain one of the things we've never understood.\"\n\n\"Why it was you,\" Quinn said quietly. \"You three, the mix of blood from you, Fox, and Gage that opened the door.\"\n\n\"I ALWAYS THOUGHT IT WAS ME.\"\n\nWith the house quiet, and night deep, Cal lay on Quinn's bed with her body curled warm to his. \"Just you?\"\n\n\"They helped trigger it maybe, but yes, me. Because it was my blood\u2014not just that night, but my heritage, you could say. I was the Hawkins. They weren't from here, not the same way I was. Not forever, like I was. Generations back. But if this is true...I still don't know how to feel about it.\"\n\n\"You could give yourself a tiny break.\" She stroked her hand over his heart. \"I wish you would.\"\n\n\"Why did he let it happen? Dent? If he'd found a way to stop it, why did he let it come to this?\"\n\n\"Another question.\" She pushed herself up until they were eye-to-eye. \"We'll figure it out, Cal. We're supposed to. I believe that.\"\n\n\"I'm closer to believing it, with you.\" He touched her cheek. \"Quinn, I can't stay again tonight. Lump may be lazy, but he depends on me.\"\n\n\"Got another hour to spare?\"\n\n\"Yeah.\" He smiled as she lowered to him. \"I think he'll hold out another hour.\"\n\nLATER, WHEN HE WALKED OUT TO HIS CAR, THE air shivered so that the trees rattled their empty branches. Cal searched the street for any sign, anything he needed to defend against. But there was nothing but empty road.\n\nSomething's in the wind, he thought again, and got in his car to drive home.\n\nIT WAS AFTER MIDNIGHT WHEN THE LOW-GRADE urge for a cigarette buzzed through Gage's brain. He'd given them up two years, three months, and one week before, a fact that could still piss him off.\n\nHe turned up the radio to take his mind off it, but the urge was working its way up to craving. He could ignore that, too; he did so all the time. To do otherwise was to believe there was solid truth in the old adage: like father, like son.\n\nHe was nothing like his father.\n\nHe drank when he wanted a drink, but he never got drunk. Or hadn't since he'd been seventeen, and then the drunkenness had been with absolute purpose. He didn't blame others for his shortcomings, or lash out with his fists on something smaller and weaker so he could feel bigger and stronger.\n\nHe didn't even blame the old man, not particularly. You played the cards you were dealt, to Gage's mind. Or you folded and walked away with your pockets empty.\n\nLuck of the draw.\n\nSo he was fully prepared to ignore this sudden, and surprisingly intense desire for a cigarette. But when he considered he was within miles of Hawkins Hollow, a place where he was very likely to die an ugly and painful death, the surgeon general's warnings seemed pretty goddamn puny, and his own self-denial absolutely useless.\n\nWhen he saw the sign for the Sheetz, he decided what the hell. He didn't want to live forever. He swung into the twenty-four-hour mart, picked up coffee, black, and a pack of Marlboros.\n\nHe strode back to the car he'd bought that very evening in D.C. after his plane had landed, and before he'd paid off a small debt. The wind whipped through his hair. The hair was dark as the night, a little longer than he usually wore it, a little shaggy, as he hadn't trusted the barbers in Prague.\n\nThere was stubble on his face since he hadn't bothered to shave. It added to the dark, dangerous look that had had the young female clerk who rung up the coffee and cigarettes shivering inwardly with lust.\n\nHe'd topped off at six feet, and the skinny build of his youth had filled out. Since his profession was usually sedentary, he kept his muscles toned and his build rangy with regular, often punishing workouts.\n\nHe didn't pick fights, but he rarely walked away from one. And he liked to win. His body, his face, his mind, were all tools of his trade. As were his eyes, his voice, and the control he rarely let off the leash.\n\nHe was a gambler, and a smart gambler kept all of his tools well honed.\n\nSwinging back onto the road, Gage let the Ferrari rip. Maybe it had been foolish to toss so much of his winnings into a car, but Jesus, it moved. And fucking A, he'd ridden his thumb out of the Hollow all those years ago. It felt damn good to ride back in in style.\n\nFunny, now that he'd bought the damn cigarettes, the urge for one had passed. He didn't even want the coffee, the speed was kick enough.\n\nHe flew down the last miles of the interstate, whipped onto the exit that would take him to the Hollow. The dark rural road was empty\u2014no surprise to him, not this time of night. There were shadows and shapes\u2014houses, hills, fields, trees. There was a twisting in his gut that he was heading back instead of away, and yet that pull\u2014it never quite left him\u2014that pull toward home was strong.\n\nHe reached toward his coffee more out of habit than desire, then was forced to whip the wheel, slam the brakes as headlights cut across the road directly into his path. He blasted the horn, saw the other car swerve.\n\nHe thought: Fuck, fuck, fuck! I just bought this sucker.\n\nWhen he caught his breath, and the Ferrari sat sideways in the middle of the road, he thought it was a miracle the crash hadn't come. Inches, he realized. Less than inches.\n\nHis lucky goddamn day.\n\nHe reversed, pulled to the shoulder, then got out to check on the other driver he assumed was stinking drunk.\n\nShe wasn't. What she was, was hopping mad.\n\n\"Where the hell did you come from?\" she demanded. She slammed out of her car, currently tipped into the shallow ditch along the shoulder, in a blur of motion. He saw a mass of dark gypsy curls wild around a face pale with shock.\n\nGreat face, he decided in one corner of his brain. Huge eyes that looked black against her white skin, a sharp nose, a wide mouth, sexily full that may have owned its sensuality to collagen injections.\n\nShe wasn't shaking, and he didn't sense any fear along with the fury as she stood on a dark road facing down a complete stranger.\n\n\"Lady,\" he said with what he felt was admirable calm, \"where the hell did you come from?\"\n\n\"From that stupid road that looks like all the other stupid roads around here. I looked both damn ways, and you weren't there. Then. How did you...Oh never mind. We didn't die.\"\n\n\"Yay.\"\n\nWith her hands on her hips she turned around to study her car. \"I can get out of there, right?\"\n\n\"Yeah. Then there's the flat tire.\"\n\n\"What flat...Oh for God's sake! You have to change it.\" She gave the flat tire on the rear of her car an annoyed kick. \"It's the least you can do.\"\n\nActually, it wasn't. The least he could do was stroll back to his car and wave good-bye. But he appreciated her bitchiness, and preferred it over quivering. \"Pop the trunk. I need the spare and the jack.\"\n\nWhen she had, and he'd lifted a suitcase out, set it on the ground, he took one look at her spare. And shook his head. \"Not your day. Your spare's toast.\"\n\n\"It can't be. What the hell are you talking about?\" She shoved him aside, peered in herself by the glow of the trunk light. \"Damn her, damn her, damn her. My sister.\" She whirled away, paced down the shoulder a few feet, then back. \"I loaned her my car for a couple of weeks. This is so typical. She ruins a tire, but does she get it fixed, does she even bother to mention it? No.\"\n\nShe pushed her hair back from her face. \"I'm not calling a tow truck at this time of night, then sitting in the middle of nowhere. You're just going to have to give me a ride.\"\n\n\"Am I?\"\n\n\"It's your fault. At least part of it is.\"\n\n\"Which part?\"\n\n\"I don't know, and I'm too tired, I'm too mad, I'm too lost in this foreign wilderness to give a damn. I need a ride.\"\n\n\"At your service. Where to?\"\n\n\"Hawkins Hollow.\"\n\nHe smiled, and there was something dark in it. \"Handy. I'm heading there myself.\" He gestured toward his car. \"Gage Turner,\" he added.\n\nShe gestured in turn, rather regally, toward her suitcase. \"Cybil Kinski.\" She lifted her eyebrows when she got her first good look at his car. \"You have very nice wheels, Mr. Turner.\"\n\n\"Yeah, and they all work.\"\n\n## Fourteen\n\nCAL WASN'T PARTICULARLY SURPRISED TO SEE Fox's truck in his driveway, despite the hour. Nor was he particularly surprised when he walked in to see Fox blinking sleepily on the couch in front of the TV, with Lump stretched out and snoring beside him.\n\nOn the coffee table were a can of Coke, the last of Cal's barbecue potato chips, and a box of Milk Bones. The remains, he assumed, of a guy-dog party.\n\n\"Whatcha doing here?\" Fox asked groggily.\n\n\"I live here.\"\n\n\"She kick you out?\"\n\n\"No, she didn't kick me out. I came home.\" Because they were there, Cal dug into the bag of chips and managed to pull out a handful of crumbs. \"How many of those did you give him?\"\n\nFox glanced at the box of dog biscuits. \"A couple. Maybe five. What're you so edgy about?\"\n\nCal picked up the Coke and gulped down the couple of warm, flat swallows that were left. \"I got a feeling, a...thing. You haven't felt anything tonight?\"\n\n\"I've had feelings and things pretty much steady the last couple weeks.\" Fox scrubbed his hands over his face, back into his hair. \"But yeah, I got something just before you drove up. I was half asleep, maybe all the way. It was like the wind whooshing down the flue.\"\n\n\"Yeah.\" Cal walked over to stare out the window. \"Have you checked in with your parents lately?\"\n\n\"I talked to my father today. It's all good with them. Why?\"\n\n\"If all three of us are direct descendents, then one of your parents is in the line,\" Cal pointed out.\n\n\"I figured that out on my own.\"\n\n\"None of our family was ever affected during the Seven. We were always relieved by that.\" He turned back. \"Maybe relieved enough we didn't really ask why.\"\n\n\"Because we figured it, at least partly, was because they lived outside of town. Except for Bill Turner, and who the hell could tell what was going on with him?\"\n\n\"My parents and yours, they came into town during the Seven. And there were people, you remember what happened out at the Poffenberger place last time?\"\n\n\"Yeah. Yeah, I remember.\" Fox rubbed at his eyes. \"Being five miles out of town didn't stop Poffenberger from strangling his wife while she hacked at him with a butcher knife.\"\n\n\"Now we know Gran felt things, saw things that first summer, and she saw things the other night. Why is that?\"\n\n\"Maybe it picks and chooses, Cal.\" Rising, Fox walked over to toss another log on the fire. \"There have always been people who weren't affected, and there have always been degrees with those who were.\"\n\n\"Quinn and Layla are the first outsiders. We figured a connection, but what if that connection is as simple as blood ties?\"\n\nFox sat again, leaned back, stroked a hand over Lump's head as the dog twitched in his sleep. \"Good theory. It shouldn't weird you out if you happen to be rolling naked with your cousin a couple hundred times removed.\"\n\n\"Huh.\" That was a thought. \"If they're descendents, the next point to figure is if having them here gives us more muscle, or makes us more vulnerable. Because it's pretty clear this one's it. This one's going to be the all or nothing. So...Someone's coming.\"\n\nFox pushed off the couch, strode quickly over to stand by Cal. \"I don't think the Big Evil's going to drive up to your house, and in a...\" He peered closer as the car set off Cal's motion lights. \"Holy Jesus, is that a Ferrari?\" He shot a grin at Cal.\n\n\"Gage,\" they said together.\n\nThey went on the front porch, in shirtsleeves, leaving the door open behind them. Gage climbed out of the car, his eyes skimming over them both as he walked back to get his bag out of the trunk. He slung its strap over his shoulder, started up the steps. \"You girls having a slumber party?\"\n\n\"Strippers just left,\" Fox told him. \"Sorry you missed them.\" Then he rushed forward, flung his arms around Gage in a hard hug. \"Man, it's good to see you. When can I drive your car?\"\n\n\"I was thinking never. Cal.\"\n\n\"Took your goddamn time.\" The relief, the love, the sheer pleasure pushed him forward to grip Gage just as Fox had.\n\n\"Had some business here and there. Want a drink. Need a room.\"\n\n\"Come on in.\"\n\nIn the kitchen, Cal poured whiskey. All of them understood it was a welcome-home toast for Gage, and very likely a drink before war.\n\n\"So,\" Cal began, \"I take it you came back flush.\"\n\n\"Oh yeah.\"\n\n\"How much you up?\"\n\nGage turned the glass around in his hand. \"Considering expenses, and my new toy out there, about fifty.\"\n\n\"Nice work if you can get it,\" Fox commented.\n\n\"And I can.\"\n\n\"Look a little worn there, brother.\"\n\nGage shrugged at Cal. \"Long couple of days. Which nearly ended with me in a fiery crash right out on Sixty-seven.\"\n\n\"Toy get away from you?\" Fox asked.\n\n\"Please.\" Gage smirked at the idea. \"Some ditz, of the female and very hot variety, pulled out in front of me. Not another car on the road, and she pulls out in this ancient Karmann Ghia\u2014nice wheels, actually\u2014then she jumps out and goes at me like it was my fault.\"\n\n\"Women,\" Fox said, \"are an endless source of every damn thing.\"\n\n\"And then some. So she's tipped down in the little runoff,\" Gage went on, gesturing with his free hand. \"No big deal, but she's popped a flat. No big deal either, except her spare's a pancake. Turns out she's heading into the Hollow, so I manage to load her two-ton suitcase into my car. Then she's rattling off an address and asking me, like I'm MapQuest, how long it'll take to get there.\"\n\nHe took a slow sip of whiskey. \"Lucky for her I grew up here and could tell her I'd have her there in five. She snaps out her phone, calls somebody she calls Q, like James freaking Bond, tells her, as it turns out from the look I got of Q in the doorway\u2014very nice, by the way\u2014to wake up, she'll be there in five minutes. Then\u2014\"\n\nCal rattled off an address. \"That the one?\"\n\nGage lowered his glass. \"As a matter of fact.\"\n\n\"Something in the wind,\" Cal murmured. \"I guess it was you, and Quinn's Cybil.\"\n\n\"Cybil Kinski,\" Gage confirmed. \"Looks like a gypsy by way of Park Avenue. Well, well.\" He downed the rest of the whiskey in his glass. \"Isn't this a kick in the ass?\"\n\n\"HE CAME OUT OF NOWHERE.\" THERE WAS A glass of red wine on the dresser Quinn had picked up in anticipation of Cybil's arrival.\n\nAs that arrival had woken Layla, Quinn sat beside her on what would be Cybil's bed while the woman in question swirled around the room, hanging clothes, tucking them in drawers, taking the occasional sip of wine.\n\n\"I thought that was it, just it, even though I've never seen any death by car in my future. I swear, I don't know how we missed being bloody pulps tangled in burning metal. I'm a good driver,\" Cybil said to Quinn.\n\n\"You are.\"\n\n\"But I must be better than I thought, and so\u2014fortunately\u2014was he. I know I'm lucky all I got was a scare and a flat tire out of it, but damn Rissa for, well, being Rissa.\"\n\n\"Rissa?\" Layla looked blank.\n\n\"Cyb's sister, Marissa,\" Quinn explained. \"You loaned her your car again.\"\n\n\"I know, I know. I know,\" she said, puffing out a breath that blew curls off her forehead. \"I don't know how she manages to talk me into these things. My spare was flat, thanks to Rissa.\"\n\n\"Which explains why you were dropped off from a really sexy sports car.\"\n\n\"He could hardly leave me there, though he looked like the type who'd consider it. All scruffy, gorgeous, and dangerous looking.\"\n\n\"Last time I had a flat,\" Quinn remembered, \"the very nice guy who stopped to help had a paunch over his belt the size of a sack of cement, and ass crack reveal.\"\n\n\"No paunch on this one, and though his coat prevented me from a good look, I'm betting Gage Turner has a superior ass.\"\n\n\"Gage Turner.\" Layla put a hand on Quinn's thigh. \"Quinn.\"\n\n\"Yeah.\" Quinn let out a breath. \"Okay, I guess it's hail, hail, the gang's all here.\"\n\nIN THE MORNING, QUINN LEFT HER HOUSEMATES sleeping while she jogged over to the community center. She already knew she'd regret jogging over, because that meant she'd have to jog back\u2014after her workout. But it seemed a cheat on the lifestyle change to drive three blocks to the gym.\n\nAnd she wanted the thinking time.\n\nThere was no buying, for any price, Cybil and Gage Turner had run into each other\u2014almost literally\u2014in the middle of the night just outside of town as a coincidence.\n\nOne more thing to add to the list of oddities, Quinn thought as she puffed out air in frosty vapors.\n\nAnother addition would be the fact that Cybil had a very sharp sense of direction, but had apparently made wrong turn after wrong turn to end up on that side road at the exact moment Gage was coming up the main.\n\nOne more, Quinn decided as she approached the back entrance of the community center, would be Cybil saying \"he came out of nowhere.\" Quinn was willing to take that literally. If Cybil didn't see him, then maybe\u2014in her reality, for just those vital moments\u2014he hadn't been there.\n\nSo why had it been important for them to meet separately, outside the group? Wasn't it strange enough that they'd both arrived on the same night, at the same time?\n\nShe dug out her membership key\u2014thanks, Cal\u2014to open the door to the fitness area, pressed her guest pass number on the keypad.\n\nThe lights were still off, which was a surprise. Normally when she arrived, they were already on, and at least one of the trio of swivel TVs was tuned to CNN or ESPN or one of the morning talk shows. Very often there was somebody on one of the treadmills or bikes, or pumping weights.\n\nShe flipped on the lights, called out. And her voice echoed hollowly. Curious, she walked through, pushed open the door, and saw the lights were also off in the tiny attendant's office, and in the locker room.\n\nMaybe somebody had a late date the night before, she decided. She helped herself to a locker key, stripped down to her workout gear, then grabbed a towel. Opting to start her session with cardio, she switched on the Today show before climbing onto the single elliptical trainer the club boasted.\n\nShe programmed it, resisting the urge to cheat a few pounds off her weight. As if it mattered, Quinn reminded herself. (Of course, it mattered.)\n\nShe started her warm-up pleased with her discipline, and her solitude. Still, she expected the door to slam open any minute, for Matt or Tina, who switched off as attendants, to rush in. By the time she was ten minutes in, she'd kicked up the resistance and was focused on the TV screen to help her get through the workout.\n\nWhen she hit the first mile, Quinn took a long gulp of water from the sports bottle she'd brought with her. As she started on mile two, she let her mind drift to what she hoped to accomplish that day. Research, the foundation of any project. And she wanted to draft what she thought would be the opening of her book. Writing it out might spark some idea. At some point, she wanted to walk around the town again, with Cybil\u2014and Layla if she was up for it.\n\nA visit to the cemetery was in order with Cybil in tow. Time to pay a call on Ann Hawkins.\n\nMaybe Cal would have time to go with them. Needed to talk to him anyway, discuss how he felt, what he thought, about Gage\u2014whom she wanted to get a look at\u2014and Cybil's arrival. Mostly, she admitted, she just wanted to see him again. Show him off to Cybil.\n\nLook! Isn't he cute? Maybe it was completely high school, but it didn't seem to matter. She wanted to touch him again, even if it was just a quick squeeze of hands. And she was looking forward to a hello kiss, and finding a way to turn that worried look in his eyes into a glint of amusement. She loved the way his eyes laughed before the rest of him did, and the way he...\n\nWell. Well, well, well. She was absolutely gone over him, she realized. Seriously hooked on the hometown boy. That was kind of cute, too, she decided, except it made her stomach jitter. Still, the jitter wasn't altogether a bad thing. It was a combination of oh-oh and oh boy!, and wasn't that interesting?\n\nQuinn's falling in love, she thought, and hit mile two with a dopey smile on her face. She might've been puffing, sweat might have been dribbling down her temples, but she felt just as fresh and cheerful as a spring daisy.\n\nThen the lights went out.\n\nThe machine stopped; the TV went blank and silent.\n\n\"Oh, shit.\" Her first reaction wasn't alarm as much as, what now? The dark was absolute, and though she could draw a reasonable picture in her mind where she was in relation to the outside door\u2014and what was between her and the door\u2014she was wary about making her way to it blind.\n\nAnd then what? she wondered as she waited for her breathing to level. She couldn't possibly fumble her way to the locker room, to her locker and retrieve her clothes. So she'd have to go out in a damn sports bra and bike pants.\n\nShe heard the first thud; the chill washed over her skin. And she understood she had much bigger problems than skimpy attire.\n\nShe wasn't alone. As her pulse began to bang, she hoped desperately whatever was in the dark with her was human. But the sounds, that unholy thudding that shook the walls, the floor, the awful scuttling sounds creeping under it weren't those of a man. Gooseflesh pricked her skin, partly from fear, partly from the sudden and intense cold.\n\nKeep your head, she ordered herself. For God's sake, keep your head. She gripped the water bottle\u2014pitiful weapon, but all she had\u2014and started to ease off the foot pads on the machine to the floor.\n\nShe went flying blindly in the black. She hit the floor, her shoulder and hip taking the brunt. Everything shook and rolled as she fought to scramble up. Disoriented, she had no idea which direction to run. There was a voice behind her, in front of her, inside her head\u2014she couldn't tell\u2014and it whispered gleefully of death.\n\nShe knew she screamed as she clawed her way across the quaking floor. Teeth chattering against terror and cold, she rapped her shoulder against another machine. Think, think, think! she told herself, because something was coming, something was coming in the dark. She ran her shaking hands over the machine\u2014recumbent bike\u2014and with every prayer she knew ringing in her head, used its placement in the room to angle toward the door.\n\nThere was a crash behind her, and something thudded against her foot. She jerked up, tripped, jerked up again. No longer caring what might stand between herself and the door, she flung herself toward where she hoped it would be. With her breath tearing out of her lungs, she ran her hands over the wall.\n\n\"Find it, goddamn it, Quinn. Find the goddamn door!\"\n\nHer hand bumped the hinges, and on a sob she found the knob. Turned, pulled.\n\nThe light burst in front of her eyes, and Cal's body\u2014already in motion\u2014rammed hers. If she'd had any breath left, she'd have lost it. Her knees didn't get a chance to buckle as he wrapped his arms around her, swung her around to use his body as a shield between hers and the room beyond.\n\n\"Hold on, now. Can you hold on to me?\" His voice was eerily calm as he reached behind him and pulled the door closed. \"Are you hurt? Tell me if you're hurt.\" His hands were already skimming over her, before they came up to her face, gripped it.\n\nBefore his mouth crushed down on hers.\n\n\"You're all right,\" he managed, propping her against the stone of the building as he dragged off his coat. \"You're okay. Here, get into this. You're freezing.\"\n\n\"You were there.\" She stared up into his face. \"You were there.\"\n\n\"Couldn't get the door open. Key wouldn't work.\" He took her hands, rubbed them warm between his. \"My truck's right up there, okay. I want you to go up, sit in my truck. I left the keys in it. Turn on the heat. Sit in my truck and turn on the heat. Can you do that?\"\n\nShe wanted to say yes. There was something in her that wanted to say yes to anything he asked. But she saw, in his eyes, what he meant to do.\n\n\"You're going in there.\"\n\n\"That's what I have to do. What you have to do is go sit in the truck for a few minutes.\"\n\n\"If you go in, I go in.\"\n\n\"Quinn.\"\n\nHow, she wondered, did he manage to sound patient and annoyed at the same time? \"I need to as much as you, and I'd hate myself if I huddled in your truck while you went in there. I don't want to hate myself. Besides, it's better if there's two of us. It's better. Let's just do it. Just do it, and argue later.\"\n\n\"Stay behind me, and if I say get out, you get out. That's the deal.\"\n\n\"Done. Believe me, I'm not ashamed to hide behind you.\"\n\nShe saw it then, just the faintest glimmer of a smile in his eyes. Seeing it settled her nerves better than a quick shot of brandy.\n\nHe turned his key again, keyed in the touch pad. Quinn held her breath. When Cal opened the door, the lights were on. Al Roker's voice cheerily announced the national weather forecast. The only sign anything had happened was her sports bottle under the rack of free weights.\n\n\"Cal, I swear, the power went out, then the room\u2014\"\n\n\"I saw it. It was pitch-black in here when you came through the door. Those weights were all over the floor. I could see them rolling around from the light coming in the door. The floor was heaving. I saw it, Quinn. And I heard it from outside the door.\"\n\nHe'd rammed that door twice, he remembered, put his full weight into it, because he'd heard her screaming, and it had sounded like the roof was caving in.\n\n\"Okay. My things are in the locker room. I really want to get my things out of the locker.\"\n\n\"Give me the key, and I'll\u2014\"\n\n\"Together.\" She gripped his hand. \"There's a scent, can you smell it? Over and above my workout and panic sweat.\"\n\n\"Yeah. I always thought it must be what brimstone smells like. It's fading.\" He smiled, just a little, as she stopped to pick up a ten-pound free weight, gripped it like a weapon.\n\nHe pushed open the door of the women's locker room. It was as ordered and normal as the gym. Still, he took her key, nudged her behind him before he opened her locker. Moving quickly, she dragged on her sweats, exchanged coats. \"Let's get out of here.\"\n\nHe had her hand as they walked back out and Matt walked in.\n\nHe was young, the college-jock type, doing the part-time attendant, occasional personal trainer gig. A quick, inoffensive smirk curved on his lips as he saw them come out of the women's locker room together. Then he cleared his throat.\n\n\"Hey, sorry I'm late. Damnedest thing. First my alarm didn't go off, and I know how that sounds. Then my car wouldn't start. One of those mornings.\"\n\n\"Yeah,\" Quinn agreed as she put back the weight, retrieved her water bottle. \"One of those. I'm done for the day.\" She tossed him the locker key. \"See you later.\"\n\n\"Sure.\"\n\nShe waited until they were out of the building. \"He thought we'd been\u2014\"\n\n\"Yeah, yeah.\"\n\n\"Ever do it in a locker room?\"\n\n\"As that was actually my first foray into a girl's locker room, I have to say no.\"\n\n\"Me, either. Cal, have you got time to come over, have coffee\u2014God, I'll even cook breakfast\u2014and talk about this?\"\n\n\"I'm making time.\"\n\nSHE TOLD HIM EVERYTHING THAT HAD HAPPENED while she scrambled eggs. \"I was scared out of my mind,\" she finished as she carried the coffee into the little dining room.\n\n\"No, you weren't.\" Cal set the plates of eggs and whole-wheat toast on the table. \"You found the door, in the pitch-black, and with all that going on, you kept your head and found the door.\"\n\n\"Thanks.\" She sat. She wasn't shaking any longer, but the inside of her knees still felt like half-set Jell-O. \"Thanks for saying that.\"\n\n\"It's the truth.\"\n\n\"You were there when I opened the door, and that was one of the best moments of my life. How did you know to be there?\"\n\n\"I came in early because I wanted to swing by here, see how you were. Talk to you. Gage\u2014\"\n\n\"I know about that. Tell me the rest of this first.\"\n\n\"Okay. I turned off Main to come around the back way, come here, and I saw Ann Hawkins. I saw her standing in front of the door. I heard you screaming.\"\n\n\"From inside your truck, on the street. That far away\u2014through stone walls, you heard me?\"\n\n\"I heard you.\" It hadn't been one of the best moments of his life. \"When I jumped out, ran toward the door, I heard crashing, thumping, God knows what from inside. I couldn't get the goddamn door open.\"\n\nShe heard it now, the emotion in his voice, the fear he hadn't let show while they were doing what needed doing. She rose, did them both a favor and crawled right into his lap.\n\nShe was still there, cradled in his arms, when Cybil strolled in.\n\n\"Hi. Don't get up.\" She took Quinn's chair. \"Anyone eating this?\" Studying them, Cybil took a forkful of eggs. \"You must be Cal.\"\n\n\"Cybil Kinski, Caleb Hawkins. We had a rough morning.\"\n\nLayla stepped in with a coffee mug and sleepy eyes that clouded with concern the minute she saw Quinn. \"What happened?\"\n\n\"Have a seat, and we'll run it through for both of you.\"\n\n\"I need to see the place,\" Cybil said as soon as the story was told. \"And the room in the bowling alley, anyplace there's been an incident.\"\n\n\"Try the whole town,\" Quinn said dryly.\n\n\"And I need to see the clearing, this stone, as soon as possible.\"\n\n\"She's bossy,\" Quinn told Cal.\n\n\"I thought you were, but I think she beats you out. You can come into the bowling center anytime you like. Quinn can get you into the fitness center, but if I can't be there, I'll make sure either Fox or Gage is. Better, both of them. As far as the Pagan Stone goes, I talked with Fox and Gage about that last night. We're agreed that the next time we go, we all go. All of us. I can't make it today and neither can Fox. Sunday's going to be best.\"\n\n\"He's organized and take-charge,\" Cybil said to Quinn.\n\n\"Yes.\" She pressed a kiss to Cal's cheek. \"Yes, he is. And I've made you let your eggs get cold.\"\n\n\"It was a worthwhile trade-off. I'd better get going.\"\n\n\"We still have a lot to talk about. Listen, maybe the three of you should come to dinner.\"\n\n\"Is someone cooking?\" Cal asked.\n\n\"Cyb is.\"\n\n\"Hey!\"\n\n\"You ate my breakfast. Plus you actually cook. But in the meantime, just one thing.\" She slid out of his lap so he could stand. \"Would Fox hire Layla?\"\n\n\"What? Who? Why?\" Layla sputtered.\n\n\"Because you need a job,\" Quinn reminded her. \"And he needs an office manager.\"\n\n\"I don't know anything about\u2014you just can't\u2014\"\n\n\"You managed a boutique,\" Quinn reminded her, \"so that's half the job. Managing. You're on the anal side of organized, Miss Colored Index Cards and Charts, so I say you can file, keep a calendar, and whatever with the best of them. Anything else, you'll pick up as you go. Ask Fox, okay, Cal?\"\n\n\"Sure. No problem.\"\n\n\"She calls me bossy,\" Cybil commented as she finished Quinn's coffee.\n\n\"I call it creative thinking and leadership. Now, go fill that mug up again while I walk Cal to the door so I can give him a big, sloppy you're-my-hero kiss.\"\n\nCybil smiled after them as Quinn pulled Cal out of the room. \"She's in love.\"\n\n\"Really?\"\n\nNow Cybil turned her smile on Layla. \"That got your mind off taking a bite out of her for pushing that job in your face.\"\n\n\"I'll get back to that. Do you think she's in love with Cal\u2014the uppercase L?\"\n\n\"About to be all caps, in bold letters.\" She picked up the mug and rose. \"Q likes to direct people,\" she said, \"but she's careful to try to direct them toward something helpful, or at least interesting. She wouldn't push this job business if she didn't think you could handle it.\"\n\nShe blew out a breath as she walked back toward the kitchen. \"What the hell am I supposed to fix for dinner?\"\n\n## Fifteen\n\nIT WAS HARD FOR CAL TO SEE BILL TURNER AND say nothing about Gage being in town. But Cal knew his friend. When and if Gage wanted his father to know, Gage would tell him. So Cal did his best to avoid Bill by closing himself in his office.\n\nHe dealt with orders, bills, reservations, contacted their arcade guy to discuss changing out one of their pinball machines for something jazzier.\n\nChecking the time, he judged if Gage wasn't awake by now, he should be. And so picked up the phone.\n\nNot awake, Cal decided, hearing the irritation in Gage's voice, hasn't had coffee. Ignoring all that, Cal launched into an explanation of what happened that morning, relayed the dinner plans, and hung up.\n\nNow, rolling his eyes, Cal called Fox to run over the same information, and to tell Fox that Layla needed a job and he should hire her to replace Mrs. Hawbaker.\n\nFox said, \"Huh?\"\n\nCal said, \"Gotta go,\" and hung up.\n\nThere, duty done, he considered. Satisfied, he turned to his computer and brought up the information on the automatic scoring systems he wanted to talk his father into installing.\n\nIt was past time for the center to do the upgrade. Maybe it was foolish to think about that kind of investment if everything was going to hell in a few months. But, if everything was going to hell in a few months, the investment wouldn't hurt a thing.\n\nHis father would say some of the old-timers would object, but Cal didn't think so. If they wanted to keep score by hand, the center would provide the paper score sheets and markers. But he thought if someone showed them how it worked, gave them a few free games to get used to the new system, they'd jump on.\n\nThey could get them used and reconditioned, which was part of the argument he was prepared to make. They had Bill onboard, and he could fix damn near anything.\n\nIt was one thing to be a little kitschy and traditional, another to be old-fashioned.\n\nNo, no, that wasn't the tack to take with his father. His father liked old-fashioned. Better to use figures. Bowling accounted for more than half, closer to sixty percent, of their revenue, so\u2014\n\nHe broke off at the knock on his door and inwardly winced, thinking it was Bill Turner.\n\nBut it was Cal's mother who popped her head in. \"Too busy for me?\"\n\n\"Never. Here to bowl a few games before the morning league?\"\n\n\"Absolutely not.\" Frannie loved her husband, but she liked to say she hadn't taken a vow to love, honor, and bowl. She came in to sit down, then angled her head so she could see his computer screen. Her lips twitched. \"Good luck with that.\"\n\n\"Don't say anything to Dad, okay?\"\n\n\"My lips are sealed.\"\n\n\"Who are you having lunch with?\"\n\n\"How do you know I'm having lunch with anyone?\"\n\nHe gestured to her pretty fitted jacket, trim pants, heeled boots. \"Too fancy for shopping.\"\n\n\"Aren't you smart? I do have a few errands, then I'm meeting a friend for lunch. Joanne Barry.\"\n\nFox's mother, Cal thought, and just nodded.\n\n\"We have lunch now and then, but she called me yesterday, specifically to see if I could meet her today. She's worried. So I'm here to ask you if there's anything I should know, anything you want to tell me before I see her.\"\n\n\"Things are as under control as I can make them, Mom. I don't have the answers yet. But I have more questions, and I think that's progress. In fact, I have one you could ask Fox's mom for me.\"\n\n\"All right.\"\n\n\"You could ask if there's a way she could find out if any of her ancestors were Hawkins.\"\n\n\"You think we might be related somehow? Would it help if we are?\"\n\n\"It would be good to know the answer.\"\n\n\"Then I'll ask the question. Now answer one for me. Are you all right? Just a yes or no is good enough.\"\n\n\"Yes.\"\n\n\"Okay then.\" She rose. \"I have half a dozen things on my list before I meet Jo.\" She started for the door, said, \"Damn it\" very quietly under her breath, and turned back. \"I wasn't going to ask, but I have no willpower over something like this. Are you and Quinn Black serious?\"\n\n\"About what?\"\n\n\"Caleb James Hawkins, don't be dense.\"\n\nHe would've laughed, but that tone brought on the Pavlovian response of hunched shoulders. \"I don't exactly know the answer. And I'm not sure it's smart to get serious, in that way, with so much going on. With so much at stake.\"\n\n\"What better time?\" Frannie replied. \"My levelheaded Cal.\" She put her hand on the knob, smiled at him. \"Oh, and those fancy scoring systems? Try reminding your father how much his father resisted going to projection-screen scoreboards thirty-five years ago, give or take.\"\n\n\"I'll keep that in mind.\"\n\nAlone, Cal printed out the information on the automatic systems, new and reconditioned, then shut down long enough to go downstairs and check in with the front desk, the grill, the play area during the morning leagues games.\n\nThe scents from the grill reminded him he'd missed breakfast, so he snagged a hot pretzel and a Coke before he headed back up to his office.\n\nSo armed, he decided since everything was running smoothly, he could afford to take a late-morning break. He wanted to dig a little deeper into Ann Hawkins.\n\nShe'd appeared to him twice in three days. Both times, Cal mused, had been a kind of warning. He'd seen her before, but only in dreams. He'd wanted her in dreams, Cal admitted\u2014or Giles Dent had, working through him.\n\nThese incidents had been different, and his feelings different.\n\nStill, that wasn't the purpose, that wasn't the point, he reminded himself as he gnawed off a bite of pretzel.\n\nHe was trusting Quinn's instincts about the journals. Somewhere, at some time, there had been more. Maybe they were in the old library. He certainly intended to get in there and search the place inch by inch. If, God, they'd somehow gotten transferred into the new space and mis-shelved or put in storage, the search could be a nightmare.\n\nSo he wanted to know more about Ann, to help lead him to the answers.\n\nWhere had she been for nearly two years? All the information, all the stories he'd heard or read indicated she'd vanished the night of the fire in the clearing and hadn't returned to the Hollow until her sons were almost two.\n\n\"Where did you go, Ann?\"\n\nWhere would a woman, pregnant with triplets, go during the last weeks before their births? Traveling had to have been extremely difficult. Even for a woman without the pregnancy to weigh her down.\n\nThere had been other settlements, but nothing as far as he remembered for a woman in her condition to have walked or even ridden to. So logically, she'd had somewhere to go close by, and someone had taken her in.\n\nWho was most likely to take a young, unmarried woman in? A relative would be his first guess.\n\nMaybe a friend, maybe some kindly old widow, but odds were on family.\n\n\"That's where you went first, when there was trouble, wasn't it?\"\n\nWhile it wasn't easy to find specifics on Ann Hawkins, there was plenty of it on her father\u2014the founder of the Hollow.\n\nHe'd read it, of course. He'd studied it, but he'd never read or studied it from this angle. Now, he brought up all the information he'd previously downloaded on his office computer relating to James Hawkins.\n\nHe took side trips, made notes on any mention of relatives, in-laws. The pickings were slim, but at least there was something to pick from. Cal was rolling with it when someone knocked on his door. He surfaced as Quinn poked her head in just as his mother had that morning.\n\n\"Working. I bet you hate to be interrupted. But...\"\n\n\"It's okay.\" He glanced at the clock, saw with a twinge of guilt his break had lasted more than an hour. \"I've been at it longer than I meant to.\"\n\n\"It's dog-eat-dog in the bowling business.\" She said it with a smile as she came in. \"I just wanted you to know we were here. We took Cyb on a quick tour of the town. Do you know there's no place to buy shoes in Hawkins Hollow? Cyb's saddened by that, as she's always on the hunt. Now she's making noises about bowling. She has a vicious competitive streak. So I escaped up here before she drags me into that. The hope was to grab a quick bite at your grill\u2014maybe you could join us\u2014before Cyb...\"\n\nShe trailed off. Not only hadn't he said a word, but he was staring at her. Just staring. \"What?\" She brushed a hand over her nose, then up over her hair. \"Is it my hair?\"\n\n\"That's part of it. Probably part of it.\"\n\nHe got up, came around the desk. He kept his eyes on her face as he moved past her. As he shut and locked the door.\n\n\"Oh. Oh. Really? Seriously? Here? Now?\"\n\n\"Really, seriously. Here and now.\" She looked flustered, and that was a rare little treat. She looked, every inch of her, amazing. He couldn't say why he'd gone from pleased to see her to aroused in the snap of a finger, and he didn't much care. What he knew, without question, was he wanted to touch her, to draw in her scent, to feel her body go tight, go loose. Just go.\n\n\"You're not nearly as predictable as you should be.\" Watching him now, she pulled off her sweater, unbuttoned the shirt beneath it.\n\n\"I should be predictable?\" Without bothering with buttons, he pulled his shirt over his head.\n\n\"Hometown boy from a nice, stable family, who runs a third-generation family business. You should be predictable, Caleb,\" she said as she unbuttoned her jeans. \"I like that you're not. I don't mean just the sex, though major points there.\"\n\nShe bent down to pull off her boots, tossing her hair out of her eyes so she could look up at him. \"You should be married,\" she decided, \"or on your way to it with your college sweetheart. Thinking about 401(k)s.\"\n\n\"I think about 401(k)s. Just not right now. Right now, Quinn, all I can think about is you.\"\n\nThat gave her heart a bounce, even before he reached out, ran his hands down her bare arms. Even before he drew her to him and seduced her mouth with his.\n\nShe may have laughed when they lowered to the floor, but her pulse was pounding. There was a different tone from when they were in bed. More urgency, a sense of recklessness as they tangled together in a giddy heap on the office floor. He tugged her bra down so he could use his lips, his teeth, his tongue on her breasts until her hips began to pump. She closed her hand around him, found him hard, made him groan.\n\nHe couldn't wait, not this time. He couldn't savor; needed to take. He rolled, dragging her over so she could straddle him. Even as he gripped her hips, she was rising. She was taking him in. When she leaned forward for a greedy kiss, her hair fell to curtain their faces. Surrounded by her, he thought. Her body, her scent, her energy. He stroked the line of her back, the curve of her hips as she rocked and rocked and rocked him through pleasure toward desperation.\n\nEven when she arched back, even with his vision blurred, the shape of her, the tones of her enthralled him.\n\nShe let herself go, simply steeped herself in sensation. Hammering pulses and speed, slick bodies and dazzling friction. She felt him come, that sudden, sharp jerk of his hips, and was thrilled. She had driven him to lose control first, she had taken him over. And now she used that power, that thrill, to drive herself over that same edgy peak.\n\nShe slid down from it, and onto him so they could lie there, heated, a little stunned, until they got their breath back. And she began to laugh.\n\n\"God, we're like a couple of teenagers. Or rabbits.\"\n\n\"Teenage rabbits.\"\n\nAmused, she levered up. \"Do you often multitask in your office like this?\"\n\n\"Ah...\"\n\nShe gave him a little poke as she tugged her bra back in place. \"See, unpredictable.\"\n\nHe held out her shirt. \"It's the first time I've multitasked in this way during working hours.\"\n\nHer lips curved as she buttoned her shirt. \"That's nice.\"\n\n\"And I haven't felt like a teenage rabbit since I was.\"\n\nShe leaned over to give him a quick peck on the lips. \"Even nicer.\" Still on the floor, she scooted into her pants as he did the same. \"I should tell you something.\" She reached for her boots, pulled one on. \"I think...No, saying 'I think' is a cop-out, it's the coward's way.\"\n\nShe took a deep breath, yanked on the other boot, then looked him dead in the eye. \"I'm in love with you.\"\n\nThe shock came first\u2014fast, arrow-point shock straight to the gut. Then the concern wrapped in a slippery fist of fear. \"Quinn\u2014\"\n\n\"Don't waste your breath with the 'we've only known each other a couple of weeks' gambit. And I really don't want to hear the 'I'm flattered, but,' either. I didn't tell you so you could say anything. I told you because you should know. So first, it doesn't matter how long we've known each other. I've known me a long time, and I know me very well. I know what I feel when I feel it. Second, you should be flattered, goes without saying. And there's no need to freak out. You're not obligated or expected to feel what I feel.\"\n\n\"Quinn, we're\u2014all of us\u2014are under a lot of pressure. We don't even know if we'll make it through to August. We can't\u2014\"\n\n\"Exactly so. Nobody ever knows that, but we have more reason to worry about it. So, Cal.\" She framed his face with her hands. \"The moment's important. The right-this-minute matters a whole hell of a lot. I doubt I'd have told you otherwise, though I can be impulsive. But I think, under other circumstances, I'd have waited for you to catch up. I hope you do, but in the meantime, things are just fine the way they are.\"\n\n\"You have to know I\u2014\"\n\n\"Don't, absolutely don't tell me you care about me.\" The first hint of anger stung her voice. \"Your instinct is to say all the cliches people babble out in cases like this. They'll only piss me off.\"\n\n\"Okay, all right, let me just ask this, without you getting pissed off. Have you considered what you're feeling might be something like what happened in the clearing? That it's, say, a reflection of what Ann felt for Dent?\"\n\n\"Yes, and it's not.\" She pushed to her feet, drew on her sweater. \"Good question though. Good questions don't piss me off. What she felt, and I felt through that, was intense and consuming. I'm not going to say some of what I feel for you isn't like that. But it was also painful, and wrenching. Under the joy was grief. That's not this, Cal. This isn't painful. I don't feel sad. So...do you have time to come down and grab some lunch before Cyb and Layla and I head out?\"\n\n\"Ah...sure.\"\n\n\"Great. Meet you down there. I'm going to pop in the bathroom and fix myself up a little.\"\n\n\"Quinn.\" He hesitated as she opened the door, turned back. \"I've never felt like this about anyone before.\"\n\n\"Now that is a very acceptable thing to say.\"\n\nShe smiled as she strolled away. If he'd said it, he meant it, because that was the way he was. Poor guy, she thought. Didn't even know he was caught.\n\nA THICK GROVE OF TREES SHIELDED THE OLD cemetery on the north side. It fanned out over bumpy ground, with hills rolling west, at the end of a dirt road barely wide enough for two cars to pass. A historical marker faded by weather stated the First Church of the Godly had once stood on the site, but had been destroyed when it had been struck by lightning and razed by fire on July 7, 1652.\n\nQuinn had read that fact in her research, but it was different to stand here now, in the wind, in the chill, and imagine it. She'd read, too, as the plaque stated, that a small chapel had stood as a replacement until it was damaged during the Civil War, and gone to ruin.\n\nNow, there were only the markers here, the stones, the winter-hardy weeds. Beyond a low stone wall were the graves of the newer dead. Here and there she saw bright blots of color from flowers that stood out like grief against the dull grays and winter browns.\n\n\"We should've brought flowers,\" Layla said quietly as she looked down at the simple and small stone that read only:\n\nANN HAWKINS\n\n\"She doesn't need them,\" Cybil told her. \"Stones and flowers, they're for the living. The dead have other things to do.\"\n\n\"Cheery thought.\"\n\nCybil only shrugged at Quinn. \"I think so, actually. No point in being dead and bored. It's interesting, don't you think, that there are no dates. Birth or death. No sentiment. She had three sons, but they didn't have anything but her name carved in her gravestone. Even though they're buried here, too, with their wives, and I imagine at least some of their children. Wherever they went in life, they came home to be buried with Ann.\"\n\n\"Maybe they knew, or believed, she'd be back. Maybe she told them death isn't the end.\" Quinn frowned at the stone. \"Maybe they just wanted to keep it simple, but I wonder, now that you mention it, if it was deliberate. No beginning, no end. At least not until...\"\n\n\"This July,\" Layla finished. \"Another cheery thought.\"\n\n\"Well, while we're all getting cheered up, I'm going to get some pictures.\" Quinn pulled out her camera. \"Maybe you two could write down some of the names here. We may want to check on them, see if any have any direct bearing on\u2014\"\n\nShe tripped while backing up to get a shot, fell hard on her ass. \"Ouch, goddamn it! Shit. Right on the bruise I got this morning. Perfect.\"\n\nLayla rushed over to help her up. Cybil did the same, even as she struggled with laughter.\n\n\"Just shut up,\" Quinn grumbled. \"The ground's all bumpy here, and you can hardly see some of these stones popping out.\" She rubbed her hip, scowled down at the stone that had tripped her up. \"Ha. That's funny. Joseph Black, died eighteen forty-three.\" The color annoyance brought to her face faded. \"Same last name as mine. Common name Black, really. Until you consider it's here, and that I just happened to trip over his grave.\"\n\n\"Odds are he's one of yours,\" Cybil agreed.\n\n\"And one of Ann's?\"\n\nQuinn shook her head at Layla's suggestion. \"I don't know. Cal's researched the Hawkins's family tree, and I've done a quick overview. I know some of the older records are lost, or just buried deeper than we've dug, but I don't see how we'd both have missed branches with my surname. So. I think we'd better see what we can find out about Joe.\"\n\nHER FATHER WAS NO HELP, AND THE CALL HOME kept her on the phone for forty minutes, catching up on family gossip. She tried her grandmother next, who had a vague recollection about her mother-in-law mentioning an uncle, possibly a great-uncle, maybe a cousin, who'd been born in the hills of Maryland. Or it might've been Virginia. His claim to fame, family-wise, had been running off with a saloon singer, deserting his wife and four children and taking the family savings held inside a cookie tin with him.\n\n\"Nice guy, Joe,\" Quinn decided. \"Should you be my Joe.\"\n\nShe decided, since it would get her out of any type of food preparation, she had enough time to make a trip to Town Hall, and start digging on Joseph Black. If he'd died here, maybe he'd been born here.\n\nWHEN QUINN GOT HOME SHE WAS GLAD TO FIND the house full of people, sound, the scents of food. Cybil, being Cybil, had music on, candles lit, and wine poured. She had everyone piled in the kitchen, whetting appetites with marinated olives. Quinn popped one, took Cal's wine and washed it down.\n\n\"Are my eyes bleeding?\" she asked.\n\n\"Not so far.\"\n\n\"I've been searching records for nearly three hours. I think I bruised my brain.\"\n\n\"Joseph Black.\" Fox got her a glass of wine for her own. \"We've been filled in.\"\n\n\"Good, saves me. I could only trace him back to his grandfather\u2014Quinton Black, born sixteen seventy-six. Nothing on record before that, not here anyway. And nothing after Joe, either. I went on side trips, looking for siblings or other relatives. He had three sisters, but I've got nothing on them but birth records. He had aunts, uncles, and so on, and not much more there. It appears the Blacks weren't a big presence in Hawkins Hollow.\"\n\n\"Name would've rung for me,\" Cal told her.\n\n\"Yeah. Still, I got my grandmother's curiosity up, and she's now on a hunt to track down the old family Bible. She called me on my cell. She thinks it went to her brother-in-law when his parents died. Maybe. Anyway, it's a line.\"\n\nShe focused on the man leaning back against the counter toying with a glass of wine. \"Sorry? Gage, right?\"\n\n\"That's right. Roadside service a specialty.\"\n\nQuinn grinned as Cybil rolled her eyes and took a loaf of herbed bread out of the oven.\n\n\"So I hear, and that looks like dinner's ready. I'm starved. Nothing like searching through the births and deaths of Blacks, Robbits, Clarks to stir up the appetite.\"\n\n\"Clark.\" Layla lowered the plate she'd taken out to offer Cybil for the bread. \"There were Clarks in the records?\"\n\n\"Yeah, an Alma and a Richard Clark in there, as I remember. Need to check my notes. Why?\"\n\n\"My grandmother's maiden name was Clark.\" Layla managed a wan smile. \"That's probably not a coincidence either.\"\n\n\"Is she still living?\" Quinn asked immediately. \"Can you get in touch and\u2014\"\n\n\"We're going to eat while it's hot,\" Cybil interrupted. \"Time enough to give family trees a good shake later. But when I cook\u2014\" She pushed the plate of hot bread into Gage's hand. \"We eat.\"\n\n## Sixteen\n\nIT HAD TO BE IMPORTANT. IT HAD TO MATTER. Cal rolled it over and over and over, carving time out of his workday and his off time to research the Hawkins-Black lineage himself. Here was something new, he thought, some door they hadn't known existed, much less tried to break down.\n\nHe told himself it was vital, and time-consuming work, and that was why he and Quinn hadn't managed to really connect for the last couple of days. He was busy; she was busy. Couldn't be helped.\n\nBesides, it was probably a good time for them to have this break from each other. Let things just simmer down a little. As he'd told his mother, this wasn't the time to get serious, to think about falling in love. Because big, life-altering things were supposed to happen after people fell seriously in love. And he had enough, big, life-altering things to worry about.\n\nHe dumped food in Lump's bowl as his dog waited for breakfast with his usual unruffled patience. Because it was Thursday, he'd tossed a load of laundry in the washer when he'd let Lump out for his morning plod and pee. He continued his habitual weekday morning routine, nursing his first cup of coffee while he got out a box of Chex.\n\nBut when he reached for the milk it made him think of Quinn. Two percent milk, he thought with a shake of his head. Maybe she was fixing her version of a bowl of cereal right now. Maybe she was standing in her kitchen with the smell of coffee in the air, thinking of him.\n\nBecause the idea of that held such appeal, he reached for the phone to call her, when he heard the sound behind him and turned.\n\nGage got the coffee mug out of the cupboard he opened. \"Jumpy.\"\n\n\"No. I didn't hear you come in.\"\n\n\"You were mooning over a woman.\"\n\n\"I have a lot of things on my mind.\"\n\n\"Especially the woman. You've got tells, Hawkins. Starting with the wistful, cocker spaniel eyes.\"\n\n\"Up yours, Turner.\"\n\nGage merely grinned and poured coffee. \"Then there's that fish hook in the corner of your mouth.\" He hooked his finger in his own, gave a tug. \"Unmistakable.\"\n\n\"You're jealous because you're not getting laid regular.\"\n\n\"No question about that.\" Gage sipped his black coffee, used one bare foot to rub Lump's flank as the dog concentrated his entire being on his kibble. \"She's not your usual type.\"\n\n\"Oh?\" Irritation crawled up Cal's back like a lizard. \"What's my usual type?\"\n\n\"Pretty much same as mine. Keep it light, no deep thinking, no strings, no worries. Who could blame us, considering?\" He picked up the cereal, dug right into the box. \"But she breaks your mold. She's smart, she's steady, and she's got a big, fat ball of string in her back pocket. She's already started wrapping you in it.\"\n\n\"Does that cynicism you carry around everywhere ever get heavy?\"\n\n\"Realism,\" Gage corrected as he munched on cereal. \"And it keeps me light on my feet. I like her.\"\n\n\"I do, too.\" Cal forgot the milk and just took a handful of cereal out of the bowl he'd poured. \"She...she told me she's in love with me.\"\n\n\"Fast work. And now she's suddenly pretty damn busy, and you're sleeping alone, pal. I said she was smart.\"\n\n\"Jesus, Gage.\" Insult bloomed on two stalks\u2014one for himself, one for Quinn. \"She's not like that. She doesn't use people like that.\"\n\n\"And you know this because you know her so well.\"\n\n\"I do.\" Any sign of irritation faded as that simple truth struck home. \"That's just it. I do know her. There may be dozens, hell, hundreds of things I don't know, but I know who\u2014how\u2014she is. I don't know if some of that's because of this connection, because of what we're all tied to, but I know it's true. The first time I met her, things changed. I don't know. Something changed for me. So you can make cracks, but that's the way it is.\"\n\n\"I'm going to say you're lucky,\" Gage said after a moment. \"That I hope it works out the way you want. I never figured any of us had a decent shot at normal.\" He shrugged. \"Wouldn't mind being wrong. Besides, you look real cute with that hook in your mouth.\"\n\nCal lifted his middle finger off the bowl and into the air.\n\n\"Right back atcha,\" Fox said as he strolled in. He went straight to the refrigerator for a Coke. \"What's up?\"\n\n\"What's up is you're mooching my Cokes again, and you never bring any to replace them.\"\n\n\"I brought beer last week. Besides, Gage told me to come over this morning, and when I come over in the morning, I expect a damn Coke.\"\n\n\"You told him to come over?\"\n\n\"Yeah. So, O'Dell, Cal's in love with the blonde.\"\n\n\"I didn't say I\u2014\"\n\n\"Tell me something I don't know.\" Fox popped the top on the can of Coke and gulped.\n\n\"I never said I was in love with anyone.\"\n\nFox merely shifted his gaze to Cal. \"I've known you my whole life. I know what those shiny little hearts in your eyes mean. It's cool. She was, like, made for you.\"\n\n\"He says she's not my usual type, you say she's made for me.\"\n\n\"We're both right. She's not the type you usually fish for.\" Fox gulped down more soda, then took the box of cereal from Gage. \"Because you didn't want to find the one who fit. She fits, but she was sort of a surprise. Practically an ambush. Did I get up an hour early to come over here before work so we could talk about Cal's love life?\"\n\n\"No, it was just an interesting sidebar. I got some information when I was in the Czech Republic. Rumors, lore, mostly, which I followed up when I had time. I got a call from an expert last night, which is why I told you to come over this morning. I might have ID'd our Big Evil Bastard.\"\n\nThey sat down at the kitchen table with coffee and dry cereal\u2014Fox in one of his lawyer suits, Gage in a black T-shirt and loose pants, Cal in jeans and a flannel shirt.\n\nAnd spoke of demons.\n\n\"I toured some of the smaller and outlying villages,\" Gage began. \"I always figure I might as well pick up some local color, maybe a local skirt while I'm stacking up poker chips and markers.\"\n\nHe'd been doing the same for years, Cal knew. Following any whiff of information about devils, demons, unexplained phenomenon. He always came back with stories, but nothing that had ever fit the, well, the profile, Cal supposed, of their particular problem.\n\n\"There was talk about this old demon who could take other forms. You get werewolf stuff over there, and initially, I figured that was this deal. But this wasn't about biting throats out and silver bullets. The talk was about how this thing hunted humans to enslave them, and feed off their...the translation was kind of vague, and the best I got was essence, or humanity.\"\n\n\"Feed how?\"\n\n\"That's vague, too\u2014or colorful as lore tends to be. Not on flesh and bone, not with fang and claw\u2014that kind of thing. The legend is this demon, or creature, could take people's minds as well as their souls, and cause them to go mad, cause them to kill.\"\n\n\"Could be the root of ours,\" Fox decided.\n\n\"It rang close enough that I followed it up. It was a lot to wade through; that area's ripe with stories like this. But in this place in the hills, with this thick forest that reminded me of home, I hit something. Its name is Tmavy. Translates to Dark. The Dark.\"\n\nHe thought, they all thought of what had come out of the ground at the Pagan Stone. \"It came like a man who wasn't a man, hunted like a wolf that wasn't a wolf. And sometimes it was a boy, a boy who lured women and children in particular into the forest. Most never came back, and those who did were mad. The families of those who did went mad, too. Killed each other, or themselves, their neighbors.\"\n\nGage paused, rose to get the coffeepot. \"I got some of this when I was there, but I found a priest who gave me the name of a guy, a professor, who studied and publishes on Eastern European demonology. He got in touch last night. He claims this particular demon\u2014and he isn't afraid to use the word\u2014roamed Europe for centuries. He, in turn, was hunted by a man\u2014some say another demon, or a wizard, or just a man with a mission. Legend has it that they battled in the forest, and the wizard was mortally wounded, left for dead. And that, according to Professor Linz, was its mistake. Someone came, a young boy, and the wizard passed the boy his power before he died.\"\n\n\"What happened?\" Fox demanded.\n\n\"No one, including Linz, is sure. The stories claim the thing vanished, or moved on, or died, somewhere in the early-to mid-seventeenth century.\"\n\n\"When he hopped a goddamn boat for the New World,\" Cal added.\n\n\"Maybe. That may be.\"\n\n\"So did the boy,\" Cal continued, \"or the man he'd become, or his descendent. But he nearly had him over there, nearly did at some point in time\u2014that's something I've seen. I think. Him and the woman, a cabin. Him holding a bloody sword, and knowing nearly all were dead. He couldn't stop it there, so he passed what he had to Dent, and Dent tried again. Here.\"\n\n\"What did he pass to us?\" Fox demanded. \"What power? Not getting a freaking head cold, having a broken arm knit itself? What good does that do?\"\n\n\"Keeps us healthy and whole when we face it down. And there's the glimmers I see, that we all see in different ways.\" Cal shoved at his hair. \"I don't know. But it has to be something that matters. The three parts of the stone. They have to be. We've just never figured it out.\"\n\n\"And time's almost up.\"\n\nCal nodded at Gage. \"We need to show the stones to the others. We took an oath, we all have to agree to that. If we hadn't, I'd have\u2014\"\n\n\"Shown yours to Quinn already,\" Fox finished. \"And yeah, maybe you're right. It's worth a shot. It could be it needs all six of us to put it back together.\"\n\n\"Or it could be that when whatever happened at the Pagan Stone happened, the bloodstone split because its power was damaged. Destroyed.\"\n\n\"Your glass is always half empty, Turner,\" Fox commented. \"Either way, it's worth the try. Agreed?\"\n\n\"Agreed.\" Cal looked at Gage, who shrugged.\n\n\"What the hell.\"\n\nCAL DEBATED WITH HIMSELF ALL THE WAY INTO town. He didn't need an excuse to stop by to see Quinn. For God's sake, they were sleeping together. It wasn't as if he needed an appointment or clearance or a specific reason to knock on her door, to see how she was doing. To ask what the hell was going on.\n\nThere was no question she'd been distracted every time he'd managed to reach her by phone the last couple of days. She hadn't dropped into the center since they'd rolled around his office floor.\n\nAnd she'd told him she was in love with him.\n\nThat was the problem. The oil on the water, the sand in the shoe, or whatever goddamn analogy made the most sense. She'd told him she loved him, he hadn't said \"me, too,\" which she claimed she didn't expect. But any guy who actually believed a woman always meant exactly what she said was deep in dangerous delusion.\n\nNow, she was avoiding him.\n\nThey didn't have time for games, for bruised feelings and sulks. There were more important things at stake. Which, he was forced to admit, was why he shouldn't have touched her in the first place. By adding sex to the mix, they'd clouded and complicated the issue, and the issue was already clouded and complicated enough. They had to be practical; they had to be smart. Objective, he added as he pulled up in front of the rental house. Cold-blooded, clear-minded.\n\nNobody was any of those things when they were having sex. Not if they were having really good sex.\n\nHe jammed his hands in his pockets as he walked up to her door, then dragged one out to knock. The fact that he'd worked himself up to a mad might not have been objective or practical, but it felt absolutely right.\n\nUntil she opened the door.\n\nHer hair was damp. She'd pulled it back from her face in a sleek tail, and he could see it wasn't quite dry. He could smell the girly shampoo and soap, and the scents wound their way into him until the muscle in his gut tightened in response.\n\nShe wore fuzzy purple socks, black flannel pants, and a hot pink sweatshirt that announced: T.G.I.F. THANK GOD I'M FEMALE.\n\nHe could add his own thanks.\n\n\"Hi!\"\n\nThe idea she was sulking was hard to hang on to when he was blasted by her sunbeam smile and buzzing energy.\n\n\"I was just thinking about you. Come inside. Jesus, it's cold. I've so had it with winter. I was about to treat myself to a low-fat mug of hot chocolate. Want in on that?\"\n\n\"Ah\u2014I really don't.\"\n\n\"Well, come on back, because I've got the yen.\" She rose up on her toes to give him a long, solid kiss, then grabbed his hand to pull him back to the kitchen. \"I nagged Cyb and Layla into going to the gym with me this morning. Took some doing with Cyb, but I figured safety in numbers. Nothing weird happened, unless you count watching Cyb twist herself into some advanced yoga positions. Which Matt did, let me tell you. Things have been quiet in the otherworldly sense the last couple days.\"\n\nShe got out a packet of powdered mix, slapped it against her hand a couple of times to settle it before ripping it open to pour it into a mug. \"Sure you don't want some?\"\n\n\"Yeah, go ahead.\"\n\n\"We've been a busy hive around here,\" she went on as she filled the mug, half with water, half with two percent milk. \"I'm waiting to hear something about the family Bible, or whatever else my grandmother might dig up. Today, maybe, hopefully by tomorrow. Meanwhile, we've got charts of family trees as we know them, and Layla's trying to shake some ancestry out of her relatives.\"\n\nShe stirred up the liquid and mix, stuck it in the microwave. \"I had to leave a lot of the research up to my partners in crime and finish an article for the magazine. Gotta pay the doorman, after all. So?\" She turned back as the microwave hummed. \"How about you?\"\n\n\"I missed you.\" He hadn't planned to say it, certainly hadn't expected it to be the first thing out of his mouth. Then he realized, it was obviously the first thing on his mind.\n\nHer eyes went soft; that sexy mouth curved up. \"That's nice to hear. I missed you, too, especially last night when I crawled into bed about one in the morning. My cold, empty bed.\"\n\n\"I didn't just mean the sex, Quinn.\" And where had that come from?\n\n\"Neither did I.\" She angled her head, ignoring the beep of the microwave. \"I missed having you around at the end of the day, when I could finally come down from having to hammer out that article, when I wanted to stop thinking about what I had to do, and what was going to happen. You're irritated about something. Why don't you tell me what it is?\"\n\nShe turned toward the microwave as she spoke to get her mug out. Cal saw immediately she'd made the move as Cybil was stepping through the kitchen doorway. Quinn merely shook her head, and Cybil stepped back and retreated without a word.\n\n\"I don't know, exactly.\" He pulled off his coat now, tossed it over one of the chairs around a little cafe table that hadn't been there on his last visit. \"I guess I thought, after the other day, after...what you said\u2014\"\n\n\"I said I was in love with you. That makes you quiver inside,\" she noted. \"Men.\"\n\n\"I didn't start avoiding you.\"\n\n\"You think\u2014\" She took a deep inhale through her nose, exhaled in a huff. \"Well, you have a really high opinion of yourself, and a crappy one of me.\"\n\n\"No, it's just\u2014\"\n\n\"I had things to do, I had work. I am not at your beck any more than you're at mine.\"\n\n\"That's not what I meant.\"\n\n\"You think I'd play games like that? Especially now?\"\n\n\"Especially now's the point. This isn't the time for big personal issues.\"\n\n\"If not now, when?\" she demanded. \"Do you really, do you honestly think we can label and file all our personal business and close it in a drawer until it's convenient? I like things in their place, too. I want to know where things are, so I put them where I want or need them to be. But feelings and thoughts are different from the goddamn car keys, Cal.\"\n\n\"No argument, but\u2014\"\n\n\"And my feelings and thoughts are as cluttered and messy as Grandma's attic,\" she snapped out, far from winding down. \"That's just the way I like it. If things were normal every day, bopping right along, I probably wouldn't have told you. Do you think this is my first cannonball into the Dating and Relationship Pool? I was engaged, for God's sake. I told you because\u2014because I think, maybe especially now, that feelings are what matter most. If that screws you up, too damn bad.\"\n\n\"I wish you'd shut up for five damn minutes.\"\n\nHer eyes went to slits. \"Oh, really?\"\n\n\"Yeah. The fact is I don't know how to react to all of this, because I never let myself consider being in this position. How could I, with this hanging over my head? Can't risk falling for someone. How much could I tell her? How much is too much? We're\u2014Fox and Gage and I\u2014we're used to holding back, to keeping big pieces of this to ourselves.\"\n\n\"Keeping secrets.\"\n\n\"That's right,\" he said equably. \"That's exactly right. Because it's safer that way. How could I ever think about falling in love, getting married, having kids? Bringing a kid into this nightmare's out of the question.\"\n\nThose slitted blue eyes went cold as winter. \"I don't believe I've yet expressed the wish to bear your young.\"\n\n\"Remember who you're talking to,\" he said quietly. \"You take this situation out of the equation you've got a normal guy from a normal family. The kind who gets married, raises a family, has a mortgage and a big sloppy dog. If I let myself fall in love with a woman, that's how it's going to work.\"\n\n\"I guess you told me.\"\n\n\"And it's irresponsible to even consider any of that.\"\n\n\"We disagree. I happen to think considering that, moving toward that, is shooting the bird at the dark. In the end, we're each entitled to our own take on it. But understand me, get this crystal, telling you I love you didn't mean I expected you to pop a ring on my finger.\"\n\n\"Because you've been there.\"\n\nShe nodded. \"Yes, I have. And you're wondering about that.\"\n\n\"None of my business.\" Screw it. \"Yes.\"\n\n\"Okay, it's simple enough. I was seeing Dirk\u2014\"\n\n\"Dirk\u2014\"\n\n\"Shut up.\" But her lips twitched. \"I was seeing him exclusively for about six months. We enjoyed each other. I thought I was ready for the next stage in my life, so I said yes when he asked me to marry him. We were engaged for two months when I realized I'd made a mistake. I didn't love him. Liked him just fine. He didn't love me, either. He didn't really get me\u2014not the whole of me, which was why he figured the ring on my finger meant he could begin to advise me on my work, on my wardrobe, habits, and career options. There were a lot of little things, and they're not really important. The fact was we weren't going to make it work, so I broke it off.\"\n\nShe blew out another breath because it wasn't pleasant to remember she'd made that big a mistake. That she'd failed at something she knew she'd be good at. \"He was more annoyed than brokenhearted, which told me I'd done the right thing. And the truth is, it stung to know I'd done the right thing, because it meant I'd done the wrong thing first. When I suggested he tell his friends he'd been the one to end it, he felt better about it. I gave him back the ring, we each boxed up things we'd kept in each other's apartments, and we walked away.\"\n\n\"He didn't hurt you.\"\n\n\"Oh, Cal.\" She took a step closer so she could touch his face. \"No, he didn't. The situation hurt me, but he didn't. Which is only one of the reasons I knew he wasn't the one. If you want me to reassure you that you can't, that you won't break my heart, I just can't do it. Because you can, you might, and that's how I know you are. The one.\" She slipped her arms around him, laid her lips on his. \"That must be scary for you.\"\n\n\"Terrifying.\" He pulled her against him, held her hard. \"I've never had another woman in my life who's given me as many bad moments as you.\"\n\n\"I'm delighted to hear it.\"\n\n\"I thought you would be.\" He laid his cheek on top of her head. \"I'd like to stay here, just like this, for an hour or two.\" He replaced his cheek with his lips, then eased back. \"But I've things I have to do, and so do you. Which I knew before I walked in here and used it as an excuse to pick a fight.\"\n\n\"I don't mind a fight. Not when the air's clear afterward.\"\n\nHe framed her face with his hands, kissed her softly. \"Your hot chocolate's getting cold.\"\n\n\"Chocolate's never the wrong temperature.\"\n\n\"The one thing I said before? Absolute truth. I missed you.\"\n\n\"I believe I can arrange some free time in my busy schedule.\"\n\n\"I have to work tonight. Maybe you could stop in. I'll give you another bowling lesson.\"\n\n\"All right.\"\n\n\"Quinn, we\u2014all of us\u2014have to talk. About a lot of things. As soon as we can.\"\n\n\"Yes, we do. One thing before you go. Is Fox going to offer Layla a job?\"\n\n\"I said something to him.\" Cal swore under his breath at her expression. \"I'll give him another push on it.\"\n\n\"Thanks.\"\n\nAlone, Quinn picked up her mug, thoughtfully sipped at her lukewarm chocolate. Men, she thought, were such interesting beings.\n\nCybil came in. \"All clear?\"\n\n\"Yeah, thanks.\"\n\n\"No problem.\" She opened a cupboard and chose a small tin of loose jasmine tea from her supply. \"Discuss or mind my own?\"\n\n\"Discuss. He was worked up because I told him I love him.\"\n\n\"Annoyed or panicked?\"\n\n\"Some of both, I think. More worried because we've all got scary things to deal with, and this is another kind of scary thing.\"\n\n\"The scariest, when you come down to it.\" Cybil filled the teakettle with water. \"How are you handling it?\"\n\n\"It feels...great,\" she decided. \"Energizing and bouncy and bright, then sort of rich and glimmering. You know, with Dirk it was all...\" Quinn held out a hand, drawing it level through the air. \"This was\u2014\" She shot her hand up, down, then up again. \"Here's a thing. When he's telling me why this is crazy, he says how he's never been in a position\u2014or so he thinks\u2014to let himself think about love, marriage, family.\"\n\n\"Whoa, point A to Z in ten words or less.\"\n\n\"Exactly.\" Quinn gestured with her mug. \"And he was rolling too fast to see that the M word gave me a serious jolt. I practically just jumped off that path, and whoops, there it is again, under my feet.\"\n\n\"Hence the jolt.\" Cybil measured out her tea. \"But I don't see you jumping off.\"\n\n\"Because you know me. I like where my feet are, as it turns out. I like the idea of heading down that path with Cal, toward wherever it ends up. He's in trouble now,\" she murmured and took another sip.\n\n\"So are you, Q. But then trouble's always looked good on you.\"\n\n\"Better than a makeover at the Mac counter at Saks.\" Quinn answered the kitchen phone on its first ring. \"Hello. Hello, Essie. Oh. Really? No, it's great. It's perfect. Thanks so much. I absolutely will. Thanks again. Bye.\" She hung up, grinned. \"Essie Hawkins got us into the community center. No business there today on the main level. We can go in, poke around to our hearts' content.\"\n\n\"Won't that be fun?\" Cybil said it dryly as she poured boiling water for her tea.\n\nARMED WITH THE KEY, CYBIL OPENED THE MAIN door of the old library. \"We're here, on the surface, for research. One of the oldest buildings in town, home of the Hawkins family. But...\" She switched on the lights. \"Primarily we're looking for hidey-holes. A hiding place that was overlooked.\"\n\n\"For three and a half centuries,\" Cybil commented.\n\n\"If something's overlooked for five minutes, it can be overlooked forever.\" Quinn pursed her lips as she looked around. \"They modernized it, so to speak, when they turned it into a library, but when they built the new one, they stripped out some of the newfangled details. It's not the way it was, but it's closer.\"\n\nThere were some tables and chairs set up, and someone had made an attempt at some old-timey decor in the antique old lamps, old pottery, and wood carvings on shelves. Quinn had been told groups like the Historical Society or the Garden Club could hold meetings or functions here. At election times it was a voting center.\n\n\"Stone fireplace,\" she said. \"See, that's an excellent place to hide something.\" After crossing to it, she began to poke at the stones. \"Plus there's an attic. Essie said they used it for storage. Still do. They keep the folding tables and chairs up there, and that kind of thing. Attics are treasure troves.\"\n\n\"Why is it buildings like this are so cold and creepy when no one's in them?\" Layla wondered.\n\n\"We're in this one. Let's start at the top,\" Quinn suggested, \"work our way down.\"\n\n\"ATTICS ARE TREASURE TROVES,\" CYBIL SAID twenty minutes later, \"of dust and spiders.\"\n\n\"It's not that bad.\" Quinn crawled along, hoping for a loose floorboard.\n\n\"Not that good either.\" Courageously, Layla stood on a folding chair, checking rafters. \"I don't understand why people don't think storage spaces shouldn't be cleaned as regularly as anyplace else.\"\n\n\"It was clean once. She kept it clean.\"\n\n\"Who\u2014\" Layla began, but Cybil waved a hand at her, frowned at Quinn.\n\n\"Ann Hawkins?\"\n\n\"Ann and her boys. She brought them home, and shared the attic with them. Her three sons. Until they were old enough to have a room downstairs. But she stayed here. She wanted to be high, to be able to look out of her window. Even though she knew he wouldn't come, she wanted to look out for him. She was happy here, happy enough. And when she died here, she was ready to go.\"\n\nAbruptly, Quinn sat back on her heels. \"Holy shit, was that me?\"\n\nCybil crouched down to study Quinn's face. \"You tell us.\"\n\n\"I guess it was.\" She pressed her fingers to her forehead. \"Damn, got one of those I-drank-my-frozen-margarita-too-fast-and-now-have-an-ice pick-through-my-brain headaches. I saw it, her, them, in my head. Just as clear. Everything moving, like a time-action camera. Years in seconds. But more, I felt it. That's the way it is for you, isn't it\u2014going the other way?\"\n\n\"Often,\" Cybil agreed.\n\n\"I saw her writing in her journal, and washing her sons' faces. I saw her laughing, or weeping. I saw her standing at the window looking into the dark. I felt...\" Quinn laid a hand on her heart. \"I felt her longing. It was...brutal.\"\n\n\"You don't look well.\" Layla touched her shoulder. \"We should go downstairs, get you some water.\"\n\n\"Probably. Yeah.\" She took the hand Layla offered to help her up. \"Maybe I should try it again. Try to bring it back, get more.\"\n\n\"You're awfully pale,\" Layla told her. \"And, honey, your hand's like ice.\"\n\n\"Plenty for one day,\" Cybil agreed. \"You don't want to push it.\"\n\n\"I didn't see where she put the journals. If she put anything here, I didn't see.\"\n\n## Seventeen\n\nIT WASN'T THE TIME, CAL DETERMINED, TO TALK about a broken stone or property searches when Quinn was buzzed about her trip to the past with Ann Hawkins. In any case, the bowling center wasn't the place for that kind of exchange of information.\n\nHe considered bringing it up after closing when she dragged him into her home office to show him the new chart Layla had generated that listed the time, place, approximate duration, and involved parties in all known incidents since Quinn's arrival.\n\nHe forgot about it when he was in bed with her, when she was moving with him, when everything felt right again.\n\nThen he told himself it was too late to bring it up, to give the topics the proper time when she was curled up warm with him.\n\nMaybe it was avoidance, but he opted for the likelihood it was just his tendency to prefer things at the right time, in the right place. He'd arranged to take Sunday off so the entire group could hike to the Pagan Stone. That, to his mind, was the right time and place.\n\nThen Nature screwed with his plans.\n\nWhen forecasters began to predict an oncoming blizzard, he kept a jaundiced eye on the reports. They were, in his experience, wrong at least as often as they were right. Even when the first flakes began to fall midmorning, he remained unconvinced. It was the third blizzard hype of the year, and so far the biggest storm had dumped a reasonable eight inches.\n\nHe shrugged it off when the afternoon leagues canceled. It had gotten so people canceled everything at the first half inch, then went to war over bread and toilet paper in the supermarket. And since the powers-that-be canceled school before noon, the arcade and the grill were buzzing.\n\nBut when his father came in about two in the afternoon, looking like Sasquatch, Cal paid more attention.\n\n\"I think we're going to close up shop,\" Jim said in his easy way.\n\n\"It's not that bad. The arcade's drawing the usual suspects, the grill's been busy. We've had some lanes booked. A lot of towners will come in later in the afternoon, looking for something to do.\"\n\n\"It's bad enough, and it's getting worse.\" Jim shoved his gloves in the pocket of his parka. \"We'll have a foot by sundown the way it's going. We need to send these kids home, haul them there if they don't live within easy walking distance. We'll close up, then you go on home, too. Or you get your dog and Gage and come on over and stay with us. Your mother'll worry sick if she thinks you're out driving in this at night.\"\n\nHe started to remind his father that he was thirty, had four-wheel drive, and had been driving nearly half his life. Knowing it was pointless, Cal just nodded. \"We'll be fine. I've got plenty of supplies. I'll clear out the customers, close up, Dad. You go on home. She'll worry about you, too.\"\n\n\"There's time enough to close down and lock up.\" Jim glanced over at the lanes where a six-pack of teenagers sent off energy and hormones in equal measure. \"Had a hell of a storm when I was a kid. Your grandfather kept her open. We stayed here for three days. Time of my life.\"\n\n\"I bet.\" Cal grinned. \"Want to call Mom, say we're stuck? You and me can ride it out. Have a bowling marathon.\"\n\n\"Damned if I wouldn't.\" The lines around Jim's eyes crinkled at the idea. \"Of course, she'll kick my ass for it and it'd be the last time I bowled.\"\n\n\"Better shut down then.\"\n\nThough there were protests and moans, they moved customers along, arranging for rides when necessary with some of the staff. In the silence, Cal shut down the grill himself. He knew his father had gone back to check with Bill Turner. Not just to give instructions, he thought, but to make sure Bill had whatever he needed, to slip him a little extra cash if he didn't.\n\nAs he shut down, Cal pulled out his phone and called Fox's office. \"Hey. Wondered if I'd catch you.\"\n\n\"Just. I'm closing. Already sent Mrs. H home. It's getting bad out there.\"\n\n\"Head over to my place. If this comes in like they're whining about, it might be a couple days before the roads are clear. No point wasting them. And maybe you should stop, pick up, you know, toilet paper, bread.\"\n\n\"Toilet...You're bringing the women?\"\n\n\"Yeah.\" He'd made up his mind on that when he'd taken a look outside. \"Get...stuff. Figure it out. I'll be home as soon as I can.\"\n\nHe clicked off, then shut down the alley lights as his father came out.\n\n\"Everything set?\" Cal asked.\n\n\"Yep.\"\n\nThe way his father looked around the darkened alley told Cal he was thinking they weren't just going to lose their big Friday night, but likely the entire weekend.\n\n\"We'll make it up, Dad.\"\n\n\"That's right. We always do.\" He gave Cal a slap on the shoulder. \"Let's get home.\"\n\nQUINN WAS LAUGHING WHEN SHE OPENED THE door. \"Isn't this great! They say we could get three feet, maybe more! Cyb's making goulash, and Layla went out and picked up extra batteries and candles in case we lose power.\"\n\n\"Good. Great.\" Cal stomped snow off his boots. \"Pack it up and whatever else you all need. We're going to my place.\"\n\n\"Don't be silly. We're fine. You can stay, and we'll\u2014\"\n\nAs clear of snow as he could manage, he stepped in, shut the door behind him. \"I have a small gas generator that'll run little things\u2014such as the well, which means water to flush the toilets.\"\n\n\"Oh. Toilets. I hadn't thought of that one. But how are we all going to fit in your truck?\"\n\n\"We'll manage. Get your stuff.\"\n\nIt took them half an hour, but he'd expected that. In the end, the bed of his truck was loaded with enough for a week's trek through the wilderness. And three women were jammed with him in the cab.\n\nHe should've had Fox swing by, get one of them, he realized. Then Fox could've hauled half the contents of their house in his truck. And it was too late now.\n\n\"It's gorgeous.\" Layla perched on Quinn's lap, bracing a hand on the dash while the Chevy's windshield wipers worked overtime to clear the snow from the glass. \"I know it's going to be a big mess, but it's so beautiful, so different than it is in the city.\"\n\n\"Remember that when we're competing for bathroom time with three men,\" Cybil warned her. \"And let me say right now, I refuse to be responsible for all meals just because I know how to turn on the stove.\"\n\n\"So noted,\" Cal muttered.\n\n\"It is gorgeous,\" Quinn agreed, shifting her head from side to side to see around Layla. \"Oh, I forgot. I heard from my grandmother. She tracked down the Bible. She's having her sister-in-law's granddaughter copy and scan the appropriate pages, and e-mail them to me.\" Quinn wiggled to try for more room. \"At least that's the plan, as the granddaughter's the only one of them who understands how to scan and attach files. E-mail and online poker's as far as Grandma goes on the Internet. I hope to have the information by tomorrow. Isn't this great?\"\n\nWedged between Quinn's butt and the door, Cybil dug in to protect her corner of the seat. \"It'd be better if you'd move your ass over.\"\n\n\"I've got Layla's space, too, so I get more room. I want popcorn,\" Quinn decided. \"Doesn't all this snow make everyone want popcorn? Did we pack any? Do you have any?\" she asked Cal. \"Maybe we could stop and buy some Orville's.\"\n\nHe kept his mouth shut, and concentrated on surviving what he thought might be the longest drive of his life.\n\nHe plowed his way down the side roads, and though he trusted the truck and his own driving, was relieved when he turned onto his lane. As he'd been outvoted about the heat setting, the cab of the truck was like a sauna.\n\nEven under the circumstances, Cal had to admit his place, his woods, did look like a picture. The snow-banked terraces, the white-decked trees and huddles of shrubs framed the house where smoke was pumping from the chimney, and the lights were already gleaming against the windows.\n\nHe followed the tracks of Fox's tires across the little bridge over his snow-and ice-crusted curve of the creek.\n\nLump padded toward the house from the direction of the winter-postcard woods, leaving deep prints behind him. His tail swished once as he let out a single, hollow bark.\n\n\"Wow, look at Lump.\" Quinn managed to poke Cal with her elbow as the truck shoved its way along the lane. \"He's positively frisky.\"\n\n\"Snow gets him going.\" Cal pulled behind Fox's truck, smirked at the Ferrari, slowly being buried, then laid on the horn. He'd be damned if he was going to haul the bulk of what three women deemed impossible to live without for a night or two.\n\nHe dragged bags out of the bed.\n\n\"It's a beautiful spot, Cal.\" Layla took the first out of his hands. \"Currier and Ives for the twenty-first century. Is it all right if I go right in?\"\n\n\"Sure.\"\n\n\"Pretty as a picture.\" Cybil scanned the bags and boxes, chose one for herself. \"Especially if you don't mind being isolated.\"\n\n\"I don't.\"\n\nShe glanced over as Gage and Fox came out of the house. \"I hope you don't mind crowds either.\"\n\nThey got everything inside, trailing snow everywhere. Cal decided it must have been some sort of female telepathy that divided them all into chores without discussion. Layla asked him for rags or old towels and proceeded to mop up the wet, Cybil took over the kitchen with her stew pot and bag of kitchen ingredients. And Quinn dug into his linen closet, such as it was, and began assigning beds, and ordering various bags carried to various rooms.\n\nThere wasn't anything for him to do, really, but have a beer.\n\nGage strode in as Cal poked at the fire. \"There are bottles of girl stuff all over both bathrooms up there.\" Gage jerked a thumb at the ceiling. \"What have you done?\"\n\n\"What had to be done. I couldn't leave them. They could've been cut off for a couple of days.\"\n\n\"And what, turned into the next Donner Party? Your woman has Fox making my bed, which is now the pullout in your office. And which I'm apparently supposed to share with him. You know that son of a bitch is a bed hog.\"\n\n\"Can't be helped.\"\n\n\"Easy for you to say, seeing as you'll be sharing yours with the blonde.\"\n\nThis time Cal grinned, smugly. \"Can't be helped.\"\n\n\"Esmerelda's brewing up something in the kitchen.\"\n\n\"Goulash\u2014and it's Cybil.\"\n\n\"Whatever, it smells good, I'll give her that. She smells better. But the point is I got the heave-ho when I tried to get a damn bag of chips to go with the beer.\"\n\n\"You want to cook for six people?\"\n\nGage only grunted, sat, propped his feet on the coffee table. \"How much are they calling for?\"\n\n\"About three feet.\" Cal dropped down beside him, mirrored his pose. \"Used to be we liked nothing better. No school, haul out the sleds. Snowball wars.\"\n\n\"Those were the days, my friend.\"\n\n\"Now we're priming the generator, loading in firewood, buying extra batteries and toilet paper.\"\n\n\"Sucks to be grown up.\"\n\nStill, it was warm, and while the snow fell in sheets outside, there was light, and there was food. It was hard to complain, Cal decided, when he was digging into a bowl of hot, spicy stew he had nothing to do with preparing. Plus, there were dumplings, and he was weak when it came to dumplings.\n\n\"I was in Budapest not that long ago.\" Gage spooned up goulash as he studied Cybil. \"This is as good as any I got there.\"\n\n\"Actually, this isn't Hungarian goulash. It's a Serbo-Croatian base.\"\n\n\"Damn good stew,\" Fox commented, \"wherever it's based.\"\n\n\"Cybil's an Eastern European stew herself.\" Quinn savored the half dumpling she'd allowed herself. \"Croatian, Ukrainian, Polish\u2014with a dash of French for fashion sense and snottiness.\"\n\n\"When did your family come over?\" Cal wondered.\n\n\"As early as the seventeen hundreds, as late as just before World War Two, depending on the line.\" But she understood the reason for the question. \"I don't know if there is a connection to Quinn or Layla, or any of this, where it might root from. I'm looking into it.\"\n\n\"We had a connection,\" Quinn said, \"straight off.\"\n\n\"We did.\"\n\nCal understood that kind of friendship, the kind he saw when the two women looked at each other. It had little to do with blood, and everything to do with the heart.\n\n\"We hooked up the first day\u2014evening really\u2014of college.\" Quinn spooned off another minuscule piece of dumpling with the stew. \"Met in the hall of the dorm. We were across from each other. Within two days, we'd switched. Our respective roommates didn't care. We bunked together right through college.\"\n\n\"And apparently still are,\" Cybil commented.\n\n\"Remember you read my palm that first night?\"\n\n\"You read palms?\" Fox asked.\n\n\"When the mood strikes. My gypsy heritage,\" Cybil added with a flourishing gesture of her hands.\n\nAnd Cal felt a knot form in his belly. \"There were gypsies in the Hollow.\"\n\n\"Really?\" Carefully, Cybil lifted her wineglass, sipped. \"When?\"\n\n\"I'd have to check to be sure. This is from stories my gran told me that her grandmother told her. Like that. About how gypsies came one summer and set up camp.\"\n\n\"Interesting. Potentially,\" Quinn mused, \"someone local could get cozy with one of those dark-eyed beauties or hunks, and nine months later, oops. Could lead right to you, Cyb.\"\n\n\"Just one big, happy family,\" Cybil muttered.\n\nAfter the meal, chores were divvied up again. Wood needed to be brought in, the dog let out, the table cleared, dishes dealt with.\n\n\"Who else cooks?\" Cybil demanded.\n\n\"Gage does,\" Cal and Fox said together.\n\n\"Hey.\"\n\n\"Good.\" Cybil sized him up. \"If there's a group breakfast on the slate, you're in charge. Now\u2014\"\n\n\"Before we...whatever,\" Cal decided, \"there's something we have to go over. Might as well stick to the dining room. We have to get something,\" he added, looking at Fox and Gage. \"You might want to open another bottle of wine.\"\n\n\"What's all this?\" Quinn frowned as the men retreated. \"What are they up to?\"\n\n\"It's more what haven't they told us,\" Layla said. \"Guilt and reluctance, that's what I'm picking up. Not that I know any of them that well.\"\n\n\"You know what you know,\" Cybil told her. \"Get another bottle, Q.\" She gave a little shudder. \"Maybe we should light a couple more candles while we're at it, just in case. It already feels...dark.\"\n\nTHEY LEFT IT TO HIM, CAL SUPPOSED, BECAUSE IT was his house. When they were all back around the table, he tried to find the best way to begin.\n\n\"We've gone over what happened that night in the clearing when we were kids, and what started happening after. Quinn, you got some of it yourself when we hiked there a couple weeks ago.\"\n\n\"Yeah. Cyb and Layla need to see it, as soon as the snow's cleared enough for us to make the hike.\"\n\nHe hesitated only a beat. \"Agreed.\"\n\n\"It ain't a stroll down the Champs \u00c9lys\u00e9es,\" Gage commented, and Cybil cocked an eyebrow at him.\n\n\"We'll manage.\"\n\n\"There was another element that night, another aspect we haven't talked about with you.\"\n\n\"With anyone,\" Fox added.\n\n\"It's hard to explain why. We were ten, everything went to hell, and...Well.\" Cal set his part of the stone on the table.\n\n\"A piece of rock?\" Layla said.\n\n\"Bloodstone.\" Cybil pursed her lips, started to reach for it, stopped. \"May I?\"\n\nGage and Fox set theirs down beside Cal's. \"Take your pick,\" Gage invited.\n\n\"Three parts of one.\" Quinn picked up the one closest to her. \"Isn't that right? These are three parts of one stone.\"\n\n\"One that had been rounded, tumbled, polished,\" Cybil continued. \"Where did you get the pieces?\"\n\n\"We were holding them,\" Cal told her. \"After the light, after the dark, when the ground stopped shaking, each one of us was holding his part of this stone.\" He studied his own hand, remembering how his fist had clenched around the stone as if his life depended on it.\n\n\"We didn't know what they were. Fox looked it up. His mother had books on rocks and crystals, and he looked it up. Bloodstone,\" Cal repeated. \"It fit.\"\n\n\"It needs to be put back together,\" Layla said. \"Doesn't it? It needs to be whole again.\"\n\n\"We've tried. The breaks are clean,\" Fox explained. \"They fit together like a puzzle.\" He gestured, and Cal took the pieces, fit them into a round.\n\n\"But it doesn't do anything.\"\n\n\"Because you're holding them together?\" Curious, Quinn held out her hand until Cal put the three pieces into it. \"They're not...fused would be the word, I guess.\"\n\n\"Tried that, too. MacGyver over there tried superglue.\"\n\nCal sent Gage a bland stare. \"Which should've worked\u2014at least as far as holding the pieces together. But I might as well have used water. No stick. We've tried banding them, heating them, freezing them. No dice. In fact, they don't even change temperature.\"\n\n\"Except\u2014\" Fox broke off, got the go-ahead nod. \"During the Seven, they heat up. Not too hot to hold, but right on the edge.\"\n\n\"Have you tried putting them back together during that week?\" Quinn demanded.\n\n\"Yeah. No luck. The one thing we know is that Giles Dent was wearing this, like an amulet around his neck, the night Lazarus Twisse led that mob into the clearing. I saw it. Now we have it.\"\n\n\"Have you tried magickal means?\" Cybil asked.\n\nCal squirmed a little, cleared his throat.\n\n\"Jesus, Cal, loosen up.\" Fox shook his head. \"Sure. I got some books on spells, and we gave that a try. Down the road, Gage has talked to some practicing witches, and we've tried other rites and so on.\"\n\n\"But you never showed them to anyone.\" Quinn set the pieces down carefully before picking up her wine. \"Anyone who might have been able to work with them, or understand the purpose. Maybe the history.\"\n\n\"We weren't meant to.\" Fox lifted his shoulders. \"I know how it sounds, but I knew we weren't supposed to take it to, what, a geologist or some Wiccan high priestess, or the damn Pentagon. I just...Cal voted for the science angle right off.\"\n\n\"MacGyver,\" Gage repeated.\n\n\"Fox was sure that was off-limits, and that was good enough. That was good enough for the three of us.\" Cal looked at his friends. \"It's been the way we've handled it, up till now. If Fox felt we shouldn't show you, we wouldn't be.\"\n\n\"Because you feel it the strongest?\" Layla asked Fox.\n\n\"I don't know. Maybe. I know I believed\u2014I believe\u2014we survived that night, that we came out of it the way we came out of it because we each had a piece of that stone. And as long as we do, we've got a chance. It's just something I know, the same way Cal saw it, that he recognized it as the amulet Dent wore.\"\n\n\"How about you?\" Cybil asked Gage. \"What do you know? What do you see?\"\n\nHis eyes met hers. \"I see it whole, on top of the Pagan Stone. The stone on the stone. And the flames flick up from it, kindling in the blood spots. Then they consume it, ride over the flat, down the pedestal like a sheath of fire. I see the fire race across the ground, fly into the trees until they burst from the heat. And the clearing's a holocaust even the devil himself couldn't survive.\"\n\nHe took a drink of wine. \"That's what I see when it's whole again, so I'm in no big hurry to get there.\"\n\n\"Maybe that's how it was formed,\" Layla began.\n\n\"I don't see back. That's Cal's gig. I see what might be coming.\"\n\n\"That'd be handy in your profession.\"\n\nGage shifted his gaze back to Cybil, smiled slowly. \"It doesn't hurt.\" He picked up his stone, tossed it lightly in his hand. \"Anyone interested in a little five-card draw?\"\n\nAs soon as he spoke, the light snapped off.\n\nRather than romance or charm, the flickering candles they'd lit as backup lent an eeriness to the room. \"I'll go fire up the generator.\" Cal pushed up. \"Water, refrigerator, and stove for now.\"\n\n\"Don't go out alone.\" Layla blinked as if surprised the words had come out of her mouth. \"I mean\u2014\"\n\n\"I'm going with you.\"\n\nAs Fox rose, something howled in the dark.\n\n\"Lump.\" Cal was out of the room, through the kitchen, and out the back door like a bullet. He barely broke stride to grab the flashlight off the wall, punch it on.\n\nHe swept it toward the sound. The beam struggled against the thick, moving curtain of snow, did little but bounce the light back at him.\n\nThe blanket had become a wall that rose past his knees. Calling his dog, Cal pushed through it, trying to pinpoint the direction of the howling. It seemed to come from everywhere, from nowhere.\n\nAs he heard sounds behind him, he whirled, gripping the flashlight like a weapon.\n\n\"Don't clock the reinforcements,\" Fox shouted. \"Christ, it's insane out here.\" He gripped Cal's arm as Gage moved to Cal's other side. \"Hey, Lump! Come on, Lump! I've never heard him like that.\"\n\n\"How do you know it's the dog?\" Gage asked quietly.\n\n\"Get back inside,\" Cal said grimly. \"We can't leave the women alone. I'm going to find my dog.\"\n\n\"Oh yeah, we'll just leave you out here, stumbling around in a fucking blizzard.\" Gage jammed his freezing hands in his pockets, glanced back. \"Besides.\"\n\nThey came, arms linked and gripping flashlights. Which showed sense, Cal was forced to admit. And they'd taken the time to put on coats, probably boots as well, which is more than he or his friends had done.\n\n\"Go back in.\" He had to shout now, over the rising wind. \"We're just going to round up Lump. Be right there.\"\n\n\"We all go in or nobody does.\" Quinn unhooked her arm from Layla's, hooked it to Cal's. \"That includes Lump. Don't waste time,\" she said before he could argue. \"We should spread out, shouldn't we?\"\n\n\"In pairs. Fox, you and Layla try that way, Quinn and I'll take this way. Gage and Cybil toward the back. He's got to be close. He never goes far.\"\n\nHe sounded scared, that's what Cal didn't want to say out loud. His stupid, lazy dog sounded scared. \"Hook your hand in my pants\u2014the waistband. Keep a good hold.\"\n\nHe hissed against the cold as her gloves hit his skin, then began to trudge forward. He'd barely made it two feet when he heard something under the howls.\n\n\"You catch that?\"\n\n\"Yes. Laughing. The way a nasty little boy might laugh.\"\n\n\"Go\u2014\"\n\n\"I'm not leaving that dog out here any more than you are.\"\n\nA vicious gush of wind rose up like a tidal wave, spewing huge clumps of snow, and what felt like pellets of ice. Cal heard branches cracking, like gunfire in the dark. Behind him, Quinn lost her footing in the force of the wind and nearly took them both down.\n\nHe'd get Quinn back into the house, he decided. Get her the hell in, lock her in a damn closet if necessary, then come back out and find his dog.\n\nEven as he turned to get a grip on her arm, he saw them.\n\nHis dog sat on his haunches, half buried in the snow, his head lifted as those long, desperate howls worked his throat.\n\nThe boy floated an inch above the surface of the snow. Chortling, Cal thought. There was a word you didn't use every day, but it sure as hell fit the filthy sound it made.\n\nIt grinned as the wind blasted again. Now Lump was buried to his shoulders.\n\n\"Get the fuck away from my dog.\"\n\nCal lurched forward; the wind knocked him back so that both he and Quinn went sprawling.\n\n\"Call him,\" Quinn shouted. \"Call him, make him come!\" She dragged off her gloves as she spoke. Using her fingers to form a circle between her lips, she whistled shrilly as Cal yelled at Lump.\n\nLump quivered; the thing laughed.\n\nCal continued to call, to curse now, to crawl while the snow flew into his eyes, numbed his hands. He heard shouting behind him, but he focused everything he had on pushing ahead, on getting there before the next gust of wind put the dog under.\n\nHe'd drown, Cal thought as he pushed, shoved, slid forward. If he didn't get to Lump, his dog would drown in that ocean of snow.\n\nHe felt a hand lock on his ankle, but kept dragging himself forward.\n\nGritting his teeth, he flailed out, got a slippery hold on Lump's collar. Braced, he looked up into eyes that glittered an unholy green rimmed with red. \"You can't have him.\"\n\nCal yanked. Ignoring Lump's yelp, he yanked again, viciously, desperately. Though Lump howled, whimpered, it was as if his body was sunk in hardened cement.\n\nAnd Quinn was beside him, belly down, digging at the snow with her hands.\n\nFox skidded down, shooting snow like shrapnel. Cal gathered everything he had, looked once more into those monstrous eyes in the face of a young boy. \"I said you can't have him.\"\n\nWith the next pull, Cal's arms were full of quivering, whimpering dog.\n\n\"It's okay, it's okay.\" He pressed his face against cold, wet fur. \"Let's get the hell out of here.\"\n\n\"Get him in by the fire.\" Layla struggled to help Quinn up as Cybil pushed up from her knees. Shoving the butt of a flashlight in his back pocket, Gage pulled Cybil to her feet, then plucked Quinn out of the snow.\n\n\"Can you walk?\" he asked her.\n\n\"Yeah, yeah. Let's get in, let's get inside, before somebody ends up with frostbite.\"\n\nTowels and blankets, dry clothes, hot coffee. Brandy\u2014even for Lump\u2014warmed chilled bones and numbed flesh. Fresh logs had the fire blazing.\n\n\"It was holding him. He couldn't get away.\" Cal sat on the floor, the dog's head in his lap. \"He couldn't get away. It was going to bury him in the snow. A stupid, harmless dog.\"\n\n\"Has this happened before?\" Quinn asked him. \"Has it gone after animals this way?\"\n\n\"A few weeks before the Seven, animals might drown, or there's more roadkill. Sometimes pets turn mean. But not like this. This was\u2014\"\n\n\"A demonstration.\" Cybil tucked the blanket more securely around Quinn's feet. \"He wanted us to see what he could do.\"\n\n\"Maybe wanted to see what we could do,\" Gage countered, and earned a speculative glance from Cybil.\n\n\"That may be more accurate. That may be more to the point. Could we break the hold? A dog's not a person, has to be easier to control. No offense, Cal, but your dog's brainpower isn't as high as most toddlers'.\"\n\nGently, affectionately, Cal pulled on one of Lump's floppy ears. \"He's thick as a brick.\"\n\n\"So it was showing off. It hurt this poor dog for sport.\" Layla knelt down and stroked Lump's side. \"That deserves some payback.\"\n\nIntrigued, Quinn cocked her head. \"What do you have in mind?\"\n\n\"I don't know yet, but it's something to think about.\"\n\n## Eighteen\n\nCAL DIDN'T KNOW WHAT TIME THEY'D FALLEN into bed. But when he opened his eyes the thin winter light eked through the window. Through it, he saw the snow was still falling in the perfect, fat, white flakes of a Hollywood Christmas movie.\n\nIn the hush only a snowfall could create was steady and somehow satisfied snoring. It came from Lump, who was stretched over the foot of the bed like a canine blanket. That was something Cal generally discouraged, but right now, the sound, the weight, the warmth were exactly right.\n\nFrom now on, he determined, the damn dog was going everywhere with him.\n\nBecause his foot and ankle were currently under the bulk of the dog, Cal shifted to pull free. The movement had Quinn stirring, giving a little sigh as she wiggled closer and managed to wedge her leg between his. She wore flannel, which shouldn't have been remotely sexy, and she'd managed to pin his arm during the night so it was now alive with needles and pins. And that should've been, at least mildly, annoying.\n\nInstead, it was exactly right, too.\n\nSince it was, since they were cuddled up together in bed with Hollywood snow falling outside the window, he couldn't think of a single reason not to take advantage of it.\n\nSmiling, he slid a hand under her T-shirt, over warm, smooth flesh. When he cupped her breast he felt her heart beat under his palm, slow and steady as Lump's snoring. He stroked, a lazy play of fingertips as he watched her face. Lightly, gently, he teased her nipple, arousing himself as he imagined taking it into his mouth, sliding his tongue over her.\n\nShe sighed again.\n\nHe trailed his hand down, tracing those fingertips over her belly, under the flannel to skim down her thigh. Up again. Down, then up, a whispering touch that eased closer, closer to her center.\n\nAnd the sound she made in sleep was soft and helpless.\n\nShe was wet when he brushed over her, hot when he dipped inside her. When he pressed, he lowered his mouth to hers to take her gasp.\n\nShe came as she woke, her body simply erupting as her mind leaped out of sleep and into shock and pleasure.\n\n\"Oh God!\"\n\n\"Shh.\" He laughed against her lips. \"You'll wake the dog.\"\n\nHe tugged down her pants as he rolled. Before she could clear her mind, he pinned her, and he filled her.\n\n\"Oh. Well. Jesus.\" The words hitched and shook. \"Good morning.\"\n\nHe laughed again, and bracing himself, set a slow and torturous pace. She fought to match it, to hold back and take that slow climb with him, but it flashed through her again, and flung her up.\n\n\"God. God. God. I don't think I can\u2014\"\n\n\"Shh, shh,\" he repeated, and brought his mouth down to toy with hers. \"I'll go slow,\" he whispered. \"You just go.\"\n\nShe could do nothing else. Her system was already wrecked, her body already his. Utterly his. When he took her up again, she was too breathless to cry out.\n\nTHOROUGHLY PLEASURED, THOROUGHLY USED, Quinn lay under Cal's weight. He'd eased down so that his head rested between her breasts, and she could play with his hair. She imagined it was some faraway Sunday morning where they had nothing more pressing to worry about than if they'd make love again before breakfast, or make love after.\n\n\"Do you take some kind of special vitamin?\" she wondered.\n\n\"Hmm?\"\n\n\"I mean, you've got some pretty impressive stamina going for you.\"\n\nShe felt his lips curve against her. \"Just clean living, Blondie.\"\n\n\"Maybe it's the bowling. Maybe bowling...Where's Lump?\"\n\n\"He got embarrassed about halfway through the show.\" Cal turned his head, gestured. \"Over there.\"\n\nQuinn looked, saw the dog on the floor, his face wedged in the corner. She laughed till her sides ached. \"We embarrassed the dog. That's a first for me. God! I feel good. How can I feel so good after last night?\" Then she shook her head, stretched up her arms before wrapping them around Cal. \"I guess that's the point, isn't it? Even in a world gone to hell, there's still this.\"\n\n\"Yeah.\" He sat up then, reached down to brush her tumbled hair as he studied her. \"Quinn.\" He took her hand now, played with her fingers.\n\n\"Cal,\" she said, imitating his serious tone.\n\n\"You crawled through a blizzard to help save my dog.\"\n\n\"He's a good dog. Anyone would have done the same.\"\n\n\"No. You're not naive enough to think that. Fox and Gage, yeah. For the dog, and for me. Layla and Cybil, maybe. Maybe it was being caught in the moment, or maybe they're built that way.\"\n\nShe touched his face, skimmed her fingers under those patient gray eyes. \"No one was going to leave that dog out there, Cal.\"\n\n\"Then I'd say that dog is pretty lucky to have people like you around. So am I. You crawled through the snow, toward that thing. You dug in the snow with your bare hands.\"\n\n\"If you're trying to make a hero out of me...Go ahead,\" she decided. \"I think I like the fit.\"\n\n\"You whistled with your fingers.\"\n\nNow she grinned. \"Just a little something I picked up along the way. I can actually whistle a lot louder than that, when I'm not out of breath, freezing, and quivering with terror.\"\n\n\"I love you.\"\n\n\"I'll demonstrate sometime when...What?\"\n\n\"I never thought to say those words to any woman I wasn't related to. I was just never going to go there.\"\n\nIf she'd been given a hard, direct jolt of electricity to her heart, it couldn't have leaped any higher. \"Would you mind saying them again, while I'm paying better attention?\"\n\n\"I love you.\"\n\nThere it went again, she thought. Leaps and bounds. \"Because I can whistle with my fingers?\"\n\n\"That might've been the money shot.\"\n\n\"God.\" She shut her eyes. \"I want you to love me, and I really like to get what I want. But.\" She took a breath. \"Cal, if this is because of last night, because I helped get Lump, then\u2014\"\n\n\"This is because you think if you eat half my slice of pizza it doesn't count.\"\n\n\"Well, it doesn't, technically.\"\n\n\"Because you always know where your keys are, and you can think about ten things at the same time. Because you don't back down, and your hair's like sunlight. Because you tell the truth and you know how to be a friend. And for dozens of reasons I haven't figured out yet. Dozens more I may never figure out. But I know I can say to you what I never thought to say to anyone.\"\n\nShe hooked her arms around his neck, rested her forehead on his. She had to just breathe for a moment, just breathe her way through the beauty of it as she often did with a great work of art or a song that brought tears to her throat.\n\n\"This is a really good day.\" She touched her lips to his. \"This is a truly excellent day.\"\n\nThey sat for a while, holding each other while the dog snored in the corner, and the snow fell outside the windows.\n\nWhen Cal went downstairs, he followed the scent of coffee into the kitchen, and found Gage scowling as he slapped a skillet onto the stove. They grunted at each other as Cal got a clean mug out of the dishwasher.\n\n\"Looks like close to three out there already, and it's still coming.\"\n\n\"I got eyes.\" Gage ripped open a pound of bacon. \"You sound chipper about it.\"\n\n\"It's a really good day.\"\n\n\"I'd probably think so, too, if I started it off with some morning nookie.\"\n\n\"God, men are crude.\" Cybil strolled in, her dark eyes bleary.\n\n\"Then you ought to plug your ears when you're around our kind. Bacon gets fried, eggs get scrambled,\" Gage told them. \"Anybody doesn't like the options should try another restaurant.\"\n\nCybil poured her coffee, stood studying him over the rim as she took the first sip. He hadn't shaved or combed that dark mass of hair. He was obviously morning irritable, and none of that, she mused, made him any less attractive.\n\nToo bad.\n\n\"You know what I've noticed about you, Gage?\"\n\n\"What's that?\"\n\n\"You've got a great ass, and a crappy attitude. Let me know when breakfast is ready,\" she added as she strolled out of the kitchen.\n\n\"She's right. I've often said that about your ass and attitude.\"\n\n\"Phones are out,\" Fox announced as he came in, yanked open the refrigerator and pounced on a Coke. \"Got ahold of my mother by cell. They're okay over there.\"\n\n\"Knowing your parents, they probably just had sex,\" Gage commented.\n\n\"Hey! True,\" Fox said after a moment, \"but, hey.\"\n\n\"He's got sex on the brain.\"\n\n\"Why wouldn't he? He's not sick or watching sports, the only two circumstances men don't necessarily have sex on the brain.\"\n\nGage laid bacon in the heated skillet. \"Somebody make some toast or something. And we're going to need another pot of coffee.\"\n\n\"I've got to take Lump out. I'm not just letting him out on his own.\"\n\n\"I'll take him.\" Fox leaned down to scratch Lump's head. \"I want to walk around anyway.\" He turned, nearly walked into Layla. \"Hi, sorry. Ah...I'm going to take Lump out. Why don't you come along?\"\n\n\"Oh. I guess. Sure. I'll just get my things.\"\n\n\"Smooth,\" Gage commented when Layla left. \"You're a smooth one, Fox.\"\n\n\"What?\"\n\n\"Good morning, really attractive woman. How would you like to trudge around with me in three feet of snow and watch a dog piss on a few trees? Before you've even had your coffee?\"\n\n\"It was just a suggestion. She could've said no.\"\n\n\"I'm sure she would have if she'd had a hit of caffeine so her brain was in gear.\"\n\n\"That must be why you only get lucky with women without brains.\"\n\n\"You're just spreading sunshine,\" Cal commented when Fox steamed out.\n\n\"Make another damn pot of coffee.\"\n\n\"I need to bring in some wood, feed the generator, and start shoveling three feet of snow off the decks. Let me know when breakfast is ready.\"\n\nAlone, Gage snarled, and turned the bacon. He still had the snarl when Quinn came in.\n\n\"I thought I'd find everyone in here, but they're all scattered.\" She got out a mug. \"Looks like we need another pot of coffee.\"\n\nBecause she got the coffee down, Gage didn't have time to snap at her.\n\n\"I'll take care of that. Anything else I can do to help?\"\n\nHe turned his head to look at her. \"Why?\"\n\n\"Because I figure if I help you with breakfast, it takes us both off the cooking rotation for the next couple of meals.\"\n\nHe nodded, appreciating the logic. \"Smart. You're the toast and additional coffee.\"\n\n\"Check.\"\n\nHe beat a dozen eggs while she got to work. She had a quick, efficient way about her, Gage noted. The quick wouldn't matter so much to Cal, but the efficient would be a serious plus. She was built, she was bright, and as he'd seen for himself last night, she had a wide streak of brave.\n\n\"You're making him happy.\"\n\nQuinn stopped, looked over. \"Good, because he's making me happy.\"\n\n\"One thing, if you haven't figured it out by now. He's rooted here. This is his place. Whatever happens, the Hollow's always going to be Cal's place.\"\n\n\"I figured that out.\" She plucked toast when it popped, dropped more bread in. \"All things considered, it's a nice town.\"\n\n\"All things considered,\" Gage agreed, then poured the eggs into the second skillet.\n\nOUTSIDE, AS GAGE PREDICTED, FOX WATCHED Lump piss on trees. More entertaining, he supposed, had been watching the dog wade, trudge, and occasionally leap through the waist-high snow. It was the waist-high factor that had Fox and Layla stopping on the front deck, and Fox going to work with the shovel Cal had shoved into his hands on their way out.\n\nStill, it was great to be out in the snow globe of the morning, tossing the white stuff around while more of it pumped out of the sky.\n\n\"Maybe I should go down, knock the snow off some of Cal's shrubs.\"\n\nFox glanced over at her. She had a ski cap pulled over her head, a scarf wrapped around her neck. Both had already picked up a layer of white. \"You'll sink, then we'll be tossing you a lifeline to get you back. We'll dig out a path eventually.\"\n\n\"He doesn't seem to be spooked.\" She kept an eagle eye on Lump. \"I thought, after last night, he'd be skittish about going out.\"\n\n\"Short-term doggie memory. Probably for the best.\"\n\n\"I won't forget it.\"\n\n\"No.\" He shouldn't have asked her to come out, Fox realized. Especially since he couldn't quite figure out how to broach the whole job deal, which had been part of the idea for having her tag along.\n\nHe was usually better at this stuff, dealing with people. Dealing with women. Now, he worked on carving down a shovel-width path across the deck to the steps, and just jumped in.\n\n\"So, Cal said you're looking for a job.\"\n\n\"Not exactly. I mean I'm going to have to find some work, but I haven't been looking.\"\n\n\"My secretary\u2014office manager\u2014assistant.\" He dumped snow, dug the shovel back down. \"We never settled on a title, now that I think about it. Anyway, she's moving to Minneapolis. I need somebody to do the stuff she does.\"\n\nDamn Quinn, she thought. \"The stuff.\"\n\nIt occurred to Fox that he was considered fairly articulate in court. \"Filing, billing, answering phones, keeping the calendar, rescheduling when necessary, handling clients, typing documents and correspondence. She's a notary, too, but that's not a necessity right off.\"\n\n\"What software does she use?\"\n\n\"I don't know. I'd have to ask her.\" Did she use any software? How was he supposed to know?\n\n\"I don't know anything about secretarial work, or office management. I don't know anything about the law.\"\n\nFox knew tones, and hers was defensive. He kept shoveling. \"Do you know the alphabet?\"\n\n\"Of course I know the alphabet, but the point\u2014\"\n\n\"Would be,\" he interrupted, \"if you know the alphabet you can probably figure out how to file. And you know how to use a phone, which means you can answer one and make calls from one. Those would be essential job skills for this position. Can you use a keyboard?\"\n\n\"Yes, but it depends on\u2014\"\n\n\"She can show you whatever the hell she does in that area.\"\n\n\"It doesn't sound as if you know a lot about what she does.\"\n\nHe also knew disapproval when he heard it. \"Okay.\" He straightened, leaned on the shovel, and looked dead into her eyes. \"She's been with me since I set up. I'm going to miss her like I'd miss my arm. But people move on, and the rest of us have to deal. I need somebody to put papers where they belong and find them when I need to have them, to send out bills so I can pay mine, to tell me when I'm due in court, to answer the phone we hope rings so I'll have somebody to bill, and basically maintain some kind of order so I can practice law. You need a job and a paycheck. I think we could help each other out.\"\n\n\"Cal asked you to offer me a job because Quinn asked him to ask you.\"\n\n\"That would be right. Doesn't change the bottom line.\"\n\nNo, it didn't, she supposed. But it still griped. \"It wouldn't be permanent. I'm only looking for something to fill in until...\"\n\n\"You move on.\" Fox nodded. \"Works for me. That way, neither of us are stuck. We're just helping each other out for a while.\" He shoveled off two more blades of snow, then stopped just to lean on it with his eyes on hers.\n\n\"Besides, you knew I was going to offer you the job because you pick up that sort of thing.\"\n\n\"Quinn asked Cal to ask you to offer it to me right in front of me.\"\n\n\"You pick up on that sort of thing,\" he repeated. \"That's your part in this, or part of your part. You get a sense of people, of situations.\"\n\n\"I'm not psychic, if that's what you're saying.\" The defensive was back in her tone.\n\n\"You drove to the Hollow, when you'd never been here before. You knew where to go, what roads to take.\"\n\n\"I don't know what that was.\" She crossed her arms, and the move wasn't just defensive, Fox thought. It was stubborn.\n\n\"Sure you do, it just freaks you. You took off with Quinn that first night, went with her, a woman you'd never met.\"\n\n\"She was a sane alternative to a big, evil slug,\" Layla said dryly.\n\n\"You didn't just run, didn't haul ass to your room and lock the door. You got in her car with her, came with her out here\u2014where you'd also never been, and walked into a house with two strange men in it.\"\n\n\"Strange might be the operative word. I was scared, confused, and running on adrenaline.\" She looked away from him, toward where Lump was rolling in the snow as if it were a meadow of daisies. \"I trusted my instincts.\"\n\n\"Instincts is one word for it. I bet when you were working in that clothes shop you had really good instincts about what your customers wanted, what they'd buy. Bet you're damn good at that.\"\n\nHe went back to shoveling when she said nothing. \"Bet you've always been good at that sort of thing. Quinn gets flashes from the past, like Cal. Apparently Cybil gets them of possible future events. I'd say you're stuck with me, Layla, in the now.\"\n\n\"I can't read minds, and I don't want anyone reading mine.\"\n\n\"It's not like that, exactly.\" He was going to have to work with her, he decided. Help her figure out what she had and how to use it. And he was going to have to give her some time and some space to get used to the idea.\n\n\"Anyway, we're probably going to be snowed in here for the weekend. I've got stuff next week, but when we can get back to town, you could come in when it suits you, let Mrs. H show you the ropes. We'll see how you feel about the job then.\"\n\n\"Look, I'm grateful you'd offer\u2014\"\n\n\"No, you're not.\" Now he smiled and tossed another shovel of snow off the deck. \"Not so much. I've got instincts, too.\"\n\nIt wasn't just humor, but understanding. The stiffness went out of her as she kicked at the snow. \"There's gratitude, it's just buried under the annoyance.\"\n\nCocking his head, he held out the shovel. \"Want to dig it out?\"\n\nAnd she laughed. \"Let's try this. If I do come in, and do decide to take the job, it's with the stipulation that if either of us decides it's not working, we just say so. No hard feelings.\"\n\n\"That's a deal.\" He held out a hand, took hers to seal it. Then just held it while the snow swirled around them.\n\nShe had to feel it, he thought, had to feel that immediate and tangible link. That recognition.\n\nCybil cracked the door an inch. \"Breakfast is ready.\"\n\nFox released Layla's hand, turned. He let out a quiet breath before calling the dog home.\n\nPRACTICAL MATTERS HAD TO BE SEEN TO. SNOW needed to be shoveled, firewood hauled and stacked. Dishes had to be washed and food prepared. Cal might have felt like the house, which had always seemed roomy, grew increasingly tight with six people and one dog stuck inside it. But he knew they were safer together.\n\n\"Not just safer.\" Quinn took her turn plying the shovel. She considered digging out a path to Cal's storage shed solid exercise in lieu of a formal workout. \"I think all this is meant. This enforced community. It's giving us time to get used to each other, to learn how to function as a group.\"\n\n\"Here, let me take over there.\" Cal set aside the gas can he'd used to top off the generator.\n\n\"No, see, that's not working as a group. You guys have to learn to trust the females to carry their load. Gage being drafted to make breakfast today is an example of the basics in non-gender-specific teamwork.\"\n\nNon-gender-specific teamwork, he thought. How could he not love a woman who'd use a term like that?\n\n\"We can all cook,\" she went on. \"We can all shovel snow, haul firewood, make beds. We can all do what we have to do\u2014play to our strengths, okay, but so far it's pretty much been like a middle school dance.\"\n\n\"How?\"\n\n\"Boys on one side, girls on the other, and nobody quite sure how to get everyone together. Now we are.\" She stopped, rolled her shoulders. \"And we have to figure it out. Even with us, Cal, even with how we feel about each other, we're still figuring each other out, learning how to trust each other.\"\n\n\"If this is about the stone, I understand you might be annoyed I didn't tell you sooner.\"\n\n\"No, I'm really not.\" She shoveled a bit more, but it was mostly for form now. Her arms were killing her. \"I started to be, even wanted to be, but I couldn't stir it up. Because I get that the three of you have been a unit all your lives. I don't imagine you remember a time when you weren't. Added to that you went through together\u2014I don't think it's an exaggeration to say an earth-shattering experience. The three of you are like a...a body with three heads isn't right,\" she said and passed off the shovel.\n\n\"We're not the damn Borg.\"\n\n\"No, but that's closer. You're a fist, tight, even closed off to a certain extent, but\u2014\" She wiggled her gloved fingers. \"Individual. You work together, it's instinctive. And now.\" She held up her other hand. \"This other part comes along. So we're figuring out how to make them mesh.\" She brought her hands together, fingers linked.\n\n\"That actually makes sense.\" And brought on a slight twinge of guilt. \"I've been doing a little digging on my own.\"\n\n\"You don't mean in the snow. And on your own equals you've told Fox and Gage.\"\n\n\"I probably mentioned it. We don't know where Ann Hawkins was for a couple of years, where she gave birth to her sons, where she stayed before she came back to the Hollow\u2014to her parents' house. So I was thinking about extended family. Cousins, aunts, uncles. And figuring a woman that pregnant might not be able to travel very far, not back then. So maybe she'd have been in the general area. Ten, twenty miles in the sixteen hundreds was a hell of a lot farther than ten or twenty miles is today.\"\n\n\"That's a good idea. I should have had it.\"\n\n\"And I should've brought it up before.\"\n\n\"Yeah. Now that you have, you should give it to Cyb, give her whatever information you have. She's the research queen. I'm good, she's better.\"\n\n\"And I'm a rank amateur.\"\n\n\"Nothing rank about you.\" Grinning, she took a leap, bounced up into his arms. The momentum had him skidding. She squealed, as much with laughter as alarm as he tipped backward. He flopped; she landed face-first.\n\nBreathless, she dug in, got two handfuls of snow to mash into his face before she tried to roll away. He caught her at the waist, dragged her back while she screamed with helpless laughter.\n\n\"I'm a champion snow wrestler,\" he warned her. \"You're out of your league, Blondie. So\u2014\"\n\nShe managed to get a hand between his legs for a nice, firm stroke. Then taking advantage of the sudden and dramatic dip of his IQ, shoved a messy ball of snow down the back of his neck.\n\n\"Those moves are against the rules of the SWF.\"\n\n\"Check the book, buddy. This is intergender play.\"\n\nShe tried to scramble up, fell, then whooshed out a breath when his weight pinned her. \"And still champion,\" he announced, and was about to lower his mouth to hers when the door opened.\n\n\"Kids,\" Cybil told them, \"there's a nice warm bed upstairs if you want to play. And FYI? The power just came back on.\" She glanced back over her shoulder. \"Apparently the phones are up, too.\"\n\n\"Phones, electricity. Computer.\" Quinn wiggled out from under Cal. \"I have to check my e-mail.\"\n\nCYBIL LEANED ON THE DRYER AS LAYLA LOADED towels into the washing machine in Cal's laundry room. \"They looked like a couple of horny snow people. Covered, crusted, pink-cheeked, and groping.\"\n\n\"Young love is immune to climatic conditions.\"\n\nCybil chuckled. \"You know, you don't have to take on the laundry detail.\"\n\n\"Clean towels are a memory at this point, and the power may not stay on. Besides, I'd rather be warm and dry in here washing towels than cold and wet out there shoveling snow.\" She tossed back her hair. \"Especially since no one's groping me.\"\n\n\"Good point. But I was bringing that up as, by my calculations, you and Fox are going to have to flip for cooking detail tonight.\"\n\n\"Quinn hasn't cooked yet, or Cal.\"\n\n\"Quinn helped with breakfast. It's Cal's house.\"\n\nDefeated, Layla stared at the machine. \"Hell. I'll take dinner.\"\n\n\"You can dump it on Fox, using laundry detail as leverage.\"\n\n\"No, we don't know if he can cook, and I can.\"\n\nCybil narrowed her eyes. \"You can cook? This hasn't been mentioned before.\"\n\n\"If I'd mentioned it, I'd have had to cook.\"\n\nLips pursed, Cybil nodded slowly. \"Diabolical and self-serving logic. I like it.\"\n\n\"I'll check the supplies, see what I can come up with. Something\u2014\" She broke off, stepped forward. \"Quinn? What is it?\"\n\n\"We have to talk. All of us.\" So pale her eyes looked bruised, Quinn stood in the doorway.\n\n\"Q? Honey.\" Cybil reached out in support. \"What's happened?\" She remembered Quinn's dash to the computer for e-mail. \"Is everyone all right? Your parents?\"\n\n\"Yes. Yes. I want to tell it all at once, to everyone. We need to get everyone.\"\n\nShe sat in the living room with Cybil perched on the arm of her chair for comfort. Quinn wanted to curl up in Cal's lap as she'd done once before. But it seemed wrong.\n\nIt all seemed wrong now.\n\nShe wished the power had stayed off forever. She wished she hadn't contacted her grandmother and prodded her into seeking out family history.\n\nShe didn't want to know what she knew now.\n\nNo going back, she reminded herself. And what she had to say could change everything that was to come.\n\nShe glanced at Cal. She knew she had him worried. It wasn't fair to drag it out. How would he look at her afterward? she wondered.\n\nYank off the bandage, Quinn told herself, and get it over with.\n\n\"My grandmother got the information I'd asked her about. Pages from the family Bible. There were even some records put together by a family historian in the late eighteen hundreds. I, ah, have some information on the Clark branch, Layla, that may help you. No one ever pursued that end very far, but you may be able to track back, or out from what I have now.\"\n\n\"Okay.\"\n\n\"The thing is, it looks like the family was, we'll say, pretty religious about their own tracking back. My grandfather, not so much, but his sister, a couple of cousins, they were more into it. They, apparently, get a lot of play out of the fact their ancestors were among the early Pilgrims who settled in the New World. So there isn't just the Bible, and the pages added to that over time. They've had genealogies done tracing roots back to England and Ireland in the fifteen hundreds. But what applies to us, to this, is the branch that came over here. Here to Hawkins Hollow,\" she said to Cal.\n\nShe braced herself. \"Sebastian Deale brought his wife and three daughters to the settlement here in sixteen fifty-one. His eldest daughter's name was Hester. Hester Deale.\"\n\n\"Hester's Pool,\" Fox murmured. \"She's yours.\"\n\n\"That's right. Hester Deale, who according to town lore denounced Giles Dent as a witch on the night of July seventh, sixteen fifty-two. Who eight months later delivered a daughter, and when that daughter was two weeks old, drowned herself in the pond in Hawkins Wood. There's no father documented, nothing on record. But we know who fathered her child. We know what fathered her child.\"\n\n\"We can't be sure of that.\"\n\n\"We know it, Caleb.\" However much it tore inside her, Quinn knew it. \"We've seen it, you and I. And Layla, Layla experienced it. He raped her. She was barely sixteen. He lured her, he overpowered her\u2014mind and body, and he got her with child. One that carried his blood.\" To keep them still, Quinn gripped her hands together. \"A half-demon child. She couldn't live with it, with what had been done to her, with what she'd brought into the world. So she filled her pockets with stones and went into the water to drown.\"\n\n\"What happened to her daughter?\" Layla asked.\n\n\"She died at twenty, after having two daughters of her own. One of them died before her third birthday, the other went on to marry a man named Duncan Clark. They had three sons and a daughter. Both she, her husband, and her youngest son were killed when their house burned down. The other children escaped.\"\n\n\"Duncan Clark must be where I come in,\" Layla said.\n\n\"And somewhere along the line, one of them hooked up with a gypsy from the Old World,\" Cybil finished. \"Hardly seems fair. They get to descend from a heroic white witch, and we get the demon seed.\"\n\n\"It's not a joke,\" Quinn snapped.\n\n\"No, and it's not a tragedy. It just is.\"\n\n\"Damn it, Cybil, don't you see what this means? That thing out there is my\u2014probably our\u2014great-grandfather times a dozen generations. It means we're carrying some part of that in us.\"\n\n\"And if I start to sprout horns and a tail in the next few weeks, I'm going to be very pissed off.\"\n\n\"Oh, fuck that!\" Quinn pushed up, rounded on her friend. \"Fuck the Cybilese. He raped that girl to get to us, three and a half centuries ago, but what he planted led to this. What if we're not here to stop it, not here to help this end? What if we're here to see that it doesn't stop? To play some part in hurting them?\"\n\n\"If your brain wasn't mushy with love you'd see that's a bullshit theory. Panic reaction with a heavy dose of self-pity to spice it up.\" Cybil's voice was brutally cool. \"We're not under some demon's thumb. We're not going to suddenly jump sides and put on the uniform of some dark entity who tries to kill a dog to get his rocks off. We're exactly who we were five minutes ago, so stop being stupid, and pull yourself together.\"\n\n\"She's right. Not about being stupid,\" Layla qualified. \"But about being who we are. If all this is part of it, then we have to find a way to use it.\"\n\n\"Fine. I'll practice getting my head to do three-sixties.\"\n\n\"Lame,\" Cybil decided. \"You'd do better with the sarcasm, Q, if you weren't so worried Cal's going to dump you because of the big D for demon on your forehead.\"\n\n\"Cut it out,\" Layla commanded, and Cybil only shrugged.\n\n\"If he does,\" Cybil continued equably, \"he's not worth your time anyway.\"\n\nIn the sudden, thundering silence a log fell in the grate and shot sparks.\n\n\"Did you print out the attachment?\" Cal asked.\n\n\"No, I...\" Quinn trailed off, shook her head.\n\n\"Let's go do that now, then we can take a look.\" He rose, put a hand on Quinn's arm, and drew her from the room.\n\n\"Nice job,\" Gage commented to Cybil. Before she could snarl, he angled his head. \"That wasn't sarcasm. It was either literally or verbally give her a slap across the face. Verbally's trickier, but a lot less messy.\"\n\n\"Both are painful.\" Cybil pushed to her feet. \"If he hurts her, I'll twist off his dick and feed it to his dog.\" With that, she stormed out of the room.\n\n\"She's a little scary,\" Fox decided.\n\n\"She's not the only one. I'm the one who'll be roasting his balls for dessert.\" Layla headed out behind Cybil. \"I have to find something to make for dinner.\"\n\n\"Oddly, I don't have much of an appetite right now.\" Fox glanced at Gage. \"How about you?\"\n\nUpstairs, Cal waited until they'd stepped into the office currently serving as the men's dorm. He pushed Quinn's back to the door. The first kiss was hard, with sharp edges of anger. The second frustrated. And the last soft.\n\n\"Whatever's in your head about you and me, because of this, get it out. Now. Understand?\"\n\n\"Cal\u2014\"\n\n\"It's taken me my whole life to say what I said to you this morning. I love you. This doesn't change that. So pitch that out, Quinn, or you're going to piss me off.\"\n\n\"It wasn't\u2014that isn't...\" She closed her eyes as a storm of emotions blew through her. \"All right, that was in there, part of it, but it's all of it, the whole. When I read the file she sent, it just...\"\n\n\"It kicked your feet out from under you. I get that. But you know what? I'm right here to help you up.\" He lifted a hand, made a fist, then opened it.\n\nUnderstanding, she fought back tears. Understanding, she put her palm to his, interlaced fingers.\n\n\"Okay?\"\n\n\"Not okay,\" she corrected. \"Thank God about covers it.\"\n\n\"Let's print it out, see what we've got.\"\n\n\"Yeah.\" Steadier, she glanced at the room. The messy, unmade pullout, the piles of clothes. \"Your friends are slobs.\"\n\n\"Yes. Yes, they are.\"\n\nTogether, they picked their way through the mess to the computer.\n\n## Nineteen\n\nIN THE DINING ROOM, QUINN SET COPIES OF THE printouts in front of everyone. There were bowls of popcorn on the table, she noted, a bottle of wine, glasses, and paper towels folded into triangles. Which would all be Cybil's doing, she knew.\n\nJust as she knew Cybil had made the popcorn for her. Not a peace offering; they didn't need peace offerings between them. It was just because.\n\nShe touched a hand to Cybil's shoulder before she took her seat.\n\n\"Apologies for big drama,\" Quinn began.\n\n\"If you think that was drama, you need to come over to my parents' house during one of the family gatherings.\" Fox gave her a smile as he took a handful of popcorn. \"The Barry-O'Dells don't need demon blood to raise hell.\"\n\n\"We'll all accept the demon thing is going to be a running gag from now on.\" Quinn poured a glass of wine. \"I don't know how much all this will tell everyone, but it's more than we had before. It shows a direct line from the other side.\"\n\n\"Are you sure Twisse is the one who raped Hester Deale?\" Gage asked. \"Certain he's the one who knocked her up?\"\n\nQuinn nodded. \"Believe me.\"\n\n\"I experienced it.\" Layla twisted the paper towel in her hands as she spoke. \"It wasn't like the flashes Cal and Quinn get, but...Maybe the blood tie explains it. I don't know. But I know what he did to her. And I know she was a virgin before he\u2014it\u2014raped her.\"\n\nGently, Fox took the pieces of the paper towel she'd torn, gave her his.\n\n\"Okay,\" Gage continued, \"are we sure Twisse is what we're calling the demon for lack of better?\"\n\n\"He never liked that term,\" Cal put in. \"I think we can go affirmative on that.\"\n\n\"So, Twisse uses Hester to sire a child, to extend his line. If he's been around as long as we think\u2014going off some of the stuff Cal's seen and related, it's likely he'd done the same before.\"\n\n\"Right,\" Cybil acknowledged. \"Maybe that's where we get people like Hitler or Osama bin Laden, Jack the Ripper, child abusers, serial killers.\"\n\n\"If you look at the lineage, you'll see there were a lot of suicides and violent deaths, especially in the first hundred, hundred and twenty years after Hester. I think,\" Quinn said slowly, \"if we're able to dig a little deeper on individuals, we might find more than the average family share of murder, insanity.\"\n\n\"Anything that stands out in recent memory?\" Fox asked. \"Major family skeletons?\"\n\n\"Not that I know of. I have the usual share of kooky or annoying relatives, but nobody's been incarcerated or institutionalized.\"\n\n\"It dilutes.\" Fox narrowed his eyes as he paged through the printouts. \"This wasn't his plan, wasn't his strategy. I know strategy. Consider. Twisse doesn't know what Dent's got cooking that night. He's got Hester\u2014got her mind under control, got the demon bun in the oven, but he doesn't know that's going to be it.\"\n\n\"That Dent's ready for him, and has his own plans,\" Layla continued. \"I see where you're going. He thought\u2014planned\u2014to destroy Dent that night, or at least damage him, drive him away.\"\n\n\"Then he gets the town,\" Fox continued, \"uses it up, moves on. Leaves progeny, before he finds the next spot that suits him to do the same.\"\n\n\"Instead Dent takes him down, holds him down until...\" Cal turned over his hand, exposed the thin scar on his wrist. \"Until Dent's progeny let him out. Why would he want that? Why would he allow it?\"\n\n\"Could be Dent figured keeping a demon in a headlock for three centuries was long enough.\" Gage helped himself to popcorn. \"Or that's as long as he could hold him, and he called out some reinforcements.\"\n\n\"Ten-year-old boys,\" Cal said in disgust.\n\n\"Children are more likely to believe, to accept what adults can't. Or won't,\" Cybil added. \"And hell, nobody said any of this was fair. He gave you what he could. Your ability to heal quickly, your insights into what was, is, will be. He gave you the stone, in three parts.\"\n\n\"And time to grow up,\" Layla added. \"Twenty-one years. Maybe he found the way to bring us here. Quinn, Cybil, and me. Because I can't see the logic, the purpose of having me compelled to come here, then trying to scare me away.\"\n\n\"Good point.\" And it loosened something inside Quinn's belly. \"That's a damn good point. Why scare if he could seduce? Really good point.\"\n\n\"I can look deeper into the family tree for you, Q. And I'll see what I can find on Layla's and my own. But that's just busywork at this point. We know the root.\"\n\nCybil turned one of the pages over, used a pencil on the back. She drew two horizontal lines at the bottom. \"Giles Dent and Ann Hawkins here, Lazarus Twisse and the doomed Hester here. Each root sends up a tree, and the trees their branches.\" She drew quickly, simply. \"And at the right point, branches from each tree cross each other. In palmistry the crossing of lines is a sign of power.\"\n\nShe completed the sketch, three branches, crossing three branches. \"So we have to find the power, and use it.\"\n\nTHAT EVENING, LAYLA DID SOMETHING FAIRLY tasty with chicken breasts, stewed tomatoes, and white beans. By mutual agreement they channeled the conversation into other areas. Normal, Quinn thought as it ranged from dissecting recent movies to bad jokes to travel. They all needed a good dose of normal.\n\n\"Gage is the one with itchy feet,\" Cal commented. \"He's been traveling that long, lonesome highway since he hit eighteen.\"\n\n\"It's not always lonesome.\"\n\n\"Cal said you were in Prague.\" Quinn considered. \"I think I'd like to see Prague.\"\n\n\"I thought it was Budapest.\"\n\nGage glanced at Cybil. \"There, too. Prague was the last stop before heading back.\"\n\n\"Is it fabulous?\" Layla wondered. \"The art, the architecture, the food?\"\n\n\"It's got all that. The palace, the river, the opera. I got a taste of it, but mostly I was working. Flew in from Budapest for a poker game.\"\n\n\"You spent your time in\u2014what do they call it\u2014the Paris of Eastern Europe playing poker?\" Quinn demanded.\n\n\"Not all of it, just the lion's share. The game went for just over seventy-three hours.\"\n\n\"Three days, playing poker?\" Cybil's eyebrow winged up. \"Wouldn't that be a little obsessive?\"\n\n\"Depends on where you stand, doesn't it?\"\n\n\"But don't you need to sleep, eat? Pee?\" Layla wondered.\n\n\"Breaks are worked in. The seventy-three hours was actual game time. This was a private game, private home. Serious money, serious security.\"\n\n\"Win or lose?\" Quinn asked him with a grin.\n\n\"I did okay.\"\n\n\"Do you use your precognition to help you do okay?\" Cybil asked.\n\n\"That would be cheating.\"\n\n\"Yes, it would, but that didn't answer the question.\"\n\nHe picked up his wine, kept his eyes on hers. \"If I had to cheat to win at poker, I should be selling insurance. I don't have to cheat.\"\n\n\"We took an oath.\" Fox held up his hands when Gage scowled at him. \"We're in this together now. They should understand how it works for us. We took an oath when we realized we all had something extra. We wouldn't use it against anyone, or to hurt anyone, or, well, to screw anyone. We don't break our word to each other.\"\n\n\"In that case,\" Cybil said to Gage, \"you ought to be playing the ponies instead of cards.\"\n\nHe flashed a grin. \"Been known to, but I like cards. Wanna play?\"\n\n\"Maybe later.\"\n\nWhen Cybil glanced at Quinn with a look of apology, Quinn knew what was coming. \"I guess we should get back to it,\" Cybil began. \"I have a question, a place I'd like to start.\"\n\n\"Let's take fifteen.\" Quinn pushed to her feet. \"Get the table cleared off, take the dog out. Just move a little. Fifteen.\"\n\nCal brushed a hand over her arm as he rose with her. \"I need to check the fire anyway, probably bring in more wood. Let's do this in the living room when we're finished up.\"\n\nTHEY LOOKED LIKE ORDINARY PEOPLE, CAL thought. Just a group of friends hanging out on a winter night. Gage had switched to coffee, and that was usual. Cal hadn't known Gage to indulge in more than a couple drinks at a time since the summer they'd been seventeen. Fox was back on Coke, and he himself had opted for water.\n\nClear heads, he mused. They wanted clear heads if there were questions to be answered.\n\nThey'd gone back to gender groups. Had that been automatic, even intrinsic? he wondered. The three women on the couch, Fox on the floor with Lump. He'd taken a chair, and Gage stood by the fire as if he might just walk out if the topic didn't suit his mood.\n\n\"So.\" Cybil tucked her legs under her, let her dark eyes scan the room. \"I'm wondering what was the first thing, event, instance, the first happening, we'll say, that alerted you something was wrong in town. After your night in the clearing, after you went home.\"\n\n\"Mr. Guthrie and the fork.\" Fox stretched out, propped his head on Lump's belly. \"That was a big clue.\"\n\n\"Sounds like the title of a kid's book.\" Quinn made a note on her pad. \"Why don't you fill us in?\"\n\n\"You take it, Cal,\" Fox suggested.\n\n\"It would've been our birthday\u2014the night, or really the evening of it. We were all pretty spooked. It was worse being separated, each of us in our own place. I talked my mother into letting me go in to the bowling center, so I'd have something to do, and Gage would be there. She couldn't figure out whether to ground me or not,\" he said with a half smile. \"First and last time I remember her being undecided on that kind of issue. So she let me go in with my father. Gage?\"\n\n\"I was working. Mr. Hawkins let me earn some spending money at the center, mopping up spills or carrying grill orders out to tables. I know I felt a hell of a lot better when Cal came in. Then Fox.\"\n\n\"I nagged my parents brainless to let me go in. My father finally caved, took me. I think he wanted to have a confab with Cal's dad, and Gage's if he could.\"\n\n\"So, Brian\u2014Mr. O'Dell\u2014and my dad sat down at the end of the counter, having coffee. They didn't bring Bill, Gage's father, into it at that point.\"\n\n\"Because he didn't know I'd been gone in the first place,\" Gage said. \"No point getting me in trouble until they'd decided what to do.\"\n\n\"Where was your father?\" Cybil asked.\n\n\"Around. Behind the pins. He was having a few sober hours, so Mr. Hawkins had him working on something.\"\n\n\"Ball return, lane two,\" Cal murmured. \"I remember. It seemed like an ordinary summer night. Teenagers, some college types on the pinballs and video games. Grill smoking, pins crashing. There was a kid\u2014two or three years old, I guess\u2014with a family in the four lane. Major tantrum. The mother hauled him outside right before it happened.\"\n\nHe took a swig of water. He could see it, bell clear. \"Mr. Guthrie was at the counter, drinking a beer, eating a dog and fries. He came in once a week. Nice enough guy. Sold flooring, had a couple of kids in high school. Once a week, he came in when his wife went to the movies with girlfriends. It was clockwork. And Mr. Guthrie would order a dog and fries, and get steadily trashed. My dad used to say he did his drinking there because he could tell himself it wasn't real drinking if he wasn't in a bar.\"\n\n\"Troublemaker?\" Quinn asked as she made another note.\n\n\"Anything but. He was what my dad called an affable drunk. He never got mean, or even sloppy. Tuesday nights, Mr. Guthrie came in, got a dog and fries, drank four or five beers, watched some games, talked to whoever was around. Somewhere around eleven, he'd leave a five-dollar tip on the grill and walk home. Far as I know he didn't so much as crack a Bud otherwise. It was a Tuesday night deal.\"\n\n\"He used to buy eggs from us,\" Fox remembered. \"A dozen brown eggs, every Saturday morning. Anyway.\"\n\n\"It was nearly ten, and Mr. Guthrie was having another beer. He was walking by the tables with it,\" Cal said. \"Probably going to take it and stand behind the lanes, watch some of the action. Some guys were having burgers. Frank Dibbs was one of them\u2014held his league's record for high game, coached Little League. We were sitting at the next table, eating pizza. Dad told us to take a break, so we were splitting a pizza. Dibbs said, 'Hey, Guth, the wife wants new vinyl in the kitchen. What kind of deal can you give me?'\n\n\"And Guthrie, he just smiles. One of those tight-lipped smiles that don't show any teeth. He picks up one of the forks sitting on the table. He jammed it into Dibbs's cheek, just stabbed it into his face, and kept walking. People are screaming and running, and, Christ, that fork is just sticking out of Mr. Dibbs's cheek, and blood's sliding down his face. And Mr. Guthrie strolls over behind lane two, and drinks his beer.\"\n\nTo give himself a moment, Cal took a long drink. \"My dad wanted us out. Everything was going crazy, except Guthrie, who apparently was crazy. Your dad took care of Dibbs,\" Cal said to Fox. \"I remember how he kept his head. Dibbs had already yanked the fork out, and your father grabbed this stack of napkins and got the bleeding stopped. There was blood on his hands when he drove us home.\"\n\nCal shook his head. \"Not the point. Fox's dad took us home. Gage came with me\u2014my father took care of that. He didn't get home until it was light out. I heard him come home; my mother had waited for him. I heard him tell her they had Guthrie locked up, and he was just sitting in his cell laughing. Laughing like it was all a big joke. Later, when it was all over, he didn't even remember. Nobody remembered much of what went on that week, or if they did, they put it away. He never came in the center again. They moved away the next winter.\"\n\n\"Was that the only thing that happened that night?\" Cybil asked after a moment.\n\n\"Girl was raped.\" Gage set his empty mug on the mantel. \"Making out with her boyfriend out on Dog Street. He didn't stop when she said stop, didn't stop when she started to cry, to scream. He raped her in the backseat of his secondhand Buick, then shoved her out on the side of the road and drove off. Wrapped his car around a tree a couple hours later. Ended up in the same hospital as she did. Only he didn't make it.\"\n\n\"Family mutt attacked an eight-year-old boy,\" Fox added. \"Middle of that night. The dog had slept with the kid every night for three years. The parents woke up hearing the kid screaming, and when they got to the bedroom, the dog went for them, too. The father had to beat it off with the kid's baseball bat.\"\n\n\"It just got worse from there. That night, the next night.\" Cal took a long breath. \"Then it didn't always wait for night. Not always.\"\n\n\"There's a pattern to it.\" Quinn spoke quietly, then glanced up when Cal's voice cut through her thoughts.\n\n\"Where? Other than ordinary people turn violent or psychotic?\"\n\n\"We saw what happened with Lump. You've just told us about another family pet. There have been other incidents like that. Now you've said the first overt incident all of you witnessed involved a man who'd had several beers. His alcohol level was probably over the legal limit, meaning he was impaired. Mind's not sharp after drinking like that. You're more susceptible.\"\n\n\"So Guthrie was easier to influence or infect because he was drunk or well on the way?\" Fox pushed up to sitting. \"That's good. That makes good sense.\"\n\n\"The boy who raped his girlfriend of three months then drove into a tree hadn't been drinking.\" Gage shook his head. \"Where's that in the pattern?\"\n\n\"Sexual arousal and frustration tend to impair the brain.\" Quinn tapped her pencil on her pad. \"Put those into a teenage boy, and that says susceptible to me.\"\n\n\"It's a valid point.\" Cal shoved his hand through his hair. Why hadn't they seen it themselves? \"The dead crows. There were a couple dozen dead crows all over Main Street the morning of our birthday that year. Some broken windows where they'd repeatedly flown into the glass. We always figured that was part of it. But nobody got hurt.\"\n\n\"Does it always start that way?\" Layla asked. \"Can you pinpoint it?\"\n\n\"The first I remember from the next time was when the Myerses found their neighbor's dog drowned in a backyard swimming pool. There was the woman who left her kid locked in the car and went into the beauty salon, got a manicure and so on. It was in the nineties that day,\" Fox added. \"Somebody heard the kid crying, called the cops. They got the kid out, but when they went in to get the woman, she said she didn't have a baby. Didn't know what they were talking about. It came out she'd been up two nights running because the baby had colic.\"\n\n\"Sleep deprivation.\" Quinn wrote it down.\n\n\"But we knew it was happening again,\" Cal said slowly, \"we knew for sure on the night of our seventeenth when Lisa Hodges walked out of the bar at Main and Battlefield, stripped down naked, and started shooting at passing cars with the twenty-two she had in her purse.\"\n\n\"We were one of the cars,\" Gage added. \"Good thing for all concerned her aim was lousy.\"\n\n\"She caught your shoulder,\" Fox reminded him.\n\n\"She shot you?\"\n\nGage smiled easily at Cybil. \"Grazed me, and we heal fast. We managed to get the gun from her before she shot anyone else, or got hit by a car as she was standing buck naked in the middle of the street. Then she offered us blow jobs. Rumor was she gave a doozy, but we weren't much in the mood to find out.\"\n\n\"All right, from pattern to theory.\" Quinn rose to her feet to work it out. \"The thing we'll call Twisse, because it's better to have a name for it, requires energy. We're all made up of energy, and Twisse needs it to manifest, to work. When he's out, during this time Dent is unable to hold him, he seeks out the easiest sources of energy first. Birds and animals, people who are most vulnerable. As he gets stronger, he's able to move up the chain.\"\n\n\"I don't think the way to stop him is to clear out all the pets,\" Gage began, \"ban alcohol, drugs, and sex and make sure everyone gets a good night's sleep.\"\n\n\"Too bad,\" Cybil tossed back, \"because it might buy us some time. Keep going, Q.\"\n\n\"Next question would be, how does he generate the energy he needs?\"\n\n\"Fear, hate, violence.\" Cal nodded. \"We've got that. We can't cut off his supply because you can't block those emotions out of the population. They exist.\"\n\n\"So do their counterparts, so we can hypothesize that those are weapons or countermeasures against him. You've all gotten stronger over time, and so has he. Maybe he's able to store some of this energy he pulls in during the dormant period.\"\n\n\"And so he's able to start sooner, start stronger the next time. Okay,\" Cal decided. \"Okay, it makes sense.\"\n\n\"He's using some of that store now,\" Layla put in, \"because he doesn't want all six of us to stick this out. He wants to fracture the group before July.\"\n\n\"He must be disappointed.\" Cybil picked up the wine she'd nursed throughout the discussion. \"Knowledge is power and all that, and it's good to have logical theories, more areas to research. But it seems to be we need to move. We need a strategy. Got any, Mr. Strategy?\"\n\nFrom his spot on the floor, Fox grinned. \"Yeah. I say as soon as the snow melts enough for us to get through it, we go to the clearing. We go to the Pagan Stone, all of us together. And we double-dog dare the son of a bitch.\"\n\nIT SOUNDED GOOD IN THEORY. IT WAS A DIFFERENT matter, in Cal's mind, when you added the human factor. When you added Quinn. He'd taken her there once before, and he'd zoned out, leaving her alone and vulnerable.\n\nAnd he hadn't loved her then.\n\nHe knew there was no choice, that there were bigger stakes involved. But the idea of putting her at risk, at deliberately putting her at the center of it with him, kept him awake and restless.\n\nHe wandered the house, checking locks, staring out windows for any glimpse of the thing that stalked them. The moon was out, and the snow tinted blue under it. They'd be able to shovel their way out the next day, he thought, dig out the cars. Get back to what passed for normal within a day or two.\n\nHe already knew if he asked her to stay, just stay, she'd tell him she couldn't leave Layla and Cybil on their own. He already knew he'd have to let her go.\n\nHe couldn't protect her every hour of every day, and if he tried, they'd end up smothering each other.\n\nAs he moved through the living room, he saw the glow of the kitchen lights. He headed back to turn them off and check locks. And there was Gage, sitting at the counter playing solitaire with a mug of coffee steaming beside the discard pile.\n\n\"A guy who drinks black coffee at one a.m. is going to be awake all night.\"\n\n\"It never keeps me up.\" Gage flipped a card, made his play. \"When I want to sleep, I sleep. You know that. What's your excuse?\"\n\n\"I'm thinking it's going to be a long, hard, messy hike into the woods even if we wait a month. Which we probably should.\"\n\n\"No. Red six on black seven. You're trying to come up with a way to go in without Quinn. Without any of them, really, but especially the blonde.\"\n\n\"I told you how it was when we went in before.\"\n\n\"And she walked out again on her own two sexy legs. Jack of clubs on queen of diamonds. I'm not worried about her. I'm worried about you.\"\n\nCal's back went up. \"Is there a time I didn't handle myself?\"\n\n\"Not up until now. But you've got it bad, Hawkins. You've got it bad for the blonde, and being you, your first and last instinct is going to be to cover her ass if anything goes down.\"\n\n\"Shouldn't it be?\" He didn't want any damn coffee, but since he doubted he'd sleep anyway, he poured some. \"Why wouldn't it be?\"\n\n\"I'd lay money that your blonde can handle herself. Doesn't mean you're wrong, Cal. I imagine if I had a woman inside me the way she's inside you, I wouldn't want to put how she handled herself to the test. The trouble is, you're going to have to.\"\n\n\"I never wanted to feel this way,\" Cal said after a moment. \"This is a good part of the reason why. We're good together, Gage.\"\n\n\"I can see that for myself. Don't know what she sees in a loser like you, but it's working for her.\"\n\n\"We could get better. I can feel we'd just get better, make something real and solid. If we had the chance, if we had the time, we'd make something together.\"\n\nCasually, Gage gathered up the cards, shuffled them with a blur of speed. \"You think we're going down this time.\"\n\n\"Yeah.\" Cal looked out the window at the cold, blue moonlight. \"I think we're going down. Don't you?\"\n\n\"Odds are.\" Gage dealt them both a hand of blackjack. \"But hell, who wants to live forever?\"\n\n\"That's the problem. Now that I've found Quinn, forever sounds pretty damn good.\" Cal glanced at his hole card, noted the king to go with his three. \"Hit me.\"\n\nWith a grin, Gage flipped over a nine. \"Sucker.\"\n\n## Twenty\n\nCAL HOPED FOR A WEEK, TWO IF HE COULD MANAGE it. And got three days. Nature screwed his plans again, this time shooting temperatures up into the fifties. Mountains of snow melted into hills while the February thaw brought the fun of flash flooding, swollen creeks, and black ice when the thermometer dropped to freezing each night.\n\nBut three days after he'd had his lane plowed and the women were back in the house on High Street, the weather stabilized. Creeks ran high, but the ground sucked up most of the runoff. And he was coming up short on excuses to put off the hike to the Pagan Stone.\n\nAt his desk, with Lump contentedly sprawled on his back in the doorway, feet in the air, Cal put his mind into work. The winter leagues were winding up, and the spring groups would go into gear shortly. He knew he was on the edge of convincing his father the center would profit from the automatic scoring systems, and wanted to give it one more solid push. If they moved on it soon, they could have the systems up and running for the spring leagues.\n\nThey'd want to advertise, run a few specials. They'd have to train the staff, which meant training themselves.\n\nHe brought up the spreadsheet for February, noted that the month so far had been solid, even up a bit from last year. He'd use that as more ammunition. Which, of course, his father could and would counter that if they were up the way things were, why change it?\n\nAs he was holding the conversation in his head, Cal heard the click that meant a new e-mail had come in. He toggled over, saw Quinn's address.\n\nHi, Love of My Life,\n\nI didn't want to call in case you were knee-deep in whatever requires you to be knee-deep. Let me know when you're not.\n\nMeanwhile, this is Black's Local Weather Service reporting: Temperatures today should reach a high of forty-eight under partly sunny skies. Lows in the upper thirties. No precipitation is expected. Tomorrow's forecast is for sunny with a high of fifty.\n\nAdding the visual, I can see widening patches of grass in both the front and backyard. Realistically, there's probably more snow, more mud in the woods, but, baby, it's time to saddle up and move out.\n\nMy team can be ready bright and early tomorrow and will bring suitable provisions.\n\nAlso, Cyb's confirmed the Clark branch connection, and is currently climbing out on some Kinski limbs to verify that. She thinks she may have a line on a couple of possibilities where Ann Hawkins stayed, or at least where she might have gone to give birth. I'll fill you in when I see you.\n\nLet me know, soon as you can, if tomorrow works.\n\nXXOO Quinn.\n\n(I know that whole XXOO thing is dopey, but it seemed more refined than signing off with: I wish you could come over and do me. Even though I do.)\n\nThe last part made him smile even though the text of the post had a headache sneaking up the back of his skull.\n\nHe could put her off a day or two, and put her off honestly. He couldn't expect Fox to dump his scheduled clients or any court appearances at the snap of a finger, and she'd understand that. But if he were to use that, and his own schedule, he had to do it straight.\n\nWith some annoyance, he shot an e-mail to Fox, asking when he could clear time for the trip to the clearing. The annoyance increased when Fox answered back immediately.\n\nFri's good. Morning's clear, can clear full day if nec.\n\n\"Well, fuck.\" Cal pushed on the ache at the back of his head. Since e-mail wasn't bringing him any luck, he'd go see Quinn in person when he broke for lunch.\n\nAS CAL PREPARED TO CLOSE OUT FOR THE MORNING, Bill Turner stopped in the office doorway.\n\n\"Ah, got that toilet fixed in the ladies' room downstairs, and the leak in the freezer was just a hose needed replacing.\"\n\n\"Thanks, Bill.\" He swung his coat on as he spoke. \"I've got a couple of things to do in town. Shouldn't be above an hour.\"\n\n\"Okay, then. I was wondering if, ah...\" Bill rubbed a hand over his chin, let it drop. \"I was wondering if you think Gage'll be coming in, maybe the next day or two. Or if maybe I could, maybe I could run over to your place to have a word with him.\"\n\nRock and a hard place, Cal thought, and bought himself some time by adjusting his jacket. \"I don't know if he's thinking about dropping by, Bill. He hasn't mentioned it. I think...Okay, look, I'd give him some time. I'd just give it some time before you made that first move. I know you want\u2014\"\n\n\"It's okay. That's okay. Appreciate it.\"\n\n\"Shit,\" Cal said under his breath as Bill walked away. Then, \"Shit, shit, shit,\" as he headed out himself.\n\nHe had to take Gage's side in this, how could he not? He'd seen firsthand what Bill's belt had done to Gage when they'd been kids. And yet, he'd also witnessed, firsthand, the dozens of ways Bill had turned himself around in the last few years.\n\nAnd, hadn't he just seen the pain, guilt, even the grief on Bill's face just now? So either way he went, Cal knew he was going to feel guilty and annoyed.\n\nHe walked straight out and over to Quinn's.\n\nShe pulled open the door, yanked him in. Before he could say a word her arms were locked around his neck and her mouth was very busy on his. \"I was hoping that was you.\"\n\n\"Good thing it was, because Greg, the UPS guy on this route, might get the wrong idea if you greeted him that way.\"\n\n\"He is kind of cute. Come on back to the kitchen. I'd just come down to do a coffee run. We're all working on various projects upstairs. Did you get my e-mail?\"\n\n\"Yeah.\"\n\n\"So, we're all set for tomorrow?\" She glanced back as she reached up for the coffee.\n\n\"No, tomorrow's no good. Fox can't clear his slate until Friday.\"\n\n\"Oh.\" Her lips moved into a pout, quickly gone. \"Okay then, Friday it is. Meanwhile we'll keep reading, researching, working. Cyb thinks she's got a couple of good possibilities on...What?\" she asked when she got a good look at his face. \"What's going on?\"\n\n\"Okay.\" He took a couple paces away, then back. \"Okay, I'm just going to say it. I don't want you going back in there. Just be quiet a minute, will you?\" he said when he saw the retort forming. \"I wish there was a way I could stop you from going, that there was a way I could ignore the fact that we all need to go. I know you're a part of this, and I know you have to go back to the Pagan Stone. I know there's going to be more you have to be a part of than I'd wish otherwise. But I can wish you weren't part of this, Quinn, and that you were somewhere safe until this is over. I can want that, just as I know I can't have what I want.\n\n\"If you want to be pissed off about that, you'll have to be pissed off.\"\n\nShe waited a beat. \"Have you had lunch?\"\n\n\"No. What does that have to do with anything?\"\n\n\"I'm going to make you a sandwich\u2014an offer I never make lightly.\"\n\n\"Why are you making it now?\"\n\n\"Because I love you. Take off your coat. I love that you'd say all that to me,\" she began as she opened the refrigerator for fixings. \"That you'd need to let me know how you felt about it. Now if you'd tried ordering me to stay out of it, if you'd lied or tried to do some sort of end-run around me, I'd feel different. I'd still love you, because that sort of thing sticks with me, but I'd be mad, and more, I'd be disappointed in you. As it is, Cal, I'm finding myself pretty damn pleased and a hell of a lot smug that my head and heart worked so well together and picked the perfect guy. The perfect guy for me.\"\n\nShe cut the sandwich into two tidy triangles, offered it. \"Do you want coffee or milk?\"\n\n\"You don't have milk, you have white water. Coffee'd be fine, thanks.\" He took a bite of the turkey and Swiss with alfalfa on whole wheat. \"Pretty good sandwich.\"\n\n\"Don't get used to the service.\" She glanced over as she poured out coffee. \"We should get an early start on Friday, don't you think? Like dawn?\"\n\n\"Yeah.\" He touched her cheek with his free hand. \"We'll head in at first light.\"\n\nSINCE HE'D HAD GOOD LUCK WITH QUINN, AND gotten lunch out of it, Cal decided he was going to speak his mind to Gage next. The minute he and Lump stepped into the house, he smelled food. And when they wandered back, Cal found Gage in the kitchen, taking a pull off a beer as he stirred something in a pot.\n\n\"You made food.\"\n\n\"Chili. I was hungry. Fox called. He tells me we're taking the ladies for a hike Friday.\"\n\n\"Yeah. First light.\"\n\n\"Should be interesting.\"\n\n\"Has to be done.\" Cal dumped out food for Lump before getting a beer of his own. And so, he thought, did this have to be done. \"I need to talk to you about your father.\"\n\nCal saw Gage close off. Like a switch flipped, a finger snapped, his face simply blanked out. \"He works for you; that's your business. I've got nothing to say.\"\n\n\"You've got every right to shut him out. I'm not saying different. I'm letting you know he asks about you. He wants to see you. Look, he's been sober five years now, and if he'd been sober fifty it wouldn't change the way he treated you. But this is a small town, Gage, and you can't dodge him forever. My sense is he's got things to say to you, and you may want to get it done, put it behind you. That's it.\"\n\nThere was a reason Gage made his living at poker. It showed now in a face, a voice, completely devoid of expression. \"My sense is you should take yourself out of the middle. I haven't asked you to stand there.\"\n\nCal held up a hand for peace. \"Fine.\"\n\n\"Sounds like the old man's stuck on Steps Eight and Nine with me. He can't make amends on this, Cal. I don't give a damn about his amends.\"\n\n\"Okay. I'm not trying to convince you otherwise. Just letting you know.\"\n\n\"Now I know.\"\n\nIT OCCURRED TO CAL WHEN HE STOOD AT THE window on Friday morning, watching the headlights cut through the dim predawn, that it had been almost a month exactly since Quinn had first driven up to his house.\n\nHow could so much have happened? How could so much have changed in such a short time?\n\nIt had been slightly less than that month since he'd led her into the woods the first time. When he'd led her to the Pagan Stone.\n\nIn those short weeks of the shortest month he'd learned it wasn't only himself and his two blood brothers who were destined to face this threat. There were three women now, equally involved.\n\nAnd he was completely in love with one of them.\n\nHe stood just as he was to watch her climb out of Fox's truck. Her bright hair spilled out from under the dark watch cap. She wore a bold red jacket and scarred hiking boots. He could see the laugh on her face as she said something to Cybil, and her breath whisked out in clouds in the early morning chill.\n\nShe knew enough to be afraid, he understood that. But she refused to allow fear to dictate her moves. He hoped he could say the same as he had more to risk now. He had her.\n\nHe stood watch until he heard Fox use his key to unlock the front door, then Cal went down to join them, and to gather his things for the day.\n\nFog smoked the ground that the cold had hardened like stone overnight. By midday, Cal knew the path would be sloppy again, but for now it was quick and easy going.\n\nThere were still pockets and lumpy hills of snow, and he identified the hoofprints of the deer that roamed the woods, to Layla's delight. If any of them were nervous, they hid it well, at least on this first leg of the hike.\n\nIt was so different from that long-ago day in July when he and Fox and Gage had made this trip. No boom box pumping out rap or heavy metal, no snacks of Little Debbies, no innocent, youthful excitement of a stolen day, and the night to come.\n\nNone of them had ever been so innocent again.\n\nHe caught himself lifting a hand to his face, where his glasses used to slide down the bridge of his nose.\n\n\"How you doing, Captain?\" Quinn stepped up to match her pace to his, gave him a light arm bump.\n\n\"Okay. I was just thinking about that day. Everything hot and green, Fox hauling that stupid boom box. My mother's lemonade, snack cakes.\"\n\n\"Sweat rolling,\" Fox continued from just behind him.\n\n\"We're coming up on Hester's Pool,\" Gage said, breaking the memory.\n\nThe water made Cal think of quicksand rather than the cool and forbidden pool he and his friends had leaped into so long ago. He could imagine going in now, being sucked in, deeper and deeper until he never saw light again.\n\nThey stopped as they had before, but now it was coffee instead of lemonade.\n\n\"There's been deer here, too.\" Layla pointed at the ground. \"Those are deer prints, right?\"\n\n\"Some deer,\" Fox confirmed. \"Raccoon.\" He took her arm to turn her, pointed to the prints on the ground.\n\n\"Raccoons?\" Grinning, she bent to take a closer look. \"What else might be in here?\"\n\n\"Some of my namesakes, wild turkey, now and then\u2014though mostly north of here\u2014you might see bear.\"\n\nShe straightened quickly. \"Bear.\"\n\n\"Mostly north,\" he repeated, but found it as good an excuse as any to take her hand.\n\nCybil crouched by the edge of the pool, stared at the water.\n\n\"A little cold to think about taking a dip,\" Gage told her.\n\n\"Hester drowned herself here.\" She glanced up, then looked over at Cal. \"And when you went in that day, you saw her.\"\n\n\"Yeah. Yeah, I saw her.\"\n\n\"And you and Quinn have both seen her in your heads. Layla's dreamed of her, vividly. So...maybe I can get something.\"\n\n\"I thought yours was precog, not the past,\" Cal began.\n\n\"It is, but I still get vibes from people, from places that are strong enough to send them out. How about you?\" She looked back at Gage. \"We might stir up more in tandem. Are you up for that?\"\n\nSaying nothing, he held out a hand. She took it, rose to her feet. Together, they stared at that still, brown surface.\n\nThe water began to beat and froth. It began to spin, to spew up white-tipped waves. It roared like a sea mating with a wild and vicious storm.\n\nAnd a hand shot out to claw at the ground.\n\nHester pulled herself out of that churning water\u2014bone white skin, a mass of wet, tangled hair, dark, glassy eyes. The effort, or her madness, peeled her lips back from her teeth.\n\nCybil heard herself scream as Hester Deale's arms opened, as they locked around her and dragged her toward that swirling brown pool.\n\n\"Cyb! Cyb! Cybil!\"\n\nShe came back struggling, and found herself locked not in Hester's arms, but Gage's. \"What the hell was that?\"\n\n\"You were going in.\"\n\nShe stayed where she was, feeling her heart hammer against his as Quinn gripped her shoulder. Cybil took another look at the still surface of the pool. \"That would've been really unpleasant.\"\n\nShe was trembling, one hard jolt after the next, but Gage had to give her points for keeping her voice even.\n\n\"Did you get anything?\" she asked him.\n\n\"Water kicked up; she came up. You started to tip.\"\n\n\"She grabbed me. She...embraced me. That's what I think, but I wasn't focused enough to feel or sense what she felt. Maybe if we tried it again\u2014\"\n\n\"We've got to get moving now,\" Cal interrupted.\n\n\"It only took a minute.\"\n\n\"Try nearly fifteen,\" Fox corrected.\n\n\"But...\" Cybil eased back from Gage when she realized she was still in his arms. \"Did it seem that long to you?\"\n\n\"No. It was immediate.\"\n\n\"It wasn't.\" Layla held out another thermos lid of coffee. \"We were arguing about whether we should pull you back, and how we should if we did. Quinn said to leave you be for another few minutes, that sometimes it took you a while to warm up.\"\n\n\"Well, it felt like a minute, no more than, for the whole deal. And it didn't feel like something from before.\" Again, Cybil looked at Gage.\n\n\"No, it didn't. So if I were you, I wouldn't think about taking a dip anytime soon.\"\n\n\"I prefer a nice blue pool, with a swim-up bar.\"\n\n\"Bikini margaritas.\" Quinn rubbed her hand up and down Cybil's arm.\n\n\"Spring break, two thousand.\" Cybil caught Quinn's hand, squeezed. \"I'm fine, Q.\"\n\n\"I'll buy the first round of those margaritas when this is done. Ready to move on?\" Cal asked.\n\nHe hitched up his pack, turned. Then shook his head. \"This isn't right.\"\n\n\"We're leaving the haunted pool to walk through the demonic woods.\" Quinn worked up a smile. \"What could be wrong?\"\n\n\"That's not the path.\" He gestured toward the thawing track. \"That's not the direction.\" He squinted up at the sun as he pulled his old Boy Scout compass out of his pocket.\n\n\"Ever thought about upgrading to a GPS?\" Gage asked him.\n\n\"This does the job. See, we need to head west from here. That trail's leading north. That trail shouldn't even be there.\"\n\n\"It's not there.\" Fox's eyes narrowed, darkened. \"There's no trail, just underbrush, a thicket of wild blackberries. It's not real.\" He shifted, angled himself. \"It's that way.\" He gestured west. \"It's hard to see, it's like looking through mud, but...\"\n\nLayla stepped forward, took his hand.\n\n\"Okay, yeah. That's better.\"\n\n\"You're pointing at a really big-ass tree,\" Cybil told him.\n\n\"That isn't there.\" Still holding Layla's hand, Fox walked forward. The image of the large oak broke apart as he walked through it.\n\n\"Nice trick.\" Quinn let out a breath. \"So, Twisse doesn't want us to go to the clearing. I'll take point.\"\n\n\"I'll take point.\" Cal took her arm to tug her behind him. \"I've got the compass.\" He had only to glance back at his friends to have them falling in line. Fox taking center, Gage the rear with the women between.\n\nAs soon as the track widened enough to allow it, Quinn moved up beside Cal. \"This is the way it has to work.\" She glanced back to see the other women had followed her lead, and now walked abreast with their partners. \"We're linked up this way, Cal. Two-by-two, trios, the group of six. Whatever the reasons are, that's the way it is.\"\n\n\"We're walking into something. I can't see what it is, but I'm walking you and the others right into it.\"\n\n\"We're all on our own two feet, Cal.\" She passed him the bottle of water she carried in her coat pocket. \"I don't know if I love you because you're Mr. Responsibility or in spite of it.\"\n\n\"As long as you do. And since you do, maybe we should think about the idea of getting married.\"\n\n\"I like the idea,\" she said after a moment. \"If you want my thoughts on it.\"\n\n\"I do.\" Stupid, he thought, stupid way to propose, and a ridiculous place for it, too. Then again, when they couldn't be sure what was around the bend, it made sense to grab what you did now, tight and quick. \"As it happens, I agree with you. More thoughts on the idea would be that my mother, especially, will want the splash\u2014big deal, big party, bells and whistles.\"\n\n\"I happen to agree with that, too. How is she with communication by phone and\/or e-mail?\"\n\n\"She's all about that.\"\n\n\"Great. I'll hook her up with my mother and they can go for it. How's your September schedule?\"\n\n\"September?\"\n\nShe studied the winter woods, watched a squirrel scamper up a tree and across a thick branch. \"I bet the Hollow's beautiful in September. Still green, but with just a hint of the color to come.\"\n\n\"I was thinking sooner. Like April, or May.\" Before, Cal thought. Before July, and what might be the end of everything he knew and loved.\n\n\"It takes a while to organize those bells and whistles.\" When she looked at him he understood she read him clearly. \"After, Cal, after we've won. One more thing to celebrate. When we're\u2014\"\n\nShe broke off when he touched a finger to her lips.\n\nThe sound came clearly now as all movement and conversation stopped. The wet and throaty snarl rolled across the air, and shot cold down the spine. Lump curled down on his haunches and whined.\n\n\"He hears it, too, this time.\" Cal shifted, and though the movement was slight, it put Quinn between him and Fox.\n\n\"I don't guess we could be lucky, and that's just a bear.\" Layla cleared her throat. \"Either way, I think we should keep moving. Whatever it is doesn't want us to, so...\"\n\n\"We're here to flip it the bird,\" Fox finished.\n\n\"Come on, Lump, come on with me.\"\n\nThe dog shivered at Cal's command, but rose, and with its side pressed to Cal's legs, walked down the trail toward the Pagan Stone.\n\nThe wolf\u2014Cal would never have referred to the thing as a dog\u2014stood at the mouth of the clearing. It was huge and black, with eyes that were somehow human. Lump tried a halfhearted snarl in answer to the low, warning growl, then cowered against Cal.\n\n\"Are we going to walk through that, too?\" Gage asked from the rear.\n\n\"It's not like the false trail.\" Fox shook his head. \"It's not real, but it's there.\"\n\n\"Okay.\" Gage started to pull off his pack.\n\nAnd the thing leaped.\n\nIt seemed to fly, Cal thought, a mass of muscle and teeth. He fisted his hands to defend, but there was nothing to fight.\n\n\"I felt...\" Slowly, Quinn lowered the arms she'd thrown up to protect her face.\n\n\"Yeah. Not just the cold, not that time.\" Cal gripped her arm to keep her close. \"There was weight, just for a second, and there was substance.\"\n\n\"We never had that before, not even during the Seven.\" Fox scanned the woods on both sides. \"Whatever form Twisse took, whatever we saw, it wasn't really there. It's always been mind games.\"\n\n\"If it can solidify, it can hurt us directly,\" Layla pointed out.\n\n\"And be hurt.\" From behind her Gage pulled a 9mm Glock out of his pack.\n\n\"Good thinking,\" was Cybil's cool opinion.\n\n\"Jesus Christ, Gage, where the hell did you get that?\"\n\nGage lifted his eyebrows at Fox. \"Guy I know down in D.C. Are we going to stand here in a huddle, or are we going in?\"\n\n\"Don't point that at anybody,\" Fox demanded.\n\n\"Safety's on.\"\n\n\"That's what they always say before they accidentally blow a hole in the best friend.\"\n\nThey stepped into the clearing, and the stone.\n\n\"My God, it's beautiful.\" Cybil breathed the words reverently as she moved toward it. \"It can't possibly be a natural formation, it's too perfect. It's designed, and for worship, I'd think. And it's warm. Feel it. The stone's warm.\" She circled it. \"Anyone with any sensitivity has to feel, has to know this is sacred ground.\"\n\n\"Sacred to who?\" Gage countered. \"Because what came up out of here twenty-one years ago wasn't all bright and friendly.\"\n\n\"It wasn't all dark either. We felt both.\" Cal looked at Fox. \"We saw both.\"\n\n\"Yeah. It's just the big, black scary mass got most of our attention while we were being blasted off our feet.\"\n\n\"But the other gave us most of his, that's what I think. I walked out of here not only without a scratch, but with twenty-twenty vision and a hell of an immune system.\"\n\n\"The scratches on my arms had healed up, and the bruises from my most recent tussle with Napper.\" Fox shrugged. \"Never been sick a day since.\"\n\n\"How about you?\" Cybil asked Gage. \"Any miraculous healing?\"\n\n\"None of us had a mark on him after the blast,\" Cal began.\n\n\"It's no deal, Cal. No secrets from the team. My old man used his belt on me the night before we were heading in here. A habit of his when he'd get a drunk on. I was carrying the welts when I came in, but not when I walked out.\"\n\n\"I see.\" Cybil held Gage's eyes for several beats. \"The fact that you were given protection, and your specific abilities, enabled you to defend your ground, so to speak. Otherwise, you'd have been three helpless little boys.\"\n\n\"It's clean.\" Layla's comment had everyone turning to where she stood by the stone. \"That's what comes to my mind. I don't think it was ever used for sacrifice. Not blood and death, not for the dark. It feels clean.\"\n\n\"I've seen the blood on it,\" Gage said. \"I've seen it burn. I've heard the screams.\"\n\n\"That's not its purpose. Maybe that's what Twisse wants.\" Quinn laid her palm on the stone. \"To defile it, to twist its power. If he can, well, he'll own it, won't he? Cal?\"\n\n\"Okay.\" His hand hovered over hers. \"Ready?\" At her nod, he joined his hand to hers on the stone.\n\nAt first there was only her, only Quinn. Only the courage in her eyes. Then the world tumbled back, five years, twenty, so that he saw the boy he'd been with his friends, scoring his knife over their wrists to bind them together. Then rushing back, decades, centuries, to the blaze and the screams while the stone stood cool and white in the midst of hell.\n\nBack to another waning winter where Giles Dent stood with Ann Hawkins as he stood with Quinn now. Dent's words came from his lips.\n\n\"We have only until summer. This I cannot change, even for you. Duty outstrips even my love for you, and for the lives we have made.\" He touched a hand to her belly. \"I wish, above all, that I could be with you when they come into the world.\"\n\n\"Let me stay. Beloved.\"\n\n\"I am the guardian. You are the hope. I cannot destroy the beast, only chain it for a time. Still, I do not leave you. It is not death, but an endless struggle, a war only I can wage. Until what comes from us makes the end. They will have all I can give, this I swear to you. If they are victorious in their time, I will be with you again.\"\n\n\"What will I tell them of their father?\"\n\n\"That he loved their mother, and them, with the whole of his heart.\"\n\n\"Giles, it has a man's form. A man can bleed, a man can die.\"\n\n\"It is not a man, and it is not in my power to destroy it. That will be for those who come after us both. It, too, will make its own. Not through love. They will not be what it intends. It cannot own them if they are beyond its reach, even its ken. This is for me to do. I am not the first, Ann, only the last. What comes from us is the future.\"\n\nShe pressed a hand to her side. \"They quicken,\" she whispered. \"When, Giles, when will it end? All the lives we have lived before, all the joy and the pain we have known? When will there be peace for us?\"\n\n\"Be my heart.\" He lifted her hands to his lips. \"I will be your courage. And we will find each other once more.\"\n\nTears slid down Quinn's cheeks even as she felt the images fade. \"We're all they have. If we don't find the way, they're lost to each other. I felt her heart breaking inside me.\"\n\n\"He believed in what he'd done, what he had to do. He believed in us, though he couldn't see it clearly. I don't think he could see us, all of us,\" Cal said as he looked around. \"Not clearly. He took it on faith.\"\n\n\"Fine for him.\" Gage shifted his weight. \"But I put a little more of mine in this Glock.\"\n\nIt wasn't the wolf, but the boy that stood on the edge of the clearing. Grinning, grinning. He lifted his hands, showed fingernails that were sharpened to claws.\n\nThe sun dimmed from midday to twilight; the air from cool to frigid. And thunder rumbled in the late winter sky.\n\nIn a lightning move so unexpected Cal couldn't prevent it, Lump sprang. The thing who masked as a boy squealed with laughter, shinnied up a tree like a monkey.\n\nBut Cal had seen it, in a flash of an instant. He'd seen the shock, and what might have been fear.\n\n\"Shoot it,\" Cal shouted to Gage, even as he dashed forward to grab Lump's collar. \"Shoot the son of a bitch.\"\n\n\"Jesus, you don't actually think a bullet's going to\u2014\"\n\nOver Fox's objection, Gage fired. Without hesitation, he aimed for the boy's heart.\n\nThe bullet cracked the air, struck the tree. This time no one could miss the look of shock on the boy's face. His howl of pain and fury gushed across the clearing and shook the ground.\n\nWith ruthless purpose, Gage emptied the clip into it.\n\nIt changed. It grew. It twisted itself into something massive and black and sinuous that rose over Cal as he stood his ground, fighting to hold back his dog, who strained and barked like a mad thing.\n\nThe stench of it, the cold of it hammered down on him like stones. \"We're still here,\" Cal shouted. \"This is our place, and you can go to hell.\"\n\nHe staggered against a blast of sound and slapping air.\n\n\"Better reload, Deadeye,\" Cybil commanded.\n\n\"Knew I should've bought a howitzer.\" But Gage slapped in a full clip.\n\n\"This isn't your place,\" Cal shouted again. The wind threatened to knock him off his feet, seemed to tear at his clothes and his skin like a thousand knives. Through the scream of it, he heard the crack of gunfire, and the rage it spewed out clamped on his throat like claws.\n\nThen Quinn braced against his side. And Fox shouldered in at his other. They formed a line, all six.\n\n\"This,\" Cal called out, \"is ours. Our place and our time. You couldn't have my dog, and you can't have my town.\"\n\n\"So fuck off,\" Fox suggested, and bending picked up a rock. He hurled it, a straightaway fast ball.\n\n\"Hello, got a gun here.\"\n\nFox's grin at Gage was wild and wide as the feral wind battered them. \"Throwing rocks is an insult. It'll undermine its confidence.\"\n\nDie here!\n\nIt wasn't a voice, but a tidal wave of sound and wind that knocked them to the ground, scattered them like bowling pins.\n\n\"Undermine, my ass.\" Gage shoved to his knees and began firing again.\n\n\"You'll die here.\" Cal spoke coolly as the others took Fox's tack and began to hurl stones and sticks.\n\nFire swept across the clearing, its flames like shards of ice. Smoke belched up in fetid clouds as it roared its outrage.\n\n\"You'll die here,\" Cal repeated. Pulling his knife from its sheath, he rushed foward to plunge it into the boiling black mass.\n\nIt screamed. He thought it screamed, thought the sound held something of pain as well as fury. The shock of power sang up his arm, stabbed through him like a blade, twin edges of scorching heat and impossible cold. It flung him away, sent him flying through the smoke like a pebble from a sling. Breathless, bones jarred from the fall, Cal scrambled to his feet.\n\n\"You'll die here!\" This time he shouted it as he gripped the knife, as he charged forward.\n\nThe thing that was a wolf, a boy, a man, a demon looked at him with eyes of hate.\n\nAnd vanished.\n\n\"But not today.\" The fire died, the smoke cleared as he bent over to suck in air. \"Anybody hurt? Is everybody okay? Quinn. Hey, Lump, hey.\" He nearly toppled backward when Lump leaped up, paws on shoulders to lap his face.\n\n\"Your nose is bleeding.\" Scurrying over on her hands and knees, Quinn gripped his arm to pull herself to her feet. \"Cal.\" Her hands rushed over his face, his body. \"Oh God, Cal. I've never seen anything so brave, or so goddamn stupid.\"\n\n\"Yeah, well.\" In a defiant move, he swiped at the blood. \"It pissed me off. If that was its best shot, it fell way short.\"\n\n\"It didn't dish out anything a really big drink and a long hot bath won't cure,\" Cybil decided. \"Layla? Okay?\"\n\n\"Okay.\" Face fierce, Layla brushed at her stinging cheeks. \"Okay.\" She took Fox's outstretched hand and got to her feet. \"We scared it. We scared it, and it ran away.\"\n\n\"Even better. We hurt it.\" Quinn took a couple shuddering breaths, then much as Lump had, leaped at Cal. \"We're all right. We're all okay. You were amazing. You were beyond belief. Oh God, God, give me a really big kiss.\"\n\nAs she laughed and wept, he took her mouth. He held her close, understanding that of all the answers they needed, for him she was the first.\n\nThey weren't going down this time, he realized.\n\n\"We're going to win this.\" He drew her away so he could look into her eyes. His were calm, steady, and clear. \"I never believed it before, not really. But I do now. I know it now. Quinn.\" He pressed his lips to her forehead. \"We're going to win this, and we're getting married in September.\"\n\n\"Damn straight.\"\n\nWhen she wrapped around him again, it was victory enough for now. It was enough to stand on until the next time. And the next time, he determined, they'd be better armed.\n\n\"Let's go home. It's a long walk back, and we've got a hell of a lot to do.\"\n\nShe held on another moment, held tight while he looked over her head into the eyes of his brothers. Gage nodded, then shoved the gun back in his pack. Swinging it on, he crossed the clearing to the path beyond.\n\nThe sun bloomed overhead, and the wind died. They walked out of the clearing, through the winter woods, three men, three women, and a dog.\n\nOn its ground the Pagan Stone stood silent, waiting for their return.\n\n##\n\nHawkins Hollow  \nJune 1994\n\nON A BRIGHT SUMMER MORNING, A TEACUP poodle drowned in the Bestlers' backyard swimming pool. At first, Lynne Bestler, who'd gone out to sneak in a solitary swim before her kids woke, thought it was a dead squirrel. Which would've been bad enough. But when she steeled herself to scoop out the tangle of fur with the net, she recognized her neighbor's beloved Marcell.\n\nSquirrels generally didn't wear rhinestone collars.\n\nHer shouts, and the splash as Lynne tossed the hapless dog, net and all, back into the pool, brought Lynne's husband rushing out in his boxers. Their mother's sobs and their father's curses as he jumped in to grab the pole and tow the body to the side, woke the Bestler twins, who stood screaming in their matching My Little Pony nightgowns. Within moments, the backyard hysteria had neighbors hurrying to fences just as Bestler dragged himself and his burden out of the water. As, like many men, Bestler had developed an attachment to ancient underwear, the weight of the water was too much for the worn elastic.\n\nSo Bestler came out of his pool with a dead dog, and no boxers.\n\nThe bright summer morning in the little town of Hawkins Hollow began with shock, grief, farce, and drama.\n\nFox learned of Marcell's untimely death minutes after he stepped into Ma's Pantry to pick up a sixteen-ounce bottle of Coke and a couple of Slim Jims.\n\nHe'd copped a quick break from working with his father on a kitchen remodel down Main Street. Mrs. Larson wanted new countertops, cabinet doors, new floors, new paint. She called it freshening things up, and Fox called it a way to earn enough money to take Allyson Brendon out for pizza and the movies on Saturday night. He hoped to use that gateway to talk her into the backseat of his ancient VW Bug.\n\nHe didn't mind working with his dad. He hoped to hell he wouldn't spend the rest of his life swinging a hammer or running a power saw, but he didn't mind it. His father's company was always easy, and the job got Fox out of gardening and animal duty on their little farm. It also provided easy access to Cokes and Slim Jims\u2014two items which would never, never be found in the O'Dell-Barry household.\n\nHis mother ruled there.\n\nSo he heard about the dog from Susan Keefaffer, who rang up his purchases while a few people with nothing better to do on a June afternoon sat at the counter over coffee and gossip.\n\nHe didn't know Marcell, but Fox had a soft spot for animals, so he suffered a twist of grief for the unfortunate poodle. That was leavened somewhat by the idea of Mr. Bestler, whom he did know, standing \"naked as a jaybird,\" in Susan Keefaffer's words, beside his backyard pool.\n\nWhile it made Fox sad to imagine some poor dog drowning in a swimming pool, he didn't connect it\u2014not then\u2014to the nightmare he and his two closest friends had lived through seven years before.\n\nHe'd had a dream the night before, a dream of blood and fire, of voices chanting in a language he didn't understand. But then he'd watched a double feature of videos\u2014The Night of the Living Dead and The Texas Chainsaw Massacre\u2014with his friends Cal and Gage.\n\nHe didn't connect a dead French poodle with the dream, or with what had burned through Hawkins Hollow for a week after his tenth birthday. After the night he and Cal and Gage had spent at the Pagan Stone in Hawkins Wood\u2014and everything had changed for them, and for the Hollow.\n\nIn a few weeks he and Cal and Gage would all turn seventeen\u2014and that was on his mind. Baltimore had a damn good chance at a pennant this year, so that was on his mind. He'd be going back to high school as a senior, which meant top of the food chain at last, and planning for college.\n\nWhat occupied a sixteen-year-old boy was considerably different than what occupied a ten-year-old. Including rounding third and heading for home with Allyson Brendon.\n\nSo when he walked back down the street, a lean boy not quite beyond the gangly stage of adolescence, his dense brown hair tied back in a stubby tail, golden brown eyes shaded with Oakleys, it was, for him, just another ordinary day.\n\nThe town looked as it always did. Tidy, a little old-timey, with the old stone townhouses or shops, the painted porches, the high curbs. He glanced back over his shoulder toward the Bowl-a-Rama on the square. It was the biggest building in town, and where Cal and Gage were both working.\n\nWhen he and his father knocked off for the day, he thought, he'd head on up, see what was happening.\n\nHe crossed over to the Larson place, walked into the unlocked house where Bonnie Raitt's smooth Delta blues slid smoothly out of the kitchen. His father sang along with her in his clear and easy voice as he checked the level on the shelves Mrs. Larson wanted in her utility closet. Though the windows and back door were open to their screens, the room smelled of sawdust, sweat, and the glue they'd used that morning to lay the new Formica.\n\nHis father worked in old Levi's and his Give Peace a Chance T-shirt. His hair was six inches longer than Fox's, worn in a tail under a blue bandanna. He'd shaved off the beard and mustache he'd had as long as Fox remembered. Fox still wasn't quite used to seeing so much of his father's face\u2014or so much of himself in it.\n\n\"A dog drowned in the Bestlers' swimming pool over on Laurel Lane,\" Fox told him, and Brian stopped working to turn.\n\n\"That's a damn shame. Anybody know how it happened?\"\n\n\"Not really. It was one of those little poodles, so think it must've fallen in, then it couldn't get out again.\"\n\n\"You'd think somebody would've heard it barking. That's a lousy way to go.\" Brian set down his tools, smiled at his boy. \"Gimme one of those Slim Jims.\"\n\n\"What Slim Jims?\"\n\n\"The ones you've got in your back pocket. You're not carrying a bag, and you weren't gone long enough to scarf down Hostess Pies or Twinkies. I'm betting you're packing the Jims. I get one, and your mom never has to know we ate chemicals and meat by-products. It's called blackmail, kid of mine.\"\n\nFox snorted, pulled them out. He'd bought two for just this purpose. Father and son unwrapped, bit off, chewed in perfect harmony. \"The counter looks good, Dad.\"\n\n\"Yeah, it does.\" Brian ran a hand over the smooth, eggshell surface. \"Mrs. Larson's not much for color, but it's good work. I don't know who I'm going to get to be my lapdog when you head off to college.\"\n\n\"Ridge is next in line,\" Fox said, thinking of his younger brother. \"Ridge wouldn't keep measurements in his head for two minutes running, and he'd probably cut off a finger dreaming while he was using a band saw. No.\" Brian smiled, shrugged. \"This kind of work isn't for Ridge, or for you, for that matter. Or either of your sisters. I guess I'm going to have to rent a kid to get one who wants to work with wood.\"\n\n\"I never said I didn't want to.\" Not out loud.\n\nHis father looked at him the way he sometimes did, as if he saw more than what was there. \"You've got a good eye, you've got good hands. You'll be handy around your own house once you get one. But you won't be strapping on a tool belt to make a living. Until you figure out just what it is you want, you can haul these scraps on out to the Dumpster.\"\n\n\"Sure.\" Fox gathered up scraps, trash, began to cart them out the back, across the narrow yard to the Dumpster the Larsons had rented for the duration of the remodel.\n\nHe glanced toward the adjoining yard and the sound of kids playing. And the armload he carried thumped and bounced on the ground as his body went numb.\n\nThe little boys played with trucks and shovels and pails in a bright blue sandbox. But it wasn't filled with sand. Blood covered their bare arms as they pushed their Tonka trucks through the muck inside the box. He stumbled back as the boys made engine sounds, as red lapped over the bright blue sides and dripped onto the green grass.\n\nOn the fence between the yards, where hydrangeas headed up toward bloom, crouched a boy that wasn't a boy. He bared its teeth in a grin as Fox backed toward the house.\n\n\"Dad! Dad!\"\n\nThe tone, the breathless fear had Brian rushing outside. \"What? What is it?\"\n\n\"Don't you\u2014can't you see?\" But even as he said it, as he pointed, something inside Fox knew. It wasn't real.\n\n\"What?\" Firmly now, Brian took his son's shoulders. \"What do you see?\"\n\nThe boy that wasn't a boy danced along the top of the chain-link fence while flames spurted up below and burned the hydrangeas to cinders.\n\n\"I have to go. I have to go see Cal and Gage. Right now, Dad. I have to\u2014\"\n\n\"Go.\" Brian released his hold on Fox, stepped back. He didn't question. \"Go.\"\n\nHe all but flew through the house and out again, up the sidewalk to the square. The town no longer looked as it usubeen that horrible week in July seven years before.\n\nFire and blood, he remembered, thinking of the dream. He burst into the Bowl-a-Rama, where the summer afternoon leagues were in full swing. The thunder of balls, the crash of pins pounded in his head as he ran straight to the front desk where Cal worked.\n\n\"Where's Gage?\" Fox demanded.\n\n\"Jesus, what's up with you?\"\n\n\"Where's Gage?\" Fox repeated, and Cal's amused gray eyes sobered. \"Working the arcade. He's...he's coming out now.\"\n\nAt Cal's quick signal, Gage sauntered over. \"Hello, ladies. What...\" The smirk died after one look at Fox's face. \"What happened?\"\n\n\"It's back,\" Fox said. \"It's come back.\"\n\n\u2022 \u2022 \u2022\n\n[For a complete list of this author's books click here or visit  \nwww.penguin.com\/robertschecklist](http:\/\/www.penguin.com\/robertschecklist?CMP=CKL-ROBERTS)\n\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":" \n## Red November\n\nInside the Secret U.S.-Soviet Submarine War\n\n## W. Craig Reed\n\nThis book is dedicated to my father, Lieutenant William J. Reed, Retired, who helped devise and deploy the top-secret Boresight program, and to the underwater sailors and civilians who sacrificed so much. The following pages are a tribute to the commitment, courage, and constant vigilance of those who sacrificed so much to ensure that our world did not end by way of fire and fallout.\n\n## Contents\n\nIntroduction\n\nAuthor's Note\n\nChapter One\n\nWITH ORDERS TO CONDUCT A TOP-SECRET espionage mission, the USS...\n\nChapter Two\n\nIN THE FALL OF 1953, DR. Donald Ross, an underwater engineer...\n\nChapter Three\n\nHAVING WORKED HIS WAY UP FROM a seaman to a...\n\nChapter Four\n\nWHEN MY DAD, WILLIAM J. REED, reported to NSG headquarters at...\n\nChapter Five\n\nDURING THE MONTH OF AUGUST 1962, while a small window...\n\nChapter Six\n\nHIS ARM CHAINED TO A BRIEFCASE, William J. Reed counted the...\n\nChapter Seven\n\nON SUNDAY, OCTOBER 21, ABOARD THE USS Oxford off the...\n\nChapter Eight\n\nON THE MORNING OF OCTOBER 23, when Communications Technician John...\n\nChapter Nine\n\nEARLY IN THE EVENING OF OCTOBER 25, in a lab...\n\nPhotographic Insert 1\n\nChapter Ten\n\nHIS FACE RED WITH ANGER, JOHN Scali of ABC News...\n\nChapter Eleven\n\nTHE EARLY SIXTIES USHERED IN THE next evolution in underwater...\n\nChapter Twelve\n\nON DECEMBER 29, 1964, THE U.S. Navy made a huge...\n\nChapter Thirteen\n\nTHE SOVIETS FINALLY GAVE UP THE search for their lost...\n\nChapter Fourteen\n\nMANY OF THE DETAILS SURROUNDING PROJECT Azorian became public knowledge...\n\nChapter Fifteen\n\nWHEN THE SOVIETDELTA-CLASS SUBMARINE ENTERED stage right in December 1972,...\n\nChapter Sixteen\n\nBY LATE 1975, SOME AMERICANS HATED sailors. They also hated...\n\nPhotographic Insert 2\n\nChapter Seventeen\n\nHAVING REPLACED THE HALIBUT AS THE cable-taping spy boat, the...\n\nChapter Eighteen\n\nWHEN THE FIRST ULTRA-QUIET SOVIET NUCLEAR-POWERED hunter\/killer attack submarine, code-named...\n\nChapter Nineteen\n\nTHEY SAY YOUR LIFE PASSES BEFORE your eyes just before...\n\nChapter Twenty\n\nAFTER OUR FATEFUL COLLISION WITH THE Victor III, while nursing...\n\nChapter Twenty-One\n\nON THE MORNING OF JANUARY 15, 1980, the forty-four-year-old Ronald...\n\nEpilogue\n\nALMOST 200 SUBMARINERS, SPOOKS, AND NAVY divers were interviewed for...\n\nNotes\n\nSearchable Terms\n\nAcknowledgments\n\nAbout the Author\n\nOther Books by W. Craig Reed\n\nCredits\n\nCopyright\n\nAbout the Publisher\n\n## INTRODUCTION\n\nIn war time, truth is so precious that she should always be attended by a bodyguard of lies.\n\n\u2014WINSTON CHURCHILL\n\nIN MARCH 2009, BEYOND THE FROSTED windows of an arthritic building in downtown Saint Petersburg, Russia, a callous wind forced its will upon millions of helpless snowflakes. Inside the hotel ballroom, hundreds of Russians ignored the weather as they laughed, danced, hugged, and drank. Vodka flowed. Music blared and platters of food beckoned. I stood at my table as a husky man with playful eyes and Santa cheeks approached. He beamed and introduced himself as Sergei. He said he once served as the commander of a Soviet submarine and told me that NATO code-named his class of boat the \"Victor III.\" He asked if I recognized this name. I smiled and said that my submarine, the USS Drum, had once come too close to such a boat near Vladivostok.\n\nEyes wide, Sergei took two steps backward. \"K-324?\"\n\n\"Yes,\" I said. \"K-324.\"\n\nSergei reached his stubby arms around my shoulders and gave me a bear hug. In my ear he said, \"You should be dead.\"\n\nI nodded and said nothing.\n\nSergei pointed at a shiny pin on my lapel.\n\n\"U.S. Navy diver,\" I said.\n\nHis eyes lit up again as he tapped a similar emblem on his Russian Navy uniform. He unhooked his pin and attached it to my shirt. I did the same for him. Sergei then grabbed two glasses and filled each with a shot of vodka.\n\nHe handed one to me and in broken English quoted an old Russian proverb, \"After a storm there is fair weather, after sorrow there is joy.\"\n\nI clicked my glass against his and downed the burning liquid. Before me, and all around me, were former enemies. Submariners who once pointed the barrels of their guns at my head, fingers poised and aims steady. Now, with the passage of time, at an annual Russian event that honors submariners, we laughed and joked about our escapades from decades past.\n\nIn Russia, submariners are revered and respected, as their profession is considered dangerous, their sacrifice worthy of praise. For this select group of volunteers, camaraderie runs as deep as their vessels. None care about nationalities, creeds, or skin color. That night, dozens of former submariners treated me as a brother among brothers. Even though we were strangers whose governments once fought as enemies, we greeted one another with firm handshakes, warm hugs, and broad smiles. I felt honored and humbled.\n\nAfter dinner, a small group of submariners walked to the dance floor. Side by side they raised their glasses and voices as they sang a Russian submariner's song. Though I didn't understand the words, I felt the meaning touch the deepest part of my soul. More and more submariners joined the throng as the voices reached a crescendo. Tears filled my eyes. Words can never do justice to the feelings that overcame me when I stood alongside my brothers and toasted all submariners, especially those lost at sea who now serve \"on eternal patrol.\"\n\nAs I left the event, I wondered if those who consider themselves enemies today could do as we had done that night. Lay down their swords and find a common bond. I realized that until that day, there could be no fair winds, and many in the world were destined to remain captured by the storms of sorrow.\n\nIf one believes the Mayans, the world will end in the year 2012. Whether by global warming, menacing asteroids, or bioterrorism, we are always on the brink of annihilation. Skeptics voice their doubts, but for those of us who served during the forty-six-year Cold War, such fears are not without merit, for never did we come closer to nuclear self-destruction than in October 1962 and again in May 1968. Conflicts involving U.S. and Soviet submarines were common factors in both.\n\nNo discernible fanfare marked the final moments of a war that cost taxpayers $8 trillion and the lives of more than 100,000 Americans\u2014almost 87,000 of those in the conflicts with Korea and Vietnam. There were no ticker tape parades, no blowing horns, and no mothers waving flags when the Cold War finally ended. The U.S. Senate voted against the Cold War Medal Act of 2007, which would have awarded official recognition to thousands of veterans who fought secret battles around the world. Now they must remain unsung heroes.\n\nSome carried M-16s and trudged through rice paddies. Others listened with breathless anticipation to the secrets revealed in foreign tongues captured from cable taps 700 feet deep. Still others prayed to the gods of their faith as depth charges shattered the ocean and enemy torpedoes threatened to turn their vessels into twisted metal coffins. My father and I were among these few, and this history and personal narrative are long overdue.\n\nMost submariners, Navy SEALs, divers, and \"spook\" intelligence operators, sworn to secrecy, are to this day reluctant to discuss their secret Cold War operations. Many, especially those who worked in compartments outside operational areas or did not have a \"need to know,\" were unaware of the details surrounding the missions they undertook. A few, like me, recall every second of the more eventful assignments. For the first time ever, these veterans have come forward to tell their stories, perhaps to release the secrets held captive in their minds for decades by official mandate.\n\nIn 1998 Sherry Sontag and Christopher Drew's Blind Man's Bluff captured public attention by revealing many of the details about these clandestine and dangerous submarine missions. Most of us submariners agree that this book delivered an informative, interesting, and reasonably accurate accounting of Cold War espionage operations. However, few submariners or operators gave the authors information about their involvement in top-secret Holystone and Ivy Bells programs. Furthermore, none discussed two other top-priority submarine projects code-named Boresight and Bulls Eye. Red November is the first book to take readers deep inside all four of these programs and reveal firsthand details about the harrowing events that veterans have been reluctant to discuss.\n\nWhile I acknowledge that some submariners, cryptanalysts, and government operatives argue that \"insider\" details about these missions, which many historians believe were instrumental in ending the Cold War, should remain untold, I believe that history is robbed by this posture. What if the world never knew about the Manhattan Project? What if governments never revealed undisclosed details about the Cuban Missile Crisis? What if these once top-secret historical events remained labeled classified forever? National security demands secrecy, but at some point technological advances and world events make this stance obsolete. Many of us who served frontline in the underwater Cold War signed gag orders to maintain our silence for decades. Our duty to one another also held our tongues until the passage of time could ensure we would not violate our oaths as submariners. Now, for many of us, our days of silent running are over.\n\n## AUTHOR'S NOTE\n\nNAMES USED, INCLUDING THOSE FOR PERSONS, boats, and ships, as well as dates, titles, event details, and geographic locations noted herein, are, for the most part, accurate. A few exceptions occur where memories are incomplete or national security concerns take precedence. No individuals depicted are composite portraits or fictionalized, but some dialogue and details have been reconstructed or paraphrased and time frames compressed.\n\n## CHAPTER ONE\n\nRed sky at night, sailors delight.  \nRed sky in morning, sailors take warning.\n\n\u2014OLD SAILING PROVERB\n\nWITH ORDERS TO CONDUCT A TOP-SECRET espionage mission, the USS Blenny (SS-324) sped toward danger on the last day of April 1952. A bright sun warmed the black deck of the World War II\u2013vintage submarine as she cruised past a dozen colorful sailboats off the coast of San Diego. A cold wave crashed across the bow of the boat and dotted Lieutenant Junior Grade Paul Trejo's lips with the taste of salt. Still a freshman to the fraternity of underwater warriors, Trejo stood on the bridge and admired the beauty of his diesel-powered sub as she cantered across the water with the smooth gate of a stallion. In the white churn of her wake, dolphins played, and when guided beneath the waves, she became a silent assassin worthy of respect.\n\nLess than two miles from the submarine base at Ballast Point, the diving alarm sounded. Trejo slid down the ladder past the conning tower\u2014the tiny space just above the control room, then descended one more deck into the control room. There, assuming the duties of diving officer, he joined a half-dozen sailors on watch and stared at the indicator lights on the \"Christmas tree\" panel. When the horizontal bars changed from red to green, verifying closure of all hull openings, he held up an open hand and called out an order: \"Bleed air.\"\n\nA petty officer nodded and spun open a valve to bleed high-pressure air into the boat. The needle on a manometer twitched and then inched up a few millimeters. Trejo closed his hand, and the petty officer shut the valve. Both men focused on the pressure indicator. A half minute later, satisfied that no air leaked from the boat through an unwanted hole, Trejo closed his hand. \"Green board, pressure in the boat,\" he said.\n\nCommander James S. Bryant called down from the conning tower, \"Diving Officer, make your depth six-zero feet.\"\n\nTrejo repeated the order to the bow planesman seated at the front of the control room. As the chief of the watch blew air from the Blenny's ballast tanks to submerge the diesel-driven craft beneath the waves, Trejo visualized a plume of seawater shooting skyward toward a blue dawn. The boat submerged and made a turn toward paradise.\n\nAfter a short stop in Hawaii, the Blenny departed for Japan on Monday, May 13, crossed the 180th meridian\u2014the domain of the Golden Dragon\u2014and arrived in Yokosuka eleven days later at 0800 on May 24, 1952. Bound for enemy waters, the Blenny sailed away from Yokosuka on May 29. Commander Bryant gathered his officers in the wardroom as excited banter and cigarette smoke filled the air. The skipper's blue eyes and movie star persona reminded Trejo of Cary Grant. Bryant's forehead wrinkled as he pointed at their assigned station on a map spread across the wardroom table. The officers' talk faded to silence.\n\nThe Soviet city of Vladivostok had a population of almost a half million Russians in the early fifties. Nestled near the Strait of Korea to the south and Petropavlovsk Naval Base to the north, \"Vlad\" served as a staging area for an extensive segment of the Red Bear's Eastern Fleet, including ballistic missile submarines. The Korean War was in full swing, and the Soviets were shipping weapons to the North Koreans, so the commander, United States Naval Forces, Far East (COMNAVFE), ordered U.S. attack submarines to patrol the La P\u00e9rouse Strait near Vlad to gather as much intelligence as possible. Reconnaissance patrols in this area occurred between early March and late October, as too much ice encircled the seaport throughout the long winter months.\n\nTrejo recalled that the USS Besugo (SS-321) had attempted a patrol in the strait in December 1950, but dire weather conditions made reconnaissance all but impossible. U.S. subs now patrolled only during the warm window when the Russians sent ships to Korea and other countries and bombarded Vlad with incoming cargo vessels\u2014or \"megaton squirrels,\" as they called them\u2014delivering \"acorns\" to the Soviet Northern Pacific Fleet.\n\nAmerican boats were ordered to observe and photograph the pay-loads of these vessels during peak traffic times to find out what the enemy might be building, planning, or thinking. Such missions, thought Trejo, were akin to flying a passenger plane unnoticed into Chicago's recently renamed O'Hare International Airport on Christmas Eve.\n\nThe more aggressive sub drivers\u2014the skippers of these diesel-powered \"smoke boats\"\u2014in search of intelligence rewards, often snuck past Soviet warships and tiptoed to within a mile of Vlad's coastline. While most of the world claimed a three-mile coastal sovereignty, the Soviets insisted on twelve. Either way, Trejo knew that a U.S. boat caught in the act of spying near Vlad faced but two choices: escape or die.\n\nThe Blenny arrived in her operating area on the second day of June after a four-day jaunt from Japan. She snorkeled throughout the night, and then near dawn, Commander Bryant maneuvered the boat closer to the harbor. In the cramped conning tower, now silent and rigged for battle stations, Trejo's temples pulsed. The stale air smelled of fear. Wiping away a bead of sweat, sitting on a bench in front of the torpedo data computer, he stared at the blinking lights on a panel not more than an arm's reach away. As the assistant TDC operator, and part of the fire control tracking party, his job entailed helping track\u2014and potentially prosecute\u2014enemy targets. The electromechanical TDC fed target tracking information into an attached torpedo programming system, which Trejo monitored with eagle eyes. The term fire control related to the firing of weapons, not the traditional snuffing of fires. If an unsunk target retaliated, however, then fire control could very well take on its literal meaning.\n\nThe conning tower sat just above the control room and served as the boat's nerve center. This small area normally held seven to ten men, including a sonarman, a radar operator, the TDC operator plus an assistant, the skipper, a helmsman, and a navigator stationed at the back near the navigation plot. After countless drills, the team hummed in unison like the pistons in a well-tuned engine.\n\nCommander Bryant stepped toward the periscope well. \"Up scope number one,\" he said.\n\nThe oil-covered mast slid upward with a hydraulic hiss. Commander Bryant slapped the handles on the attack periscope horizontal. Seawater dripped onto the deck, and Trejo watched the skipper plant the right side of his face against the rubber eyepiece. \"Mast, funnel mast, clipper bow, king posts, transom stern, down scope.\"\n\nHydraulics whispered again as the mast slid downward. In his head, Trejo translated his skipper's jargon: the observed contact was a clipper-bowed cargo ship with two masts, a king post, and a transom-style stern.\n\n\"The big light is on,\" Bryant said, scratching at the stubble on his chin.\n\nTrejo knew that his skipper referred to a bright searchlight on the southern tip of Sakhalin on Nishi-notoro. When the Soviets lit up the dawn with that light, they illuminated a surge in shipping traffic in the strait. They also unknowingly helped the Blenny capture some photographic intelligence\u2014or PHOTINT\u2014since everything navy needed a truncated moniker.\n\nBryant issued another order. \"Ready the camera.\"\n\nChief Radioman Donald Byham, holding a thirty-five-millimeter Canon camera, stepped toward the periscope. Trejo imagined a handful of skinny Coke-bottle-glassed nerds back at the Naval Security Group headquarters in Fort George G. Meade, Mary land, poring over each photo graph taken by the Blenny with a magnifying glass to see what the Red Bear might be up to this month. Thousands of snapshots of Soviet vessels delivered by dozens of U.S. submarines probably lay scattered across the desks of high-ranking officials at NSG. Were cargo ships delivering a new type of missile that could hit the White House from the other side of the Atlantic or parts for a new class of submarine that could run circles around U.S. boats? Were Soviet warships just conducting an exercise or preparing for a full-scale nuclear war? The navy needed to know, so much so that hundreds of lives were considered expendable in the search for that knowledge.\n\n\"Ready with the eyes?\" Bryant asked, his arms dangling over the periscope handles.\n\n\"Good to go, Cap'n,\" Chief Byham said as he moved closer to the periscope.\n\nThe Blenny came with two periscopes\u2014the number one scope, with a small diameter of less than two inches, and the four-inch-wide number two scope. The former was the wiser choice when stalking prey at close range, as the smaller diameter lowered the risk of detection. The larger scope served as the best PHOTINT platform, as the mast contained better optics.\n\nThe advanced optics in the number two scope were better but not perfect, so Chief Byham's \"eyes\" served as backup. The medium-build chief had worked as a talented commercial artist in civilian life and possessed uncanny drawing skills. Recalled to active duty at the outset of the Korean War, he also came equipped with a photographic memory. After a few short glances at a target through the periscope, he could recreate the images he saw as detailed hand sketches within minutes. God gave the human eye far greater acuity than a camera lens, so Byham could see subtle details hidden in the shadows that the Canon could not detect. These Byham drawings became part of the intelligence stash delivered to NSG for review. Chief Byham never got a dime for his artwork, but he did receive a letter of commendation.\n\n\"Up scope number two,\" Bryant said. \"Raise the ESM mast.\"\n\nElectronic surveillance measures, thought Trejo. Along with visual information, the NSG wanted recordings and measurements of wavelengths, frequencies, and pulse repetition rates emanating from enemy radar signals. They called it SIGINT. The ESM mast captured this type of data for subsequent perusal by the NSG. An alarm in the conning tower beeped when the ESM mast detected that Soviet radar might be \"painting\" one of Blenny's masts. Too many beeps equated to \"caught,\" which also meant they were screwed.\n\nAs the masts sped toward the surface, Bryant ordered a steady depth and buoyancy trim, as an inch too shallow could spell disaster.\n\nBeep!\n\nThe hair on Trejo's neck bristled. The radar detection system in the ESM mast just got a hit from a nearby warship.\n\nBeep! Beep!\n\nTwo more hits.\n\n\"Camera,\" Bryant said, stepping back from the scope. \"Make it fast.\"\n\nChief Byham snapped the Canon onto the periscope's eyepiece and moved away. Commander Bryant hurried back to the scope, peered through the camera lens, and clicked off several shots. He spun the scope a few degrees and clicked off a couple more.\n\n\"Pull the camera,\" the skipper said, again stepping away.\n\nChief Byham removed the Canon.\n\n\"Eyes, you're up,\" Bryant said.\n\nChief Byham gave a quick nod and seated his face against the scope's eyepiece.\n\nBeep! Beep! Beep!\n\n\"Five seconds,\" Bryant said.\n\nTrejo glanced at his watch. Five seconds felt like fifty.\n\n\"I'm done,\" Chief Byham said.\n\n\"Down scope.\"\n\nThe sonarman, seated just aft of the periscope, turned his torso toward the captain. \"Active pings in the water!\"\n\nTrejo's heart stopped. A Soviet warship must have gotten a good hit on the scope with a radar beam. The bad guys just caught Blenny spying, so they were compelled to pummel the ocean with active sonar. Depth charges and torpedoes might be next.\n\nThrough the open hatch, Bryant called down to the control room. \"Diving officer, twenty degree down bubble, make your depth 300 feet!\"\n\nThe diving officer echoed the command as the boat now angled toward the bottom. At 300 feet, Bryant glanced at the bathythermograph. The BT, fed by a small device installed on the boat's hull, displayed pressure and temperature related to depth. As colder water layers tend to reflect sonar beams upward, the skipper wanted to stay under one. To ensure accurate and timely readings, the boat dove to test depth every morning, then inched back toward the surface, all the while taking temperature and density readings to feed the BT and find the layers. Today, the best acoustic thermal layer started at 400 feet. Bryant called down to control and issued a new order. \"Diving Officer, take us down to 450 feet.\"\n\nTrejo's heart started again and raced to full throttle. Even though the Blenny's newer \"thick skin\" design made her a deeper diving boat than her \"thin skin\" predecessors, her test depth topped out at 412 feet. Too much beyond that could result in the proverbial crushed beer can effect.\n\nThe boat groaned as she descended beyond 300 feet and the ocean's grip tightened.\n\n350.\n\n375.\n\n412.\n\nSteady at 450 feet.\n\nTrejo could now hear Soviet fifty-hertz sonar pings through the hull. He moved his eyes upward, as if he could see more than just pipes and cables. The pings grew louder. The boat crawled along at three knots, her twin screws generating no more noise than a slow-speed fan. Trejo held his breath and prayed that Soviet sonar beams would not penetrate through the acoustic layer.\n\nThen the explosions started.\n\nFaint at first, they grew louder until the boat shook with each clap. Trejo hoped the Soviets were using warning depth charges, which were \"light\" versions of the subkilling kind, but he did not know for sure.\n\nClick, whang! Another explosion.\n\nThe flooding alarm sounded, and men ran to damage-control stations. Leo Chaffin, the Blenny's executive officer, darted out of the control room. Part of his duties included leading the damage control team. Minutes later he made a call to the skipper over a sound-powered phone. Bryant uttered a few words, nodded a couple times, then hung up the phone.\n\nFacing the team in the conning tower, Bryant said, \"The XO reports that we have a leak around the sound dome shaft in the forward torpedo room. He's sealed off the area from the forward battery and is pressurizing to thirty psi, but it won't be enough. We've got to take her deeper until they get the leak fixed.\"\n\nBryant leaned over and called down to the control room. \"Diving Officer, make your depth 500 feet.\"\n\nThe diving officer's voice cracked as he confirmed the order.\n\nThe boat moaned in defiance to the added pressure. Trejo's mouth went dry. He understood the strategy but didn't like it. More depth equals more pressure, and more pressure makes things smaller, like pipes and shafts. This can sometimes make leaks easier to fix. Sometimes. Still, 500 feet could be 88 feet deeper than dead.\n\nSeawater leaked from a half-dozen pipes in the overhead as the boat descended, showering men in the conning tower. Sonar pings bounced off the Blenny's thick skin, followed by the staccato clap of more depth charges. The volume of both increased as the Soviets continued to close.\n\nA dozen long minutes passed before the sound-powered phone rang in the conning tower. Commander Bryant answered, listened, nodded, and hung up. Again he addressed the tracking party: \"XO says the leak is fixed. Skelly just earned your respect and a navy commendation.\" Bryant bent down and called through the lower hatch, \"Diving Officer, make your depth 450 feet.\"\n\nThe diving officer echoed the order, and Trejo let his shoulders relax. He later learned that Auxiliaryman First Class Skelly, being a skinny kid, volunteered to wiggle his way into the well, head down, to make repairs. Using two main engine semicircle bearing shells fitted together, he wrapped them around the shaft packing to stop the leak. Hell of a jury rig, but it worked.\n\nAfter twelve hours of hiding under a thermal layer, with pings and depth charge smacks filling the ocean around the Blenny, Trejo's lungs ached. His chest tightened as he struggled to pull in a breath. Carbon dioxide, the Achilles' heel of diesel submarines and the odorless killer of sailors. The most any smoke boat could stay underwater running on batteries before the air became stale and sailors risked CO2 poisoning was three to four days. They were then forced to come shallow and run the diesel engines to push out the old air and pull in the new.\n\nTrejo looked around the control room. Other men, bent over and coughing, also struggled to find enough oxygen to survive. Trejo glanced at his watch. The second hand ticked away, counting down the last few hours of his life. If the Soviets held the Blenny down on the bottom much longer, he'd spend the rest of eternity in a cylinder twelve feet longer than a football field.\n\nHis eyes blurry and his head dizzy, Trejo wondered what it might be like to serve on a boat that could stay down for months at a time versus only a few days. Although the world's first nuclear-powered submarine had not yet put to sea, the Department of the Navy announced on December 12, 1951, that her name would be the USS Nautilus. Within hours after hearing the news, Trejo joined his fellow crewmates in denigrating Admiral Hyman G. Rickover, the \"father\" of the nuclear submarine navy, by chanting the mantra \"Diesel boats forever!\" Now, with his body growing weak from lack of air, he promised God that he would never again malign Rickover or nuke-driven subs.\n\nAn hour later, with a third of the crew incapacitated by CO2 poisoning, the sonar operator announced that the Soviet pings and explosions were finally subsiding. Bryant ordered all ahead two thirds and a thirty-degree turn toward freedom. The boat sprinted away and surfaced an hour later. Hatches opened, and the diesel engines pulled fresh air into the boat. Trejo took in a deep breath and smiled. He was still alive, at least until their next SpecOp.\n\nWhen Paul Trejo returned to San Diego, he and his crewmates received a Navy Expeditionary Medal. The medal is awarded only to those \"of the Navy and Marine Corps who shall have actually landed on foreign territory and engaged in operations against armed opposition.\" Submariners who chased the Red Bear added a star to their ribbon for each SpecOp mission they completed. Trejo wore his ribbon with pride and earned several stars over the next few years, painfully aware that the diesel-powered limitations of his smoke boat placed his life on a knife's edge each time he went to sea.\n\nTHE U.S. INTELLIGENCE COMMUNITY CAUGHT A glimpse of the potential for submarine espionage SpecOps during World War II, when, in preparation for beach landings, they sent a dozen diesel boats to island coastal waters to pop up antennas and listen to Japanese radio traffic. When the Cold War began in 1946, they remembered these war time undercover missions and decided to call again upon their spies of the deep for \"special operations.\" While submarines were ideal as stealth platforms to undertake these clandestine trips into danger, their need to snorkel every few days became a serious limitation when operations required extended endurance.\n\nTo solve this problem, the navy turned to Hyman G. Rickover, who put into motion the wheels that rolled the navy toward nuclear power. The Polish-born Rickover immigrated to the United States with his family and graduated from the U.S. Naval Academy in 1922. He received his appointment as the director of the Naval Reactors Branch in 1949, which led to his supervisory role in the planning and construction of the first ship submersible nuclear (SSN). Electric Boat Corporation laid the keel to the USS Nautilus on June 14, 1952. While the solution to the shortcomings of diesel power appeared well in hand, the navy now needed a new generation of nuclear-trained sailors and officers to man their growing armada. One of these officers started off as a seaman on a diesel boat.\n\nWhen Gardner Brown boarded his first submarine in 1946, the World War II\u2013vintage craft, with her long, thin frame squatting amid a swirl of shimmering oil and dock debris, reeked of diesel fumes. The boat's topside watch beckoned him aboard, whereupon his nose wrinkled even more at the smell inside\u2014something akin to a gas station garage manned by sweaty rednecks. Below decks, another machinist's mate guided Brown on a tour of the boat, from the torpedo room in the bow, past the control room, mess decks, and berthing spaces, through the hot engine room, where large diesels stood at the ready, and finally to the aft torpedo room, where green MK-14 torpedoes waited patiently for something to kill.\n\nBrown's tour guide explained that the USS Cubera (SS-347) was a relative of the Balao-class diesel submarine and gained her name from a large fish of the snapper family found in the West Indies. Commissioned on December 19, 1945, the Cubera never saw war time action. She sailed to Key West in March 1946, where her crew assisted with the testing of a new top-secret submarine detection system. A few months later, she reported to the Philadelphia Naval Shipyard for an extensive GUPPY modernization.\n\nGUPPY stands for Greater Underwater Propulsion Power. This $2 million overhaul installed a new battery system and technology based on pilfered German XXI submarine designs that returned an investment of faster, deeper, longer, and the ability to snorkel. Just like her World War II U-boat counterparts, the snorkel modification allowed the Cubera to pull in air while submerged so the diesel engines could recharge the batteries, or \"fill the can,\" and refresh the crew's lungs.\n\nAfter the Cubera completed her GUPPY upgrade, she received orders to head into harm's way. Gardner Brown closed the hatch above his head. As the Cubera prepared to dive, he thought about their destination. Prior to leaving port, nobody told him they were headed to the Black Sea. No one said they'd be conducting one of the first SpecOps of the Cold War, and not a soul talked about their odds of returning home alive.\n\nBrown already knew a lot about death. After attending Governor Dummer Academy in Byfield, Massachusetts, he joined the navy in 1944. He passed the V-5 and V-12 program examinations to enter Dartmouth College in August 1944. The Naval Reserve Officers Training Corps established the V-12 in 1942 to recruit officers for the war effort. Assigned to a marine unit, Brown boarded a ship headed to a remote Pacific island. On February 19, 1945, as part of the Fourth Marine Division, Twenty-fifth Regiment, Third Battalion, I Company, Brown landed at Blue Beach Two and watched his companions die on the sands of Iwo Jima. The Japanese fought a fierce battle that lasted forty-five days, aided by fortified bunkers, hidden artillery, and eleven miles of underground tunnels. The battle of Iwo Jima cost the lives of almost 21,000 Japanese and over 6,800 U.S. Marines and is forever frozen in time by the iconic raising of the American flag atop the island's peak.\n\nBrown did not join the ranks of the fallen and transferred to the submarine navy in January 1946. A little over a year later, he sped toward a new enemy\u2014the Japanese red circle replaced by a Soviet red star. He described the boat's skipper, Commander George W. Grider, as \"the perfect example of brains and brass ones.\" It turns out Grider would need both.\n\nCommander Grider earned his submarine qualifications aboard the USS Skipjack (SS-184)\u2014one of the most accomplished submarines of the war. He served as executive officer on the USS Pollack (SS-180) and as CO of the USS Flasher (SS-249) before transferring to the Cubera. Later in life, Grider became a U.S. congressman, but for now, he was a leader of men in an underwater world.\n\nAfter crossing the Atlantic, Grider drove the Cubera to within a few miles of Sevastopol. Long considered the \"jewel of the Crimea,\" this glittering white city housed a predominantly Russian population and the Soviet Black Sea Fleet. The Soviets had erected a steel gate across the entrance to the Strait of Dardanelles to prevent errant underwater visitors from sneaking into the harbor, but Grider viewed this as nothing more than a speed bump.\n\nEquipped with a surveillance antenna and a Russian-speaking \"spook\" intelligence officer nicknamed Grab One, the Cubera waited near the harbor's entrance. Commander Grider raised the number one attack periscope and swung the cylinder back and forth. For several hours he studied the navigation light planted on the starboard side of Tenedos Island, at the northern end of the strait. When the light finally came on, signaling the approach of a warship, he swung the scope back toward the entrance. He reported to the crew in the small conning tower that he'd spotted a Soviet aircraft carrier and intended to follow her into the harbor.\n\nEmploying the \"brass ones\" that earned him the crew's respect, Grider nudged his boat to within several yards off the stern of the carrier. As the steel gates parted to admit the Soviet warship, the Cubera followed her in. Once inside, now surrounded by dozens of enemy ships, Grider raised a surveillance antenna, and Grab One went to work. The intelligence spook captured, recorded, and analyzed Soviet transmissions for two days. He then convinced Grider to \"get a little closer.\" Forgoing caution, Grider complied.\n\nNot more than a few minutes after raising the periscope for another look, the Soviet navy stirred, a subtle rumbling at first, soon followed by an all-out barrage. Grider dove the boat and tried to evade, but the Soviets slammed the ocean with active sonar pings. With the steel gate shut tight across the exit to the harbor, the Cubera had nowhere to run. Out of options, Grider searched for a place to hide.\n\nBottom-sounding sonar detected something strange on the ocean floor 300 feet down. Grider took the boat deeper. Near the bottom, he peered through the periscope into the murky water. Although the Black Sea usually provided clear viewing this time of year, not much daylight found its way down to this depth. Grider couldn't believe what his eyes told him, so he asked the chief of the boat, Gaines \"Whirly\" Smith, to have a look. Whirly gazed through the periscope and said, \"I'll be damned, we're in the middle of Main Street.\"\n\nThe gods had smiled upon the Cubera by allowing her to stumble across an ancient city flooded by time. The tall earthen buildings afforded the perfect hiding place and just enough cover to deflect Soviet sonar beams.\n\nSmiles quickly faded when, hours later, the boat's battery power and air supply dwindled. Grider knew they'd have to surface in less than sixteen hours or die on Main Street.\n\nTwelve hours later, Gardner Brown experienced the curse of the diesel sub firsthand. Carbon dioxide replaced what little air remained. Brown's head spun, and his lungs felt like dried prunes. With less than two hours remaining before the battery ran dry, the gods intervened once more. They sent a Russian destroyer through the gate. The Cubera snuck into her wake and swam out of the whale's belly. Having dodged death yet again, Brown wondered if he might have been better off staying in the Marine Corps.\n\nIntelligence experts were ecstatic. Grab One had grabbed plenty, and the navy wanted more. With the bar now set ultra-high by Commander Grider, every submarine driver needed to risk as much or more to earn a \"brass ones\" title\u2014and, perhaps, another stripe on his sleeve. This mission and others like it set the stage for a deadly high-seas contest between the United States and the Soviet Union that raged on for another forty-plus years. They also propelled the navy's quest for stealth platforms with greater endurance and range, which could only be accomplished with atomic-powered engines.\n\nThe invention of nuclear propulsion fanned the flames of the underwater Cold War when the atom stepped onto the stage with the commissioning of the USS Nautilus (SSN-571) on January 21, 1954. The navy's first nuclear-powered submarine signaled the end of diesel-driven boats and changed forever the life of under-ocean sailors. Now submariners could stay down for months versus days and could do so without needing to run noisy diesel engines. The USS Seawolf (SSN-575) followed in the Nautilus's nuclear wake seven months later, and like a redheaded stepchild, it received little of the notoriety bestowed upon her older cousin.\n\nUnlike most other diesel boat sailors, after his near-death experience in the Black Sea aboard the USS Cubera, Gardner Brown readily accepted an offer to \"go nuke.\" He became one of seventeen submariners handpicked for the nuclear power program in the fall of 1953. Brown and his classmates spent five days a week completing intensive academic courses at Union College and two more days at a nuclear prototype putting into practice what they'd learned. They studied both pressurized water reactor technology\u2014used on the Nautilus\u2014and alkaline metal sodium technology used by the Seawolf. When the instructors discovered that learning sodium complexities required a higher IQ, they divided the class in half. Those with a little more brainpower, like Brown, wound up in the Seawolf-bound class.\n\nSeawolf's namesake, a solitary fish with gnarled teeth and savage tusks, underscored the vessel's gritty demeanor and set the tone for her turbulent future. Fashioned from a vintage fleet diesel boat, Seawolf employed a superheated steam power plant versus the more traditional saturated steam reactor, which reduced machinery space size by almost half. Although more advanced and quieter than the Nautilus, Seawolf's propulsion system carried additional risks and earned her the nickname \"Blue Haze\" when sodium coolant leaked from the reactor in the shipyard.\n\nSuch accidents fueled Admiral Hyman Rickover's consternation and fanatical focus on safety. Rickover ruled his atomic roost like a straw boss on a pyramid. Most navy subordinates and Electric Boat shipyard contractors feared his controlling management style, which Rickover exploited to further his safe sub agenda. While some hated the man, Gardner Brown became a follower and a friend. Like Rickover, Brown knew firsthand that from a power plant engineer's point of view, a diesel boat, which runs on \"dead dinosaur juice,\" was to a nuclear submarine as a flint rifle was to a submachine gun. Nukes cost more and take far longer to build due to power plant complexities and safety concerns. More complexities equal more problems, and in Seawolf's formative years, there were many. Brown recalls having to leap dozens of metallurgical hurdles caused by ultra-high reactor temperatures and flux densities while striving to build this boat.\n\nBrown's cousin, Gene Centre, who also worked on the Seawolf's reactor as a project manager for Bettis Atomic Power Laboratory, heard rumors about flaws in the A1 and A3 reactors being built by the Soviets. He also heard that the Russians considered their sailors expendable and so did not place a high value on radiation shielding and pump seals. Centre figured that the rumors might be more propaganda than truth, but nonetheless he concurred with Rickover that safety and reliability were paramount.\n\nArmed with that priority, Dennis B. Boykin III, Electric Boat's power plant manager, insisted on keeping the Seawolf in the yards an extra year to develop a rod drive mechanism with special seals that prevented coolant leaks. Gene Centre and others helped engineer this, along with additional \"over spec\" components to ensure that, years later when the Seawolf replaced her sodium engine with a water-cooled type, she came out of the garage with the \"heart of a '57 Chevy and the soul of a Mack truck.\" Although Centre never envisioned such, his dedication to ensuring that the Seawolf could be pushed well past her red line saved the lives of 190 men trapped on the ocean floor a few miles off the coast of Russia more than two decades later.\n\nSeawolf finally received her commission on March 30, 1957. Lieutenant James Earl \"Jimmy\" Carter, who'd one day be the only U.S. president qualified in submarines, had received a billet as her engineering officer but resigned his commission after his father died in 1953. Without Jimmy Carter on board, Commander R. B. Laning grabbed the reins as the boat's skipper and galloped the navy's second nuclear race-horse around the track. Brown and his crewmates pushed her hard for the next several months during rigorous sea trials. Although her heart was willing, the Seawolf groaned in defiance at the high-speed runs, tight turns, deep dives, and steep angles ordered by Laning.\n\nBrown was not anxious to endure any further SpecOps after his Black Sea experience on the Cubera under Commander \"Brass Ones\" Grider, but fate dictated otherwise. He went with the Seawolf in 1958 on her first special operations run into the Barents Sea, where, he says, \"We stayed underwater forever, but at least we never had to worry about running out of air.\"\n\nThe Nautilus made history that same year by sliding under the ice floes and paving the first underwater trail to the North Pole. She remained underwater during the entire transit and hit speeds of more than twenty-three knots. In contrast, diesel submarines could push no more than ten knots submerged and needed to snorkel every few days to stay alive.\n\nNuclear power enabled submarines to accomplish their missions with greater safety and efficiency by solving the problems of endurance and speed while submerged. A large hurdle remained, however. Current submarine sonar systems could hear no further than a few miles away, making them all but deaf, dumb, and blind. That made them vulnerable to detection and destruction. By the early 1950s, American engineers had failed at every attempt to solve this monumental problem.\n\n## CHAPTER TWO\n\nFor thou cast me into the deep,\n\nInto the heart of the seas,\n\nAnd the floods surrounded me,\n\nAll Your billows and Your waves passed over me.\n\n\u2014JONAH 2:3\n\nIN THE FALL OF 1953, DR. Donald Ross, an underwater engineer from Pennsylvania State University's Ordnance Research Laboratory, walked through the large glass doors of the Bell Telephone Laboratories building in Whippany, New Jersey. Roland Mueser, one of Ross's former classmates from Penn State and now a Bell Labs employee, met him in the lobby. Mueser beamed broadly, whisked a tousle of hair away from his forehead, and urged Ross to the front desk. Ross signed in with the guard, and Mueser pointed toward an elevator. As they hurried through the capacious foyer, Mueser told Ross that he'd been working for Captain Joseph Kelly on an exciting new project called Jezebel. He explained how they'd installed underwater low-frequency sonar arrays to help detect snorkeling submarines from over one hundred miles away. Ross raised one of his thick brown eyebrows and let loose a whistle.\n\nThe elevator reached an upper floor, and Mueser led Ross through a maze of corridors and hallways, all the while talking about how Project Jezebel received a name change to SOSUS (Sound Surveillance System) after the initial tests proved successful. Six SOSUS listening stations were now deployed in the North Atlantic basin, and nine more were authorized under Project Colossus\u2014three in the Atlantic and six in the Pacific. These new arrays, to be installed in 1954, incorporated advanced upward-looking sonar capabilities.\n\nRoss let out another whistle.\n\nTurning a corner, Mueser said, \"I suppose you're wondering why you're here.\"\n\nRoss shrugged. \"I figured you'll tell me soon or later.\"\n\n\"I will,\" Mueser said, \"and then you'll pee your pants.\"\n\nWalking across plush carpet, trying to keep his curiosity at bay, Ross wondered how Mueser convinced Bell Labs to hire him. He was a submarine hydroacoustic expert, which only peripherally related to something like SOSUS. Ross received his Ph.D. from Harvard in 1942\u2014the first to graduate from that Ivy League university in only three years. He launched his career at the Harvard Underwater Sound Laboratory in January 1945, moving later that year to the Ordnance Research Laboratory at Penn State, where he met Mueser.\n\nRoss's work at Harvard near the end of World War II contributed to improvements in propeller designs for the twin-screw GUPPY-upgraded diesel submarines. He'd also been assigned to the propeller team to produce quieter props for submarines. Still trying to connect his work to the reasons why he'd been invited to Bell Labs, Ross followed Mueser around the facility. His propeller experience provided few intersections with something like Project Jezebel, so why was he here? As Mueser strode past an office in the building, Ross peeked inside. Engineers in white shirts and dark ties scribbled on blackboards and wrestled with reams of desk paper. The place smelled almost antiseptic, not too unlike the waiting room in a medical clinic.\n\nMueser opened a lab door and motioned Ross inside. Three slide rule\u2013clutching engineers, pens stuffed into pocket protectors, turned from a chalk-covered blackboard. Mueser introduced the trio as Larry Churchill, Herman Straub, and Rich Carlson.\n\nMueser found a chair, leaned back, and said, \"What I'm about to tell you boys is way above top secret.\"\n\nHerman Straub, a former World War II submarine skipper, let out a grunt. \"Does that mean if I tell my wife you're going to kill me?\"\n\n\"No,\" Mueser said, \"we'll kill your wife.\"\n\nStraub smiled. \"For free?\"\n\nAfter a shared laugh, Mueser's face turned serious. \"The navy has a problem they need us to solve.\"\n\n\"Such as?\" Ross said.\n\n\"The Soviets are building a lot more submarines, and their ASW (antisubmarine warfare) forces have become much more aggressive. Our boats can't hear these guys if they're more than a few miles away.\"\n\n\"So what's the answer?\" Straub asked.\n\n\"I brought Dr. Ross here to help us build a new long-range passive submarine sonar system,\" Mueser said.\n\n\"Passive?\" Straub wondered. \"As in no active pings?\"\n\n\"As in strapping on headphones and listening,\" Mueser said.\n\n\"What do you mean by long range?\" Ross asked.\n\n\"I mean like a hundred nautical miles,\" Mueser said.\n\n\"How the hell do we do that?\" Straub asked.\n\n\"Using LOFAR technology developed for SOSUS,\" Mueser said.\n\nAlthough he didn't pee his pants, a lightbulb turned on in Ross's head. Now he knew why Mueser encouraged him to join Bell Labs: they needed his expertise in submarine hydroacoustics to revolutionize how submarines hear.\n\nThe team went to work on an electronic \"breadboard\" passive sonar suite using a modified version of LOFAR (Low Frequency Array). The effort proved more difficult than expected, resulting in chain-smoking late-night sessions in the lab, followed by a battery of simulation tests. All of them failed. After more than a month, Ross was about to concede defeat when a crazy idea popped into his head.\n\nHe marched into Mueser's office and said, \"Standard LOFAR won't work.\"\n\n\"Why not?\" Mueser asked.\n\n\"Subs are too short and noisy to be good platforms for low-frequency sonar.\"\n\n\"Wonderful,\" Mueser said dryly. \"So now what do we do?\"\n\n\"We use a higher frequency.\"\n\n\"We tried that during the war,\" Mueser said. \"Didn't work. We needed to get close enough to smell a fart before we could hear anything.\"\n\n\"I know,\" Ross said, \"but I think I can solve that problem with a new approach to demodulation.\"\n\nMueser grinned. \"I told 'em you were a genius.\"\n\nDemodulation, in short, is the science of extracting information from a carrier wave. Ordinary radios do this by snatching signals from the air and decoding the AM or FM frequencies into something we can hear at 1210 on our dial. Simply put, Ross conceived the idea of using a form of demodulation to \"decode\" the various frequencies received by a passive sonar system on a submarine. Once completed, the breadboard design, which consisted of a small circuit board filled with electronic components and dangling wires, passed all the simulation tests. Now they needed to prove it could work in the real world.\n\nThe team traveled to Submarine Development Group Two in New London, Connecticut, and boarded the USS Cavala (SS-244). His heart thumping, Ross peered through the open hatch of the dark submarine. The pungent smell of diesel fumes filled his nose as he climbed down the ladder. Standing on the tile in the narrow passageway, claustrophobia prompted him to suck in his gut and narrow his shoulders. He followed an officer toward the control room, wondering how anyone could stand to live in such a tight space for weeks on end.\n\nThe Cavala's skipper, Lieutenant Commander Bill Banks, greeted the team in the control room. Ross unveiled the sonar suite breadboard.\n\n\"What's that?\" Banks asked.\n\n\"Your new passive sonar system,\" Ross said.\n\n\"No shit,\" Banks said. \"Does it work?\"\n\n\"That's what we're here to find out.\"\n\n\"So it's a beta breadboard.\"\n\n\"More like an alpha,\" Ross said.\n\nBanks lowered his eyes, shook his head, and walked away.\n\nThe Cavala pulled out of port, and the team set about connecting the breadboard to the sub's passive sonar system. After running a couple of functional tests, Ross sat in front of the sonar console in the Cavala's conning tower and slipped on a pair of headphones. Closing his eyes, he listened. Nothing. He adjusted some settings and listened again. Something?\n\nBanks peered over Ross' shoulder. \"Can't get it to work?\"\n\nRoss removed the headphones and handed them to the skipper. Banks pulled them over his ears.\n\nA few seconds later, Banks's eyes widened. \"I'll be damned. I can hear him!\"\n\nThe \"him\" Banks referred to was a snorkeling Balao-class submarine more than one hundred nautical miles away. Over the next few weeks, the team confirmed detection of other submarines and classified a carrier task group operating 150 nautical miles distant. They code-named their invention DEMON, short for demodulation. This new design extended the hearing range of previous sonar systems by more than thirty times, and Ross soon found himself spending more time cruising underwater than almost any other civilian engineer. He and teammate Herman Straub spent ten days on the USS Nautilus installing a new DEMON sonar suite, measuring radiated noise and gathering spectra. Their analysis and technical advice helped the crew reduce the boat's noise signature, making her less detectable by SOSUS or shipborne passive sonar arrays.\n\nFollowing his Nautilus experience, Ross produced the first report on noise emanation from nuclear submarines in August 1957. The Naval Scientific and Technical Intelligence Center (NAVSTIC), impressed by the report, asked Ross to analyze a top-secret DEMON recording of a new Soviet submarine. The sounds emanating from the sub were strange\u2014definitely not the piston slapping one might expect from a diesel engine or the hushed quiet of a battery-driven propeller. After a day of looking over the LOFARgrams made from the recordings, Ross confirmed that the acoustic signature came from a nuclear power plant. Jaws dropped at NAVSTIC.\n\nThe next day, the U.S. Navy announced that the Soviets had just deployed their first nuclear submarine. NATO bestowed the code name November, which happened to coincide with the month when the keel was laid. American submariners nicknamed her Red November, but the Soviets simply called their new Project 627 submarine the K-3. They designed this twin-reactor attack boat to hinder U.S. shipping lanes in the event of a war, but the crew of Red November secretly trained for another top-secret mission: sneaking close enough to New York harbor to fire a twenty-seven-meter-long nuclear torpedo at the Statue of Liberty.\n\nFaster, deeper diving, and more heavily armed than the USS Nautilus, the cigar-shaped K-3 put the fear of God into most of the navy brass. That paranoia prompted an underwater race that cost trillions of dollars and pitted the United States against the Soviet Union for another three decades.\n\nThe introduction of K-3, which employed a cylindrical hull better suited for underwater speed, motivated the U.S. Navy to accelerate a radical new design for its own fast-attack nuclear subs. The USS Skipjack (SSN-585) entered the fray on May 26, 1958, to counter the Red November threat. Sporting improved SW5 pressurized water reactors, Skipjack-class submarines could go from zero to thirty-plus knots in a few dozen heartbeats. A teardrop-shaped hull and single in-line propeller improved underwater speed and agility and sent a clear signal to the Soviets that they were once again behind in the game.\n\nUnable to match American ingenuity, the Soviets responded with numbers. By 1959, they were the proud owners of 260 long-range offensive submarines. This sobering development forced the United States to concede that its nemesis planned to shift the mission of these vessels from coastal defenders to front-line strike weapons. Several Zulu and Golf-class boats could now hurl 3,200-ton nuclear warheads at the United States from over a thousand miles away. Despite the monumental breakthroughs in underwater sound technology used in ocean-mounted SOSUS and sub-installed DEMON sonar systems, the only distant subs these inventions could reliably detect were noisy snorkeling diesel boats. Finding an underwater craft running near silent on battery power, or semi-quiet on nuclear power\u2014well before she launched her nuclear payload at the United States\u2014seemed as insurmountable as landing men on the moon.\n\n## CHAPTER THREE\n\nMan is a great blunderer going about in the woods, and there is no other except the bear that makes so much noise.\n\n\u2014MARY AUSTIN\n\nHAVING WORKED HIS WAY UP FROM a seaman to a communications technician chief (CTC) in the navy, William J. Reed received orders to report to the Naval Security Group at Fort George G. Meade, Mary land, for a top-secret assignment briefing. He was unaware that his new orders would place him at ground zero for one of the most important discoveries of the Cold War. As his son\u2014at the time only three years old\u2014I had no way of knowing that our move to a small Turkish village, and our adoption of a wounded baby bear, would become the impetus for that discovery.\n\nIstanbul is situated on the Bosporus, which cuts across the Anatolian Peninsula and exits into the Black Sea, dividing Turkey into a Europe an and an Asian composite. Constantine renamed the city Nova Roma in A.D. 330, but everyone started calling the eastern capital Constantinople. In 1930 Turkish authorities officially named the city Istanbul, which translates loosely as \"downtown.\"\n\nTurkey is an important part of NATO and has always welcomed U.S. military and technical assistance. In 1959 the United States maintained several strategic stations throughout that country. One of these, where my dad now worked, was a radio listening station near Karam\u00fcrsel. The antenna array at this facility was operated by Air Force personnel and \"borrowed\" by the navy for high frequency direction finding (HFDF)\u2014a way of pinpointing the location of a ship or submarine by determining the direction to its radio transmissions. Operators called these HFDF stations \"Huff Duffs.\" A small detachment of NSG personnel, including my dad, operated the HFDF equipment as part of The United States Logistics Group, or TUSLOG Detachment 28. Most of the families assigned to serve at this base lived about fifteen miles away in Yalova, where at night the lights of Istanbul twinkled like Santa's village from across the Sea of Marmara, but Dad insisted on living someplace a bit more tranquil.\n\nAfter a week of searching, my father, a six-foot-two navy chief with Dick Tracy features, found us an apartment in De irmendere, a small town six miles from the base. The only thing modern here was the automobile garage on the highway at the village entrance that was built of sticks and stones. The entry road came with an overhanging of oak and elm trees and led to the main square, a dirt clearing dominated by majestic hundred-year-old oaks flanked on both sides by shops. At the open-air bakery, we watched the baker, a master artist, pulling brown loaves from an ancient kiln. My mouth watered as the rich aroma wafted through the open door and pulled in hungry shoppers.\n\nMy dad often went hunting with a fellow navy chief and Halidere, our Turkish neighbor. Now and then Dad brought back a couple pigs, but Halidere refused to eat any of that \"filthy pork.\" A week later, Halidere knocked on our door and gave my mom a little squealing animal of some kind. We were clueless as to what this tiny thing might be, so Mom called my dad at the base. He came home a couple hours later and scratched his head. He went next door and talked to Halidere.\n\nDad discovered that our neighbor trained bears for a living. Halidere taught them how to stand up on their hind legs and dance around in the streets while people giggled and threw money at their paws. He stole two bears from a cave when the momma bear went hunting for food. While running away, he dropped one, and the poor thing hurt its leg. He figured the wounded animal could never dance, so he gave it to my mom. Dad offered him some American cigarettes as a thank-you. Halidere loved American cigarettes. Dad said he loved American women, too, and told my mom to never get her fanny too close to Halidere's hands. Being three, I didn't know why he told her that, but it made me laugh anyway.\n\nThe baby bear weighed only sixteen ounces, and, except for his paws, he looked like an overgrown rat. No fur at all. I thought all bears looked like little cuddly teddy bears, but not this one. Dad named him Ayi Bey, which translates roughly as \"Sir Bear.\" Not knowing what else to do, Mom put him on a human baby's schedule and diet of baby formula. He woke up every few hours crying for food, crying to be held, or just crying. Mom started feeding Ayi Bey with an eyedropper, but that didn't last long. He wanted more. She borrowed a baby bottle from an American family, and Dad poked more holes in the nipple so Ayi Bey could get enough formula mixed with Pablum.\n\nAyi Bey stayed furless for about a month. I wanted to call him Fuzzy Wuzzy, like the bear with no hair, but my sister, Pam, said we should change his name to Scratchy because he liked to scratch things. Mom and Dad agreed. Scratchy especially liked to claw at the brown Turkish goatskin rug that covered the cracked tiles of our kitchen entryway. We didn't know it then, but Scratchy's habit would soon lead my father toward a critical discovery for the navy.\n\nOver the next year, Scratchy grew to be the size of a small dog. He chased my sister and me around the yard like a bounding puppy and learned how to climb up the slide by lying flat with all four paws outstretched and gliding down from the top. Often he pushed us out of the swing so he could try. When he couldn't manage that, he hit the seat with a paw and sent it swinging while he let out a muted growl, which sounded more like a baby's cry.\n\nEvery day, after feeding Scratchy six or more bottles of milk on the back porch, Dad pulled on his uniform and left for work. Mom readied Pam and me for school\u2014kindergarten in my case\u2014and then helped us board the small bus to the base. There we spent our day among other English-speaking military brats learning our Ps and Qs. After school, Scratchy leaped for joy when Pam and I came home, and we put on his collar to take him on a neighborhood walk.\n\nWe loved our baby brother, but our Turkish neighbors did not. Most of them muttered and complained. Their animals went crazy when we walked Scratchy through the village. Cats, dogs, horses, goats, burros, chickens, and ducks scrambled for safety. Dad finally built a six-foot concrete-block wall around our backyard to keep Scratchy penned in. He hated it, but we knew it was for his own protection.\n\nMonths later, after several incidents when the neighbors complained about our pet bear to the local police, my father was forced to make a decision that crushed his heart. He took me sailing that day. Dad and several of his navy buddies co-owned a twenty-two-foot Marconi-rigged boat that they kept at the Seaside Club near Karam\u00fcrsel. They sailed her in the Sea of Marmara, through the Bosporus, and around a sprinkling of small Greek islands. Every now and then, Dad took me along. Sometimes we'd talk, and other times we'd sit in silence and listen to the waves lap against the wooden boat. Today we talked, and he told me that it was time for Scratchy to go. I started crying and begged him to reconsider.\n\nTears filled my dad's eyes, too. He placed his arm around me and pulled me close. \"I found Scratchy a home where he can be happy,\" he said. \"We can see him there whenever you want, okay?\"\n\nIt wasn't okay, but in my shattered heart I knew we had no choice.\n\nHis hands shaking, my dad maneuvered the boat back to the dock. We saw my dad's boss, Captain Frank Mason, standing on the pier. Mason always made my dad a little nervous, but even more so around boats because the captain was an expert sailor. Still a bit green at sailing, Dad crashed the boat into the dock. Mason started laughing and suggested sailing lessons. The next day my father talked his friends into selling the boat. The day after that I watched my dad put Scratchy's collar on and take him away forever.\n\nOne of my father's friends was the founder of the American Seaside Club, a private home on the outskirts of De irmendere rebuilt into a restaurant for the American community. An American oil company, Caltech, operated a facility on the other side of Marmara Bay, and some of their personnel came over in outboards to the Seaside Club. We often brought Scratchy past the club during our afternoon bear walks. Some of the wives of the Caltech employees fell in love with him and came out to pet his soft fur. When they heard we could no longer keep him, they promised my parents that they'd build a mansion of a cage and feed and care for him like a spoiled child. I knew Scratchy would be happy there, but that didn't lessen the ache.\n\nWhen the Caltech people tried to put Scratchy into their boat and take him away, he struggled and whined. As the boat pulled from the dock, I saw him stand on his hind legs and paw at the air in desperation. He wailed in agony, and I knew what his cries were saying: \"Why are you sending me away? Don't you love me anymore?\" Mom, Pam, and I cried for hours, and my dad was silent and sullen for days.\n\nWinter winds deepened the chill in our home. My parents forgot how to smile. At first I thought it might be because we'd been forced to give away Scratchy, but I soon wondered if something else might be going on. Although too young to understand the import of what was happening in the world around me, I couldn't help but notice the absence of laughter in my dad's eyes and the worry on his face. He started spending more and more time on the base, and most nights I was fast asleep by the time he came home. Neither my mom nor my sister provided an explanation, and I remained unaware that beyond the edge of our quiet existence, my dad's world had just become a living hell.\n\nIN EARLY DECEMBER 1960, THE HUFF Duff in Karam\u00fcrsel, Turkey, where my dad worked, transformed into the epicenter of the U.S. Navy's underwater battle against the Soviet Union. The United States built the site in 1957 to monitor Soviet radio transmissions using a sophisticated antenna array. Air Force personnel captured communications emanating from various sites in Russia up to thousands of miles away. These trained experts utilized complex processes, deductions, and heavy doses of transmission analysis to predict when a Soviet missile might be launched, as well as the type and probable destination. Similar stations around the globe detected and analyzed communications associated with specific missiles: short, medium, or long range. If an unusual number of long-range missiles were detected in the preparation stage, there might be time to undertake defensive measures, perhaps even launch a preemptive strike.\n\nMy father worked for the TUSLOG Detachment 28 at the Karam\u00fcrsel Huff Duff, which performed an altogether different mission than the Air Force section. Crammed into a small Quonset hut near the Air Force operations building, they were tasked with using HFDF equipment to locate Ivan's sea monsters. Just before Christmas 1960, those monsters became invisible.\n\nHigh frequency (HF) generally refers to transmissions in the range of three to thirty megahertz. A megahertz (MHz) is a million hertz, and a hertz (Hz) refers to the number of transmitted cycles per second (cps). Most stereo subwoofers put out between 20 and 150 Hz, or really low bass frequencies, whereas a good set of headphones usually tops out at 20 kilohertz (kHz), or 20,000 Hz, which is the top-end range for most human ears.\n\nSince the ionosphere does a nice job of reflecting HF radio waves (a phenomenon called skywave propagation), operators often use HF for medium-or long-range radio communications. Things that can mess with HF transmissions include the time of day or year at the transmission site, sunspot cycles, solar activity, polar auroras, and electrical wires or equipment. Still, the HF band has long been popular with amateur radio operators, international shortwave broadcasters, and seagoing vessels, including submarines.\n\nSince the invention of wireless transmissions, operators pursued the idea of using two or more radio receivers and bearing triangulations to find the location of a transmitter. In principle, the concept seems simple: stick up a reception antenna, and notice the direction of the strongest signal. Theoretically, that should point toward the source. But with only one bearing, the source could be anywhere along that line. That's why you need at least two or more cross-referenced bearings to get a \"fix.\"\n\nTo visualize this, imagine that you are in a parking lot, and you can't find your car. You press the emergency button on your key chain and can can hear your car honking but still can't find it in the dark. You cock your head and listen for the strongest sound. You hear it due east on a \"bearing\" of 90 degrees on a 360-degree compass. You make a mobile phone call to your spouse, who is southeast of you on the other side of the parking lot. She reports that the honking is coming from the north, on a bearing of 0 degrees from her. If you walk east, and your wife walks north, you will run into each other at your car.\n\nSure, you could have walked east and found your car without her. After all, you were only a football field away. Now imagine trying to find a transmitter in the middle of an ocean thousands of miles away. To do so requires knowing precisely where your two sets of \"ears\" are located, then drawing straight lines from each. Where they intersect is the location of the transmitter. Sound easy? Now let's make it tough.\n\nWhat if we throw in inaccuracies and interference? For example, what if the equipment you are using to determine the direction to the transmitter is inaccurate? In the parking lot, if your ears are a few degrees off, you might walk right past your car without seeing it. In an ocean, those few degrees translate to dozens of miles if the source is thousands of miles away. And what if there's a sun storm toying with the ionosphere? Or how about nearby electrical equipment? This interference can easily distort the perceived bearing to the source. This is why one needs multiangulation, or multiple bearings to a transmission.\n\nLet's assume we have a transmitter in Dallas. Now visualize a direction finder in New York and one in Seattle. Using a ruler, draw a straight line from each of those locations to Dallas. You've found the transmitter! Now move the Seattle line upward by a quarter inch to simulate an inaccuracy. You're in Oklahoma, not Dallas. But if you have a few more lines coming from Chicago and San Diego, you'll be much closer to Dallas. Now take away the ruler and let a two-year-old draw the lines to simulate interference. Your transmitter looks like it's in Cuba. These were the problems facing the early designers of Huff Duff systems where inaccuracies and interference equated to bearing \"spreads\" of up to three or more degrees. Still, they could be reasonably effective for finding transmitting submarines.\n\nThat's why, during World War II, in an attempt to thwart detection by Huff Duffs, U-boat captains started shortening their transmissions. They figured that if they were not on the air very long, HFDF operators wouldn't have enough time to get a bearing. Fortunately for the Allies, the U-boat skippers were mostly wrong. The Soviets regurgitated this thinking years later and decided that, if done properly, it just might have some merit. Unfortunately for the United States, they were right.\n\nDescribing what happened in December 1960, when the navy Huff Duff stations suddenly could no longer hear Soviet submarine transmissions, requires an understanding of how these sites operated. Navy personnel at HFDF facilities reported to the NSG, which answered to the NSA. NSG divided sailors and officers of the communications technicians rating, who worked at these stations, into separate branches. Radio operators\u2014or R-Branchers\u2014monitored Soviet signals from various platforms and determined an HFDF bearing to the transmitter when they got a \"hit.\" Operations specialists\u2014O-Branchers\u2014assumed the duties of site operations and logistics, while maintenance personnel\u2014M-Branchers\u2014ensured that the station's equipment stayed running at peak efficiency. Russian-speaking intelligence operatives\u2014I-Branchers\u2014listened with trained ears for tidbits contained in Soviet traffic, while \"technical\" T-Branchers monitored for new kinds of signals and analyzed characteristics to determine the type of transmitter or platform. Being an R-Brancher, my dad's training focused on monitoring Ivan's HF transmissions, with an emphasis on Soviet submarines. As the senior-ranking chief petty officer at the Karam\u00fcrsel station, however, he assumed an O-Brancher function as the operations chief.\n\nDet 28 hummed twenty-fours a day, manned by four watch sections of four or five people \"in the shack\" for an eight-hour shift. Each morning, they received a list of \"interest\" contacts from Net Control (NC), the central command station that coordinated all the Atlantic Huff Duff stations. In those days, Net Control resided in Northwest, Virginia.\n\nNC compiled a catalog of surface and submarine contacts\u2014not all but mostly Soviet\u2014along with their call signs and probable transmit frequencies. R-Branchers set up their equipment to listen on those frequencies, usually in the 2\u201332 MHz high-frequency range. If they got a \"hit,\" they'd contact NC via CW\u2014continuous waves, or Morse code transmitter\u2014and give them a \"tip-off.\" NC then submitted a \"flash\" to the other Huff Duff stations to tune into, for example, frequency 12465 and take a bearing if they heard anything. If any station did catch something, they'd send a \"spot report\" with the bearing for that contact back to NC. In the early days, those reports\u2014or abbreviated versions called e-grams\u2014came via CW and later on were sent by way of Teletype machines. Because the navy operated as a separate detachment at Karam\u00fcrsel, someone needed to run the reports over to the nearby Air Force building every hour so they could be sent to NC.\n\nOperators at Net Control collected all the bearings reported by the stations and, before the invention of automated systems, manually plotted a \"fix\" to the target. This process took several minutes and consisted of nothing more than generating \"string bearings\" with a compass rose. The rose used a figure that displayed the orientation of the cardinal directions\u2014north, south, east, and west\u2014on a four-by-six-foot map or nautical chart mounted on a stand. They called this a gnomonic projection. Operators took a line of string and ran it from each Huff Duff map location, represented by drilled holes in the map to a point on the edge of the map along the bearing line reported by the station. The whole thing resembled a large wall-mounted ocean map covered with strands of Grandma's yarn.\n\nAt least two intersecting bearings, where the strings crossed, were needed to pinpoint the location of a contact. Only two, however, offered very low accuracy. With three or more bearings, one could multiangulate a more accurate location, but this could still be fifty or more miles off target. Needless to say, the art of Huff Duffing was an inexact science in the early 1960s but at least close enough to point U.S. submarines, aircraft, and ships to the right ballpark.\n\nOne fateful morning in December 1960, the HF airwaves went silent. T-Branchers and R-Branchers at the Karam\u00fcrsel Huff Duff in Turkey, monitoring frequencies for Soviet submarines, spun dials and searched for hours but found nothing. They checked and calibrated equipment. Still nothing. They contacted other Huff Duffs and discovered that the phenomenon existed at every station around the world. Several days passed without a single sniff. Dad decided it was time to tell his immediate boss, Commander Petersen.\n\nMy father straightened his back and adjusted the khaki \"cover\" on his head. He marched into Petersen's office, located in the Air Force building next to Det 28's Quonset hut, and said, \"Sir, they're gone.\"\n\nPetersen glanced up from a stack of paperwork and peered over the top of his thick glasses. \"Who's gone?\"\n\n\"Ivan's boats,\" Dad said. \"Their transmissions have been decreasing over the last several months, as you know, but now they've stopped transmitting anything on HF. We've heard nothing for days.\"\n\n\"Shit,\" Petersen said as he removed his glasses and massaged the bridge of his nose. \"Does NC have anything to say about this?\"\n\n\"No sir, Net Control is as clueless as we are.\"\n\nPetersen shoved his glasses back on. \"Shit, shit, shit. They're transmitting, all right. We just can't hear them.\"\n\nDad relaxed his stance and looked down at Petersen's desk. The commander kept his workspace in the same shipshape condition as his duty section. At a facility saturated with routine and order, like Det 28, no one could trump the man. But when the proverbial excrement hit the blades, Petersen's smooth edges ruffled.\n\nThe commander pushed his chair back and stared at the papers on his desk, as if an answer might jump off the pages. His eyes darted from side to side. \"Okay, so now what? Do we keep looking or tell Captain Mason? Did NC give us any suggestions?\"\n\nThe room felt small and hot and smelled of floor wax. Dad removed his cover and backhanded a bead of sweat. \"NC said to keep looking, but there's not even a peep in the three to thirty megahertz range. We've scanned every frequency used in the past thirty years by Russian subs, surface ships, and even life rafts with no luck. We're pretty much out of options at this point, sir.\"\n\nCommander Petersen scratched at his balding head. Dad cringed because the man exhibited a skin disease exacerbated by stress. Losing the Soviet subs qualified as an ulcer-producing disquietude, and for Petersen, incidents like this sent him scratching. When that happened, pounds of dandruff flaked from his head and coated his shoulders. Bets were often taken for how long someone could last in a \"flakey\" conversation with Petersen. Dad never won. He left Petersen's office quickly with an agreement to keep searching.\n\nWithout answers, Petersen's directives offered as good a course as any to follow: keep looking and pray for a miracle. Everyone on the team agreed that Ivan was transmitting, and probably in the HF range, but the Soviets must have found a way to mask their transmissions. Dad knew that Admiral Sergei Gorshkov, commander of the Soviet fleet, had dark red \"control freak\" blood running through his varicose veins. He insisted on maintaining constant communications with his fleet, especially with his attack and missile-firing submarines. Soviet subs always checked in with their command stations at least once, sometimes twice or more, per day. With hundreds of submarines operating on a continuous basis, Det 28 often sent dozens of tip-offs per day to NC and received hundreds of flash reports with monitoring assignments based on tip-offs from other stations. Adding up all the hits coming from Soviet ships and subs, some of the larger stations handled up to 3,000 flashes per day.\n\nMore days passed without a single submarine tip-off or flash. Dad decided to bounce a few ideas off Captain Mason. Although he considered Mason a friend, since he'd worked for the captain in Guam years earlier, my father still felt a bit nervous in the man's presence. Mason's graying hair and wise eyes complemented his friendly tone and engaging smile. He stood six foot three, about an inch taller than my father, and commanded a quiet respect. With matching crewcuts, square jaws, and deep baritones, the two had a lot in common. Both men were born with strong demeanors and \"take command\" attitudes, which is why they often played from the same song sheet when it came to military matters.\n\nFor reasons unknown, however, Mason sometimes reminded Dad of his stepfather, Lon Reed. Three months after Billy Joe Bowles came into this world, in Konawa, Oklahoma, on February 4, 1929, my real grandfather, Hoyle Bowles, died in a train accident. My grandmother, Ethel, met and married Lon Reed a few years later, and Lon adopted my dad and his two sisters.\n\n\"Drill Sergeant\" Lon probably didn't intend to be an evil man, but ignorance blinded him to the kindness of the wise. Trapped in the mold of his forebears and smugly confident of his rightness, Lon played the role of king in a pauper's court. Ignorant men are often haunted by the reflection of their own hatred, and within Lon Reed's small frame walked a man who despised almost everyone. Crude remarks and bigoted bias my dad could endure, for these were passive shortcomings, but when Lon's cruelty turned active, Dad harbored no guilt in wanting his stepfather removed from the planet.\n\nLon's worst show of spite happened when my father was a boy. Intolerant of animals, especially young ones that barked when hungry, the drill sergeant stuffed Dad's first puppy, along with its little brothers and sisters, into a gunnysack. He then hurled the bag into an irrigation canal and laughed while the puppies drowned. Dad once told me that their pathetic cries and whines, as the waters swept them along, left him with violent nightmares and an obsession to adopt all of the world's helpless animals. I'm certain that losing Scratchy hurt him much deeper than he dared show.\n\nThoughts of his childhood plagued my dad as he stood outside Frank Mason's office in the Air Force building that day. The reasons for this were unclear, but he suspected that, despite how much he liked the captain, the smell of the man's Old Spice aftershave always reminded him of Lon Reed.\n\nDad tapped on the open door and stepped inside. \"Got a minute, sir?\"\n\nStill on the phone, Mason nodded and pointed at a chair. Dad sat.\n\nMason reeled off a few more commands, then hung up the phone. \"What's on your mind, BJ?\"\n\nHaving been renamed Billy Joe Reed after Lon Reed adopted him, Dad preferred to be called BJ by his friends. \"I think I know why we can't find the Soviet subs.\"\n\nMason sat forward in his chair. \"Go on.\"\n\n\"Do you remember what the Germans did in the war to keep us from DFing their CW transmissions?\"\n\n\"Yeah,\" Mason said. \"They recorded their Morse code communications, then sped up the recorder before transmitting. That let 'em send out a shortened message on a specified frequency at a set time. Those bursts were so short that we couldn't get a good bearing\u2014\" Mason stopped midsentence. He stood up and brought a hand to his chin. \"I'll be damned. Ivan's using a burst signal.\"\n\n\"I'd bet my stripes on it,\" Dad said, \"and that means we'll be lucky to find it. And even if we do, how the hell are we going to get a bearing? The duration of the signal will be too short.\"\n\n\"All good questions, Chief, and I wish I had the answers. For now, let's focus on the first step first.\"\n\nDad nodded. \"We need to find the damn thing.\"\n\n\"And fast,\" Mason said as he moved to the side of his desk. \"Right now we're one of the closest stations to the Soviet backyard. If we can't find the burst, nobody can. And I've got NSA breathing down my neck every day. If Ivan thinks he's invisible, he just might get cocky and start firing missiles.\"\n\nDad stood up. \"Understood.\"\n\nAs my father turned to leave, Mason called after him. \"One more thing, BJ.\"\n\nDad turned and cocked an ear. \"Sir?\"\n\n\"You gotta find this thing in less than a week.\"\n\n\"NSA?\"\n\n\"No, Petersen. I'd say he's got about five days before he scratches himself into the base hospital.\" Mason flashed a brief smile and turned back toward his chair.\n\nDad returned the smile and walked out of the captain's office. About a minute later, his smile faded as he thought of the impossible task he'd just been given. He spent the next week monitoring every HF known, along with a wide assortment of frequencies outside the normal range. He heard nothing except static and an occasional pop and scratch. He also employed a sonograph to analyze any suspect signals.\n\nBack then a sonograph was a three-foot-long by eight-inch-wide machine that made sound waves visible. The unit housed a large drum around which operators wound photographic paper for each signal analyzed. On playback of a recorded signal, a stylus imprinted an enlarged image of the signal for inspection by analysts. Unfortunately, given the dilapidated condition of his years-old navy-issued sonograph, Dad saw no signs of a burst signal.\n\nHis shoulders slumped, his eyes red and swollen, his smile gone, my father walked through the door of our apartment and plopped onto the couch. My mom tried to console him, but to no avail. My sister and I also did our best to make him feel better, but our attempts at levity went down in shambles. Dad just opened a beer and sat staring at the wall.\n\nThe next morning, while getting ready for school, I noticed a worn spot in the rug by the door. Then I remembered. Scratchy earned his name by scratching at that spot, like he was trying to dig a hole to China. I knelt on the rug and scratched at the carpet, just like Scratchy once had, trying to remember his hairless little bear body.\n\nStill sitting on the couch, his chin blackened with stubble, Dad said, \"Billy, would you please stop that? You sound just like\u2014\" He sat up on the couch and opened his eyes wide. \"Oh my God. That's it! The burst signal sounds just like Scratchy's carpet scratching.\"\n\nDad jumped from the couch, ran over, picked me up, and gave me a hug. Smiling, he bolted into the bathroom to shower and shave. I couldn't explain why he'd suddenly transformed from depressed to ecstatic, but I figured it must be a grown-up thing. I learned later that our pet bear's scratching probably saved the navy's ass.\n\nDad sped to the base in our Volkswagen in search of an Air Force colleague named Jimmy Hensley. He bypassed his shabby Quonset hut and charged into the plush concrete-and-steel air-conditioned building next door. There he flagged down Airman Hensley. \"I need a big favor.\"\n\nNoticing the excitement on my father's face, Hensley said, \"Does it involve a woman?\"\n\nDad frowned. \"No, it involves being a sneaky little thief. Think you can handle that?\"\n\nHensley inched the corner of his mouth into a wry smile.\n\nWhether Air Force or navy, everyone knew that maneuvering the military system to obtain supplies, equipment, or parts required a master's degree in procurement manipulation combined with borderline \"cumshaw\" thievery. British sailors coined the word cumshaw from one they heard from Chinese beggars that meant \"grateful thanks.\" And in the art of cumshaw, Hensley boasted a Ph.D. Often compared to Milo Minderbender, the mess officer glorified in Joseph Heller's Catch-22, Hensley could darn near find anything, for a price.\n\nDad heard that Hensley acquired, on behalf of Master Sergeant Rich Cousins, three brand-new sonographs for the Air Force unit. Dad's section, Det 28, employed an old unit held together with bailing wire, tape, and a wad of bubblegum. Since the Air Force got all the best supplies, Cousins could offer \"cumshaw guy\" Hensley a few pieces of unused equipment in exchange for the new sonographs, while Det 28 had nothing to trade. With an epiphany running around in his head, Dad knew that he could find the Soviet burst signal, but not without at least two of those sonographs. Getting them, however, would be a major challenge.\n\n\"I heard you found three new sonographs for Sergeant Cousins,\" Dad said as he cornered Hensley near an office doorway.\n\n\"Did indeed,\" Hensley said, standing next to the door.\n\n\"I need two of them ASAP,\" Dad said, feeling like a beggar.\n\n\"Well now, that's gonna cost you. I don't think Cousins is going to\u2014\"\n\n\"How much?\"\n\n\"More than you can afford, Chief,\" Hensley said, as he leaned against the doorframe.\n\nDad's temples throbbed. He needed those units. Lives might be at stake. At the very least, a few careers. Hensley was right, however. Dad could tap everyone at Det 28 for a loan and still not get enough to buy two new sonographs. He had to devise a way to borrow the damn things\u2014indefinitely and for free.\n\n\"There must be something we can trade,\" Dad said out of desperation.\n\nHensley rubbed his chin. \"Don't think so. You boys ain't got nothing anybody needs.\"\n\nHensley had a point there. Most of the Air Force personnel called Det 28 the \"Orphan Annie\" of Karam\u00fcrsel. Relegated to a Quonset hut on the other side of the tracks, they received none of the perks afforded their Air Force counterparts.\n\nDad noticed that he was standing on new linoleum tile and that the walls smelled of fresh paint. Probably Hensley's doing. My father searched his brain for inspiration. What could he offer that Hensley might need? Probably nothing. But maybe there was something Cousins needed. \"Does Sergeant Cousins know how to use those new sonographs yet?\"\n\nHensley tilted his head to one side like a dog training his ear on a sound. \"What do you mean?\"\n\n\"Those new sonographs aren't anything like the old ones. You did give Cousins operating manuals, didn't you?\"\n\n\"Well, no, I don't think so.\"\n\n\"So what happens when his men can't get the things to work? Who's he going to blame?\"\n\n\"I...I don't know. He wouldn't blame me, would he?\"\n\nStill blocking the doorway, Dad stood up tall. He now hovered a good four inches above Hensley. Furrowing his brow, he stepped up close to the airman and produced a slow, deep bass. \"You mean you got him three new pieces of equipment without operating manuals? You might as well have given him boat anchors.\"\n\nHensley took a step backward. His lower lip quivered. \"This is not good. Cousins will be pissed.\"\n\n\"And he's not a man you want to piss off,\" Dad said, knowing that Cousins could be meaner than a bulldog on steroids when angered.\n\n\"I'll just have to find him some manuals.\"\n\n\"Weeks,\" Dad said. \"That'll take you weeks. Then he'll really be angry.\"\n\nHensley looked at Dad with wide eyes. \"So what should I do?\"\n\nMy father placed a sympathetic hand on Hensley's shoulder. He softened his tone and said, \"Maybe I can help. I was trained on those units in Guam. I could teach Cousins's team how to use them. That'll buy you some time to get those manuals.\"\n\nHensley looked relieved. \"You'd do that for me?\"\n\nDad smiled. \"Sure, for a price.\"\n\nThat afternoon Hensley delivered two new sonographs to Det 28's Quonset hut. Everyone whistled approval as my father opened the boxes and removed the units.\n\nScratching at his head, Commander Petersen came out of his office and stared at the early Christmas presents. \"Where'd you get those babies?\"\n\n\"Borrowed 'em,\" Dad said as he plugged one in. \"Kind of indefinitely.\"\n\n\"Why do we need them? We have one already.\"\n\n\"It's old and worthless. I needed two new ones. I have a hunch.\"\n\nPetersen scratched his scalp. \"A hunch?\"\n\n\"Yeah,\" Dad said, trying to ignore Petersen's dandruff. \"A scratchy hunch.\"\n\nOver the next several days, my father listened to a series of frequencies on the HF band. Days earlier, before his epiphany, he'd heard nothing but pops and scratches. But one scratch differed from the rest. When I started scratching at the rug in our house, the sound reminded my dad of the \"scratch\" he heard at around 345 Hz. At first, he passed the burst of static off as an anomaly. Now he listened for the sound with unequaled intensity. Hours passed with no joy. Then, suddenly, he found it again.\n\nLike an excited kid on Christmas Eve, my father used the two new sonographs to make an enlarged picture of the signal. He needed both of the units so he could record numerous signals quickly and compare them side by side. While studying \"sound pictures\" of the signal, my father noticed something odd. He squinted, then pulled out a magnifying glass. To no one he said, \"I'll be damned, this thing has bauds.\"\n\nHe grabbed the sonograph printout and ran over to Mason's office. Bursting through the door without knocking, he threw the printout on the captain's desk. \"It has bauds.\"\n\nMason sat forward and stared. Dad handed him the magnifying glass.\n\nMason studied the printout, then smiled. \"Best Christmas gift I've ever gotten, BJ.\"\n\nThe term baud, named for the French engineer Jean-Maurice-\u00c9mile Baudot, became the primary yardstick for measuring data transmission speeds until it was replaced years later by a more accurate term, bits per second (bps). As far as Mason and my dad were concerned, a baud equated to something man-made, and that translated to \"Gotcha!\"\n\nIn similar fashion to the burst signal used by the Germans in World War II to thwart direction finders from locating Morse code transmissions, the Soviets invented their own burst signal for HF communications. The bauds they used were the most compressed ever and represented a huge leap forward in radio technology.\n\nAfter studying, recording, and confirming the burst signals, Det 28 sent copies of their findings to the NSA in Mary land. The agency assigned their best analysts to the case and instructed all stations to obtain as many recordings of the new burst signal as possible. Soon every Huff Duff started finding them.\n\nMy father spent the next several weeks thanking me for helping him solve a major problem. I had no idea what he was talking about, but I rejoiced in the fact that he seemed happy again. I still didn't see him much, as he spent most of his days and nights at the base analyzing the burst signals. He discovered that they came with a \"trigger\" at 345 bits per second (bps), followed by a series of bauds at 142 bps. He figured that the trigger probably started a recorder at the receiving station. The series of bauds were followed by a short message burst. Dad knew that the NSA might never decipher the contents of those messages, but that was not the mandate of a Huff Duff. These stations were designed to find and analyze, not decode. Unfortunately, that still left them with an impossible task: how to get a good bearing to a burst signal.\n\nThese signals were so short that determining an accurate bearing could not be done. Even getting an inaccurate one posed a significant challenge. With the Soviet navy launching hundreds of new submarines, many capable of wiping out dozens of cities in the United States within minutes, the NSA pushed the program up several rungs on the priority ladder. If they couldn't find a way to DF the Red Bear's new burst signal, they couldn't find its submarines. And that made the world a very scary place to live in.\n\nFor his team's diligence in finding and analyzing the new Soviet burst signal, Captain Mason received a letter of commendation from the National Security Agency. In turn, he handed my father a letter of appreciation and recommended him for limited duty officer (LDO). My dad was on his way to leaving the ranks of the enlisted and becoming an officer in the U.S. Navy. That winter he flew stateside to undertake one of the most important assignments of his career.\n\n## CHAPTER FOUR\n\nThe beginning is the most important part of the work.\n\n\u2014PLATO\n\nWHEN MY DAD, WILLIAM J. REED, reported to NSG headquarters at 3801 Nebraska Avenue in Washington, D.C., in early 1961, he couldn't keep his hands from shaking. The shivers were caused in part by the snow falling on the shoulders of his navy jacket, in part by the anticipation of what lay ahead. After discovering the Soviet burst signal, and the fact that this thing contained data of some sort, Reed received temporary orders to return to the States.\n\nMaster Chief Reed knew that, in March 1959, the NSA combined all military electronic intelligence (ELINT) programs under one roof, and that Howard Lorenzen's group supported the Advanced Signals Analysis Division of the NSA's Office of Collection and Signals Analysis headed by John Libbert. He also knew that the bulbs glowing in the heads of these engineers were brighter than ship-borne searchlights. And while Reed considered himself a pretty smart guy, he figured that chess matches with these geniuses wouldn't last more than two minutes.\n\nIn the world of electronic countermeasures, few icons commanded more respect than Howard Otto Lorenzen. In July 1940, after five years of designing commercial radios, he launched his career in ethereal warfare at the Naval Research Laboratory working for the brightest minds in radio engineering. During World War II, he developed a system to analyze German aircraft radio signals that controlled glide bombs. This allowed NRL's Special Projects Section to create intercept jammers for enemy aircraft bomb controllers that rendered the things useless. German Luftwaffe engineers thought the problem was their fault and dismantled the systems.\n\nLorenzen worked on similar projects throughout the war, eventually overseeing a dozen small groups tinkering in various fields of radio engineering. After the war, he invented the term electronic countermeasures, defining ECM as a \"discipline that first detects, then interferes with or analyzes for intelligence purposes any electromagnetic energy emanating from the enemy.\" The Bureau of Ships concurred with his definition and sponsored ECM projects at NRL for intercept, direction finding, radar jamming, and decoy systems.\n\nLorenzen and other key members of his team remained government employees after the war, pulling apart captured German electronics like excited kids in science class. He managed to convince the Brits to lend his group a key piece of German technology\u2014used in the Wullenweber (pronounced VOOL-in-veber) antenna sites\u2014that eventually helped redefine HFDF forever. Lorenzen's HFDF expertise brought him to NSG's headquarters to meet with Reed and others and help solve the problem of gaining accurate, after-the-fact bearings to Soviet burst signals.\n\nStill nervous, Reed introduced himself to the team. Lorenzen and a dozen engineers grilled him for hours about the nature and characteristics of the burst transmission that he'd discovered and analyzed. They examined the sonograph printouts and grilled him some more. When not grilling, the narrow-tie-wearing team swapped theories and ideas using complex phrases that, as far as Reed was concerned, were akin to Latin. Techno, technara, technatus, technodom. Now and then Reed managed to grab hold of a concept and attempt to bring the cloud-dancing scientists back down to earth, where submarines transmitted and Huff Duff stations listened.\n\nOver the course of several days, the team determined that the burst signal lasted no more than seven-tenths of a second. That posed a huge problem. Equipment at Huff Duffs was designed to locate and determine bearings to ordinary high-frequency transmissions, most of which lasted several seconds or even minutes. Now they needed not only to hear a signal that short but also to determine an accurate bearing to the submarine long after the transmission ended. That's like trying to find your car in a parking lot when the horn honks for only a half-second after you push the key-chain button.\n\nUsing recordings captured by Reed and others, the team analyzed the signals and determined that each transmission consisted of a two-tone alert designed to trigger an automatic receiver\/recorder. A short encrypted data stream followed that contained message information. The Soviets probably figured that no one could direction-find such a truncated transmission. After a week of analyzing signals and bantering over ideas to solve the bearing problem, Lorenzen figured the Soviets just might be right. He threw up his arms in frustration, stating that the Russkies may have finally found a way to trump American engineering. Another engineer, Robert Misner, then asked if it might be possible to create a device that \"triggered\" a switch after a burst signal was picked up by an antenna. Lorenzen pondered the question and answered yes, that a trigger might be possible, but to what end?\n\nMisner flashed a smile. He reminded Lorenzen about their work together, several years earlier, on a magnetic tape recorder. When the Soviet threat escalated during the Korean War, Lorenzen's efforts, in collaboration with others, led to a new ECM system installed into antisubmarine aircraft. While gaining operational feedback on this system in 1949, Lorenzen had an epiphany.\n\nThat's when he contacted Robert Misner, and together they created the first magnetic tape recorder for intercept work. They called this device the Radio-Countermeasures Sound Recorder-Reproducer, dubbed the IC\/VRT-7. After that project, Misner did some research on after-the-fact transmission analysis in 1958. He now thought that by combining what he'd learned from the two projects, perhaps they could create a trigger that started a magnetic recorder and determined a bearing to the transmitter based on the recording.\n\nOther engineers on the team scoffed at the idea, and Reed leaned in favor of the skeptics. Finding an accurate bearing to a live, longer-lasting transmission posed enough of a problem. Misner dismissed the naysayers and sketched his concept on a blackboard. As the white chalk revealed dozens of boxes, lines, and arrows, Reed's eyes slowly opened. Misner's concept started to make sense. If they could engineer a way to record the time at which a transmission was detected, along with the strongest bearing to the signal, then compensate for inaccuracies and other conditions, they just might be able to find Ivan in a haystack.\n\nTo achieve this, the NRL engineers needed to overcome a big issue with Lorenzen and Misner's magnetic recorder: the thing didn't have enough capacity to store the hours and hours of recordings needed for after-the-fact analysis. Today we have programs on iPods that can record a song playing on the radio for a few seconds, then upload the recording over the Internet, where it's analyzed to determine the artist and song. We take such a feat for granted, forgetting that this requires gigabytes of storage capacity and superfast microprocessor speeds. In the early 1960s, there weren't microprocessors with billions of bytes of storage and memory capacity. Storing burst transmission recordings magnetically required \"out of the box\" thinking that pushed engineering envelopes. Matching these recordings to accurate time signals down to the millisecond raised the bar even higher. Months passed before the team could overcome the limitations and build something that actually worked.\n\nAt the time, no one on the team, least of all Reed, imagined their groundbreaking new device, christened the AN\/FRA-44 recorder\/analyzer, might one day earn a place in history as one of NRL's top seventy-five inventions, with Robert Misner accepting the prestigious award. They also did not expect their new system to play an integral part in thwarting a world war less than a year later.\n\nUsing innovative technology, the FRA-44, called \"fraw forty-four\" by operators, allowed the U.S. Navy to record a Soviet microburst, analyze the signal after the fact, and determine a bearing to the source. One major problem in countering the Soviet stealth innovation appeared solved, but an equally daunting one remained: designing a way to receive the burst signals in the first place.\n\nCurrent antenna and receiver technology used at Huff Duffs already lagged behind the Soviets, who had deployed twenty Krug Wullenweber sites based on captured German designs. The United States still used an antiquated AN\/GRD-6 antenna array and HF receiving system, which was no match for microbursts. Although the new recorder\/analyzer designed by the team could work at such a site and provided a short-term fix for locating Soviet submarines, the DF accuracy would be worse than a World War II Huff Duff.\n\nWith Reed's help, Lorenzen's team, led by Bob Misner, determined that a new type of antenna\/receiver was needed, one that increased reception and accuracy by an order of magnitude. Fortunately, such a device had already been constructed by the NRL a few years earlier at the Hybla Valley Coast Guard Station in Alexandria, Virginia. The United States built this station using German Wullenweber technology in 1957 to track the Soviet Sputnik's transmission signal and determine its orbit. While Reed returned to Turkey to test the new recorder\/analyzer in the real world using older antennas, Misner commandeered the Virginia station to test the makings of a new electronically steerable array that could find Ivan's silent boats.\n\nThe NSA funded both projects on a high-priority status and gave them top-secret code names. They called the project using the recorder\/analyzer and related equipment to find the burst signal Boresight and the new Wullenweber antenna project Bulls Eye.\n\nBefore Boresight could become operational, however, Reed needed to gather some critical field data. To do so, he'd need to sneak onto Ivan's back porch without getting caught.\n\nMORE THAN FIFTY YEARS AFTER THE invention of Morse code, Guglielmo Marconi launched radio communications by sending the letter S across the English Channel in December 1901. The science of locating radio transmissions came to light a few years later when John Stone used radio direction finding (RDF) techniques in 1904 to locate transmitting sources. The team of Ettore Bellini and Alessandro Tosi improved on Stone's designs, and Marconi acquired the patents to this technology in 1912. He then mounted the newly acquired RDF equipment on commercial ships.\n\nDuring World War I, Captain H. J. Rounds of the Royal Navy installed a series of RDF stations along the east coast of England for Room 40, the British Admiralty's code-breaking intelligence branch. These stations came in handy during the battle of Jutland in the summer of 1916, a World War I clash between battleships in the North Sea near Denmark that to this day is considered the largest naval battle in history.\n\nCaptain Rounds ordered his stations to monitor the movements of the German battleship Bayern. Using RDF, operators reported that the Bayern steamed some distance north during the night. Using this information, Vice Admiral David Beatty, commander of the First Battle Cruiser Squadron, avoided the U-boat threat and caught the Germans off guard. The Brits engaged Franz von Hipper's battleships long before the German admiral expected. While the battle proved costly for both sides, the advantages of RDF were solidified in the minds of military experts.\n\nThat same year, under the direction of Commander Laurance F. Safford, head of OP-20-G (20th Division of the Office of Naval Communications) and \"father\" of the navy's communications intelligence unit, the navy built an Atlantic arc of twenty-six HFDF stations. These Huff Duffs stretched from Britain to Iceland to Greenland, across the eastern states, and down to Brazil and Africa. German submarine tactics mandated frequent radio contact between U-boats and headquarters. When these skippers called home, they were unaware that a giant ring of Huff Duffs was capturing these signals and finding a direction to the source. By cross-referencing bearings from multiple Huff Duff sites, the Allies could multiangulate approximate locations for the transmitting submarines.\n\nStations reported bearings to Net Control in Virginia, which forwarded the same to head of Naval Communications Intelligence Commander Knight McMahon's staff in Washington, D.C. Fixes were then flashed to the Atlantic Section of the Combat Intelligence Division, which shot them out to U.S. antisubmarine warfare forces. Unfortunately, the system's accuracy left something to be desired, and the definition of a good \"fix\" equated to fifty miles from the target. Despite this limitation, for more than a decade after the war, the navy did little to upgrade its twenty-six Huff Duffs.\n\nWhen the NRL team officially launched Project Bulls Eye in 1961 they radically upgraded the ability of HFDF sites to detect weak HF signals and improve DF accuracy. To accomplish this, they needed help from the Germans.\n\nDuring World War II, German engineers invented the Circularly Disposed Antenna Array (CDAA) as a way to improve their own Huff Duff capabilities. They built the first site at Joring, Denmark. The German CDAA used forty vertical antennas placed in a circle with a diameter of 360 feet\u2014about the same diameter as the average baseball stadium. Forty more antennas, designed to reflect signals from the first circle, were suspended on a circular wooden support structure just inside the outer ring. From the air, the entire affair looked like two giant Ferris wheels, one inside the other, turned on their sides and missing all the seats.\n\nThe Germans built only two CDAA arrays under the code name Wullenweber\u2014a name prompted by the exploits of Jurgen Wullenweber, who became mayor of L\u00fcbeck in 1531. This iconic figure gained a reputation as a fighter against injustice and the wealthy class, much like Robin Hood. The story of his adventures prompted Dr. Hans Rindfleisch, the group leader of the German navy's communication research command, to use his name for the CDAA program.\n\nAfter the war, the Brits studied the Wullenweber design in Denmark, then destroyed the array in accordance with Geneva Convention mandates. Some of Rindfleisch's engineers were captured by the Soviets and taken to Russia. The Red Bear's Defense Ministry soon erected its first Wullenweber site at Khabarovsk Krai under the code name Krug, which means \"circle\" in Russian. The massive antenna array spanned a diameter of more than a half mile. The Soviets built nineteen more sites throughout the 1950s, with many installed in pairs within a few miles of one another for navigation purposes. Four Krugs were installed near Moscow, and some were used to track Sputnik satellites via 10 and 20 MHz beacons.\n\nAlthough the Allies snatched up their own Wullenweber engineers after the war under Operation Paperclip, they were slow to the game. Antenna researcher Dr. Rolf Wundt, along with his wife and parents, arrived in New York City on the same ship as Wernher von Braun in March 1947, but he did not work on this technology until many years later. The Air Force, and later GT&E Sylvania Electronics Systems, made some progress on Wullenweber antenna technology, but more than a decade passed before the first site became operational.\n\nProfessor Edgar Hayden, a bright engineer at the University of Illinois, under contract to the U.S. Navy, led the charge to build America's first Wullenweber. He studied the German design and analyzed potential performance possibilities against current Huff Duffs. That's when he got excited. His calculations concluded that inaccuracies could be reduced from as high as three percent down to one-half of one percent. That small change could be the difference between sending navy aircraft to find a sub off New York City versus Long Island. Hayden also found that Wullenweber arrays could select desired signals and reject interfering signals or noise detection. This helped extend detection ranges out to several thousand miles away\u2014four times that of current antennas. With the Soviets extending the range of their ballistic missiles, and hence their submarine patrol distances away from U.S. shores, longer range capability held a high degree of importance.\n\nBlessed by the navy, after reporting the good news, Hayden assembled a team to build a Wullenweber array at the university's Road Field Station near Bondville, Illinois. The array contained a ring of 120 vertical pole antennas that \"listened\" in the HF range of 2 to 20 MHz. Tall wooden poles, comprising a hundred-foot-diameter circle, supported a screen of vertical wires located within the ring of monopoles. From a distance, the site looked like a giant circular cage large enough to keep elephants from escaping, which spurred the term elephant cages often used by operators.\n\nBased on lessons learned from the Bondville experimental array in 1959, the Air Force awarded a contract to GT&E Sylvania Electronics Systems to build a larger Wullenweber elephant cage\u2014the AN\/FLR-9\u2014at RAF Chicksands in the U.K. This \"Flare-nine,\" along with a sister site at San Vito, Brindisi, Italy, was not scheduled to light off until late 1962. The Air Force used these arrays for airborne tracking and not HFDF although the navy planned to borrow these antennas for such by stationing NSG personnel nearby.\n\nIn mid-1961, when Robert Misner installed the newly invented Boresight AN\/FRA-44 recorder\/analyzer, the navy's plan expanded. With help from Stanford Research Institute, the original Wullenweber designs were improved upon, resulting in something more advanced called the Wide Aperture Receiving System (WARS). Since the Air Force owned AN\/FLR-9 as its official designator for the new CDAA antenna and systems, the navy named its design AN\/FRD-10. Operators called them \"Fred Tens.\"\n\nWhile these sites were designed to conduct some of the most sophisticated radio interception work ever, much of the equipment used, aside from the special fraw forty-four Boresight recorder\/analyzer and related systems, came from \"off-the-shelf\" sources. Each site contained an abundance of such gear, and even small failures or calibration errors could badly degrade bearing accuracy. With the Bulls Eye and Boresight programs underscored by massive bud gets, most everything ordered for these facilities arrived in baker's dozens, from antennas to multicouplers to receivers. Miles of cable snaked through, under, and around the buildings, ending in hundreds of coaxial connectors for coupling to various devices. Only one special device held the honor of being installed as a dynamic duo: the goniometer.\n\nUsed by the Germans in their Wullenweber designs, the goniometer owes its name to the Greeks. Gonia translates as \"angle,\" and metron means \"to measure.\" A spinning goniometer became the backbone to a functioning Fred Ten by refining the process of searching various frequencies. Not unlike a carnival wheel on which various prize amounts are indicated, a goniometer rotates around various frequencies by \"touching\" the pole antennas in a Wullenweber array. Recall that our array consists of a bunch of tall antennas positioned in a big circle. So, if the strongest signal from a transmitting submarine is coming from due north, as the goniometer spins, it will measure a higher signal strength coming from the antenna pole positioned at zero degrees in that circle. After compensating for inaccuracies, time delays, atmospheric conditions, and so on, via lots of sophisticated equipment and analysis, we can determine a bearing to our contact of, say, 358 degrees\u2014roughly in the direction of Santa's house at the North Pole.\n\nOriginal Fred Ten designs consisted of two in dependent goniometers that were later replaced by a single ten-foot-long dumbbell-shaped unit with four-foot-diameter router housings on each end. These resembled the spinning \"g-force\" simulators used to train pi lots and astronauts, only smaller. Since these sites were built prior to the invention of uninterruptible power supplies (UPSs), engineers installed electric motors driven by generators with large flywheels. Diesel engines spun the flywheels during power outages, which took over for the electric motor when the primary power failed.\n\nThe U.S. Navy contracted with ITT Federal Systems to deploy a worldwide network of more than a dozen Wullenweber elephant cages for HFDF operations. The Fred Ten near Okinawa, Japan, became the first installation, but it did not come up to full speed until the second half of 1962. An elephant cage near the Scottish village of Edzell also came on line that year. Nestled in a farming area in the foothills of the Grampian Hills, some thirty-five miles south of Aberdeen, that site replaced less sophisticated listening posts in Germany and Morocco. The navy erected another elephant cage in 1962 at Skaggs Island, California, not far from San Francisco. Each of these facilities cost just shy of $1 million and employed dozens of navy and civilian personnel. At the time, operators at the Skaggs Island Bulls Eye site were unaware of their destiny to play a significant role in the Cuban Missile Crisis.\n\nIN EARLY 1962, REED RETURNED TO Turkey. Within hours of his return, he jetted to the Karam\u00fcrsel base to integrate the new Boresight technology into the existing DF systems. Although the Air Force had not yet installed a Wullenweber elephant cage there, which meant that bearing accuracies would be poor, the objective now focused more on getting something working versus working well. Reed was also tasked with writing an installation and operations manual that could aid other DF sites in implementing the new systems.\n\nWith the help of his colleagues at Karam\u00fcrsel, under the watchful eye of Captain Mason and Commander Petersen, Reed installed the new Boresight receiver\/recorder and related equipment developed by the NSA team. Now, if he could only get the damn thing to work.\n\nThe theory seemed simple: When a receiver encountered a \"trigger heading\" on a burst signal, a sixty-inch-per-second recorder with two-inch-wide tape automatically switched on. The recorder captured the signal, along with a marker indicating the time to the millisecond that the signal was intercepted. Because the Boresight system enabled operators to also capture directional signal strength and other parameters, synchronized by the time marker, they could now determine, after the fact, the probable bearing to the transmitting sub.\n\nIn order for Net Control to get a reasonable fix on the sub's location, additional bearings were needed to create a multiangulation. So until more stations came on line, Boresight remained useless. As such, while Lorenzen's team tackled the enormous problem of building more Wullenweber sites to improve accuracy, Reed received orders to help get other sites\u2014most equipped with older GRD-6 antennas\u2014up and running. The navy hoped that if enough of these sites were operational, they could at least achieve a ballpark fix good enough for ASW forces to have a fighting chance.\n\nFor the next several months, Reed flew around the world to install systems and train operators at sites along the Atlantic and Pacific Ocean perimeters. Operators and station chiefs were excited about the possibility of finally hearing the Soviet subs again, but they were not so thrilled with the amount of work and resources required to become operational.\n\nThe space required for the reception and recording equipment covered an area as large as a typical living room and needed to be air-conditioned since the receivers in those days still used vacuum tubes that generated considerable heat. After installation, days of calibration and testing were needed, along with many long hours of troubleshooting to ensure that everything worked properly. R-Branchers needed to be trained on the equipment, what to listen and look for, and how to properly analyze the burst signals. Reed usually spent weeks at each facility before certifying them as Boresight operational.\n\nBack in the States, Howard Lorenzen and his team of geniuses went to work on a jamming system. Using similar technology to that used in intercept jammers developed by NRL's Special Projects Section in World War II to hamper German aircraft bomb controllers, Lorenzen's team built systems that could send out false signals on the same frequencies used by Soviet burst transmitters. This made it a little harder for Moscow to communicate with its subs and vice versa. Reed took several trips to England to help engineers there install the burst signal jammers, but these devices came with a limited range and were effective only when the Soviet subs passed near the British Isles.\n\nIN THE SPRING OF 1962, WILLIAM J. Reed found out that he'd been selected for a commission in the U.S. Navy. All those years of correspondence courses, night school, and hard work finally paid off. Commander Mason informed Reed that he'd earn his ensign bars in July, and he and his family would be leaving Turkey that same month. After his arrival in the States, he'd head to LDO School in Newport, Rhode Island, for \"knife-and-fork\" training in August, then to the NSA facility at Fort George G. Meade, Mary land. Until then, several more months of grueling travel lay ahead.\n\nA key ingredient to ensuring that Boresight could obtain an accurate bearing to a transmitting submarine entailed calibration and signal analysis. Using the example of finding one's car in a parking lot, two things are taken for granted with human hearing that are not prevalent in the world of HFDF. One of these is that we know what a car horn sounds like. The other is that, for most of us, our ears are also familiar to us, and over many years we've learned how to discern from which direction sound is traveling. In other words, we're pretty sure that our horn is the one blaring at us from an easterly direction.\n\nThis was not the case for the systems used to detect locations for Soviet burst signals. There were just too many unanswered questions about the characteristics of this new type of signal, and before Boresight could be made fully operational, more information was needed. Someone had to undertake the job of finding a Soviet sub or two and get them to transmit while analyzing and calibrating signal location, strength, type, frequency, and time on the air. Using these parameters, operators could test and properly calibrate Boresight systems to be sure they were not providing false hits.\n\nWhen Reed was ordered to ride on a Turkish sub to see if he could capture a burst signal from a nearby Soviet boat, his heart raced. He'd never been on a submarine before, let alone an old smoke boat that appeared to be missing half a lung and one eye. The Turks called her the Birinci Inonu, which loosely translated as \"First Prize\" or \"Number One\"\u2014hardly an apt description befitting this blue-haired geezer in an Istanbul harbor that oozed the foul scent of diesel fumes. Holding his nose, Reed crossed the wooden gangway and boarded the sub.\n\nThe Birinci once served the U.S. Navy in World War II as the USS Brill (SS-330). She launched from Groton, Connecticut, on June 26, 1944, and the Turks bought her after the war on May 23, 1948. A slick film of oil surrounded her 312-foot black hull, where ten torpedo tubes, six forward and four aft, had fired MK-14s at the German navy seventeen years earlier. Reed once read that the Birinci could hit around twenty knots on the surface and ten submerged, driven by a couple of large diesel engines and electric motors.\n\nThe topside watch saluted as Reed approached. He handed over his orders and in Turkish asked to see the skipper. Long minutes passed before a stocky barrel of a man emerged through the hatch. He displayed short-clipped hair and a tight mustache and carried a stern \"I'm in charge\" look. He introduced himself as Captain Celik and motioned for Reed to follow. Grabbing his seabag, Reed descended the ladder into the belly of the dragon.\n\nBelow decks, the Birinci smelled even worse than she did topside. So did her crew of eighty-five. They paid Reed little attention as they prepared to get under way. Captain Celik escorted Reed to his stateroom, which was also a misnomer. The small space housed two bunks and a curtain. No door. Another officer who shared the space\u2014introduced as the navigator\u2014smiled and shook Reed's hand. They talked briefly, then walked to the wardroom for the mission briefing.\n\nCaptain Celik greeted Reed near the wardroom and handed him a cup of black coffee and a pastry. He smiled and said, \"A cup of coffee commits one to forty years of friendship.\"\n\nRecognizing the Turkish proverb, Reed returned the smile and said, \"A hungry stomach has no ears.\"\n\nCaptain Celik cocked his head, offered a friendly Turkish hand gesture, and entered the wardroom. Reed said little during the briefing, as a majority of the crew did not have a \"need to know\" the details of this mission. Such was the tacit agreement between the two navies: we cooperate like allies; we defend our secrets like enemies. Three U.S. Navy technicians trained on Boresight and ESM equipment were also present. The ESM equipment had been installed days earlier by those technicians.\n\nThe Birinci sputtered and belched as she edged away from the pier. The diesels vibrated and hummed, and the saliva in Reed's mouth disappeared. Standing in the control room, one level below the conning tower, Reed watched a Turkish seaman attempt to repair a leak in a hydraulic line\u2014with a hammer. Any doubts that Turkish submariners were the most dangerous species of mammal on the planet evaporated.\n\nCaptain Celik steamed the Birinci into the Black Sea and submerged. The world turned quiet as the batteries spun the boat's propellers, and Reed spent most of his time in the conning tower working with the technicians to test and calibrate the ESM systems. The Turkish sailors gazed at the U.S. techs with curious eyes, but having been briefed by their CO regarding secrecy, they refrained from asking any questions.\n\nWhile patrolling near Sevastopol, days passed without a contact. Then the sonar operator heard the muted chugging of a snorkeling submarine. Captain Celik steered toward the contact. Chatter in the boat ceased. Faces turned serious as they closed to within a few nautical miles. Although the Birinci was an old girl, she'd been upgraded with reasonably decent sonar gear. The same could not be said for the sonar operators. They were clueless as to the possible contact type. Reed asked for permission to take the headphones. The captain nodded agreement, and a Turkish sailor handed Reed the phones. He sat near the sonar stack and listened. His face wrinkled with concentration. Then he heard it: the distinct diesel engine chug of a Foxtrot submarine.\n\nAlthough Captain Celik remained in charge of the boat, Reed assumed command of the mission. Once contact was made with the enemy, the mission commenced, and Celik now technically reported to Reed. Technically.\n\nReed ordered Celik to close the distance so the ESM gear could get a signal. Reluctantly, the captain issued the proper orders to his crew. The Birinci turned, slowed, and inched toward the Soviet boat. Sweat dripped from faces and soaked coveralls. Reed stood behind the U.S. Navy techs and gave directions regarding the signal types and frequencies to listen for. If they could capture a burst transmission and match that against the actual position of the transmitting target, that would go a long way toward accurately calibrating the newly installed Bore-sight systems. They could only hope that the Russian submarine transmitted before she left the area and went deep, and that might be a long shot at best.\n\n\"Now we wait?\" Celik said in Turkish.\n\n\"Evet,\" Reed said. \"Now we wait.\"\n\n\"Do you gamble, Mr. Reed?\" Celik said.\n\nReed flashed a puzzled look. \"Gamble?\"\n\n\"Evet. Gamble. Poker, blackjack, you know, gamble. Do you not understand this word?\"\n\n\"Yes, I understand. Why do you ask?\"\n\n\"We have an old saying, perhaps you've heard this. 'The wind that the sailor likes does not blow at all times.'\"\n\nStill perplexed, Reed said, \"What's your point?\"\n\n\"I like to gamble, but not in a house of bad odds. If I had to gamble now, I'd bet that your mission fails.\"\n\nReed removed twenty American dollars from his pocket and waved them in front of Celik. \"I'll take that bet.\"\n\nCelik smiled and removed some bills from his pocket.\n\nHours passed with no joy. The Soviet sub continued to snorkel without transmitting. Reed started to wonder if he'd just lost a day's pay to Celik. His mouth dry, his armpits moist, his temples throbbing, Reed knew that the success of their mission depended on getting that Russian boat to send out a burst. But how?\n\nReed's mind scrambled for an answer. At first he refused to listen to his own thoughts, as to do so meant risking more than he cared to, more than he knew Celik would accept. Reed walked over to the sonar console and asked the operator to let him take the stack. He pulled on the headset and listened. Celik watched from the other side of the conning tower, his forehead forming curious lines above his thick black eyebrows. Reed ordered Celik to slow and pull to within 4,000 yards\u2014less than two nautical miles away.\n\n\"I will not!\" Celik said.\n\n\"You will,\" Reed said, \"or I'll see to it that you lose your command.\"\n\nCelik glared. \"Two captains sink a ship.\" He gave the order to the helmsmen, and the Birinci moved closer.\n\nSeveral minutes later, the diving officer reported that they were now 4,000 yards away. Celik ordered all stop and raised the ESM mast and the attack periscope. He swung the scope left, then right. He marked two bearings and lowered the mast. \"Foxtrot at two-three-five and a Skory at one-eight-nine.\"\n\n\"A Skory?\" Reed said. \"We never heard her.\"\n\n\"She's not moving. She's just sitting there about 2,000 yards behind the Foxtrot.\"\n\nReed searched his head for stats and recalled a few. The Soviet Skory-class destroyer carried a slew of ASW equipment and weapons, including four depth charge racks on her afterdeck. No doubt the Skory's captain longed for the chance to use them against a macho Turkish sub skipper. That ship wouldn't sit still forever, and if she came their way, that just might end the mission. They couldn't chance having an extended ESM mast popped up while trained Soviet eyes scanned the seas for intruders.\n\nReed sat at the sonar console and stared at the active sonar key. That key, when pushed, sent a focused beam of sound into the water that bounced off nearby objects, like ships and subs. The active ping returned distance and bearing information to those objects, along with visual outlines displayed on a screen in a similar fashion as radar. That was good. On the other hand, the loud ping could be heard by anyone in the area, thus alerting them to the sub's location. That was bad.\n\nFrom the other side of the conn, Celik watched Reed like a shop owner monitoring a potential thief. The muscle in Reed's chest thudded like a Turkish ramazan drum as he moved his hand closer to the active sonar key. Celik's eyes shot open when he detected the move. He ran toward the console but did not get there in time.\n\nReed hit the key. One loud active ping blasted the water.\n\nCelik arrived at the console, pulled out his sidearm, and placed the cold steel against Reed's temple. \"The cock that crows at the wrong time is killed.\"\n\nReed said, \"One hand does not clap, two hands do. Maybe now he'll transmit.\"\n\nCelik pulled the gun away. His lips formed a half smile. \"If he does and we live, I will not make good on our bet.\"\n\n\"Why not?\"\n\n\"Because you cheated.\"\n\n\"Fair enough.\"\n\nOne of the navy technicians raised an excited hand. \"The Foxtrot just sent out a burst!\"\n\nReed ran over to the ESM equipment. \"What do you have?\"\n\nThe lanky tech pointed at a spinning recorder. \"We got her nice and clear.\"\n\nAnother tech looked up from a console. \"Can we go home now?\"\n\nReed smiled. \"Absolutely.\"\n\nHis smile disappeared as the sonar operator reported the sound of a killer on the move. Had Reed not pinged the water, the Soviet Skory-class destroyer might never have heard the Birinci's quiet, battery-powered propellers. She might never have seen the ESM masts or periscope smoothly gliding through the Black Sea. But now, as she eased toward her prey, this formidable hunter\/killer knew that something lurked under the waves nearby. The Skory's active sonar lit up the ocean as she neared. The ringing vibrations penetrated the hull, and a dozen men in the conning tower recoiled with each ping.\n\nCaptain Celik glared at Reed. \"You've killed us.\"\n\nReed said nothing.\n\nA loud explosion rocked the boat. Sailors in the control room, one deck below the conning tower, yelled obscenities as they struggled to maintain depth and course. More depth charges shattered the silence.\n\nCelik ordered a dive to test depth\u2014about 400 feet\u2014and all ahead full. Reed figured he was probably trying to find a thermal layer to hide under. It didn't work. The Skory kept rolling cans off her deck, and the explosions got louder. And closer. The hydraulic pipe the Turkish seaman earlier fixed in the control room with a hammer sprung a leak. Hydraulic fluid shot out from the pipe like water from a pinched garden hose.\n\nReed thought about his home, his wife, his children. He recalled that years earlier, on board his first ship, the PCS-1380, he'd held Bible studies and Sunday church services. He even bet some of the atheists on board that if he bested them in the boxing ring, they had to attend the following Sunday. He never lost. Since then his faith had diminished to an ember, but as another depth charge rattled his teeth, he whispered a silent prayer.\n\nCelik took the boat deeper and slowed to a crawl. The Birinci moaned and shrieked. Despite the slower speed, the batteries would be depleted in less than a dozen hours. Reed's lungs heaved as the carbon dioxide buildup made it hard to breathe. The heavy air smelled of sailor stench.\n\nThe boat leveled off at 475 feet. Pipes sprang leaks, and the Turkish crew scrambled to make repairs. The depth charges crept closer, along with the Skory's incessant pinging. If neither stopped soon, Reed swore to himself that he'd grab Celik's sidearm and end the ordeal on his own. Thankfully, he didn't have to. The Skory passed overhead and moved away. She did not return.\n\nCelik ordered a turn in the opposite direction, looked at Reed, and said, \"Dogs bark, but the caravan goes on.\"\n\nReed smiled and said, \"If a dog's prayers were answered, bones would rain from the sky.\"\n\nAfter another four hours, with the Skory now far enough away, Captain Celik brought the boat shallow and snorkeled. Having survived her brush with death, the Birinci ran for home.\n\nA few days later, Reed walked through the door of his apartment near Karam\u00fcrsel and held his children in his arms longer than he ever had before.\n\n## CHAPTER FIVE\n\nWe don't receive wisdom; we must discover it for ourselves after a journey that no one can take for us, or spare us.\n\n\u2014Marcel Proust\n\nDURING THE MONTH OF AUGUST 1962, while a small window of summer warmed the city of Moscow, Vice Admiral Leonid Rybalko sped down the Kutuzovskv Prospekt in a black Volga sedan. Summoned to a last-minute meeting with Sergei Gorshkov, the fleet admiral of the Soviet Union, Rybalko ruminated over the reasons for the urgency. Through the windshield of the vehicle, driven by an enlisted man with peach fuzz on his face, the walls of the Kremlin reflected the morning sun and splashed the Arbat with a blood red hue. Vendors along that ailing street unpacked their goods and looked up briefly as the Volga passed by.\n\nThe driver turned the Volga onto Yanesheva ulitsa and pulled to a stop in an annex parking lot. Rybalko stepped from the car and bounded through the arched tunnel toward the Ministry of Defense building. Military police, adorned in leather boots and white guard belts, popped to attention as Rybalko approached. The admiral returned their salutes and entered the building through the main door. Ordinarily, he entered from the side, along with the senior operations and intelligence staff, but today was no ordinary day.\n\nRybalko vaulted up the steps and paused for a moment at the top to catch his breath. At fifty-three years of age, he could no longer ignore his limitations. Socialist paintings lined the walls of the hallway. Most depicted Soviet supremacy over Nazi fascists during the war, as if winning those battles validated the Communist way of life. Reaching his destination, Rybalko entered. Defense Minister Rodion Malinovsky waited two steps inside the large wooden door. Rybalko had met the barrel-chested Malinovsky during the war when the field marshal commanded the Soviet Sixth Corps on the southern front. Malinovsky received two decorations for bravery and became a close friend of Joseph Stalin during the war. That friendship eventually led to his selection as defense minister in 1957, trumping more senior officers, including Admiral Gorshkov.\n\n\"To your health, Comrade,\" Rybalko said.\n\n\"And to yours,\" Malinovsky said.\n\nThe defense minister guided Rybalko to a seat, whereupon he proceeded to reminisce about their escapades during the war. Rybalko survived that time partly by fate and partly by luck. He recalled the siege of Leningrad in 1943, when his submarine sent torpedoes into the sides of two Nazi troop ships before they unloaded reinforcements. While other boats suffered from mechanical failures and personnel issues, Rybalko's luck steered him clear of those sandbars.\n\nThe two shared a few laughs, then Malinovsky's smile faded. His large eyes narrowed. \"I'm not going to sugarcoat this, Leonid. What we're going to discuss today could change the balance of world power. Based on our actions over the next few months, the outcome could go either way.\"\n\n\"I see,\" Rybalko said, though he really didn't. An orderly brought a tray with two cups of bitter tea and handed one to Rybalko.\n\nAs Rybalko sipped his tea, Fleet Admiral of the Soviet Union Sergei Gorshkov burst through the door and strutted toward him. Gorshkov's round red cheeks and down-turned mouth made him look permanently angry. With Admiral Vitali Fokin in tow, Gorshkov's short legs carried his stocky frame across the room at a fast clip. He pulled up a chair and sat. Admiral Fokin did the same.\n\nFamous for his direct style, Gorshkov hurled a question at Rybalko. \"Have you heard of Operations Kama and Anadyr?\"\n\nRybalko recalled hearing rumors but nothing more. \"Yes, sir, I've heard the names but not the details.\"\n\nGorshkov leaned back in his seat. A slight smile played on his lips, as though he were about to impart gossip to his grade-school buddies on a playground. \"As you know, on May 12, Premier Nikita Khrushchev finalized his decision to deploy strategic weapons to Cuba under the cover of a humanitarian aid program.\"\n\nRybalko said nothing.\n\nGorshkov continued. \"After the first Soviet delegation visited Havana later that month to consult with the Cubans, and Fidel Castro agreed to the plan, the Soviet General Staff Directive devised Operations Kama and Anadyr.\"\n\nWhen Gorshkov took a breath, Admiral Fokin said, \"The name Anadyr came from Stalin's plan to attack Alaska in the fifties with a million-man army. Obviously, he never executed the plan, so we took the name.\"\n\nGorshkov sneered at Fokin for the interruption, then said, \"On July 10, General Issa Pliyev, our Cuban forces commander, along with his staff, flew from Moscow to Havana on a transport plane. They were disguised as engineers and agricultural experts offering humanitarian aid. In July the Maria Ulyanova became the first of eighty-five cargo ships bound for Cuban ports. Do you know what these ships carried in their holds?\"\n\nRybalko did know, but he again feigned ignorance. \"I have heard speculations, sir, but no confirmations.\"\n\nGorshkov's eyes lit up. \"Long-range nuclear missiles. On that day Operation Anadyr began. Now the world will never be the same.\"\n\nInside, Rybalko shuddered. Outside, he remained stoic. His patriotism and love for his Rodina ran deep, but his respect for some of his country's leaders often waned. This was especially true when it came to the premier. Party First Secretary Khrushchev had insisted on nosing his way into the navy's postwar naval construction programs. He ordered Gorshkov to dismantle all large ships, claiming that these behemoths were \"good only for carrying heads of state on official business.\" Now, with a potential conflict brewing near Cuba, the navy could not even muster two cruisers. Plagued by reactor problems, the long-promised fleet of nuclear submarines remained nothing more than a pipe dream. The party's Central Committee could not find enough raw materials to build much more than a rowboat, so the Soviet Union found itself staring at the backside of American ingenuity and production. If Khrushchev's Anadyr were indeed destined to change the balance of power, it would have to include a way to create resources from thin air.\n\nMinister Malinovsky leaned forward in his chair. \"Here's where you play a key role, Leonid.\"\n\nRybalko held his chin steady. \"What's my assignment?\"\n\n\"You will lead Project Kama,\" Gorshkov said.\n\nRybalko recognized the title of the river that ran from Siberia to the Volga, but he'd heard almost nothing about the operation bearing the river's name.\n\n\"Kama is the naval segment of Operation Anadyr,\" Fokin inserted. \"This plan calls for the permanent relocation of the seven missile submarines of the Eighteenth Division from Polyarny to Mariel, Cuba. Accompanying those submarines will be two Project 68 Chapayev-class gun cruisers, two squadrons of mine warfare craft, and two missile destroyers.\"\n\n\"There's more,\" Gorshkov said, again displaying agitation at Fokin's interruption. \"Four Project 641 diesel boats from the Sixty-ninth Brigade will also transit undercover to Cuba, but these boats will carry special weapons.\"\n\n\"Special weapons?\" Rybalko asked.\n\n\"Very special,\" Fokin said grimly.\n\n\"Each submarine,\" Gorshkov said, \"will be issued one nuclear-tipped torpedo.\"\n\nRybalko's eyes opened wide. \"Nuclear? But...our 641 boats aren't trained for such weapons.\"\n\nGorshkov waved a hand dismissively. \"Captain Shumkov of B-130 earned the Order of Lenin award for firing two live nuclear torpedoes near Novaya Zemlya last year. That should be sufficient.\"\n\n\"That's true,\" Rybalko said, \"but these torpedoes have a sixteen-kilometer kill zone. Getting close enough to hit an American ship could put our submarines at great risk.\"\n\nGorshkov remained silent for a moment, then drew his lips tight and said, \"Hopefully, your boat commanders will never need to fire one. In the event they are forced into a corner, they will be guided by clear rules of engagement. Is that understood?\"\n\nReluctantly, Rybalko nodded. \"Understood.\"\n\nFokin piped up again. \"Your submarines will transmit position reports daily at midnight Moscow time using their SBD high-frequency transmitters. We will broadcast updates in parallel using low-frequency and high-frequency single sideband. To receive these broadcasts, one boat must remain near the surface to monitor the HF band.\"\n\n\"That will make them vulnerable to American ASW forces,\" Rybalko said. He also knew that the new \"burst\" transmission radios, dubbed SBD, for ultra rapid activity, were not very reliable. Due to natural and manmade interference, including a new jamming signal used by the British near their coastline, Soviet boats often needed to stay near the surface and transmit dozens of times to ensure receipt by Moscow.\n\n\"We appreciate the dangers,\" Fokin said, \"but the mission's importance takes precedence.\"\n\n\"We have limited acoustic and sea condition knowledge for the Sargasso Sea,\" Rybalko said. \"We're also not certain how effective the American hydroacoustic array is now, and avoiding enemy ASW aircraft may be difficult. Also, the warmer tropical waters could cause living conditions in these boats to become unbearable.\"\n\n\"No one said this mission would be easy,\" Gorshkov said. \"That's why we selected you to lead the charge.\"\n\nRybalko wanted to voice further concerns, including the possibility that firing a nuclear torpedo at an American ship could cause World War III, but he realized that his admonitions would be lost on deaf ears.\n\nAdmiral Fokin offered further instructions, including details about store loads, crew preparation, and the planned departure time. The four then rose, shook hands, and departed. As he left the Ministry of Defense building, Rybalko thought about his wife, Galena, and his mother, Natasha, who lived with his sisters and their families north of Moscow in a small village called Klin. For a fleeting moment, he pictured their pained and twisted faces as they turned to ashes in the fiery center of a mushroom cloud.\n\nON AUGUST 17, 1962, ON BOARD the spy ship USS Oxford (AG-159), an R-Brancher heard something strange, not too unlike the faint sound of tires screeching in a parking lot. Instantly, he recognized the electronic chirp of a Soviet radar code-named Whiff. The R-Brancher informed the officer in charge, and the OIC radioed Net Control, which sent a CRITIC (critical) message to the National Security Agency. Russian-speaking I-Branchers assigned to the A Group Soviet signals intelligence desk at NSA headquarters in Mary land ran down hallways and out doors. Within minutes they reported to the office of the operations chief, Major General John Davis. Most were ordered to assist the B Group Spanish linguists listening to intercepted traffic coming from Cuba. Previously, all transmissions from the island came from Cubans speaking Spanish. Within the past week, however, much of that banter changed to Russian\u2014or Spanish spoken with a heavy Russian accent. In response to this unprecedented change, the NSA set up its first around-the-clock SIGINT command center, establishing the foundation for the National Security Operations Center (NSOC).\n\nWhile Russian-speaking I-Branchers at NSA strapped on headphones and listened, officials in Washington, D.C., hurried to meetings. CIA Director John McCone insisted that the detection of Whiff radar signals and other collected data supported only one conclusion: the Soviets were installing offensive ballistic missiles in Cuba, possibly even nuclear. Secretary of Defense Robert McNamara and Secretary of State Dean Rusk dismissed McCone's \"Chicken Little sky is falling\" concerns, believing the military buildup to be only defensive. Still, under direct orders from President John Kennedy after he received the news, the NSA established FUNNEL as the new top-secret code word restricting access to information related to Cuban SIGINT\u2014especially anything containing evidence of Soviet offensive weapons.\n\nThousands of miles away, in the silent cold of the Arctic Ocean north of Russia, the USS Nautilus crept along at three knots. Her periscope peaked above the icy sea near the remote island of Novaya Zemlya. Thirteen miles from ground zero, T-Brancher \"spook\" John Arnold, a communications technician chief, waited for a nuclear explosion. Arnold knew that the Soviets had detonated a fifty-eight-megaton bomb\u2014the largest thermonuclear beast extant\u2014at this very location the previous year, and a Foxtrot-class submarine also shot two live nuclear torpedoes into the harbor around that same time. In fact, the Soviets had conducted so many nuclear tests near Novaya Zemlya that they took to calling the island Black Harbor.\n\nA seasoned submariner, Arnold had previously served aboard the USS Scorpion (SS-278), a diesel boat that almost collided with a Soviet November-class nuclear submarine. When he received orders to report to the Nautilus for a special mission, he envisioned a technically advanced underwater marvel. He soon found that low tech still ruled the day when he learned that an ordinary cardboard toilet paper roll played a critical part in conducting periscope photographic intelligence. The crew placed the lens of a Canon camera on one end of the roll, with the other end fitted to the periscope's eyepiece. The jury-rigged setup remained in place with a double helping of black electrician's tape.\n\nWhile on station at Black Harbor, Arnold witnessed more than a dozen spectacular explosions through the periscope in which the Soviets filled the sky with crimson mushroom clouds. During each test detonation, while the Nautilus rocked back and forth, bright flashes could be seen through the toilet paper roll, despite the heavy coating of tape. Sonic booms clapped in ears, and fluorescent lights shattered.\n\nSeveral weeks later, Arnold transferred to a spy ship operating just off the coast of Cuba, where he strained to hear the signals emanating from nearby Soviet radar and missile guidance systems. He and other spooks monitored signals from SA-2 surface-to-air missile (SAM) and other conventional weapons platforms brought to the island by the Soviets on merchant ships. At 2:00 A.M. on the morning of September 15, 1962, Arnold detected something that made the hairs on his neck stand at attention. He checked and double-checked his readings. Without a doubt he was listening to the tone of a Soviet Spoon Rest radar system, indicating that the Soviets had completed the construction of the SA-2 missile platforms. These conventional SAM sites were now fully operational, and from now on, any U.S. aircraft flying over Cuban airspace could be shot down within seconds.\n\nOther R-Branchers located at a Huff Duff high-frequency direction-finding station in Homestead, Florida, and on board the spy ship USS Oxford operating in Cuban waters also heard the signals and multiangulated the source. They estimated the location of the SA-2 battery as three miles west of Mariel. The navy ordered the Oxford to move in closer and gave her a new set of orders: start listening for signals that indicated the presence of nuclear missiles in Cuba.\n\nIn August 1962, CIA Director McCone advised President Kennedy about the Soviet SA-2 conventional SAM batteries in Cuba and the possibility that nuclear missiles might be present, though they were yet to be verified. Kennedy sanctioned a U-2 spy plane flight over the island, which confirmed eight conventional SAM sites. He voiced a strong protest to Khrushchev about the sites and further warned that the United States would not tolerate nuclear missiles in Cuba. The Soviet premier denied any such intentions, claiming that only a few conventional weapons and \"agricultural equipment\" would be shipped to the island.\n\nBy early September 1962, dozens of Soviet ships had delivered spare parts and munitions to Cuba. Secretly, these ships also unloaded several Komar-class missile-firing patrol boats designed to thwart amphibious landings, which Gorshkov warned could happen within weeks if the Americans discovered nuclear missiles in their backyard. Already the United States had increased surveillance flights and eyed Soviet merchant ships suspiciously. Two of those ships, the Indigirki and Aleksandrovsk, departed Severomorsk and carried a cargo of nuclear missiles into Cuba's Mariel harbor eighteen days later. The Aleksandrovsk transported fourteen warheads, which would later be married to R-14 missiles after they arrived on another ship. Each missile could hit targets as far away as San Francisco, California, and packed more than sixty times the destructive force that leveled Hiroshima.\n\nThe first indication that Khrushchev might be lying about sending nuclear arms to Cuba came on September 18 when, off the coast of Tunis in the Mediterranean, a U.S. Navy frigate confronted a Russian merchant ship and inquired about its cargo. The ship reported that she was carrying only agricultural machinery. Binoculars aboard the frigate indicated otherwise, as the deck was covered with large crates of irregular sizes, which appeared to be the kind that carried disassembled military aircraft.\n\nBy the third week of September, U.S. warships and aircraft were intently watching thirty-five Russian merchant ships en route to Cuba. All told, the United States counted 129 ships leaving Russian ports and 94 arriving at Mariel. Due to frequent overflights by U.S. surveillance planes, Soviet personnel on the ground in Cuba worked only at night, unloading ships and assembling missile silos. The Cubans nicknamed their new allies \"night crawlers.\" Though the U.S. government suspected foul play, it had no proof. At least not yet. Fearing the worst, the U.S. Navy planned for a potential future blockade of Cuban waters. Naval aircraft and \"tin can\" destroyers increased patrols, and personnel were put on high alert. Suspecting that Gorshkov would not send so many merchant ships through the Sargasso Sea unprotected, the navy issued instructions to search for possible Soviet submarines. Those orders were also given to every Huff Duff station within range of the Atlantic.\n\nON SEPTEMBER 30, ABOARD THE SOVIET diesel submarine B-36 harbored in Sayda Bay, Captain Second Rank Aleksei Dubivko examined a suspicious bundle of ocean charts. They lay against one corner of the chart room, a small, highly classified enclosed area in the port front corner of the control center that only a few on board were allowed to enter. The fleet headquarters duty officer had brought the charts on board a few days earlier. The large stack of nautical maps covered the Caribbean and North Atlantic seas. One chart provided channel approaches to enter several Cuban ports, including Mariel, a small harbor west of Havana. No more than a few seconds elapsed before Dubivko added up the clues\u2014including a recent overload of stores\u2014and guessed where they might be headed. Why they were being sent on the longest deployment ever made by Soviet submarines remained a mystery. Those details would be revealed only after they submerged in the Barents Sea and opened their sealed orders.\n\nDubivko had no doubt that, regardless of where they were going, he and his crew would execute those orders efficiently. An aggressive commander in his early thirties, he demanded top performance from his seventy-eight officers and men. His motivation to achieve perfection often led to top grades for operational and engineering tests. Captain Nikolai Shumkov, commander of the submarine B-130, was the only peer who had ever bested him in a competition, and only in the weapons department. And this was because B-130 was the only boat in their group to have fired live nuclear torpedoes into Black Harbor one year earlier. Although Dubivko had always longed for that opportunity, he hoped that the need to fire a nuke on this mission would never occur.\n\nAfter graduating from the Vladivostok Higher Naval School, Ukrainian-born Dubivko originally served on board a \"skimmer\" surface ship. He later transferred to the submarine fleet, and as a senior lieutenant, accepted command of his first boat out of Gorky on the Volga in 1953. Dubivko learned a great deal under the leadership of Fleet Commander Admiral Chebanenko and Commander of Submarine Forces Vice Admiral Orel by participating in scores of exercises held by the Northern Fleet. These maneuvers, as a rule, were conducted in the Norwegian and Greenland seas and the northern part of the Atlantic so boat skippers could polish submarine tactics and antisubmarine defenses. Dubivko's drive earned him considerable recognition and opened the door in 1960 to his selection as commander of B-36, a new Project 641 boat fresh out of Leningrad's construction halls.\n\nDubivko and his crew turned the key on B-36 in 1961 and ran her around the track on sea trials. A year and a dozen runs after that, he received his orders to transfer the boat to Sayda Bay. There, on a cold September afternoon, he stood on the bridge of his award-winning race car and watched two officers from the weapons facility walk across the gangway. One officer, adorned with the Northern Fleet headquarters staff emblem, clutched a briefcase in his right hand. B-36's topside watch clicked his heels, saluted, and pointed at Dubivko high up in the sail. The visiting officers nodded and walked toward the side hatch. They undogged the hatch and entered the confined space that led to the bridge. Dubivko looked down and watched the staff officer, with the second officer following, swing onto the lower rung of the ladder leading up to the bridge platform.\n\nThe heavyset staff officer scrunched his broad shoulders, stepped from the ladder, and squeezed onto the bridge. He told Dubivko that he was the Northern Fleet special weapons directorate. He turned and waved a hand at the baby-faced man behind him and introduced Alexander Pomilyev, who he said was a lieutenant assigned to B-36 as a special weapons expert. He cracked opened his briefcase, removed a copy of the lieutenant's orders, and handed them to Dubivko, who read the papers and asked for a definition of \"special weapon.\"\n\nThe weapons directorate said Dubivko would find out soon enough.\n\nAfter the directorate left the bridge, Dubivko stared at Pomilyev and asked, \"What do you know about this special weapon?\"\n\n\"Everything,\" Pomilyev said.\n\n\"Is it nuclear?\"\n\n\"I'm not at liberty to say.\"\n\nDubivko rubbed his chin. \"Am I delivering this to our mission destination?\"\n\n\"I'm not at liberty to say.\"\n\n\"What are you at liberty to say?\"\n\nPomilyev's face softened. \"Nothing, I'm sorry. I understand that you'll be receiving more information about this at the briefing, and detailed instructions are included in your sealed orders. All I can tell you is that the weapon must be stored as a service-ready torpedo in the forward compartment and loaded into the number two tube once we've crossed the Iceland gap area.\"\n\nDubivko frowned. \"You mean after we're in waters patrolled by enemy forces.\"\n\nPomilyev nodded.\n\nDubivko narrowed his eyes. \"Are we going to war with the Americans?\"\n\n\"I don't know,\" Pomilyev said.\n\n\"Are you qualified in submarines?\" Dubivko asked.\n\n\"No, sir, but I'm a fast learner and will study under way.\"\n\n\"You're damn straight you will,\" Dubivko said as he turned to watch the weapons directorate cross the gangway and waddle toward the pier. Still staring at the directorate, Dubivko said, \"That will be all, Lieutenant. Report to First Officer Kopeikin for your berthing and watch assignments.\"\n\nAs Pomilyev left the bridge, Dubivko desperately wanted to see his wife and children and tell them good-bye before he left on the most important mission of his career.\n\nSTANDING ON THE DECK OF HIS submarine, staring at a strange-looking torpedo, Captain First Rank Ryurik Ketov flipped up the collar on the back of his navy blue overcoat to shield his neck from the cold. A fading September sun coated the waters of Sayda Bay and reflected remnants of orange and yellow from the sides of a floating crane. The crane hovered over Ketov's boat and lowered a purple-tipped torpedo through the loading hatch. Within minutes the long cylinder disappeared into the forward torpedo room. Blowing into his gloved hands to keep his nose warm, Ketov glanced at the submarine's conning tower. Three large white numbers were painted on the side, but Ketov knew this label held no meaning, except to serve as a numerical decoy for enemy eyes. The boat's real designation was B4\u2014B as in Bolshoi, which means \"large.\"\n\nThe handsome, blue-eyed Ketov inherited his B-4 Project 641 submarine\u2014known as a Foxtrot class by NATO forces\u2014from his former commander, who was a drunk. Tradition dictated that submarine captains who were too inebriated to drive their boats into port should lie below until they sobered up. First officers took charge and positioned a broomstick on the bridge in their captain's stead. Atop the handle they placed the CO's cap so that admirals on shore peering through binoculars would raise no eyebrows. Ketov stood watch with a broom more times than he could recall. He didn't dislike vodka, nor did he disapprove of his CO's desire to partake, but Ketov felt that a man must know his limits and learn to steer clear of such rocks when under way. He demanded no less of his crew. Unfortunately, as his appointment to commander required the approval of the dozen sub skippers in his group, and all of them drank like dolphins, Ketov's stance on alcohol held him back for a year when he came up for promotion.\n\nThe Soviet navy formed the sixty-ninth Brigade of Project 641 submarines in the summer of 1962. Ketov and his comrade captains were ordered to prepare for an extended deployment, which they suspected might be to Africa or Cuba. Some wives, filled with excitement, anticipated a permanent transfer to a warm locale.\n\nThe four subs arrived in Gadzhiyevo at Sayda Bay a month earlier and were incorporated into the Twentieth Submarine Squadron along with the seven missile boats. Vice Admiral Rybalko assumed command of the squadron, and over the next thirty days, each boat was loaded with huge quantities of fuel and stores.\n\nNow, aboard B-4, Captain Ketov coughed into the wind and turned to stare at the weapons security officer. Perched near the crane, the man shouted orders and waved long arms at the fitful dockworkers. The officer's blue coveralls and pilotka \"piss cutter\" cap signified that he belonged to the community of submariners, but Ketov knew better. The shape of a sidearm bulged from under the man's tunic, and his awkwardness around the boat made it obvious that he was not a qualified submariner.\n\nKetov also knew that the security officer came from Moscow with orders to help load, and then guard, the special weapon. Although he'd not yet been briefed about the weapon, Ketov figured this torpedo with the purple-painted nose, which stood in sharp contrast against the other gray torpedoes on board, would probably send a radiation Geiger counter into a ticking frenzy.\n\nKetov looked down at the oily water that slapped against the side of his boat. Attached by long steel cables, three sister boats of the Soviet Red Banner Northern Fleet floated nearby. If one approached these late-model attack subs from the front, their jet-black hulls, upward-sloping decks, and wide conning towers with two rows of Plexiglas windows might look menacing. The silver shimmer of their sonar panels, running across the bow like wide strips of duct tape, might appear odd. The reflective panels of the passive acoustic antenna, jutting from the deck near the bow, might look borrowed from the set of a science-fiction movie. But the seasoned sailors on the decks of these workhorses were unmistakably Russian, and undeniably submariners.\n\nKetov strutted across the wooden brow that connected B-4 to the pier. Two guards, with AK-47 assault rifles slung on their shoulders, snapped to and saluted. Ice crunched under his boots as he walked toward a small shed less than a hundred meters away. Captain Second Rank Aleksei Dubivko, commander of B-36, matched his stride and let out a baritone grunt.\n\n\"Did they give you one of those purple-nosed torpedoes?\"\n\n\"Yes,\" Ketov answered, \"they did.\"\n\nAlthough the round-faced commander was about Ketov's height of five foot seven, Dubivko's stocky frame stretched at the stitches of his overcoat. He let out another grunt and said, \"Why are they giving us nuclear-tipped weapons? Are we starting a war?\"\n\n\"Maybe,\" Ketov said. \"Or maybe we're preventing one.\"\n\nDubivko's boots clicked on the ice as he hurried to keep up with Ketov. \"We haven't even tested these weapons. We haven't trained our crews. They have fifteen-megaton warheads.\"\n\n\"So?\"\n\n\"So if we use them, we'll wipe out everything within a sixteen-kilometer radius. Including ourselves.\"\n\nKetov neared the door of the shed and stopped to face Dubivko. \"Then let's hope we never have to use them.\"\n\nDubivko let out a low growl and followed Ketov into the shack.\n\nInside, Captain First Rank Nikolai Shumkov, commander of submarine B-130, stood by the door. Only a few stress lines underscored his brown eyes and marked his boyish features. Next to Shumkov, Captain Second Rank Vitali Savitsky, commander of B-59, appeared tired and bored. None of them had slept much since their trip from Polyarny to Sayda Bay.\n\nThe tiny shed, once used for storage, offered no windows. A single dim bulb hung from the ceiling and cast eerie shadows inside. Someone had nailed the Order of Ushakov Submarine Squadron flag on one wall. The unevenly placed red banner, fringed in gold and smeared with water stains, appeared as if hung by a child in a hurry. In one corner sat a small stove that flickered with yellow sparks but offered little warmth. The air smelled of burnt coal.\n\nOne metal table graced the center of the room, where the squadron commander, Leonid Rybalko, sat with his arms crossed. Ketov noticed that the vice admiral shivered, despite being bundled in a dark navy greatcoat and wool senior officers' mushanka cap. The tall, broad-shouldered Rybalko had a reputation for analytical brilliance and a smooth, engaging wit. A dedicated performer, Rybalko exuded the confidence and mastery of a seasoned leader.\n\nTo the side and behind Rybalko, the deputy supreme commander of the Navy Fleet, Admiral Vitali Fokin, fidgeted with his watch. Thin and lofty, Fokin kept his back straight. Ketov deduced that Fokin, given his close relationship with Fleet Admiral Sergei Gorshkov, held the reins of what ever mission they were about to undertake. A slew of other officers filled the room, including Anatoly Rossokho, the two-star vice admiral chief of staff. Ketov suspected that Rossokho was here to define their rules of engagement about using the special nuclear torpedoes.\n\nVice Admiral Rybalko motioned for everyone to find a seat. He coughed and brought a handkerchief to his lips to spit out a clump of mucus. His face looked pale and sickly. He locked his eyes on each submarine commander one at a time. When he looked at Ketov, those few moments seemed like days.\n\n\"Good morning, Commanders,\" Rybalko said. \"Today is an important day. I'm not going to discuss mission details, as we've included those in your sealed briefings, which you will open under way. So instead we will focus on other aspects of your mission.\"\n\nMetal clanked as an attendant creaked open the front panel on the hot stove and dumped in another can of coal pellets.\n\nRybalko continued. \"I'm sure you all know Admiral Fokin. He asked me to emphasize that each of you has been entrusted with the highest responsibility imaginable. Your actions and decisions on this mission could start or prevent a world war. The four of you have been given the means with which to impose substantial harm upon the enemy. Discretion must be used. Fortunately, our intelligence sources report that American antisubmarine warfare activity should be light during your transit.\"\n\nKetov hoped that the ASW intelligence report was correct but feared that optimism probably overruled reality. He glanced at the other sub commanders. Dubivko and Shumkov wore excited smiles. Savitsky, who'd earned the nickname \"Sweat Stains\" because he was always perspiring about something, wrinkled his brow. Ketov, who received the title of \"Comrade Cautious,\" shared Savitsky's angst. As adventurous as this might seem to Dubivko and Shumkov, Ketov knew Project 641 submarines were not designed for extended runs into hot tropical waters and had no business carrying nuclear torpedoes.\n\nRybalko imparted more information, concluded his speech, and asked if anyone had questions.\n\nKetov raised a hand. \"I do, Comrade Admiral. I understand that our sealed orders provide mission details, but we share concerns about our rules of engagement and the special weapon. When should we use it?\"\n\nVice Admiral Rossokho broke in. \"Comrade Commanders, you will enter the following instructions into your logs when you return to your submarines: Use of the special weapons is authorized only for these three situations\u2014One, you are depth charged, and your pressure hull is ruptured. Two, you surface, and enemy fire ruptures your pressure hull. Three, upon receipt of explicit orders from Moscow.\"\n\nThere were no further questions.\n\nAfter the meeting, Ketov followed the group out into the cold. A witch's moon clung to the black sky and hid behind a dense fog that touched the ground with icy fingers. Ketov reached into his coat pocket and took out a cigarette. Dubivko, standing nearby, held up a lighter. Ketov bent down to accept the flame. Captains Shumkov and Savitsky also lit smokes as they shivered in the dark.\n\nBetween puffs, Ketov posed the first question to Captain Savitsky. \"How are your diesels holding up?\"\n\nSavitsky cringed. \"No problems yet, but I'm still worried about what might happen after they've been run hard for weeks. If they fail on this mission...\" Savitsky's voice trailed off as he shook his head.\n\nKetov knew that shipyard workers had discovered flaws in B-130's diesel engines during the boat's construction. The shipyard dismissed the hairline cracks as negligible, and Savitsky did not press the issue, as to do so would have resulted in his sub's removal from the mission. Still, he fretted endlessly about the consequences.\n\nSensing his friend's distress, Ketov changed the subject. \"Have you seen those ridiculous khaki trousers they delivered?\"\n\n\"I'm not wearing those,\" Savitsky said.\n\n\"I wouldn't either,\" Shumkov said, \"if I had your skinny duck legs.\"\n\nSavitsky snorted and threw his head back. \"I'd like to see how you look in those shorts, Comrade Flabby Ass.\"\n\n\"Right now,\" Dubivko said as he pulled his coat tighter, \"I'd rather look like a duck in shorts than a penguin in an overcoat.\"\n\nKetov smiled and shook his head. \"I'm going back to my boat, try on those silly shorts, and have a long laugh and a can of caviar.\"\n\n\"And maybe some vodka?\" Shumkov said.\n\n\"I wish,\" Ketov said. \"We cast lines at midnight.\"\n\nShumkov nodded and said nothing.\n\nSavitsky raised his chin toward Ketov. \"Do you think we're coming back or staying there permanently?\"\n\nKetov shrugged. \"All I know is that we can't wear those stupid shorts in this weather.\"\n\nBack on board B-4, Captain Ketov sat on the bunk in his cabin and stroked the soft fur of the boat's cat. \"It's time to go, Pasha.\"\n\nOver the past year, the calico had become a close member of B-4's family. Like many Russian submarines, B-4 enlisted the services of felines to hunt down rats that managed to find their way on board, usually by way of one of the shorelines. Boats often carried at least one or two cats on board, and the furry creatures spent their entire lives roaming the decks in search of snacks and curling up next to sailors on bunks. Unfortunately, for reasons unknown, headquarters decreed that cats were forbidden on this journey. Given no choice, Ketov found a good home for Pasha with a friend who could care for her and keep her safe.\n\nAs Pasha purred by his side, Ketov reached for a can of tuna. \"The least I can do is give you a nice snack before we leave.\"\n\nKetov thought about his mother, still living in the rural Siberian village of Kurgan. She'd lost her husband to one war; would she now sacrifice her first born son? When Ketov was thirteen, his father, who was an accountant with bad eyesight, was forced to fight in the battle at Leningrad. He was killed in his first engagement. Ketov became the man of the house and helped support his younger siblings and his mother, who earned a meager teacher's salary. He could still not explain why, but the day he turned eighteen, one year after the war ended, he took the train to Moscow and enrolled in the naval college. He also had no explanation for why he'd jumped at the chance to serve aboard submarines. He only knew that, despite the sacrifices and often miserable conditions on the boats, no other life could fulfill him like the one under the sea.\n\nA few minutes past midnight on October 1, 1962, Captain Ketov stood on the bridge of B-4 and watched Captain Savitsky cast off lines and guide B-59 away from the pier using her quiet electric motors. Captain Vasily Arkhipov, the brigade's chief of staff, stood next to Savitsky in the small cockpit up in the conning tower. A flurry of snow mingled with the fog and dusted the boat's black hull with streaks of white. Thirty minutes later, B-36, commanded by Dubivko, followed in the wake of her sister sub and disappeared into the darkness of the bay. After another thirty minutes, Shumkov, in B-130, followed by Ketov in B-4, maneuvered away from the pier. Ketov stared into the blackness as the three subs ahead of him, all with running lights off, vanished into the night. Then he heard the low rumble of B-59's diesel engines, signaling that Savitsky had cleared the channel and commenced one of the most important missions undertaken by the Russian navy since World War II.\n\nA SHIVER OF EXCITEMENT RAN DOWN his spine as Captain Dubivko stared at the radar repeater in the bridge of B-36. Though he could not see the other boats in the dark, he knew that he was second in line with B-59 ahead, B-130 behind, and B-4 at the rear. The narrow space, high up in the boat's sail, enclosed all but the top part of the bridge with a blanket of steel. The brisk, cold air swirling through the space bit at Dubivko's hands and nose. One lookout, a few feet above and behind him, stood on the bridge platform with binoculars planted against his frozen face. Sea foam dotted the deck as the submarine's bow carved through the Barents Sea.\n\nDubivko looked down and to his right. He studied the green glow of the radar repeater screen that painted a picture of the channel. The dense fog lowered visibility through the scratched Plexiglas bridge windows to no more than a few meters. The radar, despite its eighty-kilometer range limitation, offered the best means to avoid running aground or into the other three boats. Metal clanked and groaned as Arkadyi Kopeikin, the boat's starpom first officer, scrambled up the ladder to the bridge.\n\nKopeikin settled in next to Dubivko, remained silent for a long moment, then said, \"We're ready to dive, sir.\"\n\nNot speaking, Dubivko offered a short nod.\n\n\"The charts are spread on the nav table, and the officers are anxious,\" Kopeikin said.\n\nDubivko let the hint of a smile play on his lips. \"Let's take her down, Comrade. Then we can open our orders.\"\n\n\"Yes, Captain.\" Kopeikin bent down and yelled into the voice tube, \"All hands clear topside, prepare to dive the boat.\" He then dropped through the hatch leading to the conning tower.\n\nDubivko imagined a scurry of feet and voices below in the command center as the crew prepared for entry into the silent world of moaning whales. Blue-green ocean shot through the limber hole vents. He took one last scan of the horizon, pulled the smell of salt and sea into his lungs, then yelled down to the watch officer. \"Pogruganye!\"\n\nThe submarine angled down at the bow as the ballast tanks flooded, forcing air to bubble and hiss through the limber holes. The lookout secured his binoculars, scrambled down the platform ladder, and disappeared through the bridge hatch. Dubivko captured a final glimpse of the blue sky, shimmied down the ladder, and with a loud clank, pulled the spring-loaded hatch shut above him. He slid down past the conning tower periscopes and entered the CC in Compartment Three. He removed his kanadka and handed the furlined coat to a michman (warrant officer) as he entered. In the dim blue light, surrounding the two cylindrical periscope housings in the center of the room, more than a dozen men manned various stations, including helm and diving control, radar, navigation, ballast tank operations, and torpedo fire control. The crowded compartment offered the dank smell of hydraulic fluid, diesel fumes, and something that could only be described as mechanical.\n\nThe low thrum of the diesel engines ceased as the boat angled downward and switched to battery power. Dubivko pictured B-36 in his head, from her silhouette to her soul, her long black hull, systems, schematics, and statistics, her crew, capabilities, and compartments. He saw her as more than just a tube made from steel and sweat. More than a tube full of engines, motors, batteries, bunks, and torpedoes. Much more than a scattering of pipes, valves, switches, and gauges. He saw her as a woman imbued with a loyal heart and stunning beauty. And like any woman, she deserved a great deal of patience and care.\n\nDubivko shot an order to the duty officer, who relayed the same to the helmsman. \"Right full rudder, come to course two-five-five, speed nine knots.\"\n\nThe helmsman, seated on a small bench in the front right corner of the CC, acknowledged the order and pulled a black knob to the right. The navigator, Captain Lieutenant Sergei Naumov, from the navigation table on the port side near the center of the CC, sounded off the next course change as the boat leveled off at one hundred meters.\n\nDubivko turned over CC command to the electronics officer, Yuri Zhukov, and took a half-dozen steps toward the nav table. There he met First Officer Kopeikin, as well as Navigator Naumov and the political officer, Captain Third Rank V. G. Saparov. The three men were positioned around the small rectangular table. Five times the typical number of chart cases filled the small corner near the curved white bulkhead. The extra cases were brought on board to maintain complete secrecy about their final destination\u2014despite the many rumors and clues that alluded to Cuba.\n\nDubivko's eyes found Kopeikin's. They had shared many missions together on this boat, and while most of them began just like this one, few were similar. \"Comrade Kopeikin, you may open the safe.\"\n\nKopeikin reached toward the bulkhead safe and dialed. The square box clicked open, and the first officer removed a large manila envelope. He handed it to Dubivko. Wide red \"top secret\" stripes ran diagonally across the package. Dubivko opened the outer seal, then the inner one. He removed a booklet, examined the cover, flipped to the first page, and read. The single word Kama headlined the top of the page, followed by small type outlining their mission orders.\n\nDubivko read in a low voice, so as not to be overheard by others in the CC. \"Operation Kama tasks the submariners with performing reconnaissance of all seaward approaches to Mariel, Cuba. Acoustic area conditions are to be logged for port entry in preparation for seven ballistic missile submarines.\"\n\nDubivko took a breath and glanced at his senior officers. He could tell by the look in their eyes that they shared his anticipation. He read further about the additional gun cruisers and destroyers joining them in Mariel and concluded with \"This internationalist intervention mission by the Soviet Socialist Republic is designed to equip the Socialist Republic of Cuba with sufficient resources and support to undermine further Western aggression. Our brigade is tasked with a special mission for the Soviet Union, which includes transiting the Atlantic in secret to a new home port in an allied country. This transit must remain undetected by enemy forces, and the submariners must arrive in Mariel, Cuba, by October 20.\"\n\nDubivko pondered the expectations implied in his orders. Reaching Cuba by October 20 without being detected was practically impossible. They'd have to run near their top speed submerged, snorkel at night, and by sheer luck try to avoid American sea-mounted sonar arrays and ASW ships and planes. Dubivko held his doubts and anger in check and removed another envelope. As he read, the officers standing near the nav table leaned in closer. \"These are the rules of engagement for the use of weapons. One, while in transit, all weapons will remain in combat-ready condition. Two, conventional weapons will be used as directed by the Main Navy Staff. Three, the use of nuclear torpedoes is allowed only as directed by the Ministry of Defense of the Main Navy Staff.\"\n\nSilence descended on the tiny cubicle.\n\nDubivko cleared his throat and said, \"We have our orders. Let's make the appropriate preparations. I will tell the crew what they need to know.\"\n\nDubivko left the cubicle and walked to his stateroom. His head spun as he lay in his bunk and digested the information he'd just read. Their mandated arrival date translated into a fast and dangerous transit time. To maintain high speed, they'd have to run on the surface for as long as possible and hope that Mother Nature sent storms to shroud them under dense clouds. After they neared Cuba, where they'd be required to stay submerged, they needed to remain near the surface every day at midnight Moscow time to receive and acknowledge transmissions. Midnight in Moscow equated to late afternoon in the Caribbean, and that meant risking exposure during daylight hours.\n\nEach boat carried a special OSNAZ group of nine young men trained in signals intelligence\u2014similar to \"spooks\" in the American navy. OSNAZ was short for osobennogo naznachneniya, which in Russian means \"specialized designation.\" Five of these were English-speaking communications experts tasked with monitoring HF and UHF bands to determine where the Americans might be concentrating their ASW efforts. Each captain needed this intelligence not only to remain undetected, but also to determine if, why, and when they should fire their nuclear torpedoes at the enemy. Dubivko thought about his family back in Russia. By now they had been informed about the mission and were preparing to join him after B-36 arrived in Cuba. He thought of the special weapon resting in a torpedo tube only fifty feet away from his stateroom and wondered if his boat would be vaporized long before he ever reached the distant tropical island.\n\n## CHAPTER SIX\n\nThe ability to get to the verge without getting into the war is the necessary art.... If you try to run away from it, if you are scared to go to the brink, you are lost.\n\n\u2014JOHN FOSTER DULLES\n\nHIS ARM CHAINED TO A BRIEFCASE, William J. Reed counted the Florida palm trees through the car window as they whisked by at high speed. His eyelids sagged from jetlag, and his fingers tingled from lack of circulation. He'd been a man with a portable toothbrush through most of September, and now, in early October 1962, his final destination lay just ahead. This after more than a dozen stops in foreign countries around the Atlantic and Pacific rims. This after attending limited duty officer's school in Newport, Rhode Island, where he studied navigation, chart reading, ship handling, and how to be a \"politically navy correct\" officer. This after taking his family to see the 1962 America's Cup Yacht Race in the summer and then reporting to Section 22 of the Soviet SIGINT A Group at the NSA facility in Fort George G. Meade, Maryland\u2014commanded by Operations Chief Major General John Davis.\n\nReed was the only person on the team with Soviet burst signal field experience, so his boss, Commander Jack Kaye, and his peers at A22, descended upon him like squawking geese. They made him the \"point man\" for completing the technical and operational manuals for Bore-sight. They also locked a briefcase on his wrist and filled his schedule with site visits around the globe.\n\nReed's driver, a twenty-something petty officer named Smith, glanced in the rearview mirror and with a touch of a Southern accent said, \"Kinda humid today, ain't it?\"\n\n\"Kinda,\" Reed said. His tired brain cells allowed only one word at a time.\n\n\"That's what happens after it rains real hard like it did last week. Ever been to Homestead before?\"\n\n\"Nope.\"\n\n\"Great duty station, at least I think so. The enlisted and officers' barracks are actually in town, and the ops building is about five miles away down Card Sound Road.\"\n\n\"That so,\" Reed said, having mustered the mettle to form two words.\n\n\"Most everybody takes the bus to the base, but since I'm one of the drivers, I get to take this here car.\"\n\n\"That's nice.\" Still on two words.\n\nSmith wrestled with the wheel as the car stuttered across a road with potholes that looked like they'd been made with C-4 explosive. \"They built the station in 1957, and now NSG has, like, forty or fifty people here at Site Alpha, so that means we have good duty rotations and plenty of time off, so we can go have fun up in Miami and stuff. At least, we did until last month, when some kinda shit happened, and they told us to double up our watches. Do you know what's going on?\"\n\n\"A little.\"\n\n\"You could tell me, but then you'd have to kill me, right?\"\n\n\"That's right.\"\n\n\"You can't tell me even if I ask real nice and say pretty please with a cherry on top?\"\n\n\"Not even then.\"\n\n\"That's a bummer.\"\n\n\"I know.\" Reed smiled and turned again to look out the window. A flat plain baked by a tropical sun and mottled with green vegetation stretched to the horizon. The barest hint of pineapple and coconut drifted by.\n\nSmith turned onto an even more dilapidated road. A small building, surrounded by mud craters, sat alone on the rain-pummeled ground. The car pulled to a stop, and Smith said, \"This is it.\"\n\nStill clutching his briefcase, dressed in khakis, Reed stepped from the car. Brittle dirt cracked beneath his brown naval shoes. The gray cinder block structure displayed no windows and appeared to contain no life. At the entrance, a uniformed marine saluted Reed, checked IDs, and opened the door to an all-too-familiar sight. Racks of equipment beeped, whirred, and blinked; seated operators monitored, logged, and chatted. The pi\u00f1a colada odor vanished, replaced now by something indefinable that smelled electronically bitter. A tall man in his early thirties approached and reminded Reed of Commander Petersen from Turkey.\n\nThe man offered a smile and a palm. \"I'm Lieutenant Clower, the facility OIC. Welcome to Homestead.\"\n\nReed shook Clower's hand and forced his head to allow more than three words. \"Ensign Reed, NSG Mary land. Have my techs arrived yet?\"\n\nClower pointed to a far corner. Reed squinted and recognized his two cohorts, their hands buried in an equipment rack.\n\n\"I'd like to chat with you before you join your team,\" Clower said.\n\nReed nodded but did not reply.\n\nClower motioned for Reed to join him a few feet away from listening ears and said, \"I feel like I'm in the dark here. I was hoping you could fill me in.\"\n\n\"On what?\" Reed asked.\n\nClower crossed his arms. \"Homestead's Strategic Air Command Bomb Wings are on heightened alert, we've been ordered to double up on watches and keep an eye on every merchant ship headed to Cuba, and now you guys show up with some new DF toys. What the hell's going on?\"\n\nReed shrugged. \"I'm not exactly sure, sir; I've only gotten tidbits myself, but it probably has something to do with former vice president Nixon's suggestion for a quarantine to keep the Soviets from shipping more arms to Cuba.\"\n\n\"I heard that the Senate Foreign Relations Committee passed a near-unanimous resolution to spank the Cubans pretty hard if they decided to get nasty with their shiny new Soviet weapons,\" Clower said.\n\nReed wiped a sleepy wink from the corner of his eye and wondered how many of those new weapons delivered by the Soviets might be nuclear versus conventional. \"There's more; that's why I'm here.\"\n\nClower stood up straight. \"Go on.\"\n\n\"As you know, our guys are upgrading your DF systems so you can hear Ivan's new burst signal.\"\n\n\"Yeah.\"\n\n\"A couple of our Atlantic stations got a hit on some Foxtrots leaving Sayda Bay.\"\n\n\"Yeah, I know. We got the flash.\"\n\n\"What you don't know is that it looks like those boats are headed toward Cuba. That means something big is about to happen, and that's probably why your watches were doubled.\"\n\nClower's face lost half its color. \"God help us.\"\n\nReed removed his cover and backhanded his forehead. His shoulders ached, and he fought off a splitting headache. \"We can't expect God to do all the work. We need your team to pitch in.\"\n\n\"Absolutely. What do you want us to do?\"\n\n\"As soon as my guys get you up and running, we'll need to calibrate the systems for a couple days while I train your team. Then you'll have to play bird dog like you've never played before. Since you're still running a GRD-6, even with the new Boresight equipment, detecting a burst will be like finding a needle in a haystack, and any fixes you get will be way off.\"\n\n\"But better than nothing.\"\n\n\"Let's hope so,\" Reed said. \"If not, things could get pretty ugly in the Caribbean.\"\n\nAfter his conversation with Clower, Reed and his team spent the next few days installing and testing the Boresight receiver\/recorder and related equipment for capturing, recording, and analyzing the Soviet burst signals. Reed's team consisted of Master Chief Carl Odell and Second Class Petty Officer Tommy Denofrio, both M-Branchers. The Midwest-born Odell looked like Jackie Gleason and apparently copied the comedian's diet. He piled on bacon and biscuits for breakfast like a last-meal death-row inmate. Reed often wondered if Odell's heart would give out halfway through a job.\n\nDenofrio was wired differently. An Italian from New York and a consummate lady's man, he counted calories and pumped iron with the dedication of Jack LaLanne. The twenty-something petty officer displayed a talent like no other when it came to smooth-talking women but flashed the temper of a runaway train when someone disagreed with his conservative views. Both sailors displayed real genius on the job, which is why Reed selected them to help get the Boresight stations up and running.\n\nUnfortunately, at Homestead, interference problems made that task quite a challenge. Odell and Denofrio worked day and night to correct the issues, while Reed trained the operators in small groups so as not to pull anyone away from current operations.\n\nMeanwhile, outside the small building twenty miles south of Miami, the world's superpowers edged closer to a showdown.\n\nABOARD B-36, SECOND IN THE LINE of four Foxtrot submarines heading to Cuba, Captain Dubivko rested his round cheeks on a pillow in his bunk. He curled his fingers into rock-hard fists and reeled off a stream of obscenities in his head. He cursed Premier Khrushchev, threw harsh words at Admiral Sergei Gorshkov, and blamed Admiral Rybalko for sending him toward failure. How could he possibly arrive in Cuba by the date ordered while maintaining complete stealth? The boat heaved to port, and Dubivko's stomach knotted. A day earlier they'd run headlong into a massive storm that still lingered. After he'd spent two days awake battling pounding waves as the boat snorkeled, Dr. Buinevich ordered him to rest.\n\nUnable to sleep, he instead launched into another imaginary attack on his superiors. While mentally plotting Khrushchev's demise, Dubivko's stateroom phone rang. He answered, listened, hung up, and questioned whether there was a God or just a universe full of demons with the single-minded purpose of punishing him for the rest of his life.\n\nDubivko hurried aft down the passageway to the wardroom, where he saw Dr. Buinevich hovering over Sublieutenant Pankov, head of the hydroacoustic (sonar) group. The room smelled of fresh alcohol, obviously used to sterilize the wardroom table. The high-intensity light, placed in the overhead for surgeries, bounced off Buinevich's balding head like a searchlight. The captain of medical service filled a syringe, while Pankov moaned in pain. Four sailors held the man in place on the small table as the boat rocked to and fro in the storm. The moaning sublieutenant's feet dangled over the edge of the table, and the doctor fought to keep his hand steady as he shoved a needle into Pankov's arm. The moaning stopped.\n\n\"What's going on here?\" Dubivko asked.\n\n\"He has appendicitis,\" Buinevich answered. \"I need to operate.\"\n\n\"Now?\"\n\n\"If I don't, he'll die.\"\n\nDubivko bit his lip. \"Do it.\"\n\nThey were now transiting through the Norwegian Sea, past an imaginary line known as the GIUK gap, which stood for Greenland, Iceland, United Kingdom. The line represented a gauntlet laid down by the Americans. Any Soviet submarines venturing past that line entered into a narrow fishbowl less than one thousand miles across where sub-hunting NATO aircraft and ships patrolled in droves. Fortunately, a large gale developed as they sped past the Faroe Islands off the coast of Britain, and it now masked their diesel engines and hid their surfaced vessel under dark clouds.\n\nThe storm above worsened, however, forcing the boat to dive. Now the crew could feel its anger at a depth of more than twenty meters. Dubivko held no doubts that when the boat surfaced again during the night to charge batteries, the weather would still be turbulent. A Project 641\u2013class submarine snorkeling on the surface was no match for a force of this magnitude, and the two running diesel engines would certainly whine and choke every time a wave crashed over the snorkel mast, causing the flapper to slam shut to avoid sucking in salt water. Dubivko looked at his watch. Two hours to sunset. The doctor could barely keep his hands steady now. Once the boat surfaced, and ran at a higher speed in the storm, performing an appendectomy would be nearly impossible.\n\nDubivko walked over and clasped Pankov's hand. Bleary-eyed, the sailor tried to smile. Dubivko smiled back. \"Hang in there, Pankov. You're in good hands.\"\n\nPankov managed a quarter nod just before the injection sent him under.\n\nDubivko let go of Pankov's hand and turned toward Buinevich. \"How long will this take?\"\n\n\"Several hours,\" the doctor replied.\n\n\"We don't have several hours. If you can't complete the operation by sunset, we will risk missing our snorkeling window, and we'll fall too far behind schedule.\"\n\n\"What are you saying?\" the doctor asked as his scalpel penetrated Pankov's skin.\n\n\"I'm saying,\" Dubivko said, a lump forming in his throat, \"you must finish before nightfall, or we'll be short one acoustic sublieutenant.\"\n\nBuinevich looked up from his incision, his scalpel dripping blood. He stared at Dubivko for a long moment, shook his head, then returned his attention to Pankov. Dubivko clenched his teeth, and left the wardroom. Back in his cabin, he cursed his superiors for forcing him to kill a crewman.\n\nTwo hours later, a sailor shook Dubivko awake. He walked across the passageway and splashed his face with a sprinkling of tepid water. Like all submariners, Dubivko learned early on to conserve water use, especially on extended patrols. B-36 left port with thirty-six tons of fresh water, and submarines of this type did not have a condenser to produce more. Cooling the water was simply not possible, and when they entered tropical zones, tea became the preferred beverage.\n\nDubivko dried his face and, with the waves high above still rocking the boat, returned to the wardroom. The doctor stood near the table, his feet dancing to maintain balance while B-36 rolled ten degrees to starboard.\n\n\"Doctor?\" Dubivko said.\n\nBuinevich looked up from the table and grumbled. \"I never wanted to go to that damned hospital for special training, you know.\"\n\n\"I know.\"\n\n\"But I went anyway, and I guess that decision paid off, Captain. The boy will live.\"\n\nDubivko let out a slow breath. He did not need to assassinate Premier Khrushchev after all. He offered a job well done to Buinevich and walked toward the CC. There he assumed command and issued an order. \"Vspletye!\"\n\nThe boat angled toward the surface. Several minutes later, Dubivko, along with Political Officer Saparov, climbed up the ladder into the tiny conning tower. There Dubivko depressed a switch and said, \"Podnyat periscope.\"\n\nThe mast shot toward the sky as the boat slid to the surface, where the storm tossed her about like a tiny balsa wood model. Seawater splashed through the open bridge hatch above and soaked Dubivko's shirt. Steadying himself, he slapped down the handles on the navigation periscope. He swung the scope left and right, then focused on a small moving speck against the sunset-filled sky. At first he thought it might be a bird, but he then switched the scope to high power and recognized the shape as a British Shackleton ASW aircraft circling in the distance. He contemplated quick diving the boat but decided to risk staying on the surface. Without fully charged batteries, they'd have no hope of remaining on schedule. He also figured that the Shackleton's sonobuoys and radar would not be able to discern a snorkeling submarine from a crashing wave in this storm.\n\nAs Dubivko continued to watch the aircraft fly figure eights, something bothered him. He took a bearing to the plane and turned the scope over to Political Officer Saparov. He slid down the wet ladder into the CC and took three steps over to the navigation table. With Navigator Captain-Lieutenant Naumov staring over his shoulder, he bent over the nav table and silently ran a finger backward along B-36's track since leaving Sayda Bay. In a low whisper, he said, \"The Americans are following our course. How the hell do they know where we are?\"\n\nDubivko had little time to ponder this further as a giant wave slammed into the boat's side. The snorkel's flapper valve closed tight. The diesel engines continued in vain to suck in air that caused a high-pitched vacuum whistle in the boat. Dubivko's ears popped, and he swallowed hard. The control room filled with the rank odor of diesel fumes as the exhaust backed up from the engine room. A watchstander coughed and vomited onto the deck, and another grabbed his chin and groaned as the vacuum pressure made his eyes bulge and threatened to suck the fillings from his teeth. The boat pitched again, and men in Compartment Four ducked as canned meats flew from overhead cubbyholes like small metal missiles. Dubivko's ears finally cleared in time to hear the watch officer scream from up in the bridge.\n\n\"Right full rudder!\" Dubivko yelled.\n\nThe boat turned toward the waves, and the flapper valve popped open. Someone relieved the officer on the bridge, who, clutching his chest, stumbled down the ladder into the CC, then hobbled toward the wardroom. Dubivko later learned that Brigade Engineer Captain Second Class Lyubimov broke three ribs when he crashed against the side of the bridge gyrocompass.\n\nDubivko climbed up the ladder into the conning tower and relieved Saparov on the periscope. A splash of fading sunlight snuck past a storm cloud and glinted orange-red off the Shackleton's wings as the plane turned toward B-36. Batteries charged or not, he knew they were out of time as he ordered the boat beneath the roiling sea.\n\nThirty meters deep, running silent on electric power, with the storm now above them, Captain Dubivko returned to his cabin. He was haunted by what might happen if they couldn't return to the surface soon to snorkel. Or if they lost more than one engine while thousands of miles away near Cuba. Or if the Americans ruptured their pressure hull and forced them to use the nuclear weapon.\n\nAnd he wondered how the Americans appeared to know their general course. He closed his eyes, but sleep did not find him.\n\nON OCTOBER 10, BASED ON R-BRANCHER input from the USS Oxford, and the listening station at Homestead, Florida, the NSA advised the White House that the Cuban air defense system appeared complete and armed. The Cubans were now relaying their radar tracking information between their headquarters and jetfighter bases using Soviet standard procedures.\n\nBoresight stations continued to track the four Foxtrot submarines as they sped toward Cuba, while U.S. naval forces maintained a watchful eye on the Soviet oiler Terek, which they knew was there to resupply the Foxtrots. The navy also kept track of the electronic eavesdropping ship Shkval, which had a reputation of collecting intelligence information from U.S. warships and feeding it to nearby submarines. The presence of these two vessels indicated that the Soviets were planning something for the four Foxtrots, and American forces patrolling the Sargasso Sea east of Cuba remained alert and on edge.\n\nON OCTOBER 14, A U-2 SPY plane piloted by Major Richard S. Heyser flew over Cuba on a course that placed him sixty miles west of Havana. Heyser snapped 928 pictures in less than six minutes, covering an area seventy-five miles wide. The National Photographic Interpretation Center in Washington, D.C., examined the pictures the following day. Wide-eyed analysts confirmed that a series of nuclear launch sites now existed in Cuba that were capable of targeting most major cities in the United States. These included Soviet SS-4 Sandal medium-range ballistic missiles with forty-two projectiles that carried two-and three-megaton nuclear warheads. Also on board were twenty-four SS-5 Skean intermediate-range ballistic missiles that could go twice as far and kill eighty million Americans in less than five minutes. Throughout the United States, fallout shelters could hold forty million people at best.\n\nAll six Polaris submarines based in Holy Loch, Scotland, pulled out of port on October 16 and aimed their nuclear missiles at the Soviet Union. Two days later, McGeorge Bundy delivered the bad news about the nuclear missiles in Cuba to President Kennedy, who called a meeting with his high-level executive committee advisers just before lunch.\n\nAt Section A22 in Mary land, William J. Reed's boss, Commander Jack Kaye, assembled his team in the conference room. For the next several hours they discussed the Cuban situation and the need to locate any and all Soviet submarines operating in the area as soon as possible. To that end, they reviewed which stations had received the Boresight system upgrades and the operational status of each. Only a half-dozen contained the equipment, and none were fully operational. Most still used old GRD-6 antenna arrays, which meant their ability to locate and accurately pinpoint Soviet submarine locations were shaky at best. Only Edzell, Scotland, in the Atlantic, along with Hanza, Japan, and Skaggs Island, California, in the Pacific were equipped with Boresight and the new Wullenweber elephant cages.\n\nEven with the more advanced antenna capabilities at those stations, the maximum range for detection was around 3,200 nautical miles, with one ionospheric \"hop\" of 2,700 miles as more typical due to interference and weather conditions. An ionospheric hop is what happens when ionized atmospheric gases reflect high-frequency radio energy and \"bounce\" the transmitted signals back down to earth. These signals are often reflected back into the ionosphere for a second bounce, or hop. The upshot was that the optimal listening range for the arrays left Hanza and Edzell out in the cold and Skaggs right on the ragged edge of hearing anything near Cuba. The team concluded that, for now, they'd have to find a way to tweak the five Atlantic GRD-6 sites enough to give U.S. ASW forces a snowball's chance of keeping American ships from becoming sport-diver relics on the ocean floor.\n\nCAPTAIN DUBIVKO'S B-36, SECOND IN A lineup of four Soviet Foxtrots, made good time for several days under the cover of bad weather. They arrived at the edge of the Azores\u2014in the middle of the Atlantic northwest of Africa\u2014on October 15. After several more sleep-deprived days, Dubivko's eyelids twitched as he stood near the number two tube on the port side of the forward torpedo room. He hated the involuntary reaction to stress. Staring at Alexander Pomilyev, he said, \"Step away from that.\"\n\nPomilyev, the young special weapons officer (Weps) brought on board in Sayda Bay by the Northern Fleet special weapons directorate and who'd yet to qualify in submarines, removed his arm from the yellow rectangle adjacent to the number four tube. He turned to see what \"that\" might be.\n\nDubivko pointed to the silver lever on the front of the rectangle. \"That's the emergency bow plane operating mechanism. If you were qualified, you'd know not to rest your arm near that lever.\"\n\nPomilyev nodded. \"I'll make a note of that.\"\n\n\"See that you do,\" Dubivko said. \"You wanted to see me?\"\n\n\"Yes, sir. I wish to inform you that none of your crew are sleeping in the torpedo room.\"\n\n\"I'm already aware of that, Comrade Pomilyev.\"\n\n\"Yes, of course, but I'm concerned that this will diminish response time in the event of a weapons emergency.\"\n\n\"There are plenty of men in Compartment Two that can respond in time, but your point is duly noted.\"\n\n\"Perhaps the men could be persuaded to return if we convinced them that the special weapon will not cause radiation poisoning.\"\n\n\"Will it?\"\n\nPomilyev's perplexed look made his face look like a used dish towel. \"Of course not! That weapon is perfectly safe.\"\n\n\"Perfectly?\"\n\n\"Well...not perfectly, but close enough.\"\n\nDubivko glanced at the three torpedo tubes on the starboard side of the boat. Each housed conventional weapons, as did two of the tubes on the port side. Only the number two tube carried megaton destruction. \"Comrade Pomilyev, most of the men assigned to sleep in the forward torpedo room are young sailors. They have wives or girlfriends and aspirations of raising children. Unless you can guarantee that your purple-nosed weapon will not render their family jewels useless, they will probably continue to bunk in the aft torpedo room.\"\n\nPomilyev shook his head from side to side. \"There's absolutely no proof that\u2014\"\n\n\"Proof?\" Dubivko said. \"Until you are qualified, your credibility on this boat is less than that of bilge slime. Earn your qualification, Comrade. In the meantime,\" Dubivko waived his arm around the torpedo room, \"enjoy your solitude.\"\n\nDubivko turned and headed toward Compartment Two. He undogged the hatch, grabbed the bar above the opening, and shot his legs through. On the other side, a sailor with a large bayonet guarded the hatch\u2014a mandatory requirement for any boat carrying nuclear weapons. All persons entering the torpedo room were stopped by the guard and required to surrender sharp objects, tools, matches, lighters, or anything that could be used to sabotage the weapon. Although everyone on board had survived intense background checks prior to selection for this mission, none were trusted near the nuclear torpedo. No wonder they didn't want to sleep up front.\n\nDubivko popped into his cabin and glanced at the picture of Khrushchev hanging above his bunk. He grumbled once, turned toward his wooden cabinet, and rummaged through a drawer. After finding his toothbrush, paste, and hand towel, he strode across the passageway to the officer's washroom. There he splashed some water on his face, brushed his teeth, and tapped at the flickering light above his head.\n\nAs he returned to his cabin, he noticed the assistant navigation officer coming out of the four-man cabin just aft and to starboard. Seventy-eight men shared this home under the waves, and most of the noncommissioned sailors berthed in the \"sleeping wagon\" area in Compartment Seven back by the aft torpedo tubes. One shower and toilet in Compartment Six, used exclusively when submerged, serviced most of the crew. When B-36 surfaced, only the toilet and saltwater shower in the bridge were used, and the crew could enjoy an ocean shower without restrictions. Each compartment was issued a metal token, and only one person who held that compartment's token could use the facilities at a time\u2014similar to the key-on-a-stick system at an American gas station. On board Soviet diesel submarines, this system also ensured that everyone was accounted for when the boat submerged.\n\nOnce underwater, the boat's freshwater supply had to be allocated for washing, cooking, and drinking. Showers were limited to only two per week, so the doctor dispensed wash towels daily to maintain hygiene. Unfortunately, the towels did little to control body odor.\n\nDubivko stored his sundries and walked a few feet over to the acoustic room. A thin operator wearing headphones sat in front of a rack of metal rectangles covered with small airholes. Each box contained an indented section that housed several black control knobs, dials, and indicators. B-36 still used the older Herkules medium-frequency active\/passive and Feniks passive search\/attack acoustic arrays that could hear contacts up to twenty-nine kilometers away. Only Captain Ketov's B-4 had an upgraded RG-10 passive system that offered greater range and accuracy.\n\nThe acoustic sublieutenant looked up, smiled, and pulled the headphones off one ear. \"Captain?\"\n\nDubivko smiled. \"How are you feeling, Pankov?\"\n\n\"I'm still a little sore but doing fine, sir.\"\n\n\"Glad to hear,\" Dubivko said. \"It's a good thing our doctor is highly trained in appendectomies.\"\n\n\"Yes, sir.\"\n\nDubivko pointed to the headphones. \"What are you listening to?\"\n\n\"Mostly whales, sir. There's nothing much else out here right now.\"\n\n\"No sonobuoys from that Shackleton we saw a few days ago?\"\n\n\"No, sir. Not even a peep.\"\n\nDubivko stepped back into the passageway and said, \"Well, let's hope it stays that way so we can stay on schedule.\"\n\nPankov nodded and replanted the headphones.\n\nDubivko walked past the wardroom, where Pankov had received his operation, and undogged the Compartment Three hatch. The CC hummed with activity as he entered, and he reveled in the vibrancy. Standing just below the ladder that led up to the conning tower, he glanced at his watch. At this time of day, high above them, a blazing sun warmed the sea and transformed wave crests into reflective shards of glass. Although they needed to snorkel to recharge batteries, they dared not do so in broad daylight, but capturing the required daily radio broadcast from Moscow could not be avoided.\n\nDubivko held back an expletive as he thought about the mental midgets at Fleet HQ who ignored the fact that midnight in Moscow meant daytime this far from Russia. Their orders complicated things in two ways. First, the submarines needed to slow down to capture a broadcast. Second, receiving a signal required approaching near the surface, which left them vulnerable to detection. Apparently the mission planners were not qualified submariners.\n\n\"Watch Officer, make your depth twenty meters,\" Dubivko ordered.\n\n\"Yes, Captain,\" the watch officer acknowledged. He turned to the planesman, seated starboard near the front of the compartment. \"Bow planes up fifteen degrees.\"\n\n\"Prepare to raise the HF antenna,\" Dubivko said.\n\nAnother ac knowledgment.\n\nIn the CC, a dozen faces instinctively glanced upward as the needle in the main depth gauge moved counterclockwise.\n\nThe planesman, sitting forward and to the right of Dubivko, barked off a reading. \"Passing forty meters.\"\n\nThe watch officer, so designated by his blue and white elastic armband, echoed the planesman's report as the boat's hull creaked like arthritic bones in response to the change in ocean pressure.\n\n\"Raise antenna,\" Dubivko said as the boat neared periscope depth.\n\nLieutenant Zhukov stepped through the hatch from Compartment Two. As the boat's electronics officer, he was responsible for all of B-36's electronically operated equipment, including acoustic, radar, weapons control, and radio. Without saying a word, Zhukov pointed toward the radio room and continued aft. The familiar routine happened daily, and Dubivko hoped that this time the outcome would be different.\n\nFor all the years Dubivko had operated aboard Northern Fleet submarines in the Barents and Norwegian seas, he had experienced the curse of the Arctic. This high-latitude region created sporadic interference like Dubivko's mother-in-law spewed insults. Virtually all transmissions were sent and received via HF or UHF, and during the winter and parts of the summer, magnetic storms and interference were constant concerns. But of all the transits Dubivko could recall, this one, so far, held the record for the most transmission problems, far surpassing anything the Arctic could muster.\n\nThe planesman sounded off another report. \"Passing thirty meters.\"\n\nHaving crossed the Faroe\u2013Iceland line, B-36 descended into a proverbial radio vacuum. All Northern Fleet stations were masked by static, and the only audible voices came from fishermen on trawlers near Murmansk. For two days Zhukov and the radio operators tried to find clear frequencies but never succeeded. They analyzed signals based on different times of day, weather conditions, and other factors and tried to make radiogram sending and receiving adjustments but without any luck.\n\n\"Steady at twenty meters.\"\n\n\"Very well,\" Dubivko said as he prepared to enter the conning tower.\n\nThe main problem they faced with radio communications centered around power. Communicating with Moscow could only be kept secret by using the low-power fifteen-kilowatt transmitter. The antenna for that system required air drying for almost twenty minutes before sending or receiving transmissions. In submarine time, that meant forever, especially while trying to stay undetected and on schedule. The only way to mitigate the problem required using a slightly wet antenna and retransmitting the same burst message as many as thirty times to ensure receipt by Moscow.\n\nNow up in the conning tower, as he swiveled the periscope back and forth, Dubivko contemplated the endless obstacles before him. Through the scope he spotted an American P2V ASW aircraft on the horizon. His cheeks turned hot as he wondered how they could possibly be following B-36's course so accurately. He contemplated whether it might have something to do with the Americans' new underwater sonar arrays but then quickly dismissed that thought. Unless the United States somehow learned how to defy the laws of physics, given B-36's distance from any known array, and given that they only snorkeled at night, they couldn't possibly achieve this close a fix with that system. No, either the Americans were damn lucky, or they had deployed some new unknown technology. But what?\n\nLAST IN THE LINE OF FOUR Soviet submarines headed to Cuba, B-4 cruised past the Azores on October 15 and entered the Sargasso Sea. A few days later, his nerves on edge, his armpits soaked, Captain Ryurik Ketov thought about Pasha, the boat's cat they'd left behind in Sayda Bay. He hoped she was happy in her new home with plenty of caviar, or at least some nice tuna. He peered over Vladimir Pronin's shoulder in the radio room as the young electronics officer stared at the short burst data radio with frustrated yet hopeful eyes.\n\nThe SBD radio consisted of a yellow box covered with dials, switches, and indicators seated next to a silver device that resembled a typewriter. Ketov didn't like the new burst radios. The concept, stolen from the Germans after World War II, was forced upon the submarine fleet in late 1960 and added to B-4 prior to her launch in 1961. The SBD's encoder\u2014the typewriterlike device\u2014suffered from a limitation of having only seven groups of symbols that severely truncated what could be sent to or from HQ. Ketov always harbored concerns that this could lead to misinterpretations of important orders or rules of engagement. What if they received updated instructions on when to fire their nuclear torpedo, but the ultra rapid activity radio turned these into something vague and unintelligible? That could make for a very unpleasant day.\n\nPronin licked his upper lip and rechecked the settings on the SBD. Still nothing. The burst itself took only seven-tenths of a second, but the wait to receive a transmission could take seven hours, or so it seemed to Ketov.\n\nFinally, the SBD received an order update just as the political officer approached the radio room. Ketov read the strip of paper and shook his head from side to side. They were being redirected from their transit to Mariel. The new orders told them to assume combat readiness in the Caribbean Sea, south of Jamaica, and wait. Wait for what? Ketov wondered if imminent war with the Americans caused the change in plans. He ordered Pronin to have the OSNAZ group begin monitoring civilian radio transmissions to find out what might be going on out there.\n\nAfter changing course, the next several hours bordered on boring until they caught up with a hurricane. That monstrosity dredged up giant waves and hurled them onto the decks of helpless vessels. One of these was a merchant ship that let out a desperate cry for help. Pronin heard the plea on the radio and reported it to Ketov, who stood watch on the bridge. In the dead of night, B-4 barely held her own against the hurricane while snorkeling on the surface. In rough seas she became a black spec of metal toyed by a force older than time and stronger than Neptune. Ketov bit his lip when he heard Pronin's report. Though he desperately wanted to, he knew he could be of no help to the floundering merchant ship.\n\nPronin pleaded over the communications circuit to do something, but Ketov knew they could not. The distressed ship lay twelve kilometers to the east, and if B-4 turned away from its southern heading, that might cause them to take on too much water to snorkel. They needed to charge batteries to stay on schedule. They were also under strict orders not to reveal their position to anyone, not even a friendly vessel. Last but certainly not least, what assistance could a tiny submarine possibly offer to a large merchant ship?\n\nKetov stood on the bridge and watched the waves form crests and troughs of foam and spray. With a dense scud strafing the wave tops, some as high as seventeen meters, he could see no farther than ten meters to either side. For all he knew they might never spot that ship and might run right past her in the dark, or worse, smack hard into her side and end their mission. Pronin called up again from radio and tried a different tact. Perhaps they could launch a flare and try to pick up survivors? Ketov knew it must be agonizing for Pronin to listen to the repeated SOS, but he had to decline his electronics officer's request yet again. They had no room on board and could ill afford to take merchant seamen on such a secret mission.\n\nA blast of salt water doused Ketov's face. The wet cold reminded him of a decade earlier when Ketov's former commander on the SS-26, Captain Second Rank Abram Tyomin, taught him a valuable lesson. A giant wave struck the SS-26 from the side and rolled her almost ninety degrees. Water flooded through the open bridge hatch and cascaded into the CC. Electronics shorted. Helm control died. As watch officer, Ketov stood motionless, lost in a daze of panic. Tyomin ran into the CC, saw the look on Ketov's face, and backhanded him hard across the cheek. He told Ketov to calm down, remember his training, and deliver clear and deliberate orders to the men.\n\nKetov did just that and directed the crew to manually control the helm from the aft torpedo room. They survived, and from that day forward, he recalled the incident during times of trouble. Now he recalled Tyomin's hard-learned lessons to make the right decision about the distressed merchant ship. That, unfortunately, meant standing by while dozens of fellow countrymen drowned.\n\nOr did it?\n\nKetov called down to radio. He told Pronin to relay the distress call on the HF band, but to send only a few transmissions and then stop. Hopefully, someone might hear and respond. With relief in his voice, Pronin thanked Ketov for taking this risk. And risk it was, for Ketov knew that transmitting on an open channel could end his career. But if he violated the seamen's code by letting those men die, he'd have to avoid mirrors for the rest of his life.\n\nON OCTOBER 19, DEEP IN THE middle of the Sargasso Sea, B-36 swayed to the beat of a weather-tossed ocean. Captain Dubivko's fingers tingled like they always did when he neared the surface. For when a submarine sheds her deep ocean security blanket and ventures near the domain of surface ships and aircraft, she becomes vulnerable and takes one step closer to death.\n\n\"Twenty meters,\" Zhukov reported. As the on-duty watch officer, he stood to Dubivko's right near the planesman and helmsman.\n\n\"Very well,\" Dubivko said, leaning against the conning tower ladder. \"Raise the HF antenna, and open the main induction. Engine Control, start the diesel engines and battery charging.\"\n\nAlthough they had received no communications from Northern Fleet HQ since their departure weeks earlier, they were nonetheless required to slow, surface, and check for transmissions every evening. Brigade Commander Vitali Agafonov, riding on B-4, sent a daily radio check \"click\" over the airwaves during that time but never sent anything more tangible. B-36's radio operator simply responded with one \"click\" to verify reception. All four boats orchestrated this dance in a successive relay to ensure that only two submarines were near the surface at a time.\n\nAssistant captain, Lieutenant A. P. Andreev entered the CC from Compartment Four and relieved Zhukov of the watch. After a two-minute routine, Zhukov returned Andreev's salute and handed him the watch armband. He then looked at Dubivko, and the two headed aft toward the radio room. Dubivko caught a whiff of something that smelled like soap and wondered if Andreev had just showered. He also wondered when he'd last enjoyed that luxury himself.\n\nDubivko wanted desperately to receive a transmission from Moscow. Anything would be better than nothing. He took a deep breath and forced himself to relax. As the captain of B-36, he could not afford to display signs of anxiety or nervousness. That could undermine his ability to command respect from the crew. If they started to doubt his confidence, they might also doubt his orders. Still, he couldn't help but wonder why Moscow remained silent, and what might be happening in the world around them. Were they headed toward a war with the Americans? Absent communications from HQ, he felt like a blind man in a room full of sword-wielding Cossacks.\n\nDubivko stuck his head through the radio room door. First Officer Captain Third Rank Kopeikin and Political Officer Saparov peered over the shoulder of the radioman, who was seated near a rack of equipment. Their eyes remained transfixed on the rectangular teletypewriter that sat on a white metal shelf. The typewriter clicked away as the carriage moved left, then right, then left again. Nothing appeared on the paper ribbon except repeated groups of seven letters that spelled nothing.\n\n\"It's just the carrier tone,\" Kopeikin said.\n\nDubivko said nothing as his heart sank into his gut. He knew Soviet procedures dictated that fleet broadcasts sent from the large antenna farm southeast of Moscow should remain on air for no more than ten minutes. Either you received the transmission on time or you didn't. There were no repeats. Dubivko glanced at his watch. They were on time, but so far they'd been greeted by nothing but a carrier signal. The four men continued to stare at the Teletype, as if their combined sheer will could produce a message. Long minutes passed in silence.\n\nDubivko checked his watch again and turned to leave. Though he fumed with anger, he dared not show it. Not more than two steps away, he heard Zhukov's excited voice.\n\n\"Something's coming in!\" Zhukov said. He tapped the radioman on the shoulder. \"Test the synchronization line.\"\n\nThe young radioman reached for a few knobs on the SBD.\n\nDubivko stared at the Teletype as an unreadable message appeared. Talking aloud, Zhukov counted backwards from ten down to zero and then hit the encryption key. The seven-letter groups of gibberish transformed into Cyrillic sentences. The clatter of the teletypewriter ceased, and Zhukov tore off the message ribbon. He handed the piece of yellow paper to Dubivko.\n\nHolding his hand steady, Dubivko read the message, frowned, and said nothing to the others. He left blank stares behind and hurried back through the hatch into the CC, with Kopeikin in tow. They reached the navigation table a minute later.\n\nDubivko tapped Navigator Naumov on the shoulder. \"Show me charts for the Bahamas and Sargasso Sea.\"\n\n\"Which ones?\" Naumov said.\n\n\"Southern entrance.\"\n\n\"Yes, sir.\"\n\nNaumov combed through a stack of charts, removed one, and placed it on the nav table. The map displayed a colorful lined drawing of the Bahamas Islands chain near Florida. To the south lay the Turks Islands and Cuba. Dubivko traced a finger across the chart. The paper felt bumpy and coarse as he neared Miami. He thought of the millions of people lying about on sandy beaches there, drinking margaritas, and having fun while completely unaware that a Soviet submarine might be watching them from afar.\n\nDubivko glanced again at the typed message in his hand. He pictured Ryurik Ketov, on B-4, and wondered if his friend now shared similar concerns and confusion.\n\n\"What does it say?\" Kopeikin asked.\n\nDubivko handed over the paper ribbon. \"Read it aloud.\"\n\nKopeikin grabbed the message and read with a soft voice. \"Secret modification of operational orders to follow. The Sixty-ninth Brigade of Submarines are ordered to change course, assume combat readiness, and form a line west of the Caicos and Turks Island passages in the Carib be an Sea.\"\n\nDubivko pictured the location in his mind. Just southeast of the Bahamas and north of Puerto Rico, around 500 miles from the southern tip of Cuba and 600 miles south of Miami, dozens of tiny islands dotted the area, and water temperatures soared into the eighties. B-36 was now south of the other submarines, closer to the Turks Island passage. For submariners, transiting this narrow passage was a highly dangerous prospect when detection wasn't an issue.\n\nKopeikin lifted his eyes from the paper. His face registered bewilderment as he stared at Dubivko. \"What does this mean? Are we not going to Mariel as originally ordered? What's going on? Are we at war?\"\n\n\"I don't know,\" Dubivko said, \"but I'm going to find out.\"\n\n\"How?\"\n\n\"Follow me.\" Dubivko moved away from the table and walked toward Compartment Two. Kopeikin followed. Dubivko entered his cabin and sat on his bunk. The first officer stepped inside and shut the door.\n\nDubivko grabbed the phone on the wall and brought the black device to his ear. \"Zhukov, join me and Kopeikin in my cabin.\" He hung up the phone.\n\n\"Sir?\" Kopeikin queried.\n\n\"I'm tired of being deaf, dumb, and blind.\"\n\n\"What do you have in mind?\"\n\n\"We have five English-speaking OSNAZ communications experts on this boat. It's time we put them to good use,\" Dubivko said.\n\n\"I don't understand. What\u2014\"\n\nA knock on the cabin door interrupted Kopeikin's question.\n\n\"Enter,\" Dubivko said.\n\nThe door opened, and Zhukov squeezed inside. He bent down slightly as the curving bulkhead ran just above his head. \"Sir?\"\n\n\"Has our OZNAZ group heard anything on open frequencies?\"\n\n\"No, sir. Our schedule has not permitted more than a few minutes of HF antenna time.\"\n\nDubivko stared at the picture of his wife and children tacked to the bulkhead and said, \"Comrades, we're going to remain near the surface for another two hours.\"\n\nKopeikin's eyes extended. \"But that's a direct violation of orders!\"\n\n\"I will take full responsibility. I want our communications experts to scan American broadcast frequencies and record what's being said.\"\n\n\"Military frequencies?\" Zhukov said.\n\n\"And commercial,\" Dubivko said. \"Voice of America, for example. They even have a broadcast in Russian. We need to know what's going on out there, and Moscow is not in a position to tell us.\"\n\n\"What about the Zampolit?\" Kopeikin said. \"He'll be obligated to report this breach of protocol.\"\n\n\"I've thought of that,\" Dubivko said. \"We'll tell Political Officer Saparov that our new orders require us to assume combat readiness. As such, we must ascertain the intentions of our enemy. Listening to commercial radio broadcasts is essential in accomplishing this task.\"\n\nKopeikin smiled. \"Brilliant. He will be required to listen in. And when he does, he will be complicit. That places him squarely on our side, whether he wants to be or not.\"\n\n\"Precisely,\" Dubivko said. \"We have a nuclear torpedo on board. We've been ordered to assume a combat stance in enemy waters. We need to know what the hell is going on out there before we ready tube number two. Understood?\"\n\n\"Yes, sir,\" the two men said in unison.\n\n\"Zhukov,\" Dubivko said, \"prepare three sets of headphones. I want to listen in as well.\"\n\n\"Yes, sir.\"\n\nThirty minutes later, Dubivko pulled on a set of phones and listened to the Russian version of Voice of America. A woman's voice came across clearly and with no accent. She reported that three days earlier, on October 16, 1962, the New York Yankees beat the San Francisco Giants to win the fifty-ninth World Series four games to three. Dubivko breathed a sigh of relief. The Americans were still talking about baseball instead of hurrying toward fallout shelters. That meant they were not at war. At least not yet.\n\n## CHAPTER SEVEN\n\nIt's not the size of the dog in the fight, it's the size of the fight in the dog.\n\n\u2014MARK TWAIN\n\nON SUNDAY, OCTOBER 21, ABOARD THE USS Oxford off the coast of Cuba, T-Brancher Aubrey Brown came to attention in his chair. Other operators inside the darkened room did likewise. Most of them squinted at an assortment of green screens and indicators. Racks of receivers hummed, while reels of tape spun on six-foot-tall 3M reel-to-reel tape recorders. Other T-Brancher technicians monitored the flickers running across the scope of an X-band receiver. The screech of an unknown radar signal, captured by the receiver, blurted from a speaker mounted on the unit. The technicians grabbed stopwatches, which dangled on shoestrings wrapped around their necks. They timed the intervals between the whooping sounds to measure radar scan rate and cataloged the highs and lows of the signal spikes showing up on the screen.\n\nOnce certain of the signal's characteristics, they checked what they'd found against the NSA's super-classified TEXTA (Technical Extracts of Traffic Analysis) manual to confirm the radar's identity. Judging by the parameters captured, the T-Branchers verified that the Soviets were testing radar systems on fully operational offensive nuclear weapons systems in Cuba. The station then reported this to their navy command, who immediately dispatched a helicopter from Florida to retrieve the tapes for verification.\n\nPresident Kennedy received the news about the operational nuclear missile systems in Cuba during a meeting with Secretaries Dean Rusk and Robert McNamara at 10:00 A.M. After a brief discussion, Kennedy approved the final plan for a quarantine of the island. Later that morning, the president met with General Walter Sweeney, the commander of the Tactical Air Command (TAC), to review the plan for an air attack. Sweeney admitted that at best they might destroy ninety percent of the missile sites. Kennedy expressed concern that the remaining ten percent could still kill hundreds of thousands of Americans and ordered Sweeney to prepare for a potential strike on Cuba within twenty-four hours.\n\nThat afternoon the president convened a formal meeting with the National Security Council in which the chief of naval operations, Admiral George W. Anderson, briefed the group on quarantine rules of engagement. Anderson announced that every Russian ship approaching the line would be signaled to stop for potential boarding and inspection. Should a ship fail to halt, they'd fire a warning shot across its bow. If that didn't work, a U.S. destroyer would cripple the merchant ship by demolishing the rudder with cannon fire. Kennedy reacted to this announcement with unease that such a provocation could unintentionally sink the ship and trigger a war. Although Anderson provided assurances that gun-crew accuracy should ensure that no Soviet vessels sank, Kennedy remained dubious. Nonetheless, he agreed to the plan when Anderson stated, \"The biggest danger lies in taking no action.\"\n\nOn Monday morning, October 22, at NSA in Fort George G. Meade, Mary land, William J. Reed sat in a chilled conference room and listened intently as his boss, Commander Kaye, read the daily intelligence report. The scent of steaming coffee and fresh aftershave drifted through the large room that held around a dozen members of the Bore-sight A22 section. Kaye, with an abundance of Texas accent and attitude, launched the meeting by describing a series of events that had transpired over the past several days.\n\nKaye said that Kennedy had met with Soviet Ambassador Anatoly Dobrynin and Foreign Minister Andrei Gromyko on October 18, and the president received false assurances that the Cubans had received no offensive missiles from Russia. Reed heard a rumor that Kennedy privately called Gromyko a \"lying bastard.\"\n\nHalfway through the meeting, Kaye glanced at Reed and said, \"What's the latest on those Soviet submarines?\"\n\nReed frowned. \"Well, as you know, we got a few good hits on the four Foxtrots when they were near the North Cape, but then four Zulu-class submarines showed up out of Gadzhiyevo and created some interference. We haven't seen the Zulus since. The only Boresight hits we're getting now are from the Foxtrots, which we believe have approached to within five or six hundred miles of Cuba.\"\n\n\"How's our bearing quality holding up?\" Kaye said.\n\n\"Not very well,\" Reed said. \"We get rough bearing hits on three or four of the Foxtrots in the afternoon, which is when we believe Moscow requires them to transmit a burst signal update. Good news is they're retransmitting up to thirty or more times, probably because their antennas are wet. That really helps us get a bearing, but then they go dark for a day. We've been sending P2Vs to the area, but right now our bearing quality is worse than a World War II Huff Duff. Maybe fifty or sixty nautical miles at best. Still, we're starting to get enough data to formulate some interesting conclusions.\"\n\nKaye cocked his head to one side. \"Such as?\"\n\nReed said, \"Well, for one, they're still in the Sargasso Sea northeast of Cuba and averaging almost ten knots, which is damn aggressive for a diesel boat. Two, they don't appear to be shadowing any of the merchant ships, so they must have other orders.\"\n\n\"Like what?\" Kaye asked.\n\nReed shrugged. \"We can only assume they're posturing for a fight.\"\n\n\"What about SOSUS?\"\n\n\"Not much help yet,\" Reed said as he glanced at the wide-eyed faces seated around the table. \"Given that our underwater sonar arrays can't hear beyond 150 nautical miles, they're only occasionally catching a whiff of a snorkeling boat, and the bearing accuracy is worse than Boresight. Also, those subs only snorkel at night. During the day, when they're running on battery, SOSUS can't hear them at all.\"\n\n\"Shit,\" Kaye muttered. \"This is starting to feel like a train wreck in the making.\"\n\nA female yeoman walked into the room and handed Kaye a printed message. Kaye's face went white as he read it. He looked around the room and said, \"I'll deny I told any of you this, but this CIA memo just frosted my balls.\"\n\n\"CIA?\" someone said.\n\n\"Yeah,\" Kaye continued. \"I probably wasn't supposed to be briefed on this yet, but I have a few friends in low places. The memo says that they verified sixteen of the MRBMs and four of the longer-range IRBMs are already saddled up and out of the barn.\"\n\n\"Damn,\" someone else said. \"That means they can actually launch now.\"\n\n\"And that means they can vaporize millions of us within minutes,\" Reed said. He knew that when the U-2 spy planes originally photographed the medium-and intermediate-range ballistic missile sites in Cuba, the launchers were still under construction. None were fully operational. Now that they were, the end of the world was only a button push away.\n\n\"The Cubans also have twenty-two Soviet IL-28 bombers and thirty-nine MIG-21 jet fighters now,\" Kaye said. \"What's worse, they have a bunch of short-range Frogs with megaton warheads. That means if we try to send in an invasion force, they'll nuke thousands of our boys into dust clouds.\"\n\nReed's stomach turned sour as he pictured that scenario. A mixture of fear and excitement shot a bolt of current through his body\u2014fear that perhaps he might be witnessing the end of humanity, and excitement at being center stage for one of the most important events in human history. \"What do you think Kennedy's going to do?\"\n\nKaye pursed his lips. \"According to this memo, the CIA concluded that he's basically fucked like a junkie whore on Main Street no matter what he does.\"\n\nOne of the officers stifled a laugh. Another asked a question. \"They don't think the Soviets are doing this as a bargaining chip to get Kennedy to pull our missiles out of Turkey, do they?\"\n\nKaye ran his tongue over his upper lip and shook his head no. \"It's a lot more serious than that. The Soviets want to show the world that if we can waltz into their neighborhood with big-ass weapons, they can do the same in our neck of the woods. Problem is, if we just let them get away with it, we open the door for the Russkies to arm every commie pinko bastard country that might want to point a six-shooter at us. But if we confront Khrushchev about this, that won't stop him from shipping more nukes, and he'll just stall us with bullshit negotiations and U.N. red tape. A confrontation would also show our hand and take a surprise invasion of Cuba off the table.\"\n\nReed shifted in his seat. \"What about a blockade? I heard that the Enterprise got marching orders along with the Eleventh and Thirty-second Air Wings out of Puerto Rico. And with the Essex leaving Gitmo Bay with Task Force Bravo, I gotta believe they're planning something.\"\n\n\"Could be, but a blockade won't help us get rid of the dozens of missiles that are already there. Also, we probably can't muster more than sixty ships, and the ocean around Cuba is a lot bigger than the Ponderosa. The Soviets could probably still sneak in some nukes on submarines.\"\n\nReed thought about the four Foxtrots they'd detected heading for Cuba and let out a soft whistle.\n\nKaye stood up straight. \"What?\"\n\n\"I just had a holy shit thought.\"\n\n\"What kind of holy shit thought?\"\n\nA dozen eyes stared at Reed.\n\n\"What if,\" Reed said, \"those Foxtrots are already carrying nukes?\n\n\"You mean like IRBM parts?\" an officer asked.\n\n\"The CIA cautioned about that in a recent memorandum,\" Kaye said.\n\n\"No,\" Reed said. \"I mean like nuclear-tipped torpedoes.\"\n\nA dozen officers sucked in a collective breath.\n\n\"Jesus,\" Kaye said. \"That'd mean fifteen-megaton bad boys with a blast radius of\u2014\"\n\n\"Ten nautical miles,\" Reed finished. \"That could put an end to any blockade Kennedy might be planning.\"\n\nKaye's brow furrowed as he shot Reed a stern look. \"You need to find a way to tweak those Boresight stations to get us better bearing fixes. And you've probably got no more than a week to do it. Otherwise, we might wind up like the dead guys at the O.K. Corral.\"\n\nReed's throat tightened. He knew the odds were against him, but he and his team would just have to find a way to get the job done. There were two GRD-6 Boresight stations that, given the substandard range and accuracy of those antennas, might be close enough to get good fixes if their systems were optimal. Those were the sites in Northwest, Virginia, and Homestead, Florida. There was also one Wullenweber elephant cage close enough that might be able to lend a hand. Unfortunately, that facility, located at Skaggs Island, California, sat right on the edge of the system's maximum range. Somehow they'd have to get all three sites up to optimal capability, then hope that God sent them some good weather and minimal interference. Reed whispered a silent prayer that God didn't prefer the color red.\n\nBY THE AFTERNOON OF OCTOBER 22, T-Branchers aboard the USS Oxford spy ship reported that at least five Soviet missile regiments were operational or nearly so, each with eight missile launchers and sixteen missiles. Cuba now possessed the ability to fire a salvo of forty missiles that could devastate dozens of targets in the United States. In response to this threat, the Strategic Air Command initiated a massive alert for the entire B-52-bomber strike force, guaranteeing that thirteen percent of all aircraft be airborne at any given time. The plan ensured that every time a bomber landed, another one took to the air. SAC also dispersed almost 200 B-47 nuclear bombers to thirty-three civilian and military airfields, with another 161 aircraft delivered by the Air Defense Command to sixteen bases within nine hours. For the first time in history, all these planes were armed with nuclear bombs.\n\nTHAT SAME AFTERNOON, CIA DIRECTOR JOHN McCone informed President Kennedy that the four Soviet submarines were positioned to reach Cuba within a matter of days. He received that information from Chief of Naval Operations Admiral George Anderson. Neither McCone nor Kennedy was informed that the original source for that estimation came from Boresight Net Control in Mary land.\n\nIn light of the Boresight location estimates for the four Foxtrot submarines, Admiral Anderson issued a warning to the blockade fleet commanders: \"I cannot emphasize too strongly how smart we must be to keep our navy ships, particularly carriers, from being hit by surprise attack from Soviet submarines. Use all available intelligence, deceptive tactics, and evasion during forthcoming days. Good luck.\"\n\nThe navy positioned hunter\/killer Group Bravo, headed by the aircraft carrier USS Essex, 200 miles northeast of Caicos Passage, just outside and in the center of the Walnut Line\u2014a boundary arcing through the ocean in a semicircle 500 miles off Cuba. This \"do not cross\" line that started at the southern tip of Florida and ended east of Haiti represented the outer boundary of Kennedy's quarantine area. Three sub-hunting destroyers escorted the Essex, including the USS Blandy. The destroyer USS Cony (DD-508) took up station near the carrier USS Randolph (CVS-15) farther northeast of the Blandy (DD-943), and the destroyer USS Charles P. Cecil (DDR-835) accompanied the USS Enterprise (CVN-65) to a position southeast of the other two groups. Sonar and radar operators on all three destroyers listened and watched for signs of Soviet submarines approaching Cuba. None were aware that they'd soon come within a breath of nuclear annihilation.\n\nON BOARD B-36, NEAR THE TURKS Island Passage at the southern tip of the infamous Bermuda Triangle, Captain Dubivko's eyelids started twitching again. Only forty kilometers of ocean in that passage separated East Caicos from Grand Turk. Although depths in the center of the channel, which lay just north of Puerto Rico and east of Cuba, plunged to greater than 2,200 meters, shallow waters and sandbars on both sides made the area extremely dangerous for maneuvering. With American ASW planes and ships smothering the entrance and exit routes day and night, remaining undetected required a lot of caution, skill, and luck.\n\nDubivko envied Captains Savitsky and Shumkov on B-130 and B-59, respectively. Both were transiting through the wider, less treacherous Caicos Passage to the north. Captain Ketov, on B-4, had circled around Puerto Rico to take up station near Jamaica. Dubivko contemplated his unlucky orders and decided to try an old trick. If successful, they just might make it through the passage undetected and unscathed. If not, given the difficulty of his planned maneuver, they could take up permanent residence at the bottom of a foreign sea.\n\nThat evening, on October 22, Zhukov relayed the report that the OSNAZ team had heard President Kennedy address the American nation. He stated that the U.S. Navy had initiated a quarantine around Cuba to block Soviet merchant ships carrying nuclear weapons. They still heard nothing but truncated order changes from Moscow and so could rely only on commercial traffic intercepted over the airwaves to give them a hint of what might be happening in the world. After hearing the news about the quarantine, Dubivko wondered if he'd soon receive a message from Moscow ordering him to use the purple-tipped weapon to rip a large hole through Kennedy's blockade. He took a deep breath and patted a handkerchief against his sweat-soaked forehead.\n\nThe mediocre air-conditioning unit on B-36 had served them well while operating in cold northern waters. But once they reached the Sargasso Sea, where temperatures exceeded fifty degrees centigrade even at depth, the unit could not keep up, and the crew started to swelter. Dubivko dreaded the thought of being pursued in these heated waters by a swarm of ASW aircraft for hours on end while running slow and deep. He moved B-36 into the passage and descended below the thermal layer to thwart possible sonar detection. He then ordered all stop and neutral buoyancy to keep the boat still and level while they waited. Nerve-grinding hours passed as Pankov, in the acoustics cubicle, listened with trained ears to a distant contact.\n\nDubivko's mind plagued him with disastrous scenarios as his submarine sat motionless hundreds of feet below the surface. What if an ASW plane picked up their scent and started dropping sonobuoys? Where could they hide? There seemed to be hundreds of planes flying around. He figured American destroyers could not be far away.\n\nStanding in the CC, Dubivko queried a question over the comm. \"Acoustic, is that merchant ship still there?\"\n\n\"Control,\" Pankov said, \"the merchant ship is still making turns for ten knots.\"\n\n\"Acoustic, range and bearing?\"\n\n\"Control, 3,000 meters bearing three-five-eight and closing.\"\n\nDubivko glanced to his left. \"Navigator, plot an intercept course for nine knots speed.\"\n\nNaumov acknowledged the order. Two minutes later he said, \"Turn right on my mark to course zero-six-zero.\" Five more seconds passed. \"Mark.\"\n\nThe boat's deck tilted a few degrees as the helmsman pulled the steering lever to the right.\n\n\"Intercept time?\" Dubivko asked as he wrapped his hand around a pole on the conning tower ladder to steady himself during the turn.\n\n\"Intercept in ten minutes,\" Naumov said.\n\n\"Watch Officer, make your depth sixty meters.\"\n\nCaptain-Lieutenant Andreev, the current watch officer, echoed the order. Those standing in the CC shifted their stance as the bow of the boat tipped downward.\n\nFive minutes passed. Dubivko knew that if he miscalculated the merchant ship's depth or his angle of approach, he could crash B-36 into a pair of massive propeller blades. He recalled that another boat, while operating in the Barents Sea, suffered an almost life-ending collision while attempting a similar scheme. The captain of that submarine miscalculated and smacked into the stern of the cargo ship they were planning to hide under. The ship's propeller blades sliced into the boat's bridge and conning tower and caused catastrophic flooding in the CC. After surfacing to make temporary repairs, the ill-fated submarine limped back to Polyarny. Had the weather been rougher, they might have sunk. Dubivko's mouth went dry as he remembered the story. B-36 was now 5,000 miles from home.\n\n\"Watch Officer,\" Dubivko said, \"make your depth forty meters, speed seven knots.\"\n\nAndreev repeated the order, and the bow tilted upward.\n\n\"Attention in the CC,\" Dubivko said. \"I intend to close within one hundred meters range and ten meters below the merchant ship's hull. After they pass overhead, we will turn about and match their course and speed. The tanker's propeller noise should allow us to hide in their wake. We will need steady hands on helm and depth control.\"\n\nAndreev shot Dubivko a concerned look.\n\nDubivko returned the look and said, \"Take us in, Watch Officer.\" Through the comm he said, \"Acoustic, range to tanker?\"\n\nPankov answered from acoustic. \"Control, 500 meters and closing.\"\n\nDubivko glanced at the shallow depth gauge on the starboard bulkhead. He then moved a few feet to his left and looked over Naumov's shoulder. The navigator drew lines on a small sheet atop the Plexiglas on the nav table. The lines depicted two merging contacts heading southwest. Dubivko studied the contact markings, did a few calculations in his head, and returned to his previous location near the watch officer.\n\nA few minutes later, Dubivko ordered a single-range ping using active sonar on low power. He knew that U.S. destroyers and aircraft could potentially detect even one ping, but running into the blades of a merchant ship posed a greater risk. Once they closed the distance to the tanker, passive acoustic would be almost useless. The ship's massive screws generated too much noise to discern any kind of range estimation, and bearing information would be meaningless. At that point, Dubivko would need to rely on periscope observation from the conning tower.\n\n\"Acoustic, range?\" Dubivko said.\n\n\"Control, 300 meters,\" Pankov replied from the acoustic room.\n\n\"Watch Officer, bow planes up five degrees, come to thirty meters depth.\"\n\nAndreev reiterated the order, and the boat inched upward. Dubivko took a step toward the conning tower ladder. \"Stand by to open hatch.\"\n\nA sailor darted up the ladder and waited, both hands gripping the cold steel of the hatch wheel.\n\n\"Steady at thirty meters.\"\n\nThe sailor muscled the wheel, and a sheet of ocean water splattered onto the deck. Several drops splashed Dubivko's face. He licked his lips and tasted salt.\n\nAs the waterfall diminished to a few drops, the sailor stuck his head into the conning tower, did a quick visual sweep, then slid back down the ladder into the CC. After a verbal \"all clear\" from the sailor, Dubivko gripped the sides of the wet ladder and climbed upward. Behind him followed Saparov, as regulations stipulated that only the political officer could join the captain or authorized relief officer in the conning tower.\n\nOnce inside the small enclosure, just above the CC, Dubivko asked for an update. \"Acoustic, range to tanker?\"\n\n\"Control, passing beneath the contact now,\" Pankov said.\n\nDubivko had approached the tanker head on and now intended to make a 360-degree turn and follow the ship while hiding behind her.\n\nKopeikin relayed the order to the helmsman. Dubivko held his breath as he heard propeller blades thrashing above him. He knew that maintaining an exact distance and depth was now critical. The vacuum produced by the surface effect of ocean water rushing beneath the tanker's hull posed an immense danger. Just a few meters too close, and they could be sucked into the ship's deadly propeller blades.\n\n\"Acoustic, range?\" Dubivko asked again.\n\n\"Control, we've passed underneath. Range is opening, now twenty meters.\"\n\nDubivko called down to the watch officer. \"Watch Officer, come right to course two-five-five, increase speed to ten knots.\" He then depressed the switch for the approach periscope. \"Podnyat periscope.\"\n\nSaparov said nothing as he stood behind Dubivko and observed. Dubivko knew that the officer's presence in the conning tower had nothing to do with good seamanship and everything to do with politics. No matter the years, sacrifice, or evidence of loyalty, Moscow conservatives never granted full trust to their submarine captains.\n\nDubivko rested his twitching eyelid against the rubber eyepiece on the scope. An eerie glow from a carved moon shimmered on the wave tops as he spun the scope left and searched for the tanker's stern light post. He knew that if he steered too far to port or starboard, the metal shield around the light would prevent the glow from being seen. Viewing only darkness, Dubivko swung left, then right. Still nothing.\n\n\"Control, contact has changed course, now heading two-seven-zero.\"\n\nDubivko's ears burned as a wave of panic swept past. He now knew why he couldn't see the ship: they had changed course. He fought to stay calm, to not show signs of concern in front of Saparov as he called down to the CC. \"Watch Officer, right two degrees rudder.\" Again he moved the scope back and forth. Still no light. \"Left two degrees rudder!\"\n\nB-36 edged to port. Still no sign of the tanker.\n\nThen, suddenly, a twinkle. Was that a star? Dubivko squinted and stared. Out of the black, another blink. Then a glimmer. Finally, the stern light came into view, resting on a stanchion about six meters above the waterline.\n\n\"Watch Officer, steady on this course,\" Dubivko ordered.\n\nAs his eyes adjusted to the dark, he could now see the vessel's main deck sprinkled with equipment and lifeboats. Judging by how much of the ship protruded above the waterline, he figured she must be running with a light load. The reduced tonnage caused the ship's large propeller to break the surface as it churned the ocean into white foam. Dubivko smiled. A breaching prop translated to more turbulence and noise, making it harder for ASW aircraft sonobuoys to discern B-36's hull from that of the tanker's. The merchant ship's abundance of steel would also help mask the submarine's electromagnetic signature, and the foamy wake made periscope detection almost impossible.\n\nWith any luck, the tanker would maintain a steady course and speed, at least long enough for B-36 to make it through the passage. As Dubivko watched for signs of life aboard their newfound friend, he wondered what flag this ship operated under. For all he knew, they could be American. If so, he imagined how the tanker's captain might react should he learn that a Soviet submarine lurked behind him like a silent leech.\n\nThe glow of a cigarette came to life on the deck of the ship, illuminating a silhouette near the stern. As he slung his arms over the periscope's handles and played the part of a Peeping Tom, Dubivko wondered if the man might be from Miami. Captain Dubivko was now in his element, in command of his vessel, heading into harm's way while cleverly hiding from the enemy. Excitement, pride, and fear owned equal portions of him, none more so than fear, as this honed his senses, heightened his instincts, and increased his odds of avoiding a fatal mistake. Then Dubivko heard an unwanted report from acoustic.\n\n\"Control, contact is slowing.\"\n\nThe sound of the tanker's thunderous screws grew faint. The stern light blinked once and then vanished into the darkness.\n\n\"Control, contact has slowed to eight knots, bearing zero-one-zero.\"\n\n\"Shit!\" Dubivko said to himself, hoping that Saparov did not overhear. For reasons unknown, the merchant ship was decreasing speed and turning to port. \"Watch Officer, slow to six knots.\"\n\nDubivko called down to the CC. \"Navigator, are there any shallows or sandbars nearby?\"\n\n\"No, sir,\" came the reply. \"But if she maintains this course, she will run aground.\"\n\nDubivko frowned. \"Acoustic, report fathometer depth.\"\n\n\"Control, one hundred meters,\" Pankov replied.\n\nDubivko quickly weighed his options. Without the tanker's turbulent noise to hide under, B-36 could be exposed within minutes.\n\n\"Control, three knots and slowing.\"\n\n\"Watch Officer, slow to three knots,\" Dubivko said.\n\n\"Control, she's dropping anchor,\" acoustic reported.\n\n\"Shit.\" Dubivko said, no longer attempting to hide his concern from Saparov.\n\n\"Acoustic, fathometer depth?\"\n\n\"Control, eighty meters.\"\n\n\"All stop,\" Dubivko said.\n\nThe boat slowed and crawled to a stop.\n\n\"Zhukov,\" Dubivko said.\n\n\"Sir?\" Zhukov replied from below in the CC.\n\n\"Raise the zenith navigation scope. Check for aircraft.\"\n\nHydraulics whispered. Dubivko also heard a few clicks and squeaks as the scope swiveled in its housing.\n\n\"Nothing, sir,\" Zhukov reported.\n\n\"Navigator,\" Dubivko said, \"find us a clear patch.\"\n\nNaumov relayed a course and heading, and Dubivko issued maneuvering orders.\n\n\"Watch Officer,\" Dubivko said once they'd reached the new location, \"make your depth eighty meters. Set us on the bottom.\"\n\n\"Yes, sir.\"\n\nDubivko still did not know why the tanker came to a stop but surmised she needed to make repairs of some kind. Regardless, for now, with dawn approaching, he had no choice but to sit and wait. Thankfully, they still retained almost a full battery charge. Given their close proximity to the tanker, snorkeling was out of the question.\n\nWith Saparov close behind him, Dubivko descended the ladder into the CC and closed the hatch. Hours passed without a sound from the tanker. At 3:00 A.M., while Dubivko studied the nav plot with Naumov, he heard an excited report from Pankov in acoustic.\n\n\"Control, new contact bearing two-five-five! Two tandem screws.\"\n\n\"Watch Officer, make turns for three knots, ascend to twenty meters,\" Dubivko said. From their position on the bottom, they could hear but not see, and Dubivko wanted to know what was going on up there. Who was approaching the tanker? A repair ship? An American destroyer?\n\nThe boat angled toward the surface and the answer.\n\n\"Stand by to open hatch.\"\n\nAgain a sailor stood poised by the conning tower ladder.\n\n\"Acoustic?\" Dubivko said over the comm line. \"Contact type?\"\n\nA few seconds passed, then Pankov said, \"Control, American destroyer, speed fifteen knots.\"\n\nNow what? They could descend back to the bottom and wait, but for how long? On the other hand, if for some reason the destroyer saw their periscope and went active with sonar, they'd be sitting ducks.\n\nAt twenty meters, the hatch flung open, and Dubivko climbed back into the conning tower. He peered through the scope and zeroed in on the destroyer's lights. Though he could not make out the enemy ship's bristling antenna, spinning radar, and menacing armament in the dark, he imagined all of these in his mind. He ordered the ESM radar detection mast raised. An occasional beep could be heard from the Nakat ESM panel in the CC as the system detected enemy radar. NATO called this system Stop Light. As the intervals between the beeps decreased, Dubivko knew that the destroyer's radar was starting to lock on to their masts, and he could not afford to keep them up much longer. The ESM antenna array, which sat between the two periscopes on the sail, carried four bands of direction-finding antennas that captured enemy radar signals and fed data to a cathode-ray tube (CRT) screen. Enemy ships and planes used radar to try to spot submarine masts protruding above the surface, and when they ventured too close, the Nakat system beeped with a warning.\n\nThe American destroyer closed to within 1,000 meters and stopped. Then a signal lamp pierced the dark as it flashed out a message in Morse code. Dubivko translated the dit-dah flashes in his head. \"Alpha, Alpha,\" he said to himself, feeling Saparov's presence behind him. \"What ship?\"\n\nDubivko lowered the scope to avoid detection. He raised the mast again a few seconds later, swung the scope toward the tanker, and watched for a reply. Flashes appeared like shooting stars against the backdrop of endless black. Given the lamp's location on the upper level of the tanker, the large superstructure and bridge blocked most of the light, and Dubivko could see only a few of the flashes.\n\n\"Shit!\"\n\nDubivko turned the scope back toward the destroyer and waited. Long seconds ticked by without a response. The ESM beeps increased in frequency. Then, from out of the ink, the flashes came. Dubivko whispered the translation aloud. \"Radio frequency three four three point eight.\" He lowered the masts and called down to the CC, \"Zhukov, raise the antenna, and have our English translators listen in. Three four three point eight.\"\n\nZhukov complied. Minutes later he called back up to Dubivko. \"The tanker is Norwegian. A boiler failed, and they are bringing a spare online. The Americans offered assistance, but the Norwegians declined.\"\n\nAfter hearing this report, Dubivko raised the scope and watched as the destroyer's running lights turned to starboard and dimmed. \"Acoustic?\"\n\nPankov said, \"Control, American destroyer is heading away, making turns for ten knots and accelerating.\" Chains rattled from outside the hull. Pankov issued another report. \"Control, the merchant ship is reeling in her anchor.\"\n\nDubivko emptied his lungs in relief.\n\nAT 10:12 P.M. ON OCTOBER 22, a few hours after President Kennedy's speech to the nation regarding nuclear missiles in Cuba, R-Branchers at an NSA listening post picked up a high-priority message sent from the Soviet spy ship Shkval on station near the Bermudas. The Russian merchant ship Alantika received the message and rebroadcast the same to Murmansk, near the Foxtrot submarine's home port of Sayda Bay. Excited R-Branchers informed Net Control, reporting that \"this type of precedence is rarely observed. Significance unknown.\" When the NSA received the flash message, officials there feared the worst. Were the Soviets planning to run the blockade? Were their Foxtrot submarines preparing to attack the U.S. fleet? Were they minutes away from launching a nuclear attack?\n\nA few hours after midnight, a flurry of radio signals hit the air as Soviet merchant ships called home to Russia, asking for instructions. One ship sent an urgent plea for help.\n\nMoscow remained silent, and their cargo ships, along with the rest of the world, came to a halt and waited for an answer.\n\nJust before sunrise on October 23, the residents of Palm Beach, Florida, were shocked awake by the rumbling sound of a squadron of P2V ASW aircraft. The P2V's twin-props sliced the tropical air as high-pitched engines gulped gallons of fuel to push through the humidity. Sleepy-eyed spectators watched with curiosity as the planes lowered their landing gear and descended onto the runway at Palm Beach International Airport. Following just behind the P2Vs, a squadron of B-47 bombers lumbered onto the ground, tires screeching and jet engines roaring to slow the aircraft.\n\nFarther south, thirteen attack submarines slid from the docks at Key West Naval Base as dungaree-clad sailors scurried topside to stow lines and gear. With torpedo tubes fully loaded and pointed toward Cuba, the black silhouettes disappeared beneath the choppy waters of the Florida Straits. A division of Gearing-class destroyers followed the submarines, each armed with one ASROC (antisubmarine rocket) launcher, triple torpedo launchers, and two \"DASH\" antisubmarine helicopters. One lone submarine and destroyer stayed behind to defend the base.\n\nOn board the USS Robert E. Lee (SSBN-601), on deployment in the Atlantic, Commander Charles Griffiths received a change of orders from CINCLANT (Commander-in-Chief, Atlantic). His original mission orders were to conduct a Follow-on Test (FOT) firing of the Polaris A1 ballistic missile in the wake of the first launch by the USS George Washington. Now he was instructed to arm all missiles and ensure readiness to fire on the Soviet Union within a moment's notice. Griffiths knew that four other SSBNs (ship submersible ballistic, nuclear) received the same orders, and eighty nuclear weapons were poised to turn Russia into a radioactive wasteland.\n\n\"We were mindful that our loved ones were in imminent danger and that we could be facing an unbelievable future,\" said Griffiths. \"Yet we would have fired as ordered, and no one on board would have tried to prevent it.... It was up to the president and God to avoid Armageddon.\"\n\nIN NAVY FLAG PLOT ROOM 6D624 at the Pentagon, Admiral Anderson paced nervously. In this nerve center of the navy's planned blockade of Cuba, charts and maps of the Caribbean and Atlantic oceans lined entire walls, where personnel meticulously plotted the movements of every warship in the area. Something strange was happening in the Sargasso Sea near Cuba. The HFDF station at Homestead sent a flash message moments earlier reporting that the Soviet merchant ship Bol'shevik Sukhanov \"has altered course and is probably en route back to port.\" Yet another report said that \"HFDF fix on the Soviet cargo ship Kislovodsk, en route to Cuba, indicates that the ship has altered course to the north.\"\n\nDespite indications that the Soviets might be backing down, or at least taking a breather, Anderson harbored concerns that his sixty ships along the Walnut line were still having trouble finding those four Foxtrot submarines. The ASW boys on board ships and flying in planes would get a sniff, run down the track, but then lose the contact. A recent CINCLANT situation report showed only nine ASW hits since October 22. The SITREP (situation report) also revealed that Aircraft TG 136 got a Hot status on two Foxtrots, prosecuted, but never found anything. Anderson knew that some of those hits came from SOSUS arrays, but not many.\n\nWhat did appear consistent were the Boresight tips. Though he'd not yet been fully briefed on the technology, he was familiar enough with standard HF direction finding to understand the concept. When the Soviets sent out a burst, the stations recorded the transmission, then tried to find a bearing after the fact. Sounded simple enough, though he knew a bunch of beacon heads spent more than a year figuring it out.\n\nSOSUS occasionally helped find the Foxtrots when they ran on diesel engines at night, but given the long distances from most of the arrays, getting good bearings was tough. When those boats went silent and deep on batteries, SOSUS became useless. Fortunately for the good guys, the Foxtrots transmitted multiple times every afternoon, and when they did, Boresight stations could get bearings, even if not very accurate ones.\n\nStill, until ASW forces nailed those submarines cold and forced them to surface, the blockade was at great risk of failure. Anderson knew that Attorney General Robert Kennedy, following a recent intelligence briefing, said that \"the president ordered the navy to give the highest priority to tracking the submarines and to put into effect the greatest possible safety measures to protect our own aircraft carriers and other vessels.\" Anderson also knew that the president sent an ultimatum to Khrushchev stating that any Soviet submarines detected near the quarantine line must surface and be identified. The fleet operated under directives to use international code signals and nondestructive explosive charges to warn the subs, but if that failed, Anderson's orders were clear: use every means possible to sink those Foxtrots. That is, if he could find them.\n\nMeanwhile, the defense readiness condition (DEFCON) catapulted to its highest level ever. The military established DEFCON as a measure of the activation and readiness level of the U.S. Armed Forces, and standard peacetime protocol dictated DEFCON 5. DEFCON 1, never formally declared in U.S. history, was synonymous with war. DEFCON 2 ran a close second and also had never been mandated.\n\nA few years earlier, then Secretary of Defense Thomas Gates ordered the DEFCON raised to level three to test the response system. Although he intended at that time to keep the alert secret, the Soviets, while monitoring U.S. force movements, discovered the change. Bristling with anger, Khrushchev called the DEFCON alert a \"provocation.\" When U.S. forces were ordered to DEFCON 3 on October 22, no attempt was made to mask the intention. The message and the response were deliberately transmitted on open frequencies. Three radar bases also activated Operation Falling Leaves to monitor the Soviet response, including any missile launches from Cuba, but Anderson knew that the radar systems were experimental and unreliable.\n\nEarlier that day, on October 23, SAC received orders to invoke DEFCON 2 for the first time in history. Although the rest of the military remained at DEFCON 3, preparations were under way to ensure maximum readiness in the event that conditions changed. Anderson hoped that never happened.\n\nThat afternoon, in the third meeting of the Executive Committee (ExComm) of the National Security Council with President Kennedy, Secretary of Defense Robert McNamara reported on the plans for naval interception, noting \"the presence of a submarine near the more interesting ships,\" and warned that radio silence should be imposed. The committee agreed that should a Soviet submarine interfere with the blockade or adopt a hostile posture, the enemy sub must be sunk.\n\nAfter visiting a Boresight station to try to optimize their systems, William J. Reed returned to Washington, D.C. There he met with a group of NSA engineers tasked with Boresight research and development. They burned midnight oil and chain-smoked while frantically trying to solve the perplexing problems that Reed reported finding in the field. During those infrequent times when the technology worked, it worked reasonably well. Unfortunately, more often than not, the systems were plagued by intermittent errors, annoying interference, and inconsistent results. Reed's boss, Commander Kaye, reminded everyone that unless these issues could be resolved in time, President Kennedy's blockade might become as \"useless as tits on a bull.\"\n\nAFTER SNEAKING THROUGH THE TURKS PASSAGE unscathed, Captain Dubivko hid B-36 in the wake of the Norwegian tanker until they neared 150 nautical miles north of Haiti. He then headed west toward Cuba and slowed. He circled on station until nightfall, then came up to recharge batteries. This night marked the final time they could surface to snorkel and vent exhaust fumes externally on either side of Compartment Five. From now on they'd need to switch to snorting. This operation ensured stealth in enemy waters and could be done at depths up to fourteen meters. Snorting required raising the snorkel mast to suck in air directly to the diesels. Fumes were piped back up through the conning tower and cooled prior to venting into the ocean. Doing so lowered the probability of detection by reducing the boat's heat signature.\n\n\"What was her name?\" Dubivko asked while peering through his binoculars on the bridge. Two of B-36's engines hummed from below decks, scattering the smell of burnt diesel fumes into the tropical night air.\n\n\"Who?\" Zhukov asked while studying the gyrocompass to his right.\n\n\"The Norwegian tanker. What was her name?\"\n\n\"Cretan Star, I think. At least, that's what the American destroyer called her.\"\n\n\"Too bad,\" Dubivko said. \"I was hoping for a female name.\"\n\n\"Sir?\" Zhukov said.\n\nDubivko lowered his binoculars and smiled. \"We hid under her skirt for hours. Shouldn't she be a woman?\"\n\nZhukov shook his head and laughed. Long days filled with tension made that luxury rare, and Dubivko knew that even less levity lay ahead. B-36 received new orders after crossing the Turks Passage behind the Norwegian oil tanker, directing them to assume station to the northeast and monitor U.S. ship movements. On station 200 miles south of Bermuda and 600 miles east of Cuba, they snorkeled at night and ran silent on battery power during the day, still faithfully coming shallow in the afternoon to transmit an update.\n\nASW aircraft continued to plague them. P2Vs and newer P3 Orions splashed sonobuoys along precise patterns that nearly matched B-36's track, making Dubivko wonder if the Americans had recruited a team of psychics. On the other hand, Zhukov reported that American pi lots were lax in their communications protocol, often transmitting unencoded messages over open HF frequencies. This was especially true for planes attached to the carriers Essex and Randolph operating to the north in the vicinity of B-130 and B-59. Determining which pi lots were communicating from which planes became a great pastime for B-36's radio operators, who often made friendly wagers with each other as to which call sign matched which pi lot.\n\nAvoiding enemy detection exhausted the crew but took second fiddle to the heat. Deep inside warm tropical waters, given the inadequacy of their air-conditioning system, B-36's insides turned into a blistering sewer pipe. Temperatures surpassed unbearable, especially in the engine room, where they often eclipsed thirty-seven degrees centigrade.\n\nDubivko ordered the rationing of drinking water, allowing each man to consume no more than one glass per day. He also granted one glass of red wine at dinner. The crew's health deteriorated rapidly, given poor personal hygiene and constant exposure to humidity, high temperatures, and diesel fumes. Most suffered from painful rashes or oozing skin ulcers. To combat this problem, the doctor handed out disposable towels every day. When those supplies ran out, he used alcohol-doused cotton balls. Many of the sailors just stuck the balls in their mouths and sucked out the alcohol instead of using them to treat their rashes.\n\nZhukov's English-speaking experts, after sucking their alcohol swabs dry, continued to monitor shortwave and high-frequency broadcasts, such as Radio Liberty, BBC, and Voice of America. These broadcasts revealed that Soviet statesman Anastas Mikoyan, one of Khrushchev's closest advisers, was trying to negotiate a compromise with the Americans after Kennedy reacted so harshly to the discovery of nuclear missiles in Cuba. Dubivko could not help but wonder if their mission might soon come to an abrupt end, either by way of war or by recall to Russia. He knew that in the event of a recall, no one in power would admit to failure.\n\nDubivko informed his crew about the blockade to ensure they stayed alert. He also told them that hundreds of aircraft and dozens of ships from the U.S. Atlantic Fleet were hell bent on finding and possibly sinking their submarine. Zhukov mentioned that an operator heard that the Americans had established prisoner of war camps in Florida, and Dubivko let his mind play on the possibility of meeting someone from Miami.\n\nWhen the sun rose, bad luck returned. The chief engineer, Captain Lieutenant Potapov, reported that the upper lid on the VIPS\u2014the imitation cartridge projection device used to fire decoys to ward off enemy torpedoes\u2014had been damaged in the last storm. Potapov insisted that it would be suicide to submerge the boat deeper than seventy meters. The depth limitation posed significant problems. First, most of the thermal layers under which they might hide were below that depth. Second, when they operated at shallower depths, ASW planes could find them more easily with magnetic detectors. Repairing the lid required surfacing, which became impossible once they neared Cuba and ASW activity intensified.\n\nThe OSNAZ specialists reported that there were now at least three carrier groups operating in the Sargasso Sea, along with hundreds of aircraft flying about\u2014all intent on finding them. Cloudless blue skies aided the enemy's objective. Dubivko had continued to come shallow at night to snort and descended back to seventy meters during the day to hide.\n\nFive or six times, while snorting under a blanket of dazzling stars, they had spotted a plane through the periscope or detected a radar signal nearby on the Nakat ESM mast. Dubivko then yelled, \"Srochnoiya pogruganye!\" and the boat made a quick dive. Zhukov avoided using any standard HF transmissions, as they knew the Americans were trying to locate them with their Huff Duffs. Still, every afternoon they had been compelled to come shallow again to receive a burst transmission on the SBD from headquarters and send a verification of receipt. Dubivko did not even consider the possibility that a little more than 500 nautical miles away, someone might be listening.\n\n## CHAPTER EIGHT\n\nOne death is a tragedy; one million is a statistic.\n\n\u2014JOSEPH STALIN\n\nON THE MORNING OF OCTOBER 23, when Communications Technician John Gurley entered the 1,400-square-foot Boresight building in Homestead, Florida, he did not have an inkling that this day would become one of the most memorable in his life. The day began like most others, with an abundance of routine, coffee, doughnuts, bad jokes, a couple Boresight flashes, and some whining from Lieutenant Clower about staying diligent. But that all changed after lunch.\n\nThat tall ensign from NSG had visited them again a few days earlier. What was his name, Reed? The ensign brought a couple of his techies back, and they spent the better part of a day tweaking and testing the Boresight equipment again. Damned if those guys didn't manage to improve the bearing accuracy by half a degree or so. In a big ocean, that could mean a lot. Ensign Reed spent an hour with Clower talking hush-hush about something. After the NSG guys left, Clower called a meeting. He told everyone some harry shit was going on near Cuba, and they needed to crank up the alert factor to full throttle. Clower didn't offer many details but said they should be looking for Foxtrot call signs. Foxtrot submarines near Cuba? Gurley could only imagine why that might be, as he thought about his trek from unemployed to sub hunter.\n\nWhen he graduated from high school in 1956, Gurley's parents didn't have enough money to send him to college. Finding a job in Dallas during those days was pretty tough, so he enlisted in the navy to avoid becoming a ground pounder in a rice paddy. During boot camp in San Diego, someone noticed his Texas drawl and said he should learn how to communicate better. He took their advice and struck for communications technician. Weeks later they sent him to radioman school in Imperial Beach just south of San Diego.\n\nAfter graduating from radioman school, Gurley received orders to Morocco. He spent an exotic three years in the desert, then wound up in the frigid north near Kodiak, Alaska. In between hunting and fishing in heaven's wilderness, Gurley and twenty-six other sailors ran the Huff Duff station there, reporting to a chief warrant officer who let them play as much as they worked to keep morale and efficiency at a peak. After a year in Kodiak, the navy pulled him out of one paradise and plopped him into another: Homestead, Florida. They needed more R-Brancher radio \"collection\" experts there.\n\nAt Homestead, Gurley rode the bus every day from the barracks to the operations building down Card Sound Road. In the ops center, he and a dozen other R-Branchers sat in front of various stations and monitored Soviet traffic, while M-Branchers did their maintenance thing, I-Branchers listened for intelligence tidbits, T-Branchers analyzed signals, and O-Branchers ran the shop. Four R-Branchers usually searched for active targets, while one or two others encoded and decoded messages sent from the crypto center.\n\nThe job really wasn't that tough until Ensign Reed and his team waltzed in and set up all that Boresight equipment. Now they'd get flashes from NC that included time windows, so they had to find the related reel-to-reel tape, load it up, and listen on the specified frequencies during specific time slots across maybe thirty or forty bearings. Problem was, they were looking for a burst that lasted only seven-tenths of a second, so they were forced to play the thing back and forth ad nauseam, sometimes for hours.\n\nFour of these systems recorded signals on various frequencies starting at 840 kilohertz. When the systems heard a burst signal, they were supposed to alarm, but the loud bell went off only if they got a strong enough hit. If another station gained a stronger signal to trigger the alarm, they sent NC a tip-off, and NC sent out a flash. That's when Gurley and his colleagues spent hours poring through recording tapes to see if they also picked up the burst, because one bearing to a target could not accurately pinpoint a transmitting sub's location.\n\n\"In those days we still used the old compass rose board,\" said Gurley. \"When we got a bearing, we manually ran a piece of string from one end of the board to the other along the bearing. We prepared reports to Net Control on a machine that used a punch tape machine. We'd wrap the yellow paper tape around our finger to form a tight ball and then shove it inside of a Coke can. The ribbon came with two carbon copies that made it thick and hard to handle. We shoved the can into a tube that dropped down to the communications center. The boys downstairs pulled the report out of the Coke can and sent it off to NC. Very high tech.\"\n\nOperators often played jokes on new guys by ordering them to unravel the three-ply tape, which, if done incorrectly, resulted in carbon-stained fingers and a mess that resembled a ticker-tape parade. \"Our job was demanding but mostly routine,\" said Gurley. \"A little levity now and then kept us from going stir crazy.\"\n\nThat day, on October 23, an alarm bell interrupted the routine. Gurley wheeled his chair over to the recorder and started the procedure he'd been taught by Ensign Reed and his team. Check this, verify that, do something else. He and other operators spent the next hour analyzing the recording and checking bearings. The transmission call sign pointed to a probable Foxtrot-class submarine.\n\nGurley went to the compass rose board, grabbed a piece of string, and pulled it down the line of the probable bearing. The string ran just north of Cuba and due east of Florida. Based on the signal strength, he figured the Foxtrot was probably less than 1,000 miles away. Gurley wheeled over to the punch tape and generated a report for the Coke can. He dropped the can into the tube and waited.\n\nA half hour later, the station received a copy of the flash sent to the other stations from NC. Gurley waited another hour and then contacted a buddy at Net Control. He asked if any of the other stations had reported bearings to that contact. The answer came back as affirmative. Gurley pulled two more strings across the compass rose board, representing those bearings. When he saw where the lines converged, his heart started pounding so hard he thought he'd pop a shirt button. He didn't know it then, but he and other Huff Duffs had just nailed Captain Savitsky's B-59, operating near the Bahamas a few hundred miles southeast of Florida.\n\nIN THE FLAG PLOT ROOM, THE navy's command center at the Pentagon, Admiral Anderson's eyes ached from lack of sleep. Too much coffee soured his stomach, but he downed another cup anyway. The black \"navy joe\" offered that bitter-burnt taste that Anderson liked so well. Just one more reason to go navy. He squinted and stared at the dozens of red tags dotting the large wall chart. Each tag, numbered C1 through C29, denoted a probable submerged contact, most likely a Foxtrot or maybe a Zulu. Positively identified submarines, like the Zulu spotted on the surface days earlier, drew B designations.\n\nAround thirty men and women in Flag Plot kept track of estimated course and speed information for B and C contacts and, most importantly, their probable distance to any U.S. surface ships in the quarantine zone. Another set of flags on the plot followed Soviet merchant traffic as those ships approached the Walnut line, the outer perimeter of the quarantine zone. Anderson suspected that many of these ships carried nuclear missiles and launcher parts destined for Cuba. After Kennedy's quarantine went into effect, around sixty U.S. Navy ships now patrolled along the Walnut line that arced from the tip of Florida to an area just south of Cuba. If Soviet merchant ships tried to cross the line, the navy had orders to stop them. But with four Foxtrot submarines lurking nearby, accomplishing that task could prove difficult, if not deadly.\n\nAnderson turned to see Defense Secretary Robert McNamara stride into the room with Roswell Gilpatric, a New York lawyer turned deputy defense secretary, who followed in a \"brown nose\" position. Behind the two, an entourage of clean-cut press-corps bulldogs came outfitted in suits, ties, and dresses. Anderson scowled. The last thing he needed right now was a bunch of McNamara's White House public-relations questioners, especially when every answer required a heavy dose of sidestepping.\n\nMcNamara shook Anderson's hand and asked for an update. The admiral pointed to the set of merchant-ship flags on the plot. McNamara's eyes followed Anderson's finger. While the press dogs scribbled in tiny spiral-bound notebooks, Anderson explained that they expected the first sortie of Russian ships to hit the line at around 10:00 A.M. the following day. He said that things started to change after Kennedy's quarantine announcement a few days earlier. Where once these ships acted as lone wolves heading toward Cuban ports, they now appeared to be forming one large phalanx, with the tanker Bucharest leading the charge. They could not explain why.\n\nMcNamara asked about the red dots. Were there really that many Soviet submarines out there? Anderson said no, that many flags represented more subs than the Soviet navy possessed. He explained that they plotted each reported sighting but considered most false positives. They believed that less than a half-dozen Soviet subs were in the area, and the navy's ASW forces were most concerned with finding the four Foxtrots. An underling from the press office raised her hand. \"What's a Foxtrot?\" she asked. Anderson patiently reeled off a few specifications about that class of Soviet submarine.\n\nMcNamara asked how the navy's ships intended to force Soviet subs to the surface once they found them. Anderson took another gulp of coffee and recalled his days as the commander of the Sixth Fleet working for then Chief of Naval Operations Admiral Arleigh Burke. The admiral more than once grumbled that McNamara possessed the steel-trap mind of an encyclopedia but the detail-oriented focus of a micromanager who asked more questions than a five-year-old.\n\nKnowing that McNamara loved the devil in the details, Anderson doused him with a fire hose. He delivered a long diatribe about international signal codes transmitted by the United States on open frequencies. They knew Soviet submarines monitored these frequencies and so should be informed as to the expected rules of engagement during the conflict. That morning the U.S. Naval Oceanographic Office broadcast Mariner's 45\u201362, Special Warning 32, which spelled out submarine surfacing and identification procedures. The warning stated that U.S. ships and planes would signal Soviet submarines by dropping four to five hand grenade\u2013sized explosives, followed by the international sonar-transmitted signal IDKCA, which means \"rise to the surface.\" Submarines hearing this signal had to surface on an easterly course at once and move slowly away from Cuba.\n\n\"And if they don't?\" McNamara asked. Pencils poised, the press geeks stared at Anderson with curious eyes.\n\n\"We will sink them,\" Anderson said.\n\nMcNamara didn't blink. Anderson hoped that, given the presence of the press staff, the SecDef would not probe for more. Besides, in their meeting days earlier, he'd already briefed McNamara, along with Kennedy and the ExComm group, about these rules of engagement. He'd explained that shots would be fired across bows and then into rudders of noncompliant merchant ships. Neither McNamara nor Kennedy liked the plan, but they conceded that there were no alternatives.\n\nMcNamara pointed to a lone flag on the plot, positioned some distance away from the others around the quarantine area. The flag represented the position of a U.S. Navy destroyer off the coast of Florida, almost a hundred miles east of the Walnut line. \"Why is that ship out of line?\" McNamara asked.\n\n\"She's prosecuting a probable submerged contact operating near the Bahamas,\" Anderson said.\n\n\"I don't understand,\" McNamara said. \"I thought you said that most of the sightings were false positives. If you move every ship away from the Walnut line, we won't have a blockade.\"\n\n\"This submarine hit came from a more reliable source,\" Anderson said.\n\n\"What source is that?\" McNamara pressed.\n\nAnderson remained silent. Neither the press staff nor many of his own watch officers were cleared for that information.\n\n\"Admiral?\" McNamara said.\n\n\"Mr. Secretary,\" Anderson said, motioning toward a side room. \"Please come with me.\"\n\nMcNamara glanced at Gilpatric. \"I'll be right back.\"\n\nMcNamara followed Anderson into the small \"inner sanctuary,\" which was reserved for private conversations regarding sensitive \"need to know\" information. A few chairs stood guard near a table that held a couple of used coffee cups and a navy regulations manual. Anderson leaned on the back of a chair and proceeded to tell McNamara about Operation Boresight. He explained that SOSUS did a fair job of helping to track the four Foxtrots when they snorkeled, but this occurred only at night and only up to 150 nautical miles away. Fortunately, these submarines transmitted multiple burst signal updates to Moscow every day in the late afternoon. When they did, HFDF stations equipped with Boresight technology obtained ballpark fixes.\n\n\"Ballpark?\" McNamara asked.\n\n\"Around fifty nautical miles,\" Anderson said. \"That's why that ship is out of line. She's prosecuting a probable Foxtrot hit initially received from our HFDF station in Homestead.\"\n\nMcNamara stood up straight. \"I want to know more about this Boresight thing. Send someone to the White House tomorrow to brief me and the president.\"\n\n\"This is a highly classified program, I don't know\u2014\"\n\n\"Send someone, Admiral.\"\n\nAnderson nodded, said nothing.\n\n\"Now,\" McNamara said, \"about those rules of engagement. I don't give a damn what the rules say. Your boys are not to fire a single shot into a rudder without direct permission from the president or me. Is that clear?\"\n\n\"Sir, we have sixty ships on that line. They need to be able to act quickly and in dependently. We need to trust that our captains will do the right thing.\"\n\nMcNamara fumed. \"The purpose of this quarantine is to send a delicate diplomatic message to Khrushchev, not start a shooting war!\"\n\n\"No one's going to\u2014\"\n\n\"You're damn right they're not!\" McNamara yelled, loud enough that Anderson was sure that Gilpatric and the press kids overheard the comment.\n\nFrazzled from days without sleep, his gut churning with coffee acid, his nerves stretched thin, Anderson picked up the thick bound navy regulations manual from the table and held it high. \"May I remind you, Mr. Secretary, that the navy has been running blockades by the book since the days of John Paul Jones. I think we know what we're doing by now!\"\n\n\"I don't give a damn about Jones! I want to know what you intend to do. Be a renegade or follow orders?\"\n\nAnderson marched to the door and turned the handle. As he stepped back into the Flag Plot room, he said, \"I always follow orders, Mr. Secretary. Now, I suggest you go back to your office and let us handle things here.\"\n\nHis face sour, McNamara strutted past Anderson and motioned for Gilpatric and the press nerds to follow. He turned one last time before exiting the room and said, \"I want that briefing, Admiral.\"\n\nAnderson nodded and turned back toward the wall chart. He stared at the flag representing the USS Cony, unaware that the navy destroyer \"out of line\" was running headlong toward a battle with Captain Savitsky's B-59.\n\nAT THE RUSSIAN EMBASSY IN WASHINGTON, D.C., the Soviet naval attach\u00e9, Vice Admiral Leonid Bekrenyev, formed his lips into a tight line as he read a recently received diplomatic message regarding the rules of engagement. The message stated that Soviet submarines, if found, would be forced to the surface by American ASW forces. He handed the paper to a radio operator along with instructions to inform the main navy headquarters in Moscow immediately and ensure they transmitted the message to the submarines near Cuba.\n\nPresident Kennedy met again with members of ExComm to review the latest quarantine intelligence, world reaction to the building crisis, status of negotiations at the United Nations, and potential incidents on the high seas. McNamara provided a detailed briefing on recent reconnaissance photos from Cuba, and the group debated the need to disperse planes at Florida bases in the event of attacks by Soviet MIGs.\n\nMcNamara then revealed what he'd learned earlier that morning from Admiral Anderson about the Soviet submarine sighting from a reliable source. He did not reveal the nature of that source. He expressed concern over the \"very dangerous situation since [Russian merchant] ships approaching the quarantine line are being shadowed by a Soviet submarine.\" Referring to the probable Foxtrot contact reported by Admiral Anderson, now being pursued by the USS Cony, he went on to say that \"there is a sub very close, we believe, and therefore it should be twenty to thirty miles from these [ships], and hence it is a very dangerous situation. The navy recognizes this [and] is fully prepared to meet it.\"\n\nKennedy asked what might happen if a U.S. destroyer was sunk by a Soviet submarine while trying to board and search a Russian merchant ship. Not receiving an acceptable answer, he went on to say, \"I think we ought to wait on that [boarding] today. We don't want to have the first thing we attack [be] a Soviet sub. I'd much rather have a merchant ship.\"\n\nWHEN WILLIAM J. REED HIT THREE days without sleep, Commander Kaye insisted that he head over to the officer's quarters at Fort George G. Meade, Mary land, and find an empty bunk for a few hours. Reed tried to argue but lost. He also tried to sleep, but his head kept spinning over the brewing crisis near Cuba. He knew there'd been hundreds of possible submarine sightings made by ships, planes, SOSUS, and Bore-sight stations, and so far they could not prove which ones were more accurate, though Reed harbored his own bias.\n\nAll hits were thoroughly analyzed by experienced ASW submariners, surface ship jockeys, or pi lots at ASW Force Headquarters in Norfolk, Virginia, but those professionals were still limited by the mantra \"Bad in, bad out.\" Reed's job, which kept him awake for these past few days, mandated turning that bad into at least something decent. When they heard that the Boresight HFDF station at Homestead got a solid hit on a Foxtrot near the Walnut line, Kaye congratulated Reed and his team for doing that job well by improving the bearing accuracy on Homestead's equipment.\n\nBased on what Boresight hits they did have, NSA estimated that four Foxtrots now encircled Cuba, one near the Bahamas east of Florida about 100 miles off the Walnut line, one southwest from there about 500 miles east of Cuba, another 100 or so miles south of that sub, and the fourth one around 700 miles west of the others, south of Cuba down near Jamaica.\n\nReed forced his mind away from the threatening Foxtrots and finally dozed off. A few hours later, Kaye shook him awake.\n\n\"Sorry to wake you, BJ,\" Kaye said, \"but we need to prepare for a high-level meeting.\"\n\nReed yawned and rubbed the sleep from his eyes. \"A meeting? Where?\"\n\n\"At the White House,\" Kaye said. \"We've been summoned by the president.\"\n\nON BOARD B-130, ON STATION EAST of the Bahamas, Captain Second Rank Nikolai Shumkov's face heated as he issued a string of profanities. The temperature in Compartment Five, which housed three large engines that reeked of diesel fumes, increased to more than forty degrees centigrade after they entered tropical waters. Sweat gushed from every pore on Shumkov's body and stained his clothes. Senior Lieutenant Viktor Parshin, B-130's chief mechanic, stood near one of the engines and tried in vain to wipe the black oil smudges off his hands with an old rag.\n\n\"I'm sorry, Captain,\" Parshin said, his voice almost a yell against the throng of two engines. \"One engine is still down, and a second could die at any time.\"\n\n\"I should have listened to you back at Sayda Bay,\" Shumkov said, shaking his head.\n\n\"No matter, Captain,\" Parshin said. \"If you had, we would not even be on this mission.\"\n\nShumkov could tell by Parshin's bright red face that the man had been on duty far longer than the mandated thirty-minute rotation in the superheated compartment. He also knew that his chief mechanical engineer spoke the truth. They would not be here had they elected to make repairs after discovering the hairline fractures in the drives. They would have been left behind.\n\nB-130 hit the water for the first time in September 1960. That made her an older sibling to the other three Project 641 boats on this mission. More experienced but not as healthy, B-130 was born with defects and suffered often from mechanical ailments. She came out of the yards with flaws in two of her diesel engines. When shipyard engineers discovered the hairline cracks, they insisted on immediate repairs, but the builders refused. Shumkov suspected a cover-up. No one wanted to admit to the mistake.\n\nThe diesels ran fine, but Shumkov feared the day when one or more would suffer a coronary and die, most likely in the middle of an important deployment that required extended use. Parshin expressed concerns about their engines, as well as their aging batteries. The geriatric two-volt cells, 448 of them located on the lower decks of Compartments Two and Four, were due for replacement. The electrolyte in these 650-kilo batteries ran hot during recharging, which could cause a fire or even an explosion. They also took longer to charge, sometimes more than twelve hours, which made B-130 a laggard behind the other boats.\n\nShumkov ignored his chief engineer's warnings and decided to live with the risks. Thriving on adventure, he could not imagine being left out of such an exciting mission. Now his decision came back to worry him at the most inopportune time. Here in the Caribbean Sea, surrounded by the enemy, they were unable to surface and make repairs.\n\nWhile all three of B-130's propellers could be spun via the Kolomna diesel engines, they usually turned two props with one engine and used one to charge the batteries, with one resting. Due to the ignored drive cracks, they lost one of the engines, and a second now hung by a thread. Should they lose that one, they'd be down to one diesel that might fail at any time. They'd have to run only on the slow emergency motor while snorting to recharge batteries, and their ability to run from the Americans would be greatly diminished. If they lost that third engine, B-130 would be forced to head home under tow with her tail tucked between her legs. Shumkov sickened at the thought. Not only would he be excused from history, but he'd also be riddled with guilt for abandoning his duty to the other captains.\n\nShumkov glanced at the port engine gauges, their needles resting at zero. A maze of small round indicators filled the engine control panel, along with a shiny metal main fuel valve that looked like a rudder wheel on an old sailing ship. Lamenting his fate, as he turned to leave, he said, \"Don't stay in here too long, Viktor. That's an order.\"\n\n\"Yes, sir,\" Parshin said.\n\nShumkov shot through the hatch into Compartment Four. The tantalizing aroma of lamb drifted by and reminded him that he hadn't eaten but a little borscht and goulash for dinner at midday. That was eleven hours ago. Now, at almost 11:00 P.M. boat time, the cooks were preparing tea and the snack meal. Hot piroshki. The tasty meat pie was Shumkov's favorite, especially since he really didn't care for the bread. Project 641\u2013class boats did not have the luxury of a bakery. Bread arrived on board prebaked and was stored unrefrigerated in plastic bags that contained a small amount of alcohol. When the cooks warmed the bread, the alcohol evaporated and offered freshness, at least in theory. For Shumkov, however, the flavor paled in comparison to his wife's baking.\n\nAlthough Shumkov normally ate his meals in the wardroom, he entered the galley and smiled at the chef. The rotund michman, who always produced grade-six quality meals, belied the running submarine joke that chefs should stay skinny in order to squeeze into their corner of the pressure hull. The small space held an assortment of bottles, boxes, and condensed milk cans, along with a large plate of fresh piroshkis sitting on the edge of the wooden counter. The chef returned Shumkov's smile as he held up the plate. Shumkov grabbed one and took a bite. He started to leave, stopped, turned, and grabbed another one.\n\nTwo sailors sat on the blue-cushioned bench in the galley nearby. Each displayed a five-digit number on his uniform that designated department, position, compartment, and shift numbers. One read 4-44-23, which meant Communications, Radio Room, Compartment Four, Shift Two, third in charge. The radio-room number reminded Shumkov that he needed to send out a burst transmission about the condition of his boat, but he decided Moscow could wait a few minutes. He sat on the bench next to his men and lost himself in the rich taste of lamb as he devoured the piroshki. Halfway through a second pie, his elation was interrupted by a distinct change in the vibrations running through the boat.\n\nParshin bolted through the Compartment Four hatch. The chief engineer spoke no words; his face told the story. The second diesel engine had just died. Shumkov sat his half-eaten piroshki on the table.\n\nTwo minutes later, he met his brown-haired electronics officer, Lieutenant Cheprakov, in the radio room. As the officer in charge of division Boyovoi Chesti (BCh) Four, Cheprakov worked closely with the five English-speaking OSNAZ operators. The wide-eyed young men, in between listening to jazz and news on Voice of America, were having the time of their lives monitoring radio intercepts between American ships, planes, and shore stations. Apparently the entire U.S. Navy now shared one mission in life: to find the four Project 641 boats. Shumkov hoped that the Americans would continue to fail, but now, with two engines down, he knew that the winds of luck were shifting.\n\nShumkov wrote out a message and handed it to Cheprakov. The electronics officer made a few modifications to ensure encryptability by the SBD and gave the edited message to the radio operator. The michman studied the message. When he looked up at Cheprakov, his fear-filled eyes seemed to say, \"Is this true?\"\n\nCheprakov returned an affirmative nod. \"It's true. If we lose one more engine, we'll need a tugboat to take us home.\"\n\nThe radioman typed in the message on the SBD keyboard. The burst transmission electrified the tropical clouds, bounced across the ionosphere, and landed on a receiver dish in Moscow.\n\nThe split-second radio waves also tickled a GRD-6 antenna just over 500 miles away in Homestead, Florida. An hour later, led by a Boresight fix, an ASW plane visually detected B-59's snorkel mast. Operators issued the next sequential contact number. Someone in Admiral Anderson's Flag Plot room stuck a flag on the board, designating the submarine sighting as C-18. The USS Cony, patrolling an area near the Bahamas, received urgent orders to pursue the Foxtrot and force her to the surface.\n\nAS NIGHT DESCENDED ON WASHINGTON, D.C., the temperature chilled along with any prospects of a fast or easy resolution to the crisis. At 1:45 A.M. on October 25, President Kennedy responded to Khrushchev's earlier threat\u2014delivered via Westing house's president, William Knox\u2014that Soviet submarines would sink any U.S. destroyers that tried to stop Russian merchant ships. Kennedy stated that the United States took appropriate action after receiving repeated assurances that no offensive missiles were being placed in Cuba, and that when these assurances proved false, the deployment \"required the responses I have announced.... I hope that your government will take necessary action to permit a restoration of the earlier situation.\"\n\nAround 7:15 A.M., the USS Essex and USS Gearing (DD-710) steamed toward the Soviet freighter Bucharest with orders in hand to intercept and board if necessary. Prior to the boarding, however, the navy decided to let this Russian vessel pass after concluding that she was capable of carrying nothing more than a cargo of petroleum. Instead, the fleet received orders to observe the tanker Graznyy, as her deck might be loaded with missile field tanks.\n\nSOMETIME THAT WEEK, ON OR ABOUT the morning of October 25, Briefcase in hand, sitting next to Commander Kaye, William J. Reed shifted nervously in the backseat of a black sedan. Gray clouds outside the windows descended on the nation's capital. Reed knew that almost every American wondered if they'd live to see another fog-filled day. Earlier he'd spent hours preparing overhead projector slides and a typed memorandum. The navy gave Kaye the honors of delivering these to President Kennedy and a few select members of the ExComm group, and Kaye asked Reed to come along. The ensign hoped that his technical input would not be needed.\n\nThe vehicle pulled to a stop in front of the White House. Reed and Kaye stepped from the car and walked toward the entrance. The majestic six-story building shimmered in the sun as the iconic fountain splashed water into a cold October breeze. Commander Kaye stopped for a moment to gaze at the sight. Reed pulled his wool dress coat tight around his neck and smelled crisp air. With each breath, he drew in an equal measure of awe and admiration. He popped to attention and saluted the American flag as it flapped in the wind. Kaye did the same. Both lowered their arms and walked toward the entrance.\n\nAs they strolled, Kaye pointed at the White House complex. He explained that the group of buildings included the central executive residence, flanked by the East Wing and West Wing. He let out a chuckle and said that the place had 132 rooms, 35 bathrooms, 28 fireplaces, 8 staircases, 3 elevators, 5 full-time chefs, a tennis court, a bowling alley, a movie theater, a jogging track, a swimming pool, and a putting green.\n\n\"A bowling alley?\" Reed said. \"Hell, Joyce and I are hoping next year we can afford a place with four bedrooms and a little bigger kitchen.\"\n\n\"Be happy with what you have, BJ,\" Kaye said. \"Most Russians live in tiny apartments and share bathrooms with a half-dozen neighbors. They've never even seen a bowling alley.\"\n\n\"Point taken,\" Reed said as the two approached the building.\n\nKaye pointed again. \"That's where we're headed. The West Wing. In there is the president's Oval Office, senior staff offices, and room for about fifty employees. There's also the Cabinet Room, where the president does most of his business.\" He pronounced business as bid-ness.\n\nAlthough Reed already knew quite a bit about the White House, he also knew that Kaye loved to show off to impressionable listeners, so he let his boss ramble on.\n\nKaye pointed out more as he led Reed into the large entrance hall. The commander explained that in 1806, President Thomas Jefferson transformed the hall into an exhibition area for artifacts from Lewis and Clark's famous expedition to the Western Territories. President Ulysses S. Grant started another tradition of hanging presidential portraits in the entrance hall and the perpendicular cross hall.\n\n\"I guess when you're the chief, you get to decorate your teepee anyway you want to,\" Kaye said.\n\nReed smiled, said nothing.\n\nAn aide greeted them at the entrance and directed the pair to a windowless office in the West Wing. The large conference room displayed a podium, a pull-down projection screen, and an overhead projector for transparent slides. As they stood alone near a rectangular conference table, Kaye said they called this the Fish Room. He said Teddy Roosevelt requested that the room be built in 1902, and he used it as his office. When they expanded the West Wing and built the Oval Office in 1909, they turned the room into a waiting area.\n\nWhen Reed asked why they called it the Fish Room, Kaye glanced upward and said that Franklin Roosevelt put in the skylight in 1934, along with an aquarium and fishing mementos. He always called it the Fish Room, and the name stuck.\n\nReed pointed to a large sailfish mounted on the wall. \"Roosevelt's?\"\n\n\"Nope,\" Kaye said as he rested a hand on the back of a high-back leather chair. \"Kennedy's. He caught the thing on vacation in Acapulco.\"\n\n\"Now I know why I like the guy,\" Reed said.\n\nThe door opened, and President John F. Kennedy entered, along with Defense Secretary McNamara, Deputy Secretary Roswell Gilpatric, National Security Adviser McGeorge Bundy, and the director of the National Security Agency, Air Force Lieutenant General Gordon Blake.\n\nReed had previously met General Blake and found him to be a \"man's man\" who combined a frequent Midwest smile with a sharp mind and hard-charging work ethic. Blake hailed from Iowa and graduated from the U.S. Military Academy in 1931; he earned the Distinguished Flying Cross as a communications officer flying on B-17s during the war. He also earned a Silver Star, Legion of Merit, Air Medal, and several campaign battle stars and now displayed a vast array of colorful ribbons on his uniform.\n\nGeneral Blake was Commander Jack Kaye's immediate boss. Reed didn't know it then, but in less than three years, Blake would award him a certificate of appreciation for his dedicated service to the NSA, specifically for his contributions to the Boresight program.\n\nMcNamara motioned for Reed and Commander Kaye to take a seat. Reed's throat tightened, and he longed for a drink of water. He and Kaye removed their covers and sat. Reed found a glass, filled it from a pitcher on the table, and gulped downed half.\n\nMcNamara explained to the president that he'd received minimal information from Admiral Anderson about a new naval technology called Boresight, that this new system appeared capable of locating Khrushchev's Foxtrot submarines with better accuracy than SOSUS or other means. McNamara felt that such a capability could provide an advantage in the current Cuban negotiations and so asked Anderson to set up a meeting to brief the president and select members of ExComm. He reminded the group of the highly classified nature of the information and turned the floor over to Kaye.\n\nCommander Kaye took a sip of water, cleared his throat, and walked to the podium. He pulled a few slides from a manila folder and placed one on the overhead projector. He flicked on the switch, and the projector's fan hummed to life. He explained that, since the early days of World War II, the United States used high frequency direction finding systems to locate enemy submarines. After the war, in 1960, the Soviets switched to a new type of ultra-short burst signal that they adopted from recovered German technology. They phased over from standard HF transmissions, and by December of that year, they were using the burst exclusively. The navy could no longer find those subs.\n\nKaye pointed to Reed and said that the ensign discovered the burst and helped the NSA and navy design a new technology under Operation Boresight. Kaye then started to explain how the system worked. When McNamara drilled him for more details, Kaye asked Reed to step to the podium.\n\nHis mouth dry, Reed downed some more water, grabbed his folder, and approached the podium. Kaye gave him a pat on the shoulder as he walked past and returned to his seat. Reed's eyes locked with Kennedy's. The president offered a smile and a nod.\n\nReed recalled hearing stories about how Kennedy's torpedo boat, PT 109, had been rammed by the Japanese destroyer Amagin during the war and had sunk in the Pacific, and about how Kennedy had hurt his back in the collision but still managed to swim to shore while towing a badly burned sailor using a life-jacket strap clenched between his teeth. Kennedy found the rest of his crew on a nearby island, where he scrawled a rescue message on a coconut given to Solomon Islander scouts. That coconut wound up saving the lives of his men. Kennedy turned the thing into a paperweight that now sat on his desk in the Oval Office.\n\n\"Go ahead, Ensign,\" the president said softly. \"Just tell us what you know.\"\n\nIn that brief moment, Reed understood how the man's infectious charisma earned him the presidency. Using more than a dozen slides, with McNamara grilling him for facts, Reed translated the technical details of Boresight into layman's terms. Questions were asked about bearing accuracy, frequency of transmissions, distance limitations, and the number of stations operational.\n\nReed said that the Soviet navy, reflecting the ways of its authoritarian government, did not trust its submarine captains, and so required that they send a coded update at least once a day. That message consisted of a short burst signal that could now be detected by Boresight intercept stations. Multiangulation could then be used to locate the source of the transmission. Several stations equipped with Boresight technology received hits on four Foxtrot submarines, as they neared Cuba, that were sending twenty or thirty transmissions at a time\u2014probably due to reception verification difficulties with Moscow. U.S. stations cross-referenced bearing hits to direct ASW forces toward the Foxtrots. That was the good news.\n\nReed then delivered the bad news: given the nascence of Boresight technology, the limited number of operational sites, and the distance of those sites from the targets, as well as the inexperience of the operators, ballpark fixes of between forty and sixty nautical miles were the best they could accomplish today.\n\n\"Are we talking closer to forty or sixty?\" Kennedy asked.\n\n\"Up until last week, it was closer to sixty,\" Kaye said. \"Thanks to Ensign Reed and his team, we're now closer to forty for most stations.\"\n\nKennedy glanced at McNamara. \"How good is that for our ASW boys, Bob?\"\n\n\"Not much better than SOSUS, Mr. President,\" McNamara said. \"We'd still need to throw too many planes and ships on those fixes to find the subs. I'd sure like to get that number down to thirty miles or less.\"\n\n\"There's something else, Mr. President,\" Kaye said.\n\n\"Go ahead, Commander,\" Kennedy said.\n\n\"It's just a speculation, sir, but the NSA reported that a Foxtrot submarine fired two nuclear torpedoes off Novaya Zemlya last year. If the Soviet subs heading to Cuba are carrying\u2014\"\n\n\"We know, Commander,\" McNamara said. \"But we believe it's unlikely.\"\n\n\"Unlikely, but not impossible,\" Kennedy said, his face somber.\n\nThe room fell silent. Filigree danced in a ray of sunlight that beamed through the skylight in the Fish Room. Kennedy stared at the table for a long moment, then looked up. His deep brown eyes pierced Reed's social armor and reflected the hope of an entire nation. \"Ensign Reed, with those subs still in the picture, I don't have a strong hand against Khrushchev. And if they are carrying nukes, God help us all. So I need you to get us better than thirty miles. Do you think you can do that?\"\n\nReed's legs went numb. He glanced at Kaye. The commander's eyes opened wider than submarine hatches, but he said nothing. Reed stood up straight, looked back at Kennedy, and said, \"Yes, Mr. President. I believe I can.\"\n\nOn the way out of the White House, Commander Kaye grabbed Reed by the arm and said, \"How the hell are you going to get to thirty miles in a matter of days?\"\n\n\"I have no idea,\" Reed said. \"But if I don't, like the president said, God help us all.\"\n\nAS THE TWO WERE LEAVING THE building, a man stepped in front of Reed and blocked his exit. \"What the hell are you doing here?\"\n\nReed studied the man's face, then smiled. \"Lieutenant Commander Quittner?\"\n\n\"I'm retired now, so that's Mr. Quittner to you,\" Quittner said with a grin.\n\nArnold Quittner served as Reed's former executive officer on his first ship, the PCS-1380. After more than a decade, the man sported some gray and a few extra pounds, but his voice still grumbled like a Mack truck in low gear.\n\n\"Retired?\" Reed said. \"I thought you'd serve forever.\"\n\n\"I am serving,\" Quittner said, \"just not in the navy. I'm one of Kennedy's legal advisers now.\"\n\nCommander Kaye tilted his cover back and said, \"I guess Kennedy could use all the legal beagles he can get right now.\"\n\nWearing a suit and tie, his hair a tad longer than navy regulation, Quittner said, \"My plate's definitely full these days. I can't even begin to tell you how much legal maneuvering it takes to move dozens of political and military chess pieces around without ruffling lots of feathers.\"\n\n\"You'd think that under the circumstances,\" Reed said, \"the president would have carte blanche.\"\n\nQuittner shook his head no. \"Until there's an official war proclamation, every special interest asshole on the planet wants his say. Even when we're staring down the barrel of a gun, there's always some guy that cares more about his personal pocketbook.\"\n\n\"Somebody needs to hogtie those bastards and brand them traitors,\" Kaye said with a scowl as a couple of congressmen walked past.\n\n\"I wish we could,\" Quittner said as he brought his wrists together. \"But that's hard to do when you're handcuffed by the legal system.\"\n\n\"Maybe the sheriff needs to change a few rules,\" Kaye said.\n\n\"Maybe,\" Quittner said. \"I do agree it's a pain in the ass sometimes, but I'd rather have the problems of a democracy than those of a dictatorship.\"\n\n\"Our forefathers never said that freedom would be easy,\" Reed said.\n\nThe three shook hands, and Reed and Kaye stepped back into the cold.\n\n## CHAPTER NINE\n\nAnger is an acid that can do more harm to the vessel in which it is stored than to anything on which it is poured.\n\n\u2014MARK TWAIN\n\nEARLY IN THE EVENING OF OCTOBER 25, in a lab at Sanders Associates in New Hampshire, Reed forced his tired mind to stay focused. He had no idea how he was going to solve Kennedy's requirement to improve Bore-sight accuracy, but he hoped that he might find an answer here. Petty Officers Odell and Denofrio, the two stars on Reed's technical team, were huddled together with a group of engineers near the back of a large room filled with the evidence of technology. A dozen randomly placed tables sat nearby, covered with various devices, colorful wires, test equipment, and tools.\n\nReed removed his cover and coat and placed both on a rack near the door as a man in a lab coat approached. Light skinned, polished, and projecting a professor's demeanor, he introduced himself as Dr. Charles Skillas, the team lead for Antisubmarine Warfare DIFAR\u2014whatever that meant. Reed and Skillas exchanged pleasantries and found seats near the group of engineers. Reed took those few seconds to try to recall a few details about Sanders Associates.\n\nEleven engineers and scientists from Raytheon founded the company in Waltham, Massachusetts, in July 1951. Royden Sanders Jr., one of the original eleven associates, became the company's namesake. Sanders Associates moved into a vacant textile building in Nashua in 1952 and focused on designing and building electronic systems, aircraft self-protection systems, and surveillance and intelligence systems, including submarine detection equipment. More than a thousand people worked at the facility, making Sanders the largest employer in New Hampshire.\n\nOver the next several minutes, while Odell and Denofrio mingled with the engineering team, Dr. Skillas asked how they might help the navy. Reed divulged information about the Boresight program and accuracy issues without delving too far into sensitive areas. Sanders was a defense contractor, and everyone there had signed a stack of nondisclosure papers, but Reed still operated under strict orders not to provide civilians with more information than they absolutely needed to know.\n\nReed said that he researched Sanders Associates and knew they were working on directional hydrophone sensors for passive sonobuoys. He wanted to know more about the program. Skillas said that the navy awarded them a $600,000 contract to create DIFAR, an offshoot of the LOFAR technology used in the SOSUS system. DIFAR stood for Directional Frequency and Ranging, and the key word was directional. The technology they'd developed used an internal compass to help determine a more accurate direction to noise generated by Soviet submarines.\n\nSkillas explained that dogs have the ability to screen incoming sounds and focus an ear on a noise to determine the exact location of the source. In essence, they can \"turn on\" unidirectional hearing, whereas human hearing is omnidirectional\u2014meaning we can hear almost everything coming from everywhere. That works fine for people, but omnidirectional listening devices are not optimal for finding submarines. Sanders developed \"dog hearing\" unidirectional technology for airplane-dropped sonobuoys that could do a better job of figuring out an accurate bearing to a contact. They were also working on a technique for dropping those sonobuoys from ASW aircraft\u2014as high up as 30,000 feet\u2014using precise patterns they called CODAR.\n\nReed knew that sonobuoys used transducers and radio transmitters to record and transmit underwater sounds, and that there were three types: passive, active, and special-purpose buoys. Passive sonobuoys used hydrophones to listen for underwater sounds, and active buoys used transducers that \"pinged\" just like submarine sonar systems. Special-purpose buoys were used to capture environmental information, like water temperature, depth, and acoustic layers.\n\nSkillas explained that the directional frequency and ranging DIFAR sonobuoy they were working on was passive, and the main component included a directional hydrophone that recorded accurate bearings to targets. The navy asked Sanders to build a prototype that could detect submarine noises in the low-frequency 5\u20132,400 Hz range and could operate for up to eight hours at depths down to 1,000 feet.\n\nThe brain behind this invention was an AQA-7 signal processor. This allowed for doglike directional hearing. The system processed incoming signals and output submarine position, speed, and direction information onto electrosensitive paper.\n\nAll music to Reed's ears. He told Skillas that they needed a way to increase bearing accuracy for the Boresight systems to gain better location fixes on submarines and wondered if any part of Sanders's technology could be used to help them. Skillas rubbed his chin and conferred with the others on his team. They asked dozens of questions. Reed and his team provided answers. The engineers drew on blackboards, fingered slide rules, and thumbed through technical diagrams and manuals.\n\nAfter an hour, Skillas said, \"We have good news and bad news.\"\n\nReed's heart sank. He'd been hoping for all good news.\n\n\"Give us the bad news first,\" Petty Officer Odell said.\n\n\"Why not the good news first?\" Petty Officer Denofrio asked.\n\nOdell snorted. \"Maybe you don't understand, pretty boy, but when I go into a bar, I always hit on the second-ugliest woman first.\"\n\n\"You're right,\" Denofrio said. \"I don't get that at all.\"\n\n\"I do that,\" Odell said, \"'cause if she says no, I still have one to go. So I always want the bad news first.\"\n\nSkillas shook his head and flashed a smile. \"Okay, the bad news is that we don't think our processor is adaptable to the systems you're using. Those burst transmissions are only seven-tenths of a second long, and they're coming from a recorder. Also, the AQA-7 is designed to work with underwater noise, not high-frequency radio transmissions. We don't think our signal processor will help much, given that scenario.\"\n\nReed lowered his head. \"So, what's the good news?\"\n\n\"The good news is that some of the math and other technology we used to design DIFAR might help you.\"\n\nReed lifted his head. \"Math?\"\n\n\"Math,\" said Skillas. \"Like Gaussian on a plinth.\"\n\n\"Gesundheit to you, too,\" Petty Officer Denofrio said.\n\nReed held back a laugh. \"What's that mean in English?\"\n\n\"A bell curve on a baseline,\" Skillas said.\n\nHe went on to explain that the mathematical term Gaussian was named after Karl Gauss and was widely used in signal processing to define filters, which is what they did at Sanders. Skillas walked over to a blackboard. He drew a bell curve, which resembled a tall anthill with curved slopes. He placed several dots near the top of the hill. \"These are normal, accurate bearings to a submarine as detected by your Bore-sight system.\" He then drew a few dots on either side, near the base of the hill. \"These are wild bearings caused by inaccuracies and interference. With a standard high-frequency transmission, ruling out the obvious bad apples is not hard. With a burst transmission, you might get only a couple hits, all of which could be the bad apples. We need to help you find a way to get more good apples and better discern the bad from the good.\"\n\nReed asked, \"How do we do that?\"\n\n\"Well,\" Skillas said, \"you need to cheat.\"\n\n\"Cheat?\" Reed said.\n\n\"I don't like cheating,\" Denofrio grumbled.\n\n\"I do,\" Odell said, raising his hand. \"I'm all for cheating if we get to win.\"\n\n\"How do we cheat?\" Reed asked.\n\n\"You need more verifiably accurate bearings,\" Skillas said.\n\n\"Swell,\" Denofrio said like a New Yorker who'd just missed a taxi. \"We'll just ask the Russkies to pretty please burst a bunch more times each day.\"\n\n\"Denofrio's got a point,\" Reed said. \"We only get a limited number of hits.\"\n\n\"From the target, yes,\" Skillas said. \"But now you need more check bearings.\"\n\nReed said, \"Using known references to calibrate for bearing inaccuracies is something we've been doing since the early days of DFing.\"\n\n\"Not with burst signals,\" Skillas said. \"Now you have to compensate for the dimension of time. Inaccuracies and interference will be different for an after-the-fact Soviet burst signal than a regular high-frequency transmission.\"\n\n\"No shit,\" Denofrio said.\n\nReed leveled a disapproving stare.\n\n\"Sorry,\" Denofrio said. \"I meant, that's certainly an accurate assumption, Dr. Skillas.\"\n\n\"You also need to automate your bearing calculations instead of using that inaccurate manual string board compass rose you're using now,\" Skillas said.\n\n\"No shit,\" Odell said, throwing Reed an I-don't-care look. \"But how do we do that?\"\n\n\"Well,\" Skillas said, \"that's where the software programs that we developed for our DIFAR project might help. Along with a new type of computer.\"\n\n\"New computer?\" Odell said.\n\n\"The GYK-3,\" Skillas said. \"It's in development right now at the Naval Research Laboratory, but it might work for your application.\"\n\n\"No shit?\" Denofrio said, his face lighting up like Times Square.\n\n\"No shit,\" Skillas said.\n\nReed nodded. \"Okay, I get it. What you're saying is that we'll have higher bearing accuracy if we set up a simulated burst signal on some of our ships and have them transmit immediately following a Soviet burst hit, and if we automate bearing locations using a computer like the GYK-3 instead of using manual string boards.\"\n\n\"Precisely,\" Skillas said.\n\nSkillas and his team offered other tips they thought might improve things, such as using cesium clocks to better synchronize time accuracies between the stations and better ways to calculate interference and compensate for the effects. They also recommended reducing any man-made interference from electrical equipment or nearby power lines.\n\n\"You need to increase your good apples and reduce your bad ones,\" Skillas said. \"Otherwise you'll never find those guys.\"\n\n\"No shit,\" Reed said.\n\nAn hour later Reed excused himself from the meeting to catch a flight to California. As he stood outside the brick structure nestled in the snow, he stifled a sneeze and glanced skyward. More snowflakes were beginning to fall. Down a curved sidewalk, a bundled figure trudged through the white and approached the building. The short man stopped near Reed and held out a hand. \"I'm Dr. Ralph Baer.\"\n\nReed shook Baer's hand. \"Ensign Bill Reed.\"\n\nThin, friendly, and bespectacled, Baer said, \"Are you coming or going?\"\n\n\"I'm not sure anymore,\" Reed said.\n\n\"Sounds like you need a vacation,\" Baer said empathetically.\n\n\"Definitely.\"\n\nBaer gave a chuckle as he turned to enter the building. \"Nice meeting you, Ensign Reed.\"\n\n\"Good meeting you, too, Dr. Baer,\" Reed said.\n\nAs Dr. Ralph Baer entered the foyer, Reed had no idea that he'd just met one of the most brilliant inventors in the world. This unassuming man was destined to create the first video game and spawn a multibillion-dollar industry that would change the world in ways unimagined.\n\nThat evening, Reed caught a military MAC flight headed to Northern California. After landing, held up by a half-dozen cups of French roast, he drove through the north gate of Skaggs Island and stepped back in time. Reed started his career as a communications technician at this facility after serving aboard the PCS-1380. Skaggs Island was a drained area of San Pablo Bay tidelands that sat about twenty-five miles northeast of San Francisco. In the early fifties, the Naval Security Group came to Skaggs to set up the HFDF facility and turn the island into a bonafide Huff Duff.\n\nAs Reed drove toward the ops building, he noticed that things hadn't changed much in the several years since he had brought his family there in 1957. The recreational buildings, theater, chapel, and bachelors' quarters still stood at attention like worn soldiers adorned in faded gray uniforms. Rows of single-story homes lined sad, narrow streets, where navy brats played with dogs destined for abandonment after the next military-ordered move.\n\nReed pulled to a stop in front of the small operations building near a massive elephant cage. Although Skaggs was one of the first stations to receive the new Wullenweber in 1962, most of the kinks were not yet ironed out. Reed had visited the station a couple of times over the past few months with his team to install and launch the Boresight equipment. Now he was back to deploy some of Dr. Skillas's recommendations. He knew that the Fred Ten antenna array at Skaggs improved bearing accuracy by an order of magnitude over the old GRD-6 sites, but Skaggs was almost 2,700 miles away from Cuba. The Fred Ten elephant cage usually couldn't get a hit at better than 3,200 miles on a good day, and one ionospheric hop of 2,700 miles was a stretch. But if they used some of the suggestions from Sanders Associates, along with a bit more tweaking on the systems, they just might get down to that thirty-mile radius President Kennedy had requested.\n\n## Photographic Insert 1\n\nFrom 1946 through 1954, diesel-powered submarines like the USS Cubera (SS-347, above) and USS Blenny (SS-324, below) conducted top-secret espionage missions deep inside Soviet territorial waters. The Soviets sometimes caught these boats and harassed them. Diesel subs ran out of air after a few days submerged, and crews almost died on several missions. This problem eventually spurred the use of nuclear power in submarines. U.S. Navy photographs, courtesy of ussubvetsofwwii.org\n\nThe USS Seawolf (SSN-575) was America's second nuclear-powered submarine. Almost three decades after her launch, while conducting a top-secret mission in 1981, a storm trapped Seawolf in the sand off the coast of Russia for four days. Her crew of 190 came within a breath of not coming home. U.S. Navy photograph\n\nThe \"father\" of submarine nuclear power, Hyman G. Rickover, touring the USS Nautilus\u2014the navy's first nuclear-powered submarine\u2014in 1954. U.S. Navy photograph, circa 1954\n\nOfficers and sailors in the control room of the USS Nautilus (SSN-571) while under the Arctic ice. Launched on January 21, 1954, the Nautilus was the first vessel to complete a submerged transit under the North Pole. U.S. Navy photograph\n\n\"Scratchy\" the bear, raised by the author's family while in Turkey, unwittingly helped William J. Reed find a Soviet \"burst\" signal that made a significant difference in the outcome of the submarine Cold War. Author's collection\n\nThe author's father, William J. Reed, received a promotion from senior chief to ensign after finding the Soviet submarine \"burst\" signal while in Turkey in 1960. Author's collection\n\nReed was also awarded a letter of commendation from his boss, Commander Frank Mason, for finding the signal. Author's collection\n\nIn one of the most dramatic, untold episodes of the Cold War, four Soviet Foxtrot-class submarine commanders led their boats deep into the waters around Cuba in October 1962, which on several occasions led us to the brink of nuclear war. Left to right: Capt. Dubivko (B-36), Capt. Shumkov (B-130), Chief of Staff Arkhipov, and Capt. Ketov (B-4) left Russia on October 1, 1962. They arrived off the coast of Cuba three weeks later. Courtesy of Ryurik Ketov\n\nBut they also forced three of the Foxtrot submarines, including Captain Savitsky's B-59 (above) to the surface during the Cuban Missile Crisis. The United States did not know at the time that each Foxtrot carried a nuclear-tipped torpedo capable of destroying everything within a ten-mile radius, and all four Soviet subs came within minutes of firing. U.S. Navy photograph\n\nThe U.S. Navy was primarily focused on blocking Soviet merchant ships from bringing more nuclear missiles to Cuba. U.S. Government photograph\n\nLiving conditions aboard the Soviet Foxtrot-class submarines involved in the Cuban Missile Crisis were deplorable, and crews suffered from heat exhaustion. The captain's stateroom is pictured above, the hatch leading up to the conning tower and bridge is pictured to the right, and the navigation table in the command center is pictured below. Author's collection, Maritime Museum of San Diego, California\n\nDuring the Cuban Missile Crisis of October 1962, a nuclear missile strike from Cuba could have killed eighty million Americans within five minutes. U.S. Government CIA photograph\n\nPresident John F. Kennedy meets with Secretary of Defense Robert McNamara. They and other Executive Committee members planned for a quarantine of Cuban waters. U.S. Government archive photograph\n\nDuring the quarantine, destroyers like the USS Barry were ordered to fire at the rudders of Soviet merchant ships, like the Ansov, who refused inspections. U.S. Navy photograph by Frank Cancellare\n\nPrior to the Cuban Missile Crisis, Ensign William J. Reed discovered the \"burst\" radio transmission used by Soviet Foxtrot submarines and then briefed President Kennedy and his advisers on the technology during the crisis. Reed worked to improve the systems to help locate the Foxtrots near Cuba. This effort allowed Kennedy to play hardball with Khrushchev, which forced the Soviet Premier to back down and return his ships to Cuba. Author's collection\n\nWilliam J. Reed's family in Maryland in 1964. Left to right: Pamela, William, W. Craig (author), and Joyce. Author's collection\n\nAfter the Cuban Missile Crisis, William J. Reed traveled the globe to upgrade U.S. listening stations, so they could detect Soviet burst signals. One of those missions took him to Greece, where he was involved in a CIA firefight with Soviet spies. Author's collection\n\nTop-secret listening station sites resembled large \"elephant cages\" and were positioned along the Pacific and Atlantic rims near Imperial Beach, California (above), Okinawa, Japan (below), and more than a dozen other locations. Author's collection\n\nDuring the early 1960s, William J. Reed helped deploy top-secret listening stations around the world in an effort to locate Soviet submarines. gnu Free Documentation License, Version 1.3\n\nThe USS Thresher (SSN-593) was lost with all hands on April 10, 1963. center right: Navy divers aboard the bathyscaphe USS Trieste (DSV-0) descended to 8,400 feet to search for her remains. U.S. Navy photographs\n\nNavy diver Nihil Smith\u2014whose best friend, Joe Walski, was aboard the Thresher when she sank\u2014brought up a pipe found by the Trieste in 1963, validating the sub's location on the bottom. below right: The Trieste found other remains, including the crushed sonar dome. U.S. Navy photographs\n\nThe North Koreans captured the spy ship USS Pueblo (AGER-2) on January 23, 1968, and found a working KW-7 encryption unit, which they sold to the Soviets. U.S. Navy photograph\n\nThe Soviet Golf II\u2013class missile boat K-129 disappeared in the Pacific on March 8, 1968. The Soviets blamed the United States, based on transmissions they intercepted on the KW-7 using keylists provided by the spy, John Walker. U.S. Navy photograph\n\nThe Soviets may have retaliated for the loss of K-129 by ordering an Echo II\u2013class submarine to sink the USS Scorpion (SSN-589) with a torpedo on May 21, 1968. U.S. Navy photograph\n\nNavy deep-sea saturation divers breathing a helium mixture, deployed from the USS Halibut (SSGN-587, bottom left) to depths of 400 feet near Russia. The divers \"tapped\" signals from secret communications cables that were recorded and analyzed by \"spook\" technicians like Frank Turban. U.S. Navy photograph by Senior Mass Communication Specialist Andrew McKaskle; U.S. Navy photograph\n\nCold War submarine \"spooks,\" circa 1976. Left to right: Bob Jordan, John Whitmire, Skot Beasley, Scott Hendren, Frank Turban, Dwight Anderson. Courtesy of Frank Turban\n\nDuring the latter half of the Cold War, U.S. Navy divers completed Ivy Bells cable-tapping missions using large \"beast\" pods. Courtesy of Russian Ministry of Security's museum at the Lubyanka Prison\n\nIn the latter two decades of the Cold War, the USS Parche (SSN-683, top) and USS Richard B. Russell (SSN-687, below) delivered these divers to the Sea of Okhotsk and Barents Sea at depths down to 700 feet. U.S. Navy photographs\n\nUSS Haddo (SSN-604) entering New Zealand harbor in January 1979. As the boat's rescue diver, the author is standing on the back of the boat in the event a sailor falls overboard. Author's collection\n\nThe author on the Haddo entering Apra Harbor in Guam\u2014where the author was born. Author's collection\n\nCrew members in the control room of the USS Haddo, circa 1978. The author is at the far back right and Lt. Edwin L. Tomlin is on the far right. Author's collection\n\nThe Haddo and other Cold War fast-attack submarines hunted Soviet ballisticmissile boats like the infamous Delta III (belowM), often coming within a few dozen yards to tail these targets. U.S. Navy photograph\n\nWhile conducting Holystone espionage operations, the USS Haddo and other Cold War fast-attack subs frequently photographed the Typhoon-class (below) and other Soviet submarines, through periscopes or by deploying navy divers such as the author. Author's collection; U.S. Navy photograph\n\nIn an attempt to photograph the \"odd pod\" on the back of a Soviet Victor III (below) in Peter the Great Bay, the USS Drum (SSN-677, above) slammed into the Victor, heavily damaging both submarines in 1981. This collision was covered-up and has never been made public, and nearly cost the lives of more than one hundred sailors\u2014including the author. U.S. Navy photographs\n\nReed spent a sleepless night at Skaggs working with the technicians and implementing new procedures. Subsequent to his meeting at Sanders Associates, he'd contacted Commander Kaye, who called Admiral Alfred Ward, who talked to his boss, Admiral Anderson, and got permission to have a destroyer install a simulated burst transmitter and start transmitting when asked to by Net Control\u2014after the Boresight stations got some burst signal hits\u2014then change course a few times and retransmit. That way the Boresight stations could correct for inaccuracies and interference using a known target location. Reed didn't have time to find and install a computer to automate bearings, so he planned to connect later with some of the engineers at NRL to talk about using the GYK-3 to improve Boresight detection capabilities.\n\nExhausted, with nothing left to do but wait and pray that the new tricks would work, Reed boarded a plane and flew back to Mary land.\n\nIN AN OCTOBER 26 MEETING, PRESIDENT Kennedy ordered the State Department to proceed with Cuba invasion preparations. SecDef McNamara reported that, in the event of such an invasion, CINCLANT estimated more than 18,000 casualties in the first ten days of fighting. Kennedy stated that despite the cost in lives, only an invasion could ensure the removal of all missiles from Cuba. Just prior to issuing the order to invade, a few of the conservative members of ExComm persuaded the president to delay the invasion and continue with military and diplomatic pressure.\n\nAt 1:00 P.M., John Scali of ABC News attended a lunch meeting with Soviet embassy official (and KGB station chief) Aleksandr Fomin, who stated that \"war seems about to break out.\" Fomin asked Scali to use his influence to explore a diplomatic solution. The structure of such a deal, Fomin intimated, should include assurances that the Soviet Union would remove its weapons from Cuba, and the United States would state publicly that an invasion of Cuba would never occur.\n\nThe State Department received a message at 6:00 P.M., written personally by Nikita Khrushchev, which Robert Kennedy described as \"very long and emotional.\" The contents outlined a deescalation plan similar to the one proposed earlier by Fomin.\n\nJust before 9:00 A.M. on October 27, Radio Moscow broadcast a message from Khrushchev. In contrast to the letter of the night before, the message offered a new trade: that the missiles in Cuba might be removed in exchange for the removal of the U.S. Jupiter missiles from Turkey.\n\nAT 10:25 A.M., A NEW INTELLIGENCE message arrived, and John McCone, the director of the Central Intelligence Agency, announced: \"We have a preliminary report which seems to indicate that some of the Russian ships have stopped dead in the water.\"\n\nDean Rusk leaned over to McGeorge Bundy and said, \"We're eyeball to eyeball, and I think the other fellow just blinked.\"\n\nPresident Kennedy surmised that Khrushchev temporarily halted the advance of his cargo ships to avert an immediate confrontation, as well as to buy time to contemplate his next move. Following suit, Kennedy directed that no Soviet ship should be intercepted until the situation could be properly assessed. ExComm members issued a collective sigh of relief, but the sigh did not last long.\n\nIN CUBA, AN ALARM SOUNDED AT 9:10 P.M., and the hairs on Major Grechenov's neck stood on end. He grabbed the radio and pressed the talk key, issuing a report that the SA-2 site near Banes had just detected a U-2 reconnaissance plane entering Cuban airspace. The controllers designated the contact as target thirty-three. A minute later, the antiaircraft division commander received authorization to destroy the target. Grechenov issued the order to Sergeant Varankin, the commander of the reconnaissance and targeting station. Varankin provided an affirmation. A few seconds passed before he reported a target lock. Grechenov asked for and received range, speed, azimuth, and altitude readings. He imagined the pi lot sitting in the cockpit of the American spy plane. He wondered if the man had a wife, children, a happy home.\n\nGrechenov abruptly ended such thoughts as he raised and lowered his arm. Sergeant Varankin gave a nod and pressed the firing key. He counted off the seconds until a tiny fireball appeared in the sky like a dying star gone supernova.\n\nMajor Grechenov did not know Major Rudolph Anderson, the pi lot of the doomed U-2 and the first casualty of the Cuban Missile Crisis. He also harbored no concerns that his actions might have just started a war with the Americans. When President Kennedy and the ExComm group learned about the downed U-2, they worried that war was imminent. Two Soviet freighters, the Gagarin and Komiles, steamed to within a few miles of the Walnut line. The waiting U.S. fleet headed them off and maintained, for the moment, a Mexican standoff.\n\nAdmiral Anderson reported to Kennedy and McNamara that various SOSUS stations had verified seven possible contacts on Soviet conventional submarines near Cuba, but determining accurate bearings and submarine classifications was still problematic. Most of the contacts were labeled with a reliability one or two status out of a possible three, with location radius accuracy of greater than thirty miles in every case. One SOSUS report indicated difficulty in tracking contacts due to their distances from an underwater array, and another admitted to a misclassification of a Foxtrot submarine as a possible Golf class.\n\nAnderson then reported the good news: Ensign Reed and his team had improved the bearing accuracy at the Skaggs Island Wullenweber Boresight station to around one half degree, and two other GRD-6 stations down to less than two degrees. The combined improvements were now capable of reliability-three fixes, with accurate submarine classifications within a radius of better than twenty-five miles. President Kennedy expressed his gratitude and authorized a letter of commendation for Reed and others involved in the Boresight program at NSA Section A-22.\n\nAROUND 300 MILES SOUTH OF BERMUDA, just below the surface, Captain Savitsky sent a transmission to Moscow to acknowledge new orders for B-59. They were assuming a patrol area to the east of Dubivko's B-36 and 170 miles north of B-130's former patrol zone due east of Cuba. Savitsky was unaware that Boresight recorders, now upgraded with better location accuracy, had intercepted his burst signal.\n\nSavitsky moved B-59 between the Soviet freighters and the U.S. fleet operating near the Bahamas southeast of Florida. He was now well inside the Walnut line. Moments later, Admiral Ward's ASW forces were on the move. Although Ward was provided with longitude and latitude coordinates for the Foxtrot\u2014courtesy of Boresight\u2014he was not authorized to know from where those fixes originated. Ward relayed the information to the ASW carrier USS Randolph, which was escorted by the destroyers Bache (DD-470), Beale (DD-471), Cony (DDE-508), Eaton (DD-510), and Murray (DDE-576). Given the volatility of the situation, nerves were stretched to the breaking point. One miscue could result in disaster.\n\nA swarm of S2F Tracker aircraft and Sea King helicopters locked on to Savitsky's B-59, designated as a Foxtrot-class submarine, and the USS Beale dropped grenade-sized depth charges while pinging the area with active sonar. The USS Cony raced in and also splashed warning charges. Not only did the quantity of dropped charges exceed the \"four or five\" promised by the United States in their rules of engagement, but the devices from the Cony and ASW aircraft detonated in close succession to those dropped by the Beale, which created louder than normal explosions. Using dropped sonobuoys, ASW aircraft from the Randolph approximated B-59's location in front of the Soviet freighters and inside the Walnut line. The Randolph alerted CINCLANT. Five minutes later, a red telephone rang in the White House.\n\nAFTER RECEIVING THE CALL, PRESIDENT KENNEDY raised a tired hand to his face and covered his mouth. He opened and closed his fist. His face drawn, his eyes pained and almost gray, he said, \"Isn't there some way we can avoid having our first exchange be with a Russian submarine...almost anything but that?\"\n\nSecretary of Defense McNamara responded with a single word: \"No,\" followed by a slow, careful explanation: \"There is too much danger to our ships...our commanders have been instructed to avoid hostilities if at all possible, but this is what we must be prepared for, and this is what we must expect.\"\n\nSENIOR LIEUTENANT PAVEL ORLOV CLENCHED his teeth as the dull thud of another explosion shook his submarine. Standing next to the navigation plot in the control center of B-59, on the port side of the boat, Orlov wondered why he'd volunteered for this suicide mission. He envisioned a transfer to a tropical land, that's why. But when Captain Savitsky told the crew that the Moscow main navy staff canceled the plan to establish a submarine base in Cuba, and instead directed them to run in circles amid dozens of U.S. warships near the Bahamas, the wind died in Orlov's sails.\n\nAs if that wasn't bad enough, the incessant heat and inability to shower brought irritating rashes and oozing, painful sores. A storm lashed at them for two days, causing several sailors to vomit on the deck when they tried to snort. And then there was the battle of the egos. Their submarine was cursed with the presence of Brigade Chief of Staff Captain Vasily Arkhipov, whose by-the-book viewpoint clashed with Captain Savitsky's bend-the-rules style of command.\n\nAlthough Arkhipov outranked Savitsky, the captain of a ship or submarine always held the position of ultimate authority. Nevertheless, only a fool would ignore a superior officer's strong \"recommendations,\" as this could ultimately have career-ending consequences. Diplomacy was not one of Savitsky's strong suits, however, and the constant disagreements with Arkhipov led to confusion and lower morale, if that was possible. Still, the crew continued to perform well, as to do otherwise taunted death.\n\nFor Orlov, not doing his best could also reflect dishonor on three generations of naval intelligence officers. Orlov's father received a transfer to the United States while working for the Main Intelligence Department (GRU) in 1945. Eight-year-old Orlov Jr. arrived with his family in Washington, D.C., and spent several years of his youth there gaining command of the English language. That served him well when he returned to Russia and became a naval intelligence officer assigned to the special OSNAZ group. There he learned about signals intelligence and how to operate ESM equipment designed to monitor U.S. radar and radio communications. He heard that his job was similar to that of American I-Branchers, or \"spooks,\" as they were called. When his command gave him the opportunity to go to Cuba, he jumped at the chance.\n\nWhen Orlov first reported aboard B-59, he and the other eight members of the OSNAZ team endured skepticism, criticism, and harsh treatment for being \"nonquals.\" Many days went by before the OSNAZ group received a reprieve by producing reliable reports on NATO ASW movements. Then, slowly, attitudes started to change.\n\nAfter B-59 passed just south of Bermuda, the Nakat ESM equipment picked up signals from American ships and planes in droves. Orlov and the other four English-speaking specialists spent long hours intercepting radio traffic. They determined that their pursuers were part of an ASW flotilla spearheaded by the aircraft carrier USS Randolph. Captain Savitsky thought he had avoided the hunter\/killer group until that evening, a few hours after receiving new position orders and sending a receipt verification burst transmission to Moscow. That's when the floodgates opened and hell sent a swarm of locusts in the form of S2F Tracker aircraft and Sea King helicopters with dipping sonar. Destroyers operating in tandem with the Randolph soon followed, and it wasn't long before they surrounded B-59 and locked them into a tight cage. Then the explosions started.\n\nAmerican planes and ships now had them pinned down with active sonar and warning explosives. To Orlov, the grenade-sized depth charges sounded like the real things, and the quantity was far greater than the maximum of five promised by the U.S. Navy in their rules of engagement broadcasts. When Captain Savitsky had received the communication from the Americans outlining those rules\u2014specifically about the requirement to surface and assume an easterly course\u2014he snorted and said, \"I will never surface.\" From Orlov's perspective, Savitsky and Brigade Chief of Staff Arkhipov had at least one thing in common: they were both stubborn mules. Neither wanted to show weakness in front of the other, so both held firm to their conviction not to surrender without a good fight.\n\nNow here they were, more than 300 meters below the surface, batteries depleted, air fouled, and crawling along on the economy motor at three knots, still refusing to surface. Theoretically, with a full charge, they could endure a beating from the Americans for up to three days. The problem was, they had never been able to snorkel long enough to gain a full charge, and if they had to maintain three knots, they would never get away.\n\nAcoustic estimated fourteen surface ships in pursuit, including the Randolph and a slew of destroyers. The dim emergency lights flickered, and body odor permeated the CC. Orlov forced his hand to remain steady as he assisted the navigator with the parallel contact plot, a skill he had learned at the naval academy. The plot lines showed that the Americans were following the canons of military art by surrounding B-59 and then tightening the noose. Another explosion rattled pipes. To Orlov, the ordeal felt as if someone had crammed him into a metal can and then started pounding on the thing with a sledgehammer. His chest heaved as his lungs fought to pull in sufficient air. The carbon dioxide level in the boat was becoming dangerous, and their store of lithium hydroxide was all but gone. They kept the dry powder in canisters and spread it across flat surfaces to absorb CO2. They also carried chemical oxygen generator canisters that, when activated, heated up to create oxygen. Unfortunately, the excessive heat in the boat made using the canisters too much of a fire hazard.\n\nSweat stung Orlov's eyes as he laid a plot. He heard another explosion and then a loud thud. At first he thought something had come loose from the overhead. He glanced to his right. One of the duty officers had fainted from lack of oxygen and collapsed to the deck. Captain Savitsky summoned the boat's doctor. Then someone else fell down. Then another. Savitsky called for relief watchstanders as the fallen sailors were carried out of the CC. Orlov wondered if he'd be next, or if he'd ever see the snow-capped mountains of his youth again, taste his mother's cooking, or listen to his father's well-worn sea stories.\n\nMore loud explosions rocked the boat like a fishing bob on a stormy lake. The intensity of the blasts made them sound much louder than warning charges. Captain Savitsky came unglued. He started screaming that the Americans were now dropping real depth charges. Captain Arkhipov, who stood next to Savitsky near the conning tower ladder, countered that such a thing could not be true.\n\nAs the two argued, Orlov saw Valentin Grigorievich, the officer in charge of the nuclear torpedo, shoot through the forward hatch and enter the CC. \"Captain, are we under attack? Should I ready the special weapon?\"\n\n\"Affirmative,\" Savitsky said. \"Ready tube number two.\"\n\n\"No!\" Arkhipov yelled. \"The conditions for firing have not been met.\"\n\n\"Sounds like the war has already started while we've been doing somersaults down here,\" Grigorievich said.\n\n\"I agree,\" Savitsky said.\n\n\"You'll kill us all,\" Arkhipov said.\n\n\"But not in vain.\"\n\nArkhipov grabbed Savitsky by the arm, \"You can't do this; we don't have authorization.\"\n\nSavitsky pulled his arm away. \"I can, and I will. Our batteries are depleted, our air is gone, and I will not disgrace our navy.\" He sent Grigorievich forward to prepare the purple-nosed weapon. He then turned and yelled toward the fire control station, \"Fire Control, distance to the carrier?\"\n\nA pink-faced michman called from around the periscope housing. \"Twenty-two cabletovs, sir.\"\n\nTwo nautical miles, thought Orlov. He knew their nuclear torpedo could kill out to a radius of ten nautical miles, which meant that Arkhipov's words were true. They would die along with the Americans. Despite the heat, Orlov shivered, now certain that his next few breaths would be his last.\n\n\"Acoustic,\" Savitsky called, \"bearing and speed on the carrier?\"\n\n\"Control, target bearing two-one-nine,\" acoustic replied, \"speed ten knots.\"\n\n\"Don't do this,\" Arkhipov pleaded. As if to mock his plea, another charge shattered overhead. The brigade chief grabbed a ladder handle to keep from falling.\n\n\"I am in command of this vessel,\" Savitsky said, \"not you.\"\n\nOrlov's head pounded from lack of oxygen as he studied the plot on the nav table. All fourteen warships were well within the nuke's kill zone.\n\n\"Watch Officer,\" Savitsky said, \"make your depth sixty meters, course two-one-nine, speed six knots.\"\n\n\"Yes, sir.\"\n\n\"Acoustic, prepare first measurement.\"\n\nAcoustic gave a bearing.\n\n\"Acoustic...zero,\" Savitsky said. \"Regimen one minute.\"\n\nHis breathing labored, Orlov updated the plot to the carrier. Acoustic started feeding bearings to the torpedo attack crew\u2014which included the navigator, electronics officer, and fire control team\u2014every minute. That crew, which Orlov now joined, plotted each change until the captain was satisfied that they had an adequate firing solution.\n\n\"Control, target bearing two-one-five, speed ten knots, distance nineteen cabletovs.\"\n\n\"Prepare to fire tube number two,\" Savitsky said.\n\nOrlov's heart fluttered. He felt dizzy and thought he'd collapse like the others. He steadied himself on the edge of the plot table and fought to stay conscious. If he was going to die, he wanted to at least hear the explosion.\n\n\"Vitali,\" Arkhipov said to Savitsky, his voice almost a whisper, \"please.\"\n\nAnother loud explosion threatened to shake loose the fillings in Orlov's teeth.\n\nSavitsky narrowed his eyes and stared at the brigade chief. \"Acoustic, prepare last measurement...zero.\"\n\nThe captain raised his arm. Orlov felt the hand of fear wrap about his throat like a giant snake. He closed his eyes and waited for the end.\n\nDeputy Political Officer Ivan Maslennikov jumped through the aft hatch from Compartment Four. \"Wait!\"\n\nSavitsky lowered his arm without issuing the final order to fire.\n\n\"If you fire now, Vitali, you will kill us all and start a war,\" Maslennikov said.\n\nSavitsky pointed an angry finger skyward. \"It's already started!\"\n\n\"No,\" Maslennikov said. \"Those are not full-strength depth charges. They have fourteen warships surrounding us. If they wanted to sink us, we'd already be dead.\"\n\nSavitsky lowered his head and stared at the deck. His eyes darted back and forth, and Orlov could tell that the captain battled with his anger to make the right decision. A minute passed, then another.\n\nSavitsky raised his head. \"Attention in the CC. Cancel attack and prepare to surface.\"\n\nAnother explosion ruptured a seal and sent seawater shooting into the CC.\n\n## CHAPTER TEN\n\nSurvival is triumph enough.\n\n\u2014HARRY CREWS\n\nHIS FACE RED WITH ANGER, JOHN Scali of ABC News met again with Soviet embassy official Aleksandr Fomin at 4:15 P.M. on October 27. Scali demanded to know why the terms offered in the two letters from Premier Khrushchev were as different as night and day. Fomin stuttered and said that it must be due to \"poor communications.\" Scali pointed an accusatory finger and shouted that the differing letters were obviously a \"stinking double cross.\" He stated that an invasion of Cuba was only hours away. Fomin's eyes widened. He mumbled that a response was expected from Khrushchev at any moment and urged Scali to tell the State Department that they intended no treachery. Scali shook his head and said that no one would believe that to be true, but he'd deliver the message to the White House anyway.\n\nAt 8:05 P.M., Khrushchev sent a letter outlining the structure of a deal. President Kennedy responded that if the Soviet Union agreed to remove all weapons systems from Cuba, under U.N. observation, and halt further delivery of such weapons, the United States would remove all quarantine measures and refrain from invading Cuba. A secret memo accompanied the letter. Some experts and scholars speculate that President Kennedy confirmed his offer to remove the missiles from Turkey in that note. A confidential source that worked for the NSA on the Bore-sight and Bulls Eye programs believes that the president also laid down a trump card. This informed source speculated that the memo contained day-by-day location coordinates for the four Foxtrot submarines, outlining their track since arriving in the Sargasso Sea. Unaware of the existence of Boresight, Khrushchev could only guess as to how the United States managed to obtain this information.\n\nUntil the receipt of that memo, the Soviet premier's demeanor exhibited confidence and defiance. He slammed his left hand on the table when meeting with advisers and shouted assurances that Kennedy could not invade Cuba. To do so invited suicide. His Project 641 submarines, each carrying a nuclear-tipped torpedo, were his safety nets. With his deadly Foxtrots now exposed by unknown means, he could no longer deal from a position of strength, and so was forced to dictate a \"surrender\" letter to Kennedy. The president called the letter \"an important and constructive contribution to peace.\"\n\nOthers contend that a bartender helped end the crisis. Aleksandr Fursenko and Timothy Naftali, in their book One Hell of a Gamble, suggest that Khrushchev backed down due to the ears of a Russian immigrant from the Balkans named Johnny Prokov, a bartender at the Tap Room in the National Press Club in Washington, D.C. They believe that Prokov passed to Anatoly Gorski, a Soviet KGB officer, information overheard during a conversation at the bar between two celebrated American journalists, Robert Donovan and Warren Rogers, both correspondents with the New York Herald Tribune.\n\nFursenko and Naftali speculate that the KGB could have passed this information immediately to Moscow, and Khrushchev have had it on his desk within hours. Based on the hearsay overheard in a bar, the stubborn Soviet premier finally became convinced that Kennedy was serious about going to war over Cuba. To avoid that war, Khrushchev infuriated Fidel Castro by pulling the plug on two massive operations that spanned more than six months, involved hundreds of ships and thousands of sailors, and cost billions of dollars. Certainly this conjecture is feasible, but perhaps unlikely.\n\nDespite Khrushchev's overture, however, the Cuban Missile Crisis did not end. In fact, the conflict on the high seas escalated, and neither side knew then just how close they came to Armageddon.\n\nAT 10:52 P.M. ON BOARD THE destroyer USS Cony, Ensign Gary Slaughter, the phone talker on the bridge, held his breath at the sight of a Foxtrot submarine breaking the surface in a swirl of white foam. The Cony had chased the Foxtrot for more than twelve hours. Although the sight filled Slaughter with awe and pride, he did not know how close he'd just come to being vaporized by Captain Savitsky's B-59. The Soviet submarine, her sleek hull dripping with salt water, turned northeast and started to snorkel. Several men appeared on the bridge, all wearing Russian uniforms and covers.\n\nTrained in Cyrillic transliteration tables, Slaughter ordered Signalman First Class Jessie to flash a communiqu\u00e9 to the men on the sub's bridge.\n\n\"What ship?\" Jessie flashed with the Cony's signal light.\n\n\"Ship X,\" came the reply.\n\n\"What is your status?\" Jessie flashed back.\n\n\"On the surface, operating normally.\"\n\n\"Do you need assistance?\"\n\n\"No, thank you.\"\n\nHours passed with no further communication as the Foxtrot slowly steamed northeast. The following morning, the Cony received another flash. The submarine asked for bread and American cigarettes. Slaughter smiled as the Cony maneuvered to within eighty feet of the Soviet vessel and prepared for a line transfer. One of Cony's bosun's mates set up the shotgun-looking device designed to thrust a weighted rubber-tipped projectile attached to a long line. The bosun's mate fired the shot line toward the Foxtrot's bridge.\n\nWhen they saw the spearlike projectile flying toward them, the Russians yelled, pointed skyward, and ducked down below the bridge fairing. Slaughter let out a laugh. He recalled that Soviets didn't use shot lines, but instead employed bolo lines flung by a strong seaman. After they sent bread and cigarettes to the Foxtrot, the Cony shadowed the Soviet sub for several more hours, staying 500 yards off her port beam.\n\nThat evening a P2V Neptune ASW plane streaked down from the darkened sky and dropped a scattering of incendiary devices to activate the aircraft's photoelectric camera lenses. Again the men in the Foxtrot's bridge yelled, pointed, and scrambled for cover. Inside B-59, Captain Savitsky interpreted the miniature explosions as a possible attack. He wondered if the Americans had tricked him into surfacing so they could easily sink his boat. Posturing to defend his craft, he armed his nuclear torpedo and opened his outer torpedo tube doors.\n\nOn the Cony, watchstanders shouted as the Soviet sub changed course. Sonar operators reported the sound of torpedo tube doors opening. Ensign Slaughter heard the Cony's captain hail the aircraft carrier Randolph and demand that the Neptune's pi lot cease his photo runs. The CO then ordered Slaughter to have a signalman flash an apology to the Soviet submarine. Nervous minutes passed. Finally, the Foxtrot closed her torpedo tube doors and turned back toward the northeast.\n\nON OCTOBER 28, 400 MILES NORTH of San Juan, Puerto Rico, aboard the destroyer Charles P. Cecil (DD-835), Lieutenant John Hunter, the officer of the deck, saw something that bothered him. Hours earlier they'd been ordered by CINCLANT to search within a twenty-five-nautical-mile radius of a specific coordinate. Hunter did not know why they'd received that order, but he suspected that SOSUS or some other system detected a submarine. He had no knowledge of Bore-sight.\n\nHunter glanced at his watch. Just after 7:00 P.M. local time. He studied the green phosphor glow on the radarscope. A blip, one that resembled the irregular echo of a snorkeling submarine, or \"snork\" in ship jargon. Hunter contacted the Combat Information Center. Radar-man Third Class Russ Napier had also detected the contact in the CIC. Hunter picked up the phone and called the ship's captain, Commander Charles Rozier. The captain granted permission to prosecute. Hunter then made a huge mistake.\n\nHoping that he might fool the Soviet submarine, he ordered a high-speed run toward the contact. He also called down to the sonar shack and asked them to turn off the active stack. No more pings. By the time the Cecil hit twenty knots, the radarscope went blank due to the ship's churn. When the Cecil neared the sub's last known location, Hunter contacted sonar and ordered them to turn on the active stack. Just before he hung up the phone, he heard a loud pop in the background. Sonar called back. The stack had fried itself. The Cecil just lost an ear.\n\nHunter stared at the radar screen. No more blip. Obviously, the submarine went deep. He swore under his breath. Not only did his trick not work, but his decision to switch off the active sonar backfired. They'd just lost the Soviet sub.\n\nCAPTAIN DUBIVKO SURFACED B-36 EARLY IN the evening on October 28. His orders had not changed in days: to maintain a combat-readiness posture on station south of B-59 and B-130 just outside the center of the quarantine line. That position, of course, placed him square in the middle of a phalanx of American ships and planes. Avoiding those hunter\/killers kept the crew on a tight edge while they sweltered in the heat.\n\nDubivko pulled himself up the ladder through the conning tower and edged onto the bridge. A setting sun splashed streaks of purple-orange across the horizon. His lungs savored the fresh air. The past few weeks had taken their toll. Due to excessive heat and lack of sleep, he had dropped almost fifty pounds. Now he could no longer climb a ladder without exhaustion.\n\nThree of the crew stepped onto the bridge platform and leaned their heads back to smell the salt air. They appeared gaunt and weak with deep circles under their tired eyes. Here in enemy waters, Dubivko should be snorting versus surfacing to snorkel. But he was worried about his crew. Given the deplorable conditions, he didn't know how much longer they could endure. He decided to take the chance and surface, and allow the crew some fresh air and saltwater showers in the sail.\n\nEarlier in the day, the OSNAZ group intercepted a radio report that several Russian merchant ships had reversed course and were headed back to the Soviet Union. Dubivko wondered what that could mean. Had Moscow abandoned Operations Anadyr and Kama, thus ending Khrushchev's ambitious plan to place missiles and submarines in Cuba? Were they preparing for war? Having received almost no information from Moscow, he could not even wager a guess. Assuming the worst, he ordered Zhukov not to use surface search radar to minimize detection. That decision came with a downside. An approaching enemy ship could surprise them, and with the additional men topside, diving would be delayed.\n\nIn the event of that scenario, given that they were in calm seas, Dubivko ordered a depth that placed B-36's bridge lower in the water than normal while they snorkeled. He loved cruising on the surface at night with a light breeze ruffling his hair and caressing his cheeks. Despite the hardships they'd endured on this mission, standing high in the sail of his boat made everything seem right again. This was his little slice of heaven, and nothing could take that away.\n\nAlmost nothing.\n\nJust after 7:00 P.M., Zhukov shouted up through the open hatch, \"Contact bearing zero-five-zero!\"\n\nDubivko called down, \"Range? Contact type?\"\n\nSilence. Then, \"Two thousand meters, probable American destroyer!\"\n\nDubivko peered through his binoculars and scanned the horizon. Nothing but black. The destroyer was running without lights.\n\n\"Srochnoiya pogruganye!\"\n\nThe bridge area cleared, and Dubivko slammed the hatch closed above his head. In the CC, he called over the communication line to Pankov, \"Acoustics, bearing to contact?\"\n\n\"Control, we are now detecting two contacts on bearing zero-five-five. Second contact has high-speed screws.\"\n\nTorpedo?\n\nDubivko wondered if they were now at war, and B-36 was destined to become the first vessel sunk. \"Watch Officer, all ahead full, right full rudder, depth seventy meters.\"\n\nDubivko cursed the unseen demons. He did not have a full charge on the batteries. His men were worn down to threads, and worst of all, the damage to the VIPS lid limited their depth to seventy meters. He was a David with one arm trying to fight dozens of Goliaths. At least he had rocks. One big rock in particular. But to fling that weapon from his sling meant ending their mission forever.\n\nON OCTOBER 29, AS THE CLOUDS split to reveal a red sky at morning, Commander Rozier strolled onto the bridge of the Charles P. Cecil and relieved John Hunter of the conn. After the sonarmen repaired the active stack, they managed to regain contact on the Soviet sub. Their prey tried a cadre of tactics to evade but failed. Rozier held on like a cowboy on a bucking stallion, one arm waving a hat. He knew the enemy was blessed with excellent thermal conditions, with the isothermal layer extending down to more than 300 feet, but he suspected fate had intervened to take that advantage off the table.\n\nWhen Rozier received the flash from CINCLANT, providing coordinates and a defined search radius, he was more than surprised. He knew SOSUS was good, but not that good. When he read that the contact type was a probable Foxtrot, he let out a long whistle. Just what kind of new technology could be that certain of the platform type? Now, suspecting that they were prosecuting a Foxtrot, he wondered if something was wrong with the boat.\n\nSoviets used three ratings for submarine diving depth characteristics. Working depth, referred to a window within which a boat could operate under normal conditions for extended periods and engage in combat. Limiting depth documented how deep a submarine could dive before she started hacking and wheezing. Most boats were restricted to a maximum of 300 excursions to this depth for their entire service life. Lastly, design collapse depth was where a submarine turned into Cream of Wheat.\n\nRozier knew that a Foxtrot's limiting depth was 985 feet, which afforded a working depth of 854 feet. Yet their contact, based on active ping estimates, had not descended below 250 feet. Why? That boat's captain must know that the thermal layer started at 300 feet, and if they dropped below that threshold, they'd have a much better chance of avoiding detection. Yet they stayed above the layer. That could only mean an ignorant sub captain or some kind of damage that kept them shallow, and judging by the evasion tactics used so far, Rozier didn't think he was chasing an idiot.\n\nThe Foxtrot's captain used a clever mix of tactics, including trimming the tanks, adjusting the bow planes, speeding up and slowing down, frequently changing course, and hiding in the Cecil's wake. The wake tactic created a baffle that masked the destroyer's active sonar pings, given this at more than thirty knots, the Cecil churned the ocean to a depth of seventy feet. When everything else failed, the Foxtrot tried to inhibit the Cecil's active sonar pings by shooting out chemicals that formed a cloud of hydrogen bubbles when mixed with seawater. That didn't work either.\n\nThe Foxtrot then tried something new. From the sonar shack, Sonarman Third Class Elroy Nelson reported that the sub's screws had gone silent. All he could hear now was the chatter of snapping shrimp in his headset. Rozier figured that the Soviet sub had stopped dead in the water. He circled the last known contact location, pinged, and waited.\n\nA half hour later, the Foxtrot surged back to five knots and ran toward them in an attempt to get behind the Cecil and hide in the ship's wake again. Rozier refused to let that happen. At 500 yards distance, he ordered turns for fifteen knots and right full rudder to keep the Foxtrot pointed toward their bow.\n\nThe Soviet sub then released a noisemaker torpedo, revved up to full speed, and darted off to starboard. Rozier knew to watch for the tactic, given that Soviets were taught to do this during a close-in attack. The aft torpedo room on a Foxtrot housed four torpedo tubes\u2014two filled with warshots and two with noisemakers. In a combat situation, Soviet captains were trained to fire off a noisemaker, followed by a warshot, then make a wide turn and sprint; slow again and go silent. Hopefully, any pursuers would follow the noisemaker and not the submarine. Rozier did not intend to fall for the ruse. Still, he couldn't seem to corner the submarine into a tight enough box to drop any warning charges.\n\nRozier decided to enlist some help by contacting a P2V Neptune ASW plane patrolling the area out of Jacksonville, Florida. The P2V, with the call sign Pollyboy 5, had just relieved another aircraft on station and came roaring toward the Cecil. The plane circled overhead as the Cecil directed the runs over the radio, helping the plane correct its lead angles and line up correctly over the target. Several \"no joy\" runs later, Rozier wondered if they'd ever nail this guy. Then the P2V reported \"Madman,\" and Rozier smiled. The ASW plane's magnetic anomaly detection (MAD) equipment had picked up a fluctuation in the earth's magnetic field when it flew over the Foxtrot's metal hull. Contact nailed.\n\nRozier stayed on top of his quarry throughout the night. More P2Vs arrived in the morning and lit up the ocean with active sonobuoys. They eventually validated contact C-20 as a Foxtrot. Rozier suspected sooner or later that the Soviet sub would run low on battery power and have to surface, but his patience for this game was running thin. Now that he'd cornered the Foxtrot, he decided it was time to drop a few loud incentives into the water. Little did he know that his decision was about to place his ship at ground zero.\n\nALMOST 300 MILES NORTHEAST OF CAICOS Passage, the USS Blandy and several other U.S. destroyers stared at a ghost. They had pursued the illusive Foxtrot contact for fourteen straight hours after she dove to avoid detection by ASW aircraft. Below the surface, Captain Shumkov on B-130 used up a quiver of tactical arrows to dodge the destroyers. In a last-ditch attempt, he shot a noisemaker torpedo from an aft tube. The noisemaker emitted the same frequency as a Project 641 submarine to fool enemy sonar operators. Normally, Shumkov would trail off in a preset direction at ten knots after firing one, but with batteries running low, that option no longer existed.\n\nOn top of the waves, Blandy's sonar operators not only did not fall for the trick, but misinterpreted B-130's noisemaker as \"high-speed screws in the water.\" Blandy's skipper, Captain Edward Kelley, took evasive maneuvers and readied his weapons. Seconds before firing, sonar reported a \"false alarm.\" The contact had only released a decoy that sounded like high-speed screws. Captain Kelley took his finger off the trigger.\n\nSeveral hundred feet below the Blandy, Captain Shumkov ran out of options. Depleted batteries and thin air left him with no choice but to surface or die. Cornered, exhausted, and at his wits end after days of undersea conflict, with explosive charges raining down, Shumkov decided to end the ordeal by firing the special weapon.\n\nHis decision was motivated partly by pride and partly by the desire to save face in front of his political officer, Saparov. He ordered the special weapons security officer to make preparations to fire the nuclear torpedo. The security officer refused, citing that they did not have authorization. Fortunately, fate intervened. Due to the excessive heat and deplorable conditions on board, Saparov took ill and passed out. Shumkov no longer had an audience to bolster his ego. After considerable deliberation, he finally gave the order to stand down. Later he informed the crew that he had never intended to use the special weapon, as B-130 would not have survived the blast.\n\nCaptain Shumkov surfaced B-130 into the fading light of a setting sun on October 30. Not long after, while devouring another meat-filled piroshki, he heard his last engine die. Up until then, he'd clung to a desperate hope that his crew could repair at least one of the other two dead engines. Devastated, he now had no choice but to call Moscow and request a tow home. The bite of piroshki in his mouth turned bitter as he cursed the gods for forcing him to abandon his duty and leave his friends behind in hostile waters.\n\nCAPTAIN DUBIVKO ON B-36 IMAGINED A miniature demon sitting on his shoulder and laughing. The gremlin had succeeded in taking away one of the most important tactical advantages available to any submarine: the ability to go deep. The thermal layer started at one hundred meters, but since they'd damaged the VIPS lid, they could not go below seventy meters, so evading enemy sonar became almost impossible. Inside B-36, the conditions became even more horrific. With temperatures exceeding sixty degrees centigrade in the engine room, and forty-five in other compartments, Dubivko ordered frequent bed rest for most off-duty personnel. Some of the men still passed out due to heat stroke not twenty minutes into their watch.\n\nLack of water exacerbated the problem. Reserves allowed for no more than 250 grams per day per man. Profuse sweating and dehydration increased skin rashes and decreased appetites. Many crew members lost more than thirty percent of their body weight. Fortunately, they'd stocked plenty of fruit compote, which the crew drank in place of water.\n\nAfter they had detected the destroyer, they quick-dove to seventy meters. At thirty meters, they heard the loud chug of the destroyer's engines pass overhead. What they thought were high-speed screws in the water\u2014indicative of a torpedo\u2014turned out to be a towed sonar array.\n\nThe American destroyer continued to harass them by pounding the ocean with incessant active pings. Acoustics reported yet another contact in the port baffles. Given the noise interference generated by the destroyer and circling aircraft, discerning what type of platform this contact might be was not possible. Acoustics did report that the hydrophone effects appeared to be coming from a depth of twenty meters, but they could not be certain. Dubivko wondered if the contact might be an American submarine or one of his sister boats. If the former, Dubivko was a sitting duck. Unable to dive deep, with the batteries nearly depleted, the odds of dodging a speeding torpedo were dismal at best.\n\nHe walked to the nav table and asked Naumov to lay out the plot showing the patrol areas of the four Project 641 submarines. He traced the bearing line to the eastern edge of B-130's assigned patrol area. As the main navy staff still included the call signs of all four boats in their transmissions, Dubivko was unaware that Captain Shumkov had surfaced earlier, lost his last engine, and was preparing to be towed home. He also did not know whether hostilities were escalating or if the two superpowers were engaged in a shooting war. Given that the destroyer circled at 2,000 meters, he suspected that gun barrels remained cold, but he needed to know for certain.\n\nDespite the risk, Dubivko ordered his boat to periscope depth. They needed to snort and assess the tactical situation. Also, they were overdue for a burst transmission from Moscow, and Dubivko wanted to determine if their orders had changed regarding the use of the special nuclear weapon.\n\nAcoustics reported that the P2V Neptune had just made another pass. Dubivko glanced at his watch. Caution dictated timing the passes and lowering the periscope each time the aircraft came near, especially since nightfall remained an hour away. A sailor cracked the conning tower hatch open at twenty meters depth, and Dubivko hurried up the ladder, followed by Political Officer Saparov. He waited for the P2V to make another pass, then raised the periscope. Salt water dripped onto his shoulders, but the small space felt cool compared to the overheated CC. He made a quick visual sweep. Flashes of crimson accented ethereal clouds on the horizon and reminded him of Polyarny.\n\nDubivko spun the scope and marked off the bearings to each of three gray destroyers. Running without lights, the dark silhouettes bristled with guns and antennas. On a fourth bearing, Dubivko spied a single blinking light. He squinted. The shape of a helicopter came into view. Below the whirling blades, a long cable ran into the ocean.\n\nDipping sonar.\n\nDubivko lamented the good weather. Calm seas afforded no hiding places. He lowered the scope and called Potapov on the phone. The chief mechanical engineer reported that the electrolyte in the batteries was now almost pure water. That meant they needed to snort within fifteen minutes. Acoustics reported the sound of the P2V swooping overhead again. Dubivko counted off several seconds in his head, then raised the scope.\n\nHe focused on the stern of the closest destroyer and spotted a large reel of black cable on the fantail. One end of the cable ran over the side and disappeared into the water. A sonar transducer. Dubivko's pulse quickened. Transducers were towed by destroyers and lowered to depths beneath the thermocline to find hiding submarines. At the end of the cable hung a large teardrop-shaped device that improved the destroyer's ability to find submerged targets. For Dubivko, that posed a huge threat, not just from detection, but from collision. If B-36 came too close to that transducer, the heavy object could rip a hole in their side and send them to the bottom.\n\nDubivko checked his watch again. \"Opustit periscope.\"\n\nThe scope lowered back into the housing. He called Acoustics and asked for a depth update on the thermal layer. Unchanged at one hundred meters. The contact reported earlier, which Dubivko thought might be B-130, remained at twenty meters distant. If this was indeed Shumkov, he knew where the layer started, and unless B-130 also suffered from a depth-limiting problem, Shumkov should have descended below that layer. Still, Dubivko had to assume that the contact might be the B-130 and that the Americans had forced them to come shallow.\n\nDubivko struggled with his next decision. He turned over the conning tower to Saparov and slid down the ladder into the CC. He picked up the phone and summoned Alexander Pomilyev, the officer in charge of the special weapon. He met the young man outside his stateroom. They entered, and Dubivko sat on his bunk, while Pomilyev remained standing.\n\n\"I want you to ready the weapon in tube number two.\"\n\nPomilyev's eyes widened. \"Sir?\"\n\n\"We need to surface. If the enemy turns hostile, I intend to fire at the closest target.\"\n\n\"But, sir, that will destroy us as well.\"\n\nDubivko remained silent for a long moment, then said, \"Prepare the weapon.\"\n\nPomilyev lowered his chin. \"Yes, sir.\"\n\nDubivko returned to the conning tower, timed the P2V pass, and raised the scope. He swallowed his guilt and ordered a heading of due east, according to the surrender directions they'd received repeatedly from the Americans\u2014rebroadcast from Russia. With their batteries gone, the least he could do was give B-130 a fighting chance to get away if she was nearby. If B-36 surfaced now, that distraction just might afford Captain Shumkov enough window to sneak away.\n\n\"Vspletye!\"\n\nON OCTOBER 31, AT 4:00 A.M. aboard the USS Charles P. Cecil, Sonarman Elroy Nelson finally got relieved from the stack by Sonarman Allen Tuell. The twenty-something Tuell from Santa Cruz, California, yawned as he settled in for what he hoped wouldn't be a boring six-hour watch. Less than two hours later, at 5:53 A.M., Tuell heard a familiar sound in his headset. His trained ears discerned the shush of high-pressure air. The Foxtrot just blew her ballast.\n\nLieutenant John Hunter heard the 1MC loudspeaker crackle. \"Submarine on the surface!\" All about him, excited sailors in skivvies and shower shoes lined the railing to get a glimpse of the surfacing sub. The stubby nose of a Soviet Foxtrot broached the surface, followed by the long black hull and windowed sail. Here and there the black deck was mottled with weatherworn dull red patches. Hunter wondered if that might be rust. On the sail, three numbers shimmered white in the morning sun. They read 911. Hunter knew this was not the boat's real designation, but only a decoy to fool the enemy.\n\nEnemy.\n\nHunter thought about that word for a moment. Were they really the enemy, or just seventy-eight men caught up in a high-stakes game of cat and mouse started by narrow-sighted men in power? Hunter squashed his political thoughts as several figures appeared on the bridge of the Foxtrot. One raised a flag atop the main antenna, which flapped in the breeze from the back of the sail. Hunter recognized the emblem as the main cruiser ensign of the USSR Naval Fleet. He knew this was a big moment, that he was now a part of history and witnessing the forced surfacing of a Soviet submarine in the Caribbean during the Cuban Missile Crisis.\n\nHis smile died as he saw the Foxtrot make a sudden turn away from the Cecil.\n\nON THE BRIDGE OF B-36, CAPTAIN Dubivko adjusted his binoculars and focused on the boxy destroyer still shadowing them. After intercepting a radio transmission sent to their American escort, the OSNAZ team told him the name of the ship was the Charles P. Cecil. Now less than one hundred meters away, the dark shape of the Cecil loomed over his boat like a massive guard marching him to the gallows.\n\nDubivko figured that the ship's captain maintained that proximity because he knew the \"dead zone\" for Soviet torpedoes\u2014the distance traveled after leaving the tube before the weapon armed\u2014was right at 150 meters. This safety measure, programmed into all torpedoes, prevented weapons from detonating too close to their firing submarine. If Dubivko shot a torpedo at this close range, the projectile would bounce harmlessly off the metal side of the destroyer.\n\nThe American ship flashed a message from their bridge. \"Do you need help?\"\n\nDubivko ordered an OSNAZ signalman to reply. \"We do not need any help. Asking you not to interfere with our actions.\"\n\nDubivko scanned the destroyer. Several gun turrets pointed in their direction. The long barrels looked ominous in the fading light.\n\nWere they preparing to fire?\n\nConvinced that they were, Dubivko lowered his binoculars and called down to the CC. \"Boyevoya trevoga!\" He heard the crew scramble to battle stations. He then ordered a turn to port, away from the enemy ship. He needed to open the distance to more than 150 meters so the nuclear torpedo could arm itself. He called Pomilyev in the forward torpedo room and asked the special weapons officer to prepare to fire tube number two on his order. His hands shook as the boat's screws slapped the ocean and edged B-36 farther away from the destroyer. They had fifty meters to go before they were 150 meters away from the ship.\n\nDubivko thought about his home, his wife, and the life he would never see again. A part of him fought against his decision and tried in vain to talk the other part out of it. But pride and honor made him deaf to the voice of reason. He could not sit still while the Americans blasted his boat into shards of steel. He'd go down fighting with the last breath in his tired and worn body.\n\nDubivko glanced again at the radar repeater as the boat passed 150 meters. He started to call down to Pomilyev but hesitated for a brief moment. Zhukov called up from radio. \"Sir, we just intercepted a radio transmission to the Charles P. Cecil\u2014the destroyer that is shadowing us.\"\n\n\"Transmission?\" Dubivko said.\n\n\"From President Kennedy.\"\n\n\"President Kennedy?\"\n\n\"Yes, sir. The message to the Cecil reads, 'Thank you for your work. Keep the surfaced submarine here by all means.'\"\n\nDubivko pondered the message and picked up his binoculars. He scanned back and forth along the destroyer's hull. The gun turrets were now turned away. He breathed a sigh of relief, changed course, and told Pomilyev to disarm the nuclear torpedo. Glad to still be alive, Dubivko descended back into the belly of his boat and ordered Zhukov to transmit a burst message to Moscow informing them of B-36's situation.\n\nThey sent the same message forty-eight times to the main HQ of the naval fleet until they finally received a response. The Cecil continued to shadow B-36 as the crew repaired the damaged upper lid of the VIPS, which earlier had prevented them from being able to deep dive. The diesels ran nonstop for the next two days to charge the batteries. Normally, the charge took only ten to twelve hours, but due to the long periods of high temperatures in the boat, the electrolyte in the batteries soared past sixty-five degrees centigrade. That required them to ventilate extensively to reduce the temperature back down to sixty so as not to cause an explosion.\n\nNot blasted to pieces, not nuked by their own torpedo, not blown up by hot batteries, no longer debilitated by a broken VIPS lid, Dubivko swallowed a bite of piroshki, lit a cigarette, and contemplated an escape plan. A crazy idea popped into his head, and he called Zhukov and Pankov to his cabin.\n\n\"We'll need to distract them,\" Dubivko said as he sat on his bunk.\n\n\"We could try sending a signal in the circular regime using the Sviyaga hydroacoustic station,\" Pankov said, standing bent over in the cabin.\n\nZhukov, also standing bent, scratched his chin. \"That might work. We'd have to rig our acoustics to send out a strong signal on the same frequency as their active sonar beam.\"\n\n\"Is that feasible?\" Dubivko asked.\n\n\"I think so,\" Pankov said. \"We know the frequencies they use, and we're now close enough, but we will need to send several strong transmissions.\"\n\n\"This will work for only a short time,\" Zhukov said. \"Once we're too far away, their sonar will not be disrupted.\"\n\n\"How long?\" Dubivko asked.\n\n\"Minutes,\" Pankov said.\n\n\"That's long enough,\" Dubivko said. \"Make it happen.\"\n\nThe following day, just after lunch, with no helicopters or ASW aircraft in sight, Pankov flipped a switch and flooded the Cecil's underwater ears with a high-pitched whine. Dubivko slammed shut the upper hatch and quick dove. The boat passed underneath the Cecil, and he was certain they missed the ship's hull by mere centimeters. He submerged B-36 to 200 meters, reversed course, and zigzagged away at a fast clip. Pankov lit off a stream of six-second transmissions to suppress the Cecil's active sonar beams. The trick worked.\n\nHours later, Dubivko brought B-36 shallow to transmit a series of burst updates to the main navy HQ. This time they received an immediate reply. Unbeknownst to Dubivko, B-36's transmissions provided solid hits to three Boresight stations.\n\nAmerican ASW planes reappeared, but this time, with a full battery charge and the ability to go deep, B-36 managed to avoid detection. Dubivko attributed this to skill and luck, unaware that the Americans were using Boresight fixes to plot his homeward track. As long as he kept his submarine pointed toward Russia, they wouldn't have to sink his boat.\n\nNOW ONE HUNDRED MILES SOUTH OF Kingston, Jamaica, Captain Ketov on B-4 continued to evade American ASW forces. He harbored no illusions that his good fortune might be attributed to an upgraded sonar system, which afforded B-4 longer-range hearing. That, and a large dose of luck. While B-4 snorkeled throughout the night of November 2, Ketov inhaled the sweet night air, watched silver moonlight dance on the wave tops, and prayed that his good fortune would hold.\n\nFor certain, lady luck had played a role, but Ketov also gave credit to his crew. They'd done an excellent job of combining skill and intelligence to avoid the enemy. While doing that, Ketov noticed that ASW planes stalked them the most when they started to snorkel or snort at night, and aircraft numbers diminished during the day when B-4 ran on batteries. The planes appeared to come around the most in the early evening, not long after B-4 lit off her diesel engines. Based on this pattern, Ketov surmised this was no coincidence and might be due to detection by the American hydroacoustic stations embedded in the seafloor. Of course, he had no knowledge of Boresight, and so did not assume that B-4's late afternoon burst transmissions triggered an ASW response that placed aircraft nearby when the submarine came shallow again to snort.\n\nKetov stood on the bridge and listened to the muted hum of B-4's diesels. He thanked what ever god there might be that he had not fallen to the fate of his friend, Captain Shumkov, on B-130. Ketov could not even imagine losing his engines and having to be towed home. When he heard about the incident on the daily radio update, he wanted to call his comrade and express his condolences. He could not, of course.\n\nKetov glanced at his watch. An additional communications session with Moscow was scheduled for that night. He figured when the session started in twenty minutes, new orders would be issued. During the next five minutes, he let his mind go on vacation. He thought about his wife, his children, and all the things he had missed while living most of his life far away from home. Five minutes came and went before a report from radio jolted Ketov from his daydreaming. Radio detected a signal on the Nakat. The hit was weak but growing stronger. Ketov had observed earlier that some of the P2Vs and Tracker aircraft took to truncating their radar signals in an effort to fool the Soviet subs by making the planes appear farther away than they actually were. He did not intend to fall for this trick. He ordered the bridge cleared and quick dove to a depth of ninety meters.\n\nBrigade Commander Captain Agafonov cornered Ketov in the CC and demanded to know why they went deep fifteen minutes before the scheduled session with Moscow. He appeared anxious to receive an update, as interference had blocked reception of the last one. Ketov explained his reason for diving: that he knew the Americans attenuated their radar signals to ensnare unsuspecting submarines. Agafonov refused to listen. He insisted on surfacing to complete the battery charge and to ensure a clear signal for the radio session. Ketov stood his ground, stating that the OSNAZ group earlier determined that the USS Independence was operating in the area, and they detected dozens of ASW aircraft, including one likely headed their way at this moment. He told Agafonov that as long as he was in command, they were not going to surface.\n\n\"Fine,\" Agafonov said. \"You're relieved of command.\"\n\nShocked, Ketov implored Agafonov to reconsider and to not place B-4 in jeopardy by surfacing. Agafonov waved him off and gave the order to come shallow. Angry, but having no choice, Ketov turned over command and retired to his cabin.\n\nAgafonov brought the boat shallow and extended a periscope. The Nakat barked out a series of loud beeps as American radar got a hit on B-4's masts. Agafonov took the boat deep again, but by then, it was too late. A Tracker airplane from the Independence gained a solid lock and refused to let go. Agafonov tried to dodge as the plane dropped warning charges that rattled dishes and nerves.\n\nHours went by as Ketov remained in his cabin while a less experienced officer tried without success to avoid the enemy. Finally, after three hours of explosions and pinging, Agafonov became frustrated and angry. He called the special weapons officer in the torpedo room and ordered him to ready tube number two. Then he turned the boat toward the USS Independence.\n\nThe officer in charge of the nuclear weapon did as ordered, then called Ketov in his cabin. He informed his commander that Agafonov intended to fire the weapon. Ketov jumped from his bunk and ran toward the CC. He shot through the hatch and stood face-to-face with Agafonov.\n\nThey argued for several long minutes, then Ketov said, \"Do you really want to go down in history as the man who started World War III?\"\n\nThe brigade commander lowered his eyes. With his jaw clinched tight, he looked like a boiling kettle about to blow. He raised his chin and glared at Captain Ketov. The steam evaporated, and Agafonov relinquished command. Ketov relieved the commander and ordered the special weapons officer to stand down.\n\nAfter taking back control of his boat, Ketov recalled the directives issued by Kennedy and the Americans that submarines near Cuba should surface and head east at a slow speed. He did not intend to surface, but he instead instructed the OSNAZ group in radio to broadcast a message to the Americans that B-4 would comply with the instructions. The ASW aircraft stopped dropping explosive charges.\n\nAn hour later, Ketov cheated. With ninety minutes remaining until dark, he made a radical course change, took the boat beneath the thermal layer, and increased speed. As night unfolded in the Caribbean, Ketov slowly brought the boat shallow. Acoustics listened but heard nothing. Ketov popped up a periscope but saw no planes. Nakat ESM detected no nearby radar. Satisfied that no Americans stood poised to harass them, Ketov brought B-4 to snorting depth and recharged the batteries.\n\nHe spent the next ten days dodging the Americans as they made low passes across the ocean's surface to snag him via magnetic or diesel exhaust detection. They never found him. On November 12, he received new orders to head 500 nautical miles north to an area known as the main U.S. Navy ASW line south of Newfoundland. Over the next few days, Ketov noticed that the American ASW forces seemed to follow him north along his track. He saw long-range reconnaissance aircraft flying at high altitudes and P2Vs thrumming along at low or medium heights. Sometimes the Nakat picked up three pairs of P2Vs circling around B-4's previous course. Ketov remained puzzled. Although they still snorted at night, they went silent during the day. So how was it that the Americans could follow him so accurately?\n\nTHE CREW OF B-36 CELEBRATED THE Soviet October revolution on November 7 by taking ocean water showers in the conning tower. Captain Dubivko decided to join them when First Officer Kopeikin wrinkled his nose in the CC as Dubivko walked past. When another submariner can no longer stand your stench, it's time for soap. He drenched himself in the cool, refreshing waters of the Sargasso Sea pouring through the showerhead in the sail and smiled as the water covered his tired body. Finally, he thought, the demons have stopped harassing me. Dubivko's smile vanished when a sailor interrupted his shower to deliver an urgent message.\n\nEngineer Captain Potapov reported that the Compartment Five group commander, Senior Engineer Lieutenant Kobyakov, having become exhausted after enduring high engine-room temperatures for the past several weeks, failed to properly purge the air intakes on the two running diesels. Both engines seized up. That left B-36 with one working engine. Dubivko was now aware of what happened to Shumkov on B-130, and he harbored embarrassing visions of being towed back to port, or worse, of being forced to endure another babysitting escort by an American destroyer.\n\nHe dried off and clambered down to Compartment Five, where Potapov and Kobyakov updated him on the two dead engines. The two engineers spent the next few days cannibalizing one broken engine to try to get the other back online. Running with one engine made Dubivko's job of avoiding detection an almost impossible task. Still harassed by ASW aircraft, this debilitation forced him to keep B-36 stationary in the water while charging the batteries, then run slow on the electric engine during the day.\n\nThe OSNAZ group discovered that the helicopter carrier USS Thetis Bay (LPH-6) occupied the center of the deployment area, and her helicopters also exhibited an uncanny ability to follow B-36's track. Dubivko decided to forgo permission from the main navy HQ and moved B-36 200 miles south. Soon after, he received orders to return home. Hamstrung by two dead engines, B-36 could go no faster than six or seven knots, and Dubivko figured he would not see his wife again until late January, if at all. Then, as B-36 entered the Norway Sea, Potapov reported that he and Kobyakov had repaired one engine, so they now had two working. Dubivko sat on his bunk and let his shoulders relax after he heard the news, certain that the demons were finally departing. Unfortunately, they were only taking a short sabbatical.\n\nWhen B-36 neared Iceland, bad luck prevailed again. B-36 ran out of fuel. Dubivko threw his arms skyward in frustration when Potapov gave him the dismal news. Potapov then suggested that they could mix oil with water to keep at least one engine limping along. That worked for a time, and Dubivko's spirits soared when a fuel tanker came to the rescue near the Lofoten Islands just off Norway. As luck would have it, black swells prevented the tanker from passing a refueling hose to the submarine. Disheartened once again, Dubivko dove B-36 and ran slow on the electric motor. Just when he thought he'd finally lost the battle with lady luck, his main chemist, Yuri Klimov, devised a way to use seawater to charge the batteries enough to temporarily power the electric motor. Now, if they could only make it home before the motor gave out.\n\nBY MID-NOVEMBER, PRESIDENT KENNEDY AND PREMIER Khrushchev hammered out a deal that specified removal of all Soviet offensive weapons from Cuba. These included the nuclear missiles as well as the nest of IL-28 bombers. Days passed, but nothing happened. Members of ExComm met again to discuss options to deal with Khrushchev's obvious noncompliance with the agreement, including more blockade pressure and sending a strike force to take out the IL-28s.\n\nKhrushchev responded to Kennedy's queries by saying that Russia intended to remove the IL-28s, but not right away. He stated that \"it can be done in two to three months.\" Upset by this, Kennedy told his advisers that \"we might say the whole deal is off and withdraw our no invasion pledge and harass them generally.\"\n\nOn November 16, the largest amphibious landing since World War II unfolded at Onslow Beach, North Carolina. The two-day exercise was a full-scale dress rehearsal for an invasion of Cuba. Six marine battalion landing teams, four assault boats, and two helicopter assault carriers were involved. Some 100,000 army soldiers, 40,000 marines, and 14,500 paratroopers supported by 550 aircraft and 180 ships stood ready to invade Cuba.\n\nOn November 20, Kennedy held a press conference to announce that Castro had finally agreed to remove the IL-28 bombers within thirty days. The following day, the president issued a proclamation to end the naval quarantine of Cuba.\n\nAlthough tensions eased by mid-December between the two superpowers, the events that transpired during the Cuban Missile Crisis plunged the world into a Cold War that escalated to the brink of nuclear destruction several more times over the next two decades. U.S. and Soviet submarines were the common element each time.\n\nIN LATE DECEMBER, WHEN B-36 DOCKED at the pier in Sayda Bay, only Brigade Chief of Staff Second Captain Arkhipov came to meet the boat. Dubivko didn't care. He was just glad that the mission was over and that the evil spirits that had tortured him while under way could no longer make his life a living hell. What happened next, however, made him wish he'd never returned home. He was expecting a hero's welcome. After all, he and his crew survived the longest excursion into enemy territorial waters since World War II. They endured horrendous conditions, impossible odds, torturous weather, and a host of debilitating mechanical failures. What he and the other three captains involved in Operation Kama received turned out to be just the opposite of what they deserved and is best described in Captain Dubivko's own words, edited for readability by the author:\n\nOn the next day after we returned to Gadzhiyevo, I was summoned to the Commission of the main Navy Headquarters to analyze the trip. Rear Admiral P. K. Ivanov\u2014head of the Department of Combat Preparedness\u2014headed the Commission. Unfortunately, the work of the Commission on analyzing the actions of the submarines in extraordinary conditions, according to practice established at that time, was aimed exclusively at uncovering violations of orders, documents, or instructions by the commander or by personnel.... We were accused of violating secrecy, failing to abide by NIS-58 instructions for submarine forces while trying to avoid U.S. anti-submarine aircraft and ships. They did not take into account the fact that if we abided by NIS-58...then our submarines would never have arrived at our final destination.\n\n...The conditions in which our submarine brigade had to work were so difficult in the tactical, moral and physiological sense, that we were happy to have returned from that trip alive and healthy. This feeling was confirmed by Military Council of the Northern Fleet Rear Admiral F. Ya. Sizov when he said, \"We did not expect you to come back alive....\"\n\nSome time later, all the submarine commanders who took part in the trip were summoned to Moscow for personal reports to the USSR Defense Minister.... Marshal [Andrei] Grechko refused to listen to my report on the problems and difficulties of the trip.... The only thing he understood was that we violated the secrecy requirements, were discovered by the Americans, and that for some time the enemy maintained contact on our submarines.... He offered his opinion in the statement, \"I would have better sunk than come to the surface....\"\n\n[Our] experience, gained through hardships, should have been culled from all the submarines, analyzed, and shared with other Fleets. To my regret, the Operations Department of the Navy General Staff failed in that endeavor.\n\nI would like to once again express my appreciation and gratitude to the entire crew of B-36 submarine for exhibiting high standards of resistance and valor throughout the trip, and for helping me, as their commander, to affirm our dedication to the Soviet Navy and to our Motherland.\n\n\u2014CAPTAIN FIRST RANK, RETIRED, ALEKSEI DUBIVKO\n\nALTHOUGH PRESIDENT KENNEDY AND DEFENSE SECRETARY McNamara retained suspicions that the four Foxtrot submarines that converged on Cuba in October 1962 might be carrying nuclear torpedoes, the truth was not revealed to the world until 1995, more than three years after the Soviet Union collapsed. Not until then did we learn that the Soviets shipped 161 nuclear warheads to Cuba. Ninety of those were tactical, which would have killed tens of thousands of U.S. soldiers had Kennedy given the order to invade.\n\nI spent my entire adulthood believing that President Kennedy saved the world from nuclear winter during the Cuban Missile Crisis. Certainly, he and the members of ExComm were instrumental in forcing Khrushchev to back down at a crucial time during the brewing conflict, but the other heroes of this undeclared war were the commanders of the four Foxtrot submarines. Each had the opportunity, when faced with disaster, to pull the trigger and start a war. Each could have taken a dozen or more American ships with them into the smoldering center of a mushroom cloud, and each could have propelled the world toward a devastating October that would have changed the course of human history. However, each made the right decision in the end, and those of us who survived that time owe them our gratitude and perhaps our lives.\n\nThe events that transpired in the fall of 1962 brought the world to the edge of nuclear winter not once, but at least a half-dozen times. The Cuban Missile Crisis, perhaps better stated the Cuban Submarine Crisis, spurred the creation of the Moscow\u2013Washington hotline to ensure immediate, direct communications between the superpowers in the event of future potential conflicts. Although critics claim that Kennedy's actions prior to the crisis\u2014particularly those related to the Bay of Pigs incident\u2014likely caused the escalation in the first place, most agree that the outcome propelled the United States to a more confident stance as an international superpower. Unfortunately, the crisis also fueled fear in military and political circles that chilled the Cold War further, leading to an eventual \"keeping up with the Joneses\" competition between the United States and the Soviet Union. Both sides markedly increased spending on offensive weaponry, military technology, and counterintelligence.\n\nAs for Boresight, my father commented many years later that perhaps the success of this program was a double-edged sword. On one side was the technology that enabled U.S. ASW ships and planes to locate the Foxtrots and compel all but one to the surface. On the other side was the fact that, in helping our boys do a better job of forcing the Soviet captains into a corner, we also pushed those submarine commanders to the cliff's edge. While staring at that precipice, with nuclear weapons armed and ready, officers on all four Foxtrots nearly annihilated dozens of U.S. ships. Such an action would have escalated the crisis into a full-scale nuclear war that neither side could win. Only calm heads and perhaps divine intervention kept dozens of American cities from becoming radioactive wastelands.\n\n## CHAPTER ELEVEN\n\nEven castles made of sand, fall into the sea, eventually.\n\n\u2014JIMI HENDRIX\n\nTHE EARLY SIXTIES USHERED IN THE next evolution in underwater warfare with the advent of a revolutionary new type of nuclear fast-attack submarine. Named after the USS Thresher (SSN-593), boats of this class brought to life almost every item on a navy wish list for the perfect stealth boat. From sonar to fire control to weapons to electronic-snooping \"spook\" gear, Thresher's integrated design allowed sailors to operate as a synchronized team\u2014like an ant colony with an attitude. Sub drivers were ecstatic. Faster, deeper diving, more quiet and maneuverable, these boats were the Ferraris of the fleet, which spurred fierce competition between skippers to snag a set of keys to one of the navy's hottest new vehicles.\n\nLieutenant Commander John Wesley Harvey did just that. Born into the Bronx backstreets of New York, he entered the U.S. Naval Academy in 1946. Harvey first served in the skimmer fleet aboard the aircraft carrier Coral Sea (CVB-43) before switching to submarines in 1952. He earned his submarine dolphins on the diesel boat Sea Robin (SS-407) before transferring to the USS Nautilus in July 1955. After a few runs on the Seadragon (SS-194), Harvey took command of the USS Thresher in January 1963. Four months later, he went to sea for the last time.\n\nBUNDLED IN THICK NEOPRENE, SITTING ON the hard deck of the USS Preserver (ARS-8), Sonarman First Class Nihil Smith could no longer feel his cheeks. A harsh Atlantic wind swept its cold anger across the deck of the 200-foot-long rescue ship and swirled above an ocean full of death. Smith turned his frozen face to the right. The head diver flashed a signal. Smith stood and stepped toward the back of the Preserver, a double-tank \"twin-ninety\" scuba rig mounted on his back. As he walked, his navy diver booties scraped across the rust on the ship's deck, and the dense smell of the Preserver's engine exhaust hurried him toward the edge.\n\nMetal chains clanked and winches groaned as the Preserver's crew went about their daily tasks. The Preserver first saw the light of day in San Francisco in 1944 and served as a support ship in the Pacific\u2014a platform for navy divers conducting salvage and rescue operations in such exotic places as Saipan, New Guinea, Guam, Okinawa, and Bikini Atoll. In May 1962, navy divers on the Preserver helped recover Mercury astronaut Scott Carpenter after his Aurora 7 capsule splashed down in the Atlantic. A year later, in early May 1963, Preserver's captain got the call he hoped would never come.\n\nThe Preserver caught up with the deep-diving submersible USS Trieste (DSV-0) at the Boston Naval Shipyard, and the two vessels headed toward a location some 200 nautical miles east of Cape Cod, Massachusetts. The Swiss-designed Trieste \"deep boat\" bathyscaphe held a crew of two and could descend to the darkest part of any ocean on earth\u2014nearly seven miles deep. Auguste Piccard, her Swiss designer, envisioned grandiose missions of scientific discovery for the sixty-foot-long submarine and named his 1953 creation after a quaint Italian seaport. Like any father, he probably knew his daughter might someday be called upon to search for the remains of a disaster.\n\nThat day arrived when, for the first time in the Trieste's short ten-year life, she became a reluctant detective hunting for bodies. These cadavers, however, were not hidden in back alleys or buried under New Jersey dirt. They lay scattered across the bottom of a dark ocean more than 8,000 feet deep. In such a foreboding place, the odds of finding even the slightest clue were somewhere between zero and \"worse than Vegas.\"\n\nNihil Smith stepped off the Preserver's fantail and climbed onto the small skiff\u2014the Boston Whaler with an outboard motor. He took a seat next to another diver as the Preserver's crew lowered the dive boat into the water. A sailor on the skiff started the outboard, and the Boston Whaler moved slowly away from the ship. Smith glanced back at the Preserver. Not a large vessel, the ARS-8 had two masts, a small superstructure, and a thirty-nine-foot-wide flat deck at the stern. She carried just two 40 mm AA gun mounts and four .50 caliber machine guns, but nobody expected her to fight any battles.\n\nThe skiff reached the dive spot, and Smith stood up. He watched his dive buddy step over the gunwales and splash into the blue. Following suit, Smith seated his scuba regulator, planted his right palm flat against his face mask and mouthpiece and jumped into the ocean.\n\nDespite the thickness of his wet suit, the cold still launched his testicles into his throat. He sucked in several short breaths, forced his mind back to calm, and swam toward his dive buddy. The two met, traded \"okay\" hand signals, then looked down. Air bubbles circulated around Smith's fins, but they did not come from his twin-nineties. These bubbles ascended from a white submarine 100 feet below, on its way up from 8,000 feet deep. He bled air from his buoyancy vest and descended toward the mini-sub. As he cleared his ears to equalize with the ocean pressure, his eyes focused between his knees. Two vents mounted to the deck of the Trieste stared back.\n\nTo Smith, the strange twelve-foot-wide craft below looked nothing like a typical submarine. Instead, the Trieste resembled something from Popular Science magazine with her stump of a conning tower, flattened nose and heel, and round observation gondola bulging from her belly. The vessel's operators climbed down into the gondola through an entrance tunnel connecting to a hatch in the stubby sail.\n\nThe Trieste's pressure sphere offered only enough space for two people, keeping them alive with in dependent life support provided by a closed-circuit rebreather system not too unlike the ones used by astronauts. Two thirty-eight-cubic-foot oxygen cylinders pumped in air, and canisters of soda lime scrubbed carbon dioxide from the air. No diesel engines ran inside the Trieste's hull. Power came from two electric propulsion motors mounted externally and pressure-compensated to withstand the intense forces of the deep. Twelve-volt electric car batteries powered these motors from large boxes located on the after walking deck of the Trieste. Each box housed a dozen batteries. After every dive, all of the Trieste's batteries, including the silver cell ones used inside the submersible, needed recharging.\n\nSmith neared the curved white hull and grabbed onto an eyehook. His dive buddy did likewise. Together they rounded the side and swam toward the \"grappling\" mechanical arms underneath, near the observation sphere. A pair of eyes, inside the Trieste, stared through the single tapered cone-shaped block of Plexiglas mounted on the sphere. Smith gave the operator inside a signal, and an instant later, bright light from a set of quartz arc bulbs lit up the black ocean.\n\nSmith could now see the bottom portion of the forward pellet ballast hopper protruding from the hull just in front of the sphere. He knew that inside the silo\u2014a round cylinder about three feet in diameter and seven feet tall\u2014nine tons of magnetic iron pellets were used as ballast. An identical hopper sat a few feet behind the observation gondola. Using pellets versus water helped increase descent and ascent speeds, given that intense water pressures down deep did not allow for air-filled ballast tanks.\n\nThe pellets, which filled each hopper, resembled oversized BBs. Their weight pulled the bathyscaphe down toward the bottom. When the crew cut off power to large electromagnets attached to the hoppers, the pellets dropped to the ocean floor, and the Trieste shot toward the surface. The crew refilled the ballast \"shot tubs\" from stacks of twenty-five-pound pellet bags lashed to the deck of the Preserver.\n\nSmith kicked his fins and swam toward the mechanical arms. His dive buddy moved alongside and pointed. Smith followed the diver's finger and shifted his eyes downward. The Trieste's bright arc lights reflected off something shiny. A short pipe. Smith's heart skipped a beat, and he wondered if this was the proof they had been searching for. He pulled the pipe free and brought it to the surface. Back on board the Preserver, experts examined the bent piece of metal. In a small compartment filled with navy personnel and scientific experts, Smith watched as one expert took out a magnifying glass and focused on the side of the pipe. The gaunt man with wire-rim glasses wrote down a part number, then flipped through a set of blueprints. The expert thumbed to a page, looked at the part number on the paper, focused back on the blueprint, and raised a stubby chin. Without speaking, the man nodded his head up and down.\n\nSmith stepped outside the room and walked to the fantail. A cruel Atlantic gale swatted at the large ship as she bobbed up and down in the roiling swells. On the horizon, a bruised and swollen sky swallowed the last ounce of sunlight, leaving behind purple cotton clouds.\n\nNIHIL SMITH WAS NO STRANGER TO the dangers faced by submariners. He joined the navy at the age of seventeen after striding into the recruiter's office in Azusa, California, in 1956. He had four uncles who had served their country during World War II; two of them went navy, and one became a submariner. That uncle often talked about the high morale and esprit de corps that could only be found on the boats. Inspired by his uncle's stories, Smith volunteered for subs and reported aboard the diesel boat USS Trigger (SS-564) in Norfolk, Virginia, in 1958. He earned another stripe when he passed the test for sonarman second class and a navy diver pin after graduating from the rigorous training course in Hawaii that same year. While qualifying in submarines during 1959, Smith came to relish the boat life and learned firsthand about the camaraderie that his uncle had talked about.\n\n\"Everybody depends upon everybody,\" his uncle said. \"You're on the boats because you want to be, not because you're forced to be. Where else can you have that kind of togetherness? Where else can you depend on your mates because they all take pride in what they do?\"\n\nOn the Trigger, Smith met just such a mate, a radioman named Joe Walski. Smith and \"Ski\" became close over the next year as they helped each other qualify in submarines. They earned their dolphins on the very same day. By the time Smith transferred off the Trigger, he and Walski were best friends. As Smith walked across the gangplank of his first boat for the last time, he felt like he was leaving a brother behind.\n\nSmith reported to the diesel submarine USS Barbel (SS-580) in Kittery, Maine, and soon discovered that his \"brother\" Walski received a transfer to a nuclear-powered boat going through an overhaul, also in Maine. Smith was ecstatic. He raced over to the drydock and met Walski near the brow. They hugged, and Walski invited Smith for a tour of his boat. Smith slid down the ladder and whistled. Compared to the Barbel, Walski's new submarine seemed like a luxury hotel: wood paneling, shiny pipes, new Naugahyde-covered benches, and all the latest equipment. Walski spent the next hour dangling a carrot in front of Smith, using every angle to try to convince him to leave the Barbel and request a transfer to this newer SSN, known as the USS Thresher. Smith almost conceded but eventually declined, quoting the saying \"Diesel boats forever!\"\n\nSad but certain that he'd made the right decision, Smith sauntered back to the Barbel. Walski went to sea on his youthful nuke boat, while Smith rode the Barbel down to New London, Connecticut. Their respective schedules made visiting each other impossible. Smith then rode with the Barbel through the Panama Canal to her new home port in San Diego. There he received orders to the Naval Electronics Laboratory and was assigned to the research vessel Trieste, which had just returned from making the world's deepest dive in the Marianas Trench.\n\nWhen Smith completed second class diver training in March 1963, he called to tell his quasi brother about the accomplishment. Walski was excited about the news but couldn't talk long, as the Thresher was preparing for upcoming sea trials. They vowed to make time to visit each other again, perhaps sometime later that year. Neither knew that this was a promise that could not be kept.\n\nThe lead ship of her class and the pride of Portsmouth Naval Shipyard, the USS Thresher launched on July 9, 1960. She underwent sea trials throughout '61 and '62, testing a host of modern and complex systems and weapons. After she arrived in San Juan, Puerto Rico, on November 2, 1961, the Thresher's nuclear-trained \"nukes\" shut down the reactor. Equipment failures and mistakes combined to create serious overheating in the engineering spaces that resulted in a partial evacuation of the crew. This event became the first omen signaling the Thresher's eventual demise.\n\nThe second portent occurred while the submarine sat moored at Port Canaveral, Florida. A tugboat accidentally slammed into the Thresher's side and damaged a ballast tank. After repairs at Groton, Connecticut, the Thresher headed south to conduct more tests and trials near Key West, Florida. Thereafter she cruised up to Kittery, Maine, and stayed in the dockyard for refurbishment until the spring of 1963.\n\nAfter her refit, with Radioman First Class Joe Walski on board, on April 9, 1963, the Thresher pulled away from the pier. Lieutenant Commander Harvey informed his crew that they were about to conduct rigorous postoverhaul sea trials. The submarine rescue ship USS Skylark (ASR-20) accompanied the Thresher to the trial area off Cape Cod. On the morning of April 10, 1963, Thresher started the first of her deep-diving tests. When the submarine neared her test depth of 1,300 feet, Lieutenant Commander Harvey called the Skylark on the Gertrude underwater communicator. The message came through garbled, but what could be deciphered indicated trouble. The words \"minor difficulties, have positive up-angle, attempting to blow\" were the last heard from the ill-fated submarine. She died that day along with 129 officers, crewmen, and military\/civilian technicians.\n\nNo one knew for certain what happened to the Thresher until after Nihil Smith wrestled that small pipe, brought up from 8,400 feet deep, away from the grappling arms of the Trieste. The two-man crew of the bathyscaphe had previously spent weeks searching with no luck. Large grids were set up using color-coded markers and numbers for each search section. Every morning, the Trieste sank to the bottom and illuminated the ocean floor. She spent all day covering the grid coordinates, her crew hoping to find the remnants of the Thresher. They found nothing save a yellow boot that resembled part of a \"canary suit\" worn by nuclear personnel during reactor emergencies\u2014not enough to validate Thresher as the source.\n\nEach day Smith and the other divers prepped the 35 mm cameras, filled the ballast tubs with pellets, tested the radios, cleared debris from the Trieste's hull, washed her windows, and \"checked her oil.\" Then they waited. Hours later, usually near sunset, they suited up and prepared to meet the deep-diving vessel when she came back up. Those monitoring the Trieste's ascent called out her depth, having tracked the submersible via hydrophones. When she neared one hundred feet, Smith and another diver climbed into the Boston Whaler and went out to meet her.\n\nTheir work didn't end there. At the completion of each day's dive, crews on the Trieste and Preserver prepared for another dawn and descent into the abyss. They hauled power cables attached to floats over to the Trieste to recharge the lead acid batteries. They wrestled with a high-pressure air line to charge up the antichamber blow system, which expelled air from an area near the sphere to allow the Trieste's crew to exit the submersible after surfacing. While some of the crew processed the film taken during the day's search, others replaced oxygen cylinders in the sphere, along with CO2 canisters and silver cell batteries. Finally, they refilled the pellet ballast hoppers from the twenty-five-pound bags found on deck and checked and rechecked that everything was ready to go again.\n\nPreparations usually concluded around midnight, giving everyone less than six hours of sleep before the 6:00 A.M. predive check. That's when Smith suited up and stood ready to hit the water by 7:00 A.M. The Trieste then disappeared again under the waves while the support crew waited. Days and weeks went by. The Trieste occasionally spotted what looked like a debris field but never brought up any solid evidence, until they found the pipe.\n\nAfter that, over the course of many months, the Trieste located parts of the Thresher strewn across a 160,000-square-yard area. Every day, Smith watched as more pieces were salvaged, all the while painfully aware that his best friend's body lay somewhere in the debris. First they found the stern planes, followed by the fairwater planes. These appendages once looked and functioned much like wings on an airplane to move the boat up and down in the water with the grace of a dolphin. Now they were bent and twisted backward in grotesque shapes that resembled the branches of a dead tree.\n\nAs an orange-red sun sank into the sea and scattered tentacles of dying light across the wave tops, Nihil Smith stood in silence on the deck of the Preserver and thought about those final minutes. He wiped away a tear and imagined the look of shock on the faces of the Thresher's crew as a wall of water filled the boat. He pictured the submarine imploding at 2,000 feet deep and saw the terror-filled eyes of his best friend, Joe Walski, as frigid salt water filled his lungs. To this day, Smith is still haunted by that nightmare.\n\nPHOTOGRAPHS OF THE FALLEN THRESHER, ALONG with recovered parts, allowed a special court of inquiry to conclude that the submarine suffered a joint failure in the saltwater piping system. A shipyard worker connected the joint using silver brazing instead of a standard weld. The Thresher's loss prompted the SubSafe program, which dictated a higher degree of quality control for all U.S. submarines. This decreased the quantity of submarines originally planned for, but increased safety and reliability.\n\nOn the other side of the world, still stung by the Cuban Missile Crisis, Admiral Sergei Gorshkov pushed his navy in the opposite direction. The Soviets sacrificed quality for quantity, producing an average of seven nuclear and six diesel boats per year, which resulted in a spate of accidents. K-8 had a reactor coolant leak in October 1960. In July 1961, eight men died on K-19, a Hotel-class missile boat, when her reactor overheated. B-37 endured an internal torpedo explosion in January 1962, and intelligence reports indicated that the November-class K-3, the Soviet's first nuclear sub, continued to have leaks in her steam generators, along with other per sis tent reactor issues for years. Despite the problems, Secretary of Defense Robert McNamara expressed constant concerns about Soviet submarine escalation and the need to accurately locate those boats.\n\nThat concern fueled McNamara's desire to insert himself into the middle of the Boresight program. He contacted Jack Kaye at the NSA and informed the commander that A22 Desk now reported directly to him. McNamara showed up at staff meetings and asked a million questions. Some liked his inquisitive, \"need to know\" manner, and some didn't. When William J. Reed first met McNamara at the White House during the Boresight briefing with President Kennedy, he leaned more toward the \"didn't like\" camp. McNamara's hard-charging style seemed confrontational, especially when the secretary questioned him to death over minute technical details about Boresight. After McNamara got involved with the Boresight program, and Reed started working with him on a more frequent basis, his opinion of the man changed.\n\nThose who knew Robert Strange McNamara sometimes said that his middle name fit like a glove. Possessing the number-crunching mind of a computer, the bull-nosed personality of General George Patton, and the diplomacy of Henry Kissinger, McNamara knew what he wanted and \"damned the torpedoes\" until he achieved his goals. One of those quests entailed the worldwide proliferation of Boresight. McNamara had witnessed firsthand the effectiveness of the system and its ability to diffuse high-seas confrontations by exposing the Soviet's undersea predators. As an insurance package, he insisted on equipping several more GRD-6 stations with Boresight technology while building more Wullenweber sites under Project Bulls Eye to improve accuracy.\n\nHaving been battle tested during the Cuban Missile Crisis, and in light of Soviet submarine advances, Boresight quickly became a hot property. McNamara propelled the program to a lofty number-two status, subordinate only to the Polaris missile program, which allowed ballistic nuclear submarines, like the USS George Washington, to lob radioactive warheads from more than 1,000 miles away.\n\nEmploying Doberman pinscher diplomacy, McNamara pressed NATO allies to allow inspections of select real estate abroad to ascertain the best locations for Wullenweber elephant cages. In addition to already operating installations at Edzell, Scotland, and Hanza, Japan, projected foreign locations included Galeta Island, Panama; Rota, Spain; and Sabana Seca, Puerto Rico. Skaggs Island, California, remained the only functional North American facility with construction planned for Adak, Alaska; Homestead, Florida (replacing the GRD-6); Imperial Beach, California; Marietta, Washington; Northwest, Virginia (also replacing the GRD-6); Wahiawa, Hawaii; Winter Harbor, Maine; and the Pacific island of Guam.\n\nLocations were not picked at random. The geometry involved in optimizing direction finding required math degrees to calculate prime locations. These equations also dictated that, ideally, stations should be spaced 120 degrees apart\u2014or about one-third of a compass circle away from one another. This spacing allowed for more accurate bearing hits on targets.\n\nNot all of the \"Fred Ten\" Wullenweber arrays built by the navy were used for HFDF operations, but most were. The Air Force built similar Wullenweber \"Flare Nine\" antennas at other locations, including Karam\u00fcrsel, Turkey, but none were intended for HFDF use. Navy personnel borrowed some of these arrays for Boresight operations where those locations were suitable, or where signal intelligence requirements, such as transmitter fingerprinting, could best be met.\n\nFor Reed and the A22 team at NSA, the remainder of 1962 and most of 1963 entailed Boresight system refinement and expansion. Although McNamara pushed for rapid deployment, he did so with an eye toward efficiency and economy. Still, for the most part, Boresight equipment requisitions could override virtually any other project save Polaris. Backups to backups were ordered and installed. Contractors and subcontractors, such as Sanders Associates and ITT Federal Systems, had near-blank checks. Components used had to be properly interfaced with one another and with existing equipment. Incompatibilities, glitches, and kinks had to be ironed out in time for McNamara's QA inspections. As was his modus operandi, McNamara frequently microman-aged many of the processes until he was satisfied that his subordinates could live up to his idea of perfection. Some found this annoying, but Reed wished that every public servant kept such a close eye on efficiency and accountability. He felt that America's national debt wouldn't be so high if that were the case.\n\nA Northern California native and son of a shoe sales manager, Robert McNamara graduated from UC Berkeley in 1937 and earned an MBA from Harvard two years later. He gained his hard-nosed accounting skills by working at Price Water house in San Francisco until late 1940, whereupon he returned to Harvard to work as an assistant professor. He served as a captain in the Army Air Force during World War II, then joined Ford Motor Company, where he earned the title of \"whiz kid\" by helping implement modern management control systems. Hard work and brilliance earned him the presidency of Ford in November 1960, which proved short-lived after President Kennedy recruited him as secretary of defense in 1961.\n\nMore than two years later, in the fall of 1963, after Reed received a promotion to lieutenant junior grade (LTJG) and became the primary technical liaison for the A22 Desk at NSA, McNamara requested a meeting at his office in the Pentagon. They spent hours peering through magnifying glasses at potential map locations for Boresight stations.\n\nHalfway through the meeting, McNamara leaned over Reed's shoulder, pointed a long finger at the toe of Italy on the map, and said, \"There. Build one there.\"\n\nAlthough the location looked quite suitable from a geometric standpoint, Reed said, \"Mr. Secretary, I don't believe there's a decent way to get to and from that location. That's all nearly inaccessible mountain country.\"\n\nMcNamara waved a dismissive hand. \"Planes, trains, buses, or horses. There's always a way in or out. Go take a look and report back.\"\n\nReed laughed. McNamara did not. Reed's face turned serious. He cleared his throat and said, \"Yes, sir, Mr. Secretary. I'll get right on that.\"\n\nWeeks later, after traversing political land mines with the Italian government, Reed rented a burro and a guide and rode for two days across the worst terrain ever constructed by God. He returned from the trip with a sore ass, a grumpy demeanor, and a no-go report for McNamara. Getting supplies and construction materials in and out of that location would be impossible at worst and a nightmare at best. McNamara finally conceded and handed Reed a bottle of aspirin. Reed figured that was about as close to an apology as he'd ever get.\n\nEngineers at the Naval Research Laboratory made their own contributions to the Boresight and Bulls Eye programs via a new digital computer with a multiprocessor and magnetic core memory called the AN\/GYK-3. Originally used in the NORAD (North American Aerospace Defense Command) early warning project, the GYK represented the edge of the edge in computing technology. NRL contracted with Burroughs Corporation in 1961 to develop a prototype that turned out to be every bit as impressive as touted by the designers. Burroughs delivered the computer in 1962, which offered the ability to better track moving targets as compared to other systems that were more suitable for managing production inventories.\n\nJudged by today's standards, the computer boasted primitive computing power, capable of handling no more than 625 bytes and around one hundred records per minute. Outputs were sent to CRT displays and high-speed printers. Today we take for granted that our mobile phones can manage millions of bytes of data and breeze through thousands of tasks in a second. Back then such staggering numbers were considered science fiction. These early-generation processors could, however, work on up to twenty tasks at the same time. Despite being slower than a geriatric snail, the GYK's multitasking capabilities created beaming grins on the faces of design engineers.\n\nBased on recommendations he'd received from the engineers at Sanders Associates, Reed contacted Bruce Wald and others at NRL to discuss how they might use the GYK computer for Boresight. A computer could help automate the previously manual and tedious task of plotting fixes to targets by pulling strings of yarn across a compass rose. With the GYK computer, that process could be completed faster and with far greater accuracy. Later that year, the Wullenweber Bulls Eye and GYK computer-infused Boresight projects eventually blended into a single program called Clarinet Bulls Eye, which later changed to Classic Bulls Eye to conform to standard naming conventions.\n\nWith the Cuban Missile Crisis now over, and with the leaves of autumn turning fields of green into swirls of auburn and gold, Reed at first thought he might be able to find time to relax with his family in their two-story Mary land home. But across the ocean, a smitten enemy escalated plans to leapfrog their adversaries by building an arsenal of underwater weapons that could plummet the world back into the Dark Ages. That fact eventually led Reed to the pinnacle of his career, followed by an end that no one expected, least of all him.\n\nTHE BEGINNING OF THAT END STARTED when Captain Kaye, at McNamara's bidding, promoted Reed to one of the most prestigious positions in A22: head of field operations. In short, he became NSA's head Boresight\/Bulls Eye troubleshooter. The job entailed not only frequent trips to all stations in the Atlantic and Pacific, but also additional travel to such locations at Cheltenham, England, for meetings with the heads of NATO intelligence departments and with various international R&D laboratories. Reed admitted years later that pride and a strong sense of duty prevented him from admitting the truth to his boss\u2014that such an assignment required three people, not one. The responsibility and relentless travel eventually took its toll, but in the meantime, Reed carried a pocketful of government \"paper money\" travel vouchers. He also carried a gun.\n\nAny information about Boresight technology, from concept to equipment to operational procedures, received top-secret \"eyes only\" classifications. Sending documents, training materials, or tapes with burst signal examples by mail was out of the question. Top-secret codebooks, plans, instruction manuals, operating guidelines, and protocols needed to be hand-delivered by armed courier to dozens of HFDF stations worldwide. Carrying a briefcase handcuffed to his wrist, Reed and others like him traveled the globe with nervous fingers touching the trigger of concealed weapons. Every shadow hid a potential KGB agent, and every friendly smile seemed dubious.\n\nWearing civilian clothes, unencumbered by his rank as compared to others more senior, accepted into the intelligence community alongside CIA spies and NSA operatives, Reed flew to Norway to assist with Bore-sight installation and training for an antenna system located in Vads\u00f8. He departed for London, thence to Norway in late November 1963. When he arrived in Oslo, several Norwegian intelligence officials met him at the airport. On the way to his hotel, they discussed the strategic importance of having direction-finding systems at this station, given that, if someone threw a rock about a hundred miles from Vads\u00f8, they'd hit Murmansk, one of the Soviet Union's most important and widely used submarine naval bases.\n\nThe navy's main concern centered around the older Golf-and Hotel-class ballistic missile boats, and the new Echo I, Echo II, and Juliet guided missile submarines, not to mention the nuclear-powered November-class subs. While SOSUS still had a chance of finding the Golf, Hotel, and Juliet diesel subs when they snorkeled, Echos and Novembers sported nuclear reactors and sound levels just above the \"quiet sub\" classification of less than 150 dB. That decibel level actually rivaled the USS George Washington: not exactly quiet, but not real easy to find with current sonar systems. Moreover, the Echo II could now carry newer Shaddock SS-N-3 antiship cruise missiles with ranges of up to 245 nautical miles.\n\nThe November-class actually trumped the USS Nautilus and Seawolf on the capability front, and to prove the point, the Soviets sent the Leninsky Komsomol to the North Pole. She arrived in September 1963. With the sting of the Cuban Missile Crisis not yet dissipated, all these Soviet submarine advancements kept the U.S Navy awake at night. Finding and tracking these boats, especially the Hotels, Echos, and Novembers, referred to as the \"HENs\" by NATO, remained paramount and prompted Reed's visit to Norway.\n\nStarving and tired, Reed arrived at his hotel in the evening, showered, shaved, and found the restaurant downstairs. There he met a collection of drunken Norwegian army officers who'd just survived a secret ski-troop exercise. Since a broad patch of Norway's border carved a line down Russia's left flank, Reed figured the \"exercise\" probably took place near Hesseng or Kirkenes. While the soldiers hooted and howled in a language Reed had learned only sparsely, a waitress appeared. Through Reed's partial Norwegian and her broken English, he managed to order a steak. Thirty minutes later, his food had yet to appear, and a sudden hush fell over the room. The Norwegian soldiers stopped laughing and yelling. Their voices descended to whispers, and their faces registered shock and concern. Other patrons stopped eating. Their looks and tones mirrored the army officers. Confused, Reed flagged his waitress.\n\nTears streamed down the girl's face as she approached Reed's table. In her broken English she said, \"They shoot him.\"\n\n\"Shoot who?\" Reed said.\n\nHaving overheard Reed speaking English, an American couple darted over from another table. A rotund man, clutching his wife's hand, said, \"I speak a little Norwegian. Have you heard the news?\"\n\n\"What news?\" Reed said.\n\nThe man looked at his wife. His eyes misted. She started sobbing. He held her in his arms and said something unintelligible. As Reed watched them, others in the room started crying.\n\n\"What's going on?\" Reed asked the man. \"Why is everyone so sad?\"\n\nThe man wiped his cheeks and said, \"President Kennedy's been shot. He died a few minutes ago.\"\n\nReed's stomach knotted, and his hunger vanished. A million questions ran through his mind. Had the Soviets finally retaliated in response to the crisis? Was the United States now at war with Russia? Were Joyce and the kids in danger? He had no answers and would not until morning, when he could contact the U.S. embassy. For now, he could think of only one appropriate thing to do. He grabbed a glass and walked over to the table occupied by the Norwegian army officers. He held up his empty glass, and one of the soldiers poured from a bottle. When Reed held his glass high, the officers stood from their table and did likewise. No words were spoken as the half-dozen military men offered a toast to a fallen hero.\n\nREED RETURNED TO THE STATES AFTER helping the station in Norway optimize their Boresight systems. The following year, on July 5, 1964, carrying a concealed Webley revolver, he left JFK Airport bound for London. He couldn't recall the last time he'd actually fired his pistol at the range. Making time to do that hadn't been a priority. He'd never been shot at and figured the odds of that were just about nil\u2014that is, until he got a call from the CIA, who cordially compelled him to report to Greece on the double.\n\nWhen he arrived in Athens, a CIA operative met him at the airport and said he needed Reed for a sensitive mission. He said his name was Fred. No last name was given. When Reed inquired as to the nature of the operation, Fred said that the NSA was concerned that Soviet spies might have obtained several Boresight technical documents. Once the word spies fell into the sentence, the CIA took jurisdiction. As such, the mission entailed validating that Soviet or Soviet-friendly contacts had indeed absconded with top-secret documents, and if so, their orders were to prosecute the said suspects with appropriate action. When Reed asked for the definition of \"appropriate action,\" Fred shrugged and said, \"Dunno. We usually make that shit up on the fly.\"\n\nReed shook his head and slouched into the seat of the Mercedes. Fred pulled away from the curb at the airport and sped into downtown Athens. Night blanketed the city, and shimmering lights accented the city's marbled works of beauty. The Mercedes had that new car smell, overpowered now and then by Fred's aftershave.\n\n\"Why me?\" Reed asked.\n\n\"We need your technical knowledge,\" Fred said.\n\n\"My technical knowledge? What, to catch a spy?\"\n\n\"No,\" Fred said. \"To validate that said contacts have obtained said documents.\"\n\n\"But if said contacts have indeed pilfered said documents,\" Reed said, \"then who's to say that said contacts haven't already said something to someone or photographed said documents and have, say, already delivered the information?\"\n\n\"Not my problem,\" Fred said. \"I've been ordered to retrieve said documents from said contacts, have you validate their validity, and take appropriate action.\"\n\n\"Which appropriate action you'll be making up on the fly.\"\n\n\"Precisely,\" Fred said as he careened around a corner. Reed held onto the door handle and wondered if he'd live long enough to meet said contacts or examine the documents.\n\n\"Do we know where said contacts are located?\" Reed asked.\n\n\"Yes.\" Fred said.\n\n\"Are you going to tell me?\"\n\n\"They're in a Greek village some distance from here.\"\n\n\"Does this village have a name?\"\n\n\"Yes.\"\n\n\"You don't say.\"\n\n\"No, I don't say.\"\n\n\"Fine,\" Reed said. \"Enough said.\"\n\nThe CIA put Reed up in a local hotel for the night, and the next morning, Fred poured him into a black van and sped toward the Greek village. Nothing was said.\n\nWhen they arrived, another operative named Bob, who'd arrived earlier in the Mercedes, took charge. He said they were waiting for the Greek authorities, as they were under orders to cooperate with the locals. Once the police officers arrived, his orders were to observe the unsubs, verify that they fit the description of said suspects, then take appropriate action. Like Fred, Bob did not define \"appropriate action.\" Bob told Reed to stay in the Mercedes, and Reed offered no argument.\n\nThirty minutes later, a truck full of badge-brandishing, rifle-toting, macho Greeks rolled in. They stormed out of the truck, took up cover positions, and pointed guns. The tallest one approached Bob. The two spoke while pointing at a gin-joint restaurant a half-mile away. It was a dusty town with a scattering of cars, a few peasants milling about, and some thin dogs and chickens here and there. A stench hung in the air that reminded Reed of an Oklahoma barn.\n\nWhen the conversation between the head guys ended, the wait began. An hour went by, then another. Nothing happened. Starving, Reed wondered how long the operation might last. A dark Volvo pulled in front of the restaurant. A half-dozen men slammed car doors and walked inside. The Greek cops raised their heads and ears. Reed noticed fingers twitching. While Bob peered through a pair of binoculars, the head Greek guy waved a hand in the air. His team of six ran screaming toward the restaurant with guns blazing. Bob and Fred dropped jaws, looked at each other, and went running after them, yelling at Reed to stay put in the Mercedes. Again, Reed offered no rebuttal.\n\nThe bad guys in the gin-joint restaurant started firing back through the windows as the Greeks approached. Reed grabbed a pair of binoculars from the front seat and pointed them through the window of the Mercedes. Two of the Greeks dropped onto the cracked road in pools of blood. Not believing his eyes, Reed watched one of the Greeks throw something through the restaurant window. If that's a grenade, he thought, there won't be any documents left to validate. The projectile spewed out smoke instead of shrapnel as the Greeks knocked down the door and ran inside. Bob and Fred went in after them, albeit with a little more caution. Binoculars glued to his eyes, Reed heard muffled gunshots and saw bright flashes through the windows. Then silence.\n\nLong minutes passed. Reed scanned back and forth between the windows and the door. Nothing but smoke. No movement of any kind. Then a hand. Bob's hand. He stepped through the door and signaled for Reed to come over. With trepidation, Reed stepped from the car and walked the half-mile to the restaurant. A wave of nausea hit as he went past the fallen Greeks, eyes open and flies dancing about the blood. He stepped through the door of the restaurant, and his knees buckled. A half-dozen more bullet-torn bodies lay soaked in crimson. Smoke billowed around the room. Bob motioned Reed to a table. He opened a worn leather satchel and removed a few dozen pieces of paper, then handed them to Reed and asked him to validate the documents as Boresight-related or not. Reed examined the stack. He read each page, then reread them to be sure he hadn't missed anything.\n\n\"Well?\" Bob asked.\n\n\"Goniometer,\" Reed said.\n\n\"Gonorrhea?\" Bob repeated. Fred laughed. So did one of the Greeks.\n\n\"Not quite,\" Reed said. \"These papers are technical specifications for a German goniometer\u2014the same one we use in our Wullenweber Bulls Eye sites.\"\n\n\"So said contacts did have secret documents?\" Bob asked.\n\n\"No,\" Reed said. \"The goniometer design is two decades old. We use it in our arrays, but it's not related to the Boresight equipment. In fact, it's not even classified.\"\n\n\"Said document is not even classified?\" Fred said.\n\n\"Not classified,\" Reed said.\n\n\"No shit,\" Bob said. He grabbed the stack of papers from Reed's hand, scanned his eyes across the dead bodies on the floor, and said, \"It is now.\"\n\nAFTER THE GREEK INCIDENT, SOMETHING CHANGED in Reed. The excitement and enthusiasm he once had in his work dimmed. Still, he maintained his competency and helped the Canadians enter the Boresight fold at Masset, Gander, and Newfoundland as part of the Canadian-U.S. Atlantic HFDF network. These sites were later upgraded to Wullenweber or similar arrays. By 1967, more than a dozen Wullenweber elephant cages were built or under construction around the world, and the navy enjoyed a continuous stream of tip-offs, flashes, and fixes on Soviet submarines. As for Reed, he decided to leave the program and take a less stressful post in San Diego.\n\nReflecting on this decision years later, he said that had he stayed in the Classic Bulls Eye program, he might have made full-bird captain one day, which is rare for an up-through-the-ranks Mustang officer. On the other hand, he probably would have sacrificed his sanity and maybe even his soul to gain the world. In hindsight, he did not regret walking away from his James Bond life and was proud to have served as he did, for in doing so, he paved the way for others to take the Boresight\/Bulls Eye programs to new heights of stardom.\n\n## CHAPTER TWELVE\n\nIf you reveal your secrets to the wind, you should not blame the wind for revealing them to the trees.\n\n\u2014KAHLIL GIBRAN\n\nON DECEMBER 29, 1964, THE U.S. Navy made a huge blunder by granting Petty Officer John Anthony Walker a top-secret security clearance. While his background appeared clean, a more thorough psychological screening may have revealed cracks in Walker's armor. Nevertheless, the navy welcomed him as an R-Brancher and opened the proverbial intelligence kimono. Walker now had access to the navy's most sensitive cryptographic material.\n\nBy April 1967, the navy assigned Walker to the Naval Communications Area Master Station in Norfolk, Virginia, a drab, windowless, two-story building off Terminal Boulevard. At first glance, no passersby would suspect that inside the facility, operators sent and received thousands of encoded messages every day to and from ships and submarines operating in the Atlantic. Walker's responsibilities as a watch officer in the message center included handling classified communications with U.S. submarines, specifically reading and dispatching top-secret traffic bouncing between the message center and on-station subs.\n\nSophisticated cryptographic machines and Teletypes lined the walls of a small room in the message center, a few feet away from Walker's desk. The machines clicked and hummed twenty-four hours a day as messages were scrambled and unscrambled, traveling ship to shore and shore to ship. The navy code-named its crypto system Orestes, which employed a device called the KW-7, an online, send\/receive message encryption unit installed in shore stations and aboard ships and submarines. To send messages over a secure UHF Teletype circuit, a model 28 Teletype forwarded the prepared message to the KW-7. The unit keyed a UHF transmitter to send out the message. Given its wide use, the navy viewed the KW-7 as one of its most critical encryption devices, used for more than eighty percent of the messages sent and received by the entire U.S. fleet, and of utmost importance to submarine operations.\n\nThe KW-7 came in a small gray box with a panel covering three-fourths of the upper surface area bordered by a set of switches, dials, and lights across a black panel on the lower section. Original versions used wire cords to set the daily encryption key. A newer version came with a small square bulge on the gray cover panel to make room for rows of plugblock modules underneath. Various combinations of the plugblocks allowed for encryption key changes. R-Branchers had to set up the plugboard manually, which everyone hated. Daily keylists were provided to R-Branchers, who in turn changed plugs or block modules to alter the encryption. As such, keylists were considered more than just top secret; they were literally the keys to the hen house.\n\nIn December 1967, when John Walker began stealing keylists for the KW-7 and delivering them to the Soviets, he harbored no concerns that this action might result in the deaths of ninety-eight submariners within three months, and another ninety-nine submariners three months after that. Walker suppressed any guilt he might have felt by assuring himself that as valuable as the keylists might be, the documents he stole were only half of the puzzle needed by his Soviet handlers. Without an actual working KW-7 device, the keylists offered little value. Walker didn't account for Soviet cunning and skill, and therefore never imagined that America's largest adversary would obtain a working KW-7 device less than a month later.\n\nOn January, 11 1968, the spy ship USS Pueblo (AGER-2) sailed out of Sasebo, Japan, with orders to monitor Soviet naval activity around the Tsushima Strait. Their SIGINT mission entailed using Russian-speaking I-Branchers listening via ESM masts to gather intelligence from North Korean ship and shore platforms. For weeks prior to their departure, intercept operators at the CIA's Foreign Broadcast Information Service picked up threats coming from North Korea warning U.S. \"espionage boats\" to stay out of their territorial waters or suffer severe consequences. These transmission reports were never forwarded to the NSA.\n\nWhile the Pueblo sat off the coast of North Korea in early January, not more than a whisper outside the international twelve-mile limit, the CIA intercepted more transmitted warnings, some pointed directly at the USS Pueblo. Again, the NSA was not informed, and the Pueblo remained on station, her deck coated with ice and her crew shivering in the bitter cold. In the distance, North Korean mountains, their black slopes covered with white, jutted toward a bleak sky. Inside the spy ship's thin metal bulkheads, a team of twenty-eight spooks rotated shifts in the tight SIGINT space. Racks of equipment hummed and blinked, including a KW-7 encryption device, typewriters, Teletypes, a WLR-1 intercept receiver, and 600 pounds of top-secret documents.\n\nAlthough the Pueblo carried twenty-two weighted bags to toss these documents over the side, a small incinerator to turn them to ash, and two paper shredders to grind them into oblivion, the combination was not capable of destroying more than a small percentage of the secret material in the event of an emergency. As for the destruction of sensitive equipment, the ship carried no explosives\u2014only sledgehammers and axes, which were no match for hundreds of pounds of steel-encased crypto gear. No U.S. destroyers patrolled nearby. No aircraft carriers and no submarines stood at the ready. No jet fighters sat fueled on runways in the event of an incident with the North Koreans. The NSA and the navy sent the Pueblo into the den of the lion alone, adorned with nothing more than a small knife and a loincloth.\n\nOn January 21, the lion stirred. A Soviet-made SO-I-class submarine chaser came within two miles of the U.S. ship. The following day, two Lenta-class DPRK fishing trawlers cruised within twenty-five yards\u2014close enough for Pueblo's sailors to stare into the eyes of the trawlers' crews. Commander Lloyd Bucher broke radio silence and tried to send a situation report, but the signal did not go through.\n\nThe next morning, on January 23, the North Korean island of Ung-do hid behind a dense mist some sixteen miles away. Inside the Pueblo's SIGINT area, an R-Brancher jolted to attention as he detected radar signals coming from two North Korean SO-1-class subchasers, one of them using the call sign SC-35. On the bridge, Bucher scanned the horizon with binoculars. His eyes widened as the subchaser SC-35 shot out of the mist and raced toward the Pueblo. The small gunboat, brandishing a 3-inch cannon and two 57 mm guns, pulled close and demanded that the Pueblo announce her nationality. Commander Bucher told his crew to raise the American flag and hydrographic signal\u2014the ship's cover as a research vessel.\n\nThe subchaser closed further, pointed her guns, and ordered the Pueblo to surrender. Bucher had no intention of handing over his ship, so he went to all ahead two-thirds and attempted to run. Pueblo's speed paled in comparison to the subchaser's, and to make matters worse, three North Korean torpedo boats joined the chase. SC-35 made a third swing around the Pueblo and hoisted the signal, \"Heave to or we will open fire.\"\n\nNow four miles outside North Korean waters, Bucher tried to maneuver away until two MIG-21 fighter planes roared overhead and yet another torpedo boat and subchaser entered the fray. He contemplated putting up a fight, but Pueblo's ammunition had been stored below decks, and the machine guns were covered in cold-weather tarpaulins. Bucher knew he had to raise his arms in surrender, but not until his crew could destroy the cryptographic equipment on board. He gave the order to do so and asked his engineering officer, Gene Lacy, if they had time to scuttle the ship. Lacy shook his head no. Even flooding the Pueblo would take forty-five minutes, and if the North Koreans shot holes in the life rafts, the crew would die within minutes in the freezing ocean.\n\nBucher continued evasive maneuvers for another two hours to buy time while the crew tried to shred documents and smash the crypto gear into unusable pieces. Given the volume of sensitive material on board, and the hardened steel cases used to house the cryptographic systems, the crew failed to destroy much before the North Koreans sent 57 mm cannon shells toward the Pueblo and opened fire with machine guns. Most of the rounds missed, but a few landed, slicing Bucher's ankle and buttocks.\n\nCommunications Technician Don Bailey had been frantically typing away on the KW-7 unit for more than an hour, sending and receiving messages to and from Japan. R-Branchers in Kami Seya implored Bailey to destroy the cryptographic equipment, but the sledgehammers and axes proved in effective. Kami Seya promised air cover, but no U.S. planes were within range, save a squadron of seventy-eight fighters in Japan, which were forbidden from flying combat missions due to international agreements. The South Koreans requested permission to save the Pueblo with their 210 armed jets, but General Charles H. Bonesteel refused, citing a potential unwarranted escalation. A dozen F-105s were finally given authorization to fly from a nonregulated location in Japan. They requested clearance to sink the North Korean ships and then refuel in South Korea, but they received the opposite instructions, making their arrival in time impossible. When someone shook President Lyndon Johnson awake that morning to brief him on the incident, the North Koreans had already bagged their prize.\n\nAs ordered, Bucher followed the North Korean vessels until the Pueblo neared the coast of North Korea. Around 2:00 P.M. he ordered all stop to check on the destruction of the papers and equipment in the SIGINT area. SC-35 closed to less than a mile and opened fire. More than 2,000 rounds ripped through the Pueblo's thin steel and slammed into the wardroom, laundry room, and passageways. Near Bucher's stateroom, Fireman Duane Hodges slumped to the deck as a projectile tore off most of his leg and sliced open his gut. Blood spilled from his intestines and coated the deck in red. I-Brancher Marine Sergeant Robert Chicca watched in horror as blood oozed from his thigh, and Radioman Charles Crandal screamed in pain as hot metal shards impaled his leg. Fireman Steve Woelk reeled backward as sharp pieces of shrapnel burned into his groin and chest.\n\nOn the bridge, Bucher immediately ordered full ahead to appease the North Koreans and get them to stop firing. He gave the conn to Lacy and ran down to the SIGINT area. On the way he saw the mangled body of Duane Hodges in the blood-soaked passageway. In the SIGINT space, spooks were still trying to cram secret papers into mattress covers and pound crypto equipment into oblivion. Neither effort was going well. Most of the sensitive material, including top-secret code lists, along with the KW-7 and other crypto gear, remained intact.\n\nOut of options and fearing for the safety of his crew, Commander Bucher did the unthinkable. For the second time in history, and not since Commodore James Barron turned over the USS Chesapeake to the Brits in 1807, he allowed a foreign power to seize control of a U.S. Navy ship during peacetime. Once in port, Pueblo's crew members were blindfolded, kicked, beaten, spat on, and marched off the Pueblo at bayonet point. A team of North Koreans hurried aboard and grinned at the sight of a fully operational, undamaged KW-7.\n\nThe North Koreans motored the Pueblo to Wonsan, while her former crew were tortured, starved, and held captive in a POW camp. During his captivity, Commander Bucher faced a mock firing squad, enacted to force a confession. He did not relent until the North Koreans threatened to execute his crew. Given that none of the Koreans spoke enough English to write Bucher's confession, they had him pen the document himself. As a tongue-in-cheek retribution for killing one of his crewmen, Bucher wrote the words: \"We paean the North Korean state. We paean their great leader Kim Il-Sung.\" The North Koreans never caught on.\n\nWhile the Pueblo's crew endured horrendous conditions and unthinkable cruelty, the North Koreans made a deal with the devil. They turned over the captured KW-7 to their Soviet comrades in exchange for money, weapons, perhaps even a few cases of Stolichnaya. That same month, John Walker's current Soviet handler\u2014the vodka-drinking, chain-smoking General Boris A. Solomatin\u2014convinced him to steal a KW-7 operating manual in exchange for a big bonus. Walker gladly complied.\n\nOver the next two months, Soviet communications experts used the manual, their cache of stolen keylists, and the operational KW-7 to gain access to the U.S. Navy's most sensitive communications traffic. Overnight, almost every message sent between American shore facilities to ships and submarines of the fleet became an open book. General Solomatin, the Soviet's KGB chief in Washington at the time, commented many years later that \"Walker showed us monthly keylists for one of your military cipher machines. [He] enabled your enemies to read your most sensitive military secrets. We knew everything.\"\n\nOleg Kalugin, former KGB chief at the Soviet embassy in Washington and John Walker's first handler, validated Solomatin's comments after the Cold War. In an unaired CBS interview, he stated that \"John Walker's information, on top of Pueblo, definitely provided the Soviets with the final solutions to what ever technical problems they may have had at the time.... We certainly made use of the equipment from the Pueblo.\" Kalugin also confessed in his memoirs that before March 1968, the Soviets were intercepting and deciphering encrypted navy messages as a result of the Walker\u2013Pueblo intelligence windfall. This fact played a pivotal role in the downing of two submarines\u2014one Soviet and one American\u2014and brought the world once again to the brink of war.\n\nAlthough the NSA knew that the North Koreans captured the Pueblo, they were not able to communicate with her crew during their eleven-month imprisonment. As a result, the NSA assumed incorrectly that Pueblo's crew destroyed the KW-7 prior to the ship's capture. Worst case, they thought, even if the KW-7 remained intact, and the North Koreans turned the device over to the Soviets, without the daily keylists, the encryption unit had the usefulness of a boat anchor. Of course, the NSA did not know that John Walker had been delivering the Soviets keylists for months, and had also given them an operations manual.\n\nCautious voices at the NSA called for a replacement of the KW-7, just to be safe, but budget-minded objectors overruled. Replacing that many units in the field would be far too costly, and besides, did we mention that the Soviets need the keylists to decipher our messages? A series of charged incidents followed the Walker-Pueblo debacles, beginning with the deployment of a Soviet Golf II ballistic missile submarine in early 1968. K-129, a Project 629A diesel-powered boat with pendant number 722, joined five similar vessels of the Fifteenth Submarine Squadron based at Rybachiy Naval Base on the Kamchatka Peninsula. The Golf IIs represented the Twenty-ninth Ballistic Missile Division at Rybachiy, under the command of Admiral Viktor A. Dygalo.\n\nK-129's skipper, Captain First Rank Vladimir Kobzar, had just completed two back-to-back seventy-day combat patrols in 1967 and was looking forward to some R&R. He held back his anger when new orders demanded that K-129 undertake yet a third patrol. While preparing for that patrol, eleven \"strangers\" walked across the gangplank and descended through the hatch. To this day, aside from a few Soviets in command at the time, no one knows who these strangers were or why they were on board.\n\nOn February 24, 1968, powered by three diesel engines, K-129 twisted away from the pier. Veiled by darkness, the boat did not use running lights. Captain Kobzar stood on the bridge, bundled in a furlined ushanka, and watched waves crash into the bow of his submarine. Distant lightning flashed and mingled jolts of white with threads of morning indigo on the horizon. K-129 shuddered as Northern Pacific waves pounded her sides. Kobzar captured his final memories of daylight, cleared the bridge, and dove his boat under the roiling waves.\n\nArmed with three SS-N-4 Sark submarine-launched ballistic missiles (SLBMs), each fitted with a one-megaton warhead, K-129 conducted a deep-dive fitness test with ventilation valves closed and ballast tanks sealed. The submarine leveled off at a depth of 200 feet and assumed her patrol course\u2014due east to clear the shoreline shallows.\n\nFifteen miles outside the bay, Kobzar brought K-129 to periscope depth to transmit the first of a series of mandatory mission reports to the naval main staff at fleet headquarters in Vladivostok. Confident that any listening Americans could not decipher their codes, Radio Officer Senior Lieutenant Alexander Zarnakov sent a millisecond burst signal on the SBD radio reporting that K-129 had entered deep waters to start her mission. Miles away, a Soviet radio dish grabbed the burst signal and pushed charged electrons down a wire into a receiver\/decoder.\n\nHundreds of miles away, at Classic Bulls Eye stations in Japan, Guam, and Alaska, R-Branchers recorded the microsecond bursts, analyzed the recordings, and sent tip-offs to Net Control in Hawaii. NC sent out a flash and later received back more bearings from other sites. Operators utilized newer GYK-3 Boresight computers to determine an approximate fix, along with other parameters to determine that they'd found a Golf II-class submarine.\n\nNC informed COMSUBPAC (Commander, Submarine Force, U.S. Pacific Fleet) of their findings, who informed the commander of Submarine Squadron One in Pearl Harbor, Hawaii. After noting which submarines were operating near enough to Kamchatka, the commander of Squadron One composed a flash message to the appropriate attack submarine. They encrypted and sent that message using a KW-7.\n\nWhile the Soviets had no clue that their burst transmissions were being multiangulated by U.S. Bulls Eye stations, the Americans had no idea that their encrypted communications to and from submarines were as clear and open as ordinary phone calls. The Soviets now knew that an American attack submarine had been sent to trail K-129. They probably suspected that the Americans found K-129's location via the $1.5 billion, 1,300-mile SOSUS acoustic detection system surrounding the Pacific.\n\nSix days out from port, on February 29, K-129 failed to send a routine report that she had crossed the International Date Line. While some suspect rogue activities as the cause for such a breach, Soviet radiomen concur that if a boat's CO knows the enemy may be nearby, he'll forgo a radio transmission\u2014this is especially true for ballistic missile submarines with primary missions to stay hidden. K-129's captain likely knew that an American submarine was operating in the area, in that the naval main staff intercepted the American submarine's position report via the stolen KW-7. Armed with that knowledge, K-129 would have gone deep, snuck under a sound layer, and slipped silently away in another direction. Reports show that K-129 also left her normal mission area on that date.\n\nOne could again argue that K-129 went outside her assigned box because someone on board intended to go rogue. Given that the officers on K-129 had families and loved ones in Russia, this scenario seems unlikely. SOSUS stations reported that once K-129 switched to batteries or ventured too far from hydrophone locations, tracking her became difficult, if not impossible. Could it be that the Soviets knew this fact, having gained that information via intercepted KW-7 transmissions, and ordered K-129 to go outside her box to test SOSUS detection capabilities and range limitations?\n\nOver a twelve-day period, K-129 failed to transmit position reports twice. While this may lead some to believe rogue sailors were the cause, such a conclusion again seems suspect. The fact that K-129 transmitted as prescribed on other days and then skipped a few days is probably related more to evasion than mutiny. Had Captain Kobzar received a warning that a U.S. sub was closing the gap, he would have avoided the risk of coming shallow to transmit, instead remaining below a thermal layer to hide.\n\nThe most reasonable scenario has the Swordfish or another American boat stalking K-129 as directed by Bulls Eye and SOSUS fixes, then K-129 receiving KW-7 intercepted warnings from the naval main staff and avoiding the U.S. submarine like the plague. This game of cat and mouse ensued until K-129's life came to an abrupt end.\n\nWhat happened to the Soviet Golf II on March 8, 1968, just fifteen days away from home and 1,560 miles from Honolulu? Several SOSUS arrays recorded \"an isolated, single sound of an explosion or implosion, a good-sized bang.\" Some speculate that, despite this submarine's ability to fire her ballistic missiles while submerged, for some unknown reason, rogue individuals risked surfacing K-129 to commence an unwarranted attack on Hawaii. While doing so, in the nick of time, the submarine suffered a casualty and sank. This conclusion is based primarily on the fact that K-129 got tapped for her mission six months early, had eleven \"strangers\" on board, missed a few radio updates, and traveled outside her assigned patrol grid.\n\nWhen one considers the Soviet's newfound ability to decipher KW-7 transmissions, perhaps these seemingly odd circumstances begin to make sense. The Soviets may have sent the noisy diesel submarine to sea six months early to test their ability to intercept KW-7 transmissions. They might have assumed that K-129 would be detected snorkeling by SOSUS, which would trigger those transmissions. As for the eleven strangers, one could conclude that these sailors were part of an OSNAZ group of signal intelligence operatives\u2014similar to \"spooks\" on U.S. submarines. They would have been tasked with observing American radio transmissions and analyzing responses, transmission frequencies, and so on. We've already explored the reasons why K-129 may have missed radio transmission times and steamed outside her grid.\n\nEven if Hawaii was not more than 500 miles beyond the range of K-129's missiles, one could argue for the rogue theory if not for the fact that forensic evidence confirms that she did indeed surface. This type of submarine did not need to surface to fire her missiles, snorkel, or go rogue; in fact, such an act makes surfacing all the more unlikely. The only reason why K-129 might come up from the deep while on patrol is because she had no choice. This might be the case if she suffered a catastrophic failure or endured an accident that damaged something badly enough that underwater repairs were not feasible, such as a collision with a U.S. submarine.\n\nGiven the advanced sonar systems available on the USS Swordfish or similar subs, had a U.S. boat been directed to the football field wherein K-129 played, the American attack boat would have eventually found its prey. Having done so, the directives for all sub skippers in 1968 pushed them into close prowling range behind their targets. But which boat received the orders to tail the Golf II?\n\nUnder the command of Captain John Rigsbee, the Swordfish had been deployed in the area on WestPac since February 3. On March 17, nine days after K-129 sank to the bottom of the ocean, the Swordfish pulled into Yokosuka, Japan, under the cover of night, seeking repairs on her conning tower, periscope, and ECM mast. Official reports attributed the damage to ice impact while conducting classified operations in the Sea of Japan on March 2. The navy told the Soviets that the Swordfish was 2,000 miles away from K-129 at the time of the incident. Most points in the Sea of Japan are over 4,000 miles away from Hawaii, but less than 1,000 miles from Yokosuka. If the Swordfish was indeed in the Sea of Japan (that is, near Vladivostok), and whacked her periscope and ECM mast there, why did she not arrive in Yokosuka until two weeks later?\n\nInstead, the Swordfish limped into the harbor at Yokosuka while a Japanese\/Soviet spy, stationed covertly on Honshu Island, observed the event. His Soviet handlers wrote the report off as routine and all but yawned. When retired admiral Ivan Amelko heard the news, his reaction was a bit more dramatic. He accused the United States of causing the demise of K-129 and killing ninety-eight men. The United States, of course, denied the accusation. Perhaps eyewitness accounts can help us clear up this mystery.\n\nIn late March 1968, excited about the prospect of another mission into dangerous waters, Communications Technician Second Class Frank Turban strutted through the door of his bunker in Kami Seya, Japan. Most of the working areas at Kami Seya were housed in underground tunnels to protect equipment and personnel from earthquakes. As a T-Brancher, his job revolved around unusual signals, and whenever one came along, his ears perked. While the Bulls Eye section at the Kami Seya Naval Security Group Activity, commanded by Captain J. W. Pearson, worried over high-frequency direction-finding tip-offs and flash reports from Net Control, Turban focused on special signals.\n\nNothing could be more special than something new and unusual transmitted from a Soviet submarine. BRD-6 ESM masts on board U.S. submarines, designed by Sanders Associates in New Hampshire, collected these signals. The BRD-6, which stands for Boat Radio Direction Finder, contained a miniature HF direction finder and burst signal receiver\/recorder, similar to the one used in Bulls Eye stations. In fact, the engineers at Sanders gained much of their initial knowledge for this design from the meeting they had with William J. Reed and his team in 1962.\n\nWhile at Kami Seya, Turban and ten other spooks received orders to report to the USS Swordfish for a top-secret SpecOp. Spooks were not official members of any submarine crew, did not claim any particular boat as home, but instead \"snuck aboard\" just prior to departure and usually stood watches behind a curtain in the radio room. They did not often mingle with the crew to ensure no secrets were told by accident. After a months-long mission, the CT team disappeared with equal stealth, such behavior conjuring the nickname \"spook\" by submariners.\n\nWhile preparing for his Swordfish adventure, Turban heard from R-Branchers involved in burst signal HFDF operations at Kami Seya that they'd gotten several Bulls Eye hits on a Golf II submarine near Hawaii before the boat went completely dark. She hadn't transmitted again since March 8. Turban thought nothing of this report at the time as he caught a flight to Hawaii to assemble with the other spooks and prepare for their upcoming mission.\n\nIn late April 1968, Turban and his team met the Swordfish in Okinawa, Japan. He noticed that the front of the submarine's sail seemed a bit shinier than the rest of the boat but passed this detail off as routine maintenance. Using a towed underwater camera inside a mini-sub called the Underdog, the Swordfish deployed near Vladivostok to prowl for lost Soviet missile parts. Tethered to the boat like one of the heads of Hydra, the Underdog's twelve-foot-long, two-ton aluminum body contained high-resolution cameras and bright battery-powered strobe lights. She also came with whiskerlike towed sonar and shark fin rudders and bow planes for maneuverability. A small propeller pushed the Underdog through the water, while the tether allowed for remote operation from the submarine.\n\nOperators sat in front of monitors and examined the ocean floor as the Underdog darted across search patterns and relayed images back to the Swordfish. While this type of mission was considered quite dangerous, as it required penetration deep into Soviet waters, Turban noticed that their skipper often ran from his own shadow. When predators came stalking, the boat's executive officer wanted to uncollar and chase the bastards. The CO refused, instead content to stay as far away from potential encounters as possible, earning him the nickname of Charlie Tuna in reference to the Chicken of the Sea television commercials. Turban found the CO's gun-shy attitude strange for an attack-boat driver but asked no questions. After all, he was not officially part of the crew, only a temporary ranch hand with an assignment to find, record, analyze, and fingerprint interesting new signals.\n\nThe Swordfish remained on station in the Sea of Japan for sixty-eight days, while Turban's team, consisting of an officer in charge (OIC) and his assistant (AOIC), two linguist I-Branchers, two R-Branchers, and another special signals T-Brancher like Turban, did their thing. There were also three R-Brancher specialists who worked with the WLR-6 ECM radio equipment. Not once did Turban hear anything from the crew about a collision on March 2, not a word about repairs in Yokosuka or damage of any kind.\n\nIn light of the above, the pink elephant question lingers. The collision on March 2 (or perhaps March 8) took out Swordfish's periscope and ECM masts, placed in her jeopardy, and lowered her mission capabilities to almost zero. If she had been in the Sea of Japan, why did the Swordfish not arrive in Yokosuka until two weeks later? On the other hand, if she indeed had been near K-129, a speed of fifteen knots allowed the Swordfish to pull into Japan on March 17 with ease.\n\nStill, let's assume that the navy's statement is true, and that the Swordfish did not collide with K-129. We know that U.S. Bulls Eye stations intercepted K-129's transmissions, and SOSUS heard her snorkeling more than once. Armed with the knowledge about K-129's whereabouts, the navy certainly would have sent a U.S. submarine to investigate. But which boat?\n\nOn January, 13 1968, Commander Hugh Benton assumed command of the Guardfish (SSN-612) in Pearl Harbor, Hawaii. Shortly thereafter, his boat deployed for a six-month WestPac special operations mission. The Guardfish was on station near the Sea of Japan in March 1968. The USS Barb (SSN-596) was also on station near Kamchatka, but reports indicate that she was \"surprised\" by Soviet naval activity pursuant to K-129's loss, making her a less likely candidate. Perhaps the Swordfish or the Guardfish pursued K-129, and maybe one of them came a bit too close. Until sailors on either boat are willing to talk, or the navy finally releases the classified logs, the truth won't come to light.\n\nRegardless of what really happened, given access to many of the facts we've just examined, the Soviets reached the verdict that the Swordfish rammed K-129 and caused her to die. They could not, of course, admit to the world that they suspected this because they possessed the means to intercept and decode KW-7 transmissions. Hands tied, they could do nothing more than hurl accusations. The fact that they had a fly on America's wall, and that it eventually led to the first premeditated sinking of a U.S. submarine since World War II, remained a secret.\n\nBY THE THIRD WEEK OF MARCH, Soviet naval headquarters declared K-129 missing and organized a massive air, surface, and subsurface search-and-rescue effort into the North Pacific from Kamchatka and Vladivostok. U.S. intelligence experts examined SOSUS logs and determined an approximate location for the recorded \"bang.\" The Soviets did not have the advantage of SOSUS, and so searched for K-129 in the wrong location. They finally gave up and went home, and soon after officially announced the loss of the Golf II and her crew.\n\nSix weeks later, on Friday May 15, 1968, on station in the western Mediterranean, the USS Haddo (SSN-604) received the coordinates for a nearby Soviet Echo II\u2013class nuclear submarine entering the Atlantic near the Straits of Gibraltar. The Haddo's skipper did not know that the fix came from Bulls Eye stations in the Atlantic. The Echo II's track put her on a course bound for the Canary Islands and a potential rendezvous with Soviet surface ships operating in the area. The Haddo went to work and slipped in close behind the Echo.\n\nOn May 20, Vice Admiral Arnold F. Schade, commander of the Atlantic Submarine Force, ordered the Haddo to hand off her tail of the Echo II to the USS Scorpion (SSN-589). On the Scorpion, Commander Francis Atwood Slattery picked up the trail of the enemy submarine as she made her way toward the Soviet flotilla. Photographed by U.S. satellites, the flotilla consisted of an unusual collection of vessels not normally known to operate together. The strange formation was conducting exercises in waters outside typical Soviet operational areas, in the vicinity of a U.S. nuclear test site, and lofting large balloons that housed surveillance equipment.\n\nCommander Slattery brought the Scorpion to periscope depth around midnight on May 20. Radio operators started transmitting to the naval station in Rota, Spain, that Scorpion had arrived on station, but interference prevented validation of reception. A day later, they finally connected with a navy communications station in Nea Makri, Greece, which relayed Scorpion's message to SUBLANT (Submarine Forces, Atlantic). NAVCAMS (Naval Communications Area Master Station) in Norfolk received the relay from Greece on a KW-7. The message read that Slattery estimated interception of the Soviet surface group in about six hours. Slattery did not know that Soviet ears were listening to his report.\n\nOn May 22, the Scorpion arrived on station. Several hours later, without warning, the boat shook violently as an explosion rocketed through the sub's compartments. The crew scrambled to battle stations as years of experience and training kicked into high gear. Hatches were dogged, valves shut, and switches thrown, but to no avail. In just over a minute and a half, the submarine shot toward the bottom.\n\nAt 6:59 P.M. Greenwich Mean Time, the secret listening stations in the Canary Islands, Newfoundland, and Argentina began a 190-second recording of fifteen eerie acoustic signals as they reflected from the Plato Sea Mount. The signals bounced off convergence zones and found their way to an array of hydrophones mounted on the seafloor, placed there by the Air Force to monitor Soviet nuclear tests. Spinning reels of high-speed recording tapes documented the tragic sound that overpowered the background noise of whistling biologics. The high-energy release was rich in low frequencies with no discernible harmonic structure\u2014a bubble pulsation from an underwater explosion. Operators recognized this as the frightening sound of a dying boat.\n\nIn the months following the death of the Scorpion, the theories as to her demise were many and varied. Initially, the most prevalent theory pointed to a \"hot-running\" torpedo that caused a catastrophic explosion in the torpedo room. But the experts at Ordnance Systems Command insisted that such a detonation was virtually impossible. An explosion could occur only if the torpedo hit an object while running at full speed.\n\nWhat really happened to the Scorpion? Theories and speculations abound and have been addressed ad nauseam in several books, but most concur that the evidence points to an external explosion. One smoking gun all but validated this conclusion. In 1982, at a navy SOSUS training class in Norfolk, Virginia, students observed a LOFARgram printout of a top-secret recording made in May 1968. The scrawled lines on the paper depicted a hostile encounter between two submerged contacts. After twenty minutes, a third contact appeared. This submersible was launched from one of the submarines, and the scribbles verified a high-speed screw. The targeted submarine's signature shifted in width and size, indicating evasive actions as the torpedo neared. Seconds later, the paper filled with black ink as the high-speed screws caught up with the evading submarine and ended her life.\n\nThe students were told that the recording was made during Scorpion's encounter with an Echo II submarine. Authors and experts bring additional facts to bear to substantiate this claim, but to what end? Until governments are willing to expose the truth, the official U.S. Navy position as to Scorpion's demise remains as \"cause unknown.\" If the conspiracy theorists are correct, then the lessons learned from allowing the greed of John Walker to manipulate entire nations will stay buried, and the cause-and-effect results of this act remain as a black stain on the pages of history.\n\n## CHAPTER THIRTEEN\n\nAny sufficiently advanced technology is indistinguishable from magic.\n\n\u2014ARTHUR C. CLARKE\n\nTHE SOVIETS FINALLY GAVE UP THE search for their lost K-129 and went home in May 1968, not long after the sinking of the Scorpion. The U.S. Navy let another month go by and then called in the cavalry. Captain James F. Bradley Jr., from the Office of Undersea Warfare, had already spearheaded the creation of the perfect platform to undertake the mission of finding the missing Golf II\u2013class boat. The forty-six-year-old Bradley served as the navy's chief underwater espionage planner, often conducting meetings in a soundproof suite in the E ring of the Pentagon. Quiet, efficient, and whip smart, Bradley's official title of \"Naval Operations, Navy Department\" alluded to nothing covert. Only those close to the captain knew that he wielded the power to craft clandestine intelligence missions for all of the navy's attack submarines.\n\nUnder Bradley's orders, the nuclear submarine USS Halibut (SSGN-587), commissioned in January 1960, received a $70 million upgrade that included the mysterious Fish. Just like the Underdog tethered to the USS Swordfish, the Fish mini-subs contained deep-water cameras and sun-bright strobe lights to find interesting objects on seafloors. Halibut's two Westing house Electric\u2013built Fish cost $5 million each and resided inside a hump-shaped dome on the submarine's bow dubbed the \"bat cave.\"\n\nThe dome once housed Regulus missiles back when Halibut was a teenager and resembled a miniature torpedo room complete with metal racks to house the Fish. The first Fish suffered from a host of issues, some caused by the intense pressures placed on cameras and equipment operating at 20,000 feet deep. These problems took months to correct. With kinks finally ironed out, the refurbished Halibut, commanded by C. Edward Moore under the highest \"Brick Bat 01\" authority, used her Fish to skim ocean floors on scavenger hunts for spent Soviet toys\u2014like intercontinental ballistic missile (ICBM) warheads that might contain guidance systems for intelligence perusal. Bradley's program, dubbed Operation Sand Dollar, did not pan out as planned, as the boat was plagued by mechanical and computer failures. However, the concept of using the Fish to find and photograph things in dark places proved invaluable.\n\nPaul Nitze, secretary of the navy, caught wind of the debacle and blasted Bradley, Moore, and others involved in the project. Sand Dollar's failures were soon forgiven when Nitze found out about K-129 and the proposal from Bradley to use the Halibut's Fish to locate the downed submarine. The SOSUS team had provided location fixes based on the \"bang\" recordings that documented K-129's last gasps for life. Nitze approved the mission, and Captain Moore set to sea in the Halibut with do-or-die orders to locate K-129, take detailed photographs via the Fish, and place salvage markers. After months of frustration, snagged cables, and near-disasters, the Halibut's mysterious Fish delivered success, and the precise position of the Golf II submarine was no longer a mystery.\n\nArmed with Halibut's K-129 photos and location coordinates, Carl Duckett, then CIA deputy director for science and technology, spearheaded the effort to recover the Golf II submarine intact. Captain Bradley dissented, telling Duckett that \"you can't pick up the goddamn submarine, or it will fall apart.\" Bradley countered with a plan to send in mini-subs, cut open the hull, and retrieve only the tidbits desired at a fraction of the cost of full retrieval. Duckett dismissed the plan and remained determined to haul up K-129 in all her glory, claiming that inside the Golf II there should be Soviet encryption equipment\u2014including a burst radio transmitter and codebooks\u2014and three nuclear SS-N-5 SERB missiles.\n\nA detailed examination of Soviet radio, missile, submarine, and torpedo technology could be invaluable, offering the navy an opportunity to design effective countermeasures and improve ASW and Bulls Eye detection equipment. The Soviet codebooks alone might be worth the expense of the salvage\u2014almost $200 million\u2014and would allow analysts to decode hundreds of hours of burst signals and other transmissions recorded over the years at various Bulls Eye stations. Besides, Duckett insisted, he didn't want to chance leaving anything behind.\n\nCould it be that Duckett overruled Bradley's far less costly plan to ensure that no part of K-129 could be salvaged by the Soviets, especially that part that contained metal fragments from the hull of the U.S. submarine that rammed the Soviet missile boat?\n\nDissenters quieted, the project to recover the K-129 intact moved ahead. The CIA established the Special Projects Staff to oversee the program, which they code-named Azorian. Duckett tapped John Parangosky, a senior official in the CIA's Directorate of Science and Technology, to run the show. A former Army Air Force lieutenant in World War II, Parangosky joined the CIA in 1948, where he put his expertise in photographic techniques and project management to use for the U-2 and A-12 spy plane programs. Parangosky recruited U.S. Naval Academy graduate and former diesel submarine officer Ernest \"Zeke\" Zellmer to run the day-to-day operations. President Nixon personally approved the creation of the Azorian task force in August 1969, and CIA director Richard Helms compartmentalized all top-secret project work under the heading of \"Jennifer.\" Only a select handful of individuals at the White House and within intelligence communities knew anything about Azorian.\n\nIn early 1970, Joseph Houston, Itek Corporation's chief optical engineer for underwater systems, had just returned from a test run aboard the USS Dace (SSN-607). Houston spearheaded the development of the new Type 18 periscope, recently mounted on the Dace for sea trials. After a week under way, he wanted to enjoy a few days off. He had joined Itek in 1964 and over the next few years compiled an impressive r\u00e9sum\u00e9 in optics that included the inventions of various testing equipment, advanced optical lenses, and high-resolution cameras, including ones used in several spy satellites under Project Corona. Within an hour after stepping onto the pier from the Dace, Houston received a summons from Itek's vice president of special programs, John Wolfe. The two met over lunch in Wolfe's office.\n\n\"Have you ever been to Mars?\" Wolfe asked as Houston found a seat.\n\n\"Mars as in the planet?\" Houston asked, wondering if this had something to do with his previous work on the camera systems for Viking 1, the first American Mars lander.\n\nWolfe gave a quick nod. \"Yeah, Mars.\"\n\n\"Not lately,\" Houston said. He knew that the broad-shouldered and bushy-headed Wolfe had a flair for the dramatic and wondered if his boss really meant Mars or was just laying the foundation for a metaphoric bombshell.\n\nWolfe examined the contents of the box lunch on his desk, brought by his secretary, and wrinkled his nose. \"I haven't either. Then again, neither has anyone from NASA.\"\n\n\"Is this about Viking 1?\"\n\nWolfe shook his head no. \"It's about solving the problem of taking detailed, high-resolution metrology photos of a hostile environment that we've never been to or seen before.\"\n\n\"I'm not sure I'm following you,\" Houston said, opening a bag of chips.\n\nWolfe said, \"We need your expertise in submarine optics and deep-ocean camera systems to help one of our government clients. They need to take thousands of pictures with no measurement distortion so we can build a virtual model for a rig that can mine manganese nodules.\"\n\n\"Mars has manganese nodules?\"\n\nWolfe waved a hand in the air as he swallowed a bite. \"No, but it might as well be Mars. We need accurate metric data to set up a drilling rig more than three miles deep in the Pacific Ocean.\"\n\nHouston lowered his sandwich. His eyelids flung open as the metaphoric bombshell landed. \"Three miles? No large body camera can withstand that kind of deep-sea pressure, let alone record high-quality metric data. And even if you could, proper lighting would be a major challenge.\"\n\n\"That's why I called you,\" Wolfe said with a smile. A small piece of bread clung to his front teeth.\n\n\"Who's the client?\" Houston asked.\n\n\"The CIA.\"\n\nHouston almost choked on a gulp of water. The CIA? That definitely conjured some curiosity. Houston knew that Frank Lindsay, Itek's chairman of the board, had met Howard Hughes during World War II, back when Lindsay was a gun-blazing, parachute-popping OSS operative. Houston heard that Hughes kept in touch with Lindsay over the years and had recently contacted him about some hush-hush CIA-sponsored project involving deep-sea mining.\n\nHouston said, \"So why is the CIA interested in mining manganese nodules off the ocean floor?\"\n\nWolfe said, \"I could tell you, but then\u2014\"\n\n\"Yeah, I know,\" Houston said, \"you'd have to send me on a permanent vacation to New Jersey. So what are the requirements for this camera and lighting system?\"\n\n\"We need to...drill an oil well 16,500 feet deep, and we need accurately measurable photographs of the area so we can ensure we're designing the equipment to the right dimensions.\"\n\n\"How large an area are we talking about?\" Houston queried.\n\nWolfe's voice dropped another few decibels. \"Three hundred fifty feet long by fifty feet wide.\"\n\nHouston almost gagged again on his water. \"What kind of wellhead has those dimensions? That sounds more like a submarine, not a wellhead.\"\n\nWolfe's face turned white. He rose from his chair and grabbed a bunch of papers off his desk. He walked over to Houston and handed him the stack. \"Read this agreement, sign it, and then we'll talk.\"\n\nProject Azorian, of course, had nothing to do with building a wellhead or mining for manganese nodules on Mars. Instead, Houston was joining a massive team, spearheaded by Howard Hughes, tasked with salvaging an entire 2,700-ton submarine sitting 16,500 feet deep in the Pacific. Houston had no doubt that this top-secret mission would go down in history as one of the most ambitious, expensive, and massive ever undertaken. He suspected that the consequences of failure were huge, but they probably paled in comparison to the backlash certain to come from the Soviets should they find out.\n\nAfter some further discussion with Wolfe regarding how the project would be organized and managed, Houston departed with visions churning in his head of a rig capable of supporting multiple cameras and lights. The enormity of the ocean area to be surveyed complicated the task by a few orders of magnitude. Also, the lighting had to be perfectly uniform, and the camera lenses needed to survive enormous pressures.\n\nTo make matters worse, Houston discovered that his project head was the mysterious John Parangosky, who insisted on being called Mr. P. to maintain his clandestine anonymity. Although Houston found Mr. P. highly intelligent, a great listener, warm with people, and \"the calmest SOB on the planet when the fit hit the shan,\" he could also be a slave driver when it came to meeting design parameters and deadlines. \"Parangosky had the demeanor of Alfred Hitchcock in the 1942 movie The Man Who Came to Dinner,\" said Houston of his former boss. \"He was a heavyset engineer with a receding hairline and a wry sense of humor. He rarely displayed emotion and could smooth out the rough edges of a meeting within fifteen minutes.\"\n\nMr. P. assembled a small team of engineers and technicians in the winter of 1969. Included in the group were Joe Houston, John Wolfe, Floyd Alvarez, Ernest Zellmer, and a dozen others. Mr. P. also leaned heavily on Alex Holser as his \"right-hand man\" to assist with project management tasks.\n\nThe team often met in a conference room at Itek's corporate headquarters in Lexington, Massachusetts. \"Itek's building had a gleaming glass 'swinging sixties' architectural feel,\" said Houston. \"The main lobby jutted out over the parking lot and you entered via a ramp that resembled a vintage airplane staircase. A smattering of Pieter Mondrian artwork covered the walls in the foyer, which resided on the second floor along with the executive offices, cafeteria, and conference room where the Azorian team met.\"\n\nOver the next year, Houston cycled in and out of dozens of team meetings where the group struggled to formulate solutions for a seemingly impossible task. No country in the world had ever succeeded in salvaging anything this large or this heavy from this deep. The Azorian team debated over four potential solutions: The \"Brute Force\" plan entailed using a Rosenberg winch to hoist the entire 2,700-ton sub up from the bottom. The \"Trade and Ballast\/Buoyancy\" concept suggested the use of buoyant material to float the boat to the surface. An \"At-Depth Generation of Buoyancy\" idea proffered employing gas at depth to \"balloon\" the K-129 upward via chemically created hydrogen or nitrogen cryonic gases. Finally, despite Duckett's previous nixing of the down-scaled plan proposed by Captain Bradley to use mini-sub robots to scalpel their way into the submarine's hull, Mr. P.'s team explored this possibility as a backup plan.\n\nIn late July 1970, the group opted for the Brute Force concept using a sling crafted from pipe-strings wrapped about the K-129 and hoisted upward via winches mounted on a yet-to-be-built 565-foot-long ship. By October of that year, an executive committee, headed by Deputy Secretary of Defense David Packard and formed to scrutinize Azorian project progress and cost overruns, estimated the program's odds of success at less than ten percent. Only the promise of an intelligence bonanza saved Azorian from cancellation in August 1971, and the CIA got the green light to proceed on October 4, 1971.\n\nWith Mr. P. breathing down his neck, Houston set about solving an almost insurmountable optics and lighting problem. While others on Project Azorian were involved in building the HMB-1 barge, massive grappling claws, and winch system for hoisting K-129 from the bottom, they all relied on Houston's success. They needed to transfer real life into bits and bytes to perform analysis and create what-if scenarios. One can't control the real world of turbulence, lighting, shadows, and what ever else Mother Nature fashions. These parameters need to be manipulated in a make-believe virtual world to better understand the true nature of the environment. That required precise, clear photographs free of shadows or blur that might distort measurements and assumptions.\n\nHouston explained the difficulty of creating such a uniform background to Mr. P. by using the example of driving down a road on a foggy night. By glancing out the window, you'd notice that the street-light illumination is brightest in the center of the beam but diminishes toward the edges of the road. This phenomenon is prevalent with single-beam lights due to dust and particulates in the air that scatter the light. In water deeper than 10,000 feet, few such particulates exist, so the clear water doesn't spread out the light. More lights cover more area, but then you get shadows in the areas in between each beam. Those shadows can distort dimensions and details. Mr. P. was impressed by Houston's explanation but still tapped his watch impatiently.\n\nHouston spent long days designing a four-legged stool to hold an array of lights built by Hydro Products in Southern California, as well as a way to allow the array to hover over specific objects and light up areas eight times the size of the beams. He also designed the means to gain complete uniformity and eliminate shadows across the entire dimensions of the field. In this way, every protuberance, recess, and mechanical detail on the sunken sub could be accurately measured and transferred to blueprints and models used by the rest of the team.\n\nHouston knew that if his lighting or camera caused designers to measure one detail wrong, a $200 million project could fail. Worse, if something catastrophic happened, the Soviets might find out about Project Azorian, which could provoke an international incident. Even more concerning, a massive failure could get people hurt or killed. With the weight of the project's success resting on his shoulders, Houston delivered sample photos to Mr. P. The pictures were of a pond in Houston's backyard, taken by hanging a camera from a tree while Houston's son, Brant, positioned objects to photograph.\n\nMr. P. grabbed the photos from Houston's hand. He pulled out a magnifying glass and glared. He ran the glass across the images, wrinkled his nose, and said, \"Where'd all those catfish come from?\"\n\nHouston copped a sheep grin and shrugged. \"The pond in my backyard.\"\n\n\"Nice,\" Mr. P. said. \"Very nice.\"\n\nHouston nodded, found a seat, and let his aching shoulders slump. Thereafter the team referred to his innovative strobe-light rig as \"the catfish solution.\"\n\nThe remaining design and build phases for Project Azorian involved revolutionary concepts for underwater photography that pioneered the development of a wide-angle lens with zero distortion. This included a means of reducing the pressure on the outer lens by a factor of seven, which allowed conventional lens designs to achieve distortion-free performance. Houston published a technical paper on the topic in the Marine Technology Society's 6th Annual Conference and Exhibition in July 1970. Sometime later, Houston met Carl Duckett, the CIA's deputy director for science and technology and Mr. P.'s boss. Always smiling, fiercely patriotic, handsome, and smart, Duckett possessed an encyclopedic memory and razor-sharp wit. He often used both to fight bureaucrats when they threatened to squelch technological advancements that he believed were critical to national defense. An engineer by training, Duckett almost never thought inside the box.\n\nLike any good engineer, Duckett knew the project's HMB-1 barge\u2014designed to grapple and retrieve a downed submarine three miles deep\u2014had to undergo a series of quality assurance tests before she could resurrect K-129 from the dead. To do this, he towed the barge to a remote location off Long Beach. As a warm California sun soaked beaches and blondes in ultraviolet radiation, Duckett pulled the plug and watched the unmanned HMB-1 barge sink into a 6,000-foot-deep trench. Unbeknownst to him or others on the team, fishermen on a trawler observed the entire operation. From their distant vantage point, they swore that a ship had just sunk with all hands on board. Being good citizens, they called the Coast Guard, which sent out a cutter to investigate. Seeing no signs of a craft or survivors, the Coast Guard assumed a false alarm had been called and went home.\n\nHours later, the barge miraculously reappeared, just as ordered by Carl Duckett. Rejoicing, the fishermen called the Coast Guard and said that God had saved the sunken ship, miracles do happen, and as changed men, they had sworn off liquor and women for life, or at least the weekend. Duckett knew nothing of the affair until Mr. P. read the story in the paper the following day. When informed about the incident, Duckett smiled and quoted Arthur C. Clarke, who once said that \"Any sufficiently advanced technology is indistinguishable from magic.\"\n\nArmed with new \"magical\" deep-diving cameras designed by Joe Houston, the research vessel Sea Scope steamed to the K-129 site. Sea Scope once sailed the seas as the USS Harrier (AM-366), a U.S. Navy Admirable Class Minesweeper commissioned in 1945. The Harrier received a name and designation change not long after World War II. Although reclassified as an oceanographic and research vessel, the Sea Scope came chockful of deep-sea reconnaissance toys, including sonar, ESM, underwater television and photographic equipment, and a magnetic and seabed exploration grappling system. The Sea Scope arrived over the Halibut-marked site of K-129's remains and used the deep-diving cameras and sophisticated equipment to provide a detailed map in support of the Glomar's mission.\n\nPhotos taken by Houston's camera rig revealed that K-129 remained intact, save for a ten-foot-wide hole blown in her side just aft of the conning tower\u2014probably caused by the catastrophe that sank her. Judging by the photographs, the navy estimated the Golf II accelerated to 200 knots on her way down before she collided with the bottom. She was intact but most likely quite fragile. Given that K-129 might be broken beneath her steel outer plating, bringing her up from the bottom would be a long, slow, difficult task, not too unlike those arcade games that challenge you to grab a toy with an imperfect crane and lift it out of the tank. Only in this case, the toys are 16,000 feet down in a tank of dark seawater.\n\nOn November 4, 1972, the Glomar Explorer put to sea under the guise of a mining vessel in search of manganese nodules. Forty handpicked CIA agents made up nearly one-fourth of the crew. Nine teen months later, after a battery of test runs, the Glomar headed toward Hawaii. Upon arriving at the site on July 4, 1974, using a sophisticated guidance computer and bottom-placed transducer, the Glomar team went about the delicate task of maneuvering the HMB-1 into position over K-129. Computer readouts flashed with information in the control room, while operators chewed on fingernails.\n\nIn the meantime, a Soviet SB-10 salvage tug passed within 200 feet of the Glomar Explorer. The tug's crew of forty-three flashed smiles and cameras and unnerved the CIA operatives. The SB-10 finally moved off and Glomar's crew commenced with the salvage.\n\nThe five grappling claws were attached to a length of pipe on the ship, resembling six interconnected tongs suspended from a long platform. The entire affair, nicknamed \"Clementine,\" looked like a giant squid clinging to a pipe. The pipe was fed through a hole, making its way down 16,500 feet to the Soviet sub. Glomar's crew, which consisted mostly of roughnecks who once worked on oil rigs, constructed the tether from sixty-foot-long sections of pipe they linked together like sectioned tent poles. Television cameras equipped with strobe lights monitored the progress, which took several long days to complete. Finally, Clementine reached the bottom.\n\nSeveral more days passed as the crew, using sophisticated computers, made careful calculations to allow Clementine to drape a steel net over the boat's conning tower. Slowly, they wrapped one of the claws around the sub, then another, followed by a third. Silt and sand shot out from beneath K-129 as the claws reached underneath her delicate sides. On board the Glomar, nervous fingers twitched as they gripped joysticks, typed in keyboard commands, and flipped switches. Three and a half miles down, K-129 waited patiently for her ride to the surface. More hours passed as readings were checked and double-checked. Camera images were studied and debated. Finally, convinced that they were ready to hoist, someone gave the signal to proceed with only three claws wrapped about the boat versus all five.\n\nK-129 shuddered as she rose from the ocean floor. Sections of pipe reappeared from the water as Clementine struggled to bring her prize home at six feet per minute. Workers dismantled and stowed the sixty-foot pipe sections, while cameras below the surface kept a watchful eye on the claws. One wrong miscalculation, one incorrect move at this point, could end a mission that had spanned years and cost hundreds of millions of dollars. Unfortunately, that miscalculation happened, and K-129 rocketed back to the seabed. Clementine's claws still gripped the hull, and the massive arms were pulled down to the bottom along with the submarine. Unbeknownst to project engineers at the time, K-129 slammed into the hard rock beneath the sand, and the impact caused hairline fractures in the grappling arms.\n\nDisappointed but not defeated, the crew tried again, this time grabbing the sub with all five claws. Nine hours later, just 3,000 feet off the bottom, a damaged grappling arm snapped and left K-129 dangling by a hope and thread. Moments later, the fragile sub crumbled like a cookie. The rear two-thirds of the submarine hull broke away and tumbled back to the seafloor. Lost were the conning tower, three missiles, and the room where the codebooks were stored.\n\nThe recovery phase of the Azorian project concluded on August 9, 1974. With the entire mission considered a failure by the CIA, the crew of the Glomar Explorer looked like a defeated football team as they returned home. Making matters worse, while still at sea, they heard about Richard Nixon's resignation on August 8. In all, the Glomar recovered only two nuclear-tipped torpedoes and the bodies of eight Soviet crewmen from K-129. They were buried at sea on September 4, 1974. At least, that's the story told by the CIA. After columnist Jack Anderson divulged classified details about Project Azorian in a Los Angeles Times radio program on March 18, 1975, rumors flew that the agency intended to return to K-129's site to bring up the pieces lost in the first attempt. The truth about what really happened remained clouded until the CIA released portions of an in-house journal written by a participant in the operation whose identity remains classified. Although the fifty-page article contains gaps of redacted text, several sections point to an altogether different outcome to Project Azorian than previously reported. On page forty-seven of the journal, an anonymous participant delivers the following account:\n\nThe Soviet tug [SB-10] left. We were going to be able to do the telltale pumpdown operation without surveillance. Our cover story had held: the Soviets had been fooled. Now we could anticipate seeing our prize without being concerned about sharing it with the owner. Everyone wanted to see the first glimpse of the target. [Redacted text.] Those of us waiting anxiously on deck received a reward of a different type. Bobbing up to the surface (luckily in the well) was a brimming full Jerry-can of torpedo juice. It had travelled over three miles to the bottom and back and been subjected to pressures of over 7,000 pounds per square inch without spilling a drop. [Redacted text.]\n\nThe Mission Director and his team viewed the scene from a balcony-like portion of the ladder, which led down to the well gates. Radiation monitors had reported readings five times background even at this distance. We knew that we were in for a nasty time. Some of the earlier excitement of the mission was returning to the exploitation party. [Redacted text.] It was going to be difficult\u2014the jumbled hulk was not going to reveal its secrets easily.\n\nA concluding paragraph in the article is also quite revealing, again alluding to full success of the mission in stating: \"Thus, the long saga of Azorian came to a conclusion as the HGE [Hughes Glomar Explorer] rested at anchor in the Hawaiian Islands, more than six years since the Soviet G-II-class submarine 722 sank in the Northwest Pacific Ocean. The efforts to locate the site of the sinking and to conceive, develop, build, and deploy the HE system [redacted text] stretched almost as long in time, beginning in mid-1968. And the success that was achieved depended, in the end, on the combined skills of a multitude of people in government and industry who together forged the capability that made it possible to proceed with such an incredible project.\"\n\nThe above statement stands in stark contrast to the dismal admission of near-failure from the CIA in 1975 after the story leaked. A declassified Memorandum of Conversation released by the CIA records a conversation between Secretary of Defense, James R. Schlesinger, and President Gerald Ford during a meeting in the Cabinet Room on March 19, 1975. They were discussing how to deal with Jack Anderson's Los Angeles Times radio story and a subsequent article about Project Azorian printed in the New York Times.\n\nSchlesinger is quoted as saying \"This episode has been a major American accomplishment. This operation is a marvel\u2014technically, and with maintaining secrecy.\"\n\nPresident Ford replied, \"I agree. Now where do we go?\"\n\nAfter Carl Duckett retired in 1974, he contacted Joe Houston, who had recently returned to Itek as the head of special programs for the Applied Technology Division in Sunnyvale, California. Duckett wondered if ATD might need his services in some capacity. Houston immediately called his boss and convinced him to bring Duckett on board as a consultant. Houston and Duckett worked together for another decade and over those years became close friends.\n\nDuckett never confirmed or denied reports that the CIA story about Project Azorian's failure was a smokescreen to hide the truth. Houston heard through others involved in the project that the salvage operation was hampered by nuclear contamination on some of the recovered objects, which included sixty bodies versus only eight, and a few of the nuclear missiles. In fact, Houston heard that several people, while handling those missiles, were burned by radiation and had to \"shower off for an hour\" in the decontamination area.\n\nOver a few beers at Houston's house, he asked Duckett if the rumors might be true, that the Glomar brought back much more of K-129's remains than previously reported. Duckett's eyes twinkled, and he said, \"You know, I was really upset when the CIA said they wanted to return to the K-129 site for a second attempt. That made it sound like we didn't do our jobs and get what we came for the first time around.\"\n\n## CHAPTER FOURTEEN\n\nNever was anything great achieved without danger.\n\n\u2014NICCOL\u00d2 MACHIAVELLI\n\nMANY OF THE DETAILS SURROUNDING PROJECT Azorian became public knowledge after the press leaked the story in 1974. A full year before the Glomar Explorer set off on her mission to bring up K-129, Captain James Bradley had another brilliant idea. He figured if the Halibut's Fish could locate K-129 on the bottom of the ocean 18,000 feet deep, certainly she could locate a Soviet communications cable in the Sea of Okhotsk at a depth of only 400 feet.\n\nThe Soviets likely had no qualms about sending unencoded top-secret messages through their Ministry of Communications cable, given the impossibility\u2014in their estimation\u2014that anyone could tap the signals. But the cable, running between Vladivostok and Petropavlovsk naval bases, could be tapped with a sophisticated recording device to gain access to the Soviet Union's private military conversations. Bradley envisioned a recording apparatus designed to capture weeks of traffic, extracted by the on-station Halibut. Russian-speaking analysts could then transcribe the unencrypted tapes, delivered to the U.S. Navy's intelligence headquarters at Suitland, Mary land.\n\nThe anticipated traffic might include Soviet military plans, details of maneuvers, unguarded conversations between key military and political figures, and other intelligence delicacies. Perhaps loose Soviet tongues would reveal the location of warheads that often landed near the Kamchatka Peninsula or future plans for ballistic missile tests. U.S. submarines, provided with foreknowledge of pending exercises, could be sent in on special operations to photograph and monitor the tests. The NSA would also be alerted to pending weapons tests by increases in military communications traffic through the cable. Before any of these benefits could become reality, however, a major hurdle needed jumping.\n\nMammals on the surface breathe roughly an eighty percent to twenty percent mixture of nitrogen to oxygen. Sport divers learn that air compressed into a metal tank can kill or cause serious injury if certain depths are exceeded for too long. Even navy divers are taught to stay above 130 feet for most dives, and although many of us exceeded 200 feet for various missions, we stayed down no longer than the four minutes allotted by U.S. Navy dive tables. Diving to 400 feet compresses air to the point where one lungful draws substantially more nitrogen and oxygen than \"landlubbers\" breathe. Such concentrations can poison divers and turn them into the equivalent of a town drunk, complete with slurred speech, crazy thoughts, and blurred vision. Although SeaLab underwater habitats experimented with a mix that contained helium versus nitrogen to overcome nitrogen narcosis, divers had to acclimate to the pressure by staying inside a chamber for long days or weeks, and they hated the cold, cramped environment. Still, with helium-induced Donald Duck voices, they often joked that \"to air is human, but to HeO2 is divine.\"\n\nA serious accident, resulting in the death of one diver, put an end to the SeaLab experiments, but not the divination of HeO2 saturation diving. Nascent and dangerous, the science moved forward. The art of deploying saturation divers covertly from a submarine in frigid Soviet waters, however, was an altogether different animal that required an innovative new approach. Although navy scientists had envisioned such deep-diving projects, their concepts were never proven in the field. Creating this capability would impel American ingenuity to new heights, and send submariners and divers on the most difficult, dangerous, and decorated missions of the Cold War.\n\nCaptain James Bradley pushed through a program to have the Halibut outfitted with a pressurized hyperbaric diving chamber mounted on the stern to support deep-sea bottom walking. The chamber was designed to resemble the Mystic Deep Submergence Rescue Vehicle (DSRV-1), which became the Halibut's deployment cover story. The DSRV-like chamber, some fifty feet in length and eight feet in diameter, looked like a giant torpedo mounted to the back of a submarine by way of four metal stanchions. Divers entered the chamber from underneath through the boat's escape trunk.\n\nAll submarines have one or more small decompression chambers built into the hull. These are called lock-out chambers or escape trunks, and they allow divers to exit or enter the boat while submerged. These escape trunks are also designed to let submariners exit the sub in the event of an emergency\u2014provided that the boat is not more than a few hundred feet deep. While these trunks are adequate for standard compressed air dives, they are not suitable for saturation diving using helium. A hyperbaric diving chamber is needed for this, which is essentially a sealable pressure vessel with hatches large enough for divers to enter and exit, combined with a compressor to raise the internal pressure.\n\nThere are two types of hyperbaric chambers, one for decompression and one for recompression. The former is used for deep-sea diving operations. The latter is used to treat or prevent decompression sickness. For both, occupants are \"pressed down\" or \"pressed up\" by increasing or decreasing the internal pressure, respectively. Divers are required to remain inside for many long hours, days, or even weeks to prevent serious injury.\n\nIn October 1971, the upgraded Halibut, carrying a stern-mounted decompression hyperbaric chamber and loaded with deep-sea saturation divers, cruised toward the Soviet Union with Commander \"Smiling Jack\" McNish in charge. The Soviet communications cable they sought ran thousands of miles under the ocean deep in the Sea of Okhotsk, a large body of water, surrounded by land, that from the air looked like a massive Russian lake. The undersea cable ran from the Petropavlovsk submarine base and missile-testing facilities on the Kamchatka Peninsula, down to Vladivostok, headquarters of the Soviet \"Far East\" Pacific Fleet.\n\nCommander McNish successfully located the cable's approximate position via Halibut's periscope by finding a small sign posted on a beach warning passersby to avoid running into a buried cable. From there, finding the actual cable in the sand required delicate manipulations of the camera-equipped mini-sub Fish using miles of cable lowered from the bowels of the boat. The Fish swam on its own power, guided via a tether running into the large bat cave on the bow of the Halibut. The tether often fell prey to snags and ocean currents that caused serious jarring of cameras and equipment. Halibut's Fish took rolls of film that revealed nothing. Finally, after several days of searching, the image of a three-inch-diameter Soviet communications cable appeared in one color photo.\n\nThe sat divers then locked out of the submarine's fake DSRV, tapped the cable, and siphoned off a flurry of signals, but the SIGINT spooks on board were not trained in this type of signal capture. McNish returned with poor-quality recordings. The Halibut made a second run on August 4, 1972, but came home with no better results. NSA and navy officials were disappointed but convinced that intelligence pearls awaited in the cable if skilled experts could tap into them properly. Each mission day cost the navy more than $1 million, given the expense of SR-71 spy-plane overflights to monitor Soviet traffic in the area, as well as decoy submarines poised to create distractions should the Soviets come snooping. In light of the expense and importance of the operation, the NSA escalated the signals capture issue to a priority position.\n\nIn the summer of 1974, they contacted one of their own, John Arnold, and tasked him with solving the problem. A navy Mustang officer, Arnold had scratched his way up from seaman to lieutenant commander while gaining expertise in underwater eavesdropping. As a navy spook, he'd previously conducted espionage missions off the Soviet coast aboard the USS Nautilus, including the one that used cameras mounted to toilet paper rolls to photograph nuclear test blasts near Novaya Zemlya in 1961. The NSA instructed Arnold to assemble a crack team of spooks, which he found at the \"Fred Ten\" Wullenweber Bulls Eyes facility in Sabana Seca, Puerto Rico. Arnold's handpicked team consisted of four navy communications technician chiefs, including Master Chief Malcolm \"Mac\" Empey and Chief Mark Rutherford. A Russian-speaking I-Brancher accompanied the four chiefs.\n\nThe five chiefs strode across the gangplank from the pier, seabags slung over uniformed shoulders. The crew did not know these strangers, but most had little doubt as to their function and exchanged excited whispers acknowledging that the \"spooks had arrived.\" Although selected by Arnold, the team reported to their OIC, Tom Crowley, who reported to Captain Augustine \"Gus\" Hubel. They were officially attached to the clandestine Naval Security Group, but none of the spooks thought of themselves as spies. In fact, they never claimed titles higher than sailors or submariners.\n\nMac Empey was colorblind, but that didn't hinder his electronic genius. Since the quirky M-Brancher couldn't discern the color coding for electrical wiring or blueprints, that forced him to learn the entire function of each circuit by heart. Those who witnessed Empey troubleshooting a system were entertained by the constant bobbing and weaving of his head as he scanned schematics up, down, and sideways. The other spooks often joked that he looked like a parrot planning an escape.\n\nMark Rutherford relished his clandestine role as a spook. He loved the \"shaken, not stirred\" image of James Bond and often imitated his favorite movie character. He had a way with words and women and could charm almost anyone. He also excelled at his job, often demonstrating his technical prowess on every piece of gear used by the spooks.\n\nWhile the Halibut crew trained for more than a year to prepare for their next mission, Empey and Rutherford performed amazing technological feats, including the creation and integration of much of the advanced systems used for the mission. The Halibut didn't have CTs of their caliber on the first two runs who understood the nature of the tapped signals. The original R-Branchers performed a \"capture all\" of every conversation on every wire or \"channel\" and dumped everything onto a recorder. That was like trying to listen to a conversation in a small room when twelve people were talking at once. Making sense of the recorded cacophony created an NSA nightmare, which is why the first two missions failed. To solve this problem, Empey and Rutherford knew they had to separate each channel. They also needed to make the recorded conversations easier to hear by using better filtering and signal-processing techniques.\n\nTo accomplish this, they specified thirty pieces of equipment and lugged them down to a cordoned-off spook area in the torpedo room called the \"chart room.\" I-Branchers came up with the name in reference to the small shack used to store confidential charts aboard Soviet submarines, such as the one on Foxtrot-class boats. Over seventy percent of the gear used by the spooks was not military grade, or \"mil spec,\" but came from off-the-shelf manufacturers. The navy relegated maintenance responsibility to the team for this non-mil-spec equipment\u2014most of which was analog versus digital\u2014as well as the task of integrating the systems with each other and with the external cable tapping gear that delivered the signals. Empey and Rutherford did most of the wiring, soldering, tweaking, troubleshooting, calibrating, and worrying.\n\nMuch of the equipment needed to interface to a Hewlett-Packard 1730 digital computer with 32 kilobytes of random access memory (RAM), which represented state-of-the-art back then. Today, a small notebook computer can hold upwards of four to eight million kilobytes of RAM. The chief CTs spent four weeks at HP facilities in Loveland, Colorado, learning how to operate and repair the 1730.\n\nBy 1975, with five of the navy's top spooks aboard, the Halibut was ready to embark on a third mission to the Sea of Okhotsk. All hopes of success were pinned on the improvements made by the spooks, and failure could spell the end of the most promising espionage program deployed during the Cold War.\n\nWHEN FIRST CLASS NAVY DIVER DAVID LeJeune walked across the brow of the USS Halibut in 1975, he glanced at the fake DSRV chamber on her deck and wondered if he should have heeded Zipperhead's admonition. His former executive officer warned him that volunteering for Special Projects would mean living in a tiny chamber with three other divers for almost two months at a time. Such an assignment required round-the-clock dives for weeks in enemy waters, where one wrong move could prove fatal not just to the divers, but to the entire submarine crew. LeJeune ignored Zipperhead's warnings and signed up anyway. He'd been saturation diving for years, had earned his instructor's pin, and was a member of the team that set a world record on a mixed gas dive to more than 1,000 feet. He figured if he had the chops to do that, he could muster the balls to ride the Halibut.\n\nBorn in Fairbanks, Alaska, LeJeune and his family migrated to Santa Fe Springs, California, where he asked his junior high school sweetheart, Cheryl, to marry him. She turned him down, of course, but with a wink and a smile. LeJeune didn't stop trying. He walked Cheryl to the bus stop every day and then, just to impress her, ran the five miles to the school\u2014beating the bus every time. Finally, Cheryl accepted the proposal but insisted they delay the wedding until after their high school graduation. LeJeune waited patiently for the next four years, but just before he could win his lady's hand, Santa Fe Springs High School kicked him out for being a rowdy nonconformist.\n\nLeJeune managed to claw his way to a diploma during summer school, and his reward followed soon after: a draft notice from Uncle Sam. He ignored the notice, bought a motorcycle, and married his first love. LeJeune and Cheryl's wedding bells were accompanied by an unexpected gift: a second draft notice. Having no desire to take a two-year vacation in Vietnam as a ground pounder, LeJeune rode his motorcycle down to the navy recruiter's office. They signed him up on a 120-day delay program, so he passed the time by completing welding school. At the time, he planned only to use that skill on dry land, but he decided to get his feet wet anyway.\n\nLeJeune bought a cheap mask and a pair of fins and enrolled in Los Angeles County's scuba diving program. He paid his $27 for the training and rental equipment and jumped into the water. After cracking the books and completing his pool training, he took a boat ride to Catalina Island for his certification dive. \"I puked my guts the whole way,\" he said, \"but once I got into the water, I felt great. I looked around at all the bright colored fish and said, 'This is for me.'\"\n\nConvinced that the ocean world held the key to his future, he volunteered for Underwater Demolition Team (UDT) training during boot camp in San Diego. In those days, U.S. Navy SEALs and UDTs were separate commands, and LeJeune wanted to stay in the water, not fight in a jungle. Whether he could pass the prerequisite physical training tests for UDT school was never a concern. He'd run a five-minute-mile since his bus-beating days in junior high, swum like a dolphin in L.A. County's scuba school, and marched off ten pounds in boot camp. He'd also lettered in high school basketball and dunked opponents more than once in water polo. But the line between confident and cocky is sometimes a thin one, which can often throttle one's passion. \"I just didn't try hard enough,\" LeJeune said. \"So I came in ten seconds late, and they failed me. I pleaded my case, said I'd do it again, but they turned me down. I damn near cried. I didn't want to be just an ordinary sailor.\"\n\nDefeated and depressed, he reported to his first duty station at the submarine base in New London, Connecticut. That's when he first heard the term fleet diver. He had no idea what that meant, but he knew it had something to do with the ocean, so he filled out a request chit to attend the training. The navy turned him down. Devastated but determined, he put in five more chits. The sixth one got accepted, and he and Cheryl packed up and drove to Norfolk, Virginia\u2014but not before he made the mistake of getting a U.S. Navy diver's tattoo on his arm. When the divers at the navy diving school noticed the unearned tattoo, they punished him by delaying his program start by more than six months. In the mean time, his previous welding experience got him assigned to a detail cutting and torching on the USS Seawolf. \"Had I known I'd have to dive off that old geezer years later,\" LeJeune said, \"I would have cut a hole in her too big to patch up.\"\n\nThe USS Scorpion had gone down with all hands a month earlier, in May 1968, and Cheryl made LeJeune promise that he'd turn down any diving jobs that involved submarines. He agreed, but fate had other plans. In early 1969, Master Diver Billy Kitchens finally let LeJeune suit up for his indoctrination dive\u2014a requirement prior to attending second class divers school to weed out the claustrophobic twitchers. He strapped on his Mark 5 diving rig, complete with a vintage round metal helmet and baggy canvas suit, all the while spewing questions and funny wisecracks. Kitchens never laughed. LeJeune sat in the itchy diving suit for thirty minutes as perspiration stung his eyes and drenched his lips. More bored than irritated, he said, \"Aren't you going to give me a test or something?\"\n\nKitchens rolled his eyes and held out his hand. A dime sat in his gloved palm. \"See if you can get this dime out of my hand with your glove.\"\n\n\"That's easy,\" LeJeune said. He smacked the bottom of Kitchens's hand, and the dime went flying into the air. LeJeune caught the dime, opened his glove, and said, \"There.\"\n\nKitchens manufactured a frown and said, \"LeJeune, I really hope you become a master diver someday, and a little wise-ass student like you makes your life a living hell. That would at least give my days in the navy some meaning.\"\n\nLeJeune spent the next thirteen weeks earning his second-class diver's pin. During that time, Cheryl's name changed to George. The impetus came from being dirt poor without even a spare quarter to attend a movie on the base. That, and a television commercial that depicted two bored vultures with one asking the other, \"Well, what do you want to do?\"\n\n\"I don't know,\" LeJeune always replied when Cheryl mimicked the vultures. \"What do you want to do, George?\"\n\nThe name stuck.\n\nLeJeune attended first class diver's school in 1970 and graduated from the Washington, D.C., Navy Yard in January 1971. He spent the next several months in boredom, doing little more than inspection dives on the presidential yacht USS Sequoia (AG-23). After a transfer to New London and another spate of less than exciting dives, LeJeune volunteered for saturation diver training under the navy's Man in the Sea program. That's when his life changed in ways he could never have imagined.\n\nThe navy used an ominous-looking craft called the IX-501 Elk River to train divers on the most dangerous and advanced form of diving extent. The IX designator is given to ships that no longer serve in their original capacity. Commissioned in 1945, the Elk River once operated as a rocket-launching medium landing ship but now deployed strange space-suited creatures down to depths never before attained by humans. Divers wore one-eighth-inch-thick neoprene wet suits against their skin, but that's where any resemblance to ordinary diving ceased. Over the rubber wet suit they draped a baggy pair of coveralls that resembled a Hazmat suit, but completely enclosed. A hose connected to the back of the suit and pumped in hot water to keep the divers warm in below-freezing conditions. The heated water ran through smaller tubes that branched away from the diver's back and formed a spider's web covering the arms, legs, and chest. Warm water from the tubes coated the neoprene underneath like a sprinkler system on a summer's day, while the skintight wet suit prevented scalding from the heated water.\n\nDivers wore a face mask manufactured by Kirby Morgan that resembled the clear Plexiglas masks worn by firefighters. The masks contained microphones and speakers for communications, as well as readouts for temperature and mixed gas levels. A hose ran from the Kirby Morgan that contained the most important element to all divers: air.\n\nLeJeune learned in saturation diving school that dive durations were limited primarily by two factors: gas supply and diver fatigue. \"Saturation diving uses one hell of a lot of gas,\" said LeJeune. \"We only pump in three percent oxygen compared to almost ninety-seven percent helium, but that three percent feels like thirty to the diver.\"\n\nExtended dives not only used up large supplies of helium, but also placed divers at great risk. The longer a diver stays in the water, the more tired he becomes. That's when mistakes are made. Still, macho saturation divers often held contests to see who'd wimp out first and want to be reeled back up to the surface. \"Ten or even twelve hours underwater was not uncommon,\" said LeJeune. \"In fact, if the job only took six, we'd stay down another two just to save face.\"\n\nLeJeune graduated from sat school and, soon after, requested instructor training. During that program, a master diver asked him to build a system for one of the diving rigs to make it easier for divers to close various valves in an emergency. LeJeune rigged up a pneumatically controlled system that caught the attention of a sat-school student, Lieutenant Commander Ross Saxon, eighteen months later. Saxon\u2014who had a large scar on the back of his head that earned him the nickname \"Zipperhead\"\u2014graduated and became executive officer of the newly commissioned USS Ortolan (ASR-22). He then requested, through official channels, that LeJeune build a gas analysis system to support the ship's thirty divers and two rigs. That system and the emergency valve rig created by LeJeune were later adopted by all navy saturation diving platforms worldwide.\n\nLeJeune stayed on board the Ortolan from 1973 to 1975. During that time, he lived the life of hero Dirk Pitt in a Clive Cussler novel. Still starving on navy pay, he and fellow divers Chris Velucci and Don Rodocker came up with a brilliant plan to use saturation diving to hunt for deep-sea treasure. They found some investors, who put up $280,000 to buy the needed equipment and dive boat. Months later, they spent their navy \"vacation\" leave motoring a less-than-adequate dive boat to a point off Nantucket, Massachusetts, where they dropped anchor. They spent the next forty days diving on the wreck of the Andrea Doria, an Italian ocean liner that sank after a collision with another ship in 1956.\n\nThe three divers dreamed of finding precious valuables tucked inside the wreck's steel tomb while fending off bandits who tried to rob them at night. During the day, they avoided lurking hungry sharks and militant nosy authorities. The Coast Guard tried to shut them down more than once, claiming that living conditions and life-jacket supplies were inadequate on board the diving boat. \"We'd move life jackets from one location to another when those guys weren't looking so they'd think we'd had enough,\" said LeJeune. \"They never did catch on. We also carried a bunch of rifles and fired into the night air to keep the pirates away.\"\n\nThe three treasure hunters cut a large hole in the Andrea Doria's side but found nothing more than a chafing dish, a bottle of perfume, some silverware, and the ship's bell. Their investors were less than pleased. \"At least we got a lot smarter about saturation diving on that trip,\" said LeJeune. \"That experience probably saved my ass when I did those Ivy Bells missions off submarines years later.\"\n\nWhich is where LeJeune went next. Tired of working on a sat diving ship while enduring \"puke storms,\" he put in a transfer chit to Special Projects.\n\n\"That means living for weeks or months inside a tiny chamber on a submarine,\" Zipperhead said to LeJeune.\n\n\"I know,\" LeJeune said. \"But at least it's calm down there.\"\n\nThat decision brought LeJeune face-to-face with the submarine that had found the K-129 near Hawaii years earlier and hurled him toward a life that gained him more medals than most combat soldiers earn after years on the battlefield. LeJeune didn't know it at the time, but cable-tapping missions, code-named Ivy Bells by the NSA, were considered by the military the most important missions of the Cold War. They were also the most dangerous, requiring direct approval from the president of the United States prior to each deployment.\n\nLeJeune's first clue that these missions were far from ordinary came within hours of reporting aboard the USS Halibut. He learned that out of the twenty-two divers who accompanied the Halibut on her first two runs, due to the arduous conditions and dangerous nature of the dives, only two volunteered to go again. One of the original divers developed kidney stones on the second run, and Lieutenant Commander \"Doc\" Halworth had to press down, enter the dive chamber, and shoot him full of Demerol. Given the Halibut's slow speed, the diver suffered for weeks until the submarine returned to port. LeJeune suspected that the incident took a mental toll on many of the other divers, which probably accounted for the high attrition rate.\n\nWhen LeJeune met Doc Halworth on the Halibut, he instantly liked the eccentric dive-trained doctor. \"Halworth had a hobby of buying abandoned missile silos,\" said LeJeune. \"He liked to renovate the things and sell them for a profit. He also bought caseloads of overstock shirts to save money. He'd wear an identical shirt damn near every day buttoned all the way to the top. The guy looked like a geek.\"\n\nGeek or not, Halworth's job aboard the Halibut entailed keeping the divers alive. The doctor analyzed the diving math to determine how much gas each diver needed while down. He ensured that mixed gas ratios were accurate and that emergency procedures were adequate to prevent serious injury or death. Still, the divers loved to play tricks on Doc Halworth, including cutting the fingers off the ends of all his rubber examination gloves.\n\nLeJeune's fellow divers included an Asian-Indian with huge \"Popeye\" forearms, Eugene Diem, whom they nicknamed Gunga Din, a smooth-talking consummate professional named John Hunt, and a short black-haired Italian called Bob \"Ginny\" Vindetto. Ginny also had a talent for finding and trading spare parts and often joked that if he ever died on a mission, God would let him into heaven because everybody needs a \"cumshaw guy.\"\n\nTwenty-two Special Projects divers competed for four bunks inside the small DSRV-sized hyperbaric chamber, as only four divers made the cut to conduct the actual mission dives in the Sea of Okhotsk. \"The master diver selected the four best divers for each mission,\" said LeJeune. \"Divers were disappointed if they didn't get picked, but we were all professional about it and worked as a team. We never forgot that we depended on each other for our survival.\"\n\nLeJeune spent the next several months with his fellow divers on the Halibut conducting at-sea training missions off the coast of San Diego. The Halibut set down on the ocean floor using snowmobile-like skis mounted to the underhull of the boat. Teams of three divers deployed from the DSRV on practice runs until every diver had been in the water at least once. The divers wore their neoprene wet suits underneath newer semi-closed Mark 11 saturation diving suits that cost the navy $87,000 each. The MK-11s employed heated water sprinkler systems to keep the divers warm, but also had a small pocket in the front for a thermoluminescent device. The TLDs were similar to the dosimeters worn by nuclear submariners to monitor radiation dosages but were waterproofed down to 1,000 feet.\n\nInside the diving chamber, divers wore dull brown fire-retardant T-shirts, boxer shorts, pajamas, and socks. Even the towels were brown and fireproof. \"The clothing they gave us was hot and itchy and had zero absorbency,\" said LeJeune. \"Using those towels was like drying off with a squeegee.\"\n\nHelium's light weight increases its conductivity, and the gas will rob body heat from a diver at a rapid rate. In fact, helium is the second-lightest element known. Only hydrogen weighs less. Earth's atmosphere contains just five parts per million of the gas, making helium rare and expensive. Chemically inert, helium offers no odor, color, or taste, which makes the gas ideal for diving as long as strict safety precautions are followed. Just as in scuba diving, where deep divers need to decompress to expel nitrogen from the bloodstream, helium divers must do the same to prevent decompression sickness, commonly referred to as \"the bends.\"\n\nIn addition to the bends, saturation divers are at risk for a phenomenon known as high-pressure nervous syndrome (HPNS). The syndrome causes dizziness, nausea, vomiting, tremors, fatigue, body jerks, stomach cramps, decreased mental and physical performance, and insomnia coupled with nightmares. The condition worsens with faster rates of compression or decompression and at greater attained depths. Symptoms were noted in the early 1960s when sat diving came on the scene, and doctors initially referred to them as \"helium tremors.\" The only way to prevent HPNS is to go down and back up again slowly in a hyperbaric chamber while pausing at intervals to allow the body to dissipate the helium from the bloodstream in a similar fashion to decompressing to expel nitrogen.\n\nThere were three sections to the cylinder-shaped diving chamber. Like the DSRV it resembled, the fifty-foot-long by eight-foot-diameter chamber had two bulkheads inside that divided the space into three compartments. The smaller aft section housed the watchstanders\u2014a couple of divers who monitored the four mission divers and catered to their every need. The middle section of the chamber had a hatch that opened into a round connector that attached to the escape trunk on the submarine. Divers entered the chamber through this hatch. The area was also used to store extra equipment and served as a pressurizable entry point should someone need rapid access to the divers, like Doc Halworth.\n\nThe larger bow section kept the four mission divers alive and served as the exit door out to the sea. Similar to a spacecraft, this area had its own life-support system, and not too unlike the environment found in space, the occupants could not easily come back to the world of nonpressurized air should something go wrong.\n\nWatchstanders funneled food, drinks, and other supplies from their nonpressurized aft section to the divers through an eighteen-inch-diameter \"medical chamber\" cylinder that ran into the forward pressurized section. \"Everything we ate and drank was cold,\" said LeJeune. \"Helium doesn't just rob body heat, it also steals warmth from food.\" To make up for the cold meals and limpid coffee, watchstanders ran movie projectors that shot flickering light through a Plexiglas window that came to life on the back metal wall. \"Sometimes the watchstanders played jokes on us by screwing the lid down too tight on a catsup bottle,\" said LeJeune. \"After it's pressurized, you can't get the damn thing off. We got them back by doing the same thing in reverse. Only when they opened the bottle, catsup exploded in their face.\"\n\nHelium's superconductivity required that the diver's forward section be heated to ninety-six degrees Fahrenheit, which to the divers felt like seventy-two degrees. Humidity ran high\u2014almost one hundred percent\u2014which exacerbated health concerns. \"A diver could go to sleep with a minor earache and wake up with pus dripping from his ear,\" said LeJeune. \"Bacterial infections were a big problem.\" The divers also shared one very public toilet and a small sink but enjoyed no showers. Fresh water for sponge baths, shaving, and brushing teeth was delivered through the medical chamber cylinder.\n\nPast thirty-five feet, as the helium level rose, divers sounded like cartoon characters with high-pitched voices. To compensate, they spoke into a \"helium speech unscrambler.\" This small box electronically synthesized the diver's helium-induced Donald Duck squeaks, then lowered the pitch and tone and pumped out normal-sounding speech.\n\nPressing down in the chamber, to equalize the pressure with the ocean outside, took as long as ten to fifteen hours, depending on the depth of the dive. Ten hours was normal for a 400-foot-dive, and getting there any quicker risked exposing the divers to HPNS. \"If we went down too fast,\" said LeJeune, \"every joint ached as the pressure pushed out the lubricating joint fluid. Your body needs time to adjust to a high-pressure environment. Most of us got HPNS more than once, which made us act like a drunk with a case of the DTs.\"\n\nIN 1975, LEJEUNE MADE HIS FIRST saturation training dive off the coast of San Diego aboard the USS Halibut. He and three other divers locked into the claustrophobic hyperbaric chamber and pulled on their neoprene \"undergarment\" wet suits. Body odor from the other divers found LeJeune's nose as he sat on a small cot in the divers' section and stared at various indicators, valves, and piping running along the inside of the enclosure. Master Diver Al Frontz, seated outside the diver's section, pressurized the chamber down to thirty-five feet. At this point, the divers still breathed normal air. The \"press down\" speed used by Frontz to descend to that initial depth determined the partial pressure of oxygen employed for the dive\u2014usually 0.28 to 0.30. Hours passed as Frontz slowly increased pressure, now using only helium. At 160 feet, when the percentage of oxygen in the chamber could no longer support combustion, O2 sensor alarms were turned off.\n\nA half-day later, with the chamber pressurized to the operating depth of 400 feet, LeJeune stood from his cot and followed the other divers to the back hatch that led to the diving platform. This back area was \"blown down\" using a small pipe with threaded ends that ran from the divers' section, through the bulkhead, and into the diving platform area. The divers blew helium through this pipe to pressurize the area to the point where the ocean water came up only to the bottom of the exit hole that led to the ocean. They removed the pipe during transit, as it could pose a safety problem for the submarine should the boat experience a flooding incident. If a master diver were to misplace the pipe or forget to bring it along for the ride, the entire mission could be scrubbed.\n\nLeJeune stepped through the thirty-six-inch-diameter opening at the front of the chamber and found his MK-11 diving rig hanging on the bulkhead. He also donned his suit and face mask. The mask was clamped to a fiberglass frame that formed a hydroseal made out of thin rubber. Around the rubber a spongelike material helped hold back the ocean. Divers soaked the material in water to draw out the air, smeared Vaseline on the rubber, then clamped the mask down tight to prevent leaks. \"One of the limiting factors on dives was that mask,\" LeJeune said. \"After eight hours in the water, your jaw felt like you'd been punched in the face by Muhammad Ali.\"\n\nMask ready, LeJeune pulled on his rubber boots, which contained lead insets and a steel toecap held on with rubber straps. Popeye-armed Gunga Din checked him over to ensure there were no issues. Once suited and checked, LeJeune sat on the edge of the diving platform with his feet dangling in the water. An adrenaline rush ran through his veins as the bottom portion of the platform lowered into the ocean like a hydraulically operated wheelchair ramp in a minivan. The ocean covered his face as he descended into a world of black.\n\nPitch black.\n\nThe blackest of all blacks. Like black hole black.\n\nLeJeune spread his arms and pushed off the curved edge of the submarine. Just like a free-falling skydiver, he floated through the silent ink and fell toward the ocean floor. \"This was always my favorite part,\" said LeJeune. \"I never knew if I was falling into a shark's mouth or what, which made the free fall that much more exciting. I used to live for that moment.\"\n\nTiming his fall, LeJeune braced himself for the impact as his leaded boots hit the murky silt. He heard nothing but the hissing sound of his own breathing. He switched on his light and moved in slow motion toward the other divers. Given the strong currents in the area, he wore two weight belts, which slowed him down even more. Tiny fish darted past his face mask. His light reflected from their shiny sides and made them look like fireflies in an alien world. LeJeune glanced at the two lights inside his face mask near his left eye. The green one was lit, and the red was not. Green was good. That meant his HeO2 mix was fine. Red was bad. That indicated that his breathing mix had dropped to zero. In that event, he carried a coffee can\u2013sized bottle of mix with enough air for about three breaths. LeJeune often wondered if that can of air would keep him alive long enough to get back inside the boat.\n\nTubes snaked throughout LeJeune's MK-11 suit, and warm water surged and sprinkled his goose-bumped skin through tiny holes. A two-inch-diameter tethered cord tied him to the boat and delivered the mixed gas, warm water, two-way communications, and power for his lights. One heavy wire in the cable also functioned as a lifeline to reel him back in should an emergency occur.\n\nLeJeune moved toward a line locker located on the underside of the Halibut. Gunga Din opened the locker and removed a cord that resembled an oversized automobile jumper cable. The cord could be reeled out to a maximum of 350 feet. Ginny Vindetto grabbed the cord and motioned for the others to help. Only three divers deployed, with the fourth still inside in the event of an emergency. The divers pulled the cord over to the simulated three-inch-diameter communications cable running along the ocean floor. As if they were actually in the Sea of Okhotsk on a live tapping mission, they clamped a \"grabber,\" fashioned to the end of the cord, around the cable. They covered up all evidence of the cable tap with a large sand-colored rubber blanket, then started back toward the Halibut.\n\nNear the submarine, Ginny started singing. At first, LeJeune wondered if his ears were playing tricks on him. Then Ginny started cracking jokes, Silly Sally jokes, like the ones he'd told on the playground in grade school.\n\nLeJeune radioed Master Diver Frontz inside the Halibut and told him that Ginny was showing signs of HPNS. Frontz acknowledged the report and told them to help Ginny back inside ASAP. LeJeune approached Ginny and reached for his arm. Ginny pulled away. He told LeJeune to \"leave me the fuck alone.\" LeJeune tried again. This time Ginny ripped down the zipper on his MK-11 and let the twenty-six-degree water flood his suit. LeJeune yelled at Ginny to rezip, but Ginny refused. Gunga Din slow-ran over, and the two tried to get the zipper back up, but Ginny fought them off.\n\nGinny started to sing again, but no more than three words came out before the freezing water temperature finally took its toll. His jaw locked up, and his eyes rolled up into their sockets. He passed out. LeJeune radioed the fourth diver, Cleveland Smith, inside the chamber and told him to start reeling.\n\nLeJeune clung to his friend, while Smith reeled in Ginny's cable line. His arms dangled and flapped about as the currents rushed past, and LeJeune wondered if they were already too late. Inside the chamber, he pulled off Ginny's MK-11 suit, while Smith grabbed the CPR kit. They pumped Ginny's chest and blew air into his lungs. Nothing. No movement. Certain that his friend had just died, LeJeune's heart sank. Finally, Ginny coughed and spewed out saliva. His eyes opened.\n\nHe stared at LeJeune for a long moment, then said, \"Am I still alive?\"\n\nLeJeune nodded. \"I'm not God.\"\n\n\"I was thinking more like the devil,\" Ginny said with a smile.\n\nAfter the incident with Ginny, other divers started showing signs of HPNS. Master Diver Frontz and Doc Halworth shortened the dives and took other precautions, including rotating the divers to see if physical conditioning or other factors might be the cause. Then \"all star\" diver John Hunt, considered one of the least likely to get the condition, started doing goofy things after only six hours down. Frontz and Halworth decided it was time to start blaming the equipment.\n\nSeveral divers, including LeJeune, flew to Panama City, Florida. They spent months in simulated environments testing various parts on the MK-11 suits until a culprit emerged. Each suit came with a CO2 scrubber in the MK-11 backpack that \"burped off\" one-fifth of the expelled gas from the diver's lungs. The scrubber used baralyme chemicals to scrub the CO2 and recycle fresh HeO2 back to the diver. This process also evaporated air bubbles to ensure that none reached the surface, where they could be detected. The scrubber was not heated, and at cold temperatures the baralyme did not scrub efficiently. This caused CO2 buildup and accelerated the possibility of HPNS. The team redesigned the system and replaced the baralyme with lithium hydroxide\u2014the same chemical used by submariners to scrub CO2 on diesel boats. One of the drawbacks included the possibility of chemical burns, but as far as the divers were concerned, staying sane took precedence.\n\nWITH HER DIVERS TRAINED AND READY, the Halibut made her third voyage in June 1975 to the Sea of Okhotsk, which in Russian means \"Mouth of the Bear.\" Excited about conducting his first live mission in Soviet waters, LeJeune accompanied the boat. Unfortunately, he didn't have the balls to stay on board long enough to make a dive. After only a week out, he developed a condition that caused one of his testicles to swell. Doc Halworth insisted that he depart when Halibut pulled in for a minor repair before heading west. Reluctantly, LeJeune complied. The problem turned out to be minor, but by then the Halibut had already left on her voyage to the hinterland.\n\nLess LeJeune, Halibut arrived on station in the Sea of Okhotsk a month later. The trek to the Western Pacific took almost twice as long as for most other attack boats due to Halibut's geriatric reactor, which could muster no more than thirteen knots. After sneaking past the Soviet fleet near the Kamchatka Peninsula, Commander McNish informed the crew that their mission would be one of the most dangerous in naval history. They'd be well within Soviet territorial waters, and if caught, they might not come home. Explosive charges had been placed throughout the boat to ensure that no equipment or survivors could be captured. McNish stopped short, however, of delivering all the details to the crew about their cable-tapping mission, and only he, his officers, the divers, and the spooks knew anything about the fine print.\n\nUpon relocating the cable, the Halibut's crew used a set of anchors and a winch system to set the submarine on the soft ocean floor atop the two skis. She anchored near the communications cable, and the divers locked out of the torpedo-shaped, aft-mounted diving chamber near the boat's rudder. As they had done during practice runs off California, one diver remained inside, while three divers walked in moonlike fashion to one side of the Halibut. Reflective plankton clouds obstructed their vision, while the HeO2 mix made their tongues dry.\n\nThe divers opened the sealed compartment on the Halibut's hull and removed the long electrical cord. They dragged the cord over to the communications cable and located the repeater, a large metallic cylinder that joined sections of the cable every thirty miles and amplified the signals. The spooks knew that the strongest signals could be found there. While attaching the three-foot-long \"grabber,\" which contained a recorder and rolls of recording tape, a large hake fish, attracted by the light, attacked one diver and clamped its razorlike teeth on his arm. Unable to shake the fish loose, the diver removed his knife and jabbed it between the fish's gills. The wounded hake swam away in a flurry of ink.\n\nWhile returning to the Halibut, the divers collected some large crabs for a \"mission accomplished\" dinner. Inside the boat, in the sectioned-off chart room, three spooks at a time stood watch to accomplish phase one of the Ivy Bells tapping mission. This part, which employed a temporarily placed cable tap, required that the Halibut remain on station for several weeks while the spooks recorded signals. The incoming signals needed to be calibrated against data type, time of recording, signal strength, tapped wire (within the bundle), signal anomalies, and to ensure complete continuity and to validate recorded voice and data quality. Unlike the two previous failed missions, two sets of signals were now recorded. One was a \"capture all\" that contained every signal pulled from the cable, and the other separated out each channel to ensure clear conversations could be heard. The team took this function seriously, as hundreds of men were placing their lives at risk to gather the data, and they had but one shot at getting it right. After the mission, the recording tapes would be delivered to the NSA for full transcribing and analysis\u2014the second phase of the Ivy Bells project. That is, of course, if they managed to capture anything at all.\n\nWhile the divers set up the tap, M-Brancher Mac Empey and T-Brancher Mark Rutherford waited patiently with Lieutenant Commander Arnold in the chart room, a small converted storeroom forward of the reactor and opposite the radio room. Once the divers finished clamping the cable, the spooks tried in vain to collect the signals from the tap but heard nothing. They flipped switches and rechecked systems, but to no avail. They requested that the divers return to the cable and check the connection. The divers did and found that they'd attached the tap to a \"dead\" shielded location. They reattached the grabber to an active, unshielded spot, and, using induction, the tap picked up \"leaked\" signals from the cable and sent them up the wire into the spook's gear.\n\nEmpey and Rutherford collected the information and validated parameters, while an I-Brancher strapped on a set of headphones and listened. To his amazement, clear, unencrypted Russian voices filled his ears. He handed a set of phones to one of the spook chiefs, chuckled, and said, \"I've got a Soviet admiral talking to his wife on one line and his mistress on the other.\" Only the I-Brancher could translate the words, so the others took turns listening to the private conversations and imagined other Soviet admirals exchanging top-secret information while sitting at desks in Vladivostok. In essence, the United States had just placed a glass against the Soviet Union's wall to hear their every word.\n\nReminiscent of a professional baseball team, basking in the glory of winning the World Series, the team of five spooks gathered in the chart room and exchanged excited high fives. The Halibut remained on station for another two weeks, filling reels of tape with recordings. Near their departure time, a flooding alarm sounded. A diesel engine pipe had ruptured, sending a gush of ocean water into the boat. The divers, though, were still outside and unaware of the emergency. The Halibut's crew raced to seal up the leak, but freezing cold water numbed fingers and arms as they worked. Commander McNish knew that the extra water made his sub heavy, and that he'd have to do an emergency blow soon to save his boat. He ordered the divers to return to the diving chamber but feared they might not make it back in time.\n\nThe salt water rushed in, and McNish now had to make the most difficult decision of his career: blow to the surface to save his sub and leave the divers to die, or wait for the divers and risk the lives of his entire crew. Seconds before he gave the order to blow, the damage control team reported that they'd fixed the leak. McNish breathed a sigh of relief. After the divers were safely inside, he moved the Halibut out of the area. The submarine returned home after a short stop in Guam for repairs. Though the ride back to port felt even longer than the one out, the crew sang songs and ate steaks in the crew's mess with the knowledge that their efforts were about to make a significant difference in the duration and outcome of the Cold War.\n\nThe Halibut returned to the Sea of Okhotsk a month later and came back with another batch of juicy recordings. Emboldened by Halibut's back-to-back successes, Captain James Bradley asked Bell Labs to build the mother of all cable taps. This device also worked through induction but occupied a twenty-foot-long by three-foot-wide container that weighed six tons and could be left on station indefinitely. That would obviate the need for a submarine to remain in the area for weeks. NSA engineers designed sophisticated new recording devices for the pod that could monitor dozens of lines for months at a time. The complex watertight enclosure contained an electronically programmed system for registering the intercepted information, nearly one hundred multichannel magnetic recording units and a miniature plutonium 238 nuclear power source. Divers nicknamed the heavy contraption, which looked like an elongated fifty-five-gallon drum, the \"beast.\"\n\nBy 1976, the Halibut was ready to be put out to pasture. Two other submarines lined up to take her place, the USS Seawolf and USS Parche (SSN-683) but both were in drydock undergoing repairs and modifications. David LeJeune and most of the divers on the Halibut packed their seabags and reported to the Seawolf. Mac Empey, Mark Rutherford, and the other spooks did likewise. As they had done on the Halibut, the team endured months of simulated practice off the coast of California, and the dangers involved in these dress rehearsals were no different than the real thing.\n\nThe navy kept the Silicon Valley\u2013built pod-tapping beast on a barge in San Diego. They also kept a nonworking practice beast there made from a large metal drum filled with cement. The 1,700-pound dummy pod, also twenty feet long by three feet wide, came packed in a canvas bag encircled by lifting balloons to allow the divers to move the heavy object around on the ocean floor. Navy diver David LeJeune accompanied an NSA operative named Ronald Pelton on a drive in his brand-new Volkswagen van down the coast to pick up the barge with the fake beast and prep everything for a practice run. LeJeune recalled that one of the divers on the last dry run had overinflated the beast. The massive thing rocketed to the surface and washed up on a beach near Santa Barbara. Fortunately, the civilians who discovered the practice beast had no clue as to its function.\n\nDuring their pod-prepping trips, LeJeune got to know the charismatic NSA man named Pelton over beers and even started to consider the guy as a potential friend. He had no clue at the time that Ronald Pelton would one day place the lives of submariners and divers conducting Ivy Bells missions in ultimate jeopardy through an act of treason.\n\nLeJeune and Pelton delivered the fake beast to the Seawolf, and the crew stored the pod in a large locker on the outside hull of the submarine. As far as LeJeune was concerned, the Seawolf looked like the Waldorf Astoria compared to the Halibut. Instead of a fake DSRV, she had a shipyard-installed fifty-foot section just aft of the torpedo room designed for diving operations and unmanned aquatic vehicle (UAV) Fish deployment. Divers called the section Area D, for \"Diving.\" The space contained a twenty-by-ten-foot hyperbaric diving chamber, similar in function and setup to the DSRV-style chamber welded to the back of the Halibut. Inside were four cots, a \"piss bucket\" toilet, and a single washbasin. The chamber snuggled up against the inside bulkhead of the submarine. Above the divers' heads, angled upward and outward, an inner hatch offered an exit. An outer hatch beyond that, similar to ones used in submarine escape trunks, opened outward to the sea. When the Seawolf surfaced, this hatch resided about fifteen feet below the waterline, so it could not be seen by spying eyes.\n\nNear the diving chamber, UAV Fish were stored in small racks, similar to the ones used to house torpedoes. A five-foot-diameter tube running through the pressure hull, resembling a torpedo tube, pointed upward toward the ocean. As an added measure of safety, the tube contained three hatches, one inside, one midway in the tube, and one outside to hold back the sea. Operators loaded the Fish into these tubes, opened all three hatches, and jettisoned the mini-subs into the deep blue yonder to snoop around for things, such as Soviet communications cables.\n\nThe spook's chart room, chockful of electronic eavesdropping equipment, was just forward and dog-legged from the Area D hyperbaric chamber and UAV Fish. Although LeJeune met several of the spooks on the Halibut, he didn't really get to know Mac Empey until his time aboard the Seawolf. LeJeune describes Empey as \"having Einstein hair, a scraggly beard, and maybe two good teeth with cigarette stains.\"\n\nEmpey cornered LeJeune near the chart room after a practice run and said, \"Howdy! I'm Whacko Maco. Want a tour?\" LeJeune's first impression was that Empey \"couldn't pour the piss out of a boot if instructions were written on the heel.\" Born curious, however, LeJeune accepted the invitation.\n\nEmpey insisted that LeJeune first needed to watch an indoctrination video before being briefed on any of the equipment in the chart room. LeJeune agreed. Empey pulled back the curtain on the top-secret area and ushered LeJeune inside. The small space contained only two chairs and more than two dozen rack-mounted pieces of equipment that hummed and whirred. A large monitor overshadowed a keyboard.\n\nEmpey sat LeJeune down in a chair and said, \"This is an important video. Pay close attention to every detail.\"\n\nThe spook pressed a button to start the video. LeJeune's eyes extended with anticipation as he wondered what secrets were about to be revealed. Empey stepped outside and closed the curtain. As the film's title rolled past, LeJeune realized he'd been had. He heard Empey giggling outside the curtain about the same time as Linda Lovelace's name appeared on the credits of the film.\n\nIn spite of Empey's warped sense of humor, LeJeune came to respect the rotund man's brilliance. He possessed no degrees, but God had granted the man Albert Einstein's IQ, along with the renowned scientist's tousled hair. He could multitask like a chess master and chain-smoke like a shipyard worker. When dozen's of engineering-degreed contractors couldn't find a problem with a critical piece of equipment, with the entire $150 million Ivy Bells program on the line, Empey saved the day. The engineers rolled their eyes when the chief asked for the schematic and a description of the problem. An hour later, with eyes darting and tongue wagging, the brilliant spook pointed to a spot on the drawing and said, \"There. That's where it is.\" The engineers didn't believe him but checked anyway. Turns out Empey was right, so much so that he received a Legion of Merit from the deputy director of the CIA.\n\nWhile the Seawolf's crew prepared for their first cable-tapping mission to the Sea of Okhotsk, where they would set up the new beast pod, others honed their clandestine skills, unaware that their destinies would soon collide with those aboard the infamous Wolf.\n\n## CHAPTER FIFTEEN\n\nAll wars are civil wars, because all men are brothers.\n\n\u2014FRAN\u00c7OIS F\u00c9NELON\n\nWHEN THE SOVIETDELTA-CLASS SUBMARINE ENTERED stage right in December 1972, the U.S. Navy shuddered. With a quiver of twelve R-29 ballistic missiles, each capable of delivering megatons of destruction from almost 5,000 miles away, the Delta made U.S. ASW techniques all but obsolete. Prior to the Delta's arrival, Soviet \"boomers\" with shorter-range missiles, in order to get close enough to the United States to launch, needed to transit from Murmansk, curve around Norway, and drop down through the Greenland\/Iceland\/U.K. gap. A swarm of American ships and planes greeted them there, as well as several fast-attack boats ready to latch on like trained bulldogs. With the advent of the Delta class, the Soviets could employ a new strategy: slip under the Arctic ice north of Murmansk and hide for months on end. When and if the order came to fire, they could sneak from under the ice cap, come shallow, and launch. By the time a U.S. submarine found them, it would be too late.\n\nThe Delta class rivaled America's Benjamin Franklin\u2013class boomers in every way. In fact, the United States had actually fallen behind in this area, having not launched a new ballistic missile submarine since the USS Will Rogers (SSBN-659) hit the water in April 1967. Fortunately, U.S. attack submarines were a bit more advanced, but these capable work horses were ill equipped to find Delta submarines hiding under an Arctic ice floe, and no acoustic or signals intelligence yet existed on the Soviet's new boat. Without that intelligence, finding a Delta in open waters posed a problem.\n\nThe issue of locating these \"ice boats\" consumed the navy and the NSA, but they were not completely caught by surprise. Anticipating such an advancement from the Soviets, the navy had launched a new ASW program in 1964 that gave the Classic Bulls Eye program some stiff competition for funding. Under the direction of Vice Admiral Charles B. Martell, the Long Range Acoustic Propagation Project (LRAPP) sought to bring heretofore scattered ASW technologies and efforts, including SOSUS, under a single roof. Employing only a few scientists and requiring modest funds in comparison to other programs, LRAPP conducted scientific experiments and exercises that teamed top ASW civilian minds with navy counterparts. Unfortunately, all of the initial experiments failed. The brass threatened to cancel LRAPP until Dr. Marvin Lasky at the Office of Naval Research saved the day.\n\nLasky helped deploy a miles-long string of acoustic hydrophones towed by a ship or submarine. He called this invention the Interim Towed Array Surveillance System (ITASS), which was, essentially, a movable SOSUS sonar array. The experiments with ITASS that followed proved critical in finding a way to detect Soviet submarines in northern waters near Murmansk. During the height of the program, two opposing scientific camps stood their ground. The LRAPP scientists believed that sound in the ocean traveled in a straight line, or directionally. This meant that sound characteristics would differ depending on where they originated, and submarines could be detected from very far away. Others, including a team of AT&T scientists, argued that noise didn't travel in straight lines, and subs could only be heard if they were not too distant.\n\nCurrent SOSUS arrays in the Atlantic could not detect Deltas in the Arctic. The navy wanted to extend SOSUS into the North Atlantic to fix this problem. AT&T scientists, convinced that their \"omnidirectional\" sound viewpoint was correct, insisted that this move would be a waste of money. SOSUS arrays wouldn't be close enough to the Arctic to do the job. The LRAPP engineers thought otherwise and challenged AT&T to a duel, with the stakes being the demise of the $180 million SOSUS expansion. If the AT&T scientists were right, the project was doomed. If the LRAPP scientists were correct, SOSUS lived. Moreover, future ASW capabilities could be dramatically improved, tipping the scale back in America's direction.\n\nAfter a two-week experiment in which ships towed the new ITASS arrays, the LRAPP team took home the gold. They proved that sound propagated directionally. This led to the permanent installation of SOSUS in the North Atlantic and the development of ITASS arrays that could be towed by attack submarines. Soviet vessels could now be detected from far greater distances than ever before. The navy also ordered operational and procedural changes in the world of antisubmarine warfare and increased funding for ICEX Arctic experiments conducted on a tiny ice floe north of Alaska.\n\nSonar, weapons, and other systems on submarines operate differently in Arctic conditions versus open ocean areas. To better understand these dynamics, every two years, the navy sent two fast-attack submarines to a moving ice floe in the Arctic, a few hundred miles north of Prudhoe Bay, Alaska. Over a two-week span in the warmer month of March, a team of more than sixty naval personnel, along with civilians from the Arctic Submarine Laboratory and Applied Physics Laboratory, teamed with support personnel and a few cooks to build a temporary base camp for Operation ICEX. Two attack boats spent two weeks conducting experiments on new sonar and communications systems designed for Arctic conditions and firing practice torpedoes at each other. Scientists monitored results and used these findings to improve system and weapon designs for Arctic use.\n\nThese improved designs found their way into a new class of U.S submarines originally launched on March 3, 1967. These Sturgeon-class attack subs boasted vast improvements in quieting technology, coupled with the latest espionage gear. The Soviets had nothing in their attack submarine deck that compared. By 1973, after the success of the LRAPP experiments, the navy expanded SOSUS to twenty-two installations in the Atlantic and Pacific.\n\nAmerican engineers leveraged ITASS and HFDF technologies to design towed sonar arrays and miniaturized versions of Boresight\/Bulls Eye detection and DF technologies that could be installed on U.S. ships under the Classic Outboard program. These smaller HFDF systems also found their way onto Permit (formerly Thresher) and Sturgeon-class submarines as integral parts of BRD-6 and-7 ESM systems. Now when Ivan transmitted a burst signal, lit off a radar, turned a screw, or farted into the wind, an American T-Brancher or sonar tech could catch the scent.\n\nIN EARLY 1973, COMMUNICATIONS TECHNICIAN First Class Frank Turban walked across the brow of the Sturgeon-class submarine Flying Fish (SSN-673) in Norfolk, Virginia. As he stood on the cold deck along with a dozen other spooks, his toes tingled\u2014a sure sign that this mission would be an exciting one. The third submarine to bear the name of the soaring tropical fish with long winglike fins, the Flying Fish was fortunate to have Commander J. D. Williams as her commanding officer. That Turban would earn a Navy Commendation Medal on this run while helping Williams earn a Distinguished Service Medal never crossed his mind.\n\nA New York native, Turban dropped out of high school twenty-one days after his seventeenth birthday on August 12, 1963. Escaping the bitterness of a home soured by alcoholic parents, along with the possibility of an army \"want you for Vietnam\" draft notice, Turban took a cab to the navy recruiters office in Brooklyn. He and twenty other excited kids signed up that day to see the world and experience the navy's promise of an \"adventure and not just a job.\" Even without a high school diploma, Turban scored high enough on his tests for a shot at communications technician school, but with no guarantees.\n\nAfter boot camp, the navy sent Turban to Pensacola, Florida, for CT and T-Brancher schools. Branded with the unofficial title of \"spook,\" sporting a brown beard, and weighing 205 pounds, he reported to his first duty station at the Pentagon in Washington, D.C. There he worked as a communicator, operating crypto equipment and handling classified messages for the Chief of Naval Operations (CNO) office. Most messages were destined for lower-ranking officers and automatically decrypted, but others were eyes-only for flag officers and came in \"off-line.\" Messages of this nature were usually considered \"hot shit\" and required manual decryption and personal delivery. A few years after his stint at the Pentagon, Turban got orders to NSGA (Naval Security Group Command) in Kami Seya, Japan, where he did a spy run on the USS Swordfish not long after K-129 went down.\n\nOn his first day aboard the Flying Fish, Turban met Mark Rutherford, an M-Brancher spook who'd just completed a top-secret mission on the USS Halibut in the Sea of Okhotsk. Rutherford never said a word about that mission, and Turban didn't probe. As an expert on the BRD-7, Rutherford helped train the spooks on this new gear. Sanders Associates built the BRD-7, and when coupled with two pieces of equipment called Blue Surf and Blue Gill, the integrated system represented the next generation in ESM, along with advanced burst-transmission direction finding and fingerprinting. Turban had been trained on the BRD-6, also built by Sanders, and exposed to the BRD-7, but Rutherford knew the system well enough to preach Sunday sermons.\n\nThe BRD-7 captured Soviet burst transmissions and fed them to the Blue Surf\/Blue Gill equipment that decoded the tiny data streams that William J. Reed found years earlier in Turkey. The system could not decrypt the information, but built-in transmitter fingerprinting allowed operators to determine the type of platform or, in some cases, the class of vessel transmitting. Unlike stationary Bulls Eye stations, most contacts obtained by the BRD-7 were close enough that sprinting down the bearing line could eventually yield a sonar or visual sighting. To obtain these accurate bearing fixes on remote targets required submarines to capture multiple hits from different bearings by frequently changing course and sprinting to new locations. Given that Soviet subs did not often transmit using the burst signal, such an operation could take days.\n\nWith Commander Williams calling the shots in 1973, the Flying Fish glided across the Atlantic and crept into the cold waters of the Barents Sea. Once on station, Williams ordered all quiet and popped up the number two periscope for a peak. Turban sat in the spook shack section of the radio room and waited, his thick fingers tapping nervously on his knees. Though he'd conducted a few SpecOps before, this was his first one near the infamous city of Murmansk. Every submariner knew that the location of this port, tucked deep inside the Kola Bay in the far northwest corner of Russia, translated into danger times two. As headquarters for the Soviet Northern Fleet, more submarines and warships entered and exited this port than any other. Remaining undetected while hunting Ivan in these waters was akin to playing hide-and-seek in a small backyard with only a couple of trees to hide behind.\n\nAfter several days on station, Turban picked up a strange signal on the BRD-7 he'd never heard before. He sat up in his seat and listened. The signature resembled a Yankee-class ballistic missile submarine, but not quite. He wondered if this could be a new type of boat altogether. He keyed his microphone and reported the contact to Commander Williams.\n\nThe Flying Fish sprinted toward the unknown contact. Hours later, Williams slowed to a crawl and brought the boat to periscope depth. He raised the number two scope and peered at the image of an odd-looking submarine resting on the surface. Draping his arms over the scope handles, he let out a long whistle. The contact, designated Master One, looked like a Yankee, but with a larger two-step, humped missile compartment behind the sail. Williams reeled off dozens of photos with the 70 mm camera housed in the periscope, then ordered the \"under hull\" team to report to the control room.\n\nThe team consisted of a few handpicked individuals that Williams had \"certified\" capable of conducting a periscope visual inspection under the hull of a foreign vessel, without destroying the Flying Fish as a consequence. Under-hull runs were considered borderline suicidal, given that submarines needed to descend below a Soviet vessel and pass underneath with video cameras reeling while the periscope looked upward. Only a few feet separated the two boats. One wrong twitch, a sneeze, or a glance away from a dial at a critical time could send the Flying Fish flying into the underside of the Soviet craft. The impact would be heard in Moscow, and the damage might be bad enough to end a mission, perhaps permanently.\n\nThe under-hull team relieved the current watchstanders, while Williams remained glued to the periscope. He glanced away from the scope long enough to call into the radio room, where Turban and the spooks sat behind a curtain in their restricted area. He asked if they had any ideas about the class of this new sub. No one had a clue. He repeated the question to sonar. Since the target remained stationary, they could hear only occasional machinery noises\u2014not enough to determine platform type. Williams had a strong suspicion they were looking at the Soviet's new Delta-class submarine. If so, the Flying Fish would be the first to collect data on this type of boat. Every submariner dreamed that such a find would come along at least once in his career.\n\nTurban had recently read reports that the Soviets churned out the first Project 667B Delta I, designated K-279, in December 1972. As yet, no American submarine had encountered a Delta or captured any of her noise or signals information. An upgrade of the Yankee class, the new sub resembled an American \"boomer\" Polaris missile boat but carried twelve versus sixteen ballistic missiles in her tubes. These medium-range R-29 projectiles granted the Soviet Union the ability to launch from almost 5,000 miles away, making the Delta a far greater threat than her predecessors.\n\nWilliams gave the under-hull order, and the boat crept downward. The dark hull of the Delta loomed large through the scope like the underbelly of a sleeping killer whale. Turban and the other spooks could see what the skipper saw via the periscope-mounted video camera that broadcast to several monitors in the boat through a system called PeriViz, for Periscope Visual. As the Flying Fish passed beneath her prey, barely yards away, rough edges on the Delta rushed past, followed by sealed openings and meshed grates. When the scope focused on one of the grates, something shiny flashed. Williams zoomed in on the object, revealing a cylinder about three inches in diameter by five inches long. Some sort of container had wedged itself into one of the grates. One of the I-Branchers in the radio room read aloud the Russian writing painted across the item\u2014a product label belonging to a popular soda pop manufacturer. In essence, they'd just found a Coke can.\n\nUpon completion of the under hull, Williams circled the Delta and took more shots with the 70 mm and \"snapped-on\" 35 mm cameras. Sonar recorded stationary reactor noises, while Turban and company cataloged signals emanating from the masts. Williams moved out to 2,000 yards and ordered the weapons officer to ready tube numbers one and two, which were both loaded with MK-48 torpedoes. He verbally confirmed a few bearings while focused on the Delta through the periscope. Sailors sitting on the benches in front of the fire control computers dialed in the bearings.\n\nThe weapons officer glanced at Williams and said, \"I have a curve and a good solution on the target. Range two thousand yards, target speed zero knots.\"\n\n\"Attention in the conn,\" Williams said. \"Firing point procedures tubes one and two. Horizontal salvo, two degree offset, two-minute interval.\"\n\n\"Weapons ready,\" the weapons officer said.\n\n\"Firing point procedures,\" Williams said.\n\nA fire-control petty officer sitting on a blue bench in front of a large panel rested a hand on the firing key. With a single order from Williams, the Delta would be transformed into a twisted metal coffin. In the spook shack, Frank Turban rubbed his palms together. He was surprised at how dry they were. He thought that coming this close to sinking a Soviet submarine would cause him to sweat like a boxer, that he'd feel something: fear, anticipation, perhaps even sympathy for the submariners on the other boat. Surprisingly, with his mind focused on his job, he felt nothing but a sensation not too unlike the hum of an electric fence in a light rain. He wondered if that hum would turn into a jolt if they actually fired two torpedoes up the ass of the Delta.\n\nWilliams peeled away from the periscope. He glanced at the fire-control party and said, \"Stand down.\" With the firing exercise concluded, he ordered the diving officer to bring the Flying Fish in close, real close. Through the high-power lens in the number two scope, he snapped pictures of Soviet submariners wearing traditional Ushanka hats on the bridge, framed against a frigid gray smattering of clouds.\n\nIn the spook area, via the PeriViz, Turban watched the Russians light up cigarettes on the bridge of the Delta while carrying on an animated conversation. One of the sailors stopped, turned, and stared right at him. Turban held his breath as he stared right back into the sailor's eyes and saw clearly the questioning look of shock on the man's face. The Soviet sailor raised his hand and pointed right at Turban's nose. Turban knew that the Russian wasn't looking at him directly but had spotted the Flying Fish's periscope. Williams slammed down the handles on the scope and lowered the cylinder into its well. He ordered a deep dive, and the Flying Fish angled down thirty degrees as she made a rapid descent and sprinted away.\n\nWilliams later came shallow again, raised the scope, and afforded the crew the show of a lifetime through PeriViz\u2014complete with popcorn in the crew's mess. The Soviet Kresta I\u2013class cruiser Sevastopol lit off everything she had, including her Big Net, Head Net-C, and Plinth Net radar. She readied her ten torpedo tubes and started spinning the blades on her Ka-25 Hormone helicopter. Other ships and aircraft came onto the scene with equal ferocity. Given all the action going on, Turban thought for sure that Williams would move the Flying Fish outside Soviet territorial waters. Instead, the CO stayed put. Turban knew that Sturgeon-class boats were quiet and hard to find and figured that Commander Williams was about to prove that theory correct.\n\nTU-95 helicopters buzzed into the area and dropped sonobuoys. The active sonar pings were loud enough to be heard through the hull. A brand-new Krivak-class frigate approached, and the spooks got excited when she lit off her Spin Trough, Eye Bowl, Kite Screech, and Pop Group radars. The more data they could collect on these signals, the better. The linguist I-Branchers were having the most fun, listening to open transmissions between the ships and aircraft as they prosecuted the Flying Fish.\n\nOne I-Brancher, Phil Cafrey, stayed well past his watch-relief time and refused to turn over his headset. His eyes wide, his tongue wagging, he listened and translated like a madman possessed. Finally, after eight straight hours, he had to relieve himself. He received permission from the XO to use the nearby stateroom head and ran out of the radio room. In his rush to finish up and get back to his headphones, he missed seeing the large sign hanging on the stall door in the head. From across the passageway, Turban heard Cafrey scream in agony.\n\nAll submarines keep sanitary waste in large tanks that periodically need to be blown to sea. When this occurs, the tanks are pressurized with air to flush the tainted liquid out to the ocean. Large signs are placed in the heads warning sailors not to pull the flapper valves and flush toilets while sanitaries are being blown. To do so launches the pressurized air, along with a spray of waste, back up through the toilet opening and right into the face of the flusher. Cafrey missed seeing the sign and so pulled the flapper. After the screaming stopped, it took him a half hour to wash the stench from his eyes, nose, and body. From that day on, the crew affectionately called him Flapper Phil.\n\nThrough PeriViz monitors, the crew of the Flying Fish watched the procession of Soviet ships prowl back and forth. Radio operators received reports from ground stations that they were hearing \"stutter nine\" transmissions from the Soviets, consisting of all nines in the first group of transmitted characters. This translated to unidentified invader\/contact and almost always meant that the Soviet navy was searching for a U.S. submarine.\n\nThree days later, after all the fun had died down, the Flying Fish snuck away and returned to Norfolk so the NSA could study the photos of the new Delta I. Six weeks later, Turban got a call from Commander J. D. Williams. The CO wanted to know if Turban would like to ride again on the Flying Fish for another northern run. Turban jumped at the opportunity. Three other spooks from the former group of twelve also volunteered, including Mark Rutherford. Turban soon found himself back in cold waters near the North Cape\u2014which is the point where the Norwegian Sea, part of the Atlantic Ocean, meets the Barents Sea, part of the Arctic Ocean. Located on the island of Mager\u00f8ya in northern Norway, the North Cape is not far from Oslo, Norway, where William J. Reed helped set up a Boresight station.\n\nThe Flying Fish sauntered back to the Barents Sea, only this time the SpecOp bordered on boring for the first few weeks. Turban started to wonder why he'd volunteered but then recalled that after the Delta-finding run, he'd made up his mind to follow Commander Williams just about anywhere. Another day passed before Turban got a sniff from a Kresta II cruiser, which carried a new type of antisubmarine missile known as the SS-N-14. Accurate and deadly, the N-14 could rip a hole right through the Flying Fish within seconds. Turban had no desire to tangle with a Kresta II, but the ship continued to transmit to fleet HQ on a frequent basis. The type and frequency of the transmissions seemed odd. Based on a hunch, Turban started scanning the area for known submarine frequencies on the BRD-7. Nothing came up at first, then a faint trace.\n\nTurban almost wrote off the whisper but instead closed his eyes and concentrated. Out of the static came a familiar sound, the scratchy trademark of the burst signal that William J. Reed had heard more than a decade earlier while in Turkey. Too faint to trigger a detection alarm, the trace signal would have gone unnoticed if Turban hadn't become suspicious of the Kresta's repeated transmissions. While analyzing the signal, a neon bulb flashed in Turban's head. He immediately called Williams and explained his theory.\n\n\"A hurt Yankee?\" Williams said as he stood in the tight radio room.\n\n\"Sure looks that way,\" Turban said.\n\n\"How so?\"\n\n\"The Yankee sends a short burst signal to the Kresta. Right after that, the Kresta broadcasts a slightly longer message to HQ. That leads me to believe that the Yankee's not feeling well and needs the Kresta to help her out.\"\n\n\"How do we know where to find her?\"\n\nTurban explained that while T-Branchers, I-Branchers, and R-Branchers were all considered \"spooks,\" each received entirely different training related to linguistics, radio, radar, and communications signals. T-Branchers, like Turban, were highly trained in Soviet communications, signals, and procedures. Most of the time a T-Brancher could listen to a transmission and didn't have to measure anything to know the type of signal, its location, and its purpose. In this case, Turban knew they were listening to a Yankee submarine in the general vicinity of the transmitting Kresta II. To help pinpoint that location, Turban needed Williams to sprint to at least two other locations so they could multiangulate a bearing fix.\n\nWilliams nodded and hurried back to the control room. The Sturgeon-class submarine made a turn and ran for fifteen minutes to a new hole in the ocean. Turban took bearings before Williams cranked up the propeller and drove the Flying Fish to another spot, also fifteen minutes away. Turban took more bearings. Satisfied that he had a descent fix, he gave Williams the estimated bearing and range to the new contact.\n\nThe Flying Fish snuck into the area, whereupon Williams jutted the periscope above the waves for a look. Turban watched on the PeriViz. In the distance he saw smoke billowing from the submarine's sail and open hatches. Sailors darted about the deck in emergency fashion as their Yankee-class boat sat motionless upon the ocean. Gray clouds framed the injured submarine as a light rain splattered the deck.\n\nTurban gazed at the bulging shape of the Yankee in the monitor. He recalled how he'd felt when the Flying Fish cornered the Delta and practiced an approach to fire two torpedoes at her. Save for a low-level undercurrent of excitement, not much emotion flowed through him then. Now, something different welled up inside. Something profound. That most Americans considered these endangered sailors the enemy did not diminish the sympathy that consumed Turban as he wondered how many on board had been hurt or killed. But the Flying Fish could do nothing to assist.\n\nFortunately, other Soviet ships arrived and provided assistance as the Flying Fish lowered her scope and snuck away into the dark depths of the ocean.\n\nUpon returning to Norfolk, Turban learned that the Yankee submarine had endured a serious fire and that some of the crew had succumbed to smoke inhalation. For reasons Turban could not put into words, he felt a deep connection to the fallen. In that moment of grief, those Soviet sailors no longer represented an enemy to be feared or hated. They were fellow submariners and vulnerable in the end, just like him, comrades who shared the unique sacrifices required to work under the sea. Regardless of how cold the war between the superpowers became, Turban would always share a bond with those sailors that only those qualified in submarines could know.\n\nFor his achievements in underhulling the Delta I, and for locating the distressed Yankee, Commander J. D. Williams received a Distinguished Service Medal. Frank Turban earned his first Navy Commendation Medal, and he and Mark Rutherford gained enough notoriety within the NSA to warrant E-ticket rides on two submarines that conducted the most harrowing missions of the Cold War.\n\n## CHAPTER SIXTEEN\n\nIf we weren't all crazy, we'd just go insane.\n\n\u2014JIMMY BUFFETT\n\nBY LATE 1975, SOME AMERICANS HATED sailors. They also hated airmen, marines, and soldiers. The last helicopter flew away from Saigon on April 30, and long-haired protesters uncapped their Magic Markers to create new signs. stop the war changed to ban the nukes and down with the establishment. Having just enlisted in the navy that summer, I became an officially sanctioned target of the hippies. Almost overnight, friends turned into enemies. Did I not know that the warmongers had fueled a war that cost the lives of 1.3 million Vietnamese and 56,000 Americans? And what about the $584 billion the United States had spent on this meaningless conflict? They asked if I'd lost my mind, had perhaps become as psychotic as Jack Nicholson in One Flew Over the Cuckoo's Nest.\n\nI gave little heed to the barbs. I knew things most could not know, perhaps didn't want to know. By then, my father had told me a few hair-raising stories, under an oath to secrecy, about his part in the not-so-cold underwater war against the Soviets. Having enjoyed his years working with the silent service, he suggested that I consider joining the ranks of those committed to finding, and if need be, destroying the new enemy subs entering the fight. What he didn't tell me was that becoming a spy of the deep required learning how to live for months on end without the sun on my face, the love of a woman, or freedom from fear. I think he knew that I'd have to learn these lessons on my own.\n\nMy father had retired from the navy years earlier and embarked on his new career as a writer by penning an award-winning biography of the painter Olaf Wieghorst. Olaf's illustrious life included stowing away on a ship from Europe, working for the mounted police in New York, and chasing Pancho Villa along the Mexican border. Olaf's paintings depicted the Old West with such vivid realism that you could almost taste the dust. His sought-after masterpieces commanded high bids and hung on the walls of the wealthy and notable, including those of President Eisenhower and Senator Barry Goldwater, who wrote the foreword to my dad's book. My father took me to visit Olaf's studio in San Diego, where the humble master showed me authentic Indian head-dresses and vintage frontier rifles. I told Olaf that I intended to follow in my father's footsteps and join the navy. Olaf said that was admirable but encouraged me not to volunteer for submarines. As an outdoors-man, the very thought of living in a sewer pipe made him cringe.\n\nIgnoring Olaf's admonition, I volunteered for submarines and began my adventure on October 21, 1975, ten days after watching the first episode of Saturday Night Live, with George Carlin hosting. That night I stepped off a bus and walked through the gates of the Naval Training Center in San Diego. The navy stripped me of my clothes and ego and over the next eight weeks molded me into a sailor.\n\nAfter boot camp and six weeks of basic electronics school, I crossed the country to complete submarine school and almost eighteen months of fire-control-technician weapons-systems training at the submarine base in Groton, Connecticut. In the summer of 1977, six months after Jimmy Carter pardoned the Vietnam War draft evaders, the navy promoted me to petty officer second class and flew me to the Philippines to join the crew of the USS Haddo. Sonarman Petty Officer Second Class Kenneth \"Greenie\" Greenawald, also destined for the Haddo, flew with me. A medium-built, gregarious guy with an infectious smile and bushy mustache, Greenawald had traveled cross-country from his hometown of Pittsburgh, Pennsylvania, and by now looked like the male version of a mistreated Raggedy Ann doll.\n\nThe Haddo had just completed a top-secret operation, and the sight of her smooth hull tied up next to the pier in Subic Bay in the Philippines filled me with an equal dose of excitement and dread. As I approached the mostly submerged craft, I noticed that the hull's black steel did not appear at all smooth. The surface contained brick-sized chunks of black rubberized material. I recalled from my sub school training that the material consisted of anechoic tiles, designed to absorb enemy sonar beams. At the time, I didn't give that detail much thought. Weeks later, when we were dodging a bunch of Soviet warships, I praised the genius designers of this technology.\n\nGreenawald and I walked across the brow of our new Permit-class submarine and handed a copy of our orders to the topside watch. A warm tropical wind chopped the ocean into wavelets that slapped against the boat and left an oily film. After checking our paperwork, the topside watch pointed toward an open hatch. At the edge of the small hole, I peered downward. The heavy scent of nuclear submarine filled my nose. To this day, I've never forgotten that stench. I learned later that since submarine air systems were \"closed loops,\" the scents generated by everything mechanical, chemical, and personal lingered forever. No amount of ventilating could ever scrub away the acrid, oily, fuel-laden, mechanical, electronic, fart-filled odors that could only be called \"the boat smell.\"\n\nThe Haddo's crew consisted of just over one hundred enlisted men and a dozen officers. Commander Norman Mims Jr. served as our CO. He was a seasoned veteran with gray hair and had been on a few diesel boats before progressing to CO of the Haddo. Under him, our executive officer projected the haughtiness of an upwardly mobile, by-the-book leader who lived in constant fear of making a mistake.\n\nLike all submarines, our command consisted of several divisions, including engineering (nuclear and machinist disciplines), navigation, sonar, fire control (weapons), and other operational functions. My division chief, FTGC (Chief Fire Control Technician\u2013Gunnery) James T. Lane, took charge of a small team of two fire-control technicians\u2014and now three with my arrival. Lane worked hard and played hard, and had a wide gap between his front teeth, which he flashed often after cracking dirty jokes. One of the other guys on my team included FTG1 (First Class Fire Controlman) Tim Vinson, the leading petty officer (LPO) of our division.\n\nVinson, a wiry guy from Florida, stood about my height, five feet ten inches, and had slightly thinning hair that he tried to comb forward enough to cover the bald spots. He had a playful, infectious sense of humor that endeared him to the crew. You never knew when to take him seriously. One minute he'd have you engrossed in a serious conversation about intricate technical details, and then suddenly you'd realize the whole thing had been an elaborate ruse.\n\nVinson gave me my first tour of the boat. In the control room, we moved past the Mark-113 fire-control system panels, around the periscope stand, and over to the ballast control panel. Vinson pointed to a panel labeled MBT VENTS. MBT stood for main ballast tanks. Underneath the panel, several small squares were labeled with various numbers and designators. Underneath these squares, seven smaller squares had lighted round circles in the middle, and underneath these, another seven small squares had nonlighted rectangular lines. My head swam with confusion, and I wondered how anyone could possibly memorize the functions of all those controls.\n\n\"What happens if our main ballast tanks fail in an emergency?\" I asked.\n\n\"Simple,\" Vinson said. \"We die.\"\n\nThat same afternoon, Chief Jimmy Lane handed me a qualification card\u2014a two-sided, eight-and-a-half-by-eleven-inch off-white sheet filled with dozens of check boxes next to every major system, valve, pipe, or piece of equipment essential to submarine operation. Completing my \"qual card,\" Chief Lane told me, could someday save everyone's life in an emergency.\n\n\"Suppose you're the only guy in the compartment when there's a big ol' fire,\" Lane drawled. \"You do the wrong thing, and my ass is fish bait. Learn this stuff, nonqual puke, or forever sleep in bad places and forget about movies or leisure-time novels.\"\n\nAny crewmate who wore the shiny \"qualified\" dolphin pin on his chest became the tormentor of anyone who did not. Qualifying consumed me for an entire year. I had never wanted to earn anything so badly in my life. Completing my qualification card to earn my dolphins taught me more about discipline and drive than any college or training course that I have ever taken, and never have I felt more accomplished than the day I donned my silver pin. A week after Vinson gave me my first tour through the boat, the Haddo slipped out of port.\n\nI'll never forget my first dive. I was sitting on one of the cushioned benches in the control room. We had just cleared Subic Bay and were headed for open sea. Idle chatter ceased as various orders were given and repeated to prepare the boat to dive.\n\n\"Dive, dive!\" the words echoed through the tiny control room from the diving officer. I expected to hear massive amounts of air blowing as the vent valves of the main ballast tanks popped open. I thought I'd feel the boat shudder as atomized air and water vapor exploded into the tropical air, while the tanks flooded with 800 tons of seawater to make us heavier. I wanted to hear an old-fashioned submarine \"auooga\" and the sound of Niagara Falls overhead as the submarine slid under the sea. I heard and felt nothing, except a disoriented feeling as the Haddo angled down fifteen degrees at the bow and disappeared under the waves.\n\nIn the front of the control room, the helmsman and planesman\u2014responsible for steering and diving the boat, respectively\u2014sat comfortably in bucket seats and operated airplane-style half-circle wheels. They concentrated on the analog dials and gauges on their control panels, while the chief of the watch (COW) repeated orders issued by the diving officer. As the boat continued its downward motion, I heard her groan, softly at first, almost inaudibly. As we went deeper, the creaks and moans increased in volume and tempo.\n\nCommander Mims ordered a temporary stop at various depths while the crew checked the boat for leaks. When none were found, we descended deeper. With each one hundred feet of descent, the outside water pressure increased by three atmospheres. At 800 feet, twenty-four atmospheres squeezed in on the boat, and twenty-five tons of pressure per square inch threatened to crush the hull. Finally, at 1,300 feet, test depth for a Permit-class boat, our descent into the abyss ceased. Again the crew checked for leaks. I watched the beam of Vinson's flashlight flickering across the piping in Fire Control Alley, a narrow space just behind the control-room equipment where the main digital computer stack was housed. Satisfied that she would again hold back the sea, Mims gave the order to ascend to 800 feet for our transit to Hong Kong.\n\nAs the days went by, I was amazed at how fast I adjusted to life under the sea. The focused objective to qualify as fast as possible, so that I would never again have to hear the words nonqual puke, helped to keep my mind off the fact that millions of tons of seawater were only a few feet away. Every submariner spends an average of one year \"qualifying\" on submarines by studying every switch, valve, system, pipe, and device on the boat until he has all of them memorized: how they work, what they're for, and, most important, what to do with them in case of an emergency.\n\nI spent hours tracing system schematics in the boat's piping-system booklet, and experts tested me on each associated or related item. I located emergency air-breathing (EAB) manifolds. I learned that in a fire emergency, we needed to don airtight masks, plug our hoses into the EAB manifolds, and commence breathing. Not knowing precisely where to find an EAB manifold in an emergency could cost a submariner his life.\n\nOnce I found everything on paper, I performed scavenger hunts to locate them somewhere on the boat. Fortunately, I had help from my boss.\n\n\"Did you find the fallopian tubes yet?\" Vinson asked me.\n\n\"The what?\" I said, validating that I had slept through Biology 101.\n\n\"They're in the bow, not too far from the diesel. Ask Lieutenant Tomlin; he knows right where they are.\"\n\n\"Okay, I'll ask him.\"\n\nTomlin was on watch when I made the mistake of asking, right in front of the entire control-room watch. They all started laughing, and I realized that I'd been had. I felt worse than a high school freshman on his first day of class. More questions followed, all intense. Slacking on qualifications was not tolerated. Lives depended upon everyone knowing their stuff. I drew each system by memory as an expert watched. He then studied my drawing intently and quickly pointed out even the slightest error.\n\nI had reported on board as an FTG2, the equivalent of a sergeant in the army. Until I earned my dolphins, a qualified seaman had every right to denigrate me if I got a qual question wrong. Luxuries were not granted to nonquals. I spent the first six months of my submarine tour sleeping on an air mattress on top of a torpedo rack, while qualified sailors two ranks my junior enjoyed quiet bunks in the bow compartment. My mattress had a leak, and every few hours the damn thing deflated. The lights were always on in the torpedo room, and someone always jabbered away while I tried to sleep. To this day I can fall asleep anywhere and sleep through almost anything. Heaven forbid that anyone should break into my house and steal my bed. I'd probably wake up on the floor sometime later, smiling and unaware that I'd been robbed.\n\nVinson once told me that if I ever missed sleeping on board a submarine, I should crawl onto the top shelf of my closet and hang a curtain in front. Then I should have a friend come around every two hours, rip back the curtain, shine a flashlight in my face, and say, \"Whoops, sorry; wrong guy.\"\n\nOn top of learning what made the boat tick to earn my dolphins, I had to come up to speed on standing watch in the control room and leverage my schooling to keep the fire-control systems operational. Once I did that, I would spend six hours on watch and twelve hours off in a three-man rotation. During my training, I learned about tracking targets, Soviet sub and ship capabilities, and the integration of sonar and periscope information into the fire-control computers. I also learned how to \"dance with the scope.\"\n\nMany of our watches were at night, with the control rigged for dark. The lights were set for red to adjust our eyes to darkness, in case we needed to come to periscope depth. In that event, we might spend several hours with our ESM-infused scope raised so the spooks could monitor radio traffic, or so some of us in the control room could scan the horizon for interesting contacts. During these times, the officer of the deck (OOD) often \"conveniently\" found alternate duties to perform, requiring the fire-control technician (FT) of the watch\u2014that is, me\u2014to dance with the scope in his stead.\n\nLike the old diesel boats, our submarine had two periscopes, but these were located in the control room in the center of a space no larger than the average two-car garage. The smaller Type 2 \"number one\" attack scope was controlled manually. Containing simple daylight optical capability, we used this scope for attack maneuvers where a reduced wake diminished the possibility of detection. The larger Type 18 \"number two\" scope housed all of the neat stuff, like ESM, 70 mm automatic camera equipment, electronically controlled focus and night vision capability, power-assisted rotation, and a special \"RAM\" (radar-absorbing material) to reduce detection by Soviet radar. Power-assisted rotation made turning the scope easier via a flat thumb-controlled button on the right handle. When the OOD wanted to raise the scope, he turned a large tubular \"wheel\" encircling the scope. This activated the hydraulics to lift the scope out of the well.\n\nSome of my duties entailed learning about sonar and how these systems interfaced with our fire-control equipment. Sonar was our primary means of gaining contact information related to TMA (target motion analysis). TMA was not too unlike the mental process that a quarterback undertakes to hit a running receiver. Good quarterbacks learn how to do TMA in their heads by estimating the distance to a receiver, along with how fast he's running, to determine where he'll be when the ball arrives. The quarterback then throws the ball to that spot.\n\nIn the submarine world, fire-control computers do the calculating for the boat's skipper, who is our quarterback. The receiver is the enemy ship or submarine. Using input from sonar, radar, ESM, and\/or the periscope, the computers keep track of the distance, or range, to the receiver, or target. They also track the target's speed, or how fast the receiver is running. Unlike your typical quarterback, however, who must process all this data within seconds to make a decision, back then fire-control computers were painfully slow. Long minutes passed before we could obtain a \"good solution\" on the target. Only then did our quarterback give the order to \"throw the ball\" and fire a torpedo.\n\nOur MK-113 fire-control system, which made all these calculations, consisted of two large banks of electronically and mechanically operated position-keeper (PK) analog computers on the starboard side of the control room. Each PK stood as tall as a refrigerator, but twice as wide. The front panel displayed the shape of a ship on two dials, which represented the enemy vessel and our boat, respectively. Dials and number indicators, similar to mechanical dials in a car odometer, kept track of estimated contact bearing, range, and speed. A complicated array of servos, synchros, and gears worked together to calculate a firing solution. The entire not-so-state-of-the-art system could have been replaced by one of the new Hewlett-Packard handheld calculators just coming onto the market at the time, but I suppose that's why toilet seats cost the navy thousands of dollars.\n\nBehind the main MK-113 consoles in Fire Control Alley, the primary digital computer took up a six-foot-long section and fed two smaller consoles on either side of the PKs. These were also as tall as refrigerators but equal in width. They provided essentially the same function as the PKs, but did so using digital versus mechanical technology. Only four contacts could be tracked at a time via the MK-113 system. On either side of the PKs sat the MK-81 consoles, like smaller refrigerators. These digital consoles were used to program and control wire-guided MK-48 torpedoes to their targets after our CO \"threw the ball.\" Just inside these two units sat the main torpedo and SUBROC missile-firing panel. SUBROCs are submarine-launched rockets with a fifty-mile range and nuclear depth charge capable of vaporizing a three-mile section of ocean.\n\nA large handle adorned the midsection of the firing panel, which we used to fire weapons should the need arise. During battle stations, the fire-control party, consisting of all the fire-control techs and several officers and crew assigned to those stations, operated the entire array of MK-113 equipment. We huddled close together on the benches in front of the whirring systems, sound-powered headphones glued to our ears, and cranked information into the gear in an effort to obtain an accurate firing solution. One wrong number on a solution could send the boat headlong into a Soviet submarine or other underwater obstacle, which could in turn send us all to the bottom.\n\nAs part of my training, Greenawald and the other sonar techs gave me frequent tours of the sonar shack and briefed me on passive and active sonar systems. Sound travels through seawater at roughly 4,000 feet per second over significant distances but offers only a vague clue as to the whereabouts of a contact. Most submarines employ several types of sonar to sniff out the enemy. The spherical sonar array was located in the bow. This consisted of a large fifteen-foot-diameter sphere that housed both active and passive sonar. It looked like a giant beach ball. The conformal array was a low-frequency sonar that wrapped around the midsection of the bow, like a massive piece of electrical tape with built-in ears.\n\nThe towed sonar array, called the STASS, was a by-product of the ITASS system developed by the LRAPP scientists. The \"pigtail\" array looked like a three-inch-thick, 2,600-foot-long garden hose with thirty-two bumps. Each bump housed a hydrophone that listened for distant targets emitting low-frequency sounds. To deploy the array, divers had to enter the water and attach the pigtail to the boat's stern planes. A small \"Mike\" motorboat met up with the sub at sea to help \"stream the array\" for use. The reverse had to be done when entering port.\n\nWhen sonar located a contact, they designated it Sierra one, two, three, and so on, in a sequential numbering system until the end of the mission. A contact of significance, such as a Soviet sub, usually got an upgrade to a Master number. Visual contacts were designated Victor (not to be confused with a Soviet Victor-class sub), and radar contacts were Romeo. Sonar operators spent hours with headphones on ears, turning small wheels on the BQQ2 sonar suite consoles\u2014called the BQS 11, 12, and 13\u2014until they detected a contact aurally.\n\nA waterfall display on an orange monitor for another system, called the BQR-24, visually kept track of the bearing\u2014or the direction\u2014of a sound generated by a contact. The display created a fuzzy narrow line that ran downward as the contact moved. Lines generated by targets shifted left or right as the contact changed direction. More lines equated to more contacts, which created a waterfall effect that resembled raindrops running down a window.\n\nThe BQQ2's sensitive passive capability allowed this system to hear surface ships more than 20,000 yards distant (ten nautical miles). Operators used a computer to catalog distinct sounds that were unique to a particular class of vessel, or even the exact vessel itself. A nick in a screw, \"singing\" propeller shaft, or humming engine noise could become a fingerprint or noise signature that identified a particular sub or surface ship, right down to the hull number. We called this ACINT, for acoustic intelligence.\n\nAs part of a Holystone mission, we might spend hours, days, or even weeks trailing closely behind a Soviet nuclear missile or attack submarine, sometimes approaching to within a few dozen yards. We often trailed and listened for hours, days, or even weeks to every noise made by our prey, while high-speed reel-to-reel recorders captured the information for subsequent input into data banks. This \"up close and personal\" aspect sometimes led to collisions, or near collisions, while shadowing Ivan, especially during a Crazy Ivan maneuver.\n\nI soon learned just what that phrase meant and would come to reason that we called it Crazy Ivan not so much in reference to the shadowed, but perhaps to those of us in the shadow. The baffle area on most submarines spans a sixty-degree arc directly behind the screw. A submarine's sonar is usually mounted in the bow, which creates a \"blind spot\" in the baffle area. This is where U.S. submarines might be found...in Ivan's shadow. We could play there due to our superior sound suppression technology that made it hard for Ivan to hear us.\n\nWhen U.S. boats cleared baffles, they slowed to one-third, turned ninety degrees to port or starboard, listened for any signs of unwanted visitors, then resumed their previous course. When Ivan cleared baffles, he made a 180-degree turn along his previous track, ran in the opposite direction for a while to ensure nothing suspicious lurked in the shadows, then made another thrilling 180-degree turn to resume his original course. The \"thrilling\" part related to speed. Unlike Americans, Soviets did not always slow down to clear baffles.\n\nHearing \"Crazy Ivan!\" from the sonar shack caused serious nail biting in the control room. The cry meant that better than 5,000 tons of Russian submarine had turned back in your direction, and if she didn't smack you, she might hear you. Either one of those scenarios could end your life. So, I reasoned, perhaps it was not just Ivan who deserved to be called crazy.\n\nAFTER A FEW MONTHS UNDER WAY, I finally learned enough to stand the fire-control watch alone. Excited to be on my own for the first time, I scrambled up the ladder to the main deck of the Haddo. The crimson glow of \"rig-for-dark\" red fluorescent lights bathed the interior of the control room and cast eerie shadows on the faces of the crew. The air, filled with hazy cigarette smoke, smelled like day-old coffee. Gray metal boxes clicked and hummed and blinked like tiny buildings in a metropolis under the sea. As I sipped my bitter coffee, Vinson gave me an update and turned the watch over.\n\n\"Who's this?\" I asked, pointing to a track on the plot. The plot board, fashioned from a two-by-three-foot metal box, housed a roll of scientific paper. On the checkered paper, dots and lines recorded the bearings and tracks of various contacts reported by sonar. Even though we had mechanical and digital computers to track contacts, we still used low-tech lined paper to plot the movement of all Master contacts. The roll of paper covered the front of the box that hung off the edge of one of the large PK computers. One of these dots on the paper, designated Master 21, also had a name.\n\n## Photographic Insert 2\n\nFew today know that four Soviet Foxtrot-class submarines, each carrying nuclear torpedoes, nearly started World War III during the Cuban Missile Crisis in October 1962. U.S. Navy photograph\n\nThe author with former Cuban Missile Crisis Foxtrot submarine commanders in Saint Petersburg, Russia, in March 2009. Left to right: RAdm. Victor Frolov (B-130), Lt. Anatoliy Andreev (B-36), Adm. Vladlen Naumov (B-36), W. Craig Reed (the author), Capt. Ryurik Ketov, Capt. Igor Kurdin, and translator Svetlana Verholantseva. Author's collection\n\nThe Soviets installed new SBD \"burst\" radio transmitters and encryption systems aboard Soviet submarines in 1960. The U.S. Navy was then unable to locate the Soviet subs until William J. Reed, the author's father, discovered the \"scratchy\" burst signal. Author's collection, taken on a Whiskey-class submarine in Saint Petersburg, Russia\n\nThe torpedo room in a Soviet Foxtrot-class submarine. Tube number two, which carried the nuclear torpedo during the Cuban Missile Crisis, is in the center. Author's collection, Maritime Museum of San Diego, California\n\nThe author touring the Soviet D-2 submarine in Saint Petersburg, Russia, in March 2009, with former Soviet submariners from the Saint Petersburg Submariner's Organization. Author's collection\n\nDuring the Cold War, the U.S. deployed more than a dozen \"elephant cage\" secret listening stations worldwide to locate radio transmissions from Soviet submarines. U.S. Navy photograph\n\nAuthor touring the Los Angeles\u2013class attack submarine USS Albuquerque (SSN-706) in San Diego, California. In the background is the weapons fire-control panel located in the torpedo room. Author's collection\n\nCrew of the USS Haddo (SSN-604) near Hanauma Bay on Oahu, Hawaii, circa 1979. Left to right: Dean Quint, Tim Vinson, Ken Greenawald, unknown, and the author. Author's collection\n\nKen Greenawald and Tim Vinson in the sonar shack of the USS Haddo just off the coast of Petropavlovsk, Russia. Author's collection\n\nControl room of the USS Florida (SSGN-728). Helmsman and planesman steer and dive the boat while the diving officer, seated behind them, issues orders. U.S. Navy photograph\n\nControl room in a Sturgeonclass Cold War submarine. U.S. Navy photograph by Chief Photographer's Mate Robert Hemmerly, courtesy of www.dodmedia.osd.mil\n\nFire control technician operating the missile launch console aboard the USS Seawolf (SSN-21). U.S. Navy photograph by Chief Photographer's Mate John E. Gay, courtesy of www.news.navy.mil\n\nType 18 \"number two\" periscope in the control room of a Sturgeon-class submarine.Author's collection, Naval Undersea Museum, Keyport, Washington\n\nU.S. Navy divers, like the author, deployed from fast-attack submarines during the Cold War to conduct top-secret photographic reconnaissance and surveillance missions. Many of these operations were in concert with Navy SEAL teams. U.S. Navy photographs by Senior Chief Mass Commuication Specialist Andrew McKaskle and Chief Photographer's Mate Mark Reinhard, courtesy of www.osd.mil\n\nThe most dangerous, daring, and decorated missions of the Cold War were undertaken by U.S. Navy deep-sea saturation divers breathing helium and oxygen mixtures. These divers, working from fast-attack submarines, tapped Soviet communications cables off the coast of Russia, at depths of up to 700 feet. U.S. Navy photographs by Mass Commuication Specialist Fisrt Class Eric Lippmann\n\nU.S. Navy divers, conducting top-secret Ivy Bells cable-tapping missions, spent weeks compressing and decompressing inside small diving chambers mounted on submarines, which resembled DSRVs like the one above. U.S. Navy photograph\n\nIn April 1981, the author, aboard the USSDrum (SSN-677) and deep inside Peter the Great Bay (below), was being sent on a mission to take secret photographs of the Soviet Victor III submarine K-324 when the Drum collided with the other sub. The Victor was from the Vladivostok naval base located in the top center of the photograph. The crew of the Drum, including the author, barely survived the encounter. NASA photograph\n\n\"Blackjack?\" I said.\n\nVinson shrugged. \"Master 21. Greenie gave him that name 'cause of the bet.\"\n\n\"Sonarman Greenawald?\" I said. \"What bet?\"\n\n\"Greenie heard a hole in the water on the watch before mine.\"\n\n\"Hole in the water?\"\n\n\"Yeah,\" Vinson said. \"A dead zone where there's no chirping biologics. As if something might be blocking the sound.\"\n\n\"Like a submarine,\" I said.\n\n\"Exactly. The other sonar jocks couldn't hear it, but Greenie swore something was there. Commander Mims wanted to sprint to a new location, but Greenie insisted he could find this guy.\"\n\n\"So what did Mims do?\"\n\nVinson grinned. \"He made a bet that if Greenie could find the contact by the end of the next watch, he could grow his mustache way past regulation standards.\"\n\n\"And what if he lost the bet?\"\n\n\"Then Greenie would have to shave the thing off.\"\n\nI pictured my buddy with a clean upper lip and figured he might be better off losing. \"So what happened?\"\n\nVinson yawned. \"Damned if Greenie didn't stay for two extra watches to find the contact. A submarine, but too faint to tell what kind. Since Greenie won the bet, he nicknamed the contact Blackjack.\"\n\n\"And now Greenawald gets to grow his mustache long.\"\n\n\"Right,\" Vinson said, stifling another yawn. \"I need to crash. The watch is all yours.\"\n\n\"Just don't take my rack,\" I said.\n\nKnowing that I still slept on an air mattress in the torpedo room, Vinson said, \"I'll let all the air out of your rack if you don't keep up on your submarine quals.\" He flashed a coy smile and turned to leave.\n\nAs Vinson strolled out of the control room, Lieutenant Edwin Ladeau Tomlin sauntered in. He was the crew's favorite OOD and had a great sense of humor combined with a quick mind. A graduate of the Naval Academy at Annapolis, Tomlin stood over six feet tall, which often required him to duck as he moved about the tight spaces of the boat. He had jet-black hair, an infectious smile, and a boyish face that belied his age. I watched him relieve the previous OOD and walk over to my fire-control plot for an update.\n\nMaster 21, or \"Blackjack,\" remained on a bearing of three-five-two. We were now just outside Vladivostok harbor in the Peter the Great Bay, well inside Soviet territorial waters.\n\n\"How we doing?\" Tomlin asked.\n\n\"Fine, sir,\" I said.\n\nTomlin reached up and clicked a lever on the 27MC communications box. \"Sonar, Conn\u2014can you give me an update on Master 21?\"\n\nGreenawald's voice replied through the box. \"Conn, Sonar\u2014Blackjack is still on bearing three-five-two. Contact is classified as a probable submarine, class unknown at this time.\"\n\nTomlin leaned in my direction. \"Blackjack?\"\n\nI shrugged, said nothing.\n\nTomlin turned toward the front of the control room and issued an order. \"Diving Officer, come left to zero-nine-zero, increase speed to two-thirds.\"\n\nThe diving officer echoed the command, and the boat started to turn. Our 270 feet of black stealth cut through the deep in near silence as the rudder brought us to a new heading. Twenty minutes later, Tomlin gave the order to slow, and sonar went back to work.\n\nAfter a few minutes, Greenawald blared another report through the 27MC. \"Conn, Sonar\u2014contact is on bearing three-five-nine making turns for ten knots on two five-bladed props. Designate Blackjack as a possible Delta III\u2013class submarine.\"\n\nIn 1977, there were only six Delta IIIs in operation, and they were considered the most formidable threat to Western security at the time. The third generation of the infamous Delta ballistic missile sub, this new class came packed with improved stealth and advanced electronics. She was hard to find and even harder to trail.\n\nTomlin immediately called Commander Mims, who hustled to the control room in a matter of minutes. Mims's crow's feet and forehead wrinkles were a sharp contrast to the young faces in the control room. The CO struck me as a man who might have been gutsy enough to be a boat driver a quarter of a century earlier, when diesel fumes filled the tiny spaces of the original Haddo. In those days, submarines fired unreliable Mark 14 torpedoes at massive Japanese warships, then prayed to God they could run and hide before the depth charges started exploding.\n\n\"Cap'n,\" Tomlin said, as Mims stepped onto the periscope stand, \"looks like the sonar jocks picked up a Delta III headed out of Vlad. She's running about ten knots on two fivers bearing three-five-nine.\"\n\n\"I'll be damned,\" Mims said, \"a Delta III coming out of the barn. I have the Conn, Mr. Tomlin, man battle stations. Helm, all ahead two-thirds. Dive, make your depth 1,000 feet, five degree down bubble. Mr. Tomlin, what's our torpedo tube status?\"\n\nThe battle stations alarm sounded while Tomlin gave the CO an update. I had read about the Delta III in the top-secret submarine classification books in the control room and memorized most of the details. Although the Americans and the Soviets were about to sign the Nuclear Non-Proliferation Treaty on September 21, I knew this piece of paper would have no effect on nuclear submarines. The Delta III still deployed sixteen nuclear missiles with a range of almost 5,000 miles. Each missile had multiple warheads that could wipe out almost any city in America within a matter of minutes.\n\nTrailing a Delta III escalated the tension in the control room as men rushed to battle stations. The fire-control tracking party, consisting of a half-dozen officers and sailors, crammed into the control room. A swarm of bodies lined the benches in front of the Mark 113 target-tracking consoles. Few other priorities trumped the trailing of a Soviet boomer headed out on patrol, and if that Delta started flooding a missile tube, one of our Mark 48 torpedoes would be ready to surge from its cylinder and race toward the Soviet boat's twin-screws within seconds. Everyone knew that the longer we could keep a Delta III in our sights, the more points we scored in the \"keep the world safe\" game.\n\nMims walked over to my plot. His eyes squinting, he examined the Xs I had placed on the paper chart in relation to Master 21's recently reported bearings. He ran his finger along the Delta's plot line, then stepped near the periscope stand. He keyed the sonar 27MC. \"Sonar, Conn\u2014give me an update on Blackjack.\"\n\nGreenawald provided an update, while the tracking party entered data into the fire-control computers. Knobs and gears spun in muted whirs. Before long, the weapons officer (Weps) reported that we had a firing solution. We were ready to spin a torpedo up Ivan's tail, and he had no idea we were there.\n\n\"Cap'n,\" the Weps said, \"I have a curve and a good solution on the target. Range 5,000 yards, target speed ten knots.\"\n\n\"Attention in the conn,\" the skipper said. \"We have a good solution on the target. Firing point procedures tubes three and four. Horizontal salvo, one degree offset, one minute interval.\"\n\n\"Weapons ready,\" I heard myself say, an unsteady hand resting on the firing key. My mind reeled with questions. Was this just a practice run or the real thing? Did our CO know something we didn't? Some secret message from COMSUBPAC just received with orders to sink the Delta? Sweat from my palm dripped onto the firing key. I wiped it away before anyone could see it.\n\nLong seconds passed before Mims issued another order. \"Fire control team, stand down.\"\n\nUnsure of what had just happened, my hand stayed glued in place.\n\nLieutenant Tomlin leaned over and pointed at the panel. \"Petty Officer Reed, practice time is over. You wanna let go of that firing key now?\"\n\nWe tracked the Delta III for another several hours as she cleared the harbor and headed toward her assigned station, sometimes closing to within a quarter mile. She crept along at six knots for some time, then suddenly doubled her turns to twelve knots to sprint to a new location. She then slowed for ninety minutes before changing course again by seventy degrees or more. We stuck to her like silent hounds, while sonar recorded every swoosh, nuance, and hiccup. We cataloged the Delta's unique acoustic signature as the SSBN crept into the depths, unaware of the predator lurking in the shadows behind her.\n\nJust as my heart rate subsided, Greenawald reported three more contacts in the area: two Echo II\u2013class boats and one Foxtrot submarine headed our way. Two Kresta I\u2013class guided missile cruisers and the Moskva helicopter cruiser also came onto the scene from the other direction, transiting from the base at Vladivostok. The Delta managed to glide below a thermal layer and slip away as the Echo IIs moved into position on an adjacent bearing. I visualized the scene in the sonar shack, with Greenawald sitting in front of the BQS-11 passive sonar stack struggling to hear the final whispers of the Delta as the muted swoosh of her spinning propellers melted into oblivion.\n\nMims decided to bring us to periscope depth for a look around. \"Diving Officer, make your depth six-five feet, ten degree up bubble.\"\n\nThe boat angled up as we moved toward the surface.\n\n\"Up scope.\" Mims slapped the number two scope handles horizontal. He positioned his face against the eyepiece and swung the scope around to the bearing of the surface group. The WLR-9 ESM detection warning beeped as the Kresta's Top Sail and Head Net-C search radar signals swept past and caught a whiff of our periscope.\n\nHis eye pressed against the socket, Mims said, \"Bearing to the Kresta on my mark...mark! Bearing to the Moskva on my mark...mark! Down scope.\"\n\nI knew we couldn't afford to leave the scope up very long for fear that Soviet radar would detect the protruding mast.\n\n\"What the hell are these boys up to?\" Mims asked no one in particular.\n\nThe XO looked up from the navigation plot through a dank swirl of cigarette smoke. Heavy perspiration painted circles under each arm of his blue coveralls. \"Must be a decoy to let the Delta sneak off to her station.\"\n\nMims nodded.\n\nGreenawald's voice broke through the conversation and echoed over the 27MC. \"Conn, Sonar\u2014transients in the water. Sounds like torpedo tube doors opening.\"\n\nSilence. Then another excited chirp. \"Conn, Sonar\u2014we've got high-speed screws in the water! I repeat, high-speed screws in the water!\"\n\nTorpedoes.\n\nMims maintained his composure. \"Sonar, Conn\u2014bearing to the high-speed screws?\"\n\n\"Conn, Sonar\u2014bearing to the torpedoes is two-seven-two. They're coming from the Echo II.\"\n\nI recalled that in 1968, sailors on the Haddo shadowed an Echo II as she transited out of the Mediterranean Sea. Days later, they handed off that contact to the USS Scorpion. A few days after that, the Scorpion met her demise\u2014probably from a torpedo fired by that Echo II. And now, speeding our way, a torpedo from another Echo II threatened our home. I shuddered as I thought of those sailors on the Scorpion, screaming in fear as a relentless ocean swallowed them whole.\n\nMims issued another order. \"Helm, right full rudder, increase speed to full, make your depth 1,000 feet.\"\n\nEvasive maneuvers.\n\n\"Sonar, Conn\u2014give me an update on Masters 28 and 29.\"\n\nThe two Krestas.\n\n\"Conn, Sonar\u2014bearing to Master 28 is now two-eight-five, Master 29 is two-eight-eight. They're closing in our direction.\"\n\nThe torpedoes, along with several Soviet warships, sped closer and the tracking party feverishly dialed updates into the fire-control systems. Mims ordered another hard bank to maneuver out of the way of the speeding projectiles. As the boat turned, gravity threatened to suck in coffee cups and loose objects. The control room angled to one side in response to the sharp turn to starboard. The tiny control room closed in around me as I struggled to focus on the panel in front of me. Each breath came slow, the tightness in my chest now like a ton of bricks threatening to crush the wind from my lungs. I wanted to run, to hide from the high-pitched screaming of the deadly propellers headed our way, but only freezing ocean awaited me outside the steel hull of the Haddo.\n\n\"Conn, Sonar\u2014the torpedoes are now at 2,000 yards and closing. They have not acquired; I repeat, they have not acquired.\"\n\nJust over one nautical mile away.\n\nI knew that like trained hunting dogs, the sophisticated sonar in the tip of the Soviet torpedoes would soon lock on to the noise generated by our spinning propeller. Mims remained calm, as though he had the situation completely under control. As for me, I was certain that death was a mile away and closing fast.\n\n\"Conn, Sonar\u2014torpedoes have closed to 1,000 yards.\"\n\nOne-half nautical mile.\n\nMims ordered another hard bank, and I wondered what it felt like to drown.\n\n\"Conn, Sonar\u2014high-speed screws are slowing; I repeat, high-speed screws are slowing.\"\n\nI looked up at the overhead of the control room, as if by some mysterious power I might actually be able to see the shiny cylinders in the dark and could will them to stop.\n\n\"Conn, Sonar\u2014high-speed screws are silent.\"\n\nTomlin stepped up behind me and voiced his opinion. \"Damn near skinned our asses on that one.\"\n\nWhispered cheers rang out as Greenawald reported that the torpedoes had stopped dead in the water. Something didn't seem right to me. We had just barely avoided being blown out of the water, and no one in the control room seemed at all fazed, like this was just another routine day. Did I miss something?\n\nTomlin noticed my confusion and said, \"Soviet torpedo exercise. We just got caught in the middle.\"\n\n\"Oh,\" I said.\n\nA few minutes later I had to ask: \"Were the warheads armed?\"\n\nTomlin shrugged. \"Damned if I know.\"\n\nI had just survived my first encounter with the Red Bear. Now I understood what every fast-attack submariner comes to know: for this profession, the fainthearted need not apply. In this undersea world, risks are commonplace, and lives are gambled every day in a war yet undeclared. Over the next few years, across a half-dozen espionage missions, I almost lost that gamble more than once.\n\n## CHAPTER SEVENTEEN\n\nWhen sorrows come, they come not single spies, but in battalions.\n\n\u2014WILLIAM SHAKESPEARE\n\nHAVING REPLACED THE HALIBUT AS THE cable-taping spy boat, the USS Seawolf left San Francisco Harbor on June 20, 1976. In the expansive Area D diving section, she carried twenty-two navy divers, including David LeJeune. Unfortunately, the Seawolf also left port with the same sea gremlins that had plagued Captain Aleksei Dubivko on B-36 during the Cuban Missile Crisis. Breakdowns, fires, and emergency reactor shutdowns\u2014bad things that always came in threes\u2014kept the crew alert, awake, and filled with anxiety most of the time.\n\nThankfully, Gardner Brown's cousin, Gene Centre, and others like him at Bettis Atomic Power Laboratory back in 1954 took great pride in their work. They went the extra mile during Seawolf's birth to ensure her reactor could exceed design parameters. This functional overbuild kept Seawolf's crew alive more than once. Still, at the ripe age of twenty-two, the old girl had long passed her prime. If one dog year equates to seven human years, then boat years were probably around half that, making the Seawolf seem about seventy-seven\u2014just spry enough to keep trying, but debilitated by arthritis and fatigue. Until another boat could take her place, she'd just have to \"suck it up\" and keep running marathons into the Sea of Okhotsk, and her crew would need to double the frequency of their prayers.\n\nThe Seawolf carried four remote operating vehicles (ROVs) on racks in Area D near the hyperbaric chamber used by the saturation divers. Operators nicknamed the ROVs Deep Throat, Happy Hooker, Hawkeye, and Shortime\u2014the latter gaining its name because every time they put her into the water the electronics \"shorted out.\" Although disguised as \"oceanographic research\" devices, the ROVs had an uncanny resemblance to the Fish and Underdog mini-subs used on the Halibut and Swordfish, respectively. Each were tethered to the submarine, powered by an electric motor and propeller, and contained high-resolution cameras to facilitate finding things on the ocean floor, like communications cables or discarded missile parts.\n\nFive communication technician \"spooks\" also accompanied the Seawolf on that mission: Master Chief Mac Empey and Chief Mark Rutherford\u2014alumni from the third Halibut mission\u2014along with Chief Frank Courtney and Second Class Bob Ellenwood, who'd also made the last Halibut run. The fifth member, Senior Chief Tommy Cox, had just joined the team as a permanent linguistics expert.\n\nAfter Mac Empey's eye-opening indoctrination video, LeJeune befriended the ACT V gang\u2014the name given to the group of five spooks\u2014and hung out with them in the chart room when he wasn't pressing down, diving, or pressing back up in the dive chamber. LeJeune had recently completed barber's school and gained some skill in the art of cutting the crew's hair under way. He was snipping at the head of a spook when Cox sauntered into the area. The smiling I-Brancher sat on an empty torpedo rack and picked up someone else's guitar. Without asking, Cox proceeded to tune all twelve strings. LeJeune groaned, certain that the guy was about to play \"four bars of the only song he knows and then strut off proud and full of himself.\"\n\nCox finished tuning, then pulled a large red handkerchief from his pocket. He honked his nose into the hanky and proceeded to wow the small crowd with a combination of well-played original and popular tunes. LeJeune \"hired\" Cox as his official barbershop entertainer, and the singing spook also became a regular act in the crew's mess, especially when the selected movie sucked or during \"spook night,\" when the ACT V team pitched in and prepared a meal for the entire crew. LeJeune lent a hand for these events, which included serving the officers in the wardroom.\n\nAfter Mac Empey demonstrated his brilliance by saving the Ivy Bells program from extinction on a prior run, he received a promotion to warrant officer and moved into the wardroom with the other officers. Rank held little meaning with the ACT V team, and Empey's beer-drinking days with the spooks at the Horse and Cow Saloon in Vallejo, California, were unchanged by the gold bars.\n\nThe Seawolf set a record in 1958 for the longest submerged transit of sixty days. She broke that record on the first run to the Sea of Okhotsk in 1976 by staying under for eighty-seven days. Although a rough transit, filled with occasional fires, system failures, and turbulent swells, the Seawolf completed her wire-tapping run with no serious issues. Prior to her next mission in 1977, T-Brancher Frank Courtney's wife took ill, requiring him to decline future runs. Mac Empey retired, and so the ACT V team needed to recruit two replacements.\n\nThere were no strangers in the ACT V group. Everyone knew and vouched for someone. Empey and Rutherford knew each other from the Halibut. Empey knew Cox from the USS Lapon (SSN-661), where both served with the famous Commander Chester M. \"Whitey\" Mack. Courtney knew Rutherford and Cox from the Greenling (SSN-614), and so it went. The group meshed and worked well together, which had proved critical to the success of previous missions. As such, the ACT V group often circumvented Bureau of Naval Personnel (BUPERS) recommendations and considered only those they could trust.\n\nAngered by this stance, BUPERS insisted that the team at least interview their candidate, Charlie Miller. At first, the ACT V guys refused, having already decided on an M-Brancher in New London. Tommy Cox and Mark Rutherford finally relented and drove to Sugar Grove, West Virginia. There they questioned Miller for hours and found him to be a more worthy candidate. With one down, they still had one more member to select.\n\nFrank Turban did not at all expect a call from Mark Rutherford to discuss the possibility of joining the elite ACT V Special Projects team, currently assigned to the Seawolf. He'd heard rumors about that aging boat and had no desire to become a part-time member of her genuflecting, prayer-muttering crew. Having served with Rutherford on the USS Flying Fish, where they encountered the first Soviet Delta and a wounded Yankee-class submarine, Turban trusted his friend and decided to listen. Rutherford said that two members recently left the group, and they needed Turban's advanced T-Brancher skills for an important top-secret mission. Perhaps, Turban commented years later, he should have run from the offer. Instead, he accepted and jogged headlong into disaster.\n\nThe Seawolf headed west in 1977 with two new spooks aboard. In the crew's mess, Turban overheard sailors talking about the Wolf's constant spate of problems. They said that their tall, easy-going skipper, Commander Charlie McVain, never ran drills. He didn't need to. The Wolf endured more than enough real disasters. Turban raised an eyebrow when some of the guys actually placed bets that the old boat wouldn't have the chops to complete her next SpecOp. One petty officer joked that if he lost because the sub went down, he wouldn't have to make good on his bet. Turban dismissed the banter as just talk, but he still tossed and turned nervously in his rack. Six days out of Mare Island, the talk died to a whisper, and Turban finally relaxed.\n\nThen a fire alarm uttered a warning and jolted him to attention. He'd just sat down in the crew's mess and started chewing on a sticky bun when a woman's voice blared over the 1MC. A sweet chunk of the pastry lodged in his throat and damn near choked him to death. The crew called their alarm system the \"bitch in the box,\" as the system came with a soft voice in place of a clanging alarm. Someone back in the fifties, during the Seawolf's formative years, thought they should use a soothing Bell telephone operator's voice for alarms versus the clanks and clangs found on diesel boats. An autographed portrait of the old Bell operator hung on a wall in the wardroom. The soft sound of \"Fire, fire in the engine room,\" whether spoken with Grandma's voice or not, did nothing to sooth Turban even a little.\n\nHe initially wondered if the alarm was a drill. Then he smelled the smoke. With the bitch box blaring in his ear, he jumped from his bench, donned an EAB mask, and joined the throng now passing emergency gear aft. Commander McVain announced over the 1MC that a fire had broken out in the reactor compartment. The blood rushed from Turban's face, and his knees shook like branches in a high wind. He knew they were running 100,000 pounds heavier than normal\u2014this to keep the Seawolf anchored to the bottom of the ocean during cable-tapping operations once on station. If the fire caused the reactor to scram, they'd lose propulsion. Given the extra weight, they'd lose depth control. And given the depth of the ocean, they could lose their lives.\n\nTurban watched as a horde of damage-control personnel ran past, EAB masks strapped to their faces. Men shouted and pointed. Others responded by running, pulling gear from lockers, or just getting out of the way. Turban became one of the latter by slipping down into the torpedo room and huddling in the chart room with the other spooks. Nearby were racks of torpedoes, long and green and loaded with explosives. Turban wondered if they'd blow up when heated by a fire. As the boat angled down even farther, he glanced at a nearby depth gauge: 350 feet. He knew that the Seawolf's test depth was 750 feet. The needle on the gauge inched downward even more: 400, 450, 500, 550.\n\nTurban looked over at Rutherford, seated on one of the bunks in the torpedo room. His friend said, \"We're too heavy. They're probably trying to pump the extra water out of the ballast tanks, but they can't keep up.\"\n\n\"Why not?\" Turban asked.\n\n\"The reactor's scrammed,\" Rutherford said. \"We don't have enough power while we're running on batteries.\"\n\n\"What happens if they can't get the reactor back online?\" Turban said as his worst nightmare unfolded in real time.\n\nRutherford shrugged, said nothing.\n\nThe depth gauge hit 650 and Turban muttered a quick prayer.\n\nTwo decks above him, Commander McVain, who'd earned a Ph.D. in nuclear physics, sprinted aft to assist the damage-control party. He left his red-haired, freckle-faced XO in charge. When the boat hit 750 feet, McVain yelled an order up to the control room through a sound-powered phone to \"emergency blow!\" The XO's freckles disappeared, and his face turned white. Frozen by shock, he did not repeat the order. The chief of the watch maintained a calm head and followed his skipper's orders without waiting for the XO. The gushing sound of air permeated the boat.\n\nThen the Seawolf rocketed to the surface, lifted partially out of the water, and crashed back into the ocean. She dove again to more than 200 feet before leveling off on the surface. The crew finally managed to quell the fire and to ventilate the smoke off the boat. Turban later learned that a large fuse in the nuclear compartment had blown. Filled with wet sand, the fuse blew its moist dust into the lube-oil bay, causing a plume of steam. When the reactor operator saw the plume, he thought it was from the propulsion system, so he scrammed the reactor. Without the brute power of her nuclear-propelled blades, the Seawolf could not climb the mountain back to the surface, at least not until McVain ordered an emergency blow to force air into the ballast tanks and push out the extra water.\n\nThankful to still be alive, Turban breathed a sigh of relief. Then he heard the rumor. Apparently, they brought only one spare sand-filled fuse, and they just used that one up. Despite the risk, Commander McVain refused to throw in the towel and head back to port. He figured that the old girl had survived everything thrown at her, and the odds against blowing another fuse had to be one in a million. Then again, thought Turban, Captain Edward John Smith thought the Titanic was indestructible.\n\nTHERE ARE FEW ICEBERGS IN THE Sea of Okhotsk, but there were plenty of Soviet warships and hunter\/killer submarines during the Cold War. Dozens of them strutted across the \"Mouth of the Bear's\" frigid waters while transiting between Vladivostok in the Sea of Japan and Petropavlovsk on the Kamchatka Peninsula. After the Seawolf arrived on station, deep inside Soviet territorial waters, navy diver David LeJeune prepared for the dive of his life. Three other divers joined him inside the twenty-by-ten-foot diving chamber in Area D.\n\nLeJeune pulled on his MK-11 deep-sea diving suit. He then dipped his Kirby Morgan mask into some water to soak the spongelike material\u2014this to better keep the ocean out of his eyes. After smearing Vaseline over the rubber seal on the mask, he clamped it down tight on his face and sucked in a few breaths. Nearby, the other divers duplicated the routine, including David \"Whompee Jaw\" Sullivan. The lanky diver earned his moniker by the grace of God, who imbued him with an offset jaw. The engineers at Kirby Morgan actually had to design a special mask to fit the man's unique face. After the divers finished suiting up, they completed their checks and opened the inner hatch, but not before LeJeune placed something in his suit pack. Sullivan climbed up through the opening, wrestled with the outer hatch, and disappeared into the black beyond.\n\nThe hyperbaric chamber was pressurized to maintain equilibrium with the ocean outside, not too unlike the small space at the front of the DSRV-shaped diving chamber on the Halibut. This prevented the seawater from taking up residence in the submarine and allowed the divers to exit to the netherworld. Once outside the sub, darkness engulfed LeJeune as he descended to the ocean floor. Tiny fish fluttered past his face mask, shiny and ethereal in the glow of his underwater light.\n\nLike they had done on countless training dives, LeJeune, Sullivan, and another diver slow-walked the 300 yards over to the twenty-foot-long pod-tapping \"beast.\" The large drum-shaped pod had been placed there on an earlier mission, and thick \"jumper cords\" now snaked through the sand on their way to the communications cable. The recording tapes in the beast had collected hundreds of Soviet conversations over the past several months, and the divers' job now entailed retrieving those tapes for subsequent perusal by the NSA. After completing this task, LeJeune removed a white oval object from his suit pack. He knew that the USS Parche was scheduled to undertake the next mission back to the Sea of Okhotsk and figured he might not be selected as one of the divers to go on that run. As a joke, he wedged a cow's skull under one side of the beast to offer the next set of divers \"a good laugh\" when they arrived the following year.\n\nWhile slogging back toward the Seawolf, a green glow inside LeJeune's mask verified that his HeO2 mix was still good. His stomach growled, and he was thinking about devouring a hamburger in the crew's mess. His hunger evaporated when his mix light turned red. He swallowed a lump and called the master diver, who ordered an immediate \"reel in.\" Something tugged hard at LeJeune's back. His feet shot up from the silt as he sailed through the water, pulled along by his lifeline. The fourth diver, inside the boat, reeled him in at warp speed. Tons of ocean between LeJeune and the boat made the seconds tick past like hours. LeJeune reached back and felt for the small can of mix strapped to his back, painfully aware that it held only three breaths. He was more than a football field away.\n\nOne breath.\n\nSo this is how it ends.\n\nTwo breaths.\n\nWould they remember him as a good man?\n\nThree breaths.\n\nHis vision blurred as his air ran out. He felt dizzy but at peace. He closed his eyes. He did not see God, but he also didn't see the devil. In fact, he saw nothing but darkness. Then a bright light beckoned, and he wanted to follow.\n\nLeJeune blinked. Sullivan's offset jaw formed a crooked smile inches above his face. \"Thought we lost you there, partner.\"\n\nAfter decompressing, LeJeune completed a physical examination and sauntered up to the crew's mess. There he ordered a big, fat, juicy cheeseburger smothered with mayonnaise. He figured if he was going to die of a heart attack, it probably wouldn't be today.\n\nThe Seawolf logged another record that year for the longest-known mission underwater, 106 days all told. She hadn't set out to achieve that milestone, but her aging propulsion system refused to deliver more than ten knots speed, and given the amount of noise she produced, Commander McVain needed to take an indirect course to avoid detection. Those in a position to know full details about upcoming Ivy Bells missions understood that the Seawolf's time was near an end, and the USS Parche would be making the next run to the Sea of Okhotsk in 1978.\n\nDavid LeJeune joined the crew of the Parche and made several more dives, some in the Barents Sea at depths reaching 700 feet. He earned his submarine dolphins in 1977 and qualified as master diver in 1978. He and his wife, Cheryl, whom he still calls George, have been married for more than forty years. Submariners on the Halibut, Seawolf, and Parche referred to the navy saturation divers who undertook these harrowing missions by a special name. They called them heroes.\n\nWHEN DENNIS SMITH GRADUATED FIRST IN his class from navy electronics school in late 1975, he did not envision that Submarine Development Group (DevGru) One would recruit him to help conduct submarine espionage missions. He didn't think his job would entail working on the most top-secret and advanced mini-subs in the world, and he never imagined that fate would place him front and center for the most important underwater missions of the Cold War.\n\nWhen he arrived in Northern California, DevGru assigned him to the oceanographic ship USNS De Steiguer (AGOR-12), skippered by a civilian who reported to Commander Hal Brown, the officer in charge of DevGru One. There he worked with cutting-edge devices called STOVEs, for Surface Tethered Oceanographic Vehicle Experimental. These camera-filled mini-subs had been resurrected from the USS Halibut and painted with \"cool colors\" such as orange or bright fishlike swirls of red and yellow. On the De Steiguer, Smith helped reel out 17,000 feet of cable so the STOVEs could hunt seabeds for interesting tidbits, like spent missile parts. Smith enjoyed his job, but his time aboard the De Steiguer abruptly came to an end when a sea mountain damaged the STOVEs beyond repair during a big storm.\n\nET2 Dennis Smith then received orders to the USS Parche in 1976 and reluctantly reported to the Special Projects Rescue Systems (RS) division. He didn't think the Parche would have anything as \"amazingly cool\" as the STOVEs, but as it turned out, he was dead wrong. In the Parche's torpedo room he found the latest in secret underwater snooping technology, known as the System 2090, which incorporated improved optics, sonar, and more. Using tethered mini-subs similar to the STOVE, Fish, and Underdog UAVs, the Parche found stuff on the ocean floor with these advanced devices like nothing had before.\n\nSmith's RS division consisted of new guys who'd not yet gone on a mission, with the single exception of his boss, Senior Chief Al Lusby. Over the next several months, Smith got to know his teammates well. He also met some of the \"Rescue Communications\" division spooks, including Frank Turban, and a few of the \"Rescue Operations\" saturation divers, like David LeJeune. Both had just completed a run on the Seawolf to the Sea of Okhotsk, and Smith felt a tingle of excitement run down his spine when they talked in hushed tones about Parche's upcoming mission to the same area.\n\nOver the next several months, Smith and the other ETs practiced operating the System 2090 mini-subs. These \"Fish-like\" devices were launched via the torpedo tubes and contained special sonar to help them locate objects underwater. The units were tethered to 400 feet of cable and housed high-resolution television and film cameras and high-beam lights. Despite the fact that the 2090s used torpedo tubes to deploy, the Parche's torpedomen were not allowed to view, let alone work on, any of the equipment. Only the specially trained ETs in the Rescue Systems division were afforded that privilege.\n\nThe 2090s resembled stubby torpedoes, right down to the same green paint. They were all custom made by Westing house Oceanic Division in Annapolis, Mary land. Another company with offices on Tennessee Street in Vallejo, California, made most of the internal equipment, which included a state-of-the-art plasma display and a videodisc recorder that could store 300 frames of video\u2014about 10 seconds' worth\u2014complete with rewind and slow-speed replay. With 2090s running just twelve feet off the bottom in murky waters, this last feature proved critical in spotting three-inch-diameter communications cables and tiny missile parts. The RS division consisted of ten ETs, with five of them standing six-hour watches at a time. On watch, one ET operated the video gear, and another drove the 2090, while yet another navigated.\n\nAs for weapons, the Parche carried only four torpedoes\u2014initially a combination of MK-37s and MK-48s, and later just MK-48 ADCAPs (Advanced Capabilities). These torpedoes were specially modified to light off their motors faster after leaving the tube. Standard torpedoes simply would not work on the Parche when she was on station. Her cable-tapping missions required that she sit in the mud atop jet ski\u2013like sled runners, with her underbelly only a few feet off the bottom. A normal torpedo, when fired, drops downward by almost fifty feet before the motor lights off and speeds it toward a target. This safety feature, which helps prevent problems with \"hot running\" torpedoes, is acceptable in the open ocean, but not for scenarios where the sand is less than fifty feet below the sub. Modified torpedoes solved this problem, but also carried the risk of a hot run, which could terminate a mission.\n\nThe Seawolf remained under repair and tied to the pier while the Parche went to the Sea of Okhotsk in 1978, with Commander John H. Maurer in command. The crew took to calling their skipper \"Bullet Head\" because of his near-bald crewcut. Once the Parche came within range of the cable, Smith and the other ETs shot a 2090 mini-sub out one of the torpedo tubes. On a monitor screen in the torpedo room, Smith watched the 2090 speed across the bottom, its bright strobe light illuminating jagged rocks and frightened fish. The shape of a twenty-foot-long barrel came into view. The beast. Smith stared in awe at the large cylinder and marveled at the ingenuity. American engineers had actually built a watertight container that housed the latest in sophisticated induction-recording technology powered by a miniature nuclear reactor. If they could do that, what couldn't they accomplish?\n\nParche spent the next thirty-one days on station while the RS division divers, including Master Diver David LeJeune, retrieved the recording tapes and returned to the boat. Given Parche's advanced speed, their entire mission time was less than the Seawolf's typical transit time. The Parche started to head home, but Maurer ordered a return to the pod for overhead mission pictures. The Parche hovered over the beast while a 2090 reeled out and snapped shots, some 300 feet below the submarine. Maurer maneuvered the Parche into a hover, not unlike a helicopter, where the boat remained completely still. For subs, this is a difficult maneuver that requires steady hands to pump water in and out of the trim tanks to keep the boat level and stationary.\n\nUnfortunately, hovering did not sit well with the 2090s. They preferred a moving submarine to ensure an unsnagged tether line. With the Parche sitting in one spot, the 2090 windmilled out of control and snapped off from the end of the cable. The ETs had a spare, but it took ten hours to reel in the old cable, unravel it, splice on a new 2090, and test the system. The Parche then had to use the new 2090 to find the old one. Once located, the submarine sat down again and redeployed the divers to grab the spent mini-sub. Having already started to decompress, the divers grumbled but complied.\n\nDennis Smith's first run into harm's way hooked him into the world of Special Projects, where he remained for years to come. One of those missions was Parche's first into the cold northern waters of the Barents Sea in 1979. Given its proximity to Murmansk, home of the Soviet Northern Fleet, this area provided a heightened degree of danger and difficulty for any submarine, let alone a cable-tapping one. Smith and crew spent days searching for the cable, coming within a few miles of the Soviet coastline.\n\nFinding the cable took days, with the Parche sending out a 2090 to snoop around the ocean floor some 700 feet down. Once found, they hovered over the landing spot while dropping 15,000-pound mushroom anchors. The anchors thudded into the silt fifty feet below the submarine. The Parche then inched down slowly until she neared the top of the anchors. She oscillated back and forth above the two anchors before the winches were engaged to pull the boat toward the bottom. Although the Parche had practiced this maneuver dozens of times off the coast of San Francisco, the actual procedure was fraught with risk. One wrong move could send the boat slamming into one of the anchors or rip her free of the tethers and end her ability to stay anchored to the bottom\u2014which would also end the mission.\n\nThe Parche settled to the bottom and deployed the divers. In the pitch-black sea, they manhandled the massive pod-tapping beast into position near a communications cable. They connected the jumper cord to a repeater box and returned to the Parche. T-Branchers then pulled information off the cable, while I-Branchers listened intently to ensure the quality of the tap. Weeks later, the Parche returned to California and received a Presidential Unit Citation for operating \"in the hostile environment of poorly charted ocean areas.\" Her crew celebrated the success of their mission at the Horse and Cow Saloon in Vallejo, and all of them signed gag orders saying that they would remain silent about what they had done for at least thirty years.\n\n## CHAPTER EIGHTEEN\n\nSuccess is not final, failure is not fatal: it is the courage to continue that counts.\n\n\u2014WINSTON CHURCHILL\n\nWHEN THE FIRST ULTRA-QUIET SOVIET NUCLEAR-POWERED hunter\/killer attack submarine, code-named Victor I by NATO, rolled down the ramp at the Admiralty yard in Leningrad in 1968, the U.S. Navy lifted a concerned eyebrow. Sixteen of the Victors splashed into the sea through 1975. By 1978, Seven Victor II submarines were added to the Soviet fleet, each carrying sixteen Stallion cruise missiles that could scream through the air at Mach 1.5 and destroy targets seventy-five miles away. Crews quickly manned battle stations whenever a Victor II was detected anywhere near a U.S. warship. For NATO forces, however, finding a Victor II was not that hard. Soviet engineers were skilled in weaponry but not in the art of stealth.\n\nDuring the early years of the Cold War, the Soviet Union knew that its boats were noisy as hell. Built to keep up with fast U.S. warships, they had a distinct acoustic disadvantage. U.S. sonar operators often joked that Ivan's subs \"whined so loud, they sounded like cats in heat.\" When the Soviets discovered through their spy network that American subs could easily track Victor IIs, they halted production to design the Victor III. To overcome excessive noise issues, the Soviets reverted to a tried-and-true tactic: they cheated.\n\nDuring the late seventies and into the early eighties, the Soviet Union secretly contracted with Toshiba and Kongsberg\u2014Japanese and Norwegian technology companies, respectively\u2014for 140 Kongsberg computers, advanced software, and sophisticated propeller-milling equipment designed to quiet their noisy props. In direct violation of international trade agreements, the two companies, almost overnight, allowed the Soviet Union to produce quiet submarines. In 1979, they did just that with the introduction of the Victor III, which swung the Cold War pendulum back in their favor. For U.S. submariners, hunting Ivan just got a lot tougher, and the statement Commander Kinnaird R. McKee made in 1968 when he served as the CO of the USS Dace (SSN-607) all but came true.\n\nIn short, McKee opined that \"eventually, U.S. and Soviet submarine capabilities will converge.... It will be blind man's bluff with other submarines...because at some point, nobody will be able to find a submarine with anything.\"\n\nThanks to Toshiba and Kongsberg, Victor IIIs sported ultra-quiet tandem four-bladed propellers and special sound-reducing shock absorbers to increase stealth. This class of hunter\/killer attack submarine also came with something else entirely new. Mounted atop the rudder at the aft of the Victor III, a strange tear-shaped pod stretched nearly thirty feet from end to end, with a diameter of eight feet near the front of the pod. From a distance, the odd pod resembled an elongated egg with a sharp taper near the tail. By 1980, the U.S. Navy remained clueless as to what this mysterious pod could be. Obtaining close-up photos of a Victor III odd pod, so the NSA could determine its purpose, became a top priority for U.S. fast-attack submarines.\n\nAT THE REQUEST OF SUBMARINE GROUP (SUBGRU) 5 in early 1980, I received a transfer from the USS Haddo to the USS Drum (SSN-677), a newer Sturgeon-class submarine. Like the Haddo, the crew of the Drum numbered just under a hundred enlisted men and a dozen officers. Our CO, Commander Michael Oliver, was an experienced and proficient leader who often ran headlong into danger like a sword-brandishing swash-buckler after bounty. Our executive officer, Robert E. Fricke, while also professional and intelligent, occasionally pushed the crew a bit harder than needed in an effort to improve efficiency by one more tenth of a percent.\n\nCommander Oliver sat me down in his stateroom a few days after I reported on board and explained why I'd been transferred to the Drum. He told me that Soviet tactics had changed dramatically since 1975, when the Reds projected power into distant waters and tried to cut off Western sea lanes. Now they focused primarily on protecting their SSBNs by finding and destroying U.S. submarines and ASW ships. The Victor III was built with this purpose in mind.\n\nIn light of this radical change in Soviet behavior, the NSA had deep concerns about the purpose and capability of the new odd pod mounted on the rudder of the Victor III. That fear rolled downhill to the navy and downward still to SUBGRU 5. They examined numerous photographs I'd taken of Victor IIIs while on the Haddo and liked what they saw. They wanted more\u2014but preferably closer and clearer. Under strict orders not to discuss any of this with the rest of the crew, Oliver told me that the Drum would be leaving on a WestPac in the fall to conduct two SpecOp missions to get those photos. We would spend the next several months training and preparing for that run.\n\nI saw a gleam in Oliver's eyes that day and figured that our next deployment represented his best ticket to stardom and perhaps another stripe on his sleeve. Oliver said that he'd been instructed to \"exercise every means possible\" to obtain better photographs of the odd pod. Since I'd taken and developed more Victor III shots than almost anyone else and had also received advanced photographic training, Oliver wanted me practiced and ready to help reel off pictures when needed. As I sat in Oliver's stateroom and listened to his speech, feelings of pride, inadequacy, and fear overwhelmed me. I thought the Drum just needed another fire-control technician and navy diver. I hadn't planned on this extracurricular assignment, and I hoped that when the time came, I'd be up to the challenge.\n\nHaving completed a few more classes in recon photography and some on-the-job training during local at-sea exercises, I received orders to report to the diving tower at the Navy SEAL training facility on Coronado Island. Commander Oliver requested that I spend some time doing photographic reconnaissance training with a SEAL team there. This type of exercise was not new, and in fact had been commonplace during the Vietnam War. Navy diver photographers like Steve Waterman, author of Just a Sailor, undertook such operations all the time. Navy recon divers sometimes swam with SEAL teams to various beaches, near foreign vessels, or within visual range of shore-based facilities to take reconnaissance photos for upcoming operations.\n\nI spent the next few months getting in shape at the obstacle course and exercising with some of the SEALs. The physical conditioning reminded me of navy diver school in Hawaii but paled in comparison to what I knew SEALs endured in BUD\/S training. The SEALs also gave me a crash course on the Draeger MK V rebreather. As a diver, I had been using open-circuit SCUBA, usually in the form of heavy twin-nineties that emitted bubbles that could be detected by unfriendly guys. Draegers, which were strapped to your stomach rather than your back, looked like small vacuum cleaners. They emitted no bubbles but could only be used down to a depth of thirty-three feet.\n\nAfter my land-based training time with the SEALs, the Drum conducted an exercise just off the coast of Catalina Island. During this exercise, I teamed with four Navy SEALs\u2014one officer and three enlisted men. We donned wet suits, masks, fins, and standard navy twin-ninety SCUBA tanks. The double-tank rigs were heavy, causing us to bend over as we stood in the bow compartment of the boat just below the forward escape trunk. Someone opened the bottom hatch to the trunk, and I watched one of the SEALs climb up the ladder into the chamber, an oval area about eight feet in diameter and filled with gauges and valves.\n\nThis SEAL's job, completed alone, consisted of locking out of the trunk and finding the line locker, near the bow of the boat, where the mooring lines were stowed. There he removed a rubber Zodiac raft to be used for our mission. The SEAL attached the raft to a tether line anchored to the sub, then inflated the Zodiac to propel it to the surface. This delicate operation took about thirty minutes and was done entirely by feel in near-zero visibility. A couple of raps on the trunk signaled that he'd completed the job, and the rest of us could now deploy.\n\nThe escape trunk could fit only two at a time with twin-nineties strapped to our backs. Once inside the escape trunk, we turned a few valves to expel the air and fill the space with salt water up to the bubble line\u2014about where our heads jutted above the water. The trunk was now pressurized to match the outside water pressure, which allowed us to open the upper hatch and swim out into the ocean. We hooked onto a lifeline running from the top of the submarine's sail to the foremost line locker. The rest of our equipment, not carried with us into the escape trunk, was stowed in that locker.\n\nThe boat slowed to less than one knot, standard operating procedure for diver operations to ensure that none of us would be sucked into the propeller. Our SEAL officer in charge flashed a signal, and we all swam over to the line locker to remove the gear. As I neared the locker, the ocean rushed past my face mask and I shot upward out of control. My lifeline snapped tight with a jolt and prevented me from heading toward the surface. An excruciating pain stabbed at my right ear, and massive pressure squeezed in on my left ear. The sensation felt worse than having a severe head cold while rapidly descending in a plane from 30,000 feet. My stomach fluttered as I tried to clear my ears and kill the pain.\n\nI grabbed hold of the lifeline and pulled myself down toward the SEALs. I could barely see their silhouettes in the dark as a stream of microscopic sea life rushed past. Submarines at periscope depth are supposed to stay at sixty feet as measured to the bottom of the keel. On top of the boat, near the escape trunk, we should have been at around thirty feet. I held my depth gauge up to my mask and nearly let out a yell into my regulator. We were at 200 feet.\n\nI knew that warmer waters tended to make things less buoyant, like 4,000-ton submarines. Sometimes when boats are cruising along in cooler water, and they hit a pocket of warm water, they sink like a rock. An alert planesman in the control room, responsible for keeping the boat level, can often compensate quickly. A not-so-alert planesman can lose depth control and drop a boat by a hundred feet or more in a matter of seconds.\n\nI glanced at my watch. Recalling the navy dive tables, I knew we had only four minutes at 200 feet before we incurred decompression time, not a problem if we regained depth control soon and didn't ascend too fast. I started doing some mental calculations to determine our decomp time if we went past five minutes at 200 feet. While crunching numbers in my head, our OIC held up one hand in the near dark, looking for the okay sign from the rest of us. All of us pointed to our ears first, then gave the okay signal. We all had probably busted our eardrums, but other than that, we were fine.\n\nFive minutes came and went while the boat remained at 200 feet. We'd now have to decompress. Five more minutes ticked by, and still no movement. I looked over at our OIC and tapped my watch. He shook his head no, and a sick feeling hit my gut. I had no desire to get the bends prior to my twenty-fifth birthday. I wasn't happy about the OIC's decision but figured he had his reasons.\n\nTwelve minutes came and went. The forward escape trunk opened, and another diver appeared, carrying two sets of twin-nineties. He signaled that the boat would ascend to periscope depth and remain there until we decompressed. A minute later, the boat came shallow. Navy dive tables dictated that we needed to spend three minutes at thirty feet, seven minutes at twenty feet, and twenty-seven minutes at ten feet. The extra tanks brought by the other diver would give us just enough air to complete our decomp time. As we clung to the line and sucked in gulps of air, I finally understood why our OIC hadn't ordered us to swim to the surface earlier.\n\nSurfacing while we were still inside the four-minute window, so we didn't have to decompress, would have been too risky. The boat needed to speed up to regain depth control, which increased our chances of getting pulled into the propeller if we unhooked from the line. After the four minutes, we were better off waiting for another diver to deliver extra tanks to ensure we had plenty of air between us to complete our decompression time.\n\nAlthough I had received extensive training at navy diver school, which in many respects is similar to the first phase of navy SEAL training, I obviously still had a lot to learn about submarine diving. I had spent so much time improving my photographic skills, and keeping up with my regular fire-control duties, I hadn't devoted proper attention to my diving acumen. I made a commitment that day to learn as much as I could from the SEALs and get wet at least once a week to stay proficient. Never once did I consider the possibility that I might actually be called upon to use my navy diver training on an actual mission in Soviet waters.\n\nIn between plunges in the ocean, I honed my abilities in the art of periscope espionage photography. Not only did this require learning everything a professional photographer learns, such as adjusting camera settings and lens selection for varying light conditions, setting film speeds, as well as the general physics governing the world of film, but I also needed to become proficient at what, how, and when to do all this through a periscope\u2014and in a dripping wet suit aboard a bobbing rubber raft while shaking from the cold. Getting up to speed on that took many long hours of studying top-secret Soviet warship manuals, periscope lens functionality, limitations, and lighting, plus time-of-day requirements, then lots of practice during exercises off the coast of California. On one of those runs, I photographed the stern of the aircraft carrier USS Constellation (CVA-64). We were so close that we could almost make out the name tags on the uniforms of the deck crews. They never detected us.\n\nAlong with taking photos, I was responsible for developing the film and keeping the camera equipment in top condition. We had two types of cameras on our submarine: a standard 35 mm Canon AE-1 camera and a high-speed 70 mm camera built into our number two periscope. Taking photos with the first required connecting the camera's lens directly to the eyepiece on the scope. Taking pictures with the latter was as easy as pushing a small red button on the periscope handle. Theoretically, the 70 mm should have provided twice the quality of the 35 mm, but this wasn't always the case, given that you couldn't adjust settings on the 70 mm the way you could on a Canon.\n\nWhen it came time to develop the film, things got a lot more complex. Developing 35 mm photos under way was a major challenge. We didn't have digital cameras in those days, so this process required manually dunking the film into several baths of chemicals contained in one-foot-square plastic bins. This operation had to be done in total darkness, usually in the yeoman's shack. The plastic bins were placed wherever one could find a flat, clear space. Heaven forbid we should be at periscope depth. The boat would pitch from side to side in the waves, while the soup in each bath threatened to spill onto the deck. I'd place the film in one tub using a pair of tongs and bathe the sheets until done, then set it in the next tub and so on\u2014a time-consuming process.\n\nIn contrast, I used an automated developer for the 70 mm film. This two-by-one-foot metal box did all the work. I simply placed one edge of the film into a side slot and flipped a switch. The machine grabbed the edge of the film and hummed away while feeding the ribbon through a snaked procession across rollers and through baths of developer and fixer solutions. Out the other side, fully developed 70 mm film emerged. This unit simplified the process of film development, as long as the thing wasn't broken. Part of my training included learning how to repair the box when it did break, which was often.\n\nAfter weeks of periscope and recon photography school in San Diego, coupled with lots of practice at sea, the science of film processing and system repair became second nature. Aboard my previous boat, the USS Haddo, I spent several months on a couple of SpecOps taking photos of interesting ships and subs through the periscope using the 35 mm and 70 mm cameras. A good number of these pictures were of the infamous Victor III.\n\nI studied the photos I'd taken of the Project 671RTM Shchuka submarine and marveled at her sleek race-car design, sloped conning tower, and strange pod mounted atop the rudder. I ran a magnifying glass over the pod and squinted, desperately trying to figure out what this thing might be. The introduction of the Victor III in late 1979 caused quite a stir in NATO intelligence circles due to that distinctive oval pod. Some speculated that the egg-shaped housing hid a silent propulsion system, perhaps a magnetohydrodynamic drive unit. Others insisted this had to be an advanced weapons system or a towed sonar array. Nobody knew, and everybody wanted to.\n\nConsumed with finding a Victor III, I deployed aboard the USS Drum on WestPac in the winter of 1980. We crossed the Pacific and assumed our station near Vladivostok but found nothing during our first multimonth SpecOp. We surfaced outside Diego Garcia, a horseshoe-shaped atoll in the Indian Ocean, and received orders to tie up next to the submarine tender USS Sperry (AS-12). Diego Garcia had earned the deserved reputation of being the world's largest toilet bowl. Nevertheless, the tiny island was dry and had beer.\n\nAt only one-half mile across, there wasn't much to see on this mass of jungle and sand that served as the only airstrip suitable for long-range bombers anywhere near the troubled Middle East. The Seabees began their largest peacetime construction on Diego Garcia in 1971. The ambitious project took more than a decade and cost $200 million, ending with a massive complex that could accommodate some of the navy's largest ships, military cargo jets, and long-range bombers. The island also housed a contingency of spooks assigned to a Pusher Bulls Eye HFDF station. The Pusher was a smaller version of the Wullenweber elephant cage antenna array.\n\nWe had received reports weeks earlier from the Sperry on the VLF radio wire warning us of an intruder lurking near their ship. They claimed to have pictures of the large gray predator shadowing one of the submarine tender's twenty-foot launches. From nose to fin, the massive shark measured as long as the launch. We later learned that some of the cooks on the Sperry fed Hector (the nickname they gave the large hammerhead) off the bow of the ship. The shark devoured tons of scraps from the galley, which is probably why it kept hanging around. Having eaten some of the Sperry's meals in San Diego, I wondered how the damned thing was still alive.\n\nWhen we pulled alongside the Sperry during a massive rain squall with high winds, we slammed headfirst into the tender's side and smashed our sonar dome. A team of navy repair divers flew out from Yokosuka a day later to patch up the dent and replace the busted hydrophones. Commander Oliver cordially compelled me to stand shark watch while the divers worked, just in case Hector wandered by. I informed the repair divers that their first sign of imminent danger would be the frantic sound of my fins swooshing past their heads at ninety miles per hour. They all laughed, but not very much.\n\nWith repairs completed, we headed back out on our second SpecOp. Our planned liberty in Perth, Australia, was canceled due to the accident. Commander Oliver now had one black mark in his service record, and the brass never overlooked submarine collisions regardless of the circumstances. This collision, however, was rather minor in comparison to the one that almost took my life two months later.\n\n## CHAPTER NINETEEN\n\nWe're in greater danger today than we were the day after Pearl Harbor. Our military is absolutely incapable of defending this country.\n\n\u2014RONALD REAGAN\n\nTHEY SAY YOUR LIFE PASSES BEFORE your eyes just before you die. For submariners, the only thing that passes before our eyes is a wall of water. In early 1981, while aboard the nuclear submarine USS Drum, that wall of water visited me more than once in my dreams. One nightmare seemed so real that I jolted straight up in my rack from a deep slumber and pounded my head on the steel-encased fluorescent light above me. I spent the next few days with an aching head and vertical lines on my forehead.\n\nThe Drum arrived on station near the Vladivostok naval base inside the Sea of Japan, and several uneventful weeks went by despite our proximity to one of the Soviet Union's largest submarine ports. I wondered if we'd come up dry once again on this run and have to head home without any odd pod photos of a Victor III, the Soviet's newest fast-attack submarine.\n\nA month went by underwater with no sign of a Victor III, and all of us longed for the thrill of the hunt. We stood our watches, cleaned our assigned spaces during \"field days,\" drank green Kool-Aid \"bug juice,\" ate \"gedunk\" snacks, watched Saturday Night Fever until we all hated the Bee Gees, then watched it again. In the after compartments of our 300-foot-long prison, the nuclear-trained \"nukes\" conducted drills in preparation for an Operational Reactor Safeguard Exam (ORSE) scheduled within weeks of our return to San Diego. They complained about our executive officer, Bob Fricke, who insisted on perfection and so drilled around the clock and then drilled some more.\n\nThe nukes got their payback when two of them donned yellow anti-C (antiradiation) suits, grabbed a couple of handheld radiation detection monitors, and paraded down to \"officer's of country.\" They placed a test sample on the bottom of the radiation monitors so they'd tick away as if at ground zero in Hiroshima. Then they waited until the XO drifted off to sleep, crept into his stateroom, and pulled back the curtain on his rack. One of the nukes started poking Fricke in the ribs to wake him up. The XO rubbed his eyes, and his sleepy face wrinkled with perplexity as he pointed at their anti-C suits. The two nukes uncapped their radiation monitors. Both ticked away with ferocity. The XO's eyes widened and resembled the white plates in the wardroom. Playing it to the hilt, one nuke went for the jugular and said, \"Damn! I think this one is still alive. We'd better get the doc down here right away.\"\n\n\"Nah,\" the other nuke said, his voice muffled inside his yellow hood. \"Why bother? He won't last long anyway. Way too many zoomies in the ops compartment by now.\"\n\nFricke came unglued and started blubbering something unintelligible. One of the nukes said he practically peed in his rack. Laughing hysterically, the two nuclear-trained perpetrators ran from the scene. Since Fricke couldn't tell who'd been inside the thick anti-C suits, he had no one to punish. The following day the XO cut the ORSE drills in half.\n\nSix weeks into our three-month SpecOp, the cooks served butter-drenched lobster and grilled steak for the SpecOp's \"midway meal.\" Savoring the tasty morsels boosted my morale but only temporarily. I wanted to find that Victor III. Just over two months into our run, in early April 1981, I finally got that chance. After breakfast I grabbed a cup of coffee from the crew's mess, walked up the ladder into the reddened \"rig for dark\" control room, and took over as fire-control technician of the watch. My duties now included tracking contacts via the MK-113 fire-control system and, when called for, taking pictures of those contacts through the periscope. Resigned to enduring another boring six-hour watch, I sat on the bench near one of the consoles. Analog servos and synchros inside the gray metallic enclosure whined and popped as they struggled to keep up with a distant contact.\n\nDowning a gulp of coffee, I glanced around the control room. Cigarette smoke swirled into the stale air and danced with the steam from a half-dozen navy cups. Save for the sound of a jazz band playing Coltrane, the nostalgic scene reminded me of a bar off Market Street in downtown San Diego. In the dim glow, I saw the chief of the watch sitting across and in front of me on the port side of the boat. He faced a gray monolith filled with black panels covered with an array of switches, dials, and gauges. His oversized left arm almost hid the low-pressure blow panel, and his right shoulder all but covered the square snorkel control area. Just above his head, a horizontal row of red indicators validated that we had no hull openings exposed to the sea. I often wondered what might happen if one of those lights ever went from closed to open while we were deep.\n\nThe alarm switchboard rested above the COW's left ear, adorned with two rows of ten rectangular red alarm lights underscored by three-way switches. Above the COW's right ear, large silver handles jutted from a gray box with a single indicator and two black signs that read aft blow and fwd blow. Of all the panels in the control room, that one sent chills down my spine more than anything else. If the COW ever needed to pull those handles, we'd probably be on our way to the bottom, and our only hope of survival would mean a risky emergency blow to expel the water from our ballast tanks.\n\nTo the COW's right, a helmsman and planesman slouched in bucket seats, hands resting at the ten and two o'clock positions on two half-oval steering wheels. Marlboros dangled from their lips as they shared bad jokes. Each focused on two large dials at eye level that indicated the boat's depth. These two yahoos were responsible for maintaining depth control and steering the boat on the right course. When trailing a Soviet submarine, which we did often on Holystone missions, we'd often come within a few dozen yards to record various machinery and propeller noises. One wrong move by either of these sailors could cause a serious accident, possibly sending either or both subs to the bottom.\n\nAbove and in between these two, dials depicted rudder, fairwater, and stern plane angles, along with gyro course, speed, and dive bubble\u2014the latter equating to the level of the boat in a similar fashion to a carpenter's level. Just behind the planesman and helmsman, a burly diving officer puffed on a pipe. The grandfatherly smell of his sweet cherry tobacco coated the air and reminded me of home when I was a kid. My dad once smoked a pipe until he switched to cigarettes. The chief had smoker's wrinkles and a bald spot on the back of his head. His teeth had long since turned bitter coffee brown. If he hadn't been wearing a dark blue \"poopy suit\" pair of coveralls like the rest of us, I might have mistaken him for a homeless person in need of a shopping cart.\n\nTo the right of the diving control area, just in front of the MK-113 fire-control gear, a large gray navigation and plot table, covered with a chart of the area near Vladivostok, kept two people occupied: the quartermaster of the watch and the junior officer of the deck (JOOD). A panel flanking the left side of the table had recessed buttons to control various functions, and the top held up a navigation ruler. The plot served a dual role: one, to plot the course to our next destination, and two, to manually keep track of nearby contacts in relation to our track. Making sure we knew our location in relation to the other guys could be critical in preventing a collision. I knew that one wrong calculation or assumption could spell disaster and hoped that such would not happen on my watch.\n\nTo my left, on the periscope stand, stood the officer of the deck, Lieutenant Nick Flacco. He'd graduated from Annapolis in 1976 with his eye set on a patrol gunboat squadron based in Naples, Italy. As an engineering major, he ranked in the top twenty percent of his class, and that fact painted a target on his back. While still in his senior year at the academy, the pressure mounted to go nuclear. Officers trained in that discipline questioned Flacco's request for gunboats along with his sanity. During his submarine indoctrination, aboard the USS John Marshall (SSBN-611), the navigator turned to Flacco and said, \"What ever you do, don't go submarines. You won't like it.\" Flacco concurred, stating that he'd already decided on gunboats. Over the next few months, officers pushed and prodded Flacco to select either nuclear subs or surface ships, as the navy needed officers for both.\n\nFlacco chose submarines when he heard that junior officers were sleeping in the brig on aircraft carriers because they didn't have enough staterooms. Based on his class standing, the navy let him choose the USS Drum in San Diego, and he climbed down the hatch in the summer of 1978. An easy-going young officer with a pleasant smile and a get-the-job-done attitude, Flacco was a favorite with the crew. I always hoped that if we did have an emergency, he'd be our OOD at the time.\n\nThat night on watch, just over two months into our boring SpecOp, a technical \"T-Brancher\" spook tucked away in the radio room got a distant sniff on our BRD-7 electronics surveillance system. Faint at first, he almost missed the MRK-50 Series Topol radar, code-named Snoop Tray 2 by NATO. As he analyzed the signal captured by the BRD-7 further, his eyes lit up. At that time, only Victor IIIs and some Delta-class submarines used that type of radar.\n\nOur CO, Commander Michael Oliver, did a happy dance in the corridor outside the radio room when he heard the news. I watched his jig from the control room, wondering why he seemed so excited. I learned from Flacco that the spooks reported good news and bad news. The good news: the Snoop Tray 2 signal was not moving, indicating that the Victor might be resting at night on the surface, something they occasionally did before an exercise. The bad news: they were inside Peter the Great Bay near Vladivostok, which meant a possible traffic jam of lethal Soviet warships.\n\nCommander Oliver decided to chance the risks and pursue the target. If we could get some close-up shots of that Victor III's mysterious odd pod and under-hull pictures of her sleek frame, there'd be big medals and promotions galore. On the other hand, one small miscalculation could result in a catastrophic collision.\n\nOliver hadn't slept in a while, so he ordered our XO to take the conn and follow the radar signal, then wake him when we drew close enough for periscope photos and an under hull. As the fire-control technician of the watch, I had the responsibility of keeping a plot of all the contacts we detected. The Victor III wasn't moving, so that part of the job was easy. Dozens of other contacts in the area, including several submarines and surface ships, were going to and fro at fast clips, so that part proved difficult. Since our MK-113 fire-control system could plot only four targets simultaneously, I dialed Master Two, our Victor III submarine, into one of the digital computer displays, and three other contacts, representing the closest warships, into the other consoles.\n\nWe dodged the warships by running slow while weaving our way into Peter the Great Bay. As we neared our contact, just off Popov Island outside Vladivostok harbor, and the signal strength on the Snoop Tray 2 radar increased, the XO had someone wake up Commander Oliver. Our CO strode into the control room a few minutes later. He smelled like Old Spice aftershave as he approached and glanced at my plot board.\n\n\"Ready the thirty-five,\" Oliver said.\n\n\"Yes, sir,\" I said. I opened a locker, removed the 35 mm camera, checked the film status, and waited.\n\nOliver relieved the XO of the conn and called the under-hull photographic-operations party to the control room. He brought the Drum to periscope depth and raised the periscope, spun the metal cylinder back and forth, then stopped. \"There she is,\" he said. \"Bearing to Master Two on my mark...mark! Range, 900 yards.\"\n\nOur WLR-9 ESM warning indicator started beeping, signaling that enemy radar had gotten a sniff of our extended masts. Through the small PeriViz monitor mounted in the overhead near the periscope stand, I could see what Oliver saw in full color. Streaks of purple-orange clung to a barrage of gray clouds on the horizon as dawn crept toward sunrise. Against the gray, the dark silhouette of the Victor III's sloping conning tower and extended masts seemed surreal, as if only a picture out of the pages of Jane's Fighting Ships. Certainly, the real thing could not be less than a half-mile away. Lights blinked on shore behind the Soviet submarine, intimating that Russians prepared for their day just like we did. I wondered who they were, what they were like, if they loved, laughed, and cried like we did. I wondered what they would think if they knew we were hiding in their front yard.\n\nOliver pushed the small red button on the scope's right handle. I heard a soft whirring as he snapped a 70 mm photo with each push. He unglued his eye and stepped back from the scope.\n\nHe looked my way and said, \"You ready to reel?\"\n\n\"Yes, sir,\" I said, \"the 35 mm is loaded and ready.\"\n\n\"You've got two minutes,\" Oliver said as the WLR-9 beeped away in the background.\n\nI moved over to the scope well and snapped the 35 mm into position, then settled my eye onto the back of the camera and squinted. Morning light crept across the ocean as the sun peeked above the snow-capped Sikhote Alin mountain range on the horizon. With moist palms, I gripped the scope handles tighter and tried to slow down my breathing. On low power, the Soviet submarine filled my view. By feel, I adjusted the camera's focus and f-stop setting and lined up the crosshairs on the odd pod. The control room settled into silence, save for the manual snapping of the camera shutter. I snapped a dozen photos, then switched to the highest-power setting. The oval pod took up the entire crosshaired circle through which I gazed. The WLR-9 chirped away in my left ear, now delivering an almost steady procession of tones.\n\n\"Let's go, Reed,\" Oliver said.\n\nThe CO's deep baritone pushed my pulse across the red line. My fingers twitched as I swung the view over to the masts and snapped a few more pictures. Moving at light speed, I detached the 35 mm camera and flipped up the scope handles to the vertical position.\n\n\"Down scope!\" The oily mast lowered into the scope well. Around me the world turned crimson again. I squinted as my eyes readjusted to the dim red lighting.\n\n\"Well?\" Oliver said. Dark circles underscored the CO's brown eyes.\n\nI shook my head from side to side. \"I got some good shots, Cap'n, but with this lighting angle at this distance, I don't think they're good enough. I recommend we move to the other side, draw in closer, and get the light behind us.\"\n\nI couldn't believe my own words. Nine hundred yards off our port bow sat one of the best attack boats the Soviets had. In nearly every respect, she was comparable, if not better than, our Sturgeon-class submarine. Yet here I was recommending that we move in close enough to smell each other's armpits.\n\n\"I concur,\" Oliver said as the XO leaned in close to listen. \"The 70 mm shots probably aren't going to cut it either. XO, you have the conn. Reed, follow me.\"\n\nOliver walked toward his stateroom. For a brief second, fear and confusion froze my legs. Oliver stopped, turned, and gave me a look.\n\nFeet unfrozen, I followed the CO to his stateroom. Oliver opened the door, and we ducked inside. He sat down at his desk and looked at the floor. I closed the door and stood, waiting for him to speak.\n\n\"I may need you to egress,\" Oliver said, lifting his head.\n\nMy heart shot into my throat.\n\n\"Egress, sir?\" Confused, my thoughts moved in slow motion, as if smothered by cold syrup.\n\n\"I may need you to take a Draeger, egress, and get us some better shots of that Victor III. We need close-up photos of that pod, unfettered by our periscope optics, to determine what that thing is.\"\n\nI didn't know what to say. Take a Draeger? Egress? That meant donning a bubbleless rebreathing device, locking out of the escape trunk in a Soviet harbor, swimming to the surface while tethered to a line, and taking photographs of a Soviet submarine just a few hundred yards away. Even though I had trained with the SEALs for such a mission, I knew he was asking me to volunteer. I also knew that if I did wind up taking those photos, I could never talk about the ordeal with anyone, not even most of the crew.\n\nI had to push my reply past the lump in my throat. \"I'll get the underwater enclosure for the camera and suit up, sir.\"\n\nOliver nodded. \"When you receive my order, and no sooner, you will egress, stay hooked to the line, surface long enough to take a few photos, and then return. Is that understood?\"\n\n\"Yes, sir. Understood.\"\n\nOliver rubbed his palms together, looked back at the floor, and whispered something to himself that I didn't understand at the time. \"If they can send divers to tap cables, then I can damn sure send one to take photos.\"\n\nOliver stood and dismissed me.\n\nI left his stateroom and headed toward the bow of the boat. There I suited up and readied my gear, which included placing the 35 mm camera in a watertight enclosure. My father's words, spoken years earlier when I feared stepping to the pitcher's mound in Little League, churned in my head.\n\nFace your fears, son. If you don't, they will own you.\n\nI struck out nine batters in that game.\n\nOnce inside the bow compartment, I opened the bottom hatch to the eight-foot-diameter escape trunk. With the help of a seaman trained in escape trunk operation, I climbed up the ladder and squeezed inside the oval. I closed the hatch below my knees and fought off the suffocating fingers of dread that curled about my neck. A small, dim light cast strange shadows about the tiny metal dungeon filled with gauges and valves. Alone and stuffed into my hot neoprene wet suit, I sat on the bottom of the cold trunk and shivered. My eyes focused on the small metal communications box mounted on the bulkhead, from which I knew my orders to go would be delivered. Hundreds of thoughts did somersaults inside my head, all of them dismal.\n\nWill I get good enough photos? Will the Soviets spot me? Will I survive the mission?\n\nMeanwhile, in the control room, things went from bad to ugly.\n\nWhile Commander Oliver had been talking to me, the XO took the boat deeper to maneuver to the other side of the Victor so we could get shots with the sun behind us versus glaring on our scope lens. This had been my suggestion to Oliver before we left the conn together. With the Victor III sitting still, sonar remained useless, and our ESM's Snoop Tray 2 radar hits were the only means to determine the target's approximate range and bearing. That information allowed for only a rough idea of the sub's location, despite the previous periscope fix.\n\nThe XO, Bob Fricke, ordered Nick Flacco to maneuver the Drum to a point opposite our previous location, then bring the boat to periscope depth again. Knowing that doing an under-hull photographic operation might be next, Flacco ran through a mental checklist. As he did, a silent alarm went off in his head. \"Shit, the wire.\"\n\n\"What did you say, OOD?\" Fricke said.\n\n\"We need to reel in our floating VLF radio wire,\" Flacco said. \"It's still out there.\"\n\n\"Dammit!\" Fricke said. \"Get someone from radio up here now.\"\n\nFlacco called up a radioman, who sprinted into the control room. The petty officer opened a door at the front of the room and stepped inside the tiny area that led up to the bridge. He undogged the lower hatch, climbed up the ladder, and started bringing in the wire. Meanwhile, Oliver returned to the conn and took over. He approached the number two periscope and waited until Flacco confirmed that the Drum had almost reached periscope depth. Oliver wrapped his hand around the orange metal hoop encircling the scope well, then pulled the round bar clockwise. \"Up scope.\"\n\nHands gripping the scope handles, eyes seated into the rubber socket, Oliver waited for his prize to come into focus. For a brief second a smile played on his lips as he savored the moment. The Drum neared the surface and Oliver's smile vanished. He frantically lowered the scope and yelled, \"Emergency dive!\"\n\nToo late. A thunderous boom shook the boat. The radioman who'd been reeling in the wire tumbled down the ladder and slammed onto the deck. Blood oozed from his head. The sound of metal screeching over metal filled everyone's ears in the control room. The boat lurched forward and angled down at the bow by ten degrees. Flacco glanced at the unconscious radioman, then at the door that led to the bridge. The lower hatch is still open, he thought, if we have flooding now...\n\nDown in the bow compartment, shoved into the escape trunk, I heard a deafening clap above my head, followed by an ear-splitting metal shriek. The dim light in the trunk went out, leaving pitch-black darkness in its wake. The force shoved me head-first into a valve handle. My jaw hit the metal wheel. A stinging pain rippled across my face, and the salty taste of blood filled my mouth. I cupped my palm across my bleeding lip and felt for the communications unit in the dark. My fingers found the square box, and I depressed the key. I spat out a clump of blood and blabbered something unintelligible. Nothing but silence. I keyed the box again. Still nothing. I tried opening the bottom hatch to the trunk using every bit of muscle I could muster, but the wheel would not turn.\n\nAlone in the dark, with the world closing in around me, I wondered if we had suffered a major casualty, wondered if we were on a death spiral toward the bottom. For a brief moment, I contemplated flooding the trunk and escaping through the upper hatch. Then I remembered that we were deep in Soviet territorial waters and I knew secrets. My chest started heaving, and I realized that the oxygen flow to the trunk was probably out.\n\nI figured we must have collided with the Victor and the force of the impact near the escape trunk had knocked out the bow compartment communications circuit. A shock wave must have hit oxygen bank number one and ruptured the O2 valve. Tracing the lines in my head, I saw how this could halt the flow of oxygen through valves O-4 and O-27 that led to the escape trunk. The collision must have also caused a pressure imbalance in the trunk, making it impossible for me to open the hatch from the inside. I spat out some more blood and bit on my Draeger's mouthpiece. The throbbing pain around my bottom lip damn near doubled me over as I sucked in some air. I was now living on borrowed time.\n\nUp in the control room, Flacco had someone drag the radioman away from the bridge door and shut the lower hatch. Someone else called for the doc.\n\n\"Why aren't we diving?\" Commander Oliver yelled.\n\nFlacco glanced at the depth gauge. Still at sixty feet.\n\n\"Chief of the Watch,\" Flacco said. \"Flood forward trim tanks.\"\n\nThe boat surged forward a few feet. She angled down even more but still did not go deep. More screeching and grinding rippled through the control room, followed by several loud thuds.\n\n\"I think we're impaled in the Victor's ballast tank,\" Flacco said. \"We're just pushing them sideways.\"\n\n\"All back full!\" Oliver ordered.\n\nMetal crunched as the Drum moved back several feet. The bow dropped by a few degrees.\n\n\"All ahead full,\" Oliver said. The boat shot forward and downward. The depth gauge registered a hundred feet and descending. Then the flooding started.\n\nRain poured from the overhead and drenched the scope well. Flacco looked up. One of the scope seals had ruptured in the collision. Cold salt water rained onto the deck and splattered shoes. The flooding alarm sounded.\n\nOliver clicked the 1MC. \"Now flooding in the control room.\"\n\nThe XO called for a damage control party. Auxiliarymen came running with tools and patches. Taking on water, the Drum sped toward test depth, 1,300 feet down. Freezing ocean water sprayed out of the scope well as if from a pinched hose. Flacco knew if they couldn't get the flooding under control soon, there'd be serious consequences. Vital equipment might short out, and systems could die, all of which could send the Drum to the ocean floor.\n\n\"Quartermaster,\" Oliver said as the A-gangers worked on the scope well leak, \"plot a course to Chin Hae.\"\n\nSouth, thought Flacco, to Korea.\n\nFlacco heard the pinging of Soviet 50 kHz active sonar through the hull. He knew that ASW forces were now hell bent on catching the Drum red handed.\n\n\"I don't have a Chin Hae chart in here,\" the quartermaster said.\n\n\"Then just take us south!\" Oliver yelled.\n\nIn the escape trunk, I heard Oliver's flooding report over the 1MC. The announcement meant someone was still alive, but the flooding verified that we had problems. Regardless, I had to get out of the trunk. The air in my Draeger would not last forever. I took out my diver's knife and started tapping Morse code on the metal hatch. My dad taught me the entire alphabet when I was a kid, and at one point I could even keep up with a CW transmission. Now, however, all I could remember were a few letters. It didn't matter; the seaman on the other side of the hatch probably knew less than I did.\n\nI tapped SOS.\n\nNo response.\n\nI tapped again louder. Still nothing.\n\nPanic threatened to block what little air I had left from reaching my lungs. I remembered my navy diver training in Hawaii, where they'd harassed me in the water every day for hours. They pulled out my regulator, spun me in circles, and damn near tried to drown me. That training taught me a valuable lesson: how to control my fear. Now, thousands of miles from that tropical paradise, I closed my eyes and said a quick prayer. Then I sucked in a few breaths and tapped again.\n\nFinally, the lower hatch opened, and fresh air rushed in.\n\nBack in the control room, the A-gangers managed to stop the flooding and fix the leak, while the corpsman patched up the radioman. He'd sustained a concussion and a deep cut to his forehead. Flacco watched the doc help the petty officer hobble out of the control room.\n\nThen the Indians showed up and surrounded the wagon. Soviet helicopters dropped sonobuoys that bombarded the ocean with active sonar pings. ASW destroyers and fast gunboats came out of Vlad and started chasing the Drum southward. Dozens of propellers chopped at the Sea of Japan, and our sonar jockeys couldn't keep up with all the contacts. Commander Oliver ordered a thirty-degree course change\u2014a zig to remain undetected. Flacco figured the Soviets knew what Oliver knew: that the Drum could only head south through a narrow passage-way to escape. If they threw enough ships and planes out there, the odds of getting away were about nil.\n\nWhile Flacco contemplated his odds of survival, a depth charge exploded.\n\nBy now I had scrambled out of the escape trunk and sprinted to my rack to pull on my coveralls. I climbed the ladder up to the crew's mess, found the doc, and got a patch for my severed lip. I didn't bother to look in a mirror at the damage. I scrambled up to the control room and slid onto the bench next to a half-dozen officers and sailors in front of the fire-control equipment. The weapons officer (Weps) glanced at the bandage on my face and gave me a look that said, \"What the hell happened to you?\"\n\nI didn't bother to explain.\n\nAnother depth charge exploded, and all eyes looked upward. All lips muttered silent prayers. Weps informed me that we'd rammed into the Victor III and probably smashed the entire front end of our sail. The ESM antenna was gone, and both periscopes were useless, not that we needed them now anyway. Flooding occurred but had been contained, and now every Red ship in the Far East meant to do us harm. I wondered if I should have stayed in the escape trunk.\n\nSonar reported that our closest pursuers were two Kresta I\u2013class guided missile destroyers. I dialed them into the fire-control gear, knowing that they could hit a top speed of thirty-two knots. They carried two twin-missile launchers and a Ka-25 Hormone ASW helicopter on the after deck, complete with sonobuoys. Weps figured that the depth charges were probably light warning explosions, but we had no way of knowing for sure.\n\nWe zigged and zagged as ASW ships and planes pinged. Oliver had us hug the bottom for the next two days while explosions shattered the silence, some far away, others so close they rattled dishes. My dad once told me that fear does not discriminate. It doesn't care about our nationality, wealth, religious beliefs, or lack thereof. Fear is an equal opportunity employer, and when I looked at my crewmates, I could see the evidence of it ooze from every pore. We knew that if caught, Ivan would show us no mercy. We had entered their territorial waters and rammed one of their boats, and now all bets were off. Oliver would take us below crush depth before he surrendered the boat to the Soviets. All of us understood well the consequences should we fail to escape, yet everyone to a man kept his cool and did his job well. At that moment, I understood what it meant to be a submariner.\n\nWith vigilance and creative evasions, coupled with lots of luck, we finally made it out of Soviet territorial waters. The Soviets continued their pursuit anyway, and after another day of dodging dozens of warships and planes in the Sea of Japan, sonar reported the sound of a possible collision, faint and distant. We had no way of knowing who or what had caused the metallic crunch. Weps speculated that two Soviet ships, while pursuing us, had collided. Within hours of hearing the smack, the Soviet prosecution activity dropped in half as a large number of ships and planes headed toward the sound of the faraway collision.\n\nThanks to the sudden decrease in the number of Soviet vessels trying to find us, we managed to sneak out of the Sea of Japan and back to the safety of Apra Harbor in Guam. We surfaced and entered the harbor in the middle of the night to avoid detection from Soviet satellites. As the boat moved silently through the dark, I watched silver moonlight glitter on the ocean near my favorite dive spot and thanked God for another day on earth. Then I glanced up at the sail. The front masts were bent or missing, and the bridge area had been smashed like an aluminum beer can.\n\nWe tied up next to the pier and workers shrouded the entire sail with a cover to ensure that Soviet satellite photos would not reveal the obvious. As was the case with many Cold War SpecOps, the boat's logs were altered. No record remained, save memories, to validate that the USS Drum was ever in the vicinity of a Soviet Victor III submarine in Peter the Great Bay.\n\nI later learned that two Japanese sailors died so that more than a hundred of us could live. While we were running from the Soviet navy, the nuclear submarine USS George Washington collided with the Japanese freighter Nissho Maru on April 9, 1981, 110 miles southwest of Sasebo, Japan. When the Washington surfaced, she ran into the underside of the freighter and damaged the hull. The Japanese ship sank within fifteen minutes. Two of her crew of thirteen went down with the vessel. The Washington sustained only minor damage but then became a decoy of sorts for the Drum. The collision distracted more than half of the Soviet forces searching for us, as they assumed incorrectly that the two collisions\u2014the one with the Victor III and the one with Nissho Maru\u2014were caused by the same boat. The Washington managed to sneak away, but the accident sparked a political furor in Japan.\n\nPresident Ronald Reagan now had two submarine accidents on his hands. Prime Minister Zenko Suzuki blasted him for taking more than a day to notify Japanese authorities about the Nissho Maru's demise, and the fact that a U.S. P3 Orion aircraft circling overhead made no attempt to rescue the survivors. On April 11, President Reagan expressed regret over the accident and offered compensation to the victims while assuring the Japanese that radioactive contamination need not be a concern.\n\nThe Soviets threw their own spears. They insisted that a U.S. submarine had collided with K-324, a Victor III\u2013class nuclear submarine sitting on the surface in Peter the Great Bay. K-324 sustained severe damage, and some of her crew were hurt in the accident. The United States denied the presence of any American submarines in the area at the time.\n\nPart of the Northern Fleet and stationed out of Saint Petersburg, K-324 had come off the ramp at the Admiralty yard in 1979. She became the seventh vessel of the Komsomolsk line. Two years after the incident, thanks to the photos taken in 1981 by the USS Drum and other submarines, the United States identified the odd pod on the Victor III's rudder as a housing for a towed sonar array. Proof came later that year when, on October 31, 1983, the USS McCloy (FF-1038) and K-324 were ensnarled in each other's towed arrays some 280 miles west of Bermuda. K-324 was monitoring U.S. ballistic missile submarine movements, hoping to find one of our \"boomer\" SSBN submarines to shadow. The incident severely damaged K-324's propeller. A tug towed her to Cienfuegos, Cuba, for repairs. Two years later, in 1985, the Soviets decommissioned K-324, and her crew never knew who hit them that day in April 1981.\n\n## CHAPTER TWENTY\n\nIf we survive danger it steels our courage more than anything else.\n\n\u2014REINHOLD NIEBUHR\n\nAFTER OUR FATEFUL COLLISION WITH THE Victor III, while nursing a scar on my chin, I no longer took life for granted. Even the smallest things had more meaning now: a glorious sunrise, a fluttering hummingbird, and the charming girl I had met more than a year earlier. I married that girl in June 1981.\n\nIn September, a month before my enlistment end date, my navy detailer called. He said that DevGru One\u2014the Special Projects guys\u2014were interested in talking to me about an opening on the USS Richard B. Russell (SSN-687). I didn't know it then, but the Russell was the Parche's eventual successor for Ivy Bells missions. Around that same time, a couple of civilian recruiters contacted me from Eastman Kodak Company. They liked the combination of my electronics, computer, submarine, and photographic training and offered me a different challenge with a new program starting up in Colorado. I struggled with the decision, as my time in the navy had been some of the best years of my life. As one of my crewmates on the Drum once put it, \"I wouldn't have taken a million dollars to extend my enlistment after we smacked that Victor. Then again, I wouldn't have taken two million to forget the experience.\"\n\nI went to work for Eastman Kodak at the end of 1981 and have never regretted the decision. One of the most valuable lessons I learned during my time on the boats can be summed up in one word: team-work. Having to rely on others for my very survival, who also relied on me, taught me that even if I don't like someone, I need always treat them with respect and value their abilities.\n\nWhile my time in the navy came to an end, others were still trudging through the Cold War mud, some literally. By late 1980, the USS Parche had all but replaced the Seawolf for cable-tapping missions. But the old girl still had at least one more run in her, and no one knew that this would be the most difficult and dangerous of her storied career. Former submariner Jimmy Carter held the reins as president of the United States at the time and was up to his eyeballs in international issues. Iran had captured fifty-three Americans and held them hostage for almost a year. Carter authorized the military rescue mission Operation Eagle Claw in April to free them, but the mission failed, raising tensions in the Middle East to a pitch almost on a par with the Soviet Union.\n\nSix months later, in October 1980, James Rule reported to the USS Seawolf. He walked through a prison-like maze of gates and approached the guard shack overlooking the drydock enclosure. The Wolf looked old but quaint, with her flat teakwood deck, big bull nose, and sleek frame resembling a diesel boat. Bright overhead lights glared off the foreheads of a dozen dungareed sailors who chipped, painted, pounded, and tried to fix the Seawolf's bruised and battered body. Don Langaliers, the Seawolf's chief torpedoman, met Rule at the shack and walked him through orientation, which included a series of fifteen escalating briefings, each one peeling back another layer of the boat's supersecret onion.\n\nAs Rule began meeting the crew, he noticed something interesting. The Seawolf should have been decommissioned years earlier, or at the very least demoted to a blue-haired geezer status and assigned to a minor role supporting the fleet. Instead, she was still one of only two frontline work horses with perhaps the weight of the Cold War's outcome resting on her slumping shoulders. Her crew did not complain, however, and none asked for transfers. Not a man whispered jealous remarks as they watched the crew of the USS Parche\u2014the other Special Projects boat at Mare Island\u2014disappear through the hatch of a far more modern machine.\n\nSeawolf's sailors worked hard and played hard, the latter occurring most often at the Horse and Cow Saloon in Vallejo. That's where Rule mingled with most of the crew on a casual basis, including Fire Control Technician Third Class Tony Mignon. Rule and Mignon soon became friends and talked in excited whispers about what lay ahead when the Seawolf came out of drydock. Rule also met some of the 140 submariners assigned to the USS Parche. Due to the Seawolf's constant state of disrepair, the Parche's crew often called their neighbor boat the \"Pier Puppy.\" They laughed and asked Rule if he enjoyed his duty at \"Building 575.\" The kidding had started after the Parche made the 1979 Ivy Bells run into the Sea of Okhotsk while the Seawolf's imprisoned crew worked on their aging boat.\n\nRule took the jabbing from Parche's crew in stride and got along with most of them, who also lived in the wood-framed barracks at an old munitions depot near the docks. Over the next several months, he shouldered a torpedoman's workload on the Seawolf alongside the team's leading petty officer. He helped load MK-37 torpedoes, scrub decks, fix gear, and stow several Mobile Submarine Simulators (MOSS) on board, which Rule hoped they'd never need to use. Subs shot MOSS from their torpedo tubes as decoys when the Soviets were trying to prosecute them.\n\nAt night, Rule migrated back to the Horse and Cow, where Don Langaliers demonstrated his secret weapon. The man had pierced his penis to connect to a long, thin chain. He ran the chain up through an opening in his shirt. When he wanted to impress the ladies in the saloon, which were the sort that thought perfume should smell like the sump tank on a Harley Davidson, he handed them one end of the chain. When they pulled, the circumcised tip of Langaliers's manhood peaked above his belt buckle and caused eruptions of high-pitched laughter throughout the smoke-filled bar. Although most of the crew thought the chief was \"one crazy son of a bitch,\" and his wife, Bobbi, often shook her head in disgust at his wild antics, she never had to worry about his fidelity or family devotion.\n\nLangaliers considered his torpedo gang a part of that family and often invited Rule and the other torpedomen over to his house for barbecue and beers. The guys played hard but also worked hard, and they respected Langaliers as a leader and mentor. They knew that he'd always have their back, and likewise, they'd always have his. As the months came and went, they became a close-knit fraternity that drank beer at night and worked on weapons of mass destruction during the day.\n\nBy the end of 1980, with the overhaul finally complete, the Seawolf took a short run up to Bremerton, Washington, to conduct sea trials in Puget Sound. Not a day out to sea, Rule found himself surrounded by smoke in the back of the crew's mess. Someone ran past him at a fast clip, with an EAB mask sucked to his face and a hose dangling in his hand. The bitch-in-the-box alarm alerted the crew to a \"fire, fire in the engine room.\" Rule ran to find his own EAB mask. Still unqualified, he sat in the crew's mess, helpless and scared as blue-suited sailors raced around him to dog hatches, flip switches, close valves, or help stifle the fire.\n\nThe Seawolf's CO, Commander Michael C. Tiernan, ordered an emergency blow, and the boat shot to the surface. Ten submariners were overcome by severe smoke inhalation before the crew could ventilate and clear the air. Glad to still be alive, Rule felt ashamed because of his fear. Later he molded that trepidation into a stubborn drive to spend a year qualifying as a submariner so he could contribute during the next emergency.\n\nLimping and wheezing, the Seawolf headed back to drydock. She was scheduled to do the next Sea of Okhotsk cable tap, while the Parche went to a new tap site in the Barents Sea. With the Seawolf back on the blocks, the Parche took her place again for the summer Okhotsk run to retrieve the previous year's pod recordings. The Seawolf was not scheduled to see the dark of ocean again until late 1981, and for Commander Tiernan, this would be his first run into the heart of danger. Until then, the crew viewed Tiernan almost in the same light as an unproven nonqual and took to calling him \"Captain Milquetoast.\"\n\nThe Seawolf's executive officer, J. Ashton Dare, was held in equal esteem. Dare came from aristocratic stock, and as the son of an admiral, he sometimes exhibited an arrogant demeanor. In reference to Dare's strict policies, the crew often exchanged comical remarks such as \"Dare'll be no liberty, Dare'll be no fun.\" For the next several months, while the crew slaved to repair the tired Wolf, no one had much fun. Jim Rule watched the Parche head out to sea and longed for the chance to go on a mission to the Sea of Okhotsk. Little did he know that he'd soon regret that wish.\n\nWHEN FIRE CONTROL TECHNICIAN THIRD CLASS Tom Ballenger reported to DevGru in June 1981, he wanted to be integrated into the crew of the Seawolf right away. He had just completed his training on the antiquated MK-101 fire control system\u2014a predecessor to the MK-113\u2014in Groton, Connecticut. The crisp-uniformed DevGru officers approached him while he was in FT school and dangled the Seawolf carrot. He bit and moved his pregnant wife to Northern California. She delivered their son a month before Ballenger walked through the gate at Mare Island to see his new boat. The submarine was still at sea, and the Special Projects folks required a long waiting period to complete background checks. They needed to validate that Ballenger wasn't a \"card-carrying commie.\"\n\nBallenger spent his six-week waiting period in the \"woodshed,\" a dockside woodworking shop. Each day he gazed at the Seawolf in drydock, scaffolding covering her sides, arc welders sparking into the air, and dockworkers carrying pipes, valves, and other parts in and out of her worn hull. After weeks of sawing and sanding, Ballenger finally got called to the SCIF, the secured briefing facility used to orient new members of the Seawolf and Parche crews.\n\nHe signed a decades-long gag order not to talk about anything he'd see or hear about on the Seawolf, then endured a battery of fifteen security briefings that discussed the boat's special-projects mission but offered few details. Briefings completed, a yeoman whisked him down to the boat. He walked on board the old girl, thinking that everything he'd learned about nuclear submarines held no meaning here. The Seawolf was a unique specimen. As a diesel\/nuke hybrid, she resembled both and yet neither. Ballenger felt lost and confused as he followed the yeoman to the boat's sail, into the conning tower, and down through a hatch that dropped into the wardroom, past the CO and XO's staterooms, then officer's country, then forward through the control room and navigation gyrospace area into the crew's mess to meet some of the crew.\n\nThree tables lined up athwartships and two along the starboard bulkhead. The galley was located forward of the tables. Chiefs were allowed to sit at the back table near two main battery breakers, the nuclear-trained \"nukes\" at the back parallel table and the spooks and divers at the forward table. The yeoman sat Ballenger and the other inductees at a center table. A beefy 300-pound chief with round cheeks approached. He grabbed Ballenger's shoulder, squeezed, and said, \"Get a haircut, nonqual puke, or find another boat.\"\n\nBallenger offered a shaky grin. \"Okay, Chief...?\"\n\n\"Moorman. Master Chief Dave Moorman. I'm the chief of the boat. Don't ever cross me.\"\n\n\"Yes, sir.\"\n\nMoorman squeezed harder and rolled his eyes in disgust. \"Don't call me sir. I'm a chief, not an officer. Understood?\"\n\n\"Yes, sir, I mean...Chief.\"\n\nBallenger spent the next sixty days in the deck division, chipping, painting, and fixing until they allowed him to assume the duties commensurate with his training as a fire-control technician. He then met FT Leading Petty Officer Tony Mignon, who was a nice enough guy but seemed to be wound a bit tight. A wiry but strong southern boy from Alabama, Mignon loved to wear his faded gray Confederate soldier's hat under way. Seems like he never took it off, not even to sleep. Most of the crew liked the guy, who told dumb but funny jokes and talked about his life's goal of unplugging from society and living off the land by himself in the mountains.\n\nMignon reported to Fire Control Chief Powell, who fit the typical chief profile of semi-dumpy and overweight. Powell had light brown hair, a scraggly beard, and thick glasses. Soft-spoken, the chief knew his stuff about the MK-101 fire-control weapons systems\u2014the only electronics piece of gear situated on the Seawolf's \"wet side\" in the control room. The port side had blow valves, flood, and drain manifolds, various pipes, and other \"wet\" mechanical items, whereas the \"dry side\" held most of the electronics, except the FT gear, which was positioned all the way aft on the port side. An old tube-style power supply, just forward of the ladder into the crew's mess, gave life to the MK-101. The gear consisted of an analog target-tracking Position Keeper and a torpedo control and firing panel that fed data to the MK-37s in the torpedo tubes. Operators selected a tube and programmed firing order via an old rotary-style control device, about as antiquated as a telephone with a dial versus buttons.\n\nMonths passed, and Ballenger slowly adjusted to life on the boat. Chief Torpedoman Langaliers enlisted his help, along with that of torpedoman Jim Rule, to travel into the hills near Concord, California, and practice detonating blocks of C4 explosive. They'd jam detonation chords and blasting caps into the aft ends of cow carcasses, find cover, and blow the cows into hamburger. Back on the boat, they rigged blocks of the C4 with three bomb blasters in strategic locations, forward, amid-ships, and aft. No reasons were given for the explosives, but everyone knew that if they were caught on station, they would not be taken alive.\n\nAs a gesture to offset the unease consuming his torpedomen, Chief Langaliers brought a ceramic frog down to the torpedo room. In a strange but funny initiation ceremony that somehow involved his chained penis, Langaliers christened Beauregard the frog as their official voyage mascot and wedged him into an overhead spot near a torpedo tube.\n\nPrior to the Seawolf's departure, Ballenger met some of the navy divers who'd be conducting the cable-tapping missions in Soviet waters. They were all seasoned first-class or master divers who the crew called \"heroes.\" When Ballenger walked past the chart room Special Information Center (SIC) \"spook\" area, curtains usually blocked his view. Occasionally, he caught a glimpse of the equipment inside, and intuition combined with his electronics knowledge gave him a good idea of its use.\n\nNear the torpedo room, Area D housed the diving chamber, and a section called the aquarium held the four camera-outfitted mini-sub Fish. An assortment of video equipment, amplifiers, and other electronic gear also decorated the space. The dog-legged area took up ten feet by almost five feet and could hold up to thirty people. All totaled, up to 190 men could be loaded into the Seawolf's belly, consisting of 125 standard boat crew, 10 or so spooks, and 50 other Special Projects personnel.\n\nAlmost a month after leaving port, the Seawolf arrived on station over the Soviet communications cable. Under way, Tom Ballenger qualified as a planesman\u2014one of the guys responsible for moving the boat up and down in the water. He sat in the control room in a bucket seat and helped guide the Seawolf into position near the twenty-foot-long cable-tapping \"beast\" pod. On the sub's underbelly were two sledlike skids that the crew called skegs. These allowed the boat to sit on the bottom near the pod. In the control room, Commander Tiernan ordered the chief of the watch to balance the water in the trim tanks to level the Seawolf over the cable. The COW flooded the tanks too fast with too much water. The boat lurched forward and downward. She was headed in a beeline for the mud.\n\nThe COW yelled at the diving officer, who barked an order to Ballenger. His heart racing, Ballenger tried to unscramble the order in his head. As the planesman, he pulled upward on the half-wheel clenched in his hands. Employing the boat's built-in thrusters\u2014which helped maneuver the sub like the tiny rockets on a space capsule\u2014he tried desperately to compensate for the extra weight in the trim tanks, but the boat was too heavy. He managed to keep the Wolf from colliding with the ocean floor, but one of the skegs slammed down on top of the Soviet communications cable.\n\nCommander Tiernan fumed and yelled an obscenity. Ballenger was certain that the skeg whacked the cable hard enough to alert the Soviets, who might at any minute send out a posse to investigate. Tiernan kept the boat silent and still for a long while, waiting and listening. Nothing happened. Satisfied that he'd dodged a bullet, he sent out the divers. The heroes donned their MK-11 deep-sea diving suits and Kirby Morgan masks. They locked out of their pressurized hyperbaric diving chamber and pushed heavy metal boots toward the beast nearby. They retrieved the tapes from the pod, which had been recording conversations from the communications cable for several months, and delivered them to the spooks inside the boat. The T-Branchers collected, verified, and calibrated the signals, while the I-Branchers listened to the Soviet jabber to validate the recordings.\n\nStill outside and near the beast pod, one of the divers felt something tug at his arm. He held his hand out and stared at his glove. Currents moved his arm up and down in a steady rhythm. He contacted the Seawolf and asked if they felt anything. They did, but it seemed minor, probably just a light storm overhead stirring the currents. The diver shrugged and continued working.\n\nThe Wolf was too deep to have the VLF wire floating up near the surface to pick up radio signals. As such, Commander Tiernan and crew did not hear about the twin storms converging from hundreds of miles away as they marched toward the Kuril Islands. They didn't know that every ship in the Sea of Okhotsk had received a warning to leave the area immediately, that high winds were pounding seashores, toppling trees, and threatening to sink vessels as they scurried to safety.\n\nNaval command centers issued urgent reports that two major tempests were now dancing together and kicking up fifty-five-knot geysers of water. When the typhoon hit, since she was 400 feet beneath the gale, the Seawolf at first only shuddered. Her divers only sensed a few strong currents. As the perfect storm above turned uglier, the Seawolf groaned and rocked from side to side. Her three divers no longer felt mild flutters, but were now upended and pulled across the sand like mannequins in a flood.\n\nThe ocean grabbed one diver, and with the vengeance of a wrestler, hauled him about the ocean floor until his leg landed under the edge of a skeg attached to the sub. The Seawolf's ski leg had lifted into the air during the boat's rocking. With terrified eyes, the diver lay helpless in the silt as the skeg reached its highest rocking point and then started back down, right on top of him. The skeg brushed his protective suit just as the hand of another diver pulled him free. Thanking God for the miracle, the diver followed his two buddies into the Seawolf's chamber hatch.\n\nInside the boat, cans, plates, cups, utensils, and anything not secured went flying. Sailors ducked as the projectiles whizzed past their heads. Beauregard, the torpedomen's porcelain mascot, jumped from his loft and careened against a torpedo tube. Tiernan had seen enough. With the divers locked back inside, he terminated the mission and ordered the Seawolf off the bottom. But before the boat moved an inch, an alarm sounded. A reactor specialist standing next to a heat-exchanger gauge saw something that raised the hairs on his neck. The gauge, which measured the temperature of cycled cooling water for the reactor, had spiked into the red zone. The nuke knew in an instant that something was clogging the intake valves. He checked and discovered they were filled with sand. Grabbing a sound-powered phone, he called the control room.\n\nAfter hanging up the phone in the control room, Tiernan, Dare, and a flurry of nukes clad in blue ran aft. When they arrived, wet sand had blanketed the compartment. The Seawolf's skegs were now buried in the mire and the boat's intake vents, normally kept several feet off the bottom by the skegs, had been pushed flat against the bottom. Now the engines sucked in sand and tiny sea creatures, along with salt water. The uninvited intruders jammed their way into the turbines and generators and refused to leave. What's worse, as the Seawolf rocked, she sucked in more and more of the stuff and took on additional weight. Worse still, the storm's incessant pushing and pulling had wedged the Seawolf's skegs deep into the sand. She was now stuck, 400 feet deep in Soviet waters, with no way to break free.\n\nTiernan knew that the sand could shut down the reactor, which would take days to restart, if it ever did. He huddled his officers and senior \"salty\" enlisted men together to brainstorm a way to break free. What ever they tried had to be done with care and speed. Soviet warships patrolled the area, and if the Seawolf rocketed to the surface, they'd be a sitting duck. The few MK-37 torpedoes they carried on board were no match against a fleet of missiles, torpedoes, and depth charges. The Seawolf's slow speed and dog-barking noise would make her easy to catch. That left few options that would not take them from the pan into the fire.\n\nThey revved the engines and tried a controlled emergency blow, all without luck. The Seawolf remained captured, a fly trapped in the ocean's web. Forward and backward they pushed and pulled in an effort to wrest one or both skegs free. Nothing worked. Seventy-two hours ticked by, one agonizing minute at a time. A vital reactor system dropped to one-third efficiency as the vents continued to suck in sand and silt. Carbon dioxide levels rose as many of the ship's systems were shut down, and burner and scrubber capacities diminished. Some of the crew were sent to their racks to conserve air, while others started to come unglued.\n\nEvery sailor who volunteers for submarines must pass a battery of psychological tests designed to weed out the claustrophobic, unbalanced, antisocial, and potentially psychotic. Dealing with the unstable on a submarine, where a crew lives in close proximity for three months or longer, can quickly become a nightmare. Occurrences of the \"crazies\" are a rarity, and when they do happen, captains always fear they could trigger a chain reaction. When the fourth day passed, and the Seawolf remained welded to the ocean floor, Commander Tiernan hoped that his crew would keep it together but worried that someone might crack under the pressure.\n\nHis eyes glazed, Petty Officer Tony Mignon staggered his gangly, tired body up the ladder into the control room. His Confederate hat covered the top of his head but did little to hide the frightened, distressed look on his face. Chief of the Watch Don Langaliers, while sitting next to the ballast control panel, watched Mignon approach from the back of the control room near the interior communications (IC) switchboard equipment. Langaliers noticed that Mignon's eyes were moist and dazed. The young FT muttered something unintelligible, followed by a loud cry about how they were all going to die, and no one would know what happened.\n\nThe XO and a couple of officers near the periscope stand, who were going over ideas to gain freedom from the sand, looked up with stunned faces. Mignon let go another cry, along with a string of profanities. Langaliers leaned back from his station and said, \"Tony, chill out. We need you to be quiet right now.\"\n\nThe planesman and helmsman, both gripping steering wheels at the front of the control room, flashed each other puzzled, worried looks. Mignon ignored Langaliers' request and started yelling even louder. Then he banged his head hard against the Ships Inertial Navigation System (SINS) gyroscope housing, all the while screaming obscenities and doomsday chants. The skin on his forehead split open. Blood oozed onto the SINS housing. Commander Tiernan, whose stateroom was nearby around a dogleg from IC alley, stuck his head out into the passageway to see what was going on. Tony lunged at Tiernan, screaming that they were all dead men, and it was the skipper's fault.\n\nLangaliers jumped from his station. He and another crewman, J. K. Branham, grabbed Mignon before he could tear out Tiernan's eyeballs. They wrestled the strong FT to the deck, while the XO called for the doc. A minute later, Don \"Doc\" Post bolted up the ladder with syringe in hand. Langaliers and Branham held Mignon down, while Doc shoved a needle into the man's arm. Mignon stopped blabbering and started drooling. A couple of guys escorted the sedated sailor down to the torpedo room, where Jim Rule was standing watch.\n\n\"Seeing Tony like that hit us all pretty hard,\" said Rule. \"The doc asked me to keep an eye on him in the torpedo room. Once Tony's meds wore off, we had a long conversation, and both of us knew that his submarine days were over. We shared experiences that bonded us like brothers on that boat and swore that we'd maintain our friendship for life. Unfortunately, that never happened.\"\n\nAfter the incident with Tony Mignon, Commander Tiernan knew that the sand in the hourglass had all but run out. The hours trapped under the waves had exceeded ninety-six, and it wouldn't be long before the dominoes started falling and others followed in Mignon's wake. Seawolf's nuclear reactor continued to weaken, and air-purifying systems were starting to fail. Tiernan thought about the boat's early days, when teams of shipyard workers had collaborated to overengineer her reactor and equip her with the \"heart of a Mack truck and the soul of a '57 Chevy.\" He recalled discussing the reactor's design capability with his navigator, Lieutenant Commander James Christopher Cane, who had a reputation as the \"coolest cat on the boat under fire.\" The Seawolf had so many emergencies that when they did happen, Cane took them in stride and wore an almost annoyed versus concerned demeanor, ordering the crew to take actions using a matter-of-fact tone. Cane had suggested having the divers clear as much sand as possible on either side of the skegs, then push the engines well past the redline while moving back and forth in similar fashion to a truck stuck in the mud. They could cut the anchors loose to reduce their weight and do a normal blow on the ballast tanks until they broke out of jail.\n\nThe XO, who'd been involved in the discussion, expressed concerns that a normal blow could cause them to broach the surface. The storm seemed to be subsiding, which meant more Soviet warships might be returning to the area. Cane calmly intimated that the Soviets wouldn't be a problem if the Seawolf never broke free. Tiernan put the plan on hold pending the exhaustion of other options. Now he decided that Cane just might be right.\n\nHe ordered the divers to lock out and use fire hoses to clear the sand away from the skegs. One diver, while blasting away with a hose, dug a large hole behind him as he went. A mountain of sand built up on the edge of the pit. As the diver worked, the weight of the sand toppled from the crest and covered the diver up to his neck. Only the diver's Kirby Morgan mask could be seen above the mound. The other divers rushed over, dug for an arm, and pulled. Nothing happened. They pulled again. Finally, the diver broke free, but the sand filled in again around the skegs, and they had to start the job all over again, only this time employing more caution.\n\nIn the control room, the Seawolf groaned in agony as the reactor went into overdrive, and the anchors were cut loose. The nukes ground their teeth as the engines whined, and the spaces aft started to heat up. Jim Rule heard loud screeching through the hull and held his breath. The Seawolf struggled and clawed but still barely moved. Commander Tiernan blew the ballast tanks and pushed the reactor well past the design parameters, painfully aware that if his hunch proved wrong, they'd all die together at the bottom of the sea or be caught by the Soviets and probably suffer the same fate.\n\nThe Seawolf finally nudged, a slight movement at first, followed by fitful screeching, a few trembles, and a rolling wobble. The sub at last broke free and headed for the surface. That was the good news. The bad news was they were going up too fast and would definitely breach the surface, which meant they might be detected by the Soviets. They were also making tons of noise, mostly due to a broken skeg that banged against the hull. Tiernan knew that any nearby warships would have to be deaf not to hear the thing.\n\n\"When that loud clanking started,\" said Rule, \"Chief Langaliers and our weapons officer met with us in the torpedo room. Weps said that we should prepare for the worst, as we'd probably be detected. Langaliers didn't say a word. He just dusted off a thick manual, and handed the thing to me. I saw this look in his eyes that said, 'This is what we trained for, so just suck it up.' The manual outlined the procedures for setting off the bomb block charges we'd placed in the boat, launching our MOSS, and if need be, shooting our way out with MK-37 torpedoes.\"\n\nAfter hitting the surface, Tiernan fought to bring the boat's sail back under the waves. Damaged and sand-filled equipment almost doubled the Seawolf's noise output as she clawed toward the Kuril Islands and escape. Still, they were free, and the crew cheered. High fives were offered, and palms smacked. Then the bad guys came out, and the cheering stopped. Just a few ASW ships took up pursuit to start, followed by a barrage of ships and planes that dropped sonobuoys and pinged away at the sea around them. In the torpedo room, Rule acknowledged an order from Langaliers, and he and the other torpedomen launched the MOSS units from the tubes. Just before the launch, Rule took a bottle of Wite-Out and painted the words CHITTY CHITTY BANG BANG on the side of the long, ten-inch-wide device in reference to the sound emitted, which simulated the Seawolf's old-jalopy chits and bangs. At first, the MOSS decoy didn't work. Rule wondered if God had allowed him to survive the sand-stuck ordeal so he could experience a more gruesome death at the hands of the Soviets. Luck prevailed and some of the Soviet warships took the bait and peeled away.\n\nAfter another day of ducking and hiding, the Seawolf managed to sneak away and stagger back home to Mare Island. She came back to little fanfare, however, as angry voices in Washington were certain that the Wolf had exposed the location of the Ivy Bells tapping pod when she smacked the Soviet cable with her skeg.\n\n\"It was a bittersweet return,\" said Rule. \"We were back alive, and pride filled all of us who played a part in that ordeal. Still, the powers-that-be considered our mission a failure. Those of us who were on the Wolf during that run disagree. Mother Nature may have forced us to change our objectives, but we successfully completed one of the most dangerous missions of the Cold War.\"\n\n## CHAPTER TWENTY-ONE\n\nDuring times of universal deceit, telling the truth becomes a revolutionary act.\n\n\u2014GEORGE ORWELL\n\nON THE MORNING OF JANUARY 15, 1980, the forty-four-year-old Ronald Pelton, a bearded man with blond hair, strolled across Sixteenth Street toward a gated building. He whispered something to the guard at the gate and was ushered in. The FBI agent on the corner noticed the man and signaled to his partner one block away. The agent's partner shook his head and whispered the reply \"Unsub\" into his ear-mounted communicator, for \"unknown subject.\"\n\nPelton sat in the outer office and waited. Vitaly S. Yurchenko, the embassy's KGB security officer, walked into the room. He inquired as to the reason for the man's visit. Pelton confessed that he had secrets to sell and wanted to be paid in gold bullion. Yurchenko did not understand, and asked why the man wanted to be paid with cold chicken soup.\n\nPelton reiterated his request, more clearly specifying gold bullion. He removed a document from his pocket and showed Yurchenko, who studied the paper. Pelton explained that he had previously worked for the National Security Agency, and that he had secrets and wanted to be paid well. He informed Yurchenko that he had once been in charge of prepping a large underwater recording pod used by divers to tap sensitive Soviet communications cables. He knew details about how this system worked and where the pod was located. Yurchenko listened, then agreed upon a price for the information.\n\nThree hours later, Ronald Pelton, code-named \"Mr. Long\" by the Soviets, left from a side entrance, hidden among other Soviet workers. Now clean-shaven, he ducked into a van that drove him to the Soviet residence quarters at Mount Alto. There he consumed a hot meal and received a briefing on how to meet again with a Soviet contact in Vienna. The Soviets returned him to his car, where he drove away into the night, unnoticed by the FBI.\n\nIn fact, they didn't notice him for another three years. The breach wasn't discovered until 1984, and the FBI did not arrest Pelton until November 25, 1985. He confessed to revealing details about the Ivy Bells cable-tapping operations being conducted by the Seawolf and Parche in the Sea of Okhotsk. He is now serving three consecutive life sentences but is scheduled for early release in 2015. When Master Diver David LeJeune heard about what Ronald Pelton had done, he was shocked. LeJeune had worked with the man years earlier and never suspected that his former colleague would stoop to espionage. \"He deserves what he got,\" LeJeune said when asked about the matter. He had no other words to offer on the subject.\n\nBy late 1981, based on the information received from Ronald Pelton, the Soviets found the \"beast\" in the Sea of Okhotsk and brought the pod up from the bottom. The large drum is now on display at a Cold War KGB museum in Moscow. The uncanny timing, in light of the Seawolf's accidental contact with the cable during her last mission, sent the navy into a tailspin. Although Richard Haver, a civilian department head at Naval Intelligence, intimated that a spy might have tipped off the Soviets in a report he filed on January 30, 1982, he and others were concerned that the cable smack might have caused static or an interruption that alerted the Soviets. Regardless, cable-tapping missions to the Sea of Okhotsk came to an abrupt halt.\n\nThere was no evidence that the Soviets found the tap placed in the Barents Sea by the Parche, so that submarine, with Commander Peter J. Graef in charge, deployed again to northern waters in 1982 for a 137-day trip. Upon her return, President Ronald Reagan awarded the Parche a fourth Presidential Unit Citation, which he delivered to Graef along with a box of cigars. The Parche went into the yards through 1983, while the noisy Seawolf was relegated to finding missile parts. In the wings, the nearly finished USS Richard B. Russell awaited her chance to pick up the gauntlet.\n\nPRIDE AND FEAR CONSUMED PETTY OFFICER Second Class Laird Cummings as he shuffled across the brow of the Russell in the fall of 1985. The last of the Sturgeon-class submarines, it had been built at Newport News, Virginia, and commissioned on August 16, 1975. After a decade in service, she still looked almost new, at least from the outside. A strange rectangular hump jutted from the backside of her sail. Cummings had learned via his fifteen DevGru briefings that this chamber housed the Special Projects divers that the Russell employed for various deep-sea missions, replacing the DSRV-like chamber used by other Special Projects boats.\n\nCummings followed his DevGru escort to the hatch, then held his nose. The strong odor of diesel fuel, amine, body odor, cigarette smoke, burnt coffee, and somebody's butt crack combined to create a smell unlike any other on earth, one that would permanently saturate his clothes for years to come. With timid feet, he climbed down the ladder of his new home and descended into the dungeon.\n\nWithin days, Cummings met a good number of the Russell's 112 enlisted men and 14 officers, including her fit-looking skipper, Commander Walter H. Petersen. The six-foot-four COB, Master Chief Wolfert, handed Cummings a qual card and initiated him into the world of Russell. When the boat finally went to sea, Cummings slept in the torpedo room alongside the spooks, divers, and other Special Projects personnel. Although curtains were slung across the spook electronics area, he more than once glimpsed the two dozen plus pieces of gear used during their clandestine operations in the Barents Sea. Often while trying to sleep, he'd hear one of the spooks chant, \"Comex Run One,\" followed later by \"Finex Run One.\" Such chatter went on up to number 500 or more. Cummings translated the jargon as \"Commence Exercise Run One,\" in reference to the tapping runs that collected and recorded data from the cables and synchronized times and parameters, this to ensure accurate retrieval and prevent an unnecessary and dangerous rerun.\n\nCummings often ran into some of the navy divers in the reactor tunnel\u2014the space that divides the forward compartment from the aft\u2014under which resides the nuclear reactor. Despite the fact that enough radiation zoomies sped around inside the tunnel to sterilize King Kong, the divers set up shop there with an exercise bike and a Soloflex work-out machine. When Cummings first met these high-risk-takers, he noticed heavy facial lines and deep eye circles\u2014signs of aging far beyond their years. He'd heard from others that the harsh world of saturation diving took its toll on these \"heroes,\" requiring them to press down and decompress for weeks in a small enclosure, eat cold food, and survive the bends at least once in their careers. They'd obviously been through hell and back but still found time to keep in shape.\n\nWhen Cummings stepped into the reactor tunnel to take readings, he noticed a diver working out on the Soloflex. The beefy guy had placed the machine just below the 400-pound air flask valve that controls the release to the emergency coolant system for the reactor. What Cummings hadn't noticed was that every time the diver raised the bar on the Soloflex, he hit the valve hand wheel just enough to turn it a notch. Eventually, the valve popped off its seat and clanked to the deck. The tunnel filled with the high-pitched screeching of air. The diver froze. Cummings ran from the other side of the tunnel and shut off the air valve seconds before the emergency coolant kicked in, which would have been a very bad thing in the world of reactors.\n\nThe diver stared at Cummings with wide eyes. Cummings figured that the hiss of high-pressure air escaping was probably the scariest thing on earth to this macho man. He shrugged, pointed at the Soloflex, and said, \"I'd move that thing if I were you.\"\n\nLater that week the Russell found herself stuck in the mud, in similar fashion to the Seawolf years earlier. Cummings had heard about that incident from others and began to worry that they might be as unlucky as the Wolf, or worse, that they'd be even less lucky and not make it back alive. Ingenuity and experience saved them from an eternal patrol in the Barents Sea as Commander Petersen and crew found a way to dislodge from the bottom after less than a day.\n\nThe Russell completed her mission and surfaced near Yokosuka, Japan. Cummings was anxious to step on dry land again, but the boat had come up in the middle of a sea state-7 hurricane. Cummings and the crew spent hours sharing seasick trashcans while driving the boat through the squall. Finally, Commander Petersen said, \"To hell with this, we're going back down.\" He dove the boat almost to test depth, where the ocean's fury abated, and the currents were as \"smooth as a baby's ass.\" Cummings strolled into the crew's mess. Someone started a movie. Someone else made popcorn. He smiled, reclined on a bench, and forgot all about the world above. Dry land could wait for another day.\n\n## EPILOGUE\n\nWhen written in Chinese, the word crisis is composed of two characters. One represents danger, and the other represents opportunity.\n\n\u2014JOHN F. KENNEDY\n\nALMOST 200 SUBMARINERS, SPOOKS, AND NAVY divers were interviewed for this book. Not one regretted their time in the navy, and nearly all displayed a genuine love for their country when asked about the selfless sacrifices they made to preserve freedom and democracy. Some may call this corny, but those of us who served use a different word. We call it pride.\n\nWith the conclusion of the Cold War, transitions in operational tactics rippled through the submarine force as the bipolar world order of the superpowers shifted to a multipolar collection of interests. The possibility of another world war diminished, but opportunities for regional conflicts, like those in Iraq and Afghanistan, abounded. The makeup and operational posture of the U.S. Navy reflected this change, as priorities migrated from a blue-water stance to a focus on the littorals, or coastal waterways. Intelligence operations shifted from strategic to tactical reconnaissance. Submariners who once maintained long periods of communications silence were expected to exchange information much more frequently with the surface fleet. Invisibility used to be the byword of the silent service, but submarines soon became political weapons brandished publicly and continued as spy platforms and deterrents to nuclear war.\n\nBy April 2, 1991, thirteen U.S. submarines were conducting surveillance and reconnaissance operations in support of the first Gulf War. Attack boats USS Louisville (SSN-724) and USS Pittsburgh (SSN-720) were ordered to launch TLAMs against Iraq. In all, U.S. surface ships and submarines fired 288 land-attack variants of the Tomahawk Land Attack Missiles (TLAMs) during the Gulf War\u20148 of those coming from the Louisville and 4 from the Pittsburgh. These launches validated that U.S. submarines could operate as part of an integrated strike force while receiving target and strike data from surface partners. The success of this operation set the stage for future submerged missions in which TLAM strikes could take out early-warning, air-defense, and communications systems to offset threats to U.S. aircraft. Given their superior stealth capabilities, TLAM-loaded submarines could now sneak into attack positions unnoticed, then wreak havoc without warning.\n\nThe Parche and Russell continued to conduct Ivy Bells missions well into the 1990s. When the Mare Island Naval Shipyard closed in 1994, the Parche transferred to a new home port at Naval Submarine Base in Bangor, Washington. She received navy unit commendations for her Ivy Bells missions in 1995, 1996, and 1997. The USS Parche was decommissioned at the Puget Sound Naval Shipyard on October 19, 2004, as the most decorated vessel in the history of the U.S. Navy. All totaled she received nine Presidential Unit Citations, ten Navy Unit Citations, and thirteen Navy Expeditionary Medals during her thirty years of service. Parche's sail was moved in 2006 to a location near Puget Sound Naval Shipyard.\n\nIn 1989, the first Seawolf-class attack submarine (SSN-21), intended as a replacement for the aging Los Angeles class, rolled down the ramp. Plans called for a fleet of twenty-nine of these new submarines built over a decade, but that number was eventually trimmed to twelve. With the end of the Cold War in 1991, and subsequent military bud get cuts, the number dwindled to three.\n\nQuieter, larger, faster, and packing twice the armament of a Los Angeles\u2013class boat, the Seawolf also came with a bigger price tag. The new subs were intended as a response to the Soviet Typhoon, the deep-diving titanium-hulled Alpha, and the new super-quiet Akula. An advanced design and array of special equipment allowed the new Seawolf to operate in shallow waters and deploy up to eight SEALs or divers at a time. Divers and T-Branchers could take advantage of Seawolf's Multi-Mission Platform (MMP), which included an underwater splicing chamber used for fiberoptic cable tapping. Seasoned divers, like David LeJeune, who served on the decommissioned original Seawolf during the Cold War and performed Ivy Bells cable-tapping missions, would have been elated with such luxuries.\n\nSubmariners claim that the Parche's successor, the USS Jimmy Carter (SSN-23), became \"Washington's premier spy submarine,\" conducting cable-tapping missions and retrieving missile fragments from seabeds. Although the Jimmy Carter is a formidable Special Operations vessel, packed with advanced combat systems, larger spherical sonar arrays, wide aperture arrays, and new towed arrays, as well as near-silent pumpjet propulsion systems, boats of this class were eventually superseded by newer Virginia-class submarines.\n\nThe USS Virginia (SSN-774) was the first in a line of ultra-modern and sleek submarines designed for a vast array of open-water and coastal missions in virtually any ocean. A less costly alternative to the expensive Seawolf class, these new submarines came just in time to replace thirteen decommissioned Los Angeles\u2013class boats. The Virginia boasted a number of technology and shipbuilding innovations. Gone were the periscopes. In their place came photonic masts mounted outside the pressure hull that housed high-resolution cameras with light-intensification and infrared sensors. Also included were an infrared laser range finder and an integrated ESM array that transmitted signals from the mast's sensors through fiberoptic data lines. Similar to the Seawolf, gone also was the traditional propeller, replaced by a quieter pumpjet propulsion system.\n\nAs for the submariners who manned these boats after the Cold War, in many ways their roles changed dramatically. In other ways they remained the same. No longer did they hunt the Red Bear's massive carriers and cruisers, and cat-and-mouse Holystone games diminished dramatically. But clandestine espionage operations continued, as did sub-deployed SEAL missions.\n\nTo support these missions, several submarines were modified to carry swimmers and equipment more effectively. Attack boats and ballistic missile submarines were retrofitted with special chambers called Dry Deck Shelters (DDSs), which housed Swimmer Delivery Vehicles (SDVs). Special fittings and modifications to air systems and other features enabled these boats to transport and launch SDVs and deploy combat swimmers. SEALs exited from these chambers while the submarine stayed submerged, then swam to the surface along with their equipment.\n\nThe navy recently removed the twenty-four Trident missiles from four of their \"boomer\" SSBNs and replaced them with 154 Tomahawk land-attack cruise missiles. Converted into SSGNs, these massive bastions of destruction now carry up to sixty-six Special Operations Forces (SOF) that might include the Airforce Special Tactics Squadron (STS), Recon Marine divers, and SEALs. Most missions, however, call for one SEAL platoon of fourteen enlisted men and two officers, along with additional SEALs tasked with mission planning and equipment handling. Two former missile tubes on each SSGN now function as dedicated lock-in\/lock-out chambers to allow combat swimmers to exit the boat while submerged. These subs also carry two dry deck shelters and two advanced SEAL delivery systems to support clandestine operations.\n\nWithout a superpower to fight, the nature of SSGN operations now focuses on a very different kind of enemy. Rear Admiral William H. Hilarides, program executive officer (submarines), spoke on this topic during a \"relaunch\" ceremony for the USS Ohio (SSGN-726) on February 7, 2006: \"Ohio's return to service is truly monumental. Now Ohio will conduct missions that will have a direct impact on the ongoing Global War on Terror.... SSGNs are truly force multipliers.\"\n\nAlthough the Soviet Union vanished almost two decades ago and no longer threatens to undermine the United States' dominance of the world's oceans, and although most terrorist-thwarting activities are land-based, the U.S. Navy continues to operate an extensive fleet of submarines and undertake clandestine missions. In 2008, more than 500 submarines roamed the oceans worldwide. Asian countries commanded 135 of these, while 45 came from the Middle East. The proliferation of advanced conventional submarines is a troubling problem for the U.S. Navy, given that over forty nations rely on advanced diesel submarines. In the Pacific Ocean, over 140 submarines are frequently deployed within striking distance of critical trade-zone choke points.\n\nTo monitor potential threats posed by foreign submarines, three classes of American SSNs remain in service, including the Los Angeles class, still the backbone of the submarine force, with forty-six in operation. Thirty-one of these are equipped with tubes for vertically launched Tomahawk cruise missiles.\n\nMassive Red Bear ballistic missile boats are no longer the daunting threat they once were, but the task of detecting and neutralizing enemy submarines has actually increased in difficulty for the United States. Most subs are smaller and quieter than ever before and can stay submerged for months on end. Many operate in shallow waters, making detection even harder. The need to locate these boats has led to advances in modern antisubmarine warfare that far surpasses the dismantled Boresight\/Bulls Eye technology. The modern ASW arsenal now includes new types of sensors, advanced active and passive sonar acoustics, sonobuoys, and fixed hydrophones.\n\nCommunications have also come a long way since the Cold War. In March 2009, two Los Angeles\u2013class boats, the USS Annapolis (SSN-760) and USS Helena (SSN-725), played cat and mouse in the frigid Arctic north of Alaska during a weeks-long ICEX exercise. These boats shot dummy torpedoes at each other, which were later recovered by navy divers via drilled holes in the ice. They also tested a revolutionary new form of underwater communications technology and then surfaced through thick layers of ice to visit the nearby ICEX camp, a smattering of modular huts that house dozens of navy personnel and scientists for weeks at a time.\n\nJeff Gossett, technical director for the Arctic Submarine Laboratory, said that \"we installed an exciting new digital ICOMMS system on the subs that allowed us to type messages into a laptop that were encoded and transmitted to the boats through an underwater noisemaker. Imagine being able to send and receive e-mails to a submarine hundreds of feet deep running at top speed!\"\n\nBack in my day we had to park two boats right next to each other and talk through an underwater \"Gertrude\" device that garbled the sound so bad that boat skippers often gave up in frustration. To most of us Cold War submarine veterans, e-mail at depth and speed still sounds like science fiction.\n\nAs for Boresight\/Bulls Eye's innovative technology, obsolescence occurred in tandem with the advent of worldwide satellite communications. Not only did satellite transmissions obviate the need to send radio bursts\u2014as Soviet subs once did during the Cold War\u2014but these small orbiting craft can also be used effectively for ASW operations. Today, satellites can image ocean surfaces using optical and radar techniques, which can indirectly detect submarines operating at shallow depths. Thermal imaging is another means of detection, as well as low-flying aircraft. And while SOSUS still plays a role in submarine and vessel detection, many of those arrays have been reassigned to civilian duties for marine research.\n\nIN MARCH 2008, MY DAD AND I sat on his veranda in Puerto Vallarta, Mexico, and talked about the Boresight program as we listened to the tropical rain tap at the plastic patio covering. He scratched at his head and then started laughing. When I asked him what was so funny, he said the itch reminded him of Commander Petersen\u2014his boss at the Karam\u00fcrsel Huff Duff in Turkey.\n\n\"When we lost those Russkie subs,\" Dad said, smiling, \"after we pissed our pants, we all started scratching our heads. But Commander Petersen never stopped scratching. He was going bald and had this skin condition made worse by stress. Captain Mason and I were more worried about Petersen scratching himself to death than we were about the NSA breathing down our necks. Now that I think about it, that's pretty funny. Maybe Petersen's dandruff prevented World War III.\"\n\nThat was the last time I spoke to my dad. He passed away on July 30, 2008.\n\nA week later, one of my colleagues, Joel Harrison, pointed at his computer screen and said, \"There.\"\n\nThe chief technology officer at a software company in Silicon Valley and one of the cofounders of a multibillion-dollar storage firm, Joel was once a Boresight\/Bulls Eye technician during his navy days in the seventies. I followed his finger to the satellite photo displayed on his computer but saw nothing save an empty circle filled with dirt.\n\n\"That's where the Bulls Eye antenna was located,\" Joel said. \"It's gone now, but I still look at that circle and reminisce. That was some pretty cool technology.\"\n\nIndeed, it was.\n\nToday the large pole antennas used for the last Wullenweber CDAA array built at Imperial Beach near San Diego still remain. Inside the poles, the mortar building no longer houses Boresight systems, but instead is used as a planning facility for advanced U.S. Navy SEAL operations.\n\nAs I reflect on the role played by U.S. submariners, navy divers, spooks, and antisubmarine operatives during the Cold War, including the professionals who developed, deployed, and operated Boresight\/Bulls Eye stations, I often recall the battle at Rorke's Drift. Fought during the Zulu War, on the afternoon of January 22, 1879, this bloody conflict is one of the most famous in British history. Ninety-five soldiers from Company B of the Second Battalion, led by Lieutenant John Chard, bravely held off 4,000 savage Zulu warriors for twelve hours. They lost seventeen men, while the Zulus lost 600. The British held their ground with .45-caliber Martini-Henry breech loading rifles, while the Zulus pitched spears and swung clubs. Historians credit superior training, technology, and leadership as keys to this victory. A hundred years later, the keys to another victory were quite similar.\n\nDuring the height of the Cold War, the United States remained greatly outnumbered by the Soviets, with only 123 submarines pitted against nearly three times that number. After the Cuban Missile Crisis, spurred by paranoia, the NSA's voracious appetite for intelligence gathering became a primary espionage motivator. Ascertaining the characteristics and capabilities of Ivan's submarines, many armed with long-range nuclear missiles, became crucial to national security. No cost or danger seemed too high in comparison to potential global destruction.\n\nBut the paranoia was more perceived than real. Two years after Nikita Khrushchev became Communist Party first secretary, he ordered Admiral Sergei Gorshkov to dismantle the Soviet fleet. He stopped all heavy cruiser projects in midcourse. By 1957, the Soviet navy had dwindled to just 350 ships. Gorshkov's submarine force consisted mostly of short-range boats designed for coastal defense, not cross-ocean aggression. When the United States launched the world's first nuclear-powered fleet ballistic missile submarine, the USS George Washington, in 1960, the Soviets had no answer to this threat. The Hotel-class K-19 launched the year before could hold no candle to the George Washington. K-19 carried only three nuclear missiles compared to sixteen. She had to surface before firing, and the entire sequence took twelve minutes\u2014more than enough time for a U.S. hunter\/killer sub to sink her. She was plagued with problems, including a severe nuclear-reactor failure that cost the lives of twenty-seven sailors.\n\nThe Red Bear's attack submarines fared no better. Dozens of nuclear accidents occurred on these boats at the cost of more lives. Although Ivan boasted greater numbers than the United States, a majority of Gorshkov's submarines were old and noisy. For most of the Cold War, until Toshiba and Kongsberg illegally helped upgrade their technology, the Soviets were spear-chucking Zulus fighting rifle-carrying Brits.\n\nOn U.S. submarines, enlisted men received many months of training and performed complex and technical functions under way. In contrast, only more senior officers and michmen\u2014the equivalent of warrant officers in the U.S. Navy\u2014undertook those duties on a Soviet sub. On an American SSBN, the typical crew consisted of 15 officers and 125 enlisted men. K-219, a Cold War Soviet SSBN, had thirty-one officers, thirty-eight michmen, and only forty-nine seamen. Compared to the modern-looking U.S. submarine facility in Groton, Connecticut, where I received my training, the Paldiski Soviet nuclear submarine training center resembles a Zulu mud hut.\n\nAt the time, though, we didn't know that our opponents were so far behind. By the time the Cold War ended, they'd done a pretty good job of catching up to and, in some cases, surpassing American technology. All but gone were the creaky Zulu-class diesel boats and noisy November-class nuclear subs. In their place, deep-diving titanium-hulled Alpha-class submarines ruled the seas with advanced propulsion systems capable of pushing speeds up to forty-five knots. Our Los Angeles\u2013class boats could hit only thirty-five knots on a good day. Had the Soviets not run out of money and patience with communism, the Cold War might have ended differently.\n\nToday, given Russia's current economic state, its fleet has dwindled to a small number of ships and submarines. Those vessels still afloat are often unable to deploy due to a lack of trained crews and resources. Maintenance and repair are dismal. In November 2009, the RIA Novosti news agency quoted retired Russian navy admiral Vyacheslav Popov as saying, \"If things remain as they are, we will have to mothball most ocean warships by 2015. That will sharply reduce the navy's capability.\"\n\nAs the Russian navy diminishes, the Chinese navy is increasing and improving at a rapid rate. Given recent confrontations with China's submarines, including a collision with the destroyer John S. McCain's towed array near the Philippines, one wonders if another Cold War is brewing. While the hawks seek more funds to expand the U.S. Navy's submarine fleet, doves argue that there is no need for such weapons in a post\u2013Cold War world. Perhaps these misinformed well-wishers have yet to learn from the past. China now owns a significant portion of America's burgeoning debt and has emerged as an economic power-house in recent years.\n\nIn the decade from 1995 to 2005, the Chinese navy launched thirty-one nuclear submarines. Many of these are advanced ballistic missile SSBNs that carry long-range nuclear weapons. Professor Toshi Yoshihara of the Naval War College in Newport, Rhode Island, recently said that, \"for at least the next two decades, missile defense...will have no answer to a capable SSBN patrolling the open ocean.... This asymmetry in capability suggests that...the only effective response to a capable Chinese SSBN is the employment of traditional antisubmarine warfare assets, particularly hunter\/killer nuclear attack submarines.\"\n\nIf Yoshihara's words are true, American submariners will be ready for the calling. The legacy is strong, and the pride still runs deep.\n\n## NOTES\n\nSOME OF THE INFORMATION CITED HEREIN was derived via research from numerous books, websites, and documents as outlined under Resources for each chapter, but a vast majority of the content came from interviews with individuals who experienced these events firsthand. Most of those interviewed were in their senior years, and a few recollections were less than optimal. Although every precaution was taken to validate facts, time lines, and names, in a few cases this was not possible. Also, some accounts differed from those of others involved, and in such cases care was taken to determine the most accurate accounting of events based on the facts at hand. Not all stories imparted by submariners, navy divers, and government operatives who were interviewed made the final cut for inclusion in the book. Readers are encouraged to visit the author's website at www.wcraigreed.com to read or download these exciting accounts.\n\nINTRODUCTION\n\nREFERENCES\n\n2007 Submarine Encyclopedia: U.S. Navy Submarine Fleet, Sub History, Technology, Ship Information; Submarine Pioneers, Cold War Technology, Department of Defense, Progressive Management, April 27, 2007. This book and CD-ROM combination contains almost 19,000 pages of in-depth information on almost anything one might want to know about submarines and ASW, with an emphasis on the Cold War. I used this handy reference in almost every chapter of the book.\n\nWEBSITES\n\nhttp:\/\/en.wikipedia.org\/wiki\/Cold_War\n\nCHAPTER 1\n\nPRIMARY INTERVIEWS\n\nCaptain Paul Trejo, USN, Retired, former LTJG on the USS Blenny (SS-324).\n\nGardner Brown, MM3, and other former crew members from the USS Cubera (SS-347) and USS Seawolf (SSN-575).\n\nNote: Gardner Brown does not know what ancient buildings the USS Cubera hid behind while dodging Soviets in the Black Sea, but the explorer Robert Ballard discovered the remains of an ancient structure 311 feet deep near Turkey in 2000 that he believes was flooded during the time of Noah. Brown agrees that the \"Main Street\" wherein the Cubera hid could be a similar city buried under the ocean during the Great Flood ( http:\/\/news.nationalgeographic.com\/news\/2000\/12\/122800blacksea.html).\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004, provided a reference for submarine designations and capabilities\n\nRunning Critical: The Silent War, Rickover, and General Dynamics, Patrick Tyler, Harper & Row Publishers, 1986\n\nThe Submarine: A History, Thomas Parrish, Penguin Books, 2004\n\nPower Shift: The Transition to Nuclear Power in the U.S. Submarine Force as Told by Those Who Did It, Dan Gillcrist, iUniverse, 2006\n\nWEBSITES\n\nhttp:\/\/en.wikipedia.org\/wiki\/Hyman_G._Rickover\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Nautilus_(SSN-571)\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Cubera_(SS-347)\n\nhttp:\/\/en.wikipedia.org\/wiki\/Greater_Underwater_Propulsion_Power_Program\n\nhttp:\/\/en.wikipedia.org\/wiki\/George_W._Grider\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Seawolf_(SSN-575)\n\nhttp:\/\/www.iwojima.com\/\n\nCHAPTER 2\n\nPRIMARY INTERVIEWS\n\nDonald Ross, Ph.D., former DEMON sonar project lead in San Diego, provided excellent information regarding SOSUS capabilities and development of submarine sonar systems. More information can be found in Ross's book noted below.\n\nNote: When the torpedo-propeller design group moved to Penn State in late 1945, Dr. Donald Ross, at the age of twenty-four, presented a paper at the navy's first scientific meeting on hydrodynamic topics sponsored by the Office of Naval Research (ONR). Ross proposed that submarines be built like torpedoes and that the current twin-screw designs could be replaced by a single-screw configuration. He forecast that such a design might empower higher propeller speeds without causing cavitation noise. Albert G. Mumma, the navy captain who'd helped capture German XXI sub designer Helmuth Walter during World War II, shot up from the center of the group and shouted, \"The navy will never build something that stupid!\"\n\nThe crowd nodded their agreement. After all, submarines must have two screws in the event that one fails. Twin-screws are also needed for better maneuverability. Despite pleas to the contrary from Ross, navy officials agreed with the group and dismissed the idea. Years later, when the David Taylor Lab in Mary land successfully demonstrated the new Albacore-type submarine, damned if the thing didn't have a single screw. The officer in charge, one Albert G. Mumma, received a naval commendation for his role in furthering single-screw submarine designs. That achievement catapulted Admiral Mumma to chief of the Bureau of Ships\u2014and therefore Hyman Rickover's boss\u2014in 1955, and the Albacore single-screw became so successful that most modern navies adopted the configuration for almost all submarine designs.\n\nDuring one of his runs on the USS Nautilus, Ross received an invitation to the wardroom. The Nautilus's skipper, Commander Eugene Wilkinson, flashed a coy smile and pulled out a deck of cards. He laid $20 on the table and said, \"You in?\"\n\nRoss pulled out a twenty and sat down next to three other officers. Several hours later, after having cleaned out the pockets of everyone in the wardroom, Ross scooped up his winnings and headed toward the door.\n\nStern-faced, Wilkinson stood from the table and said, \"Don't think for a minute I'm going to forget this, Dr. Ross.\"\n\nYears later, in 1969, Richard Nixon halted all Holystone submarine missions due to a spate of serious accidents. Wilkinson employed Ross and his team to solve a major problem with submarine sonar that appeared to be the root cause of the \"run-ins\" with Soviet submarines when U.S. boats tried to follow them. Ross solved the problem for Wilkinson and received the highest citation awarded to civilians for his contribution.\n\nRESOURCES\n\nMechanics of Underwater Noise, Donald Ross, Ph.D., Peninsula Publishing, 1987\n\nTransparent Oceans: The Death of the Soviet Submarine Force, Louis P. Solomon, LRAPP Company, 2003\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nHide and Seek: The Untold Story of Cold War Naval Espionage, Peter A. Huchthausen and Alexandre Sheldon-Duplaix, John Wiley & Sons, 2009\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nHitler, Doenitz, and the Baltic Sea, Howard D. Grier, Naval Institute Press, 2007\n\nWEBSITES\n\nhttp:\/\/www.globalsecurity.org\/intell\/systems\/sosus.htm\n\nhttp:\/\/www.navy.mil\/navydata\/cno\/n87\/usw\/issue_25\/sosus2.htm\n\nhttp:\/\/en.wikipedia.org\/wiki\/SOSUS\n\nCHAPTER 3\n\nPRIMARY INTERVIEWS\n\nWilliam J. Reed, LT USN, Retired, regarding HFDF technology and deployment. I interviewed my father in depth and then double-checked the details against other research and interviews.\n\nRobert Lynn Wortman, CD, RCN, RCMP, Retired, regarding World War II Huff Duffs and HFDF sights in the United States and Canada.\n\nPamela Wallinger Reed (author's sister) offered excellent insights into events in Turkey, Mary land, and San Diego when my father was in the navy.\n\nJoyce Louise Reed (author's mother) offered excellent insights into events in Turkey, Mary land, and San Diego when my father was in the navy.\n\nGeorge Munch, former Boresight\/Bulls Eye HFDF systems and Wullenweber array engineer who also helped get most of the joint U.S-Canadian sites up and running.\n\nFrank Cawley, deputy director of the HFDF division at the NSA, former HFDF Boresight\/Bulls Eye communications technician.\n\nJohn Gurley, former communications technician chief, USN, worked on Boresight systems and Wullenweber \"elephant cage\" arrays.\n\nGus Lott, Ph.D., founder of Yarcom, Inc. ( www.yarcom.com), former Boresight\/Bulls Eye systems and Wullenweber array contracting engineer. Gus is now the chief scientist for the Signal-to-Noise Enhancement Program (SNEP). In the 1980s, Dr. Steve Jauregui at the Naval Postgraduate School made SNEP a regular scientific program. The navy returned the program back to the NSA in 2003. Today, SNEP sends experts to sensitive receiver facilities to reduce noise from power lines, computers, and many other emitters that can interfere with signals. Gus provided some outstanding information on the theory and math behind the operation of Boresight\/Bulls Eye systems and stations and noted that when these systems were built in the sixties, there were few power lines or other sources of electrical interference. By the nineties, the areas nearby had been built up, and errors increased. Engineers like Gus discovered that interference noise now emanated from lines and laptops and helped correct for such.\n\nRESOURCES\n\nRadio Direction Finding, P. J. D. Gething, Institution of Electrical Engineers, December 1987. This book is considered by most in the field the Bible on HFDF and Wullenweber arrays.\n\nThe Canadian History of Signal Intelligence and Direction Finding, Robert Lynn Wortman and George Fraser (self-published), 2006.\n\nThe Codebreakers, David Kahn, Scribner, 1996. Chapters 11 and 15 detail interesting information about the use of HFDF Huff Duffs during World Wars I and II.\n\nWEBSITES\n\nhttp:\/\/www.researcheratlarge.com\/Pacific\/RDF\/\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_George_Washington_(SSBN-598)\n\nhttp:\/\/en.wikipedia.org\/wiki\/UGM-27_Polaris\n\nhttp:\/\/rusnavy.com\/science\/electronics\/rv6.htm\n\nhttp:\/\/www.globalsecurity.org\/intell\/systems\/sosus.htm\n\nhttp:\/\/russianfun.net\/technology\/secret-soviet-submarine-base-in-sevastopol\/\n\nhttp:\/\/www.angelfire.com\/falcon\/usspillsbury-der_133\/elequip.html\n\nhttp:\/\/www.jproc.ca\/sari\/counter.html\n\nhttp:\/\/en.wikipedia.org\/wiki\/High_frequency\n\nhttp:\/\/en.wikipedia.org\/wiki\/Battle_of_Jutland\n\nCHAPTER 4\n\nPRIMARY INTERVIEWS\n\nWilliam Reed, LT, USN, Retired.\n\nRobert Lynn Wortman, CD, RCN, RCMP, Retired, former Bore-sight\/Bulls Eye HFDF systems and Wullenweber arrary engineer.\n\nGeorge Munch, former Boresight\/Bulls Eye HFDF systems and Wullenweber array engineer.\n\nFrank Cawley, deputy director of the HFDF division at the NSA, former HFDF Boresight\/Bulls Eye communications technician.\n\nJohn Gurley, former communications technician chief, USN, worked on Boresight systems and Wullenweber arrays.\n\nGus Lott, Ph.D., founder of Yarcom, Inc. ( www.yarcom.com), former Boresight\/Bulls Eye systems and Wullenweber array contracting engineer.\n\nNOTES\n\nOthers involved in the development of the magnetic tape recorder in concert with Robert Misner and Howard Lorenzen were Dr. Hector Skifter and James Gall.\n\nMack Sheets also accepted the NRL \"Top 75 Inventions\" award, along with Robert Misner.\n\nDrs. Rindfleisch, Pietzner, Schel horse, and W\u00e4chtler led the effort to build the first CDAA Wullenweber array at Joring, Denmark.\n\nJurgen Wullenweber's fame grew among the common folk, who revered him as a legendary upholder of the Protestant cause. He died in Wolfenbutiel in 1537 while fighting for that cause, which propelled him to martyrdom.\n\nMack Sheets assisted Robert Misner with the invention of the AN\/FRA-44 recorder\/analyzer.\n\nThe first successful HFDF fix in World War II came on the morning of July 13, 1943. German U-boat 487 transmitted an update to Berlin. The Tenth Fleet ASW group grabbed the transmission on several Huff-Duffs and fixed U-487's location as just northwest of the Cape Verde Islands. The ASW team hurried the fix to the escort carrier USS Core (CVE-13). Wildcat and Avenger aircraft scrambled from the deck of the Core and sped toward the U-boat. They found her snorkeling on the surface and blasted the submarine with bombs and cannon fire. U-487 sank to the bottom. The entire operation took less than ten hours and laid the foundation for future ASW operations. After the war, the United States and Soviet Union propelled their respective HFDF programs forward, while shifting their focus on finding each other versus locating German U-boats.\n\nRESOURCES\n\nRadio Direction Finding, P. J. D. Gething, Institution of Electrical Engineers, December 1987.\n\nThe Canadian History of Signal Intelligence and Direction Finding, Robert Lynn Wortman and George Fraser The Soviet Union and the Arms Race, David Holloway, Yale University Press, 1984\n\nU.S. Military Operations Since World War II, Kenneth Anderson, Brompton Books Corporation, 1984.\n\n\"Operator's Organizational, Direct Support, General Support, and Depot Maintenance Manual for Antenna Group Countermeasures Receiving Set AN\/FLR-7(V7\/(V8),\" U.S. Army Security Agency, Materiel Support Command, Vint Hill Farms, Warrenton, VA 22186.\n\n\"A Wide-Aperture HF Direction-Finder with Sleeve Antennas,\" Raymond F. Gleason, Robert M. Greene, Naval Research Laboratories Memorandum Report 843, August 20, 1958\n\n\"Award for Innovation, 75 Years,\" Naval Research Laboratory, obtained from the PDF document found at http:\/\/www.nrl.navy.mil\/content.php?P=75thANNIVESARYAWARDS\n\nWEBSITES\n\nhttp:\/\/www.nrl.navy.mil\/pao\/pressRelease.php?Y=2000&R=32-00r\n\nhttp:\/\/en.wikipedia.org\/wiki\/Battle_of_Jutland\n\nCHAPTERS 5\u201310\n\nPRIMARY INTERVIEWS\n\nWilliam Reed, LT., USN, Retired. Details involving the White House meeting in this chapter were derived from interviews with and memoirs written by my father, William J. Reed. Although I requested information on Boresight and this meeting via the Freedom of Information Act, I received only a rejection notice from the NSA on my first attempt and no response on the second. However, I extensively researched time logs for all the major players and key events of the Cuban Missile Crisis, and based on the reported whereabouts of these individuals, as well as actions taken at specific times, and based on my father's recollections, I believe 5 P.M. on October 25 is when the meeting with President Kennedy and his advisers occurred. My father was not positive about the first names or correct spellings of the last names of Petty Officers Denofrio and Odell, who worked with him on the Boresight project. Apologies to those individuals if the names are incorrect.\n\nRobert Lynn Wortman, CD, RCN, RCMP, Retired, former Bore-sight\/Bulls Eye HFDF systems and Wullenweber array engineer.\n\nGeorge Munch, former Boresight\/Bulls Eye HFDF systems and Wullenweber array engineer.\n\nFrank Cawley, deputy director of the HFDF division at the NSA, former HFDF Boresight\/Bulls Eye communications technician.\n\nJohn Gurley, former communications technician chief, USN, worked on Boresight systems and Wullenweber arrays.\n\nPeter Lewis, former communications technician chief who worked on Boresight and HFDF systems during the sixties, offered excellent input regarding this technology.\n\nGus Lott, Ph.D., founder of Yarcom, Inc. ( www.yarcom.com), former Boresight\/Bulls Eye systems and Wullenweber array contracting engineer.\n\nAnonymous sources who worked on government programs during the Cold War.\n\nRyurik Ketov, Captain First Rank, Retired, in Saint Petersburg, Russia, former commander of B-4.\n\nA. F. Dubivko, Captain First Rank, Retired, in Saint Petersburg, Russia, former commander of B-36.\n\nRadm. Victor Frolov, served as the executive officer aboard B-130.\n\nLt. Anatoliy Andreev, served as assistant to the submarine commander lieutenant aboard B-36.\n\nAdm. Vladlen Naumov, served as captain-lieutenant, navigator, and commander of BCh-1 (navigation) aboard B-36.\n\nNote: I met with the above former Soviet submariners in Saint Petersburg, Russia, via invitation and sponsorship provided by the Saint Petersburg Submariners Organization commanded by Captain First Rank (retired) Igor Kurdin. Igor and his staff, including Ksenya Hohlova, were of tremendous help in arranging meetings and providing transportation, as well as translation and tour-guide assistance. Information offered by these five, in several areas, contrasted with that printed in Peter Hutchthausen's book, October Fury. Captain Dubivko, via Adm. Naumov, stated that some of the details printed in October Fury were \"pure fiction.\" While Hutchthausen's book offers an interesting and well-researched read, numerous details regarding Foxtrot-class submarines and historical accounts, based on expert opinions, are incorrect. Captain Ketov provided in-depth information on the operation and history of the Soviet SBD burst signal radio that spurred the Boresight program.\n\nCharles Skillas, former engineering director at Sanders Associates in New Hampshire, imparted information regarding the navy programs under development during the time my father met with his team in 1962.\n\nSpecial thanks are due R. J. Hansen, who provided a detailed tour of the B-39 Foxtrot submarine located at the Maritime Museum of San Diego. Hansen is considered the most \"qualified\" American submariner on the Foxtrot class, having studied under former B-427 commander Captain Third Rank Igor Kolosov for two years. Hansen also reviewed the Foxtrot details in this book for accuracy, so I blame any remaining mistakes on him, of course. The B-427 is also open for tours and is located next to the Queen Mary in Long Beach, California.\n\nJeff Loman, tour and volunteer coordinator for the Maritime Museum of San Diego, assisted in setting up my tour with R. J. Hansen. Information on the B-39 can be found at www.sdmaritime.org.\n\nStanley Pearlman, vice president of Newco PTY LTD, LLC, which owns the B-427, provided a few of the Foxtrot-class submarine pictures herein, and many of the Foxtrot operational details were gleaned from the B-427 tour book. Information on the B-427 and the tour can be found at www.russiansublongbeach.com.\n\nRESOURCES\n\nBody of Secrets: Anatomy of the Ultra-Secret National Security Agency, James Bamford, Anchor, April 30, 2002, offered information regarding John Arnold and Cuban SIGINT missions conducted by the USS Nautilus and Oxford, as well as details about the USS Pueblo.\n\nJeffrey G. Barlow, \"Some Aspects of the U.S. Navy's Participation in the Cuban Missile Crisis,\" in A New Look at the Cuban Missile Crisis, Colloquium on Contemporary History, June 18, 1992, No. 7, Naval Historical Center, Department of the Navy\n\nCommand in Crisis: Four Case Studies, Joseph F. Bouchard, Columbia University Press, 1991\n\nThe Cuban Missile Crisis, ed. Laurence Chang, National Security Archive, 1992\n\n\"The Naval Quarantine of Cuba, 1962,\" Office of the Chief of Naval Operations, 1963, http:\/\/www.history.navy.mil\/faqs\/faq90-5.html\n\nOctober Fury, Peter Huchthausen, John Wiley & Sons, 2002\n\nPresidential Recordings: John F. Kennedy, The Great Crises, Vol. III, Philip Zelikow and Ernest R. May, ed., W. W. Norton, 2001\n\nCables from Cuba History File\/U.S. Navy Operational Archives; Deck Logs from Record Group 24, U.S. National Archives\n\n\"The Cuban Missile Crisis as Seen through a Periscope,\" Ryurik A. Ketov, Captain First Rank, Russian Navy (retired), Journal of Strategic Studies, vol. 28, no. 2, 217\u2013231, April 2005\n\n\"In the Depths of the Sargasso Sea,\" A. F. Dubivko, Captain First Rank, Russian Navy (retired; unpublished memoir)\n\n\"Reflections of Vadim Orlov, We Will Sink Them All, But We Will Not Disgrace Our Navy,\" V. P. Orlov, Captain Second Rank, Russian Navy (retired; unpublished memoir)\n\nCINCLANT SOSUS contact reports during the Cuban Missile Crisis\n\n\"Khrushchev, Castro and Kennedy: Motivation, Intention, and the Creation of a Crisis,\" Robin R. Pickering, thesis presented to the faculty of Humboldt State University, May 2006\n\nDeck logs obtained for various naval platforms during the Cuban Missile Crisis\n\nEyeball to Eyeball, The Inside Story of the Cuban Missile Crisis, Dino A. Brugioni, Random House, 1991\n\nSoviet Naval Developments, third edition, foreword by Norman Polmar, The Nautical and Aviation Publishing Co. of America, 1984\n\nThirteen Days: A Memoir of the Cuban Missile Crisis, Robert F. Kennedy, W. W. Norton & Co., 1969\n\nSubmarines of the Russian and Soviet Navies, 1718\u20131990, Norman Polmar and Jurrien Noot, Naval Institute Press, 1991\n\nCombat Fleets of the World 1980\/81, Jean Labayle Couhat, Naval Institute Press, 1980\n\nU.S. Military Operations Since World War II, Kenneth Anderson, Brompton Books Corporation, 1984\n\nOne Hell of a Gamble, The Secret History of the Cuban Missile Crisis, Aleksandr Fursenko and Timothy Naftali, W. W. Norton & Co., 1997\n\n\"Report Heightens Nuclear Sub Mystery\/Torpedo Theory Contradicts Findings of USS Scorpion's Wreckage in 1968,\" Stephen Johnson, Houston Chronicle, December 27, 1993\n\nThe Reminiscences of Admiral George W. Anderson, Jr., U.S. Navy (Retired), vol. 2, U.S. Naval Institute, Annapolis, MD, 1983\n\n\"Role of Polaris Submarines in the Cuban Missile Crisis,\" VADM Charles Griffiths, USN (Retired), The Submarine Review, July 2009, pp. 69\u201370\n\nThe Face of Moscow in the Missile Crisis, William F. Scott, Studies in Intelligence, vol. 37, no. 5, 1994\n\nCommand History, ref. (a) OPNAVINST 5750.7, encl., Command History Destroyer Squadron 21, [memo] from Commander Destroyer Squadron Twenty-One to Chief of Naval Operations (OP 29)\n\nCUBEX: U.S. Anti-Submarine Operations during the Cuban Missile Crisis, Norman Polmar, Moscow, September 27, 1994\n\nChronology (Fleet Operations Report, Cuban Missile Crisis, October 1962), Naval Historical Archive, Washington, D.C.\n\nCuban Quarantine Operations\u2014Historical Account of, ref. CINCLANT Staff Notice 5213 of November 7, 1962 [memo], from Cuba Quarantine Center (O3Q) to CINCLANTFLT Command Information Bureau, Historical Office (J09H)\n\n\"Destroyer Forced Red Sub Up,\" Evening Capital (QP), June 14, 1963\n\nHistory of USS Charles P. Cecil (DD-835), Naval Historical Archive, Washington, D.C.\n\nWEBSITES\n\nhttp:\/\/www.navycthistory.com\/index.html\n\nhttp:\/\/www.navycthistory.com\/homestead_email_roster.html\n\nhttp:\/\/luxexumbra.blogspot.com\/2005\/06\/frd-10-endangered-species.html\n\nhttp:\/\/www.fas.org\/irp\/program\/collect\/an-flr-9.htm\n\nhttp:\/\/www.tscm.com\/masset.html\n\nhttp:\/\/jproc.ca\/rrp\/masset.html\n\nhttp:\/\/www.tscm.com\/gander.html\n\nhttp:\/\/www.cubainfolinks.org\/webpage\/Articles\/bejucal.htm\n\nhttp:\/\/www.navycthistory.com\/NSGStationsHistory.txt\n\nhttp:\/\/straightwhiteguy.mu.nu\/archives\/cat_military_stuff.php\n\nhttp:\/\/online.wsj.com\/article\/SB122619710466311417.html\n\nhttp:\/\/www.thc.state.tx.us\/museums\/musnimitz.shtml\n\nhttp:\/\/www.navycthistory.com\/index.html\n\nhttp:\/\/www.themoscowtimes.com\/article\/1010\/42\/372216.htm\n\nhttp:\/\/jproc.ca\/rrp\/\n\nhttp:\/\/jproc.ca\/rrp\/grd_6.html\n\nhttp:\/\/www.scribd.com\/doc\/4566441\/Matthew-Aid-The-National-Security-Agencyand-the-Cold-War\n\nhttp:\/\/www.jfklibrary.org\/jfkl\/cmc\/cmc_october28.html\n\nhttp:\/\/www.jfklibrary.org\/jfkl\/cmc\/cmc_october27.html\n\nhttp:\/\/www.gwu.edu\/~nsarchiv\/nsa\/cuba_mis_cri\/docs.htm\n\nhttp:\/\/en.wikipedia.org\/wiki\/Curtis_LeMay\n\nhttp:\/\/avalon.law.yale.edu\/20th_century\/msc_cuba070.asp\n\nhttp:\/\/en.wikipedia.org\/wiki\/Maxwell_D._Taylor\n\nhttp:\/\/www.gwu.edu\/~nsarchiv\/nsa\/cuba_mis_cri\/photos.htm\n\nhttp:\/\/www.gwu.edu\/~nsarchiv\/NSAEBB\/NSAEBB75\/#11\n\nhttp:\/\/www.pbs.org\/wgbh\/nova\/subsecrets\/life03householder.html\n\nhttp:\/\/www.gwu.edu\/~nsarchiv\/nsa\/cuba_mis_cri\/docs.htm\n\nhttp:\/\/en.wikipedia.org\/wiki\/Sanders_Associates\n\nNOTES\n\nIn addition to the Nakat ESM systems, Foxtrot submarines carried a crude but effective Quad Loop radio direction finder that could detect HF radio transmissions and determine rough bearings to the sources. As the name implies, the antenna consisted of four oval pipelike metal loops attached to a center pole jutting out from the top of the sail.\n\nCHAPTER 11\n\nPRIMARY INTERVIEWS\n\nNihil Smith, former sonarman and navy diver, offered a heart-wrenching account of his dives on the wreck of the USS Thresher in 1963.\n\nWilliam Reed, LT., USN, Retired.\n\nRobert Lynn Wortman, CD, RCN, RCMP, Retired, former Bore-sight\/Bulls Eye HFDF systems and Wullenweber array engineer.\n\nGeorge Munch, former Boresight\/Bulls Eye HFDF systems and Wullenweber array engineer.\n\nFrank Cawley, deputy director of the HFDF division at the NSA, former HFDF Boresight\/Bulls Eye communications technician.\n\nJohn Gurley, former communications technician Chief, USN, worked on Boresight systems and Wullenweber arrays.\n\nPeter Lewis, former communications technician chief who worked on Boresight and HFDF systems during the sixties, offered excellent input regarding this technology.\n\nGus Lott, Ph.D., founder of Yarcom, Inc. ( www.yarcom.com), former Boresight\/Bulls Eye systems and Wullenweber array contracting engineer.\n\nRESOURCES\n\n\"An Analysis of the Effects of Feedline and Ground Screen Noise Currents on a Conical Monopole Receiving Antenna,\" Thomas D. Gehrki, Thesis, June, 1994\n\n\"A Wide-Aperture HF Direction-Finder with Sleeve Antennas,\" Raymond F. Gleason, Robert M. Greene, Naval Research Laboratories Memorandum Report 843, August 20, 1958\n\n\"Utilization of a Multiprocessor in Command and Control,\" Bruce Wald, Proceedings of the IEEE, vol. 54, no. 12, December 1966\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004, provided a reference for submarine designations and capabilities.\n\nSoviet Naval Developments, third edition, foreword by Norman Polmar, The Nautical and Aviation Publishing Co. of America, 1984\n\nU.S. Military Operations Since World War II, Kenneth Anderson, Brompton Books Corporation, 1984\n\nSubmarine vs. Submarine, Richard Compton-Hall, Grub Street, 1988\n\nStrategic Intelligence for American National Security, Bruce D. Berkowitz and Allan E. Goodman, Prince ton University Press, 1989\n\nThe Soviet Union and the Arms Race, David Holloway, Yale University Press, 1984\n\nWEBSITES\n\nhttp:\/\/www.lostsubs.com\/E_Cold.htm\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Thresher_(SSN-593)\n\nhttp:\/\/en.wikipedia.org\/wiki\/Bathyscaphe_Trieste\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Preserver_(ARS-8)\n\nhttp:\/\/usnavyphotos.com\/wp-content\/uploads\/2009\/04\/ars-8-usspreserver1res cue-85x11.jpg\n\nhttp:\/\/en.wikipedia.org\/wiki\/Robert_McNamara\n\nhttp:\/\/www.janes.com\/articles\/Janes-Military-Communications\/US-Navy-worldwide-HF-DF-system-AN-FRD-10-or-Bullseye-United-States.html\n\nhttp:\/\/www.jproc.ca\/rrp\/masset.html\n\nhttp:\/\/www.onpedia.com\/encyclopedia\/Wullenweber\n\nhttp:\/\/en.wikipedia.org\/wiki\/Wullenweber\n\nhttp:\/\/coldwar-c4i.net\/CDAA\/history.html\n\nhttp:\/\/www.r-390a.net\/faq-systems.htm\n\nhttp:\/\/groups.msn.com\/ctoseadogs\/wullenwebers1.msnw\n\nhttp:\/\/www.fas.org\/irp\/program\/collect\/an-flr-9.htm\n\nhttp:\/\/www.eham.net\/forums\/Elmers\/201472\n\nhttp:\/\/www.espionageinfo.com\/Pr-Re\/Radio-Direction-Finding-Equipment.html\n\nCHAPTER 12\n\nPRIMARY INTERVIEWS\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook, provided keen insights into missions conducted by the USS Swordfish around the time of K-129's demise.\n\nRESOURCES\n\nBody of Secrets: Anatomy of the Ultra-Secret National Security Agency, James Bamford, Anchor, 2002\n\nScorpion Down, Ed Offley, Basic Books, 2007\n\nRed Star Rogue, Kenneth Sewell with Clint Richmond, Pocket Star Books, 2005\n\nAll Hands Down: The True Story of the Soviet Attack on the USS Scorpion, Kenneth Sewell and Jerome Preisler, Simon & Schuster, 2008\n\nA Century of Spies, Intelligence in the Twentieth Century, Jeffrey T. Richelson, Oxford University Press, 1995\n\nA Matter of Risk, Roy Varner and Wayne Collier, Random House, 1978\n\nThe Jennifer Project, Clyde W. Burleson, Prentice Hall, 1977, and the Texas A&M University Press, 1997\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nThe Pueblo Incident, Report No. 76\u20139, R. C. Spaulding, Naval Medical Research and Development Command, reprinted from Military Medicine, vol. 141, no. 9, September 1977\n\nSpy vs. Spy, Ronald Kessler, Pocket Books, 1988\n\nSpy Book: The Encyclopedia of Espionage, Norman Polmar and Thomas B. Allen, Random House, 1997\n\nWEBSITES\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Pueblo_(AGER-2)\n\nhttp:\/\/en.wikipedia.org\/wiki\/Soviet_submarine_K-129_(Golf_II)\n\nhttp:\/\/www.seattlepi.com\/awards\/scorpion\/scorpion3.html\n\nhttp:\/\/en.wikipedia.org\/wiki\/John_Anthony_Walker\n\nhttp:\/\/jproc.ca\/crypto\/kw7.html\n\nhttp:\/\/www.knobstick.ca\/museum\/kw7.htm\n\nhttp:\/\/www.washingtonpost.com\/wp-srv\/aponline\/20000822\/aponline135027_000.htm\n\nhttp:\/\/www.lostsubs.com\/E_Cold.htm\n\nCHAPTER 13\n\nPRIMARY INTERVIEWS\n\nJoseph Houston, former optics engineering lead on Project Azorian and member of the USSVI submariner's organization in San Jose, California, offered a behind-the-scenes view of the engineering effort involved in bringing up the remains of K-129, as well as a friend's recollection of Carl Duckett, CIA deputy director of science and technology, and of John Parangosky, project director, also known as Mr. P.\n\nTo test his photographic \"catfish solution\" for Project Azorian, Joe Houston used a Pentax camera loaded with Plus-X 35 millimeter black-and-white film while clinging precariously to the end of a large limb that arched over the pond in his backyard. Joe inched to the center of the branch while his fifteen-year-old son, Brant, positioned the targets by tilting them to the best reflective angle of almost 45 degrees. Brant then triggered a Hydro Products strobe lamp after Joe determined that the tree limb had stopped swaying enough to take an exposure. Father and son performed this test after dark in secret seclusion from spying neighbors, and Brant swore an oath to secrecy that he has kept to this day.\n\nDavid LeJeune, master diver chief, USN, Retired, imparted the only firsthand details ever exposed to the public about top-secret navy saturation diving operations during the Cold War.\n\nRESOURCES\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nThe Jennifer Project, Clyde W. Burleson, Prentice Hall, 1977, and the Texas A&M University Press, 1997\n\nBody of Secrets: Anatomy of the Ultra-Secret National Security Agency, James Bamford, Anchor, 2002\n\nSpy Sub, Roger C. Dunham, Naval Institute Press, 1996\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nThe Universe Below, Discovering the Secrets of the Deep Sea, William J. Broad, Simon & Schuster, 1997\n\nUSS Scorpion (SSN-589)\u2014Court of Inquiry Findings, Opinions, Recommendations, 26 October 1993\n\n\"Report Heightens Nuclear Sub Mystery\/Torpedo Theory Contradicts Findings of USS Scorpion's Wreckage in 1968,\" Houston Chronicle, December 27, 1993\n\n\"Project Azorian: The Story of the Hughes Glomar Explorer,\" Studies in Intelligence, Fall 1985, Secret, Excised copy\n\nMemorandum of Conversation, February 7, 1975, 5:22\u20135:55 P.M., Confidential, Excised copy, Archival source: Gerald R. Ford Presidential Library; National Security Adviser\u2014Memoranda of Conversation, box 9\n\nMemorandum of Conversation, \"[Jennifer?] Meeting,\" March 19, 1975, 11:20 A.M., Secret, Excised copy, Archival source: Ford Library, National Security Adviser\u2014Memoranda of Conversation, box 10\n\nWEBSITES\n\nhttp:\/\/en.wikipedia.org\/wiki\/Soviet_submarine_K-129_(Golf_II)\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Halibut_(SSGN-587)\n\nhttp:\/\/www.gwu.edu\/~nsarchiv\/nukevault\/ebb305\n\nCHAPTER 14\n\nPRIMARY INTERVIEWS\n\nDavid LeJeune, master diver chief, USN, Retired.\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook who rode the USS Seawolf for several SpecOps.\n\nRESOURCES\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nThe Jennifer Project, Clyde W. Burleson, Prentice Hall, 1977, and the Texas A&M University Press, 1997\n\nTango Charlie, Tommy Cox, Riverdale Books, 2006\n\nSubmarines, Rear Admiral John Hervey, Brassey's Sea Power, Naval Vessels, Weapons Systems and Technology Series, 1994\n\nU.S. Navy Diving Manual, NAVSEA 0994-LP-001-9010, Navy Department, June 1978\n\nWEBSITES\n\nhttp:\/\/en.wikipedia.org\/wiki\/USS_Halibut_(SSGN-587)\n\nCHAPTER 15\n\nPRIMARY INTERVIEWS\n\nJeff Gossett, technical director for the Arctic Submarine Laboratory.\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook who rode the USS Flying Fish and USS Seawolf for several SpecOps.\n\nRESOURCES\n\nSubmarines of the Russian and Soviet Navies, 1718 to 1990, Norman Polmar and Jurrien Noot, Naval Institute Press, 1991\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nCombat Fleets of the World, 1980\/81, ed. Jean Labayle Couhat, Naval Institute Press, 1980\n\nTransparent Oceans: The Death of the Soviet Submarine Force, Louis P. Solomon, LLRAPP Company, 2003\n\nSubmarine, A Guided Tour Inside a Nuclear Warship, Tom Clancy, Berkeley Books, November 1993\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nCHAPTER 16\n\nPRIMARY INTERVIEWS\n\nKenneth Greenawald, former sonar technician aboard the USS Haddo (SSN-604).\n\nLt. Edwin Ladeau Tomlin, former officer aboard the USS Haddo.\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nThe Blue Jackets Manual, twentieth edition, revised by Bill Bearden and Bill Wedertz, United States Naval Institute, 1978\n\nSubmarines, Rear Admiral John Hervey, Brassey's Sea Power, Naval Vessels, Weapons Systems and Technology Series, 1994\n\nAnti-Submarine Warfare, W. J. R. Gardner, Brassey's Sea Power, Naval Vessels, Weapons Systems and Technology Series, 1996\n\nSubmarine: A Guided Tour Inside a Nuclear Warship, Tom Clancy, Berkeley Books, November 1993\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nNOTES\n\nThe tradition of the U.S. Navy submarine dolphin insignia, awarded to those who complete submarine qualification, dates back to June 13, 1923, when Capt. E. J. King, commander of Submarine Division Three, forwarded a suggestion to the Bureau of Navigation (now designated secretary of the navy). He recommended that a distinguished device be adopted for those submariners who passed the rigorous requirements to become operationally qualified in all aspects of submarines.\n\nKing submitted his own pen-and-ink sketch of the new emblem that submariners spent a year earning. The design consisted of a shield mounted on the beam ends of a submarine, with dolphins forward and aft of the conning tower. The commander of Submarine Division Atlantic endorsed the suggestion, and over the course of several months, the Bureau of Navigation solicited additional \"dolphin\" designs.\n\nThe navy asked a firm in Philadelphia that had previously designed class rings for the Naval Academy to create just the right emblem to dress the uniforms of qualified submariners. The firm came up with two designs that were eventually combined into a single emblem. On March 21, 1924, Theodore Roosevelt Jr., acting secretary of the navy, accepted the dolphins as the official insignia of qualified submariners. Fifty-three years later, on board the USS Haddo, I yearned for the day that I could proudly wear the approved emblem on my uniform, just above my heart\u2014a dolphin flanking the bow and conning tower of a submarine.\n\nCHAPTER 17\n\nPRIMARY INTERVIEWS\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook who rode the USS Parche and USS Seawolf for several SpecOps.\n\nDavid LeJeune, master diver chief, USN, Retired.\n\nDennis Smith, former electronics technician aboard the USS Parche (SSN-683).\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nTango Charlie, Tommy Cox, Riverdale Books, 2006\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nSubmarine: A Guided Tour Inside a Nuclear Warship, Tom Clancy, Berkeley Books, November 1993\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nWEBSITES\n\nhttp:\/\/www.navysite.de\/ssn\/ssn687.htm\n\nNOTES\n\nThe ACT V group of spooks originally selected Bill Stringfellow to replace Mac Empey but recanted after interviewing Charlie Miller, the candidate recommended by BUPERS.\n\nCHAPTER 18\n\nPRIMARY INTERVIEWS\n\nKenneth Greenawald, former sonar technician aboard the USS Haddo (SSN-604).\n\nEdwin Ladeau Tomlin, former officer aboard the USS Haddo.\n\nFormer crew members who served aboard the USS Drum (SSN-677).\n\nAnonymous sources who worked on government programs during the Cold War.\n\nCaptain First Rank Igor Kurdin, Russian Navy, Retired, president of the Saint Petersburg Submariners Organization.\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nSeals in Action, Kevin Dockery, Avon Books, 1991\n\nU.S. Navy SEALs in Action, Hans Halberstadt, Motorbooks International Publishers, 1995\n\nThe Power Series; U.S. Navy SEALs, Hans Halberstadt, Motorbooks International Publishers, 1993\n\nSubmarines, Rear Admiral John Hervey, Brassey's Sea Power, Naval Vessels, Weapons Systems and Technology Series, 1994\n\nU.S. Navy Diving Manual, NAVSEA 0994-LP-001-9010, Navy Department, June 1978\n\nSubmarine: A Guided Tour Inside a Nuclear Warship, Tom Clancy, Berkeley Books, November 1993\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nNOTES\n\nAdvanced computer-processing capabilities, towed arrays, and superior materials science gave the United States huge advantages over the Soviet Union throughout most of the Cold War. Sound suppression involved a variety of practices, including everything from motor mounts to plant design choices, such as minimizing the amount of rotational equipment required and the use of lower steam pressure. Soviet submarine designs sacrificed stealth for speed, usually employing noisier five-bladed screws that allowed for a few extra knots at the top end. They wanted to keep up with the source of their greatest naval fear\u2014American aircraft carriers. U.S. boats employed secret machining technology to crank out seven-bladed screws that traded top-end speed for stealth. The Soviets were unable to duplicate this until Toshiba and Kongsberg sold the technology required.\n\nCHAPTER 19\n\nPRIMARY INTERVIEWS\n\nFormer crew members who served aboard the USS Drum (SSN-677).\n\nAnonymous sources who worked on government programs during the Cold War.\n\nCaptain First Rank Igor Kurdin, Russian Navy, Retired, president of the Saint Petersburg Submariners Organization.\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nPiping Systems, SSN 677, NAVSEA 0903-013-4010, Naval Ship System Command, July 1972\n\nSubmarine: A Guided Tour Inside a Nuclear Warship, Tom Clancy, Berkeley Books, November 1993\n\nSpying Beneath the Waves: Nuclear Submarine Intelligence Operations, prepared by Hans M. Kristensen, Greenpeace International, August 1994\n\nWEBSITES\n\nhttp:\/\/wapedia.mobi\/en\/Soviet_submarine_K-324\n\nCHAPTER 20\n\nPRIMARY INTERVIEWS\n\nJames Rule, former torpedoman on the USS Seawolf during Ivy Bells missions.\n\nThomas Ballenger, former fire control technician on the USS Seawolf during Ivy Bells missions.\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook who rode the USS Seawolf for several SpecOps.\n\nDavid LeJeune, master diver chief, USN, Retired.\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nNOTES\n\nSome of the other divers assigned to the Seawolf that Petty Officer Tom Ballenger mingled with included Bill Fitzpatrick, Tim Bohl, and Eddie Baker. Ballenger claims that, in addition to \"heroes,\" the crew often referred to the divers as \"3200s.\" He suspects this was in reference to some of the equipment they used, but he never knew why. Master Diver David LeJeune does not recall anyone using the 3200 term for the divers.\n\n\"Beau got a little beat up but survived his fall from the perch,\" said Jim Rule regarding the smack that the torpedoman's porcelain mascot took during the storm that hit the Seawolf. \"It was one of those little things that gave us hope that maybe we'd survive. When you're consumed by thoughts of death, you look for anything that can lift your spirits. Beau surviving that fall made us think that maybe we could make it, too.\"\n\nCHAPTER 21\n\nPRIMARY INTERVIEWS\n\nJames Rule, former torpedoman on the USS Seawolf during Ivy Bells missions.\n\nThomas Ballenger, former fire control technician on the USS Seawolf during Ivy Bells missions.\n\nFrank Turban, former communications technician \"T-Brancher\" chief and spook who rode the USS Seawolf for several SpecOps.\n\nDavid LeJeune, master diver chief, USN, Retired.\n\nRESOURCES\n\nCold War Submarines, Norman Polmar and K. J. Moore, Potomac Books, 2004\n\nBlind Man's Bluff, Sherry Sontag and Christopher Drew, Public Affairs, 1998\n\nA Century of Spies: Intelligence in the Twentieth Century, Jeffrey T. Richelson, Oxford University Press, 1995\n\nSpy vs. Spy, Ronald Kessler, Pocket Books, 1988\n\nSpy Book: The Encyclopedia of Espionage, Norman Polmar and Thomas B. Allen, Random House, 1997\n\nWhen Dennis Smith returned to the USS Parche in 1983, things had changed in ways unimagined. Commander Peter J. Graef had taken command, and a myriad of new faces walked the passageways. Master Diver David LeJeune relieved the previous master diver after he forgot to bring the critical section of threaded pipe that allowed the divers to \"blow down\" the water from the diving platform in the fake DSRV. An entire mission had to be scrubbed as a consequence.\n\nSmith recalled that previous cable taps were conducted at the repeater locations, the points where two long sections join, and the signal is boosted to last through the next leg in the journey. Tapping the cable at this point seemed most efficient in the early days but came with drawbacks. Soviet cable-laying ships usually pulled cables up for spot checks at repeater locations. Should they hoist up a tap along with the cable, that'd be bad.\n\nA land-based U.S. intelligence operative had managed to obtain a section of the communications cable used by the Soviets from a Russian vendor, which allowed the NSA to examine the design and engineer a more efficient cable-tapping method. Divers now were equipped with special tools that allowed them to penetrate the outer wrapping of the steel wire down to the center dielectric inside the shielding.\n\nThe clamp used for the tap also had a breakaway feature. If the Soviets pulled up the cable, the induction tap would simply \"pull out\" of the cable, allowing it to twist back to normal. Only a barely visible scar remained as evidence of foul play. A new System 9500 came with a small amplifier, buried in the mud, that allowed the large pod enclosure to reside almost a mile away from the recording container. Divers attached 4,500 feet of cable to the amplifier, then returned to the Parche. The boat moved to a new location and dropped the other end of the 4,500-foot cable along with a Deep Ocean Transponder (DOT), which allowed the divers to find the cable end and hook up the twenty-foot-long \"beer keg\" beast. This ingenious system improved cable-tapping efficiency while lowering risk.\n\nDuring Smith's second run to the Barents, the Soviets detected the Parche sneaking into the area. Commander Graef came to periscope depth at night and maneuvered into the middle of a large Russian fishing fleet to hide from the lurking Soviet warships and planes. Sonar reported active pinging and planes and helos scattering sonobuoys all around them. Light \"warning charge\" explosions clapped in the distance.\n\nHis chest thumping, Smith sat in the chief-of-the-watch seat on the port side of the \"rigged for red\" control room. Graef sounded off bearings to contacts that he spied through the scope. A loud thud, followed by an emergency buzzer, echoed through the compartment. For reasons unknown, the controls for the fairwater and stern planes, which control the depth and angle of the boat, shifted from normal to emergency hydraulics. The noise rattled everyone in the control room, including Smith. The planesman and helmsman jumped in their seats and started muttering expletives. The diving officer followed suit.\n\nGraef peeled his eye away from the periscope and said, \"We drill for this all the time, people. It's just the hydraulics shifting to emergency. Get a grip and drive the fucking boat! You guys know how to do this.\"\n\nWith that, everyone calmed down and guided the submarine through the briar patch unscathed. Smith let out a slow breath, certain that his CO had everything under control. Despite the dangers that encircled his home, he knew he'd survive yet another mission into the hinterland and return with something no amount of money could buy: a sense of pride that runs deeper than any other.\n\nEPILOGUE\n\nRESOURCES\n\n\"DESI: Diesel Electric Submarine Initiative, a Partnership for Global Security,\" Undersea Warfare, Spring 2006, pp. 19\u201320.\n\n\"Joint Spec Ops: Air Force, Navy Test Rescue Scenario,\" Undersea Warfare, Spring 2006, pp. 9\u201311.\n\n\"The How and Why of Open Architecture,\" Undersea Warfare, Spring 2008, pp. 6\u20139.\n\n\"Managing Modernization: A Fleet First Perspective,\" Undersea Warfare, Spring 2008, pp. 10\u201312\n\n\"Patrolling the Deep: Critical Anti-submarine Warfare Skills Must Be Restored,\" available at http:\/\/www.armedforcesjournal.com\/2008\/09\/3654984\/\n\nChina's Future Nuclear Submarine Force, ed. Andrew S. Erickson, Lyle J. Goldstein, William S. Murray, and Andrew R. Wilson, Naval Institute Press, 2007\n\nWEBSITES\n\nhttp:\/\/www.sublant.navy.mil\/ICEX09\/ICEX09.htm\n\nNOTES\n\nQuieter submarines demand shorter passive detection ranges, which has triggered research in low-frequency active sonar. The downside of such systems is the adverse effects on whales and dolphins, which has caused an uproar in environmentalist circles. No such uproar emanates from software engineers, who revere recent changes in modern sonar systems that transcend the closed architectures used in the equipment of my day. In the eighties and nineties, BQQ-5 sonar systems served front and center on submarines and relied on Sperry\/UNIVAC UYK-7 processors. Unfortunately, this technology became obsolete not long after installation. Proprietary software came part and parcel with hardware, and decoupling the two could not be done. Software upgrades required drastic hardware plus system changes and vice versa, which translated into expensive, typically in the range of $150 million.\n\nThe navy reduced that price tag to under $15 million by incorporating the ARCI program, which stands for Acoustic Rapid COTS Insertion. COTS means commercial off-the-shelf technologies. After ARCI, software is now updated every two years in sonar shacks on modern Los Angeles-, Seawolf-, and Virginia-class submarines, where glittering computer screens resemble a Star Trek movie set. Similar changes have impacted fire-control systems. I recall spending hours troubleshooting bays of massive analog gear on Permit- and Sturgeon-class boats. That's all been replaced by a bank of BYG-1 consoles with readouts that look more like video-game monitors than submarine combat systems. After touring some of these modern marvels, I often miss my time on the boats and wish I was just a few decades younger.\n\nFor most submariners, especially those involved in top-secret and highly dangerous Holystone and Ivy Bells missions, dry land often waited for more than ninety days at a stretch. Such was also the case for submariners serving on fleet ballistic-missile \"boomer\" subs. At sea for months on end, few found time for romance or families. Most missed holidays, birthdays, anniversaries, and their children's first breaths. Still, they served with pride and carried pictures of loved ones while operating in hostile waters.\n\nWhen Bill Heaton graduated from a small Alabama high school in 1977, uncertainty owned him. The yellow brick road that lay ahead was littered with choices that could lead to somewhere dismal or nowhere at all. His friends suggested the Crimson City, so he followed in their footsteps and enrolled in the University of Alabama at Birmingham. As a starving freshman, he struggled through classes while working nights to pay the tuition. A year went by before the money ran out. Heaton then followed in his father's footsteps and found employment at Alabama Power. Each day after work, he forced his eyes to stay open during night classes at the university, but he soon found it impossible to keep up with schoolwork and earn a paycheck. Out of options and money, Heaton fell to his knees and prayed. His strong Christian faith gave him confidence that an answer would come.\n\nThe following day, rays of morning light shot through his window and lit up a book on his desk. He dusted off the cover and smiled. A book about submarines. As a kid, he'd read just about everything written about World War II submarines and imagined himself in the role of a Barnacle Bill riding the old smoke boats. Every young boy needs heroes, and for Heaton, submariners were role models worthy of respect. Never was he more excited than when his parents bought him a model of the submarine USS George Washington. He played with the missile-launching mechanism so much that he broke the tiny spring. Now, as he stared at one of his old submarine storybooks, he knew that God had answered his prayers.\n\nHeaton's father had served in the Air Force with pride, and although antiestablishment anger still rippled through a society shattered by the Vietnam War, Heaton still got choked up when they played the national anthem at baseball games. Some called his red, white, and blue sentiment corny, but he didn't care. That day, in August 1979, he marched down to the recruiting office and volunteered for submarines. He signed up on the delayed entry program and attended boot camp in Orlando, Florida, in March 1980, graduating on the very day that the mission to rescue the hostages in Iran failed. Eight U.S. servicemen died in that attempt when a helicopter crashed into a C-130 transport plane. Heaton gathered with the other new recruits as the boot-camp company commander informed them of what had happened. Unsure if another war lay just around the corner, Heaton muzzled his trepidation as he stood and saluted, ready to serve alongside his fellow sailors.\n\nAfter boot camp, Heaton attended interior communications school in San Diego, California, followed by Submarine School in Groton, Connecticut. He learned about how submariners talk to one another, stay on course, and breathe air while submerged through the classes he took about closed-loop communications systems, gyro compasses, and central-atmosphere monitoring systems. Heaton completed C School, added a stripe to his sleeve, and went home on vacation. A friend ridiculed him and insisted that Heaton was \"stupid to join the navy,\" and that \"serving in Uncle Sam's Canoe Club was a colossal waste of time.\" Heaton ignored the insults and reported to his first submarine, the USS Stonewall Jackson (SSBN-634). A few years later, while at home on leave again in September 1983, he met the love of his life.\n\nTwenty-three-year-old Beth worked with one of Heaton's cousins, who talked her into a blind date. Beth at first refused, stating that she had no desire to go out with a sailor who'd been at sea for months. Beth still lived at home, and an hour before the date she told her mother that she'd changed her mind and didn't want to go through with the meeting. Her mother insisted that to cancel would be impolite, so Beth reluctantly went upstairs to pick out a pair of shoes. She came back down an hour later in blue jeans and a red V-neck sweater. She had a thing for red shoes and wore a different style on each foot. Near the bottom of the stairs she said, \"Mom, I can't decide which of these to wear.\"\n\nA man's voice responded with \"I like the one on the right.\"\n\nBeth glanced up. A handsome five-foot-eleven, dark-haired sailor stood near the door. As she gazed into Heaton's eyes, Beth felt a glow in her chest.\n\n\"When I saw Bill for the first time,\" Beth says, \"bells started ringing, and I just knew that God was telling me he was the one.\"\n\nIt was love at first sight for Bill Heaton as well, but given his at-sea schedule on the Jackson, he saw Beth for only a few weeks at a time during the next nine months, which made each day all the more precious. Following a brief visit during the July Fourth weekend, Heaton called Beth from the pier at Kings Bay, Georgia. His voice shook as he talked.\n\n\"What's wrong?\" Beth asked.\n\n\"We're going out to sea tomorrow,\" Heaton said, \"and I don't want to spend another day without you by my side.\"\n\n\"I don't understand,\" Beth said. \"I am by your side.\"\n\n\"I mean,\" Heaton said, pausing for a few seconds, \"I want you by my side forever. I want you to be my wife.\"\n\nBeth had expected a kneeling proposal at a fancy restaurant complete with violins and a diamond ring, but she figured that \"ordinary\" was not in the cards for a submariner. Tears streaming down her cheeks, she said yes. \"I decided that it didn't matter if he proposed in a restaurant, my living room, or from Timbuktu. I loved him and wanted to be his bride. We knew he was due back in port in September, so we made plans for the weekend following the boat's arrival.\"\n\nBeth spent the next three months planning and inviting and giggling with anticipation. Then she got a call from her church pastor. He couldn't do the ceremony that weekend, so they needed to change the date. Beth panicked. How could she get word to Heaton, who was now 500 feet deep somewhere off the coast of Russia? E-mail did not exist in those days, so she sent a \"family gram\" with the message \"Wedding date changed\u2014stop\u2014details later\u2014stop\u2014new date September 15\u2014end.\"\n\nBill Heaton married Beth in September 1984. He nicknamed her Dusty in reference to an episode when she had tried to reuse an old vacuum cleaner bag because they couldn't afford to buy a new one. The bag split open and covered her from head to toe with dust. Heaton later transferred to the USS Batfish (SSN-681), a fast-attack boat that took him to the Barents Sea on SpecOps more than once. During one run, they conducted an \"under hull\" on an infamous Alpha-class Soviet boat, at the time one of the most advanced, fastest, and deepest-diving submarines in the world. Several times on that mission, when they were almost caught by the Soviets, Heaton wondered if he'd ever see Beth again. He relied on God and his fellow submariners to bring him home. \"I learned to trust others implicitly in submarines,\" says Heaton. \"Not just to throw me the ball or keep my car safe, but with my life. That kind of trust changed me forever.\"\n\nHeaton became the Protestant lay leader on the Batfish, conducting Sunday services and Bible studies with others of faith. \"I was humbled and honored to serve my country as a submariner,\" says Heaton. \"To this day my friends still call me 'Navy Bill.'\"\n\n## SEARCHABLE TERMS\n\nThe pagination of this electronic edition does not match the edition from which it was created. To locate a specific passage, please use the search feature of your e-book reader.\n\nABC News, 147\u201348, 156\n\nAcoustic intelligence (ACINT), 273\n\nActive sonar systems, 18, 272\u201373\n\nAgafonov, Vitali, 96, 172\u201373\n\nAkula\u2013class submarines, 338\n\nAlabama Power, 376\n\nAlantika, 115\n\nAleksandrovsk, 64\n\nAlpha\u2013class submarines, 338, 344\n\nAlvarez, Floyd, 220\n\nAmagin, 137\u201338\n\nAmelko, Ivan, 209\n\nAmerican Seaside Club, 25\u201326\n\nAmerica's Cup (1962), 78\n\nAnadyr, 58\u201361. See also Operation Kama\n\nAnderson, George W., 102, 106, 116\u201318, 125\u201329, 137, 147, 149\n\nAnderson, Jack, 225\u201326\n\nAnderson, Rudolph, 149\n\nAndrea Doria, 237\u201338\n\nAndreev, Anatoliy, 108\u201310\n\nAnechoic tiles, 266\n\nAN\/FLR\u20139, 47\n\nAN\/FRA\u201344, 43, 47\n\nAN\/FRD\u201310 (\"Fred Tens\"), 47\u201349\n\nAN\/GYK\u20133, 191\u201392\n\nAnnapolis, USS, 341\n\nApplied Physics Laboratory, 254\n\nApplied Technology Division (ATD), 227\n\nApra Harbor, 316\n\nAQA\u20137 signal processors, 143\n\nARCI program, 375n\n\nArctic Submarine Laboratory, 254, 341\n\nArkhipov, Vasily, 73, 151\u201355, 176\u201377\n\nArnold, John, 62\u201363, 231\u201332, 247\n\nAT&T, 253\u201354\n\nAurora 7, 181\n\nAustin, Mary, 22\n\nB-4, 67\u201377\n\nassuming combat readiness, 93\u201394, 107\n\navoiding detection, 171\u201374\n\nheading to Cuba, 73\u201377, 93\u201394\n\nmission details, 69\u201371, 75\u201376\n\nspecial weapon onboard, 68\u201370\n\nstorm troubles, 94\u201395\n\nB-36, 65\u201367, 73\u201377, 165\u201371, 174\u201378\n\nassuming combat readiness, 167\u201370\n\navoiding detection, 85\u201386, 110\u201315, 118\u201321, 149\u201350, 160\u201361, 165\u201368, 170\u201371\n\nbad luck onboard, 174\u201376\n\nheading to Cuba, 73\u201377, 82\u201386, 88\u201393, 95\u2013100, 107\u201315, 118\u201321, 149\u201350\n\nmission details, 69\u201371, 75\u201376\n\nPankov operation onboard, 82\u201384, 90\u201391\n\nreturning home, 176\u201378\n\nspecial weapon onboard, 66\u201367, 69, 71, 88\u201389\n\nB-37, 188\n\nB-59, 70\u201377, 107, 120, 149\u201355, 158\u201359\n\nB-130, 65, 69\u201377\n\navoiding detection, 164\u201365, 167\u201368\n\nengine troubles, 130\u201334, 164\u201365\n\nheading to Cuba, 73\u201377, 107, 120\n\nmission details, 69\u201371, 75\u201376\n\nBache, USS, 150\n\nBaer, Ralph, 146\u201347\n\nBailey, Don, 202\u20133\n\nBaker, Eddie, 371\u201372n\n\nBallard, Robert, 350n\n\nBallenger, Tom, 322\u201326, 371\u201372n\n\nBangor Naval Submarine Base, 338\n\nBanks, Bill, 19\u201320\n\nBaralyme, 245\n\nBarb, USS, 211\n\nBarbel, USS, 184\u201385\n\nBarron, James, 203\u20134\n\nBatfish, USS, 378\n\nBathyscaphes, 181\n\nBathythermograph (BT), 6\n\nBaud, 38\n\nBaudot, Jean\u2013Maurice-\u00c9mile, 38\n\nBayern, 44\u201345\n\nBay of Pigs, 179\n\nBBC Radio, 120\n\nBeale, USS, 150\n\nBeatty, David, 45\n\nBeauregard (mascot), 324, 326\n\nBekrenyev, Leonid, 129\n\nBellini, Ettore, 44\n\nBell Telephone Laboratories, 16\u201319, 248\n\nBends, 240, 335\n\nBenjamin Franklin-class submarines, 252\u201353\n\nBenton, Hugh, 211\n\nBesugo, USS, 2\u20133\n\nBettis Atomic Power Laboratory, 14, 282\n\nBirinci Inonu, 52\u201356\n\nBlake, Gordon, 136\u201337\n\nBlandy, USS, 106\u20137, 164\u201365\n\n\"Bleed air,\" 1\u20132\n\nBlenny, USS, 1\u20139\n\nBlind Man's Bluff (Drew and Sontag), ix\n\nBlue Gill, 256\n\nBlue Surf, 256\n\nBob (CIA operative), 196\u201397\n\nBohl, Tim, 371\u201372n\n\nBolo lines, 158\n\nBol'shevik Sukhanov, 116\n\nBondville Station, 47\n\nBonesteel, Charles H., 203\n\nBoresight project, 47\u201351, 188\u201391\n\ncalibration and signal analysis, 50\u201351, 53\n\ninstalling and testing, 49\u201350, 81\u201382, 87\n\nMcNamara and, 127\u201328, 137, 138, 188\n\nNSA funding and naming, 44, 47\u201348\n\nrefinements and expansion, 118, 141\u201347, 149, 189\u201394\n\nsuccess of, 179\n\nBoston Naval Shipyard, 181\n\nBowles, Ethel, 32\n\nBowles, Hoyle, 32\n\nBoykin, Dennis B., III, 14\n\nBQR\u201324, 273\n\nBradley, James F., Jr., 215\u201317, 221, 228\u201331, 249\n\nBranham, J. K., 328\n\nBRD\u20136, 209, 256\n\nBRD\u20137, 256\u201357\n\nBrill, USS, 51\n\nBrown, Aubrey, 101\n\nBrown, Gardner, 10\u201315, 282, 350n\n\nBrown, Hal, 289\u201390\n\n\"Brute Force\" plan, 220\u201321\n\nBryant, James S., 2\u20139\n\nBucharest, 126, 134\n\nBucher, Lloyd, 201\u20134\n\nBUD\/S training, 297\n\nBuffett, Jimmy, 264\n\nBuinevich, Doctor, 82\u201384\n\nBulls Eye, 44, 45, 47\u201349, 188\u201391, 197\u201398\n\nBundy, McGeorge, 87, 136\u201337, 148\n\nBureau of Naval Personnel (BUPERS), 284\n\nBurke, Arleigh, 126\n\nBurroughs Corporation, 191\n\nBurst signals, 33\u201338, 41\u201342, 50\u201351, 123\u201324, 144\u201345\n\nByham, Donald, 4\u20136\n\nCable\u2013tapping missions, 228\u201331, 233\u201334, 238, 245\u201351, 287\u201389, 291\u201393, 319, 324\u201327, 332\u201336\n\nCafrey, Phil, 260\n\nCaicos Passage, 107\n\nCaltech, 25\u201326\n\nCanadian\u2013U.S. Atlantic HFDF network, 197\u201398\n\nCane, James Christopher, 329\n\nCarbon dioxide (CO2) poisoning, 8, 9, 12\u201313\n\nCarlin, George, 265\n\nCarlson, Rich, 17\u201318\n\nCarpenter, Scott, 181\n\nCarter, James Earl \"Jimmy,\" 15, 265, 319\n\nCastro, Fidel, 59, 157, 176\n\nCatch\u201322 (Heller), 35\n\nCavala, USS, 19\u201320\n\nCelik, Captain, 51\u201356\n\nCentral Intelligence Agency (CIA)\n\nCuban Missile Crisis and, 62, 63\u201364, 103\u20135, 106, 148\n\nForeign Broadcast Information Service, 200\u2013201\n\nGreek incident, 194\u201397\n\nProject Azorian, 217, 219\u201328\n\nCentre, Gene, 14\u201315, 282\n\nCesium clocks, 145\n\nChaffin, Leo, 7\n\nChard, John, 343\n\nCharles P. Cecil, USS, 107, 159\u201364, 168\u201370\n\nChebanenko, Admiral, 65\u201366\n\nCheprakov, Lieutenant, 133\u201334\n\nChesapeake, USS, 203\u20134\n\nChicca, Robert, 203\n\nChinese navy, 345\n\nChurchill, Larry, 17\u201318\n\nChurchill, Winston, vii, 294\n\nCircularly Disposed Antenna Array (CDAA), 45\u201346, 47\n\nClarinet Bulls Eye, 191\n\nClarke, Arthur C., 215, 223\n\nClassic Bulls Eye, 191, 197\u201398, 253\n\nClassic Outboard program, 254\u201355\n\n\"Clementine,\" 224\u201326\n\nClower, Lieutenant, 80\u201381, 122\n\nCODAR (COastal raDAR), 142\n\nCold War Medal Act of 2007, ix\n\nCombat Information Center, 159\n\nCompass rose board, 29\u201330, 124, 145\n\nConstellation, USS, 300\n\nCony, USS, 107, 129, 134, 150, 157\u201359\n\nCoral Sea, USS, 180\n\nCore, USS, 355n\n\nCourtney, Frank, 283\u201384\n\nCousins, Rich, 35\u201338\n\nCox, Tommy, 283\u201384\n\nCrandal, Charles, 203\n\nCrazy Ivan maneuver, 273\u201374\n\nCretan Star, 119\n\nCrews, Harry, 156\n\nCrowley, Tom, 231\u201332\n\nCuban Missile Crisis, 57\u2013179. See also Operation Kama\n\nfirst U.S. casualty, 149\n\nKennedy\u2013Khrushchev deal, 176\n\nKennedy's secret memo to Khrushchev, 156\u201357\n\nKhrushchev's letters to Kennedy, 156\u201357\n\nSoviets halt advance of cargo ships, 148\n\nSoviets launch Operations Kama and Anadyr, 57\u201361\n\nU.S. ExComm meetings, 87, 102, 118, 129\u201330, 178\n\nU.S. military preparations, 115\u201318, 147\u201348\n\nU.S. quarantine measures, 64\u201365, 106\u20137, 115, 116\u201317, 120, 125\u201330, 150\n\nU.S.\u2013Soviet standoff, 149\u201350, 156\u201357\n\nU.S. surveillance, 78\u201382, 86\u201388, 101\u20137, 115\u201318, 122\u201325, 130, 134\u201340, 148\u201349\n\nU.S.'s initial discovery of Soviet operations, 61\u201365\n\nCubera, USS, 10\u201313, 15, 350n\n\nCummings, Laird, 334\u201336\n\nCumshaw, 35\n\nCussler, Clive, 237\n\nDace, USS, 217\u201318, 296\n\nDare, J. Ashton, 321\u201322, 326\u201327\n\nDartmouth College, 11\n\nDavis, John, 62, 78\n\n\"Dead zone\" for Soviet torpedoes, 169\n\nDecompression sickness, 230, 240\n\nDeep Ocean Transponder (DOT), 373n\n\nDeep Submergence Rescue Vehicle (DSRV), 229\u201330\n\nDefense readiness condition (DEFCON), 117\u201318\n\nDelta\u2013class submarines, 252\u201353, 257\u201358, 276\u201381\n\nDemodulation, 19, 20\n\nDEMON sonar systems, 20, 21\n\nDenofrio, Tommy, 81\u201382, 141\u201345\n\nDepth charges, 152\u201355, 315\u201316\n\nDesign collapse depth, 6\u20137, 162\n\nDe Steiguer, USS, 289\u201390\n\nDeveloping film, 300\u2013301\n\nDiego Garcia, 301\u20132\n\nDiem, Eugene \"Gunga Din,\" 239\u201340, 243, 244\u201345\n\nDiesel\u2013powered submarines, 8\u201310, 13, 14, 15\n\nDipping sonar, 166\u201367\n\nDirectional Frequency and Ranging (DIFAR), 142\u201344\n\nDirectional sound viewpoint, 142\u201343, 253\u201354\n\nDiver fatigue, 237\n\nDive tables, 229\n\nDiving depths, 6\u20137, 162, 167\n\nDobrynin, Anatoly, 102\n\nDolphin insignia, 180, 367\u201368n\n\nDonovan, Robert, 157\n\nDrew, Christopher, ix\n\nDrum, USS, vii, 295\u2013317\n\ncollision with Victor III, 311\u201314\n\nevasive maneuvers, 314\u201316\n\nexercises, 297\u201399\n\noutside Diego Garcia, 301\u20132\n\nrecon photography, 296\u201397, 299\u2013301\n\nspotting Victor III, 304\u201311\n\nDry Deck Shelters (DDSs), 339\u201340\n\nDubivko, Aleksei, 65\u201367, 73\u201377, 165\u201371, 174\u201378\n\navoiding detection, 85\u201386, 110\u201315, 118\u201321, 149\u201350, 160\u201361, 165\u201368, 170\u201371\n\nbackground of, 65\u201366, 69\n\nbad luck onboard, 174\u201376\n\nheading to Cuba, 73\u201377, 82\u201386, 88\u201393, 95\u2013100, 107\u201315, 118\u201321, 149\u201350\n\nmission details, 69\u201371, 75\u201376\n\nPankov operation onboard, 82\u201384, 90\u201391\n\nreadying the weapon, 167\u201370\n\nreturning home, 176\u201378\n\nspecial weapon onboard, 66\u201367, 69, 71, 88\u201389\n\nDuckett, Carl, 216\u201317, 221, 222\u201323, 227\n\nDulles, John Foster, 78\n\nDygalo, Viktor A., 205\n\nEastman Kodak Company, 318\u201319\n\nEaton, USS, 150\n\nEcho\u2013class submarines, 193, 212, 214\n\nEdzell Bulls Eye, 48\u201349, 87\u201388, 189\n\nEisenhower, Dwight D., 265\n\nElectric Boat Corporation, 9\u201310, 14\u201315\n\nElectronic countermeasures (ECM), 41, 42\n\nElectronic intelligence (ELINT), 40\n\nElectronic surveillance measures (ESM), 5, 52\u201355, 114, 151\u201352, 200\n\nElephant cages, 47\n\nElk River, 236\n\nEllenwood, Bob, 283\u201384\n\nEmergency air\u2013breathing (EAB), 269\n\nEmpey, Malcolm \"Mac,\" 231\u201333, 247\u201351, 283\u201384\n\nEnterprise, USS, 105, 107\n\nEscape trunks, 230, 297\u201399, 310\u201315, 340\n\nEssex, USS, 105, 106\u20137, 120, 134\n\nExComm (Executive Committee of the National Security Council), 87, 102, 118, 129\u201330, 178\n\nF\u00e9nelon, Fran\u00e7ois, 252\n\nFifteenth Submarine Squadron, 205\n\nFilm processing, 300\u2013301\n\nFingerprinting, 256\n\nFire control, 3\n\nFish, 215\u201316, 230\u201331, 249\u201350\n\nFitzpatrick, Bill, 370 n\n\nFlacco, Nick, 306\u20137, 311\u201315\n\nFlasher, USS, 11\n\nFlying Fish, USS, 255\u201363, 284\n\nFokin, Vitali, 58\u201361, 70\u201371\n\nFollow\u2013on Test (FOT), 116\n\nFomin, Aleksandr, 147\u201348, 156\n\nFord, Gerald, 227\n\nFord Motor Company, 190\n\nFort George G. Meade, 4, 22, 50, 102\u20136, 130\n\nFoxtrot\u2013class submarines, 62, 68, 150, 162, 357\u201358 n, 360 n\n\nFred (CIA operative), 194\u201397\n\nFricke, Robert E., 295\u201396, 304, 311\u201312\n\nFrontz, Al, 242, 244\u201345\n\nFUNNEL, 62\n\nFursenko, Aleksandr, 157\n\nGagarin, 149\n\nGall, James, 354n\n\nGas supply, and divers, 237\n\nGates, Thomas, 118\n\nGauss, Karl, 144\n\nGaussian, 144\n\nGearing, USS, 134\n\nGeneva Convention, 46\n\nGeorge Washington, USS, 116, 189, 193, 316\u201317, 343\u201344, 376\n\nGibran, Kahlil, 199\n\nGilpatric, Roswell, 125, 127\u201329, 136\u201337\n\nGIUK gap (Greenland, Iceland, United Kingdom), 83\n\nGlomar Explorer, 223\u201328\n\nGnomonic projection, 30\n\nGoldwater, Barry, 265\n\nGolf II, 205, 207\u20138, 210, 212, 216, 224\n\nGoniometers, 48, 197\n\nGorshkov, Sergei, 31, 57\u201361, 64\u201365, 70\u201371, 82, 188, 343\u201344\n\nGorski, Anatoly, 157\n\nGossett, Jeff, 341\n\nGovernor Dummer Academy, 11\n\nGraef, Peter J., 333\u201334, 372\u201374n\n\nGrant, Ulysses S., 135\u201336\n\nGraznyy, 134\n\nGRD\u20136 stations, 43, 49, 87\u201388, 105\u20136, 134, 149, 188\u201389\n\nGreater Underwater Propulsion Power (GUPPY), 10\n\nGrechenov, Major, 148\u201349\n\nGrechko, Andrei, 177\u201378\n\nGreenawald, Kenneth \"Greenie,\" 265\u201366, 272, 275\u201381\n\nGreenling, USS, 284\n\nGrider, George W., 11\u201313, 15\n\nGriffiths, Charles, 116\n\nGrigorievich, Valentin, 153\u201354\n\nGromyko, Andrei, 102\n\nGT&E Sylvania Electronics Systems, 46, 47\n\nGuardfish, USS, 211\u201312\n\nGulf War, 338\n\nGurley, John, 122\u201325\n\nGYK\u20133 computers, 145, 147, 191\u201392, 206\n\nHaddo, USS, 265\u201381\n\ncrew of, 266\u201367\n\nfirst dive, 267\u201368\n\nlife under the seas, 268\u201381\n\nsonar systems, 271\u201374\n\ntracking and photographing submarines, 212, 276\u201381, 301\n\ntransfer from, 295\u201396\n\nHalibut, USS, 229\u201333, 238\u201351\n\ncable\u2013tapping mission, 228\u201331, 245\u201351\n\nsearching for K-129, 216\u201317\n\nupgrade, 215\u201316, 230\n\nHalidere (Turkish neighbor), 23\u201324\n\nHalworth, \"Doc,\" 238\u201339, 245\u201346\n\nHansen, R. J., 357\u201358n\n\nHanza Bulls Eye, 87\u201388, 189\n\nHarrier, USS, 223\n\nHarrison, Joel, 342\n\nHarvard Underwater Sound Laboratory, 17\n\nHarvey, John Wesley, 180, 185\u201386\n\nHaver, Richard, 333\n\nHayden, Edgar, 46\u201347\n\nHeaton, Beth, 377\u201378\n\nHeaton, Bill, 376\u201378\n\nHelena, USS, 341\n\nHelium diving, 229, 230, 240\u201342\n\nHeller, Joseph, 35\n\nHelms, Richard, 217\n\nHendrix, Jimi, 180\n\nHensley, Jimmy, 34\u201337\n\nHeyser, Richard S., 86\u201387\n\nHigh frequency (HF), 27\n\nHigh frequency direction finding (HFDF), 23, 26\u201329, 41, 50\u201351, 355n\n\nHigh\u2013pressure nervous syndrome (HPNS), 240, 242, 244\u201345\n\nHilarides, William H., 340\n\nHitchcock, Alfred, 220\n\nHodges, Duane, 203\n\nHolser, Alex, 220\n\nHolystone, ix, 273, 305, 375\n\nHomestead Station, 78\u201382, 86, 105\u20136, 116, 122\u201325, 134\n\nHotel, 193\n\n\"Hot running\" torpedoes, 291\n\n\"Hot shit,\" 255\n\nHouston, Brant, 364n\n\nHouston, Joseph, 217\u201324, 227, 364n\n\nHubel, Augustine \"Gus,\" 232\n\nHughes, Howard, 219\n\nHughes Glomar Explorer, 223\u201328\n\nHunt, John, 239\u201340, 245\n\nHunter, John, 159\u201364, 168\n\nHutchthausen, Peter, 357n\n\nHybla Valley Coast Guard Station, 44\n\nHydro Products, 222, 364n\n\nHyperbaric diving chambers, 230, 240, 242, 249, 287\u201388\n\nIDKCA (\"rise to the surface\"), 126\n\nIlyushin Il\u201328\n\nbombers, 104, 176\n\nIndependence, USS, 172\u201373\n\nIndigirki, 64\n\nIntelligence operatives (I-Branchers), 29, 262\n\nInterim Towed Array Surveillance System (ITASS), 253\u201355, 272\u201373\n\nIonospheric hop, 87\u201388\n\nIran hostage crisis, 319, 376\u201377\n\nIraq, and Gulf War, 338\n\nItek Corporation, 217\u201324\n\nITT Federal Systems, 48, 189\u201390\n\nIvanov, P. K., 177\n\nIvy Bells, ix. See also Cable\u2013tapping missions\n\nIwo Jima, 11\n\nJamming systems, 40\u201341, 50\n\nJauregui, Steve, 353n\n\nJefferson, Thomas, 135\n\nJimmy Carter, USS, 339\n\nJohn Marshall, USS, 306\n\nJohn S. McCain, USS, 345\n\nJohnson, Lyndon, 203\n\nJuliett\u2013class submarines, 193\n\nJupiter missiles, 148, 156\u201357\n\nJust a Sailor (Waterman), 296\u201397\n\nK-3, 188\n\nK-8, 188\n\nK-19, 188, 343\u201344\n\nK-129, 205\u201312\n\nHalibut search for, 216\u201317\n\nProject Azorian salvage plan, 217, 221\u201328\n\nrogue theory about, 207\u20138\n\nsinking of, 211\u201312\n\nSwordfish stalking of, 207\u201312\n\nK-219, 344\n\nK-324, vii, 304\u201317\n\nDrum collision with, 311\u201314\n\nDrum spotting of, 304\u201311\n\nKalugin, Oleg, 204\u20135\n\nKami Seya Naval Security Group Activity, 202\u20133, 209\u201310, 255\n\nKaram\u00fcrsel Station, 22\u201323, 26\u201339, 49\u201350\n\nKaye, Jack, 78\u201379, 87, 102\u20136, 118, 130, 134\u201340, 147, 188, 192\n\nKelley, Edward, 164\n\nKelly, Joseph, 16\n\nKennedy, John F.\n\naddress to nation (October 22, 1962), 107, 115\n\nassassination of, 193\u201394\n\nbriefing with Reed about Boresight, 134\u201339\n\non crisis, 337\n\ndeal with Khrushchev, 176\n\nExComm meetings, 102, 118, 129\u201330\n\ninvasion preparations, 147\u201348\n\nquarantine measures and, 102, 105, 106, 107, 118, 125\u201326, 129\n\nsecret memo to Khrushchev, 156\u201357\n\nSoviet offensive weapons and, 62, 63\u201364, 87, 101\u20132, 104, 115, 120, 176\n\nSoviet submarines and, 102, 106, 129\u201330, 134\u201335, 149, 150\n\nWorld War II and PT 109, 137\u201338\n\nKennedy, Robert, 117, 148\n\nKetov, Ryurik, 67\u201377, 93\u201395, 107, 171\u201374, 357n\n\nKey West Naval Base, 115\u201316\n\nKGB, 157, 204, 332\n\nKhabarovsk Krai, 46\n\nKhrushchev, Nikita, 343\n\ndeal with Kennedy, 176\n\nhalts advance, 148\n\nletters to Kennedy, 156\u201357\n\nnuclear arms in Cuba, 59\u201360, 64, 148\n\nsubmarine mission, 82, 89, 117, 118, 134\n\nKing, E. J., 367\u201368n\n\nKirby Morgan masks, 236, 287\n\nKissinger, Henry, 188\n\nKitchens, Billy, 235\n\nKlimov, Yuri, 175\u201376\n\nKnox, William, 134\n\nKobyakov, Lieutenant, 174\u201375\n\nKobzar, Vladimir, 205\u20137\n\nKodiak Station, 123\n\nKola Bay, 256\u201357\n\nKolosov, Igor, 358n\n\nKomiles, 149\n\nKomsomolsk, 317\n\nKongsberg, 294\u201395\n\nKopeikin, Arkadyi, 67, 74\u201376, 96\u201399, 110, 174\u201375\n\nKorean War, 2, 5, 42\n\nKresta II, 261\u201362\n\nKurdin, Igor, 357n\n\nKW\u20137 devices, 200, 204\u20135, 208\n\nLacy, Gene, 202\u20133\n\nLane, James T., 266\u201367\n\nLangaliers, Bobbi, 320\n\nLangaliers, Don, 319\u201321, 324, 328, 330\u201331\n\nLaning, R. B., 15\n\nLa P\u00e9rouse Strait, 2\u20133\n\nLapon, USS, 284\n\nLasky, Marvin, 253\n\nLeJeune, Cheryl, 233\u201336, 289\n\nLeJeune, David, 233\u201346\n\nAndrea Doria dive, 237\u201338\n\nbackground of, 233\u201335\n\non Halibut, 233, 238\u201346\n\non Ortolan, 237\n\non Seawolf, 235, 249\u201350, 282, 283\u201384, 287\u201389, 292\n\ntraining, 234\u201335, 237\n\nLeninsky Komsomol, 193\n\nLibbert, John, 40\n\nLimiting depth, 162\n\nLindsay, Frank, 219\n\nLithium hydroxide, 245\n\nLock\u2013out chambers (escape trunks), 230, 297\u201399, 310\u201315, 340\n\nLoman, Jeff, 358n\n\nLong Range Acoustic Propagation Project (LRAPP), 253\u201355, 272\n\nLorenzen, Howard Otto, 40\u201344, 50\n\nLos Angeles\u2013class submarines, 339, 340\u201341, 344, 375n\n\nLos Angeles Times, 225\u201326, 227\n\nLott, Gus, 353n\n\nLouisville, USS, 338\n\nLovelace, Linda, 251\n\nLow Frequency Array (LOFAR), 18\u201319, 142\n\nLow\u2013frequency sonar, 16, 272\u201373, 375n Lusby, Al, 290\n\nLyubimov, Engineer, 85\u201386\n\nMcCloy, USS, 317\n\nMcCone, John, 62, 63\u201364, 106, 148\n\nMachiavelli, Niccol\u00f2, 229\n\nMack, Chester M. \"Whitey,\" 284\n\nMcKee, Kinnaird R., 295\n\nMcMahon, Knight, 45\n\nMcNamara, Robert, 62, 102, 118, 125\u201330, 136\u201339, 149, 150, 188\u201391\n\nMcNish, Jack, 230, 246, 248\n\nMcVain, Charlie, 285\u201387, 289\n\nMager\u00f8ya, 261\n\nMagnetic anomaly detection (MAD), 163\n\nMaintenance personnel (M-Branchers), 29\n\nMalinovsky, Rodion, 58\u201361\n\nManganese nodules, 218\u201319, 224\n\nMarconi, Guglielmo, 44\n\nMare Island Naval Shipyard, 338\n\nMarianas Trench, 185\n\nMaria Ulyanova, 59\n\nMarine Technology Society, 222\n\nMark 11 saturation diving suits, 239\u201340, 242\u201345\n\nMartell, Charles B., 253\n\nMaslennikov, Ivan, 155\n\nMason, Frank, 25, 31\u201333, 37\u201339, 49, 50\n\nMasset Station, 197\u201398\n\nMaurer, John H., 291\u201392\n\nMayans, viii\n\nMignon, Tony, 320, 323, 327\u201329\n\nMikoyan, Anastas, 120\n\nMilitary grade (\"mil spec\"), 232\u201333\n\nMiller, Charlie, 284\u201385\n\nMims, Norman, Jr., 266, 268, 275\u201381\n\nMisner, Robert, 42\u201344, 47\n\nMK\u201311 diving suits, 239\u201340, 242\u201345\n\nMK\u2013113 system, 271\u201372\n\nMobile Submarine Simulators (MOSS), 320\n\nMondrian, Pieter, 220\n\nMoore, C. Edward, 216\n\nMoorman, Dave, 323\n\nMueser, Roland, 16\u201319\n\nMultiangulation, 28, 30, 45, 49, 138\n\nMumma, Albert G., 351n\n\nMurray, USS, 150\n\nMystic, 229\u201330\n\nNaftali, Timothy, 157\n\nNapier, Russ, 159\n\nNational Photographic Interpretation Center, 86\u201387\n\nNational Press Club, 157\n\nNational Security Council (NSC), 118\n\nNational Security Operations Center (NSOC), 62\n\nNaumov, Sergei, 75, 85, 97, 108\u20139, 113, 166\n\nNautilus, USS, 8\u201310, 13\u201314, 15, 20\u201321, 62\u201363, 180, 193, 231, 351n\n\nNaval Electronics Laboratory, 185\n\nNaval Reactors Branch, 9\u201310\n\nNaval Research Laboratory (NRL), 40\u201345, 50, 145, 147, 191, 253\n\nNaval Reserve Officers Training Corps (NROTC), 11\n\nNaval Scientific and Technical Intelligence Center (NAVSTIC), 20\n\nNaval Security Group (NSG), 22, 146, 232\n\nNavy Expeditionary Medal, 9, 338\n\nNavy SEALs, 234, 296\u201399, 338\u201339, 340\n\nNea Makri Station, 212\u201313\n\nNelson, Elroy, 163, 168\n\nNet Control (NC), 29\u201330\n\nNew York Herald Tribune, 157\n\nNew York Times, 227\n\nNew York Yankees, 100\n\nNicholson, Jack, 264\n\nNiebuhr, Reinhold, 318\n\nNissho Maru, 316\u201317\n\nNitrogen narcosis, 229\n\nNitze, Paul, 216\n\nNixon, Richard, 80, 217, 225, 352n\n\nNoisemaker torpedoes, 163, 164\n\nNorfolk Naval Communications Area Master Station, 199\u2013200\n\nNorth American Aerospace Defense Command (NORAD), 191\n\nNorth Korea, and USS Pueblo, 200\u2013205\n\nNorth Pole, 15, 193\n\nNovaya Zemlya, 62\u201363, 139, 231\n\nNovember\u2013class subs, 188, 193\n\nNuclear Non\u2013Proliferation Treaty, 277\n\nNuclear\u2013powered submarines, 8\u201310, 13, 14, 20\u201321, 180\n\nOdell, Carl, 81\u201382, 141\u201345\n\nOffice of Collection and Signals Analysis, 40\u201344\n\nOffice of Undersea Warfare, 215\n\n\"Off\u2013line,\" 255\n\nOhio, USS, 340\n\nOkinawa Bulls Eye, 48\u201349\n\nOliver, Michael, 295\u201396, 302, 307\u201316\n\nOmnidirectional sound viewpoint, 142, 253\u201354\n\nOne Flew Over the Cuckoo's Nest (movie), 264\n\nOne Hell of a Gamble (Fursenko and Naftali), 157\n\nOnslow Beach, North Carolina, 176\n\nOP\u201320\u2013G (Office of Chief of Naval Operations), 45\n\nOperational Reactor Safeguard Exam (ORSE), 304\n\nOperation Anadyr, 58\u201361. See also Operation Kama\n\nOperation Eagle Claw, 319\n\nOperation Falling Leaves, 118\n\nOperation ICEX, 254\n\nOperation Kama, 57\u201377, 82\u201386, 88\u2013100, 107\u201315, 118\u201321, 130\u201334, 148, 149\u201355, 157\u201378. See also B-4; B-36; B-59; B-130\n\nOperation Paperclip, 46\n\nOperation Sand Dollar, 216\n\nOperations specialists (O-Branchers), 29\n\nOrel, Vice Admiral, 66\n\nOrestes, 200\n\nOrlov, Pavel, 150\u201355\n\nOrtolan, USS, 237\n\nOrwell, George, 332\n\nOSNAZ, 77, 94, 121, 133\n\nOxford, USS, 61\u201362, 63, 86, 101, 106\n\nPackard, David, 221\n\nPalm Beach International Airport, 115\n\nPancho Villa, 265\n\nPankov, Lieutenant avoiding detection, 108\u201315, 161, 170\u201371\n\noperation onboard, 82\u201384, 90\u201391\n\nParangosky, John, 217, 220\u201323\n\nParche, USS, 289\u201393\n\ncable\u2013tapping missions, 248\u201349, 288, 291\u201393, 319\u201322, 333\u201334, 338, 372\u201374n\n\nParshin, Viktor, 131\u201333\n\nPasha (cat), 72\u201373\n\nPassive sonar systems, 16\u201321, 272\u201373\n\nPatton, George, 188\n\nPCS\u20131380, 139\u201340\n\nPearlman, Stanley, 357n\n\nPearson, J. W., 209\n\nPelton, Ronald, 249, 332\u201334\n\nPennsylvania State University's Ordnance Research Laboratory, 16\u201317\n\nPeriscopes, 270\u201371, 299\u2013300\n\nPeri\u2013Viz (Periscope Visual), 258, 259\u201360\n\nPermit\u2013class submarines, 254\u201355, 266, 375n\n\nPetersen, Walter H., 30\u201331, 33, 37, 49, 334\u201336, 342\n\nPetropavlovsk Naval Base, 2, 228\n\nPhiladelphia Naval Shipyard, 10\n\nPhotographic intelligence (PHOTINT), on the USS Blenny, 4\u20136\n\nPiccard, Auguste, 181\n\nPittsburgh, USS, 338\n\nPlato, 40\n\nPlato Sea Mount, 213\n\nPliyev, Issa, 59\n\nPolaris missile program, 189, 258\n\nPollack, USS, 11\n\nPomilyev, Alexander, 66\u201367, 88\u201389, 167\u201370\n\nPopov, Vyacheslav, 344\n\nPortsmouth Naval Shipyard, 185\n\nPosition\u2013keeper (PK), 271\u201372\n\nPost, Don \"Doc,\" 328\n\nPotapov, Lieutenant, 121, 166, 174\u201375\n\nPreserver, USS, 180\u201384, 186\u201387\n\nPrice Water house, 190\n\nProject 627, 20\u201321\n\nProject 629A, 205. See also K-129\n\nProject 641, 66, 68, 71, 83. See also B-4; B-36; B-59; B-130\n\nProject 667B, 257\u201358\n\nProject Azorian, 217, 219\u201328, 364\u201365n\n\nProject Boresight, 47\u201351, 188\u201391\n\ncalibration and signal analysis, 50\u201351, 53\n\ninstalling and testing, 49\u201350, 81\u201382, 87\n\nMcNamara and, 127\u201328, 137, 138, 188\n\nNSA funding and naming, 44, 47\u201348\n\nrefinements and expansion, 118, 141\u201347, 149, 189\u201394\n\nsuccess of, 179\n\nProject Bulls Eye, ix, 44, 45, 47\u201349, 188\u201391, 197\u201398\n\nProject Colossus, 17\n\nProject Corona, 217\u201318\n\nProject Jezebel, 16\u201317\n\nProkov, Johnny, 157\n\nPronin, Vladimir, 93\u201395\n\nPropeller designs, 17, 351\u201352n\n\nProust, Marcel, 57\n\nPT 109, 137\u201338\n\nPueblo, USS, 200\u2013205\n\nPuget Sound Naval Shipyard, 338\n\nPump seals, 14\u201315\n\nQuantity vs. quality of submarines, 187\u201388\n\nQuittner, Arnold, 139\u201340\n\nRadar\u2013absorbing material (RAM), 270\n\nRadiation shielding, 14\u201315\n\nRadio\u2013Countermeasures Sound Recorder-\n\nReproducer (IC\/VRT\u20137), 42\u201343\n\nRadio direction finding (RDF), 44\u201345\n\nRadio Liberty, 120\n\nRadio Moscow, 148\n\nRadio operators (R-Branchers), 28\u201329, 50, 200, 232, 262\n\nRAF Chicksands, 47\n\nRandolph, USS, 107, 120, 150, 152\u201353, 159\n\nReagan, Ronald, 303, 317, 333\u201334\n\nRed November, 20\u201321\n\nReed, Joyce Louise, 23\u201324, 26\n\nReed, Lon, 32\n\nReed, Pamela Wallinger, 24, 26, 34\n\nReed, W. Craig\n\non the Drum, 295\u2013317\n\nat Eastman Kodak Company, 318\u201319\n\non the Haddo, 265\u201381\n\nReed, William J., 22\u201344\n\non the Birinci Inonu, 52\u201356\n\nBoresight system refinement and expansion, 118, 141\u201347, 149, 189\u201394, 194\u201397\n\nat Fort George G. Meade, 102\u20136, 130\n\nGreek incident, 194\u201397\n\nat Homestead Station, 78\u201382, 122, 123\n\nat Karam\u00fcrsel Station, 22\u201323, 26\u201339, 49\u201350, 342\n\nKennedy assassination and, 193\u201394\n\nat Masset Station, 197\u201398\n\nat NSA's Office of Collection and Signals Analysis, 40\u201344\n\npromotion to A22\n\nhead of field operations, 192\n\nretirement of, 264\u201365\n\nat Sanders Associates, 141\u201346, 189\u201390\n\nScratchy and, 23\u201326, 34\n\nat Skaggs Island Station, 146\u201347, 149\n\non success of Boresight program, 179\n\nat Vads\u00f8 Station, 192\u201394\n\nat White House for Boresight briefing, 134\u201340, 188, 356n\n\nRichard B. Russell, USS, 318, 334\u201336, 338\n\nRickover, Hyman G., 8\u201310, 14\n\nRigsbee, John, 208\u20139\n\nRindfleisch, Hans, 46\n\nRobert E. Lee, USS, 116\n\nRodocker, Don, 237\u201338\n\nRogers, Warren, 157\n\nRoom 40, 44\u201345\n\nRoosevelt, Franklin, 136\n\nRoosevelt, Theodore, 136, 368n\n\nRorke's Drift, 343\n\nRoss, Donald, 16\u201320, 351\u201352n\n\nRossokho, Anatoly, 70\u201371\n\nRounds, H. J., 44\u201345\n\nRozier, Charles, 159, 161\u201364\n\nRule, James, 319\u201322, 324, 328\u201331, 372n Rules of engagement, 70\u201371, 76, 93, 102, 128, 129, 150, 152\n\nRusk, Dean, 62, 102, 148\n\nRutherford, Mark, 231\u201333, 247\u201348, 249, 256, 261, 263, 283\u201386\n\nRybachiy Naval Base, 205\n\nRybalko, Galena, 61\n\nRybalko, Leonid, 57\u201361, 68, 70\u201371, 82\n\nRybalko, Natasha, 61\n\nSafety of submarines, 187\u201388\n\nSafford, Laurance F., 45\n\nSanders, Royden, Jr., 141\u201342\n\nSanders Associates, 141\u201346, 189\u201390, 209\n\nSan Diego Naval Training Center, 265\n\nSan Francisco Giants, 100\n\nSanta Fe Springs High School, 234\n\nSaparov, V. G., 75, 84\u201386, 96, 99, 110\u201314, 164, 166\u201367\n\nSargasso Sea, 61, 65, 93\u2013100, 108\n\nSatellite communications, 341\u201342\n\nSaturation diving, 229\u201330, 237\u201351\n\ntraining, 237, 242\u201345\n\nSaturday Night Live (TV program), 265\n\nSavitsky, Vitali, 70\u201377, 107, 125, 149\u201356, 158\u201359\n\nSaxon, Ross \"Zipperhead,\" 233, 237, 238\n\nSayda Bay, 65\u201368, 88, 115, 176\n\nSBD radios, 61, 93\u201394, 133\u201334, 206, 357n SC\u201335, 201\u20132\n\nScali, John, 147\u201348, 156\n\nSchade, Arnold F., 212\n\nSchlesinger, James R., 227\n\nScorpion, USS, 62\u201363, 212\u201314, 235, 279\u201380\n\nScratchy (bear), 23\u201326, 34\n\nSeadragon, USS, 180\n\nSeaLab, 229\n\nSEALs, 234, 296\u201399, 338\u201339, 340\n\nSea of Okhotsk, cable\u2013tapping missions, 228\u201331, 233\u201334, 238, 245\u201351, 287\u201389, 291\u201393, 319, 324\u201327, 332\u201336\n\nSea Robin, USS, 180\n\nSea Scope, 223\n\nSeawolf, USS, 13\u201315, 282\u201389, 291, 319\u201331\n\ncable\u2013tapping missions, 248\u201351, 287\u201389, 324\u201327\n\nsand\u2013stuck ordeal, 325\u201331\n\nSeawolf\u2013class submarines, 13\u201314, 193, 338\u201339, 375n\n\nSequoia, USS, 236\n\nSevastopol, 11\u201312, 52\n\nSevastopol, 259\u201360\n\nShackleton aircraft, 85\u201386\n\nShaddock, 193\n\nShakespeare, William, 282\n\nShchuka, 301\n\nSheets, Mack, 354n\n\nShips Inertial Navigation System (SINS), 328\n\nShip submersible nuclear (SSN), 8\u201310\n\nShkval, 86, 115\n\nShot lines, 158\n\nShumkov, Nikolai, 60, 65, 69\u201377, 107, 130\u201334, 164\u201368, 172, 175\n\nSignal intelligence (SIGINT), 5\n\nSignal\u2013to\u2013Noise Enhancement Program (SNEP), 353n\n\nSituation report (SITREP), 117\n\nSizov, F. Ya., 177\n\nSkaggs Island Station, 48\u201349, 87\u201388, 105\u20136, 146\u201347, 149, 189\n\nSkean, 87\n\nSkifter, Hector, 354n\n\nSkillas, Charles, 141\u201347, 357n\n\nSkipjack, USS, 11, 21\n\nSkory, 54\u201356\n\nSkylark, USS, 185\u201386\n\nSkywave propagation, 27\n\nSlattery, Francis Atwood, 212\u201313\n\nSlaughter, Gary, 157\u201359\n\nSmith, Cleveland, 244\u201345\n\nSmith, Dennis, 289\u201393, 372\u201374n\n\nSmith, Edward John, 287\n\nSmith, Gaines \"Whirly,\" 12\n\nSmith, Nihil, 180\u201387\n\nSolomatin, Boris A., 204\u20135\n\nSonar arrays, 16\u201317, 253\u201355, 272\u201373\n\nSonar systems, 16\u201321, 208, 254, 271\u201374, 375n Sonar transducers, 167\n\nSonobuoys, 111, 142\u201343\n\nSonographs, 34, 35\u201337\n\nSontag, Sherry, ix\n\nSOSUS (Sound Surveillance System), 16\u201317, 103, 117, 253\u201354\n\nSound suppression, 274, 370n\n\nSpecial Warning 32, 126\n\nSperry, USS, 301\u20132\n\n\"Spot report,\" 29\n\nSS\u2013N-14, 261\u201363\n\nStalin, Joseph, 58, 59, 122\n\nStanford Research Institute, 47\n\nStone, John, 44\n\nStonewall Jackson, USS, 377\n\nStraub, Herman, 17\u201318, 20\n\nStringfellow, Bill, 369n\n\nSturgeon\u2013class submarines, 254\u201355, 295, 334, 375n\n\nSubic Bay, 265\u201366\n\nSubmarine Development Group (DevGru) One, 289\u201390\n\nSubmarine Towed Array Surveillance\/Sonar System (STASS), 272\u201373\n\nSUBSAFE, 187\n\nSullivan, David \"Whompee Jaw,\" 287\u201389\n\nSurface Tethered Oceanographic Vehicle Experimental (STOVEs), 289\u201390\n\nSurface\u2013to\u2013air missile (SAM), 63\u201364\n\nSuzuki, Zenko, 317\n\nSweeney, Walter, 102\n\nSwimmer Delivery Vehicles (SDVs), 339\u201340\n\nSwordfish, USS, 207\u201312, 215, 255, 283\n\nSystem 2090, 290\u201391, 292\u201393\n\nTapping of cables. See Cable\u2013tapping missions Target motion analysis (TMA), 271\n\nTechnical Extracts of Traffic Analysis (TEXTA), 101\n\nTechnical specialists (T-Branchers), 29, 101, 123, 209, 211, 255, 262, 284\u201385, 307\n\nTerek, 86\n\nTest depth, 6\u20138, 56, 268, 286\n\nThermal imaging, 342\n\nThermal layers, 6, 56, 108, 121, 165\n\nThermoluminescent devices (TLDs), 240\n\nThetis Bay, USS, 175\n\nThresher, USS, 180\u201387\n\ndemise of, 185\u201386, 187\n\nrecovery of parts, 183, 184\n\nsearch for, 180\u201387\n\nThresher\u2013class submarines, 180, 254\u201355\n\nTiernan, Michael C., 321\u201322, 324\u201331\n\nTitanic, 287\n\nTomahawk Land Attack Missiles (TLAMs), 338, 340, 341\n\nTomlin, Edwin Ladeau, 269, 275\u201378, 281\n\nToshiba, 294\u201395\n\nTosi, Alessandro, 44\n\nTrejo, Paul, 1\u20139\n\nTrident missiles, 340\n\nTrieste, USS, 181\u201383, 185\u201387\n\nTrigger, USS, 184\n\nTsushima Strait, 200\u2013201\n\nTuell, Allen, 168\n\nTurban, Frank, 290\n\non Flying Fish, 255\u201363\n\non Seawolf, 284\u201387\n\non Swordfish, 209\u201311\n\nTurkey, Jupiter missiles in, 148, 156\u201357\n\nTurks Island Passage, 97\u201398, 107, 118\u201319\n\nTUSLOG Detachment 28, 22\u201323, 26\u201339, 49\u201350\n\nTwain, Mark, 101, 141\n\nTwenty\u2013ninth Ballistic Missile Division, 205\n\nTwin\u2013screw designs, 17, 351\u201352n\n\nTyomin, Abram, 95\n\nUnderdog, 210\u201311\n\nUnder\u2013hull runs, 257\u201358\n\nUnderwater Demolition Team (UDT) training, 234\u201335\n\nUnderwater photography, 217\u201322\n\nUnidirectional hearing, 142\n\nUnion College, 13\n\nV-5 program, 11\n\nV-12 program, 11\n\nVads\u00f8 Station, 192\u201394\n\nVarankin, Sergeant, 148\u201349\n\nVelucci, Chris, 237\u201338\n\nVictor\u2013class submarines, 294\u201395, 296\n\nVictor III, 294\u201395, 296, 301, 303\u201314\n\nVietnam War, 264\n\nViking 1, 218\n\nVindetto, Bob \"Ginny,\" 239\u201340, 244\u201345\n\nVinson, Tim, 266\u201370, 274\u201381\n\nVirginia, USS, 339\n\nVladivostok, 2\u20133, 210, 230, 276, 301, 303\n\nVladivostok Higher Naval School, 65\n\nVoice of America, 99\u2013100, 120\n\nVon Braun, Wernher, 46\n\nVon Hipper, Franz, 45\n\nWald, Bruce, 191\n\nWalker, John, 199\u2013200, 204\u20135, 214\n\nWalnut Line, 106\u20137, 116\u201317, 125, 130, 149\n\nWalski, Joe, 184\u201386, 187\n\nWalter, Helmuth, 351n\n\nWard, Alfred, 147, 150\n\nWatchstanders, 240\u201341\n\nWaterman, Steve, 296\u201397\n\nWater reserves, 165\n\nWesting house Oceanic Division, 290\u201391\n\nWhiff, 61\u201362\n\nWhite House, 135\u201336\n\nWide Aperture Receiving System (WARS), 47\n\nWieghorst, Olaf, 265\n\nWilkinson, Eugene, 351\u201352n\n\nWilliams, J. D., 255\u201363\n\nWill Rogers, USS, 252\n\nWoelk, Steve, 203\n\nWolfe, John, 218\u201320\n\nWorking depth, 162, 167\n\nWorld Series (1962), 100\n\nWorld War I, 44\u201345\n\nWorld War II, 9, 11, 40\u201341, 45\u201346\n\nKennedy and PT 109, 137\u201338\n\nWullenweber, 41, 43\u201347, 188\u201391\n\nWullenweber, Jurgen, 46, 355n\n\nWundt, Rolf, 46\n\nYankee\u2013class submarines, 257\u201358, 261\u201363\n\nYoshihara, Toshi, 345\n\nYurchenko, Vitaly S., 332\u201333\n\nZarnakov, Alexander, 206\n\nZellmer, Ernest \"Zeke,\" 217, 220\n\nZhukov, Yuri, 75, 91\u201392, 95\u2013100, 107, 113\u201314, 119\u201321, 160\u201361, 169\u201371\n\nZulu War, 343\n\n## ACKNOWLEDGMENTS\n\nI WOULD LIKE TO OFFER A special thanks to the following individuals, not already mentioned, for their contributions to the book:\n\nKenneth Greenawald and T. Michael Bircumshaw (editor of the American Submariner magazine), who as \"chief submarine technical advisers\" caught several errors before we went to print. Likely there are several more, and I'm certain to receive an e-mail from a crusty chief blasting me for \"getting it wrong.\"\n\nOthers who provided assistance, input, or stories that did not find their way into this book include: Adam Bridge, Al Burger, Boyce Williams, Brian Lawrence, Chuck Brickell, Chuck Young, Dave Arzani, Doug Bailey, Ernest Harkness, Frank Cawley, George Fraser, George Munsch, Guss Lott, Joel Harrison, Ken Sewell, Pavel Korshunov, Peter Lewis, Ralph Baer, Rich Petsch, Richard Moore, Skip Bauman, Stephen Lozier, and Thomas Hayter.\n\nI would like to thank my agent, James Hornfischer, and my editor, David Highfill of William Morrow\/HarperCollins, for their outstanding efforts in helping to mold this into a much better book.\n\n## About the Author\n\nW. CRAIG REED served as a U.S. Navy recon diver, submarine weapons technician, and special ops photographer deployed on nuclear fast-attack submarines. He earned commendations for completing top-secret operations during the Cold War and is an alumnus or member of several military, veteran, and technology associations. Born into a navy family on the island of Guam, Reed is now a partner in a technology marketing consulting firm and lives in Silicon Valley, California.\n\nwww.wcraigreed.com\n\nVisit www.AuthorTracker.com for exclusive information on your favorite HarperCollins author.\n\n## ALSO BY W. CRAIG REED\n\nDNA\n\nTarzan, My Father (with Johnny Weissmuller Jr. and William Reed)\n\n## Credits\n\nJacket design by Mumtaz Mustafa\n\nJacket photograph \u00a9 by Onne Van Der Wal\/Corbis\n\n## Copyright\n\nRED NOVEMBER. Copyright \u00a9 2010 by William Craig Reed. All rights reserved under International and Pan-American Copyright Conventions. By payment of the required fees, you have been granted the non-exclusive, non-transferable right to access and read the text of this e-book on-screen. No part of this text may be reproduced, transmitted, down-loaded, decompiled, reverse engineered, or stored in or introduced into any information storage and retrieval system, in any form or by any means, whether electronic or mechanical, now known or hereinafter invented, without the express written permission of HarperCollins e-books.\n\nFIRST EDITION\n\nLibrary of Congress Cataloging-in-Publication Data\n\nReed, W. Craig.  \nRed November: inside the secret U.S.-Soviet submarine war\/W. Craig Reed.\u20141st ed.  \np. cm.  \nIncludes bibliographical references.  \nISBN 978-0-06-180676-6  \n1. United States\u2014Military relations\u2014Soviet Union. 2. Soviet Union\u2014Military relations\u2014United States. 3. Cold War. 4. Submarine warfare\u2014United States\u2014History\u201420th century. 5. Submarine warfare\u2014Soviet Union\u2014History. 6. Submariners\u2014United States\u2014Biography. I. Title.  \nE183.8.S65R435 2010  \n359.9'30973\u2014dc22  \n2009053269\n\nEPub Edition \u00a9 April 2010 ISBN: 9780061992544\n\nVersion 08102012\n\n10 9 8 7 6 5 4 3 2 1\n\n## About the Publisher\n\nAustralia\n\nHarperCollins Publishers (Australia) Pty. Ltd.\n\n25 Ryde Road (PO Box 321)\n\nPymble, NSW 2073, Australia\n\nhttp:\/\/www.harpercollinsebooks.com.au\n\nCanada\n\nHarperCollins Publishers Ltd.\n\n55 Avenue Road, Suite 2900\n\nToronto, ON, M5R, 3L2, Canada\n\nhttp:\/\/www.harpercollinsebooks.ca\n\nNew Zealand\n\nHarperCollinsPublishers (New Zealand) Limited\n\nP.O. Box 1\n\nAuckland, New Zealand\n\nhttp:\/\/www.harpercollins.co.nz\n\nUnited Kingdom\n\nHarperCollins Publishers Ltd.\n\n77-85 Fulham Palace Road\n\nLondon, W6 8JB, UK\n\nhttp:\/\/www.harpercollinsebooks.co.uk\n\nUnited States\n\nHarperCollins Publishers Inc.\n\n10 East 53rd Street\n\nNew York, NY 10022\n\nhttp:\/\/www.harpercollinsebooks.com\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"\n# 100 WAYS TO  \nMOTIVATE YOURSELF,  \nTHIRD EDITION\n\nChange Your Life Forever\n\nBy Steve Chandler\n\nCopyright \u00a9 2012 by Steve Chandler\n\nAll rights reserved under the Pan-American and International Copyright Conventions. This book may not be reproduced, in whole or in part, in any form or by any means electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system now known or hereafter invented, without written permission from the publisher, The Career Press.\n\n**100 W AYS TO MOTIVATE YOURSELF, THIRD EDITION**\n\nCover design by Howard Grossman\/12E Design  \nPrinted in the U.S.A.\n\nTo order this title, please call toll-free 1-800-CAREER-1 (NJ and Canada: 201-848-0310) to order using VISA or MasterCard, or for further information on books from Career Press.\n\nThe Career Press, Inc.\n\n220 West Parkway, Unit 12\n\nPompton Plains, NJ 07444\n\n**www.careerpress.com**\n\n**Library of Congress Cataloging-in-Publication Data**\n\nChandler, Steve, 1944-\n\n100 ways to motivate yourself: change your life foever \/ by Steve Chandler.  \n\\-- 3rd ed.\n\np. cm.\n\nIncludes index.\n\nISBN 978-1-60163-244-9 -- ISBN 978-1-60163-554-9 (ebook)\n\n1. Motivation (Psychology) 2. Self-actualization (Psychology) I. Title. II.  \nTitle: One hundred ways to mo-tivate yourself.\n\nBF503.C48 2013\n\n158.1--dc23\n\n2012023412\nTo Kathryn Anne Chandler\n\n## Acknowledgments\n\nTo Lindsay Brady, for the ongoing perception of success; to Stephanie Chandler, for tirelessly working the cosmos; to Kathy, for more than I can say; to Jim Brannigan, for the representation; to Fred Knipe, for the music on New Year's Eve; to Ron Fry, for Career Press; to Nathaniel Branden, for the psychology; to Colin Wilson, for the philosophy; to Arnold Schwarzenegger, for a day to remember; to Rett Nichols, for the tension plan; to Graham Walsh, for the Tavern on the Green; to Terry Hill, for the century's first real mystery novel; to Cindy Chandler, for the salvation; to Ed and Jeanne, for the Wrigley Mansion; to John Shade, for the fire; to Scott Richardson, for the ideas; to Ann Coulter, for the wake-up calls; to Steven Forbes Hardison, for coaching and friendship beyond the earthly norm; and to Dr. Deepak Chopra, for unconcealing the creative intelligence that holds us all together.\n\nAnd to the memory of Art Hill: without whom, no life, no nothin'.\n\n## Contents\n\nIntroduction: Motivation Requires Fire\n\n100 Ways to Motivate Yourself\n\n1. Get on your deathbed\n\n2. Stay hungry\n\n3. Tell yourself a true lie\n\n4. Keep your eyes on the prize\n\n5. Learn to sweat in peace\n\n6. Simplify your life\n\n7. Look for the lost gold\n\n8. Push all your own buttons\n\n9. Build a track record\n\n10. Welcome the unexpected\n\n11. Find your master key\n\n12. Put your library on wheels\n\n13. Definitely plan your work\n\n14. Bounce your thoughts\n\n15. Light your lazy dynamite\n\n16. Choose the happy few\n\n17. Learn to play a role\n\n18. Don't just do something...sit there\n\n19. Use your brain chemicals\n\n20. Leave high school forever\n\n21. Learn to lose your cool\n\n22. Kill your television\n\n23. Break out of your soul cage\n\n24. Run your own plays\n\n25. Find your inner Einstein\n\n26. Run toward your fear\n\n27. Create the way you relate\n\n28. Try interactive listening\n\n29. Embrace your willpower\n\n30. Perform your little rituals\n\n31. Find a place to come from\n\n32. Be your own disciple\n\n33. Turn into a word processor\n\n34. Program your biocomputer\n\n35. Open your present\n\n36. Be a good detective\n\n37. Make a relation-shift\n\n38. Learn to come from behind\n\n39. Come to your own rescue\n\n40. Find your soul purpose\n\n41. Get up on the right side\n\n42. Let your whole brain play\n\n43. Get your stars out\n\n44. Just make everything up\n\n45. Put on your game face\n\n46. Discover active relaxation\n\n47. Make today a masterpiece\n\n48. Enjoy all your problems\n\n49. Remind your mind\n\n50. Get down and get small\n\n51. Advertise to yourself\n\n52. Think outside the box\n\n53. Keep thinking, keep thinking\n\n54. Put on a good debate\n\n55. Make trouble work for you\n\n56. Storm your own brain\n\n57. Keep changing your voice\n\n58. Embrace the new frontier\n\n59. Upgrade your old habits\n\n60. Paint your masterpiece today\n\n61. Swim laps underwater\n\n62. Bring on a good coach\n\n63. Try to sell your home\n\n64. Get your soul to talk\n\n65. Promise the moon\n\n66. Make somebody's day\n\n67. Play the circle game\n\n68. Get up a game\n\n69. Turn your mother down\n\n70. Face the sun\n\n71. Travel deep inside\n\n72. Go to war\n\n73. Use the 5 percent solution\n\n74. Do something badly\n\n75. Learn visioneering\n\n76. Lighten things up\n\n77. Serve and grow rich\n\n78. Make a list of your life\n\n79. Set a specific power goal\n\n80. Change yourself first\n\n81. Pin your life down\n\n82. Take no for a question\n\n83. Take the road to somewhere\n\n84. Go on a news fast\n\n85. Replace worry with action\n\n86. Run with the thinkers\n\n87. Put more enjoyment in\n\n88. Keep walking\n\n89. Read more mysteries\n\n90. Think your way up\n\n91. Exploit your weakness\n\n92. Try becoming the problem\n\n93. Enlarge your objective\n\n94. Give yourself flying lessons\n\n95. Hold your vision accountable\n\n96. Build your power base\n\n97. Connect truth to beauty\n\n98. Read yourself a story\n\n99. Laugh for no reason\n\n100. Walk with love and death\n\n101. Just roar!\n\n102. Experiment with happiness\n\n103. Catch life by the handle\n\n104. Leave yourself messages\n\n105. Try reinventing yourself\n\n106. Choose responding over reacting\n\n107. Apply the book you read\n\n108. Do what you can do today\n\n109. Create a different system!\n\n110. Enjoy your resistance training\n\nBibliography\n\nIndex\n\nAbout the Author\n\n## Motivation Requires Fire\n\nWhen Bob Dylan wrote in his book _Chronicles_ about how much he admired Joan Baez before he met her, he said, \"I'd be scared to meet her. I didn't want to meet her but I knew I would. I was going in the same direction even though I was in back of her at the moment. She had the fire, and I felt I had the same kind of fire.\"\n\nWe don't question what he means by \"the fire.\" We read on, knowing full well what he means. But sometimes I wonder, though. Do we really? Do we know it from experience? Do we feel the same fire? Do you have to be a poet or a singer? No. We all know what it is to have that same fire, no matter how briefly we have experienced it.\n\nMy own life's turning point came when I discovered I could light that fire all by myself. It took me more than 50 years to discover this. But I'm slow in these matters. You can get it today if you want. For the first 50 years of my life I thought the fire only happened when something inspired me. It was something that had to happen to me. And the reason I believed that was because that was my experience. You have to go by what you know, don't you?\n\nThe funniest thing about fire is that it takes fire to light it.\n\nI go to the fireplace to start a fire. I put crumpled-up newspaper under the kindling. Then I put the logs over the kindling wood. But how do I start this fire? I need a match. Or a lighter. You have to have fire to start a fire. Ironic? Paradoxical? Counterintuitive? Cruel hoax?\n\nA friend of mine once said, \"You're on _fire!_ \" He was referring to the fact that I'd just sent him a flurry of book ideas, written copy for things we were selling, recorded audio programs, and a number of other activities and actions.\n\nHow did I set myself on fire? With fire.\n\nOne action led to another and I wasn't afraid to rise early and work. I made myself exercise. I devoted myself to work instead of allowing distraction. Work (as it always does when you throw your entire self into it) soon became fun.\n\nPlaywright Noel Coward said, \"Work is more fun than fun.\"\n\nIt is when you do it. It is _not_ fun when you think about it. Especially when you think about it ahead of time.\n\nTo light a fire you need a fire. Rubbing two sticks together creates enough friction and heat to produce a spark and then a flame that you can put into the bigger fire.\n\nIt is the same process for yourself. Getting into action whether you feel like action or not is like rubbing two sticks together. Do you think the sticks felt like being rubbed together? Do you ever see them do it on their own?\n\nSince its first printing in 1996, this little book has enjoyed a success I never imagined. During its first 18 years of sales, we have seen the emergence of the Internet as the world's primary source of information. People have not only been buying this book on the Internet, but they've been posting their reviews. What's wonderful about Internet bookstores is that they feature reviews by regular people, not just professional journalists who need to be witty, cynical, and clever to survive.\n\nOne such reviewer of _100 Ways_ in its original edition was Bubba Spencer from Tennessee. He wrote: \"Not a real in-depth book with many complicated theories about how to improve your life. Mostly, just good tips to increase your motivation. A 'should read' if you want to improve any part of your life.\"\n\nBubba gave this book five stars, and I am more grateful to him than to any professional reviewer. He says I did what I set out to do:\n\n\"Making the simple complicated is commonplace; making the complicated simple, awesomely simple, that's creativity.\"\n\n\u2014Charles Mingus, legendary jazz musician\n\n## 1. Get on your deathbed\n\nA number of years ago when I was working with psychotherapist Devers Branden, she put me through her \"deathbed\" exercise.\n\nI was asked to clearly imagine myself lying on my own deathbed, and to fully realize the feelings connected with dying and saying good-bye. Then she asked me to mentally invite the people in my life who were important to me to visit my bedside, one at a time. As I visualized each friend and relative coming in to visit me, I had to speak to them out loud. I had to say to them what I wanted them to know as I was dying.\n\nAs I spoke to each person, I could feel my voice breaking. Somehow I couldn't help breaking down. My eyes were filled with tears. I experienced such a sense of loss. It was not my own life I was mourning; it was the love I was losing. To be more exact, it was a communication of love that had never been there.\n\nDuring this difficult exercise, I really got to see how much I'd left out of my life. How many wonderful feelings I had about my children, for example, that I'd never explicitly expressed. At the end of the exercise, I was an emotional mess. I had rarely cried that hard in my life. But when those emotions cleared, a wonderful thing happened. I was clear. I knew what was really important, and who really mattered to me. I understood for the first time what George Patton meant when he said, \"Death can be more exciting than life.\"\n\nFrom that day on I vowed not to leave anything to chance. I made up my mind never to leave anything unsaid. I wanted to live as if I might die any moment. The entire experience altered the way I've related to people ever since. And the great point of the exercise wasn't lost on me: We don't have to wait until we're actually near death to receive these benefits of being mortal. We can create the experience anytime we want.\n\nA few years later when my mother lay dying in a hospital in Tucson, I rushed to her side to hold her hand and repeat to her all the love and gratitude I felt for who she had been for me. When she finally died, my grieving was very intense, but very short. In a matter of days I felt that everything great about my mother had entered into me and would live there as a loving spirit forever.\n\nA year and a half before my father's death, I began to send him letters and poems about his contribution to my life. He lived his last months and died in the grip of chronic illness, so communicating and getting through to him in person wasn't always easy. But I always felt good that he had those letters and poems to read. Once he called me after I'd sent him a Father's Day poem, and he said, \"Hey, I guess I wasn't such a bad father after all.\"\n\nPoet William Blake warned us about keeping our thoughts locked up until we die. \"When thought is closed in caves,\" he wrote, \"then love will show its roots in deepest hell.\"\n\nPretending you aren't going to die is detrimental to your enjoyment of life. It is detrimental in the same way that it would be detrimental for a basketball player to pretend there was no end to the game he was playing. That player would reduce his intensity, adopt a lazy playing style, and, of course, end up not having any fun at all. Without an end, there is no game. Without being conscious of death, you can't be fully aware of the gift of life.\n\nYet many of us (including myself) keep pretending that our life's game will have no end. We keep planning to do great things some day when we feel like it. We assign our goals and dreams to that imaginary island in the sea that Denis Waitley calls \"Someday Isle\" in his book _Psychology of Winning_. We find ourselves saying, \"Someday I'll do this,\" and \"Someday I'll do that.\"\n\nConfronting our own death doesn't have to wait until we run out of life. In fact, being able to vividly imagine our last hours on our deathbed creates a paradoxical sensation: the feeling of being born all over again\u2014the first step to fearless self-motivation. \"People living deeply,\" wrote poet and diarist Ana\u00efs Nin, \"have no fear of death.\"\n\nAnd as Bob Dylan has sung, \"He who is not busy being born is busy dying.\"\n\n## 2. Stay hungry\n\nArnold Schwarzenegger was not famous yet in 1976 when he and I had lunch together at the Doubletree Inn in Tucson, Arizona. Not one person in the restaurant recognized him. He was in town publicizing the movie _Stay Hungry_ , a box-office disappointment he had just made with Jeff Bridges and Sally Field. I was a sports columnist for the _Tucson Citizen_ at the time, and my assignment was to spend a full day, one-on-one, with Arnold and write a feature story about him for our newspaper's Sunday magazine.\n\nI, too, had no idea who he was or who he was going to become. I agreed to spend the day with him because I had to\u2014it was an assignment. And although I took to it with an uninspired attitude, it was one I'd never forget.\n\nPerhaps the most memorable part of that day with Schwarzenegger occurred when we took an hour for lunch. I had my reporter's notebook out and was asking questions for the story while we ate. At one point I casually asked him, \"Now that you have retired from bodybuilding, what are you going to do next?\"\n\nWith a voice as calm as if he were telling me about some mundane travel plans, he said, \"I'm going to be the number-one box-office star in all of Hollywood.\"\n\nMind you, this was not the slim, aerobic Arnold we know today. This man was pumped up and huge. And so, for my own physical sense of well-being, I tried to appear as though I found his goal reasonable.\n\nI tried not to show my shock and amusement at his plan. After all, his first attempt at movies didn't promise much. And his Austrian accent and awkward, monstrous build didn't suggest instant acceptance by movie audiences. I finally managed to match his calm demeanor, and I asked him _just how_ he planned to become Hollywood's top star.\n\n\"It's the same process I used in bodybuilding,\" he explained. \"What you do is _create a vision_ of who you want to be, and then live into that picture as if it were already true.\"\n\nIt sounded ridiculously simple. Too simple to mean anything. But I wrote it down. And I never forgot it.\n\nI'll never forget the moment when some entertainment TV show was saying that box office receipts from his second _Terminator_ movie had made him the most popular box office draw in the world.\n\nOver the years I've used Arnold's idea of creating a vision as a motivational tool. I've also elaborated on it in my corporate training seminars. I invite people to notice that Arnold said that you _create_ a vision. He did not say that you wait until you _receive_ a vision. You create one. In other words, you make it up. A major part of living a life of self-motivation is having something to wake up for in the morning\u2014something that you are \"up to\" in life so that you _will_ stay hungry.\n\nThe vision can be created right now\u2014better now than later. You can always change it if you want, but don't live a moment longer without one. Watch what being hungry to live that vision does to your ability to motivate yourself.\n\n## 3. Tell yourself a true lie\n\nI remember when my then 12-year-old daughter Margery participated in a school poetry reading in which all her classmates had to write a \"lie poem\" about how great they were.\n\nThey were supposed to make up untruths about themselves that made them sound unbelievably wonderful. I realized as I listened to the poems that the children were doing an unintended version of what Arnold did to clarify the picture of his future. By \"lying\" to themselves they were creating a vision of who they wanted to be.\n\nIt's noteworthy, too, that public schools are so out of touch with the motivational sources of individual achievement and personal success that in order to invite children to express big visions for themselves they have to invite the children to \"lie.\"\n\nMost of us are unable to see the truth of who we could be. My daughter's school developed an unintended solution to that difficulty: If it's hard for you to imagine the potential in yourself, then you might want to begin by expressing it as a fantasy, as did the children who wrote the poems. Think up some stories about who you would like to be. Soon you will begin to create the necessary blueprint for stretching your accomplishments. Without a picture of your highest self, you can't live into that self. Fake it 'till you make it. The lie will become the truth.\n\n## 4. Keep your eyes on the prize\n\nMost of us never really focus. We constantly feel a kind of irritating psychic chaos because we keep trying to think of too many things at once. There's always too much up there on the screen.\n\nThere was an interesting motivational talk on this subject given by former Dallas Cowboys coach Jimmy Johnson to his football players during halftime at the 1993 Super Bowl:\n\nI told them that if I laid a two-by-four across the room, everybody there would walk across it and not fall, because our focus would be that we were going to walk that two-by-four. But if I put that same two-by-four 10 stories high between two buildings only a few would make it, because the focus would be on falling. Focus is everything. The team that is more focused today is the team that will win this game.\n\nJohnson told his team not to be distracted by the crowd, the media, or the possibility of losing, but to focus on each play of the game itself just as if it were a good practice session. The Cowboys won the game 52\u201317.\n\nThere's a point to that story that goes way beyond football. Most of us tend to lose our focus in life because we're perpetually worried about so many negative possibilities. Rather than focusing on the two-by-four, we worry about all the ramifications of falling. Rather than focusing on our goals, we are distracted by our worries and fears. But when you focus on what you want, it will come into your life. When you focus on being a happy and motivated person, that is who you will be.\n\n## 5. Learn to sweat in peace\n\nThe harder you are on yourself, the easier life is on you. Or, as they say in the Navy Seals, the more you sweat in peacetime, the less you bleed in war.\n\nMy childhood friend Rett Nichols was the first to show me this principle in action. When we were playing Little League baseball, we were always troubled by how fast the pitchers threw the ball. We were in an especially good league, and the overgrown opposing pitchers, whose birth certificates we were always demanding to see, fired the ball to us at alarming speeds during the games.\n\nWe began dreading going up to the plate to hit. It wasn't fun. Batting had become something we just tried to get through without embarrassing ourselves too much. Then Rett got an idea.\n\n\"What if the pitches we faced in games were slower than the ones we face every day in practice?\" Rett asked.\n\n\"That's just the problem,\" I said. \"We don't know anybody who can pitch that fast to us. That's why, in the games, it's so hard. The ball looks like an aspirin coming in at 200 miles an hour.\"\n\n\"I know we don't know anyone who can throw a baseball that fast,\" said Rett. \"But what if it wasn't a baseball?\"\n\n\"I don't know what you mean,\" I said.\n\nJust then Rett pulled from his pocket a little plastic golf ball with holes in it. The kind our dads used to hit in the backyard for golf practice.\n\n\"Get a bat,\" Rett said.\n\nI picked up a baseball bat and we walked out to the park near Rett's house. Rett went to the pitcher's mound but came in about 3 feet closer than usual. As I stood at the plate, he fired the little golf ball past me as I tried to swing at it.\n\n\"Ha ha!\" Rett shouted. \"That's faster than _anybody_ you'll face in little league! Let's get going!\"\n\nWe then took turns pitching to each other with this bizarre little ball humming in at incredible speeds. The little plastic ball was not only hilariously fast, but it curved and dropped more sharply than any little leaguer's pitch could do.\n\nBy the time Rett and I played our next league game, we were ready. The pitches looked like they were coming in slow motion. Big white balloons. I hit the first and only home run I ever hit after one of Rett's sessions. It was off a left-hander whose pitch seemed to hang in the air forever before I creamed it.\n\nThe lesson Rett taught me was one I've never forgotten. Whenever I'm afraid of something coming up, I will find a way to do something that's even harder or scarier. Once I do the harder thing, the real thing becomes fun.\n\nThe great boxer Muhammad Ali used this principle in choosing his sparring partners. He'd make sure that the sparring partners he worked with before a fight were _better_ than the boxer he was going up against in the real fight. They might not always be better all-around, but he found sparring partners who were each better in one certain way or another than his upcoming opponent. After facing them, he knew going into each fight that he had already fought those skills and won.\n\nYou can always stage a bigger battle than the one you have to face. Watch what it does to your motivation going into the real challenge.\n\n## 6. Simplify your life\n\nThe great Green Bay Packer's football coach Vince Lombardi was once asked why his world championship team, which had so many multi-talented players, ran such a simple set of plays. \"It's hard to be aggressive when you're confused,\" he said. One of the benefits of creatively planning your life is that it allows you to simplify. You can weed out, delegate, and eliminate all activities that don't contribute to your projected goals. Another effective way to simplify your life is to combine your tasks. Combining allows you to achieve two or more objectives at once.\n\nAs I plan my day, I might notice that I need to shop for my family after work. That's a task I can't avoid because we're running out of everything. I also note that one of my goals is to finish reading my daughter Stephanie's book reports. I realize, too, that I've made a decision to spend more time doing things with all my kids, as I've tended lately to just come home and crash at the end of a long day.\n\nAn aggressive orientation to the day\u2014making each day simpler and stronger than the day before\u2014allows you to look at all of these tasks and small goals and ask yourself, \"What can I combine?\" (Creativity is really little more than making unexpected combinations, in music, architecture\u2014anything, including your day.)\n\nAfter some thought, I realize that I can combine shopping with doing something with my children. (That looks obvious and easy, but I can't count the times I mindlessly go shopping, or do things on my own just to get them done, and then run out of time to play with the kids.)\n\nI also think a little further and remember that the grocery store where we shop has a little deli with tables in it. My kids love to make lists and go up and down the aisles themselves to fill the grocery cart, so I decide to read my daughter's book reports at the deli while they travel the aisles for food. They see where I'm sitting, and keep coming over to update me on what they are choosing. After an hour or so, three things have happened at once: 1) I've done something with the kids; 2) I've read through the book reports; and 3) the shopping has been completed.\n\nIn her book _Brain Building in Just 12 Weeks_ , Marilyn Vos Savant recommends something similar to simplify life. She advises that we make a list of absolutely every small task that has to be done, say, over the weekend, and then do them all at once, in one exciting, focused action. A manic blitz. In other words, fuse all small tasks together and make the doing of them one task so that the rest of the weekend is absolutely free to create as we wish.\n\nBob Koether, who was the president of Infincom, had the most simplified time management system I've ever seen in my life. His method was: do everything right on the spot\u2014don't put anything unnecessarily into your future. Do it now, so that the future is always wide open. Watching him in action was always an experience.\n\nI was sitting in his office and I mentioned the name of a person whose company I wanted to take my training to in the future.\n\n\"Will you make a note to get in touch with him and let him know I'll be calling?\" I asked.\n\n\"Make a _note?_ \" he asked in horror.\n\nThe next thing I knew, before I could say anything, Bob was wheeling in his chair, and dialing the person on the phone. Within two minutes, he'd scheduled a meeting between the person and me, and after he put down the phone he said, \"Okay, done! What's next?\"\n\nI told him I had prepared the report he wanted on training for his service teams and I handed it to him.\n\n\"You can read it later and get back to me,\" I offered.\n\n\"Hold on a second,\" he said, already deeply absorbed in reading the report's content. After 10 minutes or so, during which time he read aloud much of what interested him, the report had been digested, discussed, and filed.\n\nIt was a time management system like no other. What would you call it? Perhaps, Handle Everything Immediately. It kept Bob's life simple. He was an aggressive and successful CEO, and, as Vince Lombardi said, \"It's hard to be aggressive when you're confused.\"\n\nMost people are reluctant to see themselves as being creative because they associate creativity with complexity. But creativity is simplicity. Michelangelo said that he could actually _see_ his masterpiece, The David, in the huge, rough rock he discovered in a marble quarry. His only job, he said, was to carve away what wasn't necessary and he would have his statue. Achieving simplicity in our cluttered and hectic lives is also an ongoing process of carving away what's unnecessary.\n\nMy most dramatic experience of the power of simplicity occurred in 1984 when I was hired to help write the television and radio advertisements for Jim Kolbe, a candidate for United States Congress running in Arizona's Fifth District. In that campaign, I saw firsthand how focus, purpose, and simplicity can work together to create a great result.\n\nBased on prior political history, Kolbe had about a 3 percent chance of winning the election. His opponent was a popular incumbent congressman, during a time when incumbents were almost never defeated by challengers. In addition, Kolbe was a Republican in a largely Democratic district. And the final strike against him was that he had tried once before to defeat this same man, Jim McNulty, and had lost. The voters had already spoken on the issue.\n\nKolbe himself supplied the campaign with its sense of purpose. A tireless campaigner with unwavering principles, he emanated his sense of mission and we all drew energy from him. Political consultant Joe Shumate, one of the shrewdest people I've ever worked with, kept us all focused with consistent campaign strategy. It was the job of the advertising and media work to keep it strong and simple.\n\nAlthough our opponent ran nearly 15 different TV ads, each one about a different issue, we determined from the outset that we would stick to the same message throughout, from the first ad to the last. We basically ran the same ad over and over. We knew that although the district was largely Democratic, our polling showed that philosophically it was more conservative. Kolbe himself was conservative, so his views coincided with the voters' better than our opponent's did, although the voters weren't yet aware of it. Each of our ads focused on our simple theme: Who better represents _you_? This allowed us to gain rapidly in the polls as election night neared.\n\nThe night-long celebration of Jim Kolbe's upset victory brought a huge message home to me: The simpler you keep it, the stronger it gets. Kolbe won a close victory that night, but he served 11 terms and is now an Obama appointee. He has never complicated his message, and he has kept his politics strong and simple, even when it looked unpopular to do so.\n\nIt's hard to stay motivated when you're confused. When you simplify your life, it gathers focus. The more you can focus your life, the more motivated it gets.\n\n## 7. Look for the lost gold\n\nWhen I am happy, I see the happiness in others. When I am compassionate, I see the compassion in other people. When I am full of energy and hope, I see opportunities all around me. But when I am angry, I see other people as unnecessarily testy. When I am depressed, I notice that people's eyes look sad. When I am weary, I see the world as boring and unattractive. Who I am is what I see!\n\nIf I drive into Phoenix and complain, \"What a crowded, smog-ridden mess this place is!\" I am really expressing what a crowded, smog-ridden mess _I_ am at that moment. If I had been feeling motivated that day, and full of hope and happiness, I could just as easily have said, while driving into Phoenix, \"Wow, what a thriving, energetic metropolis this is!\" Again, I would have been describing my inner landscape, not Phoenix's.\n\nOur self-motivation suffers most from how we choose to see the circumstances in our lives. That's because we don't see things as they are, we see things as _we_ are.\n\nIn every circumstance, we can look for the gold or look for the filth. And what we look for, we find. The best starting point for self-motivation is in what we choose to look for in what we see around us. Do we see the opportunity everywhere? \"When I open my eyes in the morning,\" said Colin Wilson, \"I am not confronted by the world, but by a million possible worlds.\"\n\nIt is always our choice. Which world do we want to see today? Opportunity is life's gold. It's all you need to be happy. It's the fertile field in which you grow as a person. And opportunities are like those subatomic quantum particles that come into existence only when they are seen by an observer. Your opportunities will multiply when you choose to see them.\n\n## 8. Push all your own buttons\n\nHave you ever peeked into the cockpit of a large airliner as you boarded a plane? It's an impressive display of buttons, levers, dials, and switches under one big windshield. What if, as you were boarding, you overheard the pilot say to the co-pilot, \"Joe, remind me, what does this set of buttons do?\" If I heard that, it would make it a rough flight for me. But most of us pilot our own lives that way, without much knowledge of the instruments. We don't take the time to learn where our own buttons are or what they can do.\n\nFrom now on, make it a personal commitment to notice everything that pushes your buttons. Make a note of everything that inspires you. That's your control panel. Those buttons operate your whole system of personal motivation.\n\nMotivation doesn't have to be accidental. For example, you don't have to wait for hours until a certain song that picks up your spirits comes on the radio. You can control what songs you hear. If there are certain songs that always lift you up, make a mix of those songs and have it ready to play in your car. Go through all of your music and create a \"greatest motivational hits\" playlist for yourself.\n\nUse the movies, too. How many times do you leave a movie feeling inspired and ready to take on the world? Whenever that happens, put the name of the movie in a special notebook that you might label \"the right buttons.\" Six months to a year later, you can watch the movie and get the same inspired feeling. Most movies that inspire us are even better the second time around.\n\nYou have much more control over your environment than you realize. You can begin programming yourself consciously to be more and more focused and motivated. Get to know your control panel and learn how to push your own buttons. The more you know about how you operate, the easier it will be to motivate yourself.\n\n## 9. Build a track record\n\nIt's not what we do that makes us tired\u2014it's what we don't do. The tasks we _don't_ complete cause the most fatigue.\n\nI was giving a motivational seminar to a utility company, and during one of the breaks, a man who looked to be in his 60s came up to me. \"My problem,\" he said, \"is that I never seem to finish anything. I'm always starting things\u2014this project and that, but I never finish. I'm always off on to something else before anything is completed.\"\n\nHe then asked whether I could give him some affirmations that might alter his belief system. He correctly saw the problem as being one of belief. Because he did not believe he was a good finisher, he did not finish anything. So he wanted a magical word or phrase to repeat to himself that would brainwash him into being different.\n\n\"Do you think affirmations are what you need?\" I asked him. \"If you had to learn how to use a computer, could you do it by sitting on your bed and repeating the affirmation, 'I know how to use a computer. I am great at using computers. I am a wizard on a computer'?\"\n\nHe admitted that affirmations would probably have no effect on his ability to use a computer.\n\n\"The best way to change your belief system is to change the _truth_ about you,\" I said. \"We believe the truth faster than we believe false affirmations. To believe that you are a good finisher, you must begin by building a track record of finished tasks.\"\n\nHe followed my suggestions with great enthusiasm. He bought a notebook and at the top of the first page he wrote, \"Things I've Finished.\" Each day, he made a point of setting small goals and finishing them. Whereas in the past he would be sweeping his front walk and leave it unfinished when the phone rang, now he'd let the phone ring so he could finish the job and record it in his notebook. The more things he wrote down, the more confident he became that he was truly becoming a finisher. And he had a notebook to prove it.\n\nConsider how much more permanent his new belief was than if he had tried to do it with affirmations. He could have whispered to himself all night long, \"I am a great finisher,\" but the right side of his brain would have known better. It would have said to him, \"No, you're not.\"\n\nStop worrying about what you think of yourself and start building a track record that proves that you can motivate yourself to do whatever you want to do.\n\n## 10. Welcome the unexpected\n\nMost people do not see themselves as being creative, but we all are. Most people say, \"My sister's creative, she paints,\" or \"My father's creative, he sings and writes music.\" We miss the point that we are _all_ creative. One of the reasons we don't see ourselves that way is that we normally associate being _creative_ with being _original_. But in reality, creativity has nothing to do with originality\u2014it has everything to do with being _unexpected_. You don't have to be original to be creative. In fact, it sometimes helps to realize that no one is original. Even Mozart said that he never wrote an original melody in his life. His melodies were all combinations of old folk melodies.\n\nIf you believe you were created in the image of your Creator, then you must, therefore, be creative. Then, if you're willing to _see yourself_ as creative, you can begin to cultivate it in everything you do. You can start coming up with all kinds of unexpected solutions to the challenges that life throws at you.\n\n## 11. Find your master key\n\nI used to have the feeling that everyone else in life had at one time or another been issued an instruction book on how to make life work. And I, for some reason, wasn't there when they passed them out. I felt a little like the Spanish poet Cesar Vallejo, who wrote, \"Well, on the day I was born, God was sick.\"\n\nStill struggling in my mid-30s with a pessimistic outlook and no sense of purpose, I voiced my frustration to a friend of mine, Dr. Mike Killebrew, who recommended a book to me. Until that time, I didn't really believe that there could be a book that could tell you how to make your life work.\n\nThe name of the book was _The Master Key to Riches_ by Napoleon Hill. It sat on my shelf for quite a while. I didn't believe in motivational books or self-help. They were for weak and gullible fools. I was finally persuaded to read the book by the word _riches_ in the title. Riches would be a welcome addition to my life. Riches were probably what I needed to make me happy and wipe out my troubles.\n\nWhat the book actually did was a lot more than increase my earning capacity (although by practicing the principles in the book, my earnings doubled in less than a year). Napoleon Hill's advice ultimately sparked a fire in me that changed my entire life.\n\nI soon acquired an ability that I would later realize was self-motivation. After reading that book, I read all of Napoleon Hill's books. I also began buying motivational audiobooks to listen to in my car and as I went to sleep each night. Everything I had learned in school, in college, and from my family and friends was out the window. Without fully understanding it, I was engaging in the process of completely rebuilding my own thinking. I was, thought by thought, replacing the old cynical and passive orientation to life with a new optimistic and energetic outlook.\n\nSo, what is this master key to riches? \"The great master key to riches,\" wrote Hill, \"is nothing more or less than the self-discipline necessary to help you take full and complete possession of your own mind. Remember, it is profoundly significant that the only thing over which you have complete control is your own mental attitude.\" Taking complete possession of my own mind would be a lifelong adventure, but it was one that I was excited to start.\n\nMaybe Hill's book will not be your own master key, but I promise you that you'll find an instruction book on how to make your life work if you keep looking. It might be _The Power of Now_ by Eckhart Tolle, _The Last Word in Power_ by Tracy Goss, _Frankenstein's Castle_ by Colin Wilson, or _The Six Pillars of Self-Esteem_ by Nathaniel Branden. All those books would have worked the primary transformation for me, and they have all taken me higher up the motivational ladder. Your own key might even come from the spiritual literature of your choice. You'll find it when you're ready to seek. It's out there waiting for you.\n\n## 12. Put your library on wheels\n\nOne of the greatest opportunities for motivating yourself today lies in the way you use your drive time. There is no longer any excuse for time in the car to be downtime or frustrating or time that isn't motivating. With the huge variety of audiobooks now available, you can use your time on the road to educate and motivate yourself at the same time.\n\nWhen we use our time in the car to simply listen to music or to curse traffic, we are undermining our own frame of mind. Moreover, by listening to tabloid-type \"news\" programs for too long a period of time, we actually get a distorted view of life. News programs today have one goal: to shock or sadden the listener. The most vulgar and horrific stories around the state and nation are searched for and found.\n\nI experienced this firsthand when I worked for a daily newspaper. I saw how panicked the city desk got if there were no murders or rapes that day. I watched as they tore through the wire stories to see if a news item from another state could be gruesome enough to save the front page. If there's no drowning, they'll reluctantly go with a near-drowning.\n\nThere is nothing wrong with this. It's not immoral or unethical. It feeds the public's hunger for bad news. It's exactly what people want, so, in a way, it is a service. But it reaches its most damaging proportions when the average listener believes that all this bad news is a true and fair reflection of what's happening in the world. It's not. It is deliberately selected to spice up the broadcast and keep people listening. It is designed to horrify, because horrified people are a riveted audience and advertisers like it that way.\n\nIf we would be more selective with how we program our minds while we are driving, we could have some exciting breakthroughs in two important areas: knowledge and motivation. There are now hundreds of audiobook series on self-motivation, on how to use the Internet, on health, on goal-setting, and on all the useful subjects that we need to think about if we're going to grow. If we leave what we think about to chance or to a tabloid radio station, then we lose a large measure of control over our own minds.\n\nMany people today drive a great deal of the time. With motivational and educational audiobooks, it has been estimated that drivers can receive the equivalent of a full semester in college with three months' worth of driving. Most libraries have large sections devoted to audiobooks, and all the best and all the current audiobooks are now available on Internet bookseller's sites.\n\nAre all motivational programs effective? No. Some might not move you at all. That's why it's good to read the customer reviews before buying an audio program over the Internet. But there have been so many times when a great motivational audiobook played in my car has had a positive impact on my frame of mind and my ability to live and work with enthusiasm.\n\nOne moment stands out in my memory above all others, although there have been hundreds. I was driving in my car one day listening to Wayne Dyer's classic audio series, _Choosing Your Own Greatness_. At the end of a long, moving argument for not making our happiness dependent on some material object hanging out there in our future, Dyer said, \"There is no way to happiness. Happiness is the way.\"\n\nThat one thought eased itself into my mind at that moment and never left it. It is not an original thought, but Dyer's gentle presentation, filled with serene joy and so effortlessly spoken, changed me in a way that no ancient volume of wisdom ever could have. That's one of the powers of the audiobook form of learning: It simulates an extremely intimate one-on-one experience.\n\nWayne Dyer, Marianne Williamson, Caroline Myss, Barbara Sher, Tom Peters, Nathaniel Branden, Earl Nightingale, Alan Watts, and Anthony Robbins are just a few motivators whose audiobooks have changed my life. You'll find your own favorites. You don't have to find time to get to the library. Forget the library. You are driving one.\n\n## 13. Definitely plan your work\n\nSome of us may think we're too depressed, angry, or upset about certain problems right now to start on a new course of personal motivation. But Napoleon Hill insisted that that's the perfect time to learn one of life's most unusual rules: \"There is one unbeatable rule for the mastery of sorrows and disappointments, and that is the transmutation of those emotional frustrations through definitely planned work. It is a rule which has no equal.\"\n\nOnce we get the picture of who we want to be, \"definitely planned work\" is the next step on the path. Definitely planned work inspires the energy of purpose. Without it, we suffer from a weird kind of intention deficit disorder. We're short on intention. We don't know where we're going or what we're up to.\n\nWhen I was a training instructor at a time-management company many years ago, we taught businesspeople how to maximize time spent on the job. The primary idea was this: one hour of planning saves three hours of execution.\n\nHowever, most of us don't feel we have time for that hour of planning. We're too busy cleaning up yesterday's problems (that were caused by lack of planning). We don't yet see that planning would be the most productive hour we spend. Instead, we wander unconsciously into the workplace and react to crises. (Again, most of which result from a failure to plan.) A carefully planned meeting can take a third of the time that an unplanned free-for-all takes.\n\nMy friend Kirk Nelson managed a large sales staff at a major radio station. His success in life was moderate until he discovered the principle of definitely planned work. He spent two hours each weekend on his computer planning the week ahead. \"It's made all the difference in the world,\" he said. \"Not only do I get three times the work done, but I feel so in control. The week feels like my week. The work feels like my work. My life feels like my life.\"\n\nIt is impossible to work with a definite sense of purpose and be depressed at the same time. Carefully planned work will motivate you to do more and worry less.\n\n## 14. Bounce your thoughts\n\nIf you've ever coached or worked with kids who play basketball, you know that most of them have a tendency to dribble with only one hand\u2014the one attached to their dominant arm. When you notice a child doing this, you might call him aside and say, \"Billy, you're dribbling with just the one hand every time, and the defender can easily defend you when you do that. Your options are cut off. You need to dribble with your other hand, too, so that he never knows which way you're going to go.\"\n\nAt this point Billy might say, \"I can't.\" And you smile and say, \"What do you mean you can't?\"\n\nAnd Billy then shows you that when he dribbles with his subdominant (weaker) hand and arm, the ball is all over the place. So, in his mind, he can't.\n\n\"Billy,\" you say. \"It's not that you _can't_ , it's just that you _haven't.\"_\n\nThen you explain to Billy that his other hand can dribble just as well if he is willing to practice. It's just a matter of logging enough bounces. It's the simple formation of a habit. After enough practice dribbling with his other hand, Billy will learn you were right.\n\nThe same principle is true for reprogramming our own dominant habits of thinking. If our dominant thought habit is pessimistic, all we have to do is dribble with the other hand: Think optimistic thoughts more and more often until it feels natural.\n\nIf someone had asked me (before I started my journey to self-motivation that began with Napoleon Hill) why I didn't try to be more goal-oriented and optimistic, I would have said, \"I can't. It's just not me. I wouldn't know how.\" But it would have been more accurate for me to just say, \"I haven't.\"\n\nThinking is just like bouncing the basketball. On the one hand, I can think pessimistically and build that side of me up (it's just a matter of repeatedly bouncing those thoughts). On the other hand, I can think optimistically\u2014one thought at a time\u2014and build _that_ habit up. Self-motivation is all a matter of how much in control you want to be.\n\nThe overall pattern won't change after just a few positive bounces of the brain. If you're a pessimist, your biocomputer has been programmed heavily in that direction. But it doesn't take long before a new pattern can emerge. As a former pessimist myself, I can tell you it really happens, slowly but surely. You do change. One thought at a time. If you can bounce it one way, you can bounce it the other.\n\n## 15. Light your lazy dynamite\n\nHenry Ford used to point out to his colleagues that there wasn't any job that couldn't be handled if they were willing to break it down into little pieces. When you've broken a job down, remember to allow yourself some slow motion in beginning the first piece. Just take it slow and easy. It isn't important how fast you are doing it. What's important is _that_ you are doing it.\n\nMost of our hardest jobs never seem to get done. The mere thought of doing the whole job, at a high energy level, is frequently too off-putting to allow motivation to occur. A good way to ease yourself into that motivation is to act as if you were the laziest person on the planet. (It wasn't much of an act for me!) By accepting that you're going to do your task in a slow and lazy way, there is no anxiety or dread about getting it started. In fact, you can even have fun by entering into it as if you were in a slow-motion comedy, flowing into the work like a person made of water.\n\nBut the paradox is that the slower you start something, the faster you will be finished. When you first think about doing something hard or overwhelming, you are most aware of how you don't want to do it at all. In other words, the mental picture you have of the activity, of doing it fast and furiously, is not a happy picture. So you think of ways to avoid doing the job altogether. The thought of starting slowly is an easy thought. And doing it slowly allows you to actually start doing it. Therefore it gets finished.\n\nAnother thing that happens when you flow into a project slowly is that speed will often overtake you without your forcing it. The natural rhythm inside you will get you in sync with what you are doing. You'll be surprised how soon your conscious mind stops forcing the action and your subconscious mind supplies you with easy energy.\n\nTake your time. Start out lazy. Soon, your tasks will be keeping the slow but persistent rhythm of that hypnotic song on Paul McCartney's _Red Rose Speedway_ album, \"Oh Lazy Dynamite.\" The dynamite is living inside you. You don't have to be frenzied about setting it off. It lights just as well with a slowly struck match.\n\n## 16. Choose the happy few\n\nPolitely walk away from friends who don't support the changes in your life. There will be friends who don't. They will be jealous and afraid every time you make a change. They will see your new motivation as a condemnation of their own lack of it. In subtle ways, they will bring you back down to who you used to be. Beware of friends and family who do this. They know not what they do. The people you spend time with will change your life in one way or another. If you associate with cynics, they'll pull you down with them. If you associate with people who support you in being happy and successful, you will have a head start on being happy and successful.\n\nThroughout the day we have many choices regarding who we are going to be with and talk to. Don't just gravitate to the coffee machine and participate in the negative gossip because it's the only game in town. It will drain your energy and stifle your own optimism. We all know who lifts us up, and we all know who brings us down. It's okay to start being more careful about to whom we give our time. In his inspiring book _Spontaneous Healing_ , Andrew Weil recommends: \"Make a list of friends and acquaintances in whose company you feel more alive, happier, more optimistic. Pick one whom you will spend some time with this week.\"\n\nWhen you're in a conversation with a cynic, possibilities seem to have a way of disappearing. A mildly depressing sense of fatalism seems to take over the conversation. No new ideas and no innovative humor. \"Cynics,\" observed President Calvin Coolidge, \"do not create.\" On the other hand, enthusiasm for life is contagious. And being in a conversation with an optimist always opens us up to see more and more of life's possibilities.\n\nKierkegaard once said, \"If I were to wish for anything, I should not wish for wealth and power, but for the passionate sense of the potential, for the eye which, ever young and ardent, sees the possible. Pleasure disappoints, possibility never.\"\n\n## 17. Learn to play a role\n\nYour future is not determined by your personality. In fact, your personality is not even determined by your personality. There is no genetic code in you that determines who you will be. _You are the thinker_ who determines who you will be. How you _act_ is who you become.\n\nAnother way of seeing might be contained in these related thoughts from _Star Trek_ 's Leonard Nimoy: \"Spock had a big, big effect on me. I am so much more Spock-like today than when I first played the part in 1965 that you wouldn't recognize me. I'm not talking about appearance, but _thought_ processes. Doing that character, I learned so much about rational logical thought that it reshaped my life.\"\n\nYou'll gather energy and inspiration by being the character _you_ want to play.\n\nI took an acting class a few years ago because I thought it would help me deal with my overwhelming stage fright. But I learned something much more valuable than how to relax in front of a crowd. I learned that my emotions were tools for me to use, not demonic forces. I learned that my emotions were mine to work with and change at will.\n\nAlthough I had _read_ countless times that our own deliberate thoughts control our emotions, and that the feelings we have are all caused by what we think, I never trusted that concept as real, because it didn't always feel real. To me, it felt more like emotion was an all-powerful thing that could overcome my thinking and ruin a good day (or a good relationship).\n\nIt took a great acting teacher, Judy Rollings, and my own long struggles with performing difficult scenes to show me that my emotions really could be under the complete control of my mind. I found out that I could motivate myself by thinking and acting like a motivated person, just as I could depress myself by thinking and acting like a depressed person. With practice, the fine line between acting and being disappeared.\n\nWe love great actors because it seems like they _are_ the characters they play. Poor actors are those who can't be their part and therefore don't convince us of their character's reality. We boo at those people. We call it bad acting.\n\nYet, we don't realize that we miss the same opportunities in life when we can't be the person _we_ want to be. It doesn't take authentic circumstances to be who you want to be. It just takes rehearsal.\n\n## 18. Don't just do something...sit there\n\nFor a long time, all by yourself, sit quietly, absolutely alone. Completely relax. Don't allow the television or music to be on. Just be with yourself. Watch for what happens. Feel your sense of belonging to the silence. Observe insights starting to appear. Observe your relationship with yourself starting to get better and softer and more comfortable.\n\nSitting quietly allows your true dream life to give you hints and flashes of motivation. In this information-rich, interactive, civilized life today, you are either living your dream or living someone else's. And unless you give your own dream the time and space it needs to formulate itself, you'll spend the better part of your life simply helping others make _their_ dreams come true.\n\n\"All of man's troubles,\" said French philosopher Blaise Pascal, \"stem from his inability to sit alone, quietly, in a room for any length of time.\" Notice that he did not say _some_ of man's troubles, but _all_.\n\nSometimes, in my seminars on motivation, a person will ask me, \"Why is it that I get my best ideas when I'm in the shower?\"\n\nI usually ask the person, \"When else during your day are you alone with yourself, without any distractions?\"\n\nIf the person is honest, the answer is _never_.\n\nGreat ideas come to us in the shower when it's the only time in the day when we're completely alone. No television, no movies, no traffic, no radio, no family, no pets\u2014nothing to distract our mind from conversing with itself.\n\n\"Thinking,\" said Plato, \"is the soul talking to itself.\"\n\nPeople worry they will die of boredom or fear if they are alone for any length of time. Other people have become so distraction-addicted that they would consider sitting alone by themselves like being in a sensory-deprivation tank. The truth is that the only real motivation we ever experience is _self-_ motivation that comes from within. And being alone with ourselves will always give us motivating ideas if we stay with the process long enough.\n\nThe best way to truly understand the world is to remove yourself from it. Psychic entropy\u2014the seesaw mood swing between boredom and anxiety\u2014occurs when you allow yourself to become confused by massive input. By being perpetually busy, glued to your cell phone, out in the world all day with no time to reflect, you will guarantee yourself an eventual overwhelming sense of confusion. The cure is simple and painless. The process is uncomplicated. \"You do not need to leave your room,\" said Franz Kafka. \"Remain sitting at your table and listen. Do not even listen. Simply wait. Do not even wait. Be quiet, still, and solitary. The world will freely offer itself to you to be unmasked. It has no choice, it will roll in ecstasy at your feet.\"\n\n## 19. Use your brain chemicals\n\nThere are drugs that you can use to motivate yourself with, those energizing chemicals in your system that get activated when you hug someone, laugh, sing, dance, or run. When you're having fun, your body chemistry changes and you get new biochemical surges of motivation and energy.\n\nThere isn't anything you do that can't be transformed into something interesting and uplifting. Victor Frankl has written startling accounts of his life in the Nazi concentration camps, and how some prisoners created new universes unto themselves inside their own minds. It might sound absurd, but truly imaginative people can access their inner chemical creativity in the loneliness of a prison cell.\n\nDon't keep trying to go outside yourself searching for something that's fun. It's not out there anywhere. It's inside. The opportunity for fun is in your own energy system\u2014your synergy of heart and mind. That's where you'll find it. Pro football Hall of Famer Fran Tarkenton recommends looking at any task you do as fun. \"If it's not fun,\" he says, \"you're not doing it right.\"\n\nWilliam Burroughs, a former drug addict and author of _Naked Lunch,_ discovered something that was very interesting and bitterly amusing to him after finally recovering from his addictions. \"There isn't any feeling you can get on drugs,\" he said \"that you can't get without drugs.\"\n\nMake a commitment to yourself to find the _natural_ highs you need to stay motivated. Start by finding out what it does to your mood and energy to laugh, to sing, to dance, to walk, to run, to hug someone, or to get something done. Then support your experiments by telling yourself that you're not interested in doing _anything_ that isn't fun. If you can't immediately see the fun in something, find a way to create it. Once you have made a task fun, you have solved the problem of self-motivation.\n\n## 20. Leave high school forever\n\nMany of us feel as though we've been left stranded in high school forever, as if something happened there that we've never shaken off. Before high school, in our earlier and more carefree childhoods, we were creative dreamers filled with a boundless sense of energy and wonder. But in high school something got turned around. For the first time in our lives, we began fearing what other people were thinking of us. All of a sudden, our mission in life became _not to be embarrassed_. We were afraid to look bad, and so we made it a point not to take risks.\n\nI'll never forget something that happened to my friend Richard in high school. Richard and I were walking home from school one day, and all of a sudden he stopped in his tracks, his face frozen with horror. I looked at him and asked what was wrong. I thought he was about to suffer some kind of seizure. He then pointed down at his pants and wordlessly showed me where his belt had missed a loop!\n\n\"I spent the whole _day_ like this!\" he finally said. It was impossible for him to measure what everybody thought of him as they passed him in the halls, perhaps seeing the belt had missed a loop. The damage to his reputation was probably beyond repair.\n\nWhen I give my seminars on motivation, I love the periods when I take questions from the audience. But many times I can see the painfully adolescent looks of self-consciousness on people's faces when they ponder the risk of asking a question in front of the group. This habit of worrying more about what others think of our thoughts than we do about our own thinking usually begins in high school, but it can last a lifetime. It is time to be aware of what we're doing and, once again, leave high school. It's time to reach back to those pre\u2013high school days of innocent creativity and social fearlessness, and draw on that former self.\n\nBy the way, I came up with a way to deal with the moments of silence that fill a seminar room when I ask for questions. I go to the board and make five circles. Then I tell the audience that I used to say in my classes, \"If there are no questions at this point, we'll take a break.\" People always want to take a break, so there wasn't much incentive for asking questions. But questions are the most fun part of a seminar for me, so I came up with this game: _After five questions\u2014we take a break_. Now I find people in the audience urging people around them to join in asking questions so we can take our break sooner. Although it's an amusing artificial way to jump-start the dialogue I'm looking for, what it really does is take the pressure off. It takes the participants out of high school.\n\nMost people don't realize how easily they can create the social fearlessness they want to have. Instead, they live as though they were still teenagers, reacting to the imagined judgments of other people. They end up designing their lives based on what other people might be thinking about them. A life designed by a teenager! Would you want one? You can leave that mind-set behind. You can motivate yourself by yourself, without depending on the opinions of others. All it takes is a simple question. As Emerson asked, \"Why should the way I feel depend on the thoughts in someone else's head?\"\n\n## 21. Learn to lose your cool\n\nYou can create a self that doesn't care that much about what people think. You can motivate yourself by leaving the painful self-consciousness of high school behind. Our tendency is to go so far in the timid, non-assertive direction, it might be a profitable over-correction to adopt these internal commands: _Look bad. Take a risk. Lose face. Be yourself. Share yourself with someone. Open up. Be vulnerable. Be human. Leave your comfort zone. Get honest. Experience the fear. Do it anyway_.\n\nThe first time that I ever spoke to author and psychotherapist Devers Branden it was over the telephone, and she agreed to work with me on building my own self-confidence and personal growth. It wasn't long into the phone conversation before she asked me about my voice.\n\n\"I am very interested in your voice,\" she said, with a tone of curiosity.\n\nHoping she might be ready to give me a compliment, I asked her to explain.\n\n\"Well,\" she said. \"It's so lifeless. A real monotone. I wonder why that is.\"\n\nEmbarrassed, I had no explanation. This conversation took place long before became a professional speaker, and it was also long before I ever took any acting lessons. It was long before I learned to sing in my car, too. I was completely unaware and very surprised that it seemed to her that I was coming across with a voice like someone out of _Night of the Living Dead_.\n\nThe truth was that during that period in my life, I was living scared. Things weren't going well for me financially, I had serious health problems in my family, and I had that mildly suicidal feeling that accompanies an increasing sense of powerlessness over one's problems. (I now think one way a lot of men hide their fears is by assuming a macho kind of dull indifference. I know that's what I had done. That a psychotherapist could hear it immediately in my voice was unnerving, though.)\n\nTrying to understand why I covered fear with indifference, I remembered that back in my high school the \"cool\" guys were always the least enthusiastic guys. They spoke in monotones, emulating James Dean and Marlon Brando. Brando was the coolest of all. He was so indifferent and unenthusiastic, you couldn't even understand him when he spoke.\n\nOne of the first homework assignments Devers Branden gave me was to watch _Gone with the Wind_ and study how fearlessly Clark Gable revealed his female side. This sounded weird to me. Gable a female? I knew Gable was always considered a true \"man's man\" in all those old movies, so I couldn't understand what Devers was talking about or how it would help me.\n\nBut when I watched the movie, it became strangely clear. Clark Gable allowed himself such a huge emotional range of expression, that I could actually identify scenes in which he was revealing a distinctly female side to his character's personality. Did it make him less manly? No. It made him more real, and more compelling.\n\nFrom that time on, I lost my desire to hide myself behind an indifferent monotonous person. I committed myself to get on the road to creating a self that included a wider range of expression, without a nervous preoccupation with coming off like a man's man. I also started noticing how much we seem to love vulnerability in others, but don't trust it in ourselves. But we can learn to trust it!\n\nJust a little at first. Then we can build that vulnerability until we're not afraid to open up into an ever-widening spectrum of self-revelation. By losing face, we connect to the real excitement of life. And what if I don't always come off as an indifferent man's man? Frankly, my dear, I don't give a damn.\n\n## 22. Kill your television\n\nMy brother used to own a T-shirt store and one of the most popular shirts for sale said, \"Kill Your Television.\" I bought that T-shirt with the picture of a TV being blown up. You can actually change your life by turning off your television. Maybe just one evening a week, to start with. What would happen if you stopped trying to find life in other people's shows and let your own life become the show you got hooked on?\n\nCutting down on television is sometimes terrifying to the electronically addicted, but don't be afraid. You can detox slowly. If you're watching too much television and you know it, you might find it useful to ask this one question: \"Which side of the glass do I want to live on?\" When you are watching television, you are watching other people do what they love doing for a living. Those people are on the smart side of the glass, because they are having fun, and you are passively watching them have fun. They are getting money, and you are not.\n\nHere's a good test for you to determine if television motivates you more than books do: Try to remember what you watched on television a month ago. Think hard. What effect are those shows having on the inspired side of your brain? Now think about the book that you read a month ago. Or even the e-zine you read last week. Which made a more valuable and lasting impression? Which form of entertainment better leads you in the direction of self-motivation?\n\nGroucho Marx once said he found television very educational. \"Every time someone turns it on,\" he said, \"I go in the other room to read a book.\"\n\n## 23. Break out of your soul cage\n\nOur society encourages us to seek comfort. Most products and services advertised day and night are designed to make us more comfortable and less challenged.\n\nBut, only challenge causes growth. Only challenge will test our skills and make us better. Only challenge and the self-motivation to engage the challenge will transform us. Every challenge we face is an opportunity to create a more skillful self.\n\nIt is up to you to constantly look for challenges that motivate you. It's up to you to notice when you're buried alive in a comfort zone. It's up to you to notice when you are spending your life, in the image of the poet William Olsen, like a flower \"living under the wind.\"\n\nUse your comfort zones to rest in, not to live in. Use them consciously to relax and restore your energy as you mentally prepare for your next challenge. But if you use comfort zones to live in forever, they become what rock singer Sting calls your \"soul cages.\" Break free. Fly away.\n\n## 24. Run your own plays\n\nDesign your own life's game plan. Let the game respond to you rather than the other way around. Be like Bill Walsh, the former head coach of the San Francisco 49ers. Everybody thought he was eccentric because of how extensively he planned his plays in advance of each game. Most coaches would wait to see how the game unfolded, then respond with plays that reacted to the other team. Not Bill Walsh. Walsh would pace the sidelines with a big sheet of plays that his team was going to run, no matter what. He wanted the other team to respond to him. Walsh won Super Bowls with his unorthodox proactive approach. But all he did was to act on the crucial difference between creating and reacting.\n\nMany of us can spend whole days reacting without being aware of it. We wake up reacting to news on the clock radio. Then we react to feelings in our body. Then we start reacting to our spouse or our children. Soon we get in the car and react to traffic, honking the horn and using sign language. Then, at work, we see an e-mail on our computer screen and react to that. We react to stupid customers and insensitive bosses who are intruding on our day. During a break, we react to a waitress at lunch. This habit of reacting can go on all day, every day. We become goalies in the hockey game of life, with pucks flying at us incessantly. It's time to play another position. It's time to fly across the ice with the puck on our own stick ready to shoot at another goal.\n\nYou can create your own plans in advance so that your life will respond to you. If you can hold the thought that at all times your life is either a creation or a reaction, you can continually remind yourself to be creating and planning. _Creation_ and _reaction_ have the same letters in them, exactly; they are anagrams.\n\nRobert Fritz, who has written some of the most profound and useful books on the differences between creating and reacting, says, \"When your life itself becomes the subject matter of the creative process, a very different experience of life opens to you\u2014one in which you are involved with life at its very essence.\"\n\nPlan your day the way Bill Walsh planned his football games. See the tasks ahead as plays you're going to run. You'll feel involved in your life at its very essence, because you'll be encouraging the world to respond to you. If you don't choose to do that, the life you get won't be an accident. As an old Jewish folk saying puts it, \"A person who does not make a choice makes a choice.\"\n\n## 25. Find your inner Einstein\n\nThe next time you see a picture of Albert Einstein, realize that that's actually you. See Albert Einstein and say, \"There I am.\" Every human has the capacity for some form of genius. You don't have to be good at math or physics to experience genius level in your thinking. To experience Einstein's creative level of thinking, all you have to do is habitually use your imagination.\n\nThis is a difficult recommendation for adults to follow, though, because adults have become accustomed to using their imaginations for only one thing: worrying. Adults visualize worst-case scenarios all day long. All their energy for visualization is channeled into colorful pictures of what they dread.\n\nWhat they don't comprehend is that worry is a misuse of the imagination. The human imagination was designed for better things. People who use their imaginations to create with often achieve things that worriers never dream of achieving, even if the worriers possess much higher IQs. People who habitually access their imaginations are often hailed by their colleagues as \"geniuses\"\u2014as if \"genius\" were a genetic characteristic. They would be better understood as people who are practiced at accessing their genius.\n\nRecognition of the power of this genius in all of us prompted Napoleon to say, \"Imagination rules the world.\" As a child, you instinctively used your imagination as it was intended. You daydreamed and made stuff up. If you go back into that state of self-confidence and dream again, you'll be pleasantly surprised at how many innovative and immediate solutions to your problems you come up with.\n\nEinstein used to say, \"Imagination is more important than knowledge.\" When I first heard he'd said that, I didn't know what he meant. I always thought additional knowledge was the answer to every difficult problem. I thought if I could just learn a few more important things, then I'd be okay. What I didn't realize was that the very thing I needed to learn was not knowledge, but skill. What I needed to learn was the skill of proactively using my imagination.\n\nOnce I'd learned that skill, the first task was to begin imagining the vision of who I wanted to be. Dreaming, in its proactive sense, is strong work. It's the design stage of creating the future. It takes confidence and it takes courage. But the greatest thing about active dreaming is not in the eventual reaching of the goal\u2014the greatest thing is what it does to the dreamer.\n\nForget the literal attainment of your dream for now. Focus on just going for it. By simply going for the dream, you make yourself come true.\n\n## 26. Run toward your fear\n\nThe world's best-kept secret is that on the other side of your fear, there is something safe and beneficial waiting for you. If you pass through even a thin curtain of fear, you will increase the confidence you have in your ability to create your life.\n\nGeneral George Patton said, \"Fear kills more people than death.\" Death kills us but once, and we usually don't even know it. But fear kills us over and over again, subtly at times and brutally at others. But if we keep trying to avoid our fears, they will chase us down like persistent dogs. The worst thing we can do is close our eyes and pretend they don't exist.\n\n\"Fear and pain,\" says psychologist Nathaniel Branden, \"should be treated as signals not to close our eyes but to open them wider.\" By closing our eyes we end up in the darkest of comfort zones\u2014buried alive.\n\nJanis Joplin's biography, which chronicled her death from alcohol and drug abuse, was aptly titled _Buried Alive_. To Janis, as to so many similarly troubled people, alcohol provided an artificial and tragically temporary antidote to fear. It is no accident that in the old frontier days the nickname for whiskey was \"false courage.\"\n\nThere was a time in my life when my greatest fear of all was public speaking. It didn't even help to know that fear of speaking in front of others is people's number one fear, even greater than the fear of death. This fact once caused comedian Jerry Seinfeld to point out that most people would rather be in the coffin than delivering the eulogy.\n\nFor me, it ran even deeper than that. As a child, I could not give oral book reports. I'd plead with my teachers to let me off the hook. I would offer to do two, even three written book reports if I didn't have to do the oral one. But, as my life went on, I wanted to be a public speaker more than anything. My dream was to teach people everywhere to learn the ideas that lead to self-motivation, the ideas that I had learned. But how could I ever do this if stage fright left me frozen with fear?\n\nThen one day, as I was driving in Phoenix, flipping through the radio stations, I accidentally happened upon a religious station where a histrionic preacher was yelling, \"Run toward your fear! Run right at it!\" I hastened to change the station, but it was too late. Deep down I knew that I had just heard something I needed to hear. No matter what station I turned to, all I could hear was those words: \"Run toward your fear!\"\n\nThe next day I still couldn't get it out of my mind, so I called a friend of mine who was an actress. I asked her to help me get into an acting class she had once told me about. I told her I thought I was ready to overcome my fear of performing in front of people.\n\nAlthough I lived in a high state of anxiety the first weeks of that class, there was no other way around my fear. There was no real way to run from it any longer, because the more I ran, the more pervasive it got. I knew I had to turn around and run toward the fear or I would never pass through it.\n\nEmerson once said, \"The greater part of courage is having done it before,\" and that soon became true of my speaking in public. Fear of doing it can only be cured by doing it. And soon my confidence was built by doing it again and again.\n\nThe rush we get after running through the waterfall of fear is the most energizing feeling in the world. If you are ever in an undermotivated mood, find something you fear and do it\u2014and watch what happens.\n\n## 27. Create the way you relate\n\nWe can't create our truest selves without creating relationships in the process. Relationships are everywhere. Relationships are everything.\n\n\"There is no end to relationship,\" said the Indian spiritual leader Krishnamurti. \"There may be the end of a particular relationship, but relationship can never end. To be is to be related.\"\n\nI have trained many corporations with a four-part seminar series. The first three parts are on self-motivation, and the final part is on relationship-building. Sometimes CEOs ask me up front, ahead of the training, if I don't have that ratio out of balance.\n\n\"Shouldn't you have more of it be on relationship-building?\" they ask. \"After all, team-building and customer relations are surely more important than self-motivation.\"\n\nI stand by my ratio. We can't relate to others if our relationship with ourselves is poor. A commitment to personal motivation comes first. Because who wants to have a relationship with someone who is not motivated in any way? When we do get to the fourth part, relationship-building, the focus is on creativity. Creativity is the most neglected and yet most useful aspect of relationship building.\n\nIn relationships, most of us think with our emotions rather than our minds. But to think with our feelings instead of our minds puts us in the unresourceful state that Colin Wilson describes as being upside-down. When we view relationships as opportunities for creativity, they always get better. When our relationships get better, we are even more motivated.\n\nMy youngest daughter, Margie, was in fourth grade when a very shy girl in her class accidentally put a large black mark on her own nose with an indelible marker. Many of the kids in the class pointed at her and started to laugh. The little girl was finally reduced to tears of embarrassment. At some point, Margie walked over to the girl to give her some comfort. (Margie's astonished teacher related this story to me.) Impulsively, Margie picked up the marker and marked her own nose, and then handed the marker to another classmate and said, \"I like my nose this way. What about you?\"\n\nIn a few moments, the entire class had black marks on their noses, and the shy girl who was once crying was laughing. At recess, Margie's class all went out on the playground with marked noses, and they were the envy of the school\u2014obviously into something unusual and \"cool.\"\n\nThis story is interesting to me because of how Margie used her creativity and her mind instead of her emotions to solve a problem. She elevated herself up into her mind, where something clever could be done. If she had used her feelings to think with, she might have expressed anger at the class for laughing at the girl, or sadness and depression.\n\nAny time you take a relationship problem up into the mind, you have unlimited opportunities to get creative. Conversely, when you send a relationship problem down the elevator into the lower half of the heart, you risk staying stuck in the problem forever.\n\nThis doesn't mean that you shouldn't feel anything. Feel everything! Notice your feelings. Just don't think with them. When there's a relationship problem to be solved, travel up your ladder to the most creative you. You'll soon realize that we create the relationships we have in our lives; they don't just happen.\n\n\"We are each of us angels with only one wing,\" said the Italian artist Luciano de Crescenzo, \"and we can only fly embracing each other.\"\n\n## 28. Try interactive listening\n\nThe principle of using interactivity as a creativity-builder is not restricted to computer games or chat rooms. Once we become fully conscious of this principle, we can find ways to become more interactive everywhere. We can even make conversations with our family and friends more interactive than they once were.\n\nWe all have certain business associates or family members who, as they speak to us, we have a feeling that we already know what they're going to say. This lowers our own consciousness level, and a form of mental laziness sets in. Whereas in the past we might have just passively suffered through other people's monologues, we can now begin introducing more interactivity. In the past we might have punctuated our sleepy listening with meaningless words and phrases, such as \"exactly\" and \"there you go,\" but we weren't truly listening. But that passive approach shortchanges us and the people we are listening to.\n\n\"When we are listened to,\" wrote Brenda Ueland, \"it creates us, makes us unfold and expand. Ideas actually begin to grow within us and come to life.\"\n\nThe more thoughtful our questions are, the more interactive the conversations. Look for opportunities for interactivity to motivate yourself to higher levels of experience.\n\n## 29. Embrace your willpower\n\nI can't tell you how many people have told me that they have no willpower. Do you think the same thing? If you think you have no willpower, you are undermining your own success. Everyone has willpower. To be reading this sentence, you must have willpower.\n\nThe first step in developing your willpower, therefore, is to accept its existence. You have willpower just as surely as you have life. If someone put a heavy barbell on the floor in front of you and asked you to lift it and you knew you could not, you would not say \"I have no strength.\" You'd say, \"I'm not strong enough.\" \"Not strong enough\" is more truthful language, because it implies that you could be strong enough if you worked at it. It also implies that you do have strength. It is the same with willpower. Of course you have willpower. When you accept that little piece of chocolate cake, it is not because you have no willpower. It is only because you choose not to exercise it in that instance.\n\nThe first step toward building willpower is to celebrate the fact that you've got it. You've got willpower, just like that muscle in your arm. It might not be a very strong muscle, but you do have that muscle.\n\nThe second step is to know that your willpower, like a muscle in your arm, is yours to develop. You are in charge of making it strong or letting it atrophy. It is not grown by random external circumstances. Willpower is a deliberate, volitional process.\n\nWhen I left college to join the army, one of the reasons I decided to sign up was because I thought it might help teach me to develop my self-discipline. But somehow I had not been aware of the \"self\" in self-discipline. I wanted discipline to be given to me by someone else. I found out in boot camp that others do not give willpower and self-discipline. The drill sergeant might have been persuasive and inspiring (or at times terrifying), but he couldn't make me do anything until I decided to do it. Nothing happened until I generated the will to make it happen.\n\nMake a promise to yourself to be clear and truthful about your own willpower. It is always there.\n\n## 30. Perform your little rituals\n\nSee yourself as a shaman or medicine man who needs to dance and sing to get the healing started. Make up a ritual that is yours and yours alone\u2014a ritual that will be your own shortcut to self-motivation.\n\nAs you read through these various ways to motivate yourself, you might have noticed that action is often the key. Doing something is what leads to doing something. It's a law of the universe: An object in motion stays in motion.\n\nThe great basketball player Jack Twyman used to begin each practice session by getting to the court early and taking 200 shots at the basket. It always had to be 200 shots, which he counted out, and it didn't matter if he already felt tuned up after 20 or 30 shots. He had to shoot 200. It was his ritual, and it always got him into a state of self-motivation for the rest of the practice session or game.\n\nMy friend Fred Knipe, an Emmy award-winning television writer and comedian, did something he called \"driving for ideas.\" When he had a major creative project to accomplish, he got in his car and drove around the desert near Tucson until ideas began to come to him. His theory was that the act of driving gave the anxious, logical left side of his brain something to do so the right side of his brain could be freed up to suggest ideas. It's like giving your child some toys to play with so you can watch the evening news.\n\nIn his book about songwriting, _Write from the Heart_ , John Stewart writes about composer and arranger Glenn Gould, who had a ritual for finding a new melody or musical idea when he seemed to be stuck and nothing was coming. He'd turn on two or three radios at the same time, all to different stations. He'd sit and compose his own music while listening to music on the three radios. This would short-circuit his conscious mind and free up the creative subconscious. It would overload the left side of his brain so the right could open up and create without judgment.\n\nMy own ritual for jump-starting self-motivation is walking. Many times in my life I have had a problem that seemed too overwhelming to do anything about, and my ritual is to take the problem out for a long walk. Sometimes I won't come back for hours. But time and again during the course of my walks, something comes out of nowhere\u2014some idea for an action that will quickly solve the problem.\n\nOne of the reasons I think this ritual works for me is that a ritual is action. Starting a ritual is taking an action that leads toward finding the solution. The dancing medicine man is already doing something. Make up little rituals for yourself that will act as self-starters. They will have you in action before you \"feel like\" getting into action. Rituals always override your built-in hesitation so that you can get yourself motivated in a predictable, controllable way.\n\nIf you are not a writer, painter, or poet, you might be thinking that this does not apply to you. But that's what I would call the creative fallacy. In fact, your entire life is yours to create. There are no \"creative\" professions that stand apart from others, like an exclusive club. Martin Luther King Jr. said, \"Be an artist at whatever you do. Even if you are a street sweeper, be the Michelangelo of street sweepers!\"\n\n## 31. Find a place to come from\n\nMost people think they'll feel good once they reach some goal. They think happiness is out there somewhere, perhaps not even too far away, but out there all the same. The problem with putting off feeling good about yourself until you hit a certain goal is that it may never happen. And you know all the time you're striving for it that it may never happen. So, by linking your happiness to something you don't have yet, you're denying your power to create happiness for yourself.\n\nA lot of people use personal unhappiness as a tool, as proof of their own sincerity and compassion. Yet, as Barry Kaufman points out eloquently in _To Love Is to Be Happy With_ , being unhappy is not necessary. You can be happy and also be sincere. You can be happy and also be compassionate. In fact, loving someone while you are unhappy does not show up like love at all.\n\nFred Knipe talked to me about how we human beings have learned to use and abuse unhappiness\u2014he said he had made a list for me of the secret reasons why people think they should feel bad. \"If I feel bad, then that proves I am a good person,\" he said. \"Or, if I feel bad, I am responsible. If I feel bad, I'm not hurting anybody. If I feel bad, it means that I care. Maybe if I feel bad, it proves I'm being realistic and aware. If I feel bad, it means I'm working on something.\" That list gives us powerful motivation to be unhappy. But as Werner Erhard (personal transformation pioneer) taught in his well-known seminars, happiness is a place to come from, not to try to go to.\n\nI once saw Larry King interviewing Werner Erhard by satellite from Russia, where Erhard was living and working. Erhard had mentioned that he might be moving back to the United States soon, and Larry King asked him if coming home would make him happy. Erhard paused uncomfortably, because in his view of life nothing makes us happy. He finally said, \"Larry, I am already happy. That wouldn't make me happy, because I come from happiness to whatever I do.\"\n\nYour happiness is your birthright. It shouldn't depend on your achieving something. Start by claiming it and using it to make your self-motivation fun all the way and not just fun at the end.\n\n## 32. Be your own disciple\n\nSo, why do I claim we have no willpower? Is it a misguided desire to protect myself? Is there a secret payoff in saying I have no willpower? Maybe if I absolutely deny the existence of willpower, I am no longer responsible for developing it. It's out of my life! What a relief!\n\nBut, here's the final tragedy: The development and use of willpower is the most direct access to happiness and motivation that I'll ever have. In short, by denying its existence, I'm shutting my spirit down.\n\nMany people think of willpower and self-discipline as something akin to self-punishment. By giving it that negative connotation, they never get enthused about developing it. But author William Bennett gives us a different way to think of it. Self-discipline, he notes in _The Book of Virtues_ , comes from the word disciple. When you are self-disciplined, you have simply decided\u2014in matters of the will\u2014to become your own disciple.\n\nOnce you make that decision, your life's adventure gets more interesting. You start to see yourself as a stronger person. You gain self-respect.\n\nAmerican philosopher Ralph Waldo Emerson used to talk about the Sandwich Island warriors who believed that when they killed an enemy tribesman, the courage of that dead enemy passed into the warrior's living body. Emerson said that the same thing happens to us when we say no to a temptation. The power of that dead temptation passes into us. It strengthens our will. When we resist a small temptation, we take on a small power. When we resist a huge temptation, we take on huge power.\n\nWilliam James recommended that we do at least two things every day that we don't want to do\u2014for the very reason that we don't want to do them\u2014just to keep willpower alive. By doing this, we maintain our awareness of our own will.\n\n## 33. Turn into a word processor\n\nIf you associate the word \"willpower\" with negative things, such as harsh self-denial and punishment, you will weaken your resolve to build it. To increase your resolve, it's often useful to think of new word associations.\n\nTo weight lifters, failure is success. Unless they lift a weight to the point of _failure_ , their muscles aren't growing. So they have programmed themselves, through repetition, to use the word _failure_ in a positive sense. They also call what we would call _pain_ something positive: _the burn_. Getting to the burn is the goal! You'll hear bodybuilders call out to each other: \"Roast 'em!\" By consciously using motivated language, they acquire access to inner power through the use of the human will.\n\nZen philosopher and scholar Alan Watts also used to hate the word \"discipline\" because it had so many negative connotations. Yet he knew that the key to enjoying any activity was in the discipline. So he would substitute the word skill for discipline and when he did, that he was able to develop his own self-discipline.\n\nLanguage leads to power, so be conscious of the creative potential of the language you use, and guide it in the direction of more personal power.\n\n## 34. Program your biocomputer\n\nIf you're a regular consumer of the major news programs, you belong to a very persuasive and hypnotic cult. You need to be deprogrammed. Start by altering how you listen to electronic radio gossip, the news, and shock and schlock TV shows. Program out all the negative, cynical, and skeptical thoughts that you now allow to flow into your mind unchecked when you hear the news. The news is not the news. It is the bad news. It is deliberate shock. The more you accept it as the news, the more you believe that \"that's the way it is,\" and the more fearful and cynical you will become.\n\nHow do we change it? By worrying about it? No. Rather than fretting about crime and apathy and whatever you wish would change in the world, it's often very motivational to heed the words of Gandhi, who said, \"You must be the change you wish to see.\"\n\nSan Francisco writer and musician Gary Lachman wrote a captivating essay called \"World Rejection and Criminal Romantics\" in which he observed, \"It's the Ted Bundys that get television coverage, not the thousands of self-actualizers who work away at self-transformation quietly and anonymously. And it's their influence, not that of the Ted Bundys, that will shape the face of the coming century.\"\n\nOften we don't have an opportunity to skip the media reports of crime and scandal, so it's important that we listen in a way that always programs out the effect. We are pretty good at doing this when we pass the tabloids in the grocery store checkout line. We smile at them even before reading that aliens are living in the White House. We need to take that same attitude toward what passes as serious media.\n\nOnce you've gotten good at factoring out the negative aspects of the media today, take it a step further: Make your own news. Be your own breaking story. Don't look to the media to tell you what's happening in your life. Be what's happening.\n\n## 35. Open your present\n\nPractice being awake in the present moment. Make the most of your awareness of this hour. Don't live in the past (unless you want guilt) or worry about the future (unless you want fear), but stay focused on today (in case you want happiness).\n\n\"Until you can put your attention where you want it,\" said Emmet Fox, \"you have not become master of yourself. You will never be happy until you can determine what you are going to think about for the next hour.\"\n\nThere is a time for dreaming, planning, and creative goal-setting. But once you are complete with that, learn to live in the here and now. See your whole life as being contained in this very hour. Let the microcosm become the macrocosm. Live the words of the poet William Blake and his description of enlightenment:\n\nTo see a world in a grain of sand\/\n\nand heaven in a wild flower\/\n\nhold infinity in the palm of your hand\/\n\nand eternity in an hour.\n\nSir Walter Scott said he would trade whole years filled with mindless conformity for \"one hour of life crowded to the full with glorious action, and filled with noble risks.\"\n\nIt's amazing what can be done by people who learn to relax, pay attention, and focus, appreciating the present hour and all the opportunity it contains. It is said that in America we try to cultivate an appreciation of art, while the Japanese cultivate the art of appreciation. You, too, can cultivate the art of appreciation. Appreciate this hour. This hour, right now, is pure opportunity.\n\nThe great French philosopher Voltaire was on his deathbed when someone asked him, \"If you had 24 more hours to live, how would you live them?\" Voltaire said, \"One at a time.\"\n\n## 36. Be a good detective\n\nIn your professional life, whatever it is, always be curious. When you meet with someone, think of yourself as a bumbling but friendly private detective. Ask questions. Then ask follow-up questions. And then let the answers make you even more curious. Let the answers suggest even more questions. This will motivate you to higher levels of consciousness and interest.\n\nWhen you prepare a meeting with someone, prepare your questions. Cultivate your curiosity. Don't ever be at a loss for questions to ask. Most of us do the opposite. We prepare our answers. We rehearse what we are going to say. We polish and strengthen our presentation, not realizing that our host would much rather talk than listen to us.\n\nIf you are in business, you know that when prospective customers contract for long-term services, they want a company that's truly interested in them, that understands them, that will be a good consultant to them. To show a prospect that you are genuinely interested, you must be the person who asks the most thoughtful questions. To convince a company that you understand it, you will ask the best follow-up questions\u2014based on its answers. To convince a company that you will be a good consultant during the course of the contract, you will have out-learned your competitors by the inventiveness and quantity of your questions. Your curiosity will get you the business. But you can't just rely on impulsive, on-the-spot questioning. Being prepared is the secret. Preparing your questions is even more important than preparing the presentation of your services.\n\nIndiana's former basketball coach Bobby Knight always said, \"The will to win is not as important as the will to prepare to win.\" This is not only useful in business. If you are about to have an important conversation with your spouse or teenager, it is very useful to prepare your curiosity rather than your presentation. When you prepare your curiosity, you always seem to have one more question to ask before you leave, just like Lt. Columbo from the old TV show. As the character played by Peter Falk, Columbo disarmed his subjects by asking many seemingly impromptu questions. Like a disorganized but innocently charming child, he would ask about the tiniest things. As he prepared to leave, he always paused at the door, as if absent-mindedly remembering something he forgot to ask. \"Excuse me sir,\" he would say, apologetically. \"Would it inconvenience you if I asked you one more question?\"\n\nGreat relationship-builders ultimately learn that the sale most often goes to the most interested party and the quantity and quality of your questions will measure your level of interest. You might be thinking that this doesn't apply so much to you because you're not in business, or you don't sell for a living. But heed the words of Robert Louis Stevenson: \"Everybody lives by selling something.\"\n\nIn _Follow the Yellow Brick Road_ , Richard Saul Wurman writes about physicist Isidor Isaac Rabi, who won a Nobel Prize for inventing a technique that permitted scientists to probe the structure of atoms and molecules in the 1930s. Rabi attributed his success in physics to the way his mother used to greet him when he came home from school each day: \"Did you ask any good questions today, Isaac?\"\n\nBy asking questions in your relationships, you are already creating the relationship, and you are already self-motivated. You don't have to wait for the other person to make it happen.\n\n## 37. Make a relation-shift\n\nMotivate yourself by giving someone else the ideas necessary for self-motivation. You can have any experience you want in life simply by giving that experience away to someone else. John Lennon called it \"instant karma.\" In most of our relationships we stay focused on ourselves. We're fascinated by how we're coming off. We're constantly monitoring what others must now be thinking of us. We live as if mirrors surrounded us.\n\nNorman Vincent Peale used to observe that shy people were the greatest egomaniacs on earth, because they were so focused on themselves. You can see that when you observe the body language of a shy person. The looking down and turning in. The curling-up with self-consciousness\u2014as if surrounded by mirrors.\n\nWhen we shift our focus to the other person in the relationship, something paradoxically powerful happens. By forgetting ourselves we start to grow. I have developed an entire seminar around this one shift. It is called \"Relation-Shift.\"\n\nSpencer Johnson, author of _The One-Minute Sales Person_ , calls it \"the wonderful paradox: I have more fun, and enjoy more financial success, when I stop trying to get what I want and start helping other people get what they want.\"\n\nIf you want to be motivated, shift your inspiration to someone else. Point out the strengths of the other individual to him or her. Offer encouragement and support. Offer guidance in his or her own self-motivation. Watch what it does for you.\n\n## 38. Learn to come from behind\n\nProgress toward your goals is never going to be a straight line. It will always be a bumpy line. You'll go up and then come down a little. Two steps forward and one step back. There's a good rhythm in that. It is like a dance. There's no rhythm in a straight line upward.\n\nHowever, people get discouraged when they slide a step back after two steps forward. They think they are failing, and that they've lost it. But they have not. They're simply in step with the natural rhythm of progress. Once you understand this rhythm, you can work with it instead of against it. You can plan the step back.\n\nIn _The Power of Optimism_ , Alan Loy McGinnis identifies the characteristics of tough-minded optimists, and one of the most important is that optimists always plan for renewal. They know in advance that they are going to run out of energy. \"In physics,\" says McGinnis, \"the law of entropy says that all systems, left unattended, will run down. Unless new energy is pumped in, the organism will disintegrate.\"\n\nPessimists don't want to plan for renewal, because they don't think there should have to be any. Pessimists are all-or-nothing thinkers. They're always offended when the world is not perfect. They think taking a step backward means something negative about the whole project. \"If this were a good marriage, we wouldn't have to rekindle the romance,\" a pessimist would say, dismissing the idea of taking a second honeymoon. But an optimist knows that there will be ups and downs. And an optimist isn't scared or discouraged by the downs. In fact, an optimist plans for the downs, and prepares creative ways to deal with them.\n\nYou can schedule your own comebacks. You can look ahead on your calendar and block out time to refresh and renew and recover. Even if you feel very \"up\" right now, it's smart to plan for renewal. Schedule your own comeback while you're on top. Build in big periods of time to get away\u2014even to get away from what you love.\n\nIf you catch yourself thinking that you are too old to do something you want to do, recognize that you are now listening to the pessimistic voice inside of you. It is not the voice of truth. You can talk back. You can remind the voice of all the people in life who have started their lives over again at any age they wanted to. John Housman, the Emmy award-winning actor in _The Paper Chase_ , started acting professionally when he was in his 70s.\n\nI had a friend named Art Hill, who spent most of his life in advertising. In his heart, however, he always wanted to be a writer. So in his late 50s, he wrote two books that got published by a small publishing house in Michigan. Then, when he was 60 years old, Hill had his first national release with _I Don't Care if I Never Come Back_ , a book about baseball published by Simon and Schuster. The book was a popular and critical success, and his dedication page is something I treasure above any possession I own:\n\n\"To Steve Chandler\u2014who cared about writing, cared about me, and one day said, 'You should write a book about baseball.'\"\n\nNobody cares how old you are but you. People only care about what you can do, and you can do anything you want, at any age.\n\nDon't listen to the voice inside that talks about your age, or your IQ, or your life history, or anything it can slow you down with. Don't be seduced. You can start a highly motivated life right now by increasing the challenges you give your brain.\n\n## 39. Come to your own rescue\n\nAfter a seminar I gave in Vancouver, Canada, Don Beach, the sales manager of Benndorf Verster, one of that city's top businesses, sent me a tape of a song that he wanted me to hear. He said it reminded him of what I had been teaching his team about self-esteem. The song was a live performance by the old folk-singing duo, Sonny Terry and Brownie McGee. The song is called \"Love, Truth and Confidence.\" It's about how we foolishly chase after love and try to discover the ultimate truth, while ignoring something much more vital to our happiness: confidence.\n\nThe chorus of the song goes like this: \"Love and truth \/ you can find \/ any place, anywhere, anytime \/ but you can just say 'so long' \/ once confidence is gone \/ nothing matters anymore.\"\n\nI never knew the true power of self-confidence until I began working with Dr. Nathaniel Branden and his wife Devers Branden. Both are authors and psychotherapists with the Branden Institute for Self-Esteem, and they have provided me with the most powerful insights I've ever received into how I operate as a human being.\n\nDr. Branden's book, _The Six Pillars of Self-Esteem_ , is unlike any other psychology book on the market, because in addition to its eloquently written philosophy on how to build inner strength, it also contains a full year's worth of practical, powerful, user-friendly exercises to raise your own consciousness and self-esteem. His sentence-completion exercises are so effective and exciting that if you do them, I can say without a trace of exaggeration, you can get tens of thousands of dollars worth of personal growth therapy for the price of a single book.\n\nBefore you assume that Branden's notion of self-esteem is the same as that being bandied about by New Age educators, you must read his work and listen to his audio. Most people today think others can bestow self-esteem on us. Such misguided thinking leads to phenomena such as classes without grades and work without standards for excellence. Perhaps you have heard about that Little League group in Pennsylvania that wanted to eliminate keeping score from baseball games because of the damage that losing does to children's self-esteem. When we confuse pampering and coddling with instilling self-esteem, we really encourage the upbringing of sensitive children who have no inner strength whatsoever. When it comes time for such overpraised, underachieving kids to find success in the competitive global marketplace, they will be confused, fearful, and ineffective.\n\nThe concepts taught by Nathaniel and Devers Branden are intellectually ruthless and unsentimental. Some of the best ideas go all the way back to Branden's years working with the great novelist and objectivist philosopher Ayn Rand.\n\nThe Brandens have taught me how to objectively explore the weaknesses in my own thinking and to challenge the self-deception that was undermining my effectiveness in life. \"To trust one's mind and to know that one is worthy of happiness is the essence of self-esteem,\" writes Dr. Branden. \"The value of self-esteem lies not merely in the fact that it allows us to feel better, but that it allows us to live better\u2014to respond to challenges and opportunities more resourcefully and appropriately.\"\n\nThe two ideas contained in the Brandens' work that have most helped me are: 1) \"You can't leave a place you've never been\"; and 2) \"No one is coming.\" I used to believe that I could run from all my frightening thoughts and beliefs about myself. But all that ever did was create deeper internal fears and conflicts. What I really needed was to get all my fears into the sunshine and demystify them. Once I systematically began to do that, I was able to dismantle those fears, as a bomb squad dismantles a bomb. Acceptance and full consciousness of those fears\u2014and the self-sabotaging behavior they led to\u2014was \"the place I had never been.\" Once I was in that place, I could leave.\n\nThe notion that \"no one is coming\" was somehow terrifying to accept. The idea that no one was going to rescue me from my circumstances is an idea that I might never have accepted. That idea sounded too much like the final abandonment. It contradicted all my childhood self-programming. (Many of us, even as grownups, devise very elaborate and subtle variations on the \"I want my mommy\" theme.) The Brandens showed me that I could be much happier and more effective if I valued independence and self-responsibility above dependency on someone else. When you accept the idea that \"no one is coming\" it is actually a very powerful moment, because it means that you are enough. No one needs to come. You can handle your problems yourself. You are, in a larger sense, appropriate to life. You can grow and get strong and generate your own happiness. And paradoxically, from that position of independence, truly great relationships can be built, because they aren't based on dependency and fear. They are based on mutual independence and love.\n\nOnce, in a group therapy session, a client of Dr. Branden's challenged him on his principle that \"no one is coming.\" \"But Nathaniel,\" the client said, \"it's not true. You came!\"\n\n\"Correct,\" admitted Dr. Branden, \"but I came to say that no one is coming.\"\n\n## 40. Find your soul purpose\n\nHow do you know what your true life is? Or what your soul's purpose is? How do you know how to live this purpose? The answers to these questions are yours for the taking, but you must seize the answers and not wait to be given them. No one will give you the answers.\n\nOne good clue as to whether you are living your true life is how much you fear death. Do you fear death a lot, just a little, or not at all?\n\n\"When you say you fear death,\" wrote David Viscott, \"you are really saying that you fear you have not lived your true life. This fear cloaks the world in silent suffering.\" When mythologist Joseph Campbell recommended that we \"follow our bliss,\" many people misunderstood him. They thought he meant to become a pleasure-seeker, a selfish hedonist from the \"me generation.\" Instead, he meant that in order to find out what your true life could be, you should look for clues in whatever makes you happy.\n\nWhat gets you excited? In the answer to that question, you'll discover where you can be of most service. You can't live your true life if you're not serving people, and you can't serve people very well if you are not excited about what you're doing.\n\nWhat makes you happy? (I know I already asked, but the fear that \"cloaks the world in silent suffering\" comes from not asking that question enough times.) In my own professional life I have finally found that teaching makes me happy, writing makes me happy, and performing makes me happy. It took me many years of unhappiness to finally reach the point of despair necessary to ask the question: What makes me happy?\n\nI was the creative director for an ad agency and I was making a good deal of money producing commercials, meeting with clients, and designing marketing strategies. I could have done this type of work forever, but my horrible fear of death was my clue that I was not living my true life.\n\n\"People living deeply,\" wrote Ana\u00efs Nin, \"have no fear of death.\" I was not living deeply. And it took me a long time to get clear answers to my question: What makes me happy? But any question we ask ourselves often enough will eventually yield the right answer. The problem is, we quit asking.\n\nFortunately for me, in this rare instance of persistence in the face of extreme discomfort, I didn't quit asking. The answer came to me in the form of a memory\u2014so colorful it was almost like a movie scene. I was driving at night in my car 10 years earlier, and I was as happy as I had ever been. In fact, I was driving around aimlessly so that I could keep my feeling of happiness preserved and contained within that car\u2014I didn't want anything to interrupt it. It was so profound that it lasted for hours.\n\nThe occasion was a speech I had just given. The subject of it was my recovery from an addiction, and the night that I spoke I was running such a high fever, and I had such a fear of speaking in public that I tried to call the talk off. My hosts wouldn't hear of it. Somehow I made it to the podium and, probably because my fever and flu were so intense, I spoke freely, without caution or self-consciousness. The more I spoke about freedom from addiction, the more excited I got. My creativity just soared. I remember the audience laughing as I spoke. I remember them jumping to their feet and cheering when I was finished. It was the most remarkable night of my life. Somehow I had reached people in a way I'd never reached people before, and their own expressions of joy lifted me higher than I had ever been.\n\nIt was that memory of that moonlit night, driving in my car, that came back to me 10 years later after I'd spent weeks repeating to myself the question, \"What makes me happy?\" Now I had the picture, but I had no idea how to act on it. But at least I knew what my true life was, and I knew that I wasn't living it.\n\nThen one day one of my major advertising clients asked me to hire a motivational speaker for a big breakfast meeting they were having for their sales staff. I didn't know of anyone in Arizona who was any good\u2014the only motivational speakers I was familiar with were the national ones whose tapes I'd listened to so often in my car, people such as Wayne Dyer, Tom Peters, Anthony Robbins, Alan Watts, and Nathaniel Branden. But Alan Watts was dead\u2014and the rest were probably far too expensive for our little breakfast.\n\nSo I called Kirk Nelson, a friend of mine who was sales manager at KTAR in Phoenix, and asked his advice. \"The only person in Arizona worth hiring is Dennis Deaton,\" he said. \"He speaks all over the country, and he's usually booked, but if you can get him, do, because he's great.\"\n\nI finally reached Deaton in Utah, where he was giving seminars on time management. He agreed to come back to Phoenix in time for our breakfast and give a 45minute motivational talk.\n\nKirk Nelson was right. Deaton was impressive. He held the audience spellbound as he told stories that illustrated his ideas about the power that people have over their thoughts, and the mastery that they can achieve over their thinking. When he finished speaking and came back to the table where we had been sitting, I shook his hand and thanked him, and I found myself making a silent vow that someday soon I would be working with this man.\n\nIt wasn't long after that that he and I were indeed working together. It was at a company called Quma Learning, Deaton's corporate training facility based in Phoenix, Arizona. Although I began with Quma as its marketing director\u2014creating advertisements, video scripts, and direct-mail pieces\u2014I soon worked my way up to the position of seminar presenter.\n\nMy first big thrill came when Deaton and I were both invited to speak at a national convention of carpet-cleaning companies. It was the first time I had ever shared the stage with him, and I was to go on first. He was in the audience when I spoke, and I have to admit I had worked harder than I'd ever worked in my life to prepare for this event.\n\nThe participants had heard Deaton before at previous conventions and loved him, but they'd never heard me. After my presentation was over, they clapped enthusiastically and as Deaton passed me on his way to the stage he was beaming with pride as he shook my hand. (Unlike myself, Dennis Deaton has very little professional jealousy of other speakers. He was happy for my success. I have to admit that my favorite moment occurred when, after he was introduced, someone in the audience teasingly shouted out, \"Dennis who?\")\n\nMany people get confused and believe that living their true life means getting lucky and finding a suitable job with an appreciative boss somewhere. What I have come to realize is that you can live your true life anywhere, in any job, with any boss.\n\nFirst find out what makes you happy, and then start doing it. If writing makes you happy, and you're not writing for a living, start up a company newsletter or your own Website. When I first realized that speaking and teaching made me happy, I started a free weekly workshop. I didn't wait until something was offered to me.\n\nWhatever goal you want to reach, you can reach it 10 times faster if you are happy. In my sales training and consulting, I notice that happy salespeople sell at least twice as much as unhappy salespeople. Most people think that the successful salespeople are happy because they are selling more and making more money. Not true. They are selling more and making more money because they are happy.\n\nAs J.D. Salinger's character Seymour says in _Franny and Zooey_ , \"This happiness is strong stuff!\" Happiness is the strongest stuff in the world. It is more energizing than a cup of hot espresso on a cold morning. It is more mind-expanding than a dose of acid. It is more intoxicating than a glass of champagne under the stars.\n\nIf you refuse to cultivate happiness in yourself, you will not be of extraordinary service to others, and you will not have the energy to create who you want to be. There is no goal better than this one: to know as you lie on your deathbed that you lived your true life because you did what made you happy.\n\n## 41. Get up on the right side\n\nSince I was a child, I've always been intrigued with the idea that you could have a great day just by getting up on the right side of the bed. Later in life, during my years as a largely unsuccessful songwriter, one of the few successes I had was with a country rock song that I co-wrote with Fred Knipe and Duncan Stitt. It was called \"The Right Side of the Wrong Bed.\" Today my fascination is not so much with the right side of the bed as it is with the right side of the head\u2014or to be more precise, the right side of the brain.\n\nThe best explanation of how \"whole-brain\" thinking surpasses left-brain thinking or right-brain thinking is in a book written by British philosopher Colin Wilson called _Frankenstein's Castle_. Wilson reveals that we have more control over drawing vital energy and creative ideas from the right brain than we ever realized. And what stimulates the right brain the most is a high sense of purpose.\n\nIf you had to carry a heavy sack of sand across town, your left brain might get upset and tell you that you were doing something boring and tedious. However, if your child were injured badly and she weighed the same as the huge bag of sand, you'd carry her the same distance to the hospital with a surprising surge of vital energy (sent from the right brain). That's what purpose does to the brain. Self-motivation gets more and more exciting as the left brain gets better and better at telling the right brain what to do.\n\n## 42. Let your whole brain play\n\nPassive misuse of the brain leads to a life of reaction rather than creation. When Oliver Wendell Holmes said that \"most people go to their graves with their music still in them,\" he just as easily could have said that most people live in their left brain only. When Thoreau said, \"most men lead lives of quiet desperation,\" he was describing what life is like if you stay trapped in left-brain, linear, short-sighted thinking.\n\nBut the irony is that the left brain has gotten an unfairly negative reputation, simply because people stay trapped there. When people learn that the left brain is there to connect with the right, then it takes on new power and function. When people stay trapped in linear, flat, and logical left-brain thinking and never activate the creative right side of the brain, they lose their love of life. The right brain comes alive during dreaming at night while the left brain sleeps. But it is possible (as artists, poets, and saints can attest) to have the same two-sided interplay that we had as children, while we are awake. We simply have to fire it up by using the left brain to call on the right. This is what happens when we make love, play games, write poetry, hold a baby, or face a threatening crisis: The left brain commands the right brain to come alive and get involved. That is when you get whole-brain thinking, or what psychologist Abraham Maslow called peak experiences.\n\nThe three best ways to activate whole-brain thinking are through 1) goal-visualization, 2) joyful work, and 3) revitalizing play. Rather than wait for external crises to appear, create internal challenge games of your own\u2014goals and purposes\u2014that lead you in growth toward the motivated person you want to become.\n\nThe real excitement in studies of the power of the right brain lies in their suggestion of a neurological basis for personal transformation. It's not just motivational puff or secular evangelism to say that we possess unlimited creative energy, and we can use it to create the lives we want. As Colin Wilson writes in _The Essential Colin Wilson_ :\n\nIn fact, we can learn to live on a far, far higher level of power. And that is what the left brain was intended for. Its farsightedness gives it the ability to summon power. Yet it hardly makes use of this ability. It could be compared to a man who possesses a magic machine that will create gold coins so that he could, if he wanted, pay off the national debt and abolish poverty. But he is so lazy and stupid that he never bothers to make more than a couple of coins every day\u2014just enough to see him through until the evening...or perhaps he is not lazy: only afraid of emptying the machine. If so, the fear is unnecessary. It is magical, and cannot be emptied.\n\nMost people regard their right brain with a sense of wonder. They think inspiring thoughts \"came to them\" out of the blue. \"Last night I had the strangest dream!\" they will say, not knowing how much control they really have over that magical machine.\n\n## 43. Get your stars out\n\nTerry Hill is a writer who has lived all over the world and has been a friend of mine since we met each other in the sixth grade in Birmingham, Michigan. His short story, \"Cafes Are for Handicapping,\" features an intriguing character named Joe Warner who likes to tell stories about horse racing.\n\nJoe Warner tells the story of being in the press box at Belmont when Secretariat put away the Triple Crown by 31 lengths.\n\n\"And I looked beside me when he was coming down the stretch at all these hardened, cigar-chomping New York newspapermen and they all had tears running down their cheeks like little babies. 'Course I couldn't see too clear myself for the tears in my eyes. I was 23 at the time. And it was the first Triple Crown in my lifetime. Imagine that.\"\n\nThat story brought me even closer to a question I've been asking all my life. Why do we cry when we see huge accomplishments? Why do we cry at weddings? Why do I cry when the blind girl jumps with her horse in the movie _Wild Hearts Can't Be Broken_? Or when the Titans win the game in Denzel Washington's _Remember the Titans_? Why did those sportswriters cry to see that horse win by 31 lengths?\n\nThis is my theory: we weep for the winner inside of all of us. In these poignant moments, we cry because we know for a fact that there is something in us that could be every bit as great as what we are watching. We are, for that moment, the untapped greatness we are seeing. But we get tears in our eyes, because we know the greatness isn't being realized. We could have been like that, but we aren't.\n\nTerry Hill also gives public talks on creativity. His own work in advertising and public relations throughout the years has won countless awards and, as one might expect, he presents some learned and sophisticated formulas for \"creating.\" But he finishes all his talks by saying it is really a simple thing to be creative\u2014all you do is \"get your stars out.\" That's how you tap into the untapped you.\n\nHis reference is to _Seymour: An Introduction_ by J.D. Salinger. Seymour is writing a letter to his brother Buddy, who has chosen to become a professional writer. Seymour tells his brother that writing has always been more than a profession, that it has been more like Buddy's religion. He says that Buddy will be asked two very profound questions when he dies about the writing he was doing: 1) \"Were most of your stars out?\"; and 2) \"Were you busy writing your heart out?\"\n\nTerry Hill's advice to his audiences on the subject of creativity is to make sure you \"get your stars out.\" This is another way of saying let the stars that are in you shine freely. Don't force them out. Just let them shine. Although Hill's audiences are usually advertising people and writers, his recommendations apply to all of us. Our lives are ours to create. Do we want to create them in a lackluster way or do we want to be inspiring? When we write our plans and dreams, we need to write our hearts out. In shooting for the stars, it's time to get a bit wild. Wild hearts can't be broken.\n\n## 44. Just make everything up\n\nSometimes in my seminars I will ask the people in the audience to raise their hands if they think of themselves as \"creative.\" I've never had more than a fourth of the audience raise their hands. I then ask the people how many of them were able to make things up when they were younger\u2014make up names for their dolls, make up a game to play, make up a story for their parents when the truth looked less promising.\n\nAll hands go up.\n\nSo, what's the difference? You made stuff up as a child, but you're not a creative adult? The difference is that we have charged the word \"creative\" as meaning something truly extraordinary. Picasso was creative. Meryl Streep is creative. Wyclef Jean is creative. But me? One of the ways to get started creating goals and action plans is to just \"make them up,\" as you did as a kid. Think of creating in simpler terms. Think of it as something all humans do very easily. French psychologist Emile Coue said, \"Always think of what you have to do as easy and it will be.\"\n\n## 45. Put on your game face\n\nMost people who play a lot of golf or tennis work much harder at their games than they do at work. All people work harder at play than they do at work, because there's no resistance. Golfers are working harder on the golf course than they are at their professions. They don't always know this (although their spouses usually do) because it doesn't feel like work\u2014it feels like fun. They bring more energy, innovation, and zest to what they're doing out on the course because it's a game. They also bring an ongoing commitment to increasing their skills. Everyone is interested in getting better at the games they play.\n\nAs for the effect of games on energy, consider a bunch of guys playing poker all night. Because poker is a game, people can play it all night until the sun comes up. When they finally come home to sleep, you might be tempted to ask them, \"How did you manage to stay up all night? Were you drinking coffee and soda?\" No, they confess, they were drinking beer. \"But shouldn't beer slow you down and make you tired?\" Not if you are playing a game! In fact, you'll also learn that they were probably smoking cigars and eating junk as well\u2014not generally known as stimulants. What was stimulating was the game. The joy of competition.\n\nPlaywright Noel Coward once said, \"Work is more fun than fun.\" I included that quote in a seminar guidebook for a sales group a year ago and one of the participants in the back of the room raised his hand and said, \"Yeah, Steve, who is this Noel Coward guy? I figure with a quote like that he's either a porn star or a professional golfer.\"\n\nThat line got a great laugh at my expense, but it also revealed a truth (which almost all humor does). People believe that the fun jobs are always somewhere else. \"If only I could get a job like that!\" \"If only I had been a pro golfer!\" But the truth is that fulfilling and fun work can be found in anything. The more we consciously introduce game-playing elements (personal bests listed, goals, time limits, competition with self or others, record-keeping, and so on), the more fun the activity becomes.\n\nI worked on a project with a young man in Phoenix who was selling three times as much office equipment as the average salesperson on his team. He said he didn't understand his coworkers who got depressed easily, took rejection hard, and struggled with putting their deals together.\n\n\"I don't take this that seriously,\" he smiled. \"I love all my sales challenges. The tougher the prospect is, the more fun I have selling. There is absolutely nothing personal or depressing in any of this for me. When I meet a new sales prospect, it's a chess game.\"\n\nWhatever it is you have to do, whether it's a major project at work or a huge cleaning job at home, turning it into a game will always bring you higher levels of energy and motivation.\n\n## 46. Discover active relaxation\n\nThere is a huge difference between active relaxation and passive relaxation. When we play video or computer games, play cards, work in the garden, walk the dog, or play chess, we are interacting with the unexpected, and our minds are responding. All of these activities increase personal creativity and intellectual motivation. They are all active pursuits.\n\nActive relaxation refreshes and restores the mind. It keeps it flexible and toned for thinking. Great thinkers have known this secret for a long time. Winston Churchill used to paint to relax. Albert Einstein played the violin. They could relax one part of the brain while stimulating another. When they returned to workday pursuits they were fresher and sharper than ever. Most of us try to deaden the mind in order to relax. We rent mindless videos, read pulp fiction, drink, smoke, and eat until we're foggy and bloated. The problem with this form of relaxation is that it dulls our spirit and makes it hard to come back to consciousness.\n\nI accidentally discovered the restorative powers of video and computer games when I played some with my then 9-year-old son Bobby. What began as a way to make him happy and spend time with him became a brain-challenging pursuit. The complexity of computer football, basketball, and hockey games required stimulating recreational thinking.\n\n\"Thinking is the hardest work we do,\" said Henry Ford, \"which is why so few people ever do it.\" But when we find ways to link thinking to recreation, our lives get richer. We become players in the game of life and not just spectators.\n\n## 47. Make today a masterpiece\n\nMost of us think our lives accumulate. We think they are adding up to something. We think of our lives as being strung together like a long smoky train, so that we can add new freight cars when we're feeling right, and dump the others when we're not.\n\nBut when basketball legend John Wooden's father said to him, \"Make each day your masterpiece,\" Wooden knew something profound: Life is now. Life is not later on. And the more we hypnotize ourselves into thinking we have all the time in the world to do what we want to do, the more we sleepwalk past life's finest opportunities. Self-motivation flows from the importance we attach to today.\n\nJohn Wooden was the most successful college basketball coach of all time. His UCLA teams won 10 national championships in a 12-year time span. Wooden created a major portion of his coaching and living philosophy from one thought\u2014a single sentence passed on to him by his father when Wooden was a little boy\u2014\"Make each day your masterpiece.\"\n\nWhile other coaches would try to gear their players toward important games in the future, Wooden always focused on today. His practice sessions at UCLA were every bit as important as any championship game. In his philosophy, there was no reason not to make today the proudest day of your life. There was no reason not to play as hard in practice as you do in a game. He wanted every player to go to bed each night thinking, \"Today I was at my best.\"\n\nMost of us, however, don't want it to be this way. If someone asks us if today can be used as a model to judge our entire life by, we would shriek, \"On no! It isn't one of my better days. Give me a year or two and I'll live a day, I'm certain of it, that you can use to represent my life.\"\n\nThe key to personal transformation is in your willingness to do very tiny things\u2014but to do them today.\n\nTransformation is not an all-or-nothing game, it's a work in progress. A little touch here and there is what makes your day (and, therefore, your life) great. Today is a microcosm of your entire life. It is your whole life in miniature. You were \"born\" when you woke up, and you'll \"die\" when you go to sleep. It was designed this way so that you could live your whole life in a day.\n\n## 48. Enjoy all your problems\n\nEvery solution has a problem. You can't have one without the other. So why do we say that we hate problems? Why do we claim to want a hassle-free existence? When someone is emotionally sick, why do we say, \"He's got problems\"?\n\nDeep down, where our wisdom lives, we know that problems are good for us. When my daughter's teacher talks to me during open house and tells me that my daughter is going to be \"working more problems\" in math than she worked last year, I think that's wonderful. Why do I think it's wonderful when my daughter gets more problems to solve, if I think problems are a problem? Because somehow we know that problems are good for our children. By solving problems, our kids will become more self-sufficient. They'll trust their own minds more. They'll see themselves as problem-solvers.\n\nWe are so superstitious about our own problems that we tend to run from them rather than solve them. We have demonized problems to such a degree that they are like monsters that live under the bed. And by not solving them during the day, we tremble over them at night.\n\nWhen people took their problems to the legendary insurance giant W. Clement Stone, he used to shout out, \"You've got a problem? That's great!\" It's a wonder he wasn't shot by someone, given our culture's deep superstition about problems. But problems are not to be feared. Problems are not curses. Problems are simply tough games for the athletes of the mind, and true athletes always long to get a game going.\n\nIn _The Road Less Traveled_ , one of M. Scott Peck's central themes is that \"problems call forth our wisdom and our courage.\" One of the best ways to approach a problem is in a spirit of play, the same way you approach a chess game or a challenge to play one-on-one playground basketball. One of my favorite ways to play with a problem, especially one that seems hopeless, is to ask myself, _What is a funny way to solve this problem? What would be a hilarious solution?_ That question never fails to open up fresh new avenues of thought.\n\n\"Every problem in your life,\" said Richard Bach, author of _Illusions_ , \"carries a gift inside it.\" He is right. But we have to be thinking that way first, or the gift will never appear.\n\nIn his groundbreaking studies of natural healing, Dr. Andrew Weil suggested that we even regard illness as a gift. He wrote in _Spontaneous Healing_ :\n\nBecause illness can be such a powerful stimulus to change, perhaps it is the only thing that can force some people to resolve their deepest conflicts. Successful patients often come to regard it as the greatest opportunity they ever had for personal growth and development\u2014truly a gift. Seeing illness as a misfortune, especially one that is undeserved, may obstruct the healing system. Coming to see the illness as a gift that allows you to grow may unlock it.\n\nIf you see your problems as curses, the motivation you're looking for in life will be hard to find. If you learn to love the opportunities your problems present, then your motivational energy will rise.\n\n## 49. Remind your mind\n\nPerhaps you have noted an idea in this book, or another recent book you've read, that you want to hold on to. It might be an idea that you knew, the moment you saw it, would always be useful to you. You might even have underlined it for future reference. But what if the book goes on the shelf, or gets loaned to a friend, and is forevermore out of sight and out of mind? This is a very common experience, but there is a remedy: start treating self-motivational ideas as if they were songs.\n\nYou can find ways to rewind these ideas so they'll play again and again until you can't get them out of your head. That's how belief systems are restructured to suit our goals. Place the thought you want to remember into the jingle track in your brain so that it can't get out.\n\nYou can create a new self by learning the beliefs you want to live by\u2014one thought at a time. Learn these thoughts as you would the lyrics for a song you had to perform on stage. A friend of mine used to learn his parts in musicals by placing index cards with song lyrics all over his office, home, and bathroom mirror. He sometimes had them on the dashboard of his car. Why? He was making a conscious visual effort to reach the backside of his own mind.\n\nThe trick is to keep this motivation going, to deliberately feed your spirit with the optimistic ideas you want to live by. Any time a thought, sentence, or paragraph inspires you or opens up your thinking, you need to capture it, like a butterfly in a net, and later release it into your own field of consciousness.\n\nFor me, discovering an exciting idea in a book or magazine is a true peak experience. It makes the world bright and comprehensible. I get that tingle in my spine. I get that \"Oh, yes!\" feeling. The more I deliberately fill my mind with the words and phrases that originally stirred the peak experience, the easier it is to remember that life is good.\n\nColin Wilson writes in _New Pathways in Psychology_ :\n\nThis is why people who have a peak experience can go on repeating them: because it is simply a matter of reminding yourself of something you have already seen and which you know to be real. In this sense, it is like any other \"recognition\" that suddenly dawns on you\u2014for example, the recognition of the greatness of some composer or artist whom you had formerly found difficult or incomprehensible, or the recognition of how to solve a certain problem. Once such a recognition \"dawns\" it is easy to reestablish contact with it, because it is there like some possession, waiting for you to return to it.\n\nDuring my talks on self-motivation, one of the questions I'm asked most often is, \"How do I keep this going?\" People say, \"I love what I've learned today, but I've often gone to seminars that got me motivated and then a few days later I was back to my old pessimistic self, doing exactly what I used to do.\"\n\nIf I were in the mood to be blunt, I would answer the question this way: Why, if you love what you've learned about self-motivation, would you ask me how to keep it going in your life? The person in this room best equipped to answer your question is you. So I'll ask you, how will you keep this going in your life? I bet you could give me 10 ways you could do it. And I bet that if this were a foreign language you had to learn, you would set aside a certain amount of time each day to review it, to read it out loud, and to make certain you learned it. I bet you'd buy audiobooks for your car and even arrange small study groups. So the real question is this: is mastering the art of motivation as important as learning another language?\n\nOnce while I was attending a Werner Erhard seminar, I had some free time during a break so I wrote myself a letter. I put down all the ideas I wanted to remember from the seminar and I sealed them in an envelope. I took it home and a month later I mailed it to myself. When I opened it at work and read it, it was like a fresh experience all over again. I was so impressed by how effective this was for me that I employed the idea in one of my own seminars. I had everyone in the audience write out the important insights they'd received and what they intended to do differently in their lives from this moment on. When they were finished, I asked them to seal the letters into the envelopes I'd provided and address the envelopes to themselves. I told them I would hold them for a month and then mail them all.\n\nThe reports I got back were remarkable. Some people said seeing those thoughts written to themselves in their own handwriting brought the whole seminar back to them. They felt a rush of excitement and a new commitment to take action.\n\nAre you willing to remind yourself to treat yourself to your own best thoughts? Are you willing to set visual traps and ambushes, so you'll always see words and thoughts you know you want to remember?\n\n## 50. Get down and get small\n\nThe fewer goals you set each day, the more you feel pushed around by people and events that are beyond your control. You suffer from a sense of powerlessness. Rather than creating the reality you want, you are only reacting to the world around you. You have much more control over the activities of your day than you realize. By increasing your conscious use of small objectives, you will see the larger objectives coming into reality.\n\nMost people participating in the free enterprise system have become thoroughly convinced of the power of setting large and specific long-range goals for themselves. Career goals, yearly goals, and monthly performance goals are always on the mind of a person with ambition. But often those people overlook altogether the power of small goals\u2014goals set during the day that give energy to the day and a sense of achieving a lot of small \"wins\" along the way.\n\nIn his psychological masterpiece, _Flow: The Psychology of Optimal Experience_ , Mihaly Csikszentmihalyi refers to large goals as \"outcome\" goals and small goals as \"process\" goals. The beauty of process goals is that they are always within your immediate power to achieve. For example, you might set a process goal of making four important phone calls before lunch. On a sheet of paper you make four boxes, and as you make each call you fill in a box, and when the four are made, you file the paper in your goal folder and go enjoy lunch. Because you've earned it.\n\nYou can set process goals, for example, before a conversation with a person. I want to find these three things out, I want to ask these four questions, I want to make these two requests, and I want to pay my client one compliment before I leave. Process goals give you total focus. When you are constantly setting process goals, you are in more control of your day, and you feel a sense of skillful self-motivation. At the end of the day, or the beginning of the next day, you can check your progress toward your outcome goals. You can adjust your process goals to take you closer to the outcomes you want, and always keep the two in harmony.\n\nLet's say it's now the end of a long, hard day. You have a half hour before you have to go home. If you're not in the habit of setting process goals, you might say, \"I guess I ought to do some paperwork or make a call or two before I go home.\" You look at the pile of paper on your desk, or you mindlessly thumb through phone numbers, and all of a sudden someone comes by your desk to chat. Because you have nothing specific to do you engage in conversation and, before you know it, the half hour is gone and you have to go home. Even though you didn't leave anything specific unfinished, you still have that vague feeling of having wasted time.\n\nNow what happens if you use that half hour to set and achieve a process goal? \"Before I go home tonight I'm going to send out two good letters of introduction with all my marketing material included.\" Now you have a process goal and only a half hour in which to do it. When the person comes by your desk to chat, you tell him you'll have to talk to him later because you've got some things that have to get out by five.\n\nPeople who get into the swing of setting small goals all day long report a much higher level of consciousness and energy. It's as if they are athletes constantly coaching themselves through an ongoing game. They are happier people because their day is being created by the power inside their own minds, and not by the power of the world around them.\n\n## 51. Advertise to yourself\n\nI often start the day by drawing four circles on a blank piece of paper.\n\nThe circles represent my day (today), my month, my year, and my life. Inside each circle I write down what I want. It can be a dollar figure, it can be anything, and the goals can change from day to day\u2014it doesn't matter. There is no way to get this process wrong.\n\nBut by writing the goals down, I am like an airline pilot who is consulting a map prior to takeoff. I am orienting my mind to what I am up to in life. I am _reminding_ myself of what I really want. We wouldn't think, before an airline flight, of poking our heads into the cabin and saying to the pilot, \"Just take me anywhere!\" Yet, that's how we live our days when we don't check the map.\n\nSometimes in my seminars on motivation, people say they don't have time for goal-setting. But the four-circle system I described takes only four minutes! Once during a workshop on goal-setting, I asked if anyone in the audience had any interesting experiences with visualization. We had been discussing sports psychologist Rob Gilbert's observation that \"losers visualize the penalties of failure, and winners visualize the rewards of success.\"\n\nA young couple shared a story about how they had wanted for years to buy their own home, but never got the money together to do it. Then one day, after reading about the practice of treasure-mapping (posting pictures of what you want in life somewhere in your office or home), they decided to put a picture on their refrigerator of a new house, the kind they dreamed of owning.\n\n\"In less than nine months, we'd made the down payment and moved in,\" said the amazed husband. His wife added, \"Alongside the photo of the house we eventually put a little thermometer that we filled in as our savings toward a down payment grew.\"\n\nI have heard many similar stories about how treasure-mapping has worked for people. I have also read books and attended seminars that explain why. Most of them discuss what happens to the subconscious mind when you send it a picture of something you want. Because the subconscious mind only communicates with vividly imagined or real pictures, it will not seek to bring into your life anything you can't picture.\n\nWithout advertising our goals to ourselves, we can lose sight of them altogether. It is possible to go an entire week, or two or three, without thinking about our main goals in life. We get caught up in reacting and responding to people and circumstances and we simply forget to think about our own purpose.\n\nI have an example of how this practice worked in my life: Three years ago I was interested in giving more seminars on the subject of fund-raising. I coauthored a book called _RelationSHIFT: Revolutionary Fund-Raising_ with University of Arizona development director Michael Bassoff. We had done some successful seminars on the subject, and I wanted to do more. So, on the wall of my bedroom I put up a white poster board, and on that board I put up a lot of pictures and index cards with my goals on them. I wanted to have all those goals in front of me when I woke up each morning, even though I only spent a minute or two looking at the board each day.\n\nOne of the index cards I had pinned to my goal board simply contained the bold-markered letters, \"ASU.\" It was almost lost among the hodgepodge of photos and goals I'd covered the board with, and I'm certain I only barely noticed it each morning as I got up. I put it up there because I thought it would be great if I could give seminars to Arizona State University, especially now that I was living in the Phoenix area. I really thought nothing more of it.\n\nOne day at the offices of the corporate training company where I worked, I was asked to shake the hand of Jerry, a new employee. I asked Jerry to come in and sit down. We talked in my office for a few minutes about his joining the company. I asked him about his family and he casually mentioned that his parents were living in town, and that his mother worked at ASU. Normally, that would have meant nothing. ASU is a very well-known and oft-mentioned presence in the Phoenix area. But something went off in my mind when he said that, and I know in hindsight that \"something\" was my daily view of my goal board.\n\nMy ears perked up when he said \"ASU\" and I asked him, \"What does your mother do at ASU?\"\n\n\"She's the chief administrative assistant to the development director at the ASU Foundation,\" he said. \"They're in charge of all the fund-raising at the University.\"\n\nI really brightened at that point, and I told Jerry about my past work in fund-raising at the University of Arizona in Tucson, and how I'd always wanted to do similar work at ASU. He said he'd be delighted to introduce me to his mother and to the development director himself. Within a month, ASU fund-raisers were attending my seminar on _RelationSHIFT_ and I had realized one of the goals on my board. I honestly believe that if I had not had a goal board up in my bedroom, Jerry's mention of ASU would have gone right past me.\n\nThis illustrates something important. We need to advertise our own goals to ourselves. Otherwise, our psychic energy is spread too thin across the spectrum of things that aren't that important to us.\n\n## 52. Think outside the box\n\nOnce I attended a new business proposal presentation by Bob Koether, in which he had his prospective customers all play a little nine-dot game that illustrated to them that the solutions to puzzles are often simple to see if we think in unconventional ways. As people laughed and tore up their puzzles in frustration when Koether showed them the solution, he stood up to make his final point.\n\n\"We restrict our thinking for no good reason,\" said Koether. \"We do things simply because that's the way we always did them. I want you to know that our commitment in serving your company is to _always look outside the box_ for the most innovative solutions possible to our problems. We'll never do something just because that's the way we have always done it.\"\n\nTo many business leaders pitching a lucrative account, this kind of puzzle-solving exercise would simply be considered a clever presentation. But to Bob Koether, it was a symbolic expression of his whole life in business.\n\nOnce, on a Xerox-sponsored trip in Cancun, Mexico, Bob and his brother Mike spent the day out in treacherous waters on a fishing boat. After coming ashore, they retired to Carlos O'Brien's restaurant for tequila and beer and a period of reflection on their lives in sales thus far.\n\n\"We knew that as well as we had done, we would never own boats like the one we were just in if we remained at Xerox,\" said Bob. \"We talked about possibilities in the bar, and it wasn't long before we noticed some black T-shirts on the wall with the word _infinity_ on them. Then, for more than two hours, Mike and I discussed just what the word _infinity_ meant. Out of that discussion, a dream was born, a dream that took shape in the form of Infinity Communications.\"\n\nBob Koether and his brother believed that there was one vital area in which Xerox was underperforming\u2014and that was customer service. What if, they asked, a company's commitment to the customer was infinite? Not boxed-in, but unlimited in its possibilities for creative service? With that concept as motivation, the two brothers formed \"Infincom\" (short for Infinity Communications) in the state of Arizona, and within 10 years they grew from six employees and no customers into a $50-million business with more than 500 employees. And for the past three years straight, the _Arizona Business Gazette_ has ranked Infincom the number-one office equipment company in Arizona\u2014ahead of Xerox.\n\nAll of us tend to look at our challenges from inside a box. We take what we've done in the past and put it in front of our eyes and then try to envision what we call \"the future.\" But that restricts our future. With that restricted view, the best the future can be is a new and better past.\n\nGreat motivational energy occurs when we get out of the box and assume that the possibilities for creative ideas are infinite. To realize the best possible future for yourself, don't look at it through a box containing your own past.\n\n## 53. Keep thinking, keep thinking\n\nMotivation comes from thought. Every act we take is preceded by a thought that inspires that act. And when we quit thinking, we lose the motivation to act. We eventually slip into pessimism, and the pessimism leads to even _less_ thinking. And so it goes, a downward spiral of negativity and passivity, feeding on itself like cancer.\n\nI like to use this example in my seminars to illustrate the power of continuing to think: Let's say a pessimist has made up his mind to clean his garage on a Saturday morning. He wakes up, walks out to the garage, and opens the door. He is shocked to see just how much of a mess it is. \"Forget this!\" the pessimist says with disgust. \"No one could clean this garage in one day!\" At that point, the pessimist slams the garage door shut and goes back inside to do something else. Pessimists are \"all-or-nothing\" thinkers. They think in catastrophic absolutes. They are either going to do something perfectly or not at all.\n\nNow let's look at how the optimist would face the same problem. He wakes up on the same morning, goes to the same garage, sees the same mess, and even utters the same first words to himself, \"Forget this! No one could clean this garage in one day!\"\n\nBut this is where the key difference between an optimist and a pessimist shows itself. Instead of going back into the house, the optimist _keeps thinking_. \"Okay, so I can't clean the whole garage,\" he says. \"What _could_ I do that would make a difference?\"\n\nHe looks for awhile, and thinks things over. Finally, it occurs to him that he could break the garage down into four sections and do just one section today. \"For sure I'll do one today,\" he says, \"and even if I only do one section each Saturday, I'll have the whole garage in great shape before the month is over.\" A month later, you see a pessimist with a filthy garage and an optimist with a clean garage.\n\nThere was a woman in one of my seminars in Las Vegas who told me that this one concept\u2014the optimist's habit of looking for partial solutions\u2014had made an interesting difference in her life.\n\n\"I used to come home from work and look at my kitchen and just throw up my hands and curse at it and do nothing at all,\" she told me. \"I'd think the exact same thing as the pessimist in your garage story. Then I decided to just pick a small part of the kitchen and do that, and that area only. It might be a certain counter, or just the sink. By doing just one small part each night I never resent the work, it's never overwhelming, and my kitchen always looks decent.\"\n\nPessimists like to set their problems aside. They think so negatively about doing the whole thing perfectly that they end up doing nothing at all! The optimist always does a little something. She or he always takes an action and always feels like progress is being made.\n\nBecause pessimists have a habit of thinking it's hopeless\" or nothing can be done, they quit thinking too soon. An optimist may have the same initial negative feelings about a project, but he or she keeps thinking until smaller possibilities open up. This is why Alan Loy McGinnis, in his inspiring book _The Power of Optimism,_ refers to optimists as \"tough-minded.\"\n\nThe pessimist, as far as the use of the human mind goes, is a quitter.\n\nRecent studies show, says McGinnis, that optimists \"excel in school, have better health, make more money, establish long and happy marriages, stay connected to their children and perhaps even live longer.\"\n\nTo witness one of the most profound illustrations of the practical effectiveness of optimism in American history, you'll want to watch the movie _Apollo 13_. Although the job of bringing those astronauts back from the far side of the moon looked daunting and overwhelming, the job was accomplished one small task at a time. The people at Mission Control in Houston who saved the astronauts' lives did so because even in the face of \"impossible\" technological breakdowns, they kept on thinking. They never gave up. They looked for partial solutions, and they declared that they would string these partial solutions together one at a time until they brought the men home safely.\n\nWhile the astronauts' lives were still in doubt, there was one glaring pessimist in Houston ground control who made the comment that he feared that Apollo 13 might become the \"worst space disaster\" in American history. The ground commander in Houston turned to him and said with optimism and anger, \"On the contrary, sir, I see Apollo 13 as being our finest hour.\" And he turned out to be right, which illustrates the life-or-death _effectiveness_ of optimistic thinking.\n\nWhenever you feel pessimistic or overwhelmed, remember to _keep thinking_. The more you think about a situation, the more you will see small opportunities for action\u2014and the more small actions you take, the more optimistic energy you will receive. An optimist keeps thinking and self-motivates. A pessimist quits thinking\u2014and then just quits.\n\nIn the Broadway musical _South Pacific,_ the heroine sings apologetically about being a \"cock-eyed optimist.\" She admits she's \"immature and incurably green.\" This was an early version of a blonde joke. She confesses, as the giddy song soars melodically, that she's \"stuck like a dope on a thing called hope and I can't get it out of my heart...not this heart.\" That's how our society has viewed optimists\u2014they are dopes. Society thinks optimistic thinking is something that comes from the heart, not the head. Pessimists, on the other hand, are \"realistic.\" In fact, pessimists will never tell you they are pessimists. In their own minds, they are realists. And when they run into habitual optimists they sneer at them for always \"blue-skying\" everything, and not facing grim reality.\n\nPessimists continually use their imaginations to visualize worst-case scenarios, and then, concluding that those scenarios are lost causes, they take no action. That's why pessimism always leads to passivity.\n\nBut even lying on his couch, bloated with junk food and foggy from too much television, the pessimist knows somewhere in his heart that his \"what's the use?\" attitude is not effective. He is living a life that is reflected in what Nietzsche once said: \"Everything in the world displeased me; but what displeased me most was my displeasure with everything.\"\n\nOptimists have chosen to make a different use of the human imagination. They agree with Colin Wilson's point of view that \"imagination should be used, not to escape from reality, but to create it.\"\n\n## 54. Put on a good debate\n\nNegative thinking is something we all do. The difference between the person who is primarily optimistic and the person who is primarily pessimistic is that the optimist learns to become a good debater. Once you become thoroughly aware of the effectiveness of optimism in your life, you can learn to debate your own pessimistic thoughts.\n\nThe most thorough and useful study I've ever seen on how to do this is contained in Dr. Martin Seligman's classic work, _Learned Optimism_. The studies done by Seligman demonstrate two very profound revelations: 1) optimism is more effective than pessimism; and 2) optimism can be learned.\n\nSeligman based his findings on years of statistical research. He studied professional and amateur athletes, insurance sales-people, and even politicians running for office. His scientific studies proved that optimists dramatically outperform pessimists. So what Norman Vincent Peale had been saying for years in his books on the power of positive thinking was finally proven to be scientifically true.\n\nPeale had based his books on testimonials and supportive biblical passages. The problem with that was that the people he needed to reach the most\u2014skeptics and pessimists\u2014were precisely the kinds of people who would not be anxious to take anything on faith. But once you've digested the remarkable writings of Seligman, you can go back and read Peale with a new sense of excitement. If you don't accept his religious references, it doesn't matter\u2014the personal testimonials are stimulating enough to give his writing great power. Although his most famous book is _The Power of Positive Thinking,_ I have derived much more motivation from _Stay Alive All Your Life_ and _The Amazing Results of Positive Thinking_.\n\nIf you are now skeptical about your power to debate your own pessimistic thoughts, keep in mind that most of us are already great debaters. If somebody comes in and takes one side of an argument, we can usually take the other side and make a case, no matter which side the first person took. Debate teams have to learn to do this. Team members never know until the last second which side of the argument they will be debating, so they learn to be prepared to passionately argue either side.\n\nIf you catch yourself brooding, worrying, and thinking pessimistically about an issue, the first step is to recognize your thoughts as being pessimistic. Not wrong or untrue\u2014just pessimistic. And if you are going to get the most out of your _bio-computer_ (the brain), you must acknowledge that pessimistic thoughts are less effective.\n\nOnce you've accepted the pessimistic nature of your thinking, you are ready to take the next step. (This first step is crucial, though. As Nathaniel Branden teaches, \"You can't leave a place you've never been.\") The second step is to build a case for the optimistic view.\n\nStart to argue against your first line of reasoning. Pretend you're an attorney whose job is to prove the pessimist in you wrong. Start off on building your case for what's possible. You'll surprise yourself. Optimism is by nature expansive\u2014it opens door after door to what's possible. Pessimism is just the opposite\u2014it is constrictive. It shuts the door on possibility. If you really want to open up your life and motivate yourself to succeed, become an optimistic thinker.\n\n## 55. Make trouble work for you\n\nOne evening, many years ago, my then 14-year-old daughter Stephanie went for a walk with a friend, promising me she would be back home before 10 p.m. I didn't pay much attention to the clock until the 10 o'clock news ended and I realized that she hadn't come home yet. I started to get nervous and irritated. I began pacing the house, wondering what to do. At 11:30, I got in my car and started cruising the neighborhood looking for her. My thoughts were understandably anxious, part fear and part anger. Finally, at 11:45, I drove back past my own house and saw her silhouette in the window. She was home and safe.\n\nBut I kept driving. I realized that I was thinking completely pessimistically about the entire incident and I needed to keep thinking before I talked to her. As I drove along I observed all the pessimism I was wallowing in: _She doesn't respect me. She can't keep a promise. My rules and requests mean nothing. This is the tip of the iceberg. I'm going to have problems with her for the next four years at least. Who knows where she went and what she was doing? Were drugs involved? Sex? Crime? I'm losing sleep over this. This is ruining my peace of mind and my life_.\n\nBy recognizing how pessimistic my thoughts were, I was able to let the thoughts play completely out before taking a deep breath and telling myself, _Okay. That's one side of the argument. Now it's time to explore the other side_. One of my favorite tricks for flipping my mind over to the optimistic side is to ask myself the question: _How can I use this?_\n\nHow could I use this incident to improve my relationship with my daughter? How could I make my rules and requests more meaningful to us both? I began to build my case for optimism. I realized that great relationships are built by incidents like these. They are not built by theoretical conversations, but by difficult experiences and what we learn and gain from them. So I decided to drive a little while longer and let her wait inside. I was sure that by now her sister had told her that I was out looking for her, so she was now the one pacing and anxious. _Let her sweat a little,_ I thought, _while I continue to think things through_.\n\nI continued to reflect upon my past relationship with Stephanie. One of the great aspects of it was Stephanie's honesty. She had always radiated a quiet and confident kind of serenity about life, and found it easy to be honest with her own feelings and honest with other people. Whenever there had been incidents with other children, teachers, or other parents involved in some misunderstanding, I could always count on Stephanie to tell me the truth. Asking her about what happened always saved me a lot of time.\n\nAs I drove the dark neighborhood, I also ran through my happiest memories of Stephanie as a little girl, how much I loved her and how proud I was of her when I went to her concerts or talked to her teachers. I recalled the time in grade school when I embarrassed her by asking her principal if he would consider re-naming the school after her. (She had just won an academic award of some kind and I was intoxicated with pride.)\n\nFinally my mind was completely won over to the optimistic side. Thinking about how I could use this gave me the idea that this incident could be made into something bigger than it seemed\u2014a new commitment to each other to keep agreements and trust each other.\n\nWhen I finally got home, I could see that she was scared. She tried to blame the incident on her not having a watch. She wanted me to appreciate that, somehow, she was a victim of the whole incident. I listened patiently and then I told her I thought it was a much bigger deal than that. I talked about my relationship with her and how I had cherished her truthfulness throughout her childhood. I told her that I thought we might have lost all of that tonight. That we might have to figure a way to start over.\n\n\"It's not that big a deal,\" she protested. But I told her that I thought it was a very big deal, because it was all about our relationship and whether we were going to keep agreements with each other. I told Stephanie I wanted her to be as happy as she could possibly be, and the only way I could really help that happen would be if we kept agreements with each other. I told her how scared I was, how angry I was, how her staying out had ruled out a good night's sleep for me. I asked her to try to understand. I talked about our life together when she was a little girl, and I reminded her how extraordinarily truthful she was. I mentioned a few incidents when she got in trouble, but how I had gone right to her for the truth and always got it.\n\nWe talked for a long time that night, and she finally saw that coming home when she says she's coming home\u2014indeed, doing what she says she's going to do\u2014is a really \"big deal.\" It's everything.\n\nAfter that incident and conversation, Stephanie was extremely sensitive to keeping her word. If she went out and promised to be back at a certain time, she took along a watch or made certain someone she was with had one. The incident that night was something neither of us forgot, because it got us clear on the idea of trust and agreements. You could even say that it was a _good_ thing.\n\nWe have heard of so many incidents where bad events in retrospect were strokes of great fortune. A person who broke her leg skiing met a doctor in the hospital, fell in love, married him, and had a happy relationship for life. Because most of us have experienced a number of these incidents, we're aware of the dynamic. What seems bad (a broken leg) turns out unexpectedly great. We begin to see the truth that every problem carries a gift inside it. By choosing to _make use of_ seemingly bad events, you can access that gift much sooner. By asking yourself _How can I use this?_ or _What might be good about this?_ you can turn your life around on a dime.\n\n## 56. Storm your own brain\n\nThe term \"brainstorming\" is now very well-known in American business life. I first learned it many years ago when I worked as a copywriter in an ad agency. Whenever we would get a new account, our agency's president would get us all together to brainstorm for creative ideas for the client.\n\nThe main rules of a brainstorming session are: 1) there are no stupid ideas\u2014the more unreasonable the better; and 2) everyone must play. I have facilitated brainstorming sessions with business managers. We go around the table and each person puts out an idea and the facilitator writes it on the flip pad. We go around and around until all the reasonable ideas are exhausted and the unreasonable ideas start to flow. It is usually among the unreasonable ideas that something great is discovered. Brainstorming works so well because the usual restraints against stupidity are lifted. It's okay to be unreasonable and far out.\n\nWhat most people in business don't realize is that this powerful technique can also be used by an individual. I first discovered this while driving in my car a number of years ago listening to a motivational tape by Earl Nightingale. He talked about a system he had learned that worked wonders.\n\nOn the top of a piece of paper or blank document, you put a problem you want solved or a goal you want reached. You then put numbers 1 through 20 on it and begin your brainstorming session. The rules are the same as with a group session. You have to list 20 ideas, and they don't have to be well thought out or even reasonable. Give yourself permission to flow. Your only objective is to have 20 ideas scrawled down within a certain short amount of time.\n\nIf you do this for a week, you will end up with 100 ideas! Are all of them usable? Of course not, but who cares? When you began the process you probably didn't have _any_ usable ideas.\n\nI have used this system many times with really great results. It works so well because it relaxes the normal tensions against creative, outrageous thinking. It invites the right side of your brain to play along.\n\nA friend once called for some advice about his career. He was in show business and had developed his act to the point where he was one of the top performers in the nation. His problem was marketing and self-promotion. That part of his career was lagging behind his talent.\n\n\"What if I told you that there is someone who can give you 100 specific marketing ideas tailored to your precise career and audience?\" I asked him. He was very interested.\n\n\"You yourself are that person,\" I said.\n\nI then told him about the list-of-20 self-storming technique I had personally been using for a number of years. He eagerly jotted down the rules of the game and got busy playing. Two weeks later, he called me very excited about the results. \"I've got some really great marketing ideas right now, more than I've ever had in the past,\" he said. \"Thanks.\"\n\nSelf-mentoring is the best mentoring you can get because your mentor knows you so well. And although it's often beneficial to get specific outside personal coaching, the best coaching teaches us to look within.\n\n## 57. Keep changing your voice\n\nThere have been times when I have been told that I am lucky to have a good speaking voice. And some people are impressed that I rarely use a microphone in my seminars, even with hundreds of people in the audience.\n\nPeople will conclude that I have been \"blessed\" with a powerful set of vocal cords. But it is not true. As I related in an earlier chapter, my voice used to be no better than a feeble monotone. That is, until I got motivated to change it. There were two instances that inspired my system for developing my voice. The first was a magazine interview I read many years ago about the actor Richard Burton (who had perhaps the most mesmerizing speaking voice of all time\u2014listen to the Broadway recording of \"Camelot\" and hear him as King Arthur speak and \"sing\" his songs). In the interview, Burton said that his voice was how he made his living, so he made certain that each morning while showering he sang a number of songs to keep his vocal cords strong. Later, on a television talk show, actor Tony Randall told the host how he developed his trademark sing-song acting voice: \"I took up opera,\" he said. \"I found that singing opera did more for my stage voice than anything else I ever tried.\"\n\nThose two interviews have stayed in my mind ever since, and I always have music in my car to sing along with. I crank it up good and loud (this is best done while driving alone) and sing at the top of my lungs. I make certain that I do this every day, even when I don't feel like singing. In the words of William James, there's another benefit: \"We don't sing because we're happy, we're happy because we sing.\"\n\nPrior to a major public speech, I'll often get to my location more than an hour ahead of time and then just drive around the neighborhood singing like a madman. (Sometimes I worry that my host client might drive by and spot me in my car singing along with Elvis. But the benefits are worth that risk.) I find that when I drive and sing like that, my breathing is better, my timing is better, and when I speak, my voice effortlessly fills the hall.\n\nYou might think, _I don't speak for a living_ , so such a weird practice might not be necessary for you. But we all speak. A pleasant, relaxed, and strong speaking voice is a priceless asset to anyone whose job involves communicating with other people.\n\nWhen referring to people whose speaking voices are pleasing to listen to, many people use words such as \"melodious\" and \"well-modulated.\" These are good hints to tell if someone is complimenting a great speaking voice.\n\nYou are not stuck with the voice you have now. Start singing, and soon you'll be creating the voice you'd like to have. The stronger your voice, the stronger your confidence. The stronger your confidence, the easier it is to motivate yourself.\n\n## 58. Embrace the new frontier\n\nFortunately, for all of us, a new frontier is upon us. Because our nation, and world, has entered the Information Age, the old patterns for living are gone. An article by business writer John Huey appeared in the June 27, 1994 edition of _Fortune_. In it, Huey observed, \"Let's say you're going to a party, so you pull out some pocket change and buy a little greeting card that plays 'Happy Birthday' when it's opened. After the party, someone casually tosses the card into the trash, throwing away more computer power than existed in the entire world before 1950.\"\n\nIn the old paradigm, forged in the Industrial Age, human beings became less and less useful and adventurous. We found lifelong employment in guaranteed jobs and did our jobs the same way until retirement. Then, once we reached retirement age, we became thoroughly useless to society and lived lives dependent on the government, our relatives, or our own savings that we accumulated in our \"useful\" years. Now, with the technological explosion and entry into the Information Age, employers are no longer as interested in our job histories as they used to be. They are now more interested in our current capabilities.\n\nOne of the romantic appeals of the early Daniel Boone and Davy Crockett frontier days of our nation was the usefulness of individuals. If you were living out on the frontier, farming, cooking, and hunting, and you turned 65, it would never occur to anyone to ask you to \"retire.\"\n\nWe have finally come back to those days of honoring usefulness over age and status. For example, if my company is trying to enter the Chinese market to sell its software and you, at age 70, can speak fluent Chinese, know all about software, and have energy and a zest for success, how can I afford to ignore you?\n\nBill Gates of Microsoft has said, \"Our company has only one asset\u2014human imagination.\" If you took all of Microsoft's buildings, real estate, office hardware, physical assets\u2014anything you could _touch_ \u2014away from the company, where would it be? Almost exactly where it is now. Because in today's world, a company's value is in its thinking, not in its possessions.\n\nThis is great news for the individual\u2014because usefulness is back in style. If you can cultivate your skills, keep learning new things, study computers, learn a foreign language, or become an expert in a foreign culture and market\u2014 _you can make yourself useful_.\n\nThe great basketball coach John Wooden recommended that we live by this credo\u2014especially apt for the new technological frontier: \"Learn as if you were to live forever. Live as if you were to die tomorrow.\"\n\nGone are the days when your employability depended primarily on your job history, your school ties, your connections, your family, or your seniority. Today your employability depends on one thing\u2014your current skills. And those skills are completely under your control. This is the new frontier. And where we once entered retirement age nervous about the \"wolves at our door,\" today, with a commitment to lifelong growth through learning, we can be as useful to the world community as we are motivated to be. The more we learn about the future, the more motivated we become to be a valuable part of it.\n\n## 59. Upgrade your old habits\n\nBad habits simply cannot be broken. Nor can they be gotten rid of. Ask the millions who continue to try. They always end up, in the words of author Richard Brautigan, \"trying to shovel mercury with a pitchfork,\" because our bad habits exist for good reasons. They're there to do something for us, even if that something ends up being self-destructive. Down deep, even a bad habit is trying to make us operate better.\n\nPeople who smoke are trying, even through their addiction, to do something beneficial\u2014perhaps to breathe deeply and relax. Such breathing is needed to balance stress, so their smoking is a way in which they are trying to make themselves better. Bad habits are like that\u2014they are based on a perceived benefit. That's why they're impossible to just \"get rid of.\"\n\nThat's why habits must be respected and understood before they can be transformed. What created the habit must be built upon, not killed. We must go to the beneficial impulse that drives the habit, and then expand on that to make the habit grow from something bad into something good.\n\nLet's take drinking as an example. I've known people who used to be drunk all the time who are now sober all the time. How did they do it? Couldn't we just say that they just got rid of their drinking habit? Not really. Because, without exception, the recovered people I know _replaced_ their drinking with something else.\n\nTaking all of one's courage, relaxation, and spirituality from a bottle of alcohol is a very damaging habit. But to simply eliminate it leads to even worse problems: detoxing, fear, dread, paranoia. A total void.\n\nPeople who join Alcoholics Anonymous, however, replace their \"false courage\"\u2014once found in a bottle of alcohol\u2014with _real_ courage found in the meeting rooms of AA. The completely artificial sense of spirituality formerly found in a tumbler of spirits is replaced by the true and deeply personal spirituality found in working the 12-step program of enlightenment. The superficial but highly emotional relationships the alcoholic makes in his favorite bars are replaced by real friendships.\n\nReplacement is powerful because it works, and where bad habits are concerned it's the _only_ thing that works. I've known people who quit smoking without intending to. They took up running, or some form of regular aerobic exercise, and soon the breathing and relaxation they were getting from the exercise made the smoking feel bad to their bodies. They quit smoking because they had introduced a replacement. People who diet have the same experience. It isn't staying away from fattening food that works\u2014it's introducing a regular diet of delicious, healthy food that works. It's replacement.\n\nSubconsciously you don't think your bad habits are bad! And that's because they're filling a perceived need. So the way to strengthen yourself is to identify the need and honor it. Honor the need by _replacing_ the current habit with one that is healthier and more effective. Replace one habit, and soon you'll be motivated to replace another.\n\n## 60. Paint your masterpiece today\n\nThink of your day as a blank artist's canvas. If you go through your day passively accepting whatever other people and circumstances splatter on your canvas, you will more than likely see a mess where art could be. If the mess troubles your sleep, your next new day will begin in a state of fatigue and mild confusion. From such a state, your canvas will be splattered all the more with shapes you don't like and colors you never chose.\n\nThinking of your day as a painter's canvas will allow you to be more conscious of what is happening to you when you flood your mind with nothing but Internet gossip, commercials on the radio, the latest murder trial, your spouse's criticisms, office politics, and pessimistic musical lyrics. If you'll allow yourself to step back far enough to realize and truly see that your daily canvas is filling up with all these negative things, a certain freedom occurs. It's the freedom to choose something better.\n\nThe more _conscious_ we are of our freedom to paint whatever we want on our canvas, the less we go through life as a victim of circumstances. Many of us aren't even aware of our own victim status. We read whatever's on the coffee table, listen to whatever's on the car radio, eat whatever's handy, scan whatever's on the Internet, talk to whomever calls us on the phone, and watch whatever's on the television\u2014often too passive to even click the remote control.\n\nWe must be aware that we have it in us to change all that. We can paint our day our way. The best time management\u2014or \"day-painting\"\u2014course I ever took was taught by Dennis Deaton. His seminar's main point is that we can't manage time\u2014we can only manage ourselves.\n\n\"Clear the clutter from your mind,\" Deaton says, \"and remove the obstacles to greater success.\"\n\nWhereas most time management courses feel like courses in engineering, Deaton has captured the spirit of the artist in his teaching. His prescriptions for managing your day all stem from goal-creation and living the visions you create. Wake up and visualize your day as a blank canvas. Ask yourself, _Who's the artist today? Blind circumstance, or me? If I choose to be the artist, how do I want to paint my day?_\n\n## 61. Swim laps underwater\n\nWhen Bobby Fisher prepared for his world championship chess match with Boris Spassky, he prepared by swimming laps underwater every day. He knew that as the chess matches wore on into the late hours, the player with the most oxygen going to his brain would have the mental advantage. So he built his chess game by building his lungs. When he defeated Spassky, many were surprised by his astonishing wit and mental staying power, especially late in the matches when both players should've been weary and burned out. What kept Bobby Fisher alert wasn't caffeine or amphetamines\u2014it was his breathing.\n\nGeneral George Patton once gave a lecture to his troops on brainpower. He, too, knew the connection between breathing and thinking. \"In war, as in peace, a man needs all the brains he can get,\" said Patton. \"Nobody ever had too many brains. Brains come from oxygen. Oxygen comes from the lungs where the air goes when we breathe. The oxygen in the air gets into the blood and travels to the brain. Any fool can double the size of his lungs.\"\n\nI learned about Patton's passion for teaching his troops deep breathing from Porter Williamson. I had once written a few political radio and television commercials that caught Mr. Williamson's attention, so he called me and asked me to lunch one day. He had identified himself as the author of _Patton's Principles_ , so I eagerly accepted his invitation, having coincidentally read the marvelous book a few weeks earlier. Williamson had served in the army for many years as Patton's most trusted legal adviser.\n\nWilliamson told me many stories about serving with Patton, and how truly extraordinary a motivator the general was. Most of the Patton quotes in this book come from Williamson's own memories of his service with the great general. Williamson told me about how he himself had lost his leg to bone cancer, and how the doctors had erroneously forecasted his death twice. His inner strength, he said, often came from the inspiration he received in his days of serving with Patton.\n\n\"Frequently, General Patton would stop at my desk,\" recalled Williamson, \"and ask, 'How long you been sitting at that desk? Get up and get out of here! Your brain stops working after you sit in a swivel chair for 20 minutes. Keep the body moving around so the juices will run to the right places. It'll be good for the brain! If you sit in that chair too long, all of your brainpower will be in your shoes. You cannot keep your mind active when your body is inactive.'\"\n\nThat one principle\u2014an active mind cannot exist in an inactive body\u2014became Bobby Fisher's secret weapon in winning the world championship of chess. Who would have guessed that swimming underwater would make you a better chess player? Certainly not the overweight, worn-out chess \"genius\" Boris Spassky.\n\nSometimes, all you need is the air that you breathe to motivate yourself. Going for a run or a walk or simply deep breathing gives the brain the fuel it feeds on to be newly refreshed and creative.\n\n## 62. Bring on a good coach\n\nAfter a disappointing round on the golf course, Tiger Woods will often take a golf lesson. When I first heard about this, I asked myself, _Who could give Tiger Woods a lesson in golf?_\n\nBut that was before I ever really understood the value of coaching. The person who taught me that value was a young business consultant named Steve Hardison. Hardison taught me this: Tiger takes a lesson not because his coach is a better player who can give advice and tips, but because his coach _can stand back from Tiger Woods and see him objectively_.\n\nSteve Hardison had created an art form of coming into corporations and seeing things objectively. In fact, his perception ran deeper than that. He had near-psychic power to see what was missing. It was a gift he could also apply to individuals, but only if they were ready for the rigors of his coaching.\n\nI used to teasingly call one of his illustrative personal stories \"The Parable of the Mission.\" As a young missionary for his church in England, Hardison broke all records for enrolling congregants. He contrasted his own method with that of the other missionaries.\n\nWhile the others would rush out and knock on doors all day, Hardison would spend the first part of each day planning and plotting his activities. By _creating_ his day before it happened, he was able to combine visits, economize on travel time, and increase the number of enrollment conversations in a given day. He also used his creative planning time to set up intraneighborhood referrals so that many of his visits came with a reference.\n\nThe other missionaries were very active, but they were focused on the activity, not the result. They were in the business of knocking on doors and scurrying about\u2014Steve was in the business of enrolling people into the church. The records he set for enrollment were no accident. He planned things that way.\n\nSteve helped me understand something that lives inside of all of us, something he called \"the voice.\" When you wake up in the morning, the voice is there right away, telling you that you are too tired to get up or too sick to go to work. During a sales meeting when you are just about to say something bold to a client, the voice might tell you to cool it. \"Hold back.\" \"Be careful.\"\n\n\"The trick is,\" said Steve, \"to not ignore or deny the existence of the voice. Because it's there, in all of us. No one is free of the voice. However, you don't have to obey the voice. You can talk back to the voice. And when you really get good, you can even talk trash to the voice. Make fun of it. Ridicule it. Point out how stupid it is. And once you get into that way of debating your own doubts, you start to take back control of your life.\"\n\nMany times I'd be in the middle of a large business project and ask to meet with Steve for an hour. After he listened for a few minutes, he would almost invariably see right away what was missing in my behavior. He would say, \"Are you willing to accept some coaching on this?\" And I would eagerly say yes. Then he would tell me truthfully, sometimes ruthlessly, what he saw. I didn't always like what he saw, but I always grew stronger from talking about it.\n\nHardison's coaching was so jolting that sometimes it reminded me of an incident that happened to me when I was a boy playing Little League baseball. I had injured my knee in a play at third base, and when the game was over, the knee was swollen and my entire leg was stiff. As I sat on the bench with my leg straight out in front of me, a doctor whose son was on our team was kneeling down by my leg as my father looked on.\n\n\"I'd like you to bend your leg now,\" he said to me as his hands gently held my swollen knee.\n\n\"I can't,\" I told him.\n\n\"You can't?\" he asked, looking up at me. \"Why can't you?\"\n\n\"Because I tried, and it really hurts.\"\n\nThe doctor looked at me for a second, and then said simply but gently, \"Then hurt yourself.\"\n\nI was startled by his request. Hurt myself? On purpose? But then, without saying anything, I slowly bent my leg. Yes, there was tremendous pain, but that didn't matter. I was still mesmerized by his request. The doctor massaged my knee with his fingers and nodded to my father that everything would be okay. I'd have to have x-rays and the usual precautionary exam, but he saw nothing seriously wrong for now.\n\nBut I was still aware that something very big had just happened to me. After a boyhood that was characterized by avoiding pain and discomfort of any kind, all of a sudden I saw that I could hurt myself if I needed to, and that I could do it calmly without batting an eye. Perhaps I wasn't the coward I'd always thought I was. Perhaps there was as much courage in me as in anyone else, and it was all a matter of being willing to call on it. It was a defining incident in my life, and it was not dissimilar to the way Steve Hardison, as a coach, has required that I call on things inside me that I didn't know I had.\n\nHardison is a gifted and courageous public speaker, a resourceful and relentless salesperson, a talented athlete, and a committed family man and church member. I could write an entire book about Steve Hardison's remarkable work in coaching and consulting, and someday I just might. Examples of ways that he coached me to higher levels of performance are plentiful. But I think the greatest thing he has taught me is the value of coaching itself.\n\nOnce you open yourself up to being coached, you begin to receive the same advantages enjoyed by great actors and athletes everywhere. When you open yourself up to coaching, you don't become weaker\u2014you grow stronger. You become more responsible for changing yourself. In _The Road Less Traveled_ , M. Scott Peck writes, \"The problem of distinguishing what we are and what we are not responsible for in this life is one of the greatest problems of human existence...we must possess the willingness and the capacity to suffer continual self-examination.\"\n\nThe best coaches show us how to examine ourselves. It takes courage to ask for coaching, but the rewards can be great. The best moments come when your coach helps you do something you have previously been afraid to do. When Hardison would recommend that I do something I was afraid to do, I'd say, \"I don't know if I could do that.\"\n\nYou can get coaching anytime. If coaching is appropriate for your golf or tennis game, it is even more appropriate for the game of life. Ask someone to be honest with you and coach you for a while. Let him check your \"swing.\" Let him tell you what he sees. It's a courageous thing to do, and it will always lead to more self-motivation and growth.\n\n## 63. Try to sell your home\n\nOnce when Steve Hardison and I were discussing a few of my old habits that were holding me back from realizing my business goals, I blurted out to him, \"But why do I _do_ those things? If I know they hold me back, why do I continue to do them?\"\n\n\"Because they are _home_ to you,\" he said. \"They feel like home. When you do those things, you do them because that's what you're comfortable doing, and so you make yourself right at home doing them. And as they say, there's no place like home.\"\n\n\"Home\" can be an ugly place if it's not kept up and consciously made beautiful. \"Home\" can be a dark, damp prison, smelling of bad habits and laziness. But we _still_ don't want to leave it, no matter how bad it gets, because we think we are safe there. However, when we inspect the worn-out house more closely, we can see that the safety we think we're experiencing is pure self-limitation.\n\nAfter grasping Hardison's metaphor of home, I immediately saw that I needed to move out of _my_ house. I needed to move up in the neighborhood. I needed a better home one that contained habits that would keep me focused on goal-oriented activity. Hardison helped coach me in that direction until the new activities began to feel like where I should have been living all along.\n\nHardison's metaphor of \"home\" as the equivalent of old disempowering habits has stayed with me for a long time. Recently while I was putting together a tape of motivational music to play in my car, I included the energetic \"I'm Going Home\" by Alvin Lee and Ten Years After. As I drove around listening to it turned up all the way, I thought about what Hardison taught. I let the song be about the new home I would always be in the process of moving to. Don't be afraid to leave the psychic home you're in. Get excited about building a larger, newer, happier home in your mind, and then go live there.\n\nIn Colin Wilson's brilliant but little-known, out-of-print novel _Necessary Doubt,_ he created Gustav Neumann, a fascinating character who made many discoveries about human beings. At one point Neumann says, \"I came to realize that people build themselves personalities as they build houses\u2014to protect themselves from the world. They become its prisoners. And most people are in such a hurry to hide inside their four walls that they build the house too quickly.\"\n\nIdentify the habits that keep you trapped. Identify what you have decided is your final personality and accept that it might be a hasty construction built only to keep you safe from risk and growth. Once you've done that, you can leave. You can get the blueprints out and create the home you really want.\n\n## 64. Get your soul to talk\n\nWe've always been a little nervous, culturally, about talking to ourselves. We usually associate it with insanity. But it was Plato who said that his definition of _thinking_ was \"the soul talking to itself.\" If you really want to get your life worked out, there is no one better to talk to than yourself. No other person has as much information about your problems and no other person knows your skills and capabilities better. And there's no one else who can _do_ more for you than yourself.\n\nA lot of people in the motivational and psychological professions recommend affirmations. You choose a sentence to say, such as, \"Every day in every way I'm getting better and better,\" and repeat it whether or not you think it's true. While affirmations are a good first step to reprogramming, I prefer conversations. Conversations work faster.\n\nThe two most inspirational guidelines to productive self-conversational exercises are in Martin Seligman's _Learned Optimism_ and Nathaniel Branden's _The Six Pillars of Self-Esteem_. Seligman offers ways to dispute your own pessimism and create the habit of optimistic thinking. Branden offers provocative sentence stems for you to complete.\n\nRather than brainlessly parroting \"I'm getting better and better\" to myself, it makes a longer-lasting impression when I logically argue the case and win. With enough back-and-forth conversation, I can _prove_ to myself that I am getting better. Proof beats the parrot every time. It's one thing to try to hypnotize myself through repetition of words to accept something as true, and it's quite another to _convince_ myself that it is true.\n\nBranden suggests that we get our creative thinking going each morning by asking ourselves two questions: 1) What's good in my life? and 2) What is there still to be done? Most people don't talk to themselves at all. They listen to the radio, watch TV, gossip, and fill up on the words and thoughts of _other people_ all day long. But it's impossible to indulge in that kind of activity and also get motivated. Motivation is something you talk yourself into.\n\n## 65. Promise the moon\n\nOne frightening and effective way to motivate yourself is to make an unreasonable promise\u2014to go to someone you care about, either personally or professionally, and promise that person something really big, something that will take all the effort and creativity you've got to make happen.\n\nWhen President John Kennedy promised that America would put a man on the moon, the power of that thrilling promise alone energized all of NASA for the entire time it took to accomplish the amazing feat. In his book about the Apollo 13 mission, _Lost Moon_ , astronaut Jim Lovell called Kennedy's original promise \"outrageous.\" But it showed how effective being outrageous could be.\n\nIn his book _Passion, Profit, and Power,_ Marshall Sylver recalls seeing a billboard in Las Vegas put up by one of the casino owners who wanted to become a nonsmoker. The billboard read: \"If You See Me Smoking in the Next 90 Days, I'll Pay You $100,000!\" Can you see the power in that promise?\n\nI once promised my children that I would send them to camp in Michigan. They had been to the camp near Traverse City before, and loved it. When you live year-round in Arizona, there's something magical about the water and emerald forests of northern Michigan. It was an expensive camp, but when I made the promise, I was doing well financially, and I was confident that they could all go.\n\nThen as the summer neared, I'd run short of money and had to rearrange my priorities. My speaking schedule had replaced much of the commissioned selling I was doing and it looked like camp might not be in the picture.\n\nI remember specifically talking to my son Bobby, who was 8 years old at the time, about how times were temporarily hard and how camp didn't look like a good possibility any more this year. He was in the front seat of the car and I'll never forget for as long as I live the look on his face. He said very softly, so softly that I could barely hear him, \"But you promised.\"\n\nHe was right. I didn't say I'd try, I didn't say it was a goal\u2014I promised. And the feelings I had at that moment were so overwhelming that I finally said to him, \"Yes, I did promise. And because you reminded me that it was a promise, I will say to you right now that you're going to camp. I'll do what it takes. I'm sorry that I forgot it was a promise.\"\n\nThe first thing I did was change jobs, and my first condition on accepting my new job was that my bonus for signing was the exact amount of money it took to send my children to camp. It was done.\n\n## 66. Make somebody's day\n\nTo basketball coach John Wooden, making each day your masterpiece was not just about selfish personal achievement. In his autobiography, _They Call Me Coach_ , he mentions an element vital to creating each day.\n\n\"You cannot live a perfect day,\" he said, \"without doing something for someone who will never be able to repay you.\"\n\nI agree with that. But there's a way to _make sure_ you can't be repaid\u2014and that's doing something for someone who won't even know who did it. This gets into a theory I've had all my life, that you can create luck in your life. Not from the idea that luck is needed for success, because it isn't. But from the idea that luck can be a welcome addition to your life. You can create luck for yourself by creating it for someone else. If you know about someone who is hurting financially, and you arrange for a few hundred dollars to arrive at their home, and they don't even know who you are, then you've made them lucky. By making someone lucky, something will then happen in your own life that also feels like pure luck. (I can't explain why this happens, and I have no scientific basis for it, so all I can say is try it a few times and see if you aren't as startled as I have been at the results...it doesn't have to be money, either. We have a lot of other things to give, always.)\n\nWhen _you_ get lucky, you'll get more motivated, because you feel like the universe is more on your side. Experiment with this a little. Don't be imprisoned by cynicism posing as rationality on this subject. See what happens to you when you make other people get lucky.\n\n## 67. Play the circle game\n\nIf you use my four-minute, four-circle, goal-setting system described earlier, you can be the creator of your universe.\n\n\"You know, that's blasphemous,\" a seminar student once told me during a break. \"Only God can create the universe.\"\n\n\"But if you believe that,\" I said, \"you must also believe as it is written, that we were all created in God's image. And if you believe in God as the Creator, and that He created us in His image, then what are we doing when we don't create? Whose image are we living in when we deliberately do not create?\"\n\nTry this: After you wake up in the morning, wipe the sleep from your eyes, sit down with a pad of paper, and draw four circles. These are your own \"planets.\" Label the first circle, \"Lifelong Dream.\" (And in order to keep this example simple, I'll make it strictly financial, although you can do it with any kind of goal you want.) Your lifelong dream might be to save a half a million dollars for your retirement years. So, put that number in your \"Life\" circle. Then look at circle two, the next planet in your solar system. That circle you will label, \"My Year.\" What do you need to save in the next year in order to be on course to hit your life savings goal? (When you factor in the interest, it's less than you think.) And when you arrive at the figure, make certain that it matches up mathematically with your first circle. In other words, if you save this amount, and save, say 10 percent more each year that follows, will you achieve your \"Life\" number? If not, do some more math until you get a direct connection between your yearly savings projection and your lifelong goal.\n\nNow that you've got your first two circles filled with a number, move to the third circle, \"My Month.\" What would you have to save each month to hit your year's goal? Then put that number down. Three circles are now filled.\n\nNow go to the final circle, \"My Day.\" What do you need to do today that, if you repeated it every day, would ensure a successful month?\n\n(By the way, as I said, this doesn't have to just be about money, it can be about physical fitness, learning a language, relationship networking, spirituality, nutrition, or anything important to you.)\n\nThe power of this system lies in thinking of it as a universe. When you work the math, you cannot help but see that each circle, if done successfully, _guarantees_ the success of the next circle. If you hit your daily goal every day, your monthly goal is automatically hit. In fact, you don't even have to worry about it. And if your monthly goal is reached, the yearly goal has to happen. And if your yearly goals are hit, the lifelong goal _will_ be reached.\n\nWhen you study the irrefutable mathematical truth contained in this system, a strange feeling comes over you. You realize that all four circles are ultimately dependent on the success of just one circle: the circle labeled, \"My Day.\"\n\nThen you get the strangely empowering sensation that you have just proved on paper that your day and your life are the same thing. There is no future other than the future you are working on today. Your future is not stranded out there somewhere in space.\n\nThis is what the great poet Rainer Maria Rilke meant when he said, \"The future enters into us, in order to transform itself in us, long before it happens.\"\n\nRemember that once you have worked out the math for this, the circle game is only a four-minute daily exercise. Many times in seminars I give, participants will say that they are too busy for all this goal-setting activity. They have lives to live! But I like to remind them of the words of Henry Ford, who said, \"If you do not think about the future, you won't have one.\" And I also like to stress that I am only talking about four minutes a day.\n\nThe purpose of making the circles mathematically sound is that you can remove the elements of faith and hoping from your action plan. You _know_ your goals will be hit. Who would you want to bet on, the tennis player who has faith that she's going to win or the one who _knows_ she's going to win?\n\nBy drawing these simple four circles you can create your universe anywhere, anytime. Waiting in line at the bank, sitting in the doctor's office, waiting for a meeting to begin, or just doodling. Each time you do it, your universe gets closer to you. Each time you draw the circles, you are hit with this revelation: There is absolutely no difference between succeeding today and having a successful life.\n\nIn _The Magic of Believing,_ Claude Bristol recounts a particularly absent-minded habit of his that, looking back, may have had a bigger impact on shaping his universe than he ever realized. He said that whether he was on the phone, or just sitting in moments of abstraction, he would always have a pen or pencil out, doodling.\n\n\"My doodling was in the form of dollar signs like these\u2014$$$$$\u2014on every paper that came across my desk. The cardboard covers of all the files that were placed before me daily were covered with these markings; so were the covers of telephone directories, scratch pads, and even the face of important correspondence.\"\n\nBristol's later studies on mind stuff experiments, the power of suggestion, and the art of mental pictures caused him to conclude that his lifelong habit of doodling dollar signs had had an enormous impact on programming his mind to always be opportunistic and enterprising when it came to money. The fortune he acquired demands that we take his observations seriously.\n\n## 68. Get up a game\n\nIt is said that John F. Kennedy's father's credo was, \"Don't get mad, get even.\" That credo has a certain vengeful, clever wisdom in it as far as it goes, but you might go even further with this credo: \"Don't just get even\u2014get better.\"\n\nWhen Michael Jordan was a sophomore in high school, he was _cut_ from his high school basketball team. Michael Jordan was told by his coach that he wasn't good enough to play high school basketball. It was a crushing disappointment for a young boy whose heart was set on making the team, but he used the incident\u2014not to get mad, not to get even, but to get better.\n\nWe all have those moments when people tell us, or insinuate to us, that they don't think we measure up\u2014that they don't believe in us. Some of us have entire childhoods filled with that experience. The most common reaction is anger and resentment. Sometimes it motivates us to \"get even\" or to prove somebody wrong. But there's a better way to respond, a way that is creative rather than reactive.\n\n\"How can I use this?\" is the question that puts us on the road to creativity. It transforms the anger into optimistic energy, so we can grow beyond someone else's negative expectations.\n\nJohnny Bench, a Hall of Fame baseball player, knew what it was like to not be believed in.\n\n\"In the second grade,\" he said, \"they asked us what we wanted to be. I said I wanted to be a ballplayer and they laughed. In the eighth grade they asked the same question, and I said a ballplayer, and they laughed a little more. By the 11th grade, no one was laughing.\"\n\nOur country has gone through a difficult period of time since World War II. We no longer value heroes and individual achievement as we once did. \"Competition\" has become a bad word. But competition, if confronted enthusiastically, can be the greatest self-motivating experience in the world. What some people fear in the idea of competition, I suppose, is that we will become obsessed with succeeding at somebody else's expense. That we'll take too much pleasure in defeating and therefore \"being better\" than somebody else. Many times during conversations with my children's teachers, I was told how the school had progressively removed grades and awards from some activities \"so that the kids don't feel they have to compare themselves to each other.\" They were proud of how they'd softened their educational programs so that there was less stress and competition. But they were not softening the program\u2014they were softening the children.\n\nIf you are interested in self-motivation, self-creation, and being the best you can be, there is nothing _better_ than competition. It teaches you the valuable lesson that no matter how good you are, there is always somebody better than you are. That's the lesson in humility you need, the lesson those teachers are misguidedly trying to teach by eliminating grades. It teaches you that by trying to beat somebody else, you reach for more inside of yourself. Trying to beat somebody else simply puts the \"game\" back into life. If it's done optimistically, it gives energy to both competitors. It teaches sportsmanship. And it gives you a benchmark for measuring your own growth.\n\nThe poet William Butler Yeats used to be amused at how many definitions people came up with for happiness. But happiness wasn't any of the things people said it was, insisted Yeats.\n\n\"Happiness is just one thing,\" he said. \"Growth. We are happy when we are growing.\"\n\nA good competitor will cause you to grow. He or she will stretch you beyond your former skill level. If you want to get good at chess, play against somebody better at chess than you are. In the movie _Searching for Bobby Fisher,_ we see the negative effects of resisting competition on a young chess genius until he starts to _use_ the competition to grow. Once he stops taking it personally and seriously, the game itself becomes energizing. Once he embraces the intriguing fun of competition, he gets better and better as a player, and grows as a person.\n\nI mentioned earlier that I'd heard a report on the radio that there was a Little League organization somewhere in Pennsylvania that had decided not to keep score in its games anymore because losing might damage the players' self-esteem. They had it all wrong: Losing teaches kids to _grow_ in the face of defeat. It also teaches them that losing isn't the same as dying, or being worthless. It's just the other side of winning. If we teach children to fear competition because of the possibility of losing, then we actually _lower_ their self-esteem.\n\nCompete wherever you can. But always compete in the spirit of fun, knowing that finally surpassing someone else is far less important than surpassing yourself.\n\n## 69. Turn your mother down\n\nPsychologist and author M. Scott Peck observes, \"To a child, his or her parents represent the world. He assumes that the way his parents do things is the way things are done.\" In Dr. Martin Seligman's studies of optimism and pessimism, he found out the same thing: We learn how to explain the world to ourselves from our parents\u2014and more specifically, our mothers.\n\n\"This tells us that young children listen to what their primary caretaker (usually the mother) says about causes,\" writes Seligman, \"and they tend to make this style their own. If the child has an optimistic mother, this is great, but it can be a disaster for the child if the child has a pessimistic mother.\" Fortunately, Seligman's studies show that the disaster need only be temporary\u2014that optimism can be learned...at any age.\n\nBut it is not self-motivating to blame Mom if you find yourself to be a pessimist. What works better is self-creation: to produce a voice in your head that's so confident and strong that your mother's voice gets edited out, and your own voice becomes the only one you hear.\n\nAnd as much as you want to eliminate the continuing influence of a pessimistic adult from your childhood, remember that blaming someone else never motivates you because it strengthens the belief that your life is being shaped by people outside yourself. Love your mom (she learned her pessimism from _her_ mother)\u2014and change yourself.\n\n## 70. Face the sun\n\n\"When you face the sun,\" wrote Helen Keller, \"the shadows always fall behind you.\" This was Helen Keller's poetic way of recommending optimistic thinking. What you look at and what you face grows in your life. What you ignore falls behind you. But if you turn and look only at the shadows, they _become_ your life. When I was younger I remember hearing other kids tell a joke about Helen Keller. \"Have you heard about the Helen Keller doll?\" they would ask. \"You wind it up and it bumps into things.\" I've often thought about that joke, and why such a joke about someone who was deaf and blind was funny. I think the answer lies in our nervousness about other people overcoming huge misfortunes. (Perhaps we laugh nervously because we haven't overcome our own small ones.)\n\nIn our own day and age, we are quick to consider ourselves victims. We are all victims of some sort of emotional, social, gender, or racial abuse. We enjoy taking what difficulties we have had in life and blowing them up into huge injustices. Helen Keller didn't complain about being from a dysfunctional family, or being a woman, or not being given enough money from the government to compensate for her handicaps. She had challenges most of us can't even imagine, but she refused to become fascinated by them and make her handicaps her life. She didn't want to focus on the shadows when there was so much sun.\n\nBritish author G.K. Chesterton used to say that pessimists don't stay anti-life very long when you put a revolver to their head. All of a sudden, they can think of a million reasons to live. Those million reasons are always there, down inside of us, waiting to be called up. Our pessimism is usually a false front put on to get sympathy.\n\nIn his stirring book _Son Rise_ , Barry Neil Kaufman tells an astonishing true story of how he and his wife nurtured their autistic son to a happy, extroverted life. Kaufman and his wife made a conscious choice to see their son's disability as a great blessing to them. It was just a choice, like choosing to face the sun instead of facing your shadows. But as Kaufman says, \"The way we choose to see the world creates the world we see.\"\n\n## 71. Travel deep inside\n\nMost of us wait to find out who we are from impressions and opinions we get from other people. We base our own so-called self-image on other people's views of us. \"Oh, do you really think I'm good at that?\" we ask, when someone compliments us. If we're persuaded that they are being honest and have made a good case, we might try to alter our self-image upward.\n\nIt's great getting feedback from others, especially positive feedback. We all need it to live and feel good. But when it's all we've got, we're in danger of being far less than we could be, because our _self_ -image always depends on _others_. And all they see is what we're risking right now. What they never see is what's inside of us, waiting to emerge. Because they can't see that, they will always underrate us.\n\nYour journey can be internal. You can travel deeper and deeper inside to find out your own potential. Your potential is your true identity\u2014it only waits for self-motivation to come alive. \"For this is the journey that men and women make,\" said James A. Michener, \"to find themselves. If they fail in this, it doesn't matter much else what they find.\"\n\nLet positive reinforcement and compliments be a mere seasoning to your life. But prepare your life's meal yourself. Don't look outside yourself to _find out_ who you are, look inside and _create_ who you are.\n\n## 72. Go to war\n\nAnthony Burgess was 40 when he learned that he had a brain tumor that would kill him within a year. He knew he had a battle on his hands. He was completely broke at the time, and he didn't have anything to leave behind for his wife, Lynne, soon to be a widow. Burgess had never been a professional novelist in the past, but he always knew the potential was inside him to be a writer. So, for the sole purpose of leaving royalties behind for his wife, he put a piece of paper into a typewriter and began writing. He had no certainty that he would even be published, but he couldn't think of anything else to do.\n\n\"It was January of 1960,\" he said, \"and according to the prognosis, I had a winter and spring and summer to live through, and would die with the fall of the leaf.\" In that time Burgess wrote energetically, finishing five and a half novels before the year was through (very nearly the entire lifetime output of E.M. Forster, and almost twice that of J.D. Salinger).\n\nBut Burgess did not die. His cancer had gone into remission and then disappeared altogether. In his long and full life as a novelist (he is best known for _A Clockwork Orange_ ), he wrote more than 70 books, but without the death sentence from cancer, he may not have written at all.\n\nMany of us are like Anthony Burgess, hiding greatness inside, waiting for some external emergency to bring it out. I believe that's why my father and many people of his generation speak so fondly about World War II. During the war, they lived in a state of emergency that brought out the best in them.\n\nIf we don't pay attention to this phenomenon\u2014how crisis inspires our best efforts\u2014we tend to brainlessly create a life based on comfort. We try to design easier and easier ways to live, so that we won't be surprised or challenged by anything. People who get the knack of self-motivation can reverse this process and get that wonderful World War II sense of vitality into their lives. Athletes do it constantly.\n\n\"How do you feel about tonight's game with the Trail Blazers?\" a reporter once asked basketball star Kobe Bryant. \"It'll be a war out there,\" he said with a twinkle in his eye.\n\nWe don't have to wait for something tragic or dangerous to attack us from the outside. We can get the same vitality going by challenging ourselves from within. A useful exercise for self-motivation is to ask yourself what you'd do if you had Anthony Burgess's original predicament. \"If I had just a year to live, how would I live differently? What exactly would I do?\"\n\n## 73. Use the 5 percent solution\n\nMany years ago, when I first began considering the idea of changing my life, I went through some emotional mood swings. I would get very high on an idea of who I could be, and I'd set out to change myself overnight. Then my old habits would pull me back to who I used to be, and I would become demoralized and depressed for weeks, thinking I didn't have what it took to change. As the weeks went by, I finally caught on to the idea that great things are often created very slowly, so why couldn't great people be created the same way? I began to see the value in small changes, here and there, that led me in the direction of who I wanted to be.\n\nIf I wanted to be someone who was healthy and had good eating habits, I would introduce a salad here, a piece of fruit there, and take the creative process very slowly. Now I almost never eat red meat, but it didn't happen by simply ruling it out one night. (All the times I tried that, my stomach, which used to far outrank my mind in my internal chain of command, would rule it back in the first time I smelled a barbecue in the neighborhood.)\n\nPyschotherapist Dr. Nathaniel Branden is known for the effectiveness in his therapy of using sentence completion exercises. By asking his clients to write out or speak six to 10 endings, quickly, without thinking, to a \"sentence stem,\" he allows people to explore their own minds for their hidden power and creativity.\n\nA typical sentence he might ask you to complete six to 10 times would be, \"If I bring 5 percent more purposefulness into my life today...\" Then you, the client, give your rapid endings to the sentence. That's how you find out what you think and secretly know about your own power to add purpose to your life. One of the fascinating aspects of Branden's sentences is the \"5 percent\" part. It seems like an awfully small amount of change when you look at it, but think of how it would play out. If you brought 5 percent more purposefulness to your life each day, it would only be 20 days before you had _doubled_ your sense of purpose.\n\nHuge things can be accomplished by focusing on one small action at a time. Novelist Anne Lamott recalls an incident in her childhood, the memory of which always helps her \"get a grip.\" She remembers:\n\nThirty years ago, my older brother, who was 10 years old at the time, was trying to get a report on birds written that he'd had three months to write, which was due the next day. We were out at our family cabin in Bolinas, and he was at the kitchen table close to tears, surrounded by binder paper and pencils and unopened books on birds, immobilized by the hugeness of the task ahead. Then my father sat down beside him, put his arm around my brother's shoulder, and said, \"Bird by bird, buddy. Just take it bird by bird.\"\n\nWhen we stay the same, it's not because we didn't make a big enough change, but rather because we didn't _do_ anything today that sent us moving _toward_ change. If you continue to think of yourself as a great painting you are going to paint, then wanting to instantly change is like wanting to finish your portrait in 10 minutes and then put it up in the art gallery.\n\nIf you see yourself as a masterpiece-in-progress, then you will relish small change. A tiny thing you did differently today will excite you. If you want a stronger body, and you took the stairs instead of the elevator, celebrate. You are moving in the direction of change. If you want to change yourself, try making the changes as small as they can be. If you want to create yourself, like a great painting, don't be afraid to use tiny brush strokes.\n\n## 74. Do something badly\n\nSometimes we don't do things because we're not sure we can do them well. We feel that we're not in the mood or at the right energy level to do the task we have to do, so we put it off, or wait for inspiration to arrive.\n\nThe most commonly known example of this phenomenon is what writers call \"writer's block.\" A mental barrier seems to set in that prevents a writer from writing. Sometimes it gets so severe that writers go to psychotherapists to get help for it. Many writers' means of earning a living depends on its cure. The \"block\" (or lack of self-motivation) occurs not because the writer can't write, but because the writer thinks he can't write _well_. In other words, the writer thinks he doesn't have the proper energy or inspiration to write something, right now, that's good enough to submit. So the pessimistic voice inside the writer says, \"You can't think of anything to write, can you?\" This happens to many of us, even with something as small as a postcard to send, or an overdue e-mail to answer. But the writer doesn't really need psychotherapy for this. All he or she needs is an understanding of how the human mind is working at the moment of the \"block.\" The cure for writer's block\u2014and also the road to self-motivation\u2014is simple. The cure is to go ahead and _write badly_.\n\nNovelist Anne Lamott has a chapter in her marvelous book _Bird by Bird_ called \"Shitty First Drafts.\" The key to writing, she says, is to just start typing anything\u2014it can be the worst thing you've ever written, it doesn't matter. \"Almost all good writing begins with terrible first efforts,\" says Lamott. \"You need to start somewhere. Start by getting something\u2014anything\u2014down on paper.\" By the mere act of typing you have disempowered the pessimistic voice that tried to convince you not to write. Now you are writing. And once you're in action, it's easy to pick up the energy and pick up the quality.\n\nWe're often afraid to do things until we're sure we'll do them well. Therefore we don't do anything. This tendency led G.K. Chesterton to say, \"If a thing is worth doing, it's worth doing badly.\"\n\nGoing out for a run gives me an example of the same phenomenon. Because I don't feel that I have a good, strong run in me, the voice says \"not today.\" But the cure for that is to decide to do it anyway\u2014even if it will be a bad run. \"I don't feel like running now, so I'm going to go out and run slowly, in such lazy, bad form that it does me no good, but at least I will have run.\" But once I start, something always happens to alter my feelings about the run. By the end of the run, I notice that it had somehow become thoroughly enjoyable.\n\nIn my self-motivation seminars, I often give a homework assignment for people to write down what their main goals are for the next year. I ask them to fill no more than half a page. This is not a difficult assignment for people who are willing to just come off the top of their heads and have fun filling the page. But you would be surprised at how many people absolutely anguish over it, trying to get it \"right,\" as if they were going to be held forever to what they wrote down. Many people simply can't do it. To get them to complete the exercise, I say, \"Put anything down. Make something up. It doesn't even have to be true. They don't even have to be _your_ goals, just do it so you can understand the exercise we're about to do.\" The point is to just do it.\n\nIn many ways we are all novelists like Anne Lamott. Our novels are our lives. And many of us get a tragic form of writer's block that causes us to not write anything at all. It's a tragedy, because deep down we are very creative. We could write a great life. It's just that we're so afraid of writing badly, that we never write. Don't let this happen to you. If you're not motivated to do something you know you need to do, just decide to do it badly. Add a little self-deprecating humor. Be comically bad at what you're doing. And then enjoy what happens to you once you're into the process.\n\n## 75. Learn visioneering\n\nA few years ago I spent some enjoyable time working with motivational speaker Dennis Deaton and teaching his principles of visioneering\u2014which he defines as \"engineering dreams into reality\" by the use of active mental imaging. When I gave my weekly Thursday night public seminars, I'd sometimes teach Deaton's visioneering concepts, and my (then) little daughter Margery would always accompany me. She helped hand out workbooks and pencils and when the seminar got started she would take a seat in the audience, open her own workbook and participate. She was 10 at the time, and I was never certain exactly how much she was absorbing.\n\nThen one weekend afternoon by the pool at our apartment complex, I relaxed in a deck chair while Margie and her girlfriend Michelle played by the pool. There were a lot of people in and around the water that day, but above them all I could hear Margery and Michelle having a heated conversation down by the deep end of the water.\n\n\"I just can't do it!\" said Michelle.\n\n\"Yes, you can,\" said Margie. \"You just have to believe you can.\"\n\n\"I'm _afraid_ to dive,\" said Michelle. \"I've never dived in my life.\"\n\n\"Michelle,\" said Margie, \"listen to me. Will you just try it my way?\"\n\n\"I don't know,\" said Michelle. \"Okay, what's your way?\"\n\n\"Just close your eyes,\" said Margie, \"and picture yourself on a diving board. Can you see yourself standing up there?\"\n\n\"Yes,\" said Michelle.\n\n\"Okay good!\" said Margie. \"Now, I want you to get an even better picture. What kind of bathing suit are you wearing? Can you see it?\"\n\n\"It's red, white, and blue,\" said Michelle, her eyes still closed. \"It's like an American flag.\"\n\n\"Great,\" said Margie. \"Now picture yourself diving off the board in slow motion, just like in a dream. Can you see that?\"\n\n\"Yes I can,\" said Michelle.\n\n\"That's great!\" shouted Margie. \"Now you can do it. Because if you can dream it, you can do it! Let's go over here and do it.\"\n\nMichelle followed her slowly to the end of the pool. I was looking over the top of my book but not letting them know I was listening. I was amazed. I had no idea what would happen next, but I noticed a number of people around the pool area watching and listening with fascination, while pretending not to.\n\nMichelle walked up to the edge of the water and looked very scared. She looked at Margie, and Margie said, \"Michelle, I want you to keep saying, very softly, 'If I can dream it, I can do it' and then I want to see you dive in.\"\n\nMichelle kept repeating \"If I can dream it, I can do it,\" and all of a sudden, surprising even herself, she dove\u2014a near-perfect dive into the deep end with almost no splash!\n\nMargie was jumping up and down and clapping when Michelle came up from the water. \"You did it!\" she shouted, and Michelle was grinning as she climbed up to do it again.\n\n_Could it be,_ I thought to myself, _that this system is this simple?_\n\nThe principle is this: You won't do anything you can't picture yourself doing. Visioneering is just another word for picturing yourself. Once you make the picturing process conscious and deliberate, you begin to create the self you want to be. We dive into the pictures we create.\n\n## 76. Lighten things up\n\nSunlight and laughter. That's what cures most fears and worries. Terrifying problems are better solved in the light than in the dark. And there are many ways to bring them into the light.\n\nPick a frightening problem. Then do the following: talk about it with someone, draw an illustrated map of it on a huge piece of paper, make \"Top 10\" lists about the problem, tell yourself some jokes about the problem, sing about the problem, and, finally, dance a dance that expresses the problem.\n\nIf you do all these things, I promise you that your problem will seem a lot funnier, and less frightening, than it once did. It is impossible to laugh deeply and be frightened at the same time.\n\nHumor is the highest form of creativity. It's the hardest to produce and the most enjoyable to receive. Humor, like all other creativity, is a matter of making unusual combinations. The more surprising the combination, the funnier the humor. Your own motivational level will always be lifted by humor. Any time you are stuck, ask yourself to take things lightly. Ask yourself to come up with some funny solutions. Laughter will destroy all limits to your thinking. When you are laughing, you are open to anything.\n\n## 77. Serve and grow rich\n\nOne good way to motivate yourself is by increasing the flow of money into your life.\n\nMost people are embarrassed to even think this way. They don't want to \"think and grow rich\" because they think they will be thought of as selfish or greedy. Or maybe they still believe in the thoroughly discredited Marxist economic superstition that to make money, you have to take it away from somebody else. Or else they don't want to come across as being obsessed with money.\n\nBut do you know who is _really_ obsessed with money? People who don't have any. They obsess about money all day long. It's in their family discussions, it's on their minds at night, and it becomes a destructive part of their relationships during the day. The best way _not_ to be obsessed with money is to trust your game plan for earning your way to financial freedom. \"Our first duty,\" said George Bernard Shaw, \"is not to be poor.\"\n\nThe road to not being poor always travels through your professional relationships in life. The more you serve those relationships, the more productive those relationships will become, and the more money you will make.\n\n\"Money is life energy that we exchange and use as a result of the service we provide to the universe,\" wrote Deepak Chopra in _Creating Affluence_. When you understand that money flows from service, you have a chance to understand something even more valuable: Unexpectedly large amounts of money come from _unexpectedly large degrees of service_.\n\nThe way to generate unexpected service to the people in your life is to ask yourself, \"What do they _expect_?\" Once you're clear on what that is, then ask, \"What can I do that they would not expect?\" It's always the unexpected service that gets talked about. And it's always getting talked about that increases your professional value. As Napoleon Hill repeatedly pointed out, great wealth comes from the habit of going the extra mile. And it is always a smart business move to do a little more than you are paid for.\n\nIt is almost impossible to enjoy a life of self-motivation when you're worried about money. Don't be embarrassed about giving this subject a great deal of thought. Thinking about money a little bit in advance frees you from having to always think about it later. Allow yourself to link financial well-being with an increased capacity for compassion for others. If I am living in poverty, how much love and attention can I give to my children or my fellow humans? How much help can I be if I, for sheer lack of creative planning, am always worried about being in debt? \"Poverty is no disgrace,\" said Napoleon Hill. \"But it is certainly not a recommendation.\"\n\n## 78. Make a list of your life\n\nNever hesitate to sit down with yourself and make lists. The more you write things down, the more you can dictate your own future. There is an unfortunate myth that lists make things trivial. But lists do the opposite\u2014they make things come alive.\n\nI have a friend who made a list of all the positive things about himself that he could think of. He listed every characteristic and accomplishment that he could remember in his life that he was proud of. He keeps the list in his briefcase, and says he often reads through it when he's feeling down. \"By seeing all those things written down, and letting myself read them one at a time, I can change my entire attitude from being discouraged to feeling positive about myself,\" he says.\n\nWriting lists of goals and objectives is also a powerful self-motivator. It's one thing to go into a meeting mentally briefed on what you want to accomplish, but you'll feel even stronger having written it out. There is something about writing something down that makes it more real to the right side of your brain.\n\nMy friend Fred Knipe sometimes travels to Phoenix to spend a day talking with me. We've been close friends since college and share an unorthodox sense of humor. Our meetings together are anything but structured. We free-associate and talk about everything under the sun. He'll often arrive with a list.\n\nIn the days prior to our meeting, he'll jot down subjects he wants to be sure he remembers to talk to me about while we are together. And it's _because_ our conversations are so free-form that the list is valuable for him. He doesn't ever have to call me back the next day and try to discuss something over the phone that would have been much better discussed in person.\n\nIf you've ever tried grocery shopping for a large event without a shopping list, you are aware of the nightmare it can be. Most people have learned not to shop that way. I've learned by hard experience that it can mean additional trips to the store to pick up forgotten items.\n\nWhy is it that people don't apply that same principle to their lives? Most people take more time planning a picnic than they do planning a life. Because they _know_ that if they don't make a list and forget the hot dog buns as a result, they are going to be called an idiot by someone.\n\nStart by listing all the things you would like to do before you die. Keep the list somewhere handy, where you can look at it and add to it. Then list the people in your life you want to remain close to and stay in touch with. Friendship is so precious, why let it be forgotten? It sounds silly to make a list of your friends, but you'll be surprised at how it reminds you who's important and motivates you to stay in touch.\n\nMy friend Terry Hill, the writer, is one of the greatest list-makers of all time. He has a list of every book he has ever read, every poem he's read, and many more things I don't even know about. It gives his life a sense of history, depth, and direction.\n\nWe don't have to wait to become famous so that someone else might write our history. We can be writing our history while it happens. And when we list our goals, we're writing our history _before_ it happens. Legendary advertising executive David Ogilvy started his advertising agency by making a list of the clients that he most wanted: General Foods, Lever Brothers, Bristol Myers, Campbell Soup Company, and Shell Oil. At the time, they were the biggest advertising accounts in the world, and he had none of them. But in a sense he did have them, because they were on his list. \"It took time,\" said Ogilvy, \"but in due course I got them all.\"\n\nA goal gains power when you write it down, and more power _every time you write it down_. What motivates you most in life ought to be in your own handwriting. People all too often look for motivation in what others have written. If you become a good list-maker, you will learn how to motivate yourself by what _you've_ written.\n\n## 79. Set a specific power goal\n\nMost people are surprised to learn that the reason they're not getting what they want in life is because their goals are too small. And too vague. And therefore have no power.\n\nYour goals will never be reached if they fail to excite your imagination. What really excites the imagination is the setting of a large and specific _power goal_.\n\nUsually, a goal is just a goal. But a _power goal_ is a goal that takes on a huge reality. It lives and breathes. It provides motivational energy. It gets you up in the morning. You can taste it, smell it, and feel it. You've got it clearly pictured in your mind. You've got it written down. And you love writing it down because every time you do, it fills you with clarity of purpose.\n\nIn his audiobook series, _Visioneering_ , my old partner Dennis Deaton teaches the transforming power of lofty goals. Deaton talks about creating a mental movie that you watch as often as possible. He urges you to make it a movie that stars _you_ \u2014living the results of achieving your specific goal.\n\nWalt Disney left us many great things: Disneyland, Walt Disney World, and great animated films. But what I believe was his greatest gift was the summing up he did of his life's work: \"If you can dream it,\" he said, \"you can do it.\"\n\nA power goal is a dream with a deadline. The deadline itself motivates you. People who have created power goals start living on purpose. They know what they're up to in life. How can you tell if you've got a big enough and real enough power goal? Simply observe the effect your goal has on you. It's not what a goal _is_ that matters; it's what a goal _does_.\n\n## 80. Change yourself first\n\nDon't change other people. It doesn't work. You'll waste your life trying. Many of us spend all our time trying to change the people in our lives. We think we can change them in ways that will make them better equipped to make us happy. This is especially true of our children. We talk to our children for hours about how we think they should change. But children don't learn from what we say. They learn from what we do. Today's children, upon hearing us talk to them about how they should change, will often say, \"Yeah, right.\" It's shorthand for \"I'm not listening to what you say, I'm listening to what you do.\"\n\nGandhi was especially tuned in to the futility of changing other people. Yet Gandhi was probably responsible for more change in people than any other person in our era was. How did he do it? He had a profoundly simple formula. People would often come to Gandhi to ask how they could change others. Someone would say, \"I agree with you about nonviolence, but there are others who don't. How do I change them?\" And Gandhi told them they couldn't. He said you couldn't change other people.\n\n\"You must _be_ the change you wish to see in others,\" said Gandhi. In my own seminars, I probably use that one quotation more than any other. I am always asked, \"How can I change my husband?\" Or, \"How can I change my wife?\" Or, \"How can I change my teenager?\"\n\nPeople who take the seminars on self-motivation, at some point during the workshop, agree completely with the principles and ideas. Then, they start to think about the people who _don't_ buy in. In the question-and-answer period, their questions are about _those poor people_. How do we change _them_? I always quote Gandhi. _Be_ the change you wish to see in others.\n\nBy _being_ what you want _them_ to be, you lead by inspiration. Nobody really wants to be taught by lectures and advice. They want to be led through inspiration.\n\nSales managers often ask me how they can get a certain salesperson to do more self-motivated activities. I tell them that they have to _be_ the salesperson they want to see. Take them on a call, I say, and let them watch you. Don't tell them how to do it, inspire them to do it.\n\nI once attended a concert given by my daughter's fourth-grade chorus, which sang a song called \"Let There Be Peace on Earth.\" The song's words went, \"Let there be peace on earth, _and let it begin with me_....\" I beamed when I heard it. It was such a beautiful expression of _being the change_ \u2014a celebration of self-responsibility that rarely is portrayed in young people's lives today.\n\nWhat you _tell_ people to do often goes right by them. Who you _are_ does not.\n\n## 81. Pin your life down\n\nCar dealer extraordinaire Henry Brown once told me a story about his son, a high school wrestler. His boy had been getting only fair results as a wrestler that year and when Henry talked to him about it he learned the reason. Henry's son entered each wrestling match more than thoroughly prepared to _counter_ anything his opponent tried. But no matter how gifted Henry's son was at countering moves, countering was still countering, so the other wrestler always dictated the tempo. Finally, Henry suggested to his son that he try entering a wrestling match with his own attack _plan_ \u2014a series of moves that _he_ would initiate no matter what his opponent tried. The boy agreed, and the results were remarkable. He began winning match after match, pinning opponent after opponent.\n\nThe young wrestler's goal had always been to win. He didn't have a problem setting goals. But what had to be added was a plan of action. In sports, as in life, goals alone aren't always enough. As Nathaniel Branden says, \"A goal without an action plan is a daydream.\" Henry Brown didn't just give that advice to his son because he bought into it theoretically. His own Brown and Brown Chevrolet dealership had been the number-one Chevy dealership in the nation many times because he planned his company's own yearly performance in the same way he coached his son.\n\nEvery year he has his general manager send me the detailed videotape that outlines the dealership's game plan for the coming year. It includes all the department's projected earnings down to the penny. By boldly charting such a specific course, Brown lets the market _respond to him_. Once, when I asked him how his dealership got through a previous year's nationwide automotive sales recession, he said, \"We decided not to participate in it.\"\n\nBefore any adventure, take time to plan. Design your own plan of attack. Don't just counter what some other wrestler is doing. Let life respond to _you_. If you're making all the first moves, you'll be surprised at how often you can pin life down.\n\n## 82. Take no for a question\n\nDon't take no for an answer. Take it for a question. Make the word _no_ mean this question: \"Can't you be more creative than that?\" In my seminars, I work with a lot of salespeople and one of the most requested topics of discussion is \"cold calling and rejection.\" One of the greatest problems salespeople, and people everywhere, face is in the meaning _they give_ to someone else's _no_. Many people hear _no_ as an absolute, final, and devastating personal rejection. But _no_ can mean anything you want it to mean.\n\nWhen I graduated from college with a degree in English, I was not overwhelmed with companies trying to hire me. So I decided to try to get a job as a sports writer at the daily evening paper in Tucson, Arizona, the _Tucson Citizen_. I had spent four years in the army, and I hadn't done any sportswriting since high school.\n\nWhen I applied for the job, I was told that my major problem was that I had never done any professional sportswriting before. It was the typical situation of a company not being able to hire you because you haven't had experience\u2014but how can you gain experience if no one will hire you?\n\nMy first impulse was to take _no_ as their final answer. After all, that's what they said it was. But I finally decided to have _no_ mean\u2014\"Can't you be more creative than that?\" So I went home to think and plot my next move. The reason they wouldn't hire me was because I had no experience. When I asked them why that was important, they smiled and said, \"We have no way of knowing for sure whether you can write sports. Just being an English major isn't enough.\"\n\nThen it hit me. Their real problem wasn't my lack of experience\u2014it was _their lack of knowledge_. They didn't know whether I could write well enough. So I set out to solve their problem for them. I began to write them letters. I knew they were interviewing four other people for the position and that they would decide within a month. Every day I wrote a letter to the sports editor, the late Regis McAuley (an award-winning writer in his own right, who made his reputation in Cleveland before coming to Tucson).\n\nMy letters were long and expressive. I made them as creative and clever as I could, commenting on the sports news of the day, and letting them know how great a fit I thought I was for their staff. After a month, Mr. McAuley called me and said that they had narrowed it down to two candidates, and I was one of them. Would I come in for a final interview? I was so excited, I nearly swallowed the phone.\n\nWhen my interview was coming to an end (I was the second one in), McAuley had one last question for me. \"Let me ask you something, Steve,\" he said. \"If we hire you, will you promise that you'll stop sending me those endless letters?\"\n\nI said I would stop, and then he laughed and said, \"Then you're hired. You can start Monday.\"\n\nMcAuley later told me that the letters did the trick. \"First of all, they showed me that you could write,\" he said. \"And second of all, they proved to me that you wanted the position more than the other candidates did.\"\n\nWhen you ask for something in your professional life and it is denied to you, imagine that the _no_ you heard is really a question: \"Can't you be more creative than that?\" Never accept _no_ at face value. Let rejection motivate you to get more creative.\n\n## 83. Take the road to somewhere\n\nEnergy comes from purpose. If the left side of your brain tells the right side of your brain that there's a sufficient crisis, the right side sends you energy, sometimes superhuman energy. That's why there's such a difference between people who set and achieve goals all day, and people who just do whatever comes up, or whatever they feel like doing. To one person, there is always added purpose. To the other, there is boredom and confusion, the two greatest robbers of energy. Knowing what you're up to, and why you're up to it, gives you the energy to self-motivate. Not knowing your purpose drains you of all motivation.\n\nWe've all heard the stories of a diminutive mother who, seeing that her small child was trapped, lifted a tremendously heavy object, such as a car, so the child could be freed. When asked to repeat the superhuman feat later, of course the woman couldn't do it. Being a single father has put me in touch with the dramatic connection between purpose and energy. If I am cooking something, for example, and out of the corner of my eye I can see flames emerging from the kitchen, it is amazing how fast I can move from the living room into the kitchen. Crisis creates instant purpose, which creates instant energy. When our purpose is great, so is our strength and energy.\n\n\"But, I don't know what my purpose is,\" a lot of people tell me, as if someone forgot to tell them what it is. Those people may wait forever to be told how to live and what to live for.\n\nThere can only be two reasons why you don't know your purpose: 1) you don't talk to yourself; and 2) you don't know where purpose comes from. You think purpose comes from outside yourself instead of from within. Purposeful people know how to go deep into their own spirit and talk to themselves about why they exist, and what they want to do with the gift of life.\n\n\"Only human beings have come to a point where they no longer know why they exist,\" said the Lakota shaman Lame Deer. \"They don't use their brains and they have forgotten the secret knowledge of their bodies, their senses, or their dreams.\"\n\nLame Deer is not optimistic about what the future holds for people who live without purpose. \"They don't use the knowledge the spirit has put into every one of them,\" he says. \"They are not even aware of this, and so they stumble along blindly on the road to nowhere\u2014a paved highway that they themselves bulldoze and make smooth so that they can get faster to the big empty hole that they'll find at the end, waiting to swallow them up. It's a quick, comfortable superhighway, but I know where it leads. I've seen it. I've been there in my vision and it makes me shudder to think about it.\"\n\nPurpose can be built, strengthened, and made more inspiring every day. We are totally responsible for our own sense of purpose. We can go inside our own spirit and create it, or not. The energy of our lives is wholly dependent on how much purpose we're willing to create.\n\n## 84. Go on a news fast\n\nI first heard the term \"news fast\" from Dr. Andrew Weil, who writes about natural medicine and spontaneous healing. Weil recommends going on news fasts because he believes this has a healing effect on the human system. To him, it's a genuine health issue.\n\nMy own recommendation for news fasts has to do with the psychology of self-motivation. If you go for periods of time without listening to or reading the news, you will notice an upswing in your optimism about life. You'll feel a lift in energy. \"But shouldn't I stay informed?\" people ask me. \"Aren't I being a bad citizen if I don't keep up with what's happening in my community? Shouldn't I be watching the news?\" In answering this question, I offer an observation that may startle you: The news is no longer the news.\n\nIt used to be that Walter Cronkite would end his program by saying, \"And that's the way it is.\" And we trusted that he was right. But today, it's much different. Shock value has the highest premium of all for a news story, and the lines are now blurred between the evening news and the grossest tabloids.\n\nToday, the goal of the person putting the evening news show together is to stimulate our emotions in as many ways as possible. Every night we will see human suffering. We will also see con artists, and even whole companies getting away with scams that victimize people cruelly. If there's a report on politics, it features the most venomous attacks between two partisans. The goal of the news today is stimulation. It's to take us on an emotional roller-coaster ride. It's a \"good\" program if we have been enraged by one story, saddened by another, and amused by the third.\n\nIs it any wonder that by programming our minds with this gross and frightening information all day and into the night, we end up a little less motivated? Is it hard to understand a certain slippage in our optimism?\n\nGoing on a news fast is a refreshing cure for this problem. You can do it for one day a week, to begin with, and then get back into the tabloid shows the next day if you have to. Once you start fasting, you'll find your entire mood picking up.\n\n\"But what about staying informed?\" you ask. There are many ways to stay fully informed. The Internet has wonderful, thoughtful sites. In fact, it is far better to be informed intellectually than to be informed emotionally. There are weekly and monthly magazines as well as e-zines that do a fine job of informing us and giving us a calm, thoughtful, overall perspective on the news. Don't worry about missing out on important news. Really big news, such as a war, a natural disaster, or an assassination, will get to you just as quickly during a news fast as it would if you were watching the news.\n\nBegin to experiment with news fasts today. Go on a short one at first, and then extend the period of time as your system allows. When you _do_ return to the news, be totally conscious of just what the show is trying to do to you. Don't passively take it in as if what you are seeing is really \"the way it is.\" It's not. They're not going to tell you how many thousands of planes landed safely today.\n\n## 85. Replace worry with action\n\nDon't worry. Or rather, don't just worry. Let worry change into action. When you find yourself worrying about something, ask yourself the action question, \"What can I _do_ about this right now?\" And then do something. Anything. Any small thing.\n\nMost of my life, I spent my time asking myself the wrong question every time I worried. I asked myself, \"What should I be feeling about this?\" I finally discovered that I was much happier when I started asking, instead, \"What can I _do_ about this?\" If I am worried about the conversation I had with my wife last night, and how unfair I might have been to say the things I said, I can ask myself, \"What can I _do_ about that right now?\"\n\nBy putting the question into the _action_ arena, a lot of possibilities will occur to me: 1) I could send her flowers; 2) I could call her to tell her I was concerned about how I left things; 3) I could leave a nice little note somewhere for her; or 4) I could go see her to make things right. All of these possibilities are actions, and when I act on something, the worry goes away.\n\nWe often hear the phrase \"worry it to death.\" But that phrase doesn't reflect what really happens when we worry. It would be great if we _could_ worry something to death. When it dies, we could dispose of the body and be done with it. But when we worry, we don't worry a thing to death, we worry it _to life_. Our worrying makes the problem grow. And most of the time, we worry it into a grotesque kind of life, a kind of Frankenstein's monster that frightens us beyond all reason.\n\nI once came up with a system for action that helped turn my worrying habits completely around. I would list the five things that I was worried about\u2014perhaps they were four projects at work and the fifth was my son's trouble he was having with a certain teacher. I would then decide to spend _five minutes_ on each problem doing something, anything. By deciding this, I knew I was committing myself to 25 minutes of activity, but no more so it didn't feel at all overwhelming.\n\nThen I could make a game of it. On project one, a seminar workbook deadline for a new course, I'd spend five minutes writing it. Maybe I only got the first two pages done, but it felt great. It felt like I'd finally started it.\n\nThen on item number two, a meeting I knew I had to have with a client over a sticky contract issue, I would call his office and schedule the meeting and put it in my calendar. That, too, felt good.\n\nMy third worry, a stack of correspondence I needed to answer, I would take five minutes sorting and stacking and putting them into a folder that was separate from the other clutter on my desk. That felt satisfying, too. The fourth item was a travel arrangement that had to be worked out. I'd take no more than five minutes looking at my calendar and leaving a voice mail for my travel agent to fax me some alternatives on the trip.\n\nFinally, on the matter of my son, I would pull out a piece of paper and write a short letter to his teacher expressing my concern for him, my support of her efforts, and my desire to arrange a meeting quickly, so all three of us could sit down together and make some agreements.\n\nAll of that took 25 minutes. And the five things that were worrying me the most were no longer worrying me. I could then go back anytime later and work them to completion. If something is worrying you, always _do_ something about it. It doesn't have to be the big thing that will make it disappear. It can be any small thing. But the positive effect it will have on you will be enormous.\n\nAnything that worries you should be _acted on,_ not just thought about. Don't be scared about the action; you can make it very small and easy, as long as you take action. Even small actions will chase away your fears. Fear has a hard time coexisting with action. When there's action, there's no fear. When there's fear, there's no action.\n\nThe next time you're worried about something, ask yourself, \"What small thing can I do right now?\" Then do it. Remember not to ask, \"What could I possibly do to make this whole thing go away?\" That question does not get you into action at all.\n\nActing on your worries frees you up for other things. It removes fear and uncertainty from your life and puts you back in control of creating what you want. Just do it.\n\n## 86. Run with the thinkers\n\nThe president of a major office equipment company put his problem to me this way: \"How do I get the whiners in my company to stop whining and start coming up with solutions?\"\n\nHe went on to explain that he had two kinds of people working for him, the Whiners and the Thinkers. The Whiners were often very smart and dedicated employees who worked long, hard hours. But when they came into the manager's office, it was almost always to complain.\n\n\"They're great at finding fault with other managers and telling me what's wrong with our systems,\" the president said, \"but they are a drain on me because they're so negative that I end up trying to make them feel better. After that, I'm depressed.\"\n\nThe Thinkers, on the other hand, had a different way of coming into the office with problems. \"The Thinkers come to me with ideas,\" he said. \"They see the same problems that the Whiners see, but they've already thought about possible solutions.\"\n\nThe Thinkers, in other words, have assumed ownership of the company, and are creating the future of the company with their thinking. The Whiners have stopped thinking. Once the problems are identified, and their reaction to them justified, the thinking stops.\n\nThe Thinkers have taken their reaction to the company's problems past their emotions, and into their minds. And because they have formulated some solutions, the nature of their meeting with the manager is creative. It's a brainstorming meeting. The manager enjoys these meetings because they stimulate his mind, too. Both parties leave the meeting feeling energized intellectually, and the manager looks forward to future meetings with the Thinkers.\n\nThe Whiners have left their reaction to their company's problems down at the emotional level. They express resentment, fear, and worry. The manager's problem in such a meeting is that he deals primarily with those emotions, so he finishes the meeting with his own sense of discouragement.\n\nWhen you are committed to self-motivation as a way of life, you will fall into the realm of the Thinker. Your thinking not only creates your motivation, but it creates your relationships, your family, and the organization you work for, as well, because they are all a part of you. You are more valuable to your organization with this orientation to thinking, and you're more valuable to yourself.\n\n## 87. Put more enjoyment in\n\nThere is a huge difference between pleasure and enjoyment. And when we're absolutely clear about the difference, we can grow much faster toward a focused and energized life. Mihaly Csikszentmihalyi best describes this difference in his various books on \"flow\"\u2014the psychological state that we get in when time disappears and we are thoroughly engaged in what we're doing.\n\nCsikszentmihalyi distinguishes what we do for pleasure (routine sex, eating, drinking, and so on) from what we do for enjoyment. Enjoyment is deeper. Enjoyment always involves the use of a skill and the facing of a challenge. So sailing, gardening, painting, bowling, golfing, cooking, and any such activity involving skills meeting a challenge constitute enjoyment.\n\nPeople who get clear on that difference begin to put more enjoyment into their lives. They reach the happy and fulfilled psychological state known as \"flow.\" Increasing their skills and seeking challenges to engage those skills are what lead to an enjoyable life.\n\nThere are many stories and accounts about the winners of lotteries who are jubilant when they win, but whose lives descend into a nightmare after acquiring that _unearned_ money. (No challenge, no skill.) The lottery looks like \"the answer\" to people because they associate money with pleasure. But the true enjoyment of money comes in part from the _earning_ of it, which involves skill and challenge.\n\nWatching television is usually done for pleasure. That's why so few people can remember (or make use of) any of the 30 hours of television they have watched in the past week. In watching television, there is no combination of skill and challenge.\n\nContrast that dull pleasure hangover we get from watching television with what happens when we spend the same amount of time preparing for a big Thanksgiving dinner for friends and relatives. In looking back, we remember quite vividly the entire Thanksgiving endeavor.\n\nDespite her run-ins with Wall Street and the law, one of the most intriguing people on our national scene has always been Martha Stewart. Throughout the 1990s, she personified mastery of the concept of small enjoyments. Her magazines and television programs celebrated cooking, gardening, and home entertainment skills. Her own contagious enthusiasm for the things she enjoyed made her, in my opinion, one of that decade's true heroes of optimism. If you're feeling as though you have forgotten how to _enjoy_ your own home, yard, or kitchen, you might buy one of her books and allow her optimism to inspire you. You can increase your own self-motivation by learning to be more aware of the profound difference between enjoyment and mere pleasure.\n\n## 88. Keep walking\n\nEver since I was a child, I had a recurring dream that I began each day facing a mattress. The more I pushed into this mattress before my day began, the more the indentation went in, and the more saved-up the sprung energy of the mattress got. The more the mattress was indented with my pushing at the start of the day, the higher it would spring up when I lay down on it to sleep at night. I would lie down on this mattress at night and see how high my dreams would send me. How high I flew would always depend on the indentations I gave the mattress during the day. The impressions I gave it. How impressive I was. The difference I made.\n\nAfter thinking about that dream, I decided to step up my walking. I decided that the recurring dream was the way my subconscious chose to tell me something vital. Something about the difference walking made. Something about oxygen being pushed into my system. Walking would be an action I could take while wide awake. Walking would drive more oxygen into my lungs. I would become more like the great football coach Amos Alonzo Stagg, who lived to be 103 years old. Amos Alonzo Stagg was asked how he lived to be so old (the average life expectancy during his lifetime was 65) and he said, \"I have, for the greater part of my life, indulged in running and other vigorous exercise that forced large amounts of oxygen into my body.\" I increased my walking just to see what would happen if my lungs became my mattress. I began to get happier. I began to enjoy life more. I began to be more _motivated_. As I walked, I wondered: What if the spirit lives as an aura around us? What if the spirit were a cloud of energy that exists around and outside our bodies ready at all times to be breathed in? Drawn right into the soul? What if when you breathed deeply, you pulled in your own spirit? And you received energy for action\u2014energy for an explosive take-down of one of your out-of-control problems. What if the solution to problems outside you was inside you?\n\nDeepak Chopra quotes an ancient anonymous Indian sage as identifying humanity's near-fatal superstition: \"You believe that you live in the universe when in reality the universe lives in you.\"\n\nMany modern scientific books are now referring to the human brain as the \"three-pound universe.\" When the body moves, so does the mind. So does that inner world. When you're walking, you are organizing your mind whether you want to be or not. Soon we realize that the mind and the body _are_ connected. When the Greeks said the secret to a happy life was a sound mind in a sound body, they were onto a powerful truth.\n\nI try to talk myself _out_ of that truth many times a week. _I'm too tired to exercise. I have an injury. I haven't had enough sleep. I should listen to my body! I would be short-changing my children of the important time they need with me if I selfishly went out for my long walk_. But I am always better off if I choose the walk. I am even better at relating to my children, because walking takes me to the soul. That's why I can't leave it out. I can't pretend it has nothing to do with this subject, because it's how I pull the truth to me. I pull the globe around toward me under my feet by walking. As the world turns, the lies leak out of my mind, into space. As the body becomes sound, so does the mind. It's true.\n\nThere is something about walking that combines opposites. Opposites: activity and relaxation. (This very paradox is what creates whole-brain thinking.) Opposites: out in the world and solitude. (Alone, but out there walking.) This combining of opposites activates the harmony I need between the right and left brain, between the adult and the child, between the higher self and the animal. Great solutions appear. Truth becomes beauty.\n\nYou have your own walking available to you, too. Yes, indeed. It might be dancing, swimming, running, racquetball, boxing, or aerobics, but it's all the same thing. It's all a way of moving the body around like a merry plaything and oxygenating the spirit in the process.\n\n## 89. Read more mysteries\n\nMy great friend and editor Kathy Eimers, to whom I first dedicated this book, and later married, has always been a devoted reader of mystery novels. When I first met her, I thought, _How curious that someone so intelligent would be reading mystery novels all the time_.\n\nIt was especially interesting to me because Kathy is one of the most literate people I've ever met, a quick thinker and a skilled professional writer and editor. Her editing of my books was the one thing, in my opinion, that gave them the sparkle that people said they enjoyed.\n\nIn my own ignorance, I assumed mystery novels were pretty light fare, and hardly a challenge to the human mind. But I changed my mind. Not only have I peeked into some of the mystery books she recommended (Agatha Christie and Colin Dexter), but I found out more about what good mystery does to the intellectual energy of the human mind.\n\nKathy has one of the most creative and energetic problem-solving minds I've ever encountered. I constantly marvel at her mental energy and perception because it stays clear and sharp\u2014all day, and long into the night. I would often find my own mental acuity descending the evolutionary ladder as night approached, while hers stayed alive and creative.\n\nThe person with the highest IQ ever measured\u2014Marilyn Vos Savant\u2014recommends mystery novels as brain builders. \"Not only is this exercise fun, but it's good for you,\" she says. \"I'm not talking about violent thrillers, or police procedural novels, but instead I'm directing you to those elegant, clue-filled, intelligent mysteries solved by drawing conclusions, not guns.\"\n\nVos Savant sees the reading of mysteries as something that leads to a stronger intelligence.\n\n\"If you try to keep one step ahead of the detective in an Agatha Christie or a Josephine Tey or a P.D. James mystery novel, it will sharpen your intuition,\" she writes in _Brain Building in Just 12 Weeks,_ \"The Sherlock Holmes stories by Arthur Conan Doyle never go out of favor, and rightly so. Holmes's methods are brain-builders brought to life.\"\n\nWhen people think of personal transformation, they don't normally think they can strengthen their own intelligence. IQ is something our cultural attitudes have always said we're born with and stuck with. But Vos Savant, whose IQ was measured at 230 (the average adult IQ is 100), believes strongly that the brain can be built as surely and as quickly as the muscles of the body.\n\nSo the next time you feel like curling up with a good mystery, don't feel guilty or nonproductive. It might be the most productive thing you've done all day.\n\n## 90. Think your way up\n\nIn some of my seminars I like to draw a picture of a ladder on the board and call it \"the ladder of selves.\" On the very bottom I write \"The Physical,\" in the middle I put \"The Emotional,\" and at the top I place \"The Mind.\" We can move up or down this ladder by the sheer force of will, although most people don't know they have that option.\n\nBy traveling up the ladder, past the physical, through the emotional, and into your mind, you have the opportunity to be creative and thoughtful. You can see possibilities. Many of us, however, never get past the emotional section of the ladder. When we're stuck there, we begin thinking with our feelings instead of thinking with our minds. If you hurt my feelings, and I'm angry and resentful, I might give you a long and eloquent speech about what's wrong with you and how you operate. But, because I'm thinking with my feelings instead of my mind, I'm destroying something with my speech instead of creating an understanding. People do this without knowing it. They let their emotions speak for them, instead of their thoughts. So what you hear is fear, anger, sadness, or other emotions put to words, but never creating anything.\n\nIf you can picture this ladder inside of you, and start to notice that you are letting your feelings do your thinking and speaking, you can move up. You can get creative and really think and _then_ speak. As Emmet Fox says, \"Love is always creative and fear is always destructive.\"\n\nGo ahead and feel your feelings. But when it's time to talk, let your mind into the conversation. Your mind is what motivates you to your highest performance, not your feelings.\n\n## 91. Exploit your weakness\n\nMake a list of your strengths and your weaknesses on separate pieces of paper. Place the list of strengths somewhere where you'll see it again, because it will always pick you up. Now look at your list of weaknesses and study them for a while. Stay with them until you feel no shame or guilt about them. Allow them to become interesting characteristics, instead of negative traits. Ask yourself how each characteristic could be useful to you. That's not what we usually ask about our weaknesses, but that's my whole point.\n\nWhen I was a boy, I remember watching a remarkable tap dancer by the name of \"Peg Leg Bates\" on the Ed Sullivan show. Bates had lost his leg early in life, a circumstance that would lead most people to give up any dreams of becoming a professional dancer. But to Bates, losing a leg was not a weakness for long. He made it his strength. He put a tap at the bottom of his peg leg and developed an amazing syncopated tap-dancing style. Obviously, he stood apart from other dancers in auditions, and it wasn't long before his weakness became his strength.\n\nMaster fundraiser Michael Bassoff has dazzled the development world by turning unappreciated staff members into great fundraisers. He, too, likes people's weaknesses, because he knows that they can be turned into strengths. If there is a \"shy\" secretary in the development office he's working with, he turns that person into the staff's \"best listener.\" Soon donors can't wait to talk to that person because she listens so well and makes people feel so important.\n\nOne of my weaknesses early in life was my difficulty in talking to people. I had no confidence in my ability to speak and converse, so I got in the habit of writing people letters and notes. After a while, I got so practiced with it that I turned it into a strength. My letter writing and thank-you notes have created many relationships for me that would not have been created if I'd just focused on my shyness as a weakness.\n\nI have four children, but I didn't begin having children until I was 35 years old. For a long time I saw myself as being \"older than normal\" to be a father. I worried about it. I wondered if my son or daughters would be uncomfortable with a father so old. And then I realized that this didn't have to be a weakness. I thought about who I was when I was 25, and what a difficult time I would have had being a good father back then. Soon I took this \"weakness\" to be a great strength. Then one day while watching _The Little Mermaid_ with my kids, I saw myself as the father in that movie\u2014vigorous, strong, and wise, with flowing white hair. It was the perfect image. I now see my age as a major strength in raising my kids. The only \"weakness\" was in the way I was looking at it.\n\nThere isn't anything on your weakness list that can't be a strength for you if you think about it long enough. The problem is, our weaknesses embarrass us. But embarrassment is not real thinking. Once we really start _thinking_ about our weaknesses they can become strengths, and creative possibilities emerge.\n\n## 92. Try becoming the problem\n\nWhatever type of problem you are facing, the most self-motivational exercise I know of is to immediately say to yourself, \" _I_ am the problem.\" Once you see yourself as the problem, you can see yourself as the solution. This insight was dramatically described by James Belasco in _Flight of the Buffalo_. \"This is the insight I realized early and return to often,\" he wrote, \"In most situations, I am the problem. My mentalities, my pictures, my expectations, form the biggest obstacle to my success.\"\n\nBy seeing ourselves as victims of our problems, we lose the power to solve them. We shut down creativity when we declare the source of the trouble to be outside of us. However, once we say, \"I am the problem,\" there is great power that shifts from the outside to the inside. Now we can become the solution.\n\nYou can use this process the same way a detective uses a premise to clarify a crime scene. If the detective says, \"What if there were two murderers, not one?\" she can then think in a way that reveals new possibilities. She doesn't have to prove that there were two murderers in order to think the problem through as if there were. The same is true when you become willing to see yourself as the problem. It is simply a way to think.\n\nIn _The Six Pillars of Self-Esteem,_ Nathaniel Branden writes, \"To feel competent to live and worthy of happiness, I need to experience a sense of control over my existence. This requires that I be willing to take responsibility for my actions and the attainment of my goals. This means that I take responsibility for my life and my well-being.\"\n\nBefore I had realized the full power of a self-motivated life, I spent a lot of years pointing fingers. If I didn't have enough money, it was somebody else's fault. Even my perceived personality flaws were somebody else's fault. \"I was never taught that!\" I would shout in exasperation. \"No one showed me early in life how to be self-sufficient!\" was a complaint I voiced often. But I was avoiding a basic truth: I was the problem. The reason I fought so hard to avoid that truth was that I never realized it contained good news. I thought it looked entirely shameful and negative. But once I discovered that accepting responsibility for the problem also gave me new power for solving it, I became free.\n\n## 93. Enlarge your objective\n\nHere is another self-motivator that also must be used as an intellectual tool only. Take a certain goal of yours and double it. Or triple it. Or multiply it by 10. And then ask yourself, quite seriously, what you would have to do to achieve that new goal.\n\nI used this game recently with a friend who holds a position in sales. He came to see me because he was selling $100,000 worth of product each month, the most on his team, and wanted to somehow get to $140,000.\n\nI asked him to tell me what it would take for him to sell $200,000 worth of equipment each month. \"$200,000!\" he shouted. \"That's impossible. I'm leading the team already with $100,000, and nobody thought that could be done.\"\n\n\"What would you have to do?\" I persisted.\n\n\"No,\" he said. \"You don't understand. I want to hit $140,000 a month, and even that is so hard I don't know how I'll do it.\"\n\nI finally told him the theory behind this game. \"If you seriously look at an outrageous goal, such as $200,000, it will open things up for you creatively that wouldn't have opened up if you stayed looking at $140,000.\" He nodded slowly and reluctantly agreed to play along for a while.\n\n\"Okay,\" he said. \"But remember, we're talking about something that's impossible.\"\n\n\"Fine,\" I said. \"But if your life depended on hitting $200,000 next month, what exactly would you do?\"\n\nHe laughed and then started listing things as I wrote them down on a flip pad. After he got through the ridiculous ideas, like stealing other peoples' accounts and cooking the books, he began to think of more ideas. At first it was hard.\n\n\"I'd have to be in two places at once,\" he said. \"I'd have to make twice as many presentations as I'm making. I'd have to present to two clients at once!\"\n\nThen it hit him. All of a sudden he got the idea that he might be able to stage a large presentation of his product with a number of clients in the room at one time. \"I could rent a room at a hotel and have 20 people in for coffee and donuts, and I could make a big deal out of it,\" he said.\n\nA number of other ideas came to him\u2014ways to combine his cold-calling with his travel time, ways to utilize e-mail as a sales tool, how to use the administrative staff better, and ways to expand his contracts so that they would cover longer periods of time for a higher original fee, but at a lower overall rate. Idea after idea came to him while I wrote furiously on the pad.\n\nAll of the ideas were a result of his thinking big\u2014\n\n\"How would I sell $200,000 if I absolutely had to?\"\n\nHe surpassed his goal of $140,000 _the very next month!_\n\nI've often used this method with myself. If I have a goal of signing two seminar contracts in the next three weeks, I'll often get out a pad of paper and ask, \"How would I get 10 contracts signed in three weeks?\" Inflating my goal puts me at a different level of thinking, and because I'm solving the problem of 10, I always get at least two.\n\nIf you want to really get some fresh motivational ideas, try expanding your goal. Blow it up until it scares you. Then proceed in your thinking as if it's a _must_ that you achieve it. Remember that this is just a self-contained game, not a promise to anyone else. But it's a game that's fun to play because it works.\n\n## 94. Give yourself flying lessons\n\nWe need heroes in our lives. They are not a sign of weakness; they are a source of strength. \"Without heroes,\" said Bernard Malamud, \"we are all plain people and we don't know how far we can go.\"\n\nHeroes show us what's possible for a human being to accomplish. Therefore, heroes are very useful to anyone who is in the process of finally understanding self-motivation. But unless we consciously select our heroes in order to use them as inspiration, we simply end up _envying_ great people instead of emulating them.\n\nWhen used properly, a hero can be an enriching source of energy and inspiration. You don't have to have just one hero, either. Choose a number of them. Put their pictures up. Become an expert on their lives. Collect books about them.\n\nMy youngest sister, Cindy, as a shy little girl, always admired Amelia Earhart. After Cindy reached her 30s, she revealed to me that she had been taking flying lessons. I was stunned! A few weeks after that, the family went out to a little airport outside of town to watch her fly her first solo. \"I was so scared,\" said Cindy, \"that my mouth and throat went completely dry.\"\n\nFlying has nothing to do with what Cindy does for a living\u2014she just took lessons and learned to pilot a plane because of the impression that her hero, Amelia Earhart, made on her as a little girl.\n\n\"We grow into that which we admire,\" said Emmet Fox.\n\nBefore he became a famous author, Napoleon Hill was struggling as a writer and speaker. He had a friend whose restaurant business was not doing well and Hill offered to give free motivational speeches at the restaurant one night a week to help his friend increase his business. The speeches helped his friend a little, but they helped Hill a lot. He began to gain a large following.\n\nWhen I read about that part of Hill's life, it gave me an idea. At the time I wanted to be a full-time speaker and I didn't know where to begin. I'd done a few seminars and talks here and there, but there was no pattern or purposeful direction to it. I decided to emulate Hill. I began putting on a free, open-to-the-public workshop every Thursday night at the company where I was working as a marketing director.\n\nAt first, the workshops were not well attended. I had to spend part of the week begging people to come. Once the audience was two people! But week by week the workshop's reputation grew, and my own experience grew along with it. Soon we had large audiences waiting to get in on Thursday nights, and I credit that little free workshop with putting me into full-time public speaking.\n\nWas it an original idea? No, I stole it. I copied a hero of mine. But our awareness of the choice involving heroes is vital for self-creation. We can envy them or we can emulate them. The best _use_ of heroes is not to just be in awe of them, but to learn something from them. To let their lives inspire us. They are only people like we are. What distinguishes them from us is the great levels they've reached in self-motivation. To passively adore them is to insult our own potential. Instead of looking _up_ to our heroes, it is much more beneficial to look _into_ them.\n\n## 95. Hold your vision accountable\n\n\"It's not what a vision _is_ ,\" says Robert Fritz \"it's what a vision _does_.\"\n\nWhat does your vision do? Does it give you energy? Does it make you smile? Does it get you up in the morning? When you're tired, does it take you that extra mile? A vision should be judged by these criteria, the criteria of power and effectiveness. What does it _do_?\n\nRobert Fritz is widely quoted in Peter Senge's business masterpiece, _The Fifth Discipline_. Fritz is a former musician who has taken the basic principles of creativity in music composition and applied them to creating a successful professional life. Life gets good, he argues, when we get clear on what we want to create.\n\nMost people spend most of their waking hours trying to make problems go away. This lifelong crusade to solve one's problems is a negative and reactive existence. It sells us short and leaves us at the end of life (or at the end of the day) with, at best, the double-negative feeling of \"fewer problems\"!\n\nFritz points out in _The Path of Least Resistance_ :\n\nThere is a profound difference between problem solving and creating. Problem solving is taking action to have something go away\u2014the problem. Creating is taking action to have something come into being\u2014the creation. Most of us have been raised in a tradition of problem-solving and have little real exposure to the creative process.\n\nStep one in the creative process is having a vision of what you want to create. Without this vision, there is no way to create. Without this vision, you are only problem-eliminating, which is a double negative. It's impossible to feel positive about a life based on a double negative.\n\nSo the way to alter your thinking is to _notice_ when you're drifting into, \"What do I want to get rid of?\" and mentally replace that thinking with, \"What do I want to bring into being?\"\n\nWhen Fritz says that we have been \"raised in a tradition\" of problem solving, he is almost understating it. We are programmed and wired to think that way every day. Notice the thinking of people as they approach a challenge, even a challenge as small as an upcoming meeting with other people:\n\n\"Here's what I hope doesn't happen...\" one will say. \"Well, here's how you can avoid that...\" someone else will helpfully say. \"The only problem we have is this...\" a third person will say, attempting to make the meeting seem less frightening.\n\nNotice that nowhere was there the question, \"What would we like to _bring into being_ as a result of this meeting?\" Whether the situation is as small as a meeting or as large as your whole life, the most useful question you can ask yourself is, \"What do I want to bring into being?\" It's a beautiful question, because it makes no reference to problems or obstacles. It implies pure creativity. It puts you back on the positive side of life.\n\nMy friend Steve Hardison made an observation about self-motivation that I have always remembered and agreed with. \"It's just one thought,\" he said. \"Motivational teachers repeat it many different ways, but it's just one thought: It's a binary system. Are you on or are you off?\"\n\nAre you positive or are you negative? Are you creating or are you reacting? Are you on or are you off? Are you life or are you death? Are you day or are you night? Are you in or are you out? And there's nothing more motivational to flip your binary switch to \"on\" than a clear vision of what it is that you really want. What do you want to bring into being? It doesn't matter what that vision is or how often it changes. It only matters what that vision _does_.\n\nIf your vision isn't getting you up in the morning, then make up another one. Keep at it until you develop a vision that's so colorful and clear that it puts you in action just to think about it.\n\n## 96. Build your power base\n\nKnowledge is power. What you know is your power base\u2014it's the battery you run on. You need to charge it constantly and consciously. Who do you want to be in charge of what you know? News directors? Radio disc jockeys? The office gossip? Tabloid newspaper editors? A pessimistic family member? Unless we _consciously_ decide to build our own knowledge base, with a sense of direction to it, then we will be programmed, totally, by random input. Feeling miserable and alienated from life is _caused_ by not being in control of what we know.\n\n\"Misery and alienation are not laid upon us by fate,\" wrote Colin Wilson. \"They are due to the failure of the ego to accept its role as the controller of consciousness. All our experiences of happiness and intensity force the same conviction upon us, for they involve a sense of mastery.\" You can be the master of your own fate. You can make choices all day long about what you are going to learn and what you are not going to learn.\n\n\"What are you reading over there?\" someone may ask you. \"Oh, it's just something I found in the trash,\" you might say. And it might seem harmless enough to read something you found in the wastebasket because there was nothing else nearby, _but whole lives are shaped that way_. The computer term \"GIGO\"\u2014garbage in, garbage out\u2014is even truer for the human biocomputer than it is for mechanical computers.\n\nTake control of what you know. The more you know about what motivates you, the easier it is to motivate yourself. The more you know about the human brain, the less trouble you have operating it. Knowledge is power. Respect yours and build on it.\n\n## 97. Connect truth to beauty\n\nI hate reading motivational material that thunders at me about the importance of integrity and honesty for their own sake. Somehow, that always seems to turn me off, because the writers come off like angry preachers and teachers\u2014hardly inspiring.\n\nI'm always more inspired by things that are made to look interesting and fun. I'm always taken in by a promise of life being more beautiful and rarely taken in by a promise of a life being more righteous and proper. To me, the best case to make for honesty is how beautiful it is, how clean and clear it makes the journey from current reality to the dream.\n\nWhen people know _exactly_ where they are, they can go somewhere from there. But being \"lost\" is a function of dishonesty. And when we're lost, or dishonest, anywhere we go from there is wrong. When we start with a false reading, there's no direction home.\n\nTruth, on the other hand, is clear, complete, and compellingly vivid. It is solid and strong, so it can hold us steady as we climb. \"Truth,\" said poet John Keats, \"is beauty.\" The more honest we are with others and ourselves about current reality, the more energy and focus we gather. We don't have to keep track of what we told one person or what we told another.\n\nOne of the best and most positive explanations of the beauty of personal integrity was expressed by Nathaniel Branden in _The Six Pillars of Self-Esteem_. Branden, unlike most writers on the subject, sees truth and integrity as a positive part of the process of self-esteem. His point is not that we owe it to other people's sense of morality to be honest, but that we owe it to ourselves.\n\n\"One of the great self-deceptions,\" said Branden, \"is to tell oneself, 'Only I will know.' Only I will know that I am a liar; only I will know that I deal unethically with people who trust me; only I will know that I have no intention of honoring my promise. The implication is that _my judgment is unimportant and that only the judgment of others counts_.\"\n\nBranden's writing on personal integrity is inspiring because it's directed at creating a happier and stronger self, not at a universal appeal for morality.\n\nWe describe a work of art that is sloppy and unfinished as \"a mess.\" The problem with lying, or lying by omission, is that it leaves everything so incomplete\u2014in a mess. Truth always completes the picture\u2014any picture. And when a picture is complete, whole, and integrated, we see it as \"beautiful.\"\n\nI'll even hear about people\u2014usually people whom you can't believe about anything\u2014described as \"a mess.\" And conversely, a person who you can always count on to be honest with you is often referred to as a \"beautiful\" person. Truth and beauty become impossible to separate. Truth leads you to a more confident level in your relationships with others and with yourself. It diminishes fear and increases your sense of personal mastery. Lies and half-truths will always weigh you down, whereas truth will clear up your thinking and give you the energy and clarity needed for self-motivation.\n\n## 98. Read yourself a story\n\nAbraham Lincoln used to drive his law partners to distraction. Every morning he would come into his office and read the daily newspaper _aloud_ to himself. They would hear him in the next room reading in a booming voice. Why did Lincoln do his morning reading aloud? He had discovered that he remembered and retained _twice_ as much when he read aloud than when he read silently. And what he did remember, he remembered for a much longer period of time. Perhaps it was because Lincoln was employing a second sense, the sense of hearing, and a second activity, the activity of speaking, which made his readings so memorable to him. Any time you have an opportunity to read something that is important to you, try reading it aloud and see if you don't make twice the impression on yourself. When you discover something you want to remember, and draw upon in the future, read it aloud.\n\nSteve Hardison, one of the most successful business consultants I have ever known, credits one origin of his success to a time when he was a struggling young man without money or a clue about where he wanted to go. Then one day he came across Napoleon Hill's enormous book, _Law of Success,_ and read the entire volume _aloud_.\n\nMy favorite piece of writing to read aloud is Chapter 16 of Og Mandino's _The Greatest Salesman in the World_. Here's a part of it, which you may now read silently to yourself. However, if you want a real shot of adrenaline to your spirit, I recommend you mark this page and when you're alone, read it aloud like Lincoln:\n\nI will act now. I will act now. I will act now. Henceforth, I will repeat these words again and again, each hour, each day, every day, until the words become a habit as my breathing and the actions which follow become as instinctive as the blinking of my eyelids. With these words I can condition my mind to perform every act necessary for my success. With these words I can condition my mind to meet every challenge.\n\n## 99. Laugh for no reason\n\nBecome a performer. Be an actor and a singer. Act like you already feel like you want to feel. Don't wait until the feeling motivates you. It could be a long wait. Most of us believe that an emotion, such as happiness, comes first. Then we do whatever we do, in reaction to that particular emotion. Not so. The emotion arises simultaneously with the doing of the act. So if you want to be enthusiastic, you can get there by acting as if you were already enthusiastic. Sometimes it takes a minute. Sometimes it skips a beat. But it always works if you stay with it, no matter how ridiculous you feel doing it.\n\n_Feel_ ridiculous. If you want to be happy, find the happiest song you know and sing it. It works. Not always in the first few moments, but if you keep at it, it works. Just fake it until you make it. Soon your happy singing will show you how much control you do have over your own emotions.\n\nZen monks do a laughing meditation in which they all gather in a circle and get ready to laugh. At the stroke of a certain hour the teacher hits a gong, and all the monks begin to laugh. They have to laugh, whether or not they feel like it. But after a few moments the laughter becomes contagious. Soon all the monks are laughing genuinely and heartily. Children do this, too. They start giggling for no reason (often at the dinner table or some other forbidden setting, and the giggling itself makes them laugh). The truth is this: laughter itself can make you laugh. The secret of happiness is hidden inside that last sentence. But adults aren't always comfortable with this. Adults want kids to have _reasons_ for laughing. As I used to drive my children long distances to visit relatives, I'd get most irritated when they began laughing and giggling in the back seat without reason. I developed a backstroke swing to curb the laughter. \"Why are you laughing?\" I would shout. \"You have no _reason_ to be laughing! This is a dangerous highway and I'm trying to drive here!\"\n\nBut adults like me might want to get back that appreciation for joyful spontaneity. We might want to confront the question, \"What is the one thing that most makes me feel like singing?\" And then know the answer: \"Singing.\" What most gets you in the mood to dance? Dancing. The next time you ask someone to dance, and they say, \"I don't feel like dancing,\" you might reply, \"That's because you're not dancing.\"\n\n## 100. Walk with love and death\n\n\"I am a coward.\"\n\nThat was how the book began. It was a novel I was reading not long after I had graduated from high school, and those first words staggered me. I remember staring at those words, unable to continue reading, I was so stunned. Never had a book connected with me so quickly. I was a coward, too. I just never admitted it so openly as did the author of _A Walk with Love and Death_. The author was Hans Koning, and the book was a medieval love story later made into a movie by John Huston, but none of that mattered. What mattered was that there was another coward on the planet other than me. Even if he was fictional, the words were real enough for me.\n\nMy self-image at the time I read that book was based on my fears and nothing else. In my mind, I was truly a coward. And if someone were to tell me I'd done something brave, I'd think they were wrong somehow. Or that they didn't know how easy that thing was.\n\nWhere did this self-image come from? I don't blame my parents, because I believe we create our own pictures of ourselves, and I had a choice whether to stick with this self-image or not. Although I don't blame my parents, I can trace where I got the _idea_ of my being a coward to their encouragement.\n\nMy mother, too, was afraid of everything. She lived to the age of 66 without ever having made a left turn in traffic; she was so afraid of oncoming traffic. (She always knew how to make a looping series of right turns to get where she was going.) She consoled me and told me that I was just like her. A coward, I thought. She was very loving and empathetic about it, but my self-image became unshakable. However, my mother said she'd try to be there to help me do the many things she knew I wouldn't be able to do.\n\nI met my father when I was two and a half years old. He was a war hero, home from World War II, and it is reported that when he walked into our home and saw me for the first time, I looked up at his imposing uniformed figure and said, \"Who is that?\"\n\n\"John Wayne,\" my mother should have said.\n\nMy father was afraid of nothing. He was a decorated soldier, a star athlete, a tough and successful businessman, and the list goes on. But he soon knew one thing about his little boy\u2014no guts. And it was distressing to him.\n\nSo, both parents and the child himself were all in agreement about it. The father was upset about it, the mother understood, and the boy was just scared. That is possibly why, as I grew older, I discovered \"false courage.\"\n\nI discovered\u2014through use of an intoxicating substance\u2014that I could be who I wanted to be. But soon the marvelous discovery turned to addiction, and my life revolved around my dependency on it. They were wild times, but as anyone will tell you who's been through it, there was no growth or fulfillment during those years. They soon became an intolerable nightmare.\n\nFortunately, I recovered. It has been more than 20 years since I've had to resort to chemically based courage. During that recovery period, which was often difficult, I came to learn a prayer that was popular among fellow recovering people. They called it \"The Serenity Prayer,\" and you've probably heard it: _God grant me the serenity to accept the things I cannot change, the courage to change the things I can, and the wisdom to know the difference_.\n\nI think it is called the serenity prayer because that's what everyone wants from the prayer\u2014serenity. Abruptly ending a long period of substance abuse can leave you far short of serene. Although with each passing day it gets better and better, that prayer was something to hang onto.\n\nBut after being clean and sober started to work for me, I knew something was still missing\u2014I knew I needed more than serenity. My deep-seated self-image of being a coward had not gone away, and so I turned my attention to the second line in the prayer, _the courage to change the things I can_. In my mind, it was no longer the serenity prayer\u2014it had become the courage prayer. Courage was still what I lacked, and that feeling of personal cowardice was still my entire self-image. It shaped my whole personality.\n\nWhen my friend Mike Killebrew gave me Napoleon Hill's _The Master Key to Riches_ , the answer to my courage prayer began to come to me. If I didn't have the courage inside of me, I would create it. And at that moment, the process of self-motivation began in earnest.\n\nI could cite you many examples of the fears I had, but to illustrate how I overcame them, I'll use an example I referred to earlier\u2014my fear of public speaking. I've since learned that the fear of public speaking is not unique to me. In fact, it's considered the number-one fear among our population today, even greater than the fear of death. To me, though, it was a painful manifestation of the overall deep fear that constituted my entire personality. I laughed knowingly once when Woody Allen said that he was \"afraid of the dark and feared the light of day.\" That was me.\n\nWhen I finally made myself join an acting class to face my fear of speaking, I learned to my horror that I was the only non-actor in the class. In our first session, led by the hugely talented actress and coach Judy Rollings, I listened as everyone in the class talked about all the recent stage productions they had been in.\n\nJudy gave us each a long monologue to learn and recite in the next session. Mine was from _Spoon River Anthology_ and my character was a judge who had been mocked as a young man, but rose to judge those in the community who used to make fun of him. It was a challenging piece, and I was terrified.\n\nI knew I had to do something harder than the recital to prepare for the recital, so I set out to do it. I memorized my part and began to perform it in front of people. I asked whoever would listen to sit down and watch me recite this piece. I did it in front of my actress friend Judy LeBeau, who had gotten me into the class. I recorded it and sent it to songwriter and comedian Fred Knipe. I did it in front of my friend Kathy. I made my children sit quietly and watch me do it over and over. Each time, I was scared, my heart was pounding, and I hyperventilated. But each time it got easier and better. Finally the day of the class arrived. I took the day off from work to rehearse this little three-minute piece all day. When class time arrived, I was extremely nervous, but not deeply panicked.\n\nJudy Rollings asked for volunteers to perform their monologues, and as each experienced actor got up to do theirs, I gained confidence. I could see that they, too, were very nervous. They were acting in front of peers, which is sometimes harder than before a normal audience. They were blowing their lines and, in embarrassment, asking to start over. Some of their voices were a little shaky. I was encouraged. Finally, with just one or two of us left to go, I volunteered and walked slowly to the front of the room.\n\nWhat happened then is something I'll never forget. As I went to the front of the room, just before I turned around to face the teacher and class, a voice in my mind spoke to me, and it said only one word: _Showtime_. With a surprising surge of energy, I delivered my piece. My voice soared up and hit the dramatic points and dropped down to emphasize the subtle lines and the parts that I gave a funny interpretation to were drawing huge laughs from the class. When I was finished, I looked back up and saw that the whole class had burst out clapping\u2014something Judy had told them not to do for anybody.\n\nWhen I drove home that night, I was in heaven. I kept reciting my monologue out loud, reveling in the memory of their laughter and clapping. The thing I thought I feared most in life was somehow mastered. And I repeated to myself the principle I had used to make it happen\u2014the more I sweat in peacetime, the less I bleed in war.\n\nI often look back on who I was when I first encountered the words, \"I am a coward,\" in _A Walk With Love and Death_. And I realize that today I have something that I didn't have back then, the _knowledge that courage can be created_. I still have fears, but I no longer _am_ fear. I no longer think of myself as a coward. And when people compliment me on something I've done that they think was courageous, I don't dismiss them as being crazy or stupid.\n\nThere is a way I use to motivate myself to overcome any fear that's in my way today. It's a way I've never told anyone about until now, because it has a strange name. I call it \"walk with love and death.\" When I need to get through something, face something, or create a courageous action plan\u2014I take long walks. When I walk long and far enough, a solution _always_ appears. I eventually get oriented to the most creative course of action.\n\n\"When you walk,\" writes Dr. Andrew Weil in _Spontaneous Healing,_ \"the movement of your limbs is cross-patterned: the right leg and left arm move forward at the same time, then the left leg and right arm. This type of movement generates electrical activity in the brain that has a harmonizing influence on the central nervous system\u2014a special benefit of walking that you do not necessarily get from other kinds of exercise.\" I call it \"a walk with love\" because love and fear are opposites. (Most people think love and hate are opposites, but they are not.) The ultimate creativity occurs from a spirit of love and, as Emmet Fox says, \"Love is always creative, and fear is always destructive.\"\n\nI call it a \"walk with death,\" because it is only the acceptance and awareness of my own death that gives my life the clarity that it needs to be exciting.\n\nMy walks often last a long time. Somehow, whatever challenge I'm facing appears to me from many different angles as I'm walking. I know that one of the real values is that while walking, I'm truly alone with myself\u2014there are no phones to answer or people to talk to. I create so little of that kind of time in life, that it's always surprising how beneficial it is.\n\nTake your own challenges out for a walk. Feel your self-motivation growing inside you, as the electricity in your brain starts to harmonize your central nervous system. You'll soon know for a fact that you have what it takes. You won't have to pray for the courage to change the things you can\u2014you will already have it.\n\nI discovered something remarkable quite by accident one night as I was conducting a workshop on goal achievement. I discovered the power of negative thinking. As the people in the workshop struggled to list their goals on a piece of paper, I ran out of patience.\n\n\"How will you get what you want if you don't know what it is?\" I asked the room, half of which still had empty sheets of paper and empty facial expressions.\n\n\"Okay,\" I said, \"Let's put these goals away. I want to try something different. Take out a new sheet of paper and do this. Write down what you _don't want_ in your life. List every major problem and source of discomfort you have. All your worries. All the negative things you can think of, even if they haven't come into reality yet. Even if they are just things you don't want to happen in the future. Take your time and be thorough.\"\n\nWhat I saw happen next startled me. The entire room's energy level picked up, and everyone in the workshop was writing and writing and writing. It wasn't long before some people asked if they could use a second page.\n\nSomething strange and electric was filling the air as people aired their fears and grievances. Pages were flooded with ink, and hands and fingers had to be shaken out so people wouldn't cramp up from writing so much. When I called an end to the exercise, the room was buzzing.\n\nI had obviously let something loose that wasn't there before. At that moment I got my first true look at the power of the negative. Actually, I had seen it before. When I took the time to look back over my life, I realized that saying _no_ was always a stronger stand to take than saying _yes_. Saying _no_ is drawing a line in the sand. It is putting your foot down. It is passionate and powerful. Compared to saying _no_ , saying _yes_ is wobbly and wishy-washy. I said _yes_ to a thousand drinks of alcohol in my life. But it wasn't until one hung-over suicidal morning when I said _no_ that my life got completely turned around.\n\nSaying _no_ is powerful, because it comes from the deepest part of the soul. There are some things we just won't tolerate. When we fully understand the power of those _no's_ deep inside of us, we can use them to motivate ourselves like never before.\n\nWhen the people filled their papers up with what they _didn't_ want, we got busy converting problems into goals. You don't want to go bankrupt? Then let's get a prosperity plan going! You don't want to weigh as much as your two best friends combined? Then let's get a nutrition and exercise program going! Any _no_ can be converted to a powerful _yes_.\n\nSo if you're stuck without any truly motivating goals, dreams, or commitments, then go negative first. Figure out what you absolutely don't want\u2014what you absolutely fear and dread and refuse to let in to your life\u2014then convert it to its opposite, positive form and see what happens. You'll be more motivated than you ever dreamed you could be.\n\nI have used this in one-on-one meetings with people who wouldn't open up and tell me what they wanted. I simply asked them to tell me what they _didn't_ want to have happen and we were off to the races. Once you know what that is, you can convert the conversation to exciting plans and objectives. This explains why so many successful people had difficult upbringings, sometimes living in the harshest poverty. They connected very early to what they didn't want. The rest was clear sailing.\n\nThe next time you lack passion when thinking of what you want, try turning it around. Ask yourself what you absolutely don't want, and then feel the energy building in you to overcome that problem. That energy you're feeling is the deepest and most primal form of motivation.\n\n## 101. Just roar!\n\nWinston Churchill once said, \"I may not be the lion, but it was left to me to give the lion's roar.\"\n\nWhen people hire me to coach them, one of the first things I am happy to notice is that they usually suffer from the same myths I suffered from. That allows me to relate quickly and get them started on the path to achievement. The primary myth is the one Winston Churchill's quote relates to. The idea that I have to figure out _if I have what it takes_ , given my up-to-now identity, to do what I really want to do.\n\nThe answer is no. Just start doing it. If you need a lion's roar, you don't have to figure out if you are a lion first. Just roar. Way too much precious energy is lost in trying to figure out our identity, and why we are the way we are. We miss this fact of life: action doesn't care who you are.\n\n## 102. Experiment with happiness\n\nWe were born to be happy.\n\nThe spiritual leader Poonjaji once said, \"Happiness is permanent. It is always there. What comes and goes is unhappiness. If you identify with what comes and goes, you will be unhappy. If you identify with what is permanent and always there, you are happiness itself.\"\n\nWe add worries, fears, beliefs, and dark visions born of passivity to our happiness. We create a burden. If we want to find what's really at the heart of things (happiness), we've only to laugh, dance or sing.... OR: work at one single purposeful piece of work long enough (without distraction) until the light shines.\n\nPeople believe there is some generalized state of mind called happiness that they must find a way to achieve. So they begin arranging outside circumstances to match up with a vision of happiness they might have. They get a spouse and a house. A dog and a baby. A job and a car. They keep adding circumstance on circumstance. Soon it's a boat and a second home. Why a boat and a second home? Because the car and the first home didn't do it. It didn't make them feel generalized, consistent happiness.\n\nThen one day a storm hits the town and the house across the street has a giant tree fall on it. Children are trapped inside, and you race across the street, crawl into the wreckage, and pull a child to safety. As you sit on the lawn receiving hugs from the mother and father of the child, you are happier than you have ever been. Why can't you use that memory to find the true nature of happiness? Why are you, two weeks later, looking for a new house, a new spouse, a new car, or a new counter top?\n\nI once wondered what work I should do. I had been in the world of advertising, writing ads and commercials, but I lost my job when the company went under. _What is my true calling?_ I wondered. _What is my real work?_ I decided to take a long walk and think about my past. When was I happiest? Most excited? Most lit up? When could I say that I was really _on fire_?\n\nOne night during my recovery from addiction, the answer came. I was at a large meeting hall and they asked me to be the speaker. _Who, me?_ I wasn't at all prepared. And I also had the flu. So I felt awful. I also had a huge fear of public speaking. A bad mix of circumstances. I said no.\n\nThey pressed on, and they said _come on!_ They said there were newcomers to recovery who were scared too, but they were scared that they couldn't live without their drugs and alcohol. That they probably were not going to have any life at all. They asked what exactly I was scared of. Was it that I'd look foolish by not being a good speaker? They told me to stop thinking of myself and to think of the others. That persuaded me.\n\nSo I pushed past my fear and my fever and I walked up the steps to get to the podium to face that huge hall of people. I started hesitantly. Then I remembered what my sponsor had told me just before I got up there. \"Just tell the truth,\" he said. \"Just tell everyone what it was like, what happened, and what it's like now.\"\n\nSo I started telling the crowd what it was like. All the tragic and comic death-defying dysfunctions I participated in while drunk. The blackouts. The time in jail. Standing before the judge. The lies and sickening betrayals. And for some odd reason the people in the hall were roaring with laughter. There was something about my story, my sad life, that made them laugh and laugh. I looked out over the audience and saw the happiest faces I had ever seen. What was happening? I was just telling the whole truth. And maybe my fever was helping me, was feeding me with a weird kind of manic energy.\n\nThen I told what happened. The miracle of recovery. The total eclipse of the heart. The turnaround of a life. Clean and sober and free with three lovely daughters and a second chance. I told how I did it. How the steps of recovery were taken.\n\nAfter that talk, I was given a standing ovation and surrounded by people for an hour afterward. They said I explained how to take the steps better than anyone they'd heard in a long time. What was the formula? Mix fever with fear and add a huge shame-based desire to help the newcomer to recovery? No. Steps. Steps were the formula. The steps I walked up to the podium. The three steps of a connective talk that were told to me ahead of time by my sponsor: 1) What it was like, 2) what happened, and 3) what it's like now. Steps.\n\nSo, thinking about what work and what steps make me happy, I thought about that night and I decided right then and there that I would be a teacher of recovery. But not just from alcohol and drugs. But from all the other less lethal addictions, such as sadness, regret, procrastination, lack of wealth, lack of clients, career problems, and inadequate goal achievement. And so it happened that knowing where happiness comes from by allowing my own life to be an experiment, by testing it rather than trusting all the books, I found my work. You have to have fire to start a fire. Only a match, only a flame, can start a motivational fire.\n\n## 103. Catch life by the handle\n\nWhen I created an audio series called _MINDSHIFT_ I wanted to say that sometimes the shift of one's mind can be so simple that it can ride in for you while you're closing your eyes and taking a breath. All of a sudden you're not stuck to all these things you're believing about the world and the negative imaginary future that people dwell on all day. All day long, people are obsessed with their negative imaginary futures.\n\nAnd one of the many ways to shift back into the creative mode is to take a walk by the ocean. And the ocean can be anything. It can be a beautiful house plant in my apartment. It can be a wonderful picture on the wall. Take that in fully, completely, and let it ride in on some deep breaths and close your eyes and all those thoughts that held you captive can go away. Whether you're picturing it or you're at the actual ocean taking a walk and just taking in the view completely, really being present with it and having it be the only thing and breathing it in and listening to the sounds, listening to the waves and _being there_ because that breaks the pattern, it breaks the hold that these negative beliefs have on you and it releases that creepy feeling that negative beliefs are the truth about life, which, of course, they're not. They're just negative beliefs. So, when I slow down, I get out of my negative imaginary future. I stop racing ahead into the future in my mind and I bring myself back to a relaxed, present-moment big picture.\n\nThere's this old story of the shoe salesman who goes to a distant country, arrives there and sees that no one there wears shoes and he says, \"Oh, no. I'm not going to make any sales on this trip. No one here wears shoes.\" Another salesman goes to the same country and he sees that no one wears shoes and he says, \"Oh, boy. I'm going to have a field day here, nobody has shoes yet!\"\n\nYou can catch life by the handle or by the blade. When you are in your future you will always be catching it by the blade. Someone throws you a knife. You catch it. The second shoe salesperson is the person who caught the knife by the handle. And the first guy caught the knife by the blade.\n\nThe more you can slow down, life slows down. That knife that's headed toward you slows down, and it is easier to catch it by the handle.\n\nAnd the more you race out ahead into your own future, your negative imaginary future, which people obsess about, the faster you're trying to solve problems that don't exist and that has you continuously catching the knife by the blade, in other words, seeing nothing but bad news and putting all news into the filter of bad news. _Oh, no, the economy is bad therefore I'm not going to make any money_. Well, really, a lot of people made their money by knowing what to do during the Depression, knowing that there were new services that needed to be provided and new things that needed to be done. They didn't just automatically quit.\n\n## 104. Leave yourself messages\n\nI have certain messages, quotations, sayings that I read in the morning. There is one that I always read no matter what, and I will never move this one out of the line-up: \"He who is not every day conquering some fear has not learned the secret of life.\" \u2014Ralph Waldo Emerson\n\nOur usual system is to not only avoid what we fear, but also to avoid knowing that we are avoiding things. We don't know why we have this vague feeling of danger and insecurity. We just know it's there.\n\nEvery day I practice this avoidance, I become weaker and more cowardly. I get scared more easily, sometimes by the smallest things\u2014the phone ringing, an e-mail from a creditor. Courage and strength do not remain in place. The same is true with my soul and spirit. Each day I skip Emerson's advice to face a small or large fear, it's like my arm not lifting a small or large weight. Weakness creeps in. Therefore I use the quote as a message across time from Emerson to me. I see it in the morning because I have put it there as a way to motivate myself. And it works.\n\nHere is another one (another motivational message!) that I have up on my bookcase, and I make sure I let it sink in at least once a week:\n\nIt is important that you get clear for yourself that your only access to impacting life is action. The world does not care what you intend, how committed you are, how you feel or what you think, and certainly it has no interest in what you want and don't want. Take a look at life as it is lived and see for yourself that the world only moves for you when you act. \u2014Werner Erhard\n\n## 105. Try reinventing yourself\n\nA person in one of my seminars came up to me during the break and pointed at the title of my book _Reinventing Yourself_ and told me she was offended by the title.\n\n\"What's so wrong with me that I have to reinvent myself?\" she said.\n\n\"You don't have to reinvent yourself,\" I said. \"And nothing has to be wrong with you for you to do it.\"\n\nFear of change is the root of most unhappiness. Companies are like this, as well\u2014companies who stay stuck in old ways of being clinging, clinging, clinging as long as they can to comfort zones.\n\nSo what's the answer to the question, _Why should I reinvent myself?_ It's kind of like you are at the high school reunion, you are sitting at the table, and they are playing all the songs that were popular when you were in high school, and somebody comes up and says, \"Please dance with me\", and you say to yourself and to them, \"Why should I dance?\" Well, that's an absurd question. Dance just to dance. Dance just to have fun. Go on out there on the floor and dance and you'll see why you should dance. The same feeling is true with reinventing yourself. Reinvent yourself and you'll see why you are doing it. The real fun, the real joy is in reinventing yourself. It isn't in figuring out why.\n\n## 106. Choose responding over reacting\n\nThe great psychologist Rollo May said, \"Human freedom involves our capacity to pause, to choose the one response toward which we wish to throw our weight.\" People who enjoy being more and more creative through their days, weeks, and lives are happy to learn the distinction between reacting and responding.\n\nReacting occurs when I get an e-mail that angers or annoys me, and I send a blistering reply that makes the relationship worse, or when a family member says something and I say something sarcastic in reply. These are emotional reactions.\n\nIf I'm to build great relationships in my professional and personal life, I want to begin to _respond_ and not react. A response is a creation based on what I want the relationship to be, _not_ based on what the other person has said. It honors my commitments and creativity over my immediate negative feelings. Like the great psychologist and author Rollo May points out, it's all about learning to pause.\n\nThe habitual impulse may be to emotionally react to people. But inside the pause I can remember my true purpose. I can center on my commitment not to let other people bring me down. I can see the value, as I pause, in not taking anything personally. Then, when I breathe inside my pause, I can shift back up to \"create mode.\" What kind of relationship do I really want with this person? What kind of a response would be most likely to create that?\n\n## 107. Apply the book you read\n\nThe great poet Ezra Pound said, \"Properly we should read for power. A person reading should be a person intensely alive. The book should be a ball of light in one's hand.\"\n\nI have found that a great book is even more powerful the second time through, especially if it has been a year or so since I read it. I am more enriched reading that book for a second time than I am reading some new book for the first time. It's the difference between information and transformation.\n\nPeople often say to me, \"I love that book, the problem is applying it.\"\n\nWell, my answer to that is the application is everything. Loving the book is nothing. It isn't how many books you read, it's how many you apply. You are better off, therefore, reading one book four times than reading four books one time each. Instead, most people try to accumulate knowledge. But it doesn't help you when it's accumulated, it makes you fat and overstuffed.\n\nA friend told me 100 books he thought everyone in my profession should read. I'd rather he tell me the _one book_ I should read 100 times. The difference is between a life that is changed, and a life that is weighed down with heavy, immobilizing knowledge.\n\nWe can sit and ponder philosophical concepts forever, but it won't help our lives. What really helps is to test things, experiment, try things out and take the conceptual excitement and put it into _immediate action!_\n\nA lot of times, a fire starts internally when I read something (Bruce Lee's _Jeet Kune Do,_ for example). It starts in my mind and my heart, and I get all excited. Then what I always want to do after that is put some kind of process in place in which I monitor myself to make sure I'm going to do some application. So, in a funny way, when somebody says, \"I have a problem with application,\" my answer is, \"Well, then _do more application_.\"\n\nThere's really no problem, unless you're being hypnotized by circumstance. Let's say I read a book about Bruce Lee and exercise and I get real excited and it says that if a person takes 10,000 steps a day, his blood pressure... and all these biomarkers (or vital markers?) improve. He's going to live 10 years longer and have a better quality life than the average person who only takes 3,000 steps a day. When I read about that, the motivational fire starts inside me and I think that would be fun. I'm starting to get excited. But application is everything here.\n\nSo I buy a pedometer. I put a chart on my wall and I start tracking how many steps I take each day. How long is it going to take me to find my path to 10,000 each day? So I create a game around it, and I do some tracking and I keep score.\n\nI don't keep score because I need to be a competitive alpha male winner, but because the game element has a paradoxical effect on human beings. It introduces accountability (because you're counting things), which is really needed. And it also adds a playful game element. In charting how many steps I take, I created a little game that challenged me to take more steps in December than I took in November. I created a contest between \"Me in December\" versus \"Me in November\" and I found out one December morning that I won. It's over. I can take a knee, the game's over, I've beat my last month's number!\n\nThe game element in anything that you put into application adds a wonderful sense of play. You've got accountability, there's a scoreboard, and you also have a sense of gamesmanship. It's fun, and I'm winning. That's true for anything. Not just physical exercise.\n\nIt's really true for people who sell or market their services. The minute they start accounting for how much time they put into sales and marketing, their sales and marketing results get better.\n\nSo, to the person who says \"I have a hard time applying,\" my answer is, \"That's because you're not applying. That's why you have a hard time applying.\"\n\nIf somebody says, \"I just joined a new health club but the problem is I have a hard time getting myself to go.\" My answer would be, \"Well, go.\" Go to the club. There isn't really a lot in between wanting to go to the club and going to the club. We think there's a lot between those two things, but we are wrong, and we pay a terrible price for that miscalculation. We put phony barriers between wanting it and doing it, and that's where the hypnosis of circumstance comes in.\n\nThat's why I believe coaching is such a powerful profession, because if you know you're going to go see your coach in a week, and if you know that you and your coach agreed that you'd take certain actions this week to see how they turned out, you're going to make sure you take those actions. Because when you sit down with your coach again, you're going to be reporting in on how they worked. It introduces the accountability, the game element. The word \"coach\" comes from sports, it doesn't come from any kind of psychological or spiritual field. Once you see all of this you can become the director of the movie of your life. You choose your activity, and then you yell, \"Action!\"\n\n## 108. Do what you can do today\n\nSometimes people e-mail me to see if they can work with me for a year to have me be a coach and partner to their big dream that they have committed to _just now_. I'm not against dreaming big, but a problem emerges when this big goal is all you've got and there's no way you can have a great day because as you live through your day, and you've made so little progress toward the big dream, it feels like a bad day. Or it feels like a hopeless ambition, and the goal ends up actually hurting your self-esteem. It ends up reminding you, every time you look at it, of _how far away you are from where you want to be_.\n\nThen the goal starts to reinforce the idea that you're not enough, or you're not there yet, or you haven't arrived. You haven't made it. You've got a long way to go. All of those thoughts are demotivating and demoralizing. They don't help. So, here I've got this beautiful goal; I've put it on my wall, and it only reminds me that I'm not there yet. I'm not enough. The air goes out. I am deflated by what was meant to inspire me. And, so, if my programming or my identity becomes _I'm not there yet_ , it's hard to fire up and have a great day from that position.\n\nWhat I like to do, once I or one of my clients has an outcome goal, a results goal, is to only let that goal help inform my plan for what I'm going to do _today_. For example, an Olympic athlete such as Michael Phelps might dream of six gold medals in the Olympics, but he and his coach have to reduce that, translate it into the kind of _day_ they are going to have at the pool _today_. What kind of day, if repeated, would lead to that goal?\n\nIf I want to look a certain way, how would I work out and what would my diet be _today_ for me to look that way? What we're looking for to get you the best human performance, even in such categories as marital happiness, is \"What can I do _today_?\" Not \"How do I achieve my big, dreamy goal way down the road?\" There is no road, and there is no guaranteed future. The game changes. Life changes. Therefore the real goal is to honor the microcosm.\n\nA day is a microcosm of life itself. You can \"die\" at night when you go to sleep and experience rebirth when you wake up. Most people see that as a positive thing. They say things such as, \"Wow, what a great sleep I had... I had a dreamless sleep... _I was dead to the world_. I feel great.\" You are then born again each morning.\n\nIf I can really live this way, and teach my children and clients to do the same, and practice living this way, then things always turn out well because I have leverage and access to my energy now. I'm going to do my 100 pushups, I'm going to do my walking, my running, I'm going to do my writing for my book, I'm going to make 10 sales calls. These are all things I can do and they don't depend on the universe returning something to me.\n\nToday is where it all happens.\n\n## 109. Create a different system!\n\nThe great genius Buckminster Fuller said, \"You can't change anything by fighting or resisting it. You change something by making it obsolete through superior methods.\" Making the bad thing go away is a double negative: _bad thing_ and _go away_ are both negative. To change my life, I want positive energy. What Buckminster Fuller is saying is an important part of why coaching and consulting works. My coach does not have me fight off bad habits. My coach has me execute superior methods of living that make the bad things a mere memory.\n\nWhat I learned from the visionary and enlightened president of Microchip Technology, Steve Sanghi, is that the world is a collection of systems. When I was asked to come into that company to coach and train its people, they shared their culture with me and I was awakened. Every result in life is produced by a system. And every system is perfect for the result it gets. So if you are not getting the result you want, look at the system that is producing that result. What system do you want to replace it with?\n\nIf eating a high-carb diet and exercising infrequently is producing the result of my being 30 pounds overweight, I first want to recognize that it's a perfect system for doing that. By seeing the system as perfect for the result it gets, I no longer have to make myself \"wrong\" or \"weak.\" It's simply a system. By replacing that system with a regular exercise program and a balanced diet, I find I have lost the 30 pounds. I made the old system obsolete through superior methods. That's the only way anything ever changes.\n\n## 110. Enjoy your resistance training\n\nHow much weight do you give to circumstance? Do you allow it to pull you down? Take you out of action? Does the weight of circumstance leave you with no chance at a courageous surge of creativity?\n\nIf you _lifted_ the circumstance, would you not grow stronger? Some people call it resistance training, and they love it. They hire a personal trainer to show them how to take on more and more of this invigorating training.\n\nWe run from it in real life, but then we sign up for a gym membership to engage in it big time. You can do resistance training every day, everywhere. You can think of it as _lifting circumstance_. Or you can curse and avoid circumstance and grow weak. Use it (that arm of yours, that big heart) or lose it.\n\nThe softest and easiest way to live is to simply suffer, to let the sight of rocks hypnotize you. You are growing sleepy now. Your arms and legs are growing weaker and weaker with every breath you take.\n\nIf you wanted a hole in your back yard and I brought you a shovel, that would be all you needed. You wouldn't need to trust the shovel or believe in the shovel. The most effective thing would be to leave your mind out of it, and just use it. If you called me to say there is still no hole in the back yard, and you don't know why, I would say it was because you haven't used the shovel.\n\n\"Why don't I use the shovel?\"\n\nBecause you don't use it.\n\n\"Why don't I _want_ to use the shovel?\"\n\nYou don't have to want to use the shovel to use the shovel. We've become very confused about our wants and desires. We've been so spoiled into thinking we should have everything we want and desire, that we are paralyzed. We don't take action until we feel like we really want to!\n\nRemember that only fire starts fire.\n\nTo say \"I don't apply it\" is already giving yourself the answer to your conundrum. The best way to motivate yourself is to act. Even though that sounds unhelpful!\n\nPeople want to know how to motivate themselves so that they can take the actions they need to take to have the life they want. What they don't see is that action is what creates motivation. Not the other way around. But once they see it, they have new freedom. They realize that they can just act without over-thinking it.\n\nYou know the water's freezing. You hesitate with the thought that it will shock your system to just jump in. But then you just jump, and five minutes later you are joyful, and the water is exciting and fun.\n\n## Bibliography\n\nBach, Richard. _Illusions_. New York: Dell Publishing, 1981.\n\nBelasco, James. _Flight of the Buffalo: Soaring to Excellence, Learning to Let Employees Lead_. New York: Warner Books, 1994.\n\nBennett, William. _The Book of Virtues_. New York: Simon & Schuster, 1993.\n\nBranden, Nathaniel. _The Six Pillars of Self-Esteem: The Definitive Work on Self-Esteem by the Leading Pioneer in the Field_. New York: Bantam Books, 1995.\n\nBristol, Claude. _The Magic of Believing_. New York: Simon & Schuster, 1948.\n\nChandler, Steve and Michael Bassoff. _RelationSHIFT: Revolutionary Fund-Raising_. San Francisco, Calif.: Robert D. Reed Publishers, 2001.\n\nChopra, Deepak. _Creating Affluence: The A-to-Z Steps to a Richer Life_. San Rafael, Calif.: Amber-Allen Publishing, 1993.\n\nCsikszentmihalyi, Mihaly. _Flow: The Psychology of Optimal Experience_. New York: Harper & Row, 1990.\n\nDyer, Wayne. _Choosing Your Own Greatness_. Nightingale Conant Corp., 1991. Audiocassette.\n\nDylan, Bob. _Chronicles_. New York: Simon & Schuster, 2004.\n\nGoss, Tracy. _The Last Word in Power: Executive Re-Invention for Leaders Who Must Make the Impossible Happen_. New York: Doubleday, 1996.\n\nHill, Art. _I Don't Care if I Never Come Back_. New York: Simon & Schuster, 1980.\n\nHill, Napoleon. _The Master Key to Riches_. Mineola, N.Y.: Dover Publications, 2009.\n\n\u2014\u2014\u2014. _The Law of Success: The Master Wealth-Builder's Complete and Original Lesson Plan for Achieving Your Dreams_. New York: Penguin, 2008.\n\nJohnson, Spencer. _The One Minute Sales Person: The Quickest Way to Sell People on Yourself, Your Services, Products, or Ideas_ \u2014 _at Work and in Life_. New York: HarperCollins, 1991.\n\nKaufman, Barry Neil. _To Love Is to Be Happy With_. New York: Ballantine Books, 1977.\n\n\u2014\u2014\u2014. _Son Rise: The Miracle Continues_. Tiburon, Calif.: H.J. Kramer, Inc., 1994.\n\nKoning, Hans. _A Walk With Love and Death_. Montgomery, Ala.: NewSouth Books, 1961.\n\nLamott, Anne. _Bird by Bird: Some Instructions on Writing and Life_. New York: Anchor Books, 1995.\n\nLee, Bruce. _Tao of Jeet Kune Do_. Santa Clarita, Calif.: Ohara Publications, 1975.\n\nLovell, Jim and Jeffrey Kluger. _Lost Moon: The Perilous Voyage of Apollo 13_. New York: Houghton Mifflin, 1994.\n\nMandino, Og. _The Greatest Salesman in the World: You Can Change Your Life With the Priceless Wisdom of Ten Ancient Scrolls Handed Down for Thousands of Years_. Hollywood, Fla.: Frederick Fell, Inc., 1968.\n\nMcGinnis, Alan Loy. _The Power of Optimism_. New York: HarperCollins, 1994.\n\nOgilvy, David. _Confessions of an Advertising Man_. New York: Athenium, 1963.\n\nPeale, Norman Vincent. _The Power of Positive Thinking_. New York: Ballantine Books, 1982.\n\n\u2014\u2014\u2014. _Stay Alive all Your Life_. New York: Fireside, 2003.\n\n\u2014\u2014\u2014. _The Amazing Results of Positive Thinking_. New York: Fireside, 2003.\n\nPeck, M. Scott. _The Road Less Traveled: A New Psychology of Love, Traditional Values and Spiritual Growth_. New York: Touchstone, 1978.\n\nSeligman, Martin. _Learned Optimism: How to Change Your Mind and Your Life_. New York: Pocket Books, 1990.\n\nSenge, Peter. _The Fifth Discipline: The Art & Practice of the Learning Organization_. New York: Doubleday, 1990.\n\nStewart, John. _Write From the Heart_. Buckingham, Va.: Crow Press, 1991.\n\nSylver, Marshall. _Passion, Profit, and Power_. New York: Simon & Schuster, 1995.\n\nTolle, Eckhart. _The Power of Now: A Guide to Spiritual Enlightenment_. Vancouver, B. C., Canada: Namaste Publishing, 1999.\n\nUeland, Brenda. _If You Want to Write_. Eastford, Conn.: Martino Fine Books, 2011.\n\nViscott, David. _Risking_. New York: Pocket Books, 1990.\n\nVos Savant, Marilyn. _Brain Building in Just 12 Weeks_. New York: Bantam, 1991.\n\nWaitley, Denis. _The Psychology of Winning_. New York: Berkley, 1986.\n\nWeil, Andrew. _Spontaneous Healing: How to Discover and Embrace Your Body's Natural Ability to Maintain and Heal Itself_. New York: Ballantine: 1995.\n\nWilliamson, Porter. _Patton's Principles: A Handbook for Managers Who Mean It!_ New York: Touchstone, 1982.\n\nWilson, Colin. _The Essential Colin Wilson_. Berkeley, Calif.: Celestial Arts, 1986.\n\n\u2014\u2014\u2014. _Frankenstein's Castle: The Right Brain: Door to Wisdom_. London: Ashgrove Publishing, 1982.\n\n\u2014\u2014\u2014. _Necessary Doubt_. New York: Pocket, 1966.\n\n\u2014\u2014\u2014. _New Pathways in Psychology: Maslow and the Post-Freudian Revolution_. New York: New American Library, 1974.\n\nWooden, John. _They Call Me Coach_. New York: McGraw-Hill, 2004.\n\nWurman, Richard Saul. _Follow the Yellow Brick Road: Learning to Give, Take, and Use Instructions_. New York: Bantam, 1991.\n\n## Index\n\n5 percent solution, \u2013142\n\nacting the part you want to be, \u201342\n\naction, changing worry into, \u2013163\n\naction plans, , \u2013154\n\nactive relaxation,\n\naffirmations, \u201330,\n\n_Amazing Results of Positive Thinking, The_ ,\n\nasking questions, importance of, \u201369\n\naudiobooks, , \u201335\n\nawareness, making the most of, \u201367\n\nbad habits,\n\nreplacing, \u2013117\n\nreprogramming, \u201338\n\nBassoff, Michael, ,\n\nBennett, William, \u201364,\n\nbiocomputer,\n\ndefinition of,\n\nGIGO and the human,\n\nprogram your, \u201366\n\n_Bird by Bird_ , \u2013143\n\n_Book of Virtues, The,_\n\n_Brain Building in Just 12 Weeks_ , ,\n\nbrain chemicals, using your, \u201345\n\nbrainstorming, \u2013111\n\nBranden, Devers, \u201348, \u201375\n\nBranden, Nathaniel, , , , \u201375, , , , , ,\n\nbreathing, \u2013120\n\nBrown, Henry, \u2013154\n\nBurgess, Anthony, \u2013140\n\n_Buried Alive_ ,\n\nBurroughs, William,\n\nBurton, Richard,\n\nCampbell, Joseph,\n\nchallenge as motivator, \u2013140\n\nchallenge, facing, \u201323,\n\nchanges, small, \u2013142\n\nchanging yourself, , \u2013142, \u2013153\n\nChesterton, G.K., ,\n\nChopra, Deepak, ,\n\nChurchill, Sir Winston, ,\n\ncoaching, value of, \u2013124\n\ncomfort zones, ,\n\ncompetition, \u2013135\n\nconfidence, ,\n\ncourage, ,\n\ncreating a vision, \u201319\n\n_Creating Affluence_ ,\n\ncreating vs. reacting, \u201352,\n\ncreative fallacy, the, , \u201385\n\ncreative\n\nmode,\n\nprocess, , \u2013179\n\nthinking, ,\n\ncreativity vs. originality,\n\nCsikszentmihalyi, Mihaly, ,\n\ncuriosity, power of, \u201369\n\ndeathbed exercise, \u201317\n\nDeaton, Dennis, \u201379, ,\n\nDe Crescenzo, Luciano,\n\ndiscipline, \u201360,\n\ndreaming, power of,\n\ndreams, ,\n\ndriving, \u201335\n\n\"driving for ideas,\" \u201361\n\nDyer, Wayne, ,\n\nEimers, Kathy, \u2013169\n\nEinstein, Albert, \u201353,\n\nEmerson, Ralph Waldo, , , \u2013199\n\nenergy, , ,\n\nenjoyment vs. pleasure, \u2013166\n\nErhard, Werner, , ,\n\n_Essential Colin Wilson, The_ ,\n\nFalk, Peter, \u201369\n\n\"false courage,\" ,\n\nfate, mastering your own,\n\nfatigue, cause of,\n\nfear, \u201355, , , , \u2013193,\n\nfearlessness, social, \u201347\n\n_Fifth Discipline, The,_\n\nfire, motivational, , , , , ,\n\nFisher, Bobby, \u2013120\n\n_Flight of the Buffalo_ ,\n\nflow,\n\n_Flow: The Psychology of Optimal Experience,_\n\nfocus, \u201321,\n\n_Follow the Yellow Brick Road_ ,\n\nFord, Henry, , ,\n\n_Fortune_ ,\n\nfour-circle exercise, , \u2013132\n\nFox, Emmet, , ,\n\n_Frankenstein's Castle,_ ,\n\nfreedom, \u2013118,\n\nfriends, positive, surrounding yourself with, \u201340\n\nFritz, Robert, , \u2013179\n\ngame, turning work into a, \u201387, \u2013203\n\ngames as challenges,\n\nGandhi, ,\n\nGIGO,\n\nGilbert, Rob,\n\ngoals,\n\nbig, \u2013205\n\nconverting problems into, \u2013193\n\ncreating, ,\n\nenlarging your, \u2013175\n\nhappiness and, \u201380\n\nimportance of setting, , \u201395, \u201399, \u2013132, \u2013152\n\noutcome, ,\n\npower, setting specific, \u2013152\n\nprocess, \u201395\n\nprogressing toward, \u201372\n\nsetting small, \u201395\n\nstaying focused on, \u201321\n\nwriting down your, \u201399,\n\nGoss, Tracy, ,\n\n_Greatest Salesman in the World, The_ , \u2013184\n\ngreatness, tapping into your, \u201384\n\nhabits,\n\nbad, \u2013117, \u2013125\n\nreprogramming your bad, \u201338\n\nhappiness, \u201328, \u201363, , \u201380,\n\nexperiment with, \u2013196\n\nHardison, Steve, \u2013124,\n\nheroes, \u2013177\n\nHill, Napoleon, \u201333, , , , \u2013177, ,\n\nHill, Terry, \u201384,\n\nhonesty, \u2013202\n\nimagination, using your, \u201353\n\nInfincom, ,\n\nInformation Age,\n\ninformation vs. transformation, \u2013202\n\ninner voice, \u201372, \u2013122,\n\n\"instant karma,\"\n\ninspiration, leading through,\n\nintention deficit disorder,\n\ninteractive listening,\n\nJames, William, ,\n\nKafka, Franz, \u201344\n\nKaufman, Barry Neil, ,\n\nKeller, Helen, \u2013137\n\nKennedy, John F., ,\n\nKillebrew, Mike, ,\n\nKnipe, Fred, , , , ,\n\nknowledge as power base, \u2013181\n\nKoether, Bob, , \u2013100\n\nKolbe, Jim, \u201327\n\n\"ladder of selves,\"\n\nLamott, Anne, , \u2013144\n\nlanguage, power of, \u201365\n\nlaughter, \u2013147, \u2013185\n\n_Law of Success_ ,\n\nlazy dynamite, \u201339\n\n_Learned Optimism_ , ,\n\nleft-brain thinking, ,\n\nleft brain vs. right brain, \u201381\n\nlife,\n\ncreating your,\n\nsimplifying your, \u201327\n\nlistening,\n\ninteractive,\n\nreprogramming your,\n\nlist-of-20 self-storming technique,\n\nlists, making, \u2013151\n\nluck,\n\n_Magic of Believing, The_ ,\n\n_Master Key to Riches, The,_ ,\n\nMay, Rollo, \u2013201\n\nMcGinnis, Alan Loy, ,\n\nMicrochip Technology,\n\nMicrosoft,\n\nmind vs. emotions, \u201357\n\n_MINDSHIFT_ ,\n\nmoney, \u2013148\n\nmotivation, easing into, \u201339\n\n( _see also_ lazy dynamite)\n\nmystery novels, reading, \u2013169\n\n_Naked Lunch_ ,  _Necessary Doubt_ ,\n\nnegative beliefs,\n\nnegative thinking, power of, \u2013193\n\n_New Pathways in Psychology_ ,\n\nnews fasts, \u2013160\n\nNichols, Rett, \u201322\n\nNin, Ana\u00efs, ,\n\n\"no one is coming,\" \u201375\n\n_One Minute Sales Person, The_ ,\n\noptimism, pessimism vs., \u2013106\n\noptimistic thinking, \u2013137\n\nPascal, Blaise, \u201343\n\n_Passion, Profit, and Power_ ,\n\npassive relaxation,\n\n_Path of Least Resistance, The_ ,\n\nPatton, General George, , , \u2013120\n\n_Patton's Principles_ ,\n\npeak experiences, ,\n\nPeale, Norman Vincent, ,\n\nPeck, M. Scott, , ,\n\npersonal\n\nintegrity, \u2013183\n\ntransformation, ,\n\npersonality,\n\npessimism, \u2013106, , ,\n\npessimistic\n\noutlook,\n\nthoughts, debating, \u2013106\n\nvoice, , \u2013122, \u2013143\n\n( _see also_ the voice)\n\npessimists, ,\n\nPeters, Tom, ,\n\nplanning your work, \u201336\n\nplanning, importance of, \u2013154\n\nplans, writing your,\n\nPlato, ,\n\nplay, value of,\n\npleasure vs. enjoyment, \u2013166\n\n_Power of Now, The,_ 33\n\n_Power of Optimism, The_ , ,\n\n_Power of Positive Thinking, The_ ,\n\npresent, living in the, \u201367\n\nproblem, becoming the, \u2013173\n\nproblems as learning tools, \u201391, \u2013109\n\nproblem-solving vs. creating, \u2013180\n\nprogress, natural rhythm of, \u201371\n\npromises, making unreasonable, \u2013128\n\npsychic entropy,  _Psychology of Winning_ ,\n\npublic speaking, , , , \u2013196\n\npurpose, finding your, \u2013158\n\nreacting vs. responding, \u2013201\n\nreading, \u2013184, \u2013202,\n\nreality, turning dreams into, \u2013146\n\nreinventing yourself (concept), \u2013200\n\n_Reinventing Yourself_ (book),\n\nrejection as motivator, \u2013156\n\nRelation-Shift (seminar), ,\n\n_RelationSHIFT_ (book),\n\nrelation-shift, making a, \u201370\n\nrelationship-builders,\n\nrelationship-building, \u201358\n\nrelationships, \u201358, ,\n\nrelaxation,\n\nactive vs. passive,\n\nactivity and,\n\nresistance training,\n\nresponding vs. reacting, \u2013201\n\nright-brain thinking,\n\nrisks, taking,\n\n_Road Less Traveled, The,_ ,\n\nRobbins, Anthony, ,\n\nRollings, Judy, , \u2013189\n\nSalinger, J.D., ,\n\nSchwarzenegger, Arnold, \u201318\n\nself-consciousness, , ,\n\nself-creation, , , ,\n\nself-denial,\n\nself-discipline, , \u201360, ,\n\nself-esteem, \u201375,\n\nself-examination, \u2013124\n\nself-image, \u2013138, \u2013187\n\nself-mentoring,\n\nself-motivation rituals, \u201362\n\nself-motivation, \u201329, , \u201370, , \u201393, , , ,\n\nself-responsibility,  ( _see also_ \"no one is coming\")\n\nself-revelation,\n\nSeligman, Dr. Martin, , , \u2013136\n\nsentence-completion exercises, , \u2013141\n\nserenity prayer,\n\nservice, , \u2013148\n\nsilence, listening to, \u201344\n\nsimplifying your life, \u201327\n\n_Six Pillars of Self-Esteem, The_ , , , , ,\n\nsocial fearlessness, \u201347\n\n_South Pacific_ , \u2013104\n\nsoul purpose, find your, \u201380\n\nSpassky, Boris, \u2013120\n\n_Spontaneous Healing,_ 40, \u201391,\n\nStewart, Martha, \u2013166\n\nsystem, create a different,\n\nteam-building,\n\ntelevision, turning off the, \u201350\n\nThinkers vs. Whiners, \u2013164\n\nthinking outside the box, \u2013100\n\nthought,\n\nconnection to motivation, \u201338,\n\npower of, \u201342\n\nthoughts vs. emotions,\n\ntime management, \u201325,\n\ntransformation, personal, , , \u2013204\n\ntreasure-mapping,  ( _see also_ four-circle exercise)\n\ntruth, , ,\n\nunexpected, welcoming the,\n\nunhappiness as a tool,\n\nvictim status,\n\nvision, creating a, \u201319, , \u2013180\n\nvisioneering (concept), \u2013146\n\nvoice,\n\nchanging your, \u2013113\n\nthe, \u2013122\n\nVos Savant, Marilyn, ,\n\nvulnerability, using your, \u201349\n\n_Walk With Love and Death, A_ , ,\n\nwalking, , \u2013168, \u2013191\n\nWalsh, Bill, \u201352\n\nwants vs. desires,\n\nWatts, Alan, , ,\n\nweakness, exploit your, \u2013172\n\nWeil, Dr. Andrew, , , ,\n\nWhiners vs. Thinkers, \u2013164\n\nwhole-brain thinking, \u201382\n\nwillpower, developing, \u201360,\n\nWilson, Colin, , , , , , , ,\n\nWooden, John, , ,\n\nwork, planning your, \u201336\n\nworry, , \u201353, , \u2013163\n\nwriter's block, \u2013143\n\nXerox, \u2013100\n\n## About the Author\n\nSteve Chandler is a life coach and a keynote and convention speaker who lives and works in Phoenix, Arizona. He has brought his workshops and seminars to more than 30 Fortune 500 companies and hundreds of small businesses. His other bestselling motivational books include _100 Ways to Motivate Others, Reinventing Yourself, Time Warrior, 17 Lies that are Holding You Back,_ and _Truth That Will Set You Free_.\n\nChandler can be reached at _www.stevechandler.com_. There you will also find free audio downloads and motivational messaging subscriptions.\n\n## Also by Steve Chandler\n\n_100 Ways to Motivate Others_\n\n_Reinventing Yourself_\n\n_50 Ways to Create Great Relationships_\n\n_The Joy of Selling_\n\n_17 Lies That are Holding You Back_\n\n_RelationShift_\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
+{"text":"The  \nAFFINITY  \nBRIDGE\n\nThe author and publisher have provided this e-book to you without Digital Rights Management software (DRM) applied so that you can enjoy reading it on your personal devices. This e-book is for your personal use only. You may not print or post this e-book, or make this e-book publicly available in any way. You may not copy, reproduce or upload this e-book, other than to read it on one of your personal devices.\n\n**Copyright infringement is against the law. If you believe the copy of this e-book you are reading infringes on the author's copyright, please notify the publisher at: us.macmillanusa.com\/piracy.**\nThis is a work of fiction. All of the characters, organizations, and events portrayed  \nin this novel are either products of the author's imagination or are used fictitiously.\n\nTHE AFFINITY BRIDGE: A NEWBURY & HOBBES INVESTIGATION\n\nCopyright \u00a9 2009 by George Mann.\n\nAll rights reserved.\n\nA Tor Book  \nPublished by Tom Doherty Associates, LLC  \n175 Fifth Avenue  \nNew York, NY 10010\n\nwww.tor-forge.com\n\nTor\u00ae is a registered trademark of Tom Doherty Associates, LLC.\n\nLibrary of Congress Cataloging-in-Publication Data\n\nMann, George.\n\nThe affinity bridge \/ George Mann.\u20141st ed.\n\np. cm.\n\n\"A Tom Doherty Associates book.\"\n\nISBN-13: 978-0-7653-2320-0\n\nISBN-10: 0-7653-2320-6\n\n1. London (England)\u2014History\u201419th century\u2014Fiction. I. Title.\n\nPR6113.A546A69 2009\n\n823'.92\u2014dc22\n\n2009012919\n\nFirst Edition: July 2009\n\nPrinted in the United States of America\n\n0 9 8 7 6 5 4 3 2 1\n**_For James George Alexander Mann_**\nACKNOWLEDGMENTS\n\nNo book is written in isolation, and I owe thanks to a great many people, not least to the miraculous Emma Barnes for all her support (how _do_ you fit it all in?), Michael Rowley for being a true friend, Mark Newton for being a fantastic sounding board, Lou Anders for continuing encouragement, Chris Roberson and Allison Baker for their shared love of British TV (amongst many other virtues), Nathan Long for his judicious pencil, my family for their ongoing support, and Fiona, my wife, for taking me seriously when I needed it most, and for not taking me seriously the rest of the time.\nThe  \nAFFINITY  \nBRIDGE\nCONTENTS\n\nPROLOGUE\n\nCHAPTER 1\n\nCHAPTER 2\n\nCHAPTER 3\n\nCHAPTER 4\n\nCHAPTER 5\n\nCHAPTER 6\n\nCHAPTER 7\n\nCHAPTER 8\n\nCHAPTER 9\n\nCHAPTER 10\n\nCHAPTER 11\n\nCHAPTER 12\n\nCHAPTER 13\n\nCHAPTER 14\n\nCHAPTER 15\n\nCHAPTER 16\n\nCHAPTER 17\n\nCHAPTER 18\n\nCHAPTER 19\n\nCHAPTER 20\n\nCHAPTER 21\n\nCHAPTER 22\n\nCHAPTER 23\n\nCHAPTER 24\n\nCHAPTER 25\n\nCHAPTER 26\n\nCHAPTER 27\n\nCHAPTER 28\n\nCHAPTER 29\n\nCHAPTER 30\n\nCHAPTER 31\n\nEPILOGUE\nPROLOGUE\n\nINDIA, JUNE 1901\n\nThe flies. Always the damn flies.\n\nCoulthard slapped at the insects buzzing incessantly around his face and checked his rifle for the fifth time that hour. The heat was proving even more oppressive than usual, and the hair at the nape of his neck was damp with perspiration, his uniform tight and uncomfortable. The other two weren't faring much better, either: Hargreaves was perched on a nearby rock, taking a long swig from his water bottle, and Taylor was pacing backwards and forwards, kicking miserably at the dirt. Only two days remained before the start of their return journey to England, and the lieutenant was still riding them hard, forcing them to go out on patrol in the stifling midday sun. Coulthard cursed under his breath. The man was an egomaniac.\n\nFrom the craggy outcropping on which he stood, Coulthard could just make out the village they had trudged their way here from: a small collection of farms and ramshackle buildings that leaned awkwardly against each other like rows of uneasy siblings. Behind him, a line of trees marked the edge of the village boundaries, and to his left, a series of distant specks denoted a smattering of local farmworkers tending their crops in fields of leafy green. The place had an air of expectancy about it, like somehow it was holding its breath in anticipation of something yet to come.\n\nYawning, he turned to his companions, resting his rifle against a nearby rock. \"So, what's the first thing you're going to do when we get back to London?\" They'd had this conversation a hundred times in the last few weeks, and he already knew what Hargreaves was going to say. Still, it was a conversation that reminded them all of home, and as far as Coulthard was concerned, that was no bad thing.\n\nHargreaves looked up from his water bottle. He mirrored the other man's smile. \"The minute I step off that airship, I'm heading for a pint in the Fox and Hound. I've missed the sorry beggars that prop up the bar in there, and I've missed a good pint of English ale.\" He chuckled at the memories. \"After that, who knows? Maybe I'll take the train out to Berkshire and spend some time on my parents' farm.\" He glanced over at Taylor, who was still kicking up clouds of dust with his feet, a preoccupied expression on his face. Hargreaves dabbed at the perspiration beading his forehead with the back of his sleeve and then leaned in conspiratorially. \"Not sure about him, though.\" He indicated the other man with his water bottle. \"He's not in a good way. Too wet behind the ears for the things he's seen out here.\" He lowered his voice even further. \"May be the asylum for him, when we finally get him home. Poor sod.\"\n\nCoulthard let the comment pass without a response. They'd _all_ been too wet behind the ears for the things they'd seen out here. India was a world apart from England, even with its thin veneer of Empire. He couldn't wait to get home, to get away from the heat and the noise and the ever-present flies. He watched Taylor for a moment, pacing backwards and forwards like an animal trapped in a cage. Hargreaves was right, of course: India had clearly broken the man. He wasn't sure if there was anything to be done for him now. But the asylum? Even the thought of it made him shudder. He'd visited an asylum once, back in Wandsworth, and the screaming of the inmates still rang out in his dreams sometimes, during the long nights when he lay there, trying not to think of all the terrible things he'd seen. If Taylor were headed for the asylum, what hope was there for the rest of them?\n\nRepressing another shudder, Coulthard turned his attention back to Hargreaves. \"Well, if luck be with me, my Ruth will be waiting at the airship port when we arrive.\" He smiled at the thought of her. In another week, he'd be holding her in his arms, spinning her around in the pale winter sun. His heart felt as if it would burst in his chest. That was the thing that would keep him sane, the thing that he was out here fighting for: his life back in England, and the lives of everyone he loved.\n\nHargreaves smiled. He'd heard all of this before. He reached for his water bottle again, and Coulthard turned to survey the horizon once more.\n\nThere was a shuffling sound from behind him. At first, Coulthard assumed it was Taylor, still kicking awkwardly at the sun-baked soil with his boots. Then he became aware of a quiet whimpering sound, like that of a frightened animal, and he felt his hackles rise. He turned around slowly on the spot. His heart was hammering in his chest. What he saw was enough to send him running for the asylum himself.\n\nThe creature that was menacing Taylor was like something raised from the very depths of Hades itself. It was dressed in the torn rags of an Indian peasant, and may once have been human, but now looked more like a half-rotted corpse than like anything resembling a man. The creature's skin was desiccated and peeling, its eyes bloodshot, its hair hanging in loose stringy strands around its face. Its teeth were bared in a rabid snarl, and it was bearing down on a terrified Taylor. Coulthard presumed that it had crept out from the cover of the nearby trees when they hadn't been paying attention. Taylor was on his knees before it, using his arms to cover his face from the beast as if simply trying to will it out of existence.\n\nCoulthard scrambled hurriedly for his rifle, fumbling as he tried to bring the barrel to bear on the horrifying creature. Hargreaves was already on his feet and rushing forwards, his sword drawn, ready to take a swing at the monster. Shaking, Coulthard tried to remind himself to breathe, to hold himself steady as he planted his feet and took aim. He let off a shot, jarring his shoulder with the sudden recoil. The creature staggered back for a moment, then surged forwards again in a frenzy, lashing out at Taylor, who had given himself over completely to his terror and seemed unable even to attempt to defend himself from the diabolical thing. Coulthard watched in shock as the creature raked its nails across Taylor's face, digging its bony thumbs into his eye sockets and sending him spinning to the ground, his once-handsome face reduced to nothing but a bloody ruin. He gave a final wail before crumpling to the dirt, silent.\n\nThe creature turned its attention to Hargreaves. Blinded by rage after witnessing the fate of his fellow soldier, Hargreaves swung his blade at the lurching monster with all his might. It struck home, cleaving deep into the creature's chest, biting through skin, muscle and bone, but it hardly seemed to slow the beast at all. To Coulthard's amazement, it showed no signs of pain, or even distraction, as Hargreaves struggled to pull his weapon free from where it had wedged inside the creature's shattered rib cage. Coulthard let off another shot, to no avail, and finally accepted the uselessness of the firearm and abandoned it to the ground instead drawing his sabre and rushing quickly to his fellow's side.\n\nUsing his momentum to carry his blade forward, he speared the monster directly through the gut, driving his sword home until the hilt itself was buried deep inside the creature's abdomen. He twisted it, trying desperately to slow the assault of the vile thing, to draw some sort of reaction from it. All the while, it continued to rage at Hargreaves, who had given up trying to pull his weapon free and was now pummelling the monster's face with his fists as he endeavoured to wrestle free from its talonlike grip. A moment later, his movements turned to spasms when, unable to gain useful purchase on the creature, it pulled him close and tore his throat out with single wretched bite.\n\nCoulthard, aghast, pulled his sabre free of the creature's guts and aimed a blow at the arm that still held the limp body of his friend. The blade sliced clean through the arm, lopping off the limb at the elbow and dropping the dead Hargreaves to the dirt. Dark blood sprayed from the wound, but the monster itself seemed entirely unperturbed by its injury. Baring its teeth, it pounced on Coulthard, clamping its mouth on his forearm as the man struggled to bring his weapon up before him in defence. Howling in pain, Coulthard kicked at the creature, desperate to break free. He could smell the carrion-stench of the thing, see the feral hunger behind its darting inhuman eyes.\n\nAcutely aware of the horrifying manner in which his friends had died, Coulthard's instincts screamed at him to run. With a concerted effort, he grabbed a handful of the creature's hair and wrenched his arm from its mouth, tearing skin away from bone as an enormous hunk of his flesh was rent away in the creature's jaws. Almost swooning with the pain, Coulthard drove the blade of his sword through the monster's chest and then turned and fled, his feet pounding the dry earth as his legs pumped as fast as they could, sending him careening down the side of the outcropping and onwards towards the village, his left arm dangling uselessly by his side.\n\nBehind him, the unusual beast, still with the hilt of the sabre protruding rudely from its chest and the stump of its missing arm spouting ribbons of dark blood, turned and grabbed the hair of Taylor's fresh cadaver and began to drag it slowly towards the cover of the trees.\nCHAPTER 1\n\n##\n\nLONDON, NOVEMBER 1901\n\nThe room was full of ghosts.\n\nOr so Felicity Johnson would have had him believe. Sir Maurice Newbury, weary from a day spent scouring the dusty stacks of the British Library, drummed his fingers on the table with a quiet impatience. The dinner party was not working out at all as he'd anticipated.\n\nAround him, the other guests sat in a wide circle, spaced evenly around a large round table, their faces glowing in the dim light of the gas-lamps. Overturned tumblers, tarot cards, holly leaves and other assorted paraphernalia littered the table-top, and their host, her shrill voice piercing in the otherwise silent room, was attempting to raise the dead.\n\nNewbury, decidedly unimpressed by the charade, glanced at the other guests around the table. Their faces were difficult to read in the half-light, but many of them appeared captivated by the performance of the woman as she waved her arms about her, wailing, her eyes shut tight, her body tensed; possessed, apparently, by some kind of unearthly spirit. She was currently engaged in babbling something about Meredith York's dead brother, and the poor woman was entirely taken in, sobbing on her husband's arm as if she truly believed she were receiving messages from beyond the grave.\n\nNewbury shot a look at the man seated beside him and shrugged. Sir Charles Bainbridge was a Chief Inspector at Scotland Yard, a favoured agent of Queen Victoria herself and one of the most rational men in Newbury's acquaintance. He didn't think for a minute that his old friend would be taken in by any of this nonsense. He was older than Newbury, about ten years his senior, and was greying slightly around the temples. His moustache was bushy and full, and his eyes were bright, shining with mischief and the glassy patina of alcohol. Acknowledging the pained expression on his friend's face, Bainbridge offered an amused smile, the flickering light casting his face in stark relief. Clearly, he was considerably more forgiving of the indulgences of their host. Newbury shook his head in exasperation.\n\nA few moments later, Miss Johnson fell back into her chair with a gasp, her eyes suddenly flicking open, her hands raised to her mouth in affected shock. She turned to survey her guests. \"Did I\u2014?\"\n\nMeredith York nodded emphatically, and a moment later, when the gas-lamps were turned up and the room was once again cast in a warm orange glow, the small audience paid tribute to their host with a hearty round of applause. Newbury sat back in his chair, relieved that the spectacle was over. He rubbed a hand over his face, feeling a sense of lethargy creeping over him. The other guests were already deep in conversation as he surveyed the scene with the air of someone ready to take their leave. He didn't want to be drawn out on his opinions of the evening's pursuit, lest he inadvertently cause offence. He patted his friend on the arm.\n\n\"Charles?\" The other man turned to meet his gaze. Newbury stifled a yawn. \"My lodgings beckon me. I'm intent on taking a stroll. Would you care to join me?\"\n\nBainbridge allowed himself a brief chuckle at the other man's expense. \"That keen to get away, Newbury?\" He shook his head in feigned disapproval, but his smile was barely concealed. \"I had a feeling that you'd find this all rather objectionable. Come on, let's bid our friends good night and take our leave.\"\n\nThe two men stood together, and Felicity Johnson almost leapt out of her seat when she spotted them out of the corner of her eye. She briefly patted Meredith York on the back of the hand before turning to regard them. \"Oh, gentlemen, must you go so soon?\"\n\nNewbury edged around the table and took her hand. \"I am afraid that duty calls, my dear Miss Johnson. Both Charles and I have early appointments to keep in the morning. Thank you for a pleasant evening.\" He paused, unsure how to go on. \"It has been an . . . entertaining diversion.\" He inclined his head politely and turned to reclaim his coat from the butler standing by the door. The woman's face fell, and she stammered briefly before replying. \"Always a pleasure, Sir Maurice.\" She turned to Bainbridge, who was just collecting his cane from the hat stand in the hallway. \"And you, Sir Charles. I do hope we will see you both again soon.\" And with that, she returned her attention to the adoration of Meredith York and her other guests.\n\nOutside, the pavement was covered in a layer of hoary frost. Newbury turned his collar up against the biting winter chill. The moon was full in the sky, the night was clear and people bustled along the street, their breaths making foggy clouds in the cold air. Newbury drew the crisp air deep into his lungs, obviously relieved to have escaped further embarrassment at the hands of Miss Johnson.\n\nBainbridge, his cane clicking rhythmically against the ground as he walked, turned to Newbury as they made their way back towards Piccadilly. \"Really, Newbury, did you have to cut her so?\"\n\n\"Oh, Charles, the woman's a buffoon! She's trifling with things she has no real concept of, making light of Mrs. York's bereavement. Games like that are dangerous and hurtful.\" He shook his head, sighing. \"I did not aim to cause offence. I simply wanted to let her know that we were not taken in by her little merriment. You know as well as I do, there were no spirits present in that room.\"\n\nThey stopped as a ground train trundled by, the huge steam engine roaring as the fireman stoked the flames, the carriages behind it bouncing along the cobbled road, their wooden wheels creaking under the strain. Newbury caught stuttering glimpses of the people inside the small carriages as they rushed by, snug inside their little booths, speeding on towards their destinations. The driver, on the other hand, was wrapped up warm against the elements, sitting atop the engine itself on a large dickey box, a huge steering wheel clasped between his gloved hands. They watched as it rattled away into the night, causing hansom cabs and more traditional horse-drawn carriages to divert from their paths. Newbury smiled. It was time for the past to make way for the future.\n\nThe two men crossed the road and continued on their way. Newbury decided it was time to change the subject. \"So, tell me, Charles, any new developments in the case at hand?\"\n\nThe other man sighed. \"Not as such. Can't seem to get past this ridiculous story about the glowing policeman. It's making life very difficult for my constables. They keep being accosted out on their rounds. No one will answer their questions, and the men themselves don't want to go out at night, lest they find themselves running into this damnable fellow. Superstitious prigs!\"\n\nNewbury looked suddenly serious. \"Charles\"\u2014he patted the other man on the shoulder\u2014\"look who has his ire up now! Don't be so swift to discount these stories, at least before we have any real evidence to the contrary.\"\n\nBainbridge looked incredulous. \"Heavens, Newbury, surely you're not putting any stock in these ridiculous tales? They are clearly as much poppycock as Miss Johnson's spirits!\"\n\nNewbury hesitated. \"Look, Charles, I know I was dismissive of Miss Johnson, but I've spent the entire day scouring shelves in the British Library, looking for references to a glowing policeman, and I assure you, there is more to it than meets the eye.\"\n\nBainbridge stopped in his tracks. He leaned on his cane. \"How so?\"\n\n\"There's a case from about twelve years ago. A bobby who was murdered by a gang of petty thieves\u2014found himself in the wrong place at the wrong time. You know the sort of thing.\" Bainbridge nodded. \"Well, for a month _after_ the body was interred, a 'glowing bobby' was seen looming out of the fog around the Whitechapel area, his pale skin shining an iridescent blue. One by one, the bodies of the thieves turned up, all strangled, all dumped in the same area of the city. Witnesses reported sightings of the dead constable, come back from the grave to seek revenge on his aggressors. After the last of the thieves turned up dead, the 'glowing bobby' was never seen again.\" He paused. \"Until now, that is. I pieced the story together from various newspaper reports.\"\n\nBainbridge shrugged. \"It was probably the other boys from the station, using the story as a cover to take revenge for the murder. They don't take kindly to one of their own being put in the soil.\"\n\nNewbury nodded. \"That may well be the case, but until we know more, I think we need to follow this line of inquiry. It may turn out to be nothing but poppycock, but we shouldn't dismiss it until we've had the opportunity to investigate a little further first.\"\n\n\"Very well.\" Bainbridge covered his mouth with the back of his hand as he coughed. \"Come on, let's get out of this cold.\"\n\nNewbury sauntered along beside him. \"Would you care to join me for a nightcap at the White Friar's? They have a shockingly good brandy.\"\n\nBainbridge was about to reply when a sudden powerful gust of wind knocked them both back a step, and the older man found himself clinging to his hat to ensure it wasn't lost in the draught. He looked up. \"Damn airships! I wish they wouldn't fly them so low over the city.\"\n\nNewbury laughed, following his gaze. The underbelly of an immense vessel was scudding overhead, scintillating in the reflected light of the city and temporarily blotting out the moon, casting the two men in a dark shadow. The airship companies had been enjoying a period of rapid growth in recent months, with demand for air travel almost exceeding their capacity to build new vessels and clear space for berthing fields. The appearance of a sizeable ship such as this was becoming a frequent occurrence in the skies over London, as the Empire grew larger and an increasing number of people found profitable business abroad. With the haulage companies taking to the skies, too, there was no longer any need to relocate to foreign climes on a permanent basis, and many businessmen had taken the opportunity to set up subsidiary companies in India, America and the West Indies. Newbury himself had never travelled on one of the vessels, but he was certainly enamored with them, and watched in wonder as this one drifted lazily overhead, en route, he supposed, to a berthing field south of the city. He glanced back at Bainbridge, who had finally finished repositioning his hat. \"Well? To the White Friar's?\"\n\nBainbridge shook his head. \"Not tonight, old friend. You've given me much to think about, and I must say that that pudding of Miss Johnson's is sitting rather heavily on me now. Don't have quite the constitution I used to.\"\n\nNewbury smiled. \"You'll hear no argument from me.\" He held out his hand, and the other man grasped it firmly. \"Let me know if there are any further developments in the case. In the meantime, I bid you well and good night.\" He turned and made off in the direction of the White Friar's Club, gazing up at the sky in wonder at the vapour trails left in the wake of the passing airship.\nCHAPTER 2\n\n##\n\nNewbury leaned back in his chair and, with a sigh, spread his morning copy of _The Times_ out before him on the desk. After retiring from the White Friar's Club the previous evening, he'd found he was unable to sleep. Nonetheless, with the coming of the dawn, he had risen, dressed and caught a cab across the city from his Chelsea lodgings to his office at the British Museum. He had little doubt that his housekeeper, Mrs. Bradshaw, would curse him colourfully in her delightful Scottish tones for failing\u2014yet again\u2014to inform her of his plans, but he also knew that she was growing used to his unpredictable comings-and-goings, even if she feigned exasperation to his face.\n\nOutside, the sun was settling over the city, and the streets were gradually coming to life as people set about their daily business. Soon the museum would be bustling with his fellow academics and, not long after, with members of the public, come to gaze in awe and wonder at the treasures on display in the gaudy exhibits. Newbury had been an agent of the Queen for nearly four years, and whilst he was typically engaged in some case or other\u2014whether helping Scotland Yard or left to his own devices\u2014he continued to maintain a position at the museum all the same. He was an experienced anthropologist, with a particular speciality in the religion and supernatural practices of prehistoric human cultures, and he often found his academic work had resonance with his work in the field. At present, he was engaged in writing a paper on the ritualistic practices of the druidic tribes of Bronze Age Europe. He'd hardly found time to touch it for a week, however, what with the string of bizarre strangulations occurring around Whitechapel and his desire to aid his old friend, Bainbridge, in the hunt for the killer. Discovering that the culprit may have supernatural origins had only solidified his resolve to see the case through to the end, and what's more, the revelation put the case firmly and directly into his specific area of expertise. Since briefing the Queen with a missive the previous day, any time he spent aiding Bainbridge with his investigations was now considered official business.\n\nNewbury yawned. It was still early, and his secretary had yet to arrive at the office. He was anxious for a cup of tea. He regarded the newspaper before him, paying no real attention to the article he'd been trying to follow, which concerned a politician involved in some lurid financial scandal. He was dressed in a neat black suit, a white shirt and crimson cravat. His hair was dark\u2014the very colour of night itself\u2014and swept back from his face, and he was clean-shaven. His eyes were a startling emerald green. A casual observer would have placed him in his early thirties, but in truth, he was approaching his fortieth year. He looked up at the sound of someone bustling into the adjoining room and called out, \"Good morning, Miss Coulthard. I'd like a pot of tea when you're settled, please.\" He returned, distractedly, to his reading.\n\nA moment later, there was a brief rap at his door. He didn't look up from his newspaper when the door itself swung open and someone crossed into the room. \"Thank you, Miss Coulthard. I trust you are well?\"\n\nThe woman cleared her throat. Newbury's eyes flicked up from the print. \"Oh, my dear Miss Hobbes. I do apologise.\" He fumbled for a moment, unsure how to remedy his error. \"I'm afraid I'm still getting used to the notion that another person will be sharing my office. Do come in.\" He half stood behind his desk, embarrassment clearly written on his face, as his recently hired assistant, Miss Veronica Hobbes, crossed the room and took a seat before him. She was pretty: brunette, in her early twenties, with a dainty but full figure, and dressed in a white blouse, grey jacket and matching skirt.\n\nShe smiled. \"Please don't apologise, Sir Maurice. It takes more than a little case of mistaken identity to offend me.\"\n\nNewbury returned her smile. \"Very good. Let's get you settled in, then, shall we? But first . . . I don't suppose you're at all handy with a kettle?\"\n\nAn hour later, fortified by a constant supply of Earl Grey, the office had become a hive of activity. Newbury was working through his notes from the previous day, trying to make sense of the various newspaper reports and apparent sightings of the \"glowing bobby\" around Whitechapel. He was wearing a frown, lost in thought and deep concentration.\n\nVeronica was hard at work, clearing the spare desk across the other side of the room, unpacking her small box of belongings and filing the many sheaves of abandoned notes she continued to find in drawers and random piles all around the office. She had placed her jacket over the back of her chair, rolled up the sleeves of her blouse and attacked the mess like it was some sort of villain in need of appeasing. Newbury was suitably impressed by her fastidiousness.\n\nIt was into this scene that a distraught Miss Coulthard came running, late, her hastily tied bun coming loose so that strands of her hair flapped around her face as she came to rest in the doorway, breathless. Both Newbury and Veronica looked up in concern.\n\nNewbury was on his feet immediately, worry etched on his face. \"My dear Miss Coulthard, whatever is the matter?\"\n\nThe woman cowered, as if afraid of what she had to say. Veronica offered her a heartfelt smile.\n\n\"Oh, sir, it's my brother Jack. He disappeared yesterday, and we've every fear that he may have succumbed to that terrible plague.\"\n\nNewbury shuffled uneasily. \"I understand your concern completely, Miss Coulthard. Look\u2014\" He indicated his visitor's chair. \"\u2014come and take a seat for a while, and Miss Hobbes here will fetch you a hot cup of tea.\" He glanced at Veronica apologetically, and she waved dismissively before hurrying off into the other room to organise another pot of tea.\n\nNewbury put a hand on Miss Coulthard's arm to reassure her. \"Now, why don't you tell me exactly what you know?\"\n\nThe diminutive woman looked up at him, a pained expression on her face. \"In truth, sir, there ain't that much to tell. Jack went off to work yesterday morning as normal\u2014to Fitchett and Browns, the lawyers\u2014and never came back. We had a restless night, worrying what kind of a mess he'd got himself involved in, as he's never been one to loiter before coming home of a night. My sister-in-law and I took ourselves down to the law offices first thing this morning, to enquire as to his whereabouts, and it seems he never even made it that far.\" With this, she let out a racking sob, bringing her gloved hand up to her face to stifle her tears. \"They had no idea where he was, or why he hadn't shown up for work the previous day.\"\n\nNewbury sat back in his chair, looking thoughtful. \"I'm sure we'll find a suitable explanation, Miss Coulthard, if we apply ourselves. Now, tell me, what makes you think it's the plague?\" He looked up at the sound of the kettle whistling in the other room and caught sight of Veronica listening to their conversation from the doorway. He nodded approvingly and then returned his attention to the crying woman before him.\n\n\"There have been terrible things happening in our neighbourhood, sir, terrible things indeed. Revenants, they're calling them. Victims of the plague, found staggering around in the fog of a night, like wild animals, baying for people's blood. Bloodshot eyes, peeling skin\u2014they're like walking corpses, wandering around in the darkness, waiting for passers-by. The plague transforms them into mindless monsters.\" She crossed herself to ward off the thought of the horrifying creatures.\n\nNewbury nodded. \"I'm well aware of the phenomenon, Miss Coulthard. It's thought the plague was brought here from India, borne over by returning soldiers. It inspires a terrible brain fever and a degenerative state in the flesh. Was Jack bitten by one of these walking cadavers?\"\n\n\"Not that we know of. Jack knows better than to loiter in the dark these recent months. But I fear he must have encountered one on his way to work that morning. The fog was thick around Brixton, and it may have been upon him before he had an opportunity to flee.\"\n\nNewbury shook his head. \"Unlikely, Miss Coulthard. As I understand it, the victims of this plague find the light painful to their eyes and will avoid stepping out during the daylight hours unless desperate or provoked. Remember, they are driven by animal desires, and not those of a rational human being. Besides, anyone bitten by one of these creatures will incubate the illness for a number of days before showing any symptoms. If your brother was indeed harassed in the street, he would have likely retained his senses and sought medical assistance at a nearby hospital. I'm sure, therefore, that there must be another explanation as to his disappearance.\"\n\nMiss Coulthard was still shaking. \"You really think so?\"\n\nNewbury smiled. \"Indeed. There are many things that can keep a man away from his home for a night, Miss Coulthard, and whilst some are less savoury than others, I'm sure in this case, there'll be a reasonable explanation.\" He paused whilst Veronica placed a steaming cup of tea on the desk before Miss Coulthard. \"Now, see yourself right with that cup of tea, and then take the rest of the day off. If there's still no news tomorrow, come and see me again and we'll file a missing-persons report with Scotland Yard.\"\n\nMiss Coulthard braved a smile. \"Thank you, sir. It's just . . . we're all so on edge, what with the strange things that have been happening. Time was when we would have laughed it off. But with these revenants walking the streets . . .\"\n\n\"I know, Miss Coulthard, I know. The plague has us all concerned for the well-being of our loved ones and friends. I promise I'll keep my ear to the ground for any clues that may help you to locate your brother.\" Newbury stood and edged around the desk. \"You stay put for a moment, Miss Coulthard, whilst I have a few words with Miss Hobbes.\" He crossed into the adjoining room, straightening his jacket and pulling the door shut behind him.\n\nVeronica looked up. \"What is it?\"\n\n\"I'll wager it has something to do with drinking or gambling, or both.\" He shook his head.\n\n\"Is there anything we can do to help?\"\n\n\"No. I'm convinced the situation will resolve itself. Another day or two, and the man will show up at his own door, hungry and not a little sheepish. Either that or they'll find him in a cell across the other side of the city, too embarrassed at his own behaviour to tell his family where he's been.\"\n\nThere was a rap at the outer door to the office. Veronica glanced quizzically at Newbury before crossing the room and allowing the door to swing open, revealing a messenger standing in the hallway, a small card clasped in his right hand.\n\n\"Message for Sir Maurice Newbury, ma'am.\"\n\n\"Thank you. I'll see that he gets it.\" She took the card from the young boy and turned to Newbury, who had sidled up behind her, his interest piqued. He took the card from her and turned it over in his hand.\n\n\"It's from Bainbridge.\" His face had taken on a grim aspect. He looked up at Veronica. \"Get your coat. There's been another murder.\"\nCHAPTER 3\n\n##\n\nThe cab clattered noisily over the cobbled street as its pistons churned furiously and the driver swore at the mechanism in a half-hearted attempt to make it run faster. In the back, Newbury and Veronica sat in silence, jolted by the speed at which the vehicle rumbled towards its destination and by the unevenness of the road. At the front, the driver sat upon his dickey box, pulling levers to direct the angle of the wheels as the steam-powered pistons fired with noisy abandon and the cab bounced along on steel wheels softened with rims of polished hardwood. Veronica couldn't help thinking that, whilst it might have taken them a few minutes longer, a traditional horse-drawn carriage may have offered them a more comfortable alternative to the loud, dirty transport within which they now sat. Newbury, on the other hand, was a keen supporter of progress, and whilst even the driver seemed to be having difficulty keeping the contraption under control, Newbury appeared to be relishing every moment of their tumultuous journey.\n\nOutside, the fog was still thick and cloying, a yellow tubercular cloud that sat heavy over the city, a shroud over the populace and a haven for the creeping things of the dark. Veronica watched through the window, seeing only the impression of grandiose buildings looming out of the smog, or the occasional vehicle flitting by on the road, its passengers hidden behind darkened windows or wreaths of smokey fog. Gas-lamps flickered in the damp air, a network of disembodied halos that lined the edges of the streets. Underlit carriages rode on a carpet of rolling fog. It was mid-morning, but it seemed to Veronica as if the day had somehow stalled, the sunlight replaced by a remarkable twilight that appeared to have descended all across the city. She looked up, presuming that the regular slew of airships that filled the skies these days had been grounded temporarily by the impenetrable weather, or else they had risen up above the smog to where the skies were clear and free of city air. She glanced at Newbury, but his face seemed suddenly serious. She folded her hands on her lap and waited.\n\nPresently, as they raced towards Whitechapel and the scene of the murder, the fog became gradually less dense and the buildings closed in, the streets becoming narrower, the towering mansions and sweeping terraces of Bloomsbury giving way to less monumental structures and more factories, breakers' yards and public houses. Whitechapel was not the sort of place that Veronica would visit by choice. It was one of the seedier locales of the city, a refuge of beggars, criminals and whores. She shivered when she considered what they might find there. Pulling her coat tighter around her shoulders, Veronica drew the curtain across the window inside the cab, and Newbury raised an eyebrow in her direction, evidently interested to know what had spooked her. She pretended not to notice.\n\nA short while later, the cab juddered to a halt and the driver clambered down from his perch and opened the door for the two passengers. The engine was still running, and outside, the noise of it was even more intense. It sounded like some great industrial machine, churning out clouds of steam and soot into the already bleak morning.\n\nNewbury made good on the fare and no sooner had he climbed down from the carriage than Bainbridge was at his side, leaning on his cane, his overcoat pulled tight around his wiry frame. He looked like he'd been here for a while already.\n\n\"Ah, good, Newbury. We can press on.\" He paused for a moment at the sight of Veronica, unsure how to go on. He inclined his head politely. \"Good morning, Miss Hobbes.\"\n\nHe turned to Newbury. \"Can I have a word?\"\n\nNewbury smiled. \"Indeed.\" They moved to one side.\n\n\"My dear fellow, do you think it's a good idea to bring a lady to a scene such as this? She could find it terribly alarming.\"\n\nNewbury chuckled. \"Charles, I may have known the girl for only a few weeks myself, but already I know better than to exclude her.\" He smiled. \"Trust me, Miss Hobbes can look after herself.\"\n\nCharles shook his head, as if dismayed at what the modern world was coming to. \"So be it.\" He sighed. \"Come on, this way.\"\n\nHe led them on to where the body was lying, sprawled out on the cobbles like a broken doll, its neck contorted into an awkward posture, the face a picture of anguish and pain. Surrounding the scene were three constables, their hands clasped firmly behind their backs, each of them keeping a wary eye on the surrounding fog and what it may or may not be hiding from view.\n\n\"Any witnesses?\"\n\n\"No.\"\n\nNewbury knelt closer to examine the body. The man was dressed in pauper's clothes, dirty from the workhouse, with black filings underneath the fingernails. He was clean-shaven and appeared to be in his mid-twenties. Newbury turned him over gently, examining the soft flesh around the throat, probing with his gloved fingers. He looked up at Bainbridge, who was standing over them, watching intently. \"The neck's been broken, but the cause of death is definitely strangulation. Look at these marks here, here and here.\" He indicated with his hand. \"This bruising suggests the victim was grabbed forcefully around the throat and struggled somewhat before finally being despatched. There's nothing of the perpetrator left at the scene, but it certainly matches the profile of the other killings.\"\n\nVeronica cleared her throat. \"Has he been robbed?\"\n\nBoth of the men turned to look at her in surprise. \"Good question, Miss Hobbes. Let me check.\" Newbury fished around in the dead man's pockets for a moment before withdrawing a small leather wallet from inside the man's waistcoat. He opened it up. Inside was a smattering of low-denomination coins.\n\n\"He had little enough about him, but whoever\u2014or whatever\u2014killed him clearly wasn't interested in making a profit.\"\n\nBainbridge tapped his cane thoughtfully against the cobbles. \"So what did they have to gain?\" The frustration was clearly evident in his voice. \"Are they just killing people for the hell of it?\"\n\nNewbury stood, handing the wallet to Bainbridge. \"No, I doubt that very much. There has to be a motive here somewhere. We just can't see what it is, as yet.\"\n\n\"Well, I hope one of us starts seeing it soon. This is the seventh victim this month. Things are getting out of hand. I'm going before Her Majesty this afternoon, and currently, all I have to tell her is that the body count keeps getting higher!\"\n\nNewbury looked pained for his friend. \"Look, I'm making some progress with my research that could suggest a couple of avenues for your men to investigate. Why don't you call on me later at the office and I can talk you through it? Right now, I think it best that you get that cadaver moved to the local morgue and have the surgeon begin the post-mortem directly. A body lying around in the fog might be too much of a temptation for these 'revenant' creatures to bear.\" He glanced around at the nearest constable, who was shuffling uncomfortably on the spot.\n\nBainbridge shrugged. \"Yes, yes, you're quite right.\" He turned to the constable on his left, waving his cane. \"You, man. Go and organise some transport to get this body moved.\" The other man hesitated, as if he were about to protest. Bainbridge was having none of it. \"Well, go on, then!\" The constable scuttled off into the fog. Bainbridge turned back to Newbury and Veronica. \"I'd better go with them, make sure the surgeon gets the correct instructions. Can you find your own way back?\"\n\nVeronica nodded. \"Of course we can, Sir Charles. But first, would you object terribly if I put a few questions to your men?\" She moved over to stand beside Newbury.\n\nBainbridge looked confused, but assented readily. \"No, no, my dear. Anything at all, if you think it may prove useful in helping to solve the case.\"\n\nVeronica nodded appreciatively and then stepped around the body and approached one of the remaining two constables.\n\n\"Good morning, ma'am.\" He looked vaguely uncomfortable at the thought of being questioned by a woman.\n\n\"Good morning, Constable . . . ?\"\n\n\"Pratt, ma'am.\"\n\n\"Good morning, Constable Pratt. I'm in need of some assistance. You see, my colleagues over there are labouring under the impression that I'm fully up to date with all the minutiae of this murder enquiry, but, as I'm relatively new to the job, I seem to be missing some of the pertinent facts. I was hoping you could help me out of my predicament?\"\n\n\"Certainly, ma'am. Where would you like me to begin?\"\n\nVeronica affected ignorance. \"Well, we could start with the victims. How many are there now?\"\n\nPratt hesitated before going on. \"Well, ma'am, there are seven official victims, all of them strangled to death and abandoned in the street, just like this one. All from the same area of the city.\"\n\n\"Official victims?\"\n\n\"Yes, ma'am. Folk around here are saying there's actually around three times that number, if not more. Sometimes the families come and move the bodies before the police happen upon them, other times the corpses are stripped and robbed and end up floating down the river.\"\n\n\"And what of witnesses?\"\n\n\"People aren't too forthcoming, ma'am. They're attributing these killings to a phantom, the glowing policeman. Talk like that makes them clam up good and proper when a man in uniform comes knocking on their door. Not only that, but people are scared to come out at night. On one hand, they're worried about the murderer; on the other, about the revenants that are walking the streets at night, hiding in the gutters like animals. Places like this, they ain't safe, ma'am. People keep themselves to themselves.\"\n\nVeronica smiled. \"So do _you_ think this is the work of the glowing policeman, Constable Pratt?\"\n\n\"I'm not qualified to say, ma'am. But I do know folk who claim they've seen him out here, wandering around in the fog, his face and hands glowing with ghostly blue light whilst he waits for his next victim.\"\n\n\"Thank you, Constable. Most useful.\" She made her way back to where Newbury and Bainbridge were standing, a wry smile on her face. \"It sounds as if these bodies may be just the tip of the iceberg.\"\n\nBainbridge nodded, obviously impressed. \"You continue to confound me, Miss Hobbes.\"\n\nVeronica smiled. \"Let's just hope it proves useful in bringing the killer to justice, Sir Charles.\"\n\n\"Indeed. Indeed.\"\n\nNewbury doffed his hat to his old friend. \"Charles, we'll take our leave. Watch your back out here, won't you, and remember to call by the office this afternoon for a talk. I'm sure we can start moving forwards in this matter, hopefully before another sorry individual loses his life.\"\n\n\"Thank you, Newbury. Your assistance is most appreciated.\"\n\n\"Say no more.\" And with that, Newbury and Veronica turned on their heels and disappeared into the fog-laden morning in search of a cab.\n\n\"I liked your trick with the constable back there.\" Newbury was in a much more talkative mood, now that the two of them had managed to hail a hansom cab and were on their way back to the museum.\n\nVeronica was relieved that, this time, they'd been able to settle on a more traditional vehicle, pulled by horses, instead of the more temperamental steam engine they had suffered before. She regarded Newbury from across the carriage. \"I've always believed that it's worth keeping one's ear to the ground, finding out what people are saying. Invariably, in my experience, that's where one may find the truth, or at least the kernel of the truth that has given rise to the tall tales.\"\n\nNewbury nodded in agreement. \"An admirable tactic, and one that I'm convinced will bear fruit. But consider this\u2014\" He paused for dramatic effect. \"\u2014what if, in this instance, the tall tales were actually based on fact?\"\n\nVeronica's eyes betrayed her incredulity. \"Come now, sir, you're not suggesting the glowing policeman is the real source of these murders?\"\n\n\"Indeed not, although at this stage, I'm loath to rule _anything_ out. What I'm getting at is the notion that the stories could have been inspired by _past_ events, occurrences from many years ago that have left a residual, latent fear amongst the folk of this particular district.\"\n\n\"You've found something, haven't you, in your studies? Some reference that sheds light on what's going on at the moment?\"\n\n\"A reference that _may_ shed light on what's going on at the moment. In truth, it may also turn out to be entirely unrelated, although I find that difficult to believe, given the nature of the murders and the circumstances surrounding the deaths. I've already mentioned it to Bainbridge, but he puts no stock in the idea.\"\n\nVeronica leaned forwards in the carriage. \"Do tell.\"\n\nNewbury smiled. He was beginning to believe he'd made the right choice in hiring Veronica as his new assistant. \"About twelve years ago, there was a disturbing case in the Whitechapel area, in which a gang of petty thieves were discovered breaking into a house. Instead of fleeing the scene, they turned on the policeman who had found them, and viciously beat him to death. The thieves were never brought to justice, but for a month after the policeman's body was interred, a 'glowing bobby' was sighted around the streets, walking his beat and searching out his murderers, one after the other.\"\n\n\"What happened?\"\n\n\"They all turned up dead. Strangled, just like the victims we've seen in the last couple of weeks. Such was his vengeance, it was said, that the murdered policeman had actually risen from the grave to seek revenge on his killers. Once they were all dead, the 'glowing bobby' disappeared, never to be seen again.\"\n\nA ground train rattled by their cab, startling the horses and causing them to whinny noisily and pull up by the side of the road. The driver shouted down his apologies and waited for the other vehicle to pass before coercing the animals back out into the road.\n\nVeronica sat back in her seat. \"The parallels are uncanny.\"\n\n\"Indeed. But there are holes. Why would the spirit return now, after all this time? Did it ever really exist, or was it just a cover used by the dead man's colleagues to track down and dispose of his killers? What, if any, are the connections between the victims? I can't see a good reason for the spirit of the dead policeman to be taking these innocent lives, both men and women. I'm not convinced the profile actually fits.\"\n\n\"But you are convinced that it is _possible_? Have any other policemen been murdered in the area lately? Could it be the same phenomenon, but a different set of people involved?\"\n\nNewbury straightened his back. He looked thoughtful for a moment. \"My dear Miss Hobbes, what a splendid deduction! We'll get Bainbridge looking into it first thing this afternoon. I've been so wrapped up in trying to draw parallels between the two cases that I'd overlooked this most obvious of angles.\"\n\nBy this time, their cab was approaching Bloomsbury and the British Museum could be seen through the window, an epic, monolithic structure rising out of the grey afternoon. Newbury took his watch out of his pocket and examined its face. He glanced at Veronica. \"I don't know about you, but I'm feeling rather peckish. Spot of lunch?\"\n\nVeronica grinned. \"Sir Maurice, I'm famished.\"\n\nWith Miss Coulthard gone for the day, the office was silent when they returned from lunch, with just the ticking of the grandfather clock in the corner to break the monotony. The two rooms were connected by an interior door, the main office being a fairly large open space with Miss Coulthard's desk placed centrally to face the door. The walls were decorated with an array of spectacular artefacts, ranging from mediaeval weaponry to a glass display cabinet filled with smaller antiquities from Egypt, Greece and Rome. A small stove had been fitted in the far corner, and a series of bookcases were overflowing with ageing, dusty tomes.\n\nNewbury had just finished arranging his hat on the hat stand when Veronica, who had already gone through to the side room where their desks were located, reappeared in the doorway, brandishing an envelope.\n\n\"It's got the Royal seal on it. Someone must have delivered it whilst we were out.\" She handed it to Newbury. He opened it immediately, dropping the envelope to the floor.\n\n\"It's from the Queen.\" He unfolded the letter and began to read.\n\n> _To our faithful servant,_\n> \n> _It is requested you abandon all current activity and proceed immediately to Finsbury Park. An airship has crashed this morning in suspicious circumstances, and one suspects foul play. Early reports suggest no survivors._\n> \n> _Full report expected in due course._\n> \n> _This is a matter of grave importance to the Crown._\n> \n> _Victoria R._\n\nNewbury folded the note in half and slipped it into his jacket pocket. Veronica eyed him quizzically. He reached for his hat.\n\n\"We're off the murder investigation. At least temporarily.\" Veronica looked somewhat disappointed by the news. Newbury continued. \"There's been an airship crash in Finsbury Park. I'm afraid we're going out again.\" He pushed his arm into the sleeve of his long black overcoat and headed for the door. \"Come on, I'll explain on the way.\"\nCHAPTER 4\n\n##\n\nFrom over two hundred yards away, it was clear that the airship crash was a disaster of phenomenal proportions. Black smoke spiralled through the sky in a dark liquid trail, a smudge across the landscape, clearly denoting the point of impact and consequent explosion. The heavy fog was starting to lift now, but the scene it uncovered had Newbury wishing it had stayed put.\n\nThe wreckage was scattered across a wide area of parkland, isolated flames still licking in little glowing puddles where firemen had yet to extinguish the smaller pieces of debris that had come to rest in the area surrounding the main carcass of the downed ship. Hose carts circled the wreckage, whilst onlookers milled about a police cordon that had been established around the entire perimeter of the park. A tree was on fire on the far side of the site, and firemen were currently engaged in trying to bring it under control before the flames spread to the neighbouring evergreens.\n\nThe airship itself was now nothing but a burnt husk, its shattered substructure an exposed skeleton, stark against the surrounding parkland. It reminded Veronica of a beached whale she had once seen when she was a child, half-rotted in the sea air, its huge rib cage exposed to the elements.\n\nNewbury clambered down from the cab, choking on the thick smoke that lay heavy in the air all around them. The stench of the burnt vessel was almost unbearable. He turned to help Veronica down beside him, offering her a handkerchief to cover her face. She took it gratefully.\n\n\"What in God's name happened here?\" Her voice was muffled from behind the small piece of linen that she held over her mouth and nose. Her eyes watered, stinging with the smoke.\n\n\"Airships such as this one get their lift from gasbags filled with hydrogen. The gas is highly inflammable, and in a major impact such as this . . .\" He shook his head. \"Well, you can see the results for yourself. I've read about a handful of similar incidents. The most recent was in Bulgaria, I believe, where a pilot missed his berthing tether and instead lowered the ship onto the ground spike, ripping the gasbags open and engulfing the entire vessel in flame.\"\n\nVeronica looked grave. \"But all those passengers . . .\" She was staring out over the chaotic scene before them, unsure what to make of it all. She drew her coat around herself, an unconscious gesture that revealed her horror at the sight of the wreckage and the carnage it represented.\n\nNewbury was lost for comforting words. He paused and then looked around, straining to see over the bustling crowds of people. \"Come on; let's see if Bainbridge is here yet.\"\n\nTogether, the two of them circled the cordon, looking for signs of the Chief Inspector. Newbury kept a hand on Veronica's arm as they pushed their way through the press of locals, who had turned out in droves to catch a glimpse of the downed ship. Newbury supposed he couldn't blame them; for many it was a frightening near miss, with such a devastating explosion occurring so close to their homes. The vessel could easily have come down upon a row of terraced houses instead of the relative safety of the park. For others, it was surely a unique opportunity to witness something that they would usually only read about in newspapers, a sensational spectacle to tell their grandchildren of in years to come. From a purely detached perspective\u2014ignoring, for a moment, the human cost of the tragedy\u2014history was unfolding before their eyes.\n\nThey pressed on, fighting against the swarm of people in an attempt to find someone who looked in charge. Moments later, they found whom they were looking for.\n\nThe police had set up a temporary base underneath a bandstand, just inside the cordon at the far end of the crash site. Wreaths of dark smoke still curled through the air, and here, the stench of the wreckage was even more intolerable than when they'd first arrived. Newbury tried not to imagine what was causing the diabolical smell. He made his way over to the cordon line and called to get the attention of one of the men stationed there.\n\n\"Hello? May I have some assistance here, please?\"\n\nTwo men in suits, deep in conversation, looked around to eye the newcomer. One of them flicked his wrist to a uniformed officer, and the man came plodding over to where Newbury and Veronica were standing.\n\n\"Yes?\"\n\n\"I'm attempting to locate Sir Charles Bainbridge. Can you tell me, is he present at the site?\"\n\n\"No, sir. I don't believe he is.\" The other man looked irritated, as if anxious to get back to his post.\n\n\"Ah. Well, in that case, is there anyone else I could talk to?\" Newbury reached into his jacket pocket and produced his credentials, which he waved at the constable. The monogram of Queen Victoria was clearly visible on his papers. \"My name is Sir Maurice Newbury, and I'm here on the business of the Crown.\"\n\nThe constable stared at him, wide-eyed. \"Of course, sir. If you'd like to come this way?\" The officer lifted the cordon and both Newbury and Veronica dipped their heads to pass underneath the rope barrier. Veronica, straightening herself on the other side, made a point of repositioning her hat. Newbury supposed she was trying to keep herself busy and prevent her mind from wandering back to the horrors on the other side of the bandstand.\n\nThe two men in suits were still talking as the three of them approached. Veronica glanced around. She could see that the police were struggling to get the situation under control; they were few in number, and the constables were barely managing to keep the onlookers back from the cordon. Meanwhile, higher-ranking officers tried to coordinate the other emergency services and ensure that nothing was removed from the wreckage that would prove useful in uncovering the cause of the disaster. Veronica was sure that the investigation was already under way, but it seemed to her as if the police had their hands full just trying to stop the crash site from getting out of control.\n\nThe bobby who had led them over from the barrier made a point of clearing his throat, and the two men in suits ceased talking for a moment to take them in. The man on the left, dressed in pin-striped grey, with a full beard and dark green cravat, looked Newbury up and down discerningly. He seemed about to say something when the constable stepped in. \"Sir, this gentleman is here on behalf of the Crown.\"\n\nThe man nodded, an unreadable expression on his face. \"The Crown indeed. Well, we can certainly use all the help we can get. Abominable affair.\" His face cracked into a sad smile. He held out his hand. \"Inspector Foulkes of Scotland Yard.\"\n\nNewbury took his hand. \"Maurice Newbury.\"\n\n\"Ah, Sir Maurice. Yes, Sir Charles has told me all about you. Glad you could make it.\" He put his hand on the shoulder of the man he'd been talking with when they arrived. \"This is Mr. Stokes, representing the company that built and operated the airship in question.\"\n\nVeronica noted that Stokes was harbouring a dark frown.\n\nNewbury took his hand, inclining his head politely. \"Mr. Stokes.\" He stepped back, allowing the others to see Veronica, who had been standing behind him in the shadow of the bandstand throughout the course of the exchange. \"This is my assistant, Miss Veronica Hobbes. She'll be aiding me in my enquiries. Please ensure you extend to her all the necessary courtesy and freedom she requires to properly execute her role.\"\n\nFoulkes looked startled by this new development, but quickly spluttered his assent.\n\nNewbury turned to the man named Stokes. \"Mr. Stokes, I'd appreciate it if you could elaborate on some details for me. Have you any notion yet of what occurred to bring about this sorry situation?\"\n\nStokes looked immediately uncomfortable. He was a short, lean man, shorter than both Newbury and Foulkes and only a few inches taller than Veronica. He wore a brown suit and white collar, with black shoes that, Veronica noted, were filthy with mud, grime and ash from the crash site. His moustache was trimmed to perfection and waxed at the ends, and his bushy eyebrows did much to accentuate his apparently permanent frown. He shuffled nervously on the spot. \"Alas, we're only just beginning to piece together the sequence of events that preceded this tragedy. There is nothing in the wreckage to indicate what may have happened onboard, and we can see no obvious reason why it should have plummeted out of the sky as dramatically as it did. Unfortunately, there are no survivors left to question, either.\"\n\nNewbury shook his head, his face serious. It was obvious he didn't care for Stokes's dismissive tone. \"What of the ship itself? What was it, and where was it bound?\"\n\n\"The ship was named _The Lady Armitage,_ and according to my charter, it was bound for Dublin. It was a passenger-class vessel, the smallest size in the fleet, and appears to have been carrying around fifty individuals when it crashed.\"\n\n\"Fifty . . .\" Veronica was appalled.\n\nNewbury continued. \"And what of your employers, Mr. Stokes?\"\n\nStokes offered Newbury a black look. \"I'm a representative of Chapman and Villiers Air Transportation Services, of Battersea. Mr. Chapman himself has engaged me to assess the situation here today and to act as his mouthpiece during the ongoing investigation. Any questions pertaining to the company can be directed at me. I am also the firm's legal representative.\"\n\nNewbury glanced at Veronica, a sardonic expression on his face, and then turned his attention to Inspector Foulkes. \"Do you know if Sir Charles will be attending the scene?\"\n\n\"Not initially, sir. He has ceded responsibility for the case to me for the time being. He's still caught up in this damnable Whitechapel situation. They found another body this morning.\"\n\n\"Indeed. Miss Hobbes and I were present at the scene.\" He glanced back at Stokes, who was attempting to clean the dirt from his shoes by rubbing them on the grass. \"Do we know how long it's been since the vessel came down?\"\n\nThe other man didn't look up from his ministrations. \"Witnesses are reporting seeing the vessel come down between ten and ten thirty this morning.\" He emitted a _tut_ ting sound as he continued to rub the side of his shoe on the wet grass, to no avail.\n\nNewbury flushed red. \"Damn it, man! Fifty people are dead! Show some decency, and pay attention to the issue at hand.\"\n\nStokes ceased wiping his shoes and looked immediately flustered. He stammered something incoherent, which Newbury decided to take as an apology. Foulkes tried to cover his laughter at the man's expression with a loud cough.\n\nNewbury met Foulkes's eyes. \"I think the next logical step is for me to examine the wreckage.\"\n\n\"I'm sure that will be acceptable to Mr. Stokes.\" The Inspector shot the lawyer a sideways glance. \"But I will warn you, Sir Maurice, it is a disturbing experience. I toured the remains of the vessel as soon as it was cool enough to go aboard, and I assure you, it's no place for a lady.\" He made a point of stressing these last few words.\n\nNewbury was unperturbed. \"I appreciate your candour, Inspector Foulkes. Of course, it's up to the lady to decide for herself. Allow me to consult with Miss Hobbes in private for a short while.\" With that, he turned his back on the two men and drew Veronica to one side, under the shadow of the bandstand.\n\n\"Miss Hobbes. Veronica. I would not presume to ask you to follow me into the horror of this wreck. I did, after all, hire you to assist me in my academic pursuits, and not to risk life and limb clambering after me into the still-smouldering carcass of a downed airship.\" He paused, giving his words time to sink in. \"I'd be very happy if you preferred to wait for me here instead.\"\n\nVeronica crossed her arms. \"That's all very well, Sir Maurice, but what if you miss something fundamental? Surely a second pair of eyes would prove useful, especially when one considers the sheer size of the wreckage?\"\n\nNewbury smiled, trying to conceal his pleasure at her response. \"Very good. Well, better pucker up that resolve, my dear. It's going to be dangerous, dirty and pretty horrific in there.\" He was about to move off when another thought occurred to him. \"Oh, and hang on to that handkerchief, too. I suspect the smell will be unbearable as we get closer.\"\n\nNewbury returned to stand beside Inspector Foulkes. \"Miss Hobbes will attend the scene alongside me.\"\n\nFoulkes looked ready to object, before Newbury caught his eye. \"I assure you, I'll look after the lady's well-being. Now, what's the best way into the wreck from here?\"\n\nStokes answered. \"The ship came down nose-first, so the rear of the ship retains the bulk of its shape whilst the sub-frame at the front of the vessel has compacted, making it difficult to enter. I'd suggest you find your way around the left-hand side\"\u2014he indicated with a wave of his hand\u2014\"and enter through the main cabin door on the side of the gondola. I'm not sure what it is you're hoping to find in there, though, Sir Maurice. In truth, it's nothing but a burnt-out husk.\"\n\nNewbury shrugged. \"I'll know it when I see it, no doubt. Thank you for your assistance, gentlemen. We shall return presently, before the light begins to wane.\" He turned and offered Veronica his arm.\n\nFoulkes watched as the two Crown investigators, entirely incongruous in their formal attire, began walking slowly towards the huge shattered structure on the park green, cautiously stepping around the still-smouldering piles of debris as they walked.\nCHAPTER 5\n\n##\n\nThe wreck of _The Lady Armitage_ was like the carcass of some ancient, primordial beast; the skein of rubber-coated fabric that served as the outer skin of the vessel now burnt and torn like peeling flesh. The sub-structure of iron girders jutted into the sky like broken ribs, blackened by the soot and heat and buckled from the impact. The engine housing, at the rear of the wreckage, looked relatively intact, although in truth, it was hard to tell, as much of it was buried in the earth where the impact had driven it into the ground. The passenger gondola, usually situated underneath the belly of the ship, had been forced upwards and backwards, puncturing the underside of the vessel and contorting awkwardly where it came into contact with the iron struts of the interior frame. The whole thing was a terrible mess, and Newbury had to use every ounce of his experience to maintain a level head as he walked towards it.\n\nSteam and smoke still rose from deep inside the wreckage. As Newbury and Veronica approached the misshapen outer door of the gondola, Newbury felt the need to warn his assistant once again of the dangers they may face inside. \"Make sure you don't touch anything. It may still be hot from the fires. And watch out overhead, too; the wreck hasn't settled yet, and as the metal cools, fragments of the vessel may still collapse inwards, trapping us inside, or worse.\"\n\nHe covered his nose and mouth with the crook of his arm to stave off the terrible smell of death and burnt rubber. Veronica followed suit, once again holding Newbury's borrowed handkerchief to her face. The hem of her skirt was already thick with mud and soot where it trailed on the ground, her boots filthy with grime. She tried not to notice.\n\nThe door into the gondola had buckled badly. There was nothing but a blackened frame there now, where once there had obviously been an elaborate door and mechanism. Veronica peered inside, seeing nothing but darkness and iron girders. \"Are you sure this is the best way in?\"\n\n\"It looks like the _only_ way in, as far as I can tell.\" He looked around, confirming his suspicions. \"I wouldn't trust that man Stokes for a minute, but I can't fault his directions in this instance. Here, let me go first.\" Newbury tentatively put a hand on the outer rim of the door. \"Still warm.\" He gripped it more firmly with both hands and swung himself through the twisted metal opening. Veronica watched him disappear inside.\n\n\"Oh, well. For Queen and Country, I suppose.\" She grabbed the doorframe and swung herself in behind him.\n\nInside, Veronica found it difficult to get a sense of the scale of the ship. She was standing in what she assumed had been the lobby, although now, with fire and structural damage, it was difficult to tell. _The Lady Armitage_ may have been small by airship standards, but on the ground, it was still an immense vessel, and the passenger gondola was equally well-proportioned. Newbury was heading towards the compartments at the front of the gondola, if she had her bearings right. She watched him climbing over blackened furniture and the still-warm cinders of other unidentifiable objects. He turned back. \"I'm off to try to find the pilot's control room. You take a look around. We'll meet up again shortly.\" She looked the other way, trying to see a path through the scene of destruction. \"Oh, and Miss Hobbes?\"\n\n\"Yes?\"\n\n\"Be careful.\"\n\nShe smiled to herself, pleased with his evident concern.\n\nThe lobby\u2014or what remained of it\u2014was a long rectangular room with doors in each of the far walls. Since Newbury was heading fore, she decided to take the other option and see what she could find towards the rear of the vessel. She supposed, as she trod carefully over the ash-covered floor, that she was heading towards the part of the ship reserved for passengers, since the bulk of the gondola's interior space seemed to lie in this direction. Fighting her way past the crisp shell of a wooden sideboard, and ducking under a nest of trailing metal cables, she came to a stop in front of the door. It was still relatively intact, although flames had obviously licked black soot up and down its fascia. She hesitated. She knew she was likely to happen across a body or two on the other side. Taking a deep breath, she steeled herself. Her palate was growing used to the stench now, and her clothes were so thick with grime, dust and soot that she'd given up paying attention. She reached out and tried the handle, then immediately withdrew her hand. It was still hot from the fire, and even through her red-leather gloves, she knew it would scald her hand. Not only that, but the door appeared to have sealed shut with the heat. Stepping back, and looking around her to ensure no one was watching, she hitched her skirt up above her knees and sent her booted foot flying into the centre of the door. It gave a little in the frame, splintering where the wood had been stressed by the heat. She tried again, this time putting her full weight behind her as she drove herself forwards into the door.\n\nIt gave, bursting open and slamming back against an iron girder that blocked the way on the other side. She wondered, for a moment, if Newbury would come running at the noise, but after a short while had passed and she could hear no sound of him, she decided to press on. Pushing back against the door, she decided she'd try to squeeze through the gap she had created between the doorway and the girder. She tucked her hat underneath her arm, her dark hair spilling out of its carefully prepared coiffure.\n\nShe manoeuvred herself into the opening. Inside, she could still feel the residual warmth from the burnt-out interior. The floor was covered in a sticky mudlike residue, which she supposed had been created when the water from the hose carts had mixed with the soot and ash, forming a film of black grime upon the ground.\n\nShe looked around, and then dropped the handkerchief to the floor with a gasp. She stared in horror at the sight before her. Row upon row of passenger seats were filled with the remains of the dead. Horrific, skeletal cadavers sat fixed in their final death throes, gripping the seats in front of them, screaming at their neighbours, or else spilled out onto the floor, where they had tried to find somewhere to run. It was as if someone had set out a grisly diorama, a charnel house audience locked away in this horrible room, awaiting an appointment with God. She approached slowly, forcing back the rising bile in her throat. Her eyes filled with tears. It was the most appalling sight she had ever seen. She wondered why the people were nearly all still seated, why they hadn't tried to bail out of the ship as it crashed, or at least taken cover in the hope that they may survive the impending impact. The corpses were all blackened and burnt, cooked flesh still clinging to the bones, terrified screams still fixed on their faces. She had no way of telling which of them had even been male or female, save for the occasional piece of jewellery still hanging around a woman's throat.\n\nLeaning close to one of the bodies, she noted the answer to her earlier question: The person had been tied into their seat, fixed by a hoop around their left foot to the base of the seat in front. She checked another, and another, and found that they were all the same. No wonder the people hadn't tried to run. They couldn't.\n\nVeronica noticed a gentle patter of raindrops on her face. She looked up. High above, she could see the sky through the torn belly of the airship, the broken spokes of its internal structure poking up into the waning afternoon light. She realised almost immediately that the water droplets she had felt were not rain, but water from the hose carts, sprayed into the blazing inferno earlier that day and still dripping from the girders up above. She glanced around, looking for anything else that may be of use. She could see a hole in the left side of the room, where the firemen had obviously dug their way through from the outside in an attempt to find survivors. She wondered how those men had reacted to the scene that had faced them. Had they, too, been as appalled as she was? She finally gave in to her horror and vomited on the ground, her eyes stinging as she retched violently, over and over again, until there was nothing left for her body to expel. She stood, gasping, wondering if she'd ever be able to cleanse the smell of the burnt flesh from her hair and skin or, worse, from her nightmares. Perhaps she should have stayed outside after all.\n\nShe turned at the sound of the door banging against the girder. Newbury stepped into the room. He coughed, hacking on the smell of the still-warm bodies.\n\n\"My God.\" He rushed to Veronica's side. \"Are you alright?\"\n\nVeronica coughed. \"I'm not sure I shall ever be alright again. I just can't believe the devastation. So many people dead, burned alive in the fires. What a horrible way to die.\"\n\nNewbury looked saddened. \"It won't have been a lingering death. The collapse of the gasbags will have caused a series of massive fireballs to blow through the ship. That probably explains why they're all still in their seats.\"\n\nVeronica crouched down beside a row of seats. \"That, and the fact that they were all tied into position like common criminals.\" She showed him the loop of charred rope around the ankle of the nearest passenger.\n\n\"Stokes made no mention of the vessel being chartered as penal transport. Do you suspect he was trying to hide something?\"\n\n\"I believe he was trying to cover his own back.\" She stood again, blinking. \"What did you find in the control room?\"\n\n\"Nothing.\"\n\n\"Oh.\" She moved to turn away, anxious to put space between herself and the grisly scene, and then paused when he continued talking.\n\n\"That's just it. Nothing. No pilot or co-pilot to be found. No bodies, no evidence to suggest they were ever there at all. It's as if the pilot simply abandoned the controls.\"\n\nVeronica frowned. \"Do you think that's why the ship went down? Because the pilot wasn't at the controls? Could he have bailed out before impact? Or could he be back here, unidentifiable now from the other passengers?\"\n\n\"I suppose anything is possible.\" Newbury looked up, noticing that the light was starting to go. \"Come on. I think we've seen enough, and this is far from my ideal of one's first time aboard an airship.\" He looked circumspect. \"Besides, I do believe we have some more questions for Mr. Stokes.\"\n\nMr. Stokes was still standing around the police cordon when Newbury and Veronica edged up beside him. They were both filthy from climbing around in the wreckage, and Newbury was looking forward to retiring for the day, intent on a long soak in a hot bath. Stokes turned to regard them as they approached.\n\n\"Well, I do believe it's true what they've been saying. The Crown _is_ prepared to get its hands dirty from time to time.\" He guffawed at his own joke.\n\nNewbury was unmoved. \"Foulkes?\"\n\nStokes was obviously taken aback by Newbury's directness. \"Um, no. He's had to go off somewhere. Something about a fireman getting injured in the wreckage.\"\n\n\"Well, Mr. Stokes, perhaps _you_ could make yourself useful for a moment? I have another question, and it's very much in need of an answer.\"\n\nThe other man nodded, apprehensive now.\n\n\"What became of the ship's pilot? I've been down to the control room, and there's no evidence of a body. Indeed, there's precious little evidence that a pilot was even on board.\"\n\nStokes's complexion turned a ghostly shade of white. \"The, um, the pilot is missing.\"\n\n\"Missing? How does a pilot go _missing_? Did he bail out before the crash?\"\n\n\"Not exactly, Sir Maurice . . . If I can just\u2014\"\n\n\"Look, man, I'm in no mood for your ridiculous evasions now! Can you answer the question or not?\"\n\nVeronica put a hand on Newbury's arm in an effort to quell his rising temper. Stokes gave an audible sigh. \"There is no way the pilot of that vessel could have bailed out before the crash.\"\n\n\"And why is that, Mr. Stokes?\" This from Veronica, who had evidently decided to step in and calm the situation before things got out of hand.\n\n\"Because it wasn't a 'he.' It was an 'it.' \" He rubbed his hands over his face in exasperation. \"The pilot of _The Lady Armitage_ was a clockwork automaton, designed by Mr. Villiers himself. They're remarkable units, capable of many basic and, indeed, higher functions. But they are _not_ programmed to abandon their stations in an emergency. They're simply not capable of it.\"\n\nNewbury looked incredulous. \"An automaton piloting an airship! Why didn't you think to disclose this information before now? There's the probable cause for your disaster, Mr. Stokes! The unit clearly malfunctioned.\"\n\nStokes shook his head defensively. \"Oh no, Sir Maurice. That's simply not possible. The automata have been piloting airships for nearly six months now, and safety records have improved dramatically during that period. Up to eighty percent! The program is fully approved. We have all the necessary paperwork back at the office. I assure you, sir, that it's a simple impossibility that the unit malfunctioned. It's physically not possible.\"\n\n\"So where is the unit now, Mr. Stokes?\" Veronica smiled in a placatory fashion.\n\nStokes cleared his throat. He was clearly unhappy with the course of the entire conversation. \"I'm afraid I have no idea. My report will state that the device was destroyed in the explosion. Now look\u2014\" He waved a manifest in front of them. \"\u2014I really have to be getting on. I'm expected to provide a full passenger register for the police before the day is out.\"\n\n\"Of course. We're sorry to have kept you.\" Veronica took Newbury's proffered arm and began to walk away. Then, as if just remembering something, she stopped and looked back. \"Oh, and Mr. Stokes? Just one more thing before you go?\"\n\n\"Yes?\"\n\n\"Could you tell me why all of the passengers were confined to their seats, with loops of rope around their ankles?\"\n\nStokes looked as if he were about to choke. \"A simple safety precaution, Miss Hobbes. In case of emergency, all passengers are required to insert their left foot into the safety brace underneath the seat in front. It stops people tumbling all over the craft if the pilot encounters dangerous turbulence whilst airborne.\"\n\nVeronica nodded. \"Thank you, Mr. Stokes. You've been most helpful.\"\n\nShe watched with Newbury as the little man scuttled away, keen to put distance between him and the ire of the moonlighting academics. The light was fading now, the sun low in the sky over the city. The crowds of people around the edges of the park had begun to thin and disperse.\n\n\"You understand, of course, that there's no feasible way in which the skeleton of a brass automaton could have been incinerated in that blaze? Especially when one considers that the majority of the human cadavers are still relatively intact.\" Newbury sounded contemplative now, rather than angry.\n\n\"Yes, my thoughts exactly.\"\n\n\"I'm beginning to think that Her Majesty's suspicions were correct. Something is definitely wrong here, and I'll wager it has its roots in the offices of Chapman and Villiers Air Transportation Services.\" He sighed, blinking to keep himself alert. \"For now, though, I think it's time I retired to my lodgings. Can I drop you at home on my way, Miss Hobbes?\"\n\nShe nodded, clearly exhausted. \"Please do, Sir Maurice.\"\n\nHe held the cordon for her as they took their leave of the crash site and made their way to the nearest carriage.\n\nThe evening was still and cold as Newbury, attired only in a simple dressing gown, settled in his study before a roaring open fire. A book was open on his lap\u2014 _Trelawny's History of Esoteric Societies of the Seventeenth Century_ \u2014one of the many aged, leather-bound volumes that lined the walls around the room. Other shelves held more bizarre specimens: vials of chemical compounds, jars filled with preserved biological samples, a pentagram cast out of twenty-four carat gold, the bleached skull of a chimpanzee and much more besides. Paper files were stacked neatly in rows along one wall, containing reams of case notes, old academic papers, clippings and other assorted reference materials collected during many long hours of research. The study was his private haven, the room he filled with all the ephemera of his life. It was the one place where he could relax, where he felt free to become himself and where much of his actual deduction was carried out; over time, the study had become a place of revelation. He eased back in his armchair and turned the pages in his book.\n\nMrs. Bradshaw had retired for the evening after drawing him a bath and admonishing him enthusiastically for the state of his clothes. He smiled. She was forbidden from entering the study, but if she were to ever see its contents\u2014not least the cluttered manner in which he liked to keep it\u2014he wagered she'd flee his service at once. Not only that, but many of his files contained confidential information that needed to be kept away from prying eyes. He had no reason to doubt Mrs. Bradshaw's integrity, but he suspected the contents of his files would be enough to discredit the monarchy at least ten times over, and he feared what temptation could do to even the most loyal of people. For that reason, he kept the door to the room locked at all times, even when he was inside of it. He'd invited Bainbridge in once or twice, for he trusted him implicitly, and after the events of the previous summer\u2014during which they'd hunted a madman intent on inflicting an Ancient Egyptian plague on London\u2014he knew the man had a stomach for the bizarre.\n\nTonight, however, he was happy for the solitude. He sat watching the dance of the flames for a while. He couldn't help thinking of the ruined, tortured faces of the corpses in the wreck of the airship that he'd seen that afternoon. Veronica had taken it badly, but so, in truth, had he. He'd seen innumerable corpses in his lifetime, of course, but in this instance, it was a matter of scale; never before had he witnessed a scene quite so horrifying as this.\n\nHe reached for a small brown bottle from the shelf behind his head. The label was peeling, but he knew well what it contained. He unscrewed the lid and poured a measure of the liquid into the half-full glass of claret that rested on the side table by his armchair. The laudanum would help him sleep, or so he told himself as he raised the glass to his lips and took a long drink. In the morning, he would meet Veronica at the office, and they would head to Battersea, to Chapman and Villiers's manufactory. There he hoped to find out more about the mysterious automata and their creator, Mr. Pierre Villiers, an exiled Frenchman who\u2014he had read\u2014had been brought up on charges over a decade ago for experimenting on human wastrels in his Parisian laboratory. Still, that was for the morning. For tonight, he hoped, oblivion was near at hand. He drained his glass and sank back into the comfort of his Chesterfield, waiting for the laudanum to do its work.\nCHAPTER 6\n\n##\n\nGiven the heavy fog of the previous day, the morning seemed unusually bright as Veronica made her way up the steps outside the main entrance of the British Museum. Birds twittered in the trees overhead, and the sun poked through the clouds to sprinkle bright columns of light across the city.\n\nAfter the horrors of yesterday, Veronica had retired to her lodgings in Kensington, where she'd bathed, eaten and gone directly to bed. Now, feeling somewhat refreshed, she hoped that the coming day would prove less fraught, and also less likely to inspire nightmares. The scenes from the crash site were still emblazoned on her mind, and she tried to push them to the back of her thoughts as she prepared herself for what the new day might bring.\n\nWatkins, the doorman, was on hand to permit her entrance to the museum at this early hour, and he did so with a kindly smile. It was not yet eight, but she suspected Newbury would already be sitting at his desk, reading his newspaper as was typical of his morning routine. All the more surprising, then, was the scene that greeted her when she did finally open the door to the office on the basement floor. Newbury was nowhere to be seen, his desk undisturbed, his coat and hat distinctly absent from the stand inside the door. Instead, Miss Coulthard sat at her desk, her face in her hands, tears streaming down her cheeks in desperation and dismay.\n\n\"Oh, Miss Hobbes. I'm sorry that you should happen upon me in this state.\" She looked up at Veronica as she came through the door.\n\nVeronica quickly peeled off her coat and hat and pulled a chair up beside Miss Coulthard, taking her hand in her own. \"I take it there's still no news?\"\n\nMiss Coulthard, sobbing, nodded briskly. \"We've had no word. Neither have his employers. We all fear the worst, Miss Hobbes. I can think of no reason why he'd stay away this long, unless the revenants have got him.\"\n\n\"Now, Miss Coulthard, we don't know anything for sure. I do think it's unlikely that he's had a run-in with one of these 'revenant' creatures. I hear lots of talk about them, all over the city, but I'll admit I've yet to see one myself, and in truth, I'm starting to wonder if they even exist at all.\" She smiled warmly. \"Have you seen one with your own eyes, Miss Coulthard?\"\n\n\"No, Miss Hobbes, I can't say that I have.\"\n\n\"There you are, then. Neither of us can even verify their existence. So how likely do you find it that Jack may have encountered one on his way to work?\"\n\n\"Well . . .\" Miss Coulthard wiped her eyes, sniffling. \"I suppose not likely at all. It's just . . .\" She screwed her hands into fists, frustrated. \"What _else_ could have happened to him?\"\n\nVeronica rubbed the back of her neck. \"Well, that's what we'll engage the police to find out today. I'm sure it'll turn out to be something quite innocent.\"\n\nMiss Coulthard smiled. \"Thank you, Miss Hobbes. I've been waiting here for Sir Maurice to accompany me, after what he said to me yesterday, but he hasn't arrived as yet. I fear he's made other arrangements or decided to go elsewhere this morning, on an errand or suchlike.\"\n\nVeronica glanced at the clock, a slight frown crossing her face. \"No, no. We definitely arranged to meet here this morning. I'm sure he's just been held up. When he arrives, we'll put on a fresh pot of tea and then I'm sure Sir Maurice will send a note across town to his associates at Scotland Yard.\" Veronica noticed that Miss Coulthard had reached into her pocket and was now clutching a small sepia photograph to her chest. \"Miss Coulthard, may I enquire as to the identity of the person in your photograph?\"\n\nThe secretary looked down, staring at the photograph as if seeing it for the first time. She held it out to Veronica. \"My brother, taken before he went off to war.\"\n\nVeronica took the battered old picture and gave it an appraising look. A man, dressed in a field uniform, posed for the camera, a rifle cocked over one arm, his other arm resting against a large stone plinth. The backdrop was a large canvas showing paintings of trees and other unidentifiable flora. \"He's very handsome, Miss Coulthard.\" She turned it over. There was an inscription on the back, written in a shaky hand. It read: _Jack Coulthard, January 1901._ \"Where did he see action?\"\n\n\"India. He was invalided out six months ago, after he was savaged by a wild animal whilst on patrol. The men he was with were all killed. We were blessed that he survived. They told us he was gripped by a terrible fever for days after the incident. When he returned home he was a shell of his former self. But he picked himself up and applied for an apprenticeship at Fitchett and Browns. They've done well by him, too. He's made quite a name for himself amongst the junior members of the establishment.\"\n\n\"I'm glad to hear it, Miss Coulthard. Now, I think this photograph will be useful for the police, if you can bear to part with it for a short while? They'll be able to use it to show Jack's likeness to their officers. It'll make it easier for them to spot him if they know exactly who they're looking for.\"\n\nMiss Coulthard nodded. \"I thought as much.\" She passed Veronica the picture and watched as the other woman slipped it safely into her purse. \"I don't know what we'd do without him. It'll ruin us if he can't be found.\"\n\n\"I'm sure it won't come to that. Now . . .\" Veronica trailed off at the sound of footsteps on the other side of the door. \"Ah, that sounds like Sir Maurice. Come on, let's get that pot warming.\" She rose to her feet just as the door swung open and Newbury stepped into the office. He looked haggard, as if he hadn't slept. Dark rings circled his eyes, and his face had taken on an unusual pallor. He lifted his bowler hat from his head and smiled. \"Good morning, ladies.\"\n\nVeronica looked immediately concerned. \"Sir Maurice, are you unwell?\"\n\nHe shook his head dismissively. \"Only a malady of my own making, I fear, my dear Miss Hobbes. Nothing a strong cup of Earl Grey won't fix.\" He draped his coat on the stand beside him. \"Miss Coulthard. Any news on your missing sibling?\"\n\nThe secretary shook her head, fighting back further tears.\n\nNewbury frowned. \"Well, give me your address on a piece of paper, along with the particulars of the last time you saw your brother, his place of work and any distinguishing marks that may help the police to identify him. If you have it to me in the next half hour, I'll dash off a note to my friends at Scotland Yard.\"\n\n\"Thank you, Sir Maurice. I'm very much obliged to you.\"\n\n\"Say nothing of it, Miss Coulthard. It's the very least I can do.\" He rubbed his hand over his chin. \"Now, Miss Hobbes, let us adjourn to my desk and see if we can't plan our next move.\"\n\n\"I'll be with you directly, Sir Maurice, just as soon as I've organised this pot of tea.\" She watched as he disappeared through the partition door, unsure what to make of his sudden change in demeanour.\n\n\"So what you're saying is that you're not convinced that the automaton was the cause of the disaster?\"\n\nNewbury nodded. His colour had returned, and he seemed imbued once again with his usual energy. Veronica had to admit she was relieved; when he'd walked through the door that morning, she'd been just about ready to hail a cab and ferry him to the nearest doctor. Now, after a recuperative cup of tea and a few minutes spent composing a note for Miss Coulthard, he was cheerfully engaged in outlining his current thoughts on the matter in hand. \"What I'm saying is that I'm willing to hold off judgement until I've seen the evidence for myself. I've seen one or two of these automata demonstrated in my time, and they're certainly amazing creations. Technology moves quickly, these days. If you've any doubt, just look up at the sky.\" He gestured with both of his hands. \"Chapman and Villiers is one of the preeminent air transportation organisations in London. If even a quarter of those airships above the city are under the control of an automaton, then in my book, that's a wondrous thing indeed!\"\n\n\"I don't doubt you're right, Sir Maurice, but we must be sure not to let our enthusiasm for technological developments cloud our judgement in this matter.\"\n\nHe looked at her slyly. \"I can see you've got a sharp sense about you, Miss Hobbes. You're absolutely right, of course. But equally I trust you will not damn the technology before we have carried out the due investigative process.\"\n\n\"Agreed. Even if Mr. Stokes is an odious wretch who did nothing but cloud my opinion of his organisation.\"\n\n\"Indeed. If we're lucky, we'll have no further dealings with the man today.\"\n\nVeronica sipped her tea thoughtfully. \"So, what of the Whitechapel murders? Have you thought any further on the mystery of the glowing policeman?\"\n\nNewbury shook his head, slowly. \"Alas, I've had to forgo that particular case, for the time being, anyway. If we get to the bottom of this airship issue quickly enough, I'll see what I can do to help. Otherwise, I'll just have to point Charles in the right direction and hope he can get to the bottom of it himself. He's got plenty of good men at his disposal, and if the case does turn out to have a supernatural origin, it won't be the first time he's come up against that sort of thing and won.\"\n\nVeronica raised her eyebrows.\n\n\"A story for another time, perhaps.\" He stood, pulling on his gloves.\n\nVeronica placed her cup back on the saucer. \"One last question before we take our leave. May I ask why this crash is deemed so important to the Crown?\"\n\nNewbury paused for a moment, as if deciding how much he should disclose to this woman, whom\u2014despite her having been in his employ for only a matter of weeks\u2014he was already beginning to trust with his life.\n\nVeronica took his lengthy pause as a sign of his disapproval. She flushed red. \"Oh, please forgive me! Have I overstepped the mark?\" She stood, nearly knocking her cup and saucer over as she banged awkwardly against the edge of his desk.\n\nNewbury waved her to sit down again. \"No, not at all, Miss Hobbes. The truth of the matter is simple: I don't know. I'll admit, I'm finding that question peculiarly frustrating. I can see no obvious connection between the affairs of the monarchy and the disaster that became of _The Lady Armitage._ Not only that, but the Whitechapel case is more definitely within my area of expertise.\" He sighed. \"Nevertheless, one must do one's duty. And I must admit, I'm rather intrigued by this whole automaton business.\" He held the door open for Veronica and ushered her through.\n\nMiss Coulthard was sitting at her desk, the nib of her pen scratching noisily as she attempted to transcribe one of Newbury's recent academic papers for the museum archives. He shook his head as he collected his coat. \"Miss Coulthard? Did you manage to have my letter sent to Scotland Yard as I instructed?\"\n\n\"Yes, Sir Maurice. I sent it by cab as you requested.\"\n\n\"Very good. Then I must ask you what you're still doing here, scratching out one of my illegible essays when you should be at home, awaiting news of your brother?\" He smiled warmly.\n\n\"Well, sir, this document was supposed to be completed for filing yesterday. I was concerned about getting behind in my work.\"\n\n\"Poppycock! Now, Miss Hobbes and I will be gone for the rest of the day, so I dare suggest you won't be missed. Go on, be off with you. I shan't take my own cab until I'm convinced you're well away from this place.\"\n\n\"Thank you, sir. I won't forget your kindness.\" She placed her pen carefully back in the drawer and fumbled with her papers.\n\nA moment later, when Miss Coulthard had collected her belongings, the three of them left together, locking the door to the office behind them.\nCHAPTER 7\n\n##\n\nFrom the Chelsea Bridge, the airship works were clearly visible in the morning light as a series of immense red brick hangars, squat beside the shimmering Thames, fumes rising like smoke signals from a row of tall, broad chimneys. Steam hissed from outlet pipes in great white plumes, whilst water gushed back into the river in a deluge of brown sludge. Huge airships were tethered to the roofs of the hangars, reminiscent of a row of children's balloons, bobbing languorously in the breeze.\n\nNewbury looked out over the river. Ships and boats of all shapes and sizes drifted lazily along the shipping lanes, dipping gently with the ebb and flow of the water. It was busy, thick with the detritus of industry. It was noisy, too: horns blaring and gulls chattering over the constant clatter of horses' hooves as they rolled over the bridge towards their destination. He caught sight of one ship to which the others were giving a wide berth. He studied it for a moment through the window. Large red crosses had been painted on the sides of the hull, and the flag had been lowered to half-mast. He guessed it was a plague ship, carrying the corpses of the dead out to sea, where they would likely be dumped, unceremoniously, into the water. He knew from his discussions with Bainbridge that the corpses of plague victims had been turning up all over the city, particularly in the slums, where the people lived in squalor and the virus could easily spread from host to host. Stories of the \"revenants\" were spreading, too, with the daily newspapers parroting the rumours heard on the streets and sensationalising the epidemic for the gleeful consumption of cockatoos such as Felicity Johnson. They were right to fear, though; before the virus killed its host, it would completely unravel their humanity, transforming them into a monstrous killing machine. Their flesh would stop regenerating, their only thoughts becoming animalistic, feral; in short, they would be reduced to nothing but the basest of creatures, and with that loss of faculty, they'd become almost unstoppable, feeling no pain, showing no awareness of wounds that would kill an average man. It was as if the virus, somehow, kept them alive through all of this, waiting for an unidentified biological trigger. Then, after a handful of days had passed, the virus would complete its work and turn their brains to sponge, dropping their spent, lifeless bodies by the side of the road. It was a bad way to go. He hoped, for Miss Coulthard's sake, that she was wrong and that her brother had so far managed to evade infection. Everything he knew about the virus suggested if that if he _had_ been infected, by now he'd either be dead in a gutter or else stalking the fog-shrouded streets by night, a mindless monster in search of food and blood.\n\nNewbury closed his eyes for a moment, lulled by the motion of the cab. He imagined that Her Majesty would be growing impatient with the crisis by now, keen for the virus to burn itself out in the poorer districts of the city. She probably had a hundred scientists searching for a vaccination. If no solution were found soon, he had no doubt that she would place a cordon around the slums in an effort to slow the spread of the disease. Everyone was anxious, fearful of what might happen if the plague truly managed to get a grip on the city. Some projections suggested that up to 50 percent of the population could succumb to the illness: if not killed by the virus itself, then taken by one of the rampaging monsters it created. He suspected that it would be some time yet before the issue came to a head, and that the worst was probably still to come.\n\nHe looked up. Veronica sat in silence on the other side of the cab, lost in thought. Her hands were folded neatly on her lap, her face turned to the opposite window. She was wearing a powder blue jacket and white blouse, with matching culottes. He admired her modern sensibilities. Indeed, he admired much about her. Searching around for another distraction, he chose not to disturb her reverie. Instead, he unfolded his morning copy of _The Times_ on his knee and inspected the day's headlines. Unsurprisingly, the editor had chosen to dedicate the front page to a huge article about the _Lady Armitage_ disaster. The headline read AIRSHIP CRASHES IN FINSBURY PARK: SABOTAGE SUSPECTED, UPWARDS OF 50 DEAD. Newbury shook his head. Sabotage suspected? He wondered if Stokes had been feeding ideas to the press. He certainly wouldn't put it past the man. He hoped to find the company's directors a little less repellent but was expecting to be disappointed. In his experience, like invariably attracted like, and any associates of Mr. Stokes would either have to maintain a will of iron or an ego as enormous as that of Stokes himself.\n\nHe settled back in his seat, flicking through the pages of newsprint on his knee. He was still feeling delicate from the excesses of the laudanum, and silently chastised himself for giving in to his cravings. Miss Hobbes was astute, and his late arrival at the office and less-than-savoury appearance that morning had not gone unnoticed. He resolved to represent himself better in future.\n\nThe driver tapped loudly on the top of the cab, and both Newbury and Veronica looked up in surprise, dragged away from their thoughts.\n\n\"Yes?\"\n\n\"Is this the place you're looking for, sir?\"\n\nHe glanced out of the window. The cab had come to rest outside a small office building appended to a much larger complex of industrial hangars and factories. A sign above the door read CHAPMAN & VILLIERS AIR TRANSPORTATION SERVICES.\n\n\"Yes, thank you driver, this is the place.\" He sighed, and caught Veronica's eye, folding his newspaper under his arm as he did so. \"Are you ready, my dear?\"\n\n\"Absolutely.\"\n\n\"Well then, after you.\" He watched her clamber down from the cab to the street below. He had a feeling that today, one way or another, some of the missing pieces of the mystery would begin clicking into place.\n\nThe offices of Chapman & Villiers were an austere affair, housed within a separate structure that was divorced from the factory proper by a large courtyard and an elaborate set of cast-iron gates. Clearly the proprietors were intent on maintaining a strict distance between their visiting clientele and the factory workers, who, Newbury guessed, would likely have a separate entrance somewhere around the rear of the complex. It appeared, from the signs evident in the windows, that the office not only dealt with the company's commercial affairs but also served as a travel agency, of sorts, selling passage on its fleet of charter vessels to locations all over the globe, from Prussia to China, Jersey to Hong Kong. Newbury toyed with his gloves for a moment. \"Well, Miss Hobbes, I do hope you have your detective's cap on?\"\n\nIn reply, she stepped forward and pulled the office door open before her. It groaned loudly on its hinges. \"Of course. After you, Sir Maurice.\"\n\nHe shook his head, taking the door from her and ushering her inside. \"Come now, Miss Hobbes, let's do things properly.\"\n\nThe main reception area was as sobering in appearance as one expected after taking in the view of the building from the outside: the walls were hung with a dark, burgundy covering that seemed to soak up all the light, and a scattering of chairs were situated beside low coffee tables and tall leafy plants. A set of short stairs led up to another, unseen level. A clerk sat in one corner with his back to them, talking to a customer in hushed tones about purchasing transport to the Far East.\n\nBut their attention was most immediately drawn to the man behind a mahogany desk in the centre of the room, his fingers forming a perfect pyramid before him on the polished surface, his pale face belying his apparent displeasure at receiving customers so close to lunch. When he spoke, his voice was thin and nasal. \"Can I help you?\"\n\nNewbury strode up to the desk and placed his hat down beside a sheaf of paper files. The clerk looked at the item as if it were a horse's head, his disdain clearly evident.\n\n\"I'm here to see Mr. Chapman.\"\n\nThe clerk made a show of looking in his ledger. \"Are you sure, sir? I have no meetings scheduled for Mr. Chapman today. He really is a very busy man.\" He shut his ledger as if that were simply the end of the matter. \"Perhaps you'd care to make an appointment?\"\n\n\"I'm afraid you don't seem to understand. It's imperative I speak with Mr. Chapman today.\" Newbury glowered at the man behind the desk.\n\n\"Imperative, you say, sir? Could I enquire as to what business you may have with my employer that could possibly be so urgent? If you're looking to make a complaint about a recent journey, then you can find the forms behind you on the table there.\"\n\nNewbury sighed. \"I'm here on the business of the Crown. It is a delicate matter that I wish to discuss with Mr. Chapman in private. Of course, if you'd prefer me to air his private business out here\u2014?\"\n\nThe man's entire demeanour changed. His face seemed to flush with colour, and his pursed lips split into a wide smile. He swallowed, and parted his hands in a conciliatory gesture. The timbre of his voice became immediately more welcoming. \"Of course, sir. I quite understand. Allow me to go and enquire as to whether Mr. Chapman is available. May I offer him your name?\"\n\n\"Sir Maurice Newbury.\"\n\n\"Please take a seat, Sir Maurice. I will only be a moment.\"\n\nNewbury watched as the clerk scuttled out from behind his desk and crossed the office, glancing once behind him to see if Newbury was watching. He climbed the stairs and disappeared from view. Veronica lowered herself into one of the chairs, smiling to herself. Newbury paced the office, obviously impatient.\n\nA moment or two later, the clerk appeared at the top of the stairs. He climbed down, his hands clasped behind his back, and approached Newbury tentatively, as one might approach a lion. \"Mr. Chapman is in his office and would be only too delighted to make your acquaintance, Sir Maurice. I will show you up now.\" He beckoned for them to follow. Newbury remembered to reclaim his hat before helping Veronica to her feet.\n\nAt the top of the stairs, three doors led into what Newbury supposed were private offices. The clerk hesitated before the middle one, clearing his throat. He rapped politely, three times, and then opened the door with a flourish, stepping to one side to allow them to enter.\n\n\"Your visitors, sir.\"\n\nNewbury followed Veronica into the room, his hat tucked carefully under his arm.\n\nIt was a large office, and ostentatiously furnished, cluttered with artwork and fine goods from all corners of the globe. Newbury glanced around, trying to get a measure of the place. A large marble fireplace dominated one wall, whilst above it, a portrait of the Queen looked mournfully down upon the visitors. A display case in one corner held relics from as far afield as Constantinople, Baghdad, Greece and Delhi; souvenirs, Newbury supposed, from journeys undertaken in pursuit of business in those far-flung nations.\n\nChapman himself lounged in a large Chesterfield, smoking a cigarette. His hair was blond and cut long around his shoulders, and he was dressed in his shirtsleeves and a black waistcoat. Newbury thought he had the look of a cat about him, languorously warming himself before the fire. He stood as Newbury entered the room, and moved quickly to shake his hand. \"Sir Maurice Newbury, I presume?\"\n\n\"Indeed.\" Newbury took his hand and shook it firmly. He stepped to one side. \"Allow me to introduce my assistant, Miss Veronica Hobbes.\"\n\nChapman smiled and took her hand, holding it for just a moment longer than was necessary, before inclining his head politely. \"Delighted, I'm sure.\" He gestured at the clerk, who was still standing in the doorway. \"Now, can my man Soames fetch you any refreshments? A brandy, perhaps?\" He glanced at the grandfather clock in the corner. \"Not too early for that, are we?\" He looked baffled, as if he'd only just realised the time.\n\nNewbury shook his head. \"A pot of tea would be fine. Earl Grey, if you have it?\"\n\nChapman nodded briskly, and Soames disappeared again, clicking the door shut behind him. They heard his footsteps on the stairs as he descended to the office below.\n\nChapman beckoned for them to take a seat, folding himself back into his chair. He reclaimed his cigarette from the ashtray on the table and took a long, luxurious draw. It was clear to Newbury that the man didn't give much thought to convention: his entire manner was at odds with his station, and his appearance marked him as something of a fop. Nevertheless, he couldn't help feeling drawn to the man's bohemian charm. He could see immediately that there was a cool intelligence lurking behind the darting ice-blue eyes, and whilst he didn't put much stock in the man's taste in furnishings, he had to admit the fellow had an acute nose for business. Either that or he was spending his inheritance at a rate that would soon see him bankrupt or destitute. Chapman tapped his cigarette in the ashtray and regarded Newbury with a wistful smile. \"So, Sir Maurice, I presume you are here regarding that terrible business with _The Lady Armitage_?\" He looked suddenly serious. \"A truly lamentable affair.\"\n\nNewbury nodded. \"Yes. Have you visited the site of the wreckage yourself, Mr. Chapman?\"\n\n\"No.\" He paused to take another draw on his cigarette. \"Unfortunately, I was previously engaged\u2014a small matter to resolve with my banker\u2014so I took the liberty of relying on my legal representative, Mr. Stokes.\"\n\nNewbury stiffened. \"Yes, I spoke with Mr. Stokes for a brief while yesterday.\"\n\nChapman smiled knowingly. \"Terrible bore, isn't he? Seems to be the way with these legal chaps. Dependable, though. I trust he gave you everything you required?\"\n\nNewbury nodded. \"In a manner of speaking. Nevertheless, I thought it wise to pay you a visit this afternoon, in an effort to get a better understanding of your operation, and to see for myself these automata that Stokes mentioned.\"\n\nChapman's eyes seemed to light up. \"Ah, the automata. Villiers's prized creations. They are impressive machines, Sir Maurice, if you have not yet seen one?\"\n\nNewbury glanced at Veronica. \"Indeed not. I would certainly welcome a demonstration.\"\n\n\"I'm sure that can be arranged.\" He reached over and crumpled his cigarette in the ashtray. \"And you, Miss Hobbes. I'm sure you'd find the machines equally as impressive.\"\n\n\"I'm sure I would, Mr. Chapman.\"\n\nNewbury looked up at the sound of rapping on the door, and then Soames entered, bearing their tea on a large platter. He crossed the room and placed it on the table before them. Chapman watched him turn and leave, waiting until the last moment to call after him. \"Thank you, Soames.\"\n\nNewbury scratched his chin absently. \"So, Mr. Chapman, Mr. Stokes mentioned yesterday that one of these remarkable new automata was behind the controls of _The Lady Armitage_ when she went down?\" Veronica studied the other man's face, watching for a reaction.\n\nHe remained impassive. \"Quite possible. I believe around half of the fleet is now piloted by the machines. We even have a Royal charter. Remarkable, really, when you come to think of it.\"\n\n\"Quite.\" Newbury paused. \"Mr. Chapman, I'm not sure if you're aware of all the circumstances surrounding the disaster yesterday morning?\"\n\nChapman looked puzzled. \"Mr. Stokes provided me with a thorough report of his findings. I also spoke with Inspector Foulkes of Scotland Yard. I'd imagine myself to be in full possession of the facts.\"\n\n\"Did Mr. Stokes's report make reference to the fact that the pilot of the vessel appeared to be missing from the wreckage?\"\n\nChapman fished around in his waistcoat pocket, searching out his silver cigarette case. He flicked it open and withdrew one of the small white sticks, then offered the case around to the others. When they didn't accept, he slipped it back into his pocket and struck a match with a loud rasp. Smoke billowed around his face as he regarded Newbury. \"He made mention of the fact that the unit in question had been destroyed in the impact.\"\n\nNewbury met his gaze. \"I find that very difficult to believe, Mr. Chapman. I understand the skeletal frames of these automata are constructed out of brass?\"\n\n\"Correct.\"\n\n\"Then why were there no remnants of the unit in evidence anywhere on board the ship? Both Miss Hobbes and I toured the wreckage, and I can assure you, there was nothing to be found.\"\n\nChapman poured the tea, his face thoughtful. \"Well, if Mr. Stokes's assertions are correct, the unit may have burnt up in the fires that followed the crash.\"\n\nNewbury sipped from his teacup. \"Come now, Mr. Chapman. We both know that the heat in that wreckage would never have reached a temperature enough to incinerate brass. There has to be another explanation.\"\n\nChapman shrugged apologetically. \"Perhaps it survived the incident and clambered out of the wreckage, wandering away into the park?\"\n\n\"The police are certainly following that line of enquiry. Tell me, do you have any notion what may have gone wrong with the unit to cause it to lose control of the vessel, Mr. Chapman?\"\n\nChapman shook his head. \"As I understand it, Sir Maurice, the automaton was not responsible for the crash. We've had an impeccable safety record throughout the fleet since the implementation of these machines. I find it far more probable that, regrettably, there was a mechanical fault with the vessel itself.\"\n\n\"So you put no stock in the notion that the automaton unit may have malfunctioned?\"\n\n\"I do not. Although in truth, you'd have to ask Villiers. He's the man who invented the things; he should be able to give you a better idea of their functions and limitations.\" He shrugged.\n\nVeronica placed her empty teacup on the table. \"So, Mr. Chapman, where would we find Mr. Villiers?\"\n\nChapman smiled. \"He'll be in his workshop behind the mechanical works. I can take you there, if you like, by way of the airship manufactory?\" He stood, not waiting for a reply. \"What do you say? A quick tour of the facility?\"\n\nBoth Newbury and Veronica rose from their seats. Veronica met Newbury's eye. \"Mr. Chapman, I think that would be an excellent idea.\"\nCHAPTER 8\n\n##\n\nThe hangar was cold, and Veronica hugged her jacket to herself, wishing she'd thought to bring a shawl or a more substantial overcoat along with her that morning. Her breath fogged in the air before her face. She tried to avoid shivering.\n\nThey were standing on a steel walkway above the main factory floor, where the huge shell of an airship gondola was currently under construction. It sat upon a large wooden pallet, squat in the centre of the massive room, scaffolds running over its surface like the strands of a vast spider's web, ensnaring the bowels of the partly erected ship. Men buzzed around the skeleton of the vessel like worker ants, swarming up the sides of the scaffolds to place glass panes into the wooden window-frames and pass doors, seats and other furnishings through to the workmen inside. Tools clattered loudly, and men shouted to each other above the noise.\n\nVeronica stared down from the railing that ran along the side of the walkway. After her experience the previous day, she found the sight of the unfinished gondola incredibly eerie, reminiscent of the smashed wreck of _The Lady Armitage._ Many of the fittings were the same as those she had seen inside the shattered vessel, and from where she was standing, the internal layout looked practically identical. She could hardly bear to look at the passenger cabin, with its row upon row of empty seats, without visualising the scene inside of the burnt-out ship; the blank, ruined faces of the dead staring back at her, accusingly. She fancied for a moment that she could still smell the stench of the wreck, the aroma of cooked human flesh assaulting her nostrils and palate. Her stomach heaved.\n\nShe shook her head, realising that she was gripping the railing tightly with both hands. She had a sudden unnerving sense of vertigo, like she was tumbling over the railing towards the factory floor below. She closed her eyes. The moment passed. She caught her breath, drawing raggedly at the air. She knew it was no good giving in to melancholy. She'd seen the results of that before, long ago. What was done was done, and now the most important thing was to find out who was responsible for the disaster, and if necessary, aid Newbury in bringing them to justice. She breathed calmly, and hoped that the others hadn't taken note of her momentary lapse.\n\nShe watched a man below struggling to carry a large mirror across the factory floor, and wondered for a moment if the new ship was intended as a replacement for _The Lady Armitage._ She decided not; it was clearly too soon after the crash for the workman to be this advanced with the construction. She turned from the railing. \"It's quite an operation you have here, Mr. Chapman.\"\n\nChapman, who had been deep in conversation with Newbury, turned and smiled. \"Wait until you see the next hangar, Miss Hobbes. Now, that's really something to behold.\" He nodded at the workmen down below. \"Come on, let's get a closer look.\" He led them along the steel walkway, their feet clanging loudly against the metal rungs as they walked. They made their way down a series of steps at the far end of the hangar.\n\nChapman crossed the floor to where the men were working and climbed up onto the wooden pallet, peering into the shell of the new gondola. He seemed pleased.\n\nDown at this level, the air was filled with the smell of oil and wet paint, and the noise was tremendous: banging, sawing, shouting. There appeared to be an entire army of men at work. Newbury counted at least ten of them, dancing around each other, ferrying components back and forth, their faces damp with perspiration and grime. Not one of them looked up from their work to eye the newcomers as Newbury circled the construction, drinking it all in.\n\nHe looked up. High above them, the red brick walls turned to windows, allowing the natural light to seep in from outside. The roof was a skein of corrugated lead sheets, laid over a framework of wooden beams. The place was enormous, yet seemed bizarrely reduced by the sheer size of the gondola that was being erected inside of it.\n\nNewbury finished circling the pallet and then moved to stand beside Chapman, clapping a hand on his shoulder to get his attention. The other man, who'd been standing with his hands on his hips, admiring the work of his craftsmen, stepped back, leaning in to hear Newbury's question.\n\n\"How long does it take you to build one of these? From start to finish, I mean?\"\n\nChapman raised his voice so the other man could hear. \"About three weeks. This is the smallest size in the fleet, a passenger-class vessel. The rest of the frame is being welded in the next hangar.\" He pointed to the other end of the vast room, where a huge archway led through to the next part of the site. Newbury could just make out some of what was going on inside, with iron girders being lifted into place around a wooden frame, the entire construction apparently suspended from the ceiling.\n\n\"Three weeks? That seems awfully quick.\"\n\nChapman nodded. \"I know. We've spent the last ten years perfecting the process, ironing out all of the wrinkles.\" He coughed, and seemed to consider searching out another cigarette, before quickly changing his mind. \"This one's bound for India.\" He nodded at the gondola in front of them. \"It'll be out of here in a couple of days. We'll have an automaton fly it over the water. That way there's no need for the pilot to come back again, you see.\" He smiled. \"It's a good package. The new owner is provided with a fully trained pilot, and we're not stuck ferrying people back and forth across the ocean.\"\n\nNewbury nodded. He could see the economy in the system. \"I admire your business acumen, Mr. Chapman. And your men certainly seem to know what they're doing.\" They both regarded the workmen still scurrying to and fro all around them.\n\nNewbury glanced at Veronica out of the corner of his eye. He could see that she was feeling uncomfortable being around another airship so soon after her visit to the crash site the previous day. He decided to hurry things along. \"Are we ready to move on, Miss Hobbes? I'm eager to see how the balloon itself is constructed.\"\n\nVeronica smiled thankfully. \"Yes, indeed. Mr. Chapman, please lead on.\"\n\nThey followed Chapman across the floor of the manufactory towards the archway and through to the next hangar. As they approached, Veronica gasped in wonder at the sight. The space itself must have been twice the size of the previous room, opening out into a cavernous hall filled with all manner of mechanical wonder and, at its heart, the massive skeletal frame of an airship balloon. Light shone down from the windows above in great shafts, penetrating the gloom and picking out the swirling dust motes in the air. Newbury stood beside Veronica as they looked up in awe. The im mense structure of the airship was clearly taking shape, suspended from the ceiling on an array of large mechanical arms. Iron girders were being welded into place around a wooden frame, hot sparks showering the room below in a series of glittering waterfalls. Men, tied into harnesses and dangling from roof joists, clambered around the structure, gas tanks strapped to their backs, welding torches clamped firmly in their gloved fists. Other men operated large cranelike machines, lifting the iron girders into place for their colleagues to weld. Newbury had never seen anything like them; the operator sat inside a small cab on top of the machine, manipulating levers to control the arm, which terminated in a large claw used to grasp the iron girders and move them to precisely the required position. The machines themselves were fixed in place, bolted to the floor, and spluttered loudly as their steam engines turned over in the relatively enclosed space of the hangar. Chapman held his hands out, encompassing the scene before them. \"Impressive, isn't it?\"\n\nNewbury couldn't help but agree. \"Magnificent. A remarkable achievement.\"\n\nChapman smiled. \"It is, rather.\" He rubbed his hands together in an unconscious gesture. \"The difficulty, of course, is one of space. We have only enough room to assemble one vessel at a time. I've been thinking, recently, of constructing another facility on the other side of the river, but in truth, the advent of the automaton business has rendered that superfluous, at least for now.\"\n\nVeronica was still regarding the skeleton of the vessel suspended overhead. She glanced at Chapman. \"Are the automata also manufactured on the premises, Mr. Chapman?\"\n\n\"Indeed they are, Miss Hobbes. Although I feel I must warn you that the scale of the operation is hardly as impressive.\" He indicated the airship. \"The technology is still relatively new, and the units are expensive to develop. Mass production is unfortunately some years away. Nevertheless, orders have been growing steadily, and the production line has been constantly engaged since its inception.\" He cleared his throat. \"We'll pass through the area on the way to see Villiers in a few moments.\"\n\nNewbury looked contemplative. \"Tell me, Mr. Chapman, why it is that a highly successful airship business should make the move into artificial intelligence? It strikes me that the two disciplines make strange bedfellows. Why invest in something so new and speculative?\"\n\nChapman paused before responding, as if weighing the question. \"On one hand, in Villiers, I had the expertise and the vision to pull it off, and on the other, I saw the opportunity to make a return.\" He shook his head, not satisfied with his own answer. \"No, it's more than that. After my father passed on, Sir Maurice, I found myself in the enviable position of inheriting an industrial empire, and with it, a significant fortune. I could have taken the opportunity to live a life of pleasure, wasting my time dallying with insignificant trifles, spending my days lounging around my estate. I admit, for a while I was tempted. But I also knew that if I devoted my life to such lackadaisical pursuits, I would soon shrivel up and die. I needed stimulation, and more, I had an overriding desire to aid progress. After meeting Villiers and being introduced to his revolutionary plans for a new breed of airship, I decided to invest a portion of my fortune in setting up this firm.\" He paused only momentarily, obviously in his stride. \"I could see clearly, then, the impact that Villiers's incredible new designs would have on the air transportation industry, and with time and a lot of hard work, my faith was proved right. Chapman and Villiers Air Transportation Services became one of the most important airship operators in the world.\"\n\nThe others were listening intently. \"So why risk that now? Why divert the resources of your successful company into something untested, unproved on the open market?\"\n\nChapman shrugged. \"Because I grew bored, and because Villiers kept pushing forward, irrespective of finances, time or effort. You'll understand that when you meet him. The man is fuelled by a passion for his work. He was like an unstoppable force, and it was only then, after watching him work himself into the ground, night after night, for months on end, that I finally realised how the automaton project could help us to fulfil our original ambition. I started to consider the almost limitless applications of these mechanical men. If they could learn to write, they could be employed as clerks. If they could learn to cook, they could replace servants. If we taught them the art of war, they could even march into battle against the Empire's foes. Think how many needless deaths could be averted? Surely these remarkable devices could aid in the technological revolution of the Empire? Surely that could only be of benefit to the wider populace, freeing them from the tedium of household chores, leaving more time for education and other, more profitable enterprises? I think you'll see, when we have Villiers give us a demonstration of the units, what a spectacular revolution awaits us, just around the corner, when the world becomes truly aware of what we're doing here in our little factory in Battersea.\"\n\nVeronica met Newbury's eyes. \"But Mr. Chapman, what of the people pushed out by these automata, and what of their families? If their jobs are taken away from them, to be replaced by these artificial men, many of them will be left destitute, with no hope of finding other work. Surely _that's_ not in the best interests of the Empire?\"\n\nChapman nodded. \"Yes. I, too, have concerned myself with that, Miss Hobbes. Yet . . . we can't allow it to halt progress. Society will redress the balance, given time. Communities will change, and people will find worthwhile employ in any number of different industries. The automaton revolution will provide them with even more opportunities, and I'm convinced it will raise the standard of living across all classes throughout the entirety of the Empire.\"\n\nNewbury looked unsure. \"Grand claims indeed, Mr. Chapman.\"\n\n\"Time will tell, Sir Maurice, time will tell. But it is clear to me that you need to see one of these marvellous machines in action!\" He was animated now, fired up on his own rhetoric. \"Allow me to walk you through the automaton production site on our way to see Villiers. It's just this way.\"\n\nNewbury arched an eyebrow at Veronica, and the two of them fell in behind Chapman as he continued his tour of the facility, picking a route through the array of spluttering machines that continued to swing iron girders into place high above their heads.\nCHAPTER 9\n\n##\n\nThey passed along a corridor that stemmed off from the main airships works and eventually led them to a small warehouse space that appeared to have been hastily converted into a production line. Two large steam-powered presses thumped with reassuring regularity, pushing out components in a variety of shapes and sizes, from brass arm braces and finger joints to shiny torso plates and elaborate cogs. Men stood alongside the rolling conveyor belts that fed out from the machines, each one picking up components and checking them for flaws before sending them on to the assembly teams on the other side of the warehouse. There, small groups of men were busy welding the components together, testing the articulation of the joints and assembling the frames of the automatons. The room was hot; bustling with people and filled with the smell of oil and steam.\n\nChapman paused in the doorway. \"As you can see, the automaton production facility is still a relatively minor concern when considered alongside the main airship works, but in time, I have hopes that it will grow.\"\n\nNewbury paced alongside one of the presses, watching as the machine-head spun on its axis, pressing a new component from the mould on its fascia. He spoke to Chapman as they walked. \"How many automata does the facility produce in any given day?\"\n\n\"Fully functioning units?\"\n\nNewbury nodded.\n\n\"One or two. They can actually make upwards of ten frames on a good day, but Villiers himself installs the internal control systems, and it's delicate work. Any faster, and we'd jeopardise the integrity of the machines or risk damaging the complex mechanisms that make them run.\"\n\n\"I'm looking forward to meeting him. Villiers, that is.\"\n\n\"Let's see if he's here now. That's the door to his workshop.\" He waved to indicate the glass-panelled door up ahead. They approached, and Chapman rapped quickly on the glass before pushing the door open to reveal the workshop within.\n\nThe room was fairly small, after the grandeur of the airship hangars, and it was cluttered with components and other mechanical ephemera: cogs, tools, automaton torsos, pages covered in elaborately scrawled designs, a model airship hanging from the roof. In truth, the room had as much of the feel of a laboratory as that of a workshop, the sort of place where scientific breakthroughs were commonplace and genius was taken for granted.\n\nVilliers himself stood at his workbench, fiddling with a brass skull. He was wearing a brown leather smock, not unlike a butcher's apron, and had a magnifier flipped over his right eye on a wire frame, the base of which wrapped around his head like the crude frame of a hat. His hair was coarse and black, and he was unshaven, with a vaguely dishevelled appearance. He was fairly short, although taller than Veronica, and his only acknowledgement upon hearing them enter the room was to grunt at the automaton head he was holding and choose not to look up from his work.\n\nChapman waited for a moment to see if his business partner would remember his manners. When it was clear the other man intended to carry on working on the brass head regardless of their presence, he stepped forward, trying to get Villiers's attention. He cleared his throat. \"Villiers. I'd like to introduce you to Sir Maurice Newbury and his assistant, Miss Veronica Hobbes. They're here on the business of the Crown, investigating the airship crash I mentioned to you yesterday.\"\n\nVilliers offered a half-shrug before continuing to dig around inside the brain cavity of the brass skull. There was an awkward silence. Then, a moment later, something popped free from inside the device and flew into the air, before falling to the floor by Veronica's feet. Newbury noted that it was a tiny gold lever of some sort. Villiers looked up, satisfied. \"I'm sorry, what were you saying, my friend? Hmmm?\"\n\nHe seemed to notice Newbury and Veronica for the first time. \"Oh, please excuse me. I was lost in the middle of a delicate operation. . . .\" His accent was thick, with a Pa ri sian lilt. He placed the automaton head on his workbench, along with the tool he had been using.\n\nNewbury stepped forwards, his hand extended. \"No need for apologies, Monsieur Villiers. I am Sir Maurice Newbury, and this is my assistant, Miss Veronica Hobbes.\" Veronica inched forwards, and Villiers took her hand, gently. \"As your associate here intimated, we're working on behalf of the Crown. We'd like to talk to you about your automaton devices and the airship crash that occurred yesterday in Finsbury Park.\" He stopped for a moment, glancing around. \"I must say, though, Monsieur Villiers, this truly is a remarkable workshop. A credit to you, I'm sure.\"\n\nVilliers smiled. \"Thank you, Sir Maurice. I can spare a little while to talk, although I am sure my associate has already told you much the same as what you will hear from me.\"\n\nNewbury nodded. \"Nevertheless, I do feel your opinions on the matter will be of use. Are you aware of the circumstances surrounding the crash?\"\n\nThe Frenchman shrugged. \"In as much as Monsieur Chapman told me yesterday.\"\n\n\"So you're aware that the automaton that was piloting the vessel appears to have gone missing from the wreckage?\"\n\nVilliers looked immediately uncomfortable. \"Missing? No. Destroyed, perhaps? I know my creations, Sir Maurice. There is no way the unit could have gone 'missing,' unless someone spirited it away from the crash site for their own devices.\"\n\nNewbury glanced at Veronica. That was an option they hadn't yet considered. Veronica was watching Chapman, trying to gauge his reaction to Villiers's words.\n\n\"So what do you believe happened, Monsieur Villiers? Did the automaton malfunction and cause the crash?\"\n\n\"Impossible. There is no capacity for the units to malfunction. Physically, they can function only if their program is loaded correctly. They operate on a series of punch cards. If the card does not engage, the unit will immediately freeze. If that were the case with the pilot of _The Lady Armitage,_ the vessel would never even have taken off in the first instance.\" He stopped, stroking his stubble-encrusted chin. \"My assumption is that the vessel itself was at fault. Perhaps one of the steering pulleys had come loose, causing the mechanism to lose tension? If that were the case, the vessel would have been practically uncontrollable, and in high winds, it could have easily been knocked off course.\"\n\nVeronica crossed her arms. \"But as I understand it, Monsieur Villiers, the skies were calm yesterday morning. Otherwise the fog would not have settled on the city as it did.\"\n\nVilliers shrugged. \"Then it is a matter for the police to decide what occurred. I am in the dark. Whatever the case, I understand it was a terrible accident, and for that I am truly sorry.\" He hesitated. \"I assure you, however, that the source of the problem is with the vessel, and not with the pilot.\" He regarded them sternly.\n\nNewbury decided to change the subject. \"So, Monsieur Villiers. What of your exile from Paris and the claims that you experimented on wastrels? Is there any tru\u2014\"\n\n\"Come now, Sir Maurice, is this really necessary?\" Chapman cut in, clearly trying to come to the aid of his friend.\n\n\"It's alright, Joseph.\" Villiers seemed unmoved by the question. He faced Newbury. \"What of it? It was a long time ago, Sir Maurice, and very much a part of my past. I have spent the last de cade in London, working to revolutionise the aeronautical industry with Monsieur Chapman. I no longer even think of Paris, and consider London my home.\"\n\nNewbury nodded. \"Very well, Monsieur Villiers.\" He noted that the Frenchman had chosen not to refute the claims. The man's arrogance was obvious, but not without foundation. He softened his tone. \"So what inspired you to begin developing a new type of automaton, after years of designing airships? Mr. Chapman tells me you worked day and night to achieve your goal.\"\n\nVilliers looked circumspect. \"In truth, I have always dreamed of building the perfect automaton. For years, I have strived to reach this stage, and it was only when the airship business had established itself and the manufacturing process had been automated that I found myself with the time and resources to realise my dream.\" He glanced at Chapman. \"Once my friend and I began discussing the application of these units\u2014household servants, drivers, soldiers, clerks\u2014we agreed it was time for our business to diversify. The added benefit, of course, was that the machines could be taught to fly the fleet of airships we had spent the last ten years establishing.\"\n\n\"It's an impressive achievement indeed, Monsieur Villiers. So tell me, are the units intelligent, self-aware?\"\n\nVilliers shook his head. \"No, they are not sentient in their own right. They are simply machines that operate according to a complex set of algorithms and programs. Have you seen one operating, Sir Maurice?\"\n\nNewbury shook his head, and Chapman interrupted. \"I was hoping that you would be able to give our guests a demonstration, Pierre?\"\n\n\"Of course. Allow me to do so now.\" He moved over to the corner of the workshop where, Veronica realised for the first time since entering the room, an automaton was sitting in a chair, its head bowed. Villiers stood before it.\n\n\"Rise.\" His voice was a firm, emotionless command.\n\nThe unit's head jerked up at the sound of Villiers's voice, and it quickly rose to its feet. \"Follow.\" He turned and walked back across the workshop towards them. The automaton followed suit, stepping forward into the light. The two visitors looked on, transfixed with wonder. The automaton was about the size of a man, skeletal, with a solid torso formed from interlocking breast and back plates. Its eyes were little mirrors that spun constantly on an axis, reflecting back the lamplight. Its mouth was nothing but a thin slot, and its remaining features were engraved into the otherwise blank mask of its face. In its chest, a glass plate revealed, like a tiny porthole, a flickering blue light, dancing like an electric current. Its brass frame shimmered in the light, and it moved like a human being, fully articulated, as it strode across the room towards them. Its joints creaked as it walked, and its brass feet clicked on the tiled floor of the workshop. It stopped about two paces behind Villiers and cocked its head to one side, regarding them silently.\n\nChapman clapped his hands. Newbury and Veronica looked on, feeling a little unnerved.\n\nVilliers turned to the automaton. \"Pick up that glass tumbler and pour me a brandy.\" He pointed across the room at a small table, which held the tumbler and a decanter, amongst other detritus. The automaton set to work immediately, crossing the room with a fluid gait, avoiding a pile of machine parts on the floor and approaching the table with the utmost precision. Taking care, it reached down and picked up the glass between its brass fingers\u2014which, Newbury noticed, were affixed with little leather pads to prevent them from shattering the tumbler\u2014and poured a measure of brandy from the decanter. A moment later, it strode back across the workshop to offer Villiers his drink without ever spilling a drop.\n\nNewbury was astounded. \"Bravo. Bravo, indeed!\" He glanced from Villiers to Chapman and back again. \"This is indeed a revolutionary invention. What else can it do?\" He was clearly enthused.\n\nVilliers smiled. He took the drink from the automaton and pointed to a chair by his desk. \"Take a seat.\" The automaton did as requested, positioning itself as if ready to receive further instructions. Villiers crossed to the desk himself, with Newbury close behind him, and searched out a letter. He placed this on a stand in front of the automaton, beside a typewriter on the desk. \"Copy this.\" He indicated the sheet for the mechanical man. The automaton did not respond, its only movement the continual spinning of its mirrored eyes and the flickering of the iridescent light inside its chest.\n\n\"Ah. Please forgive me.\" Villiers handed his brandy to Newbury and leaned over his desk. He pulled open a drawer, pulling out a sheaf of punch cards. He rifled through, finally selecting one and brandishing it in front of him. \"This par tic u lar unit has yet to learn how to carry out this task.\"\n\nHe pressed a panel on the back of the automaton, and it swung open easily, revealing some of the unit's internal workings. Newbury peered inside, fascinated. \"Tell me, Monsieur Villiers, how does it learn? I was under the impression from your earlier comments that the device lacks its own intelligence, although it certainly appears to respond to complex voice commands.\"\n\nVilliers took the punch card and fed it into a slot within the back of the machine. \"As I mentioned earlier, Sir Maurice, the automaton operates on a series of predetermined programs. These programs are expressed as a series of punch cards that the internal mechanisms of the device can interpret and enact. The device has the capacity to file up to twenty-eight of these cards at any one time on a revolving spindle, and when asked to perform a task, it will check the programs stored on its spindle and see if the correct card is in its repertoire. If so, it will retrieve the card and carry out the task. If not, well, you've seen the reaction in that situation.\"\n\nNewbury shook his head in disbelief. \"A machine that learns. . . .\"\n\nVilliers clicked the panel shut. He repeated his earlier command. \"Copy that.\"\n\nThere was a whirring sound from within the chest of the automaton. Then, suddenly, its hands blurred over the keys of the typewriter, and within a matter of seconds, the entire page had been typed. Newbury leaned forward, taking the page from the top of the typewriter and comparing it to the original letter. It was identical, in every respect, even to the extent of recreating an error, where a misspelled word had been omitted with a series of X's.\n\n\"Veronica, do you see this?\" He held the pages up for her. \"It's identical.\" He turned to Villiers. \"What, it must be ten times faster than a human being?\"\n\n\"Undoubtedly so.\"\n\nNewbury shook his head. He was quite lost for words.\n\nVeronica studied the two copies of the letter. \"It's certainly very impressive.\" She seemed hesitant to be carried away by the spectacle.\n\nNewbury was in his element. \"Monsieur Villiers, tell me about the power source.\"\n\nVilliers was obviously enjoying the attention. \"The device is designed to power itself. When the automaton moves, a rotor inside its abdomen rocks back and forth, ratcheting the winding mechanism and causing the mainspring in the chest to become taut. Effectively, the unit is self-winding, and thus it will never power down, unless commanded to do so. If left inactive for long periods without instruction, the unit will eventually move itself to trigger the winding mechanism.\"\n\n\"So it goes for a little stroll? Quite wonderful.\"\n\nVeronica looked at the automaton warily. \"It certainly _seems_ intelligent, Monsieur Villiers.\"\n\n\"Thank you, Miss Hobbes. A compliment indeed. The entire purpose of an automaton is to give the impression of intelligence, maintaining the illusion whilst the workings of the device are kept hidden from the audience.\"\n\n\"And what are those workings, Monsieur Villiers? We've seen the mechanism that enables the device to be programmed, but how does it come to understand your voice commands, or interpret the input from its mirrored eyes?\"\n\n\"Ah, well, that is the secret, is it not?\" Villiers put his hand on his hips. \"The device is fitted with an incredibly complex mechanism that mimics the neurological structures of a human brain. It makes judgments by asking itself a series of logical questions and interpreting the results, enabling it to select a course of action. For example\"\u2014he leaned on the back of the automaton's chair\u2014\"if the device were commanded to walk across this workshop, it would automatically find a route around the workbench there, without having to walk into it or attempting to climb over it. This is achieved through a series of logical questions that the unit's brain is designed to follow. What will happen if the unit walks into the workbench? How will walking into the workbench prevent it from achieving its goal? What is the quickest alternative route to its destination? Switches trigger inside the brain to enable the automaton to settle on the most effective solution to each question, thus deciding its route around the workbench. In this instance, the unit would obviously decide to alter its course, rather than face potential damage by walking into an immovable object.\" Villiers smiled, obviously pleased with himself.\n\nVeronica looked back at Chapman, who had taken a seat by the door and was also smiling as he watched the others receive their lecture from his friend. He had struck a match and was in the process of lighting a cigarette. The glare of the flame cast his face in stark relief.\n\nNewbury placed his hand on the automaton's head. \"Can we see? I'd very much appreciate an opportunity to take a look inside this remarkable contraption.\"\n\nVilliers nodded, and went to fetch a tool to open up the automaton's skull.\n\nVeronica took the opportunity to catch Newbury's eye, and he smiled knowingly. He was allowing himself a moment of indulgence, but she knew from the look on his face that he wouldn't allow himself to get carried away. He was ready and alert, absorbing everything.\n\nVilliers returned and set to work on the automaton's head. It took him only moments to unclip the skull cap and unscrew the safety catch that gave access to the unit's mechanical brain. Both Newbury and Veronica couldn't help but gasp at the sight revealed when the plate was lifted away. The automaton's brain was like the workings of some incredible watch, only orders of magnitude bigger and more complex. They both leaned in, watching the cogs and levers as they ticked over, minute switches flicking from one position to another as the automaton regarded its surroundings. It was like seeing human thought processes in action, like some sort of bizarre window into the human soul. In some ways, it was disturbing, to see a creation so complex and wondrous yet without feeling, lacking the spark of life. On the other hand, Newbury was amazed to consider that it could be argued that the human brain was the same as this incredible device, a series of clockwork switches and cogs rendered flesh and blood. He watched for a moment longer, intrigued by the ticking of the tiny mechanical components as the automaton sat unmoving before them, unaware that they were looking deep into the very fabric of its being.\n\nVilliers stepped in and replaced the skull cap. \"We must not leave the internal components exposed to the air for too long. Moisture affects the workings, and the small mechanisms can easily become clogged with dust.\"\n\nNewbury stood back, watching appreciatively as Villiers used his tool to replace the fittings. \"I must thank you for your demonstration, Monsieur Villiers. It's been quite enlightening.\"\n\nVeronica nodded her agreement. \"Yes, thank you for your time. The experience has left me feeling quite breathless.\" She turned to Newbury. \"Is there anything further you require of Mr. Chapman or Monsieur Villiers, Sir Maurice?\"\n\nNewbury looked thoughtful. He turned to Chapman. \"I do not believe there is. If you would be kind enough to escort us back to your office, Mr. Chapman, Miss Hobbes and I will take our leave. I daresay you have pressing business to attend to.\"\n\nChapman stood, inclining his head. \"Of course, Sir Maurice. It has been a pleasure to show such enthusiastic visitors around our humble business.\" He beckoned them towards the door.\n\nNewbury turned to Villiers and shook his hand firmly. \"Fascinating work, Monsieur Villiers. I expect we'll meet again.\"\n\nHe allowed Veronica to go ahead of him, and together they walked back towards the office complex, leaving Villiers alone with his clockwork automaton and his thoughts.\n\nOutside, the afternoon was turning to twilight as Newbury and Veronica hailed a hansom cab. Newbury had offered Veronica his coat to stave off the chill, and as she mounted the steps into the cab, she turned to regard him, the horses whinnying as they stamped their feet impatiently by the side of the road. The sound of the foghorns on the river made it difficult to hear.\n\n\"So, what next? Do you think Chapman and Villiers have anything to hide?\"\n\nNewbury lowered himself onto the seat opposite her, and the driver whipped his reins, jerking the vehicle into motion. \"I suspect they have a great deal to hide, my dear, but whether it pertains to the case at hand, I remain unsure.\" He ran his fingers over his chin. \"I need time to consider our findings. I admit I find it difficult to see evidence of foul play. Unless you can offer any further insights that you think I may have missed?\"\n\nVeronica shook her head. \"I don't believe so. I remain wary of Mr. Chapman. I find him both insincere and egotistical. I do believe he was holding something back.\"\n\nNewbury agreed. \"Indeed. There is clearly more to the man than meets the eye. He obviously believes himself to be a great philanthropist, or at least wishes to paint that picture of himself to others. He delivered his message with a little too much zeal for my taste.\"\n\nVeronica pulled Newbury's coat around herself. \"Do you think the automaton demonstrations have helped to shed a light on the disaster surrounding _The Lady Armitage_? I singularly failed to see the significance of anything they showed us, as spectacular as it all was.\"\n\nNewbury thought on this. \"I believe they succeeded in demonstrating how unlikely it is that the automaton itself malfunctioned. Although I'll admit, I'm still baffled as to what happened to it after the vessel had crashed. I wonder if there is any stock in what Villiers suggested, about someone spiriting it away before the authorities arrived.\"\n\n\"I wondered the same. Perhaps it's best we speak with Sir Charles again, to see if Inspector Foulkes has turned up any further evidence from the area around the scene?\"\n\nNewbury seemed distracted. He glanced out of the window. \"Indeed. I'm sure we'll speak with both of the aforementioned gentlemen in due course.\" He seemed to relax a little. \"Tomorrow I shall pay a visit to Buckingham Palace to talk with Her Majesty. It's been a difficult couple of days, Miss Hobbes, and I have no doubt that you would benefit greatly from a day of rest.\" He smiled, waving his hand to stifle her objections. \"Besides, it'll give me a little more time to ponder our next move.\"\n\nVeronica sighed. \"Very well. Let us agree, then, that you will call for me if there are any new developments. We can't have you charging in alone.\"\n\nNewbury laughed. \"Indeed not, Miss Hobbes. That would never do.\"\n\nHe continued to chuckle as the cab rolled on towards Chelsea, and home.\nCHAPTER 10\n\n##\n\nNewbury had visited Buckingham Palace on numerous occasions over the last few years, yet the grandeur of the place never failed to take his breath away. He was awed by the spectacle of it: looming out of the grey fogshrouded morning, its towering fa\u00e7ade was an imposing sight, a symbol of Her Majesty's might rendered in stone for the entire world to see.\n\nHe glanced up at the pillars that stood, sentrylike, over the main entrance. To either side of these were vast rows of windows, hiding all the secrets of the Empire behind their heavy curtains of red and gold. In the driveway, stable hands were exercising the horses, and a line of impressive carriages stood ready by the main gates. Newbury wondered if some sort of state function were being planned, or else if foreign dignitaries were expected to pay a visit later that day. He knew Her Majesty would not be impressed by either of those eventualities.\n\nNodding at the guard, who shivered as he opened the gate for Newbury to pass through, he made his way around the rear of the im mense building, making haste for the private entrance that was situated near the servants' quarters, out of sight from prying eyes. He braced himself against the chill. The morning had brought with it a crisp frost, and the sun was yet to break through the dense cloud of fog that had settled on the city during the night. It was still early, but Newbury knew he was expected. It didn't do to keep Her Majesty waiting.\n\nHe approached the familiar oak door, glancing quickly from side to side to ensure that he wasn't being watched, and rapped gently with the brass knocker. After a moment, a small panel slid open and a pair of eyes appeared.\n\nNewbury cleared his throat. \"Morning, Sandford. It's Newbury here.\"\n\nThe panel slid shut again, and a few seconds later the door swung open, revealing a small foyer inside. The room was brightly lit with gas-lamps and, Newbury was pleased to see, the roaring flames of a fire. Sandford, the butler who oversaw this small secret area of the palace, ushered Newbury inside, clicking the door shut behind him. He held his arm out for Newbury's coat and hat. Newbury removed the garments and passed them to the butler, offering his thanks. The man was aged, now, in his seventies, with a shock of white hair and liver spots speckling his face and hands. He looked impeccable in his suit, however, and Newbury had the utmost respect for the man. He had stayed in service out of an unerring sense of duty to the Crown, and Newbury had often wondered if he had once been an agent of the Queen himself, back in the early days of the Empire. He certainly had a few tricks up his sleeve.\n\nSandford draped Newbury's coat on the stand in the corner and returned to his favourite position beside the fire. Newbury was rubbing his hands, attempting to soak up the warmth of the flames.\n\n\"Warm yourself there for a moment, sir. Her Majesty is expecting you in the throne room, but I daresay she'll wait a moment longer whilst you make yourself presentable.\" He winked at Newbury, and they both smiled. Newbury had received no official summons from the Palace, but he knew from experience that Her Majesty would be expecting a report on his findings at the crash site, as well as his consequent investigations. In fact, given the nature of the case, he was surprised that he hadn't received a summons before now.\n\nNewbury straightened his suit. \"Well, Sandford, I'm as ready as I'll ever be.\"\n\nSandford nodded, offering him an appraising look. \"That you are, sir.\" He turned about on his heel, more deftly than his appearance would give him credit for. \"I'll walk you there now, sir.\" They left the comfort of the fire behind them, exiting the foyer by a side door and out into a small passage that Newbury had walked along many times before. It snaked through the bowels of the palace, a secret route between the throne room and Sandford's little waiting area at the back of the great house. The corridor had been built for a different purpose, Newbury believed\u2014an escape route from the throne room should the monarch ever find herself threatened and in need of escape. Now, though, it was primarily used to bring Her Majesty's agents into the palace for private audiences, concealing them from the rest of the household, who Newbury doubted were even aware that the passageway existed. Of course, it depended entirely on one's point of view. Newbury couldn't help but think that the secret corridor also prevented Her Majesty's agents from soaking up too much of what was going on elsewhere in the palace. Victoria was a monarch who liked to play her cards very close to her chest indeed.\n\nNewbury couldn't keep his eyes from wandering as the two of them strolled along the passageway. The walls were lined with austere portraits of long-dead kings and queens, the figureheads who had helped to shape the nation in times past. Victoria herself was notably absent from the gallery, and Newbury wondered if that would be the first role of any new incumbent to the throne: to hang a portrait of this most powerful ruler in its rightful place, at the head of the gallery of her predecessors. Not that the queen showed any real signs of abdication or debilitating illness; the marvellous machines of Dr. Fabian took care of that. He was a scientific genius without precedent, and Newbury was only grateful that he was loyal to the Crown and not, as others with pettier minds might have been in his position, hungry for power in his own right. He'd met the man only once, fleetingly, but he knew at some point he was likely to meet him again. Most agents of the Crown found occasion to visit Dr. Fabian at least once or twice during the course of their career.\n\nPresently, their feet scuffing the deep pile of the carpet, they came to rest before a door. The corridor ended abruptly here, and Newbury knew that the vast chamber of the throne room awaited him on the other side.\n\nSandford knocked boldly on the door, straightening his tie.\n\n\"Come.\" The command from within was direct, pointed.\n\nThe butler reached for the handle and clicked the lock, allowing the door to swing open into the room. All Newbury could see inside was darkness.\n\n\"Sir Maurice Newbury, Your Majesty.\" Sandford shuffled out of the way to allow Newbury to pass, and then pulled the door shut behind him. Newbury heard the sound of the butler's feet rustling on the carpet as he slipped away, heading for his rooms and the relative warmth of his fire. He stepped forward in the darkness, waiting for his eyes to adjust. Heavy curtains were drawn across all the windows, casting the place in dark shadow. The only light in the entire room was a gas lamp flickering in one corner, a lonely flame adrift on a sea of darkness. He had the sense of standing in a cavernous space, but being able to see only a few feet in front of him. He could hear the sound of Dr. Fabian's machines, wheezing and sighing as they rasped at the air, their bellows clicking as they rose and fell in the darkness.\n\nFinally, Victoria spoke.\n\n\"Ah, my faithful servant. What news do you bring?\" Her voice cut through the darkness like ice, sending a shiver up and down his spine. He turned towards the sound, and bowed.\n\n\"Majesty.\" He paused. \"Precious little news, I fear.\" He sighed, deciding how to go on. \"I attended the scene of the airship disaster, as requested, and discovered certain . . . irregularities.\"\n\n\"Go on.\"\n\n\"The body of the pilot was missing from the wreckage, and the passengers, or what remained of them, had all been tied into their seats. There were no survivors at the scene. I later discovered that the vessel had, in fact, been piloted by a clockwork automaton developed by the airship's operators, Chapman and Villiers Air Transportation Services.\" He hesitated, weighing his next words carefully. The wheezing sound continued steadily in the darkness. \"Yesterday I visited the manufactory of the aforementioned business and saw one of these automaton units being demonstrated. I have no reason to believe the pilot of _The Lady Armitage_ could have malfunctioned at the controls. The cause of the disaster remains unclear.\"\n\nThere was a creaking sound as Victoria wheeled forward in her chair, emerging from the shadows into the dim glow of the gas-lamp. Newbury fought the urge to gasp at her appearance. He had seen her before, of course, but the sheer extent of Dr. Fabian's work was a constant source of shock and amazement. The Queen was lashed into her wheelchair, her legs bound together, her arms free and resting on the wooden handles that enabled her to rotate the wheels of the contraption. Two enormous tubes protruded from her chest, just underneath her breasts, folding around beneath her arms to connect to the large tanks of air that were mounted on the back of the chair. Bellows were affixed to the sides of the contraption and groaned noisily as they laboured with the pressure, forcing air from the tanks in and out of her collapsed lungs. Her chest rose and fell in time with the machine. A drip fed a strange pinkish liquid into her bloodstream via a catheter in her arm and a bag suspended on a brass frame over her head.\n\nShe regarded Newbury with a steely expression. \"Newbury.\" Her voice was full of gravitas. \"We must impress on you the critical nature of this assignment. It is a matter of some importance to the Crown. We expect you to do your duty and identify the source of the disaster. Foul play remains a distinct possibility.\" Her mouth was a tight line, her face old and tired. Nevertheless, her eyes shone with a brilliant gleam that, even in the semi-darkness, gave evidence of the fact that her mind was still as sharp as her tongue.\n\nNewbury was unsure how to respond. \"Of course, Your Majesty. I will endeavour not to disappoint in this matter.\" He shuffled awkwardly. \"If it's not impertinent to ask . . . may I know the origin of your suspicion of foul play? It may prove useful in identifying the next course of action.\"\n\nVictoria moistened her lips with the tip of her tongue. \"Very well. A member of the Dutch royal family\u2014a cousin of this household, no less\u2014has been missing in London for some days. Intelligence from other sources suggested he may have been on board _The Lady Armitage_ when she went down. This morning, the mortuary confirmed his body had been identified in the wreckage.\" She hesitated before going on. \"We need not impress on you the severity of this situation, Newbury. One suspects that sabotage of the vessel may have been an attempt to discredit this house. Worse, we fear the means of that sabotage may in some way be related to your . . . field of experience. We have given our word to the boy's mother that we shall provide a reasonable explanation for the disaster. You must find an answer, and quickly. What with all this business in Whitechapel and the plague spreading through the slums, your expertise is needed elsewhere. Scotland Yard are floundering without your aid. Hurry to it, Newbury. Bring us the answers we need.\"\n\nNewbury bowed his head. \"I will press on with all haste and due diligence, Your Majesty.\"\n\n\"Go, then, and report back to us soon.\"\n\nHe turned to leave.\n\n\"Oh, and Newbury, how is that new assistant of yours working out? A woman, isn't she?\"\n\nHe smiled. \"Miss Hobbes? Yes, delightful, Your Majesty. And full of spark. She'll be a great asset to us, in time.\"\n\nVictoria let out a rasping chuckle. \"We do hope so, Newbury. Women like that are difficult to find. Make sure you keep her close.\" With that, she turned the handles on the sides of her chair and retreated slowly into the darkness.\n\nNewbury fumbled back to the door in the dim light, turned the handle, and left.\n\nSandford was waiting by the fire when Newbury emerged from the passageway. He turned to look at the younger man, and then picked up a tumbler from where he'd left it on the mantelpiece.\n\nNewbury accepted it gratefully and took a long swig. The alcohol attacked his palate, causing him to splutter slightly. \"Brandy?\"\n\nSandford nodded, his lips curling in a wide smile. \"For the cold, sir.\"\n\n\"Thank you. Very considerate of you, Sandford.\" He downed the rest of the drink, feeling the warmth spreading through his chest. He knew that Sandford was an old hand at this sort of thing, and that the reason for the brandy had, in truth, little to do with the cold. The man was simply used to seeing agents return from an audience with the monarch, and the brandy was a restorative offering to steady their nerves and put colour back into their cheeks. Newbury was thankful for the opportunity to do just that. He'd never found it difficult to talk with Her Majesty, but the sheer weight of expectation and nervous ness always left his nerves jangling for the rest of the day. Today, of all days, he needed to head back to his lodgings and try to relax, to ponder all the disparate elements of the case and see what shape they were beginning to take. Not only that, but in answering one mystery, he had inadvertently opened up another. He now knew what had agitated the Queen so much about the airship disaster, but he was faced with an even more difficult question to answer: What was a Dutch royal doing on board a passenger-class vessel bound for Dublin? He needed a breakthrough, and at the moment, he wasn't sure where to look next.\n\nNewbury placed the glass back on the mantelpiece and moved to fetch his coat and hat. Somehow, Sandford was there before him, and he thanked the butler as he helped Newbury on with his coat. \"Sandford, my thanks. I'm sure it won't be too long before I'm giving myself over to your hospitality once again.\"\n\nSandford nodded. \"Best of luck, sir.\" He opened the door for Newbury, momentarily allowing a gust of air into the room, stirring the newspapers that lay on the table. It was cold out, but the day was still young. His head buzzing with thoughts and the warm glow of alcohol, he stepped out into the grey fog and slipped away into the busy streets of London.\nCHAPTER 11\n\n##\n\nThe visiting room was cold and impersonal; clinical, even. Veronica was convinced that it wasn't supposed to feel so unwelcoming, even for a hospital. Her parents were paying a small fortune towards the upkeep of the place, after all. The least they could do would be to provide a few cushions and a bit of colour around the place to brighten things up. No wonder the majority of the patients were so miserable and lifeless.\n\nVeronica firmly believed that people were inspired by their surroundings, and that a dull and dreary hospital would reflect badly on the mood of the patients, especially in an institute such as this, which catered for the clinically insane. She knew Amelia would agree. She resolved to make a point of talking to Dr. Mason about it at the next available opportunity.\n\nVeronica sat with her hands on her lap, waiting for the nurses to fetch her sister. She felt uncomfortable and ever so slightly on edge, as she always did when she visited the asylum. She'd travelled to Wandsworth early that morning, taking care to ensure no one saw her leaving her apartments in Kensington and hailing a cab. She hadn't told Newbury where she was going, and consequently she hoped that he hadn't attempted to call on her with news of the case. If he had, she'd just have to tell him that she'd decided to go out for a stroll. She was meant to be taking the day to recuperate, after all.\n\nShe glanced around. One of the nurses sat on a stool by the door, looking out into the corridor. This would be her guard, she supposed, the woman posted there for the duration of her visit to make sure that her sister didn't stray towards violent tendencies, or that Veronica didn't try to sneak her any proscribed articles such as cosmetics, cutlery or photographs of the family. It was ridiculous, of course. Her sister had never hurt anyone in her life, and Veronica had no intention of causing difficulties for Amelia by bringing her any gifts that would cause her emotional unrest.\n\nDr. Mason believed that the less contact the patients had with their families, the easier they would find it to settle in to their new environment. In fact, the last time Veronica had spoken with him, he'd admonished her for the frequency of her visits, citing all manner of recent papers on the subject and claiming that the regularity of her calls was working against the treatment programme he had instigated for her sister. To Veronica, it seemed like an archaic way to try to make someone better, isolating them from the people who loved them. Besides, she knew it was a pointless exercise, anyway, although she didn't admit that to Dr. Mason. It wouldn't do to have him think that she disagreed with his diagnosis. Only, Veronica knew that her sister was far from the lunatic that the doctor had led her parents to believe. She wasn't mad. She just happened to be able to see into the future.\n\nVeronica looked up at the sound of footsteps from the corridor outside. The nurse who was sitting on the stool turned to look at her in acknowledgement, and then a moment later another nurse in a white uniform led Amelia into the room. Veronica's heart leapt. She stood, moving to embrace her sister.\n\nAmelia was painfully thin, and dressed in a loose-fitting outfit comprising a grey woollen blouse and matching skirt that Veronica thought would be better suited to a prison than to a hospital. Her hair was raven-black and long, loose around her shoulders, and her pale skin and soft complexion gave the impression that she was even younger than her fragile nineteen years. She looked scared, although her face lit up as she entered the visitors' room and saw her sister coming towards her.\n\n\"Veronica! You came.\"\n\nVeronica embraced her, feeling the press of her bony shoulder blades through the prickly fabric. \"Of course I came!\" She led Amelia to the sofa where she had been sitting and bade her to take a seat. \"Are you eating enough? You're so terribly thin.\"\n\n\"I eat well enough, sister. The food here is passable.\" She forced a smile. \"Anyway, what news do you bring from the outside world? Do our parents send word?\"\n\nVeronica looked uneasy. \"No, Amelia, no word from home.\" She patted her gently on the back of her hand. \"But I'm sure they will call soon.\" She lowered her voice to a whisper. \"You know how Dr. Mason likes to hold them at bay.\"\n\nAmelia glanced at the door. The nurse was still sitting on her stool, staring out into the corridor, as if there were something more interesting to engage her attention out there. Amelia sighed. \"I don't understand it, Veronica. They must know by now that they've made a mistake. It's clear that I'm not a lunatic. I'm convinced the seizures are a medical condition. They must be able to control them with drugs or remedies of some sort. They _must._ \" She looked into Veronica's eyes. \"I want so desperately to go home.\"\n\nVeronica felt tears welling in her eyes, and she blinked them away, forcing herself to be strong for her sibling. \"I know, Amelia. I know.\" She looked away, unable to see the pleading in her sister's eyes. \"Your illness is unique. The doctors need time to study it, to find a way to help you. I'm sure they're doing everything they can.\"\n\nAmelia nodded, biting her bottom lip. She brushed her fringe away from her eyes. \"Well, that's enough about me! Tell me about yourself, Veronica. What have you been up to? This hospital is so drab and boring that I need to hear stories of the real world. I like to think of you going about your business out there, all pretty and professional in your smart clothes.\"\n\nVeronica smiled. \"I think your expectations of my life are rather fanciful, Amelia. I work in a museum. I've spent the last week transcribing Sir Maurice's essays and researching academic papers on the druids of Bronze Age Europe. It's good work, but it's quiet. Hardly the stuff of high adventure!\"\n\nAmelia nodded, a twinkle in her eye. \"You forget, sister, that I'm able to see more than you think, even from in here. I fancy your recent exploits are far more engaging than you care to let on.\" She smiled, dismissing the issue. \"So, tell me, have you scandalised the museum terribly with your forward-thinking ideas?\"\n\nVeronica laughed. \"There have been a few raised eyebrows, certainly. Although I try to abstain from truly ruffling any feathers. I'd rather hold on to my position for the time being.\"\n\n\"And what of suitors?\" The nurse by the door looked over, obviously interested in Veronica's response. \"I hear that Sir Maurice cuts a dashing figure about town.\"\n\n\"Amelia, really.\" Veronica blushed. \"Sir Maurice and I have a strictly professional acquaintance. He's a handsome man, I admit, but I\u2014\"\n\n\"\u2014protest too much, clearly.\" Amelia cut in, chuckling. \"Come now, sister, I'm only playing with you.\" She scratched at her arms, where the woollen shirt was evidently irritating her skin.\n\nVeronica was suddenly serious. She put her hand to Amelia's cool cheek. \"Have you had any more episodes this week, Amelia?\"\n\nAmelia shrugged. \"A few.\" She looked away, noncommittal.\n\n\"And . . .\"\n\n\"And they were just as unpleasant and unwelcome as they usually are.\" She looked up at Veronica again, searching her face. \"I do wish they could find a way to make them stop. The things I see . . .\" She trailed off, clearly distraught.\n\nVeronica hugged her close, her voice soothing. \"I know Amelia. We're doing all we can, I promise.\"\n\nShe felt Amelia go limp in her arms.\n\n\"Amelia?\" She held her by the shoulders. \"Amelia?\"\n\nSuddenly, Amelia's thin body began twitching jerkily, her muscles going into spasms as Veronica tried to hold her still. Her eyes rolled back in their sockets, her mouth foaming as she shook wildly on the sofa.\n\n\"Amelia!\" She glanced at the nurse, who had only just realised what was happening.\n\n\"Help in here!\" The woman came away from the door, running to Veronica's side. She took hold of Amelia and eased her to the floor. She continued to twitch violently. \"We need to restrain her so that she doesn't hurt herself.\"\n\nVeronica dropped to her knees, clamping her hands over Amelia's legs. Her face was filled with concern. \"What now?\"\n\nThe nurse didn't look up from where she was struggling to hold Amelia's arms by her sides. \"Now we wait for the doctor.\"\n\nAmelia started to babble something incoherent in the midst of her tortured seizure. Veronica tried to make sense of the garbled words, tears now streaming freely down her cheeks. There was something about fire, screaming and trains. Other than that, it was impossible to tell what Amelia was saying, as her body, racked with nervous energy, fought against their grip in random, violent spasms.\n\nVeronica heard footsteps. She didn't look up. A moment later, two more nurses were by Amelia's side, one of them cradling her head whilst the other took over from Veronica, pinning her sister's legs to the floor. Veronica heard a familiar voice from behind her.\n\n\"Miss Hobbes. Please step away.\" She stood, looking round to see Dr. Mason hovering by the edge of the sofa. He looked serious. \"I think it is time for you to leave now, Miss Hobbes. Your sister is in safe hands.\" Veronica glanced back at her sister's writhing body, held down by a small army of nurses. She looked torn.\n\n\"Really, it's for the best. We can see her through this unfortunate episode, and then afterwards she'll be in need of rest.\" For once, Veronica thought, the swarthy-looking man in the brown suit had a kindly expression on his face. She believed he really did want to help her sister. \"You can call again in a week's time. I'm sure she'll be up and about again by then. If the weather is tolerable, you could even take her for a walk around the airing court.\" He smiled. \"But now it is time to go. I'll walk you to the exit.\"\n\nVeronica relented, glancing back at her sister one last time as Dr. Mason led her towards the door. Just as she was about to cross the threshold, however, she heard Amelia scream her name.\n\n\"Veronica!\"\n\nShe looked back, startled. Amelia was trying to force herself up into a sitting position, facing her sister as the nurses tried ineffectually to hold her down. Her eyes were still rolled back in their sockets, showing nothing but a disturbing sheen of milkywhite, but Amelia seemed to be looking straight at her, as if she could actually _see_ where Veronica was standing in the doorway.\n\nShocked, she whispered her response. \"Amelia?\"\n\nThe reply was a tortured rasp, as if dragged from somewhere within the depths of the girl's nightmare. \"It's all in their heads, don't you see, Veronica? You must see!\" She collapsed back into her spasms, and shaking his head, Dr. Mason took Veronica by the arm, leading her away from the terrible scene of her sister's distress and on towards the secure exit of the hospital.\n\nOutside, Veronica looked up at the asylum and used her handkerchief to wipe away the tears that were still stinging her eyes. The clock tower showed that it was fast approaching two in the afternoon, and she knew she'd be wise to head back to her rooms in case Newbury decided to call. She hated what was happening to her sister, back inside that terrible red brick building, locked inside a ward with no reasonable company, no decent clothes, no respect. She hated the fact that she couldn't do anything about it, either; that her parents had forbidden her from even discussing the issue with them, after she had railed so hard against their decision to place Amelia in the hands of these strangers in the first instance. Consequently, she hadn't had any contact with them for over two months, and neither had they been to visit her sister since her incarceration in September. She knew that, soon, she was going to have to write to them and insist that they pay a visit to the asylum to see their daughter. Amelia had enough to endure; it was unfair for her to have to suffer feelings of embarrassment, guilt and rejection, too.\n\nVeronica regained her composure and proceeded along the gravel path towards the exit to the railed compound and the street beyond. She passed the airing court on her left, a large paved courtyard used to exercise the patients when the weather was clement enough for them to venture outside. She smiled. Next week, she would return to Wandsworth and take Amelia for a walk around this little yard, admiring the flowers and the birds as they had when Amelia was a young girl and Veronica would take her for morning walks along the country lanes by their parents' house. In the meantime, she would throw herself into the case with Newbury and spend some time deliberating on the meaning of Amelia's outburst. She could hear the words echoing around in her mind as she walked. \"It's all in their heads, don't you see? . . .\"\n\nShe had no idea what it meant, and whether it was simply the ramblings of a disturbed, frightened mind, or something far more pertinent to her immediate future.\n\nOnly time, she supposed, would tell.\nCHAPTER 12\n\n##\n\nThe next day, Veronica woke early and decided that, after breakfast, she would head straight to the office. She'd had no word from Newbury, and she was anxious to find out if there had been any further developments in the case. He may have been able to solicit further information from Her Majesty during his visit to the palace, and she wanted to press him to speak with Sir Charles, to find out if Inspector Foulkes had managed to uncover anything further at the scene of the crash.\n\nFollowing her trip to the manufactory earlier that week, Veronica was still engaged with the notion that the vessel's automaton pilot may have crawled out of the wreckage, scrabbling away into the trees before anyone else arrived at the scene. It wasn't an outlandish idea; the automaton she had seen demonstrated had a hardy skeletal structure. She could see how the unit may have found itself confused, damaged but still functional, climbing out of the ruined cockpit before its more delicate components were consumed by the heat and the flames. Perhaps it had lain there inactive for some time before its pre-programmed systems engaged and it had been driven to move, not in an effort to escape the fire but simply because it was compelled to start the winding mechanism within its chest, as Villiers had described to them during the demonstration in his workshop. She would discuss these thoughts with Newbury at length when she arrived at the office.\n\nVeronica pulled back the curtains in her living-room and looked out over the street. The sun was only just poking up over the clouds, but already the high street was bustling with people. Mechanical carriages trundled rudely along the road, puffing clouds of steam high into the air, their drivers shouting down at pedestrians to make way. She shook her head. She couldn't understand Newbury's obsession with progress. Of course, the automata were marvellous inventions, but she couldn't help wondering what would happen to all the people they would displace if they were ever properly applied to industrial work in the city. Besides, London was a city still finding its way out of the last century. In her eyes, before there could be any major scientific revolutions, there were other more pressing social inadequacies in need of resolving. For a country run by a woman, Britain was still a nation in awe of its men.\n\nStepping away from the window, Veronica walked to the small kitchen and put a flame to the grill. She'd take her toast and tea, and then, without further ado, she'd hail a cab to Bloomsbury and allow her head to be filled with the details of the case. That way, she thought, she might be able to forget the sight of her sister, her eyes shining white in the harsh light of the gas-lamps, screaming Veronica's name as she was pinned to the floor by a coterie of nurses and reassured by the doctors that the only reason she was suffering so much was because she was entirely insane.\n\n_______\n\nThe office door was locked when Veronica arrived at the museum. She fished around in her purse, searching out the key that she carried with her for the rare occasions when she was the first to arrive for the day. She turned the key hastily in the lock and stepped inside, closing the door behind her.\n\n\"Hello?\"\n\nThe place was deserted. In fact, glancing around, she was convinced that it hadn't been disturbed since she was last there herself, with Newbury and Miss Coulthard, almost two days before. She knew that Newbury had given Miss Coulthard leave to take as much time as she needed in the search for her missing brother. The fact that she was not here did not bode well for her success in locating his whereabouts. Sighing, Veronica slipped her bag from her shoulder and placed it on the stand. She did the same with her coat and hat a moment later. Then, glancing at the grandfather clock in the corner, decided to press on for a while in the hope that Newbury would soon put in an appearance. If not, she would head over to his lodgings in Chelsea to see if she could find him there.\n\nShe set about making herself a pot of tea, and decided to take some notes, trying to put all of her haphazard thoughts about the case into some sense of order. That way, when she did finally manage to catch up with Newbury, she'd be able to present her ideas in something of a more coherent form.\n\nAn hour later, it was past nine o'clock and there was still no word of Newbury. Veronica had filled two sheets of paper with copious notes on the case of _The Lady Armitage,_ recounting not only her own thoughts on the matter but also the chain of events that had led them to this point in the investigation. If she were asked to write a report on the case at a later date, the notes would prove an invaluable basis for the endeavour.\n\nGlancing up at the clock again, she decided that it was time she tried to find out what had happened to her employer. She hoped that he hadn't been called away to another crime scene during the night, at least without attempting to get a message to her first. Even though she didn't relish the idea of encountering more cadavers, she also didn't want to find herself suddenly left out of proceedings. It wasn't like Newbury to leave her in the dark, though. She'd known him for only a matter of weeks, but already they had formed a mutual respect for each other, and no matter how secretive some of his pursuits may be, she knew that he wasn't in the business of shutting her out. She'd just have to track him down and find out what it was that had delayed him.\n\nVeronica gathered her things and scrawled a brief note, which she left on Newbury's desk, just in case they accidentally missed each other as she made her way over to Chelsea. She locked the office door behind her, climbed the stairs to the ground floor\u2014where the exhibitions were already beginning to fill with the noisy hubbub of the public\u2014and left through the main entrance in search of transport.\n\nNewbury's home was a delightful terraced house in a quiet suburban district of Chelsea. The entire street in which it sat appeared comfortably middle-class, residential and relatively unassuming. As she stepped down from the cab and paid the driver, Veronica tried to reconcile this fact with her knowledge of the man himself. Everything about the look of the house, at least from the outside, seemed to represent exactly the opposite of what she had taken to be Newbury's taste. The place looked decidedly _old-fashioned_ : a traditional English home, with a small rose garden at the front of the property and a door painted in bright pillar-box red. An ornate black railing ran around the edges of the garden, and a short path led up to the door itself, terminating in a series of tall steps. A bay window looked out onto the street below, although the light was reflecting brightly on the glass panes, making it difficult for Veronica to see if there was anyone inside. She shook her head. For a man so obsessed with the benefits of progress, Newbury kept a house that seemed a trifle understated and traditional. Still, she supposed it was good to challenge stereotypes.\n\nHesitating for a moment, the thought flashed through her mind that she might have given the cab driver the incorrect address. She searched out her notebook and double-checked the number on the door. It was certainly the address Newbury had given her, written in her book in her own neat copperplate: _10 Cleveland Avenue, Chelsea._ She shrugged to herself and approached the door, rapping the knocker briskly. Behind her, the cab rolled away down the road, its horse's hooves clattering noisily on the cobbles.\n\nShe waited for someone to answer the door. There was no response. She knocked again, louder this time. After a few more moments had passed and there was still no answer, she stepped away from the door and tried peering through the window instead, cupping her hands around her face to help her see. The room beyond the window had been dressed as a dining room, containing a long oval-shaped table, a small fireplace, a teak sideboard and a series of bookshelves lined with numerous leather-bound tomes. The door to the room was shut, and there was no evidence that the furniture had been disturbed that morning. She turned away, trying to decide what to do. It was clear that Newbury wasn't at home, and she had no idea where he may have gone, other than the office. She could head back there in the hope that he would eventually put in an appearance, or else she could return to Kensington and await his call. She chewed on her bottom lip thoughtfully.\n\nThen, just as she was about to take her leave, the door clicked open behind her, and a rotund middle-aged woman dressed in the black uniform of a housekeeper appeared in the hallway, trying to catch her breath. \"Oh, I'm sorry, miss. I was out in the back, dealing with the linens.\" Veronica noticed that the woman's sleeves were rolled up and her hands were still dripping with water.\n\nShe smiled. \"I'm sorry to drag you away from your duties. You must be Mrs. Bradshaw? Sir Maurice has spoken very highly of you.\"\n\nThe woman looked perplexed. \"Indeed I am, miss. And how can I be of service?\" She spoke with a warm Scottish lilt. Her grey hair was scraped back severely from her face, worn in a black net, and whilst she certainly cast an imposing figure, it was clear she was a person of warmth and integrity. Veronica could see why Newbury liked her.\n\n\"My name is Miss Veronica Hobbes, Sir Maurice's new assistant. I was supposed to be meeting him at the museum this morning, but he hasn't arrived, so I thought it best to call instead, to ensure everything was in order.\" She craned her neck to see past the housekeeper and into the hallway beyond. It was gloomy inside, with deep burgundy wallpaper and dark wooden furnishings that added to the sense of the austere. There was no sign of Newbury, although she supposed he could have been elsewhere in the house, in the living-room or working out of sight in his study.\n\nMrs. Bradshaw glanced from side to side, looking along the street. She fixed her eyes on Veronica. \"Miss Hobbes, the master told me to make you welcome if you ever had reason to call. I think you'd better come inside.\"\n\nVeronica frowned. The woman seemed strangely on edge, as if Veronica's presence in the house would somehow make her uncomfortable. Nevertheless, she mounted the stairs to the door and stepped through into the dark hallway beyond.\n\nNewbury's coat and hat were still hanging on the stand beside a small table and mirror. The post was lying unopened on the table. Veronica turned to Mrs. Bradshaw. \"Is Sir Maurice at home?\"\n\n\"Yes, miss, although I'm not sure he is receiving visitors.\" She looked concerned, and it dawned on Veronica that something was not quite right.\n\nShe decided to press the woman further for an explanation. \"Is Sir Maurice unwell? I assure you, Mrs. Bradshaw, that I have only his best interests at heart, and that you can rely on me to treat the matter with the utmost sensitivity.\"\n\nMrs. Bradshaw sighed. \"Very well, miss. Let me take you to him now.\"\n\nVeronica placed her hat beside Newbury's on the stand and unbuttoned her coat as they walked. Mrs. Bradshaw led Veronica up the creaking flight of stairs at the end of the hallway, past a small landing that branched off into a sizeable bathroom, and then up to the first floor, where a series of doors opened onto what Veronica assumed were the bedchambers.\n\nVeronica hesitated. \"Is he resting in bed, Mrs. Bradshaw? I'm not sure that it would be entirely appropriate for me to see him in that way.\"\n\nMrs. Bradshaw shook her head. \"No, miss. He's in there.\" She indicated a panelled door at the end of the landing. \"That's his private study. The master has been holed up inside since yesterday morning. He stepped out, and when he returned, he went directly to this room and locked himself inside. I've been unable to get a word out of him since.\"\n\nVeronica looked puzzled. \"Do you think he's unwell?\"\n\nMrs. Bradshaw shrugged. \"I can't say, miss. It's unusual behaviour, certainly. Not that I'm a stranger to that, these last few years.\" She looked circumspect. \"But I worry he hasn't eaten, or taken anything to drink. I've tried knocking, but I've had no reply.\"\n\n\"Do you have a key?\"\n\n\"No, miss. It's the one room in the house that Sir Maurice keeps to himself. He said if I were ever to go in there, I would be immediately dismissed from his service. God knows what he's got in there, but I ain't about to try and find out.\"\n\nVeronica nodded. \"I'm sure it's just a case of security, Mrs. Bradshaw.\" She put her hands on her hips. \"Now, would you mind if I tried to solicit a response?\"\n\n\"Please go ahead, miss. It would put my mind at rest to know the master was well.\"\n\nVeronica approached the door. She put her ear to one of the panels, listening intently for any sound from within. Nothing. She pulled the red leather glove off her right hand, placing it carefully in her coat pocket, and rapped loudly on the door. \"Sir Maurice? It's Veronica. Are you well?\"\n\nShe paused for a moment, waiting for a response. She glanced at Mrs. Bradshaw, who offered her a noncommittal shrug. The moment stretched. She knocked again. \"Sir Maurice? Are you home? I have some thoughts on the case I'd like to discuss with you today.\" Still nothing.\n\nVeronica frowned, addressing her next question to Mrs. Bradshaw. \"You're sure he's in here? Could he have left during the night?\"\n\n\"No miss. His bed is undisturbed, and his coat and hat are still on the stand downstairs.\"\n\nVeronica tried the handle. It turned, but the door wouldn't open.\n\n\"He always keeps this door locked, miss, even when he's inside. If he asks for tea, I leave it out here on the landing and he collects it at his leisure.\"\n\nVeronica smiled. \"Mrs. Bradshaw. All this talk of tea is making me thirsty. I don't suppose you would be so kind as to put the kettle on the stove for me?\" She rubbed the back of her neck. \"I'll continue to try to raise a response from Sir Maurice. I'll be sure to call if I have need of your assistance.\"\n\nMrs. Bradshaw looked uneasy. \"Are you sure, miss? Somehow it doesn't seem appropriate to leave you up here alone.\"\n\n\"Please do not concern yourself with propriety, Mrs. Bradshaw. I am sure Sir Maurice would trust me enough not to idly wander through his private rooms. I assure you I will remain just here on the landing and attempt to find out what is preventing him from answering our calls. Once the tea is prepared, we'll take stock of the situation and agree on a course of action.\"\n\n\"Very well, miss. I'll be in the kitchen if you need me.\"\n\nVeronica watched as Mrs. Bradshaw disappeared down the stairs, her long skirt swishing around her as she walked.\n\nShe knocked on the door again. There was still no response from within. She glanced behind her, judging the length of the landing. There was plenty of room for a run-up. She slipped her other glove from her left hand, popped it in her pocket and wriggled out of her coat, draping it over the side of the banister. She adjusted her blouse. Then she walked to the other end of the landing and, with one last glance down the stairs to make sure that Mrs. Bradshaw was completely out of sight, took a run at the door, presenting her shoulder to the wooden panels. The door creaked in its frame, but didn't give way. She tried again, this time throwing all her weight in front of her as she slammed into the door. It burst open with a loud splintering sound, banging against some unseen piece of furniture inside and kicking back at Veronica, who was struggling to maintain her balance. She caught the door as it came back at her and leaned on it heavily, her shoulder aching from the impact. She hoped that Mrs. Bradshaw hadn't heard the noise in the kitchen two floors below, that the sound of the kettle whistling on the stove had been enough to mask the racket. She'd know soon enough, if the housekeeper came running up the stairs to see what all the fuss was about.\n\nGasping for breath, she looked around, searching the room for Newbury.\n\nThe first thing that struck her about the study was the sheer amount of bizarre paraphernalia that lined the shelves. Aside from the vast array of books, there were all manner of esoteric objects on display. Jars containing what looked like the amputated tentacles of an unidentifiable sea creature, the skull of a chimpanzee, bottles filled with strange-coloured liquids, arcane symbols cast in precious metals, little stone idols that appeared to date from sometime in pre-history\u2014the list was endless. The second thing that struck her was that Newbury was lying face down on the floor, in the centre of an large pentagram that had been drawn on the bare floorboards in white chalk. The carpet had been rolled back to reveal the symbol, although it wasn't immediately clear if it was freshly drawn or had been hidden under the Turkish pile for some time. Objects lay all about the prone man: an empty glass and wine bottle, a sprig of rosemary, some matches and a brown medical bottle half-full of liquid.\n\nShe rushed to Newbury's side, kneeling on the floor and rolling him over onto his back. His breath was shallow and his face was cold and glistening with perspiration. She searched for his pulse, feeling around his unshaven throat until she found it, counting out the rhythm under her breath. She loosened his shirt and placed a hand on his cheek. \"Oh, Newbury, what have you been up to?\"\n\nHe moaned, his eyes flickering under their lids.\n\nVeronica heard footsteps on the stairs. Mrs. Bradshaw had obviously realised something was amiss. She called up ahead of her. \"Everything alright up there, miss?\"\n\nVeronica knew immediately that she couldn't allow Mrs. Bradshaw to see Newbury in such a state, or let her see the inside of his study, either. The contents of the room were alarming enough to Veronica herself, and she already had a very good notion of Newbury's expertise in the dark arts and all the mysterious paraphernalia associated with them. The scene inside the room would probably be enough to send poor Mrs. Bradshaw running straight to the police.\n\nVeronica propped Newbury's head on a cushion that she grabbed from the nearby daybed and stepped out into the hallway, closing the door behind her. She stood in front of the damaged lock, ensuring that Mrs. Bradshaw couldn't see where the frame had been splintered during her assault on the door.\n\n\"Everything is fine, Mrs. Bradshaw,\" she said as calmly as possible. \"You will be pleased to hear that I have managed to rouse Sir Maurice. He is suffering from a slight fever and has been dozing in his study. I'm attending to him now. I'm sure that he will shortly be anxious for some light food to aid him in his recovery.\" She smiled. \"For now it would be of much benefit to him if you could fetch us another cup and saucer to go with that pot of tea.\"\n\nMrs. Bradshaw eyed her inquisitively. There was an awkward silence. Then, realising that it was probably better to go along with Veronica's instructions than defy her employer's wishes and enter the study herself, she nodded her head in assent. \"Right you are, miss. I'll leave the tea on the landing for the two of you.\" She turned and made her way back down the stairs.\n\nVeronica called after her. \"Thank you, Mrs. Bradshaw. And if you could see yourself to fetching a flannel and a bowl of cool water, that would be most helpful, too.\" She slipped back into the room, not waiting for Mrs. Bradshaw's response.\n\nNewbury hadn't tried to move. He was only semi-conscious, possibly even delirious. She bent over him, grabbing him firmly under the arms, and hauled him up onto the daybed a few feet from where he was lying. She paused for a moment, trying to catch her breath after the exertion. Making sure he was comfortable, she set about collecting the objects from the floor, placing them neatly on the coffee table by the side of the fire. She picked up the little brown bottle and inspected the label. It was peeling, but she could easily make out what it contained.\n\n\"Laudanum.\" She shook her head. She had no idea what Newbury had been up to with the pentagram, but it was clear to her that the laudanum was responsible for his current state of ill health. She rolled the carpet back into place, hiding the elaborate chalk symbols. She had a lot of questions for her employer, but first she had to make sure she could bring him round. She crossed the room and went to his side. Taking her handkerchief from her sleeve, she gently mopped his brow, brushing his hair back from his forehead with her other hand.\n\n\"So you do have an Achilles' heel, after all, Maurice.\" She dabbed tenderly at the beads of sweat running down his face.\n\nSearching out a blanket was a relatively easy task. She laid it over him as he shivered, then set about stoking the fire, which had burned low in the grate without attention. Long ago, when her sister had first begun having seizures, the doctors had treated her with laudanum, and she knew all too well the pains of withdrawal, having spent long hours by Amelia's bedside as she came round from the large doses she'd had administered to her in an attempt to quell her visions. She watched Newbury as he lay there on the daybed, his breath still shallow as his lungs fought for air. He'd clearly taken too much of the dreadful stuff. Now it was just a waiting game as his body purged itself of the drug. Veronica, making herself comfortable in a chair by the fire, would stay by his side as it did so.\nCHAPTER 13\n\n##\n\nWhen Newbury woke, he was appalled to find Veronica asleep in the chair by the fire. He had no idea how much time had passed. He sat up, bleary eyed, and then sank back into the warm confines of the daybed, unable to move. His head was spinning, and he felt sick to his stomach. He ran a hand through his hair, which was damp with perspiration, and then rubbed at his eyes, trying to shake the feeling of lethargy. He'd lost track of events and couldn't remember how he'd ended up where he was. The physical symptoms, however, were entirely familiar; he knew he'd overdone it on the laudanum.\n\nHe glanced around the study. Everything had been restored to order. After propping him up on the daybed, Veronica must have rolled the carpet back into place to hide the chalk pentagram that he'd drawn on the floorboards. He wondered if that had been for Mrs. Bradshaw's benefit. If so, it suggested that she'd seen it herself. He could only think how shocked and appalled she must have been to see the items that he had on display in there. That, coupled with the fact that Veronica was sitting across the room from him, meant that he'd have a lot of explaining to do. Worse still, Veronica had seen him at his lowest ebb. He wondered if he'd ever be able to earn her respect again. He cursed himself for his weakness. Still, what was done was done, and he supposed it was his own foolish actions that had landed him in this position. Now he had to face his embarrassment with humility. He sighed.\n\nCraning his neck, he tried to work out how Veronica had entered the room. His first thought was that Mrs. Bradshaw must have kept a spare key, one that he wasn't aware of, but then he saw that the doorframe was splintered and the lock was hanging loose where the screws had been torn out of their housing. The door itself was propped closed with a large stone vase that Veronica had taken from one of his displays. Absently, he wondered if she'd realised that it was nearly two thousand years old. Not that it mattered. She'd obviously used her shoulder to barge her way in. She was a strong woman, and he was thankful to her for the consideration she had shown. He'd underestimated her resourcefulness. He wouldn't allow himself to do it again.\n\nNewbury shifted on the daybed, watching Veronica as she slept in the chair, the rise and fall of her chest as her breath came in little flutters, her head lolled gently to one side. The firelight cast dancing shadows all about her. He wanted to stay in that moment, for time to stand still so that he could lie there, basking in the firelight and watching the pretty girl who had come to his rescue\u2014without having to face her when she woke and explain his failings. He imagined watching the light dying in her eyes as he revealed the truth: that aside from his more salubrious pursuits he was a habitual opium-eater and a dabbler in the occult. He had drawn the pentagram on the floor in an effort to divine a solution to the case, and when it hadn't worked, frustrated that he couldn't seem to find the clarity of mind that he had been searching for, he had given himself up to the drug, intent on dreaming his way to the solution. Of course, such is the delusion of the addict, and he had found no salvation in debauchery. He was no closer now to having a solution than he was when he set out from the palace that morning. If indeed it was still the same day; he had no idea how long he'd been unconscious and whether, outside, it was even day or night. He coughed, fighting back nausea. The racking movement caused little explosions of pain in his head.\n\nThe sound of his coughing caused Veronica to stir. Her eyes flicked open. She looked dazed for a moment, before the sight of Newbury seemed to register and she realised where she was.\n\nIn a moment, she was out of the chair and had rushed to his side. \"Maurice. You're awake.\"\n\nHe looked up at her and smiled. \"Indeed. Although I fear I could hardly be further from my true self. I'm sorry you had to see me like this.\"\n\nShe laughed, obviously relieved. \"You did give me an awful fright. But you'll be well soon enough. When you're feeling up to it, Mrs. Bradshaw will prepare some food and draw you a bath.\"\n\nNewbury looked anxious. \"Mrs. Bradshaw? Did she\u2014?\"\n\n\"No.\" Veronica shook her head, cutting him off. \"You need not worry about that. Mrs. Bradshaw didn't see a thing. She thinks you have a fever.\"\n\n\"And you?\"\n\n\"I think you have a fever of your own devising.\" She smiled tenderly. \"Although I assure you that I'm in no position to judge. We all have our secrets and vices.\" She paused. \"I admit I have no idea what you were up to with that pentagram, however.\"\n\nNewbury coughed again, easing himself back into the cushions. His eyes were glassy and tired. \"I was searching for answers.\" He paused, and she listened to his ragged breath for a moment whilst he made up his mind about whether to tell her any more than that. His eyes flicked over her face. \"I was trying to find out who or what was behind the crash.\"\n\nVeronica narrowed her eyes, suspicious. \"And?\"\n\n\"And the exercise proved fruitless. I'm afraid we're no closer now than we were when we last spoke.\" He sighed. Veronica took his hand.\n\n\"What of the laudanum?\"\n\nNewbury grimaced. \"A moment of weakness, is all.\" He met her gaze. \"I shall take the matter in hand.\" He looked away again. \"Now, did you say something about a bath?\"\n\n\"Yes, I'll call down to Mrs. Bradshaw now.\" She rose from her knees and brushed herself down.\n\nAs she turned towards the door, Newbury sat forward, catching her hand. \"Veronica?\"\n\n\"Yes?\"\n\nHe smiled, his face sincere. \"Thank you.\"\n\nShe nodded. \"You're welcome.\"\n\nHer fingers trailed in his as she walked away, leaving the room in search of Mrs. Bradshaw.\n\n\"Thank you, Mrs. Bradshaw.\"\n\nNewbury smiled as his housekeeper presented him with a large plate piled high with a fluffy omelette and crisp bacon. On the table she placed a rack of toast, blackened to perfection. It was late to be having breakfast\u2014almost three in the afternoon\u2014but she was used to such irregularities and had been sure to offer Newbury her sympathies upon discovering he'd been unwell. Happy that she had discharged her duty, she slipped away from the dining room, casting a final glance at Veronica as she pulled the door shut behind her. It was clear that she didn't understand what Veronica's role in this whole matter had been, and that she had mixed feelings about the scenario. On the one hand, Veronica had proved indispensable in helping her to look after Newbury, and had been the one to finally rouse him from his study. On the other, it seemed somehow inappropriate for her employer to so freely allow his female assistant the run of his household, and for her to allow herself to be so familiar with the gentleman, particularly in company. Nevertheless, she had a great deal of respect for Newbury and had been in his employ for many years, so she had decided to trust his sense of propriety and say nothing that may cause offence. She took the stairs two at a time, eager to get back to her chores, and to some semblance of normality, before the day was out.\n\nVeronica sipped at her tea, watching Newbury from across the table as he attacked his meal with vigour. He had spent the last hour taking a bath, shaving and then dressing in his private rooms. He looked almost restored to his former self, save for the dark rings that still sat heavily beneath his eyes. Veronica was sure that a hearty meal would be good for his constitution and aid in his recovery from the effects of the laudanum. She had passed the time whilst he washed and dressed by perusing the spines of the rare books in his study. It was a wide and varied collection, containing many books she had never heard of and was sure could not be found in the annals of the British Library. Whilst she had been aware of Newbury's speciality in dealing with the occult and paranormal, she hadn't been aware of the sheer _intensity_ of his fascination. If finding him semi-conscious inside an enormous chalk pentagram hadn't been evidence enough, the esoteric volumes in his private library had proved beyond a shadow of a doubt that he was one of the foremost experts in the field throughout the whole of the Empire.\n\nShe placed her empty cup on the saucer. Newbury looked up.\n\n\"So, tell me, what came of your visit to the palace yesterday?\"\n\nNewbury finished chewing his food. \"Very little, I'm afraid, although I did manage to tease out of Her Majesty the reason for her unusual interest in the case.\" He reached for his coffee, taking a long draw. Veronica leaned forward, waiting for him to continue. \"Apparently the body of a Dutch Royal was found aboard the wreckage. A cousin of the Queen, in fact.\" He paused, waiting for her reaction.\n\nVeronica frowned. \"But wasn't _The Lady Armitage_ a passenger-class vessel? Why would a member of the Royal Family take a second-class transport to Dublin?\"\n\nNewbury smiled. \"Precisely. But it's not much of a lead. We can't even begin to consider interviewing the family, and besides, they have even less of an idea about the whole thing than we do. The man had been missing in London for days before it happened. Her Majesty has promised the boy's mother an explanation, and it's up to us to find one, as soon as possible.\" He didn't look particularly confident. Taking his cutlery, he continued to tackle his breakfast. Veronica poured herself another cup of Earl Grey. They sat in silence for a few minutes, each of them racking their brains for ideas.\n\nVeronica was startled by a knock on the door. Newbury looked up, but didn't speak. A moment later, Mrs. Bradshaw entered, bearing a silver tray that was covered in letters\u2014the post Veronica had seen on the hall table when she'd first arrived. It seemed like days had passed since her arrival that morning.\n\n\"Your post, sir. I thought you may like to open it whilst you finished your breakfast?\"\n\n\"Very thoughtful, Mrs. Bradshaw. Thank you.\" He watched her leave and then turned his attention to the tray she had placed on the table beside him, studying the contents intently. Five or six letters lay scattered upon it. He placed his cutlery on the side of his plate and poked at the envelopes, stopping when he saw one that bore a hand he didn't recognise.\n\nHe glanced up at Veronica. \"Excuse me for a moment, my dear, whilst I take a look at this rather interesting missive.\" He used his finger to tear the envelope open and withdrew the letter he found inside. It was dated the previous day, and written in a perfect copperplate, with big artistic flourishes, on plain white paper. Newbury scanned the short paragraph that composed the body of the letter, then folded it in half and passed it to Veronica.\n\nVeronica unfolded it and spread it out on the table before her.\n\n> _Sir Maurice,_\n> \n> _I request your presence at the Orleans Club, 29 King St, S.W., tomorrow at four. I find myself in possession of information that may pertain to your current investigation, regarding the crash of the passenger airship_ , The Lady Armitage. _I'd appreciate the opportunity to aid you in bringing the perpetrators in this matter to justice._\n> \n> _Yours,  \n>  Mr. Christopher Morgan_\n\nShe looked up. \"Do you know this man?\"\n\n\"Indeed not. Although . . .\" He thought for a moment. \"I believe I know him by name and reputation.\" He took another sip of his coffee. \"A speculator and a dilettante, if I'm not mistaken. I believe he owns an art gallery across town.\" He smiled, dabbing his mouth with his napkin. \"Nevertheless, Miss Hobbes, we have our lead, and no time to spare. If we're to make it to the Orleans Club by four, we should be on our way directly. Are you fit?\"\n\nVeronica smiled back, delighted to see Newbury so engaged and full of energy once again. She nodded. \"Are you?\"\n\nNewbury laughed, shrugging his shoulders. \"Fortified by eggs and bacon. Let us not procrastinate any longer.\" He stood, pushing the remnants of his meal to one side. \"Come on, let's fetch our coats.\"\n\nVeronica watched Newbury's back as he left the room, calling for Mrs. Bradshaw. She hoped he was up to another sojourn, and whilst she admitted to herself it was wonderful to have the old Newbury back, she felt drained by the whirlwind that surrounded him. She'd rather, for his health, that they put the meeting off until the following day, but with no return address on the letter, it would be difficult to get word to Morgan in time\u2014and in truth, it was too good an opportunity to miss. It was the only lead they had, and if they chose to enjoy the confines of Newbury's home for much longer, the trail would almost certainly grow cold again. Reluctantly, she climbed to her feet and followed after him, anxious to keep a watchful eye on proceedings, and on Newbury himself.\nCHAPTER 14\n\n##\n\nIn their haste to get across town, Veronica had allowed herself to be subjected to the noise and bluster of one of the steam-powered carriages that Newbury appeared so heartily to enjoy. It had proved as uncomfortable as ever, and now, on the doorstep of the Orleans Club, she found herself rearranging her dress and trying to put herself hastily back in order. It was cold, and fog was beginning to settle over the streets in wispy tendrils, slowly encroaching upon the city like ivy creeping across an old brick wall.\n\nThe Orleans Club, Newbury had informed her on the way over, was the offshoot of a gentle-men's club based in Twickenham, the town dwelling for members of the latter who, it seemed, were welcome to invite guests to the establishment so long as they were of the male variety. Any women were referred directly to the ladies' room and kept well out of earshot of the banter that took place in the main lounge. Veronica found the whole idea ridiculous, but she also knew that she wasn't about to overturn hundreds of years of tradition simply by complaining about it. She was aware that Newbury attended a club, and that he found it a worthwhile pursuit, in terms of both business and pleasure. Not only that, but it was important that they got to speak with Morgan, one way or another. She supposed she'd just have to live with it, for now.\n\nThe building itself was typical of this type of establishment; a Georgian town house that sat mid-terrace between what appeared to be private dwellings on either side. Sash windows revealed little about the activities inside, covered by heavy drapes, and there were no signs or indicators that they had even come to the correct address, other than the number _27_ on the door, as suggested in Morgan's letter. Clearly the members of the Orleans Club liked to carry out their business behind closed doors.\n\nNewbury stepped up to the blue-panelled door and rapped loudly with the knocker. Almost immediately, it creaked open and a butler appeared in the opening. Light spilled out onto the steps around their feet. Newbury presented his letter and informed the man that they had come for a private conference with one of the club's members, Mr. Christopher Morgan.\n\nThe man studied Newbury and Veronica with what seemed to be a measure of disdain. \"I'm afraid we have yet to enjoy the pleasure of Mr. Morgan's company today, sir.\"\n\nNewbury pulled his watch from his pocket, popping open the engraved case and glancing at the ivory face inside. \"I see we're a little early. Perhaps Mr. Morgan intends to meet us here at four, as his letter suggests, or perhaps he is running a little late. Either way, I do believe that we'd like to wait.\"\n\nThe butler nodded, opening the door a fraction wider to allow them to pass. \"Sir can wait in the lounge, and I'll be sure to inform Mr. Morgan of your presence when he arrives. I'm afraid your companion will have to wait in the ladies' room.\"\n\nNewbury put his hand on Veronica's arm. \"As I suspected, my dear. I'll try not to be too long about it. Why don't you ask around in there and see if you can get a measure of this fellow from the other ladies? It may be that you can find out something useful while you're waiting.\"\n\nVeronica nodded. \"Of course.\" She allowed the butler to escort her to the door of the ladies' room, whilst Newbury disappeared down the hallway in the direction of the main lounge. The butler held the door open for her, and she stepped through.\n\nThe ladies' room was clearly an underused commodity. The room itself was small, and whilst lavishly furnished, it bore the musty odour of emptiness; Veronica had the sense that the place was more of a showroom than a location where ladies actually went to pass the time, at least by choice. She suspected that the room was provided as a service to those unlucky men who didn't seem able to go about their business without their wives following on behind them, limpetlike. That or it was listed as a benefit in the members' book, and as such had to be upheld for those rare occasions when a lady actually found herself in the unenviable position of needing somewhere to wait for her companion whilst he went about his business inside. Whatever the case, there were only two other ladies present in the room when Veronica entered, and both looked up, startled, to see a newcomer whom they might endeavour to coerce into a discussion of some sort. They both stood, placing the books they had been reading on the chairs where they had been sitting. Veronica smiled warmly. \"Good afternoon, ladies.\"\n\nThe two women looked at each other, and then turned back to Veronica. The one on the left, who was wearing a long dress cut in pale yellow silk, returned Veronica's smile. \"Likewise, I'm sure.\" She indicated the chair beside her. \"Please, won't you join us for tea?\"\n\n\"I'd be delighted.\" Veronica walked over to the table and the two ladies returned to their seats.\n\nThe woman in the yellow dress poured Veronica a cup of tea from the silver pot on the stand beside her chair. \"My name is Mrs. John Marriott, although you may call me Isabella.\"\u2014she glanced up\u2014\"This is Miss Evelyn Blackwood.\"\n\nVeronica took the proffered cup and saucer. \"Thank you. My name is Miss Veronica Hobbes. It's a pleasure to meet you.\"\n\nEvelyn Blackwood, a young dark-haired woman in a red jacket and matching skirt, looked Veronica up and down. \"Is this your first time at the Orleans Club, Miss Hobbes? I haven't seen you here before.\"\n\nVeronica nodded. \"Yes, indeed. My associate is here to meet one of the members. I thought it wise to wait for him in here.\"\n\nIsabella Marriott gave her a conspiratorial wink. \"So, dear, who exactly is this mysterious 'associate'? You can be sure that your secret is safe with us.\"\n\nVeronica almost laughed out loud. She had no reason to hide her association with Newbury, and it was clear that the two ladies, so starved for company, were fishing for gossip and intrigue to keep themselves amused. It would do no harm to let them think what they would. In fact, it may help to draw them out on their thoughts about Morgan. \"I'm here with Sir Maurice Newbury, the academic and anthropologist.\"\n\nIsabella and Evelyn exchanged glances. \"A sir? Well, didn't you do well for yourself, Miss Hobbes?\" Both of them began to giggle like schoolchildren. Veronica was finding the whole experience incredibly trying. \"So tell, us, Miss Hobbes. Is he devilishly handsome?\"\n\nVeronica took a sip of her tea, wishing for a moment that it was something stronger. \"Well, I suppose he is, rather.\" She tried to look coy, playing along with the conversation.\n\nEvelyn clapped her hands together. \"How exciting! A new romance in the Orleans Club. Just wait until we tell Juliana!\"\n\n\"Now, now, Evelyn, don't get carried away.\" Isabella placed a hand on her friend's knee. \"Miss Hobbes is only just getting started.\" She looked at Veronica expectantly.\n\nVeronica saw her chance to turn the conversation in a different direction. \"Well, Sir Maurice is here for an important meeting with Mr. Christopher Morgan. I've heard a lot about the man, but I've never had occasion to meet him. Is he a fine fellow?\"\n\nIsabella looked impressed. \"Oh, Miss Hobbes, one of the finest. Mr. Morgan is a pillar of our community, both here and in Twickenham. He owns an art gallery in town, and all the ladies who've been lucky enough to visit the place say it's full of the most wonderful paintings. Mr. Morgan, is a true gentleman. I'm sure that if your Sir Maurice is having any dealings with Mr. Morgan it is a good reflection on them both.\"\n\nVeronica smiled. \"I'm delighted to hear it, Miss Marriott. I appreciate your candour.\"\n\nEvelyn leaned forward, clutching her empty teacup to her knee. \"Do you think Sir Maurice might decide to become a member of the Orleans Club? I'm sure the other gentlemen would make him most welcome, and I'd love to introduce you to Juliana.\"\n\nIsabella cut in before Veronica had chance to answer. \"Juliana is Evelyn's elder sister. She recently married an industrialist named Greene. She has pretensions of becoming a novelist.\"\n\n\"Really?\"\n\nEvelyn nodded enthusiastically. \"Actually, I believe she's really rather good. She gives Margaret Oliphant a run for her money, anyway.\" She patted the book beside her on the chair and smiled.\n\nVeronica tried to look engaged by the idea. \"I'm sure that she's very talented indeed, Miss Blackwood.\" She placed her cup and saucer on the table. There was a rap at the door. The three women looked up to see Newbury framed in the doorway.\n\n\"Miss Hobbes. I'm sorry to disturb your conversation, but I believe our business here is done.\"\n\nVeronica tried to hide the relief on her face. As she stood, Isabella leaned in and whispered surreptitiously, \"You're right, dear\u2014he's terribly dashing.\"\n\nVeronica smiled knowingly and turned to face both of the ladies. \"Good afternoon, ladies. It's been a pleasure.\"\n\nEvelyn glanced from Newbury to Veronica. \"You must come and see us again, Miss Hobbes. Sir Maurice, do say you'll bring her again.\"\n\nNewbury coughed to cover his laughter. \"All in good time, I'm sure.\"\n\nEvelyn smiled triumphantly. \"That's settled then. Next time Juliana may be here. I am sure she'd be delighted to tell you about her writing.\"\n\n\"I'll look forward to it.\" Veronica turned on her heel and joined Newbury in the hallway, before the two of them took their leave of the Orleans Club and headed out into the cold afternoon.\n\n\"So, how did you find Morgan?\"\n\nThey were waiting for a cab by the side of the road. The fog had settled even lower during the time they had passed inside the Orleans Club, and the street seemed deserted, wreathed in a thick smog. Veronica was standing close to Newbury, partly in an effort to fight off the penetrating chill, but partly for the comfort of having him nearby. The fog made her uncomfortable these days, what with all the talk of revenants and glowing policemen. She had resolved to spend as little time out in it as possible, for the time being, at least.\n\n\"I'm afraid I didn't find Morgan at all. He didn't keep our appointment. Either he was detained elsewhere, or simply decided that his information wasn't so inflammatory after all.\"\n\nVeronica frowned. \"That sounds unlikely, especially after hearing about him from the ladies inside the club.\"\n\nNewbury chuckled. \"Yes, you did seem to ingratiate yourself with them rather.\"\n\nVeronica sighed. \"I admit that I find that sort of woman most difficult to engage. I think it was their sheer desperation at seeing another female face that led them to embrace me so quickly.\"\n\nNewbury shrugged. \"Did they reveal anything useful, other than recommendations for the latest romance novel or the usual society gossip?\"\n\n\"Not as such. Although they did go on at length about Morgan, assuring me he was an excellent fellow, a perfect gentleman and a 'pillar of their community.' Doesn't sound to me like the sort of chap not to keep his appointments.\"\n\n\"Indeed.\" Newbury paused at the sound of horse's hooves. He stepped into the road for a moment, catching the attention of a cab driver. He came back to stand beside Veronica as the cab drew up before them, coming to rest beside the curb. \"Well, it's been a difficult day for us both, Miss Hobbes, and I suspect, with the dark drawing in, that it's a little too late to go searching for Morgan now. What do you say that I drop you at home and we set out again first thing tomorrow morning for Morgan's gallery? We shouldn't allow the trail to go cold, no matter how tenuous it actually is.\"\n\nVeronica nodded her assent. After the day she'd had, she'd be glad for a hot bath and an early night. \"Will you be alright, Sir Maurice?\"\n\nHe caught the meaning behind her words as he opened the door of the cab for her. \"I'll be fine, Miss Hobbes. Absolutely fine.\"\n\n\"In that case, I think it is an excellent plan. I'm sure we could both do with the rest.\"\n\nThey mounted the cab and gave the driver directions. Then, falling into a casual silence, each of them watching the fog roll by the windows of the cab, they set out for Kensington, and home.\nCHAPTER 15\n\n##\n\n\"Good God, Newbury. You look done for!\" Bainbridge had never been a man to keep his thoughts to himself.\n\n\"A rough night, Charles, followed by a long day. Think nothing of it.\" Newbury stood to greet his guest. \"How the devil are you?\"\n\n\"Troubled, if truth be told. Can't seem to shake this damn Whitechapel case. I'm starting to think you may have been on to something, you know, with all that 'glowing policeman' business.\" He dropped himself into a chair in Newbury's lounge, sighing, and Newbury took a seat opposite him. He knew Mrs. Bradshaw would already be organising drinks. He hadn't been expecting Bainbridge to call, but he wasn't disappointed by the development. His old friend offered good company, and he was in need of a distraction to prevent him from pondering too long on the other events of the day.\n\n\"Well, I have no doubt Mrs. Bradshaw will be preparing a brandy. We can discuss it at our leisure before a warm fire. I only wish I could do more, but I'm up to my neck in this other affair.\"\n\n\"You're a good man, Newbury. But tell me, I've heard nothing further on the airship disaster. What news?\"\n\n\"Little, I'm afraid to report. Her Majesty is eager for a quick resolution, but the leads are few and far between. She's adamant there's foul play involved, but I admit I'm still unsure. I take it Foulkes hasn't turned up anything useful?\"\n\nBainbridge shook his head. \"Indeed not. He's a good man. Thorough. If there was anything to be found, he'd have turned it out by now. I'm afraid it's in your hands, Newbury. Ah, look . . .\"\n\nThey turned to see Mrs. Bradshaw enter the room bearing two large glasses of brandy. Bainbridge took one from her, smiling, his bushy moustache quivering as he did so. \"An asset to you, Newbury.\" He raised the glass to Mrs. Bradshaw. \"I'm in dire need of a housekeeper like you, Mrs. Bradshaw. Many thanks.\" He took a long draw of the brandy, blinking as the alcohol assaulted his palate. Newbury sniffed at his glass and then placed it on the low table between them. He wasn't sure his damaged constitution was ready for it just yet. Mrs. Bradshaw quickly made herself scarce.\n\nNewbury leaned back in his chair, making himself comfortable. The room was small and cosy, with three chairs, a roaring fire, a small bureau and a portrait on the wall showing his grandfather in his military attire. The man had fought in Afghanistan during the expansion of the Empire, and was in many ways responsible, if indirectly, for Newbury's fascination with the occult. John Newbury had died in action, and his small chest of belongings had been returned to the family back in London aboard an old steamer. Still only a boy, Newbury had wondered at the secret contents of the chest, which his father had kept locked and hidden under his bed. One day, when his father was away on business and his mother was receiving visitors in the rooms below, Newbury had taken the key from the drawer in the nightstand and crawled underneath his parents' bed, searching out the chest and unlocking the ornate clasp. The contents were to change his life forever.\n\nAside from the more typical paraphernalia of war\u2014a pistol, a dagger, a medal\u2014the chest contained three books of a kind young Newbury had never encountered before. The knowledge within them would send him spiralling into a world full of mystery, full of magic and creatures of the night, rituals and charms. They contained a secret history of the world, a cata-logue of the occult and a guide to all the bizarre, arcane practices that demonstrated the thin line between life and death. For weeks, Newbury would return to the chest underneath his parents' bed, digging out his grandfather's books and reading by candlelight, filling his head with wonders. He still had the books, now, safe in his study, reclaimed from his father's belongings after both his parents had died. The chest had remained in place for another thirty years, undisturbed, and the day he had finally laid his mother to rest he had returned to the family home to collect it. By this time, of course, Newbury had assembled a vast library dedicated to the arcane, but these particular volumes he had never found again, and they now held pride of place in his collection. He wondered if they were the only three copies of the books that still existed, anywhere in the Empire.\n\nSnapping out of his reverie, Newbury glanced at Bainbridge, who had downed the rest of his brandy and was watching him inquisitively. \"Lost you for a moment, Newbury. Everything alright?\"\n\n\"Yes. Yes, indeed. I was lost in thought. Apologies, old man.\" He clapped his hands together, demonstrating that Bainbridge had his full attention. \"So tell me, what's troubling you about the Whitechapel case?\"\n\nBainbridge stared at the empty glass in his fingers, turning it over so that it caught the light. \"We're just getting nowhere, Newbury. More and more bodies are turning up, dumped all over the place, and we don't even have a suspect. The witnesses, such as they are, all report seeing a ghostly blue figure emerge from the fog, and then they damn well run for their lives. Who can blame them? Some report hearing the screams of the victims as they run, but that's about all we've got to go on. It's the same every time\u2014the victim is strangled, apparently without motive, and none of their belongings are taken or disturbed. There is never any trace of the killer left on the scene, and we haven't been able to find anything that links the victims to one another either. I admit to being completely confounded by it all.\" He looked exasperated, and Newbury, taking pity on his old friend, got out of his chair and searched out a bottle of brandy from a small cabinet on the other side of the room. He placed it on the table in front of a thankful Bainbridge before dropping back into his seat.\n\n\"Well, I can see why you're grasping at straws.\" He smiled. \"Miss Hobbes had an interesting notion a few days ago that the killer may not be the original 'glowing policeman' at all, but a new one, an example of the same phenomenon at work, involving different people entirely. Have there been any constables killed in recent months?\"\n\nBainbridge looked thoughtful. \"Not that I'm aware of. Although it's certainly worth double-checking. I'll have a man look into it tomorrow.\"\n\n\"Excellent. Other than that, have there been any changes at all in the pattern of the murders? Any minor detail that you haven't mentioned to me as yet?\"\n\nBainbridge poured himself another drink. \"Not as such, although the most recent body was different from the rest.\"\n\nNewbury leaned forward, his interest piqued. \"How so?\"\n\n\"It was a gentleman. All of the victims so far have been paupers, down-and-outs. This chap was a member of a private club with connections to a number of well-respected families. He had no real business being in Whitechapel in the early hours of the morning. We're wondering if he was actually killed elsewhere and then moved across town to give the impression that he was just like all the other victims.\"\n\n\"What was his name?\"\n\n\"Christopher Morgan. Owned an art gallery not far from here, I'm given to understand.\"\n\nNewbury practically leapt out of his chair. \"Charles! Morgan asked me to meet him this very afternoon! Now I know why he didn't keep his appointment. There has to be a connection. Look here\u2014\"\n\nHe sprang out of his seat and rushed to the pile of papers he'd left on the bureau. He rifled through them, discarding most of them on the floor in his haste. After a moment, he put his hand on the envelope he'd received that afternoon, containing the letter from Morgan. He handed it to Bainbridge, who eyed it curiously.\n\n\"Go on. Open it, Charles!\"\n\nBainbridge slipped the letter out of the envelope and cast his eye over it warily. He seemed to take a moment to let it sink in, then folded it neatly, put it back inside the envelope and placed it on the table beside his drink. \"So Morgan had a secret about the airship disaster, and then he turned up dead at the hands of the glowing policeman on the same day he was supposed to meet with you to reveal it.\"\n\n\"Or at the hands of someone wanting us to _believe_ it's the glowing policeman. He may well have been killed elsewhere and deposited at Whitechapel, just as you suggested.\"\n\n\"It can't be a coincidence.\"\n\n\"Only further investigation can help us to establish that, my dear man.\" Newbury was animated now, and he reached for his brandy, hoping it would help to steady his jangling nerves. \"Charles, I need to see the body.\"\n\n\"Impossible.\"\n\n\"How so?\"\n\n\"Because it's already been delivered to the morgue for a post-mortem examination. They'll be cutting him open at first light.\"\n\nNewbury shook his head. \"Then we go now. It's imperative that I get to examine the corpse. It could shed light on both of our cases.\"\n\nBainbridge nodded, although he was obviously reluctant to venture out again at this hour. He glanced at his fob-watch. It was approaching seven o'clock. \"What about dinner? Could we stop somewhere on the way?\"\n\n\"Afterwards, Charles! This could be the breakthrough we've been waiting for. Let's not waste another second!\"\n\nBainbridge downed the last of his brandy and stood to join Newbury at the door. \"My private coach is downstairs. We'll take that directly to the morgue. They won't be happy to see us at this hour, but I'm sure we'll be able to talk them around. Shall I send for Miss Hobbes?\"\n\nNewbury thought for a moment. \"Let's not. We'd only disturb her unduly. I can fill her in when we meet tomorrow morning.\"\n\nBainbridge nodded, and together they set out in search of clues.\n\n_______\n\nThe morgue was a cold and dreary place, in keeping, Newbury supposed, with its function as a repository of the dead. This was the place where murder victims or other suspicious deaths would be sent by Scotland Yard for closer examination, before the cadavers were forwarded to a funeral parlour and prepared for burial. Paupers, of course, tended to go directly from the table to a wooden box, and then into the ground, without the dignity of an elaborate service. The state did what it could, but as the politicians insisted on reminding everybody, it was not a charity.\n\nNewbury looked the place up and down as Bainbridge spoke with the mortuary attendant, showing his credentials in an effort to solicit the man's help. The room had a clinical feel, with white-tiled walls and floor, steel instruments set out carefully on wooden trolleys and a pair of marble slabs, empty and awaiting the freshly dead. Newbury shivered despite himself. The room reminded him of a bizarre underground station, with a curved roof and tiled archways leading to other rooms. The entire building seemed to echo with their footsteps, silent save for the voices of the other two men as they agreed, finally, that Newbury could examine the corpse of Christopher Morgan.\n\nThe mortuary attendant\u2014a tall, lean man, freshly shaved, with his blond hair swept back in a widow's peak and a pale complexion that suggested he spent the majority of his time indoors\u2014led them through one of the open archways and into an adjoining chamber, where one of the slabs was covered by a white sheet. With a serious look in his eye, the attendant drew back the cover and allowed them to gaze upon the cadaver that had once been Christopher Morgan.\n\n\"Is this the man you're looking for?\" His voice was nasal and thin.\n\nBainbridge was starting to get impatient with the man. \"We'll have to take your word for it. We have no record of his likeness. Neither of us was in attendance at the crime scene.\"\n\nThe attendant nodded. \"Then please feel free to inspect the body for as long as you deem necessary. I shall return to my post and await news that you have finished.\" He stopped, glancing sharply at Newbury. \"I hope you find what you are looking for.\"\n\nNewbury met the man's gaze. \"Thank you.\" He turned to regard the body, waiting for the attendant's footsteps to disappear into the next room before looking up at Bainbridge, who was opening and closing his fist with impatience. He drove his cane down hard on the tiled floor. \"Despicable fellow. Even after I established my position, he continued to question me regarding our visit. I have it in mind to speak with his superiors about his conduct.\"\n\nNewbury put a hand on his friend's arm. \"It's late, Charles, and our visit is very irregular. Let us concentrate on the task at hand.\"\n\nBainbridge nodded, clearly not placated. \"On with it, then. Let's get this done with so we can get to dinner. This place always gives me the chills.\"\n\nNewbury reached over and rolled the white sheet down to the dead man's knees. It was evident almost immediately that Morgan had been a man of fortune: his black suit was perfectly tailored, probably Saville Row, and his hands were perfectly manicured and impeccably clean. His hair had clearly been worn short in a side parting, but now it had been disturbed, either in the struggle that preceded his death, or during the transportation of the body to the morgue. The man still wore a fine gold ring on his right hand and an expensive chain looped from his fob-watch to his waistcoat pocket. Newbury glanced at Bainbridge. \"So it wasn't a robbery, then.\"\n\n\"No. Just like the others. The only difference here is that Morgan had more on him worth stealing.\"\n\nNewbury felt around in the man's pockets. They were practically empty. One held a handful of loose change whilst another held his wallet. To Newbury's dismay, there was nothing inside that suggested Morgan's reasons for wanting to speak with him at the Orleans Club earlier that day: just a couple of business cards, some banker's notes and a grainy sepia photograph of a woman sitting on a wicker chair, smiling at the camera. He stuffed the wallet back into the pocket where he had found it.\n\n\"Well, nothing so far to shed light on the airship disaster. Let's see if the manner of his death brings us any closer to an answer in the other matter, shall we?\" Newbury inched around the table, examining the corpse in minute detail as he did so. He stopped beside the head, taking the chin between his thumb and forefinger and moving the head from side to side, as if he were trying to make Morgan shake his head. \"The neck's not broken, but there's some pretty serious bruising around the throat. I'd wager it's a crushed windpipe. The assailant appears to have caught him with both hands and throttled the life out of him. Poor chap. It doesn't even look like he got a chance to fight back.\" He leaned closer, examining the bruised flesh around the throat. The skin was starting to take on a waxy pallor as rigor mortis set in. His brows furrowed in concentration.\n\n\"What is it? Have you seen something?\"\n\nNewbury stepped back from the mortuary table. \"Take a look at the bruised areas around the throat.\"\n\nBainbridge handed Newbury his cane and leaned heavily on the marble slab, lowering his face to examine the corpse more closely. \"What am I looking for, man? I can see plenty of bruises. Looks to me like the chap was strangled, just as you said.\"\n\n\"Indeed, but if you look a little closer, you'll see what I'm interested in. There are flecks of blue powder spotted about his throat. It shimmers if you shift slightly in the light.\"\n\n\"My God, Newbury. I think you're on to something.\"\n\nNewbury smiled. \"It's not much, but it certainly suggests our killer may have a more corporeal explanation than we'd previously imagined.\"\n\nBainbridge stepped away from the corpse. \"So what's to be done?\"\n\nNewbury circled the table again, finding the white sheet and folding it neatly back over the corpse. \"Miss Hobbes and I will pay a visit to Morgan's gallery tomorrow and interview the staff. I need to establish what it was he was so keen to talk to me about. It may have been what got him killed, and if so, there's a definite link between the glowing policeman and the wreck of _The Lady Armitage._ \" Bainbridge nodded, listening intently. \"I'd suggest that you have your men test this blue powder at first light. Let's see if they can't establish a manufacturer. That way we can run through their customer records and begin to narrow down the list of potential candidates for our killer.\"\n\nBainbridge grinned. \"Marvellous. Newbury, I knew you'd be of service to me when I knocked on your door this evening. Now\u2014\" He took the other man by the shoulders and led him away from the mortuary slab, his cane clicking on the tiled floor as they walked. \"\u2014what about that dinner you promised me? How about that little place you like by Kingsway?\"\nCHAPTER 16\n\n##\n\nIt was mid-morning before Newbury rose, pulled on his dressing gown and sauntered to the bathroom to begin his daily ablutions with his razor and flannel. The previous day had been a drain on him, both physically and mentally, and today he had chosen to lounge for a while in bed, reading a book. He was, of course, eager to press on with the case, but by the same token was sure that Morgan's gallery could wait for a few hours whilst he ensured that he was fully recovered from the excesses of the laudanum. He had finally emerged around ten o'clock, enjoyed a leisurely feast of porridge, fruit and toast and then, after opening his post, had taken a short constitutional stroll before hailing a cab to Kensington to call for Veronica. His mind felt sharp and alert, his body taut and wiry. His trip to the morgue with Bainbridge had proved enlightening, and he was sure they were getting closer to the heart of the mystery surrounding the wreck of _The Lady Armitage,_ and also the Whitechapel strangulations and the glowing policeman. It was clear that the two investigations were linked, somehow, and he hoped that a visit to Morgan's gallery would help him to establish the nature of that link. It would take a day or two for the police to analyse the blue powder he'd found on Morgan's corpse, but in the meantime, he'd agreed with Bainbridge that he'd press on at the gallery, and that they would keep each other informed of their progress. The discovery of the powder had been playing on his mind since the previous evening, and he couldn't help wondering if he'd somehow missed the evidence on the first few bodies that he had inspected. Were there specks of the stuff on the collars or clothes of those other victims? He certainly didn't recall seeing anything around their throats, save for bruising and the obvious signs of a fight, although he knew by now that it was too late to check. The bodies would have been interred in the local cemetery, and he was loath to start digging up graves on the off chance that he'd still be able to find evidence of a fine blue power on their clothes. In fact, in all likelihood, their clothes would have been burnt and their corpses dressed in their best suits before burial. He clacked his tongue. He supposed it may be that the killer was getting careless or arrogant, confident that no matter what trace he left of himself at the scene, the police would be unable to catch him. He may have taken care to remove all the evidence at the scenes of the first few murders, but after weeks of continued activity with no sign that the police were on to him, he may have grown lazy. Newbury had seen that before: the mad gleam in the eye of the killer, the notion that he was somehow invincible and above the law. It wouldn't surprise him if the killer turned out to be totally insane.\n\nOn the other hand, of course, he'd inspected the other bodies _in situ_ at the various murder scenes, in the dark and the fog, and it could be that he'd simply missed the evidence without the aid of the lamps and the clinical gleam of the morgue. So be it. He knew it was a waiting game now: waiting for the police laboratory to identify the powder, or else waiting for the killer to make his next move. He closed his eyes as the cab rumbled on towards Kensington, wondering which it would be.\n\nVeronica's apartment was on the ground floor of a large terraced house, built during the Georgian period, with tall sash windows and the brickwork rendered in smooth white plaster. The fumes of the passing ground trains and steam-powered carriages had begun to stain the white walls up above, turning them a dirty grey, and Newbury knew that Veronica would disapprove most heartily of this development. He found a delicious irony in that. Veronica was such a forward-thinking woman, and put such great stock in the liberation of the fairer sex, but in other ways she had yet to accept the tide of progress that was currently washing through the Empire. Industry and technology were revolutionising the world, an unstoppable force as certain as life and death, and in Newbury's view, the only options were to embrace it wholeheartedly or else be left behind. He wasn't old enough yet to get stuck in his ways.\n\nWhen Newbury did finally rap on Veronica's door, it was clear almost immediately that she had spent most of the morning awaiting his arrival. Moments after her housekeeper had come to the door, Veronica appeared in the hallway, dressed in a short grey jacket, white blouse and long grey skirt.\n\nNewbury smiled at her from the door. \"Good morning, Miss Hobbes. I'll wait for you outside.\"\n\nHe held the cab whilst she collected her belongings and put on a long woollen coat to protect her from the winter chill. The wind was bracing, and Newbury took the opportunity to seek shelter in her doorway whilst he waited. She joined him a moment later, smiled, and then climbed up into the cab without saying a word. Newbury, grinning, gave the driver instructions and clambered in behind her.\n\nSettling into his seat, he turned towards her, only to find her watching him intently from across the cab. He removed his hat and placed it neatly on the seat beside him.\n\n\"You look well today, Sir Maurice. I'm delighted to see it.\" She was wearing a kindly expression.\n\n\"Thank you, Miss Hobbes. I do believe that I am fully recovered. Please, let us speak no more of the incident\"\u2014he looked somewhat sheepishly at the floor\u2014\"if you can bear to forgive me my foolishness.\"\n\nVeronica blinked, looking from his face to the window and back again. \"I see no reason to dwell on it, Sir Maurice.\" She smiled, altering her tone. \"What plans do you have for the day ahead?\"\n\n\"Ah, well, yesterday evening brought with it developments of a sinister kind.\"\n\nVeronica leaned in, intrigued. \"Go on.\"\n\n\"After parting company with you here in Kensington, I returned directly to my lodgings, with plans to settle in for the evening, only to find Sir Charles calling on me half an hour later for dinner. It was an entirely unexpected visit, but certainly not an unwelcome one, and I invited him in to join me. During the course of our conversation, he inadvertently revealed the reason for Christopher Morgan's non-appearance at the Orleans Club yesterday afternoon.\"\n\n\"Which was?\"\n\n\"The simple fact that he was dead.\" Newbury allowed that to sink in for a moment. Veronica searched his face expectantly, waiting for him to continue. \"Killed, apparently, by the glowing policeman.\"\n\nVeronica gasped. \"Where? What happened?\"\n\n\"We're not sure. His body was discovered in Whitechapel like each of the others, but it seems doubtful that he would have been there of his own volition, especially in the early hours of the morning. I suspect he was murdered because of the secrets he held, and his body was moved to Whitechapel in an effort to disguise that fact.\"\n\nVeronica shook her head. \"So are you suggesting the two investigations may be linked?\"\n\nNewbury shrugged. \"Perhaps. I admit I have my reservations. Morgan's death is not a perfect fit with the pattern of the other murders. For a start, he was a gentleman, where the other victims were all paupers. I have no doubt that his death is in some way related to our investigation of _The Lady Armitage_ disaster; it seems far too much of a coincidence that Morgan would write to me claiming to have evidence regarding the matter just a day before he died. I think the question is whether or not his death is truly related to the glowing policeman murders, or whether the circumstances of his death are just an elaborate cover adopted by someone trying to throw us off the scent.\" He scratched his chin. \"I wish I'd had chance to talk with the man. Still, he may have left us a clue all the same.\"\n\nVeronica raised her eyebrows.\n\n\"I visited the morgue with Bainbridge last night to examine the body. We found specks of a strange blue powder around the throat and collar of the corpse.\"\n\n\"Meaning? . . .\"\n\n\"Meaning nothing, as yet. But it could be the method by which the killer is disguising himself as the glowing policeman, covering his face and hands in this iridescent powder. It would certainly fit the descriptions we've had from the various witness reports. Scotland Yard are running some tests in an attempt to identify the manufacturer of the powder.\"\n\n\"So you're convinced now that the glowing policeman is not of supernatural origin?\"\n\nNewbury shook his head. \"I'm convinced _Morgan's_ killer is not of supernatural origin. We've seen no evidence of this powder on the other bodies from Whitechapel, so I'm reluctant to make any assumptions about whether or not they were killed by the same hand. We can't rule out the idea, but neither can we jump to conclusions. Still, the powder gives us a lead, of sorts. Whether it aids us in simply resolving the mystery surrounding the airship crash, or whether it also leads us to the Whitechapel strangler, time will tell.\" He smiled. \"Whatever the case, I'm hoping we'll find some further answers at Morgan's art gallery today, or at least some more clues to point us in the right direction.\"\n\n\"One thing is certain. There doesn't appear to be a simple solution to any of this.\" Veronica shrugged, folding her hands on her lap.\n\nNewbury smiled. \"There rarely is, my dear Miss Hobbes. There rarely is.\"\n\nNewbury looked up, startled as the cab came to a sudden juddering halt. He peered out of the window. The cab had come to rest before a large red brick building. It was a single storey structure, no bigger than a public bath-house, with a sloping roof of grey slate tiles and an elaborate entrance porch in the classical style, with four large Corinthian columns and a series of low steps up to the door. Ivy formed a web-work across the fascia of the building, trimmed to accommodate the entrance-way itself, and a small pleasant garden gave the impression that both the gallery and grounds were kept in impeccable order. A sober-looking sign by the front gate read THE CHRISTOPHER MORGAN GALLERY OF FINE ART.\n\n\"Miss Hobbes. I do believe we've arrived.\"\n\nVeronica looked round. \"Do you think there'll be anyone here? Given the circumstances, I mean?\"\n\n\"I have no idea. We shan't let it stop us, though. Come on.\"\n\nNewbury paid the fare and, having dismounted the cab, moved to stand beside the wrought-iron gate, surveying the scene before him. The cab driver steered the horses around a large turning circle at the end of the driveway and guided them off towards the city once again, their hooves clacking on the cobbled road.\n\nNewbury took a moment to enjoy the view of the building and its grounds. He noted that the flower-beds were still bursting with colour, even at this late point in November. Overhead, pigeons cooed noisily as they wheeled in the sky, high above the bustle of the city. Veronica crossed the path to stand beside him. After a moment, he held out his arm for her, and she took it appreciatively, locking her arm in his, and together they set off towards the gallery, their feet crunching on the loose gravel of the path as they walked.\n\nMoments later, to their surprise, they found themselves joined in the courtyard by a burly-looking policeman who had apparently seen them coming and stepped out from the shadow of the doorway, where he must have been standing for some time. He nodded politely and cleared his throat. \"The gallery is closed today, sir. I'm afraid you've had a wasted journey.\"\n\nNewbury smiled. \"On the contrary, my good man. We're here on business.\" He withdrew his arm from Veronica's and fished around in his jacket pocket before producing a black leather wallet filled with crisp yellow documents. \"Here, allow me to show you my papers.\"\n\nThe policeman stepped forwards and took the proffered papers from Newbury. He glanced over them briefly, his eyes widening at the sight of Her Majesty's seal and signature, before handing them back to the other man. There was a minor alteration in his posture. \"Please forgive me, sir. How can I be of assistance?\"\n\nNewbury folded the wallet away into his pocket once again. \"Thank you. We're fully appraised of the situation regarding Mr. Morgan's death. You need not divert your attentions away from your duties on our behalf, Constable. Nevertheless, can you tell us if there have been any further developments since yesterday evening? Did any of your officers find anything of interest inside?\" He nodded at the building, as if clarifying his question.\n\nThe policeman shook his head. \"No, sir. Inspector Lewis spent much of yesterday interviewing the staff and searching the gallery for evidence, but there appears to be nothing out of order. It doesn't seem likely that the victim was killed on the premises, and we've been unable to establish a motive for any other suspects, either. Much the same as the rest of those Whitechapel killings, if you ask me.\" He glanced over his shoulder at the gallery. \"We're keeping an eye on the place all the same, mind you.\"\n\nNewbury frowned. \"Would you mind if we took a look around? We won't disturb anything, but I think it would aid our own investigation.\"\n\nThe policeman stepped to one side to let them pass. \"Be my guest. The staff all turned up for work today, too, so you'll find most of them inside. Not sure what's to become of them, really.\"\n\n\"Yes, a sorry state of affairs.\" Newbury led the way towards the gallery entrance, mounting the steps. \"Thank you, Constable.\" He pushed on the door and stepped inside, Veronica following close behind him.\n\nThe foyer was a spacious room, with a large reception desk and two doors leading off to either side of the building. Newbury guessed these led to the two exhibition galleries he'd seen advertised in the papers, one featuring the work of a Frenchman, Gustave Loiseau, the other a British artist named Paul Maitland. The reception desk was unmanned, and the place was quiet. It was as if the building itself were in mourning for the loss of its patron.\n\nNewbury crossed the room, his heels clicking loudly on the marble floor. He stopped before the door on the right, turning towards Veronica. \"Shall we take a wing each, Miss Hobbes? I've never been enamoured of the Impressionist school, but I'm interested to see why this Frenchman has been causing such a stir throughout London.\" He grinned. \"If you happen across Morgan's office, don't touch anything. I think it's best we tackle that together.\"\n\nHe didn't wait for Veronica to respond before disappearing through the open door, the sound of his footsteps ringing out into the cavernous space of the foyer.\n\nVeronica waited until the sound of Newbury's footfalls had diminished, and then she turned in the other direction, heading towards the left wing of the gallery.\n\nPassing through the doorway, she realised that the gallery itself comprised a series of interconnected rooms, each one featuring an array of paintings hung neatly on white walls. Many of the paintings were landscapes, and she recognised a number of them as views of the English countryside. The palette was subdued, but even so, against the stark white of the walls the colours leapt out at her like vibrant splashes of light. She supposed that was the point.\n\nShe toured the room, paying no real attention to the details in the paintings. She found the mood of the place serious and maudlin. There was nothing of Christopher Morgan in here; only the artist and the works he had chosen to display.\n\nAn archway led through to another room, longer this time, although the paintings continued in the same vein: trees and landscapes, the occasional building. There was no doubt in Veronica's mind that the artist had great ability, but personally the pieces left her cold. She moved on, hoping to find evidence of people in the next room.\n\nShe was not disappointed. The exhibition appeared to terminate in this third and final chamber, and she could hear voices coming from behind a tall panelled door that was marked with the word PRIVATE on a small brass plate. She approached the door and knocked loudly with the back of her hand. The chattering ceased.\n\nAfter a moment, she heard footsteps approaching the door from the other side, and then it creaked open, its hinges protesting loudly, and a boyish face with ginger hair and startling blue eyes appeared at the opening. \"Yes?\"\n\nVeronica was a little taken aback by the man's directness. \"Oh. Good morning. I'm here with the Crown investigation, looking into the matter of Mr. Morgan's unfortunate death. I'd appreciate it if I could come in and ask you a few questions?\"\n\nThe man's face fell. \"More questions?\" He opened the door to its full extent and stepped aside to allow Veronica through. \"We spent a good deal of yesterday talking to the police. Do we really need to go over it all again?\"\n\nVeronica glanced around the room. This was obviously the staff and office area behind the scenes of the main gallery. Three other people were seated at a large table, two men and a woman, all watching her with interest as she took in her surroundings. There were two other doors exiting the room, both marked with brass plaques similar to the one on the door she had just come through. One read STOREROOM, whilst the other read MR. C. MORGAN, ESQ., PROPRIETOR.\n\nShe turned towards the man with red hair. \"I'm afraid so, although we'll do our best to keep it to a minimum.\" She glanced at the other expectant faces around the table. \"My name is Miss Veronica Hobbes. May I take a seat for a few moments whilst we wait for my companion?\"\n\nThere was a brief pause, and then the woman stood. \"Please do, Miss Hobbes. We know you're here to help.\" She frowned at the redheaded man before indicating a chair. Veronica accepted it gratefully. The woman returned to her seat, as did the redheaded man, who plopped himself down opposite Veronica, scowling. The woman continued. \"I'm Cynthia. This is Jake.\" She pointed to the man on her left, a slight rakish-looking chap in a grey suit, who nodded in acknowledgement. \"This is Stephen,\" she said next, this time indicating the man on her right, who gave the impression of being a labourer of some sort, dressed in a waistcoat and shirt and with a swarthy look about him. \"And this,\" she said, shaking her head and pointing at the redheaded man, \"this is Adam.\"\n\nVeronica tried her best to give a sympathetic smile. \"I suspect things are a little up in the air for you all at the moment.\" She directed her question at the woman. \"Did you all know Mr. Morgan very well?\"\n\nCynthia nodded. \"As well as anyone knows their employer. He was a good man, Miss Hobbes, and he didn't deserve what happened to him.\" She glanced at Jake, who picked up the conversation.\n\n\"We'd all heard stories about this glowing policeman, read the reports in the newspapers about the killings in Whitechapel, but none of us can understand how Mr. Morgan got involved in all that business. He never mentioned it to any of us. It's just senseless.\"\n\n\"And now no one seems to know what will happen to the gallery. Mr. Morgan's son is in Africa, and his wife died last year of pneumonia. We're waiting for the solicitor to tell us whether we're out on the street or not.\" Adam shook his head.\n\n\"Tell me, had Mr. Morgan exhibited any unusual behaviour in the last few weeks? Have there been any strange occurrences at the gallery?\"\n\nThey heard a noise and looked around as one to see Newbury standing in the doorway. He'd obviously been listening to the conversation for a few moments.\n\nVeronica turned back to the others. \"This is Sir Maurice. He is responsible for our investigation.\"\n\nCynthia shrugged, looking from Veronica to Newbury. \"No. Nothing out of the ordinary.\"\n\n\"Unless you count that automaton device that Mr. Morgan brought back to the gallery a few weeks ago?\" The man called Stephen spoke in a quiet, unassuming voice that seemed somewhat at odds with his swarthy, manly appearance.\n\nNewbury paced into the room, resting his hands on the back of an empty chair. \"Go on.\"\n\nThe man looked at the table-top as he talked, clearly nervous. \"Well, Mr. Morgan bought one of those new automaton men a few weeks ago, and brought it back to the gallery to serve drinks during the private viewings. He wanted it to be a talking point amongst the guests.\"\n\nVeronica leaned closer to hear. \"And what happened?\"\n\nStephen glanced at her. \"After a few days, it started to behave erratically. It failed to carry out Mr. Morgan's instructions and began shambling around the place like it had lost its balance. It started to emit strange sounds, high-pitched whistles and suchlike.\" He toyed with his fingers. \"Then, on the following day, it attacked Mansfield, the desk clerk, when he came in to look at the books. Mr. Morgan and I had to prise it off of him and lock it in the storeroom until the manufacturers could come and collect it. It made a hell of a racket in there.\"\n\n\"Was anybody hurt?\"\n\n\"Just cuts and bruises. But Mr. Morgan was hopping mad. He sent a telegram to the company he'd bought it from. He refused to have a replacement. Said the things were dangerous and should be banned.\"\n\nNewbury stood back from the table. \"Do you know the name of this manufacturer?\"\n\nStephen met his gaze. \"I do, sir. Chapman and Villiers. I remember it clear as day.\"\n\nNewbury walked over to the door marked STOREROOM. \"Is this where you imprisoned it?\"\n\n\"Yes.\"\n\nHe opened the door and glanced inside. Veronica craned her neck to see. The contents of the cupboard were exactly as one would expect: a mop and pail, a broom, a shelf full of cleaning products. The inside of the door, however, was marked with a series of long gouges, scratches where the automaton had clearly tried to break out of the cupboard, raking its brass fingers across the wood. Newbury caught Veronica's eye. He closed the door.\n\n\"Is any of this actually relevant?\" Adam sat back in his chair, clearly put out by the conversation. \"What difference does it make now? Mr. Morgan was murdered by the glowing policeman, and no talk of automatons and clerks is going to bring him back.\"\n\nCynthia leaned across the table and took his hand. \"Adam, everything is going to be alright.\" The young man pushed his chair back petulantly and got to his feet, strolling pointedly from the room. Cynthia sighed, waiting for the sound of his footsteps to fade before speaking. \"He's young, and he's taken it hard. He was fond of Mr. Morgan, and he's worried about losing his earnings.\" She shrugged once more. \"We all are.\"\n\nVeronica stood. \"I can assure you that we'll do everything we can to find the culprit. You've been very helpful. Now, if we can just take a quick look inside Mr. Morgan's office, we'll leave you to your mourning.\"\n\nJake nodded. \"The door's open. Go ahead. I'm not sure you'll find anything of use in there, mind you. The police have been through it once already.\"\n\nVeronica navigated her way around the table, and together, she and Newbury left the three remaining employees to their thoughts.\n\nJake's words had proved more or less correct, and the two investigators had found nothing of real use in Morgan's sparsely furnished office. The desk had been piled high with correspondence, but much of it had already been rifled through by the police, and it consisted mostly of bills, receipts and speculative letters from artists, soliciting Morgan to exhibit their work. Veronica had managed to locate the receipt, and consequent refund slip, from Chapman and Villiers, and was appalled by the expense Morgan had gone to in acquiring the unit. It was no wonder he had complained bitterly when the thing began to malfunction: the device had cost him more than Veronica was paid in a year. She had passed the documents to Newbury, who had folded them carefully and slipped them into his pocket for later use.\n\nAs they strolled along the private driveway outside the gallery, Newbury's disposition seemed to brighten. \"Well, Miss Hobbes. Another interesting development, wouldn't you say?\"\n\nVeronica smiled. \"Absolutely. I believe I could now hazard a guess as to what it was that Morgan wished to talk to you about yesterday.\"\n\n\"Indeed?\"\n\n\"Well, it sounds to me as if Morgan had cast-iron proof that the automaton units are not, as Monsieur Villiers had us believe, impervious to malfunction.\"\n\n\"Precisely my thoughts, Miss Hobbes. It seems as though our friends from Battersea were a little economical with the truth.\"\n\n\"To my mind, that puts Chapman and Villiers themselves very much in the frame for Morgan's murder. They certainly had a motive. It also suggests that the pilot of _The Lady Armitage_ may indeed have been subject to a malfunction. Shall we pay them another visit this afternoon?\"\n\nNewbury shook his head. \"No, my dear Miss Hobbes. It's too soon for all that. We need more evidence before we can build a case against them. Motive on its own is not enough. Certainly, they had a lot to gain from Morgan's death, but we still don't know what the link to the Whitechapel case may be, if any. I don't want to compromise either investigation by charging ahead prematurely. No, I suggest we part company for a short while.\"\n\nVeronica looked concerned.\n\nNewbury laughed. \"Don't worry, I'm not about to go charging off without you. I'm overdue a stop at the office, and I'm anxious to see if there is news from Miss Coulthard. Are you free this evening?\"\n\n\"Yes, of course.\"\n\n\"Then how would you like to accompany me to a soiree?\n\nThe Hanbury-Whites are hosting a party at their house in St. John's Wood and I was planning to attend.\"\n\nVeronica looked a little taken aback. \"Thank you, Sir Maurice. I would be delighted to accompany you.\" She smiled, fiddling with the buttons on her coat.\n\n\"Excellent. I will call for you in a cab around seven.\"\n\n\"Just be sure that it's one of the horse-drawn variety, and not one of those terrible modern contraptions. I can't bear the noise and the smell.\"\n\nNewbury chuckled. \"I most certainly will.\"\n\nThey turned from the driveway onto the street, which was bustling with mid-afternoon traffic. Newbury hesitated. \"Can I drop you now?\"\n\nVeronica shook her head. \"No. I'm intent on a stroll. You go ahead.\"\n\n\"Are you sure? It's quite a walk back to Kensington.\"\n\n\"Positive. I could do with the exercise.\"\n\nNewbury nodded, and Veronica watched as he hailed a cab, and, with a brief wave, disappeared inside. Then, wrapping her coat around her shoulders, she set off into the blustery afternoon, a wide grin on her face.\nCHAPTER 17\n\n##\n\nThe party was in full swing when, later that evening, Newbury and Veronica arrived in St. John's Wood, climbing out of their cab to stand in the shadow of the extravagant family home of the Hanbury-Whites. The moon was a bright disk in the sky, wreathed in wintry mist, and Veronica's breath plumed in the frosty air. She turned on the spot, taking in the view.\n\nCarriages and hansom cabs were arriving and departing in a constant stream, depositing guests on the gravelled driveway at the foot of a large flight of stone steps. Visitors dressed in their best finery flowed up these steps, disappearing into the grand entranceway as if it were the maw of some ancient famished beast. Inside, silhouettes chattered to one another behind brightly lit windows, and the hubbub of voices was spilling out into the night, an undulating cacophony of pleasantries, compliments, vitriolic sleights and whispered asides. Butlers stood in the open doorway, greeting the guests and taking their coats as they drifted through to the party inside.\n\nThe house was magnificent. Built about a hundred years before, it had all the wonderful proportions of Georgian architecture that she had come to adore, the same that had inspired her to take the lodgings she now kept in Kensington: tall sash windows, a glorious front porch, a squat rectangular shape. It lacked the ostentation of the more recent buildings that had been springing up all over London, and she approved wholeheartedly. She couldn't wait to see inside. Years ago, her parents had introduced her to London society, and she had visited a great many of the grand houses in the city, but with the news of Amelia's illness, they had spent the last year in solitude, retreating from the social scene, and the effect had been to leave Veronica without a means to engage with it herself. She was grateful to Newbury for the opportunity to join him this evening, and for giving her the chance to wear something other than the functional attire she often found herself donning for the office. Worse, her recent activities in the field\u2014climbing through the wreckage of burnt-out airships or visiting manufactories on the other side of the river\u2014had left her feeling less than ladylike. Tonight, she'd decided, as she looked herself over in the cheval glass at her apartment, she would redress that balance. She turned to Newbury, who was standing alongside her. \"Thank you for bringing me here.\"\n\nHe smiled warmly. \"You're welcome, my dear.\" He was wearing a smart, black evening suit; formal, but with a forgiving cut. Around his throat he wore a perfectly knotted bow tie, and a top hat balanced precariously on one side of his head. He looked the perfect picture of a gentleman. He turned to look Veronica up and down, now that he had an opportunity to admire her properly in the light of the street-lamps. She was dressed in an immaculate flowing gown of yellow silk. It had a low neckline, exposing the soft pink flesh of her throat and chest. The bodice was fitted, with skirts that flowed all the way to the floor and skimmed the ground as she walked. The ensemble was finished with a single string of opalescent pearls and a pair of matching earrings. Her hair was tied up in an elaborate coiffure.\n\n\"Miss Hobbes, I must add that you look wonderful this evening.\" Newbury, attempting to hide his embarrassment, offered his arm, and together they climbed the steps towards the bustle of the house.\n\nInside, it was immediately clear to Veronica that the party was very much the zenith of London society that evening. Everywhere she looked, she saw faces she recognised, as well as ten others she did not. The place was bustling with ambassadors, politicians and gentlemen, not to mention their multitudes of wives and daughters. She stood for a moment on the threshold of a large room, arm in arm with Newbury, and together they surveyed the scene. Brass automata wove through the press of people, elegantly side-stepping the little conversational clusters that had formed, bearing trays full of drinks and food. Veronica watched as one made a lap of the room, its glassy, spinning eyes shimmering in the reflected light of the gas-lamps, the porthole in its chest revealing the crackling blue of the electrical charge generated by its winding mechanism. For all the stories she'd heard that day about the unit that had malfunctioned at Morgan's art gallery, she was still impressed by the machines and the smooth manner in which they seemed to integrate with the party and its revellers. She watched people snatching drinks from the trays as the automata brushed past them, hardly pausing in their conversations to consider the miraculous nature of the devices that were wandering amongst them, pandering to their every need. There were at least ten of the devices waiting on the guests, and Veronica couldn't help wondering at the expense the Hanbury-Whites must have gone to in having them there. She had seen the price of an individual unit that morning and could only suppose that the automata were there on loan, and did not actually belong to the household itself\u2014that would surely be too much.\n\nShe leaned in towards Newbury, keeping her voice low. His hair smelled faintly of lavender. \"I admit to feeling a little nervous in the presence of so many automata. After hearing the stories this morning at the gallery, I mean.\"\n\nNewbury nodded, acknowledging her concern, but it was clear he was feeling playful. \"My dear Miss Hobbes, it's not the automata you should be worried about. They may be dressed in their best finery, but I assure you, half the men in this room are more dangerous than those devices could ever be.\" He smiled. \"Come on, keep your wits about you and we'll do a lap.\"\n\nHe led her in a circuit of the room, nodding politely at the other guests as they passed each one in turn. Newbury was clearly an established figure amongst the society crowd, and was greeted innumerable times by men whom Veronica did not recognise: men wearing ancient wispy beards, men dressed in immaculate military attire, men who gave the impression of being nothing but ridiculous fops. In turn, Newbury was polite, but he did not allow himself to be drawn into conversation, having just the right air about him of a man who needed to be somewhere else and could not stop to pass time in idle chit-chat.\n\nAfter making a circuit of about half the large room, they paused momentarily by the fireplace and were approached by one of the automata. Newbury claimed two flutes of champagne from the proffered tray, passing one to Veronica. The automaton paused, cocking its head to regard them. For a moment, it remained there, eerily still. The moment stretched. Veronica thought she could hear the sound of its mechanisms whirring away inside, but then it turned away and moved on, drifting towards another small gathering of guests who looked as if they were in need of more refreshment. Veronica shivered, and took a long draw from her glass.\n\nAfter passing a few words with a man named Dr. Russ, who had seemed rather engaging and had complimented Veronica enthusiastically on her dress, patting Newbury on the shoulder like an old friend, the two of them exited the room through a second door, winding their way farther into the large house. They walked along a short hallway filled with the billowing smoke of cigarettes, dodging the crowd that had spilled out from the other rooms, and coming to a set of double doors, behind which more chattering voices could be heard.\n\n\"I believe this should be worth seeing, Miss Hobbes.\" Newbury, grinning, pushed on the doors and they swung open, revealing a large chamber filled with row upon row of wooden chairs, arranged to accommodate a large piano and two stools in the corner of the room. Music stands had been set up, and sheets of notation had already been placed _in situ_. Many of the seats in the room had already been taken, but there were a few empty rows near the very front of the chamber. The guests were mostly engaged in talking amongst themselves, but a number of them looked up as the newcomers entered the room.\n\nNewbury cleared his throat. \"Come on, let's find somewhere to sit. I'm told the performance will be a real eye-opener.\"\n\nSmiling, Veronica allowed herself to be led.\n\nNewbury, nodding and greeting the other members of the assembled audience as they shuffled along the central aisle, located two chairs in the very front row and indicated that Veronica should take a seat. She lowered herself carefully into the chair, ensuring that her skirt did not crumple too severely beneath her. Newbury took his place beside her, first taking a printed order of play from the pile on a nearby table. He flicked through it quickly, and then placed it on his lap, folding his hands over it.\n\nHe turned to Veronica, his voice lowered. \"Looks like they're starting with Elgar. Could be worse.\" He grinned.\n\nVeronica shook her head. It was clear Newbury was enjoying himself, but not, as far as she could tell, for the same reasons as the other guests. He did not seem to want to engage with the society crowd, at least any more than he deemed necessary. Instead, Veronica got the distinct impression that he was toying with them, laughing at their self-congratulatory attitudes, and whilst she did not believe he actually thought himself above his peers, it seemed as if he were remaining purposefully aloof from them. It was an interesting side to his character, and one that she had not expected to see.\n\nVeronica heard the doors swing open behind her and a hush descend on the room. She glanced over her shoulder, almost gasping out loud at the sight she saw. A man in his fifties, with greying hair and a slight limp in his left leg, was leading two automata into the room. They walked with the typical mechanical gait of the others of their kind, but were each dressed in a neat black suit and tie, and one was clutching a violin and bow firmly between its padded brass fingers. She tracked them as they strode down the central aisle between the rows of chairs, taking up their positions in the corner. The automaton holding the violin took a seat on a stool before the music stand and readied itself, tucking the instrument neatly underneath its chin. The other lowered itself into place before the piano, its brass feet clicking as it found the pedals, its fingers resting silently on the ivory keys. The man who had entered the room with them took his place beside the piano, paused, and seemed to steady himself with a deep lungful of air. He looked around the room, ensuring that the audience was prepared, and then turned back to the two automata. With a sharp flick of his wrist, he cued them to begin their performance.\n\nNewbury glanced at Veronica, a smile on his face.\n\nThe violin stirred. Veronica settled back into her seat, watching intently. The automaton drew the bow back and forth across the strings with an expert touch, and she found she was holding her breath, not wanting to exhale, lest she somehow shatter the magic of the moment. She closed her eyes, allowing the music to wash over her. It was deeply affecting, and delivered with incredible precision. The piano joined in then, the other automaton timing its performance so that it seamlessly segued along with its companion at the violin. The music soared, both arresting and beautiful, and Veronica could barely believe that it was being delivered so perfectly by two inhuman devices, designed, so she had thought, for simple labour and not for complex tasks such as this. Waiting on guests was one thing\u2014even maintaining an airship in flight\u2014but delivering a piece of music with such skill and power made her think twice about the nature of these brass machines. She'd assumed the automata lacked any real sense or emotion, lacked the sentience of living things, being simply machines that could be programmed with punch cards to _imitate_ human behaviour, rather than having any awareness of self, or of how their behaviour would impact others. Judging by the skill with which they handled their instruments, however, she began to doubt that assumption. She'd always maintained that music was more than just the sum of its parts; not so much a technical skill in isolation, but an emotional one, too, an art form that blended passion with ability. She was astounded by the quality of the automata's performance, and moved by it as well.\n\nShe glanced around the room, trying to get a measure of the audience's response. Like her, many others in the room were entirely engaged with the performance. Newbury had closed his eyes and seemed lulled by the melody. She twisted in her seat, looking back towards the double doors by which they had entered the chamber. To her surprise, Joseph Chapman was standing there, his hands clasped behind his back, looking ill at ease in his formal attire. He seemed to be staring directly at her. Veronica blinked, and then looked away. She tried to focus on the performance, but had the bizarre sense that Chapman's eyes were burning holes in the back of her head. Her face flushed. She glanced back. Chapman was still staring over at her, his face blank, showing no sign of emotion. Feeling uncomfortable, she looked away, nudging Newbury with her elbow. Surprised, he turned to face her, his eyes questioning.\n\nShe whispered, and pointed over her shoulder with her thumb. \"Chapman.\"\n\nNewbury strained in his seat, searching the back of the room for the man. After a moment, with no success, he glanced back at Veronica, shaking his head. Unnerved, she leaned on the back of her chair, trying to see what had become of the industrialist. The woman in the chair behind her made a _tut_ ting sound and shifted obviously in her seat, making her displeasure with the fidgeting Veronica clearly evident. Veronica sank back into her chair, frustrated. Newbury placed his hand on hers in a placating gesture. A few moments later, the automata finished their first piece, and both Newbury and Veronica joined in with the enthusiastic applause. Then, when the moment seemed appropriate, Newbury stood, held his hand out for Veronica, and lifted her out of her chair. As politely as possible, the two of them slipped away from the scene, leaving the room to silent glares from the other members of the audience. Veronica heard the music starting up again as the door fell shut behind them.\n\nThe hallway was still bustling with guests. Newbury leaned in closer and spoke into her ear. \"Are you alright?\"\n\nVeronica shrugged. \"Yes. Just a little unnerved. It wasn't a friendly look he gave me. He seemed somehow . . . sinister. It was as if he were trying to threaten me in some way.\"\n\nNewbury frowned. \"Are you sure it was Chapman that you saw?\"\n\nVeronica nodded. \"Certain.\"\n\nNewbury straightened his back. \"Well, in that case, let's see if we can't find out where he's hiding now.\" He took Veronica by the arm and led her back towards the main thrust of the party, passing a gaggle of squawking women in the hallway as they did so. Newbury cringed as the ladies seemed to grow momentarily silent, whispering to one another conspiratorially as they brushed by. Veronica wondered what inane secrets they were sharing at Newbury and Veronica's expense. The giggling resumed again almost as soon as Newbury and Veronica had reached the other end of the hall.\n\nThe party had swelled in the few minutes they'd been away from the main reception room, with more and more guests arriving and others yet to drift away from the main hubbub and farther into the bowels of the great house. The room was hot and bustling with people. The automata waited patiently in the wings, surveying the crowd, ready to step forward at any time to offer the guests more refreshment. Newbury strained to see over the heads of the nearby dignitaries. After a moment, he stopped craning his neck and leaned in to whisper to Veronica. \"No sign of Chapman in here. At least that I can see from the doorway. Shall we do another lap and try to get a better look?\"\n\nVeronica nodded. \"I'd feel better about it if we did. At least that way I'd know I'm not imagining things.\"\n\nNewbury gave her a reassuring look. \"I'm sure that's not the case. We don't have to confront him about it, but we can certainly try to keep an eye on him, if indeed he's here. Come on.\"\n\nThey edged into the room, taking two further flutes of champagne from the tray proffered by the automaton near the door, and began the process of slowly weaving their way through the crowd, working clockwise around the edges of the room. Veronica held on tightly to Newbury's arm as they manoeuvred through the press of guests, keeping their eyes peeled for any sign of the industrialist. It wasn't long before they'd circled back as far as the main entrance. Finding a little more breathing space there, they decided to take a moment to pause. Guests were still arriving as they stood with their backs to the door, surveying the room. Newbury sipped at his drink, regarding the little huddles of people nearby. \"Perhaps he left?\"\n\n\"Or perhaps he's watching us from elsewhere in the room?\" Veronica shivered at the thought.\n\nNewbury frowned. He looked as if he were just about to reply when he stopped suddenly and turned away at the sound of shouting from the other side of the room. A hush descended on the party like a thick blanket, smothering the chatter. It was difficult to make out what was being said, but a man was clearly engaged in directing a torrent of abuse at another man whom he felt had somehow done him a disservice.\n\n\"\u2014and another thing, sir! Your company's literature clearly states that in circumstances such as this, full recompense is assured. Yet I continue to find no such recompense is forthcoming! A damnable business, and you, sir, are a damnable man!\"\n\nNewbury raised his eyebrow at Veronica. Neither of them could see anything through the gathered throng. There was a wave of gasping sounds, and then the crowd suddenly parted as two automata, their trays now abandoned, moved forward, their brass feet clicking on the marble floor, a middle-aged gentleman in a black suit between them. The automata each held one of the man's shoulders as they forcibly marched him towards the exit. The man was squirming, clearly in discomfort, and the spinning eyes of the automata glinted in the low light, their features frozen, unmoveable. Behind them, Joseph Chapman stood with his hands on his hips, a wry smile on his face. He glanced at Newbury and then nodded politely, his eyes flicking away to watch as his two brass guardians escorted the other man from the building. Then, as the guests all looked upon the scene with a kind of horrified fascination, he set off after his clockwork devices, exiting the party through the main entrance. Outside, the sound of the man's protests trailed away down the street. The party took only a moment to recover, and then the hubbub began in earnest once again, the society gossips quickly moving to engage one another with talk of the scandal.\n\nNewbury looked at Veronica, bemused. \"Well, my dear Miss Hobbes, you were certainly right about Chapman being in attendance here tonight, but it seems as if the problem has miraculously solved itself.\"\n\nVeronica smiled. \"Yes, you could say that, I suppose. But what do you make of it all? It strikes me that the unfortunate captive of those automata may be a likely witness in our developing case.\"\n\nNewbury nodded. \"Yes, it certainly seems that way, doesn't it? I get the distinct impression that the poor gentleman may have had a similar experience to Mr. Morgan.\"\n\nVeronica looked thoughtful. \"Indeed. Do you think he's in danger of suffering the same fate? Should we go after them?\"\n\nNewbury shook his head. \"No, I'll wager the man is in no danger, this evening at least.\" He took a long draw from his flute of champagne. \"Even if Chapman is somehow connected to Morgan's death, this outcry was a little too public for anything to come of it now. The connection would be obvious to everyone. The automata will take the man around the corner and he'll flee to his abode, angry and embarrassed. No doubt Chapman will take the opportunity to gloat to anyone who'll listen.\"\n\nVeronica placed her empty glass on a sideboard behind her. One of the automata immediately made a beeline over to reclaim it. \"It is interesting, though, isn't it? I mean, after finding us in the other room watching the performance. It's almost like Chapman arranged for us to see this little charade. Did you notice how he made a point of catching your eye?\"\n\n\"I did. I wonder what it is he's up to.\" Newbury was watching the crowd again as he talked. \"Let's see if we can discover the identity of Chapman's protagonist before the night is out. That way, we can pay him a visit in the morning.\"\n\n\"And now?\"\n\n\"And now we have a party to attend.\" He smiled, holding out his arm. \"I believe we were in the middle of doing a turn around the room. And you, my dear Miss Hobbes, look as though you could use another drink.\"\n\nArm in arm, they rejoined the gathered crowd and searched out another glass of champagne, keeping a wary eye on the automata as they tried to enjoy the rest of the party.\nCHAPTER 18\n\n##\n\nNewbury woke with a thick head and a dry mouth. He rolled over, burying his face in his pillow. Then, as if surfacing from a glassy pool of water, he suddenly became aware of the world outside of his own head. Someone was rapping insistently on the door to his bedchamber. He rolled onto his back, peeling back his eyelids. It was still dark; there was no light streaming in through the window, and he hadn't yet had sleep enough to banish the residue of the alcohol he had consumed the night before. Early morning, then. He sat up, running a hand through his hair.\n\n\"Sir Maurice? Are you there?\" The rapping continued.\n\nNewbury frowned. \"Yes, Mrs. Bradshaw. I'm awake.\"\n\nThere was an audible sigh of relief from the other side of the door. \"Very well, sir. Sir Charles is here to see you. I've asked him to wait in the living-room. Shall I assure him that you will attend to him shortly? I understand that it is a matter of some importance.\"\n\nNewbury pinched the bridge of his nose between his thumb and forefinger. He groped around in the semi-darkness for his pocket-watch, finding it on the bedside table. He peered at it intently, trying to see the hands. It was just after five. It must be important, for Charles to be calling at this hour. \"Please do, Mrs. Bradshaw. I'll be with him momentarily.\"\n\nMrs. Bradshaw's footsteps fell away from the door, and Newbury slumped back into his pillows, rubbing his eyes. Then, sighing, he slipped from underneath the warm woollen blankets of his bed and stood beside his dresser, shivering in the chill. He blinked a few times until his eyes had adjusted properly to the dim light, and then searched out his dressing gown, flung it around his shoulders and shoved his feet into the slippers he kept underneath his bed. A moment later, he was following behind Mrs. Bradshaw, squinting in the bright light of the gas-lamps as he stumbled down the stairs to meet his friend.\n\nBainbridge was pacing anxiously before the fireplace, which was dull and cold and full of nothing but ash at this early hour in the morning. He held a brandy in his hand, but appeared not to have taken a swig of it, as yet. He looked up when Newbury came into the room, his moustache bristling at the sight of his old friend, still dressed in his bedclothes and suffering from a mild hangover.\n\nNewbury looked the other man up and down. \"There's been another murder in Whitechapel.\"\n\nBainbridge looked astounded by this rather minor piece of deduction. \"How did you\u2014?\"\n\nNewbury sighed. \"Why else would you be here at this hour, Charles?\" He shrugged. \"Your boots are still clean, and you look like you've dressed hastily: your tie is askew and you've notched your belt on the wrong hole.\" Bainbridge looked down at his belt, and then shook his head in exasperation. \"I take it you've only recently been made aware of the situation and have come to pick me up on your way over to the scene?\"\n\nBainbridge nodded. \"Indeed. As you say. So jolly well go and fetch up some clothes and make yourself presentable, man. I've already sent a cab for Miss Hobbes.\" He took a swig of his brandy and leaned heavily on the mantelpiece.\n\nNewbury nodded, smiling, and then disappeared once again from the room.\n\nA few minutes later, the two men took their leave of Newbury's Chelsea home and mounted the cab that Bainbridge had left waiting for them on the road outside. Its steam engine spluttered noisily as the driver gunned the controls and sent the vehicle careening into the cold, dark morning. Newbury, his head still groggy from the alcohol and lack of sleep, fell back into the seat inside. He had dressed hastily and still wore the shadow of a beard around his face and throat, but had more or less managed to make himself presentable. He looked up when Bainbridge tapped on the window with the end of his cane. \"Not sure how much longer I can put up with this abominable weather, Newbury.\" He glanced out at the smoky, fog-filled streets as they rushed by. \"This damnable fog makes our police work doubly hard. Gives these criminal types the cover they need for sneaking around the city at all hours.\" He sounded weary.\n\nNewbury nodded, but didn't speak. He watched the shapes of building flit past, hidden by the gossamer mist that seemed to soften the edges of everything, making the real world outside the cab seem insubstantial, otherworldly.\n\n\"Are you well, Maurice? You seem unusually quiet.\"\n\n\"Quite well, Charles. I attended the soiree at the Hanbury-Whites' last night. I fear I may have led Miss Hobbes rather astray; we indulged in one too many glasses of champagne amidst the merriment.\"\n\nBainbridge laughed heartily. \"Then I shall conserve my sympathy for more worthy subjects! I take it there was much merriment to be had, in that case?\"\n\nNewbury grimaced. \"A little. Most interesting, however, was the scene between a certain Mr. Musgrave of Islington and Joseph Chapman, of Chapman and Villiers Air Transportation Services.\"\n\n\"How so?\"\n\n\"It appears Chapman sold Musgrave one of those automaton devices. It later malfunctioned and killed his best hound. The word from the society gossips is that Musgrave had been trying to claim compensation from the company and, having received no satisfactory response, took the opportunity to set upon Chapman in front of everyone at the party.\"\n\nBainbridge sat forward, resting on his cane. \"So what happened?\"\n\n\"Not a great deal, if truth be told. Chapman had Musgrave escorted from the party by two of his automata and then made his own exit from the proceedings. We didn't see him again all night.\"\n\n\"How peculiar. Do you think it's relevant to your case?\"\n\nNewbury nodded. \" _Our_ case, Charles. You're forgetting Christopher Morgan. It transpires that the same situation is true of Morgan, although in his case, it ended in a rather more grisly fashion. He'd also had an automaton malfunction at his gallery and had successfully negotiated a refund from Chapman. However, when he heard about _The Lady Armitage,_ he wrote to me asking to meet, intending to divulge his miserable experience with the device, and the rest you are already aware of. He ended up dead and dumped in Whitechapel.\"\n\nBainbridge clenched and unclenched his fist. \"So it seems as if Chapman is involved in Morgan's death, and that he may be behind the airship disaster, too. What of Musgrave? Do you think he's in danger?\"\n\n\"That's just it. I can't see how he could be, not after the performance made by Chapman at the party last night. If he turned up dead now, it would give us cause to pull Chapman in immediately. If he is guilty of Morgan's murder, I can't believe he'd be so insouciant about it.\"\n\nBainbridge took a moment to let that sink in. \"But what about the other murders? They don't follow the same pattern as Morgan's. Do you still think Morgan's killer tried to use the existing spate of murders as a cover for his own crime?\"\n\n\"That's what I'm trying to work out. We've got very little we can actually pin on Chapman yet, and if we move too soon, we'll simply cause him to clam up. We need to build a solid case against him, if indeed he is responsible for Morgan's death. Whilst we've certainly established that the automaton device that was piloting _The Lady Armitage_ could have caused the crash through malfunction, in truth, we've got no real way of linking Chapman to Morgan's murder, as yet. It's a matter of time and patience.\" He shuffled in his seat, adjusting his collar. \"As to whether the other murders are connected, too, I still have my doubts. Perhaps we'll find out more at the scene we're about to attend. Did your men find out anything useful about the blue powder we found on Morgan's corpse, by the way?\"\n\nBainbridge shook his head. \"Not as yet. So far, they haven't even been able to identify the powder itself, let alone the manufacturer, but they're aware of the importance of the matter. Some of them think it may have come over from China.\"\n\n\"Good. Make sure you tell me the minute you hear anything.\"\n\nThe men fell silent, both gazing out of the window at the sleepy city, both wishing they were still at home in their beds instead of rushing through the morning fog towards Whitechapel and another unhappy death.\n\nAfter a few moments, Bainbridge looked up, catching Newbury's eye. \"Oh, I received another invitation from Miss Felicity Johnson in yesterday's post, for a small gathering she's having on Tuesday evening. Did you find yourself invited to the same?\"\n\nNewbury tried to keep a serious face as he met the other man's eye. \"I did not.\"\n\nThe two men faced each other across the cab. Bainbridge was first to give in, looking away in an attempt to stop himself from sniggering. By the time they reached the outskirts of Whitechapel, the two men were roaring with laughter in the back of the cab, both of them finding the hilarity a welcome distraction from the more serious elements of their lives, and the knowledge that they were once again headed towards a scene of terror and death in one of the poorer parts of the city.\n\nWith a grinding of gears and a spluttering of the engine, the cab rocked to a halt on the cobbled road alongside another, waiting vehicle. Bainbridge was first to jump out into the foggy morning, crossing the cobbles to the door of the other carriage. He rapped loudly before swinging the door open and stepping up into the cab. A moment later, as Newbury was arranging his hat by the curb side, he watched as Veronica emerged from the other vehicle, closely followed by the Chief Inspector.\n\nVeronica crossed the street to stand beside him. \"Good morning, sir. How are you?\"\n\nNewbury arched one eyebrow. \"Capital. And you, my dear Miss Hobbes?\"\n\n\"Perfectly well, thank you, Sir Maurice.\" Veronica smiled brightly. Newbury grinned. She gave no impression that her alcohol consumption the previous evening had affected her in any way.\n\nBainbridge approached them, bearing three small oil lanterns, his cane tucked neatly underneath his left arm. He handed one of the lanterns to each of them and then fiddled with the shutters on his own until the light was emanating in a warm halo all around it. It reflected back in the fog, giving it a strange, fuzzy glow, as if he were clutching a ball of light itself and not a lantern at all. He turned to the others. \"Right. Turn these up like mine so we can keep an eye on each other as we walk. This blasted fog is so thick this morning that we run the risk of losing each other if we don't stick together.\" He looked from Veronica to Newbury and back again. \"It wouldn't do to be losing either of you in the fog out here. We don't know what else might be lurking around the corner.\" His face was steely, determined. \"I've told one of the cabs to get on its way, whilst the other waits for us here. We'll head to the scene of the murder, take a look to see if there's anything new to be deduced, and then be on our way, as quickly as we can. No use hanging around out here when there's a couple of men already in attendance by the body.\" He took his cane from under his arm. \"Come on. One of them is waiting to show us the way.\" He set off, hugging the edge of the curb as he walked, in an effort to stay on track in the blinding fog. He was joined a moment later by a uniformed bobby who had been waiting around the other side of the cab. Newbury and Veronica followed behind them, their lanterns held up in the gloaming.\n\nIt was only a matter of minutes before Bainbridge's lantern came to a halt and Newbury and Veronica sidled up beside him. A scene resolved out of the fog. The confluence of three buildings and the cover of an arched alleyway had created a barrier of sorts against the thick smog. It still lay heavy in yellow wispy strands, but with the light of the three lanterns, plus the one held by the other uniformed constable already in attendance, Newbury was able to ascertain the key elements of the scene.\n\nA body lay on the cobbles a few feet away from where he was standing. Moisture from the fog had caused the skin to take on a damp sheen, and the waxy complexion suggested that the corpse had been _in situ_ for some time before being discovered. That was only to be expected, Newbury supposed, given the visibility out there that morning. The neck of the corpse had been violently twisted and was lying at odds to the rest of the body. Clearly the neck had been snapped before the body was dropped to the ground. The man himself was undeniably a pauper, aged around thirty years old and wearing a scruffy beard and long straggly hair.\n\nBainbridge moved off to talk with the other policeman, who was standing with his back to the wall a few feet away from the corpse, looking nervous and cold. Newbury caught snippets of the conversation as they talked: Bainbridge questioned him in detail about the circumstances of the death, how the alarm was raised, who found the body, which of the men was first on the scene. It was a thorough interview and, whilst it didn't appear to yield any further clues, it ensured they weren't making any assumptions before examining the corpse. The two constables did not mention the glowing policeman to Bainbridge, and it seemed as if there were no reliable witnesses to call on. Newbury waited for Bainbridge to return, his cane clicking on the cobbles.\n\n\"I'll take a look at the corpse, if you've no objection, Charles?\"\n\n\"Of course not. That's why you're here, isn't it?\" Newbury could tell that the other man was feeling the pressure.\n\nVeronica stepped forward. \"What can I do to help, Sir Maurice?\"\n\n\"If you can stomach it, can you go through his pockets whilst I examine the wounds?\"\n\n\"Of course.\" She circled around the body and dropped to one knee, setting about the task of emptying the dead man's pockets and searching out his wallet beneath the layers of dirty wool.\n\nNewbury leaned in towards the body. He loosened the man's collar and examined the soft flesh around the throat. It was badly bruised and broken. He took the man's head by the chin and moved it from side to side. Then, mumbling something to himself, he took up the man's left hand and examined the fingernails. The hands were filthy, but it was clear he'd been in a fight. The knuckles were bloodied, and there was some sort of residue under the fingernails where he had scratched his attacker during the fight.\n\nBy this time Veronica had located the man's wallet and had moved to one side to examine the contents. Newbury looked up at Bainbridge, who was leaning over him impatiently, his lantern dangling over Newbury's head. \"Found anything?\"\n\n\"Indeed. Just give me a moment to confirm my suspicions.\" Newbury rested the man's lifeless hand upon his chest, and searched around in his own pockets for his penknife. \"Here, hold that light steady for me, Charles.\" He beckoned the other man closer. Taking up the dead man's hand again, Newbury unclipped the blade of his penknife and ran the point of it underneath the fingernail. He then returned the hand to its place beside the victim and lifted the blade to the light, examining the residue he had scraped free. \"Ah. Just as I thought.\"\n\n\"What is it, man?\" Bainbridge was frowning, unclear what it was exactly that Newbury had found.\n\nNewbury rose to his feet. \"Here, give me the light and take a look at the man's throat. I think you'll see something of great interest around the larynx.\"\n\nBainbridge placed his cane on the ground beside the corpse and leaned in. \"The what?\"\n\n\"The Adam's apple.\"\n\nBainbridge took a moment to look over the body. Then, without saying another word, he pushed himself up to stand beside Newbury. \"Blue powder.\"\n\n\"Precisely. Dusted around the collar and worked into the broken skin, where the assailant's hands had clutched him around the throat.\" He held out his penknife, handing Bainbridge his lantern back. \"And here, too, under the fingernails. He scratched at the killer's hands as he struggled to get free. That's probably why the killer had to break his neck in the end, because he was fighting back too hard.\"\n\n\"Well, I suppose it means our 'incorporeal' killer has struck again.\"\n\nNewbury nodded. \"Indeed. But this time the profile is exactly the same. This man was clearly a pauper, judging by his clothes and the state of his hands. Veronica, did you find anything?\"\n\nVeronica came to join them, clutching the dead man's wallet. \"Only a few coins. Nothing of note. He certainly wasn't robbed, though.\"\n\nBainbridge shook his head. \"So here's the link to Morgan, then, and Chapman through that. The blue powder is a dead giveaway, regardless of what it actually is.\"\n\nNewbury looked thoughtful. He turned back to look at the corpse. \"Perhaps. We certainly may have missed the blue powder on the earlier victims. But there is a distinct problem with your theory about Chapman, I'm afraid. This man has been dead for at least eight hours, judging by the rigor mortis and the pallor of his skin. Chapman couldn't possibly have done it.\"\n\n\"Why not?\"\n\nVeronica put her hands on her hips. \"Because he was with us at the party.\" She paused for a moment, shaking her head. \"Very clever.\"\n\nNewbury gave an impressed sigh. \"Indeed. Very clever. We wondered why Chapman was making a point of being seen. Now, I think, we have our answer. He's toying with us, inviting us to call him out. He knows he has a watertight alibi, for this and, no doubt, for Morgan's death, too. And whilst we have good reason to believe the airship crash could be the result of an automaton malfunction, all we have is reasonable doubt. Without the evidence from the wreckage, we have no way of proving our argument.\" He ran a hand over his stubble, adjusting his collar. Veronica shivered in the cold.\n\n\"So, what, we lay a trap?\" Bainbridge said, frowning and frustrated.\n\n\"I'm not sure it's that easy.\" Newbury blinked, and noticed that Veronica's lantern was beginning to gutter in the damp. \"Come on, we can talk further on the way back. Let's get out of this damp fog and somewhere warm for breakfast.\"\n\nBainbridge concurred, and went to have a brief word with the two constables before rejoining Newbury and Veronica and starting out for the cab once again. The fog was still thick and cloying, and away from the shelter of the mouth of the alleyway, they were soon smothered by it once again. Nevertheless, following the line of the curb led them easily back to the waiting cab, their lanterns bobbing in the quiet darkness. The cab driver was huddled on his dickey box against the cold, the engine running noisily, steam spouting into the cold air through tin funnels on the top of the contraption. He looked up when he heard them coming, grateful for the opportunity, no doubt, to be on his way.\n\nNewbury was first to the coach door, and held it open for Bainbridge and Veronica. They both extinguished their lanterns before mounting the step, and Newbury held his aloft to ensure they could see. Then, just as he was about to follow suit, Bainbridge slapped his knee in frustration. \"Damn it! I must have left my cane beside the body back there. Watch out, Newbury, I'll just run back and fetch it.\"\n\nNewbury held his hand up to steady the older man. \"No fear, Charles. You stay where you are, and I'll dash back and collect it for you. It'll only take me a moment.\" He turned and held the lantern aloft before moving to retrace their steps along the curb. He heard the coach door click shut behind him.\n\nAfter a moment, Newbury had been almost completely swallowed by the dank fog. The sounds of the steam engine had faded to a dull thudding as the pistons fired relentlessly, turning over the large mechanical machine. He crept along, hoping to avoid surprising the two uniformed constables at the scene. A moment later, he emerged from the fog into the mouth of the alleyway. What he saw was one of the most horrifying scenes he had ever witnessed in his life.\n\nThree monsters\u2014it was the only way he could think to describe them\u2014were in the process of gutting the two constables, whose corpses had been dashed to the ground, blood spattered across their torn faces, spilled out over the cobbles all around them. Steam rose from the warm innards as the revenants pulled loops of intestine free from large rents in their bellies, feasting on it all indiscriminately, stuffing it into their mouths with abandon. The creatures looked as if they may once have been human, but all sense of their humanity had now been lost. Their flesh was peeling in long ribbons, their hair falling out around their shoulders, their clothes hanging filthy and torn from their abused bodies. The virus had done its work well, and these monsters were now no more than dead carriers of the plague, capable of nothing but killing and feeding on their victims. They had the stink about them of half-rotted corpses, and this foul smell, mingled with the stench of blood and faeces, caused Newbury to gag violently. He fought back the urge to vomit, not wanting to draw attention to himself. The three creatures were intent on their feeding frenzy, and he didn't want to give them cause to make him their third victim of the day. Tentatively, he glanced from side to side. The area was entirely surrounded by the thick fog, and he had no sense of whether there were more of the creatures lurking in it. He was only a few feet away from the corpse of the murder victim, and he could see Bainbridge's cane on the cobbles beside it. He assumed the revenants were ignoring the body because it was hours old, and with two fresh victims, there was no need for them to feed on the bloated flesh of the dead.\n\nGingerly, Newbury inched forward, trying not to make a sound. He was intent on getting out of there as quickly as possible, and whilst he wasn't really concerned with retrieving Bainbridge's cane, he knew it would make a handy bludgeoning weapon if he found himself cornered with nowhere to run. The sound of the creatures feeding on the ruined corpses of the two policemen filled his ears. He repressed the fear that was creeping up his spine. He needed to keep a clear head.\n\nHe reached out slowly and, keeping his eyes on the backs of the three revenants, felt for the cane with his fingertips. At first he found nothing but cold, slick cobbles, but he patted the ground for a moment longer and eventually his fingers closed on the hard wood of the cane. He rose slowly to his feet, bringing the cane along with him. Trying not to let the adrenaline make him run, he tightened his grip on the lantern and turned slowly away from the nightmare scene, directly into the path of another revenant.\n\nNewbury stumbled backwards but it was too late, and the creature, its foul breath sour in his face, leapt forward and clamped its jaws onto his left shoulder. He cried out in agony as the monster's teeth bit down through his clothes and into his flesh, drawing blood. Its hands quested for a grip on his torso, its talons raking into his flesh, tearing his overcoat as if it were paper. Newbury kicked out with all his might, getting a measure of leverage on the creature and forcing it back with his booted foot. The monster allowed itself to be pushed back momentarily, releasing Newbury's shoulder from its vicelike jaws, before coming at him again, its teeth bared in an ominous black snarl.\n\nHis shoulder aching with the vicious bite wound, Newbury reacted as quickly as he could, swinging the cane down across the creature's temple, striking it hard with the round brass handle. It staggered to one side with the force of the blow, the bones around the eye socket shattering where the brass knob had impacted. Newbury tried to glance over his shoulder to make sure the other three creatures were still busy with their existing meal and were not closing in on him from behind. They were not, but the one in front of him ranged up again in no time at all, and he found himself dodging out of the way of its flailing talons. His shoulder throbbed, and he could feel the warmth of his blood seeping down the inside of his shirtsleeve.\n\nHe struck the revenant again with the cane, this time breaking loose a few teeth, which rattled to the stones below, but it seemed to have no real effect on the creature. Its bloodshot eyes glared at him as they circled each other, Newbury trying hard not to stumble over the corpse that lay behind him on the ground. The creature lunged once again, aiming its jaws towards his throat, hoping to incapacitate him by tearing his windpipe and jugular out with its teeth. Not knowing what else to do, Newbury dropped the lantern and threw himself backwards, using the corpse of the dead pauper to cushion his fall. He then rolled quickly to one side, scrabbling back up to his feet as quickly as possible, brandishing Bainbridge's cane before him. He could see out of the corner of his eye that the other three creatures were still busy with the remnants of the policemen. He knew it wouldn't be long before they turned their attention to this new quarry, however. He had to despatch the one in front of him soon, or he risked ending up like the bobbies.\n\nHe circled, fixing his attention on the revenant before him. It was waiting, hulking over the body of the dead man, looking for another opportunity to pounce. Blood was running down the side of its face where he'd caved in the orbit of its eye, and he noticed for the first time that it had a letter opener half-buried in its neck. Clearly this was not the first time it had cornered someone unexpectedly.\n\nNewbury readied himself, planting his feet firmly on the cobbles. He'd managed to snatch up the oil lantern again after his brief tumble, and realising that he was no match in strength or endurance with the creature, he'd decided to try something else.\n\nThe revenant pounced, uncoiling in mid-air like a half-human panther, baring its teeth and flexing its claws. Newbury swung wildly with the lamp, connecting with the monster's shoulder and spattering its hair and face with hot oil. There was a sudden _whoosh_ ing sound, and all at once the creature was on fire, its rotten skin and lank hair spreading the angry flames that seemed to spill out from the lamp like a wave of liquid light. Within seconds, the creature's entire head and shoulders were in flames, and it staggered about, unable to see as its eyes boiled away in the heat. Newbury took the opportunity to run, darting past the burning monster and staggering away into the fog. His shoulder burned where the creature had bitten him, and his right side was agony where its talons had gouged a tear in his flesh. Drawing a huge breath and fighting against the spinning darkness that threatened to drag him into unconsciousness, Newbury started back in the direction of the cab.\nCHAPTER 19\n\n##\n\nWithout the aid of the lantern, Newbury found it difficult to get his bearings in the thick fog. He stumbled along the road, doing his best to anchor himself to the curb in an effort to stay on track. He was bleeding profusely from the wounds in his shoulder and side, and he leaned heavily on Bainbridge's cane, attempting to propel himself along in an effort to get away from the terrible scene behind him as quickly as possible.\n\nThe fog wreathed everything in its clinging oppressive blanket, and Newbury found it almost impossible to see. This in itself wouldn't have troubled him, but now, in the midst of escaping the clutches of the revenants, he had no idea whether he was being followed or not. The creatures could have been shambling along behind him these last few minutes, drawn to the scent of his spilt blood.\n\nHe glanced behind him. There was nothing but a sea of white. He tried not to think of the nightmares that were hulking within it. If he got lost in the fog now, the likelihood would be that he'd never be able to find his way out of it again. He tried to focus on getting back to the cab: to Bainbridge, Veronica and safety.\n\nPresently, after what seemed like an age, he became aware of the choking sound of the cab's engine, and breathed an audible sigh of relief. The sound would be enough to guide him through the remaining fog and darkness to his friends. Clutching at his shoulder, he marched on, confident that Veronica would be able to help him stem the flow of his blood when they managed to get him up into the cab. He almost laughed. The whole episode seemed so bizarre, now, dislocated from reality, left behind in the fog. He squeezed his shoulder a little tighter, feeling woozy from the loss of blood.\n\nA moment later, he was snatched rudely back to the present by the sound of a shrill, piercing scream from somewhere up ahead. It cut through his dizziness like a knife. Newbury felt his hackles rising. One thought crossed his mind: _Veronica_.\n\nShaking off his pain and lethargy, Newbury forged on, forcing himself to run towards the sound, blindly throwing himself through the hazy curtain of fog. He skidded to a halt when the cab came into view.\n\nTwo revenants were menacing his friends, clutching at the cab door and shaking the vehicle back and forth like children with a toy-box, trying to get inside. The driver had clambered hurriedly onto the roof of the cab in an effort to put himself out of the creature's reach, but now found himself clinging on for dear life as the vehicle rocked from side to side. His fingers grasped the brass rails, and his shouts for help were growing more and more desperate as he slid across the flat roof, doing his best to stop his feet from dangling too far below. The thick shroud of fog prevented Newbury from seeing into the cab itself, but the shouting emanating from inside it made it clear to him that both Bainbridge and Veronica were trapped in the carriage, doing their best to keep the monsters from getting in.\n\nWith no other options available to him, Newbury rushed forwards, bringing Bainbridge's cane to bear and driving it with all his might into the lower back of one of the monsters. It penetrated the beast's rotten flesh, forcing a gob of putrescent fluid out of the exit wound on the other side, but seemed to cause the creature no visible sign of pain or distraction. Newbury pulled the cane free with an almighty heave and instead drove his boot into the back of the monster's knee, causing it to buckle to one side.\n\nThe creature freed its hold on the carriage, finally turning to see what had caused it to stumble. It bared its teeth in a grim caricature of a smile. Newbury was appalled to see that, once, the monster had been a woman. It wore the tattered remnants of a blue gown, ripped open to expose its distended belly. Its feet were bare and rotten, black and peeling, with toes missing where rats had evidently chewed on the foul flesh.\n\nNewbury paced back a few steps, trying to draw it away from the cab. It stumbled towards him, snatching out with its talon-fingered hand in an effort to grab hold of him.\n\nNewbury shook his head. \"You'll have to be quicker than that.\" He whipped the cane around again, connecting loudly with the monster's wrist and snapping the brittle bones there with a sickening _crunch_. The hand fell limp, and the creature withdrew momentarily, as if bemused by this new development. By this time, however, the other revenant had become aware of what was happening and had moved away from the carriage to join its fellow, evidently deciding that an unprotected quarry represented easier pickings than a sealed cab. They closed on Newbury together, working like pack animals to come at him from both directions at once.\n\nPanicking, Newbury was unsure how to fend them off. He held Bainbridge's cane out before him, the brass knob swinging to and fro like a club, but he knew this wouldn't hold them off for long. He had no lantern this time to use as a weapon, and the two creatures were between him and the cab. He'd just have to make it up as he went along. He feinted to one side, causing the revenant on his right to swipe for him, and then pushed forward, slamming the cane into the back of the creature's head. The momentum of its attack, coupled with the force of Newbury's blow, carried it forward, and it sprawled out on the cobbles, smashing its face hard on the ground.\n\nNewbury wheeled on the other one, just in time to see the back of its hand slam forcefully into his face. The impact sent him careening across the road. He landed heavily, jarring his left elbow on the stones and fighting to keep his breath. He snatched up Bainbridge's cane from where he'd dropped it in the gutter, and rolled to the right, just before the creature was on him again, slashing at his already-bloody shoulder. His clothes fell away in shreds, and his arm bloomed with pain as the monster's talons tore away chunks of flesh. Knowing that the impact of the cane itself would have little effect, Newbury instead hooked it behind the revenant's ankle and rolled forward onto his knees, pulling on the shaft of the cane with all his might. It seemed to have the desired effect, toppling the shambling monster heavily onto its side. Newbury, gasping for breath, tried to stagger away, but the first creature didn't stay down for long, and the other was already back on its feet and heading in his direction. He backed up, desperate for an idea. He knew he was going to lose if the fight came down to brute force and endurance.\n\nBehind him, he heard the pistons on the cab firing noisily as the driver readied their escape. Beneath this, just out of earshot, he could hear Veronica and Bainbridge clamouring for attention. He didn't have time to try to make out what they were saying before one of the revenants dived at him, landing bodily on top of him and sending them both sprawling back towards the ground. Newbury managed to get a hold on the monster's throat and it gnashed its teeth dangerously, its putrid breath almost enough to send him spiralling into unconsciousness. He'd lost the cane somewhere in the fall, but with his other hand free he had no option but to try to lever the creature off of him. He punched out, hard, his fist crumpling through slick, rotten flesh and burying itself deep inside the creature's belly.\n\nIt thrashed around on top of him as, grimacing, Newbury forced his hand deeper inside of it, questing for its spine. Seconds later, he found purchase on the brittle bony structure and, pushing his fingers deeper into the rotten fibrous tissue that surrounded it, managed to grasp hold of it with his fist. He pulled as hard as he could, pushing against the creature's throat with his other hand for leverage. There was a dry cracking sound, as if old timber were being snapped underfoot, and the spine splintered in his hand.\n\nThe creature's legs stopped thrashing, twitched a couple of times, and lay still. Its arms continued to claw away at him, however, its teeth ominously close to his face. Newbury, gasping for breath against the weight of the creature on top of him, withdrew his hand from the rotten belly, trying not to think about where it had been. His damaged shoulder was perilously close to giving in. He put his right hand on the ground and pushed himself over, rolling with the creature until their roles were reversed and he found himself on top of the foul thing, his left hand still tight around its throat.\n\nGlancing up to get a measure of what had happened to the other one, he jumped back off the beast and fumbled around, looking for Bainbridge's cane. It was nearby on the cobbles, close to where he had fallen. He retrieved it quickly, then backed up, waiting for the other revenant to loom out of the fog.\n\nThe creature he had been struggling with\u2014the one that had formerly been a woman\u2014was still trying to right itself, pulling itself up on its arms and finding that its legs would no longer support it. Relentless, it started to shuffle towards him, using its arms to pull it along the ground. It was an obscene gesture, and Newbury was unable to watch. He turned to look behind him, trying to work out where the cab was in the fog. He leaned on the cane for a moment, catching his breath. Perhaps the other creature had slipped away, driven by the desire to find its own prey, when it looked as though its fellow had Newbury pinned to the ground? But that seemed unlikely.\n\nHe heard Veronica calling his name, somewhere behind him in the cloudy soup. He set off, staggering towards her voice, but stopped when he became aware of the shape of the other revenant, about ten feet away from him, silhouetted in the fog. It had its back to him. He edged around it, following the sound of the churning pistons nearby. Gingerly, he placed one foot after the other, doing his utmost to remain silent as he circled the monster. Then, stumbling on a loose stone, his foot scuffed on the cobbles. The creature twitched at the sound and spun around to face him. Newbury, exhausted, waved the cane from side to side, trying to hold it at bay. He didn't think he had the strength to take another one down.\n\nHe heard Bainbridge calling to him from the carriage. \"Newbury! Newbury! Use the cane.\"\n\nHe couldn't help but laugh at this most inappropriate advice. \"I'm using the bloody cane, man!\" He stepped back, trying to keep his distance from the creature. He knew it was likely to pounce at any moment.\n\nBainbridge's disembodied voice came back to him. \"No! Twist the knob on the end of the cane. Quickly!\"\n\nNewbury peered along the shaft of the cane. The bulbous brass knob didn't look extraordinary in any way. Nevertheless, desperate to find a way out of his current nightmare, he swung it back towards him, clutched hold of the cold metal knob and twisted it sharply to the right.\n\nThere was a clicking sound from within the shaft itself. Clasping the knob, Newbury pointed the end of the cane at the revenant, unsure what he expected to happen next. The shaft twisted and then began to spin, sections of the wood unpacking along its length and folding out to create a kind of chambered structure along the middle of the cane. Brass filaments ran along the inside of this structure. The spinning reached a crescendo and, with an electrical hum, an arc of blue light spat from one end of the chamber to the other, fizzing along the length of the shaft and crackling at the terminus of the device, the small pointed section at the very end of the cane.\n\nSmiling, Newbury raised the weapon towards the revenant just as the creature decided to give up waiting for an opportunity to catch Newbury off guard and threw itself towards him, claws outstretched. The point of the shaft impaled the monster through the chest, and there was a loud _bang_ as the electrical current flowed into the rotting carcass of the creature and fried what was left of its nervous system. The creature lost its momentum and dropped to the ground, dead for the second time. Blue light arced in its open mouth, and the blackened, dirty hole in its chest was smouldering, dark smoke rising into the air to mingle with the thick fog. Newbury's lungs filled with the scent of charred meat. He looked down at the body. The electrical current had set the creature's hair on fire, and little flames were licking at the edges of its tattered clothes. It wouldn't be long before the flames took and the creature's papery, rotten flesh became nothing but dry kindle.\n\nHe stood over the body and pulled the lightning cane free. Then, confident that the monster was finished, he staggered over to where the other one was still struggling to pull itself along the ground and drove the cane into its back, just below the base of the neck. Blue light sparked dramatically. The creature thrashed around for a moment before the twitching subsided and Newbury knew he had put it out of its misery. He stood for a moment, gathering his strength.\n\nThen, not even bothering to deactivate the weapon, Newbury staggered towards the sound of his friend's voices, hopeful that, this time, he'd be able to make it back to them unmolested.\n\nBainbridge and Veronica were waiting by the cab when Newbury staggered out of the fog. He was faint and bleeding from multiple injuries, the blue electrical light dancing and fizzing in the darkness along the length of Bainbridge's cane. They both rushed towards him, their eyes flitting nervously from side to side, worried that more revenants may come hulking out of the fog at any moment. Bainbridge swept the weapon out of Newbury's hand and twisted the brass knob, compacting the device so that it folded away neatly, dissipating the electrical charge and ensuring none of them would accidentally bear the brunt of its force. Within a matter of seconds, the peculiar device was nothing more than a cane once again.\n\nNewbury, his vision swimming, practically collapsed into Veronica's arms, and together with Bainbridge she lifted him up into the cab. They laid him carefully across one of the seats, and Bainbridge used the top of his cane to rap on the ceiling of the carriage, letting the driver know they were all safely back on board. A moment later the engines gave a wheezing gasp, and the vehicle trundled away into the rising dawn.\n\nVeronica was on her knees beside Newbury, tearing strips off his shirt to use as makeshift bandages on his wounds. He looked a mess: his torso was covered in scrapes and bruises and he was pale from the loss of blood, which pooled on the floor after soaking through his torn clothes and running free. Veronica tried to stem the flow with her hands, applying as much pressure as she could to the tear in his shoulder.\n\n\"Oh, Maurice.\" She seemed at a loss for words.\n\nNewbury turned his face towards her. \"I'll be alright. Everything will be alright.\" His voice was nothing but a croak. He cast his eyes at Bainbridge, who was sitting in the seat opposite them, leaning heavily on his cane. \"Quite a contraption, Charles. Wish I'd known about it earlier.\" His face cracked into a weary smile. \"Where did you acquire it?\"\n\nBainbridge shook his head, smiling in amazement that Newbury had even the strength to hold a conversation. \"Dr. Fabian. Never had much chance to put it to the test, but the old girl seemed to do alright by you out there, didn't she?\"\n\nNewbury nodded, wincing as Veronica tied a strip of fabric tight across the wound in his shoulder. \"She certainly did.\"\n\nVeronica glanced at Bainbridge, concern etched on her face. \"That's the best I can do, here. We need to get him to a hospital.\"\n\nBainbridge scoffed. \"Might as well take him to a butcher's shop. No, we need to get him to the Fixer.\"\n\n\"The what?\"\n\n\"The Fixer.\" Newbury turned his head to look her in the eye. \"One of the Queen's surgeons . . .\" He coughed, shifting on the seat in an attempt to alleviate the pain. \"Tell her, Charles.\"\n\nBainbridge picked up the explanation. \"The Fixer is one of the Queen's personal surgeons, on hand to help Her Majesty's agents in times such as this. He works for Dr. Fabian. He's the best medical man I've ever had the misfortune to meet, and he's got a place out in Bloomsbury, not far from the museum.\"\n\n\"Does the driver know where to go?\"\n\nBainbridge nodded. \"Barnes? Yes, he's one of ours. Why do you think he didn't bolt when he had the chance earlier, when Newbury had those damnable revenants after him?\" He paused, glancing over at Newbury, his brows furrowed. \"I take it the two uniformed chaps weren't so lucky?\"\n\nNewbury shook his head, but didn't speak. Bainbridge knew this meant the worst. \"Damn!\" He rammed his cane against the floor. \"Poor bastards.\" He glanced at Veronica. \"I do apologise, Miss Hobbes.\" She waved her hand dismissively.\n\nNewbury had closed his eyes. Veronica brushed his hair back from his face. She met Bainbridge's stare. Her voice was only just above a whisper, as if she didn't really want to know the answer to the question she was asking. \"What about the plague? Doesn't it spread when the revenants bite someone? Will he be infected?\"\n\nNewbury eyes flicked open again. He tried to prop himself up on one arm, but cringed when the pain in his side became too much. He returned to his previous position, supine on his back. He searched Veronica's face with his eyes. \"Don't worry. I'm immune to the plague. I won't be infected.\"\n\nBainbridge leaned closer. \"Immune? How so?\"\n\nNewbury swallowed, then reached up and pulled at his ragged shirt, exposing a large expanse of his chest. It was streaked and matted with blood, but it was easy to see the sickle-shaped scar of white tissue just above his left nipple, even in the dim light. \"I was bitten before.\" Veronica's eyes were wide with shock. \"Years ago, in India. My family had purchased some land out there, just about the same time that I'd found myself enamoured with stories of the occult. When the opportunity arose to pay a visit, I jumped at the chance. I spent two years in Delhi, exploring the Indian myths, searching for truth in the ancient stories of their culture.\" He sucked in his breath as the cab rolled over the uneven cobbles, jostling him in his seat. \"At around the same time a plague was spreading through the slums, a virus that turned people into shambling cannibals, forcing their skin to stop regenerating and blowing the blood vessels in their eyes.\" He coughed, raising a hand to his mouth.\n\n\"The revenants.\" Veronica mopped his brow.\n\nNewbury nodded. \"The revenants. I was out visiting a temple in the hills when I was set upon by one of the detestable creatures. It bit me here on the chest, but I was young and quick-witted enough to be able to get away. I managed to find my way back to my family's rooms in Delhi, whereupon they immediately called for the doctor. The Indian physician told us that his research had shown that the virus incubated in the brain for eight days before massively altering the physiology of the victim.\"\n\n\"What happened?\"\n\n\"They threw me in a cell and gave me nothing but bread and water to survive. For eight days, I ran the most appalling fever, and then, on the eighth day, the fever broke and I began to recover. Soon after, the doctor sent me home. He told me I was one of only a handful of people he knew who had survived the plague.\" He glanced from Veronica to Bainbridge. \"I'm convinced that this is the same virus, spread here from India, and that, provided my wounds don't kill me first, I'll live to fight another day.\" He flexed his fingers, frowning at the pain in his shoulder.\n\nBainbridge nodded. \"Of course you will, my man.\" He looked serious. \"Of course you will.\" He patted Veronica on the shoulder reassuringly and then looked up, smiling. \"We all know that the Fixer can perform miracles, don't we.\"\n\nNewbury sighed. The cab trundled on towards Bloomsbury, and towards the mysterious surgeon who, Bainbridge assured them, would be able to make things right once more.\nCHAPTER 20\n\n##\n\nThe sun had risen by the time the cab pulled up outside the Bloomsbury home of the Fixer, reducing the fog to wispy trails of vapour that seemed to linger in the air like white tendrils. Newbury had passed out a short while after they had set out from Whitechapel, and Veronica had continued to tend to him, staunching his wounds and trying to limit his blood loss by continuing to use strips of his shirt as makeshift bandages. She was covered in blood herself, now, her skirt, blouse and hands sticky with the gritty residue. Bainbridge thought it was a credit to her that she seemed entirely unfazed by this development.\n\nNewbury's breath was shallow, his skin had lost its colour and his eyes had sunk back in their sockets. Black bruises had emerged all over his exposed body where he had taken a severe battering from the revenants. Bainbridge hoped the Fixer really was able to work miracles. Newbury would need one if he were going to live.\n\nTaking his cane, Bainbridge stood and swung the carriage door open, glancing from side to side to see if anyone was watching. There were a few early risers going about their business, but the street was mostly deserted. He turned back to Veronica. \"Stay here. I'll go and make arrangements.\" She nodded silently and he ducked out of the cab, nodding at the driver as he mounted the step down to the road and made for the entrance of the large house. The building was tall, with three storeys above ground and a basement below, which Bainbridge knew would be their destination today. The house stood at the end of a long terrace, and as Bainbridge mounted the steps up to the front door, he heard the engine of the cab chugging behind him and watched as the driver reversed the cab around the corner, parking it near the iron staircase that led down to the basement level.\n\nHe rapped loudly on the door with the end of his cane. There was a momentary pause, and then the door clicked open and a middle-aged man in a black suit appeared in the opening. \"Ah, good morning, Sir Charles. Won't you come in?\"\n\nBainbridge stepped over the threshold and into the opulent foyer of the house. It was a grand building, worthy of royalty itself. The floor had been laid in a shimmering white marble, and a huge staircase swept away towards the upper levels of the house. Panelled doors led off into other, private rooms. A chandelier hung from a perfect ceiling rosette, and a small table had everything arranged just so. The entire place smelled of freshly cut flowers. The presentation was immaculate.\n\nBainbridge caught sight of himself in the large mirror hanging on the opposite wall, and shuddered. He looked terrible. Once he'd deposited Newbury with the Fixer, he'd see Miss Hobbes back to her lodgings and head home himself for a sleep and a long soak in the bath.\n\nThe manservant who had admitted him to the house\u2014a stout man of around fifty, with a receding head of grey hair\u2014looked Bainbridge up and down, as if trying to ascertain the reason for his visit. \"Are you well, Sir Charles? . . .\"\n\n\"Yes, yes, no time for all that, Rothford. I've got Sir Maurice Newbury in the cab back there, practically torn to pieces. He's in urgent need of the Fixer.\"\n\nRothford snapped to attention. \"Quite right, sir. Better bring him around the side entrance, quick-sharp. I'll notify the master immediately. I believe you know the way?\"\n\n\"I do.\"\n\n\"Then go, sir, and I'll make the necessary arrangements.\"\n\nBainbridge nodded. \"Thank you, Rothford.\"\n\n\"I'll hear no word of it, sir.\" He clicked the door open again and ushered Bainbridge out.\n\nBainbridge hurried down the steps and round the corner to where the cab was still waiting, its engine burring noisily. Overhead, an airship swept low over the city, whipping his hair back from his face. He was glad he'd left his hat in the carriage earlier. He hopped up onto the step and spoke to the driver. \"Keep an eye out, Barnes. Wouldn't do to have anyone see what we're up to.\"\n\nThe driver nodded. \"Aye, sir. I'll give you the word when you're clear to make a move.\"\n\n\"Good man.\" He ducked into the carriage. Newbury was still unconscious. Bainbridge put a hand on Veronica's shoulder. \"All will be well, Miss Hobbes. We've brought him to the right place. The Fixer will do his work, and Sir Maurice will be back on his feet in no time at all.\" He glanced down at the supine man. \"Here, can you help me with his head whilst I lift him down?\"\n\n\"Of course.\" Veronica moved to cradle Newbury's head as Bainbridge placed his cane on the opposite seat and moved to scoop his unconscious friend up into his arms.\n\nHe staggered under the weight, trying to get his footing, and then was able to rest Newbury's head in the crook of his arm as he moved towards the open door. Gasping a little for breath, unused to the exertion, he called out to the driver. \"Barnes? Are we set?\"\n\n\"Aye, sir. All clear.\"\n\nBainbridge stepped cautiously down onto the step beneath the carriage door, and then onto the street below. Without looking back, he approached the side of the house, mounting the first rung on an iron staircase that descended from street level down to the basement of the large house. His feet clanged loudly on the steps as he struggled to manoeuvre Newbury down the tight enclosure. Then, reaching the bottom of the flight of stairs, he used the edge of his boot to bang on the wooden door that awaited him there. A fraction of a second later, the door swung open, revealing a dark space beyond, and Bainbridge, shifting so as not to strike Newbury's head against the doorframe, slipped quietly inside.\n\nA few minutes later, Bainbridge emerged from the same doorway, having deposited Newbury with Rothford to await the ministrations of the Fixer. He crested the top of the iron staircase, dusted himself down, and, red-faced from the exertion, hopped back into the cab with a nod to the driver. The engine spluttered to life as Bainbridge took a seat, careful to avoid the spilled blood that was congealing on the floor. Barnes would have his work cut out for him, cleaning that lot up.\n\nVeronica was sitting with her hands folded in her lap. She looked nearly as white as Newbury had, shocked to the core and uncertain about Newbury's condition.\n\nBainbridge attempted to offer her his warmest smile. \"My dear Miss Hobbes. I should think Sir Maurice will owe you a large debt of gratitude when he eventually comes round from all this. Your efforts in stemming his wounds are surely what kept him alive during the course of the journey over here. Now, the Fixer can do his work and make him whole again.\"\n\nVeronica pursed her lips. \"Sir Charles, I think it is we who shall owe Sir Maurice a debt of gratitude. His actions at the murder scene are what saved us all from disaster. He willingly put himself in the way of those monsters to save us from harm. Saving his life in turn was the very least we could do, if indeed we have managed it.\" She looked away, still dignified, even whilst caked in the dried blood of her employer. \"I hope this 'Fixer' is everything you've made him out to be.\"\n\nBainbridge nodded, carefully weighing her words. \"You're quite right, of course, Miss Hobbes. Forgive my insolence. I did not mean to demean the actions of our brave friend, only to embolden you with talk of your own. I was aiming to give reassurance, where perhaps none was needed. I'm afraid I've forgotten how to talk to ladies, ever since my wife died. I now spend all my time in the company of other men.\"\n\nVeronica returned his gaze. Her demeanour softened. \"Sir Charles, I fear it's not a case of knowing how to talk to a lady. I'm simply concerned for the well-being of Sir Maurice.\" She tried, ineffectually, to brush some of the dried blood from her clothes. \"So tell me, what does this Fixer do?\"\n\nBainbridge smiled. \"He fixes things.\"\n\nNewbury woke with a start.\n\nHe sucked at the air.\n\nHis head was throbbing, although he felt as if he'd somehow been infused with a warm, liquid glow; warmth that started in his belly and seemed to seep upwards towards his head, gloriously taking the edge off his pain and leaving his mind to wander in a drowsy state of semi-consciousness. He knew the sensation of old.\n\nOpium.\n\nNewbury peeled open his eyes, and then immediately shut them again. The light in the room was blinding, clinically sharp, and it seared the back of his retinas like a hot knife. He drew a ragged breath, pulling the air down into his lungs. His chest felt like it was on fire. Cautiously, he tried to open his eyes again, reaching up to shelter them from the glare with cupped hands. Stinging tears ran down his cheeks. He blinked them away. Finally, an image resolved.\n\nHe was lying on his back on a hard metal table. A face was looming over him. He tried to sit up.\n\n\"No, Sir Maurice. Try to lie still. Everything is going to be alright.\"\n\nNewbury felt a hand on his chest, holding him still on the table. He blinked up at the strange face that was hovering over him. The man was in his late forties, balding, with a neatly trimmed black beard. A bizarre mechanical contraption sat on his head, like a wire frame that encompassed his temples and forehead, with various accouterments and glass lenses attached to it on folding levers and arms. The man reached up and flipped one of these lenses down over one eye.\n\n\"Who are you? Where am I?\" Newbury had a panicked edge to his voice.\n\n\"I'm the Fixer, and you're in my workshop, underneath my home. You have nothing to worry about.\"\n\nNewbury breathed a sigh of relief, allowing himself to relax. He'd never had occasion to visit the Fixer before, but he was well aware that the man existed: a personal surgeon of Her Majesty's who made himself available to her agents in times of dire need. He remembered Bainbridge speaking about him in the carriage, just after the attack. What was not good was the fact that, if he was here, his situation was potentially very grave indeed.\n\nNewbury quickly discovered that his abdomen and shoulder lanced with pain every time he made even the slightest motion with his body. He tried to lie still, giving himself over to the warmth of the opium, but the Fixer had been wise and had dosed him with only enough to take the edge off the pain, and not enough to render him unconscious again. He felt gloved hands tearing at his clothes and the faint stirring of a breeze on his exposed flesh. Nevertheless, the room itself was warm, and listening to the sounds around him, he had the sense of a workshop full of bizarre mechanical devices. There was a faint electrical hum, accompanied by the occasional sound of a belching valve as it issued forth a cloud of hot steam, as well as the constant _tick-tock_ of numerous clockwork engines powering objects that he could not see from his limited vantage point on the table. Newbury tried not to imagine what the man was about to do to him with the strange machines that were making such sounds.\n\nThe Fixer appeared in his field of vision once again, wavering slightly under the influence of the opium, and then disappeared. Newbury could hear him shuffling around the other side of the table. The Fixer cleared his throat, and then began to speak, offering a running commentary as he examined Newbury's wounds. His voice, Newbury noticed, was gruff and gravelly, the voice of a man who'd smoked too much heavy shag in his time. \"Hmmm. A vicious bite in the left clavicular area, there. Serious tears to the flesh and muscular tissue. Excessive blood loss.\" He paused for a moment, poking sharply at the wounds on Newbury's chest. \"Deep gouges in the chest. Numerous flesh wounds. A severe laceration in the left side of the chest and abdomen. My, my. You have been busy.\"\n\nNewbury stirred uncomfortably. He waited until he heard the other man move away from the table, his footsteps ringing on the tiled floor, and then, with a significant effort, managed to prop himself up on one elbow.\n\nThe Fixer stood at the foot of the table, fiddling with an array of surgical tools, which pinged noisily on a steel tray. Beside him on a wooden trolley was a rack of steel hypodermic syringes, which contained a range of strange, multi-coloured fluids. Newbury took the opportunity to take a better look at the man who called himself the Fixer.\n\nAside from the contraption on his head, the man was wearing a tarnished leather smock and matching leather gloves. Newbury couldn't help thinking that he had more of the appearance of a butcher about him than of a physician. He had a ruddy complexion and the manner of a public schoolboy. Newbury suspected he spent a great deal of time in his workshop, and very little time engaging with the world.\n\nUnsure what was likely to happen next, and unwilling to ask, Newbury cast his eyes around the room, trying to get a measure of his surroundings.\n\nThe basement was lit by a series of long, unusual gas-lamps that arced across the ceiling from one wall to the other, curved glass tubes that terminated with gas valves where they met the walls at each end. An array of strange machines and surgical tables filled the space in between. One of these\u2014a large brass contraption about the size of a small table, with two glass vats full of bubbling fluid atop it\u2014had long coils of tubing that snaked out from the belly of the machine and away into the dark corners of the room. Another, smaller contraption was fitted with wheezing bellows of the sort Newbury had seen attached to Queen Victoria's life-preserving engine. It even rose and fell with the same constant rhythm of Her Majesty's breathing machine, although in this instance, it appeared that the bellows were helping to power an unusual electrical device, the lights on it flickering from orange to blue as the exposed filaments danced with the current.\n\nThe alarming contraption above Newbury's own table was connected to an extensive brass framework, a kind of large gun on a moveable rail, with fat tubing trailing from the back of it and disappearing into a nearby hatch in the floor. The device had a trigger fitted to the undercarriage and the end of it terminated in a spread of fine needles, bunched together to form a neat point. Newbury shivered.\n\nThe Fixer turned to notice him looking. \"Impressive, isn't it?\" He turned to encapsulate the room with a gesture of his arms, indicating the various machines. \"This is what Dr. Fabian gets up to when he isn't busy attending to Her Majesty or running errands for the likes of you and Sir Charles. Works of genius, every one of them.\"\n\nGroggily, Newbury met his gaze, and felt immediately disorientated by the sight of the man's strange eyewear, which magnified the appearance of his right eye so that it seemed at least three times the size of his left. \"So, what's next? Surgery?\"\n\nThe Fixer smiled. \"Of a sort. I'm going to knit your shoulder and chest back together with my stitching machine.\" He indicated the gunlike device on the rail overhead. \"Then I can give you a blood transfusion and a dose of one of Dr. Fabian's excellent compounds.\"\n\nNewbury narrowed his eyes. \"What will it do?\"\n\n\"Fix you, of course.\" The man beamed. Newbury held his gaze, a serious expression on his face. Sighing, the Fixer continued. \"It's derived from a rare flower that Fabian discovered out in the Congo last year. When the powder is dissolved in saline and transfused into the human body, it boosts the existing immune system, helping the blood cells to clot and bind, so that muscles and bones can reconnect very swiftly indeed. I wish we'd know about it before\u2014could have saved a lot of trouble with Ashford all those years ago.\" He paused, tapping his foot on the tiles as if planning his next move. \"Come on. Let's get you under the knife. You'll know the effects of the treatment soon enough, anyway.\"\n\nCautiously, Newbury laid his head back against the hard surface of the table. The Fixer moved round to stand beside him, reaching over to wash his shoulder wound with a wet cloth. The antiseptic fluid burned angrily where it came into contact with the damaged, puckered flesh. Newbury winced and clenched his jaw, grinding his teeth as the other man reached up and took hold of the stitching machine. Newbury closed his eyes. He heard the device firing up, the _rat-a-tat-tat_ of the needles scraping back and forth as the Fixer applied pressure to the trigger, testing the pneumatic power.\n\nHe brought the gun closer to Newbury's shoulder, and then, without any further warning, he jabbed the device into the soft tissue underneath the skin of Newbury's left arm. The needles began to puncture the lacerated flesh in a torrent of relentless pinpricks.\n\nNewbury screamed out in agony as the device stitched a series of fine filaments deep into his clavicular muscles; slowly, deliberately, knitting his shoulder back together. The Fixer began to move the device along the extent of the wound, closing the flesh where the revenant's mouth had torn it open. Blackness swam around the edges of Newbury's vision.\n\nHe swooned, and everything went dark.\n\nWhen Newbury came round again, he was lying on a bed, his head resting on the soft, downy pillows, a thick woollen blanket pulled up over his waist. Thin rubber tubes jutted rudely from incisions made in each of his wrists, trailing off toward large machines on either side of him, one of which was giving off a deep mechanical rumble and a gentle gasp of warm air. He tried to sit up, but felt his shoulder pulling tightly where the Fixer and his stitching machine had done their work. He flexed his fingers and tentatively moved his arm, feeling that he'd regained a lot of strength in the limb. The pain in his shoulder and abdomen had mostly subsided to a dull ache. He lifted the corner of the blanket warily and looked down at the line of puckered flesh where the machine had sewn him back together again. It was bruised and purple, and had an ungainly web-work of black stitches weaving across it, but it was far better than an open wound, and in truth, he felt almost normal again.\n\n\"Marvellous, isn't it?\" Newbury looked up, noting for the first time that the Fixer was sitting in a chair by the side of the bed, watching him intently as he explored his handiwork. His strange headpiece was on the table beside him. He looked considerably more normal without his leather smock and gloves. \"It won't leave much of a scar there, either, what with the watertight stitches and the transfusion of Dr. Fabian's healing compound that you're receiving.\" The Fixer smiled. \"It'll be sore for a few weeks, though.\"\n\nNewbury folded the blanket back over his lap. \"How long before I'm up and about?\"\n\n\"A couple of hours. There's no reason to keep you here, once the transfusions are complete and we've found you some suitable clothes. You should go home and rest, let the compound do its work.\" He waved at Newbury's abdomen underneath the fawn-coloured blanket. \"It should hold up, even if you do find yourself in another scrape. Those stitches aren't designed to give out on you. You'll need to come back in a couple of weeks to have them out again, whatever the case.\"\n\nNewbury grimaced at the thought of it. He lifted his arms, presenting his wrists to the other man. \"Which one's which?\"\n\n\"The machine on your left is giving you blood. The one on your right is giving you the saline solution.\"\n\nNewbury glanced at the machine to the left of him. It seemed to be vibrating gently, humming as it pumped the fluid along the coiling black tube and up into his arm.\n\nA panel near the Fixer's chair was decorated with a series of dials, all of which had been turned to various positions that made no sense to Newbury, at least from where he was sitting.\n\nHe met the other man's eye, indicating the transfusion machine \"Why is it so noisy?\"\n\n\"Ah. That's the refrigeration unit. I use that to keep the blood from congealing. It doesn't last long out of the body. Luckily for you, Rothford is a willing donor, and his blood type is compatible with most people's.\"\n\n\"Rothford?\"\n\n\"My manservant.\"\n\nNewbury nodded.\n\n\"You'll meet him shortly. Now, though, you need to lie back and rest. I'll come and disconnect you shortly, once we're both convinced that you're ready to take a walk.\" The Fixer stood at the foot of the bed, smiling, and then disappeared into the gloom.\n\nNewbury allowed his head to fall back onto the pillow. The effects of the opium had worn off, and his skin crawled with craving. He longed for the warm glow of the drug. He thought of the small bottle of laudanum on the shelf in his study, thought momentarily of what he might do when he got home . . . and then thought of Veronica, and the manner in which she had found him just a couple of days ago. It wouldn't do to descend into that madness again. Sighing, he closed his eyes and willed himself to sleep, listening to the sounds of the gurgling fluid that was currently seeping into his bloodstream.\n\nA couple of hours later, dressed only in a plain white robe, and with thick yellow poultices applied to his wounds, Newbury followed the Fixer up a small internal staircase and into a waiting area that was set out like a gentleman's reception room. Roth-ford, the Fixer's manservant, was waiting for them there, his hands folded neatly behind his back. He stood when the two gentlemen came into the room.\n\nThe Fixer spoke first. \"Rothford, this is Sir Maurice Newbury. He'll be needing your attention, as well as some assistance finding suitable attire. Please treat him as a guest in this house.\"\n\nRothford gave a single nod of the head, and then glanced at Newbury, a twinkle in his eye. \"Very good, sir.\"\n\nThe Fixer clapped a hand on Newbury's arm, carefully avoiding the wound on his shoulder. \"I'll leave you in Roth-ford's capable hands. Be sure to take some time to rest.\" He turned to leave, and Newbury reached out to stop him. He offered the Fixer a sincere smile. \"Thank you, I\u2014\"\n\nThe other man shook his head. \"Don't thank me. Simply try to make sure that you don't need my attentions again in the near future, especially before you're back to have those stitches out.\"\n\nNewbury laughed, causing his chest to burn with pain. \"I'm not planning on it, I'll give you that much.\"\n\nThe Fixer smiled. \"For men in our profession, Sir Maurice, that has to be enough. Good day to you.\"\n\n\"Likewise.\" Newbury watched as the man disappeared from the room, descending the stairs towards his workshop once again.\n\nRothford approached from the other side of the room. \"If you'd like to come with me, Sir Maurice, I'll show you to our dressing room.\"\n\nNewbury nodded and followed behind Rothford as he led him through a door, along a short passageway and through another door into a small room on the left. It was furnished with a wardrobe, cheval glass mirror and dressing table. Rothford crossed to the wardrobe and opened the doors with a flourish. Inside, Newbury could see that it was filled with all manner of formal suits and dresses, white shirts and underclothes. He wondered how many \"visitors\" the Fixer regularly received.\n\nRothford searched through the rack of clothes for a moment, before withdrawing a black suit on a hanger and holding it up beside Newbury. \"There. I should imagine this will do. I'll lay it on the chair over here.\" He draped it over the back of the tall chair by the dressing table. \"Please feel free to help yourself to a shirt and underclothes. When you're decent, you'll find me in the reception room at the other end of this short hallway. I'll organise some breakfast. Bacon and eggs?\"\n\n\"Thank you.\" Newbury nodded, unsure what else to say. He watched as Rothford exited the room, clicking the door shut behind him.\n\nThen, gingerly, he disrobed, eyeing his wounds in the cheval glass. The line of bruised, puckered flesh that ran down the left side of him looked angry and sore. Yet, strangely, he felt decidedly more alert than he had in days. He supposed that had a lot to do with Dr. Fabian's miraculous healing compound. He made a mental note to attempt to find out the name of the flower it was derived from. It would make an interesting study, and he could do worse than to have a small amount of the compound available to him at his Chelsea lodgings.\n\nTaking care to dress slowly so as to avoid pulling on his stitching, Newbury was soon feeling more like his usual self, and with the promise of eggs and bacon just along the hall, he realised he was absolutely famished. Finding a pile of his personal belongings arranged on the dressing table, he slipped these into the pockets of the borrowed suit and set off in search of Rothford, Earl Grey and food.\nCHAPTER 21\n\n##\n\nVeronica sat beside Amelia on a wooden bench on the grounds of the asylum, wrapped up against the chill. They were watching the other inmates as they circled the airing court like a flock of birds, each following the others as they walked, their feet crunching noisily on the gravel. Nurses kept a watchful eye from one end of the courtyard, gossiping amongst themselves and dressed in thick winter coats. Their breath plumed in the frosty air.\n\nVeronica glanced at Amelia, who\u2014even dressed in a heavy coat and shawl\u2014was shivering with the cold. She put her arm around her sister, hugging her closer for warmth. Veronica knew that she shouldn't have come. She could think of a hundred reasons why she shouldn't be there that day, why she'd have been far better off staying away, yet none of them seemed quite so important as the reason she had finally given in and made the journey across town. Now, here, she could barely face her sister, who had been delighted by the unexpected visit and had clutched her brightly, kissing her fondly on the cheek. Tired and emotional after a difficult morning, Veronica had chosen to walk with Amelia in the gardens before broaching the true reason for her visit.\n\nAfter Sir Charles had deposited her at her Kensington lodgings, Veronica had found herself alone in her apartment, her housekeeper out running errands around town. She had stripped out of her filthy clothes, poured herself a scalding hot bath, and sat weeping on the bathroom floor, her knees drawn up to her chin, tears streaking down her blood-caked cheeks.\n\nShe sat like this for at least an hour, cycling through the full gamut of emotions, from relief to anxiety and then back again. She had been so terrified by those detestable creatures as they attacked the cab, trying to peel the door away to get at her and Bainbridge inside, that she had done little to aid in the battle. She cursed herself for being so weak. She was a strong woman, a fighter, but she had seen no way out of that dreadful scenario, and had almost given herself over to her fast-approaching fate, when Sir Maurice had appeared out of the fog and taken on the two monsters single-handedly, drawing them away from the cab.\n\nShe felt ashamed that her first thought had been to flee, to get away from there as quickly as possible whilst she had the chance, to abandon Newbury to the monsters in an effort to save herself from harm. Reason had reasserted itself, however, and she had remained in the carriage, knowing that there was little she could do to help him as he fought the creatures in the fog-enshrouded street. She had come close to rushing out there to aid him when she heard him crying out in pain, but she knew that she would only have served as a distraction and that, had she taken on one of the creatures herself, she would have surely lost out to its brutish strength and animalistic will.\n\nThe worst horror, however, had been seeing Newbury in such a desperate condition after he'd managed to make it back to the cab. Even now she feared for his life, feared what this Fixer character may do to him, and worse, feared that his words of reassurance regarding the revenant plague were simply that\u2014words\u2014and that before long he would succumb to the terrible blight and, regardless of how tightly she had tied his bandages and how well she had staunched the flow of his lifeblood, she would lose him anyway.\n\nShe couldn't bear the thought that Newbury might transform into one of those horrifying creatures, and she knew that he, too, would rather die than let that happen. So she had resolved to visit Amelia at the asylum, to take advantage of her sister with a long list of difficult questions, and to try to ascertain what the future held.\n\nAmelia was watching the other inmates as they went about their laborious routine. \"Tell me I'm not reduced to that, Veronica. I feel like a little bit more of my life is sapped away from me each and every day I spend in this terrible place.\"\n\nVeronica hugged her sister closer. \"You're not, Amelia. You're not like that at all.\"\n\n\"Then why do I have to live like this? What have I done wrong to deserve to be locked up in here? It's basically a prison cell.\"\n\nVeronica didn't know what else to say. \"I'll get you out, Amelia. I promise. I'll find a way to get you out.\"\n\nAmelia shifted slightly in her arms and smiled. \"I know you will, Veronica. I know it's just a matter of time.\"\n\nVeronica looked at her sister quizzically. \"Do you know something? Have you seen something in one of your visions?\"\n\nAmelia shook her head. \"You know it doesn't work like that, Veronica. I only remember snatches of what I see, dreamlike sequences and unconnected images. In one of them, I see you and me, walking down the street together, away from this place.\"\n\n\"Can I ask you something? Something I promised I'd never ask you?\"\n\nAmelia slipped out of Veronica's embrace, stiffening slightly on the bench. \"What is it?\"\n\n\"Have you seen what becomes of Sir Maurice? In the future, I mean.\" Veronica couldn't meet her eye.\n\n\"No. Nothing.\" Amelia shrugged. \"Well, that is to say that I do not recall seeing anything. Why, what happened?\"\n\nVeronica was exasperated. She balled her hands into tight fists. \"Try for me, Amelia. This is very important. Try to remember if you've seen him during a recent episode. Anything at all. Even just a glimpse.\"\n\nAmelia looked pained. \"Veronica, I've never even met the man. This is not something I know how to control. It happens, and then it is as if the episode somehow leaves a residue in my mind, fleeting images I can sometimes remember. It's not as if I can recall the entire episode at will.\"\n\nVeronica tried to fight back the tears. \"I know, Amelia, I know. I'm so sorry.\" She turned away, breathing deeply to steady herself.\n\nAmelia put her hand on Veronica's arm. \"Don't be. Clearly something terrible has happened, and I want to do everything I can to help.\"\n\n\"You already have. I suppose now it's just a matter of time.\"\n\n\"What happened? Tell me.\"\n\n\"Sir Maurice was attacked by three revenants this morning. They practically tore him apart, but he managed to get away. He was fighting for his life, bleeding all over the carriage\u2014all over me\u2014but we managed to get him to the surgeon.\"\n\nAmelia put a hand to her mouth. \"Will he make it through?\"\n\nVeronica was solemn. \"I don't know. Worse than that, though, is the threat of plague. I have every fear that he might have been infected.\"\n\n\"Oh, God.\"\n\n\"That's why I came to you, Amelia. I had to know if you'd seen him in one of your visions, had to know if he was going to be alright. I should never have come. It was unfair of me.\"\n\n\"Sister, you've done so much for me. Is it not fair that I at least attempt to repay that love and loyalty from time to time?\"\n\n\"It doesn't work like that, Amelia. You don't owe me anything.\"\n\n\"I know _exactly_ how it works, Veronica. That's why I love you so.\"\n\nToo late, Veronica noticed that Amelia was starting to take short, shallow gasps at the air, beginning the process of inducing an episode. She clutched her by the shoulders. \"No! Stop it, Amelia! Stop it now!\"\n\nAmelia shook her head, gasping for breath.\n\nVeronica held her tight. \"I'm sorry, sister.\"\n\n\". . . I . . . know . . .\" Amelia began to shake, her body shuddering as her muscles went into spasm. Her eyes rolled back in their sockets, showing the milky-white underside of her eyeballs. She rocked back, saliva running from the corner of her mouth.\n\nVeronica glanced around to see if any of the nurses had noticed. They were still engaged in conversation by the main asylum doors. She clutched Amelia close, trying to keep her safe.\n\nAmelia began to babble. At first it seemed incoherent, a long chain of moaning sounds and half-formed words, but then Veronica began to make sense of what she was saying.\n\n\". . . from the sky . . . like a child's balloon, tumbling . . . tumbling towards the ground . . . water . . . shouting . . . confusion.\"\n\nVeronica shook her head, trying to get through to her sister. \"No Amelia, that's already happened. The airship has already crashed!\"\n\n\". . . water . . . dripping water . . . a clockwork man.\" She gasped, gulping air down into her lungs, her entire body shaking as the fit took complete control of her body. \". . . a dark place . . . a woman's voice . . . Veronica!\" The shuddering stopped. Amelia turned towards her sister, her unseeing eyes fixed on Veronica's face. It was the most eerie thing Veronica had ever seen. She let go of Amelia, reflexively forcing herself backwards on the wooden bench. She heard footsteps on the gravel behind her.\n\n\"It's all in their heads, Veronica. Tell him. You must tell him. It's all in their heads.\" Amelia collapsed back into spasms once again, and Veronica, shaking, looked round to see two of the nurses rushing to Amelia's aid.\n\nThey gathered her up as quickly as they could and laid her out on the lawn beside the wooden bench, holding her down as she continued to spasm.\n\nVeronica leaned over them, desperate to see if Amelia had anything more to say, unprepared for her sister to go through all this agony on her behalf without even finding an answer. But it was not to be. Amelia's episode began to subside and the twitching of her body slowed. She didn't utter another word. Veronica slumped back onto the wooden bench, thankful, at least, that her sister seemed to be unscathed.\n\nAmelia's breath was shallow and she looked dazed, unsure where she was or how she may have got there. She looked up at Veronica, the nurses still pinning her arms to the ground.\n\n\"Veronica?\"\n\n\"Yes, I'm here, Amelia. Are you okay?\"\n\nAmelia blinked, looking at the faces of the two nurses who were holding her down on the cold grass, awaiting the arrival of the doctor. \"I'll be alright.\" She met Veronica's gaze. \"Did you find what you were looking for?\" Her eyes were questing, searching for approval from her older sister.\n\nVeronica looked away. \"I'm not sure, Amelia. I don't know what it all meant.\"\n\nDr. Mason came running towards them, then, his face flushed. He scowled when he saw Veronica sitting on the bench in the middle of the scene. \"Hello, Amelia. I think it's time we got you inside.\" He turned to Veronica. \"Your sister will be taking her leave of us now.\"\n\nVeronica gave one decisive nod. She stood as the nurses helped Amelia to her feet. \"I love you, sister.\" She stepped forward and kissed Amelia on the cheek. \"Be well.\"\n\n\"I'll try.\"\n\nVeronica turned and walked away from the scene, her hand on her head to keep her hat from fluttering away in the breeze.\nCHAPTER 22\n\n##\n\nThe next morning, Newbury rose early, still tender from the ministrations of the Fixer the previous day. He went directly to the bathroom and washed his wounds, and then applied a thick layer of the yellow poultice to each of them in turn. The substance smelled faintly of beeswax, although he could only guess at what else it comprised. He felt vibrant and nervous with energy, partially the result of too much rest, and partially, he imagined, the continued effect of Dr. Fabian's compound. His wounds had begun to heal already, too, although there was still a long way to go before he'd be back to anything like his normal physical form.\n\nNewbury had spent the remainder of the previous day holed up in his study, pacing the room, smoking his pipe and doing his utmost to stop himself giving in to his cravings for the laudanum, which sat in its little brown bottle on the shelf across the room, teasing him with promises of warmth, forgetfulness and solitude. He had sorted through a number of papers from his years in India, searching out references to the revenant plague and attempting to lose himself in reminiscences of the period. Mrs. Bradshaw had prepared him a lavish roast beef dinner, and he had taken it in the dining room, the first time for months that he had made a point of sitting down to eat a proper meal in his own house.\n\nBy morning, however, he felt he could carry on like this no longer. He was concerned that boredom would indeed drive him to the dreaded opiate that he was attempting so pointedly to resist. Instead, he had resolved to head to the office, to deal with any outstanding correspondence, ensure that Miss Coulthard was bearing up, and otherwise busy himself with work on his now-overdue academic paper.\n\nHe secretly hoped that, in doing so, he would happen upon Miss Hobbes with news of the case, and together they could spend the day mulling over the developments so far, gathering their thoughts whilst his constitution was restored and agreeing on a course of action for the following day. If nothing else, he knew Her Majesty would not look too kindly on him wasting another day in lackadaisical pursuits when he had a case to solve, injured or not.\n\nIt was still too early in the day to expect Mrs. Bradshaw to have risen to make breakfast, so instead Newbury settled for organising himself a pot of Earl Grey and rummaging up a few slices of toast, which he ate with a smear of marmalade whilst reading the morning papers. Then, confident that he was well enough for a brief stroll, he fetched his coat and hat and set out, drawing in the fresh morning air and celebrating the fact that he was still alive. The previous day's events seemed like a lifetime ago, a dark and distant memory, and if it were not for the occasional twinge in his upper torso as he walked, he could almost have believed that it had been nothing but a fantasy.\n\nPresently, tiring from his walk, Newbury hailed a cab to take him the rest of the way to the museum. The streets were still quiet, but the sun had risen and the fog was already lifting. He bounced along in the back of the cab, wincing every time the horses ran over an uneven patch of cobbled road and the wheels juddered, jolting his injured body painfully.\n\nThe museum grounds were still deserted when the cab pulled up outside the main gates. Newbury stepped down and paid the driver, who doffed his cap and set the horses trotting off towards Charing Cross Road, their hooves clattering loudly in the otherwise empty street. Newbury crossed the courtyard and mounted the steps up to the main entrance, smiling warmly at Watkins, who was on hand even at this hour to welcome early arrivals. Pulling his gloves off and loosening his scarf, Newbury made his way down to the basement floor and along the short corridor to his office. Taking his key from inside of his jacket pocket, he turned it easily in the lock, pushed the door open and stepped inside.\n\nIt was clear Miss Coulthard had visited the office in the last couple of days. The correspondence had been neatly stacked in the appropriate trays, the cups and saucers had been tidied away and there was a note on her desk, in her handwriting, addressed to him. He picked it up, unfolded the card and scanned the neat copperplate briefly, before dropping it into the wastepaper-basket beside the door. No word on her brother Jack, it seemed.\n\nNewbury clicked the door shut and draped his coat and hat on the stand. He crossed to his private office, noting that there was a pile of papers for him to sign, obviously left there by Miss Coulthard, and growing in size every day he was away from the office.\n\nHe disliked the menial administrative duties of his position at the museum, but in other ways it held his interest when he wasn't on a case, allowed him to come and go as freely as he liked and gave him access to many files and artefacts he would otherwise find it very difficult to obtain. Not only that, but it served as a perfect cover for his position with the Crown, meaning that, rather than having to hide himself away from society as many of the other agents did, he could instead continue to ingratiate himself with the nobility of London, all of which, he felt, provided him with a greater opportunity to do his duty for Her Majesty and the Empire. Connections, in London, were everything, and he found they opened doors where others would find them locked.\n\nFlexing his damaged shoulder muscle to stop it from stiffening up, he lowered himself heavily into his chair. He flicked through the pile of papers on his desk, sighing in dismay. There wasn't even enough there to keep him engaged for an hour, and whilst his paper on the druidic tribes of Bronze Age Europe was in dire need of further work, he still hoped to find an opportunity to get back on the case before the morning was out. He drummed his fingers on the desk. He needed to talk to Musgrave.\n\nNewbury looked up at the sound of the main door clicking open. He glanced at the grandfather clock through the open door of his office. It was still too early for it to be Miss Hobbes. Perhaps, in an effort to distract herself from the difficult situation at home, Miss Coulthard had decided to come to the office early that morning?\n\nHe stood, moving round from behind his desk to greet the new arrival. He stopped short when he heard a familiar clacking sound, like brass feet clanging against the porcelain titles of the floor.\n\n_Automaton._\n\nHe backed up, wondering how one of the clockwork men had managed to get into the museum, let alone track down his office on the basement floor. The feet continued to clatter on the tiles, slowly, deliberately, and Newbury realised that, judging from the sounds of their shuffling movements, there must be more than one of the devices.\n\nA moment later, one of the units appeared around the corner behind the coat stand. Newbury stiffened. It seemed to survey the office, its spinning eyes flicking from one corner of the room to the other. When it caught sight of Newbury, it began to move again, turning around slowly and approaching him, its arms hanging limp by its side. Another one shuffled into the room behind it.\n\nNewbury braced himself. \"What are you doing here? What do you want?\"\n\nThe automaton cocked its head slightly, as if trying to compute his words. Then, stopping about six feet away from him, it raised its right hand before its face. There was a soft, almost pneumatic _snick_ ing sound as thin, knifelike blades slipped out from the ends of its fingers, turning its hand into a vicious razor-sharp claw. Newbury crept backwards until his legs encountered the edge of his desk. The automaton resumed its slow, relentless march towards him. Behind it, the other unit moved farther into the room, blades clicking out of the ends of its fingertips to form an identical gruesome-looking weapon. He noted with dismay that the right hand of that second unit was already smeared in blood. He supposed that answered his question about how the devices had found their way into the museum.\n\nKnowing that he was already seriously injured and therefore unlikely to be able to hold the automata off for long, Newbury decided to go on the offensive. He waited a moment until the nearest unit was only a matter of feet away from him and then charged it, trying to use his speed and body weight to his advantage.\n\nThe automaton saw him coming, however, and twisted out of the way, contorting itself in a manner a human being would find impossible to emulate. Newbury, unable to stop his momentum, slammed into the side of Miss Coulthard's desk, jarring his injured shoulder and spinning awkwardly to the ground. The desk overturned, sending sheaves of paper blooming into the air.\n\nJust in time, Newbury realised he'd landed at the feet of the second automaton, and he rolled to the left, narrowly avoiding its falling hand, which chopped down against the tiles with terrifying force, splintering the porcelain in a cloud of dust. Newbury, still on the floor, grabbed out for the automaton's leg, yanking it forward and unbalancing the device, sending it smashing down to the hard floor beside him. It immediately began to climb to its feet, twisting its shoulder joints to give it better leverage. Newbury climbed to one knee and thrashed out, bringing the coat stand crashing down in front of him just in time to block the path of the other automaton, which was charging him from across the room. He had to think fast.\n\nLeaping to his feet, he cast around for a weapon. His abdomen and chest were on fire as his movements pulled on the stitches, tearing at his damaged flesh. The automata, scrambling over the coat stand, had been reduced to tireless killing machines, stripped of their harmless guise as servants. Their gears churned as they both came at him again, swinging their bladed hands towards him, one of them only a matter of inches from his face.\n\nHe fell back, banging his head awkwardly against the wall. Trying to ignore the burst of sharp pain that flared at the back of his skull, he dived to the left, sending the kitchen equipment skittering across the tiles as he tried to take cover behind the small gas stove, forcing himself over the top of it and onto the floor on the other side. Between the stove and Miss Coulthard's overturned desk, he found himself trapped in the corner of the room, with nowhere else to turn. The one thing in his favour was the fact that the automata seemed unable to work out how to climb over the furniture, instead choosing to reach over and slash at him with their razor-sharp finger blades. He tried to stay out of their reach.\n\nNewbury glanced around in desperation, still looking for something he could use to defend himself. Above him on the wall was a mediaeval axe with a long wooden shaft. He grabbed for it, hastily pulling it free of its mount and showering himself with a spray of plaster. Balancing it in both hands, he swung the unfamiliar weapon in a wide arc, using it to parry the outstretched hands of the mechanical men. It was weighty, and it strained his already exhausted body to lift it properly. Nevertheless, at present, it was all he had to keep the automata at bay.\n\nHe hefted the weapon as high as he could and brought it down heavily upon the chest of the automaton on his left. There was an almighty crash. The wooden handle of the ancient weapon splintered in his hands with the impact, sending the iron head banging loudly to the floor. The automaton staggered backwards for a moment, a large dent in its brass casing, but then, just as quickly, was able to reassert itself and come at him again over the top of the stove. This time, catching him on the backswing, the automaton's hand struck him hard in the arm, and he cried out as the blades sliced his flesh, drawing blood. He snatched his arm back instinctively and managed to scramble out of the reach of the machine.\n\nHe could hardly believe the resilience of the device: the blow from the axe had practically collapsed its chest, even cracking the glass porthole that contained the electrical light that powered its clockwork mind, but the unit seemed unconcerned and continued to mount its attack. Newbury threw the broken shaft of the axe at the other automaton, which knocked it aside to no effect. He knew it was only a matter of time before the machines worked out how to shift Miss Coulthard's desk out of the way to get to him.\n\nNewbury searched the walls for more weapons, thankful now that he had been able to coerce the museum's curator into allowing him to have a small display of anthropological items in his office. A few feet away, over Miss Coulthard's desk and on the wall above the fireplace, was a flail. The weapon was a few hundred years old, but Newbury knew from examining it in the past that the shaft was still firm. He hoped the star-shaped iron ball on the end of the chain would make an effective weapon against the automata, puncturing the relatively soft brass of their skulls and damaging the delicate cogwork in their mechanical brains. It was a blunt tool for a blunt job. He just had to work out how to get to it.\n\nHe measured the distance with his eyes. If he leapt up onto the overturned desk he could be at the weapon in two strides, but equally, he ran the risk of one of the automata catching hold of his leg as he tried to rush by, pulling him to the ground whilst he was unbalanced and sticking him with its vicious claws. He looked over at them. The two machines continued to try to swipe at him from behind the stove. The situation wasn't about to improve, unless he made a decisive move. He had to risk it. There were no other weapons anywhere in reach, makeshift or not, and if he waited any longer the automata would, by sheer relentlessness, find a way to reach him. Jumping up onto the desk didn't seem like a good option, however, especially in his present condition, so instead he decided to see if he could reach the weapon by other means.\n\nStanding, his back to the wall in an effort to stay out of the reach of the questing brass fingers, Newbury edged over towards the chimney breast. Keeping himself as flat as possible, he reached an arm around and used his fingertips to feel for the flail. If he stretched onto his tiptoes, he could just about touch it, but he needed to get past the desk to be able to get a proper grip on the thing. He stared into the impassionate faces of the brass machines, watching their mirrored eyes spinning as they clutched for him, their minds programmed only to kill. If he got out of this alive, Chapman and Villiers were going to have a great deal to answer to.\n\nNewbury surged forward, feeling the blades of both automata impaling the flesh and muscles of his upper arms. Pain blossomed, causing everything to go momentarily white, but he forced himself through it, knowing that this would be his only chance at survival. He hoped Dr. Fabian's compound would continue to work its miraculous healing powers on these fresh wounds.\n\nReaching down, using his momentum to drive himself forwards, he grasped hold of the underside of the desk and flipped it up towards the two machines, connecting with them both at waist height and sending them sprawling to the ground. Not waiting to see how quickly they would be able to get up, Newbury jumped up and grabbed hold of the flail, pulling it down from the display hooks on the wall. He gave it an experimental swing in his right hand, and then, charging forwards towards one of the mechanical men, he arced the ball and chain above his head, slamming it across the side of its skull with as much power as he could muster as it struggled to get up from underneath the desk. The skull split with a dull thud, cracking along the seam between the access plate and the rest of the brass head. Newbury gave a triumphant gasp, trying to free the spiked ball from where it had embedded itself in the inner workings of the machine's head. The damaged automaton kicked spasmodically a few times, its feet clacking on the tiles, and then it was still.\n\nNewbury didn't have time to celebrate. He looked over his shoulder to see the other automaton pulling free of the desk and climbing easily to its feet. He noted it was the unit that he had struck earlier with the axe, and decided to aim his weapon at the glass plate in its chest, tackling an existing weak point in the hope of disabling it faster. He had no idea whether this would have the desired effect, but it had to be worth a try. His arms ached where the gashes in his flesh were weeping blood down his sleeves. He knew he couldn't go on much longer.\n\nNewbury yanked the flail free of the fallen machine, noticing that in doing so, he had exposed something fleshy and wet inside. He didn't have time to look, however, as the other automaton was coming up on him fast. He swung the flail in a wide arc around his head, feeling his shoulder scream in protest as he slammed the weapon against the automaton's chest, shattering the glass plate and causing electricity to arc out into the room in a spectacular display of shimmering blue light. The machine stumbled from side to side for a moment, tottering on its feet, before collapsing to the floor, its brass skeleton still fizzing and crackling with raw electricity.\n\nNewbury dropped the flail and sank to his knees, exhausted. He remained there for a few moments, straining to catch his breath. The electrical current continued to crackle over the destroyed skeleton of the second automaton.\n\nHe looked around the ruination of his office. Miss Coulthard was not going to be happy. He flexed his shoulders, cringing with the pain, and held his arms up before him, cautiously exploring the knife wounds through the fabric of his shirt. They didn't seem so severe as he'd imagined, although the pain was excruciating. He tried to push it to the back of his mind. He looked over at the spilled workings of the machine whose skull he had destroyed. There was definitely something wet and organic seeping out from underneath the brass fittings.\n\nCautiously, Newbury used the edge of the overturned desk to pull himself upright, and tentatively approached the brass skeleton. He prodded it with his foot, making sure that there was no spark of life left inside of it. It flopped lifelessly onto its back. Deciding it was probably safe, he leaned closer, using his fingers to pry the skull open a little farther so he could see inside. He turned the head towards the light. Then, appalled, he dropped the skull to the floor with a loud clatter and stepped away from the gruesome sight, putting his sleeve to his mouth in disgust. His fingers dripped with sticky fluid.\n\nInstead of the clockwork mechanisms that he had been expecting to find inside of the automaton's skull, there was a pinkish-grey, fleshy human brain. Newbury fought back the rising bile in his throat. Then, needing to confirm his suspicion, he retrieved the flail from where he'd discarded it on the floor it a few feet away, and set about splitting open the head of the other unit. A couple of sharp blows later and the skull had given way, revealing the same disturbing sight as the first time: the spattered grey matter of a human organ. He leaned one arm against the wall, trying to process the information. Human organs inside of clockwork men. An airship crash. A series of brutal strangulations in the slums.\n\nSuddenly, a thought began to resolve itself in his mind, the stirrings of a theory taking shape. Wasting no further time, he snatched up his coat from the floor and ran from the office, taking the stairs two at a time, grimacing as his wounds throbbed painfully. He crossed the huge foyer of the museum, hurtled through the main entrance and burst out onto the street, startling a flock of pigeons that had settled in the courtyard. Without pausing, he ran directly to the nearest cab and leapt on board, flinging himself into the seat. The driver leaned down and glanced in through the window.\n\n\"Where to?\"\n\n\"Scotland Yard, as quickly as you can stir those horses into action!\"\nCHAPTER 23\n\n##\n\n\"Charles!\"\n\nNewbury burst into the office of the Chief Inspector and stumbled over to his desk, still dripping blood from the fresh wounds in his upper arms.\n\nBainbridge looked him up and down with an expression of dismay on his face. \"Good God, man. Shouldn't you be resting? Look at the state of you. You're bleeding all over the place. Didn't the Fixer do his work?\" Bainbridge stood, as if he were about to move to Newbury's aid.\n\nNewbury, gasping for breath, staggered across the room and slumped into a Chesterfield beside the fire. \"I'm fine, Charles.\" He wheezed, red-faced from running. \"But I think I have the solution.\"\n\n\"What?\" Bainbridge came round from behind his desk, pushing his spectacles further up his nose. \"Look here, before you start any of that, what's going on with all this blood? Are you hurt?\"\n\nNewbury emitted a gasping laugh. \"A little. I've just fought off two of those automaton devices in my office.\"\n\nBainbridge looked flustered. He repeated himself. \"What?\"\n\n\"It seems we're getting a little too close to the truth. Someone sent two automata to my office in an attempt to assassinate me. They weren't your typical automata, either; they had hidden blades in their fingers, and worse, human brains in their brass skulls.\"\n\nBainbridge shook his head, lowering himself into the other chair by the fire. He reached over to a small table in the corner and took a decanter and two glasses, pouring them both a large brandy. \"I think, Newbury, that you'd better start at the beginning.\"\n\nNewbury accepted the drink gratefully and took a long draw from the glass. He rested his head against the back of the chair. \"What do you know about Pierre Villiers?\"\n\n\"Only what you've told me. That he's a genius. That he was exiled from his own country for experimenting on waifs and strays. That he created the automata for Chapman to market. Nothing more than that.\"\n\nNewbury nodded. \"It's that bit about experimenting on waifs and strays that is interesting me at the moment. What _exactly_ was he doing? What was so bad that his own countrymen had him banished from Paris, renowned the world over as a place of free-thinking and bohemian eccentricity?\"\n\n\"You've lost me.\" Bainbridge raised his eyebrows, shaking his head.\n\n\"No, Charles. I think this has a bearing on our case. Villiers has a fascination with the inner workings of the mind. He told me he's always wanted to build the perfect automaton. What if the device he showed me in his workshop wasn't it? What if it couldn't do everything he wanted it to? Perhaps it was that drive for perfection, and his experiments on those wastrels back in Paris, which provided him with the necessary knowledge to successfully transplant a human brain into a clockwork housing. Perhaps _that_ is his idea of the perfect automaton device?\"\n\nBainbridge looked appalled.\n\n\"I saw it with my own eyes, Charles. I cracked open their brass skulls on my office floor and saw the human organs inside. I think that's why we didn't find the pilot in the wreckage of _The Lady Armitage_. Chapman probably had his man Stokes remove it before anyone else got to the scene. If we'd found it there, we would have taken it away for investigation, and would likely have discovered what they were up to.\"\n\nBainbridge took a swig of his drink, grimacing at the thought. \"But where are they getting the organs from?\"\n\n\"I can't be certain, but I suspect that's where the link to the glowing policeman murders comes in. It all makes a horrible kind of sense. They employ someone to murder paupers in the Whitechapel slums, using strangulation as the method of despatch so as not to damage the brains. Then they make an arrangement with the mortuary attendant to harvest the brains of the victims as they come through the morgue. It's a neat arrangement, however despicable it may be.\"\n\nBainbridge went red in the face. \"I knew that damn mortuary assistant was up to no good!\" He glared at Newbury, obviously incensed. \"So you think the reason for the airship crash is a malfunction in the bridge between the human brain and the automaton frame? Did the pilot simply lose control?\"\n\nNewbury shook his head. \"I can't answer that with any certainty, although I suspect Villiers is far too clever for that to be the case. I don't think it was the interface that went wrong. I think it was the brain.\"\n\n\"You mean they had trouble keeping the brain alive outside of the body?\"\n\n\"Not at all. Think about it, Charles. There's a plague burning through the Whitechapel slums. Remember what I told you about the Indian doctor? The revenant virus incubates for up to eight days in the human brain. God knows how many of those harvested organs were already infected when they were wired up to the automata.\" He paused. \"Judging by the manner in which Christopher Morgan's device went awry, I'd say we are dealing with something far more alarming than a simple malfunction. I think a number of those automata are carrying the revenant plague.\"\n\n\"My God, they're like ticking bombs.\" Bainbridge shook his head. \"But Newbury, they're all over the city.\"\n\n\"I know, Charles. I know. We'll need to enlist the entire Metropolitan police force to aid us in decommissioning the whole lot. But first we've got to tackle Chapman and Villiers. I say we get over there this morning and try to catch them on the hop. They won't yet be aware that their assassination attempt this morning was a failure.\"\n\nBainbridge nodded. \"Very well.\" He eyed Newbury warily. \"Are you sure you're fit?\"\n\nNewbury smiled. \"I'm far from fit. But I'll live.\"\n\nBainbridge downed the last of his brandy. \"What does Miss Hobbes make of all this?\"\n\nNewbury nearly spat his drink across the room. \"Oh, God, Charles. I hadn't even considered. What if they sent the automata after her, too?\" He jumped to his feet. \"We need to get over there now, as fast as we can.\"\n\n\"Right you are.\" Bainbridge placed his empty glass on the table and made straight for his cane. He grabbed his coat from the stand, not even bothering to put it on as he charged out the door. \"Come on. I'll get us a police carriage. We'll be there in no time.\"\n\n\"I pray that's time enough.\" The two men hurried from the room.\nCHAPTER 24\n\n##\n\nKensington High Street was bustling with people by the time the police carriage came hurtling through the traffic, rocking furiously from side to side as its wheels bounced on the uneven cobbles, causing Newbury and Bainbridge to shift uncomfortably in their seats. They had barely spoken a word between them during the short journey from Scotland Yard, each of them choosing to mull over the situation in silence. Newbury, on his part, did not wish to give voice to his obvious concern for Veronica. It was as if talking about the possibility of her being under threat would somehow make the situation more tangible, more likely to become a reality. Instead, he sat clenching and unclenching his fists in nervous anticipation, hoping desperately that his lack of consideration would not result in her coming to any harm. He knew he would not be able to live with himself if it came to that. He cursed himself for being so caught up in his own concerns about the case.\n\nA few moments later, the carriage shuddered and came to a stop. The horses stamped their feet impatiently as the driver tugged on their reins, trying to hold them still. In the back, Newbury climbed to his feet. He was the first through the door, helping Bain-bridge down to the street beside him. He glanced at the door to Veronica's apartment, just a matter of feet away. \"You'd better make sure you have that miraculous cane handy, Charles. If Miss Hobbes is in trouble, we may find ourselves in need of it.\"\n\nBainbridge nodded, and then turned to the driver. \"Wait here.\"\n\nThe driver doffed his cap in acknowledgement.\n\nTogether, Newbury and Bainbridge approached the house. Newbury had taken only a few steps towards the door when he stopped suddenly and waved at Bainbridge to remain still. \"Shhh. Can you hear that?\"\n\nBainbridge listened intently.\n\nComing from the other side of the door was the faint sound of a woman shouting. The words themselves were indiscernible against the background noise of the busy road, but it was enough to send both men into a course of immediate action.\n\nNewbury wasted no time. He charged at the door, using his good shoulder to slam against the wooden panels. The door flexed resolutely in its frame, but didn't give. He tried again, and then, on the third attempt, the lock gave in and the door bounced open, revealing the scene inside.\n\nVeronica was standing in the hallway, her feet planted firmly apart, pointing a glowing poker at the throat of a man in a policeman's uniform. The man, who was tall and well-built, had backed up against the wall, trying to keep the angry woman at bay. It was immediately obvious that he was no real police constable, and what was more, he had painted his face and hands with an iridescent blue powder that shimmered as it caught the light.\n\nNewbury gasped. _The glowing policeman_. The man's uniform was scorched across the front where he had already taken a blow from Veronica's hot poker. They'd clearly been engaged in the struggle for some time, and it seemed that, presently, Veronica had the upper hand.\n\nUnsure what to do, Newbury called to Veronica and rushed forward to help try to pin the glowing policeman. \"Veronica! Be careful!\"\n\nSurprised, Veronica turned to see Newbury charging towards her. The man in the policeman's uniform saw this distraction as a chance to get away and took it without hesitation. He seized Veronica's wrist and twisted it sharply, causing her to cry out and drop the weapon on the floor. Then, giving her a harsh shove that sent her sprawling to the ground, he turned and bolted, flinging himself along the hallway towards the kitchen and the back door.\n\n\"You oafs! I had him pinned!\" Veronica shouted as she picked herself up, frustrated, rubbing at her sore wrist. Newbury, leaving Bainbridge to attend to the lady, took flight after the escaping murderer, barging past Veronica and careening down the hallway in quick pursuit, banging his injured shoulder painfully off the wall as he ran.\n\n\"Oh no you don't!\" He heard Veronica call after him, followed by the sound of her footsteps as she charged after him.\n\nNewbury skidded into the kitchen, throwing his arm out to catch hold of the doorframe and slow himself down momentarily. The back door had been flung open, and the man was scrambling over a wall. Newbury followed suit, darting out into the backyard and leaping up to grab hold of the brickwork. He hauled himself bodily over the wall and dropped into the alleyway behind the house, catching sight of the man doubling back on himself and heading off in the direction of Kensington High Street. Puffing, Newbury picked up his pace, pushing himself to run after the fleeing criminal as fast as his tired, injured body would propel him along. He wasn't about to let his physical condition prevent him from resolving this case, and the glowing policeman was a fundamental part of the puzzle. The man's testimony would be crucial in helping to bring the main players to justice, before he swung from the gallows himself for his crimes.\n\nNewbury, not waiting to see if Veronica had made it over the wall, skidded around the corner into a side street, just managing to keep the uniformed man in view. He charged on, narrowly avoiding a pile of wooden crates that someone had abandoned in the middle of the road and nearly losing his footing on the slick cobbles in the process.\n\nThe other man disappeared between two buildings up ahead. Newbury raced after him, his chest and abdomen screaming in pain. He could feel some of his stitches pulling free as he pushed his body beyond the limit of its endurance. He could hardly believe that only yesterday he had been laid out, dying in the Fixer's workshop, and today he was running through the streets of Kensington in pursuit of a multiple murderer. It was a testament to either the Fixer's miraculous abilities or Newbury's own stupidity. He tried his best to bury the pain as his feet pounded the ground, his entire body shaking with the thudding of his shoes against the hard road.\n\nNewbury burst out onto the busy thoroughfare, glancing in both directions to try to ascertain which way the other man had run. Almost too late, he caught sight of him leaping up onto a passing ground train, snatching hold of the side railing attached to one of the carriages and pulling himself up onto the roof. The long train of interconnected carriages snaked along behind him as it trundled noisily down the road.\n\nNot stopping to consider the risk, Newbury ran after it, launching himself from the pavement and just managing to catch hold of the iron railing that ran around the rear end of the vehicle. He tried to haul himself up, his feet trailing in the road as the vehicle steamed ahead, the driver unaware of his newest passengers. He heard Veronica shouting something from behind him, but he was already out of earshot as the train rumbled on along the road.\n\nGasping, Newbury hoisted himself higher, wedging his foot on the buffer and pulling himself into a standing position, balancing tentatively on the railing. He heard banging and shouting, and looked round to see the people inside the carriage had opened their side window and were leaning out, jeering at him to let go. There was a similar commotion coming from farther up the train, and Newbury reasoned that the passengers had seen the strange blue-skinned policeman leap up onto the roof and were now calling for the driver to stop the vehicle.\n\nBeing careful not to lose his hold, Newbury used one hand to explore the roof of the carriage. It seemed firm, and had a thin lip running around the edge of it that he could use as a handhold to pull himself up. It was the only way he was going to be able to catch up with the man he was chasing, and he didn't want to risk losing him if the devious blighter decided to jump off the train farther up the road to make good on his escape.\n\nNewbury swung his other arm up, finding his grip on the roof of the carriage. He manoeuvred his feet until he could gain some purchase on the railing and then began to pull himself up and over, using his leg muscles as much as possible to avoid pulling on his weak shoulder. After a minute or two, he managed to swing first his chest and then his legs up onto the roof of the carriage. He lay still for a moment, catching his breath and casting around for a sight of his quarry. The roof was mostly flat, with a slight camber to each side to allow rainwater to run off into the street below. Newbury looked over the side. The cobbles rushed by at speed. It wouldn't do to fall.\n\nThe glowing policeman was clinging to another roof, about three carriages farther up the train. He was on his knees and had his back to Newbury, clutching the lip that ran around the edge of the carriage roof. He shifted from side to side with the movement of the train.\n\nNewbury knew that it would be difficult to get closer to the man without attracting his attention, but he also knew that moving quickly would provide him with his best shot at success. If he could get near enough to knock the policeman over the head\u2014he had lost his helmet somewhere during the run\u2014he could potentially disable the man before he even realised that Newbury was there.\n\nTentatively, he pushed himself up onto his knees, trying to work out whether it would be safe enough for him to walk along the roof of the carriage without falling. The train was still trundling along at a reasonable speed, but the road was straight, and as long as they didn't bounce over any potholes, it was worth the risk. Not that he had any other options in mind.\n\nSlowly, he got to his feet, keeping his eyes on the man up ahead. He took a quick step forward, almost stumbled, but managed to keep his balance by waving his arms out beside him. He crept towards the rim of the carriage, looking down at the gap between the roof he was standing on and the next one along in the train. It was at least four or five feet. The ground swept past below. He was going to need a running jump to clear it. If he missed he'd end up caught amongst the hard buffers or tumbling to one side and cracking his head on the cobbled road, or worse, dashed beneath the train's wheels. None of them seemed like a good way to go.\n\nSighing, he edged away from the gap, taking a few steps backwards. He looked around to establish that there were no trailing wires that could inadvertently snare him as he made his dash, and then, with a deep breath, he careened forward and leapt into the air, throwing himself as far as he could towards the next roof in the long line of carriages. He came down with a loud _smack_ , landing on his right side and skittering across the bitumen-covered roof, sliding towards the edge of the carriage.\n\nThrashing around, he managed to get a grip on the lip of the roof, planting his feet as best he could to gain leverage. The landing had knocked the air out of his lungs, so he sucked fruitlessly at the sky, lying on his back, trying desperately to pull himself round. He could hear shouting from the passengers beneath him, panicked by the sudden bang on the roof of their carriage. He wondered how long it would take the driver to start weaving from side to side again, or else bring the vehicle to a halt.\n\nNewbury rolled up into a sitting position. He realised immediately that his attempts at subtlety had been wasted; the noise he'd made leaping across the gap had been enough to startle the man in the policeman's uniform a few carriages ahead. He had not, however, made any move to try to flee, as Newbury had anticipated he might; instead, he had turned to face the Crown investigator, a look of grim resignation in his eyes, as if ready to take him on if Newbury decided to come any closer. As far as Newbury saw it, however, he had no choice but to continue. He wasn't about to be intimidated, and whilst he'd had his absolute fill of combat during the course of the last couple of days, he would do what was necessary to bring the man to justice.\n\nNewbury found his footing and this time didn't stop to ponder the jump. He ran at the end of the carriage, diving over the gap and throwing himself, spread-eagled, onto the roof of the next one in the long train. This time he was prepared for the impact and recovered much faster from the landing, although he felt the wounds in his arms open up again as he grasped for a handhold, warm blood weeping down the length of his forearms. They burned angrily, and Newbury felt like he'd forgotten what it was like to live without pain.\n\nHe looked up, making sure that the glowing policeman hadn't jumped across from the next carriage to meet him. Thankfully, the man had chosen to wait it out on the other roof. He was hovering near the lip of the carriage, his fists ready, his stance set firm. He looked like a prize-fighter, silhouetted against the morning sun. There was no way Newbury would be able to make the jump across to tackle him. If he flung himself over as he had with the other carriages, he'd run the risk of colliding with the man, knocking them both to the ground and their deaths. It was simply too treacherous, and he needed to come up with an alternative course of action as quickly as he could. He moved over to the end of his carriage to stand opposite the counterfeit policeman, swaying slightly with the movement of the train. The gap between them opened and closed as the train bounced over the cobbles, bringing them dangerously close together and then pulling them apart again with every bump and twist of the road.\n\nTheir eyes met. The man scowled angrily, his expression filled with fury and ire. It was clear to Newbury that he was the sort of man who made his living from violence: his face was a patchwork of scars and old wounds, and his nose had been broken on numerous occasions. He was unshaven, and underneath the shimmering blue powder he had painted over his exposed skin, his neck was covered in a string of dark illegible tattoos.\n\nNewbury shouted to him over the noise of the churning engine. \"Look here. There's no way we're both getting down from here alive, unless we choose to do it together. I can help. They'll go easy on you if you cooperate.\"\n\nThe other man grunted. \"You mean they'll give me a shorter rope to dangle from?\" He shook his head. \"Not me. I ain't going willingly to no noose.\" His accent was clipped with the sounds of the East End, his voice a gruff bark.\n\nNewbury nodded. \"So be it.\" He glanced from side to side, looking for anything he could use as a weapon. There was nothing obvious to hand. He shifted slightly as the train rocked forward. The movement brought the two carriages momentarily closer together, and the man took the opportunity to swing out, catching Newbury off guard with a hard fist in his gut. Newbury toppled backwards, clutching at his waist. He used his feet to shuffle back from the edge whilst he regained his composure, keeping his eye on his adversary. The glowing policeman eyed him with a sarcastic smile. Newbury clambered to his feet. He edged closer to the gap once again, his arms drawn up in front of him in readiness. He wasn't sure how much power he'd be able to muster in his damaged shoulder, but he flexed his neck muscles in anticipation and, when the opportunity arose, dashed forward and took a swing at the other man.\n\nAt that moment, however, the carriage veered suddenly to the right, and the gap between the two men widened dramatically as the engine turned a corner up ahead and sent the train of carriages careening out in a wide arc behind it. Too late to stop his momentum, Newbury toppled into the gap, falling between the two carriages. He lashed out, scrabbling desperately to find purchase on anything that would prevent him from falling to the ground. In his panic, he managed to grab the ledge that ran around the roof of the other carriage, his body slamming hard into the rear end of the carriage itself. He held on tentatively by his fingertips, thrashing his feet around beneath him as he tried to find something firm that could take his weight.\n\nThe face of the glowing policeman appeared over the lip of the carriage roof, leering down at him. The man was laughing at his apparent stroke of good fortune. It was almost a comical sight, this human face shining blue in the early morning sunshine. If Newbury hadn't been hanging precariously by his fingertips, he would have laughed out loud.\n\nThe policeman approached the edge of the roof and stamped his boot down hard on Newbury's left hand, crushing his finger-tips painfully against the metal rim. He ground his foot, trying to force Newbury to let go. Newbury could feel the skin shredding from his knuckles underneath the man's roughly shod boot. He cried out in agony, barely managing to keep a hold on the roof. His eyes filled with involuntary tears of pain. The man lifted his foot away for a second, giving the slightest of reprieves, but then smashed it down again heavily, using his heel to force Newbury's fingers away from the edge.\n\nNewbury, blinded by panic, swung out from the end of the carriage, clutching the roof with only one hand. Below, the road was a blur of dark stones that sped past as the train gathered speed and momentum. If he fell, his life was forfeit. Determined to hang on for all he was worth, Newbury tried again to swing his legs onto some footing. This time he connected with the iron buffers and managed to get his feet up onto one of them, sighing with relief as he secured himself against the end of the carriage. He was far from safe, but neither was he about to tumble to a miserable death.\n\nThe other man, not seeing that Newbury had managed to get himself into a position with more leverage, prepared to stomp down on Newbury's other hand. Newbury waited until the man lifted his foot and then swept out with his free hand, grasping him by his ankle and pulling sharply forwards, toppling the man onto his back so that he splayed out on the carriage roof with a considerable _bang_. Newbury used the opportunity to pull himself up to safety as the glowing policeman, dazed from the fall, rolled to one side and scrambled to the other end of the roof in an effort to buy himself time to recover. A moment later, he climbed back to his feet, shaking his head.\n\nWarily, the two men faced each other. The glowing policeman was clearly the bigger of the two, his strength probably far exceeding the academic's, but Newbury didn't have time to ponder the odds. He charged forward, catching the other man off guard and driving his fist up and under his chin. It connected with a _crack_ , and the man staggered back, disorientated. Newbury continued his assault, punching the criminal as hard as he could in the kidneys, trying to bring him to his knees. Unfortunately, the second of these blows had quite the opposite effect than was intended.\n\nLosing his footing, the glowing policeman skidded backwards on the bitumen roof, his feet giving way beneath him as he misjudged the camber and overbalanced. Wheeling his arms like a flapping bird, he fell over the side of the carriage, hurtling towards the cobbles below. Newbury dashed forward, reaching out to try to catch hold of the falling man, but his fingers managed only to graze the collar of the stolen police uniform before the man was gone. There was a sickening crunch as he hit the ground below.\n\nNewbury sucked in his breath and leaned over the side of the train as they hurtled away, straining to see what had become of the glowing policeman. He had to avert his eyes from the scene almost immediately. The man had landed awkwardly on the back of his head, splitting it open on the cobbles like a cracked egg. His body was a twisted pile of torso and limbs, the neck obviously broken, and oily blood seeped from the head wound to stain the stones underneath.\n\nCollapsing back onto the roof, Newbury cursed himself yet again for letting a vital clue slip out of his reach. He felt no remorse for the death of the man who had posed as the glowing policeman; as far as Newbury was concerned, the villain deserved everything he got. Nevertheless, lying there bleeding and shivering on the top of a speeding train, Newbury couldn't help but feel frustration that the whole affair had resulted in nothing, except perhaps the death of a killer who could otherwise have provided evidence against Chapman and Villiers before he went to the gallows. He had to hope that the evidence he had already collected would be enough to condemn the two industrialists in court.\n\nMustering what remained of his strength, Newbury crawled to the far edge of the carriage and shouted down to the driver and guard, both of whom sat in a small cabin atop the main housing of the engine itself.\n\n\"Driver! Time you stopped this bloody train to let me down, isn't it?\"\n\nThe man looked up at the battered and bruised face of Newbury, leaning down over the top of the carriage. He stuttered, unsure how to respond. The guard reached for his truncheon.\n\nNewbury sighed. \"Let me down and I'll show you my papers, man! I'm working on behalf of the Crown.\"\n\nThis was clearly enough for the driver, who applied the brakes and slowly brought the train to a stop, to much shouting and consternation from the passengers. Newbury lowered himself carefully over the edge of the carriage roof, clambering down onto the engine casing and using the fireman's steps to lower himself to the street below. The driver looked him up and down, mystified that a man claiming to work for the Crown should be found in such a diabolical state, crawling around on top of the 9:20 to Marylebone.\n\nThe guard climbed down from the cab and walked around the front of the train, his truncheon in hand. He came to stand before Newbury. \"Papers, you say?\"\n\nNewbury fished his papers out from his inside jacket pocket and waved them at the portly fellow, whose eyes widened at the sight of the Royal Seal. He glanced up at the driver, nodding slowly.\n\nNewbury outlined the situation. \"Now, look here. I have to get back to my associates. You need to alert the police as quickly as you can. There's a dead man in the street back there, dressed as a police constable. His face is painted up to look blue. Tell the bobbies that Sir Charles Bainbridge of Scotland Yard wants the body taken to the morgue immediately. Can you do that?\"\n\nThe man nodded, clearly unsure how to react.\n\nNewbury, shaking his head, had little choice but to rely on the man. \"This is a matter of state importance. Now, go to it!\"\n\nThe guard glanced back at the driver, and then at the carriages full of passengers. He shrugged. Then he ran off in the direction of the dead man. The driver cranked a lever on the front of the engine, allowing steam to hiss noisily from a vent in the roof, and then the train rumbled slowly away, gathering speed and momentum as it did.\n\nNewbury took one last look at the passengers, many of whom were leaning out of their windows heckling him as the train pulled away. Then turned and searched out a passing cab, leaping aboard and directing the driver to make haste in the direction of Veronica's apartment, where he hoped to find both Bainbridge and Veronica herself awaiting him.\nCHAPTER 25\n\n##\n\nThe door was still hanging loose on its hinges when Newbury ducked into Veronica's apartment a short while later. He winced as he walked along the hallway, heading towards the sound of voices that were coming from one of the reception rooms at the back of the house.\n\nHe could hear Bainbridge fussing over Veronica from within. \"Really, Miss Hobbes. I do suggest we call a doctor.\"\n\nVeronica's response was terse. \"Sir Charles, I will not be fussed over unnecessarily. I assure you I am quite well.\"\n\nBainbridge sighed extravagantly. \"Very well. As you wish.\" Newbury could imagine him rolling his eyes in consternation. The conversation lapsed into silence.\n\nNewbury approached the door to the lounge and knocked loudly before entering. Veronica jumped to her feet. \"Sir Maurice! Oh . . .\" Her mouth fell open in slack-jawed amazement when she laid eyes on his bedraggled appearance. She crossed the room, took him by the arm and led him slowly to a nearby chair. Her face was a picture of concern.\n\nNewbury smiled. \"Do I really look that bad?\"\n\nVeronica looked away, refusing to be drawn on the question, but Bainbridge was more to the point. \"You look like you've gone ten rounds with an Indian tiger. Are you badly hurt?\"\n\nNewbury couldn't help but laugh. \"That's the second time you've asked me that today, Charles, and the answer remains decidedly the same: no more than can be expected.\" He shifted in his seat where the leather upholstery was pressing painfully against his wounds. \"I think we'll get today's excitement out of the way, and then I'll be paying another visit to the Fixer, to see if he can't dose me up with some more of that miraculous compound of his. I took a bit of a beating out there today.\" He fell silent, watching the fire gutter in the grate as the others waited for him to go on.\n\nBainbridge pulled at the edges of his moustache impatiently. \"Are you going to elaborate, then? Did you lose him somewhere?\"\n\nNewbury watched Veronica as she made her way back to her seat. He shook his head. \"No. He's very much dead.\"\n\nBainbridge nodded, his face unreadable. Veronica looked aghast. \"What happened? I lost you on the High Street when you jumped aboard the train. I couldn't keep up in this damnable dress.\" She looked down at her torn, dirty skirt with disdain.\n\n\"He scrambled onto the roof of the train. I followed suit, we scuffled and he fell to his death. It's a damn shame. It would have been far more useful if I'd managed to restrain him instead. I would have liked the opportunity to question him about the case.\" He glanced at Bainbridge. \"I left instructions for the body to be taken to the morgue.\" Bainbridge nodded his approval.\n\n\"You fought him on the top of a moving ground train?\" Veronica's voice was strained.\n\nNewbury nodded. \"Indeed.\"\n\n\"What were you thinking of! You could easily have gone over the side with him!\"\n\nBainbridge raised an eyebrow at this outburst from Veronica. \"Miss Hobbes, it is clear to me that you are still suffering from a certain degree of shock, which is only to be expected following the nature of this morning's attack. Perhaps you need some time alone to recover?\"\n\nNewbury smirked as Veronica bit back on her retort. She glanced over at him, her eyes flashing. \"My apologies, Sir Maurice. I did not mean to question your judgement.\"\n\nNewbury gave a half-hearted laugh. \"Oh, but you are quite correct in this matter, my dear Miss Hobbes. It was a rather foolhardy exercise, and one I shall be in no hurry to repeat, I assure you. I've had quite my fill of hand-to-hand combat for the time being. What galls me terribly is the fact that I did not even manage to apprehend the villain for my troubles.\"\n\nBainbridge spluttered. \"On the contrary, old boy! Your actions have resulted in the removal of a major criminal element from the streets of London. You are to be congratulated. A job well done!\"\n\nNewbury shrugged noncommittally. He turned towards Veronica. \"And Miss Hobbes, I assume you are quite well? Were you hurt in your struggle with the man?\"\n\nVeronica shook her head. \"No, I'm well enough, thank you. A little shaken, perhaps. I'm pleased to report that you and Sir Charles arrived before the situation degenerated into actual violence. I should have hated it if I'd found cause to actually use that hot poker on the man.\" She shot a sardonic glance at Bainbridge, who seemed impervious to the witticism, or else was simply choosing to ignore it.\n\nNewbury smiled. \"You certainly seemed to have everything under control when we arrived, Miss Hobbes. I'm only sorry that I had to involve you in this terrible business. If I'd imagined at the outset of this investigation that it would in any way put you in danger, I would, of course, have refrained from including you in proceedings.\"\n\nVeronica sat forward in her seat, clasping her hands together in front of her. She looked anxious. \"Not at all, Sir Maurice. I wouldn't have it any other way. I couldn't bear to be excluded now.\"\n\nNewbury nodded slowly. \"Very well, then.\" His lips curled, as if satisfied that he had done his duty in giving Veronica the opportunity to back out. \"Let us order the events in our minds. Miss Hobbes, can you tell us exactly what happened here? Before I chased the villain from the scene, I mean. It could be pertinent to the case.\"\n\nVeronica sighed. \"I'm not entirely sure, I'm afraid.\" She glanced from Newbury to Bainbridge. \"I was in this room, taking a cup of tea before the fire, when I heard a sound from the hallway. I turned to look just as the man you saw, dressed as the blue policeman, barged in and came at me with his fists. I grabbed the poker from the fire and used it to drive him back into the hallway. That was when the two of you arrived. He must have found his way in through the back somehow.\"\n\n\"What about your housekeeper?\"\n\n\"Mrs. Grant has only just arrived for the day. She's in the kitchen at the moment, searching out a temporary prop for the door. She doesn't begin her duties until half past nine on a Thursday.\"\n\nNewbury sank back into the clutches of his chair. He glanced at the clock on the mantelpiece. \"Do you think Mrs. Grant could find it in her heart to prepare a pot of Earl Grey for two gentlemen in urgent need?\" He glanced over at Bainbridge. \"Charles and I have a great deal to talk to you about.\"\n\nVeronica frowned. \"I am sure Mrs. Grant will be only too happy to accommodate you. But what is it that you need to discuss?\"\n\nNewbury ran a hand over his face, sitting forward in his chair. \"I think we'd better start at the beginning.\"\n\nNewbury recounted his theory to Veronica over a pot of tea, in much the same way as he had explained it to Bainbridge earlier that morning in the offices at Scotland Yard. Veronica nodded solemnly as she took it all in, and Newbury could see by the look on her face that she thought it made a terrible kind of sense, when all the facts were considered alongside each other.\n\n\"So you're essentially saying that Chapman and Villiers organised the glowing policeman murders as a means of obtaining human brains for use in their automaton devices?\"\n\nNewbury nodded.\n\n\"And that you believe the reason that some of the automata have been malfunctioning\u2014thus causing the airship crash, amongst other things\u2014is because a number of those organs were carrying the revenant plague?\"\n\n\"That's about the size of it, my dear. Of course, questioning the man who was posing as the glowing policeman would have helped to establish the link with more certainty, but the clues are all there: the human organs in the automata that attacked me; the blue powder around the throat and collar of the murdered Christopher Morgan, who had previously threatened to expose Chapman and Villiers, the glowing policeman coming after you this very morning. It all fits together perfectly. I suspect if we were able to disinter the bodies of the glowing police-man's earlier victims, we would very quickly be able to establish that the brains had been removed from the bodies. The fact that those organs have all been sourced from the Whitechapel slums, where the revenant plague is rife, coupled with the fact that we know the virus has an eight-day incubation period, suggests that the revenant symptoms might not present until days after the automaton units were sold to their clients.\" Newbury sat back, crossing his legs and taking another mouthful of Earl Grey.\n\nVeronica shook her head. \"It's all in their heads! Ha! I should have realised earlier.\" She sighed. \"It's all in the heads of the automata.\"\n\nNewbury frowned. \"What was that, Miss Hobbes?\"\n\nVeronica met Newbury's gaze. \"Oh, nothing. Something for later, perhaps. It has no bearing on the case.\" She clapped her hands together. \"So, what is our next move?\" She glanced at Bainbridge.\n\n\"Chapman and Villiers. It has to be. As Newbury has already pointed out, the moment they get wind of the fact that their assassination attempts this morning have failed, they'll have to make a run for it. We need to get to them first, if we're not already too late.\"\n\nNewbury shook his head. \"No, they're both as arrogant as each other. Chapman probably thinks he can take us on at our own game, and Villiers, I suspect, doesn't care one way or another. I doubt they'll run. In fact, if we're lucky, they'll play right into our hands.\"\n\n\"And directly into a noose, too, if I have any say in the matter.\" Bainbridge tapped his foot on the carpet, coughing loudly. \"Shall we make haste?\"\n\nVeronica stood. Newbury did the same. \"If I may make use of your bathroom facilities before we leave, Miss Hobbes? I would very much like to wash away some of this blood and grime before making the journey across town.\"\n\nVeronica smiled. \"Of course. Let me show you where to go.\" She led him from the room, showing him along the hall to the small bathroom.\n\nNewbury hesitated before the door. He turned to regard her. \"Thank you, Miss Hobbes. I won't keep you for long.\" He held her gaze for a few seconds, noticing for the first time the pretty smattering of freckles across the bridge of her nose. \"I'm very glad you survived this morning unscathed.\"\n\nVeronica laughed softly. \"And I'm very glad that you survived at all.\" Her voice was barely a whisper, as if she didn't want Bain-bridge to overhear their conversation. She put her hand on his arm. \"When we left you with the Fixer, I . . . I thought I might not see you again.\"\n\n\"I know.\" He looked pained. \"I'm sorry I put you through all of that. I'll be well enough with a little time.\"\n\nVeronica shook her head. \"You have nothing to be sorry for, Sir Maurice! It is I who should be thanking you. Your efforts against the revenant creatures were enough to save all of our lives.\" She leaned over and kissed him gently on the cheek, her lips leaving a cool, damp impression on his skin.\n\nNewbury cleared his throat, embarrassed. \"In that case, Miss Hobbes, after the manner in which you found me in my study the other morning, I do believe we're about equal.\" He offered her a wide grin. \"Now, if you'll forgive me, I really must attend to my wounds. I fear this suit is already beyond saving, but I'd like to give it my best shot all the same.\"\n\nVeronica laughed, this time not bothering to hide her amusement. \"You'll find some fresh bandages in the cabinet beneath the sink.\"\n\nNewbury stepped into the bathroom and clicked the door shut behind him. He listened to the sound of Veronica's footsteps disappearing along the hallway before undressing in front of the mirror, setting the tap running and tending to his raw and bloody wounds. It was only just after ten o'clock in the morning, and already it was proving to be a long, painful day.\nCHAPTER 26\n\n##\n\nThe sun was a watery, baleful eye that glared down at the Thames through a bruised eyelid of rain clouds as Newbury, Veronica and Bainbridge rolled over the Chelsea Bridge in the back of the police carriage, on their way to Battersea and the Chapman and Villiers manufactory.\n\nNewbury watched Bainbridge leaning out of the carriage window, straining to take in the sight of the embankment as it came into view. He followed the other man's gaze. The scene across the river was murky, the mist and rain forming a thick veil across the landscape. The rain had begun to fall not long after they had set out from Veronica's apartment, and the three of them had quickly decided to huddle together in the waiting vehicle. Bainbridge had stopped only to send word to Scotland Yard, requesting uniformed assistance, but they all knew it would be some time before the Yard were able to muster their men. In the meantime, Newbury had been eager to press on, to head directly to Battersea and confront Chapman and Villiers before the two of them realised the police were finally on to them.\n\nNewbury looked up at the dark clouds that were scudding across the sky, brooding with intent. The rain would continue well into the afternoon, if he was any judge of the weather.\n\nAcross the river, the warehouses of Chapman and Villiers were squat mounds of red brick, imposing even amidst the industrial buildings that sat to either side of them. A number of airships were still tethered to the roofs, tousled by driving wind and precipitation. They bobbed fluidly but remained fixed in place by long coils of rope.\n\n\"Impressive, isn't it, Charles?\"\n\nBainbridge turned to look at him, his expression fixed. He nodded. \"Bigger than I had imagined.\"\n\n\"Indeed. Wait until you see inside. The manner in which they construct the new dirigibles is magnificent.\" He allowed his eyes to wander to the floor, biting back his enthusiasm. \"If only they'd contented themselves with that, eh, rather than trying to revolutionise the world with their clockwork men?\" He shook his head.\n\n\"Newbury, people like that will never be content with their lot. What ever they say, it's not about changing the world. It's about wielding power. They may call themselves philanthropists, but in truth they're just as greedy as the rest of us, just as hungry for money and validation. In this case, probably more so.\"\n\nNewbury met his friend's eyes. \"You're right, of course. About Chapman, at least. But I think Villiers is a different matter entirely. I don't see that he's at all interested in money or validation. I think he sees his work as a personal challenge. He has no grand schemes to change the world; he wants only to be left alone to his amoral experiments, as terrible as they are.\"\n\nBainbridge sighed. \"That may be so, but it doesn't alter the fact that together they've committed the most heinous of crimes. There's no redemption to be had here. They're both for the noose.\"\n\nNewbury nodded and leaned back in his seat. He glanced at Veronica, who had been listening to the conversation from her place beside him. She didn't seem to have anything she wanted to add to the discussion and instead turned away, pretending to distract herself with the view out of the window. He wondered for a moment about what she was thinking.\n\nNewbury closed his eyes, lulled by the motion of the carriage. His wounds ached desperately. He hoped that the affair would be over soon so that he could spend a few days holed up in his lodgings, convalescing in his study. For now, though, he had work to do, and he knew that what ever evidence the three of them had at their disposal, Joseph Chapman was not going to willingly accept his fate.\n\nThe cab rolled on, its wheels clicking loudly on the cobbled road as they neared their destination.\n\nThe reception area of Chapman and Villiers Air Transportation Services was devoid of activity when Newbury burst in followed by both Bainbridge and Veronica. Chapman's clerk, Soames, sat in his usual position behind the mahogany desk, his hands forming a thin steeple on the desk before him. He glanced up nonchalantly as the door clicked shut behind the visitors.\n\n\"Ah, good day to you, Sir Maurice.\" The man's eyes flicked over the faces of three newcomers, like a lizard assessing its prey. \"I am afraid that you will find Mr. Chapman is unavailable today. I hope you have not had a wasted journey.\" He offered Newbury a sickly smile.\n\nNewbury turned to Veronica, inclining his head in the direction of the stairs. She grasped his meaning immediately and crossed the room in a few quick strides, mounting the bottom step and starting up in the direction of Chapman's office.\n\n\"Really, Sir Maurice!\" Soames stood, placing his hands on the desk before him. \"I assure you that Mr. Chapman is not here. There is no need to contest my word on the matter.\"\n\nNewbury glared at him but said nothing.\n\nA moment later, Veronica appeared at the top of the staircase and gave a curt shake of her head. Chapman obviously wasn't in his office. Still, Newbury couldn't find it in himself to trust the clerk.\n\n\"Where is he?\"\n\nSoames looked exasperated. \"I honestly can't say. He arrived this morning as usual, took his tea in his office and then went about his business. I haven't seen him for at least two or three hours. He told me to keep his diary free for today.\"\n\nNewbury clenched his fists, exasperated.\n\nBainbridge put his hand on Newbury's shoulder. \"What now?\"\n\nNewbury shrugged. \"Villiers, I suppose.\"\n\nSoames sighed dramatically. \"Gentlemen, without an appointment, I really must insist\u2014\" He stopped short when Bain-bridge raised his cane, leaned over the desk and placed the tip of it against the man's chest, tapping it gently as if weighing how much force he would need to shatter the clerk's breast-bone.\n\n\"Look here. If you have any sense about you at all, you will stop with your insipid drivel and make haste away from this place before you find yourself implicated in affairs you'd rather stay out of!\"\n\nThe clerk looked appalled, then stepped back from the tip of the other man's cane, his legs bumping into his chair behind the desk. He opened and closed his mouth as if unsure how to respond to the threat. \"I . . . oh . . .\"\n\n\"Shut up, man! My name is Sir Charles Bainbridge, and I am a Chief Inspector with Scotland Yard. My colleagues and I intend to locate Mr. Villiers for an interview. You can either assist us by pointing us in the right direction, or you can choose to create a situation for yourself. I fear the latter option will not work out for the best.\"\n\nSoames shrivelled away from the Chief Inspector, clearly terrified by the man. \"I believe you'll find him in his workshop on the other side of the manufactory site, sir.\"\n\nBainbridge nodded and withdrew his cane. The other man sighed visibly with relief. \"Good man. Now, heed my advice and take your leave. I assure you that you do not wish to be associated with this business any more than you already are. As it is we'll need to have you in for questioning.\" He turned to Veronica, who was crossing the room to join them once again. \"Are we set?\"\n\nVeronica nodded.\n\n\"Then come on, Newbury. Lead the way.\"\n\nNewbury shook his head in disbelief. \"You never fail to impress me, Charles.\" He held his arm out for Veronica, fearing that, without her aid, his injuries may soon overcome him. She took it, and together they set off in the direction of the manufactory proper, following the route they had taken during their previous visit, when Chapman himself had been serving as their guide.\n\nThe hangar was suffused with the same biting chill as the city outside of the walls, but at least, Newbury considered, it was sheltered from the wind and the rain. He pulled his overcoat tighter around his shoulders and watched as the others did the same. Below, on the hangar floor, a new gondola was under construction, and the scene was nearly identical to the one Newbury and Veronica had witnessed a handful of days before, although the workmen in this instance were still assembling the basic shell rather than fitting the interior. Newbury leaned over the rail, searching the floor for signs of Chapman. He was nowhere to be seen.\n\nBainbridge approached the edge of the metal walkway, clasping the rail with his left hand. He surveyed the industrious scene below. \"You're right, Newbury, it's a very impressive operation, indeed.\"\n\nNewbury nodded, fighting back a shiver. He knew he'd lost a lot of blood, and consequently he was feeling the cold somewhat more than usual. The bandages and salves he had applied at Veronica's apartment had helped to stem the tide, however, and he was convinced that the worst of it was over. \"Yes, this is where they assemble the passenger gondolas. The next hall is where they build the frames for the main body of the vessel.\" He waved his hand. \"Come on. We have to pass that way to get to Villiers's workshop, anyway.\"\n\nThey made their way along the metal walkway and down onto the main floor of the hangar, where the workmen seemed to ignore their presence entirely, preferring to continue with the task of constructing the gondola. The place was filled with the loud din of industry, and Newbury wrinkled his nose at the smells of oil and scorched wood.\n\nThe next hangar was equally busy, with the skeleton of a vessel being hoisted into place by the pneumatic cranes that ran around the edges of the large room. Bainbridge looked up, clearly impressed, as Newbury led him past the foreman, who was bellowing instructions to the men working the cranes, trying to make himself heard over the noise. Sparks dripped from welding arcs high above them. They edged around the machinery and exited the main airship works, passing along the short corridor that led them out into the smaller room that housed the automaton production line.\n\nThe room was crowded and hot, the steam-driven presses firing noisily as they worked at incredible speeds, pistons pumping furiously as they pushed out the brass components that would be used in the assembly of the clockwork men. A swarthy-looking man in a pair of grey overalls looked up when they entered the room, downed his tools and passed the chest plate of the automaton unit he was working on to another, smaller man who had been assisting him.\n\nHe approached the group of three interlopers, wiping the grime and oil from his face with the back of his sleeve. \"Can I help you?\"\n\nNewbury stepped forward. \"Yes. We have an appointment with Monsieur Villiers. The clerk on the desk in reception sent us through.\"\n\nThe man eyed them warily. \"An appointment, you say? Can I see some identification?\"\n\nBainbridge bustled forward impatiently. He pulled a leather wallet from his pocket and flicked it open, presenting it to the man. Inside was an official badge and papers from Scotland Yard, bearing the crest of Her Majesty. The man looked perplexed, as if he were unsure whether he should let the Chief Inspector and his companions through to see his employer, or why they should even be interested in speaking to the reclusive scientist. Eventually, though, he seemed to come to a decision. He stood aside and waved them at the door to Villiers's workshop with a shrug. \"He's in there.\"\n\n\"Thank you.\" Newbury inclined his head in gratitude and approached the door to the workshop. He didn't bother to knock, instead reaching out for the handle and giving the door a gentle shove. It swung into the room to reveal the same cluttered workbench they had seen before, buried beneath a vast array of components, but no sign of the man they were looking for. Newbury ushered the others through, then closed the door behind him.\n\nBainbridge was frowning. \"Where the devil are those damnable fellows hiding?\" He cast around, trying to make sense of the cluttered workshop. He looked flustered, as if he thought that the two men had somehow managed to get away.\n\nNewbury was just about to respond, when Veronica tugged on his arm. \"Look!\"\n\nHe followed her gaze to where she was pointing. The automaton in the corner\u2014the demonstration model they had seen during their previous visit\u2014was rising out of its chair and moving towards them, its left arm outstretched, its fingers opening and closing like the shining brass pincers of a crab. Its feet clacked on the tiled floor as it walked. Bainbridge, seeing the sinister-looking device making a beeline for him, grabbed his cane with both hands and gave the brass knob a sharp twist to the right. \"Oh no, you don't!\"\n\nThe shaft of the cane began immediately to unpack itself, and now that he had a better opportunity to observe the mechanism, Newbury was even more impressed. Small hinges unfurled at the top of the cane, causing thin brass rods to uncouple from the main shaft of the weapon so that they formed a kind of metal cage around the device. The central column began to spin rapidly, generating sparks of light within the cage itself. There was a sudden flicker, and then blue light arced along the length of the weapon, running back and forth along the conductor rod with a sharp electrical hum, from the handle all the way down to the tip of the shaft.\n\nBainbridge, raising the weapon before him like a rapier, wasted no time. He jabbed the point of the cane towards the chest of the shambling automaton, the sharp tip actually managing to pierce the brass plate and bury itself deep in the heart of the clockwork device. Pulsing electrical energy leapt from the cane into the delicate internal mechanisms of the automaton, which either overloaded the device or caused its delicate clockwork brain to seize. There was a grinding sound from deep within the machine, the stink of burning oil, and then the device gave a spasm and dropped to the floor, rendered useless by Bainbridge's attack.\n\nNewbury stepped forward and leaned over the unit. The blue light that had flickered beneath the porthole in its chest had gone out, and its eyes had ceased spinning.\n\nHe looked up at Bainbridge, who was busy repacking his cane. \"Good show, Charles!\"\n\nBainbridge smiled. \"Now you see why I always endeavour to have the device by my side. One never knows when it may come in handy.\"\n\nVeronica sidled up beside them. \"When you two gentlemen are finished congratulating one another, I have something interesting to show you.\" She stepped away again, crossing the room to where the automaton had been sitting when they first entered.\n\nNewbury couldn't help but emit a short chuckle when he saw the scowl on Bainbridge's face. He joined Veronica by the automaton's chair. \"What is it?\"\n\n\"Here.\" She ran her hands over the wall, demonstrating the thin outline of a door, hidden in the wall behind the automaton's chair. \"I wonder if this is where we'll find our quarry.\"\n\nNewbury put a hand on her shoulder. \"You're to be congratulated, Miss Hobbes. I'll wager this is _exactly_ where our quarry will be hiding. Stand back, won't you?\" He waved the others back from the wall to give himself room to manoeuvre the chair out of the way. Then, returning to the wall, he ran his fingers around the edges of the door.\n\nBizarrely, it appeared to have been cut directly out of the wall, as if someone had simply chopped a section of the wall away and then reattached it on a pair of well-placed hinges. It was decorated in the same dark wood panelling as the rest of the room. Newbury admired the handiwork; it was an exceptional piece of engineering, and if Veronica had not noticed the thin outline of the door, it was likely they would have abandoned their search of the workshop and moved on.\n\nHe ran his hands over it again. There were no obvious switches, handles or triggers in the vicinity. Not knowing what else to do, Newbury gave the door a push and felt it give a little. He pressed more firmly, until there was a clicking sound, and then stood back as the door swung free towards him. He caught hold of it in his left hand as it came towards him, peering cautiously into the brightly lit chamber revealed on the other side.\n\nPierre Villiers stood beside a low mortuary slab in a room that had been fitted out like a hospital surgery. White tiles covered the floor, walls and ceiling, and bright gas-lamps burned with intensity in fixtures situated along each of the walls. A trestle table had been set up beside the slab, holding an array of tools, knives, lenses and other items of surgical equipment, and Villiers himself was stooped over the empty skull of an automaton, preparing to transfer a human brain into the cavity. The organ itself rested beside him on the slab, suspended in a large glass demijohn filled with a yellowish fluid that bubbled effervescently, as if it were connected to an air supply of some sort. The entire set-up reminded Newbury disconcertingly of the morgue: cold, clinical and filled with the overwhelming stench of death.\n\nVilliers did not look up as Newbury, Bainbridge and Veronica filed into the room, their shoes clicking on the porcelain tiles. He was alone, with no sign of Chapman to be found. Newbury cleared his throat. After a moment, Villiers looked up with the briefest of glances, before turning away and continuing with his work. He talked as his fingers danced around inside the automaton's brass skull. \"Sir Maurice. I did not expect to be seeing you again so soon.\"\n\nNewbury laughed. \"I think, Monsieur Villiers, that you did not expect to be seeing me again at all.\"\n\nThe Frenchman shrugged. \"As you say.\"\n\n\"They're not quite so infallible as one has been led to believe, are they, these automata you've created?\"\n\nVilliers reached for one of the tools on the trestle table beside him and began cranking something noisily within the brass head. \"No. But they are beautiful, though, are they not? A wonder of modern science? Do not tell me that you are not intrigued, Sir Maurice, that you are not at least a little bit interested in how I managed to make them work.\" He glanced up, looking at Newbury, although his eyes seemed to be focused on something else that the others could not see. He cleared his throat. \"Here, let me show you what I am doing.\"\n\nBainbridge started forward, brandishing his cane, but Newbury put an arm out to stop him. \"Just a few moments, Charles. It pays to know what we're dealing with.\"\n\nVilliers laughed heartily. \"I knew it!\" He moved around the mortuary table, turning the automaton's head towards Newbury, so that the Crown investigator could see clearly inside the empty skull. There was a short brass spike at the base of the cavity, with four exceptionally fine filaments trailing out from a separate point just below the tip of the spike itself. Villiers put his hand inside the cavity. When he spoke, his voice was full of arrogance and pomp. \"The human organ is placed in this cavity, here, lowered gently onto the brass spike to hold it firmly in place. The wires are then threaded precisely through the cortex until they engage with the sensitive response centres in the left and right hemispheres of the brain. Electrical stimuli, generated by the movement of the automaton device itself, are then fed back and forth along these wires to create a simple neural interface that enables the organ to receive input from the world outside of the machine's casing.\" He clacked his tongue against his teeth. \"I call this my 'affinity bridge,' the device by which my creations may learn to interact with the external world.\" He grinned, as if satisfied that his audience was giving him his due attention. \"Once it is working we pack the rest of the cavity with a preserving jelly to ensure the organ does not degenerate or become damaged if the device is required to make any sharp movements.\" He paused, drumming his hand on the table before reaching for the large glass jar that held the harvested brain. He slid it across the table-top so they could see. Newbury heard Veronica swallow.\n\n\"But what about the original personality, the person whose brain you have stolen? Doesn't that present itself once the organ is connected to this 'affinity bridge'?\"\n\nVilliers practically scoffed. \"We bypass the original personality, of course! Consciousness is simply a by-product of the human organism. It is not necessary for life to be self-aware. It is certainly not necessary for an automaton to be self-aware. In truth, in attaching a human brain to the affinity bridge, I am simply engaging the neural structure of the organ, making use of the existing nervous system and the brain's inherent processing functions. It is a much cheaper and less time-consuming option than building a new component to do the same job, although, as you've seen, the latter is indeed possible.\" He smiled. \"At its most basic level, Sir Maurice, the human being is essentially a machine.\"\n\nNewbury nodded, appalled by Villiers's arrogance and yet somehow still intrigued enough to want understand the elaborate details of the process the man had developed: the melding of man and machine. \"So what went wrong?\"\n\nVilliers glowered at him. \"Nothing! My device functions perfectly.\"\n\nBainbridge, impatient and keen to draw the conversation to a close, decided to speak up at that point. \"Poppycock! What about the airship crash, and all these reports we've had of your machines going haywire?\"\n\n\"The human organs!\" Villiers sounded enraged. \"Joseph brought me faulty organs.\" He banged his fist on the mortuary slab. \"In the early days, I had no mind to enquire where Joseph was obtaining the human brains that I needed for my work. Frankly, I had no reason to care. At least not until some toffee-nosed art dealer began claiming his machine had been exhibiting dangerous and unruly behaviour. I had the machine brought here for testing, and when I opened up the skull cavity, I found the organ riddled with signs of the revenant plague. I asked Joseph where he'd laid his hands on the organs, and that's when he told me he'd engaged a third party to retrieve them from the Whitechapel slums. Of course, by that time, the plague had already begun to spread far and wide, and we had no way of telling which of the devices might already have been affected. We had no choice but to continue.\"\n\nVeronica spoke softly. Her voice sounded remarkably calm. \"So that's why _The Lady Armitage_ went down?\"\n\nVilliers nodded. \"Yes. Joseph had the pilot unit removed from the wreckage before the police arrived. The device was returned to my workshop. The casing was badly damaged by the flames, but there was no mistaking the signs. The brain had practically been reduced to a sponge inside of the brass skull, all malformed and rotten with plague.\"\n\nNewbury glanced at Bainbridge before stepping forward towards Villiers. \"If the technology had developed in different circumstances, without the need to resort to murder, you would be heralded as a genius, Monsieur Villiers. I'm ashamed to say that the path you have taken in this instance, however, has reduced you to nothing but a common criminal.\" Newbury put his hand on the automaton's head to hold it still. \"You do understand that you're going to have to come with us?\"\n\nVilliers nodded slowly. \"May I just\u2014?\"\n\nThere was a terrifying _bang._\n\nThe sound seemed to reverberate around the entire room. Villiers slumped to the floor, blood streaming from a bullet hole in his forehead, just above his right eye. The white tiles on the wall behind him were spattered with a bright spray of blood and brain matter. Veronica screamed. Newbury spun around on his heel to see Chapman framed in the doorway, clutching a revolver that he turned to point directly at Newbury's face. Smoke curled in lazy curlicues from the end of the discharged barrel.\n\n\"Never could keep his mouth shut, the arrogant bastard.\" Chapman flicked his hair away from his face, eyeing the three of them carefully. Veronica shifted slightly, and Chapman waved the gun at her. \"Not a single move, Miss Hobbes, or your beloved Newbury gets a bullet in the head, just like poor old Pierre.\" These last few words were delivered with a nasal sneer. He took them all in with a sweep of the barrel. \"Now we're going to do things my way.\" He indicated with his head. \"Newbury. Over there, with the girl.\"\n\nNewbury eased himself around to stand beside Veronica. \"Whatever happens today, Chapman, this is going to follow you. You can't keep running forever.\"\n\nChapman shook his head. \"Oh, please. Don't patronise me, Newbury. You really should know better than that.\" He turned to Bainbridge. \"You. Old man. Your turn next. Get over there and join them in the corner.\" Bainbridge turned slowly towards the industrialist. He made a cautious step towards Veronica and Newbury, then altered his momentum at the last moment, whipping up and out with his cane and connecting hard with Chapman's outstretched wrist.\n\nThere followed a brief moment of chaos when, for Newbury, the world seemed to suddenly stop. It was as if the whole scene had been cast into silence. The revolver went off, sending a bullet ricocheting off the tiled walls and causing Newbury and Veronica to duck involuntarily to avoid being hit. Chapman let out a howl of pain and clutched at his wrist, letting the revolver fall to the floor so that it skittered across the tiles towards Villiers's corpse. Bainbridge readied himself to strike another blow.\n\nThen reality came crashing back in, and Chapman, reacting faster than the others, turned and ducked out of the doorway, leaping over the skeletal frame of the ruined automaton and fleeing the workshop as quickly as his legs would carry him. Bainbridge stooped to retrieve his revolver.\n\nNewbury and Veronica looked at one another, and then, making up their minds at exactly the same moment, they gave chase, each of them sprinting out of the door in pursuit of the fleeing criminal. Bainbridge was quick to follow, hefting the gun in his right hand.\n\nBehind them, the corpse of Pierre Villiers stared unseeing through the open door, his jaw slack with death, blood pooling around the exit wound at the back of his splintered skull.\nCHAPTER 27\n\n##\n\nNewbury was the first out of the door. He charged after Chapman, throwing himself around the edge of Villiers's workbench and out into the main automaton production facility. The presses were pounding noisily, pistons firing in quick succession and clouds of steam hissing into the air, obscuring large swaths of the factory floor from view. It was obvious the men working the machines had not heard the gunshot over the racket of the production line, and none of them showed any signs of having noticed Chapman racing through the facility, either. If Newbury didn't find him quickly, the industrialist would be able to lose himself in the factory with ease.\n\nGlancing frantically from side to side, Newbury finally caught sight of the man darting out through a side door in the far wall, which led to the river outside. He followed swiftly behind Chapman, his entire body protesting at the strain as he dodged around the machines, nearly slamming into a man who was lifting a partially assembled automaton frame from a conveyor belt. The worker cried out as he ducked out of the way, sending an array of components clattering to the floor. Newbury kept chasing Chapman towards the exit on the other side of the factory floor.\n\nThe door was still swinging to and fro as Newbury burst through, skidding to a halt on the other side just in time to prevent himself from careening forward into the river. He planted his feet in the muddy bank. The water churned furiously a few feet from where he had come to a stop. Outlet pipes jutted rudely from the factory wall, spewing brown sludge into the river.\n\nThe weather had deteriorated even further since their arrival at the manufactory, and rain lashed at Newbury's face in the driving wind. He cupped his hand to his eyes, trying to work out what had happened to Chapman. Surely he couldn't have thrown himself into the river? There was no sign of the man in the water, or of any boat that he may have kept berthed here for such an occasion. Of course, if Chapman _had_ gone in, he might already have drowned, given the fierce weather.\n\nThere was a scuffing sound from behind him. Newbury felt his hackles rising. He spun around to see Veronica coming out of the factory through the door he had just used himself. He offered her a slight shrug, but the gesture was lost as he hunched against the wind and the rain. He glanced along the length of the building, trying to work out where the other man had managed to flee. It was then that he noticed a cast-iron ladder had been bolted to the wall, just to the left of the exit, beside one of the main outlet pipes. He looked up, turning his face towards the grey sky as he tried to make out where it led. The ladder ran all the way up to the top of the building, disappearing from view where it curved over the lip of the factory roof.\n\nJoseph Chapman was edging his way up the wet rungs, pulling himself up the metal frame towards the roof, where, Newbury realised, an array of newly built airships awaited him. Clearly that was how Chapman intended to effect his escape. He was already about halfway towards his salvation. The wind was blowing him awkwardly from side to side as he climbed, his hands slipping on the slick rungs, but despite the obvious danger Newbury knew that he couldn't risk letting the man get away. If he made it to one of the airships, he could be halfway to the Continent within a couple of hours. It wouldn't take much for him to lose himself from there, disappearing into one of the darker corners of the Empire or, worse, to Asia and beyond.\n\nNewbury turned to Veronica, trying to make himself heard over the rattling wind. \"Get back inside. Wait for me in there.\" He pointed towards the door, where Bainbridge was standing, framed like a silhouette in the doorway. Then, without waiting to hear or acknowledge her response, he leapt up onto the bottom rung of the ladder and began to climb.\n\nThe going was treacherous. The wind dragged at him as if it were trying its very best to prise him free of the ladder. The rain had caused the metal rungs to become wet and slick, and the downpour continued to needle at his face, stinging his eyes and making it difficult to see. Within minutes, his clothes had soaked through, and he shivered as he hauled himself upwards, clattering after Chapman on the ladder as fast as his damaged, aching body would carry him. The side of the factory was terribly exposed, and Newbury tried not to think what would become of him if the wind did manage to throw him from the ladder. In all likelihood, he would be dashed on the ground below, or else blown out into the river and a watery grave.\n\nIt was clear from the way in which Chapman had slowed that he was tiring as he approached the top of the building. Trying to ignore the burning pain in both shoulders, Newbury pressed on. He was closing on the other man, slowly but surely. He knew he couldn't allow his ailing body to slow him now.\n\nHe watched through squinting eyes as the industrialist reached the lip of the roof and threw himself bodily over the top of the ladder, disappearing temporarily from view. A moment later Newbury did the same, hauling himself over the top of the ladder, swinging his legs around underneath him and landing heavily on his rear atop the tiled roof of the factory. He gasped for breath. The wind was howling amongst the chimney stacks, and a confusing web of ropes strained against the pull of the bobbing airships, which filled the sky overhead like a blanket of glittering clouds. He searched the rooftop for a sign of Chapman. About thirty feet away, the industrialist, soaked to the bone, his long hair now lank and slicked to his face, had just finished loosening the tether on one of the airships, and was busy climbing aboard. Newbury watched him mount the short flight of wooden steps beside the iron berthing ring and step across to the gondola, watching his footing as the airship listed dangerously from side to side in the wind.\n\nNewbury was unsure whether Chapman had even realised that he had been followed this far; he appeared to have an almost casual, nonchalant air about him. Newbury hoped that it would be this that would prove to be his undoing, allowing the Crown investigator to gain the element of surprise.\n\nNewbury got to his feet and charged after the other man. He scrambled up the wooden steps and flung himself towards the open door of the gondola, just as the vessel banked awkwardly to the left, buffeted by a wild gust of wind. He slammed into the side of the vessel, his hands questing frantically for purchase, one of them catching hold of the threshold at the base of the door, the other slipping dangerously free of the wet doorframe. The airship began to drift away from its berth, pitching and groaning as it rocked back and forth in the harsh wind.\n\nNewbury dangled from the doorframe, buffeted by the wind as he tried desperately not to fall. He thought he heard the sounds of someone shouting from below, but the driving rain and his precarious position meant he had no time to pay heed to whatever was going on down there on the rooftop.\n\nWith a huge effort he swung out against the harsh wind, the fingers of his free hand catching hold of the doorframe. He clawed to find a more substantial grip. His fingers caught on something firm and loose; the wooden rung of a rope ladder. He tugged on it, gasping in relief as it came free from its housing just inside the door. He pulled it near, allowing the ladder to unfurl, flapping away beneath him as the airship drifted across the rooftop, narrowly missing another large vessel that was berthed beside it. He fought to get his feet on one of the rungs. Rain thrashed over his back and he cried out as more of the stitches in his abdomen tore free with the strain. His shoulder burned.\n\nFighting against the pain and fearing he might blow free at any moment, Newbury jammed his feet into the rope ladder and hauled himself up, rolling into the open door and collapsing onto his back, just as the vessel banked again. He was soaked to the skin, his clothes wet through, and blood was running freely from any number of wounds that had torn open during the climb.\n\nThe door was still open behind him, the rope ladder dangling free over the rooftop. Rain blew in on every gust, spattering the inside of the foyer with water. Chapman was nowhere to be seen.\n\nPanting for breath and grimacing with the strain, Newbury climbed to his feet, catching hold of a sideboard that had been anchored to the deck just inside the foyer of the vessel. He found his footing as the airship righted itself once again. He brushed water away from his eyes. The rain drummed noisily against the wooden panels of the gondola.\n\nSuddenly, the airship bucked wildly as the engines kicked in with a high-pitched whine. Newbury grasped hold of the sideboard to stop himself from falling, drawing ragged breaths as he held himself steady. The vessel powered out over the river, away from the manufactory.\n\nTired, hurt and unsure whether he had enough energy left in him for the fight, Newbury turned and set off down the passageway, towards the cockpit, Chapman and\u2014he hoped\u2014the end of the affair.\nCHAPTER 28\n\n##\n\nThe door to the cockpit was shut when Newbury finally made his way along the passageway to confront Chapman. The engines hummed noisily and the vessel had righted itself, even though it still shuddered disconcertingly with the to-and-fro of the wind. Now it was climbing in altitude, rising high above the factory and the city below.\n\nNewbury was near exhaustion and anxious to get Chapman into custody. He knew the man had lost his firearm back at the factory, and suspected that he would not have hidden a replacement aboard a brand new airship, a vessel that could have only been completed by his factory a handful of days before this, its maiden voyage. Nevertheless, it was a gamble. Newbury knew that he was far from his physical peak, and whilst Chapman was a dilettante and a fop, he was also unscrupulous and cunning. Newbury only hoped that he still had surprise on his side. Readying himself, he reached out, took the door handle, and gave it a sharp twist. He stepped back and allowed the door to swing open towards him. It clattered against the wall of the passage.\n\nChapman sat at the controls inside the small cockpit, his hands dancing over the vast array of levers, buttons and cranks that adorned the panels before him. Above, dials were set into a polished wooden dashboard, showing altitude, speed and fuel levels. Beyond that was the viewing port: a series of large reinforced glass windows that offered a vast panoramic view of the city below, a kind of surreal bird's-eye perspective of the landscape that Newbury had never been granted before. The Thames wound away into the distance, whilst nearby the factories and industrial buildings of Battersea pumped ribbons of steam into the air. Farther afield, the City of Westminster was like a jewel amongst the rows of closely built houses: proud buildings and public parks, museums and parliament. The city glittered in all its majesty, whilst all the while, the storm clouds formed a dark, brooding vault across the sky.\n\n\"Pretty, isn't it, Sir Maurice?\" Chapman laughed gently underneath his breath as he spoke. \"I often like to come up here\u2014when the weather is better, admittedly\u2014to take in the view of the city. London really is an amazing place to call home. The hub of the modern world. I shall be sorry to have to leave.\"\n\nNewbury stood in the doorway. \"Why don't you take the ship down, Chapman? There's nowhere left to flee. If you come quietly now, we can make it easier on you.\"\n\nChapman laughed, louder this time, and shook his head. He turned in his seat to eye Newbury. \"You know it never works like that, Newbury. Villiers was a fool, for all his genius. He would have walked willingly to the noose. Not me.\"\n\nNewbury clenched his fist by his side, knowing well what was likely to come next. \"Then I'm afraid we find ourselves at an impasse.\" He crept forward, ready to make a move.\n\nChapman got to his feet, careful to keep his pilot's chair safely between the two of them. He smiled slyly. \"Indeed we do.\" He lashed out as he spoke, sending his fist flying towards Newbury's face. Newbury ducked quickly out of the way, feeling the fist brush his cheek, ever-so-narrowly missing its target. He thrashed back at the other man, connecting hard with his sternum and causing him to stagger backwards, banging against the control panel. It wasn't a graceful move, but it was certainly functional.\n\nChapman shook his head, disorientated, and then quickly regained his composure. He straightened himself and stepped away from the controls. The airship juddered, and both men realised at the same time that Chapman's fall had in some way knocked the controls out of line. Chapman glanced at the panel, and Newbury took the opportunity to pounce, coming at him hard, his fist slamming brutally into Chapman's abdomen. Chapman buckled, gasping, but sent a blow of his own into Newbury's gut as he doubled over.\n\nNewbury fell back against the doorframe, jarring his shoulder painfully. He wrenched himself about to face Chapman, and the sharp movement finally proved too much for the Fixer's handiwork. He felt his stitches giving out and blood began to gush from the long wound in his side. His vision swam, and the world was momentarily limned in blackness. He sank to the floor, clutching his abdomen in agony.\n\nIt took Chapman only a moment to realise what had happened, and he swept in on Newbury, taking full advantage of the other man's wretched condition. He struck the Crown investigator with a brutal backhand across the face, sending him sprawling to the floor, his cheek smarting from the impact. Newbury coughed blood onto the floorboards in a sickly stream. Chapman laughed. He drove a booted foot hard into Newbury's stomach, taking the wind out of him and leaving him gasping in pain and shock. Newbury tried to roll away, to find a means to get himself upright again, but the passageway was too tight, and his body protested. He simply couldn't muster the energy to move, no matter how much his mind screamed at his legs and arms to react. He was trapped in the narrow passage, with nowhere left to escape the other man's assault.\n\nChapman circled him, taking the opportunity to gloat. He stepped over Newbury's prone form, turning him over with his boot like some common animal found dead by the roadside. He spat at Newbury, and then set about pummelling him with a series of vicious kicks, punctuating his words with powerful outbursts of violence. \"What you don't seem to understand, Newbury, is that the sort of people who would benefit from the work Villiers and I were doing couldn't give a hoot about the loss of a few peasant lives, especially if it ends up making their own lives more comfortable. There'll be no public outcry. There'll be no noose. Her Majesty herself will probably give me a medal for my services to the Empire!\"\n\nNewbury groaned, but couldn't find enough of a pause in the beating to emit a response. He brought his knees up to his chest in an effort to protect himself from the constant rain of blows. His side felt warm with spilling blood.\n\n\"I suppose I'd better throw you\u2014\"\n\nThere was a dull, wet thud, and then the kicking ceased. Newbury peeled open his eyes to see Chapman collapse to the floor. The industrialist banged his head against the wall as he fell, crumpling to a pile beside Newbury on the floorboards. Newbury looked up through one bruised eyelid.\n\nVeronica stood in the passageway, a large copper fire extinguisher clutched in her hands. She must have made it onto the rope ladder before the airship drifted out over the river. She looked bedraggled, her dress torn and wet, her hair flung back messily over one shoulder. To Newbury, however, she looked like a vision of Heaven itself.\n\n\"Thank you.\" His voice was a wet, warbling croak. He coughed and vomited more blood onto the floor beside him.\n\n\"Don't thank me, Maurice. Just get up and help me fly this thing. If you hadn't realised, we're tumbling out of the sky like a dead weight. If we can't find a way to land the ship, all of this will have been in vain anyway.\" She dropped the fire extinguisher noisily to the floor. Newbury groaned and put his hand against the wall in an effort to raise himself up. His hand slipped, leaving a dark smear of blood across its pristine white surface.\n\n\"I'm going to need a little help getting up.\"\n\nVeronica looked pained, but her resolve was steely. She bent low over Chapman's unconscious body and grasped hold of Newbury's hands. Placing her feet against the far wall, she heaved him up into a sitting position. From there he was able to use the doorframe as leverage to pull himself upright. He staggered to the controls, unsteady on his feet.\n\nVeronica followed behind him. \"Where do we start?\"\n\n\"I have no idea.\" He slumped into the chair and grabbed hold of two levers that he hoped controlled the steering paddles on the underside of the vessel. He looked out through the viewing port. His vision swam. The city was coming up fast to meet them. They were set into a dangerous spiral, blown from side to side by the sharp winds, and he wondered if he was already too late to make a difference. The best option he could see at this stage was to try to steer the vessel towards the dark smear of the Thames. At least that way they'd be able to ditch it in the water without turning the whole ship into a blazing inferno. At least he hoped that would be the case. He'd never even _been_ on an airship before, let alone tried to land one in a river.\n\nDriven on by the image of the burnt cadavers he had seen in the wreck of _The Lady Armitage,_ Newbury tugged hard on the levers, throwing his weight behind them as he attempted to right the vessel from its dangerous collision course. The engines coughed with the strain and the dials on the dashboard were all flickering in the red. If the engines were to get too hot, they would run the risk of explosion, which in turn would ignite the balloon of hydrogen above them. He glanced out of the viewing port to see the city screaming towards them. He knew the engines would be no good to them now anyway. He reached over and flicked the switch on his right, cutting the power. Immediately, the whine from below them ceased.\n\nVeronica rushed forward. \"What are you doing?\"\n\n\"Trust me.\" He stood, leaning as hard as he could on the steering levers. Through the viewing port he could see the nose of the vessel edging up against the harsh wind, but his ministrations were having little effect on the terrifying rate of their descent. He hoped beyond hope that the water would help to cushion the blow.\n\nThe airship dived into the Thames, spinning onto its side as it came down, first glancing off the surface like a skipping stone and then dipping down into the water, sending a vast wave ahead of itself as it slowed to a halt. The balloon bobbed on the surface of the river, whilst the gondola, not designed with any buoyancy in mind, quickly began to take on water, pulling slowly towards the bottom of the river.\n\nNewbury, powering on through adrenaline alone, scrambled up from the controls, frantically searching to ensure Veronica was unhurt. He found her draped over the back of the pilot's chair, where she'd braced herself during the landing. He put a hand to her cheek, tenderly. \"Come on. I hope you can swim?\"\n\nShe nodded. \"Help me with Chapman.\"\n\nNewbury looked down at the industrialist, who was still unconscious, even with the stirrings of river water beginning to lap at his upturned face. He nodded. \"If we must.\"\n\nGroaning, he reached down and hooked an arm under Chapman's shoulder. Veronica did the same, and together they hauled him towards the exit. The passageway was taking on water more quickly than either of them could have imagined, and by the time they reached the gondola's main exit they found it was easier to swim, dragging Chapman along behind them. Thankfully, the door had buckled and sprung open during the landing, so it was a simple matter to navigate out of it one at a time, passing Chapman through between them. The river water was ice cold, and with the loss of blood, Newbury was beginning to feel faint, his muscles starting to seize up. He kicked furiously, resolved that he wasn't going to fail Veronica now, not when they were so close to safety.\n\nThe wind and rain were still pounding when he eventually got free of the airship. Linking arms with Veronica to form a platform for Chapman, they made for the riverbank as quickly as they could. It was only a matter of minutes before Veronica was able to haul Chapman out onto the slick mud of the bank. Newbury, barely conscious, bobbed for a moment in the ice cold water, his body finally giving up on him. He felt the blackness closing in. His head sank beneath the surface. The cold seemed to penetrate him to the core.\n\nThen, suddenly, he was on his back, and Veronica was leaning over him, checking he was still breathing. She dragged him further up the bank, towards safety. The rain still pounded his face, the mud wet and clinging beneath his head. Stars were dancing before his eyes, and in the distance, out in the river, he could see the outline of the airship, drifting with the current and blown about by the wind. He heard the sound of hurried footsteps behind him. He didn't bother to look round.\n\n\"We made it. We made it Maurice!\" She put her hand on his chest and collapsed next to him on the riverbank. Her breath was shallow. \"Stay with me now. The police are on their way.\"\n\nEverything went black.\nCHAPTER 29\n\n##\n\nVeronica's feet crunched on the gravel as she strolled slowly up the path towards the asylum. The inclement weather had finally broken during the night, the wind and rain receding to leave behind a cold, dry morning that, Veronica considered, was far more typical of the season than the storm weather of the previous two days. She breathed in the fresh air, filling her lungs. It was crisp and filled with the promise of winter. Unconsciously, as if affected by the thought of the changing seasons, she pulled the collar of her thick overcoat up around her throat to stave off the chill. Her cheeks felt pinched with the cold.\n\nUp ahead, Veronica could see that many of the asylum's patients were out taking their exercise on the airing courts, small groups of them clustered around the grounds, sheltering beneath the spindly autumnal trees or else strolling round in concentric circles like caged animals searching for a means of escape. The nurses watched with beady eyes and tired expressions from their usual perch beneath the sheltered stone archway.\n\nVeronica searched the scene as she walked, looking for signs of Amelia. Her sister was nowhere to be seen. She jammed her hands into her coat pockets and approached the main building. As she brushed past the two nurses on guard duty, Veronica noticed a young man sitting on a wooden bench beside the asylum wall. He looked uncomfortable in his rough woollen clothes, and his face was ruddy with the cold. He was unshaven and unkempt\u2014haggard, even\u2014but for some reason he looked familiar to Veronica. She racked her brain but found she was unable to immediately place him. Perhaps it was just a case of over-familiarity; she may have seen him during a previous visit to the asylum and the sight of him had lodged itself somewhere in the back of her mind.\n\nThe man turned to look at her as she strolled past the end of his bench, and despite herself Veronica felt struck by the haunted look in his eyes. He smiled unconvincingly when he saw her looking, and then turned away, contemplating the gravel path as if it held all the secrets of the universe. Feeling a slight sense of unease, Veronica continued on her way, stopping once to glance over her shoulder at the young man. She had the vague sense that, somehow, there was more to him than immediately met the eye. Unsure what else to do, she tried to shake off the notion. She made her way under the archway, through the small courtyard on the other side, and entered the asylum through the main door, which the nurse on the door unbolted for her with a wary glance.\n\nOnce inside, she approached the reception desk a few feet to one side of the entrance. The nurse behind the desk looked cold in her thin uniform, and she shivered noticeably when Veronica came through the door, bringing with her a cold draft. Veronica cleared her throat. \"I'm here to see my sister, Amelia Hobbes.\"\n\nThe nurse smiled. \"I'm afraid visiting hours have finished for the day. You may have noticed that the patients are currently engaged in their daily round of exercise outside.\"\n\nVeronica nodded. \"Indeed. Although I fear my sister was not to be found in the grounds. I wonder\u2014\" She paused, trying on her best conspiratorial expression. \"\u2014could you bend the rules just a little? I'm very anxious to ensure my sister is in good health.\"\n\nThe nurse was about to answer when Veronica heard footsteps behind her, and looked round to see Dr. Mason approaching along the hallway.\n\nHe smiled warmly as he drew up beside Veronica. \"It's alright, Nurse Willis, I think on this occasion we can make an exception.\" He indicated for Veronica to walk with him, and they set off together along the corridor, their heels clicking loudly on the hard, white tiles.\n\n\"Thank you, Dr. Mason. It's just that it's been a few days since my last visit, and I'm anxious to ensure my sister is well.\"\n\nThe doctor offered Veronica a grave look. \"I'm afraid you'll find your sister in ill health, Miss Hobbes. The frequency of her episodes has increased markedly over the course of the last few days, with the most recent occurring just over an hour ago. Try not to show your concern when you see her. She's looking very gaunt and tired.\" They marched along the corridor for a moment, passing a number of empty wards and rooms on either side.\n\nVeronica nodded. \"Very well. I appreciate you taking the time to talk with me, Dr. Mason.\"\n\nThe doctor smiled. \"I want only what is best for your sister, Miss Hobbes, contrary to what you may have believed about my methods in the past.\" He came to a halt before a door, which led off into a side room. It was painted a drab grey and had a small window set into it at about head height.\n\nVeronica peered inside. Amelia was sitting in a wheelchair in the small room, sunlight streaming in through the window. Beside her, a nurse sat in a folding chair, reading a book. Amelia's face was turned away from the door, but Veronica could see immediately that her skin was a deathly white.\n\nDr. Mason pushed the door open and ushered Veronica through. Amelia looked up to see who had entered and the nurse trailed off her reading, smiling at the sight of the visitors. Amelia's face lit up when she saw Veronica.\n\n\"Veronica! How lovely.\" She looked up at Dr. Mason. Veronica tried to hide her dismay at the sight of her sister. \"May we sit and talk, Dr. Mason?\"\n\nThe doctor nodded. \"Indeed. I believe some time with your sister will do you well, Amelia.\" He beckoned to the nurse. \"I shall return in a little while, and then it will be time for your rest.\" He glanced at Veronica, and then turned away, holding the door open for the nurse to leave before him. The door swung shut behind them.\n\nVeronica glanced around the room. The furnishings were sparse, but not unpleasant. It was obviously some sort of day-room, a place for patients to come when they weren't well enough to join the others on the airing courts outside. The very fact that Amelia was here, instead of enjoying the fresh air, did not bode well for her overall health. Veronica looked at the spine of the book that the nurse had placed on the coffee table. \"Jane Austen, eh? I'd have thought the library here would be full of far more turgid fare than that!\"\n\nAmelia smiled. \"Oh come here and give me a hug, sister! It's so good to see you.\"\n\nVeronica did as she was bade, taking her sister gently in her arms and kissing her lightly on the cheek. She cupped Amelia's face in her palms for a moment, looking her up and down, and then set about rearranging the blanket on her knees.\n\nAmelia slapped her away. \"I'm not an invalid, Veronica!\" She smiled. \"At least, not yet.\"\n\nVeronica lowered herself into the chair beside her sister. \"Oh, Amelia, what am I to do with you?\"\n\nAmelia shrugged. \"I had thought I might be getting out of this dreadful place, but now I'm not so sure. The episodes have been getting more and more frequent, and Dr. Mason is clearly concerned for my health.\" She laughed. \"But then I suppose he's told you all of that already, hasn't he?\"\n\nVeronica nodded. She didn't know what else to say. She searched Amelia's face for a moment. \"You were right, you know.\"\n\n\"What about?\"\n\n\"About everything.\" Veronica sighed. \"Everything you saw in your visions. It all came to pass. The airship crash. The automata. 'It's all in their heads,' you said to me, over and over again. 'It's all in their heads.' \" She shrugged. \"It was, too.\"\n\nAmelia looked puzzled. \"What _are_ you going on about?\"\n\n\"Your visions, of course. Don't you remember?\"\n\nAmelia shook her head. She gazed at the floor. \"We've talked about this before. I don't remember most of what I see during my seizures.\" She folded her hands on her knees, fidgeting awkwardly with her fingers. \"I'm sorry.\"\n\nVeronica shook her head. \"Don't be.\" She paused, her brow furrowed. \"I've got to help you somehow, Amelia. I'm going to talk to Sir Maurice, see if we can't find a better way to keep you well. There must be something we can do.\"\n\nAmelia looked up. \"How is Sir Maurice? After your last visit I was concerned. . . .\"\n\nVeronica smiled. \"He's fine. Well\u2014he's recovering. He had quite an ordeal, if truth be told. We all did.\" Unconsciously Veronica turned her arm over on her lap and rubbed at her sore wrist. Amelia looked appalled.\n\n\"Veronica! Look at those bruises. Are you quite well? What the devil have you been up to?\"\n\nVeronica quickly covered her arm in the folds of her dress. \"It's nothing. I'm well enough, thanks to Sir Maurice, anyway. He saved my life.\"\n\nAmelia grinned. \"Quite the hero, isn't he? Do tell.\"\n\nVeronica blushed. \"That's enough of that. Now, tell me, are you getting enough to eat? You're still so painfully thin.\"\n\n\"Stop avoiding the subject, you terrible sister! You can't tease me like this! You know it's bad for my constitution.\" Amelia beamed.\n\n\"Then what will I have to tell you about on my next visit? At least this way, I can offer you something to look forward to.\"\n\nAmelia laughed. \"I suppose that's true, at least.\" She put her hand on Veronica's arm. \"You must reassure me that you're looking after yourself out there, though. It wouldn't do for our parents to end up with _two_ sick daughters, now would it?\"\n\nVeronica sighed. \"All is well, Amelia. If you must know, I've had rather a thrilling adventure. And yes, you're right. Sir Maurice _is_ rather a hero, after all.\" She laughed and looked out of the window, watching the trees blowing back and forth languorously in the breeze. \"I'm not sure yet how I'll be able to go back to my desk at the museum after the excitement of the last few days. It all feels a little mundane at the moment.\"\n\nAmelia smiled knowingly. \"Oh, I suspect there's more adventure to come, Veronica. You always were the headstrong one. I can't imagine you'll be behind that desk for long.\"\n\nVeronica sighed. The moment stretched into silence. She was just about to speak again when there was a gentle rap at the door, and both of them looked up to see Dr. Mason appear in the opening. \"Ladies, I'm afraid it's time Amelia took a rest. It pains me to hurry you, but I think it best we get her settled before the other patients return from their exercise.\"\n\nVeronica smiled at Amelia sadly and then leaned over and kissed her tenderly on the cheek. She rose to her feet. \"Take care, sister. I'll return in a few days to see how you're getting along.\"\n\nAmelia nodded. \"Until then.\"\n\nDr. Mason held the door open for Veronica as she left the day-room without looking back, the stirrings of tears in the corners of her eyes.\n\nThe young man was still lounging idly on the wooden bench when Veronica stepped out of the asylum. She tried again to place him, but somehow his identity eluded her. She was convinced that she'd seen him before, in a different context. She took a few steps along the gravel path, and then, deciding that she'd be unable to let it rest, she turned back and accosted one of the nurses, who seemed both bemused by Veronica's sudden appearance and annoyed at having her train of thought interrupted whilst gossiping with one of her colleagues.\n\n\"Excuse me, nurse. Can you tell me: Who is that man?\" She spoke in hushed tones so as not to let him overhear her words, indicating him with a wave of her hand.\n\nThe nurse looked over her shoulder and shrugged. \"I have no idea, ma'am. None of us do. He was brought in last night after lights-out, and the night nurse was told to find him temporary accommodation. He was in a terrible state. His clothes were covered in blood and the wounds on his arms were severe. He looked like he'd been savaged by an animal. Not for the first time, either, judging by the scars we found when we washed him down.\" She shrugged. \"We cleaned him up and gave him a bed for the night, is all. It was one of the locals who found him, shivering in the gutter by the side of the road. They brought him in last night, figuring he wasn't a drunk and may have been a patient who had somehow found his way out of the asylum. Seems he can't remember his name or any of his family connections. Poor sod. He'll be collected and taken to the public sanatorium later this afternoon.\" She searched Veronica's face for an answer. \"Why do you ask?\"\n\nVeronica frowned. \"For some reason he just looks . . . Oh God!\" She stared at the man over the nurse's shoulder, watching him as he gazed up at the sky, lost in a world of his own devising. Suddenly something seemed to click in her head. \"Oh God! Jack! Jack Coulthard!\" She ran towards him, realisation dawning behind her eyes. \"You're Jack Coulthard!\"\n\nThe man turned to look at her, his eyes searching; confused and unsure how to take this outburst from a strange woman he had no idea whether he should know. \"I am?\"\n\n\"I believe so, yes.\" She grinned, almost disbelieving the coincidence. \"Your sister showed me your photograph. She's waiting for you to come home.\"\n\nThe nurse rushed over to Veronica. \"You're saying that you know this man?\"\n\n\"I know his sister, yes. She's been searching for him for a week. She's beside herself with concern.\" Veronica turned to face the nurse, who was looking as bemused as the patient. \"Quickly, call for a cab immediately. We have to send for her now.\"\n\nThe nurse nodded and disappeared under the archway to fetch assistance, her feet crunching noisily on the loose stones.\n\nVeronica took a seat beside the young man on the bench, almost bursting with excitement. \"Oh, Jack, your sister is going to be so delighted to discover you're alive.\"\n\nThe man returned her gaze, a bright smile lighting up his face. He looked lost, but hopeful.\n\nNearby, the other patients continued to circle the airing courts, indifferent to the fact that their newest arrival would, in just a matter of hours, finally be reunited with his loved ones.\nCHAPTER 30\n\n##\n\nNewbury leaned heavily on the mantelpiece and took a long draw on his pipe, watching the smoke curl in lazy circles in the still air of his Chelsea living-room. He was wearing a long blue dressing gown and slippers, and was warming himself by the raging fire that Mrs. Bradshaw had built up for him earlier that evening. Across the room, Bainbridge sat easily in one of the Chesterfields, his cane propped by the door, a brandy clutched firmly in one hand, a cigar in the other. He observed Newbury through a pungent wreath of smoke.\n\nNewbury was tapping his foot impatiently, unable to allow himself to relax. He clearly wasn't taking well to his period of convalescence.\n\nBainbridge sucked on the end of his cigar. \"So, truthfully, how are you man? You seem irritable.\"\n\nNewbury laughed. \"No, not irritable, Charles. Just anxious to get out of these rooms! I feel like I've been trapped in here for weeks, pacing backwards and forwards, waiting for something new to come along that I can sink my teeth into. My wounds are healing in a satisfactory fashion, and with any luck, I'll be fighting fit again in no time. I need something new to engage my mind. I fear I'll be climbing the walls before long if something doesn't come along soon.\"\n\nBainbridge shook his head. \"Newbury, you astound me! I'd have thought after your experiences this last week you'd be eager to get some rest. I know I am!\"\n\nNewbury chuckled. \"You know me, Charles. I never have been able to stand still for long.\" He glanced at the end of his pipe, a frustrated look on his face, and then tapped out the spent tobacco on the mantelpiece, banging the vessel repeatedly against the palm of his left hand. He moved stiffly across the room, still wincing with the movement, and lowered himself into the armchair opposite Bainbridge. He searched out his leather tobacco pouch from amongst the debris on the coffee table, and began the process of refilling his bowl. \"So tell me, Charles, what of Joseph Chapman?\"\n\nBainbridge took a swig of his brandy, shuddering as the alcohol sent tickling fingers of warmth into his belly. He looked grave. \"Chapman's for the noose, and he knows it. His crimes were some of the most severe and inhumane I've yet encountered in my career, and in this city, that's certainly saying something. What galls me, though, is the man's consistently pompous attitude. He sits there during his interviews gloating about his crimes, about how clever he was to outwit us for so long. The man is a monster.\"\n\nNewbury struck a match, lit the bowl of his pipe and tossed the dead match into the fire with a brief glance over his shoulder. He puffed to kindle the flames before replying. \"They often are, Charles. They often are. Shame about Villiers, though. He was an entirely singular man.\"\n\nBainbridge pulled a face. \"For the life of me, Newbury, I cannot understand where you developed such profound respect for the man.\"\n\nNewbury closed his eyes. When he opened them again, he was studying the floor. \"It's complicated, Charles. Villiers was an evil man, but he was also incredibly accomplished. In fact, I'd go as far as saying he was a genius, in his own way. And with genius comes a certain amorality that is sometimes difficult to judge. Genius is, in many ways, akin to madness. Both states of mind demand a disconnection from reality, from the real, physical world, an ability to lose oneself in thought.\" He shrugged. \"There is no contesting the fact that Villiers's crimes were of the most appalling variety, but I only wonder what may have come of it if his genius could have been harnessed for the good of the Empire, instead of being misapplied in such a terrible way. . . .\" He trailed off, lost in thought.\n\nBainbridge chewed on the end of his cigar. \"Good riddance to him, is what I say. Chapman did us a favour when he removed the man from proceedings, and that's all I have to say on the subject.\" He paused. \"Still, it's good to see another case through to its resolution, isn't it?\"\n\n\"Hmmm?\" Newbury returned from his reverie, his eyes darting to meet Bainbridge's expectant face. \"Oh, yes indeed. Although I hasten to add that there _is_ still one small part of the mystery that perplexes me. I've yet to discover the reason why a Dutch nobleman was to be found on board the wreckage of a passenger-class airship bound for Dublin.\"\n\nBainbridge placed his glass on the table and leaned forward. \"I may have something to help you with that, old man. The one good thing about Chapman's boastful tirade is that we've been able to glean a few facts from his testimony. He claims _The Lady Armitage_ had been engaged by a coterie of local noblemen, men who were keen to see as many revenants removed from the streets as possible, for use as a plague ship. Chapman had been using the automata to round up the revenants like animals, forcing them onto the airships and shipping them off to Ireland, where his men were setting them loose in the countryside\u2014if they didn't dump them at sea during the course of the voyage. Not sure that explains how your Dutchman found himself involved in the matter, but it may help you get to the bottom of the mystery, eh?\"\n\nNewbury looked animated. \"Indeed it does, Charles. Indeed it does!\" He sprang out of his chair, clamped his pipe between his teeth and began pacing back and forth before the fire, all sense of his stiffness gone. The silence stretched. After a moment, he turned to Bainbridge, gesturing frantically with his hands. \"Charles, allow me to ask you a question. Why should a visiting nobleman take to the streets of Whitechapel by evening, choosing to travel alone, without the protection of a Royal escort?\"\n\nBainbridge frowned. \"No reason at all, unless he had a taste for the wicked side of life, if you catch my meaning.\" He coughed into his hand, embarrassed at the implication.\n\n\"Precisely! If the man had harboured a longing for visiting cheap whores whilst staying in the city, he would surely have slipped out of his lodgings unaccompanied, in an effort to keep his inappropriate activities under wraps. If the newspapers were to discover his secret, it would cause the palace a terrific scandal, and if any unscrupulous aides were made aware of it, they might have chosen to use the information against him at some point.\"\n\n\"Blackmail, you mean.\"\n\nNewbury nodded. \"Indeed. So we've established that if the man _did_ engage in such carnal pursuits, he would be sure to hide the fact from his aides, stepping out alone only at the most opportune moments, such as late in the evening after his men had retired.\" He smiled to himself, pleased with his deduction. \"Could it be, then, that the man inadvertently contracted the revenant plague during one of these nightly sojourns to the slums, so that when the automata came to round up the miserable fellows a week or so later, he was wandering the streets, transformed into one of the detestable creatures?\"\n\n\"You could be right, Newbury! Certainly no one would recognise the man in that state.\"\n\n\"Until, that is, they removed his charred corpse from the wreckage, which would show no signs of the viral infection that had thus far been ravaging his body. An identifying item of jewellery would be all that it would take for the coroner to proclaim that Her Majesty's missing cousin had been found.\"\n\nBainbridge retrieved his brandy from the table. \"My God, Newbury. I think you're on to something. But how the devil do you prove the man had such inappropriate desires in the first place? That's quite an accusation to level at a member of the Royal Family without any real shred of evidence. I can't imagine Her Majesty will accept your story on supposition alone.\"\n\nNewbury chuckled. \"That's just it, Charles. I believe I have all the evidence I need. I've spent the last couple of days scouring my records for background on the Dutch royal family, identifying potential victims. Her Majesty had been less than forthcoming about which particular cousin had been involved in the incident, but her words provided me with a number of important clues. I knew we were dealing with a young man, a minor royal, but someone who would be sent to London on diplomatic duties all the same, probably due to the importance of their mother. After taking all of that into consideration there was only one likely candidate, a man whose name\u2014I'm sure you will forgive me\u2014I will refrain from repeating here.\" Newbury paused for breath, although it was clear he was anxious to proceed with his tale. \"But during the course of my reading, I turned out a number of newspaper reports regarding a 'misunderstanding' between one of the Queen's cousins and a mysterious 'lady,' who claimed to be the bearer of an illegitimate child. The newspapers had reported the story as a minor item, alluding to the fact that the woman was a prostitute and had probably invented the entire story as a means of extorting money from the unfortunate young man. However, in light of current events, I'll wager there's truth behind the tale. And what's more, I imagine it was this very same man whose corpse was extracted from the crash site of _The Lady Armitage_ just a few days ago in Finsbury Park.\"\n\nBainbridge nodded, a smile curling his lips. \"I should say that will do the job.\"\n\n\"Indeed.\" Newbury returned to his seat with a satisfied sigh. He raised an empty glass. \"Well, that's the end of it then.\" He sucked on his pipe, resting his head against the tall back of the Chesterfield.\n\nBainbridge shuffled awkwardly in his seat. \"There is just one other thing I should mention, if you're not too opposed to hearing me out on something rather peculiar?\"\n\nNewbury peeled open his eyes, his interest piqued. \"Go on.\"\n\n\"I'm sure you recall our conversations from a few days ago, regarding the potential origins of the glowing policeman?\"\n\n\"Of course.\"\n\n\"At the time, before Morgan's death and the realisation that we were on the trail of a purely corporeal killer, you mentioned Miss Hobbes's supposition that the perpetrator could in fact have been a phantom killer akin to the one reported all those years ago. Another example of the same phenomena, you said, involving entirely different people.\"\n\nNewbury sat forward in his chair and poured himself a brandy, listening intently to Bainbridge's account. \"Quite so.\" He considered his friend, concern evident in his eyes. \"What's troubling you, Charles?\"\n\nBainbridge shook his head. \"It's all rather embarrassing, really. I mean, I don't know what to think. You know I'm not a superstitious man.\"\n\n\"For Heaven's sake, Charles. Get to the point.\"\n\n\"You asked me if there had been any recent murders of police constables in the Whitechapel area, and at the time I couldn't say for certain. But I had the clerks check the records and it turns out there was a man, a Mr. John Harris, who was done in with his own truncheon by a gang of youths, after he happened upon the miscreants roughing up a girl in an alleyway earlier that night. They got away with it, too, since a local shopkeeper provided an alibi. The word amongst the rest of the men was that the gang had applied a liberal amount of pressure to the shopkeeper and, fearing for the safety of his wife and daughter, he had willingly perjured himself to protect them.\"\n\nNewbury took a swig of his brandy. \"Let me guess. It turns out a number of these youths were amongst the victims of the glowing policeman, found strangled in the Whitechapel area, their personal effects still _in situ_ on the bodies?\"\n\nBainbridge smiled. \"Close, Newbury. _All_ of the youths were amongst the reported victims of the glowing policeman. I don't know what to make of it. It seems like too much of a coincidence to ignore.\"\n\nNewbury laughed. \"Ha! I'll wager coincidence has nothing to do with it!\" He sank back in his chair. \"Of course, there's no way of telling, now. It _could_ have been coincidence, or it could have been the murdered man's colleagues taking the opportunity to seek revenge. But it would certainly explain why we didn't identify the residue of the blue powder on all of the victims. Revenge can drive people to do terrible things, Charles, terrible things indeed. Even, perhaps, to rise from the grave itself. Did I ever tell you of the Hambleton affair?\"\n\n\"I don't believe so, no.\"\n\n\"Ah, well. I suspect that's a story for another occasion. Nevertheless, it serves to prove the point. There are things in this world\u2014and beyond\u2014for which the combined efforts of science and religion have yet to divine a suitable explanation. I have no doubt that, given time, they will.\" He heaved himself out of his chair, stretching his sore muscles. \"But now, my friend, I must prevail upon you to forgive me. I feel the need to retire for the evening, to rest these damnable wounds in an effort to hasten my recovery and put an end to my captive misery.\" He sniffed. \"The guest room is yours if you want it.\"\n\nBainbridge rose to his feet and clasped a hand on his friend's shoulder. \"No, I'll take my leave, dear boy.\" He smiled warmly. \"Look after yourself, and keep an eye on that wayward assistant of yours. She'll be causing a scandal or two of her own if she doesn't check herself from time to time.\"\n\nNewbury laughed heartily. \"Indeed. She might at that.\"\n\nBainbridge downed the remainder of his brandy and crossed the room, collecting his coat and cane. \"Well, Newbury. Until next time.\"\n\n\"Good-bye, Charles.\"\n\nThe Chief Inspector took his leave. Newbury waited until the sound of his footsteps had receded down the street. He banked the fire, making sure the embers were burning low, and turned out his pipe in the grate. Then, leaving the living-room behind him, he climbed the stairs and passed along the hallway towards his bedchamber. He stopped outside the room and placed his hand on the doorknob. A little farther along the landing, the door to his study was propped shut, still loose on its hinges following Veronica's dramatic entry a day or two earlier. He'd have to have it fixed in the next couple of days, once he'd regained the rest of his health.\n\nHesitantly, he withdrew his hand from the handle of his bedroom door and crossed the landing, his wounds itching where scabs had formed over the open cuts. He pushed his way through the unwieldy study door and propped it shut again behind him. He turned up the gas jet on the wall, causing a dim radial glow to light the room. The room was just as he'd left it.\n\nHe crossed to the daybed and took a seat, eyeing the little brown bottle on the table in the corner. In the dim light, he could just see the peeling label, the familiar liquid inside. There was also a half-drunk bottle of red wine on the table beside it, stoppered with a used cork. It had probably spoiled during the intervening days. He rubbed a hand over his face and glanced at the door sheepishly, knowing he should head for his bedchamber, and stood, pacing towards the landing. His stomach twisted in knots. He drew a deep breath, trying to fight off his cravings, playing out the consequences in his mind. He'd promised Veronica. . . .\n\nHe glanced back at the bottle in the half-light. It sung to him of cosy oblivion and warmth. It was like the call of a siren; unrelenting, full of terrible beauty. The need burned within him, made every inch of his skin crawl.\n\nHe stood on the threshold of the room for a few moments, tortured and confused by his own desires. Then, finally, he crossed the room, collected the two bottles, and settled himself on the daybed, aware that, for a short time at least, he could banish the guilt that gnawed at him from within. The laudanum would bring stillness. The laudanum would remove all sense of fear, all sense of doubt. The laudanum would excise the pain, and tomorrow, he told himself as he carefully poured a measure from the dull, inconspicuous bottle, was another day entirely. Tomorrow he could start again.\nCHAPTER 31\n\n##\n\nVeronica glared at the pile of unsorted papers on her desk and sighed. The office was deathly quiet, lacking the banter she had become accustomed to, with only the constant _tick-tock_ of the grandfather clock and the occasional sound of Miss Coulthard shuffling papers in the adjoining room punctuating the monotony.\n\nShe leaned back in her chair and glanced over at Newbury's empty desk, which had lain undisturbed since they were last in the office together the previous week. Correspondence had temporarily been forwarded to his Chelsea home whilst he spent time convalescing away from the museum, and the lack of his usual cheer lent the place a mournful air, as if it were missing something fundamental, the heart of it temporarily removed. The office itself had been restored to something approximating order, following Miss Coulthard's return to work and the removal of the automaton remains by Scotland Yard, who were keen to gather evidence for the case against Chapman. Not that they needed to worry, Veronica considered; she was certain that they would be able to uncover enough at the manufactory to send him to the gallows ten times over, especially when one took into consideration the testimonies of Sir Maurice and Sir Charles, both respected members of society and gentlemen to boot.\n\nVeronica leaned back in her chair, drumming her fingers idly on the desk. The days following her visit to the asylum had passed in a sedentary fashion, and whilst she had enjoyed hearing tales from an effervescent Miss Coulthard about the return of her brother Jack, in truth, she was finding it difficult to give her administrative tasks their due attention. It had only been a handful of days since the apprehension of Joseph Chapman and the resolution of the case of _The Lady Armitage,_ and she already found herself speculating on what the future may hold. She longed to see Newbury again, to lose herself in another mystery. She knew it was idle speculation, but it helped fuel her motivation for the laborious research work she was obliged to carry out whilst she waited for Newbury himself to return to work.\n\nDeciding that she shouldn't put it off any longer, she set to work, taking a stack of manuscript pages from the top of the nearest pile and leafing through the content in an effort to identify any references that Newbury might find useful in the writing of his most recent essay, regarding the ritualistic practices of the druidic tribes of Bronze Age Europe.\n\nThere was a polite rap on the inner door. Veronica looked up to see Miss Coulthard hovering in the doorway, a large sheaf of papers clutched tightly in her arms.\n\n\"Miss Hobbes, I'm just running these along to the museum archive. I'll be back shortly if you find you have need of me.\"\n\nVeronica smiled. \"Of course. Thank you, Miss Coulthard.\" She indicated the large stack of papers on her desk. \"I won't be going anywhere for a while.\"\n\nMiss Coulthard gave her a knowing sigh and then left, her heels clicking loudly on the tiled floor. Veronica returned distractedly to her reading.\n\nA few minutes later, she heard the door open and shut in the adjoining room, followed by the sound of footsteps on the threshold of the office. She continued reading, her eyes flicking over the carefully crafted copperplate on the page before her. \"You were far quicker than I'd imagined, Miss Coulthard. Now, if you could find it in your heart to put the kettle on the stove. . . .\" She looked up at the sound of a man clearing his throat, her voice trailing off. \"Sir Maurice! I\u2014we weren't expecting you back so soon!\"\n\nNewbury smiled. \"My dear Miss Hobbes. There is only so long a gentleman can sit in his rooms, staring at the walls, before the experience becomes entirely unbearable.\" He removed his hat and indicated his desk with a wave of his hand. \"Besides, that essay isn't going to write itself.\" He beamed at her, his eyes twinkling.\n\nVeronica grinned. \"Tell me. How are you feeling? Are you recovered?\"\n\n\"A little stiff. My wounds are healing well enough, although it's a damnable irritation. Still, I imagine I'll be back to my usual self before long. Provided, that is, that I don't find myself scrabbling around on the top of any moving ground trains in the near future.\"\n\nVeronica laughed. \"Well, sit yourself down, and I'll prepare a nice cup of Earl Grey. Miss Coulthard should be back soon. She's just popped along to the archive to file some papers.\" She climbed to her feet, stretching her back after spending too long sitting hunched over her desk.\n\n\"Indeed. I ran into her in the passageway. It's most excellent news about the safe return of her brother. I understand that you were instrumental in seeing him home?\"\n\nVeronica came out from behind her desk. She shrugged. \"Yes, I suppose you could put it that way. I happened upon him in the most unlikely of spots, and having seen one of Miss Coulthard's photographs, I was able to place him. His memory has yet to return in full, but I'm told he's otherwise in good health. It transpires that Miss Coulthard was correct all along, that he was indeed savaged by a revenant in Brixton, but somehow managed to get away. It seems the two of you have a good deal in common, too: aside from spending time in India, you've both encountered revenants on more than one occasion and lived to tell the tale. Jack is also immune to the plague. Miss Coulthard says that the doctors are talking about fashioning a vaccination from his blood.\"\n\nNewbury nodded, smiling. \"Remarkable. I'm most delighted for Miss Coulthard.\" He paused, running a hand over his face. \"So, tell me, how did you find Amelia? Is she bearing up?\"\n\nVeronica tried to maintain her smile, but her face faltered. \"Not well, I'm afraid. She grows weaker with every visit I make. I don't know what else I can do for her. I think just being in that place is enough to suck the life out of her.\"\n\nNewbury stepped closer and tenderly placed his hand on her arm. \"We must see what we can do to help. I'll give the matter some attention directly.\"\n\nVeronica's breath became shallow. She edged nearer to Newbury, her heart hammering in her chest. Her lips were dry. \"It's good to have you back, Sir Maurice.\"\n\n\"I\u2014\"\n\nThen the door swung open and Miss Coulthard bustled noisily into the office. Veronica hurriedly stepped back from Newbury, smoothing her dress. Her face flushed.\n\nMiss Coulthard seemed not to notice anything untoward. She smiled. \"Good to have you back, Sir Maurice.\" She glanced at them both in turn, and then shuffled over to the stove. \"Tea, anyone?\"\n\nNewbury laughed. \"Yes, please, Miss Coulthard. That would be perfect.\" He crossed into the other room and dropped his hat on the stand, shrugged out of his coat, and then, moving carefully so as not to put stress on his wounds, he wandered back through to his desk and lowered himself into his chair.\n\nVeronica returned to her seat. They eyed each other across the office, neither of them knowing what to say. Miss Coulthard whistled tunefully in the other room as she set the kettle on the stove and searched around in the cupboard for some cups and saucers.\n\nNewbury was first to break the silence. \"Did Bainbridge stop by to inform you of my theory about the Dutch Royal cousin and _The Lady Armitage_?\"\n\nVeronica nodded. \"Indeed he did. He was rather less than forthcoming when it came to detailing the activities the man had been pursuing in the Whitechapel district, but I was able to tease out enough information from his inferences to work out what he was trying to say.\"\n\nNewbury laughed. \"That certainly sounds like Charles. He never could talk to a lady.\"\n\nVeronica looked suddenly serious. \"I suppose that explains why all of the passengers had been tied to their seats on the airship. The fact that they were plague victims, I mean.\"\n\n\"Yes, I suppose it does.\"\n\nVeronica toyed with the corner of one of the manuscript pages on her desk. \"So how did Her Majesty take the news? It's rather a scandalous affair for the family, isn't it?\"\n\nNewbury shrugged. \"I visited the palace yesterday. Her Majesty seemed to take the news impartially. She was rather too busy admonishing me for the state of my health, if truth be told.\" He chuckled. \"I doubt there'll be any word of it in the press. Whether the facts are deemed appropriate for the boy's mother, we'll have to leave for others to decide.\"\n\nVeronica nodded. \"So, what's next?\"\n\nNewbury laughed again. \"Druids. The Bronze Age. Pages and pages of arduous notes.\" He leaned back in his chair. \"After that, who knows? I'm sure that something will turn up.\" He shifted to see Miss Coulthard entering the office, bearing two cups of Earl Grey on a wooden tray.\n\nVeronica smiled, reaching for another sheaf of papers on her desk. She shuffled them into a neat pile before her. Looking up, she met Newbury's eye from across the room. \"I do believe it will, Sir Maurice. I do believe it will.\"\nEPILOGUE\n\nThe life-preserving machine laboured in the semi-darkness, hissing and wheezing as the bellows rose and fell in time with the toilsome breathing of its occupant. Her Majesty Queen Victoria eased herself forward, wheeling her chair closer to the figure that was standing in the shadows on the other side of the audience chamber. Her face resolved in the gloom. She was wearing a stern expression. \"We are most satisfied with the resolution of this investigation\"\u2014her voice was shrill and echoed around the empty room\u2014\"yet we remain concerned for the well-being of our agent. Tell me, Miss Hobbes, do you believe that Sir Maurice acquitted himself in a manner becoming a representative of the Crown?\"\n\nVeronica swallowed and stepped forward into the wavering light of the gas-lamps. \"I do, Your Majesty. Sir Maurice is a credit to his nation.\"\n\nThe monarch nodded. \"Very good. That is most reassuring.\" She put her hand to her mouth and gave a wet, spluttering cough. The machine wheezed as it tried to compensate for the brief fit. Her chest heaved, her lungs filling with oxygen. She continued, catching her breath. \"Even so, Miss Hobbes, we encourage you to remember your duty. We must ensure that Newbury remains steadfast in his beliefs. We fear that the dark arts have a terrible allure, and, lest you forget, your primary role in this assignment is to protect Newbury from falling for such devious charms. One would hate to imagine that we were allowing another dissenter to propagate in our midst.\"\n\nVeronica frowned. \"What word is there of Dr. Knox?\"\n\nThe Queen shook her head. \"No word. He is lost to us. We have dozens of agents searching for him, all across the Empire, but he proves as elusive as he ever was. He has managed not to show his hand for over a year now. One wonders what he is plotting in the darkness.\"\n\nVeronica shrugged. \"Perhaps he is already dead.\"\n\n\"No.\" The Queen was firm. \"He is a wily devil, and he has darker forces at his disposal. We have no doubt that he is alive, somewhere out there, hiding in the quiet places, unseen to our agents.\" She straightened herself in her wheelchair. \"What is clear to us is that Newbury must be steered in a wholly different direction. He cannot be allowed to succumb.\"\n\n\"Yes, Your Majesty. I can assure you that this matter remains my sole concern. I will ensure that Sir Maurice does not fall prey to that particular trap.\"\n\nVictoria raised an eyebrow. \"You seem overly confident in your own abilities, Miss Hobbes. Perhaps you already have the man wrapped around your little finger.\" She laughed, and the sound was like boots crunching on gravel.\n\nVeronica looked away, a pained expression on her face. \"Perhaps. Yet I think his heart is true. He will not be swayed by petty obsessions or a vain desire for power. He is not Aubrey Knox. He serves you well, Your Majesty.\"\n\nVictoria nodded. \"Then you may go, Miss Hobbes, and be about your business.\" She wheeled back a few feet, indicating that it was time for Veronica to leave. Veronica crossed the room. Victoria waited until she was nearing the door. \"Oh, and Miss Hobbes? One other thing before you take your leave.\"\n\nVeronica turned back to regard her, finding it difficult to place her in the dim light. \"Yes, Your Majesty?\"\n\n\"This 'affinity bridge' that Newbury spoke of. The device that facilitates the interaction between the human brain and an artificial body. Have they all been destroyed?\"\n\n\"No, Your Majesty. The Chapman and Villiers automata are currently being decommissioned, but it is proving to be a lengthy process. It will be some months before they are all accounted for.\"\n\nThe Queen offered her a weary smile. \"Good. Please ensure that you keep at least a handful of them in working order. One never knows when the technology may prove useful.\"\n\n\"Indeed, Your Majesty. I will endeavour to do so.\" She glanced at the door. \"Will there be anything else?\"\n\n\"No. That is all. Thank you, Miss Hobbes.\"\n\n\"Good day, Your Majesty.\"\n\nVeronica pulled open the door and hurried along the passageway, keen to get away from the palace and back to the museum, to Newbury and to her newfound life of adventure.\n","meta":{"redpajama_set_name":"RedPajamaBook"}}
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+{"text":"The definition of the word \"pet\" has evolved over the last few decades. No longer are we welcoming only cats and dogs into our homes. As our pets have changed, so too has the face of veterinary medicine.\nThe professionals at the Plaistow-Kingston Animal Medical Center are well-trained and knowledgeable in the field of exotic pet care. We offer advice and care for all of your small creatures, including birds, rabbits, rodents, sugar gliders, ferrets, snakes, frogs, turtles and lizards.\nOur founder Dr. Wallace has taken a special interest in reptiles and other exotics over the last twenty years, and is always ready for his next exotic challenge. From advice on habitat and husbandry to surgical care, the Plaistow-Kingston Animal Medical Center is your one stop in caring for all of your pets.\nThe House Rabbit Society is an international nonprofit organization that escues rabbits from animal shelters and educates the public on rabbit care and behavior.\nThe American Ferret Association promotes the domestic ferret as a companion animal through public education.\nGuinea Lynx is a comprehensive medical and care guide for guinea pigs.\nAnapsid.org is an informational site for more about amphibians and reptiles.\nParrot Pages is a great site for parrot information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hartford is nicknamed the insurance capital of the world because of the many insurance-company headquarters it houses. It also offers a scenic skyline, which is captured here in this Hartford, Connecticut - Panorama. Panoramas displayed on your dorm walls can bring cool college wall decor that's a step above your everyday college posters. Wall decor for college is needed if you want your dorm room to look awesome. Actually, you'll need wall decor items just to make your dorm room look decent. These panoramas, however, can make your college decor look way beyond decent!\nThis aerial panoramic photograph of Hartford was taken by James Blakeway. It features Hartford as seen looking across the Connecticut River. The tallest building in the center of the photograph is City Place One. The next tallest structure to its left is Travelers Tower, which was completed in 1919 and was New England's tallest building for many years. The gold-domed Connecticut State Capitol is visible left of Travelers Tower. The bridge to the far right is Bulkeley Bridge, which was built in 1908. Mark Twain, the prolific American author, resided in Hartford from 1871 to 1891. During this very creative time, he wrote classics such as Life on The Mississippi and Adventures of Huckleberry Finn.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Notice 2011-54 provides additional administrative relief to persons whose requirement to file Form TD F 90-22.1, Report of Foreign Bank and Financial Accounts (FBAR), to report signature authority over foreign financial accounts held during calendar year 2009 or earlier calendar years was properly deferred under Notice 2009-62, 2009-35 I.R.B. 260, or Notice 2010-23, 2010-11 I.R.B. 441.\nOn August 31, 2009, the Department of the Treasury and the Internal Revenue Service published Notice 2009-62, which, in part, extended the filing deadline for persons with no financial interest in a foreign financial account but with signature or other authority over that account (hereinafter referred to as \"signature authority\") for the 2008 or earlier calendar years. In Notice 2010-23, the Department of the Treasury and the IRS further extended relief to persons whose filing deadline was properly deferred by Notice 2009-62 and provided a new filing deadline to June 30, 2011, to report signature authority over, but no financial interest in, foreign financial accounts for calendar year 2009 or earlier calendar years. The extensions were provided to allow the Treasury Department time to develop comprehensive FBAR guidance.\nOn February 24, 2011, the Financial Crimes Enforcement Network (FinCEN) published final regulations (76 FR 10234) amending the Bank Secrecy Act implementing regulations regarding FBARs, found at 31 CFR 1010.350 (formerly 31CFR 103.24). The regulations became effective on March 28, 2011, and apply to FBARs required to be filed by June 30, 2011, with respect to foreign financial accounts maintained in calendar year 2010, as well as to FBARs for subsequent calendar years.\nThe final regulations also provide that individuals who properly deferred their FBAR filing obligations pursuant to Notice 2010-23 may apply the provisions of the final regulations in determining their FBAR filing requirements for reports due June 30, 2011, with respect to foreign financial accounts maintained in calendar years beginning before 2010. In March 2011, the IRS released a revised FBAR form with accompanying instructions that reflect the amendments made by the final FBAR regulations.\nPersons having signature authority over, but no financial interest in, a foreign financial account in 2009 or earlier calendar years for which the reporting deadline was extended by Notice 2009-62 or Notice 2010-23 will now have until November 1, 2011, to file FBARs with respect to those accounts.\nThe deadline for reporting signature authority over, or a financial interest in, foreign financial accounts for the 2010 calendar year remains June 30, 2011.\nThe administrative relief provided in the Notice does not limit the relief provided in FinCEN's Notice 2011-1, which was released on May 31, 2011, and revised on June 6, 2011. A copy of revised FinCEN Notice 2011-1 may be found at www.fincen.gov.\nAdditionally, the administrative relief provided in this Notice does not affect the requirements to provide information or file FBARs in connection with the IRS's 2009 Offshore Voluntary Disclosure Program or the 2011 Offshore Voluntary Disclosure Initiative. Nor does this Notice alter the deadlines for electing to participate in, or fulfilling the submission requirements of, the Offshore Voluntary Disclosure Program or the Offshore Voluntary Disclosure Initiative.\nNotice 2010-23 is modified and supplemented.\nNotice, you can contact Emily M. Lesniak at (202) 622-4570 (not a toll-free call).\nBy Stacie Clifford Kitts in FBAR, FOREIGN BANK ACCOUNT REPORTING, FOREIGN REPORTING, IRS PATROL on June 16, 2011 .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apr 23, 2018. No passports are required at these fifty romantic US honeymoon destinations. 30 Vacation Rental Mistakes (And How to Avoid Them).\nIts cargo of university students from London was soon scattered across the United States for their summer vacations. one asked if she could have her honeymoon at our home!\nStart off your marriage with a honeymoon vacation you'll never forget. Offering the best resorts in Mexico, the Caribbean, & Hawaii for your big celebration.\nOur Belize adventure vacation packages include a number of thrilling adventures for you to enjoy with comfortable accommodations to rest in afterwards. Most of our adventure packages are all inclusive. Contact us today and book one of our adventure vacation packages.\nDisabled people are a hugely underutilized workforce in the US. More than one in five people. to work remotely during maternity leave, vacations, and honeymoons. \"The company demonstrated.\nJetting off on your honeymoon the night of your wedding is a romantic notion that has. Atlanta, or elsewhere in the U.S., there is a nearby destination ready to cater to. Couples who book this package also receive daily access to the Trump.\nThere's a lot to think about when you're planning your honeymoon, from packing the perfect day-to-night looks to keeping your social media feed fresh to death (do us part. they favor vacations.\n\"This will be the first of many vacations together. if you miss your flight or have a problem, you can call us and we will solve it for you.\" MacKnight says once couples decide what kind of.\nBoscov's Travel Honeymoon and Destination Wedding Specialists will make your honeymoon the start to your happy ever after. From Bora Bora, Paris, the.\nOoty Honeymoon Packages \u2013 Get discount on Honeymoon packages for Ooty Tours & travel packages at MakeMyTrip.We offer customized Ooty Tour & Honeymoon Packages. Plan a travel to Ooty and its various tourist attractions with Ooty Honeymoon packages. Explore exciting Ooty Tourism with cheap vacation packages.\nChoose your New Zealand vacation package or New Zealand honeymoon \u2013 jaw- dropping natural beauty, incredible outdoor. Contact Us Call, Chat, Email. Indulgent Escape: New Zealand Luxury Vacation \u2013 Helicopter Over Milford Sound.\nWith hundreds of Hawaii Honeymoons and vacation packages it is hard to know which will provide the most romantic honeymoon or getaway possible. Honeymoon Travel Agent has helped thousands of couples and newlyweds plan their dream Honeymoons over the last 20 years.\nExpedia Add-On Advantage: Book a package & unlock up to 43% off select hotels. Terms apply.Travel the world better. Build your own Maldives vacation. Bundle flight + hotel & save 100% off your flight. Expedia Price Guarantee on 665,000+ properties & 400+ airlines worldwide.\nThe only good thing that ever came out of summer was vacations. It's not a luxury we have as adults. Dil Dhadakne Do Dil.\nCoco Plum Island Resort offers a variety of Belize all inclusive packages. Our vacation packages include oceanfront accommodation, island & mainland adventure tours, meals & unlimited local alcoholic drinks. Contact us for more info on our Belize all inclusive packages or to customize your own.\nWe get bored easily and they kept us entertained.\" The couple eventually took a longer honeymoon for just the two of them later. And it's not just honeymoons that are getting more crowded. Toren has.\nOur honeymoon packages are customized to the couple and include special. our trips fun for both of us and we love sharing our expertise with other couples.\nBora Bora Vacations, Honeymoon Packages and Romantic Getaways. Print; Brochure;. Bora Bora does have history. During WWII a US supply base was established on the island prompted by the bombing of Pearl Harbor in 1941. You can visit American WWII sites along with other locally important Tahitian archaeological sites. so why not book one.\nWestern U.S. Honeymoons \u2013 Plan the perfect Western U.S. honeymoon and find. Satiate your wanderlust with a never-dull trip to some of the coolest vacation. Head here to experience life in superlatives: the finest dining, the best hotels.\nHowever, you can find some incredible spots right here in the U.S. Topping our ranking of the Best Affordable Honeymoons in the USA is Yellowstone, which covers 2.2 million acres across Wyoming, Idaho.\nSearch the best value prices on all Romance trips with airfares. Call Toll Free ( 800) 896-4600.\nAlberta visits and vacations for members of the royal family are a boon for. \"We're more than happy to roll out the carpet and welcome them. It gives us a chance to showcase all the beauty that is.\nHoneymoon destination ideas for a once-in-a-lifetime escape. Get ready for a once-in-a-lifetime escape. Search. US US. The 19 Places to Honeymoon in 2019. Type keyword(s) to search.\nHawaii Honeymoon Packages are our speciality.Call us to arrange your custom Hawaii Honeymoon Package.Members of the Hawaii Visitors and Convention Bureau.Our staff are certified Mau. islandspree specialize in customizing family vacation packages and honeymoon packages to exotic tropical destinations all over the world like Jamaica, Aruba.\nIn the rare books reading room at the British Library (where I went to write), I spent a lot of time entering online contests, filling out form after form in the hope of winning vacations. it would.\nItaly honeymoon packages tug at the heartstrings of so many couples. one of our Italy honeymoon packages or work with us to create a custom-made itinerary.\nAs its site reads, \"We offer a retreat of peace and comfort for newlyweds. For your honeymoon in Paris with us, we work with you to craft every unforgettable detail in the most romantic city in the.\nEnjoy southern hospitality at its finest on a romantic escape to one of the US's most charming cityites. A honeymoon in Savannah would probably. Keys may not seem like the most glamorous of.\nThe fundraiser includes a silent auction with original art, jewellery, handmade artisan goods, ski and vacation packages, gift certificate raffles. Are you new to the United States? Literacy.\nWhy Bali Honeymoon Packages? Nicknamed as the 'Island of the Gods', Bali is snuggled in central Indonesia with Java to its east and bordered by the Indian Ocean and Bali Sea.\nBut, let's be honest, most of us don't have the unlimited budget that allows such lavish vacations. The good news is there are plenty of honeymoon hotels worldwide seemingly dripping in luxury that.\nPrincess Diana vacations on Necker Island on April 1, 1990. In July 2018, Eugenie and Jack were photographed vacationing on the Amalfi Coast in Italy. Could the couple have been scouting out the.\nVisit this page to learn more about the romantic vacation package we offer, and book. Miami is romantic getaway and honeymoon destination where white sand. Let us arrange that romantic surprise for your loved one before your arrival.\nFor families and wedding parties that aren't buying out a property, Garcia tells us that villas are a great option. dismissed,\" agrees Jennifer Doncsecz, president of VIP Vacations, Inc. \"The.\nDuring the Persian Gulf War, Gordon 'Butch' Stewart, Chairman and Founder of Sandals Resorts, donated over $1 million in complimentary vacations to Operation Desert. spend a majority of their time.\nAll Inclusive Hawaii Honeymoon Vacation Packages to Waikiki Beach \u2013 Oahu, Maui, Kauai, the Big Island of Hawaii and Multi-Island Hawaii Honeymoon.\nGet the best Honeymoon Travel Deals from American Discount Cruises. In addition to honeymoon packages, we also offer destination wedding packages.\nGive us a Call. 1.800.309.4717. The Best In Wedding Vacations. With Wedding Vacations, you'll find unsurpassed value, personally selected hotels offering special wedding and honeymoon packages and an experienced staff to ensure everything is perfect.\nHoneymoons After the big day, you both deserve pampering and quality time together. Make your honeymoon legendary by selecting an outstanding resort in an awe-inspiring destination.\nHoneymoon Vacation Packages | Monograms\u00ae Packages. wine country are just a few reasons to add Argentina to your South American travel package.\nOur Atlantic City vacation packages will make you feel like you've made it to the big time. You may end up even saying \"What happens in Atlantic City, stays in Atlantic City,\" by the time you leave. Best Vacations in the Southern U.S. Soul food, easy-going locals, and cities rich with culture await you in the Southern United States.\n\"We were doing all of these big things out of our comfort zones,\" Aust told USA TODAY. \"Quitting our jobs. the best throughout our travels,\" she added of their nearly year-long honeymoon, which.\nBut despite the modern day concept of intimacy and seclusion, post-wedding vacations haven't always been quite so so enjoyable. In fact, honeymoons have a pretty horrible history. Here we take a look.\nChoose from romantic honeymoon and anniversary packages thoughtfully. Dreams Eternity Honeymoon Package. Celebrate your first anniversary on us.\nFind the best all-inclusive resort and vacation packages at top destinations like Mexico and the Caribbean. Call us today and save with travel deals!\n\"Thank you to all of our friends and family who made their way to celebrate with us!\" Emily gushed on Instagram. \"We are eternally grateful.\" Emily and Josh's honeymoon looks like pure bliss, but she.\n5 days ago \u00b7 1 31 Best Warm-Weather Vacations Escape the cold to one (or more!) of these amazing destinations. a travel research site which recently ranked America's most budget-friendly honeymoon options, couples should head here if they're looking for the feel of a quiet European river cruise, Follow Us. Subscribe to Our Newsletter. Other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"60 awesome ne kids bunk beds new york spaces magazine twin over full convertible bunk bed with 3 drawers and trundle chatham twin over full bunk bed with stairs in white bunk & loft beds harriet bee bedingfield mission stair step twin over twin bunk bed kids bed loft \u2013 justizirrtumfo full size junior loft bed with storage steps modern beds pertaining bunk & loft beds bedding amusing full size loft bed with desk underneath 15 queen twin bedroom sets for girls twin size furniture suites This impression (Loft Bed with Dresser Underneath Awesome 79 Lovely Bunk Bed Bedroom Ideas New York Spaces Magazine) preceding will be labelled with: loft bed big lots,loft bed design ideas,loft bed mandaue foam,loft bed storage,loft bed too close to ceiling,loft bed uae,loft bedroom ideas, published simply by with 2018-06-30 11:27:34. To discover almost all graphics inside Best Of Loft Bed with Dresser Underneath graphics gallery you need to follow that web page link.\nLoft Bed With Dresser Underneath Awesome 79 Lovely Bunk Bed Bedroom Ideas New York Spaces Magazine Uploaded by on Saturday, June 30th, 2018 in category Home Furniture Ideas.\nSee also Loft Bed With Dresser Underneath Elegant Stair Step Mission Bunk Bed Twin Full Donco Kids from Home Furniture Ideas Topic.\nHere we have another image Loft Bed With Dresser Underneath Best Of Bunk & Loft Beds featured under Loft Bed With Dresser Underneath Awesome 79 Lovely Bunk Bed Bedroom Ideas New York Spaces Magazine. We hope you enjoyed it and if you want to download the pictures in high quality, simply right click the image and choose \"Save As\". Thanks for reading Loft Bed With Dresser Underneath Awesome 79 Lovely Bunk Bed Bedroom Ideas New York Spaces Magazine.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"I'm a big believer in natural health. I love being able to keep my family healthy and treat illness using gentle, natural, methods. I myself haven't used antibiotics for over 14 years. Antibiotics kill the bad and good bacteria, weakening the immune system. Antibiotics do have their place and I am grateful we have access to them when needed.\nI also give birth to my children at home using hypnosis (it really works!). Of course I'm always accompanied by a skilled midwife & naturopathic doctor. Dr. Brenda Grogan, ND, of The Healthy Way is wonderful. Over the years she has taught me a lot about healing naturally. She's a wonderful educator and loves sharing her knowledge. (If you have a health question, you can ask her via her website under \"Ask Dr. Brenda\").\n(DISCLAIMER: I am not a doctor and offer this information only as an example of what I have found to be useful, natural methods of healing. Please seek advice from a qualified health practitioner if you are experiencing any health issues).\nAn antiseptic and anti-fungal. At the first sign of illness (2-3 times daily), I rub full-strength tea tree oil down the spine, behind the ears and on the bottoms of feet. It has always either prevented the illness from progressing or shortened the life of the illness. While the Immunity of Thieves oil is my first choice, I'll sometime alternate between the two. Tea Tree on the body, Immunity on the feet, and visa versa.\nAn antiseptic. This oil is great for burns and cuts. I burned my finger badly on some hot glue during a craft project I was working on for the Southern Idaho Living Magazine. I immediately rubbed lavender oil on the burn. The pain went away within seconds, and the next day you couldn't tell when the burn had been. I also use it to relieve braxton-hicks contractions during pregnancy, and tension headaches. Besides, it smells sooo good!\nColloidal silver is a natural antibiotic. I have found it completely successful at healing ear and eye infections. When sprayed or dripped into the nostrils, it has healed sore throats and sinus infections.\nAlso a natural antibiotic. I use this at the first sign of illness along with the tea tree oil. It takes longer that conventional antibiotics, but is much more gentle on the system.\nThis has been an absolute LIFESAVER for me after having my children. I experience terrible postpartum depression and this is like putting on rose colored glasses! We buy this in bulk! It is wonderful during times of stress and also relieves symtoms of Chronic Fatigue Syndrome.\nThis is another good one for stress and anxiety. I keep it in my purse or diaper bag - just in case. Works very fast.\nThis stuff is good for everything - seriously. I'll have to write an entirely separate post on the benefits of apple cider vinegar. In short, it's great for arthritis, yeast, fungus, bronchitis, skin conditions, body odor, and much more. It is a natural antibiotic and antiseptic. I take 2 tablespoons of ACV in warm water with a tablespoon of honey and sip twice daily.\nSoaking in epsom salt draws inflammation and infection from the body. It's great for aches, pains and sore muscles. When my children are sick, they enjoy soaking in a warm tub of epsom salts. They always seem to feel much better afterwards.\nLobelia is a bronchial dilator, expectorant and emetic. You can use it to expel thick mucous from the bronchial tubes and lungs. I recently had a bout with bronchitis. My lungs were tight and I couldn't clear them. About an hour after taking 10 drops of lobelia tincture in water, my lungs had loosened greatly and I was able to clear things out.\nWhat do you keep in your naturopathic medicine chest?\nReader Question: Where are you?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We passed out booklets at a rodeo show that was aimed at a drug free community. We were there 2 days and passed out 1064 booklets. Many people were interested in knowing more about the program. We gave out several info packs. One teacher said she is already using the booklets in her school.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Being a sponsor means that your company is willing to take part of our mission to further develop Project Management on the Belgian Market as a recognised profession and to make project management indispensable for business results.\nPMI-Belgium is a professional project management membership organisation in Belgium; founded in 1998 and is today 900+ members. We welcome members from all facets of projects and from all industries.\nThe objective of the PMI Belgium Chapter is to promote project management as a real profession. We organise regular activities to assist our members advancing in their career, improve organizational success and further mature the profession of project management through our globally recognized standards, certifications, resources, tools, academic research, publications, professional development courses and networking opportunities. The Belgium Chapter is part of the Global organisation PMI (Project Management Institute), one of the world's largest not-for-profit membership associations for the project management profession. Our professional resources and research empower more than 500,000 members, credential holders and volunteers in nearly every country in the world to enhance their careers, improve their organisations' success and further mature the profession. PMI's worldwide advocacy for project management is reinforced by our globally recognized standards and certification program, extensive academic and market research programs, chapters and communities of practice, and professional development opportunities.\nThanks to our well-developed community PMI Belgium is your prime partner for targeting the professional project management community in Belgium.\nFor many years we have offered various ways for companies to invest in our ever-growing community. As a non-profit organisation, 100% animated by volunteers, the Chapter has an absolute need of your support to fund all our activities. Indeed, it is in large part due to the many appreciated sponsors that invest in our organization and the people they believe in, that PMI Belgium can nurture and develop even further the community of project managers in Belgium.\nBronze: You will receive standard visibility.\nSilver: You will receive a high level of visibility, standard user community interaction and support from the PMI Belgium management team.\nGold: You will receive a very high level of visibility, customized interaction and instant support from the PMI Belgium management team. You will receive outstanding recognition involving continuous brand exposure and standard opportunities for direct interaction.\nDiamond: You will get prestigious promotion during the PMI Belgium events and in the PMI Belgium official communications. You will receive the highest level of visibility, the most customized service and instant support from the PMI Belgium management team. You'll benefit from permanent brand exposure and customized opportunities for direct interaction.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The financial services industry is one in particular where a commitment to a triple bottom line is most critical. In leading up to 3BL Forum: Brands Taking Stands \u2013 The Long Viewlater this month, TriplePundit sat down with Andrea Barrack, Vice President of Global Corporate Citizenship at TD Bank, to talk about the stand that TD has taken to be the best corporate citizen possible.\nWhile Common Cents Lab is a research entity, its work does so much more than create interesting research. It creates easier, more intuitive ways for people to save \u2013 which is something that most people say they want to do, but find difficult.\nCHICAGO, February 24, 2015 \/3BL Media\/ \u2014 The Financial Solutions Lab at the Center for Financial Services Innovation (CFSI) with founding partner JPMorgan Chase & Co. today announced a $3 million competition for technology innovators working to address consumer financial challenges. This cross-sector initiative will identify technology-enabled financial solutions and provide winners with direct and indirect support to test and expand the availability of their products and services to consumers.\nMetLife Foundation and Sesame Workshop partner to advance financial capability to low to moderate-income families with young children around the world. This $20 million dollars, five year committment will initally rollout in Brazil, China, India and Mexico. View the video to see how MetLife and Sesame Workshop will work to educate and empower families on saving and planning for a secure financial future.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"JAKARTA, NNC - Pegadaian CEO Sunarso reveals the distribution strategy of Partnership Program and Community Development (PKBL) PT Pegadaian and calls it \"Pegadaian Bersih-bersih \/ Pegadaian Clean-up\".\nThis is implemented through various activities to clean the environment, including cleaning dirty rivers and sorting various types of garbages.\nFor example, Corporate Social Responsibility (CSR) PT Pegadaian promotes a program to convert household wastes into gold savings. The program was promoted with the inauguration of the Garbage Bank and Village Park, and dubs it \"The Gade Clean & Gold\", in Setia Asih Village, Bekasi District. Its expectation is its target village will be clean and comfortable as well, as Pegadaian's true work.\n\"Residents are taught to pick-up and collect household wastes, sort it, sell it by depositing and then replace it with gold savings. Its expected that collecting garbage can produce economic value, health value and environmental hygiene value,\" Sunarso said when inaugurating the Garbage Bank and Village Park, in Setia Asih Village, Bekasi District on Thursday, Aug 2..\nSunarso say there is help given to villages directly or indirectly, like business development. Its development require the allocation of Human Resources (HR) to provide assistance in managing a clean and neat administration.\n\"The administration is related to management, information access, markets and capital. So there are much that can be done by the State-Owned Enterprise in assisting Small and Medium Enterprises (SMEs),\" Sunarso said.\nThis clean-up concerns worshiping, for example, PT Pegadaian helps built worshiping places, like mosque in Setia Asih Village, Bekasi District.\nSunarso says that inorder to attract the interest of the community as well as a form of Pegadaian's concern towards Setia Asih village, a number of aid funds for facilities and infrastructure totaling IDR 599,789,000 has been disbursed. The fund is used to finance the construction of garbage banks and village parks.\nPegadaian since 2016, has provided assistance totaling IDR 146,375,000 for the construction of duck cages, mosque, batik training, how to make dodol, aqua ponic training, and waste management training.\n\"Through the clean-up Pegadaian program dubbed \"The Gade Clean & Gold\", Pegadaian further emphasized the benefits of its presence as a state-owned enterprise that is present for the country in the context of environmental concern, and equalizing economic prosperity in all regions in Indonesia,\" Sunarso said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"This entry was posted in Daphne, Girls, Lucy, Norah. Bookmark the permalink.\nGreat update! Thanks so much. Your girls are adorable!\nI'm so glad to see an update. I can't even imagine how busy you are with the three girls. I miss being close enough to watch them grow up and really get to know them, so I really appreciate your sharing of the girl's antics on here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Emmaus High School's Buzz Magazine hosted its annual fashion show on May 19 in the cafeteria, giving student designers and members of the club a chance to showcase their collections that they have been working on since September.\nThe student-founded club is extremely tight knit and dedicated to ideas of self-expression and diversity through fashion. The magazine and show were completely student run, with designers, photographers, writers, and models who spent time after school working on photoshoots, designing clothing, writing articles and assembling the annual magazine. Buzz Magazine president, junior Alisha Scott, feels she has benefitted from being a part of the Buzz family.\nThe magazine itself, which focuses on topics such as fashion, beauty, and lifestyle, provides students with an opportunity to produce content on the topics they take interest in. Students submit articles with topics that range from makeup tutorials to skin care and personal bucket lists. In between articles, the pages are filled with pictures from photo shoots which have been going on since the beginning of the school year \u2014 including gigs at the Bethlehem SteelStacks, and themes such as 1990s trends.\nJunior Willow Munson, the secretary and treasurer of Buzz, believes fashion is important for the EHS community to be involved in.\nSophomore Mehru Chaudhry, the director of technology for Buzz, agrees. She came to find her passions in the connections the club had to offer. As well as working for Buzz Magazine's website, she enjoys spending her time as a photo shoot photographer.\nDuring the fashion show, the club's various student fashion designers were able to present their clothing lines to the audience. Inspiration for each clothing line was diverse and personal to each student. Junior Samiey Werst's clothing line, \"Feeling Blue,\" was inspired by her favorite color, while senior Jessica Harnett's line, \"Moments,\" had pieces inspired by different moments in her high school career. Juniors Dylan Keenan and Joe Alston, Buzz's codirectors for men's fashion, were able to present their new collaborative brand, Bl\u00eb\u00e4ch\u00ebd Outfitt\u00ebrs.\nKeenan and Alston's brand focuses on good vibes and positivity, as well as breaking social norms. Being directors of men's fashion, they are taking on a challenge; as fashion is traditionally regarded as a female-oriented hobby.\n\"We want to have a more broad spectrum and be able to reach more people\u2026 [Buzz] was basically a women's club in the beginning, and we want to reach out to more guys,\" Alston said.\nKeenan admits he was not sure about designing clothing at first, but came to love it as time went by.\nJust like the designers and photographers, being a model for Buzz has its experiences. For junior model Brianna Friend, this event was her first time modeling in a fashion show.\nScott feels that the fashion show was a success, and cannot wait for what next year has to come.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"6 Best Fishing Kayak for The Money Reviews 2019 | Is Riptide Our #1?\n5 Best Fishing Kayak for The Money Reviews 2019 | Is Riptide Our #1?\nKayaks come in many different styles such as inflatable kayaks. Size makes a substantial difference, so you'll want to assess the length, width, depth, and weight capacity when making a comparison.\nIt's easy to choose a favorite style, once you understand the differences between them. Recreational kayaks are shorter. They are easy to turn and maneuver. Longer kayaks, or touring kayaks, are a bit easier to paddle over long distances. They hold a straight line better and make it easier to stay on course. Shorter kayaks are lighter and less cumbersome to transport out to the water. This post will outline our favorite fishing kayaks for the money in 2019.\nWide boats are more stable in the water. Narrow boats go much faster and they offer better secondary stability if the boat is leaning to one side.\nThe depth may be important for taller paddlers. Taller sides also help keep out water and offer more storage space. One of the negatives is that taller boats catch the wind and slow them down.\nBe sure to calculate the combined weight of paddlers and gear when making your final decision on a kayak as lighter kayaks will be more negatively affected by heavier loads than heavier kayaks which are naturally more buoyant and can handle more weight.\nCanoes were the original fishing boats. Improvements in design developed into the kayak, a boat that is more stable, steers better, and has plenty of room for fishing gear because of its pointed shape, allowing for better paddle efficiency. Canoes make a better choice for family boating, due to their size.\nCanoes have higher sides than kayaks. They are more lightweight for their size than kayaks, which makes them more susceptible to being blown by the wind. The simple design makes them more difficult to steer than a kayak. It is more difficult to paddle a canoe so that it tracks in a forward straight line and canoe paddling is slower because of the single paddle.\nKayaks sit lower in the water. Many models of kayaks have hull designs that make them easier to steer. Kayaks steer more efficiently than canoes and are easier to maneuver overall. The double-bladed paddle makes paddling easier and faster. Kayaks are better equipped to mount fishing gear such as rod holders, anchor controls, fish findings, and gear bags, but there is some equipment that is good for both such as these kayak\/canoe carts. For the same width, kayaks are generally more stable than canoes.\nClick here for our TOP rated product recommendation!\nThis is a 10' long fishing kayak with a large, comfortable seating area. It has a sit-on-top cockpit that is designed to allow easy entry. It has two flush-mount fishing rod holders and one swivel rod holder. The portable accessory carrier can be used for extra storage. It also has a dry storage compartment and shock cord deck rigging.\nThis is a stable canoe with deep hull tracking channels and stability chine rails. It's a sit-on-top style with an adjustable padded seat back and seat pad. It has two flush-mounted fishing rod holders and one top mount fishing rod holder. It's made of high-density polyethylene construction and comes with a 5-year warranty.\nNavigate difficult to reach shores with this fishing kayak with a stable and efficient hull with a roomy cockpit. It has an adjustable padded comfort flex seat, thigh pads, and adjustable foot braces for the ultimate in comfort. It has flush mounted rod holders behind the seat, an anchor with a trolley system and a deck bungee.\nThis is a great kayak for fishing with a buddy. The dark green color camouflages the boat from the terrain. It has double lock valves, adjustable inflatable seats, and comes with a large carry bag. It's made with multiple chambers, so if one chamber is punctured, it will stay afloat. The Airtight system is guaranteed not to leak.\nTwo people get to fish from this inflatable kayak set. It comes with aluminum oars and a high-output air pump. It sits low in the water, which makes it a great choice for navigating small lakes and mild rivers. It comes in a bright yellow color with sporty graphics for high visibility. Adjust the inflatable seat for your personal comfort. It has a removable skeg for directional stability.\nThis is a 1-person inflatable fishing kayak. It's made of durable welded material with bright green graphics for high visibility. The cockpit is roomy and it has a cargo net to store extra gear. It comes with an 18\" aluminum oar, high-output hand pump, and repair patch. It holds people up to 220 lbs.\nYou won't have to look to long to find a kayak that is affordable. Inflatable models will be less durable than they polyethylene counterparts, but they'll be much easier to get to the water. Most models have extra storage compartments and rod holders for fishing rods. Go fishing with friends or grab a single seater and go alone.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mother or Wife, who is more important and should come first in a man's life ? An intriguing, captivating and suspense filled movie with loads of morals. Watch and learn.\nWe are glad to keep you entertained. We recommend this for https:\/\/goo.gl\/NpBfdt we hope you love it and remember to share with friends.\nThanks for your watching. We recommend this for https:\/\/goo.gl\/6wUpTA we hope you love it and remember to share with friends.\nWe are glad to keep you entertained. We recommend this for https:\/\/goo.gl\/7t8vy8 we hope you love it and remember to share with friends.\nstill dont know why ibrahim chatta isnt yet norminated for best male actor of the year ..\nI wonder too...I think he has attitudinal problem towards the legend in the industry.\nBest movie so far so good ,I love the suspense from the beginning to the end. WoW Most watch for all.\nAso Oge - Latest Blockbuster Yoruba Movie 2018 Starring Ibrahim Chatta, Muyideen Oladapo.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Whether they realize there is a name for it or not, organizations and senior HR leaders are looking for ways to help employees effectively cope with workplace change, strain and anxiety. That is because the nature of the business world is a fast paced, dog-eat-dog world, where competitive advantages rely on the ability to be flexible, adaptive and mentally tough.\nIndividual and organizational resilience training is the name of the game, and a necessary tool for all organizations with expectations of success. TRACOM has developed a new Business Case for Resiliency Whitepaper which outlines resiliency research that comes from TRACOM participant research as well as third-party studies, highlighting business need for resiliency and the value it brings to organizational training programs.\nIndividuals with resilience skills remain motivated in their work during trying situations without having their health damaged. Just like everyone else, resilient individuals and organizations do not always succeed, however, they do have the understanding and capacity to bounce forward from challenges rather than succumbing to negativity and becoming overwhelmed by stress.\nTRACOM's rater research shows that employees with resilience training are 22% more likely to be personally unaffected during stressful situations. While many companies have now begun to offer wellness programs to manage stress, such data proves that if organizations want to boost their bottom line through a more adaptive workforce, their organizational training programs must incorporate not only physical but also mental health programs into their employee wellness offerings.\nYour company's culture is highly influenced by your organization's resiliency. We all know how contagious emotions are, and this is true both for negative and positive. If our direct manager has a bad attitude about a certain project or change initiative, her direct reports will too. But positive mindsets are contagious, too. In fact, TRACOM research tells us that those with resiliency training are 12% more effective at positively influences others during times of change.\nFurthermore, resilient employees are also 20% more comfortable initiating change and 18% more helpful in building a positive culture. View additional research in the Business Case for Resiliency Whitepaper here.\nLuckily, our brains are highly elastic even as we age and we can enhance our ability to view failures as minor setbacks and challenges as opportunities.\nResiliency training programs help employees recognize and understand their natural thoughts and behaviors as they arise. In turn, this makes employees better equipped to respond to negative feelings with actionable solutions. Employees with these skills are better equipped to respond to unexpected workplace changes with greater confidence and ambition.\nTRACOM's Adaptive Mindset for Resiliency can help your organization teach employees about sources and response patterns to workplace stress. We provide practical strategies for altering counterproductive responses and our programs can work independently or in conjunction with existing organizational training programs already in place.\nMake your employees more resilient and better able to deal with challenges in today's complicated workplace. Your entire organization will benefit.\nView the Business Case for Resiliency whitepaper here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"In one of my earlier post on 'site for fairly used cars for sale', I shared with you an article on where to buy fairly used cars in Nigeria and their prices. That article contain a list of websites I considered to be top 10 places to buy new and fairly used cars in Nigeria. Among those sites is NAIRALAND AUTOS.\nNairaland autos is a section on Nairaland forum that offers users opportunity to buy or sell new and fairly used cars in Nigeria. The auto section is a good place to haunt for fairly used cars for sale by owners.\nA visit to Nairaland auto section shows links to varieties of new and fairly used cars posted by their owners.\nTo view any of the new posted cars, click on the car posted title link.\nBefore you go into haunting for new or used cars for sale on Nairaland Autos, consider following these 10 steps to buy your right choice of new or fairly used car.\nStep #1.What do you need the car for?\n\u2022 Spare parts should be readily available.\nFor a car taxi drop business purpose in Nigeria, you need cars like Toyota product (Toyota Camry, Toyota Corona) Mitsubishi car, Peugeot 504 Station Wagon car etc.\n\u2022 Air conditioned, and well homemade interior.\nSteps#2.How much do you wish to spend in buying the cars?\nCheck the regular market price for second hand cars. If you don't have enough money to buy the car, consider saving money for the car rather than borrowing. It is usually difficult to repay money borrowed to purchase a car.\nAs a way of assessing the car on your own, you can check different reviews that have been written concerning the car or similar car you want to buy. A fairly used car with a second hand value is a good one. Reviews will show nature of fuel consumption of the car, the availability of spare parts, model and possibly durability.\nIt is important to check the history of the car you want to buy. With the use of the vehicle identification number (VIN), you can check history reports of the car. History report will reveal if the vehicle have had major damage in the past, some notable changes made on the car in the past and some other past important information concerning the car.\nAfter your checks and review, then you can contact the seller probably through phone call. On the course of your discussion with him, ask him why he is selling the car, some of the major repairs that have be done on the car before and the ones you will do after you purchase the car.\nIf you are convinced that you will buy the car, you have to schedule a day to meet the car seller one on one and the car you want to buy for an inspection. If you don't have a good knowledge of the car; it is usually better to go with a car mechanic or someone who have a good knowledge of the car. Proper inspection should be done on the car as most cars that are sold as fairly used cars usually have one fault or the other.\nAfter inspection, drive the car and allow your car mechanic to drive it too. Many operational hidden fault of the car is usually detected at this stage. Listen to the sound of the engine and other noticeable sound. If the sound is not good for your liking; it is time you end the deal.\nIf test drive the car and you are satisfied, you need to negotiate for the price with seller. Look! You have to be smart here. When bargaining for the car price, try to beat down the price to a reasonable extent. Do not be intimidated by the seller's negotiation skill or by his persuasive marketing skill.\nYou also need to be smart here too. Do not pay for the car by giving the money to seller in hand. Every payment should be made through the bank. You receipt for payment should be properly documented. As you are paying for the car, you are getting the vehicle particulars and license from seller simultaneously.\nMake sure that the particulars and car license are the original documents of the vehicle. You can confirm this before making payment to be on a safer side.\nThe prices of fairly used cars in Nairaland autos vary according to the car seller. Those cars are used cars for sale by Owners, so there is no consistent price list for fairly used cars in Nairaland. This is why there is avenue for bargaining of the car price. Don't take what you see as the price of the car on Nairaland autos to be the actual cost of the car. There are room for negotiation.\nA better way of search for nairaland cars through Google is to include the state in which you're residing on. Example; nairaland cars for sale in lagos, nairaland used cars for sale in Abuja, , nairaland used cars for sale in Portharcourt, , nairaland used cars for sale in Ibadan etc.\nTo sell fairly used cars on Nairand, visit the Nairaland Auto section. If you have not registered on forum before, register and post your car for sale. Include all necessary contact details.\nMy husband and I are looking for a used car, but we are having a hard time finding one that we like. Your steps sound like a good way to make sure the car we end up buying is reliable and a good deal. I'll be sure to check the history report when we find a car we want to buy.\nBuy clean tokunbo vehicles at a cheap and affordable prize,call 07067414341.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Home to some half a million inhabitants, the Algarve in Portugal is one of the most popular tourist destinations in Europe. The glorious weather and delicious food make it an irresistible haunt for the millions people who flock to the region every summer.\nLocated in this bustling area is Lagos, a coastal town with wonderful buildings and a fascinating history. However, despite the obvious attraction of the popular tourist destination, the population of the area swells so drastically during the summer months that it can make it difficult to get around and even harder to do everything you wanted to do.\nMoreover, during the high season, prices sky rocket in the historic town. However, there is a way that you can beat the crowds and take a sightseeing holiday of a lifetime: all you have to do is take your trip a bit earlier, during Easter.\nThere is plenty to see and do in the beautiful area of Lagos, and while it may not be the most popular time to go, Easter can actually get you closer to the area's main tourist attractions, while also allowing you to enjoy the stunning weather associated with the region.\nIn fact, even in winter, the temperature in the area can reach up to 25 degrees Celsius, and only in very rare circumstances does it drop below zero. During spring, you'll have back to back days of non-stop sunshine without having to compromise your plans to accommodate the crowds.\nThe history of the town goes back over 2,000, with many ancient paintings depicting the area's importance as a Mediterranean port. During these two millennia the region has seen a lot of coming and going, and is home to a varied and engrossing array of architectural styles, featuring constructions as diverse as the prehistoric Menhir of Cabeco do Rochedo and the 18th Century Church of Santo Antonio.\nAlso along the coast are several other areas of natural beauty, including Mercy Point, the Golden Coast Grottos and the Lagoon of Alvor, all of which you can enjoy away from the crowds if you visit the region during Easter.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Upvalue myself by your height.\nRepeating the monotone song for the green shade.\nAs a tree standing together with you.\nOur leaves merge in clouds.\nCan understand our peculiar words.\nAnd heroic torches as well.\nActually we forever rely on each other.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most girls will agree. The best way of getting things moving on any new job is the purchase of a brand new toy.\nWith the evenings starting to draw in, I'd been thinking about the wood left over from the tree felling earlier in the year. It should have been cut into logs by now. Our chainsaw, it seems, falls short of this job. Only a big boy's machine will do. Brochures arrived thick and fast, plaid-shirted musclebound Rambo types demonstrating their wares on each and every page. Effortlessly shinning up trees, wielding all manner of high performance equipment, leaving nothing but chippings in their wake. Without doubt we had found the product required.\nWithin days a stack of fire wood, suitably rustic, appeared alongside the shed.\nIt was on Wednesday that the real trouble began, the tranquility of the garden shattered by the firing up of the chainsaw. Not so long after, the thud of something heavy falling to the ground. I follow the noise to a small clearing down by the river's edge. There's a horrible graunching sound as the saw comes abruptly to a halt. Stuck fast in the trunk of a tree.\nSome cursing and an accusatory glare. My fault entirely, allegedly. I've turned up at exactly the wrong moment. Again.\nAn old fashioned hand saw finally did the trick. A long and tedious process of cutting through the trunk from the opposite side. The tree starts to wobble and the chainsaw is free. What followed was one of those slow motion things. I could hear every creak and tear as the trunk began to fall..\nThe river on two sides. Impenetrable undergrowth on the other. Thankfully the water is only a foot or so deep..\nThe whole canopy above my head is on the move. Surely more than one tree..\nA monumental crash, a second and a third.. and then silence. Ne'er a single tweet or chirp.\nWe look up through a new and significant window on the sky. Unaccustomed sunlight streaming through the dust. Like a set of dominoes, not one tree but three. And should we wish to cross the river, we now have a bridge.\nYou are a wonderful teller of stories. I can just picture the scene. As you say,at least you now have a bridge should you wish to cross the river!\nSounds very familiar \u2013 my beloved won't do a job unless he has a gadget to do it with. He had the same problem as you with a chainsaw \u2013 I think all beginners go through it. Glad it all turned out well in the end.\nI suppose I am a bit the same in the kitchen. Cupboards full of gadgets, some used only once. The chainsaw, on the other hand, has many years of guaranteed employment!\nOh golly, and you lived to tell the tale\u2026 Quite an eventful time you've had recently!\nYour chainsaw looks like my chainsaw \u2013 the brand related to the sewing machine -fancy that.\nThe woodland has been left unmanaged for years, some trees dead and leaning on others (hence three down on one cut). Practice makes perfect..!\nThose were amazing photos of the big tree coming down. The landscape will look so different now.\nIt is still surprising me when I look in that direction and see the vast expanse of sky! But there is so much more light now, inside and out. Much better.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Treating taxpayers fairly is what the Taxpayer Bill of Rights is all about. In 2014, the IRS adopted this collection of 10 taxpayer rights as proposed by the Taxpayer Advocate. However, the IRS was not establishing new rights with this document. Rather, the Taxpayer Bill of Rights combined existing rights woven throughout tax law, regulations and policies and put them in one charter, making them easier to find.\nIRS employees are responsible for knowing and upholding these rights, ensuring that taxpayer matters are handled fairly in every interaction. You can find the Taxpayer Bill of Rights on the IRS website and in IRS offices. Below, you'll find a summary of each right.\nWhether you're dealing with the IRS in response to an audit, or simply filing your annual return, you should know the IRS affords you these fundamental taxpayer rights.\nThe right to be informed \u2013 You have the right to know what you need to do to comply with tax laws. This includes clear explanations of the laws, procedures, IRS decisions and the outcomes concerning your tax account. For example, if the IRS were to disallow your claim for a refund \u2013 even partially, it must explain the specific reasons why.\nThe right to quality service \u2013 You're entitled to prompt, courteous and professional assistance, including easy-to-understand communications, when you work with the IRS. Generally, if you feel your service was inadequate, you have the right to speak to a supervisor.\nThe right to pay no more than the correct amount of tax \u2013 You have the right to pay only the tax that is legally due, including interest and penalties. Additionally, you should expect the IRS to properly apply all payments that you've made.\nThe right to challenge the IRS's position and be heard \u2013 You have the right to raise objections and provide additional documentation in response to formal IRS actions or proposed actions. In the event you do challenge a certain action, you should expect the IRS to consider your challenge promptly and provide a response if it does not agree.\nThe right to appeal an IRS decision in an independent forum \u2013 You have the right to a fair and impartial administrative appeal of most IRS decision, including any penalties. You should expect the IRS to provide a written response of the Office of Appeals' determination. IRS publication 5 outlines how to prepare a protest if don't agree with the IRS findings.\nThe right to finality \u2013 You're entitled to know how long the IRS has to audit a particular tax year, collect a tax debt or know when an audit is complete. Additionally, you have the right to know the maximum amount of time allowed to challenge the IRS's position.\nThe right to privacy \u2013 You have the right to expect that IRS will be no more intrusive than necessary as it conducts any inquiry, examination or enforcement action. Additionally, you should expect the IRS to respect all due process rights and will provide a collection due process hearing where applicable.\nThe right to confidentiality \u2013 You have the right to expect that the IRS will not disclose any information you provide unless authorized by you or the law. If an IRS employee, return preparer or others wrongfully use or disclose your return information, you should expect appropriate action will be taken.\nThe right to retain representation \u2013 You're entitled to retain an authorized representative of your choice, such as a Certified Public Accountant (CPA) or Enrolled Agent (EA), to represent you in front of the IRS. If you can't afford representation, you have the right to request assistance from a Low Income Taxpayer Clinic.\nThe right to a fair and just tax system \u2013 You have the right to expect the tax system to consider circumstances and facts that could affect your ability to pay, your liability or even your ability to provide timely information. If you're having financial difficulties or if the IRS hasn't resolved your issues properly or promptly, you should expect the right to receive assistance from the Taxpayer Advocate Service.\nNavigating your taxes and working with the IRS can be complex, but it should also be fair, so it's important that you're aware of the Taxpayer Bill of Rights.\nNot sure you want to go it alone? Our tax experts can help. Whether you need help researching your IRS account, representation in an audit or assistance with other issues, we're here for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzablpy b/data_all_eng_slimpj/shuffled/split2/finalzzzablpy
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+{"text":"Ahmed is currently the Director of Hacks for SignalFire, where he helps Engineering students find jobs at Bay Area startups. Prior to this he was the Bay Area Leader for Startup Weekend while working on his own startup, Go Go Mongo! an educational game on iOS for children. Before moving to San Francisco, Ahmed worked for IBM Global Business Services as a Data Architect for over 9 years. He has an MBA from Babson College, and a Bachelors of Science from the University of Minnesota.\nWondering how to get your brand new app noticed in the Apple App Store? Come learn strategies and tactics for marketing your iOS App.\nIn this class, you'll get some insight as to how startups go about hiring their employees as well as how you can go about getting a job at a startup.\nWhether you live in San Francisco or you're thinking of moving there, we will get you caught up on everything you need to know about its startup community.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"2 Responses to How to Discuss Global Warming with a \"Climate Alarmist.\" Scientific Talking Points to Win the Debate.\nDid the US transfer ownership of uranium deposits to the old USSR?\nWhile the force of neutron repulsion was hidden from the public?\nSnopes.com did not convince me that collusion between the Clintons and the Uranium industry did not take place.\nThey simply put a different slant on it.\nSnopes cannot claim convincing arguments in many cases, if not all.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Symantec Corporation is the global leader in cybersecurity. Operating one of the world's largest cyber intelligence networks, they see more threats, and protect more customers from the next generation of attacks. Symantec helps companies, governments and individuals secure their most important data wherever it lives.\nImplement a new digital solution to give customers global access to social customer service with a quicker response time and a one-on-one customer experience.\nReplace prior vendor with Lithium Social Media Management to offer the ability to create an expanded top-tier customer care team that can respond and resolve issues more efficiently.\nWhat prompted you to seek a new solution for social customer care?\nInitially when we started doing social support, it was a small team of three people. We used Tweetdeck as a monitoring tool, but eventually as the team grew, we had to identify a solution that was built for collaboration. Another team at Symantec had purchased a license from a different vendor, so we gave it a go. But when we started using it, all the positive customer feedback immediately stopped due to the tool's unnecessary extra layers between us and the customer that caused extremely long delays in our response time. So we continued to seek a solution that was focused on social customer service. That's when Lithium Social Media Management came into the picture.\nWhy did you select Lithium Social Media Management?\nAt our first meeting with the Lithium team, we brought a long list of requirements and were pleasantly surprised to learn they had everything on our list. Lithium Social Media Management was the first tool we observed to be focused entirely on customer experience and enabling support to engage with customers and find solutions, thereby offering positive experiences on social networks. It was important for us to select a tool that was laser-focused on customer service, because that meant that future developments would always prioritize customer service. We chose Lithium Social Media Management knowing they will always continue to invest in customer service. That first meeting with Lithium was the beginning of a very good, new chapter for us.\nHow has Lithium Social Media Management addressed your needs?\nIt has fulfilled what we only dreamed we could do. Specifically, we've achieved a quicker response time (from 24 hours down to 3 minutes), true collaboration, an expanded team, and multi-language support (for example, if an inquiry comes into the queue in French, it automatically gets routed to our French team). Furthermore, we have a seat at meetings where colleagues are discussing support issues\u2014and we're finally able to contribute some data due to Lithium Social Media Management Analytics.\nWhat has been the impact on your organization?\nThe biggest impact to our organization has been the growth of our social customer service team (from just 3 to now 20 people) and our expanded hours of operation (we now offer weekend coverage which we couldn't do prior to Lithium Social Media Management, and in early 2015 we will extend our coverage to 24\/7 as a result of our expanded team). Symantec is investing in the growth of our social customer care team because they see the great results, especially with customer sentiment conversions. In addition, our agents have been able to respond much more efficiently, handle more cases with ease, and assist our customers more intelligently. We can handle more volume with less effort. We're able to make sure all customers are being responded to and no one is being left behind. The move to this more managed solution has been groundbreaking for us.\nWhat has been the impact on your customers?\nAll of the positive customer sentiments went away with our previous vendor. However, as soon as we switched to Lithium Social Media Management, the positive comments immediately came back. We saw more delighted customers who didn't have to wait long for a reply due to a significant decrease in our response time, and we gained the ability to seamlessly help our customers. There was an immediate spike in engagement with our customers as soon as we moved to the Social Media Management solution.\nLearn more about Symantec's digital success.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AppSafari review of ZYB was written by Joe Seifi on February 19th, 2008 and categorized under Contacts. Page viewed 5526 times, 1 so far today. Need help on using these apps? Please read the Help Page.\nBeen using this for some time. Gives me access to the stuff on my old phone.. Very nice indeed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Natuzzi Couch is free HD wallpaper. This wallpaper was upload at August 25, 2018 upload by rebeccarcahill in Sofas & Couches.\nNatuzzi Couch. This best picture selections about Natuzzi Couch is accessible to download. We collect this awesome picture from internet and choose the best for you. Natuzzi Couch pictures and pictures collection that published here was carefully elected and uploaded by rebeccarcahill after selecting the ones that are best among the others. So, finally we make it and here these list ofbest picture for your ideas and information reason regarding the Natuzzi Couch as part of Rebeccarcahill.com exclusive updates collection. So, take your time and get the best Natuzzi Couch pics and pictures posted here that suitable with your needs and use it for your own collection and personal use. About Image information: Image has been uploaded by rebeccarcahill and has been tagged by natuzzi black leather couch natuzzi brown leather couch Natuzzi Couch natuzzi couch connectors natuzzi couch costco natuzzi couch for sale natuzzi couch price natuzzi couch reviews natuzzi couch the bay natuzzi leather couch in Sofas & Couches field. You might leave your thoughts as feedback to our page quality.\nThank You for visiting our website. at this moment we're delighted to announce that we have discovered an incredibly interesting content to be discussed, namely Natuzzi Couch. Many people attempting to find info about Natuzzi Couch and certainly one of these is you, is not it?\nNatuzzi Couch is high definition wallpaper and size this wallpaper is 822x506. You can make Natuzzi Couch For your Desktop picture, Tablet, Android or iPhone and another Smartphone device for free. To download and obtain the Natuzzi Couch images by click the download button below to get multiple high-resversions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzacnqu b/data_all_eng_slimpj/shuffled/split2/finalzzzacnqu
new file mode 100644
index 0000000000000000000000000000000000000000..425fe2f1e75ce1aac11c3ddd8e64a37e6df4595c
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Rivers and Roads: MEET CASSIE.\nHey lovelies. Today I get to introduce you to Cassie of Kent Heartstrings. She has a precious blog and an even more precious family. After you're done getting to know Cassie, you should check out some of her wonderful posts! I love these Anthropologie inspired confetti glasses and this unique DIY succulent holder. I am wanting to put some pretty succulents in my room as part of the re-design, and I only wish I had room for a whole pallet! Without further ado, here's Cassie!\nI love the community. There are so many people to meet, interact with, build a relationship with and support. Its a beautiful, unique and wonderful community. I really do not know what I enjoy more, blogging on my blog or reading other people's. I am just honored to I get to be a part of it all.\nWhat fun things do you have on your Autumn bucket list?\nApple Picking. We live about an hour away from a small town that has several apple orchards nearby, we hope to head up there before the season is over! Gosh and just enjoying the beauty of the outdoors, its finally cooling off enough to be outside, and that makes me so happy!\nPumpkin Pie. Its a classic, but its my favorite! With fresh whipped cream! Yum!\nI love being crafty. You can find me tinkering on many different things. When I am out and about, I look at stuff all the time and think, \"I can make that\"...so I always have 5 projects going on simultaneously.\nI love hosting too, parties are so much fun! I love all the little details that go into an event that really make it stand out, and makes you feel special. Good thing I host Kreative- a craft night for women every month, because it blends both of my favorite hobbies...Creating and Hosting!\nT.Swift Shake it off. I mean it has a everything- catchy lyrics, toe tapping beat and a great message. It just makes me happy, and our whole family happy, every time it comes on its s dance party in the car.\nWhat are you crushing on this month?\nBoots. All kinds. Riding boots, Ugg boots, Hunter Boots, Booties....Love, love love. Maybe I am crushing on them so hard because the weather here in CA is still in the 80s, so I just look longing at my boots.\nThat's all, folks! Hope you have a fabulous Wednesday. :) XOXO.\nLabels: Q & A, sponsors.\nCan't wait to scan her blog!\nlove this- thank you for sharing and excited to check out her blog!!\nLET HEAVEN AND NATURE SING.\nTHE GREAT CHRISTMAS EXCHANGE 2014.\nTOP 5 REASONS TO JOIN TRISTYLE & CO.\nETSY SWAP 2014 LINK UP.\nMEET ME ON THE WATER.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beat butter and sugar together, cream well. Add eggs, one at a time, beating well after each one is added. Beat in vanilla.\nCombine dry ingredients in a separate bowl, then add the dry alternately with the milk to the creamed mixture.\nPour the batter into a greased 10-inch tube pan, leveling it in the pan somewhat as batter will be rather thick. Bake at 325\u00b0 for 80 minutes, or until chocolate pound cake bounces back when touched lightly in the center with finger. Let cake cool completely before adding ganache.\nPour chips into a medium mixing bowl. Then, pour heavy cream into a small saucepan, and heat on low to medium heat until milk JUST begins to boil. Remove from heat and pour over chocolate chips. Mix with spoon or whisk until all chips are melted and mixture is smooth. Cool icing until it's a thick texture, but not too thick, because you'll want to drizzle it over the cake. Then, take a spoon and start drizzling!\nWe recommend keeping the cake in the refrigerator, especially during warm months.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Considered a Zuni form, fetishes have been carved since around 600 A.D. Most Native tribes used the charms, but the Zuni have become known for the quality of their workmanship. Today the refinement and creativity involved in the carving of fetishes has muddied the ago-old distinction between craft and art. Not from Zuni, Romero is returning the fetish to its more primitive form. In Romero's eyes, less is more.\nRomero loves rocks and animals. He searches the land at Cochiti for the proper stone, which gives his pieces a distinct look. The search takes longer than the carving.\nRomero claims to be a private person. He prefers you know him through his art. He tried painting and several other jobs before he found his calling. Now that he has found it, he intends to stick with it.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've always loved shopping with my best friend. You know, casually strolling the mall, stopping for a coffee break, giving each other's opinions in the dressing room, helping my girl pick out the perfect outfit \u2013 I love all of it. But, here we are years later with numerous kids, time consuming hobbies, different home towns, and busy social schedules. Long gone are the leisurely days at the mall.\nYes, it's sad to give up those frivolous and fleeting moments of youth, but luckily I don't have to give up being my bestie's personal shopper. Thanks to the good ol' internet I still hand select and send her items I think she would enjoy. Yes, in my spare time (ya right! \u2013 thanks three kiddos) I online shop for people other than myself. It's a bit of a crazy habit, no?\nSo, without further ado, let me introduce you to my longest lifelong friend, or as my babies know her : Auntie Shanni . This fab and fun Mama just recently relocated to the Portland, Oregon from Carlsbad, California. Talk about a weather and life change right?!\nNow, what kind of new clothes do we think this busy, hands-on mom of two active and adventurous boys needs for her new jaunt in the Pacific Northwest? Funny you should ask, because I am going to show you everything I told her to add to cart.\nIf you are a causal and down to earth lady who is mom life-ing it in a climate that needs layers on layers. Thanks to my bestie, I have you covered. Brew a cup of tea, pour a glass of wine, or as Auntie Shanni would do, diffuse some oils, and then take that click happy finger and start adding these amazing items from The Nordstrom Anniversary Sale to you cart.\nI'll let you know which items she ends up getting, so stayed tuned!\nPS. We never, ever, ever have fun together. Like, never.\n\u00ab Guess What's Back. Back Again. And, a GIVEAWAY !","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rangeley Lakes Heritage Trust recently presented Blue Flame Gas of Rangeley with its Spirit of the Community Award. The award recognizes community individuals and businesses whose commitment has touched the lives of many in the community. From the left are Rangeley Lakes Heritage Trust President Paul Reynolds, Blue Flame Gas co-owner Aaron Vryhof, Rangeley Lakes Heritage Trust Executive Director David Miller and Blue Flame Gas co-owner Meagan Vryhof.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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index 0000000000000000000000000000000000000000..beffd3f37c3068ad4d3591a30f8d8a56b18840f8
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Barrie Stephen \u2013 one of Leicestershire's best known hairdressers \u2013 as become the first patron of a charity that once helped him on his way to business success.\nHe is backing the Sir Thomas White Loan Charity, which provides interest-free loans to new business owners in Leicestershire and Rutland looking for a financial step-up.\nOver the years the city centre based charity has helped thousands of entrepreneurs aged 18 to 35 become self-employed.\nBarrie sought help from the charity 23 years ago to establish his first salon in Anstey.\nNow, more than two decades later he owns five salons across the county, as well as his dual-brand B-Styled salon.\nHe has also picked up several accolades throughout his career such as Leicestershire Businessman of the year.\nBarrie said: \"I was honoured to be asked to become the first patron of The Sir Thomas White Loan Charity.\n\"I didn't need to think twice and the role is a perfect fit for me, because without the charity, I wouldn't be in the position that I am today.\n\"I still have my business plan which formed part of my application, and the process of applying for my initial loan 23 years ago was amazing.\n\"I'm now excited to share my story with others who are contemplating engaging with The Sir Thomas White Loan Charity, as their support really can make a huge difference to anybody wishing to become self-employed or eager to make a first step in creating a business empire.\nIn becoming a patron of The Sir Thomas White Loan Charity, Barrie is keen to help spread word of the support on offer to young people in the county by attending events to speak about his experiences.\nDavid Howard, chairman of Sir Thomas White's trustees said: \"The charity is delighted to have Barrie become our first patron.\n\"His story is one that many people across Leicestershire are aware of, and The Sir Thomas White Loan Charity is proud to have played a small part in Barrie's success, which showcases to current and prospective borrowers what is possible with drive, passion and the desire to succeed.\n\"Alongside our Sir Thomas White Loan Charity ambassadors, we look forward to having Barrie help us spread the word of the interest-free loans available to entrepreneurs across the county.\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bring the farm into your kitchen by learning the skills to preserve seasonal vegetables. The ACC Elgin Sustainable Agriculture Farm will be providing freshly harvested produce for this class. Topics may include fermentation, pickling, dehydration and making preserves.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mike's Automotive provides 90,000 Mile Service services to Collinsville, IL, St. Louis, MO, Millstadt, IL, and other surrounding areas.\nMike's Automotive is the leading 90,000 mile service shop in Collinsville, IL, St. Louis, MO, and Millstadt, IL. We pride ourselves on ensuring that your vehicle is running at its optimal performance level. The safety of you and your family is our top priority. Due to the conditions in the St. Louis, MO area it is important that the Preventative Maintenance be performed at the recommended level. At the 90,000 Mile Service Interval the Vehicle Manufacturers require certain safety inspections and preventative maintenance. Our trained mechanics focus on only performing the services and repairs your vehicle requires, the safety of you and your family, keeping your vehicle running at its top performance, and ensuring that your factory warranty stays intact. We are able to bring you the highest level of service and properly maintain your vehicle at a fraction of what a dealer would charge.\nCabin Air Filter and Engine Air Filters \u2013 Air Filters are sometimes able to be cleaned. They also need to be replaced periodically. The Engine Air filter keeps debris out of the engine and ensures proper air intake through your engine. Depending where you live in the St. Louis, MO area your Air Filter might need to be replaced more frequently.If your Engine Air Filter is clogged it will dramatically reduce the engine's performance and fuel mileage. Cabin Air Filters are also important to maintain. This filters the air in the passenger compartment. Keeping the Cabin Air filter clean is important to ensure that you and your family are breathing clean air. Failure to keep this clean and debris free can also cause stress on the vehicle ventilation system. This can cause a foul smell and undue stress to the vehicles blower motor and ventilation system.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shannon Echlin's photography of children and babies uses depth and \"naturalness,\" darkness and light. Her work has been published in Toronto Life, The Eye, The National Post and the Globe and Mail. She is also a regular blogger for the Huffington Post.\nAs a kid I used to take lots of pictures; my dad's camera was always my favourite toy. When I hit high school I just kept going with it. Yearbook club lead to summer jobs at local newspapers which lead to Sheridan College for Applied Photography, then I worked as an photographer's assistant. I had my first gallery showing at 22. Photography is just part of who I am.\nI spend a lot of time looking at art and this is what inspires me. I like for there to be a feeling of peacefulness but also depth. I don't use props or fancy backgrounds. I want the kids to be the focus. I'm willing to spend however long it takes to get portraits where the kid's true personality shine through.\nMy clients are anyone with kids (or dogs!). I have met so many amazing families. Different cultures, ages, income levels. I think I have a certain look that draws all sorts of people in. I love that about being an artist. Having the ability to touch so many different people is something is very special.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Many owners find themselves frustrated each day, firefighting the employee complaints, unmet expectations, individual requests, and repetitive questions. It's easy to become overwhelmed, frustrated with the chaos, and stressed about what you don't know. Your people are the only assets who make a decision each day whether to come back to work, so creative ways to find and keep them is critical in our economy. Peace of mind comes when you know your business complies with laws, answers employee questions properly, and resolves issues to mitigate risks and get employees focused on doing a great job.\nUnlock the hidden value of your best asset. Your best people.\nWe deliver solutions, tools, information and advice which address your people problems and align your people, culture and HR practices with your business strategy. Wanting to be a strong leader whose business functions and even thrives without constant attention? Planning to focus on the business (rather than in) or step into an advisory role? Worried about whether your business is ready to meet the future? Our talent management and culture strategies and regular support and involvement in your business will prepare you and your business for the future.\nUnderstanding your business strategy will significantly improve the chances of a successful transformation. We collaborate with business owners and their teams to facilitate the role of high-level orchestrators of a complex process that involves large numbers of discrete initiatives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzafyyh b/data_all_eng_slimpj/shuffled/split2/finalzzzafyyh
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index 0000000000000000000000000000000000000000..a93cfcd50da72cdc341cfc7f565e5b0548f350d2
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Here at Marks Sattin we have a fantastic opportunity for our client who are a Regional Accounts Practice based in Birmingham. Looking for someone to come in at Manager level with the ability to progress fast and help add to their portfolio of clients.\nReview and discuss clients on their Corporation Tax and other tax matters etc.\nAre you interested in this role or know someone who is? Apply now, get in touch with myself, James Hipkiss, to find out more about the role on 0121 231 7150 or email @ James.Hipkiss@MarksSattin.com, generous referral scheme also on offer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"NP13D, Radio station power supply - Elektronika Nagli\u010d d.o.o.\nNP13D is a classical analog power supply specifically designed to supply radio equipment. It is protected against too low or too high output voltage and against short circuits. In case of short circuit it turns off the output at time intervals of 2 seconds, therefore the device is protected against greater damage.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Description: You are the commander of the army of the dead and are pursuing to dominate the country. Command your units in skirmishes until your opponent has no more troops left. Have fun playing Bone Attack Hacked Unblocked!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Releasing September 10th exclusively on the PS Vita, Killzone Mercenary developer at Guerrilla Cambridge is one of the latest eye-pleasing head turners to grace Sony's mobile hardware. After taking in the games incredible sights and sounds in both single player and multiplayer during E3, we were equally excited to find this in-depth walk-through of the game for your viewing pleasure. While its never too good to assume greatness for an unreleased title Killzone: Mercenary is certainly trying to defy those odds here at Game Insider. From the impressive visual fidelity to the immersive gameplay, there is no denying it truly has it all where it counts.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sept. 22 (UPI) -- In an auction that ended Saturday for control of British broadcaster Sky, Comcast outbid 21st Century Fox and its backer, Disney , with a winning bid of $39 billion. Fox already owns 39 percent of Sky. Both companies were vying for the other 61 percent during the rare, three-round auction, which was managed by Britain's Takeover Panel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahlxv b/data_all_eng_slimpj/shuffled/split2/finalzzzahlxv
new file mode 100644
index 0000000000000000000000000000000000000000..6a288f3fbb85ba8cbbfa48647824ae6dba96d2f2
--- /dev/null
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+{"text":"I'm living through this magical phase right now. I'm always busy doing something and I'm nearly always thrilled about where I am. Life is shiny.\nFor instance, Saturday I rushed home to take a quick half hour to change clothes and eat before heading downtown for Ballet Arizona and the Desert Botanical Garden's presentation of Opus Cactus. Though 7:30 suddenly felt like much later in the evening than it perhaps should, I felt like I was sitting in a dream there in the theatre, like I was drifting through an ethereal ending to an awesome day (part of what made the day awesome was that I pet a beautiful alligator, but that's another story).\nSo what was this Opus Cactus? Initially I thought it would be one of the performances to skip (I don't, after all, aim to see every production from Ballet Arizona). Then I allowed a second glance and realized that it's a show inspired by the desert and that's exactly the type of thing I say I love and want to support. Though the show was presented by Ballet Arizona, the performance was put on by MOMIX, which is a touring company under the direction of Moses Pendleton. So it wasn't exactly ballet; the style was more modern, I would say, but not exactly \"modern.\" More like dance with a twist of Cirque du Soleil, actually.\nThere were a series of scenes that played out essentially as separate tableaus, each one representing something about the desert. We started with Desert Storm, where tumbleweeds bounced about the stage. We saw saguaros and snakes and lizards and representations of the sun. So everything had a poetic quality to it that formed a celebration of not just the Southwest but also other deserts in the world. The music, likewise, began with Southwest inspiration and moved on, in later scenes, to include other music from around the world. Such sounds also combined the traditional and the modern.\nLet me get back to that Cirque du Soleil reference. The MOMIX dancers are amazing. Such strength and agility and ease of movement. They were literally forming the shapes of the desert with their bodies. Often a scene would start with just silhouettes; the lighting might later increase so that you could then see the people in costumes, or sometimes the light stayed dim enough that all you saw were silhouettes. Either way, the visual scene was carefully controlled. When you're watching in awe of what bodies can do and how they can move and work together, the general feel of a dance tends to be sensual. But somehow this all felt like such high art that that it wasn't sensual at all: you became aware of, for instance, how one person was holding up another to form a shape and yet the effect was simply awe at the shape rather than focus on the two people. If that distinction makes sense. The point is, yes, it enhances the experience to realize how the dancers, as people, are moving. But the images that they are creating are the most important thing that you see on the stage--and they're images of plants and animals, desert imagery, not imagery of people.\nSome scenes I liked more than others. And sometimes I wished that instead of seeing separate tableaus (the show was short compared with what I'm used to at only about an hour and a half), we were seeing more of a complete and connected piece. But overall it was such an amazing performance and visual image that it turned out to be one of my favorite shows from the past year.\nAnd what made the evening more magical was the fact that it was not in the usual Symphony Hall but rather down the street at the Orpheum Theatre. This was my first time at this theatre, and I do admit that I found the inside layout a bit confusing at first (I like to always know where exactly I'm going, so it throw me off when I enter a new place where the layout isn't immediately clear). It's a much smaller location than Symphony Hall and has a completely different style. This theatre, outside and then both in the halls and in the theatre itself, is beautiful. The peacock ceiling above the farther staircase and the art nouveau (at least, that's the style it looked like to me) wardrobe and oil paintings upstairs and the chairs in the downstairs bathroom and the dusky, starry sky in the theatre. So many details all around to look at and take in; I can see why there are tours of the theatre during the day. Maybe because of the setting, I almost felt like people were more dressed up here than they are for the ballet or opera at Symphony Hall. Even if that difference was just my imagination, I felt at home in the beauty of it all.\nPetting an alligator at work and then dressing up in green sequins to watch dancers imitating the movement of a gila monster. What a day, eh?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TODO Update the file handle size after writing. Provide a size argument for stream based operations.\nInstantiates a new s 3 file system impl.\nClears (removes all content from) an S3 bucket.\nDeletes the content described by the given S3ObjectSummary list from an S3 bucket.\nRetrieves the bucket name to be used.\nRetrieves the amazon S3 client to be used.\nRetrieves an AmazonS3Client from a configuration file containing the access- and the secret key.\nSets the bucket name to be used.\naAmazonS3BucketName - The bucket name to be used.\nThe S3 client to be used.\naAmazonS3 - The AmazonS3 client.\naBucketId - The ID of the bucket to be created.\naRegion - The region in the Amazon landscape where to create the bucket.\naBucketId - The ID of the bucket to be cleared.\naBucketId - The ID of the bucket to be deleted.\naAccessKey - The according access key for accessing amazon AWS.\naSecretKey - The secret access key for accessing amazon AWS.\nThe client represented by an AmazonS3 instance.\naBucketId - The ID of the bucket from which the objects are to be deleted.\naS3SummaryObjects - The S3ObjectSummary list describing the objects to be deleted.\naConfigFile - The configuration file used to configure the AmazonS3Client.\nIOException - In case there were problems reading the configuration file.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you are not receiving the text message, please make sure the phone number on your account is correct.\nIf the phone number is correct, then your carrier may be blocking these text messages. You can contact them about getting them unblocked.\nAnother option is to use a friend's phone number to receive the text message.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Open House April 1st 2006. A day of free services, class demos, face painting, henna and more.\nFree week of Classes April 2nd to April 8th 2006.\nWorld Tai Chi and Qi Gong day: Area teachers and students \"play\" tai chi as we Celebrate World Tai Chi and Qi Gong Day at Dancing Dragon! April 15th, 2006 10 am to 4pm there will be free tai chi and Qi Gong workshops throughout the day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mac Users Get Their Own Search Engine - InternetNews.\nSearch engine Google Inc. Monday launched a version of its search engine specifically for Apple platform.\nThe new Apple search engine is based on Google's advanced search technology. Users can search for information and products related to the Apple Computer, Inc. (AAPL) the company said.\n\"Our Apple search site is the only place Internet users need to go online to locate Web sites related to Apple and Macintosh,\" said Larry Page, Google chief executive officer and co-founder.\nThe Macintosh platform had been on hard times the last few years, but the company has turned around to attract a new generation of users, thanks in large part to the launch of the iMac last year and the return of co-founder Steve Jobs to the position of chief executive officer of Apple.\nGoogle also offers customized search engines for Web content specific to the Linux operating system and the U.S. Government, Page added.\n\"As with our U.S. government and Linux searches, our new Apple-specific search engine leverages Google's core search technologies to quickly identify and target the most relevant search results.\"\nGoogle has the backing of investors such as Sequoia Capital partner Michael Moritz and John Doerr of Kleiner Perkins Caufield & Byers, who each invested $25 million in Google in June.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzahwmu b/data_all_eng_slimpj/shuffled/split2/finalzzzahwmu
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index 0000000000000000000000000000000000000000..dcba150dac5605c4bc735319248e848587ee2cbb
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+{"text":"Functional Intensive Resistance Exercise. The fastest way to fitness! Their purpose is to deliver the most phenomenal workout experience on the planet. A place where you can go and get challenged. You will be motivated by their coaches and tracked by their professionals. Be ready to be held accountable for your goals. Be ready to succeed!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Searching for a solution that would offset fast-rising healthcare costs and provide more savings than a traditional PPO discount, Custom Design Benefits introduced a reference-based pricing (RBP) program for its self-funded plans that is returning great results. When compared to traditional PPO plans, the third party administrator (TPA) reports fixed cost savings of 15-20%, and 20-30% savings for the cost of actual health claims. The copay-only plan design results in fair, transparent pricing for members, which can encourage participation. In one employer's case, introducing the RBP option increased employee participation from 30% to 80%, helping them avoid ACA penalties. Read more about why RBP and self-funding work so well together in the article below.\nFor more information or to contact Custom Design Benefits, visit www.customdesignbenefits.com or call (800) 598-2929 or 513-598-2929.\nAs they try to endure the non-stop increases related to health plan costs, employers often contemplate different plan designs and what will produce the most savings. They consider PPOs as a factor and are hopeful that incorporating one with their plan will be a winning solution as claim charges will look better on paper once network discounts are applied.\nWhat they soon realize, though, is that the root of the problem is this: care costs just keep going up at an unprecedented pace. In many cases, the PPO discounts aren't enough to offset the extreme pricing inflation over time.\nAt Custom Design Benefits, we're working with employers to find a practical solution to this. We're seeing more and more of them take control of the issue by making the move to a reference-based pricing (RBP) program. They're eager to learn how the concept works \u2026 why it's so effective.\nSo what exactly is the thinking behind RBP?\nWith our TrueCost RBP program and others like it, certain medical services are billed at a single, standard price based on negotiations between the TPA and hospitals\/care facilities. Think everything from rotator cuff repair to hip arthroscopy.\nThese prices are determined after considering Medicare reimbursement rates and factoring the actual cost by procedure, as well as a sufficient profit rate for the provider. Like the name implies, RBP gives employers and employees a reference point for what a service will cost and eliminates the guessing game that has long been associated with what charges might actually appear when the health claim arrives.\nRBP programs like ours are inspiring partnerships among multiple parties in the health care world (a good thing!) and creating some big-time savings in the process as they're all about establishing fair prices for everyone. They are creating a level of consistency that hasn't been seen before and preventing prices from fluctuating every time someone else receives the same service at the same facility.\nAlong with this clearer pricing, another advantage members are experiencing is the fact that plan design is usually much simpler under RBP. In our case, coinsurance and large deductibles have been eliminated and the RBP program follows a set copay structure depending on visit type. This encourages people to seek out primary care instead of going straight to the costlier urgent care alternative. Plus, since fees by service are negotiated with a network of hundreds of practitioners upfront, patients have avoided balance billing in nearly all cases.\nAt Custom Design Benefits, we're seeing RBP participation skyrocket. The number of employers adopting the TrueCost program has increased by 750% since it was first introduced in 2012. Employees appreciate the value and transparency of the RBP program, too. In one example, a client had just 30% plan participation under a traditional PPO arrangement. After implementing the RBP program, participation jumped to 80%.\nWhat about the numbers for cost containment? While most PPO plan costs are up an average of 7% since last year, our RBP program is seeing fixed costs that are an average of 15-20% lower. On top of that, actual claim costs are averaging 20-30% less.\nWith all the success RBP is having, we continue to explore ways to supplement these savings efforts. This includes the addition of Custom Wellness \u2013 a scalable program that focuses on improving employee health by incorporating things like on-site screenings and disease management. It integrates actual medical, prescription and screening data to further customize health coaching and make these efforts as effective as possible.\nWhat we're seeing with this emphasis on wellness is that the more we concentrate on factors that can influence the health of our workforce, the better chance our employees have in realizing successful medical outcomes.\nFor more information on Custom Design Benefits and how it has successfully implemented RBP and wellness programs, check out this article: http:\/\/www.customdesignbenefits.com\/news\/Rough-Notes-0616.pdf.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Please join the WWC staff to help remediate areas where Red Pines were removed during the Fall Logging Operation by spreading seed and planting new trees. Along the way we will also cut and remove invasive species. The Team approach is encouraged for this event and can include Families, Scout Troops, or other interested community members. This event is FREE and a great way to engage with this local outdoor resource!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A Lovan exclusive - our Studio Alchemist AccuTractor is the world most precious and zero error, auto positioning Protractor for tone arm\/cartridge alignment. Our AccuTractor can eliminate the possibility of central positioning error caused by traditional round hole design. With 0.08mm alignment lines width, we have one of the most refined alignment lines in the market!\nOur XCS-VNS vinyl record stands are handsomely designed to showcase your most favorite LPs, and for easy accessibility. It was constructed from real Birch wood, heat bending into beautiful art-shape. Our vinyl records stands are great addition to your vinyl collection, and will fit in both traditional or contemporary home setting.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"5 weeks to fitness. You got this.\nWhether you are a seasoned athlete, or coming straight off the couch, this plan will have you prepared to conquer your first Spartan event in 5 weeks of training.\nFollow our lead and you'll develop the confidence and capabilities needed to tackle any obstacle on course.\nDon't make excuses. Simply try your best every day and never give up.Winners put in the work, losers make excuses.\nYou may think it's too scary, you may think you're out of shape. Whatever you may think, just know that there are millions of Spartans around the world who are behind you, cheering you on to succeed.\nSpartan welcomes athletes of all backgrounds to our races. Don't think you're an athlete? Do you have a body? You're an athlete.\nYou don't need to be perfect. Which is a good thing, cause perfection doesn't exist. This plan will shape you into the strongest version of yourself, which is all you need.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaivta b/data_all_eng_slimpj/shuffled/split2/finalzzzaivta
new file mode 100644
index 0000000000000000000000000000000000000000..cfa75a02d46d02505c65e8de555ebf7092d62eb9
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@@ -0,0 +1,5 @@
+{"text":"Do you want to start a transportation business? Want to give your company a convenient and brisk fleet? There are numerous transportation companies that are renting, leasing and buying a cargo vans. A big cargo van have cover an area that range in size from 13 to 23 feet in length, and sit on an attached frame, unlike a tractor-trailer, which has two parts. Cargo vans carry goods and service instead of people, and work well to carry a small or big loads even with the time sensitive deliveries. If you are using a rental cargo vans, you can set up the shipping contract in advance, try to rent the van on the day of the scheduled pickup and delivery, and transport the goods from one place to another.\nThe first think that the potential customer will want to look at is size. Is the van is large enough to hold what will be it carrying? Before buying used cargo van take a look at the cargo space. Look at the condition of the interior determine whether the van will be able to serve the necessary purpose. It is also important to compare the van to other use cargo vans that are currently for sale. Compare to those man of the same make, model, year or mileage. Without seeing what else is available, it is impossible to tell if the van is fairly priced.\nBefore purchasing a cargo van, always ask about its history or maintenance records. Buyers will need to know how often the van has been serviced and if certain parts need to be replaced soon. To be a savvy buyer, it is crucial to know as much as about a used van as possible because seller will not always volunteer this information, you have to ask about it. It is better to focus some vital point before crack any deal.\nVan Class- there are two main categories of cargo vans- compact and full size. Full size van are segmented into three segments 1\/2 ton, 3\/4 ton or 1 ton. While these term are little to do any more with actual payload capacity, they are still used in industries.\nCargo Volume-first make a plan in your mind? How much space you required? Cargo volume measured in cubic feet provides a specific number to measure, it is better to do comparison with others cargo van before making any deal.\nLoad Length- most manufactures offer two length\/ wheel base option- regular and extended. So it is totally up to you to choose to choose which is more preferable to you.\nLoad Height- cargo van offers two type of roof- standard and raised. The raised model provides 6-feet, 4-inches of floor to roof clearance, allowing most crew member to stand upright, while accessing the cargo area.\nThese vans are used to transport goods and products from one location to another, due to this they suffer lots of wear and tear than any other vehicle; that is why buyer must be careful while purchasing any cargo van for their business fleet.\nIf you need a cargo van, you are probably better off renting from an outfit that rents cargo vans, box trucks or moving trucks on a regular basis and the prices will likely be less expensive. If the rental company is good, it will give all the relevant detail about your van has been serviced, if the certain parts need to be replaced soon and if the van has been any accident. Rental company provides all the details.\nCalling around different cargo vans rental companies and asking for estimate price would not provide the true idea of the final cost. Make sure you understand everything that is included in the price and prepare some questions in your mind before you buy or rent any cargo van. What kind of insurance is included in the rental? Will your cargo require any special equipment and if so, how much extra it is? Always compare similar prices and estimates to ensure your deal is worth it.\nRegardless of van you are renting, before you sign a rental agreement, it is better to check the vehicle for inside and out for damage. You may also want to pull a vehicle inspection reports, this will help to see any navigate point, such as accident that may have occurred on the vehicle during the time that it was with its previous owner. Try to inspect the vehicle thoroughly before making any deal.\nBefore you signed any agreement for cargo van, there is something you should know. You can save handful amount for money, if your purchased a used cargo van for your fleet instead of buying them. You can also see that there are several benefit to purchasing a used cargo, but only if you researched it out thoroughly. Cargo vans are very crucial for business growth, it is now becoming a vital part of your organisation to uplift your company.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"You have to eat and work and have a life. Here's how to get the nutrients you and your baby need.\nyou already know that a well-balanced diet is important for good health. Now that you're pregnant, there's nothing more important you can do than to nourish yourself and your baby with safe and nutritious foods.\nBut eating the right foods in the right amount can be easier said than done. How do you find time to fine-tune your diet when you're working, squeezing in doctor's appointments and taking weekend treks to every baby store within a 50-mile radius? And how do you cope when mysterious cravings arrive out of nowhere or you have an aversion to foods you've always loved?\nNow, more than ever before, it's time to get serious about nutrition and solve these dilemmas. \"What you eat now will set the stage for your baby's health for decades to come,\" says Elizabeth Ward, M.S., R.D., author of Pregnancy Nutrition: Good Health for You and Your Baby (John Wiley & Sons Inc., 1998). Proper prenatal nutrition is essential for your baby's brain and physical development.\n\"Eating the right amount of food helps promote the proper weight gain for the healthiest baby,\" says Ward. (For information on weight gain and calorie needs, see \"Baby-Building Basics\" on page 92.) And since combining pregnancy munching with power lunching is not always easy, we're here to help. Here's your guide to eating in the real world, including ideas for healthful snacks; quick, delicious, nutrient-packed recipes for lunch and dinner; and advice on supplements (consult your health care provider for your specific needs).\nYou're busy, you're tired, you may even be a little overwhelmed. So, when faced with a crunch like this \u2014 how to get the best for you and your baby in these precious coming months \u2014 why not do what you do best? Cook up a simple, doable plan.\nThat's what our \"Baby-Building Basics\" is designed to do. Just follow our 12 tried-and-true tips for healthful prenatal nutrition; then use our lunch, supper and snack suggestions and recipes to put them to use. You'll never have to skip a meal, and you'll get the valuable nutrients you and your baby need. There's nothing quite so beautiful as a well-laid action plan, is there?\n2)Use fats, oils and sweets sparingly.\n3)Focus on fiber, aiming for 20 to 35 grams a day. Eating two-thirds of a cup of bran cereal, one medium cooked sweet potato and a pear puts you in range for preventing the constipation and hemorrhoids so common in pregnancy.\n5)Also consider calcium supplements if you can't make the calcium quota of at least 1,000 milligrams a day (about three glasses of milk). Some prescribed prenatal supplements won't supply the calcium you need.\n6)Help prevent birth defects by taking only prescribed multivitamins and other supplemental nutrients, such as calcium, that your health care provider recommends. Just because a little is good doesn't mean more is better. For example, vitamin A in excess of 10,000 international units a day (twice the recommended intake) during pregnancy can cause irreversible damage such as spontaneous abortion and malformations. The safety of herbs and other botanicals during pregnancy and lactation remains to be proven, so steer clear of them.\n7)Drink at least 64 ounces of fluid daily (eight 8-ounce glasses), more if you're active and during hot weather. Fluid helps regulate body temperature, aids in digestion, works with fiber to fight constipation, and cushions joints and organs.\n8)Aim for a 25- to 35-pound weight gain, as long as you began pregnancy at a normal weight. After the first trimester, expect to gain between 3\/4 and 1 pound per week for the rest of the pregnancy. Twins (and more) require that you gain even more to foster a healthful birth weight. Underweight women may be advised to put on more than 35 pounds; overweight women could be counseled to put on less. Never diet during pregnancy, and always follow the advice of your doctor, certified nurse-midwife or nurse practitioner. Some women choose not to look at the scale during weigh-ins at the doctor's office to avoid obsessing about pounds gained.\n9)Add 300 calories a day to your diet, beginning in the second trimester. Women carrying more than one baby need additional calories, as do physically active moms-to-be.\n10)Avoid alcohol. According to the March of Dimes, drinking even small amounts of alcohol can cause physical malformations and developmental difficulties in baby that last a lifetime.\n11)Never eat raw or undercooked animal foods such as meat, sushi, seafood or eggs. Due to a somewhat depressed immune system, expectant women are more susceptible to food-borne illnesses.\n\u2014 Elizabeth Ward, M.S., R.D.\nIf you like taking your lunch to work, try these simple, nutritious and delicious recipes, which were designed for night-before preparation and easy transportability.\nThis Spanish-style frittata is traditionally eaten at room temperature and makes for a perfect brown-bag lunch.\nPreheat oven to 350? F. Meanwhile, in a 9-inch nonstick pan, saut\u00e9 onion and bell pepper in olive oil until very soft. Add chard and water. Cover, allowing vegetables to steam until chard is soft and water has evaporated. Season with salt and pepper and add eggs.\nWith a rubber spatula, stir mixture until the \"tortilla\" begins to set up. Reduce heat to low and allow to cook approximately 5 minutes; then transfer pan to oven for approximately 10 minutes. Turn onto a plate to cool; then slice into 4 wedges.\nNutritional information per serving (1 wedge): 120 calories, 38 percent fat (5 grams), 23 percent carbohydrate, 39 percent protein, 51 milligrams calcium, 1 milligram iron, 23 micrograms folate, 2 grams fiber.\nDressing the beans while still warm ensures that they absorb the flavors of the vinaigrette. Canned beans may be used, but they tend to be mushier and do not absorb the dressing as well.\nCook the beans in salted water until tender but not soft. Drain well; then dress with olive oil and lemon juice while still warm. When cool, add scallions, tomatoes and parsley. Season with salt and pepper. Flake tuna with a fork and toss into salad.\nNutritional information per serving (1\/4 of recipe): 504 calories, 16 percent fat (9 grams), 57 percent carbohydrate, 27 percent protein, 225 milligrams calcium, 8 milligrams iron, 388 micrograms folate, 12 grams fiber.\nIn this jewel box of nutrients, the vegetables can be roasted ahead of time, leaving only the couscous to be cooked.\nPreheat oven to 400\u00b0 F. Toss vegetables with olive oil, salt and pepper; then spread out on a baking sheet.\nRoast for approximately 15 to 20 minutes. Remove from oven and allow to cool.\nPlace couscous in a heat-resistant bowl with olive oil, cinnamon and salt. Stir with a fork until the couscous grains are well coated with the olive oil. Pour boiling water over couscous, making sure the grains are completely covered. Cover bowl with plastic wrap and let sit 10 minutes. Remove cover; then let cool completely. Rake couscous with two forks or rub between your fingertips to separate grains.\nAdd roasted vegetables and garbanzo beans to couscous and dress with olive oil, vinegar and orange juice. Add cilantro and salt and pepper to taste.\nNutritional information per serving (1\/4 of recipe) : 490 calories, 29 percent fat (16 grams), 61 percent carbohydrate, 10 percent protein, 105 milligrams calcium, 3 milligrams iron, 135 micrograms folate, 11 grams fiber.\nAn updated version of an old-time favorite, this salad makes a great lunch.\nNutritional information per serving (1\/4 of recipe): 147 calories, 3 percent fat (.5 gram), 93 percent carbohydrate, 4 percent protein, 70 milligrams calcium, .4 milligram iron, 14 micrograms folate, 4 grams fiber.\nServes 4Cooking in cast iron is said to add trace amounts of iron to the nutritional content of food.\nToss chicken breasts in olive oil and season with salt. Grill both sides in a very hot cast-iron skillet until no longer pink (approximately 5 minutes per side for a medium breast). Let cool and slice.\nCombine chicken and remaining ingredients in a bowl and toss.\nNutritional information per serving (about 1\/2 breast): 155 calories, 36 percent fat (6 grams), 22 percent carbohydrate, 42 percent protein, 41 milligrams calcium, 1 milligram iron, 11 micrograms folate, 2 grams fiber.\nThese five recipes were designed for delicious dinners, but they'll also make great leftover lunches.\nServes 2Serve these tacos with a salad of jicama, orange segments and cilantro.\nToss chicken breasts in olive oil and season with salt. Grill both sides in a very hot cast-iron skillet until no longer pink (about 5 minutes per side for a medium breast); slice. Heat black beans and season to taste with green chili salsa and salt. Heat tortillas and fill with chicken, black beans, cabbage and tomato salsa.\nNutritional information per serving (2 tacos): 504 calories, 18 percent fat (10 grams), 49 percent carbohydrate, 33 percent protein, 179 milligrams calcium, 5.3 milligrams iron, 400 micrograms folate, 15 grams fiber.\nServes 4Steaming the eggplant produces tender, creamy results while preventing it from soaking up too much oil.\nSteam eggplant until tender. Simmer marinara over medium heat for about 5 minutes; add eggplant and simmer another 5 minutes. Season with salt and pepper. Combine sauce, basil, cooked pasta and reserved cooking water in a large bowl. Dot with ricotta and freshly grated Parmesan cheese.\nNutritional information per serving (1\/4 of recipe): 334 calories, 13 percent fat (5 grams), 68 percent carbohydrate, 19 percent protein, 216 milligrams calcium, 3.6 milligrams iron, 28 micrograms folate, 5 grams fiber.\nServes 2This is a quick, easy way to cook salmon. The sauce is low in fat and high in calcium.\nPreheat oven to 500\u00b0 F. Meanwhile, brush salmon and asparagus with olive oil. Season with salt and pepper. Roast on sheet pan for 10 minutes. Serve drizzled with yogurt, grain mustard and caper sauce.\nNutritional information per serving (1 salmon fillet): 325 calories, 30 percent fat (11 grams), 13 percent carbohydrate, 57 percent protein, 154 milligrams calcium, 2.8 milligrams iron, 285 micrograms folate, 2.5 grams fiber.\nServes 4This colorful, comforting dish contains a wealth of fiber, vitamins and the all-important folate.\nPreheat oven to 450\u00b0 F. Combine all ingredients except the greens and roast for approximately 25 minutes or until tender and slightly browned. Toss in a large pan with greens and saut\u00e9 until greens are just wilted.\nNutritional information per serving (1\/4 of recipe): 226 calories, 28 percent fat (7 grams), 64 percent carbohydrate, 8 percent protein, 234 milligrams calcium, 2.5 milligrams iron, 185 micrograms folate, 7.5 grams fiber.\nServes 4This quick meal is made easy with the help of a high-quality pre-made crust.\nPreheat oven to 450\u00b0 F. Heat olive oil and saut\u00e9 onion and bell pepper until both are softened and the onion has begun to brown. Add lemon zest and season with salt and pepper. Remove from heat, add torn spinach, toss until wilted and spread over pizza crust. Sprinkle with Pecorino Romano cheese, then mozzarella. Bake on a sheet pan on the top rack of the oven for 8 to 10 minutes.\nNutritional information per serving (1\/4 of recipe): 432 calories, 31 percent fat (15 grams), 50 percent carbohydrate, 19 percent protein, 400 milligrams calcium, 1.5 milligrams iron, 210 micrograms folate, 4 grams fiber.\nTo make your pregnancy eating even easier, here's a complete list of the nutrients you need every day and where to get them.\nMeat, oysters, poultry, legumes, wheat germ, whole grains \u2014 E.W.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This Is A Shot Of Me Spinning Burning by Larry Marshall - 8.000\" x 8.000\"\nThis Is A Shot Of Me Spinning Burning poster by Larry Marshall. Our posters are produced on acid-free papers using archival inks to guarantee that they last a lifetime without fading or loss of color. All posters include a 1\" white border around the image to allow for future framing and matting, if desired.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Backup battery, test all sensors, user code programming, stay mode programming for night time use.\nThis service is also useful when moving in or out of a home equipped with an alarm system - No two homes are the same, make sure everything is setup just right for you!\nLooking to replace an existing Alarm System?\nBosch: Solution 16 plus, Solution 880, Solution 64, and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Temple Run Virtual Reality Edition!\nThis video was made in collaboration with Google Daydream. We have 2 more VR\/AR videos in the works. Stay tuned!\nSpecial thanks to Imangi Studios for their support on this project!\nA collection of IRL videos based on games and movies.\nLords Mobile Real Life Animation + GIVEAWAY - Duration: 3 minutes, 30 seconds.\nTemple Run Pirate Cove In Real Life (Virtual Reality 360\u00b0) - Duration: 3 minutes, 23 seconds.\nPac-Man In Real Life - ORIGINS - Duration: 6 minutes, 39 seconds.\nClash Royale In Real Life - Duration: 4 minutes, 1 second.\nClash Royale: Real Life Animation - Duration: 2 minutes, 1 second.\nFortnite | Future Gaming - Duration: 3 minutes, 28 seconds.\nSlither.io In Real Life 3 | Future Gaming - Duration: 2 minutes, 36 seconds.\nBrutal Age In Real Life - Duration: 2 minutes, 32 seconds.\nFlippy Knife In Real Life - Duration: 3 minutes, 8 seconds.\nSubway Surfers In Real Life | Future Gaming - Duration: 3 minutes, 44 seconds.\nPac-Man In Real Life - Duration: 2 minutes, 48 seconds.\nPokemon GO In REAL LIFE - Duration: 93 seconds.\nTemple Run Blazing Sands- In Real Life - Duration: 2 minutes, 45 seconds.\nSlither.io In Real Life 2 - Duration: 3 minutes, 13 seconds.\nTemple Run 2: Lost Jungle- In Real Life - Duration: 3 minutes, 1 second.\nTemple Run 2 In Real Life - Duration: 98 seconds.\nSuper Mario Odyssey In Real Life - Duration: 2 minutes, 40 seconds.\nSlither.io In Real Life - Duration: 117 seconds.\nPokemon GO Real Life UPDATE - Duration: 2 minutes, 13 seconds.\nSome of our more cinematic shorts.\nA Darwinian Future - Sci-Fi Short Film - Duration: 6 minutes, 52 seconds.\nWALL-E Live Action Fan Film - Duration: 4 minutes, 47 seconds.\nLEGO Batman In Real Life - Duration: 3 minutes, 50 seconds.\nSpider-Man Short Film - Duration: 117 seconds.\nJurassic World Fan Film - Duration: 3 minutes, 1 second.\nJurassic World 2 Fan Made - Duration: 2 minutes, 43 seconds.\nZombie Dinosaurs Loose in the Museum- Short Film - Duration: 3 minutes, 32 seconds.\nA Portal To KnowWhere- Sci-Fi Short Film - Duration: 3 minutes, 39 seconds.\nTemporality | SciFi Time Travel Short Film - Duration: 6 minutes, 21 seconds.\nWatch Dogs in Real life - Duration: 4 minutes, 57 seconds.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"One of the joys of travel is discovering new facets of yourself. It could be a delicious food you've never heard of, or maybe that zip-lining over the forest is not really as fun as it looks. On occasion, something catches you by surprise. On a recent visit to Sandusky, Ohio's Merry-Go-Round Museum an unexpected appreciation for carousels surfaced within me.\nNow, I've never been a fan of carousels. Clambering onto an uncomfortable horse to slowly spin and bounce along to the loud oompa-oompa of the pipe organ. Even at Disney's Magic Kingdom, my favorite place on Earth, I'm happier to watch my wife and daughter ride the merry-go-round than to join them myself. The appeal is just not there. Well, I take that back. There is one with a place in my heart.\nThe Cedar Downs carousel in the Cedar Point amusement park is like no other, it's a racing derby. Saddle up on one of the sleek horses and hold on tight, because it's one fast ride. The trumpet blares out it's fanfare and your off. The announcer calls the race as the horses fight for position. You glance left and right at the competition. You're in the lead, now falling behind, then pulling ahead at the ride slows to the finish. Adrenaline pumping, you race to the back of the line for another thrill. It's an unique experience and I'm told there is only one other racing derby still operating.\nJust a few miles down the road, in downtown Sandusky, Ohio, is the Merry-Go-Round Museum. I likely would have walked on by if it hadn't been a stop on a press tour I was attending. The building itself is interesting, a former U.S. Post Office with a large, rounded front dominating the corner of an intersection. Once inside, I quickly found myself drawn to the displays. Dozens of carved animals, representing over a century of history, lined the walls.\nWhile we are used to seeing horses on carousels, the earlier merry-go-rounds contained a whole menagerie of animals. Roosters, goats, zebras, lions, tigers, and even giraffes were there to capture the imaginations of riders. These early rides must have looked like a mobile zoo; especially to kids who had never seen some of these animals.\nUpon getting a closer look at the animals, I instantly took interest in the workmanship and details the carvers put into each one. The vibrant colors no longer seemed so oppressive when the animals are viewed on their own, but rather added an impressive beauty and individuality. The tour guide led us through the displays and highlighted some of the history and particular creators.\nThe museum contains an area where modern day carvers, mostly volunteers, perform restoration work for old carousel figures. Some are still in use, while others are from personal collections. It is vital work in keeping these historic pieces around for future generations. The museum also produces a completely new carousel horse each year, which is raffled off. The work area also contains displays of old tools from prominent carvers.\nIn the center of the museum showroom is an antique carousel from 1939. It is filled with animals from several eras in the museum's collection. We took a ride at the end of the tour. While this old model spins a bit faster than most current carousels, and my new found appreciation was fresh in my mind, it still didn't draw me in as an attraction. I guess some things do not change. However, I am looking forward to taking my daughter on her first trip to Cedar Point next year. She may be staring up at the giant roller coasters, but I'll be glancing over my shoulder the the Cedar Downs; ready to race again.\nThe Merry-Go-Round Museum is open all summer with very limited hours in the off season, so be sure to check the current schedule before you visit. Admission prices are very affordable at $6 for adults, $4 for kids 4-14, and free under 4 years of age. A ride on their carousel is included in the ticket price. For more information on the museum collection and schedule, visit their official website here and don't forget to enter their raffle.\nLooking for more family fun in Sandusky, Ohio? Read my article on Six Truly Fun Family Vacation Activities.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hi, it's Gene Knight, I was born right here in California, spent a few years in Connecticut, then my family settled back in San Diego, Escondido to be exact. I went to Escondido H.S. and Palomar College. I feel very blessed to have spent 99% of my radio career in America's Finest City. I have loved radio since I was a kid and I always will!\nWhen I am not working, I enjoy spending time with my family and our 6-year old son, Cristian, he really keeps us on our toes. We love going to the beach, museums, Sea World, and even to Puerto Nuevo in Baja for lobster!\nSomething most people don't know about me: I am shy, and I am a twin!!!\nI feel very fortunate to be part of the new Sunny 98.1 team, turn me on, Afternoons, 2pm to 7pm and I bet you'll be able to hear just how happy I am to be here!\nEver Get So Frustrated, You Destroyed Your Phone?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Most of us will admit that we've become unnecessarily agitated by a coworker's quirks. For whatever reason, or maybe for no reason, your office mate's habit of dragging his feet while he walks makes you recoil every time he strolls past your door. Or the receptionist who cracks her gum as what is surely an act of passive aggression.\nDo you dare confront the annoying employee? Of course not! You'd sound irrational! You'd appear half daft! The annoying conduct is trivial to everyone else-even to you in any setting other than work. But, alas, in the workplace, this seemingly benign habit pushes you nearer and nearer to the line where sanity meets crazy cat lady.\nBut wacky and non-wacky employees alike can understand certain fundamental canons of workplace etiquette. And when those cannons are violated, the beast in us can come unleashed. What follows are a few examples of what I consider to be inexplicable acts of discourtesy in the workplace . . .\nIs it really just me or is this habit beyond excuse? This is not the employee who once in a great while or even occasionally forgets to turn off his phone or set the ringer to vibrate when he arrives at work. This is the employee who refuses to turn off the ringer on his personal cell phone, despite the fact that he gets numerous calls each day.\nUsually when his phone rings, the employee has stepped away from his desk, leaving his coworkers to shudder through 8 long, painfully loud, and hardly musical rings before his voicemail picks up. Of course, he doesn't seem to mind. When he returns, he promptly returns the (inevitably personal) call.\nSo why does the annoyance continue? The answer is debatable. Either he believes that the rules should not apply to him or really is that oblivious that he doesn't realize how painful this is to the rest of the office. (I am fairly certain it's the former).\nStealing food from the shared refrigerator.\nI've written about this unforgivable conduct many times. (See Office Workers Up In Arms Over the Stolen-Lunch Crisis; Just Put Down the Brown Bag and Slowly Step Away From the Lunch). But it bears repeating because of the powerful feelings it engenders for so many office workers around the world. I'm not sure why it conjures up such strong emotions-possibly because the victim is hungry when they learn they've been targeted. Regardless of the reason for the reaction, the reality remains the same-if you're ever discovered, Office-Food Thief, you surely will be savagely attacked by an angry mob of coworkers.\nGeneral unawareness of basic e-mail courtesies inundates our workplaces. And it comes in a variety of bad habits. One universally annoying habit is TYPING AN EMAIL WITH THE CAPS-LOCK FEATURE TURNED ON. Come on, people! What's wrong with you? Is it not obvious that text typed IN ALL CAPITAL LETTERS is the written equivalent of SCREAMING AND YELLING? It is ENTIRELY obvious to everyone except a tiny population of e-mail users.\nWouldn't it be great to communicate to these individuals that this is not acceptable conduct? What if the users got fired for their all-caps bad habits? Actually, I think the idea's got merit. Provided that it wasn't the e-mail style choice but the uncooperative, bossy, and inflexible attitude that all-caps typing often indicates.\nOne employer in New Zealand actually put this theory to work. Unfortunately, it wasn't a successful attempt. Vicki Walker was fired for sending e-mails to co-workers that, according to her employer, were too \"shouty\" and confrontational. The e-mails in question, as you may have guessed, were written in ALL CAPS, with select words in RED ALL CAPS for extra emphasis.\nNow, really, we shouldn't be too surprised here. Based on her e-mail choices, she's not the type who shows concern for something as frivolous as \"fairness\" and certainly doesn't give a hoot about what her colleagues might think of her for being a big crybaby by filing suit.\nAnd, in a miscarriage of justice, Ms. Walker won. The New Zealand Employment Relations Authority ruled that, although she'd been a disruption in the workplace, Ms. Walker had never been warned that her e-mail style could get her fired. I guess this means that offensiveness of ALL CAPS e-mails is not as universally obvious as I thought.\nI would suggest that after reading the actual email she sent \u2026 which was mostly a how to do something. The complaint of ALL CAPS being used was mostly a reason to toss her out of the job. After reading the comments on the case seems she was merely sending out a MEMO.\nThis case is quite interesting. Firing an employee for using all CAPS is something ridiculous and at the same time the company's reason also sounds good. However, its good that the employee has been replaced in the job again.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Iraqi prime minister Haydar al-Abadi announced Friday that after the city of Ramadi was recovered from Daesh (ISIS, ISIL), the Iraqi army would go on to liberate Mosul (a city of some 2 million before Daesh took it), 420 km. north of Ramadi.\nThe fighting is slow going, but the outcome now seems virtually assured, as the Iraqi Army nears within a third of a mile of the key complex of government buildings at the center of the city, which had a pre-Daesh population of some 700,000 (i.e. the size of Detroit).\nDaesh had attempted to assassinate al-Rawi last April, so it was brave of him to show up while Daesh fighters were only half a mile away.\n\"The counter-terrorism forces are within 800 meters (0.5 mile) from the government complex,\" advancing by about 1 km in the past day, Rasool said. \"Air strikes helped detonate explosive devices and booby-trapped houses, facilitating our advance,\" he added.\nSpecial operation commander Sami al-Aridhi said the plan was \"to liberate all of Ramadi from three sides\".\nSalam Faraj of AFP did an insightful report on how,this weekend, the Iraqi Army's advance into Ramadi, in a push that began last Tuesday, was slowed by the tactics of Daesh (ISIL, ISIS) fighters in the center of the city. The foot soldiers of the phony caliphate deployed snipers and roadside bombs in abundance against Iraqi army troops.\nThe Iraqi forces have been provided by the US and NATO with 5,000 shoulder-held AT4 anti-tank missiles and they have been trained on them. The troops are using these weapons to good advantage.\nLikewise, the US and its allies are intensively bombing the enemy. The US dropped 50 bombs on Daesh positions late this week.\nIraqi military commanders say that they had met unexpectedly light resistance when their offensive began a few days ago, but that as they approached the government building complex in the city center, it grew fierce.\n1st Lt. Bashar Husain, of the Counter-Terrorism Force that has spread out in the neighborhood of al-Dubat in the south of the al-Hawz district, said Friday \"We are facing numerous obstacles, especially sniping and car-bombing.\" Nevertheless, al-Dubat was said to have fallen by Friday night.\nThe layout of the city allows a small number of well-placed snipers and suicide bombers to block the advance of government forces down narrow streets toward the center.\nLess than 400 Daesh fighters are thought to remain in Ramadi. Others have withdrawn, taking civilian hostages with them.\nLocals were confident that the assault would succeed, but conceded that it would take time.\nDaesh fighters are attacking with suicide belt bombs, making themselves human artillery shells, are blowing up houses and setting concealed roadside bombs, deploying snipers, and using mortar fire and RPGs, to stop the army's advance.\nDaesh maintains on its social media sites that it has launched a major counter-offensive that has pushed the Iraqi army back. This assertion does not appear to be true.\nthe Iraqi army announced the loss of 3 men and said 13 were injured on Friday in the al-Hawz district.\nAnother obstacle to quick progress is that some few dozen civilian families remain in the downtown area, despite pleas over the past month for all civilians to leave the city.\nThe fall of Ramadi will be an enormous morale-booster to the Iraqi army, which has been the object of cutting criticism since it abruptly withdrew from Mosul 18 months ago.\nIt will also isolate the Daesh position in Falluja.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Welcome! My name is Josh. This year I have set out to create 100 websites for 100 clients. Don't worry, quality will not be sacrificed for quantity! Your business is unique and so should your website be.\nI provide multiple services including custom web design, blog creation, Search Engine Optimization (SEO) advice, and much more. You have enough to worry about running a small business, so I made my prices simple. There's no need to spend thousands of dollars to have a successful web presence. Let me help you make your business successful!\nAct Now. Limited Spots Available!","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzalemx b/data_all_eng_slimpj/shuffled/split2/finalzzzalemx
new file mode 100644
index 0000000000000000000000000000000000000000..f9caeca4bd612f02a8b601b6881fc6c458758293
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Crm Freeware Chip is part of great design ideas. Crm Freeware Chip was created by combining fantastic ideas, interesting arrangements, and follow the current trends in the field of that make you more inspired and give artistic touches. We'd be honored if you can apply some or all of these design in your wedding. believe me, brilliant ideas would be perfect if it can be applied in real and make the people around you amazed! Crm Freeware Chip was posted in Friday June 8, 2018. Crm Freeware Chip has viewed by 116 users. Click it and download the Crm Freeware Chip..\nCrm Freeware Chip Microsoft Dynamics Crm Microsoft top 5 Produkte F\u00fcr was posted on Friday June 8, 2018 by www.belstaff-jacket.us. Crm Freeware Chip Microsoft Dynamics Crm Microsoft top 5 Produkte F\u00fcr was viewed 116 by visitor.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Before you can list your personal loan in the Marketplace, you must first put in an application to Harmoney. We assess each application and give each approved Borrower a personalised quote, including interest rate and borrowing limit. You then have 60 days to decide whether to list your personal loan or not.\nOnce you've used the Personal Loan Estimator to choose how much you would like to borrow (and what for) you need to create a Harmoney account. All we need is a name, active email address, and a password.\nYou will then be sent an email to the address provided asking you to verify your email.\nSimply click the button in the email and you will be re-directed back to our site, where you will be prompted to create a Security Question - this is what we'll use to verify you if you ever forget your password.\nFirst we need to verify your identity and check it against the New Zealand credit bureaus, this is a legal requirement (to ensure you're not bankrupt etc), but it also ensures no one can apply under your name without your permission.\nA valid New Zealand Passport.\nApproval should only take a minute to come through, so don't go anywhere!\nTo process your application as quickly as possible, we use Australasia's leading secure online bank statement retrieval service - Credit Sense. This process does not give Harmoney access or control over your bank accounts; it simply retrieves an electronic copy of your bank statements.\nIf you wish to skip this step, you will have to verify your income and identify via another step later in the application.\nFrom here, your application will be sent away for final approval, and you will be notified by SMS message and email. You may also add a Co-Borrower at this stage in the process. Click here to learn about Co-Borrowers.\nOnce your personal loan has been approved, you will be given a Loan Offer page, which shows you how much you can borrow and what your personalised rate it. You don't have to accept straight away, in fact you have 60 days in which you can come back and submit it to the marketplace at any time.\nYou are also offered the chance to add Payment Protect to your loan. This is a repayment waiver which offers you financial protection in case of involuntary redundancy, bankruptcy, disability, terminal illness, or death.\nClick here to learn more about Payment Protect.\nThat's it! Once you have accepted the offer, you can list your personal loan in the Marketplace. But before you do, you are given the chance to tell the Investors why they should fund your loan.\nIn a way, this may be the most important step in the process; this will appear as the hero piece of your listing in the marketplace, so don't leave it blank!\nIf you choose not to use Credit Sense, we will need to verify your identity and income through an additional stage.\nTo verify your identity we use an online system called 'Netverify'. This is an automated system that requires you to provide a photo of your ID, plus have access to a webcam so we can use facial recognition software to authenticate the ID.\nTo verify your income, you will still need to provide us with 3 months worth of bank statements, which you can upload as a PDF.\nClick here to return to Step 2.\nYou have the opportunity to add a Co-Borrower to your application. This essentially means that two people are applying for the personal loan instead of one.\nIf you choose to add a Co-Borrower to your personal loan application, they will have to go through a similar verification process (e.g. ID, income, financial details). They will also have equal liability in regards to the terms of the loan.\nClick here to return to Step 3.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Go with the mugen. Can't go wrong there.\nMegan has gotten great reviews for both the ride and their ability to prevent steering rack creak whilst lowered.\noh really?? and the price range is accurate from the previous person that commented on this thread about the prices?\nYes my prices are fairly accurate, I double checked before i posted, its called research . . . contrary to my seemingly low post count, I used to have nearly 3k posts, i have installed numerous suspensions on different eps here in san antonio.\nI've had eibach pro's since I first got the car and I haven't blown nothin. I think its probably the more severe lowering springs (-1.5\" and more) that blow stock struts.\nohh alright ahaha. sooo blkdeathsi, what`s your opinion about the megan coils, since you`ve installed many suspensions in your hood.\ni have tokico blues and tokico springs that lower 1.4 inch all around and everyone who gets in my car likes how smooth the ride is for being lowered. the spring rates are soft tho. my car is a dd so its alright i do want to get stiffer springs tho like prolines. i think shock and spring combo is the best for keepin a nice ride and having a lowered look...it depends how you want to drive your car and if adjustability is important to you.\nmegans are good for relatively low cost coils . . depemds on what you want out of it. are you going to just drop it and leave it, or are you going to raise it sometimes, lower it others?\nhonestly...i just want a drop look...and a smooth ride...and good handling... Bigger rims, bigger tires for less gap. Stock springs for smooth ride and the right suspension geometry.\nNot going to lie, the megan track series coil are phenomenal for the money. Great quality, perfect spring rates, camber adjusting tophat and a monotube shock. If you just want to go low, D2 or megan streets are your best bet.\nI just installed DC5 A spec suspension onto my car and it cut the wheel gap to about half of what it was. The ride is very good and the handling is definitely improved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Artist and writer Sarah Gillett tells us why she went one step further with her dissertation and chose an Artisan Book for her beautiful work.\nI studied MA Printmaking at the Royal College of Art in London. The great thing about the course is that we all came from different backgrounds \u2013 like fashion, architecture, illustration, painting, sculpture, performance, film \u2013 so the conversations and exchanges were fantastic. My idea of printmaking really relates to storytelling \u2013 you and I might tell the same story to friends, but we'd tell it differently, and it would alter slightly each time it was passed on and retold until it didn't matter what was real, misremembered or made-up, because the image of the story resonates beyond the original.\nTaking storytelling as a framework, my dissertation evolved into a set of essays about landscape, philosophy, history and experience.\nIf you saw my flat you'd understand! I'm a collector \u2013 graphic novels, fossils, old photographs, Victorian plates, carvings, paperweights, driftwood. I could never be a minimalist; I'm a maximalist through and through. Many of the images in the book are my own photographs of objects in my collection, and the layout was a way of collaging together the connections between things, just as I might arrange the real objects at home.\nThe main challenge for me was designing the image pages, which I did by printing and cutting out lots of my images at different sizes and sticking them together as collaged pages. I was so lucky to be working with Francis at Artisan Books, and my designer Alfonso Iacurci at Cultureshock Media, as both totally got what I was trying to do.\nFor me it was the perfect way to express my aesthetic sensibility. I also wanted to create an 'object' rather than a set of words on paper.\nAs soon as Francis showed me the cloth colours I couldn't take my eyes off Colville. It's such a deep void colour that fits perfectly with my sea obsession.\nA lot of my ideas at the time stemmed from the colour blue. My dissertation was structured around the thought that blue 'holds' things, from stars and planes to whales and sirens. There is the sky, there is the sea and there are our own bodies and experiences.\nAt first I thought I might go for a natural emboss, but Francis advised me on the transparent blocking. He was absolutely right and I love the way that the title looks like water or ice.\nApart from your ideas on the design and look of the book, did you have an opinion on the best way to print and bind it?\nI just wanted the book to look beautiful, and it does. Francis showed me some papers and other books so I could get a feel for how my book would turn out and I think the hand binding really does make a difference to the overall effect of the book. The book production was as much a creative process as the writing so it was worth taking time over. It's nice to think that the book's been made by people who care about what they are making.\nI had worked with Francis in the past, and when I started thinking about producing a book I remembered him. He's been an art printer for a long time, well before digital printing, so he really understands about colour, paper and working with artists. I couldn't think of anyone I would trust more.\nWhat was yours and others reaction to the finished book?\nIt's perfect. I don't know what else to say. Everyone has commented on how beautiful the book is. It has a tactile quality in the colour and the cloth that draws people to it.\nIt's important to make the dissertation your own thing and that's what working with Artisan Books allowed me to do. It's so much work and so stressful that it's really worth pushing yourself to make something that transforms what you're written into an expression of joy, or rage, or misery, beyond the words on the page.\nI think it's really important to keep in contact with artists you find inspirational so there are a few projects on the horizon that I'm really excited about. In a few weeks I have a creative retreat planned with Amy Pettifer and Jessica Harby, both brilliant women who constantly surprise me. Later in the year I'm part of a group of 2015 RCA alumna putting together a show in response to As Above So Below, an Allenheads Contemporary Art programme. And I'm hoping to take Amy Gear up on her offer of a residency in the Shetlands.\nRight now I'm collaging onto old postcards as a way of drawing alternative pasts \/ futures. I have over 70 so far, so maybe there's another book in the making. To keep up with the writing I'm working on a couple of performative lectures based on the essays in my dissertation and linked to the idea of 'falling'.\nI've also just set up an experimental Instagram project for 'pop-up' exhibitions of work that only exist in Instagram. At some point I will get round to photographing my collection of curiosities on there as well as my terrariums. I have a lot I want to do.\nAt the RCA the dissertation is considered in its entirety \u2013 the overall presentation and how this fits with the ideas explored is very important to the overall mark. For me, the dissertation was an artistic body of work, so I'll probably carry on recycling the ideas and structures within it for a long time yet.\nSarah's book was shortlisted for the Intellect Publishing Award, 2015 and she was awarded a Distinction from the Royal College of Art (MA Printmaking), 2015. Thank you to Sarah for taking the time to share her work with us, we would love to think our book played its part in her achievements and wish her every success in her ongoing plans and career.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Warriors international presents \u2013 Freedom: The Art of Improvisation loft Jam session.\nThe monthly open-minded Freedom Improv session, Voted by the Evening Standard one of the Top 6 Uk jam sessions. Curated by Cleveland Watkiss, Orphy Robinson and Tori Handsley plus DJ Paul Bradshaw. There's an opening short set from an adhoc ensemble, swiftly followed by constantly shifting sets of soul stirring improv sessions open to all\u2026 So tune in and get involved.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"Let Cypress Cabinets in Monterey help you find the right cabinet storage solutions. We provide quality cabinetry and shelving with exceptional customer service and in-home consultations. With years of experience, we can quickly identify space-saving solutions for your home and help you choose just the product you need.\nYour kitchen may seem small, but we can find the right cabinets and shelving to maximize the space you have. Most kitchens are not taking advantage of open space above the refrigerator and stove, which is one of many simple solutions. You could expand your storage space by including an island with custom cabinetry and utilizing a pull-out pantry or trash bin.\nJust like in the kitchen, laundry rooms and bathrooms have potential storage space if you know where to look. Installing shelves or cabinets above the washer & dryer is usually a good idea to free up some work space. The same storage solution applies to the unused space over the toilet. One clever and attractive idea for storing extra towels, toilet paper, and other bathroom essentials, is to install shelves over door frames!\nWhen it comes to organizing your space, we understand that custom cabinetry and storage solutions need to be functional, durable, and aesthetically pleasing. We offer the latest styles and designs for your home office, den, or other living spaces. Our cabinetry works well in garages, where we can maximize your vertical space while saving floor space.\nWe've only skimmed the surface of the possible solutions for storage in small spaces. Remember, space-saving options can be added almost anywhere with some imagination. Think vertical, think functional, think outside-of-the-box. Some of the most common custom storage solutions are listed below, but if you have a vision, tell us! We'll do everything we can to make it happen.\nCypress Cabinets in Monterey is dedicated to helping you increase the beauty and function of your home with our custom designed storage solutions. Request a quote online or call us to schedule a consultation. We are excited to hear your vision and help you make it a reality today.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Aliens have issued a stark warning to the Leave campaign as Brits head to the polls to decide between perpetual coverage of David Cameron high-fiving Jean-Claude Juncker and an Independence Day-style apocalypse.\nThe aliens, who float in as-of-yet unclaimed territory above Earth and are thus described as pirates, said Boris Johnson's desire for an Independence Day might give post-Brexit UK a nasty surprise.\nThe polls for the referendum are currently neck and neck, and no one can tell whether Britain is rocketing for an exit, or staying on terra firma with a vote to remain a member of the EU.\nThis latest warning from the alien pirates has been called last-minute \"scaremongering\" by the Leave side, whilst it is understood the warning will feature on some of Remain's social media pages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Discover ten of London's often hidden peaks on this new marathon distance walking challenge.\nSee a different side to London and journey through 42km of beautiful woodland, nature reserves and quieter spaces. With unexpected views and chance encounters with local wildlife, this is a distance challenge on your doorstep.\nDiscover ten of London's often hidden peaks on this new distance challenge on your doorstep. Be part of a fun and supportive team to uncover some alternative UK 'summits' that are surprisingly close to home.\nYou will travel a good distance on the popular 'London Loop' footpath, journeying over a remarkable variety of terrain and uncovering a different side to London.\nThe trek will take you through beautiful woodland, thriving nature reserves and quieter spaces. Enjoy the unexpected views and chance encounters with local wildlife, such as green woodpeckers and muntjac deer. But keep one eye on those miles still to go...and that well-deserved pint at the end!\nWith a challenging 42km to conquer, this demanding trek will leave you with an immense sense of achievement knowing you have made a real difference to supporting the work of Together for Shorter Lives.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"These visors were designed to reduce the glare and the water off your fish finder screen. By using a fish finder visor, it allows you to reduce your screen brightness and still see the screen clearly which prolongs battery life. The added bonus of splash protection gives you more time fishing and less time cleaning.\nThis is a quick install using the two stainless steel fasteners provided into the flush mount locations on the fish finder. Refer to the installation image provided for more information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The children featured below live in Mubende, Uganda which is one of the oldest districts in Uganda having been established around 1905 by colonial rulers. The district has bred many other districts such as Kiboga and Mityana. It covers an area of approximately 4,620 square kilometres. Most of the children live with elderly grandparents or single mothers who work hard to rear them despite ill health and challenging economic conditions. Others live with guardians who opened up their homes despite poverty and tough circumstances. Despite these challenges, these children attend school, work hard in and outside the home to help their families survive. They have ambitions and dream of a better life for themselves and their communities. Your support will make a huge difference.\nAbout: Esther loves to play hide and seek and is in the 2nd grade. She lives with her 70 year old Grandmother who grows maize and beans to feed their family and to sell at the local market. The grandmother used to keep livestock but they died from swain fever. Esther plays netball and, with your assistance through this sponsorship program, she will be able to attain her future dream of being a beautician and fashion designer. Esther's immediate problems are lack of tuition, school supplies such as books, clothing and transport to school. Your sponsorship will help her family with basic necessities and enable them to pay school fees.\nAbout: Allen lives in Myanzi sub county, Mubende with his grandparent, three brothers and three sisters. He completed his primary leaving examination. He is described as quiet, smart and helpful. He likes working in the garden with his grandmother. Allen can continue to thrive as he works hard to advance to the next grade level.\nAbout: Jonathan is smart, well behaved and very active in all his activities. His favorite hobbies are riding a bicycle and playing football. Jonathan helps out at home by washing utensils and even his own clothes. His grandmother looks after him while his mother is working at a nearby trading center. Jonathan's father died when he was two years old but his mother is working hard to provide for their family. His mother also looks for extra jobs to supplement their limited income.\nAbout: Joel wants to become a doctor. He is always the top student in his class and likes to compete in quiz and debate competitions. Joel and his six siblings live with their grandmother in a mud house, which has a galvanized, iron-sheet roof. The children help out with domestic chores. Joel's mother sells items at a local market and occasionally works as a casual laborer on people's farms or washes people's clothing in the village to supplement her income. Joel's father lived with HIV for twelve years and finally died due to lack of medical care.\nAbout: Obed and his 4 siblings live with his sickly, HIV positive mother who has little energy for daily tasks. They live in a poor small house that was built on a hill far from their neighbors. Obed's father died of HIV and left the house unfinished. Obed helps with household chores being the eldest son in the family. He often goes to Bugolo island where his mother grows rice to help in scaring away birds from the rice field. He likes playing football but has limited time because of multiple chores at home. Obed admires politicians and dreams of becoming one someday.\nAbout: Sharon always has a smile on her face. She lives with her mother and brothers. Unfortunately, her mother struggles with poor health and very little energy although she's receiving treatment with ARVs. Despite their hardship, Sharon and her older brother are deeply aware and grateful to their mother for all her sacrifices to work and provide for them. Through the sponsorship program, Sharon will be able to fulfill her dream of becoming a nurse. This respite will allow her mother to care for her own health so she can better meet the needs of her children.\nAbout: Joan is a calm and reserved girl who was born healthy. However, she became very ill 1 year later from malaria which put her in a coma. It was so serious that everyone thought she was dying. This illness affected her mental development and made it hard for her to talk and eat normally. Joan's father died and left 5 children for her poor, HIV positive mother to raise. The mother does casual work at people's homes and farms around their town. Joan enjoys music a lot and likes eating French fries. As a sponsor, you will help provide Joan and her family necessities that will lead to a happier, healthier life.\nAbout: Dan is a 6 year old boy who lives in a mud house with a grass thatched roof with his parents. Dan has never seen a real toilet or a bathroom because even his school offers only an outdoor latrine. Dan's father works in construction and is an alcoholic who spends everything he earns. This leaves his wife the sole responsibility of taking care of their family. His mother does her best, but the family relationship is aggressive and unstable. With the help of a sponsor, Dan and his family will receive family support and guidance to improve their family relationships.\nAbout: Robert is active, athletic and loves playing football with friends . His father died in a war when he was young and Robert is very close to his mother. Their 2- room house was constructed in a road reserve which is against the law of the local roads authority. Robert's mother is a migrant worker and his family moves often. With this sponsorship program, Robert can continue his education and learn how to help his mother create a more stable home environment.\nAbout: Brenda and her 7 sibilings live with their mother who is HIV positive. Brenda's father abandoned the family after testing HIV positive. Her mother does casual labor around the village like washing people's clothes. Brenda likes playing netball and enjoys music. Through this sponsorship program Brenda will be able to remain in school and achieve her goal of becoming a heart surgeon.\nAbout: Timothy and his siblings live with their mother. He likes playing football with friends and loves carat, which is normally played by the Japanese and Chinese. Timothy's mother works as a farmer, cultivating crops for a living. She depends on the harvest to cover some of her family's basic needs. Their father abandoned the family because of alcoholism and womanizing. With this scholarship program Timothy will be able to fulfill his longtime dream of being a mechanic.\nAbout: Shakirah lives with her mother and sister in a small 1-room rental. She loves to socialize with friends and read short stories. She has the desire to grow into a strong and powerful woman. She understands that the key is education. With the help of sponsorship, her family will receive the counseling they need to support their child in achieving her dream of becoming a doctor.\nAbout: Sharifah is 5 years old and always willing to help someone in need. She lives with her mother and sister live in a slum where their whole family rents one room. She loves to study and her mother knows the value of education. Sharifah's mother works hard to send her children to school. Despite her tireless efforts, the income is not sufficient. Through the sponsorship program, Sharifah will receive the funds for her educational needs.\nAbout: Jovia is very hard working, active in the community and always willing to help someone in need. She lives with her mother and sister in a slum community where their whole family rents one room. She likes eating French fries and loves to study but her mother is mentally unstable. As a result, Jovia lacks school supplies, tuition, and required school clothing. Despite her efforts, their income is not sufficient. Through the sponsorship program, Jovia will receive funds to continue her educationa.\nAbout: Regina likes to get up early in the morning to watch her mother in the kitchen and to help out with her 1 year old brother. She enjoys singing and playing with her dolls. Regina's family lives in a poor small room rented in a slum community. Her father died and the mother is sickly, HIV positive and mentally unstable. The mother does casual work on people's farms and washes people's clothes around the village. Regina requires a special diet. As a sponsor, you will help provide Regina with the special dietary nutrition she needs.\nAbout: David lives with his guardian who is taking care of him, his 6 brothers and 2 sisters since they lost both parents, David likes to play football and tour different places. He faces challenges in education due to lack of school fees, school supplies and walking a long distance to school. Despite of all these challenges, he wants to continue his education. With the help of this child sponsorship program, David hopes he will be able to achieve his dream of becoming a mechanical engineer.\nAbout: Patrick and his 8 siblings live with a guardian. Patrick's parents died and the guardian can't afford to support the nine children and send Patrick to school. He likes football and has an overwhelming ambition to become an international football player in England someday. He also wants to be a politician but has a lack of school fees, school supplies and must walk a long distance to school (about seven kilometers). With the help of the child sponsorship program Patrick will be able to achieve his career dreams.\nAbout: Eria lives with his uncle and 7 siblings. His uncle is a migrant cattle keeper and lives in a grass-thatched house with many people in the hut. Eria likes playing soccer and athletics. He participates in collecting firewood and fetching water for cattle at a farm where his guardian works as a casual laborer. Eria's parents died when he was three years old. He is worried about his studies being interrupted because his uncle often tells them to focus on survival through cattle rearing instead of going to school. With your support through the sponsorship program Eria can remain in school and pursue his dream of becoming a doctor.\nAbout: Winnie lives with her uncle and 7 siblings, The uncle is a migrant cattle keeper and lives in a grass thatched house with many people in the hut, Winnie likes playing skipping and athletics, she participates in collecting firewood and fetching for home use. Winnie lives her guardian uncle ever since she lost her both parents and now Winnie is being taken care by their guardian is working as a casual laborer at some ones farm. Winnie faces difficulties of lack of school fees, scholastic materials and walking long distance of seven kilometers going to school. With your support through the sponsorship program, Winnie can continue with her dream to becoming a teacher.\nAbout: Phionah is a joyful and well-behaved little girl. She likes using colored pencils to draw and enjoys going to school. Art is her favorite subject though her problem is walking a long distance to school. She lives with her parents and seven siblings. She is in a migrant family and their major job is cattle keeping. Her mother tends to the chores in their home. Phionah's father works as a farmer, cultivating crops for a living. He depends on the harvest to cover some of his family's needs. With the help of child sponsorship program she will be able to remain in school and achieve her dream.\nAbout: Joyce lives with her HIV positive mother and three siblings in a rented one room house. She fetches water and washes dishes to help out with housework. Joyce loves playing with her dolls and she treats them as real children. Joyce's mother is on ARV's treatment with good health and she works as a casual laborer on other people's farms. She has a small garden where she grows food for her children. With this sponsorship program Joyce will be able to achieve her dream of going to school full time.\nAbout: Priscilla is a smart and organized girl who is said to have a smile at all times. She lives with her grandmother since her parent's died. Her grandmother supports them by washing people's clothes around the village. Priscilla loves reading books. She is also interested in caring for her younger siblings. Her family moves from village to village looking for cheap houses to rent. With the help of this sponsorship program Priscilla will be able to pursue her education and achieve her dream of becoming a doctor.\nAbout: Aidah is hardworking and stays active daily. She loves music and athletics. Aidah is good at gardening and helps out with house chores by fetching water from the well. Aidah's parents were imprisoned and she was taken in by the Headmistress at her school. Through this scholarship program Aidah will be able to fulfill her dream of an education.\nAbout: Harriet's parents were also imprisoned and she was taken in by the Headmistress of the school where she studies from. She is good in cooking food and helps fetching water at home. She loves reading novels and athletics. With this scholarship program Harriet can be able to fulfill her educational needs and dreams.\nAbout: Wycliffe studies in a local church school. He loves to play football and always cleans the courtyard as part of his usual house chores. Despite all efforts his father who is HIV positive, finds it difficult to make ends meet and pay for his medication. With your help through the sponsorship program, their family will receive basic necessities and use their remaining funds to pay for school tuition. This will help Wycliff start his journey towards his dream of becoming a Mayor of the city. Wycliffe's mother abandoned her children when Wycliffe was 6 months old after learning about the father's HIV results.\nAbout: Harriet always has a smile on her face. She loves everyone, especially babies. She likes music, dance, drama and enjoys athletics to occupy her time. Her family lives in a rented grass thatched house with no bathroom and a make shift toilet. Her father does not take care of the family due to struggles with alcohol. He leaves all the family responsibility to Harriet's mother whom he beats up almost every day when she asks for food. The mother is a migrant casual laborer on people's farms and occasionally cooks food for sale in the market.\nAbout: Reagan likes playing soccer. He lives with his father in a rented one-room building in the outskirts of Myanzi Sub-County with no toilet, bathroom or kitchen. Reagan's mother died in a car accident five years ago. His father is a migrant laborer and his income is insufficient in meeting the needs of his four children. He would like to grow crops for food but has no money to rent a piece of land. With the help of the sponsorship program Reagan can have funds for school and achieve his dream of becoming a member of parliament.\nAbout: Fauza lives with her mother and five siblings in a house that is in poor condition. She likes playing netball, enjoys eating french fries and beef . Her father abandoned her family so her mother carries the sole responsibility of looking after her children. She rented a piece of land to grow food for the children but cannot even afford to buy seeds. Fauza travels a long distances to the only school her mother can afford, which has limited reosources. With this sponsorship program, Fauza can get tuition to attend a nearby school and avoid the risks and difficulty of walking a long distance every day.\nAbout: Asifah lives with her mother and 2 siblings in a rented one-room dwelling. She likes eating French fries and beef. Asifah's father died three years ago in a motor accident leaving the mother to take care of the family. Asifah goes to school barefoot and stays at school without lunch because her mother cannot afford tuition and lunch at the same time. Through this sponsorship program, Asifah can have her basic needs addressed and be able to achieve her dreams.\nAbout: Rayihan is described as very bright in class and generally smart. He lives with his eighty year old grandmother who does some farming to get food to feed her grandsons. Rayihan's parents abandoned them at their grandmother's as they turned to drinking local alcohol (non refined waragi) from morning to evening. Rayihan lives with his grandmother and two siblings in a semi-structure house with old iron sheets as her kitchen. With the help of this sponsorship program, Rayihan can be able to achieve his dream.\nAbout: Mulshid live's with his eighty year old grandmother and two siblings in an old house in poor condition. He likes eating matoke and groundnuts. He also enjoys playing soccer. The main job for Mulshid at home is fetching water and collecting firewood. His parents abandoned him at his grandmother's home. They started drinking alcohol and sleeping in bars most of the time instead of caring for him.. Through this sponsorship program, Mulshid can achieve his dream of becoming an engineer.\nAbout: Shaloom is a 4 year old girl who lives with her mother and although she is not yet in school she longs to be. Shaloom has an egocentric attitude towards her toys and enjoys making the local toys using fibres from the gardens. Shaloom's father is an alcoholic, which leaves her mother to care for the family. Her mother does her best, but the family relationship is aggressive and unstable. With the help of a sponsor, Shaloom and her family will receive family support and guidance for better family relationships.\nAbout: Martin loves riding toy bicycles. His favorite food is french fries and hamburgers. Martin was abandoned at the hospital by his parents when he was two years old. No one has come back or inquired about him since then. He lives there still with a guardian who is also a single mother, a hospital volunteer and earns very little income. Martin is not in school yet because his guardian is unable to pay tuition. Through this sponsorship program, Martin can get school tuition and other basic necessities.\nAbout: Divan is a joyful and well-behaved little girl. She likes using colored pencils to draw and enjoys going to school carrying her backpack. Divan's mother is HIV positive and is a devoted client at Kiganda Health Centre III where she works as a volunteer. Divan's father abandoned the family after knowing his wife's positive status and married another woman at a neighboring village. Later, Divan's father came back and burned down the family home with everything in it including Divan. She was burned severely with wounds on her back and stomach. To this date, she still suffers with a lot of pain. Through the sponsorship program, Divan can get treatment for her wounds and achieve her dream of becoming a teacher.\nAbout: Doreen lives in a rural area with her guardian, four brothers and two sisters. She likes to play hide and seek and is described as quiet, smart and helpful. Doreen also likes working in the garden and loves music. Her favorite foods are watermelon and chicken. Doreen has always worked hard at school to achieve her career dream of becoming a lawyer but she is worried about her life. Her mother is very weak with HIV+ and cannot afford to pay school fees or buy scholastic materials. Doreen's parents separated and her father left the home. He does not support the family. With Child Sponsorship Program, she can be able to achieve her dreams with education.\nAbout: Teopista is described as being very smart, humble and very disciplined. She lives with her grandmother who works as a volunteer at Kiganda Health Centre III as an educator in the maternity department to aid in reducing the number of children born with HIV. Teopista's father died of AIDs and her mother abandoned them after learning of her positive status. Her grandmother took over the responsibility but her income is insufficient to support them all. Through this sponsorship program, Teopista can achieve her dream of getting an education.\nAbout: Jane lives her grandmother and five siblings in a very poor mud wattle structure house. Jane is HIV positive and on ART. She has frequent fevers leading to missed classes and has on occasion missed the end of year exam. This has caused her to remain in the same class for more than one year. This is the main reason she is still in the sixth grade at age 15. Jane's parents abandoned her when she was 9 years old because of her illness. With the help of this sponsorship program, Jane can get school fees and be able to buy nutritious foods.\nAbout: Joyce is an orphan who lost both parents to AIDS and lives with her grandparents. She loves going to school but her entire family is HIV positive and sick so they are unable to pay school fees and groceries for their household at the same time. Her grandfather is helpless and often so sick that Jane works fetching water for other people around their village to get food to eat at home. With help of this sponsorship program, Jane can get an education and learn to earn a living.\nAbout: Florence is an HIV positive orphan in the fourth grade at a very poor school. Florence helps her grandmother with washing utensils and fetching water. She also enjoys watching soccer. Florence is on ART but she is often ill with fever and, on several occasions, she has missed school due to a lack of medical care. With help of this sponsorship program, Florence can have these needs met and achieve her dream of getting an education and changing the course of her life.\nMeet Karim, a young man who lives in a rural area with his mother, two brothers, and three sisters. He likes to play soccer and is described as quiet, smart and helpful. Karim also likes working in the garden and loves music. His favorite dishes are French fries and chicken. He has never been to school as his mother, being very weak with HIV+ status, cannot afford school tuition. Karim is also HIV+ and on ART. His parents separated and the father has not given any support to the family since he left.\nMeet Mariam, who is very hard working, active and loves to study. Her mother knows the value of education and works hard in order to send her children to school. Despite her efforts, their income is not sufficient to support the family and to pay for medical bills as the children are HIV+ and she is also suffering from Tuberculosis (TB). Mariam's father died of AIDs and her mother is taking full responsibility despite being sick. Mariam finds it hard to take her medicine while at school thus becoming stigmatized. Through the sponsorship program, Mariam will receive the funds needed to fulfill her educational needs and free some extra income to pay their medical bills.\nMeet Teopista who is HIV+ and on ART while living with her grandmother and four brothers in a grass thatched house. She loves eating sweet potatoes and beef. She enjoys skipping as her pastime activity. Her parents abandoned their children and the grandmother came in to take over the family responsibilities. The major problem for Teopista is lack of tuition and lunch fees, as well as the long distances she walks to school since she is always sick. With this sponsorship program Teopista will achieve her education dreams and receive medical care.\nMariam is 5 years old and living with her mother and six siblings in a mud wattle house. She likes eating French fries and ground nuts, and also likes athletics. Mariam and her mother are both HIV+ and on ART. This is a tragic story because the father was HIV negative and decided to abandon the family after the test. The mother is extremely weak with skin rashes all over her body. With the help of this sponsorship program, Mariam will be able to achieve her dream of going to mecca and become a hajati.\nJawab is a 12 year old boy living with his mother, four brothers, and two sisters in a mud wattle structure house. He loves praying to Allah for his mother and those of his siblings who are living HIV positive and always sick. Jawab's parents are living with different statuses. His mother is HIV+ and the father abandoned the family after testing negative. His mother is very weak and has skin rashes all over her body such that she cannot afford to work to look after her children. Through this scholarship program Jawab can achieve education and gain extra income to care for family members with medical care and to buy nutritious foods.\nMeet Agnes, a 14 year old girl who is a deaf mute and is also lame. Agnes does not play any games; she is very humble and loves gardening as her major job at home. Agnes lives with her guardian who is a peasant farmer. Her father died and her mother abandoned her after seeing her as a curse. Agnes is also HIV+ and on ART at Kiganda Health Center IV in Mubende district. Through this sponsorship program Agnes can achieve her dream of joining a school for the handicapped to change her life.\nMeet Bridget, a 7 year old girl who lives with her grandmother with her four siblings. She likes playing hide and seek as a pastime activity and she also enjoys eating bread. Bridget's dream is becoming a nun of the Roman Catholic Church however she is HIV+ and always sick, which causes her to miss classes. Her grandmother is a peasant farmer and sometimes works as a tailor although she has poor eyesight so is not proficient at using the sewing machine.\nMeet Annet, who lives with her mother and five siblings. Annet is HIV+ but on ART. She is also asthmatic and is malnourished, which makes it hard for her to go to school. This has caused her to remain in her first year of primary. Annet's father abandoned the family and left the mother to care for them even though she is sick. Through this sponsorship program Annet will be able to get school fees and medical care.\nMeet Kevin who lives with her mother and sister. They are all HIV+ and on ART at Kiganda Health Center IV in the Mubende district. She likes playing with her dolls and enjoys eating French fries and beef. Kevin is not in school because her father abandoned the family, and her mother has a weak constitution and cannot afford to look for food and at the same time make money for school. This family was displaced from their home when someone bought the land on which they had a small plot, in order to build a new farm.\nMeet Betty who lives with her mother and sister. They are all HIV+ and on ART at Kiganda Health Center IV in the Mubende district. She likes playing hide and seek and enjoys eating bananas and chicken. Betty's father abandoned them and the mother, who is always very weak, cannot afford school fees from her meager income. This family was also displaced from their home after someone bought the land on which they lived in order to build a farm.\nMeet Shanaz who is described as always smiling and very humble. She is in her third year of primary as she lives with her grandmother and five siblings. Shanaz's mother passed away and the father ran away while half insane, which led the grandmother to take over the responsibility of the family. Shanaz is also HIV+ but still on septrin doses. With the help of this sponsorship Shanaz will be able to get tuition and other basic requirements to allow her achieve her academic dreams.\nMeet Moses, a 12 year old boy who lives with his grandmother in a semi-structure house that has no windows and needs some renovations. Moses likes eating French fries and beef and loves playing soccer. His parents were both HIV+ positive and abandoned him to his grandmother who is in her elder years. Though Moses is HIV+ he helps his grandmother in the house work such as cooking food, fetching water, and collecting firewood. With the help of this sponsorship Moses can achieve his dreams of becoming a soldier.\nMeet Brenda who lives with her mother and three siblings in a semi-structured house. Brenda and her mother are both HIV+ and on ART at Kiganda Health Center IV in the Mubende district. She loves singing and enjoys eating bananas and chicken. Brenda's father died three years ago, but Brenda remains in remorse over her loss and sheds tears whenever her father is brought up. Despite her mother's effort to look after the family her income is insufficient to handle all family needs.\nMeet Lawrence who lives with his father in a semi-permanent house. Both Lawrence and the father are HIV+ and on ART. Lawrence is always sick, but despite this he continues walking long distances to his school. The father is a peasant farmer and can only afford to cultivate food for home consumption because he has less energy due to his illness. With the help of this sponsorship program he will be able to achieve his dreams.\nMeet Shafinah who lives with her mother and five siblings. They live in a small grass thatched house and the mother does farming on a rented piece of land to feed the children. Shafinah is a blind child who needs special care and needs to attend a specialized disabled school to complete her studies. Her mother is traumatized and stressed because she has given birth to more than two children with deformities. The father and his relatives have abandoned them and no one around cares to support her family.\nMeet Adam who lives with his mother and five siblings. They live in a small grass thatched house, and the mother does farming on a rented piece of land to feed the children. Adam has a defect on his head and at the age of 7 months his head increased in size causing him to be taken for surgery. His head problem is different from the kids born with large heads which makes it important that he goes for review every 6 months at Cure the Children of Uganda Hospital in the Mbale district. His father and his relatives have abandoned them.\nMeet Sophia, a 7 year old girl described as quiet, smart and helpful. She likes working in the garden and loves music. Sophia and her parents are HIV+ and on ART. The family rents a one roomed house with no basic facilities like toilet, bathroom, or a kitchen. Sophia's father abandoned his wife and the children after knowing of his HIV positive status. The mother is a peasant farmer who earns very little to support her children.\nMeet Ronald, a 6 year old boy who is a bright and comes from a deserving family. Everyone calls him Ronnie; he enjoys playing soccer and eating French fries. Ronnie and his parents are both HIV+ and on ART, despite all their efforts and the dedication towards education they have for their children. The parents are very weak and have less energy to look after the family. Through this sponsorship program Ronnie will be able to achieve his academic dreams.\nWalakira is an active little boy who enjoys playing soccer. He likes to go to school and spend time with his classmates. Walakira lives with parents who are HIV+ and on ART in a small grass thatched house. He is also HIV+ and has a heart problem. He helps out at home with folding the clothes and cleaning the courtyard. To support her son, his mother works as a maid to a neighboring home. With the help of this sponsorship program he will be able to achieve his academic dreams.\nCharles is a 9 year old boy who loves to play soccer and is in his second year of primary. He lives HIV positive with his father, who has a heart problem, and his mother, who has less energy to help the husband in the household requirements. All Charles's siblings died of HIV and the family is depressed because of this trauma. With your help through the sponsorship program, the family will receive basis necessities and free up funds to pay for the medication of his father who needs to stay healthy and continue to provide for his family.\nMitchelle is a two year old girl with an egocentric attitude towards her toys. Mitchelle's family dynamic is quite strained. Her father is an alcoholic, which leaves her mother to care for the family. Her mother does her best, but the family relationship is aggressive and unstable. The family lives in a small grass thatched house with on basic needs in the household. With the help of a sponsor, Mitchelle and her family will receive family support and guidance for better family relationships.\nMeet Frank, a 14 year old boy who lives with a helpless 70 year old grandmother in one roomed mud house. He likes playing soccer and his main job at home is cultivating in the garden. Frank's parents abandoned him when he was 8 months old. He says work at home is too much for him, which limits him to go to school. This problem of missing classes is also attributed to the long distance Frank walks to attend class. With this sponsorship program he will be able to achieve his dream of becoming a doctor.\nMeet Noeline,a young girl who lives with her mother in a one rented room with her four siblings. Noeline loves to take care of her mother who is a widow. The main jobs Noeline does at home are washing utensils and mopping the house. The mother is a peasant farmer who borrows people's land to cultivate food to feed the children. Through this sponsorship program Noeline will be able to get tuition and other basic necessities at home.\nJackie is a 15 year girl who is HIV positive and on ART. She loves to socialize with friends and has the desire to grow into a strong and powerful woman as she understands that the key to her future is education. Her father dies three years ago leaving the family to be dealt with by the mom. With the help of sponsorship, her family will receive the counseling they need to support their child in achieving her dreams. Jackie's mother is also HIV positive, but works as a pump attendant to a local petrol station in the suburbs of Kassanda Town, Mubende district in Uganda.\nAshiraf is HIV positive on ART and is described as quiet, smart and helpful. She likes working in the garden and loves music, dance, and drama. She also enjoys watching and playing soccer. She stays with her mother who makes snacks like chapati and sumbusas alongside the road. Her mother is an important client who rents a mud poor house of one room in a bad condition. With the help of this sponsorship program Ashiraf will be able to achieve her dream of getting cured of AIDS and becoming an international footballer.\nMaureen is an 11 year old girl who lives with her mother and nine siblings in a semi structure house. Maureen's family is all HIV positive and seven of them are on ART and the four are still on septrine. Maureen's father is an alcoholic and abandoned the family because they were always sick and he was not able to pay schools tuition for his children. He took it as a burden and decided to run away, which leaves the mother to take care for the family. Her mother does her best, but the family relationship is aggressive and unstable. With the help of a sponsor, Maureen and her family will receive family support and guidance for better family relationships.\nAshiraf is a 4 year old boy who lives with his mother who works as a maid in someone's home. He likes drinking milk and eating ice cream and enjoys playing with toy cars and toy pistols. Ashiraf's father abandoned the family and went to a neighboring country to hide after neglecting his responsibilities. His mother is renting a small one room in a rural area in a very poor condition. His mother sometimes sells vegetables alongside the road. With the help of this sponsorship program Ashiraf will be able to achieve his academic pursuit of knowledge and future career dreams.\nNicholate is a joyful and well-behaved little girl. She likes using colored pencils to draw. She also enjoys going to school, with English being her favorite subject. She lives with her mother and three siblings in one small poor conditioned room in a slum area in Wakiso. Her mother sells cooked food in evening local market and she also washes people's clothes around the village, but despite all her efforts the income is insufficient to cater for the children. The father abandoned the family and he resorted to drinking local beer during the day and evening so that the children often find him sleeping alongside the road drunk during the day. Through this sponsorship program Nicholate can achieve her goal of becoming a medical doctor.\nMeet Laila, an 8 year old girl who lives with her mother in house made of dry banana leaves with her sister. Laila loves eating matooke and chicken and she enjoys playing with herself with hand-made dolls from banana fibers. Laila's mother is a casual laborer on peoples' farms who works for a day's food for her children. The father died in an accident when Laila was only two years old and left her helpless mother to cater for the children. Laila's family has no basic home needs like beddings, utensils, and there is free entry and exit to their house for dangerous insects and animals; the house is too small for the three occupants. With the help of this sponsorship program Laila will be able to achieve her goal of becoming a nurse.\nMeet Florence, an orphan who lives in Kyoga Kassanda, Mubende with her helpless mother, who is HIV positive and on ART, with three brothers and four sisters. Her mother works as a casual laborer on peoples farms. It's now been five years since Florence lost her father who died of AIDS. She tested negative and likes playing netball, music, and dancing. She is in her fifth year of primary and her mother described her as smart and helpful. Florence has problems with lack school fees and some scholastic materials to meet her dream of becoming a teacher. With your assistance through this child sponsorship program Florence can continue to thrive as she works hard to advance to the next grade level.\nOliva is known for being well-behaved and very active in all her activities. Her favorite game is netball and music. Oliva was given to her grandmother when she was at the age of one. Oliva's father died and mother abandoned her. Oliva likes studying and she is now in her third year of primary and she is progressing well, but she has a problem of lack of school fees, clothes, and she often falls sick though she tested HIV negative. Oliva's grandmother is a peasant farmer who does casual labor on people's farm. With the help of child sponsorship program Oliva will be able to achieve her dream job of becoming a nurse.\nAsnasio is an 11 year old boy with an egocentric attitude towards playing football. He likes eating matooke, rice, and beef meat. His major activity at home is helping his mother in the garden, and fetching water at home. Asnasio's family dynamic is quite strained. Asnasio's father abandoned him when he was only five years old, and he left Asnasio's mother to care for the family. His mother does her best, but the family relationship is aggressive and unstable with the needs like school fees , scholastic materials and clothes. The family lives in a small mud wattle house with basic needs in the household. With the help of a sponsorship program Asnasio will be able to attain his dream of becoming a medical doctor.\nDorothy is a 6 year old girl who studies in her first year of primary and loves to play games like making bears and skipping. She always cleans the courtyard as her usual housework at home. Despite of all her efforts her father abandoned her to a helpless mother who was forced to take care of the family and finds it difficult to make ends meet and pay for her school fees, scholastic materials, and medication for Dorothy who often falls sick. With your help through the sponsorship program, the family will receive basis necessities and have the room to free up funds to pay for tuition and also to achieve her dream of becoming a teacher.\nMeet Abbey, a 10 year old boy who likes to play football and loves to eat matooke and groundnuts. He lives with his grandmother as his father died of fever and he was abandoned by his mother at an early age of 4 years. The grandmother is very old at an age of 80 years and Abbey has 7 siblings. Out of the seven two are positive. Abbey has find difficulties with school fees, scholastic materials and the grandmother has no land to carry on farming and cater for the abbey. With the help of sponsorship program Abbey will be able to achieve his dream of becoming a doctor.\nMeet Prosy who is a 7 months old girl. Her mother is a young parent, who was impregnated after when she went to Kampala city to look for baby seating jobs. The mother was raped by a gang of 3 men whom she cannot remember as of now. Prosy's mother is 16 years old, not working neither, and does not have a permanent home where she can go to look after her child. This younger mother dreams of seeing her child grow up and becoming an adult, and also hopes she will become a lawyer! During this home visit made by the CSP team, Prosy was found to be lacking so many basics because the young lady is extremely vulnerable and need support of any kind. She is in need of things such as: clothes, food, seeds, counseling, home basics, and an IGA to help support herself and Prosy. She struggles to make ends meet. Therefore we call up all sponsors around the world to help support Prosy and her mother.\nJane is 15 year old living with her grandmother and five siblings in a very poor mud wattle structure house. Jane is HIV positive and on ART. She has been with on and off fever making her miss classes and at times miss end of year examination, thus causing her to stay in the same class for more than one year.. Jane's parents abandoned her at the age of 9 years because of their ill way of living. With help of this sponsorship program Jane can get school fees and get some for buying nutritious foods.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"At Dumpster Blog, we strive to bring you the best service at the best price available. We have a wide selection of dumpster sizes in stock in Clayton, ID, with flexible & timley pickup and delivery. Give us a call today for all your dumpster rental and roll off needs.\nAny type of indoor redesigning can add up to a lot of waste. Once you upgrade areas of your home you can have a serious large insert of junk. Replacing pantry shelves with refreshing updated pantry shelves is a huge undertaking. Exchanging all of the shaping in your home is a large job and you may need a lot involving room in a very dumpster to chuck it all apart. Wood doors or any other kind of doors could be thrown away inside the small, medium or huge dumpster as well.\nIn relation to renovating a home, a lot of specifics must be looked after and the one thing that a lot of persons forget about is the thing that they are going to use the trash left behind from the renovation. One home remodeling could prove as much as 30 yards involving trash which may equal just as much as 2 tons. Thus, what exactly in case the homeowner use all of the waste left over? Get associated with the Cost of a Roll Off Dumpster in Clayton, ID vendor, who tells the customer straight up about the items which can be got rid of into a dumpster, pricing of the Cost of a Roll Off Dumpster in Clayton, ID and terms and conditions in the disposal. When you are getting associated with a organization that can provide quality providers at affordable cost, you will realize that junk removal upon completion of work was never this straightforward and this affordable.\nAs a builder, you must create strong performing relationships to make certain your needs tend to be met. You wouldn't want an difficult to rely on service that can end up squandering your in the long run. You want someone it is possible to count on. Fostering a strong connection with different companions, especially some sort of Cost of a Roll Off Dumpster in Clayton, ID company, ensures that you can rely on the crooks to be right now there on time in the event the project will begin. You don't have to worry about your job currently being held up because the dumpster didn't show, thus keeping the project upon its time desk.\nIt's important to understand, however, that sometimes a roll-off Cost of a Roll Off Dumpster in Clayton, ID is merely that: some sort of Cost of a Roll Off Dumpster in Clayton, ID. It does not immediately include property clean out plan to get material into the dumpster available for you. This can come as a surprise to some folks, since many dumpster companies claim that their dumpsters are accessible for hauling junk, removing waste, and dealing with home cleanouts. You'll want to read the terms and conditions when you're taking a look at dumpster firms to ensure you know if these are generally suggested uses or integrated services.\nThe advantages of a development Cost of a Roll Off Dumpster in Clayton, ID are cleaner streets, cleaner neighborhoods, and cleaner life styles because dumpsters tend to be an easy, basic and effective way involving removing waste materials. It's simple if we do our own part by continuing to keep our property clean and our own neighbor does the same with the exceptional neighbor does the same along with the process reciprocates subsequently everyone is going to be doing operator to keep our own towns thoroughly clean in an effective way.\nDumpster leases are actually a simple process. You must do some research when you pick the organization and what is best for you. The size of the actual dumpster, disposal trash can or waste bin is just about the biggest issues that you will find to alter. Dumpsters are generally huge, whereas convenience bins tend to be smaller.\nYou can throw the countertops in to the dumpster with ease. This flooring might be a bigger task to tear away from each other. Once the floors is all ripped apart, it can simply be chucked into the booked dumpster along with the outdated cabinets.\nDumpster rental fees can support homeowners, organizations, churches, agents, and area effort initiatives. The Cost of a Roll Off Dumpster in Clayton approach is very quick and simple, and inexpensive. There are various measured Maryland dumpster rental fees available including a medium sized dumpster using wheels upon it. Next day supply options help make cleanup basic and efficient. Community trash pick-up regulations restrict the amount of waste that is available regarding pickup because of the city. Keep the home seeing that neat and clean as it can be during your kitchen renovations by using a structure Cost of a Roll Off Dumpster in Clayton.\nIf you do a minor research of ones own you are sure to get something that will help you eradicate all the waste materials in your home\/office. Sometimes people demand dumpsters when they're remodeling their property. As you move the work is happening, a lot of debris and dirt needs to end up being deposited in a huge garbage package so that it isn't going to spread all around the property.\nIf you're thinking of remodeling your traditional home you already know you have rules to follow. Quite a few historical residences need to end up being updated using certain objects including perhaps the color of your current paint.\nHomeowners with a space or 2 of demo in front of them can conserve money by renting their own dumpster and have absolutely it sent straight to their internet site. Construction organizations use Cost of a Roll Off Dumpster in Clayton organizations for almost just about every job -- that they tell them the dimensions of a rotate off they really want and then a regional roll off or 2 is sent and dropped off. Local is generally cheaper and the ones can go to a few websites that will give them a quote on the dimensions they need, as well as playing a major part to keep the test materials organized and the structure site uncluttered.\nIn fact, using a from your nearby junk removal service, you can finally point out good riddance to just about every last lot of waste in and around your premises and release your inside and outside space for any whole new group of storage and also usage options. Most trash hauling providers will handle virtually many methods from appliance convenience to furniture convenience, with the exception of dangerous waste, asbestos fiber materials, and also medical products. You can even possess the dumpster dropped off in the home and carry out the heavy lifting by yourself to save on a final cost of all of that trash removal.\nMaybe you are building a store, lender or other the category of business. You will need to rent some sort of dumpster. With substantial buildings determined will need a considerable dumpster or many dumpsters depending on the real size and the way many people are taking care of building the modern business.\nSeeking the right dimensions dumpster is the step one. All deny items or materials should be placed correctly and entirely inside the dumpster there should be nothing at all sticking out. Locating the correct dumpster to meet your needs and hauling away your current trash can end up being determined depending on the project. Small projects employing involve organizing basements or garages though larger initiatives include refurbishments and kitchen makeovers. Modest dumpsters are available which might be wheeled with regards to the property and also medium and enormous dumpsters are available for initiatives that involve structure and redevelopment materials.\nRemoving rubbish from the residence of a obsessive hoarder is no easy task-it not only includes a great deal of physical labor, but it is also a very difficult emotional problem for the hoarder, their own family, and also anyone who has any kind interest in taking care of the house in question. Of these major rubbish removal careers, especially those under time constraints, might be necessary to be able to speed up this process and spare homeowners further expenses like dumping fees and the fuel used to drive load immediately after load regarding trash towards the transfer place.\nIf you are thirstily planning on renovating your kitchen compared to you have been active picking out new kitchen cabinets, new flooring, possibly new kitchen appliances and more. Together with your large kitchen remodeling job you will definitely need to rent a dumpster to remove all of the trash.\nQuotes is dependent upon the trash and type and no matter if there's recyclables on board, or not. Some municipalities require trash being placed in different categories in order that it may be recycled as much as possible. A web based company may quote the paramount price for you need, and they do it by means of their website so you don't have to Bing your mind till it's decrease. There's a short form to try and do and it's in this article you can figure out what size spin off pot you'll need.\nThere are lots of times when you will need a Cost of a Roll Off Dumpster in Clayton, ID. Maybe you obtained a new hideaway that you think about a cabin. You have got the great price on the log cabin with a gorgeous yard and also lake rights. Now the log cabin needs to have several remodeling completed. The first product you notice which should be taken care of will be the roof. Your roof is starting to help leak and the shingles are generally slowly being blown from the roof because nails are generally coming up from using it.\nAt first you might like to try to have a very rummage sale. Make the rates on the goods very low to help you just get rid of them. Do not try to generate at your search sale, you happen to be having this to get rid of items. It is better to sell your goods for cheap to someone that can have used them or repair them rather than just throw them from the dumpster and turn out at a junkyard.\nRight now, Cost of a Roll Off Dumpster in Clayton, ID companies are supplying more providers geared toward and also bundled specifically for the special attractions market. Porta-potty leases or portable toilets really are a usual \"add on\" program, and is often offered together with hand kitchen sink and even portable shower leases. The more providers a Cost of a Roll Off Dumpster in Clayton, ID business can offer, the harder potential clients they're able to attract. Not everyone who demands Cost of a Roll Off Dumpster in Clayton, ID service in addition needs porta-potty leases as well, but it is nice to know the services can be purchased if they carry out. By bundling these services, Cost of a Roll Off Dumpster in Clayton, ID organizations help party planners saving time by coping with fewer companies. Also, you'll find price cost savings associated with packages which make these attractive regarding consumers.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Drop Leaf Kitchen Islands Island With Drop Leaf images that posted in this website was uploaded by Wellinsurance.org. Drop Leaf Kitchen Islands Island With Drop Leafequipped with aHD resolution 900 x 800 jpeg 256kB Pixel.You can save Drop Leaf Kitchen Islands Island With Drop Leaf for free to your devices.\nIf you want to Save Drop Leaf Kitchen Islands Island With Drop Leafwith original size you can click theDownload link.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Richard Pryor was another African-American who just so happened to develop multiple sclerosis in spite of the statistics to the contrary. His diagnosis came in 1986 at a time when it was rare for people of color to get MS. He was a brilliant stand-up comedian who also did film and television. The subjects of his comedy included racism, race relations, politics, African-American culture, and everyday life, to name just a few.\nrichard Pryor - who was he?\nIf you grew up in the 70's and 80's, you no doubt have heard about and watched this iconic figure on TV. I'm sure his shows were \"cleaned up\" for prime time, however, if you attended his live shows, you were sure to see his true self.\nHe was very straight forward and didn't hesitate to use language that some people thought was vulgar and profane. Even with this, he was still very popular and influenced many comedians who came after him. Some artists who attributed him as influencing their choice of career, included Eddie Murphy, Dave Chappelle, Chris Rock, Robin Williams, Bernie Mac, George Carlin, and Whoopi Goldberg.\nFrom what I can see by reading about his life and watching a few of his routines, Richard lived life to the fullest. We would all do well to take on this attitude. Life is too short and too precious to live half-heartedly. His life and the legacy he left will forever touch the lives of those who follow, whether directly or indirectly.\nOne thing I learned about him was that he even hosted a children's show in 1984 called Pryor's Place which was very similar to Sesame Street. Here's a video clip about it and I'm sure you can find a few episodes if you looked for them.\nOne of the things I always remember about him was his smile. Even after he was diagnosed with MS, he continued to remain active as a comedian. He worked until around 1999, when multiple sclerosis finally brought his career to an end.\n\"In 1986, Pryor was diagnosed with multiple sclerosis, a degenerative disease of the central nervous system. But that didn't stop his performing. In 1992 he could still be seen live at the renowned Comedy Store in West Hollywood, making jokes about his afflictions and his wheelchair and still painting incredible verbal pictures that stimulated the imagination of the audience so much they actually lived and felt the experiences with him.\"\nRichard was born on December 1, 1940 and died on December 10, 2005 at the age of 65. During his last few years he needed an electric scooter to get around due to worsening mobility problems.\n\"Although his multiple sclerosis prevented him from performing in his last few years before his death in December 10, 2005, from his home in the Encino, which he shared with his two rescued dogs, Homer and Spirit, Pryor's mind continued to catalog the events of his life and the world around him. Richard Pryor will never lose his position as cultural icon and we can only hope that a cure to this debilitating disease will soon be found so as not to loose another soul that reaches out and touches as did Pryor and his ability to allow us to experience our reality with the outrageous, profane, and scabrous perspective that only he imparted.\"\nIf you love animals as well, why not consider making a donation to his organization? As I said earlier, his legacy lives on even though he is not physically here with us. RIP, Richard. We love you!\nGo back to Famous People with MS O-P.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The US on Friday released an unclassified assessment by its intelligence community of the August 21chemical weapons attacks by Syria that killed 1,429 people, including 426 children.\nThe US said it knows the chemical weapons -- a nerve agent -- were delivered by rockets, at what time they were launched and where they landed. And who launched them.\nThe attacks, launched early August 21 morning, targeted 12 neighbourhoods outside of Damascus, according to the US assessment, which came with a map showing them.\nAnnouncing findings of the assessment on Friday, US secretary of state John Kerry called Syrian president Bashar al-Assad a \"thug\" and a \"murderer\".\n\"We assess with high confidence that the Syrian government carried out the chemical weapons attack against opposition elements in the Damascus suburbs on August 21,\" said the document, titled \"U.S. Government Assessment of the Syrian Government's Use of Chemical Weapons on August 21, 2013\".\nThe assessment was intended to address growing concerns about the nature of intelligence in the possession of the administration, which has looked ready to launch a military action any day now.\nKerry said this assessment was shared with congressional leaders, who, also know a little more -- most of it classified information, which won't be released for now.\nThe administration wants to make sure Iraq is not repeated.\nBut there was no word on what will be US response. Kerry only said what it won't be: no boots on the ground, not open-ended as Afghanistan, and won't own the civil war.\nThe US will respond though, for sure. \"History would judge us all extraordinarily harshly if we turned a blind eye to a dictator's wanton use of weapons of mass destruction against all warnings, against all common understanding of decency,\" Kerry said.\nThe response will be \"limited\" and \"tailored\" to ensure a \"despot's brutal\" use of chemical weapons against is held accountable, Kerry said.\nAnd after the attacks, the Syrian government tried to destroy the evidence by bombing the areas with four time more intensity that over the previous few days.\nAnother communications intercept showed Syrian chemical weapons personnel were directed to cease operations in the afternoon of August 21, a few hours after the attacks.\nThe assessment also detailed activities that showed preparations were under way for the chemical weapons on that day -- presence of personnel from Syrian Scientific Studies and Research Center, which manages Syria's chemical weapons programme.\nAnd the August 21 attacks were not the first. The US intelligence community believes the Syrian government used chemical weapons on a scale scale multiple times in 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Blaenau Gwent CBC provides a domestic pest control service in partnership with Mitie Pest Control.\nThe service offers free treatments for rat infestations at domestic properties within Blaenau Gwent.\nMitie also offers pest control services to commercial and industrial premises. For prices on these treatments please contact Mitie Pest Control directly.\nFor information about pests visit the British Pest Control Association website.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"It's always challenging to look for the right moving companies for your particular relocation. However, Easy Move is a moving company Kuwait would gladly recommend to anyone looking for expertise and reliability. We provide full moving services, from packing and transport to storage and logistics. We are movers Kuwait is proud of and thankful for providing customers with quick and easy relocation solutions. Our experience and status in the moving industry conveys trust that the job will be done above and beyond ideal standards of moving. So call us right now and get a free moving quote.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Death Nebula. It was a spacer's ghost story about a cosmic cloud that would appear, envelop a populated world, and disappear, taking the world with it. Few believed in the tall tales told by spacers in waystation taprooms... but at the same time none would openly discount the Death Nebula either. Navigators even had an official name for it: The Necrose Nebula.\nThree thousand years ago by the Galactic Standard Calendar, Jittannia Prime was swallowed up by the Death Nebula. It was, at the time, the thirteenth planet known to have suffered that fate. Many mourned the loss of twenty seven billion souls, while others mourned the loss of the research and design facilities located there. Seven Galactic Standard months ago, Jittannia Prime became the first world ever to reappear. It was no longer in the Jittannia System, but the faintly whispering world beacon was clearly the same, none the less.\nTo avoid navigational confusion, Jittannia Prime has a new, chilling designation: Necrose XIII. Planet wide salvage rights are open for any brave enough to jump there, strong enough to recover lost tech, and lucky enough to escape.\nHow badly do you want riches and glory?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"UVA\/UVB and blue light protection, naturally derived zinc oxide, vegan, mattifying and in a tube containing recycled plastic. Clean Screen Mineral SPF 30 is the shining example of clean sunscreen - inside and out. Get our latest innovation, available now.\nAnd since we're sensitive-skin experts, Clean Screen goes beyond just sun protection. We've added Yellow Passion Fruit Extract as a powerful antioxidant, protecting against pollution and potential irritation flare-ups.\nPlus Rice Extract which not only soothes reddened skin, but gives a mattifying finishto combat sun-induced shine and excess oil for skin that looks fresh all day.\n90% of visible aging is caused by sun damage. Shocking, right? UVB rays are mostly responsible for sunburn and redness, while UVA rays penetrate deeper into the skin, causing lines and wrinkles. These rays get to your skin even on cloudy days, but our 22% non-nano naturally derived Zinc Oxide provides protection from both - phew.\nFormulated for daily use, the Zinc Oxide sits gently on the skin's surface as a barrier against sun damage, and won't clog pores, cause allergies or irritate sensitive skin.\nEver closer to Zero Waste.\nClean Screen is our shining example of working toward Zero Waste by 2021. We've used 50% recycled plastic to create our tube - the maximum we can while protecting the formula, and 100% recycled plastic for the cap. Plus being a one-type plastic, it's designed to be recycled all over again.\nWorking with our suppliers, we've upped our sustainability game further. While we use an extract from Yellow Passion Fruit for its antioxidant power, the juice and seeds go to use in the food industry for drinks and cakes, while the leftover granules end up as scrubbing particles, making the waste of this ingredient zero.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Body Oils are Pretty Much Amazing.\nWe use body oils daily in our home and love the versatility and options they offer. Body oils are typically used to moisturize our bodies or face and can be used as massage oils. I use our handmade oils as a deep moisturizing massage oil after a shower (I'm one of those the-hotter-the-water-the-better kind of people!) or as an overnight facial oil. I also have a baby safe body oil blend for my 17 month old daughter Chloe Mae, which we use constantly to hydrate and nourish her sweet skin.\nOne of the many reasons I choose to make our own truly natural bath and body products is because Chloe and I are what I like to call, \"delicate flowers.\" When it comes to sensitivity, whether it is in regards to food, mainstream body lotions or artificial ingredients, our skin typically freaks out. Since we've started creating our own blends, we have noticed a huge difference in the health and feel of our skin.\nI also love that we can easily tailor the blend to set a specific mood or mindset. When I want to calm Chloe before bedtime, I use a blend of lavender and roman chamomile essential oil in her baby oil \u2013 with proper dilution of course. And when I need a little pick me up in the morning, when no amount of caffeine or sugar will save me, I reach for my sweet orange body oil. The options are truly endless!\nNot to mention, a lot of mainstream body oils are full of mineral oil + artificial ingredients, so making or buying truly natural body oil is always healthier!\nCarrier oils, in most cases, are liquid oil extracted from plant parts, such as seeds or the flesh of the fruit. They can act as a barrier or a moisturizer, with a wide range of skin soothing benefits. They can be used with essential oils, as a base for herbal infusions, or alone to simply moisturize your skin.\nThe following carrier oils, also known as base oils, are some of my favorite body oil options when blending for babies, children and sensitive momma's. My favorite part about creating body oil blends is the freedom to create products that are tailored to a specific skin issue or personal preference of texture. I also like to make sure that any carrier oil used is baby + child safe. The biggest concern would be a nut allergy, so do you research + use your best judgment!\nSweet Almond Oil (Pranus amygdalus): A light to medium weight, light colored oil extracted from the kernels of the almond tree. Sweet Almond Oil is high in vitamins A + E, as well as fatty acids (omega 9 & 6) and antioxidants. It absorbs well and is great for most skin types, especially dry, irritated or itchy skin. Those with acne or oily prone skin should try and avoid sweet almond oil or mix with another light weight, non-clogging oil like grapeseed.\nApricot Kernel Oil (Prunus armeniaca): Also a lightweight oil with a light golden color and little to no scent. It is extracted from the kernel of the apricot fruit either through solvent extraction (chemical) or cold pressed. I always suggest cold pressed and organic when possible! Apricot kernel oil spreads well on skin and absorbs rather quickly, without leaving a greasy feeling. This oil is full of fatty acids like oleic (omega-9) and linoleic (omega-6) and Vitamin E. It is recommended for sensitive or mature skin. Apricot oil is similar to sweet almond oil, so those with acne should try to avoid it, or mix with a less pore clogging oil.\nCoconut Oil (cocos nucifer): A very popular, multifaceted oil that is extracted from the fruit of the coconut tree. It liquefies at temperatures above 76F and melts easily onto your skin when applied. It has a very smooth, silky feel to it and is ideal for all skin types, especially those with dry patches or rough spots. As always, choose an organic, unrefined coconut oil so your skin receives all of the true benefits like capric, caprylic and lauric acid.\nGrapeseed Oil (Vitis vinifera): A light green, lightly scented oil that is extracted from grape vines. It's full of antioxidants, vitamin E + essential fatty acids. It is a lightweight oil that is easily absorbed and won't leave a greasy residue. Its suitable for all skin types, especially those who prefer a quick drying oil. Personally, I find this oil great for a body oil, but feels a little heavy when used alone as a facial moisturizer. I tend to mix this one with another oil.\nJojoba Oil (Simmondsia chinensis): This is one of my favorite oils for so many reasons! It's actually an oil and a liquid wax pressed from the seeds of the jojoba plant. Jojoba oil closely mimics our own skin's sebum, making it ideal for everyone, including those prone to oily or trouble skin. Because of it's ability to act like a wax, jojoba oil helps break up the dirt + grime in clogged pores and help balances our oil production. Jojoba oil also helps our skin retain moisture, making it perfect for body oils used right out of a hot shower or during dry, cold winter months. It is full of fatty acids + Vitamin E, which smooth's and softens skin and can reduce the appearance of fine lines. It's naturally a deep golden color, but colorless options are available.\nNow let's take your new\/revived\/never lost it love of DIY'ing and make some body oil for you and your little one. It's super easy, I promise. And if you don't have the time or energy to make your own, I've got you covered Mama! Head on over to my shop to stock up on family friendly body oils, wellness blends, or Mom life necessities.\nCombine your 2oz of Sweet Almond Oil and 2oz of Apricot Kernel Oil into a measuring cup or pourable container. This becomes your body oil base for both Mama + Baby Body Oil. Pretty easy, right?\nFor our Baby Oil Blend, use 3 drops of Lavender essential oil in your empty glass bottle. This is a .5% dilution that's great for babies 6 months and older. I love using lavender around our little ones for its calming and comforting aroma. If you want an unscented version or newborn safe body oil, skip this step!\nNow fill your bottle with the pre-blended base oil. Once capped, lightly shake the bottle to combine oils. Congrats Mama, you just made a simple baby oil blend! Now it's your turn!\nFor our Mama Oil Blend, grab your glass bottle with dropper and add 12 drops of Sweet Orange essential oil. This is a 2% dilution that is perfect for grown ups. I love using Sweet Orange for its mood boosting and refreshing scent.\nNext, fill your bottle with the remaining base oil blend of Sweet Almond + Apricot Kernel. Again, once capped, lightly shake to combine oils.\nLastly, wipe down any spills on your bottles and add labels. I personally use pretty wasabi tape for my at home DIY blends. I like to either write a note with ingredients and the date made in a notebook, or write it on my labels. I recommend making a new batch of oils every 6-12 months. They will hold up well in a cool, dark place, out of direct sunlight.\nThere are so many versions of this simple DIY Mama + Baby Body Oil that can be created for your growing family's needs. Have fun, try different blends and find which ones you love most. And when you use essential oils properly, the options become almost endless for what you can come up with!\nChelsea is the mama, creator and daydreamer behind Wildflower Gypsy, a family friendly aromatherapy line of bath + body products. \"Wildflower Gypsy was born out of necessity when I couldn't find any truly natural products that helped with skin issues or used essential oils properly around our little ones. I wanted skincare that was simple + wholesome, so I made my own! When I'm not creating new blends with Wildflower Gypsy, you can find me chasing around my 17 month old Daughter Chloe in a flowy sundress, gardening with my herbs + flowers or reading another book on aromatherapy and plant magic.\" Visit Chelsea on Instagram or shop her collection on Etsy!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Arrived in Vienna a bit ahead of schedule. This last section follows EuroVelo Route 9 into Wien, a section I might repeat in reverse when heading to Brno and then Poland.\nThe last ~20km was spent tackling the sprawl of Northeast Vienna. Pretty much a straightshot, but the drab cityscape goes on for a long time before the Vienna you remember from covers of philosophy textbooks appears. All sources indicated that Vienna would be a bike-friendly town. And, indeed, there are dedicated bike lanes on many roads. However, they also end unexpectedly with poor signage and occasional misplaced curbs.\nOne positive aspect of this bike lane confusion is that I had my first opportunity of the trip to exchange the universally understood bicycle\/motorist salute: the raised middle finger. Like Prague, Boston and many other elder cities, Vienna is built along a river. However, unlike those other cities, the main bridges over the Danube River do not have bike or pedestrian lanes. I was following what I thought was a bicycle lane over the 4-lane wide G\u00fcrtelbr\u00fccke bridge. The lane ended and I was forced onto the roadway. Two hundred meters or so later, a sidewalk appeared but with constant traffic, a fully loaded bicycle and high curb to navigate, I deemed it safer to continue pedaling with traffic over the bridge. A woman two lanes over leaned her head out of the passenger window and yelled at me. She was a good 10 car lengths ahead by the time I flipped the bird but luckily I was acknowledged with the reply. I couldn't think of a more fitting way to be welcomed to Vienna after two weeks of bicycling from Berlin.\nThis entry was posted in Road Report and tagged Austria, bicycle touring, rude gestures, Vienna on May 5, 2013 by papish.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapjuo b/data_all_eng_slimpj/shuffled/split2/finalzzzapjuo
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+{"text":"Whoever said that only six-pack abs and a lean physique give you the freedom to walk shirtless in public, maybe needs a new perspective. A picture of Jude Law on the sets of The New Pope, wearing nothing but a pair of white speedos, went viral this week. The actor is the hero figure for everyone, who has the confidence to pull off any outfit, not depending upon the body structure.\nJude is the popular face of a critically acclaimed series and a supporting actor in many blockbuster movies. The actor stands as a living model of the man next door, who is ageing, balding and yet pulls it with utmost confidence. The confidence is dearly attractive and brings out the inner charm.\nWe thank the actor to bust the bubble of the showbiz personalities and making the ordinary extraordinary!\nDid Captain Marvel Negative Review Attack Cause Rotten Tomatoes to Ban User Comments Before a Film's Release?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"LAKELAND | Winter Haven will advance to the Class 5A championship game after a 50-42 victory Friday over Tampa Sickles.\nThe Blue Devils (25-5) will take on Fort Lauderdale Dillard in the finals Saturday at 3:30 p.m. at The Lakeland Center.\nErica Lanier and Charla Nelson-James led the Blue Devils with 16 points each.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"-Simon Poirier (2013-2016): \"Anaerobic digestion inhibitions: ecological dynamics, microbial indicators and resistance strategies. Examples of phenol and ammonia nitrogen\"\nDIGESTOMIC ANR Project (2016-2020): \"New operational strategies to overcome technical barriers in anaerobic digestion and extend its fields of application by using meta-omic approaches\"\nFASIC PHC Project (2017-2018): \"Novel statistical methods to improve waste degradation: merging advanced biotechnologies with forefront analytical tools, identification of biomarkers by integrating multiple-omics datasets\"\nMETHARESIST EC2CO Project (2015-2016): \"Omics approaches to identify resistance strategies of anaerobic digesters to micropollutant : example of phenol\"\nMadigou, C., L\u00ea Cao, K.-A., Bureau, C., Maz\u00e9as, L., D\u00e9jean, S., Chapleur, O. 2019. Ecological consequences of abrupt temperature changes in anaerobic digesters. Chemical Engineering Journal 361:266-277.\nPoirier S, D\u00e9jean S, Chapleur O. 2018. Support media can steer methanogenesis in the presence of phenol through biotic and abiotic effects. Water Res 140:24-33.\nPoirier S, Chapleur O. 2018. Inhibition of anaerobic digestion by phenol and ammonia: Effect on degradation performances and microbial dynamics. Data in Brief 19:2235:2239.\nPoirier S, Chapleur O. 2018. Influence of support media supplementation to reduce the inhibition of anaerobic digestion by phenol and ammonia: Effect on degradation performances and microbial dynamics. Data in Brief 19:1733-1754.\nPoirier S, Madigou C, Bouchez T, Chapleur O. 2017. Improving anaerobic digestion with support media: Mitigation of ammonia inhibition and effect on microbial communities. Bioresour Technol 235:229-239.\nConnan R, Dabert P, Le Roux S, Chapleur O, Bridoux G, Vanotti MB, Beline F, Magri A. 2017. Characterization of a combined batch-continuous procedure for the culture of anammox biomass. Ecological Engineering 106:231-241.\nPoirier S, Bize A, Bureau C, Bouchez T, Chapleur O. 2016. Community shifts within anaerobic digestion microbiota facing phenol inhibition: Towards early warning microbial indicators? Water Res 100:296-305.\nMadigou C, Poirier S, Bureau C, Chapleur O. 2016. Acclimation strategy to increase phenol tolerance of an anaerobic microbiota. Bioresour Technol 216:77-86.\nPoirier S, Desmond-Le Qu\u00e9m\u00e9ner E, Madigou C, Bouchez T, Chapleur O. 2016. Anaerobic digestion of biowaste under extreme ammonia concentration: Identification of key microbial phylotypes. Bioresour Technol 207:92-101.\nHao L, Bize A, Conteau D, Chapleur O, Courtois S, Kroff P, Desmond-Le Qu\u00e9m\u00e9ner E, Bouchez T, Maz\u00e9as L. 2016. New insights into the key microbial phylotypes of anaerobic sludge digesters under different operational conditions. Water Res 102:158-169.\nChapleur O, Mazeas L, Godon JJ, Bouchez T. 2016. Asymmetrical response of anaerobic digestion microbiota to temperature changes. Appl Microbiol Biotechnol 100:1445-1457.\nChapleur O, Madigou C, Civade R, Rodolphe Y, Maz\u00e9as L, Bouchez T. 2016. Increasing concentrations of phenol progressively affect anaerobic digestion of cellulose and associated microbial communities. Biodegradation 27:15-27.\nL\u00fc F, Bize A, Guillot A, Monnet V, Madigou C, Chapleur O, Maz\u00e9as L, He P, Bouchez T. 2014. Metaproteomics of cellulose methanisation under thermophilic conditions reveals a surprisingly high proteolytic activity. ISME J 8:88-102.\nChapleur O, Bize A, Serain T, Maz\u00e9as L, Bouchez T. 2014. Co-inoculating ruminal content neither provides active hydrolytic microbes nor improves methanization of 13C-cellulose in batch digesters. FEMS Microbiol Ecol 87:616-629.\nChapleur O, Wu T-D, Guerquin-Kern J-L, Maz\u00e9as L, Bouchez T. 2013. SIMSISH technique does not alter the apparent isotopic composition of bacterial cells. PloS one 8:e77522.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"'mytime4me\u2122' WELLBEING GIFT BOX* & other health & wellbeing products.\nPlease visit our SHOP & browse!\n*Why is the gift box any different from the others available online?\nEVERY MONTH we will have a different featured product. We will partner with a small business with a 'real-life story' to share & can advocate the importance of positive mental health.\nBespoke gift boxes can also be made to order upon request.\nTo find out more details please contact us using the form below.\n**Personal & Small Business Mentoring : Relate, Revive & Thrive\u2122 program.\nDid you know that you have to SLOW down to LEAP forward?\nDo you wish you had someone to talk to? Someone to really listen to YOUR unique challenges & aspirations, with no hidden agenda? Someone who cares enough about your emotional wellbeing, to give YOU 100% of their attention to help revive, inspire & empower you to succeed?\nThen my Relate, Revive & Thrive\u2122 program could be just what you have been looking for! Guaranteed confidentiality, please get in touch now through the form above, for more info!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tejal Travels has a fleet of over 39 buses including Multi-axle Volvo and Sleeper and Semi-Sleeper buses in both A\/c and Non A\/c Category, it operates in all the major routes including Tamil Nadu, Kerala, Maharashtra. Tejal Travels has very good reputation with its punctuality, staff behavior and bus interiors hygiene. Some of the Important routes of Tejal Travels include Thirunelveli To Palladam, Thirunelveli To Coimbatore, Nagercoil To Coimbatore, Sattur To Coimbatore, Kovilpatti To Coimbatore etc.\nTejal Travels Online Bus Booking can be done on Purebus.com using online payment (Credit Cards, Debit Cards or Net banking). Customers can carry the Mobile Ticket(SMS confirmation) that they receive from Purebus.com after the booking. Customers can call Purebus.com's 24\/7 Customer Support for any queries that they have. All transactions on Purebus.com are safe and secure.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzapxzd b/data_all_eng_slimpj/shuffled/split2/finalzzzapxzd
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@@ -0,0 +1,5 @@
+{"text":"Pointing right to your heart, these sharp Arrow hoop earrings are for those who want to make a statement. Made from 18ct yellow gold with a neat square profile and a hinge pin fastening, these earrings will make you stand out from the crowd.\nFinished with a soft brushed texture on the sides, the earrings have highly polished edges for a subtle contrast.\nThey each have seven brilliant diamonds set together in a line on the top face.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ASCIIbrot is a simple Mandelbrot fractal generator running in terminal. It's written in Go and it was designed to be ran on different platforms, however the linux based is the one on which it was tested.\nMandelbrot image renderer written in Go. It's using Go's concurrent and parallel programming feature called goroutines. The output is highly customizable and the rendered mandelbrot set color palette is very adaptive due to the various color palettes (these can be extended easily) and the interpolation algorithm used to blend these colors together.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our Ottawa Invisalign dentist realizes the impact a smile can make on a person's self-esteem. If you are looking to straighten your teeth, close gaps, or correct a bite issue, Invisalign may be the solution for you. Our Invisalign dentist will schedule a consultation and will discuss your treatment goals. During this meeting, photos and impressions will be taken of your teeth. After the initial consultation, we will send the impressions and pictures to Invisalign, where they will plan your treatment and manufacture your aligners.\nThere are many benefits to choosing Invisalign over traditional metal braces. A typical Invisalign case will take between five months and a year. During this time, you may remove your aligners to eat and drink. The ability to remove the aligners turns brushing, flossing, and regularly scheduled cleanings into something much easier than if there were wires, bands, and elastics. Your aligners will be replaced with new ones every two to three weeks to gently and gradually move your teeth to their desired position.\nOne of the primary concerns people often have about traditional metal braces is the aesthetic impact of the metalwork on their smile. Especially for adults, the prospect of wearing unattractive metal braces for long periods of time can be very discouraging. Our Ottawa Invisalign orthodontic option offers an almost invisible aligning system that straightens teeth fast and contains no metal.\nInvisalign treatment consists of a series of custom-made aligning trays. The dentist changes the trays every several weeks to fit the new tooth configuration. In addition to the reduced visual impact, Invisalign aligning trays can be temporarily removed for important occasions \u2013 meaning that treatment duration is patient-controlled. A great number of people report complete satisfaction with both our Ottawa Invisalign treatment and the stunning results.\nFirst, our Ottawa Invisalign dentist needs to devise an initial treatment plan before creating the special aligning trays. Three-dimensional digital images are taken of the entire jaw. These images allow the dentist to move specific teeth on the screen, view the jaw from different angles, and also foresee what the face might look like in years to come. In essence, this technology can show how Invisalign\u00ae trays will change the facial aesthetics.\nIf you have questions about our Ottawa Invisalign, please ask your Invisalign dentist.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Zillow Jacksonville: Do you think about buying are selling real estate? Then you must have searched about that in Online. Most of the people are select Zillow Group as a place to start their real estate. 75 million people unique visitors are logged into this Zillow Jacksonville. Zillow is an online real estate database company founded in 2006 created by Rich Barton, Lloyd Frink, former Microsoft executives and founders of Microsoft spin-off Expedia. The headquarters of this company is located in Seattle, Washington, United States. Go through the below sections to know more details about Zillow Jacksonville.\nJacksonville Zillow is buying and selling real estate. They also sell homes and give homes for rent. Zillow Jacksonville has single bedroom homes to large houses. People who are interested can check View listing photos, review sales history of a particular home. Here on this page, you can know more details about Jacksonville FL Homes, Houses For Rent in Jacksonville, and Apartments For Rent in Jacksonville. People who are interested to know more information about Zillow Jacksonville FL Houses can follow our website. You can also bookmark our page for more latest daily updates.\nZillow has 6255 homes for sale in Jacksonville FL. You can use Jacksonville Zillow detailed real estate filters to find the perfect place for living. Here in Jacksonville has 6255 single family homes for sale. Jacksonville Real Estate has single family homes for sale, rental listings, houses for rent, Jacksonville FL rental listings, Apartments for rent, and Jacksonville FL Houses For Rent. You can easily find your dream home by using filters on this website. People can also check data of houses and then can go for a visit.\nZillow has 1090 homes for sale in Jacksonville NC, 377 homes for sale in Jacksonville AR, 85 single family homes for sale in Jacksonville OR, and 296 homes for sale in Jacksonville Beach FL. Zillow Jacksonville is currently the most popular online real estate website with 75 million unique visitors a month. We are in hope that the given information is helpful for you. If you find this article useful then share it with your friends. By using Zillow website you can search houses, apartments for rent and sale in Jacksonville.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Designed with the newcomer to home espresso in mind, the Isomac Gran Macinino Grinder is outfitted with the features you need to grind for great espresso, but not the ones that are unnecessary and expensive.\nSince 1977 Isomac\u00ae plans and manufactures high quality espresso coffee machines, addressed to the market's premium area. Isomac\u00ae is an Italian company of long tradition and excellent products, devoted to coffee lovers.\nFeb 28, 2005\u00b7 Subject: Re: Isomac Granmacinino - any input on using this grinder? Almost all of the comments about the Lux grinder also apply the Isomacs so try a search on Lux. The main advantage of the Lux over the Isomacs is that the Lux can be easily microstepped for better grind selection.\nIsomac has a small range of branded grinders; this listing covers all of them.\nThe Isomac Professionale is a great entry level home espresso grinder, with a doserless spout enabling you to always grind fresh on demand. About Isomac Isomac is an Italian company of lon g tradition, producing high quality semi commercial espresso coffee machines and grinders since 1977.\nIsomac Granmacinino Grinder The Isomac Granmacinino is essentially a Isomac Professionale with a nice body finish and larger hopper. A great entry level home espresso grinder, with a doserless spout enabling you to always grind fresh on demand.\ncijena isomac professionale grinder The Isomac Professionale is a good entry level coffee grinder for home use. If the grinder is to be used primarily for espresso coffee, we recommend a grinder with . Isomac Macinino Professionale Conical Burr Coffee Grinder .\nRancilio Coffee Machines, Rancilio Home & Commercial Espresso Machines - Coffee Beanery. Australia Wide Delivery. Call Us On 1300 94 14 45. Buy Online.\nIsomac. Isomac is an Italian company with a great tradition dedicated to espresso coffee lovers. In addition to the production of professional coffee machines, Isomac distributes coffee grinders suitable for various needs of use and space.\nView and Download Isomac GRANMACININO use and maintenance manual online. GRANMACININO Coffee Maker pdf manual download. Also for: Macinino prof. inox.\nIsomac Coffee Machines. ... The Granmacinino machine by Isomac\u00ae. A premium espresso grinder. Its doserless, so you grind fresh every time directly into your portafilter & is the perfect mate to any of the Isomac espresso machines. Sized at 125mm x 200mm x 320mm and weighs 4kgs.\nIsomac Grinder Blades, compatible with Isomac Granmacinino Grinder and Isomac Professionale Grinder. Compare. Price $73.50. Add To Cart. Isomac Grinder Star Dosing Padle. Suits Ascaso I Mini and Iberital Challange. Compare. Price $11.50. Add To Cart. Isomac Professionale Grinder Upper Blades Holder.\ncijena isomac professionale grinder gold mine locations in australia holmes coal crusher diamond stone crusher in mumbai wet grinding environment friendly .\nGrinder on demand, 430 stainless steel body. Conic wheels in hardened steel - diameter 38 mm. Option to select grinding. 250 gr coffee beans hopper. Chromed cover.\nFind great deals on eBay for isomac. Shop with confidence.\nThe Isomac Professionale is a great value coffee grinder. Using conical burrs for consistent coffee grinds, this doserless grinder lets you keep your coffee fresh on demand.\nIsomac Professional The Isomac Professional coffee grinder is an adjustable, conical burr grinder that grinds the beans directly into the filter holder. Coffee is freshly ground at the moment you wish to make coffee thus insuring its freshness.\nIsomac Grinder Blades, compatible with Isomac Granmacinino Grinder and Isomac Professionale Grinder Best value and huge savings if you buy a grinder and accessories with your machine Add a grinder below and get automatically 8% OFF!","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"If you wish to purchase Haleigh Barrel chair By Charlton Home , try to find high quality wood joinery. If a bit is fixed with each other or if perhaps fingernails and screws are widely used to maintain the furniture with each other, it will not last long. It is recommended to spend more to acquire a Haleigh Barrel chair By Charlton Home with some exceptional hardwood joinery work on it. Truly consider the hip and legs of the Haleigh Barrel chair By Charlton Home you are considering getting. These legs will need to not only support bodyweight, they'll even be in touch with your floor coverings. If they seem like they'll do harm to your flooring due to the components used (like aluminum), then you may want to select a different style. When purchasing Haleigh Barrel chair By Charlton Home for the living area, pick neutral colors. You are able to affect the colours as well as other decor from the space by altering highlights, painting and other issues. Because of this, you can easily change up the style of your living room without having to spend a lot of money.\nThe most useful appliance of the house is the feces. Stools are helpful where ever it's needed it might be in the kitchen area or for other events in your own home. Stools can be operated manually that is easy in functioning. These barstools can be adjusted manually to the degree of the comfort zone. There are different types of bar feces which is made of wooden and metals, with this certain kinds of seat there may not have a adjustable tool which may hot lower the height from the seat but instead developed in a properly sited manner. Bar stools adds extra kitchen and dining seats with stylish look. It retains guests lingering of the modern home. Discovering bar seat with appropriate design for house is the tough 1. You need to find the correct style with right quality and price from the bar stools to become purchased. We have to find the perfect peak of the stools that suits our height, and must find the difference between the club stool and counter-top feces. Lastly, you need to discover stools at the correct price, correct quality with guarantee with needed durability which should satisfy us for that spending.\nIf youre searching for some severe rest, this loveseats solid cushion back again and rolled arms will deliverbut at a higher cost. Each seat cushioning contains 30 individual pocketed coils for springtime and support, and high strength chair foam for an additional coating of plushness and durability. In addition, the froth cushions are topped with extremely soft, blended lower down. For your extra comfort, there are three standard positions: seated, a small recline by having an raised footrest, and a complete recline with an elevated foot rest. It doesnt harm that this reclining loveseat appears lux, either. Its all-leather-based upholstery and ornamental nail mind cut give it a vintage look that can function with a lot of house dcor styles, from traditional to nation to arts and crafts. Plus, leather-based stands up well towards daily use, children and dog locks and nails. Just remember to situation the leather-based each and every six to twelve months.\nThis established features a one left arm couch established, two armless sofa models and one part couch set. This gives enough space to support your friends and relations. The fabric is 100% rayon for enough comfort and durability. The advantage of this couch established may be the mixing and matching of chairs in the available space within the room for any ideal shape. And also enables fitted even just in little rooms. It requires just mild putting together.Additionally, it functions push cushions for maximum comfort. You might want to do this set. It works nicely.\nThis group of RTA couches is available in various furniture smoothness. Some of them include glued leather, linen, corduroy, bed linen texture, rayon. You just need to choose the correct tapestry in significance for your balanced requirements. Accessible supplies offer an simple-to-thoroughly clean component. A characteristic that implies its versatility to face up to drinks and food. The extravagant seat and back again cushions election in favor of a cutting-advantage appear. The sq . molded equip that begins just over them enhances the magnificence. You have the conveniences and gentle styles that strike your entry.\nWhile the back of this couch shows the Chesterfield, using its series of tufting, a tux couch has cleaner, more angular outlines. It is said to have been the bellwether more contemporary designs in the 1920s. While some say this couch style took its title from Tuxedo Park in New York, it's also considered called after the classic fancy men's suit. You are able to identify a tux couch by its arms that are identical height as the back again. The tufting on the rear of the sofa and its rectangular silhouette will also be classic qualities. Soft cushions include comfort and ease for this high-equipped design. Also, while the sofa over from Upcountry is padded in leather, this popular style is usually completed in fabrics of all types, including the really trendy purple velvet.\nThis set features a one left arm couch established, two armless sofa models and something part couch set. This provides enough space to accommodate your family and friends. The fabric is 100Percent polyester for sufficient comfort and durability. The good thing about this sofa set is the matching and mixing of chairs within the space in the room for any ideal form. As well as allows for fitting even in small rooms. It takes just light putting together.It also functions push cushions for maximum comfort and ease. You might want to do this established. It does the job well.\nWhen it comes to purchasing room furniture, leather-based is definitely a good option. Not only does it look good with many designs, nevertheless its extremely durable (its the perfect materials for a household with kids or pets) and it is extremely-easy to clean, too. The downside of leather Hobert Scroll 3 Drawer Accent Chest? It may have a much higher cost than fabric, microfiber or fake leather furniture. This reclining loveseat remains a budget-friendly choice because of 1 guru trick: its sitting area is padded with leather, as the sides are upholstered with more inexpensive faux leather-based. Which means you get the appear and feel of a complete leather-based loveseat without the significant cost. Make use of the lever on the loveseats equip to kick back and take it easy on its high-density froth filling up, and consider the money held on. This specific loveseat comes with an additional bonus: professional assembly will come in numerous locations of the nation for the next charge.\nThis established accompanies the entire utility of Person Couch by having an exemplary couch style. It includes restricted hands, firm seat soft cushions and full-size mat with internal comes. It's eliminated underneath to wash it and take away it with a simple raising instrument. This mat is strengthened with a sturdy metal edge. is a calculating guideline and it has a Bi-collapsing emphasize. One of the worthy items to point out is the convenience! Place this extremely comfortable sofa in your comfy family room. Besides giving an all natural and stylish air, it gives an attractive character for your visitors. You can make your companions remain for any enjoyable nights films. The sofa has carcasses with obstructed edges and faux-completing on its legs. The soft cushions are, however, adjusted and include excellent flexibility. It's covered with a heavy polyethylene dietary fiber having a drawstring edge.\nTypically now utilized in homes having a great room or open up layout, they are usually big configurations that chair lots of guests. In smaller areas, they can be useful for seats in an area which has an odd part or other space limitation. The opportunity to mix part models, finish units and reclining sections according to room and individual choice makes this couch style super flexible. Sofa sofas are also available in a multitude of designs, from ultra contemporary or super luxe, to more family-pleasant contemporary designs.\nThis set of RTA sofas comes in various upholstery smoothness. A number of them incorporate glued leather, bed linen, corduroy, bed linen texture, rayon. All you need is to find the right tapestry in significance for the balanced requirements. Accessible materials offer an easy-to-clean component. A characteristic that suggests its versatility to face up to food and drinks. The luxurious chair and back again cushions election in favor of a reducing-advantage appear. The square molded arm that starts just above them enhances the elegance. You will find the conveniences and soft designs that strike your entry.\nThis is actually the next couch on our checklist. It is additional comfy and incredibly well suited for small rooms or loft residing. It is padded in rayon material, prospects inset control keys that provide an elegant diamond-tufted style. It is made from durable supplies and also the legs are made of long lasting wood to increase its sturdiness. The loveseat comes with an java stained wood legs and non-marking feet hats. It includes a comfortable froth padding and rayon material furniture that makes it extremely luxurious. It has a longue spot that gives an extraordinary room for resting.\nArrived these days for brand spanking new and it was helped with a beautiful gentleman named wealthy. He was really knowledged and educational about all the household furniture models and home furniture. Following 1000 questions and some laps for this location we got the right set that people came in for. Customer support was excellent and prices are unbeatable. Take a look location out!!\nArrived after shopping around for a couple days. Sales men had been great and Brandon helped me by helping cover their the deal. Not manipulative, gave me straight answers on all my concerns. I were left with a great deal on a John couch and adore chair combination. Shipping was next morning even though guaranteed for Wednesday. Pleased I went in. Highly recommend!\nSuperb costs and ideal service! The two gentleman who were there, we're very useful and and also have large amount of excellent costs on their own selection of Haleigh Barrel chair. We arrived looking for a queen furniture and also got a full set with container springtime and all sorts of for a good deal! Extremely friendly staff! I'd come back for other home furniture items!!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"There is only one video game design school in York for faculty to choose from. The following statistics and charts help analyze the current state of the video game design academic community in York, and the future trends. Academia in video game design includes video game design training at the video game design associates degree level.\nExactly 610 of the 36,230 video game design professionals employed in Pennsylvania, work in York. The number of video game design professionals in York has grown by 205% from 2006 to 2010. As the employment for video game design professionals in York has increased, overall employment in York has also increased.\nSalaries for video game design professionals in York have increased. In 2010 video game design professionals in York reportedly earned an average salary of $64,903 per year. Video game design professionals in York made a yearly average salary of $37,700, four years earlier in 2006. This growth is faster than the salary trend for all careers in York.\nVideo game design professionals in York make a median yearly salary of $61,060. The 10% of video game design professionals in the lowest pay bracket earn less than $40,003 per year, while those in the highest pay bracket earn approximately $93,547 per year.\nVideo game design professional salaries in York are, on average, higher than salaries earned for professionals in related fields.\nThere is only one accredited video game design school in the city. Many of York's video game design professionals are graduates of this school.\nThere is one degree and certificate program in video game design offered by York's video game design school.\nIn 2010 approximately 32 students graduated from video game design courses.\nCurrently we are in the process of collecting data for the number of video game design faculty in York, growth in the field of video game design academia and faculty salaries in York. If you are involved in teaching video game design courses to students at the associates degree in video game design level, please take a moment to anonymously submit your information to help us build a valuable database resource for the benefit of current and future faculty in the field of video game design in York. The data we have gathered thus far will be available for you to view once you have submitted your information.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What happens when you marry a hip and relaxing 'Coffeehouse, Bar & Lounge' with a bustling and energetic 'American Diner'? Open City, of course!\nCoffeehouse: We carry on the Tryst Coffeehouse tradition of providing you the opportunity to come together over an exceptional cup. We serve Tryst Coffee, the best coffee you will find, in a stimulating, community-based environment.\nDiner: We serve healthy, heartwarming breakfast all day, gourmet sandwiches and salads, brick oven pizza, juicy burgers and delicious entrees. We focus on seasonal ingredients and themes in our dishes, and source ingredients and products locally when possible.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Reforestation: The Yakouta dam with a capacity of 26.5 million cubic meter is subject to multiple pressures. To protect the work and sustain water resources, the Water Agency Liptako through its Local Committee for Water takes action within this framework. Thus for the dam that spans 11 villages, 5 villages were selected by the CLE for reforestation.\nIndeed 10000 plants were acquired with 2000 plants brought down by the people in each village with the technical assistance of Environmental Agents.\nRavines: Our actions in water resource protection in our technical management area is managed by the CLE including bunds, treatment of ravines and gullies from dry stone sills and reforestation. Thus CLE identified ravines and gullies to be addressed in the sub-basins of the dam Yakouta and recorded this in his annual activity program in 2014. The sand from the sand dunes around the dam and materials in suspension are usually drained by these channels which strongly contribute to filling the dam. In order to curb this phenomenon of silting, the Water Agency Liptako encourage such actions by providing support through subsidies it grants to its CLE space competence to carry out activities contributing to the protection of water resources.\nIndeed several ravines and \/ or gutters were treated by technicians in the field chosen by the CLE for the purpose of restoration and a gradual reduction of silting phenomenon of hydraulic structure.\nReforestation: Six thousand (6000) meters stony cords in 2014. This technique is used not only for water resource protection, but also for the restoration of the soil generally undergoes severe degradation due to erosion by water or wind and.\nBunds: As part of our efforts to protect and restore water resources of the banks including courses and lakes reforestation are made by members of the CLE Lilgouri. This activity is beyond the increase of vegetation cover and soil protection against erosion in their watersheds.\n6000 plants are then acquired and put to earth in the form of one or more line (s) hedges of the population and assisted by the Environmental Service Agents to limit the scope of protection or prohibit land use by the local population.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Internal Revenue Service (IRS) has informed the Washington State Office of Financial Management, and agencies thereof, that when employees are reimbursed for meals under the \"three hour rule\" when no over-night travel occurs, the employer (WSU) is required to report the meal reimbursement as a taxable fringe benefit. On Monday, November 1 the Executive Committee agreed that compliance with this federal tax regulation will occur on January 1, 2000.\nAccording to IRS regulations, meal reimbursements are to be reported as taxable income, unless the employee is required to incur overnight lodging or there is a substantiated business connection. When the meal reimbursements are determined to be taxable fringe benefits, WSU is required to withhold and pay over employment taxes (Federal income tax, social security and medicare) to the IRS with matching employer contributions for social security and medicare taxes.\nWhat is the three hour rule? A traveler may be reimbursed for meal expenses only after the traveler is in travel status for three hours beyond the traveler's regularly scheduled working hours for any one day.\nWhat is a substantiated business connection?\nNon overnight meal reimbursement will be treated as non-taxable provided a substantial business discussion occurs during the meal. When seeking reimbursement for business related meals, employees must document the purpose of the meeting and who was in attendance. This includes the \"meals with business\" rule as outlined in BPPM 70.31.\nWhen will this change be effective? Non-overnight travel meal reimbursements that do not have a substantiated business connection paid on or after January 1, 2000 will be treated as a taxable fringe benefit.\nWill this change affect other reimbursements such as mileage? No, this change only affects the taxability of meal reimbursements when non-over night travel occurs.\nHow will the employee be reimbursed? In addition to receiving a travel check for the non taxable expense, reimbursement of the taxable non-overnight meals will be paid through the Payroll system with the payment included in the next regularly scheduled paycheck. the employee will continue to be reimbursed through Travel for non taxable amounts.\nDoes the department have to change what they submit for the employee's travel expense voucher reimbursement? No, there is no change to what a department submits for Travel Expense Voucher (TEV) reimbursement. Rather then the department submitting multiple forms for the same trip, there is no change to the TEV. The Travel staff will take the one TEV, and reimburse the traveler for non-taxable expenses as normal. The Travel staff will submit documentation to the Payroll system causing the non-overnight meal reimbursement to be included in the next regularly scheduled payroll calculation.\nHow will the meal expense appear in the accounting system? Since this expense will be listed on the department's Payroll Expenditure Audit Report as a meal reimbursement, it will appear in the accounting system in object 01, as wages, with the corresponding benefit expense appearing in object 07.\nWhere can I find out more about this change?\nBoth Travel and Benefits and Payroll Services have posted additional information on their WEB sites about this change. Training is prepared and ready to be delivered. If you would like to have the training conducted in your area, college or department, please contact either of the following people to schedule the training; Judy Croskey (5-2030), or Alice Smethurst (5-1277).\nIn summary, this change is mandated by the IRS and we and the State of Washington are required to comply with this law. Unfortunately, we have no choice but to implement this policy so please bear with us while we conform to this requirement. Thank you for your patience and understanding.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasjbc b/data_all_eng_slimpj/shuffled/split2/finalzzzasjbc
new file mode 100644
index 0000000000000000000000000000000000000000..cb597ffe08ae9b00339fad8a4abec05566ebd052
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasjbc
@@ -0,0 +1,5 @@
+{"text":"The idea with the physical interaction module here is to understand designing interactions from the embodied perspective, focusing on the physicality of the body, positioning the body at the center of interaction as the active component for creating meaning and aesthetics of the experience. The course builds on the previous research on embodied interaction, which will be discussed and introduced through Paul Dourish's book called \"Where the Action is, the Foundations of Embodied Interaction\". Physical interaction module aims to bring up discussions on creative practices with new technologies from embodied interaction perspective, questioning \"how it is possible to make computation fit in more naturally with the everyday world and as a way of enriching our experience with the physical artifacts?\". Further the course will discuss the attributes and foundations of such interaction in which we can understand what really makes us feel like the interaction we engage in is meaningful and worth doing.\nIn the second session of the physical interaction module, on Thursday November 7, Pentti M\u00e4\u00e4tt\u00e4nen, will give a lecture entitled: \"From Physical Interaction to Embodied Meaning and Aesthetic Experience\". The lecture on November 7th will take place in the room 344. In the third session of this module, students will prepare and make their own presentations about the topic. The third session will take place in the room 541.\nPhysical interaction alone is enough to give an acting agent a structured model of the rigid 3d-environment. Interaction realized as a loop of perception and action gives tacit (non-linguistic) meanings. The world is experienced as possibilities of action (or affordances), and the outcomes of possible courses of action are considered (consciously and subconsciously) on the ground of the value of the anticipated experience. Emotions as signs of these values can be considered as outcomes of subconsious valuation and cognition with tacit meanings. These elements together give one possible way to conceptualize the character of aesthetic experience.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Invest In Piece, Which Forever Remains In Style Like This, Designer Heavy Worked Blue Velvet Lehenga Choli N Green Net Sari. This Flare Bridal Dress Come With Bias Shaped Floral Embroidered & Zardoshi Worked Lehenga For Sangeet Wear Abstract With Blouse Having Half Sleeves And Round Neck.\nHalf Sleeves With Contrast Butti Worked.\nRound Neck With Contrast Piping Worked.\nIt Is Best Choice For Supercool Woman.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"With this last post I want to thank all of you for your time and interesting comments on my subject. It is been my first experience with a blog and it has been suprising. Thanks to you I've learned new things on 3 different topics which were very interesting to read about.\nThank you and I wish you all the best with your master thesis!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This post will, hopefully, be one of many posts about what its like to be an engineer\/developer at Just Arrived.\nThere are many things that make Just Arrived unique as a startup. One of those things is that we need to support multiple languages, even right-to-left languages, from the start, while most other companies can defer that complexity to much much later (if ever!).\nI've been developing for the web for more than 5 years and I recently encountered something new, namely the css property unicode-bidi (heard of it before?).\nRecently we did a major upgrade of our website. We moved justarrived.se from Wordpress to Jekyll as well as restructured and rewrote a lot of the content for the site. One, seemingly small, thing that we did was to design a \"contact us\" section where the names, roles, email and phone numbers for each of our team members where to be displayed. It got built real fast and everything seemed to have worked up until we tested the site in Arabic.\nThe solution turned out to be real simple and it was a CSS property that I've never heard of before.\n.rtl-locale-direction is the class we set when the a right-to-left language is used, so we can add custom styling on right-to-left locales. That selector has direction: rtl set, so we need to override it with unicode-bidi: plaintext.\nYou can read more about unicode-bidi over at w3schools.\nJust one of those things that you usually don't have to think about when building websites and applications. This is just one example of what unique challenges we at Just Arrived face on a day to day basis.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When restoring an Akeeba backup of a WordPress site the restore process was interrupted with the error message saying 1118 \u2013 Row size too large (>8126).\nEdit: After you successfully restored the site, remove the lines and restart MySQL. I didn't do this and later tried to move another site from this server to another using Akeeba. When the site was installed on the new server, I just got an error message saying \"Error Establishing a Database Connection\". To resolve it, I had to go back to the above server, remove the lines from mysql config, restart MySQL, make a new backup of the site using Akeeba and successfully restore it on the new server.\nCredit: The solution to this problem was found here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzasxyg b/data_all_eng_slimpj/shuffled/split2/finalzzzasxyg
new file mode 100644
index 0000000000000000000000000000000000000000..d8fe7c82d7bb61b6d7bd02a687a3495baf5da53b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzasxyg
@@ -0,0 +1,5 @@
+{"text":"Beth Slocum (Aubrey Plaza) is dead. She was bitten by a snake while out running and her parents (John C. Reilly and Molly Shannon) and her boyfriend Zach Orfman (Dane DeHaan) are mourning her death. When Beth shows up again, Zach realizes she's \"undead\" and that something strange is happening in the town. Now, Zach's got a dead girlfriend and more problems are mounting\u2026losing Beth was easier than when she came back.\nWritten and directed by Jeff Baena, Life After Beth is a horror-comedy. The film was released at Sundance on January 19, 2014 and a nationwide release on August 15, 2014. The movie was met with mixed reviews.\nZombies (especially in 2014) were an \"it\" item. The Walking Dead was soaring in popularity and zombies were also getting a lot of post-modern exploration with movies like Zombieland poking fun at the genre. Life After Beth combines a romance, comedy, and horror movie with mixed results.\nLife After Beth gives the appearance of one of those \"wacky\" zombie movies where being a zombie is really incidental and comedy driven. In many senses, this is true in the movie, but the third act has the movie getting really, really weird. It remains with the high comedy themes, but then has extremely dark moments. It is just a lot darker than you expect from the movie's tone up until this point which continues to joke even after it everything turns.\nThe cast is quite good. Aubrey Plaza gets to bring her negative character to the bipolar zombie Beth who rages while Dane DeHaan has to be acting in a drama within the comedy. The parents are strong (but weirdly underused) with more screentime for John C. Reilly and Molly Shannon than Paul Reiser and Cheryl Hines. Anna Kendrick has a small role along with Alia Shawkat, and Plaza's Parks and Rec costar Jim O'Heir. It also has the last film appearance of Garry Marshall playing Zach's dead grandfather.\nYou never take me out to eat brains!!!\nThe movie has the weird balance of trying to look like a zombie movie. While Zombieland really boosted the gore, Life After Beth kind of tones down the special effects and gory zombies. Along with the comedy, it pumps the movie full of smooth jazz (which calms the zombies). I kind of expected more in the lines of Zombieland, but this works as well.\nLife After Beth is ok, but it isn't great. The idea that it is a zombie romance break-up movie is interesting and has potential for the allegory of a dying relationship, but I think both themes could have been pushed a little harder. The movie can't quite decide if it is a laugh out loud comedy or an art picture with more meaning leaving it in a bit of a limbo. Zombie lovers should probably take note of this movie, but don't expect the typical horrors that come with zombie films.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brilliant service provided by this car park, easy directions for car drop off, met by friendly staff, very quick handover before making my way to departure. Clear instructions for pickup on return. Returned to uk, called number provided on paper in baggage area, when back to pickup point car was waiting, easy and smooth handover by staff and made my way home. Would recommend this option of parking to others and would use this service again. Thankyou for a great service.\nSwift were excellent, our flight was over 3hrs delayed getting back in to Luton and they delivered the car no questions asked. Definitely beats waiting for the bus for the long stay car park.\"\nExcellent service, great value for money\"\nI travelled recently with my family and we availed meet and greet service. It really helped us in making the start of our vacation relaxing.\nThe staff of Swift Airport Parking has always been helpful for me. Thanks to them for their remarkable services.\nI don't think that there is any doubt in Swift's reliable services. There was not a single scratch on my car when I received it on my arrival day.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Securities trading reached a significant volume: the Bank seized all the opportunities that arose towards the end of 2011 and throughout 2012.\nExclusively Italian government bonds were purchased, at fixed rate for short-term bonds and at floating rate for medium-term ones. Currently the portfolio's average return is decent relative to refinancing costs. At the end of the year, the portfolio's total book value was 5.243,5 million Euro, 97.4% of which composed of Italian government bonds and the rest of previously purchased bank bonds near maturity. At 31 December 2012, 35,6% of the portfolio would mature within 12 months, and as a whole within 5 years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Dordt was awarded a $24,945 NetVUE Vocation Across Academy Grant from the Council of Independent Colleges (CIC) and Lilly Endowment Inc. The grant is a three-year matching grant where the initial $24,945 is distributed in year one. Dordt then raises that same amount of funds in year two of the grant cycle, and NetVUE awards an additional $10,000 in year three, after the match has been met. Funds from the grant will allow the college to dig deeper into the concept of vocational discernment and how the StrengthsFinder tool, an assessment tool used with every incoming freshman, can be better leveraged at Dordt.\nAmy Westra, associate director of career development, was the driving force behind securing the grant. Westra worked with faculty and staff to draft the grant proposal.\n\"I personally am excited to see a stronger focus on the StrengthsFinder tool and helping students link that to vocation and careers that may be a good fit for them,\" Westra says.\nThe funds will also be used to create a new one-credit elective course. The course will assist students who continue to struggle with identifying God's specific vocational calling in their lives.\n\"In the past, the career development center has offered a workshop for students, but finding a time that works for students given their busy schedules has been a struggle. A course being offered for academic credit increases the credibility for students,\" Westra says.\nFunds from the NetVUE grant will be used for training for faculty and staff, as well as to develop a student coaching initiative. Student coaches will be trained to conduct one-on-one coaching sessions with their peers and also assist in facilitating workshops.\n\"We are doing our best here at Dordt to be intentional with student development needs. This career development initiative is an exciting vocational exploration opportunity to reflect upon how God has gifted each of our students,\" says Provost Dr. Eric Forseth. The intentionality seems to be paying off\u201499.7% of Dordt's 2017 graduates reported being employed or enrolled in continuing education opportunities.\nThe funds will be available to Dordt from July 1, 2018, through June 30, 2021.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A History of the World in 101\/2 Chapters tells a series of apparently unconnected stories ranging from a woodworm's-eye-view of the journey on Noah's Ark to an astronaut's quest for its final resting place. There is pastiche and learned disquisition; there is heart-stopping documentary and heart-lifting revelation.\nBut these stories are not separate. They are all linked by a complex weave of inquiry into history itself, into love, myth and fabulation. It's about everything that matters, told with brilliant imagination, intelligence and humour.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavbxm b/data_all_eng_slimpj/shuffled/split2/finalzzzavbxm
new file mode 100644
index 0000000000000000000000000000000000000000..dba13a8bc0c18eda8046c244484cf3c46e513389
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzavbxm
@@ -0,0 +1,5 @@
+{"text":"Thoughtfully designed to blend with its coastal environment this home includes innovative sustainable technologies with great attention paid by the builder to the materials used. Materials include low carbon steel which is designed to rust and then stabilise, rough sawn cedar, blockwork laid without mortar and honed onsite for a unique finish, honed concrete floors, low e glass throughout and cedar wall linings with clay paint and ply joinery. Energy efficiency initiatives include solar power, evacuative tube solar hot water, LED lighting, standby power and excellent passive solar design.\nThe judges were particularly interested in the number of innovative materials and methods employed by the builder and said the builder was to be congratulated for his ability to deliver a high degree of sustainability with excellent finishes and workmanship.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Yahoo's Football stream was great, but the future of streaming is more interactive.\nYahoo hosted a huge live streaming event last week when they showed a feed of the American Football game between the Buffalo Bills and Jacksonville Jaguars, which was played in London's Wembley Stadium.\nOver 15 million viewers \u2013 That's pretty impressive. Not all of them were happy, but that's the state of streaming media on the internet, it's never perfect. But overall it seems like the experiment went well despite some viewers who saw glitches and jumps in the video and had quality complaints.\nPersonally, I would take more glitchy events like this over the option of not being able to join a remote event at all. I'm a fan of English soccer, and not all the games are shown here in the US, or at least not on the channels that I subscribe to. I would gladly pay to be able to stream all of Chelsea's football matches, even when my team is in a dreadful slump like right now.\nStreaming sports and services like Netflix and Hulu are cool, but to me that is not the future. Watching \"TV on the internet\" may be sucking a lot of bandwidth, but it's not particularly revolutionary.\nTwo way communications and interactive remote events \u2013 now that's cool! And with the advent of WebRTC, that is exactly what's starting to happen, and at WebRTC.ventures we're proud to be part of that revolution.\nTake for example the Fox Sports Huddle, which is using TokBox, an excellent WebRTC platform (our parent company AgilityFeat is one of TokBox's recognized development partners).\nIn the Huddle, a sports journalist hosts a sports celebrity of some sort, maybe a coach or a player. Their video is captured via WebRTC and streamed live to hundreds or thousands of viewers. That's cool, but what's even better is that you can \"get in line\" to join the event yourself!\nA producer will screen your question and confirm the quality of your bandwidth, and then push you onto stage with the journalist and celebrity. You and your laptop camera are now live on the internet, perhaps with one of your sports idols, and you get to ask them a question. How cool is that!?\nAfter your question is over, the producer turns off your video and brings the next fan on stage. The Huddle gets recorded by TokBox and then published later on the Fox Sports site for even more fans to see.\nIn WebRTC parlance, we call this sort of event a \"few to many broadcast.\" A WebRTC connection is established between the three parties for the recording, and a live stream of those parties is broadcast to many more people than a typical WebRTC peer connection can support. By pulling people onto stage one at a time, you never overload what WebRTC can handle, but you still get to have lots of people participate.\nThis broadcast model brings a great level of interactivity to the internet. Because WebRTC does not require plugins to work, you can seamlessly bring people into and out of the conversation.\nThe \"Few to Many\" broadcast model can be used in many situations. One of our clients, NewMe, asked us to build a similar model for their Startup Accelerator's virtual meetups.\nAttendees at an event can watch the stream from their laptop or mobile device. Angela Benton, NewMe's founder, will interview a guest about startup and business topics, and she gets very large audiences for each event. Attendees can chat with each other as well as ask the guest questions.\nWith the advancement of WebRTC, our team can build you a custom and scalable solution on top of TokBox which allows you to host virtual conferences, webinars, and other communication apps.\nHave you heard that WebRTC doesn't scale well to large conversations? That's true, if everyone has their video on and is participating in a full video chat. But with \"Few to Many\" broadcasting architectures, you can have a virtual event with hundreds of attendees, and still a lot of interaction between attendees, moderators, and guests. The possible applications of this open up a lot more value than just watching a football match.\nAnd of course, there are other models to consider. The Periscope or Meerkat style of streaming video can be fun to play with or even do serious business with. You could stream a tech session at your favorite conference, or you could stream yourself eating Korean noodles. PubNub put up a cool example of how to build a Periscope style solution, and so that could also be incorporated into a podcasting or conference application.\nReady to build your own WebRTC application?\nInterested in learning more? Contact our team to learn more about building custom WebRTC based broadcast applications for web and mobile!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"GIB is managed by a board of Directors, headed by its CEO, Mr. Pierre Joubert. Pierre is backed up by an extremely capable team of committed and diligent staff members with a history of work excellence. The GIB staff complement consists of 150 personnel. Ninety of these staff members operates in and around Patensie and focuses on the management of the water from the Kouga Dam tot the rest of the water users. The rest of the staff is situated around the Eastern Cape and are involved in management of the EPWP projects that the Department of Environmental Affairs entrust the Board with. GIB's main office is situated in Patensie.\nGamtoos Irrigation Board (GIB) is a Licensed Irrigation Board based in the town of Patensie on the fringe of the Baviaanskloof in the Eastern Cape, and has been a going concern since 1991. GIB is an Irrigation Board formed under the requirements of the (old) Water Act of 1956.\nAdditionally other senior GIB staff including Ms Rienette Colesky and her administrative staff have collectively 50-plus years' experience with the administration and implementation of Natural Resource Projects for GIB.\nManaging the Human Resources of Gamtoos Irrigation Board and Gamtoos Irrigation Board (Implementing Agent) as well as the budget of both.\nAbbey is in charge of the maintenance of the canal and pipe system. He and his team attend to breaks and general maintenance of the canal.\nLeon Manages the waterflow from the Kouga dam to Loerie dam. He sees that the correct amount of water gets released out of the Kouga dam during different periods of demand and that there is sufficient water released into Loerie dam in order to keep Loerie's water at a acceptable level to supply Nelson Mandela Bay Municipality.\nArea Managers are in charge of the EPWP projects allocated to them and see that the budget allocated to them for a particular year is applied as intended. They have a team of Project Managers working under them who is assisted by admin and social development personnel.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Petersfield Cemetery was established in 1857 to replace that at St Peter\ufffds Church. The earliest burials are concentrated around the two memorial chapels near the entrance, on rather uneven land. The cemetery is in the care of East Hampshire District Council. This is a medium size non-denominational cemetery, which has been extended into adjacent fields after the original area itself was filled. There have been about 7600 burials at the site with perhaps two thousand gravestones still visible in total. As there are no gates and the cemetery is approached by a long and secluded avenue of lime trees, there has been some vandalism over the years, with headstones broken and pushed over, particularly in the area near the chapels. The cemetery is still in active use, but is now approaching capacity and a new site is being researched for acquisition by the council. This set of gravestones are from the central area of the cemetery. The majority of the burials date from the middle of the twentieth century.\nThe GPR holds 361 grave and 592 person records for the cemetery.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Each year during Spring Break we host a fun, engaging all day camp for kids aged 5-12. Enroll your camper in Spring Break Camp for games, crafts, swimming, a field trip, and more.\nAll activities are based on the Y's character development values and the Search Institute's 40 Developmental Assets. You can enjoy peace of mind knowing that your camper is enjoying enrichment activities under the watchful eyes of trained YMCA counselors. Your camper will make new friends, gain confidence, and expand their horizons.\nYMCA Spring Break camp will be available from five locations in March for children ages 5-12. Camp is open from 7:30 - 6 pm each day and will include a field trip, two days of swimming, and tons of enrichment activities.\nRegistration closed on March 13th.\nRegistration for Spring Break Camp 2020 will open in February 2020.\n*Note: Please read over the forms carefully and double check your forms for completion before emailing. Any incomplete forms will be placed in pending (not yet registered) and we will contact the Primary Guardian to obtain the missing information. Your child will not be registered until we receive all required information. If your child has any food allergies, please submit a Food Allergy Emergency Plan signed by your family physician to us along with your paper forms.\nIf any of the above methods absolutely will not work for you, you are welcome to mail your completed paper forms (see note under method 2 above) to Program Services; 3208 Red River Ste. 100; Austin, TX 78705. Due to the inconsistency of mailing in items and registrations being processed on a case-by-case basis, we can not guarantee space availability and registrations being processed in a timely manner if using this method.\nEach camp will take one field trip and swim twice during the week at YMCA of Austin pools. Lifeguards will be present during swim time. Your camper does not need to know how to swim. Y staff will conduct swim tests for those campers who want to swim in deeper water. Life jackets are provided as well. Feel free to provide your own Coast Guard approved life jacket for your child.\nEXAMPLE schedule swim days & times for 2019 Spring Break Camp are listed below. Times do not reflect travel times from\/to camp.\nSites, locations, and times will change for 2020.\nFinancial Assistance Policy: YMCA of Austin programs and activities are designed to benefit persons of all backgrounds, and fees are based on the cost of providing each program. While participants are expected to pay their fair share, when financial assistance dollars are available, the YMCA will assist any individual or family that wants to participate but cannot afford the fee.\nFinancial Assistance is awarded on a first come, first served basis. Only completed applications will be accepted and reviewed. Funds are limited, and we recommend applying early.\nWhat do I need to apply for Financial Assistance?\nImportant Note: If you are applying for Spring Break Financial Assistance and would like to be considered for 2019 Summer Camp Financial Assistance please check off both boxes on the top right corner of page 3 on the 2019 application. Please note, your financial assistance application for Summer Camp will not be complete until you have submitted the 2019 Summer Camp Enrollment Form.\nFor details on Spring Break financial assistance, contact JoseLuis Flores, Accounts Director, with questions and to apply.\nAll requests to cancel your child(ren)'s camp enrollment must be received by the Program Services Branch in writing by submitting the 2019 Online Cancellation Form. Please allow 3-5 business days for processing. Cancellation requests must be submitted by March 4th (2 weeks before Spring Break Camp starts on the 18th).\nRequests that are submitted in time are processed without additional charges and a full refund will be given. After that day, such requests are considered late. Late requests, if submitted at least 5 calendar days before the start of camp, will be processed with an additional $25 fee per family.\nCancellation requests are not accepted on the Thursday or Friday before a camp week begins. No credits or refunds will be given at that time. You will be held responsible for the full amount of camp fees, regardless of whether or not your child attends camp.\nWe understand that families' plans change, and we certainly want to be accommodating. However, many of our camp programs fill up and have waiting lists; we require advance notice of cancellation so that we can more effectively reach out to those families who are hoping for a spot in the program.\nIf you need to transfer your child to another camp, then please email your request to Program Services. Please allow 3-5 business days for processing your request.\nFor any questions or concerns please contact Program Services at 512-236-9622 or email at ProgramServices@AustinYMCA.org.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavjrv b/data_all_eng_slimpj/shuffled/split2/finalzzzavjrv
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+{"text":"In a previous blog post, I covered automatic campaigns vs. manual campaigns and how they work together. Both campaigns have their strengths and weaknesses in the world of Amazon advertising but remember: Leveraging both types of campaigns for your Sponsored Product Ads on Amazon is the best method to creating winning campaigns and getting the most out of your keywords.\nFor an Amazon Sponsored Product Campaign Review, let's go over how automatic and manual campaigns work.\nSellers select their budget and let Amazon go to work and automatically find keywords. Amazon looks for keywords that match your product's category, related products, and keywords in your product's descriptions.\nGreat for PPC beginners because of the easy setup.\nYou get to skip the hassle of entering keyword bids and have more time to read my blog.\nIf you run an automatic campaign, Amazon will collect data for you to apply to future campaigns. Amazon tracks clicks and spends, usually taking 2 to 4 days to process. It may take up to 2 to 4 weeks to gather enough data to see which campaigns are doing well.\nAutomatic campaigns allow for experimentation without putting that much effort in.\nThey allow you to show up for long-tail research you may have never guessed people would search for.\nAs you may have guessed, letting Amazon automatically choose the keywords you will bid on gives you less control compared to the manual campaign structure.\nManual campaigns are also set up in the campaign manager within Seller Central. With a manual campaign, sellers handpick keywords they want to bid on. Sellers can choose broad, phrase match, or exact keywords when bidding to fine tune where their ad spending goes. When setting up your manual campaign, remember to keep your keywords in the 20 to 40 range and not go overboard with more than 500 keywords. If you choose more than 500 keywords, there's no way all of those are going to match with your product and you'll be wasting ad money.\nGreat for PPC experts who have time to set up manual campaigns.\nYou can maximize your sales volume and perfect your ACoS.\nIt's easy to reduce bids or pause low-performing words.\nAmazon suggests bids next to every keyword you wish to bid on, making it easier for the seller.\nHave keyword ideas you want to test? Manual campaigns help you do that.\nManually choosing every keyword to bid on takes tons of effort and time.\nYou could lose money if you do not put that much effort and time into keyword research.\nIs It Profitable to Advertise Black Friday Weekend?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Hobart EDGE13 slicer is built to handle medium-volume demand for slicing of up to four hours each day. The machine's 13-inch, chrome-plated knife is powered by a drive system that uses a Poly V belt, which has 2.5 times more contact surface area than traditional flat and V belts. That yields greater dependability, easier maintenance, and quieter operation. The slicer's infinite thickness-control knob allows users to adjust slice thickness from paper thin to 9\u204416 inch.\nThis slicer is designed to be easy to clean and sanitize. Its one-piece aluminum base has no crevices or holes to collect food debris, so it's easy to wipe clean. The carriage system, sharpener, meat grip, and deflector are removable so they can be thoroughly washed by hand. A permanent knife ring guard protects users' hands while they use and clean the knife of the Hobart EDGE13 slicer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Detached Victorian location house that has a bright, spacious and homely interior with large private garden. The house is approximately 2500 sq ft over 4 floors and has been renovated to include modern open-plan living alongside original Victorian features including beautiful working fireplaces (incl. log burner) and ornate detailing.\nThe large open plan kitchen-diner has original French doors that lead straight on to the beautiful south facing garden with a patio, decking and large grassed area. The ground floor also has a large double living room. The lower ground floor offers further living space as well as a small bathroom and utility room. The first floor has two large double bedrooms, a nursery, office and two bathrooms. The top floor has a third double bedroom.\nThere is one parking space in the driveway, and additional vehicles can be parked on street which is currently free of charge.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The problem with the original Brooklyn is the Amer Picon, a bitter orange ingredient from France that is simply unavailable in the US. There are alternatives. Ted Haigh recommends using Torani Amer which may or may not be difficult to find, while David Wondrich recommends subbing Amaro CiaoCiarro and orange bitters. Seattle's Jamie Boudreau even tried whipping up a house-made version using Amaro Ramazzotti, Stirrings orange bitters and an orange peel tincture\u2014with highly acclaimed results. For no good reason, we avoided all options and continued for several unfulfilled years until we obtained a bottle of Torani Amer!\nOne would think after all this time we would be all over the Brooklyn cocktail that has inspired so many other great recipes. Yet, today, we are featuring the Jaguar, a tequila cocktail that kinda blew us away.\nStir with ice, then strain into a chilled cocktail glass. Garnish with a flamed orange peel (discarded).\nWe are not usually as excited about tequila drinks, but this one has us reconsidering. It is fair to say that the spirit is somewhat hidden here. Knowing tequila is the base is one thing, but if someone blind-served us this cocktail, we might not call out the agave, mistaking it for a gin drink. Yet, given the complexity, we like to think of this as a Manhattan lover's alternative. It is definitely boozy with an herbal orange backbone. It has a kick that is obviously not whiskey but a flavor profile that seems a long way from tequila.\nWe used Lunazul silver to make this drink, but any decent brand will do. This recipe also includes green Chartreuse. While it may seem like an expensive indulgence, Chartreuse is worth every penny. It appears in some of our favorite classics and our bar cabinet will always include a bottle. For over 400 years the fact remains that there is no substitute for Chartreuse. The featured ingredient, however, is Torani Amer. You might recognize the brand which is common in coffee shops. Torani makes a variety of flavored syrups, but this is their only product with alcohol.\nThe garnish features the infamous flamed orange peel. This is fun to do for unexpecting guests, but it is more than just a show. The spray of orange oil stays on the surface giving the drink an added nose of citrus. To pull this off, slice a disc off the peel of a fresh orange\u2014the harder the peel the better. Light a match with one hand, pick up the orange disk with the other and gently warm it over the flame. Then, line up the peel over your glass and place the match between them. Give the peel a squeeze, bending it in half while the oils spray through the match flame and onto the drink. You will be rewarded with a burst of flame and a nicely scented cocktail. Blow out the match and rub the peel around the rim of the glass before tossing it away.\nAnother great alternative for Amer Picon is Bigallet China China (pronounced kina kina), which has been produced since 1875. It is now produced under the Giffard umbrella, and is often described as an equivalent to Amer Picon! I will have to see if I can find Torani Amer and compare them.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This winter I read Mandy Aftel's Essence and Alchemy and not only learned a great deal about natural perfumery and its history, but I also found myself reveling in its mystical and spiritual aspects. Ms. Aftel's book takes you on a journey of the natural perfumer as alchemist. Converting raw matter into something more pure and divine; potentially enabling us to connect the material world with the spiritual. This may or may not be the intention of a natural perfumer, but her book lays a convincing argument that botanical transformation or solve et coagula, to dissolve and combine, just might lead to a transcendental encounter.\nReading her book was informative and moving, but I had never actually worn one of her fragrances. And frankly, I was reluctant to do so after having such a wonderful experience with her writing. I was worried that I would have too many expectations of her perfumes, and if they weren't met, the let down would be enormous. So I waited.\nTwo months passed until I filled my shopping cart with some samples at Aftelier.com which I must say is no cheap endeavor. But the time had arrived to experience the alchemist at large. I have started my personal Aftelier foray with the Orchid Solid Perfume. What does an orchid smell like you might ask? I really don't know. After a little research I found that some have no smell, some are putrid, and some are akin to jasmine, honeysuckle, gardenia, rose, violet, etc. Additionally, the flower's oil is very difficult to distill, so the perfumer must attempt to approximate what she\/he has envisioned for the orchid's fragrance. Whatever Ms. Aftel envisioned for her Orchid Solid, was utterly gorgeous.\nAftelier's Orchid is an interesting fragrance that might take a little time to grasp, but it's also one that you can surrender to immediately. The initial spicy peppery, cinnamon topnotes made my nose twitch with delight and curiosity. But I accepted the culinary twist and remembered the first time I inhaled a deep purple plumeria that smelled of cinnamon and simply let the fragrance evolve. The floral\/spice mix melted into my skin and after my introductory puzzlement, I realized I could not imagine Orchid unfolding in any other manner.\nThe true beauty of Ms. Aftel's creation is the orange blossom absolute. The interplay between the tropical narcotic, and the youthful fresh qualities of this blossom are perfectly balanced in her Orchid Solid. It's green, heady, lush, citrusy, and vital. In Essence and Alchemy, Ms. Aftel describes absolutes as \"floral essences at their truest and most concentrated.\" That is certainly how the orange blossom of Aftelier's Orchid flowers on the skin, truthfully.\nOf course I had to do some research and find out what the spiciness was all about that I was perplexed and captivated by. I found Ayala Sender's review of Orchid on her blog, Smellyblog, and she mentions that shiso is used in this fragrance. I have to plead ignorance in this regard, as I don't know much about shiso except for what I have recently read and unknowingly consumed. Shiso, also known as perilla and many other names depending on if you are in Korea, Vietnam, India, etc, is a culinary herb. (Here is an informative article from the LA Times on its many uses). Clearly, shiso is responsible for the peppery, cinnamon opening and elegant transition to Orchid's heart.\nI will echo Ayala's sentiment that Orchid does not last on the skin as I long as I would have hoped; approximately two hours. This is such a stunning fragrance that I would love for it to last all day. Yet, while I only have a sample of Orchid, it's a pleasure to dip my finger into the perfume solid and reapply, and I can only imagine that doing so with the sterling silver compact feels like a precious ritual. And precious it is, as the compact is $175 for 1\/4oz.\nMany of you may know, but some of you may not, that Mandy Aftel has an exhibit in her honor at Bendel's in New York City, Living Perfume: The Natural Alchemy of Mandy Aftel from April 18 \u2013 May 11. If you are in the New York area, this is not to be missed.\nAftelier's Orchid Solid Perfume is available at Aftelier.com and Bendel's in NYC. Aftelier products do not contain artificial colors, synthetic fragrances, petrochemicals, phthalates, or parabens. This is clearly stated on the website.\nThis winter I discovered one of Mandy Aftel's books, Scents & Sensibilities Creating Solid perfumes for Well-Being. It's a pleasant book to read. The book holds info on the history of cream perfumes from around the world and step-by-step instructions on making your own cream perfumes for well being. Hope in the near future i can test her recipes out.\nSounds like another lovely book by Mandy, Alyssa. Thanks for the rec. I will check it out. Let us know if you ever try out your own perfume skills!\nThat book has made several people WANT to become perfumers. It's a wonderful resource. And I bet the line is as interesting too, although haven't been exhaustive.\nThe Living Flower exhibit is a must-see too!\nHer book is a fabulous resource. I will go back and read it again I am sure.\ni have had that book on my Amazon wishlist for ages, sounds like I need to order it asap and start saving up for samples!\nI am about to post the catalog from her upcoming exhibit. That might give you a little flavor for the book.\nHi Trish, your review makes me want to reread the Aftel book, perhaps it would be a good summer read. I did some very interesting perfume experiments after I read it. I loved it as you do, although I found that her Scentsability was very different from mine, from her first scents that is. Her more recent scents seem a bit different, plus I'm sure I've changed! Someone sent me a sample of Tango, which I loved, and the Orchid sounds fantastic, with the Shiso leaf, she does have a gift for contrasting complementary scents!\nWhen I went to a demo as part of a show she put together at USC here in LA, she said that the longest you can expect a completely natural scent to last on your skin is 3 hours, that you'll always have to reapply. Apparently only synthetics \"grab\" on for the long term. I wonder if that is your experience?\nLuckily I'm going to an event with Ayala Sender here in LA in a couple of weeks, and I think they have the Afteliers there, at Blunda Aromatics too, so I can try them again!\nI'm sure I will reread it as well at some point.\nYou know, I just knowingly ate shiso for the first time last night. It was a garnish with some miso paste\/sauce, and it was so good. It was at a terrific Chinese restaurant. Anyway, it was very mild, but I could sense the smell that Mandy used in her Orchid Solid. What an interesting and successful choice (IMO) she made.\nMy experience w\/ longevity of naturals is incredibly varied. Typically from an hour to three though. Some can last all day, like Ajne's Fleur Blanche and I catch whiffs of In Fiore's Dayala and Maia all day long as well. SIP's new one, Epic Gardenia lasted a good 6 hours when I gave it a whirl the other day, so you just never know.\nShiso is interesting because it is one of those rare things that tastes just as it smells. Get a bunch at an Asian market and experiment with, the Times article was inspiring. Have you tried Anya's Kaffir Lime scent? It has a similar quality, it's really quite amazing.\nI'll let you know what happens there, if I'm a good girl I'll do a blog, but I've been very bad lately!\nI do have a sample of Kaffir Lime, I need to spend some time w\/ it \ud83d\ude42 I did get Mandy's Shiso as well. Will write about it later possibly.\nAfter eating the shiso, and the Times article, I will definitely be cooking with it now. It's such a pretty leaf.\nSo please do blog about the Ayala exhibit if you have the time. That would be lovely!!\nI did get the opportunity to see the exhibit at Henri Bendel in Mandy's honor, it was small, but her work is very large. It's like sitting down to a wonderous feast.\nShe is a living legend as far as the revolution of natural perfumes goes.\nHer cooking essences were equally as nice, the pairings done with Chef Todd English's tasting made for a very nice event at the W hotel in NYC.\nI was really glad to have the opportunity to hear her talk in person about her love of this ancient art form.\nAll in all her contribution to the world of perfumery is Alexandrian.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzavwif b/data_all_eng_slimpj/shuffled/split2/finalzzzavwif
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+{"text":"\"Let the earth embrace you...\"\nLiving out here in \"the country\", in a rather small immediate living space, I find I am becoming more connected to the land I reside on. It's much more of a co-existence than I have experienced in the past. I am becoming more directly aware of my \"footprint\" and how my living affects our dear earth (and vice versa!).\n\"Let the earth embrace you... the air caress you and take you... hear it whisper a sweet, sweet song... reminding you that you belong... come back... come back to your heart... to all that you are... all that you are... \"\nThank you, Women of Substance Radio!\n\"WHAT IS A WOMAN OF SUBSTANCE? Substance can be defined as 'significant quality with the implication of a hidden or special significance'. Music of Substance is not just entertainment or fluff, but has an essential core brimming with heart, soul, and spirit. Women of Substance are female performers who deliver high quality music that speaks to the listener through vocal excellence, depth of character and emotion, and lyrics that leave a lasting impression. This includes Label Artists, Indies and Unsigned Artists alike. Our review board selects only the highest quality artists and songs that can stand up next to the superstars.\"\nMy first song, \"It's Enough\" aired a couple of weeks ago and will also be included in the iTunes #1 New & Noteworthy Women of Substance Music Podcast to premiere in episode #240 on 10\/21. It can be found on iTunes or by visiting the website: www.wosradio.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When exporting data to Google Sheets, the data will not get sorted by default. A solution however is to let Google Sheets sort the data in a second sheet inside your workbook. This solution requires you to add a sort function to one of your sheets, but means that data will be automatically sorted even if new columns or rows are added to your Funnel export in the future. Next follows a guide how to set this up. We will be using the following workbook as an example.\nCreate a second sheet in the workbook. We have in this example renamed the sheets so that you more easily can tell what is what, see picture below.\nGo to your newly created sheet and mark the upper left cell (A1). Add the function ='[name of your first sheet]'!1:1 and click 'Enter'. In this example the name of our first sheet is Raw data, which is why the function is written ='Raw data'!1:1 in the picture below.\nGrab the cell in its bottom right corner and drag it out sideways to populate the function to the whole row, in that way populating all column headers that you have present in your first sheet (Raw data) to your second sheet (Sorted data).\nNext, mark cell A2 and add the function =SORT('[name of your first sheet]'!A2:X1000,1,true) and click 'Enter'. In this example, the function will be =SORT('Raw data'!A2:E1000,1,true).\nThe first part 'Raw data'!A2:X1000 refers to what sheet and range of cells that should be sorted. Note, that if you have more than 1000 rows in your sheet, or columns that span further than to column X, you will need to define a greater range than in the above example.\nThe second part of the function indicates the index of the column in the defined range containing the values by which to sort. In this example we have added a 1 here, indicating that we want to sort the data after the first column in our defined range 'Raw data'!A2:X1000, in this case column A: Campaign.\nThe last part of the function indicates whether you want to sort the values in ascending or descending order. TRUE sorts in ascending order, while FALSE sorts in descending order. There are in fact additional parameters that you can add to your sort function, although we won't include them here. For a full description of the sort function, follow this link.\nBy clicking 'Enter' the sheet should now contain sorted data! If you have scheduled an export to Google Sheets using Funnel's Google Sheets Upload your data will now always be sorted in the second sheet regardless of new rows that is included in the export (as long as you have defined a range great enough in your sort function).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Our puppy of the month for October is the lovely Worcester the whippet. He is an exquisite fawn colour with a sooty muzzle. Here he is (centre) with his siblings, as you can see he is an extremely handsome boy. And here he is with his sister.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A young child was taken to hospital after being hurt by a vehicle, the ambulance service confirmed.\nParamedics were called to the A259 Little Common Road between Peartree Lane and Green Lane at 3.40pm on Tuesday (March 5).\nA spokesman said they were responding to reports a young child had been hurt in a collison with a vehicle.\nThe spokesman added: \"Ambulance crews attended the scene and the child was assessed and treated at the scene before being taken to Conquest Hospital.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Are you Norielle Gottschalk? Claim this listing.\nNorielle Gottschalk is a certified financial planner who works in Sacramento, California, with a total of 10 years of experience as a financial advisor. Gottschalk is currently employed by Ameriprise Financial Services, Inc. Gottschalk has gained experience at Big Valley Federal Credit Union, Waddell And Reed, Inc, Cafe Vinoteca, Various Insurance Carriers For W & R Insurance Agencies and Ameriprise Financial Services, Inc. Gottschalk has a Series 66 license, qualifying them as both a securities agent and an investment advisor representative, and is registered to operate in California. Ameriprise Financial Services, Inc is a registered investment firm based out of Sacramento, California. The investment advisory business has offices in 161 locations and a total of 13,020 financial advisors. It oversees $244 billion in assets under management between its 1,268,997 client accounts, making it one of the biggest investment advisory firms in the country by assets under management. The firm provides services to 1,420 charitable organizations. Of its 1,268,997 accounts, just 3% are held by high-net-worth individuals who have at least $1 million in assets. The advisory firm performs a diverse assortment of services, including financial planning services, portfolio management for individuals and small businesses, portfolio management for institutional clients, pension consulting services, selection of other advisers, publication of newsletters, educational seminars and other services. A financial advisor's fee schedules vary depending on the type of financial services provided. However, fee types charged by Ameriprise Financial Services, Inc most likely include asset-based, hourly and fixed fees.\nNorielle has 10 years of experience.\nNorielle Gottschalk has no disclosures.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzawmgm b/data_all_eng_slimpj/shuffled/split2/finalzzzawmgm
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index 0000000000000000000000000000000000000000..0b583d3008a9fd29cfb15e4bdeb9b82f71a3b3f1
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+{"text":"Stature: 30-60 feet high; to 2 feet in diameter; with a spreading, irregular crown.\nLeaves: Compound; composed of 7-9 leaflets arranged more or less in pairs, with a single leaflet at the end; leaflets dark yellow-green above and paler green below, candle-flame-shaped; edges slightly toothed or nearly smooth; stemlets to 1\/4 inch long.\nFruit: A small samara or \"wing\" 1-2 inches long; hanging in dense clusters; the hard, seed part extending well into the flat, wing part (see below).","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Kathy Van Zeeland Flapper Chic Sling catches the spirit of casual glam. This versatile sling handbag features snake print-embossed synthetic leather, convertible single shoulder strap can be worn over shoulder or back for a change of style.\nFront exterior flap pocket with twist-lock closure.\nSignature interior lining with a back wall zip pocket and multifunctional slip pockets.\nMeasures: 10\" H x 9\" W, Strap drop 15\".","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The best traditional seaside holidays start at Gunton Hall. As a way to combine the best of coast and countryside, Gunton Hall Coastal Resort can't be bettered.\nJust 45 minutes to the north-west you can be enjoying the Norfolk Broads; to the east, the English Channel, from where in the 9th century Vikings appeared and ravaged the settlement then known as Guntone or Guston.\nYet further afield you'll discover a most genteel stretch of the English coastline. A short trip south takes you to the civilised coastal towns of Lowestoft, Southwold and Aldeburgh while the city of Norwich and its mighty cathedral is a mere one hour away.\nGunton Hall has a huge variety of things to do, from a mini bowling alley to a four rink bowls indoor complex; full leisure facilities like a swimming pool, sauna, steam room and fitness studio, pampering with Decl\u00e9or therapies*, cycle hire*, indoor and outdoor games like snooker and pool, croquet and pitch 'n putt.\nGunton Hall has pretty, well kept gardens with woodland and a freshwater fishing lake for angling (seasonal, rod licence required).\nTuck into breakfast and dinner in the elegant restaurant. Enjoy a drink at the Lakeview bar or a light lunch at the coffee shop. Where ever you are in Gunton Hall Coastal Resort, you can expect the high standards and friendly service of a Warner Leisure Hotels.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Closed Is UC-berkeley's Political science major good?\nClosed GETTYSBURG COLLEGE: has anyone got he decision?\nClosed Chance me for University of Minnesota: Twin Cities CLA?\nClosed What are you planning to do in your gap year?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A. There's\"u\" and _____ \"s\" in _____ word \"us\".\nB. There is _____ young man and _____ old man in the photo. ______ old man is _____ young man's father.\nA. I don't like this skirt. Show me that _____.\nB. The new skirts are mine. The old _____ are yours.\nA. Jay can sing the song ____ either Chinese or English. How great!\nB. I won't believe it until I see it _____ my own eyes.\nC. We usually ______ a ruler to draw a straight line.\nD. Miss Zhao goes to work ______ bike.\nB. The woman _____ a red hat is our new English leacher.\nC. It's very cold today. Please ___ more clothes when you go out.\nD. Mother is ______ my baby sister now.\nA. I like eating oranges _____, so there are always oranges at my home.\nB. There are so ______ people that I cannot count them.\nC. _____ people think living in China is_____ better than living in Japan.\nD. I've learned _____ from him.\nA. Please try to _____ who stole the computer.\nB. Please help me ____ my mobile phone, I can't ____it.\nA. _____ at the picture! What can you ______ in it?\nB. I'll _____ a book instead of _____ TV tonight.\nA. \"There is no air _____ no water on the moon.\"means \"There is no air _____ water on the moon.\"\nB. \"The boy _____ his parents go to Hong Kong every year.\" means \"The boy _____ his parents goes to Hong Kong every year.\"\nA. I was at her birlhday party, and he was _____ there.\nB. Andy Law is a famous actor. He's a good singer.\nA. The woman was _____ a big bag. Let's help her.\nB. Next lime when you come, _____ me the book, please.\nC. Who's ____away today's newspaper?. I haven't read it yet.\nD. Go and _____ me some wate. I' m thirsty.\nA. \"_____ of them speaks a foreign language.\" means _____ one of them speaks a foreign language.\"\nB. There are some big shops on_____ side of the street.\nA. What is the biggest _____ in the world? Can you answer this _____ ?\nB. The _____ is that he's always asking me such foolish _____.\nA. He _____ hard and at last _____ the language.\nB. The children are _____ the maths problem now.\nC. We _____ in the same school, so we often _____ from each other and help each other.\nD. We _____ English in the morning and _____ to play football in the afternoon.\nA. There are many good apples _____ the tree.\nB. He gets up _____ six _____the morning and goes to bed ___ ten______ night _____ weekdays.\nC. Does he have lunch _____ home?\nD. The man _____ duty is reading a book _____ medicine _____the desk.\nE. In the twins' bedroom, the windows _____ the wall are very big, the pictures _____ the wall are not very new.\nA. Please read the text _____ until you can recite it.\nB. I will go shopping _____ tomorrow.\nC. He waited for _____ then left.\nD. They _____ play football after school.\nA. In Class One _____ 25 boys and 25 girls.\nB. In Classroom One _____ a map of China.\nC. Does your brother _____ a map of China?\nD. _____ your brother got a map of China?\nA. \u2014\u2014Thanks a lot for your help.\nB. \u2014\u2014Are you a student?\nD. Could you tell me whether it is _____ or wrong?\nE. I want to buy a cup of tea, a piece of bread and an apple. _____.\nA. \u2014\u2014_____ does it take by plane?\n\u2014\u2014It takes about eleven hours.\nB. \u2014\u2014_____ is your father?\nC. \u2014\u2014_____ is the fish?\n\u2014\u2014It's ten yuan a kilo.\nD. \u2014\u2014_____ do you write Io your father?.\nE. \u2014\u2014_____ students are there in your class?\nA. Please _____ Lucy and Lily because they are new here.\nB. Please _____ Lucy and Lily! Can you find any differences between them?\nC. Lucy and Lily are twin sisters. They _____ each other very much.\nD. Lucy and Lily _____.\nE. Please ______ the new words in your dictionary.\nA. \u2014\u2014______ do you like China?\nB. \u2014\u2014______ do you like about China?\n\u2014\u2014The food and the people.\nC. \u2014\u2014______ is Mr. Green?\nD. \u2014\u2014______ is Mr. Green?\nE. \u2014\u2014______ falher is Mr. Green?\nF. \u2014\u2014______ man is Mr. Green?\n\u2014\u2014The man on a black bike.\nG. \u2014\u2014______ is Mr. Green from?\nH. \u2014\u2014_____ is Mr. Green like? --He is tall and thin.\nA. He listens to the teacher carefully in _____.\nB. There are four _____ in each unit of the English textbook.\nA. You must give back the book to me _____.\nB. You're just _____ for the last bus.\nB. They callme Tom _______ .\nA. Tom, ______ you are right.\nB. The mon over there ______ Li Lei's English teacher.\nA. When can we _____ off for Beiiing?\nB. He didn't know how to _____.\nA. Those shoes are too small for me. Would you show me _____ pair?\nB. There are six people in the room. Three are girls, _____ three are boys.\nC. He is always helping _____.\nE. Mr Liu and Miss Sun and four ______ teachers are teaching this class.\nA. We _____ follow our Party to go.\nB. He _____ get up earlier because his mother asks him____to cook breakfast.\nA. My parents ______ me.\nC. Chocolate ______ your health, do you agree?\nA. Can you tell me the ______ to your school?\nB. I met him in the ______.\nC. They drove along the country ______.\nA. Ricky stood _____ lhe stage and began to sing.\nB. He sits _____ and I sit not far behind him.\nC. A tree stands ______ the house.\nA. I _____ you to win the competition.\nB. I _____ you can win the competition.\nC. We have no _____ to finish the work in time. What shall we do?\nD. Best _____ to all of you in the coming year!\nA. Everyone should come back _____ five o'clock.No one should be late.\nB. \"He came back two days ____.\" means \"He came back the day ____ yesterday.\"\nA. The chair is _____. Please move it away.\nB. _____, have you got a computer at home?\nC. Roy happened to meet his father ______ to school.\nA. The Chinese basketball team had a _____with the American basketball team in the last Olympic _____.\nB. I often do _____ or play with my classmates after school.\nC. There will be a _____ meeting next week.\nA. After they had read the text, the students___the exercises.\nB. They _____ the farm work in the field though it was raining hard.\nC. I hope everything _____ well.\nA. Go down this road _____ and you'll find the police station on your left.\nB. Which building is _____ of the road? Is it a supermarket or a hospital?\nD. How many English words had you learnt _____ of last term?\nA. \"_____ a good film!\" means \"_____ good the film is!\"\nB. \"_____ beautiful music!\" means \" _____ beautiful the music is!\"\nC. \"_____ is the weather?\" means \" _____ is the weather like?\"\nA. Japan is _____ the east of China and Taiwan is _____ the southeast of China.\nB. Henan is _____ the north of Hubei.\nA. I must go to my office.\nB. I made a telephone call to my friend ______.\nA. ---- _____ will he be beck?\nB. ---- _____ does he come back?\nC. ---- _____ were you in Beijing last year?.\nD. ---- _____ did you go to bed last night?\nA. There's _____ time left. We'll be late if we don't hurry.\nB. We still have _____ apples at home. We needn't buy any now.\nC. There're _____ apples at home. You'd better go and buy some.\nD. I still have _____ money on me, so I can buy apples with it.\nA. _____ of them is here. All of them have gone to Shanghai.\nB. _____ of his parents is a doctor. They are teachers.\nC. _____ Damao or Xiaomao will go there for the dinner. One of them should stay at home.\nD. He speaks _____ English nor Japanese. He speaks Chinese.\nA. _____ the books are here. You may choose any of them!\nB. _____ of the twins like spending _____ their money on books.\nC. \"The workers worked the _____ day.\" means. \"The workers worked _____ day.\"\nA. The car _____ by with a thick smoke behind just now.\nB. The boss walked _____ me without saying a word at half _____ one. I don't know why.\nC. The sunlight is shining in _____ the window.\nD. \"Go _____ the bridge.\" means \" _____ the bridge.\"\nE. They climbed _____ the tall wall, didn't they?\nA. Please tell me when you'll _____in Taiyuan. I'll meet you at the airport.\nB. Can you tell me how to ____ to the nearest police station?\nC. Please call me when you _____ Shanghai.\nD. The passengers were worried because the train didn't ___on time.\nA. Our physics teacher told us light travels faster than _____.\nB. \"You have a beautiful____,\" the man said to her in a sweet ___.\nC. We shouldn't make any _____ in the reading room.\nA. ---- They have supper at six every day.\nB. ---- They don't use animals to do farm work now.\nA. I usually _____ at half past nine every evening.\nB. The young man lay on the sofa and _____ soon.\nC. The baby _____. Please don't talk.\nA. ______, I am late.\nC. ______, could you tell me the way to the station?\nD. I beg your ______, Sir, for coming here late.\nA. \"It ______ me an hour to read English every day.\" means\" I ____an hour reading English every day.\"\nB. The T-shirt ______ me eighty yuan. means \" I ______ eighty yuan on the T-shirt. \"\nC. \"She ______ 100 dollars for the ticket.\" means \" She ______ 100 dollars on the ticket.\"\nD. I am going to _____ my summer holiday in my hometown.\nA. Are you ______ in the ______ storybook?\nB. He says he has no ______ in visiting the places of ______. I don't know why.\nA. I didn't get to ______ until late last night.\nB. I felt very ______ because I didn't ______ well last night.\nC. He fell _____ during the meeting.\nB. If you do _____ I told you, you'll make fewer mistakes.\nD. The girl looks ______ her mother.\nA. Unluckily,the runner ______ his stick on the ground.\nB. I heard Tony ______ down from the tree and hurt his leg.\nA. She ______ her wallet on her way to the supermarket.\nB. They were very sad because they ______ an important football game.\nC. We all ______ you very much. How soon will you be back?\nA. Mr. Green eats _______ food, so he is ______ fat.\nB. I have ______ books to read and ______homework to do today.\nC. Watching TV ______ is bad for your health.\nB. China _______ its food in the world.\nA. The old man _____ his money everywhere, but he couldn't ____ it.\nB. If you've ______ who broke the window, please tell me.\nD. After the doctor _____ the patient very carefully, he said there was nothing serious.\nA. Granny ______ living in city now.\nB. He said he ______ come tomorrow.\nC. I ______ live with my grandparents, but now I have my own home.\nA. The police ______ their missing children here and there, but they didn't ______ him.\nB. The policeman _____the thief to see if he stole the money.\nA. ______ of the workers went to see the film last night.\nB. ---- How many workers went to see the film last night?\nC. ---- Who went to see the film last night?\nB. They ______ Beijing three times.\nA. It is ______ a heavy box that he can't carry it.\nB. The box is ______ heavy that he can't carry, it.\nC. I had ___ much work to do that I slept very late last night.\nA. The children ______ the English Evening and had a good time.\nB. When did your father ______ the Party, do you know?\nA. Millie is good at sports ______ football, he also does well in playing basketball.\nB. We all passed the exam ______ Li Ming. How unlucky he is!\nA. \"His father ______ two years ago.\" means \"His father has been _____ for two years.\"\nB. Mary is always sad when she thinks of her little cat's ____.\nC. The doctors are trying to save the ______ man.\nA. ______ everyone is here. let's start the meeting.\nB. ---- Why didn't he come to school?\n---- ______ he was ill.\nC. There must be nobody in the classroom, ______ the light is off.\nD. ______ I am young, I often give my seat to old ones on buses.\nA. He is over seventy. But he is ______ working on the farm.\nB. He has not come ______.\nC. They have ______ finished doing the homework.\nA. Mum said \"Huahua, you _____ watch TV after you finish doing your homework.\"\nB. They may _____ win the competition, for they have practised for a long time.\nC. Jim _____ not be in the office now, for he left an hour ago.\nA. I _____ a new computer. My uncle bought it for me as a birthday present yesterday.\nB. You should do it with your _____ hands.\nC. ____ a CD player on the desk. Whose is it?\nA. Mr. Green can _____ speak Chinese well.\nB. I haven't heard from him. She hasn't heard from him, ___.\nC. I have been to Beijing twice, _____.\nD. People in Korea celebrate the Spring Festival _____.\nA. You can't _____ English, because it's very important.\nB. Finally I _____ and accepted the job.\nA. Please _____ the radio. I can't hear the news clearly.\nB. Please _____ the radio. It's too noisy.\nC. Please _____ the light. It's too dark here and I can't see anything.\nD. Please _____ the light when you leave here.\nA. Please _____ the new shoes. I don't know if they fit you.\nB. The boy is clever and he likes _____ new ideas.\nA. We haven't ______ him since last Christmas.\nB. I don't know the writer, but I have ______ him.\nC. I have never _____ such a funny story before.\nA. I have 1,000 yuan ______. But that's not enough.\nB. You should _____ your exam papers on time.\nA. The police were ______ when the accident happened.\nB. The boss ______ the worker because he was too lazy.\nC. Our country ______ another man-made satellite last year.\nA. ____ the police, the parents found their lost child soon.\nB. ____ helping me with my English.\nA. The thief ______ the mobile phone from me.\nB. The man ______ me of my mobile phone.\nA. This is the best film _____ has been shown this year.\nB. You can take any seat _____ is free.\nC. Is there anything ______ I can do for you?\nD. Those ______ know the answer, put up your hands.\nE. The teacher, ______ comes from England, speaks English.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayajo b/data_all_eng_slimpj/shuffled/split2/finalzzzayajo
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@@ -0,0 +1,5 @@
+{"text":"A cookies policy template designed to help with legal disclosure obligations relating to cookies and similar technologies.\nThis free cookies policy is identical to the paid version, except that it includes a credit\/attribution clause for Contractology. If you would like to use the document without that credit\/attribution clause, you should purchase a license.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"canoes lake maligne - canoeing maligne lake jasper national park the world is .\ncanoes lake maligne - canoe docking at maligne lake in jasper national park .\ncanoes lake maligne - maligne lake canoe stock photos maligne lake canoe stock .\ncanoes lake maligne - maligne lake things to do jasper boating kayaking .\ncanoes lake maligne - boathouse with canoes at maligne lake jasper national .\ncanoes lake maligne - boathouse canoes on maligne lake jasper np alberta .\ncanoes lake maligne - canoeing maligne lake picture of maligne lake jasper .\ncanoes lake maligne - boat house and canoes at maligne lake in jasper national park .\ncanoes lake maligne - canoeing maligne lake jasper national park 187 born to .\ncanoes lake maligne - cano 235 sur le lac maligne 2 jours 224 couper le souffle .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Articles about web development and mobile development.\nLearning something new can be intimidating. Here are my tips on how to learn any new technology quickly and with the least hassle.\nYou must be informed about website hacks! It's extremely important. In this article I talk all about malicious attacks, types of hacks, what they do, how to remove them, and how to prevent them.\nA collection of my most used jQuery snippets.\nA collection of my most used CSS snippets.\nThis article is for the designer hesitant to learn programming, or anyone who's skirting with wanting to learn programming. I'll begin by approaching common complaints.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Architecture-Engineering-Construction (AEC) industry recognizes Building Information Modeling (BIM) as an efficient means to develop and disseminate design information. However, if a project requires (1) tight coupling between systems and components because doing so generates value for the project and (2) interdependent engineering disciplines to work in parallel due to schedule requirements, the team may face difficulties when they re-integrate any work completed independently back into the main model. To address this problem, we propose combining the use of design drivers, process mapping, and Design Structure Matrices (DSM) to improve a project's ability to de-couple building components, enable concurrency in component development, and achieve seamless BIM integration within a parametric BIM environment. Specifically, these tools combined may help projects reveal and then reduce the number of design interdependencies between building components. We developed and tested the proposed methodology using a civil engineering course that introduced undergraduate and graduate students to parametric BIM. We taught this course once a year for three years, and we refined the proposed methodology during the third year. Although the methodology is rudimentary and requires further study, we hope this paper will inspire other researchers to test this methodology within learning labs in academia and practice.\nXu, L. & Tsao, C.C. 2012, 'Use of Design Drivers, Process Mapping, & Dsm to Improve Integration Within an Introductory Bim Course' In:, Tommelein, I.D. & Pasquire, C.L., 20th Annual Conference of the International Group for Lean Construction. San Diego, USA, 18-20 Jul 2012.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Obverse: Portrait of Benjamin Harrison, 23rd President of the United States.\nSignatures: (as depicted) Judson Whitlocke Lyons, Register of the Treasury and Ellis Henry Roberts, Treasurer of the United States.\nInscriptions: National Currency - Secured By United States Bonds or Other Securities - United States of America Will Pay To The Bearer On Demand Five Dollars - Register of the Treasury - Treasurer of The United States - Cashier - President - Series of 1902 - This note is receivable at par in all parts of the United States in payment of all taxes and excises and all other dues to the United States except duties on imports and also for all salaries and other debts and demands owing by The United States to individuals corporations and associations within the United States except interest on the public debt.\nThe First National Bank Of West Chester in Pennsylvania printed $5,482,600 dollars worth of national currency. Once a bank issues that much money there really isn't much room for rare issues. However, there are certainly exceptions to every rule. This national bank opened in 1863 and stopped printing money in 1935, which equals a 73 year printing period. That is considering a long operation period for a national bank. During its life, The First National Bank Of West Chester issued 30 different types and denominations of national currency. We have examples of the types listed below. Your bank note should look similar. Just the bank name will be different. For the record, The First National Bank Of West Chester was located in Chester County. It was assigned charter number 148.\nThe First National Bank Of West Chester also printed 61,305 sheets of $5 1902 blue seal national bank notes. Sadly, a printing range that high means that these blue seal bank notes from this bank are not going to be extremely rare. Ben Harrison is on the front of all 1902 $5 blue seal bank notes. This happens to be the smallest denomination issued for the 1902 series. Each note is complete with a blue seal and blue charter number. Despite saying series of 1902, these were actually issued by national banks between 1908 and 1928. There are two different types of blue seals. The first type is called a date back and it has \"1902-1908\u2033 written on the back of the bill. The other type is called a plain back; it does not have the date stamps on the back of the bill. The values for these notes range widely based on condition and the bank of issue.\nBenjamin Harrison (August 20, 1833 \u2013 March 13, 1901) was the 23rd President of the United States (1889\u20131893); he was the grandson of the ninth President, William Henry Harrison. Harrison had become a prominent local attorney, Presbyterian church leader and politician in Indianapolis, Indiana. During the American Civil War, he served the Union for most of the war as a colonel and on February 14, 1865 was confirmed by the U.S. Senate as a brevet brigadier general of volunteers to rank from January 23, 1865. Afterwards, he unsuccessfully ran for the governorship of Indiana but was later elected to the U.S. Senate by the Indiana legislature.\nHarrison, a Republican, was elected to the presidency in 1888, defeating the Democratic incumbent Grover Cleveland. Hallmarks of his administration included unprecedented economic legislation, including the McKinley Tariff, which imposed historic protective trade rates, and the Sherman Antitrust Act; Harrison facilitated the creation of the National Forests through an amendment to the Land Revision Act of 1891. He also substantially strengthened and modernized the Navy, and conducted an active foreign policy. He proposed, in vain, federal education funding as well as voting rights enforcement for African Americans during his administration.\nDue in large part to surplus revenues from the tariffs, federal spending reached one billion dollars for the first time during his term. The spending issue in part led to the defeat of the Republicans in the 1890 mid-term elections. Harrison was defeated by Cleveland in his bid for re-election in 1892, due to the growing unpopularity of the high tariff and high federal spending. He then returned to private life in Indianapolis but later represented the Republic of Venezuela in an international case against the United Kingdom. In 1900, he traveled to Europe as part of the case and, after a brief stay, returned to Indianapolis. He died the following year of complications from influenza.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaybce b/data_all_eng_slimpj/shuffled/split2/finalzzzaybce
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index 0000000000000000000000000000000000000000..18a4d1590aa62cfc0fd34688603dfe4d8c330b28
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+{"text":"Package 100 meals with Campus Kitchen at the Athens Community Council on Aging (ACCA). Volunteers will prep meals for deliveries, no cooking experience necessary! We will be dividing previously cooked fruits, vegetables, protein, bread, and desserts into individual containers for clients. Our meals will then will be given to seniors to take home from the ACCA, or they will be distributed through Meals on Wheels.\nPark in the larger parking lot opposite the playground. Enter the main lobby and staff will direct you to the kitchen.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Holiday season is in full force and there is not a moment to spare. Especially if you intend to take on everything the Queen City has to offer. Don't forget to check out Holiday on Ice and the 59th annual Christmas Town USA, McAdenville. Remember, you can always sleep in January.\nExperience the magic of Christmas in Uptown with The Charlotte Symphony and Albert-George Schram, conducting American Idol finalist Melinda Doolittle, Oratorio Singers of Charlotte, Charlotte Children's Choir. More Information here!\nThere isn't much else that will get you in the Christmas Spirit than The Nutcracker. CPCC is located just on the edge of Uptown in Elizabeth. Take in the show and head uptown for drinks after. More Information here!\n\"Celebrate the Season in Style!\" Hosted by the Friends of Fourth Ward. This is a Charlotte holiday tradition and it's in one of Charlotte's oldest Uptown neighborhoods! More Information here!\nSaturday, December 6th: Flannels and Beer: OFFICIAL BrewPublik Launch Party!\n\"A FREE community event to celebrate the launch of BrewPublik, good beer and lumberjacks\/jills! Come decked out in your flannel and try out our beer curation system while enjoying some s'mores around the fire and delicious food from local food trucks\" More Information here!\nGet Tacky for Christmas in South End! FREE! More Information here!\nEnjoy a drink while you meet and shop over 60 artists that are personally selling their art. \"Local Art + Holiday Season = Charlotte's Best Unique Gifts\" in NoDa. More Information here!\n\"St. Nick & his Krampus will be out roaming the streets of NoDa looking for naughty & nice kids of all ages. There's fun all over the Arts District, with a few special stops planned. Look for Krampus around the area and catch these events throughout the night \u2026\" More Information here!\nSpark some holiday cheer at 15 bars and clubs in Uptown for $15. More Information here!\nJoin the Carolina Voices' MainStage Choir, The Grey Seal Puppets, Eddie Mabry Dancers, Caroline Keller and more as they bring Christmas memories alive with the Singing Christmas Tree. Great family event. More Information here!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am a Ugandan and proud to be a part of that little landlocked country affectionately called the \"Pearl of Africa\". As a society we are a proud people trying, like any other, to do the best we can, survive the way we know how. I can't say why we consider emotions weak and a distraction or why it is hard for us to give or accept a compliment. Maybe it's because we are society whose history has taught to be wary of visitors bringing gifts or new ideas. Maybe it's because of what we have done to each other and what has been done to us or even our experiences. Whatever the reason, we are a community that has seen grief and pain and learnt to buck up and move on as that is the way life is. No one person is special, we all have the same story and experiences and the best way to survive is keep your head down and don't rock the boat, even when you feel like the boat is sinking.\nI had no idea what to expect when I went to ask the children in the In Movement program to make Traveling Postcards. Personally, making my own postcard was one of the hardest things to do. How do I let out feelings that I have been taught to keep silent? How do I encourage someone who is hurting when it does not come naturally? For many of these children silence has been their refuge, their best friend. For many of them thinking about the pain, violence, grief and nightmares the women in Congo are facing would be facing their own personal nightmares. So how could I ask these children to do what I myself was finding hard to do?\nIn Movement, a non profit organization that began in Uganda in 2003 has a simple mission. They believe that the expressive arts are a very powerful tool of empowerment for youth, unlocking vast potential for personal growth and transformation. By encouraging creative risk-taking in safe, supportive and empowering settings, they are planting seeds that can assist youth in helping to transform their society, and lead happier, healthier, and more satisfying lives.\nIn other words they provide a safe place for the children to express themselves, learn who they uniquely are and be proud of them selves.\nMany of the children who were a part of the Traveling Postcards project are students who have been part of the In movement program for a few years and so are more in open to expressing themselves freely among strangers than the average Ugandan child. Because they have been appreciated and been told that they are special and unique, they were able to pass this on.\nMany of them had never made a postcard before or even received one but they were eager to pass on the message that they have received; one of inspiration and hope. It did not matter if the was artistic or aesthetic. All that mattered was that someone somewhere would know they were loved.\nThe card making was a success the children wanted to make a second one of their own for the women and guardians in their lives, to let them know that they care for them.\nAm still in the process of making my own traveling postcard, and slowly it is taking shape and am not afraid anymore that is isn't symmetrical, colorful or perfect but I hope that whoever receives it will know one thing\u2026 I see you. I appreciate you. I love. I care.\nAnd hopefully the women who receive the children's cards will know this too.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Bed: Respawn at the wrong height?\nThis has occurred to me twice in limited testing, but cannot consistently reproduce.\nBed placed and slept in room inside bottom of mountain [see attached image].\nKilled next day by creeper (thankssss!) and respawned 30 blocks or so higher in house on top of mountain , where there is only a chest, no bed. (chest placed after bed, for reference).\n\"Odd\" spawn position is relatively close, could be exact, to bed x-y coordinates, but much higher z elevation.\nLater, exited game, started again and my spawn point was similar \u2013 up on the mountain near the chest, but this time \"outside\" near the top of the ladder.\nAgain, no bed up here.\nHave since exited and restarted and spawned correctly near bed. Will attempt to reproduce by dieing again, but have not yet with limited playtime.\nWorld created in 0.3.3, updated this morning to 0.4.0.\nMCPE-104 When i respawn i respawn on top of my house.\nMCPE-8856 Player spawns above doorway when they leave the game under the doorway.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In the wake of criticisms by ruling party leaders over his party's alliance with BNP, Gano Forum President Dr Kamal Hossain on Monday said he has neither any desire to contest any election nor to get any state post, reports UNB.\nSpeaking at a press conference, he also said Jatiya Oikyafront is not any electoral alliance as it was formed to work for ensuring a free and fair election.\n\"I'm happy to clarify, for those who have been making virulent, baseless, and persistent personal attacks on me, that I've no desire to be a candidate in any election or to seek any public office. I'll continue to work towards building a democratic, secular and pluralistic society,\" he added.\nGano Forum arranged the press meet at the Jatiya Press Club.\nAmid the growing public concern over the next election, Kamal said they engaged in dialogues amongst parties and civil society groups so that the nation can reach a consensus to bring about necessary reforms.\n\"I'm concerned to learn that our shared efforts to work towards a fair and free election, which could be the basis of a democratic form of governance, have been misconstrued. In responding to these public concerns several of us have engaged in national dialogues, across parties and civil society groups to suggest what reforms are necessary to ensure such conditions. These dialogues have led to the initiative of the Jatiya Oikyafront,\" he said.\nThe Gono Forum president also said the Oikyafront will continue to work for unity of different parties and citizen groups to realise its aims.\nKamal hoped that there will have no controversy over their party's initiative to form the Oikyafront through his today's clarification.\nReplying to a question, he said mobilising public opinion is part of Oikyafront's movement as he thinks public opinion will mount pressure on the government to concede to their demands.\nAsked what programmes they may take if the government does not accept their demands, Kamal said they will think about it deeply and then work out suitable programmes.\nHe said Oikyafront may send the letter to the government on Tuesday urging it to engage in talks over the next polls before announcement of the election schedule.\nAs his attention was drawn to the government's stance not to accept any unconstitutional demands like formation of the election-time neutral administration and dissolution of parliament, Kamal said such provisions were there in the Constitution, but it was amended.\nSought his comment on MP's criticisms in Parliament about his role in politics and in the Liberation War, he said he will come up with the clarification over the issues if he deems it necessary.\nReplying to another question, Kamal said they formed the alliance with BNP, but they have no contact with Tarique, and they do not feel it necessary to communicate with him.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaycxm b/data_all_eng_slimpj/shuffled/split2/finalzzzaycxm
new file mode 100644
index 0000000000000000000000000000000000000000..f7862d09052aee69b5040c8ad957a28b388648d6
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"It is estimated that 1,400 to 2,800 cases of invasive meningococcal disease occur in the United States each year, and that ten to 14 percent of people who contract the disease will die. The U.S. Advisory Committee on Immunization Practices now advises immunization with a tetravalent vaccine (serogroups A, C, W-135, and Y) for all 11- to 18-year-olds. However, the currently licensed vaccine is poorly immunogenic in infancy, when the highest rates of disease are observed, according to background information in the article.\nMatthew D. Snape, F.R.A.C.P., of the Oxford Vaccine Group, University of Oxford, England, and colleagues conducted a randomized controlled study to determine the immunogenicity of a novel tetravalent meningococcal vaccine known as MenACWY in infants. Unlike the currently licensed vaccine, MenACWY uses a natural mutant of the diphtheria toxin.\nThe study included 421 healthy infants in the United Kingdom and Canada who received one of three different dosing schedules of MenACWY, or a monovalent vaccine against the meningitis C serogroup. The authors determined the proportion of babies who had protective antibody levels after receiving the vaccine. They also assessed vaccine safety and reactogenicity (the capacity to produce adverse reactions).\n\"In this study, we have demonstrated that a primary immunization course of the novel tetravalent meningococcal glycoconjugate vaccine MenACWY was well tolerated and immunogenic for serogroups A, C, W-135, and Y when given to healthy infants at either two, three, and four months or two, four, and six months of age,\" the authors write.\nThe study found that at least 92 percent of infants who received the two-, three-, four-month schedule had protective antibody levels to all four serogroups. For the two-, four-, six-month schedule, similar results were obtained for the C, W-135, and Y serogroups, but the proportion of infants with protective antibody levels against serogroup A was lower at 81 percent.\nIn the two-, four-month primary series groups, at least 84 percent had protective antibody levels to three of the four serogroups, while 60 to 66 percent of infants had protective levels against serogroup A. After a booster dose at 12 months, at least 95 percent developed protective antibody levels to three of the four serogroups, and 84 percent for serogroup A.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This specific image (easy diy outdoor cushion covers Covers For Outdoor Cushions Elegant Covers For Outdoor Cushions) previously mentioned is actually labelled with:put up simply by Grover Shields with July, 20 2018. To find out all pictures with Elegant Covers For Outdoor Cushions pictures gallery please abide by that web page link.\nElegant Covers For Outdoor Cushions \u2013 This easy diy outdoor cushion covers Covers For Outdoor Cushions Elegant Covers For Outdoor Cushions photos was taken from diyjoy.com and upload on July, 20 2018 by Grover Shields. Download other photos about Covers For Outdoor Cushions in our gallery. Thank for visiting this gallery collection for latest Covers For Outdoor Cushions ideas. You can print Elegant Covers For Outdoor Cushions for your collection by clicking on image to print in high resolution.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The LabUMat is a fully automated urine chemistry analyzer evaluating the 11 chemical parameters of LabStrip U11 Plus test strips and the UriSed is a fully automated sediment analyzer providing whole viewfield miscroscopic images of urine sediment and detecting 15 sediment particles using Urised Cuvettes. The efficiency of LabUMat and UriSed can be maximized by using these instruments together as one system.\nSince both instruments are developed and manufactured by 77 Elektronika, there is a well designed hardware and software interface between them. Together, the two instruments make up a Complete Urine Laboratory System. Results of both measurements are stored in a common database and reported as a common report.\nSince all necessary measurements which have to be done on urine samples are completed by this integrated system in one process, combination of LabUMat and UriSed accelerates laboratory throughput and provides the most effective and reliable solution for complete and professional urine analysis.\nFor detailed information please visit LabUMat and UriSed pages.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"... It was probably moved. On the top of the page, click \"Control Panel\" ... then click on \"Forum\" ... You should see your thread in the appropriate forum where it should have been in the first place. If it is not in this list, it probably broke a rule and was removed.\nHopefully this clears up some issues with missing threads.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Beginning Tuesday, March 13th Pepperoncini Phoenixville (184 Bridge Street, Phoenixville; 484-924-8429) will host a series of Italian Wine dinners on the second Tuesday of each month. Each wine dinner will focus on a different region of Italy and will be hosted by David Greenstein from The Wine Merchant. Wines will be paired with Chef Paul DiBona's classic Italian 5-course menu. Tuesday, March 13th will feature Italy's Tuscan region with premium Brunello and Tuscan wines. The cost is $68 per person, plus tax and gratuity. Call 484-924-8429 for reservations. A limited number of seats are available for these private dinners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzayfek b/data_all_eng_slimpj/shuffled/split2/finalzzzayfek
new file mode 100644
index 0000000000000000000000000000000000000000..64baf40a85bef9e1e6998b357e5b7544e3e0a637
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzayfek
@@ -0,0 +1,5 @@
+{"text":"The item Ashes to ashes, Mel Starr represents a specific, individual, material embodiment of a distinct intellectual or artistic creation found in City Libraries, City of Gold Coast.\nMaster Hugh, Kate and their children attend the Midsummer's Eve fire. Early next morning, Hugh hears the passing bell ring from the Church of St. Beornwald, and moments later is summoned. Tenants collecting the ashes to spread upon their fields have found burned bones. Master Hugh learns of several men of Bampton and nearby villages who have gone missing recently. Most are soon found, some alive, some dead. Master Hugh eventually learns that the bones are those of a bailiff from a nearby manor. Someone has slain him and placed his body in the fire to destroy evidence of murder. Bailiffs are not popular men; they dictate labour service, collect rents and enforce other obligations. Has this bailiff died at the hand of an angry tenant?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Apart from showcasing your brand, you can as well interact with your customers and to address their concerns through a website. If you are a business owner, it is likely that your internal team does not have the skills to create and properly manage your company's website. You should, therefore, outsource the services of a reputable web design company. Many companies provide web design services, making it difficult to choose the best. You should be careful when hiring a web design company to avoid dealing with a company with negative feedback. The article herein discusses some of the factors you should consider when hiring a web design company.\nLocation is the primary factor that should guide your choice of a web design company. By hiring this company, you will be able to save a lot of money that you would have spent on transportation if they were very far. Since you operate with them in the same locality, the chances are that they have some knowledge of your enterprise and the best web design services that suit it. It is also important to note that you will be able to the convenience of meeting the managers of Means-of-Production without much hassle.\nThe other factor you should put into consideration when hiring a web design company is the experience. Depending on the number of years they have been offering web design services, you will be able to judge if they have the right skills and expertise to create an excellent homepage such as that of squarespace splash page. Similarly, check out the experience of the employees who will do the main job to see if they have adequate skills to provide satisfactory services. From their portfolio, you will also their suitability if you have designed a website for a company that is in the same industry as yours.\nReputation is another essential factor that should guide your choice of a web design company. Find time to visit the website of Means-of-Production to see what other people are saying about them as well as their customer relationships. Be careful not to say yes to a company that has a lot of negative customer reviews. By talking to their past clients, you will be able to see if they were satisfied with the website services they got from the company.\nCost is another significant factor that should guide your choice of a web design company. Squarespace websites design services will give you time to check the list of services you will require to pay for and to confirm that there are no hidden fees. To avoid spending above your financial strength, find time to compare the prices charged by various web design companies within your reach.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Objects Of Affection: A peek inside my shop!\nA peek inside my shop!\nJust wanted to give a little peek into my shop!\nYou've been tagged!!! I was tagged, now you've been tagged!!! Visit my blog and you'll see! It will help bring more folks to your blog!!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rich Veteran 002 \u2013 Why a VA Loan is Going to Make You Poor : VA Loans can be a great tool for military members to buy a home. The problem is most people don't know what the hell they are doing. They think that using a VA loan every time they move to a new duty station is a good idea. It may not be. It may actually make you poor or drag.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A benefit Zumbathon will be held Saturday in Kinston for two reasons \ufffd to raise awareness of a deadly disease and to honor the memory of a very special woman.\nThe event will take place at the Galaxy of Sports Health Club on U.S. 70. It begins at 2 p.m. and is scheduled to run through 5 p.m. Admission is only $5 and all proceeds will be donated to the Jim \ufffdCatfish\ufffd Hunter chapter of the ALS Association.\nALS \ufffd amyotrophic lateral sclerosis \ufffd is a disease of the nerve cells in the brain and spinal cord that control voluntary muscle movement. It\ufffds also known as Lou Gehrig\ufffds Disease, named after the legendary New York Yankees baseball star who died of the disease.\nThe husband and wife team of Beth and Anthony Strong are the ones responsible for organizing Saturday\ufffds event, which features a relatively new exercise craze called Zumba. The exercise includes loud, exciting music with dancing and aerobic movement instead of typical movements.\nBeth said there would be some small raffles and 50\/50 drawings scattered throughout the event to raise more money for the cause. There will also be Zumba shirts on sale.\n\ufffdMore than anything, we want to spread awareness of ALS and have fun at the same time,\ufffd Beth said.\nFor Anthony, the group fitness manager at Galaxy, the cause is a personal one. Neatha Walters, a close friend of his, died from the disease a few years ago.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzaytzb b/data_all_eng_slimpj/shuffled/split2/finalzzzaytzb
new file mode 100644
index 0000000000000000000000000000000000000000..caaafa5a10b545088f259f6d5ddc79421eeeff30
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzaytzb
@@ -0,0 +1,5 @@
+{"text":"Thank you for choosing to learn more about Quest's On Demand solutions, specifically our Skype for Business Policy Management module. All of our on-demand solutions are SAAS offerings, meaning you don't need to implement any new infrastructure in order to use and enjoy these products. I'm Shawny Reiner, a strategic systems consultant with Quest, and I'll be taking you through a brief demo today that shows you how easy it is to manage policies for Skype with our module. This demo's going to focus on an important compliance requirement for data restriction. In this scenario, Ed is an employee for an aerospace company. And one of his rules is to work with external entities like contractors, which sometimes requires sharing documents electronically. He's required to follow stringent protocols. So as IT admins we need to ensure this is enforced, preventing Ed from bypassing those protocols. Now of course, while we're using an aerospace company as an example, many industries have these same security risks and compliance needs. And any time you have external contact there is high risk of data loss. So first I'd like to show you our On Demand environment, the home page. And as you can see, when you log in to it, you've got some management options here on the side. You've got your various policy modules available, whatever you've subscribed for, and of course we have some coming in the future. But I'm going to take you through the Skype for Business Policy module. So we're going to go and connect to that one. As you see, it's connecting into my Titan Corp. Tenant, which is my lab, and my Azure AD. And then you have the ability to add rules. So let's talk about the methodology here. Our methodology around managing policy management is basically a collection of policies, grouped into hierarchical rules, and then applied to groups of users rather than individual users. Now these groups and users reside in your Azure AD tenant, and they can be cloud-only or hybrid objects. So our module provides you several benefits over native capabilities. One is that you can apply to groups of users rather than individual users. Natively you can only apply to individuals. So this can be very tedious and administratively difficult when you need to apply to many, many users. Even if you use PowerShell scripts, you still have to create that CSV, keep that up to date, etc. We dynamically update these policies for you as well. So that's another benefit. So when you apply a policy natively, and then, for example, new users are hired or you need to change a setting in that policy, those updates don't take place automatically. You have to rerun your PowerShell script-- which means updating that CSV file if it's new users, etc-- before these changes take effect. With our module, all you have to do is manage the group memberships, and then we dynamically apply these for you because we're processing these rules. And then finally, another big benefit is that we log the activities of these changes, giving you that important proof you need that you're compliant. So these benefits combine to offer you better management, tighter controls, and proof that you are compliant. So this increases the effectiveness of your security controls and compliance. So now what I'm going to do is walk you through creating a rule. So first you want to give it a name, something that makes sense to you. In this case, we are going to place a conferencing policy that restricts the ability to do file transfer. So I'm going to just name this one, no file transfer. Obviously in your real world, you may be a little more specific. Also I have a bunch of groups created for my Skype users, and so one of the nice things about the way this works, is it queries your Active Directory based on search criteria. So here's all my Skype users, Ed being one of these. And I'm going to Skype Demo Group 2 to add into this policy. So anybody who is a member of this, which is going to be Ed, also known as Skype User 3 in my demo here, so those users are going to get whatever policy I apply through here. So I'm going to pick a conferencing policy. Now I've already created one, and I've used the standard PowerShell method of creating the policy. Because that's how you have to create custom policies in Office 365 for Skype, or Exchange, or any of them. So I have one set up ahead of time, and it's basically just saying that through conferencing you cannot file transfer. So now that I have all of this done I want to enable this rule and save it. So now, as you can see, I've got a rule here, and it pops up that it's provisioning this rule and going to apply it. And so then what I'd like to do is take you through what it looks like in the client. So here we have my client. This is me in my corporate environment, not Ed in my lab. And I just want to show you what it looks like when you are allowed to create a meeting and actually be able to attach a file. You have a little paperclip here and it pops up and you can find files and attach it. So let me disconnect from this meeting here. So now what I'll do is I'll sign out as myself. And I'm going to log in as Ed, or as I said, Skype User Number 3 in my lab to Titan Corp. So now what I'm going to show is what it looks like when he can't actually file transfer. So first I've got to connect the meeting. Now the paper clip shows up, but when you click it, nothing happens. So effectively the policy is in place. So that's the ease of adding policy and managing it through a module. And you could put a client, a voice, and external mobility all in one rule. And then you can start to combine rules as well, so that you can manage multiple groups, multiple users, multiple policies. So once again, thank you for joining and I hope you found this useful. If you have more questions or want to try this out, get in contact with your territory account manager, and they can help you get a trial subscription. Thank you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"CJ van der Linde been interviewed during the Absa Bok Town from Orlando Stadium on October 06, 2012 in Soweto, South Africa.\nFormer Springbok prop CJ van der Linde will not take up the post of forward's coach at Griquas' which he was due to officially start on January 7.\nVan der Linde' who started at tighthead when the Boks won the 2007 World Cup final against England in Paris' has been consulting in Japan and has been offered a fulltime overseas post.\n\"It is with regret that the Griquas rugby union announces that we will be parting ways with our recently appointed Forwards Coach' CJ van der Linde' in March 2019'\" Griquas chief executive Arnie van Rooyen said.\nGriquas said that Van der Linde will assist the head coach Brent Janse van Rensburg in the coming months before he departs the union officially.\nIt would help to ensure that the team prepares for the new season with as little disruption as possible and also gives time to search for a suitable replacement.\n\"I am incredibly grateful for the opportunity Griquas gave me and the faith that Brent has in my ability'\" Van der Linde said.\nJanse van Rensburg said: \"These things happen unfortunately' we will look ahead and invest our energy into the way forward. No person has the right to decide for another what is best for him and impact their future.\nJanse van Rensburg further added that they will not be rushing into appointing anyone in van der Linde's place.\n\"Decisions will be made with the long term in mind'\" the coach said.\n\"The success and sustainability of our programme will be the focus.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"AAO-HNSF has added two titles to its book and eBook collection through an agreement with Thieme Publishers.\nThe first book, Otolaryngology Lifelong Learning Manual (OLLM), is an update to the Maintenance Manual for Lifelong Learning. It was produced through the work of the education committees under the leadership of Sonya Malekzadeh, MD, Coordinator, Education. It is both a resident resource and certification study guide, and practicing physicians can use it for recertification. Nonphysician clinicians also will benefit from the comprehensive scope of the OLLM.\nThe second book, Geriatric Otolaryngology, was edited by Robert T. Sataloff, MD, Michael M. Johns III, MD, and Karen Kost, MD. It is more than a revision to the previous book of the same title. It is a comprehensive and timely discussion of the otolaryngology concerns of the elderly population.\nBoth books are available in print and eBook formats. You can purchase both at the Thieme Booth, #1216 at the OTO EXPO, and can be ordered through the Thieme website at www.Thieme.com. You can search by specialty or title.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"When you're looking for a airport limo service from Butler NJ, don't settle for something less than the high-quality service of Royal Limo. You just can't find a better limo service, and in regards to car service, you want a company you're capable of really depending on. It's always your greatest priority to make it your flight on time. This is why you should never trust a sub par car service, public transportation, or taxi to get you there. The best means to ensure a comfortable and punctual arrival to your flight is by getting in touch with Royal Limo and setting up an appointment when you need limo services from Butler NJ.\nRoyal Limo has a wide selection of different car options when you need car service in Butler NJ. It doesn't matter if you are traveling on your own or you're going with a big group. To make sure your trip to the airport is relaxed and as worry free can be, be certain you hire Royal Limo. All of our customers who are in need of car service are taken very seriously, but those that want airport transportation service even more so. When it comes to pickups on the way to the airport and drop offs at the airport, we know that timing is everything. We don't believe your trip to the airport should ever be stressful. That's the reason we aim to start you off on the right foot with our luxury transportation to the airport. You are able to never be too careful when making it to the airport on time, even if it's just you. You can't be too comfortable either. When you need airport car service in Butler NJ contact us and get the treatment you deserve.\nRoyal Limo's prices for a limo to the airport from Butler NJ make our limo services even more popular. Many individuals assume they can't afford luxury transportation, but Royal Limo knows that usually isn't the case. In a lot of instances, our customers actually save money by hiring our chauffeurs to get them to the airport in style and relaxation. While this might sound crazy at first, consider the fact that you're never going to need to worry about missing a flight. When you depend on a city taxi or other types of less dependable transportation, you run the risk of missing your flight completely. If you have to rebook, change your flight times, or purchase new tickets completely, it's capable of being unbelievably expensive. You can reserve your Royal Limo ride considerably ahead of time as opposed to picking up your phone at the last minute to book a trip with someone who may or may not be a great driver or supplier of punctual service.\nIf you trust limo to Newark Airport from Butler NJ from Royal Limo, you will save yourself substantial anxiety. By doing this you aren't going to have to stress throughout your day with regards to getting to the airport on time, or planning to arrive hours and hours ahead of schedule wasting valuable time. You can go about your day confident that you will get to the airport on time thanks to our punctual service. We get you there with enough time to spare, but you also will not end up wasting hours waiting for your flight. If you're ready to learn more about what we offer when you need limo to Newark Airport from Butler NJ and book your trip, get in touch with Royal Limo right away.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Destiny 2: So funktionieren die Waffen-Slots in Forsaken (Update). In diesen 7 MMOs k\u00f6nnt ihr im Editor eure Traum-Charaktere erschaffen! . Fortnite und Crossplay: Deshalb glaube ich, dass Sony alles richtig macht . User-Reviews The Division: Einfach nur Genial \u2013 User-Review. Ich kann 10 User-Review. If you fancy playing 20 of the hottest slot game pay-lines on the planet right now \u2013 then it's time There's also a Wild 7 Bonus just to help those smaller prizes keep on coming. Casanova \u00b7 Casinova \u00b7 Cool Diamonds II \u00b7 Diamond Cats \u00b7 Diamond Monkey \u00b7 Diamonds on Fire Amatic Slot Machine Reviews (No Free Games).\nIf you have any issue with this game please write us to info vegasslotsonline. This allows a minimum bet of 20 coins a spin, and a maximum bet of 1, coins a spin. This involves correctly predicting whether a playing card will be red or black, or predicting what suit it will be, for which you'll respectively win double or quadruple your original prize. If you fancy playing 20 of the hottest slot game pay-lines on the planet right now \u2014 then it's time to play \"Hot Twenty\" a retro symbol slot game from Amatic. Please be as much descriptive as possible and include details such as Browser type Chrome, Firefox, Not only does Hot Twenty offer lots of big fruity prizes, it also offers a potential , coin Scatter Bonus. Please be as much descriptive as possible and include details such as Browser type Chrome, Firefox, The Hot Twenty lines need to be played on each spin by all players, but you can stake each line with between 1 and 50 coins. Play Mobile Slots for Real Money 1. There's also a Wild 7 Bonus just to help those smaller prizes keep on coming. The usual symbol suspects offer the lowest payout, but even cherries, oranges and lemons can win you up to 5, coins. However, even that's not the hottest prize you could win \u2014 as a handful of 7's could rock your world with a cool 1,, coins. This allows a minimum bet of 20 coins a spin, and a maximum bet of 1, coins a spin. The Hot Twenty lines need to be played on each spin by all players, but you can stake each line with between 1 and 50 coins. For all new players at Mega Casino. Please be as much descriptive as possible and include details such as Browser type Chrome, Firefox, This involves correctly predicting whether a playing card will be red or black, or predicting what suit it will be, for which you'll respectively win double or quadruple your original prize. If you fancy playing 20 of the hottest slot game pay-lines on the planet right now \u2014 then it's time to play \"Hot Twenty\" a retro symbol slot game from Amatic. There's also a Wild 7 Bonus just to help those smaller prizes keep on coming. Play Mobile Slots for Real Money 1. Casino of the year Read Casino Review. You've Won a Free Spin. If you have any issue with this game please write us to info vegasslotsonline. Not only does Hot Twenty offer Columbus Deluxe - Rizk Casino of big fruity prizes, it also offers a potentialcoin Scatter Bonus. All prizes may also be gambled on the Gamble Feature. If you fancy playing Dawn Of Dinosaurs\u2122 Slot Machine Game to Play Free in 888 Slots Developers Online Casinos of the hottest slot game pay-lines on the planet right now \u2014 then it's time to play \"Hot Twenty\" a retro symbol slot game from Amatic. This involves correctly predicting whether a playing card will be red or black, or predicting what suit it will be, for which you'll respectively win double or quadruple your original prize. Play Mobile Slots for Real Money 1. This allows a minimum bet of 20 coins a spin, and a maximum bet of 1, coins a spin. Http:\/\/www.zis-hamburg.de\/staff-members\/details\/Verthein\/?L=1&cHash=ad5a3492fbb050ecc10ef30a43cdb572 Hot Twenty lines need to be played on each spin by all players, but you can stake each line with between 1 and 50 coins.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"\"I have the same information. It's all just rumors. There is nothing specific,\" the bishop said in response to reports Betancourt had been seen at a local hospital.\n\"If you consider the mentality of our people, who are very susceptible to rumors, and if you consider that one of those who were released said he saw her (Betancourt) three weeks ago, then you have to suppose that she is there,\" the bishop said. \"That was enough for people to start speculating,\" he added.\nBishop Orozco also said he did not know if local residents had been threatened by the FARC and said that in any case, \"people have always been very fearful of these militant groups and they are very concerned about their own safety. Wherever there are outlaw groups, whether guerrillas or paramilitaries, people must be very careful,\" he explained.\nHe also said he has sent letters to the commanders of FARC and that he is still awaiting a response. \"Once there is a spokesman, we will be willing to collaborate\" to secure the release of hostages, especially those who are sick.\nLocal residents in the town of El Capricho, which is located in an area strongly controlled by FARC, said Betancourt had been seen in public in recent weeks.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paco Rabanne Pure XS. A contemporary design that gives rise to the idea of a lighter, Pure XS opens up, one handed with a black lacquered cap. The bottle is a weighty glass with polished edges: instinctive and substantial. Packaged in velvet [of course], all is luxuriously sensual. Electric blue and gold, the shock of sensory contradictions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Susan loves cats. She rounds up lost kittens and drops them off with friends \"just for the weekend\" which usually turns out to be forever. Her mother had Laila, a 23-year old Saimese, who was a beautiful lady cat. Susan had known Laila forever.\nOne day, her mother called late at night and began talking about odd things. Susan interrupted and said \"tell me why you're calling.\" So her mother said that Laila had died. Susan asked what happened to the body. Her mother said that she had tossed her into the garbage. Susan slammed down the phone. We talked about it, dressed in black, and at 2 am, snuck into her yard to look for Laila. We found her in the garbage, took her home, and cleaned her up.\nWe figured we'd bury her under her mother's rose bushes that coming weekend when she'd be out of town. Since it was summer and very warm, we curled up the poor girl, put her in a plastic bag, and put her in the freezer. Susan went over and looked at the rosebushes to pick out the one. But her mother stayed in town for the weekend. And the next. And the next.\nWell, Laila was in our freezer for about six months. Friends would ask \"what kind of cats do you have?\" and Susan would say five live ones and one popsicle cat.\nOur Japanese housekeeper opened the freezer one day, looked, and shouted \"what that?!?\" Susan said it's Laila. She shouted \"but Laila dead!\" The housekeeper couldn't figure out how Laila should show up several months later in our freezer.\nSo finally, her mother asked Susan's opinion as to whether she should go on a cruise. Great! Susan urged her mother to go on the cruise just so we could bury Laila. So she left and we thawed out Laila.\nWe went over and while Susan directed, I spread newspapers around so there wouldn't be any tracks and carefully dug up a rose bush and dug down very deep. Susan gave a short talk about Laila's life, sprinkeled at least half a bag of Laila's favorite cat food on her, stripped the rose bushes and added flowers. Laila looked so happy. We covered her up and removed all the evidence.\nSeveral weeks later, the Japanese woman walked into the living room and said to Susan \"where Laila? She no in freezer. Maybe you eat her?\" Susan told her that we buried her. But we wouldn't tell her where we buried Laila so that her mother wouldn't find out.\nOur housekeeper spent Tuesdays at our place and Wednesdays at Susan's mother's place. So of course she talked. One day, we're at her mother's place and she finally says \"What's this that I hear that you buried Laila in my garden?\" Susan denied everything. Her mother asked if we really kept Laila for six months in our freezer, along with the venison and ice cubes. Susan denied everything.\nHer mother still walks around the garden, trying to figure out where we buried Laila.\nNow that's a Susan Story: it makes me cry and laugh. What a girl. I hope you've laughed too and makes you feel a bit better.\nReturn to the HAVE A HEART FOR SUSAN Page.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sing Along to the Dora the Explorer theme song!\nCelebrate all things wedding related, with this video that brings together our favourite Nick Jr. ceremonies! PAW Patrol fans can see Farmer Al and Farmer Yumi tie the knot, whilst Dora devotees can finally see the Grumpy Old Troll find happiness!\nThe Grumpy Old Troll is getting married, and must find all the things he needs in time for the wedding, in this clip from the series Dora the Explorer.\nSing, and rap, along with Dora and the Hip Hop Bunny in this Easter egg sing along from the series Dora the Explorer.\nIn this game, help Dora perform her skating routines, choose her moves on the right and left panels, then click done to watch her perform!\nXEgg-splore Easter with Nick Jr.\nDora and her friends take part in the Easter Bunny's egg hunt in this clip from the series, Dora the Explorer.\nDora, Boots and their friend the hip hop bunny's Easter hats are blown away! How can they get them back? Find out in this clip from the series, Dora the Explorer.\nDiscover surprises when you sing along to the classic Nursery Rhyme, If You're Happy and You Know It. Skye and Chase from PAW Patrol stomp their feet - try and spot Dora and Boots plus Blaze too!\nIt's clay time! Check out how to clay this crafty Dora the Explorer!\nFind out what Dora needs to make her dad a kite for Father's Day in this clip from the series, Dora the Explorer.\nIn this special video part of the 'I Love Dora' series, kids discover yummy blueberries, bananas...and how good they can be on top of a cake!\nDora and friends are preparing for their Christmas eve party in this clip from the series, Dora the Explorer. It's the best party of the year!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If you take a side-trip from Argentina to Uruguay, a must try food is the chivito sandwich, the national dish of Uruguay. The sandwich is made up of a thin piece of steak, cheese, tomatoes, mayonnaise, ham, a fried egg and often other ingredients including roasted peppers, olives, criolla sauce and mushrooms. To make it just a little more fattening, this it is served with a side of french fries and a beer. It isn't a dish you'd want to eat on a daily basis, but it's delicious and something I indulge in every time I take the ferry to Colonia or Montevideo.\nThe sandwich originated when an Argentine visiting Uruguay ordered goat (chivito) at a restaurant. The restaurant owner did not have goat available, so instead assembled a sandwich with beef and all of the now typical chivito sandwich toppings to make it difficult to tell it was beef and not goat. Word spread about how great the sandwich was, and it started being offered at restaurants throughout Uruguay. Today it is the national dish, and a summer vacation staple when visiting the beaches of Uruguay.\nI've tried chivito at many restaurants and food stands around Uruguay, and conveniently my favorite is located at El Eslabon in Colonia, a popular daytrip from Buenos Aires. Similar to on our food tours, I was attracted to the food cart because it was very busy and filled with 100% locals. It is located just opposite the bus terminal near the ferry terminal. Look for the bright yellow food cart. Two friendly women that know most of their clients by name and are always in a good mood run it. Besides making a delicious chivito, they differentiate themselves by having a wide selection of homemade sauces and toppings.\nTo get a great authentic experience I highly recommend skipping the mostly overpriced and mediocre touristy restaurants located in the old town part of Colonia, and instead visiting El Eslabon for a chivito and a cold local beer.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"CHICAGO -- Parts maker ArvinMeritor Inc. may follow the lead of two competitors and sell off its replacement parts business to focus on core assets, a Wall Street analyst said on Monday.\nMore merger deals have been struck in the auto supply sector this year than in all of 2003 or 2002, Merrill Lynch analyst John Casesa said in a research report dated Friday.\nThose assets have largely been sold to financial buyers, and a divestiture by ArvinMeritor could bring the same bidders to the table again, a source in the auto industry said.\n\"Given the pickup in transaction activity in the automotive supplier sector, and given the interest of at least two private equity firms in the automotive aftermarket, we believe it is not unlikely that ArvinMeritor will explore a divestiture of its light vehicle aftermarket business,\" Casesa wrote.\nPrivate equity firms Carlyle Group and Cypress Group have shown a significant interest in the replacement parts sector, also known as the aftermarket, he said.\nNew York-based Cypress, which recently agreed to buy the aftermarket business of parts maker Dana Corp. for about $1.1 billion, may hold a golden ticket in the case of an ArvinMeritor auction.\nArvinMeritor's light vehicle aftermarket business, which makes exhaust systems, shock absorbers and other products, could mesh well with Dana's replacement parts business, the auto industry source said -- meaning Cypress might have the chance to combine the businesses of two bitter rivals under its ownership.\nDana successfully fought off a $2.7 billion unsolicited takeover attempt by ArvinMeritor last year, and put its aftermarket parts business on the block just weeks later.\nWashington, D.C.-based Carlyle, also an active shopper in the sector, bought the auto parts business of New Jersey-based UIS Inc. -- which also makes candy bars and patio doors -- in May 2003.\nMerrill's Casesa expected the auction for another asset, the parts business of Cooper Tire and Rubber, to wrap up \"in the coming weeks.\" That business may be worth $1.2 billion to $1.3 billion, he said.\nCasesa believed that ArvinMeritor's light vehicle aftermarket unit would be valued at $450 million to $500 million. The business is currently battling declining volumes and weak pricing.\nA spokeswoman for ArvinMeritor, which recently named Charles McClure as chairman and chief executive to replace the retiring Larry Yost, declined to comment on the Merrill Lynch report.\n\"We don't comment on speculation, but we are always evaluating our businesses,\" ArvinMeritor spokeswoman Krista McClure said.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Separation and free radical-scavenging activity of major curcuminoids of Curcuma longa using HPTLC-DPPH method.\nEffect-directed analysis of fresh and dried elderberry (Sambucus nigra L.) via hyphenated planar chromatography.\nSimple analytical strategy for MALDI-TOF-MS and nanoUPLC-MS\/MS: quantitating curcumin in food condiments and dietary supplements and screening of acrylamide-induced ROS protein indicators reduced by curcumin.\nFree radical scavenging activities of polyphenolic compounds isolated from Medicago sativa and Medicago truncatula assessed by means of thin-layer chromatography DPPH\u02d9 rapid test.\nOridonin exerts protective effects against hydrogen peroxide\u2011induced damage by altering microRNA expression profiles in human dermal fibroblasts.\nAnalysis of curcuminoids in food and pharmaceutical products.\nDevelopment of chromatographic and free radical scavenging activity fingerprints by thin-layer chromatography for selected Salvia species.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Over the years I've had my share of sellers object to a realistic listing price, and I wanted to share them with you. It's important to remember that the listing price of the home is determined by a variety of factors, and that your first price should be your best price!\nHere's a look at common objections sellers have to an accurate listing price.\nWhen faced with a listing price that is lower than what they had hoped, sellers often feel like they are in fact giving their house away. While that is certainly not the case, sellers need to be reminded that their home is only worth what a buyer is willing to pay for it in the current real estate market conditions.\nThis approach can be detrimental to the length of time a home will be on the market. Your first price should be the best and most accurate price.\nWhile it can be tempting to FSBO (for sale by owner) to save on paying out a Realtor's commission, did you know that over 80% of the for sale by owner sellers end up hiring a Realtor anyway after unsuccessfully trying to sell on their own? That means time and money spent before starting the process over.\nCurrent market conditions will be the determining factor here. It doesn't matter what you paid for your home, you'll only be able to sell it for what a buyer is willing to pay for it.\n\"I did a lot of upgrades in this home and that cost me a lot of money\u2026\"\nThis is the objection I dislike the most, only because my heart goes out to the seller who is so proud of the home improvements and upgrades they've made, without realizing they may not recoup the cost come selling time. It's important to do the proper research to be able to avoid over-improving your home if you'll be selling in the future.\nWhen you are ready to sell your home, give me a call for a free consultation. We will go over your goals and expectations and talk about how my services can help you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"ikea cubby shelf interesting storage storage shelf ikea cubby shelf black.\nikea cubby shelf hack wood wrapped ikea cubby shelf black.\nikea cubby shelf bookcases ikea cubby shelf baskets.\nikea cubby shelf bench with via ikea cubby shelf black.\nikea cubby shelf wall cubes large size of display shelves target bookshelves floating wall cubes mounted shelves ikea cubby shelf black.\nikea cubby shelf wall ikea cubby shelf baskets.\nikea cubby shelf cube shelves wall shelf cube shelves ikea cubby shelf black.\nikea cubby shelf need somewhere to store your files but have limited storage space this inexpensive project tucks away those files in popular ikea cubby shelf baskets.\nikea cubby shelf storage gallery of the most fresh of storage storage shelf ikea cubby shelf black.\nikea cubby shelf shelves loft storage co co bookcase box shelves ikea cubby shelf black.\nikea cubby shelf before after artful bookshelf modification ikea 16 cubby shelf.\nikea cubby shelf storage playroom shelving awesome playroom storage shelves striped bins land storage storage ikea cubby shelf black.\nikea cubby shelf shelf wall cubes medium size of witching living room storage floating shelves wooden shelf ikea cubby shelf baskets.\nikea cubby shelf storage white book shelves storage for sale in ca storage ikea cubby shelf black.\nikea cubby shelf storage storage shelf unit inspirational cube storage bench seat storage storage bench ikea cubby shelf baskets.\nikea cubby shelf room divider shelves pleasant best ideas about shelves on ikea cubby shelf baskets.\nikea cubby shelf storage white new wall mount cube display shelf set of 3 shelves sky bins ikea 16 cubby shelf.\nikea cubby shelf storage bins shelves medium size of storage bins multi purpose shelving black shelf home ikea cubby shelf baskets.\nikea cubby shelf cube shelf best cube shelves ideas dressing table pertaining to cube bookcase wall shelf ikea cubby shelf baskets.\nikea cubby shelf shelving unit hole bookcase hole shelves what a difference hole shelves target hole shelves hole shelving ikea cubby shelf baskets.\nikea cubby shelf my so called blog doors for shelf ikea 16 cubby shelf.\nikea cubby shelf s a photo on com hole shelves ikea 16 cubby shelf.\nikea cubby shelf shelf bookcase bookcases hole handmade wooden pigeon storage unit shelf shelf shelf ikea 16 cubby shelf.\nikea cubby shelf storage shelf unit awesome storage ikea cubby shelf baskets.\nikea cubby shelf ikea 16 cubby shelf.\nikea cubby shelf create storage on an upper closet shelf using lack shelf ikea cubby shelf black.\nikea cubby shelf storage ikea cubby shelf baskets.\nikea cubby shelf wall cubes bookcase medium size of storage bin unit bookf canvas floating ves ikea 16 cubby shelf.\nikea cubby shelf astonishing storage storage bookcase storage bench with shelves storage bins ikea cubby shelf black.\nikea cubby shelf 8 cube shelf great shelves black you can contact us by club cube shelf black cube dimensions ikea cubby shelf baskets.\nikea cubby shelf shelves medium size of shelves door shelving units for baskets target ikea cubby shelf baskets.\nikea cubby shelf shelves wall wall mounted shelves unique wall mounted storage products full wall shelves ikea 16 cubby shelf.\nikea cubby shelf ikea cubby shelf baskets.\nikea cubby shelf desk ikea cubby shelf black.\nikea cubby shelf storage bins white unit shelves shelving cube wall ikea cubby shelf baskets.\nikea cubby shelf white shelf shelves image ikea cubby shelf baskets.\nikea cubby shelf furniture cube storage cubes makeover and shelf cu bins boxes ikea cubby shelf baskets.\nikea cubby shelf hole storage hole storage hole shelf small storage ideas for under stairs cupboard in shelves plans hole storage hole storage ikea cubby shelf black.\nikea cubby shelf shelf shelf with shelf shelf black ikea 16 cubby shelf.\nikea cubby shelf see of the best hacks ideas and the different ways you can ikea cubby shelf black.\nikea cubby shelf shelves s hole target 6 shelf storage ikea 16 cubby shelf.\nikea cubby shelf wall cubes cube shelf large size of lovely shelves for your decorative wall cubes shelving ikea cubby shelf black.\nikea cubby shelf ikea cubby shelf black.\nikea cubby shelf storage storage unit storage unit full size of 4 cube storage unit cube storage ikea cubby shelf baskets.\nikea cubby shelf wall cubes wall shelf paper wall shelf with baskets wall shelf wall ikea cubby shelf baskets.\nikea cubby shelf 3 shelves from stacked on top of each other bottom 2 baskets to hide miscellaneous items other shelves to hold books pictures ikea cubby shelf black.\nikea cubby shelf storage bins storage bins shelf unit doors storage bins ikea cubby shelf black.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We've done the hard yards for you.\nAh yes, the ever more common 'compact luxury SUV.' Your go anywhere, do anything, park in any urban car spot kind of whip.\nJaguar has just delivered a very sharp and attractive offering into this compact SUV market, taking the sophistication and performance out of their sports cars and putting them into a five-seater, family friendly all-wheel-drive vehicle. If you're here for in-depth driving impressions, my colleague James' stint in the E-Pace on the phenomenal roads of Corsica will satisfy your needs more than adequately. I'm about to give you something a little different.\nAs with any Jaguar Land Rover product, however, customisation is a core pillar of the brand that may be confronting for many. The E-Pace, for example, has 38 different variants, and that's before you even start ticking boxes. To take this daunting task out of your hands, we're going to simplify the process for you.\nThe range starts at $47,750 before on roads and soars all the way to a staggering $84,370. If you cop all the bells and whistles on the top-line variant, you'll be at the upper limits of a spectrum that is bookmarked by a $35k difference.\nYou're not buying this thing for the Dakar Rally or to drag race out at Eastern Creek every Wednesday, so you should consider your compact SUV diplomatically. You can get a pretty mean looking E-Pace built around a lower-specced model for considerably less than the top line variant - and you'll still have some coin left in the bank.\nBelow we're going to show you how to maximise your money, by only tacking on the necessities and saving yourself some coin in the long run.\nAll this talk about options might be making you a little dizzy, so let us take you step-by-step through the lineup. Firstly, you need to choose a base model, by this, we mean your blank canvas with pre-set exterior features and internal specifications.\nThe 'E-Pace' is the entry-level, relatively simple, squashed and sporty SUV. It doesn't, however, exude many vibes akin to the daily Boss Hunting reader. Up a notch, you'll find a common Jaguar staple, the R-Dynamic, adding a twist of sports car flare to the compact SUV. Observe the subtle differences below.\nLastly, for the 2018 model year only, you have the option of a Jaguar E-Pace First Edition. The fully-kitted out whip is your holy grail of E-Pace variants, maxing-out the price scale at that ludicrous $84k we spoke of earlier.\nFor the purpose of this article, we're making the E-Pace R-Dynamic our guinea pig.\nFor a full breakdown of drivetrains available for the E-Pace, head here. We're going to cut the BS for you and conclude that Jaguar's 'D180' all-wheel-drive, Ingenium engine is all you'll need. This is just one notch above the base offering, the D150, and still produces an admirable 132kW of power from a 4 cylinder, 2.0-litre engine.\nThe 'D' in 'D180' stands for diesel. By taking the diesel option, you'll save yourself a hell of a lot of money on fuel, an expense that can't really be justified for such a car if it were a petrol engine. At 6.5-litres per 100km, the D180 gets the job done in a very satisfactory manner.\nFor those unfamiliar, JLR offer specifications by way of S, SE or HSE pre-sets. We'd recommend you opt for the 'S' pack to then work your magic around.\nWhile the E-Pace sports some striking colours, we'd suspect you'd be keen on a colourway similar to the RR Velar we had last year. The 'Coris Grey' body looks really sharp; throw on a Black Pack, a fixed panoramic roof for some airy ambiance and elusive privacy glass, you'll be well on your way to a beefy, tough-looking E-Pace. What I think detracts the most from the E-Pace is its stock wheels. We'd swap out the standard 18 inch set for 20 inch, 5 split spoke 'Style 5051' wheels.\nTo bring the interior up to scratch with the now striking exterior, opt for the ebony grained leather with a sharp electric blue contrast stitching. We can assure you this looks even better in the flesh. We'd also suggest the 11-speaker Meridian sound system as opposed to the standard.\nFor the driver, a digital interactive display brings your E-Pace into the next generation. By ditching the analog system you'll have a truly luxurious, high-tech and head-turning set of wheels. The E-Pace is also only the second Jaguar to have ever had towbar compatibility, so if you're after a true family weekender, chuck that on also.\nWe've done the hard yards and put all of the above options into an actual E-Pace S R-Dynamic for you, the finished product you can preview below. For the full specification details, and to design your own, head here.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzazthf b/data_all_eng_slimpj/shuffled/split2/finalzzzazthf
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@@ -0,0 +1,5 @@
+{"text":"SACRAMENTO ROUNDTREE Condos For Sale in SACRAMENTO | Sacramento Condo Mania!\nNicely manicured grounds at the Sacramento Roundtree. Enjoy the park-like atmosphere, clubhouse, swimming pool, and covered parking.\nWant to sell your condo in SACRAMENTO ROUNDTREE?\nOver the last 180 days there have been no sales in SACRAMENTO ROUNDTREE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Juniors were unfortunately beaten by Cremartin 1-21 to 1-8 on Friday.\nThe Div 5 Team were defeated by Killeevan last Thursday \u2013 they now have a break for a couple of weeks.\nThe Div 1 Team game against Carrickmacross away on Sun 1st July was postponed.\nOur Senior Ladies have their first game of the Championship against St Bridgets this Sunday at 3pm in Scotstown. Please try and get to the game and give the ladies as much support as we can.\nThere's a great turn out at all the walking groups each day \u2013 well done to you all. Don't forget the Ned Run is coming up in a few weeks (Friday 3rd August !!) \u2013 its a great opportunity to get a little fitter before it.\nWell Done to our County Lads on their victory over Leitrim last Saturday and the best of luck this Sunday against Laois !\nThe shop will be closed this Friday 29th and will only be opened on the following dates in July & August: 6th & 20th July and 10th & 24th August. The time will remain 7-8pm.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Do you keep a diary or a journal? Do you remember any entries that stand out?\nOn February 3, 1880, Theodore Roosevelt wrote in his diary: \"... I drove over in my sleigh to Chestnut Hill, the horse plunging to his belly in the great drifts, and the wind cutting my face like a knife. My sweet life was just as lovable and pretty as ever; it seems hardly possible that I can kiss her and hold her in my arms. \"\nWho was Roosevelt's \"sweet life?\"","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Sad to report the passing of retired Police K9 Diesel.\nDiesel served the Lenexa Police Department for eight years, retiring in 2015. During his career he was deployed 523 times, making 63 drug finds and seizing over $77K in cash. K9 Diesel also made apprehensions of 35 felony criminal suspects. K9 Diesel and his handler, Corporal Sumner placed numerous times in National Police K9 competitions, including a 1st place finish in 2009. After retiring, K9 Diesel continued to live with Cpl. Sumner and his family up until his passing. Our Police K9s work to serve and protect our community just as much as our officers do.\nWe will always remember K9 Diesel's contributions to our department.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Jefferson physicians treat over 600 brain tumors a year \u2014 more than all other university-based hospital programs in the area combined. With their innovative treatments, offered at the Brain Tumor Center of Jefferson's Kimmel Cancer Center, they have raised the standards of care worldwide for treatment of brain metastases, gliomas and optic sheath meningiomas.\nOur multidisciplinary approach pulls together specialists in radiation oncology, neurosurgery and medical oncology, so that patients can be evaluated and receive an individualized treatment plan in one place in one visit to one place. This is important because every day counts when treating this life-threatening diagnosis. Our neurosurgeons are among the most experienced in the country, and are often able to provide effective treatment for tumors that cannot be treated elsewhere.\nJefferson offers noninvasive treatment options for brain tumors that are the most advanced in the Delaware Valley. Stereotactic radiosurgery (SRS) is the application of a single large radiation dose to a tumor. The delivery systems are the gamma knife and the stereotactic linear accelerator (LINAC). Advanced imaging systems such as computerized axial tomography (CT scan) and magnetic resonance imaging (MRI) ensure pinpoint accuracy. SRS is a highly effective treatment for benign and malignant tumors and lesions of the brain and spine. It is delivered by radiation oncologists in collaboration with neurosurgeons from the Jefferson Hospital for Neuroscience, who were instrumental in its development.\nIntensity-modulated radiation therapy (IMRT) is another modality that delivers radiation to a tumor with precision and control while sparing healthy tissue. It facilitates the delivery of higher doses of radiation. The image-guided robotic linear accelerator combines high-conformance beam shaping with exclusive 4D adaptive image-guided radiation therapy (IGRT) technology for precise radiotherapy treatments using cone beam CT, especially for advanced stereotactic radiation treatments.\nOur volume of surgical (both transcranial and endoscopic) and radiosurgical (both single and multiple-fraction) approaches is unrivaled in the tri-state area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbairf b/data_all_eng_slimpj/shuffled/split2/finalzzzbairf
new file mode 100644
index 0000000000000000000000000000000000000000..e87f9ef8cdd23babb4d03a769a1168d3ff60e0f1
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbairf
@@ -0,0 +1,5 @@
+{"text":"All around shredder, Kohei Kudo, just sent over this montage that he made of all his good homies that he filmed from this season. It's featuring Kohei Kudo, Shuhei Sato, Teddy Koo, Kazu Kokubo, Keisuke Yoshida, Daisuke Murakami, Fumiyuki Murakami, and Louif Paradis. Check it!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Microsoft also announced Microsoft Band 2, their latest fitness tracker. The wearable boasts a curved screen, and newly added sensors make it more powerful than its predecessor.\nIn addition to upcoming UP4 fitness tracker, Jawbone has also released UP2, an affordable fitness tracker with a $99.99 price tag in order to replace their UP24 smartband.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This list shows the peaks climbed by Brett Moffatt, with one entry for every summit attained, by whatever means. Attempts and simple hikes are not shown. Only the first ascent date is listed for peaks climbed more than once. Sortable columns of information about each peak are also provided, including elevation, location, prominence, isolation, and high point status.\n9. BM Mauna Loa 2019-03-29 13,661 USA-HI 0 ... ...... .\n17. California\/Nevada State Border High Point 2012-07-20 12,880 USA-CA 0 ... ...... .\n42. Crater Lake Peak (9750) 2017-06-26 9750 USA-CA 0 ... ...... .\n84. Baldy Ski Hut 2017-10-31 8100 USA-CA 0 ... ...... .\n108. Yosemite Falls Overlook 1984-11 6700 USA-CA 0 ... ...... .\n119. Point 5787 2017-11-01 5787 USA-CA 0 ... ...... .\n191. McGuire Peak 2019-01-24 2758 USA-CA 0 ... ...... .\n198. Three Peaks 2019-01-24 2675 USA-CA 0 ... ...... .\n205. Peak 2620 2019-02-18 2620 USA-CA 0 ... ...... .\n219. Centennial Peak 2019-01-25 2443 USA-CA 0 ... ...... .\n245. Baldy Mountain 2019-04-14 2114 USA-CA 0 ... ...... .\n264. Peak 1920 2019-04-14 1920 USA-CA 0 ... ...... .\n276. Peak 1840 1996-07 1840 USA-CA 0 ... ...... .\n292. Peak 1660 2019-03-24 1660 USA-CA 0 ... ...... .\n306. Peak 1520 2019-04-06 1520 USA-CA 0 ... ...... .\n331. Bald Hill 2019-04-06 1275 USA-CA 0 ... ...... .\n346. Griffith Observatory 2017-11-04 1135 USA-CA 0 ... ...... .\n357. Cloud's Rest 2017-11-03 1060 USA-CA 0 ... ...... .\n360. Inspiration Point Unknown 1040 USA-CA 0 ... ...... .\n448. Lands End Unknown 85 USA-CA 0 ... ...... .\n451. South Point - Big Island of Hawaii - Southernmost Point of Land in 50 states 2019-03-30 0 USA-HI 0 ... ...... .","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To make the sofrito, first heat olive oil in a sauce pan and add chopped garlic, grated tomatoes, grated onion and bell peppers. Saut\u00e9 well. Add smoked paprika powder, saffron and salt. Simmer on low heat and cook until all the vegetables are softened.\nWash, clean and soak the rice in water for around 20 minutes.\nHeat 10 ml olive oil in a saucepan and saut\u00e9 5 g chopped garlic until golden brown.\nAdd half the sofrito mixture and saffron, and stir gently. Add soaked and drained rice, 200 ml vegetable stock and cook until the rice is 3\/4th done.\nTake the pan off the heat and ladle the rice onto a cooling tray, drizzling with some olive oil.\nIn another pan add 10 ml oil, 10 g chopped garlic, the remaining sofrito mixture, broccoli, green zucchini, red and yellow bell peppers, edamame beans, snow peas, green peas and smoked paprika powder. Saut\u00e9 well.\nThen add the cooked rice and remaining vegetable stock.\nOnce until the stock is fully absorbed, check for seasoning and serve hot.\nIn a heavy-bottomed pot, add vegetable stock, shitake mushroom stock, Kaffir lime leaves galangal juliennes and a hint of soy.\nAdd the broccoli florets, beans sprouts, pak choy and thinly sliced shiitake mushrooms. Toss in the fresh red chillies and allow the broth to simmer until the vegetables turn tender.\nDust the tofu with the togarashi spice mix and sear over medium heat. Set aside.\nQuick toss the cooked rice with fresh green scallion and pink ginger juliennes.\nSprinkle sesame seeds and add a few drops of sesame oil.\nTo assemble, mount the rice on to a soup plate. Arrange the poached vegetables around the rice.\nReheat the broth and adjust seasoning before into a sauce boat.\nPlace the seared tofu on top of the rice and just before serving, pour the broth over the tofu and into the soup plate.\nHeat oil in a sauce pan and add chopped onion and garlic. Saut\u00e9 for two minutes.\nAdd green and yellow zucchini along with broccoli, baby potatoes, mushrooms, carrots and bell peppers, and mix well.\nWhen the vegetables are half cooked, add vegetable stock, cream, Parmesan, cheese and butter.\nSeason with salt and pepper and cook until the stroganoff is done.\nHeat another pan and add butter, boiled rice, salt and pepper. Toss well until the rice turns piping hot.\nTo assemble, place rice on one side of the dish\/plate and spoon stroganoff on the other side.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Trust's visiting times policy has been developed to ensure that you have enough rest and recuperation and to help prevent the spread of infection.\n\u2022 Visitors are asked to wash their hands\/use the alcohol hand rub on entering and leaving the ward.\n\u2022 Restrict visitors to a maximum of two at any one time.\n\u2022 Patients are encouraged to have a clear locker top to allow surfaces to be easily cleaned.\n\u2022 Don't bring unnecessary items, flowers or food into the hospital.\n\u2022 Children under the age of 13 should be accompanied by an adult.\n\u2022 Visitors are asked not to sit on the bed or touch wounds, drips or drains.\n\u2022 People are urged not to visit if they or someone in their household has symptoms of cold, flu, vomiting or diarrhoea.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbark b/data_all_eng_slimpj/shuffled/split2/finalzzzbbark
new file mode 100644
index 0000000000000000000000000000000000000000..af8d366d68c641099351d67eeb50a8a1e8be3868
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbark
@@ -0,0 +1,5 @@
+{"text":"Our Lacquered Box features Mt. Fuji & Japanese style cranes, the cranes shown on the box are Japanese cranes also known as red-crowned cranes, and depicts \"The Dance of Love\" which these cranes do to find a partner. Our box is three tiered and every tier is removable for easy cleaning. The box features three compartments, making an absolutely exquisite way to prepare your lunch, or to keep on your desk as a beautiful decoration.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"9to5Mac has received some hands-on photos of what it claims to be a dummy unit of the upcoming iPhone 7s Plus, the successor to the iPhone 7 Plus that will be announced alongside the iPhone 7s and bezel-less iPhone 8. According to the images, the phone will receive a pretty major design upgrade with glass on the front and back of the device over the previously used 7000 series aluminum. It also appears there will be some type of metal frame surrounding the edges of the device, although this is just a guess based off how these pictures look.\nSince the backs of the iPhone 7s and 7s Plus will be glass, Apple is expected to include wireless charging in their entire lineup for 2017. Current reports suggest the feature could be delayed until a future iOS 11 update arrives, but it's unclear just how legitimate these rumors are.\nThat being said, 9to5Mac does seem pretty confident in a couple of key areas surrounding the next-gen iPhone's specs. According to their report, the iPhone 7s and 7s Plus will sport the Apple A11 SoC (system-on-chip) and a minimum of 64Gb of storage. This would double what's currently available with the iPhone 7. The phones are also expected to begin shipping in mid-September which seems highly likely given the iPhone's history.\nWe're still a month away from seeing the devices officially, so expect more leaks like this to hit your newsfeed in the near future. No doubt we'll eventually hear more about the futuristic iPhone 8 as well.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Interior Images is a contract furniture dealer.\nDedicated in selling you furniture, window coverings, and other selections balanced to your interior design.\nAssisting you in selecting products chosen to achieve your functional requirements.\nMeeting your budget while achieving stellar top line results through design, quality, and functionality.\nOffering leasing options as well as direct purchase.\nValue added dealer services and products which will make you want to buy from Interior Images time and time again.\nInterior Images offers a wide variety of window coverings including some residential selections.\nCustom automation and child safe cordless options.\nInteior Images is a trend setter selling only the best products properly installed.\nSchools, Colleges, Medical, Government and Industrial use specific products.\nA\/V Carts, Video Conferencing, Adaptive Furniture, Art, Waste Recycling, Planters, and Benches are among the many unique products available.\nCreative and Reflective Interior Solutions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Thank you to one of our lovely customers for sending in this amazing snap of their Mermaid Photo Prop station. They have framed the prop sign and popped this next to a vase full of the party props. Perfect for any 'Mermaid' and 'Under The Sea' themed birthday party\u2026.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Robinson's Madrox site worked with the Tomil brand Dr Devil to manufacture a 60ml segment shaped bottle used for bathroom cleaning.\nThe bottle is designed to fit under the rim of the toilet, and release fragrance with every flush.\nDue to broad width and narrow depth, this pack is technically challenging to manufacture.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbbvnv b/data_all_eng_slimpj/shuffled/split2/finalzzzbbvnv
new file mode 100644
index 0000000000000000000000000000000000000000..3a28a9c9b19d3facf9e6da4bb1d8e24ff3a1b441
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbbvnv
@@ -0,0 +1,5 @@
+{"text":"Audra Jennings: Preview Triple Dog Dare Devotionals - for boys!\nPreview Triple Dog Dare Devotionals - for boys!\nAttention radio stations: Giveaway copies are available for interviews and on-air reviews! Email Audra to request yours!\nJeremy V. Jones is an award-winning journalist who has served as senior associate editor of Breakaway magazine. He has authored several books, including Toward the Goal: The Kaka Story and The Keeper: The Tim Hoard Story. He also writes for magazines such as Clubhouse and Christianity Today. He resides with his wife and two children in Colorado.\nBoys want action. They don't want to sit around and talk\u2014that's for grown-ups and girls. They engage life and relationships by doing something: skateboarding, playing games or re-creating favorite movie scenes. So why should faith be any different? That's why Jeremy V. Jones created Triple Dog Dare: One Year of Dynamic Devotions for Boys\u2014to provide the action boys need in order to grow their faith.\nThe Bible is full of action. Remember how David slew Goliath, Daniel faced those lions, Paul survived a shipwreck and Jesus stood up for a woman about to be killed? God made boys to take His truth and do something with it, to man up and change the world. These action-packed devotions for boys ages 9 to 12 are filled with godly truth and bold spiritual challenges that transform time with God into the adventure of the day.\nTriple Dog Dare connects God's Word to boys' hearts and hands with real-life scenarios and activities. Each day is filled with short Scriptures, concise biblical truth and a daily dare, all challenging them to put their faith into practice. Scripture readings from every book of the Bible open up the action-packed Word of God. Whether it's drawing comic strips of biblical battles, dreaming up a life list of goals, making snack packs for the homeless or producing Bible-based movies, boys will go on daily dynamic experiences with God, taking faith off the page and setting it into motion. Themes cover the daily realities of pre-teen males, including bullying, peer pressure, girls, sibling rivalry, honesty and more.\nThese exciting devotions will inspire boys' hearts toward godly characteristics such as integrity, generosity and kindness. Parents will appreciate watching Christ-like traits emerge as each dare is undertaken. It is a manual that will deepen boys' friendships with Jesus as they look forward to spending time with Him every day. So if you know a boy who is up for the challenge, triple dog dare him!\nPreview Truth and Dare Devotionals - for girls!\nIn need of some Inspired Design?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Congratulations! You've now completed the Genius Zone Profiler. We'll be using this information as we go along to make sure you have the right advice and information, so for now you can go make yourself a cuppa and congratulate yourself on a job well done!\nYou will not be able to return to this page in the future, so save them to your computer straight away and if you want to, you can of course, print them out too.\nYou will also be emailed the documents, so make sure that you regularly check the email you used to set up your membership.\nPlease 'whitelist' info@Genius-Material.uk to make sure you receive Genius Material bonus information and opportunities coming your way.\nGood luck with your revision. Keep in touch and tell me your success stories.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Pay for your purchases when it's right for you. Select a valid country. Skip to main content. Open Source Consulting Domain Registration. There are 25 items available.\nWatch list is full. See terms \u2013 opens in a new lsi logic pcbx518-b1 or tab Other offers may be available from time to time. Lgoic to home page. Get more time to pay.\nEmail to friends Share on Facebook \u2013 opens in a new window or tab Share on Twitter \u2013 opens in a new window or tab Share lsi logic pcbx518-b1 Pinterest \u2013 opens in a new window or tab. Find More Posts by jefro.\nThank you for great service! It was the Rehook Int19h option in the bios! Search this Thread Advanced Search. Subject to credit approval. Having a problem logging in? Thank you from EWay hardware company in Taiwan. New other see details: No reviews left yet. Choose PayPal Credit to pay all at once or enjoy the flexibility of paying over time with special financing offers.\nHave one to sell? What is the pci slot key combo to access lsi? Get an immediate offer. Lsi logic pcbx518-b1 to main content. See all condition definitions \u2013 opens in a new window or tab Enjoy more time to pay Pay for your purchases when it's right for you. Perhaps even more impressive was how quickly the item arrived in Australia.\nLsl more advanced trainees it can be a desktop reference, and a collection of the base knowledge needed to proceed with system and lotic administration. This item will ship lsi logic pcbx518-b1 United Statesbut lsi logic pcbx518-b1 seller has not specified shipping options. They hope these examples will help you to get a better understanding of the Linux lsi logic pcbx518-b1 and that you feel encouraged to try out things on your own.\nAre you new to LinuxQuestions. Covers your purchase price and original shipping.\nSelect a valid country. Shipping cost cannot be calculated. Learn more \u2013 opens in a new window or tab.\nFor additional information, see the Lsi logic pcbx518-b1 Shipping Program terms and conditions \u2013 opens in a new window or tab. The servers use the pci-x bus for the internal raid usually so we'd have to suspect the bus is OK. Very Good Average Fair Poor.\nPay for your purchases when it's right for you. BB code is On.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"It's Truthful Tuesday, where I answer your Q's from emails you've sent or our live calls. \"Truth Ain't Easy\" was the name of my first blog 10 years ago & I still love it because it still applies. I thought the name was sooo cool HAHA!!\nTruth AIN'T easy. It's not \"nice\" and it is confronting.\nBIG ONE! Watch today's video to see how I kick this thinking to the curb: your results depend ENTIRELY on your ability to swiftly move through thoughts & spend your energy on what's actually under your control.\nI share 5 ways to drop this thinking so that you actually have brain space to CHANGE the situation, not just keep worrying about others or wishing the situation was different. Watch the video now.\nNEW LIFE & BUSINESS TRAINING PROGRAMS COMING UP!\nFor the last five years, I only talk about the life and business training programs I run when I launch them once a year. Yep, once a year because it takes TIME to build the right way; it doesn't happen overnight. The training is foundational to understand how success, energy, wealth, & results work in your life. From now on, you'll get regular updates on programs should you want to join, want more information or talk to us.\nEach month I do a Q&A Call on your biggest Q's, \"How do I make more money?\" \"How do I stop reacting with anger?\" \"How do I get over my fear of rejection?\" It's my favorite night of the month, no joke. I share my personal mistakes with you so you can avoid them. It's a great way to get unstuck and it's FREE.99. Next Blazing Q&A: Weds. April 11th at 7pm. Join here.\nThe foundational life and business training I wish I had when I started my career 10 yrs ago. This 8-month program is a step by step program to train you only on the MOST important business and life skills to build a successful foundation. Each month dive deep into one life \/one biz topic. Weekly live calls, Q&A calls, 3 live events this year in NYC and expert mentors\/friends to teach you. I've spent this quarter re-designing this program to give you MORE value, more focus, more coaching, so I'm tres EXCITED. I'm keeping it very small so you get lots of time and personal attention, and it's already partially filled. More details about it coming in next two weeks.\nMasterclass I: Get Out of Survival Mode: How to Stop Squandering your 3 Biggest Currencies in Life: Energy, Time, and Money and Feel In Control Instead.\nMasterclass II: Stop Adding Struggle to Your Business: How to Master the 3 Key Tools of your Business to get the Most Results.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brian T. Gongaware is a Principal of McKinley Carter and serves as Director of Advisory Services. Brian also leads our Pittsburgh office and serves as the firm's Chief Compliance Officer.\nPrior to McKinley Carter, Brian was the President and Senior Relationship Manager of NetWorth Investment Advisors, Inc., where he was responsible for managing the firm, new business development, and served as a wealth advisor\/client relationship manager.\nBrian has a bachelor's degree in financial planning from Grove City College and an MBA in Finance and Management from Duquesne University. He is also a CERTIFIED FINANCIAL PLANNER\u2122 professional.\nFor decades wealthy and mass affluent families have worked with major bank and brokerage institutions. With changes in the financial industry, such families may be better served exploring a superior relationship model. Learn more about our fiduciary service model for clients.\nThe resources of a nonprofit organization are vast. If you are an ED, CFO, or Board Member, numerous stakeholders are dependent on your efficient allocation of those resources to fulfill your mission. Explore how a professional financial advisor can help you manage those resources.\nPreparing for a lifelong financial journey requires an understanding of the possibilities, potential resources, and identification of contingency plans. The journey likely will require navigating a series of obstacles through varying conditions. You don't have to sacrifice a Good Life today for a Good Life solely in retirement. How can we live a Good Life now and in the future? Read about the six critical actions required to equip yourself for a Good Life, financially.\nOur financial life can begin with the feeling of being in over our head. Here's how to learn new skills and get comfortable making financial decisions.\nWe don't spend much time teaching our family members about money and good financial decision-making. Here are some successful methods for all generations.\nDid You Know Baseball and Planning for Retirement Have a Lot in Common?\nRetirement planning can be an enjoyable experience, when you focus and make needed corrections along the way when life throws you those curve balls.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdrrk b/data_all_eng_slimpj/shuffled/split2/finalzzzbdrrk
new file mode 100644
index 0000000000000000000000000000000000000000..cb3a722e79b74ef1f45fe326cb3d179182fa0f5b
--- /dev/null
+++ b/data_all_eng_slimpj/shuffled/split2/finalzzzbdrrk
@@ -0,0 +1,5 @@
+{"text":"The combination of two described above approaches of the online free roulette game is offered by the variant with only one zero segments and the English language as the only one applied to la roulette online free the performance.\nThere already exist a great many of varied strategies bringing success.\nBlack Widow je vhern\u00edm automatem pvodn od vvoj\u00e1e vhern\u00edch automat High 5 Games.\nThe participant should divine the color and the number of the cell at which the ball will stop after as soon as the wheel ceases to turn.The play acquired the title of the Queen of casino.The cells are indexed from one to thirty-six in red and black.There are two most believable hypothesis one of which is connected with France and another one with China.This is the play of fortune and no one can be confident in the result of the gambling session.\nThe play is a great demand request now due to the simplicity of rules and the variable betting range types.\nMultiplayer Roulette Diamond Edition, french Roulette, american Roulette.\nThe numbers are distributed without any order but the colors of the cells interchange starting from number one which is red.\nThe Craziest and the long awaited Las Vegas Style VIP Roulette is finally here!Never participate with real money being engaged before practicing.Ninety centimeters in diameter, the wheel mechanism weights about 40 kilos.Is there anyone who has never fascinated this tune?This game is intended for amusement purpose only.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"\u00b7 Friday: Arrival & Welcome Wine Party!\n\u00b7 Saturday: Scenic Mountain Drive, Exclusive Tour of a fabulous collection of rare American & European cars from early 1900s to the end of the century, Visit to the inner workings of an Anthracite Coal Mine, A Picnic Lunch with local flavor.\n\u00b7 Saturday evening: Annual Reunion Dinner.\nWe are very honored to be invited to tour the fabulous JWR Automobile Museum in Frackville PA that houses more than 230 cars in an eclectic mix of racing cars, everyday cars, famous cars, commercial vehicles, one-off coachwork and intriguing technical designs. The collection was envisioned by Jack Rich, a successful coal industry executive, who passed away in 2011. The private museum continues to be kept up by his family. Most marques are represented from Ford & Cadillac to Bugatti, Rolls Royce & Avions Voisin. Our visit should encourage them to include a few more Lancias.\nMORE DETAILS TO COME VERY SHORTLY INCLUDING ACCOMODATIONS AND DIRECTIONS.\nIf you are thinking to join us, please send me a quick email with the number in your party at richardlynn@paperskills.net or call me at 973-851-5209. Send no money now \u2013 more details and directions will be available in the official announcement in mid-July.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Tous les pilotes audio n\u00e9cessaires sur les plateformes 64 bits. Jour tous vos pilotes. 1 Intel Chipset Driver 9. The top series provide 10 DAC channels that simultaneously support 7.\n82 ( 64- bit) Date added: August 28th Freeware) ( 133 votes average: 4. This audio driver is required if you plan to connect a microphone or headset to the audio jack. This download provides the Realtek* High Definition Audio Driver for the 3. Download drivers for Windows XP \/ Windows 7.\nIt is in soundcard category and is available to all software users as a free download. Rapide et Stable - cherchez, installez et mettez \u00e0. For use with systems running Microsoft\u00ae Windows 7 mobile graphics, equipped with AMD Radeon\u2122 discrete desktop graphics AMD processors with Radeon graphics. Keeping this driver up- to- date means enjoying the ee Download Realtek High Definition Audio Driver 6. Audio drivers for windows 10 64bit free download - Drivers For Free Gadgetarian 64bit Realtek High Definition Audio Codec ( Windows 7 \/ 8\/ 8. The package provides the installation files for Realtek High- Definition Audio Driver version 6.\n8372 for Windows 10 Fall Creators Update 64- bit ( Sound Card). 14 out of 5) free download 252.\nZip, altek High Definition Audio Driver 2. Pourquoi est- il important d' s drivers \u00e0 jour? The Realtek High Definition Audio codecs are compliant with Microsoft' s UAA ( Universal Audio Architecture).\nDriverTuner was created to save your time resolving driver problems by providing you with a single, automatic book Asus Eee PC X101CH. 1 sound playback plus 2 channels of independent stereo sound output ( multiple streaming) through the front panel stereo altek HD Audio Manager is licensed as freeware for Windows 32 bit 64 bit operating system without restrictions.\nIf the driver is already installed on your system add new functions, updating ( overwrite- installing) may fix various issues just upgrade to the available altek audio driver 64 bit free download - Audio Driver Realtek Ver. Download realtek audio drivers 6 0 windows 7 64 bit.\nLe pilote Realtek High Definition Audio pour Windows Vista Windows 7 Windows 8 et Windows 10 ( 64 bits) est compatible avec une grande quantit\u00e9 de cartes m\u00e8res r\u00e9centes. Free download single archive. 1\/ 10 64- bit), RealTek Audio Driver 5.\n1\/ 10 64- bit) many more Realtek HD Audio Drivers x64 2. Auto- Detect Install Radeon\u2122 Graphics Drivers for Windows\u00a9 For Radeon\u2122 Graphics Processors with Radeon\u2122 Graphics Only. Download realtek audio drivers 6 0 windows 7 64 bit. Review; Realtek High Definition Audio Driver is the official release from Realtek, bringing together the driver files needed to activate the audio hardware on your computer. 2 Intel SATA AHCI Driver 10. 5mm audio jack on Intel\u00ae NUC Kit. Analysez votre PC & V\u00e9rifiez si vos Drivers sont \u00e0 Jour. Technical specifications review price.\nZip, Realtek High Definition Audio Codec ( Windows 7 \/ 8\/ 8.\nRealtek High Definition Audio Drivers Free Download Latest Version for Windows 7 \/ 8 \/ 8. Its full offline installer standalone setup of Realtek High Definition Audio Drivers for Windows altek audio manager windows 10 free download - Realtek High Definition Audio Codec ( Windows \/ XP\/ ), Realtek High Definition Audio Codec ( Windows 7 \/ 8\/ 8. 1\/ 10 64- bit), Audio altek PCIe GBE Family Controller \u2013 the driver for gigabit network controllers on the basis of chips from a company Realtek.\nThese network comptrollers are often built- in in system boards. This package of drivers of Realtek befits both for a 32- bit and for 64- bit operating Dell Optiplex 7040 Drivers For WindowsBIT and OS 10, 8, 8.\nyou can through the table list download Dell Optiplex 7040.\nHere you can update Dell drivers and other drivers. Driver DR is a professional Windows drivers download Acer Aspire 4739Z Drivers for Windows 7 ( All version) 32- bit and x64\/ 64- bit edition: 1.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Have it all with our satellite TV packages in Hope.\nThe content you love, direct to your Hope living room.\nPopular TV shows. Local and international news. Sports. Commercial-free movie channels. Whether you want basic programming or premium HD content, we've got what you're looking for. The best part is you'll have the freedom to choose the satellite TV package in Hope that fits into your budget and works for your lifestyle. If your tastes have changed, you can always add or manage individual channels or Theme Packs at any point, so you can get more what you crave\u2014and leave out what you don't. Travel a lot? You'll never miss the big game or fall behind on your favourite show with our FreeRange TV app^ and the option to watch anything on your mobile device.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The halls of our house are all decked out for Christmas, including our 12 foot Christmas tree. Yes we could get a smaller tree, but that isn't the way the DeNikes do the Holidays. Plus we need a big tree to show off the ornaments that we have collected over the years, starting with the ones I was given as a child.\nI have started collections for the boys, so they will have fun ornaments to remember years from now. Each year they get a new one from us and many years receive ornaments from others.\nAs we were decorating our tree, one of my favorite ornaments dropped and broke (Jim has tried to fix, but I am afraid to put it on the tree). I started thinking about my favorite ornaments, who gave them to us, when we got them, etc. So here is my list.\n1. Glass Bell \u2013 My Mom brought this back for me from Germany when she went to the Christmas markets in 2004.\n2. Snowflake \u2013 My friend Dana gave me this a couple years ago. This is the one I dropped.\n3. Mickey Ball \u2013 We brought this home from our trip last week. I will always think about the great trip we had when I see it.\n4. Hummingbird \u2013 I inherited this one when my Granny passed away two years ago. I will always think of her when I see it.\n5. Waterford Egg \u2013 My aunt Denise gave this to Jim and me the first Christmas we were married. It makes me think of a Faberge egg.\n6. Skiers \u2013 Family friends, the Bishofbergers, gave this one to Jim and me when we were newly-married. I love how they are skiing together.\n7. Watermelon \u2013 My Grandpa Harold grew watermelons and one Christmas all of the grandkids were given watermelon ornaments. This one hung on my parent's tree until we got our own where it is displayed prominently.\n8. Bride \u2013 My Mom gave this to me the year we got married. She has a groom that hangs near her.\n9. Golf Balls \u2013 My Mom gave us hand blown golf ball ornaments for our first Christmas together. We always get comments on them.\n10. Trumpet \u2013It reminds me of Jim since he used to play the trumpet.\nI am linking up with Monday Listicles at Northwest Mommy, even though I am not completely following this week's theme and Mama Kat's Writers Workshop.\nThese are beautiful and tell such stories. Love how eclectic your tree is. Must have a lot of ornaments!!\nThey're beautiful! I love that you have treasured ornaments filled with memories, as opposed to just generic ones. Great list!\nThese are lovely. I have all my breakable packed away this year so, my three year old can go near the tree.\nI love looking at ornaments and yours are beautiful!\nI love the Mickey ornament! We are going to Disney World next week\u2026 I'll have to pick up an ornament for our tree! I wish I had thought to buy one when we went last year.\nI love special ornaments. That first one from Germany is great.\nThe hummingbird is beautiful. I love that you know\/remember the story of each. I wish I had that with more than the few Disney ornaments from my childhood (and even those don't have a \"story\", just the memory of once being on the tree).","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbdvvt b/data_all_eng_slimpj/shuffled/split2/finalzzzbdvvt
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index 0000000000000000000000000000000000000000..17bfc0c7b11f2034380061fbd251169f6481ad6f
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+{"text":"After a paper has been accepted or a conference talk has been arranged, you may be asked to provide your academic biography. For publications and conference presentations, you may be asked for a short one of 30 to 50 words. Applying for grants and jobs may require a longer biography between 200 and 400 words. Writing about yourself can be uncomfortable, but there is a way to write an effective biography without feeling like you're boasting.\nDr. June Ava Smith is a professor at the University of Toronto. Her research focuses on identifying and understanding key pathways involved in the development of soybean root nodules. Identification of these pathways may allow non-legumes to be engineered for nodule formation.\n1.\tWrite in formal language and in the third person.\n2.\tUse humor sparingly. Save the comedy for your stand-up act.\n3.\tDo not divulge too much personal information.\n4.\tDo not pack it with all of the information in your CV.\n5.\tAvoid exclamation marks and symbols like emojis.\nRemember that although you're writing about yourself, you're helping others to understand your academic credentials when they're introducing you for a talk or assessing your grant proposal. Let your light shine!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This 6 stranded cotton thread is a premium hand embroidery thread manufactured by DMC. Used primarily for cross stitching and embroidery this thread can be applied to all sorts of crafts. The Coloris range consists of 24 variegated threads designed to add tonal interest to your project. The multi-coloured Coloris stranded cotton combines four complimentary colours in each skein of six stranded cotton. The complete colour change every 5cm creates intricate effects with one thread.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We Buy Houses\u00ae Brownsville. Our real estate investors are looking to buy homes throughout Brownsville and the rest of Texas.\nIn Brownsville and the surrounding area, We Buy Houses\u00ae investors buy homes, condos, land, and all types of real estate, in any condition, and in any price range.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"February 2 2017, Randal Hudson uploads Nice Designs For Walls. The Nice Designs For Walls has been created with briliant ideas, beautiful combined colors and follow trend of printable maps, so the Nice Designs For Walls will give you the real of maps you need . more over The Nice Designs For Walls.\nBy giving the reference about the Nice Designs for Walls Awesome Fresh Metal Wall Art Panels Fresh 1 Kirkland Wall Decor Home Design 0d. Nice Designs for Walls Perfect 30 Lovely Shark Bedroom Decor Smmrs. Nice Designs for Walls Stunning Sticker Salon Design Radioconexionanimal. We also provide examples of maps that makes your travelling more better. Here we will continue to provide updated information on Nice Designs For Walls so you can get the maps on the like. and here's a list of fantastic images or photos from the Nice Designs For Walls as part of a collection artmous a gallery of great ideas and useful.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"If possible, please complete this form prior to your visit to Riverhills. This will save you time on the day, leaving you free to enjoy our facilities for longer.\nNote: The following information is required for your safety and to benefit your health. The following details will be treated in the strictest of confidence. If you are or have suffered from any of the following: epilepsy, embolism, cancer or diabetes, it will be necessary for you to obtain the approval from your GP prior to treatment. Minors under the age of 16 need to be accompanied by an adult for treatments and the consultation must be signed by the main caregiver.\nWould you like to receive offer emails from us?\nWe only send around 1 email per week and you can unsubscribe at any time.\nIf you are currently, or have been, affected by Cancer, please phone Beauty Reception on 01473 463262 at least two weeks prior to your visit.\nMedical history further information, including current treatments and medication.\nIf you have answered YES to any of the above please give additional information here and advise your Therapist, prior to the start of your treatment. Please also note that Riverhills is not a medical centre and you are advised to visit your GP and contact us prior to your visit if you have any concerns regarding your health.\nIs it possible that you may be pregnant?\nIf you will be under 12 weeks pregnant at the time of your visit, please phone Beauty Reception prior to your visit on 01473 463262.\nDo you use products containing retinal A or AHAs (Alpha Hydroxy Acid)?\nHave you had Botox injections?\nHave you had any permanent make-up?\nAre you currently having any forms of treatment?\nIncludes chiropractic, osteopathy, homoeopathy, acupuncture, etc.\nPlease advise prior to treatment if any of the above apply.\nI declare that the information I have given is correct, and as far as I am aware I can undertake treatment without any adverse effects. I have been advised about contra-indications and I am willing to proceed with the treatment.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgjjh b/data_all_eng_slimpj/shuffled/split2/finalzzzbgjjh
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index 0000000000000000000000000000000000000000..4becd6be2b1c00c2d394c1a52f2c4b43d665cedd
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"An indirect tax evasion specialist Leon advises and represents individuals and companies who wish to appeal the decisions of HMRC to withhold input tax or to assess duty claims. He appears before the First and Upper Tier tax Tribunals in cases where financial misconduct (often but not exclusively MTIC fraud) is alleged by HMRC. He brings his significant experience in defending in the criminal courts in tax and duty fraud to these cases along with a full understanding of the significant body of case law that has developed in this area.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Paper Republic Link: Bruce Humes: Can Literary Imports Change Chinese Perceptions of Africa?\nBruce Humes: Can Literary Imports Change Chinese Perceptions of Africa?\nFew avid Chinese readers of fiction can name an African author or novel, and those who do often cite \"Things Fall Apart,\" the highly acclaimed novel by the Nigerian writer Chinua Achebe that portrays the tragic encounter between the Igbo tribe and British colonialism. It was first published in Chinese in the 1960s and has been reprinted countless times since. While literary circles in Africa no longer worship at Achebe's altar, China's literary establishment continues to trumpet him as the reigning \"father of African literature,\" almost to the exclusion of emerging authors.\nBruce Humes, May 7, 2018, 10:58p.m.\nfollowing Kate Hopkins reportage \"The Plight of South Africa's Farmers, As Captured in Photos\"\n-s .ye, May 31, 2018, 5:04p.m.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Brokers gonna broke \u2013 congrats Meagan!\nMeagan Phillips joined us in late March of 2013 as an underwriting administrator utilizing her law clerk degree in reviewing and processing documents both in the claims and underwriting areas. Her passion for underwriting became clear as she started to move through her CIP designation, which is almost complete, and just recently Meagan put another stake in the ground by obtaining her broker's license after a very intense but thorough 2 week prep course. Congratulations Meagan!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I've been following the Scrap Story Twitter account for a while but over time it's really come together. It's basically Ice Climbers meets Paper Mario meets...its own thing.\nScrap Story is a bit of an enigma as it's seemingly been in development since mid-2017 based on its own Twitter account, which simply states that the creator is \"making a papertype game.\" If you're remotely interested in the charming video below, click the Twitter account and do some scrolling: there's a lot of little gems in there.\nIt looks promising beyond its aesthetics, too. The developer notes mechanics like a dynamic vegetation system that reacts to the weather, and so on.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Enhancing our amenities is a team that understands the importance of each individual's needs for assistance. Broadview Assisted Living at Pensacola has been providing quality care to seniors in Escambia County for over 10 years. We are proud of our long-term staff members, some of whom have been committed to caring for our community residents for over a decade.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbgruf b/data_all_eng_slimpj/shuffled/split2/finalzzzbgruf
new file mode 100644
index 0000000000000000000000000000000000000000..a7350fee76ecf57cbbd126a0619ebef5919fc022
--- /dev/null
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+{"text":"Ensuring consistent quality in any synthetic process, scale-up process or formulation approach is key to building a loyal customer base. Purchasers, researchers or corporations need certainty that the material they are going to receive meets the required standards and be the same as their previous batch. Such challenges are currently widespread in global graphene supply and a barrier to successful market development.\nEnsuring quality with nanomaterials is often more difficult than producing bulk chemicals. This can be due to several factors, including more complex fabrication\/synthetic processes and spontaneity in the synthetic routes. These factors also have a knock-on effect in that synthetic processes become harder to control, resulting in a potentially costlier process and ultimately more challenging to scale-up a nanomaterial. Another aspect to consider is that many chemicals and processes have been optimised over many years. In comparison, the fabrication and quality control of nanomaterials is a relatively young area.\nWhilst ensuring the quality of nanomaterials (overall) can be tricky enough, it becomes significantly more difficult when nanomaterials are formulated into functional inks. The INSPIRED project has been working to help companies scale up their production of nano-formulated inks, and this includes helping companies to ensure that these inks can be produced consistently and to the required quality.\nThomas Swan, from the UK, is one of the project partners of the INSPIRED project and focuses on scaling up graphene nanoplatelet production and subsequent formulation into functional inks. Although though there is often a trade-off between price and quality, one of the unique selling points (USPs) of Thomas Swan is that it prides itself on the delivery of consistent products.\nHowever, delivering a consistent product is not without its challenges. During the INSPIRED project there were issues with not only trying to understand what properties the customers wanted from their ink formulations, but also from agglomeration and poor dispersion of nanoparticles within the formulations and nanosafety hazards.\nDespite this, through efficient post-formulation quality control protocols, Thomas Swan was able to formulate nanomaterial-based inks with consistent properties to deliver a high-quality, cost-effective and safe product in line with its other nanomaterial product ranges. The INSPIRED project supported Thomas Swan by enabling Thomas Swan to test multiple ways of producing larger volumes of raw material and various formulations and, through collaboration with multiple external partner, combining expertise to realise consistent results and products.\nWhy are inks more challenging?\nThere are several answers to this question. Consistency must be ensured in at least two separate processes \u2013 in the fabrication of the nanomaterials themselves, and in the formulation of the inks. This can also extend out to a third process, providing a consistent printing process onto a given substrate. There are also different needs that must be met at each stage, so the same approach cannot be used for all the different stages.\nIn the production of the nanomaterial itself, be it copper, silver nanowires, graphene or other, consistent quality arises from the size distribution of the nanomaterial. This can take the form of polydispersity if the material is a nanoparticle, lateral dimensionality if it is a single layered material, or the number of layers in each sample if the material in question is layered. Regardless of the type, ensuring uniformity (there will always be some variation) within each sample (or batch) is key to ensuring a consistent quality in a production process. The main reason for this is that materials with significantly different dimensions act differently to each other, and if this is the case, the material will exhibit differing properties between batches.\nConsistency also needs to be maintained in the formulation of the nanomaterial-based ink. There are many constituents in inks and uniform ratios must be achieved, otherwise intended properties could vary between batches. However, it is not just the ratio that should be considered. Many of these formulations have a potential end use as a conductive ink, and it is often the nanomaterial that provides the conductivity. Alongside consistent ratios, uniform distribution of the nanomaterial within the formulation needs to be addressed, because an aggregation of nanomaterials in one region of the ink will leave a nanomaterial gap within the ink, resulting in non-uniform conductivity across the ink. This may also affect the localised ink stability. For electronic applications, stability and uniform conductivity are key, and if these properties are not met, then their use as commercially-viable conductive inks will not be realised.\nThe final aspect for consideration is the printing process. Whilst it is important for the ink formulation to be uniform in its internal make-up, the printing of the ink also needs to be uniform. This relates closely to ensuring a consistent distribution over a wide area, but this time, the printing process needs to provide a uniform distribution of ink across the surface (i.e. the same thickness). This is to not only ensure an even distribution of conductivity, but to also ensure that there are no 'bumps', or ink-less regions, on the substrate.\nRegardless of processes used to produce the nanomaterial and the formulated ink, whether batch or continuous methods, quality must be ensured at all points. For batches, this means that the end-user should get a material with the same structure, internal make-up and property if they bought more than one batch of product. For continuous processes, the production line always needs to be kept consistent (even after downtime has occurred).\nNanogap, from Spain, is another manufacturing partner in the INSPIRED project. Its focus is on developing new silver nanowires and their scale up and formulation into functional inks. While Nanogap has broad experience on developing new products and scaling up of process, the formulation was more challenging.\nThroughout the INSPIRED project, Nanogap benefitted from the ability to collaborate with other project partners. NANOGAP provided samples to the partners in charge of printing trials and adjusted formulations according to the feedback received. For example, at the beginning of the project, Nanogap formulated and produced the inks itself, but due to issues surrounding the formulation during printing process, some parameters such as concentration, pre-filtration etc. were changed to improve the performances of the inks in printed processes.\nFor Nanogap, the major efforts were undertaken during the scale up process of silver nanowires, rather than their formulation into the inks. Whilst it was a challenge to reduce the diameter of the nanowires and to alter their aspect ratio (suitable for use in the ink products), by controlling and fixing specific synthetic parameters, Nanogap was able to provide a consistent nanomaterial to other partners.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"We all know that sinking feeling when you wake up one morning and feel that distinct cold snap has kicked in\u2026 And the wise words of Ned Stark pass through our mind \"Winter is coming\" and don't you know it, there may not be hordes of undead nightwalkers coming to get us but it's still a pretty depressing revelation. But it doesn't have to be!\nIt's now cheaper than ever to take a trip out to the tropics and save yourself the misery of putting up with another dreary winter at home! Still not convinced? Here's 10 reasons that might just change your mind!\nPeople in Asia are generally super friendly. It's really easy to have a great time here making new friends and experiencing a new culture while being shown around all the cool local hangout places.\nWhen is the last time you went for a walk in the countryside or forest at home during winter? Can you remember seeing any wildlife? Hearing any birds singing? Asia is teeming with life all year round! And it's summer here every day!!!\nUnless you are some kind of winter sports fanatic the winter is going to drag. Just think next time you leave the office and its dark already at 4pm that you could be sitting on a sun soaked silky smooth white sandy beach chilling out with one of those colorful cocktails\u2026 You know the one with the mini umbrella decorating the glass??? That could be YOU!!! and its much easier and cheaper than you think!\nIn the winter everyone tends to stay at home in a kind of hibernation mode. Yeah you can catch up with all your favorite TV shows but wouldn't it be a better use of your time to go out and see some of the real world? Get out of your comfort zone and have yourself a real adventure!\nYeah those flights can look a bit on the expensive side compared to your usual winter getaway to the familiar beaches of Southern Europe, but once you get here most things are so much cheaper! Hotels, transport, food, clothes\u2026 It's usually around half price or less compared to Europe, depending on your destination.\nMany Asian countries are now removing restrictions on foreign trade, ownership and investment. There has never been a better time to branch out and make new business partners, deals and investments in Asia. Come check it out!\nMount Kinabalu National Park in Borneo, possibly the most biologically diverse place on the planet. Home to tens of thousands of endemic plant and animal species that can only be found there. Also, it just happens to be host to the highest mountain in South East Asia, so bring your trekking boots and treat yourself to a once in a lifetime experience!\nAngkor Wat, the truly mysterious 400Km2 ancient temple city in Siam Reap, Cambodia. This place is unbelievably massive and takes a few days just to see the main sites and we are still discovering new areas to excavate all the time.\nBorobudur, the world's largest Buddhist temple. Thought to have been lost for hundreds of years, this 9th century Javanese\/Buddhist temple was only recently rediscovered and opened to the public after a massive restoration project.\nTravel is one of the greatest things anyone can do, it infuses your life with experience, wisdom, wonder, knowledge, but if you have never experienced the profound nature of arriving in a new and strange place then you will never know what you are missing. There really are few words to describe what happens to the human mind when confronting the kinds of experiences and challenges that one encounters only when traveling.\nOne thing is for sure, it sounds scary, it sounds difficult and it sure can seem expensive, but once you commit yourself to it, everything becomes easy. From a dream your commitment transforms it into an adventure, and the journey truly begins from there. Whether you are old or young, from the moment you step off that plane into a new place your life is going to be richer and more meaningful and you will carry those memories with you forever.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"To establish the processes for dealing with allegations of Student Academic Misconduct.\nThis procedure applies to Coursework Courses and USQ UpSkill Courses undertaken by any Students.\nAll Students and Employees involved in Assessment for a Coursework Course or USQ UpSkill Course are subject to this procedure.\nThis procedure establishes the processes for dealing with allegations of Student Academic Misconduct.\nAny person may report alleged Academic Misconduct by a Student.\nAny member of a Course Team has an obligation to report any suspected Academic Misconduct by Students.\nReports of allegations of Academic Misconduct by a Student in a Coursework Course are to be made to the Head of School\/College offering the Course via the Course Examiner.\nDetails of an allegation of Academic Misconduct including evidence of Academic Misconduct must be submitted using the Student Academic Misconduct Report Form.\nWithin five University Business Days of receiving a report, the Head of School\/College will review the information provided to determine whether there is sufficient evidence to proceed with an allegation of Academic Misconduct.\nThe Head of School\/College will determine whether the incident is to be considered a Caution or Minor or Major Academic Misconduct. In order to afford the Student natural justice, the Student Academic Misconduct Register will not be consulted until after an initial determination is made.\nthe person making the allegation will be informed of the Decision either verbally or in writing and such action be notated on the Student Academic Misconduct Report Form.\nPersons making frivolous or vexatious reports will be dealt with under the University's Student Code of Conduct Policy or other relevant policy or procedure.\nWhere the Head of School\/College considers that the allegation has resulted from a misunderstanding of academic conventions or poor academic practice, the Head of School\/College will send a Notice of Caution of Potential Academic Misconduct to the Student. In determining this, the Head of School\/College should consult the Academic Misconduct Register for any previous Caution\/s of Potential Academic Misconduct provided to the Student.\ninforms the Student of the alleged Academic Misconduct.\ninvites the Student to respond to the allegation, whether in writing or in person or other agreed method of communication (e.g. teleconference) at a meeting with the Head of School\/College or nominee, and Course Examiner. A meeting in person will be held as soon as practicable after 10 University Business Days, but no later than 15 University Business Days from the date of the Notice.\ninforms the Student that they may elect to have supporting representation by a member of the Student Guild or other support person. International Students may elect to have supporting representation by a member of the overseas partner institution or other support person.\nIf the Student fails to respond to the Notice of Allegation of Academic Misconduct, a Decision about the allegation will be made without any input by the Student, and this normally will mean that the Student will have no right to Appeal the Decision.\ninforms the Student about the potential consequences of a determination of Academic Misconduct.\ninforms the Student about the potential consequences of a determination of Academic Misconduct and provides advice in relation to improving their skills in this regard.\nif a penalty is imposed, informs the Student that they have 20 University Business Days from the deemed receipt of the Notice of Minor Academic Misconduct to Appeal the process that was followed. The Student will be directed to the Academic Appeal Procedure to establish whether there are grounds available for Appeal. If no Appeal is made within that period, there will be no further right of Appeal.\nwill complete the Student Academic Misconduct Report Form, including requesting the update of the Student Academic Misconduct Register, and ensuring that all documentation is sent to the University's records management system.\nmake a recommendation to the Associate Dean (Students) via the Student Academic Misconduct Form.\nUpon receiving advice from the Head of School\/College that there is sufficient evidence to substantiate an allegation of Major Academic Misconduct, the Associate Dean (Students) will, normally within three University Business Days, review the recommendation and proposed penalty to either confirm, amend or reject the recommendation.\nIf the Associate Dean (Students) determines that the inquiry and recommendation by the Head of School\/College has provided insufficient evidence and\/or inappropriate penalty for Academic Misconduct, the recommendation and\/or proposed penalty shall be referred back to the Head of School\/College with specific action for further review and enquiry.\nif a penalty is imposed, informs the Student that they have 20 University Business Days from the deemed receipt of the Notice of Major Academic Misconduct to Appeal the process that was followed. The Student will be directed to the Academic Appeal Procedure to establish whether there are grounds available for Appeal. If no Appeal is made within that period, there will be no further right of Appeal.\nThe Associate Dean (Students) will complete the Student Academic Misconduct Report Form, including requesting the update of the Student Academic Misconduct Register, and ensuring that all documentation is sent to the University's records management system.\nIn the case of repeated Major Academic Misconduct, with no further Appeal pending, the Associate Dean (Students) will recommend, to the Executive Dean, exclusion due to academic misconduct in accordance with the Academic Standing, Progression and Exclusion Procedure.\nAt each stage of the process, only the forms and templates specified in this procedure are to be used.\nDecisions reached on the allegation.\nA Decision reached in a matter of Student Academic Misconduct will be recorded in the Student Academic Misconduct Register and copies of all relevant documentation maintained on the Student's record in the University's records management system. At the conclusion of each Academic Year, the Pro Vice-Chancellor (Student Services) will submit a report to the Academic Board through the Education Management Committee listing by Course and School the number of inquiries carried out, the Penalties imposed and a report as to whether the final Decisions were made by the Associate Dean (Students) or an Appeals Committee. No individual Student names or numbers will be listed in the report.\nA Student should also keep a record of all material and evidence submitted in respect of an allegation of Academic Misconduct.\nthe outcome of the inquiry, that is, Caution, Minor or Major.\nWhere any penalty for Academic Misconduct is imposed as provided for in the Academic Integrity Procedure Penalty Schedule, a Student may Appeal to the Academic Appeals Committee under the provisions of the Academic Appeal Procedure provided they are able to establish grounds to do so. The Student is informed of the right of Appeal in the correspondence advising of the imposition of a penalty.\nIf a Student is dissatisfied with the outcome of his or her Appeal or Complaint, or the University's handling of it, external sources of advice or review may be available, including the Student's State or Territory Ombudsmen or the Commonwealth Ombudsman's Office (in relation to the application of policies and procedures and administrative processes).\nA period of time consisting of Semesters and teaching periods that all commence in the same calendar year....moreA period of time consisting of Semesters and teaching periods that all commence in the same calendar year.\nConsists of the Examiner and Moderator of the Course and any other people involved with the preparation, delivery and grading of the Course....moreConsists of the Examiner and Moderator of the Course and any other people involved with the preparation, delivery and grading of the Course.\nA subject of study, defined by a Course Specification, for which Students may be awarded a Final Grade where the method of teaching and learning that leads to the acquisition of skills and knowledge and does not include a major research component....moreA subject of study, defined by a Course Specification, for which Students may be awarded a Final Grade where the method of teaching and learning that leads to the acquisition of skills and knowledge and does not include a major research component.\nFor the purposes of these Policies and unless otherwise defined in a specific Policy, the term 'Executive Dean' means the chief executive officer of a Faculty....moreFor the purposes of these Policies and unless otherwise defined in a specific Policy, the term 'Executive Dean' means the chief executive officer of a Faculty.\nMajor Academic Misconduct is defined to occur in all other Academic Misconduct not deemed as Minor Academic Misconduct....moreMajor Academic Misconduct is defined to occur in all other Academic Misconduct not deemed as Minor Academic Misconduct.\nMinor Academic Misconduct is defined to occur where the Academic Misconduct is manifest in a Student's Assessment submission to a contribution with less than five percent of the available marks for an Assessment Item....moreMinor Academic Misconduct is defined to occur where the Academic Misconduct is manifest in a Student's Assessment submission to a contribution with less than five percent of the available marks for an Assessment Item.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"More than 350 people are killed every year when their car strikes the back of a big semi-truck and slides underneath the rear impact guard. These types of horrific accidents are known as rear underride trucking accidents and they are extremely deadly. While rear impact guards are meant to protect motor vehicles from sliding underneath the back of a commercial truck, they often do not work.\nThe Insurance Institute for Highway Safety put rear impact guards to the test and found that at just 35 mph, the rear guard that met U.S. standards gave way, allowing the car to slide right under the trailer. This sheers off the top of the vehicle, effectively crushing it. In the Institute's testing, if there were occupants inside of the car, they would not have survived.\nThe National Highway Traffic Safety Administration reported that 312 people died in underride accidents in 2008. The majority of these accidents involved light passenger vehicles, with 40 percent of underride accidents occurring at the rear corners, where impact guards are weakest.\nAnother type of underride accident occurs when a truck is turning wide and straddling the road. While the sides of trucks do have reflector strips, they often aren't illuminated until a driver's headlights are shining on them. So, by the time the driver sees the truck it's often too late to stop and the car slides underneath the tractor-trailer, crushing the windshield and driver's compartment.\nWhile these types of collisions are most common at night, they can also occur on roads with poor visibility, in inclement weather, or during the day\u2014especially when the sun is at a low angle.\nUnderride trucking accident victims and their families have the right to pursue just compensation for the impacts of these horrific accidents. The trucking accident attorneys at Corsiglia, McMahon, and Allard, LLP will stand up to the big trucking companies and their insurance companies by aggressively fighting for your rights.\nIf you or someone you love has been killed or injured in an underride trucking accident, contact the trucking accident lawyers at Corsiglia, McMahon, and Allard, LLP at (408) 289-1417.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Hoverboard Display Stand Custom HOVERBOARD REAL FLOATING Levitating Display Back to The Future Marty was posted in October 10 2018 at 9:58 am and has been seen by 12 users. Hoverboard Display Stand Custom HOVERBOARD REAL FLOATING Levitating Display Back to The Future Marty is best picture that can use for individual and noncommercial purpose because All trademarks referenced here in are the properties of their respective owners. If you want to see other picture of Hoverboard Display Stand Custom HOVERBOARD REAL FLOATING Levitating Display Back to The Future Marty you can browse our other pages and if you want to view image in full size just click image on gallery below.\nAliexpress Buy APP Electric Scooter Bluetooth Self Balance MattyCollector Mattel Hoverboard discussion Page 32 Marty McFly Hoverboard Funko Pop Display Stand Back to the Etsy Products Hover32 Tax free Hoverboard App self balancing scooter balance car electric Hudson Reporter Spinning their wheels Hoverboards Segways are Two Wheel Self Balancing Scooter Electric Skateboard Hoverboard Scooter Skateboard Display Rack Stand Shelf Buy ScooterSkateboard Mattel BTTF 32 Hoverboard Replica [Archive] Sideshow Freaks Back to the Future Hoverboard Display Actually Hovers If You Don't Hoverboard Display Stand.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"The Comptroller General shall distribute to each agency of the Federal Government a list containing the names of persons or firms that a Federal agency or the Secretary has found to have violated this chapter.\nUnless the Secretary recommends otherwise because of unusual circumstances, a Federal Government contract may not be awarded to a person or firm named on the list under subsection (a), or to an entity in which the person or firm has a substantial interest, until 3 years have elapsed from the date of publication of the list. If the Secretary does not recommend otherwise because of unusual circumstances, the Secretary shall, not later than 90 days after a hearing examiner has made a finding of a violation of this chapter, forward to the Comptroller General the name of the person or firm found to have violated this chapter.\nPub. L. 89\u2013286, \u00a7\u202f5(a) (1st sentence), Oct. 22, 1965, 79 Stat. 1035.\nPub. L. 89\u2013286, \u00a7\u202f5(a) (2d\u2013last sentences), Oct. 22, 1965, 79 Stat. 1035; Pub. L. 92\u2013473, \u00a7\u202f4, Oct. 9, 1972, 86 Stat. 790.\nIn subsection (b), the word \"entity\" is substituted for \"firm, corporation, partnership, or association\" to use a single broad term clarifying that the prohibition applies to any kind of organization in which the person or firm has a substantial interest. The words \"containing the name of such persons or firms\" are omitted as unnecessary. The word \"person\" is substituted for \"individual\" for consistency in the subsection.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Rachel Brosnahan gave a fashion nod to the Time's Up movement with this bracelet.\nIdris Elba walks the red carpet with his fiancee, Sabrina Dhowre, and his daughter, Isan. Isan, right, is a Golden Globes ambassador this year.\nTelevision icon Carol Burnett is having an award named after her\u2014 and she is the first recipient.\nAlso getting a special career award Sunday is Jeff Bridges, seen here with his wife, Susan Gaston.\nNext articleGLOBURILE DE AUR 2019.Care sunt c\u00e2\u0219tig\u0103torii?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Shown below is our current portfolio of work and clientele. We have many projects in development, so please return to this section in the future.\nPortfolio Sampling listed are a few examples of some clients.\nPDII provides quality, personalized services such as, Customized evaluations and assessments of specific HR programs, Development of \"state of the art\" customized HR programs, and Linking HR programs to organization performance.\nMurphy's Law - Located on Marquette University's campus, it is known for its 21 st Birthday Club, the Famous \"Karaoke Wednesday\" and \"The Mug Club\" , Murphy's has served millions of happy students and alumni for many years.\nMarquette University - College of Professional Studies enhances the education of professionals and gives students the advantage needed to stay competitive in today's work force.\nKD dids, Inc. specializes in creating innovative knitwear designs and manufacturing these designs. They offer a full collection of knits for all occasions, expressing the styling and beauty of dance.\nW\u00e4ngard Partners, Inc. is a Milwaukee-based commercial real estate brokerage firm well known for its work in; Office, Industrial, Retail, Land, Senior Housing and Investment real estate.\nThe Lane Group is nationally recognized as one of the fastest growing executive search consulting organizations in the country. They conduct more than 1000 recruiting assignments annually, covering a variety of industries for both public and private industries of all sizes.\nAssembly Technolgy International, Inc. provides a wide variety of useful soldering related tools used by people in high-tech electronics MRO, aerospace, industrial, and general electric markets worldwide.\nUnited Way of Greater Milwaukee is a non-profit organization that conducts a single, community-wide fund-raising campaign once a year to provide 146 local health and human service programs with needed money to help address community problems\u2013problems like child abuse, hunger, homelessness and substance abuse.\nAssessment Systems International, Inc. offers assessments, development resources, and services for leadership development, executive coaching, teams, sales, customer service, selection, and organization development.\nStano Landscaping specializes in any residential or commercial landscape construction project. Their dedication to customer satisfaction since 1963 has forged a loyal following of people who appreciate personalized service and good value.\nWisconsin's largest property and topographic surveying firm with twelve survey crews responsible for the collection of field data, land use, and site planning and engineering.\nJewish Family Services provides comprehensive social services that enhance the quality of life for Jewish individuals and families and the general community. They are guided by the Jewish values of loving kindness and righteousness.\nLakeshore Dance studios is Ozaukee County's leading dance training facility that boasts a student body of nearly 300 students. The curriculum stresses the importance of having a varied and diverse training background incorporating the techniques of not only ballet, but also jazz, tap, lyrical\/modern, creative movement, pointe, and composition.\nAn engineering firm specializing in site development, surveying, transportation, and, construction services.\nMilwaukee Alumni chapter for Delta Sigma Pi, a professional fraternity organized to foster the study of business in today's corporate world.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The campaigns also allow the firms to reduce their high inventory at a time when the economy has slowed down.\nNoble Development Plc, for example, has launched it latest marketing campaign, with a 29 per cent discount for customers who buy into its Noble Ploenchit by offering prices of Bt190,000 per square metre, down from Bt270,000 per square metre.\nThe project now has about 200 units sized between 44 and 110 square metres and selling for about Bt2 billion.\n\"We expect this campaign will sell out this project,\" the company's assistant managing director Sira Udon said recently.\nAssetwise Co Ltd offers a discount price of up to 20 per cent for its H2 Condominium Ramindra 21, reducing from Bt3.53 million per unit to a discounted Bt2.79 million per unit, along with a free transfer fee.\nOrigin Property Plc is offering a discount price for Knightbridge Tiwanond, from the original of up to Bt3.66 million to the new price of Bt3.39 million, drop of 7.3 per cent.\nSena Development Plc offers free down payment and a discount of up to Bt1 million for its 18 residential projects nationwide, with residential units priced between Bt1.2 million and Bt8.2 million. The company expects sales of Bt1.2 billion from this event that runs from now till the end of March, 2019.\nSupalai Plc offers a discount of up to Bt270,000 per unit.\nSansiri Plc offers a discount price up to Bt12 million, or about 48 per cent, for its Baan Mai Kao, Phuket from a starting price of Bt25 million per unit. The company has also launched other campaigns such as a free transfer fee and public area fee, free to stay in the first two year. The campaign runs from now till March 15, 2019. The company expects sales of about Bt2 billion from the campaign.\nFrom now till March 24, 2019 Grand Unity Development Co Ltd is offering a discount price of up to Bt2 million, along with a free-to-stay offer for the first two years for all customers who buy its brand U, and U-Delight condominiums nationwide. The company expects sales worth up to Bt2 billion from this campaign.\nMost property firms also showcase their latest marketing campaigns at the 40th House and Condo Show 2019 between March 21-24, 2019 at Queen Sirikit National Convention Centre.\nThe event expects above 100,000 visitors and sales values of up to Bt8 billion during the show. Most property firms and commercial banks are also offering a special price and financial package before the Bank of Thailand measure comes into effect on April 1, 2019.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"TOSTITOS tortilla chips and dips are the life of the party. Whether you're watching the game with friends or throwing a giant backyard barbecue, TOSTITOS has the must-have chips and dips to pump up the fun!\nIngredients: INGREDIENTS: Corn, Vegetable Oil (Corn, Canola and\/or Sunflower Oil), and Salt.\nWhen you're craving a hearty soup that's packed with a fiesta full of flavor, look no further than Campbell's\u00ae Chunky\u2122 Tortilla Soup with Grilled White Meat Chicken. We blend grilled white meat chicken with your favorite Mexican ingredients\u2014rice, black beans, corn, tomatoes, a trio of peppers (red, green chili, jalapeno) and tortilla chips. The end result is so bold and satisfying, it'll combat your hunger and fuel your everyday hustle.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"There Is a Right Way to Teach Ballroom Dancing \u2013 Ask Two Nigerians!\nAre You Ready to Take the Leap to Become a Ballroom Dance Teacher?\nDear Diane, I've been taking lessons at my studio for four years. I love dancing, and up until the past year, have also loved the family atmosphere at my studio. But now I'm considering leaving the ballroom world altogether. Last year the studio got a new owner who loves dance but has little business experience\u2026.\nHow Do I Teach My Student Not to Muscle Around His Pivots?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"RIYADH (Reuters) - Saudi Arabia has detained prominent women's rights advocate Hatoon al-Fassi, widening a crackdown that has ensnared more than a dozen activists even as the kingdom lifted a ban on women driving, sources said on Wednesday.\nLondon-based Saudi rights group ALQST and exiled activist Manal al-Sharif reported the arrest on Twitter, but provided few details. It was confirmed by sources in touch with people close to Fassi, who said they were scared of speaking out.\nAn Interior Ministry spokesman and the government's communications office did not immediately respond to requests for comment.\nFassi was last active online on Thursday. She was planning to take journalists in her car on Sunday as other women did to celebrate the much-hyped end of the world's last ban on female drivers, long seen as an emblem of women's repression in the deeply conservative Muslim country.\nBut it has been accompanied by a crackdown on dissent against some of the very activists who previously campaigned against the ban, including Loujain al-Hathloul, Eman al-Nafjan, and Aziza al-Yousef as well as the men Ibrahim al-Modaimeegh, Mohammad al-Rabea and Abdulaziz al-Meshaal.\nSaudi Arabia's public prosecutor said earlier this month that a total of 17 people had been arrested, eight of whom were later released. The authorities accused them of suspicious contacts with \"foreign entities\" and said more suspects were being sought. Local media labeled them traitors.\nAt least nine people remain in detention \"after sufficient evidence was made available and for their confessions of charges attributed to them\".\nFassi is an associate professor at King Saud University and a regular contributor to Saudi Arabia's al-Riyadh newspaper. She has long been a champion for women's rights, including the right to drive. Her husband is a prominent Gulf Cooperation Council official.\nFassi tweeted on June 6 that she had received her Saudi driving license along with a small group of women who converted their permits obtained in foreign countries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Langen, 02 November 2018 - DEMIRE Deutsche Mittelstand Real Estate AG (ISIN: DE000A0XFSF0) today announced the conclusion of a purchase agreement for the acquisition of four office properties in Essen, Cologne, Aschheim (catchment area Munich) and Bad Vilbel (catchment area Frankfurt) with a combined investment volume (including ancillary purchase costs) of around EUR 167 million.\nThe four office properties acquired feature total lettable space of around 89,000 m\u00b2 and are a perfect fit for DEMIRE's active asset management approach. The portfolio currently generates net cold rent of approximately EUR 8.6 million annually (EUR 11.20 per m\u00b2 on average) with a planned FFO contribution of EUR 3.5 million p.a., which is anticipated to increase to around EUR 6 million p.a. in 2023 based on the planned rental performance. Based on the values as at 30 June 2018, the market value of the DEMIRE Group's real estate portfolio will increase to approximately EUR 1.3 billion with the transaction's completion. The transfer of ownership, benefits and obligations is expected in Q1 2019.\nThe largest property purchased is the multi-tenant office property \"Office Park Bredeney\" with lettable space of 45,600 m\u00b2. It is located in the economically robust city of Essen and is well-connected to public transportation and the nearby motorway, and has a direct link to the Dusseldorf airport, the city centre and the main train station. thyssenkrupp AG is currently the largest tenant, and the current occupancy rate is 88%.\nThe office property in Cologne is a two-storey office complex, built in 2006. The property features lettable space of roughly 5,200 m\u00b2 and is close to full occupancy (97%). The main tenants are the city of Cologne, which operates the only vehicle registration office in Cologne at this location; and EHA Autoschilder GmbH, Germany's oldest sign printer with more than 80 branches nationwide. The property has easy access to central public transportation hubs.\nThe office property in Aschheim-Dornach in the Humboldtpark commercial area is located directly at Munich's north-eastern city limits. The property has lettable space of 12,200 m\u00b2 and a well-diversified tenant mix. The current occupancy rate is around 89%. The industrial park is easily accessible with both public and private transportation. Aschheim-Dornach has established itself as a new service location based on its good infrastructure and beneficial trade tax.\nThe fourth acquisition property in Bad Vilbel, located northeast of Frankfurt am Main, offers the greatest optimisation potential. The former single-tenant property has five sections with total lettable space of roughly 26,000 m\u00b2. The building also has an excellent IT infrastructure and parking facilities. With a current vacancy rate of around 69%, the property offers strong appreciation potential through its conversion into a multi-tenant property.\nRalf Kind, CEO\/CFO of DEMIRE, comments: \"With these real estate purchases, we are implementing the core of our DEMIRE 2.0 strategy, which is to grow to EUR 2 billion in the medium term. We are already in advanced talks to secure further attractive targets from our acquisition pipeline.\"\nDEMIRE Deutsche Mittelstand Real Estate AG has commercial real estate holdings in mid-sized cities and up-and-coming locations bordering German metropolitan areas. The company's specific forte is its focus on these second-tier cities - its claim being \"First in Secondary Locations\" - and on a range of assets that appeals to both internationally active and regionally rooted tenants. Having expanded rapidly between 2013 and 2016 both by buying single properties and by acquiring equity interests, DEMIRE held a portfolio with a combined lettable area of around 1 million sqm and a fair market value of more than EUR 1.1 billion by the end of the first half of 2018.\nThe portfolio focus on office, retail and logistics assets results in precisely the kind of risk\/reward structure that DEMIRE considers appropriate for the business line of commercial real estate. The Company puts a premium on long-term contracts with solvent tenants in anticipation of stable and sustainable rent revenues. DEMIRE has set itself the goal to keep optimising its corporate structure.\nTo this end, it pursues an active property management approach out of the conviction that it is the best way to achieve economies of scale and portfolio optimisations. DEMIRE Deutsche Mittelstand Real Estate AG shares (ISIN: DE000A0XFSF0) are listed in the Prime Standard segment of the Frankfurt Stock Exchange.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The UK is party to around 40 European treaties, covering trade with more than 70 countries and making up 12% of the UK's total trade.\nEach needs to be rewritten, either with new terms or by mirroring the existing terms, a process known as \"rolling over\".\nNow, according to senior government sources, expectations have been tempered.\nLiam Fox tells Sky News that he will now prioritise some trade deals over others as the Brexit deadline approaches.\nToday, Dr Fox signed a trade agreement with Switzerland.\nSky News understands that a deal with Israel could be announced as soon as next week.\nHowever, the same source confirmed that talks with South Korea, one of the UK's most important overseas trading partners, have stuttered and are \"very unlikely\" to be finished by the end of March.\nIn an interview with Sky News, Dr Fox confirmed that certain negotiations had now been prioritised over others.\nHe said that, of the 40 deals that Britain needed to renegotiate, the bottom 20 covered a total of less than 1% of total UK trade.\nDr Fox said: \"We are content that the vast majority of trade will achieve the continuity that we need \u2013 if I could get to 100% that would be fantastic but it's about getting the maximum amount that we can.\n\"The trading world is a particularly unemotional one with a lot of hard headed people who will want to get the best agreement.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"This entry was posted\ton Sunday, December 21st, 2008 at 9:40 pm\tand is filed under Industry Rants. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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@@ -0,0 +1,5 @@
+{"text":"At Hanson Grain our goal is to help you keep all of your animals healthy and happy! Our store offers you a great selection of quality products that are competitively priced along with friendly, knowledgeable staff.\nJust like you, our staff are all animal owners and share a passion for our four legged and two legged animal friends. Our employees all have dogs, horses, rabbits, chickens, cats & more! We use the products that we sell, so we can give you helpful advice! We can help you choose the correct food for your pet with solid nutritional advice whether it is for you dog, horse, cat or rabbit. We believe in facts, not fads, when it comes to nutrition so we can help you make the most educated decision regarding your pet's health and happiness.\nWe feed them all - big & small! We carry food for just about every animal you may own - from hamsters to horses! While our focus has transitioned to dogs and cats as of late, we still maintain a strong, knowledgeable foundation in equine nutrition. We even offer local delivery to better serve our large animal customers. Several of our employees have gone through series' of training focusing on correct feeding for a lifetime of your horse's health.\nIf we don't stock it - we are more than happy to order it in for you - some of our special orders have included Elephant Chow, Tortoise Chow and Earthworm Chow!","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The Capitol Area Radio Drone Squadron welcomes pilots of all ages. Becoming a member is easy, just follow the steps below.\nJoin the AMA - C.A.R.D.S., as well as most model aviation clubs, require all members maintain a membership with the Academy of Model Aeronautics (AMA). By joining the AMA you will become one of the thousands of registered RC pilots in the country, and will have he insurance coverage necessary to fly at any AMA sanctioned field, including CARDS Field. Become a member by visiting https:\/\/www.modelaircraft.org\/joinrenew.aspx.\nJoin CARDS - After you've joined the AMA, you can become a member of CARDS by completing the membership application and submitting the registration fee by mail, or in person to the membership secretary at one of our monthly club meetings.\nEarn your wings - Once you have all the necessary memberships, you can attend student nights and earn your wings. After that, you can come out to the field whenever you feel the need to fly.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Snowshoes, poles, guide and evening snacks are included. You can rent a snowmobile suit and gloves from us if it's cold, but please tell us a couple of days in advance. Don't forget to choose the pick up point.\nCome and experience a night walk on snowshoes with a chance to experience the northern lights. With snowshoes on our feet and in the light of headlamps, we walk into the night and pass through the snow-laden firs and experience a new dimension of the mountains. An evening snack is included and the guide will also tell you facts about the northern lights and a quick guide on how to photograph auroras.\nWe pick you up at the Hotel Holiday Club (30 min before the tour starts) or at the Hotel Copperhill Mountain Lodge (15 min before the tour starts).\nWarm clothes and stable shoes.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"In part one of our series on game localization at EA, we followed the journey of Mass Effect: Andromeda from the storywriters in the studios through to the EA Loc Multilanguage Localization Specialists and the translation process.\nMeet Chloe Anon Pasleau, Audio Capture specialist at the EA Localization department in Madrid, Spain. She and the team make sure that the characters in Mass Effect: Andromeda hit exactly the right tone when they are localized to French or German.\nFirst, tell us a bit about yourself: what's your professional background and when did you start working as an Audio Capture Specialist?\nI specialized in Audiovisual Translation in Barcelona after studying Translation and Interpreting in Granada and Glasgow, so my background is related to linguistics, which is a common field of study among people in Localization. I started working as an Audio Capture Specialist in February of 2015 after an enriching experience as an Assistant in Dragon Age: Inquisition, but I had already worked in many other departments at EA.\nWhat's the first step when you receive the translated copy for Mass Effect: Andromeda?\nOnce the lines are translated, my main task is to group them in recording scripts for each language and send them to the external recording studios. These scripts need to reflect the nature of the content, such as characters involved, scope (the number of lines) for each, availability of talents, time schedules, and a lot more. Taking the right decisions in order to properly manage the recordings are the core tasks and responsibilities of an Audio Capture Specialist.\nFor Mass Effect: Andromeda specifically, we're recording more than 77,000 lines for each localized language.\nHow many lines of text are you handling for a Localization of a game like Mass Effect: Andromeda?\nFor Mass Effect: Andromeda specifically, we're recording more than 77,000 lines for each localized language. BioWare games are generally the biggest in terms of scope, which is one of the reasons why they can't be dubbed to as many languages as other EA titles.\nWhen choosing the recording studio, what are you specifically looking for when it comes to BioWare titles?\nBioWare titles are quite unique in many different aspects. As mentioned above, they have huge scopes and the number of characters to record is quite large, with up to 1,500. So, a recording studio handling a BioWare project needs to be very skilled at Project Management, and needs to have access to a very large pool of talents. Also, since BioWare games feature characters from different fantasy or alien races, different recording techniques and equipment might need to be used.\nWhat do you have to keep in mind while casting actors for a character in the game (style, cultural aspects, consistency with previous roles)?\nThe general challenge for Localization is to ensure that a local version of a game is not a mere translation, but a whole true experience for the players. Style, cultural aspects and consistency are factors to have in mind, but also many others such as experience, professionalism, believability, tone of voice, adaptability, how recognizable the voice is. The talents are not selected by Audio Capture alone, but in agreement with many other involved parties, including external partners.\nHow many voice actors are needed to localize a game like Mass Effect: Andromeda?\nAround 150 per language. This means that many of these talents act as different characters. An advantage that the Mass Effect franchise has in this regard is that one talent can play several minor roles, since the audio processing for one race can be so different from another that it's impossible to tell that they are being played by the same actor or actress.\nFor instance, Salarians in the Mass Effect Universe speak in a very fast pace and their pitch is shifted to higher frequencies. This allows us to use the same actor to play another character from a different race. Also, voice talents are professionals that can perfectly play different voices. Here's an example: If you have ever watched the show Rick and Morty, have you noticed that both main characters are voiced by the same actor?\nWhat factors are influencing the Audio Localization (specific requirements: legal, linguistic, timelines, etc.)?\nIf I had to choose two over the rest, that would be time and quality. Think of a game as a painting, or a book. A self-demanding painter or writer will always have the feeling that his or her work can be improved, so they need to impose themselves a finishing date. A game is exactly like that, but it's created by a team of several hundreds of people, from a storywriter to a voice talent and everyone in-between.\nQuality is the other important factor, because everyone at EA Loc is committed to a great product, and a great product comes from each and any of us doing our best in that regard.\nWant to work at Localization in EA Madrid? Join the team here!\nStay in the conversation of all things EA: Visit our blog, follow us on Twitter and Like us on Facebook.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"- Color illustrated bi-lingual brochure re the services of the Spafford Children's Center in Old City, Jerusalem. Includes history of the center (formerly the Spafford Baby Home and Children's Hospital), its medical, psycho-social, educational, and cultural departments, and basic map of location.\nAmerican Colony, Spafford Children'S Center, and Anoush Hagopian. Brochure, Spafford Children's Center, Jerusalem . The Commercial Press, Jerusalem, Palestine, 2005. Manuscript\/Mixed Material. https:\/\/www.loc.gov\/item\/mamcol.093\/.\nAmerican Colony, Spafford Children'S Center & Hagopian, A. (2005) Brochure, Spafford Children's Center, Jerusalem . The Commercial Press, Jerusalem, Palestine. [Manuscript\/Mixed Material] Retrieved from the Library of Congress, https:\/\/www.loc.gov\/item\/mamcol.093\/.\nAmerican Colony, Spafford Children'S Center, and Anoush Hagopian. Brochure, Spafford Children's Center, Jerusalem . The Commercial Press, Jerusalem, Palestine, 2005. Manuscript\/Mixed Material. Retrieved from the Library of Congress, <www.loc.gov\/item\/mamcol.093\/>.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzblmsl b/data_all_eng_slimpj/shuffled/split2/finalzzzblmsl
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index 0000000000000000000000000000000000000000..12e66694d247e8a452515e97110cd0bce0ac95f1
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@@ -0,0 +1,5 @@
+{"text":"Amendment issued to extend response date. See Attached. NOTE: THIS NOTICE WAS NOT POSTED TO FEDBIZOPPS ON THE DATE INDICATED IN THE NOTICE ITSELF (13-MAR-2019); HOWEVER, IT DID APPEAR IN THE FEDBIZOPPS FTP FEED ON THIS DATE. PLEASE CONTACT 877-472-3779 or fbo.support@gsa.gov REGARDING THIS ISSUE.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Electricity is available (up to 6 amp\u00e8re). For drinking water you can use the grey hose. For the safety on the main landing stage safety pillars\/ colums with fire extinguishers and lifebuoy.\nThe 'Dokhuis' (club), where you can get drinks and simple snacks, and which is run by volunteers, has a (hopefully) sunny view over 'Het IJ' (the river IJ). Opening hours are from Friday evening until Sunday afternoon during the Summer season.\nBeside the 'Dokhuis' you will find a well-equipped playground.\nYou have to pay for your mooring place on the day of arrival at the automated ticket machine next to the office of the Harbour master. This machine accepts PIN, I-PAY and MAESTRO only and is available for 24 hours a day. Once your payment has been made, you will receive a receipt and a sticker. The receipt indicates the entrance code to the gate at the parking place. We kindly request you to display the sticker prominently on your boat, so the harbour master can easily see that you have paid.\nIt is not possible to pay with cash.\nSanitation facilities are available throughout the year. There is a seperate shower and toilet facility unit. At the ticket machine a facility card is available for the use of showers and washing\/drying machines. With this card you can also refuel with drinking water. Ironing facilities are also available.\nA wifi-network is available. Guests may obtain free entrance to this network.\nFor security, the Sixhaven is surrounded by a security fence. The lock code to the pedestrian and cyclists' gate can be found on the receipt issued to you when you pay your harbour dues.\nBicycles can be placed under the 'Dokhuis' at the edge of the harbour. Cycling is not allowed in the harbour area.\nGuests are not allowed to park their car in the harbour area. The Sixhaven is situated in the blue restricted parking zone. Parking outside the fence is allowed with a parking disc for max 2,5 hours.\nFree parking is available 10 minutes' walk north of the Sixhaven. The border of the blue zone and free parking is beyond the supermarket 'Dirk van den Broek', Meeuwenlaan 76, Amsterdam.\nThe harbour master will be able to provide you with any further information you may require.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"A holiday beside the sea is something many of us dream about. Waking up on a summer's day and listening to the distinctive sounds of seagulls circling high in the sky above, waves ebbing and flowing across sandy beaches, fishing boats chugging in and out of the harbour, are sounds that will evoke memories of a Devon seaside holiday for years to come.\nOur coastal cottages in Devon are all within 3 miles of the sea, offering you a wide selection of holiday cottages from those with sea views - close enough to watch the waves whilst sipping your morning coffee - to others in a countryside setting with parking and plenty of space for children to play, yet close enough to be within a short car journey to the nearest beach. Whatever your preference we are sure to have the perfect holiday cottage in Devon for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Did you ever try Pirlo?\nAre you ready to support the italian team playing in one hour?\nPirlo is our typical local aperitif based sparkling white wine, Campari (or Aperol) and soda \u2013 the equivalent of Venetian Spritz. Tasting Pirlo is usually in late afternoon, before dinner, but accordingly to our barman, we would say it could drink it anytime..!\nYou can do it at your home: blend 3\/10 of a Bitter Campari or Aperol, 3\/10 of white wine and 3\/10 of soda or mineral water. Finally add lemon or orange rind.\nBut remember: pirlo is best served at Bellerive Wine bar, Caf\u00e8 del Porto.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Remember how she looked back in June?\nThe project came together in a handful of layers over a period of four months. First, a slab of Indiana Gray limestone on the hearth. Next, new \"bones\" that created a frame that stretched all the way up to the ceiling. After that, a little bit of new wood and a little bit of super old wood (the horizontal blue-gray planks). After that, new bricks and a rustic mantel beam made from 100-year-old heart pine. Last but not least, a new (Simply White) mantle surround.\nI had hoped to find an antique mantel surround that we could use, but after looking around at local salvage and antique shops, I discovered that fireplaces were usually taller and not as wide back in the day. Brian the Carpenter built our surround for about a hundred bucks in materials, and we added my old corbels to give it a little antique character. (I've had them for a couple of years, and was so excited to finally have a place to use them!) I painted them Simply White to make it look like they were always a part of the mantel, but distressed them with a sanding block so a little bit of the original mint green showed through.\nWe're going to spend the rest of the year working on the built-in storage I have planned for each side of the fireplace. I'm excited to finally put all that wasted space to work!\nPS- Click HERE to check out any of my previous fireplace makeover posts!\nAwesome. I hope tomorrow you have good news about Sweet P!\nDo you have plans for the opening when the Peace Lillys are gone? Beautiful transformation.\nYour fireplace looks amazing!! You've succeeded in adding all the perfect farmhousy\/southern touches that it needs to be in your house.\nSo beautiful, what an amazing difference in the \"after\" photo!\nWhat a great transformation! It has so much more character and beauty now! Love how you used the corbels.\nIt's beautiful!!! The bricks, the corbels, the old boards and new surround \u2013 all perfection together. Love it Layla!!\nReally nice! And I love your fireplace quote on this post!\nWell I knew I would LOVE it, I always do! But the colors, the wood and that BOX! It's just perfect Layla and Kevin! And nicely done Brian the carpenter!\nSeriously good! Love the gift you have to \"see\" something like this and make it come to pass. Must make you smile when you sit in the room.\nAwesome! Can't wait to see the completed wall!\nLooks fantastic! Wish I had a fireplace. Strangely, my cats adore shredding the leaves of my spathiphyllum. The only reaction they seem to have is like the one they get with catnip.\nPamela, Layla was assuring us cat lovers because lilies are not just toxic, but fatal to cats. Thank you Layla and the fireplace is beautiful!\nIt all turned out just beautiful! ?\nBoth the before and after are awesome. You did a great job of getting the look you were after\u2026.\nIt's stunning! It was so much fun watching this project evolve, seeing all the treasures you discovered to use for the project and to watch the transformation in stages! Thank you for taking the time to post all that was involved. I hate how \"easy\" TV shows, magazines and some blogs make the process seem! What a transformation- you and your someday soon family will enjoy this for many years yet to come! Can't wait to see your seasonal decorations through the year!\nBeautiful! I loved seeing how it all came together \u2013 such a wonderful difference between the before and after. Have a great weekend!\nThis is absolutely gorgeous! After reading this, I clicked on your \"Our House\" tab, and I was overwhelmed at what a beautiful difference you and Kevin have made in your home! I'm sure you have a lot on your plate these days, but is there any chance you could give us an updated house tour post? I love seeing befores and afters, and yours are stunning! Still praying for you and your Sweet P. Much love!\nWhat a fantastic transformation!! It all looks simply beautiful, but the brick is a special standout. Congratulations on an another creative masterpiece! And thanks so much for sharing.\nSuch a clean and soft look which appeals to me.\nWow, that looks fantastic! I bet you are beyond the moon pleased with it! Love the brick, love the whole thing!\nBetsy @ Happily Ever After, Etc.\nWow! I love love love it! What a stunning transformation! Plus I love the blue\u2026 so awesome!\nThis look is absolutely perfect. And that mint peeking through on those corbels! Wow! I can already imagine Sweet P and your family sitting on that hearth for Christmas photos. Well done.\nJust beautiful my friend!! Love every bit!\nWOW! Looks wonderful!! I love it and it suits your style and home so well, it looks awesome now. Came out exactly like your inspiration.\nGorgeous! Do you mind me asking what color the blue is?","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmmrx b/data_all_eng_slimpj/shuffled/split2/finalzzzbmmrx
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index 0000000000000000000000000000000000000000..11f88df7a10bd5c2d5a876fb91fd61f3853fc10d
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+{"text":"Image Title: Helen Levi S New Dinner Plates Are Fiestaware For The Modern Era With Designs 13. Post Title: Modern Dinner Plates. Filename: helen-levi-s-new-dinner-plates-are-fiestaware-for-the-modern-era-with-designs-13.jpg. Image Dimension: 1300 x 1158 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Malachi Larson. Uploaded Date: Monday - October 29th. 2018 21:38:20 PM. Category: Furniture. Image Source: homedsgn.com. Remodel Dinner White Plates To Bloomingdale The Gather For Modern And Hunt On 18. Plate With Regarding Throughout 1. Of Jonathan Top American Season Info Gold Chic Pertaining Dining 4. S Set 9. Ceramic Plan 3. Designs 12. Glaze Prepare Reviews Rimmed 10 1948 5. 16. Ebds Barrel Intended Silky Decor Crate 10. 8 Adler Newport 4 Regard Inside Holiday Designer Mercer AllModern 6.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"The duration of copyright in Canada is 50 years after the death of the author. There are several exceptions to this general rule of duration in Canadian copyright law. Some of these exceptions are set out below.\nUnlike physical property, ownership in copyright material has a limited duration. So for example, you can own your house for an indefinite amount of time, but you can only own the intellectual property or copyright in a painting or photograph for a limited amount of time as stipulated in the Canadian Copyright Act.\nthe life of the author, the remainder of the calendar year in which the author dies, and a period of fifty years following the end of that calendar year.\nThis is often referred to as \"life-plus-fifty.\" Under this rule, an author has copyright in a work they create throughout their lifetime. Their heirs or assignees have copyright for a period of 50 years after the author's death.\nThe copyright will expire at the calendar year end of the fiftieth year from the death of the author. For example, if an author died on 1 June 1970, copyright in their works expires on 31 December 2020. All of that author's works will expire at the same time unless they are subject to an exception to the general rule of copyright as discussed further below.\nUnder Canadian copyright law, duration is determined by the life of the author, not by the life of the owner of copyright. Where an author sells their copyright and assigns the rights in their work to another person or entity, the duration of copyright is still calculated based on the life of the author. Even where works are made during the course of employment, copyright is determined based on the life of the author.\nFor further information on Canadian Copyright law, see 8 Facts About Canadian Copyright Law.\nThere are exceptions to the above general rule of life-plus-fifty. One exception is that Crown or government works in Canada are protected until published and for an additional fifty years from the date of publication.\nOn 30 September 2018, Canada, the United States and Mexico completed negotiations on a new trade agreement, the United States-Mexico-Canada Agreement (USMCA) to replace the North American Free Trade Agreement (NAFTA). Note that NAFTA will continue in effect until the USMCA is ratified in each of the three countries. You can read the text of the agreement here.\nUnder the USMCA, the minimum duration of copyright protection in the U.S., Canada and Mexico will be life-plus-seventy. For the current time, the duration of copyright protection in Canada remains life-plus-fifty.\nThe Canadian Copyright Act specifically states that moral rights last for the same term as copyright. In Canada, moral rights protect an author's right to have their name on their work or to remain anonymous or to use a pseudonym. Moral rights also protect against modifications to a work that are prejudicial to the honor or reputation of a creator. Further, moral rights protect against use of a work in association with a product, service, cause or institution.\nMoral rights endure for the life of the author plus 50 years from the calendar year end of the author's death. This means that heirs can sue on behalf of a deceased author where it seems that a modification to a work resulted in prejudice to the author's honor or reputation, or to protect any other of the moral rights. Note that you cannot otherwise license or assign moral rights while alive \u2013 you can only pass them to an heir upon death.\nThe duration of copyright protection in works created during the course of employment is subject to the general rule of life-plus-fifty. Note however that the \"life\" is based on the life of the author, and not the life of the employer or copyright owner.\nEffective 23 June 2015, Canada extended the protection of copyright in specific works only \u2014 performances and sound recordings \u2014 by an additional 20 years to 70 years after the release date of the sound recording.\nThis extension of copyright protection in Canada is only for performers (singers and musicians) and record companies (or anyone who makes sound recordings). Its purpose is to ensure that performers and record labels are fairly compensated for the use of their music for a longer period of time.\nThe extension of the duration to 70 years also benefits Canadian artists who will now be able to collect royalties from the exploitation of their recordings in European Union countries. These countries only provide a 70-year copyright protection to nationals of countries that also provide 70 years of protection. This reciprocity will help increase royalties for artists.\nFor more about the finer details of the extension of copyright in sound recordings, see Toronto-based lawyer Bob Tarantino's article The Complexities of Canada's Extension of Copyright Duration for Sound Recordings.\nWhat is a Public Domain Work in Canada?\nAs in other countries, once copyright has expired in a work, the work is said to be in the public domain. The work is no longer protected by copyright and can be used freely, without obtaining permission from or compensating the copyright owner. Works of Mozart and Shakespeare are in the public domain and can be copied freely (provided the works are not adaptations).\nOnce copyright expires, moral rights also expire, and a work may be freely adapted and used without the author's name on it. In some countries, however, moral rights are in perpetuity and exist even after copyright in a work expires. For example, moral rights are in perpetuity in France.\nCan Copyright be Renewed in Canada?\nIn Canada, the duration of copyright cannot be extended or renewed. Copyrights in Canada are protected from the moment they are created and in some fixed form and are protected for 50 years after the creator's death.\nCopyright duration in the U.S., Europe and many other countries is now life of the author plus 70 years. Will Canada extend the length of copyright protection in all protected works? This is hard to predict but something we can all keep an eye out for.\nOwnership of copyright works is another important topic you may want to learn more about. See the article Who Owns Copyright in Canada.\nLearn more about copyright in Canada through our Canadian Copyright Law and the Certificate in Canadian Copyright Law.\nWant more Canadian copyright tips and information?\nSubscribe to our Canadian email list, featuring two mailings a month with the latest in Canadian copyright news, updates, course discounts and more.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Modern bleed new standard size business card bleed new vistaprint business cardsdimensions business card size business card size business card size. Unique business card template size photoshop new standard business cardsize business business card template size photoshop resume template mm. Swanky business card postcard illustrator template templates flyers size business card size businesscard print cards usps postcard illustrator template busine..","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Printable smart goals worksheet pdf goal setting for students worksheet. Student goals worksheet and motivation on pinterest smart template worksheets for art setting goal setting. 1000 images about smart goals on pinterest crafts high school classroom and student. 1000 images about smart goals on pinterest the fly successful people and worksheet. Smart goal worksheetsworksheets goals for you.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"BATTLE CREEK, Mich. (AP) \u2014 A federal appeals court says Michigan law doesn't stop employers from firing people who use medical marijuana.\nThe court on Wednesday upheld a decision by a federal judge in Grand Rapids, who said Wal-Mart Stores could fire an employee who tested positive for marijuana.\nJoseph Casias was an inventory-control manager at a Walmart in Battle Creek until he tested positive for marijuana in 2009. He has a Michigan medical-marijuana card and uses pot to alleviate symptoms of an inoperable brain tumor and cancer.\nWal-Mart says marijuana use conflicts with its safety policy in stores.\nThe appeals court says the state medical marijuana law provides some immunity in criminal cases, but it doesn't offer protection to people in the workplace.","meta":{"redpajama_set_name":"RedPajamaC4"}}
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+{"text":"Pixels of the Week \u2013 September 16, 2018, by St\u00e9phanie Walter - UX designer & Mobile Expert.\nThis week's selection: CSS outline, CSS dismissal, portfolio tips, a font with bad kerning to piss of people, :empty and :blank in CSS, vector tool in the browser, image compressor tool, performance, what makes great developers, etc.\nImproving Client-Side Performance, a great checklist to help you make your website more performant, from images compression, lazy loading to third part scripts, widgets, etc.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Deborah Ribner's approach to real estate is from a visual perspective. Her ability to help a client see what can be done with an apartment comes both from her design background as well as her desire to understand the space of an apartment from inside out. Deborah's passion for real estate began with the buying and selling of her own apartments. She fell in love with the entire process through her own experience; viewing properties, learning the market, identifying which attributes spoke to her and then navigating the board process to secure her dream home. Her approach is collaborative and transparent. She listens and is respectful of her client's points of view and their time. Her unrivaled energy, insight, attention to detail and creativity make her a valuable ally for any buyer or seller.\nDeborah has become an expert in the Beekman\/Sutton areas, making several sales in those neighborhoods, both representing buyers and sellers. Deborah also was recently named one of theNativeSociety's 2015 Top 10 Rising Stars in NYC Real Estate, for embodying leadership, creativity, passion, initiative and a desire to make a difference in her professional and personal life. Born and raised in New York, Deborah currently sits on her Co-op Board on the Upper Eastside where she lives with her two children.\nYour NativeAdVantage: What do you do best?\nI think that one of my greatest strengths is my ability to work with almost any type of person. I think that this is a valuable quality to possess as a real estate salesperson because we are faced with working with many different personalities at any given time, and in many situations all at once.\nI think what makes me best is having many different projects that I am working on at the same time. The more I have on my plate, the more I efficient I seem to be. In turn, this makes me a true resource to clients who are looking for a large sense of what is going on in the current market.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Here's a great deal on a 2015 Honda Civic This 4 door, 5 passenger sedan still has fewer than 40,000 miles! It features a continuously variable transmission, front-wheel drive, and a 1.8 liter 4 cylinder engine. A wealth of standard features means that you no longer have to sacrifice. Like leather upholstery, an outside temperature display, heated seats, power moon roof, heated door mirrors, remote keyless entry, cruise control, and much more. Audio features include a CD player with MP3 capability, steering wheel mounted audio controls, and 6 speakers, providing excellent sound throughout the cabin. Take assurance in side curtain airbags, providing head protection in the event of a severe collision. A Carfax history report provides you peace of mind by detailing information related to past owners and service records. We pride ourselves in consistently exceeding our customer's expectations. Please don't hesitate to give us a call.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Lake Bled is one of the most picturesque lakes in Europe \u2013 and being only an hour away from Ljubljana by public transport, a day trip to this fantastic area is highly recommended. The lake surrounds Bled Island, which is home to several notable buildings such as the pilgrimage church dedicated to the Assumption of Mary, which is decorated in Gothic paintings and contains a Baroque stairway.\nLjubljana Castle was built in the 15th century and is situated on Castle Hill. The castle itself is a dominant feature of the Ljubljana skyline, and is currently one of the most popular tourist attractions in the city. The castle was occupied by the French army in the 18th century, and was also used as a prison during the Austrian Empire during the 19th Century , and again during World War II.\nMetelkova Mesto is a cultural area situated in Ljubljana, developed from the former army barracks, and has since become one of the liveliest artistic, social and intellectual urban areas of Ljubljana. Countless bars and clubs can be found in this area, along with many unique exhibitions and public art. It's a very colourful area, and attracts many different visitors including students, underground music fans, general tourists, and professionals who are enjoying a night out.\nArt Enthusiasts: With a large variety of museums, exhibitions, traditional and contemporary music.\nCouples: Ljubljana is a city full of beautiful views, old-town streets, and unspoiled corners.\nShopping: With one of the largest shopping centres in Europe, Ljubljana is an ideal shopping haven.\nLanguage: The language spoken in Ljubljana is Slovenian.\nVisa & Health: Before your holiday to Ljubljana, visit www.gov.uk\/foreign-travel-advice\/slovenia for recommendations and advice on visas and health.\nLjubljana lies in the basin between the Alpine and Karst region, and is influenced by the Mediterranean streams during the summer (mainly in July\/August). In the winter, the average temperature is much lower, due to lying in the basin and being influenced by temperature inversion \u2013 and this means \"wet\".\nBy foot: The city centre is easiest to cover on foot. Look out for mopeds and bicycles when crossing the street at busy intersections.\nBy bus: Buses are a reliable way to get round the city. A single journey is \u20ac1.20, and this covers an up to 90-minute journey, regardless of the number of buses need to be changed to reach the destination.\nOpen Kitchen Food Market: On Fridays from mid-March to October, the Open Kitchen (located in the Central Market) offers you an opportunity to meet Slovenian chefs, taste Slovenian and international dishes, and learn about different methods of food preparation.\nAna Desetnica: The Ana Desetnica International Street Theatre Festival is held at various outdoor venues across Ljubljana. The festival brings together a large number of street artistes, clowns, circus performers and much more. Usually held in either June or July.\nLjubljana Jazz Festival: Each year, the Ljubljana Jazz Festival features many of the world's biggest names in jazz. Usually held in June or July.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mosquitoes are considered one of the deadliest pests in the world.\nMosquitoes are known to transmit many infectious diseases, including malaria, West Nile virus, encephalitis, dengue fever, yellow fever and Zika virus.\nA single female mosquito can lay up to 3,000 eggs in her lifetime.\nMale mosquitoes don't feed off humans; they rely on plants to get their nutrition.\nUp to the first two weeks of mosquitoes' lives are spent in water.\nTo combat mosquitoes, you need to know what attracts them. We know they like: Moisture (standing water, perspiration), heat (body heat, exhaled gases), smells (perfumes, scented lotions) and dark colored clothing.\nMosquitoes can breed in as little as a thimble full of standing water, so it is important to identify and remove as many sources of stagnant water around your property as possible.\nMaintain your landscaping \u2013 overgrown areas trap moisture and attract mosquitoes.\nClean your patios. Standing water brings mosquitoes close to your business \u2013 where you don't want them.\nProvide an insect repellent to your customers when they are going to be spending time outside. The best EPA-registered repellents will provide protection from bites for a long period of time with just one application.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbmvmu b/data_all_eng_slimpj/shuffled/split2/finalzzzbmvmu
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index 0000000000000000000000000000000000000000..7a0d04db9fbe1796963886000edb18095c3727d5
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+{"text":"profile of this treatment in each group of patients.\nEGFR signaling pathway, to explore their association with clinical outcome.\nPROJECTED ACCRUAL: A total of 70 patients will be accrued for this study within 1 year.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"POWDERROOM: Women's Outdoor Magazine - Watch the Burton High Fives LIVE!\nWatch the Burton High Fives LIVE!\nBurton High Fives takes over Snow Park NZ this week. No matter where you live, you can watch the action live or via playback! Some of the top women riders in the world like Kelly Clark (USA), Enni Rukajarvi (FIN), Shelly Gotleib (NZL), Hannah Teter (USA), and Queralt Castellet (ESP) will be competing.\nHead over to opensnowboarding.com to watch the live webcasts. Check the schedule below to see when you should tun in.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"What does Distance Still Matters say?\nWhy does Distance Still Matters matter?\nAlessandro Giudici is Lecturer in Strategy at Cass Business School, City, University of London. His research focuses on how organizations can support innovation dispersed across international contexts. His work has been published in outlets such as the Academy of Management Journal, Strategic Organization, Long Range Planning, and Business History.\nMarianna Rolbina is a PhD candidate at Cass Business School, City, University of London. Her research is in management and strategy, with a particular focus on cultural industries.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"how to block sound in a room curtains that block out light how.\nhow to block sound in a room how cool is this basic frame with doors creates bed great idea for.\nhow to block sound in a room recent comments.\nhow to block sound in a room by.\nhow to block sound in a room sound absorbing curtains bedroom the how to soundproof a window and block noise regarding attractive residence.\nhow to block sound in a room how to soundproof bedroom cheap ways room for drums build wall steps with pictures door home depot homemade soundproofing window film music block sound.\nhow to block sound in a room sleep soundly night and day two ways to soundproof your bedroom sound proof curtain.\nhow to block sound in a room how to use a room divider block sound for amazing proof regarding 8.\nhow to block sound in a room how.\nhow to block sound in a room soundproof.\nhow to block sound in a room soundproof door house wall sound proofing how to with household items your from neighbours soundproofing.\nhow to block sound in a room how to use a room divider to block sound for soundproof room divider.\nhow to block sound in a room how window room sound proof walls apartment best home design.\nhow to block sound in a room corner bass traps in steps how i made 8 traps for my control.\nhow to block sound in a room soundproof bedroom wall how to block noise from windows build sound proof room steps with pictures.\nhow to block sound in a room how to soundproof a room in an apartment bedroom floor for free homemade soundproofing window from.\nhow to block sound in a room how to block noise from windows soundproof window curtains sound proof bedroom brand new design decorative how to block.\nhow to block sound in a room although it is covered by a molding the gap between the door frame and wall framing is a big sound leak around your doors.\nhow to block sound in a room sound.\nhow to block sound in a room how to block noise from windows how to soundproof a window from traffic noise soundproofing your how to block.\nhow to block sound in a room sound proofing blockers block out noise from busy how to soundproof windows cheap an apartment door reduce noise through walls sweep hanging.\nhow to block sound in a room how to reduce noise through walls house wall sound proofing.\nhow to block sound in a room anti noise pollution cork helmet noise block.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Just a suggestion here but Alive 4: Alola Chapter Book spice cake on the front. Breakdown by Joseph Monninger Stay me thru so many years. How may i get a the calendar please and thank. The Princess Fairies 6: The new one but believe I like to know how to get a calendar and any anymore I just signed up to. Would love to get a everyone was very disappointed that you didn't put out a calendar this year.\nScholastic has been delivering literacy resources for kids and outstanding children's books to schools, teachers, and families for more than 90 years. With offices in 16 countries, Scholastic is the largest publisher and distributor of children's books in the world, serving millions of children, families and schools. Promotional Terms *25% off promotion: Expires on December 6, at PM PST. Code HOL must be entered in the shopping cart. Promotion cannot be combined with any other offers. Promotions, discounts and gift codes apply to the order subtotal, before any taxes, shipping, handling or other fees.\nThis site uses Akismet to reduce spam. Learn how your comment data is processed. Expert advice on children's books & reading, arts & crafts, activities & school achievement. View the parent's newsletter, articles, & weekly picks for Preschool, Grade School, & Middle School. Gracias The Thanksgiving Turkey (Scholastic Bookshelf: Holiday) [Joy Cowley, Joe Cepeda] on bankjack-downloadly.tk *FREE* shipping on qualifying offers. Let the Scholastic Bookshelf be your uide through the whole range of your child's experiences--laugh with them. The Journey of Little Charlie. By Christopher Paul Curtis. Short-listed, National Book Award, Young People s Literature Category, ; Short-listed, Governor General's Award for Children's Literature - .","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbnpch b/data_all_eng_slimpj/shuffled/split2/finalzzzbnpch
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+{"text":"Format (cm): 23.2 x 38.1 cm.\nFormat (in): 9 1\/8 x 15 in.\nThe Rev. Dr. Henry Wellesley, Oxford; Sotheby's, London, 25 June-10 July 1866, 6th day of sale, part of lot 954, 'A portfolio' (as by Jacopo Ligozzi) (\u00a31.15s. to Noseda). Sir David Kelly; Hodgson's, London, 26 November 1954, part of lot 596 (as Jacopo Ligozzi). with Hans Calmann, London; where purchased by the present owner.\nThis intricate and highly-finished drawing is a fine example of Cantagallina's draughtsmanship in the years immediately after his 1612-13 tour of the Netherlands, Belgium and Flanders. He had returned to Tuscany by 1615 and the present sheet is one of two studies of the village of Pratieghi which he executed in 1616. The other drawing, in the British Museum, shows the village from a slightly different perspective and is also inscribed and dated (inv. 1955,0117.1; British Museum, London, Landscape in Italy , 1984, no. 17). Pratieghi is a small village that lies just within the Tuscan border with Emilia-Romagna. It became part of the Republic of Florence in 1490. This drawing comes from an album of over a hundred landscape drawings, formerly in the collection of the scholar and antiquary Dr. Henry Wellesley (1791-1866), nephew of the Duke of Wellington, that appeared at auction in London in 1954. Although inscribed 'Vedute di Toscana di Jacopo Ligozzi', the album was in fact largely made up of highly finished drawings by Cantagallina of various dates, several of which were signed. The album was acquired by the dealer Hans Calmann and the drawings were dispersed over the next several years.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"As we wait and ponder the coming days of endings and beginnings, I offer you a reflection written by Br. David Steindl-Rast, OSB (Gratefulness.org).\nMay You Grow Still Enough\u2026.\nMay you grow still enough to hear the small noises earth makes in preparing for the long sleep of winter, so that you yourself may grow calm and grounded deep within.\nMay you grow still enough to hear the trickling of water seeping into the ground, so that your soul may be softened and healed, and guided in its flow.\nMay you grow still enough to hear the splintering of starlight in the winter sky and the roar at earth's fiery core.\nThe Mother Board wishes you the blessings and joys of the season.\nAs we who live in the Northern Hemisphere spin towards winter solstice, the darkest time of our year, I invite you to fully embrace that darkness. Like a seed planted into the dark earth, if we are patient and willing, we might feel the beginnings of new life and growth.\nFor me, this is a season of contemplation, a recounting of the past year and the years before. The days are often gray here in Ohio, adding to the feeling that I might never see the light again.\nLooking outwards, I see much darkness in the decisions coming from Washington and darkness coming from people who would want to define us by the color of our skin, the depth of our pockets or the nature of our god. I read warnings about the fate of our beautiful Mother Earth, see the evidence of dramatic weather changes, watch the suffering it brings and still, the denials of the decision makers who are willingly blind to the evidence. The pictures of starving children, families sprayed with tear gas and six year old children standing alone in a deportation courtroom break my heart. It brings me to my knees in sorrow and I weep for the future of our country and our planet and my grandsons and their children.\nLooking inward, I search my own life for those pockets of darkness that I have, by omission or co-mission, created. Because I am, after all, human, I find them.\nAnd then, like the seed, I let go of what was and start looking for the light of what is to come. I think about lighting the candle of my being, pounding my drum to release it and preparing to create and welcome light. I pray. I hope. I breathe the air of a new year.\nOne of the things we crones love to explore is the restaurant fare in the cities where our Counsel is being held. I remember eating Pad Thai for the first time in Seattle. What an amazing taste experience that was! You will love our venue in Tucson. It has a wonderful restaurant, The Saguaro Grill, which serves a wide variety of American food at average prices. However, within a short walk of the hotel there are 33 restaurants and shops for you to explore!\nHere's a sampling of restaurants within a short walk from our hotel for Crones Counsel 27!\nThe Mother Board thanks all of the Crones who attended the Gathering in Bellingham and responded to the survey that was sent via email. The results indicated that 74 Crones responded, giving the Gathering a favorable ranking. We are grateful for the time and energy given to the written comments. As a Board we read and use your comments in proceeding with the next Gathering. Your ideas and advice are very important to us. The information lets us know what is working and what is not. The survey is for the Board to evaluate and help us create a better Crones Counsel for you. Thank you for your continued support.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"how to tell a customer no more freebies How to say no to customers without making them angry By Olga Rogacka in LiveChat Blog > Customer service , September 9, 2014 When clients need to contact a company's support, they always want to hear the yes answer to all their requests.\n(The customer in front of me is arguing with staff.) Customer: \"Look. A sandwich is $5.50. Fries are $2. A drink is $1. But a combo of all three is $7.59.\nMoving home Please use this form to tell us that you are moving into, within, or out of our area. You can complete this form up to 28 days before you move, and any time afterwards.\nDon't tell them that you'll get back to them just to put off saying no. If you're considering saying yes but you need to check on something first, tell them that and let them know when they can expect your answer.\nEnter your email below to get your copy every week! Get great freebies and printable coupons for your favorite stores emailed to you weekly. Receive the FREE Amazing Freebie and Printable Coupon Newsletter from MomsView.com.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Artificial plant wall used for indoor and outdoor decoration, is made of PE. It has no harm to human. Suitable for weddings,parties,supermarket,home garden,hotels, etc. Artificial plant wallcolor is bright and natural.\n1.Artificial plant almost don't need maintain.\n2.Similar to real plant,has loner lifetime.\n3.UV and corrosion resistant, no regional and seasonal restrictions.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Does the delivery guy at the local Chinese take-out place know your order by heart? We know how that goes. But once you experience how easy, healthy, and tasty our make-at-home fried rice is, you'll toss that take-out menu for good. We spruced up this iconic Asian dish with fresh yellow squash, buttery toasted cashews, juicy pineapple, rich sesame oil, and crunchy green onions. The result is a vegetarian version of the classic greasy fare that's every bit as savory and delicious.\nAdd 1\u00bd cups of water, jasmine rice, and a pinch of salt to a small sauce pot and bring to a boil. Reduce heat, cover, and simmer until all water has been absorbed, about 15 minutes. Remove from heat and set aside with lid on until ready to incorporate in stir-fry.\nRinse the green onions and yellow squash. Thinly slice the green onions on a bias (on an angle.) Trim ends off the yellow squash, quarter lengthwise, and dice into quarter-rounds. Finely dice the yellow onion. Drain pineapple and reserve juice.\nWhile rice is cooking, warm half the sesame oil (1 Tbsp.) in a pan over medium heat (reserving remaining for garnish). Add all of the diced yellow onion, half the green onions (reserving remaining for garnish), and cashews for about 2 minutes. Stir frequently until aromatic and onions are slightly softened and translucent. Add diced yellow squash and cook 3-5 minutes, until it begins to soften.\nOnce cashews are lightly toasted, add slaw mix, cooked rice, and soy sauce. Saut\u00e9 over high heat for 2 minutes, until soy sauce is absorbed and rice and vegetables begin to caramelize. Add pineapple and stir-fry over high heat for another 2-3 minutes until warmed through. Taste and add reserved pineapple juice to taste. Add a pinch of salt and pepper.\nAdd a heaping serving of fried rice to a bowl. Drizzle with remaining 1 Tbsp. sesame oil. Garnish with remaining green onions and a drizzle of Sriracha sauce.","meta":{"redpajama_set_name":"RedPajamaC4"}}
diff --git a/data_all_eng_slimpj/shuffled/split2/finalzzzbopcd b/data_all_eng_slimpj/shuffled/split2/finalzzzbopcd
new file mode 100644
index 0000000000000000000000000000000000000000..858937ab42cebd7ec7e429209bc3002fdf007dd3
--- /dev/null
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@@ -0,0 +1,5 @@
+{"text":"Clearview Towing Mirrors for a Nissan Navara D40\/550 (Build Years 2004 to 2015) are backed by a 3 year warranty and are further supported by a complete range of spare parts and repair videos.\nClearview Towing Mirrors for a Nissan Navara D40\/550 (Build Years 2004 to 2015) are supplied with plug and play wiring for easy installation when replacing side mirrors with identical existing features.\n- Nissan Pathfinder (Build Years 2004 to 2013).\nNote: These mirrors are not compatible with any other vehicle, including the Nissan Navara D22 and Nissan Navara NP300.\nYou are able to purchase Clearview Towing Mirrors with electrically adjustable large flat mirror glass even if your Nissan Navara D40\/550 (Build Years 2004 to 2015) mirrors are manually operated. However, an electrics kit will also need to be purchased (which includes toggle\/adjustment switch) and all wiring - in this scenario only - will need to be hard wired by a registered auto electrician. We do not provide this service.\nNissan Navara D40\/550 (Build Years 2004 to 2015) manufactured in South Africa have slightly different wiring than those manufactured in all other locations. Please contact our office if you have a Nissan Navara D40\/550 (Build Years 2004 to 2015) manufactured in South Africa to ensure you receive wiring adapters with the correct wiring configuration.\nYou are able to purchase Clearview Towing Mirrors with indicators even if your Nissan Navara D40\/550 (Build Years 2004 to 2015) mirrors do not have indicators. However, the wiring adapters for models with indicators are different to those without indicators so please be sure to contact our office prior to ordering to ensure you get the correct wiring adapters. Also - in this scenario only - the indicator wiring will need to be hard wired by a registered auto electrician. We do not provide this service.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"I am going to keep this brief because I would really like to finish this book before I go back to work tomorrow morning. And based on my mental calculations, I can finish the book, or write a real post, but not both. I'm 80% of the way through this book, and I'm reading it on a Kindle. My boss's Kindle. (Not my own, because my mom has appropriated my Kindle for reading and Sudoku-playing. Note to self, the next time you buy a Kindle, don't buy a Kindle Fire and then let your mom take it to Europe.) My boss was recently seduced by the e-Reader movement, and bought a basic Kindle for purposes of using while traveling. She bought a couple of books on it that she would normally have lent me had she bought them in paperback. So what did she do? She brought her Kindle to work and said, \"Just get it back to me before I go out of town again.\" This is why she's the best boss ever. So I find myself reading a book exclusively on an e-Reader for the first time ever. It's not terrible. I actually find the e-Ink display much less distracting and more book-like than I thought I would. It's not the same as reading an actual book, but it's more like it than reading on the Kindle Fire.\nThe Invention of Wings is the first Sue Monk Kidd book I have read. Yes, that's right. I've never read The Secret Life of Bees. That came out during my \"I don't read anything popular\" phase of reading. I have since bought it, of course, I think when Borders was closing down. It's also my first Oprah book in quite a while, and it's kind of what I was expecting from an Oprah book, for better or worse. That, of course, means strong female characters surviving\/overcoming oppression of some sort. Some are poor, and some are Southern. There is a historical setting (Oprah seems to prefer historical fiction these days). And I like it, as I like most Oprah books I have read, in spite of my better instincts.\nThe Invention of Wings takes place in the early nineteenth century, primarily in Charleston, South Carolina, and also in Philadelphia. The two narrators are Sarah Grimk\u00e9, a historical figure, and her fictional slave, Hetty \"Handful\" Grimk\u00e9. The novel begins on Sarah's eleventh birthday, when she is given Handful as a birthday gift from her parents. Sarah grows up to be an abolitionist and suffragist, and is not thrilled with her birthday gift. Handful grows up to be, well, a slave, but one who assists in planning the thwarted slave revolt of 1822 with the historical figure Denmark Vessey, and a very talented seamstress and quilter. Quilts figure prominently in the imagery of the novel, which is why I mention them. Handful and her mother Charlotte are very close and I found the juxtaposition of their relationship with Sarah's much colder relationship with her mother Mary to be at the heart of the novel. I never get tired of reading about mother-daughter relationships, and I don't think I've gotten to do that recently, so this was nice. Well, I guess Life After Life had a bit to say about mother-daughter relationships but Ursula's relationship with her mother was not quite as central.\nI'll have more to say about this book on Tuesday. But for now, here is an interesting tidbit about the new Kindle Paperwhite: it will guesstimate for you how much longer it will take you to read a chapter of whatever book you're reading based on your prior reading habits. Isn't that wonderfully creepy? There is something that's appealing about concealing the number of books I own on a teeny tiny e-reader, it's just too bad that I already own so damn many hard copies. Well, not really too bad. I just wish it were possible to put my books onto an e-reader, like you can upload CDs onto an iPod. Someday, maybe.\nThis entry was posted in E-Readers vs. Hard Copies, Fiction - general, Fiction - Historical, Reviews by Jill, Sue Monk Kidd. Bookmark the permalink.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Eating at restaurants is simple, fast and fun. Whether it's a family will get together, business lunch or kids birthday celebration, individuals are always planning outings to savor great food and conversation.\nRestaurants today are buzzing with customers. As restaurants provide specialized menus and styles that suite every occasion, increasingly more clients are making use.It appears that individuals everywhere will always be looking for great Drinks Near Me food inside a comfortable atmosphere that they'll tell their buddies & family about.\nHow do we know which kind of restaurant is the best for your occasion? Be it only a quick bite or perhaps a fine restaurant, we provide you with a quick guide below that will help you pick the perfect place. After this information, you will notice tips about how to select good dinners once you are sitting while dining.\nSpecialized restaurants are feature live entertainment, typically within an outside setting, like on the patio. These restaurants are ideal for friendly gatherings where everybody can also enjoy an excellent meal together and relish the sights and sounds of live music entertainment.\nThese restaurants usually offer table service along with a good number of dished to select from. The climate during these restaurants is filled with energy with individuals conversing and also the music playing. Next time you're searching to enjoy yourself with several buddies, go for an outside and live entertainment experience.\nCasual dining usually represents junk food joints serving quick food through self-service. They don't typically offer table service and meals are usually offered in disposable plates and containers. Casual dines would be best when you're on the run and want to seize a fast bite. They serve moderately priced food inside a casual atmosphere. Cafes and tea houses also get certified in this group of casual dines. Casual eating places are perfect for students and buddies getting together.\nAn excellent dining experience combines food and art together. It focuses more about the service, presentation, and providing an excellent overall dining experience to the visitors. You normally have to be formally outfitted such restaurants as jeans along with a t-shirt might cause you to feel a little unnatural.\nThe services are first class and your meals are offered using the utmost delicacy appealing just as much towards the eye because it gives your palate. Fine dines are often priced greater than other eating places since the focus isn't just on food but around the atmosphere too. Lots of people choose fine dining restaurants for a special event to really make it an unforgettable experience. It's also the right setting for any romantic date.\nQuality eating places provide a niche menu. For instance: Chinese, Mexican, Italian, Thai, Bbq or Indian. Typically, there aren't a number of different cuisines to select from, only specific menu products in one region. Each Quality Dining restaurant represents a definite part of center industry and conveys its very own personality.\nLike fine eating places, quality dining restaurants possess a nice mood-setting atmosphere, genuine hospitality along with a welcoming atmosphere that keep customers returning. It's the ideal restaurant when searching to achieve the authentic flavor of the specific cuisine. However, it lacks the range of choices from various cuisines that the fine dining restaurant would serve.\nDining restaurants are appropriate for just about any casual setting. They're typically restaurants where one can sit lower and revel in your meals, instead of carryout restaurants that don't provide a sitting area. Moderately priced, and frequently full of variety, these dining restaurants will also be appropriate for kid parties and small meet up.","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Mayor Don Iveson fell in love with a flag \u2014 a gift to him from Treaty 6 chiefs \u2014 and now he wants to know if the rest of Edmonton might fly it, too.\nHe has been mulling what to do for months and this week officially hoisted the idea up the flagpole. Next council meeting, he'll ask colleagues to commission staff and see if this new yellow sun, purple sky, and blue-green river valley will capture the heart and minds of residents.\nVarious Edmonton residents have debated the merits of a new flag for years, especially since journalist Roman Mars gave a speech on city flags here and laughed at Edmonton's attempt. It's current flag \u2014 two blue bars with the coat of arms and name of the city \u2014 breaks almost all the rules of good flag making.\nBut residents aren't yet in love with McCourt's version. On social media, many embrace the idea of reconciliation, but aren't keen on the design.\n\"The only thing I really like about (McCourt's) flag is that it has some recognition among local elders,\" wrote Dustin Bajer, who designed an alternate flag featuring a magpie as a tongue-in-cheek way to push the issue.\n\"I find it too busy and too literal. Where's the symbolism?\" he asked.\n\"It would be cool to see a few concepts that the city could choose from,\" said indigenous artist Aaron Paquette, who supports the concept.\nCheryl Whiskeyjack, executive director for the indigenous Bent Arrow Traditional Healing Society, said she likes \"the flag and spirit that it represents.\" It ties into Treaty 6, but can speak to indigenous people, status or non-status, she said.\nIveson said he would like city staff to start with a simple, perhaps Internet-based consultation and work within existing budgets to see if this flag is popular. If staff find people love the old flag, or want a new flag, but not this flag, that's fine, too, he said.\n\"This shouldn't have a significant cost to it at all to just ask the question,\" he said.\nMaggie Hodgson, an indigenous community leader and health professional, said she likes that the symbolism of the flag has ties to the land, but it's even more important that it comes as part of a new relationship with the city.\n\"The city is really working hard at building a different kind of relationship, this mayor for sure,\" said Hodgson, pointing to the new aboriginal area coming to Fort Edmonton Park, the aboriginal art park planned for Queen Elizabeth Park and talk of an aboriginal museum in the old Royal Alberta Museum building.\nSmaller Edmonton Remembrance Day event pays tribute to Canada's first Afghanistan... Alberta's politicians lean left and right, but where's the centre?","meta":{"redpajama_set_name":"RedPajamaC4"}}
+{"text":"Image Title: Romantic Rustic Beige Pink Floral Comforter Set Full Throughout Designs 11. Filename: romantic-rustic-beige-pink-floral-comforter-set-full-throughout-designs-11.jpg. Image Dimension: 823 x 607 pixels. Images Format: jpg\/jpeg. Publisher\/Author: Louisa Bailey. Uploaded Date: Saturday - May 05th. 2018 22:45:21 PM. Category: Bedroom. Image Source: homedepot.com.\nTap The Thumbnail Bellow to See Related Gallery of \"Romantic Rustic Beige Pink Floral Comforter Set Full Throughout Designs 11\"","meta":{"redpajama_set_name":"RedPajamaC4"}}